{ "00001453ad6771567b7ec0e7404a1e79": "P=3B_0\\left(\\frac{1-\\eta}{\\eta^2}\\right)e^{\\frac{3}{2}(B_0'-1)(1-\\eta)}", "0000239ab0143b8cd72151bf852d7af7": "\\beth_{d-1}(|\\alpha+\\omega|^{2^{\\aleph_0}})", "00004cd84d4d46d0e37b841cd7509c2c": "\\mathrm{REC}(N)", "00009654348eebd7ab85d8599c25aace": "W(2, k) > 2^k/k^\\varepsilon", "0000a3595ace35143948315a2841b307": "h=-1", "000138f6a2210ff1f9bb5eb7bc25ab6c": "(X,\\Sigma)", "0001e5b7a90547e1d7bc8be8a5c1e161": "1-\\left[\\frac{15}{16}\\right]^{16} \\,=\\, 64.39%", "00021ba3771fa2c4684c5639fecea94e": "\\tan\\frac{3\\pi}{20}=\\tan 27^\\circ=\\sqrt5-1-\\sqrt{5-2\\sqrt5}\\,", "00023b9224ac900169410ee72115cea4": "\n\\chi(T) = T^{2g} + a_1T^{2g-1} + \\cdots + a_gT^g + \\cdots + a_1q^{g-1}T + q^g,\n", "00029fdbca88b454dc6e742a8f404ca2": "(p-1)!^n", "0002a4a9343567b9a285b034b9a38ecb": "p = { E \\over c } = { hf \\over c } = { h \\over \\lambda }. ", "0002a95f3d21c0d5e5f455a832f2c17d": "\\psi\\to e^{i\\gamma_{d+1}\\alpha(x)}\\psi\\,", "0002cea3a95fae1835af3910d7ca6930": "e \\Delta\\rho \\simeq \\epsilon_0 k_0^2 \\Delta\\phi", "000303c8a822cfee55c1bd97c1d4cc4a": "f_c(z) = z^2 + c", "000334cb9f0bccc26284c6ef02725e06": "H \\rightarrow G/N \\times G'/N'", "0003cda16b8f0d055034e3c54846c175": "m_{\\text{o}}", "0003d3bfc208df07075efc742b3af376": "\\mathbf{J_2}", "0003ff41496b4d8a9a60cf3e03db80f2": "\\{ (p \\to q), (p \\to \\neg q) \\} \\vdash \\neg p", "00040a566d6ca57745bff5a2514f424c": "A_\\mu(x_i)", "00040baa2353e06d351f6c9dac889ece": " ds^2 = g_{00} \\, dt^2 + g_{jk} \\, dx^j \\, dx^k,\\;\\; j,\\; k \\in \\{1, 2, 3\\} ", "00047597e6585d2a8d77e2c4bb610401": "\n\\bar{h}(s,i;L)=\\prod_{c=1}^i\\sum_{k_c=2+k_{c-1}}^{L-1-2(i-c)}\\bar{f}_{k_c}(s)\n", "0004c246ad141d5412a457dc81323857": "H_1(\\mathrm{A}_3)\\cong H_1(\\mathrm{A}_4) \\cong \\mathrm{C}_3,", "000592a04b7c6c5cc9a9429a048b2757": "\n \\mu = 2C_1~\\sum_{i=1}^5 i\\,\\alpha_i~\\beta^{i-1}~I_1^{i-1} \\,.\n ", "0005a6b0b0b3be71744f935c4a5eeb3a": "f:\\mathcal{H}_g \\rightarrow V", "0005eff4a121d51b65af0ee36bc65e70": "q(\\mathbf{\\pi}) \\prod_{k=1}^K q(\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k)", "000643b3754284c8b2aeb53d4394f021": "(\\forall i\\in I) f[V_i]\\subseteq V_i", "0006b557602a072b21da57443b92f449": "254 = 2^8 - 2", "000723a6105c190f41462d560ad7458a": "R(X_1, \\ldots, X_{n})", "000736cda6b8807641f5244f27742f56": "\nP_{ij}(f)=\\frac{ A_{ij}(f)}\n{\\sqrt{\\mathbf{a}^{*}_j(f)\\mathbf{a}_j(f)}}\n", "00073e38a79657d8dfb58930122512ce": "A(x, y)\\,dx + B(x, y)\\,dy", "00078c12a085f724c262a7295f8d70b0": "\\frac{\\$\\text{40m}}{\\$\\text{30m}} = 1 \\frac{1}{3} \\approx 1.33", "00079c0fe89f86a710a201e0689b2172": "\\int u \\, dv=uv-\\int v \\, du.\\!", "0008510cb7881764a542e8502fc95b28": "\\Psi(w,v)=w^\\alpha \\cdot v = \\sum_{i=1}^n w_i^q v_i", "0008c41df7229f6c3753f8c45db87f04": "{f_x}(m)", "0008d640a21f52b6b7067d7b03547108": "v_i = \\frac{\\partial \\Phi}{\\partial x_i}", "000904ee9bee58b7b339bfe4b842e49a": "\\forall x \\, \\forall y \\, P(x,y) \\Leftrightarrow \\forall y \\, \\forall x \\, P(x,y)", "000931b2d65a0f6ce57156ed9e2f457e": "\\mathrm{resultant}(p, T)=0", "000945530b96364391c181a406d4fa29": "P(X_i=a)", "0009d412dbeb47c56fe78c99cfd4dc08": "p = c \\cdot u \\cdot \\rho", "0009d7ff4e372f215e5fc71b37a42038": "\\;^+R_{\\alpha \\beta} - {1 \\over 2} g_{\\alpha \\beta} \\;^+R = 0.", "000a91452ffe8335b67f0e5ff2c0a767": "\\textstyle P ( A \\Delta f^{-1}(B) ) = 0. ", "000ab33a85842800e48143f212ac5fc0": "p = 1\\; \\text{GeV}/c = \\frac{(1 \\times 10^{9}) \\cdot (1.60217646 \\times 10^{-19} \\; \\text{C}) \\cdot \\text{V}}{(2.99792458 \\times 10^{8}\\; \\text{m}/\\text{s})} = 5.344286 \\times 10^{-19}\\; \\text{kg}{\\cdot}\\text{m}/\\text{s}.", "000ad1eb8a2c2182ff048350cc9eb0e8": "\\alpha(x)", "000ae84c0190bb851b585c79e3b8449f": "\\,2", "000af2fae5bdfcd63e6dc3e5bce0dea3": "f^*(x^*) = \\sup_{x\\in X}(\\langle x^*,x\\rangle-f(x)),\\quad x^*\\in X^*", "000b1d2bea2949b83a2325c116ed0f04": "\\nabla T = \\omega\\otimes T. \\, ", "000b37155b94f927910c738a2cb82536": "f(\\lambda x + (1 - \\lambda)y)>\\min\\big(f(x),f(y)\\big)", "000b55413dd8e51c6a5331d756bb35cd": "r_{k} = \\frac{B_{0} - B_{k}}{B^{*} - B_{0}}", "000b60e64695a061524870992c804694": "\\mathfrak{H} =\n\\begin{pmatrix}\nZ_\\infty & - \\gamma_1 \\gamma_2 \\\\\n1 & - z_\\infty\n\\end{pmatrix}, \\;\\;\nZ_\\infty = \\gamma_1 + \\gamma_2 - z_\\infty.\n", "000bdb583c44e7082a31ebb9e6d3270e": "Y_{8}^{6}(\\theta,\\varphi)={1\\over 128}\\sqrt{7293\\over \\pi}\\cdot e^{6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot(15\\cos^{2}\\theta-1)", "000c0ecd3b1cdd0c543c83fb72777e40": "\\|u\\|=\\sqrt{(u|u)}.", "000c247a72b758a4a7b58c94ef5c0143": " C_T',", "000c2d05999df03021184202a05ed589": "\\frac{\\Box p}p", "000c2fdc9d5f7e0d8645da414718e55b": "(a+bi) (c+di) = (ac-bd) + (bc+ad)i.\\ ", "000c509e2ba315d93d74f4358779d6db": "V=5 (Y/19.77)^{0.426}=1.4 Y^{0.426}", "000ccb0783ce670a6c05781e17c96ac4": "H=H_e + H_h +V(r_e -r_h)", "000dd16a691352805a456b763a587df9": "E \\cup F", "000dd846c45c943c8bc9924ef48d1f0d": "e^{i\\mathbf{k \\cdot r_{12}}}", "000de4afc6a32a049d59aeacdb9ef318": "f(x) = x^2 - x + 2", "000dfe97e8b66bd454b3cee3f7fdd708": "e^{c(\\ln n)^\\alpha(\\ln\\ln n)^{1-\\alpha}}", "000e03d98da2c9a1864a463164762254": "\\frac{1}{\\ln p}", "000e18741a314511f1bc6557ae754035": " \\mbox{E} =\\frac{\\sqrt{1.64 \\cdot N} \\cdot \\sqrt{ 120\\cdot \\pi}}{2\\cdot \\sqrt{\\pi}\\cdot d} \n\n \\approx 7\\cdot\\frac{ \\sqrt{N}}{d}", "000e540b8ebc9ff725e5bb41d49be814": " \\text{Spec }B ", "000e5c1739ea28760d66f6d05f0e18d1": "\nJ_{\\alpha} = \n\\int_{0}^{\\infty} \\frac{dx}{\\left( x + b^{2} \\right) \\sqrt{\\left( x + a^{2} \\right)^{3}}}\n", "000ec8a8686baebba2fe12442b863020": "U_{11} - U_{21}", "000f32a1b8f6232759a658d470fe72c5": "y = p(x)", "000f743b3f56fd60b28545a4a844b238": "|{\\Psi}\\rangle=\\sum_{i_1,i_2,\\alpha_1,\\alpha_2}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}\\Gamma^{[2]i_2}_{\\alpha_1\\alpha_2}\\lambda^{[2]}_{{\\alpha}_2}|{i_1i_2}\\rangle|{\\Phi^{[3..N]}_{\\alpha_2}}\\rangle", "000f9bd1ad9b3b09c9aa4c60c45692fc": "e = O( n^{2/3} m^{2/3} + n + m )", "000febfeef5745a752e85b94b75cf713": "(t_2,t_1,F_{t_1,t_0}(p)) \\in D(X)", "000ff44c1346a4a8419c634aa6792a6b": "\\scriptstyle (m\\mid k)", "0010ce961820b14519f4edb042677035": "\\vec{b} \\equiv \\vec{B}/B", "0010d521b3b9b45b628e76ac7a7e0477": "\\mathit{MPC} = \\frac{\\Delta C}{\\Delta Y}", "00114d741d2031bf778fd8e43ac0cbeb": "(r,\\theta_r,\\phi_r)", "00114eb3ada60483709d9dc80af6eb9e": "\nL_\\mathrm{dB} = 10 \\log_{10} \\bigg(\\frac{P_1}{P_0}\\bigg) \\,\n", "0011faa0f320ff9b7bc5a9e9ec93bd19": "\\sqrt{\\det g}\\mathcal{D}\\Sigma.", "001222b8821d1da420dbe52f697b6ceb": "(x',y') = (x,y) A + b\\,", "00123391b9f305cfe97c99078735ae00": "\\tilde{k}\\,", "00124f922ab1a17e5e2a9a6c50b17a11": "\\displaystyle{AB=-BA,\\,\\,\\,\\,A^2-B^2 =I.}", "0012c829b2e3bbb683c9a17381e15b4e": "\\frac{\\mathbf{T}(s+\\Delta{s})-\\mathbf{T}(s)}{\\Delta{s}}=-\\mathbf{q}(s). ", "0013269ea11adb76b0e5c55c5d2da6e3": "34^2", "0013271afabc2f00efdeafe99dabfc9c": "\\; P(s_i)", "0013383b9f26d293e8432ded6c3e5520": "\\begin{align} S_1 &=& a_1& & &\\\\\nS_2 &=& a_1& {}+ a_2& &\\\\\nS_3 &= &a_1& {}+ a_2& {}+ a_3&\\\\\n\\vdots & &\\vdots & & &\\\\\nS_N &=& a_1& {}+ a_2& {}+ a_3& {}+ \\cdots \\\\\n\\vdots & &\\vdots & & &\\end{align}", "001384455f0b171fd018da65ca08ae9a": "V \\otimes V / (v_1 \\otimes v_2 + v_2 \\otimes v_1 \\text{ for all } v_1, v_2 \\in V).", "0013ada8dc886f1e875984bee5fdea27": "\n\\rho_{x^{n}\\left( m\\right) }=\\rho_{x_{1}\\left( m\\right) }\\otimes\n\\cdots\\otimes\\rho_{x_{n}\\left( m\\right) }.\n", "0013b318ce7c8b8ca29b706aaa5ec54d": " \\mathbf{A}\\mathbf{B} = \\mathbf{A} \\cdot \\mathbf{B} + \\mathbf{A} \\times \\mathbf{B} + \\mathbf{A} \\wedge \\mathbf{B}. ", "00141348cd6cabc06166525b88bb1493": "\\lim \\sup _{\\alpha} (n_{\\alpha}/m_{\\alpha}) < r", "00143ba3149a2dfac0bbad577d553b6c": " \\vec{A} = \\frac{B}{2}(x\\hat{y} - y\\hat{x})", "001462c07545b4ba9084efef2a96cf16": "\n\\begin{align}\nq &= q \\left(p + 2 q + r\\right)\\\\\n&= q p + 2 q^2 + q r\\\\\n&= q^2 + q (p + r) + q^2\\\\\n&= q^2 + q (p + r) + p r\\\\\n&= \\left(p + q\\right) \\left(q + r\\right)\\\\\n&= q_1\n\\end{align}\n", "00147bc5b79f2b9ed52b22af8d073758": "z*x\\le y", "00148c7652375ca3d73b9b13e86e6c09": "\\psi(\\hat{\\alpha}) - \\psi(\\hat{\\alpha} + \\hat{\\beta})= \\ln \\hat{G}_X", "0014c0cbfae8735b260b1d36141ba2fb": "\\lim_\\alpha \\gamma := \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{\\varphi(x,t):t 0, c \\ne 1", "001526024fa254f09f605fe336f1efb9": "\\textstyle x+C_{i}", "0015764e9f5498369d691b91d3e231a0": "{f_{xy}\\;=\\;f_{yx}}", "0015c94baa30e618e20880703cd9574e": "\\kappa( \\cdot, \\cdot)", "00160f32f654a73bc70209c66ba07704": " K = \\mathbb{Q} ", "001664050cbc76569028d6ac26295a53": "\\theta = n \\times 137.508^\\circ,", "0016dac7c84a2f7a9a5b064c68d1af56": "B^\\prime=-(n_b-n_\\bar{b})", "0017516c449d71df2d3f9b14a22cab76": "RD = \\min\\left(\\sqrt{{RD_0}^2 + c^2 t},350\\right)", "001758801bb0a24a60d89d6ed42620aa": "\\displaystyle{g^\\prime=\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix},}", "00178a6c0a72a69875dabaf4d5ccc192": "\n \\frac{1}{(p+1)\\left(b^2-4 a\\,c\\right) \\left(c\\,d^2-b\\,d\\,e+a\\,e^2\\right)}\\,\\cdot\n", "00178f10a40e91f76517d52061ef2a42": "(n+1)!", "00179f58dfc9cf36493673f0dacf255e": "s_V(\\mathcal{R})", "0017f09b2d0eb84ef7d74112761e5ca2": "\\begin{align}\n\\Phi _{1} & =\\Phi _{2}\\equiv \\Phi (x_{\\perp }) \\\\\n& =-2p_{1}\\cdot A_{1}+A_{1}^{2}+2m_{1}S_{1}+S_{1}^{2} \\\\\n& =-2p_{2}\\cdot A_{2}+A_{2}^{2}+2m_{2}S_{2}+S_{2}^{2} \\\\\n& =2\\varepsilon _{w}A-A^{2}+2m_{w}S+S^{2}, \n\\end{align}", "0017fa64796d63c8af98928a15b3662c": "-F\\mathbf{e}_y", "001803962c3d9e04abb4057c65fa219a": "d_{\\phi}=1", "00180c42d14cacb3f499b74661393fb8": "|f(s)g(s)| \\le \\frac{|f(s)|^p}p + \\frac{|g(s)|^q}q,\\qquad s\\in S.", "001848ad365fbadd5ad138e8c017229c": "c_{\\rm s}", "0018ea864cfdaca5dd616457e5376705": "X,Y,Z", "001906e750dc40c74b91cf7d58e53031": "S^k \\,", "001914d9d31353c1e3f3a0cc4f5d1b26": "\\mathbf{a}_{\\mathrm{average}} = \\frac{\\Delta\\mathbf{v}}{\\Delta t} ", "0019535400d4fd1cc406673a5c837318": " \\sum_i {}^\\phi{V}_i= q V - (q-1)\\sum_i V_i \\,", "0019561cf8dcc36cdbaef1e31544dba0": "WL", "00195c93942fa87df4fc3cc6475b99f9": "h = \\frac{1}{4} kd \\theta^2", "0019c83f9d0e4f79dbb27fa6520759ef": "\\ell(m)", "001a3615880485d99edbd2bcfd14bbd6": "id_\\tau", "001a607e35251386d2e1be0dfd149e51": " \\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} = \\mathbf{I} \\cdot \\boldsymbol{\\omega} ", "001ab4e8bcdb353a5c9bd1db301c1b29": "x+n+a = \\sqrt{ax+(n+a)^2 +x\\sqrt{a(x+n)+(n+a)^2+(x+n) \\sqrt{\\cdots}}}", "001ac223727c30afb98538642f53b42f": "\\left( \\frac{2}{3} \\right) ^3 \\times 2^2", "001ad3e03ed6e69c3304e438fa6e082b": "\\mathbb{P} (Y \\le 0.75|X=0.5) = \\int_{-\\infty}^{0.75} f_{Y|X=0.5}(y) \\, \\mathrm{d}y = \\int_{-\\sqrt{0.75}}^{0.75} \\frac{\\mathrm{d}y}{\\pi \\sqrt{0.75-y^2} } = \\tfrac12 + \\tfrac1{\\pi} \\arcsin \\sqrt{0.75} = \\tfrac56.", "001b05b435b5ca1ad78f35000decd950": "{\\log}\\circ g: x\\mapsto \\log x^2 = 2 \\log |x|", "001bae4d7ab52c8a0edd0a57e8d85701": "\\mathrm{Poi}\\left(\\frac{C(23, 2)}{365}\\right) =\\mathrm{Poi}\\left(\\frac{253}{365}\\right) \\approx \\mathrm{Poi}(0.6932)", "001bde6f639fbdb6285b504b829d3dce": "bx-x^2", "001c03be5066415d5004e2ad5cd961da": " \\mathbf{E}(z,t) = e^{-z / \\delta_{skin} } \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)})", "001c03cd18548eff08e44a1c6a40460b": "\n\\begin{bmatrix}\n0&1&0&1&0&0&0&0&0\\\\\n0&0&1&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&1&0&1&0&1\\\\\n0&0&0&0&0&1&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&1&0\\\\\n0&0&0&0&0&0&0&0&0\\\\\n0&0&0&0&0&0&0&0&0\n\\end{bmatrix}\n", "001c1c698265214507f5814c8c9bbe62": "f(x)= \\begin{cases} \n\\frac{\\nu}{x} \\left \\{ F_{\\nu+2,\\mu} \\left (x\\sqrt{1+\\frac{2}{\\nu}} \\right ) - F_{\\nu,\\mu}(x)\\right \\}, &\\mbox{if } x\\neq 0; \\\\\n\\frac{\\Gamma(\\frac{\\nu+1}{2})}{\\sqrt{\\pi\\nu} \\Gamma(\\frac{\\nu}{2})} \\exp\\left (-\\frac{\\mu^2}{2}\\right), &\\mbox{if } x=0.\n\\end{cases}", "001c5d215d3b2e814fd7cd1aa4ff25d9": "\\Sigma \\chi(n)\\,", "001c5d9c01ea2876ea70689bc638e282": "\\omega_{k}", "001c9503cb4f65ca231b9ff284672084": "\\mathbf{m}_1", "001ce3f609a62621c609e14916adfe6d": "s_2 = r_2 - cx_2 (\\mathrm{mod}\\,q) ", "001d17159eebbaefe304508512f197cc": "(-3n,5+5n)", "001d433c42ed4314705b2e49be9be3c5": " \\operatorname{Weight}(\\sigma) = \\prod_{i=1}^n a_{i,\\sigma(i)}.", "001da83ce80e2772b581b06641d3ca0c": "\\hat{U}^{\\dagger}\\hat{U} = I,", "001de956296095739ae9e0dc253c9269": "C\\ell(E) = F(E) \\times_\\rho C\\ell_n\\mathbb R", "001df96de10d73eb37ced28a37eed908": "\\theta=\\zeta_n^{a_{g,n}}", "001e2e0eb8437d7fafe16bdea61c10f3": "A/4\\ell_\\text{P}^2", "001e37a6336dbdddd5ac30dfc8964b0d": "r_{ij}", "001e7337ad903328d8889cc1ede11dc1": "h_{\\bar{a}}(\\bar{x})^{\\mathrm{strong}} = (a_0 + \\sum_{i=0}^{k} a_{i+1} x_{i} \\bmod ~ 2^{2w} ) \\div 2^w ", "001ea95cf12dc19b9749fa4c5600c6ed": " =\n\\begin{bmatrix}\nW_{11} & W_{12} & & \\\\\n & W_{22} & W_{23} & \\\\\n & & W_{33} & W_{34} \\\\\n & & & W_{44} \\\\\n\\end{bmatrix}\n", "001f090921d4950e090223a9db6fb0be": " \n\\mu_k(A-A_k)<\\epsilon,~\\forall k\\geq N.\n", "001f1531e895160d2f69783938a8d931": "\\Leftrightarrow P(B|A) \\ = \\ P(B)", "001f223d90ce21bb776d2afe729bfeac": "\\mathcal{C} = \\{ \\mathbf{q} \\in \\mathbb{R}^N \\}\\,,", "001f504393a856e45d22e00796231c32": "\\vec r (t)", "001f53b99bd91a14b91c2e4d6d62757a": " Z = \\sum_{j} g_j \\cdot \\mathrm{e}^{- \\beta E_j}", "001fb78130e343f9c200bd3aa484a3f7": "\\tau = \\int_{E_{th}}^{E'} dE'' \\frac{1}{E''} \\frac{D(E'')}{\\overline{\\xi} \\left[ D(E'') {B_g}^2 + \\Sigma_t(E') \\right]}", "001fdd3fb9e94017c83e467233ef49ec": "\\displaystyle{H=f-P(f_{\\overline{z}})}", "001fdfda5cdd7974a1f1e9f94673914b": "\nV = \\frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \\cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \\cdots + u_{s}(q_{s}) }\n", "00201b4361e4f3f5e5e6700e906ab77e": "f_1,\\dots,f_{2^n} : \\{0,1\\}^k\\to \\{0,1\\}", "002094dbb4ecaa0e1203ad652f1688dc": "\\theta_{k}-\\theta_{k-1}", "00213d222a8d87df7a615d7276c5a6cc": "s_0(1-s_0)", "0021503bde14e7a6b4016da9424dcf7d": "\\frac{e^x}{x^x}\\,", "002155c7baeb5176edda09dbdefab697": "\\frac{\\langle E \\rangle}{A} = \n\\lim_{s\\to 0} \\frac{\\langle E(s) \\rangle}{A} = \n-\\frac {\\hbar c \\pi^{2}}{6a^{3}} \\zeta (-3).", "0021c015403002b9cd758587bb4b6964": "q_2 = 1+\\frac{k+1}{6N}+\\frac{k^2}{6N^2}. ", "00222862eb12394ac0c8c08e36208b90": "R = R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} e_I^\\alpha e_J^\\beta.", "00223afcebe050cdafb431b459794ef3": " = pN(N-1)", "00225356a24bd1ec942aeca27c1a547a": " {v} \\,", "0022573b4553c3cd0fcebdfc5e357e55": " \\langle 0 | R\\phi(x)\\phi(y) + \\phi(y)R\\phi(x)|0\\rangle = 0 \\, ", "00226656ea0692401f9834fe6994da11": "S'", "0022669f61dc6da750ad3b0b6cd0ab48": "\\text{Ker} (k_* - l_*) \\cong \\text{Im} (i_*, j_*).", "0022f6407bd7dc02538291c1ffe49744": "x=\\frac{X-X_0}{\\lambda}", "00231e43bf02e01b0e106fc44adb74e5": "Y_1,Y_2,Y_3", "002326506700d44c9abb37d147e43b5b": "2v_c \\sin(\\alpha + \\beta) = c (\\cos(\\alpha - \\beta) - \\cos (\\alpha + \\beta)).\\,", "002366902dffd8673e5f838a29448df7": " e(\\mathbf{p},u)", "0023c250d7374bd8d6cec3b306e3c490": "p_1 = p_2", "002506aecf8a8eca0bddf976a3e83647": "x_r(\\theta_r(t))", "0025775d9f14d8821126387b6fa5c846": "D(G,H) = \\sum_{i=1}^{29} | F_i(G) - F_i(H) |", "0025b36cbda8365c09737acc9159df57": "\\gamma-", "0025cd57f9b2bd585ee2e2b8a93ef1ad": "P(X_1, \\ldots, X_N) = \\frac{e^{-\\frac{E}{k_{\\rm B} T}}}{\\int dX_1\\,dX_2 \\ldots dX_N e^{-\\frac{E}{k_{\\rm B} T}}}", "0025e1301274e14414e139894060dc23": "C(x_j,x_k)", "0025e75d1ffda9c4bff6b3de9560fe9d": "(gu)h = (gh^{-1})u", "00262cd78d796a5bb0baa8fd774728fd": "\\Delta^0_n,", "00267af4bf244fb88fc329938fac577c": "rK=D_K[F(K,L)]*K\\,", "00269b430e579348929cba8ca3c9990c": "p \\mid m_i", "00269e3bc1fc99fff7bc6d83b0d70bd0": "\\! t", "0026a625f7d3fd336acca8ae2bfcc06e": "\\! E_\\mathrm{h} / a_0 ", "0026b62d6355a23f08830d835b366f02": "2\\omega", "00279c44b6f5f02d0d5a761218b91ce4": " E_\\text{k} = E_t + E_\\text{r} \\, ", "0027acfd0c7490167b612c4b8b787509": "\\mathrm{ber}(x) / e^{x/\\sqrt{2}}", "0027e0646c279e8a69c9579dbef60613": "((-g)(T^{\\mu \\nu} + t_{LL}^{\\mu \\nu}))_{,\\mu} = 0 ", "002825bde096fa03b809c2b7fa66fe47": " \\sum_{g \\in G} f(g) g", "00287e7aa89ea392e3ecb9cb2837eeb9": "\\tilde{\\boldsymbol{\\Sigma}}", "002884828b36c8d042d8a853f57e5eec": "P(X > x) = Q(x) = \\frac{1}{\\sqrt{2\\Pi}} \\int_{x}^{+\\infty} e^-\\tfrac{X^2}{2}", "0028c604c387c78bc42c47b30010b464": "\\begin{pmatrix}\n-i & i\\\\\n0 & i\n\\end{pmatrix}", "00290f11d9ba0677c1614e97a3e1f097": "v(t) = \\int_{t_0}^{t} i(\\tau) d\\tau.\\,", "002917cdd4458fc6214ed9aaf24cd803": "\\frac{v^{2}}{2c^{2}}\\approx 10 ^{-10}", "0029190f5afee4bdfbdd64cd63bc229b": "\\delta^\\prime_0 \\Omega^\\prime_0 = \\left ( \\delta_0^{-1} + k^2 + kx - 1 \\right ) \\delta_0 \\Omega_0.", "002938e91e1d12948fb82e55131c99e7": "\\|Df\\|_{\\infty,U}\\le K", "00293e3339b4ec9cb5f75b6d8ad16918": "(z_0,\\dots,z_n)", "002978af538e0cb31098f49ab472ca41": "n! [z^n] Q(z).", "0029b0f2bac08e3532a265b95a74cde9": "\\lambda(L(B)) \\leq d", "0029c61e83cd7d4546a128f79bd99822": "A,A^2, A^4,...,A^{2^L}", "002a1bd731bf132e2f5b74a55b6f5c19": "R_A=R/A=5R/3", "002a358521632ae5e656e6a8b93ab594": "\\left(\\frac{\\partial \\mathbf{u}}{\\partial x}\\right)^{\\rm T}", "002ad7526d493f4eff5ee031f9462971": "PFB = \\frac{(3200)(FC)}{(FW)(MC)}", "002aeef2f67a7ab68b15f786fe0b673c": "L\\left(C\\right) \\leq L\\left(T\\right)", "002aef6e85c21276cf6521320260f5a6": " P^{\\, a} {}_{\\, ;\\tau} = (q/m)\\,F^{\\,ab}P_b", "002af1a2280bc443756033b1f386b056": "v = \\frac{c}{n}", "002b0f6cbb93d8febf576f9419105ab4": "\\eta =1-\\frac{\\mathit{u}_1 - \\mathit{u}_4 }{ \\left(\\mathit{u}_2 - \\mathit{u}_3\\right)} = 1-\\frac{(1-4)}{ (5-9)} = 0.25 ", "002b6847b0190969eb52946cc76f76ea": "\\left\\{\\begin{matrix}ax+by&={\\color{red}e}\\\\ cx + dy&= {\\color{red}f}\\end{matrix}\\right.\\ ", "002b89f0fa3e9036b33e69d614b18060": "= [P^{(\\pm)} F, G]^{IJ} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Eq.8", "002b94338d3ad1e2adc60862582ccff2": "\\text{bind}\\colon A^{?} \\to (A \\to B^{?}) \\to B^{?} = a \\mapsto f \\mapsto \\begin{cases} \\text{Nothing} & \\text{if} \\ a = \\text{Nothing}\\\\ f \\, a' & \\text{if} \\ a = \\text{Just} \\, a' \\end{cases}", "002b9647d9a7aacbaaf44a4c005c7f54": "\\Delta \\tau = \\sqrt{\\frac{\\Delta s^2}{c^2}},\\, \\Delta s^2 > 0", "002ba4169a0d47f5c24244d1f9a82cfd": " f^{*} = \\frac{bp - q}{b} = \\frac{p(b + 1) - 1}{b}, \\! ", "002c115aa5aba4aac873a44e7ec65ae1": "\\alpha_{\\tau\\tau}-\\beta_{\\tau\\tau}=e^{4\\beta}-e^{4\\alpha},\\,", "002c1f766558995e2b1166f45a9eb1b0": "\\scriptstyle w[n]", "002c5051d7053790557612d8d2ef2019": "h=C{{\\left[ \\frac{k_{v}^{3}{{\\rho }_{v}}g\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)\\left( {{h}_{fg}}+0.4{{c}_{pv}}\\left( {{T}_{s}}-{{T}_{sat}} \\right) \\right)}{{{D}_{o}}{{\\mu }_{v}}\\left( {{T}_{s}}-{{T}_{sat}} \\right)} \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{4}\\;}}", "002c8bca1a57ee65188cb4adb14e632c": "f:M \\mapsto N", "002c92314c9a6e81956f72dbe61c39b2": "F=\\overline{(A \\wedge B) \\vee (C \\wedge D)}", "002cbb90309335ef7183f232ac4bf55d": "a^2+c^2=b^2.\\quad", "002ccee36eec167b5d69bb76524b75fd": "SU_{\\mu}(2) = (C(SU_{\\mu}(2),u)", "002cddc16ea0c92f40c38202e128497f": "\\mathrm{2\\ Squares\\ of\\ Land} =(\\frac{\\mathrm{77\\ acres}}{\\mathrm{3\\ Squares\\ of\\ Land}}) \\cdot 2\\ Squares\\ of\\ Land\\ = 50.82\\ acres ", "002d84f8f9870a8115b7866dae7d6d31": "\\sigma_y^2(\\tau) = \\frac{2\\pi^2\\tau}{3}h_{-2}", "002d94ef85bc9ea2c41a550659eb05eb": "\n \\mathbf{E} = \\xi \\exp[i(kx - \\omega t)] \\mathbf{\\hat{x}}\n", "002e06e607e26e75da249f7016a07881": "(-m_i\\partial_{tt}+\\gamma_iT_i\\nabla^2)n_{i1} = Z_ien_{i0}\\nabla\\cdot\\vec E ", "002e08819822cb8016bf5d8593615452": "\\varphi = 2\\cos{\\pi\\over 5} = \\frac{1+\\sqrt 5}{2}\\qquad\\xi = 2\\sin{\\pi\\over 5} = \\sqrt{\\frac{5-\\sqrt 5}{2}} = 5^{1/4}\\varphi^{-1/2}.", "002e5e677339873ae56de031260218b0": "N / \\Gamma ", "002e83cd7e3a8308c5836320f9ac437c": "\\langle x, y \\rangle\\ M\\ N = M\\ x\\ y\\ N", "002ec7f385d551b2c31aedcf1fce7f32": "f_{k,i}", "002f374ea2a9a5316d9dc2de5ba0db82": "\\begin{align} {z \\choose k} = \\frac{1}{k!}\\sum_{i=0}^k z^i s_{k,i}&=\\sum_{i=0}^k (z- z_0)^i \\sum_{j=i}^k {z_0 \\choose j-i} \\frac{s_{k+i-j,i}}{(k+i-j)!} \\\\ &=\\sum_{i=0}^k (z-z_0)^i \\sum_{j=i}^k z_0^{j-i} {j \\choose i} \\frac{s_{k,j}}{k!}.\\end{align}", "002f4e6f409b9611d103847696ce30dd": "C_j^n", "002f4eb7b268cb63dbf1116acb66ed23": " \\Psi(x,t) = \\sum_n a_n \\Psi_n(x,t) = a_1 \\Psi_1(x,t) + a_2 \\Psi_2(x,t) + \\cdots ", "002f8af8f796e82cb12d524429901412": "\\rho: S \\times X \\rightarrow \\{0,1\\}", "002fa9e5e3ba534bf208264e185bab38": "u\\equiv\\frac{r}{\\alpha^2}", "002fde17fb2df61903a3cb830c71241b": "(a_1,\\ b_1,\\ c_1,\\ d_1) + (a_2,\\ b_2,\\ c_2,\\ d_2) = (a_1 + a_2,\\ b_1 + b_2,\\ c_1 + c_2,\\ d_1 + d_2).", "0030177130f83768f8c7205d73fdfadc": "P(y)\\,dy + Q(x)\\,dx =0\\,\\!", "00306d74825ca4c699ac02b1aa3caa18": "=2^2\\cdot5\\cdot17\\cdot3719", "00308fa277a754af480d4ed68cce2a56": "A=\\frac{2}{3}bh", "0030dd6def07c2a872c23491e5c9ac7d": "\\displaystyle{K=\\begin{pmatrix} I & 0 \\\\ 0 & -I \\end{pmatrix}.}", "0030ee1373b5795f95a2d5c2a66b49e5": "\\Delta_{\\mathrm{adv}}(x-y)", "003101e85d556302192b466977a60a8d": "\\langle M, N \\rangle = \\lambda z.\\, z M N", "00310fe1c22c34624ec5fd12b34213a3": " R_{s\\ normal} = \\sqrt{ \\frac{\\omega \\mu_0} {2 \\sigma} }", "003144652c05f21650272d2e79242048": "s, h' \\models P", "003163f025c02255900f7c4225a576b1": " ([\\mathbf{t}]_{\\times})^{T} = \\mathbf{V} \\, (\\mathbf{W} \\, \\mathbf{\\Sigma})^{T} \\, \\mathbf{V}^{T} = - \\mathbf{V} \\, \\mathbf{W} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} = - [\\mathbf{t}]_{\\times} ", "0031b1a0e5881f6b0ca5ce52f4ab1b04": "f(x)=(x + 1)^{2}(x - 1), \\,", "0031b38c8e97ea03a011524a0ea2b77f": "\\lambda (\\lambda 1 (1 ((\\lambda 1 1) (\\lambda \\lambda \\lambda 1 (\\lambda \\lambda 1) ((\\lambda 4 4 1 ((\\lambda 1 1) (\\lambda 2 (1 1)))) (\\lambda \\lambda \\lambda \\lambda 1 3 (2 (6 4))))) (\\lambda \\lambda \\lambda 4 (1 3))))) (\\lambda \\lambda 1 (\\lambda \\lambda 2) 2)", "003206d6d973a25d27d7badeae180f6a": "\\begin{align}\n Area &{}= \\frac{1}{2} * base * height \\\\\n &{}= \\frac{1}{2} * 2 \\pi r * r \\\\\n &{}= \\pi r^2\n\\end{align}", "003222c0d800ed511b981e1590fd5579": " 0.0000182\\dots,\\, ", "003248f7ade6dc2990d6ae7a805628a8": "\\frac{1,310,000\\ \\mathrm{N}}{(2,430\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=54.97", "003275472d45cd9706e6d88486831729": "\\phi_1,\\phi_2,\\phi_3 ", "0032b9d5134fe210abc9011e684a4d23": " a _{i}", "0032f418f93bbaab612a5213f21b9122": "T_r = {T \\over T_c}", "0033322e706f0c7b7dbae50459e4e1a2": "\\Pi\\,", "00338841eb1ca80fef553f18dd02d7db": "\\forall x \\Big(\\forall y (y \\in x \\rightarrow P[y]) \\rightarrow P[x]\\Big) \\rightarrow \\forall x \\, P[x]", "003395de5184f994ecb8f96a60890b6e": "\\chi_G(\\lambda) = (-1)^{|V|-k(G)} \\lambda^{k(G)} T_G(1-\\lambda,0),", "0033aa54194929b25fd3cf4bb6c7d369": "z^p\\overline{z}^q.", "0033ccc8d80038ec44629c31966dfe06": "v_{(G; c)}(\\{1,3\\})=23", "003411f88f779a77e67b7eccd9c6d41a": "\\rho _{\\alpha +} ^{i_0 } \\ge A_{\\alpha + }^{\\sigma (i_0 )} ", "00345c04233a175efdd1e2494c42a238": " \\phi_1 = -30^\\circ...+30^\\circ", "0034991f8f6e6f84b95247f345004bb4": "\\binom Sk\\,", "0034befe82b7a681848dd6ebb6634a0e": "\\begin{cases} \n 1 & (e^{-p})\\mbox{ no disaster} \\\\\n 1-b & (1-e^{-p})\\mbox{ disaster} \\\\\n\\end{cases}", "003529eda35d403c850d8aed6ca10aef": "y = \\psi^{-1}(x)", "003532a7886018f1e650314b310a3290": "x^{q^{2}}\\neq x_{\\bar{q}}", "00354ed1ef1395977fc43f8e6c9aed64": "G_{\\delta\\sigma\\delta}", "0035522d0c7bcb717f215070b1eeef30": "\\log_2 (1-p) + 1-R", "0035587f66355cdac3b284b1fd4645dd": "\\displaystyle{R(Q(b)a,a)Q(b)=Q(b)R(Q(a)b,b)=R(b,Q(a)b)Q(b),}", "00355b116feb4556455199c0b3622e04": "\\gamma_I", "00357df66075bc66d2f4339108604c92": "T \\rightarrow \\infty", "00359027c15ea5ebdf1e499d7c8bec3a": "\n\\langle \\varphi, \\varphi_j \\rangle = \\int_\\mathcal{T} \\varphi(t)\\varphi_j(t) dt, \\text{ for } j = 1, \\dots, k-1. ", "0035cdf76a30ed71e027ee0cc502d979": "1928 = [43, 36]_{44}", "0035ff7f60718d7d705c9d61c4ab5431": "\\ \\beta = \\pi - tan^{-1}(\\frac{1}{10}) - tan^{-1}(L/D) ", "003625928997e0a4a1b8483667736ec6": " \\vec X(n) = \\{ X_d(n) \\}, d = 1..D.", "003656b0a5cdfdf2326d037c9864a835": "dU=TdS-PdV + \\sum_i \\mu_i dN_i.\\,", "003695b09b8e5ddc7fcca8ee1aed316c": " S \\subseteq [n] ", "0036ac1e1ae00ff6a59a729ecdb0ca91": "T_c", "00372ba6f6a4645a32d220eb15577468": "\\mathbb{CFM}_I(R)", "0037ecfd65cf97652c38001750960741": "t^{\\mbox{th}}", "003920cd429ea833122f2971b7944ce1": "\\ P_2= x_2P^*_2f_{2,M}\\,", "00392327200f6a4d35e9c33e723c7e26": "m = n \\sqrt{2}", "003935cf7152b790d696b09642eeea6b": "r_n = (1/2) - x_n h_n", "003941bb8340136488f449dfee574111": "dn_1", "003987dd42d31ffec69d55619deb3d97": "P_1(X)=P(X)/(X-\\alpha_1)", "0039cbae10746ef0b5c1afe4589e9a3e": "(S; \\wedge, \\vee)", "0039f36e9885ebeb4de300eb0f22ebe4": "H^*_GX,", "003a5820c464d82eca6633352a4c42b9": "r_m = r_c ( 1 - t ) \\, ", "003a5ac3c6316db47dde21e454be0a6c": "S = -k_B\\,\\sum_i p_i \\ln \\,p_i,", "003a70ac099d1c13e037072a7f78ca76": "\n U = \\frac{1}{2} \\int_0^a \\int_{-b/2}^{b/2}D\\left\\{\\left(\\frac{\\partial^2 w}{\\partial x^2} + \\frac{\\partial^2 w}{\\partial y^2}\\right)^2 +\n 2(1-\\nu)\\left[\\left(\\frac{\\partial^2 w}{\\partial x \\partial y}\\right)^2 - \\frac{\\partial^2 w}{\\partial x^2}\\frac{\\partial^2 w}{\\partial y^2}\\right]\n \\right\\}\\text{d}x\\text{d}y\n", "003ab5cf816a2d6306acef92162bd5e5": "n < \\lambda \\leq n+p", "003af996ea8f154c29fdcff0f9762f62": "\\theta_k(z) = \\sum_{\\gamma\\in\\Gamma^*} (cz+d)^{-2k}H\\left(\\frac{az+b}{cz+d}\\right)", "003b112cec5f2a74b4eaafc0d1627242": "\\tfrac{\\vec x_{n+1}-\\vec x_n}{\\Delta t}", "003b125ee6a3d44d4f40c957f2611b54": "\\phi _1 , \\phi _2 , \\dots , \\phi _{n-1} \\,", "003b2ceba9c9fca8743b7ada1a22e559": "V_0 = 0,1 ", "003b435dc6f1352fe48d6ab32e5dfd2a": "\\int_{-\\infty}^0 f(x)\\,\\mathrm{d}x=\\pm\\infty", "003b627e9de797d9a9ce175fb6392235": "\\frac{d^2}{dx^2}X=-\\frac{\\omega^2}{c^2}X\\quad\\quad\\quad", "003be3626a91a1ff64ddfc5dbd4edb48": "\\|f_{\\theta}-f_{\\theta'}\\|_{L_1}\\geq \\alpha,\\,", "003c39c6732e6fff7f2947459f7fa5df": "\n\\begin{array}{l}\ns_0=1\\qquad s_1=0\\\\\nt_0=0\\qquad t_1=1\\\\\n\\ldots\\\\\ns_{i+1}=s_{i-1}-q_i s_i\\\\\nt_{i+1}=t_{i-1}-q_i t_i\\\\\n\\ldots\n\\end{array}\n", "003c664848c04c53bedfd7853a47516d": "(-\\mu_j)^{-1/2}", "003c67ab880e13638d98d028457ce502": " V_1 = k_1 [E_{1T}], ", "003ccc5b040e4941beaf0e1c7b71604c": "n \\geq n_0 ", "003d17dfe0f53c5ec3bb56ba64d54d39": "\\{a_n\\} \\subset G", "003d1b455ffe1cfd3d52390be60afabc": "\\|f\\|_{L^{p,\\infty}(X,\\mu)}^p = \\sup_{t>0}\\left(t^p\\mu\\left\\{x\\mid |f(x)|>t\\right\\}\\right).", "003d5dbcdaf031030dca9e8aeb0b7e5d": "= \\frac{k}{n}.", "003d667ac140e61d45eb1c0148ce6885": " {\\alpha \\choose k} = \\frac{(-1)^k} {\\Gamma(-\\alpha)k^ {1+\\alpha} } \\,(1+o(1)), \\quad\\text{as }k\\to\\infty. \\qquad\\qquad(4)", "003d9844a3d178796ad777fa6e22e467": "\nS_{ij} := r_{ij}^{(t)} + g_{ij}^{(t)} + b_{ij}^{(t)}\n", "003dc09bb55482b2f72537dd1850d588": "\\sigma^2_N = \\frac{(N-1) \\, \\sigma^2_{N-1} + (x_N - \\bar x_{N-1})(x_N - \\bar x_{N})}{N}.", "003dd9b388c28533104e73e1b5429c89": "(\\psi'(\\theta))^2/I(\\theta)", "003de6af834956a356ade65eef50d280": "\\Delta\\ W_{ij}(n) = \\gamma\\ \\Delta\\ W_{ij}(n-1) \\Delta\\ R(n) + r_i(n) ", "003e239d39f2c653d6e74c9ddf2f4fe4": "\\kappa = v \\frac{\\mu \\Delta x}{\\Delta P}", "003e40578e8a8611e92faedeebe7f2b8": "x_i(\\mathbf{w}, y) = \\frac{\\partial c (\\mathbf{w}, y)}{ \\partial w_i}", "003e4578d0879dbf7092d45082daf55e": "d^* = \\sup_{y^* \\in Y^*} \\{-f^*(A^*y^*) - g^*(-y^*)\\}", "003e570691573cf65b75f9d7f3d399c1": "\\alpha_c : S(c,c)\\to T(c,c)", "003e75b4ed582eaf7e6001a024932ecf": "n = \\prod_{i=1}^r p_i^{a_i}", "003eae0fd1605ab2c3d9cb22c0e610ac": "H(j \\omega) = \\mathcal{F}\\{h(t)\\}", "003ec252d81828cf0f19388f49018e57": "X_3", "003f2cd1d7c8d8357deec5a359889df5": "\nds^{2} = d\\tau^{2} - \\frac{r_{g}}{r} d\\rho^{2}\n- r^{2}(d\\theta^{2} +\\sin^{2}\\theta\nd\\phi^{2})\n", "003f38a83670c4350403298b1f4364b6": "e_{ij} = \\mathbf{e}_i\\cdot\\mathbf{e}_j.", "003f38e45eec556ade8244f8870ae85e": " {S_3 \\over S_2} = {{16\\over15} \\div {135\\over128}} ", "003f7619ae0c1da19bd1ae62e01dcd2d": "\\pi/4", "003fa3ffdad3e57a239d9a8ce9ff8556": "N=O(n)", "003fcba6cfeca74b28e6a63de15178d5": "(S^0, S^1,\\dots)", "003ffcbad12d7b85054a98ad396622b9": "A = 2\\left(6+6\\sqrt{2}+\\sqrt{3}\\right)a^2 \\approx 32.4346644a^2", "004004a61e6f526c6c2bf255a5010811": "\\mathfrak M (K)", "00400e43c571b943e3788f989b6e4f4d": "\\scriptstyle(\\lnot u)\\Rightarrow v", "00404e17a85b5f39a7eb42f087f3c3ff": "(x+y)^n = \\sum_{k=0}^n {n \\choose k}x^{n-k}y^k = \\sum_{k=0}^n {n \\choose k}x^{k}y^{n-k}.\n", "004079a9e10ff7052646221da1745005": "\\,Q", "00409987890d39631dfb17ba290a11db": "t_a = t+\\frac{|\\mathbf r - \\mathbf r'|}{c}", "0040a8d09dc53fcd583183a7b90c38eb": "\\operatorname{Ext}_R^i(M,\\overline\\Omega) = \\operatorname{Hom}_R(H_m^{d-i}(M),E(k))", "0040bc7d53402e15e76efd567502219f": " D_x = \\frac{1}{i} \\frac{\\partial}{\\partial x}. \\,", "0040ddcb1ff90a92a8701bef0dc2e6f7": "\n\\left( \\frac{dr}{d\\tau} \\right)^{2} = \n\\frac{E^2}{m^2 c^2} - c^{2} + \\frac{ r_{s} c^2}{r} - \n\\frac{h^2}{ r^2 } + \\frac{ r_{s} h^2 }{ r^3 }\n", "00410f0f22d52a5b186f73d0c721e3b2": "\\varphi = \\frac{1 - \\sqrt{5}}{2} = -0.6180\\,339887\\dots", "00415718523d2088141fa516e7cb17cb": "T_\\mathrm{W}[\\rho] = \\frac{1}{8} \\int \\frac{\\nabla\\rho(\\mathbf{r}) \\cdot \\nabla\\rho(\\mathbf{r})}{ \\rho(\\mathbf{r}) } d\\mathbf{r} = \\int t_\\mathrm{W} \\ d\\mathbf{r} \\, ,", "00417172fd9a1d80f3d7ce0d1bdbefa7": "I_{\\mathrm{center}} = \\frac{m L^2}{12} \\,\\!", "00418dc4838b3092afa6d069011fefd0": "Y_\\alpha(z)\\sim-i\\frac{\\exp\\left( i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)}{\\sqrt{2\\pi z}}\\text{ for }-\\pi<\\arg z<0", "00423a7a5fd53953495fb4aed95bc108": " h(-,Z) = d\\Delta", "00424861f5673267a2705f68bf870be6": " \\displaystyle M(f) = \\sup_{x\\in D} \\mu(f'(x)).", "00427b119652e0a312fd6a9200137efc": "\\left(\\frac{1 + \\sqrt{1-\\beta^2}}{2}\\right) T", "0042b8b4bd18cd7f590f833a653788ae": "S - S_0 = S - 0 = 0", "0042c1492109c45e812558aac1ee6599": " \nD = O^T A O = \\begin{bmatrix}\n \\lambda_{-}&0\\\\ 0 & \\lambda_{+}\n\\end{bmatrix} \n \n ", "0042d0c90d4c6cc652c0b54ce47f81a1": "f( B_1, B_2, \\ldots, B_m)\\subset B", "0043019f31c2e65deeee14435ed0c2df": " \\nabla \\cdot ( A \\nabla u ) = 0 ", "0043bfae9decf0fe362e422acefcbe4f": "\\hat{ \\textrm{d}}_j", "0043e6787bf9c93b5f9c05ea592c6ef5": " \\operatorname{Var}(X \\mid X>a) = \\sigma^2[1-\\delta(\\alpha)],\\!", "00446ccbf030e3c1559f52147c13d9e7": "(\\tfrac{q^*}{p})=1,", "00448c4852a2cc9d5da56bb6d3a53614": "\\int_{\\mathbf{R}^d}(f*g)(x) \\, dx=\\left(\\int_{\\mathbf{R}^d}f(x) \\, dx\\right)\\left(\\int_{\\mathbf{R}^d}g(x) \\, dx\\right).", "004494b2606a7adaf174db7b6dc17d14": " \\begin{cases}\n \\frac{\\partial L_2 }{\\partial w} = 0\\quad \\to \\quad w = \\sum\\limits_{i = 1}^N \\alpha _i \\phi (x_i ) , \\\\\n \\frac{\\partial L_2 }{\\partial b} = 0\\quad \\to \\quad \\sum\\limits_{i = 1}^N \\alpha _i = 0 ,\\\\\n \\frac{\\partial L_2 }{\\partial e_i } = 0\\quad \\to \\quad \\alpha _i = \\gamma e_i ,\\;i = 1, \\ldots ,N ,\\\\\n \\frac{\\partial L_2 }{\\partial \\alpha _i } = 0\\quad \\to \\quad y_i = w^T \\phi (x_i ) + b + e_i ,\\,i = 1, \\ldots ,N .\n \\end{cases} ", "00449fa9f66ff928b3c0d4f7a0bfd190": "\\Pr\\left\\{E_{a^{n}}\\right\\}", "004573673bb14177fd56ecc3a0259b49": "\\ [A]_t = -kt + [A]_0", "00460704eeb45cb43f638437da0f138c": "T_i = K_i d_i", "00463a2876f07b3e7a8c4ce619c532a5": "\\left\\{\\left(x, y\\right) \\in A \\times B : xRy\\right\\}", "004651c8ecc3cdd380d5ac44723bb634": " [x_t - x^{*}] = A[x_{t-1}-x^{*}]. \\, ", "0046849cd8f4bd8eb09652cf7151a14e": "\\mathbf{aaaaaa}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{281DAF40}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathrm{sgfnyd}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{920ECF10}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathbf{kiebgt}", "0046ab0e7bd8520919d98cc057dbff07": "\\beta_k=\\frac{\\partial S}{\\partial\\alpha_k},\\quad k=1,2 \\cdots N ", "0047362db8e80d2564e21c2adad1ca45": "q^{42}", "004789ef923dbade2d1256e476da60ba": "\\theta_1 < \\theta_2", "0047beba5dbab2fe8e288d1e9b1d5192": "R_{k,l}", "0048528384f5b1b70e8d279c559c5436": "f:I\\rightarrow \\mathbb{R}", "004875f8b2294b19c688df2856489d01": "\\alpha(d) \\le \\left(\\sqrt{3/2} + \\varepsilon\\right)^d", "00489f32547332d509d28f64be77a6c3": " \n\\begin{cases}\nN_j\\left(U^\\left(n\\right)\\right)=\\Gamma_{jk}U_k^\\left(n\\right)-U_j^\\left(n\\right) \\\\\nM_j\\left(U^\\left(n\\right)\\right)=p_i~a_{ijkl}\\frac{\\partial U_k^\\left(n\\right)}{\\partial x_l}+\n\\rho^{-1}\\frac{\\partial }{\\partial x_i}\\left(\\rho~a_{ijkl}~p_l U_k^\\left(n\\right)\\right) \\\\\nL_j\\left(U^\\left(n\\right)\\right)=\\rho^{-1}\\frac{\\partial }{\\partial x_i}\\left(\\rho~a_{ijkl} \\frac{\\partial U_k^\\left(n\\right)} {\\partial x_l} \\right)\n\\end{cases}\n", "00493a8b1b2cb014c676b1c7f2dd1af1": "c = {r \\over {1-(1+r)^{-N}}} P_0", "0049559f98dfaee50543d7d517d24204": "\\mathcal{X}(S(z;u))=\\mathcal{X}(u)+z\\ ", "00495fa4b21e827afa0a14a0556bbb4c": "P_{em} = \\frac{3R_r^{'}I_r^{'2}n_r}{sn_s}", "00496954c373cd5810ba8c18bbaec16c": "\\dot q^\\mathrm{T}", "004984cb0fbd087fc4aa5d6ba33188c2": "dE_\\theta(t+\\textstyle{{r\\over c}})=\\displaystyle{-d\\ell j\\omega \\over 4\\pi\\varepsilon_\\circ c^2} {\\sin\\theta \\over r} e^{j\\omega t}\\,", "0049ea3f4597154927b84fc6183b2ec1": "\\mathfrak{P}^{51}", "004a0f215460cccf77c5be94cd5957a4": "\\gamma=3\\Omega/4\\ ,", "004a0f66dcf0e61c0561ce8c17d34024": " f^{\\mu} = - 8\\pi { G \\over { 3 c^4 } } \\left ( {A \\over 2} T_{\\alpha \\beta} + {B \\over 2} T \\eta_{\\alpha \\beta} \\right ) \\left ( \\delta^{\\mu}_{\\nu} + u^{\\mu} u_{\\nu} \\right ) u^{\\alpha} x^{\\nu} u^{\\beta} ", "004a192738d835e7c80660759807ffb7": "= \\sum_{k=1}^{d} \\left(\\dot v_k \\ + \\sum_{j=1}^{d} \\sum_{i=1}^{d}v_j{\\Gamma^k}_{ij}\\dot q_i \\right)\\boldsymbol{e_k} \\ . ", "004a929cbdcada032006e670aec159ce": "\\qquad{\\it (Comp1)} \\quad \\frac{\\displaystyle M \\ \\rightarrow\n\\ M'} {\\displaystyle M\\|N \\ \\rightarrow \\ M'\\|N}; \\qquad \\qquad {\\it (Comp2)}\n\\quad \\frac{\\displaystyle M \\ \\rightarrow \\ M'\\qquad\\displaystyle N\n\\ \\rightarrow \\ N'} {\\displaystyle M\\|N \\ \\rightarrow \\ M'\\|N'}", "004a9f231095f3c08e2f82e54dd4643f": "\\exp\\left(\\sum_{n=1}^\\infty {a_n \\over n!} x^n \\right)\n= \\sum_{n=0}^\\infty {B_n(a_1,\\dots,a_n) \\over n!} x^n.", "004acfd27331d9504ebbf27a7a9ffcde": "(\\cdot,\\,\\cdot)", "004ad6eb8267d487727c4f2c03c5ceae": "F_0=\\left\\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\\right\\}", "004b071ceacb7dbbc6505f34eab1216d": " \\frac{D_g u_g}{Dt} - f_{0}v_a - \\beta y v_g = 0 ", "004b15ab050ca1fe6e6092337b1116a3": "(\\alpha_j - \\alpha_i)", "004b1f52d0b2112708389023597f813a": "S \\subset L\\,", "004b8fb50f7aa0ce50232bb773f5f387": "\\operatorname{E} (X_t)=\\operatorname{E} (c)+\\varphi\\operatorname{E} (X_{t-1})+\\operatorname{E}(\\varepsilon_t),\n", "004ba7069754fed522854714a8660e16": "\\overline{z} = z \\!\\ ", "004bc28bf353a7a7dae3f540aa4c86a5": "I_c", "004c00048d155c6aaeee77859a8b45a8": "\\, A \\mapsto M\\alpha(A)M^{-1} ,", "004c04db969c835339fb23593190d46f": "\nE\\bar{X}_A = \\mu_{HA}\\frac{p_{HA}}{p_{HA}+p_{LA}} + \\mu_{LA}\\frac{p_{LA}}{p_{HA}+p_{LA}},\n", "004c69ff4b40f7cceab9e42b8f7370fa": " {d^2 \\bar h^i \\over ds^2} + 2 \\Gamma^i_j {d \\bar h^i \\over ds} + {d \\Gamma^i_j \\over ds} \\bar h^j + \\Gamma^i_j \\Gamma^j_k \\bar h^k + \\bar R^i_j \\bar h^j = 0 ", "004c72301f64855e456aa920a32a1d7c": "\\tbinom24", "004cc0101dda11ac74e94adc07c9aae2": "det(A)\\ne 0", "004cf65ad83a6a03009f6629678c1bde": "i^2= -1", "004d00460322f8ea8cfce85f9084898d": "\\lim_{\\mathbf{h}\\to 0} \\frac{\\lVert f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - f'(\\mathbf{a})\\mathbf{h}\\rVert}{\\lVert\\mathbf{h}\\rVert} = 0.", "004d51a85883bac7a3bd93d24453cd39": "f(x_i) = \\sum_{f=1}^n c_j \\mathbf K_{ij}", "004d61714e5c41d0bc9aff7cb62b7259": "(a_n)_{n\\in\\N} \\times (b_n)_{n\\in\\N} = \\left( \\sum_{k=0}^n a_k b_{n-k} \\right)_{n\\in\\N}.", "004dadc66378395b6a21b73bdbab86e3": "C=\\{C_k^i\\}", "004dbfe6dc52810c3e2192e98e8edac0": "\nM(X) = \\left( {\\begin{array}{*{20}c}\n \\mu \\\\\n \\Sigma \\\\\n\\end{array}} \\right)\n", "004df6f3067e46c45e07b3e9e96f47d3": "\\sigma_\\text{l}", "004e1f9156a736730142d8026957f78e": "\\hat{\\nu}", "004e234f6cdf2e3ff6785774b71b23b2": " \\frac{\\partial F \\left( u \\left( t \\right) \\right)}{ \\partial u}. ", "004e35035b2f412209b351f3df19dbf0": " \\ddot{r} = \\frac{1}{2} \\, \\frac{d}{dr} \\left( (E^2-V) \\, (1+m/r)^4 \\right) ", "004e652b26937bc4fc57cff56c8c45c5": " f,g_1,\\ldots,g_n\\in H", "004ed4a583fb5e14530d8a50c277465f": "\n(0, 653, 1854, 4063) \\rightarrow\n(653, 1201, 2209, 4063) \\rightarrow\n(548, 1008, 1854, 3410) \\rightarrow\n", "004f13ea26fac88c1336de7014e5d86e": " (\\sqrt{2},1); \\quad (-\\sqrt{2},1); \\quad (\\sqrt{2},-1); \\quad (-\\sqrt{2},-1); \\quad (0,\\sqrt{3}); \\quad (0,-\\sqrt{3}). ", "004f36fdc2ad8de69901b2d8334cbdc4": " N_0 k_B", "004f5f4d152754122d438075e243d9fd": "\\frac{b^2}{\\sqrt{a^2-b^2}}", "004f77d74952fece0fe7da9c0e9f362d": "A \\leq_{F} B", "004f97f6e33b7a3b21d1b8ae701da2ef": "u(x,\\dot{x})", "004fb86ed073c6e27d750267bf963bf9": "c r^n \\in I^n", "004fbd61429af6ede34c05cb20415624": "(x-c_2)^2", "004ff877b585feec05fc1619795865b4": "R\\mathcal S(\\mathcal F \\ast \\mathcal G) = R\\mathcal S(\\mathcal F) \\otimes R\\mathcal S(\\mathcal G)", "005011b1c44424b4077226fb6ed12dbd": "p_\\varepsilon (x,t) = 0\\text{ for }x \\in \\partial\\Omega_a", "0050398776b0feb63e2eeb7384b6dcd7": "\\Gamma_{\\infty}", "0050e58f180026f58f4d56eef3a51021": "\\hbar {\\mathbf k'}", "005119eb2768ca72c1837f074d72d0a7": "\\phi(t) = {\\rm Tr}[f(B + tC)]", "0051740ae877c5b18dee89574732c99a": "n_2^2\\sigma_2^2-2\\sigma_2n_2^2\\sigma_\\mathrm{n}+n_2^2\\lambda=0\\,\\!", "0051788326e3478daf0813cdc52388a5": "\\mathrm{SO}(2)", "0051f0b0fff70aba89b8d5352d80722b": "N=g^{\\mu\\nu}K_\\mu K_\\nu\\;", "0052077694b84a2fbc16b07c951977a6": " W= \\frac {1}{iwc_0 Q} (D-R) \\quad (2.6)", "005259dad02c95d61a8dcba7035615ee": "f(b)-f(a)\\geq f(x_n+0)-f(x_1-0)=\\sum_{i=1}^n [f(x_i+0)-f(x_i-0)]+", "005302f209db336a7561fc004e245c6d": " y''(t) = f(t, y(t), y'(t)), \\quad y(t_0) = y_0, \\quad y'(t_0) = a ", "0053479d9005b96a7e238f3c76676ec5": "\\exp(\\lambda (e^{t} - 1))", "00535d682974b6ce2abed6e0d9e65e30": "d^2=4*x*b_{7}*c_{12}^2=", "0053a62968e1874c0e873d21cf4634fa": "\\underline{x} \\in \\R^n", "0053bd74249ba2edd4ff39532c528ca8": "c_2 = 2.04901523, \\,\\!", "00546b61d4996074c0643b1be8cf5802": "\\{| \\phi_i \\rangle\\}", "0054cb6e5b751157081556d7e575ca24": "L(w)", "0054e06028ca38fa0a1cc337ae69ed98": "\\mathrm{core}_2", "005503b59bc42d27c5c1ba90c5099d82": "\na = \\frac{a^4+b^4+c^4+a^2b^2+b^2c^2+c^2a^2}{\\left( a^2+b^2+c^2 \\right)^2} \\Delta\n", "0055139ef653b9bfbedea5d4c316a3d4": "\\mathcal{E}(\\exp)=\\{0\\}", "00552124bea53f3a68f87e28129a5903": "e^{(1)}_i = a_i", "005522a913e457a072a578ef939fb5f3": "\\sigma = 0, \\sigma = 0.2, \\sigma = 0.4, \\sigma = 0.6, \\sigma = 0.8, \\sigma = 1", "00556d8eb6763f7cab142e2c7caf0e95": "D = \\prod_{i=1}^K d_i.", "005589a38037bf9df004958bb97d463c": " I_x(a,b) = \\sum_{j=a}^\\infty \\binom{a+b-1}{j} x^j (1-x)^{a+b-1-j}. ", "0055d263238cda7b7306068f1d676b1f": " B_0 = \\frac{\\hbar^2}{2 m_0} + \\frac{\\hbar^2}{m_0^2} \\sum^{B}_{\\gamma} \\frac{ p^{y}_{x\\gamma}p^{y}_{\\gamma x} }{ E_0-E_{\\gamma} }, ", "0055e644da0728d42924ea03350ea963": "ji=-k", "005629782cc4d869040eb39436ff3edd": "\\sigma_{mk}", "0056b3d282c468d9da43689c4ea780e3": "\\mathcal{O}(x_1,\\ldots,x_n)", "0056b8fd312214ab941b8bb4997b7c96": "\\operatorname{P}(X\\leq m) = \\operatorname{P}(X\\geq m)=\\int_{-\\infty}^m f(x)\\, dx=\\frac{1}{2}.\\,\\!", "0056ed7091c7f8276cbd7eee8c0e5577": "Y = \\beta T_8 + I X", "00572f45e35e977389316f0eef29c429": "\n\\psi_0 |0\\rangle + \\int_x \\psi_1(x) |1;x\\rangle + \\int_{x_1x_2} \\psi_2(x_1,x_2)|2;x_1 x_2\\rangle + \\ldots\n\\,", "005732f2b6be3ee1f925df935f842c6f": "F = GHB", "0057531b8dfcbaf7bf5c9326914adf8d": "k_0 \\in (K_0 \\cap K_\\pm)", "00575feb2a6676e28e72b37df84a3618": "n_{2}=\\sum\\limits_{\\alpha_l=1}^{\\chi_c} (c_{\\alpha_{{{l-1}}}\\alpha_{l}})^2\\cdot({\\lambda'}^{[l]}_{\\alpha_l})^2=\\sum\\limits_{\\alpha_l=1}^{\\chi_c}(c_{\\alpha_{{{l-1}}}\\alpha_{l}})^2\\frac{(\\lambda^{[l]}_{\\alpha_l})^2}{R} = \\frac{S_1}{R}", "00576e1590136e3c819062a933b43d7c": " \\mu (A)= \\begin{cases} 1 & \\mbox{ if } 0 \\in A \\\\ \n 0 & \\mbox{ if } 0 \\notin A.\n\\end{cases}", "00578b5ebbc08a904cf34a0c1a0819ea": "\\theta = 90^\\circ", "0057a1113ace7fce93043cd1f12d3d08": "\nJ:X\\to (X'_\\beta)'_\\beta.\n", "0057baf398e7cfd6f637c36ce0d9990a": "\\ell _{({M},\\varphi )}({\\bar x},{\\bar y})=\\sum _{p=(x,y)\\atop x\\le {\\bar x}, y>\\bar y }\\mu\\big(p\\big)+\\sum _{r:x=k\\atop k\\le {\\bar x} }\\mu\\big(r\\big)", "0057d6a820d541c86b119e50682c74b9": "\\hat{x} = (A^{T}A+ \\Gamma^{T} \\Gamma )^{-1}A^{T}\\mathbf{b}", "0057d78ddfbbb18bd8cb8ff50034d770": "Ax = y.", "0057f7c40c1c3d556269650f184c5d4d": "P(k,k') = \\frac {2 \\pi} {\\hbar} \\mid \\langle k' , q' | H_{el}| \\ k , q \\rangle \\mid ^ {2} \\delta [ \\varepsilon (k') - \\varepsilon (k) \\mp \\hbar \\omega_{q} ]", "005874faf228750704e196df7b32cfb5": "\ng(s) = \\int_0^{\\infty} (st)^{-k-1/2} \\, e^{-st/2} \\, W_{k+1/2,\\,m}(st) \\, f(t) \\; dt,\n", "0058f6dc44d924d18482c23df4fba4c4": "F \\in [0,2]", "0059129c160701104ffc251a2f9a5fd6": "{D}_{4}^{(3)}", "00592dd31623e21f87c674477cadf7b3": "\\lambda_{in}", "0059bd909ff2f47bc4ab8e6cb87b199b": "(A \\vee B) \\wedge C", "0059cfbe87754367ae99f910b2e52325": "~{\\rm slog}_b(z)~", "0059d15cf2bc2d0ef806c8572c4933b4": "\\Omega^8\\operatorname{BSp}\\simeq \\mathbf Z\\times \\operatorname{BSp} ;\\,", "005a21b75723dccee94d965dce65eba8": "rpm_{motor}", "005a491cc79d4933a1bce022a2244fef": "\\frac{\\delta^3}{\\delta J(x_1)\\delta J(x_2) \\delta J(x_3)}Z[J]", "005a5a0f4c8ae71fd658bbf442c91b6a": "1 + 2\\;", "005acbb23e5b52409b16f226c75356f8": "a_{t+1} = (1 + r) (a_t - c_t), \\; c_t \\geq 0,", "005ad6c7839bc9f58a588458fb2784be": "\\Beta;\\ G;\\ \\Upsilon", "005aff1ab64bae2fbd389e08eedceaee": "g\\isin [(X\\times Y)\\to Z]", "005b295caf5cffc88b950047571a21b8": "\\underbrace{u_1(\\mathbf{x},z_1)=v_1+\\dot{u}_x}_{\\text{By definition of }v_1}=\\overbrace{-\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1(\\underbrace{z_1-u_x(\\mathbf{x})}_{e_1})}^{v_1} \\, + \\, \\overbrace{\\frac{\\partial u_x}{\\partial \\mathbf{x}}(\\underbrace{f_x(\\mathbf{x})+g_x(\\mathbf{x})z_1}_{\\dot{\\mathbf{x}} \\text{ (i.e., } \\frac{\\operatorname{d}\\mathbf{x}}{\\operatorname{d}t} \\text{)}})}^{\\dot{u}_x \\text{ (i.e., } \\frac{ \\operatorname{d}u_x }{\\operatorname{d}t} \\text{)}}", "005b5ee9184b63d5aae64f486f7762fb": "\\begin{align}\n E_{f_1 + f_2} &= k E_{f_1} E_{f_2}\\\\\n E_{f_1 - f_2} &= k E_{f_1} E_{f_2}\n\\end{align}", "005b76ddf58418b5840fbcd038a55157": "\\nabla_{\\bold u}{\\bold v}(P)", "005b859372ff66ab53af32bd3a95d44c": "\\overline{P}_+:=\\{Q\\in \\mathcal P \\ | \\ Q\\parallel_+ P\\}", "005bee71a96229dc83bdfe3e6a3acd0e": "a + b = 1 + (a + (b - 1)),\\,\\!", "005c84a6de1981ba507fc84f6d002474": "[ES] = \\frac{[E]_0 [S]}{K_m + [S]}", "005cec355090557072bc5242720c1baf": "\\Delta_x \\subset T_xM", "005cf2bd315336ccfc51a82fbc1d011b": " D[p] = [q, \\_, p]::[x, \\_, f]::\\_ ", "005cfe08ac4514176ec9114ed86f5227": " (y + [y/4] + 5(c\\mod4) -1) \\mod 7 ", "005d02c0ccb188f9ce6f80af84add7b2": "E \\left[ \\hat{\\sigma}^2\\right]= \\frac{n-1}{n} \\sigma^2", "005d3c5a843cc4afd4f9459017e79c9b": "v = \\left( \\begin{matrix} \\alpha & \\sqrt{\\mu} \\gamma \\\\ - \\frac{1}{\\sqrt{\\mu}} \\gamma^* & \\alpha^* \\end{matrix} \\right).", "005d4b56062ccf78a1b95d44a904247f": "\\begin{align} \\text{var} (a) &= \\frac{3 \\sigma^2}{2 \\sqrt{\\pi} \\, \\delta_x Q^2 c} \\\\ \\text{var} (b) &= \\frac{2 \\sigma^2 c}{\\delta_x \\sqrt{\\pi} \\, Q^2 a^2} \\\\ \\text{var} (c) &= \\frac{2 \\sigma^2 c}{\\delta_x \\sqrt{\\pi} \\, Q^2 a^2} \\end{align}", "005d5a3817f33dbd656f7b1f926c3ca9": "i/k^2", "005d5be63f060f92e94635636bf5b460": "X_1, X_2, Y_1, Y_2", "005d5f39e6da2cbf9468db66550b1eb5": " r = \\cos^3 \\theta + \\sin^3 \\theta ", "005db61459186328eb26260e77d5c924": "\\mathbb{H}P^2", "005db7c35c2fcc2802e368349fb1dbd2": " \\gamma^\\mu ", "005ddad159bdd4129d68bbf13f9b313c": "{V_{D}} = {V_{P}} + {V_{T}} \\left(\\frac{fu}{fu_{t}}\\right)", "005de217bb2d2c562ddb6ef9b2c6e6af": "a^2 + b^2 + c^2 + d^2 = 2ab + 2 a c + 2 a d + 2 bc+2bd+2cd,\\,", "005e2424c5b287b323d90c18e7d14ebe": "\\begin{cases} y = t^5, \\\\ x = t^3. \\end{cases}", "005e3511011cdc4a24614efd9d0e46eb": "\\mathsf{fv}", "005e882e411a505e927d9403fc95de5a": "\\sum_x \\sum_y I(x,y) \\,\\!", "005ea9a1faaf40201a1fd149fe0df890": "E = R(\\frac {1}{cos(\\frac {\\Delta}{2})}-1)", "005ed603f042c5daf6424e819f284c3c": "charK=2", "005f0f12a2e245b294afb991849fa7e1": "\n\\| u \\|_{L^p} \\leq C \\| u \\|_{L^q}^\\alpha \\| u \\|_{H_0^s}^{1-\\alpha},\n", "005f483aa77c88741fb6a5aca33ab88a": "z = S(r)", "005fa114e9c6b6ee16b3fbe3cd3388d4": "\\langle \\cdot,\\,\\cdot \\rangle \\, ", "005fa1cc2fa20c304d008d28eab9f654": "\\sum_{k=1}^{k=1} \\cos (-2\\pi\\frac{n(k-1)}{1})/1 = 1,1,1,1,1,1,1,1,1...", "005fa74cd2721b0e1f14c33a18a72635": "O(n^2)\\,", "0060137dcd6ebcaf2dd43e3874138898": "\\mathbf{v}=\\mathbf{v}(\\mathbf{x},t)", "00602495b14f9a5268d76e9856935c65": "\\sum_{n=1}^\\infty(\\nu+n)\\sigma_n|a_n|^2", "00602be4ce46f584276cca5f03ce4724": "\\scriptstyle k\\le 3", "006041eaed4c1e105ab451fa672c7eee": "\\boldsymbol{F}_r", "0060430b8c2b4e4aea5fe6f13f242844": "\\mu = ( \\mu_1, \\mu_2, \\mu_3, \\dots , \\mu_N )^T", "00606ffff5f0c9b9833b36681455bd31": "|A|=q", "0060811bf995ea99d0d7af0599037529": "R - R_f = 0.15 ", "0060884b4efc537e5c4e39a03a850a1c": " d=1 ", "0060a9b42c9111cf46baa1f23c60aff3": " \\gamma_1 = \\frac{ 2 \\nu^{ 3 } } { ( \\sigma^2 + \\nu^2 )^{3/2} } ", "0060b049e7e0220cdf2da68756928145": "\\forall x [\\mathrm{Proof}_T(x, \\#\\rho) \\to \\exists z \\leq x \\mathrm{Proof}_T (z,\\mathrm{neg}(\\#\\rho))].", "0060bb0858ef5d84a9930047929fe5b8": "P_{reflect} = \\frac{9.08}{R^2} cos^2 \\alpha ", "0060e120daf207e3782db6738544b75e": "\\text{Average investment} = \\frac{\\text{Book value at beginning of year 1 + Book value at end of useful life}}{\\text{2}}", "00610a4f8b4857300c196650e8badb31": "k_\\mathrm{on}", "006152f03b3939e864f9ac66565b6b58": " \\frac{\\alpha + n}{\\beta + n \\overline{x}}. ", "00617636cc05caa13d75cdc6958d47ce": "K_B", "0062510a5af85f0f1e616f850e5b4e3e": "\n\\inf_g \\sup_f \\iint K\\,df\\,dg=\\frac{3}{7}.\n", "0062c755efea0b9be6ef3dd55ccc30c6": "\\overline{I} = \\overline{\\overline{I}}", "0062d94d1a6a6962840096804a79eb6f": "\\mathcal{L}\\{f''\\}\n = s^2 \\mathcal{L}\\{f\\} - s f(0) - f'(0)", "0062df2399c2fbd55c34251620e6f357": "\n\\begin{align}\n\\boldsymbol{F_{12}} & =m_1\\boldsymbol{a_1},\\\\\n\\boldsymbol{F_{21}} & =m_2\\boldsymbol{a_2},\n\\end{align}", "0062f69c43f50b5e581711b6f431a0af": "\\textstyle 3+\\log_2(n)", "0063113efc28a4d2117081f92b8a8e22": "\n \\begin{bmatrix}\n a_{11} & a_{12} & a_{13} \\\\\n 0 & a_{22} & a_{23} \\\\\n 0 & 0 & a_{33}\n \\end{bmatrix}\n ", "00634867b24389e3680d995d91df3a9e": "0\\rightarrow B\\rightarrow A\\oplus B\\rightarrow A\\rightarrow0.", "0063518e51e9e5ee82646085312dc4ca": "L \\to \\frac{\\omega_c'}{\\omega_c}\\,L", "006352d28b12736b2039ee834b99551c": "r\\;", "00636f68c06830b056c7dc4b296df1b5": "R_T = -2 \\sqrt{\\frac{\\bar{C}'^7}{\\bar{C}'^7+25^7}} \\sin \\left[ 60^\\circ \\cdot \\exp \\left( -\\left[ \\frac{\\bar{H}'-275^\\circ}{25^\\circ} \\right]^2 \\right) \\right]", "006380ed20df9a00246c9f6175355342": "b=3\\,\\!", "0063a4e600bfbf1e870b4704eba7e3c8": "\\begin{pmatrix}\n 1 & a & c\\\\\n 0 & 1 & b\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}", "0063a9838403b9181103f102ed4f2286": "\\begin{align}\nN(x) &= [{y}_{k}]+ [{y}_{k}, {y}_{k-1}]sh+\\cdots+[{y}_{k},\\ldots,{y}_{0}]s(s+1)\\cdots(s+k-1){h}^{k} \\\\\n&=\\sum_{i=0}^{k}{(-1)}^{i}{-s \\choose i}i!{h}^{i}[{y}_{k},\\ldots,{y}_{k-i}]\n\\end{align}", "0063afecc3edf643d2ba84bad6572269": "(\\nabla_Y T)(\\alpha_1, \\alpha_2, \\ldots, X_1, X_2, \\ldots) =Y(T(\\alpha_1,\\alpha_2,\\ldots,X_1,X_2,\\ldots))", "0063b0581ca767e70c55c38053505d09": "hom_D(d_1, d_2) = hom_C(d_1, d_2)", "0063be012d6f0372fbc5275df643e0e2": " \\sum_{n \\in \\mathbb{Z}^d} |\\psi(t,n)|^2 |n| \\leq C ", "0063c4f869877e207c7899c6524d6be8": "\\{y_1, \\dots, y_n \\}", "0063d7d97893cc32e29093238de98deb": "\\begin{pmatrix} 1 & x & z \\\\ & 1 & y \\\\ & & 1\\end{pmatrix}\\Gamma", "006431705901f4b0c40c087dddfbbe25": "\\int_{t_1}^{t_2} \\sqrt{\\left(\\frac{dr}{dt}\\right)^2 + r^2 \\left(\\frac{d\\theta}{dt}\\right)^2 + \\left(\\frac{dz}{dt}\\right)^2}", "00646d01d11376afbd540912f57493e0": "v \\ll c_{a}", "0064a66eb067fc3deb0891fe68173932": "\\mu\\!\\left(X\\right) = 1", "0064ef0ce826596fc2c66bc568d1cfaf": " p = {\\frac{-x\\pm\\sqrt{x^2-4(\\frac{-gx^2}{2v^2})(\\frac{-gx^2}{2v^2}-y)}}{2(\\frac{-gx^2}{2v^2}) }}", "0064f09258ef604746b88546e170dbad": "Z(k,z)=\\cosh(kz)\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,\\sinh(kz)\\,", "0065753065aa05f26494ba26ae99b06a": "E_s[n]", "00658707cdcacc18f896c09e3708968e": "u_{c,i}=\\frac{10.872+0.404 (c_r/c_t) c_{t,i} - 4 (d_r/d_t) d_{t,i}}{16.518+1.481 (c_r/c_t) c_{t,i} - (d_r/d_t) d_{t,i}}", "0065971788f31a3645db9df9fa09b8e8": "2\\leq l", "0066b0150dd9d84ad2d7a66b9f64f64f": " H(\\omega) ~ ", "0066d1eaa4a6602f51da84c5573afd00": "A=\\operatorname{E} (\\Gamma).", "0067045a28deee4d1cc3e1100034e3b4": "\\! J = 2", "006787201e51940f0e2132a7e8c36236": "g(x)=ax^2\\,\\!", "006790d57484d7d46cec4fb2bc2f83e0": "M_{n,k} = \\{ c : P_k(c) = P_{k+n}(c) \\}\\,", "0067c21f52d6fe72e6cf2bd2fd547157": "\\alpha \\in \\Gamma^*", "0067d840510bf6084a7c967d2c0fd5ad": "\\mathrm{K_a = 10^{-4.19} = 6.46\\times10^{-5}}", "006817227c30a11f53ee96def5bcbd71": "V_A = C_A \\cdot \\exp\\!\\left[{-z\\over\\lambda_A}\\right]", "0068402f045ff74b8daa0abfa498dbb4": "\\left\\{x,y\\right\\} \\overset{\\mathrm{def.}}{=} \\left\\{z : z=x \\vee z = y\\right\\}", "0068434645ac8d5310e51e8c2277158d": "\\frac{5 \\sqrt{3\\pi}}{16}", "00684a778b4930e2e20f2bc5f0c50eb1": "0 \\div 0 = 0", "0068b8e2e9d348cbb8a0ada31556ef9e": "\\left\\{ z \\in H: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\frac{1}{2} \\right\\}", "00691626fff7a61da09dd5f51a1a4414": "\\neg p \\wedge q", "006950912b8eb67b89c69baec75894f5": "A_1V_1 = A_2V_2", "00697121901844d211b29641023e5ffe": "Rev_t", "0069a29184ac94f333c07b1dea9e3f8c": "C_2 \\le Y_2 + S_1(1 + r)", "0069cf3e398f8b96544ad051c1f41085": "dq = \\lambda_q dl", "0069eb02ccf993aec658878fb31857c6": "K^2 = {C_N^2 \\over {p_{N_2}}}", "0069fd5cf07098f5022e7b98d242e05b": "T-\\lambda I", "006a1e610fe46c7d6abaaca8a311fc11": "\\int_V e^{-\\pi\\langle \\phi,S\\phi\\rangle}\\, \\mathcal D\\phi", "006a682b5c5c4619cf07219e28a451aa": "\\frac{G^\\mathrm{ig}-G}{RT} = \\int_V^\\infty (1-Z) \\frac{\\mathrm{d}V}{V} + \\ln Z + 1 - Z", "006a6e8bcc60e65733b803f7a1f098c0": "m = p^{\\alpha}", "006a8cae222813804405593109e83c2b": "L\\to\\infty\\,\\!", "006a9988bd5ec6cc57698b026e107a6c": "\\exists a\\in A(x,G)\\colon d(x,z)0. ", "00777b09aa9274526b5613b72b177737": "\nz^{2} = \\frac{\\left( c^{2} + \\lambda \\right) \\left( c^{2} + \\mu \\right) \\left( c^{2} + \\nu \\right)}{\\left( c^{2} - b^{2} \\right) \\left( c^{2} - a^{2} \\right)}\n", "0077897d7efba5094e15cff44f8922aa": "\\scriptstyle\\varphi:T \\mapsto \\mathbb{R}", "00779058ba3e2e1281a3bec1701ddf0b": " d \\approx 1.3 ", "0077929132c8c5223d2f96f5e3e43972": "\\sqrt{\\frac{3}{8}}\\!\\,", "00779355fc7d27f81ccd426981e0b1ec": "w\\bar{y}z", "00779d89488f552b532b9648fd849d5a": " (\\partial U)_S=-(\\partial S)_U=\\frac{PC_P}{T}\\left(\\frac{\\partial V}{\\partial P}\\right)_T+P\\left(\\frac{\\partial V}{\\partial T}\\right)_P^2", "0077bac0533450e9c240f9c0b1d9c223": "F^{\\alpha \\beta} = g^{\\alpha \\gamma} F_{\\gamma \\delta} g^{\\delta \\beta} \\,.", "0077c1ecca87764343e8bd1108d65919": " \\alpha_2 = \\frac{6 G}{2 - K} - \\frac{2 G (K + 4) e^{4 \\phi_0}}{(2 - K)^2} - 1", "0077c400b0161a221aa7adb882c272d7": "{o.p.d.} =\\Delta\\,n \\cdot t", "0077e6527194ccd11d6c32c045f506f0": "\n \\tau = \\sqrt{|\\mathbf{t}|^2 - \\sigma^2}\n ", "0077ee7d7b6ea8618a0ade235c73ef68": "\\{ p_1, r_1\\}", "0078761465bd6fd6b3215e5c47313b31": " s^2 = \\frac{ w_1 } { ( 1 - w_2 )^2 } ", "00787af9608e8ef51e433a7c51d94e00": "\\omega^2 r", "00790b5b8c2899d32e6f8362444877cc": "k^{water}_{f}=400 s^{-1}, k^{water}_{u}=2*10^{-5} s^{-1}, m^{}_{f}", "00791df9a93c86a6fde17616aaf15160": "\\{u_1,...,u_n\\}", "00792c8717b528bf0a5fd2e6a5431e47": "g(p_1,p_2,\\ldots,p_n)=\\sum_{j=1}^n p_j.", "007942efad833175d711142ea4ea22ae": "A\\mapsto (B \\Rightarrow A).", "007955300525c5dcbf90db9082725d8a": " \\frac{\\partial}{\\partial x} \\Bigl( \\frac{1}{\\phi}\\frac{\\partial\\phi}{\\partial t}\\Bigr)\n= \\nu \\frac{\\partial}{\\partial x} \\Bigl( \\frac{1}{\\phi}\\frac{\\partial^2\\phi}{\\partial x^2}\\Bigr) ", "0079557046a685511fac69d35552fb03": "p + 2b^2", "00799b185302341d53d975633fa34d9e": "\\theta \\approx 0", "0079bdea239515bb75d307ab7896cfd9": "[(i\\hbar)^{2j}\\gamma^{\\mu_1 \\mu_2 \\cdots \\mu_{2j}} \\partial_{\\mu_1}\\partial_{\\mu_2}\\cdots\\partial_{\\mu_{2j}} + (mc)^{2j}]\\Psi = 0 ", "0079e61773dda1b7e7372d850ec820d3": "\n\\int_0^\\infty x^{2l+2} e^{-x} \\left[ L^{(2l+1)}_{n-l-1}(x)\\right]^2 dx =\n\\frac{2n (n+l)!}{(n-l-1)!} .\n", "007a632787fbac1c7b731d3853db5170": "y = b\\,", "007a9a1e8463ef195d0a9b1dc88e057e": "\n\\begin{bmatrix}\n V_1 \\\\\n V_0\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n z(j\\omega)_{11} & z(j\\omega)_{12} \\\\\n z(j\\omega)_{21} & z(j\\omega)_{22}\n\\end{bmatrix}\n\\begin{bmatrix}\n I_1 \\\\\n I_0\n\\end{bmatrix}\n", "007ab0be21f37e943739ddcfc116f94c": "(a+b)\\cdot c", "007aba174663400614c30e668f8d31a0": " \\text{DWF} = \\exp\\left( -\\langle [\\mathbf{q}\\cdot \\mathbf{u}]^2 \\rangle \\right)", "007ae2204727cb1c044fd7212c2a5481": "C = C_0 \\dots C_n", "007c1c9966b9f4e95a018fb4cdd39a1f": " \\phi_{hc}(r) = \\frac{ 1.5 \\left(r+\\left| r \\right| \\right)}{ \\left(r+2 \\right)} ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{hc}(r) = 3", "007c8b779fa5b2fbfa9e5808d7d7d932": "T_i = t_i \\cdot \\pi \\left[ \\alpha_i^K \\cdot \\frac{K_{i}}{K} + \\alpha_i^L \\cdot \\frac{L_{i}}{L} + \\alpha_i^S \\cdot \\frac{S_{i}}{S} \\right].", "007d19037e264909bb548db2771d0311": "\\frac{ \\partial f}{ \\partial x} = f_x = \\partial_x f.", "007d588147947102f485cf41305639e8": "\\Omega = \\Sigma_{X|Y} \\Sigma_{XX}^{-1} = I - \\Sigma_{XY} \\Sigma_{YY}^{-1} \\Sigma_{XY}^T \\Sigma_{XX}^{-1}.\\,", "007d5999b8d9d820537b24078de96cc1": "k=\\frac{f_o^2-f_e^2}{f_o^2+f_e^2}.", "007d8191fccdd53f9153ce227ad75b6a": "\\exp(-c)", "007daa94b35faa31165f05da0bd78f8b": "S=\\theta (X_H)", "007e0ecfa25a72cc2f383f39b86540d9": "N = 7", "007e27d57fcd1d2c45476345d34bab59": "SU(3)_L \\times SU(3)_R", "007ebd2662d0cb69047e6bf0843a8ad2": " \\zeta(s,a)=\\sum_{n=0}^\\infty \\frac{1}{(n+a)^s} \\!", "007ede1f44d6d865a3eea50077444c9e": "O_{9}", "007efd2017dc9af726a9fd0111631f45": "H_{p - 1} \\equiv 0 \\pmod{p^3}\\, ,", "007f1e60ec26cf5a7bfdd270125f45ba": "T(\\Delta V) \\approx \\sum_{n = 0}^N a_n (\\Delta V)^n", "007f217f136e1b043c9093734d532e13": "\\begin{align}\n\\iint_{R_C} s(x,t) dx dt &= - \\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + c u(x_i,t_i) - c f(x_i + c t_i) + c u(x_i,t_i) - c f(x_i - c t_i) \\\\\n&= 2 c u(x_i,t_i) - c f(x_i + c t_i) - c f(x_i - c t_i) - \\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx\n\\end{align}", "007f2e086cc219e6031d2b739e28790a": "\\Delta y = \\Delta X * \\frac{1}{(1 - b_C)(1 - b_T) + b_M}", "007f3d5eec88b2997787156a0da80d1b": " a^{k} = (a^{k}_{i})_{i \\in I} ", "007f6d48228b41dbfec441fdb60f208d": "\\begin{align}\n\\Gamma(z) & = \\int_0^\\infty d\\lambda e^{-\\lambda} \\lambda^{z-1} \\\\\n & = - \\int_0^\\infty d\\left(e^{-\\lambda}\\right) \\lambda^{z-1} \\\\\n & = - \\left[e^{-\\lambda}\\lambda^{z-1}\\right]_0^\\infty + \\int_0^\\infty d\\left(\\lambda^{z-1}\\right) e^{-\\lambda} \\\\\n & = 0 + \\int_0^\\infty d\\lambda\\left(z-1\\right) \\lambda^{z-2} e^{-\\lambda} \\\\\n & = (z-1)\\Gamma(z-1) \\\\\n\\end{align} ", "007f7cfb6c836265f0ee259f9795c82e": "0 \\rightarrow G / \\ker\\, f \\rightarrow H \\rightarrow \\operatorname{coker}\\, f \\rightarrow 0", "007f7db03b1721f021e315ea7df8efac": "\\Omega\\,\\!", "007f7f641efda0619b3f766fb9789e1d": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "007fa54549c55de57cdbbc180eb5dbc3": " P_x = P - \\{ a\\mid a \\geq x\\} ", "00803d249818b788bcdef1e281e2fc83": "dx = udt", "008046f09b8004ec023907a58e377465": " D_1 \\psi = \\frac{A}{\\lambda-\\alpha} \\psi ", "008047cb51b2857a9421b19533c9180f": "\\left( \\tfrac{a}{n}\\right)", "00805d6ff79c98d6900575db1265bf54": "G(A)", "008068aab035eff3a79d9645d5fcaef3": "\\left[1 + \\frac{x}{\\sigma}\\right]^{-\\alpha}", "008080e78f109a140688c229fa3545d6": "\\eta = \\frac{ work\\ done } {heat\\ absorbed} = \\frac{ Q1-Q2 }{ Q1}", "0081356314aa1829716309fc76c3ea7f": "df = {\\partial f \\over \\partial x}dx + {\\partial f \\over \\partial y}dy = pdx + vdy", "008194e7ea2ac22286d9a9c3d4abd909": "h_{r,s}", "0081ad49d57a81653cef2dff3b7f1640": "r \\ge a^{1/4}", "0081c4012db924a05de5e2a64aaf3683": "\\varphi: G \\to G^{op}", "0081dec84e8f982234193c1af00fe0f4": "L=\\frac 1 {N(N-1)}\\sum_{i=1}^{N-1}Q_i", "008200c589d4f31f1b4dd239daae3427": "s_\\lambda= \\sum_\\mu K_{\\lambda\\mu}m_\\mu.\\ ", "00821b05a1ce106b2ceefb3f2331880b": "Z_{F \\circ G}(x_1, x_2, x_3, \\dots) = Z_F( Z_G(x_1, x_2, x_3, \\dots), Z_G(x_2, x_4, x_6, \\dots), Z_G(x_3, x_6, x_9, \\dots), \\dots )", "008222f28187648a32637e1e52306723": "\\frac{\\partial E}{\\partial \\hat{h}_i} = \\frac{\\partial}{\\partial \\hat{h}_i} \\sum_{n=-\\infty}^{\\infty}[x[n]^2 - 2x[n]\\sum_{k=0}^{N-1}\\hat{h}_ks[n-k] + (\\sum_{k=0}^{N-1}\\hat{h}_ks[n-k])^2 ]", "0082381366eb7e186fe3e2b7d31b2cd4": "J_z \\to 0", "0082ab2d2297a4ea938a0d25d6dd5c9a": "h[f] = \\lim_{\\Delta \\to 0} \\left(H^{\\Delta} + \\log \\Delta\\right) = -\\int_{-\\infty}^{\\infty} f(x) \\log f(x)\\,dx,", "0082d037b3e5c48137de5c9b8591c500": " K^M_*(k) := T^*(k^\\times)/(a\\otimes (1-a)) ", "0082f7dbe06d887ba8c2fd1c7252fe18": "(C*(1-A)+G)", "008382f5c4fb4614a13b561e58ecfa66": "b(x) = x^jb'(x) \\mod (x^{2t-1}-1)", "00839570c7cb93cd4611c23bd52bbef1": "B_1+B_2a=C_re^{iak_0}+C_le^{+ak_0}", "0083b07c7fb9fba73f101e2b1eecfba3": "\\{C : K_X \\cdot C = 0\\}", "0083c4e87edab8507e96fdde5c911ab3": "q(\\mu,\\tau) = q(\\mu)q(\\tau)", "0083dcac1f5eaa37fd0eb3503722e9b2": "\\Theta \\wedge\n(d\\Theta)^n \\neq 0", "0084209ec3306ab04a193d13223f53d3": "H(p,q) = \\mathcal{F} \\left\\{ h(x,y) \\right\\} ", "00843f9d223ff4c5c126d001c62f48c3": " \n\\mathcal{P}_2(-p_2) = a_{20}(- p_2)^2 +a_{11}(- p_2) +a_{02} = 0\n", "00844945aabd62ba8956c429106513d1": "\\sum\\limits_{i=0 or 1}^{n}P_n (i) W_n (i)", "00844d9977810a20cb96afff0ba5e562": "P(x) = \\sum_{n=0}^\\infty p_n x^n", "008451b474538e1acb9f7d5d1403b167": "D\\left(\\rho u_i\\right)/Dt\\approx0", "00848e2a02240ee7911a90ba2b2495be": " \\|\\alpha^\\prime \\| = 1", "0084c491a7482112d248bc4acefe66ef": "-[OH^-]_{0^{ }}10^{b_0}", "00862d911b12b7cbe90d7a220cf173ec": "f^{-1}\\colon P(Y)\\to P(X)", "00867b570a40821310cbfddda66378f2": "n_{ij}=\\left|U_i\\cap V_j\\right|", "00870c0e8d811a41fc05bb405771d12e": "H(x+y) = H(x) + H(y)", "0087371d07e71fac449e36f68f88dc18": "10.1) \\ \\mbox{Potential adopters}\\ -= \\mbox{Valve New adopters} ", "00875f86af8c866407a4d164d5cbf7db": "z^2 + c", "00877c9ea9a300fe50856e46eb628dde": " \\mathbf{B}. ", "00879a95cd49c2d5871a2f360db7450d": "\\mathcal{M} = (r,\\mathbf{b},\\mathbf{\\delta},\\mathbf{\\sigma}, A,\\mathbf{S}(0)) ", "0087b1f1983a2b9f594fbccc653b4472": " z_{t} = \\lambda_{1}z_{t-1} + \\varepsilon_{t} ", "0087b3df9b66ced1b6c44e67e0e3ba6b": "(u^2+v^2)^n = u^n+v^n.\\,", "0087f8f28b29f87e843973201011c49b": "\\sum_{k=0}^{n-1} \\mu^{\\otimes k}(A_k(s,t))\n\\le\\sum_{k=0}^{n-1} \\frac{\\bigl(\\mu(I_{s,t})\\bigr)^k}{k!}\n\\le\\exp\\bigl(\\mu(I_{s,t})\\bigr)", "00884c5e389a26ffde2fb1e712dac2e2": "k.", "00887fabd495a45f79d1e7c9cb7c02ee": "f(R) = a_0 + a_1 R + a_2 R^2 + \\ldots", "0088aea01f674fa148b588b5b7f441a7": " \\zeta(x,y,t) ", "0088e106641908cc6bfff060e2e61501": "\\{ k x : k \\in K \\}", "00890a623786dd585b07fa38923f0392": " G_x = \\{g \\in G: g \\cdot x = x \\} ", "0089102d73f673ad70c3a48c34bfe2ec": "f\\left(r\\right) = \\frac{\\left(1 - r^2\\right)^{\\frac{n - 4}{2}}}{\\mathbf{B}\\left(\\frac{1}{2}, \\frac{n - 2}{2}\\right)},", "0089200d6d75460d55a8abd9087b580c": "V = 2\\pi^2 n R r^2 = \\left( \\pi r^2 \\right) \\left( 2\\pi n R \\right). \\,", "008929a9fceab6c14956bee05f48132b": "l_2(\\theta) = \\theta + \\alpha/2", "008953b32e8473e4f9c6e11f36a6aab8": "504 = 2^3 \\cdot 3^2 \\cdot 7", "008962134e77f17fc6b7daee74c10f90": "p = \\operatorname{char}(F)", "0089ee1cbf646ce073b7ab871f9804c2": "{-1 \\choose n} = (-1)^n", "0089fb36fd68801cf2d544380aef3c24": "((\\mathbf{a} - \\mathbf{p}) \\cdot \\mathbf{n})\\mathbf{n}", "008a0dcf2169a1219c1c35ece550f609": "\\operatorname dE_{\\text{i}}(\\omega_{\\text{i}})", "008a30ccbc263d1d59165530204391c4": "\\mathrm{[Cr] = [CrO_4^{2-}] + [HCrO_4^-]+2[Cr_2O_7^{2-}]; pCr=-log_{10}[Cr] }", "008a4740f682e2dc462f3a52807b2bdc": "\\bar{\\partial} : \\Omega^{(p,q)} \\rightarrow \\Omega^{(p,q+1)}", "008a57162fb19a48692b5111234e6b2f": "p_0>0", "008a667e4624af2f89b2a3b153b2b1af": " 2\\mu(K)=\\mu(K+v)+\\mu(K)<\\mu(U)\\,", "008a9a3318419402e1f2b47fba0f5e81": "\\begin{align}\nx:\\;\\; \\rho \\left(\\frac{\\partial u_x}{\\partial t} + u_x \\frac{\\partial u_x}{\\partial x} + u_y \\frac{\\partial u_x}{\\partial y} + u_z \\frac{\\partial u_x}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial x} + \\frac{\\partial \\tau_{xx}}{\\partial x} + \\frac{\\partial \\tau_{xy}}{\\partial y} + \\frac{\\partial \\tau_{xz}}{\\partial z} + \\rho g_x \n\\\\\n y:\\;\\; \\rho \\left(\\frac{\\partial u_y}{\\partial t} + u_x \\frac{\\partial u_y}{\\partial x} + u_y \\frac{\\partial u_y}{\\partial y}+ u_z \\frac{\\partial u_y}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial y} + \\frac{\\partial \\tau_{yx}}{\\partial x} + \\frac{\\partial \\tau_{yy}}{\\partial y} + \\frac{\\partial \\tau_{yz}}{\\partial z} + \\rho g_y\n \\\\\nz:\\;\\; \\rho \\left(\\frac{\\partial u_z}{\\partial t} + u_x \\frac{\\partial u_z}{\\partial x} + u_y \\frac{\\partial u_z}{\\partial y}+ u_z \\frac{\\partial u_z}{\\partial z}\\right)\n &= -\\frac{\\partial P}{\\partial z} + \\frac{\\partial \\tau_{zx}}{\\partial x} + \\frac{\\partial \\tau_{zy}}{\\partial y} + \\frac{\\partial \\tau_{zz}}{\\partial z} + \\rho g_z.\n\\end{align}\n", "008b1a26a2217fe7dfd19fbfb8bab404": "\\scriptstyle x_{n+1} \\;=\\; \\frac{x_n}{2} \\,+\\, \\frac{1}{x_n}", "008b977cbc0bd622e23da22e303e3107": "I_v = \\log_2", "008bc43ba83b29504bc182aa7b9357b9": " x^{(2)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.5000 \\\\\n -0.8636\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8494 \\\\\n -0.6413\n \\end{bmatrix}. ", "008bcc3802c42a9731f0425bd63421c4": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx = \n \\frac{x^{m-n+1} \\left(a+b\\,x^n\\right)^{p+1}}{b (m+n\\,p+1)}\\,-\\,\n \\frac{a (m-n+1)}{b (m+n\\,p+1)}\\int x^{m-n}\\left(a+b\\,x^n\\right)^pdx\n", "008bd2981aaddb22cd636b66bdbdb486": "t_{A/B} = {\\int_A}^B \\frac{M}{EI} \\bar{x} \\;dx", "008bda504aea93cf0967b4159571e8ed": "\\mathbf{\\mathit{F}}", "008befef6f37a943603ef39d3d673039": "M(1):=\\lbrace 1,\\dots,d\\rbrace", "008c04f9682f741da40cae14aec5d4ee": "c(x) = \\frac{1-\\sqrt{1-4x}}{2x}=\\frac{2}{1+\\sqrt{1-4x}}", "008c0b0b97f245c3871f09555aba25ef": "T ( w[t] ) \\in \\Sigma", "008c2a31b08704c913be54aa60532f1d": "\\ \\Delta H_{vH}(T)= -R\\frac{dlnK}{dT^{-1}}", "008c2d3a1ce1fe187a52d241c017876f": "\\exp y = 1 + y + {1 \\over 2!} y^2 + {1 \\over 3!} y^3 + \\dots = \\lim_{N\\to \\infty} \\sum_{r=0}^N {N! \\over r! (N-r)!} ({y \\over N})^r = \\lim_{N\\to \\infty} (1 + {y \\over N})^N.", "008c9070a3e7cec8fd2d50ac94c09e66": "\\begin{matrix}\n\\left[x_i , p_j \\right] &=& i\\hbar\\delta_{i,j} \\\\\n\\left[x_i , x_j \\right] &=& 0 \\\\\n\\left[p_i , p_j \\right] &=& 0\n\\end{matrix}", "008c9eebf46eca9e41c9013fc59cc320": "1+i = (1+r) (1+\\pi^e)", "008cb707ff5b45a885fdf6804228837d": "F_{\\alpha \\beta} = \\partial_{\\alpha} A_{\\beta} - \\partial_{\\beta} A_{\\alpha} \\,", "008cdd207cd628ca5ec8c88a91047345": "\\phi \\colon S^{2n-1} \\to S^n", "008cf3a04a0fe2cacd8cf9d6065ad0d8": "\n\\psi \\to \\psi '=U\\psi \n", "008d0eb7c4baa772b60d8c3c01730196": "y_i=\\sum_l p_l r_{li},", "008db6e1aca3b942f9ccddabf20703b9": "\\scriptstyle{| d^\\prime \\rangle , \\ | s^\\prime \\rangle}", "008dd0838341b126a7fc9f80b8e1039f": "b_3", "008e0fd066ddf76297e8ce6ca41eea1d": "C_{}^{}", "008e306ad46f9f8427da6cbaae1c04ae": "\\left\\{ \\begin{pmatrix}\n 1 & x & z\\\\\n 0 & 1 & y\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix},\\ x,y,z \\in \\mathbb{Z}\\right\\} ", "008ec35c8e46ac1830429906dd7800c7": "\\{S_k\\} \\subset G", "008edcb19e3d7caf12c7925d31018929": "L = \\frac{1}{d}[194.4 - 0.162t]", "008f42d8b190bb3f0e4986f32541c4bf": "weight_i", "008f693046e7ccc149361a57cb40596c": "| L \\rangle ", "008fd8644838f0babbadd5f639875791": "\\phi(z_i,z_{i+1})", "009071732a22f66ece9a28dc61b02e59": "\\mbox{tr} X = \\sum_{\\{i\\}} \\lambda_i e^*_i(e_i). \\,", "0090ab8183aa1428deeff6d4923b5924": "\\upsilon_r\\,", "0090f3e8899492024afb44d709081ec4": "\\mathrm{str}(X) = \\mathrm{tr}(X_{00}) - (-1)^{|X|}\\mathrm{tr}(X_{11})\\,", "009198f5d7b698ceb14b44a81b9499fa": "\n\\psi^\\dagger \\gamma^0 \\psi = 2 \\langle \\bar{\\Psi}\\Psi \\rangle_{S R} \n", "0092851ab1475289854441df38ef5638": " \\text{sara}= r\\cdot \\frac{s^2}{(2^2 - 2)r^2} - \\Big[ r\\cdot \\frac{ s^2}{(2^2-2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2} -\\Big[ r\\cdot \\frac{ s^2}{(2^2-2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2}\\cdot \\frac{s^2}{(6^2-6)r^2}-\\cdots\\Big]\\Big] ", "00928bda52f7935d12b019dcacda9fd6": "\\left\\Vert r(T) \\right\\Vert \\leq \\left\\Vert r \\right\\Vert_{X} = \\sup \\left\\{\\left\\vert r(x) \\right\\vert : x\\in X \\right\\}", "0092907fb658877fa806da38ad96fcea": "-\\lambda_{n+1}e_{n+1}", "009296bab85172292e4b1a2391aa2ca8": "\\mathfrak{I} \\vDash \\Phi ", "0092f65c01eeef8c51053657ecf2e9d4": "\n \\begin{align}\n C_{+} &= + \\frac{1}{2} C_0 \\cos{\\left(\\theta - \\frac{\\pi}{4}\\right)},\n \\\\\n C_{-} &= - \\frac{1}{2} C_0 \\sin{\\left(\\theta - \\frac{\\pi}{4}\\right)}.\n \\end{align}\n", "00931e7be1a5c92a5c4f77c90428b2ab": "\\begin{align}\n s & = g(x,u)+\\omega_s \\\\\n \\dot{{x}} & = f(x,u)+\\omega_x\n\\end{align}", "00932f79cb15cc290228b09d758bb627": "w(n,j) = g(y_n|X_P(t^{}_n,j),t^{}_n,\\theta(t_{n-1},j))", "0093a55f284a1beb41832ac953e88581": "A' \\leq B'", "0093c10abe7bc5272a6abdeb3340bf76": "L = 0, F = 0.", "009417c186afcedf81f53493793177d1": " E[Y|X] = \\Pr(Y=1|X) =x'\\beta,", "009425bc73ae2c42a73998a3bcb964d2": "\\displaystyle{{1\\over \\pi}\\left |\\int_a^b {\\sin t \\over t}\\, dt\\right|}", "0094382b75b38eb14a29da862cd12754": "\\boldsymbol\\Sigma_{22}^{-1}", "00948e47613cadeb9121a131170a6474": " \\Delta W = \\int_{V_1}^{V_2} p \\mathrm{d}V \\,\\!", "00949bada66ee0d6400e1c1436d199b1": "\n\\begin{align}\nu = \\operatorname{prox}_R(x) \\iff & 0\\in \\partial \\left(R(u)+\\frac{1}{2}\\|u-x\\|_2^2\\right)\\\\\n\\iff & 0\\in \\partial R(u) + u-x\\\\\n\\iff & x-u\\in \\partial R(u).\n\\end{align}\n", "0094a6c6ea2cc89a880d95abe9d57da5": "\\operatorname{Aff}(A) = V \\rtimes \\operatorname{GL}(V)", "009507a7597f0473f041bb1fbf6f7922": "\\dim f(Z) > n", "0095576b64d0617187cbe451a644022a": "\\mathit{W}_{1-2}+\\mathit{Q}_{2-3}+\\mathit{W}_{3-4}+\\mathit{Q}_{4-1} = 0", "009573de099f0176195cc34a1d74cd00": "\\Phi_{00}=\\frac{M(u)_{\\,,\\,u}}{r^2}", "009585f132416dd4fd5b0ecc4862da1e": "\\mbox{dim } A = n\\,", "0096064837a09b4ff9f4bf35bc16334f": " n > 0 ", "00960935edc55c1cfe049dfb9068697f": "\\tilde{E}_i^a = \\sqrt{det (q)} E_i^a", "00961e8857a00528f8585aecefe6b93a": "\\! (x_1, y_1), \\ldots, (x_m, y_m)", "00966ec70506d450552970ef0e81c794": "\\mathcal{L}=\\frac{1}{2}(\\partial_t\\phi)^2 -\\frac{1}{2}\\delta^{ij}\\partial_i\\phi\\partial_j\\phi - \\frac{1}{2}m^2\\phi^2-\\frac{g}{4!}\\phi^4.", "00968933d49233c3d97f1d0e30a6c2b1": "\\aleph_0 + 4 \\cdot \\aleph_0 = \\aleph_0 \\,.", "0097292bb6d0270b952befc3a0a95249": "\\theta\\in\\Theta\\,\\!", "0097f84e4d935108709946d992c18cd9": "{}E[X_{n+1}|X_1,\\ldots,X_n] \\ge X_n.", "009831967ffc0b0f7f70e153342ce2dc": "C_p = \\frac{(P_m - P_s)L}{AE}", "00983828282a5f9747d1c65fda904761": " \\Theta_{\\pi} ", "0098457f5b516ebc2a71dd02a6d33b57": " t_1 = t_3 = 0, \\; t_4 = t_2^2/4 ", "0098929f79427f1159a6da9916fa1347": "\\frac{d}{dt} \\log_e t = \\frac{1}{t}.", "0098ab864f39da989fe54e22e6b63380": "h_i : X \\to \\{-1,+1\\}", "0098cd3b3fba6cb3178d7b737d7f7b34": "1-\\varepsilon", "0099427a668b698dd94196acde87e495": "2\\le seqs \\le6", "0099d25546919981aca4a5225481b56a": "c_{n} = \\sum_{\\mathbf{i}\\in \\mathcal{C}_{n}} a_{k} b_{i_{1}} b_{i_{2}} \\cdots b_{i_{k}}, ", "0099f5592168eaed97eee19da22a06b1": "d_j \\,", "009a16bae546036e134016ec108f2f5e": "T^{\\mu\\nu}={1\\over 16\\pi G} (g^{\\mu\\nu}\\eta^\\xi_\\eta-g^{\\xi\\nu}\\eta^\\nu_\\eta-g^{\\xi\\nu}\\eta^\\mu_\\eta)\\Omega^\\eta_\\xi\\;", "009a19f7f4737812d772e4192a306a03": "\\varepsilon_m(\\boldsymbol{k}) = E_m - N\\ |b (0)|^2 \\left(\\beta_m + \\sum_{\\boldsymbol{R_n}\\neq 0}\\sum_l \\gamma_{m,l}(\\boldsymbol{R_n}) e^{i \\boldsymbol{k} \\cdot \\boldsymbol{R_n}}\\right) \\ ,", "009a22483ac6a78d97d501ad686a99b8": "\\beth_{0} = \\aleph_0,", "009a879ec10bd307867c8213dd430802": "Q'_{lid} = kA_{lid} \\left ( \\frac{T_b - T_{surr}}{\\Delta x} \\right ) + hA_{lid} \\left (T_b - T_{surr} \\right ) + A_{lid} \\epsilon _{p.p.} \\sigma \\left [ \\left (T_c + \\frac {T_{surr} \\Delta S_{p.p.}}{c_p^{p.p.}} \\right )^4 - T_{surr}^4 \\right ]", "009ac1e61fa36dd671c3e2ec3f322313": "A_{\\nu ; \\rho \\sigma} - A_{\\nu ; \\sigma \\rho} = A_{\\beta} R^{\\beta}{}_{\\nu \\rho \\sigma} \\,,", "009ac90597ad34875b81bebee3c5d62b": "\\left(\\frac{7}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "009aeb6f91ab0b013d6042a748040d73": "\\left( {{{\\partial s} \\over {\\partial T}}} \\right)_P = {{c_P } \\over T}, \\left( {{{\\partial s} \\over {\\partial P}}} \\right)_T = - \\left( {{{\\partial v} \\over {\\partial T}}} \\right)_P ", "009b45b6731f6596d60061cd9b8138d5": "\n\\Delta \\hat{z}\\ =\\ 2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\cos i\\ \\left(\\ e_h\\ (1-\\frac{15}{4}\\ \\sin^2 i)\\ \\hat{g}\\ -\\ e_g\\ (1-\\frac{5}{4}\\ \\sin^2 i)\\ \\hat{h}\\right)\n", "009c16baf1e28508b213dca7d341e659": "\\int_S f g \\, \\mathrm{d}\\mu = \\|f\\|_p\\,.", "009c405b8a6248a30dcbdf70d58f2757": "\\frac{\\partial z}{\\partial x} = 2x+y", "009cd0a8f8dc2ab7263d4ef99ca9715f": "s_{p-2}\\equiv0\\pmod{M_p}.", "009d31cbcf40a1cc2bab52af464ddb35": " C_{D,\\text{induced}} = \\pi A\\!R \\sum_{n=1}^{\\infty} n A_n^2 ", "009de0b1f57210af8d7b220db90ac5cc": "(\\sigma_i)", "009df7d50ec2590cb99c99b08f87a5d0": "\\tau(\\mathcal{H})\\leq 2\\nu(\\mathcal{H})", "009e5791fcbaffcdf91cac58bbf57761": "\\Pr(3;3,6,1)=\\Pr(3;1,3,6)=\\Pr(3;3,1,6)", "009efa14b5e6f3dcef2aa3d81abfe4cc": "\\rho^{\\mathrm{ent}}(X) = \\frac{1}{\\theta}\\log\\left(\\mathbb{E}[e^{-\\theta X}]\\right) = \\sup_{Q \\in \\mathcal{M}_1} \\left\\{E^Q[-X] -\\frac{1}{\\theta}H(Q|P)\\right\\} \\,", "009f00dd75b22d7d959618d5339ed742": "f(x) = \\begin{cases} +x^2, & \\text{if }x\\ge 0 \\\\ -x^2, & \\text{if }x \\le 0.\\end{cases}", "009f4fb1e0486cae6830706cbe42128d": "X_k = \\sum_{n=0}^{N-1} x_n e^{-\\frac{2\\pi i}{N} nk }\n\\qquad\nk = 0,\\dots,N-1. ", "009fbee309e9784672526a00264c93ef": " F = \\{ (x,y) : x \\in \\mathcal{R}^b,\\, y \\in \\mathcal{R}^n,\\; x=y \\}.", "009fd4b226543561f01553326ecbfee8": "T''(t) = Kc^2T(t) \\,", "00a00703dcf8dd553d1384468debf2f3": "N=\\binom{n+d}{d} -1", "00a020df6280a23ac9daf6baf98439e9": "\\omega_i , 1 \\le i \\le n", "00a02689dfc44a622ba4f7906e0469f6": "h_{\\text{out}}(G)\\le \\left(\\sqrt{4 (d-\\lambda_2)} + 1\\right)^2 -1", "00a05ca86bbbff83a90eb6ecb485762e": "(10)_{10} ", "00a10b2b00b93021fbcc49cae4bc0e7e": "(a+b\\sqrt{p^*})(a-b\\sqrt{p^*}) = a^2 - b^2p^* \\in \\beta \\cap \\mathbf Z = (q),", "00a125e0ad76f163105b9a1a97acbafe": "f^{-1}(t)", "00a149b25c8cff2d15e3bdeaef87a6bf": "n_s(\\vec r)\\ \\stackrel{\\mathrm{def}}{=}\\ n(\\vec r)", "00a16782186c3588c2313c1b11a1bcf6": "f(a\\vec{v}) = af(\\vec{v})", "00a18ba5575387e4e4edf85672346d6d": "b=2(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))", "00a1c4642424356e90a07d4bdeb3a369": "\\delta V", "00a1de149855ddb3fd113f4c7bb2e8fb": "\\hat{\\alpha}= - \\frac{1}{\\frac{1}{N}\\sum_{i=1}^N \\ln X_i}= - \\frac{1}{ \\ln \\hat{G}_X} ", "00a225db74c83931acde710cabd1020f": "\\psi(x) =\\frac{d}{dx} \\ln{\\Gamma(x)}= \\frac{\\Gamma'(x)}{\\Gamma(x)}.", "00a2658138798c6630a6c6b75896a8ea": " p \\mapsto q p q^* \\,\\!", "00a2dc92c377a5fcf3f907cf42ef0962": "\\left(\\frac{K_0+a}{1+a}\\right)^\\gamma=\\frac{K_0}{\\phi}.", "00a34cdb4626d2fd31087d976797e802": "\nV_i(\\omega_k) \\rightarrow V_{ik}\n", "00a3567e070481826aedde5b194fd120": "\\Rightarrow^*_{amb}", "00a3b8c17922464b3763885a0a072622": "\\pi_i p_{ij} =\\pi_j p_{ji}, \\,", "00a3e681e7f16483324136c5f343c197": "\\vartheta(x)", "00a3e7c4907d298e04c2705b5217de48": " \\hat{p} = \\frac{n_1}{n}", "00a4060ce409de2a13ba982d9b63055d": "G = \\frac {4 \\pi A_{eff} } { \\lambda^2 } = \\frac {4 \\pi A_{phys} e_a } { \\lambda^2 } \\,", "00a41b5522f83aa7f1dc471f9ba0051a": "c_3=c_1+c_2=|c_1|\\cdot\\left(\\alpha_1+\\alpha_2\\tfrac{|c_2|}{|c_1|}\\right)", "00a467eb4fe6caeadcab17fd68b6d169": "xz\\leq yz", "00a48783273589c885aa79c58705f779": "\nF = A * E^2 \\propto L^2\n", "00a4cfe1a4c0720a2a520713f425e0c4": "D-1", "00a5129b37e70c31ec37a9f1f3b012fe": "\\sum_{n=0}^{\\infty}(n+1)x^n={1\\over(1-x)^2},", "00a51502fb27f23962367cc2d17ce18c": "X_i(\\omega)=\\omega_i", "00a552471b51f76a41fcbc95b4938fd2": "d(\\det(A)) = \\sum_i \\sum_j \\mathrm{adj}^{\\rm T}(A)_{ij} \\,d A_{ij},", "00a554bca784b6cfea952ee3e5f75cb7": "\\operatorname{nil} \\equiv \\operatorname{false} ", "00a59b76ebf9abfb3d9fe3eefeb9e3f6": "L^{q_\\theta}", "00a59dc981df6c75b3538a4ba059633f": "\n{\\partial{L}\\over \\partial q_i} = {\\mathrm{d} \\over \\mathrm{d}t}{\\partial{L}\\over \\partial{\\dot{q_i}}}.\n", "00a5ae7ab7a84d3ba9306ecc2364d6a8": "\\hat{\\xi}^{i} \\rightarrow \\acute{\\hat{\\xi}^{i}}=\\hat{U}^{+}\\hat{\\xi}^{i}\\hat{U}.", "00a67155ff3cd8fab09e943bfe257614": "x_7", "00a67a8d2d4bf0bf959743c81b7aa446": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^3 = \\frac{1}{1^3} - \\frac{1}{3^3} + \\frac{1}{5^3} - \\frac{1}{7^3} + \\cdots = \\frac{\\pi^3}{32}\\!", "00a6d384f5362987e87b4ce8b1320bfa": "x*", "00a6dc4d3f87b23761b272ea6b80ce2d": "X_n\\,", "00a78f6c69d27a486ccb1f1d4d2bf147": "|\\psi\\rangle ", "00a793998ab632a05917678ea364f76b": "\\,dN", "00a7b393dbd294b592f68c62459fec49": "A_r \\left ( {\\rm X} \\right ) = \\frac{\\langle m \\left ( {\\rm X} \\right ) \\rangle }{m \\left ( ^{12}{\\rm C} \\right ) / 12} ", "00a7d1ba4a6e1afcbafad38a541a341e": "\\frac{n}{12}", "00a80cce08868ef13e04f34b7f3043fd": "R = Ef/(Ts + Th) ", "00a870853110df52ed102384d3708385": "\\begin{align}\n\\Delta S_F &= \\frac{s - s_i}{c_p} = ln\\left[\\left(\\frac{M}{M_i}\\right)^\\frac{\\gamma - 1}{\\gamma}\\left(\\frac{1 + \\frac{\\gamma - 1}{2}M_i^2}{1 + \\frac{\\gamma - 1}{2}M^2}\\right)^\\frac{\\gamma + 1}{2\\gamma}\\right] \\\\\n\\Delta S_R &= \\frac{s - s_i}{c_p} = ln\\left[\\left(\\frac{M}{M_i}\\right)^2\\left(\\frac{1 + \\gamma M_i^2}{1 + \\gamma M^2}\\right)^\\frac{\\gamma + 1}{\\gamma}\\right]\n\\end{align} ", "00a8c8452e84811dd3222f97d0c094e0": "\n D_j = \\frac{y_U-y_L}{r-1} \\quad (j=i+1,\\ldots,i+r-1).\n ", "00a90587036019f4279b0ec99206f3a7": "\\Pi_n", "00a9bbdd8b6b224a61ab201c9b39ed06": " \\lambda u.x ", "00a9c8c8443a68289eb90415df7d306a": "\\alpha^2,", "00a9cdb637559fcf0fa52b15f0d24067": "p\\sum_{i=1}^n \\left ( \\frac{Y_i - \\hat\\mu \\left (x_i \\right )}{\\delta_i} \\right )^2+\\left ( 1-p \\right )\\int \\left ( \\hat\\mu^{\\left (m \\right )}\\left ( x \\right ) \\right )^2 \\, dx", "00a9cfc6f644a9f8f3258b8864da1c9b": "\\sigma^2 = X^TVX,", "00aa056f604bfb7a08392d451f0a3cf6": "\\varphi_1, \\varphi_2, \\varphi_3, ...", "00aa46e6ebcd45c14cef047bb689f248": "\\gamma= 0.95 (95\\%)", "00aa7d063f46dae0935f3b140e61941d": "\\int_{\\mathbb{R}^n}f\\,dx = \\int_0^\\infty\\left\\{\\int_{\\partial B(x_0;r)} f\\,dS\\right\\}\\,dr.", "00aa8d463550a1ee7942e5dd3330f818": "f(\\gamma,u)", "00ab0b9d2bb48616a1ee5225eecd77df": "\\max(A_1(x_1, \\dots, x_{r-1}), \\dots, A_{n_A}(x_1, \\dots, x_{r-1})) \\leq \\min(B_1(x_1, \\dots, x_{r-1}), \\dots, B_{n_B}(x_1, \\dots, x_{r-1})) \\wedge \\phi", "00ab11b2c84e14c5bc0372acf71d3baf": "(a - b)(a + b) = b(a - b) \\,", "00ab188055de2af9da6158e79db624ad": "{}_{\\ 86}^{220}\\mathrm{Rn} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 84}^{216}\\mathrm{Po}\\ \\mathrm{(55\\ s,\\ 0.54\\ MeV)}", "00ab347375308522c9fc211d16779712": "\\mathbf{N} \\equiv \\mathbf{n}_0", "00ab4f6d5ab07f3403bb7b46b92fbeac": "du = -3u {da\\over a}", "00ab97e57c2e4a4589b34dfa9b6bc551": "\n{\\nabla^2 u -\\dfrac 1{c_0^2}\\frac{\\partial^2 u}{\\partial t^2} + \\tau_\\sigma^\\alpha \\dfrac{\\partial^\\alpha}{\\partial t^\\alpha}\\nabla^2 u\t- \\dfrac {\\tau_\\epsilon^\\beta}{c_0^2} \\dfrac{\\partial^{\\beta+2} u}{\\partial t^{\\beta+2}} = 0.} ", "00aba73ffc448c45f8d1122ee9b3c9d6": "\\{r_1, r_2, r_3,r_4\\}", "00abc855b0107a8e4b9c4a38af54aed6": "\n\\frac{d^{2}\\eta}{d\\tau^{2}} = \\frac{dt}{d\\tau} \\frac{d}{dt} \\left( \\frac{d\\eta}{d\\tau} \\right) = - y^{2} \\ddot{y} = -\\frac{y^{3}}{mr} F(r)\n", "00abd9a9738fa0ab1bd4fe864640ac5f": "n\\log^{O(\\log k)}n ", "00ac8c8a2346f2a39cc30536fc519d74": "\\frac{T_A}{T} = \\bigg(\\frac{P_A}{P}\\bigg)^{(k-1)/k}", "00acac7e9220e49e733584596a5f11e7": "\n\\{x, p_x\\}_{DB} = \\{y, p_y\\}_{DB} = \\frac{1}{2}\n", "00acc68c74cb8ce7b35958b2a46e1f5d": "\\epsilon_S", "00acc9ac69fc3f2d366801f96e53c565": " (\\hat{k} , \\hat{l})", "00ace3be08aece29574b1c573b12f1f0": "\\text{GF}(2)^n", "00ad2f6e3b361d991d10c82a582bcf5a": "a_2 = \\frac{-b_1 + \\sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, \\!", "00ad646ae19cb465bf7119d513412743": " (1-X) \\sim \\textrm{Kumaraswamy}(a, 1)\\,", "00ad734308e565a05b76573ee16fce9d": " A_{i_1 i_2 \\cdots } + B_{i_1 i_2 \\cdots} = C_{i_1 i_2 \\cdots} ", "00ad89837a9fd8ba452937e8cb62cb70": "\\Phi_{\\text{P}} (x)=\\frac{ m}{4 \\omega^2} \\left[g (x)\\right]^2", "00adcb82c4f67853e8c543504656cd0c": "\\frac{\\lambda_c}{d} = \\pi v Z_0 C", "00ae016ab9b477f5e9eceaa787a7be83": " [\\phi, L_z] = i \\hbar \\ \\psi(\\phi) \\quad (8) ", "00ae3c1c548819e0a5af11b628c731d7": "\\exp X = e^X = \\sum_{n=0}^\\infty {\\frac{X^n}{n!}}.", "00ae4809938cb083caa9c3b61e1fcde4": "\\tilde{\\mathit{A}}\\subseteq\\mathbb{R}", "00ae48d6eac642900416e0978697565d": "j(i) = 1728", "00ae6724da7c06588a062b10129e7c4a": "\\sum_{k=1}^n k! S_2(n,k),", "00ae73e221ae8438c7e9050b0321f9fb": "G_{k, \\sigma} (y)= 1-(1+ky/\\sigma)^{-1/k} ", "00aee84d876ee6e51ad144b53e456586": " |L| \\cdot {2^j \\choose 2} \\leq {n \\choose 2}", "00af7f6512d73e19bf172e3b9a8b875d": "g = \\frac{V}{P}", "00b02d842e499f5d430d91c9fb0e6d25": "a = \\frac {x} {d}", "00b02ebac24fbe8e7858b4e7f5cd2e98": " \\frac{1}{1-z} \\sum_{k=1}^m \\frac{z^k}{k}\n\\mbox{ and }\n[z^n] \\frac{1}{1-z} \\sum_{k=1}^m \\frac{z^k}{k} = H_m\n\\mbox{ for }\nn \\ge m\n", "00b0b9b3a532cbcdad77535a337d5005": "n_c \\sim A + B (p - p_c) + C (p - p_c)^2 + D_\\pm |p - p_c|^{2 - \\alpha}", "00b0e6bd9379de899a741e524e1efac3": "\\textstyle \\{C_{i}\\}", "00b0e846b6f072fabff3bb11adb32af5": "A'(x)u_1(x)+B'(x)u_2(x)=0.\\,", "00b13d3f8df02a71e391fce9b198d45f": "\n{\\mathcal L}_s=-\\frac{1}{2}\\left[\\sigma^2h^{\\alpha\\beta}\\partial_\\alpha\\phi\\partial_\\beta\\phi+\\frac{1}{2}\\frac{G}{l^2}\\sigma^4F(kG\\sigma^2)\\right]\\sqrt{-g},\n", "00b173f71cdad4f4e5401621a19f24cc": "g(x_i|D)", "00b1f489539a947438d556bfbc27f889": "\\Sigma^T \\Sigma", "00b1f6a425ccc8c636dda4b95ae7e6a7": "n>e^{3100}\\approx 2 \\times 10^{1346}", "00b212863f999c8af73aa32e38ae23e4": " K ", "00b281f46653d754535354d0947ebd62": "\\Psi_1 = C_\\text{Ion}\\Phi_\\text{Ion} + C_\\text{Cov}\\Phi_\\text{Cov},", "00b285739a3b02cf66484aa107d8f5da": "x_3= \\sin i \\cdot \\sin \\omega", "00b2cbab416e71fc8fef9b1d69d40f3e": "P_n' = C_{n-1}' \\oplus E_K^{-1}(C_n)", "00b3285c8751e46d6815fe231be45f26": "L(H_B) \\otimes C(X)", "00b38b79c51077f93a85760f804d9b6b": "\\frac {G(x)}{F(x)^n}", "00b3aeceaed7d552c07adacf9cf0e201": "\\forall n\\in\\mathbb N\\colon n\\cdot 1\\le\\xi", "00b463dbda2a23f566e8f81d9c0824ae": "T(s, x) = s(x)", "00b4662abd3732893063c7d52118bff1": "Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.\\;\\!", "00b4e09b9649761dc59a6883e8136a7c": "\\Phi_{abc} = x_a \\otimes x_b \\otimes x_c", "00b501613bc59cc20f9c60a2996c41c1": "\\alpha:H\\rightarrow G", "00b514f874a9bea96964c9df44eafa61": "\\mbox{affinity} = \\alpha[A][B]\\!", "00b51803048eb8b9edf4d0405bdbf331": "\n\\sum_{\\stackrel{1\\le k\\le n}{ \\gcd(k,n)=1}} f(\\gcd(k-1,n))\n=\\varphi(n)\\sum_{d\\mid n}\\frac{(\\mu*f)(d)}{\\varphi(d)},\n", "00b56b0fba4e86b56fa04b2abdc00d76": "\n \\operatorname{Var}[s^2] =\\operatorname{Var}\\left(\\frac{\\sigma^2}{n-1} \\chi^2_{n-1}\\right)=\\frac{\\sigma^4}{(n-1)^2}\\operatorname{Var}\\left( \\chi^2_{n-1}\\right)=\\frac{2\\sigma^4 }{n-1}.\n ", "00b5d8a91cc4d17d1f997fbba2dddff8": "\n(D\\nabla^2\\psi-{\\bold u}({\\bold u}\\cdot D\\nabla^2\\psi))\n", "00b6157255eee3ea721509333534bcf1": "\\mathcal{L}_X Y = [X,Y]", "00b61ea310c446de9872ca46c979294d": "Volume = \\frac{\\pi}{6} \\times L_1 \\times L_2^2", "00b679c1724ef634f99d7959237a9ee6": "G'+*m", "00b6d2f480c9b40fe618f9917868f9b5": "\\left( \n\\begin{smallmatrix}\n\\;\\;\\;1 & 0 & 0 \\\\\n\\;\\;\\;1 & 0 & 0 \\\\\n-1 & 1 & 1 \n\\end{smallmatrix}\n\\right)", "00b6d8509e28d8c213b6f79878b1c687": "\\,L \\preceq M\\,", "00b71e8d40251307824661f54fe74704": "A_m=U^\\dagger \\partial_m U.", "00b7603bca787fe483f240835e48118f": "\\alpha^p \\smile \\beta^q = (-1)^{pq}(\\beta^q \\smile \\alpha^p)", "00b77e7e542a1b87d284e0c82f74b268": "\\pi_i = 2^{-N} \\tbinom Ni", "00b7f809d353a2d63350999bc4ad696d": "~~~~~U,V,\\{N_i\\}\\,", "00b7ffe43d793c9f4a697c6f2434bdb9": " z(\\infty) = \\frac{a(1-Q)-b}{aQ} ", "00b812e8b8737c6c5e614149e14930c7": "\\,d(X_tY_t)=X_{t-}\\,dY_t + Y_{t-}\\,dX_t+\\,dX_t \\,dY_t,", "00b83964f5f4e4cc74ce5e79d08753eb": "t_0^{\\frac{n}{n+1}}=({x_1 \\cdots x_n})^{\\frac{1}{n+1}},", "00b85b6e01c4bc53e0ea8cedc1b1ba71": "I \\stackrel{\\sim}\\to A_5 < S_5", "00b8a470bc756c7d053a4711b9a7b6ee": " v = \\frac{V_{\\max}[S]}{K_{m} + [S]}", "00b8c2978829c03f88bb7e05abb4da07": "s^2=\\frac{\\varepsilon^\\prime\\varepsilon}{n - p}.", "00b8dc5c0c1b1d0e68f0937ee77cc768": "I_3=\\frac{1}{2}[(n_u-n_\\bar{u})-(n_d-n_\\bar{d})],", "00b8fc4327416d1d8dfb9ead450412fb": "{}^{\\rm T}", "00b96c916f45f7efc4cf7ef334160561": "E^{\\star}", "00b9ff0a65b3004a3824631ea96c4312": "w(X)\\triangleq\\,", "00ba8bf73fc24298f04c5ced60e538d8": "\n\\beta_{k} = \\omega_{k}^{2} + \\frac{1}{4}\n", "00ba98ef451a5b3614d4b1cd2be24e33": "\\forall x\\,(x \\neq \\varnothing \\rightarrow \\exist y \\in x\\,(y \\cap x = \\varnothing))", "00bb4fbbf623dfb74a06868dd7e77789": "\nx = a \\ \\cosh \\mu \\ \\cos \\nu \\ \\cos \\phi\n", "00bb5fcd96c00665ed7216ecfd87c760": "(\\sqrt{p_1}, \\cdots ,\\sqrt{p_n})", "00bb83d631620faae8005e751c1b09e4": "2Pt(+O_2)\\rightleftharpoons 2Pt; \\;\\; PtO (+CO)\\rightleftharpoons Pt(+CO_2\\uparrow).", "00bc06d20b176a7590ab85e4c4109ecb": "a_n = \\frac{g_{n + 1}^2}{g_n}", "00bc150a7555c2c920dabcb58aae9719": "\\begin{align}\\binom{-4}{6} &= \\frac\n{-10\\cdot-9\\cdot-8\\cdot-7\\cdot-6\\cdot-5\\cdot-4}\n{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6\\cdot7},\\\\\n&=(-1)^7\\;\\frac{4\\cdot5\\cdot6\\cdot7\\cdot8\\cdot9\\cdot10}\n{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6\\cdot7},\\\\\n&=\\left(\\!\\!\\binom{-7}{7}\\!\\!\\right)\\left(\\!\\!\\binom{4}{7}\\!\\!\\right)=\\binom{-1}{7}\\binom{10}{7};\\end{align}", "00bc6441475813a9eb844a7642eb35d2": "\\frac {H^2}{\\mu}", "00bceadfb2a61f92952f04849109c872": " = \\frac{32e^4}{4(k+p)^4} \\left( (k \\cdot k') (p \\cdot p') + (k' \\cdot p) (k \\cdot p') \\right) \\,", "00bd1daee8deee2a0aa8025a6249447d": "H_{DR}^{n+i}(M)", "00bd6540355981104f96f819fcf7f3cf": "\\mathrm{sys\\pi}_1 \\leq 6\\; \\mathrm{FillRad}(M),", "00bdbff791648cce14b0647d6f68e215": "\\Beta", "00be62a0bdb8f1ff449ffe4204e32266": "a_i=\\begin{cases} \na \\mbox{ if } a\\in \\Sigma_k \\\\\n\\varepsilon \\mbox { otherwise }.\n\\end{cases}", "00beb505483ea27413e7cff5c1ceac84": "\\rho(u)\\sim\\frac{1}{\\xi\\sqrt{2\\pi u}}\\cdot\\exp(-u\\xi+\\operatorname{Ei}(\\xi))", "00bf412dae60cc909904f011967d6c0f": "\\boldsymbol{s}", "00bf432558a83b0fff96f92cc78e5903": "[-1, 1] \\times [-1, 1]", "00bf4feecbbec752edf2af3e77445d7f": "K > 0.\\!", "00bf7884e9bb0be697e4d75d83c636e3": "a_i = \\sqrt{\\sum \\limits_{j = 1}^{3}\\left (\\frac{\\partial x_j}{\\partial u_i}\\right )^2}", "00bf8fca49478b06c2393e07bd1d6351": "\\left( {} - \\frac{1}{2} \\nabla^2 + V \\right) \\psi = E \\psi \\qquad \\mbox{with} \\qquad V = {} - \\frac{1}{r_a^{}} - \\frac{1}{r_b^{}} \\; .\n", "00bf93081482d3b84ae03460925087b5": "c' \\ll \\bar{c}", "00bfd7f2d95c69816361b32ce6b642c5": " \\phi_{r}\\,= \\phi_{N} ", "00bfe9f1e05b83d7d1d4fbde9234c054": " \\Omega_{n}=\\frac{2\\pi^{n/2}}{\\Gamma \\left (\\frac{n}{2} \\right )} \\,", "00bfea28379eb8b01462876e924f558a": "N_\\beta \\beta + N_r\\frac{d\\mu}{dt} + N_p p = 0", "00c0358623f65485416f7facdc3f0e29": "\\scriptstyle\\mathbf{X}", "00c06f5f6d455dd42012fba388a8f492": "\\pi = \\frac{4}{1.25} = 3.2", "00c0a905ca111dd2c1be3c5e7a47e645": "r=\\frac{{\\rm ln} X_2 - {\\rm ln} X_1}{\\Delta t}", "00c0bc9e189771f3338f271b094ecf1f": "\\frac{1}{\\varepsilon_0 c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{P}^{NL},", "00c0bef6100867ab83e2f807fe4e3f77": "\\boldsymbol{\\Pi}^1_{2n+1}", "00c0c1609947a2417500d95a5d8ccd32": "f(x_0), f(f(x_0))", "00c142954a1c191f60016455013875bc": "\nG(2n,2n,2n) \\, = \\, \\bigl[t_1^{2n}t_2^{2n}t_3^{2n}\\bigl] (-1)^{3n} \\bigl(t_1 t_2 + t_1 t_3 +t_2t_3\\bigr)^{3n} \\, = \\, (-1)^{n} \\binom{3n}{n,n,n},\n", "00c1d4a23bb74b1715414e4c510aede7": "\\{x \\geq 1, y \\geq 1\\} ", "00c202de02da2a860d73448b4a381129": "\\mathbf{J^TW\\ \\Delta y}", "00c2b8f91dee216f6d966d2325f5977b": "E_{5} = \\Delta x \\Delta y \\Delta z \\Delta p^{2}", "00c2c62104ec723473cccd30d67bb175": "K_R=\\frac{\\sin(\\gamma)}{\\sin(\\beta)}", "00c2caa4b42b5416f74a0ad24214444a": " \\lambda m,p,q.(\\lambda g.\\lambda n.(n\\ (g\\ m\\ n)\\ (g\\ q\\ n)))\\ \\lambda x.\\lambda y.p\\ x\\ y ", "00c3092d6cd6589202586d7237b3461e": "\\bar V^*", "00c344caab36d9481e834e98b8323acc": "{{{\\hat{\\vec{I}}}}_{\\mathit{i}}}", "00c35a4b55d8067671553d5466c7adaf": "\\mathrm{CFS} = \\max_{S_k}\n\\left[\\frac{r_{c f_1}+r_{c f_2}+\\cdots+r_{c f_k}}\n{\\sqrt{k+2(r_{f_1 f_2}+\\cdots+r_{f_i f_j}+ \\cdots\n+ r_{f_k f_1 })}}\\right].", "00c363318e77d4e6e8e7c5dc4639ded4": " \\max_w R(w)= \\max_w \\frac{w^{T}Aw}{w^{T}Bw} ", "00c36eb0977c445260330806fc4eb747": "1 - 1/e", "00c3ca8d01abf5f923081a409096570e": "\\mathcal{N}(\\theta,\\sigma_\\theta)", "00c3e4693330fbba3b5583d98916c007": "\\operatorname{Var}(Y|X=x) = \\operatorname{E}((Y - \\operatorname{E}(Y\\mid X=x))^{2}\\mid X=x),", "00c437fbd49f1bffaeb29cfee7c74828": "\nU(I|J) = \\frac{H(I)-H(I|J)}{H(I)} ,\n", "00c46f60f691d54e4c15ca82cef20abd": "\\text{refresh overhead} = \\frac {0.246\\,\\text{ms}}{64\\, \\text{ms}} =.0038 \\,", "00c48d25333e2da00ac708770f86c606": "u(\\lambda,T)\\partial \\lambda = {8\\pi h c\\over \\lambda^5}{1\\over e^{h c/\\lambda kT}-1}\\partial \\lambda.", "00c4b86ffa5024242c69ab93cf3ffd77": "\\;\\frac{(n+\\delta-1)(n+\\delta-2)\\cdots n}{(\\delta-1)!}\\;", "00c50e95caad16094592549fb9f8173b": "\\epsilon_r, \\mu_r", "00c5368656dba9dbd0a8b29cd5175cde": "L_{\\text{o}}\\,\\!", "00c5664aadb43e0c76cbddcbaeab354d": "(\\boldsymbol{\\sigma}\\cdot \\mathbf{a})(\\boldsymbol{\\sigma}\\cdot \\mathbf{b}) = \\mathbf{a}\\cdot\\mathbf{b} + i\\boldsymbol{\\sigma}\\cdot \\left(\\mathbf{a} \\times \\mathbf{b}\\right)", "00c5f2e03ecffb3bb9a4d0e23bb04433": "T = \\frac{1}{2}[abch_ah_bh_c]^{1/3},", "00c6284781367cce9c24eca48ddc6b4d": "L_{\\alpha} = \\bigcup_{\\beta < \\alpha} \\operatorname{Def} (L_{\\beta}) \\! ", "00c6591c0602abb03d5832073d15ecfd": " T_{r} ", "00c6995d19447eda6861d53156af9b8e": "y(x)=a\\,\\operatorname{cosh}(x/a)", "00c6a639415adf84772a637ad27aac19": "J^n", "00c6bc6ad287f1eceb8ee7a7159c6ad4": "\\sum_f P(h_m^y | f, m, a_1) = \\sum_f P(h_m^y | f, m, a_2).", "00c6ceeac7b79177efb24f261dc5d36f": "{\\operatorname{d}\\Gamma_{(y)}\\over\\operatorname{d}y}", "00c704c7014de82863b96e129bb84f17": "Q= \\begin{pmatrix} {*} & {\\kappa\\pi_C} & {\\pi_A} & {\\pi_G} \\\\ {\\kappa\\pi_T} & {*} & {\\pi_A} & {\\pi_G} \\\\ {\\pi_T} & {\\pi_C} & {*} & {\\kappa\\pi_G} \\\\ {\\pi_T} & {\\pi_C} & {\\kappa\\pi_A} & {*} \\end{pmatrix}", "00c726e5ea52a40b4734ac16674a1fec": "A_{22}^{-1}", "00c74ad991c8ba7cba0a93b9e3a6e7a7": " W_{cu} = W_{S} - W_{c}", "00c758eb717145e027408cbd9a7204ed": "\\min f(\\bold x) = x_1^2+x_2^4 ", "00c77f1e6530daa5e26b5f1e8707ee58": "y_k[n]", "00c7a3396464a0f586e8f19d21426030": "\\nabla\\cdot\\vec{V}=0", "00c7b1f225ffee0b32c619833a234f8c": "J(x_t,u_t)", "00c7fda347c5973fe115f30555bbce33": " \\mathbf{Y} ", "00c8046617f5ed936f0bcb8cd79a21c8": "W_C = \\frac{e^2}{2C}. \\ ", "00c83e3709ab5c1370983c0e0a4f4028": " f(\\varepsilon) ", "00c84b4621cbf4da3aeac00e1374ad7e": "F(Tr(g),\\ X)", "00c893e55fea61a9e204c59337813468": "\n\\begin{align}\n& {} \\quad L\\left( x_1, x_2, \\ldots , x_N, \\lambda_1, \\lambda_2, \\ldots, \\lambda _M \\right) \\\\\n& = f\\left( x_1, x_2, \\ldots, x_N \\right) - \\sum\\limits_{k=1}^M {\\lambda_k g_k\\left( x_1, x_2, \\ldots , x_N \\right)}.\n\\end{align}\n", "00c8b4f2f2a97f1bd9eb9a94b8ea4421": "p(n) = p(0) + K\\sum_{i=0}^{n-1}\\sin(x(i))", "00c8c65b70d5576a583742e1530223c2": " \\| \\mathbf{v} \\times \\mathbf{u} \\| \\leq \\|v\\| \\cdot \\|u\\|.\\, ", "00c90641cf5ae67fd080516a748cddae": " N_D", "00c91d4e3ddd4a273fe8af6a44db4c1b": " N_i", "00c93451009bebf6d145c17b33b0b61d": "T_a = e^\\mu_a T_\\mu \\,", "00c9633c6fc31447b561ff0cec0e8c50": "\\phi=\\tfrac12(1+\\sqrt5)", "00c974b6be3cc1121a014e27602a281e": "r = \\lim_{n\\rightarrow\\infty} \\left| \\frac{c_n}{c_{n+1}} \\right|.", "00c982a871aa5fb28aa4186582d05810": "\\frac {F_{out}}{F_{in}} = \\eta \\frac {d_{in}}{d_{out}} \\, ", "00c9c03d069959e21c69983cf6238113": " \\|v-Pv\\|\\leq (1+\\|P\\|)\\inf_{u\\in U}\\|v-u\\|.", "00ca4220c56859dd1ca71a62e2fc97c9": "\\widehat T", "00ca467dc56cecb5da4be603d7f9582f": "T_{b_1} (T_{b_2} f) = (T_{b_1} \\circ T_{b_2}) f = T_{b_1+b_2} f.", "00ca46c0aa0147bfe7d01dcf3f4657a7": "\\oint \\mathbf{F}\\cdot d\\mathbf{l}=0", "00cab8bb09fe0d5af3f8d9e2b363d1f8": "\\ S_c ", "00cb24fe95a03f1afda4697049b5d046": "Opex_t", "00cc0486308f7424f7540183f4032c16": "S_\\ell=e^{2i\\delta_\\ell}", "00cc312cb3d81d4822bf81d7f2cad8e5": "\\begin{pmatrix} 2&-2 \\\\ -2&2 \\end{pmatrix}", "00cc32f85b209a9b142e09a9274ae106": "\ny\\rightarrow y^5-10 y^3 x^2 + 5 y x^4 + y_0", "00cc3e1c11be4342a07b3de6c9960cf8": "n < 1000_b", "00cc8ba3289130d190f3412b29a48685": "\n\\begin{align}\n B_0 &= \\quad a\\left(1-n+\\frac{5}{4}n^2-\\frac{5}{4}n^3+\\frac{81}{64}n^4-\\frac{81}{64}n^5+\\cdots \\right),\\\\[8pt]\n B_2 &= - \\frac{3}{2}a\\left(n-n^2+\\frac{7}{8}n^3-\\frac{7}{8}n^4+\\frac{55}{64}n^5-\\cdots \\right),\\\\[8pt]\n B_4 &= \\quad \\frac{15}{16} a\\left(n^2-n^3+\\frac{3}{4}n^4-\\frac{3}{4}n^5+\\cdots \\right),\\\\[8pt]\n B_6 &= - \\frac{35}{48} a\\left(n^3-n^4+\\frac{11}{16}n^5-\\cdots \\right),\\\\[8pt]\n B_8 &= \\quad \\frac{315}{512} a\\left(n^4-n^5+\\cdots \\right).\n\\end{align}\n", "00cc99d8927d9d46bf01ea3b4b9b3c77": "\\Omega(n^{k/4})", "00cd6312034e4528828ad17f5cb244a4": "{{y_1} ^2 \\over 2}+{q^2 \\over g{y_1}} = {{y_2} ^2 \\over 2}+{q^2 \\over g{y_2}}", "00cd895fbdebe9bdaa2d2e00777b0fda": "Y\\,\\! ", "00cd8ac8d03943325f2d48850aae3516": "(\\mathrm{det} (q)) q^{ab} = \\sum_{i=1}^{3} \\tilde{E}_i^a \\tilde{E}_i^b,", "00cd8f3fc4f954f02edf2e9b38fc64ad": "\\Re \\left (\\langle T y - m y, z \\rangle \\right) = 0.", "00cdba28214b48a1f791d20ff3774516": " \\omega_a = \\frac{2}{T} \\tan \\left( \\omega \\frac{T}{2} \\right) \\ ", "00cdfd3eba9e2b2ca90c08411366466c": "i = 1, \\ldots, p", "00ce12eb39455e0d4e6192d551e2aa16": "\\,P_1, \\ldots,P_4\\,", "00ce6ee441322cd8fb8e36106653af4f": "\\delta\\geq \\Big(1-R-\\epsilon\\Big) H^{-1}_2\\big(\\frac{1}{2}-\\epsilon\\big) \\sim \\frac{1}{2}(1-R-\\epsilon)", "00cea663cc9f3f4477ee32a282088a0b": "((n+2^{i-1})", "00cedb9e857cecf13657fc572c4abc3d": "a_{\\mathrm{in}}", "00cf4b7ac745cb3f5f1688e17f916e9b": "\\dot{\\mathbf{x}}(t) = A \\mathbf{x}(t) - B K \\mathbf{y}(t) + B \\mathbf{r}(t)", "00cf63659905603862c27f4a1a0af03c": "\\ln f =\\ln (u\\cdot v)=\\ln u + \\ln v.\\, ", "00cf95436c7c77d27e82bc13d8c6aabc": " \\mathbf{g}_{hk\\ell} = h \\mathbf{b}_1 + k \\mathbf{b}_2 + \\ell \\mathbf{b}_3 .", "00cfb43e97ff9b34c9c9e3b7f377b854": "(\\forall x \\ \\neg \\phi(x)) \\leftrightarrow \\neg (\\exists x \\ \\phi(x))", "00cfd502e07da068aa1251041be305ad": "(x\\le y \\and y\\le x) \\rightarrow x = y.", "00cfea03d60df13c7b510407aa538de4": "\\hat x' = R \\hat x R^\\dagger = e^{-i\\hat v \\frac{\\theta}{2}} \\hat x e^{i \\hat v \\frac{\\theta}{2}} = \\hat x \\cos^2 \\frac{\\theta}{2} + i (\\hat x \\hat v - \\hat v \\hat x) \\cos \\frac{\\theta}{2} \\sin \\frac{\\theta}{2} + \\hat v \\hat x \\hat v \\sin^2 \\frac{\\theta}{2}", "00cfec326228bb38450262d954608ea5": "a,b>0", "00cff248b36cd708630d75a0f8d5578d": "\\langle ax_1+bx_2, y\\rangle = a\\langle x_1, y\\rangle + b\\langle x_2, y\\rangle.", "00d01ce332cd24bfb260d7405c784721": "f(x) = \\frac{2}{2^{k/2} \\Gamma(k/2)} x^{k-1} \\exp\\left(-\\frac{x^2}{2}\\right)", "00d02be0050fe6b53904e4a4b469d708": "\\mathrm{adj}(\\mathbf{A})_{ij} = \\mathbf{C}_{ji} \\,", "00d03569e01b8be4b186f40df949ae2d": "F(\\nu)=\\frac{8\\pi h\\nu^3}{c^3}", "00d068fab91da8db80e20baf8367ae5f": "p_i'=\\rho_i c D\\Psi_i,\\qquad i=L,G.\\,", "00d0b5c77159a2b0473eb45c80c6446f": "\\vec{v} = P - R", "00d0ce68fc33da33c1ce0e4f1d9a5066": "0 = - \\rho [\\vec{x},t] + \\epsilon_0 \\nabla \\cdot \\vec{E} [\\vec{x},t] ", "00d157cc401f53c2a7fbfb07bda65556": "\\begin{bmatrix}1 & 1 \\\\0 & 1\\end{bmatrix}", "00d17f8035a0d96ed31b6c7d4f68d407": "\\mu_R", "00d1816c30a2064d8a33fb3b72968a7b": "\\frac{\\partial N}{\\partial t} + \\nabla\\cdot\\vec J = 0", "00d183d55d2b196abb82932ba311b65f": "a\\int_{-\\infty}^\\infty e^{-y^2/c^2}\\,dy,", "00d18957cb2173f2ed89a8c17c18c6d5": "y(t) = \\int_{t_0}^{t} f(\\tau) d\\tau\\,", "00d19c4426d87e5afe94809d4244e5fb": "r(t) \\in L_1[0,T] ", "00d1b7a7930501dc59f8789c05987ea3": "\\{X_{1},\\ldots,X_{n}\\}", "00d1ca048ffe0d52e58241c23cab4edc": "\\begin{align}\nx &= r\\sin\\theta\\cos\\phi \\\\\ny &= r\\sin\\theta\\sin\\phi \\\\\nz &= r\\cos\\theta \\end{align}", "00d24f2a938028537e5ec1e402fb025e": " r = L\\cos^2\\lambda ", "00d2996c35870a082c4257b025d1e05c": "\\left\\langle Q[F]\\right\\rangle =0.", "00d2c33a3d9573d1640ecac2b9b4840e": "f^{64}(4) = G;\\, ", "00d309d510caebc30ceba5f8950bbbd3": "f= \\left( 0.79 \\ln \\left(\\mathrm{Re}_D \\right)-1.64 \\right)^{-2}", "00d325a2fdf76f62cae935baca1795c1": " x(N)={1\\over N+1}\\sum_{n=0}^N T^n(x). ", "00d336848fa1cadb1f0bc947ef5fe26f": "\\left(-4\\right), \\left(-1\\right), 1, 1, 3", "00d37c0bdd7265bef0e6c59d9ed57c7e": "\\boldsymbol{\\mu}_\\text{I} = g_\\text{I}\\mu_\\text{N}\\mathbf{I}", "00d39b47b89dbd09d391dfaf690ff54d": "g(\\lambda z,\\overline{\\lambda}\\bar{z}) = \\overline{\\lambda}^{2s} g(z,\\bar{z}).", "00d416245b926fa94db6707e1bfa26f3": "\\Lambda_n = \\Lambda \\cap \\mathrm{QSym}_n", "00d45880eeb858a9c271cdc1ee503b18": " \\mathbf{f}+\\operatorname{div}\\,\\sigma=0", "00d45e14f600ae168770d540cd1ba279": "\n\\sigma_t \\equiv \\frac{8\\pi}{3}r_e^2\n", "00d4698687efb283b8b2efc7d4eadbd7": "\\mathrm{GF}(q).", "00d4789a3dec5360bb488b32283ae6e5": "0100km s^{-1}", "00d8f4690ab8ba747fbef705e87f85ea": "{RSF} = \\frac {D_{V0.9} - D_{V0.1}} {D_{V0.5}}", "00d91888083257bc9da64df8b5b77495": "\\mathfrak M=\\langle P, G,\\textrm I\\rangle", "00d91d0dbc2f9fe1df573c3630e695da": "\\mathrm{Inv}\\langle X | T\\rangle", "00d91e801972be464fa4a166f9632c82": "\\alpha=1\\,", "00d943bb09a594302694dbd086a23e67": "CH_4 + e^- \\to CH_4^+ + 2e^-", "00d94a624b0292143baac796a7a2c061": "A = \\frac{1}{4}\\frac{N}{V} v_{avg} = \\frac{n}{4} \\sqrt{\\frac{8 k_{B} T}{\\pi m}} . \\,", "00d955c498045606e5500803af522135": "\\mathrm{Taxicab}(5, 2, 2) > 1,024,000,000,000,000,000 = 1.024 * 10^{18}.", "00da16cf18f3f2b19a5dda51c87224f1": "{ \\partial^2 \\psi \\over \\partial t^2 } = c^2 \\nabla^2\\psi ", "00da453affacc526f052e4e8e298f098": " \\delta_X(t) \\ge c \\, t^q, \\quad t \\in [0, 2].", "00da99ec5c19e6d0e85396ae7a00cbd0": "\\tfrac{n}{m}\\,", "00dacbdfd9de8e8a8f1de82579834b1a": "A = \\lbrace q : q^* = -q \\rbrace \\!", "00db0fb33c36c75487183306752b416d": "\\nabla({\\boldsymbol\\mu}\\cdot{\\boldsymbol B})", "00db1fa81e4e01b636cd8d68cab8af6b": "ab+bc+ca=s^2+(4R+r)r,", "00dba71d6100580a7bcebbaf8cbe77c5": "C = 15 d^2", "00dbc826534ab999725ea212f1c69ead": "Y^{\\mu}(\\tau)", "00dbd349ac88a050015b40f536c37b37": "V = \\frac{4}{3}\\pi r^{3}", "00dbe5b634a4e98c045d14c8e50b29a0": "{\\tilde{D}}_{5}", "00dc099636c10a19826ff7617ad552d9": " \\mathcal{F} = \\frac{\\Delta\\lambda}{\\delta\\lambda}=\\frac{\\pi}{2 \\arcsin(1/\\sqrt F)},", "00dc240282b8eb8a8da6e88a060ae253": "x \\in L_{n+1} (\\pi_1 (X))", "00dc888cd757386d5ca7fec6f428fd8f": "P(x_1,x_2) = \\frac{p_1^3-p_1p_2}{2} + \\frac{p_1^2-p_2}{2} \\,,", "00dcdbff0ef7631903745ed151e888eb": "\nH(2^1) = \\begin{bmatrix}\n1 & 1 \\\\\n1 & -1 \\end{bmatrix},\n", "00dd261574c58b34290bf82201117286": "\\cos^{-1}\\langle v_{i}, v_{j}\\rangle", "00dd2adbf272ed1c0c561673c17b0abb": "M_2(\\tau+1) = e^{-2\\times 25\\pi i/168} M_2(\\tau)", "00dd34e39b176f5b5af123e9c219d851": "(14)\\qquad \\theta_{(n)}=\\hat{h}^{ba}\\nabla_a n_b=\\bar m^b m^a\\nabla_a n_b+m^b\\bar m^a\\nabla_a n_b=\\bar m^b \\delta n_b+m^b\\bar \\delta n_b=\\mu+\\bar\\mu\\,.", "00dd434f3b19ea165df7db7617d6b649": "a d^2 + b d + c = 0", "00dd43e22370a716f4aa72e3780e1383": "M _{BC} ^f = - \\frac{qL^2}{12} = - \\frac{1 \\times 10^2}{12} = - 8.333 \\mathrm{\\,kN \\,m}", "00dd441ee2e71bf8cc375cf8676fb415": "g(f(k)) + O(|x|^c)", "00dd5e4951f7aed71b8408ed927f31d4": " y_{ij} = \\mu + \\tau_i + \\epsilon_{ij} ", "00ddcb1d5007fb9bd4f82cacfee3e2f7": "C_1, C_2, C_3, C_4", "00dde2f7a53805b6a926341e3ffe11fe": "\\exp(i\\varphi) = \\cos(\\varphi) + i\\sin(\\varphi) \\,", "00ddfe1c0682e4afa3cdfa3764c60765": "E_{1,1}= 510,260 * \\frac {260}{510,260} * \\frac {10,060}{510,260}", "00de10b46d39cabce52c002b4a33ecc9": "\\dot{\\textbf{x}}=f(\\textbf{x},u) ", "00deaa3867a2ab2e7e90ea94042ebe23": "\\{\\hat{1}, \\hat{5}\\}", "00deb5e44ecc1d9f4d0eec4311dd44e6": "\\pi_k(O)=\\pi_{k+8}(O) \\,\\!", "00debd5d6cdedd0fd8d32f39cb8c00d8": "\\frac{a^x\\Gamma(\\frac{ax+b}{a})}{\\Gamma(\\frac{a+b}{a})}\\,", "00ded4313ff02634b6674dd079500b24": "\\sqrt{|\\Delta_K|}", "00deea9376f926019407f400638e861d": "\\ln (n+1) = \\ln(n) + 2\\sum_{k=0}^\\infty\\frac{1}{2k+1}\\left(\\frac{1}{2 n+1}\\right)^{2k+1}.", "00df09b96a36904ebb578eb1f05f77a4": "cm \\cdot \\sqrt{Hz}/ W", "00df2a8a4c44eb54e671d77699afa8ef": "\nF_1(a,b_1,b_2,c; x,y) = \\frac{\\Gamma(c)} {\\Gamma(a) \\Gamma(c-a)} \n\\int_0^1 t^{a-1} (1-t)^{c-a-1} (1-xt)^{-b_1} (1-yt)^{-b_2} \\,\\mathrm{d}t, \n\\quad \\real \\,c > \\real \\,a > 0 ~.\n", "00df3631d22d38ff63d952305dfdcbf4": "\\{\\alpha_{j1}, \\ldots ,\\alpha_{jm}\\} \\subseteq \\{\\alpha_1, \\ldots ,\\alpha_n\\}", "00df8c39fad899a9c54e7bf525399a9b": "\\oint_C \\left({1 \\over z^5}+{z \\over z^5}+{z^2 \\over 2!\\;z^5} + {z^3\\over 3!\\;z^5} + {z^4 \\over 4!\\;z^5} + {z^5 \\over 5!\\;z^5} + {z^6 \\over 6!\\;z^5} + \\cdots\\right)\\,dz ", "00df972d2c271a82d92810d7c5896ebf": "d(x,y) = \\| f_x - f_y \\|,", "00dfb406b411d6b4f4747a589f08a0bd": "B_O", "00dfd04fcd66ecfaa75cbd6216f8ecfa": " \\varphi(\\mathbf{r},t) = \\frac{1}{4\\pi \\varepsilon_0} \\int\\frac{\\mathbf{\\rho}(\\mathbf{r'},t)}{R}d^3r'", "00dfd3b1d5aa76c34b086e7bd80ad512": "x_P,y_P,a", "00e00e01f453611770fe6e93d8e3a976": " \\ddot{t} + \\frac{2}{x} \\, \\dot{x} \\, \\dot{t} = 0, \\; \\ddot{x} + x \\, \\dot{t}^2 = 0, \\; \\ddot{y} = 0, \\; \\ddot{z} = 0", "00e0135b44d128f41d10a54cfa1582d7": " \\textbf{P} = [T(\\phi, \\mathbf{d})]\\textbf{p} = \\begin{bmatrix} \\cos\\phi & -\\sin\\phi & d_x \\\\ \\sin\\phi & \\cos\\phi & d_y \\\\ 0 & 0 & 1\\end{bmatrix}\\begin{Bmatrix}x\\\\y\\\\1\\end{Bmatrix}.", "00e040d159567545fcc73346bcede176": " \\mathcal{S}", "00e05f27e4728ed01881d0110e63112e": " \\mu^+(E) = \\mu(P\\cap E)", "00e07493b2a973570f63aef3d235fa02": "\\Delta\\lambda_B", "00e078273a56777927d4d1ebad370dd0": "\\bigcup_{k\\in\\mathbb{N}} \\mbox{DSPACE}(2^{n^k})", "00e0a9b8a3df6878b80a59ae9f99da2d": "\n\\int \\exp\\left[ \\int d^4x \\left ( -\\frac 1 2 \\varphi \\hat A \\varphi + i J \\varphi \\right) \\right ] D\\varphi \\; \\propto \\;\n\\exp \\left( - { 1\\over 2} \\int d^4x \\; d^4y J\\left ( x \\right ) D\\left ( x - y \\right ) J\\left( y \\right ) \\right)\n", "00e0bc6b6fa01b4434f090b3b0dc6335": "f\\colon (x,h)\\to(x',h')", "00e0dd01c0d7c832bd2d85ed799213eb": " \\frac{d}{dx}\\arccos(x)= -\\frac{1}{\\sqrt{1-x^2}}, -1 1.96) = 0.025, \\,", "00e9ff58fd26233d196727decbb8299e": "\\psi(n) = H_{n-1}-\\gamma\\!", "00ea04e63b470b5a388a603743ca5e0c": "F(X, Y)", "00ea34016973645f9300ad306688a80c": "\\lambda=1/3^n", "00ea34d26b099e9a8fcb9c46e0c53f85": "\\lambda \\in \\Lambda", "00eaea6b6d04912ef0e1d19dec0c8de6": "\\Lambda(x,\\lambda,\\nu) = f_0(x) + \\sum_{i=1}^m \\lambda_i f_i(x) + \\sum_{i=1}^p \\nu_i h_i(x).", "00eb1b2042bf13d3cd835d1322eeaf6f": "\\sqrt{s_{NN}}=200", "00eb20cabf12f793a27c2a5efc5c83e3": "F^{-1}(p;n,1)", "00eb39716d8b7640272128c3d1efcb5a": "f_\\mathbb{H}(\\alpha)=\\omega^\\alpha.", "00eb86947f7d681c7e38a469d78c4e10": "(h*g)^* = h^* * g^*", "00eb8ddc3e102a880f8830fa40184bdf": "x_n \\to 0", "00eb9bb834af2565c19f18328604c050": "a\\quad", "00ec6670f291a54bd603a01ed1b5d802": "C_{P, el} = \\gamma T = \\frac{\\pi^2}{2}\\frac{k_B}{\\epsilon_F}nk_BT", "00eca1b27a7f6fcdb1c102ad67cfa641": "p_i(s) \\ne p_j(s)", "00ecba9a4dd7bd3d2981a76e7464ea45": " \n\\nu_{k}(\\mathbf{J}) = \\frac{1}{T}\n", "00ecf52d65fb00be76ea52bbc333dd67": "y_4=y_3+h(\\tfrac14k_1 + \\tfrac34k_2)=\\underline{1.335079087}.", "00ed278ec09422df6c1b6c7544693a3a": "\n\\Delta E=\\frac{1}{2}\\alpha_0\\left(T-T_0\\right)P_x^2+\\frac{1}{4}\\alpha_{11}P_x^4+\\frac{1}{6}\\alpha_{111}P_x^6\n", "00ed3794f143bbcf0aea4a78715c707a": " \\theta(\\xi)=\\sum\\limits_{n=0}^\\infty a_n \\xi^n ", "00eda8772cea2311b2a365f89fdfcb9b": "\\mathbf{F}_{\\mathrm{net}} = m\\mathbf{a}_{\\mathrm{cm}}", "00ee06cf2adda9c1fea6cbdeb588ea2f": "\\delta(\\varnothing)=\\varnothing", "00ee2e53e92542458ff31715b7a81ebf": "lim^*", "00ee31b0657b8616be40541c4d326199": "\n\\tan \\theta = \\sin \\lambda \\tan(15^{\\circ} \\times t)\n", "00ee8205d9738aee1e2ee3086ae05f53": " y \\ge 0.398", "00ee92b891492c30771ce8b238d0e5be": "\\left(\\sqrt{\\frac{2}{5}},\\ -\\sqrt{\\frac{2}{3}},\\ \\frac{-5}{\\sqrt{3}},\\ \\pm1\\right)", "00ee9a89b1bf53def17c6ec0901ef41d": "pf = {P_a + P_b + P_c \\over |P_a + P_b + P_c + j(Q_a + Q_b + Q_c)|}", "00eeedc7b69405e57deac906e57c5f19": "j = 2, 3, \\ldots, m\\ ", "00eefb2b6b06be1004f91ffa8db3dce5": "\\tan \\gamma = \\frac {d} {R} \\,;", "00ef41d18ca5f3e9deaf55d719272b28": " W := (W_1, \\dots, W_d)", "00ef434594abd949d326cfe092280abc": "v_i : A \\longrightarrow R_+", "00ef776b74b13504b900b0e68fca544c": "\\frac{\\partial c}{\\partial x} = \\frac{\\partial c}{\\partial \\xi} \\frac{\\partial \\xi}{\\partial x} = \\frac{1}{2 \\sqrt{t}} \\frac{\\partial c}{\\partial \\xi}", "00ef987a5388b0b127138d0aef79b6f1": "\\mathcal R=(<_1,\\dots,<_t)", "00efa6a77deaafdb2502b9c077cde286": "L_{g}L_{f}^{i}h(x)", "00efd6280759fc6e3b506689467d003a": "\\chi={C \\over T}", "00f01f2e549a95f7050c54482197c866": " P(R_{NP} \\cap R_A^c, \\theta_1)= \\int_{R_{NP}\\cap R_A^c} L(\\theta_{1}|x)\\,dx \\geq \\frac{1}{\\eta} \\int_{R_{NP}\\cap R_A^c} L(\\theta_0|x)\\,dx = \\frac{1}{\\eta}P(R_{NP} \\cap R_A^c, \\theta_0)", "00f039d45804b9bcb48cda188a6dc085": "g_i(0) = \\left. \\frac{\\partial f(z)}{\\partial z_i} \\right|_{z=0}", "00f0746da2f28aa1374e48ae048cb4b5": "\\begin{align}\\binom{-r}{k} &= \\frac{-r\\cdot-(r+1)\\dots-(r+k-2)\\cdot-(r+k-1)}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}\\\\\n&=(-1)^k\\;\\frac{r\\cdot(r+1)\\cdot(r+2)\\cdots(f-2)\\cdot(f-1)\\cdot f}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}\\\\\n&=(-1)^k\\binom{f}{k}\\\\\n&=(-1)^k\\left(\\!\\!\\binom{f-k+1}{k}\\!\\!\\right)\\\\\n&=(-1)^k\\left(\\!\\!\\binom{r}{k}\\!\\!\\right)\\;,\\end{align}", "00f0ae08d8aa3c5c08a83a108da9c688": "x_{i+1},\\ldots,x_n", "00f16a47475ad1385451f8781b66a7e3": "r_i,s_i\\in \\mathbb{R}", "00f1935351a51f42498a297e61a5cacd": "\\ C-\\text{vertex}= 1 : -1 : -1 ", "00f1a523058441ae4e449e8959edc01b": "\\phi (t) \\to (\\exists x \\ \\phi (x))", "00f1c82d17358dd9b5dfc14705f26f50": " C \\subseteq \\{0,1\\}^t, |C| = n ", "00f1d2f8c59c696529d591a3d697d1e2": " \\lambda^{2p} c_H ( \\lambda^p t, \\lambda^q H) = \\lambda^d c_H(t, H) \\, ", "00f20d86ef06cc0932330c692d8027bb": "\\gamma(i_j)=\\gamma(n)", "00f21c1aafe1f46bf3844636e73bc995": "\\epsilon=0\\,\\!.", "00f27297d54b3aeba08e7ce05172c51e": "P(\\mathbf{s})", "00f2806a43b3c8c594f16bd6c54f139e": "\\tilde{\\kappa}_{tr}=\\scriptstyle -0.4\\pm0.9\\times10^{-10}", "00f2a7fb18ef9f999f11d41d5d06f6cc": "p^2-p+1", "00f2ac1cfefd7f10d8f0f8602e8ada08": "\\mathrm df_x(X) = \\langle (\\mathrm d Y)_x(X),x\\rangle + \\langle Y_x, X_x\\rangle = 0.\\,", "00f2b472121ef098a7da40fcc25bb3e0": "\\theta_{\\text{hr}} = \\frac{1}{2}M_\\Sigma = \\frac{1}{2}(60H + M)", "00f2bef40423a891f0b44fa7b5ef62be": "\\delta_{\\theta}", "00f2d62661d2ba1bfeb24b5a69831f7c": "\\pi^{-n}|F(z)|^2 \\exp(-|z|^2)", "00f2f5f0f7f040bd0228ea0b965dd0f8": "V(y) = \\sup_{\\tau \\le \\tau_\\mathcal{S}} J^\\tau (y) = \\sup_{\\tau \\le \\tau_\\mathcal{S}} \\mathbb{E}_y \\left[ M(Y_\\tau) + \\int_0^\\tau L(Y_t) dt \\right]. ", "00f2f6810ac3900653117fb397b4bcec": "F : X \\to X ", "00f2f97d990f02788d955ded67325c25": "\\displaystyle{Q_y(a)=Q(a)Q(y),\\,\\,\\, R_y(a,b)=R(a,Q(y)b).}", "00f312a0444a815e3379b768a36f9a82": "\\alpha x_i + (1 - \\alpha)x'_i >_i x_i^*", "00f322b619703b467e6a25a969fb3e69": "\\sec(M_i)\\ge -1", "00f35a9b6f60fec19b77496b2355a1a0": "(S\\otimes T)^{i_1\\ldots i_l i_{l+1}\\ldots i_{l+n}}_{j_1\\ldots j_k j_{k+1}\\ldots j_{k+m}} =\nS^{i_1\\ldots i_l}_{j_1\\ldots j_k} T^{i_{l+1}\\ldots i_{l+n}}_{j_{k+1}\\ldots j_{k+m}},", "00f3743aa47d5bf6a020ca4a31e90398": "L = D - W", "00f38015779ac8f08efec2b41add8a5b": "{y^k}'(0)", "00f39c473af512d02fb6bd50fe4f6256": "d_x(p) := d(x,p)\\,", "00f3b6143499cc3b862de3e62062daf5": "s = 2^{0} + 2^{1} + 2^{2} + \\cdots + 2^{63}.", "00f3c9966987607a99730c76bc433930": "\\Delta\\sigma", "00f4bd49d1a7004d90ea380d36c41546": "f_0,\\dots,f_m", "00f5193589c35c3beceb543b25ad3032": "k = \\log_{b} w = \\log_{b} b^k", "00f5739e4f39eed7cbbac7fac1a6117f": "F(x,y)=0\\,\\!", "00f59200f79c84fea9991cbd3819b621": " L(P, t) = \\frac{7}{4}t^2 + \\frac{5}{2}t + \\frac{7 + (-1)^t}{8}. ", "00f5a703c61aa0fe9d1d810367643f36": "x' = x_1 = v/2a ,\\ \\ y' = y_1 + v^2/4a \\ ", "00f65f89c91d577837233107e1c43638": "\\Phi : A \\rightarrow B(H),", "00f6824b92f276a2a322ca8918ac7d0c": "= (\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)))\\ g", "00f6920c4ab9951d3e65397852efa61a": "d \\Phi = d S - \\frac {T d U - U d T} {T^2}", "00f69a8f51d74253c95d4bc78917bfdf": "T \\geq T_0", "00f70e54a98dc30bff28031d0471efd8": "\\{\\phi_i\\}_{i=1}^{\\infty}=\\{\\alpha_i\\}_{i=1}^{\\infty}\\cup\\{\\beta_i\\}_{i=1}^{\\infty} ", "00f74621e6765a1bfdf213ef5caca455": "C_p = C_{p~max}", "00f7604203a5423216dc67057ce0215a": "{\\Gamma}^{*}_n", "00f7788fa5d413bf26f99d916e262801": "\nr_{1}^{2} r_{2}^{2} \\left( \\frac{d\\theta_{1}}{dt} \\right) \\left( \\frac{d\\theta_{2}}{dt} \\right) - \n2a \\left[ \\mu_{1} \\cos \\theta_{1} + \\mu_{2} \\cos \\theta_{2} \\right],\n", "00f7871570c9fbe85f3d77ce2a47ed28": "(r, \\theta, \\phi)", "00f7afd7395deaf6c4c7b1225d1be196": "\\scriptstyle M^{-T}", "00f7c047a2bb558b2d9cacf653e904f9": "g(x)\\partial_x", "00f7c9af6fe1a26a5273fd624549bd78": "\nP_j^n = \\left( \\begin{array}{l} n \\\\ j \\end{array} \\right) j! = \\frac{n!}{(n-j)!}.\n", "00f8110b1646fdf7e83e71ec60699c1c": "H_S = H_{0,S} + H_{1, S} ~.", "00f82de0b4c0784560759a470ba1e2db": "v_n\\in V_n, a(u_n,v_n) = f(v_n)", "00f8b941960594da446f06dcb43c24d5": "\\ T\\Delta G_S^\\circ = T\\Delta H_A^\\circ + T\\Delta H_B^\\circ - T\\Delta S_{AB}^\\circ ", "00f8e2c516640e6fcd650b00d542df09": "\nC_{k} = \\left( \\frac{1}{k} \\right) \n\\int d\\theta^{\\prime}\n\\int d\\rho^{\\prime} \\left(\\rho^{\\prime}\\right)^{k+1} \n\\lambda(\\rho^{\\prime}, \\theta^{\\prime}) \\cos k\\theta^{\\prime}\n", "00f90abe1ab45bcd354b79173a50be07": "D(d) \\wedge \\underline{\\neg D(f(d)) \\wedge D(f(d))} \\wedge \\neg D(f(f(d)))", "00f922907920a1c5bf1ffab1976c3ab4": "M - 1", "00f9741740f00e3a15167a9eabc1141e": "U = - m \\sum G \\frac{ M}{r} ", "00f97a8df6a3e6b2656c97f895be7cea": " \\gamma \\ \\stackrel{\\mathrm{def}}{=}\\ \\partial u_x /\\partial y ", "00f9bfef84d607575d466d8e2cf206be": "\\mbox{then} \\quad U B_1 g = B_2 U g = \\sum_{i=0}^n (B_2^*)^i A h_i.", "00f9f64d586edc538f07598e75bd7e6a": "\\{1, 5, 9, 13\\}.", "00f9f8af4014b9c9ba89a00e688d61a8": "L \\approx 4\\pi R^2\\sigma T_I^4\\frac{l}{R}\\approx \\frac{(4\\pi)^2}{3^5}\\frac{\\sigma}{k^4}G^4\\bar m^4 \\langle \\rho \\rangle l M^3", "00fa1b09d5593180d106bf84f3aeb25e": "\\boldsymbol{\\varepsilon} = \\boldsymbol{0}", "00fa91012c19f237403f36589a916e06": "n(x,y)", "00faf620268a7727621272df0cb5d004": "s_1=\\sum_{i=1}^m \\log x_i", "00fafea58bdae9fde99ae911df2dc687": "y_c = \\left({q^2 \\over g} \\right)^{1 \\over 3}", "00fb22b24186d4bec2293e66f62c28ec": "(A_1A_2)^2-r_1^2-r_2^2 \\, ", "00fb626b41cdaeed91618e2c143511ec": "\n\\begin{align}\n\\Delta \\hat{e}\\ & =\\ \\frac{P}{2\\pi}\\ \\frac {1}{V_0}\\ \\int\\limits_{0}^{2\\pi}\\left( (-\\sin(u)\\ \\hat{k}\\ +\\ \\cos(u)\\ \\hat{l}) \\ F\\ \\cos(u)\\ + \\ 2\\ (\\cos(u)\\ \\hat{k}\\ +\\ \\sin(u)\\ \\hat{l})\\ F\\ \\sin(u)\\right)\\ du \\\\\n& = P\\ \\frac{3}{2}\\ \\frac {1}{V_0}\\ \\ F\\ \\hat{l} \n\\end{align}\n", "00fc431ddc28efbd388ad723f0f0ee25": "\\psi(0,x)", "00fc606c713c687da931b916520aa0ab": "\nV(x_1 ... ,x_N) = V_{1,2}(x_1,x_2) + V_{1,3}(x_2,x_3) + V_{2,3}(x_1,x_2)\n\\,", "00fc95fa70207762082f4c24704e320d": "x^2 - Ny^2 = 1", "00fcd7684e4b7476132e4898a5e1ef1e": "a_{11} x_1", "00fcff732898300c9f752e2a5e1f933d": "2^1 \\times 0.1000_2 - 2^1 \\times 0.0111_2", "00fd1da21e8b4ef31d987665dc575099": "3/2", "00fd89f696a1863b6ca202e1cc674619": "(k+l)", "00fdabb96d5bc35cf466e45d8c0e7ea3": "1-\\left(1-\\frac{1}{d}\\right)\\left(1-\\frac{2}{d}\\right)\\cdots\\left(1-\\frac{n-1}{d}\\right)\\geq \\frac{1}{2}.", "00fe5914b3da55ad956f46423c2e2db6": " M_t = (M_{1,t}M_{2,t}\\dots M_{\\bar{k},t}) \\in R_+^\\bar{k}. ", "00fe6a8d6543b053688d56904a800884": "\\displaystyle e^{ 2\\pi iax} f(x)\\,", "00fecd587aaaf41ac1ae0de228e72700": " \\mathbf{B} = \\mathbf{v} \\times \\frac{1}{c^2} \\mathbf{E} ", "00fed98a386431f51ecf1b300fc572f9": " \\frac{\\partial^2 f}{\\partial x^i \\, \\partial x^j} = \\frac{\\partial^2 f}{\\partial x^j \\, \\partial x^i}", "00fee35d098e7ede04688e054b0bcd95": "\\int_\\gamma \\rho\\,|dz|", "00ff8b525150181f600d4d6469d72e48": "\\varphi = \\begin{bmatrix} \\varphi_{stator} \\\\ \\varphi_{rotor} \\end{bmatrix}", "00ffe4e1b0b3c2080a17caf8b4dd5ec2": "Y_\\mathrm {i \\Pi} = \\sqrt {Y^2 + \\frac{Y}{Z}}", "00fff65d34e1aa4cec757836ae3802fb": "\\mathbf{p^{n+1}=p^n +\\delta p}", "01001680de1dcb97337713b5e92dbbae": "\\neg p \\or q", "01001b39914230da09b6548877a4cb99": "135 = (1 + 3 + 5)(1 \\times 3 \\times 5)", "01002661415b311f875cbb1b0149cabf": "x\\in\\mathbb R", "010056e8dd4c8176092bfd7c448d3ef3": "\\ell_2=r'+a'", "01009cc723b713a37f31197e765611ac": "\n \\lim_{n\\to\\infty} \\frac{\\log |W_n|}{n^2} = h > 0.\n", "0100c57389c7ef9cbf33292dc5557d3f": "\\mathcal{M}_{ij} = \\begin{cases} 1 /L(p_j) , & \\mbox{if }j\\mbox{ links to }i\\ \\\\ 0, & \\mbox{otherwise} \\end{cases}\n", "0100feb2d04bb42c8d668cb8c1f745de": "\\left[ \\begin{alignat}{6}\n1 && 0 && -3 && 0 && 2 && 0 \\\\\n0 && 1 && 5 && 0 && -1 && 4 \\\\\n0 && 0 && 0 && 1 && 7 && -9 \\\\\n0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 && \\;\\;\\;\\;\\;0 \\end{alignat} \\,\\right] ", "01013feccf496a2036355f101f8262c0": "\\scriptstyle{a=6.1121\\ \\mathrm{millibar};\\quad\\;b= 17.368;\\quad\\;c= 238.88^\\circ \\mathrm{C}:\\quad\\quad\\! 0^\\circ \\mathrm{C}\\le T\\le +50^\\circ \\mathrm{C}\\;\\;(\\le0.05%)}", "0101b1db64c8b0252ec743708a73f160": "t\\mapsto (t,f(t)).", "0101d89bbbc91b5bcb24aefc9c85d788": "\\psi_l", "01026a06c53da688c72cb0a160dfbfa9": "\n\\mu(x,y) = \\begin{cases}\n{}\\qquad 1 & \\textrm{if}\\quad x = y\\\\[6pt]\n\\displaystyle -\\sum_{z : x\\leq z -1", "010312fc903173511c916ac83e307399": "(A,B;C,D) = \\frac {AC}{AD}.\\frac {BD}{BC} = -1. \\, ", "01033f6e1fef6a26df8d24ae68b5ea94": "{{I}_{OUT}}\\approx \\frac{{{V}_{CC}}-1.4}{R1}", "01034c22987fb23fe2470a98cac59a6c": "\n\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ 4 & 2 & 7 & 6 & 5 & 8 & 1 & 3 \\end{pmatrix} =\n\\begin{pmatrix} 1 & 4 & 6 & 8 & 3 & 7 & 2 & 5 \\\\ 4 & 6 & 8 & 3 & 7 & 1 & 2 & 5 \\end{pmatrix} =\n(146837)(2)(5)", "0103d9083b6297bd3abb5e70f74e36fd": "CM\\,", "0103f34b09470ebfb13324efd2ea958a": "\\scriptstyle \\dot m_{01} \\,0\\, p_{21} \\,", "010445a7575314b56e76038a7323011e": "V=U", "01044947b534a2326edc87845aaf5e73": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 \\\\\n-1 & 2 & -1 & -1 \\\\\n 0 & -1 & 2 & 0\\\\\n 0 & -1 & 0 & 2 \n\\end{smallmatrix}\\right ]", "01045123e83db59cdcae28d0568aefb7": " E'' = \\frac{E \\tau_0 \\omega}{\\tau_0^2 \\omega^2 + 1} ,", "01045a9d6f8880f1083a52e42c4fa3a2": "\n \\underline{\\underline{\\boldsymbol{A}_1}} = \\begin{bmatrix}-1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\boldsymbol{A}_2}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\boldsymbol{A}_3}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix}\n ", "010486e2b1be8fe0ef4030d3d106dc74": "\\Gamma^\\alpha_{\\beta\\gamma} = \\frac{1}{2} g^{\\alpha\\epsilon}(g_{\\beta\\epsilon,\\gamma} + g_{\\gamma\\epsilon,\\beta} - g_{\\beta\\gamma,\\epsilon}).", "0104c0311e39f9860b10d55583ae02ea": " A^i {}_{k;\\ell} = A^i {}_{k,\\ell} + A^{m} {}_k \\Gamma^i{}_{m\\ell} - A^i {}_m \\Gamma^m{}_{k\\ell}, \\ ", "0104f8d1b787cbc747f4e3be26a2f983": " W^{1,1}(\\Omega)", "01050f5a6c40e55ce8a661e9e261001c": " \\psi_i^m(0) = 0, \\quad\\qquad \\frac{d\\psi_i^m}{dx}(0) = 1. ", "01055417f16cfe6c9cdafe71d15a7601": "\n\\begin{pmatrix}\n0 & i \\bar{\\partial}\\\\\ni \\partial & 0\n\\end{pmatrix} \n\n\\begin{pmatrix}\n \\bar{\\Psi}^\\dagger P_3 \\\\ \\Psi P_3\n\\end{pmatrix}\n\n= m\n\\begin{pmatrix}\n \\bar{\\Psi}^\\dagger P_3 \\\\ \\Psi P_3\n\\end{pmatrix}\n", "01055c25e62d0efce371faeb74de6790": "p_c = cS_w^{-a}", "010573deb29a21cce0b460e782579ca7": "Z_X = \\int_0^\\infty exp\\left\\{ -\\frac{1}{2} \\left(\\Delta X^T\\left( \\frac{k_B T}{\\gamma} \\Gamma^{-1} \\right)^{-1} \\Delta X \\right) \\right\\}d\\Delta X", "0105c98648eac04e84c78046ebe79281": "\\aleph_{0} = \\omega", "01064cc0b83abd93a55e276f959997f1": "P_{4n+1}", "01066775fb82cd31f4d24ad9f105eb72": "|\\Phi\\rangle_\\nu=|\\Phi_0\\rangle_\\nu \\oplus |\\Phi_1\\rangle_\\nu \\oplus |\\Phi_2\\rangle_\\nu \\oplus \\ldots = b_0 |0\\rangle \\oplus |\\phi_1\\rangle \\oplus \\sum_{ij} b_{ij}|\\phi_{2i}, \\phi_{2j} \\rangle_\\nu \\oplus \\ldots", "01069784c44be3f6a432ae18ad52500a": "b > a", "010699bca1525939ffb3d3afc84724c7": " p <_\\mathcal{O} r ", "01072c6ff185236a9e28ab3740190dba": "h \\circ in = f \\circ Fh", "0107992ec5fe58000025a2b4678726bb": " \\langle y^2 \\rangle = \\frac{1}{P} \\int{I(x,y) (y - \\langle y \\rangle )^2 dx dy}, ", "01083b716768aff86c8863df3ec483c5": "\\frac{| \\text{median} - \\text{mode} |}{\\text{standard deviation}} \\leq \\sqrt{3}", "01087980a48f55dc43c8509c3340e6c7": "\n\\begin{align}\nL &= T-U = \\frac {1}{2} M \\dot{\\mathbf{R}}^2 + \\left( \\frac {1}{2} \\mu \\dot{\\mathbf{r}}^2 - U(r) \\right) \\\\\n &= L_{\\mathrm{cm}} + L_{\\mathrm{rel}}\n\\end{align}", "0108997daf7ca086f0286c453fbd686a": "\n\\left\\langle \\int \\phi( \\boldsymbol{x}, t ) \\, d \\boldsymbol{x} \\, dt \\right\\rangle = \\int \\langle \\phi(\\boldsymbol{x},t) \\rangle \\, d \\boldsymbol{x} \\, dt.\n", "010902b462092577c279b155d9b6c730": "\n\\sqrt{\\frac{1}{N}\\sum_{i=1}^N(x_i-\\overline{x})^2} = \\sqrt{\\frac{1}{N} \\left(\\sum_{i=1}^N x_i^2\\right) - \\overline{x}^2} = \\sqrt{\\left(\\frac{1}{N} \\sum_{i=1}^N x_i^2\\right) - \\left(\\frac{1}{N} \\sum_{i=1}^{N} x_i\\right)^2}.\n", "01092b385f6bf4c8c66a4fe0eb43fce3": "\\nabla\\times\\nabla\\times", "0109adca58e0b5448c672b496c42d700": "2I", "0109b8038d6a1ce251f1b33fc594c43b": "t = \\frac{1}{s}", "0109ff5e08bee125b08f8871f5faf5ef": "\\frac{\\mathrm{d} \\det(A)}{\\mathrm{d} \\alpha} = \\det(A) \\operatorname{tr}\\left(A^{-1} \\frac{\\mathrm{d} A}{\\mathrm{d} \\alpha}\\right).", "010a5867de53b91da45a532bba2c19f1": "\\sigma(E)", "010a602110241800fb96b131799ae444": "\\ V_c", "010a6ae3278e36a894ba2dd26eff1d38": " \\mathbf{a_{\\mathrm{Cfgl}}}", "010a783383fdb44f6c116b76d54dcac5": "\\Rightarrow P_0-M_aTe^{-rT}=0", "010a8d6811366852e1099de8bd2a17e5": "m\\left( x^\\mu \\right) =\\Omega \\tilde{m}_0,", "010aa76873e9d7e8d8f046f780325dce": " \\sum_{i \\neq j} \\pi_i q_{ij} = \\sum_{i \\neq j} \\pi_j q'_{ji} = \\pi_j \\sum_{i \\neq j} q_{ji} = -\\pi_j q_{jj}", "010ac74caa3412b1b118d4fdf7845578": "\\rho(T) = \\rho_0[1+\\alpha (T - T_0)]", "010adf4c6ced9a728df5d15df83737a9": "~A \\triangle B \\triangle C", "010b89e573d00053cdb94543806beef2": "K_\\mu-K^{(0)}_\\mu\\,", "010b9b7813c77c13706e107bc6ed3970": "g(a)", "010ba4b68d115c03803566f5fb23aa33": "\\text{and}", "010bc0d1c798e3c3ffe66e58fd8b9aa1": "({x}_{1}, {x}_{2}, {y}_{1}, {y}_{2}, z)", "010bcb271a01cbc1992ae84a01c933cd": "\\forall f,\\ \\langle \\pi_1 \\circ f,\\pi_2 \\circ f \\rangle = f", "010bd9525b51288f53aa1b96f9df78ba": "\\sum_{n=-\\infty}^{\\infty} x[n] \\cdot \\delta(t-nT) = \\underbrace{\\sum_{k=-\\infty}^{\\infty} X[k]\\cdot e^{i 2 \\pi \\frac{k}{NT}t}}_{\\text{Fourier series}} \\quad\\stackrel{\\mathcal{F}}{\\Longleftrightarrow}\\quad \\underbrace{\\sum_{k=-\\infty}^{\\infty} X[k]\\ \\cdot\\ \\delta\\left(f-\\frac{k}{NT}\\right)}_{\\text{DTFT of a periodic sequence}},", "010ce63d12e72ccf4c6b7734c013ac74": "f(x_0, ..., x_n) = 0", "010d0031e0378397227e26ac79fdbb22": " P V^{\\gamma} = \\operatorname{constant} = 100,000 \\operatorname{pa} * 1000^{7/5} = 100 \\times 10^3 * 15.8 \\times 10^3 = 1.58 \\times 10^9 ", "010d11347ba394e5de251b56ee5cffc5": "S(t) = 1 - e^{- \\rho t} \\ \\frac { \\sin \\left( \\mu t + \\phi \\right)}{ \\sin( \\phi )}\\ ", "010d198ed3e886b2bd899031be35afc8": "I = \\frac{\\pi}{2\\sqrt{2}} \\left(17 - 5^{\\frac{3}{4}} 2^{\\frac{9}{4}} \\right) = \\frac{\\pi}{2\\sqrt{2}} \\left(17 - 40^{\\frac{3}{4}} \\right).", "010d2d61606dea3f3c9ac92797b33cde": "(1,0,0)\\,", "010d67b17db62d4120254fa78329f430": "m\\frac{d^{2}\\mathbf{x}}{dt^{2}}=-\\lambda \\frac{d\\mathbf{x}}{dt}+\\boldsymbol{\\eta}\\left( t\\right).", "010d82cce5da096194db036398fa6268": " \\geq 3 ", "010da5ca94a3a09d473eede273468b57": "y = R \\sqrt{1 - {x^2 \\over L^2}}", "010ded9ac15b567d0d703ee999cda567": " (\\text{Total COE Quota})_{qy} = g.(\\text{Motor vehicle population})_{y-1} + (\\text{Projected de-registrations})_{y} + (\\text{Unallocated quota})_{qy-1} ", "010dfad868e4db1f46382a085599dcf1": "C(f)", "010dfcb5c3f2da6b3324559ac8c4a947": "v = kT + T - \\tau", "010e015cee9b35816b245769a1312f5a": " (1 2)(3 4),\\;(1 3)(2 4),\\; (1 4)(2 3)", "010e1df78a41ec6f33dc926c7e788f53": "a = d \\sin\\alpha \\text{ and }b = d \\sin\\beta. \\, ", "010e22805899e839e8ad0357d6291459": "\\begin{align}\\mathrm{d}^kX &= \\left(\\mathrm{d}x^{i_1} e_{i_1}\\right) \\wedge \\left(\\mathrm{d}x^{i_2}e_{i_2}\\right) \\wedge\\cdots\\wedge \\left(\\mathrm{d}x^{i_k}e_{i_k}\\right) \\\\\n&= \\left( e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\right) \\mathrm{d}x^{i_1} \\mathrm{d}x^{i_2} \\cdots \\mathrm{d}x^{i_k}\\end{align}", "010e2eac6374591a1bd1915c8aad321b": "\\overline{A_i(\\alpha_1, \\ldots, \\alpha_{dim(A_i)})}", "010e406df2463597c58286a93f8b3160": "5959", "010e6246a2bf3a7312443a891f0d6807": "{\\mathrm{d}H\\over \\mathrm{d}\\theta}=v^2 2\\cos(\\theta)\\sin(\\theta) /(2g)", "010ed64a18f5a752fb8dc04b4cbb15c7": "", "010ee67f9b45e754482ee25dc169e448": "\n\\begin{matrix}\n& & 0\\\\\n &0 & \\\\\n0& &B_{i-2,3}\\\\\n &B_{i-1,2}& \\\\\n1& &B_{i-1,3}\\\\\n &B_{i,2}& \\\\\n0& &B_{i,3}\\\\\n &0& \\\\\n& & 0\\\\\n\\end{matrix}\n", "010f0cc465fa1897532a16c9a7bebccf": "\n K(\\overline{\\alpha}, \\alpha' ) = \\langle \\alpha| \\alpha'\\rangle =\n \\left[{\\mathcal N}(\\vert \\alpha\\vert^2) {\\mathcal N}(\\vert \\alpha'\\vert^2)\\right]^{-\\frac 12}\n \\sum_{n=0}^\\infty \\frac {(\\overline{\\alpha} \\alpha')^n}{\\varepsilon_n!}\\; .\n", "010f1dc08f5b3205173de9b3ef97f8d5": "\\mathbf{a} \\cdot \\mathbf{b} = \\frac{1}{2}(\\mathbf{ab} + \\mathbf{ba}).", "010f45b224c66649fd24a2d41cca9077": "A_3, BC_3,", "010f64129d9fe13f5403409b74e435de": "1+k", "010f7fafaef8d2512449da2d87f661f7": " {d^2 X^\\mu \\over ds^2} = {q \\over m} {F^{\\mu \\beta}} {d X^\\alpha \\over ds}{\\eta_{\\alpha \\beta}}.", "010facc8491c6cc8f90b1b691e331eec": " \\nabla \\times \\mathbf{E} = \\nabla \\times \\left( - \\nabla \\phi - \\frac { \\partial \\mathbf{A} } { \\partial t } \\right) = - \\frac { \\partial } { \\partial t } (\\nabla \\times \\mathbf{A}) = - \\frac { \\partial \\mathbf{B} } { \\partial t }. ", "010fbe9ba54ae3ce64ecc869a1d1f16b": "A(x)=\\sum_{n=0}^{\\infty}A_{n}\\frac{x^{n}}{n!}.", "010fddfcd902a3a23f8062b501729920": "G(S,T) = \\Pi_{i=0}^n (a_iS -b_iT)", "010ff055cb8498e38fd1928cdb931835": "z^{2M}-1 = (z^M - 1) (z^M + 1) \\,", "0110381eee9e40ad90f85de1fd4b4c11": "\\scriptstyle{E}", "01104c023b0e663624f2860e3a834417": "\\mathbf{C} \\otimes \\mathbf{C} \\to \\mathbf{C}", "01108ee28e7d36d435864892ef5d7472": "P_t (f) = f \\cdot \\Omega_t .", "0110bce9efd49901c1280eb57432d9d4": "z=\\zeta^{-2}.", "01110fd744e06804b1349f3028504fb4": "\n \\boldsymbol{u}^{(0)} = \\boldsymbol{x} - \\boldsymbol{X}, \\qquad \\boldsymbol{u}^{(1)}=\\boldsymbol{x'} - \\boldsymbol{x}\n ", "011136a856e1b55439f93cddb217cd15": "Q=\\left(1+\\frac{r^2}{d^2}\\right)^{1/2} ", "011181d1c8b4d961d70145417a40cad4": "\\Phi_V(G,k)=\\min_{S\\subseteq V} \\left\\{|\\Gamma(S)\\setminus S| : |S|=k \\right\\}", "0111dc8658ba9e9ea247f960fc04d49c": "\\{\\dot{x}_1, . . . , \\dot{x}_n\\}", "011207bf80cb24795234a1ac1028d7bd": " \\frac{n! \\cdot e^{-\\tau s}}{(s+\\alpha)^{n+1}} ", "01121327b29599ef36ed6dd2721c5249": "e_1>e_2>e_3", "011216940992ec86880f2fbb4775e8a3": "\\alpha\\in\\mathbb{C}", "01122e120967df8acc9cefaa8e670083": "X \\leftarrow Y \\rightarrow Z,", "011234a9a5e2e0dee096ea7d2e3583f5": "{AE}_{6}", "01123baec994e153a1a611a7722dfd43": "\nr^m \\sin^m\\theta \\sin m\\varphi = \\frac{1}{2i} \\left[ (r \\sin\\theta e^{i\\varphi})^m \n- (r \\sin\\theta e^{-i\\varphi})^m \\right] =\n\\frac{1}{2i} \\left[ (x+iy)^m - (x-iy)^m \\right].\n", "01125bda091746a740325341056ffbb5": "\\frac{1-2p}{\\sqrt{np(1-p)}}", "01132bb6f4147773832a0f398d70b353": "\\sum_{n=1}^\\infty S_n(s)x^n = {sx(1+x) \\over (1-x)^3 + 4sx(1-x)}.", "011334d3fb3f0590045702635200c3b2": "\\alpha=1/N", "011393892db3c5f70775f612f769abe3": "W_nW_{n+1}", "0113bc090b7893e7fe1785bd13a56f66": "\\bold{g}=-\\bold{\\nabla}\\phi_g\\,\\quad \\bold{E}=-\\bold{\\nabla}\\phi_e\\quad", "0113c2181543e683a6e08f0de1b2d2c2": " and \\; E' = \\frac{E}{y_c}", "0113eea8a622904bea55f29f3b0d8b5f": " \\mathbf{M}_{\\rm orb}=\\frac{e}{2\\hbar}\\sum_{n}\\int_{\\rm BZ}\\frac{d^{3}k}{(2\\pi)^{3}}\\,f_{n\\mathbf{k}}\\;{\\rm Im}\\;\\langle\n\\frac{\\partial u_{n\\mathbf{k}}}{\\partial{\\mathbf{k}}}|\\times(H_{\\mathbf{k}}+E_{n\\mathbf{k}}-2\\mu)|\\frac{\\partial u_{n\\mathbf{k}}}{\\partial{\\mathbf{k}}} \\rangle, ", "0115006b38e647df4fd59a12e8ca5ec7": "Pj_{\\mu\\nu} = \\delta_{\\mu\\nu}\\delta_{\\mu m}", "01157d38ff33bacb82305caaf0563185": " \\mathbf{\\hat{n}} \\,\\!", "011587386377fee6fa116ed1e0a7632f": "\\ T_c", "01158c0052a10dbede5392256528da42": "\\,\\! x=x^+-x^-", "0115a1827a27116f17a185a58c8bf45d": "S = 2160 \\text{ miles}", "0115a3cd741a626fa1ccdab6e49377bf": "\\hat x = x_0 - x_1", "0115b2bcf65b76e3a6dc869dbb461f40": " \\mathcal F ", "0115b6d0e853baa84d1d57bfc6cb34d5": "[n]_q x^{n - 1}", "0115ef6c5ad3b514a2a841b68d55fb29": ".\\qquad NP/N,\\; N/N,\\; N,\\; \\underbrace{(NP\\backslash S)/NP, \\quad NP}", "011676bac198587c1ee2747ff140304e": "g(x, y, t) = g(x, t) \\, g(y, t)", "01170a7d6571521be3cca093412de98d": "= \\arctan \\frac{120}{119} + \\arctan \\frac{-1}{1}", "0117523b217d98d3af216a5eeee428bb": "\\{0, 1/(p-1), ... , 1-1/(p-1), 1\\}", "0117813d7915d44dc57392c29a517cf2": "\\begin{align}\nV_1(\\mathbb R^n) &= S^{n-1}\\\\\nV_1(\\mathbb C^n) &= S^{2n-1}\\\\\nV_1(\\mathbb H^n) &= S^{4n-1}\n\\end{align}", "0117bd8f3283f282e12383f128066e0d": "\\textstyle v^2 = \\mathbf{v} \\cdot \\mathbf{v}", "0117e5251f4d4c8d5069db88662ea843": "\\zeta_n\\in \\mathcal{O}_k ", "01184af66a83cfabcec15e5008b7b908": "v \\mapsto \\overline v", "0118c7b56b11e08311e39ddd217b13e4": " \\frac{1}{2T}\\int_{-T}^{T}\\,F(a+it)G(b-it)\\,dt= \\sum_{n=1}^{\\infty} f(n)g(n)n^{-a-b} \\text{ as }T \\sim \\infty. ", "01192796a31d5ddef12c5932427015be": "Z = \\sqrt{{R + j \\omega L} \\over {G + j \\omega C}}", "01194ee3ef2be78544698c591b41cc29": "\\mathcal O(E)", "01195b5c3c65a2e936bbc59624736582": "\\varphi(r) = {\\sin (\\ell r)\\over \\ell \\sinh r}", "0119c33834388b477ea829d9ecdd5f5b": "\\frac{1}{T(s)}\\cdot\\frac{dT(s)}{d\\varphi}=-\\frac{t}{n}.", "0119fba08ad14ba30732514039d870fa": " p_i = q_i \\,", "011a0c4f97e9e5bdb6f22186853bb8b0": "\\{v_1, v_2, \\ldots,v_k,v_{k+1},\\ldots,v_n\\}", "011a670a56fa85d571b453901af53cc2": "n_s = (1 - \\frac {\\beta}{\\beta_0}) \\frac{n_i}{n_0}", "011a673f27b86385a3a6d173aa0a72ee": "p^2=\\mu^2\\sqrt{\\frac{\\lambda}{2}}.", "011a6a6252b5fd1cda01edea029e39b5": "q = \\frac{\\pi}{4} T\\, v(\\theta)\\, \\cos^4\\theta", "011a6dbbbf4c9061f8112708331f0778": "\\ v = k[A][B]", "011a7f737228f34b4db13701be8561fb": "s(h,k)\\,", "011abb8ca80eebdf6873f48e7569541e": "(\\forall F\\subseteq U_p)(QUA(F) \\iff (\\forall x,y)(F(x)\\wedge F(y) \\Rightarrow \\neg x<_p y))", "011aff52d9f198a5ad6e9adbf8309dc2": "\n\\begin{bmatrix}\n x & 1 \\\\\n 1 & x \\\\\n \\end{bmatrix} \\times\n \n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n \\end{bmatrix}= 0\n", "011b0fa13253a12989641a4f775d6a93": "L(\\hat{y}, y)", "011b6e3f128e0de494b3cf0dbecebdb4": "u_j = | \\langle r A_j \\rangle | ", "011b76b2657900f43aeb2eb6f00f3078": "1 + z = \\left(1 + \\frac{v}{c}\\right) \\gamma.", "011ba3f0db4cd8865f15adc08b9b1e4a": " c_2 = 0.988622465, \\,\\!", "011c329b23a7ef28a2ac2e3acf831905": " \\lim_{z\\to 0} \\frac1{z}\\left\\{\\frac1{\\Gamma(1+z)} - \\frac1{\\Gamma(1-z)} \\right\\} = 2\\gamma", "011c45f9300361dab2a3178eb0de4fc1": "{\\varphi}", "011cc0e22684bf7c68fafa96e57bfea9": "\n\\Pr \\left \\{ \\lambda_\\max \\left ( \\sum_k \\mathbf{X}_k \\right ) \\geq t \\right \\}\n\\leq \\inf_{\\theta > 0} \\left \\{ e^{-\\theta t} \\operatorname{tr} e^{\\sum_k \\log \\mathbf{M}_{\\mathbf{X}_k} (\\theta) } \\right \\}\n", "011cfaea9b775115c2ed7cd4e365c19a": "\\vec{v}_p = \\frac{m}{qB^2}\\frac{d\\vec{E}}{dt}", "011d1eb205a58a64270d1f8db8d71496": "(X-\\alpha)\\cdot H)=C\\cdot P(X)", "011d265e68fcf4fcd5c8bfeadff3d883": "\\Gamma \\vdash \\psi", "011d7055b12ac9c6011b288ea4369e4c": "F(x) = \\sum^{\\infty}_{n=0}f_nx^n", "011d85cc1dadb7c594c567b1bf84ed15": "\\Delta \\chi", "011d91f0f55bdcbfab3374d21a45f206": " m_1 e^{s_1}+m_2 e^{s_2}=m_1 e^{s_3}+m_2 e^{s_4} ", "011d945eff010dfb86e59178d558599d": "O(\\epsilon)", "011db249cbb421ddbd4646f0427b875a": "\\mathrm{d}U = T \\mathrm{d}S - P \\mathrm{d}V.", "011dd4023c135f144b52f7281f0a9283": "\\partial{C}.", "011e597046539907efaf6c364d599b7d": "\\begin{align}\n\\frac{\\partial \\mathbf{u}}{\\partial t}+\\left(\\mathbf{u}\\cdot \\nabla \\right)\\mathbf{u} &= -\\frac{\\nabla p}{\\rho} + v \\nabla^2 \\mathbf{u}\\\\\n\\nabla\\cdot \\mathbf{u} &= 0\\\\\n\\mathbf{u}_\\text{bd} &=\\mathbf{u}_\\text{s}.\n\\end{align} ", "011e6b034711ad7c2533ec10a802a236": "R_a=\\sqrt{MN}=\\frac{a^2b}{(a\\cos\\varphi)^2+(b\\sin\\varphi)^2}\\,\\!", "011f09b611e8dfdf2839e129107b57cb": "\\displaystyle I_M(\\gamma,f)", "011f9f40084dbe619093c6799fd364ca": "p_t", "011fbf27c05a51fd558715cb15ca9e6c": "(a_n X'_n + b_n) \\,", "011fe1abc0dc78ffe7389e8e075b346c": "\\varphi(m, n, p) = m\\uparrow^{p - 1}n.\\,\\!", "012042451fb61a8bc8a16fc2d9496d7a": "\\begin{bmatrix}3 & 1\\\\7 & 5\\end{bmatrix}\n\\rightarrow\n\\begin{bmatrix}0.393919 & -0.919145\\\\0.919145 & 0.393919\\end{bmatrix}", "01207a4ae4426161f9a15ba082019284": "C_N", "01208a2f1c00f274d657da007e07bcad": "\\mathbf{J}^2\\Psi = \\hbar^2{j(j+1)}\\Psi", "0120bb85314e516e67fd9e122b322d02": "w=e^\\phi\\in A_{p}", "0120c11249c5dbe88939b4d3a428bfdd": "H_k^{l,p}=Z_k^l/(B_k^{l+p}\\cap Z_k^l)", "0121170b7b0a6ca554e7f22887a4bbbd": "\\prod_{i=0}^{k-1} (x - z_i)", "01218e3452eea40edd9d230ab0057bd8": "\\gamma_{\\|}", "0121be62d098e0058b48c4f32cc2e579": " \\mu=\\Lambda ", "0121cd1b8f6435a7f637b39c96f742b6": "g=\\frac{4\\pi\\hbar^2 a_s}{m}", "01224cb59366d304002144491499e8c1": " A\\ ", "0122a035f0874d830f4198e2804ccd16": "\\omega_1+\\omega_2", "0122e6feaedd1975ebdea673a294b23d": "\\csc\\left(\\frac{\\pi}{2} - A\\right) = \\sec(A)", "0123454bed8d8b55e908efad5eeae92c": "\\Omega_{\\lambda} = .0001\\ldots_2", "012388ec0c34cb5ea3af47429243ba62": "\\mathrm{V}_4 = \\langle a,b \\mid a^2 = b^2 = (ab)^2 = 1 \\rangle.", "0123a92e3e442417076106d28f7ae281": "\\lim_{q \\rightarrow 1} {}^q\\!D = \\exp\\left(-\\sum_{i=1}^S p_i \\ln p_i\\right)", "01243d3114be219db97be76d0831b7f3": " b_n ", "0124683f164f8d31d6b54164cf7dba14": "R=U\\Sigma'V^*,\\,\\!", "01246900c35b6d82eb37621d9094a5e9": "\\scriptstyle\\bar{x} = \\frac{1}{n}\\sum_{i=1}^n x_i", "01247e727fdc6aca334e4996d78b0ec6": "\\check{f} ", "01249b96b456dc3c29cf0a71502a489c": "\\liminf_{x\\to x_0} f(x)\\ge f(x_0)", "0124aa6c23fdb3fc1f3d174333d49c6a": " \\int_0^2 \\! \\int_{0}^{\\pi/2} \\! \\int_0^2 \\! \\bar{f}(r,t,h) r \\, dh \\, dt \\, dr = 16 + 10 \\pi", "0124b193fbc8b25177f41093f23080f9": "{\\color{Blue}x^2}+{\\color{YellowOrange}2x}-{\\color{OliveGreen}1}", "0124bcf3a2001bc3da170761ee0a4ba5": "w(X,\\tau')\\leq nw(X,\\tau)\\,", "01251a850a18fe5ef4a9a02076099e5e": "J^{\\star}", "01256288155bfb1804d71b253962c5e3": "df/f=dn/(n-1)=1/n", "012571aa32cea51f459f1af58b7ba349": "N \\cdot m^{-2}\\!", "01257cc3117225db04024ce9155f2ca3": "\\sin \\theta", "0125adcbb2ba01b3e0093cea861e567d": "P_\\beta(\\sigma) ={e^{-\\beta H(\\sigma)} \\over Z_\\beta},", "01260f820ff769acbea7ee0dd2d678d1": "(p \\leftrightarrow q) \\vdash ((p \\to q) \\land (q \\to p))", "012678e9bb0cf8d9740d1be60944d8cb": "\nT_T = \\sum_{i=1}^m s_i T_{T_i} + \\sum_{i=1}^m s_i \\log{\\frac{\\overline{x}_i}{\\overline{x}}}\n", "0126a60313e72eefaf6c46737d9b41a1": "d = s_1u_1 + s_2 u_2 +s_3(v_1+v_2+h)", "0126edb486b8c0b0b88b24f0440672ba": "W_{1-i} = W''_{1-i} \\cup B", "012750d4fc9e49702ad721133305438e": "c.", "012763afcb19637d2ec85a93fc8ebcc1": "10^{-12}", "0127bc801b5fc9a97fa76be519913071": "\\operatorname{Aut}_X(X_j) \\to \\operatorname{Aut}_X(X_i)", "012809c2e71817addfcf8ab58d7d62e3": " \\tilde{P}(X_1, \\ldots, X_{n-1})=\\tilde{Q}(\\sigma_{1,n-1}, \\ldots, \\sigma_{n-1,n-1})", "01283759cb5b7d72323d613004d5c6cb": "\\operatorname{pf}\\begin{bmatrix} 0 & a & b & c \\\\ -a & 0 & d & e \\\\ -b & -d & 0& f \\\\-c & -e & -f & 0 \\end{bmatrix}=af-be+dc.", "0129236b0bf87eadf6e0c48815ec29fc": "D=A \\cdot B-C \\neq 0", "0129a9ee48ce2de0728ccc23b5d32fd2": "0 \\le \\delta < 1 - \\frac{1}{q}", "012a6bf5f2d5689d4b61f63efb7d36e9": "x_3=0", "012af98c41fa64353b10d071979f4ae5": "\n \\cfrac{\\partial g}{\\partial g_{ij}} = 2~J~\\cfrac{\\partial J}{\\partial g_{ij}} = g~g^{ij}\n ", "012afeab512cdc3b69024644abf16bff": "\\nabla_{\\mathrm{X}_i} \\mathrm{X}_j = 0 \\, ,", "012b05f6f7bec834265a393fcdb608b7": "|B^*|", "012b08e1dab01b8b9706c324265ad777": "b + c", "012b29917c1c6a0e2d2171090701d548": " Tr(K) \\,\\!", "012b2b76378399778cccd4cad4146838": "d\\omega^j = \\sum_{i=1}^r \\psi_i^j \\wedge \\omega^i", "012baadc023e1e82d21fb22b1aecf7b5": "|\\psi(x,t_1)|^2 = |\\psi(x,t_0)|^2\\quad", "012be5e7056d1da507286f526e4b3bc5": "\\|x'\\| = \\sup_{x \\in X,, \\|x\\| = 1} | \\langle x', x \\rangle |", "012c71509a2548925edcec9c39967a8a": "t \\in \\{0,1,\\dots,T\\}", "012c8cccd5e31063edc5ff7db706695a": " \\mathsf{ZFC} ", "012c91f015fe9872e2612e2fb0c33f03": "[0:1:0]", "012d01b09de6abd503712ac7ab36595d": " f(x)= \\begin{cases} x^3, & \\text{if } x\\in \\mathbb Q \\\\ \n \\arctan{x} ,& \\text{if } x\\in \\mathbb R\\backslash \\mathbb Q \\\\\n \\end{cases} ", "012d35d00b383e446f3f084fa0cff8fa": "K_{-0} \\ \\stackrel{\\mathrm{def}}{=}\\ K_{--} \\cup K_{0}", "012dfd4f0d3c6100c8810ad0b61389c8": "t=D\\,T", "012e25daf4b340530125e7655d29e5b2": "(p_1, p_2, \\dots, p_n)", "012e30acfe0f610448dce473af2107a9": "x_{n+1} = \\frac{x_n}{8} \\cdot (15 - y_n \\cdot (10 - 3 \\cdot y_n)).", "012e71358bcdf91b0dd0cdeb1e887aad": " V = \\sum_i \\left. v^i \\frac{\\partial}{\\partial v^i} \\right|_{(x,v)}.", "012e794869b8318a9c5c7bc810a12fbe": "\\mathit{H}\\mathit{H}^*", "012ea3637d253a7387d80d824b8b5876": "i = 0,1", "012ea4761337a8a050b97a456aebd691": "\\scriptstyle\\boldsymbol{f}(\\boldsymbol{x}) = \\left( f_1(\\boldsymbol{x}), f_2(\\boldsymbol{x}), f_3(\\boldsymbol{x}) \\right)", "012eaa0ffeb592014ddd33f1f0a8466a": "\\displaystyle{[(a_1,T_1,b_1),(a_2,T_2,b_2)]=(T_1a_2-T_2a_1,[T_1,T_2]+L(a_1,b_2)-L(a_2,b_1),T_2^*b_1-T_1^*b_2)}", "012eb411e6f12e33648440ca8b078a34": "z_o= \\frac{{\\frac{F_o} {m}}}{\\sqrt{(\\omega_n^2 - \\omega^2) + (\\frac{\\omega_n \\omega} {Q})^2}}, \\; \\theta=\\arctan\\left [\\frac{\\omega_n \\omega} {Q(\\omega_n^2 - \\omega^2)} \\right ]\\,\\!", "012eb63873c1483f3d0c45fabeaa5392": "m_t \\; = \\; M(u_t,v_t) \\; = \\; \\mu u_t^a v_t^b", "012f8a1247eb79e8f0a2dbdf34ac7285": "T(n)\\in O(n^2) \\, ", "012f8be8085d9a15d7e98ad5095835fb": "\\mathbf{B} = \\mathbf{A}_q", "012fe8748ee1f4ab919629265a10db9a": "a^{n-1} \\equiv 1 \\pmod{n}", "01301819a754ae52e9cb29cd2f99f39f": "y = \\int_0^L \\sin s^2 ds", "0130481a486fff641d732f80c081debb": "\\ \\mathbf{A}^3 - \\mathrm{I}_A \\mathbf{A}^2 +\\mathrm{II}_A \\mathbf{A} -\\mathrm{III}_A \\mathbf{E}= 0", "01306a128b6e5bf1c6818d9e6db26151": " r=1-p, A=\\rho", "01307bec59a2a8c59ea2dee9e62884d7": "\\mathcal{M}_{fg}", "01308b69a6af75f2703b8530739d1aad": "\\scriptstyle 1 = \\sum_{i=1}^{r}S_i Q_i", "0130abb5ce2d09836b11370a1f0b9675": "P A - (P + \\text{d}P) A - (\\rho A \\text{d}h) g_0 = 0 \\,", "0130b9feeffff34774c6552e694f8dd2": "d\\geq d_c=4\\,", "0130d4b66578b7cb583e18ffbf58e966": "\\,l_{x + 1} = l_x \\cdot (1-q_x) = l_x \\cdot p_x", "0130f7556a53fd628ce6c7711a7b6741": "y(t_0)", "013184e4ae039b6ec28d676a46c91160": "t(t-1)(t-2)(t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352)", "013196c1528820c696c05fdd019f2bc1": "(g,1)(h,0)=(gh^{-1},1)", "0131b645b7fb3092f2c6185c5e574abb": "k \\approx aF^b(\\rho T_{2lm})^c", "0132354d2539ebfd5df65b84a86c147c": "\\frac{1}{2}L_1 \\rightarrow L_1 ", "01323bc7d0450c490d6e7fe0e6d834c3": " \\gamma\\dot{x}(t) = - k( x(t) - x_0 ) + \\xi(t) ", "0132501af7f43013a2238ba00589a8ea": "L(a_1,\\ldots,a_n)", "01327f8d65f79d07ca14f8009102022e": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, S, \\langle b \\rangle), \\langle a \\rangle)", "013281a45bcd3f3b0be61a2925d85467": "\n \\hat\\theta = \\operatorname{arg}\\min_{\\theta\\in\\Theta} \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)\\bigg)' \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)g(Y_t,\\theta)'\\bigg)^{\\!-1} \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)\\bigg)\n ", "01328eae0ef136dadbc4e8035cf57e95": "z \\mapsto \\frac{az+b}{cz+d}\\;\\;\\;\\;\\mbox{ (where }a,b,c,d\\in\\mathbf{R}\\mbox{)}.", "0132c942f2b5ca4b5cf2451a37f81760": "\\Phi(v_i,z)", "0132d9f378a8d8f7ecb7c048653c4f0c": "\\mathrm{not}~s", "0132dec062ea905a7a546d908745115e": "[h_i,f_j]=-c_{ij}f_j\\ ", "01335a55c757948d19b802db16cbf961": "\\cot \\theta \\,\\! .", "01336f72f56be6d66c128e12b1710ada": "D_j, j=1,\\cdots,N", "0133fbb1b33d299c11fd161f2dca2193": "\\left(\\frac{a}{n}\\right) = \n\\begin{cases}\n\\;\\;\\,0\\mbox{ if } \\gcd(a,n) \\ne 1\n\n\\\\\\pm1\\mbox{ if } \\gcd(a,n) = 1\\end{cases}\n", "013408c14b63d227243d789a3e82deb2": "s_0(t)=\\frac{\\alpha\\,e^{\\beta t}\n-\\beta\\,e^{\\alpha t}}{\\alpha-\\beta},\\quad\ns_1(t)=\\frac{e^{\\alpha t}-e^{\\beta t}}{\\alpha-\\beta}\\quad", "013430fa683e50e86ae691586e6b6348": " |x_\\theta\\rangle ", "0134475bc0c73018a4d06bb200daf95a": "\\begin{align}\n e^{ix} &{}= 1 + ix + \\frac{(ix)^2}{2!} + \\frac{(ix)^3}{3!} + \\frac{(ix)^4}{4!} + \\frac{(ix)^5}{5!} + \\frac{(ix)^6}{6!} + \\frac{(ix)^7}{7!} + \\frac{(ix)^8}{8!} + \\cdots \\\\[8pt]\n &{}= 1 + ix - \\frac{x^2}{2!} - \\frac{ix^3}{3!} + \\frac{x^4}{4!} + \\frac{ix^5}{5!} - \\frac{x^6}{6!} - \\frac{ix^7}{7!} + \\frac{x^8}{8!} + \\cdots \\\\[8pt]\n &{}= \\left( 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + \\frac{x^8}{8!} - \\cdots \\right) + i\\left( x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots \\right) \\\\[8pt]\n &{}= \\cos x + i\\sin x \\ .\n\\end{align}", "01344dfb9ae3295888fc7757943505b8": "k \\equiv (k \\hbox{ mod } 2^n) + \\lfloor k/2^n \\rfloor \\pmod{2^n - 1}.", "01345567169ac6c885df21b57c5d1b39": "\\displaystyle \\operatorname{Tr}(R(f)) = \\sum_{\\pi} m(\\pi)\\operatorname{Tr}(R(f)|\\pi)", "013464259bd9a3f765d987a56677237c": "(sa)\\div b = \\begin{cases} \ns & \\mbox{if } a=b \\\\\n(s\\div b)a & \\mbox{if } a \\ne b\n\\end{cases}", "0134ce711aba1a5d3734b9e36f77ba51": " \\mathcal{L}^* = \\{ \\mathbf{v} \\in V \\quad | \\quad \\langle \\mathbf{v},\\mathbf{v}_i \\rangle \\in R \\}.", "0134d1a09490d2d081f8ff1c72ed5668": "\\Gamma,x\\!:\\!\\sigma \\vdash t\\!:\\!\\tau", "0134f5dca9c6d943b80f334ba20d441d": "y(t)=y_0 \\left( x - \\frac{1}{5} x^2 - \\frac{3}{175}x^3 \n - \\frac{23}{7875}x^4 - \\frac{1894}{3931875}x^5 - \\frac{3293}{21896875}x^6 - \\frac{2418092}{62077640625}x^7 - \\cdots \\right) \\ \n", "0134fe896f2db72d72ee8faad50ead66": "|(a,b,c)|^2", "013555e4d53232dc5e312301b6b684f1": "\\displaystyle{e_\\alpha(z)={z^\\alpha\\over \\sqrt{\\alpha!}}}", "01356f495dd2fd66b165b161ea7acc6c": "t=t_0", "013572c04c0d30a2f5bb460305929605": "X \\mapsto \\mathcal{P}^\\perp_{n_\\infty \\wedge n_o}\\left( \\frac{X}{- X \\cdot n_\\infty}\\right)", "0135e0d854ad3f435241fd00e79366c6": "\ny = b\\ \\sinh\\ \\mu\n", "0135efc53a1ef0d8b71ebd8bd463323c": "mol Fe_2O_3 = \\frac{20.0 g}{159.7 g/mol} = 0.125 mol\\,", "0135f990c3ed5081e26a1dc50109e6b9": "\n2 \\left\\langle T \\right\\rangle_\\tau = -\\sum_{k=1}^N \\left\\langle \\mathbf{F}_k \\cdot \\mathbf{r}_k \\right\\rangle_\\tau.\n", "0136048f886a13a8dee3dbc967d039e2": "T^i_j", "013624a40fe42347cbab24b181f961d9": "U_{\\mathbf{Q}_p}^{(n_p)} \\subseteq N_{L^\\chi_\\mathfrak{p}/\\mathbf{Q}_p}(U_{L^\\chi_\\mathfrak{p}})", "0136443ddd0a3e0e1ca5e28f7e915067": "f_{xy}(a,b)=f_{yx}(a,b)=\\frac{e^x}{1+y}\\bigg|_{(x,y)=(0,0)}=1.", "0136abaee416482ed746a2d07d3381a8": "\\frac{1}{2}(b^2+c^2-a^2) = \\frac{1}{2}[d^2+c^2-(c-d)^2] = cd.", "0136deb84efa9af07881f5f481ee2151": "\\mathbf{\\theta} = \\begin{bmatrix} \\theta_1 \\\\ \\theta_2 \\\\ \\vdots \\\\ \\theta_M \\end{bmatrix},", "0136e23e1209d9507f0afe19336223e7": "T = \\sum_{j\\in J} T_j.", "0137032f8601ff0bd8f8a9c5de8c1f00": " P_{ij}|\\sigma_i \\sigma_j\\rangle = |\\sigma_j \\sigma_i\\rangle \\,. ", "01372bd994cf6aee276abce370612dda": "\\frac{9}{8}", "01373bf85c08b0b1eef230d11547c93d": "{\\tilde{I}}_{1}", "01373f77e95fc864e269d6387936e36a": "\n\\int_1^M\\frac1{x^{1+\\varepsilon}}\\,dx\n=-\\frac1{\\varepsilon x^\\varepsilon}\\biggr|_1^M=\n\\frac1\\varepsilon\\Bigl(1-\\frac1{M^\\varepsilon}\\Bigr)\n\\le\\frac1\\varepsilon<\\infty\n\\quad\\text{for all }M\\ge1.\n", "01379ffd7c9f52bc373e34fc543c4b1c": " \\frac { \\sum x } { n } > 3 ", "0137d5bdba35d4616da310282828d112": "\\|xy\\| \\leq K \\|x\\| \\cdot \\|y\\|", "0137da2584fce8c4ab6fe12b68f12778": "I(\\mathcal{B})", "013846c9405f2d183f7979fd321a1928": "\\mathrm{Mode}[X] = e^{\\mu - \\sigma^2}.", "01387e617044b4cd37c33fa98a537db7": " \\int d^2\\theta\\; \\lambda_1\\; U^c D^c D^c ", "0138f8ca532ca5d01b5a4eb3c962bdd6": "\\left[\\frac{\\hbar^2(k+K)^2}{2m}-E_k\\right]\\cdot\\tilde{u}_k(K)+\\frac{A}{a}\\sum_{K'}\\tilde{u}_k(K')=0", "013903b9bbe6818a50bc7b29512189a4": " = \\frac{1}{2} \\left[ \\int_{0}^L \\frac{x^2}{L^2} \\rho(x) \\,dx \\right] v^2", "013922ccbb127dbe27c9a978177138bd": "\\nu_2:P^n \\to P^{n^2+2n}.\\ ", "013993d94d704fb68b0ba51eb11a18b7": "U-normalized", "0139a23ca80a4549e4a2f73e25c9302a": " [-\\nabla^4] \\Phi(\\mathbf{x},\\mathbf{x}') = \\delta(\\mathbf{x}-\\mathbf{x}')", "0139c39e61d8a08307dfe610c7467571": "d\\Omega^0(S^1)", "0139d3a2f779afa6965569049f7bd6dd": "I(F)=\\theta(F)\\mathbf{Z}[G_F]\\cap\\mathbf{Z}[G_F].", "0139ed53e26900cdc7a372ed7d81be32": " \\mathbf{u}_2 ", "013a806c5b18a7014d4325dd7fc8e4dd": " c \\in \\mathbb{R} ", "013a8eb52f67ec903e6752c45adbfb33": "F_v(t)=\\frac{M_a}{r}(e^{rt}-1).", "013ac752899599fe44ebf2b906d5a864": "\\mathbf{x}_k=\\mathbf{x}(t_k)", "013b4eb45a78130f797a2dea2b68b27d": "\\nabla_{x,y} f = - \\lambda \\nabla_{x,y} g", "013b56b61e6a2523649feee13d0abe00": "N \\leftarrow pq", "013b595732673993d8f6a29fcedc9499": "\\frac{s}{H_{N,s}}\\sum_{k=1}^N\\frac{\\ln(k)}{k^s}\n+\\ln(H_{N,s})", "013b5c53cbe17e6b04407a139ef7622e": "\\mathfrak{B}(V_+)=k[x]\\qquad \\mathfrak{B}(V_-)=k[x]/(x^2)", "013b5c7f3983c6cf2eb3c287ddd40c76": "h^{-1}\\left( {d \\over dx}\\right) p_n(x) = n p_{n-1}(x).", "013b729bdc2d69ac52cbf745a00e2bb2": "\\begin{bmatrix}\n1 & 1.25 \\\\\n0 & 1 \\end{bmatrix}", "013b9a12d32bc22945f202bbd856a308": "DPO = \\dfrac{ending~A/P}{COGS/day}", "013ba350bad36a45381a4c3468c365ad": "Y_{n-1}", "013c59dc8b95b3395c38812433707626": "D \\approx \\frac{32400}{\\Theta_{1d}\\Theta_{2d}}", "013c7b0046a59ef24d47814c8160f180": "\\mathbf{\\hat{X}}_{k - 1}\n = \\mathbf{X} -\n \\sum_{s = 1}^{k - 1}\n \\mathbf{X} \\mathbf{w}_{(s)} \\mathbf{w}_{(s)}^{\\rm T} ", "013ca4d3d0ca9e09faa9a4a2f9c6ffd8": "E_2^{p,q}", "013d64b225b6fad0f99cbc59325a03c3": "\\sim, \\nsim, \\backsim, \\thicksim, \\simeq, \\backsimeq, \\eqsim, \\cong, \\ncong \\!", "013d75c88956c6d4e232c94bd3e11fcc": "\\theta_{ij}=c_1(y_{ij}^1+y_{ij}^2)", "013d7d5fdc6bc9b4d9b40de734b01a46": "f(x) = \\frac{x^3-2x}{2(x^2-5)}", "013db70019ec75bc0be1f6adebed5348": "S: Y\\to X", "013dd7a65cf9bfed91be0dbeb88c422c": "\\Theta_{\\Gamma_8}(\\tau) = 1 + 240\\,q^2 + 2160\\,q^4 + 6720\\,q^6 + 17520\\,q^8 + 30240\\, q^{10} + 60480\\,q^{12} + O(q^{14}).", "013e372dd4bf309a78c78ec451c3628e": "\\frac{d P}{d T} = \\frac {s_{\\beta} - s_{\\alpha}}{v_{\\beta} - v_{\\alpha}} = \\frac {\\Delta s}{\\Delta v}.", "013e7bc41b539a8fc5c60ec8472b7c8e": "\\theta=\\frac{s}{r}", "013e95f41d324472d1342dff611d9b64": " F_{r}-F_{l}\\,=0", "013edb7a7f459107e469450b6eda9fff": "\\mathbf{q} - \\mathbf{p} = (q_1-p_1, q_2-p_2, \\cdots, q_n-p_n)", "013eeafbc28cf4bcadc09bb91b4a7d51": "n \\quad,", "013f16caff7ad5915666e826c746b9cd": "\\operatorname{wnchypg}(x;n,m_1,m_2,\\omega) = ", "013f1769aaa2cf3168d91e6b066995f0": "R'(W) > 0", "013f5496efefa9c604dfa4a23b0f1f1a": "\\partial(\\sigma \\frown \\psi) = (-1)^q(\\partial \\sigma \\frown \\psi - \\sigma \\frown \\delta \\psi). ", "01404b28718df1e227d2530317ab93dd": "i = 0\\,\\!", "01408c57bdb6e3c5fd797ea9a1b13946": "\\mathrm{[A]}(t) = \\mathrm{[A]}_{0} \\cdot e^{-k\\cdot t}. ", "01410ef47323fdf3853b5d3786197b0f": "d(x_m, x)<\\varepsilon/2", "01410f71538f21258f58a8932aa10cb2": "\\ln B=\\ln \\big(\\lambda(I+K)\\big)=\\ln (\\lambda I) +\\ln (I+K)= (\\ln \\lambda) I + K-\\frac{K^2}{2}+\\frac{K^3}{3}-\\frac{K^4}{4}+\\cdots", "0141643b8d5556400f163c6049a0741e": "u(w_0 + WTA , 1) = u(w_0 , 0).", "01416a661c3418153eb0c0f922b5653d": "\ng_n = \\binom{N+n-1}{n}\n", "0141f319f9169fa31c63fc24d4dfeded": "\\int x^m\\arccsc(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arccsc(a\\,x)}{m+1}\\,+\\,\n \\frac{1}{a\\,(m+1)}\\int \\frac{x^{m-1}}{\\sqrt{1-\\frac{1}{a^2\\,x^2}}}\\,dx\\quad(m\\ne-1)", "01423856a0862b51452642523a8e6997": "D(u,v) = \\frac{\\sum_{i=0}^{72}D_i}{\\sum_{i=0}^{72}w_i}", "01424af9614de9aa07f3932f4576e5e6": "\n{dP_x \\over dt } = c(P_{x+1} - 2P_{x} + P_{x-1})\n\\,", "01424d0db12f64e329b4234095f9ac26": "y_j=\\beta_0+\\beta_1 x_{1j}+\\beta_2 x_{2j}+\\cdots+\\eta_j \\, ", "014264151437df888613e0559ae86350": "\n\\begin{array}{ll}\nd\\in D & \\mbox{the decision being made, chosen from space } D \n\\\\\nx\\in X & \\mbox{an uncertain state, with true value in space } X\n\\\\\nz \\in Z & \\mbox{an observed sample composed of } n \\mbox{ observations } \\langle z_1,z_2,..,z_n \\rangle\n\\\\\nU(d,x) & \\mbox{the utility of selecting decision } d \\mbox{ from } x\n\\\\\np(x) & \\mbox{your prior subjective probability distribution (density function) on } x\n\\\\\np(z|x) & \\mbox{the conditional prior probability of observing the sample } z\n\\end{array}\n", "0142f80ddbc8ec29aba02c8582b99ee3": "\\text{STr}", "014314fe17876564f0e241b4c4a11b77": " H = \\frac{h}{l_c}", "014345ae6ac2bde9bfec0158c4e850e7": " \n\\begin{align}\nR^J_{pq} & \\equiv R_{pq}(\\theta_2)\\, R_{pq}(\\theta_1),\\text{ with} \\\\[8pt]\n\\theta_1 & \\equiv \\frac{\\pi - 2\\phi_1}{4} \\text{ and } \\theta_2 \\equiv \\frac{\\phi_2}{2},\n\\end{align}\n", "01434663733c3165cd88685687e87f8c": " {\\rm full\\;red\\;circle}=\\left\\{X\\mid \\; \\left( (XC),(XD)\\right) \\; = \\theta+k\\pi\\right\\}", "0143754b30f0af816b24a5427d1e7956": "\\mathbb{S}^\\lambda E", "01438fd76ecd96ff355b59de43f5e3ec": "\n\\left\\{ D_{i}, D_{j}\\right\\} = -\\sum_{s=1}^{3} \\epsilon_{ijs} L_{s} ~.\n", "0143aaa26015fff3cefa48a7fd7fd569": "\\ell(\\theta|X,Y) = \\log L(\\theta|X,Y) = \\sum_{i=1}^m \\left( y_i \\theta' x_i - e^{\\theta' x_i} - \\log(y_i!)\\right)", "0143c5e4040a58ba580bc87c042d165d": "b=\\sum_{i=1}^{N}b_{i}<+\\infty", "014400a930b949e0295bcdedf4489bbe": "\\ln w_r^+-\\ln w_r^-", "0144332ce95ebfe5905078ab8fe7596c": "p = \\tfrac{1}{2}", "0144689cf3b51353d42d0924a5dfbd53": "[x:=x+1]x \\ge 4\\,\\!", "0144fff1444f6169eb0a57fde0a7ce17": " e_n = O(h^p) ", "0145132faf44644d666f66456a528e6e": "c,\\!\\ c_{n-1}, c_n, c_{n+1}", "01457426424450f533996762e5f70dd6": "r \\geq -1 /(K-1)", "0145b12d7f30173a17c26272f9e647f5": " m \\geq - n+1", "0145e9ad21bdf0e4020a665891253d82": "\\omega \\in W", "014630b5a2a36b9eba221efd748a5ef7": "2^{65-1}", "01464ae3746d71824e581b72b8f8d7ef": "Loves(g(x),x)", "0146716079826b80a3d251aa9c8a3a7a": "d=-3,-4,-7,-8,-11,-19,-43,-67,-163.\\ ", "014697f828a07bbea10e47ea5765e8b3": "\\circlearrowright", "0146ccb228cab83f16fd6d6c3924d625": "\\mathit{dr}(n)=0 \\Leftrightarrow n=0.", "0146d3d7054dd057cbad9fd5bf13022b": "p=1/2,", "0146d9113ce3837a4b6112b4ae1f6fc0": "b_N,", "0146f443aa1723821a8c6b5e62aef18e": "\\mathbf{k}:=2", "01474699320c261cb8187c2a811c377f": "3\\times 1 + 3\\times 2 = 9.", "0147bbbbab72e5087a9aba9244149c4d": "\\mathbb{P}(X_1 >0, X_2<0)", "0147bbd757bfe2c961495720cec9049e": "|S|\\geq\\sum_{i=0}^{r} {n\\choose i}", "0147f8c7194007e2af9895e02f6014bf": "c_\\alpha", "014827672ea8d9165a63dd11f8ff0710": "\\mathrm{wt}_y(c)=5", "0148385d6a69af88889c1eae177d300f": "m=-1", "01484cb4279fed5d58f6aa2afcdb856a": "k \\leq n", "01488e3b6f08bd71f5366a9724e63920": "(b, a) = \\{x | x \\in \\mathbb{R}, b < x\\} \\cup \\{\\infty\\} \\cup \\{x | x \\in \\mathbb{R}, x < a\\}", "01489ec50aaac6e2d55551d0f5d416cc": "\\bar{X}_n\\to\\frac{1+r}{1-r}", "0148c7ca12630d2b0eb824ea90794844": "= 21 + 15P(3,1) + 21P(2,2) + 6P(1,3) ", "0148d2ff685f599bfb7796f1b58c0c11": " r'=\\sqrt{x'^2+y'^2}", "0149071d3784d921cb7a919fc4f4005c": " \\ C_{D_i} = \\frac{{C_L}^2}{\\pi \\text{AR} e} ", "01496b07a67abfd9d5f70961c1ea489f": "\\begin{align}\n \\mathcal{L}(\\beta,\\sigma^2|X) \n &= \\ln\\bigg( \\frac{1}{(2\\pi)^{n/2}(\\sigma^2)^{n/2}}e^{ -\\frac{1}{2}(y-X\\beta)'(\\sigma^2I)^{-1}(y-X\\beta) } \\bigg) \\\\\n &= -\\frac{n}{2}\\ln 2\\pi - \\frac{n}{2}\\ln\\sigma^2 - \\frac{1}{2\\sigma^2}(y-X\\beta)'(y-X\\beta)\n \\end{align}", "0149875825e15879e6cbe7e57e49f5d6": "\\rho=\\rho(p)", "0149902c0da462b82d0ff292ba3c7172": "\\beta_\\pm", "014a2dc2cfd0a12dc810658b9101af62": "\\qquad\\qquad R = a + bv + cv^2 ", "014a6a73d02ef7e9279c17b3ce8d17d0": "\\scriptstyle {BD = {1 \\over 2}BC}", "014a78b4ae24d42b26c604ef368dc19f": "( X \\in X ) \\to Y ", "014a8e239917a19e72eef967fd050eb3": "x=\\lim_{n\\to\\infty} x_n ", "014ad654d6fdbdf7914c210f3d490eda": "E=kmc^2", "014aec3bd0d1aaa2f1ba56e8d9a928f6": "F (E/As, \\ell/A) = 0, \\!", "014bead22ed0f89588c13d5080403591": " g_1 = 1 - \\frac{L}{R_1} ,\\qquad g_2 = 1 - \\frac{L}{R_2}", "014c497b06e9c0a1db88ca51c81148a0": "w+\\lambda v", "014c5b5767dd0135cf5882331b2b7f51": "(v, u)", "014c963ac0879ae7bfdf132b685494b5": "\\mathbf{P}^1 \\times \\mathbf{P}^1", "014c9cdd3a043e564c9aec42cd9bccd0": "\\int_0^{\\theta}\\log(\\sin x)\\,dx=-\\tfrac{1}{2}\\text{Cl}_2(2\\theta)-\\theta\\log 2", "014cbcdc18b0ba7be280c53c590ab8f5": " F_\\text{n} =\\frac{l_\\text{n}}{l_\\text{m} + l_\\text{n}} (W-L) + \\frac{h_\\text{cg}}{l_\\text{m} + l_\\text{n}} \\left(\\frac{a_\\text{x}}{g} W - D + T\\right).", "014cf89bc136d25dac6c59e89aa34bd2": "a_nX^n+\\dotsb+a_1X+a_0", "014d0143c6144f9a538fa9a8b34b70e5": "\\overline{\\Delta}", "014d82f7288863e6f215c9e6b4f8d53d": "\nC_w = \\frac {A_w}{L_{pp} \\cdot B}\n", "014e14b6b5d49acd76469e6fbc7a82fd": " \\text{error} = -\\frac{(b-a)^2}{12N^2} \\big[ f'(b)-f'(a) \\big] + O(N^{-3}). ", "014e23762e9d85d7f79dc5e98920b90b": " edg=k.(p-1)(q-1) + g", "014e7d2183b5947d954bc5262a310518": "g(x)=\\left\\lbrace\n\\begin{array}{lr}\n0 & 0\\leq x\\leq \\frac{1}{\\alpha} \\\\ \n+\\infty & \\text{otherwise}, \n\\end{array} \n\\right.\n\\,", "014e864142722fcc9b02b202e48ef62f": " L_{ni} (\\beta) = {exp(\\beta z_{ni}) \\over {\\sum_{j=1}^J exp(\\beta z_{nj})}}", "014e9666a4157b13ffa22c5f9120d61a": "\\begin{pmatrix} E_{0x} e^{i\\phi_x} & E_{0y} e^{i\\phi_y} \\end{pmatrix}^\\top ", "014ef7a82d4d84a5e5340f4085b234ac": "\\gamma_\\mu a\\!\\!\\!/ \\gamma^\\mu = -2 a\\!\\!\\!/ ", "014f13dc68a1ebef4081614936b7feea": "C^* = \\left \\{y\\in X^*: \\langle y , x \\rangle \\geq 0 \\quad \\forall x\\in C \\right \\},", "014f703deb391191425c853b0250a8b1": "{\\rm cov}(I)=\\min\\{|{\\mathcal A}|:{\\mathcal A}\\subseteq I \\wedge\\bigcup{\\mathcal A}=X\\big\\}", "01502d95785ebee778cd52c94b60fce8": "a = \\sqrt{c^2 - b^2}. \\,", "0150cf714657426a7ddac6d88b8350aa": "c<1/2", "0150da4a6df78bd2c47cdb860657017b": "R_{abcd} \\, {{}^\\star \\!R}^{abcd}", "01511265bf0fcc2e6daed57b80e0e61a": "\\dot\\theta", "0151880dd88fd5d6acf8c7246832402d": "\\frac{dW}{d\\omega}=\\frac{2e^2 \\gamma}{9 \\varepsilon_0c}S(y)\\qquad (11)", "01521b06b9713104aa72e655f28da19e": "\\varphi(p^n)", "01523d5515466eff29876c2a65a92242": "\\frac{(P_{high} * T_{high})\\, + (P_{low} * T_{low})} {T_{high} + T_{low}}", "01524f01f8f58579baef14bad535a9be": "Q_i \\subset Q_{i+1}", "01526e53e2e365f14881d561f7410482": "\\sigma: V \\to V\\,", "015272f0d625851e17fb5cec814ef476": "\nz=\\frac{x+i\\gamma}{\\sigma\\sqrt{2}}.\n", "0152b5f04f0a6a89537d2ebdefb64886": "{\\rm as}_5(5,3,4,1,2) = 5, ", "0152d4f07238ad06773c417c3029fb33": "U_{12}", "0152f4a919fc7e636a2127c374c6a820": "\\textstyle \\mathcal{F}; ", "01531ef83653cd13932e4eb31d4293a9": "\\, y(k) = Cx(k) + Du(k)", "01534343fafe188300ea31bb26178936": "\\exists C\\,\\exists M\\,\\forall n\\,\\forall m\\dots", "015344b96bbd0ec4fc811ca619d41bcd": "(\\mathbf{x-v})^\\mathrm{T} A (\\mathbf{x-v}) = 1,", "0153668e4fc7c3978d2019d3c8e0def3": "\\Phi + a = \\Phi \\ ", "0153fdca964ff1733b038341e15052a9": "\nZ_\\mathrm{in} (l)=-j Z_0 \\cot(\\beta l). \\,\n", "01544c6a478e896df7db7c6dfd37d03f": "\\lambda(t|\\theta)", "0154fd706580c907e37d92635916f59c": "E(v,h)", "01551c0ff28abafe81eea0b994e71cb0": "y_{n+1}", "01558774c0f2617ae915ab0f149b9bf6": "E[\\xi]=\\int_0^1\\Phi^{-1}(\\alpha)d\\alpha", "01559153a5f303bf5e242cfeb258376a": "MN-1-s = -s \\mod (MN - 1)", "0155b7d3ab4ed3e60406a3806351083e": "6H_2O \\rightarrow 4H_3O^+ + O_2(g) + 4e^- ", "0155c14a223bc3169ca9d52b033fe1f3": " (k[V]_P)_0", "01564d7f7c423595f5790218a0a7e506": "\\delta W = -(m_1+m_2)gL_1\\sin\\theta_1\\delta\\theta_1 - m_2gL_2\\sin\\theta_2\\delta\\theta_2,", "0156971872bbc66b173c86eaec924f90": "(X, <)", "01569b69c8445b7f7c3037a09f4f81df": " {s_{\\overline{X}_1 - \\overline{X}_2}}^2 ", "01571dc3494b356ca4a3220d0e3ae28b": "\\text{Semiperimeter}=mn(m+n) \\, ", "0157cdacfd6b753d07183d1fe676230e": " [AB,C] = A[B,C] + [A,C]B", "0157d1a94cadd34686afe0c111678caa": "-2x-2x^2+2y-2y^2+y^3+4xy-2xy^2+xy^2", "0157d210fb38c21d6d8986daffcae92d": " A_s = 1 - \\left [ \\frac {sP \\left (2\\theta_i - 2\\theta_k \\right )^2}{\\tan \\theta_k} \\right ] ", "0158382c22c0d624719e46eeeca3ba67": "\\dot m_k", "0158893c0ec80b2f8844b1d1e53143b2": "(T^*T)^p \\ge (TT^*)^p", "0158990fc044957d92d255d9848c471d": "f(w) = \\langle w, v \\rangle_W", "0158a2b95ba1b791ea183477dca28334": "466/885\\cdot 2^n - 1/3 + o(1)", "0159457a3702492f06b73e3e917c3bf6": "\\sum_{n=0}^{\\infty} \\| f_n\\|_U < \\infty", "015969d3f5f4b1046ee951f3509fa327": "\\rho(z)|dz|=\\sigma(f(z))|f^{\\prime}(z)||dz|. \\, ", "01597e9f2530e86683f34ade31503009": "BV(]0,1[)", "015a01db76489cb43c8003036548e316": "\\mathfrak a = (a)", "015a2ad28d0a278bd960b1b0eefdcc00": "\\ S_e = \\alpha_e \\times S_c ", "015a3b39628d270dd345abba52744268": "A = \\dots \\to 0 \\to A^0 \\to 0 \\to \\cdots,", "015a7ef685aafdee58899abccc2e2609": "\\alpha_0 = a/D", "015ae660b57165728429854982d05b05": "D_B>0", "015b39d72d4a0fd8e22283fd59a03971": "\\dot S_i = 0 ", "015b6ed3d17e4320673a7fbb4f08a890": "G(0) = a", "015bbc9bfb8a96e6ab5b0997090b8855": "p\\cdot2^p\\le W(2,p+1)", "015bcbdeb6916e3d184715b7d8f76ddb": "1 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\ddots}}}}", "015bd78022d8197e1f89f26566ed30cb": "\n\\begin{align}\n f&=\\frac{a-b}{a}, \\qquad e^2=2f-f^2, \\qquad e'^2=\\frac{e^2}{1-e^2}\\\\\nb&=a(1-f)=a(1-e^2)^{1/2},\\qquad n=\\frac{a-b}{a+b}.\n\\end{align}\n", "015c8a7bf6cdc8659ba3da587412882f": " L=\\{p^*_1,...,p^*_n\\} ", "015e2542aa509f972b2d9cac61e0da8b": "\\begin{cases}\n f(x) \\ge 0 \\\\\n g(x) > 0\\\\\n f(x) < \\left [ g(x) \\right ]^2 \\quad\n\\end{cases}", "015e37f07fb14db21431e6f806b6f914": "\n\\rho_i ' \\simeq \\rho_j ' \\quad \\mbox{and} \\quad \\sigma_i ' \\simeq \\sigma_j ' \\quad \\mbox{for all} \\quad i,j \\;. \n", "015e537a57c7a99528d0b4be0dbff505": " f(x;\\mu,\\sigma_1,\\sigma_2)= A \\exp (- \\frac {(x-\\mu)^2}{2 \\sigma_2^2}) \\quad \\text{otherwise}\n", "015eb7acbdb731c32e8526d3b999986c": "G= \\int_0^\\infty I(\\lambda)\\,\\overline{g}(\\lambda)\\,d\\lambda", "015ec6b712f877cf4b399641b0afcfa4": "x(0) = 1,\\,", "015ed9def054d00e7c577035a29b1c6f": "c^{-2}", "015eeb1f8a5a6a7adc4a4c42bd49a653": " ~ \\bold J(x,y,z) ~ = ~ \\sum_j ~ J_j ~ \\bold J_j(x,y,z) ~~~~~~~~~~~~~~~~~(3.4) ", "015f185f6d3620e612d1d6106a17db4b": "0 \\leq n < N", "015f511f97b854214a366171a4880d0d": " \\gamma =(\\pi Q_r)^{-1} ", "015f7b01c6c52d90a663ea2cc944f8f1": "r_b", "015f801465c03bb660e00a97dbbf9996": "\\log\\left(\\frac{m_{\\rm closed}(t)}{m_{\\rm closed}(0)}\\right) \\sim -\\overline{n^2}t,", "016005918866f2507543201a199b5a9e": "K_{\\mathrm{max}} = \\frac {1} {2} m v^2_{\\mathrm{max}}", "01606d154021cfd719f7faccfbb3519b": "P(A^c) = 1 - P(A).", "0160828f5223b7e57f403a13ef5bef77": "\\displaystyle{\\mathfrak{t}_{\\mathbf{C}} \\oplus \\bigoplus_{\\alpha\\in \\Delta_1} \\mathfrak{g}_\\alpha}", "0160f522c2aab4f57bd959ae587a2b26": "\\alpha_t^j", "0161266e84b0ac7f3091917255670e9b": " \\|\\lambda\\alpha^n\\| \\to 0, \\quad n\\to\\infty. ", "0161387d2f38b3bee6e710f7bc05e728": "P_i = {\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'\\otimes P \\otimes {\\bold 1}\\otimes\\dots\\otimes{\\bold 1}", "01614247f736bfe380d93aeb04fffe8b": "a^k \\equiv 1 \\pmod p. \\,\\!", "01616afd77e7b2f30b075c048e56828d": "\\frac{V_o}{V_i}=-\\frac{V_i\\, D^2 \\, T}{2L\\, I_o}", "0161a35f5d52b351bcad5e296026ec19": "x(t)=\\frac{Q_t-Q_0}{P}.e^{ -A.t}-\\frac{R_t-R_0}{P}.e^{ -B.t}", "0161ac05b03c2fbc4c46e5942706c976": "F_{n}=\\sum_{k=0}^{\\lfloor\\frac{n-1}{2}\\rfloor} \\tbinom {n-k-1} k.", "0161b0e1647a6aae6ad875c24dce391b": "F_2(x)", "0161b664a9d7ab02df4940a7e0e8f5ea": "V_\\mu", "0161b76065b0cf6479281020f5ac4109": "y \\cdot S", "01625deea2b34a280cf53462d5cd8e89": "\\mathfrak{sl}_2(\\mathbb{C})", "0162f6fc9cba0bc4e038bb69b2fca6ba": "\n\\langle p \\rangle_{IV} = \\langle p \\rangle_{VI} \\equiv \\langle \\langle p \\rangle_I \\rangle_V\n", "01631e49d5e996818ad1e2bc639b30eb": " \\varphi: X \\rightarrow F ", "016330ad775f855967988d01493ad559": "A=\\epsilon c l", "0163630e147945512a566c81d04f5a9e": "SH_k(X) \\cong H_k(X)", "01637dc15f870d2ade39e7cccf90ceff": " F_D ", "016385857bd33bf936a2805d8224e9eb": " \\mbox{E}i(x)=-\\int_{-x}^{\\infty} \\frac{e^{-t}}{t}\\, dt", "0164f6f7287067a014f8969590008cb7": "\\frac {d}{dt} \\iint_{\\Sigma (t)} \\mathbf{F} (\\mathbf{r}, t) \\cdot d \\mathbf{A} = \\iint_{\\Sigma (t)}\\left(\\mathbf{F}_t (\\mathbf{r}, t) + \\left[\\mathrm{\\nabla} \\cdot \\mathbf{F} (\\mathbf{r}, t) \\right] \\mathbf{v} \\right) \\cdot d \\mathbf{A} -\\oint_{\\partial \\Sigma (t)} \\left[ \\mathbf{v} \\times \\mathbf{F} ( \\mathbf{r}, t) \\right] \\cdot d \\mathbf{s} ", "01650aac5453836cb8bc89e6ab9487a0": "p_G(z) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{(z-\\mu)^2}{2\\sigma^2} }", "016557a36376f71d54e07fcb6d1b06a4": "\\partial (ax+by) = a\\,\\partial x + b\\,\\partial y.", "0166611b358712a14531de7bd37b3a92": "\n 0 \\subset T^o \\subset S^o \\subset V^*.\n ", "01666a9de09edfa09d29f736d0e88e6b": "\\frac{2}{1} = 2.0,\\quad \\frac{9}{4} = 2.25,\\quad \\frac{161}{72} = 2.23611\\dots,\\quad \\frac{51841}{23184} = 2.2360679779 \\ldots", "0167263fe51c584357b10f31948924a8": "k = \\begin{cases}\n-j-\\tfrac 1 2 & \\text{if }j=\\ell+\\tfrac 1 2 \\\\\nj+\\tfrac 1 2 & \\text{if }j=\\ell-\\tfrac 1 2\n\\end{cases}", "0167e58e36e9adcdc2dcfe53ad02ceed": "\n \\frac{\\partial \\boldsymbol{A}^T}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\boldsymbol{\\mathsf{I}}^T:\\boldsymbol{T} = \\boldsymbol{T}^T\n", "01683b5e9a02503019a64e3f0e356dd4": "l_a m^a = l_a \\bar{m}^a = n_a m^a = n_a \\bar{m}^a=0\\,.", "016847c3d5354736012b76dde02f985c": "x = x_1 A_1 x_2 ... x_n A_n x_{n+1}, y = x_1 w_1 x_2 ... x_n w_n x_{n+1}", "016918640d9fe4358ca6eecdaf2eb227": "\n(\\mathbf{\\hat{f}_{0:t}})^T = c^{-1}\\mathbf{O_t}(\\mathbf{T})^T(\\mathbf{\\hat{f}_{0:t-1}})^T\n", "016951effe27b2c913c0e4a30f5e3a67": " \\frac{d}{d t} \\left(p_1+ p_2\\right)= 0. ", "01695aacbcb4d3555cf4cd3ebac79d36": " f_X(x) = F_X'(x) = \\frac{1}{2\\pi}\\int_{\\mathbf{R}} e^{-itx}\\varphi_X(t)dt,", "01699f7f623d2f8633068b70b0104fe9": "(x, t)", "016a34677eebc38601dcce71b906a414": "\\min_{w\\in\\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n (y_i- \\langle w,x_i\\rangle)^2+ \\lambda \\|w\\|_1, ", "016a6eae4c21c9122a74c4734080dfc6": "\\frac{G^{ex}}{W_wRT} = f(I) +\\sum_i \\sum_j m_im_j\\lambda_{ij}(I)+\\sum_i \\sum_j \\sum_km_im_jm_k\\mu_{ijk}+\\cdots", "016a7a4366f6d2ba6d02ba1689f6dac4": "\\;\\delta\\;=90^\\circ\\;", "016ac14c46a8e10d3bd0959c48a55a7e": "\\int\\cosh^n ax\\,dx = \\frac{1}{an}\\sinh ax\\cosh^{n-1} ax + \\frac{n-1}{n}\\int\\cosh^{n-2} ax\\,dx \\qquad\\mbox{(for }n>0\\mbox{)}\\,", "016aff365e8041c20b749e28419cb1c5": "R_C(f)= {12200^2\\cdot f^2\\over (f^2+20.6^2)\\quad(f^2+12200^2)}\\ ,", "016b05d112234eafb35fc0882c90a117": "1\\to\\Gamma(N)\\to\\Gamma\\to\\mbox{PSL}(2,\\mathbf{Z}/n\\mathbf{Z})\\to 1", "016b0d87cb90b7ccb8c076c1d8ac6cbe": " [\\varnothing]_p = \\varnothing \\! ", "016b26921e4163616d2ab88325528257": "u(x,t) =X(x) T(t) ", "016b8314d6c42adcdaabe7e963b832af": " \\delta W = \\left(\\mathbf{M}\\cdot \\frac{\\partial\\vec{\\omega}}{\\partial\\dot{\\phi}}\\right) \\delta\\phi.", "016bc796e0261b4c5d6f436c79ceb3da": "\\mathrm{Pr} \\gg 1", "016c089a7c3a63f75a2c314fb02c6b24": " \\hat{H}_{\\mathrm{el}}=\\sum_{i=1}^N\\frac{p_{i}^2}{2m}+\\sum_{i 1 ", "016ee1c2de4d8637ac7a334f5a041990": "\\mu(x,G):= B(x,1/2n(x,G))", "016f0e72e89380b5e989d40148b2cbf4": "\\mathcal{F}=\\{F\\subset E\\vert G[F]\\hbox{ has property }\\mathcal{P}\\}", "016f1ceed5059915d5942a945d89825d": "\\frac{\\partial^2\\rho}{\\partial t^2}-c^2_0\\nabla^2\\rho = \\nabla\\cdot\\left[\\nabla\\cdot(\\rho\\mathbf{v}\\otimes\\mathbf{v})-\\nabla\\cdot\\sigma +\\nabla p-c^2_0\\nabla\\rho\\right],", "016f73e62aa171292b5559ef227c2799": "B(x;r) = \\{y \\in M : d(x,y) < r\\}.", "016f8ae636ec76129cbd95f8c08d0233": " \\frac{p_e}{\\rho g} + \\frac{V_e^2}{2 g} + z_e = \\frac{p_{0}}{\\rho g} + z_{0} + h_f", "016f8b07340bbc06290df981d225ae51": "\\frac{1}{n}\\mathbb{E}(f -\\mathbb{E} f)^2", "016faeaa7ce8d1af117195f3ff1e8d5c": "+ 7 \\cdot 9^{(7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 + 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 6)} + \\cdots", "016fc51b812746e6f6bec0ceebd4024d": "F = {1 \\over 4 N_e u + 1}", "016fcb192b7b9cd0a9d3349d1adeda0d": "\n\\begin{align}\n\\left(\\begin{matrix}r_0 \\\\ r_1 \\\\ r_2 \\\\ r_3 \\\\ r_4\\end{matrix}\\right) & {} =\n\\left(\\begin{matrix}\n1 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 1 \\\\\n1 & -1 & 1 & -1 & 1 \\\\\n1 & -2 & 4 & -8 & 16 \\\\\n0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)^{-1}\n\\left(\\begin{matrix}r(0) \\\\ r(1) \\\\ r(-1) \\\\ r(-2) \\\\ r(\\infty)\\end{matrix}\\right) \\\\\n& {} =\n\\left(\\begin{matrix}\n 1 & 0 & 0 & 0 & 0 \\\\\n 1/2 & 1/3 & -1 & 1/6 & -2 \\\\\n -1 & 1/2 & 1/2 & 0 & -1 \\\\\n-1/2 & 1/6 & 1/2 & -1/6 & 2 \\\\\n 0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}r(0) \\\\ r(1) \\\\ r(-1) \\\\ r(-2) \\\\ r(\\infty)\\end{matrix}\\right).\n\\end{align}\n", "017109036d98967b18753204f3ba2212": "\\Psi_A(1,2,\\dots,N_A) \\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B)", "01710b6db82c22ba1e3978b63169aa6f": "S_3 = I p \\sin 2\\chi\\,", "01711dd7f632f52b2769dcb1287d01fc": "\\|G_n-B_n\\|_\\infty", "01716f028414939d0760ec49f73781db": "\\tfrac{1}{2j}\\left[X(z)-X^*(z^*) \\right]", "0171b613b3d21c820963ba706362f057": " e^{\\tfrac{a(1-x^b)}{b}} x^{-2+b} (1-b+a x^b) ", "0171b9d3ea18f0244b52e9734214a6e4": "\\mathbf{F} = q (\\mathbf{v} \\times \\mathbf{B})", "0171ba21e34fac98573041370c9d7a1d": "d(x, m) = d(y, m) = d(x, y)/2", "0171c40165c03cc43ac0db36b95a2fe5": "\\gamma(t) = \n\\left(\\begin{matrix}a&b\\\\ c&d\\\\ \\end{matrix}\\right) \\left(\\begin{matrix}e^{t/2}&0\\\\ \n 0&e^{-t/2}\\\\ \\end{matrix}\\right) \\cdot i\n = \\frac {aie^t +b} {cie^t +d}. ", "0171f1bbb919ffcbf791d5ad55acbbd0": "b, ab, aab, aaab, \\dots", "01723ab614766914630de2e30e351ee5": "X0", "017e2034450b47b6670afdb40731e1b0": "\\; L (H_A)", "017e328293c91381e0341ae5c4e34e90": "a_n\\equiv\\frac{\\omega_nq_n+ip_n}{\\sqrt{2\\hbar\\omega_n}}\\,.", "017e9fec61107957ffe9121078b1f7df": "\\int_0^t H\\,dX =\\int_0^t H_s\\sigma_s\\,dB_s + \\int_0^t H_s\\mu_s\\,ds.", "017f12fc82880d3915c18beb5536cfcd": "\\,2646798 = 2^1+6^2+4^3+6^4+7^5+9^6+8^7", "017f1770e5cf555aaa36edf411633b6e": "\\forall z \\exists y \\forall x [x \\in y \\leftrightarrow ( x \\in z \\land \\phi(x))].", "017f58c5378216f7df65cba52f62c15a": "I_1 - i \\, I_2 = 16 \\, \\left( 3 \\Psi_2^2 + \\Psi_0 \\, \\Psi_4 - 4 \\, \\Psi_1 \\Psi_3 \\right)", "017fd493213155b36bce0bb5acd4b4b0": "1+x(-3+x(4+x(0+x(-12+x\\cdot 2))))=1-3x+4x^2-12x^4+2x^5", "017fd7e93eb4c2c900f221d7bf7b01e2": "b^2 + c^2 = 2m^2 + 2d^2\\,", "0180116fd314296f5bce2923f3534f80": "\\scriptstyle\\hat\\theta_{(i)}", "018031bc1c840403b6fc3312c1055a50": "\\left|F_n(x) - \\Phi(x)\\right| \\le {C \\rho \\over \\sigma^3\\,\\sqrt{n}}.\\ \\ \\ \\ (1)", "01808b41247d4647bbf4ef4b1ffc3e32": "\\or~(\\neg x_1 \\and ... \\and \\neg x_n)", "0180e774f2926393f199367f3ce20eb5": "p(\\mathbf{x}) = \\prod_{u \\in U} f_u (\\mathbf{x}_u)", "0180f298c26994c189a1fc6dc264955b": "w=D_L[F(K,L)]\\,", "01811d6565a5b93b98f52c00c4d45e0d": "\\frac{c}{c_0}=\\frac{t}{t_0}=e^{-\\frac{1}{8}\\left (\\xi_0^2-\\xi^2 \\right )}.", "018147f062e207970e698fac48499e9b": " 1 \\le j \\le n, 1 \\le i \\le m ", "018149d8f8a32fa92ace0794088c0b4d": "\n\\begin{align}\n& {} \\qquad D(X_1,\\ldots,X_n) \\\\[10pt]\n& \\equiv \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) \\right] - (n-1) \\; H(X_1, \\ldots, X_n) \\\\\n& = \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) \\right] + (1-n) \\; H(X_1, \\ldots, X_n) \\\\\n& = H(X_1, \\ldots, X_n) + \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) - H(X_1, \\ldots, X_n) \\right] \\\\\n& = H\\left( X_1, \\ldots, X_n \\right) - \\sum_{i=1}^n H\\left( X_i | X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n \\right)\\; .\n\\end{align}\n", "01817fcfbf955c5fd03383d2d5346629": "J^k_0\\rho:J^k_0({\\mathbb R}^n,{\\mathbb R}^n)\\rightarrow J^k_0({\\mathbb R}^n,{\\mathbb R}^n)", "0181e6d5fbcfc15a4ad8b8859441d6f4": " {u_z}_{max}=\\frac{R^2}{4\\mu} \\left(-\\frac{\\partial p}{\\partial z}\\right). ", "01825790c8778cf3f0f332dd06e9125e": " y_t==i ", "0182ae4649da29eb355c50ee5ce8454a": "\\|Ax-b\\|_P^2 + \\|x-x_0\\|_Q^2\\,", "0182cf033974d31a4d153589c68124ec": "B_m(0)=\\sum_{k=0}^m \\frac {(-1)^k k!}{k+1}\n\\left\\{\\begin{matrix} m \\\\ k \\end{matrix}\\right\\}. ", "018360106cc490d81523fe4ec165a677": "\\!\\ c_\\mathrm w", "01837fa8d184e94e252faf806d15565e": " H ", "0183a5cbf260eb9a551564bb32d7aecb": "E_a^2", "0183d8f984ac194cefa846fbd594b5ad": "\\sigma= (\\mu/\\rho) m_a/N_A", "0183fa90419e0bc4ec7bfea03d866cbf": "a \\rightarrow \\sqrt{6}", "01849d49cfc83c0f4a80b66876178a5b": "E=\\sum_{x}\\left [F(x+h)-G(x)\\right ]^{2}.", "0184a2180ba67dc66ef0098b88df3ae6": "\\ S_{\\sigma,\\varepsilon} \\geq N_{\\varepsilon}+1", "0184a6b9877bb3fad2d6650d5e11a8d0": "\\frac{[\\Gamma(\\tfrac13)]^6\\sqrt{10}}{12\\pi^4}=\\sum_{k = 0}^{\\infty} \\frac{(6k)!(-1)^k}{(k!)^{3}(3k)! 3^{k}160^{3k}}", "0184e54b0baba3e4c86ea92a1c3d43c0": "\\delta_1", "0184e6891a43a7356a48dd6188722dc6": "O(N^{3})", "0184fa9b8ce42508657f2c6b37e58170": "L=L_1L_2", "0184ff16fdf7203d30e05788cb0e8678": "S_0(t)", "0185085d739e30630a5c731f0b2e8fb6": " \\hat {\\textbf{Q}}(t)", "0185086322cce19b502df2c6748868a1": "\\scriptstyle{Rt=g(Y,X)}", "01852a43328cfaa6fbf421f7dde01d4e": "\\scriptstyle A_{33} \\;=\\; 0", "0185ab7d0afe257f39839e75beffabe1": "\\left( e^{ix} \\right)^n = e^{inx} .", "0185c175db308f0e18002f2108d38515": "a+b\\omega", "0186254595b6cf79ca29f60c731e597b": "f(A) = \\begin{cases} \\frac1{\\det (A)}, & \\det (A) > 0; \\\\ + \\infty, & \\det (A) \\leq 0; \\end{cases}", "0186cef4e734d3086ddd8e1d98c96217": "\nL = \\frac{1}{2} \\langle F F \\rangle_S - \\langle A \\bar{j} \\rangle_S\\,,\n", "0186d0570a2ecc636534c55241780f3e": "x = 0.", "01870684c1cf92509c6d2448a3ce7c04": "\\frac {\\mu_m}{\\mu_f} ,", "018709b6fc0fe0bf0f220fbffbbf1772": "O(\\theta^n)", "01871299a7fe7070ddc52d1944caab4e": "\n\\int_{B} \\! p_{X,A,B}(x,a,b) = \\int_{B} \\! p_X(x) p_{A,B}(a,b)\n", "01871a14188995b8fe6571db67cc270c": "\\mathcal{F} = \\oint \\mathbf{H} \\cdot \\operatorname{d}\\mathbf{l}", "0187489c33857c111d84ec1dc319fd28": " z^{1-c}\\; {}_2F_1(1+a-c,1+b-c;2-c;z),", "0187756e545d7544471db750ed81ed68": "U(t_k)", "0187e2567f8db40b594ff55be6a7c5f5": " L\\left(s,\\dfrac{x}{p}\\right).\\, ", "01883cd10fb57b382f0043bb0fa82da3": "(y, z)_{x} = \\frac1{2} \\big( d(x, y) + d(x, z) - d(y, z) \\big).", "01883db4def9f5811143f99b22b6e85b": "\n [A_{\\bold{x}}, A_{\\bold{y}}] = A_{\\bold{z}}, \\quad\n [A_{\\bold{z}}, A_{\\bold{x}}] = A_{\\bold{y}}, \\quad\n [A_{\\bold{y}}, A_{\\bold{z}}] = A_{\\bold{x}}.\n", "01883f5cab3c6b493515462df4feb4e3": "\\hat{H} = \\hat{T}^{\\mathrm {translational}} + \\hat{T}^{\\mathrm {rotational}}+ \\hat{V}", "0188847e219e48a741f4df4a3976163b": "\n\\begin{array}{|rcccl|}\n\\hline\n\\color{MidnightBlue}{\\mbox{eval left}}&&(11+9)\\times(2+4)&&\\color{MidnightBlue}{\\mbox{eval right}}\\\\\n&\\color{MidnightBlue}{\\swarrow}&&\\color{MidnightBlue}{\\searrow}&\\\\\n20\\times(2+4)&&&&(11+9)\\times 6\\\\\n&\\color{MidnightBlue}{\\searrow}&&\\color{MidnightBlue}{\\swarrow}&\\\\\n\\color{MidnightBlue}{\\mbox{eval right}}&&20 \\times 6&&\\color{MidnightBlue}{\\mbox{eval left}}\\\\\n&&\\color{MidnightBlue}{\\downarrow}&&\\\\\n&&120&&\\\\\n\\hline\n\\end{array}\n", "0188a6b7a7247688f2e91ba5b50ae1ea": "P_{\\text{ph}}^{2}=0", "0188beea0e3e38c1805d75a62e67f5b2": "[X; \\mathbf{P}^\\infty(\\mathbf{R})] = H^1(X; \\mathbf{Z}/2\\mathbf{Z})", "0188fac9faa5803a2d5be6739bbdb18c": "\\vec{v} \\times \\vec{v} = V_b^2 ", "0189762ab4d0514e0168562d37157d08": "F\\Big(L_-(x),L_0(x),L_+(x),x\\Big)=0", "0189fb8f36c001bc2835b994411aa362": " V \\otimes V ", "018a00a33f83a32a23e2e7738411dc5a": " \\operatorname{Ber}_{+-} J_{\\alpha\\beta} = \\operatorname{sgn}\\, \\operatorname{det} A\\, \\operatorname{Ber} J_{\\alpha\\beta}.", "018a170ccf5243a9d91ea9ac6dda4b8a": "T \\times A", "018a4e8598e4a263216b9e2972506c3e": "\\gamma:S\\ddot\\to x", "018a51db0c95700dedde07b803d7a4e7": "\\left\\{e^{\\frac{2 \\pi i}{6}},e^{-\\frac{2 \\pi i}{6}}\\right\\}=\\left\\{ \\frac{1 + i \\sqrt{3}}{2}, \\frac{1 - i \\sqrt{3}}{2} \\right\\}.", "018a580e2e81afcf158eebfc5ca43427": "\\begin{cases}\n\\Phi(x) - \\left[ \\varphi(0) - \\varphi(x) \\right] / x & x \\ne 0 \\\\\n1 / 2 & x = 0 \\\\\n\\end{cases}", "018aadf04c43b931fc8e2a9d169fbb1b": "C(x_j,x_k) = \\left.\n\\frac{\\partial}{\\partial J_j}\n\\frac{\\partial}{\\partial J_k}\n\\log Z(\\beta,J)\\right|_{J=0}\n", "018abe6cf12c05c131fe7ecb89c3378e": "{13 \\choose 5}{4 \\choose 1} - {10 \\choose 1}{4 \\choose 1}", "018ae3eec6dac7cb676d047a674eb383": "C_{3v}", "018b01cef997a99bb8f9bbf216a8bb84": "({\\mathbf P},F_{\\mathbf P})", "018b443a60f53655f4c367a819fc1d33": "(1+\\sqrt{2})^n", "018b6427d2e0647e427fd1de26c4c7c2": "|R|=p", "018b72eff79fe262ed869e07f933ee9e": "\n\\overline{\\mathrm{Var}(z)}=1-\\overline{R}\\,\n", "018c362a6c80b84a0b7e3ed096b2947f": "\\frac{dx(t)}{dt}=a*(y(t)-x(t))", "018c7a969d34311bed9a89e7f6187eb8": "c_{\\sigma}", "018ca55ecf6e69d4925ffa6c737d4d35": "t \\ ", "018cc819d8e1e37f057e83c9ec40173e": "H_1(z)", "018cdc0ed50078c0b68acdc099e531e7": "m_i \\in \\mathcal{M}", "018cdd89d9f5a4e5292b17b162d577c3": "f_0=0", "018cf3b6c662f14fb1d227939807b1a6": "f \\cdot (g*h)=(f \\cdot g)*(f \\cdot h)", "018d226118c2140faf0afe31a03002ce": "\\Phi(x) = -\\int_c \\vec{F} \\cdot \\mathrm{d}\\vec{r}.", "018d41f1d089523907b3ec9607588ef2": "\\textstyle\\mathbf{IPC}+\\bigvee_{i=0}^n\\bigl(\\bigwedge_{j\\ne i}p_j\\to p_i\\bigr)", "018d600acb6ef28c9e56e616cd9030ae": "\\Delta_n(\\mathcal{C}, x_1, \\ldots, x_n)", "018d86e2178b32a5b9a72537a8070bf7": "^\\bullet", "018dabb09a219a855a8236d1f5c20b33": " h(f_s(z))", "018dabebcae8c7f38c369d0781e1894c": "\\phi(\\beta)=\\frac{3}{4\\beta^{2}}\\left(\\frac{1+\\beta^{2}}{2\\beta}\\lg\\frac{1+\\beta}{1-\\beta}-1\\right), ", "018dbb3fc4985ed840a2e6a8fae944fe": "P_1(u,v)=\\left\\langle \\mathbf{F}(\\psi(u,v)) \\bigg| \\frac{\\partial \\psi}{\\partial u} \\right\\rangle, \\qquad P_2(u,v)=\\left\\langle \\mathbf{F}(\\psi(u,v)) \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle ", "018dd87f3449089eb9b8d0831f337a4f": "\\mathfrak{sp}_6(\\mathbf R)", "018ddd56a4074767957790db9d01a62e": " R = \\Sigma \\, \\Phi ", "018de11eca10ed1f4438540470dbb080": "\\nabla^2 = \\partial_{\\rho\\rho}+\\frac{1}{\\rho}\\,\\partial_\\rho +\\partial_{zz}", "018de2aa4efc1422d9375ecc1c106d94": "b^j", "018e15c37302c06def03be4efe13a3fb": "|\\phi\\rangle_A \\otimes |\\psi\\rangle_B", "018e17bcd3719db8c3a20d97cf78c8c7": "\\mathrm{V}", "018e5bf2411bdca849ce5d9cbd6594be": " \\sum_{n=1}^\\infty \\frac{1}{a_n x_n} ", "018e74d99d2d488fc1a3842be6a115f9": "k^{\\prime}\\,", "018e788359518637f7fdb08b607b8193": " \n\\ln y(r_{12})=\\rho \\int \\left[h(r_{13}) - \\ln g(r_{13}) - \\frac{u(r_{13})}{k_{B}T}\\right] [g(r_{23})-1]\\, d \\mathbf{r_{3}}. \\, ", "018e94b6ffd568f76dc90f08af295dbd": "0 \\le i \\le n", "018eb0ac4a321ccaf301048b102f6286": "\\mbox{Debtor days} = \\frac {\\mbox{Year end trade debtors}} {\\mbox{Sales}} \\times {\\mbox{Number of days in financial year}}", "018ebc97d2b46fa45b3b70520e587f57": "\\|y_{n+1}-z_{n+1}\\|\\leq\\|y_{n}-z_{n}\\|", "018ee86a3d311275b87c0a5933942455": "P(s,n) = (s-2)T_{n-1} + n = (s-3)T_{n-1} + T_n\\, .", "018f06e87c6e53a96a3defa23a69a1ad": "f : X_1 \\rightarrow X_2 \\, ", "018f10d214276c7cffc662cc1da2de5f": "|\\psi\\rangle\\in \\mathbb {C}P^N", "018f1b5a032989588141548a05459a83": " q_c = 1 - \\frac {1} {R_0} ", "018f250265d9d5ddae18ce4282d77ff2": "Y_\\mathrm{sun} = 0.25 ", "018f365b18a97f02a2c5e8924fd8540b": "\\zeta \\in F^\\ast", "018feb5d830b583433f1194bc19cf790": " \\operatorname{E}[|X|] = \\operatorname{E}[X] = \\int_0^\\infty \\lbrace 1-F(t) \\rbrace \\, \\mathrm{d}t,", "018fed5a51edfe03ee3443fdc213d0d3": " \\sum_{j=0}^k \\tbinom n j", "01901f6fd51e5c28dd5dfa2e1870d592": " dY_n(t) = S_n(t)\\left[b_n(t)dt + dA(t) + \\sum_{d=1}^D \\sigma_{n,d}(t)dW_d(t) + \\delta_n(t)\\right] , \\quad \\forall 0\\leq t \\leq T, \\quad n = 1 \\ldots N. ", "019049186c3201e21d9c6f8acc6f4762": "0 = L(\\varphi_t,\\nabla_x\\varphi) = (\\varphi_t)^2 - c(x)^2(\\nabla_x \\varphi)^2.", "0190e84d88093784116c0cf414c1f684": "V = \\left(k \\right)\\sqrt\\frac{\\rho_L - \\rho_V}{\\rho_V}", "0190ea3bade1ab5ef3fb72a836b5ae92": "\\mathbf{m}=(m_1,\\ldots,m_c) \\in \\mathbb{N}^c", "01910c3f0e4afd9ab6d71da6a7559ebf": "\\mathfrak J^k(a)_n=b_n=\\sum_{i=0}^n(-k)^{n-i}\\binom{n}{i}a_i.", "0191546962f47fcb2feea1480f82d70d": "(b_0,\\dots, b_{M-1})", "0191c5dbe0a0bed1e8ea409b3ea9449b": " \\sqrt{R^2 - \\left(\\frac{h}{2}\\right)^2},\\qquad\\qquad(1) ", "0191df0133e7e83cbc8b65b67e29cd36": "\\alpha_i(1)=\\pi_i b_i(y_1)", "01922f9fafe5c08cea75ae7237b5ac8f": "\\alpha = \\frac{1}{V} \\biggl(\\frac{\\delta V}{\\delta T} \\biggr)_{P}\\ ", "019236eee89a1bcd87153e945caff4f0": "2(2j+1)", "0192481c1a5fcd1e7f4cce09def4bcb2": "\\!\\mu \\in X", "0193d3271468b5f68bdced7c336ecffb": "\\scriptstyle\\{e^{(a)} = e^{(a)}_{\\mu} dx^\\mu\\}_{a=1\\dots4}", "0193d4d3f614be7ffb688f4a5e71a62d": "|G|", "0193deabfbc61eba0387e52afe5500f0": "unroll : \\mu\\alpha.T \\to T[\\mu\\alpha.T/\\alpha]", "0193ee11c894b0d747dcc9513cbca04c": "Em = 1 \\tfrac{2}{3} 3 + 1 \\tfrac{1}{3} 1 + 0 \\tfrac{1}{3} 2 + 0 \\tfrac{2}{3} 0", "019426145271b9b66f466862efb452a1": "(U, \\phi)", "0194420c9cd297af834bc0fc68b0d0f0": "f(h,k) - f(h,0) - f(0,k) + f(0,0)", "0194949fdd2683fca054957d9a3631f8": " Fr < 1 ", "0194f157695e76edad5de7a928aa3f27": "\\{\\xi_k\\}", "0194fc10bbd26154d932af9c338fb3e8": "y \\in \\mathbb{R}^q", "019503e25e037825852e80e771d92dda": "(n - 1)! = 1 \\times 2 \\times 3 \\times \\cdots \\times (n - 1)", "0195049235f6c32595e6551efc2c4c1b": " F(X)=\\inf_{S}\\sup_{I}\\frac{|I|}{|S|}, ", "01951ec559cd6c4cdc5e189332a65175": "F_3", "019522c5b32a9528c88582d494a9bef5": "\\{(x,t): t < f(x)\\}", "01952cbf349006bc6a12c6661316b4cc": " S = \\frac{U}{T} + N * ~ S = \\frac{U}{T} + N k_B \\ln Z - N k \\ln N + Nk ~", "01952e78f364ba78bdd3844b9917fcf9": "S\\cup\\{x\\}", "01957e8e751f1d74c88c70cc8d9610d8": "f_n(x) = \\sum(x_i - \\bar x_n)^2/(n-1)", "0195a95d13d16542bc61d5cdcf36994f": "A_1=A", "0195cf3c7a5b7625e528f831c3f063ed": "{d\\alpha_j \\over dt} = -{\\alpha_j^{19}\\over \\prod_{k\\ne j}(\\alpha_j-\\alpha_k)} = -\\prod_{k\\ne j}{\\alpha_j\\over \\alpha_j-\\alpha_k} . \\,\\!", "0196026af4a0d9cef3563be5ca49e199": "P( A_K ) = \\prod_{j=1}^n ( a_{Kj} ) ^{w_j}, \\text{ for } K = 1, 2, 3, \\dots , m. ", "01960518dbc691be905340184be10534": "d\\;", "01962c356f57eec60b239226921067db": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} b(k) \\frac{z^k}{k} =\n\\frac{1}{1-z} \\frac{z^m}{m}", "01965566c95014beea4169411276fafe": "c_p = \\frac{\\gamma R}{\\gamma - 1}", "01969bfd511865852ab937396eededfb": "p_idq^i-H(t,q^i,p_i)dt", "0196b2c7974f456231a64af2c1d2d18d": "I = \\frac{m\\left(W^2D^2+L^2D^2+L^2W^2\\right)}{6\\left(L^2+W^2+D^2\\right)}", "0196dab427cb3a38ec0befeb00f22cff": "J^{\\mu} \\, = \\, \\partial_\\nu \\mathcal{D}^{\\mu \\nu} \\,", "0196f3dfcad95bb6f7a13918cb04874a": "\\sigma_N=\\frac{\\nu_L \\sigma_L - \\nu_S \\sigma_S}{\\nu_L - \\nu_S}\\,", "01973b25656d8aaeca3411bb822c6c6c": "\\mu \\ne 4", "01973d13d4ecf598fbdc26a4efd0ac17": "\ng(z) = \\sum_{n=1}^\\infty z^n \\,\n", "01974885b9d6f1a92fea39de85c21f25": "p+\\delta p", "01975388b40bd8aafd509042d627c2ca": "\\tfrac{26}{11}.", "01975e71b69de760ff53071819946a93": "\\scriptstyle W[k] = (-1)^k\\cdot W_0[k].", "019777033a1f7b80d44ed3e8c1e8fe32": "\n\\psi(x)=\\sum_{n\\le x}\\Lambda(n). \\;\n", "0197a1981e16dbe4aca1cd37b7b8207b": "x_1 = c \\times10^{b_0}", "0197ac0c144881e2e604b20b3b77cd9a": "c_i/f_\\mathrm{eq}", "0197cf1222e54259f28ec57ef981c0b0": "\\frac{1}{T}\\int_0^T \\mu_{\\max}(t)\\, dt \\leq 1 - \\varepsilon", "0197ee0201e14a0f4ce5849df7285c34": "s \\ne t", "01983a10224cc4ef657692c9bc7ee5db": " \\exists^p L := \\left\\{ x \\in \\{0,1\\}^* \\ \\left| \\ \\left( \\exists w \\in \\{0,1\\}^{\\leq p(|x|)} \\right) \\langle x,w \\rangle \\in L \\right. \\right\\}, ", "0198791ecd1dcf1db07cac8e68982320": "\\operatorname{Categorical}(\\boldsymbol\\phi_{x_{t-1}})", "01988c67f402ddd3b5b82a07e8eaa2db": "e=3", "01989ba6f1aee5e66d1a613a1965baa3": "\\Delta E_{UVW}", "0198d3a65976defb7f5c155ce7d12591": "\\bigg(\\frac{a x_a+b x_b+c x_c}{P},\\frac{a y_a+b y_b+c y_c}{P}\\bigg) = \\frac{a(x_a,y_a)+b(x_b,y_b)+c(x_c,y_c)}{P}", "0198dff3f918245c5bddc7308da0886d": " \\begin{bmatrix} 2 & 4 \\\\ 3 & -8 \\\\ 1 & 2 \\\\ 2 & 4 \\end{bmatrix}", "01991eec76ddfec566e5d7ec6dc3e9d6": "\\dot{v}(t)", "01997a8f18de1bfbddb9cfc0c93010af": "L^{3-}+3H^+\\leftrightharpoons LH_3:[LH_3]=\\beta_{13}[L^{3-}][H^+]^3", "01997e740836ece90f7b7fc509abedd2": "0 = t_{0} < t_{1} < \\dots < t_{k} = T", "0199832e9fae5d0586f0bdd9a6282656": "\\sigma_{A}(R\\cup P)=\\sigma_{A}(R)\\cup\\sigma_{A}(P)", "0199c39c779f03bea969358615a0a035": "H^2(\\mathbb{D},\\mathbb{C})", "0199c7407ecd3009ea90841e94a08477": "\\oint_{R} B ds \\cos{\\theta} = \\mu_0 I_{enc}", "019a46fcbcf20b8f001fd25085727497": "\\int_{\\mathbf{R}^n} f(\\mathbf{x})\\delta\\{d\\mathbf{x}\\} = f(\\mathbf{0})", "019a54c4cefc7364b20ea8984379b3bb": "\n\\begin{bmatrix}\n\tZ'_{11} & Z'_{12} \\\\\n\tZ'_{21} & Z'_{22}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\tZ_{11} & Z_{12} \\\\\n\tZ_{21} & Z_{22}\n\\end{bmatrix}^{-1}\n", "019a5e4654b34048ad6c2144b93e07e6": " A = LDU, \\, ", "019a98cbd788b7cd3fb93a24dd151521": "12^2+35^2=37^2", "019ac375a68a1ba2d3dfaceecb04ba36": " \\pi_2(x) =O\\left(\\frac {x(\\log\\log x)^2}{(\\log x)^2} \\right).", "019ae5e37093a2386ddbdbbd1511ac0b": "\\, TA(t_1)A(t_2)=A(t_1)A(t_2)\\!", "019b26b86ba081c79cd0e6bea83697fe": "GE(\\alpha) =\n\t\t\\frac{1}{N}\n\t\t\\sum_{i=1}^N\n\t\t\\ln\n\t\t\\left(\n\t\t\t\\frac{\\overline{y}}{y_i}\n\t\t\\right),\n\t\t\t\\quad\n\t\t\t\\quad\\quad\n\t\t\t\\text{ for } \\alpha = 0,\n", "019b681173c537822faed84e02198141": "[\\mathtt{Gen}]", "019c168d94e358f1fb02f88a964deaba": " {x^2+y^2} - L^2=0, ", "019c3374d2c4d9109cf304dccaffe1e0": "\\,\\int_0^{\\frac{\\pi}{2}}\\,\\frac{\\sin^2\\,x\\;\\mathrm{d}x}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^2}\\;=\\;\\frac{\\pi}{4\\sqrt{ab^3}}.", "019c5f010f647f3c10973259318d5764": " E_i E_j=E_j E_i, \\mathrm{if} \\left\\vert i-j \\right\\vert \\geqslant 2,", "019c7ab6604aaf7125613d86788a2b3e": "S^s_0(t)", "019c877780717c1350fd15a60edef362": "((A \\to B) \\land (B \\to C)) \\to (A \\to C)", "019c8cf9139fb7a1a644f74c07140b2a": "Q_A = qL", "019ca4f32ed99b42a77e7c9915a2e76c": "b=0.2", "019d18f1da08be3c865ae8bafbd72d8a": "(x + 5)^{2/3} = 4,\\,", "019d3a959349b8026a9f20a8ae55d8d6": "\n\\begin{array}{lcl}\nr_j & = & FI(KI_{i,j}, l_{j-1} \\oplus KO_{i,j}) \\oplus r_{j-1} \\\\\nl_j & = & r_{j-1}\n\\end{array}\n", "019df08eb3c50b2bcd1c650bf32042fe": " \\phi_{l}\\,= \\phi_{L} ", "019e218615c919ebfbe64ec3eedf30db": " =\\frac{E}{m|\\vec{q}|} \\left(\\frac{|\\vec{q}|^2}{E}, q_x, q_y, q_z \\right) \\,", "019e3232ce1861ab115fe64ee345355b": " |\\psi(t)\\rang = \\sum_n c_n(t) e^{- i E_n t / \\hbar} |n\\rang ", "019e5a89b9b9ee6e978db1badbc7386d": "L = \\int{ \\mathcal{L} \\, d x d y d z}", "019eb126c193432d1a2a2642c3f3f09b": "P(t|M_d)", "019eb4af7f580200e254f4c75953d7e7": "R = \\frac{1}{2}\\rho \\frac{f L S}{A^3} \\equiv \\frac{1}{2}\\rho\\frac{f L}{R_{h} A^2} \\equiv \\frac{1}{2}\\rho\\frac{4 f L}{D_{h} A^2} \\equiv \\frac{1}{2}\\rho\\frac{\\lambda L}{D_{h} A^2}", "019eb91545f1d3444c54206c8d8c5e16": "\\begin{align}\n ~x_k &= (2d_k - 1) + a_k\\\\\n ~y_k &= 2( Y_k - 1) + b_k\n\\end{align}", "019f15b6e3e80ff2e19818c8ea597924": "f^b(t_i, w)", "019f2057772144bbe5ec352cbb1608ff": "RC = \\frac{(H+BB-CS) \\times (TB+(.55 \\times SB))}{AB+BB}", "019fb747da2029da6d18a634727f0dc8": " R = \\Delta T/\\dot Q_A", "019ff996753dcc37e7955f547c2c5fc4": "d_{i,j} = 0 \\mbox{ if } i \\ne j\\ \\forall i,j \\in \\{1, 2, \\ldots, n\\}", "01a015c92ea04385819cf8a2965e22d1": "i,j\\in E.", "01a02c3662ecb476540620d065822d3a": "2(n-1)", "01a07b0b22723e6d6fe75e73e815ecdd": "a\\geq 0", "01a0a57943be3ebbb0a780d7b36eb6bb": "= \\hat{a}_j^\\dagger \\hat{a}_l^\\dagger\\, \\hat{a}_i \\hat{a}_k + \\delta_{il} \\hat{a}_j^\\dagger \\, \\hat{a}_k + \\delta_{kl}\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij}\\hat{a}_l^\\dagger\\hat{a}_k + \\delta_{ij} \\delta_{kl} ", "01a0a8dfe48ae3444a60e0371df84d8e": "\\nu>0", "01a0bc19715f6f2a983ee153b0470c9b": "i^2=j^2=\\eta", "01a0ddeeb3da341949a04413c40519cf": "p(c_j | x_i) \\,", "01a0ef71ac5c04bdabf03022f1a6834f": "G^p=(V,E^p)", "01a0f00f7a6a71f0b50f6450179ac3b7": "|\\mathbf x|_p := \\left(\\sum_i |x_i|^p \\right)^{1/p}", "01a0f25168bedbc4a6f70dee4398308c": "f=\\sum_{k=0}^\\infty A_kz^k.", "01a0fdfb3cd1f76eb8002eba9c586f57": "g_{bf}", "01a11a1c84ff2134d636dbb5a8c4d861": "\n f_b\\left( \\frac{F_D}{\\frac12\\, \\rho\\, A\\, u^2},\\, \\frac{u\\, \\sqrt{A}}{\\nu} \\right)\\, =\\, 0.\n", "01a120c756b1f6cf4f08e0fca0cfa6fe": "dl", "01a1def7f98541e100920165c8ab315a": " x \\rightarrow \\infty ", "01a2003b637de11f1584eddec16efd69": "\\psi(\\lambda)\\,", "01a2090d57c865bb7b277857d0659e3e": "1.\\overline{36}", "01a22ae8fbd128bb63fd9f0304c7d584": "L=\\frac{\\Theta}{2 \\pi} \\cdot 2 \\pi R \\, \\Rightarrow \\, \\Theta = \\frac{L}{R}", "01a2bd187bcc97465946cda426857db6": "H(f)(x) = \\frac{1}{i}(F_+(x) + F_-(x))", "01a2d354eeb4299748e097f987ad06a1": "k = 0,\\ldots,N", "01a303aa74d54caa7d7fd469294630a4": "\\mathbf{f}(\\mathbf{x})\\neq1", "01a35d410aebfade90b90ef175faa85d": " \\begin{bmatrix} a_{1 1}(x) & a_{1 2}(x) & \\cdots & a_{1 n}(x) \\\\ a_{2 1}(x) & a_{2 2} (x) & \\cdots & a_{2 n}(x) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n 1}(x) & a_{n 2}(x) & \\cdots & a_{n n}(x) \\end{bmatrix} ", "01a395a6f409a18d75821974a1afefbc": "3-", "01a3a511f91e8722706af52507970b22": " \\overline{u'^2} = \\overline{v'^2} = \\overline{w'^2}. ", "01a3c2cb482cb206e9e4d2776bf35a5d": " \\operatorname{let} x : (x\\ x = \\lambda f.f\\ (x\\ x\\ f)) \\operatorname{in} x\\ x ", "01a462b57cdce353e860cb43e567bd43": "K_1=\\frac{{[NO]} {[NO_3]}} {{[NO_2]}^2}", "01a4a59350f82cc6452b8fbfdc645a84": "\\Delta p = 2410 \\left( {m \\over V} \\right)^{0.72}", "01a4b0901e7094539c040c5659f2e2eb": "\n \\Delta^2 w := \\frac{\\partial^2}{\\partial x_\\alpha \\partial x_\\alpha}\\left[\\frac{\\partial^2 w}{\\partial x_\\beta \\partial x_\\beta}\\right]\n = \\frac{\\partial^4 w}{\\partial x_1^4} + \\frac{\\partial^4 w}{\\partial x_2^4} + 2\\frac{\\partial^4 w}{\\partial x_1^2 \\partial x_2^2} \\,.\n ", "01a4e7db800567fb110001e8d958a2f3": "\n\\begin{align}\n\\sum_{i=1}^6 \\tfrac{1}{6}(i - 3.5)^2 = \\tfrac{1}{6}\\sum_{i=1}^6 (i - 3.5)^2 & = \\tfrac{1}{6}\\left((-2.5)^2{+}(-1.5)^2{+}(-0.5)^2{+}0.5^2{+}1.5^2{+}2.5^2\\right) \\\\\n& = \\tfrac{1}{6} \\cdot 17.50 = \\tfrac{35}{12} \\approx 2.92.\n\\end{align}\n", "01a4f56afd9079f8140cbc858c20bcf7": " \\frac{\\delta F[\\varphi(x)]}{\\delta \\varphi(y)} = g(y) F[\\varphi(x)]. ", "01a4fc6069dbeeb5bdf837895affc245": "x \\neq y", "01a54313673b4d77705210d217a7ef37": " \\mathbf{[Z]}=\\begin{bmatrix} Z_{11} & Z_{12} & Z_{13} \\\\ Z_{21} & Z_{22} & Z_{23} \\\\ Z_{31} & Z_{32} & Z_{33} \\end{bmatrix}", "01a550404fe927b314049e8d33de9fa7": " Q^{m}u=\\int_B T_y^mu\\left( x\\right) \\psi\\left( y\\right) \\, dx, ", "01a5708bc021cd7d1eb44d684951de2f": "\\begin{matrix}(14 - x) \\times 4\\end{matrix}", "01a5bc2d7acd353afebdaa9633dffec1": " f(x) = f(a) + f'(a)(x - a) + R_2\\ ", "01a5be688e3a97ad77ab71e95f396757": "\\ R = \\sqrt{(X_{12} - X_{11})^2 + (X_{22} - X_{21})^2}", "01a5be88fb56693e02fbee27521b5063": "{\\mathcal C}_n(z) = \\sum_{k=0}^\\infty \\pi(k+n) \\frac{z^k}{k!}", "01a5e298f2604e78351a4f9efa94aeab": "~k_a", "01a634527dee50e7bd73d69a8a63110d": "\\hat{\\mathcal{O}}", "01a6819a07cedf575f0f299dc4badf1c": "\\displaystyle \\frac{\\mathrm{d}^2 x^i}{\\mathrm{d}t^2} = -\\frac{c^2}{2} \\varepsilon \\gamma_{00|i}", "01a6aba88971cca0b2f59fab085fbe80": "10000=10^4", "01a6c084f5e59595c64196d929743f4d": "\\lambda_4 = \\sqrt{2},", "01a6e061b2e927945bb4fa00e7e344a1": "=\\!\\!(t_1 \\ldots t_n)", "01a72d27fa295850a617bf49fe186a27": "v_\\mathrm{p} = \\frac{\\lambda}{T}.", "01a764cee7384d7c873165dd7c7dd066": "\\textstyle \\leq c", "01a76635894af1be7c454818e15e864d": "g(2^n,2) = 3", "01a78c2c81fa65870adab1526aa3dd6c": "\\left\\{{5'\\atop3}\\right\\}", "01a7ec08e4a4d2e1052b46850941e4e9": "0, 1, \\ldots, n-1", "01a873e523d00e4ab7d05e3b47213d08": "F_{eq}", "01a87ab17ca903f95241c866f531ba64": "(-b-h(a))^2 + h(a)(-b-h(a))", "01a8bf77078848236bfd2d223e761215": "\n \\epsilon=a\\tan\\theta\n ", "01a8c485b579cc073a33b76977543ee8": "\n\\operatorname{Li}_s(-z) + \\operatorname{Li}_s(z) = 2^{1-s} \\,\\operatorname{Li}_s(z^2) \\,.\n", "01a8f4b4ea69d5f48e5aec150b9a938a": "(\\mathbf{J}_2, \\mathbf{E}_2)", "01a92ffdaad37599e891789ee4dc6daa": "{{f}_{M}}", "01a94aa32af850c75db975e05b64e709": "T^{-1/p} + T^{-1/p^2} + T^{-1/p^3} + \\cdots ; \\, ", "01a94f41297bd40bf5881c4b69ad38c7": "\\operatorname{sqsum}(x, y) = x \\times x + y \\times y", "01a95497fd188fe421728b66ae3e94fc": "|\\mathbf{r}|", "01a979edcb34ad6b9c69310e1ba3f01d": " \\varphi_a (g) = a g a^{-1} ", "01a9f31ab16bb54eac94bffcc7fcf7e8": "s_1,s_2 \\in S, r \\in R", "01a9fe25a65a2201a9bb94d9bb9d1c98": "\n f\\;a\\;b\\;c\\;:\\;1 \\to D\n", "01aa6b95d98a04759a36335b2c7b96b0": "\\forall\\beta.\\beta\\rightarrow\\alpha", "01aa89836a9d9b1f38be001498955085": "\\begin{matrix}{4 \\choose 4}\\end{matrix}", "01aac14134bf2df7be4dc632323e4a46": "k_{1}, k_{2}\\in K \\subset A\\,", "01aadeeff6a395a7087f2ba67c85afe6": "e_4", "01aafa82db291da77997f6b1c472899f": " RSS = y^T y - y^T X(X^T X)^{-1} X^T y = y^T [I - X(X^T X)^{-1} X^T] y = y^T [I - H] y", "01abb031e803ea01a54831fbd2ac7af4": "\\sum_{n=1}^\\infty \\frac {z^n}{n!} H_n = -e^z \\sum_{k=1}^\\infty \\frac{1}{k} \\frac {(-z)^k}{k!} = e^z \\mbox {Ein}(z)", "01ac0d10469c3dfa1296b1d1bb690511": " U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\\phi_1(q^{-n}, x^{-1};0;q,qx/a)", "01ac191dd7e8a53ee4c24bdda542fec2": "f_c(k,r)\\approx f_0(E,E_{Fn},T_n)", "01ac54386a9d909da3b638139ce7966e": "\\frac{1}{(i\\omega)^2-\\xi^2}", "01acc0905f397bf2ffad7857cb5f3384": "\\|f\\|_p = \\left (\\int |f|^p\\,d\\mu\\right)^{\\frac{1}{p}}", "01ace1d9d151a6069bf22973a57eca16": "\\int_{0}^{\\infty }\\frac{f(ax)-f(bx)}{x}\\ dx=[{f(0)-f(\\infty)}]\\ln \\frac{b}{a}", "01acecf10001f0540a51caf58766b224": "g(x) = f(x + a)", "01acfbf706dd22836aba8f79192bf009": " Y^\\ast = X'\\beta + \\varepsilon, \\, ", "01ad76a59829a51dcb3b63290c1efe8c": "\\Phi(M)", "01ad985307e177d5f92f0cc6a075051d": "a_i = f^i(n)", "01ada4c6e7ff46e9d9c6fa5fb36a69cd": " \\tilde{f}", "01adb584cb3be702a413e84135e5f0df": "h_f = r \\cdot Q^{n}", "01addcd0e7e699e500b24ddb246983b9": "\\gamma(\\mathbf{h})=\\frac{1}{2N(\\mathbf{h})}\\sum^{N(\\mathbf{h})}_{i=1}\\left(Z(x_i)-Z(x_i+\\mathbf{h})\\right)^2", "01adfa4dbebcb0ee3a196cbe0b5adde0": "B = Y_2ZZ_1Z_1 = \\sqrt{3}", "01ae55de5e9698c5db36423be6c05224": "\\omega_e = \\mathrm{id} : T_eG\\rightarrow {\\mathfrak g},\\text{ and}", "01ae6d84773d96ef563f2be9dacf5e9e": "\\frac{\\mathrm{d}N_B}{\\mathrm{d}t} = -\\lambda_B N_B + \\lambda_A N_A.", "01ae83345a7d932357d44a263ec78119": " F_{Y}(y) = P(Y \\leq y) = P(\\mathrm{log}(1 + e^{-X}) \\leq y) = P(X > -\\mathrm{log}(e^{y} - 1)).\\,", "01aee35ed3a5b074de86299a81ccaa03": "\\text{Moeb}(\\mathbf{S}^1)\\subset \\text{Diff}(\\mathbf{S}^1) \\subset \\text{QS}(\\mathbf{S}^1)", "01aee9f729a432e09c332da539eeb8d3": "mn=\\mathrm{N} \\mathfrak{p} -1 ", "01aefae77efb224ae0167b114ce3556b": "\\alpha_{\\text{object}}", "01af897a5f8a1e5cd17232f87c20c21a": "r/2", "01afdbc2d4543f51eeea1f8df91ee9de": "Y = A * F(K,L) \\,", "01aff9b8c6c4296ae629c6fed72f30c7": "\\int\\frac{dx}{ax^2+bx+c}", "01b0782a0a4f89160fd5022c5284e501": "T_{T}=\\frac{2L_{T}}{\\sqrt{c^{2}-v^{2}}}=\\frac{2L_{T}}{c}\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "01b0e951629ccbc18721289f1cd8cdf3": "P\\left(k\\right)\\sim ck^{-\\gamma} \\, ", "01b0f693fc41e3fa99a9a2e13c88dbcf": " S_u = \\frac{\\hbar}{2}(u_x\\sigma_x + u_y\\sigma_y + u_z\\sigma_z)", "01b11daa5ac39cdda7f6b4aa6b489ce4": "\\mod 2^n-1", "01b1abef0f3a7d80becdbbfdccc7763d": "f^*(\\omega + \\eta) = f^*\\omega + f^*\\eta,", "01b1c662c9569eaadaeb573607ea8644": "W_{\\text{Yuk}}", "01b20c4e008e677c5b4594db36fbc925": "H_p(B(S^{-1}S)^0) = \\varinjlim H_p(BS_n) = \\varinjlim H_p(BGL_n(R)) = H_p(BGL(R)), \\quad p \\ge 0", "01b20e5cf9941b9e9034764b678beacb": "U\\,\\!", "01b265fee6a0869c2e4f9adafa319138": "\n\\frac{\\mathrm{d}^3 x}{\\mathrm{d} t^3}+A\\frac{\\mathrm{d}^2 x}{\\mathrm{d} t^2}+\\frac{\\mathrm{d} x}{\\mathrm{d} t}-|x|+1=0.\n", "01b28d0dae8226bfb1d154b5662decc5": "\\frac{P \\or Q, \\neg P}{\\therefore Q}", "01b2c8fc03e1f4bfc3606d60022ac277": "[Q^\\dagger,b^\\dagger\\}=0", "01b2cafcfc2480ecbe9b33cb21ccf6fa": "\\pi_k^n(x_1,\\dots,x_n)=x_k,", "01b2cf201aee34804cd87795fbaf6d24": "\\frac{\\omega-\\omega_o}{\\omega_o}=\\frac{\\Delta \\omega}{\\omega_o}=\\frac{\\mu HH_k}{2kL_e^2(H+H_k)},", "01b2dfca9f0240a5b8f0859525d2e570": "N\\sin \\theta ={mv^2\\over r}", "01b2f3fed2a492efd95302ad3b7b0165": "\\rho\\;\\!", "01b33ba9f800285f0859adf08818f1e7": "x = a\\ \\operatorname{arcsinh}(s/a) + \\alpha.\\,", "01b3fe9e9446fa564c2e05d03313e91c": "v_0(\\xi\\otimes e_\\alpha)=(v,\\alpha) \\xi \\otimes e_\\alpha.", "01b415a0ef6a764eb78f82f7169c051f": "\n\\int_{-1}^{+1}\\frac{T_m(x)\\log(1+x)}{\\sqrt{1-x^2}}dx = \\sum_{n=0}^{\\infty}a_n\\int_{-1}^{+1}\\frac{T_m(x)T_n(x)}{\\sqrt{1-x^2}}dx,\n", "01b449b4ef96135c708dd6a66a52ee28": " \\varphi_\\lambda(e^t i)={1\\over 2\\pi}\\int_0^{2\\pi} (\\cosh t - \\sinh t \\cos \\theta)^{-1-i\\lambda} \\, d\\theta.", "01b4a3c7c3d1f0e5303c0740eda30fd1": " \\frac{n(n-1)}{2} ", "01b59247bdf3ed5106ba8e6ac3cceef3": "\\sum_{g \\in G} r_g g", "01b6af627963b834d56274feae25b317": "\\frac{\\exp(-\\beta \\varepsilon(\\mbox{state}))}{\\mathbb{Z}}", "01b6d4d188517123c07fa5aecdad31ec": " (\\delta f)(x) = {{ f(x+h) - f(x) } \\over h }", "01b706e17abb0a059a02379fa29619c6": "\nA =\n\\begin{bmatrix}\n ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 & ~0 \\\\\n -1 & ~4 & -1 & ~0 & -1 & ~0 & ~0 & ~0 & ~0 \\\\\n ~0 & -1 & ~4 & ~0 & ~0 & -1 & ~0 & ~0 & ~0 \\\\\n -1 & ~0 & ~0 & ~4 & -1 & ~0 & -1 & ~0 & ~0 \\\\\n ~0 & -1 & ~0 & -1 & ~4 & -1 & ~0 & -1 & ~0 \\\\\n ~0 & ~0 & -1 & ~0 & -1 & ~4 & ~0 & ~0 & -1 \\\\\n ~0 & ~0 & ~0 & -1 & ~0 & ~0 & ~4 & -1 & ~0 \\\\\n ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4 & -1 \\\\\n ~0 & ~0 & ~0 & ~0 & ~0 & -1 & ~0 & -1 & ~4\n\\end{bmatrix}\n", "01b7477bcbbd2d0769a6b4d3bf074f71": "a(k)\\left |0\\right\\rangle = 0.", "01b77ce6c8f81c618bb9968aa25d7455": "\\Delta(z)", "01b7e243a05b2f042a5bc115256a7477": "y = \\frac{1}{2} \\ln \\left(\\frac{E+p_\\text{L}}{E-p_\\text{L}}\\right)", "01b810f080f6385798780e7fc3463c97": "\\pi_1:U\\times G \\to U,\\quad \\pi_2 : U\\times G \\to G", "01b859364dda742772c2f949845f4e52": "F \\times \\mathsf{S}(a) = \\mathcal{P}_B^{\\perp} (a \\cdot \\partial F),", "01b85a530f6ff948dd2ad38b65ec707e": "p_\\mathrm{int} = p_1+s_\\mathrm{int}\\cdot\\mathbf{u}", "01b881814da8e0aaa3c96b9b650e95fd": "\\dot m_{out}=K \\cdot C \\qquad (4)", "01b894dc22ae1082677a08d6b924e48d": "\\scriptstyle v ", "01b8c391a9e2849d70e4175f47a596d5": " \\Phi(y) ", "01b8d973d292f2a29aea13ee5ef47880": " t \\mapsto \\bold{X}(\\bold{u}_0) + t \\bold{A}(\\bold{u}_0). ", "01b95a3d7abea8628080371744d90d22": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0\n\\end{array}\n\\right] .\n", "01b978a6a034de8a736775b90de615dc": "S_0 = 0", "01b97ce8ecaa783b96225eadb98e51b2": "\\operatorname{d}=\\tau", "01b997311f718112df1bbbe9a5accb6d": "H(A: B |\\Lambda)=0", "01b9b96e98ff2e57ef3171d562d79a55": "r = \\sqrt{3 / \\Lambda}", "01ba19784c735ed8b3a29614ac1c98c4": "C_\\pm(j,m)", "01ba6a18d617927034af344cc636f91f": "\\Delta u \\ge 0,", "01ba77110113019916a9054319ae7c05": "f(0)", "01baae0577e052f0eb6660747516f26f": " [n_{PQ}]_{PQ \\sim QQ} ", "01bae0ecde1960fec732c7c0a51fff82": "\\widehat{U}(t - t_0) \\equiv U(t, t_0)", "01bb12a4f0524a838974075b1126f307": " \\{ fg,h \\} = f \\{ g,h \\} + g \\{ f,h \\} ", "01bbb262477c343e2c65ed5e8a4ad417": "K_- = span\\{ \\phi_- = a \\cdot e^{-x} \\}", "01bbb6d958ab9dc5347b3e281037fa00": "F(x)=E(1_{X_1\\le x})", "01bbc47541ee02be2bda7635619e5043": "\nH(\\mathbf{Y})=-\\frac{1}{N}\\sum_{t=1}^N \\ln p_\\mathbf{s}(\\mathbf{y}^t)+\\ln|\\mathbf{W}|-H(\\mathbf{x})\n", "01bbd03668954499b3f7781400b2da2f": "w, \\mathcal O_L, \\mathfrak p", "01bc44e1b633cb8d01df71ae6569036e": "\\langle \\Sigma \\rangle = \\mbox{diag}( i \\sigma_2 f_3, i\\sigma_2 f_3, i\\sigma_2 f_3, i\\sigma_2 f_2, i \\sigma_2 f_2)", "01bc48c212bc834f5225cc8dba7ee47f": "\\mathbf{\\hat f}", "01bc4ad5ea6ec57f062a5766c6bb6f3b": "\nZ=\\int e^{-\\frac{F(r)}{kT}}dr\n", "01bc6c772c8830bc450cfa7414f52319": "\\scriptstyle v_2", "01bd0aea7abc570c68a7636e14c2650c": "P = \\sum_{n=0}^p A(n)x^n = A(0)+A(1)x+A(2)x^2+ \\cdots +A(p)x^p", "01bd5c65c6572b9a691551059117c64a": "y_{1}^\\star", "01bd5fea77e2aaf26d5127ea526462f9": "\\sum_{k = 0}^{\\infty} \\frac{16^{n-k}}{8k+1} = \\sum_{k = 0}^{n} \\frac{16^{n-k}}{8k+1} + \\sum_{k = n + 1}^{\\infty} \\frac{16^{n-k}}{8k+1}. \\!", "01bd73abf03dde76e3597ba1a3373a6e": "f(x_1,x_2,x_3)", "01bdbed18e06abe6fcd6d4204c9fde4b": "\\chi \\equiv \\{\\operatorname{Tr}\\big(\\Gamma(R)\\big)\\;|\\; R \\in G\\}", "01bde7da2af3bb5b592b0bd89c8a1a84": "n = u", "01bdfb42401098f22e4e65082a25782b": "\\alpha = \\frac{\\lambda - n}{p}", "01be15b45c8c746eeab570c7f9afb5fc": "\\scriptstyle{\\pi^*}", "01be1daa1ef16435bbd120ce445acd8f": "O(1,n)/(O(1) \\times O(n)).", "01befff5e286d6de097c46c7deb5d0e1": "\\omega_L = \\frac{1}{2}\\left[ -\\omega_c + (\\omega_c^2+4\\omega_p^2)^{1/2} \\right]", "01bf691f9de147784b9aa33fc2671716": " \\cot \\theta =\\!", "01bf7c92ab2b2bc69503bac0f5f03dc4": "\\int_0^\\infty |f(t)e^{-st}|\\,dt", "01bf930c851369a27e34ba27a127a9d1": "\\operatorname{span}(\\mathbf{v})", "01bfc575b9da2f84e9e45e0538a0d95f": "T_b=\\frac{I_{\\nu}c^2}{2k\\nu^2}", "01bfd3f7a9d4b122649fac52f46f33da": "[\\omega]^{\\omega}", "01bff782684ce9a8b67e0c4858691369": " \\phi : \\mathbb{R}^4 \\rightarrow \\{ 0 \\} ", "01c047fad210fd39854ed9a0de836647": " T(n) ", "01c06a44570541f591261037bce6aebb": " \\tau_{xy} = \\frac {\\mu b} {2 \\pi (1-\\nu)} \\frac {x(x^2 -y^2)} {(x^2 +y^2)^2}", "01c07da14ee5b66ac914af46f54c98b4": "r_n = b", "01c0a6c761628d72dbdf978bda335e81": "\nL = \\frac{qB}{2c}(x\\dot{y} - y\\dot{x}) - V(x, y)~,\n", "01c0c8b1e311e98d09a5188569dfad2f": "\\frac{(f'(\\theta),\\ f(\\theta))}{|f'(\\theta),\\ f(\\theta)|} = (\\cos \\psi,\\ \\sin \\psi)", "01c0e78b5baed0feef5041fe7545dfef": "\\begin{align}\n& {} \\quad \\langle\\phi(x_1)\\phi(x_2)\\phi(x_3)\\rangle\\\\\n&=\\langle\\phi(x_1)\\phi(x_2)\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\phi(x_2)\\rangle_\\text{con}\\langle\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\phi(x_3)\\rangle_\\text{con}\\langle\\phi(x_2)\\rangle_\\text{con} \\\\\n&+\\langle\\phi(x_1)\\rangle_\\text{con}\\langle\\phi(x_2)\\phi(x_3)\\rangle_\\text{con}\n +\\langle\\phi(x_1)\\rangle_{con}\\langle\\phi(x_2)\\rangle_\\text{con}\\langle\\phi(x_3)\\rangle_\\text{con}\n\\end{align}", "01c11575ea95126fcd60f809f8da5bcf": "E_\\mathrm{v} = 10^{(-14.18-M_\\mathrm{v})/2.5}", "01c154aeb7c0087556908ee407b5d53d": "\\displaystyle\\frac{d^ns}{dt^n}", "01c160807c5af832724af0c6fc6c2ff9": "n \\log_2 n - \\frac{n} {\\ln 2}", "01c1ebc232309ad5d6535a37d3390e4d": "C\\ell_{i,j}", "01c21154df41f632afa61480f9b835f4": "k\\times n", "01c23d55901e9c54a2271c6f35213c45": "\\tilde H,", "01c23ece393899dd12ed251f005a308a": "i^2=id_A", "01c2795090f1eece4dd7433d6ba002ff": "\\theta_A=\\frac{k_1C_A\\theta_E}{k_{-1}+kC_S\\theta_B}", "01c2da53089a42414a8f92ccc46ee9a8": " t_2 = \\sum x_i^2 ", "01c3104f950e4db29466791bda1d743f": "\n d_{(ij)k} = \\alpha_i d_{ik} + \\alpha_j d_{jk} + \\beta d_{ij} + \\gamma |d_{ik} - d_{jk}|, ", "01c3ae91742e02bc43e53d948351f27b": "{\\left(\\frac fg\\right)}' = \\frac{f'g - fg'}{g^2}.", "01c3e9c1e55d5895c44543209649d809": "\n\\begin{array}\n[c]{cccccccc}\nI & X & I & Y & Z & I & I & \\cdots\n\\end{array}\n,\n", "01c3f37234550aae5c346d656a2cbe48": "\\omega = \\frac{1}{\\sqrt{LC}}", "01c3f71b1579582146a7326c7765985e": "q\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "01c3fde8626debd09db6b81b0ad7d2d3": "\\forall A\\exists b\\exists c\\exists d\\; bA\\and cA\\and dA\\and \\lnot b=c\\and \\lnot b=d\\and \\lnot c=d ", "01c48f2a351834b3e827311bbde7137d": "\\tilde{\\rho}:K[G]\\rightarrow \\mbox{End} (V).", "01c490edb230e5ce38488ed375fc43de": "r_{j-1}0", "01cd4bb547aef88a022ade3aa2751492": " B_k(j) \\!", "01cd4e3f01a20a9194cf0e90f97cb556": "\\mathbb{P}(V)", "01cdbc2e9e7b9e782f1f84c0125c7150": "\n\\Delta g\\ =\\ \\int\\limits_{0}^{2\\pi}\\bar{V} \\bar{h}\\frac{r^2}{\\sqrt{\\mu p}}d\\theta\n", "01ce38e7e92f345f1d8657e6c1167623": "\n\\begin{align}\nI =& B_{1_1} + B_{2_1} \\\\\nA =& 3/4 B_{1_1} + B_{1_2} + B_{2_1} + B_{2_2} \\\\\nA^2 =& (3/4)^2 B_{1_1} + (3/2) B_{1_2} + B_{2_1} + 2 B_{2_2} \\\\\nA^3 =& (3/4)^3 B_{1_1} + (27/16) B_{1_2} + B_{2_1} + 3 B_{2_2}\n\\end{align} ", "01ce5bc8ee09686540fea99c45d34c3e": "\\left.\\frac{{\\rm d}W}{{\\rm d}z}\\right|_{z=0}=1.", "01ce72ac07ffca2b84ce8f610856d4cd": "\\Delta \\theta = 2 \\pi \\frac{R_W} {D} \\frac{T_1-T_2} {T_R}", "01ce7f5c6112c876528db18ce012f72e": "\nf(s) =\\liminf_{n\\to\\infty} f_n(s),\\qquad s\\in S.\n", "01ceb417b02f63b4e5d46e62973cf371": "\\Delta t = R_N^{-1} ", "01cfb4708aefc4c4a31ab61902490d5a": "y_L = F(x-\\delta, \\hat\\theta)", "01d0242e64a2c042c8683e7c24984b6c": "z:=x+iy\\in\\Omega.", "01d05171f7824558056e284722f832ec": "\\bar{e}_{x}^{ch}=\\,", "01d0587525d2f3cc498963f7b7f882aa": "X \\sim \\chi^2 \\left( 2 \\right)", "01d075d91893ddcb311ee9cd943239eb": "t_2 = \\gamma \\frac{1}{f^\\prime}", "01d09dc5f46ce25eb6baf35afb266fb2": "F=\\left\\lceil\\frac{\\ln(\\epsilon^2/4)}{\\ln(1-\\epsilon/2)}\\right\\rceil", "01d0beba1ca746f01eaa89df05659ab6": "M(a,0,0) \\to S_{ah}", "01d0db18593a536cfb2695353995a6ab": "{dN \\over dt} = aN^2 - bN", "01d1026a9bcf4926e9c62684289f26b0": " \\int X \\mathrm{d}^n x \\equiv \\int X \\mathrm{d} V_n \\equiv \\int \\cdots \\int \\int X \\mathrm{d} x_1 \\mathrm{d} x_2 \\cdots \\mathrm{d} x_n \\,\\!", "01d115d57b9fecdfa8787bc3f7558428": "\\vec J_\\sigma = \\frac{ND\\Omega}{kT}\\nabla H", "01d154d178cf32c6a6cec4a660bd644f": "\\displaystyle 2\\pi f(-\\nu)\\,", "01d1664a7b946d902bed06c864bfb264": "|\\psi\\rangle \\in \\mathcal{H}", "01d1680b2dc1fd8d7d098eb724977024": "b - f(x_0)\\,", "01d17effc3caa2eb0769b9c887809b2b": "G(\\theta|\\alpha)", "01d1a9cb178333516fb523774c0365c2": "y \\,\\!", "01d1b955c6b4390f2d079bce20c322a9": "x - 1 = 1 + \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{9} + \\cdots", "01d1cc0172190c64a713cea6a1b206ca": "C_{123}=2C", "01d20d6ae7c979e280cb6fcb05563978": "IMA = \\frac {F_{out}} {F_{in}} = \\frac {V_{in}} {V_{out}}.", "01d246be21a9a22158d722e1dda3a217": "A_1, \\ldots, A_n \\vdash B_1, \\ldots, B_k", "01d278bc6a56ef0773914beb858779ce": "[(i-1)w, i w)", "01d27ce00111b7f339e9646c360e5a8f": "1/\\ell", "01d2b09e94363278c3ac681bba860bd8": "G(\\vec r,t) = \\frac{1}{4\\pi r} \\Theta(t) \\delta\\left(t - \\frac{r}{c}\\right)", "01d2b84cce753408277d414ef7185571": "Wins = 52 + fWAR", "01d2c43c8adf5d085baf21b62fa2a944": "(c,\\varepsilon)", "01d2dd56f011f08a8d2bae92b224777c": "\n<\\mid F_{in}\\cdot e_{ex}\\mid^{2}> \\propto \\int \\sin \\theta_{1}^{in}d\\theta_{1}^{in}A_{in}(\\theta_{1}^{in}) \\times \\int \\sin \\theta_{ex}d\\theta_{ex}O(\\theta_{ex})U_{ex}(\\lambda_{in},\\theta_{1}^{in}.\\theta_{ex})", "01d34ea7454b3823348e4f8abe9c5b77": "= X \\oplus N',", "01d35b53d4425d9850ede5b316e98ba2": " \\Phi = \\frac{ { k}_{ f} }{ \\sum_{i}{ k}_{i } } ", "01d37fd202fcf8fa4aba57bb5e0e69f7": "\\; P_i \\pi(a) P_i = a", "01d380085acc4d17c2e69127c713199b": "O(2^n) \\bigcup O(n^2)", "01d3a872cf541b472ef41f84273d36e8": " D_H = \\frac{4 \\cdot 0.25 \\pi (D_o^2 - D_i^2)} {\\pi (D_o + D_i)} = D_o - D_i ", "01d415b15fdb845cc85c3ed324f1fbde": "{\\rm d}N {\\rm d} x", "01d42f036bdb98bef530250193b25fa7": "\n \\begin{matrix}\n ^{^b{b}} \\bar a = & \\underbrace{a_{}^{a^{{}^{.\\,^{.\\,^{.\\,^a}}}}}} & \n\\\\ \n & {{^b} \\bar a}\\mbox{ copies of }a\n \\end{matrix}\n ", "01d43272586b34117a0e3de96023a955": "n_1 + \\ldots + n_r = n\\,\\!", "01d44451a24dd18410cb1ed7c2ba5fce": "\\,q_x = d_x / l_x", "01d47a70793565e13c6acbc537e08978": "\\frac{\\partial}{\\partial g_i}(u^{-1}) = -u^{-1}\\frac{\\partial}{\\partial g_i}(u)", "01d4a01c24e9a19b5520a3836a691600": "\\lambda g", "01d4be5fb686cc741ae27340fe0e1539": " P(X_1,\\ldots,X_n)= P_{\\mbox{lacunary}} (X_1,\\ldots,X_n) + X_1 \\cdots X_n \\cdot Q(X_1,\\ldots,X_n). ", "01d4be8c48d4c3f4c375779c2ae1fc92": "E^{\\prime}", "01d58c08ebdb5b0e80ab88f8d72caf12": " p(a,d) \\leq (1+o(1))\\varphi(d)^2 \\ln^2 d \\; ,", "01d58e290042cd241210e2f4f8bef268": "w_r^-", "01d5ec850531d49cb9513324ec9935db": "\\,x_0\\leq x\\leq x_1\\,\\,", "01d66037af654b16d04c660764651244": "\\det S''_{zz}(z^0) = 0", "01d6ff79c4bd0e0ac0d1b6dbd6680846": "n\\# = \\prod_{i=1}^{\\pi(n)} p_i = p_{\\pi(n)}\\# ", "01d71ed39474d8e2ccecd373f1808342": "A\\mathbf{x} = \\begin{bmatrix} \\mathbf{a}_1 \\cdot \\mathbf{x} \\\\ \\mathbf{a}_2 \\cdot \\mathbf{x} \\\\ \\vdots \\\\ \\mathbf{a}_m \\cdot \\mathbf{x} \\end{bmatrix}.", "01d7389dd8daac6fa380ea48d18da2e3": "\\pi_{10}", "01d776eea2c34f8eec530b7f7a7ef049": " A_o = 0.999 \\approx 8 \\ hours \\ down \\ time \\ per \\ year", "01d779db54d10909296a3e0e20fc6c3a": "V = 2\\pi^2 r^3", "01d7eff18535ee23b9a228919c186e21": "I(s) = \\frac{V_{in}(s)}{R + Ls}", "01d830dbc637ebc6eef10832e456861a": " P(X=5) = f(5;50,5,10) = {{{5 \\choose 5} {{45} \\choose {5}}}\\over {50 \\choose 10}} = {1\\cdot 1221759\n\\over 10272278170} = 0.0001189375\\dots, ", "01d84c0b06afc9c27c5264692ec2ee41": " a, b, c \\in N ", "01d85502cacd9acc332bacd50f367f00": "\\left( -\\nabla^{2}_{\\mathbf{u}}+\\frac{1}{4}ku^{2} +\\frac{1}{u}\\right)\\Phi(\\mathbf{u}) = E_{\\mathbf{u}}\\Phi(\\mathbf{u}).", "01d882ec38abc94b1064ee49b0256d5b": "\\mathbf{P}^n", "01d922986dd7527cf78bf949673bfb1f": "A,B,X,C", "01d931498a3d7b6d7e1bc6f0ed6a4a06": " \\begin{pmatrix} x & y \\\\ -y & x \\end{pmatrix}.", "01d941f4013dec7eb7aba37d2dc11780": "\\mathcal{E}(\\rho) = \\sum_{m,n} \\chi_{mn} E_{m} \\rho E_n^\\dagger", "01d9c67c97e8047bf2bfdaa2ad5c8808": "M = \\frac{-f_2}{f_1},", "01d9db4c2459ffc051305ad74e2f4256": "I(\\bold{x}, t;\\bold{\\hat{n}},\\nu)", "01d9fc38090a5436fadf1b8b06471409": "\\frac{\\mathrm{m}/\\mathrm{s}^2}{\\mathrm{Pa}}", "01da8f763bcd3533e23d82c937942e20": "\nz = \n\\frac{1}{2} \\left( A + B - \\lambda - \\mu -\\nu \\right)\n", "01daa0079732e4a1f48600a4a3251a53": "y_n(x)=(2n\\!-\\!1)x\\,y_{n-1}(x)+y_{n-2}(x)\\,", "01dab2020cf38b41842d6c211501b787": "\ns e^{i \\Delta k \\Lambda}=e^{i \\Delta k \\Lambda} -e^{i 2 \\Delta k \\Lambda n}+e^{i 3 \\Delta k \\Lambda}+...+(-1)^N e^{i \\Delta k \\Lambda (N-1)}-(-1)^N e^{i \\Delta k \\Lambda N}.\n", "01dae64584b988a11f4f653b3359640e": "ds^2 = d\\chi^2 + \\sin^2(\\chi/\\alpha) ds_{dS,\\alpha,n-1}^2,", "01daebc5b411677123fc9f4734fa8fed": "\nC_1^+(\\beta) = \\frac{\\alpha}{2} \\log \\left( 1 + (c_{31}^2 + c_{21}^2) P_1^{(1)} \\right)\n + \\frac{1-\\alpha}{2} \\log \\left( 1 + (1-\\beta) c_{31}^2 P_1^{(2)} \\right)\n", "01db16108e95588e314e7db20af284b5": "z(m,n;s,t) < (s-1)^{1/t} (n-t+1) m^{1-1/t} + (t-1)m.", "01db34fef6aa29ed0a4092f1812ca6d3": " M = \\left\\{ (a,b); a=b; a \\in A; b \\in B \\right\\} ", "01db3f2f5f32c0e5476bacd9e378b24d": "F(\\overrightarrow{x},s)", "01db8fd1c607c20f073a9e4e01267aed": " \\nabla^2 f(x) ", "01dba1731ae06e01d5e4cb38b470dbcc": "\\frac {f{(x)}-f{(-x)}}{2}", "01dbd8419c18df8b5400d24cd60ab691": "q=\\frac{\\sqrt{Fb}}{b}", "01dc19e3571d9dfc66ab0771f91f5180": "(\\alpha_0,\\beta_0,id)", "01dc1d552c5547bade52f5f9c8d22afb": " Q=Q(p) ", "01dc276d84de2a77b12d92dcd2d354b2": "2\\log k", "01dc4239e20dd0a7c6cccfd8ddf4e7f0": "[J_{ij},Q_a] = \\frac 14 (\\gamma_i\\gamma_j-\\gamma_j\\gamma_i)_{ab} Q_b,", "01dc43fb7bd88609eb84d081d609513f": "{}_1F_1(0;b;z)=1", "01dc5007081749b7a310feccf1354232": "\\lambda_1^k,\\lambda_2^k,\\dots,\\lambda_n^k", "01dc58ec3ac830294a6a937ae668cff7": "\\hat e(\\mathbf{s}_0)", "01dc619881fa5961b4ecdd8bcfe256b5": "(k_{f_1},k_{b_1},k_{f_2},k_{b_2})", "01dc735e3025852bf1c8ab7517a735d7": "r_2,\\ p_1", "01dc774e9c2c01320bd7e31b53d233f7": " \\mathbb{Q}[Y_1,\\ldots, Y_s]. \\, ", "01dc82dd0f686daf69ba2dfbc1edd95c": "\\,\\langle P_W\\rangle", "01dccf6774f6bbc2a9ae99375f9b7a91": "\\phi:\\mathcal{G}\\to\\mathcal{N}", "01dd07ab53a078a180fd9b599836ded6": "t_r", "01dd6d12c7f43ffb157c0c0a7b3ad810": "(z_1, z_2; z_4, z_3) = {1\\over\\lambda}", "01dd7550747902bc9e4a872463a3fd20": "\\lfloor p/m \\rfloor", "01ddd304f3cc045b31ac39874c845209": "C=\\frac\\sqrt{\\mu^2c^4-E^2}{\\hbar c}", "01de27534bbeea30c00ebfa9d73e5366": "so(3, 1)", "01de33337a5e025911424042a1359e86": " C = -\\frac{dC_v(K,\\sigma(K))}{dK} = -\\frac{\\partial C_v}{\\partial K} - \\frac{\\partial C_v}{\\partial \\sigma} \\frac{\\partial \\sigma}{\\partial K}", "01de5d01b9bb6c5ea26b690f212d9b04": "n(\\mu) \\propto e^{\\mu/k_B T}", "01def30326acf780125644d83affad21": "E_{yz,3z^2-r^2} = \\sqrt{3} \\left[ m n (n^2 - (l^2 + m^2) / 2) V_{dd\\sigma} +\nm n (l^2 + m^2 - n^2) V_{dd\\pi} - m n (l^2 + m^2) / 2 V_{dd\\delta} \\right]", "01df00b61d5692071c8cbfb211a02dfa": " \\sin iy = i \\sinh y. \\,", "01df03d7ac1229038aa710b8743b3fbe": " \\sin\\left( \\pi/2-\\theta\\right) = \\cos \\theta", "01df2292ddf37ed672196cd88db9568c": "\\mathrm{gain}_{\\mathit{TE}}=a_{\\mathrm{form}} \\cdot (1 - a_{\\mathit{vf}})", "01df2eeb9b53add0e1612df83eefdc35": "X_\\alpha, and X_\\delta", "01df4445f0f3bae5903887db6e0805b0": "\\partial_t i(t,a) + \\partial_a i(t,a) = s(a,t)\\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} -\\mu(a) i(a,t) - \\nu(a)i(a,t) ", "01dfd632e0c8ed78c80b807404bd8a38": " L(t) = \\mathbf{R}^* + t\\vec{k}.", "01dff6f37fbd572a4603f7672037ad3a": "[F_{\\lambda}]", "01dffc9159b2ed3efc44c711463dc491": " \\text{HSIC}(X, Y) = \\left| \\left| \\mathcal{C}_{XY} - \\mu_X \\otimes \\mu_Y \\right| \\right|_{\\mathcal{H} \\otimes \\mathcal{H}}^2 ", "01e010f88dff08d008c6d62171d214ca": "U_\\beta = \\left(U_0, U_1, U_2, U_3 \\right) = \\gamma \\left(-c, u_x, u_y, u_z \\right) \\, ,", "01e022402b9df1bc43a30582c69795f7": "p(4063467631k+30064597)\\equiv 0\\pmod{31}.", "01e0258e23148f019a0b12b93d87c9d1": "I=I_S\\left(e^{V_D/(nV_T)}-1\\right)", "01e0415e98dbe03a58400cd4f881e666": "\\boldsymbol{m}\\cdot\\boldsymbol{N}=0", "01e0b28d6603dd6d5ca5fc5502075ec9": " \\frac{a L}{R}", "01e0e8dabc4ce2ac3887ce67a33f1296": " \\Box_2 P ", "01e108d0a2ec9beb42187c4278af1be4": "P_{D-}", "01e1586a49ee5e8cb6148aaade4882f2": "\\zeta(s)=\\frac{1}{s-1}+\\sum_{n=0}^\\infty \\frac{(-1)^n}{n!} \\gamma_n \\; (s-1)^n.", "01e19fdfa95d98838f56f6a0b6f84126": "{\\tilde{BC}}_2", "01e1f9190d77917fa124a3d4ead6a8c4": "Q_{in:friction} = C_d \\rho |\\mathbf{u}|^3", "01e21538af1d452befee558b81565532": "{}^3_2\\mathrm{He} + {}^2_1\\mathrm H \\to {}^4_2\\mathrm{He} + \\mathrm p", "01e26383848cb410a14cb8b3d2b92239": " A = (a_{i,j})_{1\\leq i,j\\leq d} ", "01e2a690c2a1b53a21a63bf4493e6cc6": "[(a,b)] - [(c,d)] := [(a+d,b+c)].\\,", "01e2db5f559085fb07c80e964a30ef0e": "P_3 = (X_3,Y_3,Z_3,ZZ_3) = (48-8\\sqrt{39},296\\sqrt{3}-144\\sqrt{13},2,4)", "01e323cb5b79d946495eab0fd0f6a9c9": "\\begin{bmatrix}\n \\cos\\beta\\cos\\lambda\\\\\n \\cos\\beta\\sin\\lambda\\\\\n \\sin\\beta\n\\end{bmatrix} = \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos\\epsilon & \\sin\\epsilon\\\\\n 0 & -\\sin\\epsilon & \\cos\\epsilon\n\\end{bmatrix}\\begin{bmatrix}\n \\cos\\delta\\cos\\alpha\\\\\n \\cos\\delta\\sin\\alpha\\\\\n \\sin\\delta\n\\end{bmatrix}", "01e3cacd2b9c2b395a3126c85a799f03": "\n \\boldsymbol{\\tau} = \\varphi_{*}[\\boldsymbol{S}] = \\boldsymbol{F}\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{F}^T~.\n", "01e44b0dc54b8b019e635f7283b75df2": "\\mu^{\\otimes 0}(A_0(s,t)):=1.", "01e46395993aa5169a4f46c994e057c2": "\\textstyle(x\\pm1, y, z\\mp1)", "01e4a15095bab293c07843429213637e": " \\ddot{q} = M^{-1}Q + M^{-1/2}\\left(AM^{-1/2}\\right)^+(b-AM^{-1}Q). ", "01e4b0c7be863d94cb74865e74285978": "%B=\\frac{f_H-f_L}{f_c}=2 \\frac{f_H-f_L}{f_H+f_L} ", "01e4b9416f3b7a700735850d73bbd049": "\\lim_{n\\to\\infty} a_n = L.", "01e51ae055d2edd7e4320fe80ffe1073": "F_0(x) = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L(y_i, \\gamma),", "01e53e5c0b2839cf0c169069276f73e5": " | \\rho | = \\frac \\mathrm {VSWR-1} \\mathrm {VSWR+1}", "01e640b0a6ced27eeac99f6f1da9bb05": " X \\sim N(\\mu, \\sigma^{2}) \\!", "01e649334a2c1ed88b47ade97a8c785f": "\\Omega\\equiv\\frac{d^3\\varphi}{dz^3}+i\\alpha Re\\left[\\left(c-U\\left(z_2=1\\right)\\right)\\frac{d\\varphi}{dz}+\\varphi\\right]-i\\alpha Re\\left(\\frac{1}{Fr}+\\frac{\\alpha^2}{We}\\right)\\frac{\\varphi}{c-U\\left(z_2=1\\right)}=0,", "01e671dbd13bce2f51b07af455e57608": "x_{i,m+j}\\geq 0", "01e6ccfc99e178a8c5bc8f927841d736": "u(x,t)=\\frac{\\lambda}{4}\\int_{E_\\lambda}u(x-y,t-s)\\frac{|y|^2}{s^2}ds\\,dy,", "01e74f04804f931cc50fbfa868d0eaf6": "\\,K_{1B},\\ K_{2B}", "01e74f89e4d6421b5c028282fe6fbf4e": "[e]=\\{f\\in E|f\\leq e\\}", "01e77ba3199f76d686f03552d12c79b2": "\\vec{v}(t + \\Delta t)", "01e78043796bc55062f208abf997af9e": " \\lambda x.\\operatorname{drop-formal}[D, \\lambda o.\\lambda y.o\\ x\\ y, F] ", "01e78c601610f6c7b2a224a6cfb15dd2": "b_{\\nu, n}(x)", "01e8066e145a375d8f6910bb91bc45ec": "(s-1)\\zeta(s) = \\int_{-\\infty}^\\infty \\frac{(1/2 + i t)^{1-s}}{(e^{\\pi t}+e^{-\\pi t})^2} \\, dt.\n", "01e8153ecd79d5daf0df9fc8579edd9e": "\\mathfrak{H}_b", "01e86ced95c51596f778d74df8c8bf96": "V=1096.7 \\sqrt{H/d}", "01e9b4e5ba85de9ac8931c518c75329d": "\\scriptstyle s_{\\infty}(x)", "01ea358477bd18b369f5831702e6e4a7": "\nF_{hkl} = \\begin{cases} 4f, & h,k,l \\ \\ \\mbox{all even or all odd}\\\\\n 0, & h,k,l \\ \\ \\mbox{mixed parity} \\end{cases}\n", "01eaaa17d9dce7e235b677bc79046182": "\\sigma^2=\\lambda^{-1}", "01eadb9d7afc7f715e95d21f4ade3bb0": "(p,\\, t) = (i,\\, 2j+i)", "01eb0cdc32ef16bfba610e677a4823ca": "\\textstyle\\sqrt{e}", "01eb240e2bfb2732e6941810498adfb2": "((p_x q_w-q_x p_w)^2+(p_y q_w-q_y p_w)^2,\\mathrm{sign}(p_w q_w)(p_w q_w)^2)\\,.", "01eb3a530c08e2193c36adc9fab5107d": "y_2 = \\left. \\frac{\\partial y}{\\partial c}\\right |_{c=\\alpha} = a_0 s^{\\alpha }\\sum_{r=0}^{\\infty } \\frac{(\\alpha )_{r}(\\alpha +1-\\gamma )_{r}}{(1)_r (1)_r}\\left( \\ln(s) +\\sum_{k=0}^{r-1} \\left( \\frac{1}{\\alpha +k}+\\frac{1}{\\alpha +1-\\gamma +k}-\\frac{2}{1+k} \\right) \\right)s^r", "01ecb5cec1a178baac07c1d3161bbe12": "B_\\lambda(T) = \\frac{2 c^2}{\\lambda^5}~\\frac{h}{e^\\frac{hc}{\\lambda kT}-1} \\approx \\frac{2c kT}{\\lambda^4}", "01eccb8e17d972949e03580c41d08994": "(X,Z)", "01ecefd8e8946da30610fe9a89d437e0": "c_{T-2}(k) \\, = \\, \\frac{Ak^a}{1+ab+a^2b^2}", "01ecf76e7b919e8f093d393b99d25b96": " |x_1|=|x_2|=\\cdots|x_n|=1", "01ecfad7082922f85b35330787b6a893": " I = I_{cont} \\cdot \\frac{1 + K_n}{1 + 1.71 K_n + 1.33 {K_n}^2}", "01ed60cad5987fe9b72dfafdb6998db4": "\\mathit{ARA}(w) =-\\frac{u''(w)}{u'(w)}", "01ed7a9778d320559052bb613ab06943": "\\varphi_{X+Y}(t)=\\varphi_X(t)\\varphi_Y(t)=(1 - \\theta\\,i\\,t)^{-k_1}(1 - \\theta\\,i\\,t)^{-k_2}=\\left(1 - \\theta\\,i\\,t\\right)^{-(k_1+k_2)}.", "01edb4f49ec6aa80e62fa89946994808": " p \\times 1 ", "01edc57c51203044a554ae8a187fc31e": "X \\sim {\\rm Beta}(\\alpha, \\beta)", "01edc5ac6e7a583842e808f0ac05b1f3": "\\sum_{i \\in I} a_i X^i ", "01ee071d0ac5779eb2dd04415cac4812": "\\max_{x \\in S_{k-1}^{\\perp}, \\|x\\| = 1} (Ax, x) \\ge \\lambda_k", "01eea2c97b7016f8b1d32cec91e85538": "I,J", "01eec55e6318535a8351f82099461fc9": "H + 1 , H + 2 , H + 3 , H + 4 , ... , H + k", "01eecc08088d2dd3a1f402ce7f92772b": "\\eta =\\dfrac {\\pi Ze^{2}m^{1/2}\\ln \\Lambda } {\\left( 4\\pi \\varepsilon_{0}\\right) ^{2}\\left( k_{B}T\\right)^{3/2}}", "01eed3297ad06ed2478e3279d7c7ae69": " AH = t\\ \\text{Crd}\\ 10^\\circ \\approx t\\ \\frac{600}{3438}", "01eee35e8a902584c0b63d1d8bb80ebc": "\\sum_{n=-\\infty}^\\infty |c_n|^2 < \\infty", "01ef7a7dc58a56553149a519ca69a021": "N = \\frac{1}{\\sigma (C + D)}", "01efcc04cd663bb90911383a56399190": "X_R", "01efea36ed99f50ede86d8fdabd95ab9": "F(x): \\mathbb{R}^n \\to \\mathbb{R}^n", "01eff3a47e7e5d237fbf738a52537ca9": "R_e = \\frac{\\max\\left\\{ \\left| \\boldsymbol{U}_p - \\boldsymbol{U}_f \\right| \\right\\}\\, d_p}{\\mu/\\rho_f}", "01f01f007d2da6d2e11e1a1078602332": "D'= [P] + [R] - 2[O]", "01f0a3e33029e37179c066622a70be96": " \\int_E |f| d\\mu < \\epsilon ", "01f166c2df9b362185cbfb587b145efb": "\\frac{1}{q}=-\\exp(\\pi \\sqrt{163})", "01f19e23d7a338320ccc53e6f461c601": "f_{\\text{Aeolian}}\\ =\\ \\frac{\\alpha v}{d}", "01f1aa9773a2bc7c9abd38f608c57ae7": "xx^T - \\Delta \\in S_+", "01f1b5844156ea62392e3fe67819686a": "I(X, Y)", "01f1c233a51e9d045b83d50e5426de86": "a_{k+1},\\dots,a_n", "01f1f0ba6ee5f9907d32c0a36befefe2": " P(\\partial_t)G = 0, \\; \\partial^j_t G(0) = 0, \\quad 0\\leq j \\leq m-2, \\; \\partial_t^{m-1} G(0) = 1/a_m. ", "01f1ffe110c901fcfaefbb12c9e9960f": "\\scriptstyle \\leq1.9\\times10^{-33}", "01f266d4782c987e450bbaa0c56f9353": "1{\\to}\\tau", "01f3a391a61df4f8bf52765c05d92877": "a_{ii}", "01f3c699a2735a0d9a7311d672fd676c": "n_p", "01f414ce69bc416ef26e3b1aa09a3efc": "\\forall x, y, z\\,\\left(xFy \\wedge xFz \\to y = z\\right)", "01f41e5176fe1b6de7af480700737b0f": "E_{\\rm barrier} = W_{\\rm e}", "01f45976384f297b8e2d9f5229576785": "\\Delta = \\{\\alpha_1 \\ldots \\alpha_n\\}", "01f46efd1c4daed220ee2b124342dffa": "\\Delta g_{AB}=O_{B}^{crys}g_{B}(O_{A}^{crys}g_{A})^{-1}", "01f481a88cc19ffe6d6db95ccaa8dd92": "\\tilde{N} < N", "01f48294df6f44567d8f296da1a45a43": "F\\,= \\rho u A", "01f4e1ed87059e734917a9565f6ddf94": "\\tilde{H}_u", "01f4ea6adcaca1e0780939c0de27db0f": "\n\\begin{align}\nW' &= y_1' y_2' + y_1 y_2'' - y_1'' y_2 - y_1' y_2' \\\\\n& = y_1 y_2'' - y_1'' y_2.\n\\end{align}\n", "01f4ef7625842d787dd646758c3b1cf4": "\\operatorname{SO}(\\mathfrak{g})", "01f50df09f313f99b927b433e5677de5": "S(\\phi,\\mathbf A) = \\int {1\\over 4g^2} \\mathop{\\textrm{tr}}(F^{\\mu\\nu}F_{\\mu\\nu}) + |D\\phi|^2 + V(|\\phi|)", "01f524f17e4b4fcb00157c2698aca042": "u_{xy}=u_{yx}.", "01f53789ea66e1ffe67f4ac57dba6499": "y = \\frac {\\pm \\pi \\left(P Q - A \\sqrt{\\left(A^2 + 1\\right)\\left(P^2 + A^2\\right) - Q^2} \\right)} {P^2 + A^2}", "01f5710b2f73e55be4aadfba476a45c8": "\\Delta (x)", "01f5eaffb4b61b9fd64f74b251ece7db": "\\scriptstyle u,v\\in BV(\\Omega)", "01f628b27e8860217ef5fc754e8d60a6": "f_2(1) = 1 \\quad\\text{and for}\\quad d|k,\\; d>1,\\quad \\sum_{m|d} f_2(m) = 0.", "01f63496dff248313e3d9395692dbf61": "f_\\ell^m", "01f64d98287d4a6cfeaf14b94c993ba1": "\\partial_{-}C", "01f65fc413190c418d946b3c95119447": " u^2 - dv^2 = \\pm 4 \\, ", "01f6b15a5434d848c8b6899052b997b7": "\\scriptstyle x \\;\\in\\; W", "01f70036cfc9760ed393feb3b4fd8ad6": "\\scriptstyle \\cos \\theta_c = \\frac{c}{nv}", "01f708ec8a33bf3b68b15d3462a5fc8b": "a = \\left ({\\text{COMP} \\over \\text{ATT}} - .3 \\right ) \\times 5", "01f709eea689f82ea1ea61ca3c385613": "\\beta^{a} \\beta^{b} \\beta^{c} + \\beta^{c} \\beta^{b} \\beta^{a} = \\beta^{a} \\eta^{b c} + \\beta^{c} \\eta^{b a}", "01f70a960eb91ed4f3aadeab35b6deb4": "\\dot{z} = -2z (\\alpha + xy), \\, ", "01f788399c97985044f2437a18aab69e": "|\\Phi^-\\rangle ", "01f78be6f7cad02658508fe4616098a9": "550", "01f7c18c56f6d93726f78c234d1868da": " ((P \\or Q) \\and \\neg P) \\to Q", "01f824346fd27a8e5ae32409c29ab9e0": " a_n \\ne 0 ", "01f864dc442db64bf93663760fa8dae7": "\\begin{align}\n\\Vert\\vec a\\Vert^2 & = \\Vert\\vec b - \\vec c\\Vert^2 \\\\\n& = (\\vec b - \\vec c)\\cdot(\\vec b - \\vec c) \\\\\n& = \\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - 2 \\vec b\\cdot\\vec c.\n\\end{align}", "01f8b80e36b662229cbd834a93134c87": "\\textstyle u\\in W_p^k(\\Omega)", "01f8cede02e588da726936d313dcaa9b": "\nP(\\vec x|\\vec y) = \\frac {1} {(2 \\pi)^{m n/2} | \\boldsymbol {S_x} |}\n\t\\exp \\left [ -\\frac{1}{2} (\\vec{x} - \\widehat{x}) ^T\n\t\\boldsymbol {S_x}^{-1} (\\vec{x} - \\widehat{x}) \\right ]\n", "01f8f7e003bf6961951efb20b8a6959e": "\\gcd(a_1, a_2) = 1", "01f93ef84b3860edd2c0508453d523ee": " \\Delta_rG^{\\ominus} = -RT \\ln K_{eq} ", "01f94a2e8b3a86a1eac37f3a307d74ef": "\\left( \\lambda_i \\right)", "01f9b8831d5ce67ce115b33c7d1e9478": "Q = f(X_1,X_2,X_3,\\dotsc,X_n)", "01fa15b00eab23e5d544b290e9299048": "550 P_e = \\frac{\\eta_c H h J}{3600},", "01fa5ded58e5d08e631aba5bd2b0feb1": "\\{x\\}_{1}\\equiv \\min(x,1)", "01faaf3be3d2ed3aa7ecd4f6850926b9": "a\\frac{\\partial \\mathbf{U}}{\\partial x}", "01fae99ca641d883ac858c905d86728e": "c \\equiv z^Q \\pmod p", "01faf716f16570e46fec6b9b0d42144b": "f(x, y) = x^2+y^2 - L^2=0,", "01fb2beb7ef70ed58c2ce56badc91b74": "{\\mathfrak\ng}", "01fb56ab71a1da87b572193a63a2feba": " -\\dot{\\hat{S}}(t)=1/2 \\left(\\tau'(t)\\Psi_2(t)+\\Psi_2(t)\\tau(t) \\right),\\hat{S}(T)=0, rank(\\hat{S}(t))=n_r", "01fb78309dc15b8c8b7bf1bc935d2ee1": "\\begin{smallmatrix}M_v\\ =\\ m + 5 (\\log_{10}{\\pi} + 1)\\ =\\ 0.03 + 5 (\\log_{10}{0.12893} + 1)\\ =\\ 0.58.\\end{smallmatrix}", "01fb9a99551dc0d48536ac23ef87c14e": "\\sum_{j = 1}^n x_{ij}\\leq W_i \\text{ for }i = 1, \\ldots, m, \\, ", "01fc58c8b0da0e07d6945f090fb567a1": "P(d)=\\log_{b}(d+1)-\\log_{b}(d)=\\log_{b} \\left(1+\\frac{1}{d}\\right).", "01fcc590495900b89daf89ded70ece09": "\\frac{d}{dt}(x^2+y^2)=\\frac{d}{dt}(h^2)", "01fced4faaa49a4d66f16eb26a0f1e8c": "\\langle f,g \\rangle = \\int_0^\\infty f(x) g(x) e^{-x}\\,dx.", "01fd1ad8daecc094e7dadd6a86273241": "\\tbinom42", "01fd4990d79a022c9f0f6ddb6c474e72": "\\geq _i", "01fdc5c5a4963039312de9a5909dae41": "\\mathbb{R}^d ", "01fde0360ee4e92ea642bfb8db1c042a": " t = t_4 = 2 ", "01fde5258ca4a48d85b73df2431b1c83": "L= \\frac{\\Pr(1)}{\\Pr(-1)} = 1", "01fdf8295daff5a8c956e998c84a1ab0": "e_{(1)}=\\frac{1}{\\sqrt{4+2(x^3)^2}}\\left[ \\left(x^3-\\sqrt{2+(x^3)^2}\\right)\\partial_0+\\left(1+(x^3)^2-x^3\\sqrt{2+(x^3)^2}\\right)\\partial_1+\\partial_2\\right]", "01fe027d59aa17835a0670a9d11d416a": " M_1 = f, N_1 = q\\ q ", "01fe37d9e5cac4cfc89965f899710fa9": " |\\boldsymbol{\\Omega} | = \\frac {d \\theta }{dt} = \\omega (t), ", "01fe48e9996766b42771f70a1bddd9df": "x_1=X_1/Z_1", "01fe558ce89cef29447b50d1c9a2454d": "\\scriptstyle\\tau_s\\,\\sim\\,10^{-6}", "01fe9cac15c05ddb569271027aa28cdf": "C_{3}", "01feeca3ca3b39eaf174f3e80a0bfb08": "O_i(v)", "01ff9831f25527e34621442ec94c296f": "E = hf.", "01ffcf4a001f4377b9230f06043102af": "\\left({\\mathit{He}}_n^{[\\alpha]}\\circ {\\mathit{He}}^{[\\beta]}\\right)(x)=\\sum_{k=0}^n h^{[\\alpha]}_{n,k}\\,{\\mathit{He}}_k^{[\\beta]}(x)\\,\\!", "020018fbc60643a41b9e6556782676f7": "H=\\begin{pmatrix}0 & -i\\\\ i & 0\\end{pmatrix}", "0200643b433a73480343668a47e713b3": "2\\pi i \\xi", "0200653e29381832b95d44a03206abe1": "\\Omega(\\alpha^{-i_k}).", "02008f14e8257624a6629c3fcf01da8f": " y=\\frac{\\int xe^{-x}}{e^{-x}}", "0200bc9485f667875f6505fff4142a32": " \\alpha=\\left(\\frac{D}{R}\\right)\\left (\\frac{\\partial f}{\\partial y}\\right)", "0200cf69d44dc36712c52a3e3981910a": " \\mathbb{R}^m ", "0200dac127ea6040113c5129053902bb": "\\alpha = 2: \\quad \\operatorname{E}\\left [- \\frac{1}{N}\\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha \\partial a}\\right ] = {\\mathcal{I}}_{\\alpha, a} ", "02013e1085d9c40ceb24d4dcfe30ea95": "[P_\\mu, P_\\nu] = 0\\,", "020158f273ee5b33f137179c93aaeb98": " \\frac{ i \\Omega }{ 2\\pi} ", "02017cb282b7b8578298acc062ceb4e3": "\\Delta \\omega_2\\ =\\ -\\cos i\\ \\Delta\\Omega \\ =\\ 2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos^2 i\\,", "0201cd4a7d2672f6df21747ba08cc2db": " \\alpha = 1,2\\,,\\dot{\\alpha} = \\dot{1},\\dot{2}", "0201e01c5002bfe328e7411a47d24dfa": "b_{MP}", "0201e8827a113f4f24e40b69706103df": "\n \\dot{\\mathbf{f}}(\\mathbf{x},t) = \n \\frac{\\partial \\mathbf{f}(\\mathbf{x},t)}{\\partial t} + [\\boldsymbol{\\nabla} \\mathbf{f}(\\mathbf{x},t)]\\cdot\\mathbf{v}(\\mathbf{x},t) ~.\n", "02025a1490050e9d1a58211869ac18ad": "0 = \\tau_{0} < \\tau_{1} < \\cdots < \\tau_{N} = T \\mbox{ and } \\Delta t = T/N;", "020289bdba9a5fc746fab9a3dc637da0": "-\\frac{1}{2}\\left[N(x + \\Delta x, t) - N(x, t)\\right]", "02034ec46591073018d6dbdcf4b653c3": "[\\Sigma Z,X]", "0203ad3d019cbfbc650562b9c791af13": "S(\\widehat{g})=\\int_P R(\\widehat{g}) \\;\\mbox{vol}(\\widehat{g})\\,", "0203c4d906c07569b9177cf884cf4601": " (\\mathbf{y}')^{T} \\, \\mathbf{E} \\, \\mathbf{y} = 0", "02049ecc75727af40b1a127c3547ecad": "A= \\begin{bmatrix} 1 & 2 & 0\\\\0 & 2 & 0\\\\ 0 & 0 & 3\\end{bmatrix},", "0204a441c70e3b74dda69b9dfbe5531c": "\\kappa' = \\frac{h\\nu}{4\\pi}~(n_1 B_{12}-n_2 B_{21}) \\,", "0204bcb90858077f463000cb8d1caa7f": "1 \\times 10^{-9}", "0204fcae90c7db37cec6e71af85f4ae2": " {1\\over D_0 ... D_n } = n! \\int_{\\mathrm{simplex}} {1\\over (v_0 D_0 +v_1 D_1 ... + v_n D_n)^{n+1}} dv_1 dv_2 ... dv_n ", "0205592edc62f25eb27d8c8e385d75c1": " \\tau = 50 + 0.6\\sigma _n", "0205d7a21e1ba59606ed6215d1ba84ca": "\\Sigma X = (X\\times I)/(X\\times\\{0\\}\\cup X\\times\\{1\\}\\cup \\{x_0\\}\\times I)", "02061f46096bfadaf285ce34044c0bb6": "y \\succeq z ~\\forall z \\in B'", "02063a9756469712d13d8db5ef2b90af": "\\begin{matrix} \\frac{8}{5} \\end{matrix}", "02066b25f031c16743e7183b4f47aa32": "x^2y''+xy'+\\left (x^2-\\nu^2 \\right )y=0", "0206d8fd533aeb1efbae23598b7752c5": "\n u(x) - u_\\epsilon(x) = O(\\epsilon^2), \\quad 0 < x < 1\n", "0206f3c7893217c7e70b760392a3580a": "\\Delta\\mu = \\Delta T - e \\Delta \\phi = 0", "02076d8f964e19ea76719f7b7fa4c54b": "\\| \\cdot \\|_{L^{p} (\\Omega)}", "02079b7cc8d61d1266ecc3968c9e0b67": "h(x):=\\lim_{\\delta\\to 0} \\frac{1}{\\delta}\\Pr[x < X \\leq x + \\delta | X > x]", "0207d4c83228fd7956a87bc94fb66bc2": "B(\\rho,\\tilde{p})", "02081576a3bc4a07bd86dcbeff6dc169": "p_A = p_B", "0208ceecd9a3efb97ebc79813aa56e3f": " \\mbox{CNOT} = \\begin{bmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&0&1\\\\0&0&1&0\\end{bmatrix} ", "0209231423580dbddef190641b0dbb33": "\n\\begin{align}\n(Tf)(x) = {e^D - 1 \\over D}f(x) & {} = \\sum_{n=0}^\\infty {D^n \\over (n+1)!}f(x) \\\\\n& {} = f(x) + {f'(x) \\over 2} + {f''(x) \\over 6} + {f'''(x) \\over 24} + \\cdots ~.\n\\end{align}\n", "02096cceea00a25d136b7df9be53e74b": "E-E_{eq} = a - b \\log(i)", "0209a79418e85433101d56bf370871e8": "P(s^2+st)\\cdot P(t^2)=P(t^2+st)\\cdot P(s^2)", "0209c5dba895a295b64d5cd10e412979": "^b", "020a13ff8c9833908347dc24fcb38981": "\\sigma(K)", "020a3fee1e9ad37dd2d9f5e874cce0dd": "r=\\frac{4\\pi\\hbar^{2}n^{2}\\varepsilon \\varepsilon _{r}}{q^{2}m^{*}} \\; \\; (4)", "020a65bc08570b1375de0229ebd438c9": "\n{\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial s} = \\sum_k \\left(\\sum_{i=1}^3 h_{ki}~{\\partial q^i \\over \\partial s}\\right) \\mathbf{e}_k ~;~~ \n{\\partial \\mathbf{x} \\over \\partial q^j}{\\partial q^j \\over \\partial t} = \\sum_m \\left(\\sum_{j=1}^3 h_{mj}~{\\partial q^j \\over \\partial t}\\right) \\mathbf{e}_{m}\n", "020ab50253f4b02b502d59ed210fdfa5": "\\sigma \\left(e^{\\frac{-\\alpha+\\sqrt{\\alpha^2+\\beta\\log{16}}}{2\\beta}}\\right)", "020ab726931c19275865811cf4641d23": "AM^{-1/2}", "020b818d3824f4c951d41124d1faf01f": "F_c\\,", "020bcccbcc330eba43647a35337c3b4b": " A_{kl}[\\nabla]=\\frac{1}{\\rho} \\, \\partial_i \\, C_{iklj} \\, \\partial_j\\,\\!", "020c2838568f25652b3a81cff1c9af84": "\n\\begin{align}\nH_0&=1+\\frac{n^2}{4}+\\frac{n^4}{64}+\\cdots\\qquad\\qquad\\qquad &\nH_6&=\\frac{35}{48}n^3+\\cdots\\\\[8pt]\nH_2&=\\frac{3}{2}\\left(n-\\frac{n^3}{8}+\\cdots\\right) &\nH_8&=\\frac{315}{512}n^4+\\cdots\\\\[8pt]\nH_4&=\\frac{15}{16}\\left(n^2-\\frac{n^4}{4}+\\cdots\\right)\n\\end{align}\n", "020ce2605d01f04976dde1bf02898e01": "\\eta_h = \\frac{\\pi}{2\\sqrt{3}} \\approx 0.9069.", "020d6bb14fd92378223068b95a273811": " P_c=\\gamma\\left (\\frac{1}{R_1}+\\frac{1}{R_2}\\right)\\!", "020d7e9916dbc07f176b19924f410686": "1+r=(1+i)/(1+\\pi) \\approx (1+i)(1-\\pi) \\approx 1+i-\\pi", "020d7ea57aafa51aeb9616ae25a4deef": "z_i\\neq z_j\\quad", "020e2809fcd895b31e4d5f9942b900d1": "\\sigma_{r t} = \\frac{1}{\\mu_0} B_r B_t - \\frac{1}{2 \\mu_0} B^2 \\delta_{r t} \\,.", "020e37332fc37169cb038657a020ff1b": " a^2 = \\frac {T}{\\rho_{solution} g}", "020e7ba90d2fd623965329b56e0f6a6a": "t_o = \\frac{t}{\\gamma},", "020e8f4ceae6ff5b053183b573b5f9fa": "D = b^2-4ac", "020ea27d91a8ee4ec996c7d823adfbc3": "C:a+bx \\rightarrow a + \\omega bx\\,", "020eb6226721a0d3ded3968d8ad8165a": " \\mbox{tr}^2\\,\\mathfrak{H}= \\mbox{tr}^2\\,\\mathfrak{H}'.", "020f083998200b2c752bf46fe39dce27": "\\frac{\\Gamma \\vdash \\alpha \\rightarrow \\beta \\qquad \\Gamma \\vdash \\alpha}{\\Gamma \\vdash \\beta}\\quad\\text{Modus Ponens}", "020f335054558fdff4f387056e345abb": "\n D\\,\\left(\\frac{\\partial^4 w}{\\partial x^4} + 2\\frac{\\partial^4 w}{\\partial x^2\\partial y^2} + \\frac{\\partial^4 w}{\\partial y^4}\\right) = -q(x, y, t) - 2\\rho h\\, \\frac{\\partial^2 w}{\\partial t^2} \\,.\n ", "020f5b0b54d45f3933659d20b4b8901d": "n_1=0\\,\\!", "020fa72193e9f80f8e86a914a89ede7a": " dQ_c = T_cdS_c ", "020fce0965fdbd8eacd65d8c5e7f735f": "\\text{Qv}", "021064cb27a88a2fae850a9fe57034df": "J=1", "0210b7fe254ee8a721aaf0c418a6199b": " \\langle \\mathrm{d} f,X_{g} \\rangle = {X_{g}}(f) = \\{ f,g \\} = - {X_{f}}(g) = - \\langle \\mathrm{d} g,X_{f} \\rangle ", "0210db18cab5ffed55e6f049b2fa4f3d": "x^2-2x-2", "0211181d10cf569cc3d19a52820c511c": "I_2\\to I_1", "02115da4b2995df7446571d92f05311d": "[D] \\cdot C \\geq 0", "0211867b79048bdcee8fc6a90f152e6a": "^{*}\\!H", "0211b2e435b909cc70950f4fcc598b49": "\\displaystyle S_{PQR} = S_{ABC} - S_{ARC} - S_{BPA} - S_{CQB} ", "0211c922a052804e564f1efc1e2421c9": "S(\\boldsymbol{\\beta}) = \\sum_{i=1}^{m}\\bigl| y_i - \\sum_{j=1}^{n} X_{ij}\\beta_j\\bigr|^2 = \\bigl\\|\\mathbf y - \\mathbf X \\boldsymbol \\beta \\bigr\\|^2.", "0211e14fbe450ba44f2fb225d7d00b04": "\\sqrt{\\lambda_1}", "0211eb04eab4ff94e9660c0fd989a0a2": " x \\in S | x\\leq a ", "02121de6d4ac8dfb9e1f7f93345e0368": "\\frac{V_\\mathrm{out}}{V_\\mathrm{in}}=\\frac{10}{1}=10\\ \\mathrm{V/V}.", "021243b9d14264da9db22721350ba73b": " \\mathbf{x}_\\text{p}(t) ", "02126e65a1ef1c21549d2c40cea26d1d": " \\frac{100}{2+2} = 25 ", "02128bd13bcbf456f93f4482b09b34ea": "\\mathbf{Z}(p^\\infty)=\\{\\exp(2\\pi i m/p^n) \\mid m\\in \\mathbf{Z}^+,\\,n\\in \\mathbf{Z}^+\\}.\\;", "02129029f8a82c8440d0197aa5c9f513": "(3+2\\sqrt{2})/6 \\approx 0.971.", "02129bb861061d1a052c592e2dc6b383": "X", "0212a55821995d1dc111723616ae41d0": "V_\\mathrm{th}", "0212c3b4e7f92cca974995c579ceb1c3": " \\lim_{p \\to \\infty} \\; cr(K_p) \\; 64/p^4 = 1. ", "02133fd6a4bdbc80080ccffe4488b883": "\\chi_{mn} = \\sum_{i} a_{mi} a_{ni}^*", "0213c767132c12afbd3114964a9b195b": "e^{i(2h-1)\\theta}", "02145e0b2385830a1d7937a47f81bc6f": "\\mathrm{Ad}_{\\exp X} = \\exp(\\mathrm{ad}_X).\\,", "02146cf17db8911d232615c5935aaea8": "= 1 + 7 + 8 + 2", "0214c7818340dcf25159250a5275c7c5": "y=\\pm \\sqrt{1-x^2}.\\,", "0214e802e89a2a43a1c326b8677eecb5": "\\Delta \\boldsymbol \\beta\\,", "0214fc667b35c1eb4d80bed3631873ee": "A_n,", "02150b6afdf71a00d7a4426c11a03137": "\\lim_{t \\rightarrow \\infty} \\phi(t,i)", "021531540e1f9a1767dc972aba2ce46d": "\\mathrm{Pe}_L = \\frac{L U}{\\alpha} = \\mathrm{Re}_L \\, \\mathrm{Pr}.", "021597b33041ab03bb7d57420dbd92bb": "Q^-(5,q)", "0215f6acde9bc1c91b8536d77d2359b2": "V=w^3 \\left (h/ \\left (\\pi w \\right ) -0.142 \\left (1-10^ \\left (-h/w \\right ) \\right ) \\right ),", "0216b10bb914f682c31527a6dfa29c5a": "\\mathcal{D}\\phi e^{i\\mathcal{S}[\\phi]}", "0216c138751070dfbabb96ef5d1eb18e": "\\tfrac{1}{k}", "0216d9f13fca6900a5faa75a2641597c": "C_2 = 'la'", "02172bc6af05615d441828bb86303fe2": "\\displaystyle{K(x,y)=\\int a (t, {x+y\\over 2})e^{i(x-y)t}\\, dt.}", "0217368dd47e4d3ae870d33145d5fbea": "\\frac{u_{i}^{n + 1} - u_{i}^{n}}{\\Delta t} = \\frac{\\alpha}{\\Delta x^2} \\left(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n}\n\\right)", "0217664181f68eccfdfa54bd94f38295": "\\hat{\\mathrm{Td}}^R(E)", "021776ba3e03f3a12b76cfb6038d460f": "\\,\\!d(x,x) = 0", "0217e727821f8b6d0f4ba70aaa0b9289": "L_\\Phi", "0217f1a1daa60d4eec6e1b17556a7691": "\\gamma p", "0218284b131eb117257a718bf33f02f1": "\\boldsymbol\\Omega \\times \\mathbf{u}_{\\theta} =-\\omega \\mathbf{u}_R \\ ,", "02186e91c74c1347bf9dea47ea4d51b3": "e^{\\frac{{\\delta}}{2}F}=\\prod_{odd \\ \\ l}e^{\\frac{{\\delta}}{2}F^{[l]}}", "021879fd8c747c0eec644ff0731fdcd6": " \\frac{VK}{Y} ", "02187fbad579b9a45c66d0ddeef4dcd4": "\\ \\displaystyle \\min \\ ", "0218892c31c600419c38902a989c1080": "\\{\\varphi_m ; m=1,2, \\cdots ,p \\}", "0218956e3e9799b38ec2e73ccb0c29c3": "[0, -\\infty)", "0218ae0a0d2cfb36098a911162226efd": " v(\\sigma) ", "0218aecfd99bbe3201441c46846f8e1b": "L^\\infty (U)", "0218d7b007a1854a503622ac667d4ead": " H=\\frac{\\phi^2}{2L} + \\frac{1}{2} L \\omega ^2 Q^2", "0218ed240c075274c8bfc76ea63844dd": "a(z)", "0218f809672e55a317f05c582cb8c1f5": "S_3 \\to S_2", "021924c0f6a483b67a498c027ad1a005": "150^\\circ", "0219415e6b09c2b7b94b95529d8d248e": "s = 2^{1/12}", "02199f601cbf0f16a3bd2030f8f6732b": "1-e^{-4\\lambda}. \\, ", "0219b34b096b2e436803a6f11c17626e": "a(v) = b(v) = d(v) = 1, \\text{ and } e(v) = 0 \\,.", "0219e915c2ebd3302c323a485855264e": "\\operatorname{Li}_n(z)=\\sum_{k=1}^{\\infty} \\frac{z^k}{k^n}\\,\\!", "021a383ade6882a9507adb8eef538985": "Eq.6", "021a5393fce02c4f57c3adce8e5a8ffe": "2^{w-1} - 1 + {n \\over w}", "021a6af6071cb77c364718edc0ca959b": "A\\oplus B", "021a90bff98f6e9cd1ef938f9968fffc": "\\left \\langle v \\right \\rangle = \\sqrt{\\frac{8 k_b T}{\\pi m}}", "021ad144f1e0aae4df5d8e05c210feed": "\\mathbb P^n_{\\mathbf k}", "021ae1d076393de740cd55333757daa7": "\\pi : \\tilde{\\mathbf{C}^n} \\to \\mathbf{C}^n.", "021b2cae67f9ff7e602432fe2c468f12": "\\begin{matrix}\\mathrm{Cabtaxi}(4)&=&2741256&=&108^3 + 114^3 \\\\&&&=&140^3 - 14^3 \\\\&&&=&168^3 - 126^3 \\\\&&&=&207^3 - 183^3\\end{matrix}", "021b7f98fa40c4921966ab2f3a10c847": " x^5+320x^2-1000x+4288", "021bdc824da4b0d0db8a7001d988daef": "|W_\\alpha(x)-W_\\alpha(y)|\\le C|x-y|^\\alpha", "021c165cdf6f1229bf98835b81614e1b": "\\frac{a}{p}=0.\\overline{a_1a_2a_3\\dots a_na_{n+1}\\dots a_{2n}}", "021c34847126ffcff029c3109c6a2c94": "\\frac{1-e^{-k}}{1+e^{-k}}\\!", "021c5216fdc8ef5520e350ba1b4d04ab": "w_m(x) = w_m(\\pm r^j A) = w_m(A)", "021c663214ddb1c48b2f4caa55d303f9": "\\oint_\\gamma (u\\,dx-v\\,dy) = \\iint_D \\left( -\\frac{\\partial v}{\\partial x} -\\frac{\\partial u}{\\partial y} \\right )\\,dx\\,dy ", "021c760eb4da2c1574bae8d8224eb616": "\\bold{j}_{{\\rm m}, \\, i} = \\rho \\left ( \\mathbf{u}_i - \\langle \\mathbf{u} \\rangle \\right ) ", "021c7d1154a7ba92517fd48bf5cdfb5d": " -\\smile \\ \\mathrm{or} \\ \\smile\\smile\\smile \\ \\mathrm{or} \\ -- \\ \\mathrm{or} \\ \\smile\\smile- \\ \\ ", "021cd5b20499445d7adc8e55e46dcd37": "(x^n - \\lambda_1) \\cdots (x^n - \\lambda_k)", "021d5907c132d4a5a77d11607b940299": "\\sqrt{\\log t}", "021d5bb84628145baa4d65616d42d6d6": "o=f(d)", "021d8d9fca3bd619f7dd60d32c8fbfa3": "F_{out}", "021d90aaf328fde1c5143da6819944a3": " \\varepsilon_t = 0.5 \\left( ( + {\\Delta p}_D + \\overline {\\Delta q} ) - {\\Delta x}_{t-1} \\right) \\,,", "021dcc12da0e15851dc65ba76ab03998": "-1 < \\lambda \\le -0.75", "021dcceba82bdc9cb593fcc99c34d32b": "\\displaystyle \\Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2", "021e2a185b50c03a079d3e0c3e4cb494": "C_\\nu(x) = \\mbox{Re} \\chi_\\nu (e^{ix})", "021e2af83661cfa2eeeff8fc5786363c": "\\mathbf X = (x_{i, j})", "021e2b3f189905b173b82d764385f3d0": "\\bar{\\omega}^{\\frac{M_p+1}{4}}", "021e31c56481b62335929e55ee5cef17": "{\\color{Blue}~5.1}", "021e366c5269ccb6488fd92a2cb8d8d2": " S_r = \\frac{dQ/dT}{Q} .", "021e73b795f4ac022970b23ccbba839b": "H=\\frac{N}{N-1}(1- \\sum_{i}x_i^2)", "021eef71ae47ec077aa3a8094ad10b03": "x \\in \\{-1, 0, 1\\}", "021f05368040315edf8116f146d414ba": " w = \\frac{I_SR}{nV_T} \\left(\\frac {I}{I_S}+1\\right ) ", "021f0d1a78e8ff3d2af1c85f679c945e": "e^{e^{e^{e^{7.705}}}}<10^{10^{10^{963}}}.", "021f10c51ad1c40dd6e0d68ed8e1c041": "\\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\delta(t-n T) = \\mathcal{F}^{-1}\\left \\{X_{1/T}(f)\\right\\} \\ \\stackrel{\\mathrm{def}}{=} \\int_{-\\infty}^\\infty X_{1/T}(f)\\cdot e^{i 2 \\pi f t} df.", "021f33e28fcf3162445b4cd6c4e6db06": "L_{\\triangle}.", "021f4c71cdce422705204798c756df5b": "(x, y) \\mapsto x", "021f565e2917eb04dc9820f81ac24fe1": "\\varphi:X\\to X", "021f637f4fd183a6797d40bfbc226244": "G^o=\\sum_{i \\in S}{p_i\\log_2{(er_i)}}+(1-\\sum_{i \\in S}{p_i})\\log_2{(R(S^o))} ,", "021fcb0e87fc1b892001c1010be7b9f4": "P=P(X).", "022022f289db140169cd9514f74ee648": "[a, b]", "0220807ccee2f8fefd14155f7ac80aaa": "X_k = \\sum_{n=0}^{N-1} x[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}}.", "022087273905a69a92023e3722643f9a": "f(\\mathbf{r}) = \\frac{1}{\\left(2\\pi\\right)^{3}} \\int F(\\mathbf{q}) e^{\\mathrm{i}\\mathbf{q}\\cdot\\mathbf{r}} \\mathrm{d}\\mathbf{q}", "022132bb3ebcec11d7f81d3f504e9ee6": "y_P-y_0=R_{12} (X-X_0)+ R_{22}(Y-Y_0) + R_{32} (Z-Z_0)", "02213f99cdbec26b01922ac7c2c6a735": "\\mathrm{Ass}_R(M')\\subseteq\\mathrm{Ass}_R(M)\\,", "022174fdae6a4922a7b170c1ee094787": "n_{\\rm e}T\\tau_{\\rm E}", "02219a66af946058fd7efd21b3ee5036": " \\oint_{\\partial \\Sigma(t)}\\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{E}(\\mathbf{r},\\ t) = - \\ \\iint_{\\Sigma(t)} \\mathrm{d} \\mathbf {A} \\cdot {{\\mathrm{d} \\,\\mathbf {B}(\\mathbf{r},\\ t)} \\over \\mathrm{d}t } ", "02219e95bb4d29afb2dbd06a72de57d7": "y^2 = x^3 + x^2", "0221d4398bfb14e28b879e50c313d424": "O( |E| |V|^{1 / 2} )", "02220173c31977d9839303516a09da5b": "\n{dL\\over dt} = i[H,L] = 0\n\\, ,", "022217a91d9b643de752294096d7f6aa": "p_4, p_1", "0222491b800049563d888f2664f4a8a6": "\n\\sigma(t) = \\frac { 1 }{ b }* log {\\frac{10^{\\alpha}(t-t_n)+1}{10^{\\alpha}(t-t_n)-1}} ", "02224ce925b278fca46db66a1da98c3e": "\n\\Sigma(A\\mathbf{x}) = A\\, \\Sigma(\\mathbf{x})\\, A^\\mathrm{T}", "022307e1bd54450e4783926cdb153408": "V_v = V_r", "02230e656b591d8f31a1b7eb03dfdaab": "\\{a_1 , a_2 , a_3 , a_4 \\}", "02234033881254ba9f33e1b63e381585": "\\text{Holant}(G, f_u T^{\\otimes (\\deg u)}, (T^{-1})^{\\otimes (\\deg v)} f_v).", "022399746d452f7fe708c5414a3ab4dd": "Ac^2\\alpha\\left(-\\rho_G-\\rho_L\\right)=Ag\\left(\\rho_G-\\rho_L\\right)-\\sigma\\alpha^2A.\\,", "0223f2bdbda18a7154bf1f35126ea943": "\\bar{\\Gamma}^{\\beta}_{\\alpha \\gamma} \\, = \\, \n\\frac{\\partial \\bar{x}^{\\beta}}{\\partial x^{\\epsilon}} \\,\n\\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\,\n\\frac{\\partial x^{\\zeta}}{\\partial \\bar{x}^{\\gamma}} \\,\n\\Gamma^{\\epsilon}_{\\delta \\zeta} \\,\n+ \n\\frac{\\partial \\bar{x}^{\\beta}}{\\partial x^{\\eta}}\\, \n\\frac{\\partial^2 x^{\\eta}}{\\partial \\bar{x}^{\\alpha} \\partial \\bar{x}^{\\gamma}} \\,", "0223fb7c8a6750e68f52034474fcc627": " c_{t+1} = (1-R^{-1}) \\left[A_{t+1} + \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} E_{t+1} y_{t+j+1} \\right] \n", "02246878093cc4bb4582527127390aba": "\\operatorname{dist}", "02246d3ddf4a376189129511f7aed444": "x = \\left(\\lambda - \\lambda_0\\right) \\cos \\varphi", "0224bf3a2802504318677efcf183c5d8": "(192, 20, 64)", "02251a6d64eac16e4975615fa1729053": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{2N^{2}} \\sum_{i,j} \n\\left( \\mathbf{r}_{i} - \\mathbf{r}_{j} \\right)^{2}.\n", "02252b9c9ecfb5255c48bc40e9468ec5": "q^1 =~q", "0225837728c76ed9a4151ba7478ef822": "sw_G= a_G -r_G+1 \\, ", "02258fb9a54bb12a8fd0d91ae705c352": "T_{\\alpha=0}", "0225cf1ee782416bebc60b86c20f7391": "A \\cap B = \\emptyset", "022601f8e00084d1493d9936c5ec4e53": "\\hat{\\lambda}_x", "02260b5ffe7e69417fdffae16ddfdf4c": "\\sigma: F \\subset \\mathbf C", "02267005721eca4c8753e098ebdbea87": "(1)-(3)", "02267731f45f5958fda3e43298fa70f7": "u=(u_n)\\in \\mathbb{R}^{\\mathbb{N}}", "02268fc40c7bbedc4d1267c6e227803f": " v_{0x} = v_0\\cos\\theta", "0226a80fb3896b26afb862b440b47b44": "H(X|Y)\\leq H(P(e))+P(e)\\log(|\\mathcal{X}|-1),", "02273fbbef6b8ed7f587354c0c979f7b": "g^{(2)}(\\tau) \\leq g^{(2)}(0)", "022740cb79459ef196f8b90f51e7c189": "\\bigcap A", "022767b288e7e3aa5058ce3415b9782c": "|Q_0| = |Q_L| = \\tfrac{P}{2}", "02277c0892b59bb77a84b6acc8da10da": " dA= r^{-2}\\, dx\\,dr", "02279e280508ce5ad88446b2647ccf9b": "A = \\begin{bmatrix} 3 & 1\\\\1 & 3 \\end{bmatrix},", "0227d59d472519da01fc1193ec83f83d": " \\frac{3}{2} (n - s_3 (n)) - 2 e_3 (n) - e_3 (n-1) ", "02283262a7b9c92bc0bfe063321d535d": "p(q\\in Q)", "0228336631a10f396ac503f882dcd26a": " P_{1}{v_{1}^{\\,n}} = P_{2}v_{2}^{\\,n}= ... = C", "0228599d96fca4db83d812af38236b09": "\\ \\det(\\mathbf A) = a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + ... + a_{in}C_{in} = \\sum_{j=1}^{n} a_{ij} C_{ij} ", "02285ca77f2b48eb0afa7341dfaf9276": " \\mathbf{y}_{2} = \\mathbf{y}'_{2} ", "02288438fd1d4c7bffb4fb864c115a70": "\\Delta S^\\circ", "0228edc841b87a34088290c1a53b4356": "\\pi=\\frac{72}{Z} \\!", "022938fe967f9e5cda854d269d72d2dc": "m = \\frac{\\sqrt{1-4c}}{2}", "02294e55210a4b616cafd39611b8fc96": "\\mathbf{A}'=\\boldsymbol{\\Lambda}\\mathbf{A}\\,\\!", "02296e14035c2116e1904e948325e16c": "\n \\begin{align}\n \\boldsymbol{\\nabla}\\cdot\\boldsymbol{S} & = \\left[\\cfrac{\\partial S_{ij}}{\\partial q^k} - \\Gamma^l_{ki}~S_{lj} - \\Gamma^l_{kj}~S_{il}\\right]~g^{ik}~\\mathbf{b}^j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S^{ij}}{\\partial q^i} + \\Gamma^i_{il}~S^{lj} + \\Gamma^j_{il}~S^{il}\\right]~\\mathbf{b}_j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S^i_{~j}}{\\partial q^i} + \\Gamma^i_{il}~S^l_{~j} - \\Gamma^l_{ij}~S^i_{~l}\\right]~\\mathbf{b}^j \\\\[8pt]\n & = \\left[\\cfrac{\\partial S_i^{~j}}{\\partial q^k} - \\Gamma^l_{ik}~S_l^{~j} + \\Gamma^j_{kl}~S_i^{~l}\\right]~g^{ik}~\\mathbf{b}_j\n \\end{align}\n ", "0229715bcd0b8ee6e85eb1137020a050": " (\\beta,\\gamma) ", "0229964a1c9475bb8e607e5b9c838930": " \\lor, \\land", "0229f6d302ed458cdbc9d3bfd86ab90c": "\\varphi,\\psi\\ ", "022a32b622291f9215bb9f3e62cbe044": "k>2", "022a6d034a5abd59a24248cbb3b0941b": "\\ A = \\frac{\\partial v}{\\partial x} + \\frac{\\partial u}{\\partial y}", "022a74c2b1d6e9d7052170bc67377d01": "~A \\cap B \\cap C", "022a90134e784ec490f2f2b6d7282f9c": " \n \\nabla^2 \\varphi - {1 \\over c^2} {\\partial^2 \\varphi \\over \\partial t^2} = - {4 \\pi \\rho } ", "022ab9646a0ab3afe4b5defbe5ccfbb8": "V = an", "022b198209b9c2837ed81d53cd974382": "1 + z=\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}", "022b40c01061319faca833a52952fb3a": "J^{\\prime\\prime}{\\leftarrow}J^{\\prime}", "022b84826ecca1d4efb4a7a396d11302": "\\mathbf{V}_i=\\mathbf{V}+\\frac{d\\mathcal{R}}{dt}\\mathbf{r}_{io}", "022b951b5041e6dad209818e3e896f84": " H_p(X,X-x, G)", "022bd032b7f0ee32c7730b7644c6240e": "|\\langle x, y\\rangle| \\le \\|x\\|\\,\\|y\\|", "022bdeb8bfa3ba5a77935025118e9e2c": "\\scriptstyle{1/2}", "022c1333b68909412b5b0041396caaee": "2^{l + 1} - 1", "022c20fd59376ce997de8331ffaedbd3": "A_t = \\{x\\in X\\mid f(x)\\ge t\\}", "022c32018a8e85ae989512fd7ecec25e": "V_n(r)", "022c3b80bef2c0f17f57ed150c1f4652": " 1 + \\frac14 + \\frac19 + \\frac1{16} + \\frac1{25} + \\cdots = \\sum_{n=1}^\\infty \\frac{1}{n^2} ", "022cbd378cab471ae5be73488db3b604": "t=\\tfrac{x-x_1}{x_2-x_1}", "022cde90c52840683f79ce7a7e627c22": "d = 2\\pi / |\\mathbf{g}_{h k \\ell}|", "022d283fc823640c77ed0a4b510ed33b": "-\\frac{\\partial}{\\partial t}p(x,t)=\\mu(x,t)\\frac{\\partial}{\\partial x}p(x,t) + \\frac{1}{2}\\sigma^2(x,t)\\frac{\\partial^2}{\\partial x^{2}}p(x,t)", "022d434b912cb7fa1b0b4644e8b4e2ae": "Y\\ \\sim\\ \\mathrm{Herm}(a_1,a_2)\\,", "022d8aa2bcbc12f4324820915872f900": "\\mathbf{e}_2 \\times \\mathbf{e}_3 = \\mathbf{e}_5, \\quad \\mathbf{e}_3 \\times \\mathbf{e}_5 = \\mathbf{e}_2, \\quad \\mathbf{e}_5 \\times \\mathbf{e}_2 = \\mathbf{e}_3,", "022daeb34db6dd6d51b0de65cf250648": "\\max_{d\\in D}\\min_{s\\in S} dist(d,s)", "022dbecbb7fa5d325462bd7a0ce699d5": "\\alpha^*F:=\\{H'\\le H|\\alpha(H)\\in F\\}", "022dcce091d8dc74031d8dbf34662dab": "n = ax^2 + 2bxy + cy^2", "022e3414e6427b3cc27c5a5911fd9588": "D G(x, s)=0", "022ed48ce122fb6d02b20ffd57a86105": "\\sin (2 \\theta) = 2 \\sin \\theta \\cos \\theta\\,", "022f7d80be231d713945ca4d7beed1cf": "\\left(\n\\frac{\\pi}{6}\n\\right)^{\\frac{1}{3}} \\approx 0.806", "022f9ef548f37cc6101d5e59875cc945": "\n \\alpha_{\\rm{THz}}(\\omega) = \\mathrm{Im}\\left[ \\frac{\\sum_{\\nu, \\lambda} S^{\\nu, \\lambda} (\\omega) \\Delta N_{\\nu,\\lambda} - \\left[ S^{\\nu, \\lambda}(-\\omega) \\Delta N_{\\nu,\\lambda}\\right]^{\\star} }{ \\omega (\\hbar \\omega + \\mathrm{i} \\gamma(\\omega))} \\right]\\;.\n", "022fb3dab2be5bff82479c16cc1780ef": " a \\otimes b \\mapsto (-1)^{|a||b|} b \\otimes a ", "022fb3e873d2b56001a689daec1b9e7d": "\\lim_{x\\to c}{|f(x)|} = \\lim_{x\\to c}{|g(x)|} = \\infty,", "02301b578da6ac04d27ae1fefb9a9133": "\nX^{\\{q\\}}=\\lambda^{-1}([m-q,m])\n", "0230363ab1c553703171c76386773875": "\\psi_1=\\psi_1\\big(\\vec{\\sigma},\\vec{\\rho}\\big)=\\Big({\\textstyle\\sum\\limits_{i=1}^n\\sigma_i^2}\\Big)^{-1/2}\\cdot\\max_{1\\le\ni\\le n}\\frac{\\rho_i}{\\sigma_i^2}.", "023068de560204c0cf3f00e2e4568840": "\\begin{matrix}\\mathrm{Cabtaxi}(1)&=&1&=&1^3 \\pm 0^3\\end{matrix}", "0230ba0ac3a6fc775e42d81c10dfbbea": "0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\\cdots", "02311b615f1556b2414572ce7f4561f0": "(F,m)", "023124f437c87fccd318c8d546e8e40a": "\\textit{VERB} \\; \\textit{NOUNPHRASE}", "02312e4fc33d871628eeb7617f6beebe": "\\displaystyle{\\widehat{P_y}(t)=e^{-y|t|},}", "023142ea5abc2a69bc90e8ed3bfb1ecf": "\\tau\\,\\sim R/c_s,", "0231aa9e0f86af7105e43b77bc14b1c5": " (X+Y)_i", "0231c298d893d2f32ee58bc2edde3c2d": "\\mathbf{F}_k", "0231c2e6053495105d4154730d981449": "n_{c1}(\\mathbf{k})", "0231d206e2f7860d2ba70ffa7f9a4391": "\\langle \\psi|\\psi \\rangle = \\sum_i |c_i|^2 = 1.", "0231f3829641a85262e0bdfaf24857ed": "K^{\\ominus} = \\mathrm{\\frac{[A^-] [H_3O^+]}{[HA] [H_2O]}\\times \\frac{\\gamma_{A^-} \\ \\gamma_{H_3O^+}}{\\gamma_{HA} \\ \\gamma_{H_2O}} =\\mathrm{\\frac{[A^-] [H_3O^+]}{[HA] [H_2O]}}\\times\\Gamma}", "02322c86ce10049b1cac9d06a539264f": "\\text{subject to } \\dot{x_t} = f(x_t,u_t)", "02323b856adbbedeca994dac706eece8": "\\tfrac{863}{60480}", "0232502a9b8aa410be0731dfefa96d89": "\\left(\\mu, \\frac{\\alpha - \\frac12}{\\beta}\\right)", "023278c95dccd52f1c1ede88e3a9bbaa": "\\psi = \\sqrt{\\rho} \\; \\exp \\left( \\frac{i \\, S}{\\hbar} \\right) ", "0232941ed731510409a6f815ba885bd8": "P_3=(0,-72,2\\sqrt{3},12)", "0232b079f9617a0dd2bf92e0501a6baf": "v = Z \\alpha c", "0232e0a4d7ff7211cc29b99bfcd79c60": "\n\\begin{align}\n&{\\partial\\rho\\over\\partial t}+\n\\nabla\\cdot(\\rho\\bold u)=0\\\\[1.2ex]\n&{\\partial(\\rho{\\bold u})\\over\\partial t}+\n\\nabla\\cdot(\\bold u\\otimes(\\rho \\bold u))+\\nabla p=\\bold{0}\\\\[1.2ex]\n&{\\partial E\\over\\partial t}+\n\\nabla\\cdot(\\bold u(E+p))=0,\n\\end{align}\n", "0232f27be40b2b647f260050dd308eb8": "D'", "0232f592c77a40287056489966672f9a": "|W|^2", "023330e0f448e77e7f36d5b64003a4af": "S_L \\, \\dot= \\, 1 - \\mbox{Tr}(\\rho^2) \\,", "023332f3c3c330f5e090368eb88239de": "\\phi:S^p\\to M", "02339a5ca7d0e8ebcf600e7a71af43ac": "\\ker(\\partial_n)=Z_n(X) ", "0233a635281a006b5ef593fd13c442bb": "\\tilde{f} : A \\to B", "0233e398d6cef3db1cb3373918134e2d": "x^2 + 2ay = 0 \\,", "0234461cdd7e945e51351ff44168c87c": "\\displaystyle{G_0=K\\cdot \\exp i\\mathfrak{p} =K\\cdot P_0 = P_0\\cdot K}", "0234566f1fbd03b0d70fb63760de4af9": "F_g = m g \\, ", "0234997dc624d9faffaabdc308aaf0bf": "\\left.\\left(\\frac{d}{dt}\\exp(tY)\\cdot v\\right)\\right|_{t=0}=Y\\cdot v.", "0234d553f8a114a57c79e5a07d1b5f30": " ({\\mathcal F}_a f)(t,y) = (2\\pi)^{-n / 2} \\int_{{\\mathbf R}^n}f(x)e^{-a |x-y|^2/2} e^{-ix \\cdot t}\\, dx.", "0234f8b37d074721fc182323d786a3b8": "\\mathcal{D}_{T*}", "02359fe87f7bece7408ee4c2fb05309d": "\\frac{\\partial \\mathbf m}{\\partial t} = - |\\gamma| \\mathbf{m} \\times \\mathbf{H}_\\mathrm{eff} + \\alpha \\mathbf{m}\\times\\frac{\\partial \\mathbf{m}} {\\partial t}", "0236397aa3334d97ef48265bd70cc65c": "\\mathrm{[HA]} = C_a - \\Delta", "0236b26572404fdd74e9b216b80ec598": "\n\\sup_p h_p(x) \\ge 1\n", "0236c4fd43865dc027e02a12932a3d38": " \\mathbf{\\xi} = \\nabla \\times \\mathbf{h} \\,\\!", "0237339d7ab322085ac4d6fe016b9180": "~ G_0=\\frac{ND}{\\sigma_{\\rm ap}+\\sigma_{\\rm ep}}~", "02373c1f5bede1d3a97be96e5bc98fa2": "30~\\mathrm{dB}", "0237559e0dd6b6ad12596f53e2b0b576": "H(X) = -\\sum_{i=1}^n {p(x_i) \\log p(x_i)}.", "02375cdd0732a6d66d1e458fa1b50b80": "\\mathbf{Q}(t)", "0237a5f569f0044a5bbb8f2192c986ad": "c \\gamma^2 - (a - d) \\gamma - b = 0 \\ ,", "0237f18a6d321a7442c3fee447abeb1d": "{1\\over 2} \\sqrt{2}", "0237f6329ee550690931c6833531edfe": "\\ell_j(x) = \\frac{\\ell(x)}{x-x_j} \\frac{1}{\\prod_{i=0,i \\neq j}^k(x_j-x_i)}", "02386c655ee72999c62ae715ab5d7292": "f(\\pi , \\pi) = -1", "023873b1bddb202be30b5afdcd5749df": "T_{AMB} = 70 \\ ^{\\circ}\\mbox{C}", "02389fda3095cddda9021cb2d21e3cd2": "\n |\\mu|(\\partial B) = 0\\,.\n ", "0238bce249da3358e4f5ed91094a93e7": "ODF(\\boldsymbol{g})=\\frac{1}{V} \\frac{dV(\\boldsymbol{g})}{d g}.", "0238c5283423c18589620888e3e89f6f": "x^2 + 6x + 5 = 0,\\,\\!", "02392d528baa8b5145109fb192d3b1d8": "\\frac{\\frac{L_1}{2l}}{\\frac{L_2}{2l}}\\approx 4 {\\left ( \\frac {L_2}{L_1} \\right ) }^2 \\Longrightarrow \\,\\!", "02393ef35f0969894f61ddf410d7f06d": "x,y \\in \\mathbb{R}^\\times_{>0}", "02398fd5e498663131fd5316fe7ee86e": " E\\left[ \\Lambda(n+1) \\right] = \\Lambda(n) + E\\left[ \\left( \\frac{\\mu\\, \\left( v(n)-r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right)^H \\left( \\frac{\\mu\\, \\left( v(n)-r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right) \\right] - 2 E\\left[\\frac{\\mu|r(n)|^2}{\\mathbf{x}^H(n)\\mathbf{x}(n)}\\right]", "02399bb49108a9455c6827292a30f6ea": "\\begin{align}\n r &{}= \\sqrt{6^2 + 5^2} = 7.8102 \\\\\n c &{}= 6 / r = 0.7682\\\\\n s &{}= -5 / r = -0.6402\n\\end{align}\n", "023a62bc62f64d7623a58945e76525ed": " O_k = x_1x_5x_9 \\cdots x_{2N-3}+ x_3x_7x_{11} \\cdots x_{2N-1}", "023a6f9688af4f89a59c4ba647f93d89": "\\nabla^2\\psi=0.\\,", "023a91025f1455379b2c7b284e046e79": "\\mathrm{Hom}^{\\bullet}(\\Gamma_c(X;I^{\\bullet}_X),k)= \\cdots \\to \\Gamma_c(X;I^2_X)^{\\vee}\\to \\Gamma_c(X;I^1_X)^{\\vee}\\to \\Gamma_c(X;I^0_X)^{\\vee}\\to 0", "023adfe845a552b23bef1cb0b61328c7": "\\{x\\in F;\\,x\\Vdash p\\}\\in V", "023b183c77a3cdbe50fe1f990a63a6de": "2^T-s", "023b3db14b66f7d0ee89fdb89c64e57d": "(K,\\, \\nu)", "023b800a1806490ff857cf9d69a260df": "\\neg \\!\\,", "023bdb642d2bb73325b663deba16c00e": " Q_N \\equiv \\frac{1}{N} \\sum_{i=1}^N \\frac{f(\\overline{\\mathbf{x}}_i)}{p(\\overline{\\mathbf{x}}_i)}", "023bf722272578ea8a889efa19070288": "\n w(x,y) = \\frac{q_0}{\\pi^4 D}\\,\\left(\\frac{1}{a^2}+\\frac{1}{b^2}\\right)^{-2}\\,\\sin\\frac{\\pi x}{a}\\sin\\frac{\\pi y}{b} \\,.\n", "023c00c41b5fca2d9265161353de9776": "\\color{Magenta}\\text{Magenta}", "023c2f48e975917d540465a883af89f3": "A_n = A + \\alpha q^n, \\,", "023c91a928cb3ea433ca767d460dcbe2": "{\\rm d}A = {\\rm d}U - (T{\\rm d}S + S{\\rm d}T)\\,", "023cc810f0ec386a71e5846889b5d75e": "k(i)\\geq 1", "023ccd363bceb5d8ec574167a06d1242": "\\delta=0,w(x_1,x_2)=\\mathbb{I}(x_1+x_2 0", "024d0aedab64caf6b9088f1ea5817f0a": " H_\\xi = \\xi^i\\frac{\\partial}{\\partial x^i}\\Big|_{(x,\\xi)} - 2G^i(x,\\xi)\\frac{\\partial}{\\partial \\xi^i}\\Big|_{(x,\\xi)}. ", "024d90f2a1a1613e65739f9c2e526069": "\\mathcal{L}_{QP}=\\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t)", "024d9dbb7ceb42faf9928f6391d3aef8": "\\ddot{V}", "024e0406ca30841312cba458db27f8b9": "A_{i}\\to A_{i}", "024e5399b10635924f8ea3f5619c63da": "f(x, \\Phi_j(x)) \\leq \\Phi_i(x) \\, ", "024e9f2ae45357083043d5794ad82d19": "X_1(t)", "024ea2eb3be1b0fc776b67d7a4e6de18": " \\frac{Y}{L}=A.\\frac{K}{L} ", "024ecdaa895a112cc1ad509e2a0a27b4": " f_*(a_0 \\otimes \\cdots \\otimes a_n) = (b_0 \\otimes \\cdots \\otimes b_m) ", "024f088ed8ba29d0348d56a5728d486a": "x_j - x_m \\neq 0", "024f75b25db2a5874f7888c41f537693": "\\mathcal{O}(-1)", "024f93a30355166f71c68076fd453c28": "\\mathbb{RFM}_I(D)", "024f9bda6687751643ae724a45f345c5": "u_n = \\sum_{k=0}^n {n\\choose k} a^k (-c)^{n-k} b_k", "024fdc4704a30426ed70030fd55a7e52": "Q = 12.5", "024fe1572700dab589fc3a4eaaee0eee": " y = \\log_{10} {P}^*_i ", "02501709f35cb1e403a42cda6991af2c": "\\pi = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-3)^{-k}}{2k+1} = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-\\frac{1}{3})^k}{2k+1} = \\sqrt{12}\\left({1\\over 1\\cdot3^0}-{1\\over 3\\cdot3^1}+{1\\over5\\cdot 3^2}-{1\\over7\\cdot 3^3}+\\cdots\\right)", "0250723330e3db998ce955076784f58d": "\\vec w \\propto \\Sigma^{-1} (\\vec \\mu_1 - \\vec \\mu_0)", "02507ba5d8c3288a6d2a2979ffff4f68": "\\hbar \\Omega _m", "0250c111e54fdf6000aec02a0d851bfa": "H(\\omega)\\,", "02510a99289a433f29b5f77146a9836d": "\n \\left(\\bigcup_{i\\in I} A_i\\right)^o = \\bigcap_{i\\in I} A_i^o.\n ", "02513c497f0b6302937c7b0a7851c18d": " \\gamma = \\gamma' = \\frac{2q -d}{p} \\,", "0251597a9057a3470a7a302dcd31b56e": "V_{ion-ion}", "0251880a00c512cf394979313f3766c8": "\\hat{\\Phi}(t) ", "02521927eba8b0cb32a3cc8ff30d4c7f": "\\tau_\\mathrm{s}\\,", "02527c4a4a9931ee779fd7cf66f30eea": "\\hat{x},\\hat{y},\\hat{z}", "0252d21ed53a7d41d3db2caefda95f8b": "S=\\{ s_n \\}_{n\\in\\N},\\,", "02531c9578e100f64befba62e273b529": "20 \\times \\log_{10} \\left(\\frac{5V}{10 \\mu V}\\right) = 20 \\times \\log_{10}(500000) = 20 \\times 5.7 = 114 \\,\\mathrm{dB}", "025329063bb50ed9795e5fe74bd919e9": "\\#(n)=|B_n(G,T)|, ", "02535ae4ac19df62aea3828db87a7817": " X_C= -\\frac{1}{\\omega C}", "0253a63318b1ccb430558dcb2955a281": "A[\\Psi]=\\int\\mathrm{d}t\\ \\langle\\Psi(t)|H-i\\frac{\\partial}{\\partial t}|\\Psi(t)\\rangle.", "0253c84666685857b6ba8cdbe9d6432a": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\frac{\\mu}{3} \\nabla (\\nabla \\cdot \\mathbf{v}) + \\mathbf{f}. ", "0253d2b1cab9f2e800f7a2e06733e33e": "a,b,k", "0254081c26dc9e45ce5c215fee67ed14": "\\langle\\Delta V\\rangle=\\frac{4}{3}\\frac{e^2}{4\\pi\\epsilon_0}\\frac{e^2}{4\\pi\\epsilon_0\\hbar c}\\left(\\frac{\\hbar}{mc}\\right)^2\\frac{1}{8\\pi a_0^3}\\ln\\frac{4\\epsilon_0\\hbar c}{e^2}", "02544ffbb49928005b35b4fc1c66f9c6": "\\mathbb{P}(n \\leq n^* | n_b \\leq n^* , s+b) = \\frac{\\mathbb{P}(n \\leq n^* |s+b)}{\\mathbb{P}(n_b \\leq n^* |s+b)}\n= \\frac{\\mathbb{P}(n \\leq n^* |s+b)}{\\mathbb{P}(n \\leq n^* |b)}.", "025464d3b6a57dde173c670b334b4c7a": "\\mathbf{\\nabla}\\cdot\\mathbf{E}(\\mathbf{x})=-\\frac{i Z_0}{k}\\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x})", "0254928e844d7febdfcfccb610b43951": "1_{GX} = G(\\varepsilon_X)\\circ\\eta_{GX}", "0254ab4d45ac475dc19a0f6111a6bee7": "\\mathbf{K}q=\\mathbf{S}\\,q - \\mathbf{V}q", "0254bddbffe3291cb211dc2690d791df": "\\mathcal O_L / \\mathfrak p^{i+1}.", "0254fe457741ff2f8ac65219733d98bc": "\\digamma(\\nu)", "02553bc981384e85483e10a26c47bf1a": "\\mathfrak{R}", "0255ae6678ed9ddf1b37d7fddd7e9cfe": "\\sum_{m=0}^\\infty \\frac{65520}{691}\\left(\\sigma_{11} (m) - \\tau (m) \\right) q^{m} = 1 + 196560q^2 + 16773120q^3 + 398034000q^4 + \\cdots", "0255b016c317e4eae99aeb727b3f3e10": "\\frac{4^n}{\\Gamma(n+1)}.", "0255d35e1cc778d50d639f145ca7a5e7": "\\lfloor", "0256681cebc402c62c9107251b6e62fe": "\\frac{}{\\Gamma_1, \\alpha, \\Gamma_2 \\vdash \\alpha} \\text{Ax}", "02568e22c87a55a649d0b1b61e3529b2": "\\mathbf{e}^i (\\mathbf{e}_j) = \\delta^i_j.", "0256a4b12d15b54af18b148540113e1e": " \\sum S = (x_{1} + x_{2} + x_{3} + ... + w)(p^{0} + p^{1} + ... + p^{k-1}) = \\sigma_{1}(w)(p^{0} + p^{1} + ... + p^{k-1}) ", "02577ce019c0536fca02f2d07889e40a": " kT = \\frac{\\hbar a}{2\\pi c}", "025784302af37d32451f062663ee025c": "\\Rightarrow x=e^{W(\\ln z)}\\, ,", "0257c4faf4027f97471d14f87512c6e1": "nF^{ _{ }}/RT", "0257d237e99f9bd9830e616b6ac54595": " \\delta W = p dV\\;", "0258321027b3e0da182a33942238407b": "Q_{\\alpha \\beta}=\\int d^3\\mathbf{x'}(3 x'_\\alpha x'_\\beta - \\|\\mathbf{x'}\\|_2^2 \\delta_{\\alpha \\beta})", "0258535e986c72130f7f01840532fc24": "\\{ A , V \\}", "025890facebaed2aec288dc7bede99b1": "J_0(kr)", "0258f7634a80d517311163f85c2bc0a9": "(u,v)=(0,0)", "025943f11cd36bf8028cfdba8a40033a": " n(r) = \n \\begin{cases}\n n_1 \\sqrt{1-2\\Delta\\left({r \\over \\alpha}\\right)^g} & r \\le \\alpha\\\\\n n_1 \\sqrt{1-2\\Delta} & r \\ge \\alpha\n \\end{cases}", "02595d47e0006a3ce08238acdaa0fd6b": " \\operatorname{get-lambda}[F, G\\ V = E] = \\operatorname{get-lambda}[F, G = \\lambda V.E] ", "025a04608819638d1b3ffbed85952e1f": "a(t) = ae^{-j\\omega t} \\ ", "025a0946c92f9fba9719cc3328931e9b": "\\begin{align}\n& \\mathbf{(D-\\omega L)^{-1}[(1-\\omega )D+\\omega U]} = \\frac{1}{12} \\begin{pmatrix}\n-1.2 & 4.4 & 6.6 \\\\\n-0.33 & 0.01 & 8.415 \\\\\n-0.8646 & 2.9062 & 5.0073\n\\end{pmatrix},\n\\end{align}", "025a1d6e6a1ae5a9a00bff0dc971b1ed": "\\Delta^4 m_6 = m_6 - 4m_7 + 6m_8 - 4m_9 + m_{10} = \\int x^6 (1-x)^4 d\\mu(x) \\geq 0.", "025a36473308d14aa4c20882682656b8": "\\alpha = (Q\\times F/4)^{1/4}\\,\\!", "025ab80c795d5d2e8499b80ac2b81b60": "X \\sim \\mathrm{Rayleigh}(1)\\,", "025adbddd8ff913fc53236ff7ae8d8ba": "\\frac{8! \\times 3^7}{24}=7! \\times 3^6=3,674,160.", "025b057912045b97ef467c2c2bc9242a": "\\hat{\\bold{H}}_\\operatorname{PI} = \\begin{bmatrix}0.052 & 0.510 \\\\ 0.510 & 8.882\\end{bmatrix}.", "025b2eae2546fafa1fd6b9f756a7700d": "\\alpha_t", "025b36ac0f07709eb91d6fd2e6d704f6": "K / L", "025b3f94d79319f2067156076bf05243": "\\Sigma", "025b580a55042ccea81fbdea600770d5": "\\|u-u_N\\|_{H^1(\\Omega)} \\leqq C \\exp( - \\gamma N )", "025b98f6d511a3d7a32f9e0dcc096d84": " E[X]_{ab} = R_{ambn} \\, X^m \\, X^n", "025bdb4f9244413527859c3df03bd71a": "\nm\\rightarrow m+S~", "025c4256ddf664dffb51d5cd897eb82e": "\\beta/\\alpha", "025c8812189a2392bba31d16f753065d": "r^{n}, r^{n-1}, \\ldots ,r", "025c9146ef1e96410c26a64fdee29d95": "i_{n-2}-i_{n-3}\\,\\!", "025d0c896f43bb3cd40766c406eba75f": "\\ell_i\\,", "025d2e99c2738d5ca731f6a04ed05e1a": " \\begin{bmatrix} y_1\\\\ y_2\\\\ y_3 \\\\ \\vdots \\\\ y_n \\end{bmatrix}= \\begin{bmatrix} 1 & x_1 & x_1^2 & \\dots & x_1^m \\\\ 1 & x_2 & x_2^2 & \\dots & x_2^m \\\\ 1 & x_3 & x_3^2 & \\dots & x_3^m \\\\ \\vdots & \\vdots & \\vdots & & \\vdots \\\\ 1 & x_n & x_n^2 & \\dots & x_n^m \\end{bmatrix} \\begin{bmatrix} a_0\\\\ a_1\\\\ a_2\\\\ \\vdots \\\\ a_m \\end{bmatrix} + \\begin{bmatrix} \\varepsilon_1\\\\ \\varepsilon_2\\\\ \\varepsilon_3 \\\\ \\vdots \\\\ \\varepsilon_n \\end{bmatrix} ", "025dceb6d6fb0f273aa5fae8c6dca7c6": "e^{S}", "025e191de58cbf019d7d91e22fe94bda": "\\frac{1}{\\sqrt{n}} \\sum_{i=1}^{n} \\left [\\mathbf{X_i} - E\\left ( X_i\\right ) \\right ]=\\frac{1}{\\sqrt{n}}\\sum_{i=1}^{n} \\left [ \\mathbf{X_i} - \\mu \\right ]=\\sqrt{n}\\left(\\mathbf{\\overline{X}}_n - \\mu\\right) ", "025e8bf0eb554eb06c314ce8dffbe64a": "\\scriptstyle \\sin \\theta \\approx \\theta\\,", "025e99932b678d1f0120fe0dbe2e13cc": "\\mathbf{P} = m\\mathbf{U} \\,", "025e9e7552edc9d5c6e1ed0eba4f68fb": " \\left| x(t) - x(t + T) \\right| = 0 \\text{ for all } t. \\ ", "025f5f529b0a6dc6d3a158197ebde4cf": "a/bc", "025f6e2d7c040ef7ec04d50fa2fc2108": "(1-2x_0)^{2^{n}}", "025fc04dcc1848a7baf1b9b46fc11fbf": "f\\in \\mathcal{PC}", "026088a2c5ca5cfa2befcb3b43266009": "f(x)=\\sum_\\alpha a_\\alpha x^\\alpha\\text{, where }\\alpha=(i_1,\\dots,i_r)\\in \\mathbb{N}^r \\text{, and } x^\\alpha=x_1^{i_1} \\cdots x_r^{i_r}", "0260ab105a2f8001f01707d2d4465067": "[M]_{v\\;\\|\\;a\\;\\|\\;u} \\rightarrow [[~]_{u\\;\\|\\;x}\\;\\|\\;M]_{v\\;\\|\\;y}", "0260c684c19a0d9dce9a8da81c542162": "V\\otimes V_{II_{1,1}}", "026150509621605b486cae1a27d552c9": "\\mbox{C}_4^6", "0261592341d2501c32a6f3978b802671": "x = t, y = t^2 \\quad \\mathrm{for} -\\infty < t < \\infty.\\,", "0261a3d116001fbd03cf425823a21970": " F(\\omega) ", "026212a7ffb0ab4032a89f1f802c2838": "Z_{F+G} = Z_F + Z_G\\,", "02628a067d4b163832045a47c972c9a5": "|M|^{2}", "0262d312520c54ffaac84863f9eee4ef": "\n\\zeta=\\frac{\\alpha}{\\sigma_0}\\;\\int_K^{F_0}\\frac{dx}{C\\left(x\\right)}\n=\\frac{\\alpha}{\\sigma_0\\left(1-\\beta\\right)}\\;\\left(F_0^{1-\\beta}-K^{1-\\beta}\\right),\n", "02633821424a5649073e041c2c6ebd0f": " \\forall\\text{ internal } f: {^*\\!A}\\rightarrow {^*\\mathbb{R}} \\dots", "02644a9c0510755d4a1390e1cc07f295": " ab \\le f(a) + g(b). \\, ", "02645e252addf19388e4eb4cdf917017": "\\sum_{f_1\\ge f_2\\ge f_N\\ge 0} \\mathrm{Tr}\\Pi_{\\mathbf{f},0}(z_1,z_1^{-1},\\ldots, z_N,z_N^{-1}) \\cdot \\mathrm{Tr}\\pi_{\\mathbf{f}}(t_1,\\ldots,t_N)\n=\\sum_{f_1\\ge f_2\\ge f_N\\ge 0} \\mathrm{Tr}\\sigma_{\\mathbf{f}}(z_1,\\ldots, z_N) \\cdot \\mathrm{Tr}\\pi_{\\mathbf{f}}(t_1,\\ldots,t_N)\\cdot \\prod_{i 1.\\,", "02718a35a1d62e76d3127af4cd4f23cc": "s_\\mathrm{in}\\,", "0271a9f2d735faff963555b6df864814": "r_2 = (A \\to S, \\{r_2\\}, \\{r_1, r_3\\})", "0271cbc3a02561a58d919aecb18029ab": "m_{p}", "0271cfd20d5c7bf792c844373753b4c9": "{\\partial / \\partial r} = -{\\partial /\\partial n}.", "02721aa35b02c75d8d1f5a9d87228d0a": "\\frac{(a+b)h}{2} \\,\\!", "027226d2312eded580526508612ce832": "\\sqrt{2}+\\sqrt{3}\\,", "02724694ac3af41dd73e0fcb69ee2466": "A_0 \\to \\ldots \\to A_{i-1} \\to A_i \\to A_{i+1} \\to \\ldots \\to A_k", "027281910cf4071ee187728510baa84f": "\\sigma_2^2", "0272b29f6e7dd14e7071eb5bf61b57bb": "T=I", "0272c90422f4b23f836598dc016c9d9f": "\\frac1{137}", "0272d268d0534de5245746bcaa96c0e1": " \\sigma^* = G(F^*) ", "02737eddf8250b8f1aaa104754d37249": "\n\\begin{bmatrix}\n 1\n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 \\\\\n 2 & 1\n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1 \\\\\n 3 & 1 & 2\n\\end{bmatrix}\n", "0273a173375948ed6cc340447e4a27ed": "\\text{If }\\lim_{x \\to c} f(x) = L_1 \\text{ and }\\lim_{x \\to c} g(x) = L_2 \\text{ then:}", "02742521dd1678400280d212566bfb47": "\\langle\\phi(0,t)\\phi(0,0)\\rangle\\sim \\sum_nA_n\\exp\\left(-\\Delta_nt\\right)", "027441dff48689fb1b7fbd1cc35a5356": "g \\circ f \\colon X \\to \\mathbf{K} \\colon x \\mapsto g(f(x))", "02752b048de7a6e77676f58bb429610f": " t_1", "027543b772146bb664f61c562344bb75": "\\sum_{i=0}^n i^2 = \\frac{n(n+1)(2n+1)}{6} = \\frac{n^3}{3} + \\frac{n^2}{2} + \\frac{n}{6}", "02757c96b2a9eada766a85e99918010d": " L_{\\sigma,\\varepsilon} := \\max\\{ \\sigma(k) | k \\in I_{\\sigma,\\varepsilon} \\}", "0275a8621507190c4edc2ff72a3e4c06": "X^G", "0275ad96d859850a8883d4d869704943": "\\pi(x) \\leq x", "0275b5048a096e7776c9a2a7bf9c39ad": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n T(A-\\lambda_n I)\\mathbf{x}_n,\\ n \\ge 0.", "0275e7e544c08853c8c58bc04897645b": "\\mathbf{A}\\cdot{\\rm d}\\boldsymbol{\\ell}=-", "0275f7fb66a3fbd19097948981f29d7e": "\\lnot\\ \\forall{x}{\\in}\\mathbf{X}\\, P(x) \\equiv\\ \\exists{x}{\\in}\\mathbf{X}\\, \\lnot P(x)", "02761f43f1ceb181d2090becb35a5739": "\\left | \\mathbf{a} \\right \\vert = \\sqrt{\\mathbf{a} \\cdot \\mathbf{a}} = \\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2 + {a_5}^2 + {a_6}^2}.", "027623c36bf90a4651c4401fdb3cc270": "(x_i,x_{i+1})\\in E", "027712408326070f9db72d79a34da1c3": "\\int_0^\\infty (1\\wedge x) \\mu(dx) <\\infty.", "027721e0be74c20fbc15d6dff1e61227": "\\mathbb{C}^{2n}", "02773c881227cb8b849971bf0a8b8aa6": " \\mbox{E} =\\frac{\\sqrt{N} \\cdot\\sqrt{R}}{2\\cdot \\sqrt{\\pi}\\cdot d}", "027757997a2330c4386e56b918e88c4f": "\\lim_{h\\to 0}\\frac{f(a+h) - f(a)}{h} = {+\\infty}\\quad\\text{or}\\quad\\lim_{h\\to 0}\\frac{f(a+h) - f(a)}{h} = {-\\infty}.", "027770abe9a99f31d66cb33a30e4494c": "j=1", "0277df1ffe8c7a546f8668e2d023a508": "\\operatorname{Re}(s) ", "0277e41e188b27fc82b47423e62409fe": " [I_1 \\cdots I_r] = [J_1 \\cdots J_s] \\in Cl(R). ", "02783fff904f832fc73014e85e617ff8": "\\frac{\\alpha}{v}\\log{\\left(\\frac{c+vT}{c}\\right)}=1\\,\\!", "02784e94955679b54e8a6d68a96f5c71": "v = H_0 d", "0278850ba059525d5f0e5e514b76f459": "\n\\frac{s}{D} = \\frac{m_b}{k_B T}\n", "0279047637786c035fd0ae1abaabecf0": "s_c = e\\,\\alpha^{c\\,i}", "02790ba6054fae5179e0a8e8a4948088": "m>0\\,", "027939899d5c69c1f82c472e8671fa17": "\\phi(t) = N\\cdot 2\\pi,\\,", "02796b1bee509dbee67ee4b7a0acbeb5": "\\tfrac{1}{X} \\sim \\mathrm{Planck} ", "0279f4c24ed40a329b8ac3dd52cd8ff2": "d(O_{r}, Q)", "027a3dc0a952751d29f62a84c0d48b7a": " \\mathcal{J}_{ij} = \\begin{cases} J & \\mbox{if }i, j\\mbox{ are neighbors} \\\\ 0 & \\mbox{else.}\\end{cases}", "027a4b3e733807da32b0aec4e03387dc": "i_{\\text{1}} = I_{\\text{B}} + i_{\\text{F}}", "027a6a7dc8797392917d232f79c29137": "\\Omega_E=\\binom{N}{(N+j)/2} = \\frac{N!}{\\left(\\frac{N+j}{2}\\right)! \\left(\\frac{N - j}{2}\\right)!}.", "027a8888e2a55823e14377fc154b0f89": "g(\\lambda) = -\\tfrac{1}{2}\\lambda^{T}AQ^{-1}A^{T}\\lambda - \\lambda^{T}b", "027aef1b3ac13ece6cbeab406d386152": "wp(\\textbf{while}\\ E\\ \\textbf{do}\\ S\\ \\textbf{done}, R)", "027b33f37f2eeb78e798fe97e5b02551": " R_N =r_o = \\begin{matrix} \\frac {1/\\lambda + V_{DS}} {I_D} \\end{matrix} = \\begin{matrix} \\frac {V_{E} L + V_{DS}} {I_D} \\end{matrix}\n", "027b3e2314e62461489d1c69ad4dec6c": "\\begin{align}\n&\\deg P_n = n~, \\quad n = 0,1,2,\\ldots\\\\\n&\\int P_m(x) \\, P_n(x) \\, W(x)\\,dx = 0~, \\quad m \\neq n~.\n\\end{align}", "027b580645d6223958e406b837abb816": "\\pm\\sqrt{1 - \\cos^2 \\theta}\\! ", "027b97cb2500a918e169b01e05f1aae4": "-\\nabla\\cdot\\mathbf{g}=\\nabla^2\\Phi=4\\pi G\\rho\\!", "027b9f898690366de9b5d8b3d9e7e41a": " \\nabla^2 f = {1 \\over r^2}{\\partial \\over \\partial r}\\left(r^2 {\\partial f \\over \\partial r}\\right) \n + {1 \\over r^2\\sin\\theta}{\\partial \\over \\partial \\theta}\\left(\\sin\\theta {\\partial f \\over \\partial \\theta}\\right) \n + {1 \\over r^2\\sin^2\\theta}{\\partial^2 f \\over \\partial \\varphi^2} = 0.", "027ba77426858754748114062a46ac88": "p(x|\\overline{y})", "027bb5050684378c588a0384461002dd": "w''=0", "027bce9859bd9d3f00cedb3501833432": "\\big\\updownarrow \\Big\\updownarrow \\bigg\\updownarrow \\Bigg\\updownarrow \\dots \\Bigg\\Updownarrow \\bigg\\Updownarrow \\Big\\Updownarrow \\big\\Updownarrow", "027bfcbbe3242bea7e33988be97c2e88": "G \\to H\\backslash G", "027c3429f98f7c39bab027549e1b9c7b": "a_1", "027cfb67122353f1488768c2823ea7fb": " r_\\mathrm{ corr } = r + \\frac{ 1 }{ n }( 1 - \\frac{ n - 1 }{ N - 1 } ) \\frac{ r s_x^2 - \\rho s_x s_y }{ m_x^2 } ", "027d00a2432091cf782e1dbec39e173f": "\n\\operatorname{Li}_s(z) = {\\Gamma(1 \\!-\\! s) \\over (2\\pi)^{1-s}} \\left[i^{1-s} ~\\zeta \\!\\left(1 \\!-\\! s, ~\\frac{1}{2} + {\\ln(-z) \\over {2\\pi i}} \\right) + i^{s-1} ~\\zeta \\!\\left(1 \\!-\\! s, ~\\frac{1}{2} - {\\ln(-z) \\over {2\\pi i}} \\right) \\right] ,\n", "027d12976af94786c8f656a872dbc10b": "H_{out}\\ =\\ f(H_{in},\\ m)", "027d85e20311d606467f08fa2b3fbad8": "N = M + 1 \\, ", "027e3a0f8b7e284ab68c542a1ae3489e": "V_{GS}=V_{th}", "027e72afb96af576be811f0b0465ed0c": "f:I\\rightarrow \\mathbb{R}^+", "027ea6e711e5f2c509cc7a4e6a5b64a2": "\n\\langle z^m \\rangle = \\oint p_w(z)z^m \\, dz.\n", "027eb3c1e422b4c252a3eebfef6b7432": "\n\\begin{align}\nq_\\mu^*(\\mu) &\\sim \\mathcal{N}(\\mu\\mid\\mu_N,\\lambda_N^{-1}) \\\\\n\\mu_N &= \\frac{\\lambda_0 \\mu_0 + N \\bar{x}}{\\lambda_0 + N} \\\\\n\\lambda_N &= (\\lambda_0 + N) \\operatorname{E}[\\tau] \\\\\n\\bar{x} &= \\frac{1}{N}\\sum_{n=1}^N x_n\n\\end{align}\n", "027ec8425f9c5fa3980d0a78a6024a36": "\\begin{align}\nx&=a\\cosh\\xi\\cos\\eta\\cos\\phi\\\\\ny&=a\\cosh\\xi\\cos\\eta\\sin\\phi\\\\\nz&=a\\sinh\\xi\\sin\\eta\n\\end{align}", "027efd0609b2b1a78ea698c8088fd976": "a_i[\\mathbf{f}] = \\sum_{k=1}^n v^k[\\mathbf{f}]g_{ki}[\\mathbf{f}]", "027fbc8ed2dce4562d06aecc8a04dff8": "R= \\left[\\frac{n_o (n_2)^{2N} - n_s\n(n_1)^{2N}}{n_o (n_2)^{2N} + n_s (n_1)^{2N}}\\right]^2,", "027fe9d27e81b68b9d9ac895264bb6eb": " (\\kappa-n-1)~r^{n+1}~\\cos(n\\theta) \\,", "028016a24cad05e17d89a0634c318ad0": "Y(y)=C_{1}\\cos(k_{y}y)+C_{2}sin(k_{y}y) ", "0280995012f1d43b2acd677acdf88bd1": "\\frac{v_b}{w_b}\\ge\\frac{v_i}{w_i}\\, ", "0280c97d8a46b10a8fcd21c89a15021b": "\\frac{\\partial}{\\partial x_1} f(x_1, x_2, \\ldots, x_n)\\,,\\quad \\frac{\\partial}{\\partial x_2} f(x_1, x_2, \\ldots x_n)\\,,\\ldots, \\frac{\\partial}{\\partial x_n} f(x_1, x_2, \\ldots, x_n) ", "0280e5bde4394d3371051b15d4770877": " \\Phi = \\iint I_\\lambda \\mathrm{d} \\lambda \\mathrm{d} \\Omega", "02813d27899acc3cff6ba6747ec873cc": "v\\in TM", "0281ba5a5aa825aeead474848d07516c": " \\sigma(\\mathfrak{G}^2 \\oplus \\mathfrak{G}^2 \\oplus \\mathfrak{G}^2 \\oplus \\mathfrak{G}^2) = 1", "028246bcf34addfe79858399c1dcfbfb": " I = \\begin{cases}\n 1 &\\text{if } Y \\le 1/3,\\\\\n 0 &\\text{otherwise},\n\\end{cases} ", "02825789173cc14e46546d75a3d6383c": "\\ell(s)\\ge \\ell(t)", "0282b2607138bd84dda06decc05eacd6": "D_{a}", "0283316329c94c014e656bca7c85f6cf": "{1 \\over 168}\\left(n^7 + 21 n^5 + 98 n^3 + 48 n\\right).", "0283648e14bf01495b25d91cf4d0b645": "N^1, N^2", "0283a6960393cb45f987c35f6d59bc40": "8100a + bx ( 180 - x ).\\,", "0283df13c758ef3af6a782345aba0ebd": "\\int_{-\\infty}^0 \\big[t\\inf \\mathrm{supp}X-g'(t)\\big]dt", "0283e8b43a0c3b781521213883348597": "\\rho_w", "0283eca90e0d5c9785555060675c9983": "NL \\left [ u \\right ] (x) = {1 \\over C(x)}\\int_\\Omega e^{-{{(G_a* \\left \\vert v(x+.)-v(y+.)\\right \\vert ^2)(0)}\\over h^2}}v(y)dy.", "02844aa24ce762ec1f7385b1fefac755": " x^0 = ct = c \\gamma \\tau \\, ", "028469922269efffb745f0d802201923": "k_{xo}=\\sqrt{{k_{o}^{2}}-(\\frac{m\\pi }{a})^{2}-\\beta ^{2}}= k_{o}\\sqrt{1-(\\frac{m\\pi }{ak_{o}})^{2}-\\frac{\\beta ^{2}}{k_{o}}}", "02848f7255ed6999eee0a31a8d180d03": "x\\in\\{0,1\\}^n", "0284d692fbed76eaf34b1b7bf1306aa7": "\\int_{\\mathbf{R}} \\delta\\bigl(g(x)\\bigr) f\\bigl(g(x)\\bigr) |g'(x)|\\,dx = \\int_{g(\\mathbf{R})} \\delta(u)f(u)\\, du", "0284f082f8fa0fe0c6a181cf5be904f5": "\n \\operatorname{Div} \\boldsymbol{P}^T = \\rho_0\\ddot{\\boldsymbol{x}'}.\n ", "02850d6a647bc6cdb7f44baeb1f90089": "{}^2", "0285aa7f11df22d43c8f93a2ca31a266": "x^2+y^2,", "028628dc15d1c92860d56ab2ffe88961": "7^2<103", "028653ccb2edb9857e722606c46a7ed0": " g_D=\\frac{dI}{dV}\\Big|_Q = \\frac{I_0}{V_T} e^{V_Q/V_T} \\approx \\frac{I_Q}{V_T} ", "02865d599780a233d3765bd4587aac66": " =Z_{DP}^{2}\\frac{\\hbar \\omega}{8 \\pi ^2 \\hbar\\rho c^{2}} (N_{q}+\\frac{1}{2} \\pm \\frac{1}{2}) g(E \\pm \\hbar \\omega) \\; \\; (18) ", "0287125b21317160ff3a19b3817dfaf5": "\\partial A", "0287249202504fd9925b675320d10892": "\\scriptstyle 2\\,\\frac{7}{12}", "02873b47e4f412bd6cbcf3456b898fc6": "d=701", "0287b3f6ae84d39a46d3b20287f54922": "c_d\\;", "0287b9ac9048d5360e75da1fe4462517": "\\frac{\\delta l}{\\delta t}=\\frac{[P_A+g \\rho (h-l\\sin\\psi)+\\frac{2\\gamma}{r}\\cos\\phi](r^4 +4 \\epsilon r^3)}{8 r^2 \\eta l}", "0287d19d4b2d3fb6122a9bbff178bc63": "f(z_0)", "0287e0b7e48b39993d5c5ddaab509d7f": "M(x) < N(x)", "02881915156de35c74e826ad9e4b4e0c": "\\frac{c^2k^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}\\,\n\\frac{\\omega^2-\\omega_p^2}{\\omega^2-\\omega_h^2}", "02882117772d1d5e3a33c5a7b1d80a07": " \\frac{(\\mu_2-\\mu_1)-(\\bar X_2 - \\bar X_1)}{\\displaystyle\\sqrt{\\frac{S^2_\\mathrm{pooled}}{n_1} + \\frac{S^2_\\mathrm{pooled}}{n_2} }} ", "0288211d1f3ba68261d99ee4081e59ec": " \n\\begin{bmatrix} \n0 & 0 & 0 & \\cdots & 0 & -c_0 \\\\\n1 & 0 & 0 & \\ldots & 0 & -c_1 \\\\\n0 & 1 & 0 & \\ldots & 0 & -c_2 \\\\\n\\vdots & & & & & \\\\\n0 & 0 & 0 & \\ldots & 1 & -c_{n-1} \\\\\n\\end{bmatrix}\n", "028826fbe2034c16c6a0c342139cca8b": "\\frac{v^2}{R} cos(\\theta)", "0288777665dd9e85992878a2ed64b9f3": " \\mathbf{F} = m \\mathbf{g}", "0288bde0c2d593f2b5766f61b826a650": "nu", "0288c1e7f3264a3b7e252e64e194603e": "E_T = t \\cdot C_T = t \\cdot \\sum_{n=1}^N C_n", "0288c82f52338089d23837709db7ef1c": "v(t;\\delta) = [u_0 \\cos \\delta t + v_0 \\sin \\delta t]e^{-t/T} - \\kappa E_0 \\int_0^t dt' \\cos \\delta(t-t')e^{-(t-t')/T}", "0288d538bd34157467f10e06eb170231": "\\sum^{k}_{i=1}1/\\lambda^2_{i}", "0288d57113c941d383db2aabb92482a7": "\\mathcal F(f)", "028954e78ec6d14a116374223c3da8dc": "\\Eta \\, \\eta \\,", "0289624a13094f127d206a236f58b67e": "G = g_{64},\\text{ where }g_1=3\\uparrow\\uparrow\\uparrow\\uparrow 3,\\ g_n = 3\\uparrow^{g_{n-1}}3,", "02896fcc232388e90696f4a0d8854552": "\\underline{P}(A^c)= 1-\\overline{P}(A)", "0289b2445bb4a1bdd64b9408dd9f3c1c": "\\zeta(6,1)+\\zeta(5,2)+\\zeta(4,3)+\\zeta(3,4)+\\zeta(2,5) = \\zeta(7)", "028a0b0b638a3450112e43a6bee6f048": "_{qp+qp'=qp+q'p\\,}\\!", "028a2a14f97cd167204572e12a98662b": "\\Delta p = f \\cdot \\frac{L}{D} \\cdot \\frac{\\rho V^2}{2}", "028a4658f0b06437681dd13bec5f1f4f": "\\operatorname{erf} (-z) = -\\operatorname{erf} (z)", "028abbffb1dd2d77f036b6d6eca2fc8a": "|x^\\mathsf{T} y|\\le\\|x\\|_2\\|y\\|_2.", "028afe481ca2232314d8cb365a1d1036": "\\scriptstyle{|0\\rangle \\equiv |\\psi(t_0)\\rangle}", "028b2eff77bbc39b102bacc22f2647d3": "f\\colon X\\to X", "028b31dd2dc7f3c03703933cc3d7d225": "\\langle r, s \\mid s^2 = 1, srs = r^{-1} \\rangle \\,\\!", "028bb5478e94d5964115b1af37f9c943": "R(\\theta ) = R(0) + G \\sin^2 \\theta ", "028bb9dd25511f8cdf762d3a3ba1b106": "\\ VDOP = \\sqrt{d_{z}^2}", "028c0d0db7547cf5aa11fea485157522": "e > 0", "028cb29c5821522a92e6dea284904c39": "W_n \\propto n,", "028cb3f87cb4f887fc0cefb603c2b051": "c({\\mathbb B})", "028cf3d120efa3656bf48d357ac1db7b": "\\delta(x-\\xi) = \\sum_{n=1}^\\infty \\varphi_n (x) \\varphi_n^*(\\xi). ", "028cf8d5e59f6aba6cf4d037caae5e4b": "\\nu(H) = \\tau(H)", "028d3c56485a12720db1e16c9e5ecc4b": "\\mathcal{L}(\\phi,\\partial\\phi,\\partial\\partial\\phi, ...,x)", "028d70ffc43db3b4f608e733a123472e": "a \\in \\mathcal{U}", "028dffcf9ec5fe1c9242c20d65a37f27": "S(u) = \\mathrm{sinc}^2(u) = \\left( \\frac {\\sin \\pi u}{\\pi u} \\right) ^2 \\ ; ", "028e0e9d0bc8702bdbf96b7d5328a941": "M \\models", "028e6e505cf7e6a9ec95c45c61d40527": "x_1(t)=x_2(s), \\ y_1(t)=y_2(s) \\ .", "028e7c230401be6584e89b2d13f261d6": "P_K=32.1\\,d", "028ec436855047c2bffa0b383f7936ea": "W_2 =2\\gamma(e_{ij})A(e_{ij})", "028ec8468b090fa5c5d1de11a4fbe39c": " \\textstyle \\sigma_A = \\prod_{j \\in A} \\sigma_j ", "028ed1eec4f4304627517d9d1fb582ae": " \\Psi^{(\\operatorname{Sha}) }(w) = \\prod \\left( \\frac {w- 3 \\pi /2} {\\pi}\\right)+\\prod \\left( \\frac {w+ 3 \\pi /2} {\\pi}\\right). ", "028ee608b65c2b27cbb42c981e683264": " z^j p_1^{k_1} p_2^{k_2} \\cdots p_n^{k_n} q_1^{\\ell_1} q_2^{\\ell_2} \\cdots q_n^{\\ell_n} \\, \\mapsto \\, \\partial_{x_1}^{k_1} \\partial_{x_2}^{k_2} \\cdots \\partial_{x_n}^{k_n} x_1^{\\ell_1} x_2^{\\ell_2} \\cdots x_n^{\\ell_n}.", "028f0c8bb26e1c6568d9feab0a9aa322": "L^p(\\mathbb{R}^n)", "028f38e2f0bc026f9a896d05d579d591": "c_{f,g\\circ h} \\cdot c_{g,h}(f^*(x)) = c_{f\\circ g, h}(x)\\cdot h^*(c_{f,g}(x)).", "028f51e014d288ea668ba5bacc32b683": " v_1 ", "028fd9e7f7bd2a7b04fd2c98b58e90b6": "\\cdots \\,\\leq\\, a_3 \\,\\leq\\, a_2 \\,\\leq\\, a_1", "02900c654e9c50288d2d779994a76b8d": "\\displaystyle \\nabla^2\\omega + \\frac{f^2}{\\sigma}\\frac{\\partial^2\\omega}{\\partial p^2} ", "02907a65f48839c3ed37e198eb8c0afd": "\\tau_{zx}=-\\nu \\frac{\\partial \\rho\\upsilon_x }{\\partial z}", "0290924c27e43ac698cc8659e787a33d": "f_1(z)=\\frac{(1-i)z}{2}", "029099ba5237fd8d6213efbcf3af7836": "\\lambda\\ge 0", "0290a332e92b98cd127f2489d929ecf4": "B^\\prime", "0290a88f14fb8b753331e1f64d60cd86": "\\lim_{c\\rightarrow -m}\\frac{{}_2F_1(a,b;c;z)}{\\Gamma(c)}=\\frac{(a)_{m+1}(b)_{m+1}}{(m+1)!}z^{m+1}{}_2F_1(a+m+1,b+m+1;m+2;z)", "0290c93ee0b15b0416d8286de7bce6ed": "dA_1= \\left(\\mathbf{n} \\cdot \\mathbf{e}_1 \\right)dA = n_1 \\; dA,\\,\\!", "02910523462ad6edcc3f5a16357bded9": "{\\mathrm MinN}(L+1,D,n) \\le 2{\\mathrm MinN}(L,n,n)", "02913629d042aa4f6f8a17b3fd183ea9": "\\scriptstyle P\\,\\sim \\,\\rm{Exp}(\\alpha)", "02914341676b562677f2898686ad23a5": "\\Delta(z)=\\sum_{n> 0}\\tau(n)q^n=q\\prod_{n>0}(1-q^n)^{24} = q-24q^2+252q^3+\\cdots", "02914e47be2a3a1357d398d3ea761c04": "\\nu Z.\\phi \\wedge [a]Z", "0291c9aa28e46d45a874837ffcddc44a": "L_{t} = \\lim_{\\varepsilon \\downarrow 0} \\frac1{2 \\varepsilon} | \\{ s \\in [0, t] | B_{s} \\in (- \\varepsilon, + \\varepsilon) \\} |.", "0291f94c8ebd42d6d8c456051f0aa4f0": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & -1 \\\\\n-1 & 2 & -1 & 0 \\\\\n 0 & -1 & 2 & -1\\\\\n-1 & 0 & -1 & 2 \n\\end{smallmatrix}\\right ]", "02922b6dcee7ce520b4efa977df1ecca": "A + 2B \\rightleftharpoons AB_2; K_\\text{c} = \\frac{[AB_2]}{[A][B]^2} /\\text{M}^{-2}", "0292eaab254e16d4760c6f6bbfcdd495": " \\psi_t = K * \\psi_0 \\, .", "029300e27efb7a2ac2857174169a9d3e": "\\frac{1}{M \\cdot s}", "029303301b802635239a678d54b41738": "\n \\begin{align}\n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q & = 0\n \\end{align}\n", "029314defb85735cc6d46b8e19c0e1c9": "\nR=\\frac{-n(n-1)}{\\alpha^2} \n", "02941d9d7a8a4647de2c3487d03cc029": "p_3 = p_1", "02942b66a4d9733e165e86739d9ee08a": "\\begin{align}\n{\\mathbf{S}}_i \\cdot {\\mathbf{S}}_j \n&=& \\sqrt {\\left( {1 - \\sum\\limits_\\alpha {\\sigma ^2 _{i\\alpha } } } \\right)\\left( {1 - \\sum\\limits_\\alpha {\\sigma ^2 _{j\\alpha } } } \\right)} + \\sum\\limits_\\alpha {\\sigma_{i\\alpha } \\sigma _{j\\alpha } } \\\\ \n&=& 1 - \\tfrac{1}{2}\n\\sum\\limits_\\alpha \\left({{\\sigma ^2 _{i\\alpha }} + {\\sigma ^2 _{j\\alpha } } }\\right) + \\sum\\limits_\\alpha {\\sigma _{i\\alpha } \\sigma _{j\\alpha } } + \\mathcal{O}(\\sigma ^4 )\\\\\n&=&\n1 - \\tfrac{1}\n{2}{\\sum\\limits_\\alpha {(\\sigma _{i\\alpha } } - \\sigma _{j\\alpha } )^2 } + \\ldots \n\\end{align}\n", "029455c92e4ec0a699d96e23d763d9d5": "\\hat{R}_n(f) = \\dfrac{1}{n}\\sum_{i = 1}^n \\mathbb{I}(f(X_n) \\neq Y_n)", "0294705005b27d51c6578400f31f9dab": "u_0=1,\\;v_0=0,\\quad u_1=0,\\;v_1=1,\\quad u_{k+1}=u_{k-1}-q_ku_k,\\;v_{k+1}=v_{k-1}-q_kv_k", "02949639dff879b56cef44160bc985c7": "\\mu_G", "0294a2fe08a3f956ae3ebbb08b074ff6": "\\varepsilon \\left[ M \\right]", "0294b227d4b07bc2935e707e4fa80dd3": "V_k(\\mathbf{R}^n)", "0294c46a2098b46bb343e59580803c2e": " \\ v_{1}-v_{2} = u_{2}-u_{1}", "0294f1e2c6a908d9529b5e98da9d3692": "A = \\frac{9}{4}a^2\\cot\\frac{\\pi}{9}\\simeq6.18182\\,a^2.", "0294fa4e2ed32efe774506d31349c49b": "\\pi S \\sin_n \\theta/\\lambda =n \\pi, n = 0, \\pm 1, \\pm 2,..... ", "0294fa66e2860bf2e2edfc6c2b7c3c22": " \n(1-\\epsilon)\\int_{A_k} \\phi \\, d\\mu_k \\geq (1-\\epsilon)\\int_E \\phi \\, d\\mu_k - \\int_{A-A_k} \\phi \\, d\\mu_k.\n", "029504ab8797ac64df7858c45b1e55b7": "R^\\ast", "02950734961dab76e76bf41728978d00": " \n1) \\quad H(\\empty) = 0\n", "029551905c3f89f00943794b4a2472ed": "g(n, m) = g(n,m-1)+X_m g(n-1,m).", "029561ef26841a2f06634549502d4c5c": "n p = \\omega(\\sqrt{n \\log n})", "0295a42715cf1a1df8cbb20cadfc74f8": "m=0.1n", "0295d8022555242acf9d0af9ee886c46": "x \\le \\frac{l}{2}\\sin\\theta.", "02965244c0ce67304c8f1cb2aa6faa6a": "F = DUV^\\top", "0296682e2a43df25157421f698988f49": "A=[0,1],", "0296de2566416c7d1dfeaf80ff6f3e96": "R(\\hat{n},\\phi) \\equiv \\exp\\left(-\\frac{i}{\\hbar}\\phi\\, \\mathbf{J}\\cdot \\hat{\\mathbf{n}}\\right)", "029768646ed4136a5e6baa9fde70eb83": "{m \\choose r}_q = {m \\choose m-r}_q. ", "0297a16e0134ecb464609f6e4d9ff403": "M_{k,j}=\\mathrm{ln}\\;(M_{k,j} / b_k).", "0297d156aad07b49a45ad666f31bbc70": "S(T) = C \\left(1 + \\frac{A}{T}\\right) - B", "029801ec7f67318dff1e0adc221317e4": "H(f) = \\mathrm{rect} \\left( \\frac{f}{2B} \\right)", "02980a825b993aa7c1e70d410418471c": "\nX \\sim \\mathrm{BNB}(n,\\alpha,\\beta).\n", "02983fb8a36cec4b1d87d21cff61e331": " H^G_*(E_{FIN}(G),K^{top}_{l^1})=H^G_*(E_{FIN}(G),K^{top})\\rightarrow H^G_*(\\{\\cdot\\},K^{top})=K_*(C_r(G))", "02985e0a38eeffec4e784d6f82ff6935": "\\operatorname{Cl}_2\\left(\\frac{3\\pi}{4}\\right)=\n2\\pi\\log \\left( \\frac{G\\left(\\frac{5}{8}\\right)}{G\\left(\\frac{3}{8}\\right)} \\right) -2\\pi \n\\log \\Gamma\\left(\\frac{3}{8}\\right)+\\frac{3\\pi}{4}\\log \\left( \\frac{2\\pi}{\\sqrt{2+\\sqrt{2}}} \n\\right)", "0298986fbd7961975bbb5c7b6cc7e7c8": "N_{\\rm A} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e})}{m_{\\rm e}} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e})c\\alpha^2}{2R_\\infty h}", "02989b2a62a0a1e102b65c1794ef4d28": "\\lfloor \\frac{d-1}{2} \\rfloor ", "0299254c9469af661203d1a69d80df20": "\n\\bar{\\delta} {\\phi^A}_{,\\sigma} = \n\\bar{\\delta} \\frac{\\partial \\phi^A}{\\partial x^{\\sigma}} = \n\\frac{\\partial}{\\partial x^{\\sigma}} \\left( \\bar{\\delta} \\phi^A \\right)\n\\,.", "0299430ed9ef9635331dcdcbe5ba1cba": "p_{j}", "029945c0ee0ac4a1e09274775c84fb07": "\\|\\hat{f}\\|_{L^q}\\leq p^{1/2p}q^{-1/2q}\\|f\\|_{L^p}", "0299d91026ddf50fefe05f8f092e2b42": "F_{\\nu}(k)\\,", "0299e342cc72a82c1bab9f222b5d88eb": "f''-{1\\over z}f'+{1-z \\over z^2}f=f''-{1\\over z}f'+\\left({1\\over z^2} - {1 \\over z}\\right) f = 0", "029a7fb4d52c5e9553d51cded8ab7924": "\\begin{align}\nL_f \\equiv \\underline{\\int_{a}^{b}} f(x) \\, dx &\\quad U_f \\equiv \\overline{\\int_{a}^{b}} f(x) \\,dx\n\\end{align}", "029b00b7fa24dcb97246d2df373ef28f": "\\frac{59049}{32768}", "029b0be093d8080d1a61e22ee093c57f": "m \\mid p-1", "029b156f8ba178c2301eb71ef498be1c": "y(t) = y_{(1)}(t)+\\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)", "029b363145195144d94b9ec7a854d54c": "dF = -b_\\text{ext} F dx", "029b3cdf1db812cf8147a10c6a08ddce": " \\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n}}\\equiv\\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n-\\wp(p-1)}}\\pmod p\\!", "029b46fb564d16ed8e2249b044615d7e": "n \\ge k", "029b596e37b45dede3df04654bec7ad0": "z_j \\mapsto iz_j", "029b72733492d85b84e82a3e01a9f2d2": "\\ge i", "029b93561645fd5d2a54de2c6f1768cd": " S_n = 1,1,\\frac{1}{2},\\frac{1}{3},\\frac{5}{24}, \\frac{2}{15},\\frac{61}{720},\\frac{17}{315},\\frac{277}{8064},\\frac{62}{2835},\\ldots ", "029ba64831d61ee5b6ef200ac8e7d816": " 2ax = -b \\pm \\sqrt{b^2-4ac} ", "029bd0d5c84b6da53e6262aee62b9dd7": "\\Pr(X=k)=F_{\\chi^2}(2\\lambda;2(k+1)) -F_{\\chi^2}(2\\lambda;2k) . \n", "029be1310b8d6075d3d3e51646d05035": "\\; AP = PJ.", "029be45299faa8334bfe288dad23166f": "\\triangle\\delta\\;=\\;\\delta' - \\delta\\; ", "029c34a36a9fe3e058eaadec6db2d0ec": "\\boldsymbol{\\sigma} = -p\\mathbb{I} + \\mathbb{T}", "029c49bad246fd00fb9fe7d17da86435": "\\gamma_2\\,", "029da23ae63f51c12d40401dd23f6d72": "x^5-9x^4-81x^3+729x^2=3888", "029da49e91a3c1e1bf3aa0faa118ad77": "\\frac{d^n\\bigl(f(x)\\bigr)}{dx^n}\\text{ or }\\frac{d^ny}{dx^n}", "029deb3f9f7ca701edbbb85b7090275f": "- \\mathbf{\\hat{n}}", "029e39793c1f7b96788cabcd8b6bf878": "\\varphi_p(x)=\\frac{1-x}{1-px}.", "029e3cfe2abae8da550ed0f34d8e3d4b": "\\sqrt{a^nx^2 + \\frac{a^n - 1}{a - 1}b}", "029e82e73de3aa9bc60f6c5f3f5f69d8": "\\operatorname{E}\\bigl[(X)_r\\bigr] =\\lambda^r.", "029e9fdca2dde780d3f91df94b9e8428": " \\frac{1}{4} |\\langle (\\hat{A} \\hat{B} - \\hat{B} \\hat{A} )x | x \\rangle|^2\\leq \\| \\hat{A} x \\|^2 \\| \\hat{B} x \\|^2.", "029ea052b66d68b4bc63bafdf60f58a9": "w_{i,j} \\,\\sim\\, \\mathrm{Multinomial}( \\phi_{z_{i,j}}) ", "029f0acdea6ba0c6e8d87bba2ca9aeec": " [M]_{C}^{B} = \n\\begin{bmatrix} \\ [b_1]_C & \\cdots & [b_n]_C \\ \\end{bmatrix} \n", "029f1578c56213de6e29eb7279760254": "\\tilde{R}=\\Phi^{2/(d-2)}\\left[ R + \\frac{2d}{d-2}\\frac{\\Box \\Phi}{\\Phi} -\\frac{3(d-1)}{(d-2)}\\left(\\frac{\\nabla\\Phi}{\\Phi}\\right)^2 \\right]", "029f239d5dc25b4312dbcc33e6430b61": "\\sum_i z_i^2 = 1", "029f2a6a6b4614b43ef44e5211ccfeb8": "S = \\operatorname{Spec} A", "029f2b00d7f140a55a20f049cd6819b3": "Z \\sim \\mathrm{Binomial}(2,p) \\,\\! .", "029f47f73eb96e10b6a61db250f1c89d": "\\scriptstyle x \\,-\\, y", "029f58ea2f6c582eff6e3810e56d80e3": "\\exists\\,c>0 \\mbox{ s.t. } \\langle Au-Av , u-v \\rangle\\geq c \\|u-v\\|^2 \\quad \\forall u,v\\in X.", "029f73d4e4e0cb442c83fdf23e0739b3": "\\{b, (o_1,0);(a_1,b_1),\\dots,(a_r,b_r)\\}\\,", "029f7e7ed8abd49bef1f978b22d6d0b7": "\\mathrm{Hol}_p(\\omega) = \\{g \\in G \\mid p \\sim p\\cdot g\\}. \\, ", "029f823c27a63a7007f99583a9699f32": " Z_i= \\left( \\sum_{j=1}^k W_{i,j} \\right) \\pmod {(m_1-1)} ", "029f9d3b498c3d36842b1ef06942b43d": "{}^ap_i = K_a \\cdot H_{ba} \\cdot K_b^{-1} \\cdot {}^bp_i", "029ff1facef6c4a0f4948e8942c3799c": "\\varepsilon_{\\alpha_1 \\dots \\alpha_n} \\,", "02a05bfedf23700420abe2fc04cb2274": "\\bar F(x) = \\sum_{x_i < x}p(x_i) + \\frac 12 p(x)", "02a085c946818133e362fa13db6a2dc9": "r = m - qp_k^{\\sigma_k}", "02a0ef3f90137c9fe63bbd0f51a02934": "\\mathcal{C}_w", "02a12eda9c985f7d14689d0e8727262e": "l = 0,", "02a1a67ae77cb73f7cbcc9268a5cef91": "\\begin{align}\nH_n &= \\int_0^1 \\frac{1 - x^n}{1 - x}\\,dx \\\\\n&=-\\int_1^0\\frac{1-(1-u)^n}{u}\\,du \\\\\n&= \\int_0^1\\frac{1-(1-u)^n}{u}\\,du \\\\\n&= \\int_0^1\\left[\\sum_{k=1}^n(-1)^{k-1}\\binom nk u^{k-1}\\right]\\,du \\\\\n&= \\sum_{k=1}^n (-1)^{k-1}\\binom nk \\int_0^1u^{k-1}\\,du \\\\\n&= \\sum_{k=1}^n(-1)^{k-1}\\frac{1}{k}\\binom nk .\n\\end{align}", "02a25184b4eb25d0888e13e4571c4e73": "v_f = v_i + at\\,\\!", "02a268132f9bea64808285dd7ddfd432": "i_W", "02a27131b649c53f8a3bc5a4220d22b7": "{R_{fu}=-C_R\\rho m_{fu}\\frac{\\varepsilon}{k}}", "02a2740281b8da515d4444d976ed585a": "G^{\\mathrm{A}}(\\omega)", "02a2e0d4b8c6f07c1077ec6933a47ff3": "\\sigma_\\epsilon^2\\,\\!", "02a2f1e9d34b560b2d6578efb66625bb": "\\varepsilon_E \\,", "02a2f9ca16667bfa29f9d5ae7e78bb87": "(x-z_1)", "02a33c6048e63a2364231d9232ecd944": "< \\Psi |", "02a33ecd982937569d6efb469ed94926": "kX \\sim \\beta^{'}(\\alpha,\\beta,p,kq)\\,", "02a36e7fdef57cedde35a70770e6b12d": "\\frac{1}{\\sqrt{0.15625}} = 2.5298221281347", "02a3af029f929b552194e7edc94f90a7": "\\scriptstyle g \\colon X \\to Z^Y ", "02a3ed1c01f6b16819a27864c3db7825": "\n F^{(n)} = F \\circ F \\circ \\cdots \\circ F.\n", "02a496ef738253943cadde022a8c15ce": "\\mathcal{G}(\\omega_n) \\sim 1/|\\omega_n|", "02a4a4d2acef6c919c157f917f61ca0d": "\n\\delta_{\\mathcal{S}_1} = -x \\dfrac{\\partial}{\\partial x}+b\\dfrac{\\partial}{\\partial b},\n\\quad\n\\delta_{\\mathcal{S}_2} = t\\dfrac{\\partial}{\\partial t}-x\\dfrac{\\partial}{\\partial x}-a\\dfrac{\\partial}{\\partial a}\n\\cdot\n", "02a4a53113d4b16223166c1e3fa68968": "h_{i}(x_{k})=h(x_{k})\\,\\!", "02a50e638d6d146745e4e78d25262300": " \\mathrm{d} \\mathcal{A} = \\mathrm{d}^2 \\Sigma \\sqrt{-g} ", "02a52052ff78b9f4ed628a12c5332ab1": "r\\to 0,", "02a565347865103393c99c62bc6a4e30": "\\frac{ dx }{ dt } = y,", "02a62656269e40d2750f95eabc4e5f77": "H = k\\, \\log(1/p),", "02a6394c29f1f96eee407b857c2cc64d": "\n\tE_4 = \\begin{pmatrix}0.792608291163763585\\\\ 0.451923120901599794\\\\ 0.322416398581824992\\\\ 0.252161169688241933\\end{pmatrix}\n", "02a66174fd3b9c026b1e7c5445d8db45": "{\\nu}^{\\lambda}={\\kappa}", "02a66708e288fbc817a0ddf468f38efd": "\\Pr(X_{t_{n+1}} = i_{n+1} | X_{t_0} = i_0 , X_{t_1} = i_1 , \\ldots, X_{t_n} = i_n ) = p_{i_n i_{n+1}}( t_{n+1} - t_n)", "02a6a5c1251f306d41f15f3ef9364d03": "\\prod _x \\cot x \\tan (x+1) = C \\tan x \\,", "02a6d8f854966e30ce61aad85869d6cd": "LC_{50} \\le 3000 \\tfrac{mL}{m^3}", "02a72fcb2a78037d92876aa41ba392c3": " s = j \\omega ", "02a9502832ce7aedad99b585dcb0271d": "u'' \\equiv {{d^2 u} \\over {d z^2}}", "02a953305e87271cb9e0a038acd3fd16": "188461 = 7 \\cdot 13 \\cdot 19 \\cdot 109\\,", "02a97f679d658932e27850dac019213c": "(A\\mid(B\\mid C))\\mid[((D\\mid C)\\mid[(A\\mid D)\\mid(A\\mid D)])\\mid(A\\mid(A\\mid B))]", "02a997eeb22ff6ee536a061e69655fdf": "r + s = t", "02a9d21cec20b2cf4021f7a19c1802b5": "C_1 \\cap C_2, C_1 - C_2 ", "02a9d64ddaff7bb25d1e0c3f412a9f77": "1-P_{m1}=P_a+P_d", "02a9ef7e14fdade7b8f566b5998bc717": "p_K:X \\rightarrow [0, \\infty)", "02aa2c6cb2d9927af06c30ee957aebfa": "sim(q,d)=\\sqrt[p]{\\frac{(1-\\sqrt[p]{(\\frac{(1-w_1)^p+(1-w_2)^p}{2}}))^p+w_3^p}{2}}", "02aa3b183049f14b8b3e5a538dc66ec2": "2^5\\cdot 3^2\\cdot 5\\cdot 7", "02aa9d2075ab9913a0326bc83973377f": "\\hat{e}_{\\nu}", "02ab3cdbc6bd00bdd71ac855c4b01bf6": " S = (\\lambda x.f\\ (x\\ x)) ", "02ab61395ebb9d4dbbb80e6e83da48bd": "A_{\\text{OL}}=\\frac{V_{\\text{out}}}{\\left(V^+-V^-\\right)}", "02aba4f16e4ba0af71aa9b43d3279317": "\\Re(z_n)= -n+1/2", "02ac03c9cdbf7d67a8c49ff8f41b4883": "\\dot{a}>0", "02ad2f550bbcff52fb668d5073b4eebe": "E[F_6].", "02ad9a8cd8bd3d22af44319a20aea411": "SU(N)_L \\times SU(N)_R \\times U(1)_V \\times U(1)_A ~,", "02addf7ddfd0dd5d9906b0daf73bd117": "Rejection Region{{=}} \\frac{{t_{\\alpha/2}}{n-1}}{\\sqrt{n}\\sqrt{n-2+{t_{\\alpha/2}^2}}}\n", "02ae1a416804acd17c7081a0f762dc95": " f:X\\rightarrow Y", "02ae3b92103e815bd84faf7ce6a3bb38": " \\operatorname{sink-tran}[(\\lambda N.B)\\ Y, X] = \\operatorname{sink-test}[(\\lambda N.\\operatorname{sink-tran}[B])\\ \\operatorname{sink-tran}[Y], X] ", "02af5f7fba9c1914973ea7efb35b0492": "\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{n}})", "02af8eb3eae4a0e30c706a011d84ee57": "W_L = \\frac{L_{QA}I_B^2}{2} = \\frac{\\hbar \\omega_B}{4}. \\ ", "02af94cd045e8619f00bb5bc7f59cae3": "\\tilde{P_n}(x) = P_n(2x-1)", "02b08642ac753e4ada3a05f4844347e0": " \\{ i \\mathbf{e}_{\\rho} \\}", "02b0fd4198f0d70aef64bf4d9bf0494d": "\\sum_{n=1}^N {f(n) \\over g(n)},", "02b1330a8aa507a073ddb972722e0d5a": "n=\\,-1", "02b1609366b6412ab4943bb3f6105417": "S_4 \\times S_2", "02b16947d6295f6f8949e95d0cc21448": "n = \\frac{P}{100}(N+1)", "02b1c9b886a073fe60883860cdefec25": "\\Delta G_{v+\\frac{1}{2}}=G(v+1)-G(v)", "02b1cd66a35997d9835de799aaaf5f86": " \\mathcal{X} ", "02b1ea0129c77d6f313cef123c252948": "\\lim_{V_{m}\\rightarrow0} \\Phi_{S}\\ne0", "02b1f54a98d3481f62b0ce87972d3b66": " (a, b) \\cdot (c, d) = (ac - bd, bc + ad).\\,", "02b219f647d76be16a3327962f87714c": "\\int \\operatorname{Ei}(x) \\, dx = x \\operatorname{Ei}(x) - e^x", "02b22383bb6bbead7f811b992c9a8025": "\\bigvee A", "02b2d0cbe95af83f6677b7aef4714558": "F = F_{\\alpha\\beta}dx^\\alpha\\wedge dx^\\beta", "02b306c557e51c40c3c7a089a87263dd": "\\scriptstyle c=G=1", "02b349a495e3836e460d8df79d664c17": "\\frac{\\pi}{4} = 2 \\arctan\\frac{1}{3} + \\arctan\\frac{1}{7}\\!", "02b37cabaecc0443326393e450028761": "= \\cos(\\phi(t)) + i\\cdot \\sin(\\phi(t)).\\,", "02b39c4bea11d679ef78cad17231b4d8": "a^n", "02b3e5c70d8f5b967b06865e21354dee": " \\scriptstyle E_{\\rm C} - \\mu \\gg kT", "02b40b0b8c70fccbed1c35172fae0ddb": "x_1 = 10^{0.2192318 - 0.2706462} = 0.888353", "02b41d27bd7283f6711f3f642d4eea89": "e = \\sqrt{1+\\frac{2E\\ell^2}{m^3\\gamma^2}}", "02b425e85a51ff18085dd95dbdcd40f7": " r = e^{i\\theta} \\to", "02b43ed6a76fbc6c03fe07e36f93e15b": "\\text{Li}_n(z)=\\sum_{k=1}^{\\infty}\\frac{z^k}{k^n} \\quad \\Rightarrow \\text{Li}_n\\left(e^{i\\theta}\\right)=\\sum_{k=1}^{\\infty}\\frac{\\left(e^{i\\theta}\\right)^k}{k^n}= \\sum_{k=1}^{\\infty}\\frac{e^{ik\\theta}}{k^n}", "02b48d2f289c6fcfcf1390ea7f3c0b78": "g_{n,k}(r)=A\\rho^\\gamma e^{-\\rho/2}\\left(Z\\alpha\\rho L_{n-|k|-1}^{2\\gamma+1}(\\rho)+(\\gamma-k)\\frac{\\gamma\\mu c^2-kE}{\\hbar cC}L_{n-|k|}^{2\\gamma-1}(\\rho)\\right)", "02b4948c18ccacef4be3a4fab3fabefb": "\\angle CAD = \\angle CBD", "02b538208f7a3bfb2142ac071c421c5e": "\\operatorname{Stick}()", "02b54bbc6a2a4c9cc43f338d36eef7e6": "\\omega \\in \\Omega_{Z,[0,t]} ", "02b564d6c4362c7129de39b2869e5277": "\\hat{L} = L(\\hat{x}, \\hat{\\lambda}_x, \\hat{p}, \\hat{\\lambda}_p)", "02b5658332059fc008e0a85535226677": " \\mu_{ij} = \\left\\lbrace\\begin{matrix}\n1 & \\text{if point }m_i\\text{ corresponds to point }s_j\\\\\n0 & \\text{otherwise}\n\\end{matrix}\\right. ", "02b5bb0d9d973a41b97006936e25c039": " \\sum_{n=0}^\\infty \\pi_n x^n = \\prod_{k=1}^p (1-x^{h_k})^{-1} ", "02b5de8a4de2bc034a849e1a42563e30": " \\sum_{n=0}^\\infty z^n,", "02b5ec030f7f9ebd5047150eae4e2b9c": "\\psi: J(E) \\rightarrow E", "02b604b09e79c0129babbb011f6d5661": "\\mu=\\sqrt{2}\\,\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}", "02b62b40df26d691e9ff9341f234e122": "\\ x/y = y/(x/2)", "02b6be5adfc86aa1f46d986bdf1acd2b": "\\delta < f_P", "02b6d397e015e480420f59701c1a1d26": "\\propto \\!\\,", "02b738fcd1ab79e389bb87198fb4b0f0": "xy^2z > x^2z^2 > x^3 > z^2", "02b74a3bdf4e5db66a093867f1ff1eb2": "101011_2", "02b75d49cc982846f5bfcfdcb49bac27": "\\frac{R_{\\text{ac}}}{\\mu L} = a B_{\\text{max}} f + c f + e f^2", "02b78fcd7b8325bc8afd228b00a7e400": "\n\\leqslant \\int_{1}^{\\infty} 2 f(x) \\sqrt{1 + f'(x)^2} \\,\\mathrm{d}x\n", "02b7b8f652b409bddb9defbcd33e9f18": "\\phi^i=-E^{,i} \\, ", "02b7d06a7f926ebbdb471e187c679023": " \\theta\\,\\! ", "02b7f1e422461eb3fd9a2506826d6218": "\\beta_{k} < \\frac{1}{4}", "02b86afb8959f20906caea2f1ee51409": "0\\rightarrow T_xM \\rightarrow T_xP|_M \\rightarrow T_x^\\perp M\\rightarrow 0.", "02b8ac4a93108652a08604e595b2169e": "(a*b)*c = a*(b*c).\\,", "02b8e23f7fb1e05df6f54a70c71e9345": "\\mbox{MS}(a)=\\max_j L_j(a)", "02b8f0fbe6de66899c009ee691ebb11b": "\\mathbb{P} (x \\in X) ", "02b8f697eb1b2083507fcd85e15dc5ca": "f \\in BMO", "02b8ff40bc81a62770626767a7432b0c": "H_{eff}", "02b90b97ee9ab3050bc8933c14dc031c": "\\overline{x} = \\left ( \\alpha_{ij} \\right )^{-1} \\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 0.3013 \\\\ 0.4586 \\\\ 0.1307 \\\\ 0.3557 \\end{bmatrix}.", "02b91f833a4fd18295bd710d3cc01ef8": "\nC^{S_2}_{E_1} = \\varepsilon^{2}_1 / D\n", "02b92d672582f9ece902c8fa66467f62": " ELA\\,\\!", "02b983191ce736331e1184228497076c": "D\\in\\mathcal{D}", "02b98c25514e6be1b3c6a42e6d794aa5": "7x^2y^3 + 4x - 9", "02b99714e6fd241374ae2ac483902911": "1 \\times \\sqrt{7}", "02b9ce5a4ce10b6965558f07c7c900f1": "n_A+n_B=N", "02b9e2009c8050e8d9a6804348fb8695": "(\\mathbf{J}_1, \\mathbf{E}_1)", "02b9ea85f5337414776557bcd19e37d7": "\\cfrac{G+C}{A+T+G+C}\\times\\ 100", "02b9f50cf32fe169808f11b48f682756": "FG = 1", "02ba02bd287c46cfd61b1056c02e4b1c": "\nH(I|J) = -\\sum_{i,~j} P_{I,J}(i,~j) \\log P_{I|J}(i|j) .\n", "02ba0742fe67f11eff16e880b77226b0": "L_0\\,\\!", "02ba5c2d321b212c8e5f1ea179e1785a": " Q_A = \\mathcal{M}.Q", "02ba8cffb6c1b287f7b2e0b801c7e8cc": "g^*(\\tau) = \\left(\\frac{i}{2}\\right)^{k - 1} \\int_{-\\overline\\tau}^{i\\infty} (z+\\tau)^{-k}\\overline{g(-\\overline z)}\\,dz= \\sum_nn^{k - 1}\\overline {b_n}\\beta_k(4ny)q^{-n + 1}", "02ba92ad3d37d5e3f06a828837a17d7a": "(Fa \\or Fb) \\leftrightarrow Fd", "02bb57797ba7e0a76063c2edd9191fbb": " Z=\\frac {a_j}{\\lVert a_j \\rVert} ", "02bc005b15bc011bf2cae4b1c1a79c12": "\n\\begin{pmatrix} y_1 \\\\ y_2 \\end{pmatrix} = \\frac{1}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} ", "02bc089f3e9a73d15e8b0f8bc64051d7": "\\hat{x} \\in W", "02bc2631e660c2c59fe6f2f762de0e3a": "(1-\\frac{1}{e})", "02bc9a95236d772fe37193c3aa2c41a4": "\\ G, \\ ", "02bd1381b2387efff7922351b7ec5d8e": "\\epsilon_0 ", "02bd3005e4504960ad57347b2cd4a62e": "n > 1/d", "02be125bcc84d55428d554f276f996bb": " m(\\mathbf{f},\\mathbf{g})=\\prod_{i=1}^N (a_i - b_i +1)", "02befb6993a658e5a0ab7db18c9ddd3d": "\\scriptstyle{\\sum_{j =1}^m P_{ij} = 1}", "02bf4aa45a5414e118fc5b8836daabcf": "\\leq \\Pr[B]+\\Pr[A|B^c]", "02bf538b901eaf61cfe4a0d0b78e22a1": "\\max_{j \\neq i} b_j", "02bf98e606c3288280c9519791bc1a48": "m_0,m_3,m_5,m_6", "02c0176317f7c9c389ed000666afd07f": "A + C \\leftrightharpoons AC; K_{AC}=\\frac{[AC]}{[A][C]}", "02c0b388ae9b0a8452bd8b53c3e25707": "3(4x^2y-6y)+7x^2y-3y^2+2(8y-4y^2-4x^2y)\\,\\!", "02c193d45dbd50de7b409c5454f045d6": "\\alpha = 2 - \\frac{\\tau - 1}{\\sigma}\\,\\!", "02c1b2634f3db0d4515eee02d79b0537": "\n\\begin{align}\nE_{2n} &=(-1)^n (2n)!~ \\begin{vmatrix} \\frac{1}{2!}& 1 &~& ~&~\\\\\n \\frac{1}{4!}& \\frac{1}{2!} & 1 &~&~\\\\\n \\vdots & ~ & \\ddots~~ &\\ddots~~ & ~\\\\\n \\frac{1}{(2n-2)!}& \\frac{1}{(2n-4)!}& ~&\\frac{1}{2!} & 1\\\\\n \\frac{1}{(2n)!}&\\frac{1}{(2n-2)!}& \\cdots & \\frac{1}{4!} & \\frac{1}{2!}\\end{vmatrix}.\n\n\\end{align}\n ", "02c200d95543444ec2205ef66b757136": "\\kappa=\\frac{\\omega_r}Q", "02c2022327405565b8cca1b582733df7": "a = d \\ne b = c, \\alpha = \\zeta = 90 ^\\circ, \\beta = \\epsilon \\ne 90 ^\\circ, \\gamma \\ne 90 ^\\circ, \\delta = 180 ^\\circ - \\gamma ", "02c26e1d2ae0c94c2eb478016ccc5442": "\\, \\delta ", "02c2916b1b5886b896d8a537fe8db434": "a = \\sum_{i=0}^{n}d_{i}(-r)^{i}", "02c29cddfbb95e518fee0d87144c595c": "\\frac {\\dot{m}\\sqrt{T_{01}}}{P_{01}}\\,", "02c2a70eab25d2c4784c023a6a316659": "(\\cos\\theta,\\sin\\theta)", "02c2f8de39e5973c727a6d4858107564": "P:=\\{p_{\\vert X} \\mid p \\in P_n \\}", "02c31bde2dae72763bb7030a6836164f": "\\nabla_r", "02c38550b3c579b5cada441aa00fea85": "\n\\frac{E(u+\\tau\\psi) - E(u)}{\\tau} = \\frac{1}{\\tau} \\left( \\int_\\Omega F(u+\\tau\\psi)dx - \\int_\\Omega F(u)dx \\right)\n", "02c40ae85808d86bd7fdd50f8c36d48a": "\n\\gamma(\\mathbf{v}) = \\frac{1}{\\sqrt{1-\\frac{|\\mathbf{v}|^2}{c^2}}}\n", "02c42262fcf5769ee18cf00a44a604ad": " Lower~limit = e^{Log_e (lower~limit)} = e^{1.49} = 4.4", "02c4ceb96e7cd644ace41c8b9f652803": "b^{-(p-1)}/2", "02c52fa215bf128cd71a773caa85464d": "\\Delta u = K^\\prime e^{2u} + K(x).", "02c559a0df7dd3616a610d9033abe4e2": "\\pi_k(O)=\\pi_{k+4}(\\operatorname{Sp}) \\,\\!", "02c58813370ab922c27e5673ce949850": "K=\\frac{e B \\lambda_u}{2 \\pi \\beta m_e c}", "02c59aa8adbc1e1e4182bb76e89b602f": "x\\not =y \\in X", "02c6703935ac8fc407610edf815fa156": "MA = \\frac{F_B}{F_A} = \\frac{V_A}{V_B} = 2.\\!", "02c67906b26d7fe40fdb90adbba3c0cf": "\\scriptstyle \\partial S/\\partial t ", "02c6f0c00b0d1b69cbb4174a5984a5e3": " q=(s,t_e) \\in Q ", "02c6f235d7c1fe631555420d92b2ef2b": "\\pi\\left(10^{10}\\right)", "02c70beac20542de6b37489aaa4d2d45": "\\mathbf{L}=\\int_V dV \\mathbf{r}\\times \\rho(\\mathbf{r}) \\mathbf{v}", "02c73fe4efabc7a908c3768c18d8ffe3": "GI", "02c784ac595f0d4f08e6b274dd7ae877": "1-R-\\varepsilon", "02c78f8d711f84178689990c53db0388": "\\gamma = \\lim_{a\\to1}\\left[ \\zeta(a) - \\frac{1}{a-1}\\right]", "02c791e85339843965044c4e2ed5b0ad": " Q^T Q = I . \\,\\!", "02c7bc501ed2be649a02a9a9fd0a87e8": "(a+b)+c = a+(b+c), (ab)c = a(bc)", "02c7d5176190cc37c48af6a7f2e008b2": "\\langle f,h_k \\rangle=\\frac{a_0}{2}\\langle h_0,h_k \\rangle + \\sum_{n=1}^\\infty \\, [a_n \\langle h_n,h_k\\rangle + b_n \\langle\\ g_n,h_k \\rangle],", "02c7dcf360a6635c00b29a984e96a1b9": "\n\\mathcal{L} =\n\\frac{1}{2} \\left| \\frac{\\partial\\mathbf{n}}{\\partial t} \\right|^2\n- W(\\mathbf{n},\\nabla\\mathbf{n})\n- \\frac{\\lambda}{2} (1-|\\mathbf{n}|^2),\n", "02c810017ce31a2012fef5f3b1634458": "\\alpha = {R \\over 2L} ", "02c886dfae77c4e687a37e9179e15ed2": "\\,\\eta(s)=\\Phi (-1,s,1).", "02c8b72adc1675abd4cf2dc5cf31a7c0": "(GX,\\varepsilon_X)", "02c925889c2b342fcf5ce61a5a4dacff": "t_1^\\prime", "02c965bb60433d59c3a8f2bef9b19469": "\\phi_\\mu", "02c98a141d6feacc270a43be76d6d897": " P_{\\rm wind} = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v_1^3 ", "02c9b6540d7509e2f58bef72d1dc7ec2": " \\prod_{j=1}^n \\left ( \\alpha - \\alpha_j \\right ) = 0 \\,\\!", "02c9d63fab98237220ce40033f33ef78": "\\nabla \\cdot \\vec v", "02c9f55b2e5f5569dea8fdc7eaec7d10": "PK_R", "02ca10636c28dba407e94fc93215d004": "b_M", "02ca156e9359d57db6f6e32cd090d0fb": " a(z)= \\sqrt{\\frac{1}{\\sigma_x^2} z^2 + \\frac{1}{\\sigma_y^2}} ", "02ca38586ab165b0d09038c1e064c730": " \\hat{C}(\\mathbf{k})=\\frac{\\hat{H}(\\mathbf{k})}{1 +\\rho \\hat{H}(\\mathbf{k})} \\,\\,\\,\\,\\,\\,\\, \\hat{H}(\\mathbf{k})=\\frac{\\hat{C}(\\mathbf{k})}{1 -\\rho \\hat{C}(\\mathbf{k})}. \\, ", "02ca5b777b17efadd0adcd3bfbc0f8e9": "P(B)=0", "02ca7e35cb7137050d8d0a7d18caae3f": "\\mathbf{P}(n,r)", "02cac3592c352a9824607d3b18002406": "r^\\ell \\, Y_\\ell^m", "02cb092dd6953f1bc7c0f25ec5d100db": "p(\\theta)", "02cb939e6fb166fb503428c09a364309": "-n_2 + n_3 = 1 \\ ", "02cb9858cb1c4712568d65b854bea41b": "\n\\begin{pmatrix}\nA_1&B_1\\\\\nA_2 & B_2\n\\end{pmatrix}\n\\begin{pmatrix}\nx\\\\y\n\\end{pmatrix} = \n\\begin{pmatrix}\nC_1\\\\\nC_2\n\\end{pmatrix}.", "02cc26a3340d7fe6e3bff902f3ee1e80": " {\\cos \\gamma = \\sin \\theta_{s}\\sin \\theta \\cos \\psi + \\cos \\theta_{s}\\cos \\theta} ", "02cc7184ed2936ba6c062cd2a905f05e": "\\alpha_1, \\ldots, \\alpha_d \\in \\mathbb{R}", "02cc83f99a9b20510ca44489ebd7a36f": "+ \\frac{200}{510,260} log_2\\left(\\frac{200/510,260}{260/510,260 * 500,200/510,260}\\right)", "02cd27c1810fb9f2696e464c60bc37f8": "\\frac{\\mbox{d}}{\\mbox{d} x} ( \\alpha \\cdot f(x) ) + \\frac{\\mbox{d}}{\\mbox{d} x} (\\beta \\cdot g(x))", "02cd346db0f5d0816cdfba9e1655a21a": "P\\cap -P", "02cd503acfc44eeab79685da98bee009": "0 \\le L(M) \\le 2^{64}", "02cdd4dc0f0d9c79c76a14d95b165c76": "\\pi(x)\\sim\\frac{x}{\\ln x}.\\!", "02cdd72d2f898fb9c7e0710bab8fe5d7": "c_{ijk\\ell} = c_{jik\\ell}\\,", "02ce03b273daac91982b3767415710c1": "(n\\mapsto n\\cdot\\log n) \\in O(n\\mapsto n^2)", "02ce325d3513bcf9b951a90e86772e48": "O(n\\text{ }\\log\\text{ }n)", "02ce4923458cb5d5911064299ae41ebd": " C_p = C_p ( \\alpha , M , Re , P) ", "02ce720ac80fe82c41335fbc936122c8": "\\begin{smallmatrix} V = \\sqrt{{V_r}^2 + {V_t}^2} = \\sqrt{11.4^2 + 16.9^2} = 20.4\\, \\end{smallmatrix}", "02ce844bf2a415f3aa1ad996f28825b2": "C=I", "02cea61bce63ab878ef920e395c399a2": " \\mathbf{\\Pi} = \\{\\pi_x, \\ x \\in \\mathcal{E} \\} ", "02cf28ff66964d3f1795f42f84b0291c": "I = I_0\\exp\\left({-\\int\\mu(x,y)\\,ds}\\right)", "02cf86dc7d65bc7133c890866a1bcb66": "f^*L_1 \\cdots f^* L_m \\cdot F = L_1 \\cdots L_m \\cdot f_* F", "02cfad08a7a516a3762139bbcaf6f27c": " r < R = \\frac{1}{\\limsup\\nolimits_{\\ell\\to\\infty} |f_\\ell^m|^{\\frac{1}{\\ell}}}.", "02cff7b0593b80135cd140f4a21e88f8": "P(i) = \\sum_{n=st_i}^{n=fin_i} |S(n)|^2, i = 1, \\dots, 5 ", "02cffdcc45ccae3138c3796cd80198e3": "c^2 {\\gamma^2} - v^2 {\\gamma^2} = c^2", "02d02364ec17f438ee1f0c55e2f0c0c0": "\\textstyle x,y\\in\\mathbb{R}", "02d0505c45ef5338e56863309cb93ddb": "\\ \\displaystyle \\mathcal{Q}\\times \\mathfrak{U} ", "02d080685ff47967c7316e79e7ea9945": "\\nabla \\times \\mathbf{D} = \\varepsilon_{0} \\nabla \\times \\mathbf{E} + \\nabla \\times \\mathbf{P}", "02d082a3eecb5f89fde6a8625ea7efd1": "H\\left(A,C\\right) =\\left\\{00,01,1\\right\\}", "02d08f4ef10416a36b5f5a548095c02a": "\\approx 0.4 m_s R^2", "02d0b1e19d0037c74a070547faf3faa7": "x1\\;", "02d44c58ff40c2ddc94695ab9955e25a": "g:\\textit{George}", "02d48ef1af7c38a5a790aa75c2a922c9": "X_{5}", "02d4b74c78ad4e5700a887e322543640": "F_{A \\rarr A} = 0", "02d4d3b0044ee3603acdcd99de9dcb05": "X = A^{-1}(I-UY)", "02d4e6ea476acf8a700897f537d8731f": "\\frac{\\pi}{2\\sqrt{2}} \\approx 1.11072073", "02d4f4692cd34a7d069bf5afb42fc10b": "\nf[x_0,\\dots,x_n] = \\sum_{j=0}^{n} \\frac{f(x_j)}{q'(x_j)}.\n", "02d53af1d934122f24861e250bb37ad4": " \\max \\{ p,q \\} \\leq p +_\\mathcal{O} q ", "02d58831ed513c0dda9de14234c8d360": "K^M_2 (K) / 2", "02d5abb1b8b92bb55972cc21e78905c1": "u_i = u_{i-1}^2 - 2", "02d5db3bd7b1a41cb5312b6f011488b1": "a_1\\chi_1+a_2\\chi_2 + \\ldots + a_n \\chi_n = 0 ", "02d5ddbbef84b35fb25ef13b66f1bb41": "\\tau_{tt} \\propto N^2", "02d61f902f91f49bb386782b23cbd9bb": "\n\\begin{align}\n |V| e^{j(\\omega t + \\phi_V)} &= |I| e^{j(\\omega t + \\phi_I)} |Z| e^{j\\theta} \\\\\n &= |I| |Z| e^{j(\\omega t + \\phi_I + \\theta)}\n\\end{align}\n", "02d629ac392c3328e14f57fd55b883ff": "H_t - H_{t-1} \\in -K_t \\; P-a.s.", "02d649ffafb5cf15d2638a8d7f8d8551": "q(x)=x^n+b_1x^{n-1}+\\cdots+b_{n-1}x+b_n, \\,", "02d687535053ef9d1ece75b487e31704": "dS = \\left(\\frac{\\partial S}{\\partial E}\\right)_{x}dE+\\left(\\frac{\\partial S}{\\partial x}\\right)_{E}dx = \\frac{dE}{T} + \\frac{X}{T} dx=\\frac{\\delta Q}{T}\\,", "02d6ede6592ceeed003d45034a9dbaf6": "U = a", "02d79611778e638b14936dac9ed4b7d3": "\\gamma = \\alpha+1/2", "02d7c67a0283ebb7d5120a7a349b23d6": "\\frac{H^2}{P^2}=\\frac{2P^2}{P^2}\\, ", "02d84ce998b810b0415fdee2bb7b6b3d": "|C_{\\alpha\\beta}|^2", "02d8e21f2415d7dcdd6b48a2400de0c5": "Ba/Bb= (Pa/Pb)*(Db/Da)square ", "02d91a6161155b99ceb528d3ae00da42": "\\mathrm{DFL} = \\frac{\\mathrm{EBIT}}{\\mathrm{EBIT\\;-\\;Total\\;Interest\\;Expense}}", "02d958ce1d9f78e47423cea80b715c63": "D_k(c) = D_k(E_k(m)) = m\\!", "02d9c758323964d8c3ec18509487a742": "\\,H_s(s=1,\\ldots,S).\\; n_s", "02d9cfbc6b68bba5b015aecfb111b14d": "\\lambda_B \\approx", "02d9d27bf1b8744745ca52cd27d83f8c": "MB(A)", "02da2a736a512660a6018cc00f4993e1": "x_i \\in S_i", "02da3fe99c93e897396595edccdbd632": "Q(e,y)", "02da6b3fb681e3d0a62145c5bc85d032": "(\\log^2 N)/N", "02da8261a5f9e984bc30751c1449a475": "= \\left[\\sum_{i=1}^{N} 2 \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{x_{i2}-x_{i1}}{2} ' \\right]^{-1} \\left[\\sum_{i=1}^{N} 2 \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{y_{i2}-y_{i1}}{2} \\right]", "02db2af0d5dd6a4fab25a12e871c8af1": "\\pi_k(\\mathbb{S})", "02db6c5b194fd0a2661890405cf6b1a1": "\\mathbf{R}^+ \\to \\mathbf{R}^+ : x \\mapsto \\sqrt{x}", "02dbbfc4d4d0a67026392826f85fda83": "0 \\leq b \\leq a ", "02dbc2a63dde34b74a8f54f7d0d15603": "\\frac{d\\left(ky\\right)}{dx} = k \\frac{dy}{dx}.", "02dbc6539d3667d420c5fefe0ee0a0f4": "d_k \\isin \\{ -1, 0, 1 \\}", "02dc2a6a3c90eff032f723994a58a5b7": "U(n) = O(2n) \\cap GL(n,\\mathbf{C}) \\cap Sp(2n, \\mathbf{R}).", "02dc55860ca76ed9063448b5ddf5e65a": "\\hat{w},x_1^n(w),y_1^n", "02dc609c4d6fb1101f650b1dbd33c25a": "s=\\sqrt{\\tfrac{m-r_k^2}d}", "02dc86b381521b74d7f7e2ba46110545": "CO = \\frac{VO_2}{C_a - C_v}", "02dc987d3612a3352f010b927d0e0c6c": "S=\\phi^{-1}(\\phi(S))", "02dcc5a30425c86ce04d7f8d70af95d0": "\\textstyle \\overline{a}_{.k}", "02dd00bf549832493910b3af11394659": "\n\\frac{\\partial F^m_{~\\alpha}}{\\partial X^\\beta} = F^m_{~\\mu}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} \\qquad; ~~\n F^i_{~\\alpha} := \\frac{\\partial x^i}{\\partial X^\\alpha}\n", "02dd3f4c7c9b4b389221cd052d2591ad": "y'=f(x,y)", "02dd4a6693bb322acb9a40b57af2174c": "\\ (U,\\ E,\\ N)", "02de23ac02b08d0fd8faba9fc285105c": " z_{xx}>z_{xy}>z_x>z_{yy}>z_y>z", "02de341fdf0e3e72724ec2f1ad15cd77": "\\alpha=\\|g\\|_q^q", "02dea128ced10af253763324a0583252": "\\mathrm{Ran}(A - \\lambda I) \\cap \\mathrm{Ker}(A - \\lambda I) = \\{0\\},", "02dea2dbd4c59b6ded1f65bd0482d579": "\n\\begin{pmatrix}\n{A'}^0 \\\\ {A'}^1 \\\\ {A'}^2 \\\\ {A'}^3\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & \\cos\\theta &-\\sin\\theta & 0 \\\\\n0 & \\sin\\theta & \\cos\\theta & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nA^0 \\\\ A^1 \\\\ A^2 \\\\ A^3\n\\end{pmatrix}\\ .\n", "02df1d67c4f118aa427e34f09b750537": "x_i \\in \\mathbb{C}^m", "02df319e57e85d778ff5b31c78d39e35": "A\\to\\bot", "02df681620026e67b28f02df6909fef1": "\n Pr[C_i=C] \\geq \\left(\\frac{n-2}{n}\\right)\\left(\\frac{n-3}{n-1}\\right)\\left(\\frac{n-4}{n-2}\\right)\\ldots\\left(\\frac{3}{5}\\right)\\left(\\frac{2}{4}\\right)\\left(\\frac{1}{3}\\right).\n", "02df87d44cd6118584fa0d6a0cbe5157": "\\tan\\left(\\frac{\\Phi}{2}\\right)=-\\frac{1+\\sqrt{R}}{1-\\sqrt{R}}\\tan\\left(\\frac{\\delta}{2}\\right) ", "02dfac85105c1461af22205de6bb9430": "F(y,z)\n\n=\\int_{-\\infty}^\\infty f(\\rho,z)\\,dx\n\n=2\\int_y^\\infty \\frac{f(\\rho,z)\\rho\\,d\\rho}{\\sqrt{\\rho^2-y^2}}\n", "02e042d61b93ff101d6b317e2f36930a": " m < n \\to m - n = 0 ", "02e07bca56d1528d156c2808df1d9484": " f(x) = \\sqrt{x}", "02e089b4d472400298ab379e4d000d99": "F = f_1(0), f_2(0), f_1(1), f_2(1), \\dots,", "02e0a100969db76ab338f592fc3d211a": "[w^r]", "02e0faaa7da52b4aabdb0478360c8cd6": "\\frac 1 {b+c} \\ge \\frac 1 {a+c} \\ge \\frac 1 {a+b} ", "02e1313d8de17516c259fd87e4e4cfe6": "v = \\frac{\\lambda}{\\tau} = \\lambda f.", "02e205b6201bc8f4dfbf3b389b9f8dc0": "W_n", "02e21374c1c420aecdfcf29d81c83b3a": "\\sigma_k =\\sqrt{\\lambda}", "02e235deac445d96c2c095d0b1fddc89": "b_m = \\frac{1}{\\pi} \\int_0^{2 \\pi} f(t) \\sin (mt) \\, dt.", "02e25eaa2b39b968aac77563a6a01eda": "\\dot{r}=\\frac{G}{2}\\dot{v}", "02e294d9ded63b8b6a38cf6592d751af": "z^*\\!", "02e2a497116037a708e6b90196648df6": "\\langle F,\\le,V\\rangle", "02e2dfa1293bc1f3b376c5b3d6bc3222": "R(n) = \\frac{r(1)+r(2)+\\cdots+r(n)}{n}. ", "02e359f86df7c9657f8a41f64540e619": "-\\sum_{k=1}^\\infty \\frac{(-x)^k}{(k-1)! \\zeta(2k)}= O\\left(x^{\\frac{1}{4}+\\epsilon}\\right)", "02e3702daaabd058e3a0853e3b051ab6": "\\left (\\partial_t-k\\partial_x^2 \\right )(\\Phi*g)=\\left [\\left (\\partial_t-k\\partial_x^2 \\right )\\Phi \\right ]*g=0.", "02e37bc832b7b258de2f53958c72c84a": "\\color{Black}\\tfrac{n}{2}", "02e388f3ae24f35e4d176cd90608f435": "m_1a=m_1g-T", "02e44e162cc167619a1b9d55a3c57371": " E^{c} ", "02e44f5d8ad6859397e15c9bf505579c": "\\ J = \\frac {500}{1 + \\lambda_i^2}", "02e46e865b0432d517ca68327c3c991b": "\\pi:\\Sigma^*\\to P(A)", "02e4bc881738860c665a51428033be6b": "n_1 , n_2 , n_3,\\ldots ", "02e53a9984840ee52f4ac5b2af181b68": "B^4 = \\{\\mathbf{x}\\in \\mathbb{R}^4 \\mid |\\mathbf{x}|\\leq 1 \\}.", "02e53d2f2041f13a171dee171f905e10": " \\mathit D ", "02e542b155f6cdbecefba13ccf9781c5": " Spin(10,\\mathbb C)", "02e63cd33d8b7b7324750b90cd29360d": "\\sup_{y^* \\in Y^*} -F^*(0,y^*) \\le \\inf_{x \\in X} F(x,0).", "02e658512da0b87b1d9a3154a3ce1de5": "\nr^2 = x^2+y^2", "02e65f4b856e7e66ea3a08999d08fc45": "\\left(\\begin{array}{ccccc} 0\\\\ & 0\\\\ & & \\ddots\\\\ & & & 0\\\\ & & & & 0\\end{array}\\right)", "02e6751b8690ba6ce14a21e47c7b38ca": "\\Delta ^{\\prime }(x)=-\\frac{g(x\\mid x)}{G(x\\mid x)}\\Delta (x)+\\left(W_{2}^{A}(x,x)-W_{2}^{B}(x,x)\\right)", "02e67e02dd0c6f5f1f4e4d979d35bd60": " ^{14}\\text{N}_2\\text{O} \\rightarrow {^{14}}\\text{N}_2, ", "02e6b069dff4f9036ddd12884e55cfa9": "x=0^0", "02e707f46baf83999b1571559404afff": "A_{k}=\\left[\\sum_{i=1}^{N}\\frac{\\partial \\ln Q_i}{\\partial \\beta}(\\beta_{k})\\frac{\\partial \\ln Q_i}{\\partial \\beta}(\\beta_{k})'\\right]^{-1} .", "02e746e1beef65a29426f5d7934c6857": "\\hat{J_z}", "02e747c355fafa7779c45a9f4cabd96f": "\\, T\\!", "02e74f10e0327ad868d138f2b4fdd6f0": "27", "02e76f246d1f9f937fba84f0b5cbaeef": "C \\ge 0", "02e78bee48c90319ceeb0e0576818254": "\\vec{p}=-g\\vec{s}/\\|\\vec{s}\\|\\,\\!", "02e7ba1333997e4e96b6813efa861eb7": "w \\in \\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}", "02e7e1c29921ca9b44aa0fe0a0c65531": " r_{1}m_{1} = r_{2}m_{2} ", "02e7fde8845e3e7327b77552d992fb00": "\nV \\setminus \\{v\\}", "02e82062942814d6b745ec77ac54faee": "3 \\Rightarrow 2", "02e8631f6a646c77f58ba39cc8219ab7": "J(\\lambda)\\,", "02e88362e58867d505ddad8f3fa4a5db": "\\,e^{+{1\\over 2}i n\\theta}q {e^{-{1\\over 2}i n\\theta}}^{*}", "02e894f585c564d4db6b4f0e06efb3e7": " {\\rm coNP} = \\forall^{\\rm P} {\\rm P} ", "02e8b238d09747bf14aa6e5732af9715": "i_i(U_g)", "02e8bb51124bd7112ebf6627110845ae": "\\frac{nm}{db}", "02e8e818fae5800a439605693f5f6cb5": "\\,p(x)=q(x)=1.", "02e93736ba5f8b4b5688942188265b26": "f(S)=\\max_{i}\\left(\\sum_{x\\in S}a_i(x)\\right)", "02e953a474c2a9818c3d75ae320728fe": " S_{x,i} ", "02e96c03176f660264850cdf7abea153": "\\alpha(k+1)/(2k)", "02e9b302bf1c76b0dfb128e4b0077c7f": "t = \\frac{-1\\pm\\sqrt{1+8(D+Y)}}{2} ", "02e9e0fce35e84f2a148a722f090844f": "L_{ij}\\,\\!", "02ea4194274544c7169a06a68462022d": " e^{-\\alpha t} \\cdot u(t) ", "02ea9721d236a98ee6036f38defef4f2": "\\partial_t s(t,a) + \\partial_a s(t,a) = -\\mu(a) s(a,t) - s(a,t)\\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1} ", "02eaa3e2db7bb9679cdb0e1e964349af": "y(t) = \\sum_{k=1}^{K} r_k \\cos\\left(2 \\pi k f_0 t + \\phi_k \\right)", "02eaeaa73a579c8c677e8d57b030f7ca": "\\mathbf{F} = m \\mathbf{g}", "02eb0e03ef4fec950b287d600730da63": "R[f^{-1}]", "02eb35dfa3fb4b87ee32d8bae6104661": "\\frac{\\partial^2}{\\partial x_k \\, \\partial x_j} H = \\frac{\\partial^2}{\\partial x_j \\, \\partial x_k} H. ", "02eb3a06f2243826cca0ee012ce70dcf": "\\log^+|f(z)|", "02eb4539112d1f9338820c579a0947ab": " A = \\begin{bmatrix} 1 & 3 \\\\ 4 & 2 \\end{bmatrix}.", "02eb5f614ec7faa2aa8dbe0645e84f11": "B\\triangleleft A", "02eba0344dcb33f69cad205e517233a3": "(B\\to F)\\to F,C\\to F\\vdash(B\\to C)\\to F", "02ec3a241b9dc06b768d2ee479064fcf": "F(t)\n= 1 - \\left(\\prod_{j=1}^a \\lambda_j^{r_j} \\right)\n\\sum_{k=1}^a \\sum_{l=1}^{r_k}\n\\frac{\\Psi_{k,l}(-\\lambda_k) t^{r_k-l} \\exp(-\\lambda_k t)}\n{(r_k-l)!(l-1)!} ,\n", "02ec45c41647c927d5b9699f8e5ee4c3": "0\\to \\mathcal O_{\\mathbb P(V)} \\to \\mathcal O_{\\mathbb P (V)}(1)\\otimes V \\to \\mathcal T_{\\mathbb P (V)} \\to 0 ", "02ec7e2c7d521f73b244175365ebd463": "(12, 35, 37).", "02ecb38af12588bb04b2d6c48e04ed13": "v(t_2)-v(t_1) = \\int_{t_1}^{t_2}{a}\\, dt. ", "02ed06b11041dfa48ccdd36142cf9600": "\\mu : \\mathit{V}_o \\to \\mathit{W}", "02ed0ce07f470a60c3f421e7fb46312d": "\\lnot (G_1 \\lor G_2) \\to (\\lnot G_1) \\land (\\lnot G_2)", "02ed321b1ae36a826de55612083e3d9f": "a_{2,j}={1\\over14}(2y_{j-2} - y_{j-1} -2y_j -y_{j+1} +2y_{j+2})", "02ed9140696aa6bfbec96ccea0c47884": "\\left\\langle M_C\\right\\rangle \\sim l^{d_f}", "02ed986c1ddd4da3b052e8d837651fbb": "J^1_\\Sigma Y", "02edb5f97703fcbcdeff31cf06a5602d": "_{q'p=qp'\\,}\\!", "02edda68d05afa6c789386adba55832f": "C(u_1,\\dots,u_{i-1},0,u_{i+1},\\dots,u_d)=0 ", "02ede70b85ad095f719554400c052a57": "\\prod A_i", "02ee09f9465ada081867c32b7b1b6a4c": "r = \\sin (6 \\varphi) + 2", "02ee209363833e57ee463c2179b6affa": " \\sigma \\, ", "02ee2d78b2edbc04150e6284b67f2099": "S_l\\;", "02eed43b80646dc09ac23605719df1fb": "\\Delta k_{j}^2", "02eee65279bf8969802e50572b86fa23": "1/\\tau = c/L", "02ef45b6a6516432184cea67db307698": " \\lambda_\\mathrm{De} \\cong \\sqrt{ \\frac{ \\varepsilon_0 T_e }{ q n_0 } }.", "02ef8162dc8ba23187a9c2dcebbe8c36": "\\frac{\\pi}{d-1}", "02ef8cfa689777040dd2fd110a88ccc8": "\\pi(z) = \\frac{1}{\\Pi(z)},", "02efc196a68c60bce5943d30386ac376": "\\cdots \\to H^q_{\\mathrm c}(U) \\overset{j_*}{\\longrightarrow} H^q_{\\mathrm c}(X) \\overset{i^*}{\\longrightarrow} H^q_{\\mathrm c}(Z) \\overset{\\delta}{\\longrightarrow} H^{q+1}_{\\mathrm c}(U) \\to \\cdots ", "02efc65c60ed9284d8784b145d836446": "-\\sqrt{\\frac{6}{35}}\\!\\,", "02efcf02b0e9ffa17460ad97a04e1212": "-0.0497", "02efe698dcd85a84e109a8aa7b6f0257": "\nI_{e} = S J_{e} = S n_{e} e \\sqrt{kT_{e}/2 \\pi m_{e}}\n", "02eff460cc45158b329c8a3037fa0aa1": "\nU_{k}(\\beta) - U_{k - 2}(\\beta) = T_k (\\beta). \n\\,\\!", "02f0006b87e5158d1abfc2f7def4fddd": "\\frac{1}{X}\\sum_{i=1}^{X}\\frac{1}{2^{i-1}}=\\text{D}", "02f0127195703a7ac61a91b73419a93f": "A\\mu=\\{ a_i \\zeta_n^j\\;:\\; 1 \\le i \\le m, \\;\\;\\;0 \\le j \\le n-1\\},", "02f02bbbc58e793e57462bb70b1c4462": "P = (x_0,\\ldots,x_n) \\,\\!", "02f03834cea656c9d3eaa15323e5c21f": "z = 1.5", "02f06e1f429788c0928e3fb3a334e3a0": "\\ E = E_\\text{covalent} + E_\\text{noncovalent} \\, ", "02f07196b3f934fa452e95387eddedf4": " P_1", "02f07d2a70e478f7deabb6ec6a8193c5": "\\begin{matrix}4&4&5\\\\6&7\\end{matrix}", "02f0a792fc10f3544da8261e98039cff": "F (q,p)", "02f15f2928387ff24b84aff0f8cc118d": " v = 614.58 g c \\cdot \\frac{p + b}{b i r n}", "02f17525b5aa1c6bfab3c606434ebb2b": " E_{\\textrm i}=E_{\\textrm r} ", "02f17f9e25eac18a4b65a5f79fd128ed": "G=\\mathbf{A}\\cup\\mathbf{B}", "02f19d2e9bd994b96bfae9aac886f424": "c_\\kappa 0=0", "02f24172940929c3c895d6b5a3f88b24": "\\frac{k}{i} i.", "02f25a51be4aaf6eb72064cd2f168aab": "\\phi_T(x)=\\left(\\alpha(x), \\alpha\\left(f(x)\\right), \\dots, \\alpha\\left(f^{k-1}(x)\\right)\\right)", "02f2b48c29610c0f37753082a3853221": "\\forall (a,b)\\ S(a,b) < \\infty", "02f2b66b570e5fe0c8bba0cf56a99d02": "\\sigma^{-1}\\omega", "02f2ba35e3bcf2226b0a61b1a0bdbf55": "S(f) = \\int_{-\\infty}^\\infty R(\\tau) e^{- j 2 \\pi f \\tau} \\, {\\rm d}\\tau.", "02f2ee7a892954eef0166ec5d10ec8d5": "n = 6", "02f3421ca1b6f287a50cf2d6c4a92ff9": "E_5", "02f3933968c77f4bed9e436105dad75b": "\\partial_y\\partial_x f|_{(0,0)} = -1", "02f3d1a8f75e61c94bee74ef609259c8": "\\frac{\\mathrm{d} g_{ij}}{\\mathrm{d} t} + \\frac{1}{2} \\frac{\\partial f^{k}}{\\partial v^{i}} g_{kj} + \\frac{1}{2} \\frac{\\partial f^{k}}{\\partial v^{j}} g_{ki} = 0 \\mbox{ for } 1 \\leq i, j \\leq n, \\quad \\mbox{(H2)}", "02f3eb23895fcb08557def8670db2e7f": "~K = \\sigma/(s t_{\\rm r})~", "02f3ee4e631696fb75edfe70f4191b42": " \\mathbf{r}^\\mathrm{PR} = \\mathbf{r}^\\mathrm{PQ} + \\mathbf{r}^\\mathrm{QR}.", "02f424541a3bf7294037e7d96c538bdc": "Q_{ij}=M(3x_i x_j-\\delta_{ij})\\ ,", "02f45c6c6f3aa7d3178d4268e161b3e8": "G(n,m)", "02f4b99e9280bfcb506bf330e6cf28fb": "\\omega_{\\mathrm{p}}", "02f4d62cb2570c2e4849461111095536": " E^m", "02f4eaec0a182c70f62fc43d4075d7d5": "e^x = \\sum^\\infty_{n=0} {x^n\\over n!} =1 + x + {x^2 \\over 2!} + {x^3 \\over 3!} + {x^4 \\over 4!}+\\cdots\\!", "02f50a6557dcbcacebf468676a1ebb52": "N(r)", "02f59cb82cc3d0082e9493b86c0528c3": "f''(x) \\ge m ", "02f5d6c787071d778ac2357a445f8228": "\\sigma_{i + 1}^{2} \\leq q \\sigma_{i}^{2},", "02f641c12f8d7e031069b09df11f1c27": "\\ h = \\frac{1}{\\sqrt{s^\\mathrm{H} R_v^{-1} s}} R_v^{-1}s.", "02f6b3b7e4a207277d717543b4562e31": "S(10) > \\Sigma(10) > 3 \\uparrow\\uparrow\\uparrow 3", "02f6f42d3d308f65531d018e6142e2c5": "\\{X_1,X_2,\\ldots,X_n\\}", "02f6f7bcbd0ec8e605fe811978b79061": "\\textstyle=min_{a^{*}(\\theta_{k}w_{k}=1)}\\ W_{k}^{*}R_{k}W_{k}", "02f716902388b03e703855a549afbef1": "N = N_1 + xN_2\\,", "02f7217d2be50e1391fd60f3e462c6b9": "x_j(t)", "02f758c3d842ae6a09d7b7a117d46240": "\\kappa_\\nu B_\\nu = j_\\nu\\,", "02f75e457493ec338c3346526cd6847f": "\\left( 1 - \\sum_{i=1}^{p'} \\alpha_i L^i \\right)", "02f7aa4a2ce39f9f29fdc34c01777154": " Lm ", "02f7add3b530415b79a48cecdb94151a": "\\,\\mathrm m", "02f7dac3f892b9379e20661985ade99c": "dt=a(t)d\\tau", "02f7e1cdf7f369ee01bde5b75bcd86c2": "v \\in U(S)\\,", "02f813eccc85fc5db7514773fac25cf6": "g^{\\alpha\\beta}\\frac{\\partial S}{\\partial x^\\alpha}\\frac{\\partial S}{\\partial x^\\beta} + (mc)^2 = 0\\,,", "02f817b9d33bd6cabbc4511bb2bf1b55": "\\sec \\zeta", "02f831a9258e43b2f63c557008338437": "\\left.\\frac{\\partial}{\\partial u} g(z, u)\\right|_{u=1} =\n\\left. \\frac{\\sum_{d\\mid k} z^d}{1-z} \n\\exp\\left(\\sum_{d\\mid k} \n\\left(u^d \\frac{z^d}{d} - \\frac{z^d}{d}\\right)\\right) \\right|_{u=1} =\n\\frac{\\sum_{d\\mid k} z^d}{1-z}.", "02f8a272aa79452db15718cdded95370": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 22.11315\\log_e(T+273.15) - \\frac {13079.73} {T+273.15} + 166.0812 + 1.233275 \\times 10^{-5} (T+273.15)^2", "02f8a69b15624b58629aed898287f86d": "\\Rightarrow x\\ln x = \\ln z\\,", "02f8c68abedd8d8a4a2cbc020788b3bf": " J_\\nu^{(3)}(x;q) = \\frac{x^\\nu(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty} \\sum_{k\\ge 0}\\frac{(-1)^kq^{k(k+1)/2}x^{2k}}{(q^{\\nu+1};q)_k(q;q)_k} ", "02f8e61699aad550bc66075c76b0df14": "{\\mathbb R}^4,S^3\\times {\\mathbb R},M^4\\setminus\\{*\\},...", "02f8eac21681dabf14f7b2d25389d52b": "\\Delta p = \\frac{2 \\gamma}{R}.", "02f90dad6539792ab4cb6b7c9320a5c8": "P = f_3\\!\\left({Q\\over {ND^3}}\\right),\\,", "02f95acfafb779e2ce212284b6520ae2": "h=\\frac{p}{2}", "02f9aa7f5617898aa9489861d7fa465e": "\\cfrac{1}{(\\sigma_3^y)^2}, \\cfrac{1}{(\\sigma_2^y)^2}", "02f9e1a817c1f42f1e788c8549c78857": "\\bar{l}", "02f9ed4db8ebd67aff8c1e92c42405e1": "V=\\{V_1, V_2,\\ldots, V_C\\}", "02fa32878072963ead556d6f86c039f9": " \\coth\\left(x\\right) = \\frac{1+\\exp\\left( -2x\\right)}{1 - \\exp\\left( -2x \\right)}.", "02fa5faadf3bd895faeca53717d8b758": "R_{25} = \\sigma _{call,25} - \\sigma _{put,25} ", "02fa67b478412c4500425d2ef02b635f": " E_0( \\rho) = \\ln \\left( \\sum_{x_i} P(x_i)^{\\frac{1}{1+ \\rho}} \\right)(1+\\rho) \\, ,", "02fa70decbb4316ea8a5a882df882bc4": "\\kappa x{:}1{\\to}\\tau_1\\,.\\,e\\;:\\;\\tau_1\\times\\tau_2\\to\\tau_3", "02faadedb2a4c152940bad34dd95ed89": "\n(A|B) = \n \\left[\\begin{array}{ccc|c}\n 1 & 1 & 2 & 3\\\\\n 1 & 1 & 1 & 1 \\\\\n 2 & 2 & 2 & 5\n \\end{array}\\right].\n", "02fad5ddcac1eb8cc93e98458d70ae2a": " t_r 0", "030be7a21c42dcb01f92e00e7d84e5e2": "Z_G(t_1,t_2,\\ldots,t_n) = \\frac{1}{|G|}\\sum_{g \\in G} t_1^{j_1(g)} t_2^{j_2(g)} \\cdots t_n^{j_n(g)}.", "030c26ccc7f7cb949016a4a8a4f51f59": "\\eta_L \\, = \\, M_w e^{ \\left[ \\sum \\eta_a - 597.82 \\right] / T + \\sum \\eta_b - 11.202 }", "030c484910162ab90dec8f32b48263ef": "d(t) = a(t)d_0,\\,", "030c51923cd08fea140adaad4268787c": "\\text{CV/PS}=0.4\\times i\\times d^2\\times S", "030c6912e2e7c1098fdfe8298120ba85": "\\lambda_1\\approx\\lambda_2\\approx\\lambda_3", "030c8beff7748ff17469998a9a0251c6": "C_P=\\frac{P}{\\frac{1}{2}\\rho A V^3}", "030c9a04d6a2b608cb18ae6cc9d3ce44": " P_0(x) = 1. ", "030cca9b8a4c4a450e7214013b2635ac": " x\\,dx^2=dy^2, \\, ", "030d0afb674a89d5aae8f9d8be13ac35": "2^a", "030d4ac464f2b3ad5f7e54d8beae9c0d": "W^u(f,p) = W^s(f^{-1},p),", "030d4ae8df70d814e136970a8542bcdd": "f(x)=\\sum_{i=0}^l L_i(x^{k_i}), L_i(y) = \\sum_{j=0}^{m_j-1}f_{p^jk_i\\bmod{N}}y^{p^j},", "030d8ac281de329fb8d38a6f021b3d6e": "Q_B", "030e76a732c956a1950899118616099b": " \\iiint_V \\left( {\\partial P\\over \\partial x} + {\\partial Q\\over \\partial y} + {\\partial R\\over \\partial z} \\right) dx \\, dy \\, dz =\n\\iint_\\Sigma \\left( P + Q + R\\right) \\, d\\Sigma ", "030e9548babdfe0825542d893e7297bb": "h\\nu_m/k", "030eb5f5725d2a23d12126c34b1a6e26": "\n~\\epsilon_t = ~\\sigma_t ~\\times z_t\n", "030ec89a5bf9fbdfe59d5d43d1fe8a5a": "\\sqrt{10/[3(5-\\sqrt{5})]}", "030ed4a5d1a603b96002db8c87a864a4": " \\ M ", "030edff1943a3e6d4e8d029853bb2e67": "\\hat{H}_0 \\to \\hat{H}'_0 \\equiv U \\hat{H}_0 U^{-1} = U (\\alpha \\cdot p + \\beta m) U^{-1} = (\\cos \\theta + \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta ) (\\alpha \\cdot p + \\beta m) (\\cos \\theta - \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta ) ", "030f24479c38f531d978533a87c0116d": " \\bold{r}(\\theta(t)) =(\\ell\\sin\\theta, -\\ell\\cos\\theta)", "030f75e47654b5eed666144c2a3f774d": "C_\\text{out} = [Q]^K \\rightarrow [Q]^N, C_\\text{in} : [q]^k \\rightarrow [q]^n", "030fa134106f353689682c3cb4373243": "f(x) = \\begin{cases} \\frac{1}{x} & \\mbox{if } x > 0, \\\\ 5 & \\mbox{if } x \\le 0. \\end{cases}", "030fa229d0d6a66562d90a7101fd629c": "\\scriptstyle a \\;=\\; b \\;=\\; q \\;=\\; 1", "030fc21068588c25507efcfcfc01c7f6": "\\cos\\theta \\pm \\mathbf{i}\\sin\\theta", "031010e9f2111e92d564b794ab7b96be": "\\Sigma^k", "0310200af96f75ba3d543491548db12b": " \\dot{V}_A", "0310222002d8373cf997eb7ddee3dc52": "f(x_4)=14.1014", "03103e6579511946d8ca3d542b1440f6": "2^{355.5}", "031062ffa5fbda1b87a49e321295e1bc": "\n\\vec{\\nabla}\\cdot\\left[\\epsilon(\\vec{r})\\vec{\\nabla}\\Psi(\\vec{r})\\right] = -\\rho^{f}(\\vec{r}) - \\sum_{i}c_{i}^{\\infty}z_{i}q\\lambda(\\vec{r})e^{\\frac{-z_{i}q\\Psi(\\vec{r})}{kT}}\n", "031070c74a10b59558c7b83730c6e5f6": "\\mu(\\hat{p},\\mathbf{1},\\hat{p})=1", "0310ce0a6519debf9789f1f3c73a70b3": "\\displaystyle \\Box n = -\\Delta (|u|^2_{})", "031100983d73a2450b5544fed638b7d1": "\\aleph_\\beta", "031157f7409d26f9f5016282da47a89a": "p_2(x)=-4x+x^2;", "0311a31db8f8795194b1dc3d9da5f1e7": "2^{14_{dec}}", "0311b195732b29dc61854381569b0447": "P(o\\mid b,a) = \\sum_{s'\\in S}\\Omega(o\\mid s',a)\\sum_{s\\in S}T(s'\\mid s,a)b(s)", "0311d0522ad7308943910f7fcb6b1eb9": "s > - 2/\\Delta t", "03124b11d599daf3b1fb9b8aaf6b8c82": " F \\subset YX ", "03124d8546e8d1671120757e00607950": "h:\\mathcal A\\rightarrow\\mathcal B", "0312851f79b545a08e8363bb755d9f4f": "\\left( \\frac{8m+61}{3}, \\frac{8+m}{3}, \\frac{m^2-61}{3} \\right).", "031290e1ccd8bcd4fd9c4f42230d7cc5": " MI(row,col)= H(row) + H(col) - H(row,col) \\, ", "0312d8af026d6167c797d49441d84137": "\\Delta f := \\operatorname{div}\\; \\operatorname{grad} f.", "0312da4595e1f000b7366734bb6d8537": "\\displaystyle{f(T)\\xi=\\lim_{r\\rightarrow 1} f_r(T)\\xi.}", "0312dad48a2779999ccfc1fac186c7cd": "h(N) \\leq c(N,P) + h(P)", "0313062e85dc040c41b9a33ef924c201": "a = r, R = 2r", "0314393ec3e1463189a167a4f2c45163": "\\lambda_c\\sin\\theta = n\\lambda\\,\\!", "0314620279358b68099b802277c1ea4c": "OH=3GO.", "031468b74b375cc8ed6d70f63e2e73a6": "i = 1, 2, ..., k", "031499d95612e801ccacb94f8850bd24": "Hx = 0", "031506c1f09d2ec31c224a1f7b427673": "E^+_q", "0315213a8991b040c5c0b50c37c2fd6b": "\\sum_{n=1}^\\infty \\Pi_0(n)x^n = \\sum_{a=2}^\\infty \\frac{x^{a}}{1-x} - \\frac{1}{2}\\sum_{a=2}^\\infty \n\\sum_{b=2}^\\infty \\frac{x^{ab}}{1-x} + \\frac{1}{3}\\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\frac{x^{abc}}{1\n-x} - \\frac{1}{4}\\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\sum_{d=2}^\\infty \\frac{x^{abcd}}{1-x} + \n\\cdots ", "03153117637a3d052115e3d9cf307dc0": "{BSA}=0.007184 \\times W^{0.425} \\times H^{0.725} ", "0315513f8afde9dd3363cc1da930c1ba": " \\omega = \\frac{\\operatorname{d}\\theta}{\\operatorname{d}t}", "031551a9e052da5c2ccdb6eab96e49e2": "\\textrm{Labor~Productivity~(output~per~hour)}={\\textrm{Output}\\over\\textrm{Labor ~Inputs}}", "031590f5590f02496da38540db955827": "{{i}_{E3}}=\\frac{\\beta +1}{\\beta }{{i}_{C3}}", "0315de9e1bec426d2fdd18e8cffc516d": "\\tau:X_\\text{reg}\\rightarrow X\\times G_r^n", "0315f119d11928920959df3b5cc610e3": "E' = E/(1-\\nu^2)", "03163673c5da8149d5b745a2d34b58d7": "\\Omega = \\sum_{p \\in P} 2^{-|p|}.", "0316b16e199796da2f21f486e1ae0418": " \\begin{alignat}{4}\nf(x)&=x(\\sqrt{x+1}-\\sqrt{x})\\\\\n & =x(\\sqrt{x+1}-\\sqrt{x})\\frac{(\\sqrt{x+1}+\\sqrt{x})}{(\\sqrt{x+1}+\\sqrt{x})}\\\\\n &=x\\frac{((\\sqrt{x+1})^2-(\\sqrt{x})^2)}{(\\sqrt{x+1}+\\sqrt{x})}\n &=\\frac {x}{(\\sqrt{x+1}+\\sqrt{x})}\n\\end{alignat}", "0316c850d12c519d4a0e6ba29c718df3": "(1,4,2)", "0316e7c928254cd7a3986f7ed83ce256": "\\{x,y,z\\}", "0317100d49f06c725fa4722579232829": "\\textstyle p \\equiv 2 \\mod 3", "0317f7c498e366823c7bad03638baf3d": "\\int_V \\rho(\\mathbf{r})(\\mathbf{r}-\\mathbf{R})dV = 0.", "031801a96c6385da55f551b10027d2f4": "\\operatorname{im}\\, \\kappa", "031882c0e138764b2fd5e51ca2e686d9": "B = {h \\over{8\\pi^2cI_{\\perp}}}", "0318d3e5bc7f0dbfffb433ef97df66b0": "\\varphi_{X+Y}(t)= \\operatorname{E}\\left [e^{it(X+Y)}\\right]= \\operatorname{E}\\left [e^{itX}e^{itY}\\right] = \\operatorname{E}\\left [e^{itX}\\right] E\\left [e^{itY}\\right] =\\varphi_X(t) \\varphi_Y(t)", "03198b53127912f205d924d069a9412d": "B(x_1,y_1) B(x_2,y_2) = B(x_1 x_2 \\pm ty_1 y_2,x_1 y_2 \\pm y_1 x_2).\\,", "0319dd14cc0deff086e99c2188ae997b": "S=\\sum_{i=1}^9 j_i.\n", "0319eff6ca14880cf69584473c382251": "\\hat{z}=\\hat{c}.", "031a2d2a71d63ad0dcd9943fa3a8ad57": "\\phi: \\mathbb{R}^{2} \\rightarrow \\{0\\} ", "031a3a56765a5d292512ddab0cbef40d": "f : M \\to \\mathbb R\\,", "031a472a51c9e9fcbdd21e7dcda00203": "T_{1/2}=\\frac{-0.693\\,N}{\\frac{dN}{dt} }", "031a4a0901cfef55c54f49518f6baf29": "F_{x}", "031a5590aeeb04464126550f43aa0dcf": " \\mathbf{E} \\, \\mathbf{t} = \\mathbf{R} \\, [\\mathbf{t}]_{\\times} \\, \\mathbf{t} = \\mathbf{0} ", "031aa2ffc9b711d2e8568231f22a365d": "J \\colon \\pi_r (\\mathrm{SO}(q)) \\to \\pi_{r+q}(S^q) \\,\\!", "031ad52745d5567e6c39296eea619281": "\\mbox{QMA}(c,s)", "031b4efd8ba7d8c4e90d80a2e199e411": "\nD_\\text{KL}(\\mathcal{N}_0 \\| \\mathcal{N}_1) = { 1 \\over 2 } \\left\\{ \\mathrm{tr} \\left( \\boldsymbol\\Sigma_1^{-1} \\boldsymbol\\Sigma_0 \\right) + \\left( \\boldsymbol\\mu_1 - \\boldsymbol\\mu_0\\right)^{\\rm T} \\boldsymbol\\Sigma_1^{-1} ( \\boldsymbol\\mu_1 - \\boldsymbol\\mu_0 ) - K -\\ln { | \\boldsymbol \\Sigma_0 | \\over | \\boldsymbol\\Sigma_1 | } \\right\\},\n", "031b9bff8fd8e223ac7b3fc4a03cbd51": "C_D = 1.456 \\times 10^5 (\\frac{\\eta P}{\\sigma S V^3})", "031ba9e2f80ace02d96bc0ec234c80a4": "\\dot{x}_i = \\partial H / \\partial p_i", "031c29938cf30deaad28a1e05352c788": " G < f^{64}(4)y) = \\frac{\\int_y^\\infty x g(x) dx}{1 - F(y)} ", "032e291fd1e5592fcd3b7abe2ea18bef": "\\vec a", "032e9e63fc63d330dfec39bd6bd9e61d": "h_x(\\alpha), \\beta_x(\\alpha), \\lambda_x(\\alpha)", "032eae71d2c2662177f5536db706a47a": "G = S_n", "032ef5050a6c2c4507e9fede8c0b9be8": " \\int \\rho(y) e^{i k y} d^n y = \\langle e^{i k y} \\rangle = \\langle \\prod_{i=1}^{n} e^{ih_i y_i}\\rangle \\,", "032f0a403827291bc6d37f54cdbde8a9": "\\mathcal{L}()", "032f27fb65b3aa322bd965750957a372": " CPP = MAP - ICP ", "032f3d72715da0c48307515af3ed66e1": "number_{(base)} = \\sum_{i=0}^n {digits_i \\times base^i}", "032f3e25fb024bf84e86c774d20682a2": " \\quad 1", "032fc87d7eaf33a24c0c566729002bc8": "\nH = \\frac{1}{a^2} + \\frac{n-1}{b^2}.\n", "032fcbe05b92f61b4fd08f0722ec1cc7": "\\Gamma^i {}_{j k}", "032ff6a1dbdbb6918b10c7310cfe2be5": "(m, P)", "03302b9afd10bdb58e56b2c229a96f77": "(\\forall n\\in\\mathbb{Z}_+):A_{n+1}(x)=\\int_0^xyA_n(y)\\,dy;", "03302f88497644ad4db5e6052d763ba3": "(2^m - m - 1)/(2^m-1)", "033041bab2326817f03a5c4fcb30a589": "\\mathbf{E} [x^m] = \\frac {\\Gamma (Nk - m)} {\\Gamma(Nk)} y^m", "03304451b4f871ed5fa8b77ff7e5a355": "r^2~\\ln r", "03307643d8dd24479f0fb0d50726c9aa": "+\\hbar k_{max}/m", "0330786fa12c9466393d0c36958a5e26": "\\scriptstyle \\theta/(2 \\pi)", "03307c2a355cb08ccf414ce55fe1dd46": "= u(\\sigma(p)) = u(\\sigma(x)) = \\sigma(x) \\,", "033088a3e831ea9c495aa021f0d91f99": "\\Delta\\Delta G_{i, j}^{stat} = \\sqrt{\\sum_x (ln P_{i|\\delta j}^x - ln P_i^x)^2}", "0330e6322712b9e884f75ba8908f4bf3": "(S,S) = 0 . ", "03315ba0f07a3c0afb394d187b635e5a": "\\mathbf{r}_1-\\mathbf{r}_3", "03315f3fc919ff9c467f51369cdb0525": "2 - 1", "0331640d8f7864d0de31445dc0a005e4": "H_n(X;A)\\simeq A", "0331c2749aa13f4e61217ec85f967f29": "E_{x,z} = l n V_{pp\\sigma} - l n V_{pp\\pi}", "0331e3a1d454230fe07a41190807ce88": "\\sqrt{1+2\\sqrt{1+3 \\sqrt{1+\\cdots}}}.", "03323da5fe872cf192f04e86b4b6d097": "x = R_H", "0332a3a0fb4aa99a9f4daa8b9b306250": "\\omega_{0}=1/\\sqrt{R_2 R_4 C_1 C_2}", "0332a6e175c2ea9fab3b0e7dc3287806": " {1 \\over \\lambda} = R \\left( {1 \\over (n^\\prime)^2} - {1 \\over n^2} \\right) \\qquad \\left( R = 1.097373 \\times 10^7 \\ \\mathrm{m}^{-1} \\right)", "0332b621d562f1bc5526f4b1879c3a50": "\\partial \\omega / \\partial x = 0", "0332ed61d1dd26753da1ea5b26a23387": "\n\\nabla:TM\\times\\Gamma(E)\\to\\Gamma(E) \\quad ; \\quad \\nabla_Xv := \\kappa(v_*X)\n", "033314f8c2ab97c4d497d7ae5b0889d0": "x_{t=0}(x,p) = x", "03334a7d3910583154d57b92ff5e90ee": " \\Delta z_{\\rm{bias}}", "0333a113cad1852120b10f74af753df0": "\nP_s\\left(T \\right) = 6.1121 \\exp \\left(\\left( 18.678 - \\frac{T} {234.5}\\right)\\left( \\frac{T} {257.14 + T} \\right)\\right)\n", "0333eb5744fa9eeb4a90a746c738ac85": "f(x) \\geq f(y) + f'(y)(x-y)", "03343105f7e2274b02c3185b0bb305b0": "\\prod _x ax^2+bx = C\\,a^x \\Gamma (x) \\Gamma \\left(x+\\frac{b}{a}\\right) \\,", "03343acd809ae93effe3a7985482a132": "(r_i-i)^2", "03345fcc32c81a8fc9e3697dcac7a670": "\\chi_{0}(\\mathbf{q} | \\Gamma)", "03347a6365e48bfd261160967f23fa18": "A,B,C \\in \\mathcal{C}", "0334c669d93e04082a201fb3aa1afc4f": "f(a)-f(x)\\quad", "0334cb1648ac9ff1ee1d24f11ada8c2f": " dt = \\gamma(\\mathbf{u})d\\tau \\,.", "0334cb73be3efe3f0d830347d285179b": "f_i^{(k\\ell)}", "033546280bfbef560db2e14aca08fce8": "\\overline{\\{0,\\dots,n\\}}", "03355c9959a615b999c58afc2c9c177f": "D(X, Y) = \\sum_{i} \\sum_{j} |x_i-y_j|P(X=x_i)P(Y=y_j).\\,", "0335bb6966e9f95fffbbbdd15844f939": "\\delta(x-z)", "0335f44e12ff05d522834047bc1d8611": "E\\bigl(g(X)(X-\\mu)\\bigr)=\\sigma^2 E\\bigl(g'(X)\\bigr).", "033612701cd03227f71fd00b13d902e7": "\\!\\mathcal A \\models_Z^+ \\psi", "0336bde25c6dbb9bacfa998e2b5016c0": "(b_{14}-a_{14})+(b_{15}-a_{15})=77", "0336c6ab921432effb4a4fda380f55e4": "a^{\\dagger}a", "0336d7f982fdbf9208732a1e267584e1": "K = \\frac{1-\\left|S_{11}\\right|^2-\\left|S_{22}\\right|^2+\\left|\\Delta\\right|^2}{2\\left|S_{12}S_{21}\\right|}\\,", "0336e62688f0d883c689d6ef13e65119": "P(X_i=a \\cap X_{i+1}=b)", "03371dc87912c24e81c3d0c1a453b4b3": "P\\in\\operatorname{Hom}(H,H)^{\\times}", "0337c65c3af4b8bb8626311abef3e21a": "\\scriptstyle \\sigma \\,=\\, 0.5", "0338003e027771272ed23c1e6a62c522": "\\rho_{XY} =E[(X-E[X])(Y-E[Y])]/(\\sigma_X \\sigma_Y) \\;", "033827b39b706414493b6dfede94ee51": "{\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y = 1.", "0338c024c047094ddaeb91ab7612cdfb": "y_0=y(t_0), \\ \\ y_1=y(t_0+h), \\ \\ y_2=y(t_0+2h), \\ \\dots", "03392cc816f562c05c7cddb75f43d0ab": "d x = {\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z \\, d y + {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y \\,dz", "033946767fb687a8acafe00a11dc39ef": "X_1\\times\\cdots\\times X_n = \\{(x_1, \\ldots, x_n) : x_i \\in X_i \\}.", "03395260509a989d7c0a6ed0871e76b3": "\\mathrm{ext}[K(X)]", "033a2debd7ace8a1f02030a6418b95c7": "A = \\textstyle 2a^2 \\cot \\frac{\\pi}{8} = 2(1+\\sqrt{2})a^2 \\simeq 4.828427 a^2.", "033a59ab09479bd41cff2a9bb8a2be03": "|A\\rangle", "033a5c232f479b005190cd88d26bd326": "\\mathbf{NTIME}(f(n)) \\subsetneq \\mathbf{NTIME}(g(n))", "033a5e7c5a9565b911c8496e0ba88b23": "{\\mathcal L} = -\\frac{1}{4} F_{\\mu\\nu}F^{\\mu\\nu}-\\frac{n_f g^2\\theta}{32\\pi^2}\nF_{\\mu\\nu}\\tilde F^{\\mu\\nu}+\\bar \\psi(i\\gamma^\\mu D_\\mu - m\ne^{i\\theta'\\gamma_5})\\psi", "033abc09c3acf249153f99047185c031": "\n\\omega^{-1} = \\sqrt{2} + 1.\\,\n", "033ad9c60d57150d5fc5b9c0b201b554": "\n\\mathbf{N} = \\begin{bmatrix}\n -1 & 0 & 0 & 0 \\\\\n 1 & 1 & 0 & 0 \\\\ \n 0 & -1 & -1 & 0 \\\\\n 0 & 0 & 1 & -1 \\\\\n 0 & 0 & 0 & 1 \\\\\n\\end{bmatrix}\n", "033aeb8a43250203f680dcd041b14ea1": "6n - 1", "033b4d0d76a0a2b53b3a977e22c00e6e": "f_C = \\frac{1}{2 \\pi n\\tau_T} \\ , ", "033b571c237d78ae1c9908427fdf52ce": "\\frac{a}{b}", "033b794d576f4bbfe4f3bbee3044741b": "X_1, X_2, \\ldots, X_N", "033b7d86b0dfb4ace148e2d293734efa": " k = 2, \\ldots , r", "033bb10b01e31642bebef44240a150f9": "x(u, v) = \\left(R + r\\cos{v}\\right)\\cos{u}, ", "033bd7948909fec272d4cde4fc3a9d59": "X(0)= \\eta", "033c331ac596852538de39eb0b3f3b96": "\\int_0^1 \\left(S_n(s) - {1 \\over 2}\\right) \\left(S_m(s) - {1 \\over 2}\\right){ds \\over \\sqrt{s(1-s)}}=0.", "033c94564ba398681d4405e630ff5379": "\\scriptstyle {2\\cos{\\tfrac{2\\pi}{7}} \\approx 1.247}", "033cf64947ed82dfdd4acedc393e56b4": "\\delta Q\\ =C^{(p)}_T(p,T)\\, \\delta p\\,+\\,C^{(T)}_p(p,T)\\,\\delta T", "033d24ff17ac6a0c27b2a6ee5db80984": "p+dp", "033d252565b13f3e612b0fa5abfc05ce": "i \\hbar \\frac{d\\psi}{dt} = - \\frac{\\hbar^2}{2 L} \\nabla^2 \\psi+\\frac{Q^2}{2C} \\psi ", "033d2c3acd83e1a970dfd82cd9be15fa": "x^0 + \\Delta x^0 = x^0 + \\tfrac{1}{2} \\left ( dx^{0(2)} + dx^{0(1)} \\right ).", "033d30d82100677736d9d12f77f0de17": "\\Re\\left(\\sum b_k - \\sum a_j\\right)>0", "033d3a80af90a171e50a9601b34ea1fc": "C_\\nu", "033d7fc1e2ae2eac0742a1097f858515": "t^2 = 2\\frac{d-d_i - v_it}{a} = 2\\frac{\\Delta d - v_it}{a}", "033dd2348284fbd3b6479e6b9599faae": "\\scriptstyle \\int r\\,d\\theta = 2\\pi r", "033dff11140d43b95e2e7ae5d19c42d2": " Y_i", "033f61cdfc761a03f151199e89a1d96a": "\\Pi^1_0", "033f7f4c3eb918d9336804647277a218": "10^{8}", "033f9e7b053ff7d55c8790005fc6fdde": " \\tbinom nk ", "033fe8d62666e7954e987a7e43d72f53": "U: x\\mapsto ax\\pmod p", "034024b1ae850a9f144606083c40a03b": "\\cos\\left(\\tfrac{\\pi}{2}\\,(2k+1)\\right)=0", "0340253c8a6f812b5baaffc88152e24c": "\\ \\mathcal{L}_\\mathrm{gf} = - \\frac{1}{2} \\operatorname{Tr}(F^{\\mu \\nu} F_{\\mu \\nu}) ", "034028a4b04ca029b2eff7d3062092b5": "\\displaystyle p_n(x;a|q) = {}_2\\phi_1(q^{-n},0;aq;q,qx) = \\frac{1}{(a^{-1}q^{-n};q)_n}{}_2\\phi_0(q^{-n},x^{-1};;q,x/a) ", "03406fcfd1a0a53351ef741c5988d692": "(R_1^T)^{-1} b", "0340aa31c15462805b477ff46b680260": "F_n(\\lambda)/F_{n-1}(\\lambda)\\cong A(\\mu)", "0340c06debcea43df493c2c8130f3ef1": "Tr(t^at^b) = \\frac{1}{2}\\delta^{ab}.", "0340eae31c3aa0d609eb49e60bc1a4d9": " \\lambda_{CW}(M) = 2 \\lambda(M) ", "0340f80858ff9393d898c1c06d2bb192": "f \\colon U \\times Q \\to V", "034173eabcc634d08a0d3abe459e0c5a": "T\\in \\operatorname{Hom}\\left(\\wedge^2 TM, TM\\right).", "0341ae6f0ad16522997f8ac07e7b6b06": "\\frac{k_B T}{\\gamma} \\Gamma^{-1}", "0341af483b2c59f352de9ff7be013758": "{3\\pi\\over 5}\\ {\\pi\\over 3}\\ {\\pi\\over 2}", "03421840ed5f91b4bf9ed2c83642c61e": "B1-B2", "03421cef1aba2eedcd955e387f7abea5": "\\mathbf{P}(X > (1+\\delta)\\mu) < \\frac{\\prod_{i=1}^n\\exp(p_i(e^t-1))}{\\exp(t(1+\\delta)\\mu)} = \\frac{\\exp\\left((e^t-1)\\sum_{i=1}^n p_i\\right)}{\\exp(t(1+\\delta)\\mu)} = \\frac{\\exp((e^t-1)\\mu)}{\\exp(t(1+\\delta)\\mu)}.", "03422e6c61867719daf2bd867bbc22da": "\n \\ln\\mathcal{L}(\\mu,\\sigma^2)\n = \\sum_{i=1}^n \\ln f(x_i;\\,\\mu,\\sigma^2)\n = -\\frac{n}{2}\\ln(2\\pi) - \\frac{n}{2}\\ln\\sigma^2 - \\frac{1}{2\\sigma^2}\\sum_{i=1}^n (x_i-\\mu)^2.\n ", "0342381b29fc3888b071b4bc6fac9d5c": "P\\simeq \\frac{1}{3} \\epsilon=0.52\\times 10^{31}\\, \\mbox{bar}.\n", "034240c73f7a3435503c27a7cfb0e88e": "U_\\mathrm{in}(t)", "0342852b3b6d1dc4bd117a60e9c54334": "f(\\xi,\\rho,\\theta) = 0 \\,", "0342b646bdcb5436267848732910280c": "C = X_1-A = 1", "0343305d22cfe9adb435704d04f30c9a": "1. \\; \\; \\mathrm{NO}_2 \\; \\xrightarrow{h \\nu } \\; \\mathrm{NO + O}", "03435ed85217ab8779bbcd4f312fc645": "a_{T}^{\\pm} \\rightarrow \\gamma W_L^\\pm", "0343a6b2e82f9d4cdd49eb0d78dcd015": "A(D)=D^2+k^2", "0343ce9c1be67fe20e0d5cedb5044a6b": "\\rho_{\\text{e}}", "03441a463fb12d0679dcc77740797155": "\\tfrac{(M-\\lambda)(M+2\\lambda)}{M+\\lambda}", "034460feb7080db6f557cfd7a3780154": "\\Lambda(n)", "03449dc8a842ae7ae49e08f0af9205c2": "4a^2-4ab+4b^2=(2a-b)^2+3b^2, \\,\\!", "0344b0d9ea0118da3c511f5ccca766d4": "d\\mathbf x'=\\mathbf R \\,d\\mathbf X\\,\\!", "0344b1178a956f175a86eb6b03fd2032": "D_F^q(p, q)", "0344f847b22a6e355a58990d65b75ce2": "\nS = k N \\ln\n\\left[ \\left(\\frac VN\\right) \\left(\\frac UN \\right)^{\\frac 32}\\right]+\n{\\frac 32}kN\\left( {\\frac 53}+ \\ln\\frac{4\\pi m}{3h^2}\\right)\n", "03452f9a2b862e540da40d6e9858ab40": "U = -0.147 \\times R - 0.289 \\times G + 0.436 \\times B", "03456c32718e6804c8a92a89315cddc7": "(\\det\\Phi)'=\\sum_{i=1}^n\\det\\begin{pmatrix}\n\\Phi_{1,1}&\\Phi_{1,2}&\\cdots&\\Phi_{1,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi'_{i,1}&\\Phi'_{i,2}&\\cdots&\\Phi'_{i,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi_{n,1}&\\Phi_{n,2}&\\cdots&\\Phi_{n,n}\n\\end{pmatrix}.", "0345716ff0625c5efcca01c286036654": "k = e'\\cos\\alpha_0,", "03458f33713d60710c6aca5db16d5bfd": "\\frac D R", "0345e2efcbef367d84fc1770dbe49332": "\nH(2^2) = \\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & -1 & 1 & -1\\\\\n1 & 1 & -1 & -1\\\\\n1 & -1 & -1 & 1\\\\\n\\end{bmatrix},\n", "03463cd817eaa6de0f808b641f59a7d6": "u = x + \\frac{b}{2c} ", "03471bcbc90fb56290b25b87fffce665": " (ae - bf - cg - dh) + (af + be + ch - dg) \\mathbf{i} + (ag - bh + ce + df) \\mathbf{j} + (ah + bg - cf + de) \\mathbf{k}", "03479877a489a72aeb0d78d3bfef37c1": "\\epsilon^{-d}", "0347b4898c96cbc2a86d3e8e2dc3f2b6": "\\mbox{Tr} \\mathcal {L} = \\sum_n \\langle \\psi_n , \\mathcal{L} \\psi_n \\rangle", "0347fffa7ae2123c9fd659b954b7ae7f": "[A + m(t)]\\cdot \\sin(\\omega_c t),\\,", "03480067b9a23e49fef8dec4890d1591": "f(\\sum\\nolimits_i a_i \\sigma_i) = \\sum\\nolimits_i a_i f(\\sigma_i)", "03483c0575c70ca078b4d06c463cfd93": " (x y)^* = y^* x^*", "0348948eec49e40cc114d0052df04810": "T_f\\;", "0348b58f302d593b58c1d3bb37944b25": "0<\\alpha^{\\,}<1", "03490af427ab865371d9aa274292dd84": "\\sim 1 nm", "034912b6711e851d4b4b85c5db46db55": "\\neg P\\,", "03491dcee43720d276368a4879f55105": "M_Z=\\frac{v\\sqrt{g^2+{g'}^2}}2,", "03498f1d79bdc5f60c4a3890d104e871": "H_{m n}^{\\text{eff}}\\left(x^{\\mu }\\right)=\\langle m|H|n\\rangle +\\langle m|\\partial _{\\mu }H|n\\rangle x^{\\mu }+\\frac{1}{2!}\\sum _{l\\in\\mathcal{H}_H} \\left(\\frac{\\langle m|\\partial _{\\mu }H|l\\rangle \\langle l|\\partial _{\\nu }H|n\\rangle }{E_m-E_l}+\\frac{\\langle m|\\partial _{\\nu }H|l\\rangle \\langle l|\\partial _{\\mu }H|n\\rangle }{E_n-E_l}\\right)x^{\\mu }x^{\\nu }+\\cdots.", "034a110f06a0fb2676560eeb40e30aaa": "\\boldsymbol{u}_e=\\boldsymbol{u}-\\boldsymbol{u}_g.", "034b20078cf947f201ab4f8238c147a8": " \\Delta\\mathbf{r}_i^\\perp = (\\mathbf{r}_i-\\mathbf{R}) - (\\mathbf{S}\\cdot(\\mathbf{r}_i-\\mathbf{R}))\\mathbf{S} = [[I]-[\\mathbf{S}\\mathbf{S}^T]](\\Delta\\mathbf{r}_i),", "034b471f7c8f16bdaa36d9c8403de256": "|C_v|", "034b6293a338c9fd04e49062fcf5946e": "k\\cdot 2^{-j}", "034bcd264c3b7361e9fd371edbde15bc": "T_1[i,j]=\\max_{k}{(T_1[k,j-1]\\cdot A_{ki}\\cdot B_{iy_j})} ", "034bd8c7af8d5b7c12b76495fdadf2d9": "\\frac{\\partial u}{\\partial t} - \\alpha \\nabla^2 u=0 ", "034c0b19b15dece85c48f9d46635d5d4": "\nV = \\int_{1}^{\\infty} f(x) \\cdot \\pi f(x) \\,\\mathrm{d}x\n\\leqslant \\int_{1}^{\\infty} {M \\over 2} \\cdot 2 \\pi f(x) \\,\\mathrm{d}x\n\\leqslant {M \\over 2} \\cdot \\int_{1}^{\\infty} 2 \\pi f(x) \\sqrt{1 + f'(x)^2} \\,\\mathrm{d}x\n", "034c2f092a5f4253243eeb08480739e7": "E_A = \\frac{Q_A}{Q_A + Q_B}", "034c3f4b6980de5052e02e3aad6dcf93": "{\\color{Periwinkle}f'}(x_0) = \\frac{1}{4}", "034c48eb7f86a75a7e074b19200b2b87": "r\\approx\\frac{\\ell c}{2 \\pi f}", "034d36cd80de7dc8c6c57f505eff084d": "{2+\\|\\mu-\\nu\\|_{TV} \\over 4}", "034d4b15ad7de43ef240fdc74360fdd5": " Z_q(V_o,T)=\\int_0^{\\infty}\\sigma(E)[1+(q-1)\\beta E]^{-\\frac{q}{(q-1)}} dE\\,,", "034d4db04be85fef0334b6527626d63c": "O(M+N)", "034d7e5fee2fec89ba717192a586abe2": " \\eta_1 = \\eta_2 ", "034d8d44230f5e1974123a2fed5a38a8": "\\lim_{r \\to 0} f_r(x)", "034d93e729f59291d0aad712e0c96eb5": "(1-x^2)^{1/2}=1-\\frac{x^2}2-\\frac{x^4}8-\\frac{x^6}{16}\\cdots", "034db7fc980bc9d70e5a32c6423d0d5a": "R = \n\\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\n", "034e098033cdd65d48996c47865c7ad6": "\\ I \\cdot I^{-1} = R", "034e32ca4e515d9ff4c8ea9faebbdc81": "\\frac{a}{2^b}-\\frac{c}{2^d}=\\frac{2^{d-b}a-c}{2^d} \\quad (d\\ge b)", "034e72f956cd45baa15c8832ec645b25": "\\operatorname{Hom}_R\\biggl( \\bigoplus_{i \\in I} M_i,L\\biggr) \\cong \\prod_{i \\in I}\\operatorname{Hom}_R\\left(M_i,L\\right).", "034f01c8ffbcfb03f06bd4919ddbeb87": "T_A^1~|~T_A^2", "034f29f30627882fc04fcfcfd26c65bf": "d_i, v_f =0", "034f48302f81deed68d4491eb032f8fb": " {(0,1,1)}", "034f59279c8bfa23635cdd456fa3e8a9": "\\frac{\\pi}{\\sin \\pi z}", "03501a92ee6948b2bb347b444193e40a": "\\hat\\beta = (X'X)^{-1}X'y = (X'X)^{-1}X'(X\\beta+\\varepsilon) = \\beta + (X'X)^{-1}X'\\mathcal{N}(0,\\sigma^2I)", "0350479dcfff9ffbf51732a10e5adfab": " b^2 c = 4(a-e)e = 4ae - 4e^2.", "03505471828bba790b78b5d6ae1426e5": "\\ell = \\frac{k_{\\rm B}T}{\\sqrt 2 \\pi d^2 P}\\, , \\;\\;\\; v_T=\\sqrt{\\frac{8k_{\\rm B}T}{\\pi m}}\\, .", "03510f0873a68eddaf477e68a9191052": "f(a) = \\mu \\log_2(a)\\,", "0351223e5532e427abfc10e9e4c8770a": "p \\cdot (\\Sigma _i x'_i) \\geq r", "03517dc5bbed7c241b29b04aafd77b11": " y_s(x) = -(1/2)x^2 + (-(1/2)x)^2 = -(1/4) \\cdot x^2. \\,\\!", "0351993ccc8ddb7723ec12b0d61bf7c2": "\\upsilon_D = \\frac{Vq}{2\\pi} \\qquad(3)\n", "0351e4eef5a980a9675866d564e970c6": "X^n/G", "0352187c748afe8507513f0d16b9d224": " |\\phi (t+dt) \\rangle - |\\phi (t) \\rangle = -i\\hat{H} dt |\\phi (t)\\rangle ", "03526787395d9c9cfeeb852f1489558e": "\\pi(A)=[A]", "0352924c1493364e408c14b645a3e297": "\\scriptstyle <10^{-12}", "035295d112bec33185dba2624c9d50c6": "\\phi (\\omega)\\triangleq\\arg K(j \\omega)=\\arctan \\frac{\\omega_0}{\\omega},", "035338c2498de04a3a6f3d3dc9c456cc": "\\sum_i a_i\\sigma_i\\,", "035355567ff818400354891fead4ca0e": " \\lambda= \\frac{D_x\\Delta t}{2 \\Delta x^2}", "03536bcbf2e10f360201eccb72dd34b0": "(x)_{n+1} = \\sum_{k=0}^n\n\\frac{n+1}{k+1}\n\\left[ \\begin{matrix} n \\\\ k \\end{matrix} \\right]\n\\left(B_{k+1}(x) - B_{k+1} \\right) ", "03544dddcb8d13bd5a09e7e442e394cb": "\n H^*_{\\overline{p}}x (n) = \\prod^{\\overline{p}}_{j=1}{x(n - \\tau_{j})} \n", "035451646fcceec539999e4521091551": "da(t)=\\delta_{t}a(t)\\,dt\\,", "0354cb942e62f3909d73c66a52072437": "\\mathbb{C}^{N/2}", "0354d96c50c762db44e348bf0fe7f48b": "3\\times3", "0354f3238103bfb970d5fea51b94adeb": "\\begin{cases}\n\\text{always} \n\\begin{cases}\n\\text{always } 0 \\\\\n\\text{if } y , +1\n\\end{cases}\n\\\\\n\\text{if } x , +2\n\\end{cases}", "03551109fb26fcb802cf4443e4d0a1bc": "\\tau_{\\beta,\\alpha}", "03551e591f616e8f74eec8a006ee40fc": "\nu_\\varepsilon \\left( \\xi ,\\eta ,z\\right) = \\frac{w_{0}}{w\\left(\nz\\right) }\\mathrm{C}_{p}^{m}\\left( i\\xi ,\\varepsilon \\right) \\mathrm{C}\n_{p}^{m}\\left( \\eta ,\\varepsilon \\right) \\exp \\left[ -ik\\frac{r^{2}}{\n2q\\left( z\\right) }-\\left( p+1\\right) \\psi _{GS}\\left( z\\right) \\right] ,\n", "03555a99a410f13b428f1ae7f0b65966": "\\operatorname{MSPE}(L)=g'(I-L)'(I-L)g+\\sigma^2\\operatorname{tr}\\left[L'L\\right].", "035571d0c387810ba8c29b16f26d4873": "s(t)=\\sum_{m=-\\infty}^\\infty \\sum_{n=-\\infty}^\\infty C_{m,n}h(t-mT)e^{jnt\\Omega}", "03557f6220dd37ac9bd22a4d3c605a20": "\\sqrt{I_2} = \\lambda \\sqrt{I_1}", "0355849f21ea7d1b1ac082a874360fbd": "\n \\boldsymbol{\\sigma}_r = \\boldsymbol{Q}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{Q}^T ~;~~ \\boldsymbol{Q}\\cdot\\boldsymbol{Q}^T = \\boldsymbol{\\mathit{1}}\n", "0355c3d493eb27ac190768aae8309697": "\n (5) \\qquad \\cfrac{\\partial^3\\varphi}{\\partial x^3} = -\\cfrac{m}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} + \\cfrac{\\partial^4 w}{\\partial x^4} + \\cfrac{1}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial x^2}\n", "0355cf52d0d548ef1797bde3e95c8626": "\\begin{array}{rcl}\n C\\dfrac{d V}{d t} &=& -I_\\mathrm{ion}(V,w) + I \\\\ \\\\\n \\dfrac{d w}{d t} &=& \\phi \\cdot \\dfrac{w_{\\infty} - w}{\\tau_{w}}\n\\end{array}", "0355fdad639ee1a519a51a03d9577982": "\n1/\\eta_f = q_4 S\n", "0356051658dbbe05dfe8095bd591a425": "\n \\mathcal{I}_j = \\frac 2{\\sqrt{-\\mu_j}\\sqrt{\\lambda}} \\int_0^{\\infty} e^{-\\xi^2/2} d\\xi = \\sqrt{ \\frac{2\\pi}{\\lambda}} (-\\mu_j)^{-1/2}.\n", "0356c7833ecb6be4248c48f846b39891": " B_r =0 , \\quad B_{\\theta} =0 , \\quad B_z=a r^k ~f(\\psi)", "0356e5d88c047cd5055748098f28e8f8": " (u^2 + dv^2)^2 - d(2uv)^2 = 4. \\, ", "035721a27302ab4cb4c360e442ba1412": "H^2 = \\frac{8 \\pi G}{3} \\rho - \\frac{kc^2}{a^2}", "0357a9fb1ab694c9ed122a29c5441768": " \\nu_{\\mathrm{F}} ", "0357f7f863cd4e8ac4242f071798b6a7": "x \\mapsto x'=f(x)", "035872fd1b17cfe817547feef6286761": "s_{0} = \\sigma_{0}+iT", "03591a93124aad4e699d57c084dc5bb0": "b_n\\,", "0359270cc899a5f343ea43b879a7c757": "\\sigma_{ij} = \n\\begin{bmatrix}\n\\sigma_{11} & \\sigma_{12} \\\\\n\\sigma_{21} & \\sigma_{22}\n\\end{bmatrix} \n\\equiv \n\\begin{bmatrix}\n\\sigma_{x} & \\tau_{xy} \\\\\n\\tau_{yx} & \\sigma_{y}\n\\end{bmatrix}", "035937e14a7259dddea132a7dc81610f": "(A \\lor \\lnot A)", "03594f7a5482079c0f1f6cb2e7ba42cd": "=1/2+2\\epsilon_1\\epsilon_2\\ ", "035955e25306ff79019df1214e7e7780": "K_m=\\tfrac{1}{2} (k_1 + k_2).", "03599cb3a1147c64f7995421d197c7ea": "T(*)=B", "035a1895933f9ad2344ba70e8b3ce4a0": " \\mathbf{A} \\circ \\mathbf{B} = \\begin{pmatrix} A_{11} & A_{12} & \\cdots & A_{1m} \\\\\n A_{21} & A_{22} & \\cdots & A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{n1} & A_{n2} & \\cdots & A_{nm} \\\\\n\\end{pmatrix}\\circ\\begin{pmatrix}\n B_{11} & B_{12} & \\cdots & B_{1m} \\\\\n B_{21} & B_{22} & \\cdots & B_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n B_{n1} & B_{n2} & \\cdots & B_{nm} \\\\\n\\end{pmatrix} =\\begin{pmatrix}\n A_{11}B_{11} & A_{12}B_{12} & \\cdots & A_{1m}B_{1m} \\\\\n A_{21}B_{21} & A_{22}B_{22} & \\cdots & A_{2m}B_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{n1}B_{n1} & A_{n2}B_{n2} & \\cdots & A_{nm}B_{nm} \\\\\n\\end{pmatrix}", "035a36827f08ba9bcb795d03f1caf852": "H = \\ln(2\\pi I_0(\\kappa))-\\kappa\\phi_1 = \\ln(2\\pi I_0(\\kappa))-\\kappa\\frac{I_1(\\kappa)}{I_0(\\kappa)}", "035aa272cdc59be3d2059ed9d259fef2": "{V_s}", "035b0c810fe8694a8e962c8b902e9be5": "m(x,\\beta)", "035b5644665e0eff926385976add04d4": "\\psi(\\Omega^\\Omega)", "035b7dde426417915d7434b285015c83": "y = \\varphi-\\varphi_0 + \\cot(\\varphi) (1 - \\cos((\\lambda - \\lambda_0)\\sin(\\varphi)))\\,", "035b830fb147793943518050a9f77f23": "{dt}", "035ca07e19340efe54cf6714b3900a19": "m = 3.4~m_e", "035d005b37b1e448ca08dcff67204ba2": " I_2", "035d0437c9ab039649490a1f5da46923": " \\widehat X^\\mathrm{T} = P_Z X", "035d0f09bc271c400235212bc27f4300": " \\frac {\\tau_1} {\\tau_2} \\approx A_v \\frac {R_i} {R_i+R_A}\\sdot \\frac {R_L} {R_L+R_o} \\ , ", "035d1e2a7c93db50eb315ea047c8eb33": "T_a f(x)", "035d37a27f6a2f1387b1af89f4252aa2": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi_\\alpha(\\mathbf{r},\\,t) = \\hat H \\Psi = \\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r})\\right)\\Psi_\\alpha(\\mathbf{r},\\,t) = -\\frac{\\hbar^2}{2m}\\nabla^2\\Psi_\\alpha(\\mathbf{r},\\,t) + V(\\mathbf{r})\\Psi_\\alpha(\\mathbf{r},\\,t)", "035dbe2b93958b4c5753c67fb434e32b": "n!!!", "035de2830abae7a5890b234834d1e68a": "y(t) = (x * h )(t) = \\int_{a}^{b} x(\\tau) h(t - \\tau)\\, d\\tau", "035e2654176d12b3ebd686c286e749bc": "E_K^{-1}(C) := D_K(C) = D(K,C): \\{0,1\\}^k \\times \\{0,1\\}^n \\rightarrow \\{0,1\\}^n,", "035e6b8d56d3d2fcb2895b95238775c6": "L(x, y, t)", "035e923baf7cc63504d6e27f6afc8d55": "k \\in K", "035ebe0ca0b5b38dcb30fbbfd813e632": "K \\equiv \\prod_{\\omega \\,\\in\\, \\Omega} A_\\omega", "035f95f8636704951c2795c1d3deb25c": "L_0-R_0 = L_{n+1}-R_{n+1}", "036007e5c3f89086b75d6326fb344d74": "\\begin{array}{rcl}\n \\int_0^1 x^{1-t} y^t\\ \\mathrm{d}t\n&=& \\int_0^1 \\left(\\frac{y}{x}\\right)^t x\\ \\mathrm{d}t \\\\\n&=& x \\int_0^1 \\left(\\frac{y}{x}\\right)^t \\mathrm{d}t \\\\\n&=& \\frac{x}{\\ln \\frac{y}{x}} \\left(\\frac{y}{x}\\right)^t|_{t=0}^{1}\\\\\n&=& \\frac{x}{\\ln \\frac{y}{x}} \\left(\\frac{y}{x}-1\\right)\\\\\n&=& \\frac{y-x}{\\ln y - \\ln x}\n\n\\end{array}", "036017945dd7313c43d579e4f8ce2826": "\\varepsilon\\gamma_{\\mu\\nu}", "0360351f5986320b5342bfd2430991ad": "\np(t)\n =\\left(\\tfrac12L\\|F'(\\mathbf x_0)^{-1}\\|^{-1}\\right)t^2\n -t+\\|\\mathbf h_0\\|\n", "03603730b2b3b89b93daebe682ecc09c": "\\scriptstyle\\tbinom{-1}0=\\frac{(-1)^{\\underline0}}{0!}=1", "036045c0ca5b6b4db6970a8bfcabea54": "4 \\cdot m\\ ", "03607b43f29d20703ab22323966ca076": " w < -1", "0360936b2c68a3610f00a35a333887ec": " L_\\text{DC} = L_\\text{cen} + L_\\text{shd} + L_\\text{ext}\\, ", "0360d98007184ca17561e52001606f3c": "x[n/k] \\!", "0360ed882d872b7b815000f26aea9b5b": "\\textrm{PPPrate}_{X,i}=\\frac{\\textrm{PPPrate}_{X,b}\\cdot \\frac{\\textrm{GDPdef}_{X,i}}{\\textrm{GDPdef}_{X,b}}}{\\textrm{PPPrate}_{U,b}\\cdot \\frac{\\textrm{GDPdef}_{U,i}}{\\textrm{GDPdef}_{U,b}}}", "036128b1342fd074969bb61e2f64717e": "\\exp(\\gamma\\,t)\\;, \\qquad\\text{with}\\quad \\gamma={\\sqrt{\\mathcal{A}g\\alpha}} \\quad\\text{and}\\quad \\mathcal{A}=\\frac{\\rho_{\\text{heavy}}-\\rho_{\\text{light}}}{\\rho_{\\text{heavy}}+\\rho_{\\text{light}}},\\,", "036163a369759a544e290723ded6ac64": "-\\frac{b}{2a},", "03616a7b205bb8bae29cb8f2dd3c3672": "x^8=\\left(\\left(x^2\\right)^2\\right)^2.", "0361859e21297ddefe083b152eca8a59": "\\lfloor R^n / n \\rfloor", "036193184a8b39ee07604218efc190ca": "\\scriptstyle 1/2(1-x^2)", "0361f56d756082b809fc65c43892e692": "u: \\mathbb{R}^l \\rightarrow[0, \\infty)", "0362281c967583ca8fe3c72e5117067c": " \\{x_1,...,x_n\\}", "03627a7c6b959f06762436e5cebce5c5": "H= 13 + 6{,}93 \\cdot D", "0362846a7f7340b70c29116354f2812b": " \\lambda_2 ", "0362b84dc2a94087be5aec918183dabc": "\\boldsymbol{H}^\\prime", "0362ba7c7b21c002a805627676c67aa7": "t = \\int\\frac{dy}{iy+F}", "0362bb4e96567e31dff05cb30ebb0da5": " L(n,k+1) = \\frac{n-k}{k(k+1)} L(n,k).", "0362c8290d2dfc0d92533cea11259e76": "Loves(", "0362fd4cc3c69a4b89f60252cbec028d": "\\displaystyle \\alpha_k", "03637d758e05cd5e09ab92f25aed6305": "Cl(p,q)", "03637e55edc44c456709a0bbfe6ad999": "|\\alpha_i\\rangle, \\; a_i", "0364531c758ef179874228d39d030061": "l\\alpha_1,\\dots,l\\alpha_n", "0364911541927bcc589331b468da4b85": "\\frac{1}{24} + \\frac{1}{48} = \\frac{1}{16}", "03650ea866ab2b553590f75acb9f9163": " a = -\\log ( 1 - w_2 ) ", "0365431cc30d2b64a93712155623ba23": "dx^2 - adx + b^2c = 0", "0365ad89f3cba76103d2c14e92a69b7f": "\\frac{\\partial (\\mathbf{U} \\otimes \\mathbf{V})}{\\partial x} =", "0365dcab177cfb0b250d751763492687": "\\langle 0|\\Phi(x)\\Phi^\\dagger(y)|0\\rangle=\\sum_n\\langle 0|\\Phi(x)|n\\rangle\\langle n|\\Phi^\\dagger(y)|0\\rangle.", "03662063cfa5a0174240ca60eb9c35a3": "H,", "03663e2f2fba6abde1a8d8248e274cdd": "\\left(\\mathbb{Q},+\\right)", "0366a9adf0b7ed517a545111a592bb2b": "[ax+by,z] = a[x,z] + b[y,z]", "0366e9804d0813d197a05778bff33376": "t + C_2 = \\pm \\int \\frac{dx}{\\sqrt{2 \\int f(x) dx + C_1}}", "0366fd7f9a3a4c4b41add2c894a60ecf": "\\textstyle\\left \\lfloor {{d-1} \\over 2}\\right \\rfloor", "036761f0aa2fa0464cac43f69851e0d7": "\\,y = \\sin^2(t)", "036775ea0d50fa662df6070e56949133": "\\frac{dS}{dt} = - \\beta SI + \\mu (N - S) + f R ", "0367f3665bc4a38c20508951978243e6": "\\mu_2 = \\mu'_2 - \\mu^2\\,", "0367f40a08359f679931b045a08b1682": "\\varphi\\left(\\mathbb{E}\\left[X\\right]\\right) \\leq \\mathbb{E}\\left[\\varphi(X)\\right].", "03680e271932f7839cee3b91aad8c099": "q = q_s + \\vec{q}_v,", "03688d6eb8297d6f8e0b141fb14bc775": " \\left(\\frac{\\partial U}{\\partial V}\\right)_{T} = T \\left(\\frac{\\partial S}{\\partial V}\\right)_{T} - p = T \\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - p ", "0368a97cbd357e2b4f7789335b837e3e": "S\\subseteq \\cup_{j=1}^t T_{i_j}", "03695c25ced5019184f11784b37187e4": "x = 1 + 5u", "03696c698681b2078c7ad19a3cd30f42": " \\langle\\phi_1\\otimes\\phi_2,\\psi_1\\otimes\\psi_2\\rangle = \\langle\\phi_1,\\psi_1\\rangle_1 \\, \\langle\\phi_2,\\psi_2\\rangle_2 \\quad \\mbox{for all } \\phi_1,\\psi_1 \\in H_1 \\mbox{ and } \\phi_2,\\psi_2 \\in H_2 ", "03697d33bfdc54be41ff20d2c4b87847": "\\partial_\\mu j^\\mu = 0 \\!", "0369d9d9cb78d36ed02ef92d31d3160e": "B_1 = b_1", "0369e06b57f43e284c7cd55476a1c41f": "\\int {\\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2 }}\\; \\mathrm{d}x= \\frac{1}{2} \\left(\\operatorname{erf}\\,\\frac{x-\\mu}{\\sigma \\sqrt{2}}\\right)", "0369fb0bda2ddf27ae799bc1f11673a2": "M_\\mu", "036a38eb6d71058ff09ac9faa8013704": "\\lambda = (1/15,2/15,3/15,4/15,5/15)", "036a67b72bc85b3ad1e8c257a2c27189": "\\tfrac{1}{\\sqrt{2}} (1 - \\sigma_1 \\sigma_2) \\, \\{a_1+a_2\\sigma_1\\sigma_2\\}=\n\\frac{a_1+a_2}{\\sqrt{2}} + \\frac{-a_1+a_2}{\\sqrt{2}}\\sigma_1\\sigma_2", "036ac1c2364d728da1e49043b7b8106f": "\\mathfrak{P}^{78}", "036ad5b0da4302c0b0d5a78c05547737": " \\mathbf{U} \\cdot \\mathbf{V} = U_0 V_0 - U_1 V_1 - U_2 V_2 - U_3 V_3\\,. ", "036aec19e4bdc4d7637c0b4e0d4b9fb9": " Z_{i} ", "036af450785fc8345206720f53793be1": "\\eta_G = \\eta_C \\, (-1)^I", "036af815806a0089ffc36b2a2156e548": "\\mathrm{erfc}", "036b87bb6514b45d3b3a8150e05190a6": "F=\\rho_{air} \\Gamma (V_{\\infty}+V_{induced}) l\n", "036bab4918e8d35cf702026d645809b4": "\\begin{bmatrix} 0 & \\cdots & 0 \\\\ \\vdots\n& \\ddots & \\vdots \\\\ 0 & \\cdots &\n0\\end{bmatrix} ", "036bc3149dd61d35c0393c32613911fd": "f_k(v_1, \\cdots, v_k) = \\frac{1}{k!}\\sum_{\\sigma\\in S_k}{\\rm sgn}(\\sigma)\\, v_{\\sigma(1)}\\cdots v_{\\sigma(k)}", "036bd5766fa637f165b47eec5ef654e9": "\\hat{H}_{\\text{JC}}=\\Omega_+ \\hat{A_+}^{\\dagger}\\hat{A_+}+\\Omega_- \\hat{A_-}^{\\dagger}\\hat{A_-}+C", "036c4d51804f4cfc1f7b6be7da534417": "x\\in\\left[0,2\\pi\\right)", "036c9e68aa344d939f39628025dcabb2": " \\{y\\in\\mathbb{R}^n: y\\cdot x \\le h_A(x) \\}", "036cbde5c4a7803acf92c88221f7a5a9": "\\varphi(\\mathbf{x},t) \\triangleq [ \\varphi_1(\\mathbf{x},t), \\varphi_2(\\mathbf{x},t), \\ldots, \\varphi_n(\\mathbf{x},t) ]^{\\operatorname{T}} : \\mathbb{R}^{n+1} \\mapsto \\mathbb{R}^n", "036ce8d6961cc156ff87ea219a16e141": "h \\circ f = k \\circ g", "036d0a1262bc79e04bf6920ed13153f9": "\\mathbf{v}=\\nabla v", "036d3e0797d404a40093360cdd543cfe": "\\mathbf{C}_{ij} = (-1)^{i+j} \\mathbf{A}_{ij} \\,", "036d75d36a3508b66e2a675c91442f09": "{\\overline P}X = \\mathbb{U}", "036db09816e8def403de9d47dea610c2": "\\lim_{y \\to \\infty} t(y) = 1,", "036e030db648c47d62ddb0d4e10323b8": "\\omega = \\frac {W}{L}", "036e266a7f7a81431af068f2315d04b7": "\\kappa z + \\lambda = \\nabla \\cdot \\mathbf{\\hat{n}}", "036e345124cff1e03283e850d1de41be": "l_{11} \\cdot u_{11} + 0 \\cdot 0 = 4", "036e4d68fbe40841581387b1d11a3814": "d\\theta^i=-\\frac12 \\sum_{jk} c_{jk}^i\\theta^j\\wedge\\theta^k", "036e5d0a20bd67842cd5ee1038993706": "A_{k1},A_{k2},\\dots,A_{kn}, (k=1\\dots m)", "036e72e42c2a752b4fcdf09fd6ae1906": "x = \\sqrt[m]{a^n}", "036e8acc23240ab856207bb391337d76": "\\left( x = y \\right) \\to \\left( \\phi[z:=x] \\to \\phi[z:=y] \\right)", "036ea9d74180c5d430ffbe4a5eb6aa73": "2*10^8", "036f131b92b23cf3a3f92416421aa04d": "n=2,4,\\dots", "036f37927080836aeaa0729bd8f1f6b3": " \\operatorname{int}(A \\cap B) = \\operatorname{int}(A) \\cap \\operatorname{int}(B) \\! ", "036f3aded6a9f3a4f82302cdfb4adc9e": "\\pi\\varepsilon", "036f5baac14db26e5b399f531e73aafe": "Z(S_0) = 1", "036fb716ae56dc260376c97fdd78067d": " \\begin{align} \n \\bold{r}(t) \n & \\equiv \\bold{r}\\left(x,y,z\\right) \\equiv x(t)\\bold{\\hat{e}}_x + y(t)\\bold{\\hat{e}}_y + z(t)\\bold{\\hat{e}}_z \\\\\n & \\equiv \\bold{r}\\left(r,\\theta,\\phi\\right) \\equiv r(t)\\bold{\\hat{e}}_r(\\theta(t), \\phi(t)) \\\\\n & \\equiv \\bold{r}\\left(r,\\theta,z\\right) \\equiv r(t)\\bold{\\hat{e}}_r(\\theta(t)) + z(t)\\bold{\\hat{e}}_z \\\\\n & \\,\\!\\cdots \\\\\n\\end{align}", "036fc1649ceb56182bc4b4a7e2bd80d9": "\\vartheta^{\\perp}", "03702fc3ab6229d5ae9f464d2eec9807": "3 * \\frac{\\sin{\\pi} - 2}{e}", "037054e3d3f2ced9c1c7d049305029a2": "\\mathcal{F}_{L^1}:L^1(\\mathbb{R}^d) \\to L^\\infty(\\mathbb{R}^d)", "037056ef8b9a61d9b133d95776ab0cc9": "H^{(n,h)}_R", "03706c64687ca7cbfa954244ae7e7743": "\\pi_i F", "0370934d456d35275de618335162f5d1": " s_\\lambda(x_1,\\ldots,x_n) = \\sum_T w(T), ", "0370cd983fecaef435601e4a2e5e87b1": "E = E + \\Delta E", "03714d539da1be0f8f3208e8df276539": "\\mathcal{F}_{T}=\\mathcal{F}_{0}+\\mathcal{F}_{d}", "03717e6f404cf96b76df449185e9e5b2": "\n\\lim_{\\lambda\\to 0}W_\\lambda\\chi_E(x)=\\chi_E(x)\n", "037184bd8627aad6f55db0c18e2bdfcc": "K/9IP = 9 \\cdot \\frac{K}{IP}", "03722d47184d8f1325b819f8f8d1c13e": "{{\\left\\{ {{\\phi }_{\\gamma }} \\right\\}}_{\\gamma \\in \\Gamma }}", "03723a5ff6915f3c97d53616e7e69f14": "\\delta \\psi = u \\delta y\\,", "03726358bae9d9db14fb16f36d2406c0": "\\sigma_{ij}= s_{ij} + \\pi\\delta_{ij},\\,", "0372baedaa6d0c831c5186a5ea6a194b": " \\sum_{m=1}^{n} m k_m = n", "0372c1cbf6c34ba77fbe60249129bc39": " h \\ \\bmod q^n-1 ", "03731de8db9bfa07b88bd460a4e69725": "\\frac{e^{\\mu z}\\gamma K_1(\\delta \\sqrt{ (\\alpha^2 -(\\beta +z)^2)})}{\\sqrt{(\\alpha^2 -(\\beta +z)^2)}K_1 (\\delta \\gamma)} ", "037342e71c300f6ba4b66ba429a1906e": "\\{EG,AF,AU\\}", "037368dcfb66641f9adc3d8cf0d1449e": "\\sigma(\\varphi)", "037369c5d056ef947d47b1438cfdd880": "\\theta_s = \\arccos(v \\cos\\theta/v_s)\\,", "03737d5105885405c3250ef71d679334": "H_{nop}", "0373863add1d0e45ebdfc2d4906e2978": "\\frac{36}{p}", "037390b143773f6fb1522a11c7b75522": "{\\det}_p", "037396c55d8e4398ea0187e334dd8bd8": "\n\\xi(\\alpha) \\approx \\sqrt{1 - \\alpha^3}\t\t\t\t\t\t\n", "0373c933a885a2f6a1231a8cb5412b68": "L(q, \\dot{q}, t)", "0373d856fc469552a17c78297e391d35": "\n\\begin{align}\nF(A^1, \\dots, cA^j, \\dots) & = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) ca_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n& = c \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) a_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n&=c F(A^1, \\dots, A^j, \\dots)\\\\\n\\\\\nF(A^1, \\dots, b+A^j, \\dots) & = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma)\\left(b_{\\sigma(j)} + a_{\\sigma(j)}^j\\right)\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\\\\n& = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma)\n\\left( \\left(b_{\\sigma(j)}\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right) + \\left(a_{\\sigma(j)}^j\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right)\\right)\\\\\n& = \\left(\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) b_{\\sigma(j)}\\prod_{i = 1, i \\neq j}^n a_{\\sigma(i)}^i\\right) \n + \\left(\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n a_{\\sigma(i)}^i\\right)\\\\\n&= F(A^1, \\dots, b, \\dots) + F(A^1, \\dots, A^j, \\dots)\\\\\n\\\\\n\\end{align}\n", "0373e738ee42ca54133f6dfabc5843f5": "b\\eta e^{bx}e^{\\eta}\\exp\\left(-\\eta e^{bx} \\right)", "0373eae7aead6e972464419409a60233": "\\text{(***)}\\qquad \\left\\|\\int_a^b v(t)\\,dt\\right\\|\\leq \\int_a^b \\|v(t)\\|\\,dt.", "0373ee19f5a13f88cece7ae2f9882d67": "A_0(h) = \\frac{f(x+h) - f(x)}{h}", "0373f990d02a2e5a460fe5a515bc5ae8": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=Cu^3", "0373febfed9fdcbc4a6eaf82c5c55b90": "\\Lambda_n[V_n]/\\langle V_n^2-\\Delta\\rangle", "0374055d88f1f7d950a5d60b0583acfc": "h:x\\to e", "03746ec1664988d3d138cf8b140d356f": "\\mathrm{Tr}_3\\big((|000\\rangle + |111\\rangle)(\\langle 000|+\\langle 111|) \\big) = |00\\rangle \\langle 00| + |11\\rangle \\langle 11|", "037488506c0517b9873957758730ac8b": " \\begin{align}\n \\lambda_1 - 3 \\lambda_2 &{}= 0 , \\\\\n \\lambda_1 + 2 \\lambda_2 &{}= 0 .\n\\end{align} ", "0374b768cce8b8c2881225f6d35574a5": "d \\geq 1", "0374c643357c023c9eef0d52be33a5ed": "W' \\subseteq \\mathcal{B}'", "0374d3d4a26403ce7b7a67f21d47ce70": "\\operatorname{int}A = \\operatorname{core}A", "0375bbdf779e79128ac563157984445d": "\\left\\{\n\\begin{array}{ll}\n\\Delta \\phi + \\lambda \\phi = 0 & \\mathrm{in\\ }\\ U\\\\\n\\phi=0 & \\mathrm{on\\ }\\ \\partial U.\n\\end{array}\\right.\n", "0375c9020d5e36fb357fe60906fbc20c": "Re(K(\\omega))", "0375d6841aac6c9721a1c86a11259576": "t=\\tan^{1/3}\\theta. \\,", "0375eedd14436a3708e34f1a115e69f5": " \\tau = \\frac{\\theta}{2\\pi}+\\frac{4\\pi i}{g^2}.", "0376020ce4e93edc22fe4f0c3b0dcb24": "{D \\zeta \\over Dt} + \\beta v = f_0 {\\partial w \\over \\partial z}", "0376445b7709ebfaf078aeaa6257bde0": "\\overrightarrow{T} = \\frac{3Gm}{r^3}(C-A)\\sin\\delta\\cos\\delta\\begin{pmatrix}\\sin\\alpha\\\\-\\cos\\alpha\\\\0\\end{pmatrix}", "037647636bf1c631caf827d1421458f2": "{{\\varepsilon }_{medium}}", "037654bdad0176286f44c6244d63d6f3": " f(z) = (z - r_1)\\cdots (z-r_n),\\qquad (n\\ge 2) ", "0376892fd567d547374e1c2e56c625da": "e^{{\\rm{i}} \\theta}", "037699b6e9751320acbe2421c3267bb4": "\\sigma_y^2(\\tau) = \\frac{1}{2}\\langle(\\bar{y}_{n+1}-\\bar{y}_n)^2\\rangle = \\frac{1}{2\\tau^2}\\langle(x_{n+2}-2x_{n+1}+x_n)^2\\rangle", "0376bf22b868a6ad8a73538d359b46f7": "r_e", "0376cab0078bf0d02682cbcd698e56ca": "~T(\\gamma)=\n4\\gamma^{1/2}\n\\left(1+O(1/\\gamma)\\right) ~", "0376f2805d5898fcf697d33df97d0262": "|\\mathbf{x} \\times \\mathbf{y}| = |\\mathbf{x}| |\\mathbf{y}|~\\mbox{if} \\ \\left( \\mathbf{x} \\cdot \\mathbf{y} \\right)= 0.", "0377064bd785a7da5f8d035e6b2c1961": "\\operatorname {dn}\\; u = \\sqrt {1-m\\sin^2 \\phi}.\\,", "037714735ed79521f7c696480eaf9693": "(1+z)^u = e^{u\\log(1+z)} = \\sum_{k = 0}^\\infty (\\log(1 + z))^k \\frac{u^k}{k!},", "037744af11534dbacf73329e86342442": "q^{2}-1", "0377566f0297e84c0cdac00b4475134e": " \\nu_{\\rm yx}", "0377a708aa8dcc7522b35aabf8036027": "F(\\mu)=\\frac{m + \\delta}{n}", "03780996512787f65644298a2159d74c": "\\frac{|v-c|}{c}<2\\times10^{-9}", "037896f8e77eff0f9551b4591460d25c": "\\begin{align}\n \\cosh (2x) &= \\sinh^2{x} + \\cosh^2{x} = 2\\sinh^2 x + 1 = 2\\cosh^2 x - 1\\\\\n \\sinh (2x) &= 2\\sinh x \\cosh x\n\\end{align}", "0378c940cadd4e337b0acae7655a2d9b": "P(X^2+1) = (X^2+1)^2-1 = X^4+2X^2", "0378d5b8c35130a441bbee1e68bac4f3": "\\vec{x}_1", "0378da57d283b686b2341fde12e3dd55": "\\eta=1/P(o\\mid b,a)", "0378eab415625ea5efd4ea711c1e59ce": "y' = re^{rx} \\, ", "0378fa6ba5361846976d3dea1cc64c50": " \\alpha \\in \\mathbb{R}", "037935127383c734dcb152dc775276b2": " f(\\boldsymbol{x}) + f(\\boldsymbol{y}) \\ge f(\\boldsymbol{x} \\wedge \\boldsymbol{y}) + f(\\boldsymbol{x} \\vee \\boldsymbol{y}) ", "037997c657ca62c38f560b041de1a550": "\\frac{33}{32}", "0379bbec0e9b0605c52b2127dd734599": "\\frac{(2n)!}{2^n\\,(n!)^2}\\,", "037a233b676370a840d2f51971fd3ebb": "G_{r}^{n}", "037a38293205b83b7920a20def8d7126": "F_X(x) = \\operatorname{E} \\left [\\mathbf{1}_{\\{X\\leq x\\}} \\right],", "037a412ff67eef88af781bd5728f0689": "J_{\\mathrm{eff}}", "037a4753e17bea1ed40d64798a88cef7": "\\{\\textit{SENTENCE}, \\textit{NOUNPHRASE}, \\textit{VERBPHRASE}, \\textit{NOUN}, \\textit{VERB}, \\textit{ADJ} \\}", "037a66437e12145951fdac16d22f8374": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "037aa4171e65cc137237f2797823fe38": "\\mathrm{NPV}(R(t)) = \\langle w, R\\rangle = \\int_{t=0}^\\infty \\frac{R(t)}{(1+i)^{t}}\\,dt.", "037ab61010b38491adbf983275260958": "\nx_i = \\bigvee_{j=1}^n (g_{ij}\\wedge y_j), i = 1, 2, \\ldots, m,\n", "037afe33ba5d543202f3a709f46394c1": "\\left(\\frac{\\partial S}{\\partial T}\\right)_{P}", "037b43965d2d07927fafe6ab4fa8f84a": "|K(x-y)-K(x)| \\leq C \\frac{|y|^{\\gamma}}{|x|^{n+\\gamma}},", "037b4b461271363e5902aeffe6c25a00": " \\textbf{h} = p\\textbf{f}_q \\cdot \\textbf{g} \\pmod q. ", "037b89b5bbe84570515c008c3674fd09": "v_{\\rm e} = g_0 I_{\\rm sp} \\,", "037bc51126f8b95d163083b50e45e317": "\\{\\to,\\land,\\lor,\\bot\\}", "037c0702f308e62ee6b430717aa5007c": "\\mathcal{H}_n = ([n], \\{E \\subseteq [n] \\mid | E \\cap [2k]| = | E \\setminus [2k]|\\})", "037c0a2568e7e1416ebc8f304b58dc04": "(i\\omega-\\xi)^{-1}", "037c7474cf061e8fe282c1bef172ad40": "\\{ w \\in \\Sigma_1^* | \\exists q \\in F . (q_0,w,\\epsilon) \\vdash^* (q,\\epsilon,\\epsilon)\\}", "037c7ff9d16428cd312a69859f16b8c8": " \\varphi(\\mathbf{r},t) = \\int\\frac{\\nabla'\\cdot{\\mathbf E}(\\mathbf{r'},t)}{4\\pi R}d^3r'-\\frac{\\partial{\\psi(\\mathbf{r},t)}}{\\partial t}", "037c812791cba4937537a02529d1ac95": "\\bar{V}_i\\otimes V_j", "037d2b07b54fd4e8670c12ecdabcd7f3": "v=v_0", "037d75c55ba6dd3a38f3ef9fcd478337": "K ", "037da86913143f31402a12f5f79f2000": "\\xi _i ", "037e1daa38c30fd321c1d5fa53b5a86d": "\\mathrm{Hom}_{D(A)}(X, Y) = \\mathrm{Hom}_{K(A)}(X, Y).", "037e2e99d8c00d57092b7c7eaf086180": "G:=(V,E)", "037e3dad8c52cb42410e614ed79453aa": "\n\\left|{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}\\right| = \\left|\\left(\\sum_i {\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial s}\\right) \\times \\left(\\sum_j {\\partial \\mathbf{x} \\over \\partial q^j}{\\partial q^j \\over \\partial t}\\right)\\right|\n", "037e3f00eac3133acd34a2d34c5d8521": "S_6,", "037e711358b5150f62f4b528adaaebed": " \\Sigma = N \\, \\sigma ", "037efc824cbe954dbd5aa581e0156503": "\\,\\hat{m}_1", "037f2c9ddd738e43d71a260275ad5049": "1-g", "037f2d988c51e5390be2a0ecc20ba321": "\\ v_o = A_v v_i \\ . ", "037f54d1963f8292451829bd2a72c387": "\nD_{t}(x_i,x_j)^2 =\\sum_y (p(y,t|x_i)-p(y,t|x_j))^2 w(y)\n", "037fa690256a7bb1884c165d56a87cfb": "u = \\int \\frac{du}{dx} \\,dx", "03803c2f702255059aa8704d1a8da64d": "\\Diamond_i P", "0380957958d369064832e39c069858f0": "t_2^\\prime = 1/f^\\prime", "03809891f820600376128c1d84da3ef0": "0 < b < 1", "03809e73c75758bd32204599b06f608d": "q^{\\eta \\sum_j t_{\\lambda_j} \\otimes t_{\\mu_j}}", "0381a7f5503b5d8e3cc2998f04027717": "\\displaystyle{T_sf(x) ={1\\over 2\\pi}\\int_{\\mathbf{R}^2} {s f(x)\\over (|x-t|^2 + s^2)^{3/2}}\\, dt.}", "0381aa4bb63ef5cbc074eb74c1cb31e3": "\\sigma(V_{\\mathbb{R}})\\subset V_{\\mathbb{R}}\\,", "0381d613bf0e2d5cb45fc42bd256b6c2": "\nk=\\frac 1\\hbar (\\frac E{D_\\alpha })^{1/\\alpha },\\qquad 1<\\alpha \\leq 2.\n", "0381fa2dcaf53009ebe5ffb0d2d872c0": " \\mathrm{Ric}^{}_{}(X_p) = f(p) X_p", "038210fc308436633142e5b8e5d10f0b": "H(e^{i\\theta})=e^{ih(\\theta)},\\,\\,\\, h(\\theta+2\\pi)=h(\\theta)+2\\pi,", "038215c7e1001f5f2fb42c3e577451d0": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = y + 9x \\geq 6 \\\\\n g_{1}\\left(x,y\\right) & = -y + 9x \\geq 1 \\\\\n\\end{cases}\n", "038282908e59d4d3896567c5ba9ce4be": " (\\boldsymbol{\\sigma} \\cdot \\mathbf{p})(\\boldsymbol{\\sigma} \\cdot \\mathbf{q}) = \\mathbf{p} \\cdot \\mathbf{q} + i(\\mathbf{p}\\times\\mathbf{q})\\cdot \\boldsymbol{\\sigma} ", "0382cfd42a408734bc1b8068a658c72c": "Z = 1 + \\frac{2}{t} - \\sqrt{\\left (1 + \\frac{2}{t} \\right)^2 - 1}", "0382f49a76f265907c281437afcb4abb": "\\frac{x_1+x_2+\\cdots+x_n}{n} \\ge \\sqrt[n]{x_1x_2 \\cdots x_n}", "0382f611dfb3c7f92e157811043231f8": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\mu \\alpha}", "03834bf648aa1cc9df49b37db483c193": " U^{A}_{jj'} = H_{jj'} + \\sum^{B}_{\\gamma \\neq j,j'} \\frac{H_{j\\gamma}H_{\\gamma j'}}{E_0-E_{\\gamma}} = H_{jj'} + \\sum^{B}_{\\gamma \\neq j,j'} \\frac{H^{'}_{j\\gamma}H^{'}_{\\gamma j'}}{E_0-E_{\\gamma}} ", "0383533bb2f23d79a9552e71e8605b77": "f : X \\to Y\\,", "03835cb8c1117fd6b5484f1178c92773": "f\\left(x,u\\right) = -x + u,", "038364fa2b3d85cc9ac202f9fc9af2e5": "C_k=X\\setminus U_k=\\varnothing", "038365d99efa684daea1919d085eeeca": "K=\\mathbb{C}(x)", "0383667c0334bc7e23f0a0b76e426761": "\n\\tilde{p}(\\bar{b}) = \\frac{1}{I} \\sum_{i=1}^I E[\\delta_{b,\\bar{b}}]\n", "0383bd103c8fe4f7aaba585a82a6de6b": "V_\\Sigma^*Y", "0383d0389f143a22feb1c6753e05ace8": "\\scriptstyle \\lambda_j", "0383d6783b0d47a01a151169487f49d9": "\\left \\|\\Gamma^m \\varphi_1 - \\Gamma^m\\varphi_2 \\right \\| \\leq \\frac{L^m\\alpha^m}{m!}\\left \\|\\varphi_1-\\varphi_2\\right \\|", "03840f00a94261bd5ad1dfdd0fff9208": "{dx_1 \\over dt} = r_1x_1\\left({K_1-x_1-\\alpha_{12}x_2 \\over K_1}\\right)", "03840f4a67085a30dee8e4227ede1797": "D_{\\mathbf{v}}{f}(\\mathbf{x}) = \\lim_{h \\rightarrow 0}{\\frac{f(\\mathbf{x} + h\\mathbf{v}) - f(\\mathbf{x})}{h}}.", "038436a5a91c995fcffc56af94062aed": "\\mu(\\xi)", "03846b88d87d5b51da1bb22ca3018143": "\\mathbf{C} = (C_x,C_y)", "0384894ad2cad3e86d932dbdb7202511": "\\psi_n(x)", "0384a5f21aee7a0fd4f40bf0148fa01a": "p\\equiv l \\pmod{q}", "0384c9b28853e0bc446fd4cc1c30e479": "N_t = N_d \\quad \\frac{\\ln 10}{2 \\ln \\frac{b_n}{a_n}}", "03854265fc1eaaf28071f1456a906318": "T_{f\\cdot g} x = T_f x \\cdot T_g x", "038577fa9ec9e7ab8ded2bf564099f4b": " F(\\mathbf{x}^{(0)})=58.456 ", "0385a9fbf3fd60aef19d34c99a222e51": "\\lambda_{0i}", "0385bbcf6ef717868c25f848630bcc47": "\\mathcal{} f", "0385d388b31cb723c1c665715d753b65": "H_{\\frac{1}{a}} = \\frac{1}{a}\\left(\\zeta(2)-\\frac{1}{a}\\zeta(3)+\\frac{1}{a^2}\\zeta(4)-\\frac{1}{a^3}\\zeta(5)+\\cdots\\right)", "0385f072bb5b43c5ba07181eb9c1a71f": "\\begin{align}(a+b+c+d)(x+y+z+w)=&\\,((a+b)+(c+d))((x+y)+(z+w)) \\\\ =&\\,(a+b)(x+y)+(a+b)(z+w) \\\\ &\\,{}+(c+d)(x+y)+(c+d)(z+w) \\\\ =&\\,ax+ay+bx+by+az+aw+bz+bw \\\\ &\\,{}+cx+cy+dx+dy+cz+cw+dz+dw. \\end{align}", "03864907a9123f61c411aa972d7242d8": "density_s", "03864965bdf3283ac3eeab59ae88c7c4": "\n\\int_{\\boldsymbol{\\theta}} \\prod_{j=1}^M P(\\theta_j;\\alpha) \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) d\\boldsymbol{\\theta} = \\prod_{j=1}^M \\int_{\\theta_j} P(\\theta_j;\\alpha) \\prod_{t=1}^N\nP(Z_{j,t}|\\theta_j) \\, d\\theta_j .\n", "03864fca9d766129e5a058b5d8a57fa2": "x+y+xy", "03872bc0e3bb7b2465a7fe1660aaf724": "\\{l\\ |\\ l \\in L\\}", "038747d1464dcbcf21ba14def68dad8e": "\\Omega\\subseteq\\Omega_1", "03879ded5b5a9b27f4bb081cdcff6e8e": "\\mathcal{J}^{\\alpha\\beta\\gamma} = (X^\\alpha - Y^\\alpha )T^{\\beta\\gamma} - (X^\\beta - Y^\\beta )T^{\\alpha\\gamma} ", "0387f562901f1262ed64c811ee661c7a": "\\nu = \\frac{\\mu}{\\rho}", "038830c168c0bf1e26b9e4cca663e4e5": "\\mathcal{E}^{(0)} = \\left \\{z \\in \\mathbb{R}^n : (z - x_0)^T P_{(0)}^{-1} (z-x_0) \\leq 1 \\right \\}", "03884f8e09487d85d849360a2a492a45": "\\frac{1+2x}{(1-x)(1-4x)}", "03885d64afc508cc84f91e010ef53d4c": "d_{\\text{eff}}", "0389297e08112ab3dd87454d65e510ab": " \\frac{Q_1}{T_1}-\\frac{Q_2}{T_2}=0 .", "038935d89deb3eedd1ff79a044a67303": "J_{n\\,(\\text{base})} = \\frac{q D_n n_{bo}}{W} e^{\\frac{V_{\\text{EB}}}{V_{\\text{T}}}}", "0389392b62cc02dc8df5ebfcd3b9629e": "x_n=1", "0389c4f1ff10f51b0b157c6495a1f36f": "\\lambda_{ex}", "0389c86b454b10b0ece9a9aa04d5e305": " \\sum_{i=1}^d{x_i^k},k \\ge 1 ", "0389cc5d3438522a7bbb1dee11cb4bb8": "\\beta(z) = \\beta^* + \\dfrac{z^2}{\\beta^*}", "038a2fc2c166bd4d273bb926a4e69f5b": "(a_d,b_d,c_d)", "038a3710e4ba601e5d1e85105805dd44": "n(n-1) + \\frac{1}{4} - a = \\left(n-\\frac{1}{2}\\right)^2 - a = 0", "038a8f14beb2f599d17425044bf88299": "[2^m - 1, 2^m-m-1, 3]", "038ac5d15adf387f4e3972f82f2ef051": " \\dot{m} ", "038ad5ab428e374131b0791fc13a461e": "\\left ( \\int_{-\\infty}^\\infty e^{-x^2}\\,dx \\right )^2=\\pi,", "038ada52f52d90583a3f20612b16e70f": "1 + z = \\sqrt{\\frac{g_{tt}(\\text{receiver})}{g_{tt}(\\text{source})}}", "038b327459f71c65a49e22ae54589573": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\psi_{2} + (\\tilde{u}_{2}- E)\\psi_{2} + \\frac{\\hbar^2}{2m} [2\\mathbf{\\tau}_{12}\\nabla + \\nabla \\mathbf{\\tau}_{12}]\\psi_{1} = 0 ", "038b6ccee470797d7128ab49a3e7d9d4": "\\gamma = \\frac{1}{A} (E_{1} - E_{0})", "038b85dc1de6025556249750059aab2c": "\\int \\! L \\, \\mathrm{d} \\theta = \\frac{T}{2 \\pi} ( - ( \\cos{\\theta_1}) \\cdot R - r \\cdot \\theta_1) - \\frac{T}{2 \\pi} ( - ( \\cos { \\theta_2}) \\cdot R - r \\cdot \\theta_2 ))", "038bde9b44dcff2cf0283077ef222b24": "\\textstyle\\text{rate(propagation)} = k_p[\\text{M}^+] [\\text{M}]", "038bdf3c2dedfcdb4211b26a3f6e84b4": "\\langle A_n \\rangle", "038c5c7732d26f91aa976779587da51c": "\\scriptstyle \\dot{}", "038c70349fa67c5870b3aedd6f57ab15": "C = \\frac{\\varepsilon A}{d}", "038ca19ab913b6bcbce87bac5dc2bbc4": "ba^{RC} \\notin \\mathcal{O}", "038cc55a61f269484f124b61d5b6464d": " \\mathcal{U} \\equiv \\frac{H\\, \\lambda^2}{h^3}.", "038cd962d0b2d6a5694382619cb14319": "E_N(\\rho) \\equiv \\log_2 ||\\rho^{\\Gamma_A}||_1", "038cf6c6d831d275d680a6e60d66265e": "dh = C_pdT", "038d04516677e9fb54be37b4707517f8": "nk\\log k", "038dd71f1b87d3876200c0f8a51df38e": "\\frac{\\mbox{d}A_1 \\ \\, \\mbox{d}A_2 \\ \\cos{\\theta_1} \\ \\cos{\\theta_2}}{r^2}", "038e2c07c3e25f2d4a199d3f8c7a6665": "T = \\sqrt[4]{ \\frac{(1-a)S}{4 \\epsilon \\sigma}}", "038e51f8a130284e07e8d1d57a5a9d1d": " P_\\mu = \\left(E/c,\\vec p \\right)", "038e93d36bfcdca83a124aeb85dac36f": "=\\int_{\\mathbb{R}^n} f(x)e^{-2\\pi i x\\cdot \\nu} \\left( \\int_{\\mathbb{R}^n} g(y) e^{-2 \\pi i y\\cdot\\nu}\\,dy \\right) \\,dx", "038eeb291fdda2c431db906966293543": "\\langle \\psi |\\phi\\rangle = \\langle \\phi |\\psi\\rangle^*", "038ef3077cd8f59b36164684c2b427d5": "\\mbox{Golden rule savings rate: } s^G=\\frac{mpk^G}{apk^G}", "038efeea3253bb25ad6c3f04702bec33": "\\omega_1\\to(\\alpha)^2_k", "038f0a9c16ed5bc7b6b8d5a91a5379aa": "Y = \\{ Y_1, Y_2, \\ldots , Y_s \\}", "038f257a4df11011d255f6987b88b940": "(S)\\,", "038f2cc623f044cd4cdbafbe77e5ea11": "F^* \\tilde g = e^{2\\varphi} g", "038f2d4e909f2f51fc3d89a8def54df5": "\\alpha \\smile \\alpha", "038f81da29e7124964889dd8a1dbdaca": "\\frac{\\sqrt{a^2-b^2}}{a}", "03900ab4df72f17a1c2261a18ba70aae": "f(x, \\boldsymbol \\beta)=\\beta_1 +\\beta_2 x", "039030e4c23c03efd88de8e91f221d3f": "\\langle W,R,\\{D_i\\}_{i\\in I},\\Vdash\\rangle", "03905a878126f449a38a2d1da24ea191": "69^{7} \\approx 2^{43}", "0390a850ed66d32e5fdb51566ec563ff": "a|\\alpha\\rangle=\\alpha|\\alpha\\rangle", "0390fbf11b84d5d0a243a933d468d354": " E=E^0+\\frac{RT}{nF}\\ln{A}", "03913b430d130efa8df572544967cd4c": "0 < r < R \\leq \\rho,", "03914a417c6d4d150c99af7d069a7a44": "{ }P_x=\\frac{A_{x}}{\\ddot{a}_{x}}", "0391aa9bf208f16bf05af1ac2ded30db": "E_z", "0391e09ef9dc27ac9ad3b0df4a59be9f": "\\int_{\\Lambda^{m\\mid n}}f\\left( x,\\theta\\right) \\mathrm{d}\\theta\\mathrm{d}x=\\int_{\\Lambda^{m\\mid n}}f\\left( x\\left( y,\\xi\\right)\n,\\theta\\left( y,\\xi\\right) \\right) \\varepsilon\\mathrm{Ber~J~d}\\xi\\mathrm{d}y", "0391e1608ca99e405c9a44258dec721b": "\nw=\\sqrt{KW-UV}\n", "039200a02ae33a09f916fd490682d171": " A = {\\rm adj}(B) ", "03927152c0cd08b3bf1cfe62e032f5b8": "P(y,x)", "03931c8ee9b99a12d3351d369f078bd2": "x^2 + x + 1", "03934e7e0730754682575bf7f01b9daf": "\\frac{\\alpha}{\\beta}.\\beta = \\alpha\\beta^{-1}.\\beta=\\alpha", "03936b068bd5908d018878c57e10ad28": "T = \\frac{Rh_bh_c}{a}", "0393f0e43221b5545069d03e804595fb": "\n \\boldsymbol{S} = \\cfrac{\\partial W}{\\partial \\boldsymbol{E}} ~.\n ", "039476b65dd78b8c2638531f2b81a989": "\\Sigma_k \\hat{\\textbf{d}}_j ", "0394815b45d7615e683c44728aa19e9c": "\\forall x. A", "03949a21221acca1f2d9e53c68dea93b": "(n^2)", "0394b5c9805f0ec750e4c1d4aa92e611": "(\\mathbb{Z}/n\\mathbb{Z})^*.", "0394d2f46758455667a8e7827f20e15b": "\n\\begin{align} \n& A=-\\frac{1}{2}R_{0}\\frac{d\\Omega}{dr}|_{R_{0}} \\\\\n& B=-\\frac{1}{2}R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}-\\Omega \\\\\n\\end{align}\n", "03952d34eb58a0d6ee46a795d07f43f9": "s={1\\over{2}}(3n^2-3n+1)(3n^2-3n+2)={9n^4-18n^3+18n^2-9n+2\\over{2}}", "03952d6280fd93ee0b7fee4d8f356b8b": "|f(\\xi)| \\le M", "03953be2782e96e30064637cec023351": "~t_0~", "03955481cb22251600ac99715ce44954": "\\cos 2\\theta_{\\mathrm{eq}}=1", "03955a7f978fb28a1a5858c31b7ffdeb": "l\\ ", "039570c1315ecd56c74d5bce803cb764": "\n\\left\\langle {J(t;F_e )} \\right\\rangle = \\left\\langle {J(0)\\exp [ - \\beta V\\int_0^t {J( - s)F_e \\;ds]} } \\right\\rangle _{F_e }. \n\\,", "0395886a7bcb04205448e0c0ff07f46f": "z_3 = 0", "0395a1f3d0bccf84440ef0c88fd6f116": " C(t) - P(t) = S(t) - K \\cdot B(t,T)\\ - D(t), ", "0395e6460de9f9ccd696d7688c4e79a3": "nT\\,", "03962b2def67fd1b6e569f33ceb7e43b": "\\gamma_{00}", "03962ffa1b76426a2e565d2343eb3e39": "\\bar{H}", "03967700d8392344b52204a6b5d4a916": " \\sqrt{n} ", "03968360cd326aa15e41060ff247e95c": " xP = x(1 - x - y) = x - x^2 - xy ", "039695a9ac4d4131c8d6f032ce0e9942": "\\beta = \\alpha / \\gamma\\,", "0396ccb4fd35747b42a23349aca95c36": "A_{sn} = A_s \\left ( \\frac {\\left (1 - \\frac {25}{1000} \\right )}{\\left (1 + \\frac {\\delta^{13}C}{1000} \\right )} \\right )^2", "0396d685f66076c856145763149ca6c6": "\\Omega_\\mu =\\frac{1}{4}\\partial_\\mu\\omega^{ij} (\\gamma_i\\gamma_j-\\gamma_j\\gamma_i)", "03975ee84ae58a2154ad5571b39dfce8": " = \\frac{G_{wr}}{G_w}", "039761dd6b0195e2cc69e3b8a0ebf0c3": "\\frac{x_j-x_m}{x_j-x_m} = 1", "03980a4b987a89d69667e8ca894e25ae": "\\{ 1,~i,~\\varepsilon ,~i_0 \\}", "039860856ae56d90e8ba163a0eaaffbc": " \\forall x \\in \\mathbb{Q}", "0398646ed7de77d19d2f3390030229fa": "Coupon yield = \\frac{C}{F}", "039882931e9bc6bdeac618f686046310": "z_2=-\\sin i \\cdot \\cos \\Omega", "0398b9a6b5e23d80d8ebabc77a019add": "t_1^\\prime = t_1 + \\frac{D_L+v\\delta t \\cos\\theta}{c}", "03998219323c607ac1d77af4c8000da7": "P=NM\\bar{v^2}", "03998cf235e1a1cf413937ef0499a229": "c_{\\omega}", "03999f856bdffa949cc76f8934546e7f": " W \\approx \\left[ \\frac{2\\epsilon_r\\epsilon_0}{q} \\left(\\frac{N_A + N_D}{N_A N_D}\\right) \\left(V_{bi} - V\\right)\\right]^\\frac{1}{2} ", "0399a8b21fd225ba9f533525b72406a4": "1\\cdot 10^5", "0399b9f388741e610b515ede8934c7ad": "f(x;\\mu,\\sigma^2) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2 }\n", "0399d6bdc879a8da017d06cefc8394c8": " \\int_\\R f(x)g(x) \\,d\\mu(x) \\geq \\int_\\R f(x)\\,d\\mu(x) \\, \\int_\\R g(x)\\,d\\mu(x),", "0399e7b324d2fb169e3d5e32d60bf175": "(\\det A)^2\\det S,", "039a097480359dd46e06126cb958bd49": "\nF^{*}_{A} \\subseteq F^{+}\n", "039a7636d144be29d8a6ed15256a28a6": "Z_k=\\int_{[a,b]} X_t e_k(t) \\,dt", "039a76ecfb559a21482c32588caec25a": "\\lim_{a\\to -\\infty}\\int_a^cx\\,\\mathrm{d}x + \\lim_{b\\to\\infty} \\int_c^b x\\,\\mathrm{d}x", "039a9658e762bf76358e34540d2c836c": " g = \\frac{ \\mathrm{d}\\nu }{ \\mathrm{d}\\mu }, ", "039aa2ecb77465fea6001b3b7814e161": "C = X_1-A", "039af6f5a9f7f409d6e6313f7fd010bf": "Z(\\beta) = \\langle\\exp(-\\beta E)\\rangle,\\,", "039b1ac63174801fbffe57cfb3d978a3": " { \\partial^2 u \\over \\partial x ^2 } - {1 \\over c^2} { \\partial^2 u \\over \\partial t ^2 } = 0 ", "039b450257d5575e8a3bf64eea27daf4": "\\int_{-\\infty }^{\\infty } \\frac{1}{x^2+1} \\, dx = \\pi", "039b496636f8ab3c33af3cbed3658ec8": "x^{-1} \\in \\mathfrak{m}_R", "039ba6dcd91075aa703fe285ffd1a726": "\\mathbb{R}\\times L,", "039bad22b928082e52bea4f704b01452": "\\bigcup U_{\\alpha}", "039bdcdd943188d7b26552ea22218ee8": " \\mathrm{d}^2 \\sigma \\rightarrow \\mathrm{d}^2 \\tilde{\\sigma} = \\mathrm{J} \\mathrm{d}^2 \\sigma \\, ", "039bfefb3498857197e9fcc21bdc9632": "\n p(s) = w \\frac{1}{N} + (1-w) \\sum_{i=1}^M \\frac{1}{M} p(s|i)\n", "039c27b50473008fa98fe6152e7f2bfe": "\\gamma=\\frac{1}{2}\\frac{F}{L}", "039c3c087ca086aee04ff3db89b92004": "f'(a) := \\lim_{x\\rightarrow a} \\frac{f(x) - f(a)}{x - a} = \\lim_{x\\rightarrow a} \\frac{h(x)}{g(x)} = \\lim_{x\\rightarrow a}f'(x)", "039c7db777be9bbb8e6c6410ecbd3418": "s \\not\\in \\{0, 2^T/2, 2^T\\}", "039cf06d449c20ec023daf70c02b8097": "[a,b) \\subset \\mathbb{R}", "039cff40c15dd448289a4b84b6604d78": "\\displaystyle \\frac{1}{\\sqrt{2 \\alpha}}\\cdot e^{-\\frac{\\omega^2}{4 \\alpha}}", "039da3dbb17327ebc7e8cf893adb3f5a": "\n\\Lambda = \\frac{\\prod_{n} \\prod_{i} \\exp(x_{ni}(\\beta_n-\\delta_i))}{\\prod_{n} \\prod_{i}(1+\\exp(\\beta_n-\\delta_i))}.\n", "039decec58684bcda79db11267bd7847": " N_j = 4 \\cdot 2^{\\left \\lceil \\frac{j}{2} \\right \\rceil}", "039df402bca763321a9a067703bceba9": "\\textstyle t_1", "039e0153bfb965b25d937758aefa2524": "\\Delta f = { -\\ f_0^{3/2} ( \\eta_l \\rho_l / \\pi \\rho_q \\mu_q )^{1/2} } ", "039e09a38557a8615a87956c7fb1e5da": " \\Delta G_{\\rm em} = 3{\\gamma V\\over\\ R_{\\rm f}} ", "039e2712925b844ceba50e31574d1b49": "A, P, Q \\in \\mathbb{R}^{n \\times n}", "039ede700c93bb84c1ac5772a0b42af4": "\n\\Omega^{2}(t) = \\omega_{n}^{2} \\left[1 + f(t) \\right],\n", "039f3623ac3651c6f16a28470c652940": " SG_\\text{true} = \\frac {\\rho_\\text{sample}}{\\rho_{\\rm H_2O}} = \\frac {(m_\\text{sample}/V)}{(m_{\\rm H_2O}/V)} = \\frac {m_\\text{sample}}{m_{\\rm H_2O}} \\frac{g}{g} = \\frac {W_{V_\\text{sample}}}{W_{V_{\\rm H_2O}}} ", "039f9c9de1ed3462a1896906001c3d8f": "\n {d \\over dx}\\tan y\n = {d \\over dx}\\frac{\\sin y}{\\cos y}\n = \\frac{{dy \\over dx} \\cos^2 y + \\sin^2 y {dy \\over dx}}{\\cos^2 y}\n = {dy \\over dx} \\left (1 + \\tan^2 y \\right)\n", "039fbdf64d87e5e749f9be62f03c1ac7": "\\frac{|A(x)|}{|R|} > 1 - \\frac{1}{2^{|x|}}", "039fbf45379f8551e9ef60aed04178e4": " e_d ", "039fe118d348a89d7b553966bb4e3a92": "\\mu_i = \\left( \\frac{\\partial U}{\\partial N_i} \\right)_{S,V, N_{j \\ne i}}", "03a0000b65d4ac9a8010e24b859031a3": "A_{m,n} = A_{m,n-2}+A_{m,n-1}", "03a0241631bcde8a890989d3fe6c657e": "\\sum_{p|n} f(p)\\;", "03a04fe6fd3748e89a61b4bc79624682": "\n\\sigma_1 = \\sigma_x =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n\\,,\\quad \\sigma_2 = \\sigma_y =\n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix}\n\\,,\\quad \\sigma_3 = \\sigma_z =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "03a05c1da417a41dae0da916caedc5c2": "\n\\int (d+e\\,x)^m\\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{(d+e\\,x)^{m+1} (b+2 c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^p}{(m+1)(2 c\\,d-b\\,e)}\\,+\\,\n \\frac{2c (m+2p+2)}{(m+1)(2 c\\,d-b\\,e)} \\int (d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^pdx\n", "03a08b718087ae8e3fa114b826d96305": "|\\mathcal{U}|=9", "03a0cc58ab774f8680e9fd94d5caf7b5": "\\mathbb{F}_p", "03a0eec6c55e041c5145403b6be3b9cb": "0.33PC+0.55U+0.12EG=0.37SW+0.63BK", "03a10d5c52c6fec06b9bf6ec97a5b6b2": "\n\\begin{matrix}\nX X^T &=& (U \\Sigma V^T) (U \\Sigma V^T)^T = (U \\Sigma V^T) (V^{T^T} \\Sigma^T U^T) = U \\Sigma V^T V \\Sigma^T U^T = U \\Sigma \\Sigma^T U^T \\\\\nX^T X &=& (U \\Sigma V^T)^T (U \\Sigma V^T) = (V^{T^T} \\Sigma^T U^T) (U \\Sigma V^T) = V \\Sigma^T U^T U \\Sigma V^T = V \\Sigma^T \\Sigma V^T\n\\end{matrix}\n", "03a19315749fee66e45a008739366d39": "the : NP/N \\qquad dog : N \\qquad John : NP \\qquad bit : (S\\backslash NP)/NP", "03a22045e75b911170c35acf9d050fd7": "\\frac {d\\ln K} {d(1/T)} = -\\frac{{\\Delta H_m}^{\\Theta}} {R}", "03a235cfdf80ee75657323bbebf0e2ca": "- m c^2 \\frac{d \\tau[t]}{d t} = - m c^2 \\sqrt {1 - \\frac{v^2 [t]}{c^2}} = -m c^2 + {1 \\over 2} m v^2 [t] + {1 \\over 8} m \\frac{v^4 [t]}{c^2} + \\dots ", "03a2d642bba3f4875961243979e8c601": "1-\\frac12-\\frac14+\\frac18-\\frac{1}{16}+\\cdots=\\frac13.", "03a2feb0dba6eba8510cddeb66e8ef1f": " r' = r \\frac{1}{1- pq} > r. ", "03a3560e6753571ad048af88264c0bb9": "\\frac{a*(b+1)}{1*(2*3)}", "03a3c39aa9852a7f991d31078a07cc97": "2\\pi R", "03a3ccf388449808794c9ddaee624540": "B(t,T) = \\mathbb{E}[(1 + r(t,t+1))^{-1} \\cdots (1 + r(T-1,T))^{-1} \\mid \\mathcal{F}_t] = \\frac{1}{1 + r(t,t+1)} \\mathbb{E}[B(t+1,T) \\mid \\mathcal{F}_t]", "03a424c9a0f9fce55418280301f6553b": "a=2.1.", "03a4330f5af1bae4248d69142fb7b656": "\\mathbf{u}_x\\mathbf{v}_x\\mathbf{w}_x", "03a48bf579647edafb8fd0d2d0d6f96f": "j = l \\pm 1/2", "03a4a751eac357b8f7b952d73f1e376b": "\\int_X g\\,d\\mu=\\sup_{f\\in F}\\int_X f\\,d\\mu.", "03a4c56878ff40b6b9dfbe1cb171a96d": "\nw = f(z) = \\frac{a}{c + dz},\\,\n", "03a4db2002c4dffb4de9aabebe5bca27": " X \\times I \\to Y", "03a53ab33c76e6a15fe0dde332242c69": "2^{<\\omega}", "03a541b53b52ea0653764c1ac51f4c8a": "Q[(\\text{d} R / \\text{d} Q) (1+\\mu ) - \\mu(\\text{d} C / \\text{d} Q)] = 0,", "03a6280a5c40bd162b6a2d6d6fc8a03a": "R_n^{(l)}(\\rho) = \\sqrt{2n+D}\\sum_{s=0}^{(n-l)/2}\n(-1)^s{(n-l)/2 \\choose s}{n-s-1+D/2 \\choose (n-l)/2}\\rho^{n-2s}", "03a6467de429b3a11edbde8b6b8fbcc7": "\\frac{1}{2} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{8} \\,+\\, \\frac{1}{16} \\,+\\, \\cdots \\;=\\; 1", "03a66803bd302876b9298ec05ac0e0a6": "\\left(1 - \\frac{it}{\\lambda}\\right)^{-1}\\,\\exp \\{ i\\mu t - \\frac{1}{2}\\sigma^2 t^2 \\}", "03a6a91018a0e7992fcf2af5d2a48bc8": "\\alpha(f_n(x))=\\alpha(x)+n ~.", "03a6afa2815a8b4207ffe936b29c7421": "N^{-3}", "03a6bdcc17724a68d4ff72ae74c17cec": "a \\leq 0", "03a6e7cdccb7aeca2e1f9049927c896e": "=\\max_{\\lambda\\in\\sigma(A)} \\frac{1}{|\\lambda-\\tilde{\\lambda}|}=\\frac{1}{\\min_{\\lambda\\in\\sigma(A)}|\\lambda-\\tilde{\\lambda}|}", "03a73aaac912a3068b0525b8b5ee69b9": "k \\gg 1", "03a7baed00a6193186899c8a0c823b90": "CH_3OH", "03a7f97055201ba94c7471d1764ed4b5": "\nm(\\varphi) = b\\int_0^\\beta\n\\sqrt{1 + e'^2\\sin^2\\beta}\\,d\\beta,\n", "03a8087bd765f80deab2bb94bb5e8c53": "\\langle T x, y \\rangle = \\langle x, T y \\rangle, \\quad x, y \\in H.", "03a8296383cfef6a36b5fb4cf4b14313": "Q_A \\,", "03a87f3d2b231e4aa09ed311b752792f": "x=x", "03a8ecac6e0e640d7f5e79f9103413a2": "N_i = \\frac{g_i}{\\Phi}", "03a91572f241ff32ab94abcc18edefd1": "\\operatorname{ker}(f) \\triangleq \\{(x,x') \\mid f(x) = f(x')\\}", "03a9172e91e7a329897548b920e4b3b1": "x^2-2y^2 = -1", "03a99382839fa0dec99c9d6655bdd747": "\\psi_m(x) = \\sqrt{\\frac{2}{L}} \\sin{\\left(\\frac{m \\pi x}{L} \\right)}, \\,", "03a9c9913020201c1c16cc4806153f2f": "P_B(\\lambda_B)", "03aa4c947fa35f3863419a879ae94189": " \\psi = 0 ", "03aa8a7e9f93b174c1ac6a2ce0776774": "\\scriptstyle t_B \\;=\\; 1", "03aab29df61fc300ace2a4ae56e8b9b1": "|\\phi_{m}^{'}\\rangle", "03aadaf02ca0751fe6a467e10803e850": "\\operatorname{sh}\\,k, \\operatorname{ch}\\,l, \\operatorname{th}\\,m, \\operatorname{coth}\\,n \\!", "03aae9f1bd007e50cdb222061e3e230e": " g(\\mu_m) = \\eta_m = \\beta_{m,0} + X_1 \\beta_{m,1} + \\ldots + X_p \\beta_{m,p} \\,", "03ab00c3face1903468063ad259fa551": "\\mathbb Q/\\mathbb Z", "03ab662b71a6e9ecc0a51e8938a9f26b": "q\\in Q", "03ab79c8d0ebe3b954ef4ae63d73bfbf": "\\phi_i(x)=x^{**}(\\phi_i)", "03ab79d3d14652b6742807f5f5225cb7": "b_i(x)^m=0", "03ac0d241a6fd9fc66a77b2e7ce6db2c": "\n\\begin{align}\np(t) &= (\\cos(2t), \\sin(2t), 0)\\\\\nr(t) &= ( \\cos t \\cos 2 t , \\cos t \\sin 2 t, \\sin t )\n\\end{align}\n", "03ac1ea5f08f206e2c6999b331b58c7d": "\\theta=2\\pi ft\\,\\!", "03ac21b328dbeb50b8d8ae916394f9ef": " \\approx 2.6 \\times 10^{36,305} ", "03ace338f1e2fc15e58f72228a56d525": "A=\\frac{\\sqrt{3}}{4} a^2", "03ad1c3de00115c51e9da7a17ed99162": "\\Phi^{-1}=({\\mathrm d\\varphi_x})^{-1} \\in GL(T_{\\varphi(x)}N, T_xM).", "03ad41ed4ad52afeed2ba2bb320eb8f9": "R'(x) = H(x)\\ \\mathrm{if}\\ x \\ne 0", "03ad4e8445553bf6abad05b0c9eb8c6c": " \\mathfrak{-a} = \\mathfrak{a} \\iff \\mathfrak{a} = \\mathfrak{0}\\qquad \\forall \\mathfrak{a} \\in \\mathfrak{G} ", "03ad7a39f4cae58812e5257e3fb50b3c": "t(tx-2at)+x=0,\\ x(t^2+1)=2at^2,\\ x=\\frac{2at^2}{t^2+1}", "03ad8c5766037005c87fa2d541860ea8": "\\forall s_{-i}\\in S_{-i}\\left[u_i(s^*,s_{-i})\\geq u_i(s^\\prime,s_{-i})\\right]", "03adc5e31d061a36824fd2d2df985b11": "\\operatorname{E}(X^n)=\\mathrm{e}^{n\\mu+\\frac{n^2\\sigma^2}{2}}", "03ae56bc99fbf11c5cbdb0123aac6830": "e^{(\\theta/2)(e_i \\wedge e_j)}= \\cos(\\theta/2)+ \\sin(\\theta/2) e_i \\wedge e_j", "03aef4a1e68615e165e412c64d14399b": "(k_1+k_2+k_3+k_4)^2=2\\,(k_1^2+k_2^2+k_3^2+k_4^2).", "03af1fa0e3d985e14ab133c0d5dfcc3f": "F_{\\theta}", "03af2ad37614e9c2dde9b231e47efac0": "\\gamma \\rightarrow 0 ", "03af3c2704860a125cb8a7cb179f62ef": "\n\\begin{cases}\n\\mathrm{out}_A = 1 \\\\\n\\mathrm{out}_{RGB} = \\mathrm{src}_{RGB} \\mathrm{src}_A + \\mathrm{dst}_{RGB} (1 - \\mathrm{src}_A)\n\\end{cases}\n", "03af47296bd992a62d24d037a7cc1c63": "\\Delta U\\;", "03af5b7be9fc5d6ae89a338a73585a98": "Z = \\frac{1}{V}\\int_\\Omega e^{-\\beta H(\\boldsymbol{r})} \\, d\\boldsymbol{r}.", "03af8f387cbac79f063aeed5e31002f0": "\\vec{a} = (1/\\lambda_a)A^TA\\vec{a}", "03aff3b2154d6187c80d748b16746e7e": "\\sqrt{a ^2+ r} \\approx a + \\frac{r}{2 \\cdot a}", "03b002da7c63cabcb42234272136bc6d": "v(n) \\ne 0", "03b062a002773b4b2e3b3d46fc3a32e3": " s_1 - s_2 = 2A \\quad (4)", "03b07069253fff670c3f652b869afb60": "| \\alpha/\\sqrt{2} \\rangle", "03b0b51d46fcb7fc4289d6674ffd59a0": "\\Sigma_u \\left \\lfloor qu/p \\right \\rfloor", "03b0d432c78ae131fdc3d3ee81f1cb40": "\\mathrm{OTF}(0)=\\mathrm{MTF}(0)", "03b0e351027bad181abe45ab499a1679": "7.72\\approx\\frac{5\\pi}{2}", "03b0ec9ffa0cc8774c2ec4894a2b06c2": "n_{adatom}=n_0 e^\\frac{-\\Delta G_{adatom}}{k_BT} \\qquad (4)", "03b0f61f6a67f7c1157f3ba6976c6f3e": " \\mathbf{b} \\prec^w \\mathbf{a} ", "03b10549cef7810b87a7c29c34ad2309": "x = (x_1, x_2, \\ldots, x_n) \\in \\mathbb{R}^n", "03b174fb05c3ec6889c14cfb9f469d03": " 0< \\delta <1 (e.g. \\delta=0.97) ", "03b183e24368a6a72f056e417204c0d9": "f(k) = -\\frac{1}{2k} + \\frac{\\pi}{2}\\coth\\left(\\pi k\\right)\n", "03b196e9ce523c30d49f5d2d7bda1ed5": "0 + 0 = 0.\\,", "03b1b44a6955cf0b191b94301740d63a": "\\Gamma_+(M)", "03b1e442a01f8c4b877e9526471fd3ea": "\\mathfrak{sl}_4 \\cong \\mathfrak{so}_6", "03b1ecb60736d072099b4bea3dbbf11e": "\nK_H(x') = x' - x_0(T),\n", "03b25e2f947ca0c07d48c53d93f617fc": "N = (P, T, F)", "03b28f6abb6be057c8e59d765aaf78c4": "\\sum_{i=1}^n \\mathrm{Bernoulli}(p) \\sim \\mathrm{Binomial}(n,p) \\qquad 0s} \\, ds", "03cc1382a913a0468b9f71a6736e27c8": "\n H^{(\\lambda)}(X)\n =P_1(X)\n +O\\left(\\prod_{\\kappa=0}^{\\lambda-1}\n \\left|\\frac{\\alpha_1-s_\\kappa}{\\alpha_2-s_\\kappa}\\right|\n \\right)\n", "03cc3ae00c3c4b556427df9ecacebce0": "i_s=i_1\\sin(\\Delta\\varphi_a^*)+i_1\\sin(\\Delta\\varphi_b^*).", "03cc43f844df88ceccd395979a438084": "m_1, m_2\\,", "03cc89a6cd7e58ab528970c70bff387f": "\\ E_{+/-} = E_{(0)} + \\frac{C \\pm J_{ex}}{1 \\pm B^2}", "03ccbc87d41cf99a35e38b27fa7c32b6": "\\sigma_{ij} = \\epsilon_{0}E_{i}E_{j} + \\frac{1}{\\mu_{0}}B_{i}B_{j} - \n\\left(\\frac12\\epsilon_{0}E^2 + \\frac{1}{2\\mu_{0}}B^2\\right)\\delta_{ij} \\,.", "03ccc140b40080330afee07a5170b9d0": "M=[1.44 0; 0 2.89]", "03ccea9e9891a786a0e5f7d8bec31bd3": "A = \\bigoplus_{n\\in \\mathbb N}A_n", "03cd9af73a09b7d07621d3f80c839abc": "(10)", "03cdeac2c9e5cf309c622bb36903465e": "\\chi_6", "03ce19dab62e4b5bf1ac3e2be81be05a": "\\frac{\\gamma}{2\\alpha\\delta K_1(\\delta \\gamma)} \\; e^{-\\alpha\\sqrt{\\delta^2 + (x - \\mu)^2}+ \\beta (x - \\mu)}", "03ce6166bf66aafeb930e5b683790242": "x_D = \\begin{cases}\n0.244063 + 0.09911 \\frac{10^3}{T} + 2.9678 \\frac{10^6}{T^2} - 4.6070 \\frac{10^9}{T^3} & 4000K \\leq T \\leq 7000K \\\\\n0.237040 + 0.24748 \\frac{10^3}{T} + 1.9018 \\frac{10^6}{T^2} - 2.0064 \\frac{10^9}{T^3} & 7000K < T \\leq 25000K\n\\end{cases}", "03cecb10712a6deafefb7b008f96ee7f": " \\lambda = \\sqrt{\\frac{m}{4 \\mu_0 e^2 \\psi_0^2}}, ", "03ced8b82b8d4ff11cc137b7b22188f4": "B\\left(\\frac{\\alpha}{2}; x, n - x + 1\\right) < \\theta < B\\left(1 - \\frac{\\alpha}{2}; x + 1, n - x\\right)", "03cee26504e3be30e6a4799eaa1ebb0a": "\\frac{\\sqrt{\\frac{(d_1\\,x)^{d_1}\\,\\,d_2^{d_2}}\n{(d_1\\,x+d_2)^{d_1+d_2}}}}\n{x\\,\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)}\\!", "03cf1d3306a36f956af672193dc515a5": "y=f(x)\\,\\!", "03cf69b8e1926e9304e27bf70f136390": " kT \\gg \\varepsilon_i-\\mu ", "03cf6c16a00da319754020f534d0f73e": "x(n)=\\sum_{k=0}^q b_n(k) d(n-k)+v(n)", "03cf83e7d9741f6bf162b1fbfe0860ac": "-\\ell k\\,", "03cf8f9ecf69055531c30a1f5c0af149": "\\mathbf{p}_{\\mathrm{1}} = m\\mathbf{v} + \\mathbf{u}\\mathrm{d}m", "03cfa63594a40ccb8e679d26ff261748": "F(0)= \\sum\\nolimits_{n\\in \\mathbf{Z}} c_n", "03cfb11ec181418826af113a24111e44": "A = -kT\\log\\left(Z\\right)\\,", "03cfbeda8a8727cdc5c429ce1debc64d": "10 | 560", "03cfdf3d21a85ad10328dba37ba17761": "m\\times n\\!", "03d024b9f26631e6a453e7841f822ddb": "y = \\prod_{\\sigma} \\sigma(x)", "03d0a2ad9a3042e12a7f671666711f1b": "\\sum_{jN+1}^{(j+1)N} X_k", "03d0e2a2646d1ced604bea3690da5cc0": " \\sigma_m = 1/R ", "03d1a28c36a9ecd807f616236d2f152d": "x_t = 1/(\\eta + w_t)", "03d25453bed03733343b2e4bb36cb56f": "q=", "03d25e22170a4308bb4e1cb54a983b33": "\n \\boldsymbol{\\sigma} = \\lambda~\\mathrm{tr}(\\boldsymbol{\\varepsilon})~\\boldsymbol{\\mathit{1}} + 2\\mu\\boldsymbol{\\varepsilon}\n ", "03d2db8193aa097c9a76ecbe7056dc98": "\\delta (\\hat{M_E}) = \\lim_{T \\rightarrow \\infty} \\int_{-T}^T dt e^{i t \\hat{M}_E}", "03d2e5f5855c3331821ae802f99355c6": "g^{(N+1)}", "03d340d167c24ad78bc190e1ab0f0988": "\\vec{e}\\!", "03d35c5129ead46219fd4b22f81f5d04": "2\\pi/3", "03d3ca3fa2226c9a550d3f4cef0a1dd5": "d_1", "03d3fc8d7471cbd34af7dcc0e716e956": " E\\left[ \\Lambda(n+1) \\right] = E\\left[ \\left| \\hat{\\mathbf{h}}(n) + \\frac{\\mu\\,e^{*}(n)\\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} - \\mathbf{h}(n) \\right|^2 \\right]", "03d41346ade9e0e7d16038913451be15": " \\text{d} C / \\text{d} Q", "03d424e371484e73cbf8027cb75f646d": "[0,1]_{\\ast},", "03d48597117c2bd97aab24e7075b6b41": "FWER = P_{any}(V \\ge 1) ", "03d4ad33274debc1fa4faa5abb7715c0": "Q_{T-1}(W_{T-1})", "03d4c263ed0a4ab7c10fe02f319d64f1": "L \\, ", "03d4f5f0b9f03f5c76dc44e448eaf6d6": "\\ r ", "03d5004c82b6f47b105ca61a9e650cf6": "D_i(u,v) = w_i \\times \\frac{|log(u_i + 1) - log(v_i + 1)|}{log(max\\{u_i, v_i\\} + 2)}", "03d549124b4658a4d2ed56f1aaef6eed": "\ndW = \\sum_{r=1}^{D} Q_{r} dq_{r} = \\sum_{k=1}^{N} \\mathbf{F}_{k} \\cdot d\\mathbf{r}_{k} = \\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot d\\mathbf{r}_{k}\n", "03d5be0871da01d01a8b0c64b5638ae3": " [ d/dz - E_{11}/z , - E_{21}/z] = [ d/dz , - E_{21}/z] + [ - E_{11}/z , - E_{21}/z] = E_{21}/z^2 -E_{21}/z^2 = 0 ", "03d5db1d5b264656b1f664ca2656a118": "\\cot\\frac{\\pi}{12}=\\cot 15^\\circ=2+\\sqrt3\\,", "03d5f2201251112296c9be71d22bcf97": "\\csc \\left(\\frac {\\pi} {z}\\right)", "03d60f1de7d97a09a377389404300b6d": "b \\equiv a \\,\\bmod{f'(a)\\mathfrak m}.", "03d63acd89770450159bbe51452ac3ae": " v_{i1} ", "03d66cdf7525281b402a60c4ef007aab": "z^1 = 0.499997032420304 - (1.221880225696050\\times10^{-6})i{\\;}{\\;}{\\mathrm {(red)}},", "03d69e6e0442b1f7218f02ce91363f32": "P_A = l_A", "03d6a6d3b41fa8d2e621a564b04822c3": " \\scriptstyle \\langle ", "03d6cb574e9b5c08083092a6e6119912": "e_\\infty = \\frac{m}{(m, \\deg(f))}", "03d6d59efc67a54e5f9d03e115ab40ea": " C_\\pm (j,m) = \\sqrt{ j(j+1) - m(m\\pm 1) } ", "03d73b3a369ef15a8cc02d20249f80fc": " g\\cdot z=\\frac{az+b}{cz+d}", "03d78a18d747ce1a9b217ba0c0fb59f3": "\\scriptstyle |\\psi_n\\rang ", "03d7f263292f2c210cf729306da936ea": "M = \\max_{1 \\le i \\ne j \\le m} \\left| a_i^H a_j \\right|.", "03d81762c9fa45c6edf4a7bd0586d464": "\\sqrt{m_\\mathrm{i}/m_\\mathrm{e}}", "03d8a890f439d09045c78c4a62d5a45f": "\n\\left.\n\\begin{matrix}\nx*y*z=(x*y)*z\\qquad\\qquad\\quad\\,\n\\\\\nw*x*y*z=((w*x)*y)*z\\quad\n\\\\\n\\mbox{etc.}\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\ \\,\n\\end{matrix}\n\\right\\}\n\\mbox{for all }w,x,y,z\\in S\n", "03d8d2ba7d77a6b9d8e40cbe310326db": "u_G''(x)=A(x)u_1''(x)+B(x)u_2''(x)+A'(x)u_1'(x)+B'(x)u_2'(x).\\,", "03d934ff2e808e9b0ffcdbdb1df1d6d9": "Z_0 = \\sqrt{\\frac{L}{C}}", "03d994023612fce11d588b24854c5050": "A > B > C", "03d9c9a1ed9d2003fc9725ed0d858420": "\\beta(2)", "03d9da20066f41452bbd390bc1981167": "= (1 + i 2\\pi fT) e^{-i 2\\pi fT} \\mathrm{sinc}^2(fT)) \\ ", "03d9e27b16b8626d6f715803f24950ed": "\\sqrt{x^2}+\\sqrt{y^2}=|x|+|y|", "03da8aa755310b93007325e732dc509b": "x\\ne 1", "03daab1d4024edcc09938b1d5acab536": "F = k X,", "03dab7746e3ecc9822922b5a75b18e61": "\\lim_{z\\rightarrow 1^-} G_a(z) = \\sum_{k=0}^{\\infty} a_k,\\qquad (*)\\!", "03dadf7919b83d68f7682da0245ec658": "\\left[ B \\right]=\\left\\{ \\begin{array}{*{35}l}\n \\left[ A \\right]_{0}\\frac{k_{1}}{k_{2}-k_{1}}\\left( e^{-k_{1}t}-e^{-k_{2}t} \\right) & k_{1}\\ne k_{2} \\\\\n \\left[ A \\right]_{0}k_{1}te^{-k_{1}t}+\\left[ B \\right]_{0}e^{-k_{1}t} & \\text{otherwise} \\\\\n\\end{array} \\right.", "03db060902394497adbf217f4ae97338": "{\\delta W}=P\\mathrm{d}V.", "03db63a398f2b45c18059782c4f36dfe": "f(x) = a e^{- { \\frac{(x-b)^2 }{ 2 c^2} } }+d", "03db64206cf3a86a497c1121e7ac5d19": "M^{\\mathbf r}=\\left(I-H \\right) M \\left(I-H \\right)^{\\rm T}.", "03dbbab2600cebe8b540da30ace6353f": "| B \\rang", "03dbf0197647d1d16db44f5235b2b043": " \\frac { \\text {density of object}} { \\text{density of fluid} } = \\frac { \\text{weight}} { \\text{weight} - \\text{apparent immersed weight}}\\,", "03dbf3e1d43b8a823c32f0f134cd5d5f": "\n f = \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) + v,\n", "03dc38392b2f7a44b03e4a8f381276ec": "P= \\sum_{i,j} u_i \\left( \\langle\\mathbf{v}, \\mathbf{u}\\rangle^{-1}\\right)_{j,i} \\otimes v_j", "03dc396c1aca49f0bf4b6f361ed5c365": "p(F_a=v_i)", "03dc5d1fd972f00dcd03a56ec82313f6": "\\tan A = \\frac{3}{7},", "03dc896e15bfb09b05f1f2cb0aa67fac": "\\lambda\\colon kG \\otimes F(X) \\to F(X) ", "03dc93a786dba8c41080ea2b97c34df5": "\\ln\\left(\\left(\\frac {1}{2}\\right)^{t/t_{1/2}}\\right) = \\ln(e^{-t/\\tau}) = \\ln(e^{-\\lambda t})", "03dd2d723a4862c986838c5169342b94": "PCER = E \\left[ {\\frac{V}{m}} \\right] ", "03dd3245478c9e35f4cf9bcf89c4a226": "\\psi( \\dots, x_j=0, \\dots ) =\\psi(\\dots, x_j=L,\\dots )", "03dd4b4beb2d8bf6ce7ff194f46c1ef7": "Z(X_0, F_0, t) = \\prod_{i}\\det(1-F^* t|H^i_c(F))^{(-1)^{i+1}}", "03dd505cbfabb8a184ff4d2fca82b1ac": " c_j | N_1, N_2, \\dots, N_j = 1, \\dots \\rangle = (-1)^{(N_1 + \\cdots + N_{j-1})} | N_1, N_2, \\dots, N_j = 0, \\dots \\rangle ", "03dd6d27f3d6f4c01e7b2dec3b9ae715": "\\tfrac{4}{5} \\pi", "03ddbc0a475c1bd43b4672cafa104e47": "S \\ni x", "03ddc9b055532c322b2a9620740d6136": "a\\left(\\frac{1}{q}-\\frac{1}{r}+1\\right)=\\frac{1}{q}-\\frac{1}{p}.", "03dde93cc610c1307a6fc777f6e2b044": "CF_i", "03de0b280f757ebe94424def39e130fc": " h_k (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq i_1 \\leq i_2 \\leq \\cdots \\leq i_k \\leq n} X_{i_1} X_{i_2} \\cdots X_{i_k}.", "03de7cfc6e7b321c4c17410f62e445f9": "\n c^2 = \\frac{2\\kappa G h}{D(1-\\nu)} \\,.\n", "03dea73658b8ca48d2bab910746fed4f": "\\,F_v", "03dedeca4f11b2b3094602be33a3fd12": "\\min_x \\|x\\|_1", "03df32a80ca426f52e6e058a7fe0e3f1": "M_p(n)", "03df43b24d79f21ac2f7483bfe649a78": "\\displaystyle a^2=cd", "03df5d2a6370f99e14f97f689511ecd9": "\\Phi^a", "03df888fc1e5a7faf8843d69f7c2ff0b": "\n\\frac{I_f^0}{I_f} = 1+k_q\\tau_0\\cdot[\\mathrm{Q}]\n", "03df91cb9676e0aefdd35bbb3285ba96": "\\omega_1'", "03df96fbb957bad6c44aab5fd6cd474a": "(\\phi \\lor \\exists x \\psi) \\leftrightarrow \\exists x (\\phi \\lor \\psi)", "03dfa0f901b84523f5dbce2cb9985583": " f(x) \\approx f(a)+f'(a)(x-a). ", "03dfeab6a13f6249ccaf3f30922fdb8d": "J(u) = \\int_D |\\nabla u|^2 \\mathrm{d}x.", "03e005d925a144ea2e6f8860daa4fdca": "\\sigma(q,t)", "03e048b5b3bddc7362e6620e6dcd4d75": "p(f_{ik})\\ ", "03e074dff06fc87c2f5feb819e612bd0": "\\chi_3\\left(z\\right)=\\frac{1}{\\alpha}\\int_\\infty^z\\sinh\\left[\\alpha\\left(z-\\xi\\right)\\right]Ai\\left[e^{i\\pi/6}\\left(\\alpha Re\\right)^{1/3}\\left(\\xi-c-\\frac{i\\alpha}{Re}\\right)\\right]d\\xi,", "03e161b98da6a296b962d010a04970f6": "H_{inv}(s) = \\frac{D(s)}{A(s)}", "03e16b8c66c7a6ca01f976ae30c9586b": "\\mathbf{x}_{k+1} = \\mathbf{f}(k,\\mathbf{x}_k)", "03e18d83841f9780acb30eeb0ec7b5d7": "a_{k,\\ell}", "03e1aca4c970b3b37ad7f63c6ef6fbb7": "\n |j_1-j_2| \\leq J \\leq j_1+j_2.\n", "03e1b478601c13078e20acf0aae90f75": "I_x=E_xL+M_x", "03e1d1c59a1694c2260609f969a05aee": "S_\\mathit{wir}", "03e1d82e99b158334f241aacf764b61c": "(n-1)^2", "03e22ec25b5ec0e94a5589c25909b951": "W_{m} = diag\\{w_{m}\\}", "03e2462350c41d7ec1a5ed27576b9572": "\\gamma \\delta \\gamma^{-1} \\delta^{-1} = \\epsilon", "03e248075c04b3c9e2f9121851c92f42": " I_i \\, ", "03e256589a2c8c87cf5cc0de0c0c70a6": "\\hat{F}_\\mathrm{inconcl.}= 1-\\hat{F}_{\\psi}-\\hat{F}_{\\phi},", "03e3015e5ade7bfebee4372443308fc7": "A_m(1,2) = 1,2,3,4,5,6,7,8,9,10,\\ldots", "03e31b4413745326637d7ee75b266a25": "\\alpha\\div\\beta", "03e344c5d678f065203d644e6cd8f6a0": "q = 2", "03e350a66d4f39798189dac57cffc007": "\\forall x\\in W\\,(x\\Vdash A)", "03e35c79fbb25874863c3a9ea4f6c69a": "\\dbinom{n}{k}", "03e3810cee17a572d4ebe76a0aac1c97": "L^{1,w}", "03e3c8c207091356616e7205551325f0": " \\frac{1}{2} k_BT ", "03e42f7f0cba50e05a4ef28ef7a119ce": "\n\\frac{dT(s)}{ds}/\\frac{T(s)}{r}=-\\frac{t}{n}", "03e44f74cd340ba2739c352b535d868a": "n=\\frac{T}{\\delta}", "03e4a025a0c7424b008ba3875a2c4e8f": " Z[J] = \\sum_{x \\in \\mathcal{X}} \\exp \\left(\\sum_{k} w_k^{\\top} f_k(x_{ \\{ k \\} }) + \\sum_v J_v x_v\\right)", "03e5091cb5cd32369b9252594e4113a4": "5^6", "03e52b5f14c4a6c618eb416e99f7772b": "\\varepsilon_{\\phi_2(0)+1}", "03e55e8dc3eaea1688a82b38f2b00412": "\\ \\|y(t)\\|_{\\infty} < \\infty", "03e58c894b4026627a6c2f57dc122d9f": " f(x)=1/x^2", "03e5aff75dead8f836733b8199d63c49": "\n\\Phi(z,s,a)=\n\\frac{1}{2a^s}+\n\\int_0^\\infty \\frac{z^t}{(a+t)^s}\\,dt+\n\\frac{2}{a^{s-1}}\n\\int_0^\\infty\n\\frac{\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^2)^{s/2}(e^{2\\pi at}-1)}\\,dt\n", "03e5c4ff8dac745730829e5dc3d136da": "\n\\Delta E = E_{n+1} - E_n = {dE \\over dJ}(J_{n+1} - J_n) = {1 \\over T} \\,\\Delta J \n", "03e5cb9a4a8ab0a96d48912ee44ccb82": "\\frac{\\omega_s}{c} \\,", "03e5d5a49cc05c533f2fc8b4fabf1032": "2^b=N", "03e61b72f90197524cecd7aabf0c3b7f": "\\frac{1}{1+a} = 1 - a + a^2 - a^3 + \\cdots \\pm a^n \\mp \\frac{a^{n+1}}{1+a},", "03e647e6672941060cc02ee23aaafacb": "k^{-1}\\bmod\\,q", "03e66241d1f7bff74afcde3a56427d83": "CABED", "03e69d68bb4b8e69ad734ffe3d1595b8": "\\equiv_D", "03e70a4e0f5c37e711494be10964acba": " \\left|\\,{x\\over a}\\,\\right|^n + \\left|\\,{y\\over b}\\,\\right|^n =1 ", "03e72bde921ee249d8d5f0fcec11ac43": "{SU(3)_C\\times SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}\\over \\mathbb{Z}_6} \\rtimes \\mathbb{Z}_2.", "03e74c0242bfd5c0fa3088820daea46c": "\\overline{op_1}'", "03e7620ee41838088ae281c1602cab97": " {s_1 / \\sqrt{n_1} \\over \\sqrt{s_1^2/n_1 + s_2^2/n_2}}. ", "03e79d8069f4e38f91e49006e4259284": "\\bold{u}_1^\\prime = \\bold{u}_1 - \\bold{V} , \\quad \\bold{u}_2^\\prime = \\bold{u}_2 - \\bold{V} ", "03e7b061f9b7de024ec507077863eb49": "\\langle \\sigma_A\\rangle", "03e7e6908f866c4dab2769c4c2b87175": "a=b>c", "03e862a39055364b665a144d012f1465": "\\scriptstyle L_2", "03e86768053aae06fb07c3ec55402e83": "\\gcd(p,q)=\\gcd(p,kq)", "03e8bc5f2d83295cd14c9d24945b18ab": "A= \\dfrac{n}{n_{e}}", "03e90fb862fd9c3b022d97e17a824bc6": "x_2 = 1.000000000000000 .", "03e9615aba27e5f307db8ba3ba2107ca": "\n \\begin{bmatrix}\n 1 & 31 & 12& -3 \\\\\n 7 & 2 \\\\\n 1 & 2 & 2\n \\end{bmatrix}\n", "03e96c424485be59990cd22a79cffc64": "\\Re {zh^\\prime(z)\\over h(z)} \\ge 0", "03e97b92772cadb788968043edbab486": "\\frac{b-a}{2}", "03e98c90ab94388ee0a3fd11220908ef": "D_{ij}={\\delta_{ij}\\over (r_i,r_i)}", "03e9d588acff06de769e9d810c61c133": "x = \\cos\\theta", "03ea481a6377c6c94f3f6293db8773dd": "Y_{MIN}", "03ea4c2ef4e4da1779a45b12f5a23f64": " P = P_e + \\frac{Y-Y_n}{a} ", "03ea501e22ef0596ae87b533bdfca027": "\\Omega, \\Omega_+, \\Omega_-", "03ea9f29d543d258a40f84e07d044afd": " \\{ |e_n\\rang\\} ", "03eac388ebfcefa1384716da5ac392d9": "V_{1} (K, L) = \\lim_{\\varepsilon \\downarrow 0} \\frac{V (K + \\varepsilon L) - V(K)}{\\varepsilon},", "03eb2bb2e0599d81d80cfaa9b03c4c7b": " x : I \\mapsto X ", "03eb6d5cb381b4a0f04113069b1e2a61": " \\frac{N}{4\\cdot \\pi \\cdot d^2}= \\frac{E^2}{R}", "03eb8a3cb7a391c69fecac33667fe4eb": "\\mathbf{x}^{(n)}", "03ebd69aa4068e875588677f031bef5a": " L = d\\cos\\alpha_{crit}\\,\\!", "03ec0562facaeea4f5cc5b21b991f65e": "A=\\frac{1}{2}(20+\\sqrt{5(145+58\\sqrt{5}+2\\sqrt{30(65+29\\sqrt{5})})})a^2\\approx32.3472...a^2", "03ec4481d5dd16bd36562db5929d3a11": "\n\\begin{bmatrix}\n\\varepsilon_1\\\\\n\\varepsilon_2\n\\end{bmatrix}\n\\mid X\n\\sim \\mathcal{N}\n\\left(\n\\begin{bmatrix}\n0\\\\\n0\n\\end{bmatrix},\n\\begin{bmatrix}\n1&\\rho\\\\\n\\rho&1\n\\end{bmatrix}\n\\right)\n", "03ec8730b2ff44a8a89494b272f75d86": "\\eta = \\frac{-dW}{-dQ_h} = \\frac{-dQ_h - dQ_c}{-dQ_h} = 1 - \\frac{dQ_c}{-dQ_h}", "03ed06b5d14dff9fea8727cd4f53e63f": "\\scriptstyle {g_{\\mu\\nu}}", "03ed20965c7afae03a27561dbd28d372": "[V, W] (x) = \\mathrm{D} V(x) W(x) - \\mathrm{D} W(x) V(x),", "03ed351400e6de29ff95107a28d66c09": "x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)} = x_n - \\frac{1/x_n - b}{-1/x_n^2} = 2x_n - bx_n^2 = x_n(2 - bx_n).", "03ed8335cb4bf1e669605934b01240f1": "\\mathbf{I}^{(1)}\\cdot\\mathbf{J}^{(1)} = \\sum_{n=-1}^{+1}(-1)^nI_{n}^{(1)}J_{-n}^{(1)} = I_0^{(1)}J_0^{(1)} - I_{-1}^{(1)}J_{+1}^{(1)} - I_{+1}^{(1)}J_{-1}^{(1)},", "03ed9ef12bf7cee79361a68fea5cb8cd": "f:M\\to S^1", "03edb2b6a493e3ea759399c7c4acd4cd": "GL_2( \\Bbb{R})^m.", "03edcaedb348abb3e7085289903a7564": " P=DEC_{k_1}(DEC_{k_2}(...(DEC_{k_n}(C))...))", "03ede58af7d86c129b4edb6af3e9cb0f": "\\rho^{\\text{induced}}(\\mathbf{r})", "03edfa39384656ea5b57eea7c49f7532": "S_\\text{baker-unfolded}(x,y)=\n\\left(2x-\\left\\lfloor 2x\\right\\rfloor \\,,\\,\\frac{y+\\left\\lfloor 2x\\right\\rfloor }{2}\\right).", "03ee304aa69c486d234a44f3378b09da": "\\Delta \\mu _{H+} = -F \\Delta \\psi + 2.3RT \\Delta pH", "03ee86361185b580eb773f753586ddb7": "H(x, v)", "03ee92d0d65558637cb6d34b82d39509": "\\,\\{\\Upsilon_j \\} = \\{\\ddagger\\sigma_m,\\ddagger\\sigma_{m-1}, \\ldots, \\ddagger\\sigma_1 \\} \\in (\\ddagger\\Gamma^+)^*", "03eea484ca5b44485dd3f8d3c741cb3c": "1,x,x^2/2,x^3/3!,\\dots,x^n/n!", "03eec948e92e06a8398b5a7fdec62758": " \\operatorname{V_r}(\\theta)= Constant", "03ef420905afef95aa9d5571cd418501": "\n\\psi_T(y) = \\int_{x} \\psi_0(x) K(x,y;T) dx = \\int^{x(T)=y} \\psi_0(x(0)) e^{i S[x]} Dx\n\\,", "03ef7f06681f7649eaf5dd13c9d53f77": "\\dot{V} - \\frac{U_{osm}}{P_{osm}}\\dot{V}", "03ef81b127bfbe48fe215949105d7e28": "Z_{ij} \\, ", "03efb900cbe0906009ca8cdf2f28ee12": "x_1x_2", "03efbb3a1702295d54d47558026f336a": "N_{MSY}", "03f0d2c858daddfd2cd839e35fdd09c7": " h_A(x+y)\\le h_A(x)+ h_A(y), \\qquad x,y\\in \\mathbb{R}^n.", "03f106c3f162e380f505214595a8b110": "\\begin{align}\nR &={1\\over 2\\pi} \\int_0^{2\\pi} e^{-i\\theta} U_\\theta H^{(1)} U_\\theta^* \\, d\\theta,\\\\\nR_\\varepsilon &={1\\over 2\\pi} \\int_0^{2\\pi} e^{-i\\theta} U_\\theta H^{(1)}_\\varepsilon U_\\theta^* \\, d\\theta.\n\\end{align}", "03f110bd9e7ea18ea3d59dd66e63a23c": "\\bar r_2\\ ", "03f1267db64f5b147017f41c868a4d94": "\\begin{bmatrix} \\dfrac{1}{y_{11}} & \\dfrac{-y_{12}}{Y_{11}} \\\\ \\dfrac{y_{21}}{y_{11}} & \\dfrac{\\Delta \\mathbf{[y]}}{y_{11}} \\end{bmatrix}", "03f190eb9234c0b19deba4f7e0bb8b4c": "4(\\pi)", "03f1e37a6367be8da35b90171b743001": "F\\,.", "03f20cc4a24a90939ad2151268aa2dcd": " d = at^2\\,", "03f23727275ecf230d0235edfff68fbd": "d = c_1d_1 + c_2(v_1+v_2+ h)", "03f23f225a2c154236dfaae5a0bf7e51": "E\\left[u(w(y(e))) - c(e)\\right] \\geq \\bar{u}", "03f275c725ccf747e0b18d0e917cf240": " (e, h, f) ", "03f2803a1e4332c25240375dae0cd931": "x = r \\cos \\phi", "03f2946ba41fedfea05608b274a24e3c": "\\tau=u^\\lambda\\partial_\\lambda", "03f2aee5882c28bc87275d8661fe382b": "f=:\\sum a_{j,k}e^{i(jx+ky)}", "03f2b21268ff5b4cfb3e212a7a352e5e": " x(\\lambda) ", "03f2ee18ea6e349e91f45e9c6d4bf77a": "E(Particle_{i,j}) = k_{s}E_{s,i,j} + k_{b}E_{b,i,j} + k_{g}E_{g,i,j}", "03f33cc02be56b7c09cc5cf7442a7ea9": "\\theta = \\operatorname{atan2} \\left( \\frac{\\partial f}{\\partial y} , \\frac{\\partial f}{\\partial x}\\right)", "03f3657c7cfeab1f4c34e813583841ed": "\\begin{align}L & = \\{uvwxy : u,y \\in \\{0,1,2,3\\}^*; v,w,x \\in \\{0,1,2,3\\} \\and (v=w \\or v=x \\or x=w)\\} \\\\ & \\cup \\{w : w \\in \\{0,1,2,3\\}^*\\and \\text {precisely 1/7 of the characters in }w \\text{ are 3's}\\}\\end{align}", "03f37a2889d1ff304acb68428ed6045b": "p_\\sigma=0", "03f3bf8fecca7e1e602a83a9b7562a11": "a - b", "03f3ca9db6a166009561d00518b1049e": "\\vartheta_{01}, \\vartheta_{10}, \\vartheta_{11}", "03f3ccfc0b3e2d7093afb0146ecb3a23": "\\displaystyle{K_p= \\|z^{-1}(z-1)^{-1}\\|_q/\\pi.}", "03f53547f0d309456588e2688b239aac": "\\begin{matrix}{52 \\choose 4} = 270,725\\end{matrix}", "03f5edbac70ba21f4f43a8ed3c68c926": "\\Lambda = {{8\\pi G} \\over {3c^2}} \\rho\\!", "03f5f86eac108f38f088b7bada9f37ad": "0\\leq \\beta < 1", "03f60de2e8eec9a071f2f23d0c648367": "h_{11}(t)", "03f65e0eeb6bf535749354fd92b970dc": "v_{g} = - \\frac{1}{\\rho_{max} \\tau_{del, jam}^{(a)}}\\qquad\\qquad(1)", "03f6c7272f9a77e0c06f5fb7290a470d": "MPK=R/P", "03f6f0f1d77b4bc5af4704cac07c9681": "x'=V(x)", "03f7e107a2d26b135be2c430d2f00f20": "\\epsilon^2 \\cdot n", "03f855a103cbcabbcbdc053b2a42274a": "\\mathfrak{t}\\ominus \\mathfrak{s}", "03f90abaf79f4744b8b7b766c6df2326": "885.7\\pm0.8~s", "03f94c2d32a2e3e9dedb87522e89d573": "\\pi a^2", "03f960a96507df5ea172c666631d9f7d": "\\left.g\\right.", "03f9745b3fb68caf25bae38a9047b451": "_SM", "03f98599e7e2a6894f748aeb548e6af0": "t \\sigma_1 \\equiv t_1", "03f98cb374db9d443f57a6b3871e2aad": " \\mathbf{A} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{A}_{11} & \\mathbf{A}_{12} \\\\\n\\hline\n\\mathbf{A}_{21} & \\mathbf{A}_{22}\n\\end{array}\n\\right]\n= \n\\left[\n\\begin{array} {c c | c}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n\\hline\n7 & 8 & 9 \n\\end{array}\n\\right]\n,\\quad\n\\mathbf{B} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{B}_{11} & \\mathbf{B}_{12} \\\\\n\\hline\n\\mathbf{B}_{21} & \\mathbf{B}_{22}\n\\end{array}\n\\right]\n= \n\\left[\n\\begin{array} {c | c c}\n1 & 4 & 7 \\\\\n\\hline\n2 & 5 & 8 \\\\\n3 & 6 & 9 \n\\end{array}\n\\right]\n,\n", "03fa0933c441dc2ac015801a807d2693": "x_{11} = p_{1} q_{1}", "03fa0bc00e305c5fcfd2959a9cce90da": "\\widehat{\\sigma_e^2} = \\frac{1}{n} \\sum_{i=1}^n (x_i-\\hat{x_i})^2.", "03fa132f865cb806ec697d4984b69b1a": "Q_r=\\frac{\\prod_j a_{j(t)}^{\\nu_j}}{\\prod_i a_{i(t)}^{\\nu_i}}", "03fa28067b7f7c8257cc8700d0957e88": "\\begin{align}\n\\alpha & = \\cos a = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{x} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{x}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}} ,\\\\\n\\beta & = \\cos b = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{y} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{y}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}} ,\\\\\n\\gamma &= \\cos c = \\frac{{\\mathbf v} \\cdot \\mathbf{e}_\\text{z} }{ \\left | {\\mathbf v} \\right | } & = \\frac{v_\\text{z}}{\\sqrt{v_\\text{x}^2 + v_\\text{y}^2 + v_\\text{z}^2}}.\n\\end{align}\n", "03fa496e35fda74947e0ecf357c79f5a": "C_o", "03fa5627e5525e969a05f15229892021": "x^3-x-1", "03fa815b0b6dd461c3d05fcb636eeea8": "8x^3 - 4x^2 - 4x + 1 = 0", "03fb606e136573b6a73d962b643adf6b": "\\mathcal{F}_{\\tau} := \\left\\{A\\in\\mathcal{F}:A\\cap\\{\\tau \\leq t\\}\\in\\mathcal{F}_t, \\ \\forall t\\geq 0\\right\\} ", "03fb82180093b4b3ddca81ddebf24ac1": "\\oint_{\\Gamma} \\mathbf{F}\\, d\\Gamma = \\iint_S \\nabla\\times\\mathbf{F}\\, dS ", "03fbd188089fa3e308aa0da3890b0c54": "-b^{-1}", "03fbde77646393d7fc1446b1f79e2bfc": "\\displaystyle{\\nabla D(\\varphi)=D(\\dot{\\varphi}\\mathbf{t}) + S(\\partial_t(\\dot{\\varphi}\\mathbf{n})),}", "03fbe811e8cf5e8eb9ef932cbe6cd17a": "ROC = \\left\\{ z : \\left|\\sum_{n=-\\infty}^{\\infty}x[n]z^{-n}\\right| < \\infty \\right\\} ", "03fbf6a0135f8a1716848e343c2ab8b3": " P_y = P_{y0}(2e^{-\\frac{\\pi|\\epsilon|^2}{2\\alpha_0}}-1)", "03fc0cf8bec9b1e6b4ea35e95f590044": " P = AMB \\bmod d ", "03fcd006e9c861273d6a04e143a20d8b": "r\\arctan(\\frac{y}{x}) = \\frac{1}{1}\\cdot\\frac{ry}{x} -\\frac{1}{3}\\cdot\\frac{ry^3}{x^3} + \\frac{1}{5}\\cdot\\frac{ry^5}{x^5} - \\cdots , ", "03fce92d16e9587b8788dfff21a7abcc": " O_{fg} ", "03fd678e6a278e851da7a244a5956614": "\\varepsilon_1\\varepsilon_2", "03fd8b322be7d8e81f0420f02fe0a57d": "\n4\\pi\\varepsilon_0 V(\\mathbf{R}) \\equiv \\sum_{i=1}^N q_i v(\\mathbf{r}_i-\\mathbf{R})\n", "03fda4629b973f8ce23b7f20635dc7a7": "x_k = - \\frac{1}{3a}\\left(b\\ +\\ u_k C\\ +\\ \\frac{\\Delta_0}{u_kC}\\right)\\ , \\qquad k \\in \\{1,2, 3\\}", "03fe3cb0e67115aaaf2269c58319360d": "\\left \\{ \\sqrt[3]{x} : x \\mbox{ is constructible} \\right \\}", "03fe618e2fdb93cae336d2862b07a167": "h\\otimes v\\in V_h", "03feabf32b5ec498b9917df7f5cdb691": "c(V) = c_0(V) + c_1(V) + c_2(V) + \\cdots .", "03fec2e47d5c99405d591f252239312d": " \\left(\\frac{a}{-1}\\right) = \\begin{cases} -1 & \\mbox{if }a < 0, \\\\ 1 & \\mbox{if } a \\ge 0. \\end{cases} ", "03ff0f1aa2432df4f947e0570f58f967": "h(a)=h_0+\\sum_{i=1}^n h_ia_i\\,", "03ff61c1d4b3054b2fea1f017bf9a0f8": " \\psi^\\dagger \\sigma_j \\frac{\\partial\\psi}{\\partial t} + \\frac{\\partial\\psi^\\dagger}{\\partial t} \\sigma_j \\psi = \\frac{\\partial \\left( \\psi^\\dagger \\sigma_j \\psi\\right)}{\\partial t} ", "03ff64736f09ee66889b1e12aa6ab45a": "\n\\begin{align}\n\\frac{d E_{\\lambda}}{d\\lambda} &= \\frac{d}{d\\lambda}\\langle\\psi(\\lambda)|\\hat{H}_{\\lambda}|\\psi(\\lambda)\\rangle \\\\\n&=\\bigg\\langle\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg|\\hat{H}_{\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\hat{H}_{\\lambda}\\bigg|\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=E_{\\lambda}\\bigg\\langle\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle + E_{\\lambda}\\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\psi(\\lambda)}{d\\lambda}\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=E_{\\lambda}\\frac{d}{d\\lambda}\\bigg\\langle\\psi(\\lambda)\\bigg|\\psi(\\lambda)\\bigg\\rangle + \\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle \\\\\n&=\\bigg\\langle\\psi(\\lambda)\\bigg|\\frac{d\\hat{H}_{\\lambda}}{d\\lambda}\\bigg|\\psi(\\lambda)\\bigg\\rangle.\n\\end{align}\n", "03ff7f007de20f47914ea971fd576bb4": " f_1(x_1,\\ldots,x_n), \\ldots, f_k(x_1,\\ldots,x_n).", "03ff922d126da125152f09f9cbabcbd1": "\n\\begin{matrix}\nx + y &=& y + x\\\\\n(x+y)+z&=& x+(y+z)\\\\\nx+x&=&x\\\\\n(x+y)\\cdot z &=& (x\\cdot z) + (y\\cdot z)\\\\\n(x \\cdot y)\\cdot z &=& x \\cdot (y \\cdot z)\n\\end{matrix}\n", "03ffbe62d3a73b362ddbd6ad63e02e40": "t_A = t_B", "04000d383194855a059ae7cba74fa374": "(m)_n = m(m-1)(m-2) \\cdots (m-n+1).", "0400c0a41bae7e3c544019662d174a8c": "\\mathbb{Q}^+,\\cdot", "0400c798906749e6e1e973746d3f0d55": "\\phi ( \\bold{r} ) = \\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac { \\rho ( \\bold{ r}_0 )-\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 )} {| \\bold{ r}- \\bold{r}_0 | } d^3 \\bold{ r}_0 \\ , ", "04012bc7cc9262fca293ae5fe12e8f71": "(f_k)", "04015b657c225a30dfba8934d61e4b23": "\\chi_1(\\omega) = {1 \\over \\pi} \\mathcal{P}\\!\\!\\!\\int \\limits_{-\\infty}^\\infty {\\chi_2(\\omega') \\over \\omega' - \\omega}\\,d\\omega'", "0401a19094ed2243a12b57cb2d91899c": "F/k", "0401a41c89b868b700cb99bb29813b49": "\\lim_{N\\to\\infty}\\left(1+\\frac{r}{N}\\right)^{Nt}=e^{rt}", "0402308150bcd0917bfccf90bf221222": " R_{in} = \\frac {v_x} {i_x} = r_{\\pi} + (\\beta + 1) ({r_O} || {R_L}) ", "0402326c7b73ae0fc89190b47b957bf1": " \\gamma (t) = 4\\pi t + i\\cos(4\\pi t) 0 \\leq t \\leq 1", "04026d001c412a65713544da11c6caf6": "x \\wedge \\left( y \\vee z \\right)\n= \\left( x \\wedge y \\right) \\vee \\left( x \\wedge z \\right) ", "0402b088614dbd675146aa12c9226915": "x\\ne 1\\ ", "0402e0626e7835f8c4b12e5778648846": "\n\\nabla \\cdot \\mathbf F =\n\\frac{1}{H}\\frac{\\partial}{\\partial q^k} \\left(\\frac{H}{h_k} F_k\\right)\n", "0402e9bced3d440d72c3e362204a1255": "\\left(Ax\\right)_i", "040317e39ab6225b2f64a7b2c7012b4f": "2a_k \\ge a_{k+1} \\, \\forall \\, k \\ge 1", "040320f7a3acf4ea621f9cdab62dc440": "N=\\frac{g_0z}{1-z}+\\frac{f}{(\\hbar\\omega\\beta)^3}~\\textrm{Li}_3(z)", "04036a75e479ca9ff8489c2ae2510683": "= A_1 \\mathbf{e_1} (\\mathbf{e_2 e_3})^2 +A_2 \\mathbf{e_2} (\\mathbf{e_3 e_1})^2 +A_3 \\mathbf{e_3}(\\mathbf{e_1 e_2})^2 \\ ", "04037abd8428e254cae323da3f211bac": "VCA(64x^3-112x+56,(0,2)) \\cup VCA(64x^3+192x^2+80x+8,(2,4))", "0403f53cae7b1c1e3791cc34264bddba": " X_i(s)=x_0 + s\\sum_{j=1}^m a_{ij} f(X_j(s)),\\,\\,\\, x(s)=x_0 + s \\sum_{j=1}^m b_jf(X_j(s))", "0403f58796bae7a024a8a63dfc6cff48": "T_6( n^2 + n ) + T_5( n^2 + 3n ) + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \\le k( n^2 + n ) + k( n^2 + 3n ) + kn + 5k", "0404085b4df5835395033d5218ff0967": " \\pi_4 = L^q \\mu^r k^s \\beta^t g^u h", "040409df3b8501385ad3738fc2580981": "\\beta<\\alpha", "04047ee4aafa6ea65dbc529a47c97f69": "\\mathbf{3}\\otimes\\mathbf{3}\\otimes\\mathbf{3}=\\mathbf{10}_S\\oplus\\mathbf{8}_M\\oplus\\mathbf{8}_M\\oplus\\mathbf{1}_A", "0404ab2b2d5eae0e14317530984cd375": "\\beta(g) \\propto g^\\alpha", "0404d3f8a99190f20fca883f8fca0385": "\\mathbf{u}_k=\\left[u_0, u_1, \\dots, u_{k-1} \\right],", "040515eac86f681bafb3b7c9852a4d58": "\n\\bar{\\mu}_{\\text{min}} = \\lambda_{\\text{min}}\\left( \\frac{1}{n} \\sum_{k=1}^n \\mathbb{E}\\, \\mathbf{X}_k \\right) \\quad \\text{and} \\quad\n\\bar{\\mu}_{\\text{max}} = \\lambda_{\\text{max}}\\left( \\frac{1}{n} \\sum_{k=1}^n \\mathbb{E}\\, \\mathbf{X}_k \\right).\n", "040548e2562d68d8aba49c12072fbbff": "v_3(t) = \\int_{t_0}^{t} (K_1i_1(\\tau)+K_2i_2(\\tau)) d\\tau.", "0405c9f9d2147a9e6088cbc4a30a8707": "B=B(b, \\lambda)", "0406546e4269ae5098ed91c8999bfa5e": "\\{x \\in V \\colon x = a + n, n \\in W\\}", "0406baf1245fc32c7fcf9e6f50931e91": "\nG = G[\\tilde{S}(\\omega)] = \\int_{-\\infty}^\\infty \\eta(\\omega) \\tilde{S}(\\omega) \\, d\\omega \n", "0406fb29ebe211df5b5b03aeec27b35d": "Re = Re_c", "0407aa1318c41683bf20fd50ff5172e1": " 0.082 H_s^2 ", "0407c900cd036ac4b5e1a43acc9cab35": "\\delta^{(k)}[\\varphi] = (-1)^k \\varphi^{(k)}(0).", "0407f208210c245681a6f4ba985097f2": "y_c = {2 \\over 3} E_{lake} \\,\\!", "040892129b35344eedc8972773e4c4f4": "-(-h)", "0408a2aa720367d75c62a7526d968221": "N \\leq \\frac{Br}{r+1}", "0408e3851c8cdd649c5cee4d7cd7a0c5": "\\alpha \\in[0,1]", "04093a271f00d21635b22f616853e6d3": " A+0=A", "040947dcf5fdde48307e915d313b0839": "\\pi^{-1}\\mathcal{I}\\cdot\\mathcal{O}_{\\operatorname{Bl}_\\mathcal{I} X}", "04099cf0c261f27bc39c95cba442e0c0": " \\forall x \\in A, \\ \\exists y \\in 2^B, \\ x \\in y ", "0409bae34a658d6a4b0c560f9aafb3ac": "\n\\{a_{11}, a_{12}, a_{13}, a_{22}, a_{23}, a_{33}\\}\n", "040a0fe1af61c8a25e80b82326132bc1": "\\left\\{\n\\bar{Z}_{1},\\ldots,\\bar{Z}_{s+c},\\bar{X}_{s+1},\\ldots,\\bar{X}_{s+c}\\right\\} ", "040a906dae13f008ae8164b64adc2eec": "\\frac{3b}{4}", "040ad4c564a4398f895a2bfa60d1e23e": "\n\\begin{align}\n2 \\int \\sec^3 x \\, dx &{}= \\sec x \\tan x + \\int \\sec x\\,dx \\\\\n&{}= \\sec x \\tan x + \\ln|\\sec x + \\tan x| + C.\n\\end{align}\n", "040adb5020d648afa0b1fae88ab194d6": "\\bar{R}^ 2", "040addb211b23f8dec306ce628709283": "\\varepsilon_1''' = -\\frac{\\nu}{E}\\sigma_3", "040aea4c66b6894f163c22953d213a86": "A \\cap B = \\overline{\\overline{A} \\cup \\overline{B}} ", "040b6c2244036a6c9bc837b62ea230b5": "x \\in \\mathbb{R}^L_+ \\ .", "040c0704d1d15a4b3fba31918f2a21b7": "D(\\alpha)", "040c11cf6b8898bb86eaf8d66253d425": "e^\\cdots", "040c19fd5867974a7cea1e053feb6984": "\\{ u', u \\} \\in E \\setminus M", "040c39e51f49c201f5780618028af2ac": " {DB} \\equiv \\frac{1}{N}\\displaystyle\\sum_{i=1}^N D_i", "040c3bbfc6598ecb26e80f76230f92b1": "\\epsilon^1: \\quad 2S_0'S_1' + S_0'' = 0.", "040c456e46507d5bcb155bfcc94d261a": " I_{KAR} = (\\frac {2 Z^2}{n^2 F r})^n ", "040cabec1114ed4c6f505e979e430e5d": " a + (180", "040cdd5b0489fa26d9225262e0eb498c": "P_n = {\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'", "040cf9b47973c6fc123715d3e59a55da": "\\frac{1}{G_\\mathrm{total}} = \\frac{1}{G_1} + \\frac{1}{G_2} + \\cdots + \\frac{1}{G_n}", "040d2d4d9d9a6775698afb13b0929807": "\\Delta \\lambda", "040d391cdb42c491cc9e569cb39f6860": "{\\mathcal C}_n(z) = \\frac{1}{2 \\pi i} \\oint_C \\frac{\\exp(z+z/t)}{t^{n+1}}\\, dt = \\frac{1}{2 \\pi}\\int_0^{2 \\pi} \\exp(z(1+\\exp(-i\\theta))-ni\\theta))\\,d\\theta.", "040d65a49095e3ca05abbfe6aea6bc68": "\n N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 ~;~~\n M_{\\alpha\\beta} := \\int_{-h}^h x_3~\\sigma_{\\alpha\\beta}~dx_3~.\n", "040d891bf42b3af1a37a77b06fdf60b9": "\\oint \\mathbf{B} \\cdot d\\boldsymbol{\\ell} = \\mu_0 I_{\\mathrm{enc}},", "040e3118a4a6e49bffe502dd69465b8e": "\\ v_i = \\sqrt {2gd}\\ ", "040e60d5d63c56e5c5c0203a79d41b50": " I=(a,b)", "040e7a524dcfb640f0ad6571cb348051": "v_{\\text{in}}", "040ebb3e39938fa7bdf7d1275aabb189": "M(E)", "040ef16ee427a4f5b8955fe1d0653ce8": "QE_\\lambda=\\eta =\\frac{N_e}{N_\\nu}", "040f4e6aad36a049d12ca18e6df07c24": "\\tanh", "040f8b1063d9fe4ac7f5d765a4f561a7": "\\hat{C} = \\sum_{i=1}^r c_i\\bar{Y}_i", "040f915801fa8603100ca166fbcec507": " U(\\mathfrak{g})/I", "040ff8a72b1f900e7b36fee6bc0cf2ed": "\nE[\\Delta(t)] \\leq B - \\epsilon \\sum_{i=1}^N E[Q_i(t)]\n", "04103810029df237b1be42a58f7fda1b": "2 \\uparrow \\uparrow \\uparrow 4 = 2 \\uparrow \\uparrow 2\\uparrow \\uparrow 2 \\uparrow \\uparrow 2=2 \\uparrow \\uparrow 2 \\uparrow \\uparrow 2 \\uparrow 2=2\\uparrow \\uparrow 2 \\uparrow \\uparrow 4=2 \\uparrow \\uparrow 2 \\uparrow 2 \\uparrow 2 \\uparrow 2 = 2 \\uparrow \\uparrow 65536", "04104fe57542b5399441f651a80081c4": " = -I I' ds ds' \\left[cos(xds)cos(rds)+cos(rds)cos(xds')\\right] ", "041061f5b7aa1fa7a7a0725b9bb244a3": "1 p_{j}\\\\\n \\end{array}\n \\right.\n\\end{array}\n", "042223f2344fc81a7c09aa69b55a73cf": "X = g_{1}^{x_1}g_{2}^{x_2}", "042306651af18bcacca1f43ab885ce08": "(\\mathbf{D_1} - \\mathbf{D_2})\\cdot \\hat{\\mathbf{n}} = D_{1,\\perp} - D_{2,\\perp} = \\sigma_\\text{f} ", "042311da4bf0cfeb58499992324c9656": "\\frac{Y(z)}{z}", "0423372acc78e5e1965fadc7052d2e63": "E(-)\\,", "04235fbcb43527845cca755f3c862950": "j = H,T", "0423631118dc235bc1c532da9069e111": "f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \\cdots \\,", "04236b0dbc6277364b244d7deb26a24c": "t \\in S", "0423a27c892d4b106a01e930565cfe7e": " A = QR\\,\\!", "0423a45525cec11e3fc7df3731d804e4": " P_{\\rm fwd}, \\, P_{\\rm bwd}", "0423c9cf2fc5bae11fe3c51366abf6cf": "\\scriptstyle S ", "0423e9f4497d84a49a61aad4d9a28793": "\\Delta := \\min \\{c(i,j)-y(i)-y(j): i \\in Z \\cap S, j \\in T \\setminus Z\\}", "04244cd38e478f660ecaab328a1b0191": "|\\{(x,y)\\;:\\; \\operatorname{lcm}(x,y)=D\\}|= 3^{\\omega(D)},\\;", "0424739beee9f4d56c88daa503a7daaf": " \\left( T(n) \\right)_{n = 1}^{\\infty} ", "0424c8a3c1bc4e3b7d8d0ff7d0f61a85": "\\Delta g_{i,\\mathrm{mix}}=RT\\ln x_i", "0424d05bf07a4693eeff7999232c683f": "\\delta W = -mg\\delta y = -mgL\\sin\\theta\\delta\\theta.", "04250f98f961b75fab11084a07494a65": " \\sum_{k=0}^\\infty a_k z^k = A(z) < \\infty \\quad \\Rightarrow \\quad {\\textstyle \\sum} a_kz^k = A(z) \\,\\, (\\boldsymbol{B},\\,\\boldsymbol{wB}). ", "0425a405b5515fb35e3cffb968a7883b": "B \\supseteq \\{c\\}", "0425a6596203e91bbf992827d5b4f628": "\\mathbf v = v_1 \\mathbf e_1 + v_2 \\mathbf e_2 + v_3 \\mathbf e_3", "0425ec80bf7831d3ae52f578c64e1ae2": "\\ \\gamma \\, ", "04262cba3e5105195da110567fadb84a": "f^{-1}\\mathcal{G}", "0426798c7976774172f3b693c5f04192": " \\frac{\\mathrm{d}}{\\mathrm{d} x} \\int_{\\Omega} \\, f(x, \\omega) \\mathrm{d} \\omega = \\int_{\\Omega} \\, f_x ( x, \\omega) \\mathrm{d} \\omega ", "0426819fccb67b54198a009965df4775": "s_{ln} \\,", "04272fe09e6c1a08802e4b3cf35b7411": "10\\uparrow\\uparrow 10\\uparrow\\uparrow (10\\uparrow)^{497}(9.73\\times 10^{32})=(10\\uparrow\\uparrow)^{2} (10\\uparrow)^{497}(9.73\\times 10^{32})", "04274f736adbd0c9342ce19544b22c48": "\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix} = \\begin{bmatrix} 3.1956 & 2.4478 & -0.1434 \\\\ -2.5455 & 7.0492 & 0.9963 \\\\ 0.0000 & 0.0000 & 1.0000 \\end{bmatrix} \\begin{bmatrix} X \\\\ Y \\\\ Z \\end{bmatrix}", "042799d05b97293e7376791b08298fc4": "f_1 \\Leftrightarrow f", "0428292809fdc49a2fa94bb50d7afab4": "\\Pi_\\beta\\, ", "04282b9625be9da1a5f988133a7f400f": "\\int P\\left(A,\\tilde{A}\\right)dAd\\tilde{A} = N\\int exp\\left(L\\left(A,\\tilde{A}\\right)\\right)dAd\\tilde{A},", "042833ea03a8a157fa009a9183156145": "N\\Delta F", "04284904414567d9d27199ed98b105d9": "V(\\rho,\\varphi,z)=\\sum_n \\int dk\\,\\, A_n(k) P_n(k,\\rho) \\Phi_n(\\varphi) Z(k,z)\\,", "04286d274644a21dfaa0c7eb4dd2b3ed": "\\gamma \\in \\mathbb R", "04289e638f16b4cb648cef93380133f1": "\\Delta v \\ll v_\\text{e}", "0428ff8815ad7c4958f8ccb8fa0451ea": "F'(R:BL < Q < BU:AL < P < AU)=\\sum_{T\\!B=1}^{U\\!B=\\infty}\\left(\\sum_{T\\!A=1}^{U\\!A=\\infty}\\frac{F'(R:Q_{(tb)}:P_{(ta)})}{U\\!A}\\right)\\frac{1}{U\\!B}.\\,\\!", "042917fde562bb8fa3e30d31f8a8a6e6": " | /2 ", "042924d24de5e4ace957814fc8e65a87": "A(x, \\vec{y}", "04295addffc8ca7a550c45469b49a0f8": "(\\pm P_i,\\pm P_i)", "04298604d0fbd5d2a4f1402f79f7dacc": "\\delta_{ij}=0", "042986dd20834643d3315851ee8ef56b": "H'=1/M\\Delta\\tau", "04298c6a5f765a2ae1bcb0b60873cd6d": " R = \\frac {4 n \\rho V D \\left( 1 - a X - b X^2 - cX^3 \\right)} {\\mu_{Wall} \\left(3 n + 1 \\right) } ", "0429a04217d4df18daa15804db0d5d25": "R_i \\leftrightarrow R_j", "0429a3a73913d53ed1fa712b8687ad85": "\nw_{jj}\\sim\\sigma_{jj}\\chi^2_m", "0429f41635a9ddf84e37da9ed6d4a610": "dS_{l,t} = \\theta (S_{t}-S_{l,t})dt + \\sigma S_{l,t} \\left(\\rho dW_t + \\sqrt{1-\\rho^2}dW_t^\\bot\\right)", "0429f4e3edd6ac67209d78a7dd3d922a": "\\left(\\Lambda^k(V)\\right)\\wedge\\left(\\Lambda^p(V)\\right)\\sub \\Lambda^{k+p}(V).", "042a038f750e6fa4c18ce6a96253daaf": "I_x \\xrightarrow{\\omega \\tau I_z} \\cos (\\omega \\tau)I_x - \\sin (\\omega \\tau)I_y ", "042a8292539f0499c2cdf28a1aad6163": " Ehr_P(z) = \\frac{\\sum_{j=0}^d h_j^\\ast z^j}{(1 - z)^{n + 1}}, ", "042a8a01b244f575db31a30e7b5e388d": "M(x)=e^{\\int \\frac{-2}{x}\\,dx} = e^{-2 \\ln x} = {(e^{\\ln x})}^{-2} = x^{-2} ", "042ab9d4e16857e079311cb3883c1e59": " \\zeta_Q(s) = \\sum_{(m,n)\\ne (0,0)} {1\\over Q(m,n)^s}.\\ ", "042acfc4e101f725e2ce84bb5b669702": "\\Delta_B =\\frac{2\\textrm{arctanh}\\sqrt{(1-\\omega^{2})(1-v_\\text{K}^2)}}{\\sqrt{1-\\omega^{2}}}", "042afd40606d18eb257b24816c76e3ad": "\\Sigma_{a \\in H} W(a) \\leq k \\cdot \\Sigma_{j \\in S} p_j", "042b0dcf8f6e22d59f523af8081a2c29": "u : E \\to F", "042b362b178d92bb939071045c77d8c3": "f(x) \\ge f(x+y) - f(y)", "042b4d725d323ea7cfa7c77ee4f036c6": "\\langle \\operatorname{exp}(aX)\\rangle \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{i=0}^\\infty\\frac{a^i}{i!} X^{'i}", "042b7d5dd05ae9c5d3171737bfa67079": " L_c^-(f) = \\left\\{ (x_1, \\cdots, x_n) \\, \\mid \\, f(x_1, \\cdots, x_n) \\leq c \\right\\} ", "042c3f68e3584df2fe195249113afa9c": "UIRP: {\\Delta}E_t(S_{t + k}) = E_t(S_{t + k}) - S_t = i_\\$ - i_c", "042cb15498fff8dd0cd98aba671b0e0c": " \\tfrac{256}{81} ", "042d04c8dfc93ba83eef36b401af7875": " \\overset{d}{=}", "042d7fe5a77bd8e1275f7167c5b284e4": "M _{CB} ^f = \\frac{qL^2}{12} = \\frac{1 \\times 10^2}{12} = 8.333 \\mathrm{\\,kN \\,m}", "042dd3fd944d5f0a44723261e66bd217": "\\chi(x,X)\\triangleq\\,", "042e6c2d87f56c5d1217c6f0a2db6b06": "x \\, = \\, \\zeta_{L}\\exp \\left( \\frac {\\mu}{k_{B}T} \\right)", "042e85197d4730fb77d56abfe20074c3": "V,f", "042efe365991582b13e0ece6de5b3a44": " e_{ij} = \\begin{cases} a_i, & \\mbox{if } i \\in [1,N] \\\\ b_{ij}, & \\mbox{if }i \\in [N+1,2N] \\end{cases} ", "042f143e616aa57ea4785ce3899aeb42": "L = k\\;S\\;V^2\\;C_L", "042f14746f4d46a704fa744eaf24c57f": "\\Phi(x) =\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x e^\\tfrac{-t^2}{2}\\,\\mathrm dt = \\frac{1}{2}\\left[1+\\operatorname{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)\\right]=\\frac{1}{2}\\,\\operatorname{erfc}\\left(-\\frac{x}{\\sqrt{2}}\\right)", "042f162c686ff22bb8e0c1c671b60b78": "H_2(z)", "042f26c0a172bfee32ccf9f4c212ad6a": "\nv = B \\sin ax \\cos by \\sin cz,\n", "042f2bf0ad28bf02656cedf3586e2125": "\\Rightarrow 0=\\frac{x^2}{2}+\\frac{x}{2}-F r_1^2\\qquad a=1/2,\\;b=1/2,\\;c=F r_1^2", "042f4855a013920daafd24d2b58267f4": "\n\\mathcal{N} \\int e^{-\\beta H(p, q)} d\\Gamma = 1,\n", "042f9220e89cca453bf5b4997f7392a3": "sig=(sig'||auth_0||auth_1||...||auth_{n-1})", "042fa1acd04ab993476705b196a351a6": "{\\Lambda^\\mu}_\\nu", "043007c6e9c962c3726d72e9b02baa57": "180^\\circ", "043013e71f91933c14ea6dd85fed8c5a": "m_{k}", "0430197ca6f8a78c3702a93ac5086129": " \\rho(a, c) -\\rho(b, a) - \\rho(b, c) \\le 1,", "043067e7c691cff1f7f7ef1221121ff0": "\n w(r) = -\\frac{q}{64 D} (a^2 -r^2)^2 \\quad \\text{and} \\quad \n \\phi(r) = \\frac{qr}{16 D}(a^2-r^2) \\,.\n", "04306fafcc30958126e5d8faf2bce637": "\\eta=(| n_\\Phi-n_{\\overline\\Phi}|)/ (n_\\Phi+n_{\\overline\\Phi})", "043099855a1d6c61e5614168914a5ca1": " h \\nu = \\Delta E ", "0430c7501f1c2d1615938f64fae168c1": "\\mathrm{(A/m^2)}", "0430fbb3b8721c34af98118178317474": "\\sigma(\\hat x) = \\{ \\chi(x) : \\chi \\in \\Delta(A) \\}", "04310407c8322c3c57b1a7224d2bdb8c": "\\begin{align}\n\\partial_1 c\n&= \\partial_1(t_1 + t_2 + t_3)\\\\\n&= \\partial_1(t_1) + \\partial_1(t_2) + \\partial_1(t_3)\\\\\n&= \\partial_1([v_1, v_2]) + \\partial_1([v_2, v_3]) + \\partial_1([v_3, v_4]) \\\\\n&= ([v_2]-[v_1]) + ([v_3]-[v_2]) + ([v_4]-[v_3]) \\\\\n&= [v_4]-[v_1].\n\\end{align}\n", "0431155a0b5d633b35f5b34cd4191f8d": "\nw(r) = \\int \\frac{\\mathrm{d}r}{\\sqrt{\\frac{r}{r_{s}}-1}} = 2 r_{s} \\sqrt{\\frac{r}{r_{s}}- 1} + \\mbox{constant}\n", "0431b0eef983a554d7d7afa238d5895d": " T_{ij}^{(3)}=s_{ik}s_{kj}-s_{mk}s_{km}\\frac{1}{3} \\delta_{ij} ", "0431bb4100c94e9cff839832e9afbf02": "(cdt)^2=dx^2+dy^2+dz^2+(cd\\tau)^2", "0431c3a638a1ed25d525799cf534562c": "\\theta_i = \\frac{x_i}{x_{max}} 2 \\pi ", "0431de28c694cf6c6b13a4b3a20ccf37": "\\begin{align}\np(\\mu|\\mathbf{X}) &\\propto p(\\mathbf{X}|\\mu) p(\\mu) \\\\\n& = \\left(\\frac{\\tau}{2\\pi}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{1}{2}\\tau \\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right)\\right] \\sqrt{\\frac{\\tau_0}{2\\pi}} \\exp\\left(-\\frac{1}{2}\\tau_0(\\mu-\\mu_0)^2\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2}\\left(\\tau\\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right) + \\tau_0(\\mu-\\mu_0)^2\\right)\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2} \\left(n\\tau(\\bar{x}-\\mu)^2 + \\tau_0(\\mu-\\mu_0)^2 \\right)\\right) \\\\\n&= \\exp\\left(-\\frac{1}{2}(n\\tau + \\tau_0)\\left(\\mu - \\dfrac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}\\right)^2 + \\frac{n\\tau\\tau_0}{n\\tau+\\tau_0}(\\bar{x} - \\mu_0)^2\\right) \\\\\n&\\propto \\exp\\left(-\\frac{1}{2}(n\\tau + \\tau_0)\\left(\\mu - \\dfrac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}\\right)^2\\right)\n\\end{align}", "0431ea012994d9769b8b4f3c65c2b705": "\\tfrac{4}{256}", "0432161c17ddb5c40865fbf0e8330d4f": " S \\approx \\frac{ \\beta }{ 3 \\alpha + 0.2 } ", "04322155a0969e53dd2af27801c6f824": " \\lbrace \\bold{x}^{ \\left( k \\right) } \\rbrace _{k = 0}^{\\infty} ", "04324364800ac799e323e692305adc0c": "\\{e^{-a_0t}\\}", "043266aabb099e64fc8c537494a7aca8": "z=c \\sinh v", "0432bd149aa64859b8197adf82771c1e": "T_h", "04337624b6e95c7568e5c14798f4015d": "A, \\ ", "043398bf57dc97792d6b2447ed511c2c": " (y_1, y_2) ", "04339e3757686e989e74a5b743cf957a": "F=m(\\ddot{r}-r\\dot{\\theta }^2)=-m\\left(h^{2}u^{2}\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+h^{2}u^{3}\\right)=-mh^{2}u^{2}\\left(\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u\\right)", "0433a0b4cdc6725c6ef740321fe194b1": " \\frac{di}{dt} + \\frac{di}{da} = \\delta(a) \\lambda ( S + \\sigma R ) - \\gamma(a) i - \\mu(a) i - m_i(a) i, ", "04345179e5470e70f5f2e79e5a741a11": "n^2-n > 2\\times365\\ln 2 \\,\\! .", "0434626709a496ba269fb912232668c5": "\\# X (\\mathbf{F}_q) = q^{\\operatorname{dim}X} \\sum_{i \\ge 0} (-1)^i \\operatorname{tr}(f; H^i(X(\\mathbf{F}_q), \\mathbb{Q}_l)),", "0434ad71ea4dbab5a8fabc778c60ce13": "\\Delta t' = \\frac{\\Delta t}{\\sqrt{1-\\frac{v^2}{c^2}}}", "0434bb534c7fc17969197e1f0600f80d": "P^{a}P^{b} = \\sum_i (-1)^{a+i}{(p-1)(b-i)-1 \\choose a-pi} P^{a+b-i}P^i", "0434f618df958d57676fc580c89c7c54": "a\\, \\frac{\\sinh\\, \\bigl( k\\, (z+h) \\bigr)}{\\sinh\\, (k\\, h)}\\, \\cos\\, \\theta\\,", "04352aa9441c869cf4ba1cc540a1b71a": "\\sum_m (-1)^{j-m}\n\\begin{pmatrix}\n j & j & J\\\\\n m & -m & 0\n\\end{pmatrix} = \\sqrt{2j+1}~ \\delta_{J0}\n", "0435531cb50ee00cd4e3167505c10d6e": "\\dot{\\mathbf{x}},\\dots,\\mathbf{x}^{(n)}", "04358bea9d231b9a487564055a5ab70e": "\\gamma(k,i)\\,\\!", "0435d2bf45e2d8102d75cbcfd5f25301": " \\mathbf{s} = \\mathbf{B} \\mathbf{d}. ", "043615f47e6c4c88de50220114b1a304": "\\, K = 2n\\pi/a", "0436c6c8b55041676fb391e7ee0214ae": "r_1,r_2,\\cdots,r_a", "043747fa54887321886921e4ceef8ba3": "R_{ix}(t)=M_{i}A_{ix}(t)\\frac{}{}", "0437674141b352e9e6a80b329e9dfa93": "A_\\lambda>0", "0437d63f527b355a2f93abafb5739d1b": "i.", "04384da8fde85931f668ea7ab2435340": "HA_i", "04388a49ab38977d0ec391e4c0510877": " \\lim_{k \\to \\infty} \\| \\bold T^k \\| = 0, ", "04389dc3e787e23ef2e5982982017cfc": "T = 50 + 10 {x- \\mu \\over \\sigma}", "0438a0e8326b4167818569ed6f179378": "f : V^k \\to K\\ ", "0439479af4192d42884cc58105facddf": "\\sum_{j=1}^{n_S} \\sum_{b_j=0}^{a_j} \\sum_{ \\beta_j } x_{b_j} \\ b_j = \\sum_{h=1}^{n_P} \\sum_{ d_h=0 }^{c_h} \\sum_{ \\gamma_h } u_{\\gamma_h} \\ y_{d_h} \\ d_{h}. ", "04397443b09ae04010032ff6bbcce1c5": "f \\mapsto \\mathbb{P}_n f", "04399fe68406275419e18c0e85eab335": "\\frac{a}{x}=o(S_0')\\,", "043a0a32bc14f031f8299bcd330a0e9b": "\\hat{f}(t) = f(t) \\, ", "043a0b9537ea34a66dd44536ef1635cf": "{{\\mathbf{k}}}[\\mathbf{x}]", "043a1613656191ec43c873898661e76e": "\\mathbb{E} \\log(S_t)=\\log(S_0)+(\\mu-\\sigma^2/2)t", "043a46836b4b629ac65945ceda7d90d4": "Y'=YM_i.", "043a49f81e88957db2da952cc274bca9": "f = {1 \\over 2 \\pi \\sqrt {LC}}", "043a4ba8841199b14d188dc969115fdb": "BA = \\frac{\\pi \\times (DBH/2)^2}{144}", "043a93f86a9f805fabace17b1c6aff92": "(X^*_{b}, Y^*_{b}; Z)", "043ada99412e7bc11f2cd700d32c0917": "\\rho\\mu = 1_Y", "043b27c3e4f7d051bb8ef7131fcbc79e": "F^\\dagger", "043b3b9bfb851fabf350c5784ec38c2f": "r e^{a j}, - r e^ {a j}, r j e^{a j}, - r j e^{a j}, \\quad r > 0", "043b43b22560464bcd85b27ca7e9bffb": "x_p^2\\equiv\\frac{2\\xi^2\\sqrt{G}}{\\sqrt{8\\xi^2(\\xi^2\\!+\\!1)+12G\\xi^2-G^3}-\\sqrt{G^3}}", "043b8526035e9453eaf9471988c9bb5c": " R_{01} = \\frac {W_{cu}} {3{I_{S}}^{2}}", "043ba4b2180cec84c17497942ebfad63": "C_\\max, L_\\max, E_\\max, T_\\max, \\sum C_i, \\sum L_i, \\sum E_i, \\sum T_i", "043bdd448fd560f75d1648edf7a1a4b1": " dN_i = \\sum_k \\nu_{ik} d\\xi_k. \\,", "043c10c6bba91fd5ba82f14b1aea724f": "\\min_{\\alpha,\\,\\beta}Q(\\alpha,\\beta)", "043c183552a5d1a083989d2e2c340959": " \\scriptstyle C_c^1(\\Omega,\\mathbb{R}^n)", "043c1ee76d36717817e06c05c9e1087e": "e(S)", "043c5dd74964ad33ceff323d809cdc8b": "\\frac{d\\alpha}{dt}=q+\\frac{Z}{mU}", "043c6bc108326a3fb8ac1410de54d183": "O(\\sqrt{V})", "043c6f9f45b13e3a1395d5a31c341bad": " \\mathcal{L} \\, = \\, \\mathcal{L}_{\\mathrm{field}} + \\mathcal{L}_{\\mathrm{int}} = - \\frac{1}{4 \\mu_0} F^{\\alpha \\beta} F_{\\alpha \\beta} - A_{\\alpha} J^{\\alpha} \\,.", "043c772f6e1bdcee98556418393b3ad3": "\\displaystyle{\\mathfrak{g}=\\oplus_{i=1}^N \\mathfrak{g}_i,}", "043cb308a9d20de572bd4c1e19cc7699": "S^{n-1} \\to G", "043cedd5ff1ac6df55cef007bad07ce7": "\\varphi = \\frac11+\\frac12+\\frac19+\\frac1{145}+\\frac1{37986}+\\cdots", "043d4aa08c6b9f69d15db48f9992471a": "w\\,R\\,u\\land w\\,R\\,v\\Rightarrow u\\,R\\,v", "043d75ee748f4359e858a79b5c6a705a": "\\cos \\theta = \\frac{e^{i\\theta} + e^{-i\\theta}}{2} \\,", "043d986c307ab908b2420d8c88cad08f": "\\textstyle [x] = [\\mathbf{v}_1, \\ldots, \\mathbf{v}_m]", "043dd1f3ff7f0961963cf74d666128f5": " a_{i \\pm \\frac{1}{2}} \\ ", "043dfa0dce4e1eaa000c4ab46ff93863": "\nL_x=\n\\begin{pmatrix}\n0& 0& 0\\\\\n0& 0& -1\\\\\n0& 1& 0\n\\end{pmatrix},\nL_y=\n\\begin{pmatrix}\n0& 0& 1\\\\\n0& 0& 0\\\\\n-1& 0& 0\n\\end{pmatrix},\nL_z=\n\\begin{pmatrix}\n0& -1& 0\\\\\n1& 0& 0\\\\\n0& 0& 0\n\\end{pmatrix}.\n", "043dfb1219b6cab3960f60c45853999d": " U_\\theta =\\begin{bmatrix} e^{i \\theta} & 0 \\\\ 0 & 1 \\end{bmatrix}, ", "043dfb9dbd2157ad42c6fe313393ef24": "mv^2", "043e0e32ada017cc478a144049396d2c": " n_{\\nu_j}", "043e63402705f9ad750f879c5e552c00": "x = x_A", "043e9e440597410021257b5f9afa39d2": "|A|=|A\\times A|", "043eb6de5f4b5487d88efaa861518ec4": "f\\left(E\\right)", "043f027b254081d100d667508ddcd4b6": "\\kappa_b(k,i)\\,\\!", "043f334c7f494be53a0fd5e6e0af9bca": "ogd", "043f7fd770592fb93fc45041bfd6ba33": "f(n)=O\\left(n^n\\right)", "043fc37a324e096884d731e132cbab12": "\\frac{f(n)}{n^{\\log_b a}} = \\frac{\\frac{n}{\\log n}}{n^{log_2 2}} = \\frac{n}{n \\log n} = \\frac{1}{\\log n}", "04400f3aa13f4ee2969b1ee5599e8570": "I:f^\\infty= \\{g\\in R| (\\exists k\\in \\mathbb N) f^ng\\in I\\}", "04404eb69c936453785be20232e1d157": "N_s={120\\times{50}\\over{6}}", "04408afa08486fceff20014e0af5c106": "k=0,1,2,... ", "0440cfa6cdd7d42bd092724ef8503f2c": "m_{em}=E_{em}/c^2", "0440f47714e39f5332168d41b2abdc51": "C=\\frac{\\pi }{24}=0.131", "0440f7ae83b9eb81c046f1fb8da9960e": "{\\rm 1~Rayl = 1~\\frac{dyn \\cdot s}{cm^3}}", "04415ad3f122fd85386427796ab790c3": "{M_{2^\\infty}(\\mathbb{C})}", "0441a46f2b935dfc0b70c5760e6755a2": "z\\mapsto\\pm z", "0441cee755e1999da93fa506f511b548": "\\pi_{n-1}(Ff)", "04422f2574ca15e544d9de9538e45e3e": "\\lim_{n \\to \\infty} |{\\frac{a_{n+1}}{a_n}}| = r.", "044288e8ffc49792d28cd86657921099": " \\mathbf{B} = \\boldsymbol{\\nabla} \\times \\mathbf{A}.", "0443c0ef188080b775ff95a0103cf0d0": "L_x(x, y)=-1/2\\cdot L(x-1, y) + 0 \\cdot L(x, y) + 1/2 \\cdot L(x+1, y)\\,", "044408da84990ea6593e36887b3579c4": "f(x)=2x\\,", "044425ed6cbeb742f7e87d152a4edf4f": " g(X,\\theta) ", "04447f59c1bf03ba62ca2bbed7933c06": "K/2,", "04459080d01e3b6016b4a1f5c038ed0f": "C_n=2^{n\\choose 2} - \\frac{1}{n}\\sum_{k=1}^{n-1} k{n\\choose k} 2^{n-k\\choose 2} C_k.", "04459395e8049f50c0de25d4afa6dec3": "(\\sqrt[5]{100})^{5--1.47}\\approx 387", "0445b220b0fa7b1df97191bf5c256d76": "\\mathbf{B}(t)=(1-t)^3\\mathbf{P}_0+3(1-t)^2t\\mathbf{P}_1+3(1-t)t^2\\mathbf{P}_2+t^3\\mathbf{P}_3 \\mbox{ , } t \\in [0,1].", "0445d20b06e8f0006930d71db82bed73": "\\left(\\frac{C}{h}\\right) = {2 \\pi \\epsilon \\over \\ln(D/d)}= {2 \\pi \\epsilon_0 \\epsilon_r \\over \\ln(D/d)}", "0445eb70e4d914478364efbaacf737c7": "c_V", "04460a1d550c894c5fed25ac9ca64815": "\\int_{\\mathbb{R}^{n}} f(x) \\, \\mathrm{d} x < + \\infty.", "0446322a5f408e8fd1f22d8f5700ecd4": "\\operatorname{relint}(S)", "0446751577e6b290779b469d5dbe7331": "v^2=Q(v)", "04468d1922634dfcdc37a7c70f64af9e": "C_n^{(\\alpha)}(x) = \\frac{(-2)^n}{n!}\\frac{\\Gamma(n+\\alpha)\\Gamma(n+2\\alpha)}{\\Gamma(\\alpha)\\Gamma(2n+2\\alpha)}(1-x^2)^{-\\alpha+1/2}\\frac{d^n}{dx^n}\\left[(1-x^2)^{n+\\alpha-1/2}\\right].", "04469c5ecd43e9b9dfd4fc24d43dde7d": "f_{\\alpha} = F_{\\alpha\\beta}J^{\\beta} .\\!", "0446d05a6479f6c947639821b6c5f13a": " \\sum_{i=1}^d S_i + \\sum_{iB)=A_3 \\cdot \\overline{B}_3+x_3 A_2 \\overline{B}_2+x_3 x_2 A_1 \\overline{B}_1+x_3x_2x_1 A_0 \\overline{B}_0", "046ca8782936938ebb7b5935d7d0c664": "f\\in C^\\alpha(\\Omega)", "046cb06e29f4c1e90331985640ad776a": " \\iint_D \\ f(x,y ) \\ dx \\,dy , ", "046cfdd94af44ab54b498ffcbd636e5b": " \\epsilon_0 = E_0 - m_0 c^2 ", "046d5aa2546f969b1fb0ece5691050d1": "\\textstyle n \\le 2^{r-b+1} -1, ", "046d857e166f77713c3c68ecdbdb9a34": "\\{ \\, (1, 111)\\}", "046d9e9007d432a078332c178710a516": "\\beth_{k+1} = 2^{\\beth_k}", "046db2abf4d0adf4240409c783152fcb": "A \\rightarrow \\varepsilon", "046e9ae403c12efe619ba669e1955a2f": "\\scriptstyle\\bar\\eta", "046ea3ef22af403d11c828ec72d711a0": "m+S", "046ebb1b48895f3d72525897d595788c": "L(p;q_1)", "046ec622fe5bd72e7deacff1d2482bf4": "\n\\begin{align}\n\\varepsilon_0 & \\sim \\operatorname{EV}_1(0,1) \\\\\n\\varepsilon_1 & \\sim \\operatorname{EV}_1(0,1)\n\\end{align}\n", "046f886b34977dca56c25e836e34862e": "X=(X_1,\\ldots,X_n)", "046fb317cb80756569c408df6d76c37e": "\\int\\frac{\\sin^n ax\\;\\mathrm{d}x}{\\cos^m ax} = \\frac{\\sin^{n+1} ax}{a(m-1)\\cos^{m-1} ax}-\\frac{n-m+2}{m-1}\\int\\frac{\\sin^n ax\\;\\mathrm{d}x}{\\cos^{m-2} ax} \\qquad\\mbox{(for }m\\neq 1\\mbox{)}\\,\\!", "046fce63a6a4f7895b14e73e2f1fac79": "A+uv^T=A\\left( I+wv^T \\right)", "047018d7a66d0aefa7616a72267b0557": "\nm(\\varphi) = B_0\\varphi + B_2\\sin 2\\varphi + B_4\\sin4\\varphi + B_6\\sin6\\varphi + B_8\\sin8\\varphi + \\cdots,\n", "04703982a8c13f3e647afa36dc258a3c": "(I_n \\mid S)", "0470a15db5621100067ced7c9ad71923": "F(f):F(X) \\rightarrow F(Y) \\in D", "0470d0befff72541c46222414a829fe5": "\\operatorname{P} (Z_i=2) = \\tau_2 = 1-\\tau_1", "04715c5a2b4e62e7fb226a438528c1cb": "\\begin{align}\n(\\pi_{m,n}(J_i))_{a'b' , ab} &= \\delta_{b'b}(J_i^{(m)})_{a'a} + \\delta_{a'a}(J_i^{(n)})_{b'b},\\\\\n(\\pi_{m,n}(K_i))_{a'b' , ab} &= i(\\delta_{a'a}(J_i^{(n)})_{b'b} - \\delta_{b'b}(J_i^{(m)})_{a'a}),\n\\end{align}", "0471615797404a49fc735c65e449a7aa": "\\mathbf{C}^\\alpha\\ ", "047174dc95a12d05b955f620f3b80798": "E_{kin}=mc^2\\left(\\frac1{\\sqrt{1-\\frac{v^2} {c^2}}}-1\\right)", "04717d25637c0f10e2095645d8f35dcb": "a \\in \\mathbb{N}", "04718f70581064c7db5652ac8bacfa5f": "(c_i = C_\\text{in}(y_i'))", "0471a6b996bf88ea837c16b82f80f25e": "v=y", "047229ccf2f40a3743e0af8092077297": "r=r_c", "047238b589c5452e86b56a33d8210972": "N \\cdot N^r \\cdot S \\cdot N^l \\cdot N ~\\leq~ S \\cdot N^l \\cdot N ~\\leq~ S", "04724bbd90d9b3ede19fc46685c32688": "C_{Hb}", "047276de01eb8ede93eb68722af37dec": "deg(p)", "04728917a32ca84813f26b5ce295bb62": "\\ R_j", "0472acfadb86c949f89853252fe915a4": "F_T + A_T \\Leftrightarrow TC", "0472b8c476010287bff5fd05acec7b2a": "\\omega_0=\\gamma|\\mathbf{B}_{\\|}|", "0472fc4df375a85af48212c820aef7ba": "\\frac{(n-2)}{2n}", "04733873f4f4988b008fa55ca9dcdef5": " (\\Gamma(V,L) \\setminus \\{0\\})/k^{\\ast}, ", "04734da7ad96610d9cd72413217e28e4": "{1\\over2}\\hbar\\omega , \\quad {3\\over2}\\hbar\\omega ,\\quad {5\\over2}\\hbar\\omega \\quad ......", "047352019ed5c2f4b607eac7ba16c621": "(n+1)\\,P_{n+1}(x) = (2n+1)x\\,P_n(x) - n\\,P_{n-1}(x).\\,", "0473831900bfa6b0690f73d1d600aa94": "\\textstyle \\sum c_n = (1,1+2,1+2+3,1+2+3+4,\\dots)", "04740a16dd5a12c6c8d3dcb1388d3a11": "\\ln(2)\\,", "04742d300ace362c7f609b6e2bf98aee": "X(x)=C_{3}e^{-jk_{x}x}+C_{4}e^{jk_{x}x}", "047454141f7d3762203d9c9c0fe94068": "\n{P}=\\left[\\begin{matrix}{T}&\\mathbf{T}^0\\\\\\mathbf{0}&1\\end{matrix}\\right],\n", "047483c44de8b6bf46e64800ab13386f": "\n\\theta\\in[-U,U]\n", "047495f722547a6cabc2b7cf66b3a722": "\\sum_{p^k|n} f(p^k)\\;", "0474b45700b2a1a17ad723d4a260200f": "\n = (1-\\frac{2}{2^s})\\zeta_{2n}(s) + \\frac{2}{2^s}(\\frac{1}{{(n+1)}^s}+\\ldots+\\frac{1}{{(2n)}^s})\n = (1-\\frac{2}{2^s})\\zeta_{2n}(s) + \\frac{2n}{{(2n)}^s}\\,\\frac{1}{n}\\,(\\frac{1}{{(1+1/n)}^s}+\n \\ldots + \\frac{1}{{(1+n/n)}^s}).\n", "04751be39631e2fb1959ca2ffee461d7": "(1-x)^\\alpha(1+x)^\\beta\\,", "047537d879c0fdf93eb53abdba46c5be": "\\Lambda:=\\lbrace 1 \\rbrace", "047542a251ee3d53b0c9912009e84238": "p = r \\Bigg[ \\frac{(1+r)^{n} B_{0}-B_{n}}{(1+r)^{n}-1} \\Bigg]", "047623dba90c6f4d24876c5193f0b4bb": "\\tau(y_i;\\lambda, \\alpha) = \\begin{cases} \\dfrac{(y_i + \\alpha)^\\lambda - 1}{\\lambda (\\operatorname{GM}(y))^{\\lambda - 1}} & \\text{if } \\lambda\\neq 0, \\\\ \\\\\n\\operatorname{GM}(y)\\ln(y_i + \\alpha)& \\text{if } \\lambda=0,\\end{cases}", "04763654620554c15ae64b5aca942bc7": "\\mathbf{A}^0 = \\mathbf{I}", "047790787c56fe7b6abfc4b0aec99d0d": "s_{i_1}s_{i_2}\\dots s_{i_m}", "0477e1ecbf939462595f6bba903295c6": "\\Phi_X(f) = (Ff)u.\\,", "047805207e4e77a99e33063ff9f5ad16": "\\mu\\left(1-\\sigma\\mathrm{log}{\\tfrac{X}{\\sigma}}\\right) \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)", "04784816d74868a93174f87c0236fe76": "dS_w\\,", "04785525962c6de79ea6eaacbb289d00": "\\mathit{gl}_n \\to \\mathit{gl}_n", "04787fb006bf601b807a0d6d88daf948": " d \\theta = d\\theta_1\\cdots \\, d\\theta_n ", "047893115b0628c644d180c0034540fb": "M_\\oplus", "0478c680906e9ec42d6d9ec19c2f9a68": "\\frac{1}{2^s}\\zeta(s) = \\frac{1}{2^s}+\\frac{1}{4^s}+\\frac{1}{6^s}+\\frac{1}{8^s}+\\frac{1}{10^s}+ \\ldots ", "0478f4e13391a5b6d468b2db291a878f": "dqo", "047913a157084d7cad54db010c56d85a": "\\mathbf{X}=\\{X_1,X_2,\\ldots,X_n\\}", "0479683ca61d38fd063ec79a30e86707": " U_0(r) \\approx a(r) e^{-ikr} ", "04798ef3c700d5ba9bb0a92b0498e9b0": "\\ln K= \\sum_k \\ln {a_k}^{m_k}-\\sum_j \\ln {a_j}^{n_j};\nK=\\frac{\\prod_k {a_k}^{m_k}}{\\prod_j {a_j}^{n_j}} \\equiv \\frac{{\\{R\\}} ^\\rho {\\{S\\}}^\\sigma ... } {{\\{A\\}}^\\alpha {\\{B\\}}^\\beta ...}\n", "04798f7b3ef3f02402eeb94577aa85dc": "\\ \\mathbb{D} _X ", "0479913b6a7d32d643336fb4840b0f06": " F_{ST} = \\frac{ \\pi_\\text{Between} - \\pi_\\text{Within} } { \\pi_\\text{Between} } ", "0479b6d1d786d3ad2be4b2ed143f74be": " (A-\\lambda I)v_2 = v_1. ", "0479bbf4aba93492b525c780cddee25f": "\\overline{\\left ( \\tau_s - \\bar{\\tau}_s \\right )^2} = \\sum_{p,q=0}^{s-1} \\left ( \\overline{\\xi_p\\xi}_q - \\bar{\\xi}_p\\bar{\\xi}_q \\right ) = s \\sum_{p=0}^{s-1} \\left ( \\overline{\\xi_0\\xi}_p - \\bar{\\xi}^2 \\right ).", "047a4e1101708bfb9fd8dc21bdbf43ce": "\\mathcal S(\\gamma) := \\int_a^b L(\\gamma(t),\\dot\\gamma(t))dt", "047acde79363c5f1670a147074d84ff3": "\\{x \\mid \\phi \\}", "047afbff91f0af5c13696532a6c2c8a0": "m\\leq O(n^{(16/15)-\\epsilon})", "047b35daba9c1d2d66362745051dc5f1": "\\partial_t u=\\delta_v H(u,v)", "047bf2ef5af416c7c7b36b6a2f66edc0": "T(V)= \\bigoplus_{k=0}^\\infty T^kV = K\\oplus V \\oplus (V\\otimes V) \\oplus (V\\otimes V\\otimes V) \\oplus \\cdots.", "047c507abd502bb88a4f60732c851832": "\\operatorname{plus} \\equiv \\lambda m.\\lambda n.\\lambda f.\\lambda x. m\\ f\\ (n\\ f\\ x)", "047c508038bcdb0a82a908d184bc2002": "=2\\pi \\varepsilon a\\left\\{ 1+\\frac{1}{2D}+\\frac{1}{4D^2}+\\frac{1}{8D^{3}}+\\frac{1}{8D^{4}}+\\frac{3}{32D^{5}}+O\\left( \\frac{1}{D^{6}}\\right) \\right\\}", "047c65a315d2c3664f293e07153b2b41": " v = \\frac{(m_\\textrm{b}+m_\\textrm{p}) \\cdot \\sqrt{2\\cdot g\\cdot h}}{m_\\textrm{b}}", "047ce8f8e02e71b5b46b73258eebddf6": " \\mathbf{V} \\cdot \\mathbf{W} = \\| \\mathbf{V} \\|\\| \\mathbf{W} \\| \\cos a . ", "047e367f8518f5559adf2909b6e264e6": "(x\\pm i0)^{-k}=x_+^{-k} + (-1)^kx_-^{-k}\\pm\\pi i \\frac{\\delta^{(k-1)}}{(k-1)!},", "047e9d8fda718ccca99693995e9444cc": "L_{p,\\mathrm{loc}}(\\Omega),", "047f60eee20519278eb4e46c31c436f1": "\n \\rho~\\dot{\\eta} \\ge - \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) \n + \\cfrac{\\rho~s}{T} \n ", "047f70ea396f58388c9fa6da42fbc7fb": "\\dot\\omega", "047f9a646fcaed8d8a620b5208eb6c1b": " \\ b = \\frac12 \\times \\rho_{water} \\times S_b \\times C ", "047fd02e5f0ffeb0eea6d81c7bda7d05": "\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n=\n\\begin{bmatrix}\n 1 & 0 & 1.28033 \\\\\n 1 & -0.21482 & -0.38059 \\\\\n 1 & 2.12798 & 0\n\\end{bmatrix}\n\\begin{bmatrix} Y' \\\\ U \\\\ V \\end{bmatrix}\n", "047fd1729016dd23ae1d2a19ffd9337c": "\\phi.", "047fde3816d96e562e3871ac2f50059d": "B_1,\\dots, B_k", "047fded37529a2cf6747b2cf845182b7": "\\vdash A \\to B.", "048044ce9dd5c1f3e267135d99f723a9": "b(t)=\\frac{1}{M}\\sum_{i=0}^{i=M-1}{w_i r_i(t-t_i)}", "0480a86160daf12d942f899757a33974": "y \\succ z", "0480c7ca01d301a310b5963cdcaef5e3": "f \\in S", "0480db02c29fbaec48531cb9d43929fe": "mn \\times mn", "04810e2033e49bc7641e329bfe04ea6c": "f \\in \\overline{K}(C)^{*}", "04811704feb2abd5e747e199718b3dab": "\\int_{\\tau_1}^{\\tau_2} \\mathbf{F}_\\mathrm{rad} \\cdot \\mathbf{v} dt = -\\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d \\mathbf{a}}{dt} \\cdot \\mathbf{a} \\bigg|_{\\tau_1}^{\\tau_2} + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^2 \\mathbf{a}}{dt^2} \\cdot \\mathbf{a} dt = -0 + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\mathbf{\\ddot{a}} \\cdot \\mathbf{a} dt", "0481771e0d238c6608d2f2acaa3ea5ea": "\\sigma = \\pi R^2 P / 2 \\pi R h = R P / 2 h", "048182459491fe2e9c939465e1c541d0": " D(a,s) \\cdot D(b,s) = \\sum_{n=1}^\\infty (a*b)(n) n^{-s} \\ ", "0481d3dfd06bfbe944d6dd475fbb60cc": " ((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2 ", "0481fc89cc8b5cd263273622583380b5": "\\frac{B_1}{h_1^2}=\\frac{B_2}{h_2^2}=\\frac{\\sqrt{B_1 B_2}}{h_1 h_2} =", "0482428fffffe7022ae2cc636c2236fe": "(S)^H\\,", "0482c2d36892eb4589b30cb08c1a360d": "f(n)=\\sum_{d\\,\\mid\\, n}\\mu(d)g(n/d)\\quad\\text{for every integer }n\\ge 1", "048316cdd8ee2f8fe08bfdf69e9b8146": " U \\subset M", "0483319d6300833ac825096bfed9e32e": "1000\\sqrt{\\ell/g }", "048342b8a951b3064014559c5611e2fd": "{\\eta_N}", "048350c2d6b47a176f5d038af2465484": " 7 ", "048365e39b6afdfb2ff84dfd585e9fa1": "q_\\max = \\sqrt{{(E_u^3)(g)\\over (1.5)^3}}=\\sqrt{{(3.04^3)(32.2)\\over (1.5)^3}}=16.4\\text{ ft}^2/s", "0483e16137c69e5676d9801cdd79875b": "\\textstyle\\left(\\frac{{p}}{{5}}\\right)", "048405977db606e46a43b4816b84f43a": "\\frac{R_1}{c}", "048429d0d991b94250f92b125a63c173": "W_{1}^{A}(x,z)", "0484565f18e2eab70b9bbd55ccde7fda": "m_b = m'_b - k", "0484c29ed7efcc6b03fc1c0b6f725a19": "x_2 \\ge 0 ", "0484d7d451687f3e79f67ec3bde75b6e": " \n\\begin{align}\n&u^{0} &=& \\alpha + \\beta x + \\gamma x^{2}/2 \\\\\n&u^{1} &=& -\\frac{1}{2} L^{-1}(u^{0}u^{0''}) &=& -L^{-1} A_{0} \\\\\n&u^{2} &=& -\\frac{1}{2} L^{-1}(u^{1}u^{0''}+u^{0}u^{1''}) &=& -L^{-1} A_{1} \\\\\n&u^{3} &=& -\\frac{1}{2} L^{-1}(u^{2}u^{0''}+u^{1}u^{1''}+u^{0}u^{2''}) &=& -L^{-1} A_{2} \\\\\n&&\\cdots&\n\\end{align}\n", "04850234a56406c23418f463a67eb060": "n \\geqslant 2", "048505e7c44acdca06cbd3d5acdd7df1": "\\theta=\\theta_i", "048538144c496ea0741a737a736eb874": "\\dot{\\bold{r}}\\times\\bold{H}=\\mu\\bold{u} + \\bold{c}", "048549fa6f951b01bd4dcd6e53002584": "\\{\\{i,j\\}: a_{i,j}\\neq 0, 1\\leq i 0", "048a1002418410b27fd943f343cb1d41": "G(\\tau)=\\frac{\\langle\\delta I(t)\\delta I(t+\\tau)\\rangle}{\\langle I(t) \\rangle^2}=\\frac{\\langle I(t)I(t+\\tau) \\rangle}{\\langle I(t)\\rangle^2}-1", "048a5cf6de0c07f4751738e85a0121a9": "\\langle Tx, y\\rangle = \\int_{\\mathbb{R}} \\lambda\\,d\\langle E_\\lambda x,y\\rangle.", "048a9c3303a1553a6aafaf48914ef2be": "2^4\\cdot 3^2\\cdot 5^2\\cdot 7", "048ad459af88f782594d6a04498110ba": "91^2", "048ae06427bbb90c8c794e33b7a6a94b": "\n2\\pi\\gamma RB^{5/2}\\Sigma^2K_1\\left(\\frac{\\ell}{L_c}\\right)\n", "048af646d5b23f889b63067b9014b488": "\\textbf{t}_i", "048b3e9d2796c31a9580d700f5ca6e28": "(\\alpha A)^+ = \\alpha^{-1} A^+\\,\\!", "048b4edc73ba6a55e2f377a459bdeabd": "\\chi = \\frac{\\mathbf{M}}{\\mathbf{H}}", "048b6a58da82ee0994e07c3f235cb954": " z=r e^{\\varphi i} \\text{ with } -\\pi < \\varphi \\le \\pi, ", "048c3096809e88057149e93d08871f7d": " z_k(s) \\leftarrow x_j(s) ", "048cc2757d099299037aca88706d9e7f": "P(y, x_1, \\ldots x_n)=P(y, x_i)P(x_1, \\ldots x_n\\mid y, x_i).", "048cf7531c4567ad53512c73a9f1f870": "l = G'^{-1}(w)", "048cfececa0469f517c2806522571044": "P\\in z", "048d497b67f361d97a7a3c42fe008e19": "n^{(1)},...,n^{(q)}", "048d7a28a426531c29096ab8086f1ab0": "p(x) = 0", "048db65b0805ba9bc7b142f961d8507b": "I(s) = \\frac{1}{ R + Ls + \\frac{1}{Cs} } V(s) ", "048dc7809459e8186f3ea67285bd3140": " H_g(P,Q) ", "048e01282e67d104e634022373d1e75d": "k_2=\\sqrt{2m (E-V_0)/\\hbar^2}", "048e3e85be0499b018b06704c9e3fdf6": " U(0,1) ", "048e48f2ecf712f857fb33ca50e51e3b": " \\ \\phi(x)", "048e52e077417008ca12b5667a8836d1": " k_{GT}", "048e6eeee89b275e038da2b31b481b6f": "\\alpha(T_r) = T_r^{N \\left( M-1 \\right)} exp \\left( L \\left( 1- T_r^{ M N }\\right) \\right)", "048e9b1d644a6d551990258826f47c94": "C(d) = \\sigma^2 \\Bigg(1 + \\frac{ \\sqrt{5}d }{\\rho} +\\frac{ 5d^2}{3 \\rho^2 } \\Bigg) \\exp \\Bigg(-\\frac{\\sqrt{5}d}{\\rho} \\Bigg) \\quad \\quad \\nu= \\tfrac{5}{2}.\n", "048ebaae8fb0e589582b112ccdaf92f4": "S_a(Tr(g^b))=\\left(Tr(g^{(a-1)b}),Tr(g^{ab}),Tr(g^{(a+1)b})\\right)\\in GF(p^2)^3", "048ebefe83e4a8507c500f9bee0f2efe": " [ES] = \\frac{K_i [S][E]_0}{K_m K_i + K_i[S] + K_m[I]}", "048efa823ac43bd64960226b1668c49f": "\\operatorname{sech}\\,x = \\left(\\cosh x\\right)^{-1} = \\frac {2} {e^x + e^{-x}} = \\frac{2e^x} {e^{2x} + 1} = \\frac{2e^{-x}} {1 + e^{-2x}}", "048f1cf76c8a9280aca95ae9a90e3dbf": "z-n", "048ff3eff28beef138d3798e8b153d59": " \n \\begin{align}\n \\hat{\\mu}_1 & = m_1 \\\\\n \\hat{\\mu}_2 & =m_2\n \\end{align}\n", "04902232a7610df3d8e4f38aabc2787a": " x_N \\in X_N", "049029f82b397ae1a5055bf7f706a9ee": "T \\ \\sin \\theta_1 =F_1 \\,\\!", "0490503c6469600795c4219e09b48d4e": "\\{|S,S_z\\rangle\\}\\equiv\\{|1,1\\rangle,|1,0\\rangle,|1,-1\\rangle\\}", "049069e06486046b7174b58402be8888": "\\varphi:\\{E^a\\} \\mapsto \\{\\Phi,E^a,I^a\\}", "0490938bb5edc81e4514ea3ef4bc2f79": "\\mathbb{R}^3,", "04909457bdda7c5c3eb1b12c98278188": "At(room1)", "0491a674d60a00af1b06d832319055c1": "\\mathbf{A} = {}^{*} \\omega_{\\mathbf{A}}= {a}_{1}d{x}_{2} \\wedge d{x}_{3}+{a}_{2}d{x}_{3} \\wedge d{x}_{1}+{a}_{3}d{x}_{1} \\wedge d{x}_{2}", "0491a76b9af525e4dba9daec6c65875b": "\\pi \\gets \\mathrm{Prove}(\\sigma,y,w)", "0491d1a54522592cd19851c5e7e553c1": "W_{t}(n)", "0491f45558966441248f4d2dee9b412d": "P_{\\mathrm{error}\\ 1\\to\\mathrm{any}} \\le M^\\rho \\prod_{i=1}^n \\sum_{y_i} \\left(\\sum_{x_i} Q_i(x_i)[p_i(y_i|x_i)]^{\\frac{1}{1+\\rho}}\\right)^{1+\\rho} ", "0492037f2bf335bbb59e262f8b0da426": "\\gamma:[0, 1]\\to \\mathbb C. ", "04920982c23e776f0cd74b5e114b96c4": "K\\subseteq_s M", "04927783e15da5933265708eacf831b0": " \\prod_{r=1}^4 \\Gamma(\\tfrac{r}{5}) = \\frac{4\\pi^2}{\\sqrt{5}} \\approx 17.6552850814935242483", "04928fe1823546455f9c6b2e93967375": "\\scriptstyle\\tfrac{1}{r}+\\tfrac{1}{s}=1", "0492e77087122537e15019d02c9dc267": "\\cdots \\to \\pi_{i+1} BD \\to \\pi_i B(d \\backslash f) \\to \\pi_i BC \\to \\pi_i BD \\to \\cdots.", "049300f2155adb98dbae6855615508dc": "\\scriptstyle\\leftarrow", "04932c19f04b83ac8733455192b348ed": "A f (x) = b(x) \\cdot \\nabla_{x} f(x) + \\frac1{2} \\big( \\sigma(x) \\sigma(x)^{\\top} \\big) : \\nabla_{x} \\nabla_{x} f(x).", "04937f297c5a4b1df274359eb81322f6": "{50 \\choose 3} = 19,600", "0493b6cc2e6df03a04e667b941a12781": "L_n[1/2, 1] = e^{(1+o(1))(\\ln n)^{1/2}(\\ln \\ln n)^{1/2}}.\\,", "0493c135827d7dd68263ccf670524310": "G(f)", "0493c23f7bffb081dcaf19fae853ceba": " F(x) = f(x) + \\cdots + (-1)^j f^{(2j)}(x) + \\cdots + (-1)^n f^{(2n)}(x),\\quad x\\in\\mathbb{R},\\!", "04941746bc3fbb903fa792b044e2418a": "A\\in \\mathcal{F}", "049441c8ce08c1ec18e16794e80465e6": "d \\vec{\\ell}_2", "04947fc757b781cf6bde09b5e0647d25": "\\varphi=0", "0494e386a35fe24bfb125175572a32a2": "\\rho(\\mathbf r,t)=\\rho[v,\\Psi_0](\\mathbf{r},t)\\leftrightarrow v(\\mathbf r,t)=v[\\rho,\\Psi_0](\\mathbf r,t)", "0494ea13ecdfad57c59dfee70213f05c": " h_2 (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq j \\leq k \\leq n} X_j X_k,", "049516430ab9768d38217c5b85a4da78": "\\frac{1}{2}(k\\!-\\!\\ln(2)\\!-\\!(k\\!-\\!1)\\psi_0(k/2))", "049526a86f30148986edffdb4168e359": "a_y", "04956b031dfcb109700b760742460d48": "\\displaystyle{\\pi^\\prime_s((g^\\prime)^{-1}) F(x) =|cx+d|^{1-2s} F\\left({ax+b\\over cx +d}\\right).}", "049594571bf34b6300576cefd2297470": " \\omega_{ }^{ } = c k", "0495c0ae17755da6efa9259f9976dc72": "\\int\\frac{x\\;dx}{s^3} = -\\frac{1}{s}", "04960db2bec542bc240a4c537f2bc27c": "\\mathbb{Z}^n_q \\times \\mathbb{Z}_q", "04962fb96f35c2c9217723bc1b531c45": "\\Lambda = \\begin{pmatrix}\n\\lambda_1 & \\ldots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\ldots & \\lambda_4\n\\end{pmatrix}\\,,\n", "04964ec95cb9665e7ba6a188e4300f90": "5\\quad 1\\quad 1\\quad 5\\quad 0\\quad 3\\quad 4\\quad 2", "04973f026f5455096a643b8e6c8e7e6f": "e^{2A} - \\frac{I+A}{I-A}=- \\frac{2}{3} A^3 +\\mathrm{O} (A^4) ~. ", "049769258f04334cccfd306f91a73e38": "e^x\\log(1+y)= y + xy - \\frac{y^2}{2} + \\cdots", "0497d90d14a6754bb11533d7e46cdcff": " = \n[\\textrm{CO}_2]_{eq} \\left(\\frac{[\\textrm{H}^+]_{eq}^2 + K_1[\\textrm{H}^+]_{eq}+K_1K_2}{[\\textrm{H}^+]_{eq}^2}\\right). ", "049817e71b75872219c9769deb9e18d7": "\\scriptstyle \\hbar = \\frac{h}{2 \\pi} \\,", "0498200b37d09b47bbc8d014ad28e86b": "{D}_{8}^{(2)}", "04984b24ee286d6f5dd129c9c1cfa224": " \\Delta h = \\star d \\star d h = \\exp(-2 p) \\, \\left( h_{xx} + h_{yy} \\right)", "049874699cbc9ddacaf4d244d90d3e8d": " 0 \\leq 2 n \\sum_{j=1}^n a_j b_j - 2 \\sum_{j=1}^n a_j \\, \\sum_{k=1}^n b_k,", "0498f50bdce41bf6b06a52a836cbb96f": "E_\\theta={-iI_0\\sin\\theta\\over 4\\varepsilon_0 c r}{L\\over\\lambda}e^{i\\left(\\omega t-kr\\right)}.", "04990f5a51869124035ab5fbdeeaf677": "(p,p^2)", "049956d7e13116db00e8822cdd8244b4": " H = -\\sum\\nolimits_{j=1}^N \\partial^2/\\partial x_j^2 +2c\n\\sum\\nolimits_{1\\leq i< j\\leq N} \\delta(x_i-x_j)\\ , ", "04998b15150c44c8c5bdd75659e311a3": " h \\nu ", "0499dc8d5b79109560ec7ac8ed4ef4d3": "[I-A]", "0499ee1ba90ee7f1581090e6a5dfeda7": "\\scriptstyle \\mathcal{N}_1=\\{-2,-1,0,1,2\\} ", "0499fbc0b0f937d8f3a192a94edcd193": "x\\leq y", "049a40c20859c5b3ecd63a2c347e9b96": "\n\\mathbf{g^{(1)}} = \\begin{bmatrix} \n+5 & +5 & +5 \\\\\n-3 & 0 & -3 \\\\\n-3 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(2)}} = \\begin{bmatrix} \n+5 & +5 & -3 \\\\\n+5 & 0 & -3 \\\\\n-3 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(3)}} = \\begin{bmatrix} \n+5 & -3 & -3 \\\\\n+5 & 0 & -3 \\\\\n+5 & -3 & -3 \n\\end{bmatrix},\\ \n\\mathbf{g^{(4)}} = \\begin{bmatrix} \n-3 & -3 & -3 \\\\\n+5 & 0 & -3 \\\\\n+5 & +5 & -3 \n\\end{bmatrix}", "049a4b75a9bb72e3e3a4936fb684bc08": " \\alpha = \\frac{e^2}{(4 \\pi \\varepsilon_0) \\hbar c} = 7.297\\,352\\,5698(24) \\times 10^{-3}.", "049a6650706ebdc9c16d62aae65fa924": "|x+y| \\le |x|+|y|", "049b3498c324a73d0f0c34d9179c2030": " \\ c ", "049b4955f342c15b0dd8f3490f044788": "\\mathrm{C + \\tfrac{1}{2}O_2 \\ \\Rightarrow \\ CO}", "049be4f3f580b85dee4a1872a70e7915": "\\langle \\cdot,\\cdot\\rangle_2", "049cb7ccab6876bb9e9092849353b81b": "p_i = \\frac{1}{1+e^{-(\\beta_0 + \\beta_1 x_{1,i} + \\cdots + \\beta_k x_{k,i})}}. \\, ", "049d1e2f594e1ee91680961985844c5d": "\n W = C_{1} (I_1-3) \\,\n ", "049d39cdb7a32b75ce1cd6ded61882cc": "M\\le w", "049d670c004f59dc4843a509ce3f7127": "\\mathfrak{P}^{84}", "049d9a8da567c5f349c38ba94254918a": "k_{x}=k_{x}'+i k_{x}''=\\left[\\frac{\\omega}{c} \\left( \\frac{\\varepsilon_1' \\varepsilon_2}{\\varepsilon_1' + \\varepsilon_2}\\right)^{1/2}\\right] + i \\left[\\frac{\\omega}{c} \\left( \\frac{\\varepsilon_1' \\varepsilon_2}{\\varepsilon_1' + \\varepsilon_2}\\right)^{3/2} \\frac{\\varepsilon_1''}{2(\\varepsilon_1')^2}\\right].", "049dab7627d4081f8530e7177cad5095": "\n H_x= k_y \\sin k_x x \\cos k_y y \\cos k_z z\n ", "049db5a373f1303f5f652aaa6e00ed88": "\\int^T_0 \\frac{N_0}{2}\\delta(t-s)k(s)ds = S(t) \\Rightarrow k(t) = C S(t), 00),", "04c01f0e4919efa824b9c43529a898c0": " {h_1 \\over h_0} =\\frac{{\\sqrt{1+{{8Fr^2}}} -1}}{2}, ", "04c0375a0567dd3e84683930cb024313": " s_0 = s_n ", "04c05885c12db7ae13199140c4d54225": "\n \\cfrac{\\partial W}{\\partial I_1}\\biggr|_{I_1=3} = \\frac{\\mu}{2} \\,.\n ", "04c0b195b64a5e0327fcdbac013810d0": "p \\oplus q", "04c0d07376defa38f245802bcbd4b3bb": "~\\hat a = X+iP~", "04c110246defb7f6a694db4b679a88ed": "\\left( \\Phi \\cup \\{\\lnot\\phi\\}\\right)", "04c110b5b06c8889aed5b28c383d6e50": "K\\otimes_{\\mathbb Q}K", "04c14940ca11289c43be6206a9c1b646": "\\sigma_{ff}", "04c185182b2f87bbbd72616c17b812da": "p_3(x)=9x^2-3\\,=3(3x^2-1)\\,=3(x\\sqrt{3}-1)(x\\sqrt{3}+1)", "04c1b944cb0a850a29331752ca1bdbd6": "\\mathrm{F=C\\ V^{-1}=A^2kg^{-1}m^{-2}s^4}", "04c1d73c3888fb72cf1de41e95ac8d81": "\\textrm{NM}(k_0,\\,p)", "04c22e12f3c8c6a80f33e0ac3d25fe5b": "\\mbox{EXPSPACE} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{DSPACE}(2^{n^k}) = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NSPACE}(2^{n^k})", "04c2327dc649b2c09a324f7cae1f7d74": "85^2", "04c25020b321831974418d1de8bc2c44": " B_j = (a_j - a_j^*)/(2i) ", "04c26f1786a13ba9848cee465f3fa420": "a_0b_2", "04c314b581e276e14dfef4a0b7e02636": "(cA)_{ij}=cA_{ij}, \\qquad (Ac)_{ij}=A_{ij}c.\\,", "04c318c98c0b1586b6565fbdab350291": "\\Delta a=\\sin^{-1}\\left(\\frac {V_w\\sin(w-d)}{V_a}\\right)", "04c359cfa326b9819aa6afe5bf8c94c3": "m = 6", "04c3c996749f1f84cea72957eb9ad245": "(i_k)_{1 \\le k \\le K}", "04c3ffe149343baf59147dc0804615d0": "\\,^{249}_{97}\\mathrm{Bk} + \\,^{50}_{22}\\mathrm{Ti} \\to \\,^{295}_{119}\\mathrm{Uue} \\,+4\\,^{1}_{0}\\mathrm{n}", "04c41f2b4656b51e364061c051c9b3ec": " \\sum_n \\left( i\\hbar \\frac{\\partial c_n}{\\partial t} - c_n(t) V(t) \\right) e^{- i E_n t /\\hbar} |n\\rang = 0", "04c42d54597e72014a0777dfe9bd9545": " S : K[G] \\to K[G] ~\\text{by}~ S(g) = g^{-1} ~\\text{for all}~ g \\in G_1 ", "04c44949f777e21a9f0581a1a99b6b3e": "\\hat{S}_{i}|\\phi\\rangle = s_{i}|\\phi\\rangle, s_{i}\\in\\mathbb{C}", "04c46410d6f23482e677bb6e5f946e16": "a(b+c)=ab+ac", "04c4669a6f6b89a6f9fc05756117dc52": " v_{Water} = \\sqrt{\\frac{2 \\cdot \\left(p_{Total}-p_{Static}\\right)}{\\rho}} \\,\\!", "04c46f30b4ba78f605ff8c0d3ed1b90a": "\\omega_{p}", "04c47c2b9159dfaec59e2e21bdef8f9f": "a(u_n, e_i) = f(e_i) \\quad i=1,\\ldots,n.", "04c48e87ae666606b70484f5db48f436": "({\\sin \\theta})^2 = -\\frac{(\\mathbf u \\wedge \\mathbf v)^2}{{ \\mathbf u }^2 { \\mathbf v }^2}", "04c49c75b7a3e933536e209d4d8805af": " \\lnot \\lnot x = x,", "04c4c430df526a2f4bb6f83f9539e9d4": " Z_\\text{in} = {v \\over i} = -Z ", "04c4c9a327125dcc9336c608e6c54653": "\\varphi(x)=-\\frac{2}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}te^{-\\frac{t^2}{2}}\\ln|x-t|\\,dt.", "04c5a57d48a350f653f25d4fe36858d2": "r^2 = - \\tan(2\\theta)/2. \\,", "04c5c869a7af06b0d7b63f3085caea1e": "\\scriptstyle (y_1,\\, y_2,\\, \\ldots,\\, y_n)", "04c67acad7c66773a0881af5009e16b3": "\\wedge^{p+1}M_{q-1}\\rightarrow \\wedge^{p}M_{q} \\rightarrow \\wedge^{p-1}M_{q+1}", "04c68062041875d4ffe70413c1372f51": "\\mathsf{(CH_2CH_2)O+H_2\\ \\xrightarrow{Zn\\ +\\ CH_3COOH}\\ CH_2\\!\\!=\\!\\!CH_2+H_2O}", "04c6db9ab3f1ef6e9b80535b5fa6b17b": "2\\eta_{\\mu\\nu}A^\\mu U^\\nu = 0.", "04c7178c91de90bb2a1fea278b1c09b8": " x_{ij} \\in \\{0,1\\}", "04c75fc8c02e137ead6f3efd786e4084": "H_2^{+}", "04c7c45e4c5faecd07a56a00d3d1eaa0": "F_\\mathrm{n}\\,", "04c879a7b484925cf17dc1946509be64": " f_{y}(x,y) \\approx \\frac{f(x,y+k ) - f(x,y-k)}{2k} \\ ", "04c8ed7896b4e9a5fce36f40ce841c1c": "\\frac{1}{2}(l^2-1)", "04c8f846f8901cc910a604daa68910d5": "(r\\bar{b}+b\\bar{r})/\\sqrt{2}.", "04c8fd52917642ff9ec6b7e1ad2b711b": "S_j", "04c95c5523b64e44b5f09bb443214031": " \\tau_n = O(h^{p+1}) ", "04c96de0f1a9b2624fbefac5583c47b5": "V = \\frac{\\pi ^ {\\frac{n-1}{2}}\\, r^{n}}{\\,\\Gamma \\left ( \\frac{n+1}{2} \\right )} \\int\\limits_{0}^{\\arccos\\left(\\frac{r-h}{r}\\right)}\\sin^n (t) \\,\\mathrm{d}t", "04c976c400307140d07071312dd322a7": "\\; \\sum_i\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n", "04c9cbb2ae0a9222fe98f1c128e3567b": "S^{IJ}", "04c9e049ecdf7b697aa92a03cc5dc80b": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, conc(\\langle b \\rangle, conc(\\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle), \\langle b \\rangle), \\langle b \\rangle), \\langle a \\rangle)", "04ca2451af9f8f56281fed4c9e2566fe": "y(t)=-\\frac{1}{2}gt^2+v_{0}t+y_0", "04ca8c61e468479e7bf1d496316aa78d": "\\mbox{vec}(\\mbox{ad}_A(X)) = (I_n\\otimes A - A^T \\otimes I_n ) \\mbox{vec}(X)", "04caad811d7de70354b943c14d443caf": "U_s U_{\\omega}", "04cafb4a620e52221658357732a348c6": " 2 \\int\\limits_{-\\infty}^\\infty f(t)\\cos\\,{2\\pi \\nu t} \\,dt.", "04cb149fecba1f56811e1d6ff04dcb7d": " a + 2 = b + 2. \\, ", "04cb1ed4b24ae1d0c7051df70770ef69": " (E_{t+1} - E_{t}) y_{t+j+1}", "04cb67c92d7cec7f994a5c5a1f7d4b11": "H \\bmod N \\times 2^L", "04cb7878b651d3c480dfc4e6941d068f": " v' = -\\frac{\\partial \\psi}{\\partial x}", "04cbae5f18b69a1c89403c9af6ae4f65": " [min(r_1, r_2), min(g_1, g_2), min(b_1, b_2)] ", "04cc0e28c90a06698de8ab8ab4269bb9": "L(F, G) := \\inf \\{ \\varepsilon > 0 | F(x - \\varepsilon) - \\varepsilon \\leq G(x) \\leq F(x + \\varepsilon) + \\varepsilon \\mathrm{\\,for\\,all\\,} x \\in \\mathbb{R} \\}.", "04cc38bcb1f8bf6c0b8a797ba4244e11": "x_i \\in \\mathbb{R}^{n+1}, \\, i = 1,...,m", "04cc7eed1a40c64a7be510d9d0d6b51c": " n = \\infty\\!", "04cc91a9b5e3aadb9b97d1921bab8f81": "\\frac{d}{d x}\\left(\\frac{1}{2 - n}\\left(\\frac{d t}{d x}\\right)^{n - 2}\\right) = f(x)", "04ccfac50c13f886fd57d6102c0674c8": "\\sigma_{zz} = \\sigma_{zx} = \\sigma_{yz} = 0", "04cd0e0151f352e7fd414d694a604136": "[1, 2]", "04cd61c128b35877531bd18ad85af8d7": "\\scriptstyle{R_0^0 + R_3^3 = 0}", "04ce4598bd3f73b2b528b57e5e1af6e6": "\\langle 1\\rangle", "04cea56b0b312d7edce09d5dd7596ba9": "\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} + \\frac{\\partial w}{\\partial z} = 0", "04cec08a0e5858f7e7d7bb8028a0746d": "\\beta \\le -2", "04cf31d6ec3540fee12e8e5ed390d9ba": "A_e = \\frac{3 \\lambda ^2 }{8 \\pi} ", "04cf4ec52b62d7ce63235d8519aa5f88": "\\int_0^\\infty x^{y-1} e^{-x}\\,dx,", "04cf774444cfa3e18887ceddd932d053": " 0\\to C^0 \\stackrel{d_0}{\\longrightarrow} C^1\\stackrel{d_1}{\\longrightarrow} C^2 \\stackrel{d_2}{\\longrightarrow} \\cdots \\stackrel{d_{n-1}}{\\longrightarrow} C^n \\longrightarrow 0. ", "04d06c41023fa9b103747ebb5689f586": "x_{1,t}", "04d12470043ba1d37c0a63948d1c200b": "U=\\sigma_x", "04d15429e1cebd053387fbefbe192dc7": "p^a q^b\\ ", "04d1e7a488ce709acd268317cfb2defe": "X_z(z)=\\frac{-1}{H(1+\\phi(z)\\bar\\phi(z))^2} \\left \\{(1-\\phi(z)^2, i(1+\\phi(z)^2), 2\\phi(z)) \\frac{\\bar{\\partial\\phi}}{\\partial \\bar z}(z) \\right \\}", "04d204bbbcfb647c1628359b7d3f8ec4": "R_h", "04d223813fed198d04db780b0d506017": "A = k[t^2, t^3] \\subset B = k[t]", "04d243e0bc2764202af0a72263fb94e9": "f\\!\\left(x\\right) \\geq f\\!\\left(y\\right)", "04d25560f37662e7a63f9f37757271d2": "\\sum_{n=1}^\\infty \\frac{t^{2n}}{n} \\zeta(2n) = \n\\log \\left(\\frac{\\pi t} {\\sin (\\pi t)}\\right)", "04d265067859ccc4737cd584b0b3c99e": "\n\\hat{\\rho} (\\mathbf{r}) = N \\sum_{j=1}^n \\int_0^1 ds \n\\delta \\left( \\mathbf{r} - \\mathbf{r}_j (s) \\right).\n", "04d29d26a6f00d0951137ace61c9ff20": " |T_j| < \\frac{t}{2d}] \\leq n \\cdot n^{-2d} \\leq n^{-d}", "04d3b323a3ea25db0d1633b89147ece0": "i = 1, \\dots, n", "04d42f232c194ce477ed3d8ef88de683": "A = \\sum_i {x_i\\, A_i}", "04d444c8f2f6c71b8b5785e58eacb9eb": "c_{sound} = \\sqrt{\\left(\\frac{\\partial P}{\\partial \\rho}\\right)_{s}} = \\sqrt{\\frac{\\gamma P}{\\rho}}=\\sqrt{\\frac{\\gamma R T}{M}} ", "04d4721517edfca170ac3802db26813e": "x = 0 \\!", "04d47874f992aef898ec8e9a27bdb7da": " z \\in \\partial \\Pi_A", "04d49a21d2d751d28a93329700556599": "A'(z) = \\sum_{k=0}^{N}a_k\\gamma^kz^{-k}", "04d4a4b969a9937e007085d733918c7f": "k>0\\,", "04d4afd88b88bf73e429f0b39c6abfd3": "C = e^{- \\frac{K \\cdot t}{V} + const } \\qquad(2c)", "04d4f97b34dc23f791fe306b0e995dfc": "\\mathbf{C}\\,\\!", "04d51ab83469a9216904129f03469844": "P_{50} = 5^{50} \\cdot \\frac{\\Gamma \\left(3/5 + 50\\right) }{\\Gamma \\left( 3 / 5 \\right) } \\approx 3.78438 \\times 10^{98}. ", "04d52171a5c6eca9a3d0bbd805b2b536": "VSWR=\\frac{|V|_{max}}{|V|_{min}}=\\frac{1+|\\Gamma|}{1-|\\Gamma|}", "04d59985c001bc8f54707f446ce9fd33": "\\{x'_k:k r", "04deed9912b56be9d7f2882643ef19ca": "e=E-127", "04df27d3472a35811336a4a701d68984": "\\vdash \\dashv, \\vDash, \\Vdash, \\models \\!", "04df55da0404fcec41530fd9c731776f": "\\delta_{x}:S \\times X \\rightarrow S", "04df898e5535afa983878f0186ec6cd9": "\\int_{\\Omega} v_j v_k\\,ds", "04dfaf8dc62e04aadd3b85654d8ea067": "{{documentation}}", "04dfc1faa60e22d2c4b3f89cf549d55a": "|k|/n", "04dfc825159e549dcf4f938211d845fe": "x^n\\in\\mathcal{X}^n", "04dfca30d4aab589307ba2b8d5b82d6e": "a|n\\rangle=\\sqrt{n}|n-1\\rangle", "04dff67803d6445e3af17ecc63827668": " t \\ne t_n ", "04e03460dc9bf154dd788748a06c2472": "\\mathcal{S}^\\prime", "04e056e18f107f3b4f74fcebcb56042c": "S_m(P,T)=S_m(P_0,T_0)+C_P \\ln \\frac {T}{T_0}-R\\ln\\frac{P}{P_0}.", "04e101d162346f2087821eda0a2354fa": "\\lim_{\\delta \\downarrow 0} \\delta \\log \\mu_{\\delta} (S) = - \\inf_{x \\in S} I(x). \\quad \\mbox{(E)}", "04e210fc8bdbcf4c7a865e30d482b05a": "\n\\operatorname{Jacobian}\\left( \\frac{x, y}{A, B} \\right)\n =\\begin{vmatrix}\n -(B^2-4A)^{-\\frac{1}{2}} & \\frac{1+B(B^2-4A)^{-\\frac{1}{2}}}{2} \\\\\n (B^2-4A)^{-\\frac{1}{2}} & \\frac{1-B(B^2-4A)^{-\\frac{1}{2}}}{2} \\\\\n \\end{vmatrix}\n = (B^2-4A)^{-\\frac{1}{2}}\n", "04e22f55c7aa1d710ace6b0dc6be18de": "\\sin^2(\\theta)+\\cos^2(\\theta)=1,", "04e26e6c3597879a21c0cf8662316481": "L((1+n)^x \\mod n^2) \\equiv x \\pmod{n}", "04e2cf6db6627e8b2e12e109e878cf46": "G=\\cfrac{E}{2(1+\\nu)}", "04e2dd42a81fe5e382a2a47dda3af106": "\\lnot \\;\\exists \\;xO(x)", "04e30a138457b99329132428dcf4682c": "\\Omega=2^\\mathbb{N}=\\{H,T\\}^\\mathbb{N}", "04e340f9a578caa3bf7db69949976347": "\\pi_{xy}", "04e346a8813bd90c853d764753d1bc1a": "2 \\rightarrow 1.", "04e3ae971cea84aab401463bc236844d": "\\int_0^{\\theta} \\operatorname{Sl}_{2m+1}(x)\\,dx=\\zeta(2m+2)-\\operatorname{Cl}_{2m+2}(\\theta)", "04e3f5127b45a587cee6af90f1652ebf": "\nc_1(q) = 1, \\;\\;\n c_q(1) = \\mu(q), \\;\n\\mbox{ and }\\; c_q(q) =\n\\phi(q)\n.\n", "04e3f78844e2687a97fa0932a63c94b8": "\\mathbf{e}_i=\\mathbf{e}_{i'} (A^{-1})^{i'}_i,\\,", "04e4a643ec333306aab41015824e77b8": "\\mathcal{P} = \\mathcal{C}\\times\\mathcal{M} = \\{ (\\mathbf{q},\\mathbf{p})\\in\\mathbb{R}^{2N} \\} \\,,", "04e4bc90fc5cb5af671c8ea0303b02b2": " P\\left[ (\\tilde{X}^n,\\tilde{Y}^n) \\in A_{\\varepsilon}^n(X,Y) \\right] \\leqslant 2^{-n (I(X;Y) - 3 \\epsilon)} ", "04e4ea40b54a681cc441a335f195180d": "s_b(z)", "04e50826ed9a1064bb210b8d98d7904e": "\\rho(x_1,x_2)=0", "04e54f1f9c3733f61da4feb2f4b9dd70": " w_{ij}^{\\nu} = w_{ij}^{\\nu-1}\n\t\t +\\frac{1}{n}\\epsilon_{i}^{\\nu} \\epsilon_{j}^{\\nu} \n\t\t -\\frac{1}{n}\\epsilon_{i}^{\\nu} h_{ji}^{\\nu}\n\t\t -\\frac{1}{n}\\epsilon_{j}^{\\nu} h_{ij}^{\\nu}\n\t\t ", "04e5c0b589f343996819a788a67d2ffc": "\\text{R-X}^-\\text{C}^+\\,+\\, \\text{M}^+ \\, \\text{B}^- \\rightleftarrows \\,\\text{R-X}^-\\text{M}^+ \\,+\\, \\text{C}^+ \\,+\\, \\text{B}^-", "04e5c6491f65a8ff500707053264975b": "n=p_1+\\cdots +p_c", "04e60aceefaac27f43d9266b4e898495": "R = R_x(\\gamma) \\, R_y(\\beta) \\, R_z(\\alpha)\\,\\!", "04e6493104c14d65c65a7e3ae307874c": " \\tau=\\frac{t}{|c|} ", "04e6b5ce6f920b15c208e31017181e58": " s^2 = \\frac{ b }{ a ( a + b ) } + \\frac{ d } {c ( c + d ) } ", "04e6e4c84f34a47815aa1c74bddce026": "\\eta(0)=0", "04e6ea5a4cfc7efe45577a4968b32fb4": "H^I_p(H^{II}_q(P_\\bull \\otimes Q_\\bull)) = H^I_p(P_\\bull \\otimes H^{II}_q(Q_\\bull))", "04e7717b13155456972e9ae515c2e5df": "u(x,t)=\\int_{0}^{t} \\frac{x}{\\sqrt{4\\pi k(t-s)^3}} \\exp\\left(-\\frac{x^2}{4k(t-s)}\\right)h(s)\\,ds, \\qquad\\forall x>0", "04e7909cf29056a41c53b565a2ee68c2": "\\mathbf{v} = v(t)\\mathbf{u}_\\mathrm{t}(s) \\ ,", "04e7a342417c7b4a3fd09ac71f00b250": "m_p\\left(r\\right)\\rightarrow m_0", "04e7bce75675f3c6e2bd9cd9de82df4d": "F_\\alpha = \\sum_{\\alpha \\succeq \\beta} M_\\beta, \\, ", "04e7f66067d55d79409ae532dff606d4": "\\operatorname{E}(c) = \\frac{1}{N+1}\\sum_{i=0}^{N} i", "04e7ffbbe1a99fadc9c4cef92ec795e9": "\\overline{O} = O", "04e801a95286ebb4a962bb8f59c4073b": "\\mathcal{D}(A).", "04e83ca06b3a78ee3bda963bd4a2fd56": "\n\\operatorname{Li}_2(u) + \\operatorname{Li}_2(v) - \\operatorname{Li}_2(uv) = \\operatorname{Li}_2 \\left( \\frac{u-uv}{1-uv} \\right) + \\operatorname{Li}_2 \\left( \\frac{v-uv}{1-uv} \\right) + \\ln \\left( \\frac{1-u}{1-uv} \\right) \\ln\\left( \\frac{1-v}{1-uv} \\right),\n", "04e880bbfa917de599eeaae85bc0bc85": "e^2", "04e9155ef246bb508734c8e560f378d9": "\\pi_2 M", "04e96147afb31e6766e43593312db18d": "t=pq^{-1}=\\gamma^r\\gamma^{-s}=\\gamma^{r-s}=h^{\\alpha\\beta(r-s)}", "04e97e24c18920b8bf657dd449790432": "s = a^pb^pc^p", "04e98084ee989d47e1373fa9fddb2d74": "shared(d)", "04e999090e2c17187ef280070a248637": "{d}", "04ea2cd547a555329ca7624d1ecea049": "E^{(+)}(\\mathbf {r}, t) = i\\sum_{i}[\\frac{\\hbar\\omega_{i}}{2}]^{1/2}\\hat{a}_{i}\\mathbf{\\varepsilon}_{i}e^{i(\\mathbf {k}_{i}\\cdot\\mathbf {r} - \\omega_{i}t)}", "04ea6720d76a2f9c83ef10db3f587c23": "y_0 \\in \\{0, 1\\}^m", "04eae6d13d1528605d9dda775789745e": " \\vee: \\mathrm{Con}(\\mathcal{A}) \\times \\mathrm{Con}(\\mathcal{A}) \\to \\mathrm{Con}(\\mathcal{A})", "04eaf227fb05fa613254a4b9ba3713a6": "K(\\!(T_n)\\!)", "04eb06021221b54fb4506e7fd94fb64e": " ||y - A(x_1+x_2+ \\cdots +x_n)|| < \\delta \\, 2^{-n} \\, ; \\quad (2) ", "04eb0f45582acc6ccd08133edaadd7b9": "g(r)=\\exp[-\\beta w(r)]", "04eb7f0e9b851eed2913fd1244b6e9f2": "c \\in \\Sigma^n", "04ebd70849b60dff8a8ef599cd00d654": "v_+ = v_- = v_{\\text{out}}.\\,", "04ec04c064836ad3876f5cbfd3c2ec4f": "|a|(1+a/4)\\pi \\,", "04ec3070cbc012e2cfa4f9fe5f939abd": "v = \\frac{\\omega}{2 \\pi c} (y_1 - y_2)", "04ec42dd7a5255c85e090b33973a8ceb": "s^2 = \\frac{1}{3N} \\left\\{ \\sum_{n=1}^{N} ( x_{n,1} - \\bar{x})^2 + \\sum_{n=1}^{N} ( x_{n,2} - \\bar{x})^2 + \\sum_{n=1}^{N} ( x_{n,3} - \\bar{x})^2\\right\\} ", "04ec6053f6146c2eb3a0bd4e08578401": "(\\tfrac{p}{q})=1", "04ecb34572dfeb5b6f2032e0bfc18806": "y_c = \\frac{2}{3}\\sqrt{M_c} ", "04ecb94754fc963b1045f89f2d595c44": "X\\subseteq V", "04eccba89e407f705f8ef660d7b4d614": "g=G \\frac {m_1}{r^2}=(6.6742 \\times 10^{-11}) \\frac{5.9736 \\times 10^{24}}{(6.37101 \\times 10^6)^2}=9.822 \\mbox{m} \\cdot \\mbox{s}^{-2}", "04ecef256c10ed98b0dcffcef97251c0": "W_T^{(2)}(\\omega)", "04ed090ac7a3e0e380bca8de5f6b41ed": "\\beta \\in \\mathcal{O}_k", "04ed6b29079f24735c5b29745ef0a1b7": "a_{1}+a_{14}", "04eda5539ba8311ed9023276aaf1b885": "\n \\sum_{n=1}^{l_\\lambda} \\; \\Gamma^{(\\lambda)} (R)_{nm}^*\\;\\Gamma^{(\\lambda)} (R)_{nk} = \\delta_{mk} \\quad \\hbox{for all}\\quad R \\in G,\n", "04edb01258a81268e75b640c739649bc": "a(x-y) \\bmod 2^w", "04edf159dde4bfd8233801c022187323": "\\psi(\\alpha+1)=\\psi(\\alpha)=\\delta", "04ee0ff1daec33fb96547c3f6fdfb597": "p_w(\\theta)=\\frac{1}{2\\pi}\\,\\sum_{n=-\\infty}^{\\infty} \\phi(-n)\\,e^{in\\theta} = \\frac{1}{2\\pi}\\,\\sum_{n=-\\infty}^{\\infty} \\phi(-n)\\,z^n ", "04ee3e02987ce86c2a483f4f4cb4dcf0": "\\sigma_{Z_1}^2.", "04ee6592608fcb53fb98eb913894d483": "\nIMD_i = \\left( e_i^t - h_i^t \\right) \\times \\left( G_i - G \\right)\n", "04eed3678d03fd0b3dc1d3f672bdeae1": "e_q(x) = \\exp(x) \\text{ if } q = 1 ", "04eed96382514f7c340f4e53fe09db69": "(hkl)", "04ef1e759d5f3184342d6948487a53d5": "\n\\mathcal{L} [\\varphi (x)] =\n-{1\\over 4} F_{\\mu \\nu} F^{\\mu \\nu} + {1\\over 2} m^2 A_{\\mu} A^{\\mu} + A_{\\mu} J^{\\mu}\n", "04efa9f79c535e637f267063d5460fba": "(F \\cdot G)[A] = \\sum_{A=B+C} F[B] \\times G[C].", "04efe8396f5fc91ac3d7e5b549fcfb7d": "\\begin{matrix}{52 \\choose 5} = 2,598,960\\end{matrix}", "04f052c0bde0bdcb192ed417678a785a": "\\{ www : w \\in \\{a,b\\}^{*}\\}", "04f06ca1499e8a908f20f92cbc1cb863": "g=h^{-1}th", "04f081930149949cf30a1b9b8635c47e": "X \\in \\C", "04f084963d52d685bb83410abe86643e": "x = c_1 c_2 \\ldots p \\ldots p' \\ldots x_n ", "04f0980f6e8fa97d144641ec8b6b8ff4": " (L_0 - \\tilde{L}_0) |\\Psi\\rangle = 0 ", "04f14932ad6780bb4713155a180f0040": "\ny=\\sqrt{a^2-x^2}, \\quad\ny'=\\frac{-x}{\\sqrt{a^2-x^2}}, \\quad\ny''=\\frac{-a^2}{(a^2-x^2)^{3/2}},\\quad\nR=|-a| =a.\n", "04f188c7b8ed64c7fe2137cda960608a": "\n \\mathbf{M}_H\n =\n \\begin{bmatrix}\n \\;\\;\\,0.38971 & 0.68898 & -0.07868 \\\\\n -0.22981 & 1.18340 & \\;\\;\\,0.04641 \\\\\n \\;\\;\\,0.00000 & 0.00000 & \\;\\;\\,1.00000\n \\end{bmatrix}\n", "04f18cafc2ee54e4b6c66b4ecbd09eca": "R_2 - R_1 = R \\sqrt{1 + \\frac{x_2^2}{R^2} + \\frac{y_2^2}{R^2}} - R \\sqrt{1 + \\frac{x_1^2}{R^2} + \\frac{y_1^2}{R^2}}", "04f1a9486e22042a59277d7022778e75": "v_{3}", "04f1dab970ad559b1fe9c0a1a1bd2a38": "1\\le i, j\\le k", "04f1ef249f07bd7a5759fd398eee3f4e": "\\Gamma(s,z) = \\Gamma(s) - \\gamma(s, z)", "04f1ff7f5c9bde7065bb8bfa4ef93d41": "\n\\begin{align}\n\\mathbb{E}\\Bigl[\\liminf_{n\\to\\infty}X_n\\,\\Big|\\,\\mathcal G\\Bigr]\n&=\\mathbb{E}[X|\\mathcal G]\n=\\mathbb{E}\\Bigl[\\lim_{k\\to\\infty}Y_k\\,\\Big|\\,\\mathcal G\\Bigr]\n=\\lim_{k\\to\\infty}\\mathbb{E}[Y_k|\\mathcal G]\\\\\n&\\le\\lim_{k\\to\\infty} \\inf_{n\\ge k}\\mathbb{E}[X_n|\\mathcal G]\n=\\liminf_{n\\to\\infty}\\,\\mathbb{E}[X_n|\\mathcal G].\n\\end{align}\n", "04f236a5d65eb2902a7521e68752fd15": "\\sum _{ v \\neq v0} (q_v - q _{v \\cap w})", "04f23f77a4da740c280d3617cb0c2a1b": "L(G) = \\{ w \\in T^{*} : S \\Rightarrow_{p_1} ... \\Rightarrow_{p_n} w \\}", "04f25fc454fe9c1218a15db30e347a68": "0^\\circ", "04f2e33784b89346f3cb7a773ace6986": " \\scriptstyle p_i = p^{\\star}_i x_i", "04f3137ff098aa5741e25f9e0a30097f": "g_0=1,", "04f37063c459dc5067b3a505eb13254a": "\\scriptstyle \\phi(a)", "04f394df1d823a51f2052efd822ee5ba": "xzy^{-1}xx^{-1}yz^{-1}zz^{-1}yz\\;\\;\\longrightarrow\\;\\;xyz.", "04f4251e7aab69f16e4921ae9c10f3fa": "X_{SC}", "04f42f9c70ae2265168f604d0e77823c": "\\kappa =\\frac{1}{\\rho}=\\frac{C}{R}", "04f45059d9e134e6f04406c34a24902f": "[j]_{TOT} \\,=\\, [j] + \\sum_{i=1}^{N_S}\\, \\nu_{i,j} \\,[i] ", "04f46896df145356b2cfb916ff84bee0": " A^{-1}=\\frac{(-1)^{n-1}}{\\det(A)}(A^{n-1}+c_{n-1}A^{n-2}+\\cdots+c_{1}I_n).", "04f46ef4f610873f0b607299831248f3": " S_m = \\int_0^m {\\left( {x \\over {2\\sqrt{x^2+4}}} + {{m+2} \\over {2m}} \\right)} \\, dx. ", "04f4fbb099ecddf77a8bd49e549a4796": " \\overbrace{\\smile \\smile -\\smile}^{\\mathrm{Foot 9}} | \\overbrace{\\underbrace{-\\smile}_{\\mathrm{Brahma}}}^{\\mathrm{Foot X}} | \\overbrace{\\smile\\smile\\smile-}^{\\mathrm{Foot 11}} || ", "04f5027f7716ebfad5764a4c176a88cf": "Q_{q} = \\frac {1} {\\sqrt {N}} \\sum_{l} u_{l} e^{- i q a l } ", "04f50f7dd4b7b9a5dc57ade5af0e862d": "\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{43}+\\cdots=1,", "04f52ca3d98d9b896f04128244d4ddf1": "d\\mathbf x'=\\mathbf U \\,d\\mathbf X\\,\\!", "04f53647bec0920c0d3570f033877ffa": "D_{n,k} = {n \\choose k} \\cdot D_{n-k,0}.", "04f5934929b135782033055aefa70325": "_C^E", "04f59a19dbe6293b61640ab1810b6854": "A=\\frac {4}{3}a^2", "04f5bbaf6b93197b7c2e2061d9751f1e": "\\varphi(z) = \\int (T_z f_z)g_z \\, d\\mu_2", "04f5be0bccdbf812a6640f4b88fd67a0": "a+b+c", "04f5cb345657f4532084a899aec6339b": " \\varphi: \\mathcal F\\rightarrow \\mathcal G ", "04f5f57d53e9bffe95e1969a780328c1": "f^{*}", "04f6027f91c7dee5c61ff63a88813e6d": "P= (X_1:Y_1:Z_1)", "04f60f28c56bcac963753ada77addbb5": "\\mathbf{P} = \\mathrm{d} \\langle \\mathbf{p} \\rangle /\\mathrm{d} V \\,\\!", "04f62906170dc100289eb31b2819479c": " r \\sqrt{4-2\\sqrt{2}} = \\frac{a}{2}\\sqrt{4+2\\sqrt{2}} \\!\\, ", "04f6460714e9c2cc801ea09b76dd543d": "u^2-a_1u+\\frac{a_1^2}{4}=a_0+\\frac{a_1^2}{4} .", "04f6c757ca09e262b8f61e709cd2b567": "\\displaystyle \\partial_t u + \\beta\\, t^n\\, \\partial_x^3 u + \\alpha\\, t^nu\\, \\partial_x u= 0", "04f6ca8294b413fe37a829daee69bee3": "\\Psi_L\\left(0\\right)=\\Psi_G\\left(0\\right)+\\text{H.O.T.},\\,", "04f6ce540b30e0340b87e29e5ece08c5": " m,\\, 0x_0", "05017b16d121273397774b34532bf10b": "k^2-2\\,i\\,k\\,x-1\\,=\\,0", "0501ab330f701f2e5ddaaaa5d8cf2af2": "f(z)=(z-a)^ng(z)\\ \\mbox{and}\\ g(a)\\neq 0.\\,", "0501c25234c86d03e007782268f04893": "x^3=(0,0,1)", "0501ceca7bd0d96b04794c2a514b6f37": "\\mathcal{L} f = -\\partial_t f(t) + r(t) f(t).", "0501eedfb34554c82f3ad105604c242a": "_{\\sim}\\!", "050202b86b163e362266acc78f67be89": "\\Box\\phi", "050244e419735079939749935cfc6c78": "\\begin{align}&(1 + 2\\mu)u_{i,j}^{n+1} - \\frac{\\mu}{2}\\left(u_{i+1,j}^{n+1} + u_{i-1,j}^{n+1} + u_{i,j+1}^{n+1} + u_{i,j-1}^{n+1}\\right) \\\\ & \\quad = (1 - 2\\mu)u_{i,j}^{n} + \\frac{\\mu}{2}\\left(u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n}\\right).\\end{align}", "05027bd684c505bb972c1b177b20c56d": " P( | X | \\ge k ) \\le \\frac{ 4 \\operatorname{ E }( X^2 ) } { 9k^2 } \\quad\\text{if} \\quad k^2 \\ge \\frac{ 4 } { 3 } \\operatorname{E} (X^2), ", "0502b073ec2e7800a308776ab0811922": "\\hat\\psi(\\vec r) = \\sum\\limits_i w_i^\\alpha (\\vec r) b_i^\\alpha", "0502cd530e4f328821d546d4a0944188": "F_n(x) = \\frac{1}{n} \\left(\\frac{\\sin \\frac{n x}{2}}{\\sin \\frac{x}{2}}\\right)^2 =\n\\frac{1}{n} \\frac{1 - \\cos(nx)}{1 - \\cos x} \n", "05035d210f3d1496caf59b529bc1410a": " \\zeta = \\frac{\\delta}{\\sqrt{(2\\pi)^2+\\delta^2}} \\qquad \\text{where} \\qquad \\delta \\triangleq \\ln\\frac{x_1}{x_2}.", "0503d5a4ed130ec62d2bc8a2f654b56a": "Z^\\dagger", "05041b22d390f8f8b61338f65a1724c4": "K_{\\lceil n/r\\rceil, \\lceil n/r\\rceil, \\ldots, \\lfloor n/r\\rfloor, \\lfloor n/r\\rfloor}.", "050456a2a2c341938f221eb3e0b50372": "\\omega = \\frac{-1 + \\sqrt{-3}}{2}=e^\\frac{2\\pi i}{3}", "050458ff63a54df48024c9ebb9932d84": "\\,\\gamma", "050473e9a8a8e6fac1b0dfc8960fb55e": "\\frac{3}{8}\\sqrt{35}\\cos(4\\theta)\\cos^4(\\phi)", "0504c1d23e37f48a62ba1437e9cab3e2": " M_{i,j}", "0505018fc0ed9786c0216099fc3b789c": "a=(v^2-u^2)(v^2+u^2), \\,", "05051b000eeb56e299912b68d5c5e2c0": "y - y_1 = m( x - x_1 ),\\,", "05052479786e4f2b053609801f833d7b": "F(d, k)", "050542d2523a82915c1fdad950acdc5e": "\\mathcal{A} f (x) = 0.", "05055bfc9a5b48f205c595eb622a5fb4": "S({\\Lambda^\\mu}_\\nu) = {(\\Lambda^{-1})^\\mu}_\\nu = {\\Lambda_\\nu}^\\mu \\,", "0505676729e95ec9f4958bceb2658882": "\\bar{\\nu}_e + p \\to e^+ + n", "0505b3b8e4b450288f5985d487fd641c": "\\omega = \\frac{\\lambda \\cdot v}{r}", "05061073560f5cf8ce91f9b49a796c9a": " \\theta_A = \\frac {P}{P+P_0}", "0506af4c0ad7aa17657c8aaf095acc26": "d_\\pm", "050704d18bf227d8d89a90f3209b39bb": "\\displaystyle{u_x=-v_y,\\,\\, u_y=v_x.}", "05070e88dfde3a30bb688c009c8f6bb4": "n_z", "050710f82f53f780d2c7fd7795137c44": "Y_1, Y_2", "05073a04fe1376c3b0c45106273f9187": "a\\sim b", "0507c11a8aee36060834108d45eec574": "\\mathbb R v_1,\\dots,\\mathbb R v_6", "0507ca3317618b35b1e64a4dbc5ad5da": "\\text{MTBF} = \\theta. \\!", "0507d05470ff6520b4965cb227d62218": "10_{123}", "0507e3cf2687b0f76c74a01a26568226": "\\hat{\\lambda}_i", "050802f5a55c0af3f857280e59e25a6d": "S_x(\\omega) = \\hat{x}(\\omega)\\hat{x}^*(\\omega)", "0508352d6beb495b1dffad1f8726fb9e": "\\frac{1}{\\tau} = ar", "0508b61cf5f29dcbc2d668fa5e93fd4f": "\\frac{1}{T_2^*}=\\frac{1}{T_2}+\\frac{1}{T_{inhom}} = \\frac{1}{T_2}+\\gamma \\Delta B_0 ", "05090603b60ddc4d45703252f192d9d6": "\\chi_1^2", "05096bd9c0a26b57faa623e920635e0e": "|U| >1/2,\\ V = W = 0,", "050992bf4515002318edb223863a9ae0": "\\alpha \\in A", "05099cbdcccc7a04282d0f96c127de8a": "R_{abcd} \\, R^{abcd}", "0509b32282371643e6308a79f7d4f5dc": "f_\\ell^m=\\int_{\\Omega} f(\\theta,\\varphi)\\, Y_\\ell^{m*}(\\theta,\\varphi)\\,d\\Omega = \\int_0^{2\\pi}d\\varphi\\int_0^\\pi \\,d\\theta\\,\\sin\\theta f(\\theta,\\varphi)Y_\\ell^{m*} (\\theta,\\varphi).", "0509d73229a2e1c0ce410544d2c0c25d": "(\\partial T)_P=1", "0509f544a02d65ac9b57509058a3a05e": " X_t = c + \\sum_{i=1}^p \\varphi_i X_{t-i}+ \\varepsilon_t .\\,", "050a2bd6fe954b091760195ffaaa0808": "\\ddot x - 2n\\dot y = \\frac{\\delta U}{\\delta x}", "050a4f9d47d3514082e7fa0c2ed2da90": "\\sqrt{2} \\ln(1 + \\sqrt{2})", "050a580104aa0173c165551a3e383357": "Z = \\left(\\overline{X}_n-\\mu\\right)\\frac{\\sqrt{n}}{\\sigma}", "050a90d6a372aebd4a064da88365182c": " \\phi _1(z) = (1-z)/2 \\quad z \\in [0,1].", "050a93457b3b36469a4362c630c68575": "\n\\sum_{n=0}^{\\infin} (-1)^n\n", "050b2d78abf6b855c631c27406f6763f": "(A \\vee B \\vee C) \\wedge (\\overline{A} \\vee \\overline{C}) \\wedge (\\overline{B} \\vee \\overline{C})", "050b377515d021da5001b6ef871978a8": "\\mathcal{F}_i= - \\frac{\\partial \\mathcal{V}}{\\partial q_i}\\, ", "050b5355658ab527c84edb8f00f387d6": "\nH^\\dagger - H = 0\n\\,", "050b57a5f8f2f3a7bf5992a5f74069d3": "c_0 = S-1 \\,\\!", "050b5e0fe4d1ae8a4a9919dc545fa7e7": "e^{ar}", "050b89800d1de9d236fd5a26e225bb5b": " \\csc \\theta\\! ", "050c2a34694f64f4b312fe044bfa151f": "P^{(i)}_{0}", "050c5d1a59538341e67943d438532d5d": "na_0x^n + (n-1)a_1x^{n-1} + \\cdots + 2a_{n-2}x^2 + a_{n-1}x = 0 \\, ", "050c6f71cd07650bd1f7ae739b59ba1d": "F_{1 \\rightarrow 2}", "050c79c03277c6a6ad35246617006d32": "\n \\begin{align}\n p_0 = -\\frac{de_0}{dV} = \\frac{\\rho C_0^2}{2s^4(1-\\chi)} \\Biggl[& \\frac{s}{(1 - s\\chi)^2} \\Bigl (- \\Gamma_0^2(1 - \\chi)(1 -s\\chi) \n + \\Gamma_0 [s \\{4 (\\chi-1) \\chi s-2 \\chi+3\\}-1] \\\\\n & - \\exp(\\Gamma_0\\chi)[\\Gamma_0(\\chi-1) -1](1-s\\chi)^2(\\Gamma_0-3s) + s [3-\\chi s \\{(\\chi-2) s+4\\}]\\Bigr) \\\\\n & - \\exp\\left[-\\tfrac{\\Gamma_0}{s} (1-s\\chi)\\right][\\Gamma_0(\\chi-1) - 1](\\Gamma_0^2 - 4 \\Gamma_0 s + 2 s^2)(\\text{Ei}[\\tfrac{\\Gamma_0}{s} (1-s\\chi )] - \\text{Ei}[\\tfrac{\\Gamma_0}{s}]) \\Biggr] \\,.\n \\end{align}\n ", "050d2253a1b35110e73f5b61e3d64d28": "\\int_{ L_0 + L_1 + L_2 } \\left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \\right ) = \\iint \\limits_{R_C} s(x,t) dx dt. ", "050d2c73ed3c5b1dc2f46f8b057a9a64": "p(\\theta | y, \\xi) = \\frac{p(y | \\theta, \\xi) p(\\theta)}{p(y | \\xi)} \\, ,", "050d3009fd9b0b042952de0e4d937f19": "\\frac{L}{r^{2}} \\frac{d}{d\\theta} \\left( \\frac{L}{mr^{2}} \\frac{dr}{d\\theta} \\right) = -\\frac{{2}L^{2}}{mr^{5}} \\left( \\frac{dr}{d\\theta} \\right)^2 + \\frac{L^{2}}{mr^{4}} \\frac{d^{2}r}{d\\theta^{2}}\n", "050da762cbd9b00e5919fcc071c87259": "\\int_0^\\infty\\frac{\\sin t}{t}\\, dt=\\int_{0}^{\\infty}\\mathcal{L}\\{\\sin t\\}(s)\\; ds=\\int_{0}^{\\infty}\\frac{1}{s^{2}+1}\\, ds=\\arctan s\\bigg|_{0}^{\\infty}=\\frac{\\pi}{2},\n", "050db563e14ba6743e9ce8a9f6a9f9a3": " \\boldsymbol\\beta^{(s+1)} = \\boldsymbol\\beta^{(s)} - \\mathbf H^{-1} \\mathbf g \\, ", "050dce3386e1fefedd89e8bce5018b68": "\\left. {\\color{white}...}\\ \\omega v \\left(\\cos\\alpha + \\omega t \\sin \\alpha \\right) \\right]\\ ", "050e1820f88caa93847c1c1826795b4c": "\\frac{p_k}{p_{k-1}} = a + \\frac{b}{k}, \\qquad k = 1, 2, 3, \\dots", "050e63bdc63bf7ed99a58f1cb20b4610": "\\mathrm{Div}^0(C)", "050ed507c56e133906e661314f467dcf": "\\sup_n\\left|\\sum_{i=1}^n x_i\\right|", "050eede33f602f8ec77ef8203acb103f": "\\mathbf{\\Sigma}^1_1", "050f1cabedeafb9366261993192f1252": " \\ \\alpha_i ", "050f2343beede00d97ce19ffdd84280b": " TE_{01n} ", "050f2abc0b8bc0b355cb908860cf4119": "\\rm \\ SCl_2 + Cl_2 \\xrightarrow{193K} SCl_4", "050f771089c62964d9e54d7ae690bc6a": " \\omega_{\\mu} = e_{\\mu\\nu\\rho\\sigma}\\xi^{\\nu}\\nabla^{\\rho}\\xi^{\\sigma}", "050f89274e7d82bcb8198954f77106ba": "E_c : z \\to e^z + c \\,", "050fd765812551d51d962c54c4a8c8bb": "\\{r,s\\}", "050fdd1960dfdbab718e82719d89afa9": "R_{\\rho z}=8\\pi T_{\\rho z}", "050ffa5ef6f992064ea682bfeae6ac8b": "p^k,", "0510097af5b114125816748fa362a294": "EL(\\Gamma_1\\cup\\Gamma_2)\\ge \\bigl(EL(\\Gamma_1)^{-1}+EL(\\Gamma_2)^{-1}\\bigr)^{-1}.", "05109063960af84cd328819ba140fa94": "[\\cdot,\\cdot]\\colon\n\\mathfrak{g} \\wedge \\mathfrak{g} \\to \\mathfrak{g}", "05109474b54025b7f9e935a25b01b1e1": "exp(-z^2)", "0510e31aee49c3fbd1d39dbd5d5f84f9": "13 = (17-4) \\mod {26}", "05114f16e74e3b815b33172483b79ca2": "\\scriptstyle \\bar{X}_i=\\frac{1}{m}\\sum_{j=1}^{m}X_{ij}", "051188924a0fba93a9c1ecc164215d7d": "\\rho_h", "0511b7daa53ac131c9a6ae3d745ec8db": "\\mathfrak q", "0511fc24827cdcffed5ea19bb6124789": "\\frac{u_{i+1}-u_i}{\\Delta x}\\ f", "05120b211561ca725f9785dfbefe359f": "AX -XB = Y ", "05121e6c9874bc5a4cf7817470a670ed": "\n R_0 = \\cfrac{d\\epsilon^p_2}{d\\epsilon^p_3} = \\cfrac{H}{G} ~;~~\n R_{90} = \\cfrac{d\\epsilon^p_1}{d\\epsilon^p_3} = \\cfrac{H}{F} ~.\n ", "051230c786f41cee9ecd2f4bd8806de0": "s_1 = c_1e_1", "051315e37a1615b3dbaf5ec61fa30952": "\\tau(p^{r + 1}) = \\tau(p)\\tau(p^r) - p^{11}\\tau(p^{r - 1})", "051344f71c00744c96451a881eb6364d": "\\nabla \\times \\mathbf{E} = -\\frac {\\partial \\mathbf{B}}{\\partial t}", "0513a6272599ff46057f412f576460cd": "d(\\gamma A) = Af_{ij}d\\epsilon_{ij}", "0513acacdfeb03bc371c4ebde470299c": "y_{n} = c_{n-1}y_{n-1} + c_{n-2}y_{n-2} + \\cdots +c_{0}y_{0}.", "0514314546f794ec13e571b5c8c4c107": "\\ell = 2 a", "05143c911e7294959a8d8ca0d12c71d7": "D=\\{1,2\\}, P(1)=\\bot, P(2)=\\top, c=1", "05144cc001f66271c26c893017144baf": "\\min E_{T} = \\sum_{i}\\Big[ E_i(r_i) + \\sum_{i\\ne j} E_{ij}(r_i, r_j)\\Big] \\, ", "051452f6a6a5a155a444d89a2ca665bc": "2~\\ln r + 1", "0514845fd4a3e78213e7ab88b9dd492a": "E = \\frac{1}{4} Wkd \\theta^2", "0514afb94e82c61cbaa2a3b503a2fab4": " u(R,t) = \\frac{dR}{dt} = \\frac{F(t)}{R^2} ", "0514c16ec7e9eb98c506535d7438bc92": "\\dot{q_i} \\, ", "05150cfbe7764ff9c0bc04c8544ef7e7": "\\nu_\\mathrm{t}", "05151a93e80308a1e909cf45e63beb65": "K \\otimes_\\Q \\mathbb{R} ", "05152c21814653d312d1a9dc611f3975": "\\Delta\\, G_i \\,\\sim \\Gamma (\\Delta t_i/\\nu, \\nu)", "051535f7bc824e59e73b31aeec32d3b8": "\\mbox{female shoe size (Brannock)} = 3\\times\\mbox{foot length in inches}-21", "05158466407bde46b85a8649ade91ec8": "\\Delta_{\\textrm{B}}", "0515ba4dd3540bee4010b6e2718689a6": "K_{SV}~=~Stern-Volmer~constant~for~oxygen~quenching", "0515e1203a6da3e9b342a993d26bb494": "\\Delta _{\\mathcal{L}}(x_{\\perp }) =-1_{1}1_{2}\\frac{\\mathcal{L}(x_{\\perp })\n}{2}\\mathcal{O}_{1},\\text{scalar}\\mathrm{,} ", "0515ecca071219dfab5ed29f01652c71": " E_\\mathrm{stored} = \\frac{1}{2} C V^2,", "05161917f741c897aba47f69fe891a57": "{T_v(s) = V_{out}(s)/V_{in}(s)}\\,", "05168f730983e424739d63483138d587": "\\mathbf{x}_R = A\\mathbf{x}_L", "0516a583f096aee2d1ef45dbd10159e9": "d(\\sigma) \\geq \\frac12 (d(\\sigma 0)+d(\\sigma 1))", "0516cd87df2bccdd7d83c444138de721": " - \\frac {\\hbar ^2}{2m} \\frac {d ^2 \\psi}{dx^2} = E \\psi.", "05172cdab5fd630a4cb101b18fe4f0f3": " -e_2=<0,-1> ", "05173ca87cf4e63b6588070bdcd42071": "\\displaystyle{W(x)W(y)=e^{-{i\\over 2} \\Im (x,y)} W(x+y).}", "051747d010127b31ff30e257312eecf1": "\nRE_{\\hat g} \\,\\, = \\,\\,{{\\hat\\sigma _g \\,} \\over {\\hat g}}\\,\\,\\, \\approx \\,\\,\\,\\sqrt {\\,\\,\\left( {{{s_L } \\over {n_L \\,\\bar L}}} \\right)^2 \\,\\,\\, + \\,\\,\\,\\,4\\left( {{{s_T } \\over {n_T \\,\\bar T}}} \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {{{\\bar \\theta } \\over 2}} \\right)^4 \\left( {{{s_\\theta } \\over {n_\\theta \\,\\bar \\theta }}} \\right)^2 \\,}", "05176183bb00310d71e626f5264ff66b": "\\displaystyle{\\frac{1}{2}R(a,b)=L(a)L(b) -L(b)L(a) +L(ab),}", "0517be056a5873b94503d2bd7e5f9cc1": "\\| Mf \\|_{L^{p}} \\leq C_{p} \\| f \\|_{L^{p}}.", "0517d1f36dae4b67ae3986160d121900": "\\int_{-\\infty}^{\\infty} |\\psi (t)|^2 \\, dt <\\infty.", "0517f31b8aacd320eaf6b16b7fa435e1": "\nZ = \\sum_{n=0}^{\\infty} e^{-n\\beta h\\nu} = \\frac{1}{1 - e^{-\\beta h\\nu}}.\n", "0518a46df04592797fb11f5a9d147616": "\\Delta \\mathbf x\\,\\!", "0518a4ea135ccbe3916da92bbe8e8701": "\\Delta \\bar{e}\\ \\,", "0518cc5d7d6d3cdbb5ab9bc1dc3bf0b5": "\nRD = \\frac{W_\\mathrm{air}}{W_\\mathrm{air} - W_\\mathrm{water}}\\,\n", "0518ce225d7219b5b7b398ce8a548f57": "\\liminf_{\\varepsilon \\to +0} \n \\varepsilon^{-1} \\left\\{ \\gamma^n (A_\\varepsilon) - \\gamma^n(A) \\right\\}\n \\geq \\varphi(\\Phi^{-1}(\\gamma^n(A))),", "05195afa5f1b5313ca387bb548c25dc2": "f(n^k) = kf(n).\\,", "05197f4f8923ce9df2ad252bcdfc1343": "\\dot Q(t)\\ =C^{(V)}_T(V,T)\\, \\dot V(t)\\,+\\,C^{(T)}_V(V,T)\\,\\dot T(t)", "05198d0212461cd43f11908164f4213a": "\\Delta(t) = c_0 + c_1 t + \\cdots + c_n t^n + \\cdots + c_0 t^{2n}", "0519bc388c4b70254424e2de54e23721": "\\theta = v/c = \\kappa", "0519cf07a04ebb3ef1e2693196df08e4": "P_{2}^{1}(x)=-3x(1-x^2)^{1/2}", "0519d4dbdec5bcca4c39bcba98058239": "\\gcd(2^a-1, 2^b-1)=2^{\\gcd(a,b)}-1", "051a7eb36d169001282aa8f35dadc66e": "\nV = V(t). \n", "051ae9d0e81bebfd1186c42463742fdf": "n \\geqslant 0 ", "051b2590dc90f6478107992385384d64": "x=y=z=0,\\,s=10,\\,t=15.", "051b2cc28181aacee228bc94d47bc04c": " \\lambda(t_1) = F(t_1) x(t_1)", "051b39b0bcdd0277e6a15d127af4d094": "\\gamma' (1)", "051b65d0bc2ef8fb4cbdcbc778ea00f9": " \\hat{x} ", "051b7e26712f1115cdc466d49d6b3305": "c_m=\\frac{1}{2\\pi}\\int_\\Gamma \\ln(f_w(\\theta))e^{-i m \\theta}\\,d\\theta", "051bad0de5df71fa5a3d047779cc191d": " \\ell(\\gamma)=\\int_\\gamma \\rho(z) \\, |dz|,\\quad A(D)=\\int_D\\rho^2(x+iy) \\, dx \\, dy, \\quad z=x+iy. ", "051bae02c7c2b9c8414016a40fe8e3bf": " \\; \\text{Var}\\left(\\boldsymbol{\\varepsilon}\\right) = \\sigma^2I_{n \\times n} ", "051c2a7ff34934f6fc05c14807b02861": "\n\\begin{bmatrix}\n B_{11} & B_{12} & 0 & \\cdots & \\cdots & 0 \\\\\n B_{21} & B_{22} & B_{23} & \\ddots & \\ddots & \\vdots \\\\\n 0 & B_{32} & B_{33} & B_{34} & \\ddots & \\vdots \\\\\n \\vdots & \\ddots & B_{43} & B_{44} & B_{45} & 0 \\\\\n \\vdots & \\ddots & \\ddots & B_{54} & B_{55} & B_{56} \\\\\n 0 & \\cdots & \\cdots & 0 & B_{65} & B_{66}\n\\end{bmatrix}\n", "051c4c75a8934dd4d7ef677f2918368c": "{N_i}", "051c780fb650536715a3fcf6121dc9e8": "\\displaystyle{\\log z = \\log |z| + i\\arg z}", "051ca4949f88882b6e288b0d5ec6d5fc": "SSR = \\sum_{i=1}^n\\bigg(\\frac{\\varepsilon_i^2}{\\sigma_\\varepsilon^2} + \\frac{\\eta_i^2}{\\sigma_\\eta^2}\\bigg) = \\frac{1}{\\sigma_\\varepsilon^2} \\sum_{i=1}^n\\Big((y_i-\\beta_0-\\beta_1x^*_i)^2 + \\delta(x_i-x^*_i)^2\\Big) \\ \\to\\ \\min_{\\beta_0,\\beta_1,x_1^*,\\ldots,x_n^*} SSR", "051ca55dcaf28074e2cb5a42b1691c17": "y_p(x) = \\sum_{i=1}^{n} c_i(x) y_i(x)\\quad\\quad {\\rm (iii)}", "051d1518eda7defecc640212bc0908df": "\\scriptstyle \\bar\\psi", "051d50d5305afafd7b365b0ed61221a4": "T_i + U_{i-1} \\sqrt{x^2-1} = (x + \\sqrt{x^2-1})^i. \\, ", "051d6224436c5fc199a4c46c1aad0003": " E_{21} =\\frac{d \\ln (c_2/c_1) }{d \\ln (U_{c_1}/U_{c_2})}\n =\\frac{d \\left(-\\ln (c_1/c_2)\\right) }{d \\left(-\\ln (U_{c_2}/U_{c_1})\\right)}\n =\\frac{d \\ln (c_1/c_2) }{d \\ln (U_{c_2}/U_{c_1})}\n = E_{12}\n", "051d6422f391f4a35ceab86263d112f8": "\\int\\frac{dx}{\\sinh^n ax} = -\\frac{\\cosh ax}{a(n-1)\\sinh^{n-1} ax}-\\frac{n-2}{n-1}\\int\\frac{dx}{\\sinh^{n-2} ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "051db488801812c2b62a83559cddaea0": "W_q=\\frac{\\rho^2+\\lambda^2\\sigma_B^2}{2\\lambda(1-\\rho)}", "051dd97e3eb4556b287a44a9e427a37b": "\\sqrt{\\varphi_1^2 + \\varphi_2^2}", "051e7321f1a7ebbc27505ecd75e6bbe8": "\\mathcal{G}(p,q)", "051e8ca3a671da00d3446e6da1f5ff6e": " \\langle f_1,\\ldots, f_k\\rangle = \\left\\{\\sum_{i=1}^k g_i f_i\\;|\\; g_1,\\ldots, g_k\\in K[x_1,\\ldots,x_n]\\right\\}.", "051ee141bac36d8612e305c8beecf706": "P(t) = \\begin{cases}\n 0 & tt_o+t_p \\\\ \n\\end{cases}\n", "051ee904daa0559210339ff3c6ed52c6": " \\nabla B_z = (d B_z/dA)\\nabla A", "051f0ede9f5842fe5a8e50066845bdc7": "p=p(V,T)\\ ", "051f218870baef81059be4a102dec711": " G\\left(X'_{i}\\beta\\right) ", "051f2871fd7e787c6ec9c8be7702f7f4": "(D V_i)^2 / Z_o = \\eta V_i^2 / Z_i ", "051f58f5abb870ac348fd824566ba1b1": " \\log_{b^n} a = {{\\log_b a} \\over n} ", "051f84bf61e8e26b26ab4cc0cd4d0af6": "\\,\\lambda_i", "051ff4c2be9011cd50b03822e0fef332": "1\\over {\\sqrt 6}", "05208e2e2161e5c9451c0ad985594f0e": "\nE_{a_0} = \\frac{E_S}{4\\pi a_0^2}\n", "0520b5f0f2d99a7a28f5ca9b8ee08bf9": "\\dot{x} \\equiv \\frac{dx}{dt} = \\left(\\begin{array}{c} \\frac{da}{dt} \\\\[6pt] \\frac{db}{dt} \\\\[6pt] \\frac{dc}{dt} \\\\[6pt] \\vdots \\end{array}\\right).", "0520df0adb6ded51ed8afb052e4bded9": "\\frac{2 \\cdot 5}{7}", "0520f68ba263a7a7ec277cc0671d6b23": "V_{out}(t)", "05210b6b9045a1666d5676422477286c": "I_C=I_E-I_B\\,", "05211990618d5f6fdc2ab9065bf70066": "r\\leftarrow p", "05218ec1fe4c2fe5e117e292cb91b5c2": "\\varphi(h(y),s) = h(\\psi(y,t))", "0521b4209637a2fb3ebc86938716bc9b": " = \\int_{-\\infty}^{\\infty} \\left[ \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, e^{-j \\omega t} \\, dt \\right] \\, d\\tau ", "0521f1cdd4dd30d846d0bd2d196c5b9b": "J(\\mathbf{x}) = (\\mathbf{x}-\\mathbf{x}_{b})^{\\mathrm{T}}\\mathbf{B}^{-1}(\\mathbf{x}-\\mathbf{x}_{b}) + (\\mathbf{y}-\\mathit{H}[\\mathbf{x}])^{\\mathrm{T}}\\mathbf{R}^{-1}(\\mathbf{y}-\\mathit{H}[\\mathbf{x}]),", "052227aa30bd74eafbce3b5cde10ea9b": "\\langle l, r \\rangle_w", "05224930b0615345deb884948267a8ac": " M = 3(N-1-j) + j = 1, \\!", "05225d2f4212a8e279c90b2d9183c6fa": "t=1\\,\\!", "0522718afc8aa16a9af1dc1323991229": "\\psi(x) = C\\,\\exp\\left(-\\frac{(x-x_0)^2}{2 w_0^2} + i p_0 x\\right)", "052274267eaed4f24d4bc546decc403f": " M = \\begin{pmatrix} e(a_1,b_1) & e(a_1,b_2) & \\cdots & e(a_1,b_n) \\\\ e(a_2,b_1) & e(a_2,b_2) & \\cdots & e(a_2,b_n) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e(a_n,b_1) & e(a_n,b_2) & \\cdots & e(a_n,b_n) \\end{pmatrix} ", "052275e1c5c5fc2132c328a5a1548487": "\\boldsymbol{\\alpha}", "0522ba869064082c64d0e85ec613c34a": " CPI = \\sum_{i=1}^{n} CPI_i*weight_i ", "0522d9bcc2cdf664ad01b74ab937f24e": "\n\\begin{align}\nA_{0} &= u_{0}^{2} \\\\\nA_{1} &= 2 u_{0} u_{1} \\\\\nA_{2} &= u_{1}^{2} + 2 u_{0} u_{2} \\\\\nA_{3} &= 2 u_{1} u_{2} + 2 u_{0} u_{3} \\\\\n & \\cdots\n\\end{align}\n", "05230f95fc3a31bc3b0826c13f8f4a31": " G = \\left\\{(\\Delta ,x):{\\rm{f}}_i (\\Delta ,x) \\le 0,0 \\le i \\le k,\\Delta = xx^T \\right\\} ", "05233dd32f8cc541bd9ced9f0786cda8": "E[G^2|K]=\\int^T_0\\int^T_0k(t)k(s)E[x(t)x(s)]dtds = \\int^T_0\\int^T_0k(t)k(s)(R_N(t,s) +S(t)S(s))dtds = \\rho + \\rho^2", "05234ea4351e32f9183ca278bbf9bac6": "\\mapsto \\!\\,", "0523517f4871af8f21c7335440857928": "D^a \\,", "05235be088fe90ed01afc11dbff739dc": " \\frac {PV}{T} = \\sqrt{k_p k_v k_t} \\,\\!", "0523b7f9b83c5c7489ec4d18839c41a1": "\\mathbf{f} \\,\\colon \\,f_1\\ge f_2\\ge \\cdots \\ge f_N", "0523d5903ad11af2202b4188d345244e": "\n\\| u \\|_{L^4} \\leq C \\| u \\|_{L^2}^{1/4} \\| \\nabla u \\|_{L^2}^{3/4}.\n", "0524ac0139c5674da470ce71e9dc2998": " \\mathbb{Z}_d \\times \\mathbb{Z}_d ", "0525441782af4827a325e2fc2c934ed2": "A = Z+N\\,\\!", "052558595508698e079de150da568929": "\\sqrt 2 \\sinh u,\\,", "05258ad8d57d6ca8ec02a490c078934c": "\nI_{+} = -I_{e} e^{-e V_{+}/(k T_{e} )} + I_{ion}^{sat}\n", "0525b7ccbe9150fc1336beb2b81b5880": "\\eta_2 = \\frac{H}{m}\\, \\left( 1 - m - \\frac{E(m)}{K(m)} \\right),", "0525cccdd7123c339471e6dc1fd332a1": "\\begin{align}\n\t&\\sin(f_c\\cdot t+I\\cdot\\sin(f_m\\cdot t)) \\\\\n\t&\\quad = J_0(I)\\sin(f_c\\cdot t) \n\t+ \\sum_{k=1}^{\\infty} J_k(I)\\left[\\sin(f_c+k f_m)t+(-1)^{k}\\sin(f_c-k f_m)t\\right]\n\\end{align}", "05261178171cbde148f522fd4ce40017": "L \\propto \\log \\log N", "05266488515a4d0c92daba82dd43647f": "x_n=T_1x_n^{(1)}+T_2 x_n^{(2)}+\\dots +T_rx_n^{(r)} \\mod T", "0526732aa1201e9383e0adb4a439229b": "\\left(\\delta_S\\right)", "0526a8bf4fe60ce15b9c93314c984f53": " \\sigma >0", "05274b730a401a9c4ac31d7e4fc653ce": "G=\\langle x_1,x_2 \\mid R \\rangle", "0527514ada9c403ea469ca02ce24f292": "a = \\sqrt{2}\\, , \\quad b = \\log_2 9\\, , \\quad a^b = 3\\, .", "05276a877a3f26d3fed313cc1cadd89a": "\\bold{F}\\;", "05276cf9f4a5efb6dfd91c6b7066883c": " m=0 \\, ", "0527fb7eec99185176dddeecb4105f22": " \\mathbf{A}\\cdot\\mathbf{B} = \\mathbf{A}'\\cdot\\mathbf{B}' ", "0528481eedf437a0564f67864056d139": "\\dot{\\alpha}^*=\\alpha", "0529d8b612fce5de2462245a2978c70c": " X , Y , Z , XX , YY , ZZ , XY , YZ , XZ ", "052a0e1ff447c2e2bec8c6e49313bb2c": "\\mathbf x, \\mathbf y\\in \\mathcal A ", "052a8dde47dfc691fa05737e59c17f16": "\\sum_{i=0}^k\\frac{\\Gamma(\\alpha + i)\\beta^i\\lambda^{k-i}e^{-\\lambda}}{\\Gamma(\\alpha)i!(1+\\beta)^{\\alpha+i}(k-i)!}", "052b48ee967b71c502168486ddb54522": "\\bigl(\\tfrac12,\\tfrac12, \\ldots \\tfrac12\\bigr).", "052b801b8b515aca0898929f40f14ada": "r_c\\,\\!", "052bb990e8596e24d7948c61a3f3a8ed": "\\Delta^\\text{op}", "052be0158cc8b723e885b9b440e1083e": "f(\\phi,\\psi)=0\\,", "052c5652b42a40ec4801dee938109f88": "\nx^{- \\alpha} \\; G_{p,\\,q+1}^{\\,m,\\,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q}, \\alpha \\end{matrix} \\; \\right| \\, \\eta x \\right) = \n\\frac{1}{2 \\pi i} \\int_{c - i \\infty}^{c + i \\infty} e^{\\omega x} \\; \\omega^{\\alpha - 1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{\\eta}{\\omega} \\right) d\\omega,\n", "052c66e5fa5f447b7f2367c9b102f1d4": "(Y, \\mu, S)", "052c6b13ea3a99fa82c0e51204815d0f": "4r^2\\le e^2+f^2+g^2+h^2 \\le 4(R^2+x^2-r^2)", "052c70b3eed17d912bd74195d3477f66": "R = 45 + 48 = 93", "052ccc833647d5b81b19d27119f7979a": "\\alpha_{\\text{pump on}} (\\omega)", "052cd2ba2e36f8f6f63b1ada442105fc": "h_m(x) = \\sum_{j=1}^J b_{jm} I(x \\in R_{jm}),", "052ce045b2c96c121918eca1de8fc712": "\\ln(n!) - \\tfrac{1}{2}\\ln(n) \\approx \\int_1^n \\ln(x)\\,{\\rm d}x = n \\ln(n) - n + 1,", "052cee2ddaf572eb50e2aeebf52edb60": "a = 2 \\arctan \\left( \\frac {D/2} {f} \\right) = 2 \\arctan \\left( \\frac {D} {2f} \\right)", "052d085a5fe58905b0426ef8b85f0638": " f^+(x) = \\left\\{\\begin{matrix} f(x) & \\text{if } f(x) > 0 \\\\ 0 & \\text{otherwise} \\end{matrix}\\right. ", "052d161539044589cc32eee91b8fda6a": "\\begin{align} P(A |\\text{not }B) &= \\frac{P(\\text{not }B | A) P(A)}{P(\\text{not }B | A)P(A) + P(\\text{not }B |\\text{not }A)P(\\text{not }A)} \\\\ \\\\\n\n &= \\frac{0.01\\times 0.001}{0.01 \\times 0.001 + 0.95\\times 0.999} \\\\ ~\\\\ &\\approx 0.0000105.\\end{align}", "052d776f4e8548cffeb47a2dbd78c129": "\\textstyle \\alpha = d", "052d7db1d302c8e2e7ae04d6a5d0ef2b": "P = \\frac{\\int_0^{\\frac{\\pi}{2}} l\\cos\\theta d\\theta}{\\int_0^{\\frac{\\pi}{2}} t d\\theta} = \\frac{l}{t}\\frac{\\int_0^{\\frac{\\pi}{2}} \\cos\\theta d\\theta}{\\int_0^{\\frac{\\pi}{2}} d\\theta} = \\frac{l}{t}\\frac{1}{\\frac{\\pi}{2}}=\\frac{2l}{t\\pi}", "052df73f7c43029df9b3fcd9c4ad22fa": "\\begin{matrix}2\\end{matrix}", "052e076e9fc04db8b0a520a78c844876": "\\sin 2x = 2\\sin\\frac{x}{2}\\cos\\frac{x}{2}", "052e34d11e812d6bb5902b169db0517f": "W(S)", "052e54d580841636190e637d0333414a": "\\quad W_{2\\,p}=\\frac{2\\,p-1}{2\\,p}\\times\\frac{2\\,p-3}{2\\,p-2}\\times\\cdots\\times\\frac{1}{2}\\,W_0=\\frac{2\\,p}{2\\,p}\\times\\frac{2\\,p-1}{2\\,p}\\times\\frac{2\\,p-2}{2\\,p-2}\\times\\frac{2\\,p-3}{2\\,p-2}\\times\\cdots\\times\\frac{2}{2}\\times\\frac{1}{2}\\,W_0 = \\frac{(2\\,p)!}{2^{2\\,p}\\, (p!)^2} \\frac{\\pi}{2}", "052ee2717d0683b8ef7729b1063002a6": "C = \\text{Tr}_{\\text{CTC}}\\left[ U \\right]", "052f2bc9062738ec52049899cddaa7c0": "\np\\sigma \\xrightarrow\\alpha p'", "052f3e6f6172ebddf8d9a015db13307f": "\n \\overset{\\circ}{\\boldsymbol{\\tau}} = \\dot{\\boldsymbol{\\tau}} - \\boldsymbol{l}\\cdot\\boldsymbol{\\tau} - \\boldsymbol{\\tau}\\cdot\\boldsymbol{l}^T \n", "052f84150425938458bfcda119406ac9": "{{\\overline{P_1P_3}\\cdot \\overline{P_2P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}} =1+{{\\overline{P_1P_2}\\cdot \\overline{P_3P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}}", "052f90c64762cae32b83178a5045cd8d": "\\bold{u}_f\\;", "052f99631cd8d4bfa945c02313d18f40": "U=0", "052fd9b4b90a459ed294c0b9c2d1d4e1": "T \\subseteq [n]", "052fe85ca556dc32e605488ad5560478": "j=1,\\ldots,m\\,\\!", "05304f3e6a805e7506cb7e955b8fa969": "\\ln (1/\\Gamma(z)) \\sim -z \\ln (z) + z + \\tfrac{1}{2} \\ln \\left (\\frac{z}{2\\pi} \\right ) - \\frac{1}{12z} + \\frac{1}{360z^3} -\\frac{1}{1260 z^5}\\qquad \\qquad \\text{for}\\quad |\\arg(z)| < \\pi", "05306690c3b5fd73579ab942e82f1768": "+\\lambda^2\\sum_{m\\neq n}\\sum_{q\\neq n}\\sum_n\\frac{\\langle m|V|n\\rangle\\langle n|V|q\\rangle}{(E_n-E_m)(E_q-E_n)}|m\\rangle\\langle q|+\\ldots", "05306697eb4b61a275ac2cf33c664371": " 4 = \\operatorname{perm} \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right )\\operatorname{perm} \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\neq \\operatorname{perm}\\left ( \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\right ) = \\operatorname{perm} \\left ( \\begin{matrix} 2 & 2 \\\\ 2 & 2 \\end{matrix} \\right )= 8.", "053095516840cd071b5a3b992cd97389": "\\mathcal M_{u}", "0530bed92f633ae1bccc61c5c5d58fdb": " f_x=x/(\\lambda z)=l/\\lambda", "0530cd86ac74e46292792762625a3337": "A(0, b) = 2 b + 1", "0530f18af57b81cb2c913b0d3089f540": "\\displaystyle{\\sum b_n(\\zeta^{-1}) z^{-n} = \\exp \\sum a_m(\\zeta^{-1}) z^{-m} = {g(z)-g(\\zeta)\\over z-\\zeta},}", "0530f4c6b8b0956007dd50dcb0eb0f6c": "\\vec{v} \\vec{w}", "05310eabb8157f12c7f05bf756526726": "F:\\mathcal{P}(S \\times S) \\to \\mathcal{P}(S \\times S)", "053188dd9d2bcfdf6aee570206038125": " E = \\sqrt{\\sum P(n-X)^2} ", "05325fc96d5fe3a9abb9a9ff8ba9465a": "r(t) = q", "0532864d8a133e2306c3965faaf76a2b": "\\delta t=1.7\\pm1.4\\ (\\mathrm{stat.}) \\pm3.2\\ (\\mathrm{sys.})", "0532aa97d0f0b796383aef0266ca31d2": "\\frac{df_{a_1,\\ldots,a_{i-1},a_{i+1},\\ldots,a_n}}{dx_i}(a_i) = \\frac{\\part f}{\\part x_i}(a_1,\\ldots,a_n).", "053326edbd07a01f83831d7f82855e5b": "\\frac{\\sin \\theta}{\\theta \\cos \\theta} > 1\\,", "05332b4197c5bb54ec4d3dbc10d9eec8": "H=\\left(N+\\frac{1}{2}\\right)\\hbar\\omega,", "05337cb337140eb5fcfceb4ab7ba6184": "\\mathit{f}", "0533bcef777c92ce342ea3625c1dfb42": "|c-[c]-\\frac{1}{2}|<\\frac{1}{2}-a", "0533d8bce4d6a4b5f26caa843aec2fdd": "1.05^4-1=21.55%", "053407cd89ffc607ac5304ac11057fd0": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 128\\cdot 2.05)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 56\\cdot R_{\\bigodot}\n\\end{align}", "053456938c3e537cd4b6aad2387c484d": "\nR(\\tau) = \\frac{\\operatorname{E}[(X_t - \\mu)(X_{t+\\tau} - \\mu)]}{\\sigma^2}, \\,\n", "0534e1c6e1c58e65a298449d6def9e2a": " (n,m,l, \\epsilon)", "0534f253fa8bdb5a9a05bfae6b479256": "{{P}_{Disk}}=\\left[ \\varphi ,\\chi \\right]*\\text{ }Vector\\text{ }of\\text{ }Disk\\text{ }performance\\text{ }counter+{{\\lambda }_{constantDisk}}", "0534f91618c21c6115ee879962400bf3": "v = (x_b + k^e)\\mod N", "0535104b327e8cbeacc6b9fcfafd7e10": "\n \\varphi_\\alpha = w^0_{,\\alpha} \\,.\n ", "05355d3274aed8dc2eca2ab604bd85d3": "N_{s}", "0535634f282026214e59aa656824235c": "f(\\mathbf{z}) \\in f(\\mathbf{y}) + [J_f](\\mathbf{[x]}) \\cdot (\\mathbf{z} - \\mathbf{y})", "0535730078adc42f7af6fe8ce72846f3": "t\\geq 0", "053640ae353786b845162394037755d4": "\\mu(X)=\\mu(Z)+\\mu(X\\setminus Z)", "05364d147eb0bc88147e0ba960605f03": " C^*_{(+)}= C_{(+)} ", "0536ad4ce780b9d76a1baa3013ac1918": "\n\\boldsymbol{x}, \\boldsymbol{y} \\sim\\ \\mathcal{N}(\\boldsymbol\\mu_{X,Y}, \\boldsymbol\\Sigma_{X,Y})\n", "0536b9dcbddd3e7ed41903ab2ea8a619": "f_n(x)=x+n", "0536cb091c7b6ab01bbbbcd34f865cfa": "\\displaystyle x_{i}\\rightarrow x_{i}b^{\\left[ x_{i}\\right]}, \\phi _{i}\\rightarrow\n\\phi _{i}b^{\\left[ \\phi _{i}\\right] }. ", "0536ccb9e1940a60e0d1dfb9178ea027": "s = +j \\omega_0\\,", "0536e7e4439f94e77793561079872db8": "\n\\exp \\left\\{ - \\frac{a}{2} x^2 \\right\\} =\n\\sqrt{\\frac{1}{2 \\pi a}} \\; \\int_{-\\infty}^\\infty\n\\exp \\left[ - \\frac{y^2}{2 a} - i x y \\right] \\, dy,\n", "0537054c6c956f5ab503c9fdcd425c06": "- + -", "0537342e84ebcc0997f7ed98ef18c3da": "g(a,a+d,a+2d,\\dots,a+sd)=\\left(\\left\\lfloor\\frac{a-2}{s}\\right\\rfloor+1\\right)a+(d-1)(a-1)-1.", "05374cb757176f04bf864caa67390057": "\\partial W=M \\sqcup N", "05378ca23df01c19a92166951a7a563e": "-S = \\left(\\frac{\\partial F}{\\partial T}\\right)_{V}\\,", "0537a4dc709adae5af1e5ad54a743ec7": "B(\\boldsymbol{u},\\boldsymbol{v}) - F(\\boldsymbol{v}) \\geq 0 \\qquad \\forall \\boldsymbol{v} \\in \\mathcal{U}_\\Sigma ", "0537c34c6bcde95545ce50bb1e94f6d1": "\\mathbf{\\nabla}\\cdot(\\epsilon \\mathbf{\\nabla}\\varphi)= -4 \\pi \\rho_{f}", "0537ca2ca25661ae0d9bbec87714cc55": "e^{-\\beta_{k}\\tau}", "0537d06876f6ff79f8b419fa747a25db": "\\int x e^{c x^2 }\\; \\mathrm{d}x= \\frac{1}{2c} \\; e^{c x^2}", "0538221777a190b91f792b30a8aedad4": " \\sum_{j \\in J} a_j \\mathbf{v}_j = \\mathbf{0} \\,", "0538a1b229fa70423b107e594a783264": "\nh\\,a = \\begin{cases}\n c & \\mbox{if } p\\,a\n \\\\b \\oplus ha' & \\mbox{otherwise}\n \\end{cases}\n ", "0538b394394e22701b79c1ea9a80f9ca": "T = T_i2^{-R/C_V}.", "0539109d4289480e283c341dff4f2491": "\\lambda_p \\approx \\lambda_j", "05391104f231a1aa43ca9f8192d45ab4": "x \\in Y", "05393d10d8fe7779e4cf3c8724c53f01": "r(T) = \\lim_{n \\to \\infty} \\|T^n\\|^{1/n}.", "0539ba5df67017d6394f3669755ba31c": " \\frac{dc}{dl} = G'(l). ", "053a98124485559a12edcc8176574789": "L(\\lambda, \\alpha, s)", "053b39ebd828ace2d4f73180f53ba2a0": "\n\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=m^2\n", "053b9a3cc33be28c94b003ccf8dc0f94": "\n=[E_{12}, E_{11}] E_{22} + E_{11} [E_{12}, E_{22}] -\n[E_{12}, E_{21}] E_{12} - E_{21}[E_{12},E_{12}] +[E_{12},E_{22}]\n ", "053bbf1ecf57ac081e3f0b9156b77ec1": "\\sigma = (x~~1)(x~~2)\\cdots (x~~i)(y~~i+1) (y~~i+2)\\cdots (y~~k)(x~~i+1)(y~~1).", "053bcefa288f5670afb2b08d40a818c1": "\\omega \\in\n\\Omega_{Z,[t_l, t_u]}", "053c104ac907d191d028dc3ce0a0126d": "\\mathbb{J}", "053c115098a7fbf8f6bcdcf83197ae46": "\\scriptstyle f(x,y,z)=w^2", "053c735d694601190d0d4d634d4b9b3c": "\\begin{pmatrix}\n 0 & 2 & 0 \\\\\n 0 & 0 & 3 \\\\ \n 0 & 0 & 0\n\\end{pmatrix}", "053c73b9865caf2fb1ef485da71b9618": "\\ker \\rho = \\left\\{g \\in G \\mid \\rho(g) = \\mathrm{id}\\right\\}.", "053c82ae3b2f75e841f38f565fffbb7b": "\\partial_x", "053ca75a838fa7f797b2d812ae85a03a": "s_{n-1}", "053d1a056e387db09b31665c752971ca": "\\Pi_{(x:A)}B(x)", "053d2eb84321dcdf2526369fc086cfb1": "r < \\operatorname{diam}(\\Omega)", "053d3b5e3c21ef4f364b0b836612264e": "\\sigma_B \\geq 0", "053d472eab7e87d03b517d01f001a2ff": "\\Omega*m", "053e5921874fb15240a8b8be120c1bb0": "{x \\over {a-x}} = D {y \\over {b-y}}", "053e6723664890129ebc3c269c2371d1": "\\sqrt{\\ } \\!\\,", "053e70c93e7e72c8678df6ff21231e17": "\n w(x_1,x_2,0) = \\varphi(x_1,x_2) \\quad \\text{on} \\quad x_1 \\in [0,a] \\quad \\text{and} \\quad\n \\frac{\\partial w}{\\partial t}(x_1,x_2,0) = \\psi(x_1,x_2)\\quad \\text{on} \\quad x_2 \\in [0,b] \n", "053ea84cf5b391a6bc0e2769a337b124": "(v,v-k,v-2k+\\lambda)", "053eb11717e6a67416b0b5b369490f43": "G = \\sum_{nm} G_{nm}", "053eb3c78df7b74a18c177d5fff4640a": " k[\\Delta] = \nk\\oplus\\bigoplus_{0\\leq r\\leq d}\n\\bigoplus_{i_0<\\ldots 0,", "053fb39ecb3a197a84a1e24f7e1036c5": "\\frac{\\Delta \\hat{z}}{P}\\,", "054071638984997309331a922b0939d9": "\\lambda=(gy-u^2-v^2)/L^2", "054079603de534fdc6b53a8ebaf62a52": " h \\equiv \\frac{\\sigma_d}{\\sigma_m} ", "0541088a52783eb8184a0704d885f61a": "\\partial_k:C_k\\to C_{k-1}", "054175e12fea6ba14d68da07557fd856": "q_b = \\iiint \\rho_b dV = -", "05417c800e0b9fa8c72b54a10bf205ad": "x_{(i)}", "05417e69be53514379344dd452419664": "R[t] \\to S, \\quad f \\mapsto \\overline{f}", "05419dda5884bdd874dedbe5304f008a": "\\forall x \\, \\phi (x) \\Leftrightarrow_{\\mathrm{def}} \\forall X \\, ( \\mathrm{set}(X) \\rightarrow \\phi (X) )", "0541c06b519ac3465f57abd96c0aacde": "p^2 \\gg k^2", "0541da45a48c535528249f3115d39b0b": "S_2=52.6 \\text{ mm}", "0541df319595678b2a34001bc11b0a6e": "\\gcd{(A,B)} = 1", "0541f24a007b7b665eeee28fc06f6ea3": "h_i=a_{0,i}", "0542761b5ed427e079c3a5eb0a388bf1": " \\rho \\left( \\frac{\\part \\vec{u}_{x}}{\\part t} + \\nabla_{y}\\cdot\\vec{u}_{x}\\vec{u}_{y}\\right)= -\\nabla_{x}p+\\nu\\nabla_{y}\\cdot\\left(\\nabla_x \\left( \\rho \\vec{u}_y \\right) +\\nabla_y \\left( \\rho \\vec{u}_x \\right)\\right) \\,\\!", "0542f906a77ca3e12b89972bc173b196": "\\Gamma(s) = (s-1)!", "054314d841c8f58c84d5c92bf9af8689": "Q(V,T)\\ ", "0543614a7ded3a57d4f0d0805f7f6818": "r^{-6}", "054395ad5b295a1788d6640f63de9c88": "m_\\mathrm{s}", "0544d0f6b025998039fc986117cb5107": "A\\to\\neg\\neg A", "0544ddcc1d9f04f0b80c59c2a0640cdd": "\\begin{align}dy_{\\text{1}}\\ =\\ I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}}\\ -\\ u(I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}})\\\\\ndy_{\\text{2}}\\ =\\ I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}}\\ -\\ u(I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}})\\end{align},\\quad y_{\\text{1}}(0)\\ =\\ y_{\\text{2}}(0) = 0", "05451ef2f9f21fce1ae0956590d7dc50": "\\ P_{ij\\ldots}=P_{ij\\ldots}(\\mathbf X,t)", "05451fff8d6c48d06fae0418857ee63c": "\\mathbf{x}=(x_1, x_2, \\dots, x_n)", "054521ed7f18b89d0ee32e64fcf995bc": "\n_a I_b^{\\left( D \\right)} 1=\\frac{1}{\\Gamma \\left( {1+\\alpha } \n\\right)}\\int_a^b {\\left( {dt} \\right)^D} ", "05458ad345a3e4f6d52e91a62e829a02": "h_2=0.1935\\times Do-0.455\\times t", "0545afccd5ede14da5822029bb943006": "\\sum_{\\pi\\in S_n} \\frac{\\sigma(\\pi)}{\\nu(\\pi)+1} = \n(-1)^{n+1} \\frac{n}{n+1},", "0545c01d23dca7b361c37f12366c25a2": " f : x = \\{x_n\\} \\in \\ell^1 \\ \\rightarrow \\ \\sum_{n=0}^{\\infty} x_n,", "0545dfdc996460d9837db4932313fb76": "i(x,y)", "054601baeb96b31f4e7eb6fbdc35e3e5": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "05462613c17fd32778bb06f4e57c8c52": "\\bigl(\\tfrac12,\\tfrac12, \\ldots\\tfrac12, -\\tfrac12\\bigr)", "05462afc37afa847c14b68233c4a6770": " D_q f(x)=\\frac{f(qx)-f(x)}{(q-1)x}.", "0546416dcc8b41b6304edac92d767118": "\\int\\frac{dx}{r} = \\operatorname{arsinh}\\frac{x}{a} = \\ln\\left( \\frac{x+r}{a} \\right)", "05465f7db8e44281594795f0a743bc36": "\\overline{N}_{\\Delta f}(f)", "0546602fbc3752251622c6ad39b77b54": " u(t,x) = T(t) v(x).\\,", "0546fce8761070e7da5fdf6bc0b0bcc1": " v_0 = \\frac{V_{\\max}[\\mbox{S}]}{K_M + [\\mbox{S}]} ", "05473bec95ce3ec988da31f310a63a1d": "\n \\Beta(x,y) =\n \\int_0^\\infty\\dfrac{t^{x-1}}{(1+t)^{x+y}}\\,\\mathrm{d}t,\n \\qquad \\mathrm{Re}(x)>0,\\ \\mathrm{Re}(y)>0\n\\!", "054745ac1b7edb7c864d90535b3feba0": "- \\nabla U(X)", "0547564a05c36f9e94a0163b186dab47": "E_{K_1}(E_{K_2}(P)) = P", "0547870987b7e4e2f8782db9146b430a": "\\left( \\left( x\\ast y\\right) \\ast \\left( x\\ast z\\right)\n\\right) \\ast \\left( z\\ast y\\right) =0", "05478d83f2ad822d957bdf9dba6eff34": "(d_1,e_1) \\cdot (d_2,e_2) = (d_1^{e_2}d_2^{e_1}, e_1e_2) \\ . ", "0547c51ea114c93a980d6dc2b6f904b9": "\\nu = \\alpha c_{\\rm s} H = \\alpha c_s^2/\\Omega = \\alpha p_{\\mathrm{tot}}/(\\rho \\Omega)", "0547e443c12f4232d1d5ea6b222a0525": "\\tilde{T} = \n\\begin{bmatrix}\n0 & \\; & \\cdots & z \\\\\n\\frac{1}{2} & \\ddots & \\ddots & \\; \\\\\n\\; & \\ddots & \\ddots & \\vdots \\\\\n\\; & \\; & \\frac{1}{2} & 0 \n\\end{bmatrix},\n", "054803695498bb95cc220bee5b591ab9": "n_F(\\xi)=\\frac{1}{2}\\left(1-\\mathrm{tanh}\\frac{\\beta\\xi}{2}\\right)", "054826fb48e60800985100a1704a3d58": "\\Gamma = \\Gamma_{ab} + \\Gamma_c", "05487c8bf39af24bd9bca6d3a28aa0cc": "(\\cos(\\theta))", "0548ce06a9515c323b8e4948f12bc697": " h_{x} = -e_{y} / \\eta ", "05494d0a69bc739e85f967733bee4c00": "x_1^2 + \\cdots + x_n^2 + 2a_1x_1 + \\cdots + 2a_nx_n + c = 0", "0549a953f3ba4ef9d116cb0e1132a3bf": " \\vec{s}_a \\cdot \\vec{s}_b ", "0549c34e4484e3cc069290d888bd9e61": "{\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z {\\left ( \\frac{\\partial y}{\\partial z} \\right )}_x {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y = -1.", "0549c7cd38c25400ab340f4a9e3443db": "\\Delta q = (q-q_0) ", "054a3a4c6ee140a2c53ddaf1b6bb0e96": "|R| < |S|", "054b07d42e4f6e99e7aaac28551aff25": "\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty a_{m-n} \\lambda_m\\overline{\\lambda_n} = 2(1-|z|^2) \\,\\Re\\, f(z).", "054b1b47b17865fe755c44c54b5551a5": "\\beta_{n}^{PR} = \\frac{\\Delta x_n^\\top (\\Delta x_n-\\Delta x_{n-1})}\n{\\Delta x_{n-1}^\\top \\Delta x_{n-1}}\n", "054b6af9ec764b7ae7bdbc715df68090": "\\mathfrak{P}^{36}", "054b947d6a054d09c69bce8840ee4886": "g_N \\left (x_1,\\dots, x_N, t \\right) = G\\left(x_1, t \\right) \\cdots G\\left(x_N, t\\right) ", "054c117040ce2a6676a9e837016aa70d": "z=1\\ldots Z", "054c373fad93a5f8848feda8b6c23c5d": "\\tan a = \\frac{\\sin a}{\\cos a}", "054c8e2980502a6d65a953d78a2f9c4a": " \\rho_1 \\mathbf{v}_1 \\cdot \\mathbf{S}_1 = \\rho_2 \\mathbf{v}_2 \\cdot \\mathbf{S}_2 ", "054c9d24788c0023b22433c604b03c41": "B[u, u] + G \\| u \\|_{L^{2} (\\Omega)}^{2} \\geq C \\| u \\|_{H^{k} (\\Omega)}^{2} \\mbox{ for all } u \\in H_{0}^{k} (\\Omega),", "054cae4d1530dc60449f72b8fe9a5c6f": " \\bar r(t) ", "054cd288d706740e52dddb6f039062f0": "\\mathbf{U} = E[(\\mathbf{X} - \\mathbf{M})(\\mathbf{X} - \\mathbf{M})^{T}]", "054d457a8a66de966ff08d0b72b5b96d": "a^{th}", "054de48d0df9a4458f7703161ba8acd9": "x \\mapsto g(x; 2)", "054dfbc4cbd2f68a06c003ae77f874ea": "\nCIQ_t = \\mathcal{A} \\left ( 1 + \\mathcal{B} \\right )^t\n", "054eb2251868bb2a2cf30c3f43b48e7b": "J_F(x_n) (x_{n+1} - x_n) = -F(x_n)\\,\\!", "054eb94b485d8f79dc65c360a6a539fe": "|\\partial A|\\geq C\\left(\\min \\left( |A| , |G\\setminus A| \\right)\\right)^{(d-1)/d}. \\, ", "054ec7718dc62f34e0fcab7a6bc9e865": "\n\\int_a^{x_0-\\delta} e^{n f(x) } \\, dx + \\int_{x_0 + \\delta}^b e^{n f(x) } \\, dx\n\\le \\int_a^b e^{f(x)}e^{(n-1)(f(x_0) - \\eta)} \\, dx = e^{(n-1)(f(x_0) - \\eta)} \\int_a^b e^{f(x)} \\, dx \n", "054ee465850b8285713271cb44f8f3c9": "M/T", "054f2cc1ac6da37ad4bf572757cf7dd5": "S_{11} = {(1 - Z_0 Y_{11}) (1 + Z_0 Y_{22}) + Z^2_0 Y_{12} Y_{21} \\over \\Delta} \\,", "054f2d87ffd8835b0422c2e3d22d76ae": "\\log (1/\\epsilon)", "054f383837eb3832ac11add3cdd54472": "x_1\\le\\cdots\\le x_n\\quad\\text{and}\\quad y_1\\le\\cdots\\le y_n", "054faf8b16b83a42e23391c90bc802e9": "t=q^{-s}", "055011102cd4a7625e250b7efd8990eb": "x\\in K", "05501f6d0a6bc597511bf940eac005b8": "R_{\\text{vertical}} = \\frac{R_{12,34} + R_{34,12} + R_{21,43} + R_{43,21}}{4}", "055036089d7d464018b5dc9f3d56ed55": " \\rho = ", "055039b61f4f25f8766e98e3f0d1daac": "\\begin{align}\n\\rho_0 \\ddot{\\boldsymbol{x}'} &= \\frac{\\partial^2}{\\partial t^2} (\\boldsymbol{u}^{(0)} + \\boldsymbol{u}^{(1)} + \\boldsymbol{X}) \\\\\n &=\\frac{\\partial^2 \\boldsymbol{u}^{(1)}}{\\partial t^2}\n\\end{align}", "05508790e1f9a13201a1f9fbaabf615d": "J \\ \\stackrel{\\mathrm{def}}{=}\\ P_+ - P_-", "05508e4ff635ae65cf58468bc93febe2": " v'' = 0. \\;", "055202e88e159f6ae4c20e58150084f8": "D_{\\mu}\\tilde{F}^{\\mu\\nu}=0.", "05520c4227b2eabb934ff18323903144": "\\sigma_x \\sigma_p \\ge \\frac{\\hbar}{4}\\sqrt{3+\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-\\left(\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-1\\right)} = \\frac{\\hbar}{2}.", "055238ed1eb06ba46e9d541b8e74cca3": "k_x^2+k_y^2+k_z^2= k^2", "05524620db947b72a399c99f6a9329a2": "x_k:=x_j+ P_k A^{-1}\\left(b - A x_j \\right),", "0552ae58c274fcd873a6ce452a0c6231": "\\left|\\frac{f(z_1)-f(z_2)}{\\overline{f(z_1)}-f(z_2)}\\right|\\le \\frac{\\left|z_1-z_2\\right|}{\\left|\\overline{z_1}-z_2\\right|}.", "05536b45589d88f483e73d0ef5612dce": "\n\\sum_i n_i(\\bar{Y}_{i\\cdot} - \\bar{Y})^2/(K-1)\n", "05539ed6141a398c93b7065ad4201a6a": "f_1, f_0", "0553c8b8021e37b9e34f9af179254d52": "\\epsilon_c = f\\epsilon_f + \\left(1-f\\right)\\epsilon_m.", "0554246077f1d1b5f2c5f4ffacf646fd": "\\bar x(t)", "055492d3c623cada3d3b79809dded22c": "(|x\\rangle, |\\psi\\rangle)", "0554ac534fbee8422cedc5d3f1fac58f": " f^* = \\min_{x\\in R^n}\\{f(x)\\}", "05551124009a4be6272c004d9493bbc3": "f=f_1 e_1+f_2 e_2+f_3 e_3+\\cdots", "0555229d9f9cefecadb272f844e063bd": "2^{-b}", "055523e005e73c3ae0bd0aa9ce2da1ce": "\\varepsilon_{eff}", "05555364502329a70d4673fa5c6e402b": "M=C+D", "0555554754988fe2b2b12ace1983d53e": "\\hat{ \\theta } =T(\\hat F_n) \\,,", "05556a62b2f29e3ffa699da5ed630595": "\\pm 2x^{7/22}", "0555da2e4f0667910a6d9e9dae93bb32": " e^{M f(x)}. \\,", "0556468dd133208856883806211b55be": "\\Pi^{\\mathbb{Z}^{+}}", "05564f5751a56a8627447ad69654ae77": "\\ln \\frac{\\hat{\\beta} - \\frac{1}{2}}{\\hat{\\alpha} + \\hat{\\beta} - \\frac{1}{2}}\\approx \\ln \\hat{G}_{(1-X)} ", "05566b811fe14c932f8abade2a39dd9b": "\\textstyle b(x)", "05569328ce8674f4e3f0bcddde66e97d": "L_1 = x_2 p_3 - x_3 p_2", "0556d29890d84c63ead4d591cf8fe6c4": " \\ell \\equiv ab\\bmod m ", "055758542eb4d96345b9a2448b5ec284": "\\textstyle e_{\\lambda}(f_{ij}) = \\frac{f_{ij}}{\\lambda}\n\\frac{\\partial \\lambda}{\\partial f_{ij}}", "0557611da6e08448d16d418a19f8c6e9": "\\frac{d^2z}{dt^2} + 2i\\Omega \\frac{dz}{dt} \\sin(\\varphi)+\\omega^2 z=0 \\,.", "05576a866d96093e3b177316997766bd": "G = \\sum_i \\mu_i N_i\\,", "05577e342b9901af7a97641588e704f0": "M(H)", "0557da9106c4a4ca91ffdf200da209ce": "\\Delta = b^2 - 4ac\\,\\!", "0557e4e4424f3b9bb39ec55278ae06ed": "\\,(1-\\varepsilon)L + \\varepsilon N\\, \\prec \\,M \\, \\prec \\,\\varepsilon L + (1-\\varepsilon)N.\\,", "05581e0aec5d7df6fb8e95cfd4e38a9b": "H(Y|X)=0", "0558205a74c10475844dba3327824980": "v\\in M", "05588378ae0dc3377574679e08679a30": "\\Rightarrow \\delta^2 = n^{2\\gamma -1}", "055889aaee38b7c53f994c5e42a40994": "\\Rightarrow", "0558c6b0f1f42f903bd7421dcbe104f1": "\\int_X\\left(\\int_Y f(x,y)\\,\\text{d}y\\right)\\,\\text{d}x=\\int_Y\\left(\\int_X f(x,y)\\,\\text{d}x\\right)\\,\\text{d}y=\\int_{X\\times Y} f(x,y)\\,\\text{d}(x,y).", "0558fd5010fefb8a612245fa9a0f90ff": "\\textrm{var}(X) = \\frac{4 - \\pi}{2} \\sigma^2 \\approx 0.429 \\sigma^2", "055984811ecb3a80e01471dd107feea9": " g(x,X) = \\sqrt{n}\\frac{x - \\overline{X}}{s} ", "0559a0924e84a864de100431a3f3a920": "K_{\\rm d}", "0559ba5a8d8c38f339a9214613f3e418": "\\mbox{Golden rule for capital/labour ratio: } \\frac{ df }{ dk } = (n+d)", "0559e2fa97bfd2c0e077f36b3d65732c": "\\forall\\ x, Rx\\ \\rightarrow\\ Bx", "0559f1beb33bebaeba1551b86a80dfde": "g_2= \\tfrac{1}{\\eta}ij", "0559ff3b5ee5589a99f3d115f5f6ba92": "hmcr", "055a1394374dd31041ec2ee185ca1749": "\n\\overbrace{\\rho \\Big(\n\\underbrace{\\frac{\\partial \\mathbf{v}}{\\partial t}}_{\n\\begin{smallmatrix}\n \\text{Unsteady}\\\\\n \\text{acceleration}\n\\end{smallmatrix}} + \n\\underbrace{\\left(\\mathbf{v} \\cdot \\nabla\\right) \\mathbf{v}}_{\n\\begin{smallmatrix}\n \\text{Convective} \\\\\n \\text{acceleration}\n\\end{smallmatrix}}\\Big)}^{\\text{Inertia}} =\n\\underbrace{-\\nabla p}_{\n\\begin{smallmatrix}\n \\text{Pressure} \\\\\n \\text{gradient}\n\\end{smallmatrix}} + \n\\underbrace{\\mu \\nabla^2 \\mathbf{v}}_{\\text{Viscosity}} + \n\\underbrace{\\mathbf{f}}_{\n\\begin{smallmatrix}\n \\text{Other} \\\\\n \\text{forces}\n\\end{smallmatrix}}\n", "055a2275d6d566469f1e8d55e11773ab": "\\int f^{-1}(y)\\,dy= x f^{-1}(y)-F\\circ f^{-1}(y)+C,", "055a6233561f4f99bbe61895da792be2": "\\scriptstyle \\hat x", "055a6eacf46e89a3e550c074366c46fd": "f(x) = \\int _{-\\infty}^{\\infty} A(\\xi)\\ e^{ i(2\\pi \\xi x +\\varphi (\\xi))}\\,d\\xi,", "055a8649953c17cb8f80b461f1718e5a": " size = 1.22\\frac{\\lambda}{D}distance", "055a9f4f3e3d8618f65d3d30c1089cbf": " a_1b_2-a_2b_1=0", "055b8424d6eeb4f26e4709d51649d526": " \\Rightarrow \\mathbf{u} = \\begin{pmatrix}\n \\mathbf{u}_1 \\\\\n \\mathbf{u}_2 \n\\end{pmatrix} = \\begin{pmatrix}\n -\\mathbf{B} \\\\\n \\mathbf{I}_{n-r} \n\\end{pmatrix}\\mathbf{u}_2 = \\mathbf{X}\\mathbf{u}_2. ", "055c210a7797b4e842635accb13e32a7": "q_{i}", "055c6a0341f1cde9ac1b10222a5ffd1a": "\\frac{D}{\\beta_k}", "055c9f1f08029271f30a3a260d8adc91": "k+d \\le n+1 ", "055cbe8016aa569202dc280991a1cb7c": " W(C;0,1) = A_{0}=1 ", "055cebdfa45752b5a893ad0e84ae382f": "\\Theta\\subseteq\\mathbb{R}^k (k\\geq 1)", "055cf3ed53c85605519082bdafd3564a": "y=k^\\alpha \\,", "055d063e8aa202f11316a89d2481c165": " \\mathbf{V}_g ", "055d47751d6a5128845a7ea5c7646411": "W_0(z)=\\overline{z}\\,", "055d4b1c657a5659f94cd3be29a7eb3c": " E_2 = E_0 \\left (\\frac{4 m_x m_y cos^2 (\\theta_1)}{(m_x + m_y)^2} \\right) ", "055d8efdf967ba0b99fb155e9ba13602": " \\Phi = \\iint L\\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) \\mathrm{d} \\Omega", "055d9ebe3f02e2ecb1aea9b5288f7012": " R = \\frac {\\textrm''{Static \\,\\, pressure \\,\\, rise\\,\\, in \\,\\, rotor}''}{\\textrm''{Total \\,\\, pressure \\,\\, rise \\,\\,in \\,\\,stage}''} ", "055db5bcedf2479d313b23195ccd4dd3": "{\\mathbf D}_a\\leq \\Delta (l,p)", "055e1511ceee6cdf419b958e47302ef5": "A=LDU", "055e525da2ab4585b14653934c601607": "\\lambda_1 = e^\\varphi", "055e6eb5934e1add9077f56cf45af477": "x_{i0}", "055e70d8c3300dfdfb5dad705eec67c5": "S(y)\\equiv \\frac{9\\sqrt{3}}{8\\pi}y\\int_{y}^{\\infty}K_{5/3}(x)dx", "055ee518ef74037299fd85d27199b5e6": "x_i, x_j", "055f16d961b10c799c8d889dfdf4c780": " U \\subseteq \\mathbb{R} ", "055f3ed5b830b80653844071be81430b": "\\aleph_\\alpha=\\beth_\\alpha", "055f95f9c0b1eb17fd7304b613bf47cb": "V(s) = I(s) \\left ( R + Ls + \\frac{1}{Cs} \\right ) ", "055f9b02b3725e514f0d09cf77cf44c5": " \\Delta G < 0 \\,", "055fb5b5fe1af9b954e324a8d3476d91": "i, i = 0, 1, 2, 3", "055fcefc1ef3182b8ed1ae0cf149091b": "b=1.", "055fe14fb7b286dd73fa7f890c73ccd1": "\\lang i|j\\rang = \\delta_{ij}", "05605eb50844d7da0adfa00a79edb154": "\\theta_f\\;", "056071b398767bff60d2fa58ce678cad": "\\exp \\overline{C}", "0560d806ee2ce7a70952aae668a97853": "\\frac{3x^2 + 12x + 11}{(x+1)(x+2)(x+3)} = \\frac{A}{x+1} + \\frac{B}{x+2} + \\frac{C}{x+3}", "0560e4258b913a2bdbd319cac3071786": "x_{k-1}", "0560f8ae7f453aa55d98ef103ee55cf8": "\\mathcal{L}\\left\\{\\frac{df}{dt}\\right\\} = s\\cdot\\mathcal{L} \\left\\{ f(t) \\right\\}-f(0), ", "0560fda2aa9d1b6a5ce3e5ea42e3ecd6": " P_\\ell, ", "0561415c538ec01c092fa3149cb72cfd": "x_{0} := g_{k}(x)", "05615c41a8772058f713dc4a59bc89a4": "k=|S|", "0561871b25f674b5d87401405a71fbe4": "x = \\left \\{ x_1, x_2, \\ldots x_d \\right \\}", "0561cf6257c67b8da7d20e530b4b2854": "\\alpha = \\delta^{\\beta_1}\\gamma_1 + \\cdots + \\delta^{\\beta_k}\\gamma_k", "05621a51b634c7dced18f7261f41c999": "T_1(\\cos(x))=\\cos(x) \\, ,", "05623a342aecb7bfba450a4d6fb10c04": "\\operatorname{Pic}(X) \\to H^2(X, \\mathbb{Z}).", "05627261e44b9fe9c6696c577f1b72a1": "C\\ell_{p,q}(\\mathbf{R}) = C\\ell_{p,q}^{+}(\\mathbf{R})\\oplus C\\ell_{p,q}^{-}(\\mathbf{R})", "0562a1641aab03785af0d3ed968f0e3f": "X(e^{iw})=1-cos(w)", "0562f50892ad4889a964d403daf603e2": "\\frac{d}{ds}{\\mathbf{s}}_u=\\frac{1}{r}\\cdot\\frac{d}{d\\varphi}{\\mathbf{s}}_u=-\\frac{1}{r}\\cdot{\\mathbf{n}}_u.", "0562f79dba54f724a2e46dd90c1848c8": "\\mathbb Z[1/p]/\\mathbb Z", "0562fcb3da51e39c2c9e3680d8dbf12c": "\\implies f_X(x|Y=y) = \\frac{f_Y(y|X=x)\\,f_X(x)}{f_Y(y)}.", "0563119e25413247d5447cf3497ae6bd": "\\sum_{k=1}^\\infty\\frac{1/k^s}{\\zeta(s)}\\log (k^s \\zeta(s)).\\,\\!", "056316892087717f6b1da0054c5b0b71": "\\psi: k\\left[M\\right] \\to \\prod_{i \\in I}k", "05636e876c6535047b56fff578067c07": "D^{k+1}", "0563d3fc61798c4971e0b7150ae040ca": "{T_{cold}}", "05642005d370089d5b8157be5c1f6a19": "t_1 = \\pi \\sqrt{\\frac{a_1^3}{\\mu}} \\quad and \\quad t_2 = \\pi \\sqrt{\\frac{a_2^3}{\\mu}}", "056438d7487f684072ab843845aa6a8b": " I_{ref} = I_{C1} (1 + 1/ { \\beta}_1) \\ ,", "0564620f798e254b5b2933dc44d0b26b": "T_\\alpha^\\pi = F_{\\alpha\\beta} \\mathcal{D}^{\\pi\\beta} - \\frac{1}{4} \\delta_\\alpha^\\pi F_{\\mu\\nu} \\mathcal{D}^{\\mu\\nu}", "05647b627b7a29a511a922dafbca560a": "\\alpha = k/(\\rho c_p)", "0565181079e13a9ab934f370e98d5b6d": "A = 4 \\sin \\frac{\\pi}{4} R^2 = 2\\sqrt{2}R^2 \\simeq 2.828427\\,R^2.", "05657d9ad07e9f51b2f6f3e210e2e2c6": "\\scriptstyle{\\langle L \\rangle \\Phi}", "0565b67cb9aa47f5e9fcf825bb8d8d93": "\\vec{X} f = f_{,a} \\, X^a", "0565be088eea5995b19bf091d936eea7": "\\begin{matrix}\\underbrace{{2^2}^{{\\cdot}^{{\\cdot}^{{\\cdot}^2}}}} - 3 \\\\n\\mbox{ + 3}\\end{matrix}", "0565e48cc9230dbec676919b2d405b4a": "\\displaystyle{z}", "0565f3387aa61808aa3fc267f563fcfe": "\\lambda_m^2 + 2\\lambda_m - J_m - 3 = 0 ", "0565f7962efe7a29de4cf05523effe90": " | \\psi \\rangle = \\int\\limits_R d^3\\mathbf{r} \\, | \\mathbf{r} \\rangle \\langle \\mathbf{r} | \\psi\\rangle = \\int\\limits_R d^3\\mathbf{r} \\, \\psi(\\mathbf{r}) | \\mathbf{r} \\rangle ", "0565f7aacc902330a589569f23bc3777": "\\partial \\alpha = 0", "0566040c991ab961543164e2b6d0add4": "\\psi(b_k) = \\sum_{i+j=k}(b)_{2i}^{j+1}\\otimes b_j", "056680547cf214f9aa06ac445f46ebb1": "\n\\frac{\\partial F_x }{\\partial x} + \\frac{\\partial F_y }{\\partial y} + \\frac{\\partial F_z }{\\partial z} = 0 ", "05668a01779ec8170b9bd5eeb0e7e921": "\\operatorname{U}(n,\\mathbf{C}/\\mathbf{R})(\\mathbf{R}) = \\operatorname{U}(n)", "0566acb6948ed36d10fdd7b86b154624": "n_b", "0566be246667812ef1b9e2d8217c66a1": "t_{ij}=\\sqrt{\\overline{O_iO_j}^2-(R_i-R_j)^2}=\\frac{\\sqrt{R-R_i}\\cdot \\sqrt{R-R_j}\\cdot \\overline{K_iK_j}}{R}", "0566e1a5690a3eb3da63262a66fa0698": "\nL=\\left(\\begin{array}{cc} 1 & x \\\\ 0 & \\partial_x+1+\\frac{1}{x}\\end{array}\\right)\n \\left(\\begin{array}{c}L_1\\\\L_2\\end{array}\\right).\n", "0566e9701b441428077c015ebab72b10": "E(X)=X^q -\\gamma", "0566ebb077e0d89398d4b183b9ffbfe4": "-{1 \\over 4a}((x+c)^2 + y^2 - 4a^2 - (x-c)^2 - y^2) = \\sqrt{(x-c)^2+y^2}", "05676fd044e1b6537d129a1ce35221ac": "\\frac{\\partial u}{\\partial x},\\frac{\\partial u}{\\partial y},\\frac{\\partial v}{\\partial x},\\frac{\\partial v}{\\partial y}", "0567bc11782096059ff91f3b6ecbfe19": "\\, k_n", "0567c4efa9b4404acc969cc3305f88e2": " \\exp(\\psi(x+\\tfrac{1}{2})) = x + \\frac{1}{4!\\cdot x} - \\frac{37}{8\\cdot6!\\cdot x^3} + \\frac{10313}{72\\cdot8!\\cdot x^5} - \\frac{5509121}{384\\cdot10!\\cdot x^7} + O\\left(\\frac{1}{x^9}\\right)\\quad\\mbox{for } x>1\n", "0567ec7054caa1f3022e6ffcbf0f32e3": "(x^3 + x) + (x + 1) = x^3 + 2x + 1 \\equiv x^3 + 1 \\pmod 2", "05680cf08e27cac3ec72e1bf4d4a939e": "a_n \\,\\!", "056830395974567389aa73b5b8e3c465": "b_r / a_{cr}\\,", "05690502ce6f2f155c061072882033a8": "\\{p: f(x) \\neq 0 \\in p\\}", "0569f6de84b11c3e31f8acfd25b439b6": "z \\cdot y", "056a03fd62348998d916bb11cc2be318": "\\mathcal{L}_Y(S\\otimes T)=(\\mathcal{L}_YS)\\otimes T+S\\otimes (\\mathcal{L}_YT).", "056a4fa84dbb17f1133a0fe6af2e2e79": "M_{\\psi}", "056a64d987f4bea4c72ed4877813caf3": "\\bar{f}g", "056a69254949cc31f6cce2b2a84673cf": " A(t) ", "056aa22a39082777d9a918b5e5f781e3": "x_1^2+x_2^2+\\cdots+x_k^2-x_{k+1}^2-\\cdots-x_{k+l}^2,", "056af43822bf2a1b53146e86a0b99a87": "c^{T} x", "056b0564f5f92a6777295b9f1aad72b5": "\\Delta S_m = -k[\\,N_1\\ln\\phi_1 + N_2\\ln\\phi_2\\,]\\,", "056bd278b08b43f49b1036042801de3e": "\\delta = \\left( \\frac{2 \\pi}{\\lambda} \\right) 2 n \\ell \\cos\\theta. ", "056bffe5543d1ee0ce2bc4be836cc566": "A f(x) = r x f'(x) + \\frac1{2} \\alpha^{2} x^{2} f''(x).", "056c0bacc33c7706434191da1d12a4d5": "\\text{left} = 2i", "056c1ebee11842df114fbc54c6c9081f": "m'+\\frac{l^2}{2}", "056c28d5e04ebb0a184ec46f4218dbc6": "dp=-\\rho\\, d\\phi", "056c2ff05baecaa2d9bc281911e67be5": "k \\to \\mathit{gl}_n", "056c3719d885b88534067656768bba41": "A_t = \\{ x \\in \\Omega ~:~ \\rho(A,x) \\le t \\}", "056c6ce531c45bf819f4c2409c94fec0": "\\sum_{k=-\\infty}^{-1} a_k (z-c)^k.", "056cc60fc03db3fd4826b5d6bf8c2a90": "\\langle j||T^k||j'\\rangle", "056d7c9223e14763ef161f68f7a378f1": "f''(x) = \\frac{4}{9}x^{-\\frac{2}{3}} \\!", "056d87295ca84b3e47d233385a121a44": "I({\\mathbf{v}^K})", "056e099b0d247d31a9d840df6faa31f2": "\\frac{T_2}{T_1} =\n \\frac{p_2}{p_1}\\frac{\\rho_1}{\\rho_2}.", "056e4dede838adb3f029756e8b1d4d19": "E = \\int \\vec{F}\\cdot \\vec{dx}", "056ea57ffb8d615466b22c21ec1ec3e9": " \\mathbf{\\bar f} ", "056eb396f5d970c10a1179f85ccad787": "p^f-1", "056ec6e1e7047facb5a711ddc022dd52": "V(S,T)", "056ed43842b510bfef52c7fca7065818": "\\Psi\\;", "056f7e72d793d391b4f94f277da1d068": "\\mathbf{rank}_q", "056fc1a23d9d948fdc2bacf0369c7647": "a_i \\leq b_i", "056fcc85dc2922d5f85c85479988c69d": "\n\\dot{x}=f(x,u), \\quad x(0)=x_0, \\quad u(t) \\in \\mathcal{U}, \\quad t \\in\n[0,T]\n", "056fe0c9c2dcef04b1d833a805918990": "\\omega+\\Omega", "05701db28cca4ac8cf3bb0028784d4a9": "K[T]/(T-1) \\oplus K[T]/(T-1)", "05707f83c6ef547df16bbeae25c9c227": "dx=\\dot { x } dt", "0570a40ecae288f0da3cac967eabfc89": "\\alpha=m \\omega/\\hbar", "0570ed6fb37085a43bc2eada9939c757": "-log_{10}[H^+]_i = b_0 - b_1E_{i^{ }}", "0571057a349615a6d0c7d0eddba6244e": "\n(x,t) \\mapsto (\\epsilon x, \\epsilon t), \\qquad \\epsilon \\to 0.\n", "0571263d18a78ee05fa0bc29cc854b09": "E/n", "05713aa7c6790e4bcf7207ef58e05c91": "\\forall x_1\\dots\\forall x_n(R(x_1,\\dots,x_n)\\leftrightarrow\\phi(x_1,\\dots,x_n))", "0571754f2edf474b173a58110b284e1c": "z = w ", "0571b600ca602cea19fc3dc53d61de9f": " \\int_{-\\infty}^\\infty |f(x)|^2 \\, dx < \\infty, ", "0571fd912bb6c5ca4f7fb043722a808e": "\\dfrac{d}{dx}(u\\cdot v \\cdot w)=\\dfrac{du}{dx} \\cdot v \\cdot w + u \\cdot \\dfrac{dv}{dx} \\cdot w + u\\cdot v\\cdot \\dfrac{dw}{dx}", "0572b30c7c1461bdae9f31e98964ad41": "H^i(K,A)\\times H^{2-i}(K,A^\\prime)\\rightarrow H^2(K,\\mu)=\\mathbf{Q}/\\mathbf{Z}", "0573242c1b0fb2514cab35af5eafc629": " (\\mu^{-1})^*(q)", "05736af293901a39c6de0ddc3e82bc65": " \\tfrac{N(N-1)}{2} ", "05738dc77a464fa1c03491f72dc18291": "\\,\\frac{\\hbar}{2} |c+\\rangle = S_c |c+\\rangle = \\mbox{D}(y, t) S_b \\mbox{D}^{-1}(y, t) |c+\\rangle \\Rightarrow", "0573998e30bb1b067df261bb84e7eaab": "0<=K<=L", "0573a69296711ddc741820ffb78d9b1b": "X\\otimes B_i=X\\setminus (X\\odot B_i)", "0573c586c2ab52ee222cb359c4fec2be": "2 \\pi r = \\pi d", "0573e756682afb04864c599b3d72534a": "\\|x\\|_p = \\left( \\sum_{i=1}^n |x_i|^p \\right) ^{1/p}, ", "0574a27738923dd052ed0b873c176afc": "0.03", "0574bd365c0dd9e5387b993473af7980": "(2t)^{2n}", "0574daa93b94cb4c103ee36aa8b63570": "R_{sd,X}", "05751a6b7a52ceea27491cb8bf2c03ce": "f(p)=p^2", "057527f0300dd9eb9cb5ef4ac291aaac": " \\omega_1=-0.201, \\omega_{2/3}=-0.223 \\pm i 62.768", "057533f317f61ef1df78c0b2dceb5a3a": "\\Rightarrow_{A \\to a}\\ aAAA \\ \\Rightarrow_{A \\to a}\\ aaAA \\ \\Rightarrow_{A \\to a}\\ aaaA \\ \\Rightarrow_{A \\to a}\\ aaaa", "057570839734aa21edc27381769b7236": "\\delta_Y", "0575751be4544d418593d2c63585b1df": "\\pi : (x, v) \\mapsto x,", "05758ba4e5a9443110f6d3250672a985": "\\int x^n\\cos ax\\;\\mathrm{d}x = \\frac{x^n\\sin ax}{a} - \\frac{n}{a}\\int x^{n-1}\\sin ax\\;\\mathrm{d}x\\,= \\sum_{k=0}^{2k+1\\leq n} (-1)^{k} \\frac{x^{n-2k-1}}{a^{2+2k}}\\frac{n!}{(n-2k-1)!} \\cos ax +\\sum_{k=0}^{2k\\leq n}(-1)^{k} \\frac{x^{n-2k}}{a^{1+2k}}\\frac{n!}{(n-2k)!} \\sin ax \\!", "0575ab58409f9aac03cedd5b6338ac3a": "\\mathrm{ADC}(x,y,z)= \\ln [S_2(x,y,z)/S_1(x,y,z)]/(b_1-b_2)", "0575e80830acab1e929cf5c964e0d546": "[n:=n+1]\\,\\!", "05762f5f873ec78b0108f4864bbfd457": "b+ \\lambda b + \\lambda^2 b + ... = b/(1-\\lambda).", "0576553202f580240b1cf104dc47b948": "\ns=\\sqrt{\\ln(1/R^2)} = \\sigma\n", "0576594e182762841595fc8a2491371f": "I_k \\subset I", "0576708a3bf3b6c024636403d7bcc3ef": "x_j \\geq 0", "0576908980c395c2024cbdfd1aafe578": " \\operatorname{cov}(\\mathbf{X}_1 + \\mathbf{X}_2,\\mathbf{Y}) = \\operatorname{cov}(\\mathbf{X}_1,\\mathbf{Y}) + \\operatorname{cov}(\\mathbf{X}_2, \\mathbf{Y})", "05769dcb970800b24eca2cc69b516db5": "\\phi_2(x,z,t) = A e^{kz} \\cos(kx - \\omega t)", "0576c789af7cc797743f3f7cbad5fb80": "(i,j,k)", "05774954cef3a0e293515b97e89be98d": "\\mathrm{tr}(\\varepsilon)", "0577852e37185c6cd0c4ac6777f14a91": "\n\\begin{align}\n& \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{\\alpha_i - 1} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i} \\, d\\theta_j \\\\\n = & \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j.\n\\end{align}\n", "057796ea520ace98e007953a19207084": "N_\\text{pop}", "0577c77434821f9f34888a3b9db7a197": "D_{E}/N_{E}", "0577d9f31339603fc68203e27839154c": "\\displaystyle -\\frac{\\sqrt{\\pi / 2}}{\\left| \\omega \\right|} - \\sqrt{2 \\pi} \\gamma \\delta \\left( \\omega \\right) ", "05781736e9c5c16927ec2d12d93f3ed9": "D_X(fY) = X[f]Y + f D_XY, \\qquad \\qquad \\qquad f\\in C^\\infty(M)", "057823af195fce5ca941d996f080e228": "\\frac{4}{3}\\pi\\rho\\left(\\frac{c}{H}\\right)^3", "05782e0451ecf804ff449f8842dbd711": "P_{\\text{ph}} = P_{i} - P_{f}", "0578369344ae0c63685d01cd24cf9e75": "d({\\rm tr}(\\mathbf{X})) =", "05784229b1d380c22ed5bef087564b0f": "x' = - \\log(x) \\in \\mathbb{R}", "057847a3ccfac155db00ca47aa3a8edc": " o ", "0578ba0070874ce131a37d4bf39876ee": "\\{\\phi_n\\}_{n = 0,\\ldots,N}", "0578eb4988c85891fd365ff71c1e66d5": "x_0, x_1", "0578f7bdf6a5a97560ddef0fc8df79da": "\\scriptstyle 0 \\,\\leq\\, k \\,<\\, \\nu_j", "05791934f40a51c096001c8b416d99ee": "m, n", "05792b009c1c76032e4f0d74fc039add": "\\tfrac{n (n-3)}{2}", "057932a6583d43823847b32fbaf5b141": " { \\mathit l^{*} } ", "05794b318d1f4b3639ecf61a6a2f2b90": "\n\\sigma_1^2 = \\sigma_2^2 = \\sigma_3^2 = -i\\sigma_1 \\sigma_2 \\sigma_3 = \\begin{pmatrix} 1&0\\\\0&1\\end{pmatrix} = I", "057973b13e59ab4b88cdff34367b443e": "f'(x) > 0", "057983feab1f42353292fb9bca66f887": "|A \\times B|", "05799aff7960fb4b181ae7028f5574e8": "S(a)M = \\{s(x) | s \\in S(a), x\\in M\\}", "0579e8a56c2c3d0afbdd9af1056865ea": "d(\\lambda)\\delta(e^X)\\Phi_\\lambda(e^X)=\\sum_{\\sigma\\in W} {\\rm sign}(\\sigma) e^{i\\lambda(X)},", "0579fcddb7c1f2aa56be97d20a3a5627": "\nW=\\bigoplus_{i=1}^n x_i V.\n", "057a08003c9b7434a4f4215c423c551e": "p(x) = x^3+6x^2+5x+1", "057a0e33b12f5c7967e15f5832e3385f": "A(U_n)", "057a45f8f29fa8af85ce222327568947": "w_T\\ ", "057a4bc42b0cf828b8296e636cb6a7a4": "U(x,t)+iV(x,t) = \\sqrt \\frac{\\pi}{4t} e^{z^2} \\text{erfc}(z) = \\sqrt \\frac{\\pi}{4t} w(iz)", "057a87890570cca5bd5cef01e20e6ce7": "[x]_1", "057a9b1d97bd44392a456f60a5cbde33": "\\mbox{Vert}_pP \\subset T_pP", "057ab4d73fa0d1a004c4446be1dbd9a1": "c=5^2=25", "057afa46cd59e2df99088a5324cab268": "{{V}_{DS}}", "057b22ee5b69f16d58e5fbbec5bea5ef": " \\cong", "057b88949f199e7e691cbe9ec91c6846": " -\\frac{1+\\xi^2}{2} \\, \\partial_\\xi. ", "057b9e161e98f9412b90e36ef4d481c3": "e = \\frac{a}{d}.", "057ba1a33b85d2a85f1a6270f7910103": "(b_s)_{s\\ge 0}", "057ba3b651a36bc6493c706e135d4ce9": "S_e", "057c11c3e16e3b4182b3d3675dff0386": "\\bar L_n W(z)=0.\\,", "057c438d7d6cce182f0416037a19c28d": " \\mathrm{li}(x)\\;=\\;\\mathrm{li}(x) - \\mathrm{li}(\\mu) ", "057c60af4d800c9e42ae59d4ed84671a": "(a+bi) + (c+di) = (a+c) + (b+d)i.\\ ", "057c85faf8ce31f4f57bd127c79373a8": "\\phi^{-}(a)=\\frac{1}{n-1}\\displaystyle\\sum_{x \\in A}\\pi(x,a)", "057d3e9c71af89337100f6ccd17652b2": "F(x) = \\frac{\\Delta\\,t(i)}{f_s(i)}", "057d7fa74c18228541ece69706f4164f": "\\textstyle h(z)", "057d8c542564872299e0cb0e69aa903f": "\\int\\limits_A \\, n(\\rho u\\phi)\\,dA = \\int\\limits_A \\,n(\\Gamma\\nabla\\phi)+\\int\\limits_{CV}\\,S_\\phi \\,dV", "057de9905acef9693d8927400102d9a4": "j \\neq k \\in [n] ", "057dff4a8daa4545ffc37758c9e8704b": "\\scriptstyle{\\|\\hat{u}\\|_{L^2} = \\|u\\|_{L^2}}", "057e5d99cb244ece8533e316322ba604": "(\\mathcal{L}f)(s) = E\\left[e^{-sX} \\right] \\, ", "057e7a19450e9501183720d33f1b7532": " \\ \\psi_o (\\phi) ", "057eb8b5a5594748fb4a27c6e06ab83a": " 8\\pi^2/105 \\approx 75.2 \\% ", "057ec7fb57573ce682b9938f7dd4bb51": "\\coprod_{X \\in K}{F(X)}", "057f67a43202131848df57b01e4adb2e": "\\Omega = 2 \\pi \\left (1 - \\cos {\\theta} \\right) ", "057f761d37d0308db1e5c5cea71ad24d": " \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{21} & z_{22} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} ", "057f89ba663e2b980408c1b4b4cd15c6": "H_{n,m}= H_n^{(m)} = H_m(n).", "057f955857779bb29bd41289dc134374": "\\int_{0}^{1} \\dot{h}_{s} \\cdot \\mathrm{d} x_{s}.", "057fac8904dcbcb9c4e74fc827e80405": "\nS^m_\\ell(x,y,z) = \\left[\\frac{2 (\\ell-m)!}{(\\ell+m)!}\\right]^{1/2} \\Pi^m_{\\ell}(z)\\;B_m(x,y)\n,\\qquad m=1,2,\\ldots,\\ell.\n", "057fec8201937ea7950f7ab6bba5c451": "a \\in U", "057ff7e49c26eaac3acb319b7599dc17": "e_{q}", "05800c2d629fb6726bd3fd05ef8af782": " F_{hkl} = \\sum_{h'k'l'} F_{h'k'l'}F_{h-h',k-k',l-l'} ", "05806b61f30e53a7aa298c5df7e94b19": "_{k+1}V^i_3(x,y)=_kV^r_1(x,y+1)", "0580caa35cb38096c461dc12b333d6da": "\\frac{3}{8}", "0581045b961280329795c8c6a45486b8": "x = f(y) .", "058110797fc814035a19dc84b41ee35f": "\\operatorname{Res}(f,c) = \\frac{g(c)}{h'(c)}.", "058123e87a3a29e28ccc28b96cfbe22c": " \\ell = \\pi \\cdot 2r ", "05813e47f2e6afecad7a27b9b92aedba": " \\sqrt{4\\pi} \\left(\\mathbf{m}, \\mathbf{M}\\right) ", "058172c7f435e28a55decbd97d50a94d": "|\\uparrow_z \\rangle", "0581bf9a4c1c4efe58eb16db28d55ace": "x\\in V(S)", "0582203dbad92451e1ec7a7cbfc1d3e5": "f \\in C^{k+1} (I)", "058310a90451f6f468eed91004066cdb": "D(p||q)\\geq 0", "058316969c3fa24ba9247ba1117d33f1": "= \\frac{600!}{2} \\cdot \\frac{1200!}{2} \\cdot \\frac{720!}{2} \\cdot \\frac{2^{720}}{2} \\cdot \\frac{6^{1200}}{2} \\cdot \\frac{12^{600}}{3}", "05836f96b679b8bd7cdf135bf8242658": "RAC h.p. = (D^2 * n)/2.5 \\,", "05842111e00efaae45238f50d6f79b46": "\\lambda(x,y,z) \\equiv x^2 + y^2 + z^2 - 2xy - 2yz - 2zx", "05842c3d39e2cc3218963be659fd058e": "\\frac{\\theta}{\\theta_b}=e^{-mx}", "05844333cebabb90adb0b1ff0466149e": " \\hat{E} = i\\hbar\\frac{\\partial }{\\partial t} \\,\\!", "05844c6d990659e658f08b35c8afe3b1": " \\operatorname{perm}(A)=\\sum_{\\sigma\\in S_n}\\prod_{i=1}^n a_{i,\\sigma(i)}.", "05845e95a493130bfd283f00883865e6": " \\beta", "05848330faa279f4cb0071cb153a3534": "p(x) = \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha-1}e^{-\\beta x}.", "0584b20d5625be70127643626f43cb71": "f=\\frac{ab}{c}.", "0584ed27d4e794dad1db3f81e7bbbea8": "k^{-s} F(s;kq) = \\sum_{n=0}^{k-1} F\\left(s,q+\\frac{n}{k}\\right).", "058550c85a503e65a1b89ee16888fec4": "(\\varepsilon,\\eta):F\\dashv G", "05857d9d93a6f74ec43cfe51ec11acc6": "p_1 \\equiv \\frac{\\partial}{\\partial q_1} L_d\\left( t_0, t_1, q_0, q_1 \\right)", "058580224fb05a175bdb6d8ddf62a94c": "(a + c) \\mid b", "058611c3621fe41d898d1d9b12e1feb6": "q = \\left\\lfloor {n_1} / {n_0} \\right\\rfloor", "05863ab8b1604fb2e47ac4df8d1bb7dc": "g_y(\\mathbf{y}) \\triangleq \\begin{bmatrix} \\mathbf{0}\\\\ 1 \\end{bmatrix},\\,", "05866caf91be86c7599a6120cfdb5d70": "\\begin{pmatrix} (mc^2 - E + e \\phi) & c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) \\\\ -c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) & \\left(mc^2 + E - e \\phi\\right) \\end{pmatrix} \\begin{pmatrix} \\psi_+ \\\\ \\psi_- \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}. ", "0586c47757aa46b672cd02d11528b548": "j(\\tau) = N", "05879c918710403f506c5c45a540ed62": "\\scriptstyle \\epsilon/m_{0}\\sim1.76\\times10^{7}", "0587b2d9f6b659f62a0a82a2936f1048": " V \\ne W \\to \\operatorname{let-combine}[\\operatorname{let} V : E \\operatorname{in} \\operatorname{let} W : F \\operatorname{in} G] \\equiv \\operatorname{let} V, W : E \\and F \\operatorname{in} G ", "0587d95a881a0784ec095a16d0720b54": "g(X)=\\frac{dF_1(X)}{dX}", "058801a9a81de4377e2ef6959d0d3d89": "[N_i,P_0]=iP_i\\left(1-\\frac{P_0}{\\eta}\\right)", "058838738338afba3b46fded336a157d": "~V=\\frac{(\\sigma_{\\rm ap}+\\sigma_{\\rm ep})\\sigma_{\\rm as}}{D}~", "058849d454ab21f6fe48b4672fe81b81": "\\displaystyle{Q(a,b)a^{-1} =b.}", "0588607f74debea92ac58e9beda8c0ef": "u \\cdot u_{n}", "0588a554f880dc083ee41f783c9c2cae": "{n^{O(1/\\varepsilon^2)}}", "058912ca192d028678452b1ff1e895df": "x^md(x)", "0589446d07423f1cf786dc2db08901a7": " \\Bigl\\| \\sum_{k=0}^n \\varepsilon_k \\alpha_k b_k \\Bigr\\|_V \\le C \\Bigl\\| \\sum_{k=0}^n \\alpha_k b_k \\Bigr\\|_V ", "0589634ff96a29b5b6027675aa45f6f4": "J_+|j\\,m\\rangle = \\alpha|j\\,m+1\\rangle,\\quad", "05898dbef5eb0036dac5efc7dbc574f1": "P(x) = \\frac{-2}{x}", "0589a243a3b7f31a686ca9321aa64b64": "(|\\text{dead}\\rangle + |\\text{alive}\\rangle)/\\sqrt 2", "0589fe63259c31bc8394d0f1dbfa49b7": " S_i ", "058a0159f6993abb9800a0876f570c53": "L(s,\\pi,r_i)", "058a113e25e870d4154580c91d6ac1c3": "16C,\\;16D,\\;32A,\\;32B,\\;32C,\\;32D,\\;34A,\\;46A,\\;46B\\;", "058a1ce7c2f092541fbafe263690e611": "= \\sgn( \\sin (\\theta+ \\frac{\\pi}{2})) \\frac{\\sqrt{1 - \\sin^2 \\theta}}{\\sin \\theta}", "058a32571bab72f7af24319b1f57d425": "\\tilde \\nu", "058a46442473533fe9c79b81850d8de6": "\\,_2F_1(a,b;c-1;z)-\\,_2F_1(a+1,b;c;z) = \\frac{(a-c+1)bz}{c(c-1)}\\,_2F_1(a+1,b+1;c+1;z)", "058a682eda2aa0b1b205129e1e36c535": "E'", "058a98728120f8e485502ff4c60835c1": "\\,t\\,", "058ad826b638e617036c8e3545c7242f": "I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \\le I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\\Leftarrow~~p_{t_n} \\le p_{t_r}", "058af3bca72462c2ad6d47dbfa36aa38": "J_n = -\\frac{\\cos{ax}}{(n-1)x^{n-1}}-\\frac{a}{n-1}\\left [-\\frac{\\sin{ax}}{(n-1)x^{n-1}}+\\frac{a}{n-1}J_{n-2} \\right ]\\,\\!", "058b28fec217060370f1f651de40658b": " P = {2 \\over 3} \\frac{q^2 a^2}{ c^3} \\mbox{ (cgs units)} ", "058b2ca50228deb144d988a9561c1d18": "\\mathrm{R{^{\\cdot}} + O_2 \\ \\xrightarrow {fast} \\ ROO{^{\\cdot}}}", "058b66f546bbae3fb29a7e8259a42364": "\\Delta(a)", "058b77bb9451583a056115e5c62f2dff": "\\exp_{10}^3(2.18726)", "058bb7c4b3cb9d8b1a7bbc860efda23a": "A =\n\\begin{bmatrix}\n 5 & 4 & 2 & 1 \\\\\n 0 & 1 & -1 & -1 \\\\\n-1 & -1 & 3 & 0 \\\\ \n 1 & 1 & -1 & 2\n\\end{bmatrix}", "058bba9129ccab88e489d2730febaa0d": "\n\\left(\\frac{\\partial U}{\\partial T}\\right)_V\n= T\\left(\\frac{\\partial S}{\\partial T}\\right)_V\n- p\\left(\\frac{\\partial V}{\\partial T}\\right)_V ; C_V = \\left(\\frac{\\partial U}{\\partial T}\\right)_V\n", "058c0dbac1a605db3a931a3ad1e62048": "kx-\\omega t = \\left(\\frac{2\\pi}{\\lambda}\\right)(x - vt)", "058c0e5a0bba35fb39f56cb8261396ee": "W(s) = \\sum_{i \\in N} u_i(s),", "058c3cc810faf90fa02b532d44bd93de": "\\Phi(M,x)=n", "058c5b9d6783c45a574f3951275aa144": "\\ell^{(-1)}=\\frac{2}{1-\\alpha}p^{\\frac{1-\\alpha}{2}}=p", "058cae5e7470da022b1a0bd31cf47d57": "(T_h f)(s) = h(s) \\cdot f(s).", "058cbc8415ed139e477dc4d67365153a": "\\sigma_{r} > \\sigma_{f}", "058cde4336e8d993179632cfee6939a4": "\\Omega_\\text{rel} = \\frac {3\\pi G m}{c^2 r}.", "058cdf0a7a1ab69ef3b026214341d476": "\\lim_{t\\rightarrow 0} \\vartheta(x,it)=\\sum_{n=-\\infty}^\\infty \\delta(x-n)", "058ce521659b36dd1c773ed1563dc8a9": "\\quad (A \\cdot B) + (A \\cdot C) = A \\cdot (B + C)", "058cec60659527b3415f49c4d666261a": "p(t)=\\delta (t - \\tau )", "058d38950ec4527f6b9ed00b276195ae": "\\int \\cosh x \\, dx = \\sinh x + C", "058d470bb28dc01348bef8eed55608da": "\\|x\\|_\\infty = \\sup_n |x_n|", "058d7c0d06b525d5cca64ba2414d8579": "=6", "058e043d15d210ad7035a7c62707767c": "\\sum_s P_s = \\frac{1}{Z} \\sum_s \\mathrm{e}^{- \\beta E_s} = \\frac{1}{Z} Z\n= 1. ", "058e20c0187ab310bbfacd83dbe56743": "\\int_{-\\infty}^\\infty H_m(x) H_n(x)\\, \\mathrm{e}^{-x^2}\\, \\mathrm{d}x = \\sqrt{ \\pi} 2^n n! \\delta_{nm}", "058e46087c15f585f9dcec23ceeb8248": " \\left( \\left|x\\right|^{r} + \\left|y\\right|^{r} \\right)^{t/r} + \\left|z\\right|^{t} \\leq 1", "058e5842df9a74c8c65bc58d03e6dfad": "\\sin \\alpha \\cos \\beta = {\\sin(\\alpha - \\beta) \\over 2} + {\\sin(\\alpha + \\beta) \\over 2} \\approx {\\alpha - \\beta \\over 2} + {\\sin(\\alpha + \\beta) \\over 2} ", "058e80a4ab77f55be256e18ba64707c2": "G(s)=K_d \\frac{s^2 + \\frac{K_p}{K_d}s + \\frac{K_i}{K_d}}{s}", "058e8d0e4e13e2ad6f046c0048d08676": "\\langle\\overline{z}\\rangle=e^{i\\mu-\\sigma^2/2}. \\,", "058ea4b0b13ae669b827d4002475a648": "Q_0=m_0 s_b L_{sludge0} ", "058eaefcf0f2e16da2c9741e9cc8f340": "k_{2(3)}\\equiv k_{2(2)}", "058ed80904627b8193cd1fbfd75b502c": "\ne_i^{t+n} - e_i^t = NS_i + IM_i + RS_i + AL_i\n", "058edad9bc884a06ddd1a0290f5d61b1": "\n\\begin{bmatrix}\n Y_1 \\\\\n Y_2 \\\\\n Y_3\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n \\cos\\theta & -\\sin\\theta & 0 \\\\\n \\sin\\theta & \\cos\\theta & 0 \\\\\n 0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\n X_1 \\\\\n X_2 \\\\\n X_3\n\\end{bmatrix}\n", "058ee734a301721e209bac7eeae3eeaa": "k \\mod q \\ne 0", "058efe614790dbb05f0134a9ba1f229b": "\\frac{P(R) - P_\\infty}{\\rho_L} = \\frac{P_B - P_\\infty}{\\rho_L} - \\frac{4\\mu_L}{\\rho_LR}\\frac{dR}{dt} - \\frac{2S}{\\rho_LR} = R\\frac{d^2R}{dt^2} + \\frac{3}{2}\\left(\\frac{dR}{dt}\\right)^2", "058f18c4de16ad5e4e948d5a35a5a371": "T' = \n\\begin{bmatrix}\n0 & 0\\\\ T & 0\n\\end{bmatrix}\n\n\\quad \\mbox{and} \\quad\n\nN' = \n\\begin{bmatrix}\nN & 0 \\\\ 0 & M\n\\end{bmatrix}.", "058f1c58404718bfef62fc8469b2451f": "E^\\mathrm{damping}(\\mathbf{x}_j,t)=\\frac{E_j^\\mathrm{ret}(\\mathbf{x}_j,t)-E_j^\\mathrm{adv}(\\mathbf{x}_j,t)}{2}", "058f758c4d6146daa6ee6006adf74bc7": "\\scriptstyle {\\frac{1}{\\sqrt{12}}}\\mathrm{LSB}\\ \\approx\\ 0.289\\,\\mathrm{LSB}", "058f7763c0d8276113bd3071bc73718d": "\\omega(z) = W_{\\big \\lceil \\frac{\\mathrm{Im}(z) - \\pi}{2 \\pi} \\big \\rceil}(e^z).", "058fa67dd1928085ab61e9d09f691a8d": "\n(x+1)(x-1) = 1\\,\n", "058fa75172f003a02c43e23043dc41f7": "O(N^{1.5})", "05903eeb43b8650a76a00736fd97466e": "D_F = k_2 \\cdot\\frac{\\lambda}{{NA}^2}", "0590580c311dff2d7a5d79e10c912e16": "\\displaystyle c_f", "059081225fb1e2be67bb10d0071e1c9d": "\\ \\displaystyle \\{S(d): d\\in D\\}\\ ", "0591382720b9f82853663a2214536734": "H(n,q^2)", "059167c366dbebe947e02a226e082451": "{h_1} + \\frac{V_1^2}{2} = {h_2} + \\frac{V_2^2}{2}", "05919836f7bb584b48725d3190ee2133": "c'=c\\pm kv\\,", "0591e21a312c6022ef7bed37de8def05": "\\Phi_{ij}\\mapsto-\\Phi_{ij}", "0592074890fe569d7e99a0621d8934d2": "\\omega_M, \\omega_N", "059212990734c096c9ecbcbdb51b37d0": "\\rho = \\int^T_0 k(t)S(t)dt = \\int^T_0 S(t)^2 dt = E", "05928b13c02c0dddd7ab38de5a50cdad": "b, c", "0593076a0a8e42ddd486e980d8a7378a": "N(\\mu,1/n),", "05934a6dfdc9db4870b16992573199fa": "u(x) \\lneqq \\max_{y \\in \\partial \\Omega} u(y)", "05935d88c1621e854f158d005cfbbbf1": "\\Omega \\setminus c", "05939125bc21745ade8be1ac850190db": "p_{\\mathrm{c}}=P(\\mathrm{SINR}>t)=1-p_{\\mathrm{out}}", "0593c8f84588c06cde68d0fc8b2a3de3": "(X_n)", "0593ceb5c70d6a8078b25691ac6de147": "(\\widetilde{s}^1, \\dots, \\widetilde{s}^T, \\widetilde{o}^1, \\dots, \\widetilde{o}^T) ", "059400fcdb7b6dbd68f163355e81db6c": "|N - Z|", "0594251298a83049cd9c21646f652c51": "\n\\text{minimize} \\quad \\text{over } \\widehat D \\quad \n\\operatorname{vec}^{\\top}(D - \\widehat D) W \\operatorname{vec}(D - \\widehat D)\n\\quad\\text{subject to}\\quad \\operatorname{rank}(\\widehat D) \\leq r,\n", "05946cf357fed2266088f9437991cf89": "N_{A(i)} = 0\\,", "0594bea4ff32f27fe0c7914658e06984": "\\psi_{2n}/y", "059559a9bd812fc270276feab50a8052": " \\frac{1}{2}[(\\kappa+1) \\theta~\\sin\\theta - \\{1 - (\\kappa-1) \\ln r\\} ~\\cos\\theta] \\,", "0595815d64ecffd529fbc3f684e64c73": " \\Delta G_{\\rm em} ", "0595f7d34ddda3f9763cadecbd9f6547": "\\Gamma^I", "059619a3cebb398372a64b4c73580ada": "m I= \\int_a^b m\\varphi(t)\\,dt \\le \\int^b_aG(t)\\varphi(t) \\, dt \\le \\int_a^b M\\varphi(t)\\,dt = M I,", "05961be985b19697bc6b124e52c48a5a": "I_{n_2, k_2}", "05962269fc7570a5e4b24c53a979a2d6": "Q = \\mathrm{Ran}(A - \\lambda I) \\cap \\mathrm{Ker}(A - \\lambda I) \\neq \\{0\\},", "0596302fde1dc2cc0678f7805b205a15": "\\bar\\psi\\equiv\\psi^\\dagger\\gamma^0", "059663f6390660fe913cf64da7cff186": "u(\\vec{p}, 1) = \\sqrt{E+m} \\begin{bmatrix}\n1\\\\\n0\\\\\n\\frac{p_3}{E+m} \\\\\n\\frac{p_1 + i p_2}{E+m}\n\\end{bmatrix} \\quad \\mathrm{and} \\quad\nu(\\vec{p}, 2) = \\sqrt{E+m} \\begin{bmatrix}\n0\\\\\n1\\\\\n\\frac{p_1 - i p_2}{E+m} \\\\\n\\frac{-p_3}{E+m} \n\\end{bmatrix} ", "059665a79da90a9b27772d692d991814": " \\Delta_1 = 1 \\, ", "0596a6b7d0432b9dcb676aff1041de16": "\\mathrm{ARFCN}=\\frac{f - 300 - 0,0125}{0,025}", "0596bc73eb25f56d08b9afec41e693ae": "\\hat\\theta=\\theta^{(M+1)}", "0596bd1d976b95c82dc2e4113845ffc9": "(2k+1)", "0597078cef136f8ee64dc05937bcc759": " \\Delta = \\det(M) = \\det\\left(\\begin{bmatrix}A_1 & B_1 & B_2\\\\B_1 & A_2 & B_3\\\\B_2&B_3&A_3\\end{bmatrix}\\right) ", "0597920f4622a27a11ef2cf6e0e4a737": "\\rho_0 = F \\cot^{n} (\\frac14 \\pi + \\frac12 \\phi_0)", "0597e2cdcb0557084c936bb7d92f5815": "(2^{2n} - 1) - 2^{n + 1}", "0597ed548c0a81dfa30ae7d7ac201f31": " \\aleph_{1} ", "059806bda3bd68c6db4b7c5ce378a95e": "L_{X} = \\sum_{i} b_{i} \\frac{\\partial}{\\partial x_{i}} + \\frac1{2} \\sum_{i, j} \\big( \\sigma \\sigma^{\\top} \\big)_{i, j} \\frac{\\partial^{2}}{\\partial x_{i} \\, \\partial x_{j}}.", "059820265ec22520c46c7f6f1a6e9e49": "\nA = L_{1}^{-1} L_{1} A^{(0)}\n= L_{1}^{-1} A^{(1)} = L_{1}^{-1} L_{2}^{-1} L_{2} A^{(1)} = \nL_{1}^{-1}L_{2}^{-1} A^{(2)} =\\ldots = L_{1}^{-1} \\ldots L_{N-1}^{-1} A^{(N-1)}.\n", "059899c64f7db0368e50acdb6a707233": " \\mathbf{\\hat T^ \\dagger} (\\varepsilon) \\mathbf{\\hat H} \\mathbf{\\hat T}(\\varepsilon) = \\mathbf{\\hat H} ", "05989d00ccacfb4642b8160a44bc64d9": "\\frac{1}{[A]^{n-1}} = \\frac{1}{{[A]_0}^{n-1}} + (n-1)kt", "0598cbccdd968ba76913621e4f5088b5": "A \\neq B", "05995ce08147baeb5fe26791807d7a84": "\nk_2 \\approx \\frac{1.5}{1+\\frac{19\\mu}{2\\rho g R}},\n", "0599699a20cb859e197a4c564c4c47cd": "xp(x)", "0599b4b872f6708fb7e6347a15a11c95": "u_1=\\mbox{Re}(y_1)=\\tfrac{1}{2} (y_1+y_2) =e^{2x}\\cos(x),", "0599bfc4eb518f0b6f9d122f1c2ff42f": " \n\\Delta m = 0 \\quad\\hbox{and}\\quad \\Delta l = \\pm 1 \n", "0599bfd0ad922eed6e26d767c6ad53e2": "(e, g, e): (A, e) \\rightarrow (A, e)", "059a194fc6285d7e2f2079721ff01fff": "j_g(x)", "059a28895dc6969cc4a70a491c9fdefe": "\\mathcal S = (\\mathcal S^1,\\dots,\\mathcal S^n)", "059a4f4e34c6270dd8579e6185198f2e": "\\gamma(s)=e^{i\\phi}Q^s\n\\prod_{i=1}^k \\Gamma (\\omega_is+\\mu_i)", "059ab98a87e4a57ecf8d4c113c392b7d": "EL(\\Gamma_1)=0", "059af9424ed592bb0476be33188b38f3": "T_\\text{goal} = b \\log_2 \\left( \\frac{A}{W} + 1 \\right)", "059b3b273d826477ff79174fa4e57b02": "y^2=x^3-x,", "059b9b4866ffa54ccf9a56e7517d209e": "x'=ax.\\,", "059bc2d9bc55c7cdbb9e82a0d2023c2b": "|H|= \\sqrt{H_x^2+H_y^2+H_z^2}", "059bebfb939b7c6f14a0c74fc933dea8": "\\text{arcsin} (x) \\approx x", "059c7e548f7ffdb49cfaaba48531baa6": " \\sum_{i=1} \\left(Y_i - g\\left(X'_i \\beta\\right)\\right)^2. ", "059ccf126490e1ffb0d75267846b1ca3": "\\frac{M}{C}", "059cec77641c915e7434c0830ebe5dd9": "1200\\log_2( 3^{1/13} )= 146.3...", "059d13248aa9b3f33a9f03be87389c2d": "P_{TAF}", "059d2e1c5f7e6f4beb099432654f423c": "1\\le q, p < \\infty", "059d8050582684950696d2bf4a0a9c22": "\n\\begin{bmatrix}\n0 & -1 & 0 \\\\\n-1 & 5 & -1 \\\\\n0 & -1 & 0\n\\end{bmatrix}\n", "059dcb7018310a884c8e68f80838958c": "\\,(1 + 9 + 6 + 8 + 3)^3=19{,}683", "059e347bb40b012f97255c18b26df569": "C_{IJK}", "059e75340a274fbea9a34c246670e73f": "z_n", "059e757564c12a958d2ef2d59cfd3bec": "\n\\beta^{0} =\n\\begin{pmatrix}\n0&1&0&0&0\\\\\n1&0&0&0&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\n\\end{pmatrix}\n", "059e7593930844763fce650787af8806": "\n\\mathbf{r} = r(\\hat{ u} \\cos \\theta + \\hat{ v} \\sin \\theta) = r \\hat{ u}(\\cos \\theta + \\hat{ u} \\hat{ v} \\sin \\theta)\n", "059e93c547b1182a9c4aef775da41c5a": "\n f(I_1, J_2, J_3) = 0 \\,\n ", "059fa5813e555d8ad1d205bcd7e7edb1": "\\zeta=+1.", "059fe7b20132b9c261200a7a7bd62966": "k_i = K_i/L_i", "05a03641923964c19f02fab6c874798e": "G(x) = \\sum_{1 \\le n \\le x}F(x/n)\\quad\\mbox{ for all }x\\ge 1", "05a03beaf3b2097c77dcbbabceddbc6a": "\\begin{cases} \\dfrac{\\partial v}{\\partial t}(t, x) = A v (t, x) - q(x) v(t, x), & t > 0, x \\in \\mathbf{R}^{n}; \\\\ v(0, x) = f(x), & x \\in \\mathbf{R}^{n}. \\end{cases}", "05a0f1219d603d2c824bb08383d87c4e": "\\hat{\\mathbf{z}}\\,\\!", "05a10f7b11ab4c4d8367790cf8710ff6": "\\mathbf{E}_{\\mathbf{P}} \\left([Y_t-Y_s]\\chi_F\\right)=0,", "05a14d12d18b07a35c4a3b985bcd8360": "V(S) = \\{x \\in \\mathbb{A}^n \\mid f(x) = 0, \\forall f \\in S\\}", "05a1670a1689d4290c39177e63d72bf0": "\n \\boldsymbol{l} = \\boldsymbol{l}^e + \\boldsymbol{F}^e\\cdot\\boldsymbol{L}^p\\cdot(\\boldsymbol{F}^e)^{-1} \\,.\n ", "05a1b0fe8d72d0636008275c15fd2299": "S(q \\to 0)", "05a1bb8d7daf0ca1440d3671c888141a": " X \\equiv X \\left ( x_1, x_2 \\cdots x_n \\right ) \\,\\!", "05a22200576b61fe2ef5aa3b91e71a2b": "\\phi(x) = \n\\begin{cases} \n1 & \\text{if } x > x_0 \\\\\n0 & \\text{if } x < x_0\n\\end{cases}", "05a23464412cb303e8a83e9594770967": "\n\\begin{align}\nE_1 & = { q_1^2 - p_1 r_1} \\\\\n& = {[(p + q)(q + r)]^2 - (p + q)^2 (q + r)^2 = 0}\n\\end{align}\n", "05a23510438cdd29a1ad83afe9d9f6c2": "(Bxuv \\and Byuz \\and x \\ne u) \\rightarrow \\exists a\\, \\exists b\\,(Bxya \\and Bxzb \\and Bavb).", "05a23c52ac89bc9ebd3c184ca593617c": "\\tilde{d}=\\frac{ncp}{\\sqrt{n}}.", "05a25a16af94aad599d3e12d42ebd8b9": "(\\lambda x.E)y \\equiv E[x:= y]\\,", "05a2631bbe66935c5c813a4fde3b7ab9": "\\Pi(n,m)=\\int_0^{\\pi/2}{\\frac{1}{(1-n \\sin^2 \\theta)\\sqrt{1-m \\sin^2 \\theta }}} d\\theta.", "05a28eddeefb3e7ec9ad3d0d823e3b51": "\\mu:G\\times G\\to G", "05a2adcce0f6ea21fe1d923c764ef033": " \\left \\| \\varphi_{N,x} \\right \\| = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\left | D_N(x-t) \\right | \\, dt = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\left | D_N(s) \\right | \\, ds = \\left \\| D_N \\right \\|_{L^1(\\mathbf{T})}.", "05a2bf3560d6f10b0957cd3fba163d77": "\\eta_s", "05a2ec30cdec30ffa6c7a682c010d9a3": "\\langle u_n:n\\in\\mathbb{N} \\rangle", "05a305c8d54caaf41a3043162d35b737": "\\gamma_\\text{LG}\\ ", "05a32025b1c511ac180801de8d64bd1e": "Y_{ij}=1", "05a325ba2832c04eb0a9be1724014c00": "\n\\begin{align}\n -fv &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part x}+K_m \\frac{\\part^2 u}{\\part z^2}, \\\\\n fu &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part y}+K_m \\frac{\\part^2 v}{\\part z^2}, \\\\\n 0 &= -\\frac{1}{\\rho_o} \\frac{\\part p}{\\part z},\n\\end{align}\n", "05a3898b2b506b889a4b2f5ee0ef71ed": " \\mathrm{Tr}(\\Pi_\\alpha \\Pi_\\beta ) = \\mu^2 \\;", "05a3b69f25b972b48a814ddf8f294836": "\n \\rho_0~\\frac{\\partial\\mathbf{v}}{\\partial t} + \\nabla p = 0 ~.\n ", "05a3dbacc000a46c03ad1d7bf299b5dc": "s_A=\\sum_{i=1}^m u_i a", "05a4c3fe37dd5267d00a2f615e697e44": "f(q)", "05a4eacbd53ee615aa8bce3f4c4b2256": "\\scriptstyle F", "05a502a14e31c468da088859adb01294": " \\left (\\frac{p_2}{p_1} \\right )^\\frac {1}{\\gamma}", "05a50c8d59692e1b21ea86916ad49629": "I(p_{t_m},\\alpha \\cdot p_{t_m},q_{t_m},q_{t_n})=\\alpha \\cdot I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})", "05a52e65d34e8c97528c6325153643e5": "A/A_t", "05a531358c926fc155d2972747886f23": "z^0\\in W", "05a54f7dfbbdffc4e8ed3ecd4d196b54": "\\left\\langle E\\right\\rangle=-\\frac{d\\log\\left(Z\\right)}{d\\beta}= \\frac{\\varepsilon}{2} + \\frac{\\varepsilon}{e^{\\beta\\varepsilon}-1}.", "05a56d47bc7464213948240437d78bb9": "\\cos \\phi_0 = \\frac {\\mathbf{W}} {\\mathbf{V_1} \\mathbf{I_0}} ", "05a576cf1c7ec9ac44b971c70222084d": "P_{\\text{SEN}}", "05a57c17051ccbd74999fe193a96296f": "(M)", "05a5860dee2ffe27c4829192d7b6f653": "\n \\left( {\\frac{\\partial G / T}{\\partial T}} \\right)_P\n =\n \\frac{1}{T}\\left( {\\frac{\\partial G}{\\partial T}} \\right)_P - \\frac{1}{T^{2}}G\n =\n - \\frac{1}{T^{2}}\\left( {G - T\\left({\\frac{\\partial G}{\\partial T}} \\right)_P\n } \\right)\n = - \\frac{H}{T^{2}}\n", "05a5d2f763dd0afef27ee5372b984bef": "v(x) = au(x) + b.", "05a5d998ebc0923def6b40d8ce610619": "\\sqrt[3]{\\left(\\sqrt{2}+ \\sqrt{3}\\right)\\left(5 - \\sqrt{6}\\right) + 3\\left(2\\sqrt{3} + 3\\sqrt{2}\\right)} = \\sqrt{10 - \\frac{13 - 5\\sqrt{6}}{5 + \\sqrt{6}}}. ", "05a5deae321ac8b5499f169726704a3e": "d^*(A)", "05a640d5e5c0b96112adf4645d970cc3": "-\\boldsymbol{ \\nabla \\times}\\left( \\boldsymbol {\\nabla \\times B} \\right ) = \\nabla^2 \\boldsymbol B =\\mu_0 \\epsilon_0 \\frac {\\partial^2}{\\partial t^2} \\boldsymbol {B } = \\frac{1}{c^2} \\frac {\\partial^2}{\\partial t^2} \\boldsymbol {B } \\ , ", "05a69ce951018ee635e7904f95f1b81d": "0 = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial L}{\\partial q_j} + \\frac {\\partial D}{\\partial \\dot{q}_j}.", "05a6ab966433656751786a90220bf926": "\\eta_p", "05a6fa309cb2210dce11fff50543885d": "\\bold{D} \\cdot\\bold{\\hat{n}}dS = \\iiint \\rho_f dV ", "05a6fbb6a85505b06737daa61a7804d8": "\\frac{18}{11}", "05a74314523ecaef49f6d096dcfb2db8": "v,w\\in T_pM", "05a755e52977242b6ddc4a0be2586cf0": " \\forall x \\in \\mathbb{R} \\quad x < x+1 ", "05a7925a61f3d86855365871b926a815": "C_{QL} = e^2D_{Gr} = \\frac{e^2m^*}{\\pi\\hbar^2}", "05a7c250a7657563d94fde4d9d389be9": "+0, \\ \\times1, \\uparrow1, \\ \\uparrow\\uparrow1, ", "05a7d68f204dd5981bf9a03c5523d4e3": "\\ 1.39m+2ln(n).", "05a7e0a3c05d0ae38bea6040d5921972": "\\sum_{1\\le k\\le n \\atop \\gcd(k,n)=1}\\!\\!k = \\frac{1}{2}n\\varphi(n)", "05a8582a57ed00fc2b8a616e13d365d4": "\\theta\\sim\\pi\\,\\!", "05a86d6045bfbb3ebe58e44b3e3c6e03": "\\tfrac{\\lambda(1+\\nu)}{3\\nu}", "05a881cca6a168c1289809a58c88862d": " 2 \\times 3 ", "05a89d47da17caf1e29bd6bb8b0679b6": "(-0) + (-0) = (-0) - (+0) = -0\\,\\!", "05a93c269574b71673c27657ba128827": "f(x) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ x \\in S \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ x \\notin S\n\\end{matrix}\\right.\n", "05a9a9253de629a22b0de164f7876ec1": " | \\Psi_{E}\\rangle \\langle \\Psi_{E} |", "05a9c4872bef340f03df495e76a13ba9": " P(\\xi) \\, \\hat u (\\xi) = \\hat f(\\xi). ", "05aa01e7c266f58bb67b90354de99c43": "\nA(L(G)) = B(G)^{T}B(G) - 2I_q\\ \n", "05aa172af5cbe36e9633e1c60bc11b8b": "\\mathbf{\\Phi}_{11}= \\sqrt{\\frac{3}{8\\pi}}\\mathrm{e}^{\\mathrm{i}\\varphi}\\left(\\mathrm{i}\\,\\hat{\\mathbf{\\theta}}-\\cos\\theta\\,\\hat{\\mathbf{\\varphi}}\\right)", "05aa1b8cd5d151e87afd734a4c6edf73": "\n\\delta q_{i} = \\epsilon g_{i}(\\mathbf{q}, \\mathbf{\\dot{q}}, t)\n", "05aaf88555241a5c68c772efe40d512c": "B_2\\,", "05ab0872388a5950ce822076bd1199db": "m^*_l , m^*_t ", "05ab6c9f62cdb72589e8b74958e9e61f": " \\frac{P_f}{P}=\\theta ", "05abb692d211f6070acdca75d5371fd6": "\\bar{d}_i ", "05abc0cae09fc2d1037dea041efc2452": "\\mathbf r(s) = \\mathbf r(0) + \\left(s-\\frac{s^3\\kappa^2(0)}{6}\\right)\\mathbf T(0) + \\left(\\frac{s^2\\kappa(0)}{2}+\\frac{s^3\\kappa'(0)}{6}\\right)\\mathbf N(0) + \\left(\\frac{s^3\\kappa(0)\\tau(0)}{6}\\right)\\mathbf B(0) + o(s^3).", "05ac4ab7acee500e0844cf32d45f363f": " \\eta = \\frac{T_H - T_C}{T_H} ", "05ac7a592ac15a18edc4cd1610e87ea8": "\\int", "05ac9133d639722a044b69afebcac8b6": "\\mathbf{g} = \\mathbf{h}\\oplus\\mathbf{p}", "05aca4449050dba074fa25af215863c6": "\n\\langle A \\rangle_\\rho = \\frac{\\int D \\sigma \\; A[\\sigma] \\; \\rho[\\sigma]}{\\int D \\sigma \\; \\rho[\\sigma]} .\n", "05ad9aa7741b8dd9c8282adfbbbb1ee4": " R(r) = \\gamma J_n(\\rho), \\,", "05adeadf32234f91b991e3ffc00d4276": "(p-\\xi)(q-\\xi)=\\frac{k'}{k}(p'+\\xi)(q'+\\xi)", "05ae0c7816ab8c2dba5f57b946a5d3c3": "S=\\cup_{n=1}^\\infty S_n", "05ae0d3dee59e87b4835d0764dd91491": "\\epsilon^0: \\quad S_0'^2 = Q(x),", "05ae382ec6db2593fd16efc16e826e95": "= \\begin{matrix} \\frac{1}{2} \\end{matrix} m_0 v^2 \\ ", "05ae5dfa706b7faa27218940abbb1dd9": "p, a_1, a_2, .., a_n, q", "05ae8ae73c309f38677a320bfe49f71e": "\\alpha(a,b)", "05aeda03cea692eb54855eee884616ca": "G_0/G_1 = H_0/H_1", "05af08f63698bdb185b50d8e1bc418f1": "\\gamma := F_{12}\\,", "05af247981d724e96b1f296624e43462": "F(\\bold{x}, t;\\nu)", "05af80787d3ee70cb19fe32fa4d51a74": "\\displaystyle{z=x+iy=(x,y),}", "05afb5ff344af186af6b7ca460765724": "\\mathrm{Lan}_{Y_D}(\\hat{\\psi})", "05afd513118371378c77886746d05ea1": "\\Psi = (a \\Psi_A + b \\Psi_B)", "05afd9a4b7b92a253abbadc92b701182": "X_{(1)}=\\min\\{\\,X_1,\\ldots,X_n\\,\\}", "05afde080c3c138560fb170da360c2a0": "L^{p_0}", "05b00416881f32ac96954a465e95293b": "Z_{\\text{resistor}} = R", "05b062c7413e26761f3999e1edd9289e": "\\displaystyle{\\psi_r(e^{i\\theta})=1+ {2r\\sin \\theta\\over 1-2r\\cos\\theta +r^2}.}", "05b0da896eeeb72186d5e2fc782bebc3": "R=-8\\pi T=0", "05b0f3e7d663fe1fa32fd698f405ace1": "\\pi_1(u)\\!:\\!\\tau_1", "05b10a093bcb0061605ffe22f10a2566": "C_{out}", "05b1239c4c206422a939bb9e50df3a5b": "f = 2^{12k/12} \\times 440 \\,\\text{Hz} = 2^k \\times 440 \\,\\text{Hz}", "05b1249c37ada36961107370141e5684": "A\\circ B = (AB+BA)/2", "05b142a9273c56306a41b6d17eedee68": "\\tan\\frac{\\pi}{5}=\\tan 36^\\circ=\\sqrt{5-2\\sqrt5}\\,", "05b1bdfd3c31fc9e683a5ac5d58ba23a": " n \\cdot t ", "05b20ea7e11d9daf79426da8d4d13da1": " P_1=(x_1,y_1)", "05b22643726772c8e63a1cc59b6bbb71": "\\psi(k)", "05b2482e9246a5c5644afabf3dad0cc3": "\\mathfrak{P}^{64}", "05b25f3d33c46808e06cbb8b6e831a9a": "\\int_L f(z)\\,dz", "05b269633ce9c9458c44db3519f54255": "\\frac{d_1-2}{d_1}\\;\\frac{d_2}{d_2+2}\\!", "05b29308d9c7c60fb46a662423e01dd4": "\\gcd(p, q, r) = \\gcd(p, \\gcd(q, r)),", "05b29d062cea367973c5af8d196b444d": "Q=2 \\pi \\frac{\\frac{1} {2} k z_o^2} {\\pi D z_o^2 \\omega_n} = \\frac{1} {2\\delta}, \\; \\omega_n=\\sqrt{\\frac{k} {m}}, \\; \\delta=\\frac{D} {2\\sqrt{mk}}\\,\\!", "05b2dec6a25665b36dda44033765c7b7": "\\int g(x) T(f)(x) \\, dx = \\iint g(x) K(x-y) f(y) \\, dy\\,dx.", "05b34b0eb41a6e2a115c9ed8deafe7df": "\\frac{V_1}{V_4}=\\frac{V_1}{V_2}\\cdot\\frac{V_2}{V_3}\\cdot\\frac{V_3}{V_4}=\\sqrt{\\frac{Z_{I1}}{Z_{I4}}}e^{\\gamma_1+\\gamma_2+\\gamma_3}", "05b3be8dbcf0d9a08ea0513410ef1282": "[\\psi^{(i)}(x),\\psi^{(j)}(y)]=0", "05b3c3a2a55dc111c4ee95fcbae49a12": "a\\leq t\\leq b ", "05b435045822a31b2a64190aac7e1bde": "\\{ \\mathbb{P}_N\\}", "05b44446ffc86fb27bcbd7eb7ff3d0f1": "(n,k,d)_{\\mathcal{F}}", "05b4521ead02d78da929929258f36fe3": "\\sum_{n=0}^\\infty nf(n/N)", "05b45fb47724552e5a57963aa09bd75d": "v\\in T^k\\left(V\\right)", "05b4799795585f3c6df304abe1080109": "I_o=\\bar{I_D}=\\frac{I_{L_{max}}}{2}\\delta", "05b4fd04922f80b290ebb1d1a672281a": "{1,...,t}\\,\\!", "05b514c88c9e2123a6c8acc8729ab939": "\\begin{align}\\begin{vmatrix}a&b&c\\\\d&e&f\\\\g&h&i\\end{vmatrix} & = a\\begin{vmatrix}e&f\\\\h&i\\end{vmatrix}-b\\begin{vmatrix}d&f\\\\g&i\\end{vmatrix}+c\\begin{vmatrix}d&e\\\\g&h\\end{vmatrix} \\\\\n& = a(ei-fh)-b(di-fg)+c(dh-eg) \\\\\n& = aei+bfg+cdh-ceg-bdi-afh.\n\\end{align} ", "05b53317814fbfc9799affa3b7f0ca93": "[x:=?]p\\,\\!", "05b54b1d076865bf416da04d9117672e": "H\\left( \\frac{1}{\\sqrt{2}}|0\\rangle-\\frac{1}{\\sqrt{2}}|1\\rangle \\right)= \\frac{1}{2}( |0\\rangle+|1\\rangle) - \\frac{1}{2}( |0\\rangle - |1\\rangle) = |1\\rangle", "05b57be59730cd2e0a174ce9ea691824": "1+2\\cos(x)+2\\cos(2x)+2\\cos(3x)+\\cdots+2\\cos(nx) = \\frac{ \\sin\\left[\\left(n+\\frac{1}{2}\\right)x\\right\\rbrack }{ \\sin\\left(\\frac{x}{2}\\right) }. ", "05b6cbeb7a2a97c6cf4ab673d03b7819": "C_\\mathrm{srgb}", "05b70f0366f4ce67735118a38ae6f4eb": "(x,y,z) = (x \\triangleleft y) \\triangleleft z - x \\triangleleft (y \\triangleleft z)", "05b75f24697420391a4349928ffb07f5": "\\bar{w_m}", "05b7a8ca7581fc1774fdbd1c4816be3b": "\\cup\\,", "05b87076480fc5625941b3a378773516": "X \\xrightarrow{vu} Z \\xrightarrow{m} Y' \\xrightarrow {n} ", "05b8976e246ca198290a3c22bd7b89d8": "F^T(t,r)=e^{A(t,T)-B(t,T)r}", "05b8add3164d2677c9bc109e067cdd30": "\\frac{\\mathrm d\\mathbb P}{\\mathrm d\\mathbb Q}", "05b94098ae4bfb39766992c8b9a9cddc": "\\gamma_{wof}", "05b9b130e331d094677aa7840b3c475e": "q_\\mathrm{net} = \\sum_{i=1}^N q_i \\,\\!", "05b9be3947267b1acbe2173d1ac1afc9": "y_3=Ay_2=AA^{2}y_0=A^{3}y_0,", "05ba28454878dec6cb87da9ff1d82648": "\nP = p + \\frac{1}{2} \\rho r^2 \\Omega^2.\n", "05ba6b36bb720587ff6f7b6e7e49a318": "S_-|s,m\\rangle=\\hbar \\sqrt{s(s+1)-m(m-1)}|s,m-1\\rangle", "05ba6f3563c0b2f7ddc62879c27b5f0e": "{{i}_{IN}}-{{i}_{OUT}}=\\frac{2\\left( \\overline{{{\\beta }_{12}}}-{{\\beta }_{3}} \\right)+2}{\\overline{{{\\beta }_{12}}}{{\\beta }_{3}}+2\\overline{{{\\beta }_{12}}}+2}", "05ba731686e7ea35361ac7af1177ae3e": " \\gamma^2 - \\kappa \\le \\frac{ 186 }{ 125 } ", "05ba874764a8fac7788e7fbe836b9b1e": "\\{0, 1\\}^m", "05ba8e4d35954e38e4dd92b63892e25e": "S_1 \\subseteq S", "05bab9329dd16291f016388c68ce4ae5": " \\tfrac{1+\\text{log}(\\text{tf}_{t,d})}{1+\\text{log}(\\text{ave}_{t \\epsilon d}( \\text{tf}_{t,d}))}", "05bae3b9cf380743cb488d4e5134ef1a": "[[G_{nm}]]", "05bb0da9a9b62787ced0ceedca9849be": "V_{max}\\ \\propto\\ \\sqrt[3]{power/f}", "05bb3d22db6d6ef0482446936fbc4aee": "T \\colon R^3 \\to R \\,", "05bb91e495dcd9f39f806f82422e3b64": " T = (X_1 + X_2 + \\cdots + X_n) ", "05bbbba0f5bc6b36a223e33e2b73afeb": "X \\to p \\to q+1 = g_q^p(1)", "05bbd70b35171806bd5098e27986dcfa": "\\nabla \\cdot (\\mathbf{a} \\mathbf{b}^\\mathrm{T}) = \\mathbf{b}(\\nabla \\cdot \\mathbf{a})+(\\mathbf{a}\\cdot \\nabla) \\mathbf{b} \\ .", "05bbe30c0a898a91ecb1f91c73e5daee": "S_1 (q)", "05bc170a4703117ff03cebcb6f69bb6c": " p(x_n) = y_n, \\quad n=0,\\ldots ,N-1. \\, ", "05bc296af2ad5d1ca4efc5ccc41b0549": " v_0^2 \\left(h_0-{h_0^2 \\over h_1}\\right) + {g \\over 2} (h_0^2 - h_1^2)=0. ", "05bc49ae7d6c96bf37559b7836c96d28": "\\mathfrak{L}", "05bcc0311a05d7aed31aee8615537f89": "DP_{p,c} = \\frac{\\sum_{p,c}(DP)}{count_{p,c}(singular cases)}", "05bd2d1be0d5d9ddb208f535d1145d7c": "s = c + \\frac{2v^2}{d}.", "05bd51e02228180c67c08ace7425da07": "R = \\frac{ \\sum{||F_\\text{obs}| - |F_\\text{calc}|| } }{ \\sum{ |F_\\text{obs}|}}", "05bd6dee39277c57474ce9368b2a6d84": "S=-3N\\langle x-0\\rangle^0\\ +\\ 6Nm^{-1}\\langle x-2m\\rangle^1\\ -\\ 9N\\langle x-4m\\rangle^0\\,", "05bd6e9d7504fc2db9a92e5a207d6ba7": "d\\ln{H}=m d\\ln{\\dot{\\varepsilon_p}}+n d\\ln{h_p}.", "05bdb30cea2ff2f555ea8236e1845287": "0\\leq i\\leq n", "05be2af0da9707467a594f649c32188a": "\n\\gamma = (b^2-c^2)\\cos^2\\beta\\sin^2\\alpha-(a^2-b^2)\\sin^2\\omega\\cos^2\\alpha,\n", "05be5f075a520c2e7d39376d2bc5587f": "a\\rightarrow_W b", "05be88b119ba2a5850f4fffae769e9fc": "L_{k-1}\\wedge\\cdots\\wedge L_{2}\\wedge L_{1}", "05bec6ac2fd604dbdd5da6ed4c98c3ba": " \\lim_i \\epsilon_i = 0", "05befe8285e71329765f393e3519cc18": "X \\cup_{f} Y", "05bf0be6456738fc35ecaacf1f0dc0dc": "\\mathcal{R}\\models\\varphi[a]", "05bf0e6d74499d4ea7bcde463a10aa02": "Z(\\beta,\\mu)\\ \\stackrel{\\mathrm{def}}{=}\\ \\mathrm{Tr}\\left[ e^{-\\beta \\left(H-\\mu N\\right)} \\right]", "05bf188f15a57c458a10ce6ea7d3f30d": " \\eta \\to 3\\gamma, ~~ e^+e^-, ~~ 4e ", "05bf4d1426470438683cc492040d9cda": " i_1=E^2\\sin^2(\\omega t)\\, ", "05bf6956b879bab7b342e627222be8ef": "(N^{1/4}+1)^2", "05bf8bd2f7266b98af0dc87ea7d7e679": "1.43^{-1} + 3.9^{-1} = 0.956", "05bfe51895722cc4d344507815a72c72": "\\mathbf{S}_{i_1,i_2,\\ldots,i_N} ", "05c054bd8daeca25e6dc6b8a1db5736c": "\\| f \\|' := \\sup_{x \\in \\mathbb{R}} \\left| \\int_{- \\infty}^{x} f \\right|.", "05c08881739e0666af07526add9e2c5d": "u(x,t) = \\frac{f(x-ct) + f(x+ct)}{2} + \\frac{1}{2c} \\int_{x-ct}^{x+ct} g(s) ds", "05c0a00d7a781bcea63d9072acfaa2a7": "\n\\frac{\\pi}{2}= \n\\sum_{k=0}^\\infty\\frac{k!}{(2k+1)!!}= \\sum_{k=0}^{\\infty} \\cfrac {2^k k!^2}{(2k + 1)!} =\n1+\\frac{1}{3}\\left(1+\\frac{2}{5}\\left(1+\\frac{3}{7}\\left(1+\\cdots\\right)\\right)\\right)\n", "05c16634a2abdf6e9fb4d0593a176811": "(e^{-x^2/2}u')' + \\lambda e^{-x^2/2}u = 0", "05c1c3d13113e80e93d06955f8a0d3d9": " \\operatorname{E}( \\widehat{\\theta}(X) ) ", "05c1f1d0a83b7ada12034af7ee1d6e5a": " H = X \\left(X^\\top \\Sigma^{-1} X\\right)^{-1} X^\\top \\Sigma^{-1}, \\, ", "05c223937c098ccc7d5fe65df085b311": "P_j(0)=0", "05c2604cd81453f4c92ca92d2ebfd54e": "x^3 - 4x + 7", "05c264d438209c0f6db45dde67a2d801": "P_0^s (S) = \\lim_{\\delta \\downarrow 0} \\sup \\left\\{ \\left. \\sum_{i \\in I} \\mathrm{diam} (B_i)^s \\right| \\begin{matrix} \\{ B_i \\}_{i \\in I} \\text{ is a countable collection} \\\\ \\text{of pairwise disjoint balls with} \\\\ \\text{diameters } \\leq \\delta \\text{ and centres in } S \\end{matrix} \\right\\}.", "05c26c300015aa8a80471774a788670c": "\\sin^5\\theta = \\frac{10 \\sin\\theta - 5 \\sin 3\\theta + \\sin 5\\theta}{16}\\!", "05c287405896816565fa9d9ab0b197dc": "\\langle\\mathbf{v},\\mathbf{w} \\rangle", "05c2a754b08bf56c1e1aa415093b2dae": "G_2=\\langle L,R,F^2,B^2,U^2,D^2\\rangle", "05c331575e79832c7d767e1326882c8e": "\\sum_{i=1}^d{P_i \\cdot \\log_2{\\frac{1}{P_i}}}", "05c34b387fb3657e877769df402b3cda": "\\textstyle \\sqrt {\\frac {1} {5}} \\left ( \\frac {1} {5} - 0 \\right ) + \\sqrt {\\frac {2} {5}} \\left ( \\frac {2} {5} - \\frac {1} {5} \\right ) + \\cdots + \\sqrt {\\frac {5} {5}} \\left ( \\frac {5} {5} - \\frac {4} {5} \\right ) \\approx 0.7497.\\,\\!", "05c381c941317f383b38c16a50d8fcb6": "(t,r,\\theta,\\phi)", "05c390791fd4770b47596dc6994505ab": "\nh_R(t) = \\delta (t) - {1 \\over RC} e^{-t / RC} u(t) = \\delta (t) - { 1 \\over \\tau} e^{-t / \\tau} u(t)\n", "05c391ace5152be60b6040676e6ce83c": "(x+3)^2 = 4.\\,\\!", "05c3a6c8f942d0c0774385ee683c2fcf": "\\mathbf{w}_{(1)}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\,\\{ \\sum_i \\left(t_1\\right)^2_{(i)} \\}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\sum_i \\left(\\mathbf{x}_{(i)} \\cdot \\mathbf{w} \\right)^2 ", "05c3e32e9d53ac4884505fa674adcc5b": "{1\\over 3} + {2\\over 9}+{2\\over 27}", "05c43a234b7406db7cf291f9002d5b8d": " Y\\,\\!", "05c440b0bf58018422ebf3802967a159": "\\mathbf {q}_e", "05c46bd969ec4f85c3f245a1ecf1b75f": "{(\\bullet)}_{,j}", "05c4ab41865fa3d449fae5f24c1e8c57": "n(q_i)", "05c4c4f452db3760a7e0c7690c9473d6": "m c^2 \\frac{dt}{d \\tau [t]} = \\frac{m c^2}{\\sqrt {1 - \\frac{v^2 [t]}{c^2}}} = +m c^2 + {1 \\over 2} m v^2 [t] + {3 \\over 8} m \\frac{v^4 [t]}{c^2} + \\dots \\,.", "05c5122c969c2531c1f6764abe5c037a": "M_A = 4 \\frac{EI}{L} \\theta_A + 2 \\frac{EI}{L} \\theta_B = 4 \\frac{EI}{L} \\theta_A", "05c51dcaee2768b6a2b0906e8239e86e": "RPM = {Speed \\over Circumference}={Speed \\over \\pi \\times Diameter}", "05c544046a1e219149406b97ff19a95d": "I[f] = \\frac{1}{4\\pi}\\int \\mathrm{d}\\Omega\\ f(\\Omega) = \\frac{1}{4\\pi}\\int_0^\\pi \\sin(\\theta)\\mathrm{d}\\theta\\int_0^{2\\pi}\\mathrm{d}\\varphi\\ f(\\theta,\\varphi),", "05c5afd9da07bff6fcf6e46abe179aa8": "N_{SV}", "05c5cb92f034600b9f06e9a37b664251": "n\\in\\mathbf{Z}", "05c5d866380a56cb67ce18b5aca4c57a": "\\phi^{\\Rightarrow x} \\leq \\psi^{\\Rightarrow x}\\,", "05c5f32ce2a5ca1c969e85db436088d6": "128^4", "05c609a273b2bea4f5dc3b9537e68b83": "\n \\hat H Y_\\ell^m (\\theta, \\varphi ) = \\frac{\\hbar^2}{2I} \\ell(\\ell+1) Y_\\ell^m (\\theta, \\varphi ). \n", "05c630ec8a6acbca4c93b0ec3c5f857e": "\\int\\frac{\\cot^n ax\\;\\mathrm{d}x}{\\cos^2 ax} = \\frac{1}{a(1-n)}\\tan^{1-n} ax +C\\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "05c671f47458f9ba8a33b361a6bb73d0": "P(n) = \\ln\\left(\\frac{\\Omega}{ n} \\right) \\frac { 1}{ \\ln(\\Omega) c } ", "05c68a5b4405c118dd85a1a78e2a3946": "t = \\frac{2}{2} = 1", "05c6c30055e6f5f1f21a6952ef39bafe": "\\text{pulse dispersion} = \\frac{\\triangle\\ n_1\\ \\ell}{c}\\,\\!", "05c6d02faf6e88de12c96b8ed371ba7f": "F = \\R", "05c71188190df6096f2e8460f93db854": "\\displaystyle{E=\\bigoplus_{i\\le j} E_{ij},}", "05c71f95ec0de948798f5478018b2622": " \\nu_\\mathrm{1}= |\\nu_\\mathrm{n}-a/2|", "05c76f804f1b3dd570c74ecb38db5642": "v_C(t)\\,", "05c7759a446e834f13203a703eceb182": "f^{n+1}\\ x = f (f^n x) ", "05c794638232b29175d15651c34614c6": "\\{2,5\\}\\;", "05c7d9478d9ca050dcb2518e566ee3f2": "G\\setminus A", "05c7f10b3ba002bff3fd3050db03314b": "\\lim_{x \\to c} x^r = c^r \\qquad \\mbox{ if } r \\mbox{ is a positive integer}", "05c7f2cac0b70415e8de7a9802d2e09c": "\\kappa=1/\\sqrt{2}", "05c850ca793caf0aa51e967a2e8e0c39": "n= \\infty ", "05c8649308e29c0869d4693b4139816b": "\\tfrac{1}{2}\\text{social cost}(Z) \\leq E(Z) \\leq \\text{social cost}(Z)", "05c898fc68d9bdc0c397cd800b83a9ca": "(k,\\theta_k,\\phi_k)", "05c8aa6c5f99b293809c083655415e70": "p_{ij}(t)\\ ", "05c8aeef3fd69db58fb38ec0fe1584cb": "\\frac{1}{\\pi} \\int |\\alpha\\rangle\\langle\\alpha| d^2\\alpha = I\n\\qquad d^2\\alpha \\equiv d\\Re(\\alpha) \\, d\\Im(\\alpha)", "05c8b0dcff5822fbde7e0398f05aea00": " x^2 ", "05c8e93d5e664b4fbdd450f7ae0df317": "X=\\sum_{i=1}^N w_i Z(x_i)", "05c916460de60839f9545a943d990edd": "y_c=C_1e^{ \\left ( -b+\\sqrt{b^2 - 4c} \\right )\\frac{x}{2}} + C_2e^{-\\left ( b+\\sqrt{b^2 - 4c} \\right )\\frac{x}{2}}\\,\\!", "05c92b00d6bb7100843d0ca196c4a56a": "A=A_1\\times A_2\\times\\dotsb\\times A_N", "05c93e0130d62c75768900584e84b491": "{{P}_{T}}f(u,\\xi )=\\frac{1}{2\\pi }\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{{{P}_{V}}f({{u}^{'}},{{\\xi }^{'}})}}.{{P}_{V}}{{\\phi }_{\\gamma (u,\\xi )}}({{u}^{'}},{{\\xi }^{'}})d{{u}^{'}}.d\\xi '", "05c93f2cfb54e2a5d9a374aacb918e0b": "\\textstyle (\\Omega_2,\\mathcal{F}_2,P_2) ", "05c953543a2f0987bb8f7c87d1f22dfa": "{\\Gamma(\\alpha+\\beta+n)}", "05c9864ab6b670cdb4261c66c183c47a": "r=k \\frac{K_1K_2C_AC_B}{(1+K_1C_A+K_2C_B)^2}. \\, ", "05c9c29453269f69480a108826520535": "\\lambda x . x x x", "05ca8e7728a7f1667453038b1ad6afa3": "K \\in \\mathcal{K}.", "05cab4a49dfc9b7b1327cc15eab681d7": "+1/2\\,", "05cad9232024d1f205509f78fc65df53": "R^h = \\mathbb{H} / \\Gamma", "05caed6c68f525022399020b4bfb94d8": "KL_{i,1}", "05cb3148609df7797d7bcc4141ba84a8": "f \\in \\mathcal{H}_k", "05cb34c6f7a4b2415077be3d9283df20": "(\\phi \\lor \\psi)", "05cbc2cf48741a31cce77a970ff4131d": "J_{ij} < 0 ", "05cc05f25d900129201d72733479ed90": "\\begin{align}\n\\Omega U &\\simeq \\mathbf{Z}\\times BU = \\mathbf{Z}\\times U/(U \\times U)\\\\\n\\Omega(Z\\times BU)& \\simeq U = (U \\times U)/U\n\\end{align}", "05cc3380c308ddf2623f7e43b6a47fc3": "\\,N\\,", "05cc9bb84d986e769d9677d2f30d1c65": "\\det(A) = \\det(X)^{-1} \\det(BX) = \\det(X)^{-1} \\det(B)\\det(X) = \\det(B) \\det(X)^{-1} \\det(X) = \\det(B).\\ ", "05cd66ec0550ba0059b46a660e693e69": ",a,g(", "05cd6b1347352544845e7f45f9b73dcb": "\\mbox{QMA}({2}/{3},1/3)", "05cda3f39c61cc3cd8f9d58bb9b25616": "\\{1, 2,\\dots, k^2\\}", "05cddb1a519e4279f413146bec4ecdb6": " \\, [0, \\theta]. \\, ", "05cdeef4cdeedd41a74966eed95c31e0": "S(c, c) \\to S(c, c')", "05cdf9f7d3e7e9a701a0e383b0d21092": "(-1)^k {z \\choose k}= {-z+k-1 \\choose k} = \\frac{1}{\\Gamma(-z)} \\frac{1}{(k+1)^{z+1}} \\prod_{j=k+1} \\frac{(1+\\frac{1}{j})^{-z-1}}{1-\\frac{z+1}{j}}", "05cdfab85fed1fd5538b08c5b2534f68": "\\hat{X}^n_{k}", "05ce135f532096424b95e65d324e3066": "\\beta(g)=-\\left(11-\\frac{2n_f}{3}\\right)\\frac{g^3}{16\\pi^2}~,", "05ce27299cfa7c645dc43f5900ab7b51": "\\omega_n = \\sqrt{\\frac{K_p K_v}{\\tau_1}}", "05ce58568ba255a9b3ea387369e3b806": "\\succ_{P}", "05ce5e7e1445f7580639409d0b4b229a": "\\theta \\in \\textrm{End}(V)", "05ce62090305d8c81d8e1984a0196b9e": "m - n + k.\\ ", "05ce6b1ede4d11a9c95f3f305238c738": "i\\ne j\\ ", "05ceaa3417df09a995fe929f4e78788f": "\\frac{2^{7-k} \\mod k}{k}\\, .", "05cec527ebdab7d072cc405cf84af1fe": "\\sigma_x,\\!", "05cef6d6ee4d460288d2f29251073318": "\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n", "05cf0a41402aa254ea95ef6c4b15c6bb": "A\\cdot (c_1 X + c_2 Y)=c_1 A\\cdot X + c_2 A\\cdot Y\\,", "05cf3bf32200fd37fdf6a1d3a77ffb78": "{0, 1, 6}", "05cf6a211012e62a821e558fc09108e5": "\\varepsilon_{jmn} \\varepsilon^{imn}=2\\delta^i_j ", "05cf867d5245db00b37d986d14c94ea9": "\\dot{\\boldsymbol{e_j}} =\\sum_{k=1}^{d}\\frac {\\partial}{\\partial q_k}\\boldsymbol{e_j}\\dot q_k \\ ", "05cf8fe66345795237ac822f788ef92d": "_2^1\\text{P}", "05cfcfea3e2eb81380e5df3c9c807736": "p=\\frac{\\varepsilon _{2}}{\\sqrt{-P^{2}}}p_{1}-\\frac{\\varepsilon _{1}}{\\sqrt{\n-P^{2}}}p_{2} ", "05cfd4a8c76cc15a15f3c9ac03e73ec2": " \\mathrm{Res}(f, \\infty) = -\\lim_{|z| \\to \\infty} z^2 \\cdot f'(z)", "05d009b38560ef6a0599e43575933115": "[E_{\\beta},E_{\\gamma}]=\\pm(p+1)E_{\\beta+\\gamma}", "05d0330244ce364c66a262afdc74082a": " \\frac{\\partial |\\Psi|^2}{\\partial t}=-\\nabla \\cdot \\mathbf{j} ", "05d05e751a80db7375eae13c25f0ca13": "\\mathbf{C}", "05d07a7c15b65d6b762ec8003c66c0be": "A\\in\\mathbb{R}^{k\\times d}", "05d08383a49f0991b19264fa014a430c": "A|\\psi_n\\rangle=a_n|\\psi_n\\rangle", "05d0e694a8c918c5eb5f137840030045": "\\ i_1^2 = -1, i_2^2 = i_3^2 = +1", "05d0e9032db15f41512942654cf714a2": "\\sigma_{\\mathbf{v}} = \\sqrt{\\frac{3kT}{m}}", "05d0f6b04e586050c1e1a56428ddabee": "|\\theta|", "05d0fabf9ade9a1b534f221d937c152e": "\\binom{m}{n}\\equiv\\prod_{i=0}^k\\binom{m_i}{n_i}\\pmod p,", "05d1398bbc4b0b7e1015d32c12854012": "\\ W=PIt", "05d22b1ac0c732002507d2dedbcad08e": " \\operatorname{build-list}[\\lambda q.\\lambda x.x\\ (q\\ q\\ x), D, D[p]] \\and D[p] = L_1", "05d27529606dcd2415f3490efde4399d": "y_{ij} = x_{ij}'\\beta +\\mu_i +\\epsilon_{ij} \\,", "05d28ad1ff5ee18982d1ab0df406bc51": "\\part_{X_i}", "05d2abd793340a845e9b9d0dcfcd4ca3": "\\ G(\\tau)=G(0)\\sum_i \\frac{\\alpha_i}{(1+(\\tau/\\tau_{D,i}))(1+a^{-2}(\\tau/\\tau_{D,i}))^{1/2}} +G(\\infty)", "05d2caf1e9ce7327126c96ca23ae5520": "\\ \\mathbf a= \\dot{\\mathbf v} = \\ddot{\\mathbf x} =\\frac{d^2\\mathbf x}{dt^2}=\\frac{\\partial^2 \\chi(\\mathbf X,t)}{\\partial t^2} ", "05d2e9dda026d6b2b9d64eb5bd8c5131": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor ", "05d2f836c9edaf00687f79a06ddafa3f": "\\sin^{n} \\alpha", "05d32c428028dd422d7d0393241c095c": "\n \\begin{align}\n N_{\\alpha\\beta,\\beta} & = J_1~\\ddot{u}_\\alpha \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q(x,t) & = J_1~\\ddot{w} - J_3~\\ddot{w}_{,\\alpha\\alpha}\n \\end{align}\n", "05d3874e7243517143e0d8e86dff6d40": "= (1-2 i \\theta)^{-\\frac{p}{2}}.~~\\blacksquare", "05d39f59c1a1e6ae07acb4cb6c2fa55e": "\\quad = \\gamma \\frac{\\omega_{\\mathrm{obs}}}{c} - \\beta \\gamma \\frac{\\omega_{\\mathrm{obs}}}{c} \\cos \\theta. \\,", "05d3e17d000e32a4bbd754019fdf0c09": "x_{k+2}", "05d3e3a666490ce89fd9a7a7b0651ab5": "V = -\\boldsymbol{\\mu}\\cdot\\bold{B} ", "05d41768c61c9626a407da6464305e0f": "G\\times X\\rightarrow X", "05d4bb3cdffddbd2eb5ee961ebf4967f": "\\hat\\beta = \\big(\\tfrac{1}{n}X'X\\big)^{-1}\\tfrac{1}{n}X'y \n = \\beta + \\big(\\tfrac{1}{n}X'X\\big)^{-1}\\tfrac{1}{n}X'\\varepsilon \n = \\beta\\; + \\;\\bigg(\\frac{1}{n}\\sum_{i=1}^n x_ix'_i\\bigg)^{\\!\\!-1} \\bigg(\\frac{1}{n}\\sum_{i=1}^n x_i\\varepsilon_i\\bigg)", "05d4cc0e17ee5806e4040d34db0e53c5": "[g_{ij}] = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & r^2 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{pmatrix}", "05d4d5a45a7e8e0494c0d39e6d3b1495": "(\\mathcal{L}_Y T)_p=\\left.\\frac{d}{dt}\\right|_{t=0}\\left((\\varphi_{-t})_*T_{\\varphi_{t}(p)}\\right)=\\left.\\frac{d}{dt}\\right|_{t=0}\\left((\\varphi_{t})^*T\\right)_p", "05d4e93dfe94044737fcc6a71dc92ed8": "n(p-1)", "05d52867f9e20a380e34202c6768102d": "\\hat{y}_1 = y_1(1+\\delta_3)", "05d5449c09ffbfbe8b15b17745c5c826": "\\mathbb{H} \\oplus \\mathbb{H}", "05d56f7c3994dedec05a6b75c11b7a18": "F\\, =\\, \\underbrace{\\rho\\, C_m\\, V\\, \\dot{u}}_{F_I} + \\underbrace{\\frac12\\, \\rho\\, C_d\\, A\\, u\\, |u|}_{F_D},", "05d57c5572d005d3c21408594a152be6": " \\mu_{disp} =\\sum_k \\gamma_k A_k", "05d5d3bfc47cb6c934131f2e4aafbe1f": "\\displaystyle \\Phi = \\mu_0\\mu_rNiA/l,", "05d5da12cf320f6b8aa34e8ff21d4eee": "\\mu_t = M_t / m", "05d6052a29659a1972c00d6345dbaf3a": "\n 1 = \\int_{-\\infty}^{\\infty} f(x)\\,dx\n = \\int_{g(-\\infty)}^{g(\\infty)} f(x)\\,dx.\n \\!", "05d65eeb37dfe3eada0857a857820791": "\\tilde{\\lambda}\\notin\\sigma(A)", "05d677a5a308358279bc0ce73a5b66eb": "x^\\ast - \\frac{\\epsilon}{2} \\frac{c}{||c||} \\in P", "05d6ee53814b655a3ffa1748da94df9d": "K_{\\rm J}^2 R_{\\rm K} = K_{\\rm J-90}^2 R_{\\rm K-90}\\frac{mgv}{U_{90}I_{90}}", "05d7133d190e6f49370153a8a3654dff": "\n\\begin{align}\n\\langle d\\Psi/dt | \\hat{x} | \\Psi \\rangle + \\langle \\Psi | \\hat{x} | d\\Psi/dt \\rangle &= \\langle \\Psi | \\hat{p}/m | \\Psi \\rangle, \\\\\n\t\\langle d\\Psi/dt | \\hat{p} | \\Psi \\rangle + \\langle \\Psi | \\hat{p} | d\\Psi/dt \\rangle & = \\langle \\Psi | -U'(\\hat{x}) | \\Psi \\rangle,\n\\end{align}\n", "05d74db688ee9c38cb781c2f1d840ace": "\\mathcal{L}(\\theta \\mid x )= f(x\\mid\\theta). \\!", "05d7cd70b0eb47eca402d4b08e43c92e": "\n\\sum_{t=1}^n \\frac{FCFF_t}{(1+WACC_{g})^t} +\n \\frac{\\left[\\frac{FCFF_{n+1}}{(WACC_{st}-g_n)}\\right]}{(1+WACC_{g})^n}\n", "05d7e8e9e7a7baf15e49422aa7eeea0b": "q(\\xi) = (\\xi-x_0) \\cdots (\\xi-x_n)", "05d87631f336b199f9daae6e57824896": "x_n = z_n \\times z_{n - 1}", "05d893a914644be9aa42f1fa2804e1d3": "\\begin{align}\n\\epsilon( t, \\omega ) \n\t&= \\int x(\\tau) h( t - \\tau ) e^{ -j \\omega \\left[ \\tau - t \\right]} d\\tau \\\\\n\t&= e^{ j \\omega t} \\int x(\\tau) h( t - \\tau ) e^{ -j \\omega \\tau } d\\tau \\\\\n\t&= e^{ j \\omega t} X(t, \\omega) \\\\\n\t&= X_{t}(\\omega) = M_{t}(\\omega) e^{j \\phi_{\\tau}(\\omega)}\n\\end{align}", "05d8d8906414d4466fef2875630a776a": "x \\in \\{ INT\\_MIN , ... , INT\\_MAX \\}", "05d8e55c60e1240c8d9385ad078d8283": "\\frac{dx}{dt}(T) \\cdot (y-Y)=\\frac{dy}{dt}(T) \\cdot (x-X).", "05d90f614706bc6fafc53f97fe45853c": "R_1=mR,\\,", "05d9936b2f9e6a19a257c7444afc3f06": "\\{y_k\\}_{k=1}^M", "05d9d2d1e373b38c4d316bc905925ea9": " w\\left(\\omega\\right)", "05da02550b176b031a52dd445ed13cdd": " D=< \\frac {\\Delta x^2}{\\tau_D}> \\sim \\frac {c^2 \\delta E^2}{B^2k_{\\perp}^2\\,D} \\sim \\frac {c \\delta E}{Bk_{\\perp}} ", "05da2717763ceea3d7f31ca86ae4fb20": "\\alpha M_k-\\beta M_{k+1}=\\left\\{\\begin{array}{cc}-ve, & if\\; \\alpha<0\\\\ ve, & if\\; \\alpha>0 \\end{array}\\right.", "05da351c30b5100133e9819823891df6": "x_{n+1}=x_n^2-c", "05da8d653aade338bf5a13b0ca5f197a": "\\exp_p(z)=\\sum_{n=0}^\\infty\\frac{z^n}{n!}.", "05dab0e648f28f5051ceca435fad3552": "a_0b_3", "05dabdf6797bb34bd0c215e92a3ec706": " f: \\mathbb R_+\\to\\mathbb R; x\\mapsto x^2", "05db644d5820ac7cb87759b8fce150e3": "F^{-1}(p) = a + p (b - a) \\,\\,\\text{ for } 00", "05e53749ca3e0fbdd8ad7b0bb193db2a": "\\Lambda(A)", "05e5912146d2c3ba23e769415892616a": "\\gcd{(a^{(N-1)/p}_p - 1, N)} = \\gcd(7^{2\\cdot 25} - 1, 11351) = 1.", "05e5d4406c44c5bcff6b911a8427d630": "\\nabla\\phi", "05e603a1e451174fdf7ca07065141804": " A = QR, \\, ", "05e61e59147411910cc55ab15f423054": "\\ F = \\frac{1}{4 \\pi \\varepsilon_0} \\frac{q^2}{r^2}. ", "05e666402749e2dafc3acf1a40303ac2": "{{\\partial \\zeta_g \\over \\partial t} = {-\\overrightarrow{V_g} \\cdot \\nabla ({\\zeta_g + f})} + {f_o {\\partial \\omega \\over \\partial p}} }", "05e67768a59320d85e3ebf316725ae67": "\\mathbb{R}\\rarr \\mathbb{R}", "05e6958d35278b9f4186874c8ce2baff": "\\vec{p}^{\\,*}", "05e6a4b2796d446ae06d219936784b31": "k < 2\\times10^{-3}", "05e6aaaf68164c07f41c8b803dc47ea1": " \\Delta P = \\frac{8 \\mu L Q}{ \\pi r^4} ", "05e6b4fcbb73a01861df35fbf63b4a03": " \nW = \\begin{cases}\nmW&\\xi \\leq 1/m \\\\\n0&\\xi > 1/m\n\\end{cases}\n", "05e6e8d1b3ca6f727ff21d16d9f02a8f": "\\Delta\\epsilon\\equiv\\epsilon_\\parallel-\\epsilon_\\perp", "05e71ee635eeaa582f914152270c8d58": "\\overline{\\mathrm{Nu}}=-{{1} \\over {S'}} \\int_{S'}^{} \\mathrm{Nu} \\, \\mathrm{d}S'\\!", "05e730c6e95369f755e64f447b385a85": "f^i \\left (p \\right )", "05e757209bb816fcb90984fa2a4eafda": " 1.57 \\approx \\frac{\\pi}{2} \\leq k_{\\R} \\leq \\mathrm{sinh}(\\frac{\\pi}{2}) \\approx 2.3", "05e7ca63d7e5b2b34f1090ef28bb487b": "\\int r\\cos \\theta dm", "05e7db456cd901f5d80e881bcc27d8e9": "\\tfrac23", "05e82ad825e447fb9a23e8aa7c714fe3": "U\\colon(\\mathbf{Ab},\\otimes_\\mathbf{Z},\\mathbf{Z}) \\rightarrow (\\mathbf{Set},\\times,\\{*\\})", "05e849e1ff7eb44e943f4681b34964c3": "\\textstyle x^jb(x)", "05e8692dbb52599435d0d7f29759f335": "{\\scriptstyle\\frac{1}{120}} (-x^5+25x^4-200x^3+600x^2-600x+120) \\,", "05e88b875c72c95fcaf55ca9bfd22ede": "\\begin{matrix} {9 \\choose 1}{4 \\choose 3} \\end{matrix}", "05e893878365af1b7320f5549d71bc2f": "f(a) \\ne \\varepsilon", "05e8d5fffb9e0eb4ba401fb00c15f755": " \\psi \\ge \\frac{ 3 }{ \\sqrt{ 3 } + 1 } \\quad ( \\approxeq 1.098 ) ", "05e8fea2636da68588744dc377fce281": "f \\colon M \\rightarrow N", "05e960ec1a492cfce14fe3d8072b2b4f": "g_D", "05e9641ef840d8182e4f3c3da469acf1": " y(t) = |H(i \\omega)| \\ a(t - \\tau_g) \\cos \\left( \\omega (t - \\tau_\\phi) + \\theta \\right) \\ ", "05e99b2733f693b9998df767196a54cd": "\\varepsilon^{\\mu_1 \\cdots \\mu_n} = \\delta^{\\mu_1 \\cdots \\mu_n}_{\\,1 \\,\\cdots \\,n} \\,", "05e9ac62f51dbba005e09d300de60664": "\\omega = 2\\pi f", "05ea09c94632221ae3b86541a2ea035c": " = [F_3, S_3, A_3]::[F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_] ", "05ea312c3901219bb261e3ed52010dbc": " \\epsilon = {v^2\\over{2}}-{\\mu\\over{r}} = -{1\\over{2}}{ \\mu^2\\over{h^2}}\\left(1-e^2\\right) = -\\frac{\\mu }{2a} ", "05ea41806d96ec5cb8f44a2da8405f3e": "r_s\\,\\!", "05ea433257df6dc6c44f7152684deb88": "t^3+pt+q=0 ", "05ea673eea6b99802aa0524f719f51ff": "\\frac{d}{dt}\\langle A(t)\\rangle = \\left\\langle\\frac{\\partial A(t)}{\\partial t}\\right\\rangle + \\frac{1}{i \\hbar}\\langle[A(t),H]\\rangle", "05ea7a1d9defbaf990e9eba60e1bcb2b": "r_1,\\ldots,r_k", "05ea91282cb0b72b3b928c2a6ffe9af7": "x^{q^{n_i}}-x \\bmod f", "05eaa64619424b6173e145259a040d6b": "\\boldsymbol\\epsilon_i \\sim N(0, \\boldsymbol\\Sigma_{\\epsilon}^2).", "05eb0089b956a39fe3bc207e4d6a7013": "\\langle\\mu\\mu|\\lambda\\lambda\\rangle", "05eb1dee4e92fb17bf8d6ddfc587a387": "T^*(x_1, x_2, x_3, \\dots) = (0, x_1, x_2, \\dots).", "05eb2a561d1784325ed89cc26246cb9a": "g(x) \\le 0", "05eb726fee1bdb3bf8f6e44507de3cb4": " PV = \\frac {FV} {\\left( 1+i \\right)^n}\\,", "05ebaa7f76fedca83ba62c60d06094ed": "T_{\\Phi} := \\{ \\; \\overline t \\; |\\; t \\in T^S \\} ", "05ebf1eb685f0543a778bf06239aff7f": " |B| \\geq \\binom{n_i}{i-r}+\\binom{n_{i-1}}{i-r-1}+\\ldots+\\binom{n_j}{j-r}. ", "05ec70e3150e60a283d05a974e47b16a": " g_{2m+1} = (2m+1) g_{2m}\\,.", "05ecbc5d16691a7e8d021d4e3e941e5e": "\\mathbb C^m", "05ecfb7f01a85bf497a522d8b6470404": "(X_0, X_1)_{\\theta,1} \\subset X \\subset (X_0, X_1)_{\\theta,\\infty},\\,", "05ed12f72126f770fa209561c79dc1ab": " \\lambda_1 \\simeq \\lambda_2 \\gg \\lambda_3 \n", "05ed131f606dc23ff7455a4c2b68d667": "a_i \\in A_i", "05ed36d37a7191b670f20c048bee54fb": "H(p,m) = H(p) + D_{\\mathrm{KL}}(p\\|m),", "05ed5d9424c97dc464245040474a92cf": " \\cup", "05edcf086e29f22c22810eeaf4c1fff2": " \\operatorname{de-lambda}[x\\ x] = \\operatorname{de-lambda}[f\\ (x\\ x)] ", "05ee121ad3a56550731350e3f6a1b768": "q \\geq p", "05ee3cd596266e4d18eec9b47db9924d": "\n \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0 = d\\mathbf{f} \n", "05ee6c3c79b2396c35dd23c5e78a511c": "A \\in \\mathbb{C}^{n \\times n}", "05ee6e4e0ded01cf48b7400aaf57c2c8": "\n\\lambda'_k = \\begin{cases}\n4 \\lambda_k - 2 m_k, \\, \\text{ if } 0 \\leq k < n \\\\\nL, \\, \\text{ otherwise }\n\\end{cases}\n", "05ee960dda2b244449b33d7306c450fd": "\\zeta = \\sum_{j=1}^N \\sqrt{2 S(\\omega_j) \\Delta \\omega_j}\\; \\sin(\\omega_j t - k_j x \\cos \\Theta_j - k_j y \\sin \\Theta_j + \\epsilon_{j}).", "05eec65bbbf300943c6628e620d68c44": "\\operatorname{Spec} R[x] \\to \\operatorname{Spec} R", "05eef43d884d5cad5a6230f1d8c963c2": "\\forall L\\in \\textrm{PSPACE}, L\\leq_p \\textrm{TQBF}", "05ef3ac55920353bc2600b4f1953f9fa": "\\displaystyle{\\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix},}", "05ef66efbdaf714b32b946c069c9273e": "\\langle(\\Delta E)^2\\rangle=\\langle E\\rangle^2/m", "05ef6956fd1187aa30f900c8668fe0ad": "a,b,c=1\\ldots N^2-1.", "05ef7a0353032e90e6e93136804f9dc9": "e(\\mathbb Z^2)=1.", "05ef9e5e9e92371c226768239fd9905d": "\\begin{align} (Q^T)^T (Q^T) &{}= Q Q^T = I\\\\ \\det Q^T &{}= \\det Q = +1. \\end{align}", "05efc84b2e65c00051a28dc40391de6e": "\\{ x_0, x_1, \\ldots, x_n\\}", "05efc9164c4af919bc6100c7b8c98280": "\\Psi(x) = \\int_0^x (\\Phi')^{-1}(t)\\, dt.", "05f004482ab1df787ffae85b2eda7c27": "h(R)=|a_0|\\,R^{-k}+\\cdots+|a_{k-1}|\\,R^{-1}-|a_k|+|a_{k+1}|\\,R+\\cdots+|a_n|\\,R^{n-k}", "05f0621535b4e916222d563fe82fc59c": "\\Psi\\left(\\mathbf{r},t\\right) \\,\\!", "05f091ec9712d065e1604d2f58048333": "P(\\ell)", "05f0da43780aff7c4fcda9e4381a9b29": "w(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta) = \\alpha x \\gamma \\widehat{y} \\delta \\beta", "05f113a1f78b0dddf389345d9a093844": "j = 0 ", "05f1327cae46de4114ff4ff01c274eec": "C'_{Op}", "05f1400dc41cdacc6e26c688a0573502": "V_t ", "05f146b6cb758883507b70d1df4c8c8f": "\\mathfrak{h}^*", "05f1b458bef42023cc00cfb97f231e71": " = n\\,c_P\\,\\Delta T ", "05f1c3c597476cc48987621dad890333": " n = ( t / D )^2 a m^{( b - 2 )} ", "05f1fc8045e32ac73c7514938f72e5ca": "M_{y} =\\int c_{y}d\\dot{m}", "05f219042d02159d7978ee14b2b9b44f": " L = V - \\{s,t\\} ", "05f2812f1ead7ec30a37ddbc1c0f1f6d": " = 2 + \\frac{2}{3} + \\frac{4}{15} + \\frac{4}{35} + \\frac{16}{315} + \\frac{16}{693}\n+ \\frac{32}{3003} + \\frac{32}{6435} + \\frac{256}{109395} + \\frac{256}{230945} + \\cdots\\! ", "05f297b7a9c0d1251633dc3be8dc3817": "p^{*} = 0.528 p_0", "05f2c01a0a6c2339d4f459ce6354e581": "\n\\begin{align}\n\\boldsymbol\\beta'_1 &= \\boldsymbol\\beta_1 - \\boldsymbol\\beta_K \\\\\n\\cdots & \\cdots \\\\\n\\boldsymbol\\beta'_{K-1} &= \\boldsymbol\\beta_{K-1} - \\boldsymbol\\beta_K \\\\\n\\boldsymbol\\beta'_K &= 0\n\\end{align}\n", "05f2f1f6344d4bdb8370c1ab38b04219": "F(x,y,z)", "05f31527b8adc26005838c7536768283": "\\boldsymbol{F} = \\boldsymbol{F}(\\mathbf{X},t)", "05f315c99f1d4a4add6ed521ed43de20": "{\\Sigma^{\\infty}_{n=1}}(a_{n})", "05f3d6d19f3c15c4eaaa7477b12581ea": "P(H_1|E)", "05f4f243796cfa7f7b0c0c4469571465": "c_1 \\in C_1, c_2 \\in C_2", "05f51a092217d8e2c80ec9b79cf181be": "\\mathrm{d}U = \\delta Q - \\delta W\\,", "05f51a2cdca83e228f598eb61013653c": "(p_1,p_2,\\ldots,p_d),\n\\sum_{i=1}^d p_i=1", "05f53074d1c7f85781e7ffb09990b40a": "l_a=(-1,0,0,0)\\,,\\quad n_a=(-\\frac{F}{2},-1,0,0)\\,,\\quad m_a=\\frac{r}{\\sqrt{2}}(0,0,1,\\sin\\theta)\\,.", "05f55405e2597005963fc687fc0a397a": "H^n(A) = H^0(A[n])", "05f5a83cb1eace02ff37ab6b5ad9d92c": "a(t)=e^{t\\delta}\\,", "05f622803eb90d0598940885b74f7aff": "\n \\hat{y}_j^{(j)} - \\hat{y}_j = x'_j\\hat\\beta^{(j)} - x'_j\\hat\\beta = - \\frac{h_j}{1-h_j}\\,\\hat\\varepsilon_j\n ", "05f6292a398e85d0f6da63e624084153": "m_1 \\ne m_2", "05f6407a1cfade6431a89db24e251d94": "\\ PER = \\left ( uPER \\times \\frac{lgPace}{tmPace} \\right ) \\times \\frac{15}{lguPER} ", "05f66fa6148b8f62b8f3ce1f2639b1a3": "\\left (\\frac{E[V]}{R} \\right )", "05f6fdde4f929e7e7849e061fd649a9e": " \\cos \\theta = \\left(\\frac{\\gamma_s - \\gamma _{ws}^0 +\\frac{CV^2}{2}}{\\gamma_w}\\right) \\,", "05f72aa198a77e03b3e1af085701840b": " \\mathbf{\\bar{x}}=\\frac{1}{N}\\sum_{i=1}^{N}\\mathbf{x}_i. ", "05f74d85a66227de47682bafc2345668": "\n\\Delta \\bar{e}\\ =\\ \\frac {J_2}{\\mu\\ p^2}\\ \\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ \\left(3\\ \\sin^2 i\\ \\sin^2 u\\ -\\ 1\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\right)du \n", "05f76669a2ca87733a9fd1b530985ebb": "d_2,d_3,\\dotsb,d_m=0", "05f78fe4ec8e018b4d0486c5e98d45d4": "\\int_A f\\,d\\mu = \\int_{[a,b]} f\\,d\\mu", "05f7e498b0add2d4f3a946e530af830f": "\\forall k\\geq N_2 \\Rightarrow \\|A^k\\|^{1/k} > (\\rho(A)-\\epsilon).", "05f7e7c3f82b8ae7fc229dcf117d33ca": " 3 q _{2}q _{3} + 3q _{3}q _{4}+ 3 q _{1}q _{2}+q _{2}q _{4} + q _{1}q _{4} - q _{1} - q _{3} - q _{4}", "05f7ff803727730e147a6364410df1a2": "\\nu(x) = j", "05f846ca0e56a4867fd161ad252d994c": "\\frac{\\sigma^2}{2 a}", "05f86dee230cb1b9bb63d2159ad4449d": "P(B)=0,", "05f875bff224e8a484deca81bd4509fd": "\\frac{|\\text{actual effort} - \\text{estimated effort}|}\\text{actual effort}", "05f8831c4b653ded6674c224df25afb1": "\\left| \\mathbf{q} \\right|", "05f910f9d3bdd8ac40ec33c32909a772": "\\sigma_1(A)", "05f928dfb944e822440d7fe52821a2c3": "\na =x_1+x_2\n", "05f9516e185c7a916bc48bcc2a83f9d8": "S=\\bigcup(S_i\\mid i\\in I)", "05f98e5a81d331404f783b349fbf36f7": "y(r)=e^{\\beta u(r)}g(r)", "05f9c7abda372d91d676d45eae84a70b": " M = \\frac{p\\,(p-1)}{2}. ", "05f9d8f712baaeb6b5630bb6b919c139": "\\mathbb{S}", "05f9ed3a71241f7c04b687316fd91004": "n_s\\,\\!", "05f9f22c2944bba1198040d2a3edc044": "\\lim_{n\\to\\infty} z^{\\pm n}", "05f9f742d73d155fc3e9a8a071c7286f": "\\dot p = -\\frac{\\partial H}{\\partial q} = \\{p,H\\} = -\\{H,p\\} ", "05fa27202aff9c146e3eabdbd86d2ec5": "N = \\rho / (1 - \\rho)", "05fa7db7bce48752a8bfdb32d3b9c2c5": "\\begin{align} \n &\\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1)x^{r + c - 1} -\\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1)(r + c - 2) x^{r + c - 1} +\\gamma \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} \\\\ \n &\\qquad -(1 + \\alpha + \\beta) \\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1) x^{r + c - 1}-\\alpha \\beta \\sum_{r = 1}^\\infty a_{r - 1} x^{r + c - 1} =0\n\\end{align}", "05fa87902f89e288654d0e2752b0cefa": "M=E-\\varepsilon\\cdot\\sin E.", "05fb05070d1664acb2a478a9111e4853": "\\mathbb{Q} \\cap [0,1]", "05fb116dd77209938c1398a35cd8b116": "\n \\begin{align}\n \\frac{\\partial I_1}{\\partial \\boldsymbol{A}} & = \\boldsymbol{\\mathit{1}} \\\\\n \\frac{\\partial I_2}{\\partial \\boldsymbol{A}} & = I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T \\\\\n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}} & = \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T \n = I_2~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T~(I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T)\n = (\\boldsymbol{A}^2 - I_1~\\boldsymbol{A} + I_2~\\boldsymbol{\\mathit{1}})^T \n \\end{align}\n", "05fb99d59ad9c1d1e6e39dab062a8b33": "m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2} = {0}\\,\\!", "05fbd3a56cf9aeb0671739bab7c850f9": "t \\in [0,1],", "05fbe6c52ef5b1319c97203ef8abc1b5": "\\omega=\\omega_o\\left( 1 +\\frac{\\mu BH_k}{kL_{e}^{2}(B+H_k)}\\right)^{1/2}\\approx \\omega_o\\left( 1 +\\frac{\\mu BH_k}{2kL_{e}^{2}(B+H_k)}+...\\right)\\Rightarrow", "05fc110a8dbad659411d44f326dfbc99": "w : X \\vdash w : X", "05fc67ba192a27ca6b8ae5d30e2978e7": "\\scriptstyle \\partial \\vec{D} / \\partial t", "05fc9804f26c17af7a6f5844a7678d2d": "\\oint_S \\mathbf{B} \\cdot \\mathrm{d}\\mathbf{A} = 0,", "05fce8ddbd2502dc79f9950775edce6f": "v X = \\{ \\lambda \\in X : r ( \\lambda ) = v \\}", "05fd0c333ff9be30a0e6c163a5d092a3": "K(GL(R), 1)", "05fd1c7db07940b7a7afe1e193282045": "{\\mathfrak{m}_B}^s \\subset (y_1, \\dots, y_m) + \\mathfrak{m}_A B", "05fd43ad299946ac38adffa752f99a60": "\\Gamma:={\\Bbb Z}^3\\ltimes{\\Bbb Z}", "05fd66fa5a897319d0fb87bd24af04d3": "\\tfrac{1}{24} \\left ( (\\operatorname{tr}A)^4-6 \\operatorname{tr}(A^2)(\\operatorname{tr}A)^2+3(\\operatorname{tr}(A^2))^2+8\\operatorname{tr}(A^3)\\operatorname{tr}(A) -6\\operatorname{tr}(A^4) \\right )", "05fd7fc4c1d13bec4e3bdd8523ba2fa5": "\\overline{W}_{\\dot{\\alpha}}", "05fd9792691ff82531e230768864e180": "2^{S''}", "05fe086cb3d686ae49d586d8f95414f6": "\\left|\\widehat f(n)\\right|\\le {K \\over |n|}", "05fe12829fff81295b9bef693f5e8779": "\nv(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\int d\\mathbf{r}^{\\prime}\\, \\rho_{uc}(\\mathbf{r}^\\prime) \\ \\varphi_{\\ell r}(\\mathbf{r} - \\mathbf{r}^\\prime)\n", "05fe9c80d66c928b66cdc1e76dd5efbc": "D=\\tfrac{a}{\\sin \\alpha} = \\tfrac{b}{\\sin \\beta} = \\tfrac{c}{\\sin \\gamma}.", "05feaeeb29070ecee88283b395e32236": "\n= \\sum_{k=0}^n k! \\,S(n\\!+\\!1, \\,k\\!+\\!1) \\left({z \\over {1-z}} \\right)^{k+1} \\qquad (n=0,1,2,\\ldots) \\,,\n", "05ffb8526e575ca4f6ccdd8ff33ca71c": "\\bar{f}(s)=\\int_0^\\infty e^{-st}f(t)\\,dt", "05ffbdb4aeadaadb02ee46b499f9ce2d": "\\dot{x}(t)", "06001471dde5949692c7cf2cf7feda6b": "\\begin{matrix} {10 \\choose 1}{4 \\choose 3}{44 \\choose 1} \\end{matrix}", "060038299b950bc5d6c8e81975ed65fe": " \\Rightarrow \\frac{p(y|H2)}{p(y|H1)} \\ge \\frac{\\pi_1}{\\pi_2}", "060064589e48960b70d7488cdb0f6d66": "\\gamma_k =\\frac{1}{y_k^T s_k}.", "0600706e87ee0b62690eaac783c0a96d": "M_{\\pi_T}^2 \\propto \\langle\\bar{T}T \\bar{T}T\\rangle_{M_{ETC}}", "06009ffd2c6b4c9bee322cb86461806e": " \\tfrac5{36} - \\tfrac1{30} \\sqrt{15} ", "0600b0075f1dc8c5beeb7e0c89d1be2e": " K= C_{12}+C_{23}-C_{13} \\le 1", "0600ce9319de00e376d249db90db96eb": "\\mathbf{w}_{n}=\\mathbf{R}_{x}^{-1}(n)\\,\\mathbf{r}_{dx}(n)", "0600eb7f294010969188a9763065934e": "\\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{bar} = \\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{water}", "06010d437e589a532f147b49326d1bb0": "* \\!\\,", "0601d5da1d270f4663f165701f1c9798": "q_1 = 1+\\frac{\\sum_{i=1}^k \\pi_{i}^{-1}-1}{6N(k-1)}. ", "0601ef2dbd4b2eae873ecbaf02ba45cc": "\\delta_t, \\, t \\in G,", "06020d9ff8c01eaaf44943780aa8a89d": "x = jb\\,", "0602535e8203a4f9c0f07088182fe798": " U_s = 2 \\left|s\\right\\rangle \\left\\langle s\\right| - I", "060268a090fed8d9854efb535e06332b": "\\Delta \\vec{F} = \\vec{F}_{n} - \\vec{F}_{n-1} ", "060290f166448ac0480686e89f6a921a": " U \\cap A = T.\\,", "060301f9fe00c278acc161de360ced0d": " A: G\\times M \\to M ", "06031c8d29a41ca293d19d7d397017de": "(11, 5_2,4, 1)", "06031cba9297343eabe2961fa3da37f3": "(NB)/3", "060361fbb611719487b00f78f51cbf9b": "\\int f(x)\\sin(x)\\,dx=F'(x)\\sin(x)-F(x)\\cos(x),", "0603b8974e76582d6b317b0aa99346f7": "\\Pi_{H}(m)\\leq\\left( \\frac{em}{d}\\right)^{d}\\,\\!", "0603ba49d242efbd716fdcc687d4aaf4": "\\mathbf{c}, \\mathbf{b}", "060462a5b69fe5821f4e5c6375706bd6": "\\partial_i\\ell", "0604dfb6d9db52ca41732d4f1c82a753": " \\frac{1}{F_{max}} = \\frac{1}{F_{e}} + \\frac{1}{F_{c}}", "060562d10a1d73260d67b7623181857c": "N = R_{\\ast} \\cdot f_p \\cdot n_e \\cdot f_{\\ell} \\cdot f_i \\cdot f_c \\cdot L", "0605d13e5ecf124707bc65207eb9065c": "\np = {h \\over \\lambda}\n", "0605ec8ff0e079dfb988391330236abf": " w \\cdot ( u \\wedge v) = \\frac{1}{2}( w ( u \\wedge v) - ( u \\wedge v) w)\n", "0606585ddb28d26b541178a2fd750d74": "\\theta = \\begin{cases} \\sin^{-1} \\frac{1}{\\beta}, & \\text{if }\\beta \\ge 1 \\\\ \\pi - \\sin^{-1}{\\beta}, & \\text{if }\\beta\\le 1\\end{cases}", "06066522c496336b0fd736296a1d0d9d": "\\gamma = \\frac{(1+w)G_s\\gamma_w}{1+e}", "0606dd106fbc2416e0716e2252336df9": "\\displaystyle s_\\mu h_r=\\sum_\\lambda s_\\lambda", "0606e81e12923a421971691286f2935c": " x", "0606f21326dd8210d4402885228f181e": "M_0,M_3,M_5,M_6", "0606f740fa19e367728576ebdd03c049": "k_{\\rm{adj}} = k \\left( \\frac{\\mbox{maximum rotor-speed}}{\\mbox{actual rotor-speed}} \\right)", "0607238db7ad004ed43ca8e1dbef539d": "z_1,\\ldots,z_n", "0607262f81fbe5797380178222c0068c": "\\nabla(\\nabla\\cdot\\vec{A}+\\frac{1}{c^2}\\frac{\\partial\\varphi}{\\partial t})=\\mu_0\\vec{J}-\\frac{1}{c^2}\\frac{\\partial^2\\vec{A}}{\\partial t^2}+\\nabla^2\\vec{A} ", "0607374c116257a45022bbd802572b26": " \\gamma_3 = \\sqrt{\\frac{2}{\\pi}}(\\sigma_2-\\sigma_1)\\left[\\left(\\frac{4}{\\pi}-1\\right)(\\sigma_2-\\sigma_1)^2 + \\sigma_1 \\sigma_2\\right]", "060746f5f4519d2e745eaba4708111c1": "E - e\\phi \\approx mc^2", "0607473e9e177ff05e9a6f4d1bf1fd81": "U(s)", "06079a798ccec5f66f8ecb8704f52987": "I=q/t\\,", "0607db2a0900d1cf784c4cd826368deb": "SS_\\text{res}=\\sum_i (y_i - f_i)^2\\,", "0607db8fc32c6ebe9fe571ceec46879d": "T_{11} = \\left(2C_1 + \\frac {2C_2} {\\alpha} \\right) \\left( \\alpha^2 - \\alpha^{-1} \\right)", "060817c208d5981b1485cabd5bdb5139": "pN", "06081fb3e714ed01d831ebd8513f1822": "\\frac{Gross Profit}{Sales}", "060844221f545e5bf6862e60aaec07aa": "\nds^{2} = d\\mathbf{q} \\cdot \\mathbf{M} \\cdot d\\mathbf{q}\n", "06084b087fb41001d770760b25cbe12f": "\\left\\{\\pm\\frac{\\pi}{2}, \\pm\\frac{3\\pi}{2}, \\pm\\frac{5\\pi}{2}, \\ldots \\right\\}\\,.", "0608732f994277f423acfaef18f70d8a": " \\rho \\neq e", "0608e49f58cb46fab57a77087a85d990": "a = 2, \\, b = 2, \\, f(n) = n^2", "0609110318e4878dbb0eeb0ccf3b336e": " \\mathfrak{g} = [\\mathfrak{g}, e] \\oplus \\mathfrak{g}_f ", "06092a49718e3e55aa32259d4a1cbdc0": " \\dim \\mathfrak{d} = \\dim W - 1 ", "0609778b9c9c588b63b6a3732e9fee9d": "K = {k_1,k_2, \\dots ,k_n}", "06099bc35e1dbd78c6c50816a9cd892d": "\\overline{x}\\,", "0609d71ea290e86c7da521ed45f0de14": "\\Delta x'", "0609f218ab24220de50e4a0bca984c61": "\\ln r = x\\ln[A] + \\textrm{constant}", "060a9233be6ac589bd81a3756d5b0a4d": " \\sum{x_i} \\leq k ", "060ab80287a2426f32708c585e447161": "y\\in Q^n", "060ac1614e0969e935138d1e7dd96062": "C_{m}", "060af5819fb36c7d0154761c2b3697c4": "\n \\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c}) =\n-\\mathbf{a}\\cdot(\\mathbf{c}\\times \\mathbf{b})\n", "060afc5a1b7d77a2e221639a9fe8fee7": "\\left.u_p\\right|_{r= R}=0", "060b84c979ace117697e202f36c77586": "q\\begin{Bmatrix} p, q , r \\end{Bmatrix}", "060b92061c5b5c477d2b4fded0e27d96": " \\vec{e}_2 = \\partial_x ", "060bfd719f6fd57edd4f3521c612dbdb": "\\partial(X,f,\\alpha)=2\\pi \\sqrt{-1}\\sum_i(V_i, f_i, res_{V_i}\\,\\alpha)", "060c301f6ac199cfb7701726bef0dcf4": "\\scriptstyle r", "060c63b9bc19a48246bfbfe3435cbc3a": "A_\\lambda", "060c6d21eec9d49717d5dd5a7c768c0d": "\\mathcal{H} = L^2(\\mathbb{R})", "060cc3fcebf81f5a13d8a8de42b490f2": "(V_i, V_j,)", "060cd7165c91012d5c391e921f7a9930": " \n\\begin{align}\n&y_{0} &=&\\ y(0)+ L^{-1}(-1) &=& -t \\\\\n&y_{1} &=& -L^{-1}(y_{0}^{2}) =-L^{-1}(t^{2}) &=& -t^{3}/3 \\\\\n&y_{2} &=& -L^{-1}(2y_{0}y_{1}) &=& -2t^{5}/15 \\\\\n&y_{3} &=& -L^{-1}(y_{1}^{2}+2y_{0}y_{2}) &=& -17t^{7}/315.\n\\end{align}\n", "060d0527eba6da4161bcb4b833b41c31": "Q(\\boldsymbol{r}) = Q(F_\\boldsymbol{r}), p(\\boldsymbol{r}) = p(F_\\boldsymbol{r}),", "060d13166c74ee3cc0985680289cf42a": "u_{\\max}^{(s+1)}=\\frac{1}{x^{(s)}},\\ k^{(s+1)}=\\left[\\frac{1}{x^{(s)}}\\right].", "060d52f0747e46a84b87ab0515dfdfb1": "\\mu_1, \\mu_2, \\ldots, \\mu_r", "060d68ae440ca0f8f5b87557cefde05b": "\nT=\\frac{V}{A} \\cdot 0.161\\,\\mathrm{s}\n", "060d6ca8599c55633a112da0b64b25bf": "\\int_{-\\infty}^{\\infty} |\\psi (t)|\\, dt <\\infty", "060dc851ace6e3e11ffc450cc603ec99": " \\eta=\\frac{y}{\\delta(x)}=y\\left( \\frac{U}{\\nu x} \\right)^{1/2}", "060e428bbbf6496c2e7d9b8a308ee239": " m", "060ee93d0601609f694cfe42a429e569": "l=d+w", "060f013cc49db63b4af50b03a20996f2": "\\scriptstyle \\mathbf{D}", "060f03bd35e64518bb9744cd7aa00b5a": "R(w) =\\sum_{g=1}^G \\|w_g\\|_2,", "060f40333258faf628efce4f086a01f3": "f: \\mathbb{R} \\rightarrow \\mathbb{R}^+", "060f987de88e7c8d4afad7d4828e3f7b": "\\sup_{\\theta \\in \\Theta} R(\\theta,\\delta^M) = \\inf_\\delta \\sup_{\\theta \\in \\Theta} R(\\theta,\\delta). \\, ", "06108a0b8b6dcaa756c6c3ab6317551d": "p=\\frac{N_0-N}{N_0}", "06109000b497df97e7b4118d2b5f9c41": "\n w = 0 \\,, -D\\frac{\\partial^2 w}{\\partial y^2}\\Bigr|_{y=b/2} = f_1(x) \\,, \n -D\\frac{\\partial^2 w}{\\partial y^2}\\Bigr|_{y=-b/2} = f_2(x) \n", "0610a36cf3ae80b3045fb4b372651650": "\\mathbf{\\nabla}\\times(\\mathbf{\\nabla}\\times\\mathbf{V})=\\mathbf{\\nabla}(\\mathbf{\\nabla}\\cdot\\mathbf{V})-\\mathbf{\\nabla}^2\\mathbf{V}", "061104ac886aef675293663800232f56": " Q^{(1)}, Q^{(2)},\\ldots ", "061107504b5aa7a97959c51cb34e484f": "z\\bar{z}+w\\bar{w}=1.", "06114dd2614cc35393a7c6b2deff8e0a": "Z=\\sum_{n=0}^{\\infty } \\frac{(10n+1) \\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^3{9}^{2n+1}}\\!", "061173042b74c01eb3b2dbbec445897c": "G_{4}(\\mathbf{p}, \\mathbf{P}, t)", "06117fb16c9900d808148064b388381a": "\\bar{n}_i = \\ \\frac {1} {e^{(\\epsilon_i - \\mu)/kT }+1}", "06119872473c06fc42d6f7cf08d6aa41": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i ([\\Delta r_i]\\boldsymbol\\omega)\\cdot([\\Delta r_i]\\boldsymbol\\omega) + \\frac{1}{2}(\\sum_{i=1}^n m_i) \\mathbf{V}_C\\cdot\\mathbf{V}_C.", "0611d5ea94a9498441c4bb70af9d9b60": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{tri} \\left( \\frac{\\nu}{2\\pi a} \\right) ", "06120cc69950c1c1c2a4679a307ac149": "Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{0} + cY_{t-1} - C_{0} - cY_{t-2})", "061299ee08b59ed4968edae3ad322fc8": "S\\in W", "061339dfbd7f3c80d83c9f59490b76fb": " Vol_q(0, \\lfloor {{d-1} \\over 2} \\rfloor) \\le q^{H_q({\\delta \\over 2})n-o(n)} ", "061377df11087841d850ebdd7a81a57c": "M^{0a} = -M^{a0} = K_a \\,,\\quad M^{ab} = \\varepsilon_{abc} J_c \\,.", "0613a14e112170454dd8ee2fac200e33": "b\\cdot a", "06141a1da5d19a810187d649c248c613": "E_c = L^2/(L^2+m^2)", "061453faeff864f7eb127d98843c4c0a": "\\Delta : \\mathcal{C} \\rightarrow \\mathcal{C}^\\mathcal{J}", "0614ad79a0f78028781bb65a4665fcf7": "\\frac{\\pi r}{2}", "0615003c55d5aab471d04225e021cf7a": " Z_{eff} ", "06150743b944ae53760c95d20c1dec95": "q(x,y) = q_0", "061510548cb220ad5348824f657cffca": "k^{-m} E_m(kx)= \\sum_{n=0}^{k-1}\n(-1)^n E_m \\left(x+\\frac{n}{k}\\right)\n\\quad \\mbox{ for } k=1,3,\\dots", "061512d21c171f0f05094bc24900f4ea": "f(x) = x^3-1\\, ,", "06154eca89935c16391249e30b659550": "M_b", "06154fbf5b0d359fdabd084cd66ebc25": "\n\\frac{d}{dx}(x^2)=2x.\n", "061550e9b9bdc85c3f4a8591b42e540b": "\n \\mathbf{J}^{23}\\mathbf{A} = \\left[ \\begin{matrix} 0 & 0& 0 \\\\ a_{31} & a_{32} & a_{33} \\\\ 0 & 0 & 0 \\end{matrix}\\right]\n", "0615609bb804231ecd6e9ea7b59a5ee6": "a_0 + a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n \\leq 0", "06158b7ebe260812220d4e2b7c3ecb90": "0,x_1,\\ldots, x_n\\,", "06159f2da0ee5c095102d190ec683974": "\\ B-\\text{vertex}= 1 : -1 : 1 ", "0615d318431f10aa166cb6d492ff5de2": "d\\colon M \\to M \\colon M_i \\to M_{i+1}", "06161097402a99112a7073c0e6f25328": "H = AF_4 = 0", "06165a207796d90c74c3962037eab3da": " m\\frac{du}{dt}=X_u u -mg\\gamma", "06166b0bfc29b2d32dbc5179ebdab4e7": "[\\nu ]=\\sec ^{-\\mu }", "06168ddfffbb48e0679a58d34ca4e824": "\\pi ab.\\,", "0616964198f654b6b7402626697ec7a4": "V = \\frac {\\pi h^2}{3} (3r-h)", "0616d198ca8080fb18755a5ce61e3e31": " \\partial p/\\partial s=0", "061702b8ec8c978285ef3f1f6486484b": "j_1 ^* \\circ F^* = \\mathrm{id}, \\; j_0^* \\circ F^* = 0.", "061705237c9d8f58e5c9702b0643d447": "Y(s) = \\left( \\frac{P(s)C(s)}{1 + F(s)P(s)C(s)} \\right) R(s) = H(s)R(s).", "06174424b7644856608e3315c3ecadd6": "P_Z ", "06177336f1deb5b1e796c2e24aad5d38": "(\\ g(-\\tau)=g(\\tau)\\ ),", "06181bccf4fdac8791f2b00ede092713": "\\forall x \\in X \\, \\forall y \\in X \\, ( ( x < y \\, \\land \\, \\lnot (y < x) \\, \\land \\, \\lnot( x = y )\\, ) \\lor \\, ( \\lnot(x < y) \\, \\land \\, y < x \\, \\land \\, \\lnot( x = y) \\, ) \\lor \\, ( \\lnot(x < y) \\, \\land \\, \\lnot( y < x) \\, \\land \\, x = y \\, \\, ) ) \\,.", "0618371d023a736616c74fc01e13271b": " {\\rm vec}(\\mathbf{X}^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1} \\mathbf{X}(\\mathbf{B} - \\hat{\\mathbf{B}})) = (\\boldsymbol\\Sigma_{\\epsilon}^{-1} \\otimes \\mathbf{X}^{\\rm T}\\mathbf{X} ){\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}}), ", "06184cbb2d2852a9c1aa37c1cf47f7a2": "(-\\nabla_x^2 + t) \\langle H(x)H(y) \\rangle = 0 \\rightarrow \\nabla^2 G(x) +t G(x) = 0", "061880dceea6b778b71a416d7c1c4dda": "4. \\; \\; \\mathrm{O} + {\\mathrm{ClO} \\cdot} \\; \\xrightarrow \\; {\\mathrm{Cl} \\cdot} + \\mathrm{O}_2", "0618b0b833d1f4b07fae34fdbd2fe21e": " g(E \\cup F) = g(E) + g(F). ", "0618bec31282a3ef777041db73a0306c": "\\begin{align}\n\\frac{\\partial \\Lambda}{\\partial x} &= 2 x y + 2 \\lambda x &&= 0, \\qquad \\text{(i)} \\\\\n\\frac{\\partial \\Lambda}{\\partial y} &= x^2 + 2 \\lambda y &&= 0, \\qquad \\text{(ii)} \\\\\n\\frac{\\partial \\Lambda}{\\partial \\lambda} &= x^2 + y^2 - 3 &&= 0. \\qquad \\text{(iii)}\n\\end{align}", "0618e26d2e432a00aec6fabf72a09380": "\\Re(s)=1. \\, ", "06193df674d572760c5db06e18741194": " \n{\\eta}", "06195a4fb543bcc3edabb8d8109acf77": "\\ \\frac{S}{C}=\\frac{1}{1300}\\frac{R^3}{P_tG^2\\lambda^2}\\frac{1}{\\tau\\theta\\sec\\psi\\sigma^o}\\frac{P_tG^2\\lambda^2}{(4\\pi)^3R^4}\\sigma", "06198eabc68808757ee3fa9a544a9f9e": "-{T^a}_b \\, Y^b", "0619a61d3d587407b32bb48256faf6bd": "\\color{Black}\\tfrac{4}{m}\\tfrac{2}{m}\\tfrac{2}{m}", "0619cfe36a6a71767c71545a48967dd4": "y=2.870x - 3.000x^2 - 0.275", "061a00186250241da69478b06d200c8f": "F(x) = f(x) - i g(x)", "061a01f63f1d2ef8c5e01e4a5bca087a": " F(x;b,\\eta) = \\left(1 - e^{-bx}\\right)e^{-\\eta e^{-bx}} \\text{ for }x \\geq 0. \\,", "061a43b72d779f9cbd8d90dd97a90679": "i =j", "061a92807a3c76be3ec523cf988cae28": " Sym^k (V) \\to Sym^k (V)", "061a979350a81ea6511a7124316f5899": "\\Delta v = \\sum_{i=0}^{n-1} {Ve}_i \\cdot ln \\frac {Minitial_i} {Mfinal_i} ", "061aa0e33a22e7061ea23b9cc82e4226": "ax + by + c = 0", "061aa8475215a37cad95a2c964bf6c0e": "|x|^2", "061abf8bccd1fe41c7f24818fb10e861": "C_{n+1}(L) = \\pi_n^{-1} ( Z ( L / C_{n}(L) ) )", "061acf5b000d56fe69754ded04693653": "m = v", "061af55e64886354d06554c5b95bbb5c": "y=a\\sin\\left(u\\right)\\sin\\left(v\\right)", "061b477bc57099c87f264be0934ccd6d": "Q_i=\\{(s_i,t_{ei})| s_i \\in S_i, t_{ei} \\in (\\mathbb{T} \\cap [0, ta_i(s_i)])\\} ", "061b8d113b20df75d64f158d0c902cc5": " g'(\\nu) = {1 \\over \\pi } { (\\Gamma / 2) \\over (\\nu - \\nu_0)^2 + (\\Gamma /2 )^2 }", "061b9d3c6eeb7b63db0df009b21a3957": "\\sum \\mathbf{p}", "061be9c6449ff9eb3eb38603c3aafbc8": "\\tau~=~i~C~t ,", "061bf8988a0342a0119609f12317688d": "\\lambda=\\operatorname{E}(X)=\\operatorname{Var}(X).", "061c85ceafcd4d4e2127e7073dbc9b6c": "P_i = wl_i + (1 + r) wl_A a_i", "061c8bf9ac1c45cd53ea36123f57aa4a": "m = ", "061ca2407a364c9ed7be5e86bc31c4cb": "\n\\begin{matrix}\nQ^2+U^2+V^2 \\le I^2.\n\\end{matrix}\n", "061ca39c6c549e7b737974e06028f3f8": "\n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{j=1}^N \\mathbf{F}_{jk} \\cdot \\mathbf{r}_k.\n", "061cc8162de7c13e3efbbbbba68e7e04": "\\tfrac{m}{n}", "061d4742bddb491b499e54c32cf2b68e": "w_j = \\frac{1}{\\prod_{i=0,i \\neq j}^k(x_j-x_i)}", "061d964ac1f0e7c9bb70b6e25980ecc1": " \\langle \\mathbf{ABC} \\rangle = 0,", "061db0126fb94036224aab33e423811b": "= \\frac{D}{V_\\text{d}}", "061dd8f29ebee8ad5e2369c0fdfac9bb": " p = \\sum_\\alpha p_\\alpha X^\\alpha,\\ ", "061df79c44ad52e26e803baec66b16d6": "d_{X,Y}=1-\\rho_{X,Y}.", "061e4287d2c03e68d2f06a853c9e345c": "\\psi(\\Omega)^3", "061e663fd567a45ac9d046e78a1b3caa": "\\mathsf{I} \\left(\\sum_{i=1}^n r_i \\mathit{1}_{ [x_{i-1},x_i)}\\right) = \\sum_{i=1}^n r_i (x_i-x_{i-1})", "061e68340720962b50a1a1f6b17c14a5": "C_\\text{V}(x) = \\exp(- b_\\text{ext} x)", "061ee6c53cb1c7143dffc932f58467ec": "\\mathbf{q}_{1}", "061fac1ccb610da7c6c849fc1b85d3ed": "E(X~|~X\\le z)\\cdot Pr(X \\Delta r^2\\\\\n s^2 &< 0 \\\\\n\\end{align}", "062034e1b1e77f92fa8d08995925acf7": " s \\in (0,\\infty) ", "062040c65b51cd1374b751276f041f00": "O(n \\cdot \\log n)", "062096cbf7815cd82b26fe175a0f1fb2": "\\begin{align}\n\\mathcal{A}\\left\\{x(t-\\tau)\\right\\}\n&= \\int_{t-a}^{t+a} x(\\lambda-\\tau) \\, \\operatorname{d} \\lambda\\\\\n&= \\int_{(t-\\tau)-a}^{(t-\\tau)+a} x(\\xi) \\, \\operatorname{d} \\xi\\\\\n&= \\mathcal{A}\\{x\\}(t-\\tau),\n\\end{align}", "0620ac251740f66d804ae4a7cae323d4": "\\infty_1 ", "0620e7e777cd8b7e87bc391a420202f3": "6\\cdot 6=36 > 27", "062129255c464930f035184ee28f91d0": " P-P_{-1}\\ \\approx \\pi ", "06213f7b6eb6c9529be8a52b9e59b147": "\\ln \\ln \\frac{\\varepsilon^{(s+1)}}{\\varepsilon^{(s)}} = \\eta_s + \\sum_{p=0}^{s-1} \\xi_p, \\quad \\eta_s = \\ln \\left [ 2\\delta^{(s)} \\left ( k^{(s)} + x^{(s)} - 1 \\right ) \\Omega^{(0)} \\right ] ", "06215cc7ede143df16d1e4a54b219d39": "3n^2", "0621b31ee4fdc1084fa5d458679be123": "\n\\begin{pmatrix}\\mathrm{Cu}\\\\\\mathrm{Ag}\\\\\\mathrm{Au}\\end{pmatrix}\n\\begin{pmatrix}\\mathrm{Al}\\\\\\mathrm{Ga}\\\\\\mathrm{In}\\end{pmatrix}\n\\begin{pmatrix}\\mathrm{S} \\\\\\mathrm{Se}\\\\\\mathrm{Te}\\end{pmatrix}_2\n", "0621d7da3d66fb1c2e19a8e9ba982159": " x^\\alpha = x_1^{\\alpha_1} x_2^{\\alpha_2} \\cdots x_n^{\\alpha_n}. ", "0621fdb9b047e3455b27417e76bc6dbb": "S_k(n,r) \\cong \\mathrm{End}_{\\mathfrak{S}_r} (V^{\\otimes r}).", "0621ffe2cc7912d02595966ce1095472": " p_c ", "06228e7e11688a89df9a6ef09a3684bb": "O(\\frac{n^2}{m^2})", "0622c18f16ae745537349b8d5f629fe5": "|1\\rangle\\leftrightarrow|2\\rangle", "0622d0f2c3a2b81593b8db0108871121": " \\left(\\mathbf{A} - \\lambda_i \\mathbf{I}\\right)\\mathbf{v} = 0. \\!\\ ", "062314ee4515e21c160b657d3f3763b0": "\\langle a \\rangle", "06236523bedf0e5f6a9d963cebdd55b5": "\\!\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = \\delta_{mn} k_{B} T.\n", "0623691cb796dbcb5716c7cc29380dd2": "\\sqrt{8r\\left(\\sqrt{4R^2+r^2}-r\\right)}\\le s \\le \\sqrt{4R^2+r^2}+r", "0623bf85ab6b26f9ea9d75e605791ab6": " \\frac {d \\mathbf{u}_j (t)}{dt} = \\boldsymbol{\\Omega} \\times \\mathbf{u}_j (t), ", "0623cdec0ce139d951d43c42901b2bc7": "= \\left[ {n \\choose 0} \\cot^n x - {n \\choose 2} \\cot^{n-2} x \\pm \\cdots \\right] \\; + \\; i\\left[ {n \\choose 1} \\cot^{n-1} x - {n \\choose 3} \\cot^{n-3} x \\pm \\cdots \\right].", "0623e20e090df47dffc522dba9515e31": "f(q(\\xi ,\\tau))^{\\;}", "062404cdddf8b568d4aec12e5ba37a13": "\\int x^3 r^{2n+1} \\; dx = \\frac{r^{2n+5}}{2n+5} - \\frac{a^2 r^{2n+3}}{2n+3}", "062405359c635dc6fce4eb706b473ab8": "\\mathbf{P} \\left( \\left\\{ \\omega \\in \\Omega \\left| \\lim_{s \\to t} \\big| X_{s} (\\omega) - X_{t} (\\omega) \\big| = 0 \\right. \\right\\} \\right) = 1.", "06240fb43060edd8a406f399def4bbb1": "a\\uparrow^n\\cdots\\uparrow^na\\uparrow^n1", "062429ee30b925bb458e4649dc433a3b": "\n\\tau = \\frac{ f \\rho v^2}{2}\n", "062434cd357e12f6cb0470162bc396d4": "\\scriptstyle A_n = \\{ i \\in I \\,:\\, a_i > 1/n \\}", "062461c7b79714d39684613c8f62ee16": "\\vec{j} = \\vec{j}_{\\text{diffusion}} + \\vec{j}_{\\text{advective}} = -D \\, \\nabla c + \\vec{v} \\, c.", "062536d639888f461dddbae1c1858e50": " F_{ST} ", "06254eaadabdb05aaf423cafa36f26a0": "a b ^{-1}", "0625863f41074bc5bce3370e701b6a31": "B_n(f,f)=0", "0625f9150cc414ca567fc3a4b32adb02": "x^2+ y^2 = -1", "0626359a6d0f2e2c24ba74cc83d6e44d": "\\frac{\\partial \\varphi}{\\partial t} + \\tfrac{1}{2} v^2 + \\frac{p}{\\rho} + gz = f(t),", "06263f482c703d45549ae3fd10f2d143": "d_k = n\\sum_{i=0}^k \\frac{(n+i-1)!4^i}{(n-i)!(2i)!}", "06266b399c5eef9b6d7464698fbefefc": "\\rho_e = \\sqrt{\\frac{L_e}{C_e}} = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} = \\rho_0 = 2\\alpha R_H. \\ ", "0626bb3ceac07cb0c47c3a18731ff0e2": "L_3 + -4L_2 \\rightarrow L_3", "062762cd1071fc2b58b93de1014f67a8": "S = \\frac{r^2}{4 \\times \\text{focal length}}", "0627b03980bb3b2b488092cd5c1eb4d0": "p(\\mathbf{Z}|\\mathbf{X},\\boldsymbol\\theta^{(t)})", "0627be370a9321cc5711d301bb89d099": "k \\in \\mathbb{C}", "0627c3775170c6cd963fb32678428514": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] (\\hat{O}^\\dagger W_\\gamma [A]) \\Psi [A]", "0627f8d1a01988c1d46a6c3d86d83dd3": "\n\\frac{r_1}{A} = 0.46224\\left(\\frac{M_1}{M_1+M_2}\\right)^{1/3}\n", "06281beef0a11f0bbd026b6a910af3e7": "\\frac{\\pi}{4} = 4 \\arctan\\frac{1}{5} - \\arctan\\frac{1}{239} ", "0628666933060a11dd7308de2080b232": "\\hat{\\mu} \\pm 1.96\\hat{\\sigma}", "0628eaa3ae45bddd90c29a79e6c49e3b": "(A \\vee B \\vee \\overline{C}) \\wedge (\\overline{A} \\vee C) \\wedge (\\overline{B} \\vee C)", "062917f45123be31aae872afa8498ec0": "ds^2=\\frac{dy^2}{y^2}", "06291e2cab443a10fe3b6d9094ca6fe0": "v'=v^2 + R(x)v +S(x),\\!", "062939382c914bd39578f170b79d1b91": "|\\chi(x)| = \\chi(1)", "062984a0416fa2887b6e7445c4cc2563": "\\theta_{\\rho\\sigma}", "0629cb6dcd098c7599fbbd90349c25dc": "R_B", "062a037a163a451ebe08bf1367e6f834": " = \\frac{B_{wr}}{B_w}", "062a199ea7c8a6452693d0cdd2d9d9a3": "\\operatorname{erf}(x)\\approx 1-\\frac{1}{(1+a_1x+a_2x^2+a_3x^3+a_4x^4)^4}", "062a2b397475bd7de2c73915909b205f": "\\{\\mathbf{X}\\}", "062a2ee51454e2cb157488a562fad2bb": "P \\or R", "062a42706cf98fee006f2ad7f10482a0": "\\scriptstyle f\\,^\\prime=g", "062a53f9126f5965ec44bd5b730eed6f": "s(n)^k", "062a9c61db003c41317573201fbc8aa5": " P_k. ", "062ab0db31f98254f062e64eff695414": "A = \\sum_{n=0}^\\infty a_n X^n ", "062acb49d4a0c8289c5db26a369c1145": "\\textstyle C \\cap \\mathbb{Z}^n", "062ad19db449b50e15b179c3c6d31c1f": "c^{2} d\\tau^{2} = \\left( g_{tt} - \\frac{g_{t\\phi}^{2}}{g_{\\phi\\phi}} \\right) dt^{2} \n+ g_{rr} dr^{2} + g_{\\theta\\theta} d\\theta^{2} + g_{\\phi\\phi} \\left( d\\phi + \\frac{g_{t\\phi}}{g_{\\phi\\phi}} dt \\right)^{2}.", "062b8bd326f589ec94b970b64b7f0320": "(X_i)_{i \\in n}", "062ca4ff82afa835f107cbd32ab9c206": "x[m - l]", "062ca8518521c3130bd03e8dea036e0e": "i+j+k+l", "062cb3ee8cef408d72495aefdf4b5ef6": "\\pi_b", "062ccb1bfc2d82ee0d8f0ef0c8ecb329": "(c_1\\mid c_1 + 0 ) \\in C_1\\mid C_2", "062cd57cbca9cbc94a168e813a3bc469": "s_{\\Lambda}=\\sum_{i=1}^m x_i; s_{K}=\\prod_{i=1}^m x_i", "062d16cc5ddb949e53dc2d22874c8cd1": " \\mu_A = \\mu_{A}^{\\ominus} + RT \\ln\\{A\\} \\,", "062d41268eaebed7f369b10257c567ba": "W_k", "062d49b83e583cc240886020adfe2fd9": " E([0,N]) ", "062d769c571f8fdb77f4b1c0dbaf8b4a": "s^m", "062e24315ff4a033f03e116eae78ddb9": " \\cos \\alpha = \\cfrac{|\\mathbf{e}_1|}{|\\mathbf{b}_1|} \\quad \\Rightarrow \\quad |\\mathbf{e}_1| = |\\mathbf{b}_1|\\cos \\alpha", "062e4e19dd4b81651c2d6d1999c0c29f": "C= \\frac{L}{R_0^2}", "062e6e7cc5add0256261b3628cbefa4e": "\\begin{pmatrix}\n1 \\\\\nu \\\\\n v \\end{pmatrix}.", "062f1f11a733f565e3b29f75e34656ec": "O(n^22^{\\sqrt{\\log n\\log\\log n}})", "062f270a7d2ff2caa4245419eb108ebd": "d(E_1,E_2)>0", "062f2bd0056a5b0d04f1547095f638f4": " s = 1 ", "062f38a477980dbb65d6a1f0dde0b0ef": "O(\\textrm{polylog} n)", "062f4baeef5e0644cd7cf013b3bf6d3b": " \\mathcal{G}_\\pi ", "062f64ee5db3f26b9bacfdf42b319493": " [N_2O_2] = K_1 [NO]^2 \\,", "062fd1dbe57ab537da02a03eeb0b1dda": " \\nabla\\times\\left(\\psi\\mathbf{A}\\right)=\\psi\\nabla\\times\\mathbf{A}-\\mathbf{A}\\times\\nabla \\psi ", "0630366d4daa8203300da7fde1b23eab": "\\alpha\\to 0", "063058ce56f748f91c2e666d35d36ff7": "w = x_{i_1} x_{i_2} \\cdots x_{i_n} \\ ", "0630ca48e0da4c93bbb613b1c29eb351": "f(x):\\mathbb{R}{\\rightarrow}\\mathbb{R}", "063153bce09b19630b96a3cafebad55f": "r={{h^2}\\over{\\mu}}{{1}\\over{1+\\cos\\theta}}", "0631b96b08c4bffdfe4d9abc6d7b2970": "\\tbinom63", "0631ce95efa824617f2196451eb7ce04": " c_1 \\sqrt{\\frac{\\log N}{N}}.", "063220b1e41ecb6061d8249c3450ef1f": "P(D_x) \\, e^{i (x - y) \\xi} = e^{i (x - y) \\xi} \\, P(\\xi) ", "063256969a21b46c2fd28cf77583a2b9": " q_K : V\\otimes_L K \\to K ", "06325e5f1e0b49e9844641fd4675cd46": " V_b = V_0 + \\left (\\acute{V}_0 - V_0 \\right)\\hat{T} ", "063291bdce4eb10fd02bc01071a91ffe": "M_{00}", "0632c0a62e5cca50c3fe63e3552f25d8": " \\| r_{k} \\| \\rightarrow 0 ", "0632c3741e6b5a3fa721280b0a00ebe1": "A f (x) = \\tfrac1{2} \\sum_{i, j} \\delta_{ij} \\frac{\\partial^{2} f}{\\partial x_{i} \\, \\partial x_{j}} (x) = \\tfrac1{2} \\sum_{i} \\frac{\\partial^{2} f}{\\partial x_{i}^{2}} (x)", "0633115c0def5ce26a9247e92e2b9c19": "\\vec{F}(\\vec{r})", "06333472307bc5773235531ccafe47ea": "\\textstyle m = v \\oplus H_2\\left(e\\left(d_{ID}, u\\right)\\right)", "0633d94eb9029d87df669cc2abb9bec4": "L^2/mr^3", "0633ff34c1b67102945ea800db03c001": "[x]_P,", "0634484bb068bfd017b502939afa9f4c": "\\nu_{22}\\sigma^2", "063462b0140315805f1159656326c24e": "\\| \\|^2 ", "0634636bdf1467c4314c6ee26d0faee1": "\\mbox{Tor}_p(M,N) \\cong E^\\infty_p = H_p(T(C_{\\bull,\\bull}))", "0634db24f94b09d50d8fd583f7767149": " \\mathbf{A}\\mathbf{x} = \\mathbf{b}", "06352f73ce4d7862f4f367d1125ceec1": "A_{0}+\\displaystyle\\sum_{n=1}^{\\infty}(A_{n} \\cos{nx} + B_{n} \\sin{nx}).", "06353d181a478e98fe7d096dac2832bc": " i =1 - (1 - \\frac{1}{n})\\alpha ", "06357f4aed117fffde206b2b6b99a083": "r_{\\mathit l \\mathit l^{\\prime}} ", "0635983bc182111b1d187d32c25f1aa3": "F \\subseteq F^\\prime", "0635e7adba5b91d7bf56394edf74dee4": "\\sum\\limits_{i=1}^\\infty a_i=A \\in G", "063664973c564954cb2c5df9f51dcb51": "[A,[B,C]_D]_D=[[A,B]_D,C]_D+[B,[A,C]_D]_D.", "06369b8178473296e74994e82b560d19": "0 = \\int_{\\partial N} \\sqrt{-g} \\ \\xi^{\\mu} T_{\\mu}^{\\nu} \\ \\mathrm{d}^3 s_{\\nu} = \\int_{\\partial N} \\xi^{\\mu} \\mathfrak{T}_{\\mu}^{\\nu} \\ \\mathrm{d}^3 s_{\\nu}", "0636e6ee3577c00d82887a3ac9b7d159": "x(0)", "0636fd78a288331071f745ef8dc93b55": "\\left(\\frac3{F_n}\\right)=-1", "06373818e93a19cae644f7ae80a93704": "D_N = A_N - A_{N/2}\\,", "06375828f8aad83b0b94d0bbd47f59dc": "O(m\\sqrt n\\log n)", "06375ff0aa87888a6a812f49c284cc90": " V = n R T / P\\,", "06377dac0f6eced39d3a1e40accfd768": " \\epsilon = 23.439^\\circ - 0.0000004^\\circ n ", "06378f09ca31cdcd679a23af53f7cbbf": "\\ (u, v) \\not \\in E", "0637b03526afa1d3c13e875c1362db58": "I \\to \\infty", "0637da5f81ac21c91ad10834e8f9a76c": "M_X^c + (\\overrightarrow{XG} \\times mg - \\overrightarrow{XG} \\times ma_G - \\dot {H}_G) = 0", "06380e026e2872cb9a9d810c0faca93b": "Re[\\lambda]<0", "06382b959ddcd48f17456b523e2174be": "\\{X\\}", "0638318d29c7a65a7bb96b00246f91f0": "y^{k}", "063839ff99e1701de1eaabd3fbe2a375": "(x^2-y^2)^3 + 8y^4+20x^2y^2-x^4-16y^2=0.", "063841e36aa49cb4034c792b94bdfac0": "\\left(\\mathbf{A} + \\mathbf{UBV}\\right) \\left( \\mathbf{A}^{-1} - \\mathbf{A}^{-1}\\mathbf{UB}\\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right)^{-1}\\mathbf{BVA}^{-1} \\right) ", "0638d22c12fd921a79d40362e1145749": "\\pi_1(X)/p_{*}(\\pi_1(C))", "0638d77c443cb5665c72a4f6c778bfe3": "k\\rightarrow \\infty.\\,", "063910a53398da0628fb1aea3d87e2fd": "\n{\\langle p \\rangle} = \\beta_\\text{max}\\left (1 + \\kappa^2\\right) \\epsilon \n \\left({1 - \\epsilon_B - \\epsilon}\\right)^2 G(\\epsilon) \\left(B_\\text{max}\\right)^2.\n", "0639331d8257eed848c99b4b94feda72": "\\mathbf{P}(t)=\\varepsilon_0 \\int_{-\\infty}^t \\chi(t-t') \\mathbf{E}(t') \\, dt'.", "06394fe2c6de92e648258ceb26e00745": "f_{\\Delta E}", "0639659520fe2a5bf59a5f1a30161ad9": "\\int_X f\\, d\\mu = \\int_X f e ^{i \\theta} \\, d |\\mu|", "0639ad6e78459f1b637d5ab558a8e6b6": "\\tilde{K}_\\pm \\ \\stackrel{\\mathrm{def}}{=}\\ K_\\pm / (K_0 \\cap K_\\pm)", "0639c979079363949c258a4efb310bb2": "\\begin{matrix} {1 \\choose 1}{11 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "0639f0fbf94d30efe14c4758e1f15aed": "\\omega(y|\\alpha,\\beta) = \n\\begin{cases}|y|^\\alpha\\left[1-i \\beta(\\tan \\tfrac{\\pi\\alpha}{2})\\mathbf{sign}(y)\\right]& \\alpha \\ne 1\\\\\n|y|\\left[1+i \\beta \\tfrac{2}{\\pi} \\mathbf{sign}(y)\\ln |y|\\right] & \\alpha = 1\\end{cases}\n", "063a05f2356036fe0a763ccb577ef79a": "B^x value \\left ( O,t \\right ) = \\left [ indexpartition \\right ]_2 + \\left [ xrep \\right ]_2", "063aa99b954878147d700c12864f8c9e": "\\mu\\ne\\nu", "063ac6c62425d6ffbdfe820371e2bc37": "\\; D_{\\mathrm{REE}} (\\rho) = \\min_{\\sigma} S(\\rho \\| \\sigma)", "063ae6fb13ea090df90cdfb17ec34e4e": "\\Gamma(x)=(\\alpha^8x-1)(\\alpha^{11}x-1).", "063b55bf6e50e78c8cfcbba464256e8b": "I_A = \\frac {\\pi} {2g} \\int_0^{T_d} a (t)^2 dt ", "063b6462ec5e3632c8e2b3df3eff3d60": "\\pi \\colon E \\to B\\, ,", "063bd81d17a4f96befa771de0625676d": "J^\\mu = \\frac{i\\hbar}{2m}(\\psi^*\\partial^\\mu\\psi - \\psi\\partial^\\mu\\psi^*) \\, . ", "063c09eb13b64940701faf1a3dc98c95": "{ \\mathbf{ \\tau } = {d \\mathbf{ L} \\over dt} }", "063c2426e7dbf82691c1d8c2706ae60d": " H = \\int {\\mathbf A}\\cdot{\\mathbf B}\\,d^3{\\mathbf r} ", "063c3a99991296354fd84c499074e27a": "f_p(x)", "063c9a6662fc69d6492027c45d75758d": "\\mathbf{H}_\\mathrm{eff} = \\frac{2A}{\\mu_0 M_s} \\nabla^2 \\mathbf{m} - \\frac{1}{\\mu_0 M_s} \\frac{\\partial \nF_\\text{anis}}{\\partial \\mathbf{m}} + \\mathbf{H}_\\text{a} + \\mathbf{H}_\\text{d}", "063ca26489809732d289ee06eb8552bd": " \\scriptstyle *:A\\times A\\to \\mathfrak{G} ", "063cb646cf29e1d4518894d1cc840635": "\\mathbb{E}(W_i) = \\frac{1+\\rho_i}{2} \\mathbb{E}(C) + \\frac{(1+\\rho_i) \\text{Var}(C_i)}{2 \\mathbb{E}(C)}", "063cc80c42fafab7ea390184547cd686": "\n\\begin{align}\np(\\mathbf{X}\\mid \\mathbf{Z},\\mathbf{\\mu},\\mathbf{\\Lambda}) & = \\prod_{n=1}^N \\prod_{k=1}^K \\mathcal{N}(\\mathbf{x}_n\\mid \\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k^{-1})^{z_{nk}} \\\\\np(\\mathbf{Z}\\mid \\mathbf{\\pi}) & = \\prod_{n=1}^N \\prod_{k=1}^K \\pi_k^{z_{nk}} \\\\\np(\\mathbf{\\pi}) & = \\frac{\\Gamma(K\\alpha_0)}{\\Gamma(\\alpha_0)^K} \\prod_{k=1}^K \\pi_k^{\\alpha_0-1} \\\\\np(\\mathbf{\\mu}\\mid \\mathbf{\\Lambda}) & = \\prod_{k=1}^K \\mathcal{N}(\\mathbf{\\mu}_k\\mid \\mathbf{\\mu}_0,(\\beta_0 \\mathbf{\\Lambda}_k)^{-1}) \\\\\np(\\mathbf{\\Lambda}) & = \\prod_{k=1}^K \\mathcal{W}(\\mathbf{\\Lambda}_k\\mid \\mathbf{W}_0, \\nu_0)\n\\end{align}\n", "063cde69390fa7062c5bda566cfce138": " \\cos E = \\frac{x}{a} = \\frac{ae +r \\cos \\theta}{a} = e+ (1-e \\cos E) \\cos \\theta \\ \\to \\cos E = \\frac{ e + \\cos \\theta }{1 + e \\cos \\theta } ", "063d0555ec921f06621d113cf250fa0c": "\\int_a^b \\frac{d}{dx}\\left(u(x)v(x)\\right)\\,dx = \\left[u(x)v(x)\\right]_a^b", "063d3d08b5833130d9c19cc910b8852c": "[\\hat{x},\\hat{p}]=i\\hbar", "063d582f0b7db6d715605e5d5b186203": "X \\dot = Y", "063d7b9a3cd93212199947067732dcf1": "5. \\; \\; 2\\mathrm{O}_3 \\; \\xrightarrow{h \\nu } \\; 3\\mathrm{O}_2 ", "063d8e899312d0ab03afd72c8f1a16de": "\\mathcal{P} = \\frac{\\mu A}{\\ell}", "063d9c323d1afbafecb1fa625f1d8dbb": "f(x)=L^{-1}(4x-1).\\, ", "063da2738dd86cb86b09b041b907e19a": "\\textstyle x\\in\\left( a,b\\right) ", "063dc8f4d1520bd2822ad410a5085574": "x_i^*\\in[x_{i-1},x_i]", "063df27909b8117eeed224b2b58f2f69": "\\lim_{n \\rightarrow \\infty} \\left( \\max_{a \\leq x \\leq b} | f(x) -P_n(x)| \\right) = 0.", "063e3063e86db87974de028a9dc464d1": "F_{-n} = (-1)^{n+1} F_n. \\, ", "063e806ef9a3b14c27155fefddde363b": " \\boldsymbol{\\beta}_{ut} = \\left(\\beta_{ut}^1, \\dots, \\beta_{ut}^n \\right) ", "063e9a76af25b78ab52ccb69cf835f3c": "(y^m u^r)^{(p-1)(q-1)/r} \\equiv 1 \\mod n", "063e9b555eaa456479aa5abc50feb3e8": " \\psi\\rightarrow \\psi e^{i2ct}", "063f318a4eab2f244bd68ab13d9b64ef": "D_{kn} = \\frac{2}{N} \\cos\\left(\\frac{nk\\pi}{N/2}\\right) \\times \\begin{cases} 1/2 & n=0,N/2 \\\\ 1 & \\mathrm{otherwise} \\end{cases}", "063fa15417cbd7b6b5a5e0087e069e95": "\n p_0 = \\cfrac{2 a E^*}{\\pi R} ~;~~\n p_0' = -\\left(\\cfrac{4\\gamma E^*}{\\pi a}\\right)^{1/2}\n ", "063fa2259f768d9e76c6ded794f42d06": "v_1, v_2, \\cdots, v_m", "063fca31a7350b0628b274d7560a8876": " \\operatorname{get-lambda}[p, p = \\lambda f.\\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x)] ", "06406f401c9db3d2015a9130b3afc883": "(dx)^2=C_{KL}dX_KdX_L\\,\\!", "0640a6b2b362e804d4520747d2b41256": "u^\\prime = u\\,", "0640b6347878b4e87d5ac0838bc3bcbe": "\\mathfrak{so}_{4} \\cong \\mathfrak{sl}_2 \\oplus \\mathfrak{sl}_2 ", "0640ce2434f184907371d2ac5d917cd5": "x=\\sqrt{t}", "0640d02d6a69d9ca4f41cd8fdb8240b6": "\\mu:\\mathcal{X} \\to \\mathbb{R}", "0640d612b34d116370eafd65b020a831": "G_n^{(1)}", "0640e12048949ecf87c8ccf163d8d404": "\\{ \\tau \\leq t \\}", "0640e5ee3912c73671421e2d89a33b92": "\\theta =180{}^\\circ ", "0640eb22ee0e9aaaa5055a6daed22104": "q_x = - k \\frac{d T}{d x}", "06410e3cc2dfd60b883efabddee06c33": " f(i) = \\beta_0 + \\beta_1 x_{i1} + \\cdots + \\beta_p x_{ip},", "064118dbe891d9f20688a8bf141bd158": "\\left ( \\frac{a}{b} \\right )", "0641205e5b9c3658f86787b4ccf76ca3": "\\langle J,J_z|\\vec \\mu_J|J,J_{{z'}}\\rangle = g_J\\mu_B\\langle J,J_z|\\vec J|J,J_{z'}\\rangle", "064145950df9574945eaa3894624a044": "\\displaystyle \n= \\sum_{\\sigma, \\tau = 1}^n\\left({\\partial^3 F \\over \\partial t^\\alpha t^\\nu t^\\sigma} \\eta^{\\sigma \\tau} {\\partial^3 F \\over \\partial t^\\mu t^\\beta t^\\tau} \\right)\n", "0641727c9e3546b2b9c5b335ea73a3b5": "y_i \\succsim_i x_i", "0641f75e5ef17d434b22e363f6e14dfd": "_{p\\leftarrow q\\,}\\!", "0642209c44c24f23c9172a30286eb802": "\\ 1/x", "06424d3fefb5dbef1d4af971d1e97773": "e^{i \\pi} = (e^{i \\delta})^{\\pi / \\delta},\\,\\!", "0642796128dd4c58f27ed1bcdef2c71d": "\\Delta H^*_{ab}", "06429e91ca50db64fc70570ed66f7380": "q_3", "06429fa86d535a910037f92a580383e4": "(x^2+y^2)^2=2(x^2-y^2)\\,", "0642cc739b2f8c0404acb0c2ac9f5ace": "0,1,\\ldots,n", "064310c16ba9839ac791f793bdd726b7": "\\hbar=c=1", "06431b49bbcb1ac96205e734d1c52fb8": " p(\\theta | I_t, O_{fg}) ", "0643d743ed71e86bd64f547f6e80308a": "\\left(\\frac{d}{dx}\\right)_q f(x)=\\frac{f(qx)-f(x)}{qx-x}.", "06440725ef44eaeb882e81dacf25fb68": "q1=q", "0644122b838b7185dacf29793cade3bd": "\\theta(\\lambda)", "06448fe7815d407ad7ff6bfe9579a7d6": "(f * s)(t)", "0644b871dfafa8cd458b32f664661297": "\\,I^n(t)\\,", "06450841c23388142835699b0ac86913": "\nP_{1D}(x)dx=\\sqrt{\\frac{3}{4{\\pi}Ll_p}}\\exp{\\left(-\\frac{3x^2}{4Ll_p}\\right)}dx \\,\\,\\,;\\,\\,\\,\\,\\,\\,\nP_{2D}(R)dR=2{\\pi}R\\frac{3}{4{\\pi}Ll_p}\\exp{\\left(-\\frac{3R^2}{4Ll_p}\\right)}dR\n", "06450d9fbdf8d1f569a98d268b6d054e": "\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n\\begin{bmatrix}\n x \\\\\n y\n\\end{bmatrix}.\n", "064518314a6112b0e1a062bc7e818e0f": "A^{-1} \\cdot B^{-1}a_{i_1}^{\\varepsilon_1}B\\cdots B^{-1}a_{i_L}^{\\varepsilon_L}B", "064579fefd9a750d9cfbda846d2eb899": "\\mbox{VBN} = 14.534 \\times \\ln\\left[ \\ln(\\nu + 0.8) \\right] + 10.975\\,", "064605eeb7caaf26a5c67dc67b015871": "\\displaystyle{(g,G)\\cdot (h,H)=(gh,K),}", "064642b96331839dec506e878d02cd46": "\\rho_{\\text{Electric dipole}}(\\mathbf{x},t) = \\frac{-i k}{4 \\pi \\epsilon_0}\\frac{e^{i k r - i \\omega t}}{r}\\mathbf{n}\\cdot\\mathbf{p}", "06464d647faa419e1912c0c22bb1f263": "i\\colon A\\hookrightarrow X", "064663015cf54c0606d3673d21187cbc": "-\\overline{v'T'}", "06467448db2258a82dbf67043db3ced2": "S_0''=0=o(S_0')\\,", "064695238557467e60cb5e053aa0fe65": "Q_1 - Q_2 > 0", "0646a890a9669fb4da9d37527102896e": "u = u_{0} + \\Phi(\\pi - \\pi_{t})", "0646dadd49af164112729c1e9e880cf2": "d(O_{r}, O_{n}) <= r(O_{r})", "06471ded130d2e5ce109d3167696c49d": "\\Omega^*(M) = \\bigoplus_{k=0}^\\infty \\Omega^k(M).", "06472b40bd4e5d9619b9b1d15d33bc04": "\\operatorname{ess.inf}", "0647e246b9aa2a45be714732516cd13f": " P(\\partial_t,\\xi)G(t,\\xi) = 0, \\; \\partial^j_t G(0,\\xi) = 0 \\; \\mbox{ for } 0\\leq j \\leq m-2, \\; \\partial_t^{m-1} G(0,\\xi) = 1/a_m. ", "0647f845a1fc3218780e47d701d32dad": "g: Z\\to V", "064832debd50461e7c43ba7fbaab62d1": "dG= \\sum_{j=1}^m \\mu_j\\,dN_j = 0", "0648373fbe646cf8a58abccc71e691a0": "dE = \\delta Q - \\delta W,", "0649273b46cdfccc71cc410bfdd07c09": "\\operatorname{fnchypg}(x;n,m_1,N,\\omega) = \\operatorname{fnchypg}(n-x;n,m_2,N,1/\\omega)\\,.", "0649b1d9ebcba60c8e487b5ff077fd37": "\\mathcal{C} \\times \\mathcal{D}", "064a39ffb4ce76f9f7888c745b194ffb": "f: \\mathcal{X} \\to \\mathcal{Y}", "064a45240a7d91dd83b638733dd88f65": " u(\\theta) = \\frac{ GM }{h^2} + A \\cos(\\theta-\\theta_0)", "064a5481f3b33fe02b5af01dcfce74a3": "\\sum n_P P \\to \\sum n_P P.", "064a7413ce1f710c1ab5175077b85716": "S(t) = \\Pr(T > t)", "064a818b99190e7f13fa62e568d6ad40": "S(P) \\geq S(Q)", "064b035ed9383c2e45edb8795b459dc2": "\\bar{T}_{\\ell_1 \\ell_2 \\cdots \\ell_q}^{k_1 k_2 \\cdots k_p} = \\mathsf{L}_{i_1}{}^{k_1} \\mathsf{L}_{i_2}{}^{k_2} \\cdots \\mathsf{L}_{i_p}{}^{k_p} (\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_1}{}^{j_1}(\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_2}{}^{j_2} \\cdots (\\boldsymbol{\\mathsf{L}}^{-1})_{\\ell_q}{}^{j_q} T_{j_1 j_2 \\cdots j_q}^{i_1 i_2 \\cdots i_p}", "064b59a42ba406a97b723db2b4277085": "T(X_1^n)=\\overline{X}=\\frac1n\\sum_{i=1}^nX_i", "064b7f588deebf178356485a12196fb8": "\\sqrt{\\frac{2}{3}}\\!\\,", "064bd950f9ba6d5ad4f577ece6659e46": "\n k = \\cfrac{5 + 5\\nu}{6 + 5\\nu}\n ", "064bd958e3e9578a2676150e5e8bd6d0": "c_{jk} = \\left[W_\\psi f\\right]\\left(2^{-j}, k2^{-j}\\right)", "064c25a4ec47eb9d7119f34745f978dc": " +48(x^2 + y^2)(x^2 - 3y^2)^2 + (x^2 - 3y^2)x[16(x^2 + y^2)^2 - 5544(x^2 + y^2) + 266382] = 720^3.", "064c678d55189edf8539d54cb383f358": "i\\leq n", "064c9727b2531f2a9a150cc5a4a815d7": "\\displaystyle Wg(1^3,d) = \\frac{d^2-2}{d(d^2-1)(d^2-4)}", "064cc9865887da54d41d095b13f33d89": "ay \\bmod 2^w", "064d4bf58e4b377ddc029af6979cdca4": " \\textbf{A} ", "064d566169bd649ee5862d04310e3ff5": " \\mathit{XP} + Y \\longrightarrow \\mathit{XY} + P_i", "064d8a5614ac0a0aac343fd50a644849": "\\widehat{\\sigma}", "064dfa1b3bea950017e1f5042a957127": " \\log_b(xy) = \\log_b(x) + \\log_b(y) \\!\\, ", "064e113d61e97c3b00cd1efd7434bbe5": "x_2=3", "064ed431e31cd627e97ea3addb1493b6": "e^{-\\pi z^{2}}", "064efb9d9fad29c9d848fbb4a42ccec3": "x_{n_1,n_2}", "064f0e3bfda1a64772d3eb4307075b2c": "\\nu \\ll \\omega", "064f3ae713cdc9d4788fa99f1cbee672": "\\pi_1 (\\mathbb{H}/\\Gamma)", "064f80c126d8f90c294a28d0d5205e5b": "\n\n\\sum_{n=0}^{\\infty}\\frac{n!L_{n}^{(\\alpha)}(x)L_{n}^{(\\alpha)}(y)r^{n}}{\\Gamma\\left(1+\\alpha+n\\right)}=\\frac{\\exp\\left(-\\frac{\\left(x+y\\right)r}{1-r}\\right)I_{\\alpha}\\left(\\frac{2\\sqrt{xyr}}{1-r}\\right)}{\\left(xyr\\right)^{\\frac{\\alpha}{2}}\\left(1-r\\right)},\\quad,\\alpha>-1,\\left|r\\right|<1.\n\n", "064fcd42f1e9f514e0fd694aa5c4a2fa": "T_{ij}", "064ff7725e1de916ba94bfc251c551d5": "w_0\\left(t-\\tfrac{(N-1)T}{2}\\right)\\cdot \\operatorname{rect}\\left(\\tfrac{t-(N-1)T/2}{NT}\\right),", "065070c1a970b7bbf09e708029624ccd": "\\text{Liquid} \\xrightarrow[\\text{cooling}]{\\text{eutectic temperature}} \\alpha \\,\\, \\text{solid solution} + \\beta \\,\\, \\text{solid solution}", "065091dc156b96ef2bc9f867ea8d9a90": "\\scriptstyle x \\oplus y = XY", "0650e38c5dfbf20f831af30d7ac69f99": "L_{[\\omega]}^{n-1}: H_{DR}^1(M) \\to H_{DR}^{2n-1}", "0652402621111239f416a1862561d031": "\\ \\mathbf b=0", "0652623115b46f2a9ccf0f54129c1506": "u^{\\alpha} = (1, 0, 0, 0) \\,,", "06529432da242c4cc04f5870bab2cd37": "\\pi_1 (X\\vee Y) \\cong \\pi_1(X) * \\pi_1(Y).", "0653081814d4d647a1bafb2d96b99591": "\\Big( \\pi \\models \\phi_1 \\Rightarrow \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big(\\pi \\not\\models \\phi_1 \\big) \\lor \\big(\\pi \\models \\phi_2 \\big) \\Big)", "06531529788d2b229a9977c605e7607a": "\\frac{\\log_2 N\\,\\log_3 N\\,\\log_5 N}{6}.", "06534b4d5b8d6f8c690344a0f0ef53d3": "\\begin{align}\nL_{x}&\\approx I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta)\\,,\\\\\nI_{2}\\ddot{\\alpha}&\\approx (L_{x}\\Omega\\sin\\delta+I_{2}\\,\\Omega^{2}\\sin^{2}\\delta)\\,\\alpha\\,.\\end{align}", "065375e324898a3a1d67dab3a2452a37": " -\\dot{S}(t) = A^\\mathrm T(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^\\mathrm T(t)S(t)+Q(t),", "0653fd4adefb8c3762a1ce0e81d50d2d": "x=x_{s}(t)", "0654029124451d6c93acee2f2456e142": "C_{QY} = \\frac{\\epsilon_0}{\\lambda_0} = 3.649\\;2417\\;\\mathrm{F/m^2}", "0654865eb641896b269a22f105ba83a3": " \\tau := \\sup \\{ t \\in [0,1] : W_t = 0 \\} ", "0654ced79aa19a4f4cb123a26bef9e4e": "\\sigma:A\\rightarrow \\mathrm{End}(V)", "0654d4bb827f9e0d5e9577c634df1dc2": " p\\in [-1,1]", "06550af4bcc791b2d570e461baefba01": "\\frac{{N-K \\choose n} \\scriptstyle{\\,_2F_1(-n, -K; N - K - n + 1; e^{t}) } }\n {{N \\choose n}} \\,\\!", "06559d4446604dad285c25f86a8b505b": " \\begin{align}\n\\operatorname{E} \\operatorname{tr} e^{\\sum_{k=1}^n \\mathbf{X}_k} & = \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-1} \\operatorname{tr} e^{\\sum_{k=1}^{n-1} \\mathbf{X}_k + \\mathbf{X}_n }\\\\\n&\\leq \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-2} \\operatorname{tr} e^{\\sum_{k=1}^{n-1} \\mathbf{X}_k + \\log(\\operatorname{E}_{n-1} e^{\\mathbf{X}_n} ) }\\\\\n&= \\operatorname{E}_0 \\cdots \\operatorname{E}_{n-2} \\operatorname{tr} e^{\\sum_{k=1}^{n-2} \\mathbf{X}_k + \\mathbf{X}_{n-1} + \\mathbf{\\Xi}_n} \\\\\n& \\vdots\\\\\n& = \\operatorname{tr} e^{\\sum_{k=1}^n \\mathbf{\\Xi}_k}\n \\end{align} ", "0655a9a3d09cb12d562a6a71520bcfc6": "\\mathbf{x}_*", "0655cce591173476dca8441b58faa7c0": " A\\otimes_K K_v \\simeq M_d(K_v). ", "0655cd5b52625898a4522c700969124b": "m(x_i)=\\frac{1}{N(\\mathbf{h})}\\sum^{N(\\mathbf{h})}_{i=1}Z(x_i)", "065646299b64e9db2a42e4a70c2044e4": " \nE[L(t)] - E[L(0)] + V\\sum_{\\tau=0}^{t-1}E[p(\\tau)] \\leq (B + C + Vp^*)t\n", "06566a283d2455c50bf4e6cc2613ffa1": " f_Y(y | \\theta, \\tau) = h(y,\\tau) \\exp{\\left(\\frac{b(\\theta)T(y) - A(\\theta)}{d(\\tau)} \\right)}. \\,\\!", "06567105e3af8240346209deed923e2c": "\\sqrt {S} = \\sqrt{\\frac{\\vert S \\vert + a}{2}} \\, + \\, \\sgn (b) \\sqrt{\\frac{\\vert S \\vert - a}{2}} \\, \\, i \\,.", "065672f2121201154ac873c04e7aaf53": "if\\, (C^{cand} \\neq \\emptyset)", "0656b0158a90a29c81bde47bd93357c0": "\\!\\mu_2(v_3)", "0656eefc86eb14c91fadefc25281bb41": " F(\\epsilon) = \\frac{1}{e^{(\\epsilon-\\mu) / k T} + 1} ", "0656f87fdfc7f7b7c0a52cf06a06bf00": "r(x) = \\sum_{a} r_a \\emptyset ^a (x) = r_a \\emptyset ^a (x) = x", "06571844a2a521364f8605899e26ed06": "H = h_1 h_2 h_3", "0657241774278b7f2381b41ac559b03b": " \\text{PV} = \\text{FV}\\cdot e^{-rt} ", "0657298c5bcea3a7c5842bfc838f74c2": "K=0,", "065729da1dbd83b81b1b168c928169b9": "O(n \\log h)", "06575ab091aca463a8dc616775352dab": "\\sum_{i=1}^n (x_i-\\mu)(x_i-\\mu)^\\mathrm{T} = \\sum_{i=1}^n (x_i-\\bar x)(x_i-\\bar x)^\\mathrm{T} = S", "0657967352cd8a9c703d53b9c13fe4dd": "L_{g}L_{f}^{k}h(x) = 0 \\qquad \\forall x", "0657d4e54d9e5e0d0a393e895b261708": "{\\rm R} + {\\rm L} \\to {\\rm RL}", "065801965a23a5923991a44e5fd950ae": "\\begin{matrix}\np \\oplus q & = & (p \\lor q) \\land \\lnot (p \\land q) \n\\end{matrix}", "065821342a09d802f7b40b0f6b9a88e1": "F(k+1) = f(F(k)) = f(G(k)) = G(k+1).", "065831348fd5839996fb2e09d9e8b681": "k_r^-", "0658bbf1015e2f0d90ca6440473c08c3": "\n\\begin{matrix} X_k= \\underbrace{\\sum \\limits_{m=0}^{N/2-1} x_{2m} e^{-\\frac{2\\pi i}{N/2} mk}}_{\\mathrm{DFT\\;of\\;even-indexed\\;part\\;of\\;} x_m} {} + e^{-\\frac{2\\pi i}{N}k}\n \\underbrace{\\sum \\limits_{m=0}^{N/2-1} x_{2m+1} e^{-\\frac{2\\pi i}{N/2} mk}}_{\\mathrm{DFT\\;of\\;odd-indexed\\;part\\;of\\;} x_m} = E_k + e^{-\\frac{2\\pi i}{N}k} O_k.\n\\end{matrix}\n", "0658bdbe1fcda6797f88e35351e12b9b": "\\begin{array}{l}\nf^1\\big(\\theta^1(t)\\big)=\\cos\\big(\\omega^1 t\\big), \nf^2\\big(\\theta^2(t)\\big)=\\sin\\big(\\omega^2 t\\big) \n\\\\\nf^1\\big(\\theta^1(t)\\big)^2\nf^2\\big(\\theta^2(t)\\big)\nf^2\\big(\\theta^2(t) - \\frac{\\pi}{2}\\big) \n=\n-\\frac{1}{8}\\Big(\n2\\sin(2\\omega^2 t)\n+\\sin(2\\omega^2 t - 2\\omega^1 t)\n+\\sin(2\\omega^2 t + 2\\omega^1 t)\n\\Big)\n\\end{array}\n", "0658d2b10b2036c4126666ff7af50dbb": "a_1,\\ldots, a_n", "065932afb82fb41aaa6f6369d7bdad6b": "DPV= \\int_0^T FV(t) \\, e^{-\\lambda t} dt \\,,", "06594abcd0cdb4baeba7da6a799ec010": "\\pm\\frac{\\tan \\theta}{\\sqrt{1 + \\tan^2 \\theta}}\\! ", "06594f0602e88da2137a52fa52f46333": "M(x) \\cdot x^{n-1} = Q(x) \\cdot K'(x) + R(x)", "0659f3831ebaefaa6fc92860933b3c69": "\\mathcal{V}\\,", "065a26e3c7e0b9576696f8564ceeb46b": " \\displaystyle{(e^\\xi,e^\\eta)=e^{(\\xi,\\eta)}.}", "065a6bc421f03312e07d3cc59ef6a059": "a^2+1", "065af87168ddb3765d35cb956546ff57": "\\sum_{i=1}^k\\sum_{j\\in N_i} w_{ij} x_{ij} \\leq W,", "065b307962ae9ec00ac3c9ed98f9cd29": "E_{inc}", "065b6bee27a66e9bb1bef8853aef3946": "I_\\mathrm{ion}(V,w) = \\bar{g}_\\mathrm{Ca} m_{\\infty}\\cdot(V-V_\\mathrm{Ca}) + \\bar{g}_\\mathrm{K} w\\cdot(V-V_\\mathrm{K}) + \\bar{g}_\\mathrm{L}\\cdot(V-V_\\mathrm{L})", "065b7cd54a67f380c3d5ee0fcb6898fd": "\\omega\\in S,", "065bccda0bb62414e07c0279c1ba8d9c": "q'=h^sg^y", "065bdd18294611a861191a74e16f8502": "\\mu_m = ", "065c472ad60d01cb605c8863aac6ab62": "l_a n^a=-1=l^a n_a\\,,\\quad m_a \\bar{m}^a=1=m^a \\bar{m}_a\\,,", "065c5afdef0432fb8fa2ac1d7b4626a9": "\\,\\!\\theta_{n,k}", "065c7d5a2d7df51c6aaaf7f33e9b137d": " \\Gamma_{ij}{}^k = \\cfrac{\\partial \\mathbf{b}_i}{\\partial q^j}\\cdot\\mathbf{b}^k = -\\mathbf{b}_i\\cdot\\cfrac{\\partial \\mathbf{b}^k}{\\partial q^j} ", "065c7fe485ac4442ec7d169b329c6636": "f\\left(r\\right) = \\frac{\\left(n - 2\\right)\\, \\mathbf{\\Gamma}\\left(n - 1\\right) \\left(1 - \\rho^2\\right)^{\\frac{n - 1}{2}} \\left(1 - r^2\\right)^{\\frac{n - 4}{2}}}{\\sqrt{2\\pi}\\, \\mathbf{\\Gamma}\\left(n - \\frac{1}{2}\\right) \\left(1 - \\rho r\\right)^{n - \\frac{3}{2}}} \\,\\mathbf{_2F_1}\\left(\\frac{1}{2}, \\frac{1}{2}; \\frac{2n - 1}{2}; \\frac{\\rho r + 1}{2}\\right)", "065c8c145cdc659a52d38d79b8ad93ee": "\\scriptstyle f: [0, T]\\longrightarrow X ", "065c90b0effda6c8a4ffcb4ba5e07d3c": "\\kappa ", "065cba6033cd20888b9121ccda1c637b": "(x+1) \\ge 4\\,\\!", "065cd9c9acf9bcd888f6b0880c54b0e9": "m_n\\ :=\\ {0}^{256\\ -\\ \\mathcal{j} m_n \\mathcal{j}} \\mathcal{k} m_n", "065d060554456cdb96fd2f6413e448b0": "f(x)/g(x)", "065d544472d087acac0094a9bc44e9e8": " \\beta x f_0 \\ll 1 ", "065e1647d7aec79e787d42b59bc7ce82": " \\epsilon> 0", "065e40dd8dbb27b2a8088e3687149788": "I_x = I_y = \\frac{m r^2}{2}\\,\\!", "065e6069c6786461750b1077bb083b29": "x = \\sum_{i=1}^{N} S_i", "065ed001d5e92678b71b01fd95fc0623": "\\exists\\lambda\\in\\sigma(A): |\\lambda-\\mu|\\leq\\|\\delta A\\|_2", "065ed0c542bedddf5eecedad6cdbfdbd": "\\{A_1,\\ldots,A_n,\\neg B\\}", "065edee3cddf0c96c2358169e219412a": "m_1=(4x+m)", "065f10721e490b358283f4cda7f9cfad": "\\forall x f\\ x = x", "065f53c5ff9b8139c5562ca4dfa66ee5": "\\forall i: |\\gamma_i|\\leq i", "065f73e8b5eb80210019f2fb94218375": "\n\\operatorname{cov}(X_i, X_j) = \\frac{\\theta_i \\theta_j}{(a-1)^2(a-2)}, \\qquad \\operatorname{cor}(X_i, X_j) = \\frac{1}{a}, \\qquad i \\neq j.\n", "065f9fe034d3e8ef22183ff943ce6d2e": "\\vec w \\cdot \\vec \\mu_{y=0} ", "065fa0367342168be7eea9e42c03a454": "\\theta_1 \\in \\Theta_1", "065fffaa6113a8311f7abb23bdd15688": "z = \\exp(i t)", "066077bc473df482eb54bfbd841d892b": "\\alpha(a, \\, b) \\stackrel{\\mathrm{def}}{=} \\displaystyle\\sum\\limits_{c \\in A} f(a, \\, c, \\, b ) \\cdot \\sum _{d,\\,e \\in A} g(a,\\,d,\\,e) ", "0660be61967e90e49ec26230c2409a36": " {\\boldsymbol{L}_{k-1}} = \\left . \\frac{\\partial f}{\\partial \\boldsymbol{w} } \\right \\vert _{\\hat{\\boldsymbol{x}}_{k-1|k-1},\\boldsymbol{u}_{k-1}} ", "06610b8ddbd684f89a62227c032baf06": "\\mathbf{b_1}", "0661758bfa11e17b001b691146866560": "g: X \\to Y", "06617c3d95af696566773ac2b8a5989e": "\\sup_{\\eta>0} \\int_{-\\infty}^\\infty \\left |f(\\xi+i\\eta) \\right|^2\\,d\\xi = C < \\infty", "0661a980b1208be48c4a350b21ac02b8": "q_x = q_y\\,", "0661d9ddb8cf900ce1d2e896d3d955d0": "L = 20\\log\\left(\\frac{R}{R_0} + 1\\right)\\, \\text{dB}", "0662056719736cf5d0d5999aa4a4ee1d": "\\{ z : e^z = w \\} = \\{ v + 2k\\pi i : k \\in \\mathbb{Z} \\}", "066213b23a5e9281e0eacdc112fdb1cb": " \\|\\cdot\\|_{C^{k, \\alpha}} ", "0662605598c033c0dde3dea8e319a8b2": "{(2n)! \\over (n+1)! (n+1)!}", "0662ae0efdba6803461bd5f402f83232": "\n\\int_{-\\infty}^\\infty \\frac {\\gamma\\left(\\frac s 2, z^2 \\pi \\right)} {(z^2 \\pi)^\\frac s 2} e^{-2 \\pi i k z} \\mathrm d z = \\frac {\\Gamma\\left(\\frac {1-s} 2, k^2 \\pi \\right)} {(k^2 \\pi)^\\frac {1-s} 2}.\n", "0663c6feb5cc4ba436fd441e8858053b": "M:D\\rightarrow C", "06640de893d91e932365dc13a6399d28": "\\scriptstyle \\mathbf{\\nabla}\\cdot\\mathbf{\\sigma} \\,+\\, \\mathbf{F} \\;=\\; {\\mathbf{0}} ", "066418514c51fdc53b0e3419861a1fb5": "\\eta^{\\mu\\mu}", "06642454857236ba2faa1c45d2fb117e": "(\\gamma\\gg 1)", "066464cd43b120eef607ad821fe70b81": "\\vartheta(z|q) = \\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 + 2 \\cos(2 \\pi z)q^{2m-1}+q^{4m-2}\\right).", "0664735ed61d755a0aec900cc1e7b9f2": "\\operatorname{erf}", "0664b46389c4fcf1ba4d081976dc6cc9": "P(a_i^T(x) \\geq b_i) \\geq p, \\quad i = 1,\\dots,m ", "0664ddd782c0850852945bed0155ef58": "[*:*:0:\\dots:0]", "0665078bf784c47a904a52a59c74f033": "do(move(2,3),S_{0})", "066510e73bd9a053774f50bc7b4d0d6e": "Z_{0} \\approx 376.730\\ 313\\ 461\\ 77 \\ldots \\Omega", "06652f0828e48ad7c33fc94ecc5fcd6b": "\\textstyle \\sigma_k = M_{\\mathrm f} R^k (\\Delta)", "066557ff29a2c4b0590aba3fdd721c60": "\\frac{d}{d x}\\left(\\sin(x^2)\\right) = 2 x \\frac{d}{d u}\\left(\\sin(u)\\right) = 2 x \\cos(x^2)\\,", "066561356344af6c07cdbb6126f8d032": "\\sqrt{31}\\times\\sqrt{31}", "066563f838bfc88c66dfaf2644dd64f6": "\\vec{S}(n) = \\sum_{j=1}^8 b_j K_j^n\\vec{\\xi}_j.", "0665666dbcc9f49f1ccd97b513147dc7": "T_I", "066627838e3b7a9238618ba6dc14be3e": "{\\widehat{HV}}_3", "06666e1bb344f1eedfb6ea7ebda9f844": "Y=2k(\\phi(front)+\\phi(rear))=4k(\\theta-\\psi)+2k\\frac{(b-a)}{V}\\frac{d\\theta}{dt}+2k\\eta", "0666e08d1fd0f044f5a0b7008e449afb": "\\big\\{\\mathbf{F}_{\\alpha}\\big\\}_{\\alpha=1}^{M=N\\times{N}}.", "0666e4b33c71c6516d1b6b295f1b6d55": "\\arcsin( )", "0666fd617b7abb83c7d26d29d45d4fea": "k^{}_{}: ", "06670427b22277cd72dc40510011730d": " u_{tt} - u_{xx} = V'(u) ", "06673df311ebe5b0e2158bf8cda07674": "H=2h", "066758da027cd5480bd8a47a807632c6": "(n - 1)", "0667aa8259991dc1003ae3a7ef3ae3b8": "T= \\sigma_N/A\\sigma_0", "0667bfc2ffc9583d811ab4927f6f7dc0": "p = C^{-1}(-2 \\ln(p_1 p_2 \\cdots p_N), 2N) \\, ", "0667d925c7da0aa40a80f26cb23fcd13": "-x\\sin A + y\\cos A", "0667e4329ba13406226d5cea24b19455": "(\\nabla \\cdot \\mathbf v) f = \\left (\\frac{\\part v_x}{\\part x}+\\frac{\\part v_y}{\\part y}+\\frac{\\part v_z}{\\part z} \\right )f = \\frac{\\part v_x}{\\part x}f+\\frac{\\part v_y}{\\part y}f+\\frac{\\part v_z}{\\part z}f ", "06684da7018ebd3b6f0eae745842b787": "t \\propto x^2", "066850b2749b50590922dcb15469e15d": " {H_1 \\over H_2} = { \\left ( {D_1 \\over D_2} \\right )^2 }", "06685b068927716f1ce3908154ccb395": " \\frac{d^2\\theta}{d\\xi^2}+\\frac{2}{\\xi}\\frac{d\\theta}{d\\xi} + \\theta = 0 ", "066873313a2c794e852cf6adbd160bcf": "n_{1, t+1} = \\lambda n_{1, t}", "06688a9275a9fd584c2ba0fef2bb5a2b": "\\mathbf{x} = [x_1, x_2, \\ldots, x_{N_t}]^T", "066939d1ed61b458e6fcc10842a2d93f": " a^{m*2^k} + b^{m*2^k}.\\!\n", "06695c2683b56718e7253a916e120efc": " \\begin{pmatrix}\nF_\\text{x} \\\\\nF_\\text{y} \\\\\nF_\\text{z} \\\\\n\\end{pmatrix} = q\\begin{pmatrix}\nE_\\text{x} \\\\\nE_\\text{y} \\\\\nE_\\text{z} \\\\\n\\end{pmatrix} - q \\begin{pmatrix}\n0 & - B_\\text{z} & B_\\text{y} \\\\\nB_\\text{z} & 0 & - B_\\text{x} \\\\\n- B_\\text{y} & B_\\text{x} & 0 \\\\\n\\end{pmatrix} \\begin{pmatrix}\nv_\\text{x} \\\\\nv_\\text{y} \\\\\nv_\\text{z} \\\\\n\\end{pmatrix}", "0669b5a06de44528ae887d71ee31ad99": "(\\sin(\\alpha/2))^2\\,", "0669c91f79ea0489e96a0d277743c1bd": " x(t+1) = f \\left [ x(t) \\right ] \\approx \\varphi(t) = \\varphi \\left [ x(t)\\right ] ", "0669f5563d91e61b7916f60874730336": "\\epsilon_\\perp", "066a873d751b6c7c1227ff3ad7a3f235": "\\scriptstyle \\Delta_0", "066ab225e700949fd1fc61bd31b2ca29": "\\Omega(M,\\mathrm{T}M)", "066abd4ee06a1a15551d4d33435e9914": "t_{1} \\leq t_{2} \\implies \\mathcal{F}_{t_{1}} \\subseteq \\mathcal{F}_{t_{2}}.", "066b143b02ca298f94a0e4598f1206e9": " r = \\sqrt{\\frac{1}{2}(\\alpha^{2}-a_{21}\\alpha)} ", "066b3d5450fc5172d18d18e7ce7537f0": " z = x^{1/u} ", "066bb2257a55de86e56d9abaf7981d1d": "\\tan \\psi = -\\cot \\theta,\\, \\psi = \\frac{\\pi}{2} + \\theta, \\alpha = 2 \\theta.", "066c18b4d2049b5dded8990995d51334": "\\beta_{n}(T_e)", "066c1f04ac58a8d7633a539616d0e1a9": "\\tfrac {1}{6} \\pi^2 \\,", "066c26c3fe080fc316eb22ed44c1476a": "E_1 = E_2", "066c2b40daeec388ad222d1c338c3c83": "\\operatorname{E}_k", "066c3292096ad58a643b5c0b1e9ecb3d": " R_{i+1} ", "066ca74ca6eb93c75f218a3ff8ab5a69": "W_{1B}(y)", "066ce4431913d9f04ee5b13c9717e224": "v_d\\gg \\langle v\\rangle", "066d1b059460622cef5ed6cc1dff98ff": "Mod(\\sigma)(M')", "066d3aac7344710c04eaa756f0206458": "a_n=\\prod_{p^k \\mid n} \\frac{1}{k}=\\prod_{p^k \\mid \\mid n} \\frac{1}{k!} ", "066d6876b45d2660cd34cf9c03ee8b03": "a_0 \\leqslant a_1 < b_0", "066d77404b1f613f287dfa835100ea96": " \\displaystyle{f_\\alpha [\\pi] = [\\pi_l]} ", "066e2433c7b2ab71cc8af27d0869768e": "1-(1-\\alpha)^{1/m},1-(1-\\alpha)^{1/(m-1)},...,1-(1-\\alpha)^{1}", "066e6728e97193ce98e6102242dadbb3": "\\mathbf{f_{k}}", "066ea5ee3b091aa219300b229d553f6f": " \\delta z^{\\pm} \\ll B_0 ", "066eb2e8dd3eae537288c8ffa1c59f19": "U(P) = \\frac {1}{4 \\pi} \\int_{S} \\left[ U \\frac {\\partial}{\\partial n} \\left( \\frac {e^{iks}}{s} \\right) - \\frac {e^{iks}}{s} \\frac {\\partial U}{\\partial n} \\right]dS ", "066eb5d6feb36ae6c23ec3f8c70bb3af": "3 \\neq 0", "066ee51fb92bce3c2c2c95cf62538f5b": " \\delta\\left[ n \\right] = H[n] - H[n-1].", "066f30cec05a2c80a69ddcba5330f385": "L^2(\\mathbb R)=\\mbox{closure of }\\bigoplus_{k\\in\\Z}W_k,", "066f366637d8b734cdcb6850a51ed8ea": "f_1(X)f_2(Y)\\le f_3(X\\vee Y)f_4(X\\wedge Y)", "066fb94f6f904ffcd10ec7548b631bad": " \\left(\\frac{P_2}{P_1}\\right)^{-1 \\over \\gamma}=\\frac{V_2}{V_1} ", "066fd05ea8024436d9591d661eb6c36b": "X^2-D", "066ff3eb14fb518035dc413b97261c9f": "\\nu(\\theta)", "06700127a9e442cf05c79c15497d8963": "\\ (u + \\operatorname{d}u, v + \\operatorname{d}v)", "067036e7b5dfa9aad061ff612f962836": "{u \\over t}h \\equiv B_{(p-1)/2} \\pmod{ p}", "06703de5f6099981a749de52b9283778": "\\Box \\varphi", "0670443ef3b1c86a2459506c8098f58c": "wp(S, x=0) \\vee wp(S,x=1)", "067056bff759bf80fa0445c87a986dad": "\\ell_{(M,\\varphi)}(x,y)", "06708913c7fe58477cf8b1551e2394c8": "R^N", "06708de6599880972472ddd95342a36b": " p=u_x, \\quad q=u_y. \\,", "0670c0438d7cb43f7437da0f5bde1d78": "\\frac{1 + {\\scriptstyle\\frac{1}{3}}z}\n{1 - {\\scriptstyle\\frac{2}{3}}z + {\\scriptstyle\\frac{1}{6}}z^2}", "0670ced325ad94af3d97acf4547b6651": "{T^{\\alpha\\beta}}_{,\\beta} + F^{\\alpha\\beta} J_{\\beta} = 0", "0670fd4ad21a740df9f80fbe52d38b28": "N_k", "0670fd9dc081c99fd16be422a65c0366": " e + 2b = 180", "06710bbaf38ff33f645828396e72779e": "\\tau=\\theta", "0671570968abcb7dd61fb0eccc0d7ed7": "\\delta(n) = \\left| \\Pr_{x \\gets D_n}[ A(x) = 1] - \\Pr_{x \\gets E_n}[ A(x) = 1] \\right|.", "0671d7dfca8003422eefad9368fb8abc": "f(x) = \\omega(x) g(x)\\,", "0671dfa1ef2d24b727088aa07c2ade62": "a = x_0 < x_1 < \\cdots < x_n = b . \\,\\!", "06720f5d3d09e97e1aa9247a475d336e": "\\Gamma \\rightarrow ", "06722a5c67c658f46df95ac512149fba": "M \\oplus M^*", "067295cdea96e5c5cd7abdbf3a7443fb": "\\mathbf{w}(\\mathbf{X})", "0672f19f9a46a703e324a2238d5cf2b3": "n^{\\Theta(1)}", "06730233da1d833267e42dc715222711": "\\phi_\\omega(x)\\,", "067372b770afb538bef4527bc27a8d2d": "{\\text{PriorProbability}}(x=p;\\alpha \\text{Prior},\\beta \\text{Prior}) = \\frac{ x^{\\alpha \\text{Prior}-1}(1-x)^{\\beta \\text{Prior}-1}}{\\Beta(\\alpha \\text{Prior},\\beta \\text{Prior})}", "06737bcd735f0d3a2b534a15671e2eff": "\n\\begin{align}\n 1^{2p+1} + 2^{2p+1} &+ 3^{2p+1} + \\cdots + n^{2p+1}\\\\ &= \\frac{1}{2^{2p+2}(2p+2)} \\sum_{q=0}^p \\binom{2p+2}{2q}\n(2-2^{2q})~ B_{2q} ~\\left[(8a+1)^{p+1-q}-1\\right].\n\\end{align}\n", "06749f2a39f73bebb1d25788272f8214": "0 \\dots r", "0674bc21dbf638814d60e546889f2b8a": "e^{-\\tfrac{1}{2}\\sigma^2}", "0674dd7c8d7f30cc54d56f5ae5560879": "*(R_1,R_2,...R_n)", "0674fd93053ee8fdf28f7aaf69a00aba": "\\langle f,g\\rangle=(\\text{constant term of }f \\overline g \\Delta)/|W|", "0675098fd53d0ed69f56fe72b8740591": "\nS(\\omega) = 155 \\frac{H_{1/3}^2}{T_1^4 \\omega^5} \\mathrm{exp} \\left(\\frac{-944}{T_1^4 \\omega^4}\\right)(3.3)^Y,\n", "067510fba0d995e22350729841c910d3": "314\\,", "0675520d44ff563fc919439cc0dfa866": "i,j)\\in S", "06755e06e02c35fa4a4b189b4ba71d7b": "(S^1)^{\\wedge i} \\wedge (S^1)^{\\wedge j} \\to A", "06757e1103a40a7617c774b23ad18745": "d\\sigma", "067580ccb786f68077eb4c401c4b170b": "\\frac{\\partial g}{\\partial x}(X,Y,Z) \\cdot x+\\frac{\\partial g}{\\partial y}(X,Y,Z) \\cdot y+\\frac{\\partial g}{\\partial z}(X,Y,Z) \\cdot z=0.", "06758409cad5acbccb9b590639b9c987": "s\\in\\mathbb{C}\\setminus\\{1\\}", "0675fc293d7b86277328488400289d70": "\\tau_{ind}\\left(\\omega \\rightarrow 0\\right) = 0", "0676063a2bcdaa36ed8dc22f2d0145c4": "|\\dot{\\sigma}|", "06769d6913d96948a188f6bb10f2a7cb": "E_g=\\gamma\\left(\\frac{2a}{d_t}\\right)", "0676b1b022bc5a569c7083dc99faa8f4": "C_p(p,T)-C_V(V,T)=\\frac{TV\\,\\beta _p^2(T,p)}{\\kappa _T(T,p)}", "0677b553c54d02a66e67001fbbb60e6a": "\\!p", "0677d517e819310328d04a9e9409b1c1": "P_{-\\mu-\\frac12}^{-\\nu-\\frac12}\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)=\n\\frac{(z^2-1)^{1/4}e^{-i\\mu\\pi} Q_\\nu^\\mu(z)}{(\\pi/2)^{1/2}\\Gamma(\\nu+\\mu+1)}\n", "06780a12fc4e68987e63d6abf66989a4": "\\psi'(g)=\\psi(g)", "067816e3563071c84ee0fcc1c7c96556": "\\alpha_1 \\,\\! ", "0678665b6db497c25d835e29345c2bb8": "f(z) = q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 + \\cdots, \\qquad q = \\exp(2 \\pi i z)", "06788225244c9188ffd21bd2a5b97f14": " U = ", "067883f6a1df461f913f2a9f071d706d": "\\textstyle \\left \\lfloor \\frac{n}{p} \\right \\rfloor", "0678caa04da34220a4e8dc041488b618": "\\hat{\\theta}", "067938f8db0864ed2e506bec3acd6e85": "\\mu < \\mu_0", "0679440b99bfa58585f8a1779401cece": "\\frac{4 \\pi \\epsilon_0 G m_p m_e}{e^2} \\approx 10^{-40}.", "06798435062759cbb501c273d45a8752": "(2/1)^{31}", "0679c47aaa53066c513fade3ce974774": " x = \\mathop{\\rm sign}(x) \\sum_{i\\in\\mathbb Z} a_i\\,10^i", "067a06be194ff19d2c9d7e9e0d8ef1bc": "t = 2 i k_L e^{-i k_R L}\\left[\\frac{M_{11} M_{22} - M_{12} M_{21}}{-M_{21} + k_L k_R M_{12} + i(k_R M_{11} + k_L M_{22})}\\right]", "067a2dd2ec46d3ee5c7057bc20f2603e": "f \\left( z \\right) = \\ln{z}-z ", "067a66737160589adb79c8d2fa2ebf56": "\\mathbb{D}", "067a76ac757e97cb7ba2e8b882d01947": "011011", "067a7959aca16e6dca1750ba02e75af2": "\\mathcal{F}^{-1}g(x):=\\lim_{R\\to\\infty}\\int_{\\mathbb{R}} \\varphi(\\xi/R)\\,e^{2\\pi ix\\xi}\\,g(\\xi)\\,d\\xi,\\qquad\\varphi(\\xi):=e^{-\\xi^2}.", "067ae1930d84e342f43e13831d08d57c": "\\begin{align} 2\\cdot R_*\n & = \\frac{(51.3\\cdot 3.26\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 36\\cdot R_{\\bigodot}\n\\end{align}", "067ae35a0e786d0a745640683ea4e6c3": "U(y)\\!", "067b196a2eb6ed203f74271ad062ffb2": "\\begin{pmatrix}\n1 & 4 & 0 & 0 \\\\\n3 & 4 & 1 & 0 \\\\\n0 & 2 & 3 & 4 \\\\\n0 & 0 & 1 & 3 \\\\\n\\end{pmatrix}.", "067b7c30cc6df3dd0bcdee997c7de9fd": "S(z;u)=\\mathcal{X}^{-1}(z+\\mathcal{X}(u))", "067b926ccfc7b6a30d5dab2d30c3b4df": "Sq^i Sq^j = \\sum_{k=0}^{[i/2]} {j-k-1 \\choose i-2k} Sq^{i+j-k} Sq^k", "067ba60ad838fb71e30ef7c04220dd92": " C_0(x) ", "067bc08281a947a82a191fda0165ece3": "L_z(x)R_z(y) = B_z(xy)", "067beafbfe4c5e47df74c436264c5493": "\\Gamma_k", "067c085e0d9391d7e09b6bdb00ae9e70": "[X,Y] = XY - YX", "067c140cc51ade1b9712233649bd08cc": "r = \\alpha/\\beta", "067c1f736e72bee73cb745c45c4416ba": "\\frac{\\partial \\phi}{\\partial t} = D\\,\\frac{\\partial^2 \\phi}{\\partial x^2}\\,\\!", "067c2c0d217bdec32ae91e410f92dd83": "H_{\\text{bath}} ", "067c371e6c85d102c1598b3d6682ebdf": "h_n^{(2)}", "067c522d835f14123e6db1e083c569e7": "\\mathcal{E} = \\frac{U}{d}", "067cac656ba3bfc52d66aca056d9efe6": "\\mu_n(A) \\to \\mu(A)", "067cdb3665d1900f55d85812a3c2c6a4": "(\\Sigma^*, \\cdot, \\stackrel{*}{\\rightarrow}_R)", "067cf2572cd687b4d0da7249bc451d1a": "\\rho_M = \\Omega_M \\rho_c", "067cf3da45f5c22435e8c1ceb934e29f": "B = \\cup _j B_j", "067cf58f905a00984db1508337e09984": "N_k(A)", "067d01e6ac353c232be79dd25dc35671": "\\frac{\\beta }{k_{0}}= \\frac{c}{v_{ph}}= \\frac{\\lambda _{0}}{\\lambda _{g}}\\simeq sin\\theta _{m}", "067d1ec1784031f26a5dab89f01456ee": "r=\\tfrac12.", "067d4d071ce173a0f3778c9236cf08e4": " = \\frac{\\operatorname{P}[E_1]}{1-(1-\\operatorname{P}[E_1]-\\operatorname{P}[E_2])}\n= \\frac{\\operatorname{P}[E_1]}{\\operatorname{P}[E_1]+\\operatorname{P}[E_2]}\n", "067d5fde8f2c0731bc2f17fb68a7a23b": "A\\neq 0", "067d81383666089aca8b14e4b6ae6ab4": "(n,q,\\pi,G)\\,", "067e11f62a8a4cd7c873c289b749eeb2": " E[\\varepsilon_t]=0 \\, ,", "067e44c2ee9cb7e8ee7b12d675688024": "\\sigma = 1/\\nu d_\\text{f}\\,\\!", "067e7c5fd6c577257323affccb9f2d62": "-(1/3)E/c^2", "067e8173226f0047ad0e439500259b2f": "a=b=c", "067e9262376f55c180f13aa130ca1201": "d J/d x = d A \\rho v/dx = k r \\rho v", "067e99060907090655b8f083897609d8": "0.0\\dot{7}", "067f12a29ae2f078ce3367d6f6ec64f7": "f(\\phi)\\,", "067f4a8a461954dfa4aa5c1661495191": "P^s (S) = \\inf \\left\\{ \\left. \\sum_{j \\in J} P_0^s (S_j) \\right| S \\subseteq \\bigcup_{j \\in J} S_j, J \\text{ countable} \\right\\},", "067f56e80150f458f7f00050e947db33": "x^3 + Ax +B", "067f5b4e31caa7ea54b65c4ad196344e": "w_i = \\{f_i^1, ..., f_i^k\\}.", "068055de35dcdebbc711f4a9afaf13d3": "Pr[\\pi \\gets \\mathrm{Prove}(\\sigma,y,w) : \\mathrm{Verify}(\\sigma,y,\\pi)=\\mathrm{accept}] =1", "068059390899e3d6cf5fd5e76586865e": "Points\\ difference\\ = \\frac{14}{No\\ of\\ teams\\ -\\ 4}", "068099a91211d97da66278ce458cba93": "p = {\\delta(H) \\over l-1}p_c + o(H) p_m", "06809e93fb9570e35c55c9552ebfce10": "\\scriptstyle {L = \\lbrace (a_1,a_2,...a_n)|a_1c_1 + a_2c_2 + ... a_nc_n = d \\rbrace}", "0680a49da435b7058f15bc86ad1d61b4": "\n \\boldsymbol{\\nabla}\\times\\boldsymbol{\\omega} = -\\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon} = - \\boldsymbol{\\nabla}\\mathbf{w}.\n ", "068132e5c9db9a7bf997db2b19e1925d": " = | {v}_1 - {v}_0 |\\;", "068168112d1f7deab9e7f61ac52b7f2c": " \\int \\Phi(a+bx) \\, dx = b^{-1} \\left ((a+bx)\\Phi(a+bx) + \\phi(a+bx)\\right) + C ", "0681892c6549c3efc5cd229822e7e9bc": "Z_{\\rm can} ", "0681b5ac60d6a074cee5e6cf09e4ddda": "\\int d\\mu(\\sigma) \\prod_j \\sigma_j^{n(j)} = 0 ", "0681eb8ecfbc9b5990c22df8c6daa4fe": "\\lambda = \\frac{g}{2\\pi}T^2 \\qquad \\scriptstyle \\text{(deep water).}", "06821a4333616ec92dde102111db818e": "\\alpha_{1}(a,\\, b) \\cdot \\alpha_{2}(a)", "068270d8dcc78b8ada5a0f9fc6f6af11": " \\frac{dV}{dt} = v \\pi R^{2} = \\frac{\\pi R^{4}}{8 \\eta} \\left( \\frac{- \\Delta P}{\\Delta x}\\right) = \\frac{\\pi R^{4}}{8 \\eta} \\frac{ |\\Delta P|}{L}, ", "06829d25041f675aed53bc90d4e4569e": "P_\\mathrm{emp}(x)", "0682f62ed88d2d9e7403ac03e0f06255": "\\displaystyle u \\ ", "0682f8703a7bbe19c459acbbc3312dd5": "\\scriptstyle p=\\tfrac{b}{a}, q=\\tfrac{c}{a}\\!", "0683443785f335e228b1d8fa42899e04": "\\sigma=1/2", "0683afc6d0540771a2bbf8f8bdedf0cb": "K = z/\\sin z", "068423177eff58cdb23c0632588209f6": "\n\\int_{B_{1/2}}\\!\\!\\!\\chi_{B_1}(x-y) \\varphi_{1/2}(y)\\mathrm{d}y= \\int_{B_{1/2}}\\!\\!\\!\n \\varphi_{1/2}(y)\\mathrm{d}y=1\n", "06842a7a139712704a54a8284b5017cd": "m,n,d,c", "06844c045947ee318197c628dcc921ef": "C = -\\ln p_k .", "0684528693d692e4c88e53e6cdc01e39": "~W_{\\rm d}=\\frac{ I_{\\rm p} \\sigma_{\\rm ep}}{ \\hbar \\omega_{\\rm p} }+\\frac{I_{\\rm s}\\sigma_{\\rm es}}{ \\hbar \\omega_{\\rm s} } +\\frac{1}{\\tau}~", "06854d35d6224320ae115a5c6c62d2c5": " {\\lVert x_k-x \\rVert^2} \\leq \\left(1-\\left|\\left\\langle\\frac{x_{k-1}-x}{\\lVert x_{k-1}-x \\rVert},Z_k\\right\\rangle\\right|^2\\right){\\lVert x_{k-1}-x \\rVert^2}. ", "06859cecf6e54eeb8e0f170d13ee9901": "\\mathsf{case}\\ e\\ \\mathsf{of}\\ x \\Rightarrow e_1 | y \\Rightarrow e_2", "0685d737671d4db440ae000f13ddd614": " \\alpha= \\frac{W}{W+k \\cdot (W+M)} \\,\\ ", "0685e6b1212801a886813184201e30b7": "{}^2 E'_{pq} = H_p(Y, H_q(*, \\mathbb{Z})) \\Rightarrow H_{p+q}(Y, \\mathbb{Z}).", "06861e59a653954257336dd2becf45a8": "\\Gamma \\backslash N", "068695c2aa2969123e274e78272c42e1": "\\xi_1,\\dots,\\xi_K", "0686bc15bcef41940b5b94f503be18fc": "a_{\\ell m}^{(M)}=\\frac{-ik^{\\ell+2}}{(2\\ell+1)!!}\\left(\\frac{\\ell+1}{\\ell}\\right)^{1/2}[M_{\\ell m}+M_{\\ell m}']", "0686f68c86e63e8c126077f0c255d319": "\n\\arccsc(z)\n", "068705c208645af032cd5da5e112042d": "q = \\int \\rho \\, dV", "06870b319501adc54583033a50d63092": "a (x-y)(x-z) + b (y-z)(y-x) + c (z-x)(z-y) \\ge 0.", "068722584f10a02209cb3a22738c9e98": "P=2.\\omega_D.i, \\; A=\\xi.\\omega-\\omega_D.i, \\; B=\\xi.\\omega+\\omega_D.i", "0687357451abeef5780b4c3bd9e107a9": "\\ \\mbox{SF} = \\frac{\\mbox{chip rate}}{\\mbox{symbol rate}}", "0687510a024f6831750fafcd94541d38": "\\left | H(j\\omega) \\right | = \\left | \\frac {1}{1+\\alpha j \\omega} \\right | =\\sqrt{ \\frac {1}{1 + \\alpha^2\\omega^2}}", "06876f67e1ffe1319ce5ef2524074a4e": "\\displaystyle \\nabla^2u+\\sinh u=0", "0687a14bf9b9e2b96b257c38fb12ca1d": "\\neg a \\vee c", "0687bc8a5c3584cb26adeaa6ae986916": "U'_a", "0687d164c9a5ac835eeec8b898440f13": " l_A a_D + (1 + \\sigma) (l_A a_B + l_B) l_D ", "0687dc1974f9d312eddda27504a24a8f": "\\mathbb{P}\\left( (Z_1,\\ldots,Z_n) =\n(z_1,\\ldots,z_{i-1},z,z_{i+1},\\ldots,z_n) \\right)\\,.", "06881dc19dc00e4ef1755e9dd4b09094": "\\int_{a}^{b}\\omega(x)f(x)=\\sum_{j=1}^{N}w_{j}f(x_{j}) = w_{i} f(x_{i}).", "06887fa940c07c1481fc61748faaac5a": "(\\phi,\\lambda)", "06889b83613b0fda2a4fdd78dc68d49c": "\\displaystyle{\\Phi(1)=P.}", "0688db2962b45a1527abaf75d43d146b": "\n D = \\cfrac{Eh^3}{12(1-\\nu^2)} \\,.\n", "0689341a386da86cf9563874ad6f75b7": "u(x_1)", "068a7b56b0cdc2e58d9bed0c11515e1f": "\n\\partial_t \\hat{u}_k\n=\n- \\frac{i k}{2} \\sum_{p+q=k} \\hat{u}_p \\hat{u}_q \n- \\rho{}k^2\\hat{u}_k\n+ \\hat{f}_k\n\\quad k\\in\\left\\{ -N/2,\\dots,N/2-1 \\right\\}, \\forall t>0.\n", "068ad15d949ff53e4c05dee669d187ce": "\\mathcal{E} = - \\frac {d \\Phi_B} {dt} = -\\frac {d}{dt}\\iint_{\\Sigma (t)} d \\boldsymbol{A} \\cdot \\mathbf{B} (\\mathbf{r},\\ t) \\ , ", "068af70b4be78ed9b688a40c2d5c8a1d": "\\begin{matrix}\\frac1{128}\\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\\,", "068b01dbadf5a8f6fb36d6dd2dda2c34": "k(s) = \\det\\begin{bmatrix}\\beta''(s) & \\beta'''(s) \\end{bmatrix}.", "068b762cc7ed5622a06f8ecbc88396d0": "x^{2} \\equiv 1\\pmod{p} ", "068b85d57c0b9c89af307e6314448557": "\n \\frac{\\partial \\sigma_{xx}}{\\partial x} + \\frac{\\partial \\tau_{xz}}{\\partial z} = 0\n", "068b8ed62a82c0bfedea19a38beda42c": "\\varphi\\circ g", "068bbfb5ca1b6884fd761f3c4dc4d4da": "q_1 q_2\\cdots q_n", "068be23a9d109063b10a313e2f71f7f5": "c_n=h_0^n + h_1^n + h_2^n.", "068c00d7f35e1321fb8899bb83b8db52": " \n \\begin{bmatrix} T_1\\\\T_2 \\\\ T_3 \\end{bmatrix} = \n \\begin{bmatrix}\n \\sigma_{11} & \\sigma_{21} & \\sigma_{31} \\\\\n \\sigma_{12} & \\sigma_{22} & \\sigma_{32} \\\\\n \\sigma_{13} & \\sigma_{23} & \\sigma_{33} \n \\end{bmatrix}\n \\begin{bmatrix} n_1\\\\n_2 \\\\ n_3 \\end{bmatrix}\n", "068c063f5304c4222bac4f60474f6b5d": "\\gamma_{s}", "068c3e650dc389220a48a4be2511f186": "\\mathbb{P}[\\omega = H] = p\\in (0, 1)", "068c4dbb106c87186d1f4c4ed052a676": "A_u", "068cbacf039d96a2e80d3c510ff41f4e": "\\frac{d\\tau}{dt} = 1 - \\frac{U}{c^2} - \\frac{v^2}{2c^2} ", "068d053da819608daa8b38c9cf3118da": "q = q_1 + ... + q_r", "068d06b9ae016c763684a23a994c4c56": "Plato:c-b=1,\\quad \\quad Pythagoras:c-a=2,\\quad \\quad Fermat:\\left| a-b \\right|=1", "068d14bf79c3ee8628bb81520cfb5b27": "g(r) = \n\\begin{cases}\n 0,&r 0", "06a55db1a17d4505e2b15c4aec2d780a": "\\| f \\|_{L^{p, q}} = \\left\\{ \n\\begin{array}{l l} \n\\left( \\int_0^{\\infty} (t^{\\frac{1}{p}} f^{*}(t))^q \\, \\frac{dt}{t} \\right)^{\\frac{1}{q}} & q \\in (0, \\infty),\\\\\n\\displaystyle \\sup_{t > 0} \\, t^{\\frac{1}{p}} f^{*}(t) & q = \\infty.\n\\end{array} \n\\right.", "06a57402a19a50c01966d32ba45a13ca": "\n\\begin{align}\n\\hat{y} = &\\ 25 \\\\\n& + 6.1 \\max(0, x - 13) \\\\\n& - 3.1 \\max(0, 13 - x) \\\\\n\\end{align}\n", "06a577e4bfc6d61013b0a750e883d100": "\\mathit{k} \\in \\mathbb{Z}^+", "06a69789c3a0377cbcfbc5f9b7e25541": "\n{\\mathbb Z}\\backslash \\left(D^n\\times{\\mathbb R}\\right)", "06a69d8fac6c838029f7d39d117759b8": "\\vec{r}\\,'(t)", "06a6cc8549169648b4030dcbf8a34b9b": "\\theta_{min} \\approx \\frac {CD} {AC} = \\frac{\\lambda}{W}", "06a76ed86a8187d5ed2914ac3e56b22c": "\n {d\\vec{\\omega} \\over dt} = (\\vec{\\omega} \\cdot \\nabla) \\vec{v} + \\nu \\nabla^2 \\vec{\\omega}\n", "06a7886775c5a32da669024135dae607": "\\quad\\quad\\int \\arccos(y) \\, dy = y\\arccos(y) - \\sin(\\arccos(y))+C.", "06a7a2204d20af7ec17f478e3bfc0dc5": "Rz = 12.528\\cdot(S^{0.542})/((P^{0.528})\\cdot(V^{0.322}))", "06a7a43679442901409ceea31fd2c63d": "\nP_\\mu (n,t)=\\frac{(\\nu t^\\mu )^n}{n!}\\sum\\limits_{k=0}^\\infty \\frac{(k+n)!}{\nk!}\\frac{(-\\nu t^\\mu )^k}{\\Gamma (\\mu (k+n)+1)},\\qquad 0<\\mu \\leq 1,\n", "06a832e45f0a2df58022d7c2e13998ee": "H = \\sum_{j\\sigma}\\epsilon_f f^{\\dagger}_{j\\sigma}f_{j\\sigma} + \\sum_{\\sigma}t_{jj'}c^{\\dagger}_{j\\sigma}c_{j'\\sigma} + \\sum_{j,\\sigma}(V_j f^{\\dagger}_{\\sigma}c_{j\\sigma} + V_j^* c^{\\dagger}_{\\sigma}f_{j\\sigma}) + U\\sum_{j}f^{\\dagger}_{j\\uparrow}f_{j\\uparrow}f^{\\dagger}_{j\\downarrow}f_{j\\downarrow}", "06a85b19a6802a789494d61c25143645": " \\tau = 0.85 \\sigma _n", "06a896fa8ff8e974397a7c1d4f7e970c": " -\\left(\\eta_2 + \\frac{p + 1}{2}\\right)(p\\ln 2 - \\ln|\\boldsymbol\\Psi|)", "06a8c4a883febf3b3c1e8d98aef4380a": "\\mathbf{w}_n", "06a927df29f1b23a6ddbca364428e099": "r^s\\,", "06a96ec00c9ee7409a4fb60fa521edb5": "U \\subset \\Omega_x", "06a9739646c8387f56abe4303aa9173e": "M=\\begin{bmatrix}\n1 & 0 & 1\\\\\n0 & 1 & -1\n\\end{bmatrix}", "06a994dc739c93a51c1249e678797b3a": "m \\in \\{ 0, \\dots, n-1 \\}", "06a9b7f085a1e129cf9ed70f2fd4cfc7": "P_{\\mathrm{i}}", "06a9c832ac2806ff5170bcd4c8966904": " (s,t_s) \\in S'", "06a9fe91a949298ba79500ecd7ff46ea": "\\mathbf{1}_A(x) = \\begin{cases}\n1 & \\text{ if } x \\in A \\\\\n0 & \\text{ otherwise}\n\\end{cases}", "06aa3ae0f57e9b87bf69f3a38047ebd5": "G \\approx 1/t\\,", "06aa92d94fa8eb721ebd2c5a74e9693d": "N=S-S_0-\\int\\frac{dQ}{T}.", "06ab332b667b5fccd38be74338754cc0": "t\\otimes v\\in V_{tgt^{-1}}", "06ab38ad98e39bcacc7a8082887e64c3": "{*}", "06ab582721055f7507054a8fdd5f9034": "\\scriptstyle W_p", "06aba47b20f4437dddfa1bde395760c3": "g_{i j} = \\mathbf e_i \\cdot \\mathbf e_j", "06abc6a479b86882b9cfa3a247503a29": "a\\frac{\\partial \\mathbf{u}}{\\partial x}", "06abe4267cc458b20d5629ad6ec811d3": " \\ CVI(ESA) = A\\phi\\mu_d\\frac{\\rho_p-\\rho_m}{\\rho_m}", "06abf479832451bfce7629e688b35f86": "(1 - p)^{k-1}\\,p\\!", "06aca9d7637d292ae6f30e90d6492007": " J \\ne R", "06acaebf61246b1dfd3f106e2ab527b3": "E_A(\\log(x))\\geq E_B(\\log(x))", "06acd492d5d9c2d9ad2d2be4be0dbd9b": "d + a\\mathbf{\\hat{i}} + b\\mathbf{\\hat{j}} + c\\mathbf{\\hat{k}}", "06ad0d79511ce778229beb2e7adbbdb8": "\\mathcal{E}_{ijk}", "06adaae10024e7fa1776d1611e210975": "\\phi^2 = \\phi+1", "06add2141d8cd704911cc9766f6b3d74": "\\tilde{\\mathbf{B}}^+ = W(\\tilde{\\mathbf{E}}^+)[1/p]", "06adfd48187e054bb4439d9133640790": "1 \\mathrm{\\ rev} = 360^{\\circ} = 2\\pi \\mathrm{\\ rad}\\mathrm{, and}", "06ae0850263fd1e91f14e535544e034b": "\n\\operatorname{Li}_2 \\left(x \\right) + \\operatorname{Li}_2 \\left(1-x \\right) = \\tfrac{1}{6} \\pi^2 - \\ln(x)\\ln(1-x) \\,,\n", "06aebef211fb825da2eda40aa75499f0": " \\varepsilon_{ni} ", "06af3826fe0062389f5975927a08f573": "\\phi_{sl}=\\frac{\\rho_{s}(\\rho_{sl} - \\rho_{l})}{\\rho_{sl}(\\rho_{s} - \\rho_{l})}", "06af87fb65c8cbd4a2578d1c5a1bf356": "\\beta_n^{ }", "06afbfb6bdc88b7a434fd27845332387": "n\\equiv 1 \\pmod{2^{k}}, \\quad n\\equiv 0 \\pmod{5^{k}}\\, .", "06b048d5acd367ff91c5a6b90010b4ef": "\n\\epsilon_{abc} \\eta_{b\\mu\\nu} \\eta_{c\\rho\\sigma}\n= \\delta_{\\mu\\rho} \\eta_{a\\nu\\sigma}\n+ \\delta_{\\nu\\sigma} \\eta_{a\\mu\\rho}\n- \\delta_{\\mu\\sigma} \\eta_{a\\nu\\rho}\n- \\delta_{\\nu\\rho} \\eta_{a\\mu\\sigma}\n", "06b068f1873379c153e916e9b9211a09": "\\left\\{ S_{\\alpha}^i, \\overline{S}_{\\dot{\\beta}j} \\right\\} = 2 \\delta^i_j \\sigma^{\\mu}_{\\alpha \\dot{\\beta}}K_\\mu", "06b06b1fc241987f0246fb3a4a70fec1": "A \\otimes B", "06b0fd8f01ce256f229860bf283f3e6d": "\n \\sigma_{ij} = \\lambda~\\varepsilon_{kk}~\\delta_{ij} + 2\\mu~\\varepsilon_{ij} = c_{ijk\\ell}~\\varepsilon_{k\\ell} ~;~~ c_{ijk\\ell} = \\lambda~\\delta_{ij}~\\delta_{k\\ell} + \\mu~(\\delta_{ik}~\\delta_{j\\ell} + \\delta_{i\\ell}~\\delta_{jk})\n ", "06b137ca45622b7aec8d85f58f8d164a": "{\\frac{m}{e}}>2.35", "06b16782e68921373cf5edf0493be912": "=\\empty", "06b18d1719a5020a6cdd57551570b346": "a^2+c^2=b^2+d^2", "06b1972b2a0d710bdac4a38f8c8d9db8": "\\forall_1", "06b1a519499bbfb45c46e582db7016bb": "\\sum_{w\\in I_n} f^{1/k}(w) \\mu_n(w) = O(n)", "06b1deed07c8747ed175f7eb5d24496f": "f_{W}/f\\,", "06b1e1d2e04eb5ac021f2575ed7b64a3": "P_{A}=\\frac{|C_{A}|^2}{|C_{A}|^2+|C_{B}|^2}", "06b20283768dfa01a2508eb1553f20d6": "\n\\nabla^{2} \\Phi = \n\\frac{1}{a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right)} \n\\left[\n\\frac{1}{\\cosh \\mu} \\frac{\\partial}{\\partial \\mu} \n\\left( \\cosh \\mu \\frac{\\partial \\Phi}{\\partial \\mu} \\right) + \n\\frac{1}{\\cos \\nu} \\frac{\\partial}{\\partial \\nu}\n\\left( \\cos \\nu \\frac{\\partial \\Phi}{\\partial \\nu} \\right)\n\\right] +\n\\frac{1}{a^{2} \\left( \\cosh^{2}\\mu+\\cos^{2}\\nu \\right)}\n\\frac{\\partial^{2} \\Phi}{\\partial \\phi^{2}}\n", "06b21f7325a11c27a747ce38c164237a": "\\mathbf{k}_o", "06b2a00ff6acf41deb8ba139ccae4f6b": " L = m\\hbar", "06b2d3f8e67033ccbdea177b6735505f": "k_i\\sigma", "06b3473cc47c38570634b2fbce24af01": "\\cot A = \\frac {1}{\\tan A} = \\frac {\\textrm{adjacent}} {\\textrm{opposite}} = \\frac {b} {a}. ", "06b37022ce795d7496c4b01ad20c9a12": "\n\\hat{\\mu} \\sim IG \\left(\\mu, \\lambda \\sum_{i=1}^n w_i \\right) \\,\\,\\,\\,\\,\\,\\,\\, \\frac{n}{\\hat{\\lambda}} \\sim \\frac{1}{\\lambda} \\chi^2_{n-1}.\n", "06b398af20108b57457bddd031c674aa": "Z_\\mathrm{in}=Z_L \\,", "06b40e2b1efc279d7974e7b8fddff0b6": " [B]=-\\frac{k_1'}{k_2} ln \\left ( 1 - \\frac{[C]}{[R]_0} \\right )", "06b45a226f0a63a5bb2936785b8bf396": "C \\subseteq X", "06b4633b591c7203cf47c5655d6c763f": "p_{e}", "06b46f7f426d4227c38f05c06e398621": "f_\\text{P1,2} = f_\\text{1,2} \\left( 1- \\vec v \\ast \\frac{\\vec e_\\text{1,2}}{c}\\right)", "06b4726fb98cc908e151656bb50e8540": "(x_1, \\ldots, x_n)", "06b48e3cafb0bdfb0be0e2bf7a6c91bc": "-1 \\le \\rho_{ij}< 1", "06b4979a43155a9f4fbf1a58ee620f44": " u(t,x,y) = tM_{ct}[\\phi] = \\frac{t}{4\\pi} \\iint_S \\phi(x + ct\\alpha,\\, y + ct\\beta) d\\omega,\\,", "06b4c72d5a08353c4adf480325e491e1": "r := 0", "06b510444ae01abaf2cb66a7085a415e": " rN", "06b53cf41884e57bcaf94421e92b2f7e": "\n\\mathcal{L}\\{f(x)\\}=-\\boldsymbol{\\alpha}(sI-\\Theta)^{-1}\\Theta\\boldsymbol{1}\n", "06b56847960f2dbf2669533900541748": "f_\\ast \\colon S_\\ast(X)\\rightarrow S_\\ast(B)", "06b5772a2f9345b1359939e01a53c047": " \\mathcal{P} = \\lbrace p \\mid p <_{\\mathcal{O}} e_d \\rbrace ", "06b58525084d383b44896c04eefd5fd8": "Q \\,", "06b58921a3a45af60b12b2818b208f59": "-j2\\pi /n", "06b5db735ebb6fc09e2c7f37cf8cee27": "T=|t|^2= \\frac{1}{1+\\frac{V_0^2\\sinh^2(k_1 a)}{4E(V_0-E)}}", "06b6181ce1cf9c509c5e45f72fc2af49": " \\exists ! x_n A(x_1, \\ldots , x_n) ", "06b63628beb319968cb673bb6d9aaeb9": "\\textstyle \\oplus_{i=1}^n \\mathbb{C}^m", "06b651a692ef43f1f96580ad0471cace": "\\mu(x,G)=B(x,\\delta/2)", "06b661bd66f0d7407279b496647dfdcf": "\n \\begin{bmatrix}\n 1 & 2 & 3 & 4 & 0 & 0 & 0 \\\\ \n 0 & 3 & 2 & 1 & 1 & 0 & 10 \\\\\n 0 & 2 & 5 & 3 & 0 & 1 & 15\n \\end{bmatrix}\n", "06b6732971a098fefbf1cdcc022a0707": "\\mathcal L \\left\\{J^2f\\right\\}=\\frac1s(\\mathcal L \\left\\{Jf\\right\\} )(s)=\\frac1{s^2}(\\mathcal L\\left\\{f\\right\\})(s)", "06b6942ffb3bd3773ab7dde8bd9d463b": "j < k + m < j + 1 \\,", "06b6b26126dc65a7c0ba235188c2ee93": " Q(\\theta_1,\\theta_2,\\theta_3)= Q_{\\bold{x}}(\\theta_1) Q_{\\bold{y}}(\\theta_2) Q_{\\bold{z}}(\\theta_3) , \\,\\!", "06b6c2f1ae45d6ba5e1a9735ec1140e9": " u'(x) = \\lim_{h \\rightarrow 0} \\frac{u(x + h) - u(x)}{h} \\qquad (2)", "06b72d8385b68498eb3c2025ccafe15b": "C(P)=\\{ \\lambda_1(\\textbf{x}_1, 1) + \\cdots + \\lambda_n(\\textbf{x}_n, 1) \\mid \\textbf{x}_i \\in P,\\ \\lambda_i \\in \\mathbb{R}, \\lambda_i\\geq 0\\}.", "06b742d2a227e5c7633150c9ebaa83bd": "\n\\widehat{\\boldsymbol \\theta}_{JS+} = \n\\left( 1 - \\frac{(m-2) \\sigma^2}{\\|{\\mathbf y} - {\\boldsymbol\\nu}\\|^2} \\right)^+ ({\\mathbf y}-{\\boldsymbol\\nu}) + {\\boldsymbol\\nu}.\n", "06b7536a72d0f7c7c5af9aa188335008": "X^\\alpha _0 = \\{0\\};\\ \\ X^\\alpha_{n+1} = \\bigcup_\\gamma X^{\\beta_{\\gamma+1}}_n\\setminus \\beta_\\gamma", "06b7541704df6ffd80bfe8b062fff09b": "{{i}_{C1}}={{i}_{C2}}\\equiv {{i}_{C}}", "06b77d730793f0bc864492d8ecfeaeb0": "g=\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix}", "06b7d06f0c2dffd025c2e773acfee9f3": " (f^*g^N)(v,w) = g^N(df(v),df(w))\\,.", "06b8160063b2584d96f2a9f25140cbf1": "x^2 + y^2 + z^2 = c^2t^2.", "06b81f958e3d1c0ee06aa37721af644f": "dT_{V}=pd\\Theta", "06b8622846eb44b8fad8a1bd22c4e77f": "FX/SO(1,3) ", "06b867677f19b7e48c680dda13b99618": "\\max_{x_1,\\ldots, x_n} \\Delta_n(\\mathcal{C}, x_1, \\ldots, x_n)", "06b8bf1b0a58f1bcb74d65a37edcbe2a": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n = \n \\begin{Bmatrix}\n j_1 & j_4 & j_7\\\\\n j_2 & j_5 & j_8\\\\\n j_3 & j_6 & j_9\n \\end{Bmatrix}\n =\n \\begin{Bmatrix}\n j_9 & j_6 & j_3\\\\\n j_8 & j_5 & j_2\\\\\n j_7 & j_4 & j_1\n \\end{Bmatrix}.\n", "06b8c6ff2fc9dc85f76b034d346ebd9f": "\\frac{ \\omega }2 \\, \\partial_\\phi=\\vec{e}_3+\\vec{e}_0", "06b8e92904fdfd7fed1edbec65f6b750": "\\mathcal{T};", "06b9a76f4b9894f3f086afb18cece11f": " \\theta = \\alpha_{0} [\\tan (\\lambda L) \\sin(\\lambda y) + \\cos (\\lambda y) - 1]", "06b9c9ab3e809d880a89970d1e7e78be": "f_1(z) = \\,_1F_1(a+1;b+1;z)", "06b9d9792d77936371978d7765d73abe": "\\scriptstyle(f_n)", "06b9da224eb265a6651cfbe33634ca70": " (-r \\sin \\varphi, r \\cos \\varphi, 0) . \\,\\!", "06b9ddbf8e582b8f76a5cfbe98c05ab5": "\\tau_{\\text{rms}}=\\sqrt{\\frac{\\int_0^\\infty(\\tau-\\overline{\\tau})^2 A_c(\\tau)d\\tau}{\\int_0^\\infty A_c(\\tau)d\\tau}}", "06b9ebc07d19b6e751779c9deec2975a": "(y'_1,y'_2,y'_3,s')", "06bab0275afda592834316418089c3aa": "\\sigma\\text{'s}", "06bae13d55767350862d703a0a443d91": "\\frac{h^4}{180}(b-a) \\max_{\\xi\\in[a,b]} |f^{(4)}(\\xi)|,", "06baefddda163e47d9a815a391f33883": "\\sqrt{9}=3", "06bb25ce4bcf2a16644ef74a68d31886": "t<0 ", "06bb2827a354b8a6a491f8956ffd9c7b": "S=A/4", "06bbe522b15fd4974f2c5dd35df30232": "\n\\begin{matrix}\n7)16^27^66^32^41^60^49^0\n\\end{matrix}\n", "06bc01adbf7210620cc43b392bd15974": "z_k:=z(kh)", "06bc16d78f2bff88c09c264bcab033f5": "P^{-1}=A^{-1},", "06bc28ff8bc6ed7a28751992b6b03fb4": "\\mathcal{F}_{\\tau}", "06bc8f6595dcd604d81c645714a0cc74": "u(x,\\dot{x}) = -(|\\dot{x}|+k+1) \\underbrace{\\operatorname{sgn}(\\overbrace{\\dot{x}+x}^{\\sigma})}_{\\text{(i.e., tests } \\sigma > 0 \\text{)}}", "06bc9999ffd4b7f56d9acb4bc41b7b2b": " \\cfrac{\\Gamma \\vdash A, \\Delta \\qquad \\Sigma \\vdash B, \\Pi}{\\Gamma, \\Sigma \\vdash A \\and B, \\Delta, \\Pi} \\quad ({\\and}R)\n ", "06bcc1b573ae3e3f913f9747f154270e": "\\begin{align}\nS\\left(v\\right)&=\\int_0^T\\frac{d}{dT}E\\left(v\\right)\\frac{dT}{T}\\\\[10pt]\n&=\\frac{E\\left(v\\right)}{T}-k\\log\\left(1-e^{-\\frac{hv}{kT}}\\right)\n\\end{align}\n", "06bcd5269b2a6532a55599dd5187f116": "2^{|V|-1}-1", "06bcf75c2eabd4e09fe93db827cc7deb": "L^1(G//K)\\ni f\\mapsto \\hat{f}", "06bd1846a371cfac4ab8923b7a92d943": "T_{first}", "06bd3be4113d20cac5c1af358416abdf": "x=\\frac{v^2_{bullet}2 \\sin(\\delta\\theta)\\cos(\\delta\\theta)}{g} \\,", "06bda16a2e19377ac3bb0d4d253bc272": "\\langle E(t) \\rangle = \\frac{C}{t^3} + \\textrm{finite}\\,", "06bdaf2d7bde957d9fff2920ba9c8028": " v^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ {dx^{\\mu} \\over d\\tau} = \\left (c {dt \\over d\\tau}, { dt \\over d\\tau}{d\\mathbf{x} \\over dt} \\right ) = \\left ( \\gamma , \\gamma { \\mathbf{v} \\over c } \\right ) ", "06bde2879f214505b9808cc161a2d455": "I_k(\\mathbf{y}, t)", "06be07d35fabd5ce8f0b06d71eee740c": "f(x_1,\\ldots,x_k) \\simeq U(\\mu y\\, T(y,e,x_1,\\ldots,x_k))", "06be37efc4272118aead8209bde71ffa": "VC(C)=VC_0(C)+1.", "06be3857f24e511c1218394307f03b29": "R_{3,3} = r^3", "06be9fa0f9e14a2759fc4fb778ea7ff2": "\\mathrm{C_0 = 0}", "06bf44d7f269895f9d5f46fc5a5955e9": "v_{110}", "06bf55697e3fb2ab1ab2d08cf18a9f2d": "X^n(j)", "06bfaa8bb4bb4f1dd228210ab38fd26b": "\\operatorname{MSE} \\, \\hat{f}(\\bold{x};\\bold{H}) = \\operatorname{Var} \\hat{f}(\\bold{x};\\bold{H}) + [\\operatorname{E} \\hat{f}(\\bold{x};\\bold{H}) - f(\\bold{x})]^2", "06c04e28a21751cb1b8483f2bc2da567": "\\psi_L\\rightarrow e^{i\\theta_L}\\psi_L", "06c0858237a31c8ec7700537002f9227": "J = 4t^2/U", "06c08b6eb8ee26cabbfcd9d4c5a5941b": " |1-z|\\leq M(1-|z|) \\, ", "06c099d6b68186dd2ffad8add13d0141": "V(t) = V_0{H*h}(t) = \\frac{V_0}{\\sqrt{\\pi}}\\int\\limits_{-\\infty}^{\\frac{\\sigma t}{2}}e^{-\\tau^2}d\\tau = \\frac{V_0}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t}{2}\\right)\\right]\\Leftrightarrow\\frac{V(t)}{V_0}=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t}{2}\\right)\\right]", "06c0b29db4c8b8b8b21123a934320e3c": "\\{f^*_n\\} \\in A'", "06c107ecfc08a6ee9113a1fc02ca1f06": "\\log(X_i)", "06c1086887a6c7c5155178a1564c7095": "\\mbox{ P1 }:\\begin{cases}\nu''(x)=f(x) \\mbox{ in } (0,1), \\\\\nu(0)=u(1)=0,\n\\end{cases}", "06c12e1824a0f5ee49f8ea651a650f27": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.8951 & 0.2664 & -0.1614 \\\\\n-0.7502 & 1.7135 & 0.0367 \\\\\n0.0389 & -0.0685 & 1.0296\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "06c141702f2f0a1c813a676f56b084da": "a=\\theta", "06c1465b20e8f7941c105e39360901d1": "D\\ne 26", "06c15b82a3ad5f71a943c8809fffaeb7": "\\Delta_{n+1} \\equiv \\Omega_{n+1} - \\Omega_n = \\frac{f(u_n)}{u_n} \\delta_n\\Omega_n = \\frac{f(u_n)(1+u_{n-1})}{f(u_{n-1})u_n}\\Delta_n,", "06c18e7e7c0d23c57e7bc656c338b014": "4!/(2!2!) = 6", "06c1c5414cb3043035bfa6eb54717f57": "\\overline{c_i}=c_i", "06c1cf783b14d26057fe32bfa1217003": "\\textstyle Y(\\omega)=y ", "06c1eeb6446fcff690c856056b8a6a02": "\n\\begin{align}\n\\mathrm{ker}_n(x) & = \\frac{1}{2} \\left(\\frac{x}{2}\\right)^{-n} \\sum_{k=0}^{n-1} \\cos\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{(n-k-1)!}{k!} \\left(\\frac{x^2}{4}\\right)^k - \\ln\\left(\\frac{x}{2}\\right) \\mathrm{ber}_n(x) + \\frac{\\pi}{4}\\mathrm{bei}_n(x) \\\\\n& {} \\quad + \\frac{1}{2} \\left(\\frac{x}{2}\\right)^n \\sum_{k \\geq 0} \\cos\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{\\psi(k+1) + \\psi(n + k + 1)}{k! (n+k)!} \\left(\\frac{x^2}{4}\\right)^k\n\\end{align}\n", "06c2fa5a49cdd7947f43cc504560f878": "\\overline{\\mathbb F}", "06c34d795a2610039740c4ff5e9afd89": "A_{n-1}(1) \\int_0^\\infty \\exp\\left(-r^2/2\\right)\\,r^{n-1}\\,dr.", "06c361310c1d1c2a6d26c628aa50b14f": " \\mathcal{F} = \\frac{\\Delta\\lambda}{\\delta\\lambda}=\\frac{\\pi}{2 \\arcsin(1/\\sqrt F)}", "06c3ce92fd68d8d369f4796a74c8837e": "q_i(F_S)= F_S", "06c4496e824cc08bdb5a8eec610bef16": " \\gamma_s \\,\\!", "06c46b49f0881195c506925d90d158fb": "t_E,t_{E'} 1", "06c98e14c7c3908709f994ff68005384": "(1-R-\\varepsilon)H_q^{-1}(\\frac{1}{2}-\\varepsilon)", "06c9fd9208a35b057752d5172887d84a": "T = \\frac{ \\lambda v w }{v F w} = \\frac{1}{\\sum e_{\\lambda} ( f_{ij} )}", "06ca06eb3872382fbf005c6e44ab7f81": "\nx = R\\lambda, \\qquad\\qquad y = R\\psi,\n", "06ca59b7c7c0502d709e5b1a414fbde0": "X \\sim \\mathrm{GH}(\\lambda, \\alpha, \\beta, 0, \\mu)\\,", "06cab4a31ea57f55055e4d095dc08f6a": "h = \\frac{(v-3)(v-4)}{12}.", "06cafe5de1b67c6a71ea5c1eee766059": "b_0 . b_1 b_2 b_3 b_4 \\ldots = b_0 + b_1\\left({\\tfrac{1}{10}}\\right) + b_2\\left({\\tfrac{1}{10}}\\right)^2 + b_3\\left({\\tfrac{1}{10}}\\right)^3 + b_4\\left({\\tfrac{1}{10}}\\right)^4 + \\cdots .", "06cb240fa85a363b8dfe7dfacce57926": "O_6(2) \\cong S_8.", "06cbc6fe0922a3a82ac909a372c797fd": "(x-3)x^{14}(x+3)(x^2-x-4)^7(x^2-2)^6(x^2+x-4)^7(x^4-6x^2+4)^{14}.\\ ", "06cc6bc06c290863fe9318fabb6cc26f": "f\\colon R^r\\to R", "06cc7b48df48ae4205f45d63023d8274": "^{\\;}\\mathbb{V}", "06cc832179f822ac4714c2853115975f": "d_y", "06cd3ef006ee03dd9dee6be33b34ac95": "\\int_{-\\pi/4}^{\\pi/4} \\ln(\\sin x+\\cos x)\\,dx=-\\frac{\\pi}{4}\\ln 2.", "06cd663ed5bd4d9da3167789c48d0028": "\\begin{align}\n \\frac{\\partial}{\\partial b} \\left (\\int_a^b f(x)\\; \\mathrm{d}x \\right ) &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\left[ \\int_a^{b+\\Delta b} f(x)\\,\\mathrm{d}x - \\int_a^b f(x)\\,\\mathrm{d}x \\right] \\\\\n &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\int_b^{b+\\Delta b} f(x)\\,\\mathrm{d}x \\\\\n &= \\lim_{\\Delta b \\to 0} \\frac{1}{\\Delta b} \\left[ f(b) \\Delta b + \\mathcal{O}\\left(\\Delta b^2\\right) \\right] \\\\\n &= f(b) \\\\\n \\frac{\\partial}{\\partial a} \\left (\\int_a^b f(x)\\; \\mathrm{d}x \\right )&= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\left[ \\int_{a+\\Delta a}^b f(x)\\,\\mathrm{d}x - \\int_a^b f(x)\\,\\mathrm{d}x \\right] \\\\\n &= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\int_{a+\\Delta a}^a f(x)\\,\\mathrm{d}x \\\\\n &= \\lim_{\\Delta a \\to 0} \\frac{1}{\\Delta a} \\left[ -f(a)\\, \\Delta a + \\mathcal{O}\\left(\\Delta a^2\\right) \\right]\\\\\n &= -f(a).\n\\end{align}", "06cd70be27adef46544f64c887693177": "J_- = J_x - iJ_y,\\quad", "06cd76d1020ab27735a252602fb177fb": "V(x) = \\dfrac{1}{2}kx^2+e\\epsilon(t)x", "06cd8b34c35f763d4ee1d16e68cf4823": "w(n)= \\frac{1}{2} \\,w_r(n) -\\frac{1}{4} e^{\\mathrm{i}2\\pi \\frac{n}{N-1}} w_r(n) - \\frac{1}{4}e^{-\\mathrm{i}2\\pi \\frac{n}{N-1}} w_r(n)", "06ce256e4f7fcf6035ef0555c52ae624": "\n\\vec{C}=2.\\vec{r_2}\n", "06cea412fb13e3f307acaec972edfdc4": "X_1Y_1Z_1", "06ceef85fc5f1f79b9262f97e16620a2": "\\mathbb{D}^qf(t)=\\mathcal{L}^{-1}\\left\\{s^q\\mathcal{L}[f(t)]\\right\\}.", "06cf26fa7a959c1bc54d9696c5487a15": "q(\\alpha^i)=0", "06cf37c067a62dbcfb0edfef71db7ff9": " x = \\sum_{1\\leq{d}\\leq{D}}{q_d} + \\sum_{D+1\\leq{n}\\leq{N}}{q_n} ", "06cf3f21716ec66fe3b8a0415eca9567": "g=14", "06cf5a60d0ff83a69bf792a9392a470c": "\\sigma_e = \\frac{F}{A_0}", "06cfb7d91d13409686276ea1f8443ac9": "\\frac{4\\% - 3\\%}{3\\%} = 0.333\\ldots = 33 \\frac{1}{3}\\%.", "06cff0a2dfea0ce8968c1f57cacc978a": "\\scriptstyle \\sqrt{3}", "06d02a33a188753e4b675a7fc68c9619": " \\hat{g}(k)+\\hat{f}_{+}(k,0) = \\hat{f}_{-}(k,0)+\\hat{f}_{+}(k,0) = \\hat{f}(k,0) = C(k)F(k,0) ", "06d06445a0db1bdfe59687eb37f15370": "x_{9} \\ ", "06d125ac778c36e3c5a4e4e70a4267ee": "\n\\omega^2 = \\omega_{pe}^2 + \\omega_{ce}^2 + 3 k^2 v_{\\mathrm{e,th}}^2\n", "06d1431d41ebf019f454c85760c3cca8": "C_{70} ", "06d17b63bf91b101ee63e7baab89231f": "Z_n^m(\\rho,\\varphi) = (-1)^m Z_n^m(\\rho,\\varphi+\\pi)", "06d183f092c404a9d7ae381aa654aac0": "\\{e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\mid 1\\le i_1 < i_2 < \\cdots < i_k \\le n\\}", "06d1b20b6d623a20b32d43a41defd91e": "\\zeta=\\zeta_0 \\exp(-\\frac{\\alpha r^2}{4\\nu}),", "06d1c20fe9248c2e99b70478a991931e": "S(x)=\\sum_{i=0}^{d-2}s_{c+i}x^i.", "06d1eaab950c44221265ebd40c503472": " v_{n+1} = 0 ", "06d1f9b858a17480c87ae3c15577d0ea": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{k}) ", "06d20a4367635fb23cda65269853e538": "\\scriptstyle{ X \\in Y }", "06d221afdb2d5f56d0361f8d769f2fb7": "r_1\\,", "06d22da9e4615a96674abdfaaebce7b8": " \\int \\frac{dx}{a+bx+cx^2} = \\frac1c \\int \\frac{du}{u^2-A^2} = \\frac1c \\int \\frac{du}{(u+A)(u-A)}. ", "06d26585a3a1d268902592d27d27ef6e": "\\frac{\\partial s}{\\partial t}=(1/T)\\frac{\\partial u}{\\partial t}+(-\\mu/T)\\frac{\\partial \\rho}{\\partial t}", "06d2680d2748feb8040aff35f62670ed": "E_{obs|ref2}=E_{obs|ref1} - E_{ref2|ref1}", "06d2a1325df844c2e1f0873af6d6155c": "d \\to -\\infty", "06d2ea6cb2cc72ca2085187314e50d6b": "\n \\delta V_{\\mathrm{ext}} = \\int_{\\Omega^0} q~\\delta w^0~\\mathrm{d}\\Omega\n", "06d321586f58ac57180d3eb4cdae5ad3": "\\widehat{\\mathbf{p}}=\\frac{\\hbar }{i}\n\\frac{\\partial }{\\partial \\mathbf{r}}", "06d355dcadf9485685d2271cbf2020dd": "\\Delta \\rho_{0}=\\rho_{l,0}-\\rho_{v,0}", "06d35a9e9556c323d829d1dff476c29a": "U = \\frac{(3/5)GM^2}{r}", "06d38a545c57e06c2569f0eea1a45fd0": "S \\cdot \\{\\varepsilon\\} = S = \\{\\varepsilon\\} \\cdot S", "06d45da0e7c98c2fb2dfe85c798fc6f4": "s\\in [0, 1],", "06d49a56b1097ffa642fa7fbbb977a91": " r_E \\left( \\approx \\frac {1}{g_m}\\right) ", "06d531222c5f5fd58c59c1bb3e475329": "\\int e^{-c x^2 }\\; \\mathrm{d}x= \\sqrt{\\frac{\\pi}{4c}} \\operatorname{erf}(\\sqrt{c} x)", "06d55f5e2fe1755898c705f016cdea8a": "X \\ \\overset {P}{\\doublebarwedge} \\ Y.", "06d570cbe959835037c081ce647fefae": " \\bigcup_{i\\in\\mathcal{I}}C_i = \\operatorname{cl}(\\bigcup_{i\\in\\mathcal{I}}C_i) ", "06d59f9ae2ed151caaa258fd663de952": " \\operatorname{st}(x y) = \\operatorname{st}(x) \\operatorname{st}(y) ", "06d5e99d7165083157b74072d13eb13b": "P_k= A_k^{-1} A", "06d60c179e16986dd5326985dc670887": "1-\\frac{8}{\\pi^2}", "06d6475d55d6388c67128f65fc7007c3": "(1,1)/(1)", "06d6594a03a1e8d75ce3381314d6b7f1": "\\lim_{x \\to c}{f(x)} = f(c).", "06d677fe7d2a0321ce2495e91150acb4": "\\epsilon(p,t) =\n\\begin{bmatrix}\n\\epsilon_{1 1} & \\epsilon_{1 2} & \\epsilon_{1 3} \\\\\n\\epsilon_{2 1} & \\epsilon_{2 2} & \\epsilon_{2 3} \\\\\n\\epsilon_{3 1} & \\epsilon_{3 2} & \\epsilon_{3 3}\n\\end{bmatrix}\n", "06d67c5b693f72aa5917b99d98d06b20": " \\nabla^2 \\phi = 0 ", "06d6e1872f82cb9a1db5f9c7c64a7571": "\\chi(1)=1,\\quad \\chi(2)=3,\\quad \\chi(3)=5,\\quad \\chi(4)=6\\text{ and } \\chi(k)=6\\text{ for }k > 4.", "06d6e24415fd48d7de21b90bd3179306": "y=4-x", "06d6e2c20abc7aed22f5b1eb55c6199c": "\\sigma^2 = 3.5033e-02", "06d6f807c5685c2cba55d485275b21dd": "x\\sim y \\iff x\\,R\\,y \\land y\\,R\\,x", "06d706413765857ed4231c13dfe495aa": "x<\\mu-s", "06d71cf7be2620b331fae5cf58c948f1": "\\rho(\\mathbf{y}|\\mathbf{X},\\boldsymbol\\beta,\\sigma^{2}) \\propto (\\sigma^{2})^{-n/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)\\right).", "06d71d56f0c78aad0ce24077fe8590c9": " (x_1^2+\\cdots+x_r^2) \\cdot (y_1^2+\\cdots+y_s^2) = (z_1^2 + \\cdots + z_n^2) \\ , ", "06d7428d5398711bd4d2ff7e2a122f1b": "\\bar{D}=\\mathbf{E}^\\theta[D(\\theta)]", "06d74648983ecf54bb131566b8c5e418": "\\langle X,D,C \\rangle", "06d800b37ae1c15f719d18fcd511e768": "x=\\tfrac{\\pi}{2k}", "06d80eb0c50b49a509b49f2424e8c805": "dog", "06d8192069321dae13f673e4324cf8f6": "\\prod_{n=1}^{\\infty} \\left( 1+C\\beta_n\\right) =P", "06d843d8a9eee3a075aefeeb8178dd05": "\\Delta E_{max} = (1 - \\alpha) E", "06d8f8215a9dc7088c23faf64a73364d": "\\{x_{(1)},\\ldots,x_{(T)}\\}", "06d9276fc2a30f0b2c971565f06e7347": "\\left(\\tfrac{1}{2}z\\right)^\\nu= \\Gamma(\\nu)\\cdot \\sum_{k=0} I_{\\nu+2k}(z)(\\nu+2k){-\\nu\\choose k} = \\Gamma(\\nu)\\cdot\\sum_{k=0}(-1)^k J_{\\nu+2k}(z)(\\nu+2k){-\\nu \\choose k}", "06d96da660982f1e88498de82cde6f85": "\n\\vec k \\cdot \\vec J = -k_0 J^0 \\rightarrow 0\n,", "06d975f667e17991f33b462cef956c1e": "\\vec\\omega = (b, c, d)", "06d9766f8ef1d304b904154bd5149e18": "F \\longrightarrow E \\ \\xrightarrow{\\, \\ \\pi \\ } \\ B", "06d9ea23ffa6fc9d4f535ef2bcdb1a4e": "g_n(z)=\\frac{z^2}{n^3}", "06da660f3d03e1f9bc27af2962dcc537": "I(t)=\\int_0^{a_M}{i(t,a)da}", "06db2dd3ef340c71b2530fc72f4beff2": "N_s\\,", "06db3756924a7876fa447c44f664476f": " \\int_E w(x)\\ dx,", "06db5db6b567d8497fb8c5750e82c1d7": "p_{eq}", "06dbaccc61a71d13ff91db5c0d2705ca": "r_1 = x_1 i + y_1 j + z_1 k , \\quad r_2 = x_2 i + y_2 j + z_2 k)", "06dbbed2b1179fcff18a3f581ec4a699": "\\scriptstyle \\delta_1 ", "06dbe93abce0797f98a1206ea8edabf1": "\\begin{align}\\text{1 Ci}&=\\frac{3.7\\times 10^{10}}{(\\ln 2)N_{\\rm A}}\\text{ moles}\\times t_{1/2}\\text{ in seconds}\\\\\n&\\approx 8.8639\\times 10^{-14}\\text{ moles}\\times t_{1/2}\\text{ in seconds}\\\\\n&\\approx 5.3183\\times 10^{-12}\\text{ moles}\\times t_{1/2}\\text{ in minutes}\\\\\n&\\approx 3.1910\\times 10^{-10}\\text{ moles}\\times t_{1/2}\\text{ in hours}\\\\\n&\\approx 7.6584\\times 10^{-9}\\text{ moles}\\times t_{1/2}\\text{ in days}\\\\\n&\\approx 2.7972\\times 10^{-6}\\text{ moles}\\times t_{1/2}\\text{ in years}\n\\end{align}", "06dbf7054de09e50f2eb8d9740e39928": "E_{tgu} = 0.5 \\cdot 11.848^2 / 4.54 = \\,", "06dc1f15d5e653961721b66c2f50c546": "Z_t = \\sum_{k=0}^t X_k", "06dc81637103e72fdfa625195ea60e44": "\\beta=(\\beta_1, \\beta_2,\\cdots)", "06dca4ab9922618adfc9155350a5b70a": "1 \\in F,", "06dcebfa58fad42a3a3d8303ca2c014f": "z_0 = \\exp(i\\theta)", "06dd0da5c04f7a6c99e16857f0c29817": "A \\rightarrow A \\wedge A", "06dd7d2c0e5a9dd8e9a8e91452c8590a": "\\alpha = \\pi", "06ddaa5ef23158584ff864431938da9d": "V\\to V^*:v\\mapsto v^*", "06de3fdcff77757723e81468cfb6e1b1": "P_{\\mathrm{error}\\ 1\\to 2} = \\sum_{x_1^n(2)} Q(x_1^n(2))1(p(y_1^n|x_1^n(2))>p(y_1^n|x_1^n(1)))", "06dea3a87ee7de5c6ea467d41933b433": "I_b = -I_x \\frac{R_2}{( R_1 \\parallel r_E ) + r_{\\pi} +R_2} \\ . ", "06debf5df321963c1ff477b0de006c05": "p_i = \\left[ \\max_{a \\in A} \\sum_{j \\neq i} b_j(a) \\right] - \\sum_{j \\neq i} b_j(a^*) ", "06dee034ea49ade3d26fe1e451d96b20": "C^{1}_k", "06def552447a886e0bfa720025cef63f": "\\Rightarrow M_n=\\frac{R^n}{n!}.", "06df73567c0247dd180edd56272d3b69": "F_{12}, F_{13}, F_{23} ", "06dfc3da0a33b852be7fbefed9ef5690": "G_i+G_e=G", "06dfcf3b2c231351f49940fb9396f3dc": "\\lambda_n", "06dfe0fe70749a43c6698ae9fc719087": "\\mathcal D_m(M)", "06dfe6f5484e00e9821e49d2464df754": "K_a = \\frac{[HG]_{eq}}{[H_{eq}][G_{eq}]}", "06dfeb4bb3a0ef570bf0994f83b5ba82": "\\partial_\\mu\\left[\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\phi)}Q[\\phi]-f^\\mu\\right]\\approx 0.", "06e0d2fe4275db5bc0d5005a5e89c591": "p_1, ..., p_d, q\\in\\Z,1\\le q\\leq N", "06e109375fa004e433314744d0521158": "f(x) = \\frac{2 \\beta^{\\frac{\\alpha}{2}}}{\\Gamma(\\frac{\\alpha}{2})} x^{\\alpha - 1} \\exp(-\\beta x^2)", "06e14d1a766e16597ace30c8b513befb": "\\varepsilon ^{\\alpha \\beta }", "06e15c8cb9648247f7cf2d8393f04df6": "\\vert\\hat{f}(\\xi)\\vert \\leq \\int_{\\mathbf{R}^n} \\vert f(x)\\vert \\,dx,", "06e168bb73d2b659d657f44db7c1fc7c": "E^{p,q}_1\n= \\begin{cases}\n0 & \\text{if } p < 0 \\text{ or } p > 1 \\\\\nH^q(C^\\bull) & \\text{if } p = 0 \\\\\nH^{q+1}(A^\\bull) & \\text{if } p = 1 \\end{cases}", "06e18996a9d3c2afb1ce39d09f1e8986": "S_z = m_s \\hbar\\,\\!", "06e18ba5b85397f0770e9943a7b8a808": "f_u \\left(\\begin{pmatrix}\na & b \\\\\n0 & 1 \\end{pmatrix}\\right)=a^u,", "06e1926d5d41cef9bbec354b734e14ec": "\\quad (3) \\qquad \\qquad \\bar{\\rho}_i \\left( t_2 \\right) = \\frac{1}{x_{i+\\frac{1}{2}} - x_{i-\\frac{1}{2}}} \\int_{x_{i-\\frac{1}{2}}}^{x_{i+\\frac{1}{2}}} \\rho \\left(x,t_2 \\right)\\, dx ,", "06e24240b47b74861da0de82940a32fa": "\ne_3 =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "06e2745e0e66a583228563c795212f20": "c=\\pm1", "06e2a3dd682d5d67e6be7c625958c372": " k_{\\mathrm{H,px}} = \\frac{p}{x} ", "06e2b3b868988d6dddb72612c4af5f99": "\\frac{1}{{{D}_{Ae}}}=\\frac{1}{{{D}_{AB}}}+\\frac{1}{{{D}_{KA}}}", "06e2b3f28b474386df1ae3cf6d50cb12": "\\mathbf{i}=\\mathbf{r}_i", "06e2c91cdbf2ba72bfae3686775d5315": " \\mu_{i,j}", "06e327d1d370a4c85d1f1558d7cf4d74": "m = \\gamma m_0 \\,\\!", "06e3623afd16b07c4a3a101e51fdcbad": "\\begin{matrix}\n \\mathrm{if} & p_l=p_{1}(u) & p_m=p_{2}(u) & p_n=p_{3}(u) \\\\\n \\mathrm{then} & p'_l=p_{2}(u-1) & p'_m=p_{1}(u-1) & p'_n=p_{3}(u-1)\n \\end{matrix}", "06e39ddc7614317468f1446eb7cbaafb": "\\cosh c = \\cosh a \\ \\cosh b - \\sinh a \\ \\sinh b \\ \\cos \\gamma \\ , ", "06e3c0415761d467d709f78b6a2f39af": " \\log p_A(n) \\sim C \\sqrt{\\alpha n} ", "06e40264795ae083e71e3d43644b5566": "\\begin{align} \\hat{H} & = \\hat{T} + \\hat{V} \\\\\n & = \\frac{\\bold{\\hat{p}}\\cdot\\bold{\\hat{p}}}{2m}+ V(\\mathbf{r},t) \\\\\n & = -\\frac{\\hbar^2}{2m}\\nabla^2+ V(\\mathbf{r},t)\n\\end{align} ", "06e41b676042e7f6903beb74cfddb357": " BT^{-1}", "06e420fac994e1cab4dce5c4863f2b99": "\\mathit{H}", "06e48758f2485170f5d8a32f64c8e8f4": "\\stackrel{\\vec v}{}", "06e4ff064815e7f80da5f70841d17505": "H(x(t)) = m \\frac{d^2(x(t))}{dt^2} + kx(t)", "06e52dab3c87e4a2d735170d93008ea7": "\\begin{bmatrix}\n0 & 0 & 3 & 0\\\\\n0 & -2 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1\\end{bmatrix}.", "06e5325a33498b0229a4bddf89137d86": " (U_sU_\\omega)^r = M \\begin{pmatrix} \\exp(2rit) & 0 \\\\ 0 & \\exp(-2rit)\\end{pmatrix} M^{-1}", "06e626389f5786c0205c99697e6a294c": "Fi_{22}\\;", "06e62aac6c74d1bc3efe7fe7270c02a7": "\\begin{smallmatrix} \\mu = \\sqrt{ {\\mu_\\delta}^2 + {\\mu_\\alpha}^2 \\cdot \\cos^2 \\delta } = 1907.79\\,\\text{mas/y} \\end{smallmatrix}", "06e640a8860ee0240a0c5237354e40db": "\nP_A \\left( 1 + e^{v_A } \\right) = e^{v_A } \n", "06e680a0049734936819f48f82b575ba": " \\mathfrak{a} \\subset \\mathcal{O}_k", "06e6d4c4500e3f92118c38cc01dc8e4c": "5\\zeta(2)\\zeta(5)+2\\zeta(3)\\zeta(4)-11\\zeta(7)", "06e78e6aa0957b68ca5e1def5adec2db": "[H,\\Pi]=0", "06e7a18e76a7fee5fe83ce36965cf2a1": "\\boldsymbol{L}_{y}\\hat{f}(k,y)-P(k,y)\\hat{f}(k,y)=0,", "06e7a7a01e0bda91c7309fcee8b78a62": "F^\\%(*)\\to F(*)", "06e84b6b0430a6929023040832bbf88e": "{\\partial\\vec{B}\\over\\partial t} = 0.", "06e86219c47ebd4c614b54c0e8b79736": " \\mathit{WER} = \\frac{S+0.5D+0.5I}{N} ", "06e8627e8d832f40ecd387bbc3e69ff4": "\n \\frac{1}{T} \\sum_{t=1}^T \\mathbf{1}_{\\{X_t\\in A\\}} \\ \\xrightarrow{a.s.}\\ \\operatorname{Pr}[X_t\\in A],\n ", "06e89453aad485a4ca9d7eb0e0bd05c2": "Q_B = C_BV_B. \\ ", "06e8a474d11071362f5ff94cf9b4068b": "\\lim_{x\\to1}\\frac{\\ln(x)}{x-1}=1", "06e8e3fcf20954509c3473c4299d2536": "\\int_0^a \\sqrt{a^{2}-x^{2}} \\, dx =\\frac{\\pi a^2}{4} ", "06e96890472fe7ab1384d5fff3917118": "l>0\\,", "06e9c29be8b22b9257e57ec136590683": "f(X)\\,", "06ea2b747cafb6b4903c3acf9f83618a": ".\\qquad \\qquad\\qquad\\quad\\;\\;\\; S", "06ea6eb5ed10b6a90fd6ccf94007ae1b": "\\frac{\\partial}{\\partial t}f(x,t) = -\\frac{\\partial}{\\partial x}\\left[\\mu(x,t)f(x,t)\\right] + \\frac{\\partial^2}{\\partial x^2}\\left[ D(x,t)f(x,t)\\right].", "06eafb7f3c501c5cd3f4da601efda614": "\\langle Hu, v \\rangle \\overset{\\mathrm{def}}{=} \\langle u, -Hv\\rangle", "06eb4f9416bd48082827a6ad5f366fe2": "L\\setminus D", "06eb9a650f16ad424939b9b8dcdd3ceb": "s=(i,j)", "06ebaa2b1ba50cf8a0edb806fb1b6ff8": "D_0(f)D_0(\\hat{f}) \\geq \\frac{1}{16\\pi^2}", "06ebb4aa52a557939af15290156fa983": "s=O(n/\\epsilon^2)", "06ebcb95f371509d486f5e59255afbf4": "\n\\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\hat{\\Pi}\n_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left(\n1\\right) },\\delta}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho_{x^{n}\\left( m\\right)\n}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta\n}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\Pi_{\\rho\n_{X^{n}\\left( m\\right) },\\delta}\\right\\} ,\n", "06ebe860836e08b5e4a0ad3731cbd535": "y= \\frac{y'}{x'^{g+1}}", "06ec614a0e2ec8c27aab21d24e399139": "\n\\begin{align}\nm\\frac{d}{dt} \\langle \\Psi(t) | \\hat{x} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle, \\\\\n\\frac{d}{dt} \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | -U'(\\hat{x}) | \\Psi(t) \\rangle.\n\\end{align}\n", "06ec91b0b084c842412cdf066bc7c37c": "f^\\star : X^{*} \\to \\mathbb{R} \\cup \\{ + \\infty \\}", "06ed7e10040e3ee1b3f9a05f97de9c6e": "\\zeta:S\\ddot\\to d", "06edb6b869ad60a20f5feac501131df1": "\\frac{\\partial\\mathcal{L}}{\\partial x_i}=0~~\\forall i", "06ee4a0e90b9a11ae66330843e01977c": "\\hat{f}(x) = \\sum_{i=1}^{k} c_i B_i(x) ", "06ee531b6e99bc6853ac756441b4c77f": "b= 2 a", "06ee789a1ea5e1bc3d9991243b79a4ad": "\\sum_{i}{ q_i \\frac{\\partial f_k}{\\partial k_i} } = -\\sum_{i}{ q_i \\frac{\\partial f_k}{\\partial \\mu} \\frac{\\partial \\epsilon_k}{\\partial k_i} } = -\\sum_{i}{ q_i k_i \\frac{\\hbar^2}{m} \\frac{\\partial f_k}{\\partial \\mu}}\n", "06eec1ccfddb8a755e5938215f7a9657": "V^{-1} (x) \\approx \\sqrt (4\\pi) \\frac{d^{1/2}N(x)}{dx^{1/2}} ", "06eee942ebe1a41550d614af6ad20e90": " i_2=E^2\\sin^2(\\omega t + \\phi)=E^2(\\sin(\\omega t)\\cos(\\phi)+\\sin(\\phi)\\cos(\\omega t))^2\\,", "06ef429c6c40dcb74594090468a61d80": "\\left.\\theta_i\\right.", "06ef9747a150e2ad887581089c78680c": "MV = PT", "06efb9f55f0f8a6b9e0e143007c26d9f": "\\widehat{f^{(k)}}(n) = (in)^k \\hat{f}(n)", "06efc40eaea741de8fa51bbd983437a0": "\\mathrm{D \\cdots H{-}A}", "06efd57083242b84c3eb89bbb1425b40": "2\\,ln\\,\\gamma ", "06efdc8e798ff9ee22c6b30b921daf64": "\\int\\cosh^n ax\\,dx = -\\frac{1}{a(n+1)}\\sinh ax\\cosh^{n+1} ax + \\frac{n+2}{n+1}\\int\\cosh^{n+2}ax\\,dx \\qquad\\mbox{(for }n<0\\mbox{, }n\\neq -1\\mbox{)}\\,", "06efec6d4c0c0da25c6238aaf03fe6a8": "\\sigma(x,x') = \\frac{1}{2} \\eta_{\\alpha \\beta} (x-x')^{\\alpha} (x-x')^{\\beta}", "06f00d95d9d9f3f2c7eb2f30f4736dee": "\\kappa_0 = 0.378893+1.4897153\\,\\omega-0.17131848\\,\\omega^2+0.0196554\\,\\omega^3", "06f01daf487e9d1b5c741b192ce92e64": "-0.75 < \\beta < -0.5", "06f0a515cfb17044e3ca9ec8aa712a6b": "\\overline{p}\\, \\propto \\,V\\,\\frac{\\sigma_1 - \\sigma_e}{\\sigma_1 + 2\\sigma_e}\\,\\overline{E_0}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(4)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,.", "06f0c89efa76c089c159e40cc5eb3609": "xv = v", "06f0c8acd9064fa1d89dfb4a6ed87e16": "q(\\tilde{x},\\tilde{{u}}\\vert \\tilde{\\mu})=\\mathcal{N}(\\tilde{\\mu},C)", "06f0e1edab2cf3baff4208e0f17d4792": " \\tfrac{mg}{Ld}", "06f0f399e2282ba52bfc5a4a3010b68e": "\n \\sum F = 0 ~,~~ \\sum M_{A} = 0 \\,.\n ", "06f14cd36432d7dd4b3e006bbc820201": "\\lambda = \\ell^2.", "06f17ae1b9444097a418c5cdd9f2cdde": "\\hat n=\\textrm{const.}", "06f19e0c3c74019a6fe60e558316752d": "-\\frac{\\partial \\operatorname{cost}}{\\partial \\mu_{ij}}", "06f22bc0a0594b1b47ecd2d686e99cbe": "\\phi \\otimes \\phi^{\\Rightarrow x} = \\phi\\,", "06f234fe042ff4bb0d94cd9463dce0cb": "c[[a,b]]", "06f276fbb135db98d357b0983fd446ed": "\\mu_6=\\kappa_6+15\\kappa_4\\kappa_2+10\\kappa_3^2+15\\kappa_2^3.\\,", "06f27f3f4b6930af17c5a12ad197eaa3": "\\scriptstyle \\mathrm{E}(e^2)", "06f30b25421cacf5df4bcaecc7a8d021": "\\mu_r' = \\bigg({\\frac{1}{2}}\\bigg) \\sum_{k=0}^r \\bigg[{\\frac{r!}{k! (r-k)!}} b^k \\mu^{(r-k)} k! \\{1 + (-1)^k\\}\\bigg]", "06f32ab04e43e768fcdbce800a4054b6": "[\\mathtt{Var}]", "06f3832b60c015244731ae1d6dbc5b20": "\\frac{d}{d\\mu}\np_{\\lambda}-F_{\\lambda}=0.", "06f3c8003382ae94fcf8d0470aadd7f1": "\\forall a \\in A, L(a) = \\mathit{out}", "06f3cd5804a745e27558ff9ca765a6b3": "{\\rm Imm}_\\lambda(A)=\\sum_{\\sigma\\in S_n}\\chi_\\lambda(\\sigma)a_{1\\sigma(1)}a_{2\\sigma(2)}\\cdots a_{n\\sigma(n)}.", "06f3dfda128b2077a1c12feb5ac41d0a": "\\textstyle s^\\alpha+t^\\alpha=1 ", "06f3f5c134185227850a6fb52f7a5cfa": "\\sideset{}{^\\prime}\\sum", "06f465defe4c51ed39be1fdd33c764db": " v(x,\\tau)=\\exp(-\\alpha x-\\beta\\tau) u(x,\\tau).", "06f4797c2b7386626e512d6d2c20c09e": "E_r(r,z)", "06f49d3ac085dde14636ef63d7d311d6": "n(\\vec{r}) ,", "06f4a4eead785d13d8b51f3e7b9290e6": "2x \\in o(x^2) \\,\\!", "06f51ec47cd65ae6ed71d25574b4ade8": "\\mathbf{1}_A (\\omega) = 0.", "06f54460efb0b5c4c906147072b0eef7": "r = 0.0961 = 9.61 \\%", "06f578789605643db73b1890cf52be34": "D = x_{11} - p_1q_1", "06f59c823dd57d3a21d55798c4c302a8": "g_2^2=g_3^3= (g_2g_3)^7=-1,", "06f5d6d42ae6d5f9a9d18907e7392814": " a(x+kv)+ b(y-ku) = ax+by + k(av -bu) =ax+by + k(udv -vdu) =ax + by", "06f6247566f82e01d436b7134b5753d3": "\\scriptstyle f_s.", "06f65750206044de34d3194bf4ff1e0a": "\n \\frac{1}{R} = \\frac{1}{R_1}+\\frac{1}{R_2}\n ", "06f6761ad8398a80686ea3ac861b86c5": "\n\tT_{max} = E {4 M m \\over (m+M)^2 }\n", "06f681e831f53857b7e0edbf9eca5b39": "q_p(1) \\equiv 0 \\pmod{p}", "06f692570e6471fafea645933393cddf": "-2\\Im(\\mathit \\Gamma)=\\tan \\left (\\frac{4\\pi}{\\lambda} x\\right)", "06f695e8d632b1d99c0afb37e1e68a4c": "\\lim_{x\\rightarrow+\\infty}\\arctan(x)=\\pi/2.", "06f6a489209115c5cef3f45036aad3ec": "PA", "06f6c1b6db4342eddb0f52c714b23026": "\\Delta \\mathbf{B} \\in \\mathbf{P}_\\pm(1,0,0)", "06f6df1976c2e03ea84a9f336763f590": "\\overline{X}={X_1 + \\cdots + X_n \\over n}", "06f710cb5ada709d2d6065f0af4f4927": "B_\\infty^{p,q} = \\bigcup_{r=0}^\\infty B_r^{p,q},", "06f745c18b05f95d97cc6f6896de1ff1": "x=s-\\epsilon", "06f765a89dd281c30bd5aa2a4d90f6bc": " \\mathbf{a} = \\sum_{i=1}^N a_i\\mathbf{e}_i = a_1 \\mathbf{e}_1 + a_2 \\mathbf{e}_2 + \\cdots a_N \\mathbf{e}_N", "06f7895cd704b1cb0921cf98aec71926": "\\alpha\\,\\!", "06f7db588b7ed518b4dff3b48f834c1a": "\\int_{X_1\\times X_2} f(x_1,x_2)\\, \\mu(\\mathrm d x_1,\\mathrm d x_2) = \\int_{X_1}\\left( \\int_{X_2} f(x_1,x_2) \\mu(\\mathrm d x_2|x_1) \\right) \\mu\\left( \\pi_1^{-1}(\\mathrm{d} x_{1})\\right)", "06f7fec6a2087c3b4559cc748a44643d": "M(a,b,c) = \\prod_{i=1}^a \\prod_{j=1}^b \\prod_{k=1}^c \\frac{i+j+k-1}{i+j+k-2}.", "06f869a41aa361bf1ad9b85d303467be": "(D_0,\\epsilon)", "06f883a740bbcc55c24333ee8767e954": "M=J", "06f8f61719c9ae54bd872b2f15ac21e8": "{\\mathfrak c} \\leq \\aleph_0 \\cdot 10^{\\aleph_0} \\leq 2^{\\aleph_0} \\cdot {(2^4)}^{\\aleph_0} = 2^{\\aleph_0 + 4 \\cdot \\aleph_0} = 2^{\\aleph_0} ", "06f8f7cc9e0d46723578a08f21b1577e": "u_i = \\overline{u_i} + u_i',\\,", "06f9293d5ce55f612cb7a6ebca367aca": "\\operatorname{tr} (A^* A) = \\sum_j^n |\\lambda_j|^2.", "06f95d0d72cee463dc00300f8b935650": " p_{i} ", "06f95e2140d5bef1d3414796c7d6e0c2": "Z_I\\,\\!", "06f9a75b18c09c0c1a86f9a95630df70": "V(\\varepsilon_i)= \\sigma^2 < \\infty,", "06f9b7b1d3f141742ad1c582b55056ba": "x = \\pm 1", "06f9be585f2e7547a204207eff5fc548": " {R^{\\alpha}}_{\\beta} ", "06fa147a005a6ef2d1e4e2c11a541d97": "\\sum_{i=1}^n(x_i-\\overline{x})(\\theta-\\overline{x})=0", "06fa35c9031e823ee6cfccb5605c4eb6": " x\\mapsto (d_\\lambda f)(x)", "06fa4b907599c8a36554f23497da2208": "C_{\\text{min}, \\text{ss}}", "06fa5385239b7aaf6deb58c60cce8798": "\n\\alpha(u) =\n\\begin{cases}\n \\frac{1}{\\sqrt{2}}, & \\mbox{if }u=0 \\\\\n 1, & \\mbox{otherwise}\n\\end{cases}\n", "06fa62a7df57887836c1e22f862ae08b": "(4~5).", "06faac98935d1cf9b57d0640c6073d4f": "X \\land \\neg X", "06fab9786c1782ba7733c31a17c6c66e": "\\begin{align}a&=6.112\\ \\mathrm{millibar};\\quad\\;b&= 17.67;\\quad\\;c&= 243.5^\\circ \\mathrm{C};\\end{align}", "06fb0756cbb51cef8245388c77460834": "\\int_S F \\, dS", "06fb12324a98e8b31b2819be10b29dca": "\\begin{align}\nx' = \\gamma x - \\frac{\\gamma v}{c}ct & \\Rightarrow & x' = \\gamma(x - vt) \\\\\nct' = -\\frac{\\gamma v}{c} x + \\gamma ct & \\Rightarrow & t' = \\gamma\\left(t-\\frac{vx}{c^2}\\right) \n\\end{align}", "06fbe3c17a36710731842480e1657952": "P=\\frac{T^\\alpha}{R^\\beta}", "06fbf0791485f24f1a0df9ea75544e43": " \n\\begin{align}\ny &= y_{0} + y_{1} + y_{2} + y_{3} + \\cdots \\\\\n & = -\\left[ t + \\frac{1}{3} t^{3} + \\frac{2}{15} t^{5} + \\frac{17}{315} t^{7} + \\cdots \\right] \n\\end{align}\n", "06fbf7e846775b80c4fffad4c0b3055b": "v = \\sum_{i=1}^n v^iX_i,\\quad w = \\sum_{i=1}^n w^iX_i", "06fc0b39a9811f7e78cf9a439d4cef40": "\n\\begin{cases}\n \\frac{dx_1}{dt} = (1-x_2^2)*x_1-x_2+u \\\\\n \\frac{dx_2}{dt} = x_1 \\\\\n \\frac{dx_3}{dt} = x_1^2+x_2^2+u^2 \\\\\n x(t_0) = [0 \\ 1 \\ 0] \\\\\n t_f = 5 \\\\\n -0.3 \\le u \\le 1.0 \\\\\n\\end{cases}\n", "06fc1a78b9aaaee997b0adbfa5992f6c": "(n+1)", "06fc4a6d4d713d72b39ef424e8c7995a": "\\mathrm{Financial\\;leverage}= \\frac{\\mathrm{Total\\;Assets}}{\\mathrm{Shareholders'\\;Equity}}", "06fc5b02a356eaa5b1b33b3f5b7a711f": "\\|u+v\\|^2+\\|u-v\\|^2=2(\\|u\\|^2+\\|v\\|^2).", "06fc5b5f85dbf32589c521ca55e05e10": "\\log (\\operatorname{E}(Y|\\mathbf{x}))=\\mathbf{a}' \\mathbf{x} + b,", "06fcd3f2aa256fd816ec7081a38c30cc": "\n\\operatorname{Var}(X) = \\int_{-\\infty}^\\infty \\frac{(x - \\mu)^2}{\\sqrt{2\\pi \\sigma^2}} e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} } \\, dx = \\sigma^2.\n", "06fcd5f9f7bf19377c6f7c4560d9ddd3": "\\sqrt{12.746 \\times A_m}", "06fce68ac85e7fd4fe558639c55dff48": "\\operatorname{tanh}(z)", "06fd262059da6c3ef9aebeb89b4eae62": "\\mathbf{e}_1 ", "06fd43a831e994e442b64b77ffb70cfb": "\\begin{bmatrix}\n1 & u_{12}/u_{11} & . & u_{1n}/u_{11}\\\\\n0 & 1 & . & u_{2n}/u_{22} \\\\\n. & . & . & . \\\\\n0 & 0 & . & 1 \\end{bmatrix}", "06fd65a45b7d5147e034e4c037c6bb07": "\\|f\\| = \\max_I |a_I|", "06fd6796cdf75b8da2f096efdd36a09a": "X\\ \\sim\\ BW2(a,b)", "06fd689d7a8096ce961bd4f8a53800d1": "U_B=Q^2\\sin^2(\\omega t + \\phi)/2C\\,\\!", "06fde8f3ea98e2025590255693da5a68": "\\{\\Phi_{ij}\\hat{=}0\\,,\\Lambda_{}\\hat{=}0\\}", "06fe11932a45adb4faff9e1461556ada": "K_\\text{joint}=\\frac{2W}{\\Delta \\theta}", "06fe30b11b4e7f2b5d4ca7eff02fd65b": "\nG(k) = {1 \\over i\\omega - {k^2\\over 2m} }.\n\\,", "06fe3fd50fd4de394e13d4e6c8ca2e2b": " p_m", "06fe9eed3a7ef77fb236b4115bc813df": "L_1(B) \\subseteq V", "06fed6899c66d75d74f56fa57e2e7c97": "k'_L = 0.664{D_{AB} \\over x} Re^{1/2}_L Sc^{1/3}", "06fef6cf9cd6d4b3c27115712d7f9f89": "b\\;", "06ffd1b14a65819e385ba237fcaeeecb": " X^i Y^{n-i}, \\quad 0\\leq i\\leq n ", "06ffeaf4615e304202a27b140949c683": "T=\\{(a,v)\\colon \\|a\\|=1,\\, a\\cdot v=0\\},", "06fff7df730a38b1bce6ec8adf57cd68": "A(\\alpha_1, ..., \\alpha_n)", "06fffc7bee852c3e3a52d94e7637c348": "\\Lambda_{\\mathrm{m}} =\\Lambda_{\\mathrm{m}}^\\circ - K\\sqrt{c}", "07001c08cbfd50263d50d487c27d473f": "\\left[F\\left(-1\\right),F\\left(1\\right)\\right]", "07001e7bd5d796308250f06e997b336f": "\\begin{align}\n& \\{ \\Gamma,\\Gamma \\} =2I && \\{ \\Gamma, Q \\} =0 && \\{ \\Gamma, \\bar{Q} \\} =0\\\\\n&\\{ Q,\\bar{Q} \\}=2Z && \\{ Q, Q \\}=2(H+P) && \\{ \\bar{Q}, \\bar{Q} \\} =2(H-P) \\\\\n& [N,Q]=\\frac{1}{2} Q && [N,\\bar{Q} ]=-\\frac{1}{2} \\bar{Q} && [N-[1-q,\\Gamma]=0 \\\\\n& [N,H+P]=H+P && [N,H-P]=-(H-P) &&\n\\end{align}\n", "07002e45e18227e8552911cf43b3eb74": "ip\\,", "07004971f8da61850b3167f634758095": "\\Delta G^\\circ = -nFE^\\circ \\,", "07009d1fe5a8f3356a54628b0a9a2e2c": "=\\lim_{x\\to\\pm\\infty}\\left[\\left(x-\\frac{1}{x}\\right)-x\\right]", "07015c9bc41543737124da6128a321cf": "|1\\rangle \\otimes |1\\rangle = \\frac{1}{\\sqrt{2}} (|\\Phi^+\\rangle - |\\Phi^-\\rangle).", "07019868d7cf7840cc2569aa632692b5": "\\scriptstyle{E_{2}}", "0701d7e98e5b319a2d6eca4593dbf8ca": "\\Delta E = \\hbar \\omega", "0701e21caf8c27e9c3c3fffaddae03da": "\\gamma=\\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}", "07023e53e01690646b5b2d31d3a79551": "\\mathbf{AA3} = \\begin{bmatrix}\n-\\beta & 0 & 0 & 0 \\\\\n0 & -\\beta & 0 & 0 \\\\\n0 & 0 & -\\beta & 0 \\\\\n0 & 0 & 0 & -\\beta \\end{bmatrix}", "07024215329a71973769a3641eb82d08": "\\psi^{*}(\\theta|_{W})=0, \\forall \\theta \\in \\Lambda_{C}^{1}\\pi_{r+1,r}.\\,", "07025eb37092f002d5e6a63feb9826fa": "\\AA^{-2}", "0702636caf3faa212e5ca5901d56b7a8": "\n\\begin{align}\nA_j& = \\frac{\\sum_{i=1}^{L} x_{L(j-1)+i}}{L} \\quad \\forall j& = 1,2,\\ldots,N\n\\end{align}\n", "070268441b1ba4ce1b40b9226759b5fd": "M^{\\rm{SN}}(x) = h - eFx -e^2/(16\\pi\\varepsilon_0 x), \\qquad\\qquad (3)", "07029e1595c60a230b6af32248fbbf84": "\nf_Y(y) = f_X \\left( g^{-1}(y) \\right) \\left| \\frac{d}{dy} g^{-1}(y) \\right|\n", "0702e0a272267151de0194ef145a01ed": "\\{X_{\\alpha}\\}_{\\alpha\\in\\Alpha} \\subset L^1(\\mu)", "0702e7e54d88e1deeb932e2f20843bbe": "f: S^1 \\rightarrow \\mathbb R^3.", "0702ec22009626b16d21444a52c6cf46": "\\gamma_{13}", "0702f8fa72ff2c6ff7eb93f0ba58aeee": "\\frac{x-a}{x-c}\\cdot \\frac{b-c}{b-a}", "070322d303658ef53b56abd0278e694f": "((1 \\times 2) \\times 3) \\times 4\\dots", "070333d4041f0bc56f1494be2d8d1ef2": "\\displaystyle \\hat{f}_3(\\omega) \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{(2 \\pi)^{n/2}} \\int_{\\mathbf{R}^n} f(x) \\ e^{-i \\omega\\cdot x}\\, dx = \\frac{1}{(2 \\pi)^{n/2}} \\hat{f}_1\\left(\\frac{\\omega}{2 \\pi} \\right) = \\frac{1}{(2 \\pi)^{n/2}} \\hat{f}_2(\\omega) ", "07034752e26042109fd161506d3571d8": " \\omega_{\\rm orb} = \\frac{L}{r^2} = \\sqrt{m/r^3} ", "07039f7406f216840d06c06d80a1e13b": "Z,", "0703a367605efeb9385d8afc267f2e77": "\\ k_b M = S - \\sum_i( I_i E_i),", "0703f26e8171d3a7864cbfe3e3336935": "G_{ab} + \\Lambda g_{ab} \\, = \\kappa T_{ab}", "0703fb725f82df343b488a1b0d99e7c3": "S = A[x_0,\\ldots,x_n]", "0703fd136dc12b6e3c60af31b2003aed": "I = \\int L(\\mathbf{q}, \\dot{\\mathbf{q}}, t) \\, dt ~,", "070440c136d68b9abc282fd3ef723457": "\\frac{\\partial (\\mathbf{u} + \\mathbf{v})}{\\partial \\mathbf{x}} =", "070458900dad2a691b5356912410f346": " \\sigma_x = \\left( \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix}\\right) ", "070465b8297a9c0490a2657287978584": "\n\\|Ax\\|_{\\beta}\\leq \\|A\\|_{\\alpha,\\beta}\\|x\\|_{\\alpha}.\n", "0704911db4e3ec5f12d536fbfd7ed629": "c\\,\\!", "0704b8919da6315a296827d30201318e": "\\mathbf{H}_\\alpha(x) = \n \\frac{2{(x/2)}^{\\alpha}}{\\sqrt{\\pi}\\Gamma(\\alpha+\\frac{1}{2})}\n \\int_{0}^{\\pi/2} \\sin (x \\cos \\tau)\\sin^{2\\alpha}(\\tau) d\\tau.", "0704d96823ccea6a0bdbe0072d4aad24": "H_2^{16}O_{(l)} + H_2^{18}O_{(g)} \\rightleftharpoons H_2^{18}O_{(l)} + H_2^{16}O_{(g)}", "0704dc9bb7fc4caccdf59e000795f364": "\\left[Re(-1/2), Im(0)\\right]", "0704fa08767f443cc1448a563edbbd5d": "r=\\tfrac{1}{2}", "07055850bcbb43dcc9c8609e7cc9e31f": "\\frac{1}{S_1} + \\frac{1}{S_2} = \\frac{1}{f} ", "07059ea6785419c6f38f887b999356f2": "\\rho(A) < 1", "0705a0d1c4faa496c25a0ec3d9162e95": " \\int \\bar\\psi(\\gamma^\\mu \\partial_{\\mu} - m ) \\psi ", "0705d1be3febdcf632f0b687bc4a1e6a": "A\\equiv((B\\equiv C)\\equiv((C\\equiv A)\\equiv B))", "07062a14bdcb7842ff61a0f6e0ea15b9": "u^{\\pm i}", "07064e3e2d782d232254b31c5bbb03d6": "g \\notin F", "07067e13cc0db2b99caced6cca364657": "\\mu_{ab}^{(c)}(t) =0", "0706b9c536eb81f763daa0a36b1eb6fe": "S(E,a_E,a)= \\prod_{u=a_E}^{a-1} \\left[ 1-q(E,a_E,u)\\right]", "0706d7c09de74ed1481735753c2ad5fa": "\nP(k) = {n-1\\choose k} p^k (1 - p)^{n-1-k},\n", "070701aeaccfe5013215c9da112ceed7": "-(1/T)\\nabla\\mu_j", "0707579e62f807dd4a752af8617b0f69": "U(1) \\hookrightarrow S^{2n+1} \\twoheadrightarrow \\mathbf{CP}^n", "0707669836d19443cf6c5cc89ca963e6": "y(t)", "07078c91cc4fdcf0d49cc18bbddc12bd": " \\zeta=\\chi + i \\eta ", "0707afd12d13ec433d645854ca98b125": " I_2 = \\frac{V_2}{|Z_{total}|}\\angle (-120^\\circ-\\theta) ", "0707c48bc143637a3ae1679c42f505f7": "I\\times S^1", "0708149ad8eaaaed1072969025150497": "\\sum F_x = \\Delta (ma_x)", "0708208ffee01d6980a07141fb2ce279": "\\mathbf{A}^{\\mathrm{T}} = -\\mathbf{A} .", "070835c49d3a13f95f614c65665eeebf": "\\frac{1}{\\Gamma(z)} = z e^{\\gamma z} \\prod_{n=1}^{\\infty} \\left(1 + \\frac{z}{n}\\right) e^{-\\frac{z}{n}}", "0708572de1f982adb99029dd6bec9dba": "\\Phi(\\vec{r})=\\frac{1}{4\\pi Dr}\\exp(-\\mu_{eff}r)", "07086da1acd701594ea69101cdaba123": "{kT \\over q}", "07089d218561cbd6cd4ea199a4c78913": " |g\\rangle=|(\\hat{B}-\\langle \\hat{B} \\rangle)\\Psi \\rangle.", "0708a28c7a65508d6f7b18ee71e983dd": "x_1, x_2 \\in I", "0709039bb733667dc30f69865cdf7de2": " \\alpha_{k} = \\frac{\\mathbf{p}_k^\\mathrm{T} \\mathbf{b}}{\\mathbf{p}_k^\\mathrm{T} \\mathbf{A} \\mathbf{p}_k} = \\frac{\\mathbf{p}_k^\\mathrm{T} (\\mathbf{r}_{k-1}+\\mathbf{Ax}_{k-1})}{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{A} \\mathbf{p}_{k}} = \\frac{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{r}_{k-1}}{\\mathbf{p}_{k}^\\mathrm{T} \\mathbf{A} \\mathbf{p}_{k}}, ", "070914944ea53f62a72003d0f4842860": "\\bigl\\|\\sum_{k=0}^\\infty u_k \\bigr\\|^2 = \\sum_{k=0}^\\infty \\|u_k\\|^2.", "07091a2d49315b83c62a336e1c6c9dce": "C = \\frac{\\;Q}{u}\\cdot\\frac{\\;f}{\\sigma_y\\sqrt{2\\pi}}\\;\\cdot\\frac{\\;g_1 + g_2 + g_3}{\\sigma_z\\sqrt{2\\pi}}", "0709f02a39c8929c9e10b7e1eb005fd0": "\\left(k,n\\right)", "070b0ab70ce7186a1c9d02a1827f73da": "\\begin{pmatrix}\n\\mathbf{e}_+ \\\\\n\\mathbf{e}_{-} \\\\\n\\mathbf{e}_0\n\\end{pmatrix} = \\mathbf{U}\\begin{pmatrix}\n\\mathbf{e}_x \\\\\n\\mathbf{e}_y \\\\\n\\mathbf{e}_z\n\\end{pmatrix} \\,,\\quad \\mathbf{U} = \\begin{pmatrix}\n- \\frac{1}{\\sqrt{2}} & - \\frac{i}{\\sqrt{2}} & 0 \\\\\n+ \\frac{1}{\\sqrt{2}} & - \\frac{i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\\,,\n", "070b83683a0c810c647073f04a216534": "\\mathbf{i}=(\\mathbf{r}_i,\\Omega_i)", "070b9ebc3aca0b4c9df25a47aa63331c": "\\sum_{q^\\prime}\\left[P_a\\right]_{qq^\\prime}=1", "070bba10304c823c16dffe6457617b82": "\\omega>\\omega_p", "070bbaa9ce926608de688431864bbe8a": "\\tau_\\sigma : V^{\\otimes n} \\to V^{\\otimes n}", "070c05b4f4c6bd8d10fce8d41e488868": "s = \\dfrac {q-1} {1 - a_1} \\bmod{\\ell} ", "070c5b0034631d6d60580faa61c3dd5b": "(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)\\,", "070c5bb99cbfd08e0249b95ecb5d0daa": "I = m(L/2)^2 + m(L/2)^2 = 2m(L/2)^2 = mL^2/2\\,", "070d27d5ae9f59ac9133ce4d69ff2be6": "R_i=\\sum_{j=1}^m r_{i,j} ,", "070d31cec69ed1b7b2bd488cde6138ab": "S^{*} = \\{ (o_{i}, o_{j}) | o_{i}, o_{j} \\in X_{k}, o_{i}, o_{j} \\in Y_{l}\\}", "070d5b305cef4688fddf42beeda3ea45": "\n\\begin{align}\n \\textbf{a}^* &=\\frac{2\\pi\\textbf{b}\\times\\hat{\\textbf{n}}}{|\\textbf{a}\\times\\textbf{b}|}\\\\\n \\textbf{b}^* &=\\frac{2\\pi\\hat{\\textbf{n}}\\times{\\textbf{a}}}{|\\textbf{a}\\times\\textbf{b}|}\n\\end{align}\n", "070d881e0d5e48fdb27cbd9ac84a89f2": " \\cap A_\\alpha ", "070d8dbcdfac6ce17ad9e33703927077": "f'(x)=2x\\sin(1/x)-\\cos(1/x)", "070df022d5055dce70f882c03fa6549d": "x(t)\\in \\mathbb{R}^n", "070e04501111e2ad0b0608cb37c5d2ac": "E_{11}=e_{(\\mathbf I_1)}+\\frac{1}{2}e_{(\\mathbf I_1)}^2\\,\\!", "070e306772d9e79210ef776d8a66a8a7": "\\frac {d} {dx} \\left[x^{n+1}J_{n+1}(x) \\right] =x^{n+1}J_n(x)", "070e80302d8b5797e3cd275e5a2d10fa": " E(Q_t) = \\delta + Q_{t-4} ", "070e9826cccd011b6d5560decbbcc991": " j^{1}\\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2}) \\,", "070eddbbf27e7936e557c6e2ff2bc758": "\\frac{1}{r^4} P^1_3(\\sin\\theta) \\sin\\varphi = \\frac{1}{r^4} \\frac{3}{2}\\ (5\\ \\sin^2\\theta - 1) \\cos\\theta \\sin\\varphi\n", "070edf626867f2e74beb2a9f117e3d17": "Q=\\{(s,t_e)|s \\in S, t_e \\in (\\mathbb{T} \\cap [0, ta(s)])\\}", "070f583f36e31d4401c0aff07df3ece9": " \\mathbf{L} = m r^2 \\boldsymbol{\\omega}", "070fecf848d0313aa08c027f08f73a4f": "\\mathbf{F} = q[- \\nabla \\phi - \\frac{d\\mathbf{A}}{dt} + \\nabla(\\mathbf{A} \\cdot \\mathbf{v})]", "070ff24de65bdcfa9e46b4c5adab778a": "\\ \\alpha_i ", "0710302b40db7631dcd79f68631d2a41": "f\\colon V \\to V", "07104fe9a27abef8ae57d48ed80223a8": "\nJ", "0710afd247c0582195d440cc0e12ba43": "f_1(x)f_2(y)\\le f_3(x\\vee y)f_4(x\\wedge y)", "0710b689e4caa52df41a447f2f810891": "\\operatorname{P}(X\\leq m) \\geq \\frac{1}{2}\\text{ and }\\operatorname{P}(X\\geq m) \\geq \\frac{1}{2}\\,\\!", "0710c12e6cbddc77d94d8a03e078dd27": " \\langle \\varepsilon_q | \\psi_N \\rangle = \\langle \\varepsilon_q | \\frac{1}{\\|\\psi\\|} \\left ( \\sum_{i = 1}^n c_i | \\varepsilon_i \\rangle \\right ) = \\frac{c_q}{\\|\\psi\\|} \\,,", "07110254746dcae91bc441539c119e0e": "\\lambda \\,", "071117439d4d72b051e49c65ff9a4f02": "T_\\text{hold}=T_\\text{load}\\cdot{e}^{-\\mu\\cdot\\phi}\\quad\\text{ or } \\quad T_\\text{load} = T_\\text{hold}\\cdot{e}^{\\mu\\cdot\\phi} ", "0711354093f22350c807383161c718ad": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{k=0}^{n-1}T^kf", "071148229a4c01ef09ca2c0b77230f2c": "\\cos (\\alpha + \\beta) = OB = OA-BA = OA-RQ = \\cos \\alpha \\cos \\beta\\ - \\sin \\alpha \\sin \\beta\\,", "07119a27e858f421530bbcace27168a0": "\\Omega(G)/G", "07119d5545091b075c4d361f8488abb2": "\\dot{r}_{j} = \\lambda_j r_j + r_{j-1}, j=2,3,\\dots,n", "0711d4567b6e6bc02a1bbb73e87c497b": "\\operatorname{E}\\,\\hat\\sigma^2 = \\frac{n-p}{n} \\sigma^2", "07128930ff48fb3fc74418d68b9f4a23": " \\frac {n!} {(n-k)!k!}. ", "071302c7ae2dc33849d7424158fa7569": "P_A = A (A^\\mathrm{T} A)^{-1} A^\\mathrm{T}. ", "07138f98c839393571d4c74d772b1305": "\t\\begin{array}{rr|rr} \n 1x & \\text{-}13 & 16x & \\text{-}81 \n\\end{array}", "0713a52582a53f4d7ba16c4c6ed27031": "C_1 \\subseteq C_2", "0713a6b3411166cb06f7ab980d9f5ede": "m_{\\mathrm{TNT}}", "071433da6a0b97575672c8502b6da5e8": "01.\n\\end{cases}", "0724b2ff19e6ceca34d0d2c3314d23c3": "\\sum_k \\kappa(u_{ik}) u_{kj} = \\sum_k u_{ik} \\kappa(u_{kj}) = \\delta_{ij} I,", "0724f4e191c17098d6f6b4b92ed70159": "\\{\\max cx \\mid x \\in P\\}", "07250ba09253f459138209af2c9054f7": "\\displaystyle{z_n=re^{2\\pi i n\\over N}}", "07250d8d86eff1000acaf0522dc3ac5f": "\\forall n < t", "07252b644efc2a703f7f681fbfce9b2d": "i\\frac{\\partial}{\\partial t}\\sigma (q,t)=L\\sigma (q,t).", "0725648f358978c7534ec4f1d491abb0": " \\mathcal{O}_X ", "0725a35b3a82eb04f1c74c9d832e9028": "g_{\\mu \\nu}", "0725d5c8eb96d73dd518391b7fd06712": "S^{\\alpha }", "072632ad30a9e26b24bee86e6ce49aae": "\n p := -\\tfrac{1}{3}~\\text{tr}(\\boldsymbol{\\sigma}) ~;~~\n \\boldsymbol{s} := \\boldsymbol{\\sigma} + p~\\boldsymbol{\\mathit{1}}\n", "0726689d2e61a048c115006c9ceea155": " e,", "072693264f50b3f4201bffb3e9a3139b": "\\sigma_{rr}\\, ", "0726a69d0264198a8f1a2a9117893503": "\\frac {\\mathrm{d} \\mathbf{r}}{\\mathrm{d}t} = \\lim_{{\\Delta}t \\to 0} \\frac {\\mathbf{r}(t + {\\Delta}t)-\\mathbf{r}(t)}{{\\Delta}t} = \\frac{\\mathrm{d} \\boldsymbol{\\ell}}{\\mathrm{d}t} \\ .", "07270aab27a420bb7c3951224b597e4b": " = \\frac{1}{n(n - 1)} [(\\sum_{j=1}^k n_{i j}^2) - (n)] ", "072753aa297b0aea0e4624f94d777ff1": "\\langle U_{g} [e], [e] \\rangle_f = f(g)", "0727cde1da04a53168ad6bb916d51f52": "\\mathfrak{sl}_n", "0727ea3e3281289bd156591d3e813ad1": "\\scriptstyle\\boldsymbol{u}(\\boldsymbol{x})=(u_1(\\boldsymbol{x}),\\ldots,u_p(\\boldsymbol{x})) ", "072858900caf8f4db8959233cc538734": " u_{jj} = \\partial_j u_j", "072871070204574fc3d379c09d1b19b9": "\\mathbf{b}=(b_1,\\dots,b_n)^t", "07289bb62b68542477d5e3ecb4cfc7cc": "f(n) = 0", "0728caa5bd49579b7d719edbd2159387": "t_0 = t_f \\sqrt{1 - \\frac{3}{2} \\! \\cdot \\! \\frac{r_0}{r}}\\, .", "0729aaabbe5af8ad7331adac3ad03e01": "\\frac12 a_0-\\frac14\\Delta a_0 +\\frac18\\Delta^2 a_0 -\\cdots = \\frac12-\\frac14.", "0729d51d0f40afdc533043d47fa40214": "10^{15}", "0729ed42759d8aade6e27204f103be48": "\\ S_c", "072a73c94590db2748152d36bc19dccd": "x=2+3t,\\;\\;\\;\\;y=-1+t\\;\\;\\;\\;z=\\frac{3}{2}-4t", "072a8af452e91f60c5eba13af18cf72c": "c_0 \\frac{\\pi}{4} = \\sum_{n=1}^N c_n \\arctan \\frac{a_n}{b_n}", "072adfc230e8952b5cb62880bf69272c": "E_o(\\rho,Q)= -\\ln\\left(\\sum_{y} \\left(\\sum_{x} Q(x)[p(y|x)]^{\\frac{1}{1+\\rho}}\\right)^{1+\\rho}\\right), ", "072af4c81ba2f88175e28f970ce3d1c9": " v_{k+1} = (4 \\beta^2 - 2) v_{k} - v_{k-1}. \\,\\!", "072b01b85e9d24695f60c4ab591dfa6a": "\\frac{d}{dx}\\int_a^x f(t)\\, dt = f(x).", "072b030ba126b2f4b2374f342be9ed44": "60", "072b2f2acfebae9e9efe2b99ea972ccc": "\\displaystyle (\\beta)", "072c54ea4bb453f4eb49b830901b60c8": "Q_i[x^j_\\alpha(t)]=t \\delta^j_i. \\, ", "072cdabf57aff1d612245d5a028b91ee": "\\mathrm{kV_p}", "072ce830e5ecb5328bbda0aad2281265": " 2(\\ell w + \\ell h + wh) \\, ", "072d4b9ebb7d8b49bc729ef951f501ed": " \\mid \\langle \\psi_{x\\pm} \\mid \\psi_{y\\pm} \\rangle \\mid ^ 2 = \\mid \\langle \\psi_{x\\pm} \\mid \\psi_{z\\pm} \\rangle \\mid ^ 2 = \\mid \\langle \\psi_{y\\pm} \\mid \\psi_{z\\pm} \\rangle \\mid ^ 2 = \\frac{1}{2}. ", "072d7b0f3eb30fcd4fccdbe870fa3994": " h_L(t) = \\delta(t) -{R \\over L} e^{-t \\frac{R}{L}} u(t) = \\delta(t) -{1 \\over \\tau} e^{-\\frac{1}{\\tau} t} u(t) ", "072db38fbbae890b4bd5ae5e1ccbb022": " \\lim_{x \\to0^+} 0^x = 0. \\! ~~ (8) ", "072db40c19a10e624b1529e767914484": "U_0=\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}", "072e29653f2eef3b350835b84e6b3744": "MD(\\varphi \\rightarrow \\psi) = max(MD(\\varphi), MD(\\psi))", "072e5620f53251f5440ad8fc03bf9736": "v=\\partial/\\partial x_j", "072f40b0d1204d25dab5d17da842c1e0": "\\gamma^0 = \\begin{pmatrix} 0 & I_2 \\\\ I_2 & 0 \\end{pmatrix},\\quad \\gamma^k = \\begin{pmatrix} 0 & \\sigma^k \\\\ -\\sigma^k & 0 \\end{pmatrix},\\quad \\gamma^5 = \\begin{pmatrix} -I_2 & 0 \\\\ 0 & I_2 \\end{pmatrix}.", "072f4ed969ed563ed39bb9a512f7a7cd": "(7.f)\\quad e^\\psi =\\,\\Phi^2-2C\\Phi+1\\,.", "072f4fd15987cf40a5ba20fc14a3c165": "1 + 1 = 2", "072f7417db2400c9952b7145ed281f46": "\\bar{X}+X_\\xi", "072f7c69d1d6211625222dc08a9465c1": "z(n;2)=n^{3/2}(1+o(1)).", "072fa72ea632294b7d71df76773551f8": "\\frac{\\partial \\ln |\\mathbf{X}^{\\rm T}\\mathbf{X}|}{\\partial \\mathbf{X}} =", "072fdbe4e27341c370c721ad551af14f": "1100000 - 1011101 = 11", "072ff341eac911182c5fac7794d611e4": "Solow Residual=g_{(Y/L)}-\\alpha*g_{(K/L)}", "072ffa50d2cf35595e2754cbf0677e95": "Rd\\Omega^2/dR=-3\\Omega^2<0\\ ,", "073016c126332f131da2cb8a9a6bea1a": "a_{N-1} a_{N-2} \\dots a_0", "0730280d3582824e0ec54ddeeab7e33c": "\\alpha = E[R_p-R_b]", "07302b9c5e204ccdad2e0265cf1545a6": "d = revolutions \\times d_f", "07304a7ad2da8514893da22daea73881": "XX = X^2", "0730a3f9a9adfcb75a656c093332d36e": "l_{i}^-\\rightarrow l_{j}^{-}\\nu_{i}\\bar{\\nu_{j}}", "0730b75e96c0453b1b196be7ff4fa194": "vu", "0730f9104d1a5098c4d88fabcdbfc8b9": "q\\, =\\, \\tfrac12\\, \\rho\\, v^2", "073104b4040b9c8ddd9f09b54805cbaf": "\\ln W = \\alpha N+\\beta E\\,", "07311e20ca459e52fef6ff1ed7e74c96": "O_X(U)", "073234664a210a834b8f8d5ff279ec95": "x'=-x.\\,", "07323a39242f586b7f085dc127d795c0": "\\neg X", "0732488d61e60296c33ec87fba7602f0": "R_{ix}(t+ \\Delta t)=P_{ix}(t+ \\Delta t)+\\sum \\frac {T_m(t+ \\Delta t)}{l_m(t+ \\Delta t)} \\times (X_j(t+ \\Delta t)-X_i(t+ \\Delta t))", "0732bcf93456bd5d72097502b87fd541": "\\Delta\\Omega=(\\Omega_{0}-\\Omega_\\mathrm{pole})", "0732c8aad6337e8d8bbd515c4693c2bf": "\\mbox{Certain safety factor} = \\frac{\\mathrm{LD}_{1}}{\\mathrm{ED}_{99}}", "0732ec27993507494f2eb050f361d1fe": "\\textstyle{5 \\div {1 \\over 2} = 5 \\times {2 \\over 1} = 5 \\times 2 = 10}", "0733483cabd437eda80d9da0d3f4018a": "V_y > V_x", "0733b8464937afd4081f25c09be53fa1": "\\begin{bmatrix} c_1 & -s_1 c_3 & -s_1 s_3 \\\\\n s_1 c_2 & c_1 c_2 c_3 - s_2 s_3 e^{i\\delta} & c_1 c_2 s_3 + s_2 c_3 e^{i\\delta}\\\\\n s_1 s_2 & c_1 s_2 c_3 + c_2 s_3 e^{i\\delta} & c_1 s_2 s_3 - c_2 c_3 e^{i\\delta} \\end{bmatrix}. ", "07341b95aeda2633856303d8f9cb497d": "A = \\begin{bmatrix}\\mathbf{a} & \\mathbf{b} & \\mathbf{c}\\end{bmatrix}", "073456157e857bf7ec76ee4ea25d69d0": "-\\sqrt{-r}", "07347921990cb6f18d2e46d3212030e1": "A^D=0.", "073546f534c6e6d0a62a04eefd1aa8bf": "R_{k+1}(a, b) = 1", "07354e7d280d293e90b43918abcfecd4": "S(f) = \\frac{\\sigma_Z^2}{| 1-\\sum_{k=1}^p \\varphi_k e^{-2 \\pi i k f} |^2}.", "073575b3716398185104175a18564140": "p=18", "07357ff8e80f38aa7f7bd72df210f1b2": "\\sqrt{\\exp}", "07358ebcd581a065b84eb48b7362b09c": "D_L \\ = \\ R_0r_1(1+z) = \\frac{c}{H_0q_0^2} \\left[q_0z+(q_0-1)(-1+\\sqrt{1+2q_0z})\\right]", "0735d9aa7920768b9cfe84434d9f18c6": "a^2 k (1-a)^{k-1} \\, ", "0736059add7e8fe4e6ff63b06b623349": "~(x)_n\\equiv (xT_h^{-1})^n=x (x-h) (x-2h) \\cdots (x-(n-1)h)", "073629db8e56188b25f5fc01c858587f": "number = normalized( weight / mean packet size )", "0736a56f3d66e66ec6c1fa27886e637e": "\\varphi(n^{s+1})", "0736ace3b1f283c0190a4fdb6b4451ee": "|df_p(v)\\times df_p(w)|=\\kappa|v\\times w|\\,", "0736b52ab6852acc846c382d0a356ef6": "f(x,y) = 181.617 \\,", "0736be535ab20cdb901f3b10b3f6601c": "\\theta[\\mathbf{f}] = \\begin{bmatrix}\\theta^1[\\mathbf{f}]\\\\\\theta^2[\\mathbf{f}]\\\\\\vdots\\\\\\theta^n[\\mathbf{f}]\\end{bmatrix}.", "0736d932c4f810737387df1b18b79499": "\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n1 & - & 0 & 0 & 1 & 1 & 0 \\\\\n1 & 0 & - & 0 & - & 0 & 1 \\\\\n1 & 0 & 0 & - & 0 & - & - \\\\\n0 & 1 & - & 0 & 0 & 1 & - \\\\\n0 & 1 & 0 & - & 1 & 0 & 1 \\\\\n0 & 0 & 1 & - & - & 1 & 0\n\\end{pmatrix}", "0737179ee66c8461eeafd1b317438d93": "{n \\choose \\lfloor{n/2}\\rfloor} \\ge {n \\choose k}", "0737552727d3f52d5f6ac33e430cccf9": "\\sigma(X', X)", "073883b3807515b371c7103bcd50240f": "\\mathbb{E}\\left [ ((H\\cdot M)_t^*)^p \\right ] \\le C\\mathbb{E}\\left [(H^2\\cdot[M]_t)^{\\frac{p}{2}} \\right ]<\\infty", "07388de6996a4bcd801b6bc90aa9df6c": "(p_n)_n\\,", "0738cf6a34f09bfa105a8f9bb6bfb679": "(u_1,v_1) = (\\cos\\theta\\,w_1-\\sin\\theta\\,z_1,\\,\\sin\\theta\\,w_1+\\cos\\theta\\,z_1)\\,\\!", "0738f11969551fbb00584191dfecd4e5": "(A\\to\\neg B)\\to(B\\to\\neg A)", "07396449ec8b62cc97795b456882987d": "\\delta(g(x)) = \\sum_i \\frac{\\delta(x-x_i)}{|g'(x_i)|}", "0739768b9134483933fc1ee966f3a4cd": "g^{efghcdb}", "07399a25bdc257ad80519b3e10b08e02": "\\prod _x f(x)\\,", "0739a75a70c0fbd83ac74b5789627a2a": "\\alpha\\beta\\gamma\\cdots", "073a512b87277b08f8abadf785cbff48": "J^{\\alpha} = \\, (c \\rho, \\bold{J} ) \\,", "073a52e4766b7792036a3077e4052d23": "{{\\gamma }_{k}}(X)", "073a6593121390a4317e466933e744c6": "s(t) = A\\cdot \\cos(\\omega t + \\theta),\\,", "073a97127b4c8c67e21103fa55663f68": "\\Delta(x) = \\sum_{n=-\\infty}^\\infty \\delta(x-n),", "073aeab6305458edb9db996b527938cf": "F=\\left\\{(x,\\ y):c\\le y \\le d,\\ r(y) \\le x \\le s(y)\\right\\}", "073aef54904465bc5f2c5c88b3fa7d30": "\\{(-,+,+,+)\\,,l^a n_a=-1\\,,m^a \\bar{m}_a=1\\}", "073b23a014cdb561e726fdfd782536de": "\\sin(45 ^\\circ) = \\frac12\\sqrt{2};", "073b301e11bb8a8080bcb487c4d7b7e4": "x^3+bx^2+cx+d=0", "073b63613eef32ad23c021ccc4317e95": "\n \\cfrac{\\partial p}{\\partial t} + \\kappa\\left[\\cfrac{\\partial v_r}{\\partial r} + \\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r\\right) + \\cfrac{\\partial v_z}{\\partial z}\\right] = 0 ~.\n ", "073b6f20e1e3a7b116757dcdac5caed2": " x = (x_1, \\ldots, x_n)^\\mathrm{T} ", "073b8c7ae4ce7097c9af53d63637ce0e": "1\\leq i \\leq 2r", "073bcfbe99da78460cf5a9266da799f2": "\\frac{\\pi^2}{12}+\\frac{\\gamma^2}{2}", "073c289961a29f9be578aee613690f6d": "\\boldsymbol{\\alpha} \\leftarrow", "073c3804b1c449d744683e78a8693ab3": "\\ cos\\theta = 1 - \\beta(\\gamma_L - \\gamma_c)\\ ", "073c7b80ea477f9c5b750163479126e6": " \\displaystyle{ We^{-3/2} \\approx 0.22\\, W} ", "073c87a1b4fd09a206f70fe96c79a1cc": "t_{E}", "073cd80184e7f504f8595e8da5cf9a36": " 1 / (\\lambda T) . ", "073d06b35bd22a5c89556e597b8a557d": "s_0 \\approx S", "073d278cb1d43a03237f3a839b0a0826": "f:\\mathbb{N} \\longrightarrow\n\\mathbb{N}", "073d378533393ea8de9b8729e76a7318": " H^{s}(E\\backslash \\bigcup\\Gamma_{i})=0", "073df4e746cfe5c69d95dc4ac562bbe4": "\\Omega_M", "073e27cb1774b505ef111a366414793f": "(1-\\omega)\\phi_i + \\frac{\\omega}{a_{ii}} (b_i - \\sigma)", "073e375add3813aab90e8e4de93e8af7": " 7/4 \\times 5/6 ", "073e61f9fd390745a15dc70c6263c3ce": "X\\times X", "073e72a48c4f261575d477418fa0139e": "{\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}})^{\\rm T} (\\boldsymbol\\Sigma_{\\epsilon}^{-1} \\otimes \\mathbf{X}^{\\rm T}\\mathbf{X} ){\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}})", "073e894de339048c4adf4abc34b32783": "\\scriptstyle1\\leq j\\leq k", "073e9b97c4bc3878248d4727163b1ae2": "(v-k-1)\\mu = k(k-\\lambda-1).", "073ea3b7cec5eeeeec4164d10c217465": "\\beta^2", "073f18a5623a40477ec466172cba8054": "\\chi:V \\rightarrow \\{-1,1\\}", "073f339583a2f4da6bdad7daaa7f6f11": "F_{electrostatic} = \\frac{1}{2} \\frac{\\partial C}{\\partial z} \\Delta V^2 ", "073f95647088ae7e39f204457c32edef": "\\frac{1}{2}\\int \\frac{d^dp}{(2\\pi)^d} \\tilde{\\phi}^*(p)R_k(p)\\tilde{\\phi}(p)", "073fba3e5f887e0871c7b450e96e0c13": "(AA^*)^{-1}\\,\\!", "073fd12dab3dfe07a12152f9c9671677": "g_{k,n}(z)\\approx z", "073ff1e86bd18f4d9184ddaf913415fd": "x:(S^1)^{\\wedge i} \\to A, \\, \\, y:(S^1)^{\\wedge j} \\to A", "073ffadcdaa43aa78a7732eb3dab54c5": "\\mathbb{Q}(\\sqrt{2},\\sqrt{3})", "074002b9606b4566f3cb61a013bc8a43": "\\frac{dy}{dt} = \\frac{dy}{dx} \\cdot \\frac{dx}{dt} ", "07400b898734e4d35872d9f00e27d8e8": "1 \\cdot x ", "074016936a361c9b3ffa5491eee80e15": " \\Rightarrow 37675 = 34250 + 3425 ", "07402c69c78a7c057efa7c217c98f14e": "0 \\le t \\le 2 \\pi.", "07405e52845785645c3846f46a49323c": " a >0.\\,", "07406589807bb14817217a224a910198": "\\bar{\\theta}(\\mathbf{r},t) = t_n \\theta^n(\\mathbf{r},t)\\,,", "074097ea89225398ceb1128b5405b9fb": "x \\ge 0", "0740ae1e776b13ee7c58dbe7d28a86e0": "\\displaystyle m_1,\\,\\ldots,\\,m_N", "0740b1ad3077f5c9eea8df09f039e468": " [Fu](t,m,n) = [Fp](m,n) \\, \\cos( \\sqrt{m^2+n^2} \\, t ) + \\frac{ [Fq](m,n) \\, \\sin (\\sqrt{m^2+n^2} \\, t) }{\\sqrt{m^2+n^2}} ", "0740d56f5a5831a0fc0323b759552e8a": "f'''(x)\\geq 0", "0740ed673b4645e6673f5176495f3d96": "y = x", "074104ddab3a7e03f350918bfb6aff94": " \\frac {v_{\\ell}} {v_i} = A_v \\frac {R_L} {R_L+R_o}\\,\\!", "0741adc73b5cbce75e96a2c1cb93f96b": "(x_i,\\hat\\mu(x_i))", "0741bc843be2d7c0e6ef88dc85352a6a": "\\Omega(t^2/n^2)", "0741f8b2588f82369bb0dde8c395406c": "\\lambda \\setminus \\mu", "0741fdc506e5886e2d86a0ca9dab339b": "E_x, E_y, E_z", "074202f478d8c34356dce99c61155557": "\\sum_{b \\in B} \\left|x (b)\\right|^2 = \\sup \\sum_{n=1}^N |x(b_n)|^2", "0742460f0c51461ec49dc9faccfe0faa": " f(i) = \\cos(1) + i\\sin(1). \\, ", "0742565886f2dd30b7c53927aa007e51": "h[n] = 0 \\ \\forall n < 0,", "074288624383bc9007623912870acfe8": " V_L = V_S \\frac {T (1 - \\Gamma_S)(1 + \\Gamma_L)} { 2 ( 1 -T^2 \\Gamma_S \\Gamma_L) } \\, ", "07429da45516fb218151c6a0d153cf0b": " s", "0743075233cdfa694552767c9396f30a": "y_{isth} = \\alpha + X_{sith} \\beta + u_{sith}.\\,", "0743243d2af3dd9be8a8cee60adcb8a1": "\\varphi_{i}:M\\supset W_{i}\\rightarrow U_{i}\\subset\\mathbb{R}^{n}", "07432f3e8b4c6dbebf18ef958c46c9a3": "\\, x", "07433d49212843e7076456754ad87639": "\\mu_{z} ((t_{1}, t_{2})) = \\int_{t_{1}}^{t_{2}} 1 + \\| \\dot{z}(t) \\| \\, \\mathrm{d} t", "07434aa38f012db69b7f60d9fa8c0126": " b_0 = -\\infty", "07434fa7b23739fe350bf93e371183f7": "Y_\\beta", "0743bb0be3b046143f0a7657b412ea61": "H \\cong G", "0743bdbf9106df9455b1a340f28b6a88": " d(x) = \\int_a^b g(x,y)\\,m(y)\\,dy ", "07449a04b6a67c2ae7875cbf072c8ccd": "t \\notin \\gamma ", "0744ba02574434f6bcd2be4d203c156c": "x \\ge 3\\,\\!", "0744e0c9460149bf75ae433e78295bbf": "\\sigma \\in G", "0745076424d51f025019d732d61dc90c": "\\frac{1}{\\sqrt{3}}", "07451e54ea5aac17d40e38cad613494d": "Y_0 = K*Y", "07453e6baf8f216467f9b664de795bfc": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\pi / 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2", "074552de1c633701e0fd74715a03d7ba": "b_{i,k}", "07455460393c61f17cba114435aca24a": "[Q^\\dagger,b\\}=\\frac{dx}{dt}+i\\Re\\{W\\}", "0745566b9902ba1c640bcf6d2b22b5c1": "g \\in L^2 (X, \\mu)", "0745579204b7ebffe2fbf2da53f4fc17": "\\lim_{n \\to \\infty}\\frac1{b_n}\\sum_{k=1}^n b_kx_k = 0.", "07456f68b8b64ab4d5f2d065d8800c80": "= \\frac{2L_{L} / \\gamma (v)}{c}\\frac{1}{1-\\frac{v^{2}}{c^{2}}}", "074570409811b2cab66129c35e89ad82": "- T(\\alpha_1, \\alpha_2, \\ldots, \\mathcal{L}_YX_1, X_2, \\ldots) \n- T(\\alpha_1, \\alpha_2, \\ldots, X_1, \\mathcal{L}_YX_2, \\ldots) - \\ldots\n", "0745ba279fe361d42860dd5ee162cc0e": " \\mu \\approx 100 ", "0745dc526a8ba31c9f0b8565be0d8e94": "\n\\operatorname{sgn}(t):=\\left\\{\n\\begin{array}{ll}\n\\frac{t}{|t|},&t\\in \\mathbb{C}\\setminus\\{0\\},\\\\\n0,&t=0.\n\\end{array}\n\\right.\n", "074665b80ba529c5a84b2f98eb137e39": "\n U_{B}=\\{x\\in V:\\quad \\|\\varphi\\|_B<1\\},\\qquad B\\in {\\mathcal B}, \n ", "074713fe645ff55c3503bc5cf350d8af": "\\Theta( L_{a}+\\vert\\mathbb{C}\\vert M_{a})= \\Theta(\\vert\\mathbb{C}\\vert M_{a})", "07472f0af8b30b0ce09edd6d6246708c": "\\Psi : Y \\to (X, \\tau)',\\quad y \\mapsto (x \\mapsto \\langle x, y\\rangle).", "0747420804c56c9a02dc45ab66e3f7c4": "\\frac{2^{4031399}+1}3", "0747c16f20f48e69325d669977d9b8ae": "W=\\textstyle{\\frac{1}{2}}(-X+3Y+Z)", "074835d1d992419a421c82c1fc3f7c3b": "\\left(X, \\Sigma_X\\right)", "0748bb666d7a04bc2763837ecc623db1": "\\mu (U_i) \\leq 2^{-i}", "0748dc9d2f6b852f0060f0b3e68eb8f9": "M_{xy}(t_{SL}) = M_{xy}(0) e^{-t_{SL}/T_{1rho}} \\,", "07491314818d10c7f92ec5a22c7d7e37": "r\\mathbf{a}=(ra_1)\\mathbf{e}_1\n+(ra_2)\\mathbf{e}_2\n+(ra_3)\\mathbf{e}_3.", "07494407d8974fe8c48ef51656d886f4": "\\| (u, v) \\| := \\| u \\|_{L^{1}} + \\| v \\|_{M},", "0749a5a308ccce845eed5c7f350e799f": "{{i}_{E3}}={{i}_{C2}}+{{i}_{B1}}+{{i}_{B2}}={{i}_{C}}+2{{i}_{B}}=\\frac{\\beta +2}{\\beta }{{i}_{C}}", "0749a79fa388599db2e978b658f9081a": "H_0: \\theta=\\theta_0", "074a08f8d3aad5c15ec7217f53b3b60d": "\\Omega^7", "074a2cb57b7a7a4af8af6ceee1a1fa55": "\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 &\\qquad \\text{ ( by Lyapunov function } V_x, \\text{ subsystem stabilized by } u_x(\\textbf{x}) \\text{ )}\\\\\n\\dot{z}_1 = z_2\n\\end{cases}\\\\\n\\dot{z}_2 = z_3\n\\end{cases}\\\\\n\\vdots\n\\end{cases}\\\\\n\\dot{z}_i = z_{i+1}\n\\end{cases}\\\\\n\\vdots\n\\end{cases}\\\\\n\\dot{z}_{k-2} = z_{k-1}\n\\end{cases}\\\\\n\\dot{z}_{k-1} = z_k\n\\end{cases}\\\\\n\\dot{z}_k = u\n\\end{cases}", "074a81a1fb5d7d806f2da321fc83ffb6": " \\, {l_c} ", "074ae6dfaebaa5e5d168b78e98a18de6": "R_r^{'}/s", "074b4a7192acfe468cc1e567ad6e7725": "\\,\\Sigma_{xx}\\beta_k(k=1,\\ldots,K)", "074b4d7cb6805e2618baa131aea71638": " S \\mapsto \\nu (g^{-1} S) \\quad ", "074b87eddded6a895dedc4c361db165c": "\\frac{200-150}{100}=0.50", "074be6c7441cfe773d43ac834a1ee97a": "-E_{act}/R", "074bf1d8d0b30d80372f9918b3845727": "Q = \\left (A^T A \\right )^{-1}\n", "074c14097a4cefef444cc434b45576f1": "3 \\cdot a_n\\ \\mathrm{dB}", "074c5003b6fd3a0d3c02d956852a529c": "\\mu*N", "074c699683f5efad06712c730018baa1": " P_2=(x_2,y_2) ", "074c6a9b9b3d5052dae458c1c1cebef0": "\n\\sum_{i=1}^{n} a_i^k = b^k\n", "074c775b2a19e7ce773d8a43e6e817f0": " \\frac{\\operatorname{d}}{\\operatorname{d}t} \\left( c_1 x_1(t) + c_2 x_2(t) \\right) = c_1 x'_1(t) + c_2 x'_2(t) ", "074c7ca9d8c84795dca480f5ba6fb001": "R_p\\simeq0.3", "074c8c5de4d3a155081a95c480b70a29": "V = D/t", "074c8fa594b5d54226e76ae3b6669fc8": "\\neg\\neg A \\to A", "074cc7670157df51929ce0e6fb025d6d": "9/11 = 0.1\\ 1\\ 3\\ 3\\ 1\\ 0\\ 5\\ 0\\ 8\\ 2_!", "074cfc7052d14a7f29d1d081c6258a63": "\\displaystyle g_i", "074d1e1f2a3c12b8ad152cee342fb779": " m_i \\scriptstyle", "074e4440565d21762c26d1ecc605021f": "x_0 = 0 \\quad (11')", "074e5c7dd44f09cd16d0b75ac973bfa2": "d'_P(P_1,P_2)=(x_1-x_2)(y_1-y_2)", "074eeee3fa471a47ee383d47edca237d": "A=60^\\circ", "074ef0e78e37964f7583f7bd5568218b": "\n\\frac{\\partial S}{\\partial \\alpha_{r}} = \n\\frac{\\partial}{\\partial \\alpha_{r}} \\frac{1}{2} \\sum_{k=1}^{N} m_{k} \\left| \\mathbf{a}_{k} \\right|^{2} = \n\\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\left( \\frac{\\partial \\mathbf{a}_{k}}{\\partial \\alpha_{r}} \\right)\n", "074ef76444599eff3f9ae2b65afcca54": " T_{n} = 1,1,1,2,5,16,61,272,1385,7936,50521,353792,\\ldots \\quad (n=1,2,\\ldots) ", "074f0008987d021f1d042094caa89a13": "\\int \\frac{\\delta Q}{T} = 0", "074f5f3bd8ebf2efab25dd2b4933a662": "w\\Vdash p", "074f84aaf0e6e73802b0caf3bbb2b5bb": "r\\gg r_a", "074fb98c1ec909b43c244d8ab7a62170": "i=0 \\ldots n-1 ", "07500d4e5afe4e53a75c4b4c205459d4": "a,b\\in X", "07503ada062ead95b8ee189fa6d93c6e": "\\textstyle{A=\\frac12 log_2(RG) = \\frac12 (log_2(R) + log_2(G))}", "0750524860f0c0d1948a04fb5044a63f": "\\delta Q = \\mathrm{d}U + \\delta W", "0750607522d1f2c4e445f1bb189dd792": "\\operatorname{End}(V) \\times \\operatorname{End}(V) \\to \\operatorname{End}(V)", "07508dd8b529ec382f9c78e0438c4d08": " \\tan\\phi = \\frac{y}{-x} = \\frac{2y}{y^2 - 1} = \\frac{2e^a}{e^{2a} - 1} = \\frac{1}{\\sinh a}. ", "0750fa6c4391138d52861e02cd8f1001": "(x-3)x^{28}(x+3)(x^2-6)^{21}(x^2-2)^{27}.\\ ", "07510ea55969b82b001f63319143f3d7": "\\frac{x}{y}=\\frac{1}{\\lceil y/x\\rceil}+\\frac{(-y)\\bmod x}{y\\lceil y/x\\rceil}", "0751669a6b433328f530f4036618aa53": "\ns_t \\equiv \\vec{S}_t \\cdot \\vec{D} = \\sum_{A=1}^N M_A \\;\\vec{s}^{\\,A}_t \\cdot \\vec{d}^{\\,A} = 0\n\\quad\\mathrm{for}\\quad t=1,\\ldots, 6.\n", "07516a79c61e8deda3a488d2625e3d5c": " E_\\text{cm}", "07517907d40805f1737608a213260a88": "\\ M_{heel} = pressure \\times S \\times A {cos(\\phi)}^n ", "07517cf760c151b716b9d4e70df09cf0": "\\mu \\ = \\mu_r \\,\\mu_0 \\,\\!", "0751a0235b4732f716701e72722cdb33": "= \\int_{0}^{\\infty}C\\, \\operatorname{d}t", "0751ceccee6d9fe57a4aaf339a0e5a35": " p_{4,4}(x) = y_4 \\, ", "075215656b7a0e53567103e62b31ee31": "\n\\begin{align}\n\\overline{Y}_1 & = \\frac{1}{6}\\sum Y_{1i} = \\frac{6 + 8 + 4 + 5 + 3 + 4}{6} = 5 \\\\\n\\overline{Y}_2 & = \\frac{1}{6}\\sum Y_{2i} = \\frac{8 + 12 + 9 + 11 + 6 + 8}{6} = 9 \\\\\n\\overline{Y}_3 & = \\frac{1}{6}\\sum Y_{3i} = \\frac{13 + 9 + 11 + 8 + 7 + 12}{6} = 10\n\\end{align}\n", "0752312c070751129630dd26655e1933": "\n c=f\\lambda\n ", "07527e3ec7a12e1161e53821bc99f95a": "\\mathbf{M} = \\mathbf{F}(\\mathbf{E})", "0752f7fefb594dc5a4937ab6746900b6": " \\theta = 3 \\nu - 2 \\mu ,", "07536b2456b41acd85206334dd863837": "\\frac{q^{r+1}-1}{q-1}", "07537a4b745cadd9926a130320d1488f": "v_0 = s, E_0 = E", "0753b0785e89821ff5957766f811f56e": "N \\geq 2", "0753bc3f9919ba50a4bc8d342b40a6e0": " \\rho_0 ", "0753d9d83d2a2f563f50c4c7ae14d898": "\n\\begin{pmatrix}\n0 & i \\partial_0 + i\\nabla \\\\\ni \\partial_0 - i \\nabla & 0\n\\end{pmatrix} \n\n\\begin{pmatrix}\n \\Psi_L \\\\ \\Psi_R\n\\end{pmatrix}\n\n= m\n\\begin{pmatrix}\n \\Psi_L \\\\ \\Psi_R\n\\end{pmatrix}\n", "075406f6eceea0908aea81e296465c89": "\\psi + \\theta + \\phi = \\tfrac{\\pi}{2}\\, ", "0754dcc9db21c1211d42115b528100d5": "0 U_a) \n= \\Pr( \\alpha P_i + \\beta D_i + \\varepsilon_i > \\alpha P_a + \\beta D_a + \\varepsilon_a ) \\\\ \n& = \\Pr( \\varepsilon_i - \\varepsilon_a > \\alpha P_a + \\beta D_a - \\alpha P_i - \\beta D_i )\n\\end{align}\n", "07629bad05308a4805729a54ee888b3e": "a_6= \\lfloor 2^\\frac{1}{2} \\rfloor = \\lfloor 1.414\\dots \\rfloor = 1. ", "07629de353237b95bd44b92e2b0b5a74": "A[i,j] + A[k,\\ell] \\le A[i,\\ell] + A[k,j].\\,", "0762d4fb3790cfb0107ceddd69321346": "0 \\to \\operatorname{Der}_B(C, M) \\to \\operatorname{Der}_A(C, M) \\to \\operatorname{Der}_A(B, M).", "0762e2065b43aea6df90e731cf029b9b": "(G_n)_{n\\in\\N}", "076323ef7554f221c8070ad5de5c22b8": "[A,C]", "076381b47788f3b58e3c4e8b6804a8c5": "g(r) = h(r) = \\sinh(r)", "0763a4e21bae93349f9f2ae62bb860a9": "1 \\not\\in \\mathfrak{p}R", "0763cdc8f55ed814496308e5f4658f29": "\\Delta \\bar \\nu=0", "0763ce93e8a2656be17560c66cc1dbdb": "p_1(x),p_2(x),\\dots ,p_n(x)", "0763dc4fcdd12467c574a777667a3334": "\\scriptstyle g_k", "0763e9c6e4becc41e920c5126e588bbd": "r_{u,i} = \\frac{1}{N}\\sum\\limits_{u^\\prime \\in U}r_{u^\\prime, i}", "0763edab0eef820e06e1ef0f13de8655": "[X^n]f(X)=\\mathrm{Res}\\left(X^{-n-1}f(X)\\right).\\,", "076402d6722b347aecd7d25f2116f240": "H(X)=0", "0764226bffda183764c6261c01d7d892": "{dQ_h \\over dt} = F_h (C_{art} - {{Q_h} \\over {P_h V_h}})", "076456bd63a74f2491312e4ecf5a9863": "\\tilde{W}_t = W_t + \\frac{\\mu -r}{\\sigma}t,", "0764da0d400071e4a976e5804b264e13": "\\mathbf{P} \\big[ \\| B \\|_{\\infty} > c \\big] \\approx \\exp \\left( - \\frac{c^{2}}{2 T} \\right).", "076504e491bcaa11b7f41e85cdb74513": "14_{11} \\ ", "07654ab947447d79c73ad4a85cd17a9b": "C^{(p)}_T(p,T)=\\frac{C^{(V)}_T(V,T)}{\\left.\\cfrac{\\partial p}{\\partial V}\\right|_{(V,T)}} ", "0765f9cf030a6ef694d7f8e83a51c489": "\\limsup_{n \\to \\infty} \\frac{\\sigma(n)}{n \\ln \\ln n} = e^\\gamma,", "07661b12ffb58e544cdfe5d13653a383": "\\exp(x+y) = (\\exp x)(\\exp y)", "076673014ed0ddbe48a92b2b6617bf58": "\nq_{yy} = \\frac{\\sum (y-\\bar{y})^2 I(x,y)}{\\sum I(x,y)}\n", "0766a65e28d01fdaf213975cdc5bf122": "%C* = %C/6 \\mbox { for } %C \\ge 0.30% ", "0766ff40b4aee25394db43ccc72965d0": "b_1 \\equiv b_2 \\pmod{n}", "0767138437a99c249de200ae67000612": "a; A/\\alpha", "076745b8e5c00ecc8dfe0907506eb923": "\\mathbf{1}_A (\\omega) = 1 ", "076754254670c93760b74ca08d8d532f": " Af = \\lim_{t\\rightarrow 0} \\frac{T_tf - f}{t},", "076771b9e64ea0b1fc4b6722c4321ec3": "F_3(a, b) = a \\uparrow\\uparrow (b + 1)", "0767b602113fbc4116971faab83e9299": "n^a\\partial_a", "0767b7b70887925ed38002857226f4f2": "\\omega_2, \\omega_3, \\ldots, \\omega_\\omega, \\omega_{\\omega + 1}, \\ldots, \\omega_{\\omega_\\omega},\\ldots", "07681e835315bf610815cfbd4d1cc885": " P_{t+h}(S\\rightarrow S' | E) - P_{t+h}(S\\rightarrow S') < P_t(S\\rightarrow S' | E) - P_t(S\\rightarrow S')", "0768ca20f268ea2b57e7e1fcb1668c71": "P_{\\mu}", "076983fb4bd3ca6730ccfef16dd7e0f7": "|\\langle k | \\alpha \\rangle|^2", "076998aa7882ad367aa77901ac1e9418": "\\rho^{f}(\\vec{r})", "0769a79fb3a50f173aa673f653e7492c": "\\Im, \\imath, \\jmath, \\Bbbk, \\ell, \\mho, \\wp, \\Re, \\circledS \\!", "0769ad1292fe7abf0ed65594ac1b4b29": "\\lambda_{1,2}", "076a06c5db36dd4ea8b0b9a39e8f1be0": "T_f(\\varphi)=\\int_{\\mathbf{R}^n}f(x)\\varphi(x)\\,dx", "076a123ca2e849665659a2720ac00d24": "{p_D \\over p_B}", "076a3422506f570b6d2bb3beb560bcbd": "\\textbf{P}_0 \\textbf{X}_0 = \\textbf{P} \\textbf{H}^{-1} \\textbf{H} \\textbf{X} = \\textbf{P} \\textbf{X} = \\mathbf{x}", "076a38cdbb042d14cc1395be90288e60": "\\ V_r = R_0 I[1 + \\pi _L \\sigma _{xx} + \\pi _T (\\sigma _{yy} + \\sigma _{zz} )] ", "076a543427dc3dd3d85167d119e6f96b": "\\psi\\in\\mathrm{End}(A)\\otimes\\mathbb{Q}", "076a5d907fd387870f7cd559d302984b": " \\varphi_1 = \\underset{\\Vert \\mathbf{\\varphi} \\Vert = 1}{\\operatorname{arg\\,max}} \n\\left\\{\\operatorname{Var}(\\int_\\mathcal{T} (X(t) - \\mu(t)) \\varphi(t) dt) \\right\\}, ", "076a7392775093a4109f6d6823f82de7": "Pr(q,r^*)", "076a809092253b8be867c519f4fd165e": " Y = \\frac{1}{2} (c_{11} + 2c_{12}) \\left[ 3 - \\frac{c_{11} + 2c_{12}}{c_{11} + 2(2c_{44} - c_{11} + c_{12}})(l^2m^2 + m^2n^2 + l^2n^2) \\right]", "076a8a4792382ef936eb8d30c0344af3": " \\frac{S'(0)}{S(0)} = 1 ", "076a9536d872162b643b3c12af11462e": "OPT\\,\\!", "076ac46c2af2a73f18609004e868cc25": "\\mathbf{S_z} = \\left| \\mathbf{S} \\right| \\cos \\theta", "076ad1a05ab5a56fba3ab7c501c59247": "r_{U \\subset V} : \\Gamma(V, \\mathcal{F}) \\to \\Gamma(U, \\mathcal{F})", "076ad72025bf258254eabe7ff54d23bc": "\\scriptstyle f(x) = \\sum_{i=1}^n v_i(x)", "076b1020422cb14d6c78aa30d137e4d2": " | \\bar{\\psi} \\rangle = \\sum_{mm'} D^{(j)}_{m'm} | j , m \\rangle \\quad \\Rightarrow \\quad | \\bar{\\psi} \\rangle = D^{(j)} | \\psi \\rangle ", "076b261c5f14add8ce5c20f7d0f8e605": "\\mathrm{Conversion\\ rate} = \\frac{\\mathrm{Number\\ of\\ Goal \\ Achievements}}{\\mathrm{Visits}}", "076b5159c531e835b323a4a37292c8c7": "\\frac{\\alpha-1}{\\alpha+\\beta-2}\\!", "076bac1211cd22f8042a6388a18536da": "G^{ab}= 8 \\pi \\left( \\psi^{;a} \\psi^{;b} - \\frac{1}{2} \n\\psi_{;m} \\psi^{;m} g^{ab} \\right) ", "076bc6f390507ab9471bbdedb7a7a8a6": " t' = \\frac{t - {v\\,x/c^2}}{\\sqrt{1-v^2/c^2}}\\ ,", "076c0cdb2b88594f3c51810670c8fa00": "\\sigma_{L_x} \\sigma_{L_y} \\geq \\frac{\\hbar}{2} \\left| \\langle L_z \\rangle \\right|.", "076c1d84e4c601afa735e9f1981f1481": "\\gamma^{xy}\\text{ given }\\gamma, \\gamma^x\\text{ and }\\gamma^y", "076c5b7ef72f7fe5eef303c071611acf": "s^2=\n{n \\choose 2}^{-1} \\sum_{i < j} \\frac{1}{2}(x_i - x_j)^2 =\n\\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\bar x)^2", "076c60af2c283a925276be8d9ac48463": "u=", "076ccb9743f01c5e8092b493693e4a3b": "\\hat\\gamma", "076cd69391958c8e34d1dec5b79ed219": "g = 2 \\, dx \\, dy.\\, ", "076d689f8e234ddda25328e7eb961ce4": " \\frac{1}{2} \\ddot{h}_{\\hat{\\theta}\\hat{\\phi}} = -R_{\\hat{t}\\hat{\\theta}\\hat{t}\\hat{\\phi}} = -R_{\\hat{r}\\hat{\\theta}\\hat{r}\\hat{\\phi}} = R_{\\hat{t}\\hat{\\theta}\\hat{r}\\hat{\\phi}} = R_{\\hat{r}\\hat{\\theta}\\hat{t}\\hat{\\phi}}\\ ,", "076daa1db6a90ca612866888bb311b54": "z_\\alpha", "076db37d0fffd46b9ed7bfebf9548faa": "\\boldsymbol\\Sigma^+", "076e00399c2caed64fb2f4b94294c74a": "W = w_{i,j} \\in[0, 1],\\; i = 1, . . . , n,\\; j = 1, . . . , c", "076e7d43cb533c3d6200edc7d18bb420": "-\\frac{\\eta}{(n-1)!}\\partial_\\xi^{n-1} n_\\eta(\\xi)", "076eb7575aeb3c05b0f49e0966d8c763": "\\frac{q^2}{g}\\left(\\frac{y_2-y_1}{y_1 y_2}\\right)=\\frac{1}{2}(y_2-y_1)(y_2+y_1).", "076ebbaefa05d01247e9acf77a3ddf30": "m_i(t)", "076eecfb3ad1c653758ea0bb69c7a261": "\\int_0^\\infty \\frac {x^{n-1}}{e^{x}-1}\\ dx=\\Gamma (n)\\zeta (n)", "076f0a34cc9c4868859a1c63fa5e21b3": "\nB_m(x,y) = \\sum_{p=0}^m \\binom{m}{p} x^p y^{m-p} \\sin ((m-p) \\frac{\\pi}{2}),\n", "076f1d8cf906ddd76befeee26faf6378": "x_2=-1\\,\\!", "076f447e151637a04c93d7729fada964": "W_i = W_i - (x_i - \\mathrm{nearHit}_i)^2 + (x_i - \\mathrm{nearMiss}_i)^2", "076f7053aa11ee2e25f239fe56837022": "(M, d)\\,", "076f79eb95affcfa5564c2e7447977c6": " \\neg ", "0770acf09851f96ea1970372b019313b": "\\varphi/n", "0770e4d2141ff0645d384dfa5cba212a": "\\bar B(\\mathbf x_1,\\theta\\|\\mathbf h_0\\|)\\subset\\bar B(\\mathbf x_0,t^*)", "07710b5c43702a8bb7b9104eacc6ba71": "\\Gamma", "0771127c2a542f877110010002618345": "\n \\mathcal{L}(\\theta\\,|\\,x_1,\\ldots,x_n) = f(x_1,x_2,\\ldots,x_n\\;|\\;\\theta) = \\prod_{i=1}^n f(x_i|\\theta).\n ", "077135ee6dd7495ef979732cb77841ad": "k=1,\\dots,K", "07713f465efe7e23092a0ccf5c5e1dfa": "p \\Leftrightarrow q", "0771aa167b47724b08e1cd2007e2521d": "2~\\mu~u_r\\, ", "0771b835887c341bf6a6031dacb3cdfb": "\\eta(\\text{Earth},\\text{Be-Ti})=(0.3 \\pm 1.8)\\times 10^{-13}", "0771f479546178c069c0e216ed1286da": "X= \\left ( - \\infty, \\infty\\right )\\times \\mathbb{R}^{L-1}_+", "0772a4cddee27cf4d09e462427329124": "\\widehat{D}=\\mathcal{A}(D)\\,\\!", "0772f3768a0d5aec0631e82c594a8f1c": "K=\\frac{1}{N} \\sum_{r=0}^N \\left ( Q(t_1)Q(t_2)+Q(t_2)Q(t_3)\n-Q(t_1)Q(t_3) \\right )_r. ", "077375f6e49036ecf60254cb3db53b04": "d \\approx 1.22\\sqrt{h} \\,.", "0773a7f0d6ea3e3de6772d5011308740": "s_1,\\ldots,s_{\\ell}", "0773fb4ec69cb402146dee39c7dfe67c": "\\Phi(\\lambda x) = \\bar{\\lambda} \\Phi(x)", "07746dfcc964e0a44b73ae010adb47b6": "D_3 \\bar R", "0774d16de8a5ea7ec5f8cedcce770371": "f(x,y) = \\frac{xy}{x^2+y^2}.", "0774eb853e768d8440a5decf37352564": "\\Phi_x(t) := \\Phi(t,x)\\,", "0774f8878b7656441abeba8fcead99c1": "p(x)=\\sum_{i=0}^{n}y_i\\cdot\\prod_{0\\leq j\\leq n,j\\neq i}\\frac{x-x_j}{x_i-x_j}.", "0775088d699e8f8e39ed21939e1ea35f": "var(\\epsilon)=\\sigma^2I ", "077517b02a759ac0d88726961e1cbf62": "BGL(R)", "077542d71239fe9a28c0b81e9b943e9c": " RDF = 100 {(p-n)\\over p} ", "077548709960a44c68ae0ec66cd7c4e0": "C \\in \\mathcal{B}_B ", "07755fc3d3fd4a2a1a25d203fc4333b7": "\\forall x. P(x) \\Rightarrow Q(x)", "0775c495f3d2b269d0366b56aa4b64ce": " [\\mathbf{t}]_{\\times} = \\mathbf{V} \\, \\mathbf{W} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} ", "0775c8c1c41841b9a7e8f6ccc78cc54c": "u_n \\in \\mathrm{ker}(e_i) \\cap M_{\\lambda + n \\alpha_i}", "0775e8e8c04f8675f218725931c50dca": "x_i w_{ji} \\,", "077602586c235d52628faee997c58f2c": "\\pi+2k\\pi=\\theta_1+\\theta_2+\\cdots+\\theta_n-(\\phi_1+\\phi_2+\\cdots+\\phi_m) ", "07766e12ed7514e4f6a97b128fa3d24d": "\\sqrt{2} =\n\\prod_{k=0}^\\infty\n\\frac{(4k+2)^2}{(4k+1)(4k+3)} =\n\\left(\\frac{2 \\cdot 2}{1 \\cdot 3}\\right)\n\\left(\\frac{6 \\cdot 6}{5 \\cdot 7}\\right)\n\\left(\\frac{10 \\cdot 10}{9 \\cdot 11}\\right)\n\\left(\\frac{14 \\cdot 14}{13 \\cdot 15}\\right) \\cdots", "0776dd788bfa2eac06807230d12a51eb": " 2n^2+1 ", "077710d78d9aaf5f53a613b973a43e3f": "\\{|a_i b_j \\rangle\\}", "07776e5b55dbf4f4afc26776e279ba4e": "[S_z,S_x]=i\\hbar S_y", "077771a16d7f24cafdc89a59245a3b16": " p_r = m \\dot r \\ , \\ p_{\\theta} = mr^2 \\dot{\\theta}\\ , ", "077823f1d2b7790b923e2f811f3a50ee": " \\mathbf{W} \\, \\mathbf{\\Sigma} = \\begin{pmatrix} 0 & -s & 0 \\\\ s & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} ", "07782a7e6670123f9540cfe2cb646d3e": " F(a) = \\langle F_a\\mid F\\rangle = \\pi^{-n} \\int_{C^n} \\kappa(a,z)F(z)\\exp(-|z|^2)\\,dz,", "0778bf59ee2c9fae1d1070944026249e": "x:=?\\,\\!", "0778c85c734c50af0af297e0fc1c0d61": " Q_k ", "0778ca1c5f7b9cce86dd69e5f2287181": "u_{mn}(r, \\theta, t) = \\left(A\\cos c\\lambda_{mn} t + B\\sin c\\lambda_{mn} t\\right)J_m\\left(\\lambda_{mn} r\\right)(C\\cos m\\theta + D \\sin m\\theta)", "07790af7823cca418e70823c74f70d21": "(\\hat{b}^\\dagger)", "077932dfe335eb11060da12290cd69e7": "f_c = \\frac{1}{2\\pi RC}", "0779386a9cd0a46f0327a20912ccd06e": " G: L^2(\\mathcal{T}) \\rightarrow L^2(\\mathcal{T}),\\, G(f) = \\int_\\mathcal{T} G(s, t) f(s) ds. ", "0779458e5cb9f831cdfa2b3b7e9553e2": "m,n", "07796dea4d8c0505afef310d2328f6ec": "\\textbf{x}(k+1) = A\\textbf{x}(k) + B\\textbf{u}(k)", "0779870da1d0900176882d27b80a4d7b": "\\textstyle \\lim_{r \\rightarrow \\infty} g(\\mathbf{r}) = 1", "0779e058d2e7c0d16b352bdd33457f3b": "\\begin{align}\n f_{\\theta_1}=f_{\\theta_2}\\ \n &\\Leftrightarrow\\ \\tfrac{1}{\\sqrt{2\\pi}\\sigma_1}e^{ -\\frac{1}{2\\sigma_1^2}(x-\\mu_1)^2 } = \\tfrac{1}{\\sqrt{2\\pi}\\sigma_2}e^{ -\\frac{1}{2\\sigma_2^2}(x-\\mu_2)^2 } \\\\\n &\\Leftrightarrow\\ \\tfrac{1}{\\sigma_1^2}(x-\\mu_1)^2 + \\ln \\sigma_1^2 = \\tfrac{1}{\\sigma_2^2}(x-\\mu_2)^2 + \\ln \\sigma_2^2 \\\\\n &\\Leftrightarrow\\ x^2\\big(\\tfrac{1}{\\sigma_1^2}-\\tfrac{1}{\\sigma_2^2}\\big) - 2x\\big(\\tfrac{\\mu_1}{\\sigma_1^2}-\\tfrac{\\mu_2}{\\sigma_2^2}\\big) + \\big(\\tfrac{\\mu_1^2}{\\sigma_1^2}-\\tfrac{\\mu_2^2}{\\sigma_2^2}+\\ln\\sigma_1^2-\\ln\\sigma_2^2\\big) = 0\n \\end{align}", "077a4a4dd8cee9a21236f43448023670": "\\nabla_{[a} R_{bc]d}^{\\ \\ \\ e} = 0", "077a72a276d6314aafe95c03f531b013": "g(x_1,x_2,x_3,v_1,v_2,v_3)\\,", "077a92483fd1b690cd5582c27fd5e27b": "k \\, \\frac{p_k}{p_{k-1}} = ak + b,", "077ac601e604a3663daf0fef57d18fb7": "\n\\begin{align}\nd^4\\sigma &=\n\\frac{Z^2\\alpha_{fine}^3c^2}{(2\\pi)^2\\hbar}|\\mathbf{p}_+||\\mathbf{p}_-|\n\\frac{dE_+}{\\omega^3}\\frac{d\\Omega_+ d\\Omega_- d\\Phi}{|\\mathbf{q}|^4}\\times \\\\\n&\\times\\left[-\n\\frac{\\mathbf{p}_-^2\\sin^2\\Theta_-}{(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)^2}\\left\n(4E_+^2-c^2\\mathbf{q}^2\\right)\\right.\\\\\n&-\\frac{\\mathbf{p}_+^2\\sin^2\\Theta_+}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)^2}\\left\n(4E_-^2-c^2\\mathbf{q}^2\\right) \\\\\n&+2\\hbar^2\\omega^2\\frac{\\mathbf{p}_+^2\\sin^2\\Theta_++\\mathbf{p}_-^2\\sin^2\\Theta_-}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)} \\\\\n&+2\\left.\\frac{|\\mathbf{p}_+||\\mathbf{p}_-|\\sin\\Theta_+\\sin\\Theta_-\\cos\\Phi}{(E_+-c|\\mathbf{p}_+|\\cos\\Theta_+)(E_--c|\\mathbf{p}_-|\\cos\\Theta_-)}\\left(2E_+^2+2E_-^2-c^2\\mathbf{q}^2\\right)\\right]. \\\\\n\\end{align}\n", "077af31100254b1688d4e5fd515e4f7f": "p(z)=z^3-2z+2", "077b4344e9a2a6f67524501d81bb5f28": "U^2 V^2 + V^2 W^2 + W^2 U^2 - U V W = 0\\,", "077b4d6796ac247752630a652ee7d6a1": "P_2 = 1 - \\frac{\\mathrm{non}\\,\\mathrm{outs}}{\\mathrm{unseen}\\,\\,\\mathrm{cards}} \\times \\frac{\\mathrm{non}\\,\\mathrm{outs} - 1}{\\mathrm{unseen}\\,\\,\\mathrm{cards} - 1}", "077b7b8d72ef0127c406eb30d6fd8ef1": "\\left\\{a_m\\right\\}", "077bd014b6feaae19ad50c0177a1bc11": "\\begin{matrix}\n & & \\text{S} & \\text{E} & \\text{N} & \\text{D} \\\\\n + & & \\text{M} & \\text{O} & \\text{R} & \\text{E} \\\\\n \\hline\n = & \\text{M} & \\text{O} & \\text{N} & \\text{E} & \\text{Y} \\\\\n\\end{matrix}", "077c2aa89f728f5a7458d3ded7baa4da": "\n\\mbox{If } \\left(\\frac{\\alpha}{\\mathfrak{a} }\\right)_n =1\n\\mbox{ then }\\alpha \\mbox{ may or may not be an }n\\mbox{-th power}\\pmod{\\mathfrak{a}}.\n", "077c3744ad2e8cfbdfe217cd4e9d3701": "c_0/c_1-1", "077c62fef3fdef49e8c72e2bc9d15eef": "\\tilde{P}=[p_0(x),p_1(x),...,p_{n-1}(x)]^{T}", "077c65bdfc1bffff22f8b60fdd146757": "f \\cdot u = -g\\frac{\\partial P / \\partial y}{\\partial P / \\partial z} = -g{\\partial Z \\over \\partial y}", "077cb05b34f5d3c3f31ce0fb5c6df809": " F(x) = \\underline{\\int_{a}^{x}} f(x) \\, dx ", "077dcdb31d7ed1a0d801f2a2f0710163": "O(n^2\\ln ^{O(1)} n)", "077dea9a5816ff7eeb169b7d7389645b": "\\rho = \\frac{(1+w)G_s\\rho_w}{1+e}", "077e3451bc1e12c3eca2b923844ba92d": "\\nabla'\\left(\\frac{1}{|\\bold{r}-\\bold{r}'|}\\right) \\equiv \\left(\\bold{e}_x \\frac{\\partial }{\\partial x'} + \\bold{e}_y\\frac{\\partial }{\\partial y'} + \\bold{e}_z\\frac{\\partial }{\\partial z'}\\right)\\left(\\frac{1}{|\\bold{r}-\\bold{r}'|}\\right) = \\frac{\\bold{r}-\\bold{r}'}{|\\bold{r}-\\bold{r}'|^3}", "077e453f76eadfc3ba6d5ee80f19e450": "n = q^k", "077e8355e57bb087ba0e3b627effb8d9": "H(Y|X) = H(Y)", "077e8ade7d473a4697aa61af200f4028": "P^\\alpha g_{\\alpha\\beta}P^\\beta = (m_0 c)^2\\,.", "077ea541cceaa0a121fc034ed98a3c9d": "[\\mathcal{T}_n(f)](x)=ne^{-nx}\\sum_{k=0}^\\infty{\\frac{(nx)^k}{k!}\\int_{k/n}^{(k+1)/n}f(t)\\,dt}", "077ee94ac9eb64530fe8724b2686b402": "2\\sqrt n", "077f0f6b89b5079ea8703b3d7e3cb630": "\\displaystyle{W(x)e^{y} = e^{-\\|x\\|^2/2} e^{-(x,y)} e^{x+y}.}", "077f1307be73c6ab24fb82ff7ebd5c91": "j_Y", "077f143f43b17d11829c03ee32cd3131": " \\operatorname{Ran}(F_1)\\times\\cdots\\times \\operatorname{Ran}(F_d) ", "077f15fc341fcb199dedeafc83810044": "\\frac{\\omega - \\omega_0}{\\omega_0}\\thickapprox \\frac{(\\bar{w_m}-\\bar{w_e})\\cdot\\Delta V}{W}\\,", "077f44a5f5abf839b2b85486f6563533": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\mu D} ", "077f64801fcff049ad6261e6f58614ec": "\\Delta^k(a_n) = \\Delta^{k-1}(a_{n+1}) - \\Delta^{k-1}(a_n)=\\sum_{t=0}^k \\binom{k}{t} (-1)^t a_{n+k-t}", "077f689e2c97906d8fbb172af3100753": "\\hat{g}_n(u_n)", "077f6952e7a9fe28ddf1404e5a7438fd": " P \\left ( \\mathbf{x} \\right ) = \\int P \\left ( \\mathbf{x} \\land y \\right ) \\, dy = {1 \\over N} \\sum_{i=1}^N \\, \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big )", "077f998032173465d265b3726ab970b5": "\\mbox{Pin}_+(n) := \\mbox{Pin}(n,0) \\qquad \\mbox{Pin}_-(n) := \\mbox{Pin}(0,n)", "077fc096705bdbf04e4df374dad720c2": "a_c=0.81\\,\\;", "078010d409b4f9155fa3603c1d3468cd": "\\mbox{Intensity} \\ \\propto \\ \\frac{1}{\\mbox{distance}^2} \\, ", "07804a56b1ff9142d02458700ef10b57": " \\vdash A \\and B ", "0780928654b8d68408912cd0ab7411e7": "P-Q \\succeq 0", "0780a1a50544e424461bf141635dcc29": " B_{0} ", "0780a8f12b33e6d16ac035270fbdc88c": "I = \\sum_{n=1}^\\infty \\varphi_n \\varphi_n^\\dagger, ", "0780f3729d0863e7a57fbf560ae42288": "\\vdash \\neg (p \\land \\neg p)", "078103def5c3c8da420e74d0665bfaca": " BC \\to D", "07813d85abecae8553a456cde7fdc47c": "j(\\tau) = {1 \\over q} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \\cdots", "078229fb92933bbd30db6b783ddc6550": " d_{\\mathrm H}(X,Y) = \\max\\{\\,\\sup_{x \\in X} \\inf_{y \\in Y} d(x,y),\\, \\sup_{y \\in Y} \\inf_{x \\in X} d(x,y)\\,\\}\\mbox{,} \\! ", "0782c5509849d7802777fd9ae857bdaa": "a(Z)=2^{1-A}(1+Z)^A", "0782fd67789ce951f272d19a88258fa1": " \\delta V = \\delta W_{H,i} + \\delta W_{H,p} - \\delta W_{g,p} + \\delta W_{\\sigma} ", "0782ffe56254b5123a769b1618afe23f": "u_0 = 0 \\, ", "078376930c9985774961ee63c5615a07": "p(X)", "0783c56c8c7d602538ef5970f4d8e594": " L(p) ", "078401532866081dddda6744220b4c75": "EP", "07846d31b4a7947cf5fcaf5185ba4b65": "10^{10^{10^{10^{10^{1.1}}}}}", "07847ca5764ba07964bfa208eee3d90e": "\\operatorname{E}[g(X)] = \\int_a^\\infty g(x) \\, \\mathrm{d} \\mathrm{P}(X \\le x)= \\begin{cases} g(a)+ \\int_a^\\infty g'(x)\\mathrm{P}(X > x) \\, \\mathrm{d} x & \\mathrm{if}\\ \\mathrm{P}(g(X) \\ge g(a))=1 \\\\ g(b) - \\int_{-\\infty}^b g'(x)\\mathrm{P}(X \\le x) \\, \\mathrm{d} x & \\mathrm{if}\\ \\mathrm{P}(g(X) \\le g(b))=1. \\end{cases}", "07849574dc0001eece45c0e0b066167d": "\\int\\frac{x^4\\;dx}{s}\n= \\frac{x^3s}{4}+\\frac{3}{8}a^2xs+\\frac{3}{8}a^4\\ln\\left|\\frac{x+s}{a}\\right| ", "0784bbbafde6513ff77e1888c5fa441a": "\\mathbf{e}_2 ", "0784be20ef0e37eb46ebd36b4a2bf6dd": "\\scriptstyle 1-\\varphi=\\frac12(1-\\sqrt5)", "078526a9ab20c4b04eeb2175fde4a01e": " q^{(t)} = \\operatorname*{arg\\,max}_q \\ F(q,\\theta^{(t)}) ", "0785af11f2b9847cae144679ba6ece53": "\\oint \\frac{\\delta Q}{T} = 0", "0785e11419e998a63a3705b8a7bc84e7": "y=\\operatorname{sign}{f(x)}", "0785e7cdb8333390991176dd4ca77445": "c_{\\pm}=\\frac{c}{1\\pm\\kappa}", "07863730e8e14f03f5853cff18ee08cc": "r^2 - \\frac{2r_0}{r_0^2 - a^2} r\\cos(\\theta-\\theta_0) + \\frac{1}{r_0^2 - a^2} = 0.", "0786b3d3ff8cb066d85837afa952dd2e": "{}^{13}_{7}\\text{N}\\to{}^{13}_{6}\\text{C}", "0787060eb68af7205e261a6d1513aa89": "\\alpha < 1", "078757e10e62f005ec259835c931b771": "\\Delta H_c^\\circ", "07876a8567eb7d684ce5172df8bd487f": "\nd_2 = \\begin{bmatrix}\n-y\\\\\nx\\\\\n\\end{bmatrix}.\n", "07877f1626ec59a1150e271a430efbe4": "M\\,ds\\sqrt{v}", "07878fc6547acfb9bcc8a958e91d7bc3": " c(M,N) = \\left({ \\sum_{k=0}^{N-1} \\binom{M+k}{k} 2^k }\\right)^{-1} \\ . ", "0787dc3b9e4bf3484368a3902c5bbced": "\n \\lambda_{\\mathrm{chain}} = \\sqrt{\\tfrac{I_1}{3}} ~;~~ \\beta = \\mathcal{L}^{-1}\\left(\\cfrac{\\lambda_{\\mathrm{chain}}}{\\sqrt{n}}\\right)\n ", "07885fa4c2e009921cc1f7ebc938cb6a": "c_{A},c_{B} \\in [0,1]", "078869f7e8fdd24930b6b5e77b36dacb": "\\epsilon/\\epsilon_0", "078889072a75e391e732a9144a555c3c": "\\langle x^2 \\rangle =\\int_{-\\infty}^{\\infty}x^2\\frac{1}{\\sqrt{2 \\pi}}e^{-\\frac{x^2}{2}}=1", "0788abe959b8ff1f7c3071845bdc6a6d": "\\Gamma(a,z) \\sim z^{a-1}e^{-z}\\left(1+\\frac{a-1}{z}+\\frac{(a-1)(a-2)}{z^2}\\dots\\right)", "0789a67bdd33013a802f662f3980e22b": "(-a, 0)", "078a398ba2ed0731db1da302aacf0209": " \\zeta(-m, \\beta )-\\frac{\\beta ^{m}}{2}-i\\int_ 0 ^{\\infty}dt \\frac{ (it+\\beta)^{m}-(-it+\\beta)^{m}}{e^{2 \\pi t}-1}=\\int_0^\\infty dp \\, (p+\\beta)^m ", "078a766704afcaa594a3832203bea1cd": "A^{(a-1)/2}\\equiv +1 \\pmod a\\;", "078b07ae6be73b8e120a54b2632b6e41": "H(x^*(t),u^*(t),\\lambda^*(t)) \\equiv \\mathrm{constant}\\,", "078b8bd0aad721cae6a101460fff3766": " \\mathbb P\\big( \\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha \\big) \\le \\frac{ 1 } { k^2 }", "078b98585c52531242818511c6b154bf": "i_1< i_2< \\cdots < i_k", "078baafe2b532e3c9745aa0789297179": "S(\\mathbf{x}) = (\\mathbf{x} \\cdot \\mathbf{d})^2 - (\\mathbf{d} \\cdot \\mathbf{d}) (\\mathbf{x} \\cdot \\mathbf{x}) (\\cos \\theta)^2", "078bb829a1a71cebb8d843d700069253": "\n\\gamma^\\mu p_\\mu \\Psi = 0.\n", "078beef5c9bced4bfe33725d080816f6": "\\vec x= \\vec p + t\\vec r", "078c4da1f294a6312e132f7f0f2c0233": "\\mathbf{X} = \\{\\mathbf{x}_1,\\dots,\\mathbf{x}_n\\} \\sim \\mathcal{N}(\\boldsymbol\\mu,\\boldsymbol\\Sigma)", "078c510cd2d8847c528ea20d6ae1545d": "10^{10}{\\rm cm}", "078c521870ecf21cba86dc7d86a934e4": "\n\\frac{1}{r} = \\frac{1}{b} \\cos\\ (\\theta_1 - \\theta_0)\n", "078c555e9b1b83a63fc1985055b8307c": "b_n = r_n \\sin \\left( \\varphi_n \\right) = - \\frac{2}{T} \\int_{t_0}^{t_0+T} x(t) \\sin(2 \\pi n f_0 t) \\, dt \\ ", "078ca34fedc623cdec4dd1b75dbf9bfc": "\\mathbf{K}\\cdot\\mathbf{R}=2\\pi(k_1 n_1+k_2 n_2+k_3 n_3)", "078cc2efaf037b682afb955a44393851": "g_{s s} = 1 - \\frac{2 \\Phi}{c^2} \\,", "078ce003ac87485b8e59e9ea6fc52c96": "H^i(V, \\mathbb{Z}/\\ell^k\\mathbb{Z})", "078d9fb579d5120de13059b860540f3a": "z_0\\in\\mathbb{C}", "078dcac09b5c6acab007d0b51613421a": "\nL_i = \\int_x (\\psi^\\dagger\\psi)(x_1)(\\psi^\\dagger\\psi)(x_2)\\cdots(\\psi^\\dagger\\psi)(x_n) V(x_1,x_2,\\dots,x_n).\\,", "078dd444551348ea3f6b37f4ec305d38": "Z_{I2} = \\sqrt{\\frac{DB}{CA}}", "078df1ac9e620524e7af4fce16563234": "f_1\\mbox{ }:\\mbox{ }f_2", "078e05545233258bfef3bf484c87874b": "\\frac{\\partial (x,y,z)}{\\partial (\\rho, \\theta, \\phi)} =\n\\begin{vmatrix}\n\\cos \\theta \\sin \\phi & - \\rho \\sin \\theta \\sin \\phi & \\rho \\cos \\theta \\cos \\phi \\\\\n\\sin \\theta \\sin \\phi & \\rho \\cos \\theta \\sin \\phi & \\rho \\sin \\theta \\cos \\phi \\\\\n\\cos \\phi & 0 & - \\rho \\sin \\phi\n\\end{vmatrix} = \\rho^2 \\sin \\phi", "078e342007a7505c9ea9195cd363fe6c": " \\scriptstyle{a=-\\infty} \\ ", "078e3be4ee9973438a1776ec10d0ade5": "(\\neg A\\to B)\\to((B\\to A)\\to A)", "078e3fc4e2dfff81082c2a4a3078b5d7": "\n\\begin{align}\n \\eta &= \\varepsilon\\, \\eta_1 + \\varepsilon^2\\, \\eta_2 + \\varepsilon^3\\, \\eta_3 + \\cdots ,\n \\\\\n \\Phi &= \\varepsilon\\, \\Phi_1 + \\varepsilon^2\\, \\Phi_2 + \\varepsilon^3\\, \\Phi_3 + \\cdots \n \\quad \\text{and}\n \\\\\n \\mathbf{u} &= \\varepsilon\\, \\mathbf{u}_1 + \\varepsilon^2\\, \\mathbf{u}_2 + \\varepsilon^3\\, \\mathbf{u}_3 + \\cdots .\n\\end{align}\n", "078e4b138b54054ddd240d6fd58adba9": " e^{-1/T^{1/2}} ", "078edaa1a5ec22034aefa57ab5f8842c": "I=(a,b],\\ a\\geq -\\infty \\ ", "078f15812c1e340a21f34ec87b774c11": "T\\subseteq\\lambda\\cdot K", "078f15817acfc96b8b6ae79c436756a6": "\n\\begin{align}\n\\omega_r &= {1 \\over r\\sin\\theta}\\left({\\partial \\over \\partial \\theta} \\left( v_\\phi\\sin\\theta \\right) - {\\partial v_\\theta \\over \\partial \\phi}\\right) \\boldsymbol{\\hat r}, \\\\\n\\omega_\\theta &= {1 \\over r}\\left({1 \\over \\sin\\theta}{\\partial v_r \\over \\partial \\phi} - {\\partial \\over \\partial r} \\left( r v_\\phi \\right) \\right) \\boldsymbol{\\hat \\theta}, \\\\\n\\omega_\\phi &= {1 \\over r}\\left({\\partial \\over \\partial r} \\left( r v_\\theta \\right) - {\\partial v_r \\over \\partial \\theta}\\right) \\boldsymbol{\\hat \\phi}.\n\\end{align}\n", "078f414851ca4081636b3f54e7c19cab": "R_n(a_1,a_2,\\cdots,a_l)=(n+1-a_l,n+1-a_{l-1},\\cdots,n+1-a_1)", "078f6ad03d97c8307ecc5627f56b638e": "\\mathbf{g} = \\hat{\\mathbf{e}}\\tan\\left(\\frac{\\theta}{2}\\right)", "078f8b1d87a03db927e913543c17bca6": "\n\\delta = \\ln \\frac{R}{r}\n", "078f902f77268415c8f69289c814149a": " L =\\text{De Jan} \\text{ s}\\ddot{\\mathrm{a}}\\text{it} \\text{ das} \\text{ mer} \\text{ (d'chind)}{}^m \\text{ (em} \\text{ Hans)}{}^n \\text{ es} \\text{ huus} \\text{ h}\\ddot{\\mathrm{a}}\\text{nd} \\text{ wele} \\text{ (laa)}{}^m \\text{ (h}\\ddot{\\mathrm{a}}\\text{lfe)}{}^n \\text{ aastriiche.} ", "078fb69b545281d66be48dbd263cc910": " a = \\frac{p_1 - 1}{2} ", "078fe6201a479ae75ab26179290fcbf0": "\\sigma^2 =a_1+4a_2", "0790293fc4b123eb0f36eb0a6d78ea3c": "\\operatorname{pos}(U \\cup V) = \\max \\left( \\operatorname{pos}(U), \\operatorname{pos}(V) \\right)", "07902f881ffe6a02162aa07d4c1dfa04": "\\mathfrak{m}_1", "07904b4de470b155735daa96f3c8d4d5": "\\,\\Delta\\mathbf{w} ~ = ~ \\eta\\, y(\\mathbf{x}_n) \\mathbf{x}_{n}", "0790767d36e3688ba6d24c1c9fe61aba": "\n [\\boldsymbol{\\nabla}f(\\mathbf{x})]\\cdot\\mathbf{c} = \\cfrac{\\partial f}{\\partial q^i}~c^i = \\left(\\cfrac{\\partial f}{\\partial q^i}~\\mathbf{b}^i\\right)\n \\left(c^i~\\mathbf{b}_i\\right) \\quad \\Rightarrow \\quad \\boldsymbol{\\nabla}f(\\mathbf{x}) = \\cfrac{\\partial f}{\\partial q^i}~\\mathbf{b}^i\n ", "0790a1d764c4d067a8c698fb27272740": "V_i = M_i \\oplus E(M_i,A_i)", "0790e02160593a946b326cbd8a217cd6": "R_{ab}=8\\pi T_{ab}", "07910a4e5d2ec75aaad3b352e1d43829": "2^a3^b\\rho\\ge3", "079176c8e29d1f917b0d00985676d0bc": "M=\\left(\\widetilde{B}\\times F\\right)/\\pi_1B", "0791905f3c905a3bba5b31afbd79ea51": "(-\\infty,\\infty)", "0791a66e82c58f903c9ce378da46b6f3": "\\frac Rr\\ne \\frac{R^*}{r^*}", "0791b56c99a17e19785117b3ec8dac89": "f(x) = \\frac{(a/b)^{p/2}}{2 K_p(\\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2},\\qquad x>0,", "0791d0fd295994201f869fda975930d9": " \\omega \\,", "07924475c362d9ef2d50b8d84ec89d17": "\n p := -\\tfrac{1}{3}\\,\\text{tr}(\\boldsymbol{\\sigma}) = -\\frac{\\partial W}{\\partial J} = -2 D_1 (J-1) \\,.\n ", "0792dac93251d1a2c473f86682ca24a0": "T;Y \\,", "0792f1b63be29965d4194e3a82d304db": " f(x) = x^n+f_n x^{n-1}+...+f_1 \\in \\mathbb{Z}[x] ", "0792fbc0396ce564588008d7ddbac637": "\nQ=\\begin{pmatrix}\n0&0&-1\\\\\n0&J&0\\\\\n-1&0&0\n\\end{pmatrix}\n", "07935152f03e472299580aa15bb39322": " J \\, ", "0793554326d422491f32938dcc782c52": " p\\ K\\ (p\\ K) = K\\ (p\\ K\\ (p\\ K)) ", "07938ce9284d4ca381250ac22878f214": "\\mathbf{D} \\cdot {\\rm d}\\mathbf{A}", "0793f3f0fa1dc48a83b18ef3711fa48f": "\\Gamma(x),", "079407b01126de398d868fa81c01c73f": "\\{b_k\\}", "07942101007b8ecb11dc153c5c8c0da5": "\\Delta\\,T_m(x)=T_{mB}-T_m(x)=T_{mB}\\frac{4\\sigma\\,_{sl}}{H_f\\rho\\,_sx}", "079435779e8a9f09354627bca21b554d": "R(x_1,\\dots,x_n,f(x_1,\\dots,x_n))", "0794e8f6490e2236c2f899d1756f19ad": " u_3 ", "07952402115e4cd280dcd06fa9794ca5": "E[X_i=H\\mbox{ k out of n times}]=\nP(k,n)={n\\choose k} p^k (1-p)^{n-k}", "07953e6ef895cfce3fa72999ffa6d9c3": "(\\mathbf{A} + \\mathbf{A}^{\\rm T})\\mathbf{x}", "0795436aa1f00dc612d66a8ad74c0197": "r'\\,", "079626a29686af62428f258cbba09efe": "\\lambda=+1", "07974ea99dfe454eefeb4bcaa6083fc0": "g_m \\ r_O = \\begin{matrix} \\frac {I_C} {V_T} \\frac {V_A +V_{CE}} {I_C} \\end{matrix} = \\begin{matrix} \\frac {V_A +V_{CE}}{V_T} \\end{matrix} ", "0797d059b1316aa1f391bf60cc948b64": "(r+1)", "0797e4a661c4bb58bf65e11bc7e8fbaf": "X = (x_1, \\dots, x_n)'", "0797eddcd37cbbdbe182b2997e67186e": "f(x,y)=f(x,y+2\\pi)", "07981a091595e82db542568bb13f4064": "\\bar{A}^f_n = \\left[a A^f + (1-a) A_n^f\\right],", "079898aa8d94522b14e48505abb4231e": "v_{(G; c)}(\\{3\\})=7", "0798c843ba97cfbf97fd46dc4183c6b5": "\\overline{f}", "0798e8918be95124f6a68db54fa66f23": "MTTF = Aj^{-n} e^\\left(\\frac{Q}{kT}\\right)", "07994e43fce43f532bfe46704d0a6b30": "5 \\cdot 0 = 5 \\cdot 2 = 5 \\cdot 4 = 5 \\cdot 6 = 5 \\cdot 8 = 0 \\mod 10", "079958cd8a0faf47e193c20057f5d768": "\\{f^{(0)}_n (x)\\}", "079966c92fd1bea34a1e75a0ac35821c": "h[n]={{\\delta[n-1] + 2\\delta[n] - 3h[n-1]} \\over {4}}", "0799a469e665882648f757c5c7d455dd": "0, \\, 1=\\omega^0, \\, \\omega=\\omega^1, \\, \\omega^\\omega, \\, \\omega^{\\omega^\\omega}, \\, \\ldots \\,.", "0799a79811856765395d37a3606f5fad": "p = \\alpha\\overline{\\alpha}", "0799e0fde4ea45bd6f223e49b942fb8d": "\\partial \\subset P_n", "079a7ee3ed6ebc9231495b76ba70762c": "x_\\text{max}=\\frac{X_\\text{max}-X_0}{\\lambda}", "079b3039d1a9af18b3838c740a61d3f1": " \\qquad x_{n+1} = (\\epsilon)[r x_n (1-x_n)]_s + (1-\\epsilon)[r x_n (1-x_n)]_{s-1} ", "079b3b1d62f3e2cebd960448cef8350e": " \\Omega =\\arccos { {n_x} \\over { \\mathbf{\\left |n \\right |}}}\\ \\ (n_y\\ge 0);", "079b548a20f175cb786037c41c5a772e": "\\Delta G_i= \\sum_{j}\\gamma_j O_j~", "079bb003772b9993669167d6f942560e": " P( X < k ) = 0 \\text{ if } E( X ) > k \\text{ and } E( X^2 ) < k E( X ) + M E( X ) - kM ", "079c28d5bb1d3579df78a4ba94b7bc0b": "\\rho(\\vec{r}),\\, r\\, \\epsilon\\, \\reals^3", "079ce6bab2c2c49ca08fc38927b65c14": "\\{|\\psi_i\\rangle\\}_{i = 0,1,2, \\dots}", "079cef4b5a0aee68224afe49dee3806a": "\\int \\mathbf{1}_B \\, |f| \\ d\\mu = \\int_{B} |f(y)| \\, d\\mu(y) > \\lambda \\, \\mu(B).", "079d04bd53369a885a4b28fc72759de9": "f \\in k(x)", "079d0c77d14321410f742b4e6723f265": "\n y = 1.9,\\ 3.7,\\ 5.8,\\ 8.0,\\ 9.6\n ", "079d1298dc1aa97fb05d3e31a34e99ba": "P(x,y)=\\alpha A_{ji}/k_i", "079d1dd1a6bfa359c20a0f0f86b95244": " \\operatorname{let} x : x\\ f = f\\ (x\\ f) \\operatorname{in} x ", "079d3a60adc0db20d1548a37a9f64798": " \\binom{n-1}{n-x}.", "079e0906f54cec5f50d68cef26dcec24": "\n \\mathbf{M}_x = \\int_A \\left(-y\\sigma_{xx}\\mathbf{e}_z + y\\sigma_{xz}\\mathbf{e}_x + z\\sigma_{xx}\\mathbf{e}_y - z\\sigma_{xy}\\mathbf{e}_x\\right)dA =: M_{xx}\\,\\mathbf{e}_x + M_{xy}\\,\\mathbf{e}_y + M_{xz}\\,\\mathbf{e}_z\\,.\n ", "079e65f2e5200ed596fefbc5ca338dab": "x_1,\\,x_2", "079e90fce3aca99e1793748d8cf13797": "=\\frac{\\varepsilon\\cdot(1+\\varepsilon\\cdot\\cos \\theta)+(1-\\varepsilon^2)\\cdot\\cos \\theta}{1+\\varepsilon\\cdot\\cos \\theta}\n", "079ea1bf502c75add05019e423631989": " \\mathbb R^3", "079f3c0fef4432f1916946d862ef1bfc": "a = \\frac{1}{4p}; \\ \\ b = \\frac{-h}{2p}; \\ \\ c = \\frac{h^2}{4p} + k; \\ \\ ", "07a00cd5b0cc3bec1fd9df012b99014e": " \\left| \\int_{C_R} \\frac{f(z)}{5-z} dz \\right| \\le 2 \\pi \\rho \\frac{(3+\\frac{1}{1000})^{\\frac{3}{4}} \\rho^{\\frac{1}{4}}}{2-\\frac{1}{1000}} \\in \\mathcal{O} \\left( \\rho^{\\frac{5}{4}} \\right) \\to 0.", "07a045db2bd1e498a63749f712ea79fb": "\\left(\\tfrac{p}{5}\\right)", "07a067208a5f2d3664c63166c2d42441": "\\partial_\\hat{t} \\phi + 6\\, \\phi\\ \\partial_\\hat{x} \\phi + \\partial_\\hat{x}^3 \\phi =0", "07a0860ff99da4aae32240a53338c565": "[X]:=[X,X] \\, ", "07a0a6c0e56e693b951a065c0e60ece0": "f,g\\colon D^n \\to D^n", "07a13336865f965687fd89cef1847882": "E_5(x)=x^5-\\frac{5}{2}x^4+\\frac{5}{2}x^2-\\frac{1}{2}\\,", "07a1380ab446246937cf802ba6231205": "[a-1,a+1]", "07a13d291ced20db18b7299bcb9ca384": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0. \\, ", "07a145ff8a030ac01257a1f36db04057": "p=p_0 \\sin(\\omega t \\mp kx)", "07a166ccb950ebdb1898ff46b551a34f": " \\mathcal{F}(t) = \\sigma \\left( \\bigcup_{0\\leq s 3 \\\\ (\\frac{h_M}{3})^2 \\mbox{ if,} h_M \\le 3 \\end{cases}", "07c42fc623708dcff18cf4725c2236de": " I(x) = x \\, \\text{ln} x + (1-x) \\, \\text{ln} (1-x) + \\text{ln}2. ", "07c486fb4b0fe62b6ac0d6812622d104": "\\widehat\\beta_j = c_{1j}y_1+\\cdots+c_{nj}y_n", "07c4b1c417f00bf0185eab23d4c98e0b": "\\tilde{4}", "07c4d198de67c9a2105575ff7ad439a2": "a\\propto t^{\\frac{2}{3(1+w)}},", "07c4e50fb61792c2c6499d99dce0fb86": "rK/Y=D_K[F(K,L)]*K/F(K,L)\\,", "07c517ae18a8634ffd9e648ceebfbb5b": "A \\propto L^2", "07c546846b741996053cf2b6439fa1a0": "\\textbf{V}_O=\\dot{\\textbf{d}},", "07c572ea3a09fb7c69a6343e8e3bf4a8": "\n \\quad (4) \\qquad \\epsilon(x,t) = \\sum_{m=1}^{M} e^{at} e^{ik_m x}\n", "07c5cd354f729bbd65ca75545e335213": "\n\\begin{align}\n\\frac{d \\phi}{d t} & = -k(D-A)\\phi \\\\\n& = -k L \\phi,\n\\end{align}\n", "07c63f19fc3af989a2abc4d944cffb25": "M_{PL}=\\frac{M_{star}V_{star}}{V_{PL}}\\,", "07c66863e9b22c9997ef6cfae0734f87": " BA = q AB", "07c6b9b031c2f39202629eebedcc4fa0": " 1-ee=\\frac{1-c-cee''}{1-c}", "07c6c00f24f9522906d343bad4c19afd": "C_1: f_1(x,y)=0, \\ C_2: f_2(x,y)=0.", "07c6d4483bf3e1a60cdb9705810301bb": " \\lambda_1 = \\lambda_2 = 0 ", "07c7010061587e4178aad0eedb95a1bf": "x, y \\in fRep", "07c72b098a91f2641aa9b6627a9499f1": "y(x,t) = y_0 \\cos \\Bigg( \\omega \\left(t-\\frac{x}{c} \\right) \\Bigg)", "07c72c20b02ad827027b41c6e810155a": " \\vec{r}_1 ", "07c74fccb3a16df1a8a955eb01442dda": "\n\\begin{align}\nP_0^0(\\cos\\theta) & = 1 \\\\[8pt]\nP_1^0(\\cos\\theta) & = \\cos\\theta \\\\[8pt]\nP_1^1(\\cos\\theta) & = -\\sin\\theta \\\\[8pt]\nP_2^0(\\cos\\theta) & = \\tfrac{1}{2} (3\\cos^2\\theta-1) \\\\[8pt]\nP_2^1(\\cos\\theta) & = -3\\cos\\theta\\sin\\theta \\\\[8pt]\nP_2^2(\\cos\\theta) & = 3\\sin^2\\theta \\\\[8pt]\nP_3^0(\\cos\\theta) & = \\tfrac{1}{2} (5\\cos^3\\theta-3\\cos\\theta) \\\\[8pt]\nP_3^1(\\cos\\theta) & = -\\tfrac{3}{2} (5\\cos^2\\theta-1)\\sin\\theta \\\\[8pt]\nP_3^2(\\cos\\theta) & = 15\\cos\\theta\\sin^2\\theta \\\\[8pt]\nP_3^3(\\cos\\theta) & = -15\\sin^3\\theta \\\\[8pt]\nP_4^0(\\cos\\theta) & = \\tfrac{1}{8} (35\\cos^4\\theta-30\\cos^2\\theta+3) \\\\[8pt]\nP_4^1(\\cos\\theta) & = - \\tfrac{5}{2} (7\\cos^3\\theta-3\\cos\\theta)\\sin\\theta \\\\[8pt]\nP_4^2(\\cos\\theta) & = \\tfrac{15}{2} (7\\cos^2\\theta-1)\\sin^2\\theta \\\\[8pt]\nP_4^3(\\cos\\theta) & = -105\\cos\\theta\\sin^3\\theta \\\\[8pt]\nP_4^4(\\cos\\theta) & = 105\\sin^4\\theta\n\\end{align}\n", "07c7d5ba70a439c2672cc9f9ff7fd5c6": " \\eta \\rightarrow 1 ", "07c86837ae050de703b9b4ae927ea74f": " s \\equiv r \\,\\bmod p^k \\Rightarrow f(s) \\equiv f(r) \\,\\bmod p^{k+1}", "07c87896e2f6ba6e78a0aee9cbc12fe9": "a^{n-1} \\equiv 1\\pmod{n}.", "07c8dbeacc8116af36c1b3751d6281b6": "(X,\\mathcal{B},m)", "07c8f060c1725bba7aa487a844c9476b": "M = (M_t)_{t \\ge 0}", "07c9080b749d0c13e4d837ebbbc9e37d": "B_{\\alpha \\beta }", "07c965672f4a5f68c7bd1e6ebcd41757": "\\Psi(x,y)=xu^{O(-u)}", "07c9cc926c59abb1d9faa3929434f9ee": "(n-1)\\times 1", "07c9d9baa051a6669920ce7cfdd6cca9": "Q^{2}_{0}", "07c9df79208d745a5f7d28440089223a": "R = {{\\mathbf{k}}}[x_1, \\ldots, x_n]", "07c9f0f65f673a264a5101683e774507": "\\alpha_{i}", "07ca6bfc1ec481f65b7a5e66ad113a86": "K(-u) = K(u) \\mbox{ for all values of } u\\,.", "07ca7c9ea7468b113c70a30966addd2f": " \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K\nn_{j,(\\cdot)}^i+\\alpha_i \\bigr)}{\\prod_{i=1}^K\n\\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)} \\prod_{i=1}^K\n\\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j =1 .", "07ca9562bbb1523000132740402e0821": "\\mathbf{A} = \\left[\\begin{array}{ccc}\n 1 - 2q_2^2 - 2q_3^2 & 2(q_1q_2 - q_3q_4) & 2(q_1q_3 + q_2q_4)\\\\\n 2(q_1q_2 + q_3q_4) & 1 - 2q_1^2- 2 q_3^2 & 2(q_2q_3 - q_1q_4)\\\\\n 2(q_1q_3 - q_2q_4) & 2(q_1q_4 + q_2q_3) & 1 - 2q_1^2 - 2q_2^2\n\\end{array} \\right]", "07caab57203b6fc1892fd63ec88de3b8": "\\textrm{Bl} \\ ([D])", "07cae0c56048358e5028c12ecf5378f9": "E(x,y) + \\lambda V(y).", "07caeb770a1a54b8038e0b7f91471753": "\\{L_i(z)\\}_{i=0, 1, ..., N-1}", "07caf5113a9a1987819f000cef81323a": "c \\leq 0", "07cbc478c48c75e20e5161ce2afe38fe": " (x-1)^{-2n-2} P_{n+1}(x) = \\left( x(1-x)^{-2n-1} P_n(x) \\right)^\\prime ", "07cbcb705ecb00a73efe88560b8111d2": "H:\\mathcal{A}\\to\\mathcal{L}", "07cbd6c155424e110559a84df364be5a": "L_2", "07cbfbc8ecd30c38d7262bd4bb61b1bb": " T_n ", "07cc002e715d0752fa5c15c2b888c436": "x_1, x_2, ..., x_k", "07cc2c497da32faf7daaf07ac443db40": "7^6 = 343^2 \\equiv 5^2 \\equiv 25 \\equiv -1 \\bmod 13.", "07cc3836ebb271c10041263ecfa731fb": "K(x) \\leq K(x,S) +O(1) \\leq K(S)+K(x|S)+O(1) \\leq K(S)+\\log|S|+O(1) \\leq K(x)+O(1)", "07cc5009afa1f350aaebe36f0a3b040f": "R(n_1,\\ldots,n_k) \\Leftrightarrow \\psi(n_1,\\ldots,n_k)", "07cc6948b3ee101934f470bb101d8e0f": "\nV(r) = \\frac{mc^{2}}{2} \\left[ - \\frac{r_{s}}{r} + \\frac{a^{2}}{r^{2}} - \\frac{r_{s} a^{2}}{r^{3}} \\right]\n", "07cc694b9b3fc636710fa08b6922c42b": "time", "07cc72d1e021c27f30df1d6859ad7487": "\\scriptstyle\\star ", "07cc76f54ed9d934037070a5d38936fa": "t_i=0,", "07ccb14a3caf4b8e2a190ad94e61c477": "\\exp \\{ i\\mu t - \\frac{1}{2}\\sigma^2 t^2 \\}", "07ccc8a49ed7e50dae6493dddacb1337": "\\gcd(a, b) =\\gcd( b, a).\\;", "07cd0c9345dc0e317d87b3277fe82d33": "(1)\\Leftrightarrow(2)\\Leftrightarrow(3)\\Leftrightarrow(4)\\Leftrightarrow(5)", "07cd160f356bcd99031846437ffb6778": " R_k(x) = \\frac{f^{(k+1)}(\\xi_C)}{k!}(x-\\xi_C)^k(x-a) ", "07cd864dbf621cda99ed595a7ac398b6": "D(g \\circ f)(x) = Dg(f(x))\\circ Df(x).", "07cde9b882c862e19d4a5eb8681f70e9": "\\overline{K}:=\\{0,1,\\infty\\}", "07cdfd3f2454ba70e05c0cdc4a7854cd": "G(\\chi\\chi^\\prime)=\\chi(N^\\prime)\\chi^\\prime(N)G(\\chi)G(\\chi^\\prime).", "07ce14b7349b70c72fbd8c385c006ca3": "\\psi(x) \\rightarrow D(\\Lambda) \\psi(\\Lambda^{-1}x) ", "07ce389119f86db93f5d510d0b6d587a": "\\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|a_n|},", "07ce6c9e3ddac38a1f039aa7ba3eba7b": "f_{e,\\Gamma, R}=\\sum_{p: \\, e \\in p}{f_p}.", "07cea9830ca9474f6448c247178d5601": "F(X) = \\frac{1}{M}\\sum_{m=1}^M T_m(X) = \\frac{1}{M}\\sum_{m=1}^M\\sum_{i=1}^n W_{im}(X)Y_i = \\sum_{i=1}^n\\left(\\frac{1}{M}\\sum_{m=1}^M W_{im}(X)\\right)Y_i", "07cec40f230f56832c4f520622dbb971": "S(T) = \\frac{1}{\\pi}\\mathop{\\mathrm{Arg}}(\\zeta(1/2+iT)) =O(\\log(T)).", "07cf3bdd7fafae355d8e940d0d0c8ff3": "\\int_X p(x;\\theta)dx =1", "07cf65b648327a23d03aee1d3d01396a": "\n\\begin{align}\ns \n&=p_1 p_2 \\cdots p_m \\\\\n&=q_1 q_2 \\cdots q_n.\n\\end{align}\n", "07cf77dd31bd2f7129a37461b9117b7b": "RSTUV", "07cfb64d1763c263fff4490df998db91": "\\left ( \\phi \\to ( \\psi \\rightarrow \\xi \\right)) \\to \\left( \\left( \\phi \\to \\psi \\right) \\to \\left( \\phi \\to \\xi \\right) \\right)", "07d00fa47dad1fdd6db21a172bf289d0": "\\mathbb E[f(x_n) - f^*] = O(1/n)", "07d041038e9a835f2354401c8e2aac4a": "\\sum_{i} p^{ij} = 1, \\ ", "07d04a0ebd91ae40b0be1239f9b9d28f": "\\frac{dW}{d\\omega}\\approx \\sqrt{\\frac{3\\pi}{2}}\\frac{e^2}{4\\pi\\varepsilon_0 c}\\gamma\\left ( \\frac{\\omega}{\\omega_\\text{c}} \\right )^2 e^{-\\omega/\\omega_\\text{c}}", "07d0bf51630248ffbe90a7052bfa15e5": " Q_B (l_A a_B + l_B) l_B ", "07d148ab82f89959ab34650ead1fe3b6": "\\mathbf{w}=\n\\begin{bmatrix}\n(Q+a-1)&\\frac{1}{3}m&0&0\\\\\nm&Q&\\frac{2}{3}m&0\\\\\n0&\\frac{2}{3}m&Q&m\\\\\n0&0&\\frac{1}{3}m&Q\n\\end{bmatrix}\n", "07d15a9ed668847ae9885c2b04698bf6": "\\tau = rF\\sin\\theta,\\!", "07d1746d1d350c31d3fb0de089483818": "\\mu_{T} = \\left( \\pi_{ST} \\right)_{*} (\\mu_{S})", "07d1810b6e4a730498bf6f95abd7a7bd": "r^0", "07d1b04e8d1599a0a4256c61132b0e27": " \\omega = 2*\\pi /0.1 ", "07d1b8e0a8f21ab94d74bdcc820fac60": "{\\delta} < \\mathrm{error} ; ", "07d1deb679816938dc05177722496beb": " (K \\phi)(x) = \\sum_y K(x,y) \\phi(y) \\,", "07d20f50f5198298e034d36b7a46493d": "2 \\times \\sqrt{3}", "07d22e4f4046963f2eaf5627d0e37d04": "p_{j,t-1}", "07d25ff8ad8b1381e164770c9e90e050": "Z(t)=I(t)+jQ(t)\\,", "07d26a08be43af7cb561d6b6b8eec113": "p^2 + 2pq + 2pr + q^2 + 2qr + r^2 = 1. \\,", "07d2a5bee02b2b0042fc92d05b95818e": "\\boldsymbol{\\omega} = (\\omega_x, \\omega_y, \\omega_z) ", "07d2aa1b053b0001c46c43695eb3655d": "e=C_vT", "07d2bb600c8d9b65679ffedd1bad08bd": "\\mathbf{F}_{\\mathrm{Centripetal}} = \\mathbf{T} + \\mathbf{F}_{\\mathrm{Fict}}\\ , ", "07d3755579f31a45280dfc8ded0e80d7": "e_1,\\ldots,e_m \\in \\mathbb{T}", "07d3936feb19afdacadbe368a18ac88d": "f(xy)=f(x)+f(y), f(1)=0", "07d3a06c3b9f4fdf60055d30a5b2070b": "[\\hat{X},\\hat{P}] = \\hat{X}\\hat{P}-\\hat{P}\\hat{X} = i\\hbar", "07d3c8cf5f9b1d5f12740463fc056102": " \\mathbf{J} = \\mathbf{J_f} + \\nabla\\times\\mathbf{M} + \\frac{\\partial\\mathbf{P}}{\\partial t}", "07d3e0a0783d2d067f6fa1f93664ce1a": "\\nabla \\times \\vec{B} = \\mu_0 \\vec{J}", "07d3e5de2d131680b4ff26c328b4cc6f": "t_{mn} = \\frac{(m+n)(m+n-1)\\cdots(m+1)}{n(n-1)\\cdots 1}.\\ ", "07d41ce9ee6e308e17d75c30e4b6c000": "\\Gamma(n+1/p) = \\Gamma(1/p) \\frac{(pn-(p-1))!^{(p)}}{p^n}", "07d421ec371c7d4d836b60b5a4da084c": "\\frac{\\partial \\rho}{\\partial t} + \\vec \\nabla \\cdot(\\rho \\vec v) = 0 ", "07d422876e555cf72ff10918f1f92485": " H^{-1}(z)", "07d4d6ec5a86f2a9b56a9d012ef281fa": " \nH = \\frac{(l_1)^2}{2I_1}+\\frac{(l_2)^2}{2I_2}+\\frac{(l_3)^2}{2I_3}+ mg (a n_1 + bn_2 + cn_3),\n", "07d52077eaa5865dfc7121020bcf09c1": "\n\\begin{array}\n[c]{cccccc}\ng_{1} & = & Z & X & Z & I\\\\\ng_{2} & = & Z & Z & I & Z\\\\\ng_{3} & = & Y & X & X & Z\\\\\ng_{4} & = & Z & Y & Y & X\n\\end{array}\n", "07d5364be7263d4eaad2c3f82df50154": "K=I\\otimes T", "07d593f0b25ba1a8bf43dac9a1d4d41f": "(x_s, t_s)\\,", "07d5c7099ff999998f0068b6b34ab6d4": "\nCIQ_t = \\mathcal{A} e^{\\mathcal{B} t}\n", "07d63f12586dfdbaebc11e3311a2d36b": "F_+(H) = \\overline{S^*H}", "07d64d6c01234b60032aa525cd2c1f96": "\\mathrm{Rot}_H", "07d68e12866dda148c93268f6bf2ec95": "\\frac{\\partial^2 y}{\\partial x^2}=\\frac{\\mu}{T}\\frac{\\partial^2 y}{\\partial t^2}.", "07d73fc27d368d61cd55cb4d5e1f29e8": "\\Leftrightarrow \\!\\,", "07d7c4352aefd6bda26303c773765454": "a_0 b_n - \\tbinom{n}{1}a_1 b_{n-1} + \\tbinom{n}{2}a_2 b_{n-2} - \\cdots +(-1)^n a_n b_0 = 0", "07d8112f3cf98ff31b7aac846f90cd75": "\\varrho(T_h)", "07d86c31e7078074357f17c2fa997928": "PR(A)= \\frac{PR(B)}{2}+ \\frac{PR(C)}{1}+ \\frac{PR(D)}{3}.\\,", "07d87337f49d692cfd1c1dc4bdc54771": "w = d + m + c + y \\mod 7,", "07d8da455eb16ff3a133f69d7a2964af": "\\zeta(s)=\\frac{\\eta(s)}{1-2^{1-s}},", "07d8fbd2720f2d36a6de65b679b3adea": "\n\\begin{pmatrix} & h& \\\\[-0.9ex] v & & v'\\\\[-0.9ex]& h'& \\end{pmatrix} \n", "07d935680b6501b2e42fe4baea021389": "mk", "07d9577618053507ed710ae0be8a4705": "n-m\\ge 0", "07d9d68a024064595021c95152f318e3": "6 + \\sqrt{3}", "07d9d7cd24111b32653ded6c2e075a8c": "\\sigma = \\pi^2 k^4 / 60 \\hbar^3 c^2 ", "07d9f7a4cfc9c776b7034b04068cce16": "pf_i = C/N = 0.311\\!", "07daf1dddf3b5a8e6724497dfb74d5d6": " [U_h(\\mathrm{M}(a,b,c))]\\psi(x) = e^{i (b \\cdot x + h c)} \\psi(x+h a). ", "07daf43d269c6cb7b45c16ca4062ceb6": "\\mathbf{z} = \\left \\{(x_{i},y_{i}) \\in X \\times Y: i = 1, \\dots, m\\right\\} \\in Z^{m}", "07daff2abb9da1e1697b8a58798985ec": " V = \\pm \\frac{ f R }{2} \\pm \\sqrt{ \\frac{f^2 R^2}{4} - \\frac{R}{\\rho}\\frac{\\partial p}{\\partial n} } ", "07db2bb21ed4bca1aeef150981f8ca83": "\\mathsf C", "07db5288cfa5b7e0cb01a657c5ab31b9": "G_X(t,f) = G_x(-f,t)e^{-j2 \\pi ft} \\, ", "07db94a164b976adbf9fbd45788266b5": "d\\mathbf X\\,\\!", "07dbc28d6621cd56804fd8d4ed5a1205": "\\frac{d^2}{d\\theta^2}\\left(\\frac{1}{\\mathbf{r}}\\right) + \\frac{1}{\\mathbf{r}} = -\\frac{\\mu\\mathbf{r}^2}{\\mathbf{l}^2}\\mathbf{F}(\\mathbf{r})", "07dbfcb7ead62d17fb5e5df064d63b6e": "\\tilde{f}=\\left|\\frac{\\tilde{d}}{2}\\right|", "07dc07bc1536e975103ee20654509c29": "\\frac{\\sqrt{2}}{2} \\left(\\frac{(2m-1)\\Omega}{m}\\right)^{1/2}", "07dc172a833d6915b7c243373714b5dd": "I_{im+}", "07dc6ec99fe20876f73ca2bc44eaf4e6": "A,B \\in E", "07dddf7ed882ed38f02642e10b723f59": "\\mathbf{r}_6 = (a/4)(3\\hat{x} + \\hat{y} + 3\\hat{z})", "07de1a3d19ef9adf4304071b0922a724": "x^* = \\text{null}", "07de7dafd1a757933a70ece3441ce9b7": "\\{\\psi(w) : w\\not=v\\}", "07de97a4f99b2d930a3ab53023301768": "( u \\wedge v) w = - w \\cdot ( u \\wedge v) + w \\wedge u \\wedge v", "07deb2311d8a8b360fbf44fa38230ceb": "A(u) = \\frac{u^2 + 2}{u \\sqrt{u^2 + 4}}", "07df2900bfa96aecf5901be3829a1bdd": "M\\to H_1(M,\\mathbb{R}) /\nH_1(M,\\mathbb{Z})_{\\mathbb{R}}", "07df403dfe51db0194e6c677a582ab10": "\\int d[wx^2] = \\int x^4 dx", "07df5771b077f4a06dd347b292015939": "R(x,y_1,\\dots,y_n)", "07df8dc8930628c9016f6332f6edab8a": "\n\\widetilde{\\theta} = \\frac{\\exp {(- \\beta u)} - \\exp {(- \\beta u_0)}}{1 - \\exp {(- \\beta u_0)}}\n", "07e01a2d436b13f41b9cdf19214307d8": "\\bar{b}^2 G_C", "07e0689fd47aac7cc7b899beb05fabbd": "\\vdash \\in \\Gamma - \\Sigma", "07e1019f0737536293bb710b19de8c60": "\\Gamma^{[k-1]}", "07e10f3656d7cfd24b00d6804a1c41cb": "H_\\Lambda^\\Phi(\\omega) = \\sum_{A\\in\\mathcal{L}, A\\cap\\Lambda\\neq\\emptyset} \\Phi_A(\\omega)", "07e13d322a1dc4341d3d7c3c36993dae": "\\Phi_{1}\\left(\\mathrm{R}_{i}\\right)", "07e1a666e867b50fc7ee58bbfa4544aa": "E_\\lambda", "07e1a8662990a0595e395b6349adbc6c": "\\displaystyle{Tf_n=\\mu_nf_n}", "07e1af018de324ecf10b02348a778236": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {c_2} {c_1} = \\frac {l_2} {l_1} \\,,", "07e200bba2d90c20ed0773de03be3cd9": "\\Omega_{-}", "07e20de5d75966ec1f7ad971c27a9490": "\\phi(v_j,v_k)=\\int_0^1 v_j' v_k'\\,dx", "07e240557ba49686c97a43603c5f1193": "s=-x^3-x^2+x\\ ", "07e25498887751c397b19bb5787ed061": "\\pi \\approx {355 \\over 113}", "07e271ec125747627ea1737274501e63": "\\sqrt{I}=\\{r\\in R|r^n\\in I\\ \\hbox{for some positive integer}\\ n\\}.", "07e2ba2bd104f609d18414d2507428be": "\\{ C (\\vec{N}) , G (\\lambda) \\} = G (\\mathcal{L}_\\vec{N} \\lambda)", "07e2e9364a982bb791f5b5745b9c1d96": "\\vec{N}=\\{0;\\; 1; \\; 0\\}; \\;\\; \\vec{E}=\\{\\frac{\\sqrt{3}}{2}; \\; \\frac{1}{2}; \\; 0\\}; \\;\\; \\vec{L}=\\{-0.6; \\; 0.8; \\; 0\\}; \\;\\; n=3", "07e2ecb3228caaddeed2a9869696a507": "\nm_1\\;\\operatorname{sc}^2(u)+m_1= m_1\\;\\operatorname{nc}^2(u) = \\operatorname{dc}^2(u)-m\n", "07e2f7f391b53640f096df4a27b66ce6": "\\hat{t} = \\operatorname{argmaxminlocal}_{t}(\\nabla^2_{norm} L(\\hat{x}, \\hat{y}; t))", "07e378366ded1264c8b2a4c2fb497a10": "x_1:= x\\,", "07e3c84bd480e3f434119e3fa3b0c84d": "\\nu_x= \\frac{1}{2} \\delta_{-1} + \\frac{1}{2}\\delta_1", "07e41023fdc5086c51ea6aa944023f34": "\\mu_r''=(\\frac{\\lambda_g^2+4a^2}{16a^2})(\\frac{V_c}{V_s})(\\frac{Q_c-Q_s}{Q_cQ_s})\\,", "07e4b3f7df2f71b1c46fa47ce0f29f56": " f(\\mathbf{y}) + [J_f](\\mathbf{[x]}) \\cdot (\\mathbf{z} - \\mathbf{y})=0 ", "07e4f44940d9af1fa9cc5954202a9b9e": "\n \\boldsymbol{s} = 2 K~\\left(\\sqrt{3}\\dot{\\varepsilon}_{\\mathrm{eq}}\\right)^{m-1}~\\dot{\\boldsymbol{\\varepsilon}}_{\\mathrm{vp}}\n ", "07e4fbe766dabd2aeccdf093039dbdad": "A_{i, j, k}", "07e4fc3d3abe469b6ace7dc96abb5e95": "d (f, g) := \\| f-g \\|", "07e51bf43f0a51d6988c5a86b5b9bcc5": "\\omega^2 = \\int_{-\\infty}^{\\infty} [F_n(x)-F^*(x)]^2\\,\\mathrm{d}F^*(x)", "07e557a11281006cc8627851723d1022": "\\Box A_\\mu = \\frac{4 \\pi}{c} J_\\mu", "07e55f58e7dbe0217993dc53faa7b1b2": "f(z) = \\sum_{n=0}^\\infty c_n z^n,", "07e59ea71bf21615c75fc41b46a45d78": "q \\ge 0", "07e5a4a56a57f5c874ebf79bb67a0b18": "\\mathbb R", "07e5b3f26da912ff2b11115cfd81091d": "D_2=kTB_2(1+N_2\\frac{d ln\\gamma_2}{d lnN_2}) ", "07e5b4c482edd88ba7d1c8cfd8d63fb5": "\\frac{d A}{dz} =-i \\gamma_{\\|}|A|^2A", "07e6d08dadd5f7bd8b8b2a7aea06aaf0": "\\phi = \\pm \\pi / 2\\,", "07e7c990ae832000342d7c0251b7e594": "\\delta \\mathbf{Z}_0 \\to 0", "07e83a2180c6cc88a1926d0e7b96f29d": "\\operatorname{E}(\\mathbf{1}_A) = \\operatorname{P}(A). \\; ", "07e84ffdd5db350ab15b792a02f529b4": "\\scriptstyle \\mathfrak{X} (M)", "07e866a596b9a2ab3e7d7da99ebb774b": "\\gamma_2 = \\exp(-\\delta_1-\\delta_2)+\\exp(-\\delta_1-\\delta_3)+\\exp(-\\delta_2-\\delta_3).", "07e866dcdf6518db1b1c1fc125830bd4": "1852 = metres\\ per\\ nautical\\ mile", "07e873918decf42106f6f9d2d99d8188": "x=a(1-\\sin\\psi),\\,y=a\\frac{(1-\\sin\\psi)^2}{\\cos\\psi}.", "07e8fda49b6107fd677f5bf1e507a270": "\\Delta^n_{t\\Delta x} f", "07e922057d45dfb7eb00b9d826750685": "1-n/N", "07e9493b94f772c30c1ab8a66aa96f7b": "\\mathbf{J}(\\mathbf{r}, t) = \\rho(\\mathbf{r},t) \\; \\mathbf{v}_\\text{d} (\\mathbf{r},t) \\,", "07e95931f58a4cf5b9c090be27f0bc6e": "1-F_{Y}(q)", "07e960f2c0ed274e20c6e1c5bb5aa04c": "T_1^{(1)},T_2^{(1)},X_1^{(1)},X_2^{(1)},H^{(1)}", "07e96fc1b25d8051509ab34ac69522e7": " \\mathbf{T} = \\begin{pmatrix} \na_\\text{x} b_\\text{x} & a_\\text{x} b_\\text{y} & a_\\text{x} b_\\text{z} \\\\ \na_\\text{y} b_\\text{x} & a_\\text{y} b_\\text{y} & a_\\text{y} b_\\text{z} \\\\\na_\\text{z} b_\\text{x} & a_\\text{z} b_\\text{y} & a_\\text{z} b_\\text{z}\n\\end{pmatrix}", "07e9990f329c2a9a3d96a09c210f94e8": "v_{1,2} = 5", "07e9b2c777ce42a404f6b03885773b7f": "\n\\tanh(\\alpha + \\beta) = {\\tanh(\\alpha) + \\tanh(\\beta) \\over 1+ \\tanh(\\alpha) \\tanh(\\beta) }\n", "07ea01e1bb346a8adf43797c14bb5e5b": "\n\\frac{1}{2} \\frac{dI}{dt} = \\frac{1}{2} \\frac{d}{dt} \\sum_{k=1}^N m_{k} \\, \\mathbf{r}_k \\cdot \\mathbf{r}_k = \\sum_{k=1}^N m_{k} \\, \\frac{d\\mathbf{r}_k}{dt} \\cdot \\mathbf{r}_k = \\sum_{k=1}^N \\mathbf{p}_k \\cdot \\mathbf{r}_k = G\\,.\n", "07ea67ca0036c2ba5440bb73c375ad1a": " \\Delta\\mathbf{r}_i\\times(\\boldsymbol\\omega\\times(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i )) + \\boldsymbol\\omega\\times(\\Delta\\mathbf{r}_i\\times(\\Delta\\mathbf{r}_i\\times\\boldsymbol\\omega))=0,", "07ea97937a2b0e75b07b6a136d022618": "\\Vert f^*_n \\Vert \\le 1", "07ea9eb1f4232484e23c7ec7420df172": "\\frac{1}{a}", "07ebdda21bfd38368e5a089060b7f27b": "\\lbrack\\mathbf z\\rbrack = \\lbrack\\mathbf z\\rbrack_1 + \\lbrack\\mathbf z\\rbrack_2 = 2\\lbrack\\mathbf z\\rbrack_1 = \\begin{bmatrix} 2R_1 + 2R_2 & 2R_2 \\\\ 2R_2 & 2R_2 \\end{bmatrix}", "07ec11425cb4ccb2b0aba3c2ed074fe4": " (\\bullet\\bullet\\bullet)(\\bullet)", "07ec12590399c4f008aeb69aebdfc16c": "\n \\tau_m = \\sigma_m \\sin\\phi + c \\cos\\phi ~.\n ", "07ec3a356588619c88a8fdb0443923da": "i=1...n", "07ec5f49bbac96fc6b295696f31015df": "\\Phi(i)", "07ec7c9d1d7727d8da38fbb903501d01": "dF_\\mathrm n\\,\\!", "07ec8edd29169a1e35f05c1344c8c0ce": "\\chi_a", "07ecf9eacc6696e59015190e4684fcbe": "\\boldsymbol\\omega=\\frac{\\mathbf{r}\\times\\mathbf{v}}{|\\mathrm{\\mathbf{r}}|^2}", "07ed5b28b87ea8650dd99c429d927e28": "R_{0}=\\{(x,x):x\\in X\\}", "07ed62bddcc38436d34bcfdb378e32bc": "\n d\\mathbf{f} = \\boldsymbol{F}\\cdot d\\mathbf{f}_0 = \\boldsymbol{F} \\cdot (\\boldsymbol{S}^T \\cdot \\mathbf{n}_0~d\\Gamma_0)\n", "07ed7b7884737b80357da49facb87ff4": "n \\ge 4.", "07ed9c3f277abe2ae9ca8f83a9b87e83": "M_{2x} = \\dot{m}V_{2x} = - \\rho QV_2 \\quad and \\quad F_{P2x} = \\overline{P}_2A_2", "07ed9cd92782b85be245409f24a9b337": "y (\\theta) = r (k - 1) \\sin \\theta - r \\sin \\left( (k - 1) \\theta \\right). \\,", "07eda6dfa5faa951c2089ae9b256594b": "\n D\\,\\nabla^2\\nabla^2 w = -q(x, y, t) - 2\\rho h \\, \\ddot{w} \\,.\n ", "07ee3609f571e22755490614f22f2f3b": "e^{i \\pi}= -1.", "07ee477f13d903896289636d38728763": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 24.03761 \\log_e(T+273.15) - \\frac {7062.404} {T+273.15} + 166.3861 + 3.368548 \\times 10^{-5} (T+273.15)^2\n", "07ee4e74e6c56bd8d51ed1a555cea2bc": "_k\\mathbf{b}_{l,m,n} = \\mathbf{S}_k\\mathbf{a}_{l,m,n}", "07ee679472c3e77e252a87bcef5a40f7": "\n\\begin{align}\n\\omega_1 &= \\omega - {e^2\\over 32\\pi\\varepsilon_0 m_e\\omega Z^3},\\\\\n\\omega_2 &= \\omega - {e^2\\over 16\\pi\\varepsilon_0 m_e\\omega Z^3}.\n\\end{align}\n", "07ee9b67a0557b8f091293637b1a079b": "(S,\\Sigma)", "07ef19656e5b0e7f28762bfaa1fb9ba8": "[i_{L_1},i_{L_2}]= i_{[L_1,L_2]^\\and}", "07ef275540acce92238e509054c30393": "{\\overline{b}}=(B^{-1}a_1B,\\ldots,B^{-1}a_nB)", "07ef7a9526aa22e94314110e1f000f61": "0,\\ldots,n-1", "07ef82cb261e1d693985694652fda01b": " \\lbrace T \\rbrace ", "07ef8344f0d7f63f29cb988bff684c67": "\\Im z =0", "07efbca572b25c0069d4b524dd94a4a1": "H_{n+1}(x)=2 xH_n(x)-H_n'(x).\\,\\!", "07efc8cc2791419a300e2582688e62f5": "x_1,\\ldots,x_j", "07effcc790d2b70570b1db621da3b832": "\\frac{v_0 [Cl^-]_0-v_i[Ag^+]_0}{v_0+v_i} \\begin{cases} \n\\approx [Cl^-]_i \\text{ or } K_{sp} 10^{-b_1E_i+b_0} & \\text{ when } v_{0^{ }} [Cl^-]_0 > v_i[Ag^+]_0 \\text{ (before equivalence)} \\\\\n= 0 & \\text{ when } v_{0^{ }} [Cl^-]_0 = v_i[Ag^+]_0 \\text{ (equivalence point)} \\\\\n\\approx -[Ag^+]_i \\text{ or } -10^{b_1E_i-b_0} & \\text{ when } v_{0^{ }} [Cl^-]_0 < v_i[Ag^+]_0 \\text{ (after equivalence)} \n\\end{cases} ", "07effe5ef25cfe6080ecea6307b82361": " a_{ab} ", "07f02e761349ad9cc156d0396d6f371f": "U_n=a \\varphi^{n-1} + b \\psi^{n-1} + a \\varphi^{n-2} + b \\psi^{n-2} = U_{n-1} + U_{n-2}.\\,", "07f03b52648fd3107fc19bc5e6f42241": "\\varphi(L)", "07f046a9608ace07fb8896f3f528bf1f": "A = 1", "07f0f62493c98773cd3ff6b8157130d2": "R_0\\ .", "07f1660e1c7fc4ebff5e9335d99c10c4": " \\Delta \\mathbf{X} = \\left(c\\Delta t, \\Delta \\mathbf{r} \\right) ", "07f17b9356c3ca4fa35ea17059be8480": "[u,\\ u+du]", "07f1b37b7ad932e9db4d027659488398": "State(Do(move(box,table,floor), s)) \\circ on(box,table) = State(s) \\circ on(box,floor)", "07f1e15a0cfa370ca2ead6af6abe5d6c": "T^{-3}", "07f2592540d35ea44029a6d5bb864e64": "\\Big(\\frac{R}{M} \\Big)^{i+j}", "07f25ebff1c2f8f67f8b679f9775404e": "|\\Psi\\rangle = c_1(t)|1\\rangle + c_2(t)|2\\rangle.", "07f2e759fb2f9e8e8ed66d8e2b96645a": "X_i, ...,", "07f366f076527f9e875ff995d8ebb87a": "\\forall j \\notin S", "07f36a2e80867725481391901421d6eb": "SCM \\cdot SPE", "07f3daf594fabc1cbdf516a9badce8be": "B_{3/2}=\\{ x: |x| < 3/2 \\}", "07f3ef3bb038afea20cf429f47257089": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "07f413b41321d20d192b373c086d8404": "\\mathcal{L}(x)=\\mathcal{L}[\\phi(x), \\partial_\\mu \\phi(x),x].\\ ", "07f434fd4dcd230df79051a964a91db5": "V^\\prime=\\mathrm{Hom}(V,\\mathbf{Q}_p(1))", "07f44f22b588b14ab897df90fd17b90a": "\\exists a \\phi(a) \\rightarrow \\exists a\\, (\\phi(a) \\wedge \\forall x\\in a\\,(\\neg \\phi(x)))", "07f47497b56738455bb427f2da7d5919": "L \\otimes_k \\overline{k}", "07f49c2885a727b0c6d8861e9380bd0b": "\\Phi : \\mathbb{R}^d \\to \\mathbb{R}^N", "07f4a089bd6a73ea154c13cfecd1e8a9": "\\boldsymbol{\\varrho\\varsigma\\vartheta\\varphi} \\!", "07f4aa82ec596a2cf91a6e4b28980c34": "\\forall n:\\int_{a_{n}}^{b_{n}}\\tilde{w}_{n,i}(p_{n})\\tilde{w}_{n,j}(p_{n}) \\, dp_n=\\delta_{i,j},\\quad1\\leq i,j\\leq I_n,", "07f4cbfcb4839a6b6a6e6e1206a25fe9": "\\psi_1(x)-\\frac{x^2}{2} = \\int_0^x \\psi(t)\\,dt - \\frac{x^2}{2} ", "07f58f8add1dfd58f33e288fb79e4f4e": "w^2+x^3+y^4=0 ", "07f5c943b03f215e31fec01edbf147bb": "\\operatorname{erf}^{-1}(x)\\approx \\sgn(x) \\sqrt{\\sqrt{\\left(\\frac{2}{\\pi a}+\\frac{\\ln(1-x^2)}{2}\\right)^2 - \\frac{\\ln(1-x^2)}{a}}\n-\\left(\\frac{2}{\\pi a}+\\frac{\\ln(1-x^2)}{2}\\right)}.", "07f607059536ad6823c8fad0e9aa56b2": "\n \\nabla^2\\nabla^2 w = -\\cfrac{q}{D} ~;~~ D := \\cfrac{2h^3E}{3(1-\\nu^2)} = \\cfrac{H^3E}{12(1-\\nu^2)}\n ", "07f60dbb469e9e9e36d1cf704ee38a45": "d \\Xi = \\frac {U + P V} {T^2} d T - \\frac {V}{T} d P + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "07f616656d5b4665686f9e2c866e21e1": "V^H", "07f6459108618319abf2c6cad952185a": " L = 2\\bullet10^{-7} \\bullet \\ln \\left ( {D\\bullet e^{1 \\over 4} \\over D_{BE}} \\right )", "07f6572f593a609ba9797612c41a2be9": "X(0)=X_0", "07f6a6057feecafb2624927486cb7df0": "\\hat{F} = \\frac{\\sum_{k=1}^N (\\hat{x}_{k+1}-\\hat{F} \\hat{x}_k)}{\\sum_{k=1}^N \\hat{x}_k^{2}} ", "07f6cb9c30863d5b991fd674fdeb79e9": "{\\rho g h S}=0.06{(\\rho_s-\\rho)(g)(D)}", "07f6cfe0053dea15639b5fe17e345f85": "H = \\int (F_{12}-F_1F_2)^2 \\, dF_{12} \\!", "07f6d1c4a280a629b4b44203827a6a7f": "D(G) = 1 + \\sum_i \\left({p^{e_i} - 1}\\right) \\ . ", "07f6dbd4ebf0d41a81baca3474fe25d4": "\n X_n \\, \\xrightarrow{\\mathrm{a.s.}} \\, X.\n ", "07f71416daa6dea5b7b41941c136e328": "\\rho \\ge 0, \\; \\; \\rho + p \\ge 0 .", "07f71f05062a336104b178ab45f4a546": "H=L^2(G)", "07f723f71af59303c67d2e7acc3f578d": "a(n)=\\sum_{k=0}^{n-1} 2^k \\left\\langle\\begin{matrix} n \\\\ k \\end{matrix}\\right\\rangle=A_n(2),", "07f72d5ba169d5124719806ab8c9b41a": "\\lambda=w\\cdot\\tilde\\lambda,", "07f7326f90f457b8b54604bd0938feea": "\\{P\\in \\R^2 \\ | \\ d_P(P,M)=r\\}", "07f75d0b6a742741fa0c9cf580a93f27": "\n\\langle\\bar{r}^2(s)\\rangle = \\frac{1}{\\bar{k}_\\alpha(s)}\\langle\\bar{r}^2(s/\\bar{k}_\\alpha(s))\\rangle_\\text{nrml}.\n", "07f768b3892b58a5caf49d00c8c9cb10": "N=KM", "07f777a06f68b273ae74156423bf9bbc": "\\sqrt{\\sigma^2 + \\hbar^2/16\\Omega^2}", "07f7c5c34e21d72d3e56aebabd3f4da9": "P \\mathcal{F} P = \\mathcal{F'} ", "07f7e3e2d1c6309fe22aee868b47cb27": "\\textbf I(\\alpha)=\\int_0^{\\frac{\\pi}{2}}\\frac{\\ln\\,(1+\\cos\\alpha\\,\\cos\\,x)}{\\cos\\,x}\\;\\mathrm{d}x, \\qquad 0 < \\alpha < \\pi.", "07f7fe67f245362d980d9ec2797c42a4": "L=\\sum_{e\\in E}L_e", "07f8019cadb67675f5173fc91f7d12fc": " ax^5+bx^4+cx^3+dx^2+ex+f=0,", "07f82936a58503d65eab46a67961a033": "\\alpha(d)", "07f83e3672e2bd67852962e1cbe8f915": "\\theta=0.25", "07f83f87447d449a3fb8f39d020e4b19": "c_n = n^{1/\\alpha} \\,", "07f84114279c3cc0a8fabaf65c411fa7": "{M} = \\left[\\begin{array}{cc} {Q} & -{A}^{T}\\\\ {A} & 0\\end{array}\\right]\\,", "07f86868058a46b043166e507eb716eb": " \\lVert ", "07f8918ac334c18ca1d640e99ad9995a": " f( r_t | M_t = m^i) = \\frac{1} {\\sqrt{2\\pi\\sigma^2(m^i)}}\\exp\\left[-\\frac{(r_t-\\mu)^2}{2\\sigma^2(m^i)}\\right] .", "07f89236a63c2a0eaabc361ef22b62c7": "\\tfrac{\\lambda(1+\\nu)(1-2\\nu)}{\\nu}", "07f89df78e4991b4fdeb361e375d6da0": " q = \\min\\left(a\\ell,\\frac {k} {\\sigma}\\right)", "07f8ed15938f4f4daa46aeb7d5339d09": "\\sum_{g\\in G}f_g g,", "07f92b1474456ac3e8fcf2f80f036cbe": "a \\in A_{n-1}", "07f9b6947a6c88e5e901e4e01b9a9b1e": "\\textbf{P}_{k\\mid k-1} = \\textbf{F}_{k-1} \\textbf{P}_{k-1\\mid k-1} \\textbf{F}_{k-1}^{\\text{T}} + \\textbf{Q}_{k} ", "07f9dd1c4bf69068cebe5bd48eafdc03": "\\nabla^2\\phi=4\\pi\\rho\\;", "07fa4c077b4e8efadcad6b8c50ad86ad": "\\scriptstyle \\Gamma^k_{ji} \\;=\\; \\Gamma^k_{ij}", "07faa54195f92c9fbe1c8ce51e2fcab0": " G_Y(s) = \\sum_{n=0}^\\infty p_n s^n = \\exp(a_1(s-1)+a_2(s^2-1)) ", "07fac7020527711c1f93664a2a8cc2dd": " H = \\{f {\\in}L_2(X)\\mathrel{\\Bigg|} \\sum_{i=1}^\\infty\\frac{^2}{\\sigma_i} < \\infty\\} ", "07fb17c177b0c8b329ab14c6e50c116c": "d - S \\approx \\Delta z / \\cos \\theta - \\Delta z\\theta", "07fb67bf137540829db30aa6a3afa376": "E_{em}=\\frac{1}{2}\\frac{e^{2}}{a},\\qquad m_{em}=\\frac{2}{3}\\frac{e^{2}}{ac^{2}}", "07fbca51be8b36b7db7ad2782684cb2a": "\\psi'(g * h) = \\psi(g * h) = \\psi(h) * \\psi(g) = \\psi(g) \\mathbin{\\ast'} \\psi(h)=\\psi'(g) \\mathbin{\\ast'} \\psi'(h).", "07fc02f658d3b17c2069e849f641c065": "0 < \\delta \\le 1", "07fc397c1492f0ca4e476ff2c7bea004": "a(bc)=(ab)c", "07fc5c45178323ac61380dbd5da6b62f": "\\operatorname{var}(X) = \\operatorname{E}[(X - \\mu)^2] = \\frac{\\alpha \\beta}{(\\alpha + \\beta)^2(\\alpha + \\beta + 1)}", "07fcab2d87b5aa4534599accc14381d1": "{\\mathcal{A}}_{i_{n}=j}", "07fd33ee378880f8d7fc75b7bea8549a": " d = \\lceil \\ln{1/\\delta} \\rceil", "07fd521442a72909417714ce6598665b": " \\mathbf{r}_i = 1 ", "07fd85bb5e9f013abd836a5c4611800f": "\nk^{2} = \\frac{\\mu}{h^2} - 1\n", "07fd9f296aee66d18c6418ef9889831e": "\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v}) = 0", "07fdc7b9d4934d172afa37d71b01ff03": "\\frac{d}{dt}\\langle \\sigma_z \\rangle = -2g\\left(\\langle a^\\dagger \\sigma \\rangle+\\langle a \\sigma^\\dagger \\rangle\\right) -2\\gamma \\langle \\sigma_z\\rangle-2\\gamma ", "07fe21a915b4b6752931a2a04d55b977": "e^{-i\\int H(t) dt_{op}}\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}", "07fe268cbfb5379831451c4a1454383f": "y' = y + kx", "07fe44b2b7a4bc99918d7ead9b6628d4": "R_n(x)\\ \\stackrel{\\mathrm{def}}{=}\\ T_n\\left(\\frac{x-1}{x+1}\\right)", "07fe896419a35b754e001c99ac31b415": "\\boldsymbol{ \\cdot}\\ ", "07feceb71273d9c3afb7b4411c6a6bcb": "(a + bi,\\ c + di) \\leftrightarrow (a, b, c, d).", "07feef3766c80eca6fa372fdd0d85a74": "{x}_{i}={x}_{k}-(k-i)h", "07ff119b44e0d0b394c9ec0ea60015a5": "\\sin (\\beta) = \\sqrt {1 - Z_3^2}.", "07ff48c90138571dcde03e88b1496a94": "\\omega_1 = 1 \\,", "07ff7187591188d861ab08e40ce7da07": "d\\Omega^2", "07ffd5675f86ea627719a5078abd1233": "F\\triangle G\\in\\mathcal{A}", "07ffe7e828ae69de037252ff612c1296": " h^0(K|_D) - 1 \\le \\frac{1}{2}\\mathrm{deg}_D(K) = \\frac{1}{2}K^2.\\, ", "07ffec902fe52741a043f367f2489075": "(Y_t)_{t\\geq0}", "08001cb417ce6f4521f76272af06aa8a": "1/e.", "080035f725082c1785e2e7fb515ca7c2": "\\sigma = Y \\, \\epsilon \\,", "0800590145a98e0c3db79f9486ba4962": "u^T a u > \\alpha u^T u", "0800cb500a7f35c564a2c2470a235670": "0 \\ (0^\\circ)", "0800f1fd8d8e51a3cfc95338d90f9b9c": "Z = \\left(1 - \\frac {3}{8}n^2\\right)(p + qi)^{2/3}\\qquad\\text{ where }\\; i = \\sqrt{-1}", "0800fc577294c34e0b28ad2839435945": "hash", "08016d6af0dcd8036c15b3241df14c39": "\\lambda f.(p\\ f)\\ (p\\ f) ", "08020db13c98dd0177d79e55fdf35861": "i \\theta = \\ln \\left(ix \\pm \\sqrt{1-x^2}\\right) \\, ", "080221c3cf8912a1f1581d70d3938fea": "\\omega=\\sqrt{|\\det [g_{ij}]|}\\;\\mathrm{d}x^1\\wedge\\cdots\\wedge \\mathrm{d}x^n", "0802233eb3d016cb5bc16c0a2f2e8c83": "y_3 = \\frac{y_2y_1-z_1x_2x_1z_2}{(y_2^2+(z_1x_2)^2)}", "0802c6028987aada44d354c9956377e0": " -13 \\mathbf{e}_1\\wedge \\mathbf{e}_2 -7 \\mathbf{e}_1\\wedge \\mathbf{e}_3 +5 \\mathbf{e}_2\\wedge \\mathbf{e}_3", "0802e3a1e982590022e68ab61f70fe82": "\nS(\\theta) = \n\\begin{bmatrix}\n\\cos \\theta & \\sin \\theta \\\\\n\\sin \\theta & -\\cos \\theta \\\\\n\\end{bmatrix}\n", "0803326ac905a86dc32fd4241ce8ad64": "a + b^2x_{i1} + \\sqrt{c}x_{i2}", "080334dd7dda84677cf51ad5ef4b12b1": "\\frac{1}{j!}\\left(\\begin{matrix}j\\\\ \\alpha\\end{matrix}\\right)=\\frac{1}{\\alpha!}", "08034666a8a0b35592ad928e7a6a6566": "\n[d(\\rho, \\rho+d\\rho)]^2 = \\frac{1}{2}\\mbox{tr}( d \\rho G ),\n", "0803cddddef0f826dc277274439946bf": "M(v)_{,\\,v}>0", "0803d4c8f7ddb5848750e3d993739400": " \nP\\left(C(\\eta)=\\frac{1}{P[\\eta_t(0)\\neq \\eta_t(1)]}\\right)=1.\n ", "0803da122304c1fb30912df9af524179": "x=(x_1,x_2)\\in \\mathbb{R}^2", "0803e4218668589d5c676e448655369f": "\\Delta : \\mathcal C \\to \\mathcal C^{\\mathcal J}", "08042c60a97650d83931631359b0612a": "\\Delta F/2^N", "08046747cf9ae3433d1dc3ad5e362185": "E_k = \\gamma m c^2 - m c^2 \\,", "080496e06f129f12b22f04cc2c63aded": " p = \\frac{m_{A}ng} {A} ", "0804e38d3286e2ba6ec104414c6acf76": "\\lambda=\\operatorname{lcm}(p-1,q-1)", "08051a547149d7059ecdb09c2aced7cb": " 1 \\le \\phi(r) \\le 2, \\left( r > 2 \\right) \\ ", "08051e685083ef235b8272a896fbb30c": "sm = 0", "08052901962833a8403a21b0f8030372": "\\sigma^2=k-\\mu^2\\,", "080582af2aa04b597f3aaa921afb9034": "\\neg\\forall a, b, c: a R b \\wedge b R c \\Rightarrow a R c.", "0805acd495c11ae19f8559768abd03b4": " \\mathbf{F}' = \\mathbf{F} - \\mathbf{F}_\\mathrm{app} ", "0805d97b0541722b463aa4b421226d5c": "\\mathbf{z} = \\mathbf{a} + F(\\mathbf{b}-\\mathbf{c})", "08060285fdc836b29e6ee4d60c078b31": "\n\\lambda_k = \\min \\{ \\max \\{ R_A(x) \\mid x \\in U \\text{ and } x \\neq 0 \\} \\mid \\dim(U)=n-k+1 \\}\n", "0806162ce9bdde35a1a3993fa7952ce1": "|\\Phi^+\\rangle=\\frac{1}{\\sqrt{2}}\\left(|00\\rangle+|11\\rangle\\right)", "080638a7c78d56b009d7b2e6be392450": "f_i \\circ g", "0806690f6db9d9d8969e809422be28ff": "\\mathbf{v} = {\\mathbf{u}\\over\\|\\mathbf{u}\\|},", "0806ef51c9d2a0c9dd910c774ad73949": "\\eta, b > 0\\,\\!", "080716cb5ffc5e8e1b4e6b39a4ac6230": "\\mathcal{Y}=\\mathcal{F}({\\mathbf{x}})", "08071eb8a8034d9ec87257a3c7d59713": "HS_{A^{[d]}}(t)=t^d\\,HS_A(t)\\,.", "080723ca1bf64850a3528333030f5bbd": "(h_1,\\dots ,h_k)\\in Z^k", "0807e932413b4b3ecb21fe4a40041c61": " F( \\mathbf{p}/2-\\mathbf{k}) ", "08081904d0a5eebd7d63b92db97c84a1": "x_{n+1} = x_n Y_{n+1}", "080821376613570566c8bddf3543f70a": "y (\\theta) = (R - r) \\sin \\theta - r \\sin \\left( \\frac{R - r}{r} \\theta \\right),", "08082bd3966f0a2646cd6474adf4051b": "\\scriptstyle |x-a/q|<\\frac{1}{q^c}", "08082f345ef9eb8c4e5bd40840f833ad": "p, q \\in {\\mathcal M}", "0808e87ccbbea546976395761b08b042": "B=\n\\left[\n\\begin{array}{rrrrrrrr}\n-26 & -3 & -6 & 2 & 2 & -1 & 0 & 0 \\\\\n0 & -2 & -4 & 1 & 1 & 0 & 0 & 0 \\\\\n-3 & 1 & 5 & -1 & -1 & 0 & 0 & 0 \\\\\n-3 & 1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{array}\n\\right].\n", "080996adb536d4f975e1051aae28b43d": "s\\cdot s' = g^{xy}\\cdot g^{y(q-1-x)} = g^{xy}\\cdot g^{(-x)y} =1", "0809ce6efaef359f28799282f3ffcac2": "v_{\\rm e} \\,", "080a383e8f46dc5dbd1a70d492f585f5": "uq \\equiv 1 \\pmod{m_p}", "080a5456e612ac18b948e5b40e6fc28d": "\\frac{\\partial g(\\mathbf{u})}{\\partial \\mathbf{u}} \\cdot \\frac{\\partial \\mathbf{u}}{\\partial x}", "080a54faf6477cb1ec8f81bec6eb4a9a": "Z = {X - \\operatorname{E}[X] \\over \\sigma(X)}", "080a8aa32f1c1671f52da71131a83f0c": "y=\\alpha x + \\gamma_1\\hat{y}^2+...+\\gamma_{k-1}\\hat{y}^k+\\epsilon", "080a8de67d7e3706ec46e3c447d5ce95": "\\forall s \\in S, \\; (T_h f)(s) = \\lambda f(s).", "080a8e28e05a9eccf41684405f8bdf4c": " \\Gamma( \\tfrac{1}{4}) = \\sqrt{ 2G \\sqrt{ 2\\pi^3 } } ", "080ab0960d40c23391570bd3570dd4e5": "\\varphi^{-1}(L)", "080ae704a512940d212a42aba919fd35": "W=360/365.24", "080aea70124a782bc5f04ef69cefe7b8": "{_{metric}} \\delta_{ck}^2", "080b4ec4f1a7c297b2d5f971d101496f": "f() \\,", "080b6a2a645c08f1aeb0d32ed2ddb29f": "T_{j,i}^{(t)} := \\operatorname{P}(Z_i=j | X_i=\\mathbf{x}_i ;\\theta^{(t)}) = \\frac{\\tau_j^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_j^{(t)},\\sigma_j^{(t)})}{\\tau_1^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_1^{(t)},\\sigma_1^{(t)}) + \\tau_2^{(t)} \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_2^{(t)},\\sigma_2^{(t)})} ", "080b810bde9e0944409f5fab33466681": "\n \\operatorname{Var} (x) = \\frac{(b-a)^2(3-2\\theta^2)}{36}.\n", "080bc27fc4bde009fd8d81156bbbee28": "v_g = c \\left( n - \\lambda_0 \\frac{dn}{d\\lambda_0} \\right)^{-1}.", "080be99436600b3a521fc139be91959e": "\\begin{align}\n-i \\pi^2 &= \\left( \\int_R + \\int_M + \\int_N + \\int_r \\right) f(z) \\, dz \\\\\n&= \\left( \\int_M + \\int_N \\right) f(z)\\, dz && \\int_R, \\int_r \\text { vanish} \\\\\n&=-\\int_\\infty^0 \\left (\\frac{\\log(-x + i\\epsilon)}{1+(-x + i\\epsilon)^2} \\right )^2\\, dx - \\int_0^\\infty \\left (\\frac{\\log(-x - i\\epsilon)}{1+(-x - i\\epsilon)^2}\\right)^2 \\, dx \\\\\n&= \\int_0^\\infty \\left (\\frac{\\log(-x + i\\epsilon)}{1+(-x + i\\epsilon)^2} \\right )^2 \\, dx - \\int_0^\\infty \\left (\\frac{\\log(-x - i\\epsilon)}{1+(-x - i\\epsilon)^2} \\right )^2 \\, dx \\\\\n&= \\int_0^\\infty \\left (\\frac{\\log(x) + i\\pi}{1+x^2} \\right )^2 \\, dx - \\int_0^\\infty \\left (\\frac{\\log(x) - i\\pi}{1+x^2} \\right )^2 \\, dx && \\epsilon \\to 0 \\\\\n&= \\int_0^\\infty \\frac{(\\log(x) + i\\pi)^2 - (\\log(x) - i\\pi)^2}{(1+x^2)^2} \\, dx \\\\\n&= \\int_0^\\infty \\frac{4 \\pi i \\log(x)}{(1+x^2)^2} \\, dx \\\\\n&= 4 \\pi i \\int_0^\\infty \\frac{\\log(x)}{(1+x^2)^2} \\, dx\n\\end{align}", "080c1910f3ceddb0b77d33d1677d746f": "N = f_\\textrm{1}", "080c35b77a898fe3f4173f82c095f1e2": "\\lambda \\geqslant 0", "080c67fdb340842524d40951a9e00a01": "\t\n\\sum^n_{ j = 1}\t x_{ij} = 1 (i = 1,2,\\dots, n), \n", "080c684aa7d273801245f9403dfd0d83": "g_S(X_p,Y_p) = [SX_p,Y_p].\\,", "080c8c3cdf354e5e33b36c507a909315": " f(z) = z^2 \\ ", "080cab92cfc99a002d6a4b1dfc9ade56": "S=\\{S,I,R\\}^N", "080cb8b467fe966ceb89c5dd5640a0ec": "Tz = a\\,", "080d24284639989668e6784879144746": "\\theta_E \\,\\!", "080d41d579122ebf1e702b2b1f0ee762": "\\operatorname{GL}(\\infty,A)", "080dbeaf489228439b69b0722dbdae6b": "X(t,\\omega)", "080dc914e350ec23c584090040c21dc6": "A \\cap B\\,\\! = A \\smallsetminus (A \\smallsetminus B) = ((A \\cup B) \\smallsetminus (A \\smallsetminus B)) \\smallsetminus (B \\smallsetminus A)", "080e2919f17a0da7e3728a9c57407470": "\\text{ENTR} = - \\sum_{\\ell=\\ell_\\min}^N p(\\ell) \\ln p(\\ell),", "080e32483ae0e0bb79a46ade8d9e67d3": "\n\\frac{d}{dx}\\ln_{k+1}(x)\n=\\frac{d}{dx}\\ln(\\ln_k(x))\n=\\frac1{\\ln_k(x)}\\frac{d}{dx}\\ln_k(x)\n=\\cdots\n=\\frac1{x\\ln(x)\\cdots\\ln_k(x)},\n", "080e36beb9cfa3069a1b88f5413e3b7c": "\nH = \n\\frac{\\left| \\mathbf{p}_{1} \\right|^{2}}{2m_{1}} + \n\\frac{\\left| \\mathbf{p}_{2} \\right|^{2}}{2m_{2}} + \n\\frac{1}{2} a q^{2},\n", "080e6ed779b2550bb44cfac745578f00": "R=K[V].", "080e7e75310e7da29d363a822d78784b": "\\langle\\mathbf{u},\\mathbf{v}\\rangle = \\cos(\\theta)\\ \\|\\mathbf{u}\\|\\ \\|\\mathbf{v}\\|.", "080e9604620a20dbce9c4f12a20b75a1": "^\\circ", "080ee50acda7a3c58ac74c82aa11d878": " d = S_{k} + C_1 \\ S_{k-1} + \\cdots + C_L \\ S_{k-L}.", "080f4cf957395aaccc1ac5e5ba068128": "\\mathcal{M}_{1,1}\\to\\mathcal{M}_{fg}", "080f8172991adfc7f0e33535de92021c": "k(\\mathbf{x}_i,\\mathbf{x}_j)=\\mathbf{x}_i\\cdot\\mathbf{x}_j", "080fd23ae2ac271d16fda37d8d3cbc36": "GS_f", "080fe291f016a44034864c25ac1eae06": "\\|\\hat{r}\\|^2", "080fe98c239e1d74c1e726857c292e4d": "g_{ij} \\in R[x_1, \\ldots, x_n]", "080fef7c5f9dd8e2f29a0d12f8a53fc0": "\\sqrt{2} = 1.414213562\\ldots", "08108b0366c6bdf6c7e25dc050fafbd2": "v_1\\odot v_2\\odot\\cdots\\odot v_r := \\frac{1}{r!}\\sum_{\\sigma\\in\\mathfrak{S}_r} v_{i_{\\sigma 1}}\\otimes v_{i_{\\sigma 2}}\\otimes\\cdots\\otimes v_{i_{\\sigma r}}.", "0810f4ff59d7f20702a1cc960f24a1e6": " \\pi^{ji} = -(-1)^{(\\left|x^{i}\\right|+1)(\\left|x^{j}\\right|+1)} \\pi^{ij} ", "08116389efc47cea36a58a41961bb6a6": " \\frac{\\partial C_1}{\\partial t}=\\frac{\\partial}{\\partial x}[\\frac{C_1 + C_2}{C} D_1 \\frac{\\partial C_1}{\\partial x} -\\frac{C_1}{C}[D_1 \\frac{\\partial C_1}{\\partial x} - D_2\\frac{\\partial C_1}{\\partial x}]]", "081191eaef5bde95a7d1a30488cfa49d": "p, q, r \\in P", "0811bcd8179acefcb8acd67de7b25dbb": "S(w):=(w''/w')' - (w''/w')^2/2 =f", "081242d676ae2969a930140e1e7274a4": "n=14", "08126368219617f6b2a0d3fcaef58c6f": "\\Sigma _{XX} ^{-1} \\Sigma _{XY} b", "0812a08226756cfa6bfb3f7758aaf11e": "\\frac{0.22}{2.4234}=0.0908", "0812c05386e29f4d1393cc971a38ce2e": " 3x^2 + 4x -5 = 0 \\,", "08131b203dd9cd51e81e0d1480d2acd8": "CAS={EAS\\times[1+\\frac{1}{8}(1-\\delta)M^{2}+\\frac{3}{640}(1-10\\delta+9\\delta^{2})M^{4}]}", "081383cb72feaa3f7812dcdb9c2496eb": "p=p_i(T_p) e^{(E_i-E_{Fp})/(k_BT_p)}", "08138b63a12f3000e86fc0cfa0688955": " \\hat{s} = \\hat{k}z + \\hat{l} ", "0813909e5271885bd5aa895185f9fdcf": " G(s)= \\sum_{n=1}^{\\infty} g(n)n^{-s}. ", "08139df44c0a347f29afd2d37cd80953": "-\\frac{Nc}{4}(\\delta_1 + \\delta_2 + \\delta_3)", "0813cfff53030157c8ddc347189139ab": "\\zeta(1/2) \\approx -1.4603545\\!", "081401fe713bbb02014da353e28b08bb": "\\color{BlueViolet}\\text{BlueViolet}", "081442a592f1940a7dc02beb010e0512": "\\mathrm{Sc} = \\frac{\\nu}{D} = \\frac {\\mu} {\\rho D} = \\frac{ \\mbox{viscous diffusion rate} }{ \\mbox{molecular (mass) diffusion rate} }", "081489eb8d9388a69a4749dc37dafc0e": " \n s=(b-a)/3, \\,", "0814a92bf0ef3dd45cc6d933ad7ef89c": "N(M)=\\{m\\ge1|P_m(M)\\ge1\\},", "0814f52f09d242777fb573267881b8c2": "P = K_1 \\rho^{5\\over 3}", "08151ffc359809b80f90697c49d21a63": "\n\\theta_{eff}=\\cos^{-1}(\\mathbf{\\hat{n}} \\cdot \\mathbf{\\hat{v}}),\n", "08152191c9ff6afb0258f0cca95e8bee": "1.8304 0", "082699a68eb4e1ffa3bda5a8cb741212": "L_n^{\\alpha }(z) = \\frac{z^{-\\alpha }e^z}{n!} \\frac{d^n}{d z^n}\\left(z^{n + \\alpha } e^{-z}\\right)=\\frac{\\Gamma (\\alpha + n + 2)/\\Gamma (\\alpha +2)}{\\Gamma (n+1)} \\, _1F_1(-n,\\alpha +1,z),\n", "0826be3909e4965656ced1974ed6ca41": "\\varphi(s(x)\\cdot g) = g.", "0826d5f1bef545293a9cb5ed76cbda38": "-\\frac{ \\text{polylog}^2(2,1-p)}{\\beta^2\\ln^2 p}", "0826dd36fc273a36a97993c3382d7596": "\\mathbf{\\hat{b}_{t:T}}", "08273b5090556d1402f1105ddf5a4078": "B_{max} = \\tfrac{1}{M}\\cdot\\tfrac{1}{2T},", "082743bdd2a37f8be8077c266c1fbaa5": "N_{E}/N_{NE}\\approx H_{E}/H_{NE}.", "082795f5c36ed57a1b4346e3e867a969": "(x_1 x_2 + N y_1 y_2,", "0827d56851adbf127a6df596e9c23635": "X(t) = \\left( \\frac{\\nu}{\\nu+1} \\right)^{\\nu} K ", "08280de5642c348a3099e37d1398975d": "T(v)", "0828167ec4f6eb3a1077cbb76d4664e2": " \\tfrac{365.242\\ 190\\ 402}{366.242\\ 190\\ 402} ", "0828559d0f38e03671647e62aedded42": "\\text{return} \\colon T \\rarr S \\rarr T \\times S = t \\mapsto s \\mapsto (t, s)", "08288b8cef1b440b233e7b2aa8b74202": "\\texttt{fix}_\\alpha", "0828a0d0be30dd9628148fcafc9c67df": " \n\\sum_{n\\le \\lambda} \\left(1-\\frac{n}{\\lambda}\\right)^\\delta\n= \\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \n\\frac{\\Gamma(1+\\delta)\\Gamma(s)}{\\Gamma(1+\\delta+s)} \\zeta(s) \\lambda^s \\, ds\n= \\frac{\\lambda}{1+\\delta} + \\sum_n b_n \\lambda^{-n}.\n", "0828afe50f227250fa69a5f682bc0512": "\n\\int_1^\\infty e^{iax}\\frac{\\ln x}{x} \\, dx = -\\frac{\\pi^2}{24} + \\gamma\\left(\\frac{\\gamma}{2}+\\ln a\\right)+\\frac{\\ln^2 a}{2}-\\frac{\\pi}{2}i(\\gamma+\\ln a) + \\sum_{n\\ge 1}\\frac{(ia)^n}{n!n^2}.\n", "0829042bf44637ca470ca32478ff2b1c": "F(y_1)=F(y_2)=\\cdots=0\\ \\Rightarrow\\ F(y_1+y_2+\\cdots)=0", "082922b6768a542390e435095ceb28ec": " Ly = f", "08292eabfd0980c97be52ef60fc47f6d": "u^T \\nabla f(x)", "0829471378a17c0994cfa3d084c38ad2": "\\pi/2-\\varphi-\\theta_0", "08297f9dcc77d2e8f0a9d461fc8d29a7": "p>0", "082980c9e438c59723e0889fafc1ca87": " \n[0,1]", "0829b30266db4e8501632d3b33671a11": "\\mathrm{SU}(n)", "0829bfcd7e8d1ecee8e9cc2b579d116d": "k(\\mathbf{x}_n,\\mathbf{x}_m^j)", "0829f8fca5e0ab811b2aa5af19d80c80": "u_j^n", "082a31c2eaac4c1aae03bb98e21e5a25": "K(u) = \\frac{15}{16}(1-u^2)^2 \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "082a5766388b2cb393e8e535301c16a7": " V_0 = \\frac{V_{\\max}[S]}{K_m(1 + \\frac{[I]}{K_i}) + [S]}", "082a82b8b71f83992035b4be4c776ac8": " p(x) = {\\alpha \\over \\lambda} \\left[{1+ {x \\over \\lambda}}\\right]^{-(\\alpha+1)}, \\qquad x \\geq 0,\n", "082b0e41ac5d0f86aa6e51587580f3b7": "\\ln(x)", "082b7febbb152527ae1f05d1bbb8c49b": "\\Delta b_T", "082bd2666489a522185b37cc49581cb8": "\\frac{2}{3} \\times 2", "082be05223beba07f8c61abaf1f9f14b": "\\lambda _j", "082bf236294ff058c05fd953990bafb0": "m=2^k", "082c05cd77606b370a83c09a4a24e33e": "n_\\max", "082c80009e98668e0306f23c4ffac32a": "\n\\mathcal{G}(\\tau - \\beta) = \\zeta \\mathcal{G}(\\tau),\n", "082cc266a32b400880939844673e26b8": "e > 7.5 n,\\,", "082ccfc0d25d2ed24504b86f937e4b22": "N(0)=N_0", "082d71cfb09c97b2c7c4dbc53d80bd8a": "\ndV = \\frac{\\left( \\mu - \\lambda \\right) \\left( \\nu - \\lambda \\right) \\left( \\nu - \\mu\\right)}{8\\sqrt{\\left( A - \\lambda \\right) \\left( B - \\lambda \\right) \\left( A - \\mu \\right) \\left( \\mu - B \\right) \\left( \\nu - A \\right) \\left( \\nu - B \\right) }} \\ d\\lambda d\\mu d\\nu\n", "082d758dbed4791e7613866dcd5ec11a": "\\sgn(x) = \\frac{|x|}{x}.", "082d91fadc58f3d6b7aa476f2597c401": "\\mathbb{E}f^2", "082dccaf0370dae658dfa85c173de3c4": "I_C = C\\frac{dV_C}{dt}", "082ddaa8702fdacb652990136fb6bc1a": "\\mathrm{tr}", "082df00ed1cc1efbbe12198fb5cf2f6d": "C_2= \\left[ \\begin{array}{rrr} \n1 & 0 \\\\ \\\\\n0 & 1 \n\\end{array} \\right] - \\frac{1}{2}\\left[ \\begin{array}{rrr} \n1 & 1 \\\\ \\\\\n1 & 1\n\\end{array} \\right] = \\left[ \\begin{array}{rrr} \n\\frac{1}{2} & -\\frac{1}{2} \\\\ \\\\\n-\\frac{1}{2} & \\frac{1}{2} \n\\end{array} \\right]\n", "082e734e8aa89c91fcc1a922d9d5adca": " -[R][R]= -\\begin{bmatrix} 0 & -z & y \\\\ z & 0 & -x \\\\ -y & x & 0 \\end{bmatrix}^2 = \\begin{bmatrix}\n y^2+z^2 & -xy & -xz \\\\ -y x & x^2+z^2 & -yz \\\\ -zx & -zy & x^2+y^2 \\end{bmatrix}.", "082eb188ca14a4fabe391a68adbed0c0": "m/e", "082eec537b35280f43027e668cbbff39": " \\pi_{(t+1)} ", "082f7bbea8ce98e9c5f929f1f8c2fc5f": "Y[x,y]=y-\\frac{y'\\int_a^t \\sqrt { x'^2 + y'^2 }\\, dt}{\\sqrt { x'^2 + y'^2 }}", "082f847a67de8a9e2091c3751e10723c": "<0.58", "082fb23fc490236c1dfcf8dba5364e34": "I_{\\mathcal Q}(+)\\colon Q\\times Q\\to Q", "08301fef54b97a39b5180cf46bdb7ded": "r_1 = (S \\to AA, \\{r_1\\}, \\{r_2\\})", "083028550121a5b354ece52f326b89be": " \\ x_d = (x - x_0)/(x_1 - x_0)", "08305728f3551363b41853a6bc90f96f": "W=W_1W_2", "08308e352cf2d86d3b78ca048a75d173": "0<\\alpha<1", "083090b0d66349f0269b1f1393604346": "\\displaystyle{(H^\\varepsilon)^*=JUH^\\varepsilon U^*J.}", "08310a10e061182e64df26263c08539c": "\\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} \\rho c \\frac{\\partial T} {\\partial t}\\,dV\\,dt = \\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} \\frac{\\partial \\frac{ k \\partial T} {\\partial x}} {\\partial x}\\,dV\\,dt + \\int_t^ {t+\\Delta t} \\!\\!\\!\\int\\limits_{cv} S\\,dV\\,dt", "083151f616e8598aea21074ac234d885": "\\vert\\psi\\rangle=\\vert\\psi_A\\rangle\\otimes\\vert\\psi_B\\rangle", "083174abb898e41376cb3a38fd8b37ce": " z + pl(a - p) + t(2ap - p^2 - 1) - pm ", "0831d1ccb710165c736b75b02f24aa58": "\\varepsilon_i=X_i-\\mu,\\,", "0831f6d6a574c6fe68549253aba4d8e6": " ds^2= \\frac{1}{2\\omega^2} [ -(dt + e^x dz)^2 + dx^2 + dy^2 + \\tfrac{1}{2} e^{2x} dz^2], \\qquad\\qquad -\\infty < t,x,y,z < \\infty,", "0832093a01e09dc2dc09ef5c91b8e566": "\\left\\langle\\sqrt{R},Z_R\\right\\rangle", "083277b08629dc9df83286c5b5b43b18": "\n\\mathbf{N} = \\frac{d\\mathbf{L}}{dt} = \\dot{\\mathbf{r}} \\times \\mu\\dot{\\mathbf{r}} + \\mathbf{r} \\times \\mu\\ddot{\\mathbf{r}} \\ ,\n", "0832cc7679d7455016051857d736b9f9": "K\\subset\\mathbb{P}^3 ", "083319a7938c6e9efe4be8a7e6f2cc5e": "pV^0 = p", "083344515659c94737ef0b7d6a2bf7ff": "(\\hat{c}_P/\\hat{c}_V)", "083355a2c49ec103aa16669eedbb3d32": "\\lambda(s)", "083356749e6c1731ec6a005d841d8f37": "(1,8,1)\\rightarrow(1,3)_0\\oplus(1,2)_{\\frac{1}{2}}\\oplus(1,2)_{-\\frac{1}{2}}\\oplus(1,1)_0", "0833d3565a5f83b2ffa0779c50fc0158": "{\\epsilon} = 1", "083406ed4ca69b03b58821b806ef6a99": "\\displaystyle{Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab).}", "08349d33537fafa342d491d1d322c862": "\n\\ln\\Omega_{E,\\ell} = \n\\ln\\left(\\ell^n n\\pi^{n/2} (2E)^{\\frac{n-1}{2}} \\right) \n- \\ln\\left[ (n/2)! \\right]\n", "0834b3533011a374c3847ad2ab279f68": "\\scriptstyle \\gamma_\\mathrm{sa}", "08354324d53a50f3f2147c9798231c7e": "\n\\sum_{k_1+\\dots +k_y = 0}^x {n\\choose k_1} {n\\choose k_2} {n\\choose k_3} \\cdots {n \\choose x - \\sum_{j = 1}^y k_j } = { \\left( y + 1 \\right) n \\choose x}.\n", "0835593c0e39feb8bbf480625454a569": "\\cos\\theta = \\frac{r^2 + R^2 - s^2}{2rR}.", "083596369fc033fba72e6a4077de3370": "R_{\\mathrm{K-90}} \\,", "0835a821464e35efbf9a2a32606055ec": "M+L \\rightleftharpoons ML:\\log \\beta_{110} =\\log \\left(\\frac{[ML]}{[M][L]} \\right)", "083629f5a933f3d7a97c03d3357eacb2": "\\frac{ d}{ dt}q(t) =~~\\frac{\\partial}{\\partial p}\\mathcal{H}", "08363353e10c2a2c28ee3c648bb8cf95": "d = d_0 + 2d_1 + 2^2d_2 + \\cdots + 2^md_m", "0836499f686226206f227fa15514a074": " D = \\frac{N}{P_d} = \\frac{Np}{\\pi}= \\frac{N}{P_{nd} \\cos\\psi} ", "0836558734369e6ee1e47366bba90ecc": " \\Delta u = \\nabla^2 u = 0\\,", "083666aab531d20c3136aee3399616e8": "\\sinh{(\\chi_{nk}/2)}", "083699762748f512cc92b85e3f9ef4de": "{\\rm Tr} A^{\\dagger}_i A_j \\sim \\delta_{ij} ", "083712f0134574c67ce76b6023adbb11": "l_1 = m_1 = \\frac{1}{\\sqrt{2}}\\sin{y}, \\qquad l_2 = m_2 = \\frac{1}{\\sqrt{2}} ", "08374f72a08d06014d01e9bf8694fba4": "\\mathbf{u}^n", "083755c68cacbe992fce507c1bc2a2ed": "f(x) = x^n", "083788997e7b8122e52fb3f583094fea": "\\phi:M\\to \\prod_{i\\in I}R\\,", "0837a60eda63e2be86c97145714d016b": " \\operatorname{build-param-lists}[q, D, V, T_7] \\and \\operatorname{build-param-lists}[q, D, V, K_7] ", "0838368df43c46fcc7e86b727b6836ef": "-e^2 \\left( \\bar{v}_{k} \\gamma^\\mu v_{k'} \\right) \\frac{1}{(k-k')^2} \\left( \\bar{u}_{p'} \\gamma_\\mu u_p \\right) ", "083882cffb2ef35c88a184362f82f323": "M_k = 0", "0838e5f328b2875d3328a756d49f7cf5": "\\frac{\\mu}{\\mu^3+1} \\to \\frac{\\mu^2}{\\mu^3+1} \\to \\frac{\\mu^3}{\\mu^3+1} \\to \\frac{\\mu}{\\mu^3+1} \\mbox{ appears at } \\mu=\\frac{1+\\sqrt{5}}{2}", "08397d0e51ccf0875df8733befda49ec": " \\Phi ( \\omega) := \\begin{cases}\n\\frac {1}{\\sqrt{2\\pi}} & \\text{if } | \\omega|< 2 \\pi /3, \\\\\n\\frac {1}{\\sqrt{2\\pi}} \\cos\\left(\\frac {\\pi}{2} \\nu \\left(\\frac{3|\\omega|} {2\\pi}-1\\right) \\right) e^{j\\omega/2} & \\text{if } 2\\pi/3<|\\omega|< 4\\pi/3, \\\\\n0 & \\text{otherwise}. \\end{cases}", "0839e30ddc2fefd6f1969bed87662de6": "L_r = \\frac{\\rho}{\\pi}\\cdot \\cos \\theta_i\\cdot L_i", "083a3b265a2be7a470c7a5ed9adbe3ca": "P_\\infin", "083aad334d5cb8d6a6f2849a925a32ca": "\\alpha\\in\\pi_p(M)", "083abb9c2e30db4852bf9496d81dbd29": "\nZ(\\omega)=\\frac{R_{\\text{t}}}{1+R_{\\text{t}}\\,C_{\\text{dl}}\\,\\text{i} \\,\\omega}\n", "083b02e64459f1366c1c661bc62e6bb6": "g_p(X_p,Y_p) = g_p(Y_p,X_p).\\,", "083b179c24b4527a892dc84eea627723": "h_{\\bar{a}}(\\bar{x}) = \\left(\\big( \\sum_{i=0}^{k-1} x_i \\cdot a_i \\big) ~\\bmod ~ 2^{2w} \\right) \\,\\, \\mathrm{div}\\,\\, 2^{2w-M}", "083b51c0c079e0adf3b1cc39aee3f094": " (Bf)(z) = \\int_{R^n} \\exp[-(z \\cdot z - 2 \\sqrt{2} z \\cdot x + x \\cdot x)/2]f(x) \\, dx, ", "083bf6e702235e5aae317792c3039ec5": "\\theta(t_iht_i^{-1})", "083c122be64a93c9170c41190ee88bbb": " T = - \\frac {i_r} {i_t} \\ . ", "083c1e6df0c761e36d8f0b4101147a39": "\\mathbf{U} \\equiv \\{\\mathbf{u}_0,\\mathbf{u}_1\\dots,\\mathbf{u}_{N-1}\\}", "083c297e4f5354265325e6158b31391c": "V(\\rho,\\varphi,z)\n=\\frac{1}{R}\n=\\sum_{n=0}^\\infty \\int_0^\\infty dk\\, A_n(k) J_n(k\\rho)\\cos(n(\\varphi-\\varphi_0))e^{-k|z-z_0|}\n", "083c7e987b886df53d04d5c25bfce4ca": "\\{e^{(2+i)x},e^{(2-i)x}\\}", "083c7ee847c7db9a6eab38b6596f2346": " X = \\mathbb{R}", "083c8120d00c4c74fa7cf4dd43b85702": "(n_x,n_y,n_z)=(3,2,5)", "083cc8cbc7e1b7f714ba0183d816e728": "\\frac{dL}{dt}v+LMv=\\frac{d\\lambda}{dt}v+MLv.", "083d0215515f5e00863f7811058f7fda": "\n\\begin{align}\ng_x (m)&\\overset{\\underset{\\mathrm{def}}{}}{=} m \\cdot x \\mod (2^{32}+15)\\\\\n&=\\textstyle \\sum_{i=1}^k m_i \\cdot x_i \\mod (2^{32}+15)\n\\end{align}\n", "083d108ada4cf78366a41ee889ef5928": "x_1, x_2,\\dots, x_m", "083d1c80c7d3a4a7825bdfe66f59f876": "\n\\begin{matrix}\n\\;\\; x \\;- 10\\\\\n\\quad x^2-2x+1\\overline{) x^3 - 12x^2 + 0x - 42}\\\\\n\\qquad\\qquad \\underline{x^3 - \\;\\;2x^2 + \\;\\;x}\\\\\n\\qquad\\qquad\\qquad\\qquad -10x^2 - \\;x - 42\\\\\n\\qquad\\qquad\\qquad\\;\\;\\; \\underline{-10x^2 + 20x - 10}\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad\\;\\; -21x - 32\n\\end{matrix}\n", "083e5d4ffdeeedeb3641fd1bbf13c265": "8.314\\,472(15)~\\frac{\\mathrm{J}}{\\mathrm{mol~K}}", "083e7ada1c8323f435fbe16ebb718dd6": "\\frac{\\partial a}{\\partial \\mathbf{x}} =", "083ea38cb2452079b0c6997df773c72c": "\\partial W'", "083ef5b128bb1a4a74b04690aa8e1e2b": "F_x>0", "083f2607d761b21e45382928bd38a900": "145 = 12^2 + 1^2 = 8^2 + 9^2", "083fd908b329b68d7baceb6d862c0884": "\\mathbf F=\\langle F,\\le,V\\rangle", "083ffc82346f253eb19efba5d89fa30f": "D:C\\rightarrow J", "08400f197df75d3490dde4e0fa924b53": "(\\cos \\theta +i\\sin \\theta)^k= \\cos k\\theta +i\\sin k\\theta \\quad \\Rightarrow \\text{Li}_n\\left(e^{i\\theta}\\right)=\\sum_{k=1}^{\\infty}\\frac{\\cos k\\theta}{k^n}+ i \\, \\sum_{k=1}^{\\infty}\\frac{\\sin k\\theta}{k^n}", "0840340ef8c42b4ba4f20e835ea4fefd": "\\mathbf{F}_{\\rm R} = - \\lambda \\mathbf{v} \\, ,", "084074297524e154da07aa0f417397c7": "\\zeta(3)= 14 \n\\sum_{k=1}^\\infty \\frac{1}{k^3 \\sinh(\\pi k)}\n-\\frac{11}{2}\n\\sum_{k=1}^\\infty \\frac{1}{k^3 (e^{2\\pi k} -1)}\n-\\frac{7}{2} \n\\sum_{k=1}^\\infty \\frac{1}{k^3 (e^{2\\pi k} +1)}.\n", "08407cb51853afc254d75629bf04ae2d": "Q = 0", "08408f2e955c657229534b324d6daeac": "t = \\frac{1}{i}\\ln(iy + F) + k", "0840a5e804bda6f0bb5fb19bc26da1b7": " {\\Phi} ", "0840daf69b940521edcb979ce016caac": "d_2(f(x),f(y))=d_1(x,y)\\quad\\mbox{for all}\\quad x,y\\in M_1", "0840f67d544559e630d781f6e7a37260": "\\mathbf{h}P_\\pi\n= \n\\begin{bmatrix} h_1 \\; h_2 \\; \\dots \\; h_n \\end{bmatrix}\n\n\\begin{bmatrix}\n\\mathbf{e}_{\\pi(1)} \\\\\n\\mathbf{e}_{\\pi(2)} \\\\\n\\vdots \\\\\n\\mathbf{e}_{\\pi(n)}\n\\end{bmatrix}\n=\n\\begin{bmatrix} h_{\\pi^{-1}(1)} \\; h_{\\pi^{-1}(2)} \\; \\dots \\; h_{\\pi^{-1}(n)} \\end{bmatrix}\n", "084118391910b6cfbb23e955f4b22d3b": "(A,M)", "08416aaefd3999c7364c585a2356b21e": "\\frac{1}{\\sqrt{2\\pi}}", "08416c42cb79122fe1dc357ff747b62e": "H \\cdot t= a \\cdot b \\cdot ( e \\cdot \\sinh E-E)", "0841816b3be3adc912b9cbf086eafdee": "\\Phi^{(k+1)}(\\omega)= \\frac {1} {\\sqrt 2} H\\left( \\frac {\\omega} {2}\\right) \\Phi^{(k)}\\left(\\frac {\\omega} {2}\\right)", "08423b8a7f05f162d5c5c9a18244439a": " \\int_{E}f\\,d\\mu=\\int_{K}f\\,d\\mu,~~~\\int_{E}f_n\\,d\\mu=\\int_{K}f_n\\,d\\mu ~\\forall n\\in \\N. ", "0842704f9234ba2e15fc47efddceecc5": "t'=t-\\tfrac{vx}{V^{2}}", "0842e280cea535a1d3a1cf35ffe3ab33": "A\\circ B = (A\\ominus B)\\oplus B, \\, ", "0842f9b0000dee93b7b1847d9ee4ff10": "S_s", "084318b834ba16c555d4e360aa779fe3": "\\arccsc (-x) = - \\arccsc x \\!", "084343c957422bb56d98768da6c03fa7": "\\tau^{a}{}_{b}\\,", "084344599a70dded3295c6e46638db85": "J_2\\,", "08438591d46590c6aecfd370bec7d16a": " \\Lambda_{p \\times p} = \\text{diag}\\left[\\lambda_1,...,\\lambda_p\\right] = \\text{diag}\\left[\\delta_1^2,...,\\delta_p^2\\right] = \\Delta^2 ", "0843dae813a7fd5e76f86100feda9ad0": "M_{a}", "084423d98402d5fa10725f9077145a37": "y_{21}-y_{22}", "08443545122b333b15f6ec00f846a254": "y^{\\prime}(s) = \\cos \\frac{s}{\\alpha} \\ ; \\ x^{\\prime}(s) = -\\sin \\frac{s}{\\alpha} \\ , ", "0844d68955e074f574e9d409b6d4d824": "G(\\xi ) = \\frac{3}{{\\xi ^2}}(\\sin \\xi - \\xi \\cos \\xi )", "0845029a236f14f39e20c5ea6b6b684c": "\\sum_{k=0}^\\infty \\frac{\\sin[(2k+1)\\theta]}{2k+1}=\\frac{\\pi}{4}, 0<\\theta<\\pi\\,\\!", "084527d395401ea9842baa5edd84917e": " 0 \\le S \\le 1 - \\log_e( 2 ) ", "08453705d77fef0f311613ac0801a483": "\\nabla^2\\mathbf{B}+\\alpha^2\\mathbf{B}= \\mathbf{B}\\times\\nabla\\alpha ", "0845854e993df19bf2fcf8d8bde95597": "SU(2)_L SU(2)_R", "0845a06ae634f99a58eba196e6e625d3": "H_k(X;A)=A^{r_k}", "0845d29982824ca8a53057468e27ed4b": " X \\leq_{HYP} Y", "0845d6c99d3a90e1ec21ad8c268fba78": "H_{ij} = {- 1 \\over {s_{ij}}^{p+2}} \\begin{bmatrix} {(X_j - X_i)(X_j - X_i)} & {(X_j - X_i)(Y_j - Y_i)} & {(X_j - X_i)(Z_j - Z_i)}\\\\{(Y_j - Y_i)(X_j - X_i)} & {(Y_j - Y_i)(Y_j - Y_i)} & {(Y_j - Y_i)(Z_j - Z_i)}\\\\{(Z_j - Z_i)(X_j - X_i)} & {(Z_j - Z_i)(Y_j - Y_i)} & {(Z_j - Z_i)(Z_j - Z_i)} \\end{bmatrix}", "084604aee805ea5d2248e1c7ea23dd00": "{\\mathit{momentum} \\over N+1} = \\mathit{SMA}_\\mathit{today} - \\mathit{SMA}_\\mathit{yesterday}", "08462e114376476a4ef1bd786c212e13": "(\\cos(\\theta/2) - i \\sigma_3 \\sin(\\theta/2)) \\, \\sigma_3 \\, (\\cos(\\theta/2) + i \\sigma_3 \\sin(\\theta/2))\n= (\\cos^2(\\theta/2) + \\sin^2(\\theta/2)) \\, \\sigma_3 = \\sigma_3.", "084653f60e71e4bdbc21bf91255609bc": " \\sigma_h = K_p \\sigma_v + 2c \\sqrt{K_p} \\ ", "0846b0ed8e72421537d7de82ee54c153": " \\frac{\\partial u_i}{\\partial x_i} = 0 ", "08473ca91ebe8d021888034cd81cb7f4": "\\,^{254}_{99}\\mathrm{Es} + \\,^{48}_{20}\\mathrm{Ca} \\to \\,^{302}_{119}\\mathrm{Uue} ^{*} ", "0847587afea00f4f42715cef67daa019": "\\theta = \\theta^\\prime", "0847c8ffad3aecaedb53f0fa2fd535b3": "\\left [\\begin{smallmatrix}2&-1\\\\-5&2\\end{smallmatrix}\\right ]", "0847d8c57819029175a2455933cfc696": "{R_{abc}}^d+{R_{cab}}^d+{R_{bca}}^d = 0.", "0847df7b9c5fb53a214fc80bc5df2df3": " F(k;n,\\tfrac{1}{2}) \\geq \\frac{1}{15} \\exp\\left(- \\frac{16 (\\frac{n}{2} - k)^2}{n}\\right). \\!", "0847f615f040b9e64a573923df4ad112": "x_\\star", "08488f06ff28d094e8c234b086f25853": " U = 1/(1/h_1 + dx_w /k + 1/h_2) ", "0848995767d6cd9b895abe93ebe53dc5": "v_i(0)", "0848b32d4a785ca97d04be1e69de3936": "\\Gamma_5", "0848d20fb47e7b315af38020c6c07856": "\\lambda(y) = X_1^2(y) + \\cdots + X_k^2(y)", "0849035a0f0432ba8fb8aaeb65740b8f": " \\nabla^2 f(x) - mI", "08493db2571077516e5f8ffcbed059f3": "r_1 > 0", "08494b4722a778ebbcc25dbe73aa019d": "\n \\underline{\\underline{\\mathbf{A}}} = \\begin{bmatrix} A_{11} & A_{12} & A_{13} \\\\ A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \\end{bmatrix}~.\n ", "08497086c0bdc9930f7bd56fb588aad2": "S_{mn} = S_{nm}\\,", "0849758f0b601830f420400199146924": "\\Delta p = p_{i,x} - p_{f,x} = p_{i,x} - (-p_{i,x}) = 2 p_{i,x} = 2 m v_x\\,", "0849ede8f3d74e1e41d884bb7b22c900": "I_{L_{Max}}", "084a1377b87cd676874736d2e07744da": "\\frac{E}{m} = K\\left(\\frac{\\sigma}{\\rho}\\right)", "084a36e9759b6be6152a5494fc9f7163": "i_a(t)+i_b(t)+i_c(t)=0", "084a5d14a846449b88da90388d0d1be7": "\\ \\Delta^r(\\alpha_{i,j,k}) = \\alpha_{i+1,j,k} - \\alpha_{i,j,k} ", "084a799f353c0237d5625a23ad626f5d": "[A,B]=0", "084ad8fd849249aa258d43a5aed6914c": " L_k = R(t)e^{\\beta_k} ", "084b34dd6de2eee9941ae886d11b6eef": "(13)\\quad Z^c\\nabla_c B_{ab}=-B^c_{\\;\\;b}B_{ac}+R_{cbad} Z^c Z^d\\;. ", "084bd624309690df3a29b3fcb906d838": "S_k (n, r) = \\mathrm{Hom}_k( A_k (n, r), k)", "084c24cc32297f7667d9742433e36289": "n:=n_0", "084c32ca00e00fb3895b49d744769c4b": "S(t) = \\frac{1}{\\pi}\\arg{\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)}", "084c6df1514b475c70c295022e8919ad": "\\{P_i, y_i\\}_{i=1}^n ", "084cea1b18a3467fe33db6e9df06f713": "E = \\frac{k\\cdot P\\cdot M}{R\\cdot T_A}", "084d3e56c35a1b9fd8fe110f3c87efba": "P=\\frac{RT}{V_m-b}-\\frac{a(T)}{V_m(V_m+b)+b(Vm-b)}", "084d5ffb9a91a4f0f9b641f773f265e5": "\\max_{s\\in X} U(s)", "084d6dc7cd051708220858e4a210b1ff": "\\sigma^2_c = \\frac{f^2 \\mathcal{L}\\left(f\\right)}{f_{osc}^3}", "084d8b189a67229cdd81f3908435f717": "\\scriptstyle \\mathbb{C}^2\\equiv\\mathbb{R}^4", "084db482325b782cb718e9996b9d11cb": " x \\wedge y = - y \\wedge x. ", "084dbad49b48a53102cc8188e8b6ade0": " \\theta_{p,\\omega}^{A} = { \\mu_{p,\\omega}^{A}, \\Gamma_{p,\\omega}^{A} } ", "084dc1e813d6d2d950d528fce1a6f476": " \\mathcal{C}_{XY}: \\mathcal{H} \\mapsto \\mathcal{H} ", "084de0b9fa9f912d5fd22a4fabe760f0": " B_k r_k ", "084de299e2ce516d42cbc70e8cccdfd0": "D(s,\\mathbf{x})", "084e15e0ce97e3947839329e00df6765": "|z_k - z^*| < \\epsilon", "084e182e56a68d767225d1158ccc4b65": "\\! 1-p+pe^{it}", "084e192fa77eb12c5a06d1f38700f56a": "\n|X|E_{k}=\\sum_{i=0}^n q_k\\left(i\\right)D_i. \\qquad (7)\n", "084e2f1d2e7342f2ac07cc95c74eba59": "[\\alpha]_\\lambda^T = \\frac{ \\alpha}{l \\times c}", "084e6ec023a3270d969c54e4a1962174": "U_\\text{Inner}=U_\\text{Outer} \\, ", "084eaec7015459fe305fcfdc78d71eed": "i_{\\ast}: T_p S \\to T_p M.", "084eb83efe38c09c9af3dc570f63eca3": " \\omega \\in L(\\mathcal{G},t)", "084eddb071091218f302917204e898e3": "\\tau_e = 1/\\dot{\\gamma}_e ", "084ee992ed154852424d0e0d7029d060": "v_{xo}", "084f5c4e013ea2ab9d5e8d1933dcd5ff": "\\mu=\\pi\\left(\\sqrt{m}\\right)-n", "084f86393f8363b1a3fd75a5f11dec3b": "X \\setminus V", "08504aadd8d0c81440a057ac70fd754d": "\\Delta m^2_\\text{atm}\\simeq2.5\\times10^{-3}\\,\\mbox{eV}^2", "08507709a2cd321cc65a78f8ba1ec1ba": "E[\\xi]=\\int_0^{+\\infty}(1-\\Phi(x))dx-\\int_{-\\infty}^0\\Phi(x)dx", "0850f79692ab55eac984ab3f24553961": "\\begin{align}& j = \\ell +s \\\\\n& j \\in \\{|\\ell-s|,|\\ell-s|+1 \\cdots |\\ell+s|-1,|\\ell+s| \\} \\\\\n\\end{align}\\,\\!", "08516cd80144421850e3d9e4a1b94afe": " \\nabla F = \\mu_0 c J ", "085199cb15978a3d6a78c9457b9d493c": "\\frac{Av}{\\|Av\\|}, \\frac{A^2v}{\\|A^2v\\|}, \\frac{A^3v}{\\|A^3v\\|}, \\dots", "0852021ce522cd53d93bd5b14df57407": "\\hat{a_1}=0.0135", "0852986d0ab05f20afeb2fa9d9aefe46": "f_i(r_1,\\dots,r_{i-1},0)", "0852b97b4adc36613e04de12789470ae": "(f*\\Delta)(x) = \\sum_{n=-\\infty}^\\infty f(x-n).", "0852f6aa94a4da23a34ea914a9cf154e": "L\\psi_n(x)=\\omega_n \\psi_n(x)", "08530922c2c40dd06bb8e686fa483d8a": " \\mathrm{Eq}(f,g) := \\{x \\in X \\mid f(x) = g(x)\\}\\mbox{.}\\! ", "08533b111de373fc0db9daf66054c188": "z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 + u_1 y_5 - u_2 y_6 - u_3 y_7 - u_4 y_8", "08535a977c369045347032867917dc94": "P^{-1}\\mathcal{F}P", "08542bb73cc183c2d33c10db7cd7cabf": "\nP = \n\\begin{bmatrix}\n\\frac{2}{right-left} & 0 & 0 & -\\frac{right+left}{right-left} \\\\\n0 & \\frac{2}{top-bottom} & 0 & -\\frac{top+bottom}{top-bottom} \\\\\n0 & 0 & \\frac{-2}{far-near} & -\\frac{far+near}{far-near} \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "0854472019d887246601ca81de5b0db2": " M(bx_1, \\ldots, bx_n) = b M(x_1, \\ldots, x_n) ", "085457224bf55d74398d25780ba8fd76": "\\int_a^b \\sqrt{1+\\bigg(\\frac{dy}{dx}\\bigg)^2}\\,dx,", "08547d225b1df1e24d9be337f9e8af3e": "xy^{-1}", "0854dfc743e3e4a31e7518f7d497e948": "i = 2i + 2\\left(-\\frac{1}{2}i\\right) = 10.2_{2i}", "0855377f16232c112cc7334456e21ad1": "x_i = r_i^{ - \\beta }", "08555285a6e8334c4942b72b27d88873": "\\ f*g ", "085555ba4b282a05cc52051f8edcd94e": "G/\\tau = K_I", "085611d3a8521b914adcd17e88f6a519": "E W = 0", "0856444a7eabf3601113168835553a3e": "d\\Omega_{k^\\prime}", "08570ba4434de2bfe6476ed45505337d": "\\mathcal{H}\\subset \\mathcal{K}", "08571639e1475d1bb81ae83fd725b12b": "N\\in\\mathcal F", "08577d8512a05cc521fbf111658ff6c0": "\\! g_i", "0857bf5d4e53bc8479a34538b4bdc91e": "\\! c", "0857c3604f27a1e80c6f56addbaa2d84": " UltOsc = 100 \\times {4 \\times avg_7 + 2 \\times avg_{14} + avg_{28} \\over 4 + 2 + 1 } ", "0857c70c656c7aaf930e3277a1139c94": "\\scriptstyle dp_B(R,t)", "0857e7053d4575070702ecd46cf668f7": "\\lVert x - y \\rVert", "085827177b928f9c0eff1964846cb77b": "0 < \\alpha\\ < 2^{160}\\,", "0858850a595e42d35f5eeb6b1c650e70": "\\operatorname{Hdg}^*(X) = \\sum_k \\operatorname{Hdg}^k(X)\\,", "0858a3974acfeb642f2364ee49fb05c2": " \\Pr[c \\in B(y, pn)] = \\Pr[y \\in B(c, pn)] = \\mathrm{Vol}(y, pn) / q^n \\ge q^{-n(1-H_q(p)) - o(n)} \\, ", "0858c12dc07329bf7e1c4bad5c47cd67": "{s}", "0858c8578fe0a24f05bb5502453ed3c0": "A_{c,b}", "0858eea75fa6bc1ea46d85bbe8f3c291": "-\\omega_{n}^{2} f(t) q", "085967d7b3ee450c85364a7461e22302": "u_2(z)", "085a08ff09618180f9c43a171281093c": " - \\tilde J = M \\nabla (\\mu_a - \\mu_b) ", "085a49f2fa9d72fbe2d5d4073bfa1677": "\\tilde{S}_{2} =M_{2}-m_{2}+\\frac{i}{2}G\\gamma _{1}\\cdot {\\partial }\\mathcal{\nL}{.} \n", "085a66174db412c03f8c7e11e844c21e": "\\lim_{t \\to \\infty} \\frac{1}{t}g(t) = \\frac{\\mathbb{E}[W_1]}{\\mathbb{E}[S_1]}.", "085abc1a4ea4269aa59a73c8ecb59330": "B^\\phi_{MX}=\\beta^{(0)}_{MX} +\\beta^{(1)}_{MX} e^{-\\alpha \\sqrt I}.", "085b40c8e85992b9b35eb73c29a5798e": "\\oint_K \\kappa\\,ds > 4\\pi.\\,", "085b91511496a5a05a5ac52d40c9489e": " DL_j ", "085bee4f4ccd429ad60e9e96744839c3": "|\\Psi_{m}^{p}\\rangle = \\mathcal{A}(\\phi_{1}(\\mathbf{r}_{1}\\sigma_{1})\\phi_{2}(\\mathbf{r}_{2}\\sigma_{2})\\cdots\\phi_{p}(\\mathbf{r}_{m}\\sigma_{m})\\phi_{n}(\\mathbf{r}_{n}\\sigma_{n})\\cdots\\phi_{N}(\\mathbf{r}_{N}\\sigma_{N})),", "085c19cf432cbcb959b468a6924bcb61": "L(p;q_1,\\ldots q_n)", "085c57e6606d29c7e177ef0385d05a40": "\\Psi_A(x)=C_A \\Psi_{0}(x-x_A)", "085c9b94c5877f6967fe47e0253394c7": " \\{e_3\\equiv z_{yy}+\\frac{1}{y^2}(xy^3-x^2-y)z_y-\\frac{1}{y}(x^3-x+y)z=0, e_2=z_x+\\frac{1}{y}z_y+xz=0\\}.", "085c9f0df11642cf704f40ffdf753055": "s'", "085ca404311ed2fa47059761173a876d": "x,y\\in \\Sigma^*", "085cfa49aa86db198f20d193d5080c92": "\\phi(a\\mathbf{x}+b\\mathbf{y},c\\mathbf{x}+d\\mathbf{y}) = \\frac{1}{|ad-bc|}\\phi(\\mathbf{x},\\mathbf{y}),", "085d1f59b60f6655415fb3f3b13a4d44": "\\gamma \\in \\{ 2^{-15},2^{-13}, \\dots, 2^{1},2^{3} \\}", "085d63c9f7b2e59353b8432c9b054d04": "P \\to (f(U) = f(V))\\,", "085d6e33ee4c55f4e65be030cd507c17": "o(w)", "085da8820e7465711a2a2a5d30250753": "\\Delta(\\tilde{w}, w') \\leq t ", "085dafbb0398cf2b6ab06d53c6987444": "e(n)=d(n)-\\hat{d}(n)", "085de9be16c6e59a32235966c7b6cbe3": "\n F_L = \\textstyle{\\frac{1}{5}} k^4 \\left( 5 L_A \\right) + \\textstyle{\\frac{1}{10}} {(1 - k^4)}^2 {\\left( 5 L_A \\right)}^{1/3}\n", "085df8dcd78197333723d2f5b9024b43": "x = R\\lambda/\\sqrt 2", "085e2dcea21c77cf0eef6e999639ee33": "q<1", "085e6f6c82be4cd865387cd12742ea9b": "\\alpha _i = 2\\cdot \\pi \\frac{iK}{N}", "085f012b4860c10d49cea9e685eda831": " BD", "085f471060f928587ef77d613de7a6ab": "\nE_{CFG} = \\{\\langle G \\rangle \\mid G \\text{ is a CFG and } L \\left( G \\right)= \\empty \\}\n", "086052c76ef59b2fb7311dcf92821286": "h_{crit} = \\left(\\frac{9B^2}{4}\\,\\frac{EI }{\\rho g\\pi r^2}\\right)^{1/3}", "0860a2c408a30ca4bbdd158c6798a132": "A = \\frac {W}{(L)(U_w)}", "0860b4bbf2e14b710c80e7a4e35aada9": "\\dot{Q}^\\mathrm{T} Q + Q^\\mathrm{T} \\dot{Q} = 0", "08610ec20d070997513de6c2ac0f5571": "y = b", "08618cc058c79edd11827393563f4a4f": "P_i'", "086192d60aa50b35adfeb8d6794eb4b5": "\nR(\\vec x) \\approx \\tanh p \\,\n", "0861a74c12b96b48f283c77d29514cda": "W=A^{-1}", "0861adc3d3e17d9c558f0b8939514aee": " \\bar{n}_i = \\frac\n\n{\\displaystyle \\sum_{n_i=0} ^1 n_i \\ e^{-\\beta (n_i\\epsilon_i)} \\quad \\sideset{ }{^{(i)}}\\sum_{n_1,n_2,\\dots} e^{-\\beta (n_1\\epsilon_1+n_2\\epsilon_2+\\cdots)} }\n\n{\\displaystyle \\sum_{n_i=0} ^1 e^{-\\beta (n_i\\epsilon_i)} \\qquad \\sideset{ }{^{(i)}}\\sum_{n_1,n_2,\\dots} e^{-\\beta (n_1\\epsilon_1+n_2\\epsilon_2+\\cdots)} } ", "0861e8ef3451e1ec1608f8d18768dc74": " p \\to q ", "08622ab1ba41cb55303f381dc2f38327": "y(x^2+y^2)=b(x^2-y^2)+2cxy", "08625098ce1218d596df0cb7a2ff1f49": "\nS_0(p) = \n\\begin{bmatrix}\n(I_x(p))^2 & I_x(p)I_y(p) & I_x(p)I_z(p) \\\\[10pt]\nI_x(p)I_y(p) & (I_y(p))^2 & I_y(p)I_z(p) \\\\[10pt]\nI_x(p)I_z(p) & I_y(p)I_z(p) & (I_z(p))^2\n\\end{bmatrix}\n", "0862609ee90f693442fc43e0c9f758bb": "Z=V", "0862629191318ef2d28418cacf18bcd6": "5F_6^2=320\\equiv -5 \\pmod {13} \\;\\;\\text{ and }\\;\\;5F_7^2=845\\equiv 0 \\pmod {13}", "08627f02774fb21464ae91032340eba4": " (\\nabla_XZ + (I - \\Delta S)X) + (h(X,Z) + d_X\\Delta)\\bold{A} = 0 , ", "0862c73df2fef82eab278377b6859017": "\\eta_B", "08630453b7b0223407e9bd890d7ae35a": "H_n=\\sum\\limits_{l=1}^{N}K^{[l]}_1 + \\sum\\limits_{l=1}^{N}K^{[l,l+1]}_2.", "0863360a323a329bc6470dbba564fa56": "K\\;", "086337e68e2d113e81e0e9db7655844e": "\\frac{\\pi}{\\sqrt{12}} \\approx 0.9069.", "08634f6ade2e4ad977a9dd7fe3172646": "\n\\Delta S = \\alpha k_{B} \\ln N \\,\n", "08639b03744654d4b36f1f47b913755a": "\\bold r", "08639c77b9ec5ddb0c6e94cb46ee0eec": "\\scriptstyle A_\\parallel", "0863e4a2343c010d5ccfdde4d7a8c0f1": "\\rho_A=\\operatorname{tr}_B\\rho_{AB}", "0863f741d7093a2bec6ece488bdae3b0": "y \\cup \\{y\\}", "086403e136c6d79c35384e36319d7181": "c^2 = ac\\cos\\beta + bc\\cos\\alpha.\\,", "08641454a6623d6aaaf7653303d81903": " \\operatorname{Re}(\\epsilon (\\mathbf{r}, \\omega)) ", "08643b522426f00d876b6175c5125f5e": "S \\in \\mathcal{A} (G)", "08644c3901ca0610e482e615b7c82f5e": "\\sum_{a_i \\in A} \\phi(a_i)=0", "08647d7f90875053446ea8763b27d956": "{\\boldsymbol{\\beta}}", "08648e92de68de9b39c67d08bb3a28a3": "*_N", "0864a3f53947fcdb99026514b527fb90": "H \\left( f \\right) * \\frac{1}{T_s}\\sum_{k = -\\infty}^{+\\infty} \\delta \\left( f - \\frac{k}{T_s} \\right) = 1", "0864b70f0056f87446ef403e849a2014": "T_0,", "0864eb673fda276a4f3a8813aa282976": "\\pi_{\\omega}(x) \\xi_{\\omega} \\mapsto \\pi_{\\phi}(x) \\xi_{\\phi} \\oplus \\pi_{\\psi}(x) \\xi_{\\psi}.", "08650fba58a14f3bc4c15c1bff29986f": " \\int \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) \\, d^n\\mathbf{x} =1", "0865a01dd577912c2e853f4fa232e797": "\\mathcal{M}_1", "0865a90eb4b59b80ad40594550d43e52": "d \\approx 3.57\\sqrt{h} \\,,", "0865ac8f14c29e628449d4808fc6a2e4": " \\begin{matrix}\\frac12\\end{matrix} mv^2=gmr.", "0865b7b5d73f8d2d267b2b4d48b4d47b": "X = -\\left\\langle\\frac{dE_{r}}{dx}\\right\\rangle\\,", "0865d6c4efb551567c3c2cab22e94bd7": "\\langle introd(j, b, x), s\\rangle", "0865ee2ee41e748e0ab653b785ea92c3": "\\mathrm{high} = R_1 C \\cdot \\ln\\left(\\frac{2V_{\\textrm{cc}}-3V_{\\textrm{diode}}}{V_{\\textrm{cc}}-3V_{\\textrm{diode}}}\\right) ", "0865fa8ab44cc8cbe3f68273fd2c03c4": "|G_a(z)| < (M+1)\\epsilon \\!", "086675e64252d1197dee3eebd34dec1e": "\\hat{H} ", "0866827b2b471138b8f46081602e57ab": "\nE_1 = q_1^2 - p_1 r_1 = 0.00000 \\,\n", "0866b23e49b4cf45803e32e679ebed22": "l=\\frac{\\hbar}{mc}", "0866b33bb22a761df4df09be6ddacbae": " { 4 \\pi \\over c }j^{\\beta} = \\partial_{\\alpha} F^{\\alpha\\beta} + {\\Gamma^{\\alpha}}_{\\mu\\alpha} F^{\\mu\\beta} + {\\Gamma^{\\beta}}_{\\mu\\alpha} F^{\\alpha \\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ \\nabla_{\\alpha} F^{\\alpha\\beta} \\ \\stackrel{\\mathrm{def}}{=}\\ {F^{\\alpha\\beta}}_{;\\alpha} \\, \\!", "0866ee36a47474c5ca2fb1811dffef82": "GF(p^t)", "08670fd819285e6e4639e2541972b049": " \\quad \\beta_n \\sim f(\\beta_n | \\theta) ", "086730665882e9b8832ad63899709bf7": " D(\\xi_a)D(\\xi_b) = D(\\xi_a \\xi_b). ", "086747ee9945206e6ea6d9ab022b125f": " F = W_1 \\cdot W_3 = \\sum_{j=1,3} W_i", "08681c27ad524a8e515a19b4d3de2807": "\\int \\frac{x+A}{\\sqrt{x^4+ax^3+bx^2+cx+d}}\\, dx", "086834a5ae253549ee9026fec8d279eb": "x^{\\lambda} = e/x \\qquad x^{\\lambda}x = e", "0868420816995522683147f384d56ac3": "\\begin{align}\n & \\hat\\beta_1 = \\frac{s_{yy}-\\delta s_{xx} + \\sqrt{(s_{yy}-\\delta s_{xx})^2 + 4\\delta s_{xy}^2}}{2s_{xy}} \\\\\n & \\hat\\nu_1=\\frac{-1}{\\hat\\beta_1} = \\frac {-2 \\delta s_{xy}}{s_{yy}-\\delta s_{xx} - \\sqrt{(s_{yy}-\\delta s_{xx})^2 + 4\\delta s_{xy}^2}}, \\\\\n & \\hat\\beta_0 = \\overline{y} - \\hat\\beta_1\\overline{x}, \\\\\n & \\hat{x}_i^* = x_i + \\frac{\\hat\\beta_1}{\\hat\\beta_1^2+\\delta}(y_i-\\hat\\beta_0-\\hat\\beta_1x_i).\n \\end{align}", "08684be4f5f5c73459c888f3b47527ac": "p_{11}/(p_{11}+p_{10})", "086886c7642c35706d22fd59d30d2689": " : \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\,\\hat{b}_3 : \\,= \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\,\\hat{b}_3", "0868a10a419e0f3f0833aa59fe8330a5": " \\mu = G(m_1 + m_2)", "0868e6113e0afe135438eef879d719e6": "\\mathit{l_{j}}\\,", "0868fa8e93f62f819441deb81c05a85f": "x'_i \\rightarrow x_i", "0869167f15b8c06f68939754920c6817": "p-p_0=c_0^2(\\rho-\\rho_0)", "0869348ba87ca5f6cefc2e30ddb36314": "F(z, m) = \\sum_{k=0}^{\\infty} f(k T + m)z^{-k}", "086992e911ad666940c60fac809643a8": " V_gf\\in L^{p,q}_m(\\mathbb{R}^{2d}) ", "0869965a2e1546a7606aff31a1f767f4": "y = \\frac{1}{1+e^{-x}}", "086a625a452572a6878a544d30624348": " P(x)=\\sum_{j=0}^{n-1} u_j x^j ", "086ad3563a5e862713aaf1cd8fd6f466": "(M \\ / \\ s) \\ / \\ t = M \\ / \\ (ts)", "086adc074fc09817139a0e2cd17879c9": "p(\\vec\\theta) \\propto \\sqrt{\\det I(\\vec\\theta)}\\,", "086b8289e29be5115bbfd329f2433486": "\\text{precision}=\\frac{\\text{number of true positives}}{\\text{number of true positives}+\\text{false positives}}", "086beb6a6c8a029942238364e5a8beab": "(X, d)", "086c08f4887854fbb8ac4b1dd3d58ee1": "x=R\\cos(\\theta)", "086c1c983eef5f4ec1e9183b80dea13e": "\\tilde{x}", "086c62f0fe0c5804428eb0133152fd95": "\n\\frac{\\theta }{(I)_{J}} {J \\choose n}\n\\int_{0}^{1}(Ix)_{n}(I(1-x))_{J-n}\\frac{(1-x)^{\\theta -1}}{x}\\,dx \n", "086cebacd2b29692f317fd1d9adc9b91": "\\alpha^\\vee= {2\\over (\\alpha,\\alpha)}\\, \\alpha.", "086d4c3a2db4cd707b5e03137be03b8e": " y^G = f(k^G) ", "086e143ecfedac00e31a515d138d3a3c": " \\sum\\limits_{i=1}^\\infty \\mathrm{P}(X\\geq i) = \\sum\\limits_{i=1}^\\infty \\sum\\limits_{j=i}^\\infty P(X = j). ", "086e1709ae075fde91365e73006c78d5": "\\left(\\frac{F/\\mathbf{Q}}{(n)}\\right)=\\left(\\frac{L/\\mathbf{Q}}{(n)}\\right)\\text{ (mod }H).", "086e1e913727da2fa28e31c854b859f4": "U(x,\\omega)= P(x,\\omega)A(x,\\omega) ", "086e77c05b32e57ee30727de66af5caf": "\n\\mathrm{ERB}(f) = 6.23 \\cdot f^2 + 93.39 \\cdot f + 28.52\n", "086ee2d9b611075135d32fe2a9b5bf61": "a + b'x_{i1} + c'x_{i2},", "086efc867d6a52e34eb88795dbb61efc": "\\gamma_\\mathrm{SG}", "086f2a64fea3650776476f320fc982ec": "T(n) = 2^nT\\left (\\frac{n}{2}\\right )+n^n", "086f5e4bee2482de92c8baefce510f22": "\\mathcal{F}(t_m) ", "086f5ebd6937c1a196ff9cde7fbace81": "E > mgl", "086f75fe0a018304ff2817a784eba8ed": "\\log(\\gamma)", "086fce0618cbc52b2baecb5059e0db1f": "\\Diamond \\varphi", "086fe0b2a0abd7dd3c433deb189569f9": "\\begin{align}\n \\text{N}_\\text{s} \\omega_\\text{s} + \\text{N}_\\text{p} \\omega_\\text{p} - (\\text{N}_\\text{s} + \\text{N}_\\text{p})\\omega_\\text{c} &= 0 \\\\\n \\text{N}_\\text{a} \\omega_\\text{a} - \\text{N}_\\text{p} \\omega_\\text{p} - (\\text{N}_\\text{a} - \\text{N}_\\text{p})\\omega_\\text{c} &= 0\n\\end{align}", "086fe4af32eaa0d8e320267ad0ffefa7": "T:\\mathcal{F}\\to\\mathcal{F}", "087017cefe10d64b6f95a15506032319": "m_1 = [12.3, 7.6] + [1.697, 0] = [13.997, 7.6]", "08703596bd26613a0f2d79d0a2c03428": "f(z)=\\frac{e^{t z}-Q_t(z)}{P(z)}", "0870b9f545ddc94563bcca5270b07e1e": "a_5= \\lfloor 6^\\frac{1}{2} \\rfloor = \\lfloor 2.449\\dots \\rfloor = 2, ", "0870f159f171f243deba0d2ee71655e7": " \\tan \\frac{\\theta}{2}", "08718e42cb5b04f1489f7b5d86544ed3": "x/\\epsilon c", "087195bbcff743d55cb9fc7d061ef781": "\\widehat{\\varepsilon}_i=X_i-\\overline{X}.", "0871c793427749c626b5771230f271a8": "\\left\\{\\begin{array}{ll}1 & n = 1\\\\ 2 & \\text{otherwise}\\end{array}\\right.", "0871f51ba0068ce4fdbe0e593444563c": "X_n\\ \\xrightarrow{\\mathcal D}\\ X,", "087249e982bffce2426824d3de6de8e5": "\\mathrm{conv}", "08727052444f8cdf3e308d5863a722a9": "I_{m,n} = \\int \\sin^m{ax}\\cos^n{ax}dx\\,\\!", "08728ea65350ce111f5ae89cda8832b8": "f\\left(x_0\\right) = p x_0 + b", "0872b027f615858ceefa69208cceab47": "N_2 + 8H^+ + 8e^- \\to 2NH_3 + H_2", "0872bf33180c1f89aa2fb8a0bfb5afc9": "{EF = \\frac{Q_e}{Q_e + Q_h} = \\frac{1}{1+B}}", "0872cfce743c4aa442423f20f85c2437": " \\langle P \\rangle \\,\\!", "0872d8a6ea05decf0ebfefd6837d7da0": "x^2 + bx + c \\,=\\, \\left(x + \\tfrac{1}{2}b\\right)^2 + k,", "0872fbbc55e4fdf5b6840c958be53aa5": "\nF_{2}(r) = F_{1}(r) + \\frac{L_{1}^{2}}{mr^{3}} \\left( 1 - k^{2} \\right)\n", "087319626a720e82ab10f4322bb92895": "c_v = \\frac{R}{{\\gamma - 1}}", "0873225a6aae94dfc0be630f38c83ec3": "y^5+y^4-24y^3-17y^2+41y-13", "087322ef246341ad8463c6eb80264c15": "x_{crit}", "08732b8ff0ad81ffc9a3105e18d3e5d3": " \\sigma^\\dagger_{A,B}=\\sigma^{-1}_{A,B}:B \\otimes A \\rightarrow A \\otimes B", "087368f6ea9528420df49d2316f9b888": "\\scriptstyle \\land", "08736f4045e96abf886c45562b8e98a1": "\\scriptstyle{\\epsilon = \\sqrt{1 - v^2/c^2}}", "087396a1975686bdeec8258553c79390": "x = a_0 + \\frac{1 \\mid}{\\mid a_1} + \\frac{1 \\mid}{\\mid a_2} + \\frac{1 \\mid}{\\mid a_3},", "0873dec567d24ed7d6c14976d4a0adb4": "f: V \\to V", "0874064e73f2352985b5c8fc7331051d": "m=0, 1, \\dots,", "0874289dabdd17e6c3fdce45ccbd973f": "\\Lambda^n A: \\Lambda^n V \\rightarrow \\Lambda^n V", "0875135b6dfecc84cfb62efcdd77c8fc": "\\scriptstyle a=\\partial u/\\partial x=\\partial v/\\partial y", "0875ba40ea78e27c50780ff19d1ceb10": "z=(z_1,z_2,\\ldots, z_n)", "08760f01ea0d9b37e4e4e6f8686ebe01": "\\scriptstyle\n\\frac{\\sqrt{3}}{{9}}\\, \\sum \\limits_{n=0}^\\infty \n\\frac{(-1)^n}{27^n}\\,\\left\\{\\!\n\\frac{{18}}{(6n+1)^2} - \\frac{{18}}{(6n+2)^2} -\n\\frac{{24}}{(6n+3)^2} -\n\\frac{{6}}{(6n+4)^2} +\n\\frac{{2}}{(6n+5)^2}\\!\\right\\}\n", "0877834805793b7a7f0c4bfb9cb88b1f": " \\frac{1}{iz} \\,dz = dt.", "0877e3a4e881d97aeb3bba90f55bae18": "\\langle U(x+y)-(Ux+Uy), U(x+y)-(Ux+Uy) \\rangle = 0 ", "0877ed54238aefb676d353b0f3eb6615": "x_i \\,", "087847d30cf204fc8cdd76973b6dcc15": "k_{xo}=k_{o}\\sqrt{1-(\\frac{m\\pi }{k_{o}a})^{2}-(\\frac{k_{z}}{k_{o}})^{2}} \\ \\ \\ \\ \\ \\ \\ (26) ", "08788987bf75976c364309a002529686": "d\\tau = \\sqrt{\\left ( 1 - 1.3908 \\times 10^{-9} \\right ) dt^2 - 2.4069 \\times 10^{-12}\\, dt^2} = \\left( 1 - 6.9660 \\times 10^{-10}\\right ) \\, dt.", "08789c1c789d3a0d70d6883cb62aa942": "\\pi _{jt}", "0878be8a1c8c0a60e498a63a19d7aed5": "\\displaystyle 5^2 + 12^2 = 13^2 \\,.", "087932e8c5faf7ec468c1c2f3a058aaf": " {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {8\\over5} \\cdot {1\\over2} \\cdot {1\\over2} \\cdot {1\\over2} = {81\\over80}", "0879894f2225d8ae7eb9b1b1a3d76aaf": "d_1>d_2", "087a00dfeebaed1d79009912a028e5bc": "= \\frac{1}{331}", "087a6d79325f7cc88bdaa3ca5e84acc2": "\\ell^\\infty(\\mathbb{R})", "087ae63322dcc247674a04de75f515a8": "\\iota^* : H^* (E) \\longrightarrow H^* (F)", "087b4c104aff1c97fb230dc27131f9f8": "a_0=\\frac{\\hbar^2}{m_e e^2}", "087b7ad7a6693503cc710a0bbd54e040": "\\epsilon_H ", "087b8782dd88a72e599d2867ec8e4ecc": "\\mu_{\\max} \\leq 1 - \\varepsilon", "087b9c21b7033fa2e1a50634d25f400a": "N\\in\\mathbb{Z}^+", "087bc030c1ad70f4a5b42c443dd2a3fe": " S_0 = \\left|\\mathcal{F}\\left[\\frac{dW(t)}{dt}\\right](\\omega)\\right|^2 = \\text{const} ", "087bd5f8d197b0f1b129664107c086c2": " g_\\mathrm{e} =2.0023", "087bd6c601163c61803bff1872872a6b": " \\ c_1 = c_{01}(1 - y_d) + c_{11}y_d", "087c213852df24f142c921a142fa2f59": "M, N", "087c3e00f68962d086837cec05e6b1fb": "H^2_0(\\Omega)", "087c5d667ebfa8eb0281e46f055f2764": "\\partial_{\\mu}A^\\mu \\equiv A^\\mu{}_{,\\mu} = 0 \\!", "087cea17dd3a808fc89b1a1de530a1f4": " 2\\pi \\left (1 + \\cos {\\theta} \\right) ", "087cf94aefdf5f470e33d919960c43b5": "a=15-15i", "087d07c94a3cce22b1dd9f377e5b0315": "k,~k_e", "087d27d1a0b418fadacfdf9521391cee": "\\text{Passer Rating}_{\\text{NCAA}} = {(8.4 \\times \\text{YDS}) + (330 \\times \\text{TD}) + (100 \\times \\text{COMP}) - (200 \\times \\text{INT}) \\over \\text{ATT}}", "087d28b05c8f87a4bb020c3ccf563f23": "Y_i \\sim \\mathrm{Pois}(\\lambda \\cdot p_i), \\rho(Y_i, Y_j) = 0", "087d30f0f21740efad9b744a16ef9885": " \\begin{cases}\n\\text{Mesh 1, 2: } -V_s + R_1I_1 + R_2I_2 = 0\\\\\n\\text{Current source: } I_s = I_2 - I_1\n\\end{cases} \\, ", "087df3bfe54732c537c1397765620510": "\nLz\\equiv l_2l_1z=\\Big(\\partial_x+\\partial_y+\\frac{2}{x+y}\\Big)\\Big(\\partial_x-\\partial_y+\\frac{2}{x+y}\\Big)z=0", "087e3e311cf84d5eb4c0daefc55ab1db": " \\dot{y} ", "087eebb0f949fb53c7e4be08ecf843d3": "\\frac{30,000\\ \\mathrm{N}}{(111\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=27.6", "087efa7ff3198fd6fa7b7bae1aeb9d8b": "\\theta \\mapsto p(x\\mid\\theta)\\!", "087f30a536338c2546ba6bc229296d24": "T^{\\mathrm{SW}}_p (x,y) = \\begin{cases}\n T_{\\mathrm{D}}(x,y) & \\text{if } p = -1 \\\\\n \\max\\left(0, \\frac{x + y - 1 + pxy}{1 + p}\\right) & \\text{if } -1 < p < +\\infty \\\\\n T_{\\mathrm{prod}}(x,y) & \\text{if } p = +\\infty \n\\end{cases}\n", "087f70bfdc412cff7f905892786e7145": "f : L \\to L", "087f77f0c34b93605f9d6dde51b9278e": "\\Delta x \\left[ f( a + \\Delta x ) + f(a + 2 \\Delta x)+\\cdots+f(b) \\right].", "087f88f33ecfbfaf6b959a0ae04c58e8": "v_C = v_{ \\pi} \\left(1+ \\frac {R_f} {R_1} \\right ) -i_B r_{ \\pi 2} \\ . ", "087fa767d07fdab42a05e99efe06d8b8": "\n \\mathrm{SNR} = \\frac{\\mu}{\\sigma}\n", "087fd50acd38042b6b2178db7a78d50a": "\\frac{d}{dx}g(x)=h(x)\\cdot\\frac{df(x)}{dx}+\\frac{dh(x)}{dx}\\cdot f(x)", "087ff194b49f1dc50600a9316af323c2": "\\Sigma^f\\,", "087ff4c278fe1896c72ec5b417934be7": "\\frac{1}{1260} = \\int_0^1\\frac{x^4 (1-x)^4}{2}\\,dx < \\int_0^1\\frac{x^4 (1-x)^4}{1+x^2}\\,dx < \\int_0^1\\frac{x^4 (1-x)^4}{1}\\,dx = {1 \\over 630}.", "088036c6d173183cf632e89525ce225a": "\\frac{\\pi}{3}", "088052edf469e8126bd37dfc8878ce51": "\\xi\\rightarrow\\infty", "088053bb25455067de4867cd725c7ed1": "\\mathcal{F}_{s}=-\\oint\\frac{1}{2}W(\\mathbf{\\hat{n}}\\cdot\\mathbf{\\hat{\\nu}})^2\\mathrm{d}S", "0880841a6ce8807ae6d6b7e94cbb9c72": " \\hat{g}_{ij}(t,x^k) \\to g_{ij}(t,x^k)", "0880ba6fa331b84320f320c3364180c7": "\\ Y=Min[K,L]", "0880c290c61d2acbaf130f0f8ef83dfb": "\\ \\zeta = \\frac{\\Delta W}{W_1} ", "088144bbe97e45f4e65a31a795dff154": "w=m(\\mathsf{i})", "088165301a2d7e4e5d299fd5b4618682": " V_{j} ", "08819ffe7f6d3e55490596f39495a63c": "\\sum_{j=5}^8 f(j) = 0.17367.", "0881e2a455bfa5b6dae63afe36f581c1": "\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\left( \\rho\\, v_i \\right)}{\\partial x_i} = 0", "0882903c39f87b9eac329bacff4942c8": "f_{clk}/f_o", "08829d8c00150260e3c4e34d1286237c": "p_{p_1}", "0882f7f585c18f32f9227f88d7d90155": "\\displaystyle{ f(a) =(f,E_a)}", "08832020387001bc2e9e82836fb500e4": "| \\Psi \\rangle=\\sum\\limits_{i=1}^{M}c_{i_1i_2..i_N} | {i_1,i_2,..,i_{N-1},i_N} \\rangle", "0883422bed909a1f6799617c4d526225": "{S_i-S_j}", "088364475c1e613e1b64d49dfeee4976": "M_i\\rightarrow M_j", "08838bd3c696e0632c8c31f133af263c": "f(\\tau)=A \\tau^k (1+b\\tau ^{k_1} + \\cdots) \\text{.}", "0883aecb4ef0d44b183e543352f9a6aa": "\\hbar = h/2\\pi", "0883b6c2fe84f73b8a93a07049b7a9d4": " \\frac{\\partial^2}{{\\partial x}^2} \\phi_x(x) = \\kappa^2 N_x e^{\\kappa x} = -\\frac{2m}{\\hbar^2} E_x N_x e^{\\kappa x} ", "0883bd84a2ab66a95fc9e8b50493eb99": "S = 2", "0883c969a7f0461fdc868b48bf89e964": "S^k", "088449fb3a983cc30349e8fe4dda7681": "\\alpha_i(t),", "08846ff3d5a8264373ae22b22d77e05c": "\\mathbf{A} =\n\\begin{bmatrix}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n7 & 8 & 9\n\\end{bmatrix}.\n", "088484ecb75d0f052ea2c7e9c13ee443": "\\ell \\gg m, m^\\prime", "0884b7813582ce810d6871a1fecea9c6": "\\frac{1}{T_2} = \\frac{\\rho S}{V}", "0884c1a36661d49a5c2e43ef65956105": "\\mathbf{I} = \\mu_0 \\mathbf{M} \\,", "0884cbace74dc1bfd878c582c5ee2efb": "{1\\over P}{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "08852cee1ca1c59ff4b9f3c157f513d6": "P_1 \\triangleleft C \\triangleright P_2 \\equiv ( C \\land P_1 ) \\lor ( \\lnot C \\land P_2 )", "08852f33943b2f0869e64534ac4f170a": "k_j\\sigma", "08854a2961463c85e318c8ef3fb7ec3c": "P = L\\times R /S", "08854c771fb53ecd0372766c1158f1f4": "b_4=\\frac{r_3,c_5 - b_3\\times a_2 - b_2\\times a_3 - b_1\\times a_4}{a_1}\\mbox{ with remainder }r_4 \\dots", "08857777bdc5a26f6c329c1db3d51570": " {\\pi\\over 4} = 4 \\arctan \\left({1\\over 5}\\right) - \\arctan \\left({1\\over 239}\\right)", "08857781972f452d9e7078a6653b08a1": "L^1(\\R)\\cap L^2(\\R).", "08857a59dc02f2a4358fb4af576e7ba8": "{\\dot u} = -3(p+u)\\frac{\\dot a}{a}", "0885df44ab1609dc6f365b99785154e1": "\\ddot{x_0}=h(x_0)-\\frac{g(x_0)g'(x_0)}{2 \\omega^2}", "08862e2d685ab71496c7eab5cfd9a832": " = 2,800\\,", "08864af48bfe60fd15b59545b5202aa4": "\n \\begin{align}\n x_0 &= \\cos(\\tau) \\\\\n \\text{and }\n x_1 &= \\tfrac{1}{32}\\, \\cos(3\\tau) + \\left( \\omega_1 - \\tfrac{3}{8} \\right)\\, \\tau\\, \\sin(\\tau).\n \\end{align}\n", "0886945705284bb0bdac03c73ee74428": "\n\\bold{Q} = \\boldsymbol\\beta = \n{\\partial S \\over \\partial \\boldsymbol\\alpha}\n", "088699c6fbb45a9f800785271f5b0563": "s:\\Lambda^n_k\\rightarrow X", "0886a4b22a8c9259eb5c296a23c6dd1f": "\\frac{dH}{dt} = \n\\frac{\\partial H}{\\partial \\boldsymbol{p}} \\cdot \\frac{d \\boldsymbol{p}}{dt} + \n\\frac{\\partial H}{\\partial \\boldsymbol{q}} \\cdot \\frac{d \\boldsymbol{q}}{dt} + \n\\frac{\\partial H}{\\partial t}", "0886b0dbeac4c1355c331bb702b288fc": "B^n x < B^n (y+1)^n\\,", "0886caabbc4904e95fc0cae59756e826": "(b_{1}-d)-(a_{1}-d)", "0886d2ea8c2837496c1f6a6c073c381a": "\\operatorname{CG}()", "0886d5b4f1591e426ad21c1b72ae89c0": "[TN] + [T_{M/N}] = [TM]", "08873ca3cc7efb7f3b26fda4b77ccfac": "\\partial_a", "0888292cf8e218d6ececdb638c169df2": " \n\\nabla_{\\hat{\\mathbf{h}}^H} C(n) = \\nabla_{\\hat{\\mathbf{h}}^H} E\\left\\{e(n) \\, e^{*}(n)\\right\\}=2E\\left\\{\\nabla_{\\hat{\\mathbf{h}}^H} ( e(n)) \\, e^{*}(n) \\right\\}\n", "088835ca7e893f42b565ab2bca10e393": "\\mathrm{Res}(fg')=-\\mathrm{Res}(f'g);\\,", "08886096e0572da090f4c40fca33537c": "= {\\begin{matrix} \\frac{1}{10} \\end{matrix} + \\begin{matrix} \\frac{1}{100} \\end{matrix} + \\begin{matrix} \\frac{1}{50} \\end{matrix} + \\begin{matrix} \\frac{1}{80} \\end{matrix} + \\begin{matrix} \\frac{1}{20} \\end{matrix} + \\begin{matrix} \\frac{1}{500} \\end{matrix} \\over 6} ", "0888805f4700a2434c62aacb14080476": "\\int\\frac{\\mathrm{d}x}{\\tan ax - 1} = -\\frac{x}{2} + \\frac{1}{2a}\\ln|\\sin ax - \\cos ax|+C\\,\\!", "08893af1be1bdd9df5eadf537e3bbbe6": "\\pi_1 := \\text{Pr}[P=1,Q=1]=\\sum_{\\omega \\in S_1} \\Psi^2_\\omega", "088967dfe40ea7b71cdc3cec6a28eea7": "\\omega_{\\mu}^{\\ IJ} = e^I_\\nu \\partial_\\mu e^{\\nu J} + e^I_\\nu e^{\\sigma J} \\Gamma^\\nu_{\\sigma\\mu}", "0889ddc3444aae10a93496769c2adb62": " w_1 \\ge w_2 \\ge \\dots \\ge w_\\mu > 0", "0889f771f64a54f2f4e00c59eab935ff": "n=(n_ln_{l-1}\\dots n_0)_2", "0889fb824147f396d1a1d815553e4fe4": "H=\\frac{1}{\\sqrt{2}}a", "088a3dcfc3dd3697b2fad6ea74d45e92": "\\ltimes \\!\\,", "088a4edb5ffe3aa76b2ff5406265a8f2": " T_s = 303 ~\\mathrm{K} = 30 ~\\mathrm{C} ", "088aa0453d2418193fb066dd406ff6bd": "h = h(-,-).\\,", "088aa21ca6d89bd3e67eb5ec3d720324": " \\begin{array}{lll}\n\\eta &=& 1- \\frac{trace(W_1^TAW_1)}{trace(D_B^{-1/2}P^TAPD_B^{-1/2})}\\\\\n &=& 1- \\frac{trace(\\hat{D_B}^{-1/2}\\hat{P}^TA\\hat{P}\\hat{D_B}^{-1/2})}{trace(D_B^{-1/2}P^TAPD_B^{-1/2})}\n \n\\end{array}\n", "088aa33c190e59659b39778bd19b5986": "\\lambda < 0", "088aaa73541213fd91084b37fd0422bf": "y=C(w(t))x(t)+D(w(t))u(t)", "088b09b0ab517ad1f368845c9818c722": "\\det(E+(n-i)\\delta_{ij}) =0 ", "088b147aae80c99347aefe2dc1fefa09": "G = \\frac{2e^2}{h}MT", "088bd0a0a6ad8aa37cb69bee357297a1": "U[\\mu_1,\\mu_2] = U-\\mu_1 N_1-\\mu_2 N_2", "088bdf436f5d9d1dea35097f67b43c12": " \\sum_{v(p)=0}\\frac{P(A_p)}{\\det A_p} = \\int_M P(i\\Theta/2\\pi)", "088c3b4690d85e237c04bc75b9ace31a": "\\mathbf{R} = n_1 \\mathbf{a}_1 + n_2 \\mathbf{a}_2 + n_3 \\mathbf{a}_3,", "088c4ffb0bb4687563ca3bf314176dba": "\n\\begin{align}\n\\sin (A) & = RQ \\\\\n& = \\text{length of arc } PS \\\\\n& = \\angle POS \\text{ in radians}\\\\\n& = \\frac{\\pi}{180\\times 60}\\left( m + \\frac{s}{60}+ \\frac{t}{60\\times 60}\\right).\n\\end{align}\n", "088c77d6e29bdb1d671b2df67aca9373": "x \\cdot p + y \\cdot q = 0", "088c8b835560feac3300998ec1332a92": " |\\psi \\rang", "088d03f1074199581e9cd748695d5521": "g(z)=z+b_0 + b_1z^{-1} + b_2 z^{-2} + \\cdots", "088d15a0bdfecaccd1c62383f94b8cbc": " -\\varepsilon_i.", "088d531507cfb1a6302c6226049e80dc": "M^{\\mu\\nu} \\,", "088d68d8a312039e05c361d2341ac71c": "E_{X} = 61.5 \\ \\mathrm{mV} \\log{ \\left( \\frac{ [X^{+}]_\\mathrm{out}}{ [X^{+}]_\\mathrm{in}} \\right) } = -61.5 \\ \\mathrm{mV} \\log{ \\left( \\frac{ [X^{-}]_\\mathrm{out}}{ [X^{-}]_\\mathrm{in}} \\right) }", "088dbb6eed854331cdab6c34e5e8852e": "\\left|\\Gamma\\left(\\omega\\right)\\right|^2", "088dc58a11fd066e189ee7f8a7725546": "\\kappa(M)\\in H^4(M;\\mathbb{Z}/2\\mathbb{Z})", "088de5149c77b89b1f21d59d3df5e987": "\\mathbf{S}= \\frac{\\epsilon_0}{2i\\omega}\\int \\left(\\mathbf{E}^\\ast\\times\\mathbf{E}\\right)d^{3}\\mathbf{r} .", "088e53945d396a436984c8403162be33": "K_{\\mathrm{rot}} = \\tfrac{1}{2}I\\omega^2,", "088f2b5f22c21c3051502aa50d7597ec": "i:=1;\\qquad S:=\\emptyset,\\qquad f^*:=f;", "088f6df761802b49580d5bb103e7a428": "L_i=C_i-d_i", "088fc910583005a88db3211c8be16637": "Q=\\big(\\alpha K^\\lambda + (1-\\alpha) N^\\lambda\\big)^{1/\\lambda},\\,", "088ff137b55f0dce3cce12801b5c9125": "\\bar{V^E}_i= RT \\frac{\\partial (\\ln(\\gamma_i))}{\\partial P}", "08900515bff2bad7ba26a0b4ae1b93a5": "48+\t32+\t1+\t64+\t33+\t17+\t16+\t49\t=\t\t260", "089017e414db79a49991a6af70d24a61": "\\scriptstyle f_0\\,", "08915b56dbc2dc98df111e43d0cdcbe3": "\\int \\ln(x^2+a^2)\\; dx = x\\ln(x^2+a^2)-2x+2a\\tan^{-1} \\frac{x}{a}", "08917d48d77824f06da1c6c0ca89d811": "2^2=4,\\, 2^3=8,\\, 2^5=32,\\, 2^7=128,\\, 2^{11}=2048,\\, 2^{13}=8192,\\, 2^{17}=131072,\\, 2^{19}=524288,\\, \\dots", "0891adbe9e3ea1673fe06c84c4b13fa8": "\n\\frac{S(\\omega)}{H_{1/3}^2 T_1} = \\frac{0.11}{2\\pi} \\left(\\frac{\\omega T_1}{2\\pi}\\right)^{-5} \\mathrm{exp} \\left[-0.44 \\left(\\frac{\\omega T_1}{2\\pi}\\right)^{-4} \\right]\n", "0891c3a975872f033667b0eddf7c0ce3": "V = \\frac{\\partial}{\\partial\\theta} \\ln f(X;\\theta)", "0891c88603858c7033d48290fe995b5a": "(AB)^+ = B^+ A^+\\,\\!", "0891d9a9c062411339abf08c6b6d2179": " H_L", "0892355b40f608ef28e5f27f63f91144": "p_{k+1} \\leftarrow M^{-1} r_{k+1} + \\beta_k \\cdot p_k\\,", "089255d1a3230d76d837329d738f9e48": "c_{1}'(t) = \\dfrac{\\mu_{01}}{i\\hbar}\\epsilon(t)\\exp\\left(-i\\dfrac{E_{0} - E_{1}}{\\hbar}t\\right)\\Rightarrow \nc_{1}(t') = \\dfrac{\\mu_{01}}{i\\hbar}\\int_{0}^{t'}\\epsilon(t)\\exp\\left(-i\\dfrac{E_{0} - E_{1}}{\\hbar}t\\right)\\mathrm{d}t", "089291bee7f9ab0573749b9336958005": " A_{2N} < C < A_{2N} + D_{2N}.", "089292e1991fecada0329c30b57887c0": "A=\\frac{1}{4}\\pi\\left(r_1+r_2\\right)^2.", "0892d1397ef71a6a4de7b75a75098215": " 0 = (x+R)^2 + z^2 - r^2 \\,\\!", "0892de2ff8aea7f3d7a3e6db5735f928": "\\displaystyle AE-EC=AF-FC.", "08931263074a27512bd50843955e6aff": " C = \\int_{0}^B \\log_2 \\left( 1+\\frac{S(f)}{N(f)} \\right) df ", "089314e445e1ab7e26cdd61671f6016e": "\\delta\\,q + \\delta\\,w = \\sum_{\\alpha,\\beta,S}\\,\\left( T\\mathrm{d}S\\, - p\\mathrm{d}V\\, -\\delta\\,w_{\\text{non-pV}}\\right)\\,.", "0893161b8426cd5e20b28075dbf3f72f": "\\sin^2 x + \\cos^2 x = 1", "08933ee61a618dd9871b6089aeab493e": "\\frac{n}{100} = \\frac{K}{N} ", "089373cc2f761bbeb5f06b45603f41d4": " 2r = \\frac{a}{3}\\sqrt{3} \\!\\, ", "0893e90dbd7c653eb2ceac23880057eb": "y\\nleq w.", "0893ed55c75d7a6c25c0993b1d023964": "\\tilde\\psi(t)=C_\\psi^{-1}\\psi(t)", "089457c1ebcb84dbcb5dc46930362c6d": "\nf(\\theta, \\phi, t) = \\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} C_{lm} Y^{m}_{l}(\\theta, \\phi) e^{-t/\\tau_{l}}\n", "0894858cd2ca24e04e981e303511dc09": "a,b\\,", "08948cf9a5597c38e92f5ee2d8c9cde3": " E = -\\mu_\\mathrm{z} B_0 \\ ,", "0894cec12874d2cb1cb7f21dd087d89e": " \\ m \\ddot{x} + k x = 0.", "0894d59244c36636bf4f60db9a606540": "S_\\alpha(x) = x - \\alpha", "08950ec45604b6b600898b4ae5f44608": "\n\\; C_\\Phi = \\sum _{i = 1} ^{nm} v_i v_i ^* .\n", "089517dc1e251ebe3f3d53ece382600e": "r = {2\\mu \\over v_\\infty^2}", "0895194b16feb437e209b4f12bb9d537": "\\begin{align} & \\nabla\\cdot\\mathbf{j}(\\mathbf{r},t) = - \\frac{\\partial \\rho(\\mathbf{r},t)}{\\partial t} + \\sigma(\\mathbf{r},t), \\\\\n& \\nabla\\cdot\\mathbf{j} + \\frac{\\partial \\rho}{\\partial t} = \\sigma \\quad \\rightleftharpoons \\quad \\nabla\\cdot(\\rho \\mathbf{v}) + \\frac{\\partial \\rho}{\\partial t} = \\sigma.\\\\\n\\end{align}", "08953dcf0a515e7cff3d6c7d8683977c": "h(t)=\\mathcal{L}^{-1} \\left \\{K(p) \\right \\}=\\delta (t)-\\omega_0 e^{-\\omega_0 t}=\\delta (t)-\\frac{1}{\\tau} e^{-\\frac{t}{\\tau}}", "0895880b7b819b766849d690ad7ecf98": "\\mbox{Skew}[Y] = \\mbox{Skew}[X]/\\sqrt{n}", "0895c40bab16acf26444c072230a909e": "\\mathit{bar}", "0896910db3486c36b4e9ebde666350a3": "ZFC\\vdash \\operatorname{Con}(ZFC)\\leftrightarrow \\operatorname{Con}(ZFL)", "08973f6125605e2de0399c5626706e37": "M^2 dM = - \\frac{K_{\\operatorname{ev}}}{c^2} dt \\;", "08975895667eb87dd2a534ba79385954": "\\mathbf{Z}_4 = \\{\\overline{0},\\overline{1},\\overline{2},\\overline{3}\\}", "089777aeed0359e947eaf33e96ad1248": " \\frac{d p_{\\alpha}}{d \\tau} \\, = q \\, F_{\\alpha \\beta} \\, u^\\beta ", "0897b1e4c0c539268fd4ba6abbe20a05": "F(\\lambda) = e^{\\lambda M^*} T e^{-\\lambda N^*}.", "0897da084ee46227777f2e99f0a8c598": "\n\\begin{array}{c|cccc}\n0 & 0 & 0 & 0 & 0\\\\\n1/3 & 1/3 & 0 & 0 & 0\\\\\n2/3 & -1/3 & 1 & 0 & 0\\\\\n1 & 1 & -1 & 1 & 0\\\\\n\\hline\n & 1/8 & 3/8 & 3/8 & 1/8\\\\\n\\end{array}\n", "0897e667e0116679b830496fa73eca25": "\\varphi\\,\\!", "08980a580ec39c643d6ab2a2e5e8abbd": "i\\hbar\\frac{\\partial}{\\partial t}\\psi=-\\sum_{k=1}^{N}\\frac{\\hbar^2}{2m_k}\\nabla_k^2\\psi + V\\psi", "08980e54192c9ed6458c9ff7f566bb95": "G = eD", "089817b894737febdd472b2c9af430fe": "\\mathrm{D}_{t}", "089843f7731a27d67155ee011d14ce3c": "\\phi = \\frac{V_\\mathrm{V}}{V_\\mathrm{T}}", "089887b5053c9d8239fd1c3dc583ed86": "F_{in} > \\frac{1}{2}", "0898cff846b040b8dffbf037986688b4": "\\int_\\gamma f(z) = \\int_a^b f(z(t)) z'(t) \\, dt.", "0898efa852b099482ca09860d347f545": "{|\\mathbf{F}|+n-1\\choose n}\\ge \\frac{|\\mathbf{F}|^n}{n!}.", "0898efbeab821569bdc5ae5d3c6f3ab5": "\\sum_{j=1}^N c_j \\left( H_{kj} - \\varepsilon S_{kj} \\right) = 0 \\quad \\text{for} \\quad k = 1,2,\\dots,N. ", "0898fa79b5b86eefb8fc18aad25f635f": "\\lim_{t\\to 1^{-}} G_a(tz) = \\sum_{k=0}^{\\infty} a_kz^k\\!", "08991589b018456998d95c6ae785f81f": "\\mathbb{C}P^{1}", "0899241e18e47bf3b78e9b2a0d632872": "g\\colon \\mathbb{R}^n \\to \\mathbb{R}^n", "08993f6c65b816ce607a024c300a1f84": "\\textstyle 2^n/(n2^{l-1}+1)", "08995f987e57f81e4b5ad4c0aaee5866": "\\frac{b^2}{a}", "08996b0dbb09a185cbc0a964a70e4ea5": "\\sigma_{j,n-1}", "0899c037f2676e8ef14b5f15d3b840b6": "\\omega_k = ck", "089a4b8acabe5054582d53f836e07ddb": " (n,\\tilde{m},l,\\epsilon) ", "089a7ceda77cb567e8854fe8de175b53": " \\|X_{(1)}-x\\| \\leq \\dots \\leq \\|X_{(n)}-x\\| ", "089a91d37a3c9cfe64804a32a3ed8cfe": "\\Sigma \\subset \\mathcal{M}^3", "089b2872354cf4599b6a22e2a3f020be": "x^*\\in\\{x_1,\\ldots,x_N\\}", "089b34600dc598cce585d42cc679d061": "p(z) = a(z-\\alpha_1)(z-\\alpha_2)\\cdots(z-\\alpha_n)", "089b5f8ea190475360f8bb85d181e265": "\\omega(j)", "089b7716be9fe6981ca3351fb15479e8": "\\boldsymbol{v}=\\frac{\\text{d}\\boldsymbol{s}}{\\text{d}t}", "089b9ac86c5f461f59c9f8f674002eba": "2^{-k}\\exp\\left(-(1+o(1))\\frac{\\log x\\,\\log\\log\\log x}{\\log\\log x}\\right)", "089ba3f87c5ecee1db40e8302f0a9667": " |s\\rangle ", "089bd4f09c60d3da74d1d7b67bf8a345": "\\textstyle \\delta_\\nu = 0", "089bf6da03747ee690cfed1a43415203": "e_k(X_1 , \\ldots ,X_n)", "089c078bf9a7ea8ae5092ccede071e22": "\n \\hat{\\theta} = \\underset{\\theta\\in\\Theta}{\\operatorname{arg\\,max}} \\; S_n(\\theta),\n \\quad\\text{where }\\ \n S_n(\\theta) = \\ln\\!\\! \\sqrt[n+1]{D_1D_2\\cdots D_{n+1}}\n = \\frac{1}{n+1}\\sum_{i=1}^{n+1}\\ln{D_i}(\\theta).\n ", "089c39ac42361671a5b39e01d28be16d": "\\Phi_M \\,", "089c50d0baecda42b714997ca31d8343": " r\\ ", "089c649017c10f59ef86716ad1835192": "T^2_0(q) = \\dfrac{\\sqrt{6}}{2}q_{zz}", "089cf08b631ba1f5c463828ee196439b": "Z^{M}_{0,0} = 1", "089d079ccc7bcd387140a47ce4f24e24": "\\nu' = \\gamma \\nu - \\gamma \\beta \\nu \\cos \\theta = \\gamma \\nu \\left ( 1 - \\beta \\cos \\theta \\right )", "089d52bdbe45a24dffc733b3cfe769a1": "F(a_1,\\dots,a_n) = \\sum_{i=1}^nF(a_1,\\dots,a_i-1,\\dots,a_n).", "089d6aa9edd04167c42d45f2dae6e667": "C = S(e^{rT} - 1) \\,", "089de14241a817bc8b4b39d127f00b72": "\\sigma_\\mathrm{n} = \\sigma_1 n_1^2 + \\sigma_2 n_2^2 + \\sigma_3 n_3^2.", "089de5f0be1afcf0ec62ae9a704ad4bb": "c= 110 001 111", "089e1624882f007bba90fb810b57fe1c": "\\begin{align}\n C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\\\\n d_1 &= \\frac{1}{\\sigma\\sqrt{T - t}}\\left[\\ln\\left(\\frac{S}{K}\\right) + \\left(r + \\frac{\\sigma^2}{2}\\right)(T - t)\\right] \\\\\n d_2 &= \\frac{1}{\\sigma\\sqrt{T - t}}\\left[\\ln\\left(\\frac{S}{K}\\right) + \\left(r - \\frac{\\sigma^2}{2}\\right)(T - t)\\right] \\\\\n &= d_1 - \\sigma\\sqrt{T - t}\n\\end{align}", "089e690bafe32a984251d428be1be4f2": "\\rho_Q = \\sqrt{\\frac{L_{QA}}{C_{QA}}} = \\sqrt{\\frac{\\phi_0^2}{e^2}} = \\frac{h}{e^2} = R_H \\ ", "089e932f69f266f26e918d19d6217b09": "b=\\lambda^{-k}", "089eb6e35ca6130e89725199e9bf4c00": " \\phi^{-}(a_i) \\ge \\phi^{-}(a_j)", "089ed104ae0893c5ab1a8ee6a5797241": "\\sigma_{\\text{mean}} = \\frac{1}{\\sqrt{N}}\\sigma", "089ed62808b2967c53cea2f8804edbf7": "x\\,y'' + (\\alpha + 1-x)\\,y' + {\\lambda}\\,y = 0\\text{ with }\\lambda = n.\\,", "089eeacfc3eb9e564575aa6d835c3673": "\\log_{10}K_\\textrm{a} = \\log_{10}[\\textrm{H}^+] + \\log_{10} \\left ( \\frac{[\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "089ef266a52b9ca4e23dce607a48b556": "\\{ n_{\\alpha} \\}, \\{ m_{\\alpha} \\} \\subset \\mathbb{N}", "089f12531b1c4a0d3560fc99de22290f": "\\left|{|gA\\cap F_i|\\over|F_i|}-{|A\\cap F_i|\\over|F_i|}\\right| = \\left|{|A\\cap g^{-1}F_i|\\over|F_i|}-{|A\\cap F_i|\\over|F_i|}\\right|", "089fb2bbc2d7478893a96420ff1623f0": "i = 1, 2, ..., s", "089fef7d2053a7b26771146ccfcf5513": "H(\\alpha)=\\max\\{|x|,|y|,|z|\\}.\\,", "089ff39a17b4ec2d40f4f00851804b5b": "a_5", "089ffe1694bd5f2c3ad2d15976da5ce7": " sp(x := x - 5, x > 15)\\ =\\ \\exists y, x = y - 5 \\wedge y > 15 \\ \\Leftrightarrow \\ x > 10", "08a0097b172b5cd5c4556d2930dfa993": "P^2=I ", "08a06702edc84571e50f183d0af6d160": "\n\\bar{\\Gamma}(\\tau) = \\forall\\ \\hat{\\alpha}\\ .\\ \\tau \\quad\\quad \\hat{\\alpha} = \\textrm{free}(\\tau) - \\textrm{free}(\\Gamma)\n", "08a0ba2451427d21e13c4775c6e3dd30": "d \\mathbf S", "08a0bc384275a2e1e7e17f6b8ed21138": "\\vec{p}_2 = \\partial_r ", "08a0ff71551639cc72531876a9c21125": "\\frac{163}{\\ln 163} \\approx 2^{5}", "08a0ffadbb96a515913099edbd30c490": "J=\\begin{pmatrix}0 & I \\\\ -I & 0\\end{pmatrix}", "08a104e9574a90a74209a609d3daacc7": "T \\times \\gamma", "08a136f9796e1034de4d5b33d1dc2bb4": "m(x, y)", "08a13947c8a56d0118b31a4ffe572d7a": " \\nabla \\cdot \\mathbf{u} = 0. ", "08a13fa295805c91afc1d69f8976fab5": " \\mathbf{y}_{k} \\sim \\mathbf{\\bar C}_{k} \\, \\mathbf{\\bar x} = \\mathbf{C}_{k} \\, \\mathbf{x} ", "08a175f5bfe368e1ee60c7fec13a057e": "\\frac{\\Delta y}{\\Delta x} = f'(x)+\\varepsilon,", "08a1811bd8c272fdc022c364e723563f": "q_{r,\\nu} = w/s.\\,", "08a20142f9fd84a6f1b44094576e4e2a": "\\sum_{i=0}^{n}{i\\binom{n}{i}^2}=\\frac{n}{2}\\binom{2n}{n}", "08a22e50f7516840508a72a6481646b7": "\\sin(\\pi z) = \\pi z \\prod_{n=1}^{\\infty} \\left(1 - \\frac{z^2}{n^2}\\right)", "08a29dd11a36e96ebf5f676b6bcd793a": "f_{t,a,b}", "08a31d631f1f0e371e354b024649cfea": "color\\ ", "08a32bbc01161ef7587f6fc60892f9b8": "S(t) = \\sum^{\\infty}_{i = 1}S_i\\Phi_i(t)", "08a35e1c70e126b9d1ea1a35ccc73bc0": "{\\mathbf P} (t)", "08a3669763ee42ad3197e9bd45e004ef": "L = K", "08a37568bdfc4282dadce1dce6f48d27": "f_{(\\xi,\\mu,\\sigma)}(x) = \\frac{1}{\\sigma}\\left(1 + \\frac{\\xi (x-\\mu)}{\\sigma}\\right)^{\\left(-\\frac{1}{\\xi} - 1\\right)},", "08a386134202a72add2ca015fc6201c6": " \\theta = 2 \\pi \\nu \\,", "08a398a26cb80312d9c83cbe83379c5a": "\\tan \\frac{\\delta'}{2}=\\sqrt{\\frac{1-\\beta}{1+\\beta}}\\cdot\\tan\\frac{\\delta}{2}", "08a39da9b0aa34d4e84a4d67294811db": "G:C\\to D", "08a3fff08bec0bb773f00db8d38ec5ea": "z^2=r_1^2-x^2-y^2", "08a422c3ea7ffca07f2210324d12db75": "\\vec k\\|\\vec B", "08a4377d7ed260cbc3b26e05f2493110": "A^* A + \\Gamma^T \\Gamma ", "08a493b23d014a745d4b9daf67519802": "\\mathrm{Id}_A(a,b)", "08a4d85fdf5424e20249248d2c9cd5f9": "\\epsilon=ArcCot(\\pm AR)", "08a4eecf61f158c09769acd2e87b8591": "\nL = \\frac{1}{2} \\left(M+m \\right ) \\dot x^2 -m \\ell \\dot x \\dot\\theta\\cos\\theta + \\frac{1}{2} m \\ell^2 \\dot \\theta^2-m g \\ell\\cos \\theta\n", "08a554081bf4c684c8f4cb0cf90f78cb": "\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow {}_{\\ 90}^{233} \\mathrm{Th} \\xrightarrow{\\beta^-} {}_{\\ 91}^{233}\\mathrm{Pa} \\xrightarrow{\\beta^-} {}_{\\ 92}^{233}\\mathrm{U}", "08a5c71d6fd63e1955df451525935d1c": " a_P T_P = a_w {T_w}^0 + a_e {T_e}^0 + [ {a_P}^0 - ( a_w + a_e - S_P )] {T_P}^0 + S_u ", "08a5d5c8d39219b8b3b0a6c2f7d4a143": "P(x_1, ..., x_n)=0", "08a5e5edf4fe6f511952a8e8f27c0f65": "\\alpha=e^{\\frac{-2\\pi i}{n}},", "08a625db3223bd84256d71f8110857d1": "u_{11} = 4", "08a62bf6fcb9536ca79feb932e81dcd2": "\\int \\exp\\left[-\\theta^T M \\theta\\right] \\,d\\theta = \\begin{cases} 2 ^{n\\over 2 } \\sqrt{ \\det M } , & n \\mbox{ even} \\\\ 0 , & n \\mbox{ odd} \\end{cases} ", "08a637a7533dcdaf3e830671dc549003": "\\varepsilon _{1}\\Psi =\\frac{-P^{2}+m_{1}^{2}-m_{2}^{2}}{2\\sqrt{-P^{2}}}\n\\Psi", "08a64a2cc12e61fefbaab0023cadfca3": " f: (-2, +\\infty)\\rightarrow \\mathbb{R}", "08a6f3b40d66c7170552d2b2e87fd2c0": " |\\psi (t)\\rangle = \\sum_{n} a_n(t) |n\\rangle ", "08a75a112849152419b42feee123eb44": "x = x_1 (1, \\lambda, \\lambda^2, \\dots),", "08a792918893b79247dec464219b5d9f": "f^{abc}", "08a7a7ab25e5448135a705ebc67c67dc": "r_{t+1}", "08a7c0c5e953a77c63ff8527b4d8bd5b": "\\gamma W \\cap W\\,", "08a7d8607501784ff17a3c4e5eeb6e92": "\\mathrm{R_{B\\beta}}", "08a85f4d2b6708c8a604cb7710a29ee6": "\n(Sv)(ds) = \\lim_{\\epsilon \\to 0, \\epsilon>0} \\frac{1}{\\epsilon}\\int_0^{1-\\epsilon} (f(t+\\epsilon \\mu(ds))-f(t))dt\n", "08a877f3a35b15505a8eb7b18d5fdc73": "\\sum_{v = 0}^{r-1} {r - v}", "08a88658e7ed556dd1eecc98587f7199": "H = \\frac{1}{2m}\\left ( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right )^2 + e\\phi - \\frac{e\\hbar}{2mc}\\boldsymbol{\\sigma}\\cdot \\mathbf{B}.", "08a8c01d82983e9bdcf0e6741f5c1e75": "\\text{min:}\\operatorname{Tr} (\\sigma_B)", "08a97bcb5749c8f469c55ce3dcc470d9": "R_{max} = v^2/g\\,", "08a97ca308307132125ee096959ee1df": "\\operatorname{Li}_2(0)=0", "08a9d74163be02ea90ab7b7a0351b5f3": "G=SU(N_c)", "08aa2420736e7ec5fd436f4229f48ad4": " n_2 ", "08aa8c63acbec691aec44ddacf455fc6": "{\\Phi}", "08aab844ad8b5788a4d2e94e6124ca22": "\n\\frac{d}{dz} \\left[ z^{1-a_1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) \\right] =\nz^{-a_1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1 - 1, a_2, \\dots, a_p \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n \\geq 1,\n", "08aade75fb531a156e4329ffcfb87ceb": " t = (n+1) \\Delta t \\,", "08aaecd3a878f9134a06a38b6ac7aad9": "R_M", "08ab13e7e260a41c4e1131ccf103e57c": "A,B \\in K", "08ab14743db555f395c558fd89ee49fb": "\\alpha |f|^p = \\beta |g|^q\\,", "08ab3226355569dd4cc1e814528a2795": "(x_1)^n = \\sum_{k_1=n} {n \\choose k_1} x_1^{k_1};\\ \\ k_1, n \\in \\mathbb{N}_0", "08ab50acbda2fd422f9f2bc299e3858b": " RSI = 100 - { 100 \\over {1 + RS} } ", "08ab809371ce6537c1a7b6fa1e535dc6": " A \\otimes B", "08ab8a1a461f7f1f2aaf5e560df56c7b": "\\textstyle \\{v_1,\\dots,v_m\\}", "08abbb203e035f5e7cd178c8dc9e9bb7": "R(f) ", "08ac18c283b358ab1356c98861464d28": "\\frac{250}{50} = 5", "08ac2718fba3838072d91d1aacb83b41": "r = k C_S \\theta_A C_B", "08acb54f0d911db77c96528a6caa7189": "\\begin{bmatrix}\n1 & 0 & 0\\\\\n0 & 4 & 0\\\\\n0 & 0 & -3\\\\\n0 & 0 & 0\\\\\n\\end{bmatrix}", "08acd20803da1f3613a020ebb5e7ed8d": " (f * g) (t) = \\begin{cases} f(2t) & 0 \\leq t \\leq \\tfrac{1}{2} \\\\ g(2t-1) & \\tfrac{1}{2} \\leq t \\leq 1 \\end{cases}", "08ace0ea6db5bc67a6dbbc71ccee47e0": "\\prod_{x \\in \\Sigma} \\delta (\\hat{H} (x))", "08acee831506e12f7d064a1081242f06": "\\theta_t(t^\\prime)=\\delta_{tt^\\prime}, ", "08ad08f6491037714d09263a79bebfba": "MT", "08ad36651c8fd2c5dc250c5cc8da1c11": "6.02 \\approx 20 \\log_{10}2", "08ad4c042fe94b11e3c1f61a141fcae3": "\\sigma_p", "08ad53c0fc97bb056df5311a351bfdb4": " S_2S_1S_2 ", "08ad58beca7a2d9538f3ab070d072807": " \\ln \\Gamma(\\alpha) + \\ln \\Gamma(\\beta) - \\ln \\Gamma(\\alpha+\\beta)", "08ad5a353879b2190b43380c06834129": "\\forall x \\in a\\exists y\\, \\phi(x,y)\\rightarrow \\exists b\\forall x \\in a\\exists y\\in b\\, \\phi(x,y) ", "08ad7e857232ce296c5b63f8c7fc5b3b": " \\Delta H - T \\Delta S_{int} \\le 0 \\,", "08ae28c054879dfbcac0417af765b298": "\\delta\\; ", "08ae6dea47bd74fd7cfd3e24ee234038": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^5 = \\frac{1}{1^5} - \\frac{1}{3^5} + \\frac{1}{5^5} - \\frac{1}{7^5} + \\cdots = \\frac{5\\pi^5}{1536}\\!", "08aea994c648e6990d27839bd3c335a6": "\\mathbf{e'} = \\mathbf{J}\\mathbf{e}.", "08aecc8a1ce3b7392dbba000ac74f843": "\n \\log \\mathcal{L} = -12492.9\n", "08aee2f8fd1c4ebe83333b4910549142": "{\\mathbf q_{u}}({\\mathbf r},t)", "08aee5b6c0c5621b81882cb6639cbdc8": "\\begin{pmatrix}0& -1\\\\ 1& 0\\end{pmatrix}", "08af80b34b54c2e45bd728257f84fb40": "b =\\hat\\beta", "08af862e381d5a6ed11787a203c54175": "v_m = \\frac{2D_e-h\\nu_0}{h\\nu_0}", "08afa9f77f75b8d63c304f964c0539e5": "\\Psi(x) = C_A \\textrm{Ai}\\left( \\sqrt[3]{U_1} \\cdot (x - x_1) \\right) + C_B \\textrm{Bi}\\left( \\sqrt[3]{U_1} \\cdot (x - x_1) \\right).", "08aff5439448f31759bc5ef7e1b0a164": "Z_{n+1}", "08b0104e514f16d489cc743b6f66d906": "P_i", "08b046e112ef09823fefab0abc701f96": "S \\rightarrow ba", "08b0c7ac7fa7a06b33ef89840a3ec0ed": "\n-X_1^2-X_2^2+\\sum_{i=3}^{n+1}X_i^2=-\\alpha^2\n", "08b1a784c63207cc2f2c69e089160748": "\\langle x,Ax \\rangle = 0", "08b24ad19f2e115acbb706c256b9271e": "M_{3,X} = M_{3,A} + M_{3,B} + \\delta^3\\frac{n_A n_B (n_A - n_B)}{n_X^2} + 3\\delta\\frac{n_AM_{2,B} - n_BM_{2,A}}{n_X}", "08b29601b034f0a05b294662bbb971cb": "T \\underline{A}", "08b29e6aff4e6ec1e064d0855d652a99": "x,z,w_1,\\ldots,w_n\\!", "08b2c2214887981149ff0c4db51e48fb": "\\overline{\\overline V} \\to V", "08b2fd3b424c21a551d770c23e0ca994": " \\omega_k ", "08b327813e8ad4772779f2e81d6144f7": "m = \\sum_{i=1}^n a_i", "08b3da0bf6532d03c95780f4e504eebe": "\\mathbf{P} \\left[ \\sup_{0 \\leq t \\leq T} B_{t} \\geq C \\right] \\leq \\exp \\left( - \\frac{C^2}{2T} \\right).", "08b41e0e3b3d28607a0109f92fc960a8": "P_{ij} = \\mu + \\alpha_i + \\alpha_j + d_{ij}", "08b4271e7a9111b5ad74901134a20477": " \\Delta(x) = \\int_0^\\infty d\\tau e^{-m^2\\tau} {1\\over ({4\\pi\\tau})^{d/2}}e^{-x^2\\over 4\\tau}", "08b43a28f93233cde0befa6872957246": "b_{14}+a_{15}-c_{13}", "08b44f9bcb2425a5c61732bd5671ce38": "\\begin{vmatrix}\n0 &\n\\frac{\\lambda_2-\\lambda_1}{(\\lambda_2-\\lambda_1)} &\n\\frac{\\lambda_2^2-\\lambda_1^2}{(\\lambda_2-\\lambda_1)} &\n\\cdots &\n\\frac{\\lambda_2^{n-1}-\\lambda_1^{n-1} }{(\\lambda_2-\\lambda_1)}\n\\end{vmatrix}\n\\quad \\begin{vmatrix}\\frac{\\lambda_2^{n+m}-\\lambda_1^{n+m} }{(\\lambda_2-\\lambda_1)}\\end{vmatrix}", "08b48c475b79d192ef03ea5415b09f89": "\\begin{matrix} {4 \\choose 2}{3 \\choose 1}^2{36 \\choose 2} \\end{matrix}", "08b4c57626841de9aed16f68df25fd2c": "z_v", "08b511fff209c4e37fbffee768045446": " S_{i',j'}^t=1 ", "08b57638e0309a5476c09e84d8846ae3": "\n\\mathbb{V}\\left[\\log \\frac{\\hat p_{x-1}}{\\hat p_x}\\right] \\approx \\frac{1}{np_x} + \\frac{1}{np_{x-1}}\n", "08b5804db7683df220712df0f72937f1": "2m+2.2\\sqrt{m}", "08b5805912e3751068405dacd06fb6ab": "\\Pi^1_{n}", "08b5d77da7e35ed91a7e17de5c5a4613": "\\text{Level 5:} \\ \\ 266 = 2 \\uparrow\\uparrow\\uparrow 2 \\uparrow\\uparrow 2 + 2 \\uparrow\\uparrow\\uparrow 2 \\times 2 + 2", "08b5fb3e2d65e1c5a8d30fa4cd40cfef": "\\beta = [\\beta_0,\\beta_1]^T", "08b61afeeb4fede6cf7ff8b0cd85612a": "\n\\exp \\left( \\frac{1}{2} \\log \\frac{1+z}{1-z} \\right)\n\\prod_{m\\ge 1} \\cosh \\frac{z^{2m}}{2m} =\n\\sqrt{\\frac{1+z}{1-z}}\n\\prod_{m\\ge 1} \\cosh \\frac{z^{2m}}{2m}.", "08b6237946830fd085f7d87a758ace10": "M(L) \\sim L^{d_\\text{f}}\\,\\!", "08b62a2628cff624d81f97b92e16892a": "\\int^\\infty_0 J_\\lambda d\\lambda", "08b64845c8b030cd2a35885708c08f7d": "x = 0 ", "08b69591f93ee237dabcb49caf282947": "c_\\Lambda(\\eta,\\xi)>0", "08b743423bf9352c267d4fc151dc45ab": "\\Delta f = -\\frac{2f_0^2}{A \\sqrt{ \\rho_q \\mu_q } }\\Delta m", "08b747a01be91249f4d1fcd29f666490": "(2^4/7!!) \\pi^3 = (16/105)\\pi^3 ", "08b750e879638f8dbfbaf8b02aa9b69e": "y_3 = \\frac{(x_1^2+ y_1^2 - 2(x_1 )^2}{-4 (y_1^2-1)y_1^2+(x_1^2-y_1^2)^2}y_1 = 0", "08b79fc9a0c43abce9cbddf55408dc95": "\\begin{align}\n\\boldsymbol{M} \\colon \\vec{x_i}(t_i)&\\to \\vec{x}_{i+1}(t_{i+1}=t_i+T,\\vec{x_i}).\n\\end{align}", "08b8018af644fa58cea50b82ecffd271": "\\N\\cup\\{\\infty\\}", "08b8176cc3ff6af94f409722f33e2822": "\\left(1 - \\frac{2GM}{rc^{2}}\\right)c^{2}dt^{2} = r^{2}\\textrm{sin}^{2}\\theta d\\phi^{2}", "08b83bc14afc91e039b47d5ab69d29d9": "\\frac{\\mathrm{d}}{\\mathrm{d} t}(\\varphi(\\mathbf x, t)) = \\frac{\\partial \\varphi}{\\partial t} + \\nabla \\varphi \\cdot \\frac{\\mathrm{d} \\mathbf x}{\\mathrm{d} t}.", "08b840d813799099f64d75175669e025": "{s \\triangleleft s}", "08b85058ed80237cdd212f6c41e0ce44": "\\mathrm {DOF} \\approx \\frac {2 N c \\left ( 1 + m/P \\right )}{m^2} \\,,", "08b8810dc87a9e8dbebbfd61d2df5071": " v = \\frac {\\omega}{\\beta} .", "08b8d7d03f84ebb2675bbe9b3ce74d65": "ee_{product}=ee_{max}ee_{catalyst}", "08b8d96d5bc3fb56e1f646098021ec43": "f\\colon X\\times [0,1]\\to B\\,", "08b8e29f1416bafafa36c6ad1adbae80": "\\scriptstyle A \\;\\not \\Rightarrow\\; B", "08b91b01ae001d95cdf87ddf9e6c94ce": " C_n^{knn} ", "08b93bb791922b09b2463dc9b2c84400": "\\widehat{\\mathbf{C}} := \\mathbf{C} \\cup\\{\\infty\\}", "08b949d0ab5a03d90069b893b62e2d0c": "\n\\begin{align}\nU(x,z)\n&= aW \\frac {\\sin \\left [\\frac {\\pi Wx} {\\lambda z} \\right ]} {\\frac {\\pi Wx} {\\lambda z}}\\\\\n&= aW ~\\mathrm{sinc} \\frac {\\pi Wx}{\\lambda z}\n\\end{align}\n", "08b98c197628f1e05eb912d81617d4e8": " OPEN_d ", "08b9cef8bfb2a152bacf7fded6d24b7d": "\\kappa= (3-\\nu)/(1+\\nu)", "08b9e5eccad5ff4a01eb737ac1606e62": "m\\geq 2", "08b9f71543f531ab69671c06d811fa72": " \\bar{x}", "08ba0b6c5a1063fb3b632c78558aa40a": "\\rho\\epsilon =\\sum_i g_i\\,\\!", "08ba7ea174f14c34b7ae89cdce8b8d4b": "\\operatorname{F}(-)", "08ba969995304a7a289e3514b1cedc76": "\\prod _x a^x = C a^{\\frac{x}{2} (x-1)} \\,", "08bac121d68536b08a2a3ecc08df2ebd": "d_A, d_B", "08bad8a366dc9f2a7e981fb27f26779b": "M^{2}f=-\\triangle_{n}P(f)+\\frac{n-2}{x_{n}}\\frac{\\partial P(f)}{\\partial x_{n}}- \\left (\\triangle_{n}Q(f)-\\frac{n-2}{x_{n}}\\frac{\\partial Q(f)}{\\partial x_{n}}+ \\frac{n-2}{x_{n}^{2}}Q(f) \\right )e_{n}", "08bbdf0546471a25454ba6b44f887bdb": "U(\\alpha,y)", "08bbecb3b1bc01e463e1288ba630f556": "b\\ \\pmod{\\Phi_n(q)}", "08bbf6749490cb5d38cf9f389f415159": "\\tilde{\\mu}", "08bbfeef3cc2675c8bbf079058085fe0": " 2^* = -4, 8, -8 \\text{ according as } m \\equiv 3 \\pmod 4, 2 \\pmod 8, -2 \\pmod 8 . ", "08bc601fee971bf4823d1b75d5e289d4": " p_n x^n+p_{n-1}x^{n-1}+\\cdots p_0 ", "08bce46625100c66af1b4437a3afd1c5": " V_{\\text{max}} = \\pm \\frac{3}{4(\\kappa_0+\\kappa)}[e_{\\text{33}} - 2(1 + \\nu) e_{\\text{15}} - 2\\nu e_{\\text{31}}] \\frac{a^3}{l^3} \\nu_{\\text{max}} ", "08bd716722d5176704191cec94def68e": "g(r,r^\\prime)\\,", "08be0a16219fcd5e1108755dfd1acf40": " \\beta > -0.5", "08be3901b3bc9236982b3ce3999c8905": "tan(\\frac {a}{D})", "08be6b931af173ebda4a85fc8d65ce68": "x \\mapsto x", "08be8677937e8336887409cfc509fc9f": "b_i=\\gamma^2+\\mathrm{sin}^2(\\frac{i\\pi}{n}), (i=1,2,...n),", "08bf08d8952c48c9e8bb08ddf5626cbf": "T=300\\,K", "08bf5c38d31fc03b9e3bfebb088c5bbf": "K = \\frac {\\Delta Y}{\\Delta I})", "08bf620f629deefe2442203f052415f0": "\\mathbf{z}=[x_1\\ q_1\\ x_2\\ x_1\\ x_1\\ 0\\ 1\\ \\sin x_1]^T", "08bf9621bd9962a098052301bac589b4": " H_n^{-1}(\\beta) ", "08bf9852415b519ab9b50a3817f9e035": "R_{\\infty}", "08bfc2eff7da80e42dc3fd78a966d298": "z=z_t\\,", "08c06dc493a3191a3fdb7a101581bf8a": "|\\psi_\\alpha\\rangle", "08c08149c2d5d6f9d6f8b93f717c489e": "\n\\begin{align}\n \\mathbf{F} & = q\\left(\\mathbf{E_1}\\left(x,y,z\\right)-\\mathbf{E_2}\\left(x,y,z\\right)+\\frac{d(\\mathbf{x}_1-\\mathbf{x}_2)}{dt}\\times\\mathbf{B}\\right) \\\\\n & = q\\left(\\mathbf{E_1}\\left(x,y,z\\right)+\\left((\\mathbf{x}_1-\\mathbf{x}_2)\\cdot\\nabla\\right)\\mathbf{E}-\\mathbf{E_1}\\left(x,y,z\\right)+\\frac{d(\\mathbf{x}_1-\\mathbf{x}_2)}{dt}\\times\\mathbf{B}\\right). \\\\\n\\end{align}\n", "08c096871a3243c00055371f939c4e14": "<\\overline{16}_H>16_f \\phi", "08c11a24017ed7c2e3444f1d521216b0": "\\mu=\\frac{D}{k_B T}.", "08c18179c206656b472a78c324d00953": "{{O}}(M{\\cdot}{\\chi}^2)", "08c196210fec44d9878c8f9f67db8a21": "g=\n\\begin{array}{c}\nx \\\\\n\\longrightarrow \\\\\n\\left[\n\\begin{array}{rrrrrrrr}\n -76 & -73 & -67 & -62 & -58 & -67 & -64 & -55 \\\\\n -65 & -69 & -73 & -38 & -19 & -43 & -59 & -56 \\\\\n -66 & -69 & -60 & -15 & 16 & -24 & -62 & -55 \\\\\n -65 & -70 & -57 & -6 & 26 & -22 & -58 & -59 \\\\\n -61 & -67 & -60 & -24 & -2 & -40 & -60 & -58 \\\\\n -49 & -63 & -68 & -58 & -51 & -60 & -70 & -53 \\\\\n -43 & -57 & -64 & -69 & -73 & -67 & -63 & -45 \\\\\n -41 & -49 & -59 & -60 & -63 & -52 & -50 & -34\n\\end{array}\n\\right]\n\\end{array}\n\\Bigg\\downarrow y.\n", "08c1ce8a801d8c0ea8f51f30f03e88fd": "\n\\operatorname{cov}(\\textbf{X},\\textbf{Y})\n=\n\\mathrm{E}\n\\left[\n (\\textbf{X} - \\mathrm{E}[\\textbf{X}])\n (\\textbf{Y} - \\mathrm{E}[\\textbf{Y}])^{\\rm T}\n\\right].\n", "08c20f58365d71b8909771c1dd1f9a23": "S_{\\text{RST}} = - \\frac{\\kappa}{8\\pi} \\int d^2x\\, \\sqrt{-g} \\left[ R\\frac{1}{\\nabla^2}R - 2\\phi R \\right]", "08c2423c8ee336bbd5124362873b2a74": "\\begin{align}\np_{s,0}(z)&= p(z) \\mod \\left(z^{2^{n-s}}-1\\right)&\\quad&\\text{and}\\\\\np_{s,m}(z) &= p(z)\\mod \\left(z^{2^{n-s}}-2\\cos\\left(\\tfrac{m}{2^s}\\pi\\right)z^{2^{n-1-s}}+1\\right)&m&=1,2,\\dots,2^s-1\n\\end{align}", "08c2626783e1a71ff9462110eef97fac": "v\\in T_pM\\,", "08c2749476c6a68ec930bf745ad93f29": " \\delta_y ", "08c28d1701d799446b540cbf8c38b7cd": "s(t)=w(t)*R(t)\\!", "08c2de929cd11b682d6aef555dab2402": "F_4(a, b) = (x \\to x^x)^{\\log_2(b)}(a)", "08c33b31860c198279fe65d253985a3b": "k \\in Z_q", "08c367501a71e8bb4bd723e02823601e": "W(c) = W_+(c) + W_-(c)", "08c39ddf5b344b4efa8b0134b4c34dce": "\na_i \\cong \\frac{\\rho_{it}}{\\sqrt{1-\\rho_{it}^2}}\n", "08c3ef68c8815c26f82d2cf3f6b1655a": "\\frac{1}{T} \\int_0^T Z(t)^4 dt \\sim \\frac{1}{2\\pi^2}(\\log T)^4", "08c422e29bee5503b52e7b047d3e86a2": "C_1,C_2,\\ldots C_n", "08c48687763aba301ed30d1c33d0ea91": "\nF = \\frac{\\alpha}{r^{2}} = \\alpha u^{2}\n", "08c48857f2f798fbf60b3a200ffff49f": "VT^{\\hat{c}_v}/N", "08c48ac0fe367d30b4adb3f9132fa1f2": " B_n \\cap B_{n+1} \\, = \\, \\emptyset \\,\\forall n ", "08c4daf476bf72c27460411bcf8eedba": "\\psi = (x^\\mu,x^a)\\,", "08c4fb03f7aa53e8693a5e9f6e3097bc": "\\alpha_i ", "08c675bff461a17050f942a866b071d0": "X \\to \\mathbb{C}", "08c756222d35d5fd3aa7bbba4cb2b8ee": "\n\\begin{array}{ccc}\n\\text{Classical Maximin Format}&& \\text{MP Maximin Format}\\\\\n \\displaystyle \\max_{d\\in D} \\ \\min_{s \\in S(d)}\\ g(d,s) &=& \\displaystyle \\max_{d\\in D,\\alpha \\in \\mathbb{R}}\\{\\alpha: \\alpha \\le \\min_{s\\in S(d)} g(d,s)\\} \n\\end{array}\n", "08c7863bcb6fd69fb923572118e873bb": "c_\\mu=1", "08c796188adbfd960080a41ac99cde0e": " m=2j - 3", "08c797c3299a67b74dfa852ed16fb9bf": "\\scriptstyle a^2+b^2+c^2 \\,=\\, 1", "08c7b3665204557f7addf4503e5880d6": "L_M=kL_P", "08c7eb5ed8da2aaf4bdbc9d79ceab260": " |AB|=\\frac{|AC||FE|}{|FC|} ", "08c7eca5e4fa5d220a38a438ea19f3c1": "\nh_{\\phi} = a \\cosh\\mu \\ \\cos\\nu\n", "08c8037f3dcab378895a4256bf65c30a": "\\begin{cases}\\mu + \\sigma\\frac{\\Gamma(1-\\xi)-1}{\\xi} & \\text{if}\\ \\xi\\neq 0,\\xi<1,\\\\ \\mu + \\sigma\\,\\gamma & \\text{if}\\ \\xi=0,\\\\ \\infty & \\text{if}\\ \\xi\\geq 1,\\end{cases}", "08c8253419d3e5353ad1b1ed4764f590": "\\Omega (\\text{ })\\,\\!", "08c82a904741b3547cd07bd0133a4b1a": "p'=(R'_1, \\ldots, R'_n)", "08c8527a545b434e1ef1a2bca515f75b": " P^2 \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = P \\begin{pmatrix} x \\\\ y \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} x \\\\ y \\\\ 0 \\end{pmatrix} = P\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}. ", "08c856d8f409003cdee6c9222fa5e399": "(b \\times b) + b + b + a", "08c87a3a9844f1d117a5ebb4730602c5": "\\overrightarrow{F}_{Hp} = +q(\\overrightarrow{v}_p \\times \\overrightarrow{B}_z)", "08c885160457626604ed9c52b2a8f0a4": "( 1.00, 0.00);", "08c88ebb97d05adc97468f81c1fb528b": "\\mathrm{D}_{H} F\\;", "08c8a49d2aa38647e4353bb9efccd38e": " \\frac{d^\\alpha}{dx^\\alpha} x^\\beta = \\begin{cases} \\frac{\\beta!}{(\\beta-\\alpha)!} x^{\\beta-\\alpha} & \\hbox{if}\\,\\, \\alpha\\le\\beta, \\\\ 0 & \\hbox{otherwise.} \\end{cases}\\qquad(1)", "08c8d8c4a9434d723579ed18502c4e7c": " \\mathfrak{der}(A)\\oplus\\mathfrak{der}(B)", "08c91383a07e35c6243eaea2c0ca0f56": "\\frac{\\langle{y,Ly}\\rangle}{\\langle{y,y}\\rangle} = \\frac{\\int_a^b{y(x)\\left(-\\frac{d}{dx}\\left[p(x)y'(x)\\right]\\right)}dx + \\int_a^b{q(x)y(x)^2} \\, dx}{\\int_a^b{w(x)y(x)^2} \\, dx}", "08c9515739b61737ec2867c854042cde": " var(e)=var((I-H)Y)=(I-H)var(Y)(I-H)'=\\sigma^2(I-H)^2=\\sigma^2(I-H) ", "08c954bd6b08eeeeaed0e08fc6821089": "P \\land Q \\land x \\le P", "08c9ac1bf5d350fa6e3d49cd32fc3fac": "{\\mathcal L}_{xy}^0", "08c9c52ff003e23c56c3c4d27f19baa3": "\\{x \\in B \\mid \\phi\\}", "08c9e1162747a6a960f27c76d067aea1": "\\scriptstyle ( p^2 - q^2,\\, 2pq,\\, p^2 + q^2 )", "08c9fd72ae28c76aed5805c3549437b1": " \\phi_< = -\\frac{3}{\\kappa +2} E_{\\infty}r \\cos \\theta \\ ,", "08ca46b884a7d9c3da24a279e3a7d043": "f(n)\\sim g(n)\\!", "08ca9adbc56c52fa20b365d678e052b7": "E = \\sum_{n=-\\infty}^{\\infty}e[n]^2", "08cabaffce0531f5356640d2eb4235ec": "\\sin(\\delta_1+\\delta_2+\\delta_3\\,)", "08cb4fb03016be77d395704a252db69b": " [x_1, \\; x_2, \\; x_3, \\; x_1^2, \\; x_1x_2, \\; x_1 x_3, \\; x_2^2, \\; x_2x_3, \\; x_3^2] ", "08cb64a670be8e46fa99d9df94b9802a": "\\Omega_{B}", "08cb7a1538984812ad8215c44d24e4e9": "\\frac{\\pi}{4} \\ (45^\\circ)", "08ccbd2b322d70ceb83bea7412b75e9c": " \\sum_{k=0}^\\infty T^k ", "08ccc306d3620da015dc0d127b388cc1": "\\rho(S(\\hat{D}),S(D))\\le\\epsilon", "08ccd2f0ad5b24a408c81fced4e598a4": "{(x+y)/2+z\\over2}\\ne{x+(y+z)/2\\over2} \\qquad \\mbox{for all }x,y,z\\in\\mathbb{R} \\mbox{ with }x\\ne z.", "08ccd5825593984505f863b96864bc35": "u(1)_A\\oplus u(1)_B \\oplus u(1)_C", "08ccefdecc07c994e16cac8520b58554": "\\frac{1}{2}\\hbar \\omega_{q}", "08cdcc437830fe62c1399d3c5072678f": "logK_a + logK_b = logK_w", "08ce00ce1306d33f0f93b7a0060e8e85": "H_{k+1}=H_{k}+\\frac {(\\Delta x_k-H_k y_k)(\\Delta x_k-H_k y_k)^T}{(\\Delta x_k-H_k y_k)^T y_k}", "08ce088f83959c520215c592a78f02ec": "\\int_{0}^{\\infty} \\frac{\\sin ^{2}px}{x^{2}}\\ dx=\\frac{\\pi p}{2}", "08ce26bd38e03dfedf2ed49be057bd69": " A,B \\in \\mathcal{C}", "08ceb334207d72356e9e6909e1c14dc4": "\nS_{x_{B-l}}=\\mathbb{I}\\otimes S_{x_l}\n", "08cf1b851c60ce9b51bd7b140373baae": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 2}{40 \\choose 1} \\end{matrix}", "08cf3af0c3887289761a35076df9adf3": " {P V \\over {\\operatorname{constant}}} = T = {{2.50 \\times 10^6 * 100} \\over {3.33 \\times 10^5}} = 751 ", "08d0526ee3dfd44324b874909c5f655f": "I(t) := \\overline{g}(0,y,0)", "08d05900d6e0b782ea2e61a4d8eff2e1": "\\{ H (N) , H (M) \\} = C (K)", "08d07705270c0dbe8b96f70df98a89b6": "1 \\leq p \\leq 2", "08d095f730c930b731cb4773868b7355": "h = T^{1/\\alpha}", "08d0c59c593458184f0119269d48af3f": "Z^*_{n^{s+1}}", "08d0df5077e5dfb76cf50b5aecc1d6a3": "x_T = \\sum_{i\\in T}x_{\\{i\\}},", "08d0f85eefa9adb6af147f8109fe85a8": "c_o = \\frac{1}{\\alpha_o}\\left[C_a \\alpha_a + C_b \\alpha_b \\left(1 - \\alpha_a\\right)\\right]", "08d17855acddd72fa221f533508acae2": "\\phi=L I \\ ", "08d1afc218ec4f314312137ef8a8193c": " \\mathcal{E} ", "08d23c49ab9b985b0c3672033948c724": "k_0,k_1,k_2", "08d2ea5371ea545a6007314da073a4a9": " \\left(S(0),E(0),I(0),R(0)\\right) \\in \\left\\{(S,E,I,R)\\in [0,N]^4 : S \\ge 0, E \\ge 0, I\\ge 0, R\\ge 0, S+E+I+R = N \\right\\} ", "08d2f62b50b798eb9750f76c060727de": "\\tfrac{\\vec x(t_{n+1})-\\vec x(t_n)}{\\Delta t}", "08d335a1bfafde67d9640c6b4561ba3b": "V_S^n", "08d352e23960b64a603f36ea901d63a3": "m(X)=\\text{max}(X)", "08d42eb1eec575bbc358c513432853bd": " h_x ", "08d4418ed0d2874a95b923b8d6685fe6": "A_\\lambda := \\bigcup_{\\alpha < \\lambda} A_\\alpha", "08d457e8ccd5d50d4647ef64ee79694a": "Z = \\frac{\\zeta^N}{N!}.", "08d47bee5cd96f3c0a4b7f1c50ccb441": "\\!-\\!\\left(1\\!+\\!\\frac{\\nu}{2}\\right)\\psi\\!\\left(\\frac{\\nu}{2}\\right)", "08d4bac20c0b44fd65b7146822121760": "\\Gamma(\\tfrac14)", "08d4c589a98368dfe97e2102f0c7ebdc": " A_{fcc} = \\frac{4J_{ex}S^2}{a}", "08d4e8fa01149867c4a3b1fc8ef0519e": "\\mathrm{Nu} = 2 + 0.6\\, \\mathrm{Re}^{\\frac{1}{2}}\\, \\mathrm{Pr}^{\\frac{1}{3}}, ~ 0 \\le ~ \\mathrm{Re} <200, ~ 0 \\le \\mathrm{Pr} < 250", "08d50ae005bfbbe2b027c14c1780a45b": "\\dot{r} = \\frac {dr} {d\\theta} \\cdot \\dot {\\theta}", "08d581da776c2a7cb178e6cdc514a4b4": "\\alpha_1 = 1 - (1 - \\alpha)^{1/n}.", "08d58d7d444e14ae1eb15fd4569e3256": "\\vec{\\chi}", "08d5934ad37a87e6335ee6438c52487a": "\\frac{\\delta Z}{\\delta N}=0=\\int \\left.\\frac{\\delta I[g_{\\mu\\nu},\\phi]}{\\delta N}\\right|_{\\Sigma} \\exp\\left(-I[g_{\\mu\\nu},\\phi]\\right)\\,\\mathcal{D}\\bold{g}\\, \\mathcal{D}\\phi", "08d620536156da2a2bfe95a74d5ef516": "\\frac{\\partial}{\\partial s}p(x,s)=-\\frac{\\partial}{\\partial x}[\\mu(x,s)p(x,s)] + \\frac{1}{2}\\frac{\\partial^2}{\\partial x^2}[\\sigma^2(x,s)p(x,s)]", "08d62a4ee85116546a8eecf9f4ecf747": "f'(x)=\\lim_{h\\to 0}{f(x+h)-f(x)\\over h}.", "08d682768c9c10f0c3f69c4a2c91a6e6": "K_Y(t)=\\mbox{ln} E[e^{tY}]=\\mbox{ln} E[E[e^{tY}|N]]=\\mbox{ln} E[e^{NK_X(t)}]=K_N(K_X(t)) . \\,", "08d69c15c33bd9c730364b5f6518971b": " L_1 = T_1 - \\frac{1}{2} \\left( (V_R)^2_1 + (V_A)^2_1 \\right) ", "08d6bb270ed44924f2dc62751f4baa42": "\\{n1, n2\\}", "08d6c0c272a07fe80bbdbf7df87c50ab": "E_0 = \\omega A_0", "08d6ce9d113abdaa391d55c2f666861a": "(\\ln x, x),", "08d6d8834ad9ec87b1dc7ec8148e7a1f": "PQ", "08d6f8414f9de6c65eeb2fc47ac3e0f8": "p^*: \\check{H}^*(P_\\delta,\\partial P_\\delta) \\to \\check{H}^*(E_\\delta, \\partial E_\\delta)", "08d7697676392eb85a784c4a70455b1c": "\\sum_n \\mathbb{P} \\left(|X_n - X| > \\varepsilon\\right) < \\infty,", "08d7e6cbce5d42754894ff52cbdb394a": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = \\boldsymbol{R}\\cdot\n \\left[\\cfrac{d}{dt}\\left(\\boldsymbol{R}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{R}^{-T}\\right)\\right]\n \\cdot\\boldsymbol{R}^T \n = \\boldsymbol{R}\\cdot\\left[\\cfrac{d}{dt}\\left(\\boldsymbol{R}^T\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{R}\\right)\\right]\n \\cdot\\boldsymbol{R}^T \n", "08d802e12ff6931b47bc2378f02c10ff": "K\\le \\tfrac{1}{2}\\sqrt{(a^2+c^2)(b^2+d^2)}", "08d8044fea0d480e139acaf126e1773a": "\nV_C \\ = \\ G_{C}V_{in} e^{j\\phi_C}\n", "08d82c4f51a5000f45b30c9daf559333": "A \\;=\\; \\int_{-r}^r 2\\sqrt{r^2 - x^2}\\,dx \\;=\\; \\pi r^2", "08d83ed8382ea113854c1f7d51ce90ee": "\\big. \\frac{\\Delta Q}{\\Delta t} = \\frac{A\\,(-\\Delta T)}{\\frac{\\Delta x_1}{k_1} + \\frac{\\Delta x_2}{k_2} + \\frac{\\Delta x_3}{k_3}+ \\cdots}.", "08d84a38b4e91bc582967a44cf358ede": "d(1)=0", "08d901ef964a996c92dafaea0b1e35f6": "n(0.5;H) \\approx 1.1774 \\sqrt H. \\,", "08d94cb53e35bca6273ede095c2609a1": "F \\in S", "08d971745912cb478423a7848b89a938": "Y = Z^{-1}= \\frac{1}{R + jX} = \\left( \\frac{1}{R^2 + X^2} \\right) \\left(R - jX\\right) ", "08d9f577d9d8c84e7b863e54ce8070d6": "X_n=\\frac{(1+r)}{(1+r-2rp)}", "08da3c6c4e61f529ce1690ff8bba7e3a": "x=\\sum_{i=0}^n10^{n-i}a_i/10^n=\\sum_{i=0}^n\\frac{a_i}{10^i}", "08da9695cd4a3c791da5473a5b4a5a53": "\\psi(\\Omega)^{\\psi(\\Omega)^{\\psi(\\Omega)}}", "08dab8b0af8f3513821d06101bc32c88": "\\bigwedge L=a_1\\land\\cdots\\land a_n", "08daf53dede6ad2e156adddcc66f51cd": "\\sum_{k = 0}^{\\infty} \\frac{1}{16^k} \\left( \\frac{4}{8k + 1} - \\frac{2}{8k + 4} - \\frac{1}{8k + 5} - \\frac{1}{8k + 6}\\right)=\\pi\\!", "08db150fdf3a2ba97c9aa41f12fe4e9b": " f(u) = 0", "08db3e889f5b99601143bdab5ad9c09a": "\\gamma = \\frac\n{\\left|P_1-P_2\\right|^2 \\left(P_3-P_1\\right) \\cdot \\left(P_3-P_2\\right)}\n{2 \\left|\\left(P_1-P_2\\right) \\times \\left(P_2-P_3\\right)\\right|^2}", "08db52dff2b122f64ac22b1bf8455ca9": "\\binom{p^2}{i}", "08db6ba19dc5c27b3903d4c237b662f0": "\n0 \\leq \\lambda \\perp x_1 \\geq 0\n", "08dbbb1365a9f1fd8616be57dd6b4d46": " \\frac{\\partial \\xi}{\\partial t} + V \\cdot \\nabla\\eta - f \\frac{\\partial \\omega}{\\partial p} = \\left( \\xi \\frac{\\partial \\omega}{\\partial p} - \\omega \\frac{\\partial \\xi}{\\partial p} \\right) + k \\cdot \\nabla\\omega \\times \\frac{\\partial V}{\\partial p} ", "08dbf67ca14ddc19fc2dc46bdf70bf3e": " = r V_a + \\theta V_d,", "08dc1a0308ff0b2ed55b865f9ab9a288": "\\partial_\\mu\\partial_\\nu E_n", "08dc2c67c879d6bc197b16bd888ae776": "P_K(p^0,p^1,u)=\\frac{C(u,p^1)}{ C(u,p^0)}", "08dc4994bf9b90f91834c1122e5d1ee8": "U(\\theta, \\phi)", "08dd266487113c297ab1a73113b8b7ba": "\n-\\frac{\\partial v(p^0,w^0)/(\\partial p_i)}{\\partial v(p^0,w^0)/\\partial w}=x_i (p^0,w^0),\ni=1, \\dots, n.\n", "08dd6653d00fcf3035261c77857a5574": "{}_S", "08dda92627e6174380bae79bf9e78876": "t\\begin{Bmatrix} r, q , p \\end{Bmatrix}", "08ddeffa622d13c5c5cb4502ab3444bf": " \\mathbf{ABCD} = ((\\mathbf{AB})\\mathbf{C})\\mathbf{D}=(\\mathbf{A}(\\mathbf{BC}))\\mathbf{D}=\\mathbf{A}((\\mathbf{BC})\\mathbf{D})=\\mathbf{A}(\\mathbf{B}(\\mathbf{CD}))=(\\mathbf{AB})(\\mathbf{CD}) ", "08de2608d958ae73328adbf4d7149456": "f: X \\rightarrow \\mathbb{R} ", "08de4748517e2bf16fb6f251f2e255f3": "\\mathrm{NA} = n \\sin \\theta,\\;", "08df396a34293d78ce62abd5a8d01d25": "\\varphi (n)=\\mathcal{F} \\left \\{ \\mathbf{x} \\right \\}[1] =\\sum\\limits_{k=1}^n \\gcd(k,n) e^{{-2\\pi i}\\tfrac{k}{n}}. ", "08df6c446109617e11f8af593a09a12b": "\\scriptstyle \\psi^\\dagger", "08dfe7bed8bef41a32d8495aa7a0ddef": "( A^4 + A^3 + A^2 + A + I )b4 + 63", "08dfea5322f1c1491fcdbd43ee5b762e": "w(C_1\\mid C_2) \\geq \\min \\{ 2w(C_1) , w(C_2) \\} ", "08dfebdcd4a178d5446dfcc32049f5c4": "S_D(\\lambda)=S_0(\\lambda)+M_1 S_1(\\lambda)+M_2 S_2(\\lambda)", "08e004525d0a192ccee7fa267bc3b937": "B=-V_{XY} V_{YY}^{-1}.", "08e03148e101fdabf93fbe0c12664581": "\\tilde g:=\\lambda g_0 + (1-\\lambda)g_1,\\qquad \\lambda\\in [0,1],", "08e0cf27ef94307f45431d40de5505fc": "D^{2}=\\{z:|z|<1\\}", "08e0fa8ae305e4b7d100a092cc27378b": "dV = \\left(\\frac{\\partial V}{\\partial T}\\right)_{P}dT+\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}dP\\,", "08e12ee7ce42bda0ba715ded03878d99": "q \\times p", "08e13b1079166a55617771f7b334c6cd": " \\begin{bmatrix}\n \\cos\\delta\\cos h\\\\\n \\cos\\delta\\sin h\\\\\n \\sin\\delta\n\\end{bmatrix}= \\begin{bmatrix}\n \\sin\\phi_o & 0 & \\cos\\phi_o \\\\\n 0 & 1 & 0\\\\\n -\\cos\\phi_o & 0 & \\sin\\phi_o\n\\end{bmatrix}\\begin{bmatrix}\n \\cos a \\cos A\\\\\n \\cos a \\sin A\\\\\n \\sin a\n\\end{bmatrix}", "08e19bf3ab3ba9d80091f6f4c84d1ce4": " x = \\lambda \\sinh\\left( \\frac{ 1 }{ \\sigma } \\Phi^{ -1 }( U ) - \\gamma \\right) + \\chi ", "08e1edd9d4a32a6033d51f585d081b02": "\\omega=180^{\\circ}", "08e1f9820e9a919ef61eeaea7cc71eca": "F^\\times/ F^{\\times 2}", "08e23c9c42a120c5b625a0453617f930": "\\beta = {b+a \\over 2}", "08e27ef958c6435f3ed4d3ef20f8cb80": "\\beta(x) = \\frac{1}{16\\pi}\\int_1^\\infty u^{-3/2}e^{-xu}du", "08e2b4fddba3995e9853afd85c02088d": "\\int\\ln (ax + b)\\;dx = \\frac{(ax+b)\\ln(ax+b) - ax}{a}", "08e2c000fe3961fe1d9c8e52c87fdd99": "\\gamma (n)", "08e2dbe2f7a3e5f55706d9f3598f786c": " \\sin^2 \\theta \\!", "08e2e7391e60bde55864396663d8b388": " \\phi^0 = A \\left( \\begin{array}{c} \\Psi_1^0 \\\\ \\Psi_2^0 \\end{array} \\right) ", "08e3447e2a35226511816695287a9376": "u^2", "08e364f63399e5ab2d1aeef34e61a3b3": "{\\Bbb M}", "08e383fdce48b725492a61dc8b59a0e9": "\\neg(A \\lor B) \\to \\neg A \\land\\neg B", "08e3952227151a571c6047d62029cea6": "\\begin{align} \\pi_1 &= E/As \\\\ \\pi_2 &= \\ell / A. \\end{align}", "08e3baeb8ba48e236aaacd9da759f946": "f(\\phi)", "08e41fb36dc83be86be1dd34396f9c27": "\\cot\\frac{3\\pi}{20}=\\cot 27^\\circ=\\sqrt5-1+\\sqrt{5-2\\sqrt5}\\,", "08e484af580944a18f3169b488733888": "|\\mu| \\,", "08e513f7ded212e7c5fcaf7f9d44fe5f": "L: \\{0,1\\}^n \\rightarrow \\{0,1\\}", "08e514d2648a9a22a96ebd8241799f70": "\\mu^*=\\mu", "08e5a2b68e1046fae8bebff11af3aacb": "\\mathbf{B}' = \\mathbf{B} - \\frac{1}{{c_0}^2} \\mathbf{v} \\times \\mathbf{E},", "08e60a0397275b61f281a6eaa3d8a744": "-\\pi\\le \\arccos(u) \\le \\pi", "08e60b1b7f79153557b39e185fbcf162": "f(x)=e^x", "08e6197d77729f2c31ddf3127eda43ab": "C_{\\beta J}^{\\;\\;\\; K} e^\\alpha_I e^\\beta_K", "08e64f7a0a4d807bfac9e2f1158dd22c": " \\lambda_{\\alpha } \\,\\!", "08e65e29a9040aa76d13e56a74406807": "\\mathrm{Distance} = \\mathrm{Speed} \\cdot \\mathrm{Time}", "08e67c9ac8c32eb0b951448037073274": "[\\![x_1]\\!] \\in [\\![\\mathsf{T}_1]\\!],~[\\![x_2]\\!] \\in [\\![\\mathsf{T}_2]\\!],~\\ldots,~[\\![x_n]\\!] \\in [\\![\\mathsf{T}_n]\\!]", "08e6a0743b311f3036b4b17558812e21": " A = \\int \\lambda \\, d \\operatorname{E}_A(\\lambda),", "08e6f27bf45594a08739a6c832914f49": "\n\\begin{align}\nF & = \\frac{ \\text{lack-of-fit sum of squares} /\\text{degrees of freedom} }{\\text{pure-error sum of squares} / \\text{degrees of freedom} } \\\\[8pt]\n& = \\frac{\\left.\\sum_{i=1}^n n_i \\left( \\overline Y_{i\\bullet} - \\widehat Y_i \\right)^2\\right/ (n-p)}{\\left.\\sum_{i=1}^n \\sum_{j=1}^{n_i} \\left(Y_{ij} - \\overline Y_{i\\bullet}\\right)^2 \\right/ (N - n)}\n\\end{align}\n", "08e745a2e1847d10cb104681ed0ff38f": "T = R_{01}R_{02}\\cdots R_{0N}", "08e757e7a55156f0010fadd95bbf1781": "\\tfrac {1}{12} \\pi^2 - \\tfrac {1}{2} \\ln^2 2 \\,", "08e76895ab552c92a9cd75fb45b9c754": " \\nabla^2 \\phi = 0", "08e76d529ee95a53711040601988d621": "Z=\\sum_{n=0}^{\\infty } \\frac{(644n+41) \\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^35^n{72}^{2n+1}}\\!", "08e7930b39961cdad59c78701ed88402": "V = \\left\\{(g_i, h_j): 1\\leq i\\leq n, 1\\leq j\\leq m\\right\\}", "08e7f6feafd6d6c18b0ff35c21d29897": "\\psi \\varphi \\neq \\empty ", "08e8057498083c0f8b73ec308f18e43a": "\n\\frac{\\mathrm d \\boldsymbol{H^\\prime}}{\\mathrm d t} = -\\nabla \\boldsymbol{H^\\prime} \\cdot \\nabla \\vec v \\qquad \\boldsymbol{H^\\prime} (t=0)=\\boldsymbol{I}\n", "08e80def44c6cdec7c177f66635a0525": "s_1 = \\left[0, \\sqrt{2}\\right]^T, \\quad s_2 = \\left[-\\sqrt{3\\over 2}, -\\sqrt{1\\over 2}\\right]^T, \\quad s_3 = \\left[\\sqrt{3\\over 2}, -\\sqrt{1\\over 2}\\right]^T", "08e8326c8e0191eb1b8bf451aa73d72d": "\\mathbf{I}=\\sum_{i=1}^3\\sum_{j=1}^3 I_{ij}\\mathbf{e}_i\\otimes\\mathbf{e}_j.", "08e8441a508c8088981c7dfc9e15d5a8": "I_D \\propto V_{GS}", "08e884b06934b8b0bffe39483a339a86": " {\\partial \\rho \\over \\partial t} = - \\nabla \\cdot \\mathbf{J}. ", "08e8a9929b064bc2e3ac205901d12970": "r_{i+31} = y_{64+2i+1}", "08e8b5adf68cce4d461bb42cd893fc94": "\\ \\psi (x, t) ", "08e8b5e2173db295038b46d2c914b352": "x^2 \\cdot g(x),\\quad (x^2+1)\\cdot g(x),\\quad (x^2+x)\\cdot g(x), \\quad (x^2+x+1) \\cdot g(x).", "08e8c77df8d74a752f8c1c5ebf78ab88": "B_E:X_E\\to X^*_E", "08e90359c2a70d26e2a5a0f8caa48260": "X \\Rightarrow Y", "08e91d1ed178c767ef40109a1da1d511": "n_g=\\frac{c}{v_g}", "08e937fa71b77a8dc61e97d3b8abbad8": "\n{\\mathbf \\sigma} = (\\sigma_x , \\sigma_y , \\sigma_z) \n=\n\\left[\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix}\n, \n\\begin{pmatrix}\n0 & - {\\rm i} \\\\\n{\\rm i} & 0\n\\end{pmatrix}\n, \n\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1\n\\end{pmatrix}\n\\right]", "08e9eea18e247c6df9837a00b859452e": "DPA_{n}", "08ea2a5d8434785c5d721f12956f2c15": "\\omega_c = q e B / 2 \\pi m", "08ea6f30ade95e39c2d19cff292b14fa": "f(a) \\cdot h \\cdot (1/2)", "08eafe38f780b3fdc0cd19d06299b793": "E_0=E_k", "08eb318d1ddf22c353da60249b9bc9f0": "g(\\psi)", "08eb4315e76231f7296b0834191aa7b5": "p(\\theta, \\psi)", "08eb9f2a5d8489a805a9c51efcd74352": "\n \\sqrt{T}\\Big( \\tfrac{1}{T}\\textstyle\\sum_{t=1}^Tz_t - \\operatorname{E}[z_t]\\Big) \\ \\xrightarrow{d}\\ \n \\mathcal{CN}(0,\\,\\Gamma,\\,C),\n ", "08ebe2b53b1ea788e252b02e21e2c786": "\\left(p(e_0), \\ldots, p(e_{n+1})\\right)", "08ec3075a527c2b93364d1db50f6aab0": "\\Phi_{Y,X}(f) = g : Y \\to GX", "08ecbf4e98447b699f181a3c9a5357ce": " \\frac{U}{t} \\sim \\nu \\frac{U}{y^{2}}", "08ecc0e3d5b5840e9d211abb2ff652f3": "H(Y,p,s) \\geq H(X,p,r) + \\lambda ||X-Y||", "08ecc318fdf6cc03e3cb7b7e1870479b": " P_{j},", "08ece0b3ba7be4d6f630cc745e32d610": " \\frac{-\\sqrt{b}+a e^{\\tfrac{(-1+a)^2}{4b}} \\sqrt{\\pi} \\operatorname{erfc}\\left(\\tfrac{-1+a}{2\\sqrt{b}}\\right) }{a^2\\sqrt{b}} ", "08ecfc1ed8d5360233d57617c89c6442": "P(T|H)", "08ecfdd539947e2df450f0c1226bf8cf": "\n\\Phi_n((-1)^{\\frac{n-1}{2}}z) = C_n^2(z) - nzD_n^2(z).\n", "08ecff33bb1c40c3d72cfe1fc7da929b": "(p,\\gamma)", "08ed0c5ef3a91bdb75903bc3b7ec2189": " X = \\begin{bmatrix}0 & X_{12} & X_{13} & \\cdots & X_{1,r-1} &X_{1r}\\\\ & 0 & X_{23} & \\cdots & X_{2,r-1} & X_{2r}\\\\ & & & \\ddots & \\\\ & & & \\cdots & 0& X_{r-1,r} \\\\ & & & & & 0 \\end{bmatrix}", "08ed3f9b2550f875b183b4c1765e2b85": "{x^2 +x -1 = 0}", "08eda79b1e3a8f2682f6c3063fab9638": "p \\rightarrow \\Box \\Diamond p", "08edd9e5c20acddea49f699e8b0970f8": "\\xi_0(z)=\\pi^{-1} (1+|z|^2)^{-2}", "08edecbef79ec2cdca39e715813389ec": "B' \\subset k", "08ee50ad97f51e6782fdd8f3593453e1": "\\langle A(a) B(b) \\rangle = \\langle A(a') B(b) \\rangle = \\langle A(a') B(b') \\rangle = \\frac{1}{\\sqrt{2}}", "08ef56c3ed484b26a592570b29a8f1d8": "x = \\frac{p}{2} + \\sqrt{\\left(\\frac{p}{2}\\right)^2 - q}", "08ef5a82b09cbeaea8d09b50aa9d2076": "^{\\;}H(\\xi )=H( c(\\xi ,\\tau ))", "08ef9e22b9761a2996927ae79ea7cc1c": " I_1, J_2, J_3", "08efcbca510dcc7213342e34cdb3ba1d": " 0 \\le y \\le \\pi \\, ", "08f01b23c07ab3c3d3375bb54a7fb5b1": "A + B \\leftrightharpoons AB; K_{AB}=\\frac{[AB]}{[A][B]}", "08f04976d008faca6991d0f8c07dd154": "O(N_{k,n})", "08f056072a1eec09f51282bcdf204126": "U_{-}=\\bigcap_{n\\ge 0}\\alpha^{-n}(U)", "08f05db3265da9ffc7132056f15a212f": "\\overline{(z + w)} = \\overline{z} + \\overline{w} \\!\\ ", "08f0663a778138bd3d254e0ac4b8d901": "\\frac{(P \\leftrightarrow Q)}{\\therefore (Q \\to P)}", "08f0aa58402bc9ed27aa92b910bf91e5": "\\partial\\Phi/\\partial{y}", "08f0cd3e07ea8522b812b23e197673a3": " X = 1 ", "08f0e891b1fe8a269be49d376002fa33": "\\mathcal{H}=\\operatorname{span}\\{\\phi_i\\}", "08f11a635e5d15f16491f8e112cfa934": "\n\\frac{\\partial}{\\partial t}(\\nabla^2 \\psi)\n + (\\nabla \\times \\vec \\psi) \\cdot \\nabla(\\nabla^2 \\psi) = \\nu \\nabla^4 \\psi", "08f1245e8b0e04c1c0b8dcaea0f98a30": "f(x,y) = \\left( 1.5 - x + xy \\right)^{2} + \\left( 2.25 - x + xy^{2}\\right)^{2}", "08f1447d656403acec8fa6758b80ce07": "(\\vee,0)", "08f1503414f0c9303c9f7a17d0bf956e": "N(a + b \\sqrt{-5}) = a^2 + 5 b^2 ", "08f16ff29dca37546da4e3616abfcf14": "\\gamma_{x}^{+} := \\{\\Phi(t,x) : t \\in (0,t_x^+)\\}", "08f17254282d7207a708c0966a83803a": "dl^2 = e^{-\\lambda(r)}{dr^2} + r^2d\\theta^2 + r^2\\sin^2\\theta d\\phi^2 \\,", "08f1ae61806daf3ff87a5bfaf55ee0c4": "\\frac{\\partial^2F}{\\partial x^2} = \n\\lim_{\\epsilon \\rightarrow 0} \n \\frac{[F(x+\\epsilon)-F(x)]+[F(x-\\epsilon)-F(x)]}{\\epsilon^2}.\n", "08f2053209f10b4d7478efc6d7e86fbd": "\nr_{upb}=\\frac{M_1-M_0-1}{\\sqrt{\\frac{n^2s_n^2}{n_1n_0}-2(M_1-M_0)+1}}.\n", "08f2266fab746e856bb9d7cf8bb426eb": "\nf(z) = \\sum_{k=1}^\\infty a_kz^{\\lambda_k} = \\sum_{n=1}^\\infty b_n z^n\\,\n", "08f28ccc584b23779d02a87c7dd66bae": "\\hat{S} = \\hat{\\sigma}_+ +\\hat{\\sigma}_-", "08f28e7f4d4560325c62e65007df8c02": "\\begin{align}\n & z\\left( {x_1 \\,\\,x_2 } \\right)\\,\\,\\,\\, \\approx \\,\\,\\,z\\left( {\\bar x_1 \\,\\,\\bar x_2 } \\right)\\,\\,\\, + \\,\\,\\,\\,{{\\partial z} \\over {\\partial x_1 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_2 }}\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\,\\,\\, \\\\\n & \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, + \\,\\,\\,{1 \\over 2}{{\\partial ^2 z} \\over {\\partial x_1 \\partial x_2 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\,\\,\\, + \\,\\,\\,{1 \\over 2}{{\\partial ^2 z} \\over {\\partial x_2 \\partial x_1 }}\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\left( {x_1 - \\,\\,\\bar x_1 } \\right) \\\\\n & \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, + \\,\\,\\,{1 \\over 2}{{\\partial ^2 z} \\over {\\partial x_1 \\partial x_1 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\,\\,\\, + \\,\\,\\,{1 \\over 2}{{\\partial ^2 z} \\over {\\partial x_2 \\partial x_2 }}\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\end{align}", "08f2b9e70911eb2d15fa8370c13dfc3f": "P^S=P^D=P>0", "08f2dfea71c4a47fc51caaedb6300a23": " \\Delta\\lambda \\approx \\Delta \\theta \\left({\\partial\\Theta\\over\\partial\\lambda}\\right)^{-1}", "08f330a24acd9b122e68adc7c6387ef5": " z \\mapsto z^d + c.\\ ", "08f334f6cae066f8b601b73dc24fea15": "\\tilde{f}(\\lambda)=\\int_0^\\infty F({a^2+a^{-2}\\over 2}) a^{-i\\lambda}\\, da= \\int_0^\\infty F(\\cosh t)e^{-it\\lambda} \\, dt.", "08f3643f0984f22493e11d0f8c464f13": " \\frac{(p_{02})_{actual}}{p_{01}} = (1+ \\frac{\\eta_{stage}\\delta (T_0)_{isentropic}}{T_{01}})^\\frac{\\gamma}{\\gamma-1}\\,", "08f41e2b56730d87f1232d525303ba14": "vp", "08f4283df511534554d8ce82cb0d0941": "\\pi '\\,\\!", "08f429d0b28f1b961a3851966f2b6fc0": "t \\in [0, T]", "08f445ca40c14a51e65bad8c914d838b": "\\frac{\\mathrm{d}f}{\\mathrm{d}x} = 2 x \\neq \\frac{\\partial f}{\\partial x} = y = x", "08f46744671674fe1bf17fdd58382347": "[0, 255]", "08f4769d05723f8c9116f1368697368b": "\\bar z = a - bi", "08f4a2c8e617f89fc58744e072773f2a": "u\\Vdash A[e]", "08f4ad80a2083e76ed11bdbd090de249": "E_a", "08f4b86a3b9ffd5604ee625a45321733": "\\overline{C_2 P_n}", "08f514122fefcde5aa6d8972c196a49d": "v_{j,k}", "08f583fc3516e68625a5f2cdeb7fb21a": "BP=\\begin{pmatrix} I_r & G \\\\ 0 & 0 \\end{pmatrix}", "08f5aa8ca129fa89172b53a52f8df83a": "\\forall\\beta.\\beta\\rightarrow\\ \\mathit{int}", "08f5af0d3874b9d9388108ed70fa0102": "E_k", "08f5c03968fa3794fa9f9c09a43bf37f": "U^n = [-1,+1]^n", "08f5cb9da1875f9fc5d801ff4bc687d5": "A_{xx}", "08f5cf5212a6cc3d28fa40a8c49a7be9": "\\displaystyle \\zeta(t) = g_t(\\gamma(t)) ", "08f5f837cd0abcc5b245418c1c56fdba": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1=x^2_2\\left[\\tau_{21}\\left(\\frac{G_{21}}{x_1+x_2 G_{21}}\\right)^2 +\\frac{\\tau_{12} G_{12}} {(x_2+x_1 G_{12})^2 }\\right] \\\\\n\\\\ \\ln\\ \\gamma_2=x^2_1\\left[\\tau_{12}\\left(\\frac{G_{12}}{x_2+x_1 G_{12}}\\right)^2 +\\frac{\\tau_{21} G_{21}} {(x_1+x_2 G_{21})^2 }\\right]\n\\end{matrix}\\right.", "08f6708c744a72a9ee3275f804894fb5": "a_{5}+b_{5}+c_{5}=2a_{1}", "08f6b46a4d42bcd73f38312383ad0604": "\\varphi=2\\cos(\\pi/5)=2\\cos 36^\\circ\\,", "08f6beca8f2a7c87668da3283c059dec": "\\textstyle N^x", "08f70ccac8039a54202c8369fd67c020": "-\\lambda-e^{-\\lambda}=0.\\,", "08f722f992cb9d089321e018b1a0dba1": "G_\\infty\\simeq 20G_N", "08f73e2aec7961927cffbd6e4f119531": "\\blacksquare\\,", "08f7c2c2243e386a631d31ffbc332451": "c_1,c_2\\in\\mathbf{C}", "08f7c7586e0cd494d7649321d99680af": "\\lim_{Z,Y \\to 0}Z_\\mathrm i=k", "08f8407d9559edaedba4edf49fcf5a87": "Z + (1+Z)^2 t + (1+Z)^3 t^2 \\ . ", "08f856dbd85b3ced877a379cb806013b": "G_{vv} = C_{xy} G_{yy}", "08f88b18a8abf049883523c51807ff25": "s = r_0 \\int_{t_0}^{t} v/r \\sin \\theta\\, dt", "08f8e3f34408a8d9d0b3670ad56f3d22": "\\gammaz_j;", "08f940ddfad6c62491b79a3b697f48cc": "(k-1, k)", "08f96b806f2515d0673fb8819ede102c": "\\operatorname{tr}(T_a) = 0 \\,", "08f9c9d30750bf03ac5ff958956449c6": "H[i]", "08fa46a40b282211210ef4f3f2e38f69": "\\textstyle P({N}(b(o,r))=1\\mid o)", "08fa75039cafb2eeafa027d6384114e9": "\\frac{\\tau_b}{(\\rho_s-\\rho)(g)(D)}=f \\left(Re_p* \\right)", "08fac7bcdbacfa0e6696833e3192e2a2": "x^i,\\; i=0,1,\\dots", "08fad96e90651b7c9df05a4cb197ca09": "n_\\Sigma^2 \\cos\\theta_\\Sigma \\left (\\sin\\theta_\\Sigma d \\theta_\\Sigma d \\varphi \\right )=n_S^2 \\cos\\theta_S \\left (\\sin\\theta_S d \\theta_S d \\varphi \\right )", "08fadeb8b0203c165c724d1fbbeb89e5": "\nS = \\sqrt{(1+(p_{1\\cdot}+p_{\\cdot 1})(R-1))^2 + 4R(1-R)p_{1\\cdot}p_{\\cdot 1}}.\n", "08fb08286daf9c5c7d2e9e1203294e52": "\\mathbf{F} = \\mathrm{d}\\mathbf{A}\\,.", "08fb1d4a3481d9f764119ebdf36fe7a4": "x - x \\wedge y \\wedge x", "08fb2f24befe571d4e697653bc136e1d": " f(z)=\\frac1\\pi\\iint_{|\\zeta|<1}F(\\zeta)\\frac{dS}{(1-\\overline\\zeta z)^2},\\quad |z|<1. ", "08fb4ce0b18fafadf98deb10acb9a10e": "\n\\begin{alignat}{8}\n\\zeta(s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\\cdots& \\\\\n2\\cdot2^{-s}\\zeta(s)&{}={}& 2\\cdot2^{-s}&& {}+2\\cdot4^{-s}&&{} +2\\cdot6^{-s}+\\cdots& \\\\\n\\left(1-2^{1-s}\\right)\\zeta(s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\\cdots&=\\eta(s) \\\\\n\\end{alignat}\n", "08fbc478214d30cab2edfa3dd2a0c524": " A = \\bigoplus_{e \\in \\min A } A e", "08fbd3614d3a3f94edc844a85fe5540b": "\\sqrt{3+2\\sqrt{2}} = 1+\\sqrt{2}\\,,", "08fc3223f172dcf9af891a46d5d2beab": "f''(x)=12x^2\\ge0", "08fc9b2c8399229b288bfba89e8f3a79": " -\\frac{1}{\\rho_L}\\frac{\\partial P}{\\partial r} = \\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial r} - \\nu_L \\left[ \\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left( r^2\\frac{\\partial u}{\\partial r}\\right) - \\frac{2u}{r^2}\\right] ", "08fce8bf68b68b998c7c84655bf396aa": "{\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1} ", "08fd046194d03be4fc66f6a81cda3a37": "\\Beta \\, \\beta \\,", "08fd2ca023e23a0884c73452c55c35ba": " \n\\mathbf{R}(\\alpha,\\beta,\\gamma)\n\\begin{pmatrix}\n0 \\\\\n0 \\\\\nr \\\\\n\\end{pmatrix}=\n\\begin{pmatrix}\nr \\cos\\alpha\\sin\\beta \\\\\nr \\sin\\alpha \\sin\\beta \\\\\nr \\cos\\beta \\\\\n\\end{pmatrix},\n", "08fd5269678a63899dce46c344cbaf76": "\\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{1}{4\\pi}\\int_{-\\pi}^\\pi x^2 \\, dx = \\frac{\\pi^2}{6}", "08fdb06222897cd3f6cd1a53ba125c71": "C \\cap C = C", "08fdc98f7dfc53b6b266be475e4a8c2d": " f: V\\to W ", "08fde725d0dd6e5ac5189e097f881655": " \\varphi_{T_\\mathrm{High}} = 90^\\circ - \\tan^{-1} (f/f_1). \\ ", "08fe9ad4134c034fa936c90e9aaad316": "\\dot{s}(\\theta) = (\\dot{s}_1(\\theta),\\,\\ldots,\\,\\dot{s}_k(\\theta))", "08fed0441b8e84e35c6c3e169378c427": " \\frac{N_n(E)}{n}\\to p\\text{ as }n\\to\\infty.\\, ", "08fedbd24dea0f2476982f017b2d3363": "(a^b)^c", "08ff3f541a4e8c6dd994b18253fbbe9e": "M \\times S^k", "08ff51325c7d2bf3bcc866b81c9f6614": "\\varphi_s=\\sum C_{s,t}\\psi_t.", "08ff9c52ed95f4faf0e6908e48b82f82": "\\scriptstyle\\langle F\\mid e^{-itH}\\mid I\\rangle ", "08ffb90ef746c7d6e1d3e9284604fb1a": "-2-2\\gamma", "08ffc90bc5212727b8ea94c5788ce752": " |\\cdot\\cdot\\cdot\\cdot\\cdot|\\cdot\\cdot\\cdot\\cdot\\cdot| \\, ", "08fff5fac6a59d92d6fad7b5fcffd2f5": "\\rho^A", "090024b5d5d5259e89f900dfcc29345a": "\\partial_\\mu \\rightarrow D_\\mu = \\partial_\\mu - i e A_\\mu ", "09004f6119e485d220772bcdcd301c00": " \\dot{\\hat{P}}(t)=1/2 \\left(\\tau(t)\\Psi_1(t)+\\Psi_1(t)\\tau'(t) \\right),\\hat{P}(0)=E({\\mathbf{x}}(0))E({\\mathbf{x}}(0))', rank(\\hat{P}(t))=n_r", "09010d3a3d1916d44af57c1166000e39": "y^k", "090118cc5b7d861869e8dc2d79654cc7": "\\overline{ D }=\\{(x, y)\\in {\\mathbb R^2}: (x-a)^2+(y-b)^2 \\le R^2\\}.", "09013dbb6539a971962b368da54a79fa": "\\Omega_d = (d\\theta/dt)_{drift} ", "0902222f3889cc1f9ab4aef8c8004530": "y(z) = \\sum_{k = 0}^\\infty y_kz^k", "09022c5adf858547281f956e790976ed": "\\frac{1}{2k - 1} - \\frac{1}{2(2k - 1)} - \\frac{1}{4k},\\quad k = 1, 2, \\dots.", "09024465bb6c4592c6155672c67e68f8": "\\begin{align}\nq(x) &= \\lim_{d \\to 0} \\Big( F \\delta(x) - F \\delta(x-d) \\Big) \\\\\n&= \\lim_{d \\to 0} \\left( \\frac{M}{d} \\delta(x) - \\frac{M}{d} \\delta(x-d) \\right) \\\\\n&= M \\lim_{d \\to 0} \\frac{\\delta(x) - \\delta(x - d)}{d}\\\\\n&= M \\delta'(x).\n\\end{align}", "090256db8431378a5ffdffff00d5f23b": "h_1 = |\\mathbf{h}_1|; \\; h_2 = |\\mathbf{h}_2|; \\; h_3 = |\\mathbf{h}_3|", "09027347334d1e8527fe6d519f159a80": "(x_i^{}, x_{i+1})", "0902a406c971b15df5ed00b74a416955": "\\lambda = \\kappa ", "0902c2b382996a827924175c9b51493f": "\\nabla^2\\omega_1 + \\frac{f^2}{\\sigma} \\frac{\\partial^2\\omega_1}{\\partial p^2} =\\frac{1}{\\sigma} \\left[ \\frac{\\partial}{\\partial p} J(\\phi,\\eta) + \\frac{1}{f}\\nabla^2 J \\left(\\phi, -\\frac{\\partial \\phi}{\\partial p} \\right) \\right] - \\frac{f}{\\sigma} \\frac{\\partial}{\\partial p} \\left( \\frac{\\partial \\omega}{\\partial y} \\cdot \\frac{\\partial u}{\\partial p} - \\frac{\\partial \\omega}{\\partial x} \\cdot \\frac{\\partial v}{\\partial p} \\right) - \\frac{f}{\\sigma} \\frac{\\partial}{\\partial p} \\left( \\xi \\frac{\\partial \\omega}{\\partial p} - \\omega \\frac{\\partial \\xi}{\\partial p} \\right)", "0903077b7ebae3ba4019324b3c4f876d": "\\! \\mathcal A", "090311abe55c69d4df0815b84955e08d": "\\widehat{a}(\\phi)=\\phi(a)", "09033e60847ab2388b63e15d975ff401": "\\approx 4 m^{3}", "09034cb45fbf22cf89e5849562318705": "S's", "09037fa5ae70155b5532a471adbef010": "2 + \\sqrt{5}", "090397a5c4875be3c748c6e5a7c5a68d": " p-p_0-(\\rho_2-\\rho_1)gz=\\gamma\\left (\\frac{1}{R_3}+\\frac{1}{R_4}\\right)\\!", "09039a7ae7fc996a69186b52d5d211e4": "w = f(z) + \\bar{f}\\left( \\frac{a^2}{\\bar{z}} \\right)", "0903dc8ea9fb88e72c09961ea7c5232b": "\\Delta Y/Y = k - c \\Delta u\\,", "0903e6516263dd169dc47e9d63cb2443": "X_1,\\ldots,X_n", "0903f68a216a8d137794b64f9820702f": "\\mathfrak{P}^{30}", "0903fd81b795dd4b63fe5fbfa874b2ff": " \\operatorname{lambda-free}[\\lambda F.X] = \\operatorname{false} ", "0904322ab23e920093715fee2d6978da": "\\frac{1}{2}\\, \\left( 1\\, +\\, k\\, h\\, \\frac{1\\, -\\, \\tanh^2\\, (k\\, h)}{\\tanh\\, (k\\, h)} \\right)", "090438b81c4ea40671144d64a867cbc6": "\\frac {P'(t)}{P(t)} = r \\frac{(1 - c)}{p(t)},", "09044d7a546675c0a0e799fe0a4017f4": "\\Gamma_{12} (u, v, \\tau) = \\lim_{T \\to \\infty} \\frac{1}{2T} \\int_{-T}^T E_1(t) E_2^*(t-\\tau) dt", "0904593e6b7980b27a67975abb84fbfd": "x^2-1", "090481e5a9874bfb89e693fdfbd419d8": "~g=\\int G(x(a),y(a),z(a))~{\\rm d}a~", "09049a6b76e1d64f1b39d338c3697d93": " A = \\varepsilon \\ell c = \\alpha\\ell \\,", "0904a74176c0b12a21f36e1f5c37c382": "(W;M_0,M_1)", "0905131e539fb1ce3326f0c848a4cc51": "A=A_o \\cos[2\\pi(x/\\lambda- ft) + \\varphi]\\,", "090560794c199c84aec3a0335aeb82bf": "\\log\\left(P_{r}\\right)=\\log\\left[\\left\\langle\\exp\\left(-\\beta H - \\log\\left(Z\\right)\\right)\\right\\rangle_{r}\\right]\\geq\\left\\langle -\\beta H - \\log\\left(Z\\right)\\right\\rangle_{r}\\,", "090604235cde82e36ec6a33d4acd3881": "\n\\begin{matrix}\nQ^{2} + U^2 +V^2 = I_p^2,\n\\end{matrix}\n", "090614c64a0b7bb53db9ff6a62933770": "S_x", "090631c41da3627ed4277dc052a6fae0": " z^- ", "09068042156731af491a5c34fb003f9a": " -r \\ln (1-p)", "09070160b71b284bb5d1db079a05202b": "y=c_1y_1+c_2y_2", "0907149553ed8aca1069f30fb615f6c8": "C(u,v)w+C(v,w)u+C(w,u)v=0 ^{}_{}.", "0907236e045a531fb1788589b18f13a8": "\\scriptstyle \\leq1\\times10^{-32}", "09074aa69d9199d45e04b6dbcaed26a8": "SO \\to PSO", "09078b651072f973326ca32b8250f097": "K \\cup L", "0907ba999a01b3d8370a02def4746c33": "\\Gamma=\\frac{z_T-1}{z_T+1}\\,", "0907e0e9c988dfb54802ca60ce8ccdda": "\nS=\\sum_{i=1}^n X_i\n\\sim\nIG \\left( \\mu_0 \\sum w_i, \\lambda_0 \\left(\\sum w_i \\right)^2 \\right). ", "09080b3dcb9b69f8ad794eb2f0a7c206": "d F/ dt \\geq 0", "09082d688d90d15b3eb6c90b7c4ee670": " \\frac{3\\alpha a - 1}{4}", "0908412f11abec075891cc1356ce38bc": "\\scriptstyle Q\\,\\sim\\,\\mathcal{N}(\\mu_2,\\sigma_2^2)", "09085992776ef7750477f9ab1a811a19": "\\Gamma=C([0,1],X)", "0908874e8bc0d69cca3cbec7552687d7": " \\sum_{i\\in V} f_{iv} \\le c(v) \\qquad \\forall v \\in V \\backslash \\{s,t\\}", "0908877bc430c2539894ff49b8501430": "0\\le y \\le 5", "0908c024103c8b53923511f5ed209cb7": "\\hat{\\mathbf{a}}", "090903469044bbd8468f7540393ad95d": "( M \\circ (\\mbox {Id} \\times M)) (x,y,z) = M (x, M(y,z))", "090934bdc2ecd593440d431a42e88d78": "Y=\\textstyle\\sum_{j=1}^n X_j", "09094947d14c8f93f371c4496bd94e83": "V=\\tfrac{1}{\\sqrt{K_{0}}}", "090953475374dfc6660beacb417939f5": "\\mathbf{x}_k", "0909714bd7f63294d0ec10692e39b6da": " \\langle b_1 | ", "09097d012954ffa36ea5438198f5c662": ":\\quad T_c \\ \\sqrt{ 1 - V^2/c^2 }", "0909ce96ee22ccd9cfb5db0777f041a7": "(c/n) \\sin(\\psi)", "0909e4cab35224cfe599d8dad0b026f2": "\n\\vec x_{n+1}=2\\vec x_n-\\vec x_{n-1}+\\vec a_n\\,\\Delta t^2,\n\\qquad \\vec a_n=A(\\vec x_n).\n", "0909fbc7d9231ebf9bfee3980c866651": "\\ N=s_0=\\sum_{k=1}^N{x_k^0}.", "090a081743cd21b4e4e3b8b25b278b7a": "k_1=\\gamma/J_1", "090a47f13e1d78b36997329e4f9a8641": "\\operatorname{diag}(a)", "090a4e2c887e51b16ce80db7d33a5b9d": "p \\sim n^{\\gamma}", "090a60633f94c20645c0ec8efda7b3f7": " \\lambda h.h\\ (\\operatorname{const}\\ f) = \\lambda h.h\\ x ", "090b0c112a40ab3f181030ffc737f841": "\\mathrm{KDF}", "090b1bcf5f5afc37b39f1aa5f1447051": "-{{q^2 \\over g{y_1}}} + {q^2 \\over g{y_2}} = {{y_1} ^2 \\over 2}-{{y_2} ^2 \\over 2}", "090b4380400e0e79224039a8d9dbeaea": "\\frac{R_1+R_2}{R_1-R_2}=\\frac{2 \\left( n^2-1 \\right)}{n+2}\\left( \\frac{i+o}{i-o}\\right)", "090b44bb4cd2b003e253f9e3d7a76d44": "f^{\\mathrm{Y}}_p(x) = (1 - x)^p.", "090b4d45fceb164c69b20ef0c313a082": "\\theta (f) \\propto f", "090b612dac015543b25b65297408332e": "\\mathit{nil}", "090b7fa32ce18867bcb864421db27c83": "\\{x\\mid M,x\\Vdash p\\}", "090b93a1b8d8fa7697e4827cdd3b876c": " \\textstyle P_N=C_P+I+D_g+E_e-S_w\\,\\ ", "090b9a46571289134180e192d4f85a72": "g(x) = \\lambda_x \\,", "090c045279e0dcc8da18608d10ec145f": "|\\mathbf{a}_1|", "090c3aba4b18c527d8c73c5dffd82059": "q_{m+n} = p_m q_n + p_n q_m\\,\\!", "090cb61ede0c514c9e2ac7619337231e": "\\text{rk}(T) + \\text{nul}(T) = \\text{dim}(V).", "090d3b42b2e06b1d30e7f0ff5574f5c0": "\\lnot \\forall x \\phi", "090d65554639d85bbc53c179be37aeeb": "\\frac{Y}{X} = \\frac{s+z}{s+p} ", "090d6975eb8c8400f516a685b200c533": "\\overline{X} = \\overline{X}_n=(X_1+\\cdots+X_n)/n", "090d7079eaeb7ebd8aedc2d16e75a58b": "E_A = E_A(q_{0,A}) + 3 f_A(\\Delta q_A)^2", "090d73ded95ff5578163137441626833": "g^{ij}S_{ij}=0", "090de40fbb68d3836049ec5fdf143db7": "\n\\mathbf{C}", "090f096b160b3afe3ad09588ae776d94": "N = \\frac f d\\,;", "090f0bc1d14ee038fe2e60b2cf32a1f3": "\\frac{M_1}{(R-r)^2}=\\frac{M_2}{r^2}+\\left(\\frac{M_1}{M_1+M_2}R-r\\right)\\frac{M_1+M_2}{R^3}", "090f0cebfe76afb1815d64cbc08dd432": "2^{n+1}-1", "090f239dad336f0b2c793cc91c7d9402": "(\\gamma, \\alpha) \\ne 0 \\, \\forall \\gamma \\in \\Phi", "090f31c0a7baeb58d233964ec2bc3dda": " F_s = 2r \\pi\\gamma\\!", "090f5532e658323e653f0ab86d0f7d22": "\\psi^R", "090f6678031c36a9c6914db4b15a10f5": "i:\\lambda", "090f89368c5c383e8dbf356200c5f379": "\\mathfrak{P}^{26}", "090f92439b671c9f0666f3d1d13dd30c": "\\Delta v\\ ", "090fc3942a2606112ed6fcad2ac30187": "\\langle x, x\\rangle^{1/2}", "090ff1d7b103de4e53e2bb4e3c6fef65": "S(T) = \\frac{C}{\\exp\\left(\\frac{c_2}{\\lambda _x T}\\right)-1}", "09106688ed486be9ada25ca35b5094c4": "\n\\begin{align}\n&\\begin{cases}\n\\dot{\\hat{\\mathbf{x}}}(t) = f\\bigl(\\hat{\\mathbf{x}}(t), \\mathbf{u}(t)\\bigr), \\\\\n\\dot{\\mathbf{P}}(t) = \\mathbf{F}(t)\\mathbf{P}(t)+\\mathbf{P}(t)\\mathbf{F}(t)^\\top + \\mathbf{Q}(t),\n\\end{cases}\\qquad\n\\text{with }\n\\begin{cases}\n\\hat{\\mathbf{x}}(t_{k-1}) = \\hat{\\mathbf{x}}_{k-1|k-1}, \\\\\n\\mathbf{P}(t_{k-1}) = \\mathbf{P}_{k-1|k-1},\n\\end{cases} \\\\\n\\Rightarrow\n&\\begin{cases}\n\\hat{\\mathbf{x}}_{k|k-1} = \\hat{\\mathbf{x}}(t_k) \\\\\n\\mathbf{P}_{k|k-1} = \\mathbf{P}(t_k)\n\\end{cases}\n\\end{align}\n", "0910b0c05db4ed89df134d7ecb559bd4": "\n \\left\\lang {\\Delta S_x}^2 \\right\\rang \\left\\lang {\\Delta S_z}^2 \\right\\rang \\ge \n \\frac{1}{4} \\left|\\left\\lang \\left[S_x, S_z\\right] \\right\\rang\\right|^2\n", "0910b8087f0078b46339c4f5de5f5c08": "[ \\hat{x}, \\hat{p} ]", "0910d95ce2e6351b3a3d9beac5675583": "\\gamma_m = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L\\left(y_i, F_{m-1}(x_i) -\n \\gamma \\frac{\\partial L(y_i, F_{m-1}(x_i))}{\\partial f(x_i)} \\right).", "0910e2480ee5b08d924fb9c9b47c5ad4": " \\sum_{i=1}^n \\|x_i\\|^2 = \\left\\|\\sum_{i=1}^n x_i \\right\\|^2. ", "09111a979677cea23448e84408371b87": "\\pi_0\\,\\text{Diff}^+(D^n) \\to \\pi_0\\,\\text{Diff}^+(S^{n-1}) \\to \\Gamma_n \\to 0. \\,\\!", "0911296bb845dcfedcab7c5b305b663c": "B(x;r)=B(y;r)", "09115053006b9b3efef572492e9b0345": " p \\neq 2", "0911a8435ba395fac34959ec4936a6e4": " (\\nu x)(\\overline{x} \\langle z \\rangle.0 | x(y). \\overline{y}\\langle x \\rangle . x(y).0 ) | z(v) . \\overline{v}\\langle v \\rangle. 0 ", "0911cad5408a1512f27757e7a6a4c49c": "\\deg(a_{k+1})+\\deg(b_{k+1}) < \\deg(a_{k})+\\deg(b_{k})\\ ,", "0912ac95ff597192a02e15e971c03024": "v_3 = v_1 - v_2", "0912d3522828b666040b654d97187312": "\np_w(\\theta)=\\sum_{k=-\\infty}^{\\infty}{p(\\theta+2\\pi k)}.\n", "0913221ab6181cbea810f1d7729df659": "f(x;\\lambda) = \\lambda e^{-\\lambda x} ,\\; x \\ge 0. ", "091337f4025e3c5d7c06a2aed1f14e97": "M = 4", "09136155a08c6f0f3e6a78119ac6cdc8": " \\bar \\nu = \\omega_0 +(B ^\\prime+B^{\\prime\\prime})m +(B^\\prime-B^{\\prime\\prime})m^2-2(D^\\prime+D^{\\prime\\prime})m^3, \n\\quad \\omega_0=\\omega_e(1-2\\chi_e)\\quad m=\\pm 1, \\pm 2 \\ etc. ", "091388cbf96e1ff77f238bf8676ddd42": "\\alpha_\\tau=\\Lambda p_1,\\ \\beta_\\tau=\\Lambda p_2,\\ \\gamma_\\tau=\\Lambda p_3\\ \\mathrm{at}\\ \\tau \\to \\infty", "091391060d54f3163293f348303b488e": "f(z+1) = W(e^{z+1})-1\\,", "091392c0aaecc4e4a88d9eed5c910092": "\nF_{p-\\left(\\frac{p}{5}\\right)} \\equiv 0 \\pmod p,\\;\\;\\;\nF_{p} \\equiv \\left(\\frac{p}{5}\\right) \\pmod p. \n", "0913c4bdcd8eb434e8ec3c664c41e77a": "\\tfrac12 A_0 + \\sum_{n=1}^\\infty \\left(A_n\\cos nx + B_n \\sin nx\\right).", "0913eaf074752c0fe0b7c90f77674845": "\\, r", "0913f73fe0f9119fe826894691ee11b3": "S=I+A+A^2+\\cdots +A^n", "091442621ed7ba03dc4f1e190e229ff6": "\\langle f^* \\star f \\rangle =\\int (f^* \\star f) \\, W(x,p) \\, dx dp \\ge 0.", "09147841158d447a0f07d8cba2958953": "i \\omega V_c + \\frac{1}{RC} V_c = \\frac{1}{RC}V_s", "091482325c8967bc9aa92530f5c483d5": "\\exists b", "09149a6119fb94a9381cce68ea4883e0": " + \\frac{C_1R_1^2+C_3R_1R_3}{C_2R_2} + \\frac{C_2R_1R_3+C_1R_1^2}{C_3R_2} + \\frac{C_1R_1^2+C_1R_1R_2+C_2R_1R_2}{C_3R_3}", "0914c56bbd7b1d4950a1244946007dfa": " \\alpha \\, ", "0914cdacfd497034622eeb281289ba2a": "k_0 = 1", "09150cc48eb71ff741d74ff0cf7123e0": "S_\\eta", "091526342c696504959a405c529a1e3c": "\\{f_{j}^{\\dagger}, f_{j}\\}=1", "0915415c5fb1271071d91ec64e89f332": "N(x|\\mu,v)", "0915564dbdf6347f9e7ce597d022defe": "C(N) = \\frac{N}{1 + \\alpha ((N-1) + \\beta N (N-1))} ", "09158ac493eb982be2f92b1ac044362b": "p(x)\\,", "0916b28226473d5c571ec2c9d6df70c9": " \\left\\vert \\left( \\frac{\\text{Doppler Frequency} \\times C}{2 \\times \\text{Transmit Frequency}} \\right) \\right\\vert > \\text{Velocity Threshold}", "0916c7a201125add754533bf2d2d2753": "\\rho(\\xi) = X_{H_\\xi}", "091700d80f6fd1d6c66c3c1c5ae506e5": "\\psi^{(m)}(z) = (-1)^{m+1}\\; m!\\; \\sum_{k=0}^\\infty\n\\frac{1}{(z+k)^{m+1}}", "09174c4395232cb6f09a34a36c96387e": "\\mathcal{O} _X| _U", "09177b8a56fc7aa328c67b8dc9d9b9ce": "T=\\frac{1}{4}\\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}", "09179fb350fcf2c4ad4d4a13604e2fbb": "x^{q^i}-x \\mod f^*,", "0917f3f82c1c7b416ffa17d770347370": "G(x,y) = \\frac{1}{(2 \\pi)^4} \\int d^4p \\, \\frac{e^{-ip(x-y)}}{p^2 - m^2\\pm i\\epsilon}", "09180b2796aa3fb7217cfe4aa6b6701e": "C=\\varepsilon \\cdot { {A} \\over {d} }", "09183c0edb89036c69e82ba20a0f78a1": "P^4", "09183d46744c20de019525df2bfbaaed": "\\mathbf{V}^{\\rm T} = \\mathbf{V}.", "0918647082ead4f66cb5e373b72c094d": "\\scriptstyle \\Pi \\,\\subset\\, \\mathbb{R}^n", "09186b7f4437732f403f59e0e14094d5": " \\frac{2}{mn} = \\frac{1}{m} \\frac{1}{k} +\\frac{1}{n} \\frac{1}{k} ", "09186f6c25ca52f6cd3aaf5691ac67aa": "GVD(\\lambda_{0}) \\equiv \\left ( \\frac{1}{\\nu_{g}(\\lambda_0-\\delta\\lambda/2)} -\\frac{1}{\\nu_{g}(\\lambda_0+\\delta\\lambda/2)} \\right )", "0918afec57b0c192615d6f0f2b725b82": "0 \\le a_n + |a_n| \\le 2|a_n|", "0918b6a685fb23ad7b6df7b5e00fa538": " ax^3+bx^2+cx+ d", "0918ba105278e2b9fc166e0c5fa155bb": "L^p(d\\nu)", "0919356f7a53777bd2fcd1ce122eef41": "+ 1.737 \\times \\log_{10} [Bilirubin (\\mu mol/L)] - 1.184 \\times [Apo A1 (g/L)] + 0.301 \\times Sex (female=0, male=1)-5.54", "091947a8ee1bac70789700c52856045d": "z_3=-5", "0919631fb604e60147b9fd3ac28f72c5": "\\gamma\\ > 0 ", "0919d464d820465b51cf0c4536d1c64b": "a^1 b^1 c^1", "091a2f4d9bcd3ca7daddb5db1288606c": "\\kappa(u_{ij})", "091a36c336b9f86ac488b9e8ac0e0ffa": "X^{(0)}", "091a513a46acbcd729c737600655f9eb": "\\frac{8}{3}m^{3}", "091a64999eda86b4be1c5164f85fb338": "{\\Bbb C} \\times H/\\Gamma", "091aab6f44ab6c366341e673c52d735e": "M_{\\phi}:\\mathbb{R}^S\\rightarrow \\mathbb{R}", "091ad9dff9f1baf25af71de76ec09794": "\\frac{d \\vec \\omega}{dt} = (\\vec \\omega \\cdot \\vec \\nabla) \\vec v ", "091b1fd63ce5a18e61a0518154295f19": "\\det \\left( 1-z\\mathcal{L}\\right)=\n\\prod_i \\left(1-\\rho_i z \\right)", "091b3f5e20d3d55b356e0a885bb80303": "\\displaystyle{D(\\varphi)=D(\\varphi)|_\\Omega \\oplus D(\\varphi)|_{\\Omega^c},\\,\\,\\,\\, S(\\psi)=S(\\psi)|_\\Omega \\oplus S(\\psi)_{\\Omega^c}.}", "091b6c4afc95f4f6d9963fc09eb9ea66": "y = \\sin(t)(R + r \\cos(u)),", "091b852783c17814fe9f2137779de4a8": " \\langle\\mathbf{p}\\rangle=\\left\\langle \\psi \\left|-i\\hbar\\nabla\\right|\\psi\\right\\rangle = \\int_\\mathrm{all\\,space} \\psi^*(\\mathbf{r},t)(-i\\hbar\\nabla)\\psi(\\mathbf{r},t) d^3 \\mathbf{r} = \\hbar\\mathbf{k} ", "091beef37747159cbda93c3e0947c753": "\\frac{\\langle E(s) \\rangle}{A} = \\hbar\n\\int \\frac{dk_x dk_y}{(2\\pi)^2} \\sum_{n=1}^\\infty \\omega_n\n\\vert \\omega_n\\vert^{-s}.", "091c17327bb2a9dccebb1bd7ebd1faa5": "\\theta \\in \\mathbb{C}", "091c33b9617c06f3c8f9211fca7a951f": "\\geqslant", "091c54bf7ab3f2927aa6af06024c6071": " \\hat{\\theta}_i", "091c70939a397005815ced426a1069a6": "q\\geq cp", "091cce78f0094f4cd21f6e9e7d4ddf9d": "{\\color{Blue}~2.23}", "091ccf7c41a5b2ff7f123fb420dd9187": "\\epsilon_i(b) = \\max\\{ n \\ge 0 : \\tilde{e}_i^n b \\ne 0 \\}", "091d9c0c19b64e2ca4fc411517aa216b": " \\boldsymbol { \\mathcal{P}} = {1 \\over 4\\pi c } \\mathbf{E}( \\mathbf{r}, t ) \\times \\mathbf{B}( \\mathbf{r}, t ) ", "091dacedc989f940358e42b35434c737": "\\scriptstyle \\sum_{n=1}^\\infty \\frac{x^n}{n^s}", "091dc0e3964e60041a23eea09883ba4f": "\\sqrt{8} (3 \\rho^3 - 2\\rho) \\sin \\theta", "091dfabee698d3d25988e357afccdbbd": "d/n", "091e57d91fb91625042c6c78cda2cdf5": "\n\\begin{align}\nq_{\\text{n}}^H(k): &{}\\quad \\frac{1}{\\tau} + (d_u^2 + \\frac{1}{\\tau} d_v^2)k^2 & =f^{\\prime}(u_{h}),\\\\[6pt]\nq_{\\text{n}}^T(k): &{}\\quad \\frac{\\kappa}{1 + d_v^2 k^2}+ d_u^2 k^2 & = f^{\\prime}(u_{h}).\n\\end{align}\n", "091e7e16ecc5c72b86b33fda6cceb6d1": "\\frac{1}{d} = \\frac{1}{r'} \\left [ 1 - 2 \\cos (\\theta' - \\theta) \\frac{r}{r'} + \\left ( \\frac{r}{r'} \\right ) ^2 \\right ] ^{- \\tfrac{1}{2}}.", "091e9a7fb6b7b0e6e04327d6cd8c0776": " C(x,x'') = \\int_{x'} A(x,x')B(x',x'') \\, ,", "091ebbc1e1c0749c4df6b535fac39f05": "P_2 \\uparrow S(X, Y, \\mathfrak{E}, H)", "091edb42262c877ba740f874405bd33d": "\\alpha^{-1}(n)", "091eecf569963d2f229b88abf5675a62": "R(5, 5)", "091f2f2d60bc57704a3577cac23da4a2": "\\tfrac{dI}{dT} = \\beta SI - \\gamma I - \\mu I", "091f5aee3c7167080396ed1c86bd3740": "y_n \\to 0", "091f5b5f08472066423a2645778c3222": "\n \\boldsymbol{C} = \\boldsymbol{F}^T\\cdot\\boldsymbol{F} = (\\mathbf{G}^i\\otimes\\mathbf{g}_i)\\cdot(\\mathbf{g}_i\\otimes\\mathbf{G}^i)\n = (\\mathbf{g}_i\\cdot\\mathbf{g}_j)(\\mathbf{G}^i\\otimes\\mathbf{G}^j)\n", "091f6b5d6f09937e0570f516690cfa54": "\\ \\overline{u' w'} = \\overline{\\xi' ^2} \\left | \\frac{\\part \\overline{w}}{\\part z}\\right| \\frac{\\part \\overline{u}}{\\part z}", "091fd227c6b83e7aa1b9d0e9b3fd2d44": "2^{3/12} = \\sqrt[4]{2}", "09200a73966b5c9559e1495e92dcc7ce": "\\hat{x}\\in C(\\theta)", "0920682753de2d80a47d6fcd3c8c195a": " T_n = ( 1 - \\frac { n } { N } ) \\frac { a + 1 } { D^2 - ( 1 - \\frac { n } { N } ) \\frac { b - 1 } { n } } ", "0920856d25e3fec73110d603dd5f56cb": "\n\\begin{align}\na & = 0.14285933+0.06404502i, \\\\\nb & = 0.14362386+0.06461542i,\\text{ and} \\\\\nc & = 0.18242894+0.81957139i,\n\\end{align}\n", "09209777121192f0d82359664658b0b9": "\\begin{cases}\n -\\frac{b}{a}\\frac{\\Gamma\\left(-\\tfrac{1}{a}\\right)\\Gamma\\left(\\tfrac{1}{a}+p\\right)}{\\Gamma(p)} & \\text{if}\\ a>1 \\\\\n \\text{Indeterminate} & \\text{otherwise}\\ \\end{cases}", "0920bf711f19ec9e01fbe7f82a2bb21e": "D_1\\left(E\\right) = \\frac {1}{ \\sqrt{c_k(E-E_0)}} \\ .", "0920f444b3780881c9612b5674c99063": "g(x) = g_nx^n+g_{n-1}x^{n-1} + \\ldots + g_0,\\,", "092144927dc432cb394abb16777fe138": "\\tau = \\mu \\frac{\\partial u}{\\partial y},", "09215af2256fcc860a2cd7c773c224ed": " \\psi ", "09219171febf5fe82c2ce71e4ff3a25b": " \\mathbf{A}(\\mathbf{r},t) = \\nabla \\times\\int\\frac{ \\mathbf{B}(\\mathbf{r'},t)}{4\\pi R}d^3r'", "0921c67f29ec8472f8258719e160047b": "\\frac{1}{2\\pi i} \\int_{-i\\infty}^{i\\infty} \\frac{\\Gamma(a+s)\\Gamma(b+s)\\Gamma(c+s)\\Gamma(1-d-s)\\Gamma(-s)}{\\Gamma(e+s)}ds", "0921c77ddde769fdfaaa757b3aee9d78": "a_z\\,dx\\wedge dy + a_y\\,dz\\wedge dx + a_x\\,dy\\wedge dz.", "0921c9b64221e4678a9a050a4a5c0191": "\\partial_1(c_1) = 0", "09221170da1edff943267d36b77d0915": "(x^3+x)-(x^3+x^2)=-x^2+x", "0922219ad86894a35df46db958dc34ab": "C \\left( a_n(q),q,x \\right) = \\frac{CE(n,q,x)}{CE(n,q,0)}", "09224714f54be3f67c05ec6c983114ea": "\\vec{b^i}=(b^i_1,...,b^i_m)", "09228e6d1aa18c62dfcdad14cc61a415": "\\phi : F \\to B", "0922b63a680dc5bd8e75bf150cb2fe4f": "\\Phi: A \\rightarrow \\reals", "0922c9b5c15dac27efc38f77731b83d6": " g(E) = g(F)", "0922eb3f7b10e8d6bc4c559e4af7d6de": "e \\in E,", "09233cbe5f03d5b8a1b4daae8d5a5422": "|\\quad|_p: \\textbf{Q} \\to \\textbf{R}", "0923d20800931dbd64476c69de90eee0": "g\\colon S^2\\to g(S^2)", "0923f3540c0efb59c522fdc507bb3443": "\\eta\\colon K \\to A", "09240fc1caa97137cb5135e7ce2eff8a": "\\frac{c}{\\sqrt2\\chi}\\sqrt{(\\chi^2-2p-1)+\\sqrt{\\chi^2(\\chi^2-4p+2)+(1+2p)^2}}", "09248813a49c15ac9e7dd76614b04168": "1200 \\,", "0924922346e7a3d5709e819c477745db": "ROIC = \\frac{\\textrm{Net Operating Profit} - \\textrm{Adjusted Taxes}}{\\textrm{Invested Capital}}", "09249b0be3fab2baad2a6f48d8ea31c9": " \\|R_i\\|_0 \\leq q ", "09249e3cc2e9cab5df7a3dd31364e664": " p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 ,\\,", "0924f87e189924b1b5d9fdc82330cdc6": "\\nabla^2\\left(\\frac{1}{r}\\right)=-4\\pi\\delta(\\vec{r})", "09254c1405dd35d9599cefc5e2936364": "\\mathrm{[A_pB_qH_r] =\\beta_{pqr}[A]^{p}[B]^{q}[H]^{r}}", "0925546441796437a496ee868b2c93c4": "\\frac{d}{|\\mathbf{v}|} = \\frac{d}{\\sqrt{a^2+b^2+c^2}}", "0925e8fbbe7807223499ff6ef7beb9aa": " \\frac{R}{2} \\sqrt{2+\\sqrt{2}} = \\frac{a}{2}\\left(\\sqrt{2}+1\\right) \\!\\, ", "092623bc48b6f3f0ec15dc1a57904e85": "\\widetilde{t}", "092628c2a346f6d8880320c5a437e015": "H = U + pV,\\,", "09265569b156bfb21b7ee605767c94d3": " \\lambda < \\omega_1^{CK} ", "092673dc2898d724d94e6019fd391c01": "x\\in \\Lambda", "09269acbde301d27b2091d4edc20134c": "\\varepsilon _i", "092779faa466d9c56c44d640bb4f9ba3": "_NC_2", "0927c033adbec8f6ac2024ab9790ef2d": "\\,v_i", "0927fa3288b43c389822c7dc835fb146": "M_{n} = M_{1}^{n}", "09280fa4ae67a9e7d4df69369d8a4d74": "F(x)=\\sum_{n=0}^\\infty f_n(x),\\qquad x\\in\\mathbb{R},", "09281df7c03d16d683bb2d8b7745fa14": "y = ax^2 + bx + c \\,", "09287cfee425829c734357bf7cb658bd": "M_{[ab]} = \\frac{1}{2!}(M_{ab} - M_{ba}) ,", "092895644979bb03db1e0a4729d41175": "\\text{Percentage change} = \\frac{\\Delta V}{V_1} = \\frac{V_2 - V_1}{V_1} \\times100 .", "0928a80b08bcd2fb57a34b7d710dd541": "M_{0}^{2}", "09296204606853b14d1a5a6b4413469a": "P_c \\sim 10^5\\,\\mathrm{Mbar}.", "092962f1ddafe24557e8dd0c44b38815": "\\frac {d\\theta}{ds} = \\frac {s}{R_c s_o}", "092993b80f35b2720f5bb39d6f4161d7": "\\Pi_{i\\in I}A_i/U", "0929a96a4d662615fd83073af51b6220": "\\frac{2\\alpha^{2^n} + 2\\alpha^{-2^n} - b}{2a}", "0929ab6e77c5fbfcf609f5889f9b5eff": "S'' = 0 ", "092a6856d4af9220c0686d5209ecc781": "\\frac 1p_\\theta= \\frac \\theta {p_1}+ \\frac {1-\\theta}{p_0}.", "092a9e583eb2e54916933fcb25b728f0": "\n \\begin{align} \n N_{\\alpha\\beta,\\alpha} & = 0 \\quad \\quad N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q & = 0 \\quad \\quad M_{\\alpha\\beta} := \\int_{-h}^h x_3~\\sigma_{\\alpha\\beta}~dx_3\n \\end{align} \n", "092a9e9394d963738b1f267d01aa3cbe": "x^5+ax^2+b", "092a9f1c921d20c4f8037bb84eb3fb05": "\\phi(Q)=c|L|", "092ad01e706529b98dedece42bf96d20": "\\mathbf E\\,\\!", "092adfb87a939e758ac84c119b2248ee": "N = n \\ell A ", "092af1486cf4cdfa3abe7c3efa85f299": "\\varphi^{n+1}\n\n= \\varphi^n + \\varphi^{n-1}.", "092b1a820808e13c82521cb113834bf4": "((24\\,\\bmod\\,5)(3^{-1}\\,\\bmod\\,5)) = 4 \\cdot 2 = 8 \\neq 3", "092b1c41202287b7e2a8a4ef87815c68": "a_1+2a_2+3a_3+\\cdots+na_n=n.\\,", "092b210e890b7ea0cacda2de9f0455cc": "W_{0}^{1, p} (X) \\subseteq L^{\\varphi} (X)", "092b6b8745ddea3da8a98a16f31e4711": "\\left\\langle \\frac{1}{2} k \\left(\\hat{x} - x_0\\right)^2 \\right\\rangle \\left\\langle \\frac{1}{2m} \\hat{p}^2 \\right\\rangle \\geq \\left(\\frac{\\hbar}{4}\\right)^2 \\frac{k}{m}.", "092b77229c3b8582317d0e224fbb17a6": " \n -{1 \\over 4\\pi} \\nabla^2 \\left( {1 \\over r} \\right) \n= \\delta \\left( \\mathbf r \\right) \n", "092b88f48f20045681113a8e99c25345": "\\lambda_n=\\log n", "092be609dae29cc54c062bf0377d77be": "\\mathbf{L} = C_L q A_{ref}", "092c0630a9e58b8d7ad12263d3f937aa": " (1 2), \\;(1 3),\\; (1 4),\\; (2 3),\\; (2 4),\\; (3 4)", "092c366a0f3e517c2ed8a49e9e8e1d17": "x_{/_{\\cong_{\\mathcal{B},\\epsilon}}}", "092c4239fce753aaffbc04bcfe87810f": "= C \\int_{-\\frac{a}{2}}^{\\frac{a}{2}}e^\\frac{ikxx^\\prime}{z} \\,dx^\\prime", "092c7ba21f6ef3e3f77a03a36e21cb3d": "\\mathbf{1}_{A}", "092c91d017df5d7ca930fb161e6657c4": "\\eta(\\xi)", "092c9f8f119bd88b4e72f4e1d4f1c3a2": "\\sin y = x \\ \\Leftrightarrow\\ y = (-1)^k \\arcsin x + k\\pi ", "092caf48acbcfe27d7c3e4134d645bfe": "\nI \\in [4.769A,5.417A]\n", "092cbee0a46deeba73e1b0d8036cd6b6": " \\omega'_0 = \\sqrt \\frac {1}{LC} ", "092cd680cd1d538539d41f0b9a191245": "q(uv) = au^2", "092d21f798b50357be84affffebe5aa4": "\\Delta_2^{\\prime}F(J)^{calculated} = 2B^{\\prime\\prime} \\left(2J+1\\right)", "092d3c21515cf56d7f46acaeacebe7f4": "\\displaystyle \\det(\\partial_{ij}\\phi) = ", "092d684280262b67670001706a335165": "k_1=15.957", "092d6a697df463c3beaaeafbff30f771": " \\frac{6}{\\pi^2}e^\\gamma\\le \\limsup_{t\\rightarrow +\\infty}\\frac{1/|\\zeta(1+it)|}{\\log\\log t}\\le \\frac{12}{\\pi^2}e^\\gamma", "092d73a7de8efeaa2cb788c40d954aa1": "r=\\text{min} \\left(1, \\frac{ w(\\mathbf{y}_1,\\mathbf{x} )+ \\ldots+ w(\\mathbf{y}_k,\\mathbf{x}) }{ w(\\mathbf{x}_1,\\mathbf{y})+ \\ldots+ w(\\mathbf{x}_k,\\mathbf{y}) } \\right)", "092d8cf0e1b1919e20eb9b5b670cb7ea": "0 \\leq t < \\infty", "092d8ff63435567921b7b2139a485d70": "\\exp(\\pm kt)e_i(t)", "092da07cb35a7e571d676643fbf67db8": "\nY_{(j+\\frac{1}{2},\\frac{1}{2})jm}=\\left(\\begin{array}{c}\n-\\sqrt{\\frac{j-m+1}{2j+2}}Y_{j+\\frac{1}{2},m-\\frac{1}{2}}\\\\\n\\sqrt{\\frac{j+m+1}{2j+2}}Y_{j+\\frac{1}{2},m+\\frac{1}{2}}\\end{array}\\right)\n", "092dd4a09cd11dcb6305bcd0a92d0884": "d(v_1, v_2) = 0", "092e66f7933af9712ed95ad2cb76aa3c": "\\{H, L^2, L_z\\}", "092e7e21e24e3f1eda6c1ca932af40cd": " (M + m) \\ddot x + m \\ell \\ddot\\theta\\cos\\theta-m \\ell \\dot\\theta ^2 \\sin\\theta = 0 ", "092e86b698513c5298c66b2520ea446c": "D(f)\\leq R_1(f)R_2(f)", "092ee1e3b9ac117fc99cc2c31ad726d0": "G = S_m", "092f4e2c9a879152faab7be3496c54be": "\\begin{align}\n \\sin x &= \\sum_{n = 0}^\\infty \\frac{(-1)^n}{(2n + 1)!} x^{2n + 1},\\\\\n \\cos x &= \\sum_{n = 0}^\\infty \\frac{(-1)^n}{(2n)!} x^{2n}.\n\\end{align}", "092f71002698fa24a1010c8459148153": "2^{7 \\times 8} + 2^{8} = 72057594037928192", "092fe1dcd5865e2de38bd7f431441b44": "N((ab)\\sigma) = N(\\sigma) \\pm 1", "09304a387a9b9611762eb244c17a91db": "q < 3 ", "0930bc94c84a1bf6d78bc9c94903e7ea": "E_n^{(2)}=\\frac{ m a^2}{2 \\hbar^2 } \\sum_k \\left|\\left(\\delta_{n,1-k}+\\delta_{n,-1-k}\\right)\\right|^2/(n^2-k^2)", "09311b6a1e840b295537603664157e2c": "\\left(Y, \\Sigma_Y\\right)", "0931214e038fcf39e2141121f5bcbce4": "A:=\\{x_1,x_2,\\dots,x_k\\}", "093138125585eb86a5b026d0d4184efd": "f(x)\\le f(y)", "09316b9d3832684c56a8ba24d2a8f404": " G \\times 1 ", "0932420622ff1696520914e8552efa05": "\\mbox{MAC} = \\frac{S}{b},", "09325b10204d4297bf53de310497577a": "\\frac{1 + {\\scriptstyle\\frac{4}{5}}z + {\\scriptstyle\\frac{3}{10}}z^2 + {\\scriptstyle\\frac{1}{15}}z^3+ {\\scriptstyle\\frac{1}{120}}z^4}\n{1 - {\\scriptstyle\\frac{1}{5}}z}", "09325c95b25d932a3b059ba5d686d067": "\\ A_{\\mu} = \\sum_a A_{\\mu}^a T^a ", "093279bb243792c430c94675ef8cb8a0": "(2+x)", "0932ae8f457aa22eaca13a3ce516ccf0": "\\mathbf{p} = \\gamma(v)m \\mathbf{v}", "0932baa9d80dca487c221eb90f1f6382": "c(X^*, X)", "0932bda7173e52c1ef331b9cf15d7017": "y = f(x_1,x_2,\\dots,x_n)", "093352bd5f5cf7937624857f6d5aaff9": "P_i(x)", "09338ab83351bc720ba02756829157dd": "P(E) = \\int_{\\omega\\in E} \\mu_F(d\\omega)\\,", "09338e33293992563d2b8b7428030b9f": "W_X(t,f) = W_x(-t,-f) \\, ", "0933b9de7a6c3848ed3bb40a5f4e9c9b": "\\pi \\sim_{st} \\pi'", "0933d7dbf9b56f8c222620eedf008dd5": "p = { {\\tbinom{10}{1}} {\\tbinom{14}{11}} }/{ {\\tbinom{24}{12}} } = \\tfrac{10!~14!~12!~12!}{1!~9!~11!~3!~24!} \\approx ", "09343b9e135a7633d2bbd941140e27af": "v = \\frac{c}{n(\\lambda_0)},", "09345faacdf4f9e55a9c523272b8efbc": " \\pi^{ij}", "093465bedcb6b80aae243c0049ba49a8": "\\displaystyle u_t=6(u+\\epsilon^2u^2)u_x+u_{xxx}", "0934869ba047b11cf4094c25e60740fe": "x\\in\\mathbb{R}^d", "09348b0b972dea30099e6d228d47c246": "H|n\\rangle=E_n|n\\rangle", "0934df980baf56854b50ce790276ce00": "\n\\frac{\\partial \\Delta E}{\\partial P_x}=\\alpha_0\\left(T-T_0\\right)P_x+\\alpha_{11}P_x^3+\\alpha_{111}P_x^5 - E_x = 0\n", "0934f7e6dbac2dd708e040808412f724": "\\ {L} ", "09352585aa32af7e20ce9dd32c1d2467": "\nu = u_{0} + u_{1} + u_{2} + u_{3} + \\cdots \n", "093555e79df91ae6b7e73db054f0e17c": " \\mathbf S(p) = \\begin{bmatrix} s_{11} & s_{12} \\\\ s_{21} & s_{22} \\end{bmatrix} ", "09355d08965267aa05b5e651b748e3bd": "\\C^n", "093565b44e4b02d268e5552c2c892a55": "Z = \\frac{\\nu}{\\omega} \\ll 1", "09356d00e4cc296c78e526b0e06d708e": "(p',u)", "0935b9ae317c4f6355ae238e936416c8": "\n\\{ l_a,l_b\\} = \\epsilon_{abc}l_c, \\ \\{l_a, n_b\\} = \\epsilon_{abc}n_c, \\ \\{n_a, n_b\\} = 0\n", "0935d32289a4b25680d1ef04b4d5d0ba": "\\{\\Delta\\ X\\}", "0935d653167a2f43f6b60532caedc09c": "w^\\perp/w", "093603c12920c42248564b69bfa16890": "f_4(\\omega)\\,", "09360629420960e3f6cc8b343e048fb2": "O_{4}", "0936080d51060e42d81f404e26dbb610": "|\\pm\\rangle \\equiv |m_J = \\pm 1/2, m_I = m_F \\mp 1/2 \\rangle ", "09360b5582197fecc1a7c4eb7531f2c8": "\\textstyle (x_0, x_1, \\ldots)", "09362f0f818c3fe663de23d2d787a9cd": "x_i=x_j", "093674ccfc9d481cfc042f76831c3f56": "\n\\begin{align}\nY_i^{1\\ast} &= \\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i + \\varepsilon_1 \\, \\\\\nY_i^{2\\ast} &= \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i + \\varepsilon_2 \\, \\\\\n\\ldots & \\ldots \\\\\nY_i^{m\\ast} &= \\boldsymbol\\beta_m \\cdot \\mathbf{X}_i + \\varepsilon_m \\, \\\\\n\\end{align}\n", "0936a4bba5aa2ce62d737887a7883b52": "W^T = C^{-1}_{Y}C_{YX} .", "09372411cfdc2b47abc230c2e60059b2": " m = log_e{ \\frac{ p_1 }{ p_2 } } ", "093738e3f6e59e278474a5500092d00d": "k^i_y = \\frac{k_y}{M} ", "09373d79c9a95008a565ecb19efd58d9": " \\langle\\bar\\psi(k_1) \\bar\\psi(k_2) ... \\bar\\psi(k_n) \\psi(k'_1) ... \\psi(k_n)\\rangle = \\sum_{\\mathrm{pairings}} (-1)^S \\prod_{\\mathrm{pairs}\\; i,j} \\delta(k_i -k_j) {1\\over \\gamma\\cdot k_i - m}", "0937619de23cd643fbc344b16ab749d2": "h_n = |f_n-f|^p", "093777afbabf9bd67c9197dfea796ea2": "|v_0v_1|=|v_1v_2|=\\cdots = |v_{d-1}v_d|", "09377a48da75477893c34af1acbb27cf": "F_{p-\\left(\\frac{p}{5}\\right)} \\equiv 0 \\pmod{p^2}.", "09378bff675841379c42ee11de4a9b6b": "f(0)\\oplus f(1)=0", "0937ebf35ea28cec4b23b3108ab7e6f6": " r^2 R'' + r R' + r^2 k^2 R - n^2 R=0. \\,", "093827ded29f910ac93f96adc251ef57": "L[y]=y\\circ (t+1) - y\\circ t = \\Delta y", "0938313f7cc2962dd463e47bd554c671": "\\Delta t\\rightarrow 0", "093846933be57dbaa33db94d3041c666": "\\textstyle{\\sum_i p_i = 1}", "09387dd5f37856997143c6a1b56a0e3f": " \\int_G |f(x)|^2 \\ d \\mu(x) = \\int_{\\widehat{G}} |\\widehat{f}(\\chi)|^2 \\ d \\nu(\\chi) ", "093898651fcefd9850fd8f3f661ebbd3": "E_{1,0}= 510,260 * \\frac {260}{510,260} * \\frac {500,200}{510,260}", "0938f06a54a0e2cc6636ddf65cabfaba": "(|1\\rang - |0\\rang)/ \\sqrt{2}", "09398cf2fd5a8aa9788f4701480a00ef": "\\mathcal{F}^2(\\mathbb{C}^n)=\\{f\\colon\\mathbb{C}^n\\to\\mathbb{C}\\mid\\Vert f\\Vert_{\\mathcal{F}^2(\\mathbb{C}^n)}<\\infty\\}", "093990db1805c5a4b0fdc77165490776": "\\,p(x)", "09399c2fa2b785004547456b40180b29": "V=\\{v:[0,1] \\rightarrow \\Bbb R\\;: v\\mbox{ is continuous, }v|_{[x_k,x_{k+1}]} \\mbox{ is linear for } k=0,\\dots,n \\mbox{, and } v(0)=v(1)=0 \\} ", "0939ba99c8703268d6a7034a4b018596": " y_{2i} = \\begin{cases} \n y_{2i}^* & \\textrm{if} \\; y_{1i}^* >0 \\\\ \n 0 & \\textrm{if} \\; y_{1i}^* \\leq 0.\n\\end{cases}", "093a254955308018b03188e94a6e6936": "\\frac{dg(p)}{dp} = \\frac{1}{f''(g(p))}", "093a5e576b1d3ad7036b1fa685356f09": "\\bar{J} = \\Gamma(\\omega)* c/4\\pi", "093a6deacfacb4edd1fc263f67ce8c10": "a\\mapsto a.x-x.a", "093a74151e91c98a425681676eda6248": "\\lim_{x \\to 0^{-}}{x^{-1}} = -\\infty", "093a9ced6065e190272528e9f3dddae2": "\\lim_{n\\to \\infty}\\widehat{S}(n)=0", "093b3a3752fce3871c5b49f0d88303b0": "v^* = -\\infty", "093b5e2d3cc4d86e5bd2bc9048c22958": "\\begin{array}{rcl}\ns & = &\\displaystyle 1+2+4+8+\\cdots \\\\[1em]\n & = &\\displaystyle 1+2(1+2+4+8+\\cdots) \\\\[1em]\n & = &\\displaystyle 1+2s\n\\end{array}", "093bb55322de6df7782d6ad0deb16272": "\n[ u, v ]_{p,q} = \\sum_{i=1}^n \\left(\\frac{\\partial q_i}{\\partial u} \\frac{\\partial p_i}{\\partial v} - \\frac{\\partial p_i}{\\partial u} \\frac{\\partial q_i}{\\partial v} \\right).\n", "093bd73b2bf47508a3b37ef266bc141e": "\\phi(\\exp(x)) = \\exp(\\phi_{*}(x)).\\,", "093bebfe1d89bb89e7421efe818289e0": "\n\\mathcal{L}=\\left\\{L=L':~x^{\\{m\\}'} L x^{\\{m\\}}=0\\right\\}.\n", "093c14b44ec3998a6609d5725d2b3b9f": "D_2\\left(E\\right) = \\frac{2 \\pi}{c_k^2}\\left(E-E_0\\right) ", "093c3d0431033c8751d76e763e7c5ea0": "{^{(4)}}\\Gamma^0_{ij}", "093c3ed8582f4044c020b3bad49c3892": "z_i = (1, z_{2i}, \\dots, z_{pi})", "093c6c2ad58bb8156d94c5d102f3dc2d": "\\begin{align}\n x & = r \\sin\\theta\\cos\\varphi \\\\\n y & = r \\sin\\theta\\sin\\varphi \\\\\n z & = r \\cos\\theta\n \\end{align} ", "093c8e205679042f3dafbae84a800cc0": "h=\\sum_i dy_i \\; dy_i", "093cafe29f6d7452dec1e5eb9e01e9ed": "O(n^{d+1-\\delta})", "093cfbd44b73b914c0b5564c925438c3": "a\\equiv^\\ast\\!b\\,(\\mathrm{mod}\\,\\mathbf{p}^\\nu)\\Leftrightarrow \\mathrm{ord}_\\mathbf{p}\\left(\\frac{a}{b}-1\\right)\\geq\\nu", "093d2b0e27fbaa55989e56c93c71fd18": "\\left(\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{\\sqrt{3}\\over 2}\\right).", "093d2b892a55ce67f922c5ce1eb4e32b": "(x,y,y',y'',\\dots,y^{(k)}).", "093dd868c534d861bd8b03ff5e583ced": "\\mathcal{L}_I = -e \\bar\\psi \\gamma_\\mu A^\\mu \\psi \\, - \\, (Z_1 - 1) e \\bar\\psi \\gamma_\\mu A^\\mu \\psi", "093e377d554e2d8aec8ff3269c3a0356": "SU_{\\mu}(2) = (C(SU_{\\mu}(2),w)", "093e3b158570c405f18d08369be44de9": "\\scriptstyle d ", "093e3be7518e3afcccefe9181c4542e6": "\n\\mathbf x_1\n= \n(\\mathbf P_B^{\\perp} \\mathbf A)^{+}\\,\n\\mathbf d\n,\n\\qquad\n\\mathbf x_2\n= \n(\\mathbf P_A^{\\perp} \\mathbf B)^{+} \n\\,\n\\mathbf d\n.\n", "093e91ccce3d59664b5cc8bd7d533b87": "\\mathcal{A} = \\mathfrak{M}\\{\\mathcal{B}\\}", "093eb46b95faa8734786a59215412808": "U_t \\geq X_t", "093f686f70f93af3ca0b28e12be67463": "||\\phi(a)\\land\\psi(b,c)||=||\\phi(a)||\\ \\land\\ ||\\psi(b,c)|| ", "093f8a967400f274ba09d084a57ef1af": "\\frac{c}{a}", "093fa97e24e55eeb98f14d3f51154ad4": "{F_y}' = qE' = -q\\gamma vB.", "094035c27ab8d07bed11709557d69220": "u_{12} = 3", "09411d4d4235225bb422cacee3826d77": "\\epsilon^{abc}", "094128623422276a599624300c3908f7": "S_{40:1}=107", "09415d667c9259d5a873c275204c2c85": "\\left\\lbrace x_i \\right\\rbrace ", "09418474ca66bee017b81a3563512aca": "x^2 = 0.\\,", "0941bedf48472baa899d2e8ea1564b90": "\\epsilon_{c} = A\\ln(r_{s}) + B + r_{s}(C\\ln(r_{s}) + D)\\ ,", "0941fb30d6a8f8d602b9b3dc5a4bdff1": " \\mathbf{v} = \\nabla \\phi \\qquad \\qquad (1) ", "094214a889d7620abdcca4fa84418868": "X_1 ... X_n", "094227dfb2405ce5b48863d9f5819939": "A_{n,k}=f^{-1}(I_{n,k}) \\,", "0942735dea8794d4f2e3deaa83ae7997": "\\scriptstyle >4\\times10^{16}", "0942a5ac181bc56a8d018034c0a78251": "\\bar{X}_n=\\frac{1+r}{1+r-2rp}", "0942b63c4f0f7af563b78c11bc220ee3": "\\lambda'_k", "0942c78c287263bcca177acf87db85da": " \\vec{e}_1 = \\frac{1}{\\sqrt{2}} \\left( -\\partial_u + \\partial_v \\right) ", "0942e92262a38e93d87c6efc63ec10fc": "M \\times \\mathbf{x} = \\mathbf{r}", "09430f293c53fa1d2b50300a4c97c432": "{n-1 \\choose k}", "0943b817f7702c7884cb9a84e1f1f76d": "\\Delta l^a-D n^a=(\\gamma+\\bar{\\gamma})l^a+(\\varepsilon+\\bar{\\varepsilon})n^a-(\\bar{\\tau}+\\pi)m^a-(\\tau+\\bar{\\pi})\\bar{m}^a\\,,", "0943ce1ac70bfe14432454af0974699a": "f'(c-):=\\lim_{h \\to 0^-}\\frac{f(c+h)-f(c)}{h}\\ge0,", "09443db0f31f1eda9eb405c7bb3b2506": " {R_0}^6 = \\frac{9\\,Q_0 \\,(\\ln 10) \\kappa^2 \\, J}{128 \\, \\pi^5 \\,n^4 \\, N_A} ", "094465bf966548d186abe0852603823b": "f^{\\#}: \\Gamma(U, \\mathcal{O}_\\mathfrak{Y}) \\to \\Gamma(f^{-1}(U), \\mathcal{O}_\\mathfrak{X})", "094483aeb9acc83cc696e281ad54e71f": " Y^I", "094485f24b16909fa0ad4595e0af8cb0": "\\ K_m\\frac{\\part \\overline{u}}{\\part z} = u_*^2 ", "0944896e940927c2c8cddf4e98ba9e6e": "\n\\kappa = \\cfrac{6(1+\\nu)}{7+6\\nu}\n", "0944ba12c6859ff459b96bb3df8e4c13": "\\textstyle k=0,\\ldots,m,", "0944dccc974c8ecd2eb635e9e07249df": "x_1=g(x_2)", "0945350e808ebfc8a74d3721ffa12fe6": "\\Kappa \\,", "0945cf4b6889981499e131660b7a201b": " b = \\frac{\\sqrt{nt_2 - t_1}}{n} = \\frac{\\sqrt{na_1^2 + a_1 - 2na_2}}{n} ", "09461d4a47d5f335d7a099083bdddde8": "f''(x) > 0", "094622e462581ca208963660990c46c5": "1\\over{\\sqrt{5}}", "09466659cb844a81439d1ed468f4e8d0": "\n |(j_1j_2)JM\\rangle = \\sum_{m_1=-j_1}^{j_1} \\sum_{m_2=-j_2}^{j_2}\n |j_1m_1\\rangle |j_2m_2\\rangle \\langle j_1m_1j_2m_2|JM\\rangle,\n", "09467e327f947a9328702c0c798537c1": "\\mathrm{d}\\,U = \\delta\\,Q -\\delta\\,W", "0946ff2e62a99a1118d5b2ca3914f9a0": "b=\\frac{L^2}{12A}", "094718e343088ac76a79ade7d407761c": "L_d", "09472afa4e28f372f4e468dd1ba70728": "G^\\circ := \\{x' \\in X^* : \\sup_{x \\in G} |\\langle x', x \\rangle | \\le 1\\}", "09472d594f6078f0940ddcb4784fe3a4": "\\vert G \\vert \\leq 1.", "09473a49c3ab4d762f8532d1d99901bc": "\\mathrm{Sh} = 2 + 0.552\\, \\mathrm{Re}^{\\frac{1}{2}}\\, \\mathrm{Sc}^{\\frac{1}{3}}", "094753ba0fc36eec5e9a3c66ff174f1c": "r_n=\\sum_i^I x_{ni}", "0947580233afe28450870ca2a4865c7d": "\\neg r", "094764b8dd5d98ad5d48fd84ed303fd8": "\n\\sum^n_{i=1}F^Q_id^P_i = \\int_\\Omega \\sigma^Q_{ij}\\epsilon^P_{ij}\\,d\\Omega\n", "09478deae56637f9624b72d6a3896bd7": "{\\tilde{E}}_{n-1}", "09478ff6b31f01fd19507cab34433c16": "\\phi_1,\\dots,\\phi_{k-1}\\in H^1_0(\\Omega)", "09479a199fdef109da421b3a4d5f020f": " \\lambda_D =\n\\left(4 \\pi \\, \\lambda_B \\, \\sum_{j = 1}^N n_j^0 \\, z_j^2\\right)^{-1/2}", "0947f85161b05919d96940f3de14852e": "B ", "094831ae988bb10fad007385038a78c2": "A \\and B", "0948394bec59903800ad2992fabb0b30": "\\scriptstyle 0\\,<\\,D\\,<\\,1", "0948a636d65bdc47b8770c1b99e8f236": "\\zeta(s) \\sim \\sum_{n=1}^{N-1}n^{-s} + \\frac{N^{1-s}}{s-1} +\nN^{-s} \\sum_{m=1}^\\infty \\frac{B_{2m} s^{\\overline{2m-1}}}{(2m)! N^{2m-1}}", "0948c8c44d9f535faf8e2e0dbc712c8e": "k+\\lfloor(d-1-k)/2\\rfloor.", "094902a95b24144c9df29e7578cb9601": "c^d \\;mod\\;n^{s+1}", "09496e5f9f581bb18f582f397efba651": "a_0 = \\sqrt{\\frac{K/\\rho} {(1+V/a)[1+(K/E)(D/t)c]}}", "09497279b3ce7b167a7bdecaeba65651": " | Re(\\lambda_t) | ", "094984d07f40c2bd9f5bf69963c4fe10": "\\nabla\\cdot{\\bold u}=0", "0949e1a2b5df281c5408ad24e3da6931": "\\delta(P,Q)\\le\\epsilon", "094a23869a432b4fc6713f5fc4a63f1a": "x(S_j)", "094a4b7f64178507193890bac0caf4c8": "\\mu>\\mu_0", "094a507b0d1fcf973868aa32292bf876": "F_{a,P}", "094a56977f70cfb482781d28e7e569c7": "\\frac{}{} V=-\\cos \\phi", "094a758d2221c5331523ad29ab9bc6c3": "t \\mapsto (x(t), y(t)),", "094a7a5a07adf16ec5826539c2846cac": "c(t,s)=0", "094aa8306996bf985a4ec6eebc2a7559": "Fx = g \\ ", "094ae3d247fb35321e48fcb479f8b285": "~~~\\and~~~", "094b49264ee8427729dfcb00c8d27505": "\nL(\\lambda) = \n\\lambda\n\\begin{bmatrix}\nM & 0 \\\\\n0 & I_n \n\\end{bmatrix}\n+\n\\begin{bmatrix}\nC & K \\\\\n-I_n & 0 \n\\end{bmatrix},\n ", "094b52849e774704a0f97670237272ac": "\\rho_{\\rm total} (\\bold{ r}_0)= \\rho ( \\bold{ r}_0 )-\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 ) \\ , ", "094b80c4c580155f7ca62ea8367d09f4": "k(x,y)\\,", "094bd2fee5adc790afce9d8a740ac813": "\\Pi( T(u), (x) T(u'(x)) )", "094c00021bc538a30547d55d7eafedea": "\nS_2 = S_1 - k \\ln \\frac{c_2}{c_1}.\n", "094c47d866750d6fdc716e0852e437ef": "\\Delta u = 0", "094c67fbae9c46095a4a35113b0b6e81": "W = \\frac{1}{2}\\left[q + c\\left( e^Q -1 \\right) \\right]", "094c875557a127d33aeb67998d9c3a21": "e=H(x)-H(\\hat{x})", "094c8c2a19feab2ae05565f68406784f": "\\lim_{t\\to 0}f(t)=\\lim_{s\\to\\infty}{sF(s)}. \\, ", "094c98b7a30d55d18d41864d4d83220d": "H_1=-\\vec{d}\\cdot\\vec{E}", "094c9a72eb8a8b5945dadfed823e2029": "\\mathit{a}_\\mathrm{InPAs} = \\mathit{x}\\mathit{a}_\\mathrm{InP} + (1-\\mathit{x})\\mathit{a}_\\mathrm{InAs}", "094d18158607da9878b4c45058a581ca": "g(x)= x^3+b_2x^2+b_1x+b_0 \\,", "094d592d14f557ddfd0ba2559d45dd63": "u_\\eta = \\left( \\nu \\epsilon \\right)^{1/4}", "094db3efa7426e81b63abdbefd035742": " E=Q\\sqrt{\\frac{2U(\\varphi_0/\\sqrt{2})}{\\phi_0^2}}. \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(2)", "094de9a81e0e7af9e7a1956c2e1c5ad8": " W(x,0) \\approx W_0(x) (1-\\beta f_0 x), ", "094e9576987031cbd4bcba4500b740df": "\\{(a \\vee \\neg b) \\wedge b, \\neg a\\}", "094ecc18ebbedb0f9fd0a15c6bb5a067": "\\operatorname{Tor}_1^R(M, R/\\mathfrak{m}_R) = 0.", "094f269e7efe7e22e584f9ef7a476fba": "F_r = m \\ddot r -mr \\dot {\\varphi}^2 \\ ", "094f2e693aff125779d7580dfd75e9bc": "k^*/k^{*\\ell} \\cong H^1(k,\\mu_\\ell)", "094f314060e898c0178a790629c3da3c": "dQ \\equiv \\left ( \\frac{\\partial Q}{\\partial x} \\right )_{y,z} dx + \\left ( \\frac{\\partial Q}{\\partial y} \\right )_{z,x} dy + \\left ( \\frac{\\partial Q}{\\partial z} \\right )_{x,y} dz,", "094f317341c60034004f2dd57ee56726": " \\mathrm{Re} = {{{\\bold \\mathrm U} L} \\over {\\nu}}", "094f690fc31adc857e41175cbe31e63a": "\\delta < 1", "0950734193e735d40fa7b541922bd3cc": "X = {x_i} \\,", "0950954fe0e0b45837c415da31e1b313": " n_{f} = 2 \\sqrt { n_\\mathrm{osc} / \\omega} p cos(\\theta) ", "09509bb217a36c28fc8091bf80905aa8": " \\mathbf{x} \\cdot \\nabla f(\\mathbf{x})= kf(\\mathbf{x}).", "09510045c7bdf1dc74d9878ff851a5fd": "\\xi\\mapsto h(\\psi(\\xi))", "095105337e8db3125f8cae0140074f61": " S_1,S_2", "09511782eb08c632e717fe4deee18516": "\\ln{a}", "09511e38861116d97d796c6a9f3bce24": "u_a", "0951fbe5c8cbca415187b32a26cfbbe8": " \\int |gh|\\le \\left(\\int |g|^p\\right)^{1\\over p} \\cdot \\left(\\int |h|^q\\right)^{1\\over q},", "09521a5c8a46e66f356125b92d2d5a45": "K_j", "095233c697a7178fcbfdfe24ed52456b": "\\mathrm{RFN}_T", "09523a1cd616ab8d1bafb32ee2bd2974": "P(x, \\partial)u(x) = 0 \\mbox{ for all } x \\in W", "0952519696eb3154aac392fec57baf21": "E_{h} = E_{0} + \\frac{0.05916}{n}\\log \\left(\\frac{\\{A\\}^{a}\\{B\\}^{b}}{\\{C\\}^{c}\\{D\\}^{d}}\\right) - \\frac{0.05916 h}{n}\\text{pH} ", "09525cb1fa2ba08fe77fcb33734de90a": "\\langle s_{\\lambda/\\mu},s_\\nu\\rangle = \\langle s_{\\lambda},s_\\mu s_\\nu\\rangle. ", "095290f62fd852c7ed7b3478abd0817c": "a_0(-\\infty)^n\\,", "0952c66c88bf229d857571d4c0b4d059": "\\begin{Bmatrix} p \\ \\ \\ \\ \\ \\\\ q , r , s \\end{Bmatrix}", "0952c9ffb538736990bcb5dae0528dec": " \\mathbf{d^2F} = -\\frac{k I I'} {2r^2} \\left[ (3-k)\\hat{\\mathbf{r_1}} (\\mathbf{dsds'}) - 3(1-k) \\hat{\\mathbf{r_1}} (\\mathbf{\\hat{r_1} ds}) (\\mathbf{\\hat{r_1} ds'}) - (1+k)\\mathbf{ds} (\\mathbf{\\hat{r_1} ds'})-(1+k) \\mathbf{d's} (\\mathbf{\\hat{r_1} ds}) \\right] ", "0952d38192ddf1f35cbe57e2dbcf270c": "\\frac{\\eta(c_\\eta(0,\\xi_1)-c_\\eta(0,\\xi_2))}{\\xi_1^2-\\xi_2^2}", "0952d4d5548366cfbb0508ddef98ac7f": "X = -\\frac{1}{\\Omega\\left(E\\right)}\\sum_{Y} Y\\Omega_{Y}\\left(E\\right)\\,", "09536cf4e7246ac68ed5cbcdacc496c7": "\\{B_i\\}", "095370a99280d227a2dfcf51c7db6377": "P(E_1)=P(E_2)", "0953df157b0e3bdea30ca43e8f93c589": "{G_1}= \\frac{1}{4}", "095403dfb91e5f1a03b3921da626a07e": " \\mathbb{E}(A) = \\operatorname{Tr}(A S) = \\operatorname{Tr}(S A). ", "095456c652d061a8dcc43a3dd76190b1": "q = A \\wedge B \\wedge C", "0954a08a2c8cf0f4e7bd803270720932": "x_{k+3}", "0954ad8f140b293c7db51dd3a944f980": "\\mathbf{x}_{i}^{\\rm T}", "0954d50e959a8d0bea34b757cfb81f8a": "87^2", "0954eaab43d03a708e1aea773b0503fe": "\n \\mathbf{x} = \\boldsymbol{\\varphi}(\\mathbf{X}, t)~; \\qquad\\implies\\qquad \n \\boldsymbol{F}(\\mathbf{X},t) = \\boldsymbol{\\nabla}_{\\circ} \\boldsymbol{\\varphi} ~.\n", "0954f477e7638d610dc3ce6944a8cd82": "H( \\omega)", "0955997ef9a58a6e7c3c4c89f19d9eb8": " p(x) = \\sum_{i=0}^{20} d_i \\ell_i(x). ", "09559aac3e7165615ee2854c47ea561c": "\\tfrac{\\hbox{Volume before the event}}{\\hbox{Volume after the event}}", "0955b0c09211920de4e8edcfa209da3a": "v(p,t)", "0955b39526c10a18f246e8a2d04c8f2d": "\\scriptstyle{\\left(\\frac{dk}{dt} \\rightarrow 0\\right)}", "09562e84f3e57fe020a11623c906039a": "\\chi(T)", "09567e4092aa67f23d128e2ae835c66c": "P_3 = (P_0(1+r)^2- c(1+r)- c)(1+r) - c", "0956aa0dc3f908f6ca6850a27799b75d": "\n\\begin{align}\n\\min &\\left(f_1(x), f_2(x),\\ldots, f_k(x) \\right) \\\\\n\\text{s.t. } &x\\in X,\n\\end{align}\n", "09573235d73083444dcdace7a99d51ff": " P(S^t \\mid S^{t-1})", "095742bb8f0481559e2d3fe85c2f71b0": "\\deg^+(v)=0", "095779904ee10259ed98c005f322a8b4": "\\partial (x_1\\wedge\\cdots\\wedge x_{p+1}) = \\frac{1}{p+1}\\sum_{j<\\ell}(-1)^{j+\\ell+1}[x_j,x_\\ell]\\wedge x_1\\wedge\\cdots\\wedge \\hat{x}_j\\wedge\\cdots\\wedge\\hat{x}_\\ell\\wedge\\cdots\\wedge x_{p+1}.", "0957cd0bd5d645ae98f6d045b075e2ed": "0 \\leq i \\leq n", "0957f3abe0246a2d5ae3691f3285252c": "|a - b| = \\sqrt{(a - b)^2}.", "095854a35f357108bdd889fb3fa9c180": "\\mathcal{H} = \\hbar \\omega [a_1^2(t) + a_2^2(t)] = \\hbar \\omega |a|^2 \\ ", "095884dd846abad1353040287b637f0f": "\\tan\\theta\\approx\\sin\\theta=\\frac{h_a+h_b}{G}", "095898131dafd77eeb3371eeb43a1d4b": " p(x) = a_2 x^2 + a_1 x + a_0 \\, ", "0958bca8472d89437654500252cca8f2": "E \\psi = - \\frac{\\hbar^2}{2 L_1} \\frac{d^2 \\psi}{d Q_1^2} - \\frac{\\hbar^2}{2 L_2} \\frac{d^2 \\psi}{d Q_2^2}- \\frac{\\hbar^2}{m} \\frac{d^2 \\psi}{d Q_1 d Q_2} +\\frac{1}{2} L_1 \\omega ^2 Q_1^2 \\psi+\\frac{1}{2} L_2 \\omega ^2 Q_2^2 \\psi ", "0958ff175cfcf08f7dd21940e9ad3e48": "N_{rep}", "095909d830f978137dbdd05dd4c799c8": "X_j\\sim\\Gamma(r_j,\\lambda_j)\\!", "09590f214e0a7a442685461e7fef3c13": "Ae^{i\\omega t}.", "095927ee22e00b24e9abbe3f2919526f": "\\mathcal{X}\\times \\mathcal{U} \\rightarrow \\mathbb{R}", "09593c7ef06dd83105a50da58b009551": "F \\circ j_1 = \\mathrm{id}, \\; F \\circ j_0 = Q.", "0959426bf646ca3acbb3a35e1008f06c": " \\mu_z=-g_S \\mu_\\mathrm{B} m_s", "09596554c8152b84ed61c789abf5f508": "c(t) = A_c\\sin\\left(\\omega_\\mathrm{c}t + \\phi_\\mathrm{c}\\right).", "0959b686c09969aacd1c134f708d3edd": "\\phi=(s+d)/2", "0959d68c51153e63e94f119cf882e31a": "\\hat{\\mathbf{e}}_z", "0959e4a345fb6e190468837bc88f1617": "\\text{if } A \\subseteq B \\text{ then } A \\times C \\subseteq B \\times C,", "095a5c3d1f428a09a5d5698d62870b04": "\\varnothing = X_{-1}\\subset X_{0}\\subset \\cdots X_{n}=P.", "095abe81cfbe7bd18c9d4d2db25c704e": "a_i^j x^i = y^j", "095ad04ab63221de2e90e52c139ecdc3": "\\mathrm{C_A = [A] + \\Sigma p \\beta_{pqr}[A]^p[B]^q[H]^{r}}", "095afb28a327e39c1b8aa65dad1aa047": "p=\\rho_f g z.\\,", "095b351e35806fa18cf49a8f9f876d25": "\\varphi^2 = \\varphi + 1 ", "095b376e98c5ae57cf7f7db6b68cb33d": "{\\tan \\delta}", "095b571b26ca92845328976c579ecbef": "Y \\sim \\Gamma(n,\\tfrac{1}{\\lambda})", "095bd0c7e0196c6060c03bf7f85677b5": "T : z \\mapsto z^{-1},", "095c73b948a40367964d73e43c5d0034": "1 + \\varepsilon", "095c9baecd446ee5d348d90a78a618ac": "\\mathbf{Z} = \\mathbf{Q}\\boldsymbol{\\beta} + \\mathbf{f} ,", "095ca029065b3107721a232b440137f0": "\\begin{align}(a\\cdot\\phi)(x) &= a\\cdot\\phi(x)\\\\\n(\\phi\\cdot a)(x) &= \\phi(a\\cdot x).\\end{align}", "095cafcc7a3a21ca5b06eed0089db8f8": "\\pm 17.8455995405\\ldots", "095cb6dd9d2b7d3ccf7e5fa7c5c1c26c": " \\xi^{(k)}_\\lambda = D E(\\lambda) \\eta^{(k)},", "095cc73e369d7a4543580ecb7be622fc": "\\ H^{BM}_i(\\mathbb{R} ^n ) ", "095cfd7f7873d712a4219a2b816c7d2e": "\\varepsilon_0+1, \\qquad \\omega^{\\varepsilon_0+1}=\\varepsilon_0\\cdot\\omega,\\qquad\\omega^{\\omega^{\\varepsilon_0+1}}=(\\varepsilon_0)^\\omega,\\qquad\\text{etc.}", "095d2252d6f62e164bbabe2e7134c830": " w = d + [2.6m - 2.2] + 5R(y,4) + 3R(y,7) + 6R(c,7) \\mod 7 ", "095d2469b2b30ffee0188cd05f74140d": "D(f\\circ g ) = \\left( Df\\circ g\\right) \\cdot Dg,", "095d5e9d33719f85e7a88caa2a8380a1": "y = 2/3E", "095d62f182c280893f56fa8c5bd0beb4": "\\nabla = \\hat e_i \\partial_i", "095de5c44408794f1b2bdbd9cc73cfbf": " \\frac{P_1}{T_1}=\\frac{P_2}{T_2} \\,", "095e0b4f6b0d16656cdec03341422ee9": "f''(x_n) = \\frac{1}{n}", "095e0c08dfbc23525edd438905f70751": "\\lim_{x \\to c} ax + b = ac + b", "095e5eb9ad0d34dd1394619992bb9638": "y\\in H_q(M)", "095e67779d8ca7e28c9da82d178f032f": "X \\perp\\!\\!\\!\\perp Y \\,|\\, W ", "095e9e9f272d6de146b05cff6bcf29ec": "\\beta < \\delta", "095efc2c0cd01c7483233734c6026f31": "x \\ne \\tfrac{\\pi}{2k}", "095f79d6947db27bd085bd08c7fb74fc": "\\left(m_\\mathrm{p}+\\frac{\\rho_\\mathrm{c}V_\\mathrm{p}}{2}\\right)\\frac{\\mathrm{d}\\mathbf v_\\mathrm{p}}{\\mathrm{d}t}=\\sum\\mathbf F + \\frac{\\rho_\\mathrm{c}V_\\mathrm{p}}{2}\\frac{\\mathrm{D}\\mathbf u}{\\mathrm{D}t},", "095feae595d9a84c43b3fec5c50c4456": "\n \\cfrac{\\partial{W}}{\\partial \\lambda_1} = 2C_1\\lambda_1 ~;~~\n \\cfrac{\\partial{W}}{\\partial \\lambda_2} = 2C_1\\lambda_2 ~;~~\n \\cfrac{\\partial{W}}{\\partial \\lambda_3} = 2C_1\\lambda_3 \n ", "0960050b9e095c506fc2a530b5b9f57e": "\\Sigma_{(W) \\tau}", "0960565040e50308c60d3f31261850bf": "\\scriptstyle Q(\\sqrt{d})", "0960744036893f503a758250ed66ae44": "\\mathbf a = {d\\mathbf u \\over dt}", "09607f1e65b76e690a8c8ccd5f19c9b1": "d^{4}x", "0960f0fa6f0edf1c5617bed02d35d400": "\\overleftarrow{Y}=-jY_{\\varepsilon } cot(k_{x\\varepsilon }w)", "096132a030180689ee7dec2c0cee9b0d": " x^{\\alpha}{}_{, \\gamma} = \\delta^{\\alpha}{}_\\gamma ", "0961b7929edcd1adc1b80168efe1f1e9": "P\\times P/G", "09621f8d9b25f524f08c1c3be42f74ff": " \\mathcal K_X", "096249664ae745478eaf33d3ccf96078": "\\cap \\!\\,", "096254c7552111f593bb632a91205f32": "g(t)", "09626e60fb4ec8af0338e7c2d8ced428": "\\kappa_3", "0962b4b8c4ba6c674eba0d8a139f1847": " s_j = High[j] - High[j - 2] + Low[j] - Low[j - 2],\\!", "0962b784783898155442e71bcc2c61d0": "S:X \\times U \\times \\Omega \\to \\mathbb{R}", "0962c4a09af7ff74899dd9c28456f237": "|S| = \\gamma n\\,", "0962cbab3034401a572b6225c8adce34": "\\cos(t \\sin(x)) = J_0(t) + 2 \\sum_{k=1}^\\infty J_{2k}(t) \\cos(2kx) ", "0962d64102fe6eb19848f88d2f216ee6": "g'_{\\mu 4}= g_{\\mu 4}", "096341dbe0702c7f67fef27239d01d6b": "\\rho_d", "0963a5ceac41578b77c25c7a6bf9ff3c": "\\gamma_n \\sim \\frac{B}{\\sqrt{n}} e^{nA} \\cos(an+b)", "0963af90edef723ecf56942cfea46cc8": "R_s=\\frac{2v^2\\cos\\theta\\sin\\theta}{g}\\left(1-\\frac{\\sin\\theta}{\\cos\\theta}\\tan\\alpha\\right)\\sec\\alpha", "0963beafb48c2673a3205af9c373448f": "bb^{-1} = b^{-1}b = 1", "0963cfbb90c638cd43ca6a43115c2a2b": "\\beta _{31}", "0963fdc248538691a611f3dde51ee512": "\\Delta f = \\sum_{i=1}^n {\\frac{\\partial^2 f}{\\partial x_i^2}}", "0964764f7e9a82f9a61e983b5981755e": " T_5(x) = 16x^5 - 20x^3 + 5x \\,", "0964ed758f2f62337e158772917ea184": "P([\\omega_1, \\omega_2,\\cdots ,\\omega_n])= p^k (1-p)^{n-k}", "09650b07280b0293af1c51f1b32e2714": "I_1=[\\alpha_1,b],\\ I_2=[\\alpha_2,\\alpha_1],\\ldots,\\ I_n=[\\alpha_n,\\alpha_{n-1}],\\ldots ", "09650ffec4f07c5d7550e4d9ecff7216": "\n K_\\text{ww}(s) = \\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} c_\\text{ww}(k)\\, \\text{e}^{iks}\\, \\text{d}k.\n", "096544be8a676d2c106621014410cb9d": "\\bar y", "0965765e2fb0e390110bf394067143ec": "s \\approx 9.017.", "0965ac349d461503070235f7f7f72a67": "a + b = 101", "0965c09c0aec92659986192f7e01b563": " \\theta_C \\, ", "0965d99537bdd25a6f0e283cc2c8ae31": " V(x) \\ = \\ V_{max} (e^{-x /\\lambda})", "09678f6206c24aaec93304fa2dd73fad": "p(x+0)=(x+0)^3 -7(x+0) + 7\\Rightarrow p(x+0)=x^3 -7x + 7 , v_0=2", "0967b0a71f7dfdef1aae03b702abe6af": "g_L", "0967bfbb79279f6a37470eb9702c8c9a": " [a,a,[b,b,x]] = [b,b,[a,a,x]] ", "0967cd071190719cbd546020d108a4d9": "\\Phi_m = \\int \\!\\!\\!\\! \\int_S \\mathbf{B} \\cdot \\operatorname{d}\\mathbf S", "0967f23ea1b49959e7044d2dca5a3939": "\\epsilon=h\\nu.", "096803b2bed1cbe0fd72476a0ce77066": "AQHI=(\\frac{1000}{10.4})\\times[(e^{0.000537\\times O_3}-1)+(e^{0.000871\\times NO_2}-1)+(e^{0.000487\\times PM_{2.5}}-1)]", "09683c59a82119f510512351a6894820": "R_{2k+1}(V) = \\left(\\frac{(2k+1)!!V}{2^{k+1}\\pi^k}\\right)^{1/(2k+1)}.", "096878ac00613b27f36f59926121e828": "|\\psi'_r\\rangle=\\alpha_0|100\\rangle + \\alpha_1|011\\rangle", "0968b3b2971c0609830e848ab80d8ae7": " \\boldsymbol\\Psi + \\mathbf{C} + \\frac{\\kappa_0 n}{\\kappa_0+n}(\\mathbf{\\bar{x}}-\\boldsymbol\\mu_0)(\\mathbf{\\bar{x}}-\\boldsymbol\\mu_0)^T ", "0968c8e554f07e3dbeff90d4f29c633c": "e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \\ldots, 2n, 1, 1, \\ldots]. \\,", "0968fca9807941e80550ee99ce0c2240": "6*8=48", "096904478ccd486df77f3feb9047b943": "x + t", "09691935278a5c625d71d8ea76d5e437": " \\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\\n 0 & \\cos(\\theta) & 0 & \\sin(\\theta) \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & -\\sin(\\theta) & 0 & \\cos(\\theta) \\end{matrix} \\right]. ", "0969414f4d05136d2436d4134a68cb99": "F(x,y) = x + y + xy.\\ ", "09694f8475d80b14df6bbc83c1fb57af": " a + d + b = 180", "096971d45197ed1d9a75ce0cd47dae24": "\\{ 1, i_1, i_2, i_3 \\}", "0969898997e5522aca9617e7de94e68c": " ~\\hat b\\hat b^\\dagger -\\hat b^\\dagger \\hat b=1~", "0969b386f6f36c93005270a70dbcd718": "(a,b)\\cup (c,d)", "0969d5ec1b40b50173b662915d2969ec": "g=dz^2+dx_1^2+dx_2^2+\\cdots+dx_n^2\\,", "096a17a4aca0892fc0c5f93fc6424d76": "\\omega^{\\left( \\omega^{\\left( \\omega^7\\cdot6+\\omega+42 \\right)}\\cdot1729+\\omega^9+88 \\right)}\\cdot3+\\omega^{\\left( \\omega^\\omega \\right)}\\cdot5+65537", "096a465d7390b3d6ad85475dc16dfa5b": "m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r} = m \\sum_j \\left[ \\sum_i \\ddot{r_i} {\\partial r_i \\over \\partial q_j} \\right] \\delta q_j ", "096aa3f81d7e58d9779a0cbe73ec51f0": "\\epsilon_{\\mu \\nu \\lambda \\sigma} \\,", "096ace34b111266e5369566ddc2de1d6": " H(t) = \\int_0^{\\infty} f(E,t) \\left[ \\log\\left(\\frac{f(E,t)}{\\sqrt{E}}\\right) - 1 \\right] \\, dE ", "096afb8c4a105c41a11e79dbb2d0a86a": " \\lim_{x \\to \\infty} f(x) = \\infty, \\lim_{x \\to a^+}f(x) = -\\infty. \\, ", "096b4eceb1c9aa32f30a59eba18b9cd6": "\\scriptstyle{\\tan \\beta = \\frac{b}{a}\\tan\\phi}\\,\\!", "096b57dec2d3fdbb30eacfd5138c09bc": "\\Gamma = \\sum_{i = 1}^{N} s_{i} \\Delta x_{i}", "096b5dbdd0efe5fdefa4e357e9a7225c": "E.", "096b9d223a695b5f6bee9412cc117441": "\\mathbf{P}( X \\ge \\mu+a) \\le e^{\\frac{-2a^2}{n}}, \\qquad a > 0", "096c1dd7a54c26d0f0b35cef8f9ed3ea": "\\mathrm{\\frac{M}{L t}}", "096c5d547901e3a6fc893504e5f2db06": "Q_{\\text{s}}=325+P-30P_{\\text{rg}}", "096c98367a59dffb36f8c2bbe5fb86d4": "T [L] \\subset S \\rightarrow S/R", "096ca414092552ff722add6950d3e4e3": "\\nu(d)=2d+1", "096cb9d94c653880be7ef740c2fb0dbd": "h_2", "096ccebe745cf5215e225d1d69097027": "\n\\left |\\sum_{k=1}^n a_k b_k \\right | \\le \\operatorname{max}_{k=1,\\dots,n} |B_k| (|a_n| - a_n + a_1),\n", "096d1141a089ba9d5257120eab9c6937": "\\sigma = \\sum_{j} L_{ij} \\frac{\\partial F_i}{\\partial x_i}\\frac{\\partial F_j}{\\partial x_j} ", "096d60169c19cc448abda87fcde082ab": "y_{n}\\to y", "096d6299450f63a8c1fd81f17f031baa": "C_\\bullet:\\bold{Top}^2\\to \\mathcal{CC}", "096d66d696d9b0171337485d3c0ad5ac": "(\\neg A)\\wedge(\\neg B)", "096d8aa5f4c64cad3813a5d52e1d6b9f": "\\displaystyle{f(e^{i\\theta}) =\\sum_{m=1}^N a_m e^{im\\theta} + a_{-m} e^{-im\\theta},\\,\\,\\,\\,\\,\\,\\,a_{-m}=\\overline{a_m}.}", "096d9ba631683970176f5af9f654e4aa": "{d \\over dx} \\left[ (1-x^2) {d \\over dx} f \\right] + n(n+1)f = 0.", "096dea68262a84f3056c534dcef6f6aa": "\\mathfrak{g}_i = (0)", "096e0e6fde47c8b3321f49f7dc713146": "\nu", "096e4febe6746dcbe9598b5460874688": "\\rho_g(X) = \\rho_g(Y)", "096e85756c63167153b43c5c4e7b8be3": " \\eta_v \\approx 90 % ", "096ea0cb824615090a2302797944e5c0": "v_{\\infty}=\\,\\!", "096ea70a8a79650e3bf8e72465a2c2a0": "\\begin{align}\n\\mathbf{E}' & \\approx \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B} \\\\\n\\mathbf{B}' & \\approx \\mathbf{B}-\\frac{1}{c^2}\\mathbf{v}\\times \\mathbf{E} \\\\\n\\mathbf{j}' & \\approx \\mathbf{j}-\\rho \\mathbf{v}\\\\\n\\rho' & \\approx \\left( \\rho -\\frac{1}{c^2}\\mathbf{j}\\cdot \\mathbf{v} \\right) \n\\end{align}", "096ecc68b832602e6dc1a3286457a994": " v' = \\frac{dx'}{dt'}=\\frac{dx^\\beta}{dt^\\alpha}\\,,\\quad \\alpha,\\beta>0", "096ed71a2569492eb8aaf1db3cb4dc24": "n\\leq r", "096f040c920b4733618df7324e1675cd": " f\\left(\\frac{az+b}{cz+d}\\right) = (cz+d)^k f(z)", "096f05198f447797874c0e5c54fcf0b7": "\\begin{align}R_{\\frac{\\lambda}{2}}\n&=60\\operatorname{Cin}(2\\pi)=60\\left[\\ln(2\\pi\\gamma)-\\operatorname{Ci}(2\\pi)\\right]=120\\int_{0}^{\\frac{\\pi}{2}}\\frac{\\cos\\left(\\frac{\\pi}{2}\\cos\\theta\\right)^2}{\\sin\\theta}d\\theta,\\\\\n&=15\\left[2\\pi^2-\\frac{1}{3}\\pi^4+\\frac{4}{135}\\pi^6-\\frac{1}{630}\\pi^8+\\frac{4}{70875}\\pi^{10}\\ldots-(-1)^n\\frac{(2\\pi)^{2n}}{n(2n)!}\\right],\\\\\n&=73.1296\\ldots\\;\\Omega;\n\\end{align}\\,\\!", "096f2e30968f0d3a41237fe94c9d134d": "T=\\text{PA} \\cdot \\text{PB} = (s-a)(s-b).", "096f459f6961efcf35ad08b2a9150215": "\\frac{1}{m}\\sum_{j=1}^m 1-\\mbox{erf}(\\sqrt{c}) = 1-\\mbox{erf}(\\sqrt{c})", "096f98947922bba34654053c41d3e754": " \\mathsf{cap}(\\mathbb{Z}) \\approx 1- 0.5 \\mathsf{H}(p) \\,", "096fc7fbca9f40181b5e487c4f5ad897": "\\bar{x}_{i}= \\frac{1}{N} \\sum_{j=1}^{N}x_j ", "097009ed75ef0bb863cc82b7f8a132de": "f^\\star:\\mathcal{P}(Y)\\rightarrow\\mathcal{P}(X)", "09700b05417a749d1d44aca99c89f6e1": "\\S", "097073df6da15a0976e270a725533f9c": "[(D_w + F_w/2) + (D_e - F_e/2) + (F_e - F_w)]\\phi_P = (D_w + F_w/2)\\phi_W + (D_e - F_e/2)\\phi_E", "097089d7c14702d3841de37b9e73f981": "\\rho(p,q)", "0970993acbcca31a8ad1daf4cde9bd04": "\\log_2 (N) + 1", "0970a84585c8793159a6ed20a9d3ec28": "\\infty \\infty", "0970f1ef8543c935a4485f71f7e71f3d": "\\beta = \\frac{1}{v}\\left(\\frac{\\partial v}{\\partial T}\\right)_p =\\frac{-1}{\\rho}\\left(\\frac{\\partial\\rho}{\\partial T}\\right)_p", "0971043f6845805b93760346b10539d0": "I = \\epsilon \\sigma T^4\\,", "09712e1bc31a3cbca8b59ee62161c421": "\\,dE", "09720b310739f42f8f43f75de39d76a2": "\\operatorname{true}", "09722107a1fc3e97695db13a2c8920f5": "=\\sqrt{1^2+\\sinh^2(t)}", "09727656860d33f2e92652fad35ceebf": "\\frac{\\part C}{\\part a}", "0972a2790e564eb2c668120dd341a753": "C = \\frac{\\mu_0 \\mu_B^2}{3 k_B}N g^2 J(J+1)", "0972b0a00bae0824536f35954e43094d": "\\rho_$", "0973689429469b272bee6ad9b5c6ba0a": "z^2 + (r^2 - R^2 + a^2) = 2ax.", "097391bd3e1e7dcc273a48e568291d20": "\\|\\mathbf{v} + \\mathbf{u}\\| \\le \\|\\mathbf{v}\\| + \\|\\mathbf{u}\\| ", "0973d0605145d4dba13f07b809abbab8": "\\|\\sum_{k=1}^{n}\\mathbf{v}_k\\|^2 = \\sum_{k=1}^n \\|\\mathbf{v}_k\\|^2.", "09741544dadae6628807fe3d1bf7c5c2": "{d^n \\over dx^n} f(g(x)) = \\sum_{k=1}^n f^{(k)}(g(x)) B_{n,k}\\left(g'(x),g''(x),\\dots,g^{(n-k+1)}(x)\\right).", "09741c5ffc900a64f1974cf18780e5bb": "\\theta^*_n", "0974695ef8a05c316d5ac876ac20d024": "z+n=\\prod_{p_i\\in P} p_i^{b_i}", "09752d7b87a2851f9fc95ecea062f4ea": "M_{int}", "0975621693e34cf3b4ad2d38cd1967ac": "\\textstyle P = \\frac{1}{\\mathcal Z} e^{(\\mu N-E)/(k T)}", "0975646846c2a4df7b4a157872a6eea3": "\\begin{align}\nu_1 & = \\frac{F_1(\\kappa+1)\\ln|x_1|}{4\\pi\\mu} +\n\\frac{F_2(\\kappa+1)\\text{sign}(x_1)}{8\\mu} \\\\\nu_2 & = \\frac{F_2(\\kappa+1)\\ln|x_1|}{4\\pi\\mu} +\n\\frac{F_1(\\kappa+1)\\text{sign}(x_1)}{8\\mu} \n\\end{align}", "09756f541757a26b30c46350ebcaecac": " N_{t+1} = \\lambda \\ N_t \\ [ 1 - f(N_t, P_t) ]", "09759381df78e67a0bb93494641ed110": "E[m] \\cong (\\mathbb{Z}/m\\mathbb{Z}) * (\\mathbb{Z}/m\\mathbb{Z})", "0975ce570b45bf71988686e6cf9c31ad": "gate5", "0975d2363205bf5e1fd7faf7c0f8aeee": "m1 = 1", "0975dbd2002cc29494d43c86f625b851": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = 6.5 - \\frac{x}{6} - y \\geq 0 \\\\\n g_{2}\\left(x,y\\right) & = 7.5 - 0.5x - y \\geq 0 \\\\\n g_{3}\\left(x,y\\right) & = 30 - 5x - y \\geq 0 \\\\\n\\end{cases}\n", "0976007f6874da15abe6fa496b78f514": " h \\circ f ", "097633138af488186f3186b7b891d704": " \\sigma(M) = \\langle L_n(p_1(M), \\dots, p_n(M)), [M]\\rangle. ", "09763da86a7cd69f749f8167611472a9": "w^2+x^2+y^{n+1}=0", "0976495e501d07840e173eed15d0c8b1": "\\theta_{ab}", "09775492fcfb9eb777ac4cb1abb6321b": "C_{0}\\;", "0977848f1d538d445f8ca37db3fbacaf": "\\rho_{XY\\cdot \\mathbf{Z} } =\n \\frac{\\rho_{XY\\cdot\\mathbf{Z}\\setminus\\{Z_0\\}} - \\rho_{XZ_0\\cdot\\mathbf{Z}\\setminus\\{Z_0\\}}\\rho_{Z_0Y\\cdot\\mathbf{Z}\\setminus\\{Z_0\\}}}\n {\\sqrt{1-\\rho_{XZ_0\\cdot\\mathbf{Z}\\setminus\\{Z_0\\}}^2} \\sqrt{1-\\rho_{Z_0Y\\cdot\\mathbf{Z}\\setminus\\{Z_0\\}}^2}}.", "0978441c65781ef8858885e566c51159": " T_f ", "09784872ea44d9b0dc26740990266bf0": "\\scriptstyle\\rho(r)", "0978927b31f441753e587f876ddb9834": " \\rho", "0978c8d19fe08434e9e1f689b148d94f": "H^{n-1}", "0978fe6ad5995cfd6c1e14c4f6dc78ce": "\\frac{dm}{dt} = \\alpha_m(1 - m) - \\beta_m m", "0979d887e36a5e1e07b517a7fee02c95": "\\gcd (z,n)", "0979f533e9b621efce4a1efc0ee32001": "\\scriptstyle\\blacksquare", "097a3aff7cb38c8ea745a2906532297d": "\\nabla \\cdot \\mathbf{E} = 4 \\pi \\rho \\ ", "097a50be807bb22aed485bdee4e92d8b": "\\iiint_D f(x,y,z) \\, dx\\, dy\\, dz = \\iiint_T f(\\rho \\cos \\phi, \\rho \\sin \\phi, z) \\rho \\, d\\rho\\, d\\phi\\, dz.", "097a559ec433955b587d0a6ce5197b55": "-\\log\\text{det}(X)", "097a84ad0f37c7cab16c83a005271e8b": "v_n(\\xi\\otimes e_\\alpha)=(v_n\\xi)\\otimes e_\\alpha.", "097a98dbbca91e3b2b0e85b8d9d9523b": "\\lambda_1, \\lambda_2, \\ldots, \\lambda_r", "097a9b36222e3b07ed9c68a582c9c8d9": " \\mathrm{SML}: E(R_i) - R_f = \\beta_i (E(R_M) - R_f).~ ", "097b15b5b599c8cef7289a97dd545f49": "\\mbox{estimated median}=\\widehat{\\kappa}\\cdot 2^{1/\\widehat{\\theta}},\\,", "097bacc79ab8726dc04e7647f8d32742": "\n\\Psi (\\rho) = \\begin{bmatrix} \\langle F_1, \\rho \\rangle \\\\ \\vdots \\\\ \\langle F_n, \\rho \\rangle \\end{bmatrix} \n", "097bf9d105523a0a559358845998755b": "\\begin{align} 2\\cdot R_*\n & = \\frac{(160\\cdot 4.24 \\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 146\\cdot R_{\\bigodot}\n\\end{align}", "097bfb21de52e2233346207b45995bbd": "\n \\sigma (f, g) := \\inf_{\\lambda \\in \\Lambda} \\max \\{ \\| \\lambda - I \\|, \\| f - g \\circ \\lambda \\| \\},\n ", "097c01f668e1b9181575da3b69b94777": " E_{n,k} = \\bigcup_{m\\ge n} \\left\\{ x\\in A \\,\\Big|\\, |f_m(x) - f(x)| \\ge \\frac1k \\right\\}.", "097c0fa7da7dfe4ae638d32e1a2f7af7": "b=u^2+v^2,", "097c595ad1fdd1f412534364752b3eb6": "\\mathbb{E}[X_n] > 0", "097c7b6870e070b68a6127fa5aafc6d5": "\\mathcal{G}_2", "097ce48eb7ba9f62a32222253205111f": "T((\\lambda \\dot{q}_i)^2, (\\lambda \\dot{q}_j \\lambda \\dot{q}_k), \\mathbf{q}) = \\lambda^2 T(\\dot{q}_i, \\dot{q}_j\\dot{q}_k, \\mathbf{q})\\,,\\quad L(\\mathbf{q},\\mathbf{\\dot{q}})\\,,", "097cf59b7d9ce35264b75bf4efddd0a1": "\\begin{matrix}\\frac{\\pi}{2}\\end{matrix}", "097d48183f5023757b1e0aa89c426b67": " {\\zeta_g} = {{\\partial v_g \\over \\partial x} - {\\partial u_g \\over \\partial y} = {1 \\over f_o} ({ {\\partial^2 \\Phi \\over \\partial x^2} + {\\partial^2 \\Phi \\over \\partial y^2}}) = {1 \\over f_o}{\\nabla^2 \\Phi}} ", "097d51db028cbbe5ad2bd557eb51310c": " [\\mathbf A, \\mathbf B], \\qquad \\mathbf A \\in \\reals^{n\\times m}, \\qquad \\mathbf B \\in \\reals^{p\\times m}, \\qquad m \\geq n+p.", "097d55f668e4274885302f68024762d8": "\\frac{1}{5-\\frac{1}{z}} = -z \\left(1 + 5z + 5^2 z^2 + 5^3 z^3 + \\cdots\\right)", "097d8759686fb29d878843530d39965b": "\\nu_0'{\\sigma_0^2}'", "097da62f4a63ee6af7d9e705f624067f": "(H\\ or\\ E)", "097ded293601685da87fcd695852c05f": "z_4 = \\frac{z_1 k_1 + z_2 k_2 + z_3 k_3 \\pm 2 \\sqrt{k_1 k_2 z_1 z_2 + k_2 k_3 z_2 z_3 + k_1 k_3 z_1 z_3} }{k_4}.", "097e4dc3186bba142989fce838e56df2": "R = \\frac{1}{r},\\ \\Theta=\\theta,", "097ee796eed6c815586d9373311afaef": "pA+(1-p)C", "097f0656cb205e1963956c521fb4f408": "[1]\\,\\!", "097f20080dce1b9b0584fa371b948475": " \\ell = \\frac{d}{\\tan \\alpha} + \\frac{d}{\\tan \\beta}", "097f2e4dfb5c66ef949c438dc540d1c4": "H_5 = 0\\,", "097f8961516236e8c71a5eaf265098ee": "N(t) = N_0 e^{-t/\\tau}", "097fcb4b95369f3515606dfdc7c39acc": "\\hat{p} = -j\\hbar \\frac{\\partial }{\\partial q}", "09800103d9cdd149c7fa40ada0bd230a": " s_\\hat{\\beta} = \\sqrt{ \\frac{\\tfrac{1}{n-2}\\sum_{i=1}^n \\hat{\\varepsilon}_i^{\\,2}} {\\sum_{i=1}^n (x_i -\\bar{x})^2} }", "098009d0faadd3d7b3fd754d84183ed2": "M_{CB} = \\frac{2EI}{L} \\left( 2 \\theta_B + 4 \\theta_C \\right) = 0.4EI \\theta_B + 0.8EI \\theta_C", "0980296df0730a80f387b01bafb64e9d": "t\\colon A\\rightarrow\\ V", "09805d958fc7ab3d6c57c02f71afdc39": "(1)\\,", "09807a84ef1a3f642a4f2033a4854a8f": "P = \\tau\\omega. \\!", "0980aca54f873f12790fe3506eb3d61f": "\\Omega = 2 \\ F", "0980cf9e91aeb7ea0ddc6fddbad535ee": "\n-f(x) = f(-x), \\,\n", "0980fbfa5727b6f2ca6a7c1b17b48674": "\\exist x_1\\exist x_2(x_1 \\not =x_2)", "0980feeeaa6ae8fef931eedafc1c55b4": "A\\in\\mathbb{R}^{2\\times 2},\\textbf{b}\\in\\mathbb{R}^{2\\times 1},c\\in\\mathbb{R}", "09810d9ecc81282741b9dc8f9c503895": " \\begin{matrix}\n \\mathrm{ELEMENTARY} & = & \\mathrm{EXP}\\cup\\mathrm{2EXP}\\cup\\mathrm{3EXP}\\cup\\cdots \\\\\n & = & \\mathrm{DTIME}(2^{n})\\cup\\mathrm{DTIME}(2^{2^{n}})\\cup\n \\mathrm{DTIME}(2^{2^{2^{n}}})\\cup\\cdots\n \\end{matrix}\n", "09815b3a7e8e43b07cd97cadb9a90e18": "x \\mapsto (\\tau_{j} \\circ \\tau_{i}^{-1})_{x}", "09819451d727e92e946d4b286b90e8f2": "\\eta=1-\\left(\\frac{{1}}{{r}^{(\\gamma-1)}}\\right)", "0981db6047065ebfd3318d3ccbe75e29": " \\langle x \\rangle = \\int_{-\\infty}^{+\\infty} x |\\psi|^2 dx = \\int_{-\\infty}^{+\\infty} \\psi^* x \\psi dx ", "09820eb4bcdd600f9e532ab865563fcd": "\nc \\, \\delta \\frac{d\\tau}{dq} = - \\frac{r^{2}}{c} \\frac{d\\varphi}{d\\tau} \\delta \\frac{d\\varphi}{dq} \n= - \\frac{r^{2}}{c} \\frac{d\\varphi}{d\\tau} \\frac{d \\delta \\varphi}{dq} \n\\,.", "098242e80f673a13cffbd1503594e264": "\\Lambda: \\mathbb{R}^n \\times \\mathbb{R}^m \\times \\mathbb{R}^p \\to \\mathbb{R} ", "0982a90d01632f7f9530ba11ea06b118": " N = { \\{ x_1 , x_2 , x_3 , . . . , x_N \\} } ", "0982dadeae8a228dd4b9f70dcb434e2c": "A = \\tfrac {1}{2}\\begin{pmatrix}1 & 1 \\\\ 1 & -1 \\end{pmatrix}\\ :", "0982e34a1f03c5af8411265f9f7d7154": "\\cong F", "09830b5cfdd205dad87790289c7e9759": "h_{ab} = \\mathrm{atan2}(b^*, a^*)\\;", "09831a0e4fa9247ca945da792ebcb688": "\\lambda_{CWL}(M_1\\#M_2)=\\left\\vert H_1(M_2)\\right\\vert\\lambda_{CWL}(M_1)+\\left\\vert H_1(M_1)\\right\\vert\\lambda_{CWL}(M_2)", "0983244c70964fc8799fa40690b30455": "2x^2+x^3=0", "09835a6d51b9fdffd8d39d0c73cdec77": "\n\\begin{align}\ns_{t}& = \\alpha x_{t} + (1-\\alpha)(s_{t-1} + b_{t-1})\\\\\nb_{t}& = \\beta (s_t - s_{t-1}) + (1-\\beta)b_{t-1}\\\\\n\\end{align}\n\n", "0983c99e7e7a64f926891c02ae09517f": "%dMU = \\frac{\\frac{d^2u}{dc^2}}{\\frac{du}{dc}} = -\\frac{\\theta}{c}", "0983ca34c5e32b69c4927eb0d82f0016": "1 + f(z) + \\ln (1 + f(z)) = z.\\,", "0983f2ee5a9870a798b2e318275cfb01": " A_q(n,d) \\times m = A_q(n,d) \\times\n \\begin{matrix}\n \\sum_{k=0}^t \\binom{n}{k}(q-1)^k\n \\end{matrix}\n\\leq q^n.", "0984a5058c519be102dee9ec5d428e1b": "\\leq e^{-\\sqrt{d+1}} + \\frac{1}{\\sqrt{d+1}}.", "09852f1866f0b516b83d14195bfab1ce": " f_i x_i + \\sum_{j=1}^n{b_{ij} t_j} \\ge h_i", "09858f28dd06ae24d4e450112c317d06": "1x = x1 = x", "0985c6389e59e6d9bc65f468ba01dced": "\\int_\\gamma |z|^{-1}\\,ds \\ge \\int_\\gamma |z|^{-1}\\,d|z| = \\int_\\gamma d\\log |z|=\\log(r_2/r_1).", "0985ea2ce5a835ddd8ca34e92602ee75": "X=(X_t:t\\geq 0)", "09861baa3c8470520f191421c0e07035": "{d \\over dx}\\cot y={d \\over dx}x", "0986362c046227cbb01c6ab5a48cb4c3": " \\frac{\\operatorname{Vol} (B_\\varepsilon(p) \\subset M)}{\\operatorname{Vol} \n (B_\\varepsilon(0)\\subset {\\mathbb R}^n)}=\n 1- \\frac{S}{6(n+2)}\\varepsilon^2 + O(\\varepsilon^4).", "09866ffbc8cb073dbec02b910afdd2c1": "\\text{post-money valuation} = \\text{new investment} \\,\\cdot\\, \\frac {\\text{total post investment shares outstanding}}{\\text{shares issued for new investment}}", "0986cdefd6ee502366e89f598a303154": "J = \\begin{bmatrix}0 & -I_V \\\\ I_V & 0\\end{bmatrix}.", "0986d7afc7049e26fe594ed1dbc5e7f5": "\\left|\\alpha - \\frac{p}{q}\\right| > \\frac{C(\\alpha,\\epsilon)}{q^{2 + \\epsilon}}", "0986ea696bbd39ed9ca6bb0590272f87": " (1-R-\\epsilon)", "0987157267dae2f3bf8698474206f699": "\\cos(C) =0 \\,", "098721a2282b5c4c233baf5582a53d36": "f'(x) = f(x)\\times \\Bigg\\{h'(x) \\ln(g(x)) + h(x)\\frac{g'(x)}{g(x)}\\Bigg\\}=\ng(x)^{h(x)}\\times \\Bigg\\{h'(x) \\ln(g(x)) + h(x)\\frac{g'(x)}{g(x)}\\Bigg\\}.", "09877b064fa5471c8194cddb57d64020": " \\overline T ", "0987c1a0688089dca4c4e0fc012d44ae": "y' e^{\\int_{s_0}^{x} P(s) ds} + P(x) y e^{\\int_{s_0}^{x} P(s) ds} = Q(x)e^{\\int_{s_0}^{x} P(s) ds} ", "0987e45a4deecff31f9af6654c280af1": " p^{k + 1} = p^k + \\text{urf} \\cdot p^{'} ", "0987ee8197421cb30cc7bc847561ba6d": " \\frac{P_i}{\\varepsilon_0} = \\sum_j \\chi^{(1)}_{ij} E_j + \\sum_{jk} \\chi_{ijk}^{(2)} E_j E_k + \\sum_{jk\\ell} \\chi_{ijk\\ell}^{(3)} E_j E_k E_\\ell + \\cdots. \\!", "098818059bfd2dde17efbbde43cf8edf": "y(t) = \\frac{v(t)-v_n}{v_n} = \\frac{v(t)}{v_n}-1", "098829e32a19cef921503374218122d4": "N=pq\\,", "0988378c122ad2df86bd1b52f42438e2": "\\mu(n) = (-1)^{2x + 1} = -1.\\,", "098842703beb9d0a23797626421f2871": "\\Delta I_{L_{Off}}=\\int_{DT}^{T}\\frac{\\left(V_i-V_o\\right) dt}{L}=\\frac{\\left(V_i-V_o\\right) \\left(1-D\\right) T}{L}", "098858c2e462b2a4074f0f59317fd2b2": "p_4(x)=-512x +192x^2-24x^3+x^4;", "0988883d6d31b6e2be361660d5a75515": "A(x)=\\Omega(x^{1/3})", "0988a06cb7c934ce557c3a2e87cee7bd": "\nO\\Big(\\sum_{i0", "0995841f0f62a95f7c242f4ec4930b53": "C \\in M(r,n;\\mathbb{K})\\,\\!", "099593cda6e698b53cfbe594b61a8174": " \\alpha_{SID} = 1-(1-\\alpha)^\\frac{1}{m} ", "099621954608cb6bf333415eb4507a34": "R_{\\mu\\nu} \\cong \\frac{1}{2}\\left[\\ln(-g)\\right]_{|\\mu|\\nu|}-[\\mu\\nu,\\beta]_{|\\beta}", "09962f051b570d1498d494053eaf8026": "f_\\alpha - f_\\beta", "09965877ff74a5cb88cf265b134c7dc6": "H(\\omega)= \\frac {1} {\\sqrt 2} \\sum_{k \\in Z} h_k e^{j \\omega k}", "09967d670c98fb8f3b9be63df7a81c6a": " -\\frac{\\hbar^2}{2m}\\frac{d^2}{d x^2}\\psi(x) + V(x)\\psi(x) = E\\psi(x) ", "0996b06f4d715a0f9551e5c219fdff2f": " x_1\\in X_1 ", "0997199f820cb74dd1a3a1e6541c4748": "f,g:M \\to N", "099752f006bd3a40c926dc076fee8930": "\\rho_1 ", "0997a652bf2570678009f56f910281bd": "\\frac{\\Delta P}{\\Delta T} ", "0997d68bf07ac02c9a14ef557a6dde99": "E_{0} (\\textrm{volts}) = -\\frac{\\Delta G^\\ominus}{nF}", "0997e7d9040a21ee00bb261f76367d1b": "R^{-1}\\nabla^2R\\rightarrow\\infty.", "099834bdfc700dd0dd86dc64749ebc5b": " \\frac{1}{m^*} = \\frac{1}{\\hbar^2} \\frac{\\partial^2 E(k)}{\\partial k^2} ", "0998676d55300fb29951cd4e7100e2f8": "\n f(p,q,p_c) = \\left[\\frac{q}{M}\\right]^2 + p\\,(p - p_c) \\le 0\n ", "09989a650103e6c656dd117d1001b026": " \\lim_{y \\to q} \\big( \\lim_{x \\to p} f(x, y) \\big). \\, ", "0998aca5b089d13d466f93c28245dc7e": "[-1,1]^2 \\,", "0998afbc0acded704c1318a2ceb36e90": "\\scriptstyle f_n(x) \\leq f_{n+1}(x)", "0998d79e7094eb9a79037c47426c8aae": "d\\mathbf x^2=d\\mathbf X\\cdot\\mathbf C d\\mathbf X\\,\\!", "0999538dbc8306d9f6ca28d74d26a279": "N\\geq\\frac{1}{n} \\left(q^n-\\sum_{p|n, \\; p \\text{ prime }} q^{\\frac{n}{p}}\\right).", "09997759dc23f2fe1bae1dbbba258a9e": "T\\colon \\Omega_c^m(M)\\to \\mathbb{R}", "09999eba6c9c8f12c42e47a54c4979e7": "g(z) = \\frac{1}{f(z) - b}", "099aa13b8bb2583dcacc7fe55f54c919": "a_k= C/k", "099ab665b59216cde0d59e51ae4cbd4a": "\\,\\mu", "099b34a42fd8c504e6ccc1ee1a5d30e1": "\\text{CIRC}(T;P,Z) = \\{ M ~|~ M \\models T \\text{ and }\n\\not\\exists N \\text{ such that } N \\models T ,~ N \\cap P \\subset M \\cap P \\text{ and } N \\cap Z = M \\cap Z \\}", "099b564e6be3850fa2c3b1c16a5a47a1": "ind(P) + \\gamma^5/20", "099b5d1a5253a577bdabdd87b17e70d0": "\\operatorname{O}(M) \\times_{\\sigma_-} \\{-1,+1\\}.", "099b6c1a56664593d1b6baf5c240bd7e": "\\sup_{x \\in E} N_{r}(B_{R}(x) \\cap E) \\leq C \\left( \\frac{R}{r} \\right)^{\\alpha}.", "099b90d035d3ee7c61f2e1a1b38ae971": "\\|x\\|_{C(X)} = \\max\\nolimits_{x\\in X} |f(x)|", "099be9a554513f5cd5028350bc13b07b": " P(x)\\propto \\exp{\\left(-\\frac{\\frac{1}{2}kx^2}{K_BT}\\right)} ", "099c0558d1e4d2c531680a7b25cb24de": "f^{*}(x) = \\lim_{h \\to 0}{ \\left({f(x+h)\\over{f(x)}}\\right)^{1\\over{h}} }", "099c27c38ffdb1e023a27c1360e68671": "\\frac{\\partial^2 f}{\\partial \\sigma^2}<0", "099c5c1404941de8b00a095cbfd3d27b": " \\Pr(X - \\mu \\geq k \\sigma) \\leq \\frac{ 1 }{ 1 + k^2 }. ", "099c75fe0488e6ca9fa7616ea6ff72dc": "\\mathrm{auth}(p)=\\displaystyle\\sum_{i=1}^n \\mathrm{hub}(i)", "099cdfabb805c4d513bdf867d3de30f3": "\nX_2=\\frac{(t+1)x^2 -1}{(t-1)x^2 +1},\\qquad \nX_4=\\frac{(t_2+1)X_2^2-1}{(t_2-1)X_2^2+1},\\qquad \nX_8=\\frac{(t_4+1)X_4^2-1}{(t_4-1)X_4^2+1}.\n", "099d2e1000b227a383ce43097c4f36fa": " d \\mathbf{r}", "099d3447462e93a4f340dd930752e239": "A\\to C,(A\\to B)\\to C\\vdash C", "099d7af978b6fb5ce727211c86f5639a": " \\frac{1}{\\operatorname{Tr}((I-E) S)}(I- E) S (I- E). \\, ", "099db936cba2b046dc8c979ad61933a5": "T_{j',j}", "099dbeb744724fbdd14aa9c20458e166": " \\sigma_w=0.1W_{20} ", "099e47733b317dcafe1c45cea8a29dfe": "\\Phi:F\\dashv G", "099ee94a7a9d6a6809f685c2464c98be": "\\lambda_{11}=3.83171", "099efdd7be2c05f33125555513005dc0": "\\mathbf{\\epsilon}^\\beta\\ ", "099f1fa68187b4d345e532bbda55e66f": "((x,g_U)\\in U\\times G) \\sim ((x,g_V) \\in V\\times G) \\iff {\\mathbf e}_V={\\mathbf e}_U\\cdot h_{UV} \\text{ and } g_U = h_{UV}^{-1}(x) g_V. ", "099f92f26f75f6d919b945ce7c852b2b": "2 \\cdot 3 \\equiv 0 \\pmod{6}", "099fa53577482b5d4ff29eec93cda21d": "R \\ )", "099fc11abb2842272fd9eb824470481a": "\n\\begin{align}\n\\frac{\\theta \\Gamma(\\theta+n+\\alpha)\\Gamma(\\theta+1)}{\\alpha \\Gamma(\\theta+n)\\Gamma(\\theta+\\alpha)}-\\frac{\\theta}{\\alpha}.\n\\end{align}\n", "099fc59c3efaacedf8a1e069446342bc": "(-\\alpha, -\\alpha)", "09a038f959bd2e8ba81e1fe6eeca2853": "\\frac{[A]_{f}}{[A]_{i}}=e^{-kt}", "09a0467830b0cb52cae6bef87d434df9": "\\mathbf{v}_1", "09a09296c8c2628cd63ec2555bc1d848": "g^2(q;\\tau)", "09a0bc9555fb8df32a66cf04f113a318": "\n\\begin{array}{c}\n\\text{area of}\\\\\n\\text{rectangles}\n\\end{array}\n= 1 \\,+\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,+\\, \\cdots.\n", "09a0f8ccbe3337a8e95dd36497891ca7": "g_{kl}=2\\chi \\Delta n\\overline{n}\\, ", "09a11106d1a04300591ebc86734b9a08": "\\Psi(z) = \\frac{\\cos 2\\pi(z^2-z-1/16)}{\\cos 2\\pi z}", "09a12cb751fe5626d7c138d2ef3a928b": "X_t = \\mu + X_{t-1} + \\epsilon_t\\,", "09a14ed70f3bdce0e9288d7e5cd5994b": "abs(\\lambda)=1 \\,", "09a194a0a61c475d6260611b5255a5a2": "\\sigma\\sqrt{2}\\mbox{erf}^{-1}(1/2)", "09a1c11135b445f9a9f9e8893d08cfe7": "x_{i}\\geq x_\\min", "09a1cb4663947e85224ba06569cebebf": "\\frac{95}{48}", "09a1fd285dd3dcad58e135b2a8acc0eb": " \\mathit CF = cash flow \\,", "09a20bfbac6c09b7be92573d4379ed97": "i \\in \\lbrace 1, \\ldots k \\rbrace ", "09a26fa2581b375c5c1770a395187c29": "\\lambda_2^2 = \\lambda_3", "09a29cc014fcfc407ba0b0533f1a91f8": "a_n=\\begin{cases}1&\\mbox{if }n=2k-1, \\\\ 0&\\mbox{if }n=2k\\end{cases}", "09a3516831e4dc4bd70986f5a2ee84fb": "\\ \\|f\\|_p \\le \\mu(S)^{\\frac{1}{p} - \\frac{1}{q}} \\|f\\|_q ", "09a35bb3db23ca623e8eb2cdf7e1a960": "SH_0(\\text{point})\\cong\\mathbb{Z}", "09a3d6a4d9e07bb432d49075c87b1097": "\\scriptstyle p_\\theta = \\dot\\theta", "09a3eab23fe0510255994528a4d7299e": "\\Delta p_x \\approx 2\\frac{h}{\\lambda}\\sin\\varepsilon/2.", "09a452b6295ddeadbf1ad24de8f304c9": " \\mathbf{w}(n) = \\mathbf{w}(n-1)+\\,\\alpha(n)\\mathbf{g}(n)", "09a539a89442cc879f1cb4fd7a1b09e4": "x\\in U, y\\in V", "09a55ca392d1b3a31643fc87a6fb6c4d": " 0.886 \\langle v \\rangle = v_p < \\langle v \\rangle < \\sqrt{\\langle v^2 \\rangle} = 1.085 \\langle v \\rangle. ", "09a5631b96f33c7cd08a5f7ac94507a5": "Z_2 \\times Z_2 \\times Z_2", "09a62612c4e4f4043c87ec6d3790429d": " P_i(S_x) ", "09a715fce75d188d2129ab1ebece5336": "\\delta_\\epsilon \\Omega^{(d+2)}=d\\delta_\\epsilon \\Omega^{(d+1)}=d^2\\Omega^{(d)}(\\epsilon)=0", "09a736721e0a5081dcbbccd2c01c71ff": "p \\land q \\Rightarrow r", "09a7422672629ad14d62321da76d93ec": "ln(N)", "09a76ccc2b2efa52ac4a8976ec5792cb": "a_7' = a_3 \\oplus a_4 \\oplus a_5 \\oplus a_6 \\oplus a_7 \\oplus 0 = 1 \\oplus 0 \\oplus 0 \\oplus 1 \\oplus 1 \\oplus 0 = 1.", "09a78a9ee276650e6fd9475985427662": "a+dx=a", "09a82fea9a85ce8517c992c36fc868f2": "\nf^{(t)} = [f_{ij}^{(t)}]_{i = 1...N, j = 1...M}\n", "09a85bd3efa5f39ea790dec70d3732f3": "\\left( b_{k} \\right) ", "09a88b73190a3ef2ce7c051c9e867a8c": " \\frac{1}{2} \\sum_{i=1}^n \\sum_{j=1}^n \\left( x_i y_j - x_j y_i \\right)^2\n\n= \\sum_{i=1}^n x_i^2 \\sum_{i=1}^n y_i^2 - \\left( \\sum_{i=1}^n x_i y_i \\right)^2 . ", "09a89796812d795a2deaedc5b3f57284": "\\scriptstyle{T}", "09a8ebdc269ac1f2f9b2f53ee3fc46e3": "\\begin{matrix} {9 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "09a9a9b63f858809e165da927547a4fa": " pH = 6.1 + \\log \\left ( \\frac{[HCO_3^-]}{0.03 \\times pCO_2} \\right )", "09aa028c019b5d2d4f9a42656c40af79": "p / q", "09aa47885632645c9ac93fd74c1dfc9b": "\\alpha \\,\\rightarrow\\, \\phi_\\alpha(0)", "09aa64ff4add7aa3790267246c0d1d48": "G = (V,\\, \\Sigma,\\, ::=,\\, S)", "09aad069203af52f17d6432471a8d93f": "{\\rm E}(n,x)", "09aadd014986908dd9072062ccbc3bd0": "\\scriptstyle OA/OB=8.27", "09ab2495cb1a5b4be76ce6032d51060a": "B_{\\lambda} = B \\cap (L_{\\lambda}/qL_{\\lambda}),", "09ab69a2e511b7912d3386f5f4c60569": "\n\\rho = \\frac{r_{12} v_1 v_2 - r_{14} v_1 v_4 - r_{23} v_2 v_3 + r_{24} v_2 v_4}{\\sqrt{v_1^2 + v_3^2 - 2 r_{13} v_1 v_3} \\sqrt{v_2^2 + v_4^2 - 2 r_{24} v_2 v_4}}\n", "09ab8743a652f3ef0ea9c02772bfe183": "\\frac{16}{9}", "09abfd122217ce3f94278c0512f49774": "u \\ne v", "09ac9bbecc7d24533986b71f8fc53f0d": "q_i = \\left \\lfloor \\frac{n}{5^i} \\right \\rfloor,\\,", "09ac9d81af463f83ae29bbc63ff265b1": "\\tfrac {mg}{L}", "09acc9b155de86f6781ab94ab5a81efe": "\\scriptstyle E_K", "09ad38db86a3ffdf2b92d66d95906625": "w^R", "09ad40c39c6ed860401d70432632ae82": "\\varphi: \\vec x \\rightarrow \\vec x-2\\frac{f(\\vec p,\\vec x)}{f(\\vec p,\\vec p)}\\vec p", "09ad72048b4e4932f84109f09cffb5ec": "H_n = H_{n-1} + \\frac{1}{n}.", "09adee3111d09346fb6da046f1bd266f": "R_\\alpha:S^1\\rightarrow S^1", "09adfd442038c1cbc7686167ee7d00f5": "B(S,i,j)=\\max_{S'\\subset X_i\\atop S=S'\\cap X_j} A(S',i)", "09ae008ae4ceab83db8969dbbdaa426a": " \\Pi_5 = ", "09ae3c49e6634a769a653b51b1283ea3": "\\psi(\\zeta_0) = \\zeta_0", "09ae87bf709ae355f1af53b3e08176f2": "\n \\begin{bmatrix}\n \\epsilon_{{\\rm xx}} \\\\ \\epsilon_{\\rm yy} \\\\ \\epsilon_{\\rm zz} \\\\ 2\\epsilon_{\\rm yz} \\\\ 2\\epsilon_{\\rm zx} \\\\ 2\\epsilon_{\\rm xy}\n \\end{bmatrix}\n = \\begin{bmatrix}\n \\tfrac{1}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm zx}}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & \\tfrac{1}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm zy}}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xz}}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yz}}{E_{\\rm y}} & \\tfrac{1}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\tfrac{1}{G_{\\rm yz}} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm zx}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm xy}} \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\sigma_{\\rm xx} \\\\ \\sigma_{\\rm yy} \\\\ \\sigma_{\\rm zz} \\\\ \\sigma_{\\rm yz} \\\\ \\sigma_{\\rm zx} \\\\ \\sigma_{\\rm xy}\n \\end{bmatrix}\n ", "09aec5d2afe56504020f7bed3154173b": "\n\\begin{bmatrix}\n\\sigma_x^2 & \\sigma_{xy}^2 & \\sigma_{xz}^2 & \\sigma_{xt}^2 \\\\\n\\sigma_{xy}^2 & \\sigma_{y}^2 & \\sigma_{yz}^2 & \\sigma_{yt}^2 \\\\\n\\sigma_{xz}^2 & \\sigma_{yz}^2 & \\sigma_{z}^2 & \\sigma_{zt}^2 \\\\\n\\sigma_{xt}^2 & \\sigma_{yt}^2 & \\sigma_{zt}^2 & \\sigma_{t}^2\n\\end{bmatrix} = \\sigma_R^2\n\\begin{bmatrix}\nd_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\\\\nd_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\\\\nd_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\\\\nd_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2\n\\end{bmatrix} \\ (7)\n", "09af09bba361f9ad8f05f4af1c5d5fb4": "F_{\\Lambda,K}(\\lambda,k)=F_{\\Lambda|k}(\\lambda|k)F_K(k)=F_{K|\\lambda}(k|\\lambda)F_\\Lambda(\\lambda)", "09af2ef1d44ba3233c9e9854410dc129": "\\frac{1}{(1-p)^2}=2kt[COOH]^2+1=X^2_n", "09af4b0852cb85adfa02fd9486d130cd": "S_y(f)", "09af7cb8e7a8338187ad899971cee128": "\n\\mathrm{DR} = \\mathrm{SNR} = 20\\log_{10}{\\left(2^n\\sqrt{\\tfrac{3}{2}}\\right)} \\approx 6.0206 \\cdot n + 1.761\n", "09af8a8d97b96b0a6df07384bb8a76f9": "N(\\mathbf z)=\\#\\{k : z_k=x_k\\}", "09af8f5c75957906420936806b7e36a0": " \n p = \\frac{\\rho_0 C_0^2 \\chi\n \\left[1 - \\frac{\\Gamma_0}{2}\\,\\chi\\right]}\n {\\left(1 - s\\chi\\right)^2} + \\Gamma_0 E;\\quad\n \\chi := 1-\\cfrac{\\rho_0}{\\rho}\n ", "09af910d326766be00f054e0074b3a65": "\\operatorname{E} (X_t)", "09af935a5b2855356e62bf3ce934f9f2": "\\pi r^2 + \\pi r l ", "09afa0c0079b3c7288e66c8fcb2f1dca": "\n\\frac{\\partial y}{\\partial \\mathbf{x}} =\n\\begin{bmatrix}\n\\frac{\\partial y}{\\partial x_1}\\\\\n\\frac{\\partial y}{\\partial x_2}\\\\\n\\vdots\\\\\n\\frac{\\partial y}{\\partial x_n}\\\\\n\\end{bmatrix}.\n", "09afc5cab7cb9dafea474218a304f4d5": "g_{ij}=E[\\partial_i\\ell\\partial_j\\ell]=E[(F_i-\\eta_i)(F_j-\\eta_j)]=V[\\eta]", "09b058f7ce46d3d531434b039686fcb6": "k_R = \\frac{k_r [UA]}{[AB][RA]}", "09b078b00331fc7eb842cbea01642da1": "\\log_2 \\ln n + \\theta(m/n)", "09b08150ec41bac8fdfe2446e8ce3fa3": "E_1 = y_1 + \\frac{q^2}{2gy_1^2} = 5.0ft + \\frac{\\left(10\\frac{ft^2}{s}\\right)^2}{2\\left(32.2\\frac{ft}{s^2}\\right)(5.0ft)^2} = 5.06ft", "09b14be5d3b3a683cb9074db4a29911f": "reward = -1", "09b1722405c47b120f5070bbb0ed7884": " D = 2^\\frac{ 1 }{ 2 } \\frac{ \\left| \\mu_1 - \\mu_2 \\right| }{ \\sqrt{ ( \\sigma_1^2 + \\sigma_2^2 ) } } ", "09b2466d6c0a7409856798bb5fe09d32": "\n\\alpha_{ij}=\\alpha_{ij_{0}}+\\alpha_{ij_{1}}T", "09b26d835893491a6f700db1e668edfb": "S_a", "09b2761e082a80d565646fa5f8487cc4": "L_n, \\bar{L}_n, -\\infty 1\\\\\n\\frac{d}{dx} \\arccsc x & {}= \\frac{-1}{|x|\\,\\sqrt{x^2-1}}; \\qquad |x| > 1\n\\end{align}", "09cb61347bcd01f92e78c57f9604363e": "s=(\\ldots, (s_{i},t_{si},t_{ei}),\\ldots)", "09cb76df6751573c0507d3fda343c080": " s= \\frac{a+b+c}{2} \\, ", "09cbca1050980a56a89eb976b41c899d": "f(x)=\\begin{cases}\\exp(-1/x)&\\text{if }x>0,\\\\ 0&\\text{if }x\\le0,\\end{cases}", "09cc24bc90b0b2cf900ae6f618af13d5": "\\Psi (A) = \\sum_{i = 1}^N K_i A K_i^*", "09cc573a2780647ee7322462ab1b2582": "U_n(T)=\\sup \\{ U_n(p) : p\\in S(T) \\}", "09cce1c4ae66cffc2e2c66d6ddad4335": "\\vec{r}(s,0)", "09ccf4c384153f2029f149bb2980ea81": "\nT_{HL} = \\tau \\cdot \\mathrm{ln}\\,2. \n", "09cd9cd43514bcc2f18fa416dab5903b": "L_d\\left( t_0, t_1, q_0, q_1 \\right) = \\int_{t_0}^{t_1} dt\\, L(t,q(t),v(t)) + \\mathcal{O}\\left(t_1 - t_0\\right)^3", "09cdce8b229a87d3685ae8365da6e31b": "T(n, t) = e^{-t} I_n(t)", "09cddb6125ab410065ddc37956d54785": "\\sum_{i=1}^m w(P_i)\\geq w(C).", "09ce11e63fc7ee44fe80193b9334c43f": "\\Box A_1; \\ldots; \\Box A_n; \\neg \\Box B_1; \\neg \\Box B_2", "09ce1a7980b265974b030377b97186de": " V ", "09ce5ea37c0278ea35cdf4eff7ea6d57": "P_c^n(z)", "09ceaba2dec9f15b202d7c002a975d75": "\\psi(x,t) = A \\cos (2 \\pi (k x - \\omega t)+\\varphi)", "09ceabf864f361caaece7be6782e33f3": "\n\\mathbf{j}(\\mathbf{r},t)=\\frac{D_\\alpha \\hbar }i\\left( \\psi ^{*}(\\mathbf{r}\n,t)(-\\hbar ^2\\Delta )^{\\alpha /2-1}\\mathbf{\\nabla }\\psi (\\mathbf{r},t)-\\psi (\n\\mathbf{r},t)(-\\hbar ^2\\Delta )^{\\alpha /2-1}\\mathbf{\\nabla }\\psi ^{*}(\n\\mathbf{r},t)\\right) , \n", "09ceb85e5d5d470bf97e527eacb8d3e6": "\\delta_{0} = \\frac{a_{P}}{A^{1/2}}.", "09cfa02d6b4b9b1cec97243b8b7607f9": " \\operatorname{build-param-lists}[g\\ q\\ p, D, V, T_6] \\and \\operatorname{build-param-lists}[n, D, V, K_6] ", "09cfb0416e8bdbcd65c08db1d0c865c0": " \\operatorname{get-lambda}[x, x = \\lambda q.f\\ (q\\ q)] ", "09cfc4fa3a6c540362efd6cbb50072ca": "\n\\; = -(\\sum_x \\; ( \\sum_{y'} p(x,y') \\log \\sum_{y'} p(x,y') ) + \\sum_y ( \\sum_{x'} p(x',y) \\log \\sum_{x'} p(x',y)))", "09cfc8d62cd7b33a08908c896b3d6a53": "v = \\sum \\lambda_i b_i'", "09d021ad89f00335a8a8cc4877f2fe98": " f_i =\\sum_{j \\ne i}(F^C_{ij} + F^D_{ij} + F^R_{ij}) ", "09d0ac15b9c43792ddb3847c13e5ad66": "\\sqrt{\\sigma_S^P}", "09d0b6bad23d4e4ef211a55c136d5c3a": " \\frac {[A_{ad}]}{p_A\\,[S]} = K^A_{eq} ", "09d0b7ddacbd62f19ed54e802b583cb5": "\\mathrm{Re}(s)=1/2,3/2,\\dots,", "09d0bb0d0b379ba0f2776269f09d885b": "N_{0}=0", "09d0f4571592140e2de18d53edf41944": "{d \\sigma \\over d \\Omega}.", "09d1069b692f87841955de4c63585fb9": "\n \\begin{align}\n \\eta(x,t) &= \\eta_2 + H\\, \\operatorname{cn}^2 \\left( \\begin{array}{c|c} \\displaystyle \\frac{x-c\\,t}{\\Delta} & m \\end{array} \\right),\n \\\\\n \\eta_2 &= \\frac{H}{m}\\, \\left( 1 - m - \\frac{E(m)}{K(m)} \\right),\n \\\\\n \\Delta &= h\\, \\sqrt{\\frac{4}{3}\\, \\frac{m\\, h}{H}\\, \\frac{c}{\\sqrt{g\\, h}} } && = \\frac{\\lambda}{2\\, K(m)},\n \\\\\n \\lambda &= h\\, \\sqrt{\\frac{16}{3}\\, \\frac{m\\, h}{H}\\, \\frac{c}{\\sqrt{gh}}}\\; K(m),\n \\\\ \n c &= \\sqrt{gh}\\, \\left[ 1 + \\frac{H}{m\\, h}\\, \\left( 1 - \\frac12\\, m - \\frac32\\, \\frac{E(m)}{K(m)} \\right) \\right] && \\text{and}\n \\\\\n \\tau &= \\frac{\\lambda}{c}.\n \\end{align}\n", "09d16272a5eb774a03bd54ef17f4acec": "N(xy) = N(x)N(y)\\,", "09d16dbe76356199420f0cc04c2a7c9f": "\\rho^*", "09d1a180ff7daea0a42c505c86deac69": " C_{V} = \\partial Q/\\partial T \\,\\!", "09d1b8589ae3d19902c3d5494a993711": "dH\\left(S,p,N_{i}\\right) = TdS + Vdp + \\sum_{i} \\mu_{i} dN_{i}", "09d22ed6274ce33f310c6e4de82b9d9e": "\\Gamma^{\\alpha}_{\\beta\\gamma}+T^{\\alpha}_{~\\beta\\gamma}+S^{\\alpha}_{~\\beta\\gamma}.", "09d26432709dc3c5b05ae907c4324311": "LC_{50} (mixture) \\le 5000 \\tfrac{mL}{m^3}", "09d2788103c2c371e6fb606ad3021a9e": "\\alpha = \\arccos\\left(\\frac{\\cos a-\\cos b\\ \\cos c}{\\sin b\\ \\sin c}\\right),", "09d288b9607b2ab6024845c6279c7c28": "\\int\\frac{1}{2}\\xi_d^2\\frac{d}{d\\zeta}\\left(\\left(\\frac{d\\theta}{d\\zeta}\\right)^2\\right)+\\frac{1}{2}\\frac{d}{d\\zeta}\\left ( \\sin^2{\\theta}\\right)\\,d\\zeta \\,=0", "09d2b138724407ae8ca5490e1b6f5839": "\\sup_{y^* \\in Y^*} -F^*(0,y^*) \\le \\inf_{x \\in X} F(x,0),", "09d2c1055863bdad1eda764739bfc1ea": "\n\\begin{align}\n\\mathcal{A}^{AB}\\Psi_A(1,2,\\dots,N_A)&\\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B)\\\\\n &= \\tilde{\\mathcal{A}}^{AB}\\Psi_A(1,2,\\dots,N_A) \\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B),\n\\end{align}\n", "09d33e642b0ce4ce2b732db7468e0452": "EAP=BP=EC(I_r,G)", "09d34d017b4611c13ec1a5818df7f248": "\\kappa (R)", "09d396adb41a8af8f97997736a8e073c": "\\frac{1}{b}", "09d3c623910a65e31a8428ced4dc5c9c": "\\scriptstyle\\{u_n\\}_{n\\in\\mathbb{R}}", "09d3c9bcaa224a234df83194834f139c": "\\begin{align}\n\\int_0^\\delta t^{\\lambda + n} e^{-xt}\\,dt &= \\int_0^\\infty t^{\\lambda + n} e^{-xt}\\,dt - \\int_\\delta^\\infty t^{\\lambda + n} e^{-xt}\\,dt \\\\\n&= \\frac{\\Gamma(\\lambda+n+1)}{x^{\\lambda+n+1}} - \\int_\\delta^\\infty t^{\\lambda + n} e^{-xt}\\,dt,\n\\end{align}", "09d411ccbbb420e73808542f7e4e4f10": "y_n \\approx y(t_n)", "09d434d65384c517c8cff943c17f0986": "U_i \\subset X_i", "09d43b8d3ed520382e7ee4129065cd32": "\n\\delta m_0 =\n\\frac{1}{4\\pi G}\\left[\\frac{1}{\\rho_0 c^2}\\frac{\\partial P}{\\partial t} -\n\\left(\\frac{1}{\\rho_0 c^2}\\right)^2 \\frac{P^2}{V}\\right]\n", "09d4de471943d3408acc76800051dd20": "F_{q^s}", "09d4ee4f722c7c32ddd590a8049431f9": "\\displaystyle{\\|\\delta_h u\\|_{(k+1)} \\le C\\|\\Delta_1 \\delta_h u\\|_{(k-1)} + C \\|\\delta_h u\\|_{(k)}\n\\le C \\|\\delta_h \\Delta_1 u\\|_{(k-1)} + C\\|[\\delta_h,\\Delta_1] u\\|_{(k-1)} + C \\|\\delta_h u\\|_{(k)} \\le C\\|\\Delta_1 u\\|_{(k)} + C^\\prime \\|u\\|_{(k+1)}.}", "09d548d1045c2e0019d7008c386dcead": "MA = \\frac{F_{out}}{F_{in}} = \\frac{v_{in}}{v_{out}} \\,", "09d56ff0e3c7334978dc9e7d89cdd35d": "\\{l_1, \\ldots, l_n\\}", "09d5c372ed012b465e11a6e80e90ebdf": "\\hat{T}^{i_1\\dots i_n}_{i_{n+1}\\dots i_m}(\\bar{x}_1,\\ldots,\\bar{x}_k) =\n\\frac{\\partial \\bar{x}^{i_1}}{\\partial x^{j_1}}\n\\cdots\n\\frac{\\partial \\bar{x}^{i_n}}{\\partial x^{j_n}}\n\\frac{\\partial x^{j_{n+1}}}{\\partial \\bar{x}^{i_{n+1}}}\n\\cdots\n\\frac{\\partial x^{j_m}}{\\partial \\bar{x}^{i_m}}\nT^{j_1\\dots j_n}_{j_{n+1}\\dots j_m}(x_1,\\ldots,x_k).\n", "09d5d7aadb6a1368aa5739312a06d2f9": "n\\stackrel{\\text{def}}{=}\\lim_{x\\rightarrow a}(f(x)-mx)", "09d64b1f2d0a2c2b477cdd20332f1572": "\\left|{zf^\\prime(z)\\over f(z)}\\right|\\le {1+|z|\\over 1-|z|}", "09d66dde72d13e8a4f7660348a02e6fb": "1000^2 + 3^2", "09d6749d3ef8e02f3078ba7b8e384be8": "\\overline{X}_n \\, \\xrightarrow{P} \\, \\mu \\qquad\\text{for}\\qquad n \\to \\infty.", "09d69efcb4b1aaadda75c329f393629b": "f(x) - a f'(x) + {a^2\\over 2!} f''(x) - {a^3\\over 3!} f'''(x) + \\cdots", "09d6bd0c879b8e358c7a02853763fa54": "P(x=v|c)", "09d6d15e9836b8296c0c911144b9df2a": " 2^{2N} ", "09d6eb9f4f6ab048899f1aa9ed5f458c": "A=A_{0}\\left( e^{-k\\Delta t_{p}} \\right)^{n}", "09d6f72812aa2c9ee6f331c7a0a3528a": " \\pi\\ : \\prod_{i\\in \\mathrm{N}} \\Sigma\\ ^ i \\to \\mathbb{R}^\\mathrm{N}", "09d71b61e67754dcce6f3b2515454830": "F_N(J)", "09d763c29f4504840551407303e802de": "x = \\int_0^L \\cos s^2 ds", "09d794cd66c560bb1571a71998ec043f": " \\psi^{(m)}(1)", "09d7b5f7a7237a855b70da62770d96c5": "0 < \\left| a q_n - b p_n \\right| < 1 \\, ", "09d7c32dcbd5b25eb8bea40446e42f71": " Q = I_3 + \\frac{1}{2} (B+S+C+B^\\prime+T) ", "09d83f476c0bc42331604b8aaede7e39": "A=U^TU", "09d88a548409e850ede3c3c77fbbfe52": "i=1,...,N", "09d8bdf28f56b26306e94b1c71d38385": "\n \\mathcal{L} = \\sum_{i=1}^n \\pi_{i} + \\mu (1- \\sum_{i=1}^n \\pi_{i})-n\\tau' \\sum_{i=1}^n \\pi_{i} h(y_{i};\\theta)\n", "09d8f09c410d180586acdf584f613921": "p = \\frac{n R T}{V}.", "09d96d9708822e3f5dd4ae9a5411c5fd": "_{r=0}\\!", "09d9741f2d433a9fa7b26f06f645e665": "P(\\neg A|B) = c \\cdot P(\\neg A) \\cdot P(B|\\neg A)\\cdot ", "09d9be2b53bc6bc928726e444e6ce4e9": "\\frac{m}{s} + \\frac{n}{t} := \\frac{tm+sn}{st}", "09da455b257c7a7e3ef915407da20770": "\\mathbf{\\mu_z} = \\pm \\frac{1}{2}g{\\mu_B}", "09da71c37a9d91596ce304eb275b66bc": "\\alpha^{\\dagger B}", "09daae4fda46071d452272bef8583ea2": "r \\,", "09dacfe7401a6cca9a31a65c156803a4": "\nI_{fl}=0\n", "09dadba2958f69b94b39163b664cd5cf": " G = 1/R \\,\\!", "09daec7e238ac4770fccd55dd2d1294a": "\n \\boldsymbol{B}^{-1} = \\begin{bmatrix} 1 & -\\gamma & 0 \\\\ -\\gamma & 1+\\gamma^2 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\n ", "09dafcb3f81c6eccf2f3a989ae3bbed9": "\\mathcal{F}^3 = \\mathcal{F}^{-1} = \\mathcal{P} \\circ \\mathcal{F} = \\mathcal{F} \\circ \\mathcal{P}", "09db9abdce7ea350cea097b1d3a5ab78": " \\sum_{n=1}^\\infty \\frac{|a_n|+|b_n|}{n^2} < \\infty,\\, ", "09dbf08604b4dc478635f025988f1a26": " \\begin{align}\n& [\\hat{L}_a, \\hat{L}^2] = 0\\\\\n& [\\hat{S}_a, \\hat{S}^2] = 0\\\\\n& [\\hat{J}_a, \\hat{J}^2] = 0\\\\\n\\end{align}", "09dbf9e83adffdef3971ef9418272ff8": "M(x)=e^{\\int P(x)\\,dx}", "09dc0725384db62ec77b846d1c7ec865": "|\\phi(t)| < Ke^{bt} \\ \\forall t>0,", "09dcd14058ff41bad89ff5aa3ace012a": "\\Theta(\\tfrac{t^2}{\\log t})", "09dce38b81405727a859db4802fd7e6f": "\\mathbf \\tau= \\lim_{\\Delta S \\to 0} \\frac {\\Delta F_\\mathrm s}{\\Delta S} = \\frac{dF_\\mathrm s}{dS},", "09dd14c6fe7f59c5602f210e3142b8db": "\\check{~}", "09dd853dcb749c8d580c791b7fab2648": "T_{c0}", "09dd8cb5f9087b6119dc83c5958dbc4f": " \\mu_\\infty \\subset \\C^\\times ", "09ddcbc4a9e8848bf252ee0736084673": "w = e^{\\frac{2\\pi i}{c}}", "09de29557c6bd203a40b515cfa078e88": "t_a^2+mn = bc", "09de51c51c98cad7c39f6bc4977539e5": "\\scriptstyle P(c_t|s_t)", "09ded49103a530234c52a1504e2554c7": "\\mu(B)^{-1/q}\\|u-u_B\\|_{L^q(B)}\\le C \\text{rad}(B) \\mu(B)^{-1/p} \\| \\nabla u\\|_{L^p(\\lambda B)}.", "09dedad0ec263786be85542ea1a1345b": "x\\mapsto ax", "09def1137ecd849efa023974f3df766f": "u_{03}", "09df226ba36c784607bcd794757b54c5": "f(2j+1)=2f(j)+1\\;.", "09df569742d4fc5d2ce2489d780f277c": "a_2=-0.23\\,", "09df6b640027b6b670c9f51b1b1a192e": "T_{1,n}(z)=g_{1,n}(z)", "09dfb5998f333da54a19f4aeea972ccb": "\\frac{1}{D}", "09dfc9ac5bd61e3d3016e0f9ecbc0055": "\np(t)=c_0 + c_1 t + \\cdots + c_{n-1}t^{n-1} + t^n ~,\n", "09dfd2695f0cc23acfc6c2f0a81aadbe": " N(t) = N(0) \\ e^{rt} ", "09e02c6ffd1343d0573ac5a1d8831bc0": " 2\\cos \\lambda = 1+(l-l^{-1})/(\\lambda \\sin \\mu)", "09e07484cbb22259750f534b4bdaeec9": "\\mathbf{U} \\mathbf{V}^*", "09e0a8f5ba23ebc0a9a77636e6abe5c8": "(0,t)", "09e0df0a1f9dddadd41884ef1fb7722e": "u=\\cos\\theta_n\\,", "09e163edc3362baed4f46442bfdc008d": "\\begin{smallmatrix} m = M_v + 5\\cdot((\\log_{10} 3.64) - 1) = 2.6 \\end{smallmatrix}", "09e1e5312b29562b52c657bb9a24fd62": "dF\\ = -PdV\\ + \\gamma dA", "09e2578e5a0e9d9de4491045ed909c79": "\\Phi^{2}", "09e25e57528d466cbc406b1671ada025": " S = \\frac{s}{\\ell} L \\approx \\frac{12+1/3}{1/3} (67+1/3) = 2491 + 1/3 \\approx 2490 ", "09e2781a05bfb8eb8ea17d9fb0e81f02": "I\\ ", "09e286fd78ece730d55049353bdbadd9": " p = p^0 \\mathbf{e}_0 + p^1 \\mathbf{e}_1 + p^2 \\mathbf{e}_2 + p^3 \\mathbf{e}_3", "09e2a004a2228b5fb7494a36d5b4150a": "\\theta \\log{\\tan \\theta} - \\frac{1}{2}\\int_0^{2\\theta}\\log\\left(\\tan \\frac{x}{2}\\right)\\,dx=", "09e2ae91a9cfaf52ab6c4b3a9571b387": " \\frac{\\Gamma_w}{\\delta x_{WP}} A_w ", "09e2e4bf1bdf175b0238e47d27fa775f": "\\lim_{n\\to \\infty} \\int_{X}|f_n-f|d\\mu=0", "09e32a2e3d5e1edba5c9aa402a5e29d9": "g^{(n)}( \\mathbf{r}_1,t_1;\\mathbf{r}_2,t_2;\\dots;\\mathbf{r}_n,t_n)= \\frac{\\left \\langle E^*(\\mathbf{r}_1,t_1)E^*(\\mathbf{r}_2,t_2)\\cdots E^*(\\mathbf{r}_n,t_n)E(\\mathbf{r}_1,t_1)E(\\mathbf{r}_2,t_2)\\cdots E(\\mathbf{r}_n,t_n) \\right \\rangle}{\\left \\langle\\left | E(\\mathbf{r}_1,t_1)\\right |^2 \\right \\rangle \\left \\langle \\left |E(\\mathbf{r}_2,t_2)\\right |^2 \\right \\rangle\\cdots \\left \\langle \\left |E(\\mathbf{r}_n,t_n)\\right |^2 \\right \\rangle }", "09e33a58644a519ce7428cd3b9313ed1": "\\boxplus_c", "09e344fbe56d259a8d1a743061bb2f72": "\\beta = d_2/d_1", "09e38052b56f3b0bc2d4b39a676236db": "\\Gamma\\models\\psi", "09e391eb85c13e8a661bf3fed45bac3e": "\\int^{x_i + c t_i}_{x_i - c t_i} - u_t(x,0) dx = - \\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx.", "09e3adb135c2ea395bafc12726012c30": "x_1 \\geq \\dots \\geq x_n", "09e3ca144517bfff7b8ea1f9f50c9f33": "\\scriptstyle{\\hat{H}}", "09e445ea5e0c0ebe60cc96f14fa16029": "D = {Y_1}^2", "09e46311281e6bd1b35dab523bc9087d": "0 A, X=x) + \\frac{1}{2} \\cdot E( \\frac{A}{2} | B < A, X=x) \\\\\n&= E( A | B > A, X=x) + \\frac{1}{4} \\cdot E( A | B < A, X=x).\n\\end{align}", "09f6ff0a6bdbbaa07b0ec3f8174481af": "\\mathfrak{c}^2 = (2^{\\aleph_0})^2 = 2^{2\\times{\\aleph_0}} = 2^{\\aleph_0} = \\mathfrak{c}.", "09f73f07826c7778f677267a4dcc1e83": " h=q^{\\theta}+1 ", "09f7709f3f4571de1007b8e37b304d83": "\\mathrm{Re}_D < 2300", "09f77fca2fbed7d28524154b39b4dac9": "1 + 2\\,X + 5/2\\,X^2 + 5/2\\,X^3", "09f7984a078e7d5a126b09ffec037d59": "\\begin{bmatrix}\n-1 & 0 \\\\\n0 & 1 \\end{bmatrix}", "09f7e036d90033206ee873732268996b": "\\textstyle \\alpha", "09f7f163f069a6e1cb82c27537ac2bc0": "-\\frac{Nc}{4} ( \\delta_1 + \\delta_2 + \\delta_3 + \\delta_4 )", "09f814b92333ad08a35c39b669759e36": "[A]_p=A", "09f81a4c140a1024aeb41311877c86ac": "\\log_e", "09f82309ddacbe27fcf111fac5bc476d": "\\mathbf{t} ", "09f86c4021c0f1cf1d43f9c66150a0a3": " m_1\\bold{u}_1 = m_1\\bold{v}_1 = m_1\\bold{V}, \\quad m_2\\bold{u}_2 = m_2\\bold{v}_2 = m_2\\bold{V}", "09f8768bb972f60620857dbddb6d4297": "8 \\times 8", "09f88bd1376b437aeaa84da6eae1f1a1": "\\{R_a,R_b\\}", "09f89bf5f3d18172d1fade9c609b39a2": "O (|V|)", "09f8d711091f9e9870ddb7b53afd7216": "(x_1,\\dots,x_n,u) = (x_1(s),\\dots,x_n(s),u(s))", "09f8fa67084fd5a3d573b066e8be3c2e": " 2 (7225 - x^2) ", "09f92d607499f0c155459a4fa6760080": " [D \\rightarrow D^{'}] \\times D \\rightarrow D^{'}", "09f957d086522ac403e1dd1414ff4ffb": "\\mathfrak{a}\\subset\\mathcal{O}_k", "09f967a86658cb4b10019456579393fa": " \\mathbf{p} \\in [\\mathbf{p}]", "09f98b6136162c8370ea2ded329e6a1a": "\\phi(8)=4", "09f99ab956506103afe8ecb5e4a6d421": "\\alpha : A \\longrightarrow FA", "09f9a24fceb781ad152e890f915c50ea": "(f * g_N)[n] \\equiv \\sum_{m=0}^{N-1} \\left(\\sum_{k=-\\infty}^\\infty {f}[m+kN] \\right) g_N[n-m].\\,", "09f9a7255d2ae92bc51320ebf81a2645": "a^2-N", "09f9d712bd2972cf3391235881b47247": " \\tilde{\\chi}_i^\\pm", "09f9d8fe1db60eb5e7adcdd240ec1259": "(x_i)", "09f9ea98965d8f695524ec4167ab135e": "\\sum_\\rho\\left[1-\\left(1-\\frac{1}{\\rho}\\right)^n \\right] \\ge 0.", "09fa1d42b14a6b20cf1a3b99323815e9": "a_{14}+c_{14}", "09fac7eb351b39056841200250fa572a": "\\{p_{1},p_{2}", "09facc7b7f7e9b933b01625f3b71efce": "K(m)\\,", "09fb226b214696177c0ff3648e6e7007": "\\ell_4 = \\tfrac{1}{4} {\\tbinom{n}{4}}^{-1} \\sum_{i=1}^n \\left\\{ \\tbinom{i-1}{3} - 3\\tbinom{i-1}{2}\\tbinom{n-i}{1} + 3\\tbinom{i-1}{1}\\tbinom{n-i}{2} - \\tbinom{n-i}{3} \\right\\} x_{(i)}", "09fb6ed77cb734c6c350f1a8912d37ef": "\\chi '' = \\frac{\\mu_0 M_S^2 V}{3 k_B T} \\frac{2 \\pi f \\tau}{1+(2 \\pi f \\tau)^2}", "09fb822add16933e61d7fec595dac346": "z_T", "09fb89b1f4143a8e3ee33438495049e6": " (\\mathbf{ab})\\mathbf{c} = \\mathbf{a}(\\mathbf{bc}) ", "09fb9d212fd64d14c34373af16a0763e": "{\\rm e}^{At}", "09fbd3d593560d98414ce17325b62e93": "g = h = 1", "09fc3476985746e5bea1170831da4345": "\\psi_R\\rightarrow \\psi_R", "09fc50a95220ec28fb9f36d761d34917": "\n\\Gamma = \\Delta E \\ \\frac{\\partial \\Sigma}{\\partial E} = \\Delta E \\ \\rho(E),\n", "09fc70cff2099486202882c21866642d": "t(s)", "09fccec47018501be53791cd70dcc03c": "\n\\overset{\\leftrightarrow }{ \\mathbf{\\sigma} } = \\frac{1}{4\\pi} \\left[ \\mathbf{E}\\otimes\\mathbf{E} + \\mathbf{H}\\otimes\\mathbf{H} - \\frac{E^2+H^2 }{2} (\\mathbf{\\hat x}\\otimes\\mathbf{\\hat x} + \\mathbf{\\hat y}\\otimes\\mathbf{\\hat y} + \\mathbf{\\hat z}\\otimes\\mathbf{\\hat z}) \\right]\n", "09fd1380b47a6dea1e26cb0ab5047871": "f(\\gamma X)=f(X)^d \\mod E(X)", "09fd1833b601de4e62402ad84a8b1c2d": "\\varepsilon \\rightarrow \\varepsilon -A", "09fd1ac92b9bd17831cebbe77ad20412": "\\det(A-\\lambda B)\\neq 0", "09fd2c930e34e86a38066d7fc99bb9bf": " \\Psi = \\Psi \\big( X_1, \\dots, X_i, Y_{i+1}, \\dots Y_r \\big) \\, ", "09fd33ee440a1ee62a0a1f399b608868": "(-1)^{p_2 r_2}", "09fd3a5199c868a2bba315756cea73e2": "\\hat{H}'_0= \\beta (m \\cos 2\\theta + |p| \\sin 2\\theta)", "09fd43d6185d69c7fcf12f510cb7992e": "H^{0,0}", "09fd5b1a43aa14571a85e566434355e2": "H(\\vec{x},\\vec{x}')\\,\\!", "09fd6a333efb763f044cfbc10b1c375c": "(a+b)_q^n=\\prod_{i=0}^{n-1}(a+q^ib) .", "09fdd90aa9f138c056863e403fc9be97": "v=v_k + \\frac{P-p_k}{p_{k+1}-p_k}(v_{k+1}-v_k)=v_k+N\\times\\frac{P-p_k}{100}(v_{k+1}-v_k).", "09fe0d0edbec5a968be741af711db259": "\\frac{x^n}{n!}= \\sum_{i=0}^n (-1)^i {n+ \\alpha \\choose n-i} L_i^{(\\alpha)}(x),", "09fe11a693bfe9c88bc4a43062084528": "k=k_1+k_2", "09fe157d61946f7066d9efe1ae49901c": "p+q = n", "09fe4357f902d1530aad68f9c300eb63": " \\hat{E}\\Psi = i\\hbar\\frac{\\partial }{\\partial t}\\Psi=E\\Psi \\,\\!", "09fe862e0a60a263cafc4633b93ebb92": "C(K) = \\int_S \\frac{\\partial u}{\\partial\\nu}\\,\\mathrm{d}\\sigma.", "09feaf0a67966c222e48187b9d15faa5": " xy \\lor yz \\lor xz ", "09fed0febf1c6a84c276d92173171fca": "|\\chi 1 \\rangle= (|1,0,0 \\rangle + |2,1,1 \\rangle)/ \\sqrt{2} \\frac {}{} ", "09ff0be4d51af77c41516739b582de9a": "\\Omega^\\infty = 0", "09ff4f6721aa47fba648e7ebfa50fad8": "W.", "0a000227bbf039496d72929d60dfaf94": "\n [G^{ij}] = [G_{ij}]^{-1} ~;~~ [g^{\\alpha\\beta}] = [g_{\\alpha\\beta}]^{-1}\n", "0a005d2d57313c2c068478a85bedd715": "|x\\rang\\left(|0\\rang - |1\\rang\\right)/\\sqrt{2} \\overset{U_{\\omega}}\\longrightarrow (-1)^{f(x)}|x\\rang\\left( |0\\rang- |1\\rang\\right)/\\sqrt{2} ", "0a0072570a418dcb3826503f14d6786f": "\\hat{x}=Gy", "0a009b06af1bc64a4924f68a91ca3f21": "\\neg (a \\vee b)", "0a00c1fb93bc39f3b3b64402023bcb1b": "M(a,a+b,it)", "0a00e7d53427e1ef168679db0ddac661": "\\mathrm{Lie}_q\\,\\mathcal{F}= T_q M", "0a0141d8b9039fc98c6cb0c97d90f240": "(f^{*}>0)", "0a01d7147859b6fe005bc76ef8aa25d2": " \\mathbb{Q}(\\zeta_{p^\\alpha}+\\zeta_{p^\\alpha}^{-1}) ", "0a01eb6aedc5dab91c3cc43749bcac23": "\\{ x \\}", "0a022142514b3c1e104fa5b1cdf3fc55": "\n\\mathbf{H}(x) = \\nabla \\times \\mathbf{A}(x). \\quad\\quad\\quad\\quad (6)\n\n", "0a024764aab75341d90009565c6d4cb9": "\\begin{matrix} {48 \\choose 4} = 194,580 \\end{matrix}", "0a024a743f9863ed7550c3cafbbce117": "d\\Omega^2 = d\\theta^2+\\sin^2\\theta d\\phi^2\\ ", "0a028cc625a51dfc59809735c518368d": "\\tfrac{3(M-K)}{4}", "0a02f7be7d6ac20b97182a389777935e": "l, j, m_\\text{l} , m_s, m_j", "0a0320933ebeb7d6ab4df72acfad0336": "\n\\delta'_2(n)=\n\\frac{\\pi}{4}\n\\left(\n\\frac{c_1(4n+1)}{1}-\n\\frac{c_3(4n+1)}{3}+\n\\frac{c_5(4n+1)}{5}-\n\\frac{c_7(4n+1)}{7}+\n\\dots\n\\right).\n", "0a0393ec57d613f07fc734dd9f9a0b7b": "K^n", "0a040cfb4e98304d43e3ccd84b052be0": "Z^{(\\ell)}_{\\mathbf{x}}(\\mathbf{y})", "0a04315fff14859d66e75bebbaaa6990": "M_1", "0a049896dfb5b04d3d36f2b0ad0d3940": "y_n=x_{n+1}", "0a0499322e54c5f529336329e75e18ec": "x - c", "0a04d013f6688b54e521e8ee50bb15d4": "\\lambda \\leftarrow \\ln(2|\\mathcal U|)", "0a04da4e4e9362ea7bc5dee4b63d0712": "\\phi_x(D|m_x)", "0a04ec73d991624208cd624968c34bac": " \\mathbf{\\chi}_\\mu^A (1) \\mathbf{\\chi}_\\nu^B (1) ", "0a0530a98021395cb4224cca9fdc8bc1": "=\\int_N \\left[\\frac{\\partial\\mathcal{L}}{\\partial\\phi}-\n\\partial_\\mu\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\phi)}\\right]Q[\\phi] \\, \\mathrm{d}^nx +\n\\int_{\\partial N} \\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\phi)}Q[\\phi] \\, \\mathrm{d}s_\\mu \n", "0a053819ad3af1dda9071956e694622a": " v_0 ", "0a0562897ee1360e2dd957aa1bec6d53": "\\psi(\\alpha)", "0a0573881713e11008d46bd9b5d9727a": "\\scriptstyle p_2=(x_2,y_2)", "0a058a5d190a4921c7c6af726ba77944": " x \\sqsubseteq d ", "0a0591ded9dcf0fcf4ff693ecb4d86de": "\\frac{\\partial E}{\\partial \\hat{h}_i} = \\sum_{n=-\\infty}^{\\infty}[-2x[n]s[n-i] + 2(\\sum_{k=0}^{N-1}\\hat{h}_ks[n-k])s[n-i] ]", "0a0596a02eb219bd6336b93543a68c06": "m>0", "0a0605b51c34900c74441839273b47e3": " A[C] = \\int_{t=t_0}^{t_1} P \\cdot \\dot X \\, dt.\\,", "0a061ac6197ca0594b9287722fbba36c": "{{\\varepsilon }_{particle}}+2{{\\varepsilon }_{medium}}\\approx 0", "0a070c5763067f38ff8ee3f2dcfdf7cb": "Y_x = \\frac{P_1 + P_2 + \\cdots + P_x}{x}.", "0a07202355a4e1313a4b71f1fb1d61a4": "g=e^{\\epsilon \\theta}", "0a07aac793879300f27a7a20f159f4ba": "w,w'", "0a07bc667ce4bcae7d2a5d54b5c08e55": "\\mbox{Span}\\,(S) \\leq \\mbox{Ann}(\\mbox{Ann}\\,(S))", "0a07e2591063bc1d36dc01c4097998c3": " or \\,\\! ", "0a07f13c7b9a840b338d1961c92c8807": "\\gamma^\\mu \\rightarrow S(\\Lambda)\\gamma^\\nu S(\\Lambda)^{-1} = {{({\\Lambda}^{-1})}^\\mu}_\\nu \\gamma^\\nu := {\\Lambda_\\nu}^\\mu \\gamma^\\nu,", "0a081241e7e619f0622c185799e99544": "\\dot x^2+\\dot y^2+\\dot z^2=2U-C_J ", "0a0812d1b1e6512f0159ddf948309e05": "k \\mapsto \\begin{pmatrix}\n 0 & i \\\\\n i & 0\n\\end{pmatrix}", "0a082adf97abe109ec396bf328091144": "\\pi \\int_c^d [R(y)]^2\\ \\mathrm{d}y", "0a0837a73ffb8cb5135a9646a6dd351a": "\\| y_n - y_m \\|^2 = 2\\|y_n -x\\|^2 + 2\\|y_m -x\\|^2 - 4\\| \\frac{y_n + y_m}2 -x \\|^2", "0a083b07c1608f61022a3131aba076d6": "A_t(t,T)-(1+B_t(t,T))r-\\mu(t,r)B(t,T)+\\frac{1}{2}\\sigma^2(t,r)B^2(t,T)=0", "0a087ad70673831a93762e4721b53524": "\\frac{\\omega_2}{\\omega_1} ={\\sin \\alpha\\over \\sin \\beta}", "0a089b6f769b3241b7359f896b64894f": "z=.656747-.129015i", "0a08a2ce6ea4ef079afd59c51f31dacf": "\\sigma_k^*(n) = (-1)^{n}\\sum_{d|n}(-1)^d d^k=\n\\begin{cases}\n\\sum_{d\\,|\\,n} d^k=\\sigma_k(n)&\\mbox{if } n \\mbox{ is odd }\\\\\n\\sum_{\\stackrel{d\\,|\\,n}{ 2\\, \\mid \\,d}}d^k -\\sum_{\\stackrel{d\\,|\\,n}{ 2\\, \\nmid \\,d}}d^k&\\mbox{if } n \\mbox{ is even}.\n\\end{cases}\n", "0a08ff81588ada550c35ae81a497d6ed": " \\csc \\theta\\!", "0a09182e06899a88785977c1907c9d4c": "\\mathbf{a}_2 = \\mathbf{a} - \\mathbf{a}_1.", "0a09e28a82dcbd975816d85cac710a67": "\\textstyle \\alpha > 0", "0a0aa7d0c51fa6a5fdf68f855694cf73": "\n\\mathbf{S}= \\frac{1}{24}\n\\begin{bmatrix}\n2 & 1 & 1 \\\\\n1 & 2 & 1 \\\\\n1 & 1 & 2 \\\\\n\\end{bmatrix}\n", "0a0acc2973119b91a1e057dc53d5ff08": "Q(\\alpha_i) = P(\\alpha_i)E(\\alpha_i) = y_iE(\\alpha_i) = 0", "0a0acec7322df5bc74afe0a063cbd7d9": "c=\\sqrt{\\frac{g \\lambda}{2\\pi} \\tanh \\left(\\frac{2\\pi d}{\\lambda}\\right)}", "0a0ae5b96c69de86b336cf26626b835f": " T_c = \\frac{J z}{k_B} ", "0a0b30ecf732c9d4c7f2319172b15ea5": "L_{e+r}", "0a0b69766627682f4fe79abd0d9f599e": "\\prod _x x^x= C\\, e^{\\zeta^\\prime(-1,x)-\\zeta^\\prime(-1)}= C\\,e^{\\psi^{(-2)}(z)+\\frac{z^2-z}{2}-\\frac z2 \\ln (2\\pi)}= C\\, \\operatorname{K}(x) \\,", "0a0b803d9f6fe0824910872584f104bf": "b' = b^2, c' = \\sqrt{c},", "0a0bc17d6e6c88654d22f312cd8c44e7": " A - A(h) = a_0h^{k_0} + a_1h^{k_1} + a_2h^{k_2} + \\cdots ", "0a0bdc23ec0e4077adbe7d7f1b13d244": "\\chi^2(1)", "0a0c026ed94f063e7f1a0f7a13104f96": "E[S_j^{2+\\delta}]", "0a0c05f9888b0c45d99e1a03cc13d41a": " \\beta = \\frac{d-q}{p}\\equiv \\frac{\\nu}2(d-2+\\eta). ", "0a0c40bc46312c39bda2d9b43c967a92": "a_m ", "0a0c5f0ff0933ac16229d97240b8cce4": "\\int_S \\omega = \\int_T d \\omega ", "0a0c6eec7f717e267b12606b755f1bbd": "\\mathbf{B}_{\\mathbf{P}_0}(t) = \\mathbf{P}_0 \\text{, and}", "0a0c793df5df8098779330436e984ef4": " \\sqrt{2} ", "0a0cb81b6cd87583715b2825975836ca": "3. \\quad f\\star 1=1\\star f=f", "0a0cdd2bebf7825cfa46895160aa822e": "a,b,c,...", "0a0ceaa7cdc694b21f25e591b9cb0401": "\\Delta_2^1", "0a0cfa7a34002d212ce11c71179c9e52": " \\forall i,j \\,\\!", "0a0d174daa91c84e87a3f8c5fa8407cd": "-j1.49 = \\frac{-j}{\\omega L_1 Y_0}\\,", "0a0d1b1b07712a9f7f301f4050740790": "[\\nabla^2 + E]G(\\bold{r},\\bold{r}') = \\delta(\\bold{r}-\\bold{r}')", "0a0d2cb7add652380463f1bbb320d9f4": "\\beta = 45^\\circ", "0a0d925b0b4c4f0b84ff04bfbb84178e": "\\int p(x\\mid I)f_k(x)dx = F_k \\qquad k = 1, \\dotsc,m.", "0a0da101911f41c75d61aec64817f5ce": "\\lfloor \\log_2(n) \\rfloor", "0a0db1022a14e3de3f0fc480cded0fd8": "1-d = \\left(1-\\frac{d^{(p)}}{p}\\right)^p", "0a0deb2f3354b5de893085ec672f2273": "a_i^{-1}", "0a0e2f98a5da8fa805e23efcebe41404": "2Z_{oc}", "0a0e33f9866ac37568007ecf61d9f1d3": "\n\\sigma = \\frac {F} {A} + \\frac {M} {\\rho A} + {\\frac {M} {{I_x}'}}y{\\frac {\\rho}{\\rho +y}}\n", "0a0e3d6a910f97b9338039d9bfe659d9": "R_1 = \\frac{R_bR_c}{R_T} ", "0a0e8ada664ee9af47ca00b2f64a276b": "2 p \\sigma_{\\mu}", "0a0eeb59095b3c37820ae82916392683": "G^I", "0a0f00b9cbc0d133ff747d28924f3720": "W^{1, p}(\\Omega)", "0a0fd6f3b9e47e223f1ef377c143f1c8": " {\\Gamma}_{\\beta \\mu \\nu } = {\\frac{1}{2}} \\left ( \n{ { \\partial {g}}_{\\beta \\nu} \\over {\\partial x^{\\mu}} }\n+ { { \\partial {g}}_{\\beta \\mu} \\over {\\partial x^{\\nu}} }\n- { { \\partial {g}}_{\\mu \\nu} \\over {\\partial x^{\\beta}} }\n\\right )\n", "0a0ffdf01e297cca4f2365ba8d56b69e": " f(re^{i\\theta})\\rightarrow f_{1}(e^{i\\theta})", "0a1055edc119afe3aebd434d1467df28": "x\\succ y", "0a109b075fa56d2ab98435c19d91c57b": "\\Phi(x) + \\Phi(-x) = 100\\%", "0a10a3832eece68774016f1c564bb546": "C_{scat}=\\frac{1}{6\\pi}\\left(\\frac{2\\pi}{\\lambda}\\right)^4|\\alpha|^2", "0a10b0421b04b87997dc7360586db48b": "{\\color{Blue}~6.12}", "0a10d3013e3245e74a1a9eb214a85ca1": " H = H_0 + V(t) \\,", "0a10fd2acbe4c093cb9003d62fdfd7f8": "\\,\\mathcal{M}(n+1,n)<\\mathcal{M}(n+x,n)\\,", "0a1112aeea57dc846577716461c6ea59": "rate={k_p}\\left(\\frac{fk_d}{k_t}\\right)^{1/2}[I]^{1/2}[M]", "0a1163b70d0b6dcc052ad623f808082f": "SO_2", "0a1174d4999b5f870fb05ffc243b88b2": "Z_{in} = Z_{11} - \\frac{Z_{12}Z_{21}}{Z_{22}+Z_L}", "0a11841004e2dc9d292fa0e359e65b03": "\\begin{align}\nD_{1} &= \\operatorname{diag}\\{\\lambda_{1},\\dots,\\lambda_{m}\\}\\\\\nD_{0} &=R-D_1.\n\\end{align}", "0a1187e2367d522a94e57abca9ab8c75": " \\kappa(A) = \\left|\\frac{\\lambda_{\\max}(A)}{\\lambda_{\\min}(A)}\\right| ,", "0a11de642fc16acc02b7678fc438b50c": "A=P^{-1} B Q", "0a11de6665c74759c18c9e823843d45c": " \\frac{d}{dx}\\ \\operatorname{csch}\\,x = - \\coth x \\ \\operatorname{csch}\\,x \\,", "0a11e0c53e246a6c14dd080cfbe47964": "q_T", "0a11eb0ab11f5cfbc4163f7315bdbb74": "J(y):= \\int_a^\\infty dx \\ f(x,\\ y)", "0a12651908668b936d408556cc9f74db": "b_{i,j}", "0a12fde9c8c7181e4fc7e8f03e00ed0a": "f(x|a < X \\leq b)", "0a130617c2d7ea17d7a9b4a08f379480": "\\tan \\beta = \\frac {\\tan \\alpha}{\\sin \\gamma} ", "0a1307bf913c6975296c938a4db5a2d4": "u\\Vdash p", "0a13087e5f21203ea7785717e8a315c1": "\\{0.8, 0.8, 0.8\\}", "0a132c7df4f8a9e09f6dc3a375cbdf8d": "I={b}{v_x}", "0a134fd92f049e6b63675ebbe2a4c968": "|\\Psi\\rangle=|\\psi\\rangle_{1}\\otimes|\\psi\\rangle_{2}\\otimes\\ldots\\otimes|\\psi\\rangle_{n}", "0a1364b933e458d94e72db36d7967966": "\\hat{H}_3 = \\frac{\\mu_B}{c} \\sum_i \\frac{1}{m_i} \\mathbf{s}_i\\cdot\\left[ \\mathbf{F}(\\mathbf{r}_{ij})\\times\\mathbf{\\hat{p}}_i + \\sum_{j > i} \\frac{2q_i}{r_{ij}^3}\\mathbf{r}_{ij}\\times\\mathbf{\\hat{p}}_j \\right]", "0a138410f3f81e11efb5a7dcf9d0f33d": "\\left\\| \\mathbf{q} \\right\\|", "0a13c40d2f3d9ded6249c64dc128e3ff": "\\mathbf{S}(\\mathbf{p}(t))=\\mathcal{S}_z\\boxtimes_{n=1}^N\\mathbf{w}_{z,n}(p_n(t)),", "0a13d3b7887910a10c259f09a2646fa1": " \\sigma _c = 1 ", "0a13ec02a5dcb29dc8099db4d6eacef9": "A^{j_1} = A^{j_2}", "0a140c9a1e20994313e50835ee7e8d06": " S = \\{\\,a \\mid \\exists x_1, \\ldots, x_k[p(a,n_0,x_1,\\ldots,x_k)=0]\\,\\}.", "0a1422b05d8add93e61974d7c7e7031c": "5(x - 1)\\left(x^2 + x + 1\\right)", "0a1445becbb9bc326fce80739e4b6327": "\\ \\Gamma_a ", "0a1465333ef8e2ab6117a5633b95fdf1": "\\displaystyle x^n\\,", "0a14abb19a0c4fbaf70a5c3a1907fd29": " \\boldsymbol{P}_{k|k} = (\\boldsymbol{I} - \\boldsymbol{K}_{k} {\\color{Red}\\boldsymbol{H}_{k}}) \\boldsymbol{P}_{k|k-1} ", "0a150b071d92bb038c3bfffea4083996": "\\pi_{\\nu,k}\\begin{pmatrix}a& b\\\\ c& d\\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\\nu} \\left({cz+d\\over |cz+d|}\\right)^{-k}f\\left({az+b\\over cz+d}\\right).", "0a15207b66dc7a2e5de49762410f0b3c": " \\sum_{j=0}^m (-1)^j \\dim \\wedge^{2j} \\C^{2m+1} = (-1)^{\\frac12 m(m+1)} 2^m = (-1)^{\\frac12 m(m+1)}(\\dim \\mathrm S^2S-\\dim \\wedge^2 S)", "0a153fa2bc343e91250d2347191e7aaf": "v_{2} = \\frac{u_{2}(m_{2}-m_{1})+2m_{1}u_{1}}{m_{1}+m_{2}}", "0a155fc145e3d52afd5016ec51491704": "W(V) := T(V) / (\\!( v \\otimes u - u \\otimes v - \\omega(v,u), \\text{ for } v,u \\in V )\\!),", "0a156b115d741a8534a7fcb88c366938": "|a_n| \\ge |b_n|", "0a156fe394221ad15eba61e8c1344458": "[M_i,M_j]=i\\epsilon_{ijk} M_k ", "0a1574d202cb4d29e08def85b56ae6f3": "\\lambda y.e", "0a159364b389c3a6eb66c6b8b87268cb": "\\widetilde{K}^{n+2}(X)=\\widetilde{K}^n(X).", "0a1629d84ec3de449a6de90b2afb5662": "t=\\frac{\\overline{x}-\\mu_0} {( s / \\sqrt{n} )} ,", "0a165087348425a6196424c0cc77cf88": "K_\\mathit{row}=\\frac{(1-S_\\mathit{wn})^{L_o}}{{(1-S_\\mathit{wn})^{L_o}}+{E_\\mathit{o}}{S_\\mathit{wn}}^{T_\\mathit{o}}}", "0a165b8fb0a0e9907882c37f44669bcb": "\\alpha_{g}", "0a167e932da019919268f9c58921ded3": "\\lambda_i = (\\lambda_{1, i}, \\dots, \\lambda_{n, i})", "0a16a3f8f8f5fd3299ccfd0d5c70e487": "A = \n\\begin{bmatrix}\n\\frac {(x_1- x)} {R_1} & \\frac {(y_1-y)} {R_1} & \\frac {(z_1-z)} {R_1} & -1 \\\\\n\\frac {(x_2- x)} {R_2} & \\frac {(y_2-y)} {R_2} & \\frac {(z_2-z)} {R_2} & -1 \\\\\n\\frac {(x_3- x)} {R_3} & \\frac {(y_3-y)} {R_3} & \\frac {(z_3-z)} {R_3} & -1 \\\\\n\\frac {(x_4- x)} {R_4} & \\frac {(y_4-y)} {R_4} & \\frac {(z_4-z)} {R_4} & -1\n\\end{bmatrix}\n", "0a16ac818e03048c86de65e128453433": "p_{m}", "0a1729a453ae53351145b0216e15ac77": "B = \\frac{0.5a + 1.5}{5.5a+b+1.5}", "0a1733bd93771f3262048eb223b5034b": "u_i(\\overbrace{\\mathbf{x},z_1,z_2,\\dots,z_i}^{\\triangleq \\, \\mathbf{x}_i})\n=\n\\frac{1}{g_i(\\mathbf{x}_i)}\n\\left( \\overbrace{-\\frac{\\partial V_{i-1}}{\\partial \\mathbf{x}_{i-1} }\ng_{i-1}(\\mathbf{x}_{i-1})\n\\, - \\,\nk_i\\left( z_i \\, - \\, u_{i-1}(\\mathbf{x}_{i-1}) \\right)\n\\, + \\,\n\\frac{\\partial u_{i-1}}{\\partial \\mathbf{x}_{i-1}}(f_{i-1}(\\mathbf{x}_{i-1})\n\\, + \\,\ng_{i-1}(\\mathbf{x}_{i-1})z_i) }^{\\text{Single-integrator stabilizing control } u_{a\\;\\!i}(\\mathbf{x}_i)}\n\\, - \\,\nf_i( \\mathbf{x}_{i-1} )\n\\right)", "0a17939d6c58d4440be5331771c23f79": "(e, e') \\mapsto g(e) = g'(e')", "0a17f6cc447fa91f2569507065e93f1d": "W^4\\,", "0a183ed5142c1166275da8fb1cbbd43f": "\\rightarrow", "0a1862a6c53d19b2232845e9cd763465": " 2J(1-M^2) = kTg'(H/2JM) ", "0a186446102ecd71a5ebea951e4efbfd": " \\sum_{n=0}^{\\infty}f(n)-\\int_{0}^{\\infty}f(x)\\,dx= f(0)/2+i\\int_{0}^{\\infty}\\frac{f(it)-f(-it)}{e^{2\\pi t}-1}\\, dt ", "0a1892480841c9ee6c99ebfde012563c": " \\int\\limits_\\Omega f\\partial_{x_i}\\mathbf\\varphi_i = - \\int\\limits_\\Omega \\mathbf\\varphi_i\\partial_{x_i} f ", "0a18ab5ec27e2fd3b14e4b8b4d8dfe40": "E(v)=\\frac{1}{2}hv+\\frac{hv}{e^{hv/kT}-1}", "0a18fb53e9264af4daca7f7c683b97af": "X_n=\\sum_{v\\in T_n}1_{v\\in K}.", "0a1913fe386d80c7e18cc21f0b6d2a23": "{{dv} \\over {dt}}=\\epsilon (\\beta u-v).", "0a193563f17826c797e4fc1dcffc3f78": " |f(x)g(z-x)e^{-2\\pi i z\\cdot\\nu}|=|f(x)g(z-x)|", "0a19359a53b67d41c5bec2ab8020de4e": "\\forall m_\\bullet \\forall X_\\bullet ((\\varphi(0) \\land \\forall n (\\varphi(n) \\rightarrow \\varphi(Sn)) \\rightarrow \\forall n \\varphi(n))", "0a1967fa696523c5094b76eb31a2299b": "P(G,k)= P(G+uv, k) + P(G/uv,k)", "0a19a174c8c305734ee08cd9c59b8173": "\\textstyle \\rho(m) ", "0a19cb88a54ddc8c0ab871ea2d899da4": "\\|x\\|_{bv_0} = TV(x) = \\sum_{i=1}^\\infty |x_{i+1}-x_i|.", "0a19e88be978af42cd1c72fcc7c43272": "\\tau = F\\left( \\frac {\\partial u} {\\partial y} \\right) ", "0a19e8a90704fee00143624d2d601811": "(zI-A)^{-1}", "0a19fc74f4f19b6b29af362f2dd933df": " y^1 = r \\cos( x / r) ", "0a1a18cb25e26836240de2d3b4427a35": "y\\tan\\varphi=y\\frac{dy}{dx}.", "0a1a1fba115965b654ff0834746c6b98": "\\mathfrak{P}^{92}", "0a1a579c6cca0ed87742348865b344b5": "N \\rtimes F", "0a1a596db1fce034b1ad4ac503c0f4b6": "EAS = TAS \\times \\sqrt{\\frac{\\rho}{\\rho_0}}", "0a1a88e81e4cf6ea8911ef0aa78e0652": "a\\ \\pmod{\\Phi_n(q)}", "0a1ace4f9fb9fd22d914c7769070d597": "EIF_i", "0a1af8df9006ec8d2e60e181da9209a1": "\\psi d \\mathbf{s} = \\iiint_V \\nabla \\psi\\, dV", "0a1b186df888dd0885b9dfafa85a27e1": "P\\left[A|H\\right] \\mbox{ }=\\mbox{ } \\frac{P\\left[A\\right] * P\\left[H|A\\right]}{P\\left(H\\right)} \\mbox{ }=\\mbox{ } \\frac{P\\left[A\\right] * P\\left(H|A\\right)}{[P\\left(A\\right) * P\\left(H|A\\right) + P\\left(R\\right)*P\\left(H|R\\right)]}\\mbox{ } ", "0a1b4cb7586f33a09e22ccf7b42d3e44": "\\mathbf{J}_{n,\\text{diffusion}}/(-q) = - D_n \\nabla n, \\qquad \\mathbf{J}_{p,\\text{diffusion}}/q = - D_p \\nabla p.", "0a1b5bdfd6e0fe5f49dcfe3e4ef3dd4f": "\\eta_e\\,\\!", "0a1bb414a4b7febbfc581c32f2a7252e": "\\lambda x.x^2+2", "0a1bc7b13d508efee843b56c89707c37": "m({}^{12}{\\rm C}) = \\frac{12 M_{\\rm u}}{N_{\\rm A}}", "0a1cad5fc1dc53ca85e162eec66e3907": " \\Psi(x,t)=\\psi(x) e^{-iEt/\\hbar} \\, . ", "0a1cc9e93748b91ebcd91d264e6f8073": " L = {\\rm st}(x_H)\\,", "0a1da683b8497c3da0c344c52c5135cb": "f(x) = 6x^4 - 2x^3 +5", "0a1daa1ce4d23fac8b1d3626e6adb740": " \\gamma^k = \\begin{pmatrix} 0 & \\sigma^k \\\\ -\\sigma^k & 0 \\end{pmatrix} ", "0a1dc1fdc410edc85d748b690474c267": " Y(s) = \\mathcal{L}\\left \\{ y(t) \\right \\} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty} y(t) e^{-st}\\, dt ", "0a1e05d68b37bc1cd4778f6c5bf5a2cc": "\\mathcal{F}(\\mathbf{x})=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(x_n).", "0a1e1094647b377be14178c4cd356895": "r_1 = \\frac{\\delta}{\\sigma}", "0a1e2df73a299b0402f03d7076ac60d5": "\\sqrt{xy} = \\sqrt x \\sqrt y", "0a1eb5132685a7e2077db51c722cfbe7": "-\\frac{1}{2}+\\frac{1}{2}i \\sqrt 3", "0a1f3d8dbc79d1c9275aeb3ea50c129a": "\nc_0 \\equiv a_0+b_0 \\mod p\n", "0a1f3d9a63905b83635d9d5f43a0c508": "\\mathrm{^{244}_{\\ 96}Cm\\ \\xrightarrow [18.11 \\ yr]{\\alpha} \\ ^{240}_{\\ 94}Pu}", "0a1f64ce55f7f835e2510cc481fcb532": "r, n(i) \\in \\mathbb{Z}", "0a1fa36c48ca2abec461e7b9b57875b3": "\\frac{1}{500\\log(1/\\varepsilon)}\\log n < s_{r,\\varepsilon}(n) < \\frac{5}{\\log(1/\\varepsilon)}\\log n", "0a1fb3eb638b962c50e5e3a33cc95f89": "\\mathcal{N}_p(\\boldsymbol{\\mu},{\\mathbf \\Sigma})", "0a1fc8eb91cf4bb2280c0a7e4a1d298c": "\\mathcal{}R_*", "0a2009055e9d9210748c06dd7d87778d": "f = \\sum_{e\\in\\Gamma} c_e T^e", "0a204b7c16db604c5958922d7fd5d564": "v_{(G; c)}(\\{1,2\\})=16", "0a204bb81ec3e110c6250555adb5b2fc": "\\epsilon\\colon\\underset{\\longleftarrow}{\\mathrm{Lim}} (F/E) \\to F_e,\\qquad s\\mapsto s(e)", "0a2053358b00e620a5991b3020e30878": "O(\\log q)", "0a20ae981425113068a8691156c7aaff": "\\int_\\Omega g(x) |J_k u(x)|\\, dx = \\int_{\\mathbb{R}^k} \\left(\\int_{u^{-1}(t)}g(x)\\,dH_{n-k}(x)\\right)\\,dt", "0a20c395b3006a84fb557e209e150c19": "\\scriptstyle \\mathcal X", "0a20f8ff37a71490a400525c80201c56": "\\Delta d = - d \\cdot \\nu {{\\Delta L} \\over L}", "0a210060d6838c83dbcc3377af211eb5": "g=3N", "0a214f681774e1d756a65ddffc9bfc6e": "\\mathbf{H}_w", "0a21b7b50d65cf0b71c1d1aa0a20c3d3": " W_{n+1} = W_n - \\mu\\Delta \\varepsilon [n] ", "0a21ef1b100a7ba8890cfeab2c071c32": "f : \\mathbb{R}\\rightarrow\\mathbb{R}", "0a2200094516917c86e129afa1a4ea60": "\\mathbf F' = \\mathbf F_\\mathrm{physical} + \\mathbf F'_\\mathrm{Euler} + \\mathbf F'_\\mathrm{Coriolis} + \\mathbf F'_\\mathrm{centripetal} - m\\mathbf A_0", "0a222067e2868d4d9755518d0deff13a": "N=-{{1}\\over{2\\pi i}} \\oint_{u(\\Gamma_s)} {1 \\over u}\\, du=-{{1}\\over{2\\pi i}} \\oint_{v(u(\\Gamma_s))} {1 \\over {v+1/k}}\\, dv", "0a22604c6270cafda16d1bee51963ab4": "C_n", "0a22a24e587c35c18dc4a7894d567029": " H(t) \\left| \\psi (t) \\right\\rangle = i \\hbar {\\partial\\over\\partial t} \\left| \\psi (t) \\right\\rangle", "0a232289179fd29263a0c0c8f3965d0c": "\\displaystyle{\\|u\\|_{(1)}^2 =|(L u, u)| +\\|u\\|^2_{(0)}\\le \\|L u\\|_{(-1)}\\|u\\|_{(1)} + \\|u\\|_{(0)}\\|u\\|_{(1)}.}", "0a232fbd7b8c8710497626120fcc7050": "x\\in X(k)", "0a233f3208a4c2087053afffcea5a559": "\\begin{align}\n\\sum_{n=0}^{\\infty} 2^n a_{2^n} & = \\underbrace{a_1+a_2}_{\\leq a_1+a_1}+\\underbrace{a_2+a_4+a_4+a_4}_{\\leq a_2+a_2+a_3+a_3}+\\cdots +\\underbrace{a_{2^n}+a_{2^{n+1}}+\\cdots +a_{2^{n+1}}}_{\\leq a_{2^n}+a_{2^n}+a_{(2^n+1)}+a_{(2^n+1)}+\\cdots +a_{(2^{n+1}-1)}}+\\cdots \\\\\n & \\leq a_1 + a_1 + a_2 +a_2 + a_3 + a_3 + \\cdots + a_n + a_n + \\cdots = 2 \\sum_{n=1}^{\\infty} a_n.\n\\end{align}", "0a23437c16c2b1e852308f4de88fe61d": " BS=REL-RES+UNC", "0a2350c59d19586ac840508e23861ca8": "[\\![\\mu Z. \\phi]\\!]_i", "0a23557b016ca469764688058f344aa9": "x_{n2}", "0a23616b189666f5e4b397319ddc35d2": "\\{A, B, C\\}", "0a23867b18487c42af3f27cc561feb9c": " \\omega_{eff}^{2} = (\\omega_{1}^{2} + \\Delta\\omega^{2})^{1/2} ", "0a23b91713b56ab27054aaaa771aa4d4": "\\frac{r^2+s^2}{4r}", "0a23b9f6eb1e2996867bc4b08879589d": "\\scriptstyle x", "0a23dc895b2ad3505ffd5117be6a3cbb": "J_3(\\mathbb O)", "0a23fe65b81a08d65b8454ffa28f0787": "\\ominus\\alpha(t)=\\alpha({b}_{1}+{a}_{1}-t)", "0a241ca60122547fd31c0f79922c212d": "\\forall i \\in N : \\sum_{S \\in 2^N : \\; i \\in S} \\alpha (S) = 1", "0a2459dee3ff9de888b6b2203b5b7a0c": " V_0 = \\sqrt{{R^2 g} \\over {R \\sin 2\\theta + 2h \\cos^2\\theta}} ", "0a2494047becbd145aa981c9e2b3e7b5": "\n\\begin{align}\nU(\\theta)\n&= a \\left [e^{\\frac { i\\pi S \\sin \\theta }{\\lambda}} + e^{- \\frac { i \\pi S \\sin \\theta} {\\lambda}} \\right]\\int_ {-W/2}^{W/2} e^{ {-2 \\pi ix' \\sin \\theta}/(\\lambda)} dx'\\\\\n&= 2a \\cos {\\frac { \\pi S \\sin \\theta }{\\lambda}} W ~\\mathrm{sinc} \\frac { \\pi W \\sin \\theta}{\\lambda}\n\\end{align}\n", "0a24e06581747f0e1517fe4bdcb54a76": "f(\\mathbf{x}) < f(\\mathbf{x}_0)", "0a24fc2ccf9c0a995e1c2e40a4de862c": "1, e_1, \\dots, e_{n-1}", "0a25c540ced7de06e6fe17a4aa4eeb7f": "F_u", "0a25c6f2df018d7fb09fda8d3f31ec27": "\\ge 40", "0a2635976155a6c5ff7bd65ba0000a97": "-\\frac{dz}{z}", "0a26616a3b8b7d2c3d53c5a412262bcf": " NID(x,y) = \\frac{ \\max\\{K{(x\\mid y)},K{(y\\mid x)}\\} }{ \\max \\{K(x),K(y)\\}}, ", "0a267158028361e65b9a28b938fdb56b": "\\chi(S') = N\\cdot\\chi(S). \\,", "0a268719d39a59c4669014aad95e6e69": "|V|^{1/3-\\epsilon}", "0a2687efa8a35f9ccd6d77ff82f1389d": "\ng(t) = \\int\\limits_0^t f^1(\\theta^1(t))f^2(\\theta^2(t))dt\n", "0a26aa222ea7c3aa32828fca5277921a": "\\mid\\mid \\!\\,", "0a27138ba2a959c21fb884cebb18c288": " \\vdash \\phi \\wedge \\chi \\rightarrow \\phi ", "0a2758ecf3314d27ba3e0b55be06e0de": "GL^+(4,\\mathbb R) ", "0a27703d30fb698a0dac02f83219bfae": " \\epsilon^{\\delta\\alpha\\beta\\gamma} \\dfrac{\\partial F_{\\beta\\gamma}}{\\partial x^\\alpha} = \\dfrac{\\partial F_{\\alpha\\beta}}{\\partial x^\\gamma} + \\dfrac{\\partial F_{\\gamma\\alpha}}{\\partial x^\\beta} + \\dfrac{\\partial F_{\\beta\\gamma}}{\\partial x^\\alpha} = 0 ", "0a2791e3c5dcd530943fdcd7e9823c8b": "\\frac{1}{g} = {\\rho \\over \\gamma}", "0a27aa865dda31a239f4e31d7daa0ed0": "\nW = -\\mu_{1} \\left( \\cosh \\xi + \\cos \\eta \\right) - \\mu_{2} \\left( \\cosh \\xi - \\cos \\eta \\right) \n", "0a27ab4efbbcb36bd47a638739d1bc26": "\\frac{dP}{P} = - \\frac{dz}{H}", "0a28d5914e8d87ea9e82f2fda4d03335": "G(q) = \\sum_{n=0}^\\infty \\frac {q^{n^2}} {(q;q)_n} = \n\\frac {1}{(q;q^5)_\\infty (q^4; q^5)_\\infty}\n\t=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\\cdots \\,\n", "0a28f54614e0e21cc69b94b7a0a72409": "u_\\star", "0a29847069a9b927b5cec9260a687e11": " q_{ult} = c' N_c + \\sigma '_{zD} N_q + 0.5 \\gamma ' B N_\\gamma \\ ", "0a299f0c496a8dd70ae4b1227ae7590d": "r_{2}=(g_{31}-g_{13})/(2sin\\Theta)", "0a2a069b23d8a67797c59708b55da8e4": "[low] \\vdash\\ \\textbf{if}\\ l = 42\\ \\textbf{then}\\ h\\;:=\\; 3\\ \\textbf{else}\\ l\\;:=\\;0", "0a2a6a3d066336a46e432a96a9fa5934": "\\, h_01 -h_03 = h_02 - h_03 = ( U_2\\,V_w2- U_1\\,V_w1)", "0a2a6af37d470b213319a6fb77a333f4": "\\bigcup_{n=1}^{\\infty} A_{n} \\in \\mathcal{R}", "0a2a6b9de070a7b8a54136eb78f0582c": "\n\\begin{align}\n& {} \\quad 1 + \\cfrac{e^{-2\\pi\\sqrt{5}}}{1 + \\cfrac{e^{-4\\pi\\sqrt{5}}}{1+\\dots}} \\\\ \\\\\n& = \\cfrac{e^{-2\\pi/\\sqrt{5}}}{{}\\ \\ \\cfrac{\\sqrt{5}}{ 1+\\left[5^{3/4}( \\varphi-1)^{5/2} - 1\\right]^{1/5}} - \\varphi\\ \\ {}} = 1.000000791267\\dots\n\\end{align}\n", "0a2aa4be1667d172d02679476f7f9b36": "s\\in S_0", "0a2ad2d54d7baf7661087ab4085d9614": "O(n^3\\log{n})", "0a2b23378f9a7fb8c26c60564a751de8": " \\dot x = \\sum_{i=1}^k f_i(x) u_i(t) \\, ", "0a2bac01dd11440c606a9c87d669d452": "\\mathbf x_0\\in X", "0a2bf7cd88ad961dddd7e2c0ef01c16a": "\\tau_{ij} = \\mu \\left( \\frac{\\partial v_i}{\\partial x_j} + \\frac{\\partial v_j}{\\partial x_i} - \\frac{2}{3} \\delta_{ij} \\nabla \\cdot \\mathbf{v} \\right) + \\kappa \\delta_{ij} \\nabla \\cdot \\mathbf{v} ", "0a2cde9838dd5604ac10922e5f618b9c": "\\frac{1}{Z_{\\text{eq}}} = \\frac{1}{Z_1} + \\frac{1}{Z_2} = \\frac{Z_1 + Z_2}{Z_1 Z_2}", "0a2cf1f004d8822a9f5b2ba2731b2d1c": "\\Delta H_{mix} \\,", "0a2d4d62bafe23c91641d70117cd9472": "P_{i,i+1}=\\alpha_i", "0a2d80180ac8e08f83b17e18e470bd7c": " \\left( \\frac{\\varepsilon_{eff}-1}{\\varepsilon_{eff}+2} \\right) = \\delta_i \\left( \\frac{\\varepsilon_i-1}{\\varepsilon_i+2} \\right)", "0a2d870e41cd39fc1970ad586a02a135": "V_t^T", "0a2da9de14f28db8e5703bf9fbbdfea5": "X = \\prod X_i", "0a2ddafaad926eb4ca45cea0d2e04c04": "\\begin{align}\nR'_D &=& \\frac{255}{219}\\cdot(Y'-16) &+&&& \\frac{255}{112}\\cdot0.701\\cdot(C_R-128)\\\\\nG'_D &=& \\frac{255}{219}\\cdot(Y'-16) &-& \\frac{255}{112}\\cdot0.886\\cdot\\frac{0.114}{0.587}\\cdot(C_B-128) &-& \\frac{255}{112}\\cdot0.701\\cdot\\frac{0.299}{0.587}\\cdot(C_R-128)\\\\\nB'_D &=& \\frac{255}{219}\\cdot(Y'-16) &+& \\frac{255}{112}\\cdot0.886\\cdot(C_B-128)\n\\end{align}", "0a2e5566cb87829bfbebf083888752e7": " \\dot{W}_{net} ", "0a2e84afd5587e449c4624d49f236d35": "(\\sigma(t))_{t\\in\\Lambda}", "0a2ea04e2867762b008d434644b870d5": "n_1, n_2,\\ldots,n_k", "0a2f0b6393a88d3af070069f84377cba": "\\int_{0}^{\\infty} e^{ax}\\,\\mathrm{d}x=\\frac{1}{-a} \\quad (\\operatorname{Re}(a)<0)", "0a2f665fc840337903369b0ff94c4767": " \\dfrac{\\partial \\mathbf{q}}{\\partial t} = \\dfrac{\\partial H}{\\partial \\mathbf{p}} ", "0a2f79fc558c707173ec6aadd110f53d": "_{y_\\wedge x}\\!", "0a2fcdfb8691df0d01332bfeb72bb1c1": "0\\leq n< N", "0a2ffbbe0b8fa3c801b4c4eab4d3a562": "w \\equiv v", "0a3016839173a5802586efe4e2dd3d25": "p^{-s}", "0a303a444cfbc3a337fa9c13eb7b3562": "\\theta (u,u',\\xi ,\\xi ')=\\frac{1}{2\\pi }{{P}_{V}}{{\\phi }_{\\gamma (u,\\xi )}}({{u}^{'}},{{\\xi }^{'}})=\\frac{1}{2\\pi }{{P}_{V}}g({{u}^{'}}-u,{{\\xi }^{'}}-\\xi )", "0a303a8c501e15b31d25f8f17b081948": "|A \\setminus B| \\le 2|B \\setminus A|.", "0a3040cc25c36d563a6bd07dc7f3f1b4": "2|\\alpha|^{3s}+|\\alpha|^{4s}=1", "0a3061ea1a29ede000b8e8b5c908b51d": "(\\mu_{n_k})", "0a306ab913684a1ba3935715d3dd8ad8": "c=\\infty", "0a3081ffc9ca02edf660215fc9f43d09": "P(r|\\theta,a,x)", "0a30b4681a3cf7d32adae695801a90bb": "|\\psi(t_0)\\rangle", "0a30b975899e6aa7360802287d94ef86": "\\rho: \\mathcal{L}^p \\to \\mathbb{R}", "0a30e094f098626d3468318024fb190b": "j = J = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix},", "0a3133675cad9befa7f7f3bfc4751354": "x^3(x-3)(x-2)^3(x+1)^2(x+2)(x^2+x-4)(x^3+x^2-4x-2)^4", "0a31b7360edc977e9bcc67a9d16757ea": "m(t) = \\mathbb{E}[X_t].\\, ", "0a32128131fcb6d1c96bbda3c630d4af": "R_{0} = \\frac{(R1 + R2)}{2}", "0a32600cdacf59cd2c0910bb1d54949f": "N=\\frac{MC}{R}, ", "0a32638a1f44c1cd3a154269cbe988ef": " \\int_C f \\,ds = \\int_a^b f(\\mathbf{x}(t))\\left|{\\partial \\mathbf{x} \\over \\partial t}\\right|\\; dt", "0a3278a32578b589ad28c625ea2da5e4": "R = k[t_0, ..., t_n].", "0a33a0402f81d78e27b78eb648fce6bc": " \\rho = \\frac{3 M}{4 \\pi R^3}", "0a33c5c7d2bd6d5be93a040569ac9e72": "\\cot (\\alpha + \\beta) = \\frac{\\cot \\alpha \\cot \\beta - 1}{\\cot \\alpha + \\cot \\beta}\\,", "0a33eaeefccedd899e202eeb19628af9": " \\frac{d^2y}{dt^2} + f(t) y = 0, ", "0a34d509c2b7d3ec0c8646c7ef7891d3": "xP(L_+) + yP(L_-) + zP(L_0)=0,\\,", "0a34f75cb7d740e36ec572e0646e504d": "A-C(W)= \\frac{1}{n-1} \\sum_{j=1}^n (n - j) w_j. ", "0a3515fb8c3379cfa5afe465fb88cce1": " f(\\beta | \\theta) ", "0a3518643cdfabf1868a63b8f4b1d2ff": "\\sigma_{ab}", "0a354360956756f67eacd15a585666cf": "\\mathbf{Q}^m = 0 ", "0a35a5b4f285f64822315fdede393e0c": " Q ", "0a35c88be98fc920f259fee5ba22b369": "\n\\sigma_{rr} = \\frac{\\sigma}{2}\\left(1 - \\frac{a^2}{r^2}\\right) + \\frac{\\sigma}{2}\\left(1 + 3\\frac{a^4}{r^4} - 4\\frac{a^2}{r^2}\\right)\\cos 2\\theta\n", "0a360cce81115dd7d0eadf379c26a653": " x_0 = y_0 ", "0a362438f4c44249db5259f2ce293948": "\\lim_{n\\to\\infty}\\frac{1}{\\log n}\\prod_{p\\le n}\\frac{p}{p-1}=e^{\\gamma},", "0a365f67af7fdd5e027e368d41c9a7af": "A(\\lambda)", "0a36e68fd79ecd237e1312b8de585bb6": "\\left\\{ y~\\backepsilon~x\\succcurlyeq y\\right\\}", "0a372e04dd231b62ff576dce9ebc9f9b": "\\text{duplicate}: (C \\times A) \\rarr (C \\times (C \\times A)) = (c, a) \\mapsto (c, (c, a))", "0a37a4997591025b2ab5a3acfb8407a4": "\n\nf(Q) = {\\sqrt{2}\\over 4\\pi^2} {d\\over dQ} \\int_Q^0 {d\\Phi\\over\\sqrt{\\Phi-Q}} {d\\rho^'\\over d\\Phi},\\ \\ \\ \\ \\ \\rho^'(\\Phi) = \\left[1+r(\\Phi)^2/r_a^2\\right]\\rho\\left[r(\\Phi)\\right].\n", "0a385d59d41a2331b81d4afdfeebc2f2": "P6/mmm", "0a3895e528e72ca76a7705c686e729b0": "\\quad V = \\sum_{i=1}^n \\frac{1}{|\\mathbf{x} - \\mathbf{x}_i|}.", "0a38c22bfd89b600974f06f80cdbc7a9": "\\mu : M \\to \\mathfrak{g}^*", "0a38c541403c690be348db680f4a1de0": "MPL_{t,1} = \\int \\Delta X \\cdot \\Delta X^T p(\\Delta X)d\\Delta X=\\frac{k_B T}{\\gamma}\\Gamma^{-1} ", "0a3d4a2c3fbb7071c2288f7542dc6452": " \\big[\\partial_x+\\partial_y+\\tfrac12(y-x)\\big]\\,\\big[...\\big]", "0a3d655931e50806941c7f9edc142324": "\ni\\frac{\\partial}{\\partial t}u(x,t)=-\\frac{\\partial^2}{\\partial x^2}\nu(x,t)+g(|u(x,t)|^2)u(x,t),\n", "0a3da209c870dd96b8746e853cb6575c": "\\varepsilon_n = -\\frac{\\partial z_n}{\\partial p_n}\\frac{p_n}{z_n} ", "0a3da9973aa4bba2479d8110395a0c96": "H(s) = \\frac{G_0}{B_n(a)}", "0a3db6a040700fa21b51e59ceb52ea75": "\\langle e_n, e_n\\rangle = 1.", "0a3dd035bbc04a67ea2221a3eef66713": "L(x,y)", "0a3e16c1412b8f0d57c11d8e63862f25": "\\dot m = C A \\sqrt{k \\rho_0 P_0 \\left(\\frac{2}{k + 1}\\right)^{\\frac{k + 1}{k - 1}}}", "0a3ed5734d668253b37a5fdab6f7ae16": "\n\\operatorname{Var}_{X_i} \\left( E_{\\textbf{X}_{\\sim i}} \\left( Y \\mid X_i \\right)\n\\right)\n", "0a3ef145e2b4c50f43092d30caf5cbc9": "\\mathrm{d}S=\\frac {C_P}{T}\\mathrm{d}T-\\alpha_V V\\mathrm{d}P.", "0a3f14a03e1179ff5fcf85f56d22aefe": "\\mathrm{SNR} = \\frac{ | {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{-1/2}s) |^2 }\n { {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h) }\n \\leq s^\\mathrm{H} R_v^{-1} s.\n ", "0a3f18d6b2eb22610e573e77a96d8e2f": "X[p\\Delta_{F}]\\,", "0a3f1fdbf8d4999a6721385856cb28ba": "E_{1j} = \\frac{O_j}{N_j}N_{1j}", "0a3f6e6fcf2b2547bf675fa98d064c79": "g(t|x) = \\frac{f(x|t)g(t)}{\\int_{x}^{\\infty} f(x|t)g(t)dt} .", "0a4002d3471a20dab34fba559688fc25": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ +0.10\\ \\pm\\ 0.03\\end{smallmatrix}", "0a404084903fdc45863ca72a4c05f66a": "E[R_a-R_b]", "0a4043df7ad41149fd439dce8bc75afb": "j_m = \\lim\\limits_{A \\rightarrow 0}\\frac{I_m}{A}", "0a406eb6da79e917940143234380ba81": " N(x) ", "0a40ac116550b45e7c09756fa6356f95": " \\sin\\theta = \\frac{2x\\sin\\theta'}{(x+y)+(x-y)\\sin^2\\theta'}, ", "0a4173ea4d4535bd3068215590c761da": "T_M", "0a41b44f2382f76cb9bc852a21ac42e8": " f \\ ", "0a41e223b52e58bf4824d6d9189b5d97": "\\mathfrak{P}^{71}", "0a41f2dd923676871312de0e840d302e": "TP", "0a4240e180e6f0d9807b58cea49d1b05": "\\tan\\phi= \\frac{X_L - X_C}{R}\\,\\!", "0a42a92fa31f71446817e999772362d6": "H(e^{j\\omega})", "0a42b6293e7172cea22786b67425f43b": " \\left|\\Theta\\right\\rang ", "0a42cb1feaed1096d983d4406f88cbfa": "s_n = \\left\\lfloor E^{2^{n+1}}+\\frac12 \\right\\rfloor", "0a42fe7084bd3a52d2c7e5f77e7ccd3d": "bWAR = (P_{runs} - A_{runs}) + (A_{runs} - R_{runs})", "0a4342572c283f55ea626fe939845e4e": "\nh_i^t = e^t \\times { E_i^t \\over E^t } \n", "0a43470746f696689b3bb4673bc2c7e6": "\\textbf{P}_{k\\mid n} = \\textbf{P}_{k\\mid k} + \\textbf{C}_k ( \\textbf{P}_{k+1\\mid n} - \\textbf{P}_{k+1\\mid k} ) \\textbf{C}_k^T ", "0a434fc2469bd6166b56190158c24c75": "P = (u-v) \\cdot (s-t) + (u \\cdot t)(v \\cdot s) - (u \\cdot s)(v \\cdot t),\\,", "0a4361107b0ec4674e31bc3ea176d76a": "\\ell_j(\\xi) = \\prod_{i=0,\\, i\\neq j}^{k} \\frac{\\xi-x_i}{x_j-x_i}, ", "0a4388db865396276769880ba66c6d92": "u = \\begin{bmatrix}1\\\\3\\\\4\\end{bmatrix}\\quad\\quad\\quad", "0a43cabef5cdf4b5a97a4dcb7268f2cb": "A[Y_1,\\ldots,Y_n]", "0a43f21155e59f2ed6206ba470a89fac": "{n \\choose r}2^{1-{r \\choose 2}}.", "0a4421c54bb9409c499165e5910c54b9": "g_L = 1", "0a44261e59194cea7e82f93dc22d8a9f": "\\frac{\\mathrm{d}}{\\mathrm{d}t}", "0a443b86d7bc3f2681fd175ebc1aae68": "v_m=\\sqrt{2gD}", "0a44b2fc5e822be617ca2cb296916d9a": "c = C / N_A", "0a45051524bb6da37aa3a1f3b99d1184": " \\operatorname{Pr}\\{y\\} = \\operatorname{Tr} (S U^* \\operatorname{E}_{y} U)", "0a4528c48a099c5cffb227173fb2c503": "\\sqrt{ \\frac{7}{3} }", "0a455a9a2a862bf914721d43176f8c01": "\nE_\\mathrm{force} = \\int{F \\mathrm{d} x = F L (1 - \\cos \\theta )}\n", "0a45887b6fd3b1f83832a6dcf9039b18": "\\gamma_*: C_* \\to C_{*+1}", "0a4589e7895335316611a906d22ef9ba": "\\ \\bar{T}", "0a45a76ee727bf5b49651de72ccada7e": "R(X,Y)=\\nabla_X \\nabla_Y - \\nabla_Y\\nabla_X - \\nabla_{[X,Y]}", "0a45f5a5faba1511b18e77455330af6b": "\\Delta = \\sum_k e_{kk} = \\text{div}\\; \\mathbf{v},", "0a46638600a6198ef3916077c2d91306": "\\operatorname{core}", "0a468ad0b29472abe84e8132d50032c4": "L' \\to L= \\frac{\\omega_c' Q}{\\omega_0}L' \\,,\\,C= \\frac{1}{\\omega_0 \\omega_c' Q}\\frac{1}{L'}", "0a473bb4fe983d5d609760478314c4d8": " v=b^*+\\frac{\\Pr(b^*\\ \\textrm{wins})}{\\partial \\Pr(b^*\\ \\textrm{wins})/\\partial b} ", "0a47984f4692cab6153f38cadaf2f462": "\\mathfrak{a}+\\mathfrak{b}", "0a479924b84b40e86d990664c0afed85": " B \\subset \\mathbb R. ", "0a47ad8c1e7ebde115be82a55e91b587": "\\textstyle V_\\theta^2 ", "0a47c8f4848e78ac5d2f0c4d5bb2fe47": " n \\geq \\frac{1}{(p -\\frac{1}{2})^2} \\ln \\frac{1}{\\sqrt{\\varepsilon}}.", "0a47db5f4074a82206c4cc01f1a60d67": "C_p(p,T)-C_V(V,T)=\\left [p(V,T)\\,+\\,\\left.\\frac{\\partial U}{\\partial V}\\right|_{(V,T)}\\right ]\\, \\left.\\frac{\\partial V}{\\partial T}\\right|_{(p,T)}", "0a483b2fb471e4d19089462c552fed26": " (a\\cdot d+b\\cdot c)\\cdot 10 ", "0a491e43f5231fc38462b12ea2309d18": "g'N'", "0a4934396f709f76cf3007f2c0437159": "S(n_1,\\ldots,n_l) = \\exists m_1\\cdots \\exists m_k R(n_1,\\ldots,n_l,m_1,\\ldots,m_k)", "0a496937c8a48ab07f4313bfb4dd6f49": "\\sum_{n=-\\infty}^\\infty a^{n(n+1)/2} \\; b^{n(n-1)/2} = (-a; ab)_\\infty \\;(-b; ab)_\\infty \\;(ab;ab)_\\infty.", "0a498ea7a459d8602f53de446b0d5e67": "KE = \\frac12 I\\omega_{max}^2=\\left ( 10^{-3} \\frac{\\ell^2}{3} \\right ) \\left ( \\frac{254}{\\ell/2} \\right )^2 = 43 \\text { erg}", "0a499d901a3f2af99c4d726590ac2006": "(A\\oplus B)\\subseteq C", "0a49dd19c59685ed54a4a3abb5cac0af": "\\widehat{\\mathcal{M}}", "0a49ffa5288576b9a809b7c539eb25b2": "\\operatorname{Alph}(s)", "0a4a1ae98e9b80e839929846248e45cf": "p(n)\\approx 1-e^{- n^2/(2 \\times 365)},\\,", "0a4a1da19ac2b7e365c805b2d2bbcb04": "3^\\frac{2}{13}", "0a4a56947caf6fdcf237a5c3a5db941a": "(E, \\mathcal E)", "0a4aa2cee55c184ce52c4a2094ac6319": "\\tau_g=-\\frac{d}{d\\omega}\\arg(H(j\\omega))", "0a4aafda9ad959024478e6df5a1c2af2": "U_i(N),i=1,2", "0a4ae321700a998ee69c8d71b7f9b202": "f': X \\to S'", "0a4b018f4a1382228c85c71a987d566f": "k=n", "0a4b247283f01625b902164464fc5acd": "\\frac{d t}{d \\tau}", "0a4b7f7e00aac7ea3381e0a89d874f12": "Z_\\epsilon", "0a4bc9d99594802468034e0960f9acb7": "L=\\mathrm{DE}=L'_{0}/\\gamma=18\\ \\mathrm{cm}.", "0a4bd081c8a5481e1c8b320466b3ba96": "|\\cdot|_p", "0a4c006fca86431acd1924cdce0b3448": "P \\to Q \\vdash P \\to (P \\and Q)", "0a4c09917c25decc00d66a6ddb424731": "\\mathrm{Re}(\\tilde{n}) = \\frac{ck}{\\omega}", "0a4c316a8a007da3bb7d27aea35d6e37": "\\epsilon = (1+\\alpha){{\\sigma_0}/{E}}\\,", "0a4cf0cfd79a52b60ab6b7a20017bc94": "\\lambda_1 = \\lambda, \\lambda_2 = 0", "0a4d1af755ff4e6d0e38b74b93c35e36": "\\mathcal{L}_{X}g = \\lambda g", "0a4d3cd1b183bc23051aef1c2b51d315": " \\frac{\\overline P}{A} < \\sigma_0 ", "0a4d7ad4b2e31fbaafcb3d69a750184f": "T_1 = \\sum_F\\text{(link to the root)}", "0a4d8ebe42988bac58dc92b4e849b2d5": "\\arccos\\left({23\\over27}\\right)", "0a4dc552dd3b32a12e72ed6211f03279": "\n\\mathbf{f_{0:1}} = \\mathbf{\\pi} \\mathbf{T} \\mathbf{O_1}\n", "0a4e17d97de673d6f7034c10c7a917d8": "c_F(a,b)\\equiv c_-(a,b)", "0a4e2ef1e300d5eb178e96272ea61f7a": "23 = 10", "0a4e4f84dadbf64c5fe90b04c82326d8": "b \\geq 5", "0a4e5aa782a10295367e7ba65948e6b0": "V_{\\mathbb{R}} \\subset V", "0a4ea28d6b34a3d8aee23c8e2a9a77c6": "\\textbf{P}_{0\\mid 0} = \\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix} ", "0a4eb72f94439f4c3a6f4d0f75572a29": "\nE_{x \\in_R D_n}[\\frac{t_{A}(x)^{\\epsilon}}{n}] \\leq C\n", "0a4f65c643313b1349de59af0e7c9c01": "v_\\mathrm{x}", "0a4f773ab833d7c432c8fb9c2a336426": "[J_i,K_j]=i \\epsilon_{ijk} K_k,\\,\\!", "0a4f9cfa2118d339103529d669a0593d": "\\vec\\omega=\\frac{|\\mathrm{\\mathbf{v}}|\\sin(\\theta)}{|\\mathrm{\\mathbf{r}}|}\\,\\vec u", "0a4faaa65d0949db17f80824f26d180f": "21=3\\times7,231=3\\times7\\times11,744=24\\times 31\\,", "0a4fb033c3d6707ba1ec464b9355f69c": " = \\begin{vmatrix} \\boldsymbol{i}&\\boldsymbol{j}&\\boldsymbol{k} \\\\ 0 & 0 & \\omega \\\\ -\\omega t v \\sin\\alpha & \\omega t v \\cos\\alpha & 0 \\end{vmatrix}\\ \\ , ", "0a5000fe8b6b5570dd5a1ce00b828ef6": "\\theta \\,\\!", "0a50292b96b71152e9a1c7ccdeecd176": "\n\\left[ T_\\mathrm{n} +E_k(\\mathbf{R})+\\mathcal{T}_\\mathrm{k}(\\mathbf{R})\\right] \\; \\phi_k(\\mathbf{R}) =\nE \\phi_k(\\mathbf{R})\n\\quad\\mathrm{for}\\quad k=1,\\ldots, K,\n", "0a503598148b46a9c32f9283624e9c14": "\n\\hat{A}_{m_j}=\n\\begin{cases}\n0 & m_j \\text{ odd} \\\\\n\\frac{1}{\\pi}\\int_{-\\pi/2}^{\\pi/2}f\\bigl(\\mathbf X(s)\\bigr)\\cos\\left(m_j\\omega_js\\right)ds & m_j \\text{ even}\n\\end{cases}\n", "0a5044fd8223c03d094e80122b5dbc4d": "h^{1,1} = h^{1,1}_{+} + h^{1,1}_{-}", "0a50606f7c0591dc0dc6bbe8dd555eed": "s+\\Delta s", "0a5096e79e49cf6788475bb19ca34c68": "\\langle X^N \\rangle X ", "0a50d4e09a2c9dff3e302b50f78abbc3": "\nx = x^0 + x^1 \\mathbf{e}_1 + x^2 \\mathbf{e}_2 + x^3 \\mathbf{e}_3, \n", "0a514266c43d80e53ee002673b749ffa": "\\! p=-\\rho", "0a51463a7c4566600ed863d7e1c35f40": " \\it{\\nu} ", "0a515a686edfb83439cc39e028394a01": "I = T_1 \\cap \\dots \\cap T_n", "0a51ae5658d0346929f307121a89cb69": "V_\\mathrm T = V_\\mathrm {iL}[(1+\\mathit \\Gamma)\\cosh(\\gamma x) + (1-\\mathit \\Gamma)\\sinh(\\gamma x)]\\,\\!", "0a523d3edcc9ca50f6ba3b2d1952ab21": "\\frac{\\sqrt{D}}{D} = \\frac{1}{\\sqrt{D}} = \\sqrt{\\frac{T}{\\tau}}", "0a52b4c489ad207a5f3d40ef8247686e": "D^\\alpha_x f(x) =\n\\begin{cases}\n\\frac{d^{\\lceil\\alpha\\rceil}}{dx^{\\lceil\\alpha\\rceil}} I^{\\lceil\\alpha\\rceil-\\alpha}f(x)& \\alpha>0\\\\\nf(x) & \\alpha=0\\\\\nI^{-\\alpha}f(x) & \\alpha<0.\n\\end{cases}", "0a52c71202d682d910f2b3c756221706": "\\otimes_o", "0a52dda0fa627cd533f5065c3aec9b31": " \\lambda \\subset \\begin{pmatrix} A & B \\\\ C & -A^T \\end{pmatrix} ", "0a53440524558b8f5866430d1661b32e": "[[M]_{a\\;\\|\\;u}\\;\\|\\;N]_{\\overline{a}\\;\\|\\;v} \\rightarrow M\\;\\|\\;[N]_{u\\;\\|\\;v\\;\\|\\;x}", "0a5349c263db987e473ec20e0593428f": "\\forall x,y \\in I: x - y \\in I", "0a5380be71df066e64f061c2cd91c47a": " \\tilde \\nu_{J^{\\prime}\\leftrightarrow J^{\\prime\\prime}} = 2 \\tilde B \\left( J^{\\prime\\prime} + 1 \\right) - 4\\tilde D \\left( J^{\\prime\\prime} +1 \\right)^3 \\qquad J^{\\prime\\prime} = 0,1,2,...", "0a53a6e8d3e302eef4d3b3519cd3414a": "| s - t | \\to 0.", "0a53d5d631516bd9558f88a99c227a89": "\\left [\\begin{smallmatrix}\n\\cos 2\\pi/p & \\sin 2\\pi/p \\\\\n-\\sin 2\\pi/p & \\cos 2\\pi/p \\\\\n\\end{smallmatrix}\\right ]\n", "0a53df76736bc93a648b9f5df26a187a": "S = \\sum_{j=h}^n C_{n,j} p^j (1-p)^{n-j},", "0a53eb045ed05e22443717ebeeabac23": "D(p+x\\| p)", "0a547477d57298d8feac2fea4ccfff3d": "T_\\mathrm{total}=", "0a5508caaf3523cd69a2ffdd24a60c6d": "\\int\\frac{dx}{xR}= -\\frac{2\\sqrt{a}}{b} \\int du = -\\frac{2\\sqrt{a}}{b}u", "0a554cb3e5efaabd40bfc7a0d9eabacc": "\\scriptstyle \\leq5.9\\times10^{-35}", "0a559e47dc1acc4c8e570a52d4288ee3": " Molecule~1+ Molecule~2 \\xrightarrow{} macromolecule ", "0a55ec379e8b3cb43d7b6861be6ab73d": "\\lambda_A \\colon I\\otimes A\\cong A", "0a562b077f9747f6cc22f2eb308b09d3": "m_\\ell \\in \\{0 \\cdots n-1\\}\\,\\!", "0a563403cc1e7ce64647faa684876a38": "\\mbox{If } a \\in X \\in Con\\mbox{ then }X \\vdash a", "0a56463af1f51954a097e806eed96109": " \\| e^{X+Y} - e^X \\| \\le \\|Y\\| e^{\\|X\\|} e^{\\|Y\\|}, ", "0a56d591f6a4e9f6c320727988a55428": "\\varphi_n=\\arctan\\left(\\frac{1}{\\sqrt{n}}\\right).", "0a56e6a85c9b3a1d47edf7c33b457352": "u(t)^2/x(t)", "0a57063a19daee162bcad8e9e289f77b": "\\sum_{1\\le i_1 < i_2 < \\cdots < i_k\\le n} x_{i_1}x_{i_2}\\cdots x_{i_k}=(-1)^k\\frac{a_{n-k}}{a_n}", "0a5720eccf8aa3ca496b3a6df486df3c": "m^{k+1}/k^2.", "0a575d69e5b16ba7e834bb2744e705a3": "1+x+x^2+x^3+\\cdots x^{k-1}=x^k,", "0a576944996102bfc7f0469d9f83b9ec": " \\! P_{\\text{dias}}", "0a576df111e98cef87d719c8398c126f": "ax^2 + bxy + cy^2 + dx + ey + f,\\,\\!", "0a5784aa75138ac236c921fed44d3d02": "F \\colon \\mathbb{R}^n \\rightarrow \\mathbb{R}^m", "0a57dd72e80dd410e0b8ac1e224fa3f4": "\\tfrac{13}{25}", "0a580334796dfb43dd50e98670eb2c5a": "\\mathbf{P}_{n + 1}", "0a588d0f5f42f9905755007f389d2b24": "\n3 \\uparrow\\uparrow\\uparrow 2\n", "0a58d8fc7ed5da8e2f5b74faeb16bc80": " s^{unsigned}_{ij}=|cor(x_i,x_j)|", "0a5923f82079ed7794c069e83f8175d7": "A_0=\\mathbb R.", "0a5939b4a812482093288207e24fb6b0": "Tu = \\sum_{k=0}^n a_k(x) D^k u", "0a594759822b412110943bd8d02fe61e": "3\\times 1", "0a5964d0b537eeabda740715cebbeb11": "g_{1}\\left( p \\right) = 0,g_{2}\\left( p \\right) = 0,\\,\\,\\ldots ,g_{M}\\left( p \\right) = 0", "0a59ddca0e0f6fc286f2d4c0ecc0cb54": "{\\Bbb R}^n", "0a59ec2ebab203195860256fa83c9ae7": "N \\ln N", "0a5a0c7accbedc622b8a847033657fc0": "\\ G(0)=\\frac{1}{\\langle N\\rangle}=\\frac{1}{V_\\text{eff}\\langle C\\rangle},", "0a5aabd1437be1e3655efac93492dc14": "\\tau = R_\\text{i} \\cdot C", "0a5abbb5bfcd55fde6b2939e8e2390f7": "E_\\text{P} = m_\\text{P} c^2 = \\frac{\\hbar}{t_\\text{P}} = \\sqrt{\\frac{\\hbar c^5}{G}} ", "0a5ac298c038832b5ab7316ff72e3ad0": "\\max_{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}} b \\cdot u^1(x_{1}^{1},x_{2}^{1})+ (1-b) \\cdot u^2(x_{1}^{2},x_{2}^{2})", "0a5b4a651f194987270d92aa2c6f97c8": "\\frac{\\mathrm{\\mathrm{d}}v}{\\mathrm{\\mathrm{d}}t} = \\frac{\\mathrm{d}v}{\\mathrm{d}s}\\ \\frac{\\mathrm{d}s}{\\mathrm{d}t} = \\frac{\\mathrm{d}v}{\\mathrm{d}s}\\ v \\ . ", "0a5b5210d54aeefc7a174728d3100dbc": "C_{(-)}[t] = \\sigma[t - N_{(-)}] \\sigma [t - N_{(-)}] \\ldots \\sigma[t - 1]", "0a5bc6431d5120a92296f3b1c824f480": "\\Gamma_w = g\\, \\frac{1 + \\dfrac{H_v\\, r}{R_{sd}\\, T}}{c_{p d} + \\dfrac{H_v^2\\, r}{R_{sw}\\, T^2}}= g\\, \\frac{1 + \\dfrac{H_v\\, r}{R_{sd}\\, T}}{c_{p d} + \\dfrac{H_v^2\\, r\\, \\epsilon}{R_{sd}\\, T^2}}", "0a5c35242adf0f8f3d0bacd5cfcd8306": "\\mathrm{SNR} = \\frac{|y_s|^2}{ E\\{|y_v|^2\\} }.", "0a5c41050a92a6364509c3ac3bc9ea70": "16K^2 = (a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d) - 16abcd \\cos^2 \\left(\\frac{\\alpha + \\gamma}{2}\\right).", "0a5c5a80864a354c8fa9efca3d0c9765": " F=\\frac{1}{2} F_{\\lambda\\mu}^m \\, dx^\\lambda\\wedge dx^\\mu\\otimes {\\mathrm e}_m ", "0a5ca3ba11aa2f0acd98bdf38aebe50e": "B_{12}=\\frac{4\\pi^2 e^2}{m_e h\\nu c}\\,f_{12}", "0a5cbac1e71866378f0823109fa362ed": "R=\\sqrt{X^2+Y^2}", "0a5d770e47ff31696a645897319cb558": "G_X(t,f) = G_x(f,-t)e^{j2 \\pi tf} \\, ", "0a5d8d0857d8b61ba92f61bfa97cf6aa": "zx = e^2 + f^2 \\leq c^2+d^2 \\leq \\left(\\frac{x}{2}\\right)^2 + \\left(\\frac{x}{2}\\right)^2 = \\frac{1}{2}x^2.", "0a5da26a38768fe11e66540b55e2b545": "_{interval} \\delta_{14}^2=3^2", "0a5da28c55da60257a5f15cea75d806c": "\\lnot \\textit{par}(x_{gt},x_{me})", "0a5da2e5220730129349ea549d1b3982": "o_2", "0a5db5bd4dc59b07d0c51757e5728d1a": "\\delta = \\theta_0 - \\theta_4", "0a5defa8642de97ec5bcd8cc1e5b2187": "\\displaystyle{Ef=C_-f|_{\\partial\\Omega},\\,\\,\\,\\, (I-E)f=C_+f|_{\\partial\\Omega}.}", "0a5e31dec9f696d99980f561f4b628ef": "\\mu : \\boldsymbol{\\rm S} \\rightarrow \\boldsymbol{\\rm R} ", "0a5e61459aba57df17b5f5c3854af75a": "D(E(m_1, r_1)^{m_2}\\mod n^2) = m_1 m_2 \\mod n, \\, ", "0a5e66456e67a23dbe0ea9a422ee153f": "f a", "0a5ebcd281d1c546aa21e4b2c830aee8": " \\epsilon = \\epsilon_r \\epsilon_0\\,\\!", "0a5ef2659c0c5232f066cb908f8c9f82": "\\bar p - p = \\gamma \\nabla \\cdot \\mathbf{\\hat{n}}", "0a5f4c73a9e601859ef6bb3746e12f12": "\\alpha\\geq0", "0a5f58661bffe880da6cdd9e7c13d207": " f_x = \\frac{ \\mid \\mathbf{E} \\mid^2 \\cos^2\\theta }{ \\mid \\mathbf{E} \\mid^2 } = \\phi_x^*\\phi_x ", "0a5f7ee87603528014d17f15d1d4b7ad": "e/c^2", "0a5f8b30a4cb784acb10448eb3b5340f": "U_k=A_ke^{i\\omega_kt};\\qquad\\quad \\omega_k=\\sqrt{ {2C \\over m}(1-\\cos{kd})}", "0a5f8dc7d2bc512d716bfb39d8d484b1": "A_4 = \\begin{bmatrix}0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 0 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 0 & 0 & \\cdots & 1 & 0 \\\\\n\\end{bmatrix}", "0a5fabd2be075f9d50dfec8036196058": " U^*_t = u^*(X_t)=\\frac{r\\bar\\lambda \\sqrt{1-\\ X_t}}{2} = \\begin{cases}\n{} > \\bar{u} & \\text{if } X_t < \\bar{x}, \\\\\n{} = \\bar{u} & \\text{if } X_t = \\bar{x}, \\\\\n{} < \\bar{u} & \\text{if } X_t > \\bar{x},\n\\end{cases} ", "0a5fd752a9644d8825cd5b7a8fcce732": "q_2=y_2\\cdot\\sqrt {2g\\bigl(E_1-y_2\\bigr)}=q_g=y_g\\cdot\\sqrt {2g\\cdot h} ", "0a5ffabb24f666e8972f7ed2ed0f4853": "CH_4 + CH_3^+ \\to C_2H_5^+ + H_2", "0a60891d81e36a701c40fbb0823bae2e": "n-x", "0a60cd6434fd1f01723c45aa7576eacb": "m = \\frac{\\Delta y}{\\Delta x} = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{8 - 2}{13 - 1} = \\frac{6}{12} = \\frac{1}{2}", "0a60d72cea2f6396ba352177aece4b53": "Z'_w", "0a6101ec72d4317b4ace994ce83f8bfa": "Y_t \\,", "0a610230cbd463bfe6aa18ab3266993a": "B= \\frac {2D_{ox} C_s}{N_i} ", "0a613492a0d0d42d4dc849b0d83418ab": "\\Delta_{K_n} = (-1)^{\\varphi(n)/2} \\frac{n^{\\varphi(n)}}{\\displaystyle\\prod_{p|n} p^{\\varphi(n)/(p-1)}}", "0a61a403929176a6efd85d77d069ba64": "\\{(i,x,t) \\in \\mathbb{N}^3 | \\Phi_i(x) = t\\}", "0a61a7fefcc21f13fad4e7d38909cc7d": "2p_2(E\\oplus F)=2p_2(E)+2p_1(E)\\smile p_1(F)+2p_2(F)", "0a61cda98eef76a4da74b652bd387729": "R\\bowtie S=S\\bowtie R\\,", "0a622ad3bd0ad03e8e14a9753aa5b251": "1 + \\ln(2b) \\, ", "0a628ba9b1845b6acce4b4ff2c9ec575": "\\sigma ^2", "0a62a30803055f49cbec5088a0686d08": "\\mu\\left[\\left(1+\\frac{9 \\mu^2}{4 \\lambda^2}\\right)^\\frac{1}{2}-\\frac{3 \\mu}{2 \\lambda}\\right]", "0a63028aad661f43712b1b5e9db36109": " \n\\text{(Eq. 5)} \\qquad \\sum_{c=1}^N\\mu_{ab}^{(c)}(t) \\leq \\mu_{ab}(t) \\qquad \\forall (a,b), \\forall t \n", "0a63136247dd6edd0140d3aea242fbbb": "\\displaystyle A^{(2)}", "0a631a194dea59eee5b644ff7abb6b93": ") \\to ", "0a6389f8d48024b70b0842cd26d51e06": "\\Delta S_{i=1} = S_{i=2}-S_{i-1}= \\big(2000.0\\text{ ft}\\cdot 1\\text { ft}\\cdot 3.63\\text { ft}\\big)-7130.5 \\text { ft}^3 = 123.2 \\text { ft}^3 ", "0a63cec8a33bf055104bf41af88f757f": "\\langle \\mu \\rangle = \\frac{\\int \\mu dP}{\\int dP},", "0a63d57241466bad0086dc935d81cc9b": " ||f||^2_{\\mathcal H} = \\langle f,f\\rangle_\\mathcal H = \\sum_{i=1}^n\\sum_{j=1}^n c_ic_jK(x_i,x_j) = c^T\\mathbf K c ", "0a640a1e098c7880f9a1f6bb988cf283": "x+\\infty=\\infty", "0a645fc515fe6128f91758f996ebe307": "x_i\\,", "0a647fdd7aa9d6bbb8633f8f034bb50e": "\\xi \\,", "0a65623d37b26c0f768d4cdc8d638a5c": "u^a = \\delta^a_0 = (1, 0, 0, 0)", "0a656d0d581c435e643a9b0934173723": "\\mathrm{E}(e_t) = 0\\,", "0a6595104e3b1b4e61b4568e450766af": " S=\\sum\\limits_{i = \\lfloor \\frac{n}{2} \\rfloor + 1}^n \\binom{n}{i}p^i (1 - p)^{n - i} .", "0a65aaed8624538799b7d70b3995a053": "\\sin \\theta = \\frac {f} {J} \\,.", "0a65bf2d85a1f17f162770ba6033347d": "distance = \\frac {c \\delta t}{2}", "0a65fcf09bca7b5c622bdcda56c71a3d": "R_{\\text{vertical}} = \\frac{R_{12,34} + R_{34,12}}{2}", "0a6618e95d0397e28cda00fbb663a84f": "-R_3 i_3 - \\epsilon_2 - \\epsilon_1 + R_2 i_2 = 0 ", "0a66218865919d546ff8b3ae261d7c65": "A(0,u_k,v_k)", "0a663d246ff9370ba9ee4fec7ab4019d": " \\frac {1} {N_f} = \\frac {1} {N_f^{fatigue}} + \\frac {1} {N_f^{oxidation}} + \\frac {1} {N_f^{creep}}", "0a6652d80a542bbc53855f89bd1bb37f": "b,d,u,a \\in [0,1]\\,\\!", "0a665387bb4989331d083413c8cb4000": "T(z)=z+c\\int_0^1 tdt.", "0a672740a279b44d366aea32423472ea": "\\Omega = [0,1]", "0a6737eb40ad5cc13a887df1d1a4ece7": " \\{ \\gamma^0, \\gamma^1, \\gamma^2, \\gamma^3 \\} ", "0a67f1ca9f0df9c3b6ee44a53db67626": " u_t + au_x = 0\\,", "0a680d21adfaeb3eb50a4526856097f4": "M(\\text{H}_2\\text{O}) \\rightleftharpoons M(\\text{OH}) +H:[M(\\text{OH})]=\\beta^*[M][\\text{H}]^{-1} ", "0a681bee5f23b54137db80c1da9962eb": " u(t,r) = \\frac{1}{r} F(r-ct) + \\frac{1}{r} G(r+ct), \\,", "0a6857a4920c79ba8f989d4b829f4975": "\\frac{\\operatorname{d}I_{\\text{L}}}{\\operatorname{d}t}=\\frac{V_i}{L}", "0a685ddd0c07efb9d44fa2b2b2be41f1": "\\omega^{A}_{x\\lor y}=\\omega^{A}_{x}\\sqcup \\omega^{A}_{y}\\,\\!", "0a689066f9815ca92d65d9ca05109454": " P / {\\rm hp} = \\frac{ \\tau / {\\rm (lbf \\cdot ft)} \\times 2 \\pi \\times \\omega / {\\rm rpm}} {33,000}. ", "0a689befe02045c71043d3986ad85df3": "{\\partial x_i(\\mathbf{p}, w) \\over \\partial p_j} = {\\partial h_i(\\mathbf{p}, u) \\over \\partial p_j} - {\\partial x_i(\\mathbf{p}, w) \\over \\partial w } x_j(\\mathbf{p}, w),\\,", "0a68a7675f3f06a491925480020a14a1": "p_1=0,\\ldots,p_k=0\\,", "0a6920a9dfa5c6654196b50cb217b7af": "b_T", "0a692cc37fe6d13991a12630a7192bb2": "\\scriptstyle a \\,=\\, \\infty", "0a69534ad90cd440a667583706dfc338": "\n2y = -\\frac{x^{2}}{\\tau^{2}} + \\tau^{2}\n", "0a698956847f166a1be859c688ee56a4": " \\left\\langle \\left( \\sum_{n=0}^\\infty u^n \\langle f, \\psi_n \\rangle \\psi_n\\right), g \\right\\rangle = \\int \\int E(x, y; u) f(x) \\overline{g(y)} \\, \\mathrm{d}x \\, \\mathrm{d}y \\rightarrow \\int f(x) \\overline{g(x)} \\, \\mathrm{d} x = \\langle f, g \\rangle,", "0a69c137345c1e1f6d6db9420926f630": "V_{min}(z) = \\sum_{k=0}^p v_{min}(k) z^{-k}", "0a6a87863e4d4926f2a3142f39674144": "\\omega^{A\\underline{\\diamond} B}_{x}=\\omega^{A}_{x}\\;\\underline{\\oplus}\\; \\omega^{B}_{x}\\,\\!", "0a6a8d4c7720abc9dbb38ad6f23501e4": " f(x+h) = f(x) + f'(x)h + \\psi_1(h) \\qquad \\qquad g(x+h) = g(x) + g'(x)h + \\psi_2(h) ", "0a6a9d1e426c22b7531b48623a60a061": "F_Y", "0a6ad0e3d13e1c665f3e5ae978804976": "0 \\le p \\le 1", "0a6adf905fcb2028fad6ca232aba0356": "S_\\theta", "0a6ae37a6d4d363bb1a99e91c1eb3606": " \\text{value} = 1.25 \\times 2^{-3} = 0.15625 ", "0a6af32f750ef00668109cb8793b91ec": "\\sigma_{y'}", "0a6bda3dcba2df8b8b56579d2cfbe7b6": " \\phi_e(z)=\\frac{1}{2\\pi i}\\mathcal{P}\\int_C\\frac{\\phi(\\zeta) d\\zeta}{\\zeta-z}-\\frac{1}{2}\\phi(z). \\, ", "0a6c11d1ae791d08aa93c8a8d69c4bfa": "d(p,A) = 0", "0a6c22504db2d969c9646fb2a01d125a": "S(x,t)", "0a6c4a0d788571abfe1997ac6a5036b7": "Z_2^9", "0a6c8bc1bdbc0f9d233ef23d4a5d9096": "74^2", "0a6c98807229adaf2dd97660ed557168": " d + 2c = 180", "0a6d5c4ac7b6bdc0e2aab86e76db1c3a": " t= \\lceil 1 + n/\\sqrt 2\\rceil", "0a6d626a9902cd0fb7d84a8b4143a54e": " L(h)\\leq \\frac{1}{8}h^2=\\frac{1}{8}\\lambda^2(b-a)^2", "0a6d7ddd16fab563c4184450807d5d38": "1\\to A\\to B\\to C\\to 1\\,", "0a6db390f9b1e646c441fb088a8e955a": "U\\left(x_1 ,x_2 ,\\dots,x_n\\right)", "0a6dd1ec55a668d7e84633f49faf281e": "y\\in V", "0a6e6c60dcb0072ca15787ef4b64a5b7": "\\mathrm{SU}(2n)\\,", "0a6ec12ebd600cd251837bbe54b5d774": " \\varepsilon_{ijk}\\varepsilon_{imn} = \\delta_{jm}\\delta_{kn} - \\delta_{jn}\\delta_{km}\\,.", "0a6f0332d0ad817314fb1aa5ec002c31": "\\alpha \\varphi = {d \\varphi \\over d z} = 0", "0a6f4e954b60efed010eea2f3fdfadad": " t, \\lim_{n \\rightarrow \\infty} n_t/n = c_t > 0 ", "0a6f8c0f86fc463dd96cedfc3da0fe82": " \\sin(x) = \\sum_{n=0}^\\infty \\frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!}+\\cdots,", "0a6feeb4b14b09fd29a2f22c6b6823c1": "M = |M_{1}|e^{i\\theta_{1}}e^{i\\phi_{1}} + |M_{2}|e^{i\\theta_{2}}e^{i\\phi_{2}}", "0a6ffa5b309e21885b7586e8add2d152": "I_1+I_2+I_3=0", "0a70033be217e33353951677643105bc": " \\cdots \\to H_{n+1}( X_{n+1}, X_n ) \\to H_n( X_n, X_{n-1} ) \\to H_{n-1}( X_{n-1}, X_{n-2} ) \\to \\cdots . ", "0a701f7d5819334492641c129208568b": "{ 1 + {\\underbrace{1 + 1 + 1 + \\cdots + 1}_{b}} }", "0a70e1e06ec023f11f8060faf1806c32": "\\textstyle 2m+1", "0a7101b12e80841b388888740faa7a24": "H_i(S,T) = \\frac{G(S,T)} {(a_iS-b_iT)}", "0a7148eabe2f26dae5d42f1c2d2c41ae": "\\{0^n1^n \\mid n \\ge 0\\}", "0a715bd905c3dc34931ecad5f34e6903": "\\mathbf{\\omega}_i", "0a716816b16d082398377809ec003f3d": " V := \\bigcup_{\\alpha} V_\\alpha.", "0a71b1e3b2981ffa4ec126d033b3a84b": "F(x;k)=P(k/2,x^2/2)\\,", "0a71bb9f26976994082eb22e40d21ed1": "E_{sig}(\\omega,\\tau)", "0a71cb5e502f901c74ba4fb99204f28f": "\\lambda_s(2)=s+1\\,", "0a71d3285d990312ab1850bffa37c53b": "X\\to\\mathbb{R}", "0a72e76e86c46a3c2a5f9fdb35203737": "(1-\\epsilon)\\log n", "0a72f9801e1a0f4a03e56e2297e0c848": "x_0 \\in \\Omega", "0a730050106636e0e0bf8b35516e87eb": "U_q(\\mathfrak{g})", "0a7369114d303ba522a8ec9862141869": "|n|_\\ast=\\Pi_{i0", "0a804b1ed1ebf0230eb8783baeffb821": "G_2:\\{0,1\\}^\\ell \\to \\{0,1\\}^h", "0a80dcfba6f4ce38eb766846c8ec3dfb": "f(x) = \\frac{A(x+h)-A(x)}{h} - \\frac{(Red Excess)}{h}", "0a810140f0ba4bdac1592a894508ba5b": "x - 1 = \\log_2 3\\,", "0a81cc5c3a9950fc082db4fae414dabe": "\n W = \\int_0^\\epsilon \\boldsymbol{\\sigma}:d\\boldsymbol{\\epsilon} ~;~~ \n \\boldsymbol{\\epsilon} = \\tfrac{1}{2}\\left[\\boldsymbol{\\nabla}\\mathbf{u}+(\\boldsymbol{\\nabla}\\mathbf{u})^T\\right] ~.\n ", "0a81e71240394f1c60598f3331850cae": "\\mathrm{Le} = \\frac{\\alpha}{D} = \\frac{\\mathrm{Sc}}{\\mathrm{Pr}}", "0a821d19fda56d7f9d685aefc4c05b9e": "X \\sim \\mbox{Inv-Gamma}(\\alpha, \\tfrac{1}{2})", "0a8224e787df13e07e706dd41a254db0": " H^{BM} _i (X) =Ker (\\partial :C_i ((X)) \\to C_{i-1} ((X)) )/ Im (\\partial :C_{i+1} ((X)) \\to C_i ((X)) ). ", "0a833a08c0c5b875621902cc4823c0ed": "O_4^{(\\alpha)}(t)=\\frac {(1+\\alpha)(4+\\alpha)}{2t}+ 4\\frac {(1+\\alpha)(2+\\alpha)(4+\\alpha)}{t^3}+ 16\\frac {(1+\\alpha)(2+\\alpha)(3+\\alpha)(4+\\alpha)}{t^5}.", "0a835d61f03bea60a6b5e1427c71e7db": " (p,q \\in X) ", "0a836a2e911068a17983cb3e917e0bbf": " p(\\boldsymbol{\\theta}|\\rm{data}) ", "0a83dbb275ef74fc4b5f58024079d61a": "\n\\begin{align}\n H(z) &= Z\\{h[n]\\} \\\\\n &= \\sum_{n=-\\infty}^{\\infty} h[n] z^{-n} \\\\\n &= \\sum_{n=0}^{N}b_n\\,z^{-n}\n\\end{align}\n", "0a83fc11a7fdf57b8f0f3b74916a39c2": "a = mx \\pm c,\\qquad b = nx \\pm d", "0a841365cc7530bcafc5f2f5772c3bdf": "\nNS_i = \\sum_{k=t+1}^{t+n} \\left[ e_i^{k-1} \\left( G^k \\right) \\right]\n", "0a849de52079d2df084c26a570f3d692": "U_{iy}\\subset\\mathbb{R}^p", "0a84bea043146c2c23559f53f5bca690": "Pr[\\sigma(x)\\in R]\\leq t\\cdot max\\left(Pr[|\\frac{1}{m}\\left(\\sum_{i}w^{j}_{\\sigma_{i}}-\\sum_{i}w^{j}_{\\sigma_{m+i}}\\right)|\\geq\\frac{\\epsilon}{2}]\\right)", "0a84fa4e8c926eeee85ac67e7778455d": "i \\rightarrow j", "0a8522d7a711bc4e6a0322233052639e": "F_f", "0a854e78a3801634aafa3b8e9250806f": "N\\,l", "0a856823f9689d20db459d9393a0fb46": "\\rho_E=\\frac{4\\cdot {\\pi}\\cdot a\\cdot R_W}{1+\\frac{2\\cdot a}{\\sqrt{a^2+4\\cdot b^2}}-\\frac{a}{\\sqrt{a^2+b^2}}}\\,", "0a8610a6ac60a5cf79607be23b516f0b": " F = \\frac{\\vec{G}\\cdot(\\vec{e}-\\vec{u})}{RT}f^{eq} ", "0a86bea1739ad845794a217a35b78548": " C_{n^{*}l^{*}} ", "0a86e709eb8c2d0751d7e827a50e6c6d": "S_3(x,y) = QM(x,y,GM(x,y))", "0a874c119e759077832d65598daf9116": "W=\\int_{V_i}^{V_f} \\,P\\,dV", "0a874c974a315930ff823fd0dda9d886": "=(ac)'+(bc)'=(ac+bc)',", "0a8768b319a9f85e712dd926f3b06c08": " \\mathrm{B}_\\mathrm{2}", "0a876cb41049e2533d9bc9e878ba712f": "\\ \\rho_{water}", "0a886d8d7f69aab15bc76ccaf23ce109": "x_1,", "0a88fa202330bdab11ca18b855a91c38": "x_1, x_2, x_3, ..., x_n", "0a89563d09558842989a6d6d48cce69d": "(\\mu)", "0a8990794d6f0ef65c82e7f6e973b1b9": "\\prec_K", "0a899eced66c9882eb4ae08e037d962c": "e=1", "0a8a30cc75ba663acfb56efaf4c9d6eb": " \\tilde{\\mathbf{y}} ", "0a8ad3b9896c6c195de0243fb722fe9d": "~x~", "0a8b270d88f2be91095f4a78b91e0043": "r=e^{i\\Phi} ", "0a8b5a8efe1c2d7a9bc12fe9bbdbf2c6": "\\begin{matrix} {2 \\choose 1}^2{11 \\choose 1}{4 \\choose 3} \\end{matrix}", "0a8ba492b4583452d259471de722b717": "\\mu_{\\rm N} = \\mu_{\\rm B} \\frac{A_{\\rm r}({\\rm e})}{A_{\\rm r}({\\rm p})}", "0a8bad72209b87d0405e59af1b0f4af9": "f*g=fg+\\mathcal{O}(\\hbar),", "0a8bb3670a5fb39e668d1263facaf0be": "\\sigma_{zz} - \\sigma_{yz} + \\sigma_{xz}", "0a8ca3202af084bc79fd7067012d19e3": "\\sqrt p", "0a8cc1b87b4343c9034b76390099e82a": "\\phi\\wedge\\phi'=(-1)^{|\\phi||\\phi'|\n+[\\phi][\\phi']}\\phi'\\wedge\\phi", "0a8cdf337307f461204f9eebeb8dd35b": "1/\\gamma", "0a8ce59fb0d11e553c6c47fae50b59e1": "f_1=3x", "0a8d583f9b3016c6649985d43255e601": " f \\colon (X, \\Sigma ) \\rightarrow ( Y, T ) ", "0a8d74f7b0ba39c4d917f53a5d900a0a": "\\nabla\\cdot{\\mathbf A} + \\frac{1}{c}\\frac{\\partial\\varphi}{\\partial t}=0.", "0a8d87f95ec4e258f7cc7202d7017a14": "\\lang x|\\psi_1\\rang = \\sqrt{ \\frac{2}{L} }~{\\rm sin}\\left(\\frac{\\pi x}{L}\\right)", "0a8e77d0114866c191519d9660f13c76": "\\textstyle \\mathbb{R}^\\mathbb{R} ", "0a8e8cf8fb77f8e1282adc97cf88ee36": "\\bar{A}", "0a8e97e89ac8c137a33840d748dc678d": "\n\\left(\\frac{p}{q}\\right) \n= \n\\begin{cases}\n \\left(\\frac{q}{p}\\right) \\;\\;\\text{ if } q \\equiv 1 \\pmod{4}\n\\\\\\left(\\frac{-q}{p}\\right) \\text{ if } q \\equiv 3 \\pmod{4}\n\\end{cases}\n", "0a8e9b29b44548aeb1cc70a8bfc48d07": "B = Q (n I_p) Q^{-1} = n I_p ", "0a8f453e32cc455eeb7ed1a2bdd2bfaa": "y\\in E", "0a8f59f8f520dceede68cf8327ddec6f": "\\prod_s \\left( 1+ieA_\\mu {dx^\\mu \\over ds} ds \\right) = \\exp \\left( ie\\int A\\cdot dx \\right) . ", "0a8f733556ca682c414d403f00a4af25": "{J}", "0a8f7ce71cfbc4945b21943b018926f0": "\\omega_r = \\frac{2{\\pi}n_s}{60} = \\frac{4{\\pi}f_s}{p}", "0a8f94f3432a5152ce3bc7abf6dd44cd": "e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} = e^{\\mathbf{0} \\cdot \\mathbf{X}_i} = 1", "0a8fd40516398a9469b6c80084f07a84": "g: S \\to X", "0a90190b6f26b7c751e4da864b3ddd18": "T_{1L}", "0a902501c935ad0c486e42b88cfd1379": "Q \\ne d", "0a9037f807f4d3bc91c92d3ea20b2cdd": "x_i \\cup y_i", "0a904396c0d81e52e04289476e38ddd2": " x_1^2 + x_2^2 + \\cdots + x_n ^2 = 1", "0a90819df25db81e5acbf6c20e691e15": "75^2", "0a914118779a075496338053c599939d": "\\textrm{E}(X(X - 1) \\cdots (X - k + 1))", "0a9151c4b0c3161781098770c1ea6f4a": "{r_{\\rm c}}", "0a9165a6c7ecdbc9b81e07c105ff534e": "c \\subset X", "0a916e42ffa1a41216e11adfaa299931": "J_1^s=J_1^m", "0a918b99a6447f03255f3f92d306f9f4": "\\gamma = \\epsilon", "0a918f8974a600eb7fe0bfa01a70bcc1": "\n\\sum_{n=-\\infty}^{\\infty} x[n]\\ z^{-nL} = \\sum_{n=-\\infty}^{\\infty} x[n]\\ e^{-i\\omega Ln} = \\frac{1}{T}\\sum_{k=-\\infty}^{\\infty} \\underbrace{X\\left(\\tfrac{\\omega L}{2\\pi T} - \\tfrac{k}{T}\\right)}_{X\\left(\\frac{\\omega - 2\\pi k/L}{2\\pi T/L}\\right)},\n", "0a920178557d322b2a517b518b2eb40d": "\\theta=(\\mu,\\sigma^2)", "0a924a15542ccd8be300610b9183dad2": " \\rho(t) = \\cos(6\\pi t)[\\cos(2\\pi t) + i\\sin(2\\pi t)], 0 \\leq t \\leq 1 ", "0a926558a06e972d65972e320dca0fbf": " r \\ = \\ (1+j)^{12} - 1", "0a928ec05dba4fa8bfd87ad1e689fb35": "B_2 \\cong C_2,", "0a92bd8b9549d23c78ff57834870be04": "M^{z_+}_p X^{z_-}_q", "0a92c9a80b141a4e630b8d64d8b3fbc9": "a_B = \\gamma_{BY}\\cdot \\lambda_0, \\ ", "0a92f9148f59a4f3201a012a2ea2bd83": " f:M\\to M ", "0a9309c3100f5a94bbe74ca1c147a4e3": "P(x) \\downarrow (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\downarrow Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "0a9312a549a978ed84c7064d0d223961": "-2F\\dot{H} = \\rho_{{\\rm m}}+\\frac{4}{3}\\rho_{{\\rm rad}}+\\ddot{F}-H\\dot{F},", "0a93584edf76c7fe398aae935f870169": "\ndx\\;dy\\;dz=\\det{\\frac{\\partial(x, y, z)}{\\partial(r, \\theta, h)}} dr\\;d\\theta\\;dh =\n{r}\\; dr \\; d\\theta \\; dh \\;\n\n", "0a93c280d6342c9f31eb89c18318cf48": "\\widehat{\\widehat{\\sigma_e^2}}", "0a93d5cb5658a362c2dd2959f37882ac": "\\frac{d}{dt}e^{tX} = Xe^{tX} = e^{tX}X. \\qquad (1)", "0a942588666e9613660d4008fb645435": " \\forall{f}{\\in}L_2(X) \\ f = \\sum_{i=1}^\\infty \\phi_i ", "0a94586053aea0803c6c6568b297697b": " H(x) = \\tfrac{1}{2}(1+\\sgn(x)).", "0a947e57051daae9c0686c36b9d8fad6": "\\theta \\operatorname{E}[Z]", "0a94a3e84aa8777c3777b07079dac68f": "\\frac{287}{50} = 5.74", "0a94d0769182456bcbfc9adfaf254297": "0 \\le \\theta \\le \\pi", "0a94d21fc48e5b3dc98b40d792072125": "t(m-s+1)>\\dfrac{N(m-s+1)+s(k-1)}{s+1}", "0a94dcd9092e2d2fa316c8a63fa30572": "d_{VE}={\\sqrt{6}\\over2}a=\\sqrt{3\\over2}\\,a\\,", "0a95000e1f68f5f6cc9fbaccb19d5b1f": "e^{1/e}\\approx1.444667", "0a952f495a10afbc4457fa7f8de85758": " a_i ", "0a953d48bc6b465b4928c6660ec0589b": "g(\\epsilon)", "0a954ab13484b2e5d99c48734627c736": "3x+6=16-x", "0a95710a9af9fa8aeabdbeb961539268": " a\\cdot x^3+y^3+1=d\\cdot x\\cdot y ", "0a95e114dfef3b0d87cb5cb7dc0e734f": "\\scriptstyle \\alpha_\\Lambda", "0a960d7a39e0e682207dad4f5d56ef5f": "(I\\ nat\\ 3\\ 4) \\to \\bot", "0a96d8c7f8ed21ef9129691da22a7869": "T = \\frac{\\tan \\alpha}{4}(b^{2}+c^{2}-a^{2})", "0a96f5ac8902687fc56cb291773c7f3c": "\\,\\sin(x+y)=\\sin x\\cos y+\\cos x\\sin y,\\, ", "0a9735be087434f174ef639f54d07043": "B^2 - 4AC = 0 ", "0a97a89f8c0697a75bd39cf54c005a92": "p_{2n} = p_n^2 + S \\cdot q_n^2\\,\\!", "0a97a8c27855254be6692a23af223973": " [\\dot{S}] = \\begin{bmatrix} \\dot{\\Omega} & -\\dot{\\Omega}\\textbf{d} -\\Omega\\dot{\\textbf{d}} + \\ddot{\\textbf{d}} \\\\ 0 & 0 \\end{bmatrix} = \\begin{bmatrix} \\dot{\\Omega} & -\\dot{\\Omega}\\textbf{d} -\\Omega\\textbf{V}_O + \\textbf{A}_O \\\\ 0 & 0 \\end{bmatrix}", "0a97b1b4e725c505b64ed5d6f9b345e0": "_{s.1.left=s.8.right \\,}\\!", "0a97d621b275da59155513a87e101c48": "\\sigma_i = \\sigma_0", "0a981d7056c1b87c83ee1465ac52ac87": "(I \\rightarrow \\neg R)", "0a986c890e5f73e088e630438f5e69d9": "u(x,t)=\\sum_{n=1}^{\\infty}u_{n}(t)\\sin\\frac{n\\pi x}{L},", "0a987546c57d6264c4d1a3478e950009": "m_n\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{c}{2\\pi}}\\int_0^\\infty \\frac{e^{-c/2x}\\,x^n}{x^{3/2}}\\,dx", "0a98acf7be52f18fb4ff6d5fe1dee96f": "V_{out2}", "0a98d6b4652616a19b039e886f937347": "\\frac{\\Delta_h}{h} ~(x)_n=n ~(x)_{n-1} ~,", "0a98dfc5128dbef73fb03082c7cae8b7": "S^\\prime", "0a990cf78e97fe6d5421e6c1b3bb6199": "L[u] = \\det D^2 u - f(\\mathbf{x},u,Du)=0\\qquad\\qquad (1)", "0a9926ab592a35a9227434228c772204": " \\mathcal{D} = \\{(x,y) \\in \\R^2 : y = x^3\\} \\cup \\{(x,y) \\in \\R^2 : y = 0 \\} \\ . ", "0a99f33438b68b81e10f51f2c4399ade": "e_1^2 = e_2^2 =-1,\\ e_1 e_2 = - e_2 e_1", "0a9a2322e8e851c5dfeb2ce16a171d54": "(z_1,z_2,z_3)\\approx e^{i\\phi} (z_1,z_2,z_3) . \\,\\!", "0a9a5bf39687a3dacfe506b4a8800934": " \\langle x | \\cdots | y \\rangle ", "0a9b1d8367a5eb48b97a66146f14798c": "\\omega_\\text{res}(\\theta)=\\frac{2\\pi c}{\\lambda_\\text{res}(\\theta)}", "0a9b4e176f5fe418787c0b804e21bc35": "\\mathfrak{g}^*", "0a9b55a587051fd1b02048d5839f8b93": "C_H = [H] + [HA]", "0a9b98f4433f94fccad33fcc9d8d0cd0": " |i\\rang |\\epsilon \\rang ", "0a9bb6dce4060d8e63e8f6c92a03b820": " \\delta(\\mathbf{x}) ", "0a9bf7c54060c195d2b9dcbdd8e24ade": "a_1, a_2, \\dots", "0a9c7304273c4adc9c26d081641510a4": "v_F(\\mathfrak{D}_{F/K}) = {1 \\over e_{L/F}} \\sum_{s \\not\\in H} i_G(s).", "0a9c7ef7e5f10053d6d922098854889c": "\n\\frac{d}{dt}\\left(\\frac{\\mathbf r}{\\Vert \\mathbf r \\Vert}\\right)\n= \\frac{1}{{\\Vert \\mathbf r \\Vert}^3}\\mathbf r \\left(\\mathbf r \\wedge \\frac{d \\mathbf r}{dt}\\right)\n= \\frac{1}{{ \\mathbf r }}\\left(\\hat{\\mathbf r} \\wedge \\frac{d \\mathbf r}{dt}\\right)\n", "0a9c83410c31325de7d857efc927021a": "NEP=\\frac{S_n}{\\mathfrak{R}}", "0a9c88a25944fa0301997e445fadcdb4": " e^{i\\omega t \\boldsymbol{\\sigma}\\cdot \\mathbf{\\hat{n}}} =\n\\begin{pmatrix}\ne^{i\\omega t} & 0 \\\\\n0 & e^{-i\\omega t}\n\\end{pmatrix}.", "0a9cb88f9137d01c15d27bba942872ab": "\\mathbf{Y}(s) = C\\mathbf{X}(s) + D\\mathbf{U}(s),", "0a9cfe58a4d927dce6c7c9de47ace4e9": "\\alpha_0", "0a9d1fce409fbef1d2c676ddcdcc1cf2": "c_i, c_j", "0a9d5bfd22b632e8af15c30be5d1bf6e": "\\beta=\\,", "0a9d617b99bfd42b68fd099ebdd7d633": " M= (d\\colon H \\longrightarrow G) \\! ", "0a9e1b7a4a58bcdae6480e3e812658e8": "d(f,u)=\\frac{1}{2}d(f,g)+\\frac{1}{2(r-2)} \\left [ \\sum_{k=1}^r d(f,k) - \\sum_{k=1}^r d(g,k) \\right ] \\quad ", "0a9e26cd6e65f7315777ce26502fdfae": "E=\\frac{1}{2} L I_L^2", "0a9e4c6d789326ff9579297a1c7a91c8": "\\alef_1\\rightarrow(\\alef_1)^{\\alef_1}_2", "0a9ea40761d743bbcf3a7fbc858d6c7e": "\n\\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^p}{e(m+1)}\\,-\\,\n \\frac{p (d+e\\,x)^{m+2}(b+2 c\\,x) \\left(a+b\\,x+c\\,x^2\\right)^{p-1}}{e^2(m+1)(m+2 p+1)}\\,+\\,\n \\frac{p(2 p-1)(2 c\\,d-b\\,e)}{e^2(m+1)(m+2 p+1)} \\int (d+e\\,x)^{m+1}\\left(a+b\\,x+c\\,x^2\\right)^{p-1}dx\n", "0a9edd6479d648f2c47364595502d970": "X_{v}= ", "0a9f5fdc7c75be3ff3aea0cffaa8ef0f": "c^2 = 8.98755\\times10^{16}\\, m^2sec^{-2}", "0a9f9049fd85001ece99c93c1e74f7b1": "2n_{\\rm film}d\\cos\\big(\\theta_2)=m\\lambda", "0a9f9dd57729ca20f6e3fded066ccc54": "{\\rho}_{f} \\propto a^{-3 (1 + w_{f})} \\,.", "0a9faac9096f735c3f42c9c14414aaac": "s\\;", "0a9fc72e1e67cda37754ef8bb00c1be5": " \\frac{1}{\\sqrt{f}}= 1.1364\\ldots + 2 \\log_{10} (D_\\mathrm{h} / \\varepsilon) -2 \\log_{10} \\left( 1 + \\frac { 9.287} {\\mathrm{Re} (\\varepsilon/D_\\mathrm{h}) \\sqrt{f}} \\right)", "0aa002e9418974bbe6ec275d784dbf8b": "b=(k-1)/3", "0aa02de685fc954cffa62e48f1dc7341": "\\, g_{ab}", "0aa0515be1376978b735ce4fae64f0be": "2^{1+4}", "0aa0882275dfdd9ce70b984d3f043f99": "p^2=A(r^2-a^2)", "0aa0b66092a88ce2f3b0df720cbac3ae": "1\\;2\\;+", "0aa0f8474f5a966e62a5cb2e9dd42366": " \\sigma = E \\varepsilon ", "0aa10f27130cebfeb62737802a79fdff": "\\begin{align}\n Q_{\\bold{u}}(\\theta)\n &{}= \n \\begin{bmatrix}\n 0&-z&y\\\\\n z&0&-x\\\\\n -y&x&0\n \\end{bmatrix} \\sin \\theta + (I - \\bold{u}\\bold{u}^T) \\cos \\theta + \\bold{u}\\bold{u}^T \\\\\n &{}=\n \\begin{bmatrix}\n (1-x^2) c_{\\theta} + x^2 & - z s_{\\theta} - x y c_{\\theta} + x y & y s_{\\theta} - x z c_{\\theta} + x z \\\\\n z s_{\\theta} - x y c_{\\theta} + x y & (1-y^2) c_{\\theta} + y^2 & -x s_{\\theta} - y z c_{\\theta} + y z \\\\\n -y s_{\\theta} - x z c_{\\theta} + x z & x s_{\\theta} - y z c_{\\theta} + y z & (1-z^2) c_{\\theta} + z^2\n \\end{bmatrix} \\\\\n &{}=\n \\begin{bmatrix}\n x^2 (1-c_{\\theta}) + c_{\\theta} & x y (1-c_{\\theta}) - z s_{\\theta} & x z (1-c_{\\theta}) + y s_{\\theta} \\\\\n x y (1-c_{\\theta}) + z s_{\\theta} & y^2 (1-c_{\\theta}) + c_{\\theta} & y z (1-c_{\\theta}) - x s_{\\theta} \\\\\n x z (1-c_{\\theta}) - y s_{\\theta} & y z (1-c_{\\theta}) + x s_{\\theta} & z^2 (1-c_{\\theta}) + c_{\\theta}\n \\end{bmatrix} , \n\\end{align}", "0aa126687e2a7320ba7e1f54a3e79a03": "\\mathbf{A}+(-1)\\mathbf{B} = (A^0, A^1, A^2,A^3) + (-1)(B^0, B^1, B^2,B^3) = (A^0 - B^0, A^1 - B^1, A^2 - B^2, A^3 - B^3) ", "0aa13b137923a11b7abe08075780a0c1": "K_M^{N + 2}", "0aa1d09f0db66f42459ec42124f727c8": "\\epsilon \\circ \\nabla = \\nabla_0 \\circ \\epsilon_2 : (B \\otimes B) \\to K", "0aa1eabb9b373d56901ceefbfcdbf7cd": "\\int_a^b \\frac{d}{dx}\\left(u(x)v(x)\\right)\\,dx = \\int_a^b u'(x)v(x)\\,dx + \\int_a^b u(x)v'(x)\\,dx ", "0aa23d614ac0c1ffdf58b658286dd862": "\nj^{\\star} = \\varepsilon\\sigma T^{4}\n", "0aa261c140856613b91c3b65582470a8": "a/b.\\,", "0aa267ba142027a329ab28c424c75c06": "2^{n-2} \\equiv 1 \\pmod{n}\\,\\!", "0aa270e347c65525faa1bb7772a8383e": "\n\\frac{A\\hbox{ prop} \\qquad B\\hbox{ prop}}{A \\vee B\\hbox{ prop}}\\ \\vee_F\n\\qquad\n\\frac{A\\hbox{ prop} \\qquad B\\hbox{ prop}}{A \\supset B\\hbox{ prop}}\\ \\supset_F\n\\qquad\n\\frac{\\hbox{ }}{\\top\\hbox{ prop}}\\ \\top_F\n\\qquad\n\\frac{\\hbox{ }}{\\bot\\hbox{ prop}}\\ \\bot_F\n", "0aa27d946056cdd21388d7cd18a9f969": "\\sum_{n=1}^\\infty \\frac{1}{2n+1}[\\zeta(n)-1] = 1-\\gamma-\\frac{1}{2}\\ln 2.", "0aa283b20ca498410b8cd10e5fbcd42a": "D_4", "0aa2ebfdc74934ffe8b784eb6b5df847": "\\langle v_i \\rangle", "0aa318c99e04acabc52e39ef2f1700f2": "\\textstyle R_{k+qm} = R_k", "0aa36281330671a817e747f6271edbd1": " \\{\\langle X_1, X_2, ...., X_n \\rangle \\mid p(\\langle X_1, X_2, ...., X_n\\rangle) \\} ", "0aa3a9d67938be9eb0691442c674b5e0": "\\,r=\\sin(k\\theta)", "0aa41bd74e4bd77649484a2c86037926": "\\mathcal{E}=\\oint_{C}\\boldsymbol{ \\left[E + v \\times B \\right] \\cdot } d \\boldsymbol{ \\ell } \\ ", "0aa43d06971528b73dd777b39d508cd7": "\\delta_x", "0aa460b5aef85265cc24d65df0458c2a": "x_k(\\zeta) = \\Re \\left\\{ \\int_{0}^{\\zeta} \\varphi_{k}(z) \\, dz \\right\\} + c_k , \\qquad k=1,2,3", "0aa48d0248dcbaf032405853d1ebf0ac": "e_x=E_x-B", "0aa4b777716a125e6154c748ceb06760": "R_W=\\frac{\\rho_E}{2\\cdot \\pi\\cdot a_W}\\,", "0aa505623e91b2f462841c46926f5cfe": "\\lfloor k\\rfloor", "0aa505ea75841dbbecff1a050bc871ab": "f: \\mathbb{R}^N \\rightarrow \\mathbb{R} ", "0aa56aaa257aeb04de384bf8dfb1950f": "\\begin{align}\n F_\\vec{r} (p) dp & = \\frac{4 \\pi p^2 dp} {\\frac{4}{3} \\pi p_f^3(\\vec{r})} \\qquad \\qquad p \\le p_f(\\vec{r}) \\\\\n & = 0 \\qquad \\qquad \\qquad \\quad \\text{otherwise} \\\\\n\\end{align} ", "0aa56d842fa681d0a6781ed3d64113e8": "\\underline{\\lnot \\lnot \\varphi}\\,\\!", "0aa572144bed5ebd56399e1885058a0c": "w_{it}", "0aa5a728821daf1bb1b22e48ea6dd74b": "Q(\\xi, \\eta) = \\int \\omega_b^{n-k} \\wedge \\xi \\wedge \\eta.", "0aa5b58fc191f2a686da3948ce651e8b": " T(r,t) ", "0aa5d573ac848216e8757642c45b3d06": "i\\in \\{1,...,N\\}", "0aa618eb29ee0d363381f618364a87db": "\\lambda_1 = 1 \\,\\!", "0aa64b4d34c62565dedba0a3cc00ad56": "\\langle \\rangle ", "0aa66bb3c56357d702290d7492977fdc": "\\textstyle \\frac{\\partial C_2}{\\partial t}", "0aa673048938f656942ba7966bd4eb7e": "Usage(time)/Lifetime(time)", "0aa6a6089cef1d17c9169364a245f6a8": "b R c", "0aa6d88d57434b58f1ec59a9b300b593": "\\sum_{n=-\\infty}^{\\infty} s(nT)\\cdot \\delta(t-nT) = \\underbrace{\\int_{-\\infty}^{\\infty} \\tfrac{1}{T}\\ S_{1/T}(f)\\cdot e^{i 2\\pi f t}\\,df}_{\\text{inverse Fourier transform}}\\,", "0aa6e9efdaef42043ddd1cfd7a7570ad": " ( E - mc^2 ) \\psi_+ = \\frac{1}{2m} \\left [ \\boldsymbol{\\sigma}\\cdot \\left ( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right ) \\right ]^2 \\psi_+ + e\\phi \\psi_+", "0aa6fb1141586b1c9c383b08eea3cb19": " R \\simeq \\frac {R_1}{1-A} ", "0aa75fd6e89c9fff43198d93dd0ce521": "x[v_1\\otimes v_2]=x[v_1]\\otimes v_2+v_1\\otimes x[v_2] .", "0aa76511f8ea47f6099088bb89f41350": "{\\frac{\\Delta S}{R}=\\frac{\\gamma}{\\gamma-1}\\ln\\left(\\frac{2}{\\left(\\gamma+1\\right)M^2_x}+\\frac{\\gamma-1}{\\gamma+1}\\right)+\\frac{1}{\\gamma-1}\\ln\\left(\\frac{2\\gamma}{\\gamma+1}M^2_x-\\frac{\\gamma-1}{\\gamma+1}\\right)}", "0aa781ca2b9ef42658866b6f4202e8e6": "\\int_S f \\,dS = \\iint_T f(\\mathbf{x}(s, t)) \\left|{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}\\right| \\, ds \\, dt", "0aa78426c5b6a66a12015e8c45a199e6": "\n\\text{Tr}\\left\\{ \\Lambda\\rho\\right\\} \\leq\\text{Tr}\\left\\{ \\Lambda\n\\sigma\\right\\} +\\left\\Vert \\rho-\\sigma\\right\\Vert _{1}.\n", "0aa7c74d2c060853eb0c6fa511f1795d": " M \\mapsto M_{\\mathfrak{g}} := M / \\mathfrak{g} M.", "0aa7d7c2b205c78e1885a02371db2f90": "\\hat{C}", "0aa7dae70ca29f652d842d18b360e397": "A \\Delta B", "0aa82b70834da9cbe82c3cb48547a35f": "B_n", "0aa8725509013ca63cd2925b4642cae1": "d_{yz}", "0aa8d34dc55a888709ee80973c870293": "\\ F_{forward} = drag ", "0aa8f5a09588e383fd454d26a61841c1": " E = 1 - {F\\,'_{\\rm D}}/{F_{\\rm D}} \\!", "0aa91177aa774b550375479c883d5055": "L_g = N\\Lambda\\,", "0aa914f6f9f119c85099c1b830a85470": "\\begin{pmatrix}x \\\\ y\\end{pmatrix} \\mapsto \\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix} \\begin{pmatrix}x \\\\ y\\end{pmatrix} = \n\\begin{pmatrix}ax + by \\\\ cx + dy\\end{pmatrix}.", "0aa91cfe48cb3947e4ea10ed12d22c87": "e^2 = [7; 2, 1, 1, 3, 18, 5, 1, 1, 6, 30, 8, 1, 1, 9, 42, 11, 1, 1, 12, 54, 14, 1, 1 \\dots, 3k, 12k+6, 3k+2, 1, 1 \\dots] \\,\\!.", "0aa92911dc8219c92909933f6c71482e": "\\scriptstyle n \\geq 2 ", "0aa9cb87647412672a8dd65c5ced873b": "Z_\\mathrm{sun} = 0.02", "0aa9e64de9dc0e93ca03f21a48f6ee05": "\n(\\leftarrow/) \\quad\n{Z\\leftarrow \\Delta Y \\Delta' \\qquad X\\leftarrow\\Gamma \n \\over\n Z\\leftarrow \\Delta (Y/X)\\Gamma \\Delta'}\n", "0aaa10ceadf13196a2cfebd2223b1468": "0 0\\,", "0ab17fc73bca257d46f02f7379e36f7d": "Q_{Actual} = Q_{Fan}*\\sqrt{\\rho_{Ref} \\over \\rho_{Actual}}\\,\\!", "0ab19a1c58a36156bbb4c0300ad21cf9": " W(C;1,0) = A_{n}= 1 \\mbox{ iff } (1,\\ldots,1)\\in C\\ \\mbox{ and } 0 \\mbox{ otherwise.} ", "0ab1cbb034fc7fdede7b3a8856b87e52": "\\frac{dX_i}{dt}=\\sum_j \\mu_{ij} \\cdot \\gamma_j \\prod_k X_k^{f_{jk}}\\,", "0ab22dfd8e88a3a1b5d793bdfee0e7d4": "\\frac{d}{dt}\\langle\\mathbf{p}(t)\\rangle = q\\mathbf{E} - \\frac{\\langle\\mathbf{p}(t)\\rangle}{\\tau},", "0ab23d4889c810546d523bdf5d8248f0": "X_C = {1 \\over j \\omega C} = {1 \\over j 2 \\pi f C} \\ ; \\ X_L = j \\omega L = j 2 \\pi f L", "0ab24aaa993c96392f6ee45a11e4f28d": "\nA_v | \\psi \\rangle = | \\psi \\rangle, \\,\\, \\forall v, \\,\\, B_p | \\psi \\rangle = | \\psi \\rangle, \\,\\, \\forall p,\n", "0ab24ca116a09a877f37060009993050": "\\varepsilon _t", "0ab268dacd2dbfdb07ae780787be79bb": "\\mathbb{D}^q\\mathbb{D}^{-q} = \\mathbb{I}", "0ab277d305236c1f5274035ba1309cba": "\\because \\!\\,", "0ab2b8020f296fb27c395434c8a47cc1": "x = \\sqrt{y}", "0ab2c1f8b698090755718ab622e6d685": "\\pi_i(S^n) \\otimes \\mathbb{Q}", "0ab2c4719593d2b5fb026bbe28c3e916": "\\eta:S\\to T", "0ab2c4cd9fceb89ac2ffd80b5b39e1e1": "e_0 ^2 = e_1 ^2 = 1", "0ab2fabfc07ab3a98e7dc3c7ab92926c": "\\kappa\\Big.", "0ab31a5fed53d96bc37cd22f8061a09a": "f_1(\\text{AA}) = p^2 = f_0(\\text{A})^2", "0ab31be6a0962a3aed1868ef918a2cab": "\\sum_{k=0}\\frac{B_k(x)}{z^k}\\frac{{1-s\\choose k}}{s-1}= z^{s-1}\\zeta(s,x+z),", "0ab3ca7f3234bec5859ba72752227c05": "\\int f(x)d_hx", "0ab3f6c138cd8032ac51d26763b2bac8": "p(x|y)", "0ab3f82999513e36de6f4b0dbf5e81e0": "\\textstyle W(y|x,s)", "0ab4652fd102e74f7e19bc45fba4a79a": "\n d\\left( {\\vec x;\\vec y} \\right) = r\\left( {\\vec x - \\vec y;q} \\right)\n", "0ab492f6cbc2ecfa64374d88f7755a66": "\\textrm{coversin}(\\theta) := \\textrm{versin}\\!\\left(\\frac{\\pi}{2} - \\theta\\right) = 1 - \\sin(\\theta) \\,", "0ab4d99d9556c4727ac5b865641d1144": " \\sum_{n=0}^\\infty H_n^{(r)}z^n=-\\frac{\\ln(1-z)}{(1-z)^r}. ", "0ab5276ab725610bcf839e332f305313": "\\ f_{2,M}= exp(Ax_1^2)\\,", "0ab52d27d1ea9bfd8c482872fb504761": "=1-\\alpha x \\gamma+(1/2)(\\alpha x)^2 (\\gamma+1)\\gamma-...\\,", "0ab5516223f24f0779e99a66212e9bf9": "h = \\frac{3.5 \\hat \\sigma}{n^{1/3}},", "0ab55a10a44a3f243d05b5d6a7a9a3fb": "n=1,", "0ab6334bf1d68abf09a3756cbac5f4ae": "\\log V^2 = 2 \\log V", "0ab652abbab16f55bb79c87e548bd85a": "\\boldsymbol{\\sigma} = \\begin{bmatrix}\\sigma_1 & \\sigma_2 & \\sigma_3 & \\sigma_4 & \\sigma_5 & \\sigma_6 \\end{bmatrix}^T \\equiv \\begin{bmatrix}\\sigma_{11} & \\sigma_{22} & \\sigma_{33} & \\sigma_{23} & \\sigma_{31} & \\sigma_{12} \\end{bmatrix}^T.\\,\\!", "0ab654034920736609f88f614b9ce9f1": "\\Omega(n^{1/3})", "0ab65726c4090804a96edeaf6e93fa78": "f_k = \\sum_{j=0}^{n-1} v_j e^{\\frac{-2\\pi i}{n}jk}.", "0ab66fe892087d10e4b4ee1b1d98ec10": "\nf(n_i)=\\ln(W)+\\alpha(N-\\sum n_i)+\\beta(E-\\sum n_i \\varepsilon_i)\n", "0ab6a58d85c4548fe3c88935db2395af": "\\scriptstyle{V_c}", "0ab6a98da3418135a067dad824ddc9ae": " \\mathbb{E}[R_P] = \\sum^{n}_{i=1}x_i\\mathbb{E}[R_i] ", "0ab78ad8cc490342fe04b15966c38f54": "1093 = 1111111_3 = 3^6 + 3^5 + 3^4 + 3^3 + 3^2 + 3^1 + 3^0 \\, .", "0ab7fe497d59ec59b94eb2ea4205a9d1": "\\mathcal L= \\tfrac{1}{2}\\ \\partial^\\mu \\hat n \\cdot\\partial_\\mu \\hat n ", "0ab81192bbc2616a6f8007e05cb82572": "\\nu = \\rho \\neq \\mu", "0ab8520530f4ccebdb0a0f095263d5f7": " \\mathbf{\\bar y} = \\mathbf{T} \\, \\mathbf{y} ", "0ab8b46d861c347adde3545bd0ffd8c1": "t' \\equiv 1 \\pmod p", "0ab928ed87d8bdde337ecba2a7b2a5cd": "\\tau = T - t", "0ab979249bba64a40d6f296b462c09d3": "z_i=e^{i\\theta_i}=\\cos(\\theta_i)+i\\sin(\\theta_i)", "0ab98bc4f818e112bb3e48b331513a81": "\\begin{align}Q(AC) & \\equiv (C_x - A_x)^2 + (C_y - A_y)^2 \\\\ & = ((b\\lambda\\ + A_x) - A_x)^2 + ((-a\\lambda\\ + A_y) - A_y)^2 \\\\ & = (b\\lambda\\ + A_x - A_x)^2 + (-a\\lambda\\ + A_y - A_y)^2\\\\ & = (b\\lambda)^2 + (-a\\lambda)^2\\\\ & = b^2\\lambda^2 + (-a)^2\\lambda^2\\\\ & = b^2\\lambda^2 + a^2\\lambda^2\\\\ & = (a^2 + b^2)\\lambda^2\\end{align}", "0ab9dc6cc662d678f58b6cb2d3410560": "H = \\sum_k {\\dot q_k} p_k - L", "0ab9fe1ed9e288a89499d4b770c0b083": " \\left|\\langle A | \\Psi(t)\\rangle \\right|^2 ", "0aba042963dddbddc3ccc114ab9f6f13": "\\lambda_i \\ne 0 \\; \\forall \\,i", "0aba1ae1cc808cfd1b2b910fcd97e4b8": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = J^{-1}~\\dot{J}~\\boldsymbol{\\sigma} + \n \\boldsymbol{F}\\cdot\\dot{\\boldsymbol{F}^{-1}}\\cdot\\boldsymbol{\\sigma} + \\dot{\\boldsymbol{\\sigma}} + \n \\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{F}^{-T}}\\cdot\\boldsymbol{F}^T \n", "0aba2480809eb0eb04bf4e87f2bbbd3b": "\\frac{G}{N} = \\frac{G}{N}^\\circ + kT\\ln \\frac{p}{{p^\\circ }}", "0aba3e92d95491df32aff9dc2ca23577": "\\log_2 k", "0aba57f2b2ada99e67ec0704ee05340f": "~\\Phi_4(x) = x^2 + 1", "0aba5b1ef550dba0157bd45062077bca": "u(x)=\\int\\limits_{\\partial B(x, r)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\, u(y) \\,\\mathrm{d}S(y).", "0abaf6d8fcb1c5a5d0705d44f0b61c9c": "y_2,\\dots,y_k", "0abb5cf9d9de43ae7b97d69b767281b5": "H_L = 10 \\times \\log_{10}(A_{510} / A_{610})", "0abbd0b00b2ce865975fece655a362bf": "L^\\infty([0,1] \\cup \\mathbf{N}). ", "0abbff2aef73d6ac0227476b9763fa0b": "X^{\\beta}:=\\{x \\in X \\mid \\sum_{i=1}^{\\infty} x_i y_i < \\infty \\quad \\forall y \\in X\\}.", "0abc0c9c6c047030a0f21bad8ac9c59e": "z_0^n\\overline{p(1/\\bar{z_0})} = z_0^n\\overline{p(z_0)} = z_0^n\\bar{0} = 0", "0abc436f9f35683683660cf5ddeafa26": "\nf_X(x)= \\begin{cases}\n\\frac{1}{24}x^4 & 0\\le x \\le 1\\\\\n\\frac{1}{24}\\left(-4x^4 + 20x^3 - 30x^2+20x-5 \\right)& 1\\le x \\le 2\\\\\n\\frac{1}{24}\\left(6x^4-60x^3+210x^2-300x+155 \\right) & 2\\le x \\le 3\\\\\n\\frac{1}{24}\\left(-4x^4+60x^3-330x^2+780x-655 \\right) & 3\\le x \\le 4\\\\\n\\frac{1}{24}\\left(x^4-20x^3+150x^2-500x+625\\right) &4\\le x\\le5\n\\end{cases}\n", "0abc5149e2921fb41c2ffc3e22041930": "\\dfrac{d^{\\frac{1}{2}}}{dx^{\\frac{1}{2}}}2 \\pi^{-\\frac{1}{2}}x^{\\frac{1}{2}}=2 \\pi^{-\\frac{1}{2}}\\dfrac{\\Gamma(1+\\frac{1}{2})}{\\Gamma(\\frac{1}{2}-\\frac{1}{2}+1)}x^{\\frac{1}{2}-\\frac{1}{2}}=2 \\pi^{-\\frac{1}{2}}\\dfrac{\\Gamma(\\frac{3}{2})}{\\Gamma(1)}x^{0}=\\dfrac{2 \\sqrt{\\pi}x^0}{2 \\sqrt{\\pi}0!}=1,", "0abc701af9cfd1e2ba8ab6f1e06bcb83": "f\\in(0,1]", "0abc8da009882f9bc53c674c9b3700fd": "\\, A", "0abc8f40d3557a10146995de647b4e92": "CG = V_{rt}/Q_{rt}", "0abcc20f29c5824d20220cba6b4124cf": "\\mathbf{v}=v'^a\\mathbf{e}'_a=v^b(R^{-1})_b^a R_a^c \\mathbf{e}_c=v^b\\mathbf{e}_b.", "0abcc9598ea4cfb8a14666566cef04c4": "\\sigma(A)", "0abccdeb306ac58bca84a20aae3b6314": "\\operatorname{NWScore}", "0abccdf17d169d9ce64ab640a90a8ecb": "t=m_1^2 + m_3^2 - 2p_1 \\cdot p_3 \\,", "0abcd7271f7e8889e06d8476b46ddab1": "A = \\pi r (2h + a) \\;.", "0abd863cfdd959caa83b6068b6b000fc": "\\{x_i-b_i\\}", "0abde827b4af407fe765bd03cfe196f5": "X^L", "0abdfa3e821850c47c87bf08ab4cac16": "R=\\sigma^2 I", "0abe4605e6740e35d7daeccc1e4f00c2": "\\mu(X\\,\\triangle\\,Y) = 0", "0abe7ba39b1d683718b7c084530e24cb": "A(x)>c\\sqrt[3]{x}", "0abe80eceafe12537afc61f6b61330dd": "\\breve\\theta_{j,i}", "0abe9ad771462462c459bec7c0671601": "L(k_2)=32\\left\\lceil l/16 \\right\\rceil+40", "0abee1df9e40d481a91c2e9d80897141": "\n\\vec{v}(t+\\frac{dt}{2})\n= \\vec{v}(t-\\frac{dt}{2}) + \\vec{F}(t) \\, dt\n", "0abeec726e06d5d11b456075fac1ffeb": "R_1+R_3 = \\frac{R_b(R_a+R_c)}{R_T}.", "0abf02f1817477d9f300bc0176df38fe": "\\begin{bmatrix}0&1\\\\0&-1\\end{bmatrix}:\\mathbf b", "0abf23ff097df4c2775f0e5d365d0867": "-2\\ln(LR) = \\textstyle -2\\sum_{i=1}^k x_{i}\\ln(\\pi_{i}/p_{i}) .", "0abf379a9400e41ff5f59cf92afc12fc": " \\vec{F_a} \\cdot \\vec{x} = F_{a}\\,x .", "0abf7646f0446cd16f1f40e0315399f4": "S = {V^{1.85} A^{1.85}\\over k^{1.85}\\, C^{1.85}\\, R^{1.17}\\, A^{1.85}} = {Q^{1.85}\\over k^{1.85}\\, C^{1.85}\\, R^{1.17}\\, A^{1.85}} ", "0abf86882f5fb08f2528e412a3038d47": "f : \\mathcal{P}_{=n}(\\kappa) \\to \\{0,1\\}", "0abfa74cbcb7d42716d0250cc31ed341": " C = B \\log_2 \\left( 1+\\frac{S}{N} \\right)\\ ", "0abfd5a1bd50bac038306ec372049bb1": "p_1 = - p_2", "0abfdc7c810161cb10ca3eeb0ab4ea1d": "b = \\frac{n_{solute}}{m_{solvent}}", "0abfe9cf38ade7c5ce88b1eaebf4616f": "\\operatorname{dim}R \\ge \\delta(R).", "0ac0475c8557d20fcb473088f789b720": "x^{p^n}\\in F", "0ac0521677fdeac70934d8e03fa19a64": "\\begin{align}\n P_{t+\\Delta t} & = P_t+(rP_t-M_N)\\Delta t\\\\[12pt]\n\\dfrac{P_{t+\\Delta t}-P_t}{\\Delta t} & = rP_t-M_N\n\\end{align}", "0ac0a1e3154c59f80839598aca5acbd7": "H(S,P)\\,", "0ac11eba32233953849b5da16a58f8e5": "p\\leq2", "0ac14e7ca95955736a136a4830f2781c": " \\mathbf{p}_k = -B_k^{-1} \\nabla f(\\mathbf{x}_k) ", "0ac1a7ccf1ccf6a88d07e9e6866da2c6": "R\\bowtie S:=\\{f\\mid f \\quad (x\\cup y)\\hbox{-tuple},\\quad f[x]\\in R,\n \\;f[y]\\in S\\}.\\,", "0ac2253736858e5bfb08f67a1f7bd58b": "\\nabla \\!\\,", "0ac2662c42f9ae476827ec285805a274": "K^\\prime(x)= e^{-2u} (K(x) - \\Delta u),", "0ac2740199bc183cf310066f7c28cd71": "\\mathrm{l}", "0ac2a46e7711261101eea324418915ff": "n = 8", "0ac2a50c48210c0fa7da24740f88b0bd": " C = \\int \\Phi_a(r_1)^2 \\left(\\frac{1}{R_{ab}} + \\frac{1}{r_{12}} - \\frac{1}{r_{a1}} - \\frac{1}{r_{b2}}\\right) \\Phi_b(r_2)^2 \\, dr_1\\, dr_2", "0ac2adcf642727c46385a5ba43c4e9d9": "g(e) = (g_1(e), \\ldots , g_k(e))", "0ac2c4bc51028124710747b9983fbaa3": "[v,v']\\in\\Gamma(T)", "0ac340c58b8c0cd2b02f256f5faedc90": "\n\\max(a) = \\frac{1}{A} \\int_A \\left \\lbrace \\max_j P \\left [j |\n \\vec{y}(\\vec{r}) \\right ] \\right \\rbrace \\, d\\vec{r}\n", "0ac3c341e931a390385440690571c8e2": "\\operatorname{ad}_{\\mathfrak{g}} x", "0ac3c4ca3ed07532181422d15886b71b": " \\frac{\\partial \\psi}{\\partial t}+\\frac{\\partial^3 \\psi}{\\partial x^3}-3(u-\\lambda)\n\\frac{\\partial \\psi}{\\partial x}=C\\psi+D\\psi\\int \\frac{dx}{\\psi^2}", "0ac44265bdd3bf7252067cceed598b1f": " x^2 - ny^2 = -1. \\, ", "0ac452807ab0097a402dc73d594f0ec3": " \\sum_{n=1}^\\infty n^{-p}, ", "0ac473fc014fcd1891811192855b759c": "f_\\sim(x)=1-x,", "0ac55db71f8b7290a038d3c72fa574d3": "\nf(x) = \\frac{\\sqrt{x}+\\sqrt{\\frac{1}{x}}}{2\\gamma x}\\phi\\left(\\frac{\\sqrt{x}-\\sqrt{\\frac{1}{x}}}{\\gamma}\\right)\\quad x > 0; \\gamma >0\n", "0ac56a500561c30cac86bc43d460dd0c": "He(2^3S) + H_2O \\to H_2O^{+\\bullet} + He(1^1S) + e^-", "0ac573a76cef06d901ac68f961535fff": " u(0)=0, ", "0ac592f604247c87b2029467e777cf78": "R = \\sqrt{x^2 + y^2 + z^2}", "0ac59dee686571fd9ee649da40104aa7": "f(\\pi(X)) = \\pi(f(X)) = f(X)\\ \\bmod\\ f(X) = 0", "0ac5ab8967850137b6c5158beca7bada": " P(y) \\ ", "0ac5fd87a03f69428cbdf790c11d7a80": "\\textstyle\\varphi=\\frac{1+\\sqrt5}2", "0ac672a1322969bd6147fc2fd816893c": "n\\$=(n!)\\uparrow\\uparrow(n!). \\,", "0ac6b5246d4da4e79135e76b5eaf96e5": "v'= -h-v \\mod u'", "0ac6ef0d8fa553cb3a0fcb9ec32a23d3": "\\{1,\\alpha,\\cdots,{\\alpha}^{n-1}\\}", "0ac6f910c5b3ea8272dad86670481ce5": "R_1 = R/f_1 R", "0ac727eafb8be039ae8da9a8a77eed76": "V\\in\\mathbb{C}^{n,n}", "0ac73ece2591b7ddc5158def8b45137e": " v(s) = \\frac {\\mathrm{d}s}{\\mathrm{d}t}\\ . ", "0ac75f2375edd1cdde6c04189d334bf9": " \\frac{R}{2} = \\frac{a}{6}\\sqrt{3} \\!\\, ", "0ac764a152cd8addee84feecdfd53b50": " { \\frac{\\partial{(\\rho T)}}{\\partial t}} + { div\\, (\\rho u T )} ={div\\, (k\\, grad\\, T )} ", "0ac7a9bea3b54a075f8b578330a6e6db": " \\mathbf S = \\begin{bmatrix} 0 & \\tau & \\kappa & 0 \\\\ \\tau & 0 & 0 & \\kappa \\\\ \\kappa & 0 & 0 & \\tau \\\\ 0 & \\kappa & \\tau & 0 \\end{bmatrix} ", "0ac7c564084ad99b4ee431f67f2af6c6": "f_{\\psi(\\varepsilon_{\\Omega+1})}(n)", "0ac7f3d784bd054e6971b30ac1e1251a": "AT = TJ", "0ac8143003d388fd0ef1e531da66b31d": "\\lambda_{z,n}", "0ac83a05f1717c5f689e22d9b96476b2": "\\bar{x} = \\frac{1}{n}\\sum_{i=1}^n x_i = \\frac{1}{n} (x_1 + \\cdots + x_n)", "0ac8d8837a8bd760f91fed392f9dc9c6": "(\\alpha \\neq \\beta )", "0ac928f63faf51766bad45cbd4984295": "\\pi(x)-\\pi\\left(\\sqrt{x}\\right)+1", "0ac9f1b0c58e3860c196d17efa806bcc": "\\exp\\left(-(x-2\\theta)^2/4\\right).", "0aca355f8b843e21cb17e7c768edab18": "\\scriptstyle 6.283\\,185\\,307\\,179\\,586\\,476", "0aca76669b779dd4a3db42245be82b30": "\na_{i,n}= \\left[ \\frac{n-\\sum_{j=0}^{i-1}k_j+[L/2]-1-i}{[L/2]-i} \\right]; i>0,\n", "0aca9c0a69b59c541a40a4e48e52077c": "n<1\\,", "0ad0e36e3613657408c4d6b739302693": "\\mathrm{vec}(A) = [a_{1,1}, ..., a_{m,1}, a_{1,2}, ..., a_{m,2}, ..., a_{1,n}, ..., a_{m,n}]^T", "0ad13033291025cbcb017455d940a878": "H_\\omega(1) - 1", "0ad193a4119b84ddf0274386f762faa6": "\\Delta = \\nabla\\cdot\\mathbf{v}", "0ad1a0ea11728a00f81a93d254d466c4": "p_j\\,", "0ad246dae821be375536d8ecc4fd3b30": "E_{rot} = \\frac{l(l+1) \\hbar^2}{2 \\mu r_{0}^2} \\ \\ \\ \\ \\ l=0,1,2,... \\,", "0ad2f3cf3bc80559d3b96d9151d7d481": " V^{\\pi} (s) = E[R|s,\\pi],", "0ad2f6d09ade22978c25bb174f3122a9": " \\operatorname{Trace}(\\rho_f(\\operatorname{Frob}_p))=a_p\\ ", "0ad37bfccd3b49545be5ed21871119c8": "n_{\\mathrm{i}}", "0ad3b74694389e4dc6b9faf0477d359d": "|n|_p=p^{-v_p(n)}", "0ad3fc8da420819e2173a3de1a335668": "R^{1/2}\\,", "0ad4548531b9ee1545b5eb7827313825": "\n\\dot{x}=b-x/\\tau+\\sqrt{Dx}\\xi(t)\n", "0ad461bd015a1320ef9c9b53ad032490": "\\, x= \\theta/2\\pi \\, ", "0ad48f3dc6536fc08529e4289fd28941": "T=\\tfrac{1}{2}ab", "0ad492cef6b429ca87eb038e54617396": "\n \\begin{align}\n \\left| \\int_{C} f(x) e^{\\lambda S(x)} dx \\right| \n &\\leqslant \\int_{C} \\left|f(x)\\right| \\left|e^{\\lambda S(x)} dx \\right| \n \\equiv \\int_{C} \\left| f(x) \\right| e^{\\lambda M} \\left| e^{\\lambda_0 [S(x)-M]} e^{(\\lambda-\\lambda_0)(S(x)-M)} dx \\right| \\\\\n & \\leqslant \\int_{C} \\left| f(x) \\right| e^{\\lambda M} \\left| e^{\\lambda_0 [S(x)-M]} dx \\right| \n = \\underbrace{e^{-\\lambda_0 M} \\int_{C} \\left| f(x) e^{\\lambda_0 S(x)} dx \\right| }_{\\text{const}} \\cdot e^{\\lambda M}.\n \\end{align}\n", "0ad4966f96e4977e245ce2fe7e0dff81": "\\tau:\\; V\\otimes V\\longrightarrow V\\otimes V \\,", "0ad4d55cda7cac30484bb00deacd7253": "i^i = \\left( e^{i (\\pi/2 + 2k \\pi)} \\right)^i = e^{i^2 (\\pi/2 + 2k \\pi)} = e^{- (\\pi/2 + 2k \\pi)}", "0ad5110e61d07cd15d0e7ff5c8e20b50": "m:I\\to F^{*}J", "0ad549abfc6ed3f68e2b890d96c0fc58": "ax^2+2bxy+cy^2", "0ad54a6de1d35b7484afaf01d8c1ebfa": "\n \\frac{\\partial^3\\varphi_1}{\\partial x_1^3}+\\frac{\\partial^3 \\varphi_1}{\\partial x_1\\partial x_2^2} + \\frac{\\partial^3 \\varphi_2}{\\partial x_1^2 \\partial x_2}+\n \\frac{\\partial^3 \\varphi_2}{\\partial x_2^3}= -\\frac{q}{D} \\,.\n", "0ad574c599c118a4d1a3aea85c14c627": " \\mathrm{Re}_m = \\mathrm{Re} \\; ", "0ad5898a1dc24678b6d0646be3695306": "\\hat{\\mathbf{x}}_i'", "0ad5a5795233954acfdfed9a5275e070": "q = -k \\frac{\\partial T}{\\partial r}", "0ad5ac4713db38888edd4381d2cd724d": "0\\neq 1", "0ad5d68eca854d52a82faf85fc094ddc": "\\left| \\overline{z} \\right| = \\left| z \\right|", "0ad6195ac2881486c287c4b96340561a": "r_1,\\;r_2\\geq 0", "0ad6223bdd76dffe58cf64b43882d742": "A\\to A", "0ad66224f1d204b6bb1a6f9dcfe8c91b": "Var(A) = f(bb)a^2_{bb}+f(Bb)a^2_{Bb}+f(BB)a^2_{BB},", "0ad663559a6e673a15b7ee91d4451f29": "\n F_c = -3\\gamma\\pi R\\,\n ", "0ad666ceea810cbb9b46e5e602c1b7f3": "\\left(\\lambda+1\\right)/n", "0ad6836c5749718d50ac958a6b474df4": " F = 1- \\frac{\\operatorname{O}(f(\\mathbf{Aa}))} {\\operatorname{E}(f(\\mathbf{Aa}))} = 1- \\frac{\\operatorname{ObservedNumber}(\\mathbf{Aa})} {n \\operatorname{E}(f(\\mathbf{Aa}))}, \\!", "0ad68add8ede6dd61b7eec9e07ce231d": "x+y=2, \\,\\,\\,\\,\\, 2x+2y=4", "0ad6e304efa22426be7a8bde52422264": "\\mathcal{L} = - {1 \\over 2 \\kappa^2} e e_I^\\mu e_J^\\nu \\Omega_{\\mu \\nu}^{\\;\\;\\;\\; IJ} [\\omega] + e \\overline{\\psi} (i \\gamma^\\mu \\nabla_\\mu - m) \\psi", "0ad6fbcabc0a265328c16db7c814e3ed": "TT = \\frac{\\sum_{v=v_\\min}^{N} v P(v)} {\\sum_{v=v_\\min}^{N} P(v)}", "0ad701e82310fb9b57bd31b4046dacb2": "r=\\frac{y'(t)}{y(t)}", "0ad75fdf00a4056733b425273446c041": "\\mathbb{E}\\left[\\mbox{ Charles }|\\mbox{ calling }\\right] = \\frac{4}{42} \\cdot (P+2) - \\frac{38}{42} \\cdot 1", "0ad80da9c5d99359d7f02677f40c4412": "\\sin\\frac{\\gamma_s}{2} = \\frac{a_s}{2r_u},", "0ad81ebc1a4a0d6b0f64bfb84e0e934c": "-\\tfrac{3}{2}", "0ad8206218e987db6f1cf49e5b20b103": " \\frac{\\hbar \\omega}{2} \\left( -\\frac{d^2}{d \\chi^2} + \\chi^2 \\right) \\psi(\\chi) = E \\psi(\\chi) .", "0ad82926eed098f07871a7b712353e0e": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\frac{\\tan\\delta}{\\delta} \\times \\lim_{\\delta \\to 0} \\left( \\frac{1 + \\tan^2\\theta}{1 - \\tan\\theta\\tan\\delta} \\right) .\n", "0ad894a67715cf79e1c6e8ca60c4bd93": " B \\rightarrow A ", "0ad8b7348b790908c11239b7d0f46f6d": "t_{ff} = \\frac{1}{4}\\sqrt {\\frac{3 \\pi}{2 G \\rho}}\\simeq 0.5427 \\frac{1}{ \\sqrt {G \\rho}}\\simeq 66430\\,{\\rm s} \\frac{1}{ \\sqrt {\\rho}}", "0ad932359c1bcae7aff5609deeeb46a9": "\\mathfrak{a}, \\mathfrak{b}", "0ad939a6a380786dbbb2202b3e62f5fa": "K=ab \\cdot \\sin{A}.", "0ad93ff7bfb00f57273d1262d31ac61c": "\\eta_{therm}", "0ad9676d5d1302e127cac77bba485049": " \\psi_L \\ \\stackrel{\\mathrm{def}}{=}\\ \\left ( {\\cos\\theta +i\\sin\\theta \\over \\sqrt{2} } \\right ) \\exp \\left ( i \\alpha_x \\right ) = \\left ( {\\exp(i\\theta) \\over \\sqrt{2} } \\right ) \\exp \\left ( i \\alpha_x \\right ) ", "0ad989cf548d5faf9dcd88c4f0f3fdac": "\\int x^{n} S dx = \\frac{2}{a (2 n + 3)} \\left(x^{n} S^{3} - n b \\int x^{n - 1} S dx\\right)", "0ad9b0cbf0774ca90fc4d215a7050772": "E_{i,j}=\\frac{\\left(\\sum_{n_c=1}^c O_{i,n_c}\\right) \\cdot\\left(\\sum_{n_r=1}^r O_{n_r,j}\\right)}{N} \\, ,", "0ad9efe4940c9b7a57eefc4ba4c7b105": "Z_L", "0ada268d42742fc19970ed1d4411cfe3": "d = |E| / |V|^2", "0ada4206c2d86ceb2abc5c74d52ae4dc": "y(t) = e^{-\\frac{t}{5}}\\sin(t)", "0ada9e98ac83205ab173c7a958814284": "p_1 = \\frac{ 0+0+0+0+2+7+3+2+6+0 }{140} = 0.143", "0adaabea1e27f22df013859217cc49b4": "r=\\frac{a}{\\theta}", "0adace6e1f771a8d412057231ae9e11b": "\\Gamma(z) \\approx \\sqrt{\\frac{2 \\pi}{z} } \\left( \\frac{1}{e} \\left( z + \\frac{1}{12z- \\frac{1}{10z}} \\right) \\right)^{z},", "0adb263e90b7c77dc1ee0a994fe6ccd2": "\\displaystyle\\Psi_{\\gamma\\alpha}(u) = \\Psi_{\\gamma\\beta}(u)\\Psi_{\\beta\\alpha}(u)", "0adb3fac1ad5ad0517a3c2c32cc01424": "\nQ_{flow} = k \\frac{D}{\\tau}\n", "0adb7dafdc41b9a4760f617066612a6c": "\n[ [A , B] , C ] = [A , [B , C]] - [ B , [A , C]]\n\\,", "0adba1de183aafddd4146813f294fa2e": "\n\\begin{array}[t]{rcl}\n\\operatorname{var}(PX)\n\t&= &\\mathbb{E}[PX~(PX)^{\\dagger}]\\\\\n\t&= &\\mathbb{E}[PX~X^{\\dagger}P^{\\dagger}]\\\\\n\t&= &P~\\mathbb{E}[XX^{\\dagger}]P^{\\dagger}\\\\\n\t&= &P~\\operatorname{var}(X)P^{-1}\\\\\n\\end{array}\n", "0adbd3ca4cb163c59d16e3e875fe4e8f": "p_1=5", "0adbdcbb5329dd46c2d3bf5317afcc81": "\\mu(A)=U{\\textrm-}\\lim{|A\\cap F_i|\\over|F_i|}.", "0adc1270c320cab6abf8308fc8d3687e": " \\scriptstyle \\log_e (\\frac {760} {101.325}) - 5.381564 \\log_e(T+273.15) - \\frac {2626.728} {T+273.15} + 1.601858 \\times 10^{-05}(T+273.15)^2", "0adc1c00f3e59bea21899491bf14fdd3": "V = \\{v_1 \\ldots v_n\\}", "0adc3d60e9744ac3ed6621168c52222c": "\\frac{{}_{(1)1}\\partial x^{-1}}{\\partial x}=x^{-1}\\,\\!", "0adc4805cb939b46ae33577d014985ff": "|\\Psi\\rangle_\\nu", "0adcaf6f179bdbc47234308c993cf28a": "{C_L}", "0adcb3f03a670941fddd67415651a8bb": "\\Sigma _{i}\\cdot \\partial \\mathcal{G}\\Sigma _{i}", "0adcc9ab86b40f6eeb1da2fd4f986813": " \\alpha \\approx 0.85 ", "0adcd34bcfa72a9acb4b3a5e6e50944c": "\\sigma_n=\\frac12", "0adce572f800120bb12403ce77380bb3": "< \\alpha, \\beta >", "0adceb1a4e17e1f6982bcda5746b5f75": "\\hat{B}(\\xi )= \\int \\frac{d^{2n}\\eta }{(2\\pi \\hbar )^{n}}\n\\exp (-\\frac{i}{\\hbar }\\eta _{k}(\\xi - \\hat{\\xi})^{k}) \\in \\mathbb{V}.", "0add083d28e9974a35eaa9d9bbc32728": "\n\\tan \\theta_{1} = \\tan \\eta \\cos \\chi \\,\n", "0add35947d40601f06f88757edb8cb34": "y_1^n", "0add49170ca83ceb0e481d9b65b9c4b2": "(a_1,a_2,...,a_{10}), \\qquad a_k = k^2.", "0adda38e472ea3f2b5b6abfca71b0326": "A_n(R) = \\frac{d}{dR}V_{n+1}(R).", "0addbbfa78e302f758e311020b8bebe3": " \\mu_{\\operatorname{eff}}(\\dot \\gamma) ", "0addcf96e7f409d886f8d1e649907116": "t \\rightarrow P_tf(x)", "0ade0c77b850b0197fb9f8bcbd9b016d": " \\| Ax_n - b \\| = \\| \\tilde{H}_ny_n - \\beta e_1 \\|, \\, ", "0ade1bfcbe41fcfd15d15576a4587922": "A_{jk}=f_j(g_k)", "0ade45986bd634701e3ef6b1f42187ef": "\\Phi=e^{\\beta(\\epsilon_i-\\mu)}-1", "0aded1a42b92b3af244a984feab5f5d6": "ev=0", "0adee9c4df3e9f23d4d45d934b2595fc": "J_{ij}=\\frac{\\partial\\theta_i}{\\partial\\xi_j}.", "0adf243c7df85a9fde4f478ed121ba95": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = \\left[1 + \\left(A_{1} - B_{1}\\left(x,y\\right) \\right)^{2} + \\left(A_{2} - B_{2}\\left(x,y\\right) \\right)^{2} \\right] \\\\\n f_{2}\\left(x,y\\right) & = \\left(x + 3\\right)^{2} + \\left(y + 1 \\right)^{2} \\\\\n\\end{cases}\n", "0adfd0bb7d5a814c2f7bb3116146c882": " f((\\mathbf{v} , \\mathbf{u})) = \\sin{(\\mathbf{v} , \\mathbf{u})} ", "0adfe03b2e4c67613fa5b5e6975fc42f": "s = \\frac {2 \\pi r}{z}", "0ae00baece84cb7a8f2d1c42e71fa7c9": " \\begin{pmatrix} A & B\\\\ B^* & \\end{pmatrix} \\begin{pmatrix} x_1\\\\ x_2 \\end{pmatrix}\n = \\begin{pmatrix} b_1\\\\ b_2 \\end{pmatrix},", "0ae00d7feea9f0b0f8b0154779ef7ec8": " \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x). ", "0ae01416e90b9cd36a31269816379f88": " S= \\frac{e^{- \\beta H}}{\\operatorname{Tr}(e^{- \\beta H})}. ", "0ae0353f26d6d3c71136114942f1ff14": "\\mu_{i1} = \\mu_{i2}\\, ", "0ae0f9542148acfda68f4177c534a18a": "\\begin{align}\n\\mathbf{T}^6 = \\begin{pmatrix}\n\\frac{1}{4096} & \\frac{3}{1024} \\\\[4pt]\n0 & \\frac{1}{4096}\n\\end{pmatrix},\n\\end{align}", "0ae1285ce5610001567ddb53236e50fe": "M.", "0ae14b7377bc33c387c2661a0b65ad91": " k_x, k_y ", "0ae160e4c7d3842ee9eaed697676bb3f": "\\Omega_n(E) = \\frac{c_n}{c_k^{n/p}}\\left(E-E_0\\right)^{n/p}\\ ,", "0ae185a9a0d7167749b407efcc6a1ced": "G(\\vec{r},t) = \\langle \\frac{1}{N}\\int \\rho(\\vec{r}'+\\vec{r},t)\\rho(\\vec{r}',0) d\\vec{r}'\\rangle ", "0ae185fb3e4dc62970b6e11005be8c1d": "\\mathbf R = \\frac 1M \\int_V\\rho(\\mathbf{r}) \\mathbf{r} dV,", "0ae1f248a8ba4353c0934d07dbca3423": "k|x_{t}|<1", "0ae21678a1f069a78dab02b4c6ea17d3": "k = A + \\ N(0, s^{2}/y_k) ", "0ae2353e5d59b84f52906b7025b8e628": "{\\displaystyle}P_{2}=(x_{2},y_{2})=(\\sin{\\alpha_{2}},\\cos{\\alpha_{2}})", "0ae241997f4e8defc892e1b015cb1e7f": "\nA_{(m+r) (n+r)} - A_{mn} \\approx r\\; \\left({dA\\over dJ}\\right)_{m n}\n\\, ,", "0ae25898d277b9e1445a22ab0e83b67e": "\n\\begin{align}\n(\\gamma a_1)(\\gamma a_2)\\dots(\\gamma a_m) \n&\\equiv \n{\\zeta_n^{b(1)}a_{\\pi(1)}} {\\zeta_n^{b(2)}a_{\\pi(2)}}\\dots{\\zeta_n^{b(m)}a_{\\pi(m)}} \\\\\n&\\equiv\n\\zeta_n^{b(1)+b(2)+\\dots+b(m)}a_{\\pi(1)} a_{\\pi(2)}\\dots a_{\\pi(m)}\\\\\n&\\equiv\n\\zeta_n^{b(1)+b(2)+\\dots+b(m)} a_1 a_2\\dots a_m\n\\pmod{\\mathfrak{p}},\n\\end{align}\n", "0ae25e97958e0619d97709cbe4d8e6e1": "g(x)=f^{(0)}_n (x) + \\mathcal{O}(\\epsilon)", "0ae260b632c6ffcbd3e92c01ef262348": "t_r=t_2-t_1=2t", "0ae271f7bf529e2058de7094e9cf7978": "x_0 = r^2~mod~N", "0ae2890e26891a12d8612233d1b5b97b": "\nR(\\vec x) = P(2 | \\vec x) - P(1 | \\vec x) \n= \\frac{P(2, \\vec x) - P(1, \\vec x)}{P(1, \\vec x) + P(2, \\vec x)}\n", "0ae2c9e1494b3b6e3d7e073751f8e08d": "\\mathcal{O}(1/k)", "0ae2d2635748aba7216b75e0e6c9e9f3": " S = W(q_1,q_2 \\cdots q_N) - Et ", "0ae349e680e10cb11344b7f7fd67849d": " K_\\text{GN}(x,x') = \\sigma^2 \\delta_{x,x'}", "0ae387cd1eeb766689f643ba1231b705": " \\textstyle \\sim \\left \\langle {\\sin}^{2}x \\right \\rangle = \\frac{1}{2}", "0ae3f521fce4775f8cc83a4023c20bdc": "\\mathbf{U} = \\frac{d\\mathbf{x}}{d \\tau} ", "0ae49173b1ab7f770fc3927f1bd65c8e": "e_0 = 1,\\ e_1 = i,\\ e_2 = j,\\ e_3 = k, \\!", "0ae4ff12c49634470ce0cebe2cb78ea5": "\n\\langle L(t) \\rangle\\equiv \\int^{\\infty}_{-\\infty} L(x,t) P(x,t) dx,\n", "0ae537600491b69d6f8857b3ddbc9368": "\n1=\\sum_{n=2}^{\\infty}(\\zeta(n)-1).\n", "0ae53c4958cb19e1efe9e2638ebb1d4f": "\\scriptstyle V \\, \\overset{\\sim}{\\to} \\, V^*", "0ae57e7fd2baa3e23e6fa5d341168439": "i: X \\to \\mathbf{P}^r_A", "0ae59705a0e645366c129299debb00ef": "(3^{n+1}(3^{n+1}-1)/2,3^n(3^{n+1}+1)/2,3^n(3^n+1)/2)", "0ae5cdaf55de16ec636ac97b974aca5a": "\\int_1^\\infty \\frac{\\ln\\ln x}{x^3}\\,dx = -\\frac{1}{2}(\\gamma+\\ln 2).", "0ae5d18287a4c831fb0e8eb0bf97a046": "f(x,y,z) = x^2 + y^2 \\longrightarrow \\rho^2 \\sin^2 \\theta \\cos^2 \\phi + \\rho^2 \\sin^2 \\theta \\sin^2 \\phi = \\rho^2 \\sin^2 \\theta", "0ae61f6339e973f2a71e43f843e6bd35": " \\sgn(x) = [x > 0] - [x < 0] \\,", "0ae65f4bc72b6d04c26f108f4cd35b3b": "x (\\theta) = (R + r)\\cos\\theta - d\\cos\\left({R + r \\over r}\\theta\\right),\\,", "0ae65f853c825e3107071d966d62514f": " D^2(\\mathcal{F})\\cong \\mathcal{F}", "0ae6c26938eb6f75d3dd165425ff0ce7": "\\hat{H}_{0} = \\sum_{i}\\frac{\\hat{p}_{i}^{2}}{2m_{i}} + V", "0ae6e25fafe20bb259b6da2b4c12ea48": "\\, w = w_i e^i = \\begin{bmatrix}w_1 & w_2 & \\cdots & w_n\\end{bmatrix}\\begin{bmatrix}e^1\\\\e^2\\\\\\vdots\\\\e^n\\end{bmatrix} ", "0ae6fb7e963837b2070b887c63c117ee": "\\mathbf{Q}_p", "0ae74f20e2e511df0f36adcfd193acc4": "(\\exists y) \\mu y R(y) = \\{ \\mbox{the least (natural number)}\\ y \\ \\mbox{such that} \\ R(y)\\}", "0ae7b35ba67c9740b964635410608cbf": "W_{ijkl}", "0ae7ea772114e874d075b0beb258e48f": "\\rm J = {}\\rm \\frac{kg \\cdot m^2}{s^2} = N \\cdot m = \\rm Pa \\cdot m^3={}\\rm W \\cdot s = C \\cdot V", "0ae7f4a7afa0b92f7e129a66cc208d6b": "c_d = \\frac{1}{\\pi\\omega_{d-1}} = \\frac{\\Gamma[(d+1)/2]}{\\pi^{(d+1)/2}}.", "0ae83b9a4460c4d9ac182eb506deddb3": "\nG'_k(u) = |F(u)|e^{i \\phi_k(u)}\n", "0ae8b117e9f0a44cfda7c8d7cf8eaf2d": "\\sum_{1\\le k\\le n \\atop (k,m)=1} 1 = n \\frac {\\varphi(m)}{m} + \n\\mathcal{O} \\left ( 2^{\\omega(m)} \\right ),", "0ae914d4537770229b7846cfdcd84292": "114\\frac{1}{2}", "0ae92c14c0e9def62b2c99369d444c10": "B \\to x", "0ae9357ebdd3d570acab982976a61da9": "\\left\\{\\frac{x_1+(1+x_2)x_3}{x_1x_2},\\frac{x_1+x_3}{x_2},\\frac{(1+x_2)x_1+(1+x_2)x_3}{x_1 x_2 x_3} \\right\\},", "0ae946a990b8fb199c7f60f2bc27706b": "x \\geq 0", "0ae98f0799e3f806a4a35c66a5ccfd40": "(\\Omega, \\mathcal{F},P)", "0ae9b0da0724bc884adf9942c3c6f074": "\\begin{bmatrix}\na & b \\\\ c & d \\\\ \\end{bmatrix}^{-1} \\begin{bmatrix} 1 & 0 \\\\ 1 & 3 \\\\ \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\\\ \\end{bmatrix} = \\begin{bmatrix} x & 0 \\\\ 0 & y \\\\ \\end{bmatrix}", "0ae9d13aad0812ba6f090b3e7baf1383": "B[v'] = AB[u'] ", "0aea0f1ee68a135cbcff00702c976d7e": "q^n", "0aea24903360bfd846680a41c1291623": "O(\\log^2 p)", "0aea252426836658d80f6c6681ecd4a0": "M^*\\,", "0aea403a758eb15358c65d6a99385e2e": "\\mathbb{R}^n\\ni x\\mapsto \\Psi_r(x)=f(r^2-\\|x\\|^2)", "0aea552e9914182ab1da8e050e89082a": " \\int_{-\\infty}^\\infty x\\Phi(a+bx)\\phi(x) \\, dx = \\tfrac{b}{t}\\phi(\\tfrac{a}{t}), \\qquad t = \\sqrt{1+b^2} ", "0aea63c51c29649339627194050c41cb": "A=\\begin{pmatrix}\n2 & 1\\\\\n-1& 0\n\\end{pmatrix}.\n", "0aeab43d70097555ea9c86e0daa800fe": "{x^2 \\over a^2} + {y^2 \\over b^2} = 1 \\,", "0aeaba0177223bf6907edb6aa8811e88": "\\phi_1,...,\\phi_m", "0aeaf52485cd4a50b7ab767a3361a6cd": "w_i \\,", "0aec3c266f29dda82b652734669009af": "\\phi_{\\mu}", "0aec811717d3b2e514ec9f3639182d90": "l^a=g^{ab}l_b", "0aecdaad710b30cd6b6be0996cffe248": "a_n = a_{n-1} + a_{n-2}", "0aed50a25843e0b74b7a4b43ad0074f9": " \\operatorname{tr}[\\rho F(A)]\\, , ", "0aed50d0ba7fda9096b4b8a37447313e": "(\\lambda - k_1)(\\lambda - k_2)", "0aed8dbb14390f58a2145583d3230522": " E_{n}=E_{n}^{0}+ \\langle \\varphi^{0}_n \\vert V \\vert \\varphi^{0}_n \\rangle ", "0aee4f081c8ef3ab812df7e4bac5e97b": "i_2", "0aeec20960bed943019da0d278e35151": "p(11^3 \\cdot 13 \\cdot k + 237)\\equiv 0 \\pmod {13}.", "0aef083e866296e552fdfaab6985870b": "P\\land (Q\\lor R)=(P\\land Q)\\lor(P\\land R)", "0aef321f8dbf730edc1a9efd62d604ae": " C_i = \\sum_{j} u_{ij} B_j . \\quad", "0aef5e0e685b879acde1a40eca9a0064": " n=\\frac{1}{\\xi}(\\ln E_0-\\ln E)", "0aef8a8a93b67e85ecfdb7aa60b632ab": "E_c^{\\rm GGA}", "0aefe5e1e1d54d587fbd314141cc0804": " \\cos^{-1} \\left(-\\sqrt{\\frac{1}{15}\\left(5+2\\sqrt{5}\\right)}\\right)", "0af013a1d081a6e99a7721592d6b5555": "\\scriptstyle Z_\\theta\\,\\sim\\,\\mathcal{N}(0,\\,I^{-1}_{q(\\theta)})", "0af04b45f774ed65b2a68fed6a17daec": " \\vec R \\cdot \\vec s =0 ", "0af0e907e0626bac05dbaeb759954376": "= 2\\pi \\text{ rad}", "0af149732181b7aa0644109d228e7585": "r\\neq 1", "0af164866f37679d9b4ac8ae84fbb54f": " \\mathrm{Cu^{2+}_{(aq)} + 2 e^- \\rightarrow Cu_{(s)}\\ }, ", "0af1b25de9436c937f4fb90c3f969eb7": "M \\leftarrow M \\oplus (P_1 \\cup P_2 \\cup \\dots \\cup P_k)", "0af28a4718bf5b5f51f3f5243cb0be8a": "r_m = \\frac{a}{2} = 0.5\\cdot a", "0af2a3d0977d08a9a23b3c9103aadb2c": "HP_S(n)=HF_S(n)", "0af2c6231b40809e0630d39f985e4b33": " f:\\mathbb R \\to \\mathbb R; \\qquad f(x) = \\begin{cases} e^{-\\frac{1}{x^2}} & x>0, \\\\ 0 & x\\leq 0.\\end{cases} ", "0af32ee352608a6f3ad7fc3fe606415d": "F_{\\nu,\\mu}(x)=\\begin{cases}\n\\tilde{F}_{\\nu,\\mu}(x), & \\mbox{if } x\\ge 0; \\\\\n1-\\tilde{F}_{\\nu, -\\mu}(-x), &\\mbox{if } x < 0, \n\\end{cases}", "0af331690bd48325646f240e9e06b554": "\nV_R \\ = \\ G_{R}V_{in} e^{j\\phi_R}\n", "0af35b25271436e00741d88a7b6a6d0f": "\\nabla \\cdot \\mathbf{E} = \\rho/\\epsilon_0", "0af410b858f1e8cb088109e76b93195a": " FV = PV ( 1+i )^n\\, ", "0af4187e03f4bf1693e268465b811f3d": "1 - (37/38)^{35}", "0af521c5bbd8e2d5cc34cffd2e1b0a5a": "t = u + v j \\ ", "0af522d7c2423d04e4e821cf815d7136": "\\text{EVaR}_{1-\\alpha}(X):=\\inf_{z>0}\\{z^{-1}\\ln(M_X(z)/\\alpha)\\}. \\,", "0af52bad2bda865a62d72acc25d7d15e": "\\bar{f}", "0af576cb9c17bf59fcd123781de39882": "A, S1 (K_x (S0, response)", "0af5ddc24e0b7c7c747e11337ae5106e": "A^\\prime \\rightarrow \\epsilon\\, |\\, \\alpha_1A^\\prime\\, |\\, \\ldots\\, |\\, \\alpha_nA^\\prime", "0af5f6ade3b2bbc1d9af090a1880c854": "\\textstyle c \\in \\mathcal{C}", "0af61c2cd218420fc8773ff9ed193ead": "{(S\\ll 1)}", "0af61f6ca8fa904eacc0f2e5952c51b5": " \\and (S_6 \\implies (\\operatorname{equate}[A_6, n] \\and V[F_6] = n)) \\and D[F_6] = D[n]) ", "0af67b699bf07611f74c7577c40159e6": "\\begin{matrix} {r \\choose 1}{4 \\choose 2}{r - 1 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "0af682c0c49d58279b7f2208094cbde6": "\\sum_{n=1}^x \\tfrac1 n", "0af6b6d6fb20ccf078ac3c4efaa9aafb": " \\varepsilon = { v^2 \\over {2} } - {\\mu \\over \\left | \\mathbf{r} \\right |} ", "0af6f160224749ce767c375359dd8875": "(\\text{Gal}(K^{sep}/K),\\varinjlim C_L)", "0af74242b153e40a79fe7c46c0e8915e": "\\bigcup A \\subseteq A", "0af791063649141160e8541600194230": "G(x,Q) = \\sum_{k < Q} \\sum_{a \\bmod k} E^2(x;a,k) \\ . ", "0af7995eebdeb25623b49b2d1bce6a4b": "x_3 = \\frac{(x_1^2+ y_1^2) - (2 y_1)^2}{4(x_1^2-1)x_1^2 - (x_1^2-y_1^2)^2}x_1 = -1", "0af7b8b09e092976331a379614b1906f": "\\Delta N_{\\lambda, \\lambda}", "0af7d9f51ab88a4de70df7dd381cc568": "-x^2-680x+96000=0", "0af7e286ebf1d55836f19142ffe4c32d": "\\omega =\\omega_0 \\left[ 1 +\\frac{1}{4} \\left( \\frac{V_{ab}}{\\hbar \\omega_0} \\right)^2 \\right]", "0af80b42a67ff69ea41cade44b80951b": "\\Gamma^{'[k]i_k}_{\\alpha_{k-1}\\alpha_k}=\\sum_{j}U^{i_k}_{j_k}\\Gamma^{[k]j_k}_{\\alpha_{k-1}\\alpha_k}.", "0af828f433089744ada7372c4ccc44d6": "|k' \\rangle", "0af90d8295f0367f62ddee9a7553ce40": " \\lambda = {q \\over l}_+ - {q \\over l}_- = {q \\over l}(\\sqrt{1-v^2/c^2} - 1/\\sqrt{1-v^2/c^2}) \\approx {q \\over l}(1-0.5({v\\over c}^2) - 1 - 0.5({v\\over c}^2)) = -{q \\over l}{v\\over c}^2", "0af93a25047804006a820e8bea811de3": " \\sum_{j \\neq i} a_{ij} x_j = \\lambda x_i - a_{ii} x_i. ", "0af947790e837c1199b253f51c52ed5f": "\\mathfrak{so}(V,Q) = \\mathfrak{so}(n,\\mathbb C).", "0af94ac17566b5da4c12e08c0dc49576": "\\mathbf{s}(x)", "0af9aa8300133591b344c0cd9227f029": "s\\in\\mathcal C", "0af9ca61411c4da95162b8ff95fd8402": "e_i-e_j", "0af9f12c8548657b41134a5ec5e38075": "\\hat{\\alpha} = 1/N \\sum_1^N y_i, \\hat{f_j} \\equiv 0", "0afa3e281db513a77f0204adf2942569": " \\|x\\|^2 + \\|y\\|^2 = \\|x+y\\|^2. ", "0afa5079e7e1200e3e253d4027918f58": "\\vec\\xi = [x_1,..., x_n, v_1,...,v_n]=[\\vec x, \\vec v\\,]", "0afaa57c1dc6f8a4f014c512f3e60b30": "(\\phi',\\lambda')", "0afaafb20f551a18d0af89030a4fd52e": " \\Delta^0_2 ", "0afaed5df1696a5daba959c1d6f62856": "b_0(\\omega)=\\left.b(\\omega)\\right\\vert_{t=t_0}", "0afaf0c8a701bbac3d0f0cf86348b788": "ds^2 = \\left(v_s(t)^2 f(r_s(t))^2 -1\\right)\\,dt^2 - 2v_s(t)f(r_s(t))\\,dx\\,dt +dx^2 + dy^2 + dz^2.", "0afbf8970edabafe535647e3e34e9c92": " P(supp(\\hat{\\mathbf{b}}) = supp(\\mathbf{b})) \\rightarrow 1 ", "0afc88fbada83dff6bbe15f41712cc68": "d = \\frac{\\lambda}{2 NA}", "0afcd7155c174db5cac504e5c7fcd2d6": "i = 1,\\dots,m", "0afce10b59df85c7006c17ede18f0c26": "z=\\frac{R}{Z_0}=RY_0\\,", "0afcf299473f36e27995bc871d9ce7ca": "\\mu(n) = (-1)^{2x} = 1\\,", "0afd13b9b27529be38846c360f004b5a": "\\sum_{s\\in S_{-p}}u_p(js)x_s > \\varepsilon+\\sum_{s\\in S_{-p}}u_p(j's)x_s \\Longrightarrow x^p_{j'} = 0.", "0afd2c56d8ebe40e1e5bee5960694016": "\\frac {E\\varepsilon(t)} {\\sigma_0} ", "0afd5b870674d4efd365affbe8d23052": "\\sigma_c=\\limsup_{n\\to\\infty}\\frac{\\log|a_1+a_2+\\cdots+a_n|}{\\lambda_n}.", "0afde54cc308eaa5f496587f74b638ec": "n=x_1^k+\\cdots+x_\\ell^k. \\,", "0afe2bff60b793ff51934fd0aa5acbb6": "\\delta \\mathbf{Z}_0", "0afe7b9c1786b0de754b74542b65f408": "\\max_{0 \\le i \\le m+n} {(|u_i|,|v_i|)}\\le 2b^{9(m+n)}.", "0afe9b2a7b30304eb14249802957f8aa": "\nr'_8(n)=\n\\frac{(\\frac12\\pi)^4}{6}(n+1)^3\n\\left(\n\\frac{c_1(n+1)}{1}+\n\\frac{c_3(n+1)}{81}+\n\\frac{c_5(n+1)}{625}+\n\\dots\n\\right).\n", "0aff0e21abfc190bf229abb142b4a011": "Lu=a u_{xx} + b u_{yy}", "0aff737e794c23be86ae194d542a64be": "\\begin{align}\nt' &= t \\\\ \nx' &= x - v t\n\\end{align}", "0aff9a174a6ab7bfc3c979c9764b5874": "\\sqrt{\\hbar G/c^5}", "0affa9fd0694b61247359ab5acf58a21": "\\sum_{n=1}^\\infty q^n \\sigma_0(n) = \\sum_{n=1}^\\infty \\frac{q^n}{1-q^n}", "0affd5716e67407b2620c73849705fee": "(n,k,2t+1)_{\\mathcal{F}}", "0affe29f92bcca5fe28a947d53d7e709": "v(p + r,t) \\approx v(p,t) + E(p,t)(r) + R(p,t)(r),", "0b00abade0dfec7157808f254f284f9a": " \\ v_{1} ", "0b00ea88e6a73f75f8751591843d912f": "O(n^c),\\;00 ", "0b0b4bf40868b9a659f9d4417631dce1": "{\\rm ad} (x){\\rm ad} (x)(y) = [x,[x,y]\\,]", "0b0b4c7241b24789623828c7de52aff7": "F= F_2(q, P, t) - QP \\,\\!", "0b0b52d7a2c57333f60176e9b1891e61": "20 \\div 4=5", "0b0b556d31bcc75a0e259af55920a0ea": "K_s(A)", "0b0bb06e3244e6beeb99703fe680e1af": "\\textstyle \\rho_L ", "0b0bce7f66cd1f7fad91a9fc34772308": "\\,\\zeta(3/2)\\approx 2.6124.", "0b0c03a2f9d4447d282d026b4b11fcd4": "\\mathbf{F}_\\text{c} = \\mathbf{F}_\\text{g}", "0b0c07872c021789774d17e3a6ef13b0": " g(\\zeta)=f(\\zeta^{-2})^{-{1\\over 2}}", "0b0c1f24e1b8461dd05a0c2025a7b479": "z=\\frac{1}{2}\\cdot((a^2+1)\\cdot(a^2+b^2)^2+4\\cdot b^2)", "0b0c4ac9eb4f84101e896551bd342c40": "\\beta (b)", "0b0cb65efd353d5ee4b9f4db0f5829aa": "\\mathbf{p}(t)=m\\mathbf{v}(t)", "0b0d098dc2ae0d225a4d67eef246897e": " {\\Gamma(c=K^{-1})} = \\Gamma_{max} \\frac{K K^{-1}}{1 + K K^{-1}} = \\frac{\\Gamma_{max}}{2}", "0b0d73136bacf275838a54965c8b5aeb": "x = -a /\\omega^2 = -\\frac{eE}{m\\omega^2} \\, \\exp(-i\\omega t) = -\\frac{e}{m\\omega^2} \\sqrt{\\frac{2I_0}{c\\epsilon_0}} \\, \\exp(-i\\omega t)", "0b0d77792b0117723c77f1950e4e3487": "S = \\frac{1}{2}f(m) + f\\left(m + 1\\right) + \\cdots + f\\left(n - 1\\right) + \\frac{1}{2}f(n)", "0b0dae69b9ae2c69ce7ed51ebc2cc966": "C(\\alpha,0)", "0b0db187ca92b5f64873942717d227fe": "(S-K)^{+}", "0b0dd242bf350e6ef06f569229a0278a": "\\scriptstyle\\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} = \\sqrt{\\psi_1(\\alpha)\\psi_1(\\beta)-( \\psi_1(\\alpha)+\\psi_1(\\beta) )\\psi_1(\\alpha + \\beta)}", "0b0de107f276831321196bb7588887a1": "[fg](x) = f(x)\\cdot g(x)", "0b0e1bfa525f11f7e5d1e67eb3da2751": "\\textstyle e\\left(P, Q\\right) \\neq 1", "0b0ec3f1721a875d4742be25f204fd26": "\\mathbb{R} \\times \\mathbb{R}^d", "0b0f1e7540e2fdb8b8cd059e7e733ec0": "\\dot{\\hat{z}} = A(u(t)) \\hat{z}+ \\phi(y,u(t) ) - L(t) \\left(C \\hat{z}-y \\right) ", "0b0f8b3863d5d24a9a310a1bac13aa94": "Obs\\,", "0b1189ce7bd5f85b616515d10c845404": " f=\\frac{1}{2L}\\sqrt{\\frac{F}{\\mu}}, ", "0b11a19533849ad711d522ec0e7d02a2": " \\lim_{k\\rightarrow\\infty} \\, \\mu(T^{-k}A \\cap B) = \\mu(A) \\cdot \\mu(B) ", "0b11d212fd09a23a6854e90476d3479b": "U_F( x, 1) \\mapsto U_A(x, 1) , \\quad U_F(1, 0) \\mapsto U_A(1, 0).", "0b11d32f960aba9accc04cfa0000514c": "\\gamma=\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}", "0b120962493753d604c18b6afbf30ace": "T = \\frac{1}{2}\\left|\\det\\begin{pmatrix}x_B & x_C \\\\ y_B & y_C \\end{pmatrix}\\right| = \\frac{1}{2}|x_B y_C - x_C y_B|.", "0b1250212d4a214f1d3666745265e7d0": "\\exists x_r~S", "0b129a4834611f48173672ce51bb8811": "F=RT \\sum_i N_i \\left(\\ln\\left(\\frac{N_i}{V}\\right)-1+\\frac{\\mu^{\\ominus}_i(T)}{RT}\\right) ", "0b12bf82691ca5c6fe95e7720e56b7d0": "\\chi_N = e\\left(\\sum_{i=1}^{t} \\mu^{(g_i)}\\right), ", "0b12d021397567943d56d0b0ae311eb8": "e =", "0b1345f9433f6a6fd7a57ba8b2f3d241": "\nr\\dot{r}=-\\frac{\\kappa}{2}r^2\n", "0b135c45909648708d01224c30b7e96b": "a \\uparrow^4 b = a \\uparrow \\uparrow \\uparrow \\uparrow b", "0b1387add57864bff81a792d3d924760": "\\int_{\\Gamma} \\phi(g) = 0.", "0b13a818b2887173fbf96623d279f660": "\\aleph_{\\alpha}^{\\aleph_{\\beta}}", "0b1403b7ab144745401b0c4244ddcc6d": "\\phi(z,t_1,\\dots,t_n)", "0b14c77c94a8cc9dc95491b15e6c7bfd": "m=r", "0b14fa01ab6aa848696cce164e50621a": "\\scriptstyle 6\\times 6\\times 6", "0b156bc724c5cd9d72272527688af806": "\\bold{D} = \\epsilon_0 \\bold{E} + \\bold{P} \\,", "0b158224c3b168f9e94c1f9e67a5f673": " Z = \\frac{P\\, V_m}{R\\, T} = \\frac{1}{1-h}\\ - \\frac{A^2}{B} \\frac{h}{1+h}", "0b15aedcf31f894e268005e908009745": "0\\leq2\\beta<\\kappa", "0b15b424c248ea8ef83e0ff920366072": "\\bar{u^\\prime} = 0", "0b15c970ea8fba5c0d08b137adcd09a2": "\\Lambda^2\\mathbb C^m", "0b169eaa5bdca18c3e25237c486a9f26": "\n \\varphi_{tt} = a(\\varphi) [a(\\varphi) \\varphi_x]_x.\n", "0b16e46cbcd7a6cb6953cfdfd0e7cc35": "v.", "0b17104790bd48e99631fa516b1b6645": "(A_1 \\land \\cdots\\land A_n)", "0b17149f3dd48b885dad4c30a7fadfec": "\\tau_\\text{wind} = \\rho_\\text{air} C_D U_h^2, ", "0b176ad622b6f8b2b6183620a35df7a8": "x_i(t) \\ne 0", "0b177e9bdbcda67f6f25d3d599cfbfe1": "x = \\sqrt{c}", "0b1794d4203b483781426926429e00d8": "\\frac {P} {F} = \\frac { \\frac {1} {2} {\\dot m v^2}} {\\dot m v} = \\frac {1} {2} v ", "0b17b85ebf2a90b7dc0fde77abd8812f": "F'(\\mathbf x_0)", "0b17d47a86849b9a3f651fdbbb7ca717": " \\mathbf{d}_i^{[0]} = \\mathbf{d}_i ", "0b1807c6bd9a0a1112d19fe419539758": " e^{br \\epsilon} e^{-ar},", "0b1847e036f5a594981d76b99c6a1789": " \\langle P,Q\\rangle = \\frac{1}{2} \\bigl( \\hat h(P+Q) - \\hat h(P) - \\hat h(Q) \\bigr) .", "0b184877c6e0899328336741280a9e90": "\n\\begin{array}{lll}\n& \\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & . & . & . & . & . & . & . & . & . & . & . \\\\\n-5& . & . & . & . & . & . & . & . & . & . & . \\\\\n. &-4 & . & . & . & . & . & . & . & . & . & . \\\\\n. & . &-3 & . & . & . & . & . & . & . & . & . \\\\\n. & . & . &-2 & . & . & . & . & . & . & . & . \\\\\n. & . & . & . &-1 & . & . & . & . & . & . & . \\\\\n. & . & . & . & . & 0 & . & . & . & . & . & . \\\\\n. & . & . & . & . & . & 1 & . & . & . & . & . \\\\\n. & . & . & . & . & . & . & 2 & . & . & . & . \\\\\n. & . & . & . & . & . & . & . & 3 & . & . & . \\\\\n. & . & . & . & . & . & . & . & . & 4 & . & . \\\\\n. & . & . & . & . & . & . & . & . & . & 5 & . \n\n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n1 & . & . & . & . & . & . & . & . & . & . & . \\\\\n-5 & 1 & . & . & . & . & . & . & . & . & . & . \\\\\n10 & -4 & 1 & . & . & . & . & . & . & . & . & . \\\\\n-10 & 6 & -3 & 1 & . & . & . & . & . & . & . & . \\\\\n5 & -4 & 3 & -2 & 1 & . & . & . & . & . & . & . \\\\\n-1 & 1 & -1 & 1 & -1 & 1 & . & . & . & . & . & . \\\\\n. & . & . & . & . & 0 & 1 & . & . & . & . & . \\\\\n. & . & . & . & . & . & 1 & 1 & . & . & . & . \\\\\n. & . & . & . & . & . & 1 & 2 & 1 & . & . & . \\\\\n. & . & . & . & . & . & 1 & 3 & 3 & 1 & . & . \\\\\n. & . & . & . & . & . & 1 & 4 & 6 & 4 & 1 & . \\\\\n. & . & . & . & . & . & 1 & 5 & 10 & 10 & 5 & 1 \n\\end{smallmatrix}\n\\right ]\n.\n\\end{array}\n", "0b186e7690e82c74c7fcc8446c67a5e4": "a = u_0 < u_1 < \\cdots < u_n = b, \\ \\ t_i \\in [u_{i-1}, u_i]", "0b187a2caeb8dcb5e4e38d33793b0b9d": "S(x,y,z) = 0", "0b18bc91c985db1e22710041768fb9d5": "\\scriptstyle\\mathcal{L} \\,", "0b18d22e5edf55b62a2ec826ab799ba6": "f\\ x = y ", "0b190402dbfacc4178bb7c2bcca76cec": "\ni\\hbar \\frac{\\partial \\psi (\\mathbf{r},t)}{\\partial t}=D_{\\alpha }(-\\hbar\n^{2}\\Delta )^{\\alpha /2}\\psi (\\mathbf{r},t)+q^{2}|\\mathbf{r}|^{\\beta }\\psi (\n\\mathbf{r},t) \n", "0b191d8e502d1b16fe7fbf74b711e012": "\n \\tau_m = \\tfrac{1}{2}\\sqrt{A(\\sigma_m - \\tau_m) + B^2}\n ", "0b1926f473583e6b7c3460a01827be1c": "\\scriptstyle A \\,+\\, B", "0b19b03c6ffb49e6ac2c419bf31817a0": "|A \\rangle = A_1|1 \\rangle + A_2|2 \\rangle + A_3|3 \\rangle ", "0b19da354ed93cabf4cd7781329f3dd1": "x^{-\\alpha},", "0b1a533be95b1b1abea5c0d20cc691d3": "{2\\over21}+{1\\over6}={4\\over42}+{7\\over42}={11\\over42}", "0b1a5d6f4914c5dd650bef67457ba2ed": "\\chi(K)", "0b1b2df342f3cf67f5d967d906be56e6": "\\overline{k}[x_1, \\dots, x_n]", "0b1b59a2229ac29ef96c4b4c01758c20": "E \\in S", "0b1ba5f162a97256ee1aba011d38dd0c": "d(X, Y) := \\mathrm{e}^{\\delta(X, Y)} = \\inf \\{ \\|T\\| \\|T^{-1}\\| : T \\in \\operatorname{GL}(X, Y) \\},", "0b1bb108f11b346cf840484500226c0d": " (\\mathbf{A}-\\lambda \\mathbf{I}) \\mathbf{u} = 0 ", "0b1bb8268166ad39ceb4fdbcd7c30c49": "\\frac{1}{v}=\\frac{1}{Kv_\\mathrm{mon}}\\frac{1}{P}+\\frac{1}{v_\\mathrm{mon}}", "0b1bf50e8b457bb5238816a310cd7635": "{\\hat{q}_{{\\rm c}}}({r_{\\rm c}}),", "0b1bf8f751e8a48c9fea05c7304f54c7": "(r\\mathbf{a}) \\times \\mathbf{b} = \\mathbf{a} \\times (r\\mathbf{b}) = r(\\mathbf{a} \\times \\mathbf{b}).", "0b1c000fe85c0f3e9938e83c20f9fdd8": "m_L=\\frac{m_0}{\\left(\\sqrt{1-\\frac{v^2}{c^2}}\\right)^3},\\quad m_T=\\frac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}}, ", "0b1c0a77ea5f1a39ce043f64d46c8069": "E( B | A =a , A < B) = E( 2A | A =a , A < B) = 2a,", "0b1c1b3f6d23443e4dc99792808fe692": "\nH_{2^k} = \\begin{bmatrix}\nH_{2^{k-1}} & H_{2^{k-1}}\\\\\nH_{2^{k-1}} & -H_{2^{k-1}}\\end{bmatrix} = H_2\\otimes H_{2^{k-1}},\n", "0b1c5ac4999e9b049d613ca0ad43b59e": " {\\mathbf{e}_i}^2 = \\begin{cases} 1, & i = 1, 2, 3 \\\\ -1, & i = 4 \\end{cases} ", "0b1cd5c47bb3865e91fbb46cc8c20681": "\\mathfrak{so}_5", "0b1ce465bfa5bf6e560417157449ed11": "b = c", "0b1cfcb7584ecb0a02710f1d06f2c569": "\\mathrm{JKLMNOPQR} \\!", "0b1d4e7ae17a043fb95588eae39ea20a": ";u", "0b1d9b7f04b8175b2b2134c130293666": "\\Delta \\mathbf{L} = \\begin{bmatrix}1 & -1 & 0\\\\ 0 & 1 & -1\\end{bmatrix} \\mathbf{x} - L", "0b1dc1a409c63858538d4cddb4d6188d": " \\frac{\\partial \\langle H \\rangle}{\\partial \\pi_n} = \\frac{\\partial a_n}{\\partial t},\\quad \\frac{\\partial \\langle H \\rangle}{\\partial a_n} = - \\frac{\\partial \\pi_n}{\\partial t} ", "0b1deac5eb480e3d03542c9a319ab216": "c>s\\ge 0.", "0b1e12b1503353093abcfb4df37e6647": "\\|\\mathcal F\\|_{q,p} = \\left(p^{1/p}/q^{1/q}\\right)^{n/2}.", "0b1e1e805333aeee29d50936831b35ac": "\n{\\rm cov}(V,T)\n=\n{\\rm E}\n\\left(\n T \\cdot \\frac{\\partial}{\\partial\\theta} \\ln f(X;\\theta)\n\\right)\n", "0b1eaced739d5edca2d5694bdd8111f2": " k=1,2, 3, \\dots , N-1 \\,", "0b1f03115d54972b55cbf0f14c09b660": " G \\,", "0b1f72bba20f2d5396ec27233856c8b4": "v_1\\wedge\\cdots\\wedge v_k", "0b1f90cb65a1a1a9f2207c586e00c39e": " \\int_0^x \\frac{\\sin(\\theta)}{\\theta}\\,d\\theta = \\mathrm{Si}(x) \\,\\!", "0b1fba5cfa06e468385c1fd682070237": "x_1 \\in X", "0b2030c799283bb8285d536dba50d7e5": "x_i y_j = x_j y_i ", "0b2047f99518b5b6010d0665158c817d": "|C|", "0b2052cab85752cdcbe29fbaaf097cad": "\n\\overline{X}_1 - \\overline{X}_2 = 0.095.\n", "0b20a20785af243b973b8af20c3f80e3": "R=R_{s}", "0b20a45a98a411d873cc917db0fe4d43": "2.2 RC", "0b20c0fb32195402e02f40a20f33a1be": "F_2=\\frac{F_{load}}{\\left [\\frac{Sin(\\beta )Cos(\\alpha )}{Sin(\\alpha )}+\\frac{Cos(\\beta )Sin(\\alpha)}{Sin(\\alpha)}\\right ]} \\,", "0b214a069e513fbc215742275989029d": "A=D+R \\qquad \\text{where} \\qquad D = \\begin{bmatrix} a_{11} & 0 & \\cdots & 0 \\\\ 0 & a_{22} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\cdots & a_{nn} \\end{bmatrix} \\text{ and } R = \\begin{bmatrix} 0 & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & 0 & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n1} & a_{n2} & \\cdots & 0 \\end{bmatrix}. ", "0b2165ab0fac44657e0035f107ffcc9f": "\\mathrm{Nu}_D = 3.66", "0b21870c30fc7ebe105fd4fc5a5f80c1": "\\sigma:\\mathbb{C}\\rightarrow\\mathbb{C}", "0b219b5321ab8079aa13ea0f9349b0e9": " Q_3, R_3,\\ldots , Q_{r-2}, R_{r-2}, Q_{r-1} ", "0b21a666a81629962ade8afd967826ed": "x_{0}", "0b21b88be8cbda85d6f623b3486a48e8": "\\sum_{j=1}^J \\mu_j Q_j", "0b21e10c346c5a26e9cb6bf5c028a58c": "k(k-1)(k-2)", "0b21efde9fbdc6c709090d137e2bd8c3": "\\frac{Z}{R_0}=\\frac{R_0}{Z'}", "0b22047efa7cb78fd4d53cf973b83ba3": "\\, X_t = L X_{t+1}", "0b2209dade114962405c2de07e75bcbd": "f(x) = \\frac{1}{\\pi (1+x^2)}.", "0b2220be4d2cf621d428c7190f82354c": "l_1, l_2 \\text{ and } m", "0b226e2265605852cb52401ab8a65069": "I_{o_{lim}} = \\frac{V_i - V_o}{2L}D T", "0b22ce22af97047e180f675fea7c5657": "\\upharpoonright", "0b22db96440b0d85f4002b499b43f8ac": "\nmax_{i = 1}^{p}(w_i) + max_{i=1}^{p}(h_i g) + l \n", "0b230d76c19fa0d8afe6350e90b27891": "\\vdash \\Box P \\rightarrow P", "0b2355e380add9df4b5d69b2575f16e7": "-3*y^2+8*y-8*x+8*z=0", "0b235f7701bde6967516add87539a715": "N \\wedge M := N \\cap M", "0b23b1a155aaf147c0ab8ec166c7e9a3": "f_0(\\vec{x}) = - \\sum_{i=1}^n x_i \\log x_i ", "0b241a844bbaee5f5bb205f2cf859842": "\\sigma_y^2(n\\tau_0, N) = \\text{AVAR}(n\\tau_0, N) = \\frac{1}{2n^2\\tau_0^2(\\frac{N-1}{n}-1)} \\sum_{i=0}^{\\frac{N-1}{n}-2}(x_{ni+2n}-2x_{ni+n}+x_{ni})^2", "0b244b15ac660518a8f3bdca42320595": "\\biggl(\\int_S |g|^q\\,\\mathrm{d}\\mu\\biggr)^{1/q}.", "0b249e8fe5495ceb98df510cae5d402e": "f(q) = q^{-1}", "0b2529d7b282db3baa35e50b0cfee33d": "S' \\subseteq S", "0b253b148901d21647b9e9890fd4ca9c": " \\sum_{0\\leq k\\leq K} { {K \\choose k} { N-K \\choose n-k} \\over {N \\choose n} } = 1", "0b256f92caebeb2c8a38e5fadefbfa9a": "P = \\frac{\\omega^4}{12\\pi\\varepsilon_0 c^3} |\\mathbf{p}|^2.", "0b25be1699beddd6778cc201df0fba41": "a\\sqrt{D}", "0b264cf7209fa033b90cf8606430b55b": "a_1,\\ldots,a_n \\in A^n,", "0b268f2c5085ff9b8ed74480420e7055": "\\cot A = {\\cos A \\over \\sin A} ", "0b26e34f2880a5c5ef298bcac7b6c449": "\\operatorname{Ad}(g)", "0b2733a7de366b183dbe1e433a8ca467": "\\langle T\\varphi,\\psi\\rangle=\\left\\langle\\frac{\\partial\\varphi}{\\partial x_k},\\psi\\right\\rangle = -\\left\\langle\\varphi,\\frac{\\partial\\psi}{\\partial x_k}\\right\\rangle", "0b27ae7f4d381c49d68ead1b5f719039": "E(r)\\neq o\\left(r^{1/2}(\\log r)^{1/4}\\right),", "0b27aefb6806e8e8edc032737e607a7d": "({v_0+v_i})10^{pH_{i}} \\text{ vs. } v_{i^{ }}", "0b27bcdeadef678b4b0487d10460c0b1": "j \\ ", "0b27e1a137a4ec2b3549293e1ab5398f": "=\\left(R_\\mathrm{S}+r_{\\pi}\\right)\\left(1+\\frac{R_\\mathrm{E}}{R}\\right ) \\ ", "0b28276780ee2d33b2c731fbe038004b": "[H_\\lambda,H_\\mu] =0 \\text{ for all }\\lambda,\\mu\\in\\Delta", "0b28d0b65a19b37b3298372f6f6d4dc3": " \\int_a^b \\sqrt{E\\,u'(t)^2 + 2F\\,u'(t)v'(t) + G\\,v'(t)^2}\\, dt. ", "0b28df2945cc8dc8a74150a229542793": "\\dot{m} = \\lim\\limits_{\\Delta t \\rightarrow 0}\\frac{\\Delta m}{ \\Delta t}= \\frac{{\\rm d}m}{{\\rm d}t}", "0b28ebdcf58d96ce257dc32270881447": "f = \\frac{Gr^2}{r^4 + a^2z^2}\\left[2Mr - Q^2 \\right]", "0b2911d97f2720d8b916b1c7255dbae2": "\\hat a_i \\leq a_i + \\epsilon |a| ", "0b29236672ecf8ef0f1fefba7fdbcf03": "\\sigma_D^{(k)}", "0b2928186ff178ba04dd8784c4ed1ad5": "p(a,b)\\ ", "0b29372c93042a9778d4c1869d945604": "\\pi\\,\\mathrm{sr}", "0b2941f34bde6aca723264dc6c64094b": "\\mathbb{R}^3\\times(0,T)", "0b294e8cb1d79a1323265f5c34b4cff3": "D(af + bg) = a \\cdot Df + b \\cdot Dg", "0b2a80fa78d0d070c8411f415b431033": "h \\in C^\\infty_0(\\Omega),", "0b2ac02010ae83184d1a37e5929689f4": "\\int \\cosh (ax+b)\\sin (cx+d)\\,dx = \\frac{a}{a^2+c^2}\\sinh(ax+b)\\sin(cx+d)-\\frac{c}{a^2+c^2}\\cosh(ax+b)\\cos(cx+d)+C\\,", "0b2acc9ad0136806ac97d3cc12bfaef7": "\\mathbf{F}_{q^f}", "0b2b49e9804f7b52c60587afb300572c": " \\Phi(z) = 1-\\mbox{erf}(z/\\sqrt{c}) ", "0b2b5ea98816f56e406c3ff216cd0e6a": "{a_0\\gg 1}", "0b2ba2852a16ef75fbc23710d711e74f": "I_{L_{Max}}+\\frac{\\left(V_i-V_o\\right) \\delta T}{L}=0", "0b2c131ed63c888e310641b056a64224": "OOO\\ldots\\Diamond\\varphi", "0b2c15c15aa09b9d946fbf35557fc25c": "i=n ", "0b2c241d365373d87c151ed6b1c50a51": "|\\nu| \\geq q^{(1-\\varepsilon)k}", "0b2c4725c762c6fc48c6edfffe3567fc": "\n k_1 < f'(e)< k_2. \\,\n", "0b2c71e38fc18dc10349f40508b752ad": " \\mathbf{\\bar x}_{\\rm est} \\sim \\mathbf{T} \\, \\mathbf{x}_{\\rm est} ", "0b2ccfc2d7f24693e64cbcf1ea1071a0": "I(C) = E( -\\log (P(C))) = -\\sum_{k=1}^n P(k) \\log(P(k))", "0b2cfacb06a88f4c4c96ada16e8580d4": "(\\hat{c}-\\hat{a})", "0b2d0a60f25f2e288be79b26a25ed944": "\\pm 0.5", "0b2d18a5d69d90af5eb3db0804bbb92a": "R=3", "0b2d26737d13f4fa4b8a75846a2df0f5": "a/b \\le 0.6", "0b2d5654df91c906fc7f731dab39fb84": "G = V^\\mathrm{T} V", "0b2d59525fcbd14acf1da95562034670": "G(z) = \\operatorname{E} (z^X) = \\sum_{x=0}^{\\infty}p(x)z^x,", "0b2d6c04bfa3042333abfb7779813779": " I_{32} ~,~ \\Gamma_{a_1 a_2 a_3} ~,~ \\Gamma_{a_1 \\dots a_4}", "0b2dbd235cd4e740936f9103cf5a5e06": " \\dfrac{1}{\\sqrt{2\\pi}} ", "0b2e13518881b1b817f25f9fe8ac371a": "\\tau_p^\\mathrm{iono} = -\\kappa \\frac{\\mathrm{TEC}}{f^2} ", "0b2e182d1239ec313a46d5d8b05ef89e": "\\cos\\phi = \\frac{r^2 + s^2 - R^2}{2rs}", "0b2e4d7f11195b0b2adc3a2058dd1e6f": "J_\\nu'(z)=\\frac{J_{\\nu-1}(z)-J_{\\nu+1}(z)}{2}\\quad(\\nu\\neq 0), \\quad J_0'(z)=-J_1(z);", "0b2e80ed23221cbf673b007af29676d8": "\\lim_{x\\to\\infty} \\psi(x)/x=1", "0b2ede7c37dc72282a0531da2d02909d": "\\mathbf{a} \\times \\mathbf{b} = -\\mathbf{b} \\times \\mathbf{a},", "0b2f038ebbaa690589250668c770064c": "h, h^{p^2}", "0b2fc65104f2bce71803c9a066530ffc": "MN = \\begin{pmatrix}1 & -1 \\\\ 0 & 0\\end{pmatrix} \\begin{pmatrix}0 & 1 \\\\ 0 & 1\\end{pmatrix} = \\begin{pmatrix}0 & 0 \\\\ 0 & 0\\end{pmatrix} = 0", "0b2ffe0cb00ee49e1d122ebb2e80f784": "\\Phi=(\\Phi_A)_{A\\in\\mathcal{L}}", "0b3025a733cc51dcb352d22e28dd3566": "\n\\tau = r_{m} c_{m}\n", "0b305dcf5ee62ed1d76d1a44ae6c4ae5": "*(R_q)", "0b30634dfea3a37f1f9cc4772f74161c": "\\begin{matrix}\\frac{4 \\times 8}{1081} \\approx 0.0296\\end{matrix}", "0b308ee0f9295b615900fc3a82063522": "\\cos \\theta <0", "0b309b7db5e0cb99b8fdcbb8f71c1abd": "Y_{9}^{9}(\\theta,\\varphi)={-1\\over 512}\\sqrt{230945\\over \\pi}\\cdot e^{9i\\varphi}\\cdot\\sin^{9}\\theta", "0b30a0581f39b9fcbd67e2d38f39a725": "x^2=2x+1\\, .", "0b30addb86cc79931415ff76dff9bae0": "\\left(\\sqrt{1/21},\\ -\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "0b30d6ed670716039cab3ef99b08bcb5": "a \\in A,", "0b315a41c159493e2869b3c7d5743570": "m(f)>0", "0b315ab77a0fbabccfe3baad902f518b": "{\\Delta p}_D\\,", "0b319be72b5e417ebb4efd9ea4a20fc6": "x_1 = N_1/N = p_1 \\;\\;and \\;\\;x_2 = N_2/N = p_2\\,", "0b31d52116763ba9c239b8a0d733d4cd": "\\langle \\alpha, \\beta, \\gamma, \\delta \\rangle", "0b32123d093442bad32b4f36773bbf47": "\\mu_x + i\\mu_y", "0b322569cc973a3780b69dc16b13af02": "\\alpha, \\bar\\alpha", "0b323adbc99127564984a808b4fed4ed": " 1.5\\mu m", "0b3246ff754cb8eb28a5b9a60e3cd487": "\n \\alpha(\\omega) = \\mathrm{Im}\\left[ \\sum_\\lambda \\frac{F_\\lambda}{E_\\lambda - \\hbar \\omega - \\mathrm{i} \\gamma_\\lambda(\\omega)}\\right] \n", "0b3248d59bd55b92e5e5ca559f1913da": "\\left [t_i - \\tfrac{\\delta}{2}, x_j \\right ], \\quad \\left [x_j,t_i + \\tfrac{\\delta}{2} \\right ].", "0b327f127472e0d81eb45dfd30ab028b": "\\phi^*\\phi", "0b32ae4617c6b52a3827afa3b23dc49c": "\\frac{X^2}{2\\lambda a}+\\frac{Y^2}{2\\lambda b}+\\frac{Z^2}{2\\lambda a}=0.", "0b32b8738a92bd1d373d79c4b4aec78b": "f(x)=\\sqrt{1-x^2}", "0b32c5e7997c2b0bea66872bb01da630": " K \\partial_t^+ u_{n,i+1/2} + L \\partial_x^+ u_{n+1/2,i} = \\nabla{S}(u_{n+1/2,i+1/2}), ", "0b32e239ba54437908d9d6bce9e573f5": " l_w", "0b33216ab0930e7b71253505af6a7d24": "f_1 \\circ g", "0b3343255b4f896c721249e22ba2643e": "x' \\,", "0b338455700e4c0bbef2417dcb5fb529": "\\|u\\|_{C^{0,\\gamma}(\\mathbf{R}^n)}\\leq C \\|u\\|_{W^{1,p}(\\mathbf{R}^n)}", "0b3392aca95f6d2ad2fdbb923a565f99": "\\pm e_p\\pm e_q\\pm e_r\\pm e_s", "0b33ad1a7ecaef3570b7837bac55c923": "\\{ z \\in \\mathbb{C} | |\\phi(z)| > \\mathrm{e}^z \\}", "0b33ddd03325ac3d24e36a4c20cb02d6": "\\alpha \\! \\left( \\lambda \\right) =\\frac{\\partial \\log S \\! \\left( \\lambda \\right)}{\\partial \\log \\lambda}.", "0b33fe96f9ddcea346f1a8a850fff65b": "\\scriptstyle \\mathbf{I}_3", "0b34512ea29020e8a0bb065d93a272a4": " {q_2 \\choose 1} = k \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} {q_1 \\choose 1} ", "0b3485a4e4e29af51f0f664ace9e0527": "m_{ij} := \\mathbb{E}(x_{ij})", "0b34a4e99b8218212e4549762bf21d70": "\\mu_\\text{bkg}", "0b34af9e7e0db9ae77f28c47dfb8807c": "v_\\mathrm{N}", "0b34d783ab3b382d1d262ba78ef6ea39": "\\hat{A} \\hat{B} \\hat{C} \\hat{D} \\hat{E} \\hat{F}\\ldots ", "0b351ba74cf928742dc1137471703f6e": " \\frac{1}{2}(\\mathbf{ab} - \\mathbf{ba})", "0b352ac842981f494ea4e2412767cbaa": "i[p_0,x_i]=\\frac{p_i}{p^0}=v_i", "0b352cad00c6c287f6903f0763ea40b4": "~ d \\theta_n = n \\lambda,~ n=0,1,2,\\ldots", "0b35526473b06bf45ca6ff062a5ecc86": "\n\\begin{align}\nX^{\\rm VV} &= X ^{BS} + p_{vanna} X_{vanna}\n\\Omega_{vanna} + p_{volga} X_{volga} \\Omega_{volga} \n\\end{align}\n", "0b355326b4d0cea1c5ee21cac6cbc74f": "\\Lambda = \\frac{ch\\beta}{2\\,\\pi^{1/3}}", "0b355b7f4b9d7286c810baf9a7b152fe": " I = S \\sqrt{t}\\ ", "0b3576a0921e1559618e21809f63cfc1": "\\bigcap A_\\alpha^C = \\left(\\bigcup A_\\alpha\\right)^C", "0b35b203428ea307180fb6bad619a003": "\\dot{x}_2(t) \\neq 0 ", "0b362b7e68df25b49de5540e2be58c23": "Y^2\\sim\\lambda\\chi_1^2", "0b365b79213974e5d4470843ff99c69b": "2^{-32}", "0b3685434c1693d6ad9fbd407b57a6fa": "n(\\vec r )", "0b36b2ead10e88ae4f79925bad77ee3a": "\\, \\log 2 \\,", "0b36ee693126b34b58f77dba7ed23987": "\\textstyle i", "0b36f11bd33dd8abb296800e2f340e2a": "a^{b^c}=a^{(b^c)}.\\,", "0b3710ff4f482383b3fdfb9c08ff44f1": "\\mu(A) = \\infty\\,\\,", "0b373fc6150d7af6e2b8e3d5677ffefd": "g: S' \\to S", "0b379ea6857d56503227628fe9e5d78b": " p_n = P(Y=n) = e^{[-a_1+a_2]} \\sum_{j=0}^{[n/2]} \\frac{a_1^{n-2j}a_2^j}{(n-2j)!j!} ", "0b37ca40516473c15b5227f99db7bf3f": "m_0 = \\frac{E_0}{c^2}\\!", "0b38606109a505d6fa951887c28ce859": " e: G_1 \\times G_2 \\rightarrow G_T ", "0b38675dbc370a7e1c0abe8aae2b59a0": "\\mathbf{\\lambda}", "0b38a9b62f4c408a78614a8169b2bad6": "P=\\frac {2St} {D}", "0b390514b9b7c4aca74936148d1661fc": "q=\\sqrt{g}", "0b39aed8dcb38e489628d79d43b2a78d": "k(n)=\\Theta (\\log m'(n))", "0b39af0e243d684bd94af62623260d40": "A^B = \\bigcup_{L \\in B} A^L", "0b39cba96270c26a1425484611812e54": " [X; Gr_n]", "0b3a84d5453bb30accd25ea4fe20d064": "\\triangle ABC \\cong \\triangle DEF\\,", "0b3aed796f2182cc70231e1537b50131": "\\Im e^{i\\omega}=\\sin \\omega,", "0b3be80ee121f0032102a9aaefaf10d3": "\\frac{(-1)^{a-1}\\psi^{(-a-1)}(x)}{\\Gamma(-a)},\\,a\\in\\mathbb{Z}^-", "0b3c2b8272ca73ae55cef750a165bd89": "I(X_1; \\cdots; X_n) = I(X_1; \\cdots; X_{n-1}) - I(X_1; \\cdots; X_{n-1}|X_n),", "0b3c45e4a2d5247ce8e024fcb23a86ba": "\\sum_{n=1}^\\infty \\frac{1}{n}=\\infty.\\!", "0b3c465dd4c4bded3b4005541f0b24e0": "S_F(z)", "0b3c7780ec18ee6ee6e132740571cc25": "\n \\begin{align}\n \\boldsymbol{\\nabla}\\mathbf{v} & = \\cfrac{\\partial v_r}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_r + \n \\cfrac{1}{r}\\left(\\cfrac{\\partial v_r}{\\partial \\theta} - v_\\theta\\right)~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta + \\cfrac{\\partial v_r}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\cfrac{\\partial v_\\theta}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r + \n \\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r \\right)~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta + \\cfrac{\\partial v_\\theta}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\cfrac{\\partial v_z}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_r + \n \\cfrac{1}{r}\\cfrac{\\partial v_z}{\\partial \\theta}~\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta + \\cfrac{\\partial v_z}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\n \\end{align} \n ", "0b3d21ed288a486099e963cd716b9268": "|\\psi\\rangle = \\sum_{j=1}^{N}b_{j}|j\\rangle,\\quad b_{j}\\in\\mathbb{C}.", "0b3d7aae4fb2032a621d138e96a85ff5": "\\pi^{s}", "0b3d86190245fdcc18dfb6bc9bdbbf90": "1", "0b4370ad219b723cc1117596a6f937e2": "f_c(k,r)", "0b437bd59162b6fbe27c269ce648424d": "(\\xi,\\zeta)", "0b43a0ebe087869e7d6c770de9c77507": "x_i^{\\beta_i}", "0b43e70dc01855a0d0dffbffb9da3ca5": " Y = 1000 ", "0b448bad3764cfcb21d9270815ef81d3": "\\mathbb{Z}^d", "0b449bc28c66557d4cac6dff2f4ef27d": " \\Delta (r\\#h)=(r^{(1)}\\#r^{(2)}{}_{(-1)}h_{(1)})\\otimes (r^{(2)}{}_{(0)}\\#h_{(2)}), \\quad r\\in R,h\\in H.", "0b4538c7253805be3412ba2226faaf93": "\\Phi_{3\\times 7\\times 31}(x)", "0b455d59c5f3d4aa7111d861ea44780a": "x+y+z=1 ; \\,\\! ", "0b45f53446f36f1b1f94f63de8b50840": "1 \\over 6", "0b46807325355c03ae4b3a9cc181a459": "\\scriptstyle f,g \\in C^1(\\Omega)", "0b46e4979f9aa9cd935e01ba29bd035c": "-i(r\\bar{b}-b\\bar{r})/\\sqrt{2}", "0b46e7e78079210bcd1e16f7b0936632": "P\\in\\Pi(A),", "0b46f7199cbd1d1daf0700d93bcebcbe": "K(z)=(2\\pi)^{(-z+1)/2} \\exp\\left[\\begin{pmatrix} z\\\\ 2\\end{pmatrix}+\\int_0^{z-1} \\ln(t!)\\,dt\\right].", "0b47b135b10f2bc90fa7fcfb3c9fb999": "\\mathcal{T} = e^{-\\tau}", "0b47ee3a2a1fb6e2996273ae19a1b3e0": "\\displaystyle{\\widehat{Rf}(z)={\\overline{z}\\over |z|} \\widehat{f}(z),\\,\\,\\, \\widehat{R^*f}(z)={z\\over |z|} \\widehat{f}(z).}", "0b480b680753e6b32893cee15782b019": "\n \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) , \\left( B \\or C \\right) \\vdash \\lnot A\n ", "0b4820c7770e2712070011be456b5b48": "||{\\Phi^{[l-1..N]}_{\\alpha_{l-1}}}||", "0b483c7ddef11a9b4906fcc0b718739a": "s,t\\in B", "0b48592d55ea07026adfa4153d8ff58b": "f'(x) \\neq 0.", "0b485f9e9d945234b168961c69e322a8": "e^{i\\omega t}u,", "0b48cba4d29474c277c9654b1d414cda": "(Y_i)", "0b48d9c3258d89b22772e52a7b450263": " [\\eta,\\eta] ", "0b490cb3b069a46b5b1806fab6e72858": "1=-1", "0b497c52e8285d21db629e8a36475d36": "Ra=\\{ra \\mid r\\in R\\}\\,", "0b49aaba7dbc4888a9714a28ff623044": "\nR^\\lambda_{\\kappa \\mu \\nu} = 2 \\gamma^{\\lambda \\sigma} U_{, \\sigma [ \\mu}\\Psi_{\\nu]}\\Psi_\\kappa\n", "0b49dbb29aad13ea115b8958b56d2fdd": "\n \\begin{pmatrix}y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_m \\end{pmatrix} = \n \\begin{pmatrix}X_1&0&\\ldots&0 \\\\ 0&X_2&\\ldots&0 \\\\ \\vdots&\\vdots&\\ddots&\\vdots \\\\ 0&0&\\ldots&X_m \\end{pmatrix}\n \\begin{pmatrix}\\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_m \\end{pmatrix} +\n \\begin{pmatrix}\\varepsilon_1 \\\\ \\varepsilon_2 \\\\ \\vdots \\\\ \\varepsilon_m \\end{pmatrix}\n = X\\beta + \\varepsilon\\,.\n ", "0b49dd9c4d7c313032638ed763dbfefc": "m_{\\rm e} = \\frac{2 R_{\\infty} h}{c_0 \\alpha^2}", "0b49ef655fe7a8625dd45c3536dd2611": "\nQ_n(z) \\left(c_0 + c_1z + c_2z^2 + \\cdots + c_{m+n}z^{m+n}\\right) = P_m(z)\n", "0b4a58e47dc473931b0f945171c3ae5e": "\\begin{align}\nf_{X_1^n}(x_1^n)\n & = \\prod_{i=1}^n \\tfrac{1}{\\sqrt{2\\pi\\sigma^2}}\\, e^{-(x_i-\\theta)^2/(2\\sigma^2)}\n = (2\\pi\\sigma^2)^{-n/2}\\, e^{ -\\sum_{i=1}^n(x_i-\\theta)^2/(2\\sigma^2)} \\\\\n & = (2\\pi\\sigma^2)^{-n/2}\\, e^{ -\\sum_{i=1}^n( (x_i-\\overline{x}) - (\\theta-\\overline{x}) )^2/(2\\sigma^2)} \\\\\n & = (2\\pi\\sigma^2)^{-n/2}\\, \\exp \\left( {-1\\over2\\sigma^2} \\left(\\sum_{i=1}^n(x_i-\\overline{x})^2 + \\sum_{i=1}^n(\\theta-\\overline{x})^2 -2\\sum_{i=1}^n(x_i-\\overline{x})(\\theta-\\overline{x})\\right) \\right).\n\\end{align}", "0b4a6de4fb58a851b6b85c1830e882c1": " KE = \\frac{3}{5} E_F = \\frac{3}{5}\\frac{\\hbar ^2k_F^2}{2m_e} = \\frac{2.21}{r_s^2} \\textrm{Ryd} ", "0b4a767cc89a2f6babf4be5bf11bd46d": "\\tilde{E}_i^a \\mapsto \\tilde{E}_i^a / \\beta", "0b4aa42c8dbf765ee551b156555a8e05": "\\mathbb A^n_Y", "0b4b4f642cc158eb20ae3e76522b9580": "3\\omega", "0b4b56414ff00d2b87d0fa16dab2d83b": "\\hat k \\cdot \\nabla\\omega \\times \\frac{\\partial V}{\\partial p} = \\frac{\\partial \\omega}{\\partial y}\\frac{\\partial u}{\\partial p} - \\frac{\\partial \\omega}{\\partial x}\\frac{\\partial v}{\\partial p}", "0b4b8247106f143dbad733e6aa383a07": "z \\mapsto b z", "0b4b879fc55f9b929c29ccb03b53bbbf": "j_r = \\frac { -(1+e) \\mathbf{v}_r \\cdot \\mathbf{\\hat{n}} } { {m_1}^{-1} + {m_2}^{-1} + ({\\mathbf{I}_1}^{-1} (\\mathbf{r}_1 \\times \\mathbf{\\hat{n}} ) \\times \\mathbf{r}_1 + {\\mathbf{I}_2}^{-1} (\\mathbf{r}_2 \\times \\mathbf{\\hat{n}} ) \\times \\mathbf{r}_2) \\cdot \\mathbf{\\hat{n}} }", "0b4bb8d3371748c78de7498a9b6b87be": "z'_{k+1} = f'(z_k)z'_k", "0b4be7198491383c310e2a553098acfa": "\\langle j_1m_1j_2m_2|JM\\rangle", "0b4c6e3346811cd92440cbb7bf202fbf": "T=L_{0}/v", "0b4c9bdb81382bb918cf6d666f9d9ddb": "\\beta_2(T_e) \\approx 2 \\times 10^{-16} T_e^{-3/4} \\ \\mathrm{[m^{3} s^{-1}]}", "0b4d44d9509a20bd4952af13a21ed7d6": "F = \\frac{1}{4\\pi\\varepsilon_r\\varepsilon_0} \\frac{q_1 q_2}{r^2}", "0b4d4705a0400a9f7b0d4b0dcd7e9faf": "\\begin{bmatrix} A & U \\\\ V & C \\end{bmatrix} = \\begin{bmatrix} I & 0 \\\\ VA^{-1} & I \\end{bmatrix} \\begin{bmatrix} A & 0 \\\\ 0 & C-VA^{-1}U \\end{bmatrix} \\begin{bmatrix} I & A^{-1}U \\\\ 0 & I \\end{bmatrix}", "0b4d5ac429d333fac141e0fdddaaa11b": " Ar \\rightarrow \\exists xAx", "0b4d9331ae0dcc86b4d23eeb633c3fda": "v\\in T^0\\left(V\\right)", "0b4de37b4e42b7d92be622a3b145446e": "\n\\begin{bmatrix}\nR\\\\\nG \\\\\nB\n\\end{bmatrix}\n\n=\n\n\\begin{bmatrix}\n1.0498110175 & 0.0000000000 & -0.0000974845\\\\\n- 0.4959030231 & 1.3733130458 & 0.0982400361\\\\\n0.0000000000 & 0.0000000000 & 0.9912520182\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\\nY\\\\\nZ\n\\end{bmatrix}\n", "0b4e3a5cc248dd732bee56ba460c0b6b": "\\vert\\omega_n\\vert^{-s}", "0b4e6282f174e5939e96d35d7fc1befb": "\\varepsilon_{total}=(-1)^{n+1}\\prod_{i=1}^n \\varepsilon_i,", "0b4e73f935b9bd42529b784cdcf99784": "\\lambda_{k}\\alpha_{k}\\alpha_{k}'", "0b4e77b610ea5f8d13a1eb8730f8caa3": "P^{2m}\\,\\!", "0b4ed3dabb89aec97ad9fc9e37d7c145": " \\delta\\ ", "0b4f31fb64207229e6341ff4a9d08a6c": "m - M = 5 \\log_{10}d_{L}", "0b4f4100756a7a65689d964352244d61": "e^{{-E_a}/{(RT)}}\\ \\ \\ ", "0b4f9454cb92e7210ef2beabfbe33d26": "[micelle]", "0b4fcca6163464bf22afde8686b7c117": "\\omega,", "0b502b678c8955fca4be0e828fd25089": "m=K_1+K_2,\\ b=\\frac 1 2 \\frac {\\sum\\lambda_i} {\\sum\\lambda_i^2}", "0b5048058ec1d207710ccf28bec890e0": "\\displaystyle{P(p,q)= \\begin{pmatrix}A_{pp} & B_{pq}\\\\ 0 & D_{qq}\\end{pmatrix}}", "0b50f6f218d322b03688752cddd7a06e": "\\Delta w_{k}", "0b5102bc014ec7e93cff02cc06d3b726": " p_0 = \\left ( v \\rho \\omega \\right ) s_0\\,\\!", "0b511a3eab3f963551086c615faf80e2": " f(i)M^{k+1}(i,j) = \\sum^{N}_{n=0} M^k(n,i)(f(n)M(n,j))", "0b513d869d9c68fb6d1a500d9fc2c2dc": "V_{wd}", "0b51776edcd81842ab89bd8f0f1ebe9e": "X\\cdot\\alpha = \\begin{bmatrix}\nX_{00}\\,\\alpha & X_{01}\\,\\hat\\alpha \\\\\nX_{10}\\,\\alpha & X_{11}\\,\\hat\\alpha\n\\end{bmatrix}.", "0b51878fe9cfe8092c1aa6a72bfe5ce4": " D_2 \\psi = \\frac{B}{\\lambda+\\alpha} \\psi ", "0b5198913caacbfdc98b89688e427ab5": "C \\equiv C_i\\, \\bmod\\, N_i", "0b520835d59a4f7e03e4cdead48b4699": "x_{i+1}=y_{i+1} \\oplus y_i", "0b520e27e385332015d1e92702212db3": "\\lambda_k = \\sum_{j=0}^{N-1} c_j\\gamma^{kj}", "0b52664134c34dd9e3b7ae3525aa06c4": "L_M\\;", "0b52a7be011ff9fb54334bb6523f31fd": " d\\rho^2 = \\left( 1 + h^\\prime(r)^2 \\right) \\, dr^2 + r^2 \\, d\\phi^2, \\; r_1 < r < r_2, \\, -\\pi < \\phi < \\pi ", "0b52d9752d0a764ec95a732511164ada": "\\vec r(t)", "0b52e1a1a44a31894c9eb2851bb49b97": " time = \\frac{u_n {N_d}^2 \\ln 10}{4 \\ln \\frac{b_n}{a_n}} = \\frac{k u_n}{\\ln \\frac{b_n}{a_n}}", "0b53009eb6fc35da431ee63b576e77c1": "x_i = \\alpha z_i \\sinh(t/\\alpha) \\sinh\\xi, \\qquad 2 \\leq i \\leq n", "0b530e6fdb028c037d6b67e7166bc638": "d(x,y) = d(x+a,y+a)", "0b533ea0787ff734054ed35687d17304": " \\alpha = \\alpha^{(1)} + \\alpha^{(2)} + \\cdots + \\alpha^{(s)}", "0b5350bd741575d99a0e1baa2840bc71": "\\textstyle l \\le m", "0b5399a623ac967617a90fc060618075": "t_0 = t_f \\sqrt{1 - \\frac{2GM}{rc^2}} = t_f \\sqrt{1 - \\frac{r_0}{r}} ", "0b53a43234c81c50f99463240ae47f68": " \\mathrm{E}(T^2) = \\frac{2}{\\theta^{2}} ", "0b53aebb9f79193ab502fcbbaf438dfd": "- \\left( 2 \\dot r \\Omega \\right) \\hat{\\boldsymbol\\theta} ", "0b53fe0f482447857fb6f0c1a2121319": "V_{TS} = \\frac{\\rho_0 d_a^2 g}{18 \\eta}", "0b540cc31f049cb587b5465f6fb65fe7": "f'(t):=\\lim_{h\\rightarrow0}\\frac{f(t+h)-f(t)}{h}", "0b54404f2ddb7d5d03356d867c1745da": "k_p", "0b54410f57eda1a2fbb7c8d2f266b457": " \\frac{\\omega_s-\\omega_c}{-\\omega_c}=R, \\quad \\mbox{or}\\quad \\frac{\\omega_s}{\\omega_c}=1-R,\\quad \\mbox{so}\\quad \\frac{\\omega_s}{\\omega_c}= 1+\\frac{N_a}{N_s}.", "0b547e25226a90bd5ec6f39b54180e54": "\\gamma_\\mathrm L=\\sinh^{-1}{\\sqrt{ZY}}", "0b54a141cae1244e3e7cc5876e5fe4d4": "\\frac{D}{dt}J", "0b55181123778f7b165bd2c67171dec4": "[S]=[S]_0(1-v/[S]_0)^{t}\\,", "0b55c7c9724eaf5641500c5df814375a": "\\begin{alignat}{2}\ne_1 &= a_1^* \\\\\ne_2 &= \\left(a_1^*a_2^*a_3\\right)^*\\\\\ne_3 &= \\left(\\left(a_1^*a_2^*a_3\\right)^*\\left(a_4^*a_5^*a_6\\right)^*a_7\\right)^*\\\\\ne_4 &= \\left(\n\\left(\\left(a_1^*a_2^*a_3\\right)^*\\left(a_4^*a_5^*a_6\\right)^*a_7\\right)^*\n\\left(\\left(a_8^*a_9^*a_{10}\\right)^*\\left(a_{11}^*a_{12}^*a_{13}\\right)^*a_{14}\\right)^*\na_{15}\\right)^*\n\\end{alignat}\n", "0b55ca7e5c02e2883d6b470595d20717": "3 (d_1 + d_4 + d_7) + 7 (d_2 + d_5 + d_8) + (d_3 + d_6 + d_9) \\mod 10 = 0.\\,", "0b55f7629f65297f05e0f4b6cf7b2b11": "(-\\infty,a)", "0b560f9cf7061e89180ca4ef5732a69b": "\\mathbb{RP}^n", "0b561ee83d2021a504ecc6d70b590b94": "\\delta m^a=(\\beta-\\bar{\\alpha})m^a+\\bar{\\lambda}l^a-\\sigma n^a\\,,", "0b566ed25f29c5d76f3feb17babcb030": "\\mathcal{F}_i= - \\frac{\\partial \\mathcal{V}}{\\partial q_i}+\\frac{d}{dt}\\left(\\frac{\\partial \\mathcal{V}}{\\partial \\dot{q_i}}\\right);\\, ", "0b56a2cf40a6ac08de5e8556d874ccc9": "\\Phi(x) = \\emptyset", "0b575452d7dc1e9df545521752160a5a": "D(ab) = D(a)D(b) ", "0b579828e88795bf048c9819d87d22ed": " \\theta_0 = 2 \\psi_0. \\,", "0b57c07de6e0076273d1840bdecd2138": "f\\left( \\frac{x+y}2 \\right) \\ge \\frac{f(x) + f(y)}2", "0b5803e59a09d3fa383c8d3cd3b4024d": " N_W = {\\text{Transient Inertial Force}\\over \\text{Shear Force}}\\,\\!", "0b581483b18d4c189ca68d62d4535610": "\\begin{align}\n e_1(x_1,\\ldots,x_n) &= p_1(x_1,\\ldots,x_n),\\\\\n 2e_2(x_1,\\ldots,x_n) &= e_1(x_1,\\ldots,x_n)p_1(x_1,\\ldots,x_n)-p_2(x_1,\\ldots,x_n),\\\\\n 3e_3(x_1,\\ldots,x_n) &= e_2(x_1,\\ldots,x_n)p_1(x_1,\\ldots,x_n) - e_1(x_1,\\ldots,x_n)p_2(x_1,\\ldots,x_n) + p_3(x_1,\\ldots,x_n).\\\\ \n\\end{align}", "0b58212628b04c3c9aefdd3b9d441b8f": "M=S^3", "0b58448b0b751a3bde6ddb3fddeff234": " PLEASE ADD CATEGORY AND LANGUAGE LINKS TO THE /doc PAGE, ''not'' HERE\n", "0b58596743f39389da0f13f74d2ccdfe": "e^\\bar{\\alpha} = L^\\bar{\\alpha}{}_\\beta e^\\beta ", "0b58bc6627165700f64e10e69c59e764": "\\frac{\\left(x+\\sqrt{a^{2}+b^{2}}\\right)^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1", "0b594a8e6ab09a0103551d0aaf155aac": "\\det(cA) = c^n\\det(A)", "0b59500cb2c74858986609d125f13794": "(P,J,E)", "0b598f0913ad093d7637585a7c89e143": "\\Pi_i = \\frac{\\eta_i k_i}{\\sum_j \\eta_{j}k_j}.", "0b5aa703832eb3248df7472b3d7199a2": "\\nabla_{\\vec{e}_0} \\vec{e}_1, \\; \\nabla_{\\vec{e}_0} \\vec{e}_2, \\; \\nabla_{\\vec{e}_0} \\vec{e}_3", "0b5ab034c9c5ed41d722e7065cda0c8f": "11 ", "0b5c5e49946b8ec71fee24dabe7239ec": "\nG_T \\approx G_a + \\bar{b}^2 G_C + \\bar{C}^2 G_b\n", "0b5c82e45f633a4ca6b49dc55febcd9f": " B(x) = - \\log x +xB(1/x)", "0b5c84fc3e26501a42c5da7fde17b735": "-f(x)", "0b5ca8b5c58140508cdd62446744ed59": "X_{a,b}=Y^{1/a}_{1,b},", "0b5cb9bb91a9eb32198f4467839649c1": " 10^{21} -10^{24}", "0b5d1b8e0d8708d71a9d9899dd7d42d3": "T(t=0) = T_1\\cdot\\Theta(-z)", "0b5d5663844e90972c92593a9a98eba1": "\\mathcal{S}_{E}", "0b5d676e57a419922b786b5a4ddeec2e": "\\operatorname{codim}(N) = \\dim(M) - \\dim(N).", "0b5dbc95b984bf150d5815fec585407a": "4\\pi G=c=\\hbar=\\varepsilon_0=1", "0b5dc9a79b2441c4e04cc15ee2a3e2bc": "a_n(q), \\, b_n(q)=\\frac{2 m E l^2} {\\hbar^2}-\\frac{2 m^2 g l^3} {\\hbar^2} ", "0b5dd45570bf4d63b247b96df1293589": "O_{r}^{*} = \\min_{O_{r}\\in N_{in}} d(O_{r}, O_{n}) - r(O_{r})", "0b5df7868e30c503f4b88730220b67cb": "k \\cdot 2^n+1 ", "0b5df7f323131fea5e143e6b778c4aa3": "{\\hat{\\beta}}(q)", "0b5e023a7f4801d1a76947c84f319d42": "\\gamma \\to -\\infty", "0b5e14f4dac6e7a27ef9f5888c1ee8c3": "\\tilde{e}", "0b5e57a269146a42c483657ea3ecfec9": "C_{Di} = \\frac{C_L^2}{ \\pi e AR} ", "0b5eb6c22c5d5483a0696f1dcd6b8a35": "e=2^8+1", "0b5efb365d0830d1c5030b78ed56cb21": "\n p^H(r) = \\left(\\cfrac{3F^H}{2\\pi a^2}\\right)\\left(1 - \\cfrac{r^2}{a^2}\\right)^{1/2}\n ", "0b5f022d687df82e544eeca553fffd04": "\\sum_0^\\infty (-x)^n", "0b5f71fef68b47c962faaa14d15d9262": " D^{1/2}", "0b5ff567a1734093359d384a92bed018": "(l_1+l_2)^\\theta f(x)=f((l_1+l_2)x)=f(l_1 x)+f(l_2 x)=(l_1^\\theta+l_2^\\theta)f(x)", "0b6042563f051dfb18da92766c7d5f58": "x_0=\\frac{\\hbar k_y}{m \\omega_c}", "0b6050078de4499eb0912b8ee7027a27": " B[x / a] ", "0b605fdfebe77225086319d595c41a96": "v \\wedge w", "0b60878e9116c231af283a4395d0ca96": "Q = q^N/N!", "0b60ad9f2ad7854a48c85e3d36560585": "g = f\\circ\\psi", "0b60ccd69f728800c74993348d55560f": "\\, x \\ge 2. \\,", "0b615f2d18fce9a50f34e032b96d5273": "\n\\mathbf{u} =\n\\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle \\tilde{\\mathbf{e}}_{k}\n", "0b616b5b31e08d8ab7214ba854ff03a5": "2 \\cdot m\\ ", "0b61715d93ea950011b950efbbd3d865": "\\left[{n\\atop j}\\right]", "0b61c05a0a6fc25a93b91be417c22acf": "\\sigma \\times\\sigma", "0b61de5eb16898d82b661c53c8830b13": "\\Delta^2=0", "0b61fb4685f42449df062ba690b2a00b": " y \\in [0,1] ", "0b621240763da1148bf3bd77fe1675f6": " f(t) = \\int_a^t K(t,s)\\,x(s)\\,ds", "0b6214965799cc6f65709439e4ae45bf": "\\textstyle y_{i+1} = y_i + h \\text{Slope}_{\\text{ideal}}", "0b6261c1922d72efc5ad25e8f4ffc50b": "\\left(\\frac{-b}{2a},\\frac{-D}{4a}\\right)", "0b6264b77360383344976e727640d7a6": "H(z) = \\frac{(z+1)^2} {(z-\\frac{1}{2}) (z+\\frac{3}{4})}", "0b62b3d09050e46eb8811411db2e7747": "GCD(n,b) = 1", "0b62f4565ab2ee04fad0c8c7bbc9bf06": "\\left(a, z\\right)= \\left(c, x\\right)", "0b632de5206d36d5c867d33a9e8935b9": "M([\\Gamma(1)]) = \\mathrm{Spec}(R[j]) \\, ", "0b635bacdb4dd7d15eb8b395c9f46fab": "X = \\begin{cases}\n1 & \\text{with probability } p, \\\\\n0 & \\text{with probability } 1-p,\n\\end{cases}\n", "0b63b73a88c26b47925a9e302be22e1c": "N =", "0b64043437f8b6719eca36c49922cb52": "F_Y(y)", "0b641d1199526ae18b344c1d59ef5f81": "B_r^{p,q} = d_0^{p,q}(Z_{r-1}^{p-r+1,q+r-2}).", "0b64c093edf1d116179e53ae0f1bad3e": "{\\tilde{D}}_{6}", "0b64de3b3a7b9c3e1f7637cff37191db": "E = E_o\\;E_u", "0b64e5e33d57439d698c5f8e627f6d9c": "L^*G(x,s)=\\delta(x-s),", "0b64fc5c88797971e95f2a1c4db33015": "\\frac{ 1 }{ i }[M_{\\mu\\nu}, M_{\\rho\\sigma}] = \\eta_{\\mu\\rho} M_{\\nu\\sigma} - \\eta_{\\mu\\sigma} M_{\\nu\\rho} - \\eta_{\\nu\\rho} M_{\\mu\\sigma} + \\eta_{\\nu\\sigma} M_{\\mu\\rho}\\,", "0b654724b02f2476bb1732f98b89fc54": "Y \\subseteq X \\subseteq H", "0b65a75973388be6d2a7c13d69503df9": "\\liminf_{x \\to +\\infty} x^2 q(x) > \\tfrac{1}{4}.", "0b65d521b64b955ddb3c1db6ab4be245": "f(x) = f(p) + f'(p) ( x - p ) + o(x-p).", "0b65ea8ba9dcbae4c186be4ae3ebcc78": " (a_1 + ib_1) + (a_2 + ib_2) = (a_1 + a_2) + i(b_1 + b_2), ", "0b667a3a47daed6d7d1b1ecb6899d19b": "\n\\mu^+(E) = \\sup_{B\\in\\Sigma, B\\subset E} \\mu(B) \n", "0b668d48204bd73fe63c81584ee5c552": "\\langle n_i\\rangle =\\frac{1}{e^{(\\epsilon_i-\\mu)/k_BT}+1},", "0b66b723daa0490205546121416b5c06": "\\Pi_{H}(2m)\\,\\!", "0b66c2051c7a0333fb28047b33eeb19e": " \\mu(A) > \\mu (B) > 0. \\, ", "0b6717c5b70b6a8b60a6434579dd268a": "TM_{mnp}", "0b673d1dadfee24055406c633d5980b9": "n_{ij}=|X_i \\cap Y_j|", "0b674220ba0c38156ebcc2198df97600": "4\\sin^{-1}\\left({1\\over 3}\\right)", "0b675c6942f1cdda7c3ce4cedf3e60da": "\\mathbf{m}^\\phi", "0b67d686b322c47dbe48350b9102df4d": "\\frac1{16}\\int_0^1\\frac{x^{12}(1-x)^{12}}{1+x^2}\\,dx= \\frac{431\\,302\\,721}{137\\,287\\,920}-\\pi", "0b67d78e715d104395a4550d2565f36a": "C = \\cosh^{-1} \\frac {Q_b}{Q_t} = 3.0022", "0b67ed251272f8b32816542f9b4551bc": "\\ \\exists x \\in S", "0b6808e17f615a59c53ee277db89279d": "\\mu(t)=e^{\\int_a^t{\\Gamma_\\gamma(s)}ds}", "0b680cf4e5d86fa9fd7c424f84056884": "\\Omega^{p,q}=\\Omega^{1,0}\\wedge\\dotsb\\wedge\\Omega^{1,0}\\wedge\\Omega^{0,1}\\wedge\\dotsb\\wedge\\Omega^{0,1}", "0b6813a6b7451fa84910d4886bdec9f4": "P'Q'", "0b681556ac0e4af173508210691dc21e": " \\frac{\\partial\\phi}{\\partial t} = \\nu\\frac{\\partial^2\\phi}{\\partial x^2} + f(t) \\phi ", "0b688f26c5b321af582b842fc74dbdd5": "ds^2 = -q \\, \\sin(\\omega u)^2 \\, du^2 - 2 \\, du \\, dv + dx^2 + dy^2,", "0b68917c1a3c3f18b30876c87b711db2": "D(P_{\\mathit{max}}) = 0.", "0b689cfe83964e9a104a3235c7eb192c": " E = n_1\\epsilon_1 + n_2\\epsilon_2 + \\ldots + n_6\\epsilon_6 ", "0b693616955a1f920aa3d090850b3017": "\\nabla^4 A+\\nabla^4 B+\\nabla^4 C=3\\left(\n\\frac{\\partial^2 A}{\\partial x^2}+\n\\frac{\\partial^2 B}{\\partial y^2}+\n\\frac{\\partial^2 C}{\\partial z^2}\\right)/(2-\\nu),", "0b69e718c59eedc900655d007d0ccd13": "{f_n}(10)", "0b69ed85215d84927bc3d7f0c70a025e": "(S, \\times)", "0b6a19a2be7f050a5185b1c0b3235742": "\\hat{\\mu}_{MAP} = \\frac{n \\sigma_m^2}{n \\sigma_m^2 + \\sigma_v^2 } \\left(\\frac{1}{n} \\sum_{j=1}^n x_j \\right) + \\frac{\\sigma_v^2}{n \\sigma_m^2 + \\sigma_v^2 } \\mu_0.", "0b6a3da19681846c5ace240a86199720": "\\beta(2k+1)={{{({-1})^k}{E_{2k}}{\\pi^{2k+1}} \\over {4^{k+1}}(2k)!}}, ", "0b6a5514e51091b8fe805aa502d5282e": "\\Lambda_E", "0b6adf7a97ec05a201250417232edfdf": " 7^{7^7} ", "0b6b37065a9be1e67e3d7927eb156a68": "a_k^0=\\frac{1}{2 \\pi i} \\int_{|z|=c} f(z) O_k(z) \\,dz,\\!", "0b6b46cb74b06b90456888e1a8451871": "g(x) = \\sqrt[3]{x^2}", "0b6baecfc75c8101ee3c9f0caeb2ff00": "\\mathcal{M}_{g;k_1, \\dots, k_n}", "0b6bb2e6140acec75bc33da8062d4470": "S(\\lambda) = S_0(\\lambda) + M_1 S_1(\\lambda) + M_2 S_2(\\lambda)", "0b6bc1ef048fb3dfd69aefab41c05ded": "\\operatorname{V}_{\\mathbb{P}^n}.", "0b6bde769d746e35d252dc89cac92bb6": "\\left(\\frac{\\partial f}{\\partial r}, \\frac{1}{r} \\frac{\\partial f}{\\partial \\theta}\\right) \\cdot \\left(\\frac{\\partial}{\\partial r}(r - R), \\frac{1}{r} \\frac{\\partial}{\\partial \\theta}(r - R)\\right) = 0", "0b6c88979589e6a7ec2f6d64dd2b8339": "F_N(j)\\approx N/N\\approx 1", "0b6caf25d1ee05ee016a35872b511036": "\\limsup_{n\\to\\infty}x_n := \\inf_{n\\geq 0}\\,\\sup_{m\\geq n}x_m=\\inf\\{\\,\\sup\\{\\,x_m:m\\geq n\\,\\}:n\\geq 0\\,\\}.", "0b6ccccae14c323f947c9bfd908804a2": "\\textit{on}(1)", "0b6d0394794d86b24d024166d43159fa": "\\mathrm{slog}_a x > -2", "0b6d2e45ad5d4e6e032c3c4182efada7": "[P'_i,P'_j]=0 \\,\\!", "0b6d652bb1312a01a2433aaeee8cf45b": "i=j=k=l , \\alpha=\\beta=\\gamma=\\delta=0", "0b6dbb40ed7be2440d57913a546bd27f": "y=c \\cosh \\frac{v}{c} \\sin u", "0b6df40e3a9a5560b3b91fd653531c4f": "\\sum_{i=1}^n w_i h(x_i) = \\sum_{i=1}^n w_i r(x_i)", "0b6df66735fe57143f9af3d20fe74b0a": "\\Gamma^{[{{D]}}}", "0b6e1acfb23d9f0326d87794384872df": "\\scriptstyle{\\langle\\psi_m|\\dot{\\psi_n}\\rangle}", "0b6e1ff740e4d5e5de339a7270ee2e5a": "\\displaystyle{R^2 > 1+2 \\|H\\|_{2^n} R}", "0b6e4aed685ec6ab72efe70617e9f2d2": "q_0 > 0", "0b6e6884a8c9955bf578c90ac95c8094": "\\psi_{2,5}=1", "0b6e6cf649347bfd91ddcd676f030ab4": "\\alpha + \\beta := (\\alpha, \\beta) \\in A^{*}(X \\times X) \\oplus A^{*}(Y \\times Y) \\hookrightarrow A^{*}((X \\coprod Y) \\times (X \\coprod Y))", "0b6e80017f7bf943c4eb0cb0463aa820": "y'(t) = f(t,y(t)), \\ \\ y(t_0)=y_0,", "0b6e8bb0bcbe5c5fd93a2b945bb71c99": "Q (q) = - \\frac {\\hbar^2}{2m} \\frac {\\Delta R}{R}", "0b6f1c18d140564c9a02b66c0079751a": "S_\\rho = \\frac{1}{2}\\varepsilon_{\\lambda\\mu\\nu\\rho} U^\\lambda J^{\\mu\\nu} ", "0b6f6c0f23cf3b29f3652c7315c456aa": "\\textstyle V", "0b6f706c387e5fc4de34329b85cda7bd": "b_i^*(k)=\\frac{\\sum^T_{t=1} 1_{y_t=v_k} \\gamma_i(t)}{\\sum^T_{t=1} \\gamma_i(t)}", "0b6fff4df0982015d7628a4ad461a31b": "d\\mathbf x= d\\mathbf{X}+d\\mathbf{u}\\,\\!", "0b7006de07931805fce1f3b177cfde1c": "\\operatorname{PSL}(2,\\mathbb{R})", "0b700c6fee8029dc0635590c37e78470": " \\delta W = M\\delta \\phi,\\!", "0b7014a25bef20e32e159157ef1847c8": " S^k(U) = \\{ f \\in C^{\\infty}(M,N) : (j^kf)(M) \\subseteq U \\} . ", "0b70554efd6d108f1e593b3d4debf190": "2 : nat", "0b705b005b9a77aaac8462a1a3379c92": "X^{n-1}\\cup e^n", "0b7069acf3cda4e07c03dba565a9c3ba": "x_\\text{upper}", "0b7098a64c3819d855cc94ec732d79eb": "t_b= \\cos i \\ \\cos u\\,", "0b712cbfd644933a8a95708263367dc2": "\\exp \\mathfrak{m}_+ ", "0b7197879e05bde29b9e1ba8b41a70c9": "\\int_{-\\infty}^\\infty f_X(x) L_{X\\mid Y=y}(x)\\,dx", "0b71bbe9511a2e0f3f6b8a00ec9ea7f4": "\\lambda = 1/\\tau.", "0b723599354b5f36b6100668d08d606d": " \n L_n = \n \\begin{cases}\n 2 & \\mbox{if } n = 0; \\\\\n 1 & \\mbox{if } n = 1; \\\\\n L_{n-1} + 2L_{n-2} & \\mbox{if } n > 1. \\\\\n \\end{cases}\n", "0b725c6bb5d13b891fdec55fad18b172": "=\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)-\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)", "0b72709f27ee0cd5d5051072ad206763": " \\langle a_{i}, x_{k} \\rangle = b_i ", "0b7293253c6ac4331548948b2ab33645": " B = (b_1, b_2, \\dots , b_n)", "0b72a3031b3ddc7646abde00a44ee8a9": "\\sum_{i \\mathop =m}^n a_i = a_m + a_{m+1} + a_{m+2} +\\cdots+ a_{n-1} + a_n. ", "0b72b5bb2041cc2cc46cff117dfdf333": "g = (A \\to B, B^{*}A^{+})", "0b72f70632ea534fd5aa33b8b6e87dda": "3(2P) = 4P \\boxplus 2P", "0b731d2ebd143daed5111a68cffef1b3": "m_s", "0b7355a770d945d1a7487c6267c36192": "x = x_1x_2\\cdots x_p", "0b7360f59db9ca086f599d3cdf738fd5": "\\mathit \\Gamma = \\frac {jX_\\mathrm L - R_\\mathrm 0}{jX_\\mathrm L+R_\\mathrm 0}", "0b738bfd1c9f8e654b72d49014ae3ee0": "R_S=R_H \\left(1-\\frac{\\sin(\\alpha)\\sin(\\theta-\\alpha)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "0b739b38d8cd7055c8303f36736548e6": "E^\\perp=(\\operatorname{Sp}(E))^\\perp=(\\overline{\\operatorname{Sp}(E)})^\\perp.", "0b73e3d244b0fe91f14699654c42f621": "\n\\begin{align}\ndA_\\text{ellipse} &= \\det \\begin{pmatrix}\n \\frac{\\partial \\bold{T}}{\\partial r} & \\frac{\\partial \\bold{T}}{\\partial \\theta}\\\\\n \\end{pmatrix}\n \\,dr\\,d\\theta \\\\\n &= \\det \\begin{pmatrix}\n a\\cos\\theta & -ra\\sin\\theta \\\\ \n b\\sin\\theta & rb\\cos\\theta\n \\end{pmatrix}\n \\,dr\\,d\\theta \\\\\n &= abr\\,dr\\,d\\theta.\n\\end{align}\n", "0b742933011d816b807ddc248f5defc4": "x(\\theta)", "0b748af62424599c585ce90450e45bc7": "\\mathbf{{\\Sigma}} = \\lambda_{1}\\alpha_{1}\\alpha_{1}' + \\lambda_{2}\\alpha_{2}\\alpha_{2}' + ... + \\lambda_{p}\\alpha_{p}\\alpha_{p}'", "0b74b6b66713946fddaf3eebd0507ab5": "\\langle f^* \\star f \\rangle =\\begin{bmatrix}a^* & b^* & c^* \\end{bmatrix}\\begin{bmatrix}1 & \\langle x \\rangle & \\langle p \\rangle \\\\ \\langle x \\rangle & \\langle x \\star x \\rangle & \\langle x \\star p \\rangle \\\\ \\langle p \\rangle & \\langle p \\star x \\rangle & \\langle p \\star p \\rangle \\end{bmatrix}\\begin{bmatrix}a \\\\ b \\\\ c\\end{bmatrix} \\ge 0.", "0b74ea3628ef1e7160808180f41be5ca": "1 \\text{ rad} = 1 \\cdot \\frac {180^\\circ} {\\pi} \\approx 57.2958^\\circ ", "0b74f2b2b4353057526a60a67b01c84c": "\\sinh(\\mathrm{arsinh}(1/\\varepsilon)/n)", "0b7509c4a2f2085cd2ee34085f371766": " t_{1/2e} = \\frac{t_{1/2p}\\times t_{1/2b}} {t_{1/2p} + t_{1/2b}}", "0b754285f5d11bc7d1e3e08682f96f9e": "\\begin{align}\n y_{11} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_1}{V_1} \\right|_{V_2 = 0} \\qquad y_{12} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_1}{V_2 } \\right|_{V_1 = 0} \\\\\n y_{21} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_2}{V_1} \\right|_{V_2 = 0} \\qquad y_{22} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_2}{V_2 } \\right|_{V_1 = 0}\n\\end{align}", "0b756e98d772b24e85f9f7ecfc502035": "T_i^{(n)} =\\sigma_{ji}n_j\\,\\!", "0b7587cb6f3b2d1d5ec17aa0b2be1880": "\\tfrac34\\tbinom63", "0b75b1c42c8e3ffbda0ee14d1b055a72": " \\gamma = 1 / \\lambda", "0b75c1483fe9060ba5169500b2554896": "(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.", "0b75fbfb66a0bee29dd40f2e567fdd0e": "[f(x_1)\\dots f(x_{l+u})]", "0b769abfd7b91b02f417ef457003f14f": "C_{ij} =\\frac{\\partial Q_{i}}{\\partial V_{j}}", "0b76b99e897d365e388fc443ee3d0cf4": "\\wp'(z)", "0b76baf1bddbd75b6771ed733ad611e5": "\\hat{Y}(X_{0})=\\alpha (X_{0})+\\sum\\limits_{j=1}^{d}{\\beta _{j}(X_{0})X_{0}^{j}}", "0b76be8e813b7522591d66b7ffcf6e5c": "u_k = M^{-1} r_k,\\,", "0b76c38abc9a752466ce2f1749b19229": "\\frac{f(n+1)}{f(n)}=\\frac{1}{2}\\,", "0b7762b81b67ed61e897562de92407e2": "(x_1,x_2,x_3,x_4) = (x,y,z,ict)", "0b777b7178383d7c06b8516b53eaae5d": "R =\\left| \\frac { \\left(1 + y'^{\\,2}\\right)^{3/2}}{y''}\\right|, \n\\qquad\\mbox{where}\\quad\ny' = \\frac{dy}{dx},\\quad y'' = \\frac{d^2y}{dx^2},", "0b779dfe034244f21565f934f8e5bc13": "H^h_\\varepsilon f(\\zeta)={1\\over \\pi i} \\int_{|H(z)-H(\\zeta)|\\ge \\varepsilon} \\frac{f(z)}{z -\\zeta} dz= {1\\over \\pi i} \\int_{|H(z)-H(\\zeta)|\\ge \\varepsilon} {f(z)-f(\\zeta)\\over z -\\zeta}\\, dz + \\frac{f(\\zeta)}{\\pi i} \\int_{|H(z)-H(\\zeta)|\\ge \\varepsilon} {dz\\over z -\\zeta}.", "0b77a0c57e4d83ea934d4ef032e10c08": "[0,c]", "0b77d91370c91e27e44676e02c7678f7": " \\hat{\\sigma}^2 = 0. ", "0b77df392dd262ffe4f22b543de6bd5e": "\\mathbf{a}\\cdot\\mathbf{b} \\equiv a_i b_i", "0b780ab920f0fce6bd1baaac1e639265": "\\vert \\pi^0\\rangle = \\frac{1}{\\sqrt{2}}\\left(\\vert u\\overline {u}\\rangle - \\vert d \\overline{d} \\rangle \\right)", "0b783ec3ce75c6e1e4f7984140e672fb": " \\rho = {2r \\over {na_0}} ", "0b78429649cb061c397b9a4325f68362": "C_1:=\\{z:|z|=r_1\\}", "0b785832743f30eccf333e693a61e1d5": "\\hat F[\\{\\phi_j\\}](1)", "0b785bc8ff8d426af7a1c5f01a0937b8": " \\delta ", "0b785f8d039c5349d836f063bfa934e7": "I_P=\\frac{V_P}{|Z|}", "0b78a36f0e1ff678ddef51be448e179f": "\\pi\\models\\phi", "0b78e62c7228e3fc96d7a3fcef9c4e5e": "\\begin{align}\nf(x; \\mu,\\sigma) \n& = \\frac{1}{x\\pi\\sigma \\left[1 + \\left(\\frac{\\ln x - \\mu}{\\sigma}\\right)^2\\right]}, \\ \\ x>0 \\\\\n& = { 1 \\over x\\pi } \\left[ { \\sigma \\over (\\ln x - \\mu)^2 + \\sigma^2 } \\right], \\ \\ x>0\n\\end{align}", "0b78f77ae85a50f0ced942e1148f80ae": "PG(2,q)", "0b7982431e7175ff473e7c4d291ee43f": "F(x) = Z_G(f(x),f(x^2),f(x^3),\\ldots,f(x^n))", "0b7993280f4518243cd5d52453c7a69d": "r(f,D)=\\inf_{g\\in C}\\{\\|g\\|:f+g\\notin S(D)\\},", "0b79aed814b1fbd7a838366fb80481fb": "\\theta=\\frac{\\pi}{2}-\\phi;\\,\\!", "0b79c106592d3d9bba9529ec85298e70": "\\tan(7\\pi/16)", "0b7a6f1d3ae1e25d27ed71bb50718eff": "[f_{j}^{\\dagger},f_{k}] = 0", "0b7a7742cb4d88af0ca594abda1becad": "V_n(P,Q)\\,", "0b7a957ae7ea7b4094d82ccf74915758": "F = \\frac12 \\times \\rho \\times S \\times C \\times V^2", "0b7ab652df8f24992ff5bf2958e35191": " \\frac{1}{m(n - 1)}E(SSW) = \\sigma^2", "0b7aec5c0e36c799bc6480a697db16fc": "EL(\\Gamma_2)=0", "0b7b4ee416bb8bde0d91fa0a499bec10": "\\displaystyle{\\Delta V_{bat}}", "0b7b6ee857e6eebe95609fb5aeff1347": "\\frac{-\\partial I}{\\partial \\mathrm{z}} = 2\\alpha I= Q", "0b7bc128112f0449f6454d03f3870350": "\\, n \\times n", "0b7c0e06c0b65ed9f714989a493c1276": "P_5 < 0.00009", "0b7c3549265ec90d580049f054047889": "\n\nRl = (3,14) nl \\!\n\n", "0b7c3b28f45eb152b42b115ba89875c0": "l_\\mathrm{tot~core~run}", "0b7c4d88068200f067fe8a4df30139d2": " \\{y_i \\}^n_{i=1} ", "0b7cb79dd1db776ff58adf439972f641": "U \\circ \\text{hom}(-, -) \\simeq \\text{Hom}(-, -)", "0b7cc55e9b054c615ea097a5b07a1f5b": "\\tan{\\frac{A}{2}}\\tan{\\frac{C}{2}}=\\tan{\\frac{B}{2}}\\tan{\\frac{D}{2}}=1.", "0b7cc6cabd3ea8f34e93e77c6a5a44e6": "|V_n(i)| = \\binom{n}{i}", "0b7d3ac30d3a62125e3d564a55b96f75": "J_{k} = \\frac{\\lambda}{k} \\frac{\\sin k\\theta^{\\prime}}{\\left( \\rho^{\\prime} \\right)^{k}}.", "0b7dafc26b990d2ded1472b3337dc000": "\\scriptstyle \\cup_{i=1}^n A_i", "0b7dbfa54fa2d188a0f1dfe6968ecbf8": "\\alpha_\\xi = \\alpha_i(x,\\xi) dx^i|_x\\in T_x^*M", "0b7e1bac8263678a8aa543b985365a34": "a_R\\approx -4.5 \\times 10^{-6}(1+r)A/m", "0b7ecd901143f6121e34a8323f58b464": "{S^a}_b \\, {S^b}_a", "0b7f14da27fad5a8049557711a714987": " (E_a)^2 = ((m_k c^2)^2 + (m_t c^2)^2 + 2 (m_t c^2) (m_k c^2)) ", "0b7f5aed32134c95ca4ec4859df2b923": "\n \\gamma_{zx}^{\\mathrm{core}} = \\tfrac{2h + f}{2h}~\\gamma_{zx}^{\\mathrm{beam}}\n ", "0b7fc2ac7312ef72e7549d8a0bc98e61": "\\beta = 1 - \\tfrac12 \\gamma^{-2} - \\tfrac18 \\gamma^{-4} - \\tfrac{1}{16} \\gamma^{-6} - \\tfrac{5}{128} \\gamma^{-8} + \\cdots", "0b7fcfec239e74d9c3829893de187094": "(x_c,y_c)", "0b7ff5942359e394540ad9acc83a0a0a": " \\mathrm{sinc}(0)=1 ", "0b806bcecbe657d1d1cc139c80bd037b": "n_e < n < n_o", "0b8075730eb10568cc1cbf279b05e692": "1/r \\ll 1", "0b80ab2cbc0acd7a8415a3ba85284cef": "f_\\mathrm{red}\\,", "0b80aceff873fa66f50b5ff8c18e41be": "\\big|E(AC'|\\text{coinc.})+E(AD'|\\text{coinc.})\\big|+\\big|E(BC'|\\text{coinc.})-E(BD'|\\text{coinc.})\\big|\\le \\frac 4{\\eta} - 2", "0b80afaa5a8348cf96fdc709e5d4a1f3": "r = f\\theta_c", "0b816267900f6a2442395fc2df8bf8d9": "\\delta \\leq \\tfrac{n-1}{2}", "0b819cefce710c5133a2290b08a626c5": "e_v = H(M || r_v)", "0b81a23de1be1230650955f5f18f4954": "\\langle x,y,z \\rangle = (x \\vee y) \\wedge (y \\vee z) \\wedge (z \\vee x)", "0b81ba28f76a18d31e4ef977056d4d3e": "LOW_N(Price)", "0b8213972cb9f9322096411b8de928a2": "{5 \\choose 4} = 5", "0b822aab47b8696cafc6884ea021b37c": "\\alpha = \\gamma \\,", "0b82a7c1ad82c6280c00e30b81be916d": "yx", "0b830da82f6f54d70c4e1d3b5ee81371": "\\mathbf{P} = \\mathbf{M} \\mathbf{S}.\\,", "0b83178b8cec9e168bee6d1885952433": " \\mathsf{T}=(\\vec{\\omega}, \\mathbf{v} + \\mathbf{d}\\times \\vec{\\omega}),\\!", "0b83327e07094c25c75fbc32a9aacee5": "f(x) = C x ^ \\alpha", "0b8361b4e83f4f11f1e944c64dc60a21": "a_{14}", "0b8395a97f30d3add09584c47d734847": "\\sigma_\\theta = \\arcsin (\\varepsilon) \\left[1+\\left(\\tfrac{2}{\\sqrt 3} -1\\right)\\varepsilon^3\\right], ", "0b83c9c5a34c63d5591c3d35dc1a1239": "\\Omega-\\Omega_{0}", "0b83d951dc58a6be9b072f1f63b53d3a": "R_1(\\xi,x)=x\\,", "0b83e7957bfdd69c4bdea8c104b4618b": "d(X,Y) = \\mathrm{rank}(Y - X)", "0b83f343362828bdbbf35dc918c07982": "q \\to 0", "0b849ecb0d5a2111cbbf5f93e3e2aa8c": "m n \\equiv 1 \\mod k", "0b84cd8b045314d1fb7ca942f0d1db06": " F^{h}(t) = (1-a) k_i z(t) ", "0b84ce28a6e7a6cb4aa256b8e95e29ca": " \\sum_{n=1}^\\infty P_n(x) t^n = \\frac{t}{1-xt^2-t^3} . ", "0b84cfc3bb7538e3b4b64f09102ad6e1": "w(S,T) = \\sum_{uv\\in E\\colon u\\in S, v\\in T} w(uv)\\,,", "0b84d5b355c742c32d2e7b2e14c44aab": "N_{BOC}", "0b84e0fbd8677f8047c8941c094f8503": "\\;p(3) = p(2) r - A = P r^3 - A r^2 - A r - A", "0b84ed88c371c61221401dc2c32939fd": "r_{\\beta} ", "0b84f3b3b93a9294071089fa2ae3a2fa": "v_{i}(x)", "0b84f433965317ba73273746e23a012e": "R^{\\alpha\\gamma} = \\kappa \\left(T^{\\alpha\\gamma} - \\frac{1}{2} \\, \\mathrm{g}^{\\alpha\\gamma} \\, T \\right)", "0b85076223ef479109111bea691f91cb": "\\phi=\\phi_{in}", "0b85715a2e19f6f99f90da5dada5fdae": "|a_2|\\le 2.", "0b85981a51df9d17c449b07de89fe1df": "\\operatorname{sinc}(a+n)", "0b85c1bc2cb84a41989a63d0f0fbe7cf": "\\hat{a}_{i+} = \\sum_{j=1}^n \\hat{a}_{ij} = u_i", "0b8620395de582340a5492b319925c0a": "q_\\text{opt}=F^{-1}\\left(\\frac{7-5}{7}\\right)=\\mu e^{Z\\left(0.285\\right) \\sigma} = 50 e^{\\left(0.2 \\cdot -0.56595 \\right)} = 44.64\\approx 45.", "0b8633fe1643477bd08df0dfe4749835": "\\theta _{2}", "0b86489034212d6ccac6fdb108f7b2d7": "\\sigma \\sqrt{\\pi/2}\\,\\,L_{1/2}(-\\nu^2/2\\sigma^2)", "0b865725ae0f8bbe98b78aa5d0f54ba5": "\\{0,1,2,\\ldots\\}", "0b86bf8474ff943fd65ca8d0e1d2c20b": "\\left\\{x : x \\in I\\right\\}", "0b86d91fb6aa864c14399c769b6e9723": "r_\\mathit{outer} = \\frac{r_{s} + \\sqrt{r_{s}^{2} - 4\\alpha^{2} \\cos^{2}\\theta}}{2}", "0b873c219b6299177d80f91773b38a1d": " log(y) = mlog(x) + log(b) ", "0b8742117a471e24d998009652fca294": "R_r^'", "0b8754b55c7cc8ea595c71acd3f1d76b": "1+G(s)", "0b8755b3966b76817617359ce6fc0e2a": "\\Pi_{Z \\gamma}(q^2) = q^2 \\Pi_{Z \\gamma}^{\\prime}(0) + ...", "0b8758658d684df6c97e4e0ecf6b8434": "\\mathrm{Hom}(X\\times Y,Z) \\cong \\mathrm{Hom}(X,Z^Y).", "0b87d66b88c72957dfea8c9605016442": "Beta", "0b88330b1ad90aae4c837487a4fe419f": "\\Big( \\pi \\models \\neg\\phi \\Big) \\Leftrightarrow \\Big( \\pi \\not\\models \\phi \\Big)", "0b883581eb4f6bcdaf34c4659ea6965c": "\\frac {\\sqrt{\\lambda}} {\\sigma\\sqrt{2\\pi} } \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} \\, \\left( \\frac{1}{\\sigma^2} \\right)^{\\alpha + 1} e^{ -\\frac { 2\\beta + \\lambda(x - \\mu)^2} {2\\sigma^2} } ", "0b88407af1a4690c94366983e50ed883": " \\phi_1, \\ \\phi_2, \\ ... , \\ \\phi_n, \\ \\chi \\vdash \\psi ", "0b88a7d9e650f51e5966c0639d92e1af": " \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} = \\left \\vert {s_z = +\\textstyle\\frac 1 2} \\right \\rang = | {\\uparrow } \\rang = | 0 \\rang ", "0b88e36d11a61c1f76adfedd44a92d69": "(w/({P}.\\sqrt{T}", "0b88fa21de52ff02130eab54048104ae": "ArcCot", "0b896114208e001d359852c7e00d7391": "\\left(\\sqrt{1/45},\\ 1/6,\\ -\\sqrt{7/4},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "0b89a7dc15ee602d26a386d06ab9c4cd": "(\\$50-\\$40)/\\$40", "0b89ab744563103f98340710275df705": " ||\\cdot ||_{HS} ", "0b89f2e024547f57cb720d715c9bb459": "\\scriptstyle \\phi(nT)", "0b89f4c0080dbfdd2e2b9452c28d5f6e": "\\operatorname{E}[S_N]=\\sum_{n=1}^\\infty\\sum_{i=n}^\\infty\\operatorname{E}[X_n1_{\\{N=i\\}}]=\\sum_{n=1}^\\infty\\operatorname{E}[X_n1_{\\{N\\ge n\\}}],", "0b8a1d01c1b6bd2d24d5819c1105a17e": "\n\\frac{z^2}{S \\cdot \\mathit{near}}-\\frac{z^2}{S \\cdot \\mathit{far} }=\n", "0b8a49d10c72ff877b4efb57b89ae93c": "\\hat{f}(\\xi) = \\int_{-\\infty}^{\\infty} f(x)\\ e^{- 2\\pi i x \\xi}\\,dx", "0b8a97aedda0ccab78fce961a1f30dab": "T_G^j = \\sum_{k=1}^\\infty S_G^j(k)", "0b8ac357e679a7f816a8bce1653073fe": "(B \\nrightarrow C)", "0b8b2e26a8f38103aae99b36cf987aef": "n=\\binom{2k-1}{k}", "0b8b4020c0d5c9f450dc7a960e6484be": " \\mathcal{R}^n \\times \\mathcal{R}^n ", "0b8b4a5031b67bcb1c99ba05b2aa4567": "Qb=B", "0b8b6f6d7621d3cecb8cc3cc14e87257": "K^{\\times}/R^{\\times}", "0b8b9291fc1b4c7190e104f21f67f9ea": "Z = \\langle A, P, U, \\succsim \\rangle", "0b8bff07ea53d9df8a06df478520cdbe": "g_{\\mu\\nu} = \\eta_{\\mu\\nu} + h_{\\mu\\nu} +\\mathcal{O}(h^2)", "0b8d5dab99fe1841ed35dbdf5cfeb710": "t_{UV}\\colon U\\cap V \\to \\mathrm{GL}_k\\mathbb C", "0b8dc03b332673838f991d87db61b622": "\\operatorname{ch}(L) = e^{c_1(L)}", "0b8e388e1485a5541bebf24172e7523f": "T^{r}_{r} = T^{\\theta}_{\\theta} = T^{\\phi}_{\\phi} = -P", "0b8e3e2ab58bd8d9caa072ffa382481e": " \\mathbf Z=79", "0b8e4583d5fa7caa5e6f16821bfbb944": " e \\ne e_0^i\\rbrace", "0b8e7623d6069852da88bef89f34abad": "\\theta(a)", "0b8eee8946c0a1bd9ba955620d120e05": "\\phi(t,i)", "0b8f00cff9ce96028ce2a36179028ce9": "a = 2\\sin\\left(\\frac{\\beta}{2}\\right)", "0b8f3c9d7ae4047199cc7da8d2c35a88": "5\\times(c\\bmod 4)\\bmod 7 + \\rm{Tuesday = anchor.}", "0b8fb2e814d079329f8c2c7c94581e01": "C_{QA} = \\frac{C_{QL}}{n_B} \\ ", "0b903759e7dd1356234866065e6de58f": "\n\\left[ \\Delta - \\lambda^2 \\right] u(\\mathbf{r}) = - f(\\mathbf{r})\n", "0b905b0f1d13ad2cc3391b785b7f9d87": "\\sigma^2_y(\\tau) = \\frac{2}{\\nu_0^2} \\int^{f_b}_0 S_\\phi(f) \\frac{\\sin^4(\\pi \\tau f)}{(\\pi \\tau)^2} df", "0b905ee6b95e4a4f7a15f23e0f1414d0": " M = i \\left( E - \\varepsilon \\sin E \\right) ", "0b9067fbcf4187e7903cc0f1671a1d98": "L\\, f(k)", "0b90dadead92fafe87e8997eb94db71f": "pdet[f]<0", "0b91776179b762c2296b2605c0b21806": "\\frac{1}{C_p} \\left ( \\frac{p_\\circ}{p} \\right )^\\kappa \\left [ \\frac{\\partial}{\\partial y} \\left (\\frac{dQ}{dt} \\right ) \\right ]", "0b917b6d41669f2ef991e25e9a946461": "BJD_{TT} = JD_{TT} + \\frac{\\vec{r} \\cdot \\hat{n}}{c} + \\frac{\\vec{r} \\cdot \\vec{r} - (\\vec{r} \\cdot \\hat{n})^2}{2 \\, c \\, d}", "0b918bddb5d8f10fe36b6a76f436c66b": "f(1,0)=f(0,1)=f(0,0)=1", "0b91dffd37ebd9a87fc354ad5c3fc9cb": "\\mathbf{r} = x_1\\frac{\\mathbf{a}_{1}}{a_1}+ x_2\\frac{\\mathbf{a}_{2}}{a_2}+ x_3\\frac{\\mathbf{a}_{3}}{a_3},", "0b9248fbd6a7170b2729a5317c4ca841": "\nn^a\\partial_a r \\,=\\,-1 \\; \\Leftrightarrow\\; n^a\\partial_a \\,=\\, -\\partial_r\\,.\n", "0b927e3aa160d256a9b5b590561c9733": "\\displaystyle{f_r(\\theta)=F(re^{i\\theta})},", "0b92c5b09123e92bf345a3f42a180a16": "r_{\\pi} = \\frac{v_{be}}{i_{b}}\\Bigg |_{v_{ce}=0} ", "0b92f8c2972983f15725fd66e4a72066": "\\phi=0", "0b931f4ac8b9ae51e673c440f52af38a": "\\begin{smallmatrix} \\frac{L}{L_{sun}} = \\left ( \\frac{R}{R_{sun}} \\right )^2 \\left ( \\frac{T_{eff}}{T_{sun}} \\right )^4 \\end{smallmatrix}", "0b93a9495de1456b4889c0a5321f3650": "\\scriptstyle V_i", "0b93c179e90f6166c51de7dc76a8d022": "\\mathbf{F} = q \\mathbf{E} \\left ( \\mathbf{r} \\right ),", "0b93db949aed9916a78216bbd63f6520": "\\overrightarrow{\\phi_q}", "0b94188cbb6c13ffdbf72aca16df2d1b": "x' = x \\cos \\theta - y \\sin \\theta", "0b941f8cf3d583be85bff166b8051771": "U(L)", "0b945b9986b82aa85fd2956bb2f21d69": "I = \\frac {V_\\mathrm{in}}{Z_1+Z_2}", "0b94bf92cb2a283096173d78d3795a86": "x_j \\in A_{i(j)}", "0b959ccc59ae88d0b6d1cbe4997752a9": "F_\\ell(\\mathbf{P},\\mathbf{K})(j\\omega)", "0b95a781867a79e98ef6e3f879f5802d": "P=\\{\\langle \\sigma,\n C\\rangle\\,\\colon\\sigma", "0b95bfa03f9a6f5000bba702dde655fa": "\\displaystyle{G(c,m)=\\sum_{x\\in \\mathbf{Z}/m\\mathbf{Z}} e^{2\\pi i c x^2/m}.}", "0b95cd93d06493cf484b3eb879a5913d": "\\sqrt{2} a", "0b9632187fed0c6ecaeeea0247589b21": "n^{-1}", "0b965e94d23c3c8bcc9308825ace26db": "\\rho_n e^{j\\phi_n}", "0b967d862e2a6342029344433abbcada": "b^2 \\equiv 8^2 \\equiv -1 \\pmod {13}", "0b978534a5bff0ff8067f1b886eeb935": " V_{\\left(p-k\\right)} = \\left[\\mathbf{v}_{k+1},...,\\mathbf{v}_p\\right]_{p\\times \\left(p-k\\right)} ", "0b97b2c4c2ac3d6c6b0b56479a4237c4": "W = K \\log m", "0b97fb60c9ca782f68e6579577232bff": " T = \\int dT = \\int \\frac1K \\cos\\phi\\,d\\phi\\,d\\lambda,\n", "0b984aa5911dca5050b31a3c636a74cc": "\\sigma_\\pm", "0b9852a37e50fccb384b96e19320bba9": "C = 1+ \\frac{2\\lambda}{d}\\cdot (A_1+A_2\\cdot e^{\\frac{-A_3\\cdot d}{\\lambda}})", "0b98648ecd97fa0c480a86faf5a2ce96": "\\textstyle (\\Omega,\\mathcal{F},P) ", "0b98f2e9e681c56feff1d8530889c2fc": "\\overline{x} = x_0 + D \\,", "0b98f4aa15429bf36834791a956cb0a0": "L_0 = \\frac{g}{2\\pi}\\, T^2,", "0b98f93bff89329f1ebf155876baf140": " E = { w: (w,w') \\in U } ", "0b99205304750c192bcaa89c09284bba": " [n,k_1]", "0b9957e1f4cd80581d89334d1b0d946b": "g(x)\\,", "0b99aab32e2546bb34238e7f682b70c5": "P_3=|001\\rangle\\langle001|+|110\\rangle\\langle110|", "0b99e14ffb6ef0664c2e4f8d2888bf3d": "\\tau (i) = \\binom{d_i}{2}", "0b99ecce5134503de1cfa264b309d767": "\\left(\\frac{\\text{votes}}{\\text{seats}+1}\\right)+1", "0b9a105871ab2bc396b7145a3cb8db48": "\\psi'' + ({\\lambda}+1-x^2)\\psi = 0.\\,", "0b9a3fcbf64804763b5e781e20d0d29d": "y_i \\succ_{x} y_j\\,\\!", "0b9a5658a78abe5d7547c5fe652e6421": "y'+a_0y=0", "0b9a5991500b1b73c3f690fbcb5b041e": "ax+b=0", "0b9a5ca220b82cb5dbb8f723efb2ef8c": "x=\\tan y\\,\\!", "0b9a810ede1539979438b64f21459ceb": "\n\\begin{align}\n \\mathrm{i}\\hbar\\frac{\\partial}{\\partial t} c_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}} = & \\left(\n \\tilde{\\epsilon}_{\\mathbf{k}} - \\tilde{\\epsilon}_{\\mathbf{k'}} \\right)\\, c_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}} \n + S_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}}\n \\\\\n &+ \\Bigl( 1-f^e_{\\mathbf{k'}}-f^h_{\\mathbf{k'}} \\Bigr) \\sum_{\\mathbf{l}} V_{\\mathbf{l}-\\mathbf{k}'} \\,c_\\mathrm{X}^{\\mathbf{k},\\mathbf{l}} \n - \\Bigl( 1-f^e_{\\mathbf{k}}-f^h_{\\mathbf{k}} \\Bigr) \\sum_{\\mathbf{l}} V_{\\mathbf{l}-\\mathbf{k}'} \\,c_\\mathrm{X}^{\\mathbf{l},\\mathbf{k'}} \n \\\\\n &+ D_\\mathrm{X,\\,rest}^{\\mathbf{k},\\mathbf{k'}}+ T_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}}\\,.\n\\end{align}\n", "0b9b19d96a82439031d5aa0a8115e2dc": "x = N z \\,", "0b9b48003fed47fa48414fe15e9a8938": "w(n) = \\cos^\\alpha\\left(\\frac{\\pi n}{N-1} - \\frac{\\pi}{2}\\right)", "0b9b533d9720ef2a605bb4511eed7cda": "\\vec x_2", "0b9b9b18377bb2ccceffb5342198ecfe": "\n\\nabla ^2 x ", "0b9bf76ce80e7a0103d97ae09c53b507": "\\displaystyle{D=2A^*A+I.}", "0b9d9d311a375871d66c77fc24ce26a0": "\\operatorname{VaR}_{\\alpha}(X)", "0b9dba97fbeb04de108696cd97fe51c4": "f^{-1}(t) = \\begin{cases}\n t^3 & \\text{if } t > \\tfrac{6}{29} \\\\\n3\\left(\\tfrac{6}{29}\\right)^2\\left(t - \\tfrac{4}{29}\\right) & \\text{otherwise}\n\\end{cases}", "0b9ddea1481307ad13403f4b5a212a12": "f(\\alpha) = \\alpha + \\delta, \\delta \\neq 0", "0b9dfd5131e36b288d1a52a5c0d689fd": "\\nu+d\\nu\\,", "0b9e1db56abf9af7ff37786e6623c613": "xy \\leq 0", "0b9e478282e5034680c2ea7c97ac48cf": "n^\\text{left}", "0b9e62db7302fcf1cd21af484cb62778": "\\oplus_0^k M_j", "0b9e7a0d9b5e57d18dfd0a16ed277ffe": " S e^{-q \\tau} \\phi(d_1) \\sqrt{\\tau} = K e^{-r \\tau} \\phi(d_2) \\sqrt{\\tau} \\, ", "0b9e96409092b9369d0e24e8fedbc3b4": "BF = 5", "0b9ef3122465e76b15b0db02c07bb898": "\\ R_* = R/M_{\\mathrm{air}}", "0b9f2991087ddb13a722a3319a0bbe2e": "T_{n}", "0b9f7998ac41c70208a48a3acee941b5": "Df", "0ba0102b18fa1fe0edb6bc8981743036": "f\\left(x\\right)=1234+166x+94x^2\\,\\!", "0ba067726be762b158e3644944a702b6": " \\ N_j", "0ba11ebdc4068356183c1d861d1c0bd8": "O(h)", "0ba183661fb2d56720d7a83f9359ee95": "\\langle \\mathbf{M}, \\mathbf{N} \\rangle = \\operatorname{trace}(\\mathbf{N}^*\\mathbf{M})", "0ba193407c682efade2e58f0f5661b44": "L_n[\\alpha,c] = L_n[1, c] = e^{(c + o(1)) \\ln n} = n^{c + o(1)}\\,", "0ba1a0d58d15ffa72426b99d7b710a15": "\\sum_{\\omega\\neq0}\\frac{1}{\\left|\\omega\\right|^{3}}", "0ba1d143245d8c31a437c8d56cbc814c": "P(\\nabla)", "0ba20a4199f33137fb4c08b3360e1ea8": "A=D", "0ba20e4e3cf6179712b7ac48b0876593": "\\log_2(n)/2", "0ba23bba106c02feed2787ae4c29b6fa": "A,B \\subseteq S", "0ba26bb837fefb2b28a751cda5a80607": "L^3", "0ba2a3fab3d792c693f6d61c99755b86": "\\, \\! V_-", "0ba2b0949f5422099a2e43e25c1d76d9": "{}- \\frac{d \\boldsymbol\\Omega}{dt} \\times \\mathbf{r}_B \\ ,\n", "0ba30127b3f99aa18a8691a08bb716c8": "TC = 5000", "0ba323d6320471121122b0a1b38454a2": "|0 \\rangle", "0ba35cc02fc7944cd344bda793fbabab": "L=\\rho M", "0ba3d2cb9da846198feac5a045253dd9": "C_{4,n} = 1 + 4\\, T_{n-1},\\,", "0ba3dbb9afc91fad9c30897769ac7b59": "|P|=(s t+1)(s+1)", "0ba4a4576a6dec5580ab283d2fd10ef7": "\\textstyle \\bar{M}_{\\mathrm e} R \\bar{M}_{\\mathrm e}^{-1} = T^{-1}", "0ba4f57789ecd10b0f9bb54c80d3e5a9": "\n\\ \\Delta=r^2-r_sr+\\alpha^2+r_Q^2\\,,\n", "0ba50b63d8d29baee511b4f6c80e91a7": " \\|v(t) \\| \\le \\|u(0)\\| e^{(-2 \\alpha + \\lambda) t} ", "0ba574dcaccfc86993ea40105a40130d": "\\hat{Y}(X_0)=\\left( 1,X_0 \\right)\\left( B^{T}W(X_0)B \\right)^{-1}B^{T}W(X_0)y", "0ba59a1224d9fc2b5b158b2b2d6e3b56": " c(r,z) = \\bar{c}(z) + c'(r,z)", "0ba5d06350e48e10f43e830cdef2b249": "\\eta \\colon K\\to A", "0ba5e8b33627efd46f38aea8e8599218": "0 \\le i < n.", "0ba6046d5a8b196ec30674bfcd1217d5": " \\forall x \\, (P(x) \\rightarrow Q(x)) \\vdash \\forall x \\, P(x) \\rightarrow \\forall x \\, Q(x) ", "0ba65a0ebc962a1083c75f8b37ec412e": "\\sqrt{\\frac{8}{15}}\\!\\,", "0ba66fdb504e3b99391cefa43500fc2c": "\\displaystyle{\\alpha_0=m_1 \\alpha_1 + \\cdots + m_n\\alpha_n}", "0ba6b38fb04f9aabbf0a90866305a8e0": "G \\to H", "0ba6bb9145441912d08d5f993ca4c998": " \\displaystyle{[L_m^\\prime,L_n^\\prime]=(m-n)L_{m+n}^\\prime,\\,\\,(L_i^\\prime)^*=L_{-i}^\\prime.}", "0ba6d93c11afa1e9143781d4b44ce862": " \\frac{1}{2 \\pi}\\int_{-\\infty}^\\infty g(c+ix)e^{-ux}e^{icu} \\, dx. ", "0ba70339b1175759333f58a7a9ee24d2": "\n\\text{The congruence }x^2 \\equiv -1 \\pmod p \\text{ is solvable if and only if }p\\equiv 1 \\pmod 4.\n", "0ba7196b4140661cf128ee1704e4df74": "T = 1", "0ba74fa0adc0987a8d21b8f74d960684": " avg_7 = {bp_1 + bp_2 + \\cdots + bp_7 \\over tr_1 + tr_2 + \\cdots + tr_7 } ", "0ba7b67a514c2552b9a5b316ceb7666c": "-\\mathrm d^2/\\mathrm dx^2", "0ba81a39570b976e204464473622e33c": "\\iota(a) = \\delta_a", "0ba8219315d56b65d961176d06b70431": " \\Delta p = I \\cdot {\\dot Q} = I \\cdot {\\mathrm d Q \\over {\\mathrm d t}}", "0ba844424e7784e97fa22106b5a77621": "P(H1)", "0ba849c74778d4ea05f6b8b5595fea67": "\\nabla \\times \\mathbf{E} = - \\frac{\\partial \\mathbf{B}} {\\partial t}, ", "0ba975adfb8d2b8968f7813711c1a3d0": "\\mathbf{Y} \\approx \\mathbf{X} \\mathbf{B} ", "0ba9f47effb2442d4b840f79e1129608": "\\mu = \\sum_{i=1}^n p_i\\cdot x_i ", "0baa1ca183d6b40e4d4793b8e735df11": "\\mbox{Debt ratio} = \\frac {\\mbox{Total Debt}} {\\mbox{Total Assets}}", "0bab28a58cec8aa398832b9c37480c5c": "a^{2a} + b^{2b} + c^{2c} \\ge a^{2b} + b^{2c} + c^{2a}.\\,", "0bab36a30b983277c0f2327329866d5f": "2^{r - 1} - 1", "0bab70e54d6633f20a610781d0e395fc": "\\frac{\\mathrm{d}A}{\\mathrm{d}t} = \\frac{\\left | \\mathbf{L} \\right |}{2m} \\,\\!", "0bab71d520fdbb6da5d2e1e4ddda0679": "(1-\\nu)", "0babb717e0bc191c6b4b202d811de433": "t \\propto l^{1-k/2}.", "0babc9682e7f8d81b369e685c543f104": "|n(t) \\rangle=e^{-i\\hat H_0t}|\\hat{n}(t) \\rangle=e^{-i\\hat H_0t}\\hat{U}(t,t_0)|\\hat{n}(t_0) \\rangle", "0bac1ed4c9dca459cf01bffcc7f21359": "P_d = 1 - \\left(\\frac{3}{4}\\right)^n", "0bac22e6ebfe568d47d94d7f8857aa33": "E_{pq}^2 = \\bigoplus_{q_1 + q_2 = q} \\mathrm{Tor}^R_p(H_{q_1}(X; R), H_{q_2}(Y; R)) \\Rightarrow H_{p+q}(X \\times Y; R).", "0bac28ad961eeabec7b79305e19d4c4a": " H_1(\\mathbb{Z}S) = \\frac{\\ker \\partial_1}{\\mathrm{im} \\partial_2} = \\mathbb{Z} \\langle e \\rangle \\cong \\Z. ", "0bac4b729344774ac205c3e843d93d2d": " R(n) = \\begin{cases} \\{2n\\} & \\text{if } n\\equiv 0,1,2,3,5 \\\\ \\{2n,(n-1)/3\\} & \\text{if } n\\equiv 4 \\end{cases} \\pmod{6}. ", "0bac6cd12cadf62ebe34e04b2ff1f9ca": "A = (a_{ij})_{m\\times m}", "0baca563ee0be40a2314c2482f04aba0": "F_i(a, b) = a + b", "0bacd09e5e818719c43dcb9bb3a43c0e": "M\\ ", "0bacfebb679930e4f1effca1310ce936": " v(S)=\\max_{x\\geq 0} (c_1x_1+...+c_nx_n) ", "0bacff768343a0f729ac4b71f0e8c851": "{\\mathbf x} = x_1 {\\mathbf e}_1 + x_2{\\mathbf e}_2 + x_3{\\mathbf e}_3.", "0bad167be678207c6e724c2c3b1c1994": "h_i = j\\,", "0bad51c0b9b2ba77c19bf6bfbbf88dc3": " 0", "0bad5d67e14bbbb4170fd1440053e667": "-y-z-y^2 x-x+xyz=0", "0bad61302975f6429fcabc7ca87a7f58": "V = {E}{l} \\ \\ \\text{or} \\ \\ E = \\frac{V}{l}. ", "0bad67dc10d38a644210cbdd9be0957d": " Y_{-i} = X \\Pi + U_{-1} ", "0badf72749b98e4414a1fa2a068efa84": " K=QH^{\\mathrm{T}}\\left( HQH^{\\mathrm{T}}+R\\right) ^{-1}", "0bae05f210eb263f5dba7a5b2cb1b119": "\\int_{s_1}^{s_2}\\left|\\frac{dR}{ds}\\right| ds = |R(s_2)-R(s_1)|.", "0bae081810feb487f9f8c51d689e1a9f": "\\lim_{n \\to \\infty} \\frac{a_n}{n}", "0bae17534242bdb9d036535353ccb46c": " x \\, \\left( \\partial_t + \\partial_z \\right) + (t-z) \\, \\partial_x. ", "0bae79ab39566b9793d28221c132c625": "x_N = 1", "0baebcb88f448e19280dad5f3eac6893": "(x,y)\\in R", "0baf1b40465f62453b11be6d4a3b97a3": "\n\\begin{align}\n \\mbox{Capital account} & = \\mbox{Change in foreign ownership of domestic assets} \\\\\n & - \\mbox{Change in domestic ownership of foreign assets} \\\\\n \n\\end{align}\n", "0baf21f6ca334feab629cd9c7f31e6ed": "h_{cm}", "0baf3ca5f3308365e2a955f864736ba2": "d^2=(4*x^2+644*x+11760)^2=16*x^4+5152*x^3+508816*x^2+15146880*x+138297600", "0baf43e3943769e109159902f482d0ce": " y\\ F\\ n = F\\ (y\\ F)\\ n ", "0baf934993a57a2eb5f875db67e0e7f0": " r_{i} = \\frac{u_{i} - u_{i-1}}{u_{i+1} - u_{i}} ", "0bafc6a4c0054478ae74d684d2e210bc": "\\left(\\frac{1+\\alpha}{1-\\alpha}\\cdot\\tan\\left(\\frac{x}{2}\\right)\\right)\\,", "0bafc6b47003a1720f7366e07740cb14": " \\Lambda = (1 - e^2) \\frac { \\tan \\phi_2}{ \\tan \\phi_1} + e^2 \\sqrt{ \\cfrac {1 + (1 - e^2)(\\tan \\phi_2)^2}{1 + (1 - e^2)(\\tan \\phi_1)^2}} ", "0bafda7a0b29eeb2a2d815ad6c168550": "\\scriptstyle X_C\\,", "0bafe4d14c508e7df71ce1e19821a595": "s_3 = 1011,", "0bb03cbb0b2613dfd5a718bd636a3306": "\\gamma A", "0bb091ca4baacdc46adf23c9bbfc4099": "\\!\\mathcal A \\models_X^- t_1 = t_2", "0bb0f05e0bf985e305cf129b21faf5fa": "e_1(t)=\\dot\\gamma(t)/|\\dot\\gamma|", "0bb124a800ec449f52d2bb18760077b0": "\\mathit{e(q)}", "0bb15cef2971a51158cc90d75a17ca19": "\\begin{align}\n x &= \\frac{R\\pi\\lambda\\cos 45^\\circ}{180^\\circ} = \\frac{R\\pi\\lambda}{180^\\circ\\sqrt{2}}\\\\\n y &= \\frac{R\\sin \\varphi}{\\cos 45^\\circ} = R \\sqrt{2} \\sin \\varphi\n\\end{align}", "0bb190bfb7fc3feb0af47612a52be95f": " (T(f))^\\sharp(x) \\leq C(M(|f|^r))^\\frac{1}{r}(x) ", "0bb23fb43ca45cd5f6bc61c8e8b02ef3": "\n \\boldsymbol{u}^{(1)}(\\boldsymbol{x}, t) = \\boldsymbol{m}\\, f(\\boldsymbol{N}\\cdot\\boldsymbol{x} - ct),\n ", "0bb25b753de5226c64b16dfa9bf275f2": "60 = 2^2 \\times 3^1 \\times 5^1", "0bb290ec2c11a2c1e901db3c307b2721": "g'(c)\\neq 0", "0bb2c53ba39921625fc4045b8e2e78c5": "D_{jj}", "0bb2e99617d136301bd8f5800cfbb616": "\\text{Hom}(A\\otimes B, C) \\cong \\text{Hom}(A,\\text{hom}(B,C))", "0bb2ee0c58e2a31c750637575886f585": "\nf(z) = \\sum_{n=0}^\\infty z^{2^n} = z + z^2 + z^4 + z^8 + \\cdots\\,\n", "0bb33fe445adde8ae66ed7ca9f44f5d9": "2\\mu (K)>\\mu(U).\\,", "0bb3c2f0f0ea75b0bad599c233bc2b0e": "\\{(s_0 s_1 \\cdots s_N,s_1 s_2 \\cdots s_N s_{N + 1})| \\; s_i \\in A \\; s_i \\neq s_{i + 1} \\} \\, ", "0bb4588fefe4ed4886d2aeed785adada": "(\\tfrac{1}{2}(a+b),b)", "0bb46ea81527e6aaf89636a3181cb791": "T(S_{liquid} - S_{solid}) > H_{liquid} - H_{solid}", "0bb488c36ba76bb7cd57243ba5fd84da": "S=\\underset{i \\in D}\\times Q_i", "0bb4c734060fc454d0a1e9caa1990e07": "\\begin{align}\n\\operatorname{E}[\\ln(X)] &= \\psi(\\alpha) - \\psi(\\alpha + \\beta)= - \\operatorname{E}\\left[\\ln \\left (\\frac{1}{X} \\right )\\right],\\\\\n\\operatorname{E}[\\ln(1-X)] &=\\psi(\\beta) - \\psi(\\alpha + \\beta)= - \\operatorname{E} \\left[\\ln \\left (\\frac{1}{1-X} \\right )\\right].\n\\end{align}", "0bb4d1bfd576055d1e7004c6af6b0e23": " \\mathbf{q}=\\mathbf{q}(t) \\,\\!", "0bb4fd171cdcbdae0167f08e3ae3aae9": "{\\text{engine}}\\;\\overset{\\textstyle\\tau}{\\underset{\\textstyle\\omega}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoondown}}\\;\\text{wheel}", "0bb51c7bd27f97d725be1e502378c921": " \\otimes ", "0bb58f5bc5165ff923fa6cb33b5c7134": "\\text{Aortic valve area} \\approx \\frac{3.5}{\\sqrt {50}}\\approx 0.5\\ \\text{cm}^2", "0bb5b6d91e2278fc25029b9a99d25108": "\\,\\sim\\,", "0bb622ab2f73b87f543937a23cccd7fe": "r(0)= r(\\pi)", "0bb62a16ccfe325cf239669304dc28f8": "\\begin{align}\n\\operatorname{E}(X) &= \\mu \\quad \\quad \\quad \\text{for }\\,\\nu > 1 ,\\\\\n\\text{var}(X) &= \\sigma^2\\frac{\\nu}{\\nu-2}\\, \\quad \\text{for }\\,\\nu > 2 ,\\\\\n\\text{mode}(X) &= \\mu.\n\\end{align} ", "0bb66270cf300e2423596cf871666074": "G=\\langle X|S\\rangle", "0bb68ac9c6290a78fcae1b7818f29d5d": "\\sum_{k=0}^\\infty \\|\\mathbf{x}_k\\| < \\infty.", "0bb6bbed431441aa372baf8034bd7218": "\\mathrm{LCM}(T_{p^k}, T_{q^\\ell}, T_{r^m}, \\ldots)", "0bb75646b7cd8e490515c98a08c31c2e": "\\ k=k_1/k_2 ", "0bb76dfe7af80a332d7997e59f8af29a": "\\sum_{n=1}^\\infty \\frac{1}{n(4n^2-1)} = 2\\ln 2 -1.", "0bb771ba9cf53a2fc6e6b8f05f77c732": "0 = - \\mu [\\vec{x},t] \\delta\\zeta [\\vec{x},t] - {2 \\over 8 \\pi G} (\\nabla \\zeta [\\vec{x},t]) \\cdot (\\nabla \\delta\\zeta [\\vec{x},t]) .", "0bb782aa8f9e133e487458690d3fbfb8": "g:rcl(R)\\to A", "0bb78497083c20789a86269b118db177": "\\hat{\\mathbf{a}} = \\mathbf{e}_i \\widehat{a}_i \\,,\\quad \\hat{\\mathbf{b}} = \\mathbf{e}_j \\widehat{b}_j ", "0bb7f7258bbbe786bdfa37a71d9852e1": "x_3 = (a+3b)-(a^2+3b^2)^2, x_4 = (a^2+3b^2)^2-(a-3b)", "0bb8215c3b5e83e3d5790537bbfe5a46": "(b \\rightarrow c))", "0bb84a7003dc3d6f17a64868475384fc": "\\frac{en}{e}", "0bb85a393354d3218aeb6b9dea16d825": "\\left[x^{(1)}\\right] \\supset \\left[x^{(2)}\\right] \\supset \\cdots \\supset \\left[x^{(k)} \\right]", "0bb886f3be44195026180987e61f4940": "\\alpha_k = \\gamma_k/\\lVert g^{(k)} \\rVert_2", "0bb8ecebb3198a020ba7803ab55bc4d7": "T_{\\text{f}}", "0bb96e96279951e5c687efc5cce6d14c": "t \\rightarrow \\infty ", "0bb975e1c6680c7dbfc3d5e37e730a90": "R_{abcd} \\, R^{abcd} = 12 \\omega^4, \\; R_{abcd} {{}^\\star R}^{abcd} = 0.", "0bb97e67fb3e13e62c23ab7a73cd118a": "G(x)=F(x+h)=\\sin (x+h).", "0bb98e9f6d33ed4940a6369aa13a984d": "{d \\over a+b\\sqrt{c}} = {ad - bd\\sqrt{c} \\over a^2-b^2c}. \\, ", "0bb9adf98182c82718f846c0a2cf49d4": "0 + j (\\omega-r)", "0bb9c23f489429bcb78a03199065b77c": "\\tfrac{1}{X} \\sim \\mbox{Inv-}\\chi^2(\\nu)\\, ", "0bba2efaf800616d232acb7ce5c5e60d": "\\textstyle \\mathbf{a} = \\frac{d \\mathbf{v}}{dt} ", "0bba49c57a1b37832d4476dc22691b08": "\n\\begin{align}\n\\nabla^2 \\Phi =\n\\frac{\\left( \\cosh \\tau - \\cos\\sigma \\right)^3}{a^2 \\sin \\sigma} \n& \\left[\n\\frac{\\partial}{\\partial \\sigma}\n\\left( \\frac{\\sin \\sigma}{\\cosh \\tau - \\cos\\sigma}\n\\frac{\\partial \\Phi}{\\partial \\sigma}\n\\right) \\right. \\\\[8pt]\n&{} \\quad + \\left.\n\\sin \\sigma \\frac{\\partial}{\\partial \\tau}\n\\left( \\frac{1}{\\cosh \\tau - \\cos\\sigma}\n\\frac{\\partial \\Phi}{\\partial \\tau}\n\\right) + \n\\frac{1}{\\sin \\sigma \\left( \\cosh \\tau - \\cos\\sigma \\right)}\n\\frac{\\partial^2 \\Phi}{\\partial \\phi^2}\n\\right]\n\\end{align}\n", "0bba835a648bebcea1aad3108949ee32": "H_n(j \\omega)", "0bba885ffd7144990c41cc925a2ee79c": "a_n = C\\lambda_1^n+D\\lambda_2^n", "0bba99ffd3d7db3f2e823531e79f9254": "\\hat{X} = \\frac{\\alpha-1}{\\beta+1}", "0bbb7c3aee026872dea6bde392956454": "\\and", "0bbb7c9fd6468deb705a5b480b8128a9": "r(N)=\\sum_{k_1+k_2+k_3=N}\\Lambda(k_1)\\Lambda(k_2)\\Lambda(k_3),", "0bbb8bb1a9761d2cd8e7992ad3ea197e": "\\omega_{c}", "0bbb91e5e3b2e3285082422166cfa4b3": " H^n(R,S;M)", "0bbb956a418945e7e6d7f8a019942c20": "\\frac{d}{d\\theta}\\operatorname{Cl}_{2m+1}(\\theta) = \\frac{d}{d\\theta}\\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m+1}}=-\\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m}}=-\\operatorname{Cl}_{2m}(\\theta)", "0bbbd1765bfca7ee728a0396b8d96f09": " \\ \\textbf{f} \\cdot \\textbf{f}_q = 1 \\pmod q ", "0bbc52963f22710610ce13d699d91841": "H^i(C(f)) = 0 \\text{ for } i \\neq -1, 0.\\ ", "0bbc534b447e14a0810504b8e33c5a61": "Y_i = C + \\alpha\\sin(\\omega T_i + \\phi) + E_i ", "0bbc85f829e11c2be0ceaa95bd95d95c": "\\gamma'(t) \\cdot \\gamma''(t)", "0bbc8626b11849e7019772fdd114ea02": " \\lambda_1 = -a, \\lambda_2 = a ", "0bbcf6d6ffcb14d427333af4905ee9bd": "A_{xy}", "0bbeb807fb5ad085cbaf6d6bb3705ee0": "\\beta=n^{o(1/\\log{\\log{n}})}", "0bbed5fe35b5362d05c53b182782155e": "i:N_0 \\rightarrow S ", "0bbef43d9a302c1d65634f240e1236c0": "\\mathbf{M}_{3} := \\mathbf{A}_{1,1} (\\mathbf{B}_{1,2} - \\mathbf{B}_{2,2})", "0bbf043db89bfd3995b143df42391734": "\\{ a^n b^n c^n : n \\geq 0 \\}", "0bbf314db1695862c71b47b15d6b8ef6": "SU(2n) \\supset Sp(2n).", "0bbfa2d9e91f9be7c28d2455eb0ff26a": "\\displaystyle\\mu_c = [B_1,B_2]+IJ.", "0bbfac647b7dcd97dc59e6d426d7b79e": "E \\subseteq \\mathbb{R}", "0bc07dcc6357cbd9ebf56650693d0c9a": "\\textstyle M_{\\mathrm f} = \\left( \\begin{array}{cc} 3 & 1 \\\\ 2 & 0 \\end{array} \\right)", "0bc07fa5b3a1f1c797979b4f1208e20e": " \\frac{\\partial f}{\\partial t} + r \\cdot \\nabla_r f\n- \\frac{1}{\\hbar} \\nabla_r V \\cdot \\nabla_k f\n+ \\sum_{\\alpha = 1}^{\\infty} \\frac{(-1)^{\\alpha +1}}{\\hbar 4^{\\alpha} (2 \\alpha +1)!}\n\\times (\\nabla_r \\nabla_k)^{2 \\alpha +1} V f = \\left(\\frac{\\partial f}{\\partial t}\\right)_c\n", "0bc099db2499122f4ab36efc41ac5454": "xAB \\rightarrow A-(B-A)_{x-1}-B", "0bc0e415644021b6a85649eaa903726a": "\\scriptstyle \\rho", "0bc0e5973e45def3494453fd0af29455": "J=\\det(\\boldsymbol{F})", "0bc15763ad9f28237b6db86df863162b": "\\frac{Z_q}{mU}", "0bc17f83938237ac1e12af3b13ea7d4f": "{\\mathbb P}^{2k+1}{\\mathbb C}\\,", "0bc26165fe3c36961425216e7a080afb": "(\\delta^\\dagger \\otimes 1_A) \\circ (1_A \\otimes \\delta) = \\delta \\circ \\delta^\\dagger", "0bc2b0dbabdfc0a993ba856984d937fa": "x=c \\cosh \\frac{v}{c} \\cos u", "0bc2b3518a89b0b9834c2f36baebfcd5": "Q(x) = \\frac{1}{\\sqrt{2\\pi}} \\int_x^\\infty \\exp\\left(-\\frac{u^2}{2}\\right) \\, du.", "0bc2e48be05a5a2797a081045a86ac86": "=f(n-1)-n^2+3n-2+\\sum^{n-1}_{i=1}\\left(ni-i^2\\right).", "0bc3b26061a337cdb5e9f583194ae53d": "\\Phi_{G}", "0bc3f11ccb324d1d244c87353c3e47d5": "\\scriptstyle \\vec L = \\vec J - \\vec S", "0bc463cb43711b684210ce06466a3b04": "\\displaystyle \\zeta = (m_0 - m_f)/m_0 = m_p/(m_p + m_f) = 1 - m_f/m_0", "0bc4a76f4cc040f13e4bba0cf4f4f1f5": " \\sigma(m_1+1) < \\ldots < \\sigma(n_1) = q.", "0bc4e7d532f726745d65abccd61add65": "\\scriptstyle{R_m^0}", "0bc5273141e5f3d29344c50b379125d7": "A = \\mathcal{O}_K, B = \\mathcal{O}_L", "0bc53c1d437f2532a357082f46f603e7": "\\hat{\\theta} = \\hat{z}", "0bc57f651e8658f14da371561fc429a9": "\\begin{align}\n(a + b\\,\\mathrm i)\\cdot (c+d\\,\\mathrm i) \n & = a\\cdot c + a \\cdot d\\,\\mathrm i + b\\cdot c \\,\\mathrm i + b\\cdot d \\cdot \\mathrm i^2\\\\\n & = (a \\cdot c - b\\cdot d) + (a\\cdot d + b\\cdot c) \\,\\mathrm i\n \\end{align}", "0bc583482b159ca5300f3e65f8b8be6a": "\\lambda=\\frac{2\\pi}{k}", "0bc5f5f1569c4e6b56d9dd7f77c838c5": "s_0\\left( t \\right)=s\\left( t \\right)", "0bc661c30d2fefc01718ba2ad71a86a6": "B_C \\approx \\frac{1}{T_M}", "0bc671de69a93127085e028316c98721": "\\dot{X} = \\frac{\\partial X}{\\partial \\tau}", "0bc698bc701562e4aaea7119cdb75255": "y_U = F(x+\\delta, \\hat\\theta)", "0bc70e047b0bbb1bc42a49652812fd3e": "A_{1} \\supset A_{2} \\supset A_{3} \\supset \\cdots.", "0bc74c2667a19a3265e2df805e24eeb9": "|[A,B;C,P]| = 1.\\ ", "0bc752a29ae295f2c2c82f6c208e2d46": "s(x) = \\mathop{\\mathrm{ess\\,sup}}_{y \\in \\mathbb{R}^{n}} f \\left( \\frac{x - y}{1 - \\lambda} \\right)^{1 - \\lambda} g \\left( \\frac{y}{\\lambda} \\right)^{\\lambda}.", "0bc7f379a1459ae5bd43b131b1a7a6c9": "u= dx/dt", "0bc811a05f19125ea669f88d7579b575": "|x^\\mathsf{T} y|\\le\\| x\\|_p\\|y\\|_q\\qquad \\frac{1}{p}+\\frac{1}{q}=1.", "0bc8465c05ddf597c1c12d240735938a": "N \\in M_X", "0bc8dff343096783dbeba168596b1630": "d_2\\,", "0bc91df3b6cb92c5438e0975d295bdfd": "\\sum_{m\\ne 0} |m| \\left|\\sum_{n\\ne 0} c_{mn}\\lambda_n\\right|^2 \\le \\sum_{m\\ne 0} {1\\over |m|} |\\lambda_m|^2", "0bc9207f6cedc6b8e8623d6344be2c83": " \\tan \\left( \\frac{1}{4} \\Omega \\right) =\n \\sqrt{ \\tan \\left( \\frac{\\theta_s}{2}\\right) \\tan \\left( \\frac{\\theta_s - \\theta_a}{2}\\right) \\tan \\left( \\frac{\\theta_s - \\theta_b}{2}\\right) \\tan \\left(\\frac{\\theta_s - \\theta_c}{2}\\right)} ", "0bc938f703277381c26db47d6032d529": "\\boldsymbol k \\cdot \\boldsymbol p", "0bc95db8cd7650ad9b1972255c6c136f": "z_1,\\ldots,z_4", "0bc99c248f639ab8797e8c61fe0efc6c": "x \\in A^c\\cup B^c", "0bc99da241ced5fb23e6cdc615f2b260": "\\langle x, y\\rangle := \\sum_{i=1}^{\\infty} x_i y_i \\quad x \\in E , y \\in E^\\beta", "0bc9bba6c22030e1d4dd1dbe6415ff4f": "\\mu_{k,i}", "0bca490eb344191fa2a7a379be377d10": "s=\\sigma+ti", "0bca595265c852b0a4d3b070983bef00": "\n\\Omega_{ij} = \\frac{1}{2} ( \\partial u_i / \\partial x_j - \\partial u_j / \\partial x_i )\n", "0bcab909c1e3e7e18a1c2c898d813ff7": "\n p := -2D_1~J(J-1) ~;~~\n \\mathrm{dev}(\\bar{\\boldsymbol{B}}) = \\bar{\\boldsymbol{B}} - \\tfrac{1}{3}\\bar{I}_1\\boldsymbol{\\mathit{1}} ~;~~\n \\bar{\\boldsymbol{B}} = J^{-2/3}\\boldsymbol{B} ~.\n ", "0bcac5527a2d22d1e6c4dd5517c100c1": "A \\models R(a_1,\\ldots,a_n)", "0bcadf78891f225454549428315cd9fd": "Tg = f * g.", "0bcb57347d875cf79605789c9c8aad68": "I\\cap K[Y].", "0bcb8572d9754ce8a6a9cebc01b50402": "-c_1\\,", "0bcb965d93d54dded0e1b55d743a9fb6": "\\pm 1, \\pm\\omega, \\pm\\omega^2", "0bcbc672badf11f08d13d1be16644d04": "\\psi_{ij}(t)", "0bcc36d2da085f786426a6e93c420afe": "P(\\phi) = P_\\mu(\\phi) = P_{\\Omega,\\mu}(\\phi) = \\frac{\\mu(\\{w \\mid \\phi(w) \\text{ and } w\\in\\Omega\\})}{\\mu(\\{w \\mid w\\in\\Omega\\})}", "0bcc6cea8bbade2794131c2853c7cdff": "\\textstyle\\frac{p}{q}", "0bcc6eb07326241ec09fb86e20e4f2b1": "y(x)", "0bcc7e60f7edb910aac88601954736c3": "P_{n,\\theta}", "0bcca8722a6e60adc53ffb74024df3f4": "\\mathsf{Q}=(\\mathbf{Q}-\\mathbf{P}, \\mathbf{P}\\times\\mathbf{Q}) = ([A](\\mathbf{q}-\\mathbf{p}), [A](\\mathbf{p}\\times\\mathbf{q}) + \\mathbf{d}\\times[A](\\mathbf{q}-\\mathbf{p}))", "0bccc911690f253ff87914a518c2ae41": "(r_{1})^2", "0bccd12acbaf7f0423a2d20ef346f355": "N_1\\subseteq N_2", "0bccdd2f481faa1e28a724569e730b0f": "B_m(x)= \\sum_{n=0}^m \\frac{(-1)^n}{n+1} \\Delta^n x^m. ", "0bcd48a71b2a676420645b1e0c0c137e": "(\\tfrac{p}{b}) =(\\tfrac{p}{B}) = -1,", "0bcd55df2b8c038c0cd464d3f06e4947": "ds^2 = e^{2\\Phi}\\left(dt - w_i \\, dx^i \\right)^2 - k_{ij} \\, dx^i \\, dx^j.", "0bcd584e0b1b1d9061b6b700ac8237ea": " [a_1\\otimes \\cdots \\otimes a_n,x]=a_1\\otimes \\cdots a_n\\otimes x\\quad \\text{for }a_1,\\ldots, a_n,x\\in V.", "0bcd8bfe7251e94179d7a24236b10374": "m_{1}u_{1}^{2} + m_{2}u_{2}^{2} = m_{1}v_{1}^{2} + m_{2}v_{2}^{2}\\,\\!", "0bcd91fb6432f991d5b4cbb079e562c7": "z_t", "0bcdcc38d1d9395793a01a0dc52c2093": "\\frac{1}{((i\\omega)^2-\\xi^2)^2}", "0bce8c6df4159efc00ea47dce2236a38": "f(a_1)=a_2", "0bcebb29fb0eee8f8be42be4436a4823": "D = 2(Y_1-B) = 2(\\sqrt{13}-\\sqrt{3})", "0bcf4c6c79db1875135dc475e94e12b8": " V_1\\times V_2\\times V_3\\to \\mathbb F,", "0bcf4da023d9e69b2c555d7e5188b7be": "a_{n+1,k} = - 2 n a_{n-1,k} \\ \\ k =0", "0bcf7aa0874b301cba529c8c62e67501": " \\operatorname{tr.deg}_k k(p) + \\dim D \\le \\operatorname{tr.deg}_k K", "0bcffc346310869894990a469979e6ee": "\\frac{}{} \\Delta", "0bd010bc6889f4a443f788d8c6e055e1": "2\\times 3\\times 5 = 6\\times 5 = 30", "0bd084b7b4a41909762a1bbf1afaec13": "\\beta = \\frac{R_1}{R_1+R_2}", "0bd08649070cfe010ad66e771bd2faea": "\\tau_0 = \\frac{n}{c} \\cdot \\frac{l}{1-R+X}", "0bd0ce81fe8a365ec05d91496ed88df4": "=\\operatorname{st}\\left(\\frac{u \\cdot \\mathrm dv + (v + \\mathrm dv) \\cdot \\mathrm du}{\\mathrm dx}\\right)", "0bd10836058ca3bee3d54e3f313bf5ab": "^{\\;}W(\\xi,\\tau) = W(\\star q(\\xi ,- \\tau ),0)", "0bd1511db1dea8758b4ebbc48d8109a1": "\\frac{1}{Z_{\\text{eq}}} = \\frac{1}{Z_1} + \\frac{1}{Z_2} + \\cdots + \\frac{1}{Z_n}", "0bd160c9fcc2db715029d73211989a2d": "y(x) = A\\cdot(Ax^2+2Bx+C)", "0bd1a732c88b7ea0de3ba2c93d80a57f": "\\begin{matrix} \\frac{6}{5} \\end{matrix}", "0bd1aa585ad016db886eaad1aeec7307": "\n \\oint_{\\mathbf{X}_A}^{\\mathbf{X}_B} (\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon})\\cdot d\\mathbf{X} = \\boldsymbol{0} \n", "0bd1ecc6359ce275e3a626020b904c8c": "c_{1}", "0bd22ab446a4297363bc1fa5fbaeddd0": "d\\theta_1 = \\omega\\wedge\\theta_2, \\,\\, d\\theta_2= - \\omega\\wedge \\theta_1", "0bd257cdae1a39ee054c760b6bae9db1": "SO^+(1,3)", "0bd2778564ec3915ca420cd6ee4382fc": "\\Pr_{y\\in\\{0,1\\}^k} \\big[ (\\text{Had}(x))_y = 1 \\big] = \\Pr_{y\\in\\{0,1\\}^k} \\big[ \\langle x,y\\rangle = 1 \\big]\\,.", "0bd29a46079c8cd3b3522e581fef4356": "(y - y_0)(x_1 - x_0) = (y_1 - y_0)(x - x_0)", "0bd29f0b3414278302206922a703b271": "\\left\\|u\\right\\|^2 = \\left|\\frac{\\langle u, v \\rangle}{\\langle v, v \\rangle}\\right|^2 \\left\\|v\\right\\|^2 + \\left\\|z\\right\\|^2 = \\frac{|\\langle u, v \\rangle|^2}{\\left\\|v\\right\\|^2} + \\left\\|z\\right\\|^2 \\geq \\frac{|\\langle u, v \\rangle|^2}{\\left\\|v\\right\\|^2},", "0bd2a420fa1d46354b133aa3817af6f7": "\\Theta=\\sum_{ij} g_{ij} \\dot q^i dq^j", "0bd2a702207692aa5fbf30d6fd1a4f82": " \\lang x | x' \\rang = \\delta^3 (x - x') ", "0bd2aee67df207805b6e1cdc246e9e82": "f_1(n) = f_0^n(n)", "0bd2bf1364551fb50300e90528343264": "f:\\mathbb{H}\\rightarrow R", "0bd2c988cdd871a62c6286e6bc3e11e3": "Power loss = \\Delta p_{LS} \\cdot Q_{tot}", "0bd2d9b1b8b40f6ec5e4340e9d48ca20": "\\widehat{\\neg \\alpha}:= \\mathbb{I} - \\hat{\\alpha}", "0bd32d2ab93c4c68fc4e9f0c9d778d37": "\n\\langle 0|\\varphi_{\\mathrm{in}}(x)|p\\rangle= \\frac{e^{-ip\\cdot x}}{(2\\pi)^{3/2}}\n", "0bd3704899f112b1e2dd72dca836ed8f": "A^a_\\mu", "0bd3956bd1cae5bb15558010eed3bf43": "p\\rightarrow 0", "0bd3fedc5f9682ccb2726345b23d4b7a": "\n\\mathbb{E}_{X}\\left\\{ \\left\\Vert \\sqrt{\\Lambda}\\rho_{X}\\sqrt{\\Lambda}\n-\\rho_{X}\\right\\Vert _{1}\\right\\} \\leq2\\sqrt{\\epsilon}.\n", "0bd41f80d2a8fab5a5d8ce9317d612cc": " \\mathbf{W} \\in \\mathbb{R}^{C \\times N} ", "0bd4482e10ee6708f45f070244a7bd50": "\\begin{align}\n &m = L - \\textstyle{\\frac{1}{2}}C \\\\\n &(R, G, B) = (R_1 + m, G_1 + m, B_1 + m)\n\\end{align}", "0bd4558eaeb1eeb35fd9d6c1097c0aad": "\\forall A: A \\times \\varnothing = \\varnothing\\, .", "0bd47873c1c6dd17c6e21690f3fa043a": "H(e^{j\\omega}) = H_c(j\\omega/T)\\,", "0bd4b1c75e866ea36627f4d7ffe665bc": "\n\\sum_{A=1}^N M_A\\,\\big(\\delta_{ij}|\\mathbf{R}_A^0|^2 - R^0_{Ai} R^0_{Aj}\\big) = \\lambda^0_i \\delta_{ij} \\quad\\mathrm{and}\\quad\n\\sum_{A=1}^N M_A \\mathbf{R}_A^0 = \\mathbf{0}.\n", "0bd544bc9dfe2625b4c6fce15e8b2979": "J_{l+1/2}(\\rho)", "0bd54c5c230b4f3ccbbfb56d362fba68": "y_{\\mathrm{low}} [n] = \\sum\\limits_{k = - \\infty }^\\infty {x[k] g[2 n - k]} ", "0bd5973cce44b58ebb67f6111158b43c": "{6\\choose 2}{43\\choose 4}", "0bd5f8494fe5d199df6ce2a800f0cdf1": "\\widehat{f}", "0bd5f9ccbc354415ff0c32ca61c332d0": " |\\{d \\in D: t \\in d\\}| ", "0bd60fb8b717bc81c038596f85907717": "\\tilde{\\mathbf{M}} = \\mathbf{U}_t \\boldsymbol{\\Sigma}_t \\mathbf{V}_t^*", "0bd610c28dec19473137174ad39ba64a": "d_{1}-d_{2}", "0bd617f99d6f3fcf268a2368b340bb90": "f_T = S_T - K", "0bd6c3b263df94127bec3beb1c23e495": "[a]={1 \\over n!}\\sum_\\sigma x_{\\sigma_1}^{a_1}\\cdots x_{\\sigma_n}^{a_n},", "0bd6df5eec3eae69e41fdf2b4c262670": "m = O(n)", "0bd7637a4b50d8043cebfa0734bf51f4": " \\frac{\\partial\\phi(\\mathbf{r},t)}{\\partial t} = \\nabla\\cdot \\left[D(\\phi,\\mathbf{r})\\right] \\nabla \\phi(\\mathbf{r},t) + {\\rm tr} \\Big[ D(\\phi,\\mathbf{r})\\big(\\nabla\\nabla^T \\phi(\\mathbf{r},t)\\big)\\Big] ", "0bd7654b37bf9c6525fb12802c525268": "\\rho(g) \\mapsto \\tilde{\\rho}(e_g)", "0bd7c641c6d170178c725d4f5e10b9c0": "|k_{0}|=|k_{i}|", "0bd7d76313f69fd4b7e08d257ad02239": "(1 + x)^r \\geq 1 + rx\\!", "0bd890413bed9e9302718c8ccc74cda7": "E_g", "0bd8c9689b286a31ca75037b750a3599": "\\nabla^2\\mathbf{v} = \\nabla(\\nabla\\cdot\\mathbf{v}) - \\nabla\\times\\nabla\\times\\mathbf{v}", "0bd8f39c14d9d0684cb0def61f6e679e": " \n E = \n\\begin{cases}\n\n\\displaystyle \\sum_{n=1}^{\\infty}\n {\\frac{M^{\\frac{n}{3}}}{n!}} \\lim_{\\theta \\to 0} \\left(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}\\theta^{\\,n-1}} \\left(\n \\frac{\\theta}{ \\sqrt[3]{\\theta - \\sin(\\theta)} } ^n \\right)\n\\right)\n, & \\epsilon = 1 \\\\\n\n\\displaystyle \\sum_{n=1}^{\\infty}\n{ \\frac{ M^n }{ n! } }\n\\lim_{\\theta \\to 0} \\left(\n\\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}\\theta^{\\,n-1}} \\left(\n \\frac{ \\theta }{ \\theta - \\epsilon \\cdot \\sin(\\theta)} ^n \\right)\n\\right)\n, & \\epsilon \\ne 1\n\n\\end{cases}\n", "0bd8fd5d9ebbc10592b8b7ee70663d3f": "{\\rm E}(V)", "0bd9126c47b8120c51d3fc38383d1f32": "a,b \\in K", "0bd9522f573aabff1cef7779d290d074": "y\\in Y\\,", "0bd97e0d2983e0467b978524a19f345e": "f_{cr}\\equiv\\frac{\\pi^{2}E_T}{(\\frac{KL}{r})^{2}}\\qquad (2)", "0bd9b26f982e5198437202595f401b30": "[a+(n-1)d] r^{n-1} ", "0bda1b7ad85d8fdf990c212c8104d548": "(T,\\mu^T,\\eta^T)", "0bda24b10d7355305438917b41c4401c": "\\underline{\\mathrm{Hom}}(Y, \\Omega^{-1}(X))", "0bda386678ba6aadb3b0f8f3fd6e35cb": "0\\, ", "0bda879984a5765e34987f75b2b0a3a6": " \\mathbf{y}_{k} \\in \\mathbb{R}^{q} ", "0bdad16f8453e5f5bf2dff2e17efd057": "\\mathbf{P} = \\hbar \\mathbf{K}\\,.", "0bdb0f615b65ba86faffbe528a0292e5": "\\mathbf{t}=\\boldsymbol{\\sigma}\\cdot\\mathbf{n}", "0bdb2fe3cd993978c4fd09ae6eff2a6f": " \\sum_{k=0}^n \\tbinom n k = 2^n", "0bdb4aab659219ef245b752bb69a7359": "\\gamma \\equiv g \\frac{q}{2m}", "0bdb4b388170a062e3c8a2a944af6943": "\\varepsilon^{-1/2}", "0bdb4e12b1ab01e6471e3e0ab10057c5": " \\tfrac{25}{8} ", "0bdb5788606155ecae30d988287f8693": "\\mathit{C}_\\mathit{G}", "0bdb762ff68ee041fbf58148884827cc": "\\frac {dm} {dt} = {k_a} C_i", "0bdba9dd5fcca535a640dae7916c9265": " \\operatorname{E}_{GB1}(Y^{h}) = \\frac{b^{h}B(p+h/a,q)}{B(p,q)}. ", "0bdbc2acf29d6ee4517a0e34b96df1c5": " P_{ni} \\equiv Prob( \\text{Person } n \\text{ chooses alternative } i) = G(x_{ni}, \\;x_{nj}, \\; j \\neq i,\\; s_n, \\;\\beta), ", "0bdca44d0c3ec8d151498f4708c286e3": "f(\\varphi) = \\int_0^{2\\pi} e^{\\varphi\\cos\\theta} \\cos(\\varphi\\sin\\theta)\\;\\mathrm{d}\\theta.", "0bdcb8231d1fc88e3b054956ee8e3c15": "a_2= \\lfloor 5^\\frac{3}{2} \\rfloor = \\lfloor 11.180\\dots \\rfloor = 11, ", "0bdce0e3d67fc40224ae9c2b9aea501f": "[C_i,H]=i P_i \\,\\!", "0bdcfaa5eac1af4d3bb4e58026235984": "\\bigstar\\bigstar\\bigstar\\bigstar\\bigstar", "0bdd69d4f0d5820d0dca2a2545303644": "i\\text{ such that }\\sum_{j=1}^{i-1} p_j \\leq 0.5\\text{ and }\\sum_{j=1}^{i} p_j \\geq 0.5", "0bdd823c8102ecf09ed8510d48c0142c": "-i \\epsilon^{\\sigma 1 2 3} \\gamma_\\sigma \\gamma^5 = -i\\epsilon^{0 1 2 3}\n(\\gamma^0) (i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3) = \\epsilon^{0 1 2 3} \\gamma^1 \\gamma^2 \\gamma^3", "0bdda2c1bf4fcb1c06401a4e7e8e53ba": "M \\leftarrow \\lambda_B, i \\leftarrow 0", "0bddb2181901c5b50efce95fbe7aa4fb": "\\mathbb{Z}_3", "0bde3675f7237bb4bc2e814796a7c5a5": "E[K]=\\boldsymbol{\\tau}(I+T+T^2+\\cdots)\\boldsymbol{1}=\\boldsymbol{\\tau}(I-T)^{-1}\\boldsymbol{1}=4.5.", "0bded11b2f244789bbc090b3e5f2ebed": "q \\succ_P r", "0bdf10aa4228053856428648ff7e387c": "\\Phi_{12}(r)", "0bdf4f5af065b8936d26688308d59ac0": "M_{Y}\\ ", "0bdf7af0093ab06fee4ef824e72e6adb": "\\phi(a)(h)\\in G", "0bdfb7b5e97c75fcf87685ae72dd0ae3": "{\\mathcal M}_p\\,", "0be059b215b74db9e94d5b9ef4455b70": "(\\mathbf A^T \\mathbf P_B^{\\perp} \\mathbf A)^{-1}", "0be072b811505937e7ccfd4318fd5753": "\n\\sum_{k=0}^n \\mu_i q_k (i) q_\\ell (i)=v \\mu_k \\delta_{k \\ell}. \\quad(9)\n", "0be094014704082336d1d05a451b00e6": "\\Lambda_n(a) = \\sum_{x\\mod p}\\binom{D_{n+1}(x,a)}{p}", "0be0a0a107ade58d198608b9411f90bb": "Qp_n(x) = np_{n-1}(x)\\, .", "0be0b5c7ba61aa2ebce298f0caf055b2": "= <(A-a)\\psi|(A-a)\\psi> =0 ", "0be0c173c0ac9e0a3ce4a4c649668980": "\\chi(t-t')\\!", "0be25044b639fc8cff948ea87e363d33": "\\ddot{S} - (m^2\\sigma^2+n^2)S=0", "0be250a091dcb46adbed4d45a369cc3f": "a^2+b^2\\quad", "0be2827c27c5e28e4740fcad1427fc3d": " \\langle \\Psi_n | \\Psi_n \\rangle_\\nu = \\lim_{M\\to \\infty}\\sum_{i_1,\\ldots i_n, j_1, \\ldots j_n < M}a_{i_1,\\ldots, i_n}^*a_{j_1, \\ldots, j_n} \\langle \\psi_{i_1}| \\psi_{j_1} \\rangle\\cdots \\langle \\psi_{i_n}| \\psi_{j_n} \\rangle ", "0be2dc5a9949fb8f490a67ed32294805": "\\Psi = aC \\frac{\\sin\\frac{ka\\sin\\theta}{2}}{\\frac{ka\\sin\\theta}{2}} = aC \\left[ \\operatorname{sinc} \\left( \\frac{ka\\sin\\theta}{2} \\right) \\right]", "0be33594bf8b16a0c1f976e0e2e5ce9b": "\\begin{align}\n\\nabla^2 |\\mathbf{B}|^2 &= \\nabla^2 \\left (B_x^2 + B_y^2 + B_z^2 \\right ) \\\\\n &= 2\\left( |\\nabla B_x|^2 + |\\nabla B_y|^2 + |\\nabla B_z|^2 +B_x\\nabla^2 B_x + B_y\\nabla^2 B_y + B_z\\nabla^2 B_z \\right)\n\\end{align}", "0be36d48cd689178c7ea6769e8814a22": "i_s=i_1\\sin(\\Delta\\varphi_b^*+2\\pi\\frac{\\Phi}{\\Phi_0})+i_1\\sin(\\Delta\\varphi_b^*).", "0be3f1cb172185623d255059c7070971": "\\frac{\\partial (u+v)}{\\partial \\mathbf{x}} =", "0be3f82983571a05b83cbb79498f3b4d": "f(f^{-1}(L)) \\subseteq L", "0be401717500a5421d3097ffe98b3fe2": "s_n' = T(s_n,s_{n+1},\\dots,s_{n+k})", "0be4a8888c1e757918b2992a0c933b60": "\\alpha =5", "0be55c2ce32ce93808c3dc161a0a8198": "N=\\{ (0)=S_0, S_1, S_2, \\dots, S_{n-1}, S_n=\\mathbb{C}^n \\};", "0be67e814203369faf20e2c72ab8effb": "\\operatorname{E}(I - T) = (n - k)\\sigma^2", "0be6e2b6c489f2a3f0d575221799045b": " \\quad (1) \\qquad W = F_b + D ", "0be6edd7a6959446d28e09792bcc6e81": "(2b_{2}-a_{10})x-b_{2}a_{10}=0", "0be722e2013294ba0ad6af6e67fddc6e": "\nH_\\mu (s,t)=\\sum\\limits_{m=0}^\\infty \\frac 1{\\Gamma (m\\mu +1)}\\left( \\nu\nt^\\mu (e^{-s}-1)\\right) ^m, \n", "0be770fb0709e80e40ee66212bc4a680": "f(h(x)) = cf(x)\\,\\!", "0be7efca48ecc5518e1d90048538ab75": "\\|u\\|_{L^q(R^n)}\\leq C(p,q)\\|Du\\|_{L^p(R^n)}", "0be7fc3e54236ab0af584567f25d70bd": "\\int_{-\\infty}^\\infty \\frac{H(\\varepsilon)}{e^{\\beta(\\varepsilon - \\mu)} + 1}\\,\\mathrm{d}\\varepsilon = \\int_{-\\infty}^\\mu H(\\varepsilon)\\,\\mathrm{d}\\varepsilon + \\frac{\\pi^2}{6}\\left(\\frac{1}{\\beta}\\right)^2H^\\prime(\\mu) + \\mathrm{O} \\left(\\frac{1}{\\beta\\mu}\\right)^4", "0be818fe9598878e911b2b62b81a2089": "\\Bbb{Z}\\left[\\frac{1+\\sqrt{-19}}{2}\\right]", "0be850843cbb359206b2ec172e27bda5": "s_A=\\max\\{x_1,\\ldots,x_m\\}", "0be86ec7f5d671247d92215518ec1320": " \\left(\\frac{M_y}{m},\\frac{M_x}{m}\\right) ", "0be86f48c8ceda039ff0a3e395e0dcc1": " f_j(z)=\\gamma_j+\\frac{1-|\\gamma_j|^2}{\\overline {\\gamma_j}+\\frac{1}{zf_{j+1}(z)}}", "0be8c0ea04da4408e6a5a1a5b53295b6": "Q = \\frac{1}{(k^3 a^3)} + \\frac{1}{(k a)}", "0be8cd7defa4f8055c7e969d751ee14b": "\n\\begin{bmatrix}\na & b & c & d & e \\\\\nf & a & b & c & d \\\\\ng & f & a & b & c \\\\\nh & g & f & a & b \\\\\ni & h & g & f & a \n\\end{bmatrix}.\n", "0be8ceaec8f5f9c2458c21d42f815439": " \\textbf{c} = \\textbf{f}_p \\cdot \\textbf{b} = \\textbf{f}_p \\cdot \\textbf{f} \\cdot \\textbf{m} \\pmod 3 = \\textbf{m} \\pmod 3 ", "0be8d0bb151ec0b3b7defa606c891002": "\\sin\\left( x \\right) \\approx x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!}.\\!", "0be8d57bbd73792a5f139ea2b72dcc83": "e^{\\lambda_m}", "0be8f2cf475147f928a0d5bfdbdd6bf8": "c^2 = 2a^2 (1 - \\cos\\gamma).\\;", "0be9019520e6dd5ec7896ebc70ea68bf": " = e(p_0, u_1) - w ", "0be962bcf8fb08da7589a1568acbc3be": "\\textstyle L^1", "0be978dd4b000b0fb83e29bf0785692c": "r_{\\text{th}} = 10", "0be9945a9e1cc744aed150bf425175ed": "\\mathbf{Z}/n\\mathbf{Z}.", "0be9f616d07ebdff72dce97b88ff3e6e": "Q = -1 + \\left(\\frac {K}{Y_0} \\right)^{\\nu}", "0bea02cc5ca22b8f34fa2b7045400646": "E \\subseteq S", "0bea10a8ebfd38cd7ff87d762b1847cf": "\\overline{Q}^{\\text{day}} = -\\frac{1}{2\\pi}{\\int_{\\pi}^{-\\pi}Q\\,dh}", "0beaa0d1a955422faaf437d03628fdfa": "b^p_i=\\frac{1}{\\pi}\\int\\limits_{-\\pi}^{\\pi} f^p(x)\\cos(ix)dx,\n", "0beac0b29bf3e5e1ef26562bc3d06122": " \\frac{\\rho^4}{\\lambda^4} \\ll 8 {z^3 \\over \\lambda^3}", "0beb06a7bff537757ceb26645713f85f": "\\sqrt{ \\hbar }", "0beb2161a1cdb1a63a3fe4cfab6eea61": "[\\alpha] = \\frac{\\alpha}{c \\cdot l}", "0beb2ee17d0705977738fa9f2a5202bb": "LC_{50} \\le 5000 \\tfrac{mL}{m^3}", "0beb716c77b4b6aa97ec6fac926d81a0": "g(\\alpha_0) = k^2(\\alpha_0)", "0beb8488fe180b3f5fb245789456786a": " 2 \\cdot 5^2 + 2 \\cdot 5 ", "0bec16e17077ae64d9d742c478490e4a": "\n\\begin{align}\n\\operatorname{Var}(T)& = \\operatorname{Var}(t_1) + \\operatorname{Var}(t_2) + \\cdots + \\operatorname{Var}(t_n) \\\\ \n& = \\frac{1-p_1}{p_1^2} + \\frac{1-p_2}{p_2^2} + \\cdots + \\frac{1-p_n}{p_n^2} \\\\\n& \\leq \\frac{n^2}{n^2} + \\frac{n^2}{(n-1)^2} + \\cdots + \\frac{n^2}{1^2} \\\\\n& \\leq n^2 \\cdot \\left(\\frac{1}{1^2} + \\frac{1}{2^2} + \\cdots \\right) = \\frac{\\pi^2}{6} n^2 < 2 n^2,\n\\end{align}\n", "0bec24be5311b8514f217023c447c616": "\\sin x \\approx \\frac {p} {2 d} \\,.", "0bec5a248004dc95f57db43715fe775b": "u=D\\Psi=0,\\,\\,\\text{on}\\,z=-\\infty.", "0bec894000693179bbe9588fa3da2f28": "h^r \\not\\equiv 1\\;(\\text{mod}\\;p)", "0bec9521ce48ff0fa8bc19e15faa0aba": "A^+ + B \\to A + B^+", "0bed791c1fc42bcd9a74987cbc6fae0d": "\\frac{\\Delta_h}{h}~(1+\\lambda h)^{x/h} =\\frac{\\Delta_h}{h} ~e^{\\ln (1+\\lambda h) ~x/h}= \\lambda ~e^{\\ln (1+\\lambda h) ~x/h} ~,", "0bedd0bc04a88fcf76335f9581531c12": "f(x) = \\tfrac{1}{x} \\chi_{(0,1)}(x)\\ \\ \\text{and} \\ \\ g(x) = \\tfrac{1}{1-x} \\chi_{(0,1)}(x),", "0bedf5049f48710b4d70c2267beb0788": "0,1,\\infty.", "0bee7a098f60602e79ec7c7469352883": "\\boldsymbol{\\mathsf{S}}", "0bee836bdab775d1f62b1ff759a08971": "\\eta = \\frac{P - \\sqrt{D}}{Q}", "0bef3400253d490699af4adcd7b75a81": "\\mathrm{tr} (A^{\\rm T} B) = \\sum_j (A^{\\rm T} B)_{jj} = \\sum_j \\sum_i A_{ij} B_{ij} = \\sum_i \\sum_j A_{ij} B_{ij}.\\ \\square", "0befb670d5e59cadeaf3132bef54e5af": "I(r) = \\frac{1}{2\\pi} \\int_0^{2\\pi}\\! \\left| f(r e^{i\\theta}) \\right| \\,d\\theta", "0befc8268e9015e1607906ba3d6086b0": " \\sum_{i_1,i_2\\ldots i_k=1}^n f_{i_1i_2\\ldots i_k} dx^{i_1} \\wedge dx^{i_2} \\wedge\\cdots \\wedge dx^{i_k}", "0beff6847c91a05fc45a431b3c2003d9": "\\lambda = e^{\\pm i\\pi/3}", "0bf00bc684e8f4c1529bca65684575ae": "\\underline x =(x_1,\\ldots,x_n),\\ \\underline \\alpha = (\\alpha, \\ldots ,\\alpha)", "0bf0828dfbf843a0acb0a2695cd9ae29": "a,\\tilde a", "0bf0b649e25a6018cd243abd523b5fa5": "A_{FB} = \\frac {A_0} {1+ \\beta A_0}", "0bf10a5e9fa4b431590aa3648fd18d66": "i_{\\mathrm F_{SO}(M)} \\colon \\mathrm F_{SO}(M)\\hookrightarrow \\mathrm FM", "0bf120638d7dab77f86ba9851108018a": "F(k,m) + p^{2} = (p+1)^{2}", "0bf13937962d753883f2e6cc9cb93268": " \\chi(x) = \\kappa(e,x) ", "0bf16b86371719d532fc642050c6102e": "\\delta S= \\delta\\int_{\\mathbf{A}}^{\\mathbf{B}} n \\, ds = \\delta\\int_{x_{3A}}^{x_{3B}} n \\frac{ds}{dx_3}\\, dx_3 = \\delta\\int_{x_{3A}}^{x_{3B}} L\\left(x_1,x_2,\\dot{x}_1,\\dot{x}_2,x_3\\right)\\, dx_3=0", "0bf1dac419469aa6e01a6b04e4b93617": "(x,y)\\to(a,a)", "0bf20619ddb0605cf22b7a53a4739006": "a=\\frac{2A}{b+d}\\, ,", "0bf22b556d7943cdd4739ecf840e57d8": "F_{\\Theta|S=s}(\\theta)= F_{S|\\Theta=\\theta}(s)", "0bf2502fa37369ae3229a4683e4ecfa3": "= (2\\eta^{\\mu\\nu}-\\gamma^\\nu \\gamma^\\mu) \\gamma^\\rho \\gamma^\\sigma \\gamma_\\mu \\, \\quad", "0bf26a61be5eb20046a515d2ed602b01": "k_{t+1}=Ak^a_t - c_t", "0bf27b03635f84cd46d3580b8422ea45": "\\vec{e}_0 = \\partial_t, \\; \\vec{e}_1 = \\exp(a^2 r^2/2) \\, \\partial_z, \\; \\vec{e}_2 = \\exp(a^2 r^2/2) \\, \\partial_r, \\; \\vec{e}_3 = \\frac{1}{r} \\, \\partial_\\phi - a r \\, \\partial_t", "0bf282e11e21f9134c237b21902a406e": " w'(z)=a_0+\\frac{a_1}{z}+\\frac{a_2}{z^2}+\\dots .", "0bf2e17d7d272b1819885e99164fc494": "\\phi_{i\\alpha}(\\mathbf{r})", "0bf306110e2cd8653227037401eab0c9": "x\\in F_S", "0bf312d49b7399d15d6127e8e11ae9dd": "y(t) = \\sin(2 \\pi f_0 t) + \\sin(4 \\pi f_0 t) + \\sin(8 \\pi f_0 t)", "0bf343b7ea7fe475bf8cb5a29f00237d": "(n+m)\\times(n+m)", "0bf3689cc2f49413b379115c1a939811": "2^c", "0bf36a00ebe4d392059536bc2df48404": "\\phi:G\\to H", "0bf3901f8822d5bf17650e00d6a08d60": "\\omega \\colon \\mathbb{R}^n \\to [0,\\infty)", "0bf3b1180e13d316f28c658d97bedc39": " {= {{V \\times I \\times t} \\over {m}}}", "0bf4190f5a57125f1e62da9041c52995": "\\frac{\\delta (i)}{\\tau (i)}", "0bf442baca4055640cb0928d53dcd53f": "h^*_t", "0bf4732ab0e993b11a439cb854a5706a": " (X:Y:Z:Z^2)=(\\lambda X: \\lambda^2Y: \\lambda Z: \\lambda^2Z^2) ", "0bf48468b075e9f6517944a884c80f3d": "\\beta K", "0bf4ad0f089a845623943b3c3b7a99bd": "\\mathcal{M}(k) = \\epsilon_{\\mu}(k) \\mathcal{M}^{\\mu}(k)", "0bf4b906623f18815f2a24c8b06955f6": "\\, L X_t = X_{t-1} ", "0bf4c7adbc7438011c3f97ce6139a863": "X\\times\\mathbb{Z}", "0bf4ff425a9d1adb1c0d3fee7db725e6": "\\mathbf{q}_1 = \\sin(\\alpha/2)\\cos(\\beta_x)", "0bf5a7bac67d693ee2f3fa4e72ce7b6a": " R = \\Phi \\, \\Sigma \\, dt \\, dV ", "0bf5cee2c7cdec149e9943dfe9231821": "\\stackrel{*}{\\leftrightarrow}", "0bf66cfac4c16ac259e8c603d679de32": "\\oint_C {1 \\over z^n} \\,dz=0,\\quad n \\in \\mathbb{Z},\\mbox{ for }n \\ne 1.", "0bf6e816c4b55d78aee33db6a0a4a33c": "H_n = \\sum_{k=1}^n \\frac{1}{k}", "0bf75941d6426e5efc12c0a20858452b": "N_2", "0bf77fa1b2daab3b1be278027ed17b30": "L(G) = \\{P \\in W(V_T) | X_0 \\Rightarrow^{*} P\\}.", "0bf7edef535a681d0358425c346a5aa7": "F[\\phi]=\\frac{\\partial^{k_1}}{\\partial x_1^{k_1}}\\phi(x_1)\\cdots \\frac{\\partial^{k_n}}{\\partial x_n^{k_n}}\\phi(x_n)", "0bf7edf962425fdd790b7fa4a727106d": "(+,-,-,-)", "0bf806a941874086cac5dff3c4a390ff": "\\kappa_{(n)}", "0bf8266cd9dc09220b41cfffbe112c23": "(\\phi \\wedge \\neg \\psi) \\to \\neg (\\phi \\to \\psi)", "0bf88d4238d2d9a6d6009af79971cfcc": "a = N_A^2a' ", "0bf88e8d88ca8d43d187523d0c666d03": "f_{r}", "0bf89847eabe86b645a13a39c5bf2167": "2\\!\\cdot\\!1\\,+\\,2\\!\\cdot\\!4\\,+\\,2\\!\\cdot\\!4\\,=\\,18", "0bf8e2eb9ab89734931e9f47f7f79e3e": "\n\n\\operatorname{npmi}(x;y) = \\frac{\\operatorname{pmi}(x;y)}{-\\log \\left[ p(x, y) \\right] }\n\n", "0bf94f3700a36fbe262186493b69b3d5": "3 a_1^2 a_2^2.\\,", "0bf9a7392bec4857f3a5ffca7084d674": "\\kappa_\\nu=\\alpha/\\rho", "0bf9adf2f73f78b8c356391156e50530": "y' = y", "0bf9d515971a01f6136e697e61a5ca9d": "\\sigma (z)={\\bar z}", "0bf9d8a71d96ff89578858a5a8a86f33": "\n\\mathbf{y} = \\mathbf{W}^{\\text{T}}\\phi(\\mathbf{x}),\n", "0bf9ec326e8557d5ec2608b37589017f": "U(\\theta,\\hat{\\mathbf{e}}_j) = \\exp\\left(-\\frac{i}{2} \\sum_{n=1}^8 \\theta_n \\lambda_n \\right) ", "0bfa34edf8e1b79cd60f741f9417436b": "x_n^2=(x_0+n)^2", "0bfa3b574ad3e371956a3c4ca3e0cd28": "\\tilde{V}(\\tilde{\\Phi}) = e^{-2/\\sqrt{3}\\;\\tilde{\\Phi}}V(e^{\\tilde{\\Phi}/\\sqrt{3}}).", "0bfa4b46db86d454c7197aee11a65586": "\\partial_\\alpha S^\\alpha = \\partial_\\alpha (nU^\\alpha) = 0", "0bfaecb987c21b1a3ff6606fb9e6a3e9": "\ns_t = \\sum_{n=1}^k w_n x_{t+1-n} = w_1x_t + w_2x_{t-1} + \\cdots + w_kx_{t-k+1}.\n", "0bfb354df0a98d8f3ad77d6227bccda6": "\\mathbf{n}(\\mathbf{x},t)", "0bfb87f306c9950ce4fd244d00a59755": "\\varepsilon _1", "0bfbbfb807bb7156dbc064fd26b5517c": " P = \\int{I(x,y) dx dy} ", "0bfc221172d0996d925300b8041cf266": " W_n = \\int_0^{\\frac{\\pi}{2}} \\cos^n(x)\\,dx", "0bfc9bc32c471476e64cf87069242d67": " C = 19 ", "0bfcab08790a2d6d5e54cdba0cccb4a4": "\\alpha_\\lambda=\\epsilon_\\lambda", "0bfcf757aeeebbe71334661c2a18e29d": "\\sin(50\\tfrac58 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2-\\sqrt{2+\\sqrt{2}}}};", "0bfcfbb9e0ceab16c1b09e39737c4320": "N_t = \\frac{H}{HETP}", "0bfece9c01e3196b12c2df860459a4e4": "\\frac{\\mathrm{d}(TS)}{\\mathrm{d}t} - \\frac{\\mathrm{d}U}{\\mathrm{d}t} +P\\geq 0.", "0bfecf5bf4e5487e74782c850b5468c8": "\\cos(\\omega t) + \\cos(\\omega t + 2\\pi/3) + \\cos(\\omega t -2\\pi/3) = 0.\\,", "0bfede26834cd03854115d254dc5e951": "\\Lambda(s)", "0bff4249684c8c733114b19d37e5ca90": " \\widehat Y_i = \\widehat\\alpha x_i + \\widehat\\beta \\,", "0bff50bceaea2e80cc8af55113e59cc1": " r = 0 \\to ", "0bff6b9bfa947df2856774677426c3f6": " MPP_L.P_Q = VMPP_L ", "0bff87770d4defdec660d96abda92726": "\n\\langle N_i \\rangle = \\frac {1} {e^{(\\varepsilon_i-\\mu)/kT}} = \\frac{N}{Z}\\,e^{-\\varepsilon_i/kT}\n", "0bffa898be39285c4c7623b57bb9f897": "\\ddot{u} + (a + B \\cos t)u =0 \\ ", "0bffd0910a81504c47d514e9f3b27ccb": " \\frac{dx}{dy}\\,\\cdot\\, \\frac{dy}{dx} = \\frac{dx}{dx} ", "0bffe1f0e41cc9ce8213ef1b75a1a6ab": "P(z)=z^2-(a+d)\\ z+ ad-bc= (z-\\alpha)(z-\\beta) ~ .", "0c0040c1ebcab66bab13fe10067738da": "\\dot{Q}=\\dot{m}C_p\\Delta T", "0c005db297a554555b5494e7db746fac": "C(K) = \\{ (a,b) \\in K^2 | b^2 + h(a) b = f(a) \\} \\cup S ", "0c00a8cd9fd2d33d95494ef708d0ce6a": "\\alpha - \\beta", "0c010ed7d7f0baa5a6843251e5b1c597": "W = \\int_{t_1}^{t_2}\\tau \\dot{\\phi}dt = \\tau(\\phi_2-\\phi_1).", "0c01174090098b9eaa775947a72c7173": "\ng \\equiv \\frac{\\gamma}{1-\\kappa}\n", "0c012e25843d0eed0125df9f3dbc3852": " f \\in W^n_p(\\R) ", "0c018c5dd80380d95957c7a03cdceeaa": "d_{10}-d_{11}", "0c019dc2ea0b8647878cc5932d0b6e29": "R(A\\|B)", "0c0206b0aadf10fcd94c5c31d4b6c3f9": "u^*\\in \\mathcal{U}", "0c02434ea579e94472458aec727d480c": "\nE = \\xi^2 \\cdot \\omega^2 \\cdot \\rho = v^2 \\cdot \\rho = \\frac{a^2 \\cdot \\rho}{\\omega^2} = \\frac{p^2}{Z \\cdot c} = \\frac{I}{c} = \\frac{P_{ac}}{c \\cdot A} = \\frac{I}{f \\cdot \\lambda}\n", "0c03f6d1bd59b23cce733e46a0ac744e": "\\textstyle g_{ID} = e\\left(Q_{ID}, K_{pub}\\right) \\in G_2", "0c043d0b19e66b48586ead934677eb68": "\\Gamma \\vDash \\varphi", "0c0521d3a3601bb982e25d0ec0fd8788": "t_iht_i^{-1}", "0c05449e9b697399b845a4d7a90794aa": "g_{ab}=\\eta_{ab}+h_{ab}", "0c0585e64c757d046bf69502716cb98a": "n_\\downarrow(\\mathbf{p})", "0c05ecf0d3cddd1eafb204fea0caadab": "\\frac {\\mathrm d b} {\\mathrm d f}\n= \\frac {\\pm m_\\mathrm s ^3 x_\\mathrm d ^2}\n {N \\left [ \\left ( m_\\mathrm s + 1 \\right ) f \\pm m_\\mathrm s x_\\mathrm d \\right ]^2 }\\,.\n", "0c05f0c5fc9c43ea7a4ee9a2d655debd": "D=S(d)=\\mathbb{R}", "0c0615f83a9bc1b683656892ea45dd58": "\\widehat \\mu = \\bar X = {1 \\over n} \\sum_{i=1}^n X_i.", "0c06594985f13c0fde9a4b2796aaebda": "p_{i,t-1}", "0c06b699f994eb01d14fa3033f201b06": "\\left(\\begin{array}{c}\nX_{4}\\\\\nY_{4}\\\\\nZ_{4}\n\\end{array}\\right)=\\left(\\begin{array}{ccc}\n\\cos\\delta & 0 & -\\sin\\delta\\\\\n0 & 1 & 0\\\\\n\\sin\\delta & 0 & \\cos\\delta\n\\end{array}\\right)\\left(\\begin{array}{c}\nX_{3}\\\\\nY_{3}\\\\\nZ_{3}\n\\end{array}\\right),", "0c06dcdde3ebc4a255df92cb29bd357e": " \\bar{\\psi} ", "0c074be282fb6b643719dada065dc23a": "\\nabla \\mathbf{ \\gamma + F_{12} = 0 } \\cdot \\Longrightarrow \\cdot\\gamma\\left( \\mathbf{q}\\mid \\Gamma \\right) = -\\int_\\mathbf{q_0}^\\mathbf{q}\\mathbf{F}_{12} \\left( \\mathbf{q'}\\mid \\Gamma \\right) \\cdot d\\mathbf{q'}", "0c07a03fbc1cecd07807121a4fdf6f86": " \\qquad \\qquad \\mathrm{diatomic\\ ideal\\ gas}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ c_{v,f} = \\frac{R_g}{M} \\{ \\frac{3}{2} + (\\frac{T_{f,v}}{T})^2 \\frac{\\mathrm{exp}(T_{f,v,i}/T)}{[\\mathrm{exp}(T_{f,v,i}/T)-1]^2}+1 +\\frac{2}{15}(\\frac{T_{f,v}}{T})^2\\},", "0c07aae4fdc09c89a293dcc6d5df2f9a": "R = {D}\\cdot {P} = {D_{max}\\cdot (P_{max} - \\frac{2\\cdot P_{max}} {3})}\\cdot \\frac{2\\cdot P_{max}} {3} = {D_{max}\\cdot \\frac{2\\cdot P_{max}^2} {9}}", "0c07ccc85f62ae9a4ffb8d7eab721ebb": " \\boldsymbol{\\nabla}\\bold{F} = \\boldsymbol{\\nabla} \\cdot \\bold{F} + \\boldsymbol{\\nabla} \\wedge \\bold{F} = \\boldsymbol{\\nabla} \\cdot \\bold{F} + I \\boldsymbol{\\nabla} \\times \\bold{F}", "0c07e9d8e87234d3d808bf1eb6ee922c": "{d^2\\theta\\over dt^2} + {g\\over\\ell}\\sin\\theta = 0", "0c084c1d951e099539549c1d08ab606e": "(10 | 13)", "0c0884ea59a985c311d75e65d59e978f": "\n\\mathcal H \\psi = E \\psi\n", "0c0897c43881f30a158ed280ae89b821": "\nF(x) = \\sum_{j=0}^\\infty \\frac{1}{j!}\\left(\\frac{\\lambda}{2}\\right)^je^{-\\lambda/2}I_x(\\alpha+j,\\beta).\n", "0c08be9440a7f64331d78b7290c19da2": "U\\mathfrak g", "0c0923b635c9f8d65ef71b44c7a5f7f8": "(\\mathbf u + \\mathbf v) \\wedge \\mathbf w = \\mathbf u \\wedge \\mathbf w + \\mathbf v \\wedge \\mathbf w", "0c096c184d170d6b414f22a50a3e751a": "\\forall j=1\\ldots n", "0c09bd7a597c583474a737d09277fe21": " (\\lambda \\mathbf{A})_{ij} = \\lambda\\left(\\mathbf{A}\\right)_{ij}\\,,", "0c09f3c458e9e3dc6827de4fb450c44f": "r > 0", "0c09f7cf43477384a29c85b212d4011b": "\\lambda<\\kappa", "0c0a0efb31a051e3494731d99ef48198": "F_{rad} =\\frac{dp}{dt}=\\frac{1}{c}\\frac{dE}{dt}=\\frac{1}{c}\\sigma_t\\frac{L}{4\\pi r^2}", "0c0a8efa62790179582465b52d21ebb8": "u^s_i", "0c0a964dc284211be87e0f8e07921516": "x^5-5x^4+30x^3-50x^2+55x-21=0\\,,", "0c0a9f9a225af75271a5858c14e5151b": "\n\\begin{matrix} e^{\\frac{-2\\pi i}{N} (k + N/2)} & = & e^{\\frac{-2\\pi i k}{N} - {\\pi i}} \\\\\n& = & e^{-\\pi i} e^{\\frac{-2\\pi i k}{N}} \\\\\n& = & -e^{\\frac{-2\\pi i k}{N}}\n\\end{matrix}\n", "0c0aae11e0e021ef3794eb3fbd032516": "t_0 < t_n \\le T = t_N", "0c0ad2f13e4a6b5f14501e7dd28894c3": "e \\in B^l", "0c0ad7cfc11d107f41d2adf03e70765b": "f(M_n) \\subset N_{dn}", "0c0af21e99b7d9eb9bd8aba7e1583352": "c^{\\ominus}", "0c0b12c5271ccb75740de607c9ac19a3": " p^v_m = \\frac{z^m}{m! \\Upsilon_v} \\sum_{t=0}^\\infty \\frac{z^t}{t!} \\int \\left [ \\prod_{i = 1}^m \\prod_{j = m+1}^{m+t}\\psi^v_i\\chi^v_j \\right ] {\\mathcal B}^{(m,t)}_{1...m+t} d\\boldsymbol{r}_1...d\\boldsymbol{r}_{m+t}, ", "0c0b23138f8ec80d499b381673c72806": "n'\\approx n-{}^{\\left( \\ln Kmn \\right)}\\!\\!\\diagup\\!\\!{}_{H}\\;", "0c0b3eb515c3727a7db32a481bfd45d5": "\\hat{B}^\\dagger_\\omega", "0c0b47cc72bd584dc2108f3dd23713ae": "V(y(x))", "0c0bdf34c56de318fa9b4ff37daa2934": "F(EG, X)", "0c0c002d0b395b4028c764d36b44dea7": "-\\frac{Nc}{4} \\delta_1", "0c0c03b7a1586434bc5e59ecc0504e5a": "\n\\langle 1 \\otimes h, 1 \\otimes h \\rangle _K = \\langle V^* h, V^* h \\rangle _K \n= \\langle V V^* h, h \\rangle _K \n= \\langle \\Phi (1) h, h \\rangle _H .\n", "0c0c09ef52369107540282a7678e508a": "\\lambda =", "0c0c0fd24d52fb79c981f545f955ebda": "\\theta y + (1-\\theta) z \\succeq x ", "0c0c3c995014ec50286f2d7d7b579891": "w'\\left(1/2\\right) = \\sqrt{2\\pi}.", "0c0c9af5af71205b8e65176982f15710": " \\langle f\\ , \\ g \\rangle = \\int_\\Omega f(x) \\overline{g(x)} \\, dx,\\,\\!", "0c0d9408882db21c5482265b1e554ead": "O(kH) \\cong O(k).", "0c0d9c0ccfe6a9001bf632dfeec22a06": " \\omega_0 = \\mu B_0/\\hbar", "0c0db4091b722d7f4de68706e55e0355": "\nA_n = \\sum_m u_m \\frac{\\partial \\phi_m}{\\partial q_n}\n", "0c0e10da81daa82ba76faccdfd3b9422": "X\\in\\mathcal{F}", "0c0e1a359ea8a473d2fb0278d8647ada": " A_1 ", "0c0e74956b0a057386c1e8741c5466fc": "\n\\begin{align}\n\\langle A \\rangle &= \\langle \\Psi (t)| A(\\hat{x}, \\hat{p}) | \\Psi(t) \\rangle \n= \\int dxdp \\, \\langle \\Psi (t)| x \\, p\\rangle A(x, p) \\langle x \\, p | \\Psi(t) \\rangle \\\\\n& = \\int dxdp \\, A(x, p) \\langle \\Psi (t)| x \\, p\\rangle \\langle x \\, p | \\Psi(t) \\rangle\n= \\int dxdp \\, A(x, p) \\rho(x,p;t).\n\\end{align}\n", "0c0e77a85361f6bab6d7eec38987c98e": "\\mathrm{Spin}(n)", "0c0e8ef8644b4c52694cb24f98307549": "S^{t}", "0c0ee3cbfeec6062f3065090f380cea5": " (Z_\\max) = (E_{u/p} - E_c) = (6.04 - 2.20) = 3.84\\text{ ft} \\,\\!", "0c0f1586d1964b316f588b0d11837fa9": "I = \\int d |i\\rangle \\langle i|", "0c0f190551d6f40da548430508fffe87": " NM_{\\cong_{\\mathcal{B}}}(X,Y) = \\Biggl ( \\sum_{z_{/\\cong_{\\mathcal{B}}}\\in Z_{/\\cong_{\\mathcal{B}}}} |z_{/\\cong_{\\mathcal{B}}}| \\Biggr)^{-1} \\sum_{z_{/\\cong_{\\mathcal{B}}}\\in Z_{/\\cong_{\\mathcal{B}}}}|z_{/\\cong_{\\mathcal{B}}}| \\frac{ \\min (|[z_{/\\cong_{\\mathcal{B}}}]_X|,|[z_{/\\cong_{\\mathcal{B}}}]_Y|)}{\\max (|[z_{/\\cong_{\\mathcal{B}}}]_X|,|[z_{/\\cong_{\\mathcal{B}}}]_Y|)}. ", "0c0f5bb2224ed5ef79f1d5e2dc9dd76a": "R=\\frac{\\sqrt{3}}{3} a", "0c0fce579020b772e95d28e87e000b6a": "L= 2S-\\frac{120 + 3.5S}{A}", "0c0fde8b7d4673414f7a682e88a32fda": "\\left\\langle \\frac{\\rho[\\sigma]}{p[\\sigma]}\\right\\rangle_p \\propto \\exp(-f V/T)", "0c10658d5372fd2b18ac40b4a2457bb5": "{\\displaystyle}x^2 + y^2 = 1 + 2 x^2 y^2", "0c107a0b847837924f555bd7ffb3dca3": "V_i,V_j,V_i \\cup V_j", "0c10cdd8e7d73c155529dbf095208858": "h_L = 0.664{k \\over x} Re^{1/2}_L Pr^{1/3}", "0c10dddf0759af59d669fce9e0f12c78": "O(v)", "0c10e0221869098cd4b5731f2e5e2f14": "\\overrightarrow{q}", "0c10eb8ea451fdb97907ce6e62020bbb": "D_i D_\\ell", "0c111535b4d4317c2277da8b096e65a8": "U_{ij},\\, U_{ji}", "0c118ebebd5266d9575f0f2914eaa475": "\\mathcal{S} (X) = 0", "0c11f3ca204cd2f15bc59644f57056a9": " C \t= \t\\frac{1}{4}bk\\left(k+1\\right)^2 ", "0c120ce52e22bd84a5eed1b86ee99ba4": "\\det(A) = \\sum_j A_{ij} \\mathrm{adj}^{\\rm T} (A)_{ij}.", "0c12383ae73de5946d6cceef92ce87fd": "\\sum_{m=0}^n P(2m+1)=P(2n+4)-1", "0c12445eb8a7aa13b4c74ef46ef008a4": "V = \\begin{bmatrix} 29 & 139 & 529 & 391 & 1037 & 647 & \\cdots & 17191 \\end{bmatrix}", "0c1274619d75dc135fb153942fad5130": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) - 1 > 0 ", "0c12dbdd0dec0d13fb71ead181d2a7c4": "s_y = 0", "0c131335e1615c37f1d231c6e17b4327": "\n\\lbrace w_1, w_2,\\dots,w_k \\rbrace ", "0c1355394c2c2e76394c663a71c0f0d9": "c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}} \\ . ", "0c13642ada76dc2d40ef3ac5e7da2912": "\\langle \\alpha, \\beta \\rangle", "0c13aa1b6f9f53919ab6009029121371": "\\varepsilon_\\gamma", "0c13c702dfc8ccdce6f30103b5f8beba": "\\precsim, \\precnsim, \\precapprox, \\precnapprox \\,", "0c13f9b96bd652367361c120ce52317a": "\\tfrac 3 2 q", "0c14546d5fd1458589f4f31478067969": "\\hbar = 0", "0c14d40b843d92a11b43eb71747e4fa7": "A[n] \\times A[n] \\longrightarrow \\mu_n.", "0c157b34bbc0875a3e53e16edb8de05e": "\\frac{\\partial \\langle H \\rangle}{\\partial a_{n'}^{*}}\n= i \\hbar \\frac{\\partial a_{n'}}{\\partial t} ", "0c1619879060b38ba844e510c8e4430c": "H_0: \\theta_1 = \\theta_2 = \\cdots = \\theta_k", "0c1632efe6d8dc956be0e68daef11fce": "A'(z) = A(z/\\gamma)", "0c16c24482ddd3b621a2213b6a2407cc": "\\hat{m}_{ij} = a_i b_j", "0c16e66a0b815eb7c3781ac7a77033e3": " \\frac{1}{d} \\sum_\\alpha \\Pi_\\alpha = I", "0c16eb8e1e1c2393545fe127a7550705": "\\frac{\\partial L}{\\partial t}=0", "0c16f3856d50cb76bc98107d3c03e3ec": "\\bold{p}\\rightarrow R(\\bold{\\hat{n}},\\theta)\\bold{p}", "0c171df1127b2ac1e31f4580ba3021f2": " |\\nu_{i}(L)\\rangle = e^{ -i m_{i}^2 L/2E }|\\nu_{i}(0)\\rangle.", "0c176f787421c87d7155eacd810d561b": "L_{\\nu, p}(R^{+})", "0c17b53846ed571eb2c65aa430c050aa": "\\bar v_N := \\frac{1}{N} \\sum_{n=1}^N v_n", "0c18075a4e9e42ee0f96fe15c675b171": "\\kappa E_0", "0c1897114b3e3243f27526700c6f1f00": "(a\\cdot b)\\cdot c=(a\\cdot (b\\cdot c))\\cdot (a,b,c)", "0c189b999666c77000c1f09ae29cb7de": "y_1\\ge y_2\\ge\\cdots\\ge y_n,", "0c18b4cb842730428d3b92fba3042f6d": "\\alpha = f \\, dx_{i_1} \\wedge \\dots \\wedge dx_{i_k} + g \\, dt \\wedge dx_{j_1} \\wedge \\dots \\wedge dx_{j_{k-1}}.", "0c190c34e6cb454f22ec65fa338ca766": "\\omega\\times\\omega", "0c191749537a5ff60f927f96b504f2e0": "\\sqrt{det (q)} H", "0c195e94426b178d1e295fdf862998c2": "M=N_1\\#N_2, ", "0c197f3d26123c6fe4dd354f798891eb": "\\frac{1.4388}{1.438}", "0c19e4da62f6dd962a132438c9a7f5f6": "2g+1", "0c1a219e0036fa19beaca1c41a0c5fdc": "N = -z\\frac{\\partial\\Omega}{\\partial z} \\approx\\frac{\\textrm{Li}_\\alpha(z)}{(\\beta E_c)^\\alpha}", "0c1a5d0bc626112747fb63c296c65b0e": "{n \\choose 4} \\times 6.", "0c1a725b546068462a7dbad787e6e0ac": "\\exist W(x \\in W);", "0c1a8a8b0416b2799486361bd34b37d2": " f_{n+2} - f_{n+1} = \\sum_{k=0}^{2^n - 1} \\bigl( f(x_{n,k}) - f_{n+1}(x_{n, k}) \\bigr) s_{n, k} = \\sum_{k=0}^{2^n - 1} a_{n, k} s_{n, k} ", "0c1b346494d63e806e5ffe4f4526887c": " H = H_{\\rm entropic} + H_{\\rm external}= \\frac {1}{2}k_B T \\int_{0}^{L_0} P \\cdot \\left (\\frac {\\partial^2 \\vec r(s) }{\\partial s^2}\\right )^{2} ds - xF", "0c1b3dd11f5e83d97f632130dbca773a": "e^{\\log(z)} = z", "0c1b5d1678bea6cf742a08b99d0946bc": "am \\equiv 1 \\pmod b", "0c1ba3d8616d00752aba9843186681bd": "P_{c}(t) = \\frac{1}{2N_e} e^{-\\frac{t-1}{2N_e}}.", "0c1baa806043f51f22726a9a7768d5af": "\\cos nx = \\sum_{k=0}^n \\binom{n}{k} \\cos^kx\\,\\sin^{n-k}x\\,\\cos\\left(\\frac{1}{2}(n-k)\\pi\\right).", "0c1bc22e629b9d940664b19c6525adf4": "\\alpha \\, \\!", "0c1c3b736d4f580275f93960c47792c2": "t_r'", "0c1ce8988c239efcb2d8cc865a4c8116": "-a\\,", "0c1cf7ea98e097d69d6bf7acb1d6a01d": "\nx = \\frac{r_1^2-r_2^2}{4a}\n", "0c1d01b7ce7a8d098177395f4b31b977": "S=\\{s|s \\;is \\;a\\; source\\}", "0c1d1f0ee4fffd269dc2f032b8aa4b2c": "h(\\Beta(\\alpha, \\beta) )= h(\\Beta(\\beta, \\alpha) )", "0c1d56af4b6f9632b0c85c630b629921": "(x-3)(x^{15}-22x^{13}+x^{12}+184x^{11}-26x^{10}-731x^9+199x^8+1383x^7-576x^6-1061x^5+561x^4+233x^3-151x^2+4x+4)^2", "0c1d9062bf6be15690bdffb6543d888f": "\\frac{\\dot m_0}{\\dot m_{01}} = \\sqrt{\\frac{T_{01}}{T_0}} \\sqrt{\\frac {\\epsilon_0^2 (1 - \\epsilon_c)^2 - (\\epsilon_2 - \\epsilon_c \\epsilon_0)^2}{\\epsilon_{01}^2 (1 - \\epsilon_c)^2 - (\\epsilon_{21} - \\epsilon_c \\epsilon_{01})^2}}", "0c1da82d929af9e3fef154ae5ca7fa67": "\\!\\mu_1", "0c1db10a37337f20040d4edeae94a163": "\\alpha_2 < 4\\times 10^{-7}\\,", "0c1dc327f2873cc0b82566eab4ad1ade": "x \\geq \\theta", "0c1dc69a469ddd00e3c6b018d304cea6": "\n\\mu = \n\\sqrt {\\lambda ^2 - k^2 } \n", "0c1e1043fbe7bb2bf09a5ef1bfb46969": "-\\int_a^b \\mathbf{E} \\cdot \\mathrm{d}\\boldsymbol{\\ell} \\neq V_{(b)} - V_{(a)}, \\, ", "0c1e32860509b3d417a0130e5fe1c423": "\\frac{d u_i}{d t} + \\frac{1}{\\Delta x_i} \\left[ \nF \\left( u_{i + \\frac{1}{2}} \\right) - F \\left( u_{i - \\frac{1}{2}} \\right) \\right] =0, ", "0c1e60e464562eb6bd5f2c698ac02edf": "E[X] = E[Y] = 0", "0c1e6f7f4e55639c4927413bf8adab48": "\\lambda(x) = C (t-x)^2 \\; ", "0c1e7bab84b67daaddea150ea0bf1f66": "N_\\mu^\\perp=\\{\\sigma\\in ba(\\Sigma) : \\mu(A)=0\\Rightarrow \\sigma(A)= 0 \\text{ for any }A\\in\\Sigma\\},", "0c1eede8238255b8365fcc1adb7bfd04": "\\Pr(R \\cap B \\mid Y) = \\Pr(R \\mid Y)\\Pr(B \\mid Y),\\,", "0c1ef12c1e2845808d48b347e95f12cf": "y,z", "0c1f0fcf7fde4b7ed268de595f92dfcd": "F_{eq} = F_1 = F_2 ", "0c1f13a0859aaf0f480228f6abff028d": "F = \\{f_i\\}, \\ i=1 \\ldots n", "0c1fa56e9836372889e4687fb9815fd5": "\\bold B", "0c1fc80974b467f49ca07ae0ea6d96be": "c_V\\,", "0c1fd9ed7287bf152bc0ce6850184446": "I_i = 0^+ \\rightarrow I_f = 1^+ \\Rightarrow \\Delta I = 1", "0c1fed1ee0ea5c982b1aea22e5413c10": "x^4=x^2y-y^3. \\,", "0c1fff627dcee3b3df85c770986d9163": "\\scriptstyle\\bar \\theta", "0c20063a215893c46ff0dbc158869941": "A_{(\\alpha\\beta)\\gamma\\cdots} = \\dfrac{1}{2!} \\left(A_{\\alpha\\beta\\gamma\\cdots} + A_{\\beta\\alpha\\gamma\\cdots} \\right)", "0c203fdedbfe4c27483d4dfbc1ddc306": "5394826801 = 7 \\cdot 13 \\cdot 17 \\cdot 23 \\cdot 31 \\cdot 67 \\cdot 73\\,", "0c20a0a5b6db7c7e1912f8f8d321f00d": "A \\rightarrow S: \\left . A, B \\right .", "0c20c5eb68af7b3ffaf0ed46c392f788": "\\langle", "0c20fe4791299aca97aaebe841e23bcb": "\\mu_1 \\dots \\mu_j", "0c21007f78283a9873ead0e94f6a8abb": " \\|s\\|_n = \\sum_{j=0}^n \\sup_{x\\in M}|D^js| ", "0c21b31adf16d37874174fb589b0b454": "\\mathfrak{f}_{(p)}(\\chi)", "0c233d9a76d35bb0c0655f5d0d36e552": "\\exp(ahr) \\exp(bhr) = \\exp((a+b)hr). \\!", "0c2373886b2b0cf01f37b7681a519837": "x*(y+z)=\\overline{x(y+z)}=\\overline{xy+xz}=\\overline{xy}+\\overline{xz}=x*y+x*z,", "0c2391a2b9ae4fbc9bd219126028702c": "f^{n+1} ~ \\stackrel{\\mathrm{def}}{=} ~ f \\circ f^{n},\\,", "0c23bc8a42ed5c2babc4197a8e0de622": "\\scriptstyle \\epsilon_c \\,", "0c23ceae2a6f54e4907dd361a3494e7a": "\nF(x) = \\mu(-\\infty,x] = \\int_{-\\infty}^x f(t)\\,dt.\n", "0c23d1ca571cf2d600bea055f89a751b": "R^{(p)} = A[X_1, \\ldots, X_n] / (f_1^{(p)}, \\ldots, f_m^{(p)}).", "0c23f865dcb1eb12861e54b0c505ca4d": "\\mathrm{O}_2", "0c24169b81a093490bc0585b22aca710": "\\Gamma,\\Lambda", "0c24362f3df681547cd3e39e4d503fae": " y_l", "0c24653cc31cd12e2c68cff3f7889b5c": "\\nabla \\times \\mathbf{H} = \\frac{4\\pi}{c}\\mathbf{J}_{\\text{f}} + \\frac{1}{c}\\frac{\\partial \\mathbf{D}} {\\partial t}", "0c24b2132650458e6b7be43680d4a0e4": "Wp", "0c24f9e1d3a1d8bd98a052d6bd20c2e8": " {N}^{(n)}(B_1\\times,\\dots,\\times B_n)= \\sum_{(x_1\\neq,\\dots,\\neq x_n)\\in {N} } \\prod_{i=1}^n \\mathbf{1}_{B_i}(x_i) ", "0c24fe0412722140a94debe3cf15a57c": " M = \\sup\\limits_{x \\in C} \\Re [S(x)] < \\infty \\quad ", "0c2577954896e67091440e5e07485860": "\\forall x \\forall y \\exists z \\forall w [w \\in z \\leftrightarrow (w = x \\or w = y)].", "0c25c6ac9bc84c799b0647e64fead93b": "d_w = 0.767 + (2 * 0.1 (1 - 0.767)) = 0.813", "0c26082951aba3022da93a06b6a1965f": " \\gamma(u) = \\gamma u \\gamma^* = \\gamma^2 u\\,", "0c26b457ba9d8a105597dfd7a18616f1": "\\alpha \\colon S \\times S \\to A", "0c26ef5902519ca467f9a1880c167862": "1 + \\frac{a_1\\dots a_p}{b_1\\dots b_q}\\frac{z}{1!} + \\frac{a_1(a_1+1)\\dots a_p(a_p+1)}{b_1(b_1+1)\\dots b_q(b_q+1)}\\frac{z^2}{2!}+\\dots", "0c26f9d81df7481339ccdb89f436a653": "A, U, B \\text{and} V", "0c272339c176ecd5eb12f16a953fcb87": "\\operatorname{E}[X_{t+1} - X_t \\vert X_1,\\dots, X_t]=0\\,,", "0c272fee72d2b94860c3c747bf23b02f": "f(\\zeta) = \\frac{1}{2\\pi i}\\int_{\\partial D} \\frac{f(z) dz}{z-\\zeta} + \\frac{1}{2\\pi i}\\iint_D \\frac{\\partial f}{\\partial \\bar{z}}(z) \\frac{dz\\wedge d\\bar{z}}{z-\\zeta}.", "0c273a92c5148554044d961a49029fd6": "{\\mathbb P}^{2k+1}{\\mathbb C}\\, ,", "0c27533d5dd29798f72ffb7ac1ffc57d": "\\text{with }{_2\\text{F}_1}(a,b;c;z) = \\sum_{k=0}^\\infty[(a)_k(b)_k/(c)_k]z^k/k!", "0c275399c73ee1b70950b836943c1995": "m_i \\leq q_i\\rho, i = 1, 2, ..., N_{sd}", "0c27d6ba59972ab85de78ca644ffe9cb": "y_{\\text{ave}}", "0c2880c0faaadc90a6e329ffd33ce241": "h(k2)", "0c28aa9ccb33a2d9e8302c51f12f8131": " \\tau ", "0c28b81f87c79a4272b324a221f99848": "b_1 = V_1^-", "0c28f1f7661894e6619a2fea912dd938": "\\hat{\\boldsymbol{\\beta}} = \\underset{\\boldsymbol{\\beta}}{\\operatorname{arg\\,min}}\\,S(\\boldsymbol{\\beta}), ", "0c28f2e47422c88c275ea7ffeeee7e41": "P^{A}_\\alpha", "0c28f8287babf9fac9272dc8d1cee8ce": "\\chi \\rightarrow \\{-1, +1\\}", "0c28fd74f84a23796ccb3084e1928034": "WBGT=0.7T_w + 0.3T_d", "0c29162a8daac1a3781c97a656db5da0": "\\scriptstyle g(U) \\,\\cap\\, U \\;\\ne\\; \\emptyset", "0c2929e1f6890b113452d9bc5c896609": "f(z) = \\frac{z+2}{(z-5)^2(z+7)^3}", "0c29d9a0a7f5fb012b48025e83af4c99": "\nD \\to \\frac{k_B T}{8 \\eta_m h a} \\frac{L_{sd}}{a}\n", "0c29ec88e10b2d0655aed9f334c998c4": "q_0\\,\\in Q\\,", "0c29f1ca956ebc631b02139b4503ee14": "v,v'\\in V\\,", "0c2a24393c01f34e1ec0c3f6a9bf3391": "E_x^{\\rm PBE,SR}(\\omega)", "0c2b7be478cb3c91c3c7f50457471d8c": " \\sum_{n=1}^\\infty (-1)^{n} a_n = - a_1 + a_2 - a_3 + \\cdots \\!", "0c2b81a4d22a30289a24ed0cf4ea8e22": "\\forall y\\exists x (x^2=y)", "0c2b95541a022c82ebbf0248ee90dca2": "\\sin 87^\\circ = \\cos 3^\\circ =\\frac{\\sqrt{60 + 12\\sqrt5}+\\sqrt{20 + 4\\sqrt5}+\\sqrt{30}+\\sqrt2-\\sqrt6-\\sqrt{10}}{16}", "0c2b9ba7364d8db8614b0d602ee6de2f": "\n\n\\mathrm{Efficiency}=\\frac{\\mathrm{Useful\\ power\\ output}}{\\mathrm{Total\\ power\\ input}}\n", "0c2b9c94aa69394d0652550c8f20f64a": "(f_1 + f_2)-(f_3 + f_4)", "0c2bb2aebb69d4ace76e27deb93f2643": "\\hat{\\lambda}\\,", "0c2bbff22a71fef5b363acb30f0dc041": "\\frac{x-\\lambda-a_1}{a_2-a_1}", "0c2be49ff2494b14ce9713e5af78ea81": " ~\\epsilon_{t-1}^{-} = ~\\epsilon_{t-1} ", "0c2c06c343d5d73164aba01fd3fef344": "V^*,", "0c2c747f9ef05a0fffb04221c89403bd": " i=\\mathbf{e}_2 \\mathbf{e}_3, j=\\mathbf{e}_3 \\mathbf{e}_1, k = \\mathbf{e}_1 \\mathbf{e}_2, \\,\\, \\varepsilon = \\mathbf{e}_1 \\mathbf{e}_2\\mathbf{e}_3 \\mathbf{e}_4. \\!", "0c2cb0029c3418660112059ac23cc16f": "x\\in(1/4,1/2]", "0c2cb856cefb469efe1610c9d08a2485": "X_0=N", "0c2cc1d3937d818248ebe7a3c8858e3a": "\\widehat{K_\\alpha}(\\xi) = |2\\pi\\xi|^{-\\alpha}", "0c2d020c8454a43077e27d19dfa9f909": "\\det T_{g*h} = \\det T_g \\cdot \\det T_h \\cdot \\mathrm{res}(g_-,h)", "0c2d1663b1ccfe8134cb66c248945b06": "A = \\text{ constant}", "0c2d251e761eefb82204185f07130afa": " \\textbf{r} = -1+X^2+X^3+X^4-X^5-X^7 ", "0c2d954320884adb96cc0b8ee65596ab": "(1 - q)(1 - q + q^2)p^2 - q(1 - q)(1 + 2q)p - q^3 > 0\\,", "0c2d9b32387dd0c133969c924f6a9f20": "\\scriptstyle n_i", "0c2e12b1333082e00715b3b4f93d0278": "\\mbox{Free}(\\phi \\vee \\psi) = \\mbox{Free}(\\phi) \\cup \\mbox{Free}(\\psi)", "0c2e423c9aecb851b82593f9b4b37709": " \\omega = y_n ", "0c2e55a5f834d6666a6c66f1606127de": "\\left. {} -\n 33\\left(\\theta_2(0,q)^4+\\theta_3(0,q)^4\\right)\\cdot\n \\theta_2(0,q)^4\\theta_3(0,q)^4\n\\right]\n", "0c2e6c44260a4a2281013648a888fa43": "x \\circ y = \\frac12 (x \\cdot y + y \\cdot x),", "0c2e7b202ad7b0f4f1d92eab5ebde3f0": "A_k \\subseteq A_{k+1}", "0c2e8467d8bfa408220d11ac6afbc324": "|P| = |S| \\cos\\varphi", "0c2f20b2bd4e6fcce67554bcdad32753": " \\max \\left [ F_{l} , \\left ( D_{l} + \\frac{ F_{l}}{2} \\right ), 0 \\right ] ", "0c2f24503b43c4d42ad2c7d55e3d6acf": " x | \\sigma^2, \\mu, \\lambda\\sim \\mathrm{N}(\\mu,\\sigma^2 / \\lambda) \\,\\! ", "0c2f8cc05147d963bf195e3335014324": " \\left [ \\hat{ x }, \\hat{ p } \\right ] = \\hat{x} \\hat{p} - \\hat{p} \\hat{x} = i \\hbar. ", "0c2fbc79e19643b0dd3d5cc0f4479d54": "\\sigma(t)= \\frac { \\sigma_0 }{ 1-[1-(t/t*)(1^{1-n})]}", "0c2fd170e5e47a8e27ece0f7b6739959": "\\sigma \\in \\Delta", "0c301a02221077fa56c1922a93606f56": "\\ GL_{n}( \\mathbb{Z}_{p}) ", "0c30282696cb5839ad1ecf78035363a5": "K^{\\Dagger '} = e^{\\frac{- \\Delta G^{\\Dagger }}{RT}}", "0c316a80e3e9ab49a2585086b231ec6a": "I \\subseteq R_+", "0c31e09df9adb9361574e270ccac3457": "\\tau_c*=0.03", "0c31f2c011d37fc982efa83760ff76b7": "\\tau_{ri}", "0c32205b35444bd5bac9b1478702b651": " return: fail", "0c323b0d06b6ea86c6963fec7cd0505b": " \\frac x{\\cos\\gamma} + \\frac y{\\sin\\gamma} = 1, ", "0c328a9e72a1840913bfc6da64ebee16": "\\{U_i\\}_{i=1}^k", "0c328b77399b3a1de4a25ff8cdccb10e": "F_{Y}(y)=P(Y\\leq y)", "0c32eae5ea0138451eca1d877243b730": "f_B", "0c32f76e19392dbf43da275a97e5fc13": "P(B \\mid A) = \\frac{P(A \\cap B)}{P(A)}.", "0c332bde051526f0124f664b00d90eba": "\\textstyle i>m", "0c338bd49d71c8383995854260581ed4": "\\mathcal{H}_{Heis} = \\frac{1}{2}(-2J\\sum_{i,j} \\vec{S}_i \\cdot \\vec{S}_j\\quad) = -\\sum_{i,j}J \\vec{S}_i \\cdot \\vec{S}_j ", "0c33ab1e5d620c71ecbdc529f20ca4ec": "[t, t+h]", "0c342ec47723a3a13b624bc44c98d94e": "\\nabla f = \n\\frac{\\frac{\\partial f}{\\partial x} - \\sin(\\phi) \\frac{\\partial f}{\\partial z}}{\\cos(\\phi)^2} \\mathbf e_1 + \n\\frac{\\partial f}{\\partial y} \\mathbf e_2 + \n\\frac{-\\sin(\\phi) \\frac{\\partial f}{\\partial x} + \\frac{\\partial f}{\\partial z}}{\\cos(\\phi)^2} \\mathbf e_3.", "0c344f8ccee98f30cff89cbda229bf4d": "\\tau_\\mathrm{delay}=\\underset{t}{\\operatorname{arg\\,max}}((f \\star g)(t))", "0c3451207604f71dde486c37745fad7e": "_5^6", "0c3455048afb4a63fde93c0bc96c8b95": "\\scriptstyle\\frac{1}{2} m^2 A_{\\mu}A^{\\mu}", "0c3470f889fbae5a424685f7fcef7143": " W=\\Delta F ", "0c34afb31a801725cb54bf74e16e5b83": "(r,r-1)", "0c34d7a043bd5ea7924c45aaec774e98": " x + 1/x = - 1", "0c351e3856c4c03a080caae1cf5bca22": "E_{n}^{(1)}=\\langle\\psi^{0}\\vert H'\\vert\\psi^{0}\\rangle=-\\frac{1}{8m^{3}c^{2}}\\langle\\psi^{0}\\vert p^{4}\\vert\\psi^{0}\\rangle=-\\frac{1}{8m^{3}c^{2}}\\langle\\psi^{0}\\vert p^{2}p^{2}\\vert\\psi^{0}\\rangle", "0c353cae3830e4aa1ac6fca418796f60": "\\cos (\\arcsin x) = \\sqrt{1-x^2}", "0c355f0ebd208bc310f3494422a23908": "\\langle y|R\\rangle\\langle R|x\\rangle,", "0c361efc0d92fc39bd5527f847f69f71": "[x] = \\textrm{cl}\\{x\\} = \\bigcap\\mathcal{N}_x.", "0c362270aed94e591d1c374764a5a327": "\nf(\\varepsilon) \\sim \\varepsilon^\\gamma \\,\n", "0c364d87aabab19b046e5f5d7149d4c0": "k_1, \\ldots, k_m", "0c364fc7967fd3dd4ba9c17d9be44536": "\\langle p_{\\mathbf{R},i} | \\tilde{\\phi}_{\\mathbf{R},j} \\rangle_{r T_{setup} + T_{ko} + T_{skew}", "0c3d701f6160b320f39c480588c6ffa4": " T = \\begin{bmatrix} T_{11} & T_{12} \\\\ 0 & T_{22} \\end{bmatrix} ", "0c3d72395d7576ab13b9e9389f865960": "P(X)", "0c3e08e83a5937916dcb653ad1cb799a": "a_{i}, 1 \\leq i \\leq n", "0c3ea8f4cdbf7b626bb9b2f09477b632": " M = M_{ii} = \\frac{\\mu_0}{4\\pi} \\left ( \\oint_{C}\\oint_{C'} \\frac{\\mathbf{ds}\\cdot\\mathbf{ds}'}{|\\mathbf{R}_{ss^{\\prime }}|}\\right )_{|\\mathbf{R}| > a/2}\n+ \\frac{\\mu_0}{2\\pi}lY + O\\left( \\mu_0 a \\right ).", "0c3ec16ce2495229ab5b269d764562b4": "\\textstyle \\exp(rT) = R_0", "0c3ee0b09d3c337e066308ae140704c1": "\\frac{d\\phi(t, t_0)}{dt} = A(t)\\phi(t, t_0)", "0c3ef3f7b93ab6b21572c7134390378a": "1 - (\\lambda + \\mu\n) \\Delta t", "0c3f0e3a4608bac89f6cebab322f2d22": "b \\ne 0", "0c3f7f0c6f72effbc12058ea4a3ba10e": "M = -\\int_{\\mathcal{X}}\\left(\\frac{\\partial \\psi(x,\\theta)}{\\partial \\theta}\\right)_{T(F)}dF(x)", "0c3f8fc2259e29eee543ec9029e53603": "Y\\sim N_n(0,\\sigma^2I_n)", "0c3f980d93ca95ace0071700f6088ce3": " d\\tau^2= dt_r^2 \\left[ 1-{\\frac {2M}{r}} + 2\\sqrt{\\frac{2M}{r}} {\\sqrt{\\frac{2M}{r}}} -{\\sqrt{\\frac{2M}{r}}}^2\\right]=dt_r^2 ", "0c3fbe9776491ed80ea80723fc6503e2": "z^2+w^3=0", "0c3ff8a5bbfaeafb675fca98d0741432": "\\tau^{\\alpha \\beta} \\,", "0c4017f6a2401c4b5bd1c02adebe33da": "\\gamma^2 = \\gamma", "0c4024bf6c05fb544400bdd856c58b91": "\n \\mathcal{L}^{-1}(x) = \\begin{cases}\n 1.31\\tan(1.59 x) + 0.91 x & \\quad\\mathrm{for}~|x| < 0.841 \\\\\n \\tfrac{1}{\\sgn(x)-x} & \\quad\\mathrm{for}~ 0.841 \\le |x| < 1\n \\end{cases}\n ", "0c402d854ca6f159ecfcd2dddc05e23b": "A\\subseteq {\\rm pcf}(A)", "0c404174baa6c416a814b2486a5841d0": " f'' ", "0c40462f440862674116eec19a92ecb2": "(x-x_1)(x-x_2)\\cdots(x-x_n),", "0c4056d0f08a2d26e7374c773b39d165": " \\nabla \\times \\nabla \\times \\mathbf{E} = -\\mu_o \\frac{\\partial } {\\partial t} \\nabla \\times \\mathbf{H} = -\\mu_o \\varepsilon_o \\frac{\\partial^2 \\mathbf{E} } {\\partial t^2} ", "0c4059314ecc30ff4ebc41f6fc6138ed": "\nQ_1^{(\\text{green})}(t) - Q_2^{(\\text{green})}(t) = 2\n", "0c408f36576e603377026d3e20ce57dd": "f(x) = \\begin{cases} 0 & \\mbox{if Goldbach's conjecture is false} \\\\ 1 & \\mbox{if Goldbach's conjecture is true}\\end{cases}", "0c40923b462b71b5b7a9ccc35962430a": "\\sum\\limits_ {a_i \\in S} a_i=|S|a_j", "0c40a0a2d82dd414c5ac5deb9b589647": "0.\\overline{571428}", "0c4134fd43aada42849770ac26005c8c": " \\frac{ r }{ m } - \\ln( m ) - 1.24 = 0 .", "0c4147b63c68ff83b76223526c582702": "|0\\rangle^{\\otimes n} |1\\rangle ", "0c41d355882df8f3b078b2ce48100f31": "a,b,\\dots,k", "0c41e040e70e48def3f230eddd9001a0": "4+\\sum_{i=1}^n i\\leq 4+\\sum_{i=1}^n n=4+n^2\\leq5n^2", "0c4209a8faecbcee233e5df1e1751e24": "\\frac{6}{5}", "0c42406640583a31bf9fd0948db1b66f": "\\delta_a:Q\\rightarrow Q", "0c4266cc913627ce579cfed1d48652ed": "P_0 \\cdot C_0", "0c4281bde96d7f1b2681313f3538a989": "\\begin{alignat}{2}\n ax_1x_2+bx_1y_2+cx_2y_1+dy_1y_2&=&\\alpha\\\\\n ex_1x_2+fx_1y_2+gx_2y_1+hy_1y_2&=&\\beta\n\\end{alignat}", "0c42896a9b4e598aa2b9fab46328985a": "\\partial_a = \\frac{\\partial}{\\partial x^a}", "0c42918eb31efe0ba6711ac312f23003": "t_l=2w", "0c42a7a275ba8863cd1732e2a3be4b66": "e_q( \\ln_q(x)) = x ", "0c42b89ac347eca8baa6f35f36ec6fc1": "\\mathbb{E}[1_{D(y)\\neq m}]", "0c42e7346e7a3f464d1648dfcf625ea1": "P(78 \\le X \\le 88) = P\\left(\\frac{78 - 80}{5} \\le Z \\le \\frac{88 - 80}{5}\\right) \n= P(-0.40 \\le Z \\le 1.60) = P(Z \\le 1.60) - P(Z \\le - 0.40) = 0.9452 - 0.3446 = 0.6006", "0c42fdfcc5b5e6601e398d7c71216db3": "\\operatorname{E}[S_N]=\\operatorname{E}[N]\\, \\operatorname{E}[X_1].", "0c431b73a43e17926497f3204abccdc4": " \\frac{\\partial f}{\\partial t} + r \\cdot \\nabla_r f\n- \\frac{1}{\\hbar} \\nabla_r V \\cdot \\nabla_k f = \\left(\\frac{\\partial f}{\\partial t}\\right)_c\n", "0c432282fcdfca30f53df2a2479edc8a": "|\\lambda|< 1", "0c436009c4714ade991c78a204cc8a50": " poly( \\lambda ) ", "0c438a31479ea3ed8039f29013e82599": " A_{n} = \\frac{P}{\\sqrt{n(n+1)}} ", "0c43b5bc69ca67f4d78f82cead712168": "\\sigma_{fail}=\\sigma_Y", "0c447ddede8f365b49ffc8c963bda7d1": "\\{x_1, x_2, \\dots , x_n\\},", "0c45507111319c8b237201dca0fd2c4a": "dT_V=pd\\Theta", "0c455c9e4175f2093f9c3b61a97e40e3": "\\mu_r = [B_1,B_1^\\dagger]+[B_2,B_2^\\dagger]+II^\\dagger-J^\\dagger J,", "0c45ac606062178381e49d38b9a979b1": "\na - x_{n} = \\frac{- 2 f(x_n) f'(x_n)}{2 [f'(x_n)]^2 - f(x_n) f''(x_n)}\n- \\frac{2 f'(x_n) f'''(\\xi) - 3 f''(x_n) f''(\\eta)} {6(2 [f'(x_n)]^2 - f(x_n) f''(x_n))} (a - x_n)^3.\n", "0c45da996841cbcd3fe0838e9687efd6": "\nds^2 = - \\left( 1 - {2V(r)\\over c^2} \\right) c^2 dt^2 + dx^2 + dy^2 + dz^2\n", "0c461183e2e4008c3661a14cc34a8865": "\\sqcap X", "0c46240353e703ab4ab54295d7bc9850": "(x-1)^3x", "0c4633ddfd76a62570b420c5a7856370": " \\begin{align}\\varphi(g)\\colon V& \\to V\\\\\nv & \\mapsto \\Phi(g, v)\\end{align}", "0c4676a2a5438e2ca500a1b4cfa25964": "\\mathbf{b}: [0,T] \\times \\mathbb{R}^N \\rightarrow \\mathbb{R} \\in L_2[0,T] ", "0c467ede3869016ea059e85bc3cfb6e1": "m^2=\\eta_0^2(p_\\mu,\\eta)-\\eta_i^2(p_\\mu,\\eta)", "0c4692635510aa9eeba461cf1f217118": " V^*=\\bigcup_{i \\in \\N }V_i = \\{\\varepsilon\\} \\cup V \\cup V_2 \\cup V_3 \\cup V_4 \\cup \\ldots.", "0c46bfc49d48f31f1cdb0d6336e67ddb": " f_n = X_n + X_{n+1} + X_{n+2} + \\ldots ", "0c46ccd184562b893e8633b518624511": "X_{k_1, k_2, \\dots, k_d} = X_{N_1 - k_1, N_2 - k_2, \\dots, N_d - k_d}^* ,", "0c46cdb789fd6cab181a3d2b342dfcae": "P\\ =\\ 2\\pi\\ a\\ \\sqrt{\\frac{a}{\\mu}}\\,", "0c473235a11c4f3d289fe2eed7c5ebe6": " {\\mathit l\\over n}={1\\over 2} \\pm {1\\over 2n}", "0c473afc7ba9849eae3939c0013625b4": "r_{li}", "0c4748f3be6412c3e9b547e11d0a0131": "R_{nl}(r) = r^{l}f_{nl}(r)\\exp\\left(-\\frac{Zr}{n}\\right),", "0c474e9e82aa677acf9b6a4014372a97": "\\mathrm{^{241}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{242}_{\\ 95}Am}", "0c477408ace07085a18b6c66198492a0": "Y = \\log(X)", "0c4787b8aac88ac5488ee69523a05381": "\nK = \\frac{(1-e^2\\sin^2\\phi)^2}{b^2}\n = \\frac{b^2}{a^4(1-e^2\\cos^2\\beta)^2}.\n", "0c4795612ee01990b45d17463059ab7b": "\\textstyle\\frac{a}{b}", "0c4809ff81769d7213ad56e521c117f5": "w\\mapsto R(u,v)w", "0c480c04e486fb18c3d016dd168a5798": "Q_3 = (\\sigma \\, z_3) + X", "0c482cabe693c6a06288dbfac3527be2": "\\textstyle q^{n-r}", "0c48abe0126552b7382e70c3801b8bc2": "\\begin{bmatrix}\nu_{11} & u_{12} & . & u_{1n}\\\\\n0 & u_{22} & . & u_{2n} \\\\\n. & . & . & . \\\\\n0 & 0 & . & u_{nn} \\end{bmatrix} =LDU= \\begin{bmatrix}\n1 & 0 & . & 0\\\\\nl_{12} & 1 & . & 0 \\\\\n. & . & . & . \\\\\nl_{1n} & l_{2n} & . & 1 \\end{bmatrix}", "0c4927294c93f32340903c76ccd6210d": "\\mathcal{A}_\\mu = t_a \\mathcal{A}^a_\\mu ", "0c4937f25a689e866c45c67435cabb4c": "W(dx) = dx / (\\pi x^2)", "0c493e43973bd9cc8905d8d7c3fbacd9": "\\Gamma(z) = \\frac1z-\\gamma+\\frac16\\left(3\\gamma^2+\\frac {\\pi^2}2\\right)z+O(z^2)", "0c4949e23cea11495f9a6e94c2077a54": "\n\\theta = \\arctan \\left[ \\sqrt{\\frac{8c^2(r_1^2+r_2^2-2c^2)}{r_1^2-r_2^2}-1}\\right]\n", "0c4953b32792bf8316d92b514c4e3f48": "C \\subseteq B\\,\\!", "0c49546ecb73718a1a091ff88d5ab885": "n=0,1,2,... ", "0c4987875329472336262f519e093915": "h\\sim{2 \\times 10^{-20}/\\sqrt{\\mathrm{Hz}}} ", "0c49af1109a97c7ae982d0e122b1b72d": "\\operatorname{Dom}(n) = \\left ( \\bigcap_{p \\in \\text{preds}(n)}^{} \\operatorname{Dom}(p) \\right ) \\bigcup^{} \\left \\{ n \\right \\} ", "0c49d9c162f8bb9bcab5897a56dbc73b": "A(xy) = A(x)A(y).\\,", "0c49e677829106468cb9c7eee716a544": "M_0 \\to_{G,t_{i_1}} M_1 \\wedge \\ldots \\wedge M_{n-1} \\to_{G,t_{i_n}} M_n", "0c4a3445efd5488a8469e7b10f68d4b1": "\\gamma^0 = \\begin{pmatrix} 0 & \\sigma^2 \\\\ \\sigma^2 & 0 \\end{pmatrix}, \\quad \\gamma^1 = \\begin{pmatrix} i\\sigma^3 & 0 \\\\ 0 & i\\sigma^3 \\end{pmatrix}", "0c4aee9384242c43a99883a6b9c9190c": "x^{16} + x^{10} + x^8 + x^7 + x^3 + 1", "0c4b0c56ecf0f027224babb73cec7044": "h(\\psi(h(\\psi(0))))", "0c4b316ecf27f5494e1f5fbd74dcd75b": "Y = G_i \\exp\\left(M^{(i)} \\log(x_i)+\\sum_{j=1}^{r_i}\\frac{T^{(i)}_j}{x_i^{j}}\\right).", "0c4b3df32e5674357d7508ae3d5e706c": "M_-(z) = \\overline{M_+(z)},\\quad z\\in\\Sigma.", "0c4b3f7c6c486c4f940706dbe2a77f00": "\n \\mathcal{P} = \\big\\{ P_\\theta\\ \\big|\\ \\theta\\in\\Theta \\big\\}.\n ", "0c4b7cd6210e693a094d76bf2c06b1d1": "\\vec{c}_2(v)", "0c4ba1c2bd327fea8113992ca1f3c77a": "f(P,w) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ w\\neq1\\ \\mbox{in}\\ H \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ w=1\\ \\mbox{in}\\ H.\n\\end{matrix}\\right.", "0c4bb2e2df8049bdf0a178b17cba4c7b": "V_0 = RC\\frac{dV(t)}{dt}+V(t).", "0c4bc85518833bc2d14944c53eb6b518": "\n\\left(\\mathrm{PL}_j\\right)_i = \\frac{\\left(X_{j\\bullet[j]}\\right)_i^2}{\\sum_{k=1}^n\\left(X_{j\\bullet[j]}\\right)_k^2}\n", "0c4c72c4a523f41c517b4185683a153f": "\n Q =\n A_1\\sqrt{\\frac{2}{\\rho} \\cdot \\frac{\\left(p_1 - p_2\\right)}{\\left(\\frac{A_1}{A_2}\\right)^2 - 1}} =\n A_2\\sqrt{\\frac{2}{\\rho} \\cdot \\frac{\\left(p_1 - p_2\\right)}{1 - \\left(\\frac{A_2}{A_1}\\right)^2}}\n", "0c4c7926af504700991938b35978d4cb": "|\\mathbf{J}| = \\hbar\\sqrt{j(j+1)}\\,\\!", "0c4c9575ee93c2265f2886ffbc32e601": "E_{1},E_{2},E_{3},\\ldots, E_{N}", "0c4ca2473c7cda2aa5d6f89535728326": "\\psi_L(0)=\\psi_C(0)", "0c4d11f509846e3a2211ca4236fc6aab": "\\begin{align}\n \\int {\\frac{du}{\\sqrt{a^2 + u^2}}} & = \\sinh ^{-1}\\left( \\frac{u}{a} \\right) + C \\\\\n \\int {\\frac{du}{\\sqrt{u^2 - a^2}}} &= \\cosh ^{-1}\\left( \\frac{u}{a} \\right) + C \\\\\n \\int {\\frac{du}{a^2 - u^2}} & = a^{-1}\\tanh ^{-1}\\left( \\frac{u}{a} \\right) + C; u^2 < a^2 \\\\\n \\int {\\frac{du}{a^2 - u^2}} & = a^{-1}\\coth ^{-1}\\left( \\frac{u}{a} \\right) + C; u^2 > a^2 \\\\\n \\int {\\frac{du}{u\\sqrt{a^2 - u^2}}} & = -a^{-1}\\operatorname{sech}^{-1}\\left( \\frac{u}{a} \\right) + C \\\\\n \\int {\\frac{du}{u\\sqrt{a^2 + u^2}}} & = -a^{-1}\\operatorname{csch}^{-1}\\left| \\frac{u}{a} \\right| + C\n\\end{align}", "0c4d45a1ce65df9fb01caac3918a3fbc": "A_a", "0c4d91873d2a7a8dd1d2c0ba7f8f271e": "u_i(x)", "0c4db32718b8c53b3209df130c005d2d": "\\forall k\\ \\|\\mathbf{e}_k\\| = c ", "0c4e1df2a7e19274eea5df7c4d27ec9a": "\n\\left. + \\left| F(\\mathbf{p}/2-\\mathbf{k}) \\right|^2\n( n_{c1}(\\mathbf{k}) + n_{a2}(\\mathbf{k}) - n_{c2}(\\mathbf{k}) - n_{a1}(\\mathbf{k}) )\n\\right]\n", "0c4e660a83e8684cf55369080717c873": "A = \\frac{a}{|r_0^2 - a^2|}", "0c4ea554c815215628b2c79e48df6405": "\\sigma_{ij}", "0c4f20bc198645607ac566df18b423ba": "0< -K_X \\cdot C_i \\leq \\operatorname{dim} X +1 ", "0c4f58f68d2ca2e6273cd323c66a3aef": " OPEN_d' ", "0c4f7da8b40888d0d73de1e5e4b9927b": " \\frac{r_E}{r_O} \\ll 1", "0c4f98d7758dd7569b4e2cff204f06e3": "f:x\\mapsto h(R x)", "0c4f9ea17b32cab6493f89bf2e3978f3": " \\langle x \\rangle = \\frac{1}{P} \\int{I(x,y) x dx dy}, ", "0c4fa624c3da1689fefc711f07124f8a": " \\gamma^{\\mu \\nu} ", "0c5010870ba0629086e10e3db1038559": "\\frac{1}{FG}=\\frac{1}{2} \\left( \\frac{1}{AB}+ \\frac{1}{DC} \\right).", "0c502ee79f3b09999ba9d57212bceb67": "\\pm e_i", "0c503b5cb68185209fe963aaf64a5938": " \\left(\\mathbf{P}^T\\mathbf{A}\\mathbf{P}\\right) \\left(\\mathbf{P}^T\\mathbf{x}\\right) = \\mathbf{P}^T\\mathbf{b}.", "0c506595cbd5a3ca706f8795eed679ea": "\n \\left[\\begin{array}{ccc|c}\n 2 & 3 & 5 & 0 \\\\\n -4 & 2 & 3 & 0\n \\end{array}\\right].\n", "0c50995c83b30a81e4e4a5f09ece0cb7": "H|\\alpha\\rangle=E|\\alpha\\rangle", "0c51d1715f3246dff44a573ac21b7ca3": "x = y = 0", "0c51dde9586da2e526cb82809b896e6c": "\\mathbf{B} = \\mathbf{DN} \\qquad \\qquad \\qquad \\qquad \\mathrm{(8)}", "0c51df2af65ec969dd356261620760d4": "V_i ^* = P_i V^*", "0c51f684199167e97a4ae798262d0e68": "(1,2,\\cdots,k)", "0c51f9e080ca217ae2d6a581f142e12d": "R_m(R_n(\\xi,\\xi),R_n(\\xi,x))=R_{m\\cdot n}(\\xi,x)\\,", "0c5207fdccbbe3b48a269af808f3fc78": "\\phi_n(x) = \\phi(x-n)", "0c5212ae373f51c67b6e1a83f65e0c60": "\\approx 10^{7.8 \\times 10^{41}}", "0c5263856259f30ccd1df1fb3146055c": "F(k, x) \\leftarrow Dec(sk, c)", "0c5312947e9f4146d389419eedbc5088": "H(X) = \\operatorname{E}\\left[\\log_b \\left( \\frac{1}{p(X)}\\right) \\right] \\leq \\log_b \\left( \\operatorname{E}\\left[ \\frac{1}{p(X)} \\right] \\right) = \\log_b(n)", "0c531c88463688ae6b25af1b44a9ed3b": "\\mu = \\operatorname (m(\\vartheta))", "0c534f8590e3e5cbcdc5b1a9a52798df": "\ni{d \\over dt} U(t,t_0)\\Psi(t_0) = V(t) U(t,t_0)\\Psi(t_0).\n", "0c53694d9ea0dfd6af455ed2c85df9e3": " I_d = \\oint_S \\mathbf{J}_{\\rm d} \\cdot {\\rm d}\\mathbf{A} , \\,\\!", "0c5464c819c71febf835909a603f1d9a": "a = \\iota x (\\pi (x,b) \\land x=a)", "0c5470706c3157b1ce3605d76d00e10b": "D^* = \\frac{q\\lambda \\eta}{hc} \\left[\\frac{4kT}{R_0 A}+2q^2 \\eta \\Phi_b\\right]^{-1/2}", "0c54cc5a72c233e96ccad4105727a9b1": "\\omega=0^{\\circ}", "0c550cab08f6617ca3c13a930c4191e1": "\\mathbb{F}_q^k", "0c553bfe89889ede1798457cda24fa37": "\\Lambda^r U = \\bigoplus_{p+q=r}(\\Lambda^p S)\\otimes(\\Lambda^q T).", "0c55c3c8730eaa2047bfadb9620b9a0f": "\\partial^i\\phi=\\theta^i", "0c56a2bc23d94cfab41587837e1c431b": " \\mathbf{H}\\mathbf{C} = \\mathbf{M} \\mathbf{C} \\boldsymbol{\\Phi},\n", "0c56e17183d1902018b37da28c58d631": "m \\in \\mathbb{F}_q^k \\backslash \\{ 0\\}", "0c574dde467b9c4e59307b3663b4d5ae": "f(x) - f(u) \\geqq g(x, u) \\cdot \\nabla f(u), \\, ", "0c575a1fa28128a5a078b3757f9f1d73": " V_{\\mathrm{int}} = - \\mathbf{F}\\cdot \\boldsymbol{\\mu}.\n", "0c57a8b7a6166a2162d1670278e64e97": "s_\\infty(z)=z^\\nu (1+\\mathcal{O}(\\tfrac{1}{z})).", "0c57e67e9d7593f78c74880b9d98cef6": "H(u)(t) \\approx A \\cdot \\sin(\\omega t + \\phi_m(t))", "0c57f3025fc06db223af09ef43f26afe": "f(x) = x^5 + x^4 + x^2 + x + 2", "0c588743b8102cfe7602fcbc999d55da": "M + C_2H_5^+ \\to [M+ C_2H_5]^+", "0c58a1e9ef4203c464eb36b7c3d95a10": "a \\otimes b = b \\otimes a ", "0c58b9f000aeade0a46a9e77889fa65f": "\\boldsymbol{\\Delta}^1_1=\\boldsymbol{\\Sigma}^1_1\\cap \\boldsymbol{\\Pi}^1_1", "0c58ed46b137c32253dbde5909e691a6": "R=\\frac{A_1,\\dots,A_n}{B}", "0c595e423e56612a862efd5dee8d72a4": " M^{\\!\\!\\!\\!\\! {}^\\beta}", "0c596f5b75ceb6f44c51a8c09ee83bab": " \\mathbf{x}_B = R\\mathbf{u}_R \\ , ", "0c5987d824c0cdb4c19a2437393ae9f4": "\\mathbf{A} = \\mathbf{P} = r \\mathbf{\\hat r} + z \\mathbf{\\hat z}", "0c59de0fa75c1baa1c024aabfa43b2e3": "\\textstyle n", "0c59e3f0198f16be9ac1e499453576c9": "k=\\frac{J_{12}}{\\sqrt{b_1b_2}}", "0c59f8825ec535fc8bce1ee7249dbf24": "\\textstyle \\mathcal{N}(m_k,\\sigma_k^2 C_k)", "0c5a44a1b1a0d8803827cd2704539596": "p(x_{n+1}) \\ = \\ p_1(x_{n+1}) - p_2(x_{n+1})\\!\\cdot\\!E \\ = \\ f(x_{n+1}) - (-1)^{n+1} E", "0c5a4f80f1cbf88f90b7c2fc33bbe0b1": " \\gamma_{tx} ", "0c5a5ad9b4f40ffed32a6a66af599367": "\\tilde{S}(q,\\omega)", "0c5a7af66122a274bf6d463e213d4f04": "-(0.75)^nu[-n-1]\\ ", "0c5a91c0f09ecd88b2afb669cabf0f85": "\\int_{-a}^a f(z)\\,dz = \\oint_C f(z)\\,dz - \\int_\\text{Arc} f(z)\\,dz ", "0c5ab7bb5d9ecadc029bb27f422269a5": "\\{y_1,y_2,y_3,y_4\\}=\\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\\}.", "0c5ac033023f12586ef43a2662a663ff": "f \\approx g", "0c5ac03ccf3b49620b84d3caebc56686": " c = \\left\\lfloor\\frac{y-1}{100}\\right\\rfloor \\quad \\text{and} \\quad g = y - 1 - 100 c.", "0c5af2f63a912f189c8e67a4c8bd90e7": "(N,y) \\to (N,y).", "0c5b1102f4c09c6281c7a01967d3afe4": " \\hat q=2^{-1/2}(\\hat a^\\dagger+\\hat a)", "0c5b5d9bcbb922ae90f87cf90e7d8b44": "n \\left[{n\\atop k}\\right]", "0c5bee0a4c62c3501013220b5ff97c0d": " \\frac{MacD } { (1+y_k/k)} = - \\frac{1} {V(y_k)} \\cdot \\frac{\\partial V}{\\partial y_k} \\equiv ModD ", "0c5c092e63829f6fd5825f6b291800ca": "P_J[f](x)=P_k[f](x)+D_k[f](x)+\\dots+D_{J-1}[f](x)", "0c5c13dd202360245712aadba935119f": "\\ddot{T} - (m^2\\tau^2 -n^2)T=0", "0c5c5826298887ada15b9c8e18dd48a5": "[S_x,S_y]=i\\hbar S_z", "0c5d030d5cc98694da183cd68f7d4d48": "\\epsilon = 1/3", "0c5d1bf05f146172ec5834b147a81bba": "\n \\mathrm{P}(A=0,B=0)=P\\{1\\}=\\frac{1}{6},\\; \\mathrm{P}(A=1,B=0)=P\\{4,6\\}=\\frac{2}{6},\n", "0c5d54c3e4515867c393721f9565fd9c": "n = \\frac{p}{2 \\delta p}", "0c5df5806926312c23820e442b872b86": "Y\\in W", "0c5e844118728ac06c998f442ff26e1a": "\\pi_1(U(1))=\\mathbf Z,", "0c5ea4b2197e5749474800f0f622986f": "\nS = \\left( \\begin{array}{cc}\ni & 0 \\\\\n0 & -i \\end{array} \\right) , \nV = \\left( \\begin{array}{cc}\n0 & i \\\\\ni & 0 \\end{array} \\right), \nU = \\frac{1}{\\sqrt{2}} \\left( \\begin{array}{cc}\n\\epsilon & \\epsilon^3 \\\\\n\\epsilon & \\epsilon^7 \\end{array} \\right),\n", "0c5f6f09098a49b8143693323b2c1d58": "k=N", "0c5fad867a03ba7f8cf94149d93bb08d": "\\frac{\\Beta(\\alpha+n,\\beta+k)}{k\\Beta(\\alpha,\\beta)\\Beta(n,k)}", "0c611f83ef132f8728def8bb96fbd214": "X_1 \\sim \\mathrm{Pois}(\\lambda_1)\\,", "0c61907468e5c3d49d261330fd536e63": "I R \\subseteq I", "0c61b41995f9d2da561b12294e455c6c": "\\boldsymbol \\beta = (\\beta_1, \\beta_2, \\dots, \\beta_n),", "0c61c6dddb3585d926995b40476f94d8": "\n\\begin{align}\n\\frac{(\\mu_2-\\mu_1)-(\\bar X_2 - \\bar X_1)}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }} & = \\frac{\\mu_2-\\bar{X}_2}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }} - \\frac{\\mu_1-\\bar{X}_1}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }} \\\\[10pt]\n& = \\underbrace{\\frac{\\mu_2-\\bar{X}_2}{S_2/\\sqrt{n_2}}}_{\\text{This is }T_2} \\cdot \\underbrace{\\left( \\frac{S_2/\\sqrt{n_2}}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }} \\right)}_{\\text{This is }\\cos\\theta} - \\underbrace{\\frac{\\mu_1-\\bar{X}_1}{S_1/\\sqrt{n_1}}}_{\\text{This is }T_1}\\cdot\\underbrace{\\left( \\frac{S_1/\\sqrt{n_1}}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }} \\right)}_{\\text{This is }\\sin\\theta}.\\qquad\\qquad\\qquad (1)\n\\end{align}\n", "0c61f2d07cff0aaf902ed995f465c49f": "l_G = \\frac{1}{n \\cdot (n - 1)} \\cdot \\sum_{i \\ne j} d(v_i, v_j)", "0c6218811da6eaf8e9b3e7b74b3335a8": "J'_k", "0c62330322871b7e83daaea9322208d1": "\\xi\\simeq \\frac{2}{A+1}", "0c623332b8cb236713967189dce7dec1": "144^5 = 27^5 + 84^5 + 110^5 + 133^5", "0c6304421d16c88be1d3f8548a96e55e": " H_n ", "0c632da34337fd5fca09226f7400aedb": "\\frac{1}{2}\\,|x_1 y_2 - x_2 y_1|.\\,", "0c633d028b6f9368c1bc0bb6aedde1ea": " S g = \\{s.g\\,:\\,s \\in S\\}.", "0c637ad9003e2dc63ea25dee54540c4f": "i_G(t s t^{-1}) = i_G(s).", "0c6388b0be02ea5716b00a9918746c08": "\\begin{bmatrix} \n 0 & 1 \\\\\n 0 & 0\n\\end{bmatrix}.", "0c640324c5f7765196f2d64755d89d27": "d=\\sqrt{\\ell^2 + w^2}", "0c6491a608f3bae7bc2cd5bc636c4e32": "\\lnot (P \\Rightarrow Q) \\equiv \\lnot (\\lnot P \\vee Q) \\equiv P \\wedge \\lnot Q", "0c64e0d6b9e20a0b9e155939330804ea": "-r^{-1}\\,", "0c64f49edcfa74c8d1a99056480ef113": "\nV_{\\text{out}}\n\\approx \\frac{V_{\\text{in}}}{\\beta}\n= \\frac{V_{\\text{in}}}{\\frac{R_{\\text{1}}}{R_{\\text{1}}+R_{\\text{2}}}}\n= V_{\\text{in}} \\left( 1 + \\frac{R_2}{R_1} \\right)\n", "0c64f5b682abac8f6828e4376296d35b": "MS_2 + \\frac{1}{2} O_2 + H_2SO_4 \\rightarrow MSO_4 + S^0 + H_2O", "0c655ef77d1f0c23c345ce4ccbd4b066": "H^n(K(G,n);G) = \\mathrm{Hom}(H_n(K(G,n);\\mathbf{Z}), G) = \\mathrm{Hom}(\\pi_n(K(G,n)), G) = \\mathrm{Hom}(G,G),", "0c659059c211ded790ba512c461c543f": "X^T \\hat e = X^T [I - X(X^T X)^{-1}X^T]y= 0", "0c65fbc500c305f2e6c16a441d7866fd": "a_1a_2 + a_1b_2i + a_1c_2j + a_1d_2k", "0c666d1d54aa989cf74523eb22c998e5": "X_{\\sigma(X, Y)}", "0c667959eeceea9c31f3f104d8ad7704": "\\nabla_{\\dot\\gamma(t)}X=0\\text{ for }t \\in I.\\,", "0c67327855d5eb7ae6a279f60afd2793": "f : X \\rightarrow Y", "0c67ba5b3caaab0958a0605ff7077b37": "C^* \\cong C^* \\otimes 1 \\rightarrow C^* \\otimes (C \\otimes C^*) \\cong (C^* \\otimes C) \\otimes C^* \\rightarrow 1 \\otimes C^* \\cong C^*", "0c67f72120fe279fb4c87c4b82e06858": "\\bar{\\delta}=1-\\sqrt{2}", "0c6851e4dde516a9d8a786657323a2bd": "Probability [Parameter | Data] = \\frac{Probability [Data | Parameter ] X Probability [Parameter ]}{Pr[Data]}", "0c68620ee2ea4f1286fcd672a47ea080": "t\\,", "0c68644751a7cf4d050e06a67d3aa3b7": "\\mathbf{I} ", "0c687e4644cf41b98867b88224312874": "L_{xx}(x, y) = L(x-1, y) - 2 L(x, y) + L(x+1, y).\\,", "0c68c2ef0818a2c3943ec4050d941aef": "P dV", "0c69000328ccb9755609833ca7d5f184": "a^2+nb^2", "0c6906a2a9c5fcc2d4875a48421dcdaa": "p_1 p_2 p_3 (1-p_4)\\,", "0c6950ddc28d1b6dbfcfc93ab4352d8c": "\\omega \\rightarrow 0", "0c69bfa2c09b9a9ffea0ebf0c0b09acd": "l_iA_{ij}=m_jT'", "0c69e77866ab83d257e967e1f50f68f1": " \\tilde{\\Omega}(n) ", "0c6a4b8eb9b002fc0b4aa236d725245a": "(Df)(a,b) = \\left [ \\frac{\\partial f}{\\partial x}(a,b) \\ \\ \\frac{\\partial f}{\\partial y}(a,b) \\right ] = [2a \\ \\ 2b]", "0c6a5c48af63d4600ea68025bb66a745": "\\liminf_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{W}_{\\varepsilon} (G) \\geq - \\inf_{\\omega \\in G} I(\\omega).", "0c6a5e1b1fb691057c255e5cc342c54a": "\\langle V^{2}\\rangle=\\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}}", "0c6a885b287c52cda1cc8bf19a3a6ae0": " f_{xy}(x,y) \\approx \\frac{f(x+h,y+k) - f(x+h,y-k) - f(x-h,y+k) + f(x-h,y-k)}{4hk} ~. ", "0c6aaee0c6d12e877d64f310b51e80c5": "\\,=A(I-B)A',", "0c6af210a17e2dd1d3054491801f8488": "A_i \\ne A_j \\quad \\Rightarrow \\quad \\left|A_i \\cap A_j\\right| < \\infty.", "0c6af5c6ed61a69097374b7c77240481": "L(x,\\xi) = \\tfrac{1}{2}F^2(x,\\xi),", "0c6b37c282d00505f9af3038f67c3d3d": "\\, p_n \\,", "0c6b38f61bd6ff96449417fa4245c83c": "~x=x(t)~", "0c6b6f0cd4d1493335b02bdf0b382686": "D(\\omega) = 1-\\frac{\\omega^6}{225}+\\frac{\\omega^8}{1125}+\\cdots.", "0c6b758e919b4d692ab5bef515db4dc2": "15/16", "0c6b9ad1f6be150d79f233ae7bad21d2": " \\mathbf{t}(s) = \\mathbf{u}(s) \\times \\mathbf{T}(s), ", "0c6ba098deaec26a35b1b9aaad036eea": "\\mathrm{^{238}_{\\ 92}U\\ \\xrightarrow {+\\ 15 n, 7 \\beta^-} \\ ^{253}_{\\ 99}Es}", "0c6bd65c04a5bdf49cfffff7bf6a7996": "t_1^2 + 13u_1^2", "0c6bddefda0643f606a063cf6233c15e": "\\! -1/3", "0c6c04dc73a1cd71cbb7388a21c7a3b7": "\\nu: A \\rightarrow \\Z\\cup\\{\\infty\\}", "0c6c44fb9c7765160ae80e13f9442708": "G = (\\{S\\} , \\{a, b\\}, P, S)", "0c6c67f5a95594342cfc1d3acecaaa93": "\\textstyle \\theta", "0c6cec00fc3dea60cf65cc0454dca06b": "\\cos\\frac{\\pi}{3}=\\cos 60^\\circ=\\tfrac{1}{2}\\,", "0c6d7b7e6c7bb12759ac88371d0bc337": "\\mathrm{d}F = - p\\,\\mathrm{d}V - S\\mathrm{d}T + \\sum_i \\mu_i \\,\\mathrm{d}N_i\\,", "0c6d9ea002ab80da90453d21d27c1014": "\ny(x_0-h) = y(x_0) - hy'(x_0) + \\frac{h^2}{2!}y''(x_0) - \\frac{h^3}{3!}y'''(x_0) + \\frac{h^4}{4!}y''''(x_0) - \\frac{h^5}{5!}y'''''(x_0) + \\mathcal{O} (h^6)\n", "0c6dbfd4aaab56d84af722c986e34f19": " a^3 \\sqrt{1 - 3\\cos^2\\alpha + 2\\cos^3\\alpha} ", "0c6e333e3f3ce894d6ca6cc9c7bcefbe": "\\scriptstyle \\mathrm{E}(V_{k}-U_k)=\\infty ", "0c6e92b7fe51d53249264573b5c285d9": "\\mathbf A^1_k", "0c6eb9ecabdbc668407d83e6968c9b08": "A_d \\cong Sym^d V^*.", "0c6f16b943707092fa2868d6a73dca92": "\\,(r+e)\\cdot(r+e) \\le x", "0c6f65f0f2000c072241774b499d8868": "\n\\int \\sec^3 x \\, dx = \\int \\frac{dx}{\\cos^3 x} = \\int \\frac{\\cos x\\,dx}{\\cos^4 x} = \\int \\frac{\\cos x\\,dx}{(1-\\sin^2 x)^2} = \\int \\frac{du}{(1-u^2)^2}\n", "0c6fd96b3b6b3a5c6ffb5d7cc4efecf6": "\\ \\frac{\\partial p'}{\\partial \\varepsilon_s}=\\frac{\\partial q}{\\partial \\varepsilon_s}=\\frac{\\partial \\nu}{\\partial \\varepsilon_s}=0", "0c700189aeb81935ad53bef18a822a75": "vT_3", "0c70e652b8789f2f8ccf05ae2b0f8967": "h_J \\leftarrow \\hat h", "0c7166ce0310c4509cea3864d0bc554e": "S_1, S_2,..., S_n", "0c71948cb63010f5c951019c2c274f48": "G_q(x) = \\begin{cases} 0 & \\text{if } x < -\\nu \\\\[12pt]\n\\displaystyle \\frac{1}{c(q)}\\int_{-\\nu}^{x} E_{q^2}^{-q^2 t^2/[2]} \\, d_qt & \\text{if } -\\nu \\leq x \\leq \\nu \\\\[12pt]\n1 & \\text{if } x>\\nu\n\\end{cases}", "0c71d1ceea31a94b3011202b936ffbc0": "\\,e_x = \\sum_{t=1}^{\\infty} \\ _tp_x", "0c722fb940f4d14d9d61f35bbf43a622": "x^2 = 4.", "0c728d08a0652654851a9636e13da825": "\\{1,2,3,4,5,6\\}", "0c72974f2266ce7a3fe004260404465e": "\\delta W = \\sum_{i} \\mathbf {F}_{i} \\cdot \\delta \\mathbf r_i + \\sum_{i} \\mathbf {C}_{i} \\cdot \\delta \\mathbf r_i - \\sum_{i} m_i \\mathbf{a}_i \\cdot \\delta \\mathbf r_i = 0.", "0c72b92f7a69ca1c84cf1291884de4a0": "1-s = \\frac{(Q_1+Q_2-Q_3)^2}{4Q_1Q_2}.\\,", "0c7315b75069674e5ebb404e10137252": "V(\\phi) = 0", "0c7377fe042986b13af5e3587962e26c": "\\{s, 1, 4, t\\}", "0c73ad357d888e8dfe85892afffe64ee": "\\frac{d}{dx} \\int_0^x t^3\\, dt ", "0c73af3efe0bc8a0720b7c8b0f961ce8": " u \\in -\\mathrm{int}K_M \\Rightarrow u1 \\not\\in A", "0c73ca9ff414bc142a59798e3b5a2713": "T_G(-2,0)", "0c743cab686e268b848cf2533435165f": "\n\\mathbf{v}_\\mathrm{rot} = \\mathbf{v} \\cos\\theta + (\\mathbf{k} \\times \\mathbf{v})\\sin\\theta\n + \\mathbf{k} (\\mathbf{k} \\cdot \\mathbf{v}) (1 - \\cos\\theta).\n", "0c7481f8b50f5151a8d3a173e8a473b5": "E_{\\omega}", "0c753ac7e424d2dcd09c2aaa3e3255db": "\n \\boldsymbol{B}^{-1} = \\boldsymbol{B}\\cdot\\boldsymbol{B} - I_1~\\boldsymbol{B} + I_2~\\boldsymbol{\\mathit{1}}\n ", "0c755fb3b184017793cbc66859500e13": "Y[\\mathrm{100}] = c_{11} + c_{12} -2 \\left( \\frac{c_{12}^2}{c_{11}}\\right)", "0c757b96131f1e02288ab9af6504c115": "x_0, y_0, x_1, y_1, \\dots.", "0c75b2ec7bf52d7b5c1fbd70f4effcac": " x^4 + 2y^2 = 8 \\, ", "0c75b76d77e44aece8df88c26f5defb3": "f(z)=z+\\sum_{n\\geq 2} a_n z^n", "0c76642fe963a290459752d4cab37ab5": "(1 + j)(1 - j) = 0", "0c7669a058fb48d21246b85bb82d8cbb": "\\arccos\\left({-1\\over3}\\right ) = 2\\arctan(\\sqrt{2})\\,", "0c76e5bd96c465b86e08ee50f6225631": "\\mathbf{p} = m \\mathbf{v}", "0c7702a1bb5256ebdbaddd1f0dca2bdf": "ds = \\frac{ds}{dy} dy", "0c770b4608a4e2798039e68f1fb4d85c": "S_P^2 = \\frac{(n_1-1)S_1^2+(n_2-1)S_2^2 + \\cdots + (n_k - 1)S_k^2}{(n_1 - 1) + (n_2 - 1) + \\cdots +(n_k - 1)}", "0c77513ee982ab821316a1161b2557d6": "\n D := \\cfrac{2h^3E}{3(1-\\nu^2)} \\,.\n ", "0c77993efa79a80ed8d4e135778926d3": " \\frac{A}{P} + \\frac{B}{Q}", "0c77ce52be17ea1aec2f74045a15f964": " \\sin\\theta_3+\\sin\\theta_2\\cos(\\theta_2+\\theta_3)=\\sin(\\theta_3+\\theta_2)\\cos\\theta_2 \\, ", "0c77e14fa839780c36e6dddb6ee0d5d2": "\\Omega_{n,\\mu\\nu}=\\epsilon_{\\mu\\nu\\xi}\\,\\mathbf\\Omega_{n,\\xi}", "0c781e57035ef9349d599e89b8e864e9": "\\mathbf{x}_{0i}", "0c78565a768d5385b45c136f034a8bfa": "\\lambda_1 \\cdot \\lambda_2=\\alpha^2+\\beta^2=-B,", "0c786c118b9dc1f2f7c4356e6fb087f9": "\\alpha_2 = {{6\\alpha_0 + 1\\alpha_1} \\over 7}", "0c7889d0b29cad50525d0d86e73e9a88": "E_v = A_v + T_v", "0c7892b2bc0ab81f603b60e1fd8778f1": "\\mathbf{C}(\\mathbf{q},\\dot{\\mathbf{q}}) = 0", "0c7895ac374a5356f2b8c75e45233bf8": "2^{-L(x)} \\le \\frac 12 p(x)", "0c78a907f9fc3a8b3b3ee3f9431efa88": "\\{\\alpha, -\\alpha\\}", "0c78b1cd825b78669a9de88c4d45d8da": "H_{q/p,m}=\\zeta(m)-p^m\\sum_{k=1}^\\infty \\frac{1}{(q+pk)^m}", "0c79519cd692d16c4c0652cf742b8a43": "a_1, a_2,\\ldots, a_n \\in H", "0c7968fb20078e88450de13719977de8": " r\\frac{\\partial \\log R}{\\partial r} = \\frac{\\partial \\Phi}{\\partial \\theta},\\ \\ \\ \\ \\ \\ \\frac{\\partial \\log R}{\\partial \\theta} = -r\\frac{\\partial \\Phi}{\\partial r},", "0c7a0cbd944ab0d940d60dea9f4e2420": "R'G'B'", "0c7a131c1e0d807dcf1efb41634a6ef0": "E_1, E_2 \\in \\mathcal{E}", "0c7a2a7f55339301ed38a9be0738ec2e": "R_{12}", "0c7a2fb98cb07936bb6cba7eb8830dd1": "\\phi ^{\\mathrm{even}}(x)", "0c7aa40e42354f8320f9b4413c9b101e": "f = P", "0c7bca0498049e4fc0c97f5a8847aebf": "\\beth_n^+\\rightarrow(\\alef_1)_{\\alef_0}^{n+1}", "0c7be2564c6c563f40d7dda8a8178931": "\\psi\\in T", "0c7c0005c08a0ec74e0bac1f2eb831c8": "\\omega_2 = 1-a/c", "0c7c561b89a43c2dd995973579beb435": "|\\psi_\\text{tot}(t)\\rangle = e^{-i\\hat{H}_{\\text{JC}}t/\\hbar}|\\psi_\\text{tot}(0)\\rangle = \\sum_n C_n \\left[ \\cos \\left(\\frac{\\alpha_n}{2}\\right)|n,+\\rangle e^{-iE_+(n)t/\\hbar}- \\sin \\left(\\frac{\\alpha_n}{2}\\right)|n,-\\rangle e^{-iE_-(n)t/\\hbar}\\right].", "0c7c768ef7173df92d2246ddd211daec": "n \\times n ", "0c7c8fa329d91de0bb7e4d6aaa64ad23": "\\omega^{(2)}=\\sum_{i1", "0ca1d1ade23787214f38b27d585f12e8": "\\circ L \\circ", "0ca1d2f6c11a8727d6355f2341ec9fbc": "D \\cdot r=0", "0ca1f156cbfd5262babc97c9d8929bc1": " \\begin{align} \n\\left[\\sigma_a, \\sigma_b\\right] + \\{\\sigma_a, \\sigma_b\\} & = ( \\sigma_a \\sigma_b - \\sigma_b \\sigma_a ) + (\\sigma_a \\sigma_b + \\sigma_b \\sigma_a) \\\\\n2i\\sum_c\\varepsilon_{a b c}\\,\\sigma_c + 2 \\delta_{a b}I & = 2\\sigma_a \\sigma_b \n\\end{align}", "0ca203894b5e392d3561749af721cec2": "L_k : E[k-n/2] \\to E[-k-n/2].", "0ca2345e55c97b9511c232eb2a8c4365": "C=ND(N\\otimes N)", "0ca283c39942c267245f6dbf895f3851": "\\scriptstyle \\oint_C ", "0ca332a0f65e064b6f8bb8e214b481cc": "\\Omega^1(P,\\mathfrak g)\\cong C^\\infty(P, T^*P)\\otimes\\mathfrak g", "0ca334f046ebafdcda716447c2cee856": "\\mathcal C|\\psi\\rangle", "0ca33564341edc71a9bfba55950587fb": "\\gamma_{il}\\langle\\Xi_l(t)x_k\\rangle=\\gamma_{kl}\\langle x_i\\Xi_l(t)\\rangle", "0ca3844350590799d77b082ea2b4282f": " \\operatorname{Perf}(f,r) = \\sum_{x \\in X_n} f(x) r(x) D_n(x). ", "0ca3b782b8b2ecc6581936592d248f04": "r=\\cfrac{1}{\\cfrac{i\\hbar^2 k}{m\\lambda} - 1}\\,\\!", "0ca3cca0f40831f88fc93d5936a8e286": " j=k-p \\ ", "0ca3d493bb99dbbcdd3144cdde7a208e": "\\exp (At) = \\frac1{2 \\pi i} \\int_{\\gamma} e^{\\lambda t} ( \\lambda \\mathrm{id} - A )^{-1} \\, \\mathrm{d} \\lambda,", "0ca3ff159b485b7f01d7530242aa6995": "\\mathcal{A} = \\Theta\\mathcal{B}", "0ca405309a63a1ab5ee4975dae4d29bd": "\\! \\frac{e^{itb} - e^{ita}}{it(b-a)}", "0ca40f1a44c808e58818344026fe76ff": "\\phi: X \\to \\mathbf{P}^n_A", "0ca4689faf88795afdd993e05da65e1d": "(2j_1+1)(2j_2+1)", "0ca47d9a481af371d1210a620c1945db": "O(\\log n)", "0ca4dcc971e167bd9e798e711888e872": " =\\sum_{p=1}^n{(-1)^{(p-1)}v_p(\\mathrm{d}x^1 \\wedge \\cdots \\wedge \\mathrm{d}x^{p-1} \\wedge \\widehat{\\mathrm{d}x^{p}} \\wedge \\mathrm{d}x^{p+1} \\wedge \\cdots \\wedge \\mathrm{d}x^n)}.", "0ca52a2b0cfe5ec11dacd71abb2968cf": "\n r_{B1} \n=\n{\\sqrt{4 \\pi}m_1v_1\\over a_1 B}\n=\n\\sqrt { 2 \\hbar \\over m_1 \\omega_c }\n ", "0ca57439bc11892ba17547b9c9f4d8cf": " x' = x_0 ", "0ca5767775207095948bfc1d523d7a78": "\\|X\\|_{\\Psi_p} = c \\rightarrow \\lim_{x \\rightarrow \\infty} f_X(x) \\exp(|x/c|^p) = 0, ", "0ca595220b95e897840383f2625e2618": "\\zeta(m)", "0ca5abfa72e14bc5752540760326d97c": "\\widehat{g}", "0ca5e4e27ca8dc4c891d2c8598529ed1": "O(mn)", "0ca63f5565c65f0d7ec78e881ff06ac4": "\\ p_1 = p_x \\quad ; \\quad p_2 = p_y \\quad ; \\quad T_1 = T_x \\quad ; \\quad T_2 = T_y,", "0ca6547a21c90740045c4b22142ed8e5": "\\overline{\\phi}", "0ca683cdf5b19a16f7ff74a7b9ce8775": "\\color{Cerulean}\\text{Cerulean}", "0ca6b6b74e2fb1dfe2ebbbf1a9c5d38a": "a=\\sqrt{2Rd}", "0ca779e844ce9b79a91b286c685f6d6d": "dr_t = (\\theta-\\alpha r_t)\\,dt + \\sqrt{r_t}\\,\\sigma\\, dW_t", "0ca77c052f06eda44b2a175baeeaec70": "T = P^{-1}", "0ca786ff0bb6cb577da29861e3fcce79": "~ D=\\mathbb{C} \\backslash \\{-1,1\\}.", "0ca7c67385baae31054df97d472b8133": "(n\\times n)", "0ca8943e703b79f50f8d275ae3690c4c": "\\varphi(x)=2\\left (\\ln(x)-\\int_0^{\\infty}e^{-t}\\ln|x-t|dt\\right ).", "0ca89893c6eca17dd72d3de085e4f120": "\\sum_{j=1}^{n_S} \\sum_{b_j=0}^{a_j} \\sum_{ \\beta_j } x_{b_j} \\ {_{a_j}^{b_j}}\\text{S}_j^{\\beta_j} \\rightarrow \\sum_{h=1}^{n_P} \\sum_{ d_h=0 }^{c_h} \\sum_{ \\gamma_h } u_{\\gamma_h} \\ y_{d_h} \\ {_{c_h}^{d_h}}\\text{P}_h^{\\gamma_h}. \\qquad \\qquad (1) ", "0ca8a4bfcbd003fd6789a211b32778fc": "\\scriptstyle (p^2-q^2,2pq,p^2+q^2)", "0ca8fbd1011176d3dfce3280504e3058": "\\mathrm{0.\\overline{3}}", "0ca91cc0a1de75d66d963e7ef676c315": "\\text{Equivalent force, }F_{eq} = \\int{F\\bar{u}}dx", "0ca9259bf34924ebd68bc06113bb1aa9": "\\frac{p\\lor q}{p,q}.", "0ca94001a900e3664c8236280aa588b8": "\\sum_{p|n} \\frac{1}{p} - \\prod_{p|n} \\frac{1}{p} = 1.", "0ca97bc136074a054953da4e5beb7e32": "(\\cdot)_+", "0ca99ce67c9024c1fdd644423937c30d": "\\mathbb{E}[X_i^?] = \\Pr[X_i^? = 1] = {2\\omega_i \\over d}", "0ca9ae4fc6daec1f60a3abfb510d3772": "\\Psi_0", "0ca9b50c4f630610a4851c789b270a9a": "C_x", "0caa07bd1550b64e2cf05e3f667704e6": "(2ab)^2+(2cd)^2 = (a^2+b^2-c^2-d^2)^2", "0caa2cedb72f96156dd1a4e18260e138": "\\nabla^2 A + k^2 A = 0", "0caa58bc12643f0869d115cb897c3d48": "\\epsilon_i^\\mu", "0caa5a4a28bdb7391c0f6ca6cfa20af3": "\\mathbf{z}' = \\mathbf{X}", "0caa6bd42800ea3d156d6ea968dc65fa": "z_T = \\frac{Z_T}{Z_0}\\,", "0caa7a5e7d7c217df561cc4130aff48b": "\n\\sigma_1 = \\sin \\psi \\, d \\theta - \\cos \\psi \\sin \\theta \\, d \\phi\n", "0caac215c7b2944e3421c0ff167d24d7": "c_n = \\langle f, \\phi_n \\rangle", "0cab651f73376762d67453c4f9e81596": "\\mathbf{x} \\in \\mathcal{D}", "0cabebec8c6ee5374a14880b88361279": "\\frac{\\mathrm{opposite}^2 + \\mathrm{adjacent}^2}{\\mathrm{hypotenuse}^2}", "0cabf5dc9b6072e9af1ba86b0b7c73e3": "\\forall x,y\\in U, ~ x\\ne y", "0cabfee9eab46860c3475c180e8d2535": "\\begin{align}\n R &= \\sqrt{(X_{12} - X_{11})^2 + (X_{22} - X_{21})^2 + (X_{32} - X_{31})^2} \\\\\n L &= \\sqrt{(x_{12} - x_{11})^2 + (x_{22} - x_{21})^2}\n\\end{align}", "0cac58c3600f6525663fb73a0eed9711": "(z_1, z_4; z_2, z_3) = {{\\lambda-1}\\over\\lambda}", "0cac661a1b56040be72a916df99cae87": "\\zeta_1=e_1+ie_2,\\zeta_2=e_3+ie_4,\\zeta_3=e_5+ie_6", "0cacd044f46d4fac2da878dabb62670f": "M: (u,v) \\mapsto (u',v')\\,", "0cacf149a2873f48cefc9be8d08f9001": "n \\mapsto \\dim_k M_n", "0cad27b02b1f2e63c2183cbef9398000": "\\beta_j^- (r_m^--r_f )_t", "0cad4207285dc7e8c337e5e356b766d3": "L(\\sigma)=(L(\\sigma)_1,\\ldots,L(\\sigma)_n)\\quad\\text{where}\\quad L(\\sigma)_i=\\#\\{ j>i : \\sigma_j<\\sigma_i \\},", "0cad5cfd12c48bf9a0830159b7226234": "\\sin{5x}=16\\sin^5 x-20\\sin^3 x+5\\sin x\\,", "0cadec12065f027759a15c0ccbc217fc": "= \\left.\\frac{\\partial \\sigma}{\\partial x}\\right|_{p} = \\sigma'(x)", "0caeb8cec6aa4b6d9d92cb6db9fd39d3": "p\\left( \\mathbf{\\hat{x}}\\right) ", "0caebd4279121a0e06a1d74578b4cbd9": "E_3 \\subseteq \\mathcal{D}", "0caef1bc2e2b2924340168899c5f797c": "\\text{Let } q^* = (-1)^{\\frac{q-1}{2}}q \\;\\;\n\\text{ (in other words } |q^*|=|q| \\text{ and }q^*\\equiv 1 \\pmod 4 \\text{).}\\;\n", "0caf21d663b985d202ec4044cf359c5d": "x=-\\frac{a}{1+p}+\\frac{a(1+p)}{1-p+p^2} = \\frac{3ap}{1+p^3},\\ y=px", "0caf2ecff3d7c68b2c5f76daa64f1570": "\\eta_{\\mu\\nu}\\Lambda^\\mu_\\alpha \\Lambda^\\nu_\\beta = \\eta_{\\alpha\\beta}\\ .", "0caf4169891bbb020dd34842a374df1f": "\np = 49 + 12\\cdot\\log_2 {(f/440)}\n\\,", "0caf5a81c305a4be64739ac76f8e9ac9": " p_s(C)=H_n(C) = \\int_C d H(\\theta). ", "0cafb2d94c6dc611e7d2092ce1600cea": "c^2 = g H", "0cafb96a5935cc47dc3f03032d76b151": "[P^+ F, P^- G]^{IJ}", "0cafc962e0f1285d801261fa846ff972": "I\\Delta_0+\\exp", "0cb00aea1f952f9060869f1d6c3f7932": "{\\Psi \\in R^{\\it {N \\times N}}}", "0cb0109eec8a399e0deeadf6780ef0bc": " PV^\\gamma = \\text{constant} ", "0cb04864e158c2a33a42532e661d7330": " M_x = \\left\\{ f \\in A(\\mathbb{T}) \\, \\mid \\, f(x) = 0 \\right\\}, \\quad x \\in \\mathbb{T}. \\,", "0cb08fd8ee924890ec98dd856fbe6064": "y\\in(0,h)", "0cb0f7a63ff9aca39bdaeb1c4a31813b": "F[y]=f\\circ y", "0cb1390497ccb1f298960b08b8cc421a": "K(t,x,y) = \\sum_{n=0}^\\infty e^{-\\lambda_n t}\\phi_n(x)\\phi_n(y).", "0cb17ce8729e421ae5624ba19b67d262": "\n j_1 = |j_2-j_3|, \\ldots, j_2+j_3\n", "0cb1deb06987b513858298d18dc8d293": "\n \\underline{\\underline{\\mathsf{A}_\\varepsilon}} = \\begin{bmatrix} \n 1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 1\n \\end{bmatrix}\n ", "0cb1e8758d07263b720b83ea48ec5749": "\\forall a \\in A,\\; 0 + a = a", "0cb1fc058c4d8c109deb2f17c35ebe21": "d(G\\circ F)(u;x) = dG(F(u); dF(u;x))", "0cb23a098de6be1634d94054278db86f": "M \\times 1", "0cb284fdeb9d28eceb9fa3a80e2840da": "t>{{D+k-1}\\over{m-s+1}}", "0cb2f2aefdd662e02c0509cbfb363c60": "\\frac{1}{(2\\pi)^{n/2}}\\int\\cdots\\int \\left|\\prod_r\\frac{2(x,r)}{(r,r)}\\right|^{\\gamma}e^{-(x_1^2+\\cdots+x_n^2)/2}dx_1\\cdots dx_n \n=\\prod_{j=1}^n\\frac{\\Gamma(1+d_j\\gamma)}{\\Gamma(1+\\gamma)}", "0cb2fb86788e2104843dcc5908255f01": "{{S}_1=(\\varphi + 2)(1 + \\varphi + 1) = (\\varphi + 2)^2 = 4 + 4\\varphi + \\varphi^2 = 5 + 3\\varphi}", "0cb325cdc56e2fb74b40b3c0f2813302": "\\mathcal{O}_L", "0cb34c2b882f2f237022f9b3d9f9df04": "(-1)^3 + 7(-1)^2 + 8(-1) + 2.", "0cb36a503efa898db5a73b1121c97dab": "\\rho_{In}\\,\\!", "0cb36cdb888c9cb34281f4041ef42be4": "2n_{\\rm oil}d\\cos\\big(\\theta_2)=m\\lambda", "0cb36da122e8269ae0fb02da9afcb9d4": "h_{\\mathrm{FOH}}(t)\\,= \\frac{1}{T} \\mathrm{tri} \\left(\\frac{t}{T} \\right)\n = \\begin{cases}\n\\frac{1}{T} \\left( 1 - \\frac{|t|}{T} \\right) & \\mbox{if } |t| < T \\\\\n0 & \\mbox{otherwise}\n\\end{cases} \\ ", "0cb38ab84bc92cac3054285d6c182d52": "x=c+v\\,\\!", "0cb40ef0440b5f02229399a2a133a8c2": "RT_{60} = \\frac{24 \\ln 10^1}{c_{20}} \\frac{V}{Sa} \\approx 0.1611\\,\\mathrm{m}^{-1} \\frac{V}{Sa}", "0cb41b67784187aa25f1374f230f3fd0": " d \\propto \\left| ln \\left|t\\right| \\right| ", "0cb41fe258d5c11a07414f1aed2c0bb6": "\\vec{F}\\,(\\vec{p})\\!", "0cb4b0445f794db8609d6374fabbd217": "x=\\frac{d-c}{a-b}", "0cb4d6d7745a8a9d5efd8c7cf1afc069": "\\theta \\vdash \\theta", "0cb4eee036463a512f13b6d2fb836d92": " a^2 + b^2 + c^2 + d^2 \\ge p^2 + q^2 ", "0cb54f3b113c6c453138316cb5c9c1a1": "3\\pi/4 \\approx 2.36 \\approx", "0cb561efdc0d1822ff2e4c497ec78cd9": "A(\\mu,\\nu)=\\begin{cases}\n e^{-\\beta(H_\\nu-H_\\mu)}, & \\text{if }H_\\nu-H_\\mu>0 \\\\\n 1, & \\text{otherwise}.\n\\end{cases}", "0cb5a2a6e179aa01c6c98da94ac1b785": "\\frac{\\mbox{Net Income}}{\\mbox{Total Assets}}", "0cb635e0ec9033a24849eee750ec13b8": "\\mbox{N}\\,\\mbox{m}\\,\\mbox{radian}^{-1}\\,", "0cb63dcf644aeda4419850c0de81b363": "\\mu(x,G)\\cap\\mu(y,H)", "0cb683a8dc4215bc8de951d6fe521675": "\\psi = \\begin{pmatrix}\\psi_1\\\\ \\psi_2\\\\ \\psi_3\\end{pmatrix}", "0cb6bc99c7df8da0c95d7e046c33215d": "I(\\rho^{AB}) = S(\\rho^{AB} \\| \\rho^A \\otimes \\rho^B)", "0cb6f583427a8a958b28c43000ab816b": " \\lim_{x \\to \\infty}f(x) = L,", "0cb710e2a38f16363db88ff24efd7ce6": "\\mathbf{x}_2", "0cb72d3496c8cfdf612b8bae9b833ffa": "\nI_R = \\frac{V_{in}}{R}\\,\n", "0cb7636c2f52a60a30187b70e623b90f": "m^\\star = m + c\\left(t-\\tau\\right) \\, ", "0cb796817cd329dda950cf2f4a004851": " A \\subseteq \\operatorname{cl}(A) \\! ", "0cb7b130fdbd6c06e8c0bfc432871c26": "d_1\\times d_2\\times\\cdots\\times d_n", "0cb7b46ddeafb4808766f896b00ee896": "z \\mapsto \\alpha z + \\beta.\\,", "0cb7d7d7221392e6761d2aee98f1f715": "\\int_0^1 \\frac{\\ln x}{1-x}\\, dx= -\\frac{\\pi^2}{6}", "0cb84591243d40b0a2c5725012098c1f": "f(x_i,\\boldsymbol \\beta)", "0cb86af24539d8df8294ca3ad0cf958b": "N = 2^k-1", "0cb86f3698858595a8778d13f3ef8c36": "\n \\nabla^2 \\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) = -\\frac{2\\kappa G h}{D(1-\\nu)}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) \\,.\n", "0cb92a0b33596118301fd5b999fccae1": "L = I_1 \\supseteq I_2 \\supseteq I_3 \\supseteq \\cdots \\supseteq I_n = \\{0\\}", "0cb92d71d5382feb078d6a2a4fd9b5ac": "\\varphi_t:X\\to X", "0cb931a2f1d81ceddc8fbd878d788c0d": "R_jf := p.v. c_n \\int_{\\mathbb{R}^n} \\frac{f(y)(x_j-y_j)}{|x-y|^n} dy", "0cb93f94b03c365daf039170f339b844": " \\sum_n (m_n c^2)^2 - (M_0 c^2)^2 = 2\\sum_{n0 \\text{ for } -1 0", "0cd57b73cfe488974d599ff76802fe3f": "U = Y^{D} - D(L^{S})", "0cd57c396b4caaae14406507c3771af2": "D_{cl}", "0cd591fbeeab306f3c45fb3b4f31e4c8": " \\dot{V} ", "0cd599e76be9d96a8bc688e1b916d9dc": "{\\gamma }_{\\mathit{i}}", "0cd5c7a8ec567b2d124e40d976fc5065": "x = X/T", "0cd648fb0680d7bf9feeb3c075531437": "\\Delta^{EXP}_k=\\mathrm{EXP}^{\\Sigma^P_{k-1}}", "0cd6664dc7852f9081f95fdaf7afa4d4": "\\mathrm{HA \\rightleftharpoons A^- + H^+}", "0cd6688af29da404b8939f7a9229fa8a": "\\mathbf{p}_1-\\mathbf{p}_0", "0cd68cd7fd8e4023dfc000331849f121": "A/I", "0cd719ae39e1e4d693299105bc56424a": "\\coth\\tau", "0cd79a334f9f261e390e5c72d439c5da": " p_t := 1000(SG-1)\\, ", "0cd7fc20752e73dccc5907f9a8c553d9": "Na_2 O (SiO_2) + CO_2 \\rightleftharpoons Na_2 CO_3 + 2SiO_2 + Heat", "0cd8ae3c93c68f708ab076c7141ec22f": "\\pi_{Y\\Sigma}: Y\\to\\Sigma, \\qquad \\pi_{\\Sigma X}: \\Sigma\\to X. ", "0cd8aeec5bcd05e759e3b662cf927127": " \\delta W\\leq - \\mathrm{d}A, ", "0cd8f6c142f63b07396f872a873c85bf": "Ax-b=0.", "0cd9c6affb70b313199258f50acd512f": " \\nabla \\otimes \\nabla \\circ \\Delta_2 = \\Delta \\circ \\nabla : (B \\otimes B) \\to (B \\otimes B),", "0cda1788b34a1fa926a45c93da40c496": "\\textstyle b > \\textstyle \\delta", "0cda4520f331b5b849a10bb7ee5fc926": "f(\\textbf{x}_{r}) < f(\\textbf{x}_{1}), ", "0cda5a4ec439908ee8e4db271ce185fc": "\\mathrm{height}(v)+1", "0cda893150122ffb4acb1104d265f269": "\nX_{5}=[3,4],\n", "0cdaa661bb4e0e420be3a04f6a567a78": "Z = \\int D \\sigma \\; \\rho[\\sigma]", "0cdadcfc374aeb4c61748ca147c24904": "D_{ds}~", "0cdb18a45a5ce69d2552c60824c85262": "S(v)=\\pm \\nabla_{v}n", "0cdb42d742b481e755beea25c80a0798": "L_{\\infty}", "0cdb698aba73efb699605661253b0da7": "z \\geq 0", "0cdba785f9c4dc985def43a01b6385a0": "s\\leq m", "0cdbd3a708fe1c40aab1c7c348ea7c54": "\\mathbf{F}\\leftarrow \\frac{1}{\\lambda}\\sum_{k=1}^{\\lambda} \n\\nabla_\\theta\\log\\pi(x_k | \\theta) \n\\nabla_\\theta\\log\\pi(x_k | \\theta)^{\\top}", "0cdbe28768b586e52546acbfdff270bb": "k>>1", "0cdc061630a95c435664cc7d4cdb8918": "M_a=\\lim_{N\\to\\infty}N\\cdot x(N)=\\frac{P_T\\cdot r}{e^{rT}-1}", "0cdc140d3e335fdc677f589d200c9542": "DF=D\\cdot F+D\\wedge F", "0cdc647ef9420e43910611bbaf335a5e": "=\\det(\\Lambda-\\mu I)\\det[(\\Lambda-\\mu I)^{-1}V^{-1}\\delta AV +I]", "0cdccaa518c7cb0c6acf322db6574f0e": "V_{an} = V_{cn} = \\frac{V_{ac}}{2} = 120 V", "0cdce0dc6368a9432e47e5745eb2724d": " M_2 := M_0 + M_1 ", "0cdd76c5a6cd41df11b6b0b2095f95d3": "S = \\cfrac{bh^2}{6}", "0cdd95ec8842546fdc16096c18cb7689": "\\frac{9}{5}+\\sqrt{\\frac{9}{5}} = 3.1416^+", "0cde0a2c46558755e521eb7ccc2be233": "\\forall x \\in L, z \\in \\{0, 1\\}^{*}, \\text{View}_{\\hat V} [P(x) \\leftrightarrow \\hat V(x, z)] = S(x, z)", "0cde38afd98441cc0f0a4a638c7d8b0b": "\n =\\sum_{a^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}}}\\Pr\\left\\{ E_{a^{n}}\\right\\}\n\\Pr_{\\mathcal{S}}\\left\\{ \\exists E_{b^{n}}:b^{n}\\in T_{\\delta}^{\\mathbf{p}\n^{n}},\\ b^{n}\\neq a^{n},\\ E_{a^{n}}^{\\dagger}E_{b^{n}}\\in N\\left( \\mathcal{S}\n\\right) \\backslash\\mathcal{S}\\right\\} ", "0cdea8de9e449fc10716ff2b9a5c1990": " \\mathbf{R}'_i =\\mathbf{R}_i + \\mathbf{t} \\;\\;\\textrm{(translation)\\;\\; and}\\;\\;\n\\mathbf{R}'_i =\\mathbf{R}_i + \\frac{\\Delta\\phi}{|\\mathbf{s}|} \\; ( \\mathbf{s}\\times \\mathbf{R}_i)\n\\;\\;\\textrm{(infinitesimal\\;\\; rotation)},\n", "0cdec0f568c75a0993780c7d0c6251e7": "C\\ell(T^*M) \\otimes S(M) \\to S(M)", "0cdef127b3d44f3491d62761b9a87c63": "x-y-z=(x-y)-z\\qquad\\mbox{for all }x,y,z\\in\\mathbb{R};", "0cdf3be7f329c4e9f54870549d480181": "\\mathcal{F}^{-1}f", "0cdf4e90d139f3e620251bad3ceeb2da": "q^*(\\mathbf{\\pi}) \\sim \\operatorname{Dir}(\\mathbf{\\alpha}) \\, ", "0cdf7b47c0b409953f652a0c00b22be5": "\\rho_c^* = 0.3009 - 0.00785\\mu^{*2} - 0.00198\\mu^{*4}", "0cdf83a5640a96a3a65ba0e457f5c638": "\\gg \\!\\,", "0cdfb0d135e620979fd70e5da8f8c37d": "V = \\frac{n}{6}hs^2 \\cot\\frac{\\pi}{n}.", "0cdfb578894c146850e012b1c4a4150f": " \\therefore \\neg B", "0cdfc32e0e580878cd949792c1eda664": "\\displaystyle d", "0cdffe9a56d14cabe6c9be1bf4d7f534": "\n\\mathbf{C}^0 = \n\\frac{1}{24}\n\\begin{bmatrix}\n2 & 1 & 0 \\\\\n1 & 2 & 0 \\\\\n0 & 0 & 0 \\\\\n\\end{bmatrix}\n=\n\\frac{1}{48}\n\\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\end{bmatrix}\n\\begin{bmatrix} 1 & -1 & 0 \\end{bmatrix}^{\\mathrm{T}}\n+\n\\frac{1}{16}\n\\begin{bmatrix} 1 \\\\ 1 \\\\ 0 \\end{bmatrix}\n\\begin{bmatrix} 1 & 1 & 0 \\end{bmatrix}^{\\mathrm{T}}\n", "0ce154ab57ea6dd5e6d09024c780da6b": "f:X\\otimes I\\to Y\\otimes I", "0ce15b40c001f98bd7f3c5b14eca646d": "=2(1/2+\\epsilon_1)(1/2+\\epsilon_2)-(1/2+\\epsilon_1)-(1/2+\\epsilon_2)+1\\ ", "0ce16a4e0ae23b16c4236156542655a1": "r_e^2", "0ce16c1d11eebd4e1827918769385e60": "C=(z s^{-1}+r d_A s^{-1})\\times G", "0ce1acfc79636daed6568606e31e9c2c": "{\\omega}_c", "0ce1ae819f2406c66e42c8033d03fbc8": "\\ \\mathbb{R} \\ ", "0ce1e54d072e8ec1e740bfd0ff8769fa": "\n\\frac{{\\alpha _s }}\n{{\\alpha _n }} = \\frac{2}\n{{\\hbar \\omega }}\\int_\\Delta ^\\infty {\\frac{{\\left| {{\\text{E(E + }}\\hbar \\omega {\\text{)}} + \\Delta ^2 } \\right|[f(E) - f(E + \\hbar \\omega )]}}\n{{(E^2 - \\Delta ^2 )^{1/2} [(E + \\hbar \\omega )^2 - \\Delta ^2 ]^{1/2} }}dE} {\\text{ + }}\\frac{1}\n{{\\hbar \\omega }}\\int_{\\Delta - \\hbar \\omega }^{ - \\Delta } {\\frac{{\\left| {{\\text{E(E + }}\\hbar \\omega {\\text{)}} + \\Delta ^2 } \\right|[1 - 2f(E + \\hbar \\omega )]}}\n{{(E^2 - \\Delta ^2 )^{1/2} [(E + \\hbar \\omega )^2 - \\Delta ^2 ]^{1/2} }}dE} \n", "0ce1fbcdf3430039873847c8dc739a74": "\na = \\frac {1}{A} \\int_A P \\left[c(\\vec{r}) | \\vec{y}(\\vec{r}) \\right]\n \\, d\\vec{r}\n", "0ce1fee993f8af0c45258cd70db19018": " \\epsilon = \\Delta D / D \\,\\!", "0ce25cebede817fe94972017de2e945f": "(a,b,k)", "0ce281f67b9fb7931114920a3c3fe235": "{}_m^n \\{ x \\}", "0ce292d442ca33bdbeacbc4cf5cb0f08": " \\dim_{\\mathbb R}V = 2\\dim_{\\mathbb C}V ", "0ce339d3047a6ceaab1d86abc421f026": "\\frac{\\rho}{\\rho-1}\\,", "0ce3989150a739016aa9b8deb6e327ac": "\\frac 1 u + \\frac 1 v = \\frac 1 f \\,;", "0ce3d837d6a6f9b76b9e6be9f50ebabb": " \\frac{dy}{dx}\\,\\cdot\\,\\frac{dx}{dy} = e^x \\cdot \\frac{1}{y} = \\frac{e^x}{e^x} = 1 ", "0ce4ab360eff33a4488c3538ce2b8203": "I / I_0 = e^{-m \\tau}.\\, ", "0ce4b90297d0c9f6ba4a167f3fead9b7": "e_b(k,i+1) = e_b(k-1,i) - \\kappa_b(k,i)e_f(k,i)\\,\\!", "0ce4d53d0d87d88c9eb190ca4871a44f": "r^{J^2}=\\varepsilon r \\varepsilon^{-1}", "0ce50bb8ed733a9418321398bdb9b6da": "\\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \n \\; \\left[ \\mathbf{1} - \\mathbf{\\hat{k}} \\mathbf{\\hat{k}} \\right] \\;\n{ \\exp \\left ( i \\mathbf{k} \\cdot \\mathbf{r}\\right ) \\over k^2 +m^2 } \n= {1\\over 2} {e^{ - m r } \\over 4 \\pi r } \\left\\{ \n {2 \\over \\left( mr \\right)^2 } \\left( e^{mr} -1 \\right) - {2\\over mr} \\right \\}\n\\left[\\mathbf{1} + \\mathbf{\\hat{r}} \\mathbf{\\hat{r}}\\right]", "0ce52e93d53fab11fb3c7e2c50502cf1": "\\beta_{T} \\equiv -\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial p}\\right)_{T}\\,", "0ce530b7aa2791ceb89ce1ccc835f863": "\\omega_2 = -\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i", "0ce55abeed1f6cf4d229dc679eb8f4e1": "\\scriptstyle\\frac{c}{2a}", "0ce55bdd51579c55ec4bd9f0a5afce82": "d\\epsilon_{i,j}^p=s_{i,j}d{\\lambda}", "0ce562bf3369700559bcca78ac4f22a0": " S = \\frac{ 1 }{ 3 }\\frac{ ( \\alpha - 2 \\beta ) ( \\alpha + \\beta + 1 )^{ 1 / 2 } }{ ( \\alpha + \\beta - 2 / 3 ) ( \\alpha \\beta )^{ 1 / 2 } } ", "0ce5af08f20265b14618a82387ad2689": "\\log\\ a + \\log\\ b", "0ce6761745867b58d92a4dee05b15e3d": "x_0 \\in \\mathcal{O}", "0ce67eaa3167b284cbff224f801002d7": " \\|x'\\|_{\\theta, p; J} \\simeq \\inf \\Bigl\\{ \\Bigl( \\sum_{n \\in \\mathbf{Z}} \\bigl( 2^{\\theta n} \\max(\\|x'_n\\|_{X'_0}, 2^{-n} \\|x'_n\\|_{X'_1}) \\bigr)^p \\Bigr)^{1/p} \\!:\\, x' = \\sum_{n \\in \\mathbf{Z}} x'_n \\Bigr\\}.", "0ce69b13de14bf7732b2d5b583711949": "\\begin{pmatrix} \\varphi A_L \\\\ \\varphi A_S\\end{pmatrix} = \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} B_L \\\\ B_S\\end{pmatrix} = \\begin{pmatrix} 2 & 1 \\\\ 1 & 1 \\end{pmatrix}\\begin{pmatrix} A_L \\\\ A_S \\end{pmatrix}\\, ,", "0ce6dc8c08c27a4d34b981e23f9bab7e": "P_\\mathrm{dBm} = -174 + 10\\ \\log_{10}(\\Delta f)", "0ce6ee2bc39c562268d7d73a02024d10": "RRA=-\\frac{d(u'(c_t))}{d(c_t)}\\frac{c_t}{u'(c_t)}=-u''(c_t)\\frac{c_t}{u'(c_t)}", "0ce7b01724be7da5ee4f097d9ab42e0b": "\\lambda_1 = \\lambda_2 = k", "0ce7e12ff3f27732ff211de93411c2ae": "\\,k=4", "0ce82642b763feeb9fede2b2e912a2f8": " \\left\\{\\omega_k\\right\\} ", "0ce8e38ac84d7004f0a6a0ddfb68b284": "1 + z = \\sqrt{\\frac{1 - \\frac{2GM}{ c^2 r_{\\text{receiver}}}}{1 - \\frac{2GM}{ c^2 r_{\\text{source} }}}}", "0ce8e5baec8c4f25858c9a333a0dc930": "T_s \\approx 2 \\rightarrow q_s* = 5.7\\left(\\tau*-0.047 \\right)^{3/2}", "0ce91cce3204b6098c3bada646eb360a": "\\mathbf{r}_k = \\mathbf{r}_{k-1} - \\alpha_{k-1} A \\mathbf{p}_{k-1}", "0ce91e3f5fb94d4cac8d73e3647221c8": " ((c+n)^2+c-n)/2", "0ce936e0cbdb88b5b7c2e807585a3169": "\\omega=1", "0ce95c69019bdbc1dafb6a47238fcbcb": "d_4", "0ce970ed47a1bd5f07826b3aa1d46755": "{\\mathfrak d}({\\mathbb P})=\\min\\big\\{|Y|:Y\\subseteq{\\mathbb P}\\ \\wedge\\ (\\forall x\\in {\\mathbb P})(\\exists y\\in Y)(x\\sqsubseteq y)\\big\\}", "0ce98a3bec553daa19d96480e71b775b": "w_t=w_{t+1}", "0ce9ce2807d9033e86f21c083c29917d": "x^{16} + x^{14} + x^{13} + x^{11} + 1.\\,", "0ce9d4245034107569e3757f4b412795": "\\frac {N c m} {f}", "0cea0d63ff59608cf5af41b2abc35e2b": " x_{step} ", "0ceb25dfa3c84228ae91c44b67b648af": "C_4 = G_3 + G_2 \\cdot P_3 + G_1 \\cdot P_2 \\cdot P_3 + G_0 \\cdot P_1 \\cdot P_2 \\cdot P_3 + C_0 \\cdot P_0 \\cdot P_1 \\cdot P_2 \\cdot P_3", "0cebec6260b66a20615ea8fc50d5107f": " I_{n} = -\\frac{- e^{ax}}{(n-1)x^{n-1}} - \\frac{a}{n-1}I_{n-1} \\,\\!", "0cec1df1676582f4eb7961182f63f874": "e - \\ ", "0cec677abaabe1dad7dcb899ca2a1f33": "\\begin{cases} u_{t} = -\\Delta u & \\textrm{on} \\ \\ \\Omega \\times (0,T), \\\\ u=0 & \\textrm{on} \\ \\ \\partial\\Omega \\times (0,T), \\\\ u = f & \\textrm{on} \\ \\ \\Omega \\times \\left \\{ 0 \\right \\}. \\end{cases} ", "0cec7498c15902a77a7883ea5ab3e6b1": "\\Phi(x) = \\int_{-\\infty}^{x} \\phi(t)\\ dt = \\frac{1}{2} \\left[ 1 + \\operatorname{erf} \\left(\\frac{x}{\\sqrt{2}}\\right)\\right]", "0cec91bb2affb75cdadcc0a4a33d7e9f": " E(p_i,r-R_i) = \\frac{3(p\\cdot \\hat{r})\\hat{r} - p}{r^3} ", "0ceca3ceff4858eada85c4fcc3033858": "I_n = \\frac{2x^n\\sqrt{ax+b}}{a(2n+1)} - \\frac{2nb}{a(2n+1)} I_{n-1}\\,\\!", "0ced22ddbe3e56715933e90d766c055c": "\\sigma(Y) \\subset \\sigma(X)", "0ced288748230e00030fa95fc07a2f59": " = \\sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}.", "0ced5b2cea4dc6450d621f9e497da64a": "{1\\over k_1} \\times \\int \\limits_{[A_1]^0}^{[A_1]} {d[A_1']\\over [A_1']} = {1\\over k_2} \\times \\int \\limits_{[A_2]^0}^{[A_2]}{d[A_2'] \\over [A_2']}", "0ced9780c9b676853846b24ceea83642": "\n\\begin{bmatrix}\n\\mathbf A^T \\mathbf A & \\mathbf A^T \\mathbf B \\\\\n\\mathbf B^T \\mathbf A & \\mathbf B^T \\mathbf B\n\\end{bmatrix}^{-1}\n=\n\\begin{bmatrix}\n (\\mathbf A^T \\mathbf A-\\mathbf A^T \\mathbf B(\\mathbf B^T \\mathbf B)^{-1}\\mathbf B^T \\mathbf A)^{-1} \n & -(\\mathbf A^T \\mathbf A)^{-1}\\mathbf A^T \\mathbf B(\\mathbf B^T \\mathbf B-\\mathbf B^T \\mathbf A(\\mathbf A^T \\mathbf A)^{-1}\\mathbf A^T \\mathbf B)^{-1} \n\\\\\n -(\\mathbf B^T \\mathbf B)^{-1}\\mathbf B^T \\mathbf A(\\mathbf A^T \\mathbf A-\\mathbf A^T \\mathbf B(\\mathbf B^T \\mathbf B)^{-1}\\mathbf B^T \\mathbf A)^{-1} \n & (\\mathbf B^T \\mathbf B-\\mathbf B^T \\mathbf A(\\mathbf A^T \\mathbf A)^{-1}\\mathbf A^T \\mathbf B)^{-1} \n\\end{bmatrix}\n", "0cedb27cb5ec58f010b6760c0ecd2b5a": "\\mathfrak{g}^{\\mathrm{red}}", "0cee8a32ff09a6e7aab1926b4eee7a25": "SVI = {SV\\over BSA} = {(CO / HR) \\over BSA} = {CO \\over {HR \\times BSA}}", "0ceea09946cdce0ccecaed08641ab99d": "P(H|I,N)=P(T|I,N)=P(S|I,N)=1/3 ", "0ceee9bc38e54fbfc96e4fbd215f5a3e": "F_0 = 0", "0cef01f449f7fa2c81c6a9d3fe1afcd9": " \\Pr(Y1,~\\text{ Fib}(n):=\\text{Fib}(n-1) + \\text{Fib}(n-2).", "0cf2e3538c045cb29bf99a649413e339": "\n\\mathrm{Fr}=\\frac{u}{\\sqrt{g' h}}\n", "0cf34a6b848c5a4d7a4387a603372c9b": "\\bar \\epsilon_{sh} \\propto \\sqrt{t-t_0}", "0cf35cbac313fbd2806a836d9ecfc880": " p(\\textbf{x}_k\\mid \\textbf{Z}_{k}) = \\frac{p(\\textbf{z}_k\\mid \\textbf{x}_k) p(\\textbf{x}_k\\mid \\textbf{Z}_{k-1})}{p(\\textbf{z}_k\\mid \\textbf{Z}_{k-1})} ", "0cf3d3aa91772f039ba61463720bd429": "\\left|r_i\\frac{\\partial^2 r_i}{\\partial \\beta_j \\partial \\beta_k}\\right| \\ll \\left|\\frac{\\partial r_i}{\\partial \\beta_j}\\frac{\\partial r_i}{\\partial \\beta_k}\\right|", "0cf42fbad4728db36a804a252cf52dbe": "I_{1z}(t)=2\\sigma_{12}tI_{2z}^0+I_{1z}^0", "0cf435010a92d67f399c3db28c306076": "G^* + M \\to M^{+\\bullet} + e^- + G", "0cf4692446b962403224fc038effa58a": "\\epsilon(a)=1", "0cf47fa4cfb90d33829a9a1663e3cb8c": "n((I+J-1+\\left\\lfloor\\frac{n}{2}\\right\\rfloor)\\,\\bmod\\,n)+((I+2J-2)\\,\\bmod\\,n)+1", "0cf4a55dc8534c02b15eff8870c88c80": " C^* \\,", "0cf526df5ccfcccc04b4091962b7fcbc": "\\{b_i\\}", "0cf54197a06d7d168cf35bc8aa380c62": "E_g(f; N)", "0cf5419a0ebaf6a44e771da8e279d25d": "confidence_{i}", "0cf66929e0a12e33f474775dcc2fcfab": "\\tau = Fr \\sin \\alpha \\,", "0cf6fbb22bfbf6ef5ff74545f2062061": "\\textstyle \\lim_{p \\to -\\infty} M_p = M_{-\\infty}", "0cf7553c3b2935fb33a46f4aad4b3bdb": "O_1, O_2, O_3,...", "0cf78cf58cceb47cbc9ba2694b5afbdb": "\\scriptstyle p_\\phi=\\sin(\\theta)^2 \\dot\\phi", "0cf790fab9f788c5a45bda912bf82b8c": " \\tilde{u}({\\vec e}_j) = \\sum_i u_i (\\tilde{\\omega}^i ({\\vec e}_j)) = \\sum_i u_i \\delta^i {}_j = u_j ", "0cf7acb4860a686b81d9fd804a72a986": "(d\\mathbf{X})^{\\rm H}", "0cf81f9038402e85910cfad17d0051b3": "1s", "0cf82af57dac0e5a2406c265ddbc0da5": "\\chi_n(z)=2^{-n}z\\,\\Phi (z^2,n,1/2).\\,", "0cf8332524f1d7df0458ccaed391105d": " \nx^{(k+1)} = x^{(k)} + \\omega \\left( b - A x^{(k)} \\right),\n", "0cf86e6161c4982781a626dde8bac485": "X\\not\\Vdash A", "0cf890331835ee06782b8ec782a7e634": "H = T^{a+\\varepsilon}", "0cf8c44b1a780162367cb2338bbd2d35": "-\\textstyle\\frac{1}{3}", "0cf8caeefdfb84063b53a7a4f0bf7363": "\\begin{align}\np(\\sigma^2|D, I) \\; \\propto \\; & \\frac{1}{\\sigma^{n+2}} \\; \\exp \\left[ -\\frac{\\sum_i^n(x_i-\\bar{x})^2}{2\\sigma^2} \\right] \\; \\int_{-\\infty}^{\\infty} \\exp \\left[ -\\frac{\\sum_i^n(\\mu -\\bar{x})^2}{2\\sigma^2} \\right] d\\mu\\\\\n = \\; & \\frac{1}{\\sigma^{n+2}} \\; \\exp \\left[ -\\frac{\\sum_i^n(x_i-\\bar{x})^2}{2\\sigma^2} \\right] \\; \\sqrt{2 \\pi \\sigma^2 / n} \\\\\n \\propto \\; & (\\sigma^2)^{-(n+1)/2} \\; \\exp \\left[ -\\frac{(n-1)s^2}{2\\sigma^2} \\right]\n\\end{align}", "0cf8da118fda3210904625dd5d34d53d": "I'_R = I_R x 10^{-at}", "0cf9d66a30fbfe388026e95dd8f03f97": "c_{ad}", "0cfaa0a6b0d363502e91dc2fe6c8f1f2": "\\frac{\\mbox{Enterprise Value}}{\\mbox{EBITDA}}", "0cfab02aabd092dc431c0715cb7d0f42": "\\, c ", "0cfac76573d2b2389bf793fd1f3527f8": "\\Gamma(V)=\\{\\varphi:G\\rightarrow\\mathbb{V}\\;:\\;\\varphi(gh)=\\rho(h^{-1})\\varphi(g)\\;\\forall\\;g\\in G,\\; h\\in H\\}.", "0cfaf9758400c6a695279edb495f892b": "\\mathbf{P}^n_A = \\operatorname{Proj}A[x_0, \\ldots, x_n].", "0cfb720281d8649ae3698766c3d85b4e": "\\mathcal{N} (X) \\cong [X,G/O].", "0cfbbe316fab78bf851858700fd65e1f": "\\alpha \\in \\mathbb{F}_{q^k}", "0cfbf719aa6482f331d4479b8f666634": "\\left(\\frac{-1}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "0cfc0c41bb3ac291ba83fa1b1db6fc4d": "\\scriptstyle{h}", "0cfc0cf414540af98a0d77c6e49489ec": "\nds = - \\frac{g}{k^2} \\cos \\theta \\,d\\theta\\,\n", "0cfc3b7f5e5f43fa82a98ab3547e9850": " f\\in M^{\\ast}=\\operatorname{Hom}_R(M,R),\\quad f(m)\\ne 0.", "0cfc40cdcdcc532008908d73ece9c049": "\\exp_{10}^3(5.84259)", "0cfc7038f239995e93b5a2cb40a3d82d": "((1+\\sqrt{5})/2)^{n+m}=O(1.6180^{n+m})", "0cfc81ac400b9f2ba5839a38d0ee3823": "T\\to T_c", "0cfcbb8db350f1eac94263aea8278a04": "\\int_c \\mathrm{Hom}_{\\mathbf{X}}(F(c), G(c)) = \\mathrm{Nat}(F, G) ", "0cfce0a8b040eec0d5cecc2dffc37276": "\\,l_{x+t} = (1 - t)l_x + tl_{x+1} ", "0cfce3299466e89630a4d3ea1788e86c": "R\\mathcal{F}f", "0cfcfb0b72b5ba2a9d31a84bbad9468d": "x\\Leftrightarrow y \\equiv (x\\Rightarrow y)\\ast(y\\Rightarrow x)", "0cfd8cfdca3ab6eb1f90b6ae91a79ad9": "A,B,C,D \\in z", "0cfda2fa771f7205d2711104b4483032": "y(\\mathbf{x}) = \\sum_{i=1}^N w_i \\, \\phi(\\|\\mathbf{x} - \\mathbf{x}_i\\|),", "0cfdbdab6a5cf7327cdd7e783ad9f857": " P'=P+RQ", "0cfdcfd651bc8a9ea7b34e170fcc0cd6": "Z[J]", "0cfdf5b3acd19a5cbfdd9d92d87c5b77": "2^2 + 13^2", "0cfe589be715a942147af4a8464b82c0": "\\mathbf{x}(i), \\mathbf{x}(j), 1 \\le i \\le 49 ", "0cfe77ddc931028b21983ba34507dfb9": "\\langle v_1,...,v_g,q_1,...q_r,h| v_ih=h^{\\epsilon_i}v_i,q_ih=hq_i, q_j^{a_j}h^{b_j}=1, q_1...q_rv_1^2...v_g^2=h^b\\rangle", "0cfe817f1f47a4e716b2e97325fd08be": "\\kappa=\\frac{\\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}.", "0cfea7c0f28d70f0d2380bb277ba38f5": " P_n ", "0cff0a6b89e5b68de16a9177f78b9eab": "K=\\mathbb{Q}(\\sqrt{7})", "0cff164bc14d4ef25a639dc45b19b891": "f\\colon V \\to W\\,", "0cff42aaf3f77897c0ffd11286abedaf": "ln\\left(s\\right) + 2 = ln\\left(\\sigma\\right) + 1.404576+,", "0cff9eed0b0b0c2e271ca076bc9ca638": " \\varphi(x) = (\\sqrt(\\sigma_i)\\phi_i(x))_i ", "0cffb8a51f191cda38c44cbc688a24d9": "\n \\cfrac{\\cfrac{A \\wedge B\\hbox{ true}}{B\\hbox{ true}}\\ \\wedge_{E2} \n \\qquad\n \\cfrac{A \\wedge B\\hbox{ true}}{A\\hbox{ true}}\\ \\wedge_{E1}}\n {B \\wedge A\\hbox{ true}}\\ \\wedge_I\n", "0d00278997afc38630f338a5f3f8eeec": " j ", "0d00bad7feceeb55c817d0b581c389b4": " e^{-x}\\, ", "0d00c8234d77c4bb38d0fa90c64d19c9": " |\\psi\\rangle_A |0\\rangle_B \\rightarrow \\sum_i M_i |\\psi\\rangle_A |i\\rangle_B, ", "0d00d293426942203a2b25819fde31cf": "\n\\psi = \\sin x \\sin y F(t)\\, \\hat{\\mathbf{z}}.\n", "0d011e142283619e55af90cdb93aa7f6": "\\begin{bmatrix} \\mathbf{p}_1 & \\cdots & \\mathbf{p}_i \\end{bmatrix}", "0d017a5760ded5225e2030e8e93a8777": "T_M(d)=T_{MB}(1-\\frac{4\\sigma\\,_{sl}}{H_f\\rho\\,_sd})", "0d01867a48fee511b6f2b96c53c0c619": "F(\\cosh t)={2\\over \\pi} \\int_0^\\infty\\tilde{f}(i\\lambda)\\cos(\\lambda t) \\, d\\lambda.", "0d024a80597c3095196968fdc8587699": "2\\pi^2 r^3 \\,", "0d027f4001f1491f2cc8f28fb2f1b588": "(\\kappa_x,\\kappa_y),\\,", "0d028976429128aa4dcd80b80b13ab5e": "\\left [\\begin{smallmatrix}2&-\\phi\\\\-\\phi&2\\end{smallmatrix}\\right ]", "0d02c112f2688f2563c5c0dabcca2ac9": " Z_1,\\cdots,Z_{k-1} ", "0d038d7795f9e1102a7fe09b09f9c9fe": "1 \\leftrightarrow 1, 2 \\leftrightarrow 4, 3 \\leftrightarrow 9, 4 \\leftrightarrow 16, 5 \\leftrightarrow 25, \\ldots", "0d0392b985a1e23b0e73072bbc4a58fd": "z_3(x,y)={\\displaystyle\\int} xe^{-x}H\\big(\\bar{y}+\\frac{1}{2}x^2\\big)\n dx\\Big|_{\\bar{y}=y-\\frac{1}{2}x^2}", "0d03c8f189c2a0d5aa0341b4c08dd1c5": "\\frac{\\psi^{(m)}(n)}{(-1)^{m+1}\\,m!} = \\zeta(1+m) - \\sum_{k=1}^{n-1} \\frac{1}{k^{m+1}} = \\sum_{k=n}^\\infty \\frac{1}{k^{m+1}} \\qquad m \\ge 1", "0d03d469c53743948ce6851a4c02415b": "1\\tfrac{1}{3}\\text{ }\\xrightarrow{\\text{yields}}\\text{ }[3,4,5],\\text{ 2}\\tfrac{2}{5}\\text{ }\\xrightarrow{\\text{yields}}\\text{ }[5,12,13],\\text{ 3}\\tfrac{3}{7}\\text{ }\\xrightarrow{\\text{yields}}\\text{ }[7,24,25],\\text{ 4}\\tfrac{4}{9}\\text{ }\\xrightarrow{\\text{yields}}\\text{ }[9,40,41],\\text{ }\\ldots", "0d051830c81900841fa766716f05eb06": "PA = FO+\\text{pos}\\,TC", "0d05f91d31fef9c0a01768860bcc1962": "f\\colon X\\to Y", "0d060b66360f765db24937577f11fc97": "\\frac{1 + \\sin (\\theta)}{2}", "0d0639a1d88e1de88c3a6a8145709609": " {4 \\over 1} \\cdot {5 \\over 4} = 5. ", "0d06c13cecd572f6e3dcc12802bef855": "\\displaystyle{K_Z(x,y)=e^{-(Ax^2 +2B xy +Dy^2)}.}", "0d06e9adeb7d619dd6a583bd39e4ba33": " n=3 ", "0d06f68062b9b478a9ff91e424340b44": "\\wedge,", "0d077d0efba3f9d45d4c783944fa2036": " M = \\varepsilon \\sinh H - H ", "0d078d604e795cdd84a59f3faa798805": "b_S", "0d07e3ff354d6e7a18ebc1d38772e053": "\\in_r", "0d0806fdc14c8cc7338470ddd5c29849": "(n-1)(n-2)", "0d089509cd64838816edec60b14fcc1e": "\\Phi_0(w)\\equiv 1.", "0d08be02f8cfd0b9bbc6a00e8e4049a6": "GF(m)", "0d093ae321c2640045f76be4f1a8c47f": "\\mathcal{FSL}, or \\mathcal{FSRI}", "0d0964133fb3658ac1eae8e99649f30f": " h(X) = -\\int_X f(x) \\log f(x) \\,dx. ", "0d097894ffcc23dd17ee0faceed4b963": "f(x) = \\sum_{i=0}^n a_ix^i; g(x) = \\sum_{i=0}^m b_ix^i", "0d09ed5e9bc899f392fb4410db3f5db1": "P := \\sum_{\\chi(k)=1} |\\omega_k \\rangle \\langle \\omega_k|", "0d09ffd312d11ca8ddeca2b81765e2ff": "v=\\sqrt{2\\mu\\over{r}}", "0d0a240622524a7ad459b367a0f333d7": " {\\rm li} (x) = O \\left( {x\\over \\ln x} \\right) \\; . ", "0d0a31a6c0cf03d2b7b95e81c7d73bc7": "\\displaystyle{(f_{W_1},f_{W_2})=\\det (1-W_1\\overline{W_2})^{-1/2}.}", "0d0a33da47501a1a7132b47f9b89c78f": "n/w", "0d0a368cb1d733b9643faead9188e41f": "\\Gamma(t)=x^t \\sum_{n=0}^{\\infty} \\frac{L_n^{(t)}(x)}{t+n},", "0d0a84f60b80bd1e8cf76cd687bcaed5": "\\text{Re}[Y_{\\ell m}]", "0d0ad5c08cc077cd5406656ce138aa07": "(R : I) = \\{ x \\in K : xI \\subseteq R \\} ", "0d0ae0f1cc569f2f867bd62b25b84438": " R \\le 1- {1 \\over n} \\cdot \\log_{q} \\cdot \\left[\\sum_{i=0}^{\\lfloor {{\\delta \\cdot n-1}\\over 2}\\rfloor}\\binom{n}{i}(q-1)^i\\right]", "0d0afcdef60073f474eb7c479e606c41": "{\\rm Tr}(B^{1/2}AB^{1/2})^{rq}\\leq {\\rm Tr}(B^{r/2}A^rB^{r/2})^q,", "0d0afce1725c6280c62552ed9281eb5e": " r_N= \\left[\\left(N - {1 \\over 2}\\right)\\lambda R\\right]^{1/2}, ", "0d0b46920e711135fbeac93621d0597c": " \\Delta t = \\frac{n}{2\\omega}", "0d0b4e3b3ded537ba0e471c90904e19e": "Hy", "0d0b6b0172cfddb91c55a6d03eae6124": "\\hat{e}_{\\mu}", "0d0bb9d5a1002bcb284d4f7a6a0c4ba0": "N = 90581", "0d0bbaf42806d64474d5e606ee0007cc": "f_{illusions.hu}(a, b) = a^{( 2^{2(0.5 - b)} )}", "0d0c32d836fec88a92a66b1e092e2587": "\\neg,\\or,\\and,\\Rightarrow", "0d0c64cb51e10c8933bd814c2ac752ba": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\mu \\nu}", "0d0c7355b9b6868b9755378210f53e1f": "\\mathrm{M \\xrightarrow[]{h\\nu\\ (5\\ \\text{eV})} M^* \\xrightarrow[]{h\\nu\\ (5\\ \\text{eV})} M^{+ \\cdot}+ e^-}", "0d0cb760819acadc5040fbecaed81ca8": "M_x= \\left.\\left(\\frac{-x^4}{2}+\\frac{4x^3}{3}-20x^2+80x\\right)\\right|_0^2", "0d0d30b28ad72c5338a5fcce70e06b37": "\\tilde{\\kappa}_{o+}=\\scriptstyle 0.7\\pm1\\times10^{-14}", "0d0d525d41ae34cd02b4707000dd7a26": "{n(3n-1)(3n-2) \\over 2}", "0d0d62c2fdffa2e5cedc35e099329266": "[(\\gamma_1)_\\mu (p_1-\\tilde{A}_1)^\\mu+m_1c + \\tilde{S}_1]\\Psi=0,", "0d0d6e27773a4d8daaa2ccd06fae3849": " \\qquad \\qquad \\mathbf{j}_e = -\\frac{e_c}{\\hbar^3}\\sum_p\\mathbf{u}_ef_e^\\prime = -\\frac{e_c}{\\hbar^3k_\\mathrm{B}T}\\sum_p\\mathbf{u}_e\\tau_e(-\\frac{\\partial f_e^\\mathrm{o}}{\\partial E_e})(\\mathbf{u}_e\\cdot\\mathbf{F}_{te}),", "0d0d7fee17b1432ea07a56ec95579f96": "F=\\overline{ A \\vee (B \\wedge C)}", "0d0db343dafbe7e23849033e3346288a": "\\{x: p_{\\alpha_1}(x) < \\epsilon, \\cdots, p_{\\alpha_n}(x) < \\epsilon\\}", "0d0db7d76c9052c8ba42d17c449d117c": "m_{solvent}", "0d0dde6d755b96b85d59014e3168f700": " \\frac {x}{\\ln x + 2} < \\pi(x) < \\frac {x}{\\ln x - 4}", "0d0ddeb64a1d1e92f4adfccee1c285d8": "\\delta t=-6.5\\pm7\\ (\\mathrm{stat.}) \\pm6\\ (\\mathrm{sys.})", "0d0e02cb783b4d37bfd5da3ecd0a506f": "\\left\\Vert \\mathbf{x} - \\mathbf{c} \\right\\Vert^2=r^2", "0d0e0485539af8381bf0e653bc083810": "z_m", "0d0e7b94e4fd1fbb404e609f2d6427a9": "pV", "0d0e9e99c8639eaf90300e449a52d109": "\\mu(\\sigma) \\geq \\mu(\\tau)", "0d0ec4dfa71d81b6c06a605bf9a415a1": "\\displaystyle \\int", "0d0ee2c2ad53de6180b1ab5bc1addeda": "E=\\int_{S}\\rho e\\left( \\rho \\right) \\, dS.", "0d0ee4482f7e3d9029649a80c4cdd6f1": "k\\geq l-1", "0d0f0b11b41a7b6522da4bca1630b24a": "\\frac{d^2y}{dx^2} = \\begin{cases}\n\\mbox{unbounded} & \\mbox{if } y = 0 \\mbox{ and } x \\ne 0 \\\\\n0 & \\mbox{if } y = 0 \\mbox{ and } x = 0 \\\\\n\\frac{3a^6(y^2 - x^2)}{y^3(a^2 + 2x^2 + 2y^2)^3} & \\mbox{if } y \\ne 0 \n\\end{cases}", "0d0f13ed6bdce1de9b4b961c513509a7": "{T}_{\\mathrm{m}}", "0d0f7304637ecc05e6032ed806668242": "H(X_1) \\le H(X_1, X_2)", "0d103e597a557acac8e624df87c26cca": " 2 - \\eta = \\frac{\\gamma}{\\nu} = d \\frac{\\delta - 1}{\\delta + 1}", "0d104191c889282aa464b7fb36d6bed4": "\\bar v = \\frac{\\bar q}{\\bar k}", "0d107ece98c0895e7a3803954ed8074a": "m | \\lambda(n) ", "0d115e6b11b58cad4046c41d4bd0471e": "1,2,\\ldots,n", "0d11771ee8b0b71ee5e2679d9323f860": "\\left(\\vec \\mu_0, \\Sigma_{y=0}\\right)", "0d118be65949bca98b49c59f8b401130": "\n\\mathbf{M}_{\\rm orb}=\\frac{-e}{2m_e}\\sum_n\\int_{\\rm BZ}\\frac{d^3k}{(2\\pi)^3}\\,\\langle\\psi_{n\\mathbf{k}}\\vert\\mathbf{r}\\times\\mathbf{p}\\vert \\psi_{n\\mathbf{k}}\\rangle \\,,\n", "0d11a9233dfb621f73612b720b9ae770": "\nA_{xx} x^{2} + 2 A_{xy} xy + A_{yy} y^{2} + 2 B_{x} x + 2 B_{y} y + C = 0\\,\n", "0d11c9f5cffe121efbb4a96857a4107a": "~~~= C+\\tau +2\\sqrt{2GM\\tau} +4GM\\ln\\left(\\sqrt{\\frac{\\tau}{2GM}}-1 \\right)\n", "0d11f291eb75b1dc6cb4a8541de39433": " N_J=0\\mbox{ and }d\\omega=0 \\,", "0d11fda126df5d8404401c618ebb59ce": "x\\ \\rightarrow\\ -x", "0d12237ccc6035b81781f5621d4cadb2": "\\sum_{j = 1}^3\\partial_j\\tau_{ij} + \\rho g_i - \\partial_ip = Q,\\,", "0d127a74dba109a97b99c424ab1b89eb": "g_{n + 1} = \\sqrt{g_n \\cdot a_n} \\geqslant \\sqrt{g_n \\cdot g_n} = g_n", "0d12c259ace57e4c18cfb55619336a62": "Sq\\,60 \\times 9", "0d12e1d934c94beaf6f3803f2c236b11": "KC(x_i, x_j) = \\int K(x, x_i) \\cdot K(x, x_j) dx", "0d1321efd35b20d2a2391609a9b59dc9": "r,s \\in T", "0d1379966d5dd80e86a3044371ec5898": "W_\\theta((z_1,z_2)) = (e^{i\\theta/2} \\overline{z_2}, e^{-i\\theta/2}\\overline{z_1}). \\, ", "0d13a1b68829647af5b6b0fbc1c03da7": "V \\otimes V/(v\\otimes v \\text{ for all } v\\in V).", "0d13aa952054b8a8bf34017a75aa4f03": " f(z) = z + a_2 z^2 + \\cdots .", "0d13b875c8183e4e96486212feab69b3": "log(n),log\\frac{R}{S}(n)", "0d13b9c0b860a192d858d90f2002d01c": "E = - \\sum_i h_i S_i", "0d13cdba6cf1defa57ddb9df7ccca439": "\\frac{4bh}{3}", "0d13f7d127bd0965408c48ff337af703": "\\scriptstyle \\varphi_{AB}", "0d1401b9be38833c006a4fa098717f44": "\\int_{\\mbox{arc}}{e^{itz} \\over z^2+1}\\,dz \\rightarrow 0\\ \\mbox{as}\\ a\\rightarrow\\infty.", "0d14467f35cab959457c4b565a1471da": "C^*(\\theta) \\subset C(\\theta)", "0d149b90e7394297301c90191ae775f0": "it", "0d14aed5be8c8756e4a52158149750de": "v_{x} = \\frac{dx}{dt}", "0d14ef0f5c907a1deb056a660280790f": "{\\rm tr}(\\mathbf{A}^{\\rm T}\\mathbf{B}) = \\operatorname{vec}(\\mathbf{A}) \\cdot \\operatorname{vec}(\\mathbf{B}),", "0d1578a15ef9db477c7e63c5c62e699f": " a = \\frac{p_1 + 1}{2} ", "0d1586512455d6ef9241f6874e15a601": "\\mathbf{f}\\mapsto \\mathbf{f}' = \\left(\\sum_k X_ka_{k1},\\dots,\\sum_k X_ka_{kn}\\right) = \\mathbf{f}A", "0d1591192899a5b38c1894780efdd6cb": "\\phi_{\\Gamma}(\\omega)", "0d159fded7bade094c53802c76e57d9f": "q = 23", "0d168fa842421ee66899e756bbb4b7c5": "\\frac{\\Delta R}{R} = GF \\varepsilon + \\alpha \\theta", "0d16a6cb2ca34d65da949608a7bc01d1": "0.02", "0d16b120ce6eb62b8466601b5259fbaf": " \\begin{align} \\mathbf{u} &= \\sum_{n=-\\infty, n\\neq1}^{n=\\infty} \\left[ \\frac{(n+3)r^2\\nabla p_n}{2\\mu(n+1)(2n+3)} - \\frac{n\\mathbf{x}p_n}{\\mu(n+1)(2n+3)}\\right] + \\sum_{n=-\\infty}^{n=\\infty} [\\nabla\\Phi_n + \\nabla \\times (\\mathbf{x}\\chi_n)] \\\\\np &= \\sum_{n=-\\infty}^{n=\\infty}p_n \\end{align} ", "0d16f72296bd681931be5741bc1e2964": "\\kappa \\,.", "0d1713700b056ccc81c30672d64323af": "\\operatorname{cov}(X, Y)", "0d171d6a16ab48052105c6921f8e78d4": "y^{(n)}", "0d179d6c5cb2567c8d1f809017799eb5": "\\delta_X(X)", "0d17ae5473412be33cb013507f078adb": "b_2 \\approx \n\\left[\\begin{matrix}\n 0.64676 \\\\\n 0.40422 \\\\\n 0.64676 \\\\\n\\end{matrix}\\right], ~\\mu_2 \\approx 5.2418\n", "0d17c96f7f4592cf6458686db32b9676": " f: \\mathbb R\\to\\mathbb R; x\\mapsto x^2", "0d17ed49c7a259bba8b0ab4134cdaf84": "\\displaystyle{R_j}", "0d17f15ddb059d52d289b254bcf74f4b": "\\,O_i", "0d17f8dc895d6b2f90775b63f6db0163": "\\langle P, \\le \\rangle", "0d18110c6be9d42370e9d8ee2feab8d2": "R^{1/2}\\ ,", "0d1820932a6e23cb1a23a6602c4d5851": "\\psi_f", "0d186644a6b0048d2f4760e14fa16196": "\nV = V_1 = V_2 = \\ldots = V_n\n", "0d1878dfddb55a2ebeb3124eb58be1d9": "\\vert : A \\times A \\rightarrow A", "0d188a9c7c08a4806555b1201fefa9d8": " x(0)= \\phi (-t_0)x(t_0)-\\phi(-t_0)\\int_{0}^{t_0}\\phi(t_0 - \\tau)[Bu(\\tau)+Ew(\\tau)]d\\tau", "0d188eed59acc58f3f2da960f2a158b1": "f(\\cdot;\\mathbf{v}^{(t-1)})", "0d18e073bc56712e22cc4599b12ff989": " = a_1 e_1 + a_2 e_2 + a_3 e_3. \\,", "0d18f3d858056d92fbca83a7741fed77": "\\sigma_S", "0d1901a17a5c3234ff79910c818f4531": "\\left\\{ \\rho_{x^{n}\\left( m\\right) }\\right\\}\n", "0d192fb144e165a94ce0ad3d999e284f": "{\\rm d}(ab-ba)=0,\\ \\forall a,b\\in A", "0d194afea9be631172525ea9db6f15bb": "-\\frac{\\hbar^2}{2m}\\nabla^2\\phi = i\\hbar\\frac{\\partial}{\\partial t}\\phi.", "0d1a5a2b554fab5b3076c0152981e6e3": "u=(u_1,...,u_n)", "0d1a73530f7a59d7fc84b3a5094d09e1": "\\Delta\\lambda", "0d1a76ef259a793f7a800ae90b0870de": "D_{**}^{(p)}(\\mathbf{X}, \\mathbf{Y}) = \\left( {\\sum_i{D_{**}(X_i, Y_i)}^p} \\right)^{\\frac1p}", "0d1ab40684001e761f4fb4f2256bde93": "= \\gamma \\frac{\\omega_{\\mathrm{obs}}}{c} - \\beta \\gamma k^1_{\\mathrm{obs}} \\,", "0d1b52bc46246e7df5acc42385979615": "a\\times a", "0d1b62cac42bf3fbd446acf638ce1978": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ 0.5\\end{smallmatrix}", "0d1b80ad19cafad14503d2f2060bd38c": "(V_x,V_y)", "0d1bc873fc7a89d2a5fecab6aa8a3265": "\\tfrac{dy}{dt}", "0d1c137f5055f97e670d1393065b97f9": "5 \\times 7 = 35 \\,", "0d1c62636094364b4c5ca07192d0fd48": "f(\\xi )\\star g(\\xi )=Tr[\\hat{B}(\\xi )\\hat{f}\\hat{g}]", "0d1cfc2b565cc8ef6c591b870149fdee": "GL(R)", "0d1d0541ed5b1a49323e3e5b1a426d06": "{\\mathbf{}}P(t),S(t),\\hat{P}(t),\\hat{S}(t)", "0d1d353a531d0c830d2ee514f36b3e6d": "\\operatorname{dim}[D(\\cdot)] = q \\times p", "0d1d512c5a77deaf46d335aa42d873aa": "\nP(Y \\in S) = \\int_{\\phi^{-1}(S)} p_x(x)~dx = \\int_S p_x(\\phi^{-1}(y)) ~ \\left|\\frac{d\\phi^{-1}}{dy}\\right|~dy. ", "0d1d5de64a1eca2f5186aaf6f5529bb2": " (D-\\lambda)\\varphi_\\lambda = 0, \\quad (D-\\lambda)\\theta_\\lambda =0 ", "0d1da73a7e2cdd65c45bd490652776a3": "\\Delta L / Lo", "0d1dce0001d87680ea382492c73dedf7": "Y = mX + b", "0d1f051084914b4857972354998479d6": "\\text{Si}(z)=\\int_0^z\\frac{\\sin \\zeta}{\\zeta}\\,d\\zeta", "0d1f538321149df9161ae9bcb8321589": " \\Delta Y = V \\Delta K + W \\Delta L + \\Delta Y' ", "0d1fb4cc5a4abdb666d33e0ceb2bb04c": "A_{n}=A_{n-1}-r_{avg,n}\\Delta t_{p}", "0d1fe771863fbaaf863d6b545e5b7b1c": " (g(T)\\xi,\\eta) = \\int_0^1 g(\\lambda)\\, d(E(\\lambda)\\xi,\\eta) = \\int_0^1 g(\\lambda)\\, d\\rho_{\\xi,\\eta}(\\lambda). ", "0d20024c7340f05df7f7abdbd9858389": " n\\ge (m+1)", "0d2008537bbce6671711b431aa5bfc1e": " t_n \\leftarrow t_l + ta(s); ", "0d202ce153f5df5cb6fe44b9461a20b5": "H\\left( X\\right) \\equiv-\\sum_{x}p_{X}\\left( x\\right) \\log p_{X}\\left(\nx\\right) .", "0d207726d1b7ee91d77a749a0fe04bc8": "\\chi(R)", "0d20bff51398891724de0e5ebe4ee297": " \\ln \\left ( \\frac{1}{p} - 1 \\right ) = \\sum_{i=1}^N \\left[ \\ln(1-p_i) - \\ln p_i \\right]", "0d20c1dbe3e3a3ad0d4a224004f63af5": " \\log=\\exp^{-1} ", "0d20ca8c039a274aa997eff14b25f11c": " r = a", "0d20cd48b4784d29c208353fb3e79545": "\\sigma^2_a > 0", "0d20df5445563118b38f0a894c4f7e98": "X\\subset\\R^n", "0d20fcd3e97607f522c75657445e2ab3": "{D(R)}^{(j)}_{m'm} = \\langle j, m' | U(R) |j,m \\rangle", "0d211795903196a13bc257387de9b858": " \\scriptstyle ratio = \\frac{PV(B_2) - PV(B_1)}{PV(C_2) - PV(C_1)} ", "0d217b2d4e34f78f6c6fd2e02695c591": "k=\\frac{\\omega n_1}{c}\\sin(\\theta_I)", "0d2183bca51700162b8418f4239f35d1": "E_{\\rm g}", "0d2194541ab3767828573818fcaad8d3": "f_k(x) = e^{kx}", "0d21ed18a98150ae571d46a2d4ff254d": "\\bigoplus_{p=1}^n S^2(\\Omega^p M),", "0d21edba5c892ac3ec8de9f8d602eb34": "\\dot{q_i}", "0d2205031b1c6b2e8d118218cf68ee51": "e d \\equiv 1 \\pmod{(p - 1)(q - 1)}", "0d22066efc96d7e7d4b6c2a428d76646": "\\exp (X) = \\sum_{k=0}^\\infty\\frac{X^k}{k!} = I + X + \\frac{1}{2}X^2 + \\frac{1}{6}X^3 + \\cdots", "0d2272aed78c76153375fa478fdb5b77": "C_{D,i} = \\frac{C_L^2}{\\pi A \\epsilon}", "0d22b0574833ed0702cb4cc05c077fc6": "\\scriptstyle a \\equiv b \\pmod m", "0d22be7fc4410551eecdd682be98569f": "p \\sim \\mathrm{Beta}\\left(\\alpha+n,\\ \\beta+\\sum_{i=1}^n k_i\\right). \\!", "0d22d4f5aa39ec827e00911e251a5ae0": "\n\\vec{D}=|\\vec{C}.\\vec{X_p}(t)-\\vec{X}(t)|\n", "0d23d33bf55d973e8dc20d2434e2ebc5": "\\mathbf{V}^{-1} \\mathbf{C} \\mathbf{V} = \\mathbf{D} ", "0d23f09a073cdb873cead2349fd0748a": "\\int_{U_\\alpha} f = \\int_{\\phi_\\alpha(U_\\alpha)} t_\\alpha\\circ f\\circ\\phi_\\alpha^{-1}d\\mu", "0d24044f81ee0e3875e213e76122b1ca": "\\mathrm{Inv}^1 \\langle X | T\\rangle=(X\\cup X^{-1})^*/(T\\cup\\rho_X)^{\\mathrm{c}}.", "0d249fd47fefa390fa28d1ae2c141d9a": "{\\mathbf y} - {\\boldsymbol\\nu}", "0d24c605000ecb1ba6dc1183f36dd153": "{\\rm Vol}(M)<\\varepsilon(n)", "0d250ccf9b29327fc9cd0be9f2d1a8b3": "(9)", "0d2522f2702995f340f189e0f0b81559": "\\sqrt{e^{\\sigma^2}\\!\\!-1}", "0d25530d2daf54334b54c7ddd7dd2a81": "\\kappa(A,Q_1) \\neq 0 \\neq \\kappa(K_M,A')", "0d255fc40699018dfbc407a92e64e25d": "s_j(x) = s_i(x)\\cdot t_{ij}(x).", "0d25727aed4204122915f50b57f45e0a": "\\upsilon_p = \\upsilon_o = \\pm( \\upsilon _D -| \\upsilon _M | ) \\qquad(8)", "0d257d22b5be1adb99c63cf8207a7818": " \n c(x,\\eta)= \\left\\{\n \\begin{array}{l}\n 1 \\quad \\text{if}\\quad |\\{y\\in x+\\mathcal{N}:\\eta(y)\\neq\\eta(x)\\}|\\geq T \\\\\n 0 \\quad \\text{otherwise} \\\\\n \\end{array} \\right.\n", "0d261e8e4a3dd10b67d214d2b0d8d4ea": " D_{k-1} \\wedge ", "0d2623e10c0aea105f257c1188978a19": " B_0 = -V \\left(\\frac{\\partial P}{\\partial V}\\right)_{P = 0}", "0d265bb29d3ea13319bfc6c65d006e71": "t+k\\,\\Delta t", "0d266404d59506165f02b7f9330257af": "O(n) \\subset O(n+1)", "0d267342c9e958b2d9571c87d97604c9": "\n\\int (A+B\\,x) (a+b\\,x)^m (c+d\\,x)^n (e+f\\,x)^p dx=\n \\frac{(A\\,b-a\\,B)(a+b\\,x)^{m+1} (c+d\\,x)^{n+1}(e+f\\,x)^{p+1}}{(m+1)(a\\,d-b\\,c)(a\\,f-b\\,e)}\\,+\\,\n \\frac{1}{(m+1)(a\\,d-b\\,c)(a\\,f-b\\,e)}\\,\\cdot\n", "0d2686a84e52ad3ea4d6d16f8d2c5336": "I=\\bigcap_{i=1}^k P_i", "0d26eb023d1e859d4e55319835f5dd70": "T_{\\rm h}", "0d27c8fab6667d760378094752161e20": "c = \\mathbf{st}(x_{i_0})", "0d27d9bd6674d75a69331e42c1ac0b45": "\\mathbf{A}\\mathbf{x} = \\mathbf{b},\\, x_i \\ge 0", "0d27de988caad8fbd3c1dfd807e09332": " \\mathrm{Eu}_m = \\mathrm{Eu} \\; \\quad\\quad \\mbox{i.e.} \\quad {p_m \\over \\rho_m {v_m}^{2}} = {p\\over \\rho v^{2}} \\; , ", "0d281732e55b5b0446f4cb26f620ee3a": "V_0=2, V_1=A, V_j=AV_{j-1}-V_{j-2}", "0d28187719daa8439ef00bdce60d0cdc": " \\{y_k\\}_{k=1}^{M} ", "0d2876b89cf5452c755b6685369b8591": "x>a", "0d289ca0712b0e45db926c51f7e77153": " r v_{\\theta} = \\Gamma/(2 \\pi)", "0d28b4516f695f007e7f87f8f95529d4": "x=as+x_0=at+x_0\\,", "0d28d8ceee019945cdd265ac580aec67": "\\gamma \\in E", "0d2901ef433b8e8b8a1fe785330e4edd": "\\|\\boldsymbol{x}\\| := \\sqrt{x_1^2 + \\cdots + x_n^2}.", "0d2941a258bdb4c6652738256c19daf2": "MU*(MU)", "0d2952936deadd6f35f0ce351fa6710f": "|{\\mathcal Z}|=n(n^2+1).", "0d2990b73c38ab89d3eb9287b0432b24": "T(X_1^n) = \\left( \\prod_{i=1}^n{x_i} , \\sum_{i=1}^n x_i \\right)\\,", "0d299763a1317c3b9bf559c527627b48": "6v - 2e = 6\\sum_{i=1}^D v_i - \\sum_{i=1}^D iv_i = \\sum_{i=1}^D (6 - i)v_i = 12.", "0d29a3a7d93ff82adf7d58cea12cc7d1": "\n\\frac{dK_i}{dt} = 1 \\,\\forall\\, i \\in S\n", "0d29c6241f4e8753818cf25871124385": " b = \\sum_i x_i \\; b_i", "0d29d0111b21f34ce720c6c4a665cdea": "\\frac{\\lambda\\Delta\\Theta}{4\\pi \\Delta t}", "0d2aec71e040b90af40a5c82f1cd40f4": "\\alpha \\varphi = {d \\varphi \\over d x} = 0", "0d2af1bb9bdb0e828e0b1643ce5dfbe9": "\\scriptstyle E_0(x)=\\frac{x}{e \\sqrt{\\pi}}", "0d2af255ed8deb7614e9b97cc3e82df5": "\n \\mathcal{M}_{XY}(s) = \\mathcal{M}_X(s)\\mathcal{M}_Y(s)\n ", "0d2b3128c7ae59366163f66691bab00b": "\\nabla_{\\boldsymbol{r}} H(\\boldsymbol{r}) = \\begin{bmatrix}\n\\partial_\\boldsymbol{q}H(\\boldsymbol{q},\\boldsymbol{p}) \\\\\n\\partial_\\boldsymbol{p}H(\\boldsymbol{q},\\boldsymbol{p}) \\\\\n\\end{bmatrix}", "0d2b4014b1181599e39765af3670387d": "Q = I^2 \\cdot R \\cdot t", "0d2b4ac581f80000a4abcdc1e2115f27": "\\lambda = \\lambda_0\\,", "0d2b65af50b37263fc83b5d435647433": "\\displaystyle g", "0d2b922fc35916c85df042d4e0bb7ddc": "\\textstyle\\tilde{u}", "0d2ba1be115838af1b49c1a6139674d4": "1-\\delta/2", "0d2bbcb2cff894f24437683c56beb41c": "B/I", "0d2c25d79d691c6d36b745715404093f": "A \\subseteq_\\omega G", "0d2c5875cb4535292638d4a381417bb6": "p = -\\left(\\frac{\\partial A}{\\partial V}\\right)_T \n= \\frac{NkT}{V-Nb'}-\\frac{a' N^2}{V^2}.", "0d2cab7545531536ea36a769019f921b": "\\text{PoP} = C \\times A", "0d2d275575f90b900f8fe693c3f1e79b": "ZY = U + iV\\,\\!", "0d2d5aac0c0d0ec3ffb080cc6ed30ab2": "\\begin{pmatrix}\nA'(x) \\\\\nB'(x)\\end{pmatrix}=\n\\begin{pmatrix}\nu_1(x) & u_2(x) \\\\\nu_1'(x) & u_2'(x) \\end{pmatrix}^{-1}\n\\begin{pmatrix}\n0\\\\\nf\\end{pmatrix}", "0d2d8db530bbee3a8caf21c607110a87": " ALG(\\sigma)-c.OPT(\\sigma)\\leq\\alpha", "0d2d998328a6fe85713bedef922fa466": "\\pi(x_k|x_{0:k-1},y_{0:k})\\, ", "0d2da4e5aa8e8c7d8aaf1020ebb5fd55": " \\{ \\cdot,\\cdot \\}_{\\eta} ", "0d2dc97221a8c1d67c2cc5a89f0c4812": "g\\cdot\\left(x,y,\\frac{dy}{dx}\\right) \\stackrel{\\text{def}}{=} \\left(\\overline{x},\\overline{y},\\frac{d\\overline{y}}{d\\overline{x}}\\right).", "0d2e17ef5417e82f59050aaf118ab302": "p = vi \\,", "0d2e751ea16d8df35051ef5c09af15b4": "\\mathrm{H}^n(G; M) = \\mathrm{Ext}^n_{kG}(k, M)", "0d2e80e7b52d3ae4c59192eec337947b": "\\{ U_{\\lambda} \\vert \\lambda\\ \\in\\ \\Lambda \\}", "0d2e823943aec1ab24408beedcc9116c": " \\sigma_P ", "0d2e858bd7f89eed5461e5637d6e0a50": "\\log n", "0d2e89ded8b8c63f4ebde42591795bcb": "p^{ij}=p\\left(Q_b^{(i)}|Q_c^{(j)}\\right)", "0d2f44a7b4269b89b6b5610eabdcec76": "|p||q|", "0d2f45f312359620d82962b2ec6be309": " \\tau_0 ", "0d2f6250f9472057738f1e2f6a783f20": "a := J/M", "0d2fc0ce4b2672c236ee1e7727be2a63": "\n\\pm \\frac{z_{1-\\alpha/2}}{\\sqrt{N}}\n", "0d2fc3dce992c1a20056f3ca9d16f03b": "\\frac{\\partial^2 y}{\\partial x_i \\partial x_j} = \\sum_k \\frac{\\partial y}{\\partial u_k} \\frac{\\partial^2 u_k}{\\partial x_i \\partial x_j} + \\sum_{k, \\ell} \\frac{\\partial^2 y}{\\partial u_k \\partial u_\\ell} \\frac{\\partial u_k}{\\partial x_i} \\frac{\\partial u_\\ell}{\\partial x_j}.", "0d300d8bb92e7abcb5f1fc0d3637425e": "\\mathbb{RFM}_I(D)\\,", "0d306f1bd424d62905a90745404a17f7": "a_n \\approx \\sqrt{\\frac{\\rho+\\rho^2B'(\\rho^2)}{2\\pi}} \\frac{\\rho^{-n}}{n^{3/2}},", "0d309d7aef3404621e0719ccdbb900ea": "f_r(\\theta)=F(re^{i\\theta}),", "0d30a10e004417673985e972bf8fd97f": "\\oint_\\gamma f(z)\\,dz =0", "0d30a83d5cbeb54b7220236e608f1e90": " j = 1 \\ldots p ", "0d319acc1a9015aab6b189e45c77f7b5": " \\sigma(n) \\le H_n + \\ln(H_n)e^{H_n}", "0d31a97e764a72427abaed22f8483aba": "H_1:\\theta=0", "0d31e16b4e6149c32102911d79cf4e0f": "\\dot{M}", "0d31f258914ebf9e21fbd3cd1b55801e": "(x,\\, y,\\,z) = (1,\\,2,\\,3)", "0d32149b4ee4096ea0a8bf9a2d5d41ed": "\\mathcal{G} \\times \\mathcal{H}", "0d322a6883cd9363afefdb344596517f": "\\Psi : G \\to \\mathrm{Aut}(G)\\,", "0d3244f649381020e137893daa357eee": " {\\frac {|AB|} {|BD|} \\sin \\angle\\ BAD = \\frac {|AC|} {|DC|} \\sin \\angle\\ DAC}", "0d32c7b080b9c0ab6b422b59eac02eef": "5/4 E_{\\mathrm{h}}\\,", "0d331c82cd6dfbe6ef081a4c5facc021": "L_c \\ll \\lambda", "0d3376240ee4aa12f272f6bd4ce0a4ef": "\\Omega = 2\\pi \\nu", "0d337c802e7e5c198e21a6ab6eb17ec8": "E(\\mathbb{F}_p) \\cong \\mathbb{Z}_{2^{k}n}", "0d3381bc741b5c2ae9f71b2ad21f4058": "\\begin{align} F_x & = -q\\left(\\frac{\\partial \\phi}{\\partial x}+\\frac{\\partial A_x}{\\partial t}\\right) + q\\left[\\dot{y}\\left(\\frac{\\partial A_y}{\\partial x} - \\frac{\\partial A_x}{\\partial y}\\right)+\\dot{z}\\left(\\frac{\\partial A_z}{\\partial x}-\\frac{\\partial A_x}{\\partial z}\\right)\\right] \\\\\n& = qE_x + q[\\dot{y}(\\nabla\\times\\mathbf{A})_z-\\dot{z}(\\nabla\\times\\mathbf{A})_y] \\\\\n& = qE_x + q[\\mathbf{\\dot{r}}\\times(\\nabla\\times\\mathbf{A})]_x \\\\\n& = qE_x + q(\\mathbf{\\dot{r}}\\times\\mathbf{B})_x\n\\end{align}", "0d3385c0bc07dd05265d2bfeb61ff11d": "i\\in N.", "0d3387f183b071349623c1d18272f415": "z^* := f(x^*)\\in \\mathbb R^k", "0d33e9f5b106df3293cdf9a9ba1e4c77": "E(z,s) ={1\\over 2}\\sum_{(m,n)=1}{y^s\\over|mz+n|^{2s}}", "0d3412cfa681f1b4049bcd69b873b892": "\\overline f \\colon \\overline V \\to \\overline W", "0d342dc5b57d999d0e51e4e8e1dba385": "\\Sigma=1", "0d3438af5eafe64e58b63a68d9d73f4c": "\n\\operatorname{Ind}_H^G\\pi=\\{f:G \\rightarrow V|f(hg)=\\Delta_G^{-1/2}(h)\\Delta_H^{1/2}(h)\\pi(h)f(g) \\text{ and } f\\in L^2(G) \\}.\n", "0d348c4753b1623b79d6f1c8420766ef": "a_0x^n + a_1x^{n-1} + \\cdots + a_{n-1}x + a_n \\, ", "0d34c45e8cc424127988be70470886b2": "\\displaystyle = \\frac{f}{\\sigma}\\frac{\\partial}{\\partial p}\\mathbf{V}_g\\cdot\\nabla_p (\\zeta_g + f) + \\frac{R}{\\sigma p}\\nabla^2_p(\\mathbf{V}_g\\cdot\\nabla_p T)", "0d34c87c64f32752b59c9fc56b15ad09": "x^2+y^2-1=0", "0d34ce5114059b25027f3669291236cf": "\\nabla_{{\\mathbf e}_j} {\\mathbf u}=\\nabla_j {\\mathbf u} = \\left( \\frac{\\partial u^i}{\\partial x^j} + u^k \\Gamma^i {}_{jk} \\right) {\\mathbf e}_i ", "0d34f3f304db97277d1b5aa730a452df": "\\rho(\\mathbf{r^{\\prime}})", "0d3521d9f3df8bd3fdd04f19b3e2d9f6": "\\sigma'_{p}", "0d3562e83fb6cb804b5f73faaa1a47be": "K'_0", "0d35715a228671a7b6bf190a507c97b8": "\nT = \\frac{1}{2} \\boldsymbol\\omega \\cdot \\mathbf{I} \\cdot \\boldsymbol\\omega = \n\\frac{1}{2} I_{1} \\omega_{1}^{2} + \\frac{1}{2} I_{2} \\omega_{2}^{2} + \\frac{1}{2} I_{3} \\omega_{3}^{2}\n", "0d3596c58bb98272bcdb7e2061d0e69f": "\\mathrm{supp}\\,X = \\mathrm{supp}\\,X _* \\mathfrak P.", "0d35ec8c42f9191581d0b11cb1d1af9f": "d=\\partial+\\bar{\\partial},\\ \\ \\ \\ d^*=\\partial^*+\\bar{\\partial}^*", "0d35fc1f504d5a3ca234e17aed31b978": "\\scriptstyle \\sqrt{1-x^2}", "0d3600f23d218a8d6ef9599d23cab885": "m^2=0", "0d3626a345c75f44137598236ab85e8c": " U(P) = - \\frac{i}{2\\lambda}\\frac {ae^{ikr_0}}{r_0} \\int_{S} \\frac {e^{iks}}{s} (1+ \\cos \\chi)\\,dS ", "0d362bd3c9965386899a8e1bdc8cd27b": " X\\leftarrow Y,\\; Y\\backslash X", "0d362e64d8ac6558fa534d852165aaa3": " y \\,=\\, r_1(\\theta)\\sin(\\theta)r_2(\\phi)\\cos(\\phi)", "0d368de2a395c090b46e83e71f24ed4f": "y\\equiv 0", "0d36a44f95642058433c588e2c564d8f": "(c_i \\ne C_\\text{in}(y_i'))", "0d36b40da67336f180fd1667993048f2": "\\begin{align}\n \\rho \\left(\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} + v \\frac{\\partial u}{\\partial y} + w \\frac{\\partial u}{\\partial z}\\right)\n &= -\\frac{\\partial p}{\\partial x} + \\mu \\left(\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2}\\right) + \\rho g_x \\\\\n \\rho \\left(\\frac{\\partial v}{\\partial t} + u \\frac{\\partial v}{\\partial x} + v \\frac{\\partial v}{\\partial y}+ w \\frac{\\partial v}{\\partial z}\\right)\n &= -\\frac{\\partial p}{\\partial y} + \\mu \\left(\\frac{\\partial^2 v}{\\partial x^2} + \\frac{\\partial^2 v}{\\partial y^2} + \\frac{\\partial^2 v}{\\partial z^2}\\right) + \\rho g_y \\\\\n \\rho \\left(\\frac{\\partial w}{\\partial t} + u \\frac{\\partial w}{\\partial x} + v \\frac{\\partial w}{\\partial y}+ w \\frac{\\partial w}{\\partial z}\\right)\n &= -\\frac{\\partial p}{\\partial z} + \\mu \\left(\\frac{\\partial^2 w}{\\partial x^2} + \\frac{\\partial^2 w}{\\partial y^2} + \\frac{\\partial^2 w}{\\partial z^2}\\right) + \\rho g_z.\n\\end{align}", "0d36b70987f5e50a7c1e4c2ba5b17c85": "\\displaystyle{X=Y^*,\\,\\,\\, XY-YX=2I.}", "0d370d005e07668b18ad75cdefc288ab": "P_\\lambda", "0d37622c2f3e173c0b258e5eb164842e": "b_c = 2 \\left( \\Delta c \\right) \\tan\\phi", "0d3785ba869328d58bb93b1d2d9b1f91": "M= ", "0d37d5cb4cc927f3d9371a5b3196d83c": "\\mathcal{O}=\\begin{bmatrix} C \\\\ CA \\\\ CA^2 \\\\ \\vdots \\\\ CA^{n-1} \\end{bmatrix}", "0d37d7a6a31e214764d679d2550860a5": "\\dot{m} = \\frac{d}{dt}m = \\,", "0d37f08551c16c1b3df722f9a230f833": "\\begin{align}\nc_2&=(a_1+a_2+c_1-1)/2-K \\\\\nc_3&=(3a_1+3a_2-c_1-3)/2-K \\\\\nc_4&=2a_1+2a_2-c_1-2 \\\\\nc_5&=(3a_1+3a_2-c_1-3)/2+K \\\\\nc_6&=(a_1+a_2+c_1-1)/2+K.\n\\end{align}\n", "0d3819fcc545ed0bdf8d3fdb1362a12f": " \\rho\\vec{\\nabla} \\times \\vec{v} ", "0d384ef21957d905c670977d9b67945e": "\\bot\\rightarrow\\psi", "0d3854a0747b1e2b2e2dc4aaee53058e": "\\begin{align}\n\n\\dot{e}\n&=\n\\frac{\\operatorname{d}}{\\operatorname{d}t} H(x)\n-\n\\frac{\\operatorname{d}}{\\operatorname{d}t} H(\\hat{x})\\\\\n&=\n\\frac{\\operatorname{d}}{\\operatorname{d}t} H(x)\n-\nM(\\hat{x}) \\, \\operatorname{sgn}( V(t) - H(\\hat{x}(t)) ),\n\\end{align}", "0d386210025f2537646ea1bd9f49b2bb": "\n \\kappa_{2n} = \\frac{2^{2n-1} (2^{2n}-1) B_{2n}}\n {n\\, (3^{2n}-1)}, \\,\\!\n", "0d386aef9fd3aa282a8e0f0d56b661c4": " {= {{V \\times I} \\over {m}}}", "0d3898ab45954a4419634dad94d3185c": "(X,p)", "0d38bd1865594124901028a3e55f08a6": "\\mathbf{R}_{B/A} = \\mathbf{P}_B - \\mathbf{P}_A = (x_B-x_A,y_B-y_A,z_B-z_A).", "0d390802d6e6490916db3adb36e5513a": "\n\\langle f_{thm}(s)f^T_{thm}(t) \\rangle = -\\left(2k_B{T}\\right)\\left(\\mu \\Delta - \\Lambda \\Upsilon\\Gamma\\right)\\delta(t - s).\n", "0d390d989d38d2710c82f29a2b01fe00": "\\bar{x} = \\frac{1}{3 N} \\sum_{n=1}^{N} (x_{n,1} + x_{n,2} + x_{n,3}) ", "0d392315c70d1c7a09e50ddddb0ee8d1": "2\\otimes2", "0d393a9bd11da911fdba0436fc08b613": "\\arccos \\frac{\\langle \\mathbf{r}(\\theta), \\mathbf{r}'(\\theta) \\rangle}{\\|\\mathbf{r}(\\theta)\\|\\|\\mathbf{r}'(\\theta)\\|} = \\arctan \\frac{1}{b} = \\phi.", "0d39b1b7b912ed75ff9e9aa383b6e7e6": "\\frac{(\\gamma/\\delta)^\\lambda}{\\sqrt{2\\pi}K_\\lambda(\\delta \\gamma)} \\; e^{\\beta (x - \\mu)} \\!", "0d39bd7aba5ab8f666ff14b2b638cd86": "\\gamma_k(G)\\leq G^{p^{k-1}}", "0d39ccf34f19b82c6e63a147dcad4a38": " \\alpha_1 = 4 \\frac GK ((2K-1) e^{-4\\phi_0} - e^{4\\phi_0} + 8) - 8 ", "0d39efc22d906cf481dcfcbfece93069": "a_{11} x_1 + a_{12} x_2 \\le b_1", "0d3a377330c597627c46f63d35f04031": "\\frac{1-p}{p^2}\\!", "0d3a3aacf4c3bd97b785057b39abd0de": "\\mathbf{p} \\rightleftharpoons \\mathbf{q}, \\quad H \\rightarrow -H . ", "0d3a74d610de42789cd0ca20b2a3c320": "\\begin{matrix}2&2&4\\\\3&5\\end{matrix}", "0d3abdc1ea0c2ebffbd6b221553a0504": "\n \\sum_{R\\in G}^{|G|} \\; \\Gamma^{(\\mu)} (R)_{nm} = 0 \n", "0d3b0c01e429b0a310dc32dfb61f6c59": "\\frac{\\part \\rho\\vec{u}}{\\part t} +\\nabla \\cdot \\Pi = 0 ", "0d3b265f26ca4093a6af09e954cba72a": "f(z) = \\int_D f(\\zeta)K(\\zeta,z)\\,d\\mu(\\zeta).", "0d3b3b0a92b0da46f8e512429229e033": "\\scriptstyle\\chi_o(G)", "0d3bda1e8f0c93cb21cdb05311977fb1": "\n\\mathbf K(\\hat n, \\nu) = \\vec a(\\nu)\\mathbf I + \\int_{4\\pi} \\mathbf Z(\\hat n^\\prime, \\hat n, \\nu) \\mathrm d \\hat n^\\prime\n", "0d3c7749fc39bfa9d6d4dfba34da5891": " 1 \\leq k < p ", "0d3c81cf1a175840d0e78f4967d677d2": "t \\triangleleft S", "0d3d06761d3042ca7648b218eadf3e09": "x^5+ax+b=0", "0d3d13b2ba6e4695ac5ad4b52c3c83ec": "p_{ss'}(a)", "0d3d2408fb0714eafda37a45cd554af3": "\\{(n - 2 k)^{\\binom{n}{k}}; k = 0, \\ldots, n\\}", "0d3d51c8013241ddb98fd15c694eebf4": " \\theta_{bg} ", "0d3d59dd6d67bbb885b5b46aac08a119": " T_{\\Phi} \\!", "0d3d649a9a3fd720302497f562e20fa7": " \\forall x \\in M: \\quad {\\{ f,g \\}_{M}}(x) = \\langle (\\mathrm{d} f)_{x} \\otimes (\\mathrm{d} g)_{x},\\eta_{x} \\rangle ", "0d3d6a79611b3704809919fcef7c29fe": "f_{\\text{b}}=\\frac{1}{{2\\pi}{R_{\\text{1}}}{ C_{\\text{F}}}}", "0d3db5f33c524eb5d934356c19bee5ca": "R^{n-1} \\oplus I", "0d3ddcf99f31e92ec8056a99247666f2": "y=y(t)", "0d3de16561dba05bf7a3e43187184aa4": " \\sigma_{P}^{2}=\\sum_{i=1}^{n}x_{i}^{2}\\underbrace{\\mathbb{E}\\left[R_{i}-\\mathbb{E}[R_{i}]\\right]^{2}}_{\\equiv\\sigma_{i}^{2}}+\\sum_{i=1}^{n}\\sum_{j=1,i\\neq j}^{n}x_{i}x_{j}\\underbrace{\\mathbb{E}\\left[(R_{i}-\\mathbb{E}[R_{i}])(R_{j}-\\mathbb{E}[R_{j}])\\right]}_{\\equiv\\sigma_{ij}}", "0d3e29998370f0bd204ad0b279a588be": "V_\\mathrm{swap} = B_\\mathrm{floating} - B_\\mathrm{fixed} \\, ", "0d3e6bd656f4b72a96b665ad8e4187b2": "\nf^n(x) = \\frac{D}{1-C} + (x-\\frac{D}{1-C})C^n = C^n x + \\frac{1-C^n}{1-C}D ~,\n", "0d3e710e24ff833abe64f4e7cfeaa703": "\\pi(x)>\\operatorname{Li}(x) +\\frac13\\frac{\\sqrt x}{\\log x}\\log\\log\\log x,", "0d3ea2ad17ad549d8e81ad9150607b53": "\t\\mathrm{v}=(A^T W A)^{-1}A^T W b", "0d3ee1c744c2568cfdd269e202457839": "\\operatorname{var_{GX}} = e^{\\operatorname{var}[\\ln X]}", "0d3ee75d810bc2dfb163d0e221524bce": "\\widehat{\\varepsilon} = \\frac{D_0}{E_0} = |\\varepsilon|e^{i\\delta}.", "0d3f04f1fa0ed58189e3e643518d2147": " V_7 = \\frac{16\\pi^3 r^7 }{105} ", "0d3f18f580556b9f285a21347527e61f": " \\mathbf{u}_k^{(i-1)} ", "0d3f197ee292ba0cca526cc015bac497": "y\\preceq x", "0d3f1a2930fca5e62b5babb4769895f6": "\\begin{align}\nh_1&=1\\\\\nh_2^2&=\\frac{\\lambda^2(\\mu^2-\\nu^2)}{(\\mu^2-a^2)(b^2-\\mu^2)}\\\\\nh_3^2&=\\frac{\\lambda^2(\\mu^2-\\nu^2)}{(\\nu^2-a^2)(\\nu^2-b^2)}\n\\end{align}", "0d3f57f00e9d8fe9e759be900b06465f": "s \\cdot s^2 ,\\qquad s \\cdot s^2 \\cdot s^2 , \\qquad s \\cdot s^2 \\cdot s^2 \\cdot s^2 , \\cdot ", "0d3f9f9ac9ecc98c3f41eccea575b445": "w(abcd;ef)\\equiv\n\\sum_z\\frac{(-1)^{z+\\beta_1}(z+1)!}{(z-\\alpha_1)!(z-\\alpha_2)!(z-\\alpha_3)!\n(z-\\alpha_4)!(\\beta_1-z)!(\\beta_2-z)!(\\beta_3-z)!}", "0d3fae5c51f1845e1bc1521dc69da04f": " \\epsilon_1=\\epsilon_2=0, \\epsilon_3=\\epsilon_4=\\epsilon_5=\\epsilon_6>0", "0d3fd5ea850d4397ac9a15e17dcaa56d": "\\mu\\Psi(\\mathbf{r}) = \\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r}) + g\\vert\\Psi(\\mathbf{r})\\vert^2\\right)\\Psi(\\mathbf{r}) ", "0d401919d1a7b58fa1a7027637587d9e": "\\mathbb{F}=GF(2)", "0d4026d4486d710582f7f63dd7d8d285": "\nJ_{A} = q_{\\mathrm{A}} j_{\\mathrm{A}}\n", "0d405d76bbb1bdd28be41db327144066": "h\\ll R", "0d4083d1b03dc87a52689b8524ca6c98": "\\mathcal{S} = \\int{\\mathcal{L} \\mathrm{d}^4x}.", "0d410447eb3b039f627f6256109cb740": "\\frac{p_{01}}{p_{1}}=(1+\\frac{\\gamma-1}{2}M_{1}^2)^{\\frac{\\gamma}{\\gamma-1}}", "0d418bc6ac2910e8f42bb722a1fc219d": "y=r\\sin\\theta", "0d41a092f385d4e00d361683eddc0b23": "\\textstyle x_1", "0d41ab17705002a473b26a8376762b2c": "y'=(2-y)y", "0d423e580708b1c1f686d142a635f618": "S_\\max= 1 + \\exp \\left( - \\pi \\frac { \\rho }{ \\mu } \\right). ", "0d424c34c2273d08727fac1d6bfc4b15": "{d,e,n}", "0d426f79201ec92973381653f2d165f0": "\\scriptstyle \\pi^{\\!*}F = dA.", "0d432fa7630523417b09114e6ebcc7c0": "r(x)=r(y)+1", "0d435509b55b9239d154e1f311994b40": "\\Phi(Y_i)=", "0d436a7d5a6abc8b5a66f2a87d0b943a": " \\varepsilon_\\text{Total} = \\varepsilon_D = \\varepsilon_S. ", "0d437439f04c80a6c9fd2019664de555": " \\tilde A = {h\\over{8\\pi^2cI_C}} ", "0d43981b3f452ea51c95a8906d662d7b": "\\alpha_0 \\le \\alpha_1", "0d43dd2ee0e2ae30c5b0d08a89df4a0d": "f^{(n)}(x)=a^{2^n-1}(x-x_0)^{2^n}+x_0\\,\\!", "0d445ec375ae7f77c87bdc321004baa3": "0 0", "0d547bafcbe7b374ba8c3413a8b12d7e": "f(z) = \\frac{1}{\\sin z}.", "0d5486f654b3c2ec9836b6aaaf9453ca": "q=\\frac{\\alpha}{\\beta}", "0d5512b584084204e6230c673b27345c": " l_i\\left(A_{ij} \\frac{\\partial u_j}{\\partial t} +a_{ij}\\frac{\\partial u_j}{\\partial x} \\right)+l_j b_j=0 ", "0d5513ddae1a1b8e60cacbb09d4fb18a": "\\langle\\ ,\\ \\rangle \\!\\,", "0d559c91acaa09c111d864b27ded63c6": " \\sigma = 1-\\frac{1.98}{z} ", "0d55b3badf33bb084c8e840debf460a1": " L_n(y) \\equiv \\left[\\,D^n + A_{1}(t)D^{n-1} + \\cdots + A_{n-1}(t) D + A_n(t)\\right] y", "0d55d80d215a530ba36225030f843d3e": "\\operatorname{Ti}_2(\\tan \\theta) = \\theta \\log{\\tan \\theta} +\\frac{1}{2}\\operatorname{Cl}_2(2\\theta)+ \\frac{1}{2} \\operatorname{Cl}_2(\\pi-2\\theta)\\, . \\, \\Box ", "0d55df8967de2ccce1d9838c4d87389e": "(a_{i,j})_{i,j \\in \\mathbb{N}}", "0d57694264eb7f5424f7feff938d0c87": "\\mathrm{OPT} - c", "0d58cedbd2b300987098452ecd4fa49c": "(\\partial \\langle N \\rangle/\\partial V)_{\\mu,T} = N/V", "0d58da4f428867347c6d83af51455530": "b^{2}(-P^{2},m_{1}^{2},m_{2}^{2})=\\varepsilon _{1}^{2}-m_{1}^{2}=\\varepsilon\n_{2}^{2}-m_{2}^{2}\\ =-\\frac{1}{4P^{2}}\n(P^{4}+2P^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2})\\,.", "0d591c1b9d3cec28690e2e3a6fbce794": "Z_L = V / I = j \\omega L \\, ", "0d59ab2b1aa5dfc093b5c68bc4f60eff": "p+q-1<3 \\sqrt{N} ", "0d59b4710476ea5bc11b2e04498d8e9c": " Z= \\int e^{-S} D\\phi = e^{-HT} = e^{-\\rho V} ", "0d59e1ac2fa6f3ba813b32b95c8b0fe6": "Z_{AC} = Z_{ref} + c{dC_l\\over dC_y} + c{dC_m\\over dC_x}", "0d5a2e64d007b197c34c58e151fb7aec": "\n\\mathrm{Ec} = \\frac{V^2}{c_p\\Delta T} = \\frac{\\mbox{Kinetic Energy}}{\\mbox{Enthalpy}}\n", "0d5a8ad7bb045db82195a3f5e99ea7a5": " pI = {{pKa} + {pKb} \\over 2} ", "0d5a9c236678da2c6adcaaf02997bfcc": "e_n = \\frac{1}{\\ln{2}}\\times\\frac{s-1}{2n}", "0d5ac1b33e4733486bd3025bbbae4587": "(1 + \\lambda_2)", "0d5adcb163927d4b17d3826018322070": "R_1=\\tfrac{1}{2}(\\sigma_2 - \\sigma_3)", "0d5ae2347ec769236abb2bb54822f72a": "M \\times \\{0\\} \\cup \\partial M \\times [0,1]", "0d5b015f7055943c22d58f1610233b1e": "L_L", "0d5b3fd549d306dcd5d8f29622e7c419": "\\begin{matrix}{4 \\choose 1}{3 \\choose 1}^2\\end{matrix}", "0d5b43518d23bc0bf26e946517649a3c": "p \\land \\neg q \\rightarrow s.", "0d5b509e6b353c7cc601dbb637434e13": "T_{\\mu \\nu}\\,", "0d5b7a43c4bb1dd3dcdb03bd6ab47db8": "{\\kappa}_{I}=C \\cdot d^2", "0d5b7f3aa14fbd23e85918400bcff9dc": "\\dot{Q}=\\frac{T_1-T_2}{\\left ( \\frac{L}{kA} \\right )}", "0d5c19f4325680412cb9b347bdbecdcd": " FV(A_1) \\not \\subset \\{p, q, m\\} ", "0d5c5da5ef7534b852fc07d37d1b2aa9": "\n \\begin{array}{ccc}\n \\frac{0.693}{r - r^2/2} & = & \\frac{69.3}{R - R^2/200} \\\\ & & \\\\\n & = & \\frac{69.3}{R} \\frac{1}{1-R/200} \\\\ & & \\\\\n & \\approx & \\frac{69.3 (1+R/200)}{R} \\\\ & & \\\\\n & = & \\frac{69.3}{R}+\\frac{69.3}{200} \\\\ & & \\\\\n & = & \\frac{69.3}{R}+0.34\\end{array}\n", "0d5cda0df025be1855baffdca92a4a10": "g(6) = 73", "0d5d6e921dcfa41128697ffa75f360d0": " \\frac{\\partial}{\\partial z_i} \\left(f\\cdot g\\right)= \\frac{\\partial f}{\\partial z_i}\\cdot g + f\\cdot\\frac{\\partial g}{\\partial z_i},\\quad \\frac{\\partial}{\\partial\\bar{z}_i} \\left(f\\cdot g\\right) = \\frac{\\partial f}{\\partial\\bar{z}_i}\\cdot g + f\\cdot\\frac{\\partial g}{\\partial\\bar{z}_i}", "0d5da1fefa03f21983ee7b5a7fb61f7b": "|\\psi(t)\\rangle=c_aexp(-i\\omega_at)|a\\rangle+c_bexp(-i\\omega_bt)|b\\rangle+c_cexp(-i\\omega_ct)|c\\rangle", "0d5df42c8e9316a4178e7862647973c0": "L=\\{X+\\xi\\in (\\mathbf{T}\\oplus\\mathbf{T}^*)\\otimes\\mathbb{C}\\ :\\ {\\mathcal J}(X+\\xi)=\\sqrt{-1}(X+\\xi)\\}", "0d5df489e2f1a84135cd1b87e837f796": "\\alpha = \\frac{[X]}{K_d^R}", "0d5e6a0fb08469455a5cc24e05d38bf5": "\nt_{\\textrm{lock}} \\approx \\frac{w a^6 I Q}{3 G m_p^2 k_2 R^5}\n", "0d5e80f0ab15deab09d277f5298b5c2d": "(Y\\cap Z)\\cup\\{x\\}\\subseteq X", "0d5ea8d130e95ceaa130e27156d59746": "(\\alpha -1)E_{0}/\\hbar c", "0d5ee01bc7d547ebe25fd08f615ebbeb": "U_{\\theta}", "0d5f09252973a93666c74fa009a36445": " S+I\\xrightarrow{\\alpha}2I ", "0d5f1644d03c7307a32fd1fca52f200b": "Z \\sim \\mathrm{Normal} (\\mu, \\sigma^{2}) \\implies \\mathbb{P} (Z \\in A) = \\gamma_{\\mu, \\sigma^{2}}^{n} (A).", "0d5fa3f335333b23d4aaf795d1336587": "X_1", "0d5fad88ed0b9085be3722b4f1e9a95f": "\\beta_m>k n_1", "0d5fcac139ae50e313382ffb3afae15f": " h=A*a_i ", "0d5fff6174419b8a0e3c044e7e01e991": "\\lambda_p(x)=\\frac{1}{1+x}.", "0d6004e23db3d54429858eda9a46c71c": "J(x, t; X_0, X_1) = \\max(\\|x\\|_{X_0}, t \\|x\\|_{X_1}).", "0d603094ff1076c7047ce8c93cdd6638": "\\scriptstyle \\Omega((n/\\log n)^{1/3})", "0d60d123c0dc96377db47739457728ad": "m_n = \\int_0^1 x^n\\,d\\mu(x)\\,", "0d61138809433c4493e1b1c6c16b7fcc": " = \\frac {Rise}{Run} * 100% ", "0d61152f9ff3c169afdbeb34ca366412": " c := \\lfloor - \\log_{10} x \\rfloor", "0d612e27b8e64dd5d3aa4b4d5046113b": "\\mathbf{e}_{123} ", "0d617cf30a4f2c270fa6b0958c1e2699": "\\{T^{-1}w_j\\}", "0d619e148f335c2d2fe38adeb5b9ff9e": " \\operatorname{ad}_x (y) = [x,y]\\ ", "0d61f8370cad1d412f80b84d143e1257": "C", "0d622acc0d5a62ee206f37f487dcd945": "f(x) = \\frac{x^{\\alpha-1}(1-x)^{\\beta-1}}{B(\\alpha,\\beta)}", "0d6283de008c2ce94def5fafb1ca337a": " t > \\sqrt{\\frac{2(n+1)d^2}{d}} - \\frac {d}{2} -1 ", "0d62eef61909a38edc064cc0294a6d2f": "H(u)[n] = \\scriptstyle{DTFT}^{-1} \\displaystyle \\{U(\\omega)\\cdot \\sigma_H(\\omega)\\}", "0d62f082a0c9aec576d53956a9d4bccc": "\\dot{m} = C\\;A_2\\;\\sqrt{2\\;\\rho_1\\;\\bigg (\\frac{k}{k-1}\\bigg)\\bigg[\\frac{(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}}{(P_1-P_2)/P_1}\\bigg](P_1-P_2)}", "0d62f184812c807e551b7f617a4f23b3": "G\\backslash X", "0d6378c66cc570ece9b6f9dc50044d57": "\\mathfrak{se}(d)", "0d638921e13e353de35287c0a2745e94": "e^{j2 \\pi f_0t}x(t) \\rightarrow S_x(t,f-f_0)", "0d6395a2d0abcc5cc6aa489130bd95e1": "\\frac{\\partial }{\\partial \\tau }q^{i}(\\xi ,\\tau ) = \\{\\zeta ^{i},H(\\zeta )\\}|_{\\zeta =\\star q(\\xi ,\\tau )}", "0d63e93bad860e148833242986589815": "U\\subseteq V", "0d63f2112e7cc4c2255f14e46301a253": "W^{(\\beta)}(t)=\\tilde W(t)+\\beta t", "0d64470b28db4216d30496c046fb0e98": "\\mathbf{p}\\psi(p) = p\\psi(p)\\,,", "0d644badd96f4ef5a1681f3d94a531c6": "\\ Z_\\mathrm{in} = \\frac{R^2}{Z_\\mathrm{load}} ", "0d6488b7edb024f2ed321a3300623550": "\\frac{1+3x}{(1-x^2)(1-4x)}", "0d648f393e6994136ec46e0c2baf9a9a": "M_{m} = \\frac {D}{D_{ep}} = \\frac {254}{7} \\approx 36", "0d649f87214df1393d310af177218e94": "f\\colon X \\rightarrow S, \\,", "0d64bccf62bc5c087ff4b2017f9c98dd": "A = N + Z", "0d64bd52634bdf799ee46236a0beaf29": "U > U*", "0d64fd7602820f08b4e8d7908846253c": "\\epsilon,\\delta>0", "0d65347d215124be539bde62d863dfbe": "\\mathbb{DICKENS}", "0d654f6b8093c0cf7124cfe990f7e2f4": "(p,q,t)\\cdot(p',q',t') =\\left (p+p',q+q',t+t'+\\frac{1}{2}(p q'-p' q)\\right).", "0d6572662db30822bbed759906067c5b": "\\gamma \\lambda(L)", "0d6578e80287ec2dda7958e9436486e9": "PR(A)= PR(B) + PR(C) + PR(D).\\,", "0d65a596011d0a7f904751aaccd84258": " I = S \\sqrt{t}\\ + A_1 t", "0d65abb818f0229c460007c7df628c47": "L(\\theta,a)=L(\\bar{g}(\\theta),a^*)", "0d65abbe010477c25bc6300b8a81ef91": "x',y'", "0d65e90e51759966dd4bfd0255008e53": "p(x_i)", "0d661ed63a08ce573e30035b0f5f3c8a": "L(\\chi_2, s)=\\beta(s).\\, ", "0d66747437616002e0f421e3d3075642": "g'(\\tilde{\\theta})\\,\\xrightarrow{P}\\,g'(\\theta),", "0d66ba5ff5ba9c4d3a15a0d24e9d6da8": "\\mu_n(X+Y)=\\mu_n(X)+\\mu_n(Y)\\text{ provided }1 \\leq n\\leq 3.\\,", "0d66e857aa9252d6a8b315535559e204": "j=1,\\dots,m", "0d66ee15f1123cdb5ef6f0e7d17fdfd4": "{n\\choose k} p^k (1-p)^{n-k}", "0d66f2288c0073410c0cb98ef61c400e": " \\det : M_n (\\mathbb K) \\rightarrow \\mathbb K ", "0d675882c51ec3a94a97eef709cd620c": "\\binom{C}{A}", "0d6761e5304f1c6981198514c5e02b3e": "L_{\\rm star}", "0d67cf596e4689d0f119be28c1c8daab": "\\prod_{\\eta \\le p \\le \\xi} \\left( 1 - \\frac{w(p)}{p} \\right) ^{-1} < \\left( \\frac{\\ln \\xi}{\\ln \\eta} \\right) ^\\kappa \\left( 1 + \\frac{C}{\\ln \\eta} \\right). ", "0d67e91be40d4d654130b8b74855f0d3": "\\Phi=e^{B+i\\omega}\\Omega", "0d680ccdea0752dc4b99ccd27b37edf2": "(15)\\quad Z^c\\nabla_c \\sigma_{ab}=-\\frac{2}{3}\\theta\\sigma_{ab}-\\sigma_{ac}\\sigma^c_{\\;b}-\\omega_{ac}\\omega^c_{\\;b}+\\frac{1}{3}h_{ab}\\,(\\sigma_{cd}\\sigma^{cd}-\\omega_{cd}\\omega^{cd})+C_{cbad}Z^c Z^d+\\frac{1}{2}\\tilde{R}_{ab}\\,.", "0d686788eeefdcd1f956dcab761da7b4": "\\langle\\phi|A|\\phi\\rangle", "0d6880c00d2e3c4f03e192217b0dbe10": "y(x) = A \\sin(x) + B \\cos(x).\\,", "0d6900f7ca5e6af1058b9c74f9040852": "V_D \\approx 0.18 V", "0d69048cce41d29392b99b246139214e": "\\tfrac{347897}{7558272}", "0d69073fca9788ab650eea1ebbf8d4af": "y = C x, ", "0d693922b7d69a86092944da8979294c": " R_{xx} = R_{yy} = -\\left( p_{xx} + p_{yy} \\right). ", "0d6940f87d5c0140cb7db39082082bfb": "|x-y|<\\delta\\,", "0d694f6b87eedd8efdc17b4ca0fea10f": "Y[x,y]=y+x'\\frac{x'^2+y'^2}{y''x'-x''y'}", "0d69682f10dc5e27b194fc7ce0420293": "b < a", "0d69e110a3b6126323880ec44e8fff34": "\\partial{F}/\\partial{z}", "0d6a1ef2a7cc036259368e7b556631d2": "\n\\theta = x_w=x \\mod 2\\pi\\ \\ \\in (-\\pi,\\pi]\n", "0d6a44b16e1d5c7c646ffb42b9573e14": " GT^2 \\sigma = 3\\pi \\left( \\frac{a}{r} \\right)^3, ", "0d6a6e2d4e605fad978233281c16de0c": "n \\bar X \\sim \\operatorname{Binomial}(n, p)", "0d6af3205639e8b20fd2d8d592667175": "V_{\\mathrm{RMS}} = {V_\\mathrm{p} \\over {\\sqrt 2}}.", "0d6b4b09044f84bae2629d44efd5dd8e": "g=pu", "0d6b4f33ca32ab0c67f99ca299cd3beb": "N^{2+z}", "0d6be9255337b5b2addd204b2e448cb1": " \\le^{-1}=\\ \\ge ,~ <^{-1}=\\ > ", "0d6bff90cbf5767cb046830946cbfd62": "f \\in L(G-D)", "0d6c999ff34fa251ea9b190207e9cf58": "\\phi_{\\mathbf{R},i}", "0d6d5a256719e4e1b64b484e5a313b0a": "\\scriptstyle{\\sqrt{2}/2}", "0d6d7560650c020295aec4b16601399a": "f(x_2)=0", "0d6da45ef0794e291028c7801c723fa3": "\\sin(x) = \\frac{2t}{1 + t^2}\\text{ and }\\cos(x) = \\frac{1 - t^2}{1 + t^2}\\text{ and }e^{i x} = \\frac{1 + i t}{1 - i t}", "0d6dcb926e0374df3cfa539818cca85b": "\\operatorname{ev}_p: C(X)\\to \\mathbf{R}", "0d6e1af3345dd778a3b40359b4e06579": "\\ln\\left(-\\ln\\left(1-F)\\right)\\right)", "0d6e3de138f515b1b2012f563c73f741": "\\nabla\\cdot\\mathbf{g} = 4\\pi G\\rho \\,\\!", "0d6e9a066cd61dda70aad1dc7d8300fd": " \\bar{H}_n^{(c)}=-1+\\frac{1}{2^c}-\\frac{1}{3^c}+\\cdots", "0d6e9a28d2ce026d1366909a0e0d103f": " \\begin{align}\nx_{n+1}&=x_n-\\frac{f(x_n)}{f'(x_n)}\\\\\n&=x_n-\\frac{a-\\frac{1}{x_n}}{\\frac{1}{x_n^2}}\\\\\n&=x_n\\,(2-ax_n)\n\\end{align}\n", "0d6e9a4fb698f6ee4d436b455b22af40": "d_{12}-d_{13}", "0d6ee871ca9996cf47c5891afcb05cce": "\\scriptstyle \\dot e", "0d6f46c188dfe8d51d42dd1409bc0fae": "H = (V(G),E(G))", "0d6f80b2d562827628b98a29989f5a2e": "\n16 = 1 + 3 + 5 +7 \\;\\; \\Longrightarrow\\;\\; n=4\\;\\hbox{contains}\\; s\\oplus p\\oplus d\\oplus f.\n", "0d6fd7e4ae56eaba3eafa60955bc5e24": "r=\\min", "0d701637da3c44a8d61c8121a65e5736": "N_A", "0d70a494ea5b172de048f91098b76e9b": "B = (I - \\rho F)^{-1} E \\;", "0d70a66a036eb5d07505356a7ee4c7f4": " E_t c_{t+1} = c_t", "0d70d6897f2205642d7847aeb43dca66": "{m^+}_1 = [15.927, 0.497]", "0d70f38109b310382048347d736b8a3b": "\\mathcal{P}=\\{\\text{all distributions}\\}", "0d70f9806aaf7d48b1823eba6d92ed96": "C_{N}^{2}(h)", "0d711edb02bbc7e66e7f61722cc00e4b": " \\ c = c_0(1 - z_d) + c_1z_d .", "0d712dcf00bf69151f4fca8326ec1efa": "\nL_e \\propto \\langle I \\rangle_e \\langle I \\rangle_e^{-1.66}\n", "0d7177e2bb8b7ad65c67b20d79923cbc": "T^2 = s (s-a) (s-b) (s-c)", "0d717ba8ab3b783601f1cf4c17aefde3": "C(\\langle 1, \\omega\\rangle), \\ C(\\langle 1, \\omega^{\\omega} \\rangle), \\ C(\\langle 1, \\omega^{\\omega^2}\\rangle), \\ C(\\langle 1, \\omega^{\\omega^3} \\rangle), \\ldots, C(\\langle 1, \\omega^{\\omega^\\omega} \\rangle), \\ldots", "0d71d145ab8905cced508a3fade4dde2": "\\xi = \\frac{x}{\\lambda_\\mathrm{D}}", "0d71e2ae8a51fc92fec3e976c84d8d4d": "I(X, Y) = \\sum_{x\\in X}{ \\sum_{y\\in Y} {p(x, y)log_2\\left(\\frac{p(x, y)}{p_1(x)p_2(y)}\\right)}}", "0d7262b93d906c0a0b3a1f73d66d566e": "\\Delta_{uv}=\\pm 0.05", "0d726b1a94abbfa016ae08054fd2297c": "0\\rarr L \\rarr M \\rarr N \\rarr 0", "0d728c5911111cfbbf9cc5a8ba354140": "\\mathit{m}", "0d72ba127726c94c6e544ca5d8add32b": "{1 \\over 2} K \\rho c^4 = 4 \\pi G \\rho \\,", "0d73026876bc64c53169dd10eaf06a4f": "2 \\zeta \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{b}{\\sqrt{ac}}. ", "0d7337197d8106c809e4b387a8adc055": " \\begin{align}\n\\mathbf{u}\\times\\mathbf{v}=&(u_1\\mathbf{i}+u_2\\mathbf{j}+u_3\\mathbf{k})\\times(v_1\\mathbf{i}+v_2\\mathbf{j}+v_3\\mathbf{k})\\\\\n=&u_1v_1(\\mathbf{i}\\times\\mathbf{i})+u_1v_2(\\mathbf{i}\\times\\mathbf{j})+u_1v_3(\\mathbf{i}\\times\\mathbf{k})\\\\\n&+u_2v_1(\\mathbf{j}\\times\\mathbf{i})+u_2v_2(\\mathbf{j}\\times\\mathbf{j})+u_2v_3(\\mathbf{j}\\times\\mathbf{k})\\\\\n&+u_3v_1(\\mathbf{k}\\times\\mathbf{i})+u_3v_2(\\mathbf{k}\\times\\mathbf{j})+u_3v_3(\\mathbf{k}\\times\\mathbf{k})\\\\\n\\end{align}", "0d733b8d7b7c6f032507207dccaec2b8": "C_2\\,\\!", "0d738eb8fef2f1ceaeb9f87d11e69fc2": "\\omega_e = \\omega_r - \\omega_c", "0d7397e218aaac0ca67d4013be769177": "n_{solute}", "0d73b9977315c46391da27462b3f8f8c": "\\tilde{\\chi}(\\omega) ", "0d73fcd2920035cb8250286d82e7c5c8": "\\dim Y_r = \\dim Z_r", "0d7416c56cf275658c3ff63a1f17c03c": "A\\subset X\\times Y", "0d7442b0b30a469fe600882e9dca3675": "C \\subseteq A \\cap B\\,\\!", "0d745c26ae530b582affb288201aca65": " \\mu(B(x,2r))\\leq C\\mu(B(x,r)) \\, ", "0d74b29d914f387cef90b8b18c01db4e": "\\Theta = \\sum_i e_i \\Theta^i(\\mathbf e).", "0d74ee22e926e3475f2f694b36898c51": "\\sigma_{x'}=-10\\textrm{ MPa}", "0d750e13123b81d84808b6866686767d": "V_1 = ", "0d755e78dfe816f5a522133c47e6ec9d": "E = \\alpha - x \\beta", "0d757efa484ce21608162f2e81d28415": "\\Delta u^j = 0\\,", "0d75959d44373b17d51c95b0ba9368d5": "\\{1,2,..,n\\}", "0d75fe0afc9367ef2832fdb4fa534a71": "x (t) = e^{- \\zeta \\omega_0 t} (A \\cos\\,(\\omega_\\mathrm{d}\\,t) + B \\sin\\,(\\omega_\\mathrm{d}\\,t ))\\,", "0d7600fc9e8e0eca288e77a5d3367566": "\\begin{align}\n\\Psi_{1}(t) &= \\sqrt{2}\\pi^{-1/4}te^{(-t^2/2)}\\\\\n\\Psi_{2}(t) &=\\frac{2}{3}\\sqrt{3}\\pi^{-1/4}(1-t^2)e^{(-t^2/2)}\\\\\n\\Psi_{3}(t) &= \\frac{2}{15}\\sqrt{30}\\pi^{-1/4}(t^3 - 3t)e^{(-t^2/2)}\n\\end{align}", "0d7610ee830324f4a14318811be5dd14": "\\{\\{a,b\\},\\{a\\}\\} = (a,b) \\neq (b,a)", "0d761d5130edcc610e160ac935bd9c38": "\\alpha = 0 \\,", "0d764f666a2a99db1a46a7457c315d01": "x\\to-\\infty", "0d766c9b7c2f9fe10bc8f34c5556cf8a": "(a\\cdot a_i)^{(n-1)/2}\\not\\equiv \\left(\\frac{a\\cdot a_i}{n}\\right)\\pmod{n}.", "0d76b01c4468af710e193c26939b8f87": "n^{O(1)}", "0d76ce2a875072de1dd8b901ad2499e7": "\\Delta \\!\\,", "0d76d7f33686631db7c764ea5b0f425c": "\\bigg[\\ F \\frac{\\quad}{\\quad} H\\ {}^-\\!F \\quad \\longleftrightarrow \\quad F^- \\ {}\\!H \\frac{\\quad}{\\quad} F\\ \\bigg]", "0d772cf39de3a6a5aa23d5f89ec9f7c9": "\\partial_\\mu \\partial^\\mu A^\\nu = 0", "0d77387cf99d9bddbd3075eae2479286": "=\\left(n \\cos \\alpha_1,n \\cos \\alpha_2,n \\cos \\alpha_3\\right)=n\\mathbf{\\hat{e}}", "0d775cdaa064177df3f8d024c5de76d7": " \\nu_i~", "0d7797d3bdb64e5c98546b183e815141": "g_{jk}", "0d781ab61c6ca4cb383356a2692a0956": " a(t)=\\frac{F(t)}{m} ", "0d7857e88614b1893fe9bc955830c0ee": "l(0)=N/2K\\gg 1", "0d78924a9147e4baf0753ba46c3ad2c3": "f^{(n)}(a) = \\frac{n!}{2\\pi i} \\oint_\\gamma \\frac{f(z)}{(z-a)^{n+1}}\\, dz.", "0d78d69b27b43a0a7d4ccab265e59736": "C=\\frac{N g^2S(S+1)\\mu_B^2}{3k}", "0d798f87d5b8422df5a1f3624eb3b9c6": "\\left[J_a ,K_b\\right] = i\\varepsilon_{abc}K_c", "0d79d11581d802295d7629c3e4203a84": "r=\\frac{a_2}{\\cos (\\theta-\\alpha_2)}", "0d79f1095ebff93e416c9f3347cf71b4": " (p_x, p_y) ", "0d7a060b526554544b47c643330b80e6": "a(t)=\\sum_{i=1}^N K(i)(I(i,t-1))^2", "0d7a08596bda9d186ec59382ec8697f2": "\\scriptstyle \\bar\\eta", "0d7a08acc06406fdfb8011754580022a": "H^\\prime", "0d7a613622106410ab383f63f294a9ef": "|\\langle f|g\\rangle|^{2} = \\bigg(\\frac{\\langle f|g\\rangle+\\langle g|f\\rangle}{2}\\bigg)^{2} + \\bigg(\\frac{\\langle f|g\\rangle-\\langle g|f\\rangle}{2i}\\bigg)^{2}", "0d7a78ab723beebe175d722b94d53190": "X(fg)= (Xf)g + f(Xg).", "0d7aa8a6bcbfa03ede53f63b3483efb9": " {\\mu} = {\\mu}_0 \\frac {T_0+C} {T + C} \\left (\\frac {T} {T_0} \\right )^{3/2}.", "0d7ade4f83b26fb7a6ae6e9824e15c45": "f^{m_k-1}", "0d7ae37e75939904a44f4858f63b1437": " [x_i,x_j]=0,\\quad [x_i, t]=i \\lambda x_i ", "0d7b48fb530cfc179e59ced04d320a34": "Q(p;k,\\lambda) = \\lambda {(-ln(1-p))}^{1/k}", "0d7b5a5cc84765fb28c091e7ca468560": " \\{\\bold x|A x < b\\}\\ne\\emptyset ", "0d7b6dc12afec874363a413789dc17bc": "\\mu_i=\\mu\\text{ for }1 \\leq i \\leq K. \\, ", "0d7b7f18521046707e53552b814e3551": "\\{C_n:n \\in \\mathbb{N}\\}", "0d7b9e641ce99cf7c9c35a6aa688bb2c": "f(x_0,y_0)=0", "0d7bbe593b4a477bbf1c136f5a3ad925": "\\operatorname{E} \\left[ \\| {\\boldsymbol \\theta} - \\hat {\\boldsymbol \\theta} \\|^2 \\right].", "0d7bd8d05c82067f4a72d909599a3e89": "3 \\times 3", "0d7bdc312ceaf61e812652a0bf83605b": "R^{(3)} = h^{ij}R^{(3)}_{ij}", "0d7bdd2f8a13112eab50edc2c1140ebe": "\nQ \\ \\stackrel{\\mathrm{def}}{=}\\ Ax^2+Bxy+Cy^2+Dx+Ey+F=0. \\,\n", "0d7bf23a4af03acba81746f5d07aa709": "\\mathfrak{P}^{40}", "0d7cb24d5f9172fbb586220c30abeb6e": "N_w \\bar{r_w}", "0d7cbb92bcbd9ccde3999786867f88ef": "\\scriptstyle a^2 \\,+\\, b^2 \\,+\\, 1", "0d7cc9bf05a9dae0dae18409480f40be": "\\rho(X)=\\lim \\frac{|X\\cap B_n|}{|B_n|}.", "0d7ccb5484cba4246c37c5abc1febe2e": "\\begin{align} P(hypercalcemia~is~present~despite~no~disease~in~individual) = \\\\\n \\frac {P(hypercalcemia~WHOIFPI~by~no~disease)}{P(hypercalcemia~WHOIFPI)} = \\\\\n \\frac {0.0014}{0.00335} = 0.418= 41.8% \\end{align}", "0d7d32809f4dcb2f6764ee4bc1cf1b39": "\nS = S_1 + R , \\ \\ \\ (3)\n", "0d7d343c05df869fe377a6f304e320d5": "\\max(A_1(x_1, \\dots, x_{r-1}), \\dots, A_{n_A}(x_1, \\dots, x_{r-1})) \\leq \\min(B_1(x_1, \\dots, x_{r-1}), \\dots, B_{n_B}(x_1, \\dots, x_{r-1}))", "0d7d626ea13ca6c3be56987b5075e9c7": "(\\mathrm{surface area})^{1.5}/(\\mathrm{volume})", "0d7dcc2be3633c9f279a1b8d21fce253": "T_{U,i}", "0d7e1498ce26e771fa75c0b1c1e439eb": "E_{k'}", "0d7e16a223873cd9add2a5a6f418b615": "= \\sum_{j} (v_j^{T} x)(v_j^{T} x)\\lambda_j", "0d7e6ff89efd5920b1d3944c51c59e93": "\\nabla \\times \\mathbf{B} = \\frac{1}{c}\\frac{\\partial \\mathbf{E}}{\\partial t} + \\frac{4 \\pi}{c} \\mathbf{j}_{\\mathrm e} ", "0d7eabf56237b9ee88a6275eee456c6e": "n_{\\bullet1}", "0d7efc22617c0e771e5d6b9e5e8a6661": "\\,00", "0d828a89660289c7cf1e6dc5c0c9a299": "S,X_1,X_2,\\ldots,X_n\\subseteq \\Omega", "0d82f533ed1b5a41a1ee6ce63a53eb8e": "\\frac{1+7x+12x^2-4x^3}{(1-x^2)(1-16x^2)}", "0d836d4d1c166228e3faafc03be46f50": "O_K", "0d8373d159ea431f8f000e8f4d948ed8": " \\hat{A}", "0d8402aed8d581ac2572b976aa51bd66": "\\frac{\\partial^2u}{\\partial x_i\\partial x_j} = -R_iR_j\\Delta u + P_{ij}(x)", "0d843f33ade67aa1d186596d6ff4c82a": "\\vec x' = \\vec x + 2a(\\vec \\omega\\times\\vec x) + 2(\\vec \\omega\\times(\\vec \\omega\\times\\vec x)).", "0d846812ddc803b47558631d214b9714": "\\psi,\\bar\\psi", "0d847e21f5a2049d9348a17fd7892cce": " \\Phi", "0d849d387fc670057ed5a185f063ba97": "B^\\mu\\,", "0d84d24d7e9ca33cdf9b6334d3b8fd98": "\\omega_{\\alpha I}^{\\;\\;\\;\\; J}", "0d851c4baee08c08a36545df2c12303f": " \\gamma_{jk} = \\int_{-\\infty}^{\\infty} x(t) \\frac{1}{\\sqrt{2^j}} \\psi \\left( \\frac{t - k 2^j}{2^j} \\right) dt ", "0d8544831f20e8809b2a8dd0c1031191": "\\tilde C(\\alpha,\\beta)", "0d85553795fe9cda12a72a5b288431af": " \\frac{\\text{d}[{^{d_h}_{c_h}}P^{\\gamma_h}_h]}{\\text{d}t} = \\sum_i u_{\\gamma_{hi}} y_{d_{hi}} \\text{k}_{3(i)} C_i \\qquad \\qquad (8c) ", "0d856f7538042b8cd88ffb9e4f13f5aa": "V_n(R) = \\frac{R^n}{n} \\textstyle B(\\frac{n-1}{2}, \\frac{1}{2}) B(\\frac{n-2}{2}, \\frac{1}{2}) \\cdots B(\\frac{2}{2}, \\frac{1}{2}) \\cdot 2B(\\frac{1}{2}, \\frac{1}{2}).", "0d8687e0f639768283dc7fcdc66fa162": "\\rho_{SE} (0) \\,", "0d86d4127a593a07e5fcdba5055c879b": " \\ddot{y} + y -\\varepsilon \\left( \\frac{\\dot{y}^{3}}{3} - \\dot{y}\\right) = 0.", "0d86d76f19612258365c96121add2094": "I = \\varprojlim I_n", "0d87337780b0396d92db0f255a5684c5": " g( u, v ) = \\exp[ ( \\theta_1 - \\theta_{ 12 } )( u - 1 ) + ( \\theta_2 - \\theta_{ 12 } )( v - 1 ) + \\theta_{ 12 } ( uv - 1 ) ] ", "0d873d3ee3bd152e069d0a006a489baa": "\\tau(n)\\equiv n\\sigma_{9}(n)\\ \\bmod\\ 7^2\\text{ for }n\\equiv 3,5,6\\ \\bmod\\ 7", "0d8790697dc884fde9b378eeb376e128": "I: T \\to P", "0d879b526f3f4a257d912a3b9a4f7a64": "s_n = \\sum_{k=0}^n\\frac{x^k}{k!},\\ t_n=\\left(1+\\frac{x}{n}\\right)^n.", "0d87f4c1968c24d8491b3940378dd1e7": "\\sigma_x = \\frac{x_0}{\\sqrt{2}} \\sqrt{1+\\omega_0^2 t^2}", "0d880c5cc27fba976bdc442c95a140e9": "\\operatorname{vol}(B)=\\prod_{i=1}^n (b_i-a_i) \\, .", "0d883be6eccff1b524bc7e5f629238c2": "r = 2 \\sqrt {\\tfrac{Z_1 Z_2 e^2 L}{E_0}} ", "0d8877599db5915e1e60b80d202a2b2a": " |\\alpha^k| \\le q^{kd/2+1/2}", "0d89646470c473a5e74e47613c2716f5": "F=(1+E\\cdot N)^\\lambda.", "0d8a501a499c359df4a6bfcb3da1a508": "\\gamma\\,", "0d8a781134042c17b3787a21725e1f42": "-\\frac{\\partial T_{o}}{\\partial x}^{TM}-\\frac{k_{z}}{\\omega \\mu }\\frac{\\partial T_{o}}{\\partial y}^{TE}= -\\frac{\\partial T_{\\varepsilon }}{\\partial x}^{TM}-\\frac{k_{z}}{\\omega \\mu }\\frac{\\partial T_{\\varepsilon } }{\\partial y}^{TE}", "0d8a9a86f426f2dfb3611526ce1b4ddc": "dim \\;f_i(X_i)=k", "0d8ac044e13e73bba35fc58eec938f5e": "\\scriptstyle Z_{L}", "0d8b4abfbca5ae40fbab2837e8284d88": "\\sigma_H(\\omega) = \\begin{cases}\n i = e^{+\\frac{i\\pi}{2}}, & \\mbox{for } \\omega < 0\\\\\n 0, & \\mbox{for } \\omega = 0\\\\\n -i = e^{-\\frac{i\\pi}{2}}, & \\mbox{for } \\omega > 0\n\\end{cases}", "0d8b83a7c24f34d046f5477ff841c48c": " 2 \\omega_1 ", "0d8bcb2482102b6c838150e3a33e583f": "\\mathbb P.", "0d8bd35ed636bb503b7a7101de12f349": "\\sigma_{N+1} = \\sigma_1 ", "0d8bd3be4aad812cd2d4d7f5c982cd3d": "\\sum_{k=1}^\\infty u_k", "0d8bf374888f310d4b9577d46d3be73e": "xR + yR \\equiv zR \\pmod{N}", "0d8bfc86536b04eae0696bd8e74ee1fc": "X\\subseteq \\Omega", "0d8c70cd344458e4dfc23a334f968371": "C_{12}", "0d8c8b3fc79b5a7feed7a8d247a7152e": "Square numbers end on 7", "0d8c960b49353f0cad057a8a3c53d373": "\nV_t \\approx pq\\left(1-\\exp\\left\\{-\\frac{t}{2N_e} \\right\\}\\right).\n", "0d8cca4b3596f47e1064473dc6f6bce1": "E(x^2)=2\\sigma^2, E(\\ln(x))=\\frac{\\ln(2\\sigma^2)-\\gamma_E}{2}\\,", "0d8d3750e579ba4f5f2f90e828b5cf02": " Z=\\int d\\mu(\\sigma) e^{-H(\\sigma)} ", "0d8d7338a1a1eb83be267ee8b9071951": "\\Delta P = \\rho V v \\qquad \\text{(ignoring } \\frac{\\rho}{2}v^2),\\,", "0d8d8d31eebf4c6049d640e5d7f03530": " =_E ", "0d8e349f39c7fe8106368f0566c8c6bb": "\n\\eta^2 (2n+1) \\ll 1,\n", "0d8e3642269dde4efec6d45fad7005e5": "\n\\begin{array}{c|cc}\n\t0\\\\\n\t1& \t1 \\\\\n\\hline\n&\t1/2& \t1/2\\\\\n\t&\t1 &\t0\n\\end{array}\n", "0d8e893d539d243b9021c3756fee06f0": " \n\\langle i_1i_2\\rangle = \\lim_{T\\rightarrow\\infty}\\frac{E^4}{T}\\int^T_0 \\sin^2(\\omega t)(\\sin(\\omega t)\\cos(\\phi)+\\sin(\\phi)\\cos(\\omega t))^2\\,dt", "0d8eb21b869efd85672f2a6910284b9a": "E(R_i) = R_f + \\beta_{ip}[E(R_p) - R_f]\\,", "0d8ec496a199e12042a3566cef8e68ec": "\\pi(a_i,a_j)+\\pi(a_j,a_i)\\le 1", "0d8ec776616a87f5f6843c8a7c72755e": "\n[x_2] =[x_2]\\ \\cap \\ \\ [a]-[x_1]\n", "0d8edf7ff8f86d0f702dd560ed4e0422": "\\dot C", "0d8edfe0089b4be6b1844bafd54e3ec9": "\\operatorname{GrpRng}\\colon \\mathbf{\\operatorname{Grp}} \\to R\\mathbf{\\operatorname{-Alg}}", "0d8ef5afb6abcfc1be71d1d0d9dd9350": " N=n_{+}+n_{-}=2n_{-}+1 \\,", "0d8f078bb545686c5037c64461fa17c3": "\\vec{\\gamma} = (\\gamma_1, \\ldots, \\gamma_N)", "0d8f12173d7de1e804cb17084ab236de": " \\text{EVaR}_{1-\\alpha}(X)=\\text{EVaR}_{1-\\alpha}(Y) ", "0d8f4b55462e0c90c52e71210df9e662": "\\frac{p(\\xi)}{q(\\xi)} = \\left(t\\to\\frac{p(t)}{\\xi-t}\\right)[x_1,\\dots,x_n].", "0d8f6af4ec47b5b34b45fb130bda8473": "\\lim_{r\\to +\\infty}z(r)=\\frac{1}{\\sqrt{1-\\frac{r_s}{R^*}}}-1", "0d902d1c547b5626d16f7ded98fdec10": "banana$", "0d9036db3e0411fac61e9a0b5e3fbad1": "x^8 + u^3\\,", "0d90c489fa71b6a650bff5c4a564cf48": "\\lambda = \\frac{2}{n}\\mathrm{div}X", "0d91608af95f1972da1faa6ebf4f0670": "\\int \\mathbb{I}[\\mathbb{E}(r,a,x,\\theta) = \\max_{a'} \\mathbb{E}(r,a',x,\\theta)] P(\\theta|\\mathcal{D}) \\, d\\theta,", "0d917891a29af05b2e25833fd4e8868d": "{GOR}", "0d91790a096807684aa124eb24935e0e": "r \\ge 2 ", "0d91b663ee1745060d67671bce7efba4": "\\bar{n} \\in S \\iff (\\exists \\bar{m} \\in \\mathbb{N}^{k})(P(\\bar{n},\\bar{m})=0) ", "0d91b6a83546f93c2a3e82aa5b318a2f": " N \\to \\infty, \\quad \\text{Problem } B^{\\lambda N} \\to \\text{Problem } B^\\lambda ", "0d91c55b235fd2e0c72da0f8537f7e6a": "D = K^2\\sum C(n)", "0d921dabab09e7874da27eb1e490b62d": "D=D_{ij}", "0d92c778b34d0ee3a946196b10ab8647": "\\nu=\\log_2 N", "0d92f68ad626b36b2ddcdc3a2d7adb67": "\\omega=\\frac{\\rho g Q S}{b}", "0d931ee352169206ea51ab112d6037af": "\n\\sqrt{(1+v)(1-u) \\over (1-v)(1+u)} = \\sqrt{ 1+ (v-u)/(1-vu) \\over 1 - (v-u)/(1-vu) } \\, ,\n", "0d932117fe50aa894b84cb4606584588": "-\\frac{2}{r} \\rightarrow \\frac{1}{2}kr^{2}.", "0d9390cf6a960f7966b005c57cbb4a67": "L(p,q)", "0d93a1f2b62aad5e244f8572a8a1e15e": " \\mathfrak{F}\\{f \\cdot g\\}= \\mathfrak{F}\\{f\\}*\\mathfrak{F}\\{g\\}", "0d93daab7d03c6f24c72c9f6c75b32a0": "R(x, \\xi) = e^{-{\\langle x, \\xi \\rangle}} R (e^{\\langle x, \\xi \\rangle}).", "0d940266b39d150a39730577c04989e9": "h_v", "0d940bfebf3a61b5532643e2ae2b495d": "m := \\rho A", "0d944e10715625c7b47eacc48f058fec": "\\langle z^n\\rangle=\\int_\\Gamma e^{in\\theta}\\,f_{WE}(\\theta;\\lambda)\\,d\\theta = \\frac{1}{1-in/\\lambda} ,", "0d951cbc3b600f618691ecc971c377cd": "k_1, k_{-1}", "0d9534c0577572c7aff812faf75fb86b": "A \\subseteq \\widehat{\\mathbb{R}}", "0d958bd7b6f2a10ef8f5f555facfbb9a": " y_{n+1}^{(i)}\\equiv a_{i} (y_{n}^{(i)})^{p_{i}-2} + b_{i} \\pmod {p_{i}}\\; \\text{, }n \\geqslant 0 \\;\\text {where} \\;y_{0}\\equiv m_{i} (y_{0}^{(i)})\\pmod {p_{i}}\\;\\text{is assumed. } ", "0d95c8ca14996a01b0c3b9e22524cc06": "3 \\over 12", "0d9624f2c813b89586817e034d242a22": " T(n,k)=a(\\gcd(n,k)) ", "0d9626a634d22ccdd7bcda4c3d1a8b21": " \\tan 2 \\theta_n = + 2 \\frac{\\sqrt{ac}}{b} ,", "0d96559fc4b37fc17b7c7fdac02404ba": "d = 2e - 1", "0d96bdce1d0ec71638f06b38d425b9e7": "E_{eq} = E_1 + E_2 ", "0d979240d6a15d146b37bded6712df21": "U(\\{c_t\\}_{t=0}^\\infty)=\\sum_{t=0}^\\infty {f(t)u(c_t)}", "0d9799861c2ecca1b8430be02ca949f5": "E=E^\\ominus + \\frac{RT}{nF}\\ln\\frac{{\\{S\\}} ^\\sigma {\\{T\\}}^\\tau ... } {{\\{A\\}}^\\alpha {\\{B\\}}^\\beta ...}", "0d97dcfff837a7366366cadd81930638": "- 5^{\\frac{3}{4}} \\exp \\left (\\tfrac{\\log(2) + \\pi i}{4} \\right ) = - \\exp (\\tfrac{\\pi i}{4}) 5^{\\frac{3}{4}} 2^{\\frac{1}{4}}.", "0d980f2d8d94f8ba2c3267fc8bd120aa": " = z^2 - i^2 \\cdot 5", "0d9860954c3dc996600bc4fa045b6ccc": "\\int\\frac{x\\,dx}{r} = r", "0d98cce60df8b24994abae1c5706ff9b": "S_r(m)", "0d9935671c83cac6a1430aa071841a8a": "E_m", "0d99ebbe2fb15f25b413cf711e284518": "1 \\over 10^5", "0d9a35d1367f1de3730ff26dfdaa6790": "\\,\\zeta", "0d9a88ecae09acdc8a8d8d0a461052d8": " F + 110 = \\begin{bmatrix}\n 1 & 1 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 1 & 1\n\\end{bmatrix} = F^{(1,2)_R(1,2)_C}=F^{(1,2,3)_R(1,2,3)_C}.\n", "0d9ac7439d83b6630ffd611888cba21e": "~z=L^*", "0d9acdb3906955c73c796ee195ce9b71": "\n\\eta^{\\rho \\sigma} {\\Lambda^\\mu}_\\rho {\\Lambda^\\nu}_\\sigma = \\eta^{\\mu \\nu} ~,\n", "0d9afbdd7a18b8bec173de537a3ae6c6": " \\begin{align} a_0 & = 6 - 4\\sqrt{2} \\\\\n y_0 & = \\sqrt{2} - 1\n \\end{align}\n", "0d9b495148c73d958a19ee689fcd9657": " v = a u + x", "0d9b538e2dec4884ac972fb1a729373b": "(X_c,Y_c)", "0d9b9cee83854ae41b3597477bb6d472": "\\frac{\\pi}{4} = \\arctan\\frac{1}{2} + \\arctan\\frac{1}{3}\\!", "0d9b9de2e406366685d40fb4b36bdd39": "PE=\\frac{N}{2}4\\pi\\rho\\int^{\\infty}_0r^2u(r)g(r)dr ", "0d9ba6c9ac7634112345957eaf08f934": "\\tilde I = I", "0d9bcf457d3598211209dcca52aff02b": "\\text{SE} = \\frac{1}{\\sqrt{n - 3}}.", "0d9bfbe01f32d7330c825d66adfa4afd": "\\Delta \\bar{V} = \\sigma \\bar{V}_S + \\tau \\bar{V}_T - \\alpha \\bar{V}_A - \\beta \\bar{V}_B ", "0d9c2f943fb2edeff2d9fff3502cd455": "V(p)", "0d9c360b262efb1f99c82429864648c3": "\\hat{\\theta}=f(m_1(x_1),\\cdot,m_N(x_N))", "0d9cf0523988611bc3c0fed5302b1321": "\\tau < 0", "0d9d12abf76751ce8d6607a11f22093b": "X(s) = \\int^\\infty_{0^-} x(t)e^{-s t}\\, dt", "0d9d1ee87c3898fb1a6a179d4f55c147": "-(1 + \\log(-x^*))", "0d9d6440e765032275216c2fab5c5a51": "\\empty \\notin \\mathcal{C}", "0d9d7e71af402d4fd82b1276cf93cadb": "P(t) = e^{Qt} = e^{U^{-1} (\\Lambda t) U} = U^{-1} e^{\\Lambda t}\\,U\\,,", "0d9da8392ccebd5adff2be98fc763838": "f = \\Omega (g)\\,\\!", "0d9db82e77ec6716d3d058b95c371704": "e^{-V}", "0d9df1edc26e29d206e7d47df559bd86": " a\\ne 0 ", "0d9df69f83bf20d54eb9891850fe15c2": "\\tbinom{0}{0} = \\tfrac{0!}{0!0!} = 1", "0d9e036e7c1664d2dccf2aff96e018ab": "[x z, y] = [x, y]^z\\cdot [z, y].", "0d9e13d8b0ced0e69c66b6c622f547bb": "\\Lambda_{00} = 1 ", "0d9e2fc952809dbff3f9c576eef26bc1": "\\mid P(x)-f(x)\\mid", "0d9e4b03d57dd5a9c596ef5f2a258a0d": "\n\\left(\\frac{1}{2j}\\sum_{r=1}^{2j}\\gamma_r^\\mu \\hat{P}_\\mu - mc \\right)\\Psi =0\n", "0d9eb00907c40a8c20b5762a2ec7db33": "\\beta = v/c \\,\\!", "0d9ef4b60f7b39147365039ae7b5123d": "E(\\alpha ) = k(S^{max} - S(\\alpha))", "0d9efec78b6da7f0c74dfc6da051ca3a": "\\mathrm{lim} : \\mathcal{C}^\\mathcal{J} \\to \\mathcal{C}", "0d9f1f87399a79ef4abb5a536581b702": "\\Delta_\\infty", "0d9f75f3fbec0025079eb21e40eef01e": " p(\\theta_1, \\cdots, \\theta_n) = \\frac{1}{Z_{n,\\beta}} \\prod_{1 \\leq k < j \\leq n} |e^{i \\theta_k} - e^{i \\theta_j}|^\\beta~,", "0da020fcdf983f09bc98cbfbf05dc19c": "\\frac{1}{d} = \\frac{1}{r'} \\sum_{k=0}^\\infty P^0_k ( \\cos ( \\theta' - \\theta ) ) \\left ( \\frac{r}{r'} \\right ) ^k.", "0da048f0ce9fbba2fb5e8c872c23a09b": "\\displaystyle \\operatorname{rect}(a x) \\,", "0da05be399632160d1da0d992f2373fc": "\\cot(x)\\cot(y) + \\cot(y)\\cot(z) + \\cot(z)\\cot(x) = 1.\\,", "0da063dfa254df8ac90cbf75ce5ad8c2": "\\frac{d\\varphi}{ds} = (n+1)\\frac{d\\theta}{ds} = \\frac{n+1}{a} \\cos^{1-\\tfrac{1}{n}} n\\theta", "0da0b2834a999307db5a111af48c8bb2": "d(p,r) = d(p',r')", "0da0bce64c966b4446165bc7478dd221": "\\ell\\,", "0da18cc362f67fa7a163257e0562bfd1": " n_{i} =E_i/\\omega ", "0da1c7889d595a5a4b9689594d4addc0": "w(1,1)", "0da1d48464d74512eaefb36f29c38df2": "\\sigma_1=\\ldots=\\sigma_k=1", "0da21031745a2c46682b5a51c313a14e": " \\ | \\psi \\rangle = |0 \\rangle |f_k \\rangle |f_k \\rangle ", "0da2363a5388b925ec8fb6d80172ffe1": "\\epsilon_F = \\frac{\\hbar^2}{2m} \\left( {3 \\pi^2 n} \\right)^{2/3} \\,.", "0da2b4e34a02fad013c45f55192a77ac": "\\vartheta = T - 273.15 ", "0da2b6e86e862ad79e6056eccbc793b0": "=Q\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{at constant pressure.}", "0da2b8eb4d3e03d462fa73accfb7ca35": "\\tilde{M}", "0da2e9a4353c7198549298c2dd0c8655": " \\alpha, \\lambda, y_0", "0da32dbedf26be3584d5ff8ba55951d2": " \\hat{\\mathbf{k}} ", "0da398635d8466a41494c840cf1f2cbc": "{\\partial_t}\\left( \\begin{matrix}\n \\mathcal{P} \\\\\n \\mathcal{Q} \\\\\n\\end{matrix} \\right)\\rho =\\left( \\begin{matrix}\n \\mathcal{P} \\\\\n \\mathcal{Q} \\\\\n\\end{matrix} \\right)L\\left( \\begin{matrix}\n \\mathcal{P} \\\\\n \\mathcal{Q} \\\\\n\\end{matrix} \\right)\\rho +\\left( \\begin{matrix}\n \\mathcal{P} \\\\\n \\mathcal{Q} \\\\\n\\end{matrix} \\right)L\\left( \\begin{matrix}\n \\mathcal{Q} \\\\\n \\mathcal{P} \\\\\n\\end{matrix} \\right)\\rho.", "0da3a9e3c64456a1300679ff36d688cd": "\\lambda x", "0da3b79af62975cc3f3c2accc9c58b29": "T = t_1 \\dots t_n", "0da42889712766f450d8809bf0d1875d": "a = 14.0", "0da48b91d44acb9434b1438c54a313f2": "\\begin{bmatrix}\\Psi\\end{bmatrix}^{T}\\begin{bmatrix}M\\end{bmatrix}\\begin{bmatrix} \\Psi \\end{bmatrix}\\begin{Bmatrix}\\ddot{q}\\end{Bmatrix}+\\begin{bmatrix}\\Psi\\end{bmatrix}^{T}\\begin{bmatrix}K\\end{bmatrix} \\begin{bmatrix} \\Psi \\end{bmatrix} \\begin{Bmatrix} q\\end{Bmatrix}=0.", "0da49c5cc19c85868c31d8ed5cc6fd7f": "\\mathrm{im}(\\partial_{n+1})=B_n(X)", "0da4d6df96372cd7518279c6afe39f01": "\\| u\\|_{L^p(I)}\\le C\\| u\\|^{1-a}_{L^q(I)} \\|u\\|^a_{W^{1,r}(I)}", "0da4def82b782260dff64dd8aaf2c8b0": "\\frac{1}{2}(W,W) = i\\hbar{\\Delta}_{\\rho}(W)+\\hbar^{2}\\Delta(1) . ", "0da527be5c93b67647f268ffe648925f": "F(T,V) = U(V) + E_{ZP}(V) - T S(T,V)", "0da5371338658907fa7cd77e08119aaf": "R(w_1,w_2)\\,\\!", "0da566d4fa9895181ae9cf15980f2182": "m_{UT} = [14.545, 0.550]", "0da57e517017ed945561f9a4f0fc9dbd": "L_{\\nu, q}(R^{+})", "0da5a0710a1ea6210f08e02e8f634a38": " \\ln \\Gamma(k)+k\\ln\\theta", "0da5f128abc87b02b0668828bbcad5cd": "\\int_{M} \\mathrm{d} \\omega = \\int_{\\partial M} \\omega.", "0da6094501aee02f1211be3664432545": "\\frac {E_2} {\\omega \\eta}", "0da62f283b79985b2aa44e4b0ef5dd37": "F = \\rho \\times V^2 \\times S_{air} \\times \\cos ( \\frac{\\pi}{2} - \\alpha) ", "0da678a5e79fa2ada0a2e8fc7eec7791": " \\pi_i ", "0da67d44d2b3bbddd22750738f785645": "p_k = 1 - \\textstyle\\sum_{i=1}^{k-1} p_i", "0da6d1cb219b6c7a9a881a724a2f2039": "\\big\\{\\mathbf{R}_{r}\\big\\}.", "0da72410e1b7d155251c40d3ea979c31": "\nx = a\\sigma\\tau \\cos \\phi\\,\n", "0da7d465471c7e34046194317479a226": "c_{n-m} = \\frac{(-)^m}{m!} \n\\begin{vmatrix} \\operatorname{tr}A & m-1 &0&\\cdots\\\\\n\\operatorname{tr}A^2 &\\operatorname{tr}A& m-2 &\\cdots\\\\\n \\cdots & \\cdots & \\cdots & \\cdots \\\\\n\\operatorname{tr}A^{m-1} &\\operatorname{tr}A^{m-2}& \\cdots& 1 \\\\ \n\\operatorname{tr}A^m &\\operatorname{tr}A^{m-1}& \\cdots& \\operatorname{tr}A \\\\ \\end{vmatrix} ~.", "0da7d9cdb7f01f62e3a95ceab0209fc8": "f(f^*(x)f(y))=f(y)\\; ,\\; f(f(x)f_*(y))=f(x) ", "0da7daf20f407e77d756d0d08e402021": "\\frac{{}_1F_1(a+1;b+1;z)}{b{}_1F_1(a;b;z)} = \\cfrac{1}{b + \\cfrac{(a-b) z}{(b+1) + \\cfrac{(a+1) z}{(b+2) + \\cfrac{(a-b-1) z}{(b+3) + \\cfrac{(a+2) z}{(b+4) + {}\\ddots}}}}}", "0da880018cfc1392a545754ea89bd9ff": " I = \\mu R^2", "0da8c26a4dd78d940ea36ff164e5e389": "K = H + \\frac{\\partial}{\\partial t}G(\\mathbf{q},\\mathbf{p},t) \\,,", "0da90e4815ea9e5b0594c7fec1b9c366": "L_n=-z^{n+1} \\frac{\\partial}{\\partial z}", "0da95629853431a308d586dcc4a68593": "E_7", "0da98e1e2e047a05a87e1ceabda8ab1f": "h := -f", "0da9b2ecc02af771e08d8765136cd5dc": " G_\\lambda(x,y) = {(\\varphi_\\lambda(x) + a(\\lambda)\\chi_\\lambda(x))(\\varphi_\\lambda(y) + b(\\lambda)\\chi_\\lambda(y))\\over b(\\lambda)-a(\\lambda)} \\,\\, (x\\le y), \\,\\,\\,\\, {(\\varphi_\\lambda(x) + b(\\lambda)\\chi_\\lambda(x))(\\varphi_\\lambda(y) + a(\\lambda)\\chi_\\lambda(y))\\over b(\\lambda)-a(\\lambda)} \\,\\, (y\\le x). ", "0da9c3c46a63b7a12178032113a619cd": "dx_s/dt", "0daab5654dadefdc54d6dc85cbb5822e": "\\mathbf{L}_{\\mathrm{system}} = \\mathrm{constant} \\leftrightarrow \\sum \\mathbf{\\tau}_{\\mathrm{ext}} = 0 ", "0dab5f5bac2309806fee4f9584994c9b": "k'\\in\\{1,\\ldots,n\\}", "0dabaf9af303417f7191354b3bc5db22": "M = b_1(\\tilde G) = \\operatorname{rank}H_1(\\tilde G).", "0dabd9e30f3b361f3ca84dbf017405f6": "\\mathbb P \\left\\{ \\sup_{t\\in[0,1]} X(t) \\geq u \\right\\}", "0dabe459301bbd7c0bc8f884e11f2240": "\\varepsilon_{RF}", "0dac058af1cb61f6188658c21d74dcb2": "II_{1,1}", "0dac1450413e016baa209ceef8b3428d": "x_i = \\frac{y_i}{\\sum_{j=1}^K y_j}.", "0dac61b1a408569c7320d3598590bb38": "P = G\\left(\\frac {2} {\\sin \\theta} - 1\\right)", "0daca570410e633ef534e720f5d54852": "b=10", "0dad94963c0eb6757672437ec3c024bd": "\\begin{align}\n w_0 &= ab\\left(\\frac{\\frac{n_0^2}{2} + \\frac{1}{2}n_0n_1 + \\frac{1}{8}n_1^2}{n_0(n_0 + n_1)}\\right) \\\\\n w_1 &= b\\left(\\frac{\\frac{1}{2}n_0n_1 + \\frac{1}{4}n_1^2}{n_1(n_0 + n_1)} \\right) \\\\\n w_2 &= 0\n\\end{align}", "0dadc1427b67ba053977742620ba6f19": "X \\}", "0dae0bec3fdc41c8dec0fda153e93ce6": "\\lambda = \\sqrt{2n \\ln (2m)}", "0dae48d818b736af4db41c5c1974001f": "\\ln (\\Gamma (z)) \\sim \\left(z-\\tfrac{1}{2}\\right)\\ln(z) -z + \\tfrac{1}{2}\\ln(2 \\pi) + \\sum_{n=1}^\\infty \\frac{B_{2n}}{2n(2n-1)z^{2n-1}}", "0dae5cf7b9c7a9f10117fbbee5e04f2d": "c_{11}-b_{11}", "0dae817a798cd816ea765fd8bcdb81f1": "\\dot \\gamma", "0dae8380ec54b73025c95a45bebc317e": "\\triangle = \\sum_{i=1}^m \\nabla_{E_i}\\nabla_{E_i}", "0daf097d2ef4e94bebe5f3cf2c0e982c": "\\nabla^2 A = -\\mu_0 \\frac{d}{dA}\\left(p + \\frac{B_z^2}{2\\mu_0}\\right)", "0daf60a2fe5c7d90c0264ad25f4433a2": "\\mathcal{L}={1\\over 2}\\partial^\\mu \\phi_a \\partial_\\mu \\phi_a-{m^2\\over 2}\\phi_a \\phi_a-{\\lambda\\over 8N}(\\phi_a \\phi_a)^2", "0dafc970c030a361af5a3565ec5bb05f": "\n\\mu = \\frac{\\theta}{\\beta} \\frac{d\\theta}{d\\beta}\n", "0db00b4423a8afd05a6bfedfb70008e7": "N(uv) = N(u)N(v)", "0db0a89579a2587527b82cde1933a824": "u^+ = \\frac{1}{\\kappa} \\ln\\, y^+ + C^+,", "0db0bc0d0be0a34ccf57509591d8382c": " \\ d \\leq n ", "0db0be4dee403cec097b39a14b4688d1": "\\Delta\\lambda-\\delta\\nu=-(\\mu+\\bar{\\mu})\\lambda-(3\\gamma-\\bar{\\gamma})\\lambda+(3\\alpha+\\bar{\\beta}+\\pi-\\bar{\\tau})\\nu-\\Psi_4\\,,", "0db0e7ab626f327d36909545083b416b": "X(L)", "0db12ee81520e6f367d16d5b91d00533": "\\left(\\sum_\\alpha c_\\alpha X^\\alpha\\right)\\times\\left(\\sum_\\alpha d_\\alpha X^\\alpha\\right)=\\sum_{\\alpha,\\beta} c_\\alpha d_\\beta X^{\\alpha} \\cdot X^{\\beta}", "0db146e4fba58adca497f0a03e29ae73": "\\forall s \\in S, \\forall \\omega \\in \\Omega : s^\\omega \\in S.", "0db15aa0a9d92225cd62fa3740158f88": "\\tfrac{2\\sqrt 2}{3}", "0db17f57005873cad39c21cae1f79db0": "\\ell_1=r+a+b", "0db21945f703a223943fcd3a68bdb235": "\\frac{j R_1 X_1}{R_1 + j X_1} ", "0db277d439010bdf0dc88ed4e2225e89": "P(r)=0", "0db32c8026e059df8fcacdaf80c10cf9": "\\sqrt{15} - \\sqrt {3} + 1 = 3.140^+", "0db343165d5f00e33f35369b142d7c6d": "B = H_n", "0db3dc0d46208da95aecfa1d342b508f": "V_L=L\\frac{dI}{dt}", "0db42abf6951c0f4193c15d99a41baee": "\n\\begin{align}\nA&=\\int \\nabla I(\\mathbf{x'})\\nabla I(\\mathbf{x'})^{\\top}d\\mathbf{x'}\\\\\n\\mathbf{b}&=\\int \\nabla I(\\mathbf{x'})\\nabla I(\\mathbf{x'})^{\\top}\\mathbf{x'}d\\mathbf{x'}\\\\\nc&=\\int \\mathbf{x'}^{\\top}\\nabla I(\\mathbf{x'})\\nabla I(\\mathbf{x'})^{\\top}\\mathbf{x'}d\\mathbf{x'}\\\\\n\\end{align}\n", "0db45a9e1bff6eded6f0351ae7fa76ed": "l \\,", "0db4815a04e1eff45d5501e83fd82e78": "\n\\mathbf{p} = ~~\\frac{\\partial G_{1}}{\\partial \\mathbf{q}} = \\mathbf{Q}\n", "0db4842aaa59ee8a7365eeecadbe4207": "\\{w | w \\in L_1 \\lor w \\in L_2\\} ", "0db48c55fe4347b6145166a1b37748e8": "x(t) - x(t_{1}) = v_{g}(t-t_{1}),\\ \\ t > t_{1}\\qquad\\qquad(5)", "0db4b0393eae168851671b28e5f03c5f": "b(x)", "0db5255d6802d17c55fb65e18cb1cc59": "{\\varphi^3 = \\varphi - \\varphi^2}", "0db531324552bb9ad528b76dc082521b": "\nC_A = \\frac{C_{\\bar v}}{\\lambda_\\text{b:air} + \\dot V_A/\\dot Q_c},\n ", "0db55d39e76fa07a82d16668d0109203": "\\mathrm{response} = \\mathrm{MD5}\\Big( \\mathrm{HA1} : \\mathrm{nonce} : \\mathrm{HA2} \\Big) ", "0db5a1b30e14d296f63ec74b883b725e": "I^{+}(S)", "0db5ffcddf7c21d712643920d0ea4b74": "f(t,n) = \\sum_{i=0}^{n}{ \\binom{t}{i} }", "0db60e579b241e1b6a426329cc8315cd": "w(\\mathbf{x},\\mathbf{y})=\\pi(\\mathbf{x})Q(\\mathbf{x},\\mathbf{y})\\lambda(\\mathbf{x},\\mathbf{y})", "0db69cac78f8256f871a23aaf8695bf1": "\n \\mathrm{i}\\hbar \\frac{\\partial}{\\partial t} \\Delta \\langle\\hat{N}\\rangle = \\mathrm{T}\\left[ \\Delta \\langle\\hat{N}\\rangle \\right] + \n \\mathrm{NL} \\left[\\langle\\hat{1}\\rangle, \\Delta \\langle\\hat{2}\\rangle,\\cdots, \\Delta \\langle\\hat{N}\\rangle \\right]\n +\n \\mathrm{Hi}\\left[ \\Delta \\langle\\hat{N}+1\\rangle \\right]\\,,\n", "0db6a18b8be8e566091e4aeb7e81b3b4": "1, 2, 3, ...", "0db6bed8e27c8514a63c0f8ce62da993": "r(A^n) \\le r(A)^n", "0db6e19be8496f3f75a41c559c4fa2a6": "f:S^G \\to S^G", "0db6e7cea6e6a56ce0842aec8b1f8210": " \n \\begin{bmatrix}\n 1\\\\ 0 \n\\end{bmatrix} \n ", "0db73c5556e237fe31c84e83c328080a": "\\phi \\in \\mathcal{A}", "0db7571130969b3cbc9e6814cef41d68": "Happens", "0db7729bbe6ad3641c16988b7bada910": " \\langle \\cdot, \\cdot \\rangle \\colon \\Phi \\times \\Phi \\to \\mathbb{Z}", "0db79af945b3192786e9d27f1fdcb1bb": "x = \\left(1+\\frac{\\text{VAT}}{100%}\\right) \\cdot EC_\\text{rate} \\cdot Ex_\\text{rate}", "0db7dc2f9e7ae017e1281632cd1bfc99": " \\dfrac{dW(t)}{dt} = \\sigma _{t} ln(1-P) W(t) \\,", "0db7ff929037b95c239359017351afaf": "\\displaystyle{\\mathbf{v}(t)-\\mathbf{u}= -\\lambda\\mathbf{n}(0) + t\\mathbf{t}(0) + {t^2 \\over 2}\\kappa(0)\\mathbf{n}(0) + O(t^3).}", "0db8237cedf539d942605907e9075cb1": " [J_x,J_y] = i J_z,\\quad [J_z,J_x] = i J_y,\\quad [J_y,J_z] = i J_x, ", "0db83033b346e32af275913ef5dbbff3": "\n \\mathbf{x}(\\mathbf{X},t) = \\boldsymbol{F}(t)\\cdot\\mathbf{X} + \\mathbf{c}(t)\n ", "0db86e2288a2c79e909c4b9c1190217f": "I_\\parallel", "0db8f6db68f2cba2652164a7062bfd62": "\\frac{p}{r} = 1+ \\varepsilon \\cos\\theta.", "0db9428b706928c8ac951a858402fc4f": "\\ \\mu_\\delta", "0db99f8334d54ad40c6fd90a0c72d59a": "L = T - V", "0dba18841141c773b1aee7c9b4f86426": "(d_1e_1,\\ldots, d_ke_k)", "0dba2f26a19a1e55673e2f42abcb3886": "f: X_n \\to S_n", "0dbaa5b6a288f6484b4f1e8ef8f35130": "\\partial_t g_{ij}=-2 R_{ij} +\\frac{2}{n} R_\\mathrm{avg} g_{ij}", "0dbadaf02264f2c4bb453cc22ff82177": "(\\phi \\leftrightarrow \\chi ) \\to (\\chi \\to \\phi )", "0dbafe9cc6ef43dcd8b1fda7b4e93ff4": "\\mathfrak{g}/I", "0dbb54b92e0af3184d4fb02d2273585c": "r_1 = -p,", "0dbb78e7ff9f4f06af326d3d9aad0387": "\\textstyle l(x)", "0dbba5fecbae3204a7eda8c3d9ac3527": "\\psi(\\alpha)^{\\psi(\\alpha)}", "0dbbb6b623e7af334b6b83ffe0edf541": "\\sum_{n=1}^\\infty \\left(\\frac{f(n)}{n}\\right)^{1 + \\epsilon} \\varphi(n) = \\infty ", "0dbbd48f77affc660112daa8ca762f09": " H_{\\mathrm{Darwinian}}=\\frac{2n}{m_e c^{2}}\\,E_n^2", "0dbc2f898dd53548ed2546ca8bf2cd83": " c = A m {|S_2 - S_1| \\over S_2} \\,.", "0dbc5caee18a77db082a28b97e862f0d": " f''(x_0)<0 ", "0dbc6f19102b5fc99c196ae2ef31cad0": "\\boldsymbol{u}_x(\\boldsymbol{x},z,t)\\,", "0dbcab05347576af95b05f84e9bee5a8": " \\sigma_X ", "0dbd1d200bec7fe91f697b75038cad3f": "3 \\times 6 \\times 9 \\times 12 \\times 15 \\times 18 \\equiv 3 \\times 6 \\times 2 \\times 5 \\times 1 \\times 4 \\equiv 1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6 \\pmod 7;\\,\\!", "0dbd2a7d28db235e9a024cd1a49a0605": "R^n\\varprojlim:C^I\\rightarrow C.", "0dbda1de1761908c58141ef18cae70b2": " SubCipher_1=DEC_{b_1}(k_{b_1},s_1) ", "0dbdee735176e5313bc4632874ba03ae": "0 0", "0dcbf4aa02c6fae39b1843a1db82624f": "u(c,l) = \\frac{1}{1-\\gamma}\\left(c - \\psi \\frac{l^{1+\\theta}}{1+\\theta} \\right)^{1-\\gamma}", "0dcc320cc0864c35eb52f74cdc09c5bd": "\n\\begin{align}\n\\mathbf{P}(\\tau, \\mu | \\mathbf{X}) &\\propto \\mathbf{L}(\\mathbf{X} | \\tau,\\mu) \\pi(\\tau,\\mu) \\\\\n&\\propto \\tau^{n/2} \\exp[\\frac{-\\tau}{2}\\left(n s + n(\\bar{x} -\\mu)^2\\right)] \n \\tau^{\\alpha_0-\\frac{1}{2}}\\,\\exp[{-\\beta_0\\tau}]\\,\\exp[{ -\\frac{\\lambda_0\\tau(\\mu-\\mu_0)^2}{2}}] \\\\ \n &\\propto \\tau^{\\frac{n}{2} + \\alpha_0 - \\frac{1}{2}}\\exp[-\\tau \\left( \\frac{1}{2} n s + \\beta_0 \\right) ] \\exp\\left[- \\frac{\\tau}{2}\\left(\\lambda_0(\\mu-\\mu_0)^2 + n(\\bar{x} -\\mu)^2\\right)\\right] \\\\ \n\\end{align}\n", "0dcc375d5558262f89c8d543fed92970": " Q = 2 - \\frac{4}{p} \\sin{p} + \\frac{4}{p^2} (1-\\cos{p}),", "0dcc56e97688059028e18955bdd0df7e": "-J\\sum_{n,\\alpha}\\left(\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n+1,\\alpha}+\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n-1,\\alpha}\\right)", "0dcc9f9668d172a5523d09e35169c65e": "er_1", "0dcca3f1738b8292859653984c2c4e9a": "\\pi \\circ \\sigma = \\pi \\circ p_2 ", "0dcca5ab1eeb43a675b41ef35f3e5e91": "\\cdots\\rightarrow H^{n}(X)\\,\\xrightarrow{\\rho}\\,H^{n}(U)\\oplus H^{n}(V)\\,\\xrightarrow{\\Delta}\\,H^{n}(U\\cap V)\\,\\xrightarrow{d^*}\\,H^{n+1}(X)\\rightarrow\\cdots", "0dccc4ec9f4ac7222bbbb51cdcd6910b": "\\operatorname{cr}(G) \\geq c_r\\frac{e^{r+2}}{n^{r+1}}.\\,", "0dcce12a8089392e5d31682faf41c9fd": "\\Sigma^x", "0dccebb44d4e0544dabd7d15a26ecedf": "\\textrm{mes} E_\\lambda\\leq\\frac{\\textrm{mes} E}{2}\n", "0dcd3fec7c3c8eccf74a69d3e5e9d986": " 5x^3-5", "0dcd53ac2db434537c97ea9d1a131290": "z \\cdot x", "0dcd93864516050fea9fa81976fd799f": "\\forall a \\forall b \\exist c\\; a\\le c\\wedge b\\le c \\wedge \\forall d\\;a\\le d\\wedge b\\le d \\rightarrow c\\le d", "0dcdc726d1f8b0176f4dccda0d157652": "\\omega_ \\mathrm c = \\frac {1}{\\alpha}", "0dcddc8d5a88c7a270f0bb57892d00e5": " R \\propto \\frac{\\int{p(X,A | \\theta, O_{fg}) p(\\theta | X_t, A_t, O_{fg})} d\\theta}{\\int{p(X,A | \\theta_{bg}, O_{bg}) p(\\theta_{bg} | X_t, A_t, O_{bg})} d\\theta_{bg}} = \\frac{\\int{p(X,A | \\theta) p(\\theta | X_t, A_t, O_{fg})} d\\theta}{\\int{p(X,A | \\theta_{bg}) p(\\theta_{bg} | X_t, A_t, O_{bg})} \\,d\\theta_{bg}}", "0dcde92fd09c5978c5c2ff064d405219": "\\overline{(C \\wedge (A \\vee B))} \\wedge (C \\vee A \\vee B))", "0dce3f25b599c43e8a365cecf4c14bfd": "D_{A}^{*}D_{A}\\phi=\\nabla _A^{*}\\nabla_{A}\\phi+\\frac{1}{4}R\\phi+\\frac{1}{2}\\langle F_{A}^{+},\\phi\\rangle.", "0dce40d4e6911723f6409624cd67707b": "F_c", "0dce4e864791fb762c24fa203327c0b7": " x_n = \\frac{1}{k_{n+1} + \\frac{1}{k_{n+2} + \\cdots}}~; ", "0dce8ed5a0fc3585f99eeb61dde05f46": "_w = \\frac{\\sum_{i=1}^N w_i}{(\\sum_{i=1}^N w_i)^2 - \\sum_{i=1}^N w_i^2 } \\ . \\ \\sum_{i=1}^N \\frac{w_i . (x_i - \\overline{x}^{\\,*})^2}{(\\sigma_{x_i})^2 }", "0dce98c6a29a1f14df7e23241a7f528a": "\\sum_{i=1}^{k} p_i = 1.", "0dcef858935daa507d82b40efc97fa09": "\\displaystyle 2\\Delta=(ab)^2", "0dcf0594b30defd35403d9e4cf7263fe": "(p_x,p_y,p_z)", "0dcf2a6a16718eb52daa1e014a2e3469": "\\langle\\psi_V(t)|E_1^{(-)}(t)E_2^{(+)}(t)|\\psi_V(t)\\rangle", "0dcf345068bbe391fa3fdcf24aee3d9d": "\\mathbf{A} = \\frac{\\mathbf{J} + i \\mathbf{K}}{2}\\,,\\quad \\mathbf{B} = \\frac{\\mathbf{J} - i \\mathbf{K}}{2}\\,,", "0dcfa2b582e1560b89a1378e290d4adf": " \\Z ", "0dcfafc998e729c39c6f4f858e3374c0": "\n\\mathrm{CNR_{dB}} = 10 \\log_{10}\\left( \\frac {C}{N} \\right) = C_{dBm} - N_{dBm}\n", "0dcfdf4d0d176ca8a1b055efe367970b": " \\mathbf{x}_{k}^{T} \\, \\mathbf{H} \\, \\mathbf{x}_{k} = 0, ", "0dcffc4f63f6391d6b93de949071487e": " K_2F \\rightarrow \\oplus_{\\mathbf p} K_1 A/{\\mathbf p} \\rightarrow K_1 A \\rightarrow K_1 F \\rightarrow \\oplus_{\\mathbf p} K_0 A/{\\mathbf p} \\rightarrow K_0 A \\rightarrow K_0 F \\rightarrow 0 \\ ", "0dd031b4a27219b06f3c3f7032bc423c": "x\\le y", "0dd068bb6a336601f11f835bed96dc88": "\n\\begin{align}\n \\langle\\Phi_0 | (\\hat{H} - \\hat{F}) | \\Phi_0\\rangle &\\ne 0\n & &\\Longleftrightarrow &\n E_{\\text{HF}} &\\ne 2 \\sum_{i=1}^{N/2} \\varepsilon_i.\n\\end{align}\n", "0dd068d5ccbfb3690ce73266713a56ca": "L_{\\mathrm{z}} Y_{lm} = \\hbar m Y_{lm}, ", "0dd07ef5eda99a8c90abad049a4b43a5": "v(s) = \\exp\\biggl({-}\\int_a^s\\beta(r)\\,\\mathrm{d}r\\biggr)\\int_a^s\\beta(r)u(r)\\,\\mathrm{d}r,\\qquad s\\in I.", "0dd0c01b3941ca0423b208d457c83a04": "\\displaystyle{f^\\sim(T)=f(T^*)^*.}", "0dd0c5146a112cdefb9d9032d39c94d4": "\\tau_U", "0dd123404451caf8bcbae6021c8ed4a1": "(\\tfrac{a}{m}) = 1", "0dd129b645807e94f3d1c1758bddf07c": "1 + \\epsilon \\sum_{\\mu=1}^{\\mu=M}P_\\mu", "0dd17927c0f85a707fcef426b6d9f627": "\\mathbf{y}^*_1:=\\min\\{f_1(\\mathbf{x})\\mid\\mathbf{x}\\in X\\}", "0dd1fb01acd07995cd9a25e3f0b6a44b": "2^{-H_{\\min}(X|B)} = \\max_x P_X(x)~.", "0dd21e13c78dfd27099fa3774ec84179": "\nf \\propto \\frac{1}{l}\n", "0dd2737d5f487aa384eb6b3a8610f213": "f_d = f_r-f_t = 2v \\frac {f_t}{(c'-v)}", "0dd2ca036accb223f7e3c603b8c21932": "D_J", "0dd2e3c20f161ca67c45bf59d6ff7abe": "P = (a,b)", "0dd2fa9ccc77ba058673950ac4170883": " \\sin(n\\theta) \\, \\hbox{or} \\, \\cos(n\\theta), \\text{ and } J_n(k_{m,n}r). ", "0dd3165978b723be3b1e4b0759660f13": " Q = Q_1^T Q_2^T \\cdots Q_t^T, ", "0dd360338af997f4b181d323e119aeea": "\\mathfrak F'", "0dd3626c452207ee77bc3b375f9cd3f4": "f(k,0)=k!-\\sum_{i=1}^k(-1)^{i+1}{k \\choose i}(k-i)!", "0dd3aaeac1183872c4aaba300d6e948c": " 0.622 = {{MM_{H_2O}} \\over {MM_{dry\\, air}}} ", "0dd3fcf36f75087415064bbda0c3bee6": " b_{k} = \\dim H^{k} (V) = \\sum_{p+q=k} h^{p,q},\\, ", "0dd40a0ed1b4273dedd041e1414d9499": "X_i \\leq E(X_i)+a_i+M", "0dd431587e81bc100334b3d78d532a54": "L(A)", "0dd46cd1d97279852df5f200f0b4ddc6": "\\neg (\\overline{\\beta_1} \\wedge \\overline{\\beta_2})", "0dd4959bdd4ec461917a6fb760733cea": "\n \\operatorname{E}[|V^S - V^B|] \\approx \\alpha \\mu \\;.\n ", "0dd49b69061ee7d4332577b638a3d799": "\n\\Delta g = g - \\gamma.\\,\n", "0dd4ba47f064411e18ebd674c2a68418": "y(i)+y(j) = c(i, j)", "0dd4e75028a101ddb9d374d10280738e": "\\ MAE = \\frac{\\sum_{t=1}^{N} |E_t|}{N} ", "0dd4f39b473a17635819421778abdd49": "d\\mathbf X_1\\,\\!", "0dd51a155ce9e119a738e8358ecb6bb2": "\nI_{m, k}\n=\n\\frac{\\partial \\boldsymbol{\\mu}^T}{\\partial \\theta_m}\n{\\boldsymbol C}^{-1}\n\\frac{\\partial \\boldsymbol{\\mu}}{\\partial \\theta_k}\n+\n\\frac{1}{2}\n\\mathrm{tr}\n\\left(\n {\\boldsymbol C}^{-1}\n \\frac{\\partial {\\boldsymbol C}}{\\partial \\theta_m}\n {\\boldsymbol C}^{-1}\n \\frac{\\partial {\\boldsymbol C}}{\\partial \\theta_k}\n\\right)\n", "0dd51db8af6f06e461cd2b2796e6e165": "D(x,y)= \\frac{1}{2}", "0dd52f6d1dd6a0aa557cc1dac3101c17": "k = A \\exp \\left[-\\left(\\frac{E_a}{RT}\\right)^{\\beta}\\right]", "0dd57315448852ac5e85d1d7f4227461": "\\frac{d\\omega_r}{dt}=\\frac{1}{J(T_e-B_m \\omega_r-T_L)}", "0dd5e36179a728854337e7914b0acfab": "\\forall\\ x, \\overline{Bx}\\ \\rightarrow\\ \\overline{Rx}", "0dd70ae73c03aef43b8738b7e30e4365": "w^2x^2y^2 + w^2x^2z^2 + w^2y^2z^2 + x^2y^2z^2 + wxyzQ(w,x,y,z) = 0", "0dd77d1443b8864daa663a4dc1ac777d": "Interest = Principal \\left [ \\left ( \\frac {APY} {100} + 1 \\right )^{Days~in~term/365} - 1 \\right ]", "0dd7a2e8644f61a9cf6f8be6f07c8aab": "{M}=0.88128485\\sqrt{\\left[\\left(\\frac{p_t}{p}+1\\right)\\left(1-\\frac{1}{[7M^2]}\\right)^\\frac{5}{2}\\right]}", "0dd7a863ab4bf0f72c2d31922c4bdb08": " Z = \\frac{\\xi^N}{N!}.", "0dd7cd6051c76f98b1f7b7ee038198dc": "\\scriptstyle c^{\\underline{k-1}}", "0dd83b06966e0638ae985fb96d5cb7f3": "Q(x) = \\frac{6x}{\\pi^2} + O\\left(x^{1/2}\\exp\\left(-c\\frac{(\\log x)^{3/5}}{(\\log\\log x)^{1/5}}\\right)\\right).", "0dd85a63d2c5a98bedd56fd3f174c67a": "\nJ(x)(f)=f(x),\\qquad f\\in X',\n", "0dd864c88ec238950bdf95793fd8fc64": "F_{\\nu,max}", "0dd86a3bd9445a746e6f4255e61909ba": "EL(\\Gamma):= \\sup_\\rho \\frac{L_\\rho(\\Gamma)^2}{A(\\rho)}\\,,", "0dd86b4fc234be303d4b954116484140": "\\Pr[y_i^{\\prime\\prime} = ?] = \\Pr[\\theta \\in [0, {2\\omega_i \\over d}]] = {2\\omega_i \\over d}.", "0dd87427da2fabf8b593c8de198fa91d": "\\frac{1}{Z_{\\beta, n}} \\prod_{k=1}^n e^{-\\frac{\\beta n}{4}\\lambda_k^2}\\prod_{i 0\\!", "0ddf5b362207534fd9ec8c845018b5a6": "\\sum_{x\\in A} f_X(x) = 1", "0ddfc81bf6e635063312fc353ab81bb5": "\\Delta_m", "0ddfdbadc8663e0638229c2c60be28c9": "\nx = [\\overline{a_0;a_1,a_2,\\dots,a_m}],\n", "0de08414c38de494106254e16823f05a": " \\text{Maximize} \\,\\, pQ - wL(w) - rK \\,\\, \\text{with respect to} \\,\\, Q, \\, w, \\, \\text{and} \\, K", "0de0ca0514b0197d643ca5b3290932a5": "\\sin \\theta =\\frac{y'(s)}{\\sqrt{x'(s)^2+y'(s)^2}} = y'(s) \\ ;", "0de0d49ed704dbf3bce7b6e63ce33876": " J = \\frac{1- \\psi^'}{\\phi^'}\\,", "0de0dc6ba8b5f2802aea992a9a7782d8": "\n\\mu_{\\operatorname{eff}}(\\dot \\gamma) = \\mu_{\\operatorname{\\inf}} + (\\mu_0 - \\mu_{\\operatorname{\\inf}}) \\left(1+\\left(\\lambda \\dot \\gamma\\right) ^2 \\right) ^ {\\frac {n-1} {2}}\n", "0de14e0aed75a8dbb5d64ea6a20a721e": " \\frac{d}{2}", "0de23b9e67fe517252937091644dd937": "E[G|H] = \\int^T_0 k(t)E[x(t)|H]dt = 0", "0de31b19055091711fccd433999c3e62": "\\log K_0 = \\log 2 + \\frac{1}{\\log 2} \\left[\n\\mbox{Li}_2 \\left( \\frac{-1}{2} \\right) + \n\\frac{1}{2}\\sum_{k=2}^\\infty (-1)^k \\mbox{Li}_2 \\left( \\frac{4}{k^2} \\right)\n\\right].\n", "0de32fab8a1927bb9864cfa1dc42a480": "\\frac{dX(t)}{dt} = u[X(t),t]", "0de35b0a6b8fc32350e8c500c52089af": " \\and T_7 = [F_7, S_7, A_7]::[F_6, S_6, A_6]::K_5 ", "0de3738a81e5265ef899cc00f7d5edc2": "\n\\sigma _1 ^2 = \\sigma _3 ^2 = I, \\; \\sigma _1 \\sigma _3 = - \\sigma _3 \\sigma _1 = e^{\\pi i} \\sigma _3 \\sigma_1.\n", "0de3886d300b5c7dfc14329ba1983246": "\\scriptstyle x \\;\\in\\; [0,\\, \\infty)\\!", "0de39f4449f39bc0eeae178121bdf307": "\\mu_n = \\{1,\\zeta_n,\\zeta_n^2,\\dots,\\zeta_n^{n-1}\\} ", "0de3a8a2ea897397f77ac970ce3a4173": "\\hat{H}(\\mathbf{k})", "0de3f48e92d7e79380da650f41a56e23": "\nL_\\Sigma = 10\\,\\cdot\\,{\\rm log}_{10} \\left(\\frac{{p_1}^2 + {p_2}^2 + \\cdots + {p_n}^2}{{p_{\\mathrm{ref}}}^2}\\right)\n = 10\\,\\cdot\\,{\\rm log}_{10} \\left(\\left({\\frac{p_1}{p_{\\mathrm{ref}}}}\\right)^2 + \\left({\\frac{p_2}{p_{\\mathrm{ref}}}}\\right)^2 + \\cdots + \\left({\\frac{p_n}{p_{\\mathrm{ref}}}}\\right)^2\\right)\n", "0de47177820aa228efeb43a86c53fc5c": "L^\\prime ", "0de4ef4752110c23926d7fc377784f36": " \\mbox {Diameter in millimetres} = 2 \\times T \\times A / 100 + W \\times 25.4 ", "0de4ef9255a21226302c6def1ec1cc7b": "\\mathcal J_{p+2}", "0de5580bc106089722df88ed8fa1762f": "u = \\frac{\\omega}{2 \\pi c} (x_1 - x_2)", "0de56f5d8c7016548339ec76bca76f3d": "f = \\frac{V_a+V_w}{V_s+V_a+V_w}", "0de6b31f36ae71429eaf0e8e367eadb5": "{K-B\\over N-K}{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "0de7047912a29c5ae7004b971d26148f": "p(\\chi) = \\frac{2}{3} \\left( \\left(1 - \\frac{\\chi}{\\pi}\\right) \\cos{\\chi} + \\frac{1}{\\pi} \\sin{\\chi} \\right).\\!\\,", "0de70fe771d6e5d7bd1c7af1d6d5c9f9": " \\mid ", "0de73479fb335937ebcb9a84715982ad": " \\mbox{Exp}_{certificate of deposit} = 0.5\\times500 + 0.3\\times500 + 0.2\\times500 = 500", "0de74ceae7ca99234f0e9aae44a1af97": " Vt = \\frac{Vk}{Re}Rw ", "0de77c0791da7b4cf334d4dad8d82808": "D_o = \\sum_{u=1}^{N} \\frac{m_u}{n} D_u = \\frac{1}{n} \\sum_{u=1}^{N} \\frac{1}{m_u -1} \\sum_{i=1,j=1}^{m}{_{metric}} \\delta_{c_{iu} k_{ju}}^2", "0de7a5b26bb28e5a76e8b60a4150636f": "b = 90^\\mathrm{o} - \\lambda_\\mathrm{A}\\,", "0de7b6a61a70688b26e6eeb3113531a3": "dz", "0de9401c1235d8595cbf5e197c2b57dd": "{}E[X_t|\\{X_{\\tau} : \\tau \\le s\\}] \\le X_s \\quad \\forall s \\le t.", "0de9a79981ec28b4d341491681803722": " \\textbf{m} = c\\textbf{h}+c -q\\textbf{f}_p \\cdot K \\pmod p ", "0de9a97807dd2fd5bcbdc5ad60d34fc6": " \\!\\ {1 \\over {\\sqrt{2} - 1}} = \\sqrt{2} + 1 ", "0de9dc0f5ff52aee4ee7c9f99f1c39d2": "\\overrightarrow{Y}=Y_{o} \\ \\ \\ , \\ \\ \\ \\overleftarrow{Y}=jY_{\\varepsilon }tan(k_{x\\varepsilon }w)", "0de9f248831c58969286349a3d852f91": "\\eta = a\\, \\cos\\, ( kx - \\omega t) = a\\, \\cos\\, \\theta ,", "0dea0736fdbd2b398c7ed50f5844d075": " \\displaystyle{ 0.58\\, W } ", "0dea22f489272c79924fe58f2cd48103": "f_P (f_P (X)) = X \\text{ for all }X \\in [\\ell]", "0dea598bd161f2566f652dcbb5e9feb4": "\\Xi \\, \\xi \\,", "0dea8e7d4176801176a52c0236b6c808": "\n\\left( x^2 + y^2 \\right) +\n\\left( z - a \\coth \\tau \\right)^2 = \\frac{a^2}{\\sinh^2 \\tau}\n", "0deac00ba48cb30e568820886a46dfae": "=C_{H_2O}", "0deae5992a4b076ce59a9a31792603ee": "\\partial^{4}B/\\partial x^{4}", "0deaea42c9ef1c67ec9775c7f3b92938": "\\left(3\\sqrt{\\frac{2}{5}},\\ \\pm\\sqrt{6},\\ 0,\\ \\pm2\\right)", "0deafa7525eb2625f8d7dcaaa6f3490b": "\\overline W", "0deb0294ad9faaaf5c9024339b47581a": "x_1\\in\\mathfrak{g}_{\\lambda_1}", "0deb57a0dc714b0f8c52bcde78e8718f": "z_{match}\\,", "0debd733e70395cb0e53177ac6ac8a0c": "\\tfrac{3KE}{9K-E}", "0dec13788f1388692a2432f552b1fdfc": "c_{jk}", "0dec58ac609485be2477b46e58a7e162": "f\\circ\\Phi(x,y) = \\textstyle\\frac12 x^TBx + h(y).", "0dec6a3e3d3f26835acad2233af715df": "\n \\omega_{mn} = \\sqrt{\\frac{D\\pi^4}{2\\rho h}}\\left(\\frac{m^2}{a^2} + \\frac{n^2}{b^2}\\right) \\,.\n", "0decf9b8b8f8301b6b36efbee0f182aa": "g\\!", "0dee0fd3d3e936f88ca9e7fd45906b5e": "\\displaystyle{\\varphi_{s,t}(z)= f_t^{-1} \\circ f_s(z)= e^{s-t} (z+a_2(s,t)z^2 + a_3(s,t) z^3 + \\cdots)}", "0def02d4e7a3051a5c1de75372cae213": "\\overline A", "0def164a7fa0a232f15c93cfb75e2818": "\\lim_{w \\to 0^+} w \\log_2 w = 0", "0def4b1affe6183b9f9dd3dcf095c235": " \\lim_{x \\to p}f(x) = L ", "0def4d70826d93e077b226405da4b6a2": "\\frac{\\mu^3}{\\lambda} ", "0def5bcd31fd6c0c87ed2adda37bddc4": "\\pi_1f_1", "0defb44b8ac8c676aa39aaa503de5106": " Q^j_i = 9.6 \\text{ m}^3/\\text{s}", "0defb66af90a0ea6b261aab5f6cc54f9": "\\frac{x}{2} \\coth \\frac{x}{2} = \\sum_{n=0}^\\infty c_n x^n = 1 + \\frac{1}{12} x^2 - \\frac{1}{720}x^4 + \\cdots", "0deff6ac005223f150c6e9935a4024ad": "\\omega_m \\ll \\omega_c\\,", "0deffa862b19684c17dc2fd4fdfdcf44": " \\ln \\left ( e^{A} e^Be^{-A} e^{-B}\\right )= [A,B]+\\frac{1}{2!}[(A+B),[A,B]]+\\frac{1}{3!}\\left ( [A,[B,[B,A]]]/2+ [(A+B),[(A+B),[A,B]]] \\right )+\\cdots .", "0df02c8889f4670a3dba1d2ed8ffb4de": "\\displaystyle{\\|f\\|_{(k)}^2 = \\sum_{j=0}^k {k\\choose j} \\|\\partial_x^j\\partial_y^{k-j}f\\|^2.}", "0df0695845e62978683766baf2d7c2fc": " n = \\frac {4 \\, (z_\\alpha + z_\\beta)^2 } {d\\log^2{\\lambda}}", "0df0b267e87f9bb70729600a8b527488": "\\eta\\,\\!", "0df0f9fca8a0c95d27e0b2f4a560a310": "\\displaystyle{f(z)=\\overline{f(\\overline{z})},}", "0df14837305561d74a847771eb5f106d": "{\\bar{HP}}_3", "0df18f1eec7c67d65b06a79355d7e49e": " [r_i -S]\\mathbf{y} = (\\mathbf{r}_i - \\mathbf{S})\\times \\mathbf{y},", "0df1ba97c4e027a135a467f26561f612": " q \\overset{\\alpha}{\\rightarrow} q' ", "0df1f51621a8e294c813056976211d1c": "X^*_{\\tau(X^*, X^{**})}", "0df20f79abb6b93039da4e0cb1e6d172": "\\operatorname{Pref} (L) = \\bigcup_{s\\in L} \\operatorname{Pref}_L(s) = \\left\\{ t\\vert s=tu; s\\in L; t,u\\in \\operatorname{Alph}(L)^* \\right\\}", "0df2440ac89ecfa63a7e317ed22acba7": " J = J_1(1) \\oplus J_1(2) \\oplus J_2(4) = \n\\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & 4 & 1 \\\\ 0 & 0 & 0 & 4 \\end{bmatrix}. ", "0df26f62149f5814c6372a101993654a": "GL(\\mathbb{R}^7)", "0df2dfc600298b2706ad4ab774c858c3": "\nL_{r}=n_{e} n_{H} P(T)~~ {W~m^{-3}} \n", "0df308f3c3554c5ef08358bb562130c3": "J_{gas} = h_g (C_g- C_s)", "0df32534ae1f73ea43b99cddf272ca05": "\\lVert z \\rVert = z z^* = z^* z = x^2 - y^2.", "0df336d237764d8d6747521fa443fece": "e^{-sT}\\dot{=}\\frac{(sT)^{2}-6sT+12}{(sT)^{2}+6sT+12}", "0df338892151c4df85941524eb3a1d37": "\\neg A\\to(A\\to B)", "0df3dd737ed6ef805d378925bc123797": "V = \\frac{h_1 B_1 - h_2 B_2}{3}", "0df4015a869eb9ec434a9bc9d33a9856": "a_\\mu", "0df41b034c4ba373a6f62d6410b4874c": "a=b^c", "0df45099cc2928ae59be9c703fe1aa97": "a_{1}=320", "0df4b8c24f1b02af924df7d79c9dac91": "\\mathbb{Z}[\\sqrt{-5}].", "0df4d2f9436603c67a4edcf5d4d54ff6": "\\boldsymbol{\\sigma}* \\in \\mathbf{b}(\\boldsymbol{\\sigma}*)", "0df4dc59f99a775d94e31df99368e1c5": "\\vert n\\vert\\leq N", "0df510feb88385bdc04683b4975d2fa9": "D(x,y)", "0df5d6a0542f397b75c9db6ef0087b0c": "\\,\\,\\boldsymbol{\\sigma} = 2\\mu\\,\\boldsymbol{\\varepsilon} + \\lambda\\,\\text{tr}(\\boldsymbol{\\varepsilon})\\,\\boldsymbol{I}\\,\\,", "0df601c6accb491001c07df9c6c1c4e4": "dU = C_v dT", "0df62b1e73d01d8a354c785d6684383b": "J_w = A \\left(\\Delta \\pi - \\Delta P \\right)", "0df63f38b0a445aae34f1f5bb42f5cb7": "F_n(x) = U_n(x,-1),\\,", "0df696ed178fe22a9068eaf4ceb586c9": "\\,\\prod_n (z-c_n)", "0df6bd159e216d9baa128178f0cae69e": "z \\in \\mathbb{C}", "0df6f101fe227cb36d54334e794301d2": "Ker(df_x)", "0df719d0291aa256b61a04397d63e754": "\n\\left[ {\\mathrm{d} \\over \\mathrm{d}t}{\\partial{T}\\over \\partial{\\dot{q_i}}}-{\\partial{(T-V)}\\over \\partial q_i}\\right] = 0\n", "0df73d835f83efa414b6762e69e2fa98": "K=\\mathbb{C}", "0df757a3729a50b14003c69b4089f5b4": "r^2-R^2", "0df77b9d7105fdc39bae30bdc23addfb": "p(A)=1", "0df8444ac0c9455e206e607ab5cdff73": "\\vec{r}_v.", "0df859cc5c058f03a493b68070433539": " X | \\mu ", "0df872997025404baedfb272a051f1c8": "V = k_2T \\qquad (2)", "0df92300e9acc53840cdb5e971134346": "\\frac {dC_{AS}}{dt}= 0 = k_1 C_A C_S (1-\\theta)- k_2 \\theta C_S -k_{-1}\\theta C_S ", "0df9cc6763484db7df8a7418bff18a65": "w_k", "0df9dd2dfec0c0ded5aa8e16e0b69b5a": "e^{\\lambda(r)} - 1 = \\frac{r_s}{r - r_s} \\;", "0dfa1c3d9778bf0bd5ed348cac016a6f": "\n \\left(-\\frac{n}2f_\\mathrm{s},-\\frac{n-1}2f_\\mathrm{s}\\right)\n \\cup\\left(\\frac{n-1}2f_\\mathrm{s},\\frac{n}2f_\\mathrm{s}\\right)\n ", "0dfa7d19db7627045b6292f0fb99629c": "0=N_0\\subsetneq N_1 \\subsetneq \\cdots \\subsetneq N_n=M", "0dfa9bd2538905ec9d491e230d88dd17": " \\Lambda^3_0\\mathbb C^6 ", "0dfaca0f8aad55df388e07eeb190a086": "\\boldsymbol{K}", "0dfad386bed71d50d2656d4320dd0a61": " Y = f(X) \\ ", "0dfbf929067de6d44384294102818d6a": "\\pi=\\frac{3528}{Z} \\!", "0dfbfd3cab6588265e46ad52deb0063b": "s\\in\\{1,2,3\\}", "0dfc402e042b5d814aa39f24bbdd96d9": "1,0,-1", "0dfc516dec7cec7d54092a206af5c494": "\\scriptstyle{\\mathbf{L}}", "0dfc76213a3f46666199ce2419e22a79": "f(x) = \\frac{1}{2b} \\exp\\left(-\\frac{|x - \\mu|}{b}\\right)", "0dfc846aa4eaf09afee3fcb0340fe877": "A \\otimes_R A^\\circ", "0dfc97cb3615a7fac8eb365474728340": "(z-z_0)^{-2}-\\frac{z_0}{10}(z-z_0)^2-\\frac{1}{6}(z-z_0)^3+h(z-z_0)^4+\\frac{z_0^2}{300}(z-z_0)^6+\\cdots", "0dfc9c4427c33d64c997bd801ac82c4b": "W_X=\\left(\\frac{x/x_n}{y/y_n}-1\\right) V_J", "0dfcfec2a00afd662748a25bc26cae64": "r \\ne 0", "0dfcfecd43bec0005964b5f47653c306": "x\\neq 1", "0dfd02c657f5b499d64e0cb88f03e6a5": "g^{ab}\\theta_{ab}=\\theta", "0dfd21de8f1d0744a94d21dc2daf5f18": "L_3=1/3", "0dfd5f87577526aa7c6dd5e77595f035": "\\displaystyle{\\pi_{ij}(xy) =\\pi_{ij}(x)\\pi_{ij}(y),\\,\\,\\, \\pi_{ij}(1)= I.}", "0dfd8560df76e8c860748d9efcd07607": "R_{5,4} = 11 r^5-10 r^4", "0dfdb95b77a645c11063543994f025a9": "\\Gamma_N(w|a_1,...,a_N)=\\Gamma_{N-1}(w|a_1,...,a_{N-1})\\Gamma_N(w+a_N|a_1,...,a_N)", "0dfde751679220aa2dfabe52aa975e41": "\\frac pq", "0dfe1bcb396d9fef2614363d96980238": "d > 12", "0dfe776fedcc377da1f838a5e43d5a11": "p_{i_1, i_2, \\ldots, i_N} \\approx \\sum_t^T p_t \\, \\prod_n^N p^n_{i_n,\nt},", "0dfec86dabeaf3d3d7624b7348b75506": "y_{n}(x) = \\sqrt{\\frac{\\pi}{2x}} Y_{n+\\frac{1}{2}}(x) = (-1)^{n+1} \\sqrt{\\frac{\\pi}{2x}} J_{-n-\\frac{1}{2}}(x).", "0dfec9ab94481562873a30208a5b5c54": "\\lfloor n^2/4 \\rfloor", "0dff4b4e798fcf9b6fcbf2a08879653c": "\n \\mathcal{M} = \\mathcal{M}^K + \\frac{\\mathcal{B}}{1+\\nu}\\,q + D \\nabla^2 \\Phi\n", "0dff5d5251f5163826306c2785f8de0a": "\n\\begin{align}\n\\tan x & {} = \\sum_{n=0}^\\infty \\frac{U_{2n+1} x^{2n+1}}{(2n+1)!} \\\\\n& {} = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} x^{2n-1}}{(2n)!} \\\\\n& {} = x + \\frac{1}{3}x^3 + \\frac{2}{15}x^5 + \\frac{17}{315}x^7 + \\cdots, \\qquad \\text{for } |x| < \\frac{\\pi}{2}.\n\\end{align}\n", "0dff77b567fa34bfe97953fa0708bcae": "X = X_0, X_1, \\ldots, X_{n-1}, X_n = Y", "0dff9220b8a143e3bb43279d9bd8f62e": "y = x^{3/2}", "0e0039c3a43d982732917b456a3ff4e9": "R/I^n", "0e00f7b53d80743af1c5cc0c6eb3ba67": "U=-G\\int_0^R {\\frac{(4\\pi r^2\\rho)(\\tfrac{4}{3}\\pi r^{3}\\rho)}{r}} dr = -G{\\frac{16}{15}}{\\pi}^2{\\rho}^2 R^5", "0e0163e4ab8ec7b5c915ce81276976c5": "\n\\int\\limits_{0}^{2\\pi} t_g\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ du\\ =\\ \n-\\frac{3}{2}\\ \\int\\limits_{0}^{2\\pi}\\ \\left(1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u\\right)^2\\ \\ \\sin u\\ du\\ =\\ \n-3\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\sin^2 u\\ du\\ =\\ -2\\pi \\frac{3}{2}\\ e_h\n", "0e017fcf2d4accdc16cc6af46f76263a": "W(k) \\approx \\frac{1}{k^2 \\ln 2}. ", "0e01b96d4340e500cc02588adac68db8": "t(1/n,\\epsilon)\\,", "0e01e49a06e77b96edf6d0d45aa44cbe": "s^2 = \\frac{N \\sum_{i=1}^N n_i (\\hat{\\theta_i} - \\hat{\\mu})^2 }\n {(N-1)\\sum_{i=1}^N n_i }.", "0e021dc2488637367e3ce0a851ef89b1": "ax^3+bx^2+cx+d=0 \\qquad(1)", "0e024938216add9a87cdc7ae7570ea4a": "\\omega = \\sqrt{k/m}", "0e0275c21f40fecb1512deda66339ddc": "\\mathbf{\\gamma_0}", "0e02a725bd42081ec64219e4fec4152b": "\n\\tau_{l} = \\frac{1}{D_{\\mathrm{rot}}l(l+1)}\n", "0e02ef4a6916cb54430c8d30a044d106": "d_{2,1}^{2} = -\\frac{1}{2}\\sin \\theta \\left(1 + \\cos \\theta\\right)", "0e03245f9c26dbaf07c5f8d56e92cdec": "\\underbrace{A_i\\land\\dots\\land A_i}_{f(i)}", "0e036688ebce66792099c72a2ce03045": "L^1_{\\mathrm{loc}}\\;", "0e0375cf7a64599001ce1371ede02acd": "\\sigma^{2}_{x}/n", "0e037e4350ff36f937860a924c30d1b7": "Q \\cap I", "0e03cf7c5a5cf6ed1c8823318a60aab7": "d = \\min{\\Delta(c_1, c_2)}", "0e03fa752807abea7e0adc56c5775c85": " F = -\\frac{G M m}{r^2} - \\frac{\\Lambda M m r}{3} ", "0e0416f6936830bfe3d2f8ffae19975e": "L = 3.4~\\mathrm{a.u.}", "0e0457cd7c7a4011c0db5c2063d1dd61": "L f (x) = \\Delta f (x) - x \\cdot \\nabla f (x).", "0e045c92b526e900b3b124c807340acd": " i", "0e0480e97af34e0ffe9687ad2e7c700a": "X_1, ...,X_n", "0e04da75e1bed4e55dacfc73b528f8ab": "\nE_{0}=\\left( \\frac{Ze^{2}}{\\alpha D_{\\alpha }^{1/\\alpha }\\hbar }\\right) ^{\\alpha/(\\alpha -1)}. \n", "0e04e08cca7a0724785106f1f25f9c21": "\\scriptstyle b^4", "0e04e8cf0eed70b7effc626e02d36d60": "\\mathrm{deg}\\, C > {\\sum_{i=1}^r m_i \\over \\sqrt{r}}.", "0e04f8b48af9158b690fda9111dcb813": "n^2 = 1 - \\frac{X\\left(1-X\\right)}{1 - X - {\\frac{1}{2}Y^2\\sin^2\\theta} \\pm \\left(\\left(\\frac{1}{2}Y^2\\sin^3\\theta\\right)^2 + \\left(1-X\\right)^2Y^2\\cos^2\\theta\\right)^{1/2}}", "0e0519181090c0e90fcdb129d7ff1695": "\\sum_{i,j} r_{i,j}(x,y)\\partial_x^i\\partial_y^j", "0e053935a6d965bc16ca903289028186": "y = A_{y} \\cos \\left(\\omega_{0} t + \\phi_{y} \\right)", "0e0543aa32a4db1e5482305372fa7197": " h = \\theta_G - \\lambda_o - \\alpha", "0e054b1618f28ebe35d61832126a510c": " {\\lVert a_{i} \\rVert^2} ", "0e0558ac202bf4f6d81ad71b6ee92c11": "L(x_1,x_2,x_3) = (2x_1 + 5x_2 - 3x_3,\\; 4x_1 + 2x_2 + 7x_3)", "0e05b9ce1187fd800c4aaed120def3c2": "S^{2n+1}", "0e05bed100c5460dd523f83322b3e4d7": "f,", "0e05d4047132bf390f2a8690c09c34a9": "g_2(\\tau)=\\frac{\\langle I(t)I(t+\\tau)\\rangle_t}{\\langle I(t)\\rangle_t^2}", "0e05e60e2f4ff79d43dea8d46f0aeb13": "y(t_0) = y_0. \\,", "0e06c59c460ecd57bee11ecaacaf1751": " L = \\omega^l\\, \\cap \\, (\\omega^l)^{-1} ", "0e06d9824fe4426d1a65b6fd6300a927": "\\Delta\\lambda_0 =\n\\frac{4\\lambda_o}{\\pi}\\arcsin\\left(\\frac{n_2 - n_1}{n_2 + n_1}\\right),", "0e0709113ba0fd364d521802e28c023c": " -2 k \\cdot p' \\approx \\,", "0e0738c84dfb6eec9f26a53301899b4f": " \\frac{1}{2}-1.", "0e2ce498ed457deb087daa280b4b5d63": "H(u)(t) = -\\frac{1}{\\pi}\\lim_{\\epsilon\\downarrow 0}\\int_\\epsilon^\\infty \\frac{u(t + \\tau) - u(t - \\tau)}{\\tau}\\,d\\tau", "0e2d27af092b5d56d79c38a5e64372b6": "(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)", "0e2d27e61835a094469f4b27f1acdbbc": "\nf(t) = \\frac {1} {\\sqrt {2 \\pi} \\,i} \\int_{c - i \\infty}^{c + i \\infty} (ts)^{1/2} \\, I_{\\nu}(ts) \\, g(s) \\; ds,\n", "0e2d5fdb037e188e3e8c3fd9a96730c6": "\\frac{\\pi}{8} \\ (22.5^\\circ)", "0e2e241c1e9f5ed14c32f64521d6c661": "Pr[|\\frac{1}{m}\\left(\\sum_{i}\\left(w^{j}_{\\sigma_{i}}-w^{j}_{\\sigma_{m+i}}\\right)\\right)|\\geq\\frac{\\epsilon}{2}]=Pr[|\\frac{1}{m}\\left(\\sum_{i}|w^{j}_{i}-w^{j}_{m+i}|\\beta_{i}\\right)|\\geq\\frac{\\epsilon}{2}]\\,\\!", "0e2e83b675862caa6e233408abf829fa": " V : \\mbox{Var} \\rightarrow 2^S ", "0e2e9deb9c13e2c424fa1dc16fc36854": " \\text{(2)} \\qquad W = \\int_{V_1}^{V_2}P\\, dV ", "0e2ef5b993a4cfa66e4620c76c86d0e1": "=\\mathbf{w}_{n-1}+\\mathbf{g}(n)\\alpha(n)", "0e2efece425172fce21d95a7f4685c80": "|c_{1}(t)|^2", "0e2f5778d73f8c2ad524d627c03eef79": "f = \\frac{Nv}{2d}\\qquad\\qquad N \\in \\{1,2,3,\\dots\\}", "0e2f8bccc90ecb3f8e0390074a54e3c0": "\\begin{align}\n \\mu_X &= \\frac{ \\sum_i N_{X_i}\\mu_{X_i} }{ \\sum_i N_{X_i} } \\\\\n \\sigma_X &= \\sqrt{ \\frac{ \\sum_i N_{X_i}(\\sigma_{X_i}^2 + \\mu_{X_i}^2) }{ \\sum_i N_{X_i} } - \\mu_X^2 }\n = \\sqrt{ \\frac{ \\sum_i N_{X_i}\\sigma_{X_i}^2 }{ \\sum_i N_{X_i} } + \\frac{ \\sum_{i \\frac{4}{3}\\log g", "0e3508e72485ea6a0da30226d62f390e": "w^{(2)}(r)", "0e358e80cc416bcf774c7f622b02b660": "\nf(x; \\alpha, \\beta)\n= \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)}\nx^{-\\alpha - 1}\\exp\\left(-\\frac{\\beta}{x}\\right)\n", "0e3593936f8a03ab46ed68a4a730fa01": "I = \\frac{V}{R} \\quad \\text{or}\\quad V = IR \\quad \\text{or} \\quad R = \\frac{V}{I}. ", "0e359fff73cf5d682013ae71ecd6bced": "Z_P = \\cfrac{bh^2}{4}", "0e35cdac20c814a3bf7f9fca634e08b9": "k_0 \\frac{\\partial E_0}{\\partial z} + \\omega_0\\, \\mu_0\\, \\varepsilon_0\\, \\frac{\\partial E_0}{\\partial t} - \\tfrac12\\, i\\, \\Delta_\\perp E_0 = 0.", "0e35de83aca3aa66631bb7fa668e51fb": "\\Pr\\left(-c\\le T \\le c\\right)=0.95\\,", "0e360cc8d6e90ceb6a2f9aec2772df0a": "\\frac {de} {dt}=f(1-e)-b(e) e=f(1-e)-e^2", "0e36616592497c1ef341680f03266599": "k>p", "0e36b291b88dc4997979ea9336896bb3": "\\gtrdot", "0e36b3f6a21b235088364364e1d1079e": " X_v\\, )", "0e36e690d00f1e4f555c93794e4ec0ae": "(25)\\quad ds^2=-\\Big(1-\\frac{M}{r} \\Big)^2 dt^2+\\Big(1-\\frac{M}{r} \\Big)^{-2}dr^2+r^2d\\theta^2+r^2\\sin^2\\theta\\, d\\phi^2\\,.", "0e3750ec580b2ea02b090bd212b6dbe8": "\\omega_{R}", "0e3762719d56d05eb3314097c7994591": "\n\\begin{align}\n\\partial_t u &= d_u^2 \\,\\nabla^2 u + f(u) - \\sigma v, \\\\\n\\tau \\partial_t v &= d_v^2 \\,\\nabla^2 v + u - v\n\\end{align}\n", "0e3817aa8f75717c7cd02ccfe60c2ecf": "x_1^2+x_2^2+x_3^2=0.", "0e382264f8eaf08579b73f95e3883e36": "\\scriptstyle \\lambda_B", "0e383bb4f702745557b66ca9f69d3515": "b^{\\dagger}_i", "0e383f2c5d9881e1c8417c257c0af981": "\\mathbb{Z}^x = \\{n\\in \\mathbb{Z} : X_n(x) = 1 \\} \\, ", "0e3882068f0e733a2e6689f2eb8c7f91": "map(-, -)", "0e389fad6a525d62196c5af0f4e708d9": "n=p_1^{n_1}\\ p_2^{n_2}\\ \\cdots\\ p_k^{n_k}", "0e38d211b77423742da51f7beaa897b7": "\\boldsymbol{\\omega}_2", "0e3964e09b7a3d311d3bd7bd682592a1": "\\rm \\ 4VF_5 + 5SiO_2 \\rightarrow 2V_2O_5 + 5SiF_4", "0e39cf1e9da4fea5ef94f061c420afab": "H-\\frac{B}{8}-\\frac{B}{4}---\\frac{B}{2}-------B", "0e39f0a2b76a2c43e4801ff5518d9367": "\\operatorname{H}^*(\\mathbb{R}P^\\infty; \\mathbb{F}_2) = \\mathbb{F}_2[\\alpha]", "0e3a846357326022783cc0ae44d238ac": " k_y = k ~ \\sin \\theta ~ \\sin \\phi ", "0e3a9c695b4b12d38866e352d6147a08": "x = \\frac{a\\beta-\\alpha t^2}{a-t^2}", "0e3ad1a187abcabcc794360b5a2c1be7": " |\\psi\\rangle_A |0\\rangle_B |A'\\rangle_C = \n(\\alpha |0 \\rangle_A |0\\rangle_B + \\beta |1\\rangle_A |0\\rangle_B) |A'\\rangle_C", "0e3af61ba058fc81d64432a47685efc4": "{2n \\choose n} = \\frac{(2n)!}{(n!)^2}\\text{ for all }n \\geq 0.", "0e3b25193a349e31e044b072cde1e2fd": "(x,y) = (x_0,y_0) + t (x_1-x_0,y_1-y_0),\\,", "0e3b5eb3b41180b0bc5bbc763c6fcbe9": " a^2+b^2+c^2+d^2=8R^2. ", "0e3ba4c66cb27e6098a25bdd55659bd5": "tan \\delta=\\frac{\\epsilon_r''}{\\epsilon_r'}\\,", "0e3bcbbb056672b489498e98b5c4d234": "(x, y)\\,", "0e3c19f27936f55c4b0ce89b9a8bde47": "0.3 W ", "0e3cb4d9543a5190c02f0c39a2762414": "\\displaystyle P(I|c) = \\prod_i \\sum_j P(x_i|z_j,c)P(z_j|c)", "0e3d703d747485a13ed85f57fda2dc8f": "X \\sim \\chi^2(\\nu)", "0e3d95c373b75507711c5d58fbe36298": "\\mathbf{F}_{p^n}", "0e3df9f6b096b5005468115858a46e52": " = 2E/N = p(N-1)", "0e3dfe5af34c3d3a23ba821596ef7b49": " \\Sigma_n a_n ", "0e3e011c1151754d74a4289616db9f02": "e^{i\\theta}|1\\rangle", "0e3e572ebbd47a5e09b62ce00c97b193": "M_{e,\\lambda}(\\lambda,T)=\\frac{c_1 \\lambda^{-5}}{\\exp\\left(\\frac{c_2}{\\lambda T}\\right)-1}", "0e3ec46a8137f4bb7a0a7214dbf48d02": "\\begin{align}\nt' & = \\gamma \\left(t - \\frac{\\mathbf{r} \\cdot \\mathbf{v}}{c^{2}} \\right) \\\\\n\\mathbf{r'} & = \\mathbf{r}_\\perp + \\gamma (\\mathbf{r}_\\| - \\mathbf{v} t)\n\\end{align}", "0e3eccbc5241d6041d95e68f746abe10": "\\mathrm{P}(A \\cap B) = \\mathrm{P}(A)\\mathrm{P}(B)", "0e3f05e6d7fe07ac545afd9b0595006f": "\\mathbf{P}\\cdot\\left(\\nabla\\times\\nabla\\times\\mathbf{Q}\\right)-\\mathbf{Q}\\cdot\\left(\\nabla\\times\\nabla\\times\\mathbf{P}\\right)=\\nabla\\cdot\\left(\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}-\\mathbf{P}\\times\\nabla\\times\\mathbf{Q}\\right)", "0e3f15091d753046f2573f961351f5f0": "ax^2+bx+c=0.", "0e3f3d83e9682e71ad49c5a4dff2b19c": " U_E(r) = - \\int_\\infty^r q\\mathbf{E} \\cdot \\mathrm{d} \\mathbf{s} ", "0e3f492334b43049635142320789ffa3": "\\textstyle \\rho", "0e3f4b8fdb3385f44d93fc6ec0935ef6": "{G_\\mathrm{total}} = {G_1} + {G_2} + \\cdots + {G_n}", "0e3f7cc2ce3ee08ad40d7955f74bb0e5": " a_{30} = p_1p_4,", "0e407a544a82af598b71cdeb271c1bc8": " F = - dP \\cdot dA = \\rho \\cdot dA \\cdot dz \\cdot a", "0e408ec7923b18a8b2c4921bb6439b08": "\\begin{smallmatrix} 10^{5.96} \\approx 912,000 \\end{smallmatrix}", "0e40baa4a01c82a0c1646f71f22ae51a": "I(\\alpha h + \\beta k) = \\alpha Ih + \\beta Ik", "0e40ea9522ffef1cb516cc5cf5e3b951": "\\frac{\\partial p_1}{\\partial t}=D_p \\frac{\\partial^2 p_1}{\\partial x^2}-\\mu_p p \\frac{\\partial E}{\\partial x}-\n\\mu_p E \\frac{\\partial p_1}{\\partial x}-\\frac{p_1}{\\tau_p}", "0e4100e47300f75a7397a0950cce7d81": "y^\\star_j(\\mathbf{w}\\cdot\\mathbf{x^\\star_j} - b) \\ge 1,", "0e41624ffe7b1f9bcfdf84c51d2b3e58": "\\bar{R} = (x\\, \\bmod\\, 2^L) + 2^L", "0e417785a33f28e2e3be5a0a74b1d333": "\\displaystyle{\\mathfrak{h}=\\mathfrak{k}\\oplus\\mathfrak{m},}", "0e417e81ec79c8916b95f7e5ff02ad89": " \\int_a^b e^{nf(x)} \\, dx ", "0e419894a4c35c539999ee44e8da0101": " l = ( \\frac{ A }{ \\lambda g } )^\\frac{ 1 }{ 3 }", "0e41eeec38c96ed65649bd137a2e396c": "\\psi_n^*", "0e42174cf10d612474caa6118843c46c": "f(15x) = \\ln 15 + f(x)", "0e42ebed04c01a43fa5951d60a5532bf": "\\mathbf{u\\times v}=\\begin{vmatrix}\n\\mathbf{i}&\\mathbf{j}&\\mathbf{k}\\\\\nu_1&u_2&u_3\\\\\nv_1&v_2&v_3\\\\\n\\end{vmatrix}", "0e431002c06f00e69a9ab8a234c93fd8": "\\beta_{t,k}|\\beta_{t-1,k} \\sim N(\\beta_{t-1,k},\\sigma^2 I)", "0e43383374dc49cc0556dab28af259e7": "f(t) = u(t) + iv(t).", "0e437f4c8bb87de63c2d99d1405245a3": "\\dot{X}_a \\, X_b + X_{a;b} = \\theta_{ab} + \\omega_{ab}", "0e43bfb90514c9073b22f978c80bbd6b": "\\scriptstyle{ \\mathrm{R}}", "0e43f48d0084b9f951109beb09eeded8": "(d_i) \\neq R", "0e43f92345417252175108d320557ba7": "\\zeta^{\\prime}(-4) = \\frac{3}{4\\pi^4} \\zeta(5)", "0e44d5bc246f8a491a9f7fb02a36da4b": "| \\Omega \\rangle= \\sum_{i=1}^n | e_i \\rangle \\otimes | e_i \\rangle ", "0e4501d8227353c9bcfcec37baf96fbb": "\\beta =\\frac{(1-m)(\\alpha +1)}{m}.", "0e453023bccb249dcc1ad5c5ead0c658": "T^u_d", "0e455bd8e5345a4569d3dd8b6f70de3a": "\\boldsymbol{x}|_{{t=0}}=\\boldsymbol{x}_0", "0e459281a00a90d4a7ab45569dbaf153": "K_d = k_r / k_f", "0e4625c586d92a93fb161c1ecfd57b39": "\\textstyle D(x)", "0e4652af07e520df72d81a6c65f28a10": "E_{x}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\\frac{jk_{xo}k_{z}}{\\omega \\varepsilon _{o}\\varepsilon_{r} }(A \\ e^{-jk_{x\\varepsilon }w}+B \\ e^{jk_{x\\varepsilon }w})+\\frac{m\\pi }{a}(C \\ e^{-jk_{x\\varepsilon }w}+D \\ e^{jk_{x\\varepsilon }w})]e^{-jk_{xo}(x-w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ \\ (39) ", "0e46659fd87e5c6e6b0591ebded4ffbe": "\\nu_x", "0e472905fd2ff578258fb384fba0e9f9": "\\rho_{2k}(y):=\\rho\\left(\\frac{y-x_{2k}}{\\delta}\\right)", "0e47dfeb284dc257cafaa427ee4568c6": "\\hat{h}^{ab}=\\hat{h}^{ba}=m^b\\bar m^a+\\bar m^b m^a", "0e481442918de87548cd0cfc660f8c1c": "\\int k \\frac{dy}{dx} dx = k \\int \\frac{dy}{dx} dx. \\quad ", "0e483c1ecb634e62fd5cf16ce8ea27f5": "f(\\boldsymbol\\mu,\\boldsymbol\\Sigma|\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi,\\nu) = \\mathcal{N}\\left(\\boldsymbol\\mu\\Big|\\boldsymbol\\mu_0,\\frac{1}{\\lambda}\\boldsymbol\\Sigma\\right) \\mathcal{W}^{-1}(\\boldsymbol\\Sigma|\\boldsymbol\\Psi,\\nu)", "0e486dd0adf1b09ff606d1bc43e8ff87": "\n\\begin{align}\nx_{ji}|G_j &\\sim \\sum_{k=1}^\\infty \\pi_{jk} F(\\theta^*_k)\n\\end{align}\n", "0e487858411e8e712068596b3e346cb3": " {n \\choose i,j,k} = \\frac{n!}{i!\\,j!\\,k!} \\,.", "0e48d1d898a137f8f0af6d701499c1b5": "\\lambda_{1} + \\lambda_{2} + \\lambda_{3} = 1\\,", "0e492d0d4471ed40d44a2335a40ec3bd": "P = k_3T \\qquad (3)", "0e49b63bf7dc3972b1fb7057cbe8ddc8": " \\rho_{xx'} = \\frac{\\sigma^2_T}{\\sigma^2_X} = 1 - \\frac{ \\sigma^2_E }{ \\sigma^2_X } ", "0e4a42cd12bffa76aab9d472302e3af9": " \\frac{kT}{e} < A ", "0e4a56ebfd19f0f260d11b6ae9aa784a": "R=(N,C,V)", "0e4a76acf5eafd1678db230b86240115": "\\mathrm{cf}\\,(\\alpha)\\le\\kappa", "0e4a7c05ff1bd99a20d92475be87513d": "V = \\frac{Ze}{a} \\,\\!", "0e4a7e4cf3ed92de1e7ff20730d94253": "\\Delta_k(p)", "0e4b0be7e9f49e51a4511ce657bcc01d": "p \\leftarrow q \\and \\mathrm{not}~r", "0e4b15050d98a7cae758be7d6347c340": "\\underline{\\lnot \\psi \\lor \\lnot \\varphi}\\,\\!", "0e4b332cc0e37403dff51019ff361775": "k - 1", "0e4b9755e4c8e4b13ff29d62681c1fa1": "X_{C}\\approx 10^6 ", "0e4bb74a86e611b3a1120dc70583131d": "\\int_0^\\infty\\sin(x^a)\\ \\mathrm{d}x = \\Gamma\\left(1+\\frac{1}{a}\\right)\\sin(\\frac{\\pi}{2a})", "0e4bba6c9c90c6053a33c4e64a8372ff": "\\scriptstyle{\\Delta\\bar{H}}", "0e4be229616a81c9354adea334c3a24f": "\\approx 3.3233509704478425512\\,,", "0e4be4dda6d3207e4fbf9be288c1f86c": "a^2/4n", "0e4c46df226b9c0cb391311c54f28efe": "Ab", "0e4c8eabe1504c25033852e7d61f9aab": "\\biggl( \\sum_{i=1}^k p_i z_i \\biggr)^n\\text{ for }(z_1,\\ldots,z_k)\\in\\mathbb{C}^k", "0e4cdaf873bb3a79f278c618472f93ea": "\n \\begin{array}{lcl}\n r & = & -p \\\\\n 2s & = & b + p^2 + c/p \\\\\n 2q & = & b + p^2 - c/p\n \\end{array}\n ", "0e4d30dfe974fb513ab4cbd1cf5d6afa": " P(t) = \\big(P_{ij}(t)\\big)", "0e4d69f0e93722f23abbb67af08a56c4": "\\text{ ln}_q", "0e4d6dddfc12fec9f95e30354a365123": " 1 \\leq k < p", "0e4d8bfd0eb84d48c057b34b11b0ad5d": "|+\\rangle = (|1\\rangle+|0\\rangle)/\\sqrt{2}", "0e4d8d8f0f3e33a559808dd9c1fcf4e0": "(r, 0, 0)", "0e4d9a2433664ab0137a75104ce31e3c": "H_{so}=-(e/mc){\\vec{m}.\\vec{L}/r^3}=[(e^2/(m^2c^2r^3))\\vec{S}.\\vec{L}] ", "0e4db665d0b0382409949442c1fee583": " \\Delta^o \\overset{S^1}{\\longrightarrow} \\text{Fin}_* \\overset{\\mathcal{L}(A,M)}{\\longrightarrow} k\\text{-}\\operatorname{mod}, ", "0e4e48c91d277dc96cc2d0a647260734": "\n g_{ij} = \\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\\cdot\\cfrac{\\partial\\mathbf{x}}{\\partial q^j}\n = \\left(\\sum_{k} h_{ki}~\\mathbf{e}_k\\right)\\cdot\\left(\\sum_{m} h_{mj}~\\mathbf{e}_m\\right)\n = \\sum_{k} h_{ki}~h_{kj}\n ", "0e4e93d2722d57f6d127fcbe3df12f11": "\\bar b", "0e4e9e490e47d6637276504749d4657c": " r^{(g)} = \\frac{1}{2^{k-1}} \\left( \\sum_{ndom_k(Q^{(g+1)})} \\Delta{H}(a,ndom_k(Q^{(g+1)})) - \\sum_{ndom_k(Q^{(g)})} \\Delta{H}(a,ndom_k(Q^{(g)})) \\right)", "0e4eada83710c356246e91533864b10a": "\\,\\Delta(x-y) = -\\Delta(y-x)", "0e4ed84d54302b8c2d44268642388a02": "{d \\over dx} \\ln x = {1 \\over x },", "0e4f1d58745ec687fc7646351fa3e83c": " \\scriptstyle\\sigma_3^{2} = (\\scriptstyle\\sigma_1^{-2} + \\scriptstyle\\sigma_2^{-2})^{-1}", "0e4f7aaf3c3ca8e7e0ddad280be2c9fe": "(p,q)", "0e4fad68490aba79ee0ab0889c8a7d77": "\\mu(X)\\,", "0e4fadb193108ada369e9a569a441e6e": "\\int_0^t {\\rm e}^{As}Bu(s)\\,ds", "0e4fb8803f9a89da5375d39b82f01ffc": "\n\\Pr \\left\\{ \\lambda_{\\text{max}}\\left( \\sum_k \\mathbf{X}_k \\right) \\geq (1+\\delta)\\mu_{\\text{max}} \\right\\} \\leq d \\cdot \\left[ \\frac{e^{\\delta}}{(1+\\delta)^{1+\\delta}} \\right]^{\\mu_{\\text{max}}/R} \\quad \\text{for } \\delta \\geq 0.\n", "0e501e9fe7cc8f9bbec62b7f21ff797f": "E = E_1", "0e503674340614639eaabddba8338ee7": "d(f(x),f(y)) \\le \\rho(x,y)", "0e503e6ba8df74a61723be29415bcdc5": "x,", "0e50b98048ae2e637b76cb4b12375099": "W = \\frac{(N-k)}{(k-1)} \\frac{\\sum_{i=1}^k N_i (Z_{i\\cdot}-Z_{\\cdot\\cdot})^2} {\\sum_{i=1}^k \\sum_{j=1}^{N_i} (Z_{ij}-Z_{i\\cdot})^2},", "0e50fb319fe778603bc853bba30ecb05": "T \\cup E \\models O", "0e5105b2e1df05235b386aabbbfc9372": "\\textstyle\\frac{1}{2}n^3", "0e51a131e14659464433850813a93cbc": "\\text{extract}: ((S \\rarr T) \\times S) \\rarr T = (f, s) \\mapsto f \\, s", "0e51a5550991c4bf38cefe5a85c6f9a2": "\\scriptstyle H(t)\\,", "0e51a87ec173dd9534a056a403c85881": "E0", "0e51c1546e903370f3c6d76219162c91": "\\beta(x)", "0e520a0eeb0a8e61b3aee10e991a6527": "\\{ P_i\\} = \\{ P_1 , P_2 , ... , P_N \\}", "0e5213611c31364b5a28916bab37da60": "p(x^n,y^n)=\\prod_{i=1}^n p(x_i,y_i)", "0e5242a9f55a80e05176135f2ab2bd57": "V_i(t+ \\frac {\\Delta t}{2})\\frac{}{}", "0e53079228c5fe49ad2b2e4350e2a532": "x_3=-2\\,\\!", "0e5388b48285be3e04c8a9a0bf2355d0": "Se^{-q \\tau} \\phi(d_1) \\sqrt{\\tau} \\left[ q + \\frac{ \\left( r - q \\right) d_1 }{ \\sigma \\sqrt{\\tau} } - \\frac{1 + d_1 d_2}{2 \\tau} \\right] \\,", "0e53e2113faf3ffc62e3a278185c9fc7": "P \\leq 0.240 \\cdot \\sqrt{\\Phi} + 0.0103 \\cdot \\Phi.", "0e53f99c1d33116dcd8bd232c3b32262": " \\frac{\\sigma_u}{\\sigma_w}=\\frac{\\sigma_v}{\\sigma_w}=\\frac{1}{(0.177+0.000823h)^{0.4}} ", "0e5438e9a7edbf91053afb62491656fd": " k_\\mathrm{B} = \\frac{R}{N_{\\rm A}}.\\,", "0e544735e4c021b2721e5f4542267558": "X = x_0 + W", "0e544f26767efe60d2d1895f93131285": " \\approx 3.8 \\times 10^{695,974} ", "0e545b7b1708817953182d27e7c36c82": "\\scriptstyle{q}", "0e54826742955d53144a0ecd58720b53": "P = \\mathrm{d} Q/\\mathrm{d} t \\,\\!", "0e54c1608fc661bdb22b54c1ba1c9fb9": "TPO\\ \\times\\ loss_{feedline}\\ \\times\\ gain_{antenna}\\ =\\ ERP", "0e54c7c6f12e1bb942bd2affc76e5bec": " f^+(x) = \\max(f(x),0) = \\begin{cases} f(x) & \\mbox{ if } f(x) > 0 \\\\ 0 & \\mbox{ otherwise.} \\end{cases} ", "0e54eb737ec974bb7c0251abd16b1197": "\\mathfrak{S}_n,", "0e54f0b9318da4fa67fd7079b1ac7fcb": "\n p_c \\gets \\underbrace{(1-c_c)}_{\\!\\!\\!\\!\\!\\text{discount factor}\\!\\!\\!\\!\\!}\\, \n p_c + \n \\underbrace{\\mathbf{1}_{[0,\\alpha\\sqrt{n}]}(\\|p_\\sigma\\|)}_{\\text{indicator function}} \n \\overbrace{\\sqrt{1 - (1-c_c)^2}}^{\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\text{complements for discounted variance}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}\n \\underbrace{\\sqrt{\\mu_w} \n \\, \\frac{m_{k+1} - m_k}{\\sigma_k}}_{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{distributed as}\\; \\mathcal{N}(0,C_k)\\;\\text{under neutral selection}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}\n ", "0e552a46a02bc20a57cf11a164ab7255": "\\begin{align}\n\\mathbb{P}\\biggl(\\bigcup_{i=1}^n A_i\\biggr) & {} =\\sum_{i=1}^n \\mathbb{P}(A_i)\n-\\sum_{i0})^2\\right ]=E[X_n^2]\\,P(X_n>0).", "0e57387985e70b02d850ab34c2c20620": "\\left( \\frac{27379}{8658}\\right)^2=10+\\frac{1}{8658^2}.", "0e574efea80a3f6ab02c82627206f751": "B \\mapsto +A-BB--B-A++A+B", "0e57800c01e7644dc1ceefedbdfb37f7": "10^9", "0e57bea2518b074117a36ecce418afcd": "w_i(x) \\ := \\ \\frac{u_i(x) - m_i}{M_i - m_i}", "0e582ecca47b8f41349d57134732cfab": "s_{\\lambda}", "0e58a0b8b84bdf25dedf3d49c07a2072": "~A~", "0e58c7ec919aa81794e4b8cebd90e3bb": "\\mathcal{C} = ", "0e58d6499420076c9ca91c45cc092c61": "\\mathrm{~^{238}_{92}U}\\rightarrow\\mathrm{~^{234}_{90}Th} + \\mathrm{~^{4}_{2}He}", "0e58ed1250d654c85c529cb6fd71fdf4": "v^2 = t^3 + \\left(\\frac{3-A^2}{3B^2}\\right)t + \\left(\\frac{2A^3-9A}{27B^3}\\right)", "0e590a0b7ac9fd4d5ef629caa808f11b": "\\displaystyle{\\mu_{F\\circ h^{-1}}=0,}", "0e59586c4297b74a76f8da1396ddf153": "f(t) = \\begin{cases}\n t^{1/3} & \\text{if } t > (\\frac{6}{29})^3 \\\\\n \\frac13 \\left( \\frac{29}{6} \\right)^2 t + \\frac{4}{29} & \\text{otherwise}\n\\end{cases}", "0e596f42c8962f8ac39913dd61ef8ab8": "F_j(a, b) = a\\cdot b", "0e59a0d23041499b470c639cfd48f6ab": "\\widehat{R}_x, \\widehat{R}_y, \\widehat{R}_z ", "0e59d15dbc38599ddbf2fe1296c53fef": "\\int_{-\\infty}^\\infty x f(x)\\,dx. \\qquad\\qquad (1)\\!", "0e59daf62a798bcc308af5087b40681c": "3^4 = 81", "0e5a34fbce6d65c2e1fa0d7fa1aaecc7": "\\Omega(n) = \\Omega \\sqrt{n+1}", "0e5a52b2f19124fbec2b75a99c2a8edf": "M \\in \\left[ 0, l - 1 \\right]", "0e5aa17cc9306c5f67d6ed823a2471aa": "T = Rtan(\\frac {\\Delta}{2})", "0e5afc54aaa1828ea8d4c20428e411ed": "E_u(r,\\theta,\\phi)~=~2 \\pi j~ (k~\\cos\\theta)~ \\frac{e^{-jkr}}{r}~ E_u(k~\\sin\\theta~\\cos\\phi,k~\\sin\\theta~\\sin\\phi) ~~~~~~~~~~~~(2.2)", "0e5b2e44cbb7455368a30b018c758a42": "e^{4 \\kappa^2} + 2 e^{3 \\kappa^2} + 3 e^{2 \\kappa^2} - 6 ", "0e5b5133b4e91f22b9257469755ce2db": "Av\\hat{a}r(\\hat{\\beta}_{FD})=\\hat{\\sigma}^{2}_{u}(\\Delta X'\\Delta X)^{-1} ,", "0e5b5938e813ae78ce4e9f591c85f3af": "\\scriptstyle \\frac{1}{\\lambda}\\sqrt{\\sigma/(\\rho g)}", "0e5b98132ffabb056a20b633d72f4a4b": "|E(S, T) - \\frac{d \\cdot |S| \\cdot |T|}{n}| \\leq d \\lambda \\sqrt{|S| \\cdot |T|}", "0e5b9d285e57ab81b1406ea35c1b37da": "\\,\\Sigma_{xx}=Cov(X)", "0e5bdcf11353a8b33887cb0ab7c1c3f0": "\\scriptstyle{R_a^0}", "0e5c4a7842cdfee21f996a6590c615bc": " \\scriptstyle\\mathcal{F}_x ", "0e5c592f6179d77efcfac5a23e61e91b": "\\frac{\\partial k_{i}}{\\partial t}=\\frac{m_{}}{m_{0}+t}\\frac{k_{i}}{\\sum_{j\\in Local}k_{i}^{}}", "0e5c8d944ef669d1162dc08a4ff901a3": " Y(u,v) = r \\, v \\, \\sin 2 u, ", "0e5cc2a53fc6f8fe7d3953a8b1e3d35a": " J_f := \\left\\langle \\frac{\\partial f}{\\partial z_i} : 1 \\le i \\le n \\right\\rangle. ", "0e5ce1029c3413777fc598d3a940c156": "\\Gamma_8' = \\left\\{(x_i) \\in \\mathbb Z^8 : {{\\textstyle\\sum_i} x_i} \\equiv 0(\\mbox{mod }2)\\right\\} \n\\cup \\left\\{(x_i) \\in (\\mathbb Z + \\tfrac{1}{2})^8 : {{\\textstyle\\sum_i} x_i} \\equiv 1(\\mbox{mod }2)\\right\\}.", "0e5d00366864a19737b2d9e211fe8321": "C_p^\\ominus = A + BT + CT^{-2}", "0e5d01486e4dad226ca2dd44bd9307dd": " {dy \\over dt} = - \\alpha xy", "0e5d11d44a887140cf6871818dcca9a0": "\\Delta w_{ij} ~ \\propto ~ \\langle x_i y_j \\rangle - \\epsilon \\left\\langle \\left(c_\\mathrm{pre} * \\sum_k w_{ik} y_k \\right) \\cdot \\left(c_\\mathrm{post} * y_j \\right) \\right\\rangle,", "0e5d5cf88dcf28c29f7e6ef458c10383": " x_{k+1} = x_k - a ", "0e5e53628cd495c704f319535cd6040b": "xs = 0", "0e5ec19a3f022ce99b71fcbcf81a59ba": " \\underbrace{ (A^{T}A)^{-1}A^{T} }_{ A^{-1}_\\text{left} } A = I_{n} ", "0e5ec778b4c37d0f3d59a1f8dd690877": " \\lambda_1,\\,\\lambda_2 ", "0e5eea747ad8dc040b571d59796b4eb3": " IMM_{i-1}(S_{x,{i-1}}, a) ", "0e5f7f4c06ba1ac30c3de64f06de38b6": " (x_0,y_0,z_0). ", "0e5f8aba2621357e79b7955122bd9aad": "\\boldsymbol R = \\frac{1}{m_0} \\sum_{k=1}^N\\ m_k \\boldsymbol{x_k} \\ ; ", "0e5fc02a6555377d76c347a2315fe2f5": "n_k^{(-n)}", "0e5fc78b51ca8c03b44eb8de293a0117": "C_{yy}(y_i)", "0e5fcba9d2fe11b4e36f4f127d5afd9c": "\\mathfrak{P}^{27}", "0e5fdd452a2fe2dec8673b2a9f9490ee": "\\Gamma'(m+1) = m! \\left( - \\gamma + \\sum_{k=1}^m\\frac{1}{k} \\right)\\,.", "0e5febdf21ee53bc0903071fe3a80923": "~\\cos^{2}(x)~", "0e5fef39fa2d8f23849aedab2700298e": "(C_e)_{m+1}", "0e6000cb1205b89cb03091572d394bde": " m = p^k ", "0e6057401cf26e7c5635402da5d7d045": "k =(-3\\pm13)\\times10^{-5}", "0e6105f1f5ac555d624847439d38804a": "G = \\langle R_G \\mid S_G \\rangle", "0e610aaf7e1a947d4ad1a2fd92c4807c": "\\mathit{KP}(\\mathit{PRIMES}) \\leq 176", "0e612125b2af2057bd855505125e99c2": "f(m-h) = f(m) - f'(m)\\cdot h + f''(m)\\cdot\\frac{h^2}{2!} - f'''(m)\\cdot\\frac{h^3}{3!} + \\dots ", "0e61495531347f2ec0a4fc519312339e": "\\mathfrak{gl}(n)\\,", "0e61b502087a025beb3fb74eb31af787": ".\\qquad \\underbrace{NP/N,\\; \\quad N},\\; \\qquad (NP\\backslash S)", "0e621e0e8531edc16cf1c11d314468ec": "\\tau\\,", "0e627122246550af59bad2416c5f60a6": "\\overline{P} = (a, -b-h(a))", "0e6286f13c24ac8baf48afb5b28bfe6d": "\\frac{1}{\\rho_{max}}", "0e6359bd8d2685435913e7413a1a001c": "0 \\ge \\alpha \\ge 1", "0e63aaca16fff0c9bca8aa3a3ce88538": " \\{ e \\} ", "0e63d1d741a847802d6dffc6c1744d53": "F_{\\mu\\nu\\rho}=\\partial_\\mu A_{\\nu\\rho}+\\partial_\\rho A_{\\mu\\nu}", "0e63d3069a2a8b90b96877f9b4264fe1": "\\omega = 2 \\pi f\\,", "0e63e3f173ba95881dab44b02456e785": " d = d_1 + d_2 + \\cdots + d_n, \\; \\; d_1 \\geq d_2 \\geq \\cdots \\ge d_n", "0e63eb64a1ab439e81a2505d60a95589": "m_{00}^{S}", "0e64090fbaaf05e3ed96bebfbd1f12eb": "\\begin{align}\nd(x,y) &\\le d(x,T(x)) + d(T(x),T(y)) + d(T(y),y) \\\\\n&\\le d(x,T(x)) + q d(x,y) + d(T(y),y) \n\\end{align}", "0e640f7338892b86b49a775df12a99d4": "dS^2=\\left( 1-\\frac{\\ell^2_{P}}{r^2}\\right)c^2dt^2-\\frac{dr^2}{ 1-{\\ell^2_{P}}/{r^2}}-r^2(d\\Omega^2+\\sin^2\\Omega d\\varphi^2)", "0e6426d647a469ef35981f6791b83613": "S_{2m+1}=\\frac{\\left(1 + \\sqrt{2}\\right)^{2m+1} + \\left(1 - \\sqrt{2}\\right)^{2m+1}}{2}.", "0e642f75f7debf5032bd936f63b0134b": "\\left(A-sI\\right)^{-1}", "0e6460d58708bdc32db82dd88100f86a": "K_{i+1}", "0e6466f3433f7363e327045917bbe102": " \\Pi_P = \\{z \\in \\mathbb{C} \\, \\colon \\, Oz \\cap L_P = \\varnothing \\}, ", "0e64a72837afa7c8e4d25339d070e959": "S[k] = \\frac{1}{P}\\int_{P} s_P(t)\\cdot e^{-i 2\\pi \\frac{k}{P} t}\\, dt", "0e64ab01a3fb91403ec8e6c0e54dc736": "n = 0", "0e64d03e6dff82ca747c2e8e269fe52b": "\\scriptstyle i \\;=\\; 1,\\, 2,\\, 3,\\, 4,\\, ..,\\, n", "0e651ccb9422e0fbce88e3ba1440907a": "\\hat{\\boldsymbol {\\beta_1}} = \\mathbf {X_1}^+ (\\mathbf {y} - \\mathbf {X_2} \\boldsymbol {\\beta_2})", "0e661bfdf4ef0af694145d65d68de5fe": "P_1=(1,\\sqrt{2})", "0e664cd80eb5c9005092ef58d05238ca": "s \\in \\{ 0,0.5,1,1.5, \\ldots \\}", "0e66565dabe8def123b5db70d6675171": "f(x_1, \\ldots, x_d)\\ ", "0e667ad4f5fd2ea7b2031ae6108d39e8": "\\begin{align}\n s(t) &= \\Re \\left\\{\\left[I(t) + i Q(t)\\right] e^{i 2 \\pi f_0 t}\\right\\} \\\\\n &= I(t) \\cos(2 \\pi f_0 t) - Q(t) \\sin(2 \\pi f_0 t)\n\\end{align}", "0e66cc1365bf91848c84919f8cb72993": "\nX_\\mathrm{L} = -X_\\mathrm{S}.\\,\n", "0e66d3571abdb50c9039b3fb633f0cd5": " \\underline p ", "0e66da71e3f8b765d70cc119d9ae2b1b": "\n \\operatorname{E}[A_t] = \\operatorname{E}[S_t] + \\frac{\\mu \\alpha_t(1-\\delta_t)}{\\epsilon + \\mu \\alpha_t(1-\\delta_t)} (S_G - \\operatorname{E}[S_t]) \\;.\n ", "0e671582e02ba55e3e8bf04849361941": "Z_i(N) / Z_i(N- 1) = e^{-\\mu / kT } \\,", "0e673b3ec895573f88973739eb34edd2": " G_n(x)= \\sqrt n ( F_n(x) - F(x) ) \\, ", "0e67c4bb9bb313f3abe247da26fc8773": "\n\\Phi(\\rho, \\theta) =\n\\frac{-\\lambda}{4\\pi\\epsilon} \\left\\{ 2\\ln \\rho +\n\\ln \\left( 1 - \\frac{\\rho^{\\prime}}{\\rho} e^{i \\left(\\theta - \\theta^{\\prime}\\right)} \\right) \\left( 1 - \\frac{\\rho^{\\prime}}{\\rho} e^{-i \\left(\\theta - \\theta^{\\prime} \\right)} \\right) \\right\\}\n", "0e6858173e8c6f41ac6a1554586d75dd": "[A, B]", "0e6868f4117768e71d1fbf50900987ed": "n>1.", "0e695b4e85dfd6b87c76d4c0e79a48fe": "\\lambda_1 = \\lambda_2 = \\lambda\\,", "0e6975c9fac4c9bbfd30c785b9db0d92": "1-2 \\times 10^{8}", "0e69c1004b3b608015fff068369e54c6": "(g^a,g^b,g^{ab})", "0e6a17a54034d388482311b96343dbd1": "l < r", "0e6a2a1606c0371a8ab37e5afc257f3c": "F= 6\\pi\\,\\mu\\,a U\\left( 1 + {3 \\over 8} N_R + {9 \\over 40} N_R^2 \\ln N_R + \\mathcal{O}( N_R^2) \\right).", "0e6a363031ea544a6cc54508538980c5": "\\phi \\vee \\psi\\,", "0e6a39848c03a194febf92dade32947d": "w(x)< w(y)", "0e6af34199118a74e38af7f9d199dd12": "1-P_m\\cong 0", "0e6b301e006dc6462b9f7b2ffcb8f1ce": "\\omega_\\text {max}=\\frac{v_\\text {max}}{\\ell/2}", "0e6bfcbe684ee2d6ba4d3bc1b36f9118": "A_{\\alpha + 1,\\alpha + 1} - A_{\\alpha,\\alpha}", "0e6c2368a8c1c64888e411af0ca89620": "\\cos\\langle u,u'\\rangle = \\langle \\cos(u) , -u' \\sin(u) \\rangle ", "0e6c317be0c224aa069e0f7c8821e719": "\\hat{\\mathbf{T}} = \\mathbf{e}_i \\widehat{a}_i \\otimes \\mathbf{e}_j \\widehat{b}_j = \\mathbf{e}_i \\otimes \\mathbf{e}_j \\widehat{a}_i \\widehat{b}_j ", "0e6c98397c2394953717f20e1bf2a022": "\\boldsymbol{\\bar{v}} = \\frac{\\Delta \\boldsymbol{x}}{\\Delta t}.", "0e6ca36b9d93fb69d8a05cff1089c6a7": "\\overline{z+w} = \\bar{z} + \\bar{w}, \\,", "0e6cd02b4e09f063ba211c24e717010e": " \\frac{\\mu_0 l}{2\\pi }\\operatorname{arcosh}\\left( \\frac{d}{a}\\right)=\\frac{\\mu_0 l}{2\\pi }\\ln \\left(\\frac{d}{a}+\\sqrt{\\frac{d^{2}}{a^{2}}-1}\\right)", "0e6ce52b6252231b160de7021f9a32aa": "\\,\\phi_C(k)", "0e6cfa45f2bdef8abcd50566fa6ddb3f": "{\\tilde{A}}_6", "0e6cffd1f8b0d25850bda82c7601f289": "\\mathbb{R}^d = \\{y \\in \\mathbb{R}^n\\colon y_{d+1}=\\cdots=y_n=0\\}", "0e6d0e96960101ef8c1716c01f4e0ff2": "\\rho_{\\,\\mathrm{humid~air}} =", "0e6d79355e0a32acfffee00ec238a122": "-\\left[ \\frac{z}{2}\\log 2\\pi -\\frac{z}{2}-\\frac{z^2}{2} -\\frac{z^2 \\gamma}{2} + \\sum_{k=1}^{\\infty} \\Bigg\\{k\\log\\left(1+\\frac{z}{k}\\right) +\\frac{z^2}{2k} -z \\Bigg\\} \\right]", "0e6db5600db7911afbb2148719dbc0dd": "(A \\vee B) \\wedge (\\neg B \\vee C \\vee \\neg D) \\wedge (D \\vee \\neg E)", "0e6dd8573e210585c290be2818122b1c": "d(x_n,x) < \\epsilon ", "0e6e5a1663dcf6bb0b7cccfb88d7da8d": "e(E \\oplus F) = e(E) \\smile e(F).", "0e6e79e6a494ee519d6d41c792aeeb1a": "E^-", "0e6ee774fa2511941880427f617545ec": "O (|V|^2 \\log |V| + |V||E|)", "0e6f8e3cfdebaee37a2933d774a741c3": "\\mathbf{v} = \\left[ \\begin{matrix} \\rho \\\\ \\angle \\theta \\\\ \\angle \\phi \\end{matrix} \\right]", "0e6f9704806143947f95026c75eca98e": "\\varphi _j^{n + 1} \\ ", "0e6fa63dfd12d423f80e11128e656f65": "\\displaystyle{{d\\over ds} U(s)f=iPU(s)f,\\,\\,\\, {d\\over dt}V(t) f=iQV(t)f,}", "0e70552df00c4fde99c079d0abd5bde6": "V\\otimes\\mathbf{Q}_p(-1)^{\\otimes m}.", "0e70e546bd0faf1ce98673119640a582": " 2r \\sin \\left( \\frac{ \\theta}{2} \\right) ", "0e70f3cc7f5f3ccfb35fd4e299be7191": "\\tan \\theta (x) = \\mathfrak{Im}[f(x)]/\\mathfrak{Re}[f(x)]\\,", "0e7144e140cf5b9ab93390b2f7505acd": "|\\Lambda| ", "0e71592fc48837083a49aa8f76fde5d9": "\nr_0 = \\left [ 0.423 \\, k^2 \\, \\int_{\\mathrm{Path}} C_n^2(z') \\, dz' \\right ]^{-3/5} \n", "0e7194c2b4f230349a680fbcbd166661": "\\theta_c", "0e719634cb26f836e5efb9fa23399b10": "2^{287}.", "0e71bdd8079e40979de816daf3c87bed": "\\mathbf x,\\mathbf y\\in U", "0e71dac9bfc6ad9b27d1eb9462656a97": " p(y|\\theta)\\, ", "0e71dd06e05336399523c1d83ea589f5": "A \\cdot B \\cdot C", "0e71ed017e3f211d040bbe91d28a47f7": "\n \\Pr\\!\\left[\\,|T_n-\\mu|\\geq\\varepsilon\\,\\right] = \n \\Pr\\!\\left[ \\frac{\\sqrt{n}\\,\\big|T_n-\\mu\\big|}{\\sigma} \\geq \\sqrt{n}\\varepsilon/\\sigma \\right] = \n 2\\left(1-\\Phi\\left(\\frac{\\sqrt{n}\\,\\varepsilon}{\\sigma}\\right)\\right) \\to 0\n ", "0e7222e80a99c34e702265946cdf2d89": "a_j\\in\\{0, 1\\}^k", "0e725c779db3d4b12630dcb693bc7220": " Q = U \\times Ar \\times LMTD", "0e727e07f1a22829d8c383d1dbf6cdba": "4^n", "0e72abf74276b688bbdfa59bd0145a6f": "\\int_0^\\infty{x^2 e^{-a x^2}\\,dx} = \\frac{1}{4} \\sqrt \\frac {\\pi} {a^3} ", "0e72c5d2edf2edb01928dc5d74b6c48f": " x(t)e^{j2\\pi f_0 t}\\,", "0e72dda469b0e69153f2d21c1339b86b": "h_4", "0e72e09cafc6ecca6c0137849a77a61b": "u_i\\colon L \\rightarrow R", "0e731bc838d321aceb1c971e5abaf573": "\\frac{-\\hbar^2}{2m}(\\frac{\\partial^2 \\psi}{\\partial x^2}+\\frac{\\partial^2 \\psi}{\\partial y^2}+\\frac{\\partial^2 \\psi}{\\partial z^2})+(1/2){m\\omega^2(x^2+y^2+z^2)\\psi}=E\\psi", "0e733e5fd1232f2528da91ff849df269": "\\mathbf{u^2G+c_{s}=e}", "0e734fa649ec5c43f3e8a96ed6adb9c1": "s^2 = \\Delta r^2 - c^2\\Delta t^2 \\,", "0e735e58b73e0a5a273b8b58ee8048e9": "[-1, \\infty)", "0e73a12c1c9c3c5d8550d4662f5ff79e": "\\mathbf{f}=q\\mathbf{V}\\!", "0e73aaf203a49d3cc9353f8fc4dc7a6a": "|A| < |P(A)|", "0e73b435d1723f87d52350c151e31fc1": "23025850 \\approx 10^7 \\ln (10).", "0e73cc33c027f446f40fec4fadb2ba3f": "\\chi(M \\times N) = \\chi(M) \\cdot \\chi(N).", "0e73f768dbe11b43ac9f450a85264df9": " \\|\\cdot \\| ", "0e7416f1c0c0cf352cf1d4b821723e55": "c = 4", "0e742bc418557ddfd00d2958ba05f31e": "3^\\frac{3}{13}", "0e74c220b9a58bb2b72cc939a5a555c4": "\ne^{A t} = B_{1_1} e^{\\lambda_1 t} + B_{1_2} t e^{\\lambda_1 t} + B_{2_1} e^{\\lambda_2 t} + B_{2_2} t e^{\\lambda_2 t} ,\n", "0e74d2dbd5d1b8a82d48c2a0381bddb2": "K<0,", "0e74e4afad10a2db4f09aff522caf089": "\\chi = \\frac{x}{x_c}, \\ \\psi(x) = \\psi(x(\\chi)) = \\psi(\\chi).", "0e7513be5eb465bec0fe47175c7afb7a": "I \\cap (g_1)", "0e75147c76a2b0a3661eca1f30e7108c": "{}+6\\kappa(Y\\kappa_2(W),Y\\kappa_1(W),Y\\kappa_1(W))\n+\\kappa_4(Y\\kappa_1(W))\\,", "0e751ab18bbbdd4626edcadbe0d302a2": "n! [z^n] \\int_0^1 g(z, -1, v) dv = \n\\sum_{\\pi\\in S_n} \\frac{\\sigma(\\pi)}{\\nu(\\pi)+1}.", "0e753814d7ed0c1d83367b180f4c1f07": " \\frac{D_g v_g}{Dt} + f_{0}u_a + \\beta y u_g = 0 ", "0e7569e786f99b6bc676a30fcdb7ae11": "argmax_{y} \\sum_i w_i \\phi_i (x,y) - \\sum \\rho_i C_i (x,y)", "0e7595504877f2ef7aa7d79d26512ab2": "P_i \\not= \\overline{P_j}", "0e759ad13c17fc177622696641580de0": "\\log|1/\\Gamma(z)|", "0e75c9efc65ddf171ae92c41c9a9d811": "\\mathbf{e}_2 = {1 \\over \\sqrt{40 \\over 25}} \\begin{pmatrix}-2/5\\\\6/5\\end{pmatrix}\n = {1\\over\\sqrt{10}} \\begin{pmatrix}-1\\\\3\\end{pmatrix}. ", "0e765638d5cf299f5b83e436f21a62e7": "i_{abc}(t) = \\frac32\\begin{bmatrix} \\frac23 & 0 \\\\\n-\\frac{1}{3} & \\frac{\\sqrt{3}}{3} \\\\\n-\\frac{1}{3} & -\\frac{\\sqrt{3}}{3} \\end{bmatrix}\n\\begin{bmatrix}i_\\alpha(t)\\\\i_\\beta(t)\\end{bmatrix}.", "0e766e20ab586e5f36ec9ae316ea778d": "((w_1,z_1),\\ldots)", "0e768016cb11e4a08e198ab688569a11": "H \\times A = \\left(A, \\lbrace e_i | \ni\\in I_e, e_i \\subseteq A \\rbrace \\right).", "0e7688c66e7577ad8740cda3ba3ceee0": " e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}} ", "0e7692338d403d82a279343d538c9788": "o(n^2)", "0e769cc6011a52be937df67dbfeef446": "f = 2 \\Omega \\sin \\varphi.\\,", "0e76b7d62be0816562794b11810fd684": "\\begin{align}\n(a)_0 &= 1, \\\\\n(a)_n &= a(a+1)(a+2)...(a+n-1), && n \\geq 1\n\\end{align}", "0e771dcf56ea4607598b22b4d35337dc": "\\, L = M\\, ", "0e776b5ef74f0c97db4689b4b9e6bae6": "T = \\frac{n_1-1}{f_1} - \\frac{n_2-1}{f_2}", "0e77a1489168815d30906c13c068ea59": "F_{\\Lambda|k}(\\lambda|k)=1 - \\frac{\\Gamma(k m, \\lambda s_\\Lambda)}{\\Gamma(k m)}", "0e77a647b5a5930263d25e9d71bb6e7f": "l_\\text{P}", "0e77d46236ae7127395611b87f5607c6": "\\tfrac{1351}{780} > \\sqrt{3} > \\tfrac{265}{153}\\,.", "0e77eab22bef534de433ebdef7de8f33": "dY", "0e780e22992a79b7fa9ab719b1192869": "c_1(V) \\not= 0.", "0e7847bd0023db3a7fd5cfe1bfd2eaae": "X=\\mathcal{F}^n", "0e78d84b4c7794c8842203b24e3b128e": "\\hat{y_j}=h(x_j) \\approx y_j", "0e78ecfa41b1af1c7c1aff112932c801": "\\nabla \\cdot \\mathbf{E} = 4 \\pi \\rho_{\\mathrm e} ", "0e791b0d44ebe091f4f06082320a15b6": "T = a + b \\log_2 \\frac{D}{W}.", "0e7943219b1a626638d7aae7881e1b72": " \\text{MI} = \\frac{\\text{PNP}}{\\sqrt{F_c}},", "0e7955975eb9e08c82eb9230bd25926a": "X \\mathbf{\\operatorname{s}} Y", "0e7a2013cddaf77ef0d962b6c9f445f4": "\\Gamma=\\partial \\Omega", "0e7a22f2ba9cc6801a437bbac55a4b7c": "-302\\pm 2.8%", "0e7a2d7fe661a952bebfc4cfbfd30477": "y_3=-3", "0e7a6d507eeee6158d5811e47583f2bd": " \\; {}_2F_1(a+1,b;c;z)- \\, {}_2F_1(a,b;c;z) = \\frac{bz}{c} \\; {}_2F_1(a+1,b+1;c+1;z)", "0e7a846b2b3af579cc2a173e24fadb45": "x_1^2, x_2^2, \\ldots, x_n^2", "0e7a85794228b8f302e066b928731cc8": " G_\\text{III} = biL_\\text{p} = G_\\text{III}^\\circ e^{{-U^* / k(T-T_\\text{0})} - (K_\\text{g}/T \\Delta T)} ", "0e7a921229386a13d9aea1c8bec8f21d": "(g \\circ f)^{-1} = f^{-1} \\circ g^{-1}", "0e7a963235ccd9827b5eb34131ab7d1b": "I(f,g,h) \\leq I(f^*,g^*,h^*)", "0e7ad70b43873a81aa4a9d782b748489": "\n\\begin{matrix}\nx = \\lambda_{1} x_{1} + \\lambda_{2} x_{2} + (1 - \\lambda_{1} - \\lambda_{2}) x_{3} \\\\\ny = \\lambda_{1} y_{1} + \\lambda_{2} y_{2} + (1 - \\lambda_{1} - \\lambda_{2}) y_{3} \\\\\n\\end{matrix}\n\\,", "0e7ae4b871c5bb3a045b883dca7ff8e2": "\\sigma_{i}^{(n)}", "0e7aee4ad8d55cb6c6b95068735ef07e": "f_1=-f", "0e7af9e2cf8c217ad49efa8a35643328": "\\frac {V_\\mathrm i}{V_{xn}} = \\left ( 1 + \\frac {\\delta Z}{Z_0} + \\delta Z \\delta Y \\right)^n", "0e7b3dea352b4a1671800721d4eb2dcb": "I_{L1} - I_{L2}*0.5 - I_{L3}*0.5 + j*\\frac{\\sqrt{3}}{2}*\\left(I_{L2} - I_{L3}\\right)", "0e7b511e89a9a8d01b831fece25c94d8": "\\beth_\\alpha(\\kappa)", "0e7b85202267a353810a9053cf6e6eba": "\\omega\\notin\\mathbb{D}", "0e7bcfce42627811c1d4d16d7cd4d2d5": " \\phi(-x) \\phi(x)\\,", "0e7be073a8345e1c52de04c0948ffe10": "b_{3} ", "0e7cc0170848276a8b767d2bea2b97a4": "\\displaystyle f'(x_0) = 0.", "0e7cda6cad88d6351c1c73727a869a2e": "\\frac{H_{N,q,s-1}}{H_{N,q,s}}-q", "0e7d2878d4316ea7b314af92acb0e03a": "K_2(R)", "0e7d5a107407ddd2bf8389cb0b95d5c1": "h_t(x,y)", "0e7dacb33a49a3af1c1ec6dd2fe6409b": "\\gamma\\in\\hat\\Gamma", "0e7dccda8d9698d24d86cdaf9bf8b1aa": "\\scriptstyle \\mathcal G\\,\\subset\\,\\mathcal F", "0e7e4bf766655feac26fe2ddc47fb2dd": "\\mathbb{R}\\,", "0e7e4e8f4bed0fbebbb09c42c872b224": "A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\\cdots A_n B_n A_n^{-1} B_n^{-1} = 1", "0e7e57320b041af765bf24ac7a92a362": "P_D", "0e7e968f153c9d186c0303c4e770d4e9": "V_{Th} = I_{No} R_{No} \\!", "0e7eaf8393b157790c06d73ff40ce0c1": "\\mod \\, n", "0e7fc4ef11a3ec0a05fc14e422899621": "\\chi=-30", "0e7fc75092c22ffcaa1d4b1a6470bead": "\\chi^{(\\rho)}\\left(\\mathcal{P}\\left\\{e^{\\int_\\gamma A}\\right\\}\\right)", "0e80372b33db5f1db461b8829dea34f4": " 2 \\cdot 6^2 + 2 \\cdot 6 - 1 = 2 \\cdot 6^2 + 6 + 5 ", "0e8046f4d4248b53fb47cb9d4821a613": "\\mathbb{Z}_{12}", "0e80a87a6dfe47dd63066c421a5bef1e": "\n E (t, T) = E (a_{\\rm T}\\,t, T_0)\\,.\n", "0e80b4379d0c15b15a6ca966eeb69fcd": " T_{ij}^{(2)}=s_{ik}w_{kj} - w_{ik}s{kj}", "0e80e9d18de2044aad15a99744d2ac5b": "(x_1,y_1),\\ldots,(x_m,y_m)", "0e815a3e8a7a4ede8e1273e3d3b732e1": "\n \\sigma_a := \\sigma_g + \\frac{k_h}{\\sqrt{l}} + K\\varepsilon_{\\rm{p}}^n,\n", "0e81a4aa5171a2240b1f29856b8cd00e": "c_o", "0e81cec3a7a1611d7a0ef655e542a3ae": "M_{refl}=\\left( \\begin{matrix}\n r_{p}e^{j\\chi _{p}} & 0 \\\\\n 0 & -r_{s}e^{j\\chi _{s}} \\\\\n\\end{matrix} \\right)", "0e81e19fc5410f6fb1e99fff9e5bf0df": "A\\Rightarrow_{amb}^*B", "0e821792f27eb7013db23cc809852e55": "F(x)=\\sum _x f(x) \\,", "0e828e00b5680078364a6646db15b076": "\\tau^2", "0e82d5bfb5e62cfcc905bcabf32524f4": "\\neg A\\phi \\equiv E \\neg \\phi ", "0e836d236d498955e959318bcbdb7319": "\\lim_{x \\to \\infty} \\frac{e^x+e^{-x}}{e^x-e^{-x}} = \\lim_{x \\to \\infty} \\frac{e^x-e^{-x}}{e^x+e^{-x}} = \\lim_{x \\to \\infty} \\frac{e^x+e^{-x}}{e^x-e^{-x}} = \\dots .", "0e838abef7a32fb45ee727b9dd3a6995": "\\partial=\\frac{\\partial}{\\partial z}=\\frac{1}{2} \\left(\\frac{\\partial}{\\partial x} - i \\frac{\\partial}{\\partial y} \\right)", "0e8398a1ab8ab23b4ee460c8161887a9": "\\mathcal{O}_{\\lambda + \\rho}", "0e83d570202919199cd6861e3a4e9f82": "\\mathbf{r}_{3}\\,", "0e83e694ccff9283de93438758996837": "\\left( x_1,y_1 \\right)\\ = \\left( \\frac{-b_1m}{m^2+1},\\frac{b_1}{m^2+1} \\right)\\,", "0e83ed0ae798208d3a4015f6d02c41d9": "\\det(V(\\lambda_2,\\ldots,\\lambda_n))=\\prod_{3\\le j\\le n}(\\lambda_j-\\lambda_2)^2\\prod_{3\\le i0", "0ea40f3876f15cbbd7948bf38a0767cc": " \\sum_s \\langle X_s(z)\\Phi(z_1,v_1) \\cdots \\Phi(X_sv_i,z_i) \\cdots \\Phi(v_n,z_n)\\rangle\n= \\sum_j\\sum_s \\langle\\cdots \\Phi(X_s v_j, z_j) \\cdots \\Phi(X_s v_i,z_i) \\cdots\\rangle (z-z_j)^{-1}.", "0ea41fdeff58dccbf0cbc3363d7ed4cb": "\\mathcal{Z}(\\mu) \\, = \\, \\sum_{N=0}^{N_{S}} \\, \\exp \\left(\\, \\frac {N\\mu}{k_{B}T} \\right)\\frac {\\zeta^{N}_{L}}{N!} \\, \\frac { N_{S}!} { (N_{S}-N)!} \\,", "0ea4545a40750c2254fee418264394e5": "i=\\{1,\\ldots,n\\}", "0ea499b797fa117da837bda69406d8c3": "s_y(t) = -\\frac{mg}{k}t + \\frac{m}{k}(v_{yo} + \\frac{mg}{k})(1 - e^{-\\frac{k}{m}t})", "0ea58128218c9e6e3382c31b1b386426": "p'_{n}(x_{i})", "0ea6088dd979025a8a95bd13fb5649bb": "\\sigma_{\\text{x},\\text{y}}", "0ea66280e98f872e896d80254f66649f": "~\\hat{\\Theta}~", "0ea6e928a1a396d44cfadaa79f77aa24": " d=- \\left\\lfloor \\frac{m}{2} \\right\\rfloor ", "0ea71c8e2f6e47d54943ab5bf85c0593": "N(t) = N_0 \\left(\\frac {1}{2}\\right)^{t/t_{1/2}} = N_0 2^{-t/t_{1/2}} = N_0 e^{-t\\ln(2)/t_{1/2}}", "0ea72b4ffc1411af2cad90045be51d22": "\\begin{align}\n\\nabla \\cdot \\mathbf{E} &= 0 \\quad\n&\\nabla \\times \\mathbf{E} = \\ -&\\frac{\\partial\\mathbf B}{\\partial t},\n\\\\\n\\nabla \\cdot \\mathbf{B} &= 0 \\quad\n&\\nabla \\times \\mathbf{B} = \\frac{1}{c^2} &\\frac{\\partial \\mathbf E}{\\partial t}.\n\\end{align}", "0ea74eeb53d853501f568548900bb6a5": "V_{\\text{out}} = V_{\\text{in}} \\left( 1 + \\frac{R_2}{R_1} \\right)\\,", "0ea7641bf655d3747f7b93ee1f162888": "\\varepsilon_{FY}", "0ea77b901652b89fda0e00a6deaa5100": " ds^2 = -\\frac{4 m a \\, \\log(r)}{\\pi \\, (1+a^2 u^2)} \\, du^2\n- 2 du \\, dv + dr^2 + r^2 \\, d\\theta^2, ", "0ea77d56ce7fe8633de9e82863bf023a": "y(x) = g(f^{-1}(x))", "0ea7af9a703d10b174cec7d427ac8fd8": "\\mathfrak{P}^{1}", "0ea8222a199c773ccac25000c065b29f": "a_1\\neq 0", "0ea8370254d92d397cac00e9a24b0d5d": "f = \\frac{h}{L}", "0ea89006232ee5198a67fe066f4a4fcb": "\\sigma(q) = \\sum_{n\\ge 0} {q^{(n+1)(n+2)/2}(-q;q)_n\\over (q;q^2)_{n+1}}", "0ea8a45bfadd623a462a0ad2fa2398b4": "\\phi = \\sqrt{ \\frac{\\chi^2}{N}}", "0ea8ee7e7edc006d1506e9c91978a29b": "F_{\\overline{z}}", "0ea904611993a9c83e48f06369decb56": "\\mathbf{\\ddot r}", "0ea91bab06b6e13b1ba12c54d465881c": "\\ | \\psi \\rang", "0ea9255f2f01bb39e43b5a9c22985569": "p\\rightarrow p_\\infty ", "0ea9436afe559a8cf0358f5cd0a4b745": "(a_{n+N})", "0ea98df10970c673ab6ddd8172b4b008": "\\mathbf{y}^1", "0ea9ba1994230367d549f17d25a18fbd": "A(<\\lambda)", "0eaa1b9d4512f660f8e1eb9624672fc8": " |k_f| = |k_i|", "0eaa6c551efd2d5ff66244802addc954": "\\Delta f_{\\mathrm{actual}}", "0eab853da730fe3600c61987eee3002d": "d = \\frac{i}{(1+i)} \\,", "0eac433369f47c36e5875937ec7c17be": "x = 6 \\in Z_{11}", "0eac5ec7886d3a7d662ff0307f6cf978": "\nH_{\\mathrm{grav}} = -\\int_0^R \\frac{4\\pi r^2 G}{r} M(r)\\, \\rho(r)\\, dr,\n", "0eac7603efc07fcae78802f010c80224": "\n\\langle p \\rangle_S = \\frac{1}{2}(p + \\overline{p}),\n", "0eacb2fece59b0e087e845f1f55c7067": "C'(B,\\succeq) ~=~ \\left \\{ x \\in B : x \\succeq y ~ \\forall y \\in B \\right \\}", "0eacc199b1fc9db6afab38ccc227d72e": " X: \\C^n \\rightarrow \\C^n, \\, ", "0eaccda6096d719f3cb13763d10c297e": "d_3\\times b^3+d_2\\times b^2+d_1\\times b+d_0", "0eace2000256d1de6702664352976e11": " F = \\pi r \\lambda \\sin (2\\theta)\\,\\!", "0ead594b15aece8e5aadeb93d13b78ad": "\\theta (0)", "0ead7d3650792a3226b74adc4aae6644": "F(u)=-mh^{2}u^{2}\\left(\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u\\right).", "0ead9b4466b2633025cd99e9faedcb07": "I = \\sum_j I_j", "0eadd2bea2d38de75b20ebaeff25fad4": "D_{\\mathbf{a}}(f \\circ g) = D_{g(\\mathbf{a})}f \\circ D_{\\mathbf{a}}g,", "0eadda835ed7183e777155529bbed555": "H_{Moon}\\!\\,", "0eade14ee80fed4210e48965652baf76": "\n h = \\frac{X^2}{a^2} + \\frac{Y^2}{b^2} + \\frac{Z^2}{c^2} = 1,\n", "0eade55b64f11d713e935351b9434aaa": " {}^\\mathrm{N}\\mathbf{v}^\\mathrm{P} = \\frac{{}^\\mathrm{N}\\mathrm{d}}{\\mathrm{d}t}(\\mathbf{r}^\\mathrm{OP}) ", "0eadfb15d26ff037d56ec5650fedc41c": "\\delta:Q\\times\\Gamma\\rightarrow\\mathcal{P}(Q\\times\\Gamma\\times\\{L,R\\})", "0eae10467cf4176e3ac165c77fdce3ad": "\\scriptstyle 1/f", "0eae39a6c136aa55e51cbc81a413ce0d": "t_{th}", "0eae599a5797c4f24b9d8bde86ef434b": "q = \\infty", "0eae5cdc121f487d0428983593133c63": "\\Gamma=0", "0eaf57088b3bfe3d6038300067ed2f51": "A = b \\cdot h .", "0eaf9b08cbadf1e158810847cba8f84e": "\\overline M = 0", "0eafc8452643a846ba7b67a4f7570af6": "\\begin{align}\n \\operatorname{ev}_A\\bigl(p(t) I_n\\bigr) &= \\operatorname{ev}_A((t I_n - A)\\cdot B) \\\\\n p(A)&= \\operatorname{ev}_A(t I_n - A)\\cdot \\operatorname{ev}_A(B) \\\\\n p(A) &= (A \\cdot I_n - A) \\cdot \\operatorname{ev}_A(B) = 0\\cdot\\operatorname{ev}_A(B) = 0 .\n \\end{align}", "0eafcfd59844a23aeb530fbcf1f44346": "\\frac{N^2 + N}{2}", "0eafde8ccfb1918122abaefe9eb3e305": "\\Delta_6", "0eafee684a7fd2ca9f99533b805ba871": "\\frac{x}{y} = \\sum\\frac{1}{2a_i+1},", "0eb01a1012ba4254e5648ff757c8a8bd": "4N-1", "0eb039e9fa57bc62a2ffd651e4e06d2f": "0 = \\frac{1}{2} + \\cos x + \\cos 2x + \\cos 3x + \\cdots.", "0eb0423bffb0ce0a63efafbeab654932": "u_1(\\mathbf{q})", "0eb0b8ed9242ea5a66e657a41508339c": "x(x^2+y^2) = cx^2+dy^2. \\, ", "0eb0d988746ad9b46b3a64ffbf3af50e": "\\scriptstyle{\\hbar}", "0eb0e5797824b9acc1b6c778994a2f06": "{{\\Delta}E_t(S_{t + k})} / {S_t}", "0eb115e04e172ce70a740f80d7c89222": "\\not\\rightarrow", "0eb13082fcc7fcc7941c377a4c7a8b51": "Z_{F'}(x_1, x_2, \\dots) = \\left( \\frac{\\partial}{\\partial x_1} Z_F \\right)(x_1, x_2, \\dots)", "0eb1701a503a9c9abefb212f4c384a2b": "\\mathbf{TK}q = \\mathbf{T}q", "0eb1b3584d3fb2bf38b4efe1eb497c2d": "\\{ z : \\lVert z \\rVert = a^2 \\}", "0eb1d13b31aff26e8224411572059dba": "(x)_n=x(x-1)(x-2)\\cdots(x-n+1) \\,", "0eb1e2be5ee93cd7950a7aafdf6b2a89": "\\frac{1}{1-z}", "0eb1fbfd143fc6d52de70f0efd612cdc": " F'' + F = f.\\, ", "0eb21bb38a1b60e5754d211e8d13b0b3": "\\textstyle p(c_j) \\sum_{f_i \\in F} \\sum_{k=1}^m p(f_{ik}|c_j)^2", "0eb2980d4f992c8c1729fcdbc256f284": "\\{A_x\\}_{x\\in[0,1]}", "0eb2c84f5eca0bc90216498b830257c5": " S^{2n}", "0eb2d613657ce34a03da3119a30a594c": "U \\setminus A = A^C\\,\\!", "0eb2de6ace0d4e2a9238f4ad175420da": "\\theta([x]) = t(x)", "0eb2fa3888181ea09b14ad17ec1d9c49": " \\mu=c+\\varphi\\mu+0,", "0eb30a3d05c82c4f7fbd37fe2648841b": "m = 1/R,", "0eb32500513c88a447c2338109f2aaa4": "\\scriptstyle f (x) = K \\left(\\sum_i w_i g_i(x)\\right) ", "0eb329a32751aee64c2f7bc8ff03c566": "w(x_1,x_2)", "0eb398dd11a17572e69fa184063bcea0": "1/[y_1, y_2] = [-\\infty, \\infty]", "0eb3d001233bf2fc4d1a5540f8f5325a": "1/\\Gamma(z)=\\lim_{n \\to \\infty}\\frac 1{n!}(1-(\\ln n)z/n)^n\\prod_{m=0}^n(z+m).", "0eb41e837b0d3ad5ebf45624d37d5460": "x_n/\\|x_n\\|", "0eb43228eee2e4a595fad4870fccd8ca": "\\int_\\Omega \\mathrm{d}\\omega = \\int_{\\partial \\Omega} \\omega \\ \\ \\left( = \\oint_{\\partial \\Omega} \\omega\\right).", "0eb4d83fb3d823251fbbac5272ad5b57": "\\mu_1, \\dots , \\mu_K", "0eb4de4b5c3827fb484f61d84319dfd2": "a=a_\\mathrm{now}", "0eb4f14757345f3cd4beb47aafe6dfa4": " |\\mathbf{a} \\times \\mathbf{b}|^2 = \\sum_{1 \\le i < j \\le 3} \\left(a_ib_j-a_jb_i \\right)^2 = (a_1b_2 - b_1a_2)^2 + (a_2b_3 - a_3b_2)^2 + (a_3b_1-a_1b_3)^2 \\ . ", "0eb5283e42b54f63368336ea6e10207c": "\n\\begin{align}\n& {}\\qquad \\Pr\\left(\\limsup_{n\\to\\infty} E_n\\right) = \\Pr(E_n \\text{ infinitely often}) \\\\[8pt]\n& = \\Pr\\left(\\bigcap_{N=1}^\\infty \\bigcup_{n=N}^\\infty E_n\\right)\n\\leq \\inf_{N \\geq 1} \\Pr\\left( \\bigcup_{n=N}^\\infty E_n\\right) \\leq \\inf_{N\\geq 1} \\sum_{n=N}^\\infty \\Pr(E_n) = 0.\n\\end{align}\n", "0eb5bd083ed58ada145a9a7196faf5fb": "\\displaystyle{[L(X),L(X^2)]=0.}", "0eb5d648979927ee4d679ea0c7de75b8": " \\sum_{n \\geq 1} m_n^{-1/(2n)} = \\infty~.", "0eb61e8e89cee4ac5fa3a5164cb59da1": "e^{\\mathbf AT} \\approx \\left( \\mathbf I +\\frac{1}{2} \\mathbf A T \\right) \\left( \\mathbf I - \\frac{1}{2} \\mathbf A T \\right)^{-1}", "0eb63cdc93bb79c713c1b86349092aa4": "e=1.602", "0eb6e913c491a43adaf99142a44265f4": "M\\models\\varphi(f_{\\varphi} (a_1, \\dots, a_n), a_1, \\dots, a_n)", "0eb754a1719a4719448f41a8e873108f": "\\frac{1}{2s}\n\\left[1+\\cos\\left(\\frac{x\\!-\\!\\mu}{s}\\,\\pi\\right)\\right]\\,", "0eb7af267e698ebfe78bc1d36ec78864": " \\psi(\\bold{r}+\\bold{T})=\\psi(\\bold{r}) ", "0eb7bb2bea16904f84eef7f327b57249": "\nK = \\frac{1}{2}MV^2\n", "0eb81bd405fe4c05a74b21aeb967d63f": " p=\\sqrt{X^2+Y^2} ", "0eb8895828648ae2fa2dbe6796544711": "\\bar u = \\frac {\\sum_{i=1}^m \\sum_{j=1}^n \\mbox{no. of defects for } x_{ij}}{mn}", "0eb8b5873e73b4b256d76d73a6ec141d": "\\theta \\in \\Lambda_{C}^{1}\\pi_{r+1,r}\\,", "0eb8ee51188335d5b3ab0e7a20d06721": " u(x,t)", "0eb8f56cf96b43ee9144a1b1ae516f3c": "\\log{f\\left(r\\right)}", "0eb95a603f7419146e20d3d1d327d2ae": "g=hk", "0eb95c27a97267cf22187a1752ce15fc": "\\mathbf{B} \\cdot d\\mathbf S = 0", "0eb97b28473a065278ab4438044b2cbd": "f^{\\mathbb C}(v\\otimes z) = f(v)\\otimes z.", "0eb9d7c11d7bb3a752a70ab6770b210a": "O(3)\\,", "0eb9deaa5ba8237b13a2d795e1e735d0": "\\tau_\\mathrm n", "0eba172d4f1cb7974554f710f3c15702": "\\Omega_x", "0eba632a840229c752b62f25f548adbb": "\\mbox{sgn}(\\sigma)(e_1e_1e_2e_2\\cdots e_ne_n)", "0eba633b4a911169fe71d83cd2b11423": "\\hat n^\\prime", "0ebb16a4dff5342da84dee626d7bc3c8": "\\textbf{j}=(1, \\dots ,1)", "0ebb1da5cbe57827cbc723d4eca97bf9": "g_{i_1p_1}g_{i_2p_2}\\cdots g_{i_np_n}g^{j_1q_1}g^{j_2q_2}\\cdots g^{j_nq_m}A^{i_1i_2\\cdots i_n}{}_{j_1j_2\\cdots j_m} = A_{p_1p_2\\cdots p_n}{}^{q_1q_2\\cdots q_m}", "0ebb3263600be14283905ab9a80b608c": "\\overline{CD}", "0ebbfb9956a1db270c6abe264dcb20ff": "N_{tot}", "0ebc56270b64b4e41ce5ab48aef96eba": "(\\lambda \\setminus \\left\\{d\\right\\}) / (\\mu \\setminus \\left\\{c\\right\\})", "0ebc768a76bb99d7dede45f1b19b5db0": "\\left[ \\frac{1}{2} (n^2 + n) \\right] T_6 + \\left[ \\frac{1}{2} (n^2 + 3n) \\right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \\le cn^2,\\ n \\ge n_0", "0ebc8b4aa86723972807c3259f80081d": " \\mathbf{A}\\!\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\cdot\n\\end{array}\\!\\!\\!\n\\mathbf{B} = \\sum_j\\sum _i \\left(\\mathbf{a}_i\\times\\mathbf{c}_j\\right)\\left(\\mathbf{b}_i\\cdot\\mathbf{d}_j\\right) ", "0ebcc88f2451cd1f59df012e2ad8e66f": "f(\\pi)=\\pi-1 = VT/D - 1 = 0 \\, ", "0ebcde5ead75a432ff5bdff12be06750": "\\overline{r \\rightarrow s} \\leftrightarrow (\\overline{r} \\mathbin{\\rightarrow} \\overline{s}),", "0ebceb912a0ded74e744c896e373be54": "\\phi(x,n)", "0ebd32cedc48ff57eac9ba729f2dedbe": "B(\\mathbf x_0,t^{*\\ast})", "0ebd56200c66e74b8ac68160aa45c5e4": "g(\\lambda\\mathbf{a}+\\mu\\mathbf{a'},\\mathbf{b}) = \\lambda g(\\mathbf{a},\\mathbf{b}) + \\mu g(\\mathbf{a'},\\mathbf{b}),\\quad\\text{and}", "0ebd710ed3f4a44d1c716103072cc1d8": "{B_g}^2 = \\frac{\\nu \\Sigma_f - \\Sigma_a}{D}", "0ebdd7ecd225770f6d5cb6cf32393fdc": "\\chi_A\\,", "0ebe8373df4420551bf5a02e1c5d0589": " \\and T_{10} = [F_{10}, S_{10}, A_{10}]::[F_9, S_9, A_9]::L ", "0ebe898685cbb12412c857b0ddf0d942": " P \\,", "0ebea3e2c34b811fc470b266b0d15782": "\\vert{\\Psi_{\\mathbf{p}}^{(\\pm)}}\\rangle = \\vert{\\Psi_{\\mathbf{p}}^{\\circ}}\\rangle + G^\\circ(E_p \\pm i\\epsilon) V \\vert{\\Psi_{\\mathbf{p}}^{(\\pm)}}\\rangle", "0ebee93cf97e14f0e1607d08c88af44c": "Z^\\alpha_i", "0ebf15fab8504ae10fa58561274c01be": "Trips = a + b ln (Area) ", "0ebf7abe98f9df8368065ecbc96e11ce": "\\langle\\cdot\\rangle", "0ec0ea4784a13001945265f32cf33e77": " u (x) = \\frac{1}{(2 \\pi)^n} \\iint e^{i (x-y) \\xi} \\frac{1}{P(\\xi)} f (y) \\, dy \\, d\\xi.", "0ec10cefa3fd19348d43e5daf6aaa8f4": "\\mathbb P(T(y)>t) = \\Phi\\left(\\frac{y-\\mu t}{\\sigma t^{1/2}}\\right) -e^{2\\mu y/\\sigma^2}\\Phi\\left(\\frac{-y-\\mu t}{\\sigma t^{1/2}}\\right).", "0ec1287eff4c1d7c0a941a838addc21a": "_{p}", "0ec12bf327a3585fd5f64657b9fcb4c5": "\\lim_{x\\to\\infty}N^{-x}=\\lim_{x\\to\\infty}1/N^{x}=0 \\text{ for any } N > 1", "0ec186827925aba30f82debbc7af6d47": "D = C \\oplus \\nabla", "0ec1c590b5337e876a71ec9125484f0f": "Ob", "0ec1f9a10ddacd21beb1b90b1f78197d": "\\int xdy = xy - \\int y dx", "0ec1fd2fe41393140048a18c255a70c1": "\\sum_{k=0}^\\infty k n_k = n ", "0ec26e0b3d41f0b9abeda4cd852351a7": "s_2 \\leq t_2", "0ec2aa080ebb37b1eaa058f9bab09b66": "\\frac{p_1\\cdot V_1}{T_1\\cdot n_1}=\\frac{p_2\\cdot V_2}{T_2 \\cdot n_2} = constant", "0ec2bf8227c09c739045105578da1301": "a\\,P(X\\in[a;a+da])", "0ec2cc1749e80ae632dadacef9aca98c": "{u_{ij}^{n+1}-u_{ij}^{n+1/2}\\over \\Delta t/2} = \n\\left(\\delta_x^2 u_{ij}^{n+1/2}+\\delta_y^2 u_{ij}^{n+1}\\right).", "0ec2cfd176b6c3aad4aace67330c0d43": "\\mathfrak{k}^n,", "0ec30d3a9169df113a8f2070b84ba303": " M(M-1) d \\leq \\frac{1}{2} n M^2", "0ec32ff534e449bd0b1a54a264512207": "\\mathbf{MTF_{sensor}}(\\xi,\\eta) ", "0ec35c6714dffb0bc052982d0d6342a2": " \\and S_7 \\implies A_7 = p ", "0ec36571f70b0345be34c1d77173346c": "J_{1k}", "0ec3bdb2d0ab8fc23fa6af566ae046ab": "E(g_i^{-1}g_j) = \\delta_{ij} 1", "0ec3ec0e1b8132f23d046f7acd4e5ad7": "\\partial_t (f_t(z)) h^\\prime(f_t(z))= \\mu e^{\\mu t} h(z)=\\mu h(f_t(z)),", "0ec3fe2fd44add93e95a70a0e8ea4329": " {} = -4 \\cdot (-4) \n -(-8) \\cdot (-8) \n +(-12) \\cdot (-4) \n +(-4) \\cdot (-12)\n -(-8) \\cdot (-8)\n +(-4) \\cdot (-4) ", "0ec418f9c266219ec2fc6892197e8c4f": " e(p_0,v(p_0,w+EV) = e(p_0,u_1) ", "0ec455fcaeb0560387593e685c6e4118": "\\,\\,\\sigma_{ij} = 2\\mu\\varepsilon_{ij} + \\lambda\\varepsilon_{kk}\\delta_{ij} + \\lambda'\\,\\varepsilon_{ik}\\,\\varepsilon_{kj}", "0ec4b8a5b91e0ed523fce5e73b5f096b": "\n\\frac{A_{n-1}}{B_{n-1}} \\approx \\frac{A_n}{B_n} \\quad\\Rightarrow\\quad \n\\frac{A_{n-1}}{A_n} \\approx \\frac{B_{n-1}}{B_n} = k\\,\n", "0ec4ee2d22e6e65b90e4fa0208c90190": "\n\\begin{align}\n\\sum_{k=1}^\\infty \\frac{1}{k^2}\\left(1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{k}\\right)^2 = \\frac{17\\pi^4}{360}.\n\\end{align}", "0ec5bbdf6a503d1e92ff7962eaaac0e9": " \\iota_{[X,Y]}=\\mathcal L_X \\iota_Y-\\iota_Y \\mathcal L_X. ", "0ec6d2ba1223048e8b289d76aa74d0d7": "\\scriptstyle(x_0,x_1)", "0ec7704560dcaa2dec9da4c63ed5ad92": "AI_T = 100\\times\\frac{d}{n}", "0ec7829f6bb626aa266bc8b6c752d366": " {(3/2)^4 \\over 2^2} = {81/16 \\over 4} = {81 \\over 64} \\approx {5 \\over 4} = {5 \\cdot 16 \\over 4 \\cdot 16} = {80 \\over 64}. ", "0ec788e46e02fa5b65722bb563f2162f": " \\psi\\left(\\frac{1}{6}\\right) = -\\frac{\\pi}{2}\\sqrt{3} -2\\ln{2} -\\frac{3}{2}\\ln(3) - \\gamma", "0ec7a9b5256fbee66a791ec70c88ee6c": "\\mathbf{S}_{j}", "0ec7d815383c3810635d8782a201009c": " V(n) = (1-P)^n \\,", "0ec82a0a81033b59075155f9d0fd00c9": "{\\mathbf{v}}_n({\\mathbf{k}}) = \\frac{1}{\\hbar} \\nabla_{\\mathbf{k}} E_n({\\mathbf{k}}).", "0ec8382bc2e59b199f8d5fe125b5da64": "\\bar{\\boldsymbol\\omega}", "0ec84b534799c3dff2eb8ab8874aa716": " \\sqrt{ m_f c^2 / V} ", "0ec8900ebf65161f341ae7bb10febbff": "S(\\omega) = E(\\omega)W(\\omega) \\, ", "0ec8f2c925ac1636f04c351cb5290308": "(g^c)^a = g^{ca}", "0ec9aeabff7320cea63e64c3d9797f59": "\\operatorname{erf}^{-1}(x)", "0ec9d46d25e751c96b5b3beef5f1ac45": "f = \\frac{n_1}{n_0} = \\frac{2(1 - a)}{a}", "0ec9e6875e4c6e6702e1b81813a0b70d": "B:1", "0eca1aecaf6c0328ef50d8ab6b579376": "2 + 15\\sqrt{\\frac{2}{17}}", "0eca4aad9f6de63434e73c6a583cdb45": "O(n^{2+\\epsilon})", "0ecaa4dc9275d64ce26ff1ab4cfb2b4d": "R={(n-1) \\over \\phi},", "0ecae8568f75b2269e53630285776138": "X \\Rightarrow Z.", "0ecb2700dabebda5fccf228cb4fbf171": "\\eta_{GX}", "0ecb4b6317baf46cb7a2240545f28107": "x = \\hat x x_\\text{scale} + x_\\text{shift}", "0ecbd394fa38e6c1512f8e017d83e280": "t=t_\\mathrm{now}+\\lambda_\\mathrm{now}/c\\,.", "0ecbead79a995072ed0c664a53cef2ff": " \\left(E_n^{(0)} - E_k^{(0)} \\right) \\langle k^{(0)}|n^{(1)}\\rang = \\langle k^{(0)}|V|n^{(0)} \\rangle ", "0ecc3fda4595563dda26f7e36c9789f4": "(S_a f)(z) = f(z+a)= \\exp (a \\partial_z) ~ f(z) ", "0ecc72384da5d90e5e16bd5149acbdd1": "\\lim_{x \\to \\infty} \\frac{\\pi(x,a,q)\\phi(q)}{x/\\log x} = 1.", "0ecc934f5a973ff364f77fe46cac2ca3": "k_4(s) = u_0 + l_1 s^1 + l_2 s^2 + u_3 s^3 + u_4 s^4 + l_5 s^5 + \\cdots \\,", "0ecc9d58ea7385c1bd00a6446a707e01": "\\, \\! f=\\frac{1}{2\\ln(3)RC}", "0ecca81206d9341099b254a8ec0b69bf": " A+1=1", "0ecce93dedf82b925357742dedd3c189": "\n \\varepsilon_{xx}^{\\mathrm{topface}} = -z~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} -\\left(z - h - \\tfrac{f}{2}\\right)~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2} ~;~~\n \\varepsilon_{xx}^{\\mathrm{botface}} = -z~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} -\\left(z + h + \\tfrac{f}{2}\\right)~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2}\n ", "0ecd0835f3f532af0ce7a90a18c6fd6e": "\\displaystyle{|\\beta|={b\\over |\\alpha+\\alpha^{-1}|}={|\\alpha|^{-1} -|\\alpha|\\over 1-\\delta} > |\\alpha|^{-1} -|\\alpha|.}", "0ecda0976ce67796e1989cd9a3eb6f9f": " y \\ = \\ {\\log (x) \\over \\log(b)} ", "0ecdf23207e45eaa19859ecf6b5c49e0": " m = \\mu + \\nu ", "0ecdf30b118b10d69b3e85c36dd8e323": "1 + c_{n+1} = (1 + c_n) (1 - c_n/2)^2 \\,\\!", "0ece1664733cb0255dffbdd4d06d9654": "\\sum\\limits _{m}\\left[p_{m}\\nabla^{2}q_{m}-q_{m}\\nabla^{2}p_{m}\\right]=\\sum\\limits _{m}\\left[\\nabla\\cdot\\left(p_{m}\\nabla q_{m}-q_{m}\\nabla p_{m}\\right)\\right].", "0ece4b923fe95e8a309279f49649d168": "{n(5n^2-5n+2) \\over 2}", "0ece67573f5fc0400f2c65818b169579": " 5.6%", "0ece9c02190ce10d3600e89a796e2fd5": "\\Omega_{\\perp}\\tau=\\pi", "0ecea947b5c57ab62a754ff9286d7849": "(\\sigma >0)", "0ecf32e5d1127f2804d30f10c050eecf": " \\frac{b-a}{24} (11 f_1 + f_2 + f_3 + 11 f_4) ", "0ecfb6c09c20db0d526e72fe7c24ad2a": "\n\\hat{\\varepsilon}(\\omega) = \\varepsilon_{\\infty} + \\frac{\\Delta\\varepsilon}{1+i\\omega\\tau},\n", "0ecfd6e83303dd5b0cf3fc4c5db989e0": "\\theta_1+\\theta_2=\\theta_3+\\theta_4=90^\\circ\\;", "0ed0442ea254842e1dc623f2dbb706cb": "\nX(t) = \\sum_{n=-\\infty}^\\infty e^{2\\pi i nt / T} X_n\n", "0ed0868d9ae61951946a25ec61a58a16": "n^{\\Omega(1)}", "0ed0bebdc5b5d5c768912b4165659c61": "\\frac{\\partial}{\\partial \\tau}y^k\\frac{\\partial}{\\partial \\overline\\tau}", "0ed142e2a90bf0065034c8d2f8ac63cc": "\\lambda_+", "0ed18fe290d9a1af0845e5870a9f2d7b": "G^{\\alpha\\beta}", "0ed1aa40537c18854f570601252c351f": "\\rho_{s}\\,", "0ed1bb07211cc10f4a50ab8c1742bd6e": "\\text{max}_i \\text{ min}\n\\{d(x_i ) + 1, i\\}", "0ed26e610e062b54a93ef2723b4dee36": "\\{\\mathbf{x_i} \\leftrightarrow \\mathbf{x_i}'\\}", "0ed26fc4d8fb4f32a72dcf9ae8f58ec4": "f : X \\to \\mathbb{R}", "0ed2c2a9542b29bf43054e6df3ac4a40": "\n\\bar{\\Phi} \\left[ \\mathbf{r} \\right] = \\frac{1}{2} \\int d \\mathbf{r}\n\\int d \\mathbf{r}' \\hat{\\rho} (\\mathbf{r}) \\bar{\\Phi} \n( \\left| \\mathbf{r} - \\mathbf{r}' \\right| ) \\hat{\\rho} \n(\\mathbf{r}') - \\frac{1}{2} n N \\bar{\\Phi} (0). \\qquad (3) \n", "0ed2efea5afd7380a81da0052837e3cf": "r=1,2", "0ed37d3e2811fc9224aac747ccf2b731": "|\\mathit{before}\\rang ", "0ed3abcc3853fb85bcab9873a6aef42e": "c_K=1.151 c_B (d_{BU})^{-0.5}\\,", "0ed3f1170c3833b5f4837d47f586d80b": "\\int\\mathbf{f} = \\left(\\int f_1,\\,\\dots, \\int f_n\\right).", "0ed4237587c3da5dabdf15501e0e4e0f": "\\omega(x)=1/\\sqrt{1-x^2}\\,", "0ed45c17a2c820cd87fc9ba6dd4ccc3c": "y = \\frac{\\sqrt 2\\sin(\\phi)}{\\sqrt{1 + \\cos(\\phi) \\cos\\left(\\frac\\lambda 2\\right)}}", "0ed4fab98433d204d4710264cddd9a68": "\\partial_t \\rho = -w d\\rho_0/dz", "0ed547de0c8db395d6065e7fe6f99851": "\\nu(A) = \\sum_{n = 1}^\\infty w_n \\frac { \\mu (A \\cap V_n) } {\\mu (V_n)} ", "0ed5aea834e1a3f60b154dfa9f512882": "A_o=\\mathrm{min}_x(\\mathrm{max}_{S_o}(U(S_o,x)-T_oS_o)) = \n\\mathrm{min}_x(A_o(T_o,x))\n", "0ed5b2ae271063d16122ea149f76a85c": "\\mathcal{M}(P(x))=1.176280818\\dots \\ .", "0ed5d7bcf12581d86bf0c6ff6011f0fb": "co(ci, x, y) = M_0 M_1 M_2 M_4", "0ed646b67fe7cd6c54e32e199acdded1": "m\\frac{du^{\\tau}}{ds} = e F^{\\tau \\sigma}u_{\\sigma},", "0ed646cdd4eaecf57a0080659f51ad98": "\\mathbf{Wv}", "0ed663e2b5710eefd700a198c0752a8d": "N = M^{k}", "0ed6705acdd00af7708b96fe7f52d7b9": " C = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3{B'}^3 ", "0ed748968b473c78a269ab07e5d09bd2": "E_{\\mathrm tc}", "0ed76d1800949b99e254688920ce92e7": "L = {mg\\over{\\cos \\theta}}", "0ed775167e5c7887ef5b798190d3ed1b": "y=-1", "0ed781e2fe28c2a05577d4fcb080472f": "mol Al = \\frac{8.00 g}{26.98 g/mol} = 0.297 mol\\,", "0ed7b37cbfa9b2f866c4a6df84dd1fcd": " r = \\sqrt{(a t + b)^2 + c^2} ", "0ed814303885052c91b1ab408b9f27ff": " \\sum_{k=1}^n {n-1 \\choose k-1} = 2^{n-1}.", "0ed83380783162190fbcd90fb7aa8892": "xy - yx \\ne \\mathbf{1}", "0ed8f065ac012b25a8085814f03e2ae9": "\n\\gamma= \\lim_{n \\rightarrow \\infty} - \\frac{\\ln N(T)}{T}\n", "0ed96ca5d9cae4cb98875cb4e4b704e2": "(\\boldsymbol{\\sigma}\\times{\\boldsymbol{k}})", "0ed9f4b9d22f8074b83ad3331d54b57b": "\\frac{\\sqrt{3}}{6^5} \\sum_{k = 0}^{\\infty} \\frac{((4k)!)^2(6k)!}{9^{k+1}(12k)!(2k)!} \\left( \\frac{127169}{12k + 1} - \\frac{1070}{12k + 5} - \\frac{131}{12k + 7} + \\frac{2}{12k + 11}\\right)=\\pi\\!", "0eda0d7540824ac98a394fc759a932f4": "f(x) \\in B", "0eda365b536bec1bd78a7337561918e5": "r_2 = $36.67 * 3.9 = $143", "0eda70d6900607af8b18e75b47106df4": "!n = \\left\\lfloor(e+e^{-1})n!\\right\\rfloor-\\lfloor en!\\rfloor , \\quad n\\geq 2,", "0edac5e3cfa9812a2fb87852d30e76ed": "X=(X_t,\\ t\\in I)", "0edae8dec406acd8a0fc1b63a10810e6": "\\frac{\\Delta E_x}{\\Delta y} - \\frac{\\Delta E_y}{\\Delta x} = 2L'\\frac{\\partial H_z}{\\partial t}", "0edaf460f3bc9297904bb0d4ed85aa2b": "\\varepsilon=\\mu^{-1}\\sigma^2=\\kappa_1^{-1}\\kappa_2, \\,", "0edb12d8bef2ed0d0bab91c4ef569df4": " f(\\boldsymbol{\\sigma},\\boldsymbol{\\varepsilon}_p) = 0 ", "0edb76fe6c9280fff4b4557075f55fda": "\\gamma = (t_{ox}/\\epsilon_{ox})\\sqrt{2q\\epsilon_{si}N_A}", "0edba0d61ce70a0fd0d34a2050a5043e": "\\textstyle s>0", "0edbd92c7d47dd8bc471f3a61b2f8dff": "2^S", "0edc278bf60452487b33872bd8e32291": "\n\\frac{d\\mathbf r}{dt}=\\mathbf v,\\qquad\\frac{d\\mathbf v}{dt}=\\mathbf F(\\mathbf r,\\mathbf v,t), \n", "0edc54b87a77d6d1ee703bf8c2272213": "G= \\begin{bmatrix}\n1 & \\dots & 1 & 0 & \\dots & 0 \\\\\n\\ast & \\ast & \\ast & & G' & \\\\\n\\end{bmatrix}", "0edc68bc802e369920d69cb090a9ea4b": "\\scriptstyle p_c \\,", "0edccb953a137f27a52a69fd72a3d0a6": "\\mathbb{H}^{\\bullet}", "0edd0e9ac1e5cd291807a66a81016e7a": "\\mathbf{x}(t) \\triangleq \\begin{bmatrix}x_1(t)\\\\x_2(t)\\\\\\vdots\\\\x_{n-1}(t)\\\\x_n(t)\\end{bmatrix} \\in \\mathbb{R}^n", "0edd18309ba3f2b38f6fa8ce6be11eb2": "\\operatorname{Spec} A_i", "0eddb1c5ac349c6b0e4105604acb4905": "\\mathbf{S}\\cdot d\\mathbf{A} + \\int_V\\mathbf{J}\\cdot\\mathbf{E}dV", "0eddb5d6c96d704e54155ab661e7e308": " \\mathbf{N(s)X(s)} + \\mathbf{M(s)Y(s)} = \\mathbf{1} ", "0eddefd18f4e88079ea756e371965dcd": "S(p/q) = \\frac{I_{p/q}(2/q)}{I_{1+p/q}(2/q)},", "0eddf6cb023b65a580f569bb8cdf1001": "\n\\begin{align}\nS_B & = n(\\overline{Y}_1-\\overline{Y})^2 + n(\\overline{Y}_2-\\overline{Y})^2 + n(\\overline{Y}_3-\\overline{Y})^2 \\\\[8pt]\n& = 6(5-8)^2 + 6(9-8)^2 + 6(10-8)^2 = 84\n\\end{align}\n", "0ede64db6d7538cae16dcf2d5b59a591": " h(-r)=h(r), \\ \\vert h(r) \\vert \\leq M \\left( 1+\\vert \\Re(r) \\vert^{-2-\\delta} \\right ),", "0ede7929d658edab751c7667b22f158f": " ax=b \\,", "0edeab0f57af0df5f6c2ca0f014701fd": "\n\\frac{dr}{dt} = \\alpha r\n", "0edef6f74c6fedc43a450cd88b398866": "S_{n} = X_{1} + \\cdots + X_{n}.", "0edf461ed9d813d53b6d0a1e1efc7b32": "m \\dot x = \\frac{ \\partial H }{ \\partial v } ", "0edf97e42e0295df0ec452351d3c1c90": "\nf_i(p) \\in [y_i]\n", "0edfbaf65e24d4db501fcf3a8b2e99d6": "\\tilde a", "0ee03f592af0190f77a1274ebc5431be": "\\beta_0+\\beta_1x", "0ee0872605988295f8a5ece95a7e27b6": " \\langle A | B \\rangle = \\left( \\, \\langle A | \\, \\right) \\,\\, \\left( \\, | B \\rangle \\, \\right)", "0ee0c2a4abf7bdb1935bb2bfa867050a": "U_0\\cap\\cdots\\cap U_k\\ne\\emptyset\\,\\!", "0ee0eee0acf84ea91e59ab0803351baa": "+\\varepsilon", "0ee11ea708afd96b74b79cd2fd744853": "\\gamma \\in \\Gamma_g(N),", "0ee13b5dbfaa24408d659cbe134d5bcc": "S(y)", "0ee15d5fa9f07c1d646885626fbbd38a": "\\frac{\\overrightarrow{DC}}{\\overrightarrow{DB}} \\times \n \\frac{\\overrightarrow{EA}}{\\overrightarrow{EC}} \\times\n \\frac{\\overrightarrow{FB}}{\\overrightarrow{FA}} = 1,", "0ee1bdb71f255a6a8281017c43451f98": "\\vec{f}_0 = \\partial_T, \\; \\vec{f}_1 = \\partial_X, \\; \\vec{f}_2 = \\partial_Y, \\; \\vec{f}_3 = \\partial_Z ", "0ee2320b3516e72e3a91ab12d0b3a122": "P \\lor (\\exists x Q(x))", "0ee26194cbb26ae2beb47079f365fba3": "\\{ 1, 2, \\ldots, n \\}.", "0ee28892ec70ca17ef0e8f81f4ff294f": "e_p(x) = 0", "0ee2c3781ce63ae5c02ef7eb5cdc4873": "|| \\Phi(t) P \\Phi^{-1}(s) || \\le Ke^{-\\alpha(t - s)}\\mbox{ for }s \\le t < \\infty", "0ee2ff3e3c0e579a843b7a06e5de2544": "s \\models_K \\mathcal{P}_{\\sim\\lambda}(f_1 \\mathcal{U} f_2)", "0ee32875a74d8115b278f69ea964add1": "\\hat{X}(z)=\\hat{X}_{Bayes}\\left(\n\\Pi^{-\\top}\\mathbf{P}_Z\\odot\\pi_z \\right)", "0ee368f975a90e2cf638b1deb6837c6f": "\\Theta(\\theta)", "0ee38e81a17473b55bc575f2d550cd9a": " \\operatorname{let} x : \\operatorname{get-lambda}[x, x = \\lambda f.f\\ (x\\ f)][x:=x\\ x] \\operatorname{in} x\\ x ", "0ee446231316d6a668ea1014cb043e67": "x_{i,1}", "0ee4646802c4ab236a205caccde1f0ca": "i(e,n)", "0ee47777604e68e889857383ebc0e8fa": "\\mathbf{b_{1}}=2 \\pi \\frac{\\mathbf{a_{2}} \\times \\mathbf{a_{3}}}{\\mathbf{a_{1}} \\cdot (\\mathbf{a_{2}} \\times \\mathbf{a_{3}})}", "0ee4b1217318e94d4e6ee3dcc1653336": " \\frac {A}{H} = \\frac{ \\frac {k_c}{m+m_A}} { \\omega^2 - \\frac {k_c}{m+m_A}} ", "0ee4c8cdbf49efbb065a11e64045dddb": "\\frac{L}{L_{\\odot}} \\approx {\\left ( \\frac{M}{M_{\\odot}} \\right )}^{3.9}", "0ee4ea2d2548299e7d4d951d6e2a049b": "\n\\begin{array}{rl}\n\n\\text{Ax. 1.} & \\left\\{P(\\varphi) \\wedge \\Box \\; \\forall x[\\varphi(x) \\to \\psi(x)]\\right\\} \\to P(\\psi) \\\\\n\n\\text{Ax. 2.} & P(\\neg \\varphi) \\leftrightarrow \\neg P(\\varphi) \\\\\n\n\\text{Th. 1.} & P(\\varphi) \\to \\Diamond \\; \\exists x[\\varphi(x)] \\\\\n\n\\text{Df. 1.} & G(x) \\iff \\forall \\varphi [P(\\varphi) \\to \\varphi(x)] \\\\\n\n\\text{Ax. 3.} & P(G) \\\\\n\n\\text{Th. 2.} & \\Diamond \\; \\exists x \\; G(x) \\\\\n\n\\text{Df. 2.} & \\varphi \\text{ ess } x \\iff \\varphi(x) \\wedge \\forall \\psi \\left\\{\\psi(x) \\to \\Box \\; \\forall y[\\varphi(y) \\to \\psi(y)]\\right\\} \\\\\n\n\\text{Ax. 4.} & P(\\varphi) \\to \\Box \\; P(\\varphi) \\\\\n\n\\text{Th. 3.} & G(x) \\to G \\text{ ess } x \\\\\n\t\t\n\\text{Df. 3.} & E(x) \\iff \\forall \\varphi[\\varphi \\text{ ess } x \\to \\Box \\; \\exists y \\; \\varphi(y)] \\\\\n\t\t\t\n\\text{Ax. 5.} & P(E) \\\\\n\t\t\t\n\\text{Th. 4.} & \\Box \\; \\exists x \\; G(x)\n \n\\end{array}\n", "0ee55d794f17e382f8980ebd1adcf211": "\\Delta (t).", "0ee599876ea529b25749b16ad57e86c2": " (\\ddot r - r\\dot\\theta^2)\\hat{\\mathbf{r}} + (r \\ddot\\theta + 2 \\dot r \\dot\\theta) \\hat{\\boldsymbol\\theta} = \\left (-\\frac{\\mu}{r^2}\\right )\\hat{\\mathbf{r}} + (0)\\hat{\\boldsymbol\\theta}", "0ee5e5a0affa7b765858efea96611f62": " \nV_n(x_1,\\ldots, x_n)=\\sum_{i\\exp(1/\\mathrm{e})", "0eebbae6d53eec48e94abbd5f1051ceb": "T_\\text{s}", "0eebc6779815507fbde89d4852e65337": "P(i|s_j) = \\frac{P(i)p(s_j|i)}{p(s_j)}", "0eebef778df80f071321658c5c428db1": "\\textstyle v", "0eec1197ecf816c5bab696c64594491c": "x,y \\in M", "0eec2cc5ecada971045e34620070f7dd": "\\check{H}^{*}(X;A)", "0eede95a493fc729cc19afceefbdf4b1": "p_{B}", "0eedf80c1dae3ef6cc4996e1d2da41e0": "A_{\\gamma-norm}", "0eee2fc28c40b7b24b5514b60c6a3fa3": "a_0=-1", "0eee4b1b73a7073779fc70dfd2febf68": "(a,0) \\circ (b,1)=(a + b,1)", "0eeed4f4a72ac653419603e778a37f8f": " ds^2 = \\rho (du^2 + dv^2)", "0eef586a2d9ef3503cd41c7f4338b7f9": "\\rho_\\text{perm}(r)", "0eeff2c7b49ef6ec59883c9686d3b3fa": "\\bar{X}_P, \\bar{X}_N", "0eeff349dbe28511462ce4daa7904b4e": "\\sin x = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots ", "0ef07d555b1f79a0fd62ea633b584a8c": "F_{\\mathrm{Anchor}} = \\frac {\\mathrm{Weight}} {2 \\cos(\\frac 1 2 {\\theta_\\mathrm{Bottom}})} \\approx \\mathrm{Weight}\\times 0.5 + O({\\theta_\\mathrm{Bottom}}^2)", "0ef07d7c7e409b1e69ccdde2c9df1042": "c/R", "0ef0bf7c8ea22fb54e995ff882972e7c": "\\begin{align}\n\\left( \\int_R + \\int_M + \\int_N + \\int_r \\right) f(z) \\, dz &= 2 \\pi i \\left( \\mathrm{Res}_{z=i} f(z) + \\mathrm{Res}_{z=-i} f(z) \\right) \\\\\n&= 2 \\pi i \\left( - \\frac{\\pi}{4} + \\frac{1}{16} i \\pi^2 - \\frac{\\pi}{4} - \\frac{1}{16} i \\pi^2 \\right) \\\\\n&= - i \\pi^2. \n\\end{align}", "0ef0de55df063c5d53bea8fffbcc95ba": "x \\propto \\sqrt t", "0ef0e13322d1d789c2396e89df1ab7c3": " \\mathbf{r} = \\left( r + R_0 \\cos\\theta \\right) \\mathbf{e}_r - R_0 \\sin\\theta \\mathbf{e}_\\theta ", "0ef10eefc4d96e2911d20a2263a3180e": "a_i > 0 ", "0ef134486d3a3012f2eb99f526d1cd25": "a^{-1}+b^{-1}=c^{-1}", "0ef13611c7bb7573ea79b51dfd3cc540": "\\frac{20\\varphi^2}{3}", "0ef14bfaeb767ee868194c783d99099f": "X_n\\ \\xrightarrow{p}\\ Y", "0ef178116dd59cfc7c4a667fcd96e583": "\\breve\\theta_j= h^{-1}(s,\\breve z_1^j, \\ldots,\\breve z_m^j)", "0ef1cdb14e471d556fdeba8feb761e71": "{\\it{K \\ll N}}", "0ef2101f275ca4f95f311320a6ed7d83": "L_{\\text{r}}(\\omega_{\\text{r}})", "0ef232ece43e3aea83e7acb3a20ee377": " x^n (x^4-x^3-1) = -(x^4+x-1) ", "0ef2cb651f6d31b5d32f89c05207eb23": "\\scriptstyle\\overline{\\mathbb{Q}}", "0ef302b82db7d48f8072bca5738db8cd": "f(1) = -\\tfrac 1 2 (1-1-1-1) = 1, f(i) = -\\tfrac 1 2 (i-i+i+i) = -i, f(j) = -j, f(k) = -k ", "0ef329f38c93680ff747a65b33bc993d": "a = 2\\arctan \\left\\{ \\tan\\left(\\frac12(b-c)\\right) \\frac{\\sin \\left(\\frac12(\\beta+\\gamma)\\right)}{\\sin\\left(\\frac12(\\beta-\\gamma)\\right)} \\right\\},", "0ef3b61f17457ab10cd1d299bc448d13": " (x,y,z) = (4 \\cos \\vartheta, +3+5 \\sin \\vartheta, 3 \\cos \\vartheta) \\,\\!", "0ef3c9f2756d0f1d870db34b925037f9": "h(k_2)", "0ef488c2c0a63879ff015daf0b260196": "( \\lambda x .t ) s ", "0ef4ab58edc89f1ecd9a4241e4dc3d87": "{\\mathbf{v}}_n({\\mathbf{k}}) = \\frac{1}{\\hbar} \\nabla_{\\mathbf{k}} E_n({\\mathbf{k}}), ", "0ef4b0799abb7cbaadffb54ae5d6479f": "\n\\begin{align}\n \\qquad \\frac{x}{84} & = \\frac{3}{2}, \\\\\n {84} \\cdot \\frac{x}{84} & = \\frac{3}{2} \\cdot {84}, \\\\\n \\qquad {x} & = \\frac{3 \\cdot 84}{2}, \\\\\n \\qquad {x} & = {126} \\\\\n\\end{align}\n", "0ef50e7d25b4821a71f4bfb2f5b9606e": "\nn^2 = n_1^2 + n_2^2 + n_3^2.\n", "0ef55c818acd22814738d42716125193": " p = kT \\frac{\\partial \\ln Q}{\\partial V}\n", "0ef565c20297a1b2eb143db7936cc63d": "(x-3) (x-2)^9 (x-1)^{18} x^{10} (x+1)^{18} (x+2)^9 (x+3) (x^2-6)^{12}", "0ef5bcbd2e8dc44fee945fa8e9dbf0a0": "\n \\hat{x}_{\\mathrm{fl}}(t_j) = -\\frac{\\hat{\\phi}_j}{2 k_p}\n", "0ef648e26b0e2aa05230dba5cb097a02": " - x_1x_5x_6x_7 ", "0ef672b64161cba04d542b30995895e0": "x^2 - 2y^2 = 0. \\, ", "0ef68273cb4b7933668abaab86701d1a": "\\lambda x_s^*", "0ef6d776c2a05cd278364a75f349e622": "x+S(x)=(1,1,1,\\ldots)", "0ef6f67a7e5f705a94bc71f101921250": "h=\\{h_n\\}_{n\\in\\Z}", "0ef73008b57d4b71ddeab2efa3b39cd5": "(\\mathbb{Z}S,\\partial)", "0ef75bc3c2227172bb22d931991ed806": "U_e^2 = 0", "0ef8079040fa6b9ab7a0d248b2ef3dd9": "q(n,t)", "0ef8ae60eea281dcdb4b6016b2045016": " P = P_1 \\begin{bmatrix} I & 0 \\\\ 0 & P_2 \\end{bmatrix}\\begin{bmatrix} I & 0 \\\\ 0 & P_3 \\end{bmatrix}\\cdots \\begin{bmatrix} I & 0 \\\\ 0 & P_r \\end{bmatrix}", "0ef8f309cd59de106dce18bff6cf3998": "u=m(S\\otimes \\text{id})(\\mathcal{R}_{21})", "0ef933b93605a459e87fe63f3abf33af": "e^{ix} = \\cos x + i\\sin x \\ ", "0ef98851438e1c3b9983222f4f479fb6": "\\pi = 3.1459...", "0ef98868107285f53fc0131b51c3d232": "s_i = ln(r)", "0ef9abca75a25203ad92dc53e6a1b8c8": " \\frac{\\partial \\boldsymbol{F}}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\frac{\\partial \\boldsymbol{F}_1}{\\partial \\boldsymbol{F}_2}:\\left(\\frac{\\partial \\boldsymbol{F}_2}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} \\right) ", "0ef9b6a8231cc0f7586f7ece8e4b4b3c": " \\Delta W = P \\Delta V ", "0ef9cd6d614e159e77c82faa9e1e7793": " \\scriptstyle b(x+e)-(x+e)^2=bx+be-x^2-2xe-e^2\\; \\sim\\; bx-x^2. ", "0ef9e3eb60826d8df291b2d958435ffd": "\\Psi^k (\\rho) = N_k(\\Lambda^1\\rho,\\Lambda^2\\rho,\\ldots,\\Lambda^d\\rho) \\ ", "0ef9f5d21a51b5f2d4de1a9552522a3c": "\\scriptstyle \\{V_{k}-U_k,k\\geq1\\} ", "0efa0e81552b8709489c7abc4b7e8c30": " \\sigma = 1.661687949633594121296\\dots\\;", "0efa34689d3c4d4c40187804e30fa63c": "(*)\\qquad a_1x_1^r+\\cdots+a_nx_n^r=0,\\quad a_i\\in K,\\quad i=1,\\ldots,n", "0efa76cd29585aa1d733d872fadb52e4": "7/8", "0efa813aeb3b56b6d97cc534d229e351": "d(x)=[[x]](x)+1", "0efac8bccb03a57a4e822317e2498f47": "\\psi_{1}= \\phi_{1}+e_{B}-e_{C}", "0efaff8c84eb1a9dce594c56ffa0a1af": "\\Delta v = v_\\mathrm{rel} \\ln \\frac {m_0} {m_1}", "0efb61223cf94255be128c750a839e58": "D_4^-", "0efb66636f1cdbe8813790a141ca2c77": "CM+\\alpha G", "0efbde7501dec5cbf1cb358503cbe134": "\\Delta_d", "0efbeaf74629bc34826bae9a002ad92b": "S : H \\to H", "0efbf0885f3e6692ef3edce7f49b7cd1": "\\begin{align}\n&\\mathit{CV}_{i,j,k-\\tfrac{1}{2}} h^m_{i,j,k-1} +\n \\mathit{CC}_{i-\\tfrac{1}{2},j,k} h^m_{i-1,j,k} +\n \\mathit{CR}_{i,j-\\tfrac{1}{2},k} h^m_{i,j-1,k} \\\\\n&+ \\left(\n - \\mathit{CV}_{i,j,k-\\tfrac{1}{2}} - \\mathit{CC}_{i-\\tfrac{1}{2},j,k} - \\mathit{CR}_{i,j-\\tfrac{1}{2},k}\n - \\mathit{CR}_{i,j+\\tfrac{1}{2},k} - \\mathit{CC}_{i+\\tfrac{1}{2},j,k} - \\mathit{CV}_{i,j,k+\\tfrac{1}{2}}\n + \\mathit{HCOF}_{i,j,k}\\right) h^m_{i,j,k} \\\\\n&+ \\mathit{CR}_{i,j+\\tfrac{1}{2},k} h^m_{i,j+1,k}\n + \\mathit{CC}_{i+\\tfrac{1}{2},j,k} h^m_{i+1,j,k}\n + \\mathit{CV}_{i,j,k+\\tfrac{1}{2}} h^m_{i,j,k+1}\n = \\mathit{RHS}_{i,j,k}\n\\end{align}", "0efc06fcab5210a83d659bae0f38abd6": " K_{il(m+1)}=1 ", "0efcaa4ee592761dfbaac353df2b7e36": "[0,1).", "0efd20a8c567fb28d43428ee20e2bbcb": "\\phi_Y", "0efd61bc41390992a77f80a5a06c47bf": "x \\in {0,1}", "0efd788024567518a5b4293e0931ae9b": "n = 2k+1", "0efdc919754730e8a1d775867dd9a765": "EL(\\Gamma)=EL(\\Gamma_0\\cup\\hat\\Gamma)\\ge EL(\\Gamma_0)", "0efdd35ff55faf6c57660f8f2fabe06a": "1 \\over 1", "0efddb396a169aae735d4c6338bbd573": "\\tilde X \\to X", "0efddecf476bfe08bf5614cecb579e81": "D_{\\odot}=8 \\cdot 10^{22} T m^3", "0efe3aa0265316cfd541fa2d8ddec772": "X^*_{\\gamma}", "0efe603d75586124a0c9063f2efa1ee9": "\\operatorname{succ}(C,\\langle \\alpha,\\beta,\\gamma,\\delta \\rangle)", "0efe7438c4655ff1db613dc403c879c8": " P_{CJ} ", "0eff006a60fd5cc965f8cb0c1deb0abf": "G(z, v) = \\left.\\frac{d}{dv} (Q_1(z,v)-Q_2(z,v))\\right|_{v=1}", "0eff7f098e5d3f54f62d64451e7d73f2": "{s_4}", "0efff695b0b58001158ce4db0a4b2bcc": "\\psi_\\alpha(z) = \\left| f(z) \\right|^\\alpha", "0f00b3c10d0376443564d5f3caee0d94": "\\mathbf{p}_1", "0f00bc2c7720fa6ad525f36dcdeb1e12": "\\tbinom n r p^r q^n \\!", "0f00c9c3df8cec18099ead98b0fe798f": "R_\\odot", "0f00f1ac31b2a114d7604981bf0b4506": "(\\Theta+w_1)\\cap(\\Theta+w_2)=\\{w_1-w_2,0\\}", "0f016364aa5f908606195967b5720016": "(12)\\quad \\psi_{SS}=\\frac{1}{2}\\ln\\frac{L-M}{L+M}\\,,\\quad \\gamma_{SS}=\\frac{1}{2}\\ln\\frac{L^2-M^2}{l_+ l_-}\\,,", "0f0220afee2fda153a57d4917ccc9c73": "P_C(t) = \\sum_{j=0}^n \\dim(C^j)t^j.", "0f02415893f392ae3a49b7f094582894": "\\ A \\oplus B", "0f027be980543d6a3190da0644af525e": "\\theta= \\pi/2", "0f028a9b2ff22fc19f614142846f2b60": "p = \\dot{f}\\left(x_0\\right)", "0f02d2af8d4cb3398befb330521499aa": "T_\\text{H} = \\hbar g/(2\\pi c k_\\text{B})", "0f032e6603cd3acec91e101e4151608a": " p \\left( x \\right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 ", "0f039bc7f9fbb9d104edfd36ff8e1a8f": " y_{2c} = \\left ( \\frac{q_2^2}{g} \\right )^\\frac{1}{3} = \\left ( \\frac{50.0^2}{32.2} \\right )^\\frac{1}{3} = 4.27 \\text{ ft}", "0f040012f32ccda8df006d37de843dca": "vx/c^{2}", "0f04410c5d06eb16acee9bd6dba56da1": "\\nabla^2 \\tilde{E} + n^2 (\\omega) k_0^2 \\tilde{E} = 0", "0f044fd89a02f0faaa33f601cd4456c6": "(x_1, x^2_1),\\dots,(x_n, x^2_n)", "0f04760baf5558d4bbe75cf61baf6c3e": "\\,\\! E(p_1,p_2,m):=p_1x_1^*(p_1,p_2,m)+p_2x_2^*(p_1,p_2,m) ", "0f0479dde21b538e20613c236524e8bd": "\\alpha_{m_1, \\ldots, m_a,k}", "0f04830c34e7cc214636ea1943b847ff": "| \\psi_1 \\rangle ", "0f04f756ef2ffd214dfa965190df0ea5": "1, 4, 13, 40, 121, \\ldots ", "0f0531115e3e77f256c4793f6b5b7cba": " \\mathbf{p}_{\\rm i} = m_i \\mathbf{v}_{\\rm i} \\,\\!", "0f0548e3edbe197a085783f43c886fd7": "\\Delta f\\Delta t \\ge 1", "0f05ed8a80db39a44f2a36e3add4ce6d": "F^g = \\mathbb{I}+[\\lambda^{g}-1]f_{0}\\otimes f_{0} \\,", "0f065de05a3af4d3b9d35512c5b8fb1b": "\\epsilon = -1,0,1", "0f068c855c612be613d98074d8104729": "n_H", "0f068e12955a8bb3921ff41539198f29": "L = 20\\ \\log_{10}\\left(\\frac{4\\cdot \\pi \\cdot d \\cdot f}{c}\\right) ", "0f0692c6a127aeb0927f9560b5d0957c": "t^+(X)", "0f0732ee7778c809bc99b171b67bb183": "\\psi\\, {\\sim}\\, \\psi + 2\\pi", "0f073743b3bcca827a11bfdbbff36525": "L_{1} \\pitchfork L_{2}", "0f076a371caa07eec622be5168e5063f": "\n[u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)] =\n[u_{2}(+1,-1)+u_2(-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)] \n", "0f076a79e69ae714cb4f6e082f1452ff": " MA = \\frac{T_B}{T_A} = \\frac{r_B}{r_A}.", "0f076f8b744e97a21de9fe1a95a8472b": "43^8+96222^3=30042907^2\\;", "0f07873018bf1a703d5afadfd6970ea4": " d(t) ", "0f07c7495a7216ac745bb0e93b9042ef": "R = \\begin{bmatrix}1/2 & -1/\\sqrt{2}\\\\\n-1/\\sqrt{2} & 2\n\\end{bmatrix},", "0f07d5fcda7e3b7c2421c5abf2e3f009": " V_f", "0f081a488e5950f3db1dff195e784251": "\\frac{\\partial}{\\partial z_1}, ..., \\frac{\\partial}{\\partial z_n}", "0f092a85c54621235644012feb10b93e": "I(2\\omega,l) = I(\\omega,0)\\tanh^2{\\left(\\frac{E_0\\omega d_{\\text{eff}}l}{n_\\omega c}\\right)}", "0f095189b6b24dfc311fb7cb95426510": "\n\\begin{align}\n \\int x\\cos (x) \\,dx & = \\int u \\, dv \\\\\n & = uv - \\int v \\, du \\\\\n & = x\\sin (x) - \\int \\sin (x) \\,dx \\\\\n & = x\\sin (x) + \\cos (x) + C,\n\\end{align}\n\\!", "0f0977808653871ae4a6844920bad475": "\\langle 0|\\Phi^\\dagger(x)\\Phi(y)|0\\rangle", "0f09f0a472a0128d7648ba178a3ef497": "g(x, y, t) \\geq 0 ", "0f09f7f67aaff7917507b729ca4475fb": "\\sqrt{\\scriptstyle{R(R-2r)}},", "0f0a198fa4bfaca64e402dec5c7432c6": "(\\mathbf{A}+\\mathbf{A}^{\\rm T})\\mathbf{X}", "0f0a6fff8b02a5f891baedc44d75db3a": "\\hat{G}_j |\\psi \\rangle = 0", "0f0a803810eee22c12ffe5fda15c68fb": " M_t ", "0f0ad94c353a22cb807dec55d5804ccf": "ax^3+bx^2+cx+d=0.\\,", "0f0ae744d0a1c066d4e7909bf74b8047": "\\scriptstyle {p_i-1} \\,>\\, {p_{i-1}}", "0f0b2f09c81ae61048bb5dcbfd62f374": "\\,\\! G_D = G_m A_D \\tanh(L_D) / L_D", "0f0b647a6d76b42ee1f3c9318356cd0a": "\\ Z\\in\\,\\Gamma", "0f0ba9e9598750f90d669b0e93f473ae": "x(\\alpha - \\beta y) = 0\\,", "0f0bb9f9d04a17a54202e85cb5e2e518": "s_1=s_2=1", "0f0bc587de127b293209cbed48446924": "h=e-f", "0f0bc80015e3f6b8ec2013685387aa07": "k_B T C", "0f0c270d43cf35e63758f8e587e5d8e2": "a_1 \\equiv a_2 \\pmod{n}", "0f0c65c2fea532dde8a1fd0c56c6ccf7": "Z^0_{e}", "0f0c8fadf0bc978fc4b57422dfd06eae": "\\kappa = \\rho_0 c_0^2 ~.", "0f0d089095143c98b63de7fa90778416": "{\\mathcal{A}}_{i_{n}=i}", "0f0d472568fd30411f161b15937203c8": "\\Gamma=\\frac{V_R}{V_F}\\,", "0f0d877106d519c364414428ed6c8343": "\\textstyle x!\\{{n\\atop x}\\}\\,", "0f0d999e448e9d8ad5620a349374e05d": "\\sum_{k=0}^\\infty (-1)^k k! = \\int_0^\\infty \\left[\\sum_{k=0}^\\infty (-x)^k \\right]\\exp(-x) \\, dx", "0f0dc63e1f0389e7906126b17bd3d363": "B(a, b)", "0f0dd321a08735ebfd9622515597fc3e": "R \\times R \\stackrel{m}\\to R", "0f0dfad96415df06399d5889f925e884": "(x,z)\\in R_{i}", "0f0eef7027405e6f329f95badada1bea": "C - P = D(F - K) = S - D K", "0f0f088f38f4ff29f23b1485d1037801": "B_{\\lambda}(T) \\neq B_{\\nu}(T)", "0f0f2b960c4a5542f476c18e6ce1cf72": "\\left\\{ Q_{\\alpha i}, \\overline{Q}_{\\dot{\\beta}}^j \\right\\} = 2 \\delta^j_i \\sigma^{\\mu}_{\\alpha \\dot{\\beta}}P_\\mu", "0f0f408101834f1e9fe619a9b93fa289": "\\int_1^{\\infty} \\left( \\int_1^{\\infty}\\frac {x^2-y^2}{\\left(x^2+y^2\\right)^2}\\ dx \\right)\\ dy = \\frac{\\pi}{4} \\ .", "0f0fa540d7e1cb650a0556a76b2d6edd": "d(y,z)+d(x,y)\\geq d(x,z)", "0f102b13d5901c3496616db74fc186e3": "\\langle 1/n_{i+1} \\rangle \\subseteq \\langle 1/n_i \\rangle", "0f102ea0f824fa658f2cdb21c083c06f": "\\bar{u}(T)=\\frac{0.860117757+1.54118254 \\times 10^{-4}T + 1.28641212 \\times 10^{-7} T^2}{1+8.42420235 \\times 10^{-4}T + 7.08145163 \\times 10^{-7}T^2}", "0f1054ee3143dafa62f0ef3bd496790a": "\\scriptstyle \\varphi(\\alpha) \\,", "0f1064238c29a38c14fd70abd3efeb4f": "\\omega(\\lambda) = W(\\Phi_\\lambda,\\Chi_\\lambda),", "0f106db4509c14c554251e3dbd751c4a": "{}_pF_q(a_1,\\dots,a_p;c_1,\\dots,c_q;z) = \\sum_{n=0}^\\infty \\frac{(a_1)_n\\cdots(a_p)_n}{(c_1)_n\\cdots(c_q)_n}\\frac{z^n}{n!}", "0f10a24f958a56e8a29ea76f8d3c9d62": "\\displaystyle{L(a,b)=2([L(a),L(b)] + L(ab)).}", "0f10c55dd6b23de94ce63ac8620f6a93": "a_{r}=\\frac{(c)_r(c+1-\\gamma )_r}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}a_{0}\\quad \\forall r \\ge 0", "0f11322364697effa291c09cbb4b329b": " T^0 ", "0f113d2a51cfa347752c80b86ccdaf4c": " \\text{The particle horizon } H_p \\text{ exists if and only if } N>2", "0f11a79b80fc384f5fc8bf0c1b982015": "-\\frac{c^4}{8 \\pi G} \\frac{v_s^2 (y^2+z^2)}{4 g^2 r_s ^2} \\left(\\frac{df}{dr_s}\\right)^2,", "0f11ac423daf13c71340554654c1f774": " \n{1 \\over 2}\n\\begin{pmatrix} \n1 & 1 & 0 & 0 \\\\ \n1 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & 0\n\\end{pmatrix}\n\\quad \n", "0f11e991ac9a2e24be89fd5b6fc73522": "0 < |a| < 1", "0f120720fca7dc520b10b29c78724048": "e^{2.5}", "0f12112fad912306098877356a13460e": "R \\, exp (i S /\\hbar)", "0f1211b4dca7a3046a352d3ca0fc7922": "\\frac14\\begin{bmatrix}\n1+c&a-ib&\\pm (1+c)&\\pm(a-ib)\\\\\na+ib&1-c&\\pm(a+ib)&\\pm (1-c)\\\\\n\\pm (1+c)&\\pm(a-ib)&1+c&a-ib\\\\\n\\pm(a+ib)&\\pm (1-c)&a+ib&1-c\n\\end{bmatrix}", "0f12224ab3e145715557d1856b6e7d27": "4a^2x^2 + 4abx + 4ac = 0", "0f126900012f02725a504d249bd78a10": "(\\mathbb{Z}_6, +)", "0f1270a0b146db3c08cb6bbd545bb785": "D\\overline{D}A = -J", "0f12cd3827b8b5e1ad8ade89ead4e378": "\\langle C_\\xi : \\xi < \\alpha\\rangle \\,", "0f13444fc91bddebf3da47ff668a633b": "f^{(i)}(0)", "0f135e01a8d4d50018a46c4051f79c3e": "h(\\cdot)", "0f136d4a5ddfc2d79894a45600d12934": "\n\\begin{matrix}\n\\beta_0 & = \\beta_0^{(0)} & & & \\\\\n & & \\beta_0^{(1)} & & \\\\\n\\beta_1 & = \\beta_1^{(0)} & & & \\\\\n & & & \\ddots & \\\\\n\\vdots & & \\vdots & & \\beta_0^{(n)} \\\\\n & & & & \\\\\n\\beta_{n-1} & = \\beta_{n-1}^{(0)} & & & \\\\\n & & \\beta_{n-1}^{(1)} & & \\\\\n\\beta_n & = \\beta_n^{(0)} & & & \\\\\n\\end{matrix}\n", "0f136f00ead8866abe9a7ea038843828": "\\sqrt{\\tfrac{32}{5}} = \\sqrt{\\tfrac{16 \\cdot 2}{5}} = 4 \\sqrt{\\tfrac{2}{5}}", "0f1371682cd4c8a8e9a1be893361676a": "\\sum_{j=1}^{D-1} p_j^2 = 1.", "0f13f3c39690d785a4a1d04c50b0f1f8": "(x, y, z)", "0f141a44688b80e310d7959ec6ae93ac": " \\frac{1}{3} ", "0f1438efb3841f09489d9ab8bd19e004": "\n \\gamma = 1 + \\frac{M_\\mathrm{orb}}{(M_\\mathrm{spin}+M_\\mathrm{orb})}\n", "0f1484c7ec1f75de907fdf1d3d5a73d8": "\\Delta_{KN}\\,:=\\, r^2-2Mr+a^2+Q^2\\,,\\;\\; \\rho_{KN}\\,:=\\,r^2+a^2\\cos^2\\!\\theta\\,,\\;\\;\\Sigma^2\\,:=\\,(r^2+a^2)^2-\\Delta_{KN} a^2\\sin^2\\theta\\,.", "0f1495edf029c94f6e2d799433c86b01": "k \\to k[X_1,\\ldots,X_n].", "0f150e200abc8fd48aa0cf3adac855bc": "m \\leftarrow N", "0f1529f45f27a0a59079bba2adc07d8f": "F_\\mathrm{e} = \\frac{\\varepsilon_0\\varepsilon_rAV^2}{2d^2}", "0f158ae8922e991854741f120ada1f98": "(l_k,l_k,m_k)\\,", "0f15cf873eeb280fdc5d799347af871f": "\\gamma_{\\xi} = -\\frac{1}{4}\\xi+\\frac{1}{8}\\xi\\left (2\\chi_{\\xi}^2 + \\mathrm{ch}2\\chi + 1\\right ).", "0f15eb788b44e710faa9b3e94ad71e85": "U_{L, 0} = \\mathcal{O}_L^\\times, U_{L, i} = 1 + \\mathfrak{p}^i", "0f1632cbfe9047c43071f9922adcc684": "\\frac{1}{2} C \\omega^2 = -\\frac{1}{2} C U^2 \\left(\\frac{2k}{rd}\\right) \\theta^2", "0f1667fcf4945b65ed8e562246210463": "Y=\\{-(1+x):x\\in X\\}", "0f1670cd146f6acc74d478f311d74b0d": "n \\mathbin{:} {\\mathbb N}", "0f16aabf3789de8261bbaa408b02037c": "n_e = N_{\\rm C}(T) \\exp((E_{\\rm F} - E_{\\rm C})/kT), \\quad n_h = N_{\\rm V}(T) \\exp((E_{\\rm V} - E_{\\rm F})/kT),", "0f16ac37fade9b3e37b91246bac75b19": "\\rho _w", "0f174fa569af15a5c309f7a6ab4ae9ce": "S+E+I+R=N", "0f177369a3b71275d25ab1b44db9f95f": "SG", "0f177b64ba415650092993ad7ffd08bf": "m_{n}=\\int_I t^n\\,\\rho(t)\\,dt,", "0f17909efaa6c4600b27b53e1464afb8": " \\angle DOC = \\angle EOC - \\angle DOE. ", "0f180b6b6420caeb7a3ac04d7673d097": "\\forall i \\exists j:\\; j\\geq i \\quad \\rho(w_j) \\in F.", "0f18b0719a8a25c84045838eed640d5d": " \\operatorname{drop-param}[(g\\ q\\ p\\ n), D, V, \\_] ", "0f18d177552f76439714df1a070557ef": "n_R", "0f18e1e5b5e7572e9ae0771f74f505de": "\\psi=\\begin{pmatrix}\\psi_{1}\\\\\n\\psi_{2}\\\\\n\\psi_{3}\n\\end{pmatrix},\\overline{\\psi}=\\begin{pmatrix}\\overline{\\psi}^*_{1}\\\\\n\\overline{\\psi}^*_{2}\\\\\n\\overline{\\psi}^*_{3}\n\\end{pmatrix}\n", "0f1931a673d4a539da6ab4ebe1a89c12": "\\tilde{g}_{33}=r_{0}^2 \\sin ^2 \\theta", "0f195c56b2ea0e2d3dbbdd06985642a1": "z^n-y^n", "0f19c7a357993d0b24f33526aae81e6c": " Z_{in} = Z_C \\, ", "0f1a0bf001f82a76496bc5b40d6efaf7": "C^*", "0f1a0f019de9f84c8da32d01bdae9be1": " \\rightarrow D_n", "0f1af1f75945c10f599368811e2d8a64": "\\frac{1}{8}", "0f1af5f00bf02e3b287ede795dd97000": "x_i y_j = x_j y_i", "0f1b0622732b9a85b94f0276c68bcbe2": " T^{-1}_{\\mathbf{v}} = T_{-\\mathbf{v}} . \\! ", "0f1b46270f6500d1db212c0b97fcf23d": "\\mathcal{O}_X \\to k(x) \\xrightarrow{F} k(x)", "0f1b81d9663ffffd7db0fc133954caff": " \\frac{Y}{X} = \\frac{(s+z_1)(s+z_2)}{(s+p_1)(s+p_2)}. ", "0f1c584c74c18e33b7af55168a49c417": "I_o = \\frac{\\left(V_i - V_o\\right)DT\\left(D + \\delta\\right)}{2L}", "0f1c5de67149c89cfb36bdce0db33fdb": "L_{\\text{o}}(\\mathbf x,\\, \\omega_{\\text{o}},\\, \\lambda,\\, t) \\,=\\, L_e(\\mathbf x,\\, \\omega_{\\text{o}},\\, \\lambda,\\, t) \\ +\\, \\int_\\Omega f_r(\\mathbf x,\\, \\omega_{\\text{i}},\\, \\omega_{\\text{o}},\\, \\lambda,\\, t)\\, L_{\\text{i}}(\\mathbf x,\\, \\omega_{\\text{i}},\\, \\lambda,\\, t)\\, (\\omega_{\\text{i}}\\,\\cdot\\,\\mathbf n)\\, \\operatorname d \\omega_{\\text{i}}", "0f1c5ecfd263f374d42b3ed098cdbdbc": "N\\triangleleft\\text{mess}", "0f1cd825b8ef9dfdc6adac0a716ce7df": " c_k = \\frac{\\frac{1}{2}f(b_k) a_k- f(a_k) b_k}{\\frac{1}{2}f(b_k)-f(a_k)}", "0f1d1c919beaad5e662854164f3547ee": " v_{k+1} = U_k (\\alpha) \\,\\!", "0f1d3168bb177f371b7ae01ff6e619da": "(V_r,V_s)", "0f1d457ed91a664e4a8558572a6ca2e6": "m > \\lfloor n/2 \\rfloor", "0f1d50fd14bb478046961dbf4a645642": "\\ln(2)", "0f1d6bb4ba23b6f35e3491360c82f119": "\\sigma_N", "0f1d8f39f3821707300f85643412a8d6": "\\omega = \\frac{(6+\\sigma)^2}{24}", "0f1dc919c903c93b11a16d99c3cb2971": "F:A\\rightarrow B", "0f1e1872b95b8bbc4bf0d8a0eb159198": "\\cos x + \\cos 2x + \\cos 3x + \\cdots = \\sum_{k=1}^\\infty\\cos(kx).", "0f1e44ebca5015f56450922eee10efde": "\nV=\\frac{\\partial}{\\partial\\sigma^2}\\log L(\\sigma^2,X)\n", "0f1e914f26d9342611b4ede510773172": "\\sigma_f^2 = a^2\\sigma_A^2", "0f1f1d95b54202965f0eba7e56ee7fa8": "\\mathbf{a}\\times(\\mathbf{b}\\times\\mathbf{c}) = (\\mathbf{a}\\cdot\\mathbf{c})\\mathbf{b} - (\\mathbf{a}\\cdot\\mathbf{b})\\mathbf{c}", "0f1fbd76ac5335872f5e662dc522ed9d": "\\left(P_2-P_1\\right)", "0f1fcc336ac5b6f2dd73e6eb152031cb": "d(x, y) \\ge 0", "0f20b9bf9ac0229f73b3cafb15befa2a": "\\rho = (U \\otimes U) \\rho (U^\\dagger \\otimes U^\\dagger)", "0f20ed920f547b473a9490757c6cd0e7": " \\tilde{C} ", "0f22779771cafdad017cfaa8e4555d89": "\\vec{H}", "0f229b490c383cf89a339be0147ed7a8": " V_{\\text{out}} = \\begin{cases} V_{\\text{S}+} & \\text{if } V_1 > V_2, \\\\ V_{\\text{S}-} & \\text{if } V_1 < V_2, \\\\ 0 & \\text{if } V_1 = V_2, \\end{cases} ", "0f22f7a85eace6a190c0f03d885e43e6": "H(z_1,z_2)", "0f23029ad700531f836306808d4ee666": "X \\cong \\coprod_{\\alpha\\in A}X_\\alpha / \\sim.", "0f23807a5181625cb24b2199060779cf": "\\sup_{ \\|x\\|=1 } \\|Nx\\| = \\sup_{ \\|x\\|=1 } |\\langle Nx, x \\rangle| = \\max \\{ |\\lambda| : \\lambda \\in \\sigma(N) \\}", "0f2423ae04770ec780adb9336b2cf204": "u_2 \\ll c ", "0f2431b3c780bada1951d1298c2dd2f8": "H=P_\\theta\\dot \\theta + P_\\phi\\dot \\phi-L", "0f24536163dca5c527d38647f4ae098c": "\\! F_m(x) = F_{m-1}(x) + \\gamma_m h_m(x).", "0f2453d40c2dc445a0bb569874fe0f7c": "\\hat{a_2} = \\bar{x}+\\log(f_0)", "0f24cba8048d2c721e29b35c1f3e46e1": " b_S = \\pi_{11} / \\pi_{21} ", "0f24f466b9e8cef89353fb3bb9c0dab4": "\n\\frac{\\partial F^m_{~\\alpha}}{\\partial X^\\beta} = F^m_{~\\mu}~\\cfrac{C^{\\mu\\gamma}}{2}\\left(\\frac{\\partial C_{\\alpha\\gamma}}{\\partial X^\\beta} + \\frac{\\partial C_{\\beta\\gamma}}{\\partial X^\\alpha} - \\frac{\\partial C_{\\alpha\\beta}}{\\partial X^\\gamma}\\right) \n", "0f250b3016fa8a4537b638a45cf27cc1": "w^4 = ~-w", "0f2518c9682fcfa2fe98d2b7ec874b9f": "M_{\\mathbf{\\Xi}}", "0f25cef5194916c365c9e0faf49a85cc": "\\mathit{0123456789} \\!", "0f25e10a4df0807c6ff02b1fc01cf09e": "{{\\int_0^t \\mathrm{ROI}(\\tau) \\, d\\tau} \\over \\mathrm{ROI}(t)} = (-\\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{Q} +V_p) {{\\int_0^t C_p(\\tau) \\, d\\tau} \\over \\mathrm{ROI}(t)} + {{\\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{A}} \\over {\\mathbf{U}_n^T \\mathbf{A} + V_p C_p}}", "0f25eb45cba7dc8c684367d743271ed0": "\nJ_t = \\nabla \\cdot \\left( \\mathbf\\Sigma_i \\nabla v_i \\right) = -\\nabla \\cdot \\left( \\mathbf \\Sigma_e \\nabla v_e \\right)\n.", "0f25f733fe2a1a6091864471db745ace": "v_{2k}Sq^1v_{2k},", "0f260dffee2ee82e5d2117ba2b6ce84a": "\n\\begin{bmatrix}\n1 & 1 &\\cdots & 1\\\\\n\\end{bmatrix}\\mathbf{v}=0", "0f26205e74fb31569deb7ab1b7467a23": " \\oint_C F(s)e^{st} \\, ds \\sim M(e^{t}). ", "0f268d3b3cc0a4fbeb1e9f51e5e956e8": " TE", "0f26a3fef41d6e391d94bc6b51393ced": "\\frac{1}{\\sqrt{2^{n+1}}}\\sum_{x=0}^{2^n-1} (-1)^{f(x)} |x\\rangle (|0\\rangle - |1\\rangle )", "0f275195be149eed09134e989f001d23": "\\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}} = \\frac{\\dot{y}(t)}{\\dot{x}(t)},", "0f27700570ea3004ce2ea222d31d96fb": "\\xi,\\rho", "0f277c55987313a9238709bdcc955949": "n_{\\rm Io} - 2\\cdot n_{\\rm Eu} = 0 ", "0f278ff0e8569e34b9b69f4fc854a1f0": "J\\subset I", "0f27a1ad7136239b0e6426bda09220bb": "\\frac{1}{|G|}\\sum_{g\\in G}\\rho(g^n)", "0f27b441de8eacee243e8e9f479be8c9": "a = \\frac {1}{\\sqrt {2R_c s_o} }", "0f2830a8414078bb2cd39146da67ec8a": "a=c=1", "0f28aa8408af50619c1f9e1e68f1bcca": "\\mathcal{L}=\\mathcal{L}_w", "0f28d55798465cb3d30259441a48c367": " \\chi^2 = \\frac{ [b - (b+c)/2]^2}{(b+c)/2} + \\frac{ [c - (b+c)/2]^2}{(b+c)/2} = \\frac{(b-c)^2}{b+c} ", "0f28dfe4d2c9e391c9d8ecf65705a224": "\\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n,", "0f290393adafbe2ad3931d3a7a5e6904": " \\theta = \\theta \\,", "0f29748f7355c89e852d6fcfa26ebb82": " \\underset{\\rightharpoondown}{P} = |\\alpha \\beta\\gamma|\\,,\n\\quad \\underset{\\rightharpoondown}{Q} = |\\delta\\epsilon\\cdots\\lambda|\\,,\n\\quad\\underset{\\rightharpoondown}{R} = |\\mu \\nu \\cdots\\zeta| ", "0f2987be8a349502b8e54e2c14ba84e5": "( A, B )_{L^{2} (\\Omega)} := \\mathbb{E} ( A B ) = \\int_{\\Omega} A(\\omega) B(\\omega) \\, \\mathrm{d} \\gamma (\\omega).", "0f29cd3dda57e795a19c31921ad2850d": " q A \\,\\!", "0f29d4bd4785055b52cb8ba2b7647306": "\\overline{\\mathcal{M}}_{1,1}", "0f2a01742e8abdca6aa1eeb953fbbb17": "\n\\begin{align}\n s &= (0\\times 10) + (3\\times 9) + (0\\times 8) + (6\\times 7) + (4\\times 6) + (0\\times 5) + (6\\times 4) + (1\\times 3) + (5\\times 2) + (2\\times 1) \\\\\n &= 0 + 27 + 0 + 42 + 24 + 0 + 24 + 3 + 10 + 2\\\\\n &= 132 = 12\\times 11 \n\\end{align}\n", "0f2c04f82a1eb8e3e371366214579f5b": "\\supset ", "0f2cf44295dcc1a49d4de46477fb8abb": "\\Lambda(g\\cdot f)=\\Lambda(f)\\,", "0f2d0ff577228c5621fb0f58e2a4bd47": "\n\\delta \\varphi \\approx \\frac{6\\pi G M}{c^{2} A \\left( 1 - e^{2} \\right)}\n", "0f2db757e7e6e820d2bddfa9bb02b6e8": "(0,\\pm(P_i+P_{i-1}))", "0f2ded002cd3670af550c8a6f2daaf64": "\\int \\frac{\\mathrm{d}x}{\\sec{x} + 1} = x - \\tan{\\frac{x}{2}}+C", "0f2e3c8469103b9c39113cd88707648f": "\\begin{pmatrix} i & 0 \\\\ 0 & i \\end{pmatrix} \\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix} = \\begin{pmatrix} 0 & -1 \\\\ -1 & 0 \\end{pmatrix} .", "0f2e54b0d7e3690177e2482742f34a12": "a^b = \\left(\\sqrt{2}^{\\sqrt{2}}\\right)^{\\sqrt{2}} = \\sqrt{2}^{\\left(\\sqrt{2}\\cdot\\sqrt{2}\\right)} = \\sqrt{2}^2 = 2", "0f2e8b2c16f943cfbf9da8575deb310e": "\\tan \\vartheta_1=\\frac{m_2 \\sin \\theta}{m_1+m_2 \\cos \\theta},\\qquad\n\\vartheta_2=\\frac{{\\pi}-{\\theta}}{2}.", "0f2ef62c222e821714074d6a6b11d7c8": "{\\operatorname{d}P\\over\\operatorname{d}t_max}", "0f2f4ce74bf98b20435bd9b909283886": "\\alpha\\rightarrow\\beta_2", "0f2f560eba2860afe8b32615b0d068e3": "E(\\lambda) = {1\\over 2\\pi i}\\int _{\\gamma} (\\xi - C)^{-1} d \\xi", "0f2fbfc6c9db44646bba4d5efa0dd699": "m_{\\alpha}", "0f2fd254c697b0cbdc0a1ad9b13959d0": "c\\Delta t=-\\frac{2GM}{c^2}\\log(1-\\mathbf{R}\\cdot\\mathbf{x})", "0f301412982bd87a36a780794afdab98": "\\mathbb P^n", "0f3045256fa92f3cfc1af431ac69f32f": "{\\mathfrak M}=({\\mathcal P},{\\mathcal Z};\\parallel_+,\\parallel_-,\\in)", "0f3078a471b681e0980a3dc4ea1f4ad0": "N/12", "0f308f050ac74ca75d09cd969c91b031": "q=1/h \\, ", "0f30e6c2f865ab2364096b58f32818e7": "g_\\ast", "0f30ec69daf99e1640c0856b85200793": "\\scriptstyle 16\\times\\log_2(16) = 16\\times4 = 64", "0f310176b70474f8fcb26984da053c70": "V = Ad.\\,", "0f313509aef0c814cb9d3688cf373544": "S_1(t),S_2(t), \\ldots, S_N(t)", "0f315773a36f2ec90337b70f932ac83b": "v^{3}=-{q\\over 2} - \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}", "0f32160e7dabb64c3dec9294da1d653d": "(a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + 3 c^2 a + 3 c^2 b + 6 a b c.", "0f3221deb8ddd20d4d32624437225da3": "\\psi \\in L^2(X)", "0f323994996f9a1acc5b0ab9c55414f3": "[x,p] = i\\hbar, \\, ", "0f32c51cefc8c3b1925bb4fc124e1867": "r:=(X,Y,\\alpha)", "0f32e5f6b05aafa69ef6c5588e261c8b": "d_H(X,Y) = 1\\ ", "0f331d2e8981682cc3b484ba06270963": "00g_{-2}, 00g_{-1}, 10g_{-1}, 10g_{-2}, \\cdots, 10g_0, 11g_0, 11g_{-1}, 01g_{-1}, 01g_0", "0f334b7cbe6c32d2afa5b5a5f8a84952": "for\\, each\\, k-element\\,s \\subset cand", "0f33bbb8a98664d85ac65fa6a2967899": "\\begin{align}\n& {} \\qquad \\frac{N}{N_1} O(N_1 \\log N_1) + \\cdots + \\frac{N}{N_d} O(N_d \\log N_d) \\\\[6pt]\n& = O\\left(N \\left[\\log N_1 + \\cdots + \\log N_d\\right]\\right) = O(N \\log N).\n\\end{align}", "0f33e9bf56f9c1affcd1ac1c0fdb0092": "F(x) = P(X \\leq x) = P\\left(\\mathrm{all\\ observations} \\leq x\\right) = \\left(\\frac{x}{\\omega}\\right)^n .", "0f3421d8690ff16c369367b3e635b3d4": " u = \\sqrt{\\frac{B}{C}} ", "0f343b0f0ff47039a81cb85e14533429": "f(z) = \\frac{a z + b}{c z + d}, \\qquad ad - bc \\neq 0", "0f3451b8f08f7eb1cd5e43469d3df39b": "\\frac{d}{dx}(\\mathbf{a} \\times \\mathbf{b}) = \\frac{d\\mathbf{a}}{dx} \\times \\mathbf{b} + \\mathbf{a} \\times \\frac{d\\mathbf{b}}{dx}.", "0f3495d31be5bd5dbe0b023edb451f04": " \\lim_{k\\to \\infty} \\frac{|x_{k+1}-L|}{|x_k-L|} = \\mu.", "0f34bda01adf7e8bbeaa50542adb03fa": "\n\\begin{align}\n& \\operatorname{Prob}(14\\text{ heads}) + \\operatorname{Prob}(15\\text{ heads}) + \\cdots + \\operatorname{Prob}(20\\text{ heads}) \\\\\n& = \\frac{1}{2^{20}} \\left[ \\binom{20}{14} + \\binom{20}{15} + \\cdots + \\binom{20}{20} \\right] = \\frac{60,\\!460}{1,\\!048,\\!576} \\approx 0.058\n\\end{align}\n", "0f34fdc5621fc8c7acea8567b8fafa9c": "\\,S_b", "0f3526400dc9d9fd89be71e0074ee009": "Tm", "0f3557c6fde148f1baf83547798aba4f": "\\sigma\\in T_\\mu S", "0f356d5dd2d7c0c414edf03edb339498": "(A-\\lambda I)", "0f3575915d29fb35d631775d767a9cd4": "A = \\{e \\mid e \\in W_e\\}", "0f35cbf70c4a6a73d16efa38de80df79": "\\phi(a)\\,", "0f35e0a32fc3940c857d8ec6615c5a94": "K_2", "0f35efd263894c8aa9fe001a56273ed4": "c(E \\oplus F) = c(E) \\cup c(F)", "0f3627c098be67e4281854fccd1df2ed": "\n g_{jk} = \\frac{1}{4(1 - (x^1)^2 - (x^2)^2 - (x^3)^2)} \n \\begin{pmatrix}\n 1 - (x^2)^2 - (x^3)^2 & x^1 x^2 & x^1 x^3 \\\\\n x^1 x^2 & 1 - (x^1)^2 - (x^3)^2 & x^2 x^3 \\\\\n x^1 x^3 & x^2 x^3 & 1 - (x^1)^2 - (x^2)^2\n \\end{pmatrix}\n", "0f366a2dd1bbbafeed36851a01915575": " \\| [x] \\|_{X/M} = \\inf_{m \\in M} \\|x-m\\|_X. ", "0f36732c9e0f50e062901f5cb6578c85": "\\scriptstyle a_k", "0f367d4c8c9040e121a831d23fe270af": "p_1=p-p_0\\,,\\quad n_1=n-n_0", "0f368113bb450e46cd49a0e11e203fa5": " y_3'=3\\cdot 10^7y_2^2", "0f36b13548284b92828080c402c74806": "\\begin{align}\n\\sigma_\\mathrm{n} &= \\mathbf{T}^{(\\mathbf{n})}\\cdot \\mathbf{n} \\\\\n&=T^{(\\mathbf n)}_i n_i \\\\\n&=\\sigma_{ij}n_i n_j.\n\\end{align}", "0f36df3a43644fbe8b42bc118bbe81aa": "Gr_F C\\ell(V,Q) = \\bigoplus_k F^k/F^{k-1}", "0f374bbbc6038ec8ba5707a596827354": "[a_i,a_j]=[a^\\dagger_i,a^\\dagger_j]=0", "0f374c30e71d3a70f6d16e7eba98e46f": "MTBO = \\frac{MTBF}{1-FFAS}", "0f37746084f346f0abd765a4ab4914cd": "\\lim_{x\\to 0}{\\frac{e^x-1-x}{x^2}}\n=\\lim_{x\\to 0}{\\frac{e^x-1}{2x}}\n=\\lim_{x\\to 0}{\\frac{e^x}{2}}={\\frac{1}{2}}.", "0f3774a53e1b8ab9c88a0f0e7df3df78": "A_\\alpha", "0f37a954036311f685118508d444f655": "f : X\\rightarrow Y", "0f37b0ef3b0ccbab254bfc0e8cae938a": "\\{\\alpha_0+\\alpha_1 I\\mid \\alpha_i\\in\\mathbb{R} \\}", "0f37ea66a329b2554107927e9c3156e7": "(p-1)(q-1)/2.", "0f380d79ddc6a7bf977ee045044f3c1f": " a_{01} = \\mathcal{L}(a_{11}+a_{20} \\omega)+p_3( a_{11} + a_{20}\\omega)-\n \\omega p_6., ", "0f3818e98c4cdc22f152209641a10953": "f_2: L_2 \\to M", "0f3821382e6e5586e28ae27fb636766a": "2^{128-1}", "0f388596276c22c2572ce5ec71c8708e": "k + {a_{c} \\cdot c\\over 100} + {a_{v} \\cdot v\\over 100}", "0f38b4fb7e0098f2a6d9976ffc30bfc1": " \\int_{[0,1]^s} f(u)\\,{\\rm d}u \\approx \\frac{1}{N}\\,\\sum_{i=1}^N f(x_i). ", "0f38d7aafc5b65a72621859a9e66ee73": "f^{(-1)}(x) = \\int f(x)\\,dx", "0f38dbcf0d45f6245bc1ea34dba4dd19": "m_2 = \\overline{p_1 d_2}. \\, ", "0f39206383acb3f8feed855cb2838cef": " \\dot{M} = \\pi R^2 \\rho v ", "0f392c3addd8b044400a52603dfac1e7": "c_n =\\mathcal{O}(1)", "0f399d0785717979476e46ec94f2e04d": "\n\\begin{align}\n \\\\[8pt]\n(x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\\\[8pt]\n(x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4, \\\\[8pt]\n(x+y)^5 & = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5, \\\\[8pt]\n(x+y)^6 & = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6, \\\\[8pt]\n(x+y)^7 & = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7.\n\\end{align}\n", "0f39d9ffb6612a5371b64cf219a36f21": "T(\\cdot,\\cdot)", "0f39ebb2e514d426d7cb9ef23c226683": "\\scriptstyle \\left|I_o\\right|=\\frac{L}{T\\, V_i}I_o", "0f3ac1bb11cb34ada8999cbeb9011e87": "\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} P_{\\mathbf{k}}\n=\n\\tilde{\\varepsilon}_{\\mathbf{k}} P_{\\mathbf{k}} - \\left[ 1 - f^{e}_{\\mathbf{k}}(t) - f^{h}_{\\mathbf{k}}(t) \\right] \\Omega_{\\mathbf{k}}\n+ \\mathrm{i} \\hbar \\left. \\frac{\\partial}{\\partial t} P_{\\mathbf{k}} \\right|_{\\mathrm{scatter}}\\,,\n", "0f3adb63ac174178a08f1a8ead64d1b6": "-\\sqrt{\\frac{8}{21}}\\!\\,", "0f3ae8a2a712035b064b92d8defa9074": "P_7(x)=x^3+1 \\,", "0f3b60e2a78c45655e66a5a0dbdfa492": "r_k(x)=u_k(x)p(x)+v_k(x)q(x)", "0f3bae9abc24c3614bdb470ac29f8e41": "F(z) = \\sum_{n \\ge 1} \\sum_{G\\in \\operatorname{Cl}(S_n)} c_G Z(G)(f(z), f(z^2), \\ldots, f(z^n))", "0f3bbd2fd22c7c691a4156c77905e25f": "d^{n}", "0f3bc946df6bb696d1c0235cf8036681": "2m \\boldsymbol{\\omega}\\times \\left[ \\operatorname{d} \\boldsymbol{r}/\\operatorname{d}t \\right]", "0f3cd2722d4c94a49626605256e727f1": "{}^A\\!X^* \\to {}^A\\!Y + \\gamma ", "0f3d04bced028feb8726b0be410953a9": " E' - E = E(k') - E(k) \\pm \\hbar \\omega_q \\, ", "0f3d070ab4695fc5a4933c76f4b7f6c6": "R_{;\\varepsilon} \\, - 2R^\\gamma{}_{\\varepsilon;\\gamma} \\, = 0", "0f3d98282e1e720f30b65cd69b4414dc": "\\textstyle \\frac{N-1}{2N} - \\frac{1}{n}\\frac{l_{max}^2}{n(\\Lambda - \\Lambda_0(P))^2}", "0f3daacdf551d16d828e545747c7e63f": " C(n,d) := \\mathbf{conv}\\{\\mathbf{x}(t_1),\\mathbf{x}(t_2),\\ldots,\\mathbf{x}(t_n) \\} ", "0f3dd706edcabc4f0e642e5d0201fc4e": "q \\colon A \\times B \\to B", "0f3df9c8c81e6fb5a9f714f6ac652144": "h_*", "0f3e13391df8b1bc516adf6db3e04f78": "\\mu \\phi. \\lambda \\alpha. 1 + \\alpha \\times \\phi\\ \\alpha", "0f3e60bc04bdebcd5a2239f05d62e8f2": "L \\rightarrow \\infty", "0f3e93fd8018d2187ee6d7124b3ea12e": "\\Pr\\{CS\\}", "0f3ecdb19a9988c8ab8c78ec4a2a84d2": "I(X;Y) = \\mathbb E_{p(y)} [D_{\\mathrm{KL}}( p(X|Y=y) \\| p(X) )].", "0f3ee364495228af1e9399faaac204a0": "2\\alpha.", "0f3fdc97ba234e19266127e3d9792199": "\\overline Y\\to X", "0f4003d63f7c7511d8664df210534dfd": "\\rho = P_{L}/P", "0f40b5e1ff787b62e63375cf470ea504": "-2 < \\beta \\le -1.645", "0f4138e40cbf42b45961ef42db906844": "d^{2}", "0f41459abc933d5a1053a91c7fa07dcf": "C^1([0,1])", "0f4151dce52828d17604fc605aede3d9": " R= \\frac{u_s-u_c}{u_o-u_c}", "0f41722bdbce9c3d9c964ca21178610d": "y-X \\hat{\\boldsymbol\\beta},", "0f41a54b81d38041694d8c1c1d44fccf": "L = \\begin{cases} 1.33 \\, f^{0.284} \\, d^{0.588} \\,\\mbox{, if } 14 < d \\le 400 \\\\ 0.45 \\, f^{0.284} \\, d \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\mbox{, if } 0 < d \\le 14 \\end{cases}", "0f41b32e4c6d85530556e1ecab5f13d2": "A_c = A_c^i T_i", "0f422ff7e741adb2604d6a0225e712d5": "\\frac{4\\sqrt{\\alpha-2}}{\\alpha-3}\\!", "0f4315e21622712854f9302da6a1a459": "\n [\\boldsymbol{\\sigma}] = \\begin{bmatrix}\\sigma_{11}\\\\ \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix} \\equiv\n\\begin{bmatrix} \\sigma_1 \\\\ \\sigma_2 \\\\ \\sigma_3 \\\\ \\sigma_4 \\\\ \\sigma_5 \\\\ \\sigma_6 \\end{bmatrix} ~;~~\n[\\boldsymbol{\\epsilon}] = \\begin{bmatrix}\\epsilon_{11}\\\\ \\epsilon_{22} \\\\ \\epsilon_{33} \\\\ 2\\epsilon_{23} \\\\ 2\\epsilon_{31} \\\\ 2\\epsilon_{12} \\end{bmatrix} \\equiv\n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\end{bmatrix}\n ", "0f437c5dda614eb1bfa7ae3129ef7c6e": "\\ cos\\phi\\ = -cos(R-C-C)/[cos(1/2(R-C-R))] ", "0f438346f0a15c5f6990c3916dd2adb4": "\\chi =\\chi _{p}-\\chi _{s}", "0f43a07387957bf9345ae44f835e4d75": "(X,X^*)", "0f43d77334aa552241e9cbb7b4e1ccca": "(r_1 \\cdot r_2) \\times s = r_1 \\times (r_2 \\times s)", "0f43ea941fdf874cdf9a19aa5624cb04": "r = a \\frac{\\cos ((\\alpha+\\theta)/2)}{\\cos ((\\alpha-\\theta)/2)}", "0f43f6dc320dd6a0bd3959b7b8ec6035": "\\theta \\left( x, y \\right)", "0f441b575e0c9a0467ade92543379f2f": "T ( w ) \\subseteq \\Gamma", "0f449b09d568ac088ab865f9aa93cf58": "\\mu(A)", "0f44e32a051880be1b8a1715b9fc5c80": "\\displaystyle{\\begin{pmatrix}1 & \\beta \\\\ 0 & 1\\end{pmatrix}(a,T,b)=(a +\\beta T(1) -\\beta^2 b,T - \\beta L(a),b),}", "0f4516bfde5ec4e334446ecba949c331": "a=0.5+i0.5", "0f4526c17da78aa3e266e62b27c38c3f": "\\rho(\\mathbf{Y}|\\mathbf{X},\\mathbf{B},\\boldsymbol\\Sigma_{\\epsilon}) \\propto (\\boldsymbol\\Sigma_{\\epsilon}^{2})^{-n/2} \\exp(-\\frac{1}{2} {\\rm tr}((\\mathbf{Y}-\\mathbf{X}\\mathbf{\\mathbf{B}})^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1}(\\mathbf{Y}-\\mathbf{X}\\mathbf{\\mathbf{B}})) ) ,", "0f45789338d28eb754e62d1a80045b50": "(af)(m):=f(S(a)m)", "0f457b5a02025fc72c47e18480dd40bb": "\\frac4n = \\frac1x + \\frac1y + \\frac1z.", "0f45a5f5705dbf225da1ed056fe1550d": "\n = {k_1\\choose k_1}{k_1+k_2\\choose k_2}\\cdots{k_1+k_2+\\cdots+k_m\\choose k_m}\n = \\prod_{i=1}^m {\\sum_{j=1}^i k_j \\choose k_i}", "0f45a715f24607eb90eaa05bffb8a779": "Q^n", "0f4624a8497be6567374c5f16231f56c": "\\delta t = \\frac{\\varepsilon^2}{6 D_0}", "0f467f0ff84963121cd4f9ad4fae4ede": "\\xi(m) = m^{-\\alpha},", "0f46c77b168aeb55fc1f2b71fb2fb829": "H(\\bar{\\sigma},\\bar{\\epsilon})=\\cfrac{d\\bar{\\sigma}}{d\\bar{\\epsilon}}", "0f46db5aa8ab62b190c4ad47b834dfed": "q \\in Q_A ", "0f47a3f60d1157b69118e908dfab9dc3": "\\begin{align}\n \\mathbf{U} \\mathbf{U}^* &=\n \\begin{bmatrix}\n 0 & 0 & 1 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 \\\\\n 1 & 0 & 0 & 0\n \\end{bmatrix} \\cdot\n\n \\begin{bmatrix}\n 0 & 0 & 0 & 1 \\\\\n 0 & 1 & 0 & 0 \\\\\n 1 & 0 & 0 & 0 \\\\\n 0 & 0 & -1 & 0\n \\end{bmatrix} \\\\\n\n &= \n \\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix} \\equiv\n\n \\mathbf{I}_4\n\\end{align}", "0f47a9711b28c4acd9dcd4af29ef8066": "n_{\\rm Q}=\\left(\\frac{M k T}{2 \\pi \\hbar^2}\\right)^{3/2}", "0f47f98a556f329dffc303bafd94853e": "\\scriptstyle >1.8\\times10^{15}", "0f48087903db5651c7f25b0d254ff050": "\\mathrm{vis}", "0f482605492c028a78db0ba355d1cafb": " ([g_1,h_1] \\cdots [g_n,h_n])^s = [g_1^s,h_1^s] \\cdots [g_n^s,h_n^s]", "0f48716ddd3296ba459b851fcd483f27": "\n= {M \\over 2} \\cdot A.\n", "0f4871dd0ff9551e39ca7a2e73b37305": "\\Upsilon\\,", "0f487c94170acb324233fad4d4d18fac": "ZFC+\\lnot \\operatorname{Con}(ZFC+H)\\vdash\\lnot \\operatorname{Con}(ZFC)", "0f48a335a7c9e4528be77f00cad07d57": "Y(s) = Z(s)G(s) \\Rightarrow Z(s) = \\dfrac{Y(s)}{G(s)} ", "0f48aeabf7edfb0b62703187e367cd8c": "x^2 \\frac{d^2 y}{dx^2} + x \\frac{dy}{dx} + (x^2 - \\alpha^2)y = \\frac{4{(x/2)}^{\\alpha+1}}{\\sqrt{\\pi}\\Gamma(\\alpha+\\frac{1}{2})}", "0f48ba77a92cfc6534ac0cdaefa06495": "a_{-1}=1", "0f48da724906e6cd83a23bfc1a5e7229": "z(r,\\theta) = \\Re ( 2i(-\\ln(1-r^2e^{2i \\theta}) + \\ln(1+r^2e^{2i \\theta}) ) = 2 \\tan^{-1}\\left( \\frac{2 r^2 \\sin 2\\theta}{r^4-1} \\right)", "0f48f62c0f7893f9fb7fccc7817ee5ed": "s^\\mathfrak{n}(t)", "0f48fff9a3b17d030f4b22a90b332a89": " d^{-1} ", "0f490342247f08cfa2930b46aaaa92af": "x = ka \\sin \\theta = \\frac{2 \\pi a}{\\lambda} \\frac{q}{R} = \\frac{\\pi q}{\\lambda N} ", "0f492b6e31bca36580398bb1d0be3685": "\\displaystyle{a^{-1} = (a-b)^{-1} + (a - Q(a)b^{-1})^{-1} = (a-b)^{-1} + Q(a)^{-1}(a^{-1} - b^{-1})^{-1}.}", "0f4956319d3e1831c8d3a2a341fd0323": "\\sigma( \\mathbf{x} ) = \\mathbf{0}", "0f49ab8db7d4552fe5370831bd1ffd3d": "A \\succeq B \\succeq C", "0f4a36a71d180e1c06d76d770ce919cc": "\\Gamma (D(P), \\mathcal O_{\\mathbb P (V)})", "0f4a720d82fb0ece5e4582e3d497e773": "g\\in \\mathcal H", "0f4acbf419a68b1e231f5a17f1dfe74a": "\n\\eta_{0} \\propto \\begin{cases}\n const. & \\text{, }a < \\sqrt{a_{m}} \\\\\n \\left|t\\right|^{1/4} & \\text{, }a = \\sqrt{a_{m}} \\\\\n \\left|t\\right|^{1/2} & \\text{, }a > \\sqrt{a_{m}}\n\\end{cases}\n", "0f4ad3bfb282249852d4a44f2da783d9": "q_\\text{opt}=F^{-1}\\left( \\frac{7-5}{7}\\right)=F^{-1}\\left( 0.285 \\right) = D_\\min+(D_\\max-D_\\min) \\cdot 0.285 = 58.55\\approx59.", "0f4b5c2053084288ba37032ffc17b004": "i = f ( I\\cdot t^p )", "0f4b5cb0d3f20255da0a4f9aa377ba6d": "s = a\\tan \\varphi", "0f4baa38d51338570ac03f2cde849699": "\\operatorname{Hom}_{\\rm Schemes}(X, \\operatorname{Spec}(A)) \\cong \\operatorname{Hom}_{\\rm CRing}(A, {\\mathcal O}_X(X)).", "0f4bf550421793761fe8501bd8dcad19": "y_i \\in \\{0,1\\}^k", "0f4c1baf6f3b4b49b635036b02520aa0": "x = \\left(\\frac{1}{k}\\right)y,", "0f4c4ce0863d100a12c90c114fd9abeb": "\\vec{b}", "0f4c8bcae3d428804c468ce4c2e374c4": "K_{m,n},", "0f4cd392fdcd029636175fbee752d895": "X\\setminus Y", "0f4cd952c45670d1a7b9c0f59f1319b6": "y_3= \\sin i \\cdot \\cos \\omega", "0f4d44a6d1ab415a985235815794ffd8": " n \\approx 2.5 \\sqrt { \\frac{ c }{ 2j + 1 } }", "0f4d4d48f82159f4e29325b88c84a4a8": "\\frac{\\cos(x) x^2 - \\sin(x)2x}{x^4}", "0f4d6327a617f88e3d585b30e6eab420": "Z>m_N", "0f4d948653fa27cc90071f85d5b6d8d8": " \\frac{\\partial c}{\\partial t} + \\nabla\\cdot\\vec{j} = R, ", "0f4dab6356d1fd9de79ec5678d07fa83": "\\tau_{\\varepsilon}", "0f4e0ab41c9ac56bd3ec84bc8d93671f": "\\langle \\Phi(\\rho \\otimes \\omega), I \\otimes O \\rangle = \\langle \\rho, O \\rangle", "0f4e76b575249460870a83c151a53d67": "\n\\begin{array}{lcll}\nI_2(d^2\\theta/dt^2)&= &-K-M(d\\theta/dt) &\\cdots(Eq.7)\\\\\n(d^2\\theta/dt^2)+M/I_2(d\\theta/dt)&= &0 &\\cdots(Eq.8)\n\\end{array}\n", "0f4ec8f1bf97041b050e900f1aebf684": " {XX'' \\over YY'} = {DX \\over BY}, ", "0f4f7ffc015aa89b86f99b7d40446152": "\\{\\mu_i\\}_{i=1\\dots\\infty}", "0f4fbe4fb9f670136bd4cabbddb71a4e": "\n\\nabla^2 G = \\delta(x). \n", "0f4fdbf640f71457a4f8156b659621b6": " p=\\hbar k\\,\\!", "0f500804d115099c77b8aa3ac2fd901f": "b \\in I^j/I^{j + 1}", "0f5010640e0352579f10c06ffc92375d": "\\{x\\} \\mapsto [x]_R", "0f50309a698653a330504a19be7b3bfb": "\\begin{bmatrix} [x=1] \\\\ \\vdots \\\\ {[x=k]} \\end{bmatrix} ", "0f503ea579eaee3098543fbfcc343b03": " \\rho_2=\\rho_3 = z^{-2} + 9z^{-3} + 80z^{-4} + 965z^{-5}-\\cdots.", "0f5073c04b8fcde03f80f3e1324933e5": "Lx", "0f508098e84c6b780ca835ec5862a647": "\\scriptstyle{Z_{21}}", "0f50c9baa20cdd71cf493e06a21c5bee": "\\{\\ X^/\\}", "0f50ce1725284da939e3907c2521e787": "\\eta_{v_e}", "0f515746a540b5b8f8e2096d33a55c75": " S= \\int Tr \\partial_\\mu A_\\nu \\partial^\\mu A^\\nu + f^i_{jk} \\partial^\\nu A_i^\\mu A^j_\\mu A^k_\\nu + f^i_{jr} f^r_{kl} A_i A_j A^k A^l + Tr \\partial_\\mu \\bar\\eta \\partial^\\mu \\eta + \\bar\\eta A_j \\eta \\,", "0f52708aa0d2f74a63230012b40d1ca8": "k_q", "0f52d112f4addae265628151fa31b328": "\n\\mathbf{m}_i = \\frac{1}{l_i}\\sum_{n=1}^{l_i}\\mathbf{x}_n^i,\n", "0f52d94344143bc8b1fd698cae16e83d": "\\psi(\\beta)<\\delta", "0f52dec5c9d961f937211f6db1952f92": "\\frac{| \\triangle SCA|}{|\\triangle SDA|}=\\frac{|\\triangle SCA|}{|\\triangle SCB|}", "0f53350338713b8a22fe41905d7749c5": "\n\\kappa_{xy}(f)= \\frac{A_{xy}^2}{ \\Gamma_{xx}(f) \\Gamma_{yy}(f)} ,\n", "0f537a85a20e520b18abcce2df9eb893": "\n \\mathcal{P} = \\Big\\{\\ f_\\theta(x) = \\tfrac{1}{\\sqrt{2\\pi}\\sigma} e^{ -\\frac{1}{2\\sigma^2}(x-\\mu)^2 }\\ \\Big|\\ \\theta=(\\mu,\\sigma): \\mu\\in\\mathbb{R}, \\,\\sigma\\!>0 \\ \\Big\\}.\n ", "0f53e98758ba9b2b118b728482471323": "(a^{n}-1)^{2n}+(a^{n}-1)^{2n+1}=[a(a^{n}-1)^{2}]^{n};", "0f5478b96f3b8b3c582269a8dd1f6f4c": "[x_0, x_1,\\ldots,x_n]", "0f549e0a2668dcee3144d997fee0ae7a": "J^1_0(\\varphi\\circ f)(t)=tv^i \\frac{\\partial f}{\\partial x^i}(p)", "0f54c07dc29cc38240e9bd81c391cd8e": "Y = A \\times K^\\alpha \\times L^\\beta ", "0f54ee04ab1446600e4220c0d4232b67": "\\chi_\\lambda(e^X)={\\rm Tr}\\, \\pi_\\lambda(e^X), (X\\in \\mathfrak{t})),\\,\\,\\, d(\\lambda)={\\rm dim}\\, \\pi_\\lambda.", "0f54f750550d134b74f42ef611710b7c": " B_\\lambda(\\lambda,\\ T)\\ =\\ -\\ \\frac{\\mathrm d \\nu}{ \\mathrm d \\lambda}B_\\nu(\\nu (\\lambda),\\ T).", "0f554ea390eeb2d4997d482777249fcd": "\\text{grant}(j)", "0f555f8a6116983db9ff85a0ac6bd491": "\\mathcal{M} = (r,\\mathbf{b},\\mathbf{\\delta},\\mathbf{\\sigma},A,\\mathbf{S}(0)) ", "0f55612e676ceca86078c208ebbb02d8": "\\alpha=1/\\sqrt{\\beta_c}", "0f55c3cdf8180af20d86a06eb31954f6": "M \\sqcup (-N)", "0f55cab694c029be42a8f8a8c2e0a99e": "\\int_\\Omega v L[u]\\ d\\Omega = \\int_{\\Omega} u L^*[v]\\ d\\Omega +\\int_S \\boldsymbol{M \\cdot n } \\, dS, ", "0f55e29d9f506eb65b6a9dbcf1ebc719": "\nH(\\omega) = \\left\\{\\mathcal{H}f\\right\\}(\\omega) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty\nf(t) \\, \\mbox{cas}(\\omega t) \\mathrm{d}t,\n", "0f560a2e916b1cb8e819012fae3491ae": "\\ln(x)=\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} (x-1) ^ n", "0f560d3528ac0cf5b4bf0da4eaf64f80": "\\psi(\\Omega\\omega) = \\varepsilon_\\omega = \\phi_1(\\omega)", "0f5647baff8f89c8ba99759ee35a03bd": "K = 2 \\pi -\\alpha-\\beta-C.", "0f5670c10524c3f9ea5d7d47f6fe4ab8": "\\mathcal{B}[f](\\alpha u + \\beta v,\\dots) = \\alpha\\mathcal{B}[f](u,\\dots) + \\beta\\mathcal{B}[f](v,\\dots),\\text{ when }\\alpha + \\beta = 1.\\,", "0f571e9eadc7f8e6c36922c6ad8157dc": "(p \\land (q \\land r)) \\vdash ((p \\land q) \\land r)", "0f572d482ec9fe1a0feb546afe74cec7": "q'_\\text{P} = \\sqrt{\\epsilon_0 \\hbar c} = \\frac{e}{\\sqrt{4\\pi\\alpha}} = 5.291 \\times 10^{-19}", "0f572e78d729f39d24acc591924ebc00": "1\\,\\text{pdl} = 1\\,\\text{lb}_m \\cdot 1\\,\\frac{\\text{ft}}{\\text{s}^2}", "0f5745a39590195f97cefda202c1a00f": "{\\it{M}}=2", "0f574cafb461928fde1420211e049020": "\\textbf{F} = \\begin{bmatrix} 1 & \\Delta t \\\\ 0 & 1 \\end{bmatrix}", "0f57680523558dcf650fc1aa21528949": "F_{\\alpha \\beta} \\, = \\, \\partial_{\\alpha} A_{\\beta} \\, - \\, \\partial_{\\beta} A_{\\alpha} \\,.", "0f57acdd48e0530d2e2fda2d824788bc": "f \\ge 0", "0f57f1313a8e92722688d3aa000f2f14": "e = \\lim_{n\\to\\infty} \\left(1 + \\frac{1}{n}\\right)^n.", "0f581636dd5a8183f5a0484d96d7cb19": "D_t =", "0f581f31e63cdce64f590ceb83d7f824": "\\operatorname{Cl}_{2}(2\\pi z) = 2\\pi \\log \\left( \\frac{G(1-z)}{G(1+z)} \\right) -2\\pi \\log \\left( \\frac{\\sin \\pi z}{ \\pi } \\right) ", "0f583d100fec13c47aa4f911b268158c": "\\epsilon(\\mathbf{q}) = 1 + \\frac{k_0^2}{q^2}", "0f5853b0f8b5c20679b2c5bd78c91fef": "\\tfrac{h_i}{k_i}", "0f586042fa46296b3eb8b940f9b3a0f8": "\\Box \\mathbf{A} = \\frac{4 \\pi}{c} \\mathbf{j} ", "0f5883fcd146319d282838deb4c174ec": "f(x)=x^TAx", "0f58de596713bb42ec73c8b379d46c59": "LC_{50} (mixture) \\le 200 \\tfrac{mL}{m^3}", "0f58fac7c62960d176682b1d6404ea02": "e = 2 \\sum_{k=0}^\\infty \\frac{k+1}{(2k+1)!}", "0f5906ce29c97f3d473437f693f8addf": "\\oint_K \\kappa \\,ds \\leq 4\\pi,", "0f59120e4ede42e765b23cd5e35d32b8": " \\Lambda=\\sum\\limits_{i=1}^{\\infty}\\Lambda_i. ", "0f596901845f805cb6305b7b365cf5c4": "z <= -0.25", "0f596ef6330dafa2e50880d1b5ec22b5": " \\theta = \\arccos\\left( \\frac{\\mathrm{trace}(R) - 1}{2} \\right) ", "0f597eba7730a8d5cce3b1b23cf733e8": "A=\\mathrm d^2/\\mathrm dx^2", "0f598644b790d37b8f90de2586a0c90d": "E_{a'b'}=\\mathrm{diag}(\\lambda_1,\\lambda_2,\\lambda_3)", "0f5994a9965ffe3f1965ecf6e89abe25": "U_1(T)=1", "0f59952ffe586f13939805ca01426332": " h(n) ", "0f59a737f68757fbd97c1987919dcb9b": "\\frac{X_{\\tau_\\nu}}{\\sqrt{\\nu}}", "0f5a077a904db6bf28b28ea5957b55a1": " 3(b^2-c^2)(c^2-a^2)(a^2-b^2)(a^2yz+b^2zx+c^2xy)\\,\\, + ", "0f5a6bacf345b81f7666359b3b79e6fe": "k=37", "0f5a911c58332baa07c6d99dadd954bf": "\\{\\{64x^3+384x^2-1024x+512,(1,\\tfrac{3}{2})\\},\\{64x^3+576x^2-64x-64,(\\tfrac{3}{2},2)\\},\\{64x^3+192x^2+80x+8,(2,4)\\}\\}", "0f5aa0a8e99463f2a36455aba15bb5b5": "\\sigma_\\mathrm{oct}\\,\\!", "0f5aa3c68338c7fbb692df971a15955e": "\\ell_i \\in \\mathbb{R} \\cup \\{-\\infty\\}", "0f5b0dbee721c33ca7fe31936a32a42c": "\\sigma_{\\bar x} = \\sigma/\\sqrt n", "0f5b11f8a843355b4382ead6e182c57f": "\\begin{align}\n R_1 &= \\frac{R_bR_c}{R_a + R_b + R_c} \\\\\n R_2 &= \\frac{R_aR_c}{R_a + R_b + R_c} \\\\\n R_3 &= \\frac{R_aR_b}{R_a + R_b + R_c}\n\\end{align}", "0f5bfa684083cdd5d5a17fb3ebe51d8b": "P = \\{ 0, \\dots, 76 \\}", "0f5bfc82b754e0f20c1aec792c462dfa": "\\operatorname{tr}(\\gamma^5)=\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu\\gamma^5) = 0", "0f5bff0227f7d270d110b0fd2671a345": "SU(3)_C \\times SU(3)_L \\times U(1)_X", "0f5c02812f1eaadfdbcab8608a5e9889": "\n\\rho \\left( 1 \\right) =\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{bmatrix}\n\\qquad\n\\rho \\left( u \\right) =\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & u \\\\\n\\end{bmatrix}\n\\qquad\n\\rho \\left( u^2 \\right) =\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & u^2 \\\\\n\\end{bmatrix}.\n", "0f5ca855c830bbecb5eb874a90957d52": "\\vec{G}_m", "0f5cc5a41884b677bd2718fd26a7b78b": "\\{re^{2\\pi i \\theta} : 0 \\leq r \\leq 1, \\theta \\in \\mathbb Q\\} \\subset \\mathbb R^2", "0f5cd19852fbfa265cf774e761adb7e2": "P_0=\\frac{M_a}{r}(1-e^{-rT})", "0f5cd37268db1469d362d94329c65859": "k \\leq n-\\log(\\sum_{i=0}^t{n \\choose i})", "0f5d33b506f67ed0ab8624a34197ccba": "O\\left[m(1 + \\log n) + n\\log n\\right]", "0f5d6266c8b9e375b15b230a70c77838": "x' = x \\cos \\theta + y \\sin \\theta", "0f5e0853f82372e76e84d712f57cc02c": "s \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\mathrm{entropy}}{\\mathrm{volume}} = \\frac{p + \\rho}{T} = \\frac{2\\pi^2}{45}g_{*}(T) T^3", "0f5ec36d93ad2d1052db24fc39f55c23": " * Prob( D", "0f5f44d5b4a90b72616c4852072a34c4": "p(x) = x^3 + 2x - 3", "0f5f69e7e68e63a5d666bbed7a37c9da": "(1-p + pe^{it})^n \\!", "0f5f8024248470026e785696ce48e843": "\\ k = \\frac{k_BT}{h}e^{-\\frac{\\Delta G^\\ddagger}{RT}}", "0f5ff46bdd9b1ef57c896c74a5d5367c": " \\mathrm{Var}[X] = K''_X(0) = A''(\\theta) \\, .", "0f5ff4c85303ec0f41d1c03a6d049943": "\\mathbf{r} = r \\hat{\\mathbf{r}} ", "0f603579577c12956a74c1ce88f48220": "P_+ = {1 \\over 5}", "0f60666f0d48d92bfe0a891a311d1db6": "\\Phi \\left(\\eta,\\tau \\right) = 1", "0f6073904491b22c04a145532cd6b1e4": "J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m)=\\sum_\\mu\nJ_\\mu^{(\\alpha )}(x_1,x_2,\\ldots,x_{m-1})\nx_m^{|\\kappa /\\mu|}\\beta_{\\kappa \\mu}, ", "0f607506cdeea84e8447574da8a17a96": "\\scriptstyle{(\\lambda^+_i,\\lambda^-_i)}", "0f609244a8c2f9b549f734e7387bd13d": "\\, \\frac{(1-p)^r}{(1-pe^t)^r}", "0f60967e91d8f401f5e796b14ee0d126": "P_c \\sim 10^5\\,\\mathrm{Mbar}", "0f60a09ff0df09fb54ca7b12045f573d": "\\sigma(t) = \\left(\\frac{1}{1-t},0\\right)", "0f60f8ba7733b53856f3f587c096ccfd": "\\textstyle P \\in G_1, Q \\in G_2", "0f6189ff0cded3aab0ba434d8c325982": "\\mathrm{C_nH_m + \\frac{n}{2} \\ O_2 \\rightarrow n \\ CO + \\frac{m}{2} \\ H_2}", "0f61cd7a7973003a62651054f6f9fbf5": "c_s = (\\gamma ZkT_e/m_i)^{1/2} = 9.79\\times10^5\\,(\\gamma ZT_e/\\mu)^{1/2}\\,\\mbox{cm/s}", "0f624bdf2befce40d2eb69d434373f3e": "gfg^{-1}", "0f6286fcc6f4dc3786c2c54159d9c24f": "n \\ge \\ N", "0f62ce61480470c586f2c6c152d17100": "{1/b^2} ", "0f6307077598fc956455fbf831228f16": " \\Delta (h)=h_{(1)}\\otimes h_{(2)} ", "0f631e166d309cad911b55ed9fab0c71": "E_{image}=w_{line}E_{line}+w_{edge}E_{edge}+w_{term}E_{term}", "0f63b3432883b8eaf1d7c54f60be6110": " p_\\textrm{b} = p_\\textrm{(p + b)} \\,", "0f6422ea0ca94a25194a53c75b3f4096": "j(\\tau')", "0f64303c04289769db0982cf1bd7f148": "\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_{\\widetilde{n}} )^T", "0f6430ec8a66e7789db8f0ad21b72bde": "f_2^*(g)=\\overline{f_2(g^{-1})}", "0f648da14647174ee9df6f22e8216164": "(0,0,1)", "0f64b4f9251d46ad162ce7bb2d92d687": "2 T_G(3,3)", "0f651da3e03453030d123f907e8e7116": "\\eta_{so} = \\frac{ 8-4\\sqrt{2}-\\ln{2} }{2\\sqrt{2}-1} \\approx 0.902414 \\, .", "0f655b911040e025b2df4941f848658e": "COD = \\frac{8000 (b - s)n}{sample\\ volume}", "0f6561dff7b4d3b35589805aab6b9f1a": "1 + {1 \\over 2} + {1 \\over 3} + {1 \\over 4} + {1 \\over 5} + \\cdots = \\sum_{n=1}^\\infty {1 \\over n}.", "0f65b1418dfc6bf4395ba18b17ad596c": "f_a = e^{3.158x10^{-5} * h}", "0f65b521fb825c6ac584c26609b41e25": "\\{\\{64x^3-112x+56,(0,2)\\},\\{64x^3+192x^2+80x+8,(2,4)\\}\\}", "0f65dbbd5678818c72f1a6085ab17ff4": "\\frac{h \\nu'}{c} = \\gamma \\frac{h \\nu}{c} - \\gamma \\beta \\left \\Vert p \\right \\| \\cos \\theta = \\gamma \\frac{h \\nu}{c} - \\gamma \\beta \\frac{h \\nu}{c} \\cos \\theta", "0f65ec39a1a40a045a876f743f3db79f": " \\bar X \\xrightarrow{n \\to \\infty} N(k, 2\\cdot k /n ) = N(2, 4/3 )", "0f66112133d2bb49d036732670cb14d4": " \\text { Recovery} = \\left ( \\frac{\\text{Mass of protein in the foam}} {\\text{Mass of protein in the initial feed}} \\right) * 100% ", "0f6619145e3b27578a526a55ca52c3e6": "{}_t p_x", "0f663749d5f854395e25f43b77e222ee": "J=\\begin{bmatrix}\nJ_{m_1}(\\lambda_1) & 0 & 0 & \\cdots & 0 \\\\\n0 & J_{m_2}(\\lambda_2) & 0 & \\cdots & 0 \\\\\n\\vdots & \\cdots & \\ddots & \\cdots & \\vdots \\\\\n0 & \\cdots & 0 & J_{m_{s-1}}(\\lambda_{s-1}) & 0 \\\\\n0 & \\cdots & \\cdots & 0 & J_{m_s}(\\lambda_s)\n\\end{bmatrix}", "0f665891419cf6212ac6e273c7a526be": "F^+", "0f66bb703c825146400acc9251364cf5": "p_{k}^{(i)}", "0f66db32cdff104091dc8334eda4ec5d": " \\mathrm{For} \\quad 0 < x <(a-b)", "0f66e753921d66eb551501bd0e0b43d5": "P(x) = \\sum_k a_k x^k \\,\\!", "0f66f2cbbc1624c6540fe2b52c6d9b3a": " v = \\frac{d [P]}{d t} = \\frac{ V_\\max {[S]}}{K_m + [S]} ", "0f67201e5464f36bce2e7a4bf3cc376e": "sk \\leftarrow Keygen(msk, k)", "0f67311530100c2111600d871e0e4cf7": "\n\\left.\n\\begin{matrix} 4^{4^{\\cdot^{\\cdot^{\\cdot^{\\cdot^{4}}}}}}\\end{matrix}\n\\right \\}\n\\left.\n\\begin{matrix}4^{4^{\\cdot^{\\cdot^{\\cdot^{4}}}}}\\end{matrix}\n\\right \\}\n\\dots\n\\left.\n\\begin{matrix}4^{4^{4^4}}\\end{matrix}\n\\right \\}\n4,\n", "0f6760b92123c21d857b1888c8aaa0e3": "\n\\begin{align}\n\\Pr(Y_i=0) &= \\frac{e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i}}{e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}} \\, \\\\\n\\Pr(Y_i=1) &= \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}} \\,\n\\end{align}\n", "0f678613c7687e5252ad611dc0db6325": "p.v. \\frac{c_nx_j}{|x|^n},\\quad c_n=\\frac{\\Gamma((n+1)/2)}{\\pi^{(n+1)/2}}", "0f67a0be1f62de36d683fb987b464f85": "\\scriptstyle \\frac{\\partial R}{\\partial x}", "0f67e637ace3bf5af79a45e367857733": "\\boldsymbol{\\pi}P=\\boldsymbol{\\pi}.", "0f67e98dac57dc848c15d2bb02dba27b": "2 v_c \\sin\\gamma - c \\cos\\gamma = c\\cos (\\alpha - \\beta).\\,", "0f680f48cdd079611e306b0adee2fc85": "\\;(gh)^2=(hg)^2", "0f6821e39d796e05e5141314c93ad7d9": "1~\\mathrm{Tesla} = \\mathrm{kg C}^{-1} \\mathrm{s}^{-1}", "0f6885186415cef2a0295610c82e3f66": " C(t\\cdot x_1, t\\cdot x_2, \\dots , t\\cdot x_n) = t\\cdot C(x_1, x_2, \\dots , x_n)\\text{ for }t > 0 ", "0f68f62a556484635461c1ebc134ce7f": "2^a 3^b", "0f69fc38b4d429f823d2b6f35352aaad": " \\max_w {w^{T}Aw}", "0f6a2127a9175b5ddf193f0850dd3f52": "\\mathcal{O}_\\mathfrak{X}", "0f6a4684a0e94788263f4648b08ad591": "-\\nabla\\Phi = a", "0f6a516c14b744036ed6bdd9522de8bb": "N_0 = ", "0f6aaa23b4c4abafb8af57a96f00db52": "\\scriptstyle 1,\\dots,m", "0f6accea29c9b0cd88f02c0bb8ad2f5c": "D(x,y,z)", "0f6af10bb3ba2496ca65735fea21bf55": " \\mathrm{CV(RMSD)} = \\frac {\\mathrm{RMSD}}{\\bar x}. ", "0f6b1ff44038c4dceccde97fde4ec400": "T_p(S)", "0f6b371db1cc3b0fee9e2f437b95479b": "\\textstyle \\oint_C ", "0f6b8dae6e45b5d8e94d2eab32ccd094": "\n e^{-iHt}\\vert J, \\gamma \\rangle = \\vert J, \\gamma + \\omega t \\rangle\\; ,\n", "0f6b8e39c7c8a53464afccb74eb48c66": "\\sqrt{a^2+b^2}", "0f6b959e66ff686f45bc9c74333ab4c7": "\\xi_1 = 2/3 u_1, \\xi_2 = 2/3 u_2, \\xi_3 = -2/3(u_1 + u_2)", "0f6b9aa74b7b873b1242a04c381c15d1": "n(n+1)(n+2)(n(4n+1+1/2)+(4n+1/2))", "0f6bc7f53815aac4f372dd7aeaacc00b": " \\infty ", "0f6bd8a1491c8d8d8217074717ec5f2b": "\\approx\\int_{\\partial N} f^\\mu \\, \\mathrm{d}s_\\mu .", "0f6c1667564081fa0f06039091c3fb8e": "(z - i\\sqrt5)", "0f6c9d456bbd7638819aea3447715a81": " \\big\\{ v\\Big({\\textstyle \\sum_{k=1}^n } \\mathbf{1}\\Big) : n \\in \\mathbb{N} \\big\\} ", "0f6cd4a6cd050026e41db79e658db0d5": " Rec(w',SS(w)) = w ", "0f6ce229c71b3b00f3e58391bf4daec7": " U^{c} =\\empty.", "0f6d58e9376213cef83c595ce9d7a057": "\\frac{dr}{dt}=A r", "0f6d68c8eb550bb4361f03d69ebacefe": "\n \\langle F | \\exp\\left( {- {i \\over \\hbar } \\hat H T} \\right) |0\\rangle =\n\\left( {-i m \\over 2\\pi \\delta t \\hbar } \\right)^{N\\over 2} \n\\left( \\prod_{j=1}^{N-1} \\int dq_j \\right)\n\\exp\\left[ {i\\over \\hbar} \\sum_{j=0}^{N-1} \\delta t \\left( {1\\over 2} m \\left( {q_{j+1}-q_j \\over \\delta t } \\right)^2 - \n V \\left( q_j \\right) \\right) \\right]\n ", "0f6d87b5c4f3eb0c263bd87de4894096": "\\epsilon_F", "0f6dd6b7ce3aa84e81ec8ec2eb0eacd6": "t = 2\\frac{d}{V}", "0f6df9808fdd3fa5f2a8f534c49ed0a5": "\\mathcal{O}^\\times", "0f6e0da8f27afccc12baa42a54545e11": "w[k]", "0f6e67b0b39bda7131c549f02ccc2654": " \\operatorname{dim}_{\\mathrm{Haus}}(X\\times Y)\\ge \\operatorname{dim}_{\\mathrm{Haus}}(X)+ \\operatorname{dim}_{\\mathrm{Haus}}(Y).", "0f6e9f089a23b00f0e39ab4adce1e5f0": "\nTP+FP+FN+TN=n(n-1)/2\n", "0f6eda019c8a968ce0a901771ab35492": " a \\tfrac{b}{c} = a \\times \\tfrac{b}{c}", "0f6f045bc2bc0d0ad36fd8d538290b36": "K(u) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{1}{2}u^2}", "0f6f26ea2b255696850c427bce93fb29": "\\Delta f = r^{1-n}\\frac{\\partial}{\\partial r}\\left(r^{n-1}\\frac{\\partial f}{\\partial r}\\right) + r^{-2}\\Delta_{S^{n-1}}f.", "0f6f3c72940b1f4e44a9c456f5f7feb7": "I=c\\epsilon_0 E_a^2/2", "0f6f41df58abee1a5911cc2abe770c6e": "p_n(t^2)=(a+b)_n(a+c)_n(a+d)_n \\; {}_4F_3\\left( \\begin{matrix} -n&a+b+c+d+n-1&a-t&a+t \\\\ a+b&a+c&a+d \\end{matrix} ;1\\right).", "0f6f4e577681817576b132a249576597": "\\gamma\\cdot z = (\\sigma_1(\\gamma) z_1, \\dots, \\sigma_m(\\gamma) z_m)", "0f6f79b4e7f27028a46817ea08796b80": "\\int\\!\\!\\!\\!\\!\\int\\!\\!\\!\\!\\!\\int_{\\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\;\\;\\;\\subset\\!\\supset \\mathbf D\\cdot\\mathrm{d}\\mathbf A", "0f6f8b0c71033aca81444d3c33bf5fdc": "E^{\\alpha \\beta \\gamma \\delta} = - g^{\\alpha \\zeta} g^{\\beta \\eta} g^{\\gamma \\theta} g^{\\delta \\iota} E_{\\zeta \\eta \\theta \\iota} \\,.", "0f700bb9c48e768a657326fdcf5d01ba": "50i+10", "0f703e8a6f24bff70706f1c825e5a18f": " [I_C] = -\\sum_{i=1}^n m_i[\\Delta r_i]^2,", "0f708423ecaa127142ef36159287b73c": "E[h^2(X_1,X_2)]<\\infty, \\, E|h(X_1,X_1)|<\\infty, ", "0f70c4a9ee079ab888365a33cc36ffe3": "A_{21}g_2(e^{h\\nu/kT}-1)+ B_{21}g_2F(\\nu)= B_{12}g_1e^{h\\nu/kT}F(\\nu)\\,", "0f70db458164242f6b547fe5d7664fb8": "\\langle A \\rangle_\\rho", "0f711b414fb6ed6b3be70c83c9bbf63a": "h_j \\,", "0f71445e144d4763dc511b5a6d9a75f9": "\\operatorname{Div}(X)", "0f717f78bace9a6cfea28c74e156e770": "\\langle x, y \\rangle + \\langle y, x \\rangle \\le 2|\\langle x, y \\rangle| ", "0f718175813b99b38b10d8ed1b77239f": "X^{(3)}", "0f71e50e24e47df7de4c47cb3b30cc47": " NL = 1000*({ELA \\over A_{Floor}})*({H \\over H_{Ref}})^{0.3}\\,\\!", "0f71eb7a24845d771b41131af87b288d": "b_{i} ", "0f72416ba535f2e5ee53e26ba25388fd": "\\Psi(r)\\propto \\frac{e^{ik r}}{4 \\pi r} \\int\\!\\!\\!\\int_\\mathrm{aperture} E_{inc}(x',y') e^{-i (k_x x' + k_y y') } \\,dx'\\, dy',", "0f727abd001d9a3a2d6b720d4e8a2e15": "\nz = \\ z\n", "0f72937f28cee25bd6d096a3414d1c35": "K_b(message')=K_b(message)", "0f72d22546951413f8170379faf822d4": "\\int_{\\mathbf{R}^n} f \\,dV = \\int_0^\\infty \\int_{S^{n-1}(r)} \\exp\\left(-r^2/2\\right) \\,dA\\,dr,", "0f72fd197f769468040706159452ea25": "\\operatorname{Pic}(X) = H^1(X, \\mathcal{O}_X^*)", "0f7324a963a0272333f1d3710b76dacc": "R_iA_j \\subseteq A_{i+j}", "0f7354834f8744aeba4a32c639224701": "\n2z = \\frac{x^{2} + y^{2}}{\\sigma^{2}} - \\sigma^{2}\n", "0f73b5f9fb185c4b07daace297b590aa": "f_0 = {1 \\over 2 \\pi R_0}\\sqrt{3 \\gamma p_0 \\over \\rho}", "0f73e18171d0366d89444bb39f0192c3": " \\exists k>0 \\; \\exists n_0 \\; \\forall n>n_0 \\; f(n) \\leq g(n)\\cdot k", "0f742253a17b7057d3fb9914245a04bd": " p \\times k ", "0f74665df199b3055a82560d54109e71": "(x_1,\\ldots,x_k)\\preceq(y_1,\\ldots,y_k)\\ ", "0f75579333df0e185a4dbb96c76a535a": " \\pi(x) = \\Pi(x) + O(\\sqrt x). ", "0f75a3dad716a836e5a8a14392ec7132": "\\sin\\frac{\\pi}{5}=\\sin 36^\\circ=\\tfrac14[\\sqrt{2(5-\\sqrt5)}]\\,", "0f75ace06351b0d194a690efb2c87b45": "(x_1-x_2) = -(x_2-x_1)", "0f75be97bbec5955c71233e79d221a17": "F_{142} x - F_{141} = 0", "0f75d3ab56044b4d125acf01586561fb": "Z \\cap S", "0f763abdee89b55d8004ae55ea25a795": "|\\phi\\rangle", "0f768ac5d5dea8d93716a27da05871de": "a_{2}", "0f77141e51439de02bfaea54740504b8": "x(p, w)", "0f773e1a0284e3af2e9fef701a31b379": "n=\\int_{E_0}^{E_0+E_F}g(E)dE", "0f77e84fdaf785dac996824c9d4f7916": "m\\left( x^\\mu \\right)", "0f780cb28944a42fe461040d31cd2447": "\\frac{\\partial L}{\\partial\\dot x}", "0f789187e3943ccab592d1192aa153b1": "\\psi(\\xi)", "0f78f2358bab8308cede6a7c7c16ffa3": " -\\infty < t < \\infty, \\; 0 < r < \\infty, \\; 0 < \\theta < \\pi, \\; -\\pi < \\phi < \\pi ", "0f7910ba96dbf6e7c5b57195f78637a0": "\\Box R/R", "0f79a6cacdf411053092297d794b3670": "N = \\infty", "0f79cf36792c15af84e4507027b3ff70": "\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) u_x", "0f79e3880da261c5afdaf5aa8e2d6687": "\\,\\,\\boldsymbol{\\sigma} = 2\\mu\\,\\boldsymbol{\\varepsilon} + \\lambda\\,\\text{tr}(\\boldsymbol{\\varepsilon})\\,\\boldsymbol{I} + \\lambda'\\,\\boldsymbol{\\varepsilon}\\cdot\\boldsymbol{\\varepsilon}\\,\\,", "0f79e7d8a622bfe707c05af890b7b6eb": "\\Phi_{ij} = U_{,i j} - \\frac{1}{3} {U^{,k}}_{,k} \\, \\eta_{ij} ", "0f7aaa364d9356b7f91f3072b222a3c6": "d_H \\ge t+1 ", "0f7b32ed66568648ebae5322840ad78c": "\\mu_p(N_A) = 48 + \\frac{447}{1+(\\frac{N_A}{6.3\\times10^{16}})^{0.76}}", "0f7b3cd1526c78e15efcfcf1791d1e1f": "\\lambda_\\mathrm{ap}", "0f7ba275ca05c83d55385b55da9bbf56": "x^2+y^2=z^2", "0f7ba4ca8213c1f532be08df739b0c38": "a_i \\,\\!", "0f7bf6d9586efc297fac6722931839c1": "\\rho g H sin\\beta + \\rho_w g D sin(\\alpha + \\beta) + \\tau_b = \\frac{d}{dx}\\int_{0}^{H}\\sigma_x dz", "0f7c2934fc08c1a3a700f5c978dd1f3f": "e^z(-1)^{a-b}U(b-a,b,-z)", "0f7c48a14239fdd0e8ed6f4b2f6e218f": "i\\ne 1\\ ", "0f7c8574d2567a4eb8f69d82fcb17b14": "u_n(y,z)", "0f7c93358031a9ffbee7bbe4c380cfbf": "y_n", "0f7cca39b8bba5b389b3486e336e9a4a": "f_\\mathrm{blue}\\,", "0f7d136c8779a9b810e16e8270476c60": "v_{\\text{C}}(t) = V_p \\sin(\\omega t) \\,", "0f7d3ce22dd8e5160beebd3fe15be387": " \\langle \\mathbf{x},\\mathbf{y} \\rangle = \\eta_{ij} (e_{\\alpha}^i \\, x^{\\alpha}) (e_{\\beta}^j \\, y^{\\beta}).\\, ", "0f7d56618b7b90b9a9ea7ea38437fa9f": "r u_i + s u/u_i = 1", "0f7d626ec9d0f6e80a235cbb817b1190": " \\frac{dS_j}{dt} = f_j \\left (S_1,S_2, \\ldots, S_N \\right) ", "0f7d71091bb8212bc033d14dd8e77ee4": "\\zeta(s) \\zeta(s-a)=\\sum_{n=1}^{\\infty} \\frac{\\sigma_{a}(n)}{n^s}", "0f7eaa92dc9a05bc36e5753b4118d211": "\n\\sum_{h \\in H} \\underbrace{ p(A|X,S,h,\\Theta) }_{\\mbox{Appearance}} \\underbrace{ p(X|S,h,\\Theta) }_{\\mbox{Shape}} \\underbrace{ p(S|h,\\Theta) }_{\\mbox{Rel. Scale}} \\underbrace{ p(h|\\Theta) }_{\\mbox{Other}}\n", "0f7eab3cca94103ba848ce3d5059c34b": "m^{(t+1)}_i = \\frac{1}{|S^{(t)}_i|} \\sum_{x_j \\in S^{(t)}_i} x_j ", "0f7ec556d479b09c40a9d2c4f98fbe7a": "m_j = 0", "0f7ecebd8ba83feed39378f141b6bc62": "D>4", "0f7f03c76044a68faf851319543a29ac": "I_{r,r-1}", "0f7f1890a81292060210bbf8dcd4218a": "\\delta^n[f](x + h/2)", "0f7f23ae06b9d3ef30259d0b033e8234": "\\mathrm{Vol} \\,B(p,r) \\leq \\mathrm{Vol} \\, B(p_k,r).", "0f7f40d5077a28acd9e794cd42d7d78e": " r_\\mathrm{ corr } = r \\frac{ 1 + \\theta c_{ xy } }{ 1 + \\theta c_x^2 }", "0f7f7ab585fb1fca9e53c736a03fad97": "E \\not \\in \\operatorname{FV}[G] \\and E \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] = \\operatorname{sink-test}[G\\ \\operatorname{sink-test}[(\\lambda E.H)\\ Y, X]] ", "0f7f867274a4ad48d09c66ea6498656a": "xy = x", "0f7fd84b7c85c66f952364e4d10bab0f": " y_n \\approx y(t_n) ", "0f7ff6789afae668ff60803160b67208": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ -0.04\\end{smallmatrix}", "0f80b58328ab87252779eb2851a8ceef": "\\textstyle 2^5 - 1 = 31", "0f80cb13f63ec51875946b4ccbbeb7d2": "N = m(g + a)", "0f80d4c54b7c1098bfd0e274d5f5850d": " \\bar{x} = \\left ( \\prod_{i=1}^n{x_i} \\right ) ^\\tfrac1n", "0f80d78e8efcab9a201d22706043965b": "a=x_0 x", "0f81dc166bb16a659e0955c88a53b1eb": "V_{\\rm as} = \\rho \\cdot c^2 \\cdot S_{\\rm d}^2 \\cdot C_{\\rm ms}", "0f81fcaf86f636038761640ab3705284": "\n | x\\rangle = U(g(x))|\\psi\\rangle\\in {\\mathfrak H}.\n", "0f82555be6cd4b6d885314d1df76ac3f": "R_\\text{D}\\,", "0f82c0ed2cb6cbffade2408f25fd7b34": " \\sgn(x) = \\begin{cases}\n-1 & \\text{if } x < 0, \\\\\n~~\\, 0 & \\text{if } x = 0, \\\\\n~~\\, 1 & \\text{if } x > 0. \\end{cases}", "0f82d548df24685f230e386d952d3332": " U = \\langle E \\rangle = \\langle \\sum_{i=1}^N E_i \\rangle = \\sum_{i=1}^N \\langle E_i \\rangle", "0f82f41bd79dafae16c7d83b3176f326": "d_1,", "0f834a186f2e5eee334482f0cb70ecf9": "D\\chi_E", "0f836e48ba01cc613d9a133379003dec": "\\Delta \\cup \\{A\\} \\vdash B ", "0f837b7ae1e5c94134c017416474bcb8": "\\langle f, f \\rangle = 1\\,", "0f837d9e15f43ad43049981331acd0e7": "\\wp'(\\omega_i/2)^2=\\wp'(\\omega_i/2)=0", "0f83cabb59b8731a0b5df7a0af120654": "i(e)=e", "0f83e2f37c8d0867fa853a78a32c4c8e": "\nA=\\begin{pmatrix}a_1&\\dots&a_n\\\\b_1&\\dots& b_n\\end{pmatrix},\\quad\nB=\\begin{pmatrix}c_1&d_1\\\\\\vdots&\\vdots\\\\c_n&d_n\\end{pmatrix}.\n", "0f83e95a31c99346fb57a6bf364b8a37": "j \\omega \\ ", "0f83f2488a3704ac5e4e42c33a31833e": "n_s = \\frac{V_s}{V_v} ", "0f84b7c96c4d278db55fd56096d4fa41": " \\mathrm{stsys}_1{}^{n} \\leq \\gamma_n \\mathrm{vol}_n(M),", "0f8530c37c5027d2a2ea974460bf65fd": " {t = t_2}\\ ", "0f85394623f7bf6d1c1aaf8da61cda12": "\\mathbf{x_2}", "0f85856a4e282968d7f20a340070a3c2": " x_k^1", "0f859fba06e1f0ab16df9ef2f523fea8": "\\mathbf{j}_3", "0f85a70fb1740380f25ecd468dbd6093": "g(x,b)", "0f85b89606bc6a4d440b60da136140a4": " \\langle X^{N1} \\rangle", "0f85bc3596e7cf85b5f4101792d05b6f": "\\mathrm{Volume} = \\int_0^{2 \\pi } d \\phi \\int_0^R h \\rho \\ d \\rho = h 2 \\pi \\left[\\frac{\\rho^2}{2 }\\right]_0^R = \\pi R^2 h", "0f86346404e78b9a0e706e0e4f573f08": "tan^{2}\\theta", "0f8639ee60c9a331a1d10d1eeea27dd4": "\\textstyle (x_n - \\bar x_n) = \\frac{n-1}{n}(x_n - \\bar x_{n-1})", "0f86434a4ebb51e2fd16540da2ae1193": "\\langle {\\mathbb N}, S\\rangle", "0f8648406e1a728158ba22f115699c3a": " k = {{k_0 + k_\\inf {\\gamma^{0.5}}_r } \\over {1+{\\gamma^{0.5}}_r}} ", "0f869d71054fa3024f7693f4afb9c542": "\n\\begin{align}\np(z_i \\in \\mathbf{Q}|\\mathbf{z}_{1,\\dots,i-1},\\alpha) \n& = \\lim_{K\\to\\infty} \\sum_{u\\in\\mathbf{Q}} \\frac{\\alpha/K}{i - 1 + \\alpha} \n\\\\\n&= \\frac{\\alpha}{i-1+\\alpha} \\lim_{K\\to\\infty} \\frac{K-L}{K}\n\\\\\n&= \\frac{\\alpha}{i-1+\\alpha}\n\\end{align}\n", "0f86d0b48ede27b4e8d8ee6cd8400ce4": "\n\\frac{{e^{ik_0 r} }}\n{r} = i\\int\\limits_0^\\infty {dk_\\rho \\frac{{k_\\rho }}\n{{k_z }}J_0 (k_\\rho \\rho )e^{ik_z \\left| z \\right|} } \n", "0f86d2258150f7079f2660a8e48262f4": "\\beta> 1", "0f86d5ddc9a449255d2b09de9f58cb96": "E'_y = {E_y\\over\\sqrt{1 - v^2/c^2}} ", "0f86ef08ff272986f2b2f002a9e23d7a": "\\sum_{n=i}^\\infty \\frac{\\left[{n\\atop i}\\right]}{n (n!)} = \\zeta(i+1) ", "0f876286cee3b5efed5e0fa42ff90a73": "i_{C}", "0f8768c7908fb5025bd1ccb70d730119": "c(t)=\\frac{1}{2\\pi} \\frac{d^2\\phi(t)}{dt^2}", "0f8775338487ed50957f07771d8a8e0a": "y = Dx", "0f87a4ca5fbfa50fdc9f583068ad054b": "T_pM\\to T_pM", "0f87c81c859fa514e7097b84ea63a58f": "\\theta-\\alpha", "0f87d593cb2ab07cb1a091232e007dbb": "(b^z)^u=b^{zu}", "0f87e9db119b585ffa88f2d11d1e814a": "-r^{-2}\\,", "0f880e3e23ef56452a2e4a4ff19a20c1": " b = \\infty ", "0f888189578e25ed99e12c83011fc3f0": "\\dim\\, \\operatorname{Sym}^k(V) = {N + k - 1 \\choose k}.", "0f888640ea5fdc8569e9a5876a0134cb": " \\operatorname{de-let}[f\\ (x\\ x)] ", "0f893f2edc2b22d08c3f91ac13e77e4e": "\\mu\\left(f^n(U) \\cap U \\right) = 0.\\,", "0f8957fab6484c33b5001d50cc3130e9": "c_I", "0f897d1e8594a75f1f48d424b4641c88": "\\phi ~\\mathcal{U}~ \\psi", "0f89a646bc78da6ff833e9f337d1ffb2": "\\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}", "0f89d0a22ae8af89860425fe8b060dc3": "dG\\left(T,p,n_{i}\\right) = -SdT + Vdp + \\sum_{i} \\mu_{i} dN_{i}", "0f89fc758d1ebc6683f7bf0227582da7": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V\\nabla\\times\\mathbf{F}\\, dV =", "0f8a2408502495b1b7ad16253d2962e9": "S(x_1,x_2,\\ldots,x_n) = (x_2,\\ldots,x_n,0).", "0f8a3165ed272b22750e56b9a30cdde9": "q = m/n", "0f8b06f3561bb6f6955827c926766a86": "N.k + M_e", "0f8b1d784f2fb7550f60fc4cf5d35427": "R(q)=P(q)\\cdot q ", "0f8b57e8522423166fce1d4e0fc5a59b": "\\mathcal{F}_n^+", "0f8b7217482e5c28b4cd673605bed1c3": "\\Gamma_{a \\;\\; i}^{\\;\\; j}", "0f8b7c3dc93cfd7502480ed615925d62": "\\rho =N/L", "0f8b9b1f5e5f90f00ba08ba8e341c92b": " (v_P)_{\\mathcal{P}}", "0f8bcc58dfd18b199fd474c5120d0c46": "\n \\cfrac{\\partial^2 \\mathbf{v}}{\\partial t^2} - c_0^2~\\nabla^2\\mathbf{v} = 0\n \\qquad \\text{or} \\qquad\n \\cfrac{\\partial^2 p}{\\partial t^2} - c_0^2~\\nabla^2 p = 0,\n ", "0f8bf99dd27c2e921f76fc479848b221": "\n \\vdash A , \\lnot A\n ", "0f8ca084a5043d103e4ef5e85549c8c7": "H(x) \\triangleq\n\n\\begin{bmatrix}\nh_1(x)\\\\\nh_2(x)\\\\\nh_3(x)\\\\\n\\vdots\\\\\nh_n(x)\n\\end{bmatrix}\n\\triangleq\n\\begin{bmatrix}\nh(x)\\\\\nL_{f}h(x)\\\\\nL_{f}^2 h(x)\\\\\n\\vdots\\\\\nL_{f}^{n-1}h(x)\n\\end{bmatrix}", "0f8ca64a950171d84952b80f660ccf24": "|x|<1 ", "0f8cde1d90db67b5e9e84fddd1783dc1": " \\mathbf{S} = \\begin{pmatrix} 1 & d \\\\ 0 & 1 \\end{pmatrix} ", "0f8ce94b375579bd280ecbe491064831": "v \\in \\mathbb{R}", "0f8d2b097d78573fb144352f861e978c": "\\hat{G}(\\boldsymbol{k},\\omega):", "0f8d6f8da96e5d4e7eae1971269e2d38": "s=(\\ldots, (s_{i},t_{ei}),\\ldots)", "0f8ddd36e329e9106facd25b79876b52": " | z_k | < 1 , k=1,\\ldots,n ", "0f8e4372551617be928b5729e09a3725": "\\{a, b, c, d\\} \\in \\mathbb{R}", "0f8e959c393de05b2faafb5e92cd9edb": " F : M \\times N \\to \\mathbb{R} .", "0f8eb2799178e2e3df91d90fafb61084": "p = \\omega \\cdot \\varepsilon_r'' \\cdot \\varepsilon_0 \\cdot E^2,", "0f8ece4ba3b54c75fc3f090ddae8f901": "^{\\;}q^{i}(\\xi,\\tau)", "0f8fc9f8362111886d55e05f39729230": "\\beta=\\partial_t F_t|_{t=0}.", "0f901e21172331d03ccfe8e3553e99bb": "\\Delta=L/16", "0f90453c26a5fc908a5345e189a52440": "\\textstyle W_p^m(\\Omega)", "0f909cb7dfab5fa7b870ca9f386c910c": "\\mathrm D\\, \\mathbf F\\left(\\mathbf X\\right) = \\frac{\\partial\\, \\mathrm{vec}\\ \\mathbf F\\left(\\mathbf X\\right)}{\\partial\\left(\\mathrm{vec}\\ \\mathbf X\\right)^{\\prime}}.\n", "0f90bc5155ea84c4352bc03ba37e0517": "\\alpha > \\beta", "0f90be5d1b4c19c754d912e0729add74": " \\mathbf{\\hat{e}}_{\\bot}\\,\\!", "0f90c57425a10c0c6b841d1d82f634bb": "L_{\\frac{1}{2}}", "0f911c00e08e9f7cfeff6527ed9ef486": "c_\\text{deep}", "0f911c2e59c279f47901d4467276b92d": "\\frac{1}{H_{N,s}}\\sum_{n=1}^N \\frac{e^{nt}}{n^s}", "0f913aa03754914c7c1c442943c5b5ce": "(1,0), (0,1)", "0f916a9cab3d3315eb76b88ee8ddd16c": "\n\\nabla \\cdot \\mathbf{v}=\n\\frac{\\partial \\dot{x}_1}{\\partial x_1}\n+\\frac{\\partial \\dot{x}_2}{\\partial x_2}\n+\\frac{\\partial \\dot{p}_1}{\\partial p_1}\n+\\frac{\\partial \\dot{p}_2}{\\partial p_2}\n=\\frac{\\partial }{\\partial x_1}\\frac{\\partial H}{\\partial p_1}\n+\\frac{\\partial }{\\partial x_2}\\frac{\\partial H}{\\partial p_2}\n-\\frac{\\partial }{\\partial p_1}\\frac{\\partial H}{\\partial x_1}\n-\\frac{\\partial }{\\partial p_2}\\frac{\\partial H}{\\partial x_2}\n=0\n", "0f916aebbb8b12e8f96bf781fc854e26": "\n(x,y,z)\\rightarrow( f(x,y,z)+x_0, g(x,y,z) + y_0, h(x,y,z) + z_0 )\n", "0f91836fce3a27a5dfe0e56fc658e00d": "\\Delta u=0", "0f91906a35c86ed83e3f85281673751f": "Q_2", "0f9190e956ee7347cfd61a1f0488793f": "D = \\Gamma / \\delta x", "0f91a78f4699260331be744450bc6e06": "f: X \\to \\mathbb{R}^d", "0f920ab28129df0e212b4c561554520a": "S^{-1}S", "0f921b0605129a2e67da2b634d296c43": "C_{ML}=\\frac{1}{n}S.", "0f92446c6a38c0bc35d6003f43809059": "V_C=\\frac{F_v\\cdot a}{a+b+c}", "0f926a776da9725cf76c44c715bc70aa": "\\varphi (x) = \\mathbf{E}^{x} \\left [ f(X_{\\tau_{H}}) \\right],", "0f927dc9c2eb688d1821f31d6234c43a": "\\frac{\\Gamma_1, A, \\Gamma_2, B, \\Gamma_3 \\vdash \\Sigma}{\\Gamma_1, B, \\Gamma_2, A, \\Gamma_3 \\vdash \\Sigma}", "0f929cb64eaa6ae1dcf1267c0da63e01": "\\varphi(t) = \\int_{-\\infty}^\\infty F(x)e^{itx}\\,dx", "0f933a81fadd3cfb58a83144f1d1f784": "\n F_o=\\int (f_c+\\lambda) dA+\\gamma\\oint_C ds\n", "0f938ae0c3b91676bd112eb548fb6263": "\\frac{V_i - V_o}{L}t_\\mathit{on} - \\frac{V_o}{L}t_\\mathit{off} = 0", "0f93b9917f7d148c9156acd04c4a4768": "M \\rightarrow [~]_{n^\\searrow\n\\;\\|\\;\\overline{n}^\\nwarrow}", "0f93e4769d2b006339748a5dc99f8568": "H_*(C_*(X \\times Y)) \\cong H_*(C_*(X) \\otimes C_*(Y)).", "0f9425f5e2b2df044511faced9303ef4": "\\displaystyle{a^{-1}\\le {|f(z_1)-f(z_2)|\\over |f(z_1)-f(z_3)|} \\le a.}", "0f943fe9d35f55afd59f64c531070806": "\\tilde r", "0f944c42f590fb7f5995b075e389e60d": "\\bar{\\lambda}_e \\equiv \\tfrac{\\lambda_e}{2\\pi}\\simeq 386~\\textrm{fm}", "0f94ba5a0860b67d84f738a44150adf4": "\\{\\gamma^\\mu,\\gamma^\\nu\\} = 2 g^{\\mu\\nu} \\,", "0f94f7b73affbce3a4f189203c6db4dd": "\\alpha_n = (1-aq^{2n})\\sum_{j=0}^n\\frac{(aq;q)_{n+j-1}(-1)^{n-j}q^{n-j\\choose 2}\\beta_j}{(q;q)_{n-j}}.", "0f953f13ef3001c1c8049820623f1b68": "\\hat{R}_m = 2(\\hat{L}_m\\cdot \\hat{N})\\hat{N} - \\hat{L}_m", "0f95b1c1ead7bb1fe5a5bf09ddad8a00": "i, a, b \\in \\mathbb{Z}", "0f95d2f2265095c8032bb38dd0790e74": "\\lambda=1", "0f95fdb4329c089877c5ddfb6bfb4c82": "\\phi(t) = \\omega_0 t - kx = \\omega_0 t - \\frac{2 \\pi}{\\lambda_0} \\cdot n(I) L", "0f9604a9c8698a2fb16fdaf0012a69c1": "H_n(X) = \\tilde{H}_n(X)", "0f9611cb17be237d2866355fcaaac3bb": "n^{4/3};", "0f967f36c6c9fa31619c810c7f7cacf9": "\\partial_\\mu \\hat{A}^\\mu", "0f969be3fd4a88a56a91e846d23a7aea": "u_i(x, t, \\theta_i) = \\theta_i x_i + t_i.", "0f96abfc454fc36ff095577f9c27f967": " \\langle \\phi(k_1)\\phi(k_2)\\phi(k_3)\\phi(k_4)\\rangle = {i\\over k_1^2}{i\\over k_2^2} {i\\over k_3^2} {i\\over k_4^2} i\\lambda \\,", "0f96c8d82fcfd9adf03bd9ed16ba6496": "\\tau_E = \\frac{W}{P_{\\rm loss}}", "0f97884bd76cfbb8588c95f6faa27905": " ds^2 = -g^2 x^2 dt^2 + dx^2 + dy^2 + dz^2, \\; \\forall x>0, \\forall t, y, z", "0f97aa07d7c5242faf011e9f4bfd86cd": "k/s \\le j/r", "0f982eb4e607b6bd0e95c5c17aa061bd": " -\\frac{d^2}{dq^2} + q^2 = \\left(-\\frac{d}{dq}+q \\right) \\left(\\frac{d}{dq}+ q \\right) + 1 ", "0f988f6267e703b3e704be5c92e5928f": "\\beta*\\,", "0f98ecbc950c82f6766d05a0de81302d": "\\dot{a}=-\\frac{i}{\\hbar}[a,H_\\mathrm{sys}]-[a,c^\\dagger]\\left(\\frac{\\gamma}{2}c+\\sqrt{\\gamma}b_\\mathrm{in}(t)\\right)+\\left(\\frac{\\gamma}{2}c^\\dagger+\\sqrt{\\gamma}b^\\dagger_\\mathrm{in}(t)\\right)[a,c]\\,.", "0f9925457a8362dfbd541adaef71e405": "\\hbar\\omega", "0f9960105518def1fcd4dc394d1e608a": "\\int^T_0 R_N(t,s)k(s)ds = \\sum^{\\infty}_{i=1} \\lambda_i S_i \\int^T _0 R_N(t,s)\\Phi_i (s) ds = \\sum^{\\infty}_{i=1} S_i \\Phi_i(t) = S(t)", "0f999d9a8625b402e2d9ff7f1054a20a": " w_j \\leftarrow A v_j \\, ", "0f99d8fa133089747bcf97f037598cea": " \\pi _T = \\left ( \\frac{\\partial U}{\\partial V} \\right )_T ", "0f99ff7ba20d49484a5b9c59e2cc92b1": "{x{:}1{\\to}\\tau\\;\\in\\;\\Gamma}\\over{\\Gamma \\vdash x : 1{\\to}\\tau }", "0f9a473c826fd7b5b9dda2be2e16ace5": "\\Delta :R\\to R\\otimes R ", "0f9ac29b5a9afd7b205ae9659595546a": "\n\\log \\left| R - R_{+} \\right| - \\log \\left| R - R_{-} \\right| = -2Dk_{1}t + \\phi_{0}\n", "0f9b351a277d51977639c72a8f9ebd3f": "Q = \\frac {20,000} {10,000} \\times \\frac {20} {50} = 0.8", "0f9b4c013307df46756dc4df84c3299a": "I=\\{0,1,\\ldots n-1\\}", "0f9b5d101e34509491b9c7bcb93e2d91": "j = 1, 2, \\dots, N", "0f9ba803bfaff87945d4e4b62681181c": "\\alpha = {\\epsilon_0 k_{\\rm B}T\\over p}\\chi_{\\rm e}", "0f9bb27f4749a30ccde83856e055831c": "M_2 =\n \\frac{1}{\\sin(\\beta-\\theta)}\\sqrt{\\frac{1+\\frac{\\gamma-1}{2}M_1^2 \\sin^2 \\beta}{\\gamma M_1^2 \\sin^2 \\beta- \\frac{\\gamma-1}{2}}}.", "0f9bc1d2168f177a75610f41d437bccf": "\n\\varphi(U^0,t) = \\mbox{e}^{t\\mathcal{A}}U^0\n", "0f9bdddd9de1feabce131880c2803018": "\\scriptstyle f_s > 2B\\!", "0f9be017bd8dfde699a6a7839273faa5": "y(N) = \\sum_{k=0}^{N} x(k)e^{-2 \\pi i \\frac{k K}{N}} ", "0f9c5e421c4ba26dc4a5b39ebf8f89f7": "c_f = \\prod_{p\\ prime} x_i \\left ( 1 - \\frac{\\omega\\,\\!_f (p)}{p^{2+q_p}} \\right )", "0f9c742170552489d91dc8aa4fa58199": "PFB", "0f9c815e60b0f4a8eb2fd850c4cbf730": "F \\gtrsim 1", "0f9c960b45e724c54078fc607fe18bfd": " \\beta_m = -\\int \\varphi_m^*(\\boldsymbol{r})\\Delta U(\\boldsymbol{r}) \\varphi_m(\\boldsymbol{r}) \\, d^3 r \\ ", "0f9cdfcc8eb270ce86bfa80ca12449ee": "\\gamma^*\\,", "0f9d27408478b3ecc0c718eb7c9d3028": "\\sqrt{I} \\cdot S^{-1}R = \\sqrt{I \\cdot S^{-1}R}", "0f9d9792bf9bd583f1e954ce17dced17": "\\sqrt{2} \\times \\sqrt{3}", "0f9d9ba652e6e902ad2e6b231b09711d": "\n d(X,Y) = \\inf\\!\\big\\{ \\varepsilon>0:\\ \\Pr\\big(|X-Y|>\\varepsilon\\big)\\leq\\varepsilon\\big\\}\n ", "0f9dda58fd559ce047deddfae4f53712": "\\sigma_X^2=\\left (\\frac{\\partial f}{\\partial A}\\sigma_A\\right )^2+\\left (\\frac{\\partial f}{\\partial B}\\sigma_B\\right )^2+\\left (\\frac{\\partial f}{\\partial C}\\sigma_C\\right )^2+\\cdots", "0f9e59f0c9a528be1fc5cce7b139fe14": "\\varepsilon = \\frac{T-T_c}{T_c} ", "0f9e6cee356d9b261c5547e57347689c": " \\sigma_{log} = \\sqrt{\\ln\\!\\left(1 + \\!\\left(\\frac{0.42}{5.33}\\right)^2 \\right)} = 0.079 ", "0f9ec194e7c7f5d384944c8feca146d1": "W_{n-1}", "0f9ecbfac90ad2cf197f5f62472b49e0": " \\omega := *L^* ", "0f9eee8d9d1e1546df7290a7a181a8a3": "\nr_{d,s} = h_{d,s} x_{s} + n_{d,s} \\quad\n", "0f9f16fbacf4e36950560330937206d0": "37 - 57 = -20", "0f9f729d515983aa8a73f4718e7aee70": "a=1+i, p=z^2-1", "0f9f9cb5d8adf0a1a4e2aba7f7168da9": "y'(x)=\\frac{T_y-y}{T_x-x}", "0f9f9d2d5ae4771fed1989ac89ba1645": " 1 - \\sqrt{2R}", "0f9fc49fd9bb017e81cdd79c8d3580ef": " A_q(n,d) = \\max_{C \\in C_q(n,d)} |C|.", "0f9ff5d924d853a7a6800ab77927f905": "S(\\mathbf{r},i)", "0fa01fa203eb55df5ee784dfb5fa9351": "\\min(x,y) \\leq H(x,y) \\leq HG(x,y) \\leq G(x,y) \\leq GA(x,y) \\leq A(x,y) \\leq \\max(x,y)", "0fa10cd5e24994a12e059ce244e612c6": "Q_1 \\bot Q_2 \\equiv Q_1 \\supset \\neg Q_2 ", "0fa180793647215fc680ba3a3edcd004": "O(\\cdot)", "0fa1d9dfdeacb5e65637b90b99097ad2": "\\displaystyle{f_z(x) = e^{izx^2/2}}", "0fa2055bb25c164ee502717117e52848": "\\scriptstyle G=\\langle H, t | t^{-1}Kt=L\\rangle", "0fa2c6c326f1a09438e512daf732a44a": "\\lim_{a\\to 0;\\,a^3\\cdot Q\\to\\rm{const.}}", "0fa2ebc31b4d8afc07009d1e871a6864": "e_n \\sim \\mathcal{N}(0,1),", "0fa330514cbf4c99e71d00c822ee69f7": "\\frac{dP_{\\text{Magnetic dipole}}(\\mathbf{x})}{d\\Omega}=\\frac{Z_0}{32 \\pi^2}k^4\\|\\mathbf{m}\\|_2^2\\sin^2\\theta", "0fa37b1f811139e9bf7e1a5476784fc3": "x^5 + ax + b = 0\\,", "0fa3a2a5fd0bf9b1850bced816f95052": "\\Delta \\mathcal{F}\\propto\\Delta \\mathcal{O}^{-1}", "0fa41df8aa07b065276c83b4afd6e7ae": "E(X^{-n})", "0fa42ca38f870da9bd73afc47e8bbc9a": " \\alpha \\ominus \\beta = \\alpha - \\beta", "0fa47e711bb72a7351dfd0c1cf8ad6df": "0 < x_1 < \\frac{1}{5} < x_5 < \\frac{2}{5} < x_3 < \\frac{3}{5} < x_4 < \\frac{4}{5} < x_2 < 1.", "0fa5d837c0ea73edb459203fd808e85e": "x \\cong y", "0fa63ae080f50c214794e6b3ea8da70c": "m_0=\\frac{4}{3}\\frac{E_{em}}{c^2}", "0fa66625a016472690cfa7b1cdda64da": " a= x_0 < x_1 < \\dots < x_k =b ", "0fa67916078db2998318fd0a54fa535b": "j=1,2,3,\\dots,n", "0fa6f01d979dfba4d4e5d75640f91b4f": "\\mathfrak{sl}(2,\\mathbf{C})", "0fa751e9f3ae19a1f2c7ef27367c07eb": "(\\ast_{v\\in V} A_v)*F(E^+T)", "0fa763fa7cc2b9770388ffdb57db8127": "b e^{-bx} e^{-\\eta e^{-bx}}\\left[1 + \\eta\\left(1 - e^{-bx}\\right)\\right]", "0fa777d8e14e0434b8d5985e3fc57cf6": " F(gh) - F(g)-F(h) = \\int_{\\Delta}\\, d\\alpha,", "0fa77e82061c13e821ef9d94e228a77c": "\\scriptstyle \\varphi", "0fa78f9ce83744e1651c9d729c2d19b6": "\\textstyle -1", "0fa7b7634c1c618c7bae39410456cc08": "P_{4}^{-4}(x)=\\begin{matrix}\\frac{1}{40320}\\end{matrix}P_{4}^{4}(x)", "0fa7bc8f6062acc1e71147640c4b9da5": "C_1(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iiiint \\dfrac{\\phi_1(\\theta,\\tau)}{\\phi_2(\\theta,\\tau)}C_2(t,\\omega^')e^{j\\theta(t^'-t)+j\\tau(\\omega^'-\\omega)}\\, d\\theta\\,d\\tau\\,dt^'\\,d\\omega^'", "0fa7d12282e633da041cc172917a8108": "g(Z_{t})=\\theta Z_{t}+\\lambda(|Z_{t}|-E(|Z_{t}|))", "0fa7fe94891aebfbc82022869afb19ec": "X \\vdash Y", "0fa80e1df80e9b60b4f5fedaaf632bc8": "\\,f(x)=0", "0fa817c07564f088252888f576ede505": "\\rho\\,", "0fa8209d3a38dd94497468010350dd8b": "(rf)(x) = rf(x)\\,", "0fa8c88e540cdf7433e40beea09a0c13": "G/N\\times G'/N'", "0fa8e5c51f293235e2468959de72f8ef": "\\Phi_{Y,X}", "0fa8effe66bc1b24ddaa8af53760f5be": " \\frac{1}{2m} \\left( \\nabla S \\right)^{2} + U + \\frac{\\partial S}{\\partial t} = 0. ", "0fa907d602de6829b37c4a2c959b2230": "\\gamma(\\vec{s})", "0fa93a9953e32e816eda8522cef8e75c": "H^T_q(p_F(x;\\theta)) = \\frac{1}{1-q} \\left((e^{F(q\\theta)-q F(\\theta)}) E_p[e^{(q-1)k(x)}]-1 \\right)", "0fa94ba121f4a6985fd6f4f0678ceb6e": " \\hat{H} = \\sum_{n=1}^{N}\\frac{\\hat{p}_n^2}{2m_n} + V(x_1,x_2,\\cdots x_N) \\,,\\quad \\hat{p}_n = -i\\hbar \\frac{\\partial}{\\partial x_n}", "0fa961f5cbb625b076f8af4282ed13d8": "MMH_{32}^*", "0fa9c14da57558d44b68cefb00daf99d": "D = \\frac{d^2}{4 \\lambda}", "0fa9c623c985961d2fc8548b4891b591": " c_t^{cert}=A e^{gt} ", "0fa9e52f8f08dd60856a6130617f75c1": "\\; \\Omega _ R (s_1)", "0fa9f8f67620b91215f3d096e9e66683": "f(x;\\mu,\\sigma) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2 \\sigma^2}}.", "0faa0e31130965d6db0d79a2c7cfcf7e": " J ", "0faa1fa666ad0386281728a9602a701b": "\\mathbf{Q}_{13} (\\gamma_{13})", "0faa243d718b99b01770de6f707e18e2": "\\scriptstyle h^i(X, \\mathcal{F}) ", "0faa291106fd2ef98af2797f68188d1c": "X_n=\\frac 1 n \\sum_{i=1}^n Z_i", "0faab3988eb9146e3ced3ddbcb53abea": " \\operatorname{GM}(y) = (y_1\\cdots y_n)^{1/n} \\, ", "0faace7e07408e583cee41cf4b9c6784": "(\\overline{\\underset{=}{A}(kU)})^\\circ \\neq \\varnothing", "0faade81fe99028104d4339730d4baf9": "\\psi_k (r)", "0fab1491eb3a52781e4cd07a962e8a57": " \\omega_n = \\sqrt{\\frac{g}{L}}=\\sqrt{\\frac{mgr}{I_P}},", "0fab56088dded78ea605929b1783e3bc": "\n \\xi = \\tfrac{1}{\\sqrt{3}}~I_1 = \\sqrt{3}~p ~;~~\n \\rho = \\sqrt{2 J_2} = \\sqrt{\\tfrac{2}{3}}~q ~;~~\n \\cos(3\\theta) = \\left(\\tfrac{r}{q}\\right)^3 = \\tfrac{3\\sqrt{3}}{2}~\\cfrac{J_3}{J_2^{3/2}}\n ", "0fabb9a7b8e36470fa92474a1514ff24": "\\Re(s)=1, 2, \\dots,n-1", "0fabe2e478591dc2549ef6a1aa3b0bcd": " f(x_1) + f(x_2)+ \\cdots ", "0fac01993efb33ba4c2451ba28b809c5": "S(t)\\mathcal M\\subset\\mathcal M", "0fac0696414eae8925a2b28f43aeefc8": "\\prod_{i=0}^{n}{Y_i}", "0fac56945b1af10175c7a75ac2157d5e": "Pr(\\max_{i > 1} v_i < z)", "0fac5cd6c578fabcad86ec9e59df1cce": "qS_t\\,dt", "0faccedcba327b141e485149bb1ffd11": "1 + x", "0facd8b3513c1b32f9b336f12084558f": "D^2_x y\\;", "0fad07d84d58d9ba2bb4613993b2f2c7": "\\frac{\\partial\\Pi(n,k)}{\\partial n}=\n\\frac{1}{2(k^2-n)(n-1)}\\left(E(k)+\\frac{1}{n}(k^2-n)K(k)+\\frac{1}{n}(n^2-k^2)\\Pi(n,k)\\right)", "0fad15b15ebeafdae2a80e19431bba30": "\\begin{align}\n n = 0: B_m &= \\left[ m = 0 \\right] - \\sum_{k=0}^{m-1}\\binom mk\\frac{B_k}{m-k+1} \\\\\n n = 1: B_m &= 1 - \\sum_{k=0}^{m-1}\\binom mk\\frac{B_k}{m-k+1}\n\\end{align}", "0fad15bbe5861de44665ed0111c98d78": "\n Z = X + iY \\,\n ", "0fada3f02373d823c6bfbd7c93024900": "\n\\left(\\frac{-1}{p}\\right) \n= (-1)^\\tfrac{p-1}{2}\n=\\begin{cases}\n\\;\\;\\,1\\mbox{ if }p \\equiv 1\\pmod{4} \\\\\n-1\\mbox{ if }p \\equiv 3\\pmod{4}. \\end{cases}", "0fadd2bc14bfa7b494af9a2b262c9adc": "\\forall (g:b\\rightarrow b^\\prime)\\in \\mathrm{Mor}\\, B", "0fadd94fe09a2c520b2d11491138af2a": "\\Phi_c: \\mathbb{\\hat{C}}\\setminus Kc \\to \\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}", "0fae56862a6a9039e2ad23e96d6e1694": "\\scriptstyle G_\\sigma", "0fae5bf70e1a4323837f9611e68d4817": " x = y \\equiv (x <= y \\and y <= x) ", "0fae9c57fb62a3fb6854bdffe49d9386": "D_{S}", "0faed6b59fc67ec12910a1961c85f882": "\\frac{\\partial^2I(x)}{\\partial x^2} = \\gamma^2 I(x)", "0faed91409173fa10be6c072e5753621": "\\left(1,\\ 1+\\sqrt{2},\\ 1+1\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+3\\sqrt{2}\\right)", "0faf30a096d52cbdf1451e8b8a743c0c": " a_{00} = \\mathcal{L}(p_6)+p_3p_6, ", "0fafbf982deea62005cbbc0413b19cde": "\\mu_K", "0fb0114e75c3661441b93f8733ea1948": "+\\sqrt{2}", "0fb052c2216b120b244a2edf3bb492af": "-g ", "0fb05de73669fe8807c2a0c98d3fe496": "L = X^TVX + 2\\lambda(\\mu - X^Tr) + 2\\eta (W-X^T1),", "0fb08f9de4df17cfba73c11be7e975eb": "m=\\frac{1}{n} \\sum_{i=1}^{n} X_i \\,.", "0fb099fb9048dd6c61f5d2008b0b3706": "1.77\\overline{7}:1", "0fb0d33216ee0931c6a6f9c5fa188548": "X^R", "0fb0d433d9fad37f53c3e5a6b85312ac": "\\scriptstyle \\, X(t)", "0fb17d5f584924627b299f495d1742d9": " \\theta_t = \\alpha_0 + \\alpha_1 \\tau_{t-1} + \\cdots + \\alpha_q \\tau_{t-q} + \\beta_1 \\theta_{t-1} + \\cdots + \\beta_p\\theta_{t-p} = \\alpha_0 + \\sum_{i=1}^q \\alpha_i \\tau_{t-i} + \\sum_{i=1}^p \\beta_i \\theta_{t-i} ", "0fb188c49f071ddaeff5fc65511bdaee": "H(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)})", "0fb18979865765eb2943e77939ff963a": "\\langle f, \\chi_s\\rangle", "0fb1a5176d2147e105a96b782b4e2a0e": "p_a(p_a(s))=p_a(s)", "0fb2a6b483eb03968cb791ee29bd73df": "\nG = \\frac{\\textit{Acc}}{r}\n", "0fb35bbff1cfd94fa88a7a463e50c9c8": "e = \\frac{V_V}{V_S} = \\frac{V_V}{V_T - V_V} = \\frac{\\phi}{1 - \\phi}", "0fb361f0d05a7a1d3c91556a29e11cf9": "\\dot{x}_1 = -x_1", "0fb36bc20459efc3cdc0b6242a5944af": "n_{max}=L/2a", "0fb3e7ce5db72668d0919de9b0deab9d": "q(t)=\\Phi(p(t))", "0fb405d79a7565403f6aca2acba871a2": "v^{*}:= \\max_{d\\in D,\\,z\\in \\mathbb{R}} \\{z: z \\le f(d,s), g(d,s) \\le 0, \\forall s\\in S(d)\\}.", "0fb412fa3ccd41e38be30c1aba41efc6": " E = 1 - {\\tau_{\\rm pb}}/{\\tau'_{\\rm pb}} \\!", "0fb45f06675915eb74f124541cceb707": "(x a_1) \\cdot s_1 + (x a_2) \\cdot s_2 + \\dots + (x a_n) \\cdot s_n.", "0fb47e95382e5b81eae33aab878c94e5": " \\alpha_{k} \\, \\mathbf{x}_{k}^{T} \\, \\mathbf{H} \\, \\mathbf{x}_{k} = \\mathbf{x}_{k}^{T} \\, \\mathbf{H} \\, \\mathbf{A} \\, \\mathbf{y}_{k} ", "0fb4e0a8481b174af2ea429be9fe4139": "\\scriptstyle \\mu_0", "0fb4f1adb87326d648ebf5ea737d18ba": "\\frac{d^2}{dx^2} \\Psi(x) = \\frac{2m}{\\hbar^2} M(x) \\Psi(x) .", "0fb52b1a6ed6ffc91e39a26e47a7d82e": "\\frac{1}{Q} = \\sum_{i=1}^{r}\\frac{S_i}{(x - \\lambda_i)^{\\nu_i}}", "0fb54b585a17a9ab8b5859e08d12f999": "(1+i) = e^{\\delta}", "0fb57251ff3f352dd4217d86071407bc": " ~=~ \\epsilon - \\frac{(1-\\epsilon)\\delta}{1-\\operatorname{Tr} (Q\\rho)} ~\\leq~ \\epsilon - (1-\\epsilon)\\delta ~.\n", "0fb57aedc29ee64f059164942d51d3fa": "k[x_1, ..., x_d]", "0fb58c0edcdddccbef0e2b9c52cdf1ba": "f(x)=O(g(x))\\mbox{ as }x\\to\\infty\\,", "0fb5acc12988be06bdd2d21f81855447": "|\\!\\arg z| < \\tfrac{3}{2} \\pi", "0fb63170db40126b17ed35509205df94": " \\sin(20^\\circ) \\cdot \\sin(40^\\circ) \\cdot \\sin(80^\\circ)=\\frac{\\sqrt 3\\ }{8}.", "0fb68fce1c7758ebaef343f81071c7f9": "D_{\\mathrm N} \\approx \\frac {H s} {H + s}", "0fb6a79c6efcda7b93914dcb398f3cca": "\n(N,+,\\times,<,n_{0},n_{1}) \\equiv M, \\,\n", "0fb6dc15e6e90556610d31bd7de0a5fe": "{\\mathcal L}_{xy}^4: L=Lclm\\big(\\mathbb{L}_{x^m}(L),\\mathbb{L}_{y^n}(L)\\big);", "0fb6e523ddb0be7536d1371f7898412c": "p_{n+1}-p_n=O((\\log p_n)^2),\\ ", "0fb7327d656c4f5d5d06043855b64c14": "\\hat{a}_{i}", "0fb734d9f37222c4626ec4c6c6490ea3": "y_{j}", "0fb751e4d856cd3bac825d1214978535": "t_{1/2} = \\frac{1}{[A]_0 k}", "0fb7524c955b4879714bed2c99e720e4": "\\boldsymbol{M} = \\overline{\\int_{-h}^\\eta \\rho\\, \\boldsymbol{v}\\; \\text{d}z}\n = \\rho\\, \\left( h + \\overline{\\eta} \\right) \\overline{\\boldsymbol{v}} + \\boldsymbol{M}_w,", "0fb7555b2fcdb00cd4b0dbb1e74d2a94": "f(x, t) \\approx \\frac{\\left|F(\\omega_0)\\right|}{\\pi} \\sqrt{\\frac{2\\pi}{x \\left|k''(\\omega_0)\\right|}} \\cos\\left[k(\\omega_0) x - \\omega_0 t \\pm \\frac{\\pi}{4}\\right]", "0fb7878c9c075438f56bf019b6702ea6": "T_m", "0fb7b351a298a7a7c0e8d42307311cf0": "=|AC|+|AC'|=|BC|+|B'C|=|BC|+|BC'|.", "0fb7bd260bb9093999e073b239066edc": " Mainlobe \\ Criteria \\begin{cases} \\mathrm{ Main \\ Lobe > Constant \\times Side \\ Lobe } \\end{cases} ", "0fb7f6309aabf56c13d63ddf30256eb8": "C_{11}^\\lambda := x^\\lambda", "0fb7fe4f3a90cfb5c3e852ae216f38c2": "\\,\\mathfrak{g}", "0fb83a14032c05ee52464f257e53eaed": " n_{AB} = \\frac{n_A}{n_B} \\,", "0fb869ec3121a34c20cd3fe1bde0178d": "v_{OLS}[\\hat\\beta_{OLS}] = s^2 (\\mathbb{X}'\\mathbb{X})^{-1}, s^2 = \\frac{\\sum_i \\hat u_i^2}{n-k} ", "0fb8a446a1dda88e8eeb853f0c7936a6": "= \\frac{e^4}{(k-k')^4} \\Big( (\\bar{v}_{k} \\gamma^\\mu v_{k'} )^* ( \\bar{u}_{p'} \\gamma_\\mu u_p)^* \\Big) \\Big( (\\bar{v}_{k} \\gamma^\\nu v_{k'})( \\bar{u}_{p'} \\gamma_\\nu u_p) \\Big) \\,", "0fb8cae157e7181ee7643085bda5172c": "\\Pi^{0,Y}_n", "0fb8d05cb55fe538e6c907ff89fbb3f9": "\n\\gamma_1=\\frac{C'\\left(F_{\\text{mid}}\\right)}{C\\left(F_{\\text{mid}}\\right)}\n=\\frac{\\beta}{F_{\\text{mid}}}\\;,\n", "0fb8d7cbbc10479a5fa1b84e17912de8": "\\Delta d = \\frac{\\lambda}{\\pi \\cdot n\\cdot\\sqrt{\\frac{I}{Isat}}}", "0fb932c91ae74c269eacedb368b1c915": "\\mathit{g^{(1)}}", "0fb942c8435592280811d88cce7ba74e": "V \\sim {\\chi'}^2_k(\\lambda)", "0fb966248096f610895496745e95e368": "\n_pF_q^{(\\alpha )}(a_1,\\ldots,a_p;\nb_1,\\ldots,b_q;X,Y) =\n\\sum_{k=0}^\\infty\\sum_{\\kappa\\vdash k}\n\\frac{1}{k!}\\cdot\n\\frac{(a_1)^{(\\alpha )}_\\kappa\\cdots(a_p)_\\kappa^{(\\alpha )}}\n{(b_1)_\\kappa^{(\\alpha )}\\cdots(b_q)_\\kappa^{(\\alpha )}} \\cdot\n\\frac{C_\\kappa^{(\\alpha )}(X)\nC_\\kappa^{(\\alpha )}(Y)\n}{C_\\kappa^{(\\alpha )}(I)},\n", "0fb99c9bdd971cbc7ac97e8bf3ca38e2": "\\ Z = R + jX", "0fb99eb483202bc8ceff4467a6a78101": "\\zeta =\\prod_i \\zeta^i =\\zeta^{trans}\\zeta^{rot}\\zeta^{vib}\\zeta^{e}.", "0fb9b328b82db6997fb4a2b824955052": " R(1) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\delta(\\omega - \\omega_0) e^{i\\,\\omega} d\\omega ", "0fb9c2648be84b10fd2194ddb4e77174": "X^t = X", "0fb9ed11385a262ae353289145e773d6": "\\operatorname{Var}_{Y\\mid X}(Y|x).", "0fba181ef88c54352735b493592944b0": "W(\\alpha,\\alpha^*)= \\frac{2}{\\pi} \\int P(\\beta,\\beta^*) e^{-2|\\alpha-\\beta|^2} \\, d^2\\beta", "0fba4dc11d5071cba86fe42327c07ffb": "PC IOPS (mean)=1/[0.5/SSD IOPS (Iwrite)] + [0.5/SSD IOPS (Iread)]", "0fba620c0859484228589f5a45227fce": "\\alpha=\\frac {(y_{LK}/x_{LK})}{(y_{HK}/x_{HK})} = K_{LK}/K_{HK}", "0fbaa9e3951f9c47557bd14c53631adb": "z\\cdot\\overline{r}", "0fbabc341d97ce5bdfe1cef255fc5655": "\\mu_2=\\frac{k_0 p_2}{p_0} = \\frac{10\\times 0.1}{0.5}=2.0", "0fbafa8a78e5b8e4b060b74a5a439ccd": " \\nabla^2 \\equiv \\operatorname{div}\\ \\operatorname{grad} \\equiv \\nabla \\cdot \\nabla ", "0fbb921c724f9f860cb5c1f1bfb90cf9": "2^{10,000}", "0fbbaf0cc0e84424bca4f843cdd4f702": "a^2 = \\left[ \\frac{\\partial p}{\\partial \\rho} \\right]_S.", "0fbbc606b21134ea0c9f01a1e834043e": "b_{1}^{*}:= b_{1},B_{1}:= \\langle b_{1}^{*}, b_{1}^{*} \\rangle ", "0fbbef1e1ce114dc34416faf8eab0ec8": "u_{k}(\\mathbf{r})\\equiv\\left.\\frac{\\partial^k}{\\partial t^k}\\big(v(\\mathbf{r},t)-v'(\\mathbf{r},t)\\big)\\right|_{t=t_0}", "0fbc597b405aece846d7df0d3dfc94df": "\n\\begin{align}\n\\sum & \\Big ( Y_1Y_2\\ln \\Phi(X_1\\beta_1,X_2\\beta_2,\\rho) \\\\[4pt]\n& {}\\quad{} + (1-Y_1)Y_2\\ln \\Phi(-X_1\\beta_1,X_2\\beta_2,-\\rho) \\\\[4pt]\n& {}\\quad{} + Y_1(1-Y_2)\\ln \\Phi(X_1\\beta_1,-X_2\\beta_2,-\\rho) \\\\[4pt]\n& {}\\quad{} +(1-Y_1)(1-Y_2)\\ln \\Phi(-X_1\\beta_1,-X_2\\beta_2,\\rho) \\Big).\n\\end{align}\n", "0fbc611a81d11b3acd10004138f4184d": "\n\\text{volume} = nAr/3\n", "0fbc683f4b9203db0fd009ec23827f47": "{1 \\over 1}+{1 \\over 3}+{1 \\over 6}+{1 \\over 10}+{1 \\over 15}+{1 \\over 21}+\\cdots = 2.", "0fbcad5fadb912e8afa6d113a75c83e4": "a\\!b", "0fbce2fc7995c733099c64b7b03c9437": "x \\geq 0 \\land y \\geq 0", "0fbcf41bc04edf4d4cfc4fcb369628df": "\\;q_c=q \\left(1 + \\frac{M^2}{4} + \\frac{M^4}{40} + \\frac{M^6}{1600} ... \\right)\\;", "0fbd1776e1ad22c59a7080d35c7fd4db": "]", "0fbd7fe649111cedf3b16e0a383d02ac": "\\mathcal{N} (X)", "0fbdae42cf54b2dc02cff2eab55462e5": "\\hat{\\mathbf{e}}_1", "0fbdb60d9f64ab2f0173a41becc7cbc4": "\\phi \\lor \\neg \\phi", "0fbdddf2f1ccac4c2fade79aa78af39f": "\\mathrm{diag}\\left(J_{\\alpha,l}, J_{\\beta,m}, J_{\\gamma,n}\\right)", "0fbe0ea246a7ee254d0be8503ab0b3ec": "\\mathcal{H}=\\mathcal{H}^1\\otimes \\mathcal{H}^2\\otimes \\mathcal{H}^3.", "0fbe536b0ecd3ec51540e2bb9b1b9557": "x \\in \\widehat{\\mathbb{R}}, A \\subseteq \\widehat{\\mathbb{R}}", "0fbe9e972640b6ebe63d973f40b73d75": "R_B(f)= {12200^2\\cdot f^3\\over (f^2+20.6^2)\\quad\\sqrt{(f^2+158.5^2)}\n\\quad (f^2+12200^2)}\\ ,", "0fbedf9f1798f041a491fe19bd77c5d0": "\\frac{4+\\delta+\\delta^{2}}{2-\\delta-\\delta^{2}}", "0fbef3b2b762736dc45de3374d94ffac": "\\mathcal{F}(X)", "0fbf0728a057b8b1d9ea65f6b97477ca": "\\!a", "0fbf16349e67cb24b0634960bae7cc28": "\\begin{align}\n |V| &= |I| |Z| \\\\\n \\phi_V &= \\phi_I + \\theta\n\\end{align}", "0fbf5183bc4af7336a65ef477a06732c": "E_n(mx)= \\frac{-2}{n+1} m^n \\sum_{k=0}^{m-1}\n(-1)^k B_{n+1} \\left(x+\\frac{k}{m}\\right)\n\\quad \\mbox{ for } m=2,4,\\dots", "0fbf629057576a0f948c3b41dd6c730f": "(r\\bar{g}+g\\bar{r})/\\sqrt{2}", "0fbfba2d66eb7c1b0571c7ff7d1c09ae": "c=2\\sin\\tfrac{\\pi}{5}=\\sqrt{\\tfrac{5-\\sqrt{5}}{2}}", "0fbfe4d8e7731db2599020587eaa92b0": "t = w - \\frac{p}{3w}", "0fc047ec6ce679fea54586089980d199": "\\omega(x_i, y_j) = -\\omega(y_j, x_i) = \\delta_{ij}\\,", "0fc063d19f47aaa28eafe8145b73a9a8": "\\tilde{f}(\\lambda)=\\int_G f(g) \\varphi_{-\\lambda}(g)\\, dg.", "0fc0747d110cbc84c925e08d75c6c948": "\\tilde{g}_{lk}\\overline{n}=g_{kl}", "0fc07624dee8cd68b111a92599adb39d": "n(I) = n + n_2 I", "0fc084aaa92db3b493c21915b46d3994": "\\ x = s + v,\\,", "0fc0931f7407f09477b2f5b809be7165": "\n\te_1 = 2585.25381092892231\n", "0fc099675c6d8e7ec4c70b8fcbd9e2e2": "\\beta = i", "0fc09a771f874f23e0384316ca8e424a": "\\rho^\\prime \\equiv \\rho + \\frac{\\Lambda}{8 \\pi G}", "0fc0b8a7f599213481452408a8675c6f": " D_{\\mathrm{KL}}(P\\|Q) \\equiv \\sum_{i=1}^n p_i \\log_2 \\frac{p_i}{q_i} \\geq 0.", "0fc0cfa73492e389aa75471ad6707b30": " \\frac{\\partial C_1}{\\partial t}=\\frac{\\partial}{\\partial x}[D_1 \\frac{\\partial C_1}{\\partial x} -\\frac{C_1}{C}[D_1 \\frac{\\partial C_1}{\\partial x} + D_2\\frac{\\partial C_2}{\\partial x}]]", "0fc10db026bd13e5a0c606b5cdaa2765": "\\max_{\\alpha} \\sum_{i=1}^n \\alpha_i - \\frac12 \\sum_{i=1}^n \\sum_{j=1}^n y_i y_j K(x_i, x_j) \\alpha_i \\alpha_j,", "0fc14e67d1fbfb874e033bf12360b3cc": "\\mathbf{T}_{U,i}", "0fc22bf62c1ffe928ed175dcb02070c8": "G[\\mu_j]=U+pV-TS-\\mu_jN_j\\,", "0fc294aa66aba0ffb41529bbf232188f": "\nu \\bar{u} = 1\n", "0fc2adfe6beba9cd4bf68bd5b9f9b89b": "S=\\frac{\\sigma_{\\text{between}}^2}{\\sigma_{\\text{within}}^2}= \\frac{(\\vec w \\cdot \\vec \\mu_{y=1} - \\vec w \\cdot \\vec \\mu_{y=0})^2}{\\vec w^T \\Sigma_{y=1} \\vec w + \\vec w^T \\Sigma_{y=0} \\vec w} = \\frac{(\\vec w \\cdot (\\vec \\mu_{y=1} - \\vec \\mu_{y=0}))^2}{\\vec w^T (\\Sigma_{y=0}+\\Sigma_{y=1}) \\vec w} ", "0fc322eb39e7f8cfe71f7b568b4fba61": "\\phi (x)", "0fc38e493c1285b0ee01777c163bcc49": "\\varphi(t,x)/\\varepsilon", "0fc3a467b257571cbedb9d01bf706fd0": "\\mathbf{Ax} = {\\lambda}\\mathbf{x}", "0fc3b37acac582ac8327a49b0a9b544d": "90^\\circ/m", "0fc3d8831995c4401b8e744f8d9c42e8": " -e_2", "0fc3dfa9b5b8b7f307e246f6859bb17c": "h_{\\text{fe}}", "0fc420468409c0648c1b0a06cac24384": "\\sin^2 \\theta", "0fc4266b6f939b070809ed195d978557": "\\text{angle} = \\arctan \\left( \\frac{\\text{slope}}{100%} \\right)", "0fc4397d3642ba456c3416794c063e53": " v_L ", "0fc47a5ff0f41a4c2d286a2f66d63d8b": "|z| = |a + \\sum{b_n i_n} + \\sum{c_n \\varepsilon_n } + d| := \\sqrt[4]{ (a^2 + b_n^2 - c_n^2 - d^2)^2 + 4(ad - b_n c_n)^2 }", "0fc4a16b0f6ba85aac16d1a0bd34d732": "3.096262735\\times10^{78}", "0fc4db473d51d022ae8c7527ca72f385": "\\left\\langle hk\\ell\\right\\rangle", "0fc54c0fbc206bc462232612ba3d73d2": "\\frac{\\Omega^{(s)}}{\\Omega^{(0)}} = \\exp \\left ( \\sum_{p=0}^{s-1} \\xi_p \\right ).", "0fc588ce3c156f858cf5b0e4781a78bf": "\\tilde{K}_0\\left(A\\right) = \\bigcap\\limits_{\\mathfrak p\\text{ prime ideal of }A}\\mathrm{Ker}\\dim_{\\mathfrak p},", "0fc64d72d017c4e6de9ef38be23fe050": "\\frac{J_{2k}(n)}{J_k(n)}", "0fc691491f464ec04fefa7aa584ed639": "x_2\\rightarrow x_s(t)", "0fc6afa86487fa6984a3d272536df644": "=\\frac 1 {Z_0\\cdot\\left(\\pi\\epsilon_0\\epsilon_r\\right)\\cdot \\left[\\ln \\left({2D/d}\\right)\\right]}", "0fc72a1f05c9f8309e64772f7958e09c": "a^2\\,", "0fc7313c481025d4585c3667f306dbff": "it H", "0fc74c75e4205a0bfbf953d3d5f05dec": "U(\\lambda)", "0fc7777459eb9dbe16ecaa0511f94c10": " M_t = p ( W_t, t ) - \\int_0^t a(W_s,s) \\, \\mathrm{d}s, ", "0fc7926c975ed551728c1035c0ee12b0": "\\hat F(i) = \\hat h(i)+\\sum_{ j=1 }^{n/2}[2 \\hat J_j(i)-\\hat \nK_j(i)]", "0fc7aaa887faafddeaeeab150764302e": "(a_1b_7 + a_2b_8 + a_3b_5 - a_4b_6 - a_5b_3 + a_6b_4 + a_7b_1 - a_8b_2)^2+\\,", "0fc7d5ea1b013420e10444a51403375c": "\n\\mathrm{DQE} = \\frac{\\mathrm{NEQ}}{q}\n", "0fc7f31f553e020baf4ccc6deaf31e4c": "T = u^\\alpha \\gamma^{\\alpha\\beta}_\\mu(x-x')_\\mu \\overline{u}^\\beta.", "0fc81ae8fa1e7f4f359aee819e173686": "\\scriptstyle \\{U_n\\}", "0fc859fbb8f85ff0f76f2e63963578ca": "\\vec{e}_i = c\\times\n\\begin{cases} \n (0,0,0) & i = 0 \\\\\n (\\plusmn 1,0,0),(0,\\plusmn 1,0),(0,0,\\plusmn 1) & i = 1,2,...,5,6 \\\\\n (\\plusmn1,\\plusmn1,0),(\\plusmn1,0,\\plusmn1),(0,\\plusmn1,\\plusmn1) & i = 7,8,...,17,18 \\\\\n\\end{cases}", "0fc87d0ee76aaa7e2c012abba01dbf28": "\\mathbf{F} = m\\mathbf{a} \\quad \\to \\quad \\mathbf{a} = \\mathbf{F}/m", "0fc931801fbf2cc5230b5d966debc290": "disc(\\mathcal{H}) = O(\\sqrt{n}).", "0fc94bc17ba6ef87f29aaebfc83454c2": "a \\uparrow^n b = a \\uparrow^{n-1} \\left(a \\uparrow^{n-1}(...(a \\uparrow^{n-1}a))...)\\right)", "0fc95ba5988654853f65ca319d9dd63f": "\\delta \\sim \\frac{1}{T} \\sim \\frac{v}{L}", "0fc9a0a4bbba13ec06c952fa2c29344f": "Z' = Y/X, Y' = Z/X\\ ", "0fc9b9f524180e5d78fce0a906acc595": "r = x \\cos \\theta+y\\sin \\theta", "0fc9d065d91cefbcddf512dd10310cdc": "\nL_\\mathrm{p} = L_\\mathrm{W}+10\\, \\log_{10}\\left(\\frac{S_0}{4\\pi r^2}\\right)\\,\n", "0fc9f41c7244f732ea2ed08bc9b58edd": "x^1 \\cdot 2^0 = x", "0fca3f83324d05eae9454da929175411": "p0, \\beta\\ne1", "0fd06dc95a40a81847c6258a87fda96e": " \\mathbf X V \\Sigma^+ U^{\\rm T} = U \\Sigma \nV^{\\rm T} V \\Sigma^+ U^{\\rm T} = U P U^{\\rm T},", "0fd08afa70f7ac95d28fc1330d2ed79e": "\\scriptstyle\\leq1.6\\times10^{-14}", "0fd0ad1ca1deb2e3f3ede003ede23703": "\\mathrm{hub}(p) = 1", "0fd12e9afec1f7098461e3da25d98001": "ds^2 = -f(t,r)^2 \\, dt^2 + g(t,r)^2 \\, dr^2 + r^2 \\left( d\\theta^2 + \\sin^2\\theta \\, d\\phi^2 \\right), ", "0fd13dfb64dbf5362eacc93bac0445f8": "E(\\mathbf{\\hat f}+i\\mathbf{\\hat s})\\mathrm{e}^{i(kz-\\omega t)},", "0fd19ab2182c2be5eee79f4941a5f5a3": "R/I,", "0fd1da4a72751e2d8d60e1794c22495a": "\\alpha' = \\alpha/(4\\pi\\varepsilon_0)", "0fd1ec1b238c8d6315503d97d4825e4a": " {\\sigma_S^D}=18.0~\\mathrm{mN/m} ", "0fd24248803990e22edde58e4b456f76": "c_i = p_i \\oplus F(x_{1i}, x_{2i}, x_{3i})", "0fd2c3d3bbac540097dbc4ebc1e04bb2": " \\mathcal{H} = \\left(\\begin{array}{rr}\n1 & -1\\\\\n-1 & 1\n\\end{array}\\right)\n", "0fd2edfc8daa121e2356ebdd9e09e5d5": "\\sqrt{\\frac{g}{k}}\\, =\\, \\frac{g}{\\sigma}\\,", "0fd335ac99693b80d5f6ac83fe0bb513": "L\\rho_L\\,", "0fd752a060af3224aba5bad07de07ded": "2 \\pi f", "0fd75ec1092b8a849eb9b6400f1f2041": "0\\leq r\\leq 1", "0fd7a5887b765d548197900f109f293e": "\\nabla\\left(\\alpha f+\\beta g\\right)(a) = \\alpha \\nabla f(a) + \\beta\\nabla g (a).", "0fd7d07f37158be28be1f04a2f435de8": "f(x)=a(x-x_0)^2+x_0=h^{(-1)}(g(h(x)))\\,\\!", "0fd7f98c0bb1731fd618055871c5f973": "\\sup_n \\{a_n\\}.", "0fd85b89337c24dc207587957b8624e3": "\\langle k\\rangle * \\alpha", "0fd8ba339a14c61da1a517401157e7de": "\\frac{\\partial F}{\\partial n_1} = n_1\\sigma_1^2-2n_1\\sigma_1\\left(\\sigma_1 n_1^2+\\sigma_2 n_2^2+\\sigma_3 n_3^2\\right)+\\lambda n_1 = 0\\,\\!", "0fd96b818247d78b9375742cd6586e04": "M=\\frac{Q^2}{gA}+\\overline yA", "0fd99ff654fb7b30bfa43221653a32f9": "2^{3n/2}", "0fd9a0ad48af1c3aad45a6d8ea90101e": "dp = \\frac{\\partial p}{\\partial t}dt \n+ \\frac{\\partial p}{\\partial x}dx \n+ \\frac{\\partial p}{\\partial y}dy \n+ \\frac{\\partial p}{\\partial z}dz", "0fd9c63d545d81c4da8caa0e0d2ce436": "p_{e} = p_{amb}", "0fd9e1aafc2e37fdba1f20cec8266816": "\\mathrm{1 \\, sb = 10^4 \\, nit = 10^7 \\, millinit}", "0fda17baf45cbf6af5b6b69232abee04": " \\prod_{n=1}^\\infty (1-x^n) = \\sum_{n=0}^\\infty a_nx^n", "0fda22d202dc2502ce40892fe16b45e5": " \\tilde {\\textbf{P}}^{2n}(t) =\n[H^{2n}(t)]\\textbf{P}, ", "0fda8cca8ce1020383e61c7b2eb140a3": "(U; z^A,c^a)", "0fda905a8eaa285d890e74b373493ca7": "H_j", "0fdacd20ce254f34fbc464cc2ec3b740": "f(x) = s", "0fdae7d74b475dc1c455c0efe3590ddf": "m_0=0\\,\\!", "0fdb21e4f9bce557a6fb4cdfa4292f3d": "\n \\dot{\\lambda} \\ge 0 ~,~~ f \\le 0~,~~ \\dot{\\lambda}\\,f = 0 \\,.\n ", "0fdb8e258a6856772851a8f961e8d9a7": "\\frac{\\partial\\varphi}{\\partial t} = v|\\nabla \\varphi|.", "0fdbe2baf71976743a48d73949d22d55": "\\text{Sl}_{2m-1}(\\theta) = \\frac{(-1)^{m}(2\\pi)^{2m-1}}{2(2m-1)!} B_{2m-1}\\left(\\frac{\\theta}{2\\pi}\\right)", "0fdbe4ba43f5904b8e7cb8cdd0b78439": "\\mathbf{x} \\times \\mathbf{y} = T_{\\mathbf{x}}(\\mathbf{y}).", "0fdc25637dbf5d907528731ab295504b": "\\Omega=\\Lambda+\\Sigma", "0fdc804330bcf315883436ed70a62126": "\n \\sqrt{n}(\\hat\\sigma^2 - \\sigma^2) \\simeq\n \\sqrt{n}(s^2-\\sigma^2)\\ \\xrightarrow{d}\\ \\mathcal{N}(0,\\,2\\sigma^4).\n ", "0fdd8448f4767b2e1b1b875faf0e5f49": "\\ C_x", "0fde09bee2e3e11bb85c024acf0a2b10": "T^n \\times P^n", "0fdeb3d0d02691eee0f388da567410e0": "C_p\\left ( T_2-T_1 \\right )\\;", "0fdedd1216fd5c16a7f10583779b40e5": "\\frac{1}{2}|\\mathbf{AB}\\times\\mathbf{AC}|.", "0fdef2041c00867622e5cfb18e44590a": "P \\land\\neg P", "0fdf7a0f59f999169e75e4f41c435188": " {\\rm Li} (x) = \\int_2^x \\frac{dt}{\\ln t} \\, ", "0fdf89089449682d95e380f9a23dfe74": "\n\\begin{array}{cc}\n\\mathbf{I}=\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\n; & \n\\mathbf{I}^{-1}=\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\\\\n\\\\\n\\mathbf{R}=\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n; &\n\\mathbf{R}^{-1}=\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\\\\n\\\\\n\\mathbf{S}=\\begin{pmatrix}\n+1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & -1\n\\end{pmatrix}\n; &\n\\mathbf{S}^{-1}=\\begin{pmatrix}\n+1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & -1\n\\end{pmatrix}\n\\\\\n\\end{array}\n", "0fdfe61eb9713f3d86d27e9c2e7aba41": " \\frac{d}{dt}\\langle A\\rangle = \\frac{d}{dt}\\int \\Phi^* A \\Phi~dx^3 = \\int \\left( \\frac{\\partial \\Phi^*}{\\partial t} \\right) A\\Phi~dx^3 + \\int \\Phi^* \\left( \\frac{\\partial A}{\\partial t}\\right) \\Phi~dx^3 +\\int \\Phi^* A \\left( \\frac{\\partial \\Phi}{\\partial t} \\right) ~dx^3 ", "0fe04ccd07305d4426cb2d0fe1e06b24": "24 \\times 1,312,000", "0fe06a7178cf7287a9a4870318cd5060": "r_{SOI} = a\\left(\\frac{m}{M}\\right)^{2/5}", "0fe082d15fa88a3ae09e057d205a551e": "\\sin_k^2(i)\\equiv (2^{-1}\\bmod{p})\\cdot(1-\\cos_k(2i).", "0fe0a2514a9cf2ba888ecca28eb1b71d": "\\forall M'", "0fe1540a9034b396e8c0ca9a8d41ca84": "\n\\mathbf{m}=\\mathbf{m}_{\\rm orb}+\\mathbf{m}_{\\rm spin}", "0fe20a4f5323ed611f6a14ba590dd252": "I_2 = - {Z_{21} \\over Z_{22}} \\, I_1 ", "0fe22f29aac3cd821c4aa3c2c5a999b0": "Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0\\,", "0fe34beb493986d579b09cf842e4a7ef": " \\sum_{ij;i \\ne j} \\psi^*_i \\psi_j \\phi^*_j\\phi_i", "0fe34f3258a18e913bba77be0d89243c": "\nm_0D + m_1\\sigma(D) + \\cdots + m_{n-1}\\sigma^{n-1}(D), ~~~~\\text{where }m_i = O(\\ell^{1/(n-1)}) = O(q^g)\n", "0fe394cf6df0566fd8e4cddae78758b5": "\\tilde\\lambda_i", "0fe398c270ae0c7f308943145950d12d": " \\boldsymbol{u}_e=-\\hat{\\boldsymbol{z}}\\times\\Big(\\frac{1}{f}\\frac{\\partial \\boldsymbol{\\tau}}{\\partial z}\\Big). ", "0fe3a286ae4e08b0d994222bd306afaf": " \\frac{|\\Delta x|}{|x|} ", "0fe3d81d2b000c5f8d99b84dd98b32ab": "\\displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),\\,\\,\\,Q(a^m)=Q(a)^m \\,\\,(m\\ge 0).}", "0fe3d9c6b4bcc5525435458790c04fec": "b_{max}", "0fe42c26de1d22fba3d54abb76741191": "\\chi'(G) \\ge \\Delta(G).\\,", "0fe4669801f948c6d9240747889d269b": " \\ln (t) \\cdot u(t) ", "0fe522862f36b8bca8b846197b66e331": "\\Pi_f", "0fe58892c79e94268a59a7078a7814f4": "-\\int_a^b \\mathbf{F}\\cdot d\\mathbf{l} = P(\\mathbf{b})-P(\\mathbf{a})", "0fe5a9b80f6d0f0b299d28c2da30fa63": "\\lambda (x,y) = (\\lambda x, y) = (x, \\lambda y) \\!", "0fe5d36b3fec3f7d074868a303a99434": "\\partial F = \\mathcal{P}_B (\\nabla )F", "0fe6683a06cd0338b7ab4cd63633b78a": "\\textbf{P}_{k|k} = (I - \\textbf{K}_k \\textbf{H}_k) \\textbf{P}_{k|k-1}", "0fe6b96de49ee5f6d313c5e7b3136835": "c_{T-j}", "0fe6e35b6e6e36652c8774aae0628915": "\\scriptstyle \\Gamma_{(z_0,r)}=\\partial D(z_0,r)", "0fe6edbc6bd9c973761ea25268521545": "P\\left(t,T\\right)=e^{-\\int_t^T f\\left(t,s\\right)\\,ds}.", "0fe74d16c564008427727959c6e71222": "{n \\choose 5} {5 \\choose 3} \\times 2", "0fe74d4e0208bc631753e803f0353928": "36\\operatorname{Li}_2\\left(\\frac{1}{2}\\right)-36\\operatorname{Li}_2\\left(\\frac{1}{4}\\right)-12\\operatorname{Li}_2\\left(\\frac{1}{8}\\right)+6\\operatorname{Li}_2\\left(\\frac{1}{64}\\right)={\\pi}^2", "0fe75a5189c2ea3f123621d098ddd03e": "PR", "0fe7c0b5e58d24e3fba107270bb953e8": "\\mathcal E(s)", "0fe7ddc565ebe9d16eb3444a392f8df6": "\\lbrace q \\mid q <_{\\mathcal{O}} p \\rbrace ", "0fe86cc2b6a3198af1cc93df5d4d6193": "D_y", "0fe8a89af2c204db83ef653388ae6fbc": "c_0(\\log|\\Delta|)^2", "0fe8f38e1d2c3868701be659beaa5848": "C\\ddot \\phi + G\\dot \\phi + \\phi/L = i\\,", "0fe960ea7d4104ef9d09cbbd41307d6e": "E(\\nu )", "0fe9686935e2be0226b24959f6713536": "a \\uparrow b \\uparrow c = a \\uparrow (b \\uparrow c)", "0fe97f260eae33f063386d2443ca4e2d": "\\gamma_K=\\dot{k}/k = s.f(k)/ k - (n+\\delta)\\ ,", "0fe992a391e39803222f9713e2142be1": "r = 2a\\cos{\\theta \\over 3}", "0fe9e27a4bf412e50275321f5c5d431c": "\\lim_{n\\rightarrow\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|=1", "0fe9e816422c723d1c76b691e1d676bd": "x \\in A ", "0fea552abd4c2f09dd0ecb41dcbda411": "\n c_{pqrs} = l_{pi}~l_{qj}~l_{rk}~l_{s\\ell}~c_{ijk\\ell}\n ", "0fea78208e465b4e57cb2fa209f49f99": "E_{min} = \\alpha E", "0fea8f9347ee90ad9d31bc31d443fd0e": "\\alpha + h_x (\\alpha) = K(x)+O(1)", "0feab75901667616678429cfbe6114a3": "C_I", "0feabcc18804c47f0abde563618a85f4": "B_t = \\sum_{k=1}^\\infty Z_k \\frac{\\sqrt{2} \\sin(k \\pi t)}{k \\pi}", "0feac4661c7f76a4e0842ef697f4eb98": "\\operatorname{E}[\\cdot]", "0feb33d8c32fa19df125704507c8d2b5": "q_{1}=x_{11}+x_{21}", "0feb34f24556a69af98fb532876080b1": "C_H^d(S):=\\inf\\Bigl\\{\\sum_i r_i^d:\\text{ there is a cover of } S\\text{ by balls with radii }r_i>0\\Bigr\\}.", "0feb44d857d3510ab18975469799b2d1": "|j,m,l,1/2\\rangle", "0feb4f87e6273c51fb878ac983e01629": "\\bar{t}", "0feb6c48073bf9ee45aeebdc468d612e": " R^i_j ", "0febe3443dc70c82815095ddf6a334ae": "K_{\\lambda\\mu}= K_{\\lambda\\mu}(1)=K_{\\lambda\\mu}(0,1).\\ ", "0fec6acbfbb750f86b1469612f8d0327": " \\frac{c^2-s^2}{sc} = \\frac{a_{\\ell\\ell} - a_{kk}}{a_{k\\ell}} . ", "0fecb1d76491880ab53dea1bdedd5b66": "f^n(x)\\in U", "0fecbc6612b90bff28ec553d603dba8c": "hf_{ij} = \\text{estimated frequency of haplotype } ij = gf_i \\; gf_j = 0.0215\\!", "0fecf021df2f33adeea6a5c8501bff95": "\\vec\\mu_s", "0fecfcebc29fc9f890ec12e787727a09": "\\vartheta(x)=\\sum_{p\\le x} \\log p=\\log \\prod_{p\\le x} p = \\log (x\\#).", "0fed1bb26cfec02d08fc67fe7d1d6bb8": "\\theta(F)=\\frac{1}{m}\\underset{(a,m)=1}{\\sum_{a=1}^m}a\\cdot\\mathrm{res}_m\\sigma_a^{-1}\\in\\mathbf{Q}[G_F].", "0fed219a5c9ca170788a66534e8c7700": "P = \\exp{\\left( A + \\frac{B}{C+T} + D \\cdot T + E \\cdot T^2 + F \\cdot \\ln \\left( T \\right) \\right)}", "0fed3cdb09247f9d104ba6015df01a75": "C^{(p)}_T(p,T)\\ ", "0fed9e0fd38ec4502d9b5a064ec52c0b": "\\frac{1}{1-w}=\\sum_{n=0}^\\infty w^n.", "0fee49d4621b4f2d14111406a999e287": " a'_{kk} = a_{kk} - t a_{k \\ell} \\,\\! ", "0feea555e91f76eba0d3ac6a94022f67": "S(t) = P_2 (t) \\mbox{ , } t_1 \\le t < t_2,", "0feed645f145b95a360f8719e3b63698": "\\mathrm{Hom}(A\\otimes B,C^*)\\cong\\mathrm{Hom}(A,(B\\otimes C)^*)", "0feee6daa84b45625e975aa9208158b0": "C_{st}", "0feeeed9ae23d02b13d968bec0ac7022": "\\displaystyle (w^2-y^2)(x^2-z^2) = 0", "0fef4df83c418dff7499b5213d7bfbf2": "\n\\mathbf{H} =\n\\left(\\left.\\begin{array}{cccc}\n0 & 1 & 1 & 1\\\\\n1 & 0 & 1 & 1\\\\\n1 & 1 & 0 & 1\\\\\n1 & 1 & 1 & 0\\end{array}\\right|\\begin{array}{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\\end{array}\\right)_{4,8}\n.", "0fefd781564f52dbfe44f21e09884a24": "4) B \\rightarrow KDC: ID_B||ID_A||E_{KU_{KDC}}[N_A]", "0fefeb462cf7a96341cb609770d318d5": "\\,G_t", "0feff53d07e872f617b2573a963c82f3": "\\tfrac{26}{11} = 2.", "0ff011c94a39b73c58e7b6c89505b703": "\\mathrm{Aut}(P)\\cong\\mathrm{GL}(n,\\mathbf{F}_p),", "0ff026df29aa71d28573c55904c23c14": "\\Delta\\phi", "0ff030591fb4fe92a6043eeeb9ebbc49": "(\\overline{Y} \\Delta Y - Y \\Delta \\overline{Y})/(\\overline{Y}(\\overline{Y} + \\Delta \\overline{Y}))= c(\\Delta \\overline{u}-\\Delta u).", "0ff08a4fbccfffe26d52f3afb5794ffb": "\\Theta_{n,m}(\\tau + 1, z)= e^{\\frac{\\pi i n^2}{m}} \\Theta_{n,m}(\\tau,z)", "0ff09c9a0a5067cf4372df3fb475a52f": "\\mathbf{B}/\\mu_0 = \\mathbf{H} + \\mathbf{M}", "0ff0bab613e19680da4229cc42504e98": "\nD_\\alpha (\\hbar \\nabla )^\\alpha \\phi (x)=E\\phi (x). \n", "0ff0d0ef90337e8561c80aeea3abd776": "\\mathit{near}", "0ff0db4abb479d02a4d5f239f987fdcf": "Z'= Z \\coprod\\! D^2/\\! \\sim", "0ff128a22e698a0e776b4d462c4c9703": "\\textstyle i\\hbar\\frac{\\partial\\Psi}{\\partial t} = E\\Psi", "0ff13e89b58af6bb727987b341ccdbb8": "\\Delta h \\propto -\\Delta(v^2)", "0ff1500e6fbe70daf2cb950b85fc59a3": "\\begin{align}\\pi & \\approx 3.141\\ 592\\ 653\\ 5\\dots \\\\\n\\\\\n\\frac{355}{113} & \\approx 3.141\\ 592\\ 920\\ 3\\dots \\\\\n\\\\\n\\frac{52163}{16604} & \\approx 3.141\\ 592\\ 387\\ 4\\dots \\\\\n\\\\\n\\frac{86953}{27678} & \\approx 3.141\\ 592\\ 600\\ 6\\dots\\end{align}", "0ff1bd8a48a765143afe15a3ca7942c5": "\\left.\\frac{\\partial}{\\partial p_k}\\left\\{-\\sum_{j=1}^n p_j \\log_2 p_j + \\lambda \\left(\\sum_{j=1}^n p_j - 1\\right) \\right\\}\\right|_{p_k=p^*_k} = 0.", "0ff1be70f0e687af1b7aa293c90a1801": "\\Delta\\lambda=\\lambda_2-\\lambda_1", "0ff1c76e4bd8ebb5cce5497f2bc58641": "\\mathbf{C} = \\{ C[k, l] \\} ", "0ff1e6bcecb184973b05cd7c9edf6c38": "\\delta(q, \\varepsilon, x) \\not= \\emptyset\\,", "0ff1f84c6778a61989b6134b4a2a2496": "q_{12}q_{21}=\\pm 1", "0ff25b4c01f7eee56c6d995eafe65a1a": " \\frac{V^2}{2.g} + z +\\frac{p}{\\rho.g} = \\mathrm{constant}", "0ff2fdb1c974fa7dccf23e3871e98e10": "N >= X", "0ff30fbff074d55c7f69d5b00c010ea6": "(1+x)^p \\approxeq 1+px\\mbox{ when }x<<1", "0ff32026b7bc78e4c1ed622ae5891bde": "V(\\zeta)=\\int_0^\\infty G(|\\varphi(t,\\zeta)|)dt", "0ff33750ad45820ec1319aa926e4d788": " (\\alpha, \\beta) = (\\zeta, 1).", "0ff35e7a721728fd06d7ff81dcfead3f": "x=\\frac{p}{2^n5^m}=\\frac{2^m5^np}{2^{n+m}5^{n+m}}=\n\\frac{2^m5^np}{10^{n+m}}", "0ff36d579dbadedcab2d801a1fe9fe45": "P_{t}", "0ff38944aefc9f54e7f6c2089ae858f0": "\\forall j \\in \\{1,\\ldots, n\\}", "0ff39a3b850ebc88cd39c4618de3e411": "\\delta \\propto k^{-2}", "0ff3d3ff61190d6d56da7390e8cb1313": "t_4 = 1213121121312", "0ff4178ce42f74534ee213b6b691c76e": "k^{n-k}", "0ff428303f7c5ab220c0a4c0b97b57fb": "I_\\text{s}", "0ff45e9ec87a1bb80ddff335180180ec": "O( n \\log k )", "0ff4dee6e989771a41e8975cccd3be25": "\\rho (\\bold D)", "0ff4efd46bc3379a65250abc6d288333": " C\\subseteq X ", "0ff5464870c6c1e0a8fe6ed07d3f0666": "[ x^2 + ( \\cot \\varphi_1 - y)^2 ]^{1/2}", "0ff548518306c8ede8f90e492317ebe1": "T_{Out}\\,\\!", "0ff54d80e3d4fc62feaa8d2c376722da": "\nW(\\mathbf{C})=\\frac{\\mu}{2}(I_1^C-3)\n", "0ff592606b9f2dfa45548d2415fd8e9d": "\\mathbf{Gr}(r, \\mathcal E)_s", "0ff5e71bf4aab2580c3788cd6c9b8f39": "\\Omega\\!", "0ff60dcae60fe1793331843f23602c21": " a^{\\dagger}_{{\\mathbf{k}}_l} |n_{{\\mathbf{k}}_{1}}, n_{{\\mathbf{k}}_{2}} ,n_{{\\mathbf{k}}_{3}}...n_{{\\mathbf{k}}_{l}},...\\rangle = \\sqrt{n_{{\\mathbf{k}}_{l}} +1 } |n_{{\\mathbf{k}}_{1}}, n_{{\\mathbf{k}}_{2}} ,n_{{\\mathbf{k}}_{3}}...n_{{\\mathbf{k}}_{l}}+1 ,...\\rangle ", "0ff695b719f5589ce68ae5fad5f9b970": "f(\\sigma_\\bar{C}, \\cdot) - \\bar{C} = 0 \\,", "0ff6a5ca4fe06952d6997c564515c254": "3\\in X", "0ff760323df54c472bbe3df83eea3368": "X \\to Z", "0ff7650e1dcef535f34c1e7d9b49a55e": "A=-k_\\mathrm{B}T\\ln \\bigg( \\frac{V_\\mathrm{A}^N}{N! \\Lambda^{3N}} \\bigg)", "0ff790ce5aa9be38a3af2a6c3212a452": "V_o = V_i", "0ff7e8099b9fba40e069c21722c9ac77": "\\|f\\|_{B} = \\sup_{x\\in X}\\left|f(x)\\right|", "0ff81db00ea358a73d443a7b10ac962e": "E[p]", "0ff86ade0c00234d6a86395e7e9e7fc0": "2^2\\cdot 3\\cdot 5", "0ff870d34fce892ecf9019b59bc51c1b": "\\frac{1}{F(p)}=\\sum_{n=0}^\\infty a_n p^{-n}", "0ff886fcf5436652cb6c0f6c82ac0500": "x\\!\\!\\sim y \\text{ iff} \n\\begin{cases}\nx=y, \\mbox{ or }\\\\\nx=\\gamma (t) \\text{ and } y= g(t) \\text{ for some } t\\in [0,1]\\mbox{ or }\\\\\nx=g (t) \\text{ and } y= \\gamma (t) \\text{ for some } t\\in [0,1]\n\\end{cases}", "0ff895cc17d6587aaa31e2b5b9c43334": "x_2, \\ldots, x_{K-1}", "0ff930fda5dd413e5a5848cb2806a992": "\\pi(\\theta|x) = \\frac{p(x|\\theta) \\pi(\\theta)}{p(x)} = \\frac{f(x-\\theta)}{p(x)}", "0ff9557af1eab7c66e0fa804660ea21c": "\\textstyle{1}", "0ff95c81b92a9909c7e15bae42b2b919": "[A\\cup B]_{\\text{seq}} = \n[A]_{\\text{seq}} \\cup [B]_{\\text{seq}}", "0ff96b58177c94df21db350bc3720e21": "k( \\tau)= \\int\\limits_{- \\infty}^ \\infty S ( \\nu) \\exp \\left(-i2 \\pi \\nu \\tau \\right)d \\nu = \\exp \\left(- \\pi^2 \\tau^2 \\Delta \\nu^2 \\right) \\exp \\left(-i2 \\pi \\nu_0 \\tau \\right)", "0ff99fa99eda51bf18c8ceaa737f5b00": "Q_n(i)", "0ff9b1f42708a1392c719e29425d8322": "a_n=t_n - \\frac{1}{2(n+1)}", "0ff9dfefd9a847bab174b2b0386ee229": "\\scriptstyle \\emptyset\\in \\mathcal{F}", "0ff9e4b796741eb00f0178d7c994a8e4": "\\forall x_1 \\ldots \\forall x_n \\; \\exists y \\; P(y)", "0ffa1c61f3af78f7a579dd62f6f7e741": " \\chi(3,7) = q_1 + q_1 q_2 - q_1", "0ffac5cd4249e1754a8e1bac9497ad67": "M_1\\prec_K M_2", "0ffaf658e74aee96b2410fc4f1ab9266": "\\sum_{n\\in\\Z}\\left|x[n]\\right|^p<\\infty.", "0ffb7e5fc4f9fc7ea4d0c4ac9738e8b5": "\\iiint_D (x^2 + y^2 +z) \\, dx\\, dy\\, dz = \\iiint_T ( \\rho^2 + z) \\rho \\, d\\rho\\, d\\phi\\, dz;", "0ffbdef80663d0b5f7be54455a3f3736": "\\hat{f}(n) = c_n", "0ffbf8e663951350614606c52ea3f31c": " \\left( \\frac{ \\lambda\\ }{ 2a } \\right)^2 = \\frac{ \\sin ^2 \\theta\\ }{ h^2 + k^2 + l^2 }. ", "0ffc2fb0831add8c4d1b889f6360805a": "\\Delta v_r = \\int \\frac{qE_r(r,z)}{m_0v_z} dz", "0ffc49f6b711b28603c43491fcbc9639": "A \\equiv_T B\\,", "0ffd7c709a323baf97fe8b678f3b0114": "P_r \\approx P_t ({\\frac{\\lambda G}{4\\pi d}} ) ^2 \\times (\\frac{4 \\pi h_t h_r }{\\lambda d})^2", "0ffd89bc3eaa39ebd01f8b349e0c8d39": "\\mu(S)=\\frac1{2\\pi}m\\left(f^{-1}(S)\\right),", "0ffdf10da47650638f0b5d716a7c9407": "i\\overline{\\psi}(\\partial_\\mu+A_\\mu)\\Gamma^\\mu\\psi", "0ffdff24d8460133a3425569b27c1f5d": "P_i = \\frac{a^\\text{o}_i}{a^\\text{w}_i} = \\exp \\left[\\frac{z_iF}{RT}(\\Delta^\\text{w}_\\text{o}\\phi - \\Delta^\\text{w}_\\text{o}\\phi^\\ominus_i)\\right] = P^\\ominus_i \\exp \\left[\\frac{z_iF}{RT}\\Delta^\\text{w}_\\text{o}\\phi\\right]", "0ffe48bc7bbbfbd49b3b2ec98c888f77": "\\mathbf{Q}^m ", "0fffa731659443521b209273e179ca6d": " E = a \\cdot E_0 + B ", "0fffc4e9ea6664cdbc8a33802ccb863e": "\\hat F", "0fffe46f0dea0cc08391a462dc86f6a6": "\\{|f\\rangle=|u\\rangle\\otimes|t\\rangle\\otimes|s\\rangle\\otimes|r\\rangle\\}", "10001d1b13648c4fde049b96dcc46879": "dQ=T dS", "10004ee38a47bc9fa1c843c56c73436a": "Y(s) = \\mathcal{L}\\left\\{y(t)\\right\\}", "10006d0675a431acba482acde502d102": "F=\\begin{smallmatrix}\\frac{1}{\\sqrt{2\\pi}\\sigma}P^2\\exp(-(P-P_0)^2/\\sigma^2)\\end{smallmatrix}", "100089280ba9a89fc86b84be968c610c": "\\mathbb{E} \\big[ \\nabla f \\cdot v \\big] = \\mathbb{E} \\big[ f \\delta v \\big].", "1001858f21202f87a4dbc2ef0ecc0e33": " x \\not= y ", "10024912e355b3ef15847978acc8c900": "\\log\\zeta(s)=-\\sum_p\\log (1-p^{-s} )=\\sum_{p,n}p^{-ns}/n.", "100263ddff847c180682c3b41ae33f60": "\\frac{d\\mathbf{\\rho}}{dt} = \\mathbf{\\Omega}\\times\\mathbf{\\rho}", "100283dea92f90aa425d28ec59528000": "K = H - \\mu N", "1002b06e12aafa8dcf8b88d347f28951": " \n\\begin{align}\n\\frac{\\mathrm d F_\\varepsilon}{\\mathrm d\\varepsilon} & =\\frac{\\mathrm d x}{\\mathrm d\\varepsilon}\\frac{\\partial F_\\varepsilon}{\\partial x} + \\frac{\\mathrm d g_\\varepsilon}{\\mathrm d\\varepsilon}\\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon} + \\frac{\\mathrm d g_\\varepsilon'}{\\mathrm d\\varepsilon}\\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon'} \\\\\n& = \\frac{\\mathrm d g_\\varepsilon}{\\mathrm d\\varepsilon}\\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon}+\\frac{\\mathrm d g'_\\varepsilon}{\\mathrm d\\varepsilon}\\frac{\\partial F_\\varepsilon}{\\partial g'_\\varepsilon} \\\\\n& = \\eta(x) \\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon} + \\eta'(x) \\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon'} \\ . \\\\\n\\end{align}\n", "1002cd7181a8c9442bdc0a372e458272": " \\sum_{i=1}^k \\ln \\Gamma(\\alpha_i) - \\ln \\Gamma\\left(\\sum_{i=1}^k\\alpha_i\\right)\n", "1002f925a2f1bc9dcda108dc38692100": "c_{t}=y_{t}+(1+r)b_{t}", "100356104d78ff40589ac8dfeb5f32a0": " \\overrightarrow{\\Gamma} = \\overrightarrow{m} \\times \\overrightarrow{B} ", "1003664af1b58fe33f27838a7802c2c8": "\\Beta(k, n+1-k)", "1003b738a3a1ab33f9aeb768e36fb2db": "E(A)", "1003effa0a74afd23c46e2ac1e1864c3": "\\scriptstyle\\zeta [\\vec{x},t]", "10040034a9fa378aa45741429194af62": "x=l_0\\|r_0", "100484d176dca301883760832989c297": "\\xi_d=d^{-1}\\sqrt{\\frac{K_2}{\\epsilon_0\\Delta\\chi_eE^2}}", "10048d2e8b8a081a92405a9c2daa3203": "\\frac{d}{dx} \\left(\\prod_{i=1}^n u_i(x) \\right)= \\sum_{j=1}^n \\prod_{i\\neq j}^n u_i(x) \\frac{du_j(x)}{dx}, ", "10048e1db55647a2c536090802a2013c": "ds^2= - c^2 \\left(1 - \\frac{2GM}{rc^2}\\right)dt^2 + \\frac{dr^2}{1 - \\frac{2GM}{rc^2}} + r^2(d \\theta^2 + \\sin^2 \\theta \\, d\\phi^2).", "1004bdae5ef831a50bfee125f1342fa4": "d (x^2(x^2 + y^2))-d (a^2y^2) = 0 ", "1004dc886a9953c947eabc29e72e22d3": "I=(i_1,i_2,\\cdots,i_k) \\in \\Im", "100521c77d873735e9b77843868ce73b": "H_{ij} = \\begin{bmatrix} {\\partial^2 V_{ij}\\over\\partial x_i\\partial x_j} & {\\partial^2 V_{ij}\\over\\partial x_i\\partial y_j} & {\\partial^2 V_{ij}\\over\\partial x_i\\partial z_j} \\\\ {\\partial^2 V_{ij}\\over\\partial y_i\\partial x_j} & {\\partial^2 V_{ij}\\over\\partial y_i\\partial y_j} & {\\partial^2 V_{ij}\\over\\partial y_i\\partial z_j} \\\\ {\\partial^2 V_{ij}\\over\\partial z_i\\partial x_j} & {\\partial^2 V_{ij}\\over\\partial z_i\\partial y_j} & {\\partial^2 V_{ij}\\over\\partial z_i\\partial z_j}\\end{bmatrix} ", "100540e73a7630cf24c3e6030fd4573b": "\\forall{\\mathbf{X}, \\mathbf{Y}}\\ D_{**}^{(p)}(\\mathbf{X}, \\mathbf{Y}) = 0 \\ \\nLeftrightarrow \\ \\mathbf{X} = \\mathbf{Y} \\,", "100543673f111aba9912aaaa01cf5bf5": "C^{(T)}_p(p,T)=\\left.\\frac{\\partial U}{\\partial T}\\right|_{(p,T)}\\,+\\,p\\left.\\frac{\\partial V}{\\partial T}\\right|_{(p,T)}\\ ", "1005484cbee0886fa3105c50d997e297": "\\sum d_i^2 = 194", "1005893aad2aa15fc21e483bfa2a404e": "f=f_{y}=\\frac{R_{M}}{2}\\cdot \\frac{1}{\\cos \\theta }", "10059ba0b3d5b51f386cfd1993ef5e9a": "\\Sigma _{XY} = \\operatorname{cov}(X, Y) ", "1005bab0ce24fe12b04c23f4dbf0c9f0": "\\mathrm{not} ~p", "1005d7f37f874a7268b92222a8476894": "H_{\\frac{3}{4},3}={(\\tfrac{4}{3})}^3-27\\zeta(3)+\\pi^3", "1006079f9d88cb3b4a13fd7b8d54c8d8": "S(\\rho^{123}||\\rho^1\\otimes\\rho^{23})- S(\\rho^{12}||\\rho^1\\otimes \\rho^2)=S(\\rho^{12})+S(\\rho^{23})-S(\\rho^{123})-S(\\rho^2)\\geq 0, ", "1006649e21624f24f9fe35836a56c387": "\\gamma^\\mu\\gamma^\\nu\\gamma_\\mu=-2\\gamma^\\nu . \\,", "1006cc2b33e4c62fdba3b0dc068d08e9": "\\left|\\Gamma(\\omega)\\right|^2 = \\sin^2 \\phi_{\\Gamma}(\\omega) = 1 - \\sin^2 \\phi_{\\tau}(\\omega)", "10071bcd9da4a9cbc26be5144494ff55": "n = f(x) = A(x,x)", "10073179d2cd088b9054d5a3068f446a": "v_\\lambda", "1007377099f7b8b812f04c87a775f8de": "\\sqrt{n}{mn \\choose n} \\ge \\frac{m^{m(n-1)+1}}{(m-1)^{(m-1)(n-1)}}", "10074c8f698696461ea3375b3c9a88d7": "dz = \\left(\\frac{\\partial z}{\\partial x}\\right)_y dx + \\left(\\frac{\\partial z}{\\partial y}\\right)_x dy", "1007e5b4d0e8e4801c245326fe2d911d": "\\begin{cases} \\varphi_X\\!:\\mathbf{R}\\to\\mathbf{C} \\\\ \\varphi_X(t) = \\operatorname{E}\\left[e^{itX}\\right] = \\int_{\\mathbf{R}} e^{itx}\\,dF_X(x) = \\int_{\\mathbf{R}} e^{itx} f_X(x)\\,dx = \\int_0^1 e^{it Q_X(p)}\\,dp \\end{cases}", "1007f2eb6a35e3f978c8552d0899df30": "(x_1,y_1) + (x_2,y_2) = \\left(\\frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} , \\frac{y_1y_2-ax_1x_2}{1-dx_1x_2y_1y_2}\\right)", "10082936419735934dab86df00c4b1ab": "t_r\\!", "10084ae97f86c9321131e6c3e31a9a9a": "\\displaystyle \\frac{1}{2 \\pi} \\iint f(x,y) e^{-i (\\omega_x x +\\omega_y y)}\\, dx\\,dy", "1008ac9834bd5cefffc465a4b26a7ac7": "\\frac{1}{r_a}+\\frac{1}{r_c}=\\frac{1}{r_b}+\\frac{1}{r_d}.", "1008bc3b2ccaa1ce32a5da86f8899ccf": "A \\times P ", "10090dedbdfa037f2f0860222fdcbaa7": "\\left\\langle \\sum_i a_i \\otimes \\lambda_i, \\sum_j b_j \\otimes \\mu_j \\right\\rangle = \\sum_{i, j} (\\lambda_i)_0 (\\mu_j)_0 \\int_X a_i \\smile b_j.", "1009208bf66795bbc26f94eca59122f8": "c_k = c(x_k, u_k)", "10092d92ea7b2e67ea8cb963bd134450": "X_{i1}=1", "1009ac27f02d37e5db7f3462885a1acd": "E_{1} = \\Delta x + \\Delta y + \\Delta z + 2 \\Delta p = 0", "1009dde431503a7d94d31b1ef4516cf8": "R=mg\\plusmn\\frac{mv^2}{r}", "100a1c0d2f14ad7111e3e6ea5d887c3a": "h=H(m)", "100a6918357a83bfcf174046ccd5f870": "d\\Phi=\\omega\\wedge\\Phi", "100a7d1ed937ebe8e0db1ca02ab75f4e": "m_{\\rm u} = \\frac{N_{\\rm A}}{M_{\\rm u}} = \\frac{A_{\\rm r}({\\rm e})}{m_{\\rm e}} = \\frac{A_{\\rm r}({\\rm e})c\\alpha^2}{2R_\\infty h}", "100ac5c6c6d0b755e826bd0096f0347b": "{I_c} =", "100ae91942a961d32bebd12bbe368146": "\\eta_A:A\\to T(A)", "100b483f3fe1552228dc3b6f46f699c8": "\\overline{\\mathcal{M}}_{g, n}(X, A)", "100b54fa9bb67b6b3e2f1032dd430fff": "L(p;q)", "100b581e900eef1a0a5515e3a5d23761": " \\sim 10^{2075} \\,\\!", "100b8e815cab90d99a5855816c38ce0f": "h{(x)}", "100b908b432230b727445c2b4f09310a": "KG(n,2)", "100bb23db4f39d534c682204f8a80108": "m(m-1)\\cdots(m-N+1) + a_{N-1} m(m-1) \\cdots (m-N+2) + \\cdots + a_1 m + a_0 = 0.", "100bfde3965eb604a6606f2b4035ea94": "\n\\nu^{2} < c^{2} < \\mu^{2} < b^{2} \n", "100c1942325e7db5eb0228bb06f4f77d": " n = 3", "100c3459c9faac0632883ad1a60b67ea": " \\mathbf{P} \\left( \\sum_{i=1}^n X_i \\geq 2 t \\sqrt{\\sum_{i=1}^n R_i \\mathbf{E}\\left [ X_i^2 \\right ]} \\right) < \\exp(-t^2), \\qquad \\text{for } 0 < t \\leq \\tfrac{1}{2L} \\sqrt{\\sum_{i=1}^n R_i \\mathbf{E} \\left [X_i^2 \\right ]}. ", "100c3ff8545181a97d4da3c7875cd7a9": "G^{+}", "100c408314244120416e19def3c38c3d": "Y^\\phi=Y \\cup \\{\\phi\\}", "100c7aeb5953e1cd50448129a7a20967": "A\\mathbf{x}\\geq \\mathbf{b}", "100cb49f690d8cc325c637e37292a77b": "A = \\begin{bmatrix}-2&2&-3\\\\\n-1& 1& 3\\\\\n2 &0 &-1\\end{bmatrix} \\,,", "100d1c0bc2659e5b7ee65c8cbe764c0c": "\\dot{Q_i} = A_i(J_i - H_i)", "100d6e775ac940621db6f7268127fa23": "Holds(f, do(a, s)) \\leftarrow Poss(a, s) \\wedge Initiates(a, f, s)", "100dab574f5a1db3bb61ec4e5cd099a5": "0\\leq{i}<64", "100dc37082ef1984556e31677fdaf019": "{O}(nk)", "100e0707f7d199a661c8f428fc7ba62f": "\\int \\mathcal{D}\\phi", "100e5404f9c31a6ed76f327f6fb9c8f6": "m : P(S)\\rightarrow R", "100ea8600a75c7d1f6ecd9d54ef3aab0": "X\\le_T Y", "100ee41c122e9337ee7ae75213777f76": "f(x) = \\lceil x \\rceil - 1", "100f05a70d1c84f35a87d494273bda5a": "TL_5(\\delta)", "100f1213f2614e74a47ec8831b02bff4": "{{n^2-n} \\over 2} ", "100f1f53498dbe04a476ac1463389a54": "\\Delta S = \\frac {\\Delta H} {T}", "100fcc6bc41ec6c7aa42bc8a6129fa8f": " \\mathbb{E}(dW_{t}^i dW_{t}^j) = \\rho_{i,j} dt", "100ff7bc28bc6830ff34558f487eaf5e": "q = q_{\\text{IN}}-q_{\\text{OUT}} = C_S(V_{\\text{IN}}-V_{\\text{OUT}})\\ ", "10101e318a1a3d82757686046d7e8548": "\\dot{x} = y (z - 1 + x^2) + \\gamma x \\, ", "10102dcc248b1c254a175f15bb1696b8": "\n\\begin{matrix}\nI\\dot{\\omega}_{1} &=& N_{1}\\\\\nI\\dot{\\omega}_{2} &=& N_{2}\\\\\nI\\dot{\\omega}_{3} &=& N_{3}\n\\end{matrix}\n", "101070f2562431f38efc0a4c0ada4922": "\\sum_{(c)} c_{(1)}\\otimes c_{(2)}\\otimes c_{(3)}.", "1010b6c92178442ba651f2ddc697a253": "\nM = \\frac{4}{3} \\pi R^{3} \\rho\n", "1010fd31d0b9e49a3f3a8c6e1dc29a85": "\n \\left[\\mathbf{u},\\mathbf{v},\\mathbf{w}\\right] := \\mathbf{u}\\cdot(\\mathbf{v}\\times\\mathbf{w}).\n", "1011271522b26756c9e795b122869bc9": "\\mathbf{a} \\cdot \\mathbf{b} + \\mathbf{a} \\wedge \\mathbf{b} = \\frac{1}{2}(\\mathbf{ab} + \\mathbf{ba}) + \\frac{1}{2}(\\mathbf{ab} - \\mathbf{ba}) = \\mathbf{ab}", "101150d27a8dab82745e5a19fc277686": "\n\\epsilon = \\frac{\\overline{BC}}{r}\n", "10117357f37f011bda557b047aa53e0a": "\\psi_\\ell", "10125364a160b25a15cb760431aee050": "\\Pi = \\left[\\pi_{ij}\\right]_{1 \\leq i,j \\leq d}", "101268065830c5fbd5d8458ad04934da": "-1.7917", "1012d8c6fbf6c949b011229cb6f99053": "\\scriptstyle x(t)", "1012ef1d91b822b48445357a05b868bc": "\\begin{pmatrix} x_0 \\\\ y_0 \\\\ z_0 \\end{pmatrix}, \\begin{pmatrix} x_1 \\\\ y_1 \\\\ z_1 \\end{pmatrix}, \\begin{pmatrix} x_2 \\\\ y_2 \\\\ z_2 \\end{pmatrix}, \\begin{pmatrix} x_3 \\\\ y_3 \\\\ z_3 \\end{pmatrix}", "10130821535233756bbfdfe1eaef5c2d": "\\ln r = \\ln k + x\\ln[A] + y\\ln[B] + ...", "10130e08f6cb78188c9ab3cf9d8694b9": "A+1", "10133300e6a7dddc7ea3bf8b43a0b833": " \\hbar k_{i\\parallel}=\\hbar k_{f\\parallel}=\\sqrt{2m E_f}\\sin\\theta", "101402ccddece5cdaa75b2e8ea8f5b44": "v({\\mathbf A}_1{\\mathbf A}_2) = v({\\mathbf A}_1) v({\\mathbf A}_2).", "10145689b1ded987949529ce86b2047a": "\\Rightarrow c=\\frac{d}{2\\varphi}.", "10147211ece5173f5fa605b0b1fad8f9": "\nESP(x) = \\sum_{i=0}^{m} x^{si}", "1014f746e1bd6ca728d782df109de1ce": "\\mu(x+y), \\mu(x), \\mu(y)", "10152700de2734a855d14aacbd31748a": "\\alpha=\\beta, \\alpha'=\\beta', g=h", "101549870afb71e18f0b59ff6a5879d4": "q\\gtrsim k", "101572e8b9ad2065b5483f3225c8825b": "|a(x, y)|\\le K\\|x\\|\\,\\|y\\|", "1015842f05bbbee12a54c46107d6d053": " H^{s}(E)\\leq \\sum_{j} \\mathrm{diam} (U_{j})^{s}+\\varepsilon.", "101669d1616c4530575bb68bb8485d1b": "c: V(G) \\rightarrow \\mathbb{N}", "10166acb7a015f0e86ddcf6be9179c39": "f\\in\\mathbf{N}", "101677eecac13fc7fe9da7787fece4b4": "\nh_{LCL} = (20 + \\frac{T}{5}) (100 - RH)\n", "1016fc7d7ec9a7dec4451c173be47067": "d=2 ", "101734a77c6e813cbc1a08672c64f06d": "\\frac{F}{P} = \\frac{x_{p} - x_{t}}{x_{f} - x_{t}}", "1017525353a77d3eec6998a7519eef7c": "\\sum_{i=0}^{M-1} c_i b_i = 0", "1017a37c9d90cd8da77ac1134abb2a78": "F\\left(x_1,\\dots,x_n,u,\\frac{\\partial u}{\\partial x_1},\\dots,\\frac{\\partial u}{\\partial x_n}\\right) = 0.", "1018045904d066582b35d9fb21e9da2c": "(13) 2", "101831942f8cd71ffc8ff63a05e10814": " \\frac{r_E}{r_E + r_O}", "1018510c7f5c1246f02fef804ce8bb87": "{a}\\,\\!", "101858feca4929e73c77ec3e18256def": "F(s) = \\mathcal{L}\\left\\{f(t)\\right\\}(s) =\\int_{-\\infty}^{\\infty} e^{-st} f(t)\\,dt.", "1018f55a94318deaae1fde691977a6fb": "T^{\\mu\\nu}=T^{\\nu\\mu}", "1019805965d4d5cfd0c8c6300142fe8d": "c^2=a^2+b^2-2ab\\cos C, \\, ", "1019906e5e943cfc000037fdc2a200e6": " A \\succeq 0 ", "1019f71c1f2e61361f61be6ffe05e745": "\\mathfrak{B}(V_q)=k[x]", "101a8b0d906583b3b72e95deb6f89323": "\\vec M\\!", "101a937099de78c3921090bc174b07a6": " \\hat{S}f = QY \\, ", "101ab42c84a2e6e952e61b373e7924e6": "\\alpha_V = \\frac{1}{V_m} \\left(\\frac{\\part V_m}{\\part T}\\right)_{p}.", "101b0cd937f58d476d9f72710656dab0": " \\det\\left(R_{a, \\theta}\\right) = 1 ", "101b65d07b807bc698c83121b8ceb326": "10^{10^{10^{34}}}", "101b8d318c8f03a0f872d5bae7b5fa8b": " H( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ K( \\mathbf{w} ) + \\lambda S( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{t=1}^\\infty H_t( \\mathbf{w} ) ", "101bb4aaface0473428ef95b361ab3fa": " D_{n, m} = {n \\choose m} D_{n-m, 0} \\; \\; \\mbox{ and } \\; \\;\n\\frac{D_{n, m}}{n!} \\approx \\frac{e^{-1}}{m!}", "101bc1c68732daa1f8d224c9a811c05f": "\\frac{\\partial E}{\\partial w_i}", "101bc855e74b31124d6b7789708f3a0a": "W_\\alpha \\equiv -\\frac{1}{4}\\overline{D}^2 D_\\alpha V", "101bd2a399c67ff5a5f8f09d8b405f85": "\\frac{6,770,000\\ \\mathrm{N}}{(8,391\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=82.27", "101bdc0472aab002b435d88f669aa0bd": "\\nabla_{\\vec{e}_0} \\vec{e}_0 = 0", "101be2f87d5402bae889b8c96aa989d1": "\\theta_{i=1 \\dots N}", "101bff88738c41a75eadd39eeefc036c": "F(x) = \\exp\\left[\\sum_{r=1}^\\infty(\\kappa_r - \\gamma_r)\\frac{(-D)^r}{r!}\\right]\\Psi(x)\\,.", "101c12cc55409eaa4f55b7534550a0d2": "\\Sigma r (Q_0 + \\Delta Q)^{n} = \\Sigma r (Q_0^n + n Q_0^{n-1} \\Delta Q + ...) = 0", "101c272186e252afc449dc8d237dab15": "\n{1 \\over \\left( 2 \\pi\\right)^2\\; 2^{2n} \\; n! }\n\\int d^2z_1 \\; d^2z_2 \\; \\mid z_1 - z_2 \\mid^{2n} \\; \\exp \\left[ - 2 \\left( \\mid z_1 \\mid^2 + \\mid z_2\\mid^2 \\right) \\right] \\;\\mathcal J_0 \\left ( \\sqrt{2}\\; { k\\mid z_1 - z_2 \\mid } \\right)\n=\n", "101c9d39e3a1cec94b2011dfd1f40f30": "\\lim_{\\epsilon\\to 0^{\\pm}}\\zeta(1+\\epsilon) = \\pm\\infty", "101ca71841056332bab0792dc4fd7e40": "A=(0,0)", "101cbe29cd042b0cfc4bda2f303051bf": "l_a m^a=l_a \\bar{m}^a=n_a m^a=n_a \\bar{m}^a=0\\,;", "101cd62a5ed4aa01112ec57b614ab03b": "2^{|S|}-1", "101d23d1f67bf1f5e1b05e349b3a1753": "n=0.5", "101d534074f9a44ad29f6736b3f38871": "\\lang x, y\\rang = \\lang x^t y\\rang.", "101d938f56e9bec4816eb951f1900faa": "R = \\frac{m}{w_{r}}", "101d971d7290862710e5cae4cb0ee5f1": "\ns_{\\overline n|i} = 1 + (1+i) + (1+i)^2 + \\cdots + (1+i)^{n-1} = (1+i)^n a_{\\overline n|i} = \\frac{(1+i)^n-1}{i}\n", "101da1e64639b233c4c4437ec810d79d": " \\and T = [F, S, A]::R \\and (S \\implies (\\operatorname{equate}[A, P] \\and V[F] = A)) \\and D[F] = K ", "101e29910e5e5a213c941d04402bb0f1": "(X, \\mathcal{T})", "101e3ca6e1cc8fb165e81d5fdace94e0": " TX = \\bigcup_{x \\in X} \\{ x \\} \\times T_{x}X ", "101e52b4e380de9ac709d7bd30b7514d": "\\zeta\\left(\\tfrac{1}{2}+it\\right)", "101e5d86cc807f43c4719b5a5b83fa19": " \\hat{f}(z):=\\int_{\\mathbb{R}^{d}} f(x) e^{-iz \\cdot x}\\,dx \\rightarrow 0\\text{ as } |z|\\rightarrow \\infty.", "101e66560b32b106878b472f9a8366ab": "1/q + 1/p = 1", "101e887aceafd71b80779b55c67ea386": "\\theta_i^{(m+1)}= \\theta_i^{(m)}+V_i(t_{1})\\sum_{n=1}^N V_i^{-1}(t_{n})(\\bar\\theta_i(t_n)-\\bar\\theta_i(t_{n-1}))", "101e94baac4c70d59bcce352bf3b1345": "\\langle gv,gw\\rangle = \\langle v,w\\rangle", "101ec361e8daa1452c8e8f64fa81a4c2": " B_1 \\subset \\cdots\\subset B_m ", "101eea366df2cfc2e745004a9a5b94b2": "\\frac{\\partial c}{\\partial t} + \\boldsymbol{w} \\cdot \\boldsymbol{\\nabla} c = D \\nabla^2 c", "101effc2591bc0c647d98cbc76310437": "\\frac{1}{3} \\pi r^2 h", "101f1f9f7f2e58304b4790fd94857d7b": "T(n,k,r) \\geq \\binom{n}{r} {\\binom{k}{r}}^{-1}.", "101fe5774366427afe4dd92e9f412cf7": "\\tilde Y \\to \\mathbf{C}", "10201b736e5c84aa179e2c7f31d2db86": " -\\Gamma_a^T ", "10206cd0f4fee98ce02ded678934d0fd": "\\kappa={b \\over a}", "10206e9dcc19519691fdbcad3dd98409": " t_1^{j_1(g)} t_2^{j_2(g)} \\cdots t_n^{j_n(g)}", "10207f691e45ce6c21203d38029bbe5e": "\\{N(t)", "1021502180e9899a649eac88e05d4cf6": "\\tan(\\alpha \\pm \\beta) = \\frac{\\tan \\alpha \\pm \\tan \\beta}{1 \\mp \\tan \\alpha \\tan \\beta}", "1021616cd3324e941a9ae09c8d9cf947": "\\beta^{\\prime}(m_2+1,-m_2-m_1-1)\\!", "102195f15090f026d8317cd60dea038d": " \\sigma_z", "102231476db11d43c8dc571a1ffda61b": "\n\\phi\\ =\\ R(r)\\ \\Theta(\\theta)\\ \\Phi(\\varphi)\n", "1023119763c930712618f0f34f01132f": "Y=\\emptyset", "10232c5112c06ea83d8c23155314d7cc": "\\begin{smallmatrix}\\frac{M}{M_{\\odot}} \\cdot \\left( \\frac{R}{R_{\\odot}} \\right)^{-3} \\cdot \\rho_{\\odot}\\end{smallmatrix}", "10233e455cba656a8d3a982890708a4a": "\\frac{\\#X}{\\#G}.", "1024427dee9937f2c344d83879d55774": "Z_{\\mathrm{p}}(s)=\\frac{\\det \\mathbf{[A]}}{s \\, a_{11}}", "10244a40f23babe697c28fe646533369": "\\Gamma_{nrad}", "102452b19e738d171a19f052392e8038": " E = T \\,\\rightarrow \\,\\frac{\\hbar^2 k^2}{2m} =\\hbar \\omega ", "1024535b14e8894eae104245542d2293": " \\partial_\\mu F^{\\mu \\nu} = \\mu_0 J^\\nu ", "10247536f8d9fc00f5eec6d721ed5b36": "\\Rightarrow \\omega_s = \\frac{1}{\\sqrt{L_1 \\cdot C_1}}, \\quad \\omega_p = \\sqrt{\\frac{C_1+C_0}{L_1 \\cdot C_1 \\cdot C_0}} = \\omega_s \\sqrt{1+\\frac{C_1}{C_0}} \\approx \\omega_s \\left(1 + \\frac{C_1}{2 C_0}\\right) \\quad (C_0 \\gg C_1) ", "1024999806694509b25f976ce4e71954": "F_\\mathbf{g}", "1024acb807c16b8242f450d1db97d040": " s=(\\ldots, (s_{i}, t_{si}, t_{ei}),\\ldots)", "1024c86bac9029bee6473c1fb3c15082": "(V\\otimes W)_i = \\bigoplus_{j+k=i}V_j\\otimes W_k", "1024eea116c9ed1058523bba6ccfebad": " \\sin z = z \\prod_{n=1}^{\\infty} \\left(1 - \\frac{z^2}{n^2\\pi^2}\\right) ", "1024f32ba46288162150ebad1dfefa18": "(\\mathbf y - X \\hat{\\boldsymbol{\\beta}})^{\\rm T} X=0.", "102573c9e2aef705beb137b92ab98b04": " u(t) = g(x_0, t), \\quad x(t_0) = x_0, \\quad t_0 \\le t \\le t_1 ", "102579f5e34db0edd96c3be5448f2666": "\\scriptstyle p(13k \\,+\\, a) \\;\\equiv\\; 0 \\pmod{13}", "10259508c482500b36090ab0a79ec2aa": "I\\times I", "1025cd290de1cbf23a39090af6d36b77": "W^{1, 2}(\\Omega)", "1025dd023d6a3c47947806a8612a2ca5": "\\displaystyle \\cos^2{A}+\\cos^2{B}+\\cos^2{C}=1.", "1025fa8ce05b7547eca28aac99d3841f": "= V_0e^{-t/\\tau} +A\\frac{1}{j\\omega +1/\\tau} \\left( e^{j \\omega t} - e^{-t/\\tau}\\right).", "102615d9a43cf88c37c99c66336ce69c": "\n \\{ \\hat\\theta_\\mathrm{mle}\\} \\subseteq \\{ \\underset{\\theta\\in\\Theta}{\\operatorname{arg\\,max}}\\ \\hat\\ell(\\theta\\,|\\,x_1,\\ldots,x_n) \\}.\n ", "10263a140f90eb72cc009d63dbb5a431": " z = \\pm h", "10263b17025282037401d176edf8223b": "\n R^\\gamma_{\\alpha\\beta\\rho} := \n \\frac{\\partial }{\\partial X^\\rho}[\\,_{(X)}\\Gamma^\\gamma_{\\alpha\\beta}] -\n \\frac{\\partial }{\\partial X^\\beta}[\\,_{(X)}\\Gamma^\\gamma_{\\alpha\\rho}] +\n \\,_{(X)}\\Gamma^\\gamma_{\\mu\\rho}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} - \n \\,_{(X)}\\Gamma^\\gamma_{\\mu\\beta}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\rho} = 0\n", "10264615b7563717f5f50062ccc4e098": "P_h=h g \\rho - l g \\rho\\sin\\psi", "1026715418f54e5d8ad6c89bbf1dbccb": "\\gamma = 2\\arccot \\left\\{\\tan\\left(\\frac12(\\alpha-\\beta)\\right) \\frac{\\sin \\left(\\frac12(a+b)\\right)}{\\sin \\left(\\frac12(a-b)\\right)} \\right\\},", "10268d7b80e09da1f88dc6b3e552e53c": "\\bar{\\epsilon}", "1026fa5d8cbf459764f08fd3a83a3f95": "x(\\sigma)=\\sum_{k=0}^\\infty \\sigma_{k} 2^{-(k+1)}", "10270d931daf6d727d13d59b8bd682f4": " b>0,s=1,\\beta\\ne1", "10270df5d24fcbb9dd6a00cd96610039": "1/100000th", "102758538c3a776dbb7789b560578eb8": "f_A(x) = \\begin{cases}\n \\frac{1}{n} & \\text{if } x \\text{ is rational and } n \\text{ is minimal so that } x \\in F_n\\\\\n -\\frac{1}{n} & \\text{if } x \\text{ is irrational and } n \\text{ is minimal so that } x \\in F_n\\\\\n 0 & \\text{if } x \\notin A\n\\end{cases}", "1027d252747926648c413216c056ffbd": " f(p) = \\frac{1}{4 \\pi m^3 c^3 \\theta K_2(1/\\theta)} \\exp\\left( -\\frac{\\gamma(p)}{\\theta}\\right)\n", "1027d37c74892a9be58ce55dd3a68e5f": "Math \\geq medium", "1027d569c08705a1d176ca26904144ab": "f_\\theta(x)", "1027d86971238e2c8890888ea6757a37": " \\text{storage efficiency} = \\frac{\\text{effective capacity} + \\text{free capacity}}{\\text{raw capacity}}. ", "10283d17d27eaf286f6a21bd089b06bd": "sim(d_{j,q}) = \\frac{P(R|\\vec{d}_j)}{P(\\bar{R}|\\vec{d}_j)}", "102855b21b719ae1b495b15c22051259": "\\tbinom{n+k-1}{n-1}=\\tbinom{n+k-1}{k}", "10285e0066647eeb9add855d58741547": "q + r \\leq 1", "10285f558ab4b167191f884c9d5a06ab": "4^5=1024", "1028a7c1188bd3d81ab75fe764f5f036": "|\\psi'\\rangle= \\alpha_0 |000\\rangle + \\alpha_1|111\\rangle. ", "1028b5307b619b51cb78c0796b3ef506": "y_1 = 1.066869388", "1028f2df4211d4057f8892585c2b98b8": "\\hat{x_i}", "1029b6f472a96d3b952f1f823448c5c0": "20^2 = 400 \\equiv 15", "1029cd07e0ea07c469dcc29d97aba969": "\\left( x,t\\right) ", "1029cd3f41bd4c684ea220434bb119dd": " \\boldsymbol{\\nabla} \\cdot (\\varphi~\\mathbf{v}) = \\varphi~\\boldsymbol{\\nabla} \\cdot \\mathbf{v} + \\mathbf{v}\\cdot\\boldsymbol{\\nabla} \\varphi", "102a10cb398bd3e335fe8a0668bfd8f1": "\nv=\\sqrt{\\frac{T}{\\mu}}, \\,\n", "102a28c1c081668e97ec05cbab03a3ed": "\n \\begin{bmatrix}\n a & b/2 \\\\\n b/2 & c\n \\end{bmatrix}.\n", "102a6408ae9e7cbd8ad32c664bd9de81": "\\vec{\\tau} = I\\vec{\\alpha}", "102ab1c6749dce0c13fddf3990dfdfe7": "y_2", "102b67b7bb51826fc86b2470c1a9b531": "\\langle -\\alpha|\\hat{\\rho}|\\alpha\\rangle=\\langle -\\alpha|\\psi\\rangle\\langle\\psi|\\alpha\\rangle", "102b7e8d3f76a9c950d08bfdf98830ef": "C_n \\Bbb R^2", "102bae106140aeaf227180024a6ae2b2": "4 \\over 5", "102bb348fd2c1410e444140ec8889158": "X_{(a,b,c,d)}(u) = \\sqrt{-i} \\cdot e^{i \\pi \\frac{d}{b} u^{2}} \\int_{-\\infty}^\\infty e^{-i 2 \\pi \\frac{1}{b} ut}e^{i \\pi \\frac{a}{b} t^2} x(t) \\; dt \\, , ", "102bc530ef1253fd4431edb7880f4466": "b=1;", "102bd4c258a00714d7dd11ce3e7075e0": " (1 + year + [year/4] + [(year - 1600)/400] - [(year - 1600)/100]) \\mod 7 ", "102bdc15f139a867d8b0f3396a968e9d": "\nJ = \\Psi\\Psi^\\dagger, \n", "102c2b378276eb5b124af69720498b7a": "\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n=\n(-1)^{j_1+j_2+j_3}\n\\begin{pmatrix}\n j_2 & j_1 & j_3\\\\\n m_2 & m_1 & m_3\n\\end{pmatrix}\n=\n(-1)^{j_1+j_2+j_3}\n\\begin{pmatrix}\n j_1 & j_3 & j_2\\\\\n m_1 & m_3 & m_2\n\\end{pmatrix}.\n", "102c32b26bbc907b501556281aa53841": "f(x) = x^3 - 12x^2 - 42\\,", "102cab400d8f3c240d4e50d658da24f3": "\\mathfrak{sl}(n,\\mathbb C)", "102d2bf2ff3751ac8226713a73d8ff96": "M = \\mathrm{diag}[ m_1 I_{n_1}, m_2 I_{n_2}, \\cdots, m_N I_{n_N} ] ", "102d6049a3b8e8b67a70919709a316ed": "\\frac{1}{\\alpha}\\tanh(\\alpha g t) = v", "102d6202c25478272229f898a5365033": "R_{J}=\\int_{0}^{\\pi/2}3\\sin \\theta (\\cos \\theta)^2 R_F(\\cos \\theta)d\\theta", "102da3191257df7b8c5bc3369b4b631f": "\\int \\cos(x)\\,dx = \\sin(x) + C.", "102dc7004515c2ac64327380faa80844": "L\\,\\!", "102dd3e45c84a4142f0b2b0bf905e243": "= \\left.\\frac{\\partial \\sigma}{\\partial x}\\right|_{p} = \\dot{\\sigma}(x) \\,", "102de6bc0991c781305934d2c92a8d9f": "\nf_X(x)= \\begin{cases}\n\\frac{1}{6}x^3 & 0\\le x \\le 1\\\\\n\\frac{1}{6}\\left(-3x^3 + 12x^2 - 12x+4 \\right)& 1\\le x \\le 2\\\\\n\\frac{1}{6}\\left(3x^3 - 24x^2 +60x-44 \\right) & 2\\le x \\le 3\\\\\n\\frac{1}{6}\\left(-x^3 + 12x^2 -48x+64 \\right) & 3\\le x \\le 4\n\\end{cases}\n", "102e0c46a0c48c3d19943bae38ca1c77": "\\ln \\gamma_i = \\ln \\gamma_i^c + \\ln \\gamma_i^r", "102e39fb67fe113e0d04e5bbc5e040ce": "I \\approx \\frac{c \\, n \\, \\epsilon_0}{2} |E|^2,", "102e6cd8bf051901c3c2144d287d75a8": "\n \\varphi \\to \\varphi e^{k \\omega}.\n", "102e7a07844dffac65a4e4000cc2394a": "A = (a_1,\\dots,a_n) \\subseteq V", "102ea74c2046dc9560869cce40aec725": "P = \\mathbf{v} \\cdot \\mathbf{F}", "102ec24202515e1cd4c4556cea5da73b": "V_z", "102ed868d63f74136287b8fe0cc5d661": "V_n=\\frac{2 (2\\pi)^{(n-1)/2}}{n!!} R^n.", "102edd41a0f0268c29d257e10ca467b9": "T^{\\mathrm{H}}_p (x,y) = \\begin{cases}\n T_{\\mathrm{D}}(x,y) & \\text{if } p = +\\infty \\\\\n 0 & \\text{if } p = x = y = 0 \\\\\n \\frac{xy}{p + (1 - p)(x + y - xy)} & \\text{otherwise.}\n\\end{cases}", "102f7748c227f8ef79cd8e77c89ff989": "\\mathcal{R}^K _{1/2}", "102f7b26809a107e81be9e0371929a6e": " \\frac{\\partial \\theta }{\\partial t}= \\nabla \\cdot D(\\theta) \\nabla \\theta ", "102f8896841ad94d3cbd761d904c422f": " \\nabla^2 \\phi = \\rho (x, y, z) \\; .", "102fba291d5242c58260fa7a6ec6a869": "n=2\\,", "102fca3929563807011e10100f19722a": "\\int \\sigma_x dA = 0 ", "102fe0e8f33a162633812a71731ca902": "\\prod_{t \\in T}(1-x^t)^{-1}.", "10302ca9e8f4048ee0a24fc4d9705e81": "d_G(p,q) = d_{G'}(f(p),f(q))", "1030448c29ab3533e26c8ab0dfd0c0a1": "\n x^{k+1} \n = \n x^{k} \n + \n \\frac{b_{i} - \\langle a_{i}, x^{k} \\rangle}{\\lVert a_{i} \\rVert^2} a_{i}\n", "10304638e0cede49b46e5ec345077525": "Coenergy = area ~ OACO = W'_{stored} = \\int_{0}^{i} \\lambda(i) ~ di \\;", "10308439b407adff8dd4ce208f49302b": "1/2B", "103084ce820b78321322b7981badf2bd": "T=\\frac{m}{2}\\mathbf{\\dot{r}}\\cdot\\mathbf{\\dot{r}}", "1030dad401f7d4683087bf75e1e09de1": "a = 0.5", "1030dec6085b4ce36d0ea5d25132b0e3": "S=R-\\{0\\}", "1030e9eaa4fcba3835fb2b77d219599c": " g_m(z) =\n\\frac{1}{|S_m|} \\left( \\log \\frac{1}{1-z} \\right)^m =\n\\frac{1}{m!} \\left( \\log \\frac{1}{1-z} \\right)^m.", "103196e771d34f5b93e7f5390638f8fd": "\\Delta I = 0 \\Rightarrow", "1031b74ac2bf56c60578f00f3506b94c": "\\zeta(-k)= -\\frac{B_{k+1}}{k+1}", "1031e9537fd9a53d196fde746ac8572f": "f:E\\mapsto R", "103214e1a717b7c1a5a33e55e5757c11": "\\arcsin x = 2 \\arctan \\frac{x}{1+\\sqrt{1-x^2}}", "10326a7f44a2aed29848023d9e0ed0d1": "\n\\frac {m_2 u_2 - m_2 u_1 + m_1 u_1 + m_2 u_2}{m_1 + m_2} =\n\\frac{u_1 (m_1 - m_2) + 2m_2 u_2}{m_1 + m_2}", "1032735dcc0ca5203b269cf9a4e448bf": "(\\coth \\alpha - 1) \\left(e^{2 \\alpha} - 1 \\right ) = 2", "1032779a9946de78d71667c4eccf0941": "F^{ab} = \\partial^b A^a - \\partial^a A^b \\,\\!", "1033162fd547cdebb5f1f70c765f4286": "Tf(\\psi) = f(T^*\\psi).\\,", "103353cc9b8470d53fdcecdfd235bef1": "\\Sigma_1(L_\\alpha[B])", "10337fe2b560f392b4a56867f5c5698a": "O(c^{12} \\log{n})", "1033a62855a3339dba801a62c385cd0c": "\\displaystyle u_{xt}= \\sinh u ", "1033cf9540529a545f88b8b29f6dda13": "\n A \\subset B \\subset C \\quad\\textrm{and}\\quad |A|=|C| \\qquad\\Rightarrow\\qquad |A|=|B|=|C|\n", "1033d1b50c9529f52307e1d9b8f83550": "T |\\alpha, \\beta \\sim \\mathrm{Gamma}(\\alpha,\\beta) \\! ,", "1033ec26434eeb17994f6b4f38bd4b9b": "\\frac{V_1}{T_1} = \\frac{V_2}{T_2} \\qquad \\mathrm{or} \\qquad \\frac {V_2}{V_1} = \\frac{T_2}{T_1} \\qquad \\mathrm{or} \\qquad V_1 T_2 = V_2 T_1.", "103408b48618d71c7e1baf910b144e75": "P_n = n\\binom{n+1}{2}-\\binom{n+1}{3}.", "103418c721caddb55f16da1b7e733e91": "G^a_{\\mu \\nu} \\,", "10341cab940e4f0d0dfb4e8d0c197db0": " P \\propto 1/K^{\\beta}", "10342149cd04422c7be79506cb544ac5": "x>y", "10343a0ffa3897b059bf833c27346d2c": "\\omega^* = \\underset{\\omega \\in \\Omega}{\\textrm{argmax}} \\sum_{n=1}^{\\ell} y_n h(\\boldsymbol{x}_n;\\omega) \\lambda_n.", "10345a62ca611efe33e873521938b1d4": "\\forall x D(x)", "1034bb957d8d9ac84e4fdf700718c97b": "\\zeta^{\\prime}(-8) = \\frac{315}{4\\pi^8} \\zeta(9).", "1034e605ec7bc7833f2217ed2bb6f683": "\n\\begin{align}\nA & {} =\\begin{bmatrix}\n 1 & 2 \\\\\n 0 & 1 \n\\end{bmatrix}, & & \\mbox{SNF}(xI-A) =\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & (x-1)^2\n\\end{bmatrix} \\\\\nB & {} =\\begin{bmatrix}\n 3 & -4 \\\\\n 1 & -1 \n\\end{bmatrix}, & & \\mbox{SNF}(xI-B) =\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & (x-1)^2\n\\end{bmatrix} \\\\\nC & {} =\\begin{bmatrix}\n 1 & 0 \\\\\n 1 & 2 \n\\end{bmatrix}, & & \\mbox{SNF}(xI-C) =\\begin{bmatrix}\n 1 & 0 \\\\\n 0 & (x-1)(x-2)\n\\end{bmatrix}.\n\\end{align}\n", "103558daf835ebcf075ebdad1bf46046": "\\delta^{(s+1)} = 1 - \\frac{\\left ( k^{(s)} / x^{(s)} + 1 \\right ) \\delta^{(s)}}{1 + \\delta^{(s)} k^{(s)} \\left ( 1 + x^{(s)} + 1 / x^{(s)} \\right )},", "10357eca6dae03ef9444d6959d569d68": " (\\mathbb{Z}/p^k\\mathbb{Z})^\\times \\cong \\mathrm{C}_{p^{k-1}(p-1)} \\cong \\mathrm{C}_{\\varphi(p^k)} .", "103581bf307c91c65d905f3713cd4c3a": "\na^{\\dagger}|\\alpha\\rangle=\\left({\\partial\\over\\partial\\alpha}+{\\alpha^*\\over 2}\\right)|\\alpha\\rangle\n", "1035f984542a0935bb1b52a2107c2b55": "\\mathcal D\\phi", "1036107b80b3c20e8a4b5d0f3d435f3b": "\\psi(y_1,y_2) = A\\!\\int_{-\\infty}^\\infty dp\ne^{-p^2/4\\sigma^2}e^{-ipy_2/\\hbar} e^{i py_1/\\hbar}\n\\exp[-{(y_1+y_2)^2/16\\Omega^2}]", "1036c54502c98d1a6544ef96cbb3931d": "N^cH^k(X, \\mathbf{Z}) = H^k(X, \\mathbf{Z}) \\cap (H^{k-c,c}(X) \\oplus\\cdots\\oplus H^{c,k-c}(X)).", "10372075e5d94832483f4b3ba8003d64": "n = \\sum_i n_i", "10373f982f06c0fde45572daabc28069": "\\textstyle 1/e", "10375bb6626b2dbfa8d09354e45c8f00": "(0,1/4)", "103765e308cd130adf02b875f87339fa": "V \\otimes V \\otimes V^{\\otimes N}", "1037a5eaa16f867b3ec4d2e3719c6335": "\\Delta \\left( \\omega /2\\right) =\\frac 8{3\\rho _{\\mathrm{F}}Z_q}f_f^{\\,4}m_{ \n\\mathrm{F}}^3n^3\\pi ^2J^{\\prime \\prime }", "1037b761864b6ad19d60a53dc6e183ff": "\\frac{1}{q}=\\frac{1}{R}-j\\frac{\\lambda }{\\pi w^{2}}", "1037d8a52b9d26cb6583d4814eee6a22": " {n\\choose k} = \\frac{n!}{k! (n-k)!}.", "10383cf58902cd2c31693b239314bb31": "\\displaystyle{v_1=L^\\prime_{-1}v_0.}", "103849b2bf6419f9c09671c422720e16": "f: G \\rightarrow S", "10386b156192707a11884ed2e3a823d2": "x(t) = \\frac{\\phi(t)}{2\\pi v_n} = \\frac{\\Phi(t)}{2\\pi v_n} - t = T(t) - t ", "10387d7085857102d3ae38d479874e0e": " \\alpha_j\\over \\alpha_j-\\alpha_k ", "10388960b06dfe7a831efe3b625f0f6a": "u_y(\\mathbf{y}) \\triangleq u_1(\\mathbf{x},z_1)", "1038c2d84bf2d2e047578d70cc42f289": "a,r > 0", "103900171be17cef5009421ac3d8abb4": "h^{\\alpha\\beta ab(x-y)}=h^0=1", "103943d5a8e71e1bddd99215f5bce206": "\\|\\cdot\\|_{\\alpha}", "10395826925c4ee44ccbfd4828f72efe": "\\left(2+\\sqrt{-6}\\right)^6 = \\left(-2 + 4\\sqrt{-6}\\right)\\left(-1-3\\sqrt{-6}\\right) = 9+2\\sqrt{-6} .", "10399bd6f02aa8ee0c26c9fd85d27fcd": "E_{10}", "10399d6467fff318f95995da28a7aee0": "\\frac{p(y|H2)}{p(y|H1)}", "1039d4ecfce0a267b94a159b0c756f9f": "z \\in \\mathbb{C}^\\times", "1039d86887ddb36a4b4cbe8332e3f05e": "f = \\sum_e c_e T^e\\,", "103a0423a08cf4835cbdc9ad3990a6bd": "s^* ", "103a3afea292a4152356bb956807cf5c": " M = i\\lambda \\,", "103a6537463f3fc109fb69744b35a922": "2^{-a}\\theta_{-a}(x/2)", "103a9b591a0c13bf963ee47217364bb9": "A=\\{x\\in X|(\\exists y\\in Y)\\langle x,y \\rangle\\in B\\}.", "103af78d2723cbc7390feca98a2f6627": "S_{n p_{i+1}} = F_{p_{i+1}} [S_n \\cup S_n + n \\cup S_n + 2n \\cup ... \\cup S_n + n (p_{i+1} - 1)]", "103b219e350cd93c0481cbe2e34c9702": "\\Delta=-128p^2r^4+3125s^4-72p^4qrs+560p^2qr^2s+16p^4r^3+256r^5+108p^5s^2\n", "103b54a6c3552438e651ae24c1e05e15": "\\frac{dN}{dt} + \\lambda N = 0. ", "103bc12bcc604e062151c53eb4c4bfcc": "P>p_N", "103be57d2dd0dc101fdc0f1680a43ca9": "\\bar{H_i}=\\bar{U_i}+P\\bar{V_i},", "103be638391ff903d89fd4f0cf206271": "h_{n-1}", "103c1fa60eb3dd7a2fd25df702db837e": "S_{\\rm metal} = \\frac{\\pi^2 k^2 T}{-3 e} \\frac{c'(\\mu)}{c(\\mu)} + O[(kT)^3], \\quad \\sigma_{\\rm metal} = c(\\mu) + O[(kT)^2].", "103c4d1d4b12b42fe38ce7c10f3d1c23": "h_A(X)=\\operatorname{Hom}_\\mathcal{A}(A,X)", "103c555bb111c3ae1160080c6cbf924b": "\\Leftrightarrow \\frac{y}{x} \\cdot \\frac{2y}{2x+3c} = -1 ", "103c8952f01a7876f26a9490bb62ab3b": "\\scriptstyle Z'", "103cbc44576c4035a2df6850bb50ad60": "T( \\text{deg C}) = 16.9 - 4.0 \\times \\mathrm{\\delta^{18}O_{calcite}} - \\mathrm{\\delta^{18}O_{seawater}}", "103d195c206db9ddeeee72242a84807b": "\\boldsymbol{r} = (\\boldsymbol{q},\\boldsymbol{p})", "103d7a0c324401169a814fe6d5043396": "a_1=0.93\\,", "103daec9259d6cc712f853b6c5dde6fa": "^{\\;}f(\\xi )", "103dedc904b031e2459e88d1bd093bcb": "\\frac{1}{f} = \\frac{1}{f_1} + \\frac{1}{f_2}.", "103e0caf979e6731177b723f8edd652c": "-\\infty < t, \\, z < \\infty, 0 < r < \\frac{1}{\\omega}, \\; -\\pi < \\phi < \\pi", "103e20d90cbf0b98541b2e4e0e9901a0": "\\mathbf{w} = \\sum_{i=1}^n{\\alpha_i y_i\\mathbf{x_i}}.", "103e2ec4f880c11447aa9596b821c83c": " c\\left( \\sum_n | a_n|^2 \\right) \\leq \\left\\Vert \\sum_n a_n x_n \\right\\Vert^2 \\leq C \\left( \\sum_n | a_n|^2 \\right) ", "103f21c2c5b971f581c044b54f90a7c6": "\\operatorname{fnchypg}(x;n,m_1,N,\\omega) = \\operatorname{fnchypg}(m_1-x;N-n,m_1,N,1/\\omega)\\,.", "103f2691aeac73ca7be126270db302c5": "(\\mathbf{W}^i -\\mathbf{W}^1)\\cdot(\\frac{\\mathbf{W}^i +\\mathbf{W}^1}{2} - \\mathbf{G}) =0,\\quad i=2,\\ldots, 5,", "103f667bc70d6b2f5972ca2e5e27d09e": "f=165.4(10^\\frac{2.1*10}{35}-0.88)=513 \\ \\mathrm{Hz.}", "103f7463e1b2d2eeee76fbb24691b7eb": "\\ Z = |Z| e^{j\\theta}", "103f7841758fdc5d11a51cc2b125851e": "\n\\frac\n{\\sum_{i=1}^{n-k} \\nu_i \\xi_i^2}\n{\\sum_{i=1}^{n-k} \\xi_i^2},\n ", "103f90a4349b47fabb047deacb4549c6": "I_{j}\\left( \\Gamma ,N \\right)=\\prod\\limits_{n=0}^{N-1}{\\left\\langle \\psi _{j}\\left[ R\\left( t_{n} \\right) \\right] | \\psi _{j}\\left[ R\\left( t_{n+1} \\right) \\right] \\right\\rangle }", "103fa1bc75aecc520d5933e49e0b5549": "\\{a_{1}a_{2}...a_{n}|n\\ge0\\wedge\\forall i.a_{i}\\in\\Gamma\\}", "103fc80e8c3663040541afce940f2bed": "(ab)c\\ne a(bc)", "104066723193938c2edd888e32d870f2": " r_1 =e x + a\\,\\!", "10408c5f41240751f756006e6f7cf37e": "f\\colon\\,x\\mapsto x^2", "1040faaf1340df220c1a37c596b55d4f": "F(u) = 0", "1041225ecb84be65214b0102e74421ba": "p^d:G", "10413747457e0acff2083fecdb2920c7": "\\frac{1}{R^{3}}", "10413bfe0b5e6681771148750d25825e": "f=u_n+u_{n-1}+\\dots+u_1+u_0\\,", "10418882993ea07c4230763537f98586": "x^3 - 2x^2 + 0x - 4.", "10420df041a475a7648a272cc30d92b3": "\\mathbf A^{-1} = \\frac{1}{\\operatorname{det}(\\mathbf A)} \\mathbf C^\\mathsf{T}.", "10426de545ced1faebef461d9a384648": " \\phi_{mm} (r) = \\max \\left[ 0 , \\min \\left( 1 , r \\right) \\right] ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{mm}(r) = 1", "10427c68788765eefb18cdcff6dd0197": "c_{a}e^{a}\\int^{\\infty}_{a} f_k e^{-x}\\,dx", "1042ba18401d7d943c64063b746cc1f3": "\\mathbf{x}_B", "104303a4b2b2160d4e7c43adb3bfb099": "\\Delta E = E_2(t) - E_1(t) \\equiv \\alpha t, \\, ", "10432217f21f27d4b7b3ba19cda3786f": "A(1-\\rho)", "1043756a4177dceabbac8d1893407d04": "\\vec{a} = (\\sin \\theta \\cos \\phi, \\; \\sin \\theta \\sin \\phi, \\; \\cos \\theta)", "10438eb1d9691d1372f3f538767651a1": "\\log \\exp x = x", "1043de35feee1e77e1b541c07dcf97c1": "R(i,j) = \\begin{cases} 1 \\quad &\\text{if} \\quad \\| \\vec{x}(i) - \\vec{x}(j)\\| \\le \\varepsilon \\\\ 0 \\quad & \\text{otherwise}, \\end{cases}", "104415ce2cf1c7ff70b5dac7efe84836": "G : \\mathcal{D} \\rightarrow \\mathcal{C}", "10446d281db49f3a7a2d535c197faa2f": "L_{TX}", "10448234aa48987392431d0083a29736": "h_m", "1044ca8f8bcdc062498b6162458e15f1": "\\scriptstyle x^a", "104540420a414f7e4d8e85ce9961b640": "\\begin{align}\\Phi_{Y,X}(f) = G(f)\\circ \\eta_Y\\\\\n\\Psi_{Y,X}(g) = \\varepsilon_X\\circ F(g)\\end{align}", "10455d028dabd71a566bfd34b577ba1e": "x>T_{i-1,j}", "10456541e2e4199768f8dfbbd16ad328": "T([a_1,a_2,\\dots])=[a_2,a_3,\\dots].\\,", "1045703f180d4f164c014b7c8e7ee923": "\\operatorname{aff}(S)", "10466eac46ef87b0c7eaa7a937a50715": "R_{sens}", "104711545dac6c139387e42e0e161892": "|Df(x)|^2 \\le K(x)|J(x,f)| \\, ", "104721890407035b7824da6b9116dc52": "\\left\\lceil \\frac{-1}{\\log_2(1-p)} \\right\\rceil\\! - 1", "104748d879f2ce4e1e2766b081636757": "t' = C_n = t", "10482210c26ef1b1e7ccc8d8c810d803": "b = \\frac{2}{3} \\sqrt{-m_b^2 + 2m_a^2 + 2m_c^2} = \\sqrt{2(a^2+c^2)-4m_b^2} = \\sqrt{\\frac{a^2}{2} - c^2 + 2m_a^2} = \\sqrt{\\frac{c^2}{2} - a^2 + 2m_c^2},", "1048277d6c56a6c550f44662ab2b4432": "\\arccos\\left(-\\frac{r}{R}\\right)", "10483b9c8e8f35edc0f3825631e10fe5": " X_i = \\alpha Q_j + \\beta Q_k + . . . \\qquad i=1,2, ... I\\qquad \\qquad \\mathrm{(4)} ", "10486f564aa00a30bbfbee1f12edb357": "\n\\begin{align}\nQ(1)&=Q(2)=1, \\\\\nQ(n)&=Q(n-Q(n-1))+Q(n-Q(n-2)), \\quad n>2.\n\\end{align}\n", "1048bed8822c83d9b10093d3e41076fe": "\\Pi_{\\gamma\\gamma}(q^2) = q^2 \\Pi_{\\gamma\\gamma}^{\\prime}(0) + ...", "1048c41875dddbcb4527ba359df67eb9": "[z,x]\\subseteq B_{\\delta}([x,y]\\cup[y,z]). ", "1048ea90cbe90bc3ad306ba3345fcb8c": "x_1(t) \\,", "104907595c4752d247a512e898cd0791": "T = \\sum_{i=1}^r \\lambda_i \\, v_i^{\\otimes k}", "10490d946bbff452c7fac9a474ead806": " W_i \\,\\!", "1049776b40f2fa8b36a9d445962591fe": "k_e = 1 / (4\\pi\\varepsilon_0\\varepsilon)", "1049e1b7b43dfb935fb8aede1557b12e": "S_1(t) = S_0(t)^r", "104a3743c75e1801116e0675561ec38d": "x=\\frac{1}{a} \\int_0^{L'} \\cos s^2 \\, ds", "104a45b3f83657b3596ad131b439d49c": "C_0,\\theta", "104a7100ab2ed8107e49e48cb74b1d08": "n=i", "104a802daeb358d70acf05982a49eb85": " \\mathbb F_p ", "104a844d8b7debdd9f5c268b8b6c98d9": " = \\int_0^\\infty G(\\tau)F(t-\\tau)\\,d\\tau ", "104a9c2259ad58975903cd20b822dd18": "g^{\\alpha\\beta}P_\\alpha P_\\beta + (mc)^2 = 0 \\,, ", "104af5c0b0bd341f2537166f7f4b64fd": "c_2 = h c / k \\,", "104b150315cb3f0186f5f7a7890278af": "z=-1", "104b68522344cd182f7e6bf48597f8b4": "U(r) = \\frac {1}{4 \\pi} \\int_{S} \\left[ U \\frac {\\partial}{\\partial n} \\left( \\frac {e^{iks}}{s} \\right) - \\frac {e^{iks}}{s} \\frac {\\partial U}{\\partial n} \\right]dS ", "104b6ce5d83caf9db146524664c3bff4": "4:3 = 12:9", "104b9e820bba415dd2bd69440404dfc1": "Distance > \\left (\\frac {C}{2 \\times PRF} \\right)", "104ba9eca7bf3510ed30e77d6341954a": "\\sum_{\\boldsymbol{y\\in \\mathcal{N}(x)}}w_{\\boldsymbol{xy}}=1", "104bfa2b279ec591d6df5e56e43b7fa6": " \\gamma = ", "104c7143d90880d3190db00b6bb322c1": "\\neg B \\wedge \\neg C", "104c7ef64fbb8aef5a90561b37c2b0bb": "r = |x_1|", "104cd82faa54ffdf7e340acdcf2a122f": " n=N_D ", "104ce8c59cd154603c1a795b5eb17cc2": "\\mu(\\cdot, \\omega)", "104cf66a1e9b97dece25ef0fa9192bef": "gr E = \\bigoplus_i E_i/E_{i-1}", "104d462d2f0571247bf1c2f54241b56a": "{S^k}_h", "104da3274bc566f1a6a79f84204e4100": "s = \\frac{u^2}{2g}.", "104db75fd9c0c1d13834f8a53c765713": "v'=2t", "104e44d1d9ecfb3b95438a9dffe43cec": "\\mathit l", "104e8538e2be49fb06026350e29bab47": "1 \\mathrm{RPM} = \\pi/30", "104e9831686cbd2236305106009ebc0c": "\\mathcal{O}_{X}", "104eabba4fcc513e8a7171fe02fa056b": "\\vec{W}=\\vec{V}-\\vec{U}", "104eb81fef5e228c6b8a202b62e6f27a": "\n\\sigma(X)=\\sum_{i", "104f6adbd6ba7b01d4e5871d8b2be687": " X \\in \\mathcal{A} \\rightarrow I_{\\mathcal{A}}(X)= 1, X \\not \\in \\mathcal{A} \\rightarrow I_{\\mathcal{A}}(X)= 0 ", "104f6aea21ae8e217713eaba35450aec": "f(-x)=-(x - 1)^{2}(x + 1), \\,", "1050049992e5e1da727e42b203f22f5c": " \\left|2,H\\right\\rang ", "105031f299eba2a0412ef65a31a7bb06": "d(m,n) = n-m", "105034c0fa69d87cd61fe93e6fedfeb7": " r = \\frac{2 m cv}{eH}", "1050657ca01e3ef0856917571ac4bed1": "P = e^{\\frac{\\Omega + \\mu_1 N_1 + \\mu_2 N_2 + \\ldots + \\mu_s N_s - E}{k T}},", "10506f3ec86b0c69d40e251d78f4ab6c": "\\cos\\,{\\theta^*}= r \\cos\\,{\\theta}", "1050f432eb116ed842c1036ca0f40bf2": "|F(z)|\\le 1", "1050fb8ed5cf4421503064a55e9ffbf0": "H=-t\\sum_{\\langle ij\\rangle}c^{\\dagger}_{i\\sigma}c_{j\\sigma}+\\text{h.c.}+U\\sum_in_{i\\uparrow}n_{i\\downarrow}", "10510a06c145c567f25257c49e030903": "K(r,r^\\prime)", "10513679dc5acb961011d3d9a49bacab": "Y_{6}^{-2}(\\theta,\\varphi)={1\\over 64}\\sqrt{1365\\over \\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(33\\cos^{4}\\theta-18\\cos^{2}\\theta+1)", "10513952894e4f9a943efe7059a3974a": "\\operatorname{E}(X_i) = n p_i.\\,", "105156c37f38929fbdef4951b39c6507": "B_n(G,T) = \\{x\\in G | x = a_1 \\cdot a_2 \\cdots a_k \\mbox{ where } a_i\\in T \\mbox{ and } k\\le n\\}.", "105194c083e82084928df656ac2ae614": "\\operatorname{E} (X) = \\operatorname{E}_Y ( \\operatorname{E}_{X \\mid Y} ( X \\mid Y)),", "1051a9374ef548795bc25032fd225135": "\\lim_{t \\to \\infty} \\frac {X_t}{t} = \\frac{1}{\\mathbb{E}[S_1]}.", "1051b92c06d017dc0372c525c8f7d8b3": "\\displaystyle{{1\\over p} + {1\\over q} = 1.}", "1051e8b5104bec04e6c0393852efb4b6": "V = \\pi \\int_a^b \\vert f^2(x) - g^2(x)\\vert\\,dx", "1051f0ed74981811fc5f45a5e888de3e": "H^\\ast_{\\mathrm{dR}}(X/K)", "1051f481dcda92f5cda1c0ec389f3439": "n_j(x) := \\prod_{i=0}^{j-1} (x - x_i)", "105205252c4ef67036974ff1d4edd43a": "\\begin{align}\n\\mathbf E &= \\frac{1}{2}\\left(\\mathbf F^T\\mathbf F-\\mathbf I\\right) \\\\\n&=\\frac{1}{2}\\left[ \\left\\{ (\\nabla_{\\mathbf X}\\mathbf u)^T+\\mathbf I\\right\\}\\left( \\nabla_{\\mathbf X}\\mathbf u+\\mathbf I\\right)-\\mathbf I\\right] \\\\\n&=\\frac{1}{2}\\left[ (\\nabla_{\\mathbf X}\\mathbf u)^T + \\nabla_{\\mathbf X}\\mathbf u + (\\nabla_{\\mathbf X}\\mathbf u)^T \\cdot\\nabla_{\\mathbf X}\\mathbf u\\right] \\\\\n\\end{align}\\,\\!", "1052174039620e7fde413d924d00add2": "-623\\pm 4.1%", "10524aead52f40bf34e46d4acaccfc90": "\\left(\\rho_t\\right)_{t=0}^{T}", "10526ecf0692d922c5664a175092df1d": "\\displaystyle{\\|\\partial^\\alpha F_m \\|_\\infty \\le 2^{-m}}", "1052dedc4fe612121562cf309599dc8a": "f^\\prime\\colon M_X \\to N.", "1053297cfec39507c150a031d9807c5c": "d_Y(d_X\\Delta) = -d_Y(h(X,Z)).", "10533ef211fc97c64582c0c197c61ea7": "v_0 = f(x, y)", "10536c37f23c49d03626478ec55a5176": " \\log( m ) = \\log( a ) + b \\log( - \\log( p_0 ) ) ", "1053775fa2bb5d21ae8e025d4bedb4c9": "r_{0}\\in E(T_{1}(q_{1},\\epsilon ))\\Rightarrow r_{0}\\in E(T(q_{1},\\epsilon ))", "10539eafc7cde74d6ea0425f0f9d6d93": " z(x,y) = \\operatorname{Re} (x+iy)^3.", "1053a176d2a1b179bcf6f9ed52a9d2d5": "F_n(a, 1)", "1053bcf811bd354caf3d733292e863b5": "d\\theta = 7.2923 \\times 10^{-5}\\, dt", "10540fc9ca730bb540932da6c5ebcecc": "\\mathbf{k} = \\frac{4}{3}(\\sqrt{2} - 1) \\approx 0.5522847498", "105493c7bb9503fc8661a0d6fe1a6568": "\n H\\, =\\, \n \\frac12\\, \\rho\\, \\varphi\\, \n \\Bigl[\n w\\, \\left( 1\\, +\\, \\left| \\boldsymbol{\\nabla} \\eta \\right|^2 \\right)\n -\\, \\boldsymbol{\\nabla}\\eta \\cdot \\boldsymbol{\\nabla}\\, \\varphi \n \\Bigr]\\, \n +\\, \\frac12\\, \\rho\\, g\\, \\eta^2,\n", "1054fd36e3154250341c186305517c11": "dU= TdS-PdV+\\sum_i \\mu_i dn_i,\\,", "1055954120a6b9502509223a782e0d3f": "L = \\partial Q/ \\partial m \\,\\!", "10559a162fc88f61dfdd5eaa334ad70c": " \\lambda: S \\rightarrow Y^\\phi", "105603623a8013e2227cd34c0c85f97e": "\\sigma_{ij}=-p\\delta_{ij}.\\,", "10561afac9e2e03fc7e1c71fefd35e3d": "V_1 \\ge V_2 \\ge V_3 > 0", "10567a9251d9c6e06845a2d4b72c3db9": "\\lambda^2(x)", "1056c695268c17e80ac95c07c4f66fd6": " m_2 \\parallel m_1", "1056d96fe69e8d0e01edca1be931984a": "RPF = \\frac{\\text{effective RPF}}{\\text{extraction ratio}}", "1057341a46d972c82ed2b4f710a0de35": " \\sigma^{\\mu\\nu} = (i/2) [\\gamma^{\\mu}, \\gamma^{\\nu}] ", "10574254a52ea6c81af7e490c6526ab5": "n=N_{c}\\frac{e^{-(E_{c}-E_{Fn})}}{k_{B}T}", "1057d99dae7444eca0cdb00c7c919bae": " T_{\\alpha \\beta} {}^\\lambda ", "1057e69f9d8a63a02d170a18dd07cab0": "\\frac{1}{2}m_\\mathrm{i}\\,u(x)^2 = \\frac{1}{2}m_\\mathrm{i}\\,u_0^2 - e\\,\\varphi(x)", "105816d30833811fbe2f31ba7d322893": "H^2(z)= H_0^2 \\left( \\Omega_M (1+z)^{3} + \\Omega_k (1+z)^{2} + \\Omega_{\\Lambda} \\right).", "10587446e71952ff71043b9e01285df8": "\n EI w_{\\mathrm{max}} = \\cfrac{1}{3}\\left[\\dfrac{Pb(L^2-b^2)^{3/2}}{6\\sqrt{3}L}\\right] -\\cfrac{Pb(L^2-b^2)^{3/2}}{6\\sqrt{3}L}\n ", "10587a0bf20b915f5b40657c0d14c351": "{f_k}^{m_k}", "1058f207c5827901aacfffbddc133749": "s(mt) = (sm)t \\, \\forall s \\in S, r \\in R, m \\in M", "10590a2a921f3cfb3c78b19c8991567e": "K^{\\ominus} = \\mathrm{\\frac{[ML]}{[M][L]}\\times \\frac{\\gamma_{ML}}{\\gamma_{M}\\gamma_{L}} =\\mathrm{\\frac{[ML]}{[M][L]}}\\times\\Gamma}", "10597a859eb8f063b84dcf46a78a340e": "\\pi/4 \\approx 79%", "1059892d1a83f26026494bad45230314": "\\|\\mu\\|_{ba} = \\sup_{A\\in\\Sigma} |\\mu|(A)", "105a431e5e7208ab9a95885e4c1a6d00": "(-e a, 0)", "105a583f856c240861fdbce6a08ef950": "\\mu_3 = \\mu'_3 - 3 \\mu \\mu'_2 + 2 \\mu^3\\,", "105a61b297ec79d4df9bf0eda8711ce3": "L(s, E) = \\sum_{n=1}^\\infty \\frac{a_n}{n^s}.", "105a93ec9598b442a6dbcac9a9c51afd": "f:A\\otimes C\\to B\\otimes C", "105ac573a342bb2a73b39ce5a67ecfc7": " s = {v+u \\over 1+(vu/c^2)} . ", "105ad981a3a494ccfd19f083e3c39181": "s=(p_1+p_2)^2=p_1^2+p_2^2+2 p_1 \\cdot p_2 \\approx 2 p_1 \\cdot p_2 \\,", "105b57eb54d52c16693790affb42b3f1": "a^{2^{ \\overset{n} {}}} + 1", "105b7dc1803bed57779c3fb7113de4f0": "\\displaystyle 2", "105be3a9948bc0c843cfd71f9cee62c8": "a^{2}+b^{4}", "105bfa8f7a7c8a5c0517811f065d225e": " \\frac{q_i}{p_i} = 1", "105c380923405a73fd556a84e71ccc4a": "\\|x\\|^2 = \\sum_{y\\in A} |\\langle x,y\\rangle|^2.", "105c81bdc324722ea6882508c2f161c9": "N=\\sum_{i=0}^{n-1} 10^i {d_i},", "105cd58796c60bffd5c93d411ffec2ee": " dU = \\varepsilon \\sigma dV T^{4} + 4 \\varepsilon \\sigma V T^{3} dT ", "105d095fea9b5dc74027a60ea59480a8": "\\Theta \\subseteq \\mathbb{R}^d", "105d09df013b357a1adaf9cace814542": "\\operatorname{hypg}(x_c; x_{c-1}+x_c, m_c, m_{c-1}+m_c)\\,,", "105d335879c4dd1c2b8d6804123c75a1": "\\bigg. J = - \\frac{P(\\sqrt{p_1} - \\sqrt{p_2})}{\\delta} \\bigg. ", "105d564a2e246d83b13c6c60da5be00f": "\\lambda_{dB}", "105d791c39f39768722d45fc8de26331": "\n\\begin{array}{c||c}\n\\textit{Worst-Case\\ Pessimism} & \\textit{Best-Case\\ Optimism}\\\\\n\\hline\nMaximin \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Minimax & Minimin \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Maximax\\\\\n\\displaystyle \\max_{d\\in D}\\,\\min_{s\\in S(d)}\\,g(d,s) \\ \\ \\ \\displaystyle \\min_{d\\in D}\\,\\max_{s\\in S(d)}\\,g(d,s) & \\displaystyle \\min_{d\\in D}\\,\\min_{s\\in S(d)}\\,g(d,s) \\ \\ \\ \\displaystyle \\max_{d\\in D}\\,\\max_{s\\in S(d)}\\,g(d,s)\n\\end{array}\n", "105dbffea4bae06bb72aa655a9fd85b9": "\\left[ \\mathbf{N}\\right] :\\left( \\mathbb{Z}_{2}\\right)\n^{2n}\\rightarrow\\left[ \\Pi^{n}\\right] ", "105df97cb2b28ce762603e65e7014891": " A(a,\\theta) = 0. \\,", "105e0fd711cbf7051fc7c5b074e03dd9": "\\phi\\leftrightarrow\\psi\\in T,", "105e6cc625ee9a47b3de284e18ff0d4d": " |f_n(z)| \\le f_n^\\prime(0) {|z|\\over (1-|z|)^2}.", "105e9ffddb82970beeb9758fefa8d67c": " \\mathbf{x}^{(k+1)} = D^{-1} (\\mathbf{b} - R \\mathbf{x}^{(k)}). ", "105eff5d63f81d987ca1ef3ab59e20e4": "\\alpha > 0", "105f04ba9b57f6e5bc4384d8e2c3c60d": " \\sum_{n=1}^{\\infty} \\|v_n\\|_X < \\infty \\quad \\text{implies that} \\quad \\sum_{n=1}^{\\infty} v_n\\ \\ \\text{converges in} \\ \\ X.", "105f691465c8509b4b0c9ae338874cf8": "u=(\\frac{t}{2\\pi})^{1/4}", "105fd69314e27aad4330fb666aa8d63e": "\n\n\\mathbf{J}_{\\rm final} = \\mathbf{S},\n\n", "1060909feeb5b9fd8aa58acdbc585d98": "\\langle \\Delta \\hat{a}_1^2(t) \\rangle \\langle \\Delta \\hat{a}_2^2(t) \\rangle \\ge \\frac{1}{16} \\ ", "10609172afe82afd4f6d2430801db223": "N = \\frac{f}{D} \\ ", "1060940f9bc195152586c815b695e1c0": "g \\cdot e = q \\in Q", "1060a00607fcbcc7c5e358c396c1dad9": "\\mathbf{K} = l_{1}\\mathbf{g}_{1} + l_{2}\\mathbf{g}_{2} + l_{3}\\mathbf{g}_{3}", "1060a146fb2ebe00e174f0df8ee07b0c": "T_{i_1},T_{i_2},\\dots,T_{i_t}", "1060a8639a8eaf6a0b95b562be31054e": "\n \\mathrm{j}_\\pm |j\\,m\\rangle = \\hbar C_\\pm(j,m) |j\\,m\\pm 1\\rangle\n", "1060b3bb974263a2607cc128310f149c": "M(X) = \\left( {\\begin{array}{*{20}c}\n \\mu \\\\\n 0 \\\\\n\\end{array}} \\right)\n", "10612beb6fc538fc018be844eeb0d072": "a \\uparrow \\uparrow b", "1061c7b1f36aa2444bc0e18691b608a8": "\\operatorname{Vol}(B_{p_1, \\ldots, p_n}) = 2^n \\frac{\\Gamma(1 + p_1^{-1}) \\cdots \\Gamma(1 + p_n^{-1})}{\\Gamma(1 + p_1^{-1} + \\cdots + p_n^{-1})}.", "1061e17fcbcc8a225fe764cee9e30517": "\\{ \\alpha,\\beta,[\\alpha,\\beta]\\}", "1061e659c9b89c1363a2da84727acae0": "\\mathop{Br}(L/K) \\equiv K^*/\\mathop{N}_{L/K} L^* \\equiv \\mathop{H}^2(G,L^*) . ", "10620363f64e24c3f8939194ddb2b8ca": "{1 \\over 2} = 0.5", "10627075f5bf992d67536050b968ef46": "\\frac {3\\ \\mathrm {hours}} {90\\ \\mathrm {miles}} = \\frac {7\\ \\mathrm {hours}} {x\\ \\mathrm {miles}} \\quad", "1062845af68f1fc14fa25d718acadd1c": "\\xi=\\alpha^{-1}>0", "106293382fb634df421dcec22e2df409": "P(t)=\\sum_{i\\geq 0}t^i\\text{Sq}^i", "1062f6a24adfea22683e3b1166ef0138": "\\overline{2m-n}\\approx\\sqrt{\\frac{2n}{\\pi}},", "1063093cbd65d421dd4883ac9cb5605e": "\\int f(x)d\\alpha(x)", "10635b6b468ac98f48df3506226b35fb": " \\int_k^{k + 1} f(x)\\,dx = \\int u\\,dv", "10641b531d42332b827f9f555e148999": "A = \\begin{cases}d \\gamma \\mbox{ , frequency} < 3 GHz \\\\ R_fd \\;+\\;k[1-e^{(R_f - R_i)\\frac{d}{k}}] \\mbox{ , frequency} > 5 GHz \\end{cases}", "106493900bc7a57f4ce726dc7cc3afbd": "\\mathbf{x},\\mathbf{u}", "1064b869b64a3a85c41ceacac520bcd2": "X_2 = \\{2,4\\}", "1064bf99872d91f6bb96808c3c72d9a9": "n_1, ..., n_N", "106507a26fe9e5247042ebc0473ecd96": "R_{n+1}(x)=\\frac{2n+1}{n+1}\\,\\frac{x-1}{x+1}\\,R_n(x)-\\frac{n}{n+1}\\,R_{n-1}(x)\\quad\\mathrm{for\\,n\\ge 1}", "10652d4d853204be81405853512bef73": "\\ \\ O", "106559128efc97eaaf5bd9d4e29ca27c": " \\alpha_{k,i} = \\frac{x-u_i}{u_{i+n+1-k}-u_i}. ", "1065bba4fd73439743384ba9757c756b": "F^T(T,r)=1", "1066ab3520d801e5e652b5141851896f": "var(X)", "1067536fbc266714eb70a7abacdbb3bc": "R = \\{ (a_1,\\ldots,a_n ) \\in M^n : \\mathcal{M} \\vDash \\phi(a_1,\\ldots,a_n) \\}", "106757ca4d0cefe808860134c91ba89c": "f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\\,\\!", "10677f4164982825ebd37ae9b9c1fd84": "q_p(a) = \\frac{a^{p-1}-1}{p}.", "1067e4670b029a04916f94026f2b229c": "\\mu_{k} \\equiv E[W^k]=g_{k}(\\theta_{1}, \\theta_{2}, \\dots, \\theta_{k}) .", "106868bb6b6812fb6906433de421aebb": "\\mathbf{d}^T \\mathbf{y} + \\beta t - 1 = 0", "106886c83a135af29eedc43e92d86d97": "\\{ \\mathcal{F}_{t} | 0 \\leq t \\leq T \\}", "1068eca0030d9eb179874297e9509db2": " \\int d \\mathbf{r}_{1} d \\mathbf{r}_{2} h(r_{12})e^{i\\mathbf{k \\cdot r_{12}}}=\\int d \\mathbf{r}_{1} d \\mathbf{r}_{2} c(r_{12})e^{i\\mathbf{k \\cdot r_{12}}} + \\rho \\int d \\mathbf{r}_{1} d \\mathbf{r}_{2} d \\mathbf{r}_{3} c(r_{13})e^{i\\mathbf{k \\cdot r_{12}}}h(r_{23}). \\, ", "106920849742cae40711297a15e4813d": "\\operatorname{de-let}[\\operatorname{let} p : p\\ f\\ x = f\\ (x\\ x) \\operatorname{in} \\operatorname{let} q : q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p] ", "106924f892eb6ed989f692796d450f51": "\\begin{align}\\frac{d}{dx}\\left[\\frac{(4x - 2)}{x^2 + 1}\\right] &= \\frac{(4)(x^2 + 1) - (4x - 2)(2x)}{(x^2 + 1)^2}\\\\\n& = \\frac{(4x^2 + 4) - (8x^2 - 4x)}{(x^2 + 1)^2} = \\frac{-4x^2 + 4x + 4}{(x^2 + 1)^2}\\end{align}", "10697fb7a3928e4bb592bd7af5615318": "||X^\\alpha||^2 = \\frac{|\\alpha|!}{\\alpha!},", "10698191eebd9da71b7cdbd8216a1f07": "I[f] = \\displaystyle \\int_{X \\otimes Y} V(f(\\vec{x}),y) p(\\vec{x},y) d\\vec{x} dy", "10699ae1c07c6fec5ee1ef3213c194f2": "\\mathbf{m}_i^{\\phi}", "1069cc47df11b079df3edb6d23e3c4a0": "\\; 1", "1069fbe44be8eda06b917423d38613e5": "\\ \\frac{S}{C}=\\frac{4^4R^2}{P_tG^2\\lambda^2}(180/\\theta^o)^2\\frac{1}{\\sigma^o}\\frac{P_tG^2\\lambda^2}{(4\\pi)^3R^4}\\sigma", "106a039b82e803000fca22f2341e2cf5": "i = 0 \\ldots i_\\max, j=0 \\ldots j_\\max, t = 0 \\ldots t_\\max", "106a06f6f1a113339c50ed3982032719": "\\scriptstyle c\\eta=\\Psi\\,", "106a0c7baf3eec15560cea013971df8a": " g^M = f^* g^N\\,,", "106a7b104d047613b80a7434b5a6bd27": "\\frac{\\partial g}{\\partial x} \\cdot X +\\frac{\\partial g}{\\partial y} \\cdot Y+\\frac{\\partial g}{\\partial z} \\cdot Z=ng(X, Y, Z)=0.", "106aebe50c16f51c44fd467a2b68402a": "6 n^2 + O(n)", "106af9cc8a7e2e822f577eae0c33fa42": "\\{c_i,z_r\\}", "106b0afa62ca1964e04a1fa532b14bb9": " \\bold f(t) = \\left( \\begin{array}{c}\n0 \\\\\n0\n\\end{array} \\right), \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (12) ", "106b3b5f7073e9ac19dc62fd12cd1493": " si \\,\\ b = 0 ", "106b407530e035bbd26af3ee9bd77f73": "\\hat{c}_P=\\hat{c}_V+1", "106b582092e6ef60750cf2c00c43cbe5": "Y=y_j ", "106b58c5304459c0559f99a203384afa": "\\{a_1,a_2, \\ldots, a_n \\}", "106b70ebd6d052d831002beef1f7bd15": "e\\ =\\ 0\\,", "106c226ad079a63ed3ab090f239b3634": "newvar", "106c2ccab729dc12e57d0a929c184374": "M\\left(t;\\mu_1,\\mu_2\\right) = G(e^t;\\mu_1,\\mu_2)", "106c3e97042d56cf4370396e9ac1834c": "\n\\eta^{\\mu \\nu} = \\left(\\begin{array}{cccc} -1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right) ~.\n", "106c69553435b01f5e308a6a2f5ba4d4": "X_u", "106c8456a4332abef6463aa131812f54": "\\begin{align}\ny^1_{t+h} &= y_t + hf\\left(y_t, t\\right) \\\\\ny^2_{t+h} &= y_t + hf\\left(y^1_{t+h/2}, t+\\frac{h}{2}\\right) \\\\\ny^3_{t+h} &= y_t + hf\\left(y^2_{t+h/2}, t+\\frac{h}{2}\\right)\n\\end{align}", "106cae097fafc14b81d04dc5ddc7e80c": "bulging factor = SIF(curved)/SIF(flat)", "106d1549f7650c1526733c7fba356da2": "J^\\mu = \\left(c \\rho, \\mathbf{j} \\right)", "106d77d909081cd0d762dfd51d3e069c": "J_{\\alpha+1} \\cap \\text{Pow}(J_\\alpha) = \\text{Def}(J_\\alpha),", "106dcef9b0ca1bd9a1add24ec797b01e": "\\Omega^1", "106de3f7eb065e89e2dd8b93caa3b18b": "A(\\eta) = A+\\eta B", "106e04faf922d88c3d5b205ed14d2c70": "\\log \\,f = u", "106e17b4c00b964366ca3cc1a8690bba": " \\int_{\\mathbb{R}^n} |\\nabla u^*|^p \\, d \\mathcal{H}^n \\leq \\int_{\\mathbb{R}^n} |\\nabla u|^p \\, d \\mathcal{H}^n,", "106e4a9917d6719374df3e2eb1a00a81": " \\mu^* ", "106e7d1e8966ebd1d549b2934a58ec3d": " r_\\mathrm{O} = \\frac{v_\\mathrm{ds}}{i_\\mathrm{d}}\\Bigg |_{v_\\mathrm{gs}=0}", "106ea042a96b6ed470419b87b07ba73d": "O(\\min(m, n))", "106eda0fe8058b452c725b41a207ce86": "f(x) = \\langle A \\varphi(x), \\varphi(x) \\rangle", "106f13b1ceb9dc898019ee56397ecf33": "\\rho = \\frac{R_\\mathrm{thresh}}{R_\\mathrm{rms}}.", "106f6e95c824ca8a420b97ce357f03bf": " \\nabla \\cdot \\mathbf{D} = \\rho,\\, ", "106f7b2626462a1240e3cc721e7d784c": "A = (T, Con, \\vdash) ", "106fbabdc5e1b39ac8dd530312b0b04d": "S [\\text {Bq/g}] \\simeq \\frac{4.17\\times 10^{23}}{T_{1/2} [s]\\times m}", "106fbbc9a6e399c079525a7f16ef274d": "\\theta_n = n \\pi / N", "107058f4765614d00722900ef06d4b58": "A_{\\mu}\\ ", "1070831b69831844c8342c0c7406accd": "\\scriptstyle f : X \\rightarrow Y ", "1070e4b7755be48ca68357f6aae76639": "f_{LO}", "1070e805fbcb793a96b1fb2d34d04c67": "\\frac{\\delta F[\\rho(x)]}{\\delta \\rho(y)}=\\lim_{\\varepsilon\\to 0}\\frac{F[\\rho(x)+\\varepsilon\\delta(x-y)]-F[\\rho(x)]}{\\varepsilon}.\n", "10710f5febe38ba2da2c436922871706": " FiB_y = log((Y_y/(TE)^{TL_y})/(Y_0/(TE)^{TL_0})) ", "107294afd7a8960ca16629bbfb9daf29": "c_1^2", "1072971e5b20f51ee61526f67421d422": "\\left(p_R + \\frac{3}{v_R^2}\\right)(v_R - 1/3) = (8/3) T_R", "1072e5433390314a628706b6c54d5e03": "D\\varphi_x,\\; (\\varphi_*)_x, \\;\\varphi'(x).", "1072fc17b02aa18abd77f940b9882d69": "\n\\frac{1}{4} + \\frac{1}{4} = \\frac{1}{2}.\n", "1073454e636af61eb8b1104d87901338": "r=f_2(\\theta)-f_1(\\theta)", "10735efd8ff2ac607289435c23d52148": "\n\\begin{alignat}{2}\n \\left(\\cos x+i\\sin x\\right)^{k+1} & = \\left(\\cos x+i\\sin x\\right)^{k} \\left(\\cos x+i\\sin x\\right)\\\\\n & = \\left[\\cos\\left(kx\\right) + i\\sin\\left(kx\\right)\\right] \\left(\\cos x+i\\sin x\\right) &&\\qquad \\text{by the induction hypothesis}\\\\\n & = \\cos \\left(kx\\right) \\cos x - \\sin \\left(kx\\right) \\sin x + i \\left[\\cos \\left(kx\\right) \\sin x + \\sin \\left(kx\\right) \\cos x\\right]\\\\\n & = \\cos \\left[ \\left(k+1\\right) x \\right] + i\\sin \\left[ \\left(k+1\\right) x \\right] &&\\qquad \\text{by the trigonometric identities}\n\\end{alignat}\n", "10737fb13bacd501f750df1ed63ea912": "\\frac{}{}-I \\, \\delta", "1073878c7558aa393950b0f77b35c72c": "\\Re(z)>2", "1073a309599df6c0ae8a0db6caac31dd": " H^{'}|\\psi_{-}\\rangle=E_{-}|\\psi_{-}\\rangle ", "1073e8b593ec12cab16adcf65aed6d12": "\\textstyle Z_{3}", "1073f96f9d0f122ad8eea35c9f401b06": "1-\\frac{\\ln(1-(1-p) e^{-\\beta x})}{\\ln p}", "10742a7031c333375f78905b214665c3": "L_z(q, p) = p_i", "10744650b75b5cc2c34e541becc7a113": "s_{ln} = s_b ln(b) \\,", "10748196696737041ef54f908953b664": "\\cdot :A\\times A \\longrightarrow A", "1074b41fe631c4109245966e4475305a": "g=210=2\\cdot 3\\cdot 5\\cdot 7", "10756ac0abc0a953399aa8f5ad73ad20": "P_1 + \\frac{1}{2}\\cdot\\rho\\cdot V_1^2 = P_2 + \\frac{1}{2}\\cdot\\rho\\cdot V_2^2 ", "1075b680183f7db57fcb0ba1fec31650": " \n(p,q) \\overset{\\alpha}{\\rightarrow} (p',q)\n ", "107626a21fcebdb7d7478578f8243c83": "\n\\mathbf{\\hat{f}_{0:t}} = c^{-1}\\mathbf{\\hat{f}_{0:t-1}} \\mathbf{T} \\mathbf{O_t}\n", "10765a82dac92461a9861c2f2ba7f61d": "\\displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),}", "1076761c92b639e2dcfa04d074be11dd": "\\left(\\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "10768a2dbab107d673c70a9836cd2e5c": "y_{n+1}^{(0)} + \\tau_{n+1}^{(0)}=y(t+h)", "1076a0378f4783eaad07687b0b2ace00": "J:X\\to (X'_\\beta)'_\\beta", "1076ec62b59f3573ad57ec0d51a70642": "\\sum_{i=1}^n \\widehat{\\varepsilon}_i x_i=0.", "1076ecc0eb1f101be0acc6f812dc6e11": " L[t] = - m c \\sqrt {- g_{\\alpha\\beta}[x[t]] \\frac{d x^{\\alpha}[t]}{d t} \\frac{d x^{\\beta}[t]}{d t}} + q \\frac{d x^{\\gamma}[t]}{d t} A_{\\gamma}[x[t]].", "10773b949ef4edfc8d8205ac51e06e93": "P\\in\\mathcal{C}", "107749f8872102b27f5d67907c906870": "\\sin{5x}=1\\,", "10785587b0bcf02c1b649966d1b4f36f": " \\sigma_d ", "10785bd2cabc31918e71c6c7f0aad68b": "\\scriptstyle\\times", "1078af129de448ce026148e2ef6e9781": "x \\geq 0, y \\leq 0", "1078b711da6ad03ecfd409dd0e3bd60e": "\\mathrm{SNR} = \\frac{|y_s|^2}{E\\{|y_v|^2\\}}.", "1078f7c5c13fc2a607de3f3bf40aa185": "\\mathbf{S}_{k}", "10790bda39fe0fb4ae54f9c5bdb6f8d4": "a'\\omega r", "10790fb041dcdb02a57c3226416a7e62": "\\scriptstyle n \\;", "1079277ea10c07e866cc2077ea4e389d": "\\sigma= \\sigma_0e^{-(T_0/T)^{1/4}}", "10796f9f3cd25b66c6c7cecdcf37836c": "X = \\operatorname{spec} A, Y = \\operatorname{spec} B", "107982d4f370747346ed1f3a2149643a": "H^c(p(x)\\|m(x)) = -\\int p(x)\\log\\frac{p(x)}{m(x)}\\,dx,", "1079c61dda342bdb1baabd5946ab6c2c": "\\mu_r \\approx 1", "1079f23c89fd9a8e69f16d12de519b02": " \\displaystyle{\\langle f_1, f_2\\rangle = \\int f_1f_2 \\, dx.}", "1079f6506d6071b69e7128fd3b43dac1": "|u(x,t)| = \\frac{1}{(1+4t^2)^{1/4}}e^{-\\frac{(x-k_0t)^2}{1+4t^2}}", "107a1ab538603d47ea3b978e911b5e45": "E_{rel}", "107a22bc2dab2d4f7cfcc1f317c6b9c8": "\\pi\\colon{\\mathcal M}\\to M", "107a617402042c7e265e8157bffe7aac": "\\mu (\\Beta(\\alpha, \\beta) )= 1 - \\mu (\\Beta(\\beta, \\alpha) )", "107a7214747351a152be8f899c283787": "(p,0011,Z) \\vdash (q,0011,Z) \\vdash (r,0011,Z)", "107a739c20f7dc172b8faafa964dcf19": "\\widehat\\beta=(X'X)^{-1}X'y", "107a9b12b61ac742b9e22eefd2073de1": "A(n,m)", "107aa4b91a348e14b6d395655526ab5f": "\\sqrt{\\frac{3}{10}}\\!\\,", "107b374ab3faeef4b9fe6095835b2bb5": "\nE^2 = \\hbar^2 \\omega^2\\quad\\mathrm{and}\\quad p^2 = \\hbar^2 k^2 = \\frac{\\hbar^2 \\omega^2}{c^2}\n", "107b9515e3671f27063de1b517ee8af5": " X_k = \\alpha^{i_k}, \\ Y_k = e_{i_k} ", "107bdfd4818aea4b03788e63b4c0b1a2": "\\scriptstyle\\overline{DA}", "107c09e61217a7d4463696efda8fb5de": " \\Lambda_{CW} \\!", "107c0c78bd971e91000c8e53b24bd2b6": " \\overline{X} = \\frac{1}{n}\\sum_{i=1}^n{X_i} ", "107c630b9895142e62434b8de1d9a31d": " \\varphi(t) = \\int_0^t h_s\\, d s .", "107c69c252a32f9740a7d559b577bbf5": "c_1, c_2, \\ldots", "107c80dc7288f340848c585e41e48729": "\\vec{x}(i)\\approx \\vec{x}(j),\\,", "107c8fa0b6ea181f94fc2cebdad05c36": "\\scriptstyle{\\tau}", "107cafd46841c433f11b3fdc07a85f43": "W_C := \\mathrm{Tr}\\,(\\, \\mathcal{P}\\exp i \\oint_C A_\\mu dx^\\mu \\,)\\,.", "107d3f2106f84c08b1c6329ed7808264": "\\theta = \\tan \\theta - \\frac{\\tan^3 \\theta}{3} + \\frac{\\tan^5\\theta}{5} - \\frac{\\tan^7 \\theta}{7} + \\quad \\cdots ", "107d5913c5006af1aa9da9ffc953a5f9": "\\frac {4}{\\pi}", "107db30076dfd42abad12bcc4f2df54e": "\\ell = \\log \\log n + \\log k + O(\\log(\\epsilon^{-1}))", "107dd35529337915d96b93c468c5d2db": " {G^a}_b \\, {G^b}_a = R^2", "107e0df20ebaa373a96eba1d385b3e62": " \\mathbf{U}, \\mathbf{V} ", "107e493a70ed552c9bf06445170cb933": "m_4 = [12.3, 7.6] - [0, 2.404] = [12.3, 5.196]", "107e58a61d09bb2662af482f4854c1a6": "K(x,x';t)=\\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}dk\\,e^{ik(x-x')}e^{-i\\hbar k^2 t/(2m)}=\\left(\\frac{m}{2\\pi i\\hbar t}\\right)^{1/2}e^{-m(x-x')^2/(2i\\hbar t)}", "107eb092479031ee1b070ecb8cbde8c8": "\n \\hat\\beta^{(j)} - \\hat\\beta = - \\frac{1}{1-h_j} (X'X)^{-1}x'_j\\hat\\varepsilon_j\\,,\n ", "107ed2aa825eaf2a59619ac77900e5a1": "(\\rho u A \\phi)_e - (\\rho u A \\phi)_w = (rA\\frac{\\partial \\phi}{\\partial x})_e - (rA\\frac{\\partial \\phi}{\\partial x})_w ", "107f1d148daa2d473899b081282f4ee7": "\\bigg. J = - D \\frac{(C_2 - C_1)}{\\delta} \\bigg. ", "107f55430a19cb9f2d5fe5eaa850d266": "{A_{cross}}\\,\\!", "107f5c8797d7ab2bb7b974cbfc89179e": "\\mathbb{R}^+", "107fa32631456ce154c9be68882e2950": "\\varepsilon_{\\text{eff}}", "1080166158767d99bf842ea8f58eb3b3": " E_F^n=\\sqrt{(p_F^n)^2c^2 + m_n^2 c^4} \\,", "108033bcad1a58ce4bff250a191c0759": "d((\\varphi -1)b) = b d \\ln\\gamma", "10808e732947926e55f234f4743fb6c6": "I(t)=I_{ex}\\sin(w_{ex}t)", "10809472c9f904ea5424db169b4e6c8d": "E_v = A_v + T_v = B_v + S_v \\,,", "1080ae9dfed0a9a6dd056268b45cb8a9": "\ny = a \\sinh \\xi \\sin \\eta,\n", "1080f14dae21503e351cd5bbf7f950d7": "|a|_\\ast^{\\log_a b}\\leq1", "1080f9b5a4252bf0eaf82baeefe52701": "0 = \\cap Q_i \\Leftrightarrow \\emptyset = \\operatorname{Ass}(\\cap Q_i) = \\cap \\operatorname{Ass}(Q_i)", "1081056e04121326e85e3e087c15fb68": "{\\rho_{Out} \\over \\rho_{In}}\\,\\!", "10811bb6fe7f685d31276622b2c40bdd": "T_{\\text{C}} ", "1081205f9a79d1f491babb9abb6b6323": "(\\mathbf{E}_1, \\hat{O} \\mathbf{E}_2) = (\\hat{O} \\mathbf{E}_1, \\mathbf{E}_2)", "10812f90b6269226a99b24d864c3efa1": "[ML']=K\\frac{[ML][L']}{[L]} = K \\frac{\\beta_{ML}[M][L][L']}{[L]}= K \\beta_{ML}[M][L']: \\beta_{ML'}=K\\beta_{ML}", "10816784add7b5da0654f1ca9edd919e": "\\sigma_f = \\frac{3 F L}{2 b d^2}", "1081dae289d8b07256d132eae92792cc": "{}+ (a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2)j", "10821b05e6436e9492cf84b0379fba1b": "(n_i, n_j)", "10821f8715a1c75669f17b8ffbbaabb1": " \\mathbf{\\bar{x}} ", "10826dbd67d65ada562c2d351963d9b7": "d_{H}(x,y)=k", "1082f2f5dd560b12bf8ce758d58140f2": "\\mathcal{T} = e^{-\\alpha \\, x}", "108315147577d0f2627997312d1958e6": "\n\\operatorname{E}(s^2)=\\operatorname{E}\\left(\\frac{\\sigma^2}{n-1} \\chi^2_{n-1}\\right)=\\sigma^2 ,\n", "108324da8fa55a6a317590dd17e235d2": "\\frac{\\Delta F(P_0)}{\\Delta P}=\\frac{F(P_{\\acute{n}})-F(P_0)}{\\Delta_{\\acute{n}}P}=\\frac{F(P_1)-F(P_0)}{\\Delta _1P}=\\frac{F(P_1)-F(P_0)}{P_1-P_0}.\\,\\!", "1083a8190456fbd27d9ffb29e2411ce7": "3k-1", "1083d6c1c705a1c7074f6005f4eb38c0": "\\frac{\\partial f}{\\partial x_{1}}v_{x_{1}}+\\frac{\\partial f}{\\partial x_{2}}v_{x_{2}}+\\,\\,\\,\\cdots \\,\\,\\,+\\frac{\\partial f}{\\partial x_{N}}v_{x_{N}}=0", "1084379e31aed4fbb0d59f20eed153b5": " \\zeta \\ll 1. \\, ", "1084451459414a3fa16f40378e4378db": "\\frac{u_b-u_a}{L}", "108456c8399350ce4bd12c33f6f1f5b4": "\\ MRS_{xy}=P_x/P_y ", "108458fecbf9807d98c393c840235f11": "\\partial f\\over \\partial y", "10846c5e06dc27e8d9336af5928fe805": " f_U ", "1085b4cad24b8792a98a689c26390907": "\\ h", "1085da3130b1f6574f3a37e8f68f3eee": "\\tilde E_7", "1085ebad3828994a51d84c0387a2f7be": "\\log_2(1+m_y)+e_y=-\\frac{1}{2}\\log_2{(1+m_x)}-\\frac{1}{2}e_x", "10862040ad4ab49098ddf7a3314aefc6": "(Y,L)", "10865655faa44f74d5a18a82a5650669": "x = \\left(x_1,x_2,\\ldots,x_n\\right)^\\mathsf{T}", "1086bd456aeff33774e603f9db50ac07": "W = \\min(Y_1, Y_2, \\cdots, Y_n)", "1086d86b37eaf879ca8ab9f53a9387a6": "\\lim_{x \\to \\infty} L(x) = b \\in (0,\\infty),", "1086eda3eff89a0b05819a5ab87c6383": "\\delta W\\, ", "108702f3c756c6a444975a31a40b1329": "|w|=1.", "108734a14b5051135ab92742de362841": "x'^2 + y'^2 + z'^2 = c^2t'^2.", "108739ea7dec5ebd0e5d2fa008f31fa0": "p(y) \\propto \\left[1 + \\frac{y^2}{\\alpha^2}\\right]^{-m} \\exp\\left[-\\nu \\arctan\\left(\\frac{y}{\\alpha}\\right)\\right]", "1087cd498720f5f1389d5abaa558542c": "\n\\begin{align}\nx_\\mathrm{triangle}(t) & {} = \\frac {8}{\\pi^2} \\sum_{k=0}^\\infty (-1)^k \\, \\frac{ \\sin \\left( (2k+1) t \\right)}{(2k+1)^2} \\\\\n& {} = \\frac{8}{\\pi^2} \\left( \\sin ( t)-{1 \\over 9} \\sin (3 t)+{1 \\over 25} \\sin (5 t) - \\cdots \\right)\n\\end{align}\n", "1087db6db630cca7423ee9106de76bb4": "S \\to aSa | bSb | aa | bb", "1088093dd4df4c0f25bf46f87cc84456": "E_d = 0 ", "1088ab7f22bec82db219ebc8815799f3": "\\{X \\mid \\forall n \\exists m \\phi(X,n,m)\\}", "1088d8875db5f61d743db700b3fb1635": "\n \\sqrt{n}(\\hat\\theta_\\text{mle} - \\theta_0)\\ \\ \\xrightarrow{d}\\ \\ \\mathcal{N}(0,\\ I^{-1}),\n ", "10891edb4bb2a4636cda37168f5c49fd": "\\begin{align}\nx_{n+1}-2x_n+x_{n-1}&=h^2w^2x_n\\\\\n\\iff \\quad\nx_{n+1}-2(1+\\tfrac12(wh)^2)x_n+x_{n-1}&=0.\n\\end{align}", "10898ba36924b03a17aad6280082ca26": "\\eta = \\sqrt{\\frac{6780}{9640}}=0.8386\\ldots", "1089f5a290ca9f1743550a7abbd7c9f2": "x^{ 7 }+x^{ 6 }+1", "108a08a709a62f9915e68eb522a08c58": "{v}, {\\lambda}\\,", "108a1a65717fa3e21a6a80b49b8e4348": " q > p - 1 ", "108a20afe90bb03e3da6f0919c3915e9": "|Z|_{max} \\approx {RQ^2_L}", "108a25cab551e4360cf1c6e99aa2798c": "\\displaystyle{T(t)=e^{At},}", "108a3d41363f8f92a05573b228415938": "\\frac{U}{L^3} = \\int_0^\\infty u_\\lambda(T)\\, d\\lambda,", "108a4df21b9ff1de4cc8fd13275985b4": "\\mod 0", "108ab83f7305b42e54f202f43ef090fc": "\\displaystyle s= \\frac{n}{\\text{poly}(\\epsilon)}", "108ac618dcb9569f19bea5c82e98c6c5": " c_1^2 \\le 3 c_2\\ ", "108af287296db6e7232b21cfa1e1ea02": "dy = f'(x)\\,dx", "108af9c0e086d1fffd162b597a4350d0": "\n\\sum_{A=1}^N M_A \\mathbf{d}_A = 0 .\n", "108afc158333ae823721263268463db7": "W_{LC} = \\hbar \\omega_{LC} = \\frac{\\hbar}{\\sqrt{LC}}. \\ ", "108b73e74e6ca41dd55657b5a59a9bb3": "\\epsilon^T\\tilde{\\Lambda}\\epsilon=\\epsilon^T\\left(\\Lambda+\\Lambda^T\\right)\\epsilon/2", "108ba330334cf046276613d7c159d65a": "T(z) ", "108bc405d112e54017ab721bb3ad0db3": "\\beta \\in\\left(0, 1\\right)\\;", "108bc63c98914afd3fd36e4091bf1a9c": "EP = (\\tfrac{3}{4} + \\tfrac{1}{6} - 1) / (2\\times \\tfrac{1}{6} - 1) = \\tfrac{1}{8}", "108c73bc39aec6a533d4ce7efd8a95fe": " B_{k+1} = \\textrm{argmin}_B \\|B-B_k\\|_V ", "108c8126d7205ba92f83795580cbf308": "=\\widehat{D}(-\\alpha)(\\alpha|\\alpha\\rangle - \\alpha|\\alpha\\rangle)", "108c97a55f5eb2bf236f975bdd437a55": "D^{\\ge 1}\\subset D^{\\ge 0};", "108d00703f36e0f9943ff05832978df4": "G \\mapsto F+G", "108d2c0c1a8150e22483011555779f45": "x > n\\,\\!", "108d3dfa2b81d80b9a374b999af15d79": "c(n)=\\int_0^{2\\pi}f(t)e^{-int}\\,dt", "108d5f63d448324e2fb851304313b21a": "\\Delta\\ge\\frac{n}{2}", "108d83492dba4cd0d61c903cc5a681cd": "F_X(x) = \\int_{-\\infty}^x f_X(t)\\,dt.", "108dcb2b089e392344da55ed7c23f2d4": "s_f = \\sqrt{ \\left(\\frac{\\partial f}{\\partial {x} }\\right)^2 s_x^2 + \\left(\\frac{\\partial f}{\\partial {y} }\\right)^2 s_y^2 + \\left(\\frac{\\partial f}{\\partial {z} }\\right)^2 s_z^2 + ...}", "108e0f264777eaf47d084a72ef06a13b": " 2n + p + q + z - e ", "108e4d17a841743afae7cc214a84312d": "\\frac{1}{M_{SUSY}^2}", "108e70d04afa261d72eee192a19743f0": "\\pi_i(X,A) \\,", "108ebcc631dd4b5414fa9467201dc3f2": "F(K,L)", "108ee72fb98668534397dd40dbf4b38f": "q(x)[\\phi(y)] = \\delta^{(d)}(X-y)Q[\\phi(y)] \\,", "108f160d64f0f0d892f00b20c5d05d66": "\\textstyle \\prod_{} x \\hbox{, } (x \\hbox{ in } S)", "108f1cca0b631655b2bacc370d9060a6": "p=\\left[1-\\frac{{\\rm Tr}(\\Omega ^2)}{8 \\pi ^2}+\\frac{{\\rm Tr}(\\Omega ^2)^2-2 {\\rm Tr}(\\Omega ^4)}{128 \\pi ^4}-\\frac{{\\rm Tr}(\\Omega ^2)^3-6 {\\rm Tr}(\\Omega ^2) {\\rm Tr}(\\Omega ^4)+8 {\\rm Tr}(\\Omega ^6)}{3072 \\pi ^6}+\\cdots\\right]\\in H^*_{dR}(M),", "108f3188bced2591245d04cf8fc3576a": "\\ P(X = x \\ \\mbox{and} \\ Y = y ) = P( X = x) \\cdot P( Y = y) ", "108f5465001029d6733711abb3552989": "M\\!\\ '", "108f64135afda75d6e4c2bdc14b1c4a4": "\\sum_{n=1}^m n(-1)^{n-1}.", "108fc3080dc8491cc88ec4af36f472f7": "\\tau_{ij}^{r} = L_{ij} + C_{ij} + R_{ij}", "108fc3e6d0b52a362023a1c50501cd3e": " f(k;N,K,n) = f(K-k;N,K,N-n)", "10901d93e9f9697cbe2461384f34c262": "Z_i^' = f^{'-1}(X_i)", "10905c67c8b0fbb962072d3a8a7fd279": "\n\\Pr(Y_i=1) = \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{1 + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}} = \\frac{1}{1+e^{-\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}} = p_i", "10908f4f03deb67e0ea70b94e4084d60": "(\\pi r^2)(2\\pi R) = 2\\pi^2 Rr^2", "1090f0cb613db397ada239de8b14f38a": " G^{\\hat{a}\\hat{b}} = 8 \\pi \\epsilon \\, \\left[ \\begin{matrix} 1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&-1\\end{matrix} \\right] ", "1090fadfb43f68956350e51293aef824": "\\int_1^\\infty \\frac{1}{x^2}\\,\\mathrm{d}x", "10917eb4cde3371e74feb7c3521e3c4a": "M^{[2]}", "10919cc68ebafd9d17595d97911b6312": "C_P(T)=\\left \\{ \\lim_{\\Delta T \\to 0}\\frac{\\Delta H}{\\Delta T}\\right \\}=\\left ( \\frac{\\partial H}{\\partial T}\\right ) _p", "1091c2161de19592006170cddb32fa37": "\\vdash \\phi", "1091db94750f3580097d97db9b6a62c2": "\\vec{r}_1", "1092100a3427cacb91c4b831300fa98e": "\nS = \\frac{1}{4\\pi } \\int d^2x \\sqrt{g} (g^{\\mu \\nu} \\partial _\\mu \\phi \\partial _{\\nu} \\phi + (b+b^{-1}) R \\phi + 4\\pi e^{2b\\phi }),\n", "1092d3acd5d77a0082844280a9a32af5": "\\begin{align}\ne_1 &= (1,0,0,\\dots)\\\\\ne_2 &= (0,1,0,\\dots)\\\\\n& \\ \\ \\vdots\n\\end{align}\n", "1092fada509c1a2f2071453253ff55b4": "\\theta_\\mathrm R", "109325b4621a9f7e8fc36c9d59703437": "|\\sqrt2 - a / b|", "109345941ca96584d9d591da970533d4": "\\frac{256}{243}", "1093a6ecef8973689f5e3b132269543a": "f: G \\to A", "10940f217b9529cd1671a2f59fef3e1b": "i(t)=C\\frac{dv} {dt}", "1094190fcdae75727ef87cbfee568838": " Q^{j+1}_i ", "109425303bf0d94f257ccd70a86cadc5": "\np(t) = \\text{penalty function whose time average must be minimized}\n", "109445adb7dbe075dd0140e5705f50dd": "1/(\\exp(ar - br \\epsilon)) = ", "10945651c529b0dd04112517c0a02e95": "|S|= O(\\sqrt{N})\\,", "1094a5d3c18738997a6377d822bc4ebc": "[x:=e](b=c+x) \\equiv b=c+e\\,\\!", "1094a60a924ff4012789a5cd6fd5d7dc": "(\\theta)", "1094e05af58a9a840369c0cbceab87a5": "\nY = \n\\begin{cases}\n\\xi K, & \\xi > 0 \\\\\nY_0 \\frac{L}{L_0}\\left(\\frac {L_0}{L}\n\\frac{P}{P_0}\\right)^\\alpha, & 0 < \\alpha < 1\n\\end{cases}\n", "1094f5e49a3fa41977c0538d88af1670": "\\overline{\\mathcal{M}}_{g,n} ", "10950e4c8e9403c73a1994e7d5c66cb9": " \\Phi(e, x) = I \\quad ", "1095242412ee0efae54b5add8204474c": "\\rho (\\vartheta) = \\rho_\\parallel \\cdot cos^2 \\vartheta + \\rho_\\perp \\cdot sin^2 \\vartheta", "10952edcb44b46d9a7e0c67c7384199f": " \\Delta R \\ ", "109554ba5842b0240aa09a03b7c46f70": "p(x +a+ 1)", "109599a77cd127b7ce94528d1041a521": "CCA = tdUCC", "1095a3448c3c62a66f09fd74ecd18e1d": " A_{k+1} = R_k Q_k = Q_k^T Q_k R_k Q_k = Q_k^T A_k Q_k = Q_k^{-1} A_k Q_k, ", "10965da0183ca0c1990c8472aa5fcf3b": " T_{d} = \\frac{\\log(2)}{\\log(1+\\frac{r}{100})}", "1096671ddd8eb7963f45a6de97aaac6b": "\\ \\Delta G(T)=\\Delta H(T_d) \\frac{T_d-T}{T_d}+ \\int_{T_d}^T \\Delta C_p dT - T\\int_{T_d}^T \\Delta C_p dlnT ", "1096919679550cac1aab5503a0f8c9cf": "Z(k,z)=e^{kz}\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,e^{-kz}\\,", "1096d230bf1ee03e34f15b5cdd8cf72f": "L' = \\frac{dL}{dl} \\approx \\frac{\\mu_d}{2 \\pi} \\ln\\frac{r_{o1}}{r_i}", "1096db202e772b1231e07202c527d6d4": "(\\cdot)_k", "1096fccbfe868c49606440b743b7c442": "\nr_{1}^{2} r_{2}^{2} \\left( \\frac{d\\theta_{1}}{dt} \\right) \\left( \\frac{d\\theta_{2}}{dt} \\right) - \n2c \\left[ \\mu_{1} \\cos \\theta_{1} + \\mu_{2} \\cos \\theta_{2} \\right],\n", "109758ad89b167abc74e919edc865b3e": "\\bar{Y}=1", "10976ef65af219f5308e519dacf145b6": " \\kappa^{F(n)}_1-\\gamma_1 = 0\\,,", "1097a73ec8b15bcf86aba7fda80ebd7f": "p_1,\\dots , p_r", "109852bb99a76283a46dbe3ea5d4dff0": "\n \\cfrac{\\Gamma \\vdash A, \\Delta}{\\Gamma, \\lnot A \\vdash \\Delta} \\quad ({\\lnot}L)\n ", "1098739e1872273f2c88409e32233d44": " \\frac{1}{r} \\frac{\\partial}{\\partial r}(r E_r)= qn/\\epsilon_0 , ", "10988c3ae1eb27588bf6540cece5b83f": "B_\\max k_\\mathrm{on}", "10989df6a215e0838610a5a7b58805cc": "(\\xi,\\eta)", "1098b5481b8f62c4a6db391ff7cfa93d": "p=59", "1098e3736e4bf2238a5c58411942c41d": "\\gamma^{1,2,3} = \\begin{pmatrix} 0 & i\\sigma^{1,2,3} \\\\ -i\\sigma^{1,2,3} & 0 \\end{pmatrix}, \\quad\n\\gamma^4=\\begin{pmatrix} 0 & I_2 \\\\ I_2 & 0 \\end{pmatrix} ", "1098f88fbae465e2ceaa6e44e41743a2": "(dX)^2\\,\\!", "1098fccd0ee4c85d98ca1b7e9b5f55b1": "\\bigcup_{n=1}^\\infty A_n", "109922db3cc0e2922b5dc86bf493482f": "P(n) = \\sum_{d \\mid n} d \\sigma(d) \\mu(n/d)", "1099c661d6b9604c64fb9d1b5c56f2d9": " P_{theo,c}=A_{c,t} \\times C_y \\times h_i", "1099e9ee8b1263137e7da97f986f13a8": "\\bigvee_i X_i = \\coprod_i X_i\\;/ \\sim,\\,", "1099f15fee692bf95e4e9786f96cac35": "\\displaystyle X(t)=\\eta+tu(\\eta,0)", "109a51860c7f47a6a5118614e6c81b25": "F^{\\alpha\\beta}{}_{;\\beta} \\, = \\mu_0 J^{\\alpha}", "109a60954f638e328c89597bb27bd6d8": "\\Gamma_{ij,k}^{(\\alpha)}=\\Gamma_{ij,k}^{(0)}+\\alpha T_{ijk}", "109a87d8dec44ed62486174a7e074f7b": "\\mu_s(l,x_s,p+\\Pi)=\\mu_s^0(l,p+\\Pi)+RT\\ln\\gamma_s x_s", "109a884b34370355ceac60f83ecd5ab6": "n, f = (0, 1) : (n + 1, f \\times (n+1))", "109a99edf06bbf0cafe7955eb6ac0916": "K[T]/T^2,", "109aaf1c2c923237608aa9f6b7b91a45": "\\delta = -3^\\circ", "109ac253e71fcd17f8df1ffbafe1be85": " {\\mathbb R }/ {\\mathbb Z}", "109b1b64deed72bcd74934ef21619fe6": " a,d \\in k\\setminus\\{0\\}", "109b9d05c6bdcbff24d7987dd774eb39": "\n a-b=0 \\;", "109ba257f02a31f82155d48ddfd5eb92": "\\frac{\\partial \\mathrm{net}} {\\partial w_i}", "109ba6ed595dc1bec4f88e1ed6afc5d6": "\\rightsquigarrow 1", "109c115a8fb6e68a44a03bf67dc6f29e": " A[f] = \\int_{x=x_0}^{x_1} n(x,f(x)) \\sqrt{1 + f'(x)^2} dx, \\,", "109c1fe50f34ceea5a04dbe877e17ab6": "A \\vec e_j = \\sum a_{i,j} \\vec e_i", "109c296a3a88cce2c57abc9cd3a62b1b": " A+L -> A^+ + L -> A^{++} ", "109d065990a0f0c7cdd3d82178cfe467": " t^{*}=(\\hat{\\theta}^{*}-\\hat{\\theta})/\\hat{se}_{\\hat{\\theta}^*} ", "109d14234ca7b15e14d436df4b8a32b3": "\n\\begin{align}\n\\mathbf{y}_p'(t) & = (e^{tA})'\\mathbf{z}(t)+e^{tA}\\mathbf{z}'(t) \\\\[6pt]\n& = Ae^{tA}\\mathbf{z}(t)+e^{tA}\\mathbf{z}'(t) \\\\[6pt]\n& = A\\mathbf{y}_p(t)+e^{tA}\\mathbf{z}'(t)~.\n\\end{align}\n", "109d26a38e9a65bf49110c621f3e2b43": "x^\\alpha", "109d82045882b47372aa6669a8c65a46": "X=\\mathbb{D}.", "109da39c9ef1dc48b2c34face7f1705d": "L(g) = \\{G_{i}^{(g)}\\}", "109eca64ff76194cd2e8e9b1cff8f709": "M = \\frac{r_{screen}}{r_{tip}}. ", "109edb220aff864a2f77f003b4a64608": "\\langle F', \\varphi \\rangle = - \\langle F, \\varphi' \\rangle = - \\int_{- \\infty}^{+ \\infty} F \\varphi' = \\langle f, \\varphi \\rangle ", "109eedca4965cc4f1435b5ec4e1bc5a1": "\\rho_{\\text{isBusinessContact / isFriend} } ( \\text{addressBook} )", "109f172d0f6309e1793dee513c5c9881": "E = E_0 + c_k k^p ", "109f310698c67b59267233f30a2dbce8": "\\mathcal{C}^\\mathcal{J}", "109f6e29e4cdef0b2850bac2d6989db3": " \\sum_{i=0}^{n-1} a^i \\otimes_K \\frac{b_i}{p'(a)}", "109f9dba683e7cc9f7bd0f8aa8c16aa3": "\n\\begin{align}\n\\left\\|\\mu-m\\right\\| \n= \\left\\| \\mathrm{E} (X-m) \\right\\| \n& \\leq \\mathrm{E} \\|X-m\\| \\\\\n& \\leq \\mathrm{E} (\\left\\| X-\\mu \\right\\| ) \\\\\n& \\leq \\sqrt{ \\mathrm{E} ( \\| X-\\mu \\|^2 ) }\n= \\sqrt{ \\mathrm{trace} (\\mathrm{var} (X) ) }\n\\end{align}\n", "109fa6f56471f8b5c0a9c63236dc50b2": "L_0=\\mathrm{AB}=30\\ \\mathrm{cm}", "109fd3fb75823668d9731ce1f8eef742": "\\beta_{12}=K\\beta_{11}\\,", "109fdeb06203947183fd61e2fe859c86": "r=\\frac{1}{T}\\left (W(-se^{-s})+s\\right )\\text{ with }s=\\frac{M_at}{P_0}", "10a08e31a172a3de88d0596827571ff6": " \\frac{dT_\\text{core}}{dt}=Q_\\text{cmb}\\left[ A_\\text{c} (L+E_G)\\left(\\frac{R_i}{R_\\text{c}}\\right)^2 \\rho_i \\frac{dR_i}{dT_\\text{cmb}\\eta_\\text{c}}-\\frac{R_\\text{c}^3-R_i^3}{3R_\\text{c}^3}\\rho_\\text{c} c_\\text{c}\\right]^{-1}\n", "10a18a252ed439b206e9e09d59c31f35": "i\\in\\{0,1\\} ", "10a1a9b21789426655a7cef3eedb323d": " \n \\eta_{\\mu \\alpha} \\left ( \\partial^2 + m^2\\right ) D^{\\alpha \\nu}\\left ( x-y \\right ) = \\delta_{\\mu }^{ \\nu} \\delta^4\\left ( x-y \\right ) \n.", "10a1c62e7176cf9361da91b78993d8c0": "\\limsup_{n \\to \\infty} (a_n + b_n) \\leq \\limsup_{n \\to \\infty}(a_n) + \\limsup_{n \\to \\infty}(b_n).", "10a22a029ea33e3c0935b81cf178e77f": "w(2+i)=1", "10a234df3a29c89fd2754a22b3e5f55a": "F(\\rho, \\sigma) = \\lVert \\sqrt{\\rho} \\sqrt{\\sigma} \\rVert_\\mathrm{tr},", "10a25d4db586c0447d3182ac742895f1": "f(x)=\\Omega(g(x))\\Leftrightarrow g(x)=O(f(x))", "10a2b772e79678e1c08cb162664fa34f": " \\gamma_\\mathrm{se} ", "10a2dab3b8bb6fc3062e4ea3b86d3bfc": "\\frac{d}{dx}f(x)=\\frac{\\frac{dg(x)}{dx}\\cdot h(x)-\\frac{dh(x)}{dx}\\cdot g(x)}{h^2(x)}", "10a301200cbd1ea17a4d727e10e00d9e": "w(v)+w(u)\\geq d", "10a3236fec503e5ee001a7f87b492733": "\\Delta{}E = W + Q + E ", "10a35ce791976a5d4ee0442d5817697a": "\\displaystyle{\\|a^*\\| = \\|a\\|,\\,\\,\\, \\|\\{a,a^*,a\\}\\| =\\|a\\|^3.}", "10a365729012f70af2556be9cdf5a732": "-\\sqrt{\\frac{18}{35}}\\!\\,", "10a3b06a3657e3d1b93c145b064abbcc": " R \\sin x^\\circ = \\frac{R x(180-x)}{10125 - \\frac{1}{4}x(180-x)}", "10a3e10030df2b1c63325a75c1820253": "\\mu^- \\, / \\, \\mu^+ ", "10a3e6b75a65c3e43e9bd70a4aea933f": "\\tbinom nk", "10a40604fd5176f4dc10a23ffb6409b2": "\\scriptstyle E(w)", "10a4442c17e48d9fce1702597cb4750b": "\\left. (\\mathcal{L}_{\\!X} f)(p) \\triangleq \\frac{\\operatorname{d}}{\\operatorname{d}t} f(\\gamma(t)) \\right\\vert_{t=0}", "10a460c5cf4a0e9ce6887ccfb89e8ada": "\n \\text{at } z = 0 : \n \\quad A \\frac{\\part u}{\\part z} = \\tau^x \\quad \\text{and} \n \\quad A \\frac{\\part v}{\\part z} = \\tau^y,\n", "10a491d6e6e5da59dd7a8ade4e7d203a": " {\\frac {|AB|} {|BD|} \\sin \\angle\\ BAD = \\sin \\angle BDA}", "10a51d59d8f78ee0d8c665bbc281a995": "f^*Y\\to X'", "10a55ad0f78ce6e9dddc9f0d25692522": " (N_j,N_k)", "10a55ea60679e457a0ae710cdf5fd184": "Q = A_1\\cdot V_1 = A_2\\cdot V_2", "10a58fc621da6aa7dcba68639f7ae83a": " (a, b) = \\{ \\{ a \\}, \\{ a, b \\} \\}.\\,", "10a5e908edbb2881c5ea62363e7b8e51": "S \\equiv -k_B\\sum_s P_s\\ln P_s= k_B (\\ln Z + \\beta \\langle E\\rangle)=\\frac{\\partial}{\\partial T}(k_B T \\ln Z) =-\\frac{\\partial A}{\\partial T}", "10a5fe428ccc4739924d9299b3d3e925": " \\operatorname{drop-formal}[Z, Y, V] \\equiv Y ", "10a6066086cf9605188376749c98653a": " \\angle DOC = \\angle DOE + \\angle EOC. ", "10a65dec87bdc5afe4a4c6fc8625d4a2": "10\\uparrow^m n", "10a7360428d7a4ec4d0d5885b227054d": "\nC_{\\psi}=\\int\\limits_{-\\infty}^{+\\infty}{\\frac{|\\Psi(\\Omega)|^2}{|\\Omega|}}\\,d\\Omega<\\infty\n", "10a7491db271a208d3436f577f35576a": "\\vec{r}_{i}(t+\\delta t_{\\mathrm{MPC}}) = \\vec{r}_{i}(t) + \\vec{v}_{i}(t) \\delta t_{\\mathrm{MPC}}", "10a74e7bba9417af59929258c975355e": "M_Y\\ ", "10a77078361fbfefd59b647fc1ab06aa": "z=\\frac{(\\overline{x}_1 - \\overline{x}_2) - d_0}{\\sqrt{\\frac{\\sigma_1^2}{n_1} + \\frac{\\sigma_2^2}{n_2}}}", "10a7832cda41bfad9dec2473adf31fc4": "log(t)", "10a7ad705170d5cef996e4b6bde38618": "I\\underline{A}", "10a7bcc75aa9d193c304c1937418798d": "Cn_r", "10a7cf8ba6f8ee42519b2af39ca0b724": "\\{\\mathrm X_1(p), \\dots,\\mathrm X_4(p)\\}", "10a826b1854b836a0b481dd9d1d251bf": "\\{\\mathbf{Z}_k:k=1,2,\\ldots,n\\}", "10a8ab7ea986641a5c054200fc263648": "\\ y=(d/2) \\sqrt{(\\xi^2-1)(1-\\eta^2)} \\cos \\phi, ", "10a93856574e68c286da272962609390": "x' = x\\sqrt{1 - v^2/c^2} ", "10a93d56445a9e7abc8455e26825bed7": "\n \\xi(\\omega) =\n \\begin{cases}\n -r, &\\omega \\text{ is red} \\\\\n -r, &\\omega = 0 \\\\\n r, &\\omega \\text{ is black}\n \\end{cases},\n", "10a9537eed0c904da74fe0fde3041bfc": "\n\\left\\{\n\\begin{bmatrix} 0 \\\\ 1 \\\\ -3 \\\\ 3 \\\\ -1 \\end{bmatrix}\n\\begin{bmatrix} 1 \\\\ -15 \\\\ 30 \\\\ -1 \\\\ -45 \\end{bmatrix} \n\\right\\},\n\\left\\{ \n\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\n\\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}\n\\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ -2 \\\\ 0 \\end{bmatrix}\n\\right\\}\n", "10a95d31413b5465b8c83c30d657d6ea": "Q=kd", "10a95df6e59f7a1a40dd176bc5884cb7": "^{2}\\Sigma^{+}", "10a98d22844a50d53ed669b1a0a5d42f": "\\{ (x, z) \\ | \\ x \\, (S \\circ R) \\, z \\wedge \\forall y \\in Y \\ (y \\, S \\, z \\Rightarrow x \\, R \\, y )\\}.", "10aa51128f0a6f16f6ee6253b5b8f2d6": "a(u, v)=\\int_a^b\\! u'(x) v'(x)\\,dx", "10aa66225d91cadf44874edaee2b0244": "z_3 = \\frac{z_1z_2-ax_1y_1x_2y_2}{(y_2^2+(z_1x_2)^2)}", "10aa7b8b0b15fd27d28a6612265cba7b": "\\Gamma_7", "10aa9887049893fdb1a896fca150a797": "-\\frac{\\hbar^2}{2m}\\nabla ^2 \\delta\\psi+V\\delta\\psi+g(2|\\psi_0|^2\\delta\\psi+\\psi^2\\delta\\psi^*) = i\\hbar\\frac{\\partial\\delta\\psi}{\\partial t}", "10aaab75818f6fd1f7490ed8ca10af39": "\\kappa = \\frac{\\Pr(a) - \\Pr(e)}{1 - \\Pr(e)}, \\!", "10aaac2d30cec6245ecdb4c6d83abed2": " w_i = \\int_{-s}^s \\frac{1}{y-y_0} d\\Gamma \\qquad (7)", "10ac41137292c9de95b784464a8f33d3": "g_{uc}(\\langle sss \\rangle) = g_{uc}(\\langle \\text{sß} \\rangle) = g_{uc}(\\langle \\text{ßs} \\rangle) = \\langle SSS \\rangle", "10ac7828cbfca1c7fa78610954f47c99": "\\mathbf{H} = \\mathbf{F} / q_m \\,", "10ac9e1d972523558bc6e58a4d4efd38": "\\nu^*", "10aca448948213b86b8a1aa47d638b33": "\\gamma^\\sigma \\,", "10ad07348572f86c16caa72122080733": "\\eta_f = \\frac{\\tanh{mL_c}}{mL_c}", "10ad7f98c69273f5be1ccb8fa572f02e": "M_{T}^2 = (E_{T, 1} + E_{T, 2})^2 - (\\overrightarrow{p}_{T, 1} + \\overrightarrow{p}_{T, 2})^2", "10ad8e428e9137aa983c5dd8c252a19b": " f( \\lambda ) = \\frac{3}{ \\lambda } \\left [ \\frac{1}{tanh( \\lambda )} - \\frac{1}{ \\lambda } \\right ] ", "10adb068eb6ad18e34a61387770fc0be": " \\Phi:\\mathcal{H}\\to\\mathcal{H}^* ", "10adba9a4d1293648444f90c1f98f4af": "<1,2>", "10ae0e1dfeeadf184fab5f11d82e8271": "\\{ C_I , C_J \\} = f_{IJ}^K C_K", "10ae161e3b32f9a344ac89bb428ae2d6": "x_t=x_t(\\xi_{[t]})", "10ae23b174a87716e3840e53044d1502": "J(u) = \\int_D |\\nabla u|^2\\mathrm{d}x", "10ae2db675e5043225913ed613ad1697": " \\begin{align} \\mathbf{Y} & = \\frac{d\\mathbf{F}}{dt} = \\frac{d^2\\mathbf{p}}{dt^2} = \\frac{d^2(m\\mathbf{v})}{dt^2} \\\\\n& = m\\mathbf{j} + \\mathbf{2a}\\frac{{\\rm d}m}{{\\rm d}t} + \\mathbf{v}\\frac{{\\rm d^2}m}{{\\rm d}t^2} \\\\\n\\end{align} \\,\\!", "10ae67bf9ad12ccc730ab496b1863a09": "g_{dsF}=g_{ds}+\\frac{C_{GD}}{C_T}g_m", "10af0757ad55e23fa0cea99aafab63d7": "\\lambda x.x+y", "10af4fab96c96f4c326d3a7f0cfc8141": " : \\hat{b}^\\dagger \\, \\hat{b} : \\,= \\hat{b}^\\dagger \\, \\hat{b}. ", "10af6b7078f7aa9f6d583996bfc30d5a": "s = vt - \\begin{matrix} \\frac{1}{2} \\end{matrix} at^2", "10af71414b9126febdd74ba0d3815105": "y'' = -y\\,", "10afc8268f8964502e8824eff7aa42a0": "L_{4k}(\\mathbf{Z})", "10b001559e5ea111724784ac9136f042": "\n(x+m+1)\\sum_{i=0}^m(-1)^i\\dbinom{x+y+i}{m-i}\\dbinom{y+2i}{i}\n-\\sum_{i=0}^{m}\\dbinom{x+i}{m-i}(-4)^i=(x-m)\\dbinom{x}{m}.\n", "10b01101b78d570970fdd18bc9bca880": "\\overline{\\lim} (\\log a_n)/2^n < +\\infty.", "10b04274297b40fe360362f70661531e": "\\quad (7) \\qquad \\qquad \\frac{d \\bar{\\rho}_i}{d t} + \\frac{1}{\\Delta x_i} \\left[ \nf_{i + \\frac{1}{2}} - f_{i - \\frac{1}{2}} \\right] =0 ,", "10b09ad594da0a55c2366a2f2c1e057b": "\\Omega_{i,j} = \\Gamma_{i,i}+\\Gamma_{j,j}-\\Gamma_{i,j}-\\Gamma_{j,i} = K_iK_i^T + K_jK_j^T - K_iK_j^T - K_jK_i^T = (K_i - K_j)^2", "10b1127d5f0f8fe6e96b9f49ce875f92": " 1 \\leq k \\leq |\\mathbf{X}|", "10b112b5372daf1cc5161752bd552176": "n\\in\\{0,1, ... ,N-1\\}", "10b1161dd4156b738b506af877a96db6": "\\frac{dy}{dt} = g(x, y) = -y\\left(\\gamma - \\delta \\frac{x}{1 + x}\\right)", "10b17aad991be7b34a64cf31c763432b": "\\Lambda(x)=\\sum_{i=1}^v\\lambda_ix^i", "10b1fa405163165d5adb8d287e24a50f": "g = {1\\over 2}h_{\\alpha\\bar\\beta}\\,(dz^\\alpha\\otimes d\\bar z^\\beta + d\\bar z^\\beta\\otimes dz^\\alpha).", "10b22174c40bce87be79c006df59ceed": " \\phi_{bc} ,\\, \\phi_{ac} ", "10b23c2b5a687510af844bc0b3fc9ed8": "^{-1}:G\\to G,", "10b25f75c898d917edeb80f0d61a5e5b": "S + C \\supseteq \\{f(x): x \\in M\\}", "10b2666e21341f2b952499d4e22efbef": "\\hat{r}=\\frac{\\bar{r}}{|\\bar{r}|}", "10b3bb2bb94ad44f4cca3cc43fe84ff8": "R_{ab}l^a l^b=R_{ab}l^a l^b-\\frac{1}{2}Rg_{ab}l^al^b=8\\pi \\, T_{ab}l^a l^b", "10b3c7530c0ce99746ed4ad690ac03e3": "\\mathbb{H}^{p+1}", "10b42f67a9e763181931e1c056d1d04e": "m>\\lambda", "10b443d7f9559f9b0f14e5f71343bc1a": "\n\\begin{align}\nc & {} = \\frac{r(1+r)^N}{(1+r)^N-1}P \\\\\n& {} = \\frac{r}{1-(1+r)^{-N}}P\n\\end{align}\n", "10b4821438b54304dbdc6d8d0fd66a56": "\\begin{align}\n\\mbox{D}(\\mbox{E}(x)) &= a^{-1}(\\mbox{E}(x)-b)\\mod{m}\\\\\n &= a^{-1}(((ax+b)\\mod{m})-b)\\mod{m} \\\\\n &= a^{-1}(ax+b-b)\\mod{m} \\\\\n &= a^{-1}ax \\mod{m}\\\\\n &= x\\mod{m}.\n\\end{align}", "10b4aac1a654918f5814117468e23fd2": "{y \\in R^{\\it M}}", "10b4bf357efa33ae062b5b7202d9446a": " \\forall k \\in \\mathbb{N} : \\phi(x,k+1) \\geq \\phi(x,k) ", "10b541396eb5a9ffb8b6080f3525595b": "(u,v), \\mathrm{height}(u) \\leq \\mathrm{height}(v) +1 ", "10b5766bf11dd27b7a7a07a88588bee8": "\\frac{3}{4}V_g + V_e", "10b5ed345a5bbe2bda56fa0def78adcc": "\\hat{H} = \\frac{1}{2m}\\left[\\left(\\mathbf{p} - q \\mathbf{A}\\right)^2 - \\hbar q \\boldsymbol{\\sigma}\\cdot \\mathbf{B}\\right] + q \\phi", "10b5f67b32006b705c9bd7ff308fe91e": "M \\otimes_{{\\Bbb Q}(q)} M'", "10b623ccb527b48de06c362ce120ddc0": " A_N = \\int D\\mu \\prod_{0500^\\circ C} \\ U+2MgF_2}", "10c7f8a4b7b0a99a6532f5db55d3b665": "\\mathbf{r}\\rightarrow R(\\mathbf{\\hat{n}},\\theta)\\mathbf{r}", "10c81bf8c00e16c19dd22f8096fdd0fb": "S + \\alpha.m_0 < m_1...m_k", "10c83035651370b89637715da77fdbcc": "\\{\\{1,2\\},\\{3,4\\}\\}", "10c84eee0a16a448846a79bc76197787": "(u\\cdot v)' = v\\cdot u' + u\\cdot v'. \\,\\! ", "10c88c0e767d89d2960818c18626b48f": "\\lambda_j=\\lambda_i", "10c898dfa9b133f34799fc2c8f55a9dc": "(X, \\pi_0 \\mathcal{O})", "10c8a99ee4ef318b762f5710f584c3db": "\\frac{1}{N} (\\sin(Nt/2) / \\sin(t/2))^2", "10c8c253dfaaff044ea423f9c2dbb6e0": "p(x)y''+q(x)y'+r(x)y=0\\,", "10c9648e059bfb2cbe2508e9284c7597": "Y_t = \\mu + \\int_{A_{t}(x)} g(\\eta,s,x,t)\\sigma_{s}(\\eta) L(d\\eta,ds) + \\int_{B_t(x)} q(\\eta,s,x,t)a_{s}(\\eta) \\, d\\eta \\, ds", "10c9e3febcb081149e8bfad8fffdff20": "(M \\cup H^j) \\cup H^i", "10c9ef0f1a7e8daeac9b042247ed50d6": "\\frac{x(2x+1)}{(x-1)^4}.", "10ca6d80af5e8a84b6715b05269b4f03": "P(T) \\approx 10^{-31}~T^{2} ~~~~~~~~~(10^{4.6} < T < 10^{4.9} K) ", "10ca95f619f3ac6ff307febf7b873727": "\\lambda(n) =\n\\begin{cases}\n\\;\\;\\phi(n) &\\mbox{if }n = 2,3,4,5,7,9,11,13,17,19,23,25,27,\\dots\\\\\n\\tfrac12\\phi(n)&\\text{if }n=8,16,32,64,\\dots\n\\end{cases}\n", "10ca97af002147360939f6c37fccdd14": "\\displaystyle A_3=S(b)M", "10caa8bedd30a67c19f7bd2483b6b44f": "\\ln(n!) = n \\ln \\left( \\frac{n}{e} \\right) + \\tfrac{1}{2}\\ln(n) + y + \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)n^{k-1}} + O \\left( \\frac{1}{n^m} \\right).", "10cb006455960c569479bafab346e166": " V_{r1} ", "10cb39dd1b82a58c7f5da5938da8598b": " \\operatorname{E}(\\theta|y) ", "10cb40d0c7fa028aeff3b6024d8a8bdf": "S: A\\rightarrow A", "10cb5780c791f645a8012addef9314a0": "f(\\mathbf{v}_j)=a_{1j} \\mathbf{w}_1 + \\cdots + a_{mj} \\mathbf{w}_m.", "10cba4d01b41c5e1b4559af7ba27f573": "|I,J,m_I,m_J\\rangle", "10cba74920da7d97fe3b41eae5abeb66": " \\lor", "10cbba719a6ef66f9b28fcce378c8a77": "-I", "10cbf483119bbfb05f8037124d8498b7": "\\mathbf{I_m}", "10cc191c447d12bd0500011e999f16b0": "\\bar y_j", "10cc37772c503b0eea0e4c2bd2ea434d": "V_{\\rm w} = \\frac{\\pi V_{\\rm m}}{N_{\\rm A}\\sqrt{18}}", "10cd5253a5e5101d2917177cca849312": "\\nabla\\cdot\\mathbf{B}=0", "10cda089e29425b37fbe8434d1aa4899": "S(1853)=17\\,", "10cdb4b0e0c5e45166bf3f89d5cac2b7": "\\tau = \\frac{F}{A}", "10cdbfc90a3b4583861dac8ca8d50645": "n^2-3n+4", "10ce83d6a5b51d45c368149fd1ab24b8": " \\mathbf{y} \\sim \\mathbf{C} \\, \\mathbf{x} ", "10ced1230035869b5cafb1965328e5f3": "J_m.", "10ceddd04ecde51e2d3fd53f39f3af82": "S^2=\\frac{1}{n-1}\\sum_{i=1}^n\\left(X_i-\\bar{X}\\,\\right)^2.", "10cede95815039de9ed5d07c73e46bfb": "\\mathbf{\\sigma_\\mathrm{n}}= \\lim_{\\Delta S \\to 0} \\frac {\\Delta F_\\mathrm n}{\\Delta S} = \\frac{dF_\\mathrm n}{dS},", "10cf102ddb29667be7e8aef13b40b6dd": "(4\\pi/3)n\\lambda_D^3 = 1.72\\times10^9\\,T^{3/2}n^{-1/2}", "10cf24916dcb3a811db5997f1f462335": "y^2 = x^2(a-x)/(a+x)", "10cf8cc82c039d4e01a5370a4e4f069a": "g(z, u) = \\exp\\left(u \\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty \\frac{z^k}{k} +\n\\sum_{k=1}^{\\lfloor\\frac{n}{2}\\rfloor} \\frac{z^k}{k} \\right).", "10cff4b7a419264e8c021751dd91a76c": "\\omega_2=iR_2", "10d00a611c37ed3585b3819fedfc6f0f": "S(\\boldsymbol{\\beta}) = \\mathbf y ^{\\rm T} \\mathbf y - 2\\boldsymbol \\beta ^{\\rm T} \\mathbf X ^{\\rm T} \\mathbf y + \\boldsymbol \\beta ^{\\rm T} \\mathbf X ^{\\rm T} \\mathbf X \\boldsymbol \\beta .", "10d0392dbbc815dbd45eaa908643f589": " A \\times_{C} B = \\left\\{(a,b) \\in A \\times B \\; \\big| \\; \\alpha(a) = \\beta(b) \\right\\} ", "10d0599cc949b0f3ae1a61c7eaca8353": "P=\\rho R_s T,", "10d0ad92bb3547cb511c2c7beec53a51": "k \\leq 0", "10d112ead69f7b0aeb7bb6823f973f04": "K: G \\sim N(E,\\frac{N_{0}E}{2})", "10d17bb6648eec2409b0159ff27356a5": "V_v = \\frac{\\hbar}{2}(2\\omega_1+\\omega_2-3\\omega)= - {\\hbar e^2\\over 16\\pi\\varepsilon_0 m_e\\omega Z^3}.", "10d1c53ac47198404da42496e393662e": "E(|X_n|,|X_n|\\ge K)= 1\\ \\text{ for all } n\\ge K,", "10d1d029504c5eee87108420925b9733": "\\Phi (x) \\equiv 0", "10d1f58f0d83a8b2481c2943109fd288": "\\frac{x}{\\sigma \\sqrt{2}}", "10d20370c745ee26c5556cb817b36fd3": "X^2 = X\\times X ", "10d2053af5c2cf7f281292a067ac5380": "f: \\mathbb{Z^+} \\rightarrow \\mathbb{R}", "10d213356e630fc9a88029c082c834b1": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log [\\frac{\\varepsilon}{3.7D} - \\frac{5.02}{Re} \\log(\\frac{\\varepsilon}{3.7D} + \\frac{13}{Re})]\n", "10d22e184c2215fb046a5d3b2f26304b": "\\!V = \\frac{4}{3}\\pi r^3", "10d241651b61cfdd54169b153ab938e4": "\\varepsilon\\dot{x}_2 = f_2(x_1,x_2) + \\varepsilon g_2(x_1,x_2,\\varepsilon), \\, ", "10d27c956fff38e4b59b6486b5538bcb": "u = \\left\\lfloor \\sqrt{x}\\right\\rfloor", "10d2c843b66fe271ad3b5218b499622c": "f_1(x),f_2(x), \\dots,f_m(x)\\,", "10d301becbebc5d4edbc436250495a7a": "x + z = y", "10d37c0fcf8bfb55880f122ef7476640": "g(x)=\\frac{f(x)}{f(x)+f(1-x)},\\qquad x\\in\\mathbb{R},", "10d4b6c2f87c147e5fa757c20c29ce17": "K=uv[2t(1-uv)-(u+v)(1-t^2)][2(u+v)t+(1-uv)(1-t^2)]", "10d5099a2b6b08bf93311d42709413b9": " \\sigma_3 \\,\\!", "10d54f8ab2aa3bf7b9fbf56532bcdd92": "\\varphi=(1+\\sqrt{5})/2,", "10d559a732e1e9b07700a1fdcc1b2077": "f_1f_2 \\rightarrow g", "10d5a142e603155741af2829cef43516": " H_{j - 1, j}^{ } = H_{j, j + 1}^{ } \\equiv H_L ", "10d5a456b521b432b67ab5024ade2409": "\\displaystyle Y_t = \\begin{cases}\n \\delta_{1-t}(W_t) &\\text{for } 0 \\le t < 1,\\\\\n 0 &\\text{for } 1 \\le t < \\infty,\n \\end{cases} ", "10d5c144f59e84fff7a4a9a48c8a40e9": "T = \\tau", "10d62bf6f1e6566885e584356f173a97": "T_a(x)=x+a", "10d634548ef000d636630284051b76eb": "\\sum_{k=n}^{n+17} F_k = 76 F_{n+10}", "10d7016ae8a3134ac187ec27ab164b8e": "=32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}-\\frac{i}{2}d_{-1}-\\frac{1}{4}d_{-2}+\\frac{i}{8}d_{-3}", "10d748916f1e04efb42fb6ded622eee1": "\\begin{cases}\n a\\\\\n b\n\\end{cases}=a_0\\sqrt{\\frac{\\xi}{\\xi_0}}\\left[1\\pm \\frac{A}{\\sqrt{\\xi}}\\sin \\left(\\xi-\\xi_0\\right)\\right],", "10d785a454a472daa18581b57c13b7a7": "\\begin{array}{lcl}\n \\mathbf{A} &=& \\begin{bmatrix}\n \\cos\\theta \\cos\\psi & \\cos\\phi \\sin\\psi + \\sin\\phi \\sin\\theta \\cos\\psi & \\sin\\phi \\sin\\psi - \\cos\\phi \\sin\\theta \\cos\\psi \\\\\n -\\cos\\theta \\sin\\psi & \\cos\\phi \\cos\\psi - \\sin\\phi \\sin\\theta \\sin\\psi & \\sin\\phi \\cos\\psi + \\cos\\phi \\sin\\theta \\sin\\psi \\\\\n \\sin\\theta & -\\sin\\phi \\cos\\theta & \\cos\\phi \\cos\\theta \\\\\n \\end{bmatrix}\n\\end{array}", "10d78f524211cb318a4dad4f027664b7": " (\\partial H)_P=-(\\partial P)_H=C_P", "10d7adf5c6cf00828f5abf61c0da4347": " U x \\cdot \\operatorname{diag}(f(x_0),\\dots,f(x_n)) = T_f x \\cdot U x ", "10d7dcb7b76d35a438b7c571adc509f7": "\nx(t) = Ae^{\\gamma_+ t} + Be^{\\gamma_- t} \\, ,\n", "10d7ebe08036bf0dca2ea2aebcd6739c": "v=\\frac{\\sum_{}\\frac{\\pm e_{k}}{R_{k}}+\\sum_{}\\pm a_{m}}{\\sum_{}\\frac{1}{R_{k}}+\\sum_{}\\frac{1}{R_{i}}}", "10d839602e2bd61fd6b89395c2f2b1d8": "\\eta c_\\eta(\\xi_1,\\xi_2)", "10d867586deb88c9dff1a2bd50fe94e8": "p_2=\\frac{m_1}{1+m_1}\\ ,", "10d890fc5c36a42fdaec7e589f1c216e": "(2^{\\aleph_0})^{\\aleph_0} = 2^{\\aleph_0^2} =2^{\\aleph_0},\\,", "10d89835595e6e6f8578a5312e942cec": "\\exp(i \\omega t)", "10d8989ebc40b46b347efeb066400891": " \\left (\\frac{p_2}{p_1} \\right )^\\frac {\\gamma-1}{\\gamma}", "10d8a5f8785fe2bcae31e13760724f71": "\\sum_{k = 0}^\\infty \\left( \\frac{1}{k!} \\int_{A^k} \\rho_k(x_1,\\ldots,x_k) \\, \\textrm{d}x_1\\cdots\\textrm{d}x_k \\right)^{-\\frac{1}{k}} = \\infty", "10d8c8cc3c107342f64fb76596c0908b": "0 \\to K \\to F \\overset{u}\\to M \\to 0", "10d8fa9d04d95ac73026b8dc03d7c8c4": " P V^{\\gamma} = \\operatorname{constant} \\qquad ", "10d91283b650cd88188f1de614da1872": "\\approx 10 \\times 0.02 \\times 0.834", "10d94500d9cce400d297c5f7e179143a": "\\delta n_k", "10d9c53b46755279bbb0ec2c8dcacc49": "\\scriptstyle \\frac{\\sqrt[12]{2}}{\\sqrt[12]{2}-1}", "10d9ddbd5d2f6453db360b396af9032a": "{1\\over 2\\pi} \\int_0^{2\\pi} |e^g|^2 \\, d\\theta \\le e^A,", "10d9ee9be1f790eb1addb53458190cdd": "n-CN(0,I_{N \\times N})", "10da04269de1736c4f92c635f399f7b4": "\\psi_{7,8}=1", "10da7af65b26ed6927b624abdede8994": "x_1,\\ldots,x_m", "10dab83c45ba3ddfe99ee0a5ec537c8a": "c_{15}", "10dac876187a7e73313fc986fb5cec19": "C(K) = \\lim_{r \\to \\infty} C(\\Sigma, S_{r}).", "10db389c133e8bc28e9be66025b0064b": "\\gcd(N,du)=d", "10db4ad06b1ae4064c3c12ba4b2bab66": "R = {R^m}_{m}", "10db7c533c7259766477548ffc284fba": "\n\\bold A_z= \\left[ \n\\begin{array}{c c c c c}\n0 & 0 & 0 & 1 & 0 \\\\\n-uw & w & 0 & u & 0 \\\\\n-vw & 0 & w & v & 0 \\\\\n\\hat{\\gamma}H-w^2-a^2 & -\\hat{\\gamma}u & -\\hat{\\gamma}v & (3-\\gamma)w& \\hat{\\gamma} \\\\\nw[(\\gamma-2)H-a^2] & -\\hat{\\gamma}uw & -\\hat{\\gamma}vw & H-\\hat{\\gamma}w^2 & \\gamma w\n\\end{array}\n\\right].\n", "10db94130479c95f1e316620e654a436": "\n\\left[\\int_0^1(1-U(x))\\,dx, \\int_0^1(1-L(x))\\,dx \\right].\n", "10db9ebe9cbcc9a65480d326235caffb": "x^4+18x^3+23x^2+8x+1", "10dc276caea41ecc0e3ecf39fc3aca22": "\\Omega=d\\omega+\\omega\\wedge\\omega=0.", "10dc52167ae97abd7ec6f4d4c89a532c": "z \\mapsto u z .", "10dc75b26ed9777d36a3ba7ed338e8f9": "n=N, \\alpha,\\, \\beta\\!", "10dca1af11a65ac9a06c0ee2b56b907d": "\\scriptstyle q_e^2 q_Z^2 = q_e^4\\,", "10dd04ab5dda18d086a6048e29272b7f": "\nK_H = \\int\\limits_{-\\infty}^{x'}\\big [ k_H^{ \\widetilde{u}(x)} - k_H\\big ] dx - \\int\\limits_{x'}^{\\infty}\\big [ 1 - k_H^{\\widetilde{u}(x)}\\big ] dx\n", "10dd2476912f3b7ec3a4b86d418944c9": "~~~~~S,p,\\{N_{i\\ne j}\\},\\mu_j\\,", "10dd73f7d88ac75c34b6542921b0ae1a": "\\ B \\,", "10dd8fa7ff9196e4b5bd5e200d6d0506": "a_{m,n}+a_{m,n+1}+a_{m+1,n}+a_{m+1,n+1}=34", "10de06fa069a8d1cbcac910f71e8c40c": "[b^3,b^4]", "10de86a15a513180b9deaff8f6145372": " {\\pi }", "10df8ee196666c9e2ebea43c48cc11e3": "\\Phi(r,\\theta,z; r',\\theta',z')=a'\\Phi_{\\infty}(r,\\theta,z; r',\\theta',z')-a'\\Phi_{\\infty}(r,\\theta,z; r',\\theta',-z'-2z_b)", "10df9a6dcd5de2a5b151f831fca7759c": "\\sum_{l=0}^{N_j-1}\\left| 2^{\\frac{3j}{4}}\\hat{\\phi}_{j,0,0}(r, \\omega-\\frac{2\\pi l}{N_j}) \\right| ^2=\\left|W(2^{-j}r) \\right|^2\\sum_{l=0}^{N_j-1}\\tilde{V}^2_{N_j}(\\omega-\\frac{2\\pi l}{N})=\\left|W(2^{-j}r) \\right|^2", "10dfd3e1f8a43ead38b23b9f32f6948b": "\\mathbb{Z}/p", "10dfdcb4c0a14620873148af4214dc28": "a^{n-1}\\not\\equiv1 \\pmod n", "10dff05cc4d1d30ebe2e9ec2e4604a2c": "\\rho_A(X) = \\inf\\{u \\in \\mathbb{R}: X + u1 \\in A\\}", "10e05521d0da50a8c6a59c9bc5dc320d": "V\\rightarrow U", "10e084e983c9507caf9014c40e479cf2": "M_+^1(A) \\times M_+^1(A) \\rightarrow [0,\\infty{})", "10e0dd76c46da5fa492f40a634ec23cb": "x_{1}, x_{2} \\in X", "10e1042b56900798d95e67427511971d": "\\ln S_t", "10e1104ac9b22a46729ada710a651c31": "0\\ f\\ x = x ", "10e15af0938930ead144c97eba0e7dec": "\\sigma_{2c}=4.6", "10e16c6a764d367ca5077a54bf156f7e": "\\sigma^2", "10e196a4eee70e2d42ccfd3eeff37969": "\\sum_{c\\,\\in\\, C} (p_{c,t}\\cdot q_{c,t})=\\sum_{c\\,\\in\\, C} [(P_t\\cdot p'_{c,t})\\cdot q_{c,t}]=P_t\\cdot \\sum_{c\\,\\in\\, C} (p'_{c,t}\\cdot q_{c,t})", "10e1a6140d3569e0b5d723b335277017": " \\epsilon > \\| f(t) - f_e \\| ", "10e1ce139d2e15d736318ef7e14d3bd9": "d_X\\Delta = 0", "10e1ef4e195f389a2640d9321a0f6936": "r \\leq d", "10e22a5acb1d5cfe9ce5d3eb73884243": "F(x) = f(x) + ig(x)", "10e31ef2b67b23aed0351f7448f12369": "S^{p,q}", "10e324f2ceef507b29a7581bf01c7b12": " \ny(0) = 0.\n", "10e343d8eb2ab64dacb8ae345eb765ea": "| \\psi( \\mathbf{r}, t) |^2 \\to 0", "10e3b4c8abc4c4b7b6ea1acca1efab41": "M_{\\odot}", "10e400abc162ac9b9e0dabb6b47336f8": "Cv(b)=Cv(a)+\\int_{a}^{b}i(t)dt.", "10e4538433703868de1dd42648521159": "\\mathbf{J} = n q \\langle\\mathbf{v}\\rangle,", "10e48e317edea309efe28c0c7be85826": "\nOBV = OBV_{prev} + \\left\\{ \\begin{matrix}\nvolume & \\mathrm{if}\\ close > close_{prev} \\\\\n0 & \\mathrm{if}\\ close = close_{prev} \\\\\n-volume & \\mathrm{if}\\ close < close_{prev}\n\\end{matrix} \\right.\n", "10e4e551eda40b348b9cc09a8ccc5855": "\\vec{r}\\,'=\\vec{r}-\\vec{v}t", "10e569312a5700fac8e0228f36273347": "|\\zeta-\\zeta_0|", "10e572be9a70d49f0afa12780e63d22b": "\\epsilon=2", "10e58bbdec91cf22ec36899c07543fb5": "\\lesssim10^{-16}", "10e5968d8e1de6d745d2ba0c17217874": "\\omega = \\pm \\infty\\,", "10e5b91fea5e75ac99cf88a35e465b0c": "\\pi r l ", "10e5e4a51b90e6c1d5d57670b268ca9f": "8119+5741\\sqrt{2}=16238.00006\\ldots", "10e631acacff0b8d893e03dd585864f7": "O(1) \\to S^n \\to \\mathbf{RP}^n", "10e635474dfa3376a860cd8327e507be": " \\langle \\psi_{1}|H|\\psi_{1}\\rangle=\\langle \\psi_{2}|H|\\psi_{2}\\rangle=E ", "10e63d9753ed56c4f0c0d01246706ae3": "\\{1,2,\\ldots,\\ell-1,\\ell\\}", "10e69118f91dff63923a6770f1f7e37a": " {5 \\over 6} \\cdot {2 \\over 1} = {10 \\over 6} = {5 \\over 3}. ", "10e6b0fe5b28c773856efa0c3e2725a8": "Z = \\int_{\\mathbf{R}^{n}} \\exp ( - \\beta \\Psi (x) ) \\, \\mathrm{d} x.", "10e742fcb4eb4ae71eb3be78b96cbd50": " \\rho_{m} = r_{m} \\times 2\\pi r ", "10e78664c42c2ca0d46153ea3f1139be": "(\\mathcal C,\\otimes,I_{\\mathcal C})", "10e7b6c64043fb1f4af4678b701c896f": "f : M \\to \\mathbb R", "10e7d3592454ce8f7a26f1a09c1098b3": "D \\to D/\\mathfrak{m}_D", "10e8022e2261b17e34d1908e9bf942e6": "F_A", "10e807a5782402f73dbe1c3d76276baa": "R=\\sqrt{\\frac {-\\cos S}{\\cos (S-\\alpha) \\cos (S-\\beta) \\cos (S-\\gamma)}}.", "10e80aa30a468146c6a9b6d9b86c9c2e": "A_m,B_m,C_m,D_m", "10e8597401a83552960ca8fe539bfda8": "k(\\mathbf{x},\\cdot)", "10e89386e21ee7a7d3b69848081c370e": "-323\\pm 50", "10e8e01e61ce1264eb1a759d096d1394": "\\ell^2(I)", "10e9a2e2f33b2be8fa5fe850dd9fdb24": " (n+1)(n+2)/2 ", "10e9cbfcb777a2892d7fa27ca01edabd": "\\begin{align}X_2^{(4)} = y\\partial_x&-y'^2\\partial_{y'}-3y'y''\\partial_{y''}-(3y''^2+4y'y''')\\partial_{y'''}\\\\\n&-(10y''y'''+5y'y'''')\\partial_{y''''}\n\\end{align}", "10e9f13438437c11fcac1713dbbefffb": "F:J \\to C", "10ea4e7c161d43338f593dc396bfd0b6": "\\frac{d}{dt}\\begin{pmatrix}x - x_3\\\\y - y_3\\\\\\end{pmatrix} \\approx \\begin{pmatrix}\\alpha\\left( 1 - (1 + 2 x_3)/K \\right)&- \\alpha\\\\ \\delta (1 - \\alpha)^2 y_3 & 0\\\\\\end{pmatrix} \\begin{pmatrix}x - x_3\\\\y - y_3\\\\\\end{pmatrix}", "10ea4fff179eaea0eb63f69d6bb155f9": "\\Bigg[\\frac{\\overline{\\pi}}{\\pi}\\Bigg]=\\Bigg[\\frac{-2}{\\pi}\\Bigg](-1)^\\frac{a^2-1}{8}", "10ea60806bc3c467520cd4b981ca12a2": "S=1500\\left(1 + \\frac{0.043}{4}\\right)^{4 \\times 6} = 1938.84 ", "10eab4a19609b175205e93005e57458b": "e^{-e}", "10eb36f7e4932933f57e7dfd3787ee6f": "\\mu_{2} \\equiv E[W^2]=g_{2}(\\theta_{1}, \\theta_{2}, \\dots, \\theta_{k}) ,", "10eb6e3a5230e4e4cbd35ab90fa35350": "a_{ij}\\,", "10ebffaafbab19f123f02915563b9559": "A(x) = L(x) - \\alpha \\text{ for } \\alpha \\in F_{q^n},", "10ec56dbe0695609e64ebf7325992680": " : \\hat{f}_2 \\, \\hat{f}_1^\\dagger \\, \\hat{f}^\\dagger_2 : \\,= \\hat{f}_1^\\dagger \\, \\hat{f}_2^\\dagger \\,\\hat{f}_2 = -\\hat{f}_2^\\dagger \\, \\hat{f}_1^\\dagger \\,\\hat{f}_2 ", "10ec663bb966afd58191158c8f251a12": "|\\rho|<1", "10ecdf1b42ae44367ae6726dc658b93a": "4.5\\cdot10^5 kg^{-1} m^5 s^{-2}", "10ed06237c15c51013f8ef4ecf7d8f5c": "\\mathrm{supp}T_\\epsilon=\\mathrm{supp}(T\\ast\\varphi_\\epsilon)\\subset\\mathrm{supp}T+\\mathrm{supp}\\varphi_\\epsilon", "10ed2bb89894d1764e9151633d7aaec8": "\nP= \\frac{Li}{1-\\frac{1}{(1+i)^n}}\n", "10ed9c17e18ad7b138d3d2415fe62717": "\\mu_i^{(k)} = \\mu^{-\\tfrac{1}{2}}(\\mathbf{x}_w^{(k)}, \\sigma_I^{(k)}, \\sigma_D^{(k)})", "10edbc8479b2f7509a3409dfd3c49ed8": "\\frac{1}{48}\\left[\\begin{array}{ccccc}- & - & \\# & 7 & 5 \\\\ 3 & 5 & 7 & 5 & 3 \\\\ 1 & 3 & 5 & 3 & 1 \\end{array}\\right]", "10eddcd77a54fb161cdc76542eedb4be": "Q_a = \\frac{T_a}{T_H}Q_H + T_a S_i = Q_{a,\\mathrm{min}}+T_a S_i.", "10ee126cb0c931dbd49eadc778284aad": " \\sum_{x:\\, f(x)=f(x_0)} \\omega^{x y} = \\sum_{b} \\omega^{(x_0 + r b) y} = \\omega^{x_0y} \\sum_{b} \\omega^{r b y}.", "10ee3cc4b88cb627bf4e90d177e75483": "\\boldsymbol{A} ", "10ee4ae9b571a4a19478277dead08a36": "\\tilde{Z}[\\tilde{J}]=\\int \\mathcal{D}\\tilde\\phi e^{-\\int d^4p \\left({1\\over 2}(p^2+m^2)\\tilde\\phi^2+{\\lambda\\over 4!}\\tilde\\phi^4-\\tilde{J}\\tilde\\phi\\right)}.", "10ee9ef569900b1194540f78a0bbfd3a": "\\bar{x} = (x, y)^T", "10ef36e12bab8a7fb7848800e80a0cd9": "Wy+b", "10ef4ca8e625bc07aa4bd016fcbf2da1": " N = M ^ {-3/4} ", "10ef67aa4e722f7f7737a215071ee78b": "z\\ = 40 h", "10ef829300f0fd3d9b2ce1d0c0b24f4b": "\\frac{dV_{TBG}}{dt} = -a_T - Q V_{TBG}", "10efb574a368e9e98e089e1815b76540": "1/T\\hbar", "10f04ae626a0ffe3fec668042b67e1b4": "f'(a) = \\lim_{x\\rarr a} x^{n-1}+ax^{n-2}+ \\cdots +a^{n-2}x+a^{n-1} = a^{n-1}+a^{n-1}+ \\cdots +a^{n-1}+a^{n-1} = n\\cdot a^{n-1} ", "10f08d18ebeb32adb63d538d1ff2f03c": "\\xi_{\\times\\times}", "10f0e326e0ea81f311b251b63492b73e": "\\scriptstyle \\sum_p \\frac{1}{p - 1}", "10f12012bf1bee3166b41b57fcaf851d": " m \\sin(\\gamma - \\mu) = c \\cos(\\alpha - \\beta).\\,", "10f14d8a62e4d162c5f39490b2185f30": "\\frac{\\Sigma r Q^{n}}{\\Sigma n r Q^{n - 1}}", "10f16686b167f08f08e08c2b2bae9a42": " \\Delta\\ Unemployment= \\beta_0 + \\beta_1\\text{Growth} + \\varepsilon. ", "10f197ef50b02116326e269d0121ce57": "E(1)=E_{1,0}+d {\\epsilon} \\frac {}{} ", "10f1e8c75f65c494e8c55e30220ef4be": "\\mathit{H(p,p_n) = x K}", "10f234ad120164995786cbc8bd15bc31": "\\begin{bmatrix}\n 1&0&1\\\\\n 1&0&1\\\\\n 0&1&1\\\\\n 1&1&0\n\\end{bmatrix}", "10f25ef036400dccd7e57880db01f885": "\\hat{I}I_{1,1}", "10f263894eab24f8c1ab00da38d8cefa": "{d}h = 0=c_p{d}T", "10f2a4239294639dc2d944cc804251ec": "L(G)=MG^{1-D}", "10f2a748a6d38f4f1eb236e3e17fa58b": "f\\circ\\gamma", "10f3483117771f17427d8ebe8fd3aa66": "\\bar{g}_{Na^+}", "10f34d4bc9dbc777a96366b361b60a02": "g_t = \\left [ \\left ( 1 - \\frac{1}{2N} \\right ) g_{t-1} + \\frac{1}{2N} \\right ] \\left [ 1 - 2dF_{t-1} \\right ]", "10f3ddf70d57c1f451f2d67be2ed4154": " \\mathcal S(\\gamma) = \\frac{(b-a)\\lambda^2}{2} = \\frac{\\ell(\\gamma)^2}{2(b-a)}. ", "10f3ea90280bef706d0123792f3c8170": "\\begin{align}\\operatorname{arcosh}\\, x & = \\ln 2x - \\left( \\left( \\frac {1} {2} \\right) \\frac {x^{-2}} {2} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {x^{-4}} {4} + \\left( \\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} \\right) \\frac {x^{-6}} {6} +\\cdots \\right) \\\\\n & = \\ln 2x - \\sum_{n=1}^\\infty \\left( \\frac {(2n)!} {2^{2n}(n!)^2} \\right) \\frac {x^{-2n}} {(2n)} , \\qquad x > 1 \\end{align} ", "10f3f9c364a3d3ed7a7b6fba42b5e125": "\\begin{Vmatrix} x & y \\\\ z & v\n\\end{Vmatrix}", "10f40f397dca533b29193d407258d0da": "e^{2\\pi i/7}", "10f46008ea75bd8724e8b496799d7560": "p_{k,i}^C<\\frac{1}{2}+\\frac{1}{Q(k)}", "10f46706ddd9ad8305035939ad897d92": " d = 180", "10f495b4e951124d0018fe8d90a56ba0": " h^0(X,L)=1", "10f4a6c0fbf00aaaca912101c7c36c58": "P_{FNL} \\approx \\mathrm{exp} \\left( -{B_g}^2 \\tau_{th} \\right)", "10f4b44378472a2a883fad0f67713fbc": " n_{t+1}(x) = \\int_{\\Omega} k(x, y)\\, f(n_t(y))\\, dy,", "10f4c37af9bdc057940bbc18b032c08d": " \\langle\\sigma\\rangle ", "10f4fb3c812f816984bb8e6ad192a818": "\\mathbf{}V(t)", "10f6053ad1dd29614967d5da8ecb36c2": "X[Y/Z]", "10f625ee8d71697fee8df97b3f8dc527": "\\sum_{n=1}^\\infty \\frac{\\sigma_a(n)\\sigma_b(n)}{n^s} = \\frac{\\zeta(s) \\zeta(s-a) \\zeta(s-b) \\zeta(s-a-b)}{\\zeta(2s-a-b)}.", "10f6369053f87f9504e22666522480df": "T_i^2 - (x^2-1) U_{i-1}^2 = 1. \\, ", "10f685669f14a794377659dccc9ed427": "\\textstyle x \\in X", "10f6b407daf02fbdc1b33b14a8fa7751": "\\mathbf{\\hat p}|k\\rangle=k|k\\rangle", "10f6bf04f81a6a4db7ad206d6c8dda0a": "[x_1 : x_2]", "10f6fbd31545a0a9a19989d2bcbad7eb": "{a_1}^2+{a_2}^2+{a_3}^2 = 1", "10f71c9b2dbcab8c31fe6f4d4b98758f": "\\operatorname{ord}(\\phi\\vert_K) = [O_K/\\delta O_K : \\mathbf Z/q\\mathbf Z].", "10f76616946b0a4e9e33f1a83c4c0955": "\n\\begin{align}\n\\Delta & {} = \\frac{\\sqrt{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}}{4} \\\\\n& {} = \\frac{1}{4} \\sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}.\n\\end{align}\n", "10f7721e867ed84fdd9532ad22f63f53": "\\int_0^\\infty( \\frac {1}{e^x-1}-\\frac{e^{-x}}{x})dx=\\gamma", "10f78d77368b3170ffbac9c54786d63c": "\\mathcal{M} \\cong \\operatorname{PSL}(2,\\mathbf{C})", "10f7cf89be8ff6206bbca376a30e7930": " (\\omega_1,0,\\omega_0 + \\omega_r/2) ", "10f7db2a7ae08b93d9eefec15cd7a76a": "\\Phi_1^{-1}\\circ\\Phi_2 = \\mathrm{Hom}(\\varphi,-)", "10f7db55a055f3b0bc669d2c785cf132": "\\frac{y - Y_0}{Y_1 - Y_0} \\approx \\frac{x - X_0}{X_1 - X_0}", "10f7fec54220538ced34a65a1a2d28f6": " \\vec \\Omega = \\frac {\\vec V_r \\times \\vec R}{ \\vec R \\cdot \\vec R } ", "10f80fcece2bd8202d21042a418518ed": "D[||\\partial_j]=\\partial'_jD(p||p)=0", "10f8108e609759e20476df797b027818": "\\mathcal{L}_X Y := \\left.\\frac12\\frac{\\mathrm{d}^2}{\\mathrm{dt}^2}\\right|_{t=0} \\Phi^Y_{-t} \\circ \\Phi^X_{-t} \\circ \\Phi^Y_{t} \\circ \\Phi^X_{t} = \\left.\\frac{\\mathrm{d}}{\\mathrm{d} t}\\right|_{t=0} \\Phi^Y_{-\\sqrt{t}} \\circ \\Phi^X_{-\\sqrt{t}} \\circ \\Phi^Y_{\\sqrt{t}} \\circ \\Phi^X_{\\sqrt{t}}\\,", "10f84c85c8bc5c78a60b666fcd4d6fd9": " H_q(x)\\equiv_{def} -x\\cdot\\log_q{x \\over {q-1}}-(1-x)\\cdot\\log_q{(1-x)} ", "10f871805364e3b2edcbeee007c07ede": " \\left(\\frac{N_1}{N_2}\\right)_T", "10f8b56ea13c46a90c4180bb43498294": "\n\\Gamma_{\\varphi} \\ = \\ \\int d{q} \\int \\frac{d\\omega}{2\\pi} \\,\\tilde{S}({q},\\omega) \\, \\tilde{P}(-{q},-\\omega)\n", "10f8cdb2300466e9a4344fb2acd86552": "R(p_b) = \\frac{C}{1-H_2(p_b)} .", "10f8fde9f843ff8a9984b509036c49f1": " P, P^* ", "10f9132e586392a6513eeeb303db630c": "yi", "10f92f7e05286641e8a4baec0d3da30f": " \\pi/2", "10f937112d89ed784552c7af9ac763ec": "[x,y] = xy-yx", "10f95fd0d3e47c5973c7f562260ad572": " y[n] = \\sum_{m=-\\infty}^{\\infty} { h[n,n-m] x[n-m] } ", "10fa0278ab54eb34767f6cffab9259b3": "10 - 3 + 2 \\,", "10fa5ad9536941ac7b45268510f8b815": "\\theta_s = \\frac{L_s}{2 R_c} = L_s ^2", "10facdc361b9034de46cfc789cf18295": "\\mu _i = \\mu_i^{\\star} + RT\\ln x_i\\,", "10faea16cffd1370a2f40d8bb96f6316": "m=(1+\\epsilon)(n+1) \\log{q}", "10fb109c7f10edcbd87bf4b6d1ca7a53": "T^{\\mu \\dots}_{\\nu \\dots} \\,", "10fb2c81feb695801f7f550c18f6361b": "\\begin{bmatrix} \\alpha-1 \\\\ -\\beta \\end{bmatrix} ", "10fb2cabe67272af59733c4b404d9a90": "d(i) = \\sum \\limits_j w_{ij}", "10fb63698de8e6f0369f75eaebee667f": "e \\cdot B^{-1}", "10fc34ae7d2d44980d9a409eb1938ca6": "A_0(x) = 0", "10fc4bfdd36f48519636f1529f6abc9e": "b(x)^m", "10fc95efe0bf9dd932c53c256d5c3ad6": "\\frac{V}{l} = \\frac{I}{a}\\rho \\qquad \\text{or} \\qquad V = I \\rho \\frac{l}{a}.", "10fca3f454e370e68e5d4ef5bf3ddf62": " y'(t) = f(t,y(t)), \\quad y(t_0)=y_0, ", "10fcaa7bf3452453fd00b97960bf9c1a": "\\begin{align}\nx&=r\\cos\\phi \\\\\ny&=r\\sin\\phi \\\\\nz&=z\n\\end{align}", "10fcf43d858c03dafc054d102f147ce6": "\\bigcap_{i\\in I} C_i \\neq \\emptyset ", "10fd22543d21500e6751e6346c984f8c": "G_i = 1/R_i \\,", "10fd6a3702e89c5c360d99db2c7a4805": "\\mathbf{a}_{AB} = \\frac{d^2}{dt^2} \\mathbf{X}_{AB} ", "10fd9c5893e5e8cb5137151faeb8f3ec": "Ld = 1\\, Laborer \\cdot 0.5\\,\\frac{Ph}{Laborer} = 0.5\\,Ph", "10fdaf2ab79128a041a6403212d1c8f0": "Prob_{pure}=e^{-2G}", "10fe11584f7d5f67aa0aecca0ca95731": "{\\Delta}P", "10fe9a7e35b02412ce6ab517840ff4df": " V_y = v \\sin \\theta - \\frac{gx}{v \\cos \\theta} ", "10fea0b6eda684767263baa4da13daad": "\\nabla^2 \\mathbf{E} - \\mu_0 \\epsilon_0 \\frac{\\partial^2 \\mathbf{E}}{\\partial t^2} = 0.", "10fef7f1ece06914d356929bb2f02d91": " \\vec{F}_{12} = \\frac {\\mu_0 I_1 I_2} {4 \\pi}\\frac{2}{D}(0,-1,0) \\int_{L_1} dx_1 ", "10ff1a9e235307c647eb79d9503321da": " c = kp \\,", "10ff1b27d7648f47afcbfe2927d442ff": "L^p(\\mathbb {R})", "10ff2ff0656919d5fd52ba752f034ca5": "\\scriptstyle C_c^1(\\Omega,\\mathbb{R}^n)", "10ff3921d987215a0a44f012c768dc08": "\\langle \\partial_{t} u , e^{i k x} \\rangle = \\langle \\frac{1}{2} u^2 - \\rho \\partial_{x} u , \\partial_x e^{i k x} \\rangle + \\langle f, e^{i k x} \\rangle \\quad \\forall k\\in \\left\\{ -N/2,\\dots,N/2-1 \\right\\}, \\forall t>0.", "10ff3951f3c68b7ef1da98837f2707cb": "k_2 = f(t_0 + \\tfrac23h ,y_0 + \\tfrac23hk_1) = 2.7139", "10ff4194bb5fbd72798c3a7b057fea0d": "C (\\vec{N})", "10ffd8f26b66effcadf3af92fc9ad000": "M_{B_1}^i ,\\dots, M_{B_k}^i", "10ffe7009b277632c4f8045875566bb4": "c_f \\approx", "110022d17790fed7bc43781dfd1b7eb9": "x[0]=\\lim_{z\\to \\infty}X(z).", "110035449b34621bf2da984cd87cffcf": "a^nx + \\frac{a^n - 1}{a - 1}b", "1100361a91ac2da2bd6a80d0d1ca1636": "\\Omega^1 \\to \\Omega^2", "110052477e50e69862dce956ef5e1f89": "(n_x,n_y,n_z)=(3,5,7)", "11008e2549109c23f79fe494bd2b6ed6": "m_\\text{P} = c^{n_1} G^{n_2} \\hbar^{n_3},", "11009eff1b06cea1b8d716c777254188": " n \\, ", "1100b76bf84f16689dfc7deb6533146e": " \\delta < \\alpha", "1100d11beff5e2b62c7f2c8142232f4a": "\\mathrm{S}\\!\\!\\!\\Vert", "11010df7e53eac5d8da576b6a0ddfbf9": " LR- = \\frac{\\Pr({T-}|D+)}{\\Pr({T-}|D-)} ", "11013a75061a55751f28de2c74ade4d5": "p\\times1", "110145a930d21967d1147147ea289e66": "\\Rightarrow q_2^* = \\frac{a - 3 \\cdot \\frac{\\partial C_2 (q_2)}{\\partial q_2}+ 2 \\cdot \\frac{\\partial C_1 (q_1)}{\\partial q_1}}{4b}.", "110198bffc5514ac79c683ca681a7600": "f' \\circ f = g' \\circ g \\, ", "1101c99afd8028f653cb43f10f4f3ce6": "p_4=m_1q_3(1+m_2)\\ .", "1101ca65bea66aef754d5bd34892e220": " h(w) =\\sum_{n\\ge 1} {\\lambda_n\\over n} \\Phi_n(w) + \\sum_{n\\ge 1} {\\lambda_{-n}\\over n} \\Phi_{-n}({a\\over w}).", "1101f7f3d2b4039dea68701f38ea9ed7": "\\operatorname{etr}(A) := \\exp(\\operatorname{tr}(A))", "11020b4f21edd4f09a0ead06b6d63627": " \\Psi = e^{i(kx-\\omega t)} \\,\\!", "1102b844e752112397f05d00e91324a1": "A^\\alpha \\vec{u}=\\sum_{k=1}^\\infty \\lambda_k^{\\alpha} u_k\\vec{w_k}", "11030e148c36253940edfc08ede968d1": "\nh_{\\mu} = \\frac{1}{2} \\sqrt{\\frac{\\left( \\nu - \\mu \\right) \\left( \\lambda - \\mu \\right)}{ \\left( A - \\mu \\right) \\left( B - \\mu \\right)}}\n", "1103137ca878e48a577d7b65f17013ab": "9k \\equiv - 1 \\pmod{7}", "1104572e09c96d0ca09b566665e394ee": "\\{x\\leq10\\}\\; \\mathbf{while}\\ (x<10)\\ x := x+1 \\;\\{\\lnot(x<10) \\land x\\leq10\\}", "1104d00b81ae3af412f3d583183c1379": "2 \\cdot A_n \\to A_n.", "1105080e231ecf26cd884aea3ff8b624": "S_k = \\sum_{n=1}^k (-1)^{n-1} a_n\\!", "110555287c78c0f3e148298cda41d666": "\\Lambda_{MS} = 217^{+25}_{-23}{\\rm\\ MeV}.", "11055c7adb9c8a326c3469790173af18": "\\alpha = \\frac\\mu a", "110574a5bb89681c6bc07de2eb2af8e6": "\n\\mathcal{N} \\int e^{-\\beta H(p, q)} x_{k} \\frac{\\partial H}{\\partial x_{k}} \\,d\\Gamma = \n\\Bigl\\langle x_{k} \\frac{\\partial H}{\\partial x_{k}} \\Bigr\\rangle = \\frac{1}{\\beta} = k_{B} T.\n", "11057a773dc90dff2e5c9cd86fe68402": "R(\\theta,\\delta)=\\operatorname{E}_{F(x\\mid\\theta)}[{L(\\theta,\\delta(x))]}.\\,\\!", "110585049926feaaa6e156f2e0f2a5e5": "DP_{S}^{T}", "1105a7158f1cba5299aa791c50832c34": "u_i,~ i=1,2,3", "1105b792fa5f2e515c85aa2ea0e94418": "(-1)^m \\psi^{(m)} (1-z) - \\psi^{(m)} (z) = \\pi \\frac{d^m}{d z^m} \\cot{(\\pi z)} \n= \\pi^{m+1} \\frac{P_m(\\cos(\\pi z))}{\\sin^{m+1}(\\pi z)}\n", "1105dc5e3c5a368c1afd2c9721017a8c": "Q_{1}", "1105df8f8e65e5bbf15b59e5c08504bf": "\\displaystyle \\overline{\\hat{f}(-\\nu)}", "1105e3f63832168d478e6f8ddb440f12": "\\log c", "110629f545ec2d54c06a639d80ec39e1": " \\int f \\, \\mathrm{d}\\mu ", "11063aba125c8de2441aee5feed84ffe": "a_i = \\gamma_{xi} x_i \\,", "110657a47e7a40ff5221e5bca56ceccd": "2.00356", "11069c05d732517bd08a43546fc02a73": "\n\\approx\\frac{1}{2}\n\\begin{bmatrix}\n1\\\\\n\\delta\\mathbf{x}\\\\\n\\delta\\mathbf{u}\n\\end{bmatrix}^\\mathsf{T}\n\\begin{bmatrix}\n0 & Q_\\mathbf{x}^\\mathsf{T} & Q_\\mathbf{u}^\\mathsf{T}\\\\\nQ_\\mathbf{x} & Q_{\\mathbf{x}\\mathbf{x}} & Q_{\\mathbf{x}\\mathbf{u}}\\\\\nQ_\\mathbf{u} & Q_{\\mathbf{u}\\mathbf{x}} & Q_{\\mathbf{u}\\mathbf{u}}\n\\end{bmatrix}\n\\begin{bmatrix}\n1\\\\\n\\delta\\mathbf{x}\\\\\n\\delta\\mathbf{u}\n\\end{bmatrix}\n", "1106c567f4722709c525ecb870fcf51c": "f(x) = \\frac{1}{(x - 1)(x^2 + x + 1)}", "1106f844677df0c805a8cc002fcb1470": "\\dot{\\varepsilon}", "11073c2cfeb60b6c04ec95959ab3ca15": "\\hat{\\alpha}, \\hat{\\beta}", "11076c1c9b6b49e95f5f9e377dc42278": "t_{nm} = \\left[\\mathbf{T}\\right]_{nm}", "11076fafbd8872769c4a44689ae803d0": "\\Delta\\epsilon", "1107be8d11e3f0f354bd0ad697f7562c": "\\alpha = \\frac{2ax + b \\pm \\sqrt{(2ax + b)^2 - 16}}{4}", "1107f25512e907df730bdb847d8c75d7": "a_k=2 \\langle\\phi(x),\\phi(2x-k)\\rangle", "1108517637315bc0360bca1416cfc517": "u\\sim w_0\\,", "1108c2522505a49ba4219e2fbe6a5ca9": " \\cos a \\cos b ", "110943d95fc5ddcd8a009f6a9b63486b": " |D|>0", "11096792ca8dde823d044cc948f15534": "\\operatorname{Im}", "11096ba55e57b0ba1b35efb241f87569": "\\epsilon \\ne 0", "110972b39a69695d1bd6b63fcff9b8c9": "\\mathit{F} = \\frac{a^{2}}{L \\lambda}", "11097ba45d6b2fc241b6ffee9e3f07b4": "\\mu=a_1+2a_2", "11098b06bc5ce941f1283ed5e081ea17": "dW/dt=0", "1109a243eb6e39d5536973d96353cbb4": "\\cdot e^{-\\frac{\\nu^2}{2 a^2}} H_n\\left(\\frac \\nu a \\right)", "1109e2f82871c8b4af63678e4fae1a52": " \\mathbf{x}_{k} \\propto \\mathbf{A} \\, \\mathbf{y}_{k} ", "1109f06297279a41dedd7973cfe667a0": "-D_1\\frac{\\partial C_1}{\\partial x}", "110a38a973ecc3a6d5d642e75c3a6671": "x=x_i", "110a3ea61e602565f8caf38c0b2c2c04": " L_1 \\times L_2 \\equiv L_2 \\times L_3 \\equiv L_3 \\times L_1. ", "110a451bff14cf0501f2ebb7c418a581": "u_{0}", "110a8755d8163407ee9c371792d932ef": "C = \\frac {u_ x\\,\\Delta t}{\\Delta x} + \\frac {u_ y\\,\\Delta t}{\\Delta y} \\leq C_{max} ", "110aca91bcd7ff0e9c4a27cfc5a08ea4": "\\forall x \\forall y \\forall s [(\\langle x, s \\rangle \\in F \\and \\langle y, s \\rangle \\in F) \\rightarrow x = y])].", "110b013cca576480b40669e4d960abd5": "\nH(A:B) =H(A) + H(B) - H(A, B)\\,\n", "110b3a66217e97116fbd198a0cb1ba8e": "L(s,\\chi) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}", "110b9257f9d7d2d395edec28d272930a": "L_n^{(\\alpha)}", "110b9df0a7948a3528fe05ba12a861f5": "\\operatorname dL_{\\text{r}}(\\omega_{\\text{r}})", "110ba441223e562aa02c7c32f54f12eb": "1+r= \\frac{M_1 -C_1}{M_0}\\times\\frac{M_2 -C_2}{M_1}\\times\\frac{M_3 -C_3}{M_2}\\times...\\times\\frac{M_{n-1} -C_{n-1}}{M_{n-2}}\\times\\frac{M_{n} -C_{n}}{M_{n-1}}", "110bda9da3f5e510b3e317f0585d36f7": "y_1 = \\frac{2y_g}{-1+\\sqrt{1+\\frac{8gy_g^3}{q^2}}}=\\frac{(2)(0.30)}{-1+\\sqrt{1+\\frac{8(9.8)(0.30)^3}{3.70^2}}} = 8.04\\text{ m}", "110c2cf45ddb788fbb479bd105c78cd8": "\\delta\\rho-\\bar{\\delta}\\sigma=\\rho(\\bar{\\alpha}+\\beta)-\\sigma(3\\alpha-\\bar{\\beta})+(\\rho-\\bar{\\rho})\\tau+(\\mu-\\bar{\\mu})\\kappa-\\Psi_1+\\Phi_{01}\\,,", "110c8597512e49796437092e71b6c94d": "(1)\\; F r_1=\\frac{v}{\\sqrt{gy}}=\\frac{10[m/s]}{\\sqrt{9.81[m/s^2]*0.5[m]}}=4.5", "110ccacdf41482aa6a44192990e56bcf": "\\nabla h_i (i\\in\\mathcal{E})", "110d1ec129ecf4c227b843be04fdfc29": "\\Delta_* \\colon H_\\bullet(X) \\to H_\\bullet(X \\times X)", "110d95a683d386b31d93433baf49db52": "p(x)=p(x|\\theta)", "110d97436114572b3b0ef30429e4fe8b": "(x^n)'=nx^{n-1},", "110da4245831429b2f43162353ffafa8": "\n H_n(a, b) = \n \\begin{cases}\n b + 1 & \\text{if } n = 0 \\\\\n a &\\text{if } n = 1, b = 0 \\\\\n 0 &\\text{if } n = 2, b = 0 \\\\\n 1 &\\text{if } n \\ge 3, b = 0 \\\\\n H_{n-1}(a, H_n(a, b - 1)) & \\text{otherwise}\n \\end{cases}\\,\\!\n", "110dc2975f1834c40361303b53b676a5": "\\mu(x):\\mathbb{R}{\\rightarrow}\\mathbb{R}", "110e2bf8ead8b8997c7a14d025ce30a4": " I_\\mathrm{max} ", "110e3416e03f22194e5e9234d4a1b327": "\nN_{xx} = \\int_A \\sigma_{xx}~ \\mathrm{d}A ~;~~ M_{xx} = \\int_A z\\sigma_{xx}~ \\mathrm{d}A \n", "110ebbbe7ffaf067a5dd5dbedb369ac6": "\\operatorname{E}[g(X)] = \\int_{-\\infty}^\\infty g(x) f(x)\\, \\mathrm{d}x .", "110eee294603f883d5c15c8179d8baa0": "\\omega_0 = { 1 \\over \\sqrt{LC} } ", "110f0e7b0458f8b860f8d36a2776c659": "\\textit{ADJ} \\; \\textit{NOUNPHRASE}", "110faf0a710447a4d467cc98d586a0ce": " \\hbar \\omega_n = R \\left(\\frac{1}{2^2} - \\frac{1}{n^2}\\right) \\quad n=3,4,5,...", "110fca3e22a1dbfe1abb234ca3259ea5": "I_{L_{\\text{max}}}=\\frac{V_i\\, D\\, T}{L}", "110fd17ed759bb2d02a8e1a1e1ba93da": "\\int \\frac{dx}{\\sqrt{x^2-1}} = \\mbox{arccosh}(x) + C \\quad (x > 1)", "110ffbbb53c2696a1252640beb988e3b": "S(q_{1},\\dots,q_{N},t)= W(q_{1},\\dots,q_{N}) - E\\cdot t", "111019a279c754001bea7f9d5343b55c": "\\frac{1}{r^{2}}", "111024e02e58228a7435e46dcdf4bf96": " r(t)=r_e\\left[1-\\exp\\left(-\\left(\\frac{2\\gamma_{LG}}{r^{12}_e}+\\frac{\\rho g}{9r^{10}_e}\\right)\\frac{24\\lambda V^4 (t+t_0)}{\\pi^2\\eta}\\right)\\right]^\\frac{1}{6}", "1110544cf450124486e6a6c8a315ff70": "\\int_{-a}^a f(z)\\,dz =\\pi e^{-t}-\\int_{\\mathrm{arc}} f(z)\\,dz.", "11107bf7200e8ea8c65157a3ab5eed0e": "\\Gamma\\,", "1110948f63b45eebb689ea45bd20e33f": " n_{t+1} = f'(0) k * n_t ", "1111091518909c264d8fba533dd169da": "\\triangle{M} = M_1 - M_2 = 25.5", "11113478595309bdef22cb31bc94b0dc": "\\psi_{i+2}", "111185f76fc35ac247d23bc2ea139aea": "r = 4s(1-s)", "111196a2755fc7ac7b4fbf9c739c0422": " 0.872 ", "1111f4cf24ae298dc7a326a55c09c108": " \\ln(I_z) = - \\sigma N z + C . \\,", "1111fd11a6fa9dcb7af73680df44de80": "\\textit{dau}(m,h) \\leftarrow \\textit{par}(h,m) \\land \\textit{par}(h,t) \\land \\textit{par}(g,m) \\land \\textit{par}(t,e) \\land \\textit{par}(n,e) \\land \\textit{fem}(h) \\land \\textit{fem}(m) \\land \\textit{fem}(n) \\land \\textit{fem}(e)", "1112838385b37079efd51e340b3cbd75": "\\mathbb{C}/L", "111310acf2892086d3e75dd8c5523628": "p({\\boldsymbol{x}|\\rm label})", "11134ff62a3534476fc6cc89687d67bf": "a_t", "111355b8fd3c56f629ebb75905df6888": "\\scriptstyle \\delta x", "11136d975f9e0c5a36bbe8c6a6c30c65": " a\\neq d", "11138f63622d9aacf6ddb06b9e57b138": "\\rho = \\frac{p}{R_{\\rm specific} T} ", "1113ddb156abcd31f04d54feb03cda9a": "\\mathrm{gyr}[\\mathbf{u},\\mathbf{v}]\\mathbf{w}=\\ominus(\\mathbf{u} \\oplus \\mathbf{v}) \\oplus (\\mathbf{u} \\oplus (\\mathbf{v} \\oplus \\mathbf{w}))", "111404248a0360274a22146234c585b4": "s_{(a^p)}s_{(b^q)}", "1114799c925afee9bbda93fe79a96633": "P(a_{T+1}|\\hat{a}_{1:T}, o_{1:T}) = \\int_{\\Theta} P(a_{T+1}|\\theta, \\hat{a}_{1:T}, o_{1:T}) P(\\theta|\\hat{a}_{1:T}, o_{1:T}) \\, d\\theta", "11147b59ea9e585fca5b2314d6ecbc4a": " a y_1 v'' = 0. \\;", "1114c80010a66a42faea3d20cfa4cbd2": "|S|=n", "1114d50f217bc0281228d0c670b26336": "\\delta_k", "1114da8d3d28a20e353ebff4ec0da999": " f = -k_BT N^{-1}\\log Z.", "1114f50a824355c5a96324d9cc456dd1": "\\int\\!\\!\\frac{x-a}{(x - a_1) (x - a_2)} \\,\\mathrm{d}x = \\frac{(a_1-a)\\ln(x-a_1) - (a_2-a)\\ln(x-a_2)}{a_1-a_2} + C", "11155ed053e766272d7cbd379321ebfb": "\\frac{1}{2} m v^2", "11156ef4182af303a38779b329afb012": "\\hat f(5)", "111571612bcd52aea3c64a4121c65d3b": " 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (5 \\ ro)", "111585dd48e959cb11225784a88a06b4": "\\Omega_{p} : \\sigma \\to 2\\pi - \\sigma", "1115cc55da57b8046f88d722e9c8b8fd": "\\omega |_\\xi>0", "1115cc7289d83d9234925bdcfad2b88d": "(x^{i_1},\\sigma^{j_1})(x^{i_2},\\sigma^{j_2}) = (x^{i_1+i_22^{^{j_1}}},\\sigma^{j_1+j_2})", "111652c367c6eff15bf4836e56ecf561": "\\Pr(5\\text{ heads}) = f(5) = \\Pr(X = 5) = {6\\choose 5}0.3^5 (1-0.3)^{6-5} \\approx 0.0102 ", "1116bc7b60a1156225cbcf51ea0614f0": "\\scriptstyle x \\;=\\; x_0 \\,+\\, h, \\; y \\;=\\; 0,\\; \\; z \\;=\\; 0", "1116d784637b4b4292dd2649b22e6e6b": "x_i = \\cos\\left(\\frac{2i-1}{2n}\\pi\\right) \\mbox{ , } i=1,\\ldots,n.", "1116e1424b39c12d01de442e53799d8b": "x + y = y + x", "1116f44a804480205e0fde026b5afa4b": " V_{\\left(p-k\\right)}^{T}\\boldsymbol{\\beta}_* = \\mathbf{0} ", "11173d648d53d06aca0b1e4024515e2f": " \\frac{1}{f}=\\frac{1}{d_o}+\\frac{1}{d_i}", "11176deba9a0cdb756a12de6db30bd63": " \\begin{cases} \\frac{dx}{dt} = -y - z \\\\ \\frac{dy}{dt} = x + ay \\\\ \\frac{dz}{dt} = b + z(x-c) \\end{cases} ", "11178357bc66a35266c727b2f657438c": "\n \\sigma = \\int_{r_\\mathrm{obs}}^{r_\\mathrm{atm}} \\frac {\\rho\\, \\mathrm d r}\n {\\sqrt { 1 - \\left ( \\frac {n_\\mathrm{obs}} {1 + ( n_\\mathrm{obs} - 1 ) \\rho/\\rho_\\mathrm{obs}} \\right )^2 \\left ( \\frac {r_\\mathrm{obs}} r \\right )^2 \\sin^2 z}} \\,.\n", "1117bf4e60f49773ad7e18289e3bb115": "A \\oplus B = \\bigcup_{b\\in B} A_b", "1117cdb74845ff280dceb1c070770d44": "p_2-p_1\\,", "1118433b9ccb65b7c14e48985612938b": " \\int_0^\\infty \\frac{\\gamma x+\\log\\Gamma(1+x)}{x^{5/2}} \\, dx =\\frac{2\\pi}{3} \\zeta\\left( \\frac{3}{2} \\right) ", "11189aa6b1d37bd8d4e8fe3c79653f03": "a \\frac{x_c}{t_c^2} \\frac{ d^2 \\chi}{d \\tau^2} + b \\frac{x_c}{t_c} \\frac{d \\chi}{d \\tau} + c x_c \\chi = A f(\\tau t_c) = A F(\\tau) .", "1118a7c0f92998eae45123971960361a": "~\\omega_{\\rm s}~", "1119725f6a8281d25ce18c40a995f5eb": "Y_t = a \\cdot t + b + e_t", "1119c8d1911a1511da2c8d3c85c760cd": "f(\\partial X)", "1119e1a2b8d600202c198255528f1946": "B_\\delta", "111a2979f2509c2d09f082f039efbb12": "=\\frac{V_{nk_4}V_{k_4k_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}E_{nk_4}}-E_{n}^{(2)}\\frac{|V_{nk_4}|^2}{E_{nk_4}^2}-2V_{nn}\\frac{V_{nk_4}V_{k_4k_3}V_{k_3n}}{E_{nk_3}^2E_{nk_4}}+V_{nn}^2\\frac{|V_{nk_4}|^2}{E_{nk_4}^3}", "111a538025ee0698a7a26c922b5b9a86": "\\frac{\\partial h}{\\partial t} + C \\frac{\\partial h}{\\partial x} = D \\frac{\\partial^2 h}{\\partial x^2},", "111a63e3fd193e80d2cfd33b187bbc12": "L=F/\\sim", "111b3cda00313d80e2967f6e045ad149": " r_O ", "111bb025692e5ba52c23eae1d47b2035": "D=1", "111be2a57cccb50dece2c4598eca667b": "[T] = {}^{0}T_n = \\prod_{i=1}^n {}^{i - 1}T_i(\\theta_i),", "111be2ac0bacdca11c8d0be7c6beae80": "\\frac{\\left( \\cfrac{q^2}{4 \\pi \\epsilon_0 L_1^2} \\right)}{\\left(\\cfrac{q^2/4}{4 \\pi \\epsilon_0 L_2^2}\\right)}=\n\\frac{mg \\tan \\theta_1}{mg \\tan \\theta_2}\n\\Longrightarrow 4 {\\left ( \\frac {L_2}{L_1} \\right ) }^2= \n\\frac{ \\tan \\theta_1}{ \\tan \\theta_2}", "111c0693607c0ed88a3d6936a4434532": "j_z = m_j \\, \\hbar", "111c77400d4672c1936276d2647df165": "m(x)\\leq\\frac{f(x)-f(y)}{g(x)-g(y)}=\\frac{\\frac{f(x)}{g(x)}-\\frac{f(y)}{g(x)}}{1-\\frac{g(y)}{g(x)}}\\leq M(x)", "111cb170603f82a5ee2f584bd4eb838c": "\\dot{e}_1 = 0", "111d10ccaa2716291b402bea0680c23d": "\\Psi_2 := C_{\\alpha\\beta\\gamma\\delta} l^\\alpha m^\\beta \\bar{m}^\\gamma n^\\delta\\ , ", "111d25fdd402a4b1eba732d027332bce": " - log_{10} ( \\frac{ 1 + 2 d }{ 365 } )", "111d4300d81fac14bb67ded72ce8222a": "\\frac{\\xi}{\\xi_0}=\\frac{\\ln \\tfrac{t}{t_1}}{\\ln \\tfrac{t_0}{t_1}}.", "111d835b36d8312e993aa099aa72e62a": "(P \\leftrightarrow Q) \\leftrightarrow (Q \\leftrightarrow P)", "111d8940d1b32b4e525396ac09f9bfb5": "(V, E, A-a)", "111deeccc6453c79fbe0fcfce4e14a8f": "\\eta^{\\mu \\rho}=0", "111e07ca18e0b69d9765d5693c244e7b": "f(x +uR) = (x + u_1R, \\ldots, x + u_kR) \\quad\\mbox{ for every } x \\in R", "111e454a716b3a61f7ef4de9f14f1ba9": "J(\\mathbf{x})", "111e5751a8d46d382301a04a61f97ebe": "\\Delta E \\Delta t \\ge \\frac{\\hbar}{2} \\ , ", "111e78d1ab40407546acc0c3f2c455e7": "\\sqrt{ax^2+bx+c}", "111ea26ef5fdbd87adee0b2545a2c421": " f : X\\to Y", "111ecf951fd47af074e7209524576f85": "\\ Z = |Z| e^{j\\arg (Z)}", "111ef680298282cb7ed299f28678acd8": "x \\in \\Sigma", "111f9a7b168df5ddf62b26391cfc00c6": "\\langle\\psi|\\mathcal{T}\\{F \\phi^j\\}|\\psi\\rangle=\\langle\\psi|\\mathcal{T}\\{iF_{,i}D^{ij}-FS_{int,i}D^{ij}\\}|\\psi\\rangle.", "111fb75d0fc2d07f03ffec898598a279": "\\langle a_0,a_1,a_2,\\ldots\\rangle\\in A", "111ff7857fbb1e8261592993d61d2e47": "= \\frac12 + \\frac{\\sin(k_1 x) + \\sin(k_2 x)}{4}", "112023b22122f83cef644aabb93fd324": "L = 20\\ \\log_{10}\\left(\\frac{4\\cdot \\pi \\cdot d}{\\lambda}\\right) ", "11203164535cb8a09eb060515de2282e": "\\sqrt{\\theta_1^2 + \\theta_2^2}", "11203f393143bd6971260789730f9ed5": "\\frac{-a_0}{a_1}", "11205144e60361dced45e00c2c7d553e": "Q = \\begin{pmatrix} {-(x_1+x_2+x_3)} & x_1 & x_2 & x_3 \\\\ {\\pi_1 x_1 \\over \\pi_2} & {-({\\pi_1 x_1 \\over \\pi_2} + x_4 + x_5)} & x_4 & x_5 \\\\ {\\pi_1 x_2 \\over \\pi_3} & {\\pi_2 x_4 \\over \\pi_3} & {-({\\pi_1 x_2 \\over \\pi_3} + {\\pi_2 x_4 \\over \\pi_3} + x_6)} & x_6 \\\\ {\\pi_1 x_3 \\over \\pi_4} & {\\pi_2 x_5 \\over \\pi_4} & {\\pi_3 x_6 \\over \\pi_4} & {-({\\pi_1 x_3 \\over \\pi_4} + {\\pi_2 x_5 \\over \\pi_4} + {\\pi_3 x_6 \\over \\pi_4})} \\end{pmatrix} ", "112096cb5347f695c5271d5d23d3b4bd": "Q_n(x;a,b,N)= {}_3F_2(-n,-x,n+a+b+1;a+1,-N+1;1).\\ ", "1120d973c96b8c15fe6d1074cbf1e50a": "\n R_O + R_B - F = 0 \\quad \\text{and} \\quad -\\mathbf{r}_A\\times\\mathbf{R}_O + \\mathbf{r}_B\\times\\mathbf{R}_B = \\mathbf{0} \\,.\n ", "1120e8d3f2657a841811c6b77f21e996": "t = \\frac{x}{v \\cos \\theta}", "112123c25cf63310139f9ee076803a17": "x+1\\,\\!", "112140d8a0c2c74b7242625c70b65e83": "-v_{\\mathrm{rel}}\\frac{\\mathrm{d}m}{\\mathrm{d}t} = m{\\mathrm{d} v \\over \\mathrm{d}t}", "11217ddea497d67192e898572bd42bb0": "(1+\\epsilon)", "112200292440ad6785be80f1b9b955e6": "z = \\sqrt{ x^2 + y^2}", "112251de2fe1fa944cf29785c2b74638": " RC_{t}=\\ln\\left (\\frac{P_{t}}{P_{t-1}}\\right ).", "112301345da08130e42b524c7d8e7152": "\\mathcal{E} = q E + \\mu B + \\frac{1}{2} m v_{\\parallel}^2 + \\frac{1}{2} m v_{\\perp}^2", "1123039eb00b314adb814e533ee330a2": "\\mathrm{FillRad}(X) \\leq C_n \\mathrm{vol}_n{}^{\\tfrac{1}{n}}(X),", "11230d45736bd09b3653c926735adf7a": "\\psi^n = \\psi^{n-1} + \\psi^{n-2}\\, .", "1123194db6f75bf8855ce75e51a2d454": "\\mathbb{E}[(R_r - R_\\min)_-^2]", "1123593f39ece8e15eeacf8058003d77": "h(x,y) = f(x) + g(y)", "1123768fcd42c1b6c4cf1d6f5594e57c": "\\mathbf{i}(\\mathbf{i}x) \\leftrightarrow \\mathbf{i}x.", "11238c4a19646952d7282e5e9954a0f9": "u = v", "1123a1fc5fbf885d0ec54e0a5fcdd92b": "(X,Y,Z)=\\left\\{\\begin{matrix}\n(0,0,0) & \\text{with probability}\\ 1/4, \\\\\n(0,1,1) & \\text{with probability}\\ 1/4, \\\\\n(1,0,1) & \\text{with probability}\\ 1/4, \\\\\n(1,1,0) & \\text{with probability}\\ 1/4.\n\\end{matrix}\\right.", "1123bb63b22a9a4309ad4d49afa8b57e": " f(S_{x,i}, c) ", "1123dc128c4342924f078f79bdc82ef3": "\n \\mathcal{J}(\\hat{T}) := 1 + \\exp\\left[-\\cfrac{1+1/\\zeta}\n {1+\\zeta/(1-\\hat{T})}\\right] \\quad\n \\text{for} \\quad \\hat{T}:=\\frac{T}{T_m}\\in[0,1+\\zeta],\n ", "1123f425a881d74b45ddb30c264f0e1f": "(H_i)", "112405e9295c4b9eaf79b5dcafa7427f": "\\forall m \\, \\forall n \\, (\\varphi(n,m)\\iff\\frac{n}{m} 0 \\}.", "113653cfe1e86a56605ac2f73536f6a6": "\\forall V\\in\\mathcal{T}, f(x)\\in V\\exist U\\in\\mathcal{T}, x\\in U: U\\subseteq f^{-1}(V)", "113660501ce535ef3f7236adc13f80aa": "\\Sigma_0(A) = (A\\!\\ggg\\!2) \\oplus (A\\!\\ggg\\!13) \\oplus (A\\!\\ggg\\!22)", "11366cd4251a48c25e11d34e038655a7": "\\Re.", "1136b03089226940e1d7fb351c702626": "\n\\left [ -{\\hbar^2 \\over 2\\mu} \\left ( \\frac{1}{r} {\\partial^2 \\over \\partial r^2} r- {l(l+1) \\over r^2} \\right ) + V(r) \\right ] R(r) = E R(r)\n", "113721c395c3e9cd302a4ffb75bf0a6b": "K \\geq 2", "11373d017ce2d9410fb2b9e8f385b849": " K_p^{eff} = \\frac {1} {t_c} \\int_0^t D_0 exp \\left(\\frac {-Q} {RT(t)} \\right) dt", "11375ad1dd1e39f1d877d65c107cfd34": "2^{Rb}", "11376cc39053f7daa4e94ce0a7069a68": "\\mu(x,y)", "113780691a78e21f390ac884dd4486e9": "- \\operatorname{Tr} \\rho \\log \\rho.", "1137e2250a6eb176ce798ed682ed2411": "H^{\\rm T} + H= 2I. \\, ", "11387e8f3b98a0ff9fbb0537a2d049f2": "I_n(\\rho)=J_n(i \\rho)", "11389d2ef487b17c4f8a6dff48b55df0": "\\frac{\\partial \\theta}{\\partial t}= \\frac{\\partial}{\\partial z} \\left[ K \\left ( \\frac{\\partial \\psi}{\\partial z} + \\frac{\\partial z}{\\partial z} \\right ) \\right] = \\frac{\\partial}{\\partial z} \\left[ K \\left ( \\frac{\\partial \\psi}{\\partial z} + 1 \\right ) \\right]", "1138b6443d4c7b2d30095d5741eec4a8": "|I| := \\sum_{i=1}^{m} I(i)", "1138fb7fb1eafebb618573abbeddb7d5": "A B A^{-1} B^{-1}", "113923a67b215b851c60fc6a9e7dd56b": " \\frac{F}{\\rho_f U^2 A} = \\frac{12}{ \\mathrm{Re}} \\left( 1 + 0.15\\mathrm{Re}^{0.687} \\right) . ", "113a95b141f7314208c329cc42ab7b52": "\\parallel\\boldsymbol{\\phi}_{\\mathcal{B}}(x) -\\boldsymbol{\\phi}_{\\mathcal{B}}(y)\\parallel_2\\leq\\varepsilon", "113aa98b9c33bfd73ff331d9a59abb53": "BP^{2} = wB^{2} + Az^{2},", "113aece3fbe3acd17f86223218948956": "n = \\left\\lfloor\\frac{\\pi}{4\\theta}\\right\\rfloor\n\\approx \\left\\lfloor\\frac{\\pi}{4 \\sin(\\theta)}\\right\\rfloor\n= \\left\\lfloor\\frac{\\pi}{4} \\sqrt{\\frac{N}{G}}\\right\\rfloor = O(\\sqrt{N}).\n", "113b13f7d253420d7034f48ce979f2b0": "\nQ = \\frac{2\\pi f_o\\,\\mathcal{E}}{P}, \\,\n", "113bb1754905ffd7453260851fe208ba": "M {\\to_G}^* M'", "113bd144dfec035303bc967572eee272": "\n\\begin{align}\n \\varphi\\, =\\, &\n \\left\\{\\, \n \\varphi_b\\,\n -\\, \\frac{1}{2}\\, (z+h)^2\\, \\frac{\\partial^2 \\varphi_b}{\\partial x^2}\\,\n +\\, \\frac{1}{24}\\, (z+h)^4\\, \\frac{\\partial^4 \\varphi_b}{\\partial x^4}\\,\n +\\, \\cdots\\,\n \\right\\}\\, \n \\\\\n & +\\,\n \\left\\{\\,\n (z+h)\\, \\left[ \\frac{\\partial \\varphi}{\\partial z} \\right]_{z=-h}\\, \n -\\, \\frac16\\, (z+h)^3\\, \\frac{\\partial^2}{\\partial x^2} \\left[ \\frac{\\partial \\varphi}{\\partial z} \\right]_{z=-h}\\,\n +\\, \\cdots\\,\n \\right\\}\n \\\\\n =\\, &\n \\left\\{\\, \n \\varphi_b\\,\n -\\, \\frac{1}{2}\\, (z+h)^2\\, \\frac{\\partial^2 \\varphi_b}{\\partial x^2}\\,\n +\\, \\frac{1}{24}\\, (z+h)^4\\, \\frac{\\partial^4 \\varphi_b}{\\partial x^4}\\,\n +\\, \\cdots\\,\n \\right\\}, \n\\end{align}\n", "113be821399e699753e79ea62d6bed80": " \\frac{Q_{hot}}{T_{hot}}=\\frac{Q_{cold}}{T_{cold}}", "113c148ae8eb779638a15d7d8103b115": "\\begin{align}\n Lu & = u'' + k^2 u = f(x)\\\\\n u(0)& = 0, \\quad u\\left(\\tfrac{\\pi}{2k}\\right) = 0.\n \\end{align}", "113c1f9998f3d206f938020b1356cc53": "-j0.23 = \\frac{-j}{\\omega C_2 Z_0}\\,", "113c604e3acd7c5eccffaab50a14562f": "\\sigma:(u,v) \\mapsto (r u, v/r) ,\\quad r = e^b .", "113c78dec425577876ce0ec66c1e72f0": " P \\equiv ", "113cc4e8feb0006c5e165577550bf7eb": "G/N\\approx G'/N'", "113d0bf3c4ca86be1731bc904755e493": "\\mathbf{B}^*=\\{ \\mathbf{b}^*_1, \\mathbf{b}^*_2, \\dots, \\mathbf{b}^*_n \\},", "113d1615f9ab61862cd85415d5c99d9a": "\\begin{align}\ne_{(\\mathbf I_1)}=\\frac{dx_1-dX_1}{dX_1}&=\\Lambda_{(\\mathbf I_1)}-1\\\\\n&=\\sqrt {C_{11}} -1=\\sqrt{\\delta_{11}+2E_{11}}-1\\\\\n&=\\sqrt{1+2E_{11}}-1\\end{align}\\,\\!", "113e06af3cb1014bc4bd20479a8d58fb": "\\frac{1}{(x-1)(x^2+x+1)} = \\frac{A}{x - 1} + \\frac{Bx + C}{x^2 + x + 1}", "113e16ae4724578abc2b1330c8afed5a": "r_u = a\\, \\frac{\\sqrt{3}}{2} \\varphi", "113e2755d04d625f296bd3ab8572451a": "\\sigma \\in (0,+\\infty) \\,", "113e2b3a08df2273ee34a171678d95e3": "f(x)=\\Omega_+(g(x))", "113e592b280e3c7f6674166a55ebb3e6": "R=\\tfrac{1}{2}\\sqrt{a^2+c^2}", "113e5d0ee812c73b087d02b4780873b1": "x[n] = \\sum_{k=0}^{N-1} h_ks[n-k] + w[n]", "113e8a4a1fd4a91fd6b3664411cbfc5f": "I(\\theta)", "113ec26e14fea1e8b8ef8b9c3955eed6": " (M,m,l,t,\\epsilon) ", "113efb35df30996a5b1f383dd52759b9": "x^3 - 2x^2 + 10x -1 = 5", "113fab14368535575d8feb4a91c7177a": "A_k(x) = \\frac{1}{k!}\\sum_{m=0}^k{k \\choose m}x^m\\sum_{l=1}^{k-m}\\frac{(-x)^{l-1}}{l}", "113fb17b52cc29abd2a8fcd3f0046273": "W_D = \\frac{J_2}{2G}\\,\\!", "11404065aad86239f168f1dd0b47a639": "\\tfrac{5}{8}", "114059d52cf464dd4c2811cd718c40c8": "B \\leq_T A.", "11406a650f6017bb05db0eebd407ccae": "T \\, ", "1140ab3ef4ae67e684bb87f119e9b26e": "G_{I}", "1140b8095b48ecf78f052a86c1afe555": "d/dx", "1141633dbd7b6b138d55b000b9cd5284": "Uxy \\rightarrow \\exists z \\forall v [Ovz \\leftrightarrow (Ovx \\or Ovy)].", "11417f9bd33db4bedc068659fb106d9a": "\nF(x; a, d, p) = \\frac{\\gamma(d/p, (x/a)^p)}{\\Gamma(d/p)} ,", "1141a31b5508d2972d69626172b1e1fb": "\\oint_{\\partial \\Sigma} \\mathbf{E} \\cdot \\mathrm{d}\\boldsymbol{\\ell}, \\quad \\oint_{\\partial \\Sigma} \\mathbf{B} \\cdot \\mathrm{d}\\boldsymbol{\\ell}\\,,", "1141a52d1df57f90d0a2f75bd56126b0": "V(0)=0", "1141e020d3ce8a28aafc9fcb8ed471c1": "(xy)(xx) = x(y(xx))", "114240fd63fb6e9698ed0036541ba7fa": "B^n x + \\alpha = (B y + \\beta)^n + r'.", "11424a3fb00068cbe8a87391c99a31d6": " T_{s*t} = T_s\\circ T_t.", "1142794e9db23eddd31936c76c5a947e": " f_1 \\propto \\tfrac{1}{L}.", "1142b5d5391f559645974f438a760192": " \\frac{k}{d} = 0, \\frac{1}{5}, \\frac{29}{146}, \\frac{117}{589}, \\frac{146}{735}, \\frac{555}{2794}, \\frac{1256}{6323}, \\frac{5579}{28086}, \\frac{17993}{90581}", "1142e3219876d1ba09de22153d8a74a5": "\\chi(G_K,M)=\\left(\\#R/mR\\right)^{-1},", "1142fa173a39cc5a9ec43a1686cac3cc": "(2n+1)\\times(5n^2+5n+1)", "1143071c5dadda6abc647004a79249bc": "r_i = \\frac{\\varphi^2 a}{2 \\sqrt{3}} = \\frac{\\sqrt{3}}{12} \\left(3+ \\sqrt{5} \\right) a \\approx 0.7557613141\\cdot a", "11433acd20dbe84bf6bdfa7133f66659": "\n\\mathcal F_\\infty^* = \\sigma\\left\\{ X_s^{-1}(B) : s\\in[0,\\infty), B \\in \\mathcal E^*\\right\\}.\n", "11435743bd2a32ab5515f2a21badbf16": "\\frac{x}{\\sigma^2} e^{-x^2/2\\sigma^2}", "114390c4f3a48c39cc943d255869db29": "F_\\lambda", "1143daa3e1751d75996b0a35fc4960f5": "f(x) = f_n H_n(x)", "1143f59fb20a94b3d4a3ae79ecafa798": "z=(x-a)/(b-a)", "114432e160732831afda796d46442ff1": "2.25>2", "11443ef243378aaddce5d74f152ff6ee": "\\exists{x} \\Box Ax", "11443fad8b35176eeda4fceea98977dc": "-\\frac{\\hbar^2}{2m} \\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} \\Psi(x) + V(x) \\Psi(x) = E \\Psi(x),", "114443ed4006898ab5cc8995cabb2367": " (abc)' = a'b'c' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1}) = -ma(m^{-1}m)b(m^{-1}m)cm^{-1} = -mabcm^{-1} \\,", "114446f596abbc4ee7b2d34c6e1fb97c": "s_n = (-1)^n (\\triangle^n a)_0", "11447ab7e8f3a8c95a15c639bdb47f11": "X \\sim N(0, \\sigma^2)", "1144b75226347ac9bb7e74e9e135337e": "{\\ f(p,V,T) = 0}.", "1145493ee88586261a9948c1369c7cd1": "a^2 = 1-h^2 \\ ; \\ a = \\frac{1}{2}\\sqrt { \\frac {5+\\sqrt 5}{2}} \\ .", "11455d9cbc4864713ecafb527e30d6c7": "\\Delta Y/Y \\approx \\Delta \\overline{Y}/\\overline{Y} + c(\\Delta \\overline{u}-\\Delta u).", "1145c27a623e14e3abe85706d2794286": " X = {5^{7/4} : 16}, \\ ", "1145c6ca798ae04fe22789746bd5bdbe": "R \\approx_\\bar{x} R'", "1145cdcda0c92366117ff24212cc1ecc": "\\alpha,\\beta \\in \\Phi", "1145d8ecd66df21777bf5a6d5f6667fa": " \\{ y \\in O_L : yO_L \\subseteq O_K[\\theta]\\}; ", "1146b5f80e8ee91b4e6d838f6aef84a9": "\\lbrace z = \\sigma j e^{aj} : \\sigma \\isin R \\rbrace", "1146e78611867431398a5122a81e2a6d": "g(x) = x", "1146f1015dd4284308212916379113c1": "0 \\leq s \\le t", "1146ff7dd3bc058061f873e6154a42cb": "f_{ab}", "11474654a5e8eda0484361c96e38234c": "\\frac{\\partial P_{ij}}{\\partial s}(s;t) = -\\sum_k A_{ik}(s) P_{kj}(s;t) ", "11477a1a9ef60be0b65b4b2d5c480797": "\\square(O\\underline{A} \\to \\underline{A})", "114780cc31a9e7b5a0eb16a06a3cf004": " L(z)\\equiv \\sum L_n z^{-n-2} = {1\\over 2} \\sum_i : v^{(i)}(z)^2:", "1147c6f0ef61b104c918a7d8790d496d": "{\\chi}", "1147d0801b73fae5de3c31dba1a8fb09": "\\frac{e^{-iut}}{\\sqrt{2 \\pi}}", "11480970a23e1290174787419b11539d": "\\phi+\\delta\\phi", "114809b615bff5a770613797c7171637": " f \\to m, x \\to f", "114893c3454abd2a02cded328ca50c5f": "x^5-\\frac{4}{13}x+\\frac{29}{65}", "1148a2214c517568a6596204429024c8": "Q(p)=\\mu-\\beta\\ln(-\\ln(p)),", "1148bb577e31cb82ddcced154b0d3442": "x*y=\\alpha(x)\\cdot \\beta(y)", "1148bfaab34d174e55512b2a4cd0234a": "\\mathrm{2H}_2 + \\mathrm{4OH}^- \\longrightarrow \\mathrm{4H}_2\\mathrm{O} + \\mathrm{4e}^-", "114943323cca16302ce6e2e45151c030": "z < s+1", "1149745fb249ae37941a9bbbf6ea0df7": "T_{11}", "11498a7e5b115acea6e48b7b29bd49b7": "I = [x_0-h,x_0+h] \\subset [x_0-a,x_0+a]", "114a69505ccb944b10aa856da92ee9e9": "\\sum |c_n|", "114a8f99cd60498ce8e0c6007db3a84b": "\\psi_1(0)=\\psi_2(0)", "114aa1d043d750478f2cc33d43b2c355": "e^{ax^2}", "114aee363d8f4540b547973d7838cf5f": "\\Psi_L=A_L e^{\\alpha z},\\qquad \\Psi_G = A_G e^{-\\alpha z}.\\,", "114b0313a7f5726e617691bc38aea614": "\\hat{\\textbf{z}}_{k} = \\sum_{i=0}^{2L} W_{s}^{i} \\gamma_{k}^{i} ", "114b1a0e125d801f5cd0ec06fc77182f": "p == l", "114c0ecffdeaa88a3b807e393419ae22": "\\Gamma_{\\beta+1} [0] = \\Gamma_{\\beta} + 1 \\,", "114c6123545bd2d5986d74c989443923": "\\begin{array}{rcl}\n f(0)&=&R(0)\\\\\n f'(0)&=&R'(0)\\\\\n f''(0)&=&R''(0)\\\\\n &\\vdots& \\\\\nf^{(m+n)}(0)&=&R^{(m+n)}(0)\\end{array}", "114c70eddffff74a5c12dc48188be743": "f(x) \\approx f(a) + f'(a)(x-a) + \\frac{1}{2}f''(a)(x-a)^2.", "114c74a2cf3ec3249f8c337c06dc924b": "(2n+1)", "114c7a3168fadfd66279cbbcbf996f0a": "\\mathbf{y} = [y_1, \\ldots, y_n]", "114c7b9d4dca38426264d48a0c6ba34c": "y_n = \\begin{cases}0&\\rm{if}\\ x_n=0\\\\\nx_n^{-1}|x_n|^q &\\rm{if}\\ x_n\\not=0\n\\end{cases}", "114c921ce9a34ba23bffa07dbf5668e1": "\\,\\ -\\cos x + C", "114cd61c747449cd69f6450ebccdab8d": "P_{impact}", "114ce059f8347790fb562e845909ba19": "P = P(0) \\times P(1) = Ge^{-G} \\times e^{-G} = Ge^{-2G}", "114cfc278a1ed668107e1b24b9294840": "\nF(r) = Ar^{-3} + Br^{-2} + Cr^{-4} + Dr^{-5}\n", "114d246d9e140d52b4ec82e04328bf3b": "\\phi : H_1(S) \\times H_1(S) \\to \\mathbb Z", "114d45dc6dce7e51317326a14b540c4f": "\\frac{d}{ds}\\mathbf{T}(s)=-\\mathbf{q}(s).", "114d62f00509ae414c43a15b49ba21d4": " M \\succeq 0 ", "114d929f66c0c440bcb8665586609e19": "\\varphi^\\triangledown:X^\\triangledown\\to Y^\\triangledown", "114daa8276f488b28f02405acaaeba45": "\\mu < 0", "114dfb8097ab0fbfcbdf79bf5e87c481": " [x,p] = x p - p x = i \\hbar.", "114e1334a2de0497f2f68c1a7c620b49": "(x_2,y_2,0)", "114ea874d8c3092096b1ea71c3913d8c": "x \\rightarrow \\log\\|C(x,t)^{-1}\\|", "114efbcbe302c0b1abb62e32b45ce160": "(d\\mathbf{X})^{\\rm T}", "114f1b8b56df88e996d3cf6dd6570758": "a \\over \\sqrt 3", "114f6463c881b337e250e3f346b4aa0a": "\\Pi_{\\rho_{x^{n}\\left( m\\right) },\\delta}", "114fb67ec514be01e4f94fb04fdb39c7": "-2 \\le x,y \\le 2", "114ffb68f58546dfa45b65f36f6193d4": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta = \\lim_{\\delta \\to 0} \\left( \\frac{\\cos\\theta\\cos\\delta - \\sin\\theta\\sin\\delta-\\cos\\theta}{\\delta} \\right) = \\lim_{\\delta \\to 0} \\left[ \\left(\\frac{\\cos\\delta -1}{\\delta}\\cos\\theta\\right) - \\left(\\frac{\\sin\\delta}{\\delta} \\sin\\theta\\right) \\right] . ", "11500c461f803f48d8bbde82ec4fce92": "b \\ne 0~ mod~ p", "115057dd71634325fdd678f01166dcb4": "E[u(A)] \\geq E[u(B)]", "11510443f4c574c459afe9c2d35be0f1": "n<0", "1151091b1c39b88b8365f9c655cd600b": " F(x,v) := \\sqrt{a_{ij}(x)v^iv^j} + b_i(x)v^i", "115131a17b668ad125610dce576d0022": "\\sin x = \\frac{e^{ix} - e^{-ix}}{2i}, \\qquad \\cos x = \\frac{e^{ix} + e^{-ix}}{2}, \\qquad \\tan x = \\frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.", "11514514a8d21298372c0bc0824dae4d": "\\textstyle a a = 1 \\, , \\quad b b = 0 \\, , \\quad a b = - b a = b ", "11517763a5344ddb1bfda84939469c96": "R'(\\phi)", "1151a4311ffcc52052da2bdb670c85c3": "\\ln2+\\sum_{k=1}^\\infty \\frac{(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^2}=\\ln\\left(1+\\sqrt{1+z^2}\\right), |z|\\le1\\,\\!", "115284bb3898d791e337e6fffbcb8fdb": "a\\ne b \\ne c \\ne d, \\alpha \\ne \\beta \\ne \\gamma \\ne 90 ^\\circ, \\delta = \\epsilon = \\zeta = 90 ^\\circ", "1153dc8ed25e768969ad4b8c032fb7bf": "failures\\left(\\left(a \\rightarrow STOP\\right) \\sqcap \\left(b \\rightarrow STOP\\right)\\right) = \\left\\{ \\left(\\langle\\rangle,\\left\\{a\\right\\}\\right), \\left(\\langle\\rangle,\\left\\{b\\right\\}\\right),\n\\left(\\langle a \\rangle, \\left\\{a,b\\right\\}\\right), \\left(\\langle b \\rangle,\\left\\{a,b\\right\\}\\right) \\right\\}", "1153e8e3ebc0a037365d2b60ac79eb0c": "r_{t_1,t_2}", "115419652b1120064435f552f85851dd": "r_i(k)", "115440333e3f0fe7682228936f8f502f": "\\frac{T_\\text{hold}}{T_\\text{load}}={e}^{-\\mu\\cdot\\phi}", "11548d46621105bf1fb7c1031694d981": "\\sin^2(x) + \\cos^2(x) = \\frac{a^2}{h^2} + \\frac{b^2}{h^2} = \\frac{a^2+b^2}{h^2} = \\frac{h^2}{h^2} = 1.\\, ", "1154d0ca4ccf4b8a863efcd8128f6527": "R_{vs}=\\frac{A}{f}\\left(\\frac{1}{h}-1\\right)-R_{va}", "11557659d7304d70eb57a6fcf721c289": "x>73.2", "115576e1bea0010d2e8420e134e091b2": " \\frac{\\partial \\langle H \\rangle}{\\partial a_n}\n= - i \\hbar \\frac{\\partial a_{n}^{*}}{\\partial t} ", "11558254980c2ddf12c4c292bef600cd": "\\begin{align}\nr &= ct \\\\\nr' &= ct'\n\\end{align}", "1155cf5cac0483c06999787abb3f9c25": "L(s) = \\frac{N(s)}{D(s)}.", "115604d47d645ff5255cd29680c46b3f": "\n\\frac {m_1 u_1 + m_2 u_1 - m_1 u_1 - m_2 u_2}{m_1 + m_2} = \n\\frac {m_2 (u_1 - u_2)}{m_1 + m_2}", "11564ee93d8d0aa8a63309414c54b245": "F = \\frac{2 \\cdot \\mathrm{precision} \\cdot \\mathrm{recall}}{(\\mathrm{precision} + \\mathrm{recall})}.\\,", "115660db79e1999c639558c9b50e863a": " -\\left (R_{im}\\frac{\\partial U_{j}}{\\partial x_{m}}+R_{jm}\\frac{\\partial U_{i}}{\\partial x_{m}}\\right )", "11567450d788888c7720845b90bdb00f": "{\\underline{Z}(\\mathbf{r},\\omega) = \\frac{\\underline{p}(\\mathbf{r},\\omega)}{\\underline{v}(\\mathbf{r},\\omega)}}", "11568c5b8ff966fa1f29789a2f930912": " \\overline{\\lambda} = -1000, \\underline{\\lambda} = -1 ", "1156b75f0b2d38b4c20117280e4af3f1": "C_n.", "115706f968adfb2f1b1c453fb1f8be04": "\n\\begin{array}{cl}\n\\displaystyle\\frac{x:\\sigma \\in \\Gamma \\quad \\tau = \\mathit{inst}(\\sigma)}{\\Gamma \\vdash x:\\tau}&[\\mathtt{Var}]\\\\ \\\\\n\\displaystyle\\frac{\\Gamma \\vdash e_0:\\tau_0 \\quad \\Gamma \\vdash e_1 : \\tau_1 \\quad \\tau'=\\mathit{newvar} \\quad \\mathit{unify}(\\tau_0,\\ \\tau_1 \\rightarrow \\tau') }{\\Gamma \\vdash e_0\\ e_1 : \\tau'}&[\\mathtt{App}]\\\\ \\\\\n\\displaystyle\\frac{\\tau = \\mathit{newvar} \\quad \\Gamma,\\;x:\\tau\\vdash e:\\tau'}{\\Gamma \\vdash \\lambda\\ x\\ .\\ e : \\tau \\rightarrow \\tau'}&[\\mathtt{Abs}]\\\\ \\\\\n\\displaystyle\\frac{\\Gamma \\vdash e_0:\\tau \\quad\\quad \\Gamma,\\,x:\\bar{\\Gamma}(\\tau) \\vdash e_1:\\tau'}{\\Gamma \\vdash \\mathtt{let}\\ x = e_0\\ \\mathtt{in}\\ e_1 : \\tau'}&[\\mathtt{Let}]\n\\end{array}\n", "11570cd2d7c0faaf632db1c3ae51fb24": "\\neg (\\neg (\\neg A\\lor B)\\lor (C\\lor (D\\lor E)))\\lor (\\neg (\\neg E\\lor D)\\lor (C\\lor (A\\lor D)))", "115733207da2070a7898aac909f3e2f4": "f(x) \\in o(g(x))", "11578cced88be1dbe51af21d285c9a6e": "\\widehat{U}", "1157b715ac82804ec725517a7c91748d": "|m-1|", "1157d4a8ddde804f2f2d454828c82fda": "P_{\\mathbf k}", "1157d9753cfd628b4e46d257440ea519": "\\left[\\widehat{U}, \\widehat{H} \\right]=0 ", "1157ed1eaeceabd9c9b042937c58e410": "\\mathcal{SHIQ}^\\mathcal{(D)}", "11581a37ec230c38b480deecfe66b9fd": "3^{-1/\\beta} \\alpha", "11586ca43984fa4adce66fab55c57cf4": "\\lang 2,2,1,0,0 \\rang", "1158dc93546783a9f8a3a3a987780463": "\\omega^i \\ldots \\omega^j", "11590c5f4a09afd40aa65505af9fc952": "R_{4,0} = 5+126 r^4-280 r^3+210 r^2-60 r", "1159af01a969666a7affdeefb4a899d0": "9\\cdot 2^{3w}", "1159d1e4729abc812d241c69ffaadf78": "\\epsilon \\approx 23.4^{\\circ}", "115a813b34fe8662f323dcea4ad7d198": "\\chi(\\mathbf{R})", "115a8e5fba4b0b414f44cc1b4c9d4615": "f(t)=e^{-2\\tau_D/t}\\left(I_{0}(2\\tau_D/t)+I_{1}(2\\tau_D/t)\\right)", "115ab4e4e168872519e917b973ab21f8": "z_{0}=\\frac{\\pi W_{0}^{2}}{\\lambda}", "115acdbbdb4767f0208d686f8613fef6": "\\beta_A = \\beta_D \\left(\\frac {D}{V}\\right) + \\beta_E \\left(\\frac {E}{V}\\right) ", "115b12876ce8eb8a4d7ede1d9a802342": "P = \\operatorname{Perm}(A')\\,\\bmod\\,Q", "115b1b827ac9cfbb5bcb78595d9706f4": "\\partial\\colon \\mathcal D_{m+1} \\to \\mathcal D_m", "115b757eed221a3b5c2341b1b2222429": "\\alpha=\\theta/2", "115bdc1b9a7cab95337a886c1992c91e": "r^{ed}\\equiv r\\pmod{N}", "115c36c53126eb8ada7117af11ee4670": "\\mathbb R\\setminus\\mathbb A", "115c469411c5831217307670ea40f92a": "V = d' \\Sigma _{YY} ^{-1/2} Y = b' Y", "115c793833709b1514aceb8f8c5e7a20": "c=K_R/K_T", "115ca2c2ee963546ff5de45343b0d0e5": "\\begin{bmatrix}\n\\cosh\\phi &-\\sinh\\phi & 0 & 0 \\\\\n-\\sinh\\phi & \\cosh\\phi & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{bmatrix}= \\exp \\left( - \\phi \\begin{bmatrix}\n0 &1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{bmatrix}\\right)\\equiv \\exp (-\\phi K_x),", "115ca478164bafb41bb075eedc3bc311": "\\mathbb{C}^3", "115ca536bb0aaaff4b2eccc62af2d3a1": "\\cos{\\theta}_W* = r \\cos{\\theta}", "115cab08cf56cd0cbe43382811fe10d5": " \\frac{1}{\\sec \\theta}\\! ", "115cc42e4c85a83033ad9cf797f9db81": "m(H,t)", "115cdc52306415f28dee695e67870175": "a_i^j := (A\\mathbf{u}_i)^j \\in K", "115ce754cd3689beba99006c1070a37f": " \\!\\ \\sum_{n=1}^N n^2", "115d0271af1b2b2a7467788276944a6f": " \\overline {x} ", "115d7db3c5060afe2a39c77831ae48b9": "\\vec{e}_1, \\vec{e}_2, \\, \\vec{e}_3", "115dc1e020322b605c687d29cc08be1c": "\\hat{g}\\ ,\\ \\hat{h}\\,", "115dfe6472be5fbebe6abd2476687b55": "\\phi \\mbox{ fixes } K \\quad \\iff \\quad q \\mbox{ splits completely in } K.", "115e1db2d082962d49b2cc7e91d632b2": "x \\in \\mathrm{supp}\\,X.", "115e2c8264d9c247771fc75c882d94cd": "\\left(\\begin{smallmatrix}\\lambda \\\\ & \\lambda^{-1}\\end{smallmatrix}\\right) \\times \\{\\pm I\\}", "115ef0a4b772bc4d2c8396f2c87b07d8": "m-2,m,m+2", "115f00f446eb59162d4575ac63502376": "F(x;\\mu,\\sigma,\\xi) = \\exp\\left\\{-\\left[1+\\xi\\left(\\frac{x-\\mu}{\\sigma}\\right)\\right]^{-1/\\xi}\\right\\}", "115f5428a916422b09db1bd60a2dd67b": "\\gamma V^{2/3}=k(T_C-T)\\,\\!", "115f56d072ef0db3cfb4fddc6f534e13": "x=8", "115f8a166d3bfbee69b62c6f9830a907": " \\frac{R}{2}\\sqrt{3} = \\frac{a}{2}\\sqrt{3} \\!\\, ", "115f8f7a3e1d808db462a94f9ccc5f81": "\\tau_{ph-e}", "115fa04e79b504896a190f6a591439a8": "\\lambda_2 - \\lambda_1", "115fa499c0ef07569c045f29dbe7c40f": "\n\\tau_{ij}^\\mu(t) := \\sum_{\\nu=1}^n (\\xi_{i\\nu}(t) - \\xi_\\nu^\\mu(t))(\\xi_{j\\nu}(t) - \\xi_\\nu^\\mu(t)),\\qquad \n\\xi_{i\\nu}(t) := \\sum_{i=1}^n \\mu_i(t) \\xi_{i\\nu}(t)\n", "115fc26783cbc21f1da892eb190dbc8d": "{}^{3}i = i^{\\left({}^{2}i\\right)}", "1160a8d1145bcf5538e74df9e491f55a": "c_V = (d\\delta_{ij} - n_{ij})_{ij},", "1160daebba71f9f8a58469ac16f75af5": "\\mathbf{R}_j", "11610dcae4326433c839040e21f00907": "\\scriptstyle E(z),", "116116f8dff1f18bab37e1bf14d6d8a6": "\\sum_{j=1}^n A_{jk} = \\sum_{j=1}^n f_j(g_k) = 0", "1161217fa260fba5181ae8b30438fd2d": "|\\Psi(t)\\rangle = e^{-iE_{\\Psi}t/\\hbar}|\\Psi(0)\\rangle", "116164854da736767a9ec29a9c493b33": "(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge 0},\\mathbb{P}_x)", "1161aaaf0a1228651a7ca301949e1b7c": " \\langle \\psi(t) | = \\langle \\psi(t_0) |U^{\\dagger}(t,t_0).", "1161acc9c5893cbbb9980c230f53362b": "L_2 \\in \\mathbb{R}^{(n-1)}", "1161f2ab11758784d3b9c3961013965c": "(\\mathrm{proj}_\\theta F)\\setminus\n G_\\theta", "1161fc5ddeb59b72a5a49d5c70386aea": "r=L/10 ", "116202fcd704620524148cc876ca6648": "x \\rightarrow b \\equiv (a_1, a_2, \\ldots, a_n, b)", "11620499dadd6b0d68a72e06a8bd9bb2": "(256^{\\,\\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\\times 256^{256^{257}}}=256^{256^{257+256^{257}}}", "11628573b0b504edd4ef4ef2d4d290de": " \\sum_{n=1}^\\infty a_n =1+2+3+4+5+\\cdots ", "11628a882ce9d229db3fd6a37b343edc": "\\hat T = \\frac{\\hat p^2}{2m}.", "11629b4d41796922245a01b3463f239c": "h_k = \\max\\left\\{\\left\\lfloor 5h_{k-1}/11 \\right\\rfloor, 1\\right\\}, h_0 = N", "1162d2547e153449686efdd5fc203b57": "\nT_{ss} = \\sqrt{2}\\, T \\approx 1.41\\, T,\n", "1163720ff25bc474b03273f89160ad5a": "\\alpha_R", "1163a4f72e317a52b4c6ee4ca3a98d41": "wp(\\textbf{if}\\ E\\ \\textbf{then}\\ S_1\\ \\textbf{else}\\ S_2\\ \\textbf{end},R)\\ =\\ (E \\Rightarrow wp(S_1,R)) \\wedge (\\neg E \\Rightarrow wp(S_2,R))", "1163ecc17467b15ce8fce2a661471954": "\\phi(5)=4", "11642cfb4226577be605838141fff5b2": "A = f^{-1}(B)", "116436af151c6d132ee4d4c21880be54": "n>-\\delta", "1164446217f2d40dbed109c14ed03843": "\\left( \\alpha^m \\right)^{-1} = \\alpha^{-m}", "11644feff7f3d02f1e93c502204d8632": "P_1, ..., P_k", "11648838cb046147cc143cdc73f89fcf": "\\left \\{ f_1,\\ f_2,\\ f_3,\\dots \\right \\}", "116500e5e8f64c5dacebf52dba6fdb4b": "L(x,y) = \\int_0^1 x^{1-t} y^t\\ \\mathrm{d}t", "1165b63f4e07dd7247f469168fbe5e8d": "C_i^n", "116654a97d39dec49e9ad9310a642740": "\\beta = 1/(1-\\pi_A^2-\\pi_C^2-\\pi_G^2-\\pi_T^2)", "11667bdf2486170b6ef0d01dc6429589": "[24,12,8]_2", "116682e06468a4e1504bbf1752d36bce": " \\frac{d}{dx}e^x = e^x.", "1166c9cc5504cf193eb09412d20ab295": "= \\sum_i p_i ( \\log p_i - \\sum_j \\log q_j | v_i ^* w_j |^2)", "11676eea716a9c8a71b526204666aa0d": "\\phi\\colon \\mathfrak{U} \\to [0,+\\infty)", "1167fd98f4799a8fc909465ca24e5308": "\\mathbf{v}_T = \\frac{R}{f} \\ln \\left [ \\frac{p_0}{p_1}\\right ] \\mathbf{k} \\times \\nabla_p \\bar{T}", "11687dc6f6b86e47916885a545f2d942": " \\langle p \\rangle = -\\frac{\\partial A} {\\partial V}, ", "11688718147baa0f26bd72966b0fe42e": "\\lambda_N", "1168d34108b07c6d001252fa32bf6607": "L=-\\tilde{F}\\dot{v}^2+2\\dot{v}\\dot{r}\\,,", "1169116784fe73e9e7f4f08c2071eb6c": "H_*(Y)", "11691e618a6a432b47d7a0559b9905bc": "E^{(2)} = \\sum_{l=3}^{\\infty} C_{2 l} R^{-2 l}", "116948fc0f905f88bc2c8d2e55dd1dab": "P(n)^2-P(n+1)P(n-1)=P(-n-7).\\,", "1169726f7c7d63110123a8e83018a8f7": " |\\Psi \\rangle \\sim |\\Psi\\rangle + Q_B |\\Lambda \\rangle ", "116988c0814431969204c6aebd934c2f": "\\sum _{1\\le j\\le m, x_j\\ne y}\n\\frac {x_j - y} {\\left \\| x_j - y \\right \\|} \\le \\left|\\{\n\\,j\\mid 1\\le j\\le m, x_j= y\\,\\}\\right|.", "116a3bb7832dc0b4a701a0459b9d6039": "\\scriptstyle{(Rc,Rt)}", "116a505ed2fffa33ebb2afe22102bdd8": "{A}_{10}^{(2)}", "116a6bc7babae4e38520eee6a9b23a3b": "\\mathcal{D}^J", "116ab8beaf22b17c0eb7661719fbd20f": "(\\sigma, \\tau, z)", "116aff26524952c0187b4bd82c511e4d": "\\nabla_{\\mathbf{x}_0} f \\cdot \\mathbf{e}_i=0", "116b1fae0f0afb22dd4fa38693ba8bad": " G(s) = \\frac{Y(s)}{X(s)} ", "116c6f7e9919a523854279e9a38d1ffc": "\n-\\ln(r) = \\int^{T_f}_0 \\lambda (\\vec{p}(t)) \\, dt\n", "116c8d4537aa23b794cdfb220e537561": "\\sin 20^\\circ=\\frac{h}{d}", "116cba1e615fa0c81ef017ca0220696f": "P_3=(3/4)-\\epsilon", "116cf4bfbb0de46e5535f3eaa5925cb0": "(f * g_T)(t) \\equiv \\int_{t_0}^{t_0+T} \\left[\\sum_{k=-\\infty}^\\infty f(\\tau + kT)\\right] g_T(t - \\tau)\\, d\\tau,", "116cff2d0a695d9b64feb758cfbce710": "\\frac{\\infty}{\\infty}", "116d05e0567fd82ed85b9ea1a8209a22": "c=14", "116d23f78bdda731cd508bf1edc2fab0": "C^*(\\theta)", "116d24d9bec83e9fa98ba54fca653f20": "C\\ell_2(\\mathbb{C}) = \\mathbb{H} \\otimes \\mathbb{C}.", "116e0beaaf40538cb22ff6dd234f210e": "T_{\\epsilon} \\in \\mathcal{M}_{\\epsilon} ", "116e874e1b0a8d4fe2f6ef411cbc1725": " \\frac{dG}{dr} > 0 ", "116e8d0161aa18dcd8697c29107a5444": "\\langle x,\\,y \\rangle = \\frac{1}{4} (\\langle x+y,\\,x+y \\rangle - \\langle x-y,\\,x-y \\rangle)", "116f17dbce75f9c59e6d379785145eda": "a(u, v)=L(v)\\,", "116f1a4ea8fa990c13d061391fcdf3aa": " Car\\{f(t)\\} = G_{Car}(p)\n= p\\int_0^\\infty e^{-pt}f(t)\\,dt\\qquad(4) ", "116f482925f13a16a0f215e52151a684": "\\frac{V(t)}{V_0}=0.1", "116f721d3da983c298853c95cdc136c3": "(V^*)^{\\mathbb C} = V^*\\otimes \\mathbb C \\cong \\mathrm{Hom}_{\\mathbb R}(V,\\mathbb C).", "116f922091726325efc390df425993bd": "H_\\alpha^{(2)}(ze^{im\\pi})", "116fdf0b91652e41f66e1e8dbf32cd86": "e(E) \\cup e(E)", "117036e7c6aab20fab2a2877c4b25ac9": "\\frac{\\text{base area} \\times \\text{height}}{3},", "11706b2094097664eac87a3f06fc7ec3": "\\exp : \\mathfrak{g} \\to G", "1170bed0e4fc98b25a8d4eb45e33e368": "= 52,900 \\pi", "11710dab78027e9dfd08c7f33cb3549f": " \\vec u=\\left[\\begin{array}{c} e^{i2t}s+e^{-i2t}\\bar s\\\\\n \\frac{1-i}2e^{i2t}s +\\frac{1+i}2e^{-i2t}\\bar s\\\\\n -\\frac{i}2e^{i2t}s +\\frac{i}2e^{-i2t}\\bar s \n\\end{array}\\right]\n+{O}(\\alpha+|s|^2) ", "11712808843c7284f8321cfd1737c232": "\\frac{a^2 e^t}{\\left((a-1) e^t+1\\right)^2}", "1171607b0a6433f5c479d07bb2493c08": " v_g = {g \\over f} {\\partial Z \\over \\partial x}", "1171745fec644de49fc65b3e8decb474": " \\left |\\sum_{m=1}^N\\sum_{n=1}^N c_{mn} \\lambda_m\\lambda_n\\right|^2 \\le \\sum_{n=1}^N {1\\over n} |\\lambda_n|^2,", "1171a7669353b1c82b746d338c6243fd": "[t_v(x), 1-f_v(x)]", "1171bd68fef4916e94ad457fdc70c348": "S(X)", "1171d7db59f3494bbfedf2a9502651ff": "\\bar{n}_i", "1171f3c2e687c8f5ff49704b82fd7971": "\\frac{\\Gamma \\vdash t:\\alpha \\rightarrow \\beta \\qquad \\Gamma \\vdash u:\\alpha}{\\Gamma \\vdash t\\;u:\\beta} ", "1172181a130c2e7d7800c3af251b088a": "\\Psi_0, \\ldots, \\Psi_4", "117283af27dbb4ca45b5a6eb4b3a1dc7": "\\scriptstyle C \\;=\\; E_K(P \\,\\oplus\\, X) \\,\\oplus\\, X", "1172f0828ab9f76ef1abc90365328068": "k \\left\\{{ n \\atop k }\\right\\}", "1172f4d0764352ed3247b500c30204c7": "|x| \\geq 2|y|", "1172f77da75dafe6030cbaed4593570a": "i^2=+1=j^2=k^2", "1173055876a7b2905f697d8f9668a637": "x_{11} = x_{22} = x_{33} = 1 \\ ", "11736f4d29b55d79c10b77ecd975e9fe": "h=\\frac{P}{\\gamma}", "11739414d73d5a3aaf9d200f04d71b37": "G_{AW} = \\frac{1}{R_{AW}}", "1173e3ebd1948797d17368626c35f8a9": "F: \\mathcal{D} \\rightarrow \\mathcal{C}", "117401135dec95dc50ec8608e19b6b25": "\\frac{\\partial \\mathbf F (\\mathbf X)} {\\partial \\mathbf{X}}=\n\\begin{bmatrix}\n \\frac{\\partial f_{1,1}}{\\partial \\mathbf X} & \\cdots & \\frac{\\partial f_{1,p}}{\\partial \\mathbf X}\\\\\n \\vdots & \\ddots & \\vdots\\\\\n \\frac{\\partial f_{m,1}}{\\partial \\mathbf X} & \\cdots & \\frac{\\partial f_{m,p}}{\\partial \\mathbf X}\\\\\n\\end{bmatrix}\n", "11746bccbcf8464f21751a70089991a6": "n \\cdot k", "1174993a2831637a3259982405c06635": "{A}_{8}^{(1)}", "1174cb7011fde8918aedd7223b59169b": "\\frac{n_2}{n_1}=\\frac{g_2}{g_1} e^{-h\\nu/k_\\mathrm{B} T}", "1175a25b4597fb7f0e1943dcc8cb7f76": "1=2-\\phi(2), 3 = 9 - \\phi(9)", "1175ba5a589b300ea49f15171dab0b0b": "x\\frac{dx}{dt}+y\\frac{dy}{dt}=h\\frac{dh}{dt}", "1175d3d4ed6e02a31cf9cda3f9c21419": "\\ Ci ", "11764209c8d5f4598c920c2508390d3a": "\\mathrm{Re}=\\frac{Vd}{\\nu}\\ ", "117647c7157ba9372a6ecf8a3dc0527b": "\\operatorname{Var}[\\,\\varepsilon|X\\,] = \\sigma^2 I_n", "11765921e09fe6e829549e5dda369257": "{{S}_5 = 1 }", "1176c5a284fbbccc28d74d018b05141a": "t^* = t - \\frac{\\epsilon vx^*}{c^2}\\cdot", "1176f078de6b83917830cb6eb5dd04fa": "\\frac{dQ}{dx} = -q", "1176f5ea4e4804b2b7a60f5d637c3949": "\np_{\\mathbf{y}}(\\mathbf{y})=\\frac{p_\\mathbf{x}(\\mathbf{x})}{|\\frac{\\partial\\mathbf{y}}{\\partial\\mathbf{x}}|}=\\frac{p_\\mathbf{x}(\\mathbf{x})}{|\\mathbf{W}|}\n", "11770404062ba82c7d4c1aa072790ef1": "a_0 1 + a_1 x + a_2 x^2 + \\ldots + a_n x^n", "11774547ef4cd4c5448862aa2c0a6023": "e_0,e_1,\\ldots,e_n", "117776f7745bf3e5c5d51e88f1a23254": "\\begin{Bmatrix} 8 \\\\ 8 \\end{Bmatrix}", "11780b176b58cd6ea28dee0873293bc7": "\\beta > 0\\,", "11786ecb9f238b7b8df2d61d3eac99ba": " \\begin{matrix} g_m R_C \\end{matrix}", "11787d2fd817cd9fe9823373c8074a48": "f(t) = \\sum_{k=-\\infty}^\\infty f(t_k)\\frac{G(t)}{G'(t_k)(t-t_k)},\\qquad \\forall f\\in B^2_\\pi,\\qquad (t\\in \\mathbb{R}),", "11788ebd5725a052338cd77f6490e97c": "E = 0.0100\\, ", "117899eaf7ed6ffda5badf88104d63c0": " \\int_a^b f(x) \\, dx ", "11789c067d528fecc5ce45f9bb9c71eb": "\n\\forall x\\in X\\qquad \\|J(x)\\|''=\\|x\\|,\n", "11796cdc1086128796382de44b2507bb": " \\min(\\mathbf{x}) \\leq H(\\mathbf{x}) \\leq G(\\mathbf{x}) \\leq L(\\mathbf{x}) \\leq A(\\mathbf{x}) \\leq R(\\mathbf{x}) \\leq C(\\mathbf{x}) \\leq \\max(\\mathbf{x}) ", "1179a931d14c6bbfca61f93374717098": "t = \\frac{\\mathrm{arctanh}(\\alpha v)}{\\alpha g}", "1179b180d2fcc8046e5a1efb81fbf1e3": "K\\supseteq F", "117a094440c2eba0e6e5d0467c49c8db": "\\lambda C_1 + \\mu C_2", "117a2b6f6d6c2fdb27ca47b848e48fee": "H_2(p)", "117a45d968b83dcd7914847eb3b00b38": " T_1 ", "117a59245a2de5e897b4261659347e1d": "g(z)=G(z)e^{-z\\tau}", "117a7a6ca88ee132faef24da7ee814cb": "\\frac 1 u + \\frac 1 v = \\frac 1 f,", "117ab296ecdee1fa53eea4bf18846eb6": "V(a_0, b_0, c_0, d_0, \\dots) = 0\\,", "117b1de65c406a8c921dc07ab9c31d03": " Q = \\mathbf{F}\\cdot \\frac{\\partial \\mathbf{V}}{\\partial \\dot{q}} + \\mathbf{T}\\cdot\\frac{\\partial \\vec{\\omega}}{\\partial \\dot{q}},", "117b36b2a8dbf6b49b0ec6412754c478": "\nS = \\frac{A}{L}.\n", "117b72513d4f8f6a45e02eaa40705857": "C^TB \\leq l_{W \\times L}", "117ba7a7e6c5336d3ac4f55dde406e72": "\n \\left(\\frac{\\partial f}{\\partial t} \\right)_{\\mathrm{coll}} = \\iint gI(g, \\Omega)[f(\\mathbf{p'}_A,t) f(\\mathbf{p'}_B,t) - f(\\mathbf{p}_A,t) f(\\mathbf{p}_B,t)] \\,d\\Omega\\,d^3\\mathbf{p}_A.\n", "117c22a09a24b31fa321d92efe8542cc": "\\begin{align}\n \\phi_L &\\to 90^{\\circ} = \\frac{\\pi}{2}^{c}\\\\\n \\phi_R &\\to 0\n\\end{align}", "117c24ef265606c75bb90945ed3e13f3": "f^{(m+n)}(x)=f^m(f^n(x))", "117c2d0addfba1f25d27d36ed2d11d36": "L(e) := L(\\mathbf{Z}[e]) = L(\\mathbf{Z})", "117c33d7e9e2dc477b8404376e5e676c": "\\lambda \\in K", "117c3bf6611d125422aa42d357d4b0fa": " M(\\beta):=|\\langle \n\\mathbf{s}_i\\rangle|=0\n", "117c7c5e148e7093e155c0ef153c1a4d": "[A]_p \\subseteq [B]_p", "117c876c77fcb4838d85de387cbe3891": " S=A[x_0,\\ldots, x_n] ", "117c9316c7e7125ca3655cfd675357c4": "Y(t)", "117cd7625dcd554c0a040e910224f44e": "[A_n]", "117ced56c999973b3908400d2b2e3760": "\n\\sqrt{\\gamma}\\,\n", "117d6831cf06fe1df9285813beea3b0b": " \\{X_n\\}_{1 \\leq n \\leq \\omega} ", "117dcc0fbfbf870f9191d14870821f43": "(x, z)", "117df26a1bbe348410220d5ca5647475": "n = \\sqrt{\\frac{ G( M \\! + \\!m ) }{a^3}}\\,\\!", "117e1cd49f525ec50ceec22f12c4a6c1": "[a,a,a]=a,\\, [a,a,b]=b,\\, [b,a,a]=b,\\, [b,a,b]=a,", "117e870077ca470d0f60c7078f66dde0": "c(t,s)=d", "117ed0f9be061d102e91686c7ca0235d": "dE_\\nu = I_\\nu(\\mathbf{r},\\hat{\\mathbf{n}},t) \\cos\\theta \\ d\\nu \\, da \\, d\\Omega \\, dt", "117ed34abf0ea194a6bbd386056beaa8": "t_{0}=0", "117eeb703fa190e97a3c85a8985b657c": "\\lim_{n\\to\\infty} |f(x_n)-f(y_n)|=0.\\,", "117f0d2b15dc619b8808d30c79d0adc0": "\\{\\Omega,F,Q,\\{F^W_t\\}\\}", "117f15c8f3faf940d396ef1eeb176ee7": " R_{mn}(r) = \\sum_{s=0}^{\\frac {m-|n|}2} (-1)^s F(m,n,s,r) ", "117f2eada08195a0107d3f68b5cd63dd": "d = \\min d\\left( x_i ,x_j \\right)", "117f3f14332af756759f7baf43b54f36": "\\omega_p = -\\frac{3}{2}\\frac{R_E^2}{(a(1-e^2))^2} J_2 \\omega \\cos i", "117f5f3114e0819728f6395981e8c8ad": "\\vert a \\vert ", "117f66712cc5f8d2098910b18a6a5704": "r\\approx \\frac{d\\,\\Delta\\,E}{e\\,U}", "118033221b907a59dae49784ba28f73d": " \\langle \\phi(x) \\otimes \\phi(y), \\phi(x') \\otimes \\phi(y') \\rangle = k(x,x') \\otimes k(y,y')", "118058c316ed9495877f9723050eb264": "S = I + NFI\\,\\!", "1180b57a13e84ba8490843f79f8bcce0": " \\tau \\circ i_1 = i_2 \\circ \\sigma, \\ \\nu \\circ \\pi_1 = \\pi_2 \\circ \\tau, \\ \\tau^*\\omega_2 = \\omega_1 \\, , ", "1180e2e37a24f0571e6bbe5ea1ff681a": "\\mathcal{H}_\\mbox{accept}", "1180e8b090976498bb302f9a6c780880": " \\frac{1}{\\nu} ", "1180e9e0ae1b514839448da396de66a7": "\\ h = \\alpha R_v^{-1}s.", "118102e7c18a48d8899a119f97769f5e": "\\mathbf{u}(\\mathbf{x})", "1181773135eee2b565798627a2595adc": " [C]", "1181a08a1f8789c5818aa2684fdc8c47": "i \\frac{\\partial \\psi}{\\partial \\tau} + \\frac{1}{2} \\frac{\\partial^2 \\psi}{\\partial \\xi^2} + |\\psi|^2 \\psi = 0", "1181b70ddbc5b4d1bed0efdeb5b624d2": "\nV(r) = \\frac{k}{r}\n", "1181c0a94992c78f2d3fd3e4198a6260": " \\lambda=3 ", "1181d0815ce412228210b8274de2cbb6": "\\frac{q}{ \\sin( \\theta/2 ) } =\n\\frac{[x\\,y\\,z]^\\top}{ \\|[x\\,y\\,z]\\|}\n", "1181e002c98c3cbccaee0b35ce0e7351": "X_0 = x_0", "1181f6aaa1cea49eac0fef907785b6d0": "\\{c_r,c_i,z_r\\}", "1182188601478ec0f6faeb247b0de1c8": " Q_2,Q_4,Q_6,\\ldots", "11829544f8c367926e75bcb8511da71f": "\\lim_{z\\to i}(z-i)f(z)=\\lim_{z\\to i}(z-i){e^{itz} \\over z^2+1}=\\lim_{z\\to i}(z-i){e^{itz} \\over (z-i)(z+i)}=\\lim_{z\\to i}{e^{itz} \\over z+i}={e^{-t}\\over 2i}.", "1182c033e5b85ce38cef766da9963bad": "\\forall x,y,z: x < y \\;\\to\\; (x < z \\;\\vee\\; z < y)", "11831fd6b3946915a222a4dbb9307020": "I = I_{L} - I_{D} - I_{SH}", "11832313f3c4fcf32932a63a106ddb67": "\n\\begin{cases}\n(P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2) \\\\ \n(P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2) \\\\\n(P_1^2 \\or P_3^0) \\and (P_2^0) \\and (P_1^2 \\or P_3^0) \\and (P_1^2 \\or P_2^0 \\or P_3^0) \\and (P_2^0) \\\\\n(P_1^2 \\or P_3^0) \\and (P_2^0) \\and (P_1^2 \\or P_3^0) \\and (P_1^2 \\or P_2^0 \\or P_3^0) \\and (P_2^0) \\\\\n(P_1^2 \\or P_3^0) \\and (P_2^0) \\and (P_1^2 \\or P_3^0) \\and (P_1^2 \\or P_2^0 \\or P_3^0) \\and (P_2^0)\n\\end{cases}\n", "118337530070f44bdf9c7cdeb8e35f9a": "x = a", "118374df122056f4bf223ddbd201eba1": "\\mathrm \\frac{N_\\mathrm{atoms \\ of \\ dopant}}{N_\\mathrm{atoms \\ of \\ solution \\ which \\ can \\ be \\ substituted \\ with \\ the \\ dopant}}", "1183c5cec943644eb36781928cdd8453": "H_n(X,A) = H_n (C_\\bullet(X) /C_\\bullet(A)).", "11843073bd436a5e84f39e219779e863": "\\neg \\psi \\to \\phi", "11844cc09c2effaf69aefed3c133dd2c": "\\eta=\\frac{|B(L)|^2}{|A_0|^2}=\\frac{I_{out}}{I_{in}} =4(\\frac{\\gamma_{\\perp}}{\\gamma_{||}})^2 \\sin^2(\\gamma_{||} |A|^2L/2)", "1184997afe41683a9ec6d47acccb74c1": "\\frac{1}{n - 1}", "1184a0a082d22690584c52a457c4c1c8": "\\textstyle {\\mathrm{Cov}}([x=i],[x=j]) = - p_i p_j~~(i\\neq j)", "11850c0a7ed88adcef38f11322bd3225": "p_k = \\frac{\\partial S}{\\partial q_k}.", "11855ba52b4328850fc589b87ffff73a": "k \\in \\{0,\\dots,n\\}\\!", "11862191664eabd9e4334dbced390e99": "\\ \\Delta S(T)=\\frac{\\Delta H(T_d)}{T_d}+ \\int_{T_d}^T \\Delta C_p dlnT", "11862dfc5381aac8394f9e094300eb50": "w(f) \\ge w(e).", "1186b520e88191c68e4618419fce17bd": "\\mathcal{E}_{q}^{\\epsilon}(d):=\\,\\!", "1186c8646441c30b989f25214271293d": "\\dfrac{0.04m}{0.72m}", "1186d8481598d52c9a2453e4a6ebbc36": " \\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} \\,\\!", "1186e4aef1c32ce1cc5c7696f4e9dc63": "_{nominal} \\delta_{ck}^2 =\n\\begin{cases}\n 0 & \\mbox{iff }c\\mbox{ = k} \\\\\n 1 & \\mbox{iff }c\\mbox{ ≠ k}\n\\end{cases}\n ", "1186e50f233800a571df55fa001395c3": "\\gamma_{j,0} = \\lambda_j", "1186f97d96d6d0c702a0ab3eaef1399e": "U \\in \\mathbb{R}", "1187531411d513c19ff8851837f62eec": "\\lambda_2=\\frac{\\tau-\\sqrt{\\tau^2-4\\Delta}}{2}", "11878b79048eeda0e028e3ee0ee866d2": "(p, q) (r, s)\n = (p r - s^* q, s p + q r^*).\\,", "1187bdc2f485779ae32b44e8fdbf3fd5": "Q_{-\\mu-\\frac12}^{-\\nu-\\frac12}\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)=\n-i(\\pi/2)^{1/2}\\Gamma(-\\nu-\\mu)(z^2-1)^{1/4}e^{-i\\nu\\pi} P_\\nu^\\mu(z).\n", "1187da31fa07e5af2bf2e78083f5eda3": "\\ a = -\\infty", "1187db7b1cea6a1d2ff8797c08a3c67a": "\\dot\\gamma = \\frac{8v}{d},", "118804bc76ac774723e9983f4ad140aa": "T^{\\mu\\nu} = \\frac{1}{4\\pi} [ F^{\\mu\\alpha}F^{\\nu}{}_{\\alpha} - \\frac{1}{4} \\eta^{\\mu\\nu}F_{\\alpha\\beta}F^{\\alpha\\beta}] \\,.", "1188281cecbfc8060884ac985217a0c4": "\n \\mathbf{M} \n = \\left|\\begin{matrix}\\mathbf{e}_x & \\mathbf{e}_y & \\mathbf{e}_z \\\\ x_A - x & 0 & 0 \\\\ 0 & -F & 0 \\end{matrix}\\right| -\n \\left|\\begin{matrix}\\mathbf{e}_x & \\mathbf{e}_y & \\mathbf{e}_z \\\\ x & 0 & 0 \\\\ 0 & R_0 & 0 \\end{matrix}\\right|\n = F(x-x_A)\\,\\mathbf{e}_z -R_0x\\,\\mathbf{e}_z = -\\frac{F x_A}{L}(L-x)\\,\\mathbf{e}_z \\,.\n ", "11885ffe8a1fd720676f797a532f40cc": "x=\\frac{-b\\pm\\sqrt{b^2-4ac\\ }}{2a}.", "118865a0340a13b11558a5cc6b51b6e6": "O([2N]^{-k}/k)", "11890bd7e4666828f0cb8a2ff8a2860c": " k \\leftarrow k+1 ", "118911bae1ff55e1d578dacb6f63603a": " c_i ", "1189557f74b6ccb594a569c9a42569ff": "v_i \\otimes w_k", "1189574b165b82b24531600cee765181": "H_M := H(X) \\gets \\lg(N)", "1189c6eff38297a56004dbab0161481d": "\\int_e^w \\!\\!\\!\\int_t^ {t+\\Delta t} (\\rho c \\frac{\\partial T} {\\partial t}\\,dt)\\,dV = \\int_t^ {t+\\Delta t} [ (k A \\frac{\\partial T} {\\partial x})_e - (k A \\frac{\\partial T} {\\partial x})_w]\\,dt + \\int_t^ {t+\\Delta t} \\bar S\\Delta V \\,dt ", "118a325bb9c1ff20d069893d80f3b476": "abc u_0 \\varepsilon^{\\frac{3}{4}}= \\mathrm{const},\\ u_{\\alpha} \\varepsilon^{\\frac{1}{4}}= \\mathrm{const},", "118a5de2914c8704bfbc5745ffcd8d14": "\\mathrm{Hom}_R(-,-)", "118aeb2447c35413a3226e8c0c439431": "\\tilde{g}_{\\mu\\nu}=\\Phi^{-2/(d-2)} g_{\\mu\\nu}", "118afe337ca8f12f1d094735f758eb79": " t+\\Delta t", "118b0c4d8422b17479c8fe882102fab5": "I(k,k') = \\Omega \\int_{\\Omega} d^{3}r \\, u^{\\ast}_{k'} (r) u_{k} (r) ", "118b2c8acd478fc1d4402c17eb15d367": "2^{2n-2}(2^{2n-1}-1)B \\,\\!", "118b785f852bd069283c8ac283cd0b7d": "{\\rm d} A {\\rm d} x", "118b7d052048b2502f9863993c09757c": " \\gamma = ", "118b9237b6c109b394e4dca7a7b80bb5": "(f\\circ g)\\circ h=f\\circ(g\\circ h)=f\\circ g\\circ h", "118c0a7d7443e382c6569a94cca23cf0": "\nx\\rightarrow x^5-10 x^3 y^2 + 5 x y^4 + x_0", "118cacf9a1f66cf0b90194e0c67f2948": "\\log h(t) = f(h_0(t),\\alpha + \\beta_1 X_1 + \\cdots + \\beta_k X_k).\\,", "118d30d87063888ba97b7a68c5ca4fb5": "_1F_1(a,b;z)", "118db7cdf5cab5b09371dd9e8c3f4986": "{\\{\\ ,\\!\\ \\}} \\!\\,", "118de5727a5720638ec32410684b21f5": "(\\phi \\to \\psi) \\to ((\\chi \\to \\psi) \\to (\\phi \\lor \\chi \\to \\psi))", "118e21c9c6f6134f0f2e4b34b9d1195c": "k_{col} = K (1 - g),", "118e2671863a596b85bcec326121bc9a": " R_{\\nu \\rho} \\ \\stackrel{\\mathrm{def}}{=}\\ {R^{\\mu}}_{\\nu\\mu \\rho} ", "118e97cce8f652a8871226d1e01a8393": "{s_{ln}} \\,", "118e994444c157b1117e2ca408f679ff": "\\mbox{gl dim } A := \\sup \\{ \\mbox{pd } M \\mbox{ }|\\mbox{ } M \\mbox{ is an A-module} \\}", "118eed6ffb5d9b8f8ccec215f27fdb8f": "\\mathcal{B}[f](u_1,\\dots,u_d) = \\mathcal{B}[f]\\big(\\pi(u_1,\\dots,u_d)\\big),\\,", "118f3789ea7f096e9157b72d7bd1c02f": "f_s,\\,", "118f38d4bece1e28f04b01d65934d137": "\\mathbb{P}(F)", "118f5785bc9bc50ada6e495aa90c2ac0": "=-\\frac{d}{dk}W_k[J_k[\\phi]]-\\frac{1}{2}\\phi\\cdot \\frac{d}{dk}R_k \\cdot \\phi=\\frac{1}{2}\\left\\langle\\phi \\cdot \\frac{d}{dk}R_k \\cdot \\phi\\right\\rangle_{J_k[\\phi];k}-\\frac{1}{2}\\phi\\cdot \\frac{d}{dk}R_k \\cdot \\phi", "118f60f62585568d3e49f58093211069": "X\\subseteq Y\\Rightarrow\\Psi_k(X)\\subseteq\\Psi_k(Y)", "118f7485c73497f33155d1302dc8888d": "\\mathrm{COP}_{\\mathrm{heating}} \\equiv \\frac{Q_H}{W_{in}}\\,", "118fdbf9e089dc81c3fa8c60408e4c47": "{2.5 \\times 10^{-6} m{^2}Pa^{-2}s^{-2}}", "119029b45cd67fb1901ee7fbdb6cfdb6": "\nr_\\mathrm{A}=\\left(\\frac{\\sin^3{\\theta_\\mathrm{A}}}{2-3\\cos{\\theta_\\mathrm{A}}+\\cos^3{\\theta_\\mathrm{A}}}\\right)^{1/3}\n~;~~\nr_\\mathrm{R}=\\left(\\frac{\\sin^3{\\theta_\\mathrm{R}}}{2-3\\cos{\\theta_\\mathrm{R}}+\\cos^3{\\theta_\\mathrm{R}}}\\right)^{1/3}\n", "11906e052ab54b6dcd153d26257280ac": "\\langle \\cdot , \\cdot \\rangle", "119088c6abe67ae14d8da5a8cc3fefff": " \\overline{r_{ff}} ", "1191137a87cae17151f5c67ad2d2b753": "h^2 = a(2b\\cos\\gamma - a).\\,", "119136c31a123d8e4c25d4c3b2d4b319": "\\bar{t}\\neq 0", "119157304a8e9dccf2ae92e1a4d8d528": "Ax^2 + Cy^2 + \\cdots = 0", "11916b467befc398376eb60abec77dc2": "\\epsilon_X : Y \\otimes X \\to \\mathbf{1}", "119181510e9a00cbd5603f72cc12cfa9": "\n{u}(x,y,z) = {u}(x,z)\\, {u}(y,z),\n", "119194a8f77eb04d66f419375393ce11": "\\mathbf{X}_I=\\mathbf{X}_{i_1}+\\ldots+\\mathbf{X}_{i_p}", "1191ba3ec975d874607877ed91adb2d0": " {_2^1}\\text{S}^\\beta + \\text{E} \\underset{\\text{k}_{2(2)}}{\\overset{\\text{k}_{1(2)}}{\\rightleftarrows}} \\text{C}_2 \\overset{\\text{k}_{3(2)}}{\\rightarrow} {_2^1}\\text{P} + \\text{E}, ", "1191d469a9d7ae2305957b67e8df3caa": "W^TBW=I", "11925742bd23fdd2a53f8560a1a9b52c": "BN(N+1)", "1192c92b40460d3de6f3fecc65506fc7": " \\mu: M \\to \\mathfrak g^* ", "11931db6bcc49bf29315743cfbbc50b8": "(\\alpha, f)", "11932b4ace4a57e8ec47bfc169429bc8": "\\frac{\\sqrt{2}}{12}\\,s^3", "11932be47944815bec73a02a437be0b0": "\\log_{10} K_\\mathrm{equil} = \\frac{2408.1}{T} + 1.5350 \\times \\log_{10} T - 7.452 \\times 10^{-5} (T) - 6.7753", "1193918c26f3f5d6c8db0e86b39a0fc8": "\\hat {f} [g(x-a)] = e^{-2 \\pi i a f_x} \\hat {f} [g(x)]", "1193efd45f0148ae378e163a16550997": " \\Pi = x - x^2 - xy ", "1193fba83d5dcc239ffbc0c75cb8ca13": "\\,\\! n = {^{(-1)}1} ", "1194447a264883ed8ff5c6280b0e0f7c": "A \\rightarrow B: \\{A, K_{AB}, T_B\\}_{K_{BS}}, \\{N_B\\}_{K_{AB}}", "11949ea1e2c5e8e324429d43186a9038": "\nq = \\frac{(V-Nb') \\, e^{-\\phi/(2kT)}}{\\Lambda^3}.\n", "1194fb4cca1ff8a3adffcfc700ad18fe": " X|T \\sim N(\\mu,1 /(\\lambda T)) \\,\\! , ", "119500fc05586ad523e495ca4a027e6c": "H_0(S) \\cong (\\mathbb{Z} \\oplus \\mathbb{Z} \\oplus \\mathbb{Z})/ (\\mathbb{Z} \\oplus \\mathbb{Z}) \\cong \\mathbb{Z}", "119504800928ae0d9182475cb7fa67fc": " \\sum_k \\sum_j \\left(q^{j(m - k + j) + k(k - 1)/2} \\binom{m}{k - j}_{\\!\\!q} \\binom{n}{j}_{\\!\\!q}\\right)x^k.", "119518e1874dbc7b36c22f7420fb31c8": "1, 2, .., m", "1195336672530a34c055afaad4c006fe": "\\phi^{19}=\\frac{9349+4181\\sqrt5}{2}\\approx 9349.000107\\,", "11956d84307b6b8a91891d5c5da786e1": "(1,2,2)", "1195a1633c6755e18081a6101af0f5ad": "\\{i: x\\succ_i^p y\\} \\in \\mathcal{B}", "1195af66163d88d84868a25b7e6e99a7": "M=\\{m_{j,i}\\}_{i,j=0}^{i,j=m-1}", "1195bc83f69451df12dc4d9bd37f0344": "x'=x+a-a'", "1195e2f8b4ad75cf9051fc847f4e35e2": "\\sqrt{2n+D}", "1195f43c7377511401e172c317eda3ef": "U_k(\\omega) \\to Q^2\\left(\\frac{i\\omega}{\\omega_0}+\\frac{\\omega_0}{i\\omega}\\right)^2", "119625a954be4f6ee2ec5ee26bbf06c6": "\\displaystyle e^{- a x} u(x) \\,", "1196281900f189fbcb4354a0b0b4d6d0": "I\\!I(X,Y)", "11963c2420f12963b5139fb91a3d9309": "x_1+..+x_c=z", "119662c334115736b426092d0bf909e3": "X \\sim \\mbox{Inv-}\\chi^2(2 \\alpha)\\,", "1196653608a6341651056e03ae8a2eab": " \\frac{\\partial\\varphi}{\\partial t} + \\tfrac12 \\boldsymbol{u} \\cdot \\boldsymbol{u} + \\frac p\\rho = 0, \\qquad (2) ", "1196856c89c5401658f8e2c0ddfdebf6": "S_{k_i}, \\ 1\\leq k_i \\leq m", "1196a6f959731867f65c6458edf4400e": "p \\rightarrow \\infty", "1196f14d64e2977e5924f665787dda50": "d \\gg a", "119704da2988cb5fe3f9e59b53fa963e": "h \\in \\mathcal F", "119722c4337ba587f5c93a19385099e5": "N=\\left(\\frac{Vf}{\\Lambda^3}\\right)\\left[-\\textrm{Li}_{3/2}(-z)\\right]", "119732894aa14f3dfa5a6b0e145d7338": "f_2(z)= 1/z \\quad", "119748e51a1c033ab0467a736182d0a7": "I_{n,m} =\n\\begin{cases}\n -\\frac{1}{(n-1)(bp-aq)}\\left [ \\frac{(ax+b)^{m+1}}{(px+q)^{n-1}}+a(n+m-2)I_{m-1,n-1} \\right ] \\\\\n -\\frac{1}{(n-m-1)p}\\left [ \\frac{(ax+b)^m}{(px+q)^{n-1}}+m(bp-aq)I_{m-1,n} \\right ] \\\\\n -\\frac{1}{(n-1)p}\\left [ \\frac{(ax+b)^m}{(px+q)^{n-1}}-amI_{m-1,n-1} \\right ]\n\\end{cases}\\,\\!", "1197741356960a7d0e56de762d602233": " \\omega_a = \\frac{2}{T} \\tan \\left( \\omega \\frac{T}{2} \\right) ", "119786f5c55dd01e5337311be735b87f": "\\sigma(X, Y)", "11979b0f0922123a8e1c7f631b6b33e8": "\\,\\!\\gamma : I \\rightarrow X", "1197adede16ecd997c9063966a0c7fd0": "y[n-1]", "1197c0de2407ba38f43b6ea44e8561b0": "\\operatorname{Ric}(X,Y) = \\rho(X,JY).", "1197d76a31e7c3758a1846b3df029776": "u^2+dv^2=m", "1197f1b7d276b380f3c00ceebc2776f9": " \\ c = 0 \\pmod p, c < \\frac{q}{2} ", "1197f78f4b29db77f08551ecabfcd3c5": "f_3(z)= \\frac{bc-ad}{c^2} z \\quad", "11981fcaccb9adcc3f4d9bf4a61b0221": "\\frac{1+x^2}{1+x}", "11984da54bf16025594706567ad8dc12": "d\\omega = \\frac{\\sin\\alpha_0}{\\cos^2\\beta}\\,d\\sigma,", "1198635902a94fd6f7d84a4731b99cba": "\\Psi^* = c_a \\psi_a - c_b \\psi_b", "119865e6e4ffdb05d89d7a44a6786132": "U < U*", "11988027f67efbc8fe80945e67d70f62": "Q\\,\\,\\!", "11988708fa7cab20ffc7d0ada445a039": "k = \\partial_v", "1198cacedd924fbdf1f67a1604f070bf": "[\\mathbf A, \\mathbf B]", "1198fadeac48af02a2b6f85a5a8aaf69": " {}^\\mathrm{N}\\mathbf{a}^\\mathrm{Q} = {}^\\mathrm{N}\\mathbf{a}^\\mathrm{P} + {}^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B} \\times \\left( {}^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B} \\times \\mathbf{r}^\\mathrm{PQ} \\right) + {}^\\mathrm{N}\\boldsymbol{\\alpha}^\\mathrm{B} \\times \\mathbf{r}^\\mathrm{PQ} ", "1199112de8b60370a99dbb519424aed4": " A^{[k]} = D^\\top D/4\\cdot \\mathrm{area}(T)", "11992554b6fcc8b050bb145982ab9dd3": "\\partial \\pi", "11992c60618ea42efe5b74790f6feb35": "\\begin{align}\n n = 0: \\frac t{e^t-1} &= \\sum_{m=0}^\\infty B_m\\frac{t^m}{m!}\\\\\n n = 1: \\frac t{1-e^{-t}} &= \\sum_{m=0}^\\infty B_m\\frac{(-t)^m}{m!}.\n\\end{align}", "1199ac3fb6066166352c30708f123832": " V_n ", "1199af256b4e15dfb21723f0e200f87c": " N \\rightarrow N / 2 \\,\\!", "1199c2c8572c2d7d1f6dc23abd25c930": "\\mathbf{u} = -\\left( \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} B(\\mathbf{x},t) \\right)^{-1} \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} f(\\mathbf{x},t)", "1199ece478de3ef6bf73f15b1f2e77df": "PL\\;=P_{t_{dBm}}-P_{r_{dBm}}\\;=40 log(d)-10 log(G h_t ^2 h_r ^2) ", "119a3ce9e018065809b5093744dc656c": "V_{OC} = \\sum_{i=1}^3 V_{OCi}", "119a9067998d4df42d096fd47b77f532": "H(m) \\leq H(m+1)", "119a92921177be708dab76f123d4af7c": "(-1)^n B_n(-x) = B_n(x) + nx^{n-1}\\,", "119b0b0884e2cb8f40c4dc5230c20a3e": "\\hat H = J_1 \\sum_{\\langle ij \\rangle}\\vec S_i \\cdot \\vec S_j + J_2 \\sum_{\\langle\\langle ij \\rangle\\rangle} \\vec S_i \\cdot \\vec S_j ", "119b6064e029ee4d3f47d285dbc4c79e": "i_a", "119badd85e22f4c6b79debb4bb6f0c02": "\\frac{1}{2}\\|\\mathbf{w}\\|^2", "119c0321681990428e5952970f009c3c": "m=p_1,\\dots p_r", "119c4e0c8d184c3024e25622aee4ca6c": "\\nabla: D_X \\rightarrow End_K(M), v \\mapsto \\nabla_v", "119c5ba31c1ef98b00c6808f9813a372": " \nl_2= a_{00} - \\mathcal{L} \\left\\{\n \\frac{\\omega a_{10}+a_{01} - \\mathcal{L}(2a_{20} \\omega+a_{11})}\n{2a_{20}\\omega+a_{11}}\\right\\}+ \\frac{\\omega a_{10}+a_{01} -\n\\mathcal{L}(2a_{20} \\omega+a_{11})}\n{2a_{20}\\omega+a_{11}}\\times\n\\frac{ a_{20}(a_{01}-\\mathcal{L}(a_{20}\\omega+a_{11}))+\n(a_{20}\\omega+a_{11})(a_{10}-\\mathcal{L}a_{20})}{2a_{20}\\omega+a_{11}} =0,\n", "119cc029329fc282395091564547bd89": " 0 = F_{\\text{pressure}} + F_{\\text{viscosity, fast}} + F_{\\text{viscosity, slow}} ", "119cdc8cdd8df0534b9aef84f7df94fc": "(x^{10}-20 x^8+143 x^6-437 x^4+500 x^2-59)", "119d72374eb5b26635e37001a1257bc4": "c_{v}", "119d77f8553d28bde7c43136cc7b069e": "\\| y_2 - y_1 \\|^2 \\leq 2\\delta^2 + 2\\delta^2 - 4\\delta^2=0 \\, ", "119d7dd49c4009c728e69299ae8b21c0": "\\alpha_j \\leq \\beta_j \\leq \\alpha_{j+1}.", "119d9b05e3cbd443a992bdff078f1f9d": " h_ih_j = g_{ij} = \\mathbf{b}_i\\cdot\\mathbf{b}_j \\quad \\Rightarrow \\quad h_i =\\sqrt{g_{ii}}= \\left|\\mathbf{b}_i\\right|=\\left|\\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\\right| ", "119e142b3101dda61c15a83671abeae9": "= \\frac{a}{b} \\cdot \\left(\\frac{cf}{df} + \\frac{ed}{fd}\\right) = \\frac{a}{b} \\cdot \\frac{cf + ed}{df}", "119ecbf2a884fed34e72da319b352456": "\\log \\gamma_{\\pm} = \\frac{-A_\\gamma I^{1/2}}{1+I^{1/2} }+\\beta b.", "119f5f28a27f82e9a83136bb76697845": "\\operatorname{succ}(n)=n+1", "119f67752e260691b2709eddbd8f0225": "\\,g(aX + Y, Z) = a g(X,Z) + g(Y,Z)", "119face623137dc751fc89691385965c": "S_4/K", "119fb29e75e599daad2d71ebf27f75b9": "\\tfrac{275}{24192}", "11a045cc03a1c9d4139485d72c34c10d": "f^{\\prime\\prime} = \\frac{(f^\\prime)^2}{f} + \\frac{f^\\prime}{r}, \\; (h^\\prime)^2 + \\frac{2 h^\\prime h}{r} + \\frac{h^2}{r^2} = \\frac{4 f^\\prime}{r \\, f}", "11a052fee3cc5272375405e63eb45647": "\\omega^{ryb}", "11a07849715982d8cf6e9eceff597635": "\\|x+y\\|^2+\\|x-y\\|^2 = 2\\langle x, x\\rangle + 2\\langle y, y\\rangle = 2\\|x\\|^2+2\\|y\\|^2, \\, ", "11a0ac94a44eb0fd00fd03b96f60ba4b": "\\ E_a \\equiv -R \\left[ \\frac{\\partial \\ln k}{\\partial ~(1/T)} \\right]_P ", "11a15c8f28c6c2b4bab01103b6508f00": "H=\\alpha(2)-1", "11a166f1c369f04d3ee49a216a8fef9f": "\\varphi^2 = \\varphi + 1 = 2.618\\dots", "11a1a8160dbd92411cf5b8d37128eee8": " V_2 = Z_{21} I_1 + Z_{22} I_2 ", "11a1d4d1e90a6af06d040a7a28c35a8d": " 2 = -i(1+i)^2 ", "11a1d4e5b9afd12bc38651facdd27244": "c_q(n)=\n\\sum_{a=1\\atop (a,q)=1}^q\ne^{2 \\pi i \\tfrac{a}{q} n}\n,\n", "11a26849e307f598061d5650f570b9b1": "80^2", "11a26f4832d1a1039fd16b6bf68b3cff": "a,b\\in[\\omega]^\\omega", "11a289482e8f405fbbbc34bfd6e21b0a": "\\frac{u_{i+1} + u_{i-1}\\ -\\ 2u_i}{{\\Delta x}^2} \\rightarrow \\frac{\\partial^2 u }{\\partial x^2}", "11a2a55048ce3c5bc3bd75020698b623": "C[W^{-1}] \\to C", "11a2b4c52754dc00cf52f60e412aec85": "(\\R\\cup{\\infty})^2", "11a2c76600fbe806e0bc94ab50ccc678": "(\\nu x)(P | Q) \\equiv (\\nu x)P | Q ", "11a2ebf26959024ffbeeddc4b59359cc": "\\scriptstyle C^{1,\\alpha}", "11a397457592e391c883ab5f6882d9f3": "\\{f_{i},f^{(k)}_{j}:i 0", "11a7eabd1b1c44aa3f4c873b38028ae3": " \\psi^{(0)}_\\pm ", "11a84a75cf428a917f16ea7ff0945e8b": "\n EI~\\cfrac{\\mathrm{d}^4 \\hat{w}}{\\mathrm{d}x^4} - \\mu\\omega^2\\hat{w} = 0 \\,.\n ", "11a8d86726d302556a29dfa1368e290a": "l_m = c t / 2", "11a90372ae2b049f9adffee655e2a72a": "K'_1", "11a96c38f521725bc19c7ae083c5289b": "\\mathrm{Prog}", "11a9d0f35ce9d8e4f180c6abe854a211": "\n \\begin{align}\n & EI \\frac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} = 0 \\\\\n & w|_{x = 0} = 0 \\quad ; \\quad \\frac{\\mathrm{d} w}{\\mathrm{d} x}\\bigg|_{x = 0} = 0 \\quad ; \\quad\n \\frac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\\bigg|_{x = L} = 0 \\quad ; \\quad -EI \\frac{\\mathrm{d}^3 w}{\\mathrm{d} x^3}\\bigg|_{x = L} = F\\,\n \\end{align}\n ", "11aa0643bf16b4b9043967e8de22625f": "i \\leftarrow i+r", "11aa3248f8ca1c728ccaeb6cf8b1add0": "b+d<1\\,\\!", "11aad80233a22332f9547cb445c0b2fb": "G(L,R,E)\\,", "11aae48d8fc93f983099e28586e1a3b3": "d \\colon M \\times M \\rightarrow \\mathbb{R}", "11aaeb6ceb570fdd697907fbbf7388d5": "1-\\gamma_kdt", "11ab2b2ccbfeceba3b5595ca34baca26": "(\\pm 1, 0, 0, \\dots, 0)", "11ab4b54123ec919b1071b230866710d": "\\frac{1}{4\\pi} \\operatorname{Im} \\Bigl[ \\int d^4 \\theta \\frac{dF}{dA} \\bar{A} + \\int d^2 \\theta \\frac{1}{2} \\frac{d^2 F}{dA^2} W_\\alpha W^\\alpha \\Bigr] \\,", "11abcd51751cf261db1b87f46460f1ce": "i_n^2=-1", "11abd2f806d34ae561ca62929700cf00": "U = \\{v \\mid \\Omega(v) = p\\}", "11abfd083e6b86882c80a3378e2b3d1e": "\\psi_1(x) \\sim \\frac{x^2}{2}.", "11ac146493ae65f4fbf4f05b484e64bc": " p_0 =\\, p_1 +\\ \\tfrac12\\, \\rho\\, v^2.", "11ac209ac1341776954956d0f6554cc7": "\\operatorname{P} (Z_i = 1 ) = \\tau_1 \\, ", "11ac57ef5edcf4bbe51d6697476aa894": "\\!\\ Re < 10^5", "11ac6a2f0d3ecf155dca5dd67ec23781": "\\sigma_v^2", "11acb0d6d50d41bffdecce3e83039b09": "(A^*A)^{-1}\\,\\!", "11ad5eb0249d88065004e74b38ee9e66": "(f\\big|\\gamma)(\\tau)=(\\rho(C\\tau+D))^{-1}f(\\gamma\\tau).", "11ae002a0fb31ee6f3fd7596e4b07a4d": "t_r=\\frac{4}{\\sigma}{\\mathrm{erf}^{-1}(0.8)}\\cong\\frac{0.3394}{f_H}", "11ae07a6a8c4ff9601203bdead6014bf": "A_\\sigma^\\mu", "11ae468a3b92b84d2e6b12ae0eebb9b2": " \\frac{d}{d r} \\left( p +\\frac{B_z^2}{2 \\mu_0 } \\right) =0 ", "11ae968c4e2df747c7091ff1e6f8e2b5": "{\\tilde{B}}_{9}", "11aebcef8709da91fea76d93c5545261": "\\mathfrak{t}", "11aedd0e432747c2bcd97b82808d24a0": "FR", "11af963ffd06d004bc5daa73379f4e2f": "\\beta^{a} \\gamma + \\gamma \\beta^{a} = \\beta^{a}", "11b054f14693226433206621deab70cf": "\\hat i", "11b05d3bf74162ae4340a4e8a50ee956": " S_y = S_{zx} = \\int_{\\partial \\mathcal{V}} [(z - z_\\text{com}) T^{0x} - (x - x_\\text{com}) T^{0z} ]dxdydz ", "11b0f750d5701cde064ead807a85552a": "h_e(X)\\,", "11b1198851cf3ab82e536cd11abc14ae": "f(x \\pm h) = f(x) \\pm h f'(x) + \\frac{h^2}{2}f''(x) \\pm \\frac{h^3}{6} f^{(3)}(x) + O_{1\\pm}(h^4). \\qquad (E_{1\\pm}).", "11b179c60b7c7af416738accb6ef9b91": "\\scriptstyle p\\equiv 1\\; mod\\; 4", "11b1aa371e1b4354f6a238494c3d1fd3": "M = k", "11b1adea22b58011cd4f701393cfdc79": "\\scriptstyle V\\left(\\frac{1}{e}\\right)", "11b1b3bb43582b1f3cb0dc554b4dc7fb": "fg+gf=(f,g), \\,", "11b1c063d3bf3bbac5b0efed84f7001c": " -B_1 \\left(f(n) + f(m)\\right) = \\frac{1}{2}\\left(f(n) + f(m)\\right).", "11b228092f47746cbe2158b88f0bceb7": "x_A+y", "11b2abf03847e92e346d70d816fba149": "\n \\begin{align}\n N_{11} &= \\cfrac{Eh}{2(1-\\nu^2)}\\left[\\left(\\frac{\\partial w}{\\partial x_1}\\right)^2 \n + \\nu\\left(\\frac{\\partial w}{\\partial x_2}\\right)^2 \\right] \\\\\n N_{22} &= \\cfrac{Eh}{2(1-\\nu^2)}\\left[\\nu\\left(\\frac{\\partial w}{\\partial x_1}\\right)^2 \n + \\left(\\frac{\\partial w}{\\partial x_2}\\right)^2 \\right] \\\\\n N_{12} &= \\cfrac{Eh}{2(1+\\nu)}\\,\\frac{\\partial w}{\\partial x_1}\\,\\frac{\\partial w}{\\partial x_2} \n \\end{align}\n ", "11b34e87417616c61bbedffacac1b65f": "xy \\equiv yx\\,.", "11b399b94a25be9b0f801511afabf8e2": "f(r) = (1 - r^2)^2", "11b41fef36dd8adbde5ae9673b2b065d": "V(q) = \n \\begin{cases}\n 0 &q \\in \\Omega \\\\ \n \\infty &q \\notin \\Omega\n \\end{cases}\n", "11b4310a65d3510d0c485f8f55f6909d": " f(5) = 0.01024 \\, ", "11b49f590d903ad849cdc607e34e8d16": " Resolution = 2^{{group} + {\\frac{element - 1}{6}}} ", "11b50b8fbbee172b3204f66c98711b23": "c_{i}(k)", "11b5160b179795fe0624b937c8da905c": "(\\mathrm{id}_V)^{\\mathbb C} = \\mathrm{id}_{V^{\\mathbb C}}", "11b59e2a38012325167752b7700ebb8a": "\\dim R[T_1,\\ldots,T_n] = n + \\dim R, \\,", "11b5a76d5751498d4bd31705e9407ae1": "O(n2^n)", "11b60dff80227709a6cb11540a856c8b": "C'C''", "11b610e497c1c994acebd013e63278ae": "S = g^{ab} (\\Gamma^c_{ab,c} - \\Gamma^c_{ac,b} + \\Gamma^d_{ab}\\Gamma^c_{cd} - \\Gamma^d_{ac} \\Gamma^c_{bd})\n=\n2g^{ab} (\\Gamma^c_{a[b,c]} + \\Gamma^d_{a[b}\\Gamma^c_{c]d})\n", "11b61470da9f0be95bf73a19e9361477": "c=\\sqrt n", "11b617e9bb799b3805e664ad7faea35e": "x_{\\mathrm{i}} = \\frac{p_{\\mathrm{i}}}{p} = \\frac{n_{\\mathrm{i}}}{n}", "11b636f081d5f3565c2adb6892000a94": "k/q.", "11b668f53f9a2e32b74f2b5d837664f7": "\\mathcal{N}_R", "11b6906b668bbd84ebc34fe2f7a3d336": "\\frac {m_0} {m_1}", "11b695a31ec2e50b1f0de0245fdc05fb": "(x'_i)", "11b73dd6e8ad1c143ca5ab30fa5d6a28": "\\frac{\\partial y}{\\partial \\mathbf{X}}.", "11b76c61361bf60df3db38e37da01777": "H(s) = \\frac{ s^2 + \\omega_z^2}{ s^2 + \\frac{ \\omega_0 }{Q}s + \\omega_0^2 }", "11b78314ceb6e861cca1d51a6f20a9e4": "T = g_m \\left( R_D\\ ||r_O \\right) \\approx g_m R_D \\ , ", "11b7c96e63bc207eeb4534586c222726": "Impulsiveness", "11b7cda882c050df7c5ec8acbddc3d60": "\\varphi(t) = \\sum_{n=0}^\\infty \\frac{(it)^n}{n!}e^{n\\mu+n^2\\sigma^2/2}.", "11b7ebf120fae22e0d7aaa9628d9e3d9": "\\Delta U = N C_V \\Delta T\\,\\!", "11b7f71278d18664661fa856b8d7f973": "matrix(3, 3)", "11b814048a39d34d64a5f7607b36979d": "mN_k - N_{k+1},", "11b8294189f3a23fbe2327e2d3da5885": "\\sqrt[3]{n}_s", "11b8318688ada4ec54f10c83f608bca4": " -6\\pi r_p \\mu V_r + \\frac{4}{3}\\pi r_p^3 \\frac{V_t^2}{r}\\rho_p -\\frac{4}{3}\\pi r_p^3 \\frac{V_t^2}{r}\\rho_f =0 ", "11b83510388ffdecff441c67f6b7b90f": "(\\mathbf{a}\\otimes\\mathbf{b})\\cdot\\mathbf{n} = (\\mathbf{b}\\cdot\\mathbf{n})\\mathbf{a}", "11b8397342d93d09ee9704377150a7c7": "\\tilde{\\nu} = \\frac{\\nu}{c_\\mathrm{n}}", "11b88743ebc05a218dd129c5fb2c7d11": "\\, P(X=1)=p", "11b8dc85f0024f4753b55976d657959d": "B\\subseteq S", "11b8ed25729a8c79af1d22e958effdf1": "{B}_{3}^{(1)}", "11b90f5f8b1c7511f7881725aa078db5": "2^{-7/4}", "11b9547165d679e0d80ed6596d14faa0": "(mR)^2+(n/R)^2", "11b9732f985f005686a8b77109599314": "\\displaystyle{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).}", "11b9ccb302aa3f5f626a860c8b680bf5": "(\\alpha \\to \\beta)", "11b9f85ab74578e12373c1cbfa3ebec5": "\\frac{d u_i}{d t} + \\frac{1}{\\Delta x_i} \\left[ \nF \\left( u^*_{i + 1/2} \\right) - F \\left( u^*_{i - 1/2} \\right) \\right] =0. ", "11b9fdfa6a1f1146cd765f616a6740a7": "G_B(\\tau=0^-)=\\frac{1}{\\beta}\\sum_{i\\omega_n}\\frac{e^{i\\omega_n 0^+}}{i\\omega_n-\\xi}=-n_B(\\xi)", "11ba21c9ecec309ae0f871e1232bdb26": " \\mathbf{D}(X) ", "11bb45bf03607fa4a8780b2eb967b8f8": "\\textbf{x}_{e} = \\textbf{x}_o + \\gamma (\\textbf{x}_o - \\textbf{x}_{n+1})", "11bb48e2b88c2f4b2f122c08b6091665": "\\!Y \\subseteq X", "11bb7801e0ddd1340404d859e0028056": " C_{ijkl} \\Rightarrow C_{\\alpha \\beta} =\\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\\\\n C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\\\\n C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\\\\n C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\\\\n C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\\\\n C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \n\\end{bmatrix}.\n\\,\\!", "11bb7fe00bad88246f53d380583586cb": "\\mathbb{E}^x[X_{\\tau-}^jX_\\tau^k]", "11bba3af6b0b6e810be8a9d4b7103a39": "\\pi \\sim (\\ell)^{-1}(t)^1(\\ell/t)^1 \\sim 1", "11bbdf62da879f31eab2cd9947d2b230": "\\delta_H(d_N) = \\delta_{H0} \\frac{\\exp\\left(-d_N/\\lambda_N\\right)}{1 + d_N/d_0},", "11bc1cb1f8cbbec01a2643bbb0c314ac": "L(s_{i_k}) = L(s_{i_k+1}) = \\ldots = L(s_{i_{k+1}-1}) = L(r_{j_k}) = L(r_{j_k+1}) = \\ldots = L(r_{j_{k+1}-1})", "11bc2f7ed78af035a2b47b0adb6a0677": "1/R^4", "11bc99ce86e4f6c555e8daecfa5888dd": " R=\\frac{2 G M}{c_s^2} ", "11bcde0d2c833b7c1002feb91b6b4c93": "\\widehat{C}_{Y\\mid X} = \\boldsymbol{\\Phi} (\\mathbf{K} + \\lambda \\mathbf{I})^{-1} \\boldsymbol{\\Upsilon}^T", "11bcec96c06854bfe499bc54f4ee0da5": " \\mathfrak{q} ", "11bd10ddb1f764d6fcbded698dc86707": "\\varphi(-r) = i\\sqrt{r}", "11bd2361a6eb807f36b53586e5f80af6": " \\alpha_H = a_1x_1 + a_2 x_2 + \\cdots ", "11bd9da3b32a4786783d25184a03bd31": "r=\\left(g^{k}\\bmod\\,p\\right)\\bmod\\,q", "11bdc1e30fd3ed1b9887e8ddeb96c06e": "f(x)=g(x)(h(x))^{-1}", "11bee95572ff1b37d3161ca75fab97c3": "\\delta_g = \\frac{\\tau c_0}{2 cos\\gamma}", "11befdb7e597dc81e6b7d87cdf7b5e2d": " \\mathrm{SU}(2) = \\left \\{ \\begin{pmatrix} \\alpha&-\\overline{\\beta}\\\\ \\beta & \\overline{\\alpha} \\end{pmatrix}: \\ \\ \\alpha,\\beta\\in\\mathbf{C}, |\\alpha|^2 + |\\beta|^2 = 1\\right \\} ~,", "11bf062195375b554ddaea39d71c36aa": "\\ \\displaystyle \\varphi(q,\\alpha,u)", "11bfc01102a4c23ece55bfaef15791db": " a_b=a ", "11bff4ef546eaddaa32849dca0ecbf2f": "\\det(\\varphi I_n-A)", "11c0521e5bb89ec0ecce70100746eaf6": "m \\cdot O(1) = O(m) = O(n)", "11c05f4117923f95ac1edf6cd520e3b9": "X_1,...,X_n,Y_1,...,Y_n", "11c06d5b5312a1707492fda3188f324d": "s = a\\varphi", "11c07821cf89ac201e0641174b78632a": " \n(Eq. 9) \\text{ } \\text{Subject to: } \\lim_{t\\rightarrow\\infty} \\overline{Y}_i(t) \\leq 0 \\text{ } \\forall i \\in \\{1, ..., K\\} \n", "11c08f093cff73b3d86a5147a5b6cd28": "q^n = k^n(\\cos n \\theta + \\epsilon \\sin n \\theta)", "11c09e7863d9d3194e9096ac86bae950": "Rec(w',s)=s+dec(w'-s)=w", "11c0cc3b57d3256be2cb8ff9cdaf890b": "s(v_j, v_k) = v_j \\cdot v_k", "11c10ca094f47cfa99c7de7d4dc4eb16": " \nf_1(x; \\nu) = \\frac{2^{-\\nu/2}}{\\Gamma(\\nu/2)}\\,x^{-\\nu/2-1} e^{-1/(2 x)},\n", "11c114e7209a5960780cfb1ebd9fc9f6": "Q=c_{1}e\\left (\\int^{1}_{0} f_k e^{-x}\\,dx\\right )+c_{2}e^{2} \\left (\\int^{2}_{0} f_k e^{-x}\\,dx\\right )+\\cdots+c_{n}e^{n}\\left (\\int^{n}_{0} f_k e^{-x}\\,dx \\right ) ", "11c19357ff9addad82843b1fa7701271": "\\frac{\\partial W}{\\partial t} = -\\{\\{W,H\\}\\} = -\\frac{2}{\\hbar} W \\sin \\left ( {{\\frac{\\hbar }{2}}(\\stackrel{\\leftarrow }{\\partial }_x\n\\stackrel{\\rightarrow }{\\partial }_{p}-\\stackrel{\\leftarrow }{\\partial }_{p}\\stackrel{\\rightarrow }{\\partial }_{x})} \\right ) \n\\ H =-\\{W,H\\} + O(\\hbar^2),", "11c19633c6d7b85eec5efafdd655392d": "i + 1", "11c1c2288469326fb14435817d4d7d00": "\\textstyle d=4", "11c1ca1e01883a29c81f45fd43419fd3": "\\{x_n\\}_{n=1}^\\infty", "11c1cfdab01ae7681edfb77fea93eec0": "\n \\mathbf{x}^{(0)} = \\begin{bmatrix}\n 1 & 0\n \\end{bmatrix}\n", "11c1fc7aa258b5e0ae4870e0cc1ba60a": "\\delta \\Gamma^\\lambda_{\\mu\\nu}=\\frac{1}{2}g^{\\lambda a}\\left(\\nabla_\\mu\\delta g_{a\\nu}+\\nabla_\\nu\\delta g_{a\\mu}-\\nabla_a\\delta g_{\\mu\\nu} \\right).", "11c2228ac0051bca1eda480e726b37f9": " \\vec{p}_0 = \\frac{1}{\\sqrt{1-\\omega^2 \\, R^2}} \\, \\partial_T + \\frac{\\omega \\, R}{\\sqrt{1-\\omega^2 \\, R^2}} \\; \\frac{1}{R} \\partial_\\Phi ", "11c233a746ebb17a0e3a25cfab18831d": " J < q_{0.95}^{\\chi^2_{k-\\ell}} ", "11c2355828116823354ed45a14920c15": "p_k^\\prime(t)", "11c24d6589c55fdb677fa5fe0feda4d0": "\\pi:\\; U \\mapsto V", "11c275a6be5c8621edd125901fc045a5": "R^*_{S_j}(t)= \\frac{ \\displaystyle \\sum_{b_j\\neq0} \\sum_{\\beta_j} \\frac{b_j q}{ ^{b_j}M_{S_j}} \\ {^{b_j}_{a_j}}S^{\\beta_j}_j (t)} { \\displaystyle \\sum_{b_j\\neq\na_j} \\sum_{\\beta_j} \\frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \\ {^{b_j}_{a_j}}S^{\\beta_j}_j (t)\n} ", "11c2c887206c7ba09e1a3061863bf742": "\\exists a \\, \\exists b\\, \\exists c\\, [\\neg Babc \\and \\neg Bbca \\and \\neg Bcab].", "11c2d681fab713b6ebdd533023379447": "\\tau_x=-M_yf,\\,", "11c2e1b509bc74a92da9357c184cda7a": "(2)^e", "11c30e85ef351220045729fb5f1b9006": "\\mathcal{I}", "11c3199df2c32904b81c3e990cd325f2": "x_0, \\ldots, x_{N-1}", "11c34a9e604e466a312cdd3133c9095c": "{\\Bbb C} \\times H", "11c3a8323abe40892c9103b32870db76": "H = \\frac{n}{1/a_1 + 1/a_2 + \\cdots + 1/a_n}", "11c3d792558427ed381de89f456787f4": "\n g_1 = \\frac{m_3}{m_2^{3/2}} \n = \\frac{\\tfrac{1}{n} \\sum_{i=1}^n (x_i-\\overline{x})^3}{\\left(\\tfrac{1}{n} \\sum_{i=1}^n (x_i-\\overline{x})^2\\right)^{3/2}}\\ ,\n ", "11c40d647d3b4adb7e7ac740b2acf21f": " \\nabla^2 \\mathbf{E} \\ - \\ { 1 \\over c^2 } {\\partial^2 \\mathbf{E} \\over \\partial t^2} \\ \\ = \\ \\ 0", "11c416dfecde0dc69eb9c3775e3fec1b": " CLOSED_d ", "11c42aec546c6dc6a4e9be0c4bf4e16c": "\\left\\lang \\left( \\begin{smallmatrix} 1 \\\\ 0 \\\\ 0 \\end{smallmatrix} \\right), \\left( \\begin{smallmatrix} 0 \\\\ 1 \\\\ 0 \\end{smallmatrix} \\right), \\left( \\begin{smallmatrix} 0 \\\\ 0 \\\\ 1 \\end{smallmatrix} \\right) \\right\\rang = \\mathbb{R}^3", "11c4f2c6ef72b37ae38f31544368308a": "w_i=\\frac {w^{i-1}}{V_1},", "11c513ebb0617f88f6469422caa3c720": "(S\\cdot X,Y)_\\sigma,", "11c5ab1242f0d577a4816b31274ee30f": "g^{\\star} (g^{\\star})^* = 1.", "11c63dec07d8c80c34ff55fd7ce891a6": "y \\wedge x \\wedge y = y", "11c664622c3835a92ca7f8d3f599ba80": "I_{static}", "11c68ffa2e7eec391b4b212c2e12ec04": "\\Delta\\sigma=\\arctan\\left(\\frac{\\sqrt{\\left(\\cos\\phi_2\\sin\\Delta\\lambda\\right)^2+\\left(\\cos\\phi_1\\sin\\phi_2-\\sin\\phi_1\\cos\\phi_2\\cos\\Delta\\lambda\\right)^2}}{\\sin\\phi_1\\sin\\phi_2+\\cos\\phi_1\\cos\\phi_2\\cos\\Delta\\lambda}\\right).", "11c6ab4447bad8c7801d0654a9b21092": "\\int_{-\\infty}^\\infty x^{2(n+1)}e^{-x^{2}/2}\\,dx=\\frac{(2n+1)!}{2^{n}n!}\\sqrt{2 \\pi} \\quad n=0,1,2,\\ldots ", "11c6e015eb1d5d6185519eb1edbb1df2": "\n \\cfrac{\\partial^2 }{\\partial x^2}\\left(EI\\cfrac{\\partial^2 w}{\\partial x^2}\\right) = - \\mu\\cfrac{\\partial^2 w}{\\partial t^2} + q(x)\n ", "11c70181557b8e1c016ff5a7dc653ef1": "\\phi(\\mathbf{R})=- \\mathbf{p}\\cdot\\mathbf{\\nabla}\\frac {1}{4 \\pi \\varepsilon _0 R}\\ , ", "11c75b97ad902b8520ecea6c6f48304d": "W^\\ast(s) = \\frac{\\beta \\lambda + s \\lambda - \\beta \\mu + \\alpha \\mu - \\sqrt{4\\beta \\alpha \\mu(\\mu-\\lambda) + (s \\lambda + \\beta(\\lambda-\\mu)+\\alpha \\mu)^2}}{2 \\beta (\\lambda - \\mu)}", "11c79bfffb8bc4a72452231585cd194c": "C^2(\\Omega, \\mathbb{R}^m)", "11c82e5e053fdb847715a7aa0916406b": "J_1(x)=-4x\\cos(x)+4\\sin(x).\\,", "11c88f24866da03aaa79ac301172a694": "\\bold{P} \\cdot \\mathbf{\\hat{n}} dS = -\\iiint\\nabla\\cdot\\mathbf{P}dV ", "11c893056850d32ce5d9e5fe8cbfe599": "\\int_0^t H\\,dX = \\lim_{n\\rightarrow\\infty} \\sum_{t_{i-1},t_i\\in\\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}}).", "11c89b370fbb18501ae7081bdbd9920c": "x_0,x_1,\\ldots,x_n", "11c8abb17c2c2f91e9ff74e8b2c7ed26": " V = \\frac {3\\pi^2}{16}R_{max}^2 L ", "11c8b552200491507f82f6d877e63616": "\\lnot \\textit{par}(x_{ht},x_{me})", "11c8b861d9a9f0285a93dfeee28ec1c9": "f \\circ \\hat{f} = [n].", "11c8fdc0d184ad9aa59cb3157814353d": "SF = {\\sqrt (W) \\over R} {{(1+sin \\phi)} \\over 2}", "11c9b7d78b8053e34ca70d871418c642": "\n\\kappa(\\vec{\\theta}) = \\frac{\\Sigma(D_d\\vec{\\theta})}{\\Sigma_{cr}}\n", "11ca72722cce69807a78b61dbf7b6091": "\n \\frac{d^2 u}{d y^2} = \\frac{1}{\\mu} \\frac{dp}{dx},\n", "11cad0ca4262664138d60df9da05a0e2": "r \\frac{\\partial u}{\\partial r} = r \\frac{\\partial u}{\\partial x} \\cos \\varphi + r \\frac{\\partial u}{\\partial y} \\sin \\varphi = x \\frac{\\partial u}{\\partial x} + y \\frac{\\partial u}{\\partial y},", "11cad85f8949f41cf907f3afea547108": " \\psi (x)", "11cb2d6368c679f424dc1370b2c0ebbe": "v_\\mathrm{F}", "11cb64100d783c2a99ac646e92eb3e9d": "\\sigma_L=\\sigma_L^D + \\sigma_L^\\mp ", "11cb80b7e73f8cc6c6808d47ef105496": "|z_1|=|z_2|=|e_1|=|e_2|", "11cb8224526266beb3f41ce55c5046aa": "Tr\\rho^p K^*\\rho^{1-p}K", "11cb8e587f80d53151d9905e1bc1abaa": "f_k = p_v(\\alpha^k).\\,", "11cba06eb995822716883e40f36575b5": "f_n(x)=\\frac{1}{x^2+n^2}", "11cbc8b984bf7ff2950b4f43cbde41e6": "\\mathbf{E} = \\frac{1}{4\\pi}\\frac{q}{r^2} \\mathbf{\\hat r}", "11cbeafe0a9b2b0740d50892973a5082": "B_k:=\\left(\\bigcup_{i=1}^{k-1}C_i\\right)\\cap C_k", "11cc37d6dae6b905a0a0eb2ff087ae24": "L^2", "11cc4ff6fe25698e2cde38d320359ae9": "\\frac{\\sin A}{\\sinh a} = \\frac{\\sin B}{\\sinh b} = \\frac{\\sin C}{\\sinh c}.", "11cc662314411444210390ad93eaf12d": "-\\infty\\!\\,,", "11cc7ea0c26c5ec9d6429e9734a29c14": "\n\\bar{P}(x,s\\mid x_0) = \\frac{1}{\\bar{k}_{\\alpha}(s)} \\bar{P}_\\text{nrml}(x,s/\\bar{k}_\\alpha(s)\\mid x_0).\n", "11cca6865222dbe583174e7587a026d5": "HL \\rightleftharpoons L+H:pK =-\\log \\left(\\frac{[L]\\{H\\}}{[HL]} \\right) ", "11ccc6827cf285b8863ad2ef0a17a3e3": "\\otimes : M \\times N \\to M \\otimes_{R} N", "11ccf7a1045d5e094c756683e63ec166": "|g\\cap {\\mathcal O}|=0, \\ |g\\cap {\\mathcal O}|=1", "11cd02424617a7b8edb9d721510b64c5": "\\mathbf{Q}^*_{M \\times 1} = \\mathbf{B}_{M \\times N} \\mathbf{R}^*_{N \\times 1} \\qquad \\qquad \\qquad \\mathrm{(1)} ", "11cd21b65565867b1e742f48982e50b7": "J_\\nu(x e^{3 \\pi i/4}),\\,", "11cd913b85d6693c52b1046acb084e1b": "\nf(x)=-\\boldsymbol{\\alpha}e^{x\\Theta}\\Theta\\boldsymbol{1}\\; ,\n", "11cdc20418b4ab50c684541836e60bf2": "\\epsilon_0=\\frac{1}{4\\pi},\\quad \\mu_0=4\\pi\\,", "11ce236adfa7a1d98e1742ce7a5a967c": "Z_{F \\times G}(x_1, x_2, \\dots) = Z_F(x_1, x_2, \\dots) \\times Z_G(x_1, x_2, \\dots).\\,", "11ce4d554f7079228d8458109c94cb75": " \\frac{d P_k}{dt}(t) = \\sum_j A_{jk}(t) P_j(t);\\quad P_k(0)=\\delta_{ik}, \\qquad k=0,1,\\dots .", "11ce58e9343e4f6d55b207e2ba00fad5": " -\\Delta + |x|^2. \\quad ", "11cea0fbb03cf60d8055008ae90bda40": "I(\\omega)", "11cf8489a46f87cbe003514c55bba3ed": "\\textstyle o", "11cfaffd80021d9c7ddfa44715e4b3ba": "r_n = \\textstyle\\frac{p}{10^n}", "11cfc5deb69ff036968d4042bbda1cf4": "\\psi(\\alpha) = \\psi(\\rho)", "11cfe343ac391fa9eee7f1637655b637": "\\zeta(3)/F'(0)", "11cfee89cffcfa0bebc6d0cd85ccc73f": "\n\\prod_{i=1}^{n}\\left(1+x_{i}\\right)=1+\\sum_{i=1}^{n}x_{i}+\\sum_{i 0 \\\\\n 0 & \\rm{otherwise}\n \\end{cases}\n ", "11d1c399bc953bd57052df718c97eb28": "D_\\mathrm{p} = D_\\mathrm{maj} - 2\\cdot\\frac38\\cdot H = D_\\mathrm{maj} - \\frac{ 3 {\\sqrt 3}}{8}\\cdot P \\approx D_\\mathrm{maj} - 0.649519 \\cdot P", "11d1fe36981aa8c3eff4a88cc464b32a": "\\scriptstyle a,\\, b", "11d2160cfc9793e262568d9b9bc96b87": "m = (G/2) * u", "11d24fe702c2d1cdd62babbba5708df6": "\\forall x \\in U: Px = x.", "11d25fdc92b7462f27048eced349ab6b": "\\left.\\right.\\left. F(z)\\right.", "11d2bf2ea54935cbfc833e064e593fd4": " u = \\frac{T}{3} \\left(\\frac{\\partial u}{\\partial T}\\right)_{V} - \\frac{u}{3} ", "11d2c9e260cf131c4573d14a6fb40290": "\\frac{AF}{FB} \\times \\frac{BD}{DC} \\times \\frac{CE}{EA} = -1.", "11d2d83d2a8b74903fe01d972b7a3ccf": " F(s - a) \\ ", "11d2fe8f803bb9174e28aab2ad00c0e7": " \\kappa = \\tfrac {1}{R} = \\tfrac {\\mathrm{d}\\theta}{\\mathrm{d}t} = 2t ", "11d38009cc294822bb57ad979908a7cd": "\n\\min Z' = \\sum_{k = 1}^u {r^k L^k + \\sum_{i = 1}^n {v_i \\left( { - N_i } \\right)} } \n", "11d38807371727c1cb969272b2ca024d": "= \\widehat{U}^{\\dagger}i(\\widehat{a}^{\\dagger}\\widehat{a}\\widehat{a} - \\widehat{a}\\widehat{a}^{\\dagger}\\widehat{a})\\widehat{U} = \\widehat{U}^{\\dagger}i[\\widehat{a}^{\\dagger},\\widehat{a}]\\widehat{a}\\widehat{U} = -i\\widehat{U}^{\\dagger}\\widehat{a}\\widehat{U}", "11d39ad00e0261287376610f95b99ae0": "\\mathrm{W}(\\theta,\\Phi) = \\frac{\\mathrm{G}(\\theta,\\Phi)}{4 \\pi r^{2}} P_{t}", "11d3b633daf16f3fef0913059b75f3db": "f_c'(z) = \\frac{d}{dz}f_c(z) = 2z ", "11d3bad9bd977d95c0bffc7a473760ea": " \\exists \\delta>0 \\left [ d(x,y)<\\delta \\Rightarrow \\lim_{n\\to\\infty} d \\left(f^n(x),f^n(y) \\right)=0\\right ].", "11d46313ebd320b17294a457b363aa8b": "\\left(x(.), u(.)\\right)", "11d55c055be5f075ce61b2d2ba1f79fa": "\\operatorname{wnchypg}(x-1;n-1,m_1-1,m_2,\\omega) \\frac{m_1\\omega}{m_1\\omega+m_2} + ", "11d56074081eeffe4c63eb095a38206b": "\\mathrm{NOT} = \\lambda x^{\\mathsf{Boolean}}{.} x\\, \\mathsf{Boolean}\\, \\mathbf{F}\\, \\mathbf{T} ", "11d57ab76d6294282af585087ca7334e": "Y_b", "11d5866f1abdb1c49cac70dbbbe57902": "\\alpha = c_1 + 2r", "11d5b60ce76bbc641214acb704ab6fe6": "\\left | \\mathbf{E} \\right | = \\sqrt{\\frac{\\epsilon}{\\mu}} \\left | \\mathbf{H} \\right | \\,\\!", "11d61eaf65925f427f4dc01f1b7e84e7": " \\hat{M}=M\\otimes_R \\hat{R}. ", "11d67ea4b9fb6edd5f254b938295e3b5": "\\frac {d M_z(t)} {d t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _z", "11d6e47a6720e94b34b288c195ef5b40": "f(x) \\propto x^{\\alpha_1}", "11d77f58e03f7ed2001c6f2a0f5b2df1": "\\|\\alpha\\|_1 = \\sum_{i=1}^p |\\alpha_i|", "11d78c73e6a2eaeb07f68ceec74c2950": "\\dot{x} = f(x) + \\varepsilon^2 g(x,y,\\varepsilon)", "11d7c155b970e74f0062d495b8fd4d5a": " n_-(V) = \\operatorname{dim} \\ \\operatorname{ran}(V)^{\\perp}", "11d814dce29075fb019d4532066556e0": " L_+ ", "11d901c3f95a6b29f967667d965579b0": "i = 0, ... ,n", "11d91946425bcfec566ce02871ef6028": "V(\\boldsymbol{r})=0", "11d92354c0639da86952e51c6a7e7b88": "\\lambda = \\frac{\\lambda_0}{n(\\lambda_0)}.", "11d939a9a0deb985b2f8665979f5ef30": "\n\\begin{array}{rl}\n \\partial_t u &= d_u^2 \\Delta u + \\lambda u -u^3 - \\sigma v + \\kappa,\\\\\n\\tau \\partial_t v &= d_v^2 \\Delta v + u - v\n\\end{array}\n", "11d93c03615feab22fe36ba8d244629f": "O(1/i)", "11d93e19e9ff675a9c86b0ffa75cf786": "E = \\{e_1,e_2,e_3,e_4\\} = ", "11d9767e7ea405a2d815e0cdd080b1fc": "m^{}_{u}", "11d9ebff2514f655e2909ae8eee20ce3": "\\text{arcsin} x \\approx x", "11d9fc68eaa08665e5ce85834cf001c1": "\n\\limsup_{n\\rightarrow\\infty}\\frac{\\sigma(n)}{n\\,\\log \\log n}=e^\\gamma,\n", "11da1adc767e0c9a0ce7dfa41807a3dc": "\\frac{1}{3b^2}", "11da63597524890adae32f86363d5f5a": "z \\in \\mathrm{Im} (\\eta)", "11dad195d8e52921a73c86e2e61eb50a": "\\mathbf{C}^*", "11db12869f053838508a4fbbf89f2ddf": "\\mathbf{a} \\succ \\mathbf{b}~", "11db6ee9d7d579f4ea6b2b98bd1217f4": " \\langle \\hat{L} \\rangle = \\sum_{i=1}^{n} \\langle \\phi_i \\phi_i | \\hat{L} | \\phi_i \\phi_i \\rangle ", "11db89e9fbef3a15e4b6244be8bec816": " (S_6 \\implies (\\operatorname{equate}[A_6, x] \\and V[F_6] = A_6)) \\and D[F_6] = D[x] ", "11dbbb6333e772d15bd6c9ef2a268f90": "x_i = u_i, \\; F_i(x) \\le 0", "11dbcbacd821c1f2fd829823642ba0f2": "[a, b].", "11dbcced51c5e20bdbd0e06e218676c1": "P_{CSWD} = \\sqrt {P_C \\cdot P_{HR}}", "11dbeb470a22725eae723ee964219880": "\n\\begin{bmatrix}\n A_{11} & A_{12} & A_{13} \\\\\n A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \n\\end{bmatrix}^T\n\\begin{bmatrix}\n A_{11} & A_{12} & A_{13} \\\\\n A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \n\\end{bmatrix} =\n\\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & 1 & 0 \\\\\n 0 & 0 & 1 \n\\end{bmatrix}\n", "11dc28755d48bdd248d2e630065d5ca1": "I(X)=H(X)", "11dc32ca6523e30001129dbf8e96f2d5": "a_{i,j} = (-1)^{i+j} \\det(M_i^j)", "11dc7aa165ef6d539a3fe04b99536837": " 0 \\leq \\operatorname{tr}(A B)^2 \\leq \\operatorname{tr}(A^2) \\operatorname{tr}(B^2) \\leq \\operatorname{tr}(A)^2 \\operatorname{tr}(B)^2", "11dcb4cb1d4066a67678b6bb3e356035": "x_{0} \\in X", "11dcbfc57387afca9e1759de34f5a796": "\n{\\rm E}\\left[ {x_i - \\bar x_i } \\right]\\,\\,\\, = \\,\\,\\,{\\rm E}\\left[ {x_i } \\right]\\,\\,\\, - \\,\\,\\,{\\rm E}\\left[ {\\bar x_i } \\right]\\,\\,\\, = \\,\\,\\,\\mu _i - \\,\\,\\mu _i \\,\\,\\, = \\,\\,\\,0", "11dcecd31e731212639c8d646afb3eb7": "\\scriptstyle p \\;=\\; (xy \\,+\\, 1)x", "11dced657f158f0a6f79c137b0a15e5e": "\\Box p \\rightarrow \\Diamond p", "11dceeb5ce1e6f8c5547918b7890fe7d": "(c-\\gamma(s))\\cdot\\underline{T}(s)=(c-\\gamma(t))\\cdot\\underline{T}(t)=0,", "11dd08fe97945563a17d261788aaacaa": "e^{i(h-(-h))\\theta}=e^{2ih\\theta}", "11dd385761f0a8b2c9ee507971fc7501": " E\\Psi = -\\frac{\\hbar^2}{2m}\\nabla^2\\Psi + V\\Psi ", "11ddf3a3bacec1bc9d5b1671849981e8": "T(v) = \\lambda v", "11deb27929c1bd314c59e988aea4da64": "t = 2Nc \\left ( 1 + m \\right ) \\,.", "11dec9074649889630b840ade80c8205": "\\hat{g}(\\xi)", "11df0b7969eec89421afd102b07e604a": "\\mathcal{N}(\\mu,\\sigma^2)", "11df4671d951c3d4600067696845f99f": "f \\in W^*", "11df48d189a8241e594a1e90b2460b29": "t_{\\Delta}", "11df63dcebb430119b832c602f154893": "\\scriptstyle w(e)", "11dfa998f298ca5e7d31bf2b45f931d8": "\n {\\gamma}=\\frac{4{\\pi}^2 \\mathrm{m}{\\mathrm{V}}} {{A}^2 p {T}^2}\n", "11dfca20e8ea322b70e7b8ba7c976d5e": "P_{\\mathbf{v}^K}(\\mathbf{x}|spike) = \\langle \\prod_{i=1}^K \\delta(x_i - \\mathbf{s} \\cdot \\mathbf{v}_i) |spike \\rangle_{\\mathbf{s}}", "11dfe907f9269ce0adeddafc973d18bd": "\n\\sum_{A=1}^N \\mathbf{s}^t_{A} = 0\\quad\\mathrm{and}\\quad\n\\sum_{A=1}^N \\mathbf{R}^0_A\\times \\mathbf{s}^t_A= 0, \\quad t=1,\\ldots,3N-6.\n", "11dff278df10b879b16db44ab74f039b": "pH = -\\log \\{H^+\\}\\,", "11e018751617e4fabe33aa6c29f27abf": "\\prod_{i=0}^k\\binom{m_i}{n_i}\\pmod{p}", "11e01ade2d2e023deacc9ad5391caf82": " x^*_I(\\theta) \\in \\arg\\max_{x \\in X} \\sum_{i \\in I} v(x,\\theta_i) ", "11e02b8cb6226946f595f58d507936cb": "a+rA=\\{a+rx:x\\in A\\}.", "11e04a304f4b6423980f7555bdd4cf41": "\\frac{64}{16} = \\frac{\\!\\!\\!\\not64}{1\\!\\!\\!\\not6} = \\frac{4}{1} = 4", "11e07516c9ff1ae1d18f844093bab267": "\\nabla y_t =\\delta y_{t-1}+u_t \\,", "11e08ce9610b4561dc8a839a3846109b": "\\begin{bmatrix}\nc_3 c_1 - c_2 s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & s_2 s_3 \\\\\nc_2 c_3 s_1 + c_1 s_3 & c_3 c_2 c_1 - s_3 s_1 & - c_3 s_2 \\\\\ns_1 s_2 & c_1 s_2 & c_2\n\\end{bmatrix}", "11e0d0c606434a00ace138baef950774": "n = 77", "11e0d930026f0dbdb064a96437fbeaf8": "A = A_0\\oplus A_1", "11e10dde3dff77631a2b72f2505f4a11": "i_n i_m + i_m i_n = 0", "11e124cc17fe1f2c70ade5b9721b8f60": "\\angle QCB = \\angle QBA = \\angle QAC", "11e17fe92d980c807bdb9d613e8adb15": "Z_{ij}", "11e182d4c29bdf5bad4882f4809ff77e": "2\\pi r", "11e18821fa277ffb37661c872d2073c4": " \\sin(iy) = {e^{-y} - e^{y} \\over 2i} = - {e^{y} - e^{-y} \\over 2i} = i\\sinh(y) \\ . ", "11e1ad7ee5e0ad4c2904d90355381278": "\\tilde{\\psi}", "11e1ed0d090db03164b41f636faa5590": " [-\\nabla^2+k^2] \\Phi(\\mathbf{x},\\mathbf{x}') = \\delta(\\mathbf{x}-\\mathbf{x}')", "11e202dc62949f494c84b7271922bdb7": "T^2 = \\mathbb{R}^2 / \\mathbb{Z}^2", "11e20d73c8440be32494bdbc3d383be1": "\\sqrt{N}", "11e24123a5a9a4b549f2d008acc32892": "A=[a_{ij}] ", "11e24259e061c7dc8e029ad399fa496a": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{tri} \\left( \\frac{\\xi}{a} \\right) ", "11e24faca9d893c790a6fd17042c9a7d": " \\lambda ", "11e265b6f385b00fc20e9a7994802b70": " T(-h,a) = T(h,a) ", "11e2660de92e7bf1d36ed791aacf78bc": "\\vec{x}(t + \\Delta t) = \\vec{x}(t) + \\vec{v}(t)\\, \\Delta t+\\tfrac12 \\,\\vec{a}(t)\\,\\Delta t^2", "11e3699d824b4516fb3bc8e0c74dfc39": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge m} {k \\choose m} \\frac{z^k}{k} =\n\\frac{1}{1-z} \\frac{1}{m} \\frac{z^m}{(1-z)^m} =\n\\frac{1}{m} \\frac{z^m}{(1-z)^{m+1}}.", "11e3c76bd167154c7dd85234e874b4a5": "\\Omega(\\sqrt{n\\log n})", "11e456ec90d6ede89f6ddce301a3ebaa": "\n(Cut) \\quad\n{Z \\leftarrow \\Delta X \\Delta' \\qquad X \\leftarrow \\Gamma\n \\over\n Z \\leftarrow \\Delta \\Gamma \\Delta'}\n", "11e4ba83387d0d23a023742c0fc19876": "K(u) = \\frac{70}{81}(1- {\\left| u \\right|}^3)^3 \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "11e4c6e9d0b7973f5d0de4dc99316cf8": "\\hat{h}(\\xi) = \\overline{\\hat{f}(\\xi)} \\,\\cdot\\, \\hat{g}(\\xi).", "11e4e1075fc71fe7d00a880b73f32068": "\\sum_{n=1}^\\infty a(n) n^{-s}", "11e57ab7584e41d57386988166ea22a6": "\\left(\\begin{matrix}x_3&x_1-x_2\\\\x_1+x_2&-x_3\\end{matrix}\\right)", "11e58c902d658b7346dc294e4d616e39": "\\mathbf{k}^e", "11e59cf7b7aad86b1073be0f4ee186ff": "\\mathbf{P}^n_k", "11e5aad0c5544602a9cd3731c81e9228": "L_{f, P} = \\sum_{i=1}^n (x_{i}-x_{i-1}) m_i . \\,\\!", "11e5bc6e852aecd259ad9cb97c302d45": "\nJ_0 = \\left({{\\partial L}\\over{\\partial \\vec \\omega}}, \\vec \\omega \\right) + \\left({{\\partial L}\\over{\\partial \\vec v}}, \\vec v \\right) - L, \\quad\nJ_1 = \\left({{\\partial L}\\over{\\partial \\vec \\omega}},{{\\partial L}\\over{\\partial \\vec v}}\\right), \\quad J_2 = \\left({{\\partial L}\\over{\\partial \\vec v}},{{\\partial L}\\over{\\partial \\vec v}}\\right)\n ", "11e5f3e73b68bb531cef3a6b56ca3923": "f_* \\mathcal{O}_X", "11e60ba50ad18c52fd95e243667a62cd": "a \\cdot b", "11e6917b95ef4a0199496878138098f8": "d_\\odot = 1.58{\\times}10^{-5}\\,\\text{lyr}", "11e6af22dbabbf1de1356f75431b9ade": "\\delta = (-1)^{nk + n + 1}s\\, {\\star \\mathrm{d}\\star} = (-1)^k\\,{\\star^{-1}\\mathrm{d}\\star} ", "11e6df5ff24c641172f4c03ea5556a25": "A = \\frac{V_o}{V_i} = \\frac {\\left [ \\frac{M}{C} + \\left ( a \\frac{M}{C} \\right ) \\left ( \\beta^2 - 1 \\right ) + \\frac { \\beta^2 + 2 \\beta + 3 } {4 \\left ( \\beta^2 + \\beta + 1 \\right ) } \\right ] }{ \\left ( \\frac{N}{C} + \\frac {3 \\beta^2 + 2 \\beta + 1 } { 4 \\left ( \\beta^2 + \\beta + 1 \\right ) } \\right )} ", "11e6f967d89c6728653edad985c83678": "\\theta \\sim p(\\theta|\\alpha)", "11e71825e743d04b43fa2879fb60c35e": "P(E(s), E(s'), T)", "11e7d493ebd781f7727261a9c9627c8d": "\\omega \\in A_p", "11e83df9231f795a5ce8b3400e41ef3e": "b \\triangleleft a", "11e8b25fd8e651a8f88303c7a37ca8b7": "4^k >600 \\gamma ^2", "11e8bf6878e9108fbb653ee2f8f3d47e": "\\frac{\\beta (1-p)}{-p \\ln p}", "11e8e6dd37899edab9c3442b294a0e31": "x^{23}-1", "11e91e60f0be8349203530e87d1c4afe": "V(\\mathbf{r}) = \\frac{1}{2} kr^{2} = \\frac{1}{2} k \\left( x^{2} + y^{2} + z^{2}\\right)", "11e94c7d3fcfc2c32919b430fd1cf845": "0.167,\\ 0.177,\\ 0.181,\\ 0.181,\\ 0.182,\\ 0.183,\\ 0.184,\\ 0.186,\\ 0.187,\\ 0.189 \\, ", "11e9817dcf2067fd98b2c39842966e07": "x = \\psi(y)", "11e9855eba1e79c22f2029ba4001ea50": "A\\in U", "11e98a93b3bc26cdd4ec517c128c1ea6": "f(\\alpha \\mathbf{v})=\\alpha f(\\mathbf{v})", "11e99fdec9c9dd116bf8116a56db548e": "\\operatorname{Cov}[\\mathbf{z}]= V", "11ea44d24b9497446942458ca8f8d00c": " f_n( X_{(n)} ) = n \\frac{1}{L} \\left(\\frac{ X_{(n)} }{ L }\\right)^{ n - 1 } =n \\frac{ X_{(n)}^{ n - 1 } }{ L^n }, 0 < X_n < L ", "11ea63368bf4c6ef47ec16d6f57dfa68": "R_x(t;\\tau) = R_x(t+T_0; \\tau)\\text{ for all }t, \\tau.", "11ea83b1d5df45e60e93aaf7cebd2af4": " x = 1 ", "11eab01aa0a374d7065f943b2a76fc2f": "2^{n-2} + 2^{\\lfloor n/2 \\rfloor - 1}", "11eab9ef7c71e4c86f6234a09d63c11b": "-\\sigma", "11eb7825e9d1c3dda35c59ecf53495ad": "N_A = N_{A0}e^{-{\\lambda}t} \\,\\!", "11eb78db0da8c04cf0a499f59a95a99c": "\n(\\mathbf{a}\\ \\mathbf{b}\\ \\mathbf{c}) = (\\mathbf{c}\\ \\mathbf{a}\\ \\mathbf{b}) = (\\mathbf{b}\\ \\mathbf{c}\\ \\mathbf{a})=\n -(\\mathbf{a}\\ \\mathbf{c}\\ \\mathbf{b}) = -(\\mathbf{b}\\ \\mathbf{a}\\ \\mathbf{c}) = -(\\mathbf{c}\\ \\mathbf{b}\\ \\mathbf{a}).", "11eb9bc27d0afe163f8c11a485f22883": "\\partial^ig^{jk}=\\Gamma^{ij,k}+(\\Gamma^{(*)ik,j}=0)=\\partial^i\\partial^j\\partial^k\\phi", "11ebb311d1d7eaca87769ab0b1164f87": " 20 \\log_{10}(|M(s)|) ", "11ec0a1dcc2184d34ea19dfd599a2919": "\\ q", "11ec0d3526404a6657163b0f9563cdbf": "|X| < k", "11ec7884d9870493a65fd7ecd6c2ba2f": "a \\rightarrow 0", "11ec9a6f0b6f3973586669a13be3ef75": "\\{\\alpha_n\\}_n", "11ec9fc092ddb3b56e7c2fdce40b9001": "p(y|E(m))", "11ed0fdb63ef4dbfd07c2ddbba6e3652": "W^+_{\\mu\\nu}\\equiv(W^-_{\\mu\\nu})^\\dagger", "11edbc4bbfdd037c2f10ce9c5bbfa475": "\\ C \\in \\R^*", "11eddc46fe3658760e99fb85a0dfcd3d": " f(u * v) = f(u) \\odot f(v)", "11ee393ad69880229b97da3e08f27f0c": "k={F_w \\over P_{TMP}}", "11ee8a4d05181a112c530a059d8817c8": "\n\\mathcal{L}_2 = \\frac{1}{2} u_x^2 + w (v_t + u v_x)\n", "11eea285ba9bb401d2e68bce7f18265f": "S[A[i],A[i]+v_{max}-1]", "11eeb5219e5c805a6b78cfcbddd0f0e8": "\nU_2 \\in [14V,16V]\n", "11eec02cdaf20a44be3713c9f41ccf94": "\\mathbf{q}^m = 0 ", "11ef0abd976ed46b9b8b71f12584bbe3": "(X_1,X_2,\\dots,X_d)", "11ef15a2fb698625dd2382be2615cd16": "\\rho_{a / b}(R \\cap P) = \\rho_{a / b}(R) \\cap \\rho_{a / b}(P)", "11ef18e183e129712044dd88645381f8": "\\ln {p \\over p_0}= {2 \\gamma V_{\\rm{m}} \\over rRT}", "11ef217dffcdd0747961f9c65d017f29": "\nx_{3,4} = -\\frac{b}{4a} + S \\pm \\frac12\\sqrt{-4S^2 - 2p - \\frac{q}{S}}\n", "11ef73af549b41d4d47f4c6ee4b46e8b": "u_m(t) = m(t) + i \\cdot \\widehat{m}(t)", "11ef8752b906ada0673f9f70cb1ecc40": "O(log(p) (\\tau+\\sigma m)) ", "11efa08a69f47582d86727e38bf42693": "\\{{k_i}\\}", "11efb0768af86ff84fcd7a394d75c544": "-js=\\mathrm{cd}(w,1/\\xi)", "11efe5ed53dbca62270daabdb13c9289": "U_{ijk\\dots}=U_{(ij)k\\dots}+U_{[ij]k\\dots}.", "11f006df49bff547bfec8c9c9c99ce79": " \\cos\\left({\\theta \\over 2}\\right) \\cdot \\cos\\left({\\theta \\over 4}\\right)\n\\cdot \\cos\\left({\\theta \\over 8}\\right)\\cdots = \\prod_{n=1}^\\infty \\cos\\left({\\theta \\over 2^n}\\right)\n= {\\sin(\\theta)\\over \\theta} = \\operatorname{sinc}\\,\\theta. ", "11f03dca27429905314b38cf49e9a66a": "\\dot{\\rho_j}", "11f04d7d07da7a6e45eae2cb949aa9e7": "G = \\int_\\Sigma \\!\\int_S d^2G \\ ", "11f07ce16a4ebc09e5453838440cd107": " H_0(X) = \\tilde{H}_0(X) \\oplus \\mathbb{Z}", "11f114d28790501e7deb3ef3c84f1595": "(a \\# g_1)(b \\# g_2) = a(g_1 \\cdot b) \\# g_1 g_2", "11f11b4b1147d7c8b0317fdcbfe113cd": "(r,1)", "11f13492fabc53654d3eb69dd5233cd7": "\\mathbf{e}_1,\\,\\mathbf{e}_2", "11f16bb05854ece012e8172706fe07dc": "\n\\langle \\Psi \\rangle_V \\rightarrow\n \\begin{pmatrix}\n 0 & \\psi_{12} \\\\ \\psi_{21} & 0\n \\end{pmatrix} \n", "11f1797e2af2cfbb297739f75b21752c": "\nU(x) =-\\frac{\\lambda(\\lambda+1)}{2}\\mathrm{sech}^2(x)\n", "11f1f75d699e7f3261f7c6a488640d79": "A\\mathbf{y} - \\mathbf{b} t \\leq \\mathbf{0}", "11f1ffa651a05c024f205e4ebac51bd1": " 2n ", "11f217a7d66382c887170bd452cfdb14": "W_F", "11f22b137abbb270c8b457c7ffa1c561": "M\\int v\\,ds", "11f2cca633f8a4b03cb458ac22378bef": "\\frac{L}{2a} = \\sinh\\left(\\frac{k}{2a}\\right) ", "11f2fd4c207d1c9909f949ed0b31978e": " c^2 = a^2 + b^2 - 2 a b \\cos \\gamma ", "11f38188b9d397db6328870ef339c8d7": "\\frac{E_1}{E_2} = \\frac{k_1}{k_2} \\left(\\frac{k_2}{k_1}\\right)^2 = \\frac{k_2}{k_1} . \\,", "11f3d4b15dd0bc8728a195a01783adea": "\\begin{matrix} {11 \\choose 1}{4 \\choose 3} \\end{matrix}", "11f3e221c078482c1dc3af52c381dda3": "\nI_n^f = \\left(A_n^f\\right)^2,\n", "11f3e226b01931d56bd077e1a4fc0ca9": "m^n", "11f46319213b3f478142daa045758fd3": "\\mathit{N}", "11f470c6e9c7c5e976b770b2c734b37a": "1_{A_1\\cup\\cdots\\cup A_n}\\ge\\sum_{j=1}^k (-1)^{j-1}\\sum_{1\\le i_1<\\cdotsb\n \\end{cases} \n", "11f79cd4cbaf600d68e91e548441bb1e": "\\tau_{D,i}", "11f7c1b5e66c2424cd7260a1cf45b307": "A B A B", "11f7c5713a99d91837c961a1e25dc701": " f(x) \\sim A {(x-\\mu)}^{-\\alpha-1} \\, . ", "11f8240fd173bd97d3756ed4b1084e6a": "\\left(1+\\theta t\\right)^{-1/\\theta}", "11f84454ff4380cdb326350928d64e55": "\\begin{align}\n &\\begin{bmatrix}0&-z&y\\\\z&0&-x\\\\-y&x&0\\end{bmatrix} \\mapsto {} \\\\\n &\\quad \\frac{1}{1+x^2+y^2+z^2}\n \\begin{bmatrix}\n 1+x^2-y^2-z^2 & 2 x y-2 z & 2 y+2 x z \\\\\n 2 x y+2 z & 1-x^2+y^2-z^2 & 2 y z-2 x \\\\\n 2 x z-2 y & 2 x+2 y z & 1-x^2-y^2+z^2\n \\end{bmatrix} .\n\\end{align}", "11f873575f831054a4d9f4a5f8cde97a": "\\hat{H}(t)=\\hat{H}_0+\\hat{V}(t) \\theta (t-t_0)", "11f8b1b30d0ebbcdd70bfee12d4fd283": "h_m (X_0)=\\left\\| X_0 - X_{[m]} \\right\\|", "11f9076610e11a6f6e8ff746702402af": "\\Phi_{{\\mathrm{eE}},h}:z_k\\mapsto z_{k+1}", "11f92604bf3b962e180703b811d7b000": "\\Mu \\, \\mu \\,", "11f92a018e196a4387c248253f4d0d19": "m_k=\\int t^k\\psi(t)\\, dt.", "11f9672216276ff3d800c0693c54ce55": "\\sigma: P(U)\\times P(V) \\to P(U\\otimes V).\\ ", "11f96ce0f1569effd19ad91910fd2107": "\\frac{(\\alpha-\\beta)\\,\\Gamma(2n+\\alpha+\\beta)}{(n-1)!\\,2^n\\,\\Gamma(n+1+\\alpha+\\beta)}\\,", "11f96d4c7b035493552ad8fd95057d61": "S^\\prime(a,q,0)=1", "11f9c97a767b44a71863a0839580ba63": "\n f(k; r, p) \\equiv \\Pr(X = k) = {k+r-1 \\choose k} (1-p)^kp^r \\quad\\text{for }k = 0, 1, 2, \\dots\n ", "11fa56fcce59e3bcb7031dfaa53c6ef1": " (d + [(m + 1)2.6] + y +[y/4] + [c/4] - 2c)\\ \\bmod\\ 7 - (d + [2.6m - 0.2] + y +[y/4] + [c/4] - 2c)\\ \\bmod\\ 7", "11fa88866d4e21e2ed0c9428e47e27c4": "\\frac{d\\lambda}{dt}=0", "11fa8e5d143041a8e707991fee6ae3f4": "n_\\mathrm{tot}", "11fac24dd2b06c4ec14eb363df98b7e0": "\\epsilon_y=\\frac{\\partial u_y}{\\partial y}\\,\\!", "11fae8d7d9a95a34400eb6e6af382a1c": "\\cos\\theta ", "11fb3a578987f34f532c6fdcc5ad663e": "\\mathcal{H}\\left(q_j,p_j,t\\right) = \\sum_i \\dot{q}_i p_i - \\mathcal{L}(q_j,\\dot{q}_j,t).", "11fb9ec0afca1866ff70aa8825bc62f7": "+ \\sin \\left[ 2 \\pi \\left( \\frac {x}{\\lambda + \\Delta \\lambda } - ( f - \\Delta f )t \\right) \\right] ", "11fbf25b430a5ad659dd0ec045b88026": "1-t\\leqslant e^{-t}", "11fc3a0c8fca92ae6bc10ff8241a116e": "Hg", "11fc3d25620edbaebea78565eaee7e97": "y.", "11fcc08dd2049b49044a68ad6227eb70": " m_1 \\frac{d^2 {\\mathbf r}_1 }{ dt^2} = -\\frac{m_1 m_2 g ({\\mathbf r}_1 - {\\mathbf r}_2)}{ |{\\mathbf r}_1 - {\\mathbf r}_2|^3};\\; m_2 \\frac{d^2 {\\mathbf r}_2 }{dt^2} = -\\frac{m_1 m_2 g ({\\mathbf r}_2 - {\\mathbf r}_1) }{ |{\\mathbf r}_2 - {\\mathbf r}_1|^3}, ", "11fcc0d29b5ca02c22d2743e57f24fdc": "\\widehat{\\mu} = 1/(2\\pi)^n\\mu", "11fcdff0e4febef6282057017d633eae": "\n\\begin{array}{c|ccc}\n0 & 1/6 & -1/6& 0 \\\\\n1/2 & 1/6 & 1/3 & 0 \\\\\n1 & 1/6 & 5/6 & 0 \\\\\n\\hline\n & 1/6 & 2/3 & 1/6 \\\\\n\\end{array}\n", "11fce5b4c07fc997a961f3fdb3c3b13b": " \\lambda \\mapsto \\int^\\oplus_X \\ \\lambda_x d \\mu(x) ", "11fd48dc9847b6765280e4b59eff33d4": "|\\ell - s| \\le j \\le \\ell + s", "11fd6d28c2ef34b6bda25ecb312c0464": "\\frac{5\\pi}{12} \\ (75^\\circ)", "11fd73531d2e69e1003cbf0abc35cfe9": "\\forall n < t\\, \\phi", "11fd7b8271a668c88c7f6d95bec51aca": " \\langle x^* , y - x \\rangle \\ge 0 \\,,", "11fdc72aed23572ab4321f13f7819862": " |r| \\le |a|\\,", "11fe38b72e59349f5fea99e2c123715f": "(S\\tau)(j)=\\frac{3}{8(1-j)^2}+\\frac{4}{9j^2}+\\frac{23}{72j(1-j)}", "11fe83fb9802f44396f066d6d891815c": "A_v = ", "11fe8d1583ec9ecd46dad79bbcb5de71": "H = \\int d^dx \\left[ \\frac{1}{2}E^2 + \\frac{1}{4}B_{ij}B_{ij} - \\pi\\nabla\\cdot\\vec{A} + \\vec{E}\\cdot\\nabla\\phi + \\frac{m^2}{2}A^2 - \\frac{m^2}{2}\\phi^2\\right]", "11fed10743b680da1f65a0b4c3f69170": " \\lim_{t\\rightarrow 0} K_t(x-y) = \\delta(x-y) ~,", "11ff3a1fa4c36858687d23a45aa0537f": "\\alpha \\in \\mathbb{F}_{q^k } - \\{ 0\\} ", "11ffa68585c52d7db38fdbc8b8ad2c20": "Y=C_0 + \\sum_{i=1}^k C_i \\frac{X_i-\\bar{X_i}}{\\theta^i}", "11ffbcdf349d977c816a6db52b31e30c": "a \\ge b \\ge c", "1200212a910737a18e4f888a15214a8b": "\\Gamma \\vdash \\psi \\to \\phi", "12003fc87de5f10afe5dbfb9fc144dd5": "\\,\\Delta U = mg \\Delta h.\\ ", "1200967f9e022ae2529529f7dd29c4db": "Aa~Gradient=P_AO_2-P_aO_2", "1200b58d646a55e8bdcb7ae84df6b029": "\\left.\\tau_p\\right|_{r= R-\\delta}=\\left.\\tau_c\\right|_{r= R-\\delta}", "1200fd1ea8e3d246c19f3b740d3f450f": "H^s(\\Omega) = \\{ (1-\\Delta)^{-s/2}f | f\\in L^2(\\Omega)\\}.", "12015de8c3cdad9d013145427739cf22": "\\gamma\\in\\Lambda", "1201725b99fb1a105dea9c07f395b047": "S_{k_\\nu}\\cdot S_{k_\\mu}", "1201bcda22584f5916f144044dbd6a01": "(t - \\tau, t)", "12020cb08f5fbe152b53051e816f6633": "d = 19.3", "1202309bfb8eb839998ca502e02f1c4e": "\\scriptstyle F(\\omega).", "12024f42b9bdc8008c09eb28ca03205b": "\\Sigma=\\mathrm{E} \\left[ \\left( \\textbf{X} - \\mathrm{E}[\\textbf{X}] \\right) \\left( \\textbf{X} - \\mathrm{E}[\\textbf{X}] \\right)^{\\rm T} \\right]", "12027918376e69f2a68750e64b0ac690": " (u_{in},u_{out}) ", "1202e7d8b954110e0ebf602786846d40": "\\begin{align}\n D(x \\mapsto 1) &= (x \\mapsto 0),\\\\\n D(x \\mapsto x) &= (x \\mapsto 1),\\\\\n D(x \\mapsto x^2) &= (x \\mapsto 2\\cdot x).\n\\end{align}", "12034b79f7bf492c2453bbc319718635": "\\mathrm{Rot}_G \\circ \\mathrm{Rot}_G", "12035fa9d63859711f912394f3df0f48": "\\tilde{P_n}(x)", "1203be016a2b371f6de0c899077e4e38": "d_1'", "1203be3050d587b6f1cdd4af952ba830": "F^*_{i-\\frac{1}{2}} =\\frac{1}{2} \\left\\{\n\\left[ F \\left(u^R_{i - \\frac{1}{2}} \\right) + F \\left(u^L_{i - \\frac{1}{2}} \\right) \\right]\n- a_{i - \\frac{1}{2} } \\left[u^R_{i - \\frac{1}{2}} - u^L_{i - \\frac{1}{2}} \\right] \\right\\}. ", "1204125069f6c57338ef779aad9e0b71": "x^l", "12046f77a5b7da929baacdbff9247309": "\\beta(2)\\;=\\;G,", "12046fe3bc161f15348549727028edb6": "t_j \\,", "1204e1d9099edc5e8b60cb6d96228807": "M_X", "12050b88d8659d9293d24b0e68783cab": " S=\\frac{A(X)}{A(Y)}. ", "12050d22c64c8d6ee464989e5dbe299d": "\\widehat{a}\\widehat{D}(-\\alpha)|\\alpha\\rangle = \\widehat{D}(-\\alpha)(\\widehat{a} - \\alpha)|\\alpha\\rangle", "12050fa9468457fa2a2ed280303f5144": " \\frac{T_2}{T_1} = \\frac{\\left(1 + \\frac{\\gamma - 1}{2}M_1^2\\right)\\left(\\frac{2\\gamma}{\\gamma - 1}M_1^2 - 1\\right)}{M_1^2\\left(\\frac{2\\gamma}{\\gamma - 1} + \\frac{\\gamma - 1}{2}\\right)}", "120573ed717b9b721ee9676f6085a036": "X=s_{X}\\phi (x)^{r_{X}}", "1205b1cfee577c96bf8ef59e75bb380c": " \\mathbf{a} = \\boldsymbol \\Omega \\times \\mathbf v = \\boldsymbol \\Omega \\times \\left( \\boldsymbol \\Omega \\times \\mathbf r \\right) \\ , ", "1205c7e618ef7bfb22206012efa424b3": "J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m)=0, \\mbox{ if }\\kappa_{m+1}>0.", "1205d7a46ec30a343a54d574c0a8658c": "r(i,k) \\leftarrow s(i,k) - \\max_{k' \\neq k} \\left\\{ a(i,k') + s(i,k') \\right\\}", "120618f829de6dbd77a0a8d1a54bfa36": "f = g + b", "12063f42a48485f4dbbedfd82d00a1f2": " \\langle a^N \\rangle\\langle b^N \\rangle\\langle a^N \\rangle ", "12066669bb72df1468e41090f81e90b3": "\\frac{\\partial\\,\\textbf J}{\\partial\\,a}= \\frac{\\partial}{\\partial\\,a} \\left (\\frac{\\pi}{2\\sqrt{ab}}\\right) =-\\frac{\\pi}{4\\sqrt{a^3b}}.\\,", "1206766680720eee51aea77c5ab9f6b3": "[t_a, t_b ] = i f^{abc} t_c ", "12069ce86e3daea709688687431ab3f2": "x[\\infty]=\\lim_{z\\to 1}(z-1)X(z).", "120716a31746fb7c6f5a6efec31f2cae": "\\sin \\theta \\approx \\frac{3.83}{ka} = \\frac{3.83 \\lambda}{2 \\pi a} = 1.22 \\frac{\\lambda}{2a} = 1.22 \\frac{\\lambda}{d}", "12077168e44b7f66b87b686004c403bd": "\\phi_x \\in F", "1207df2ea399c1bb2a8a49493445c6cd": "s\\in\\{-1,1\\}", "1207f006c17fce4c5629a09a1800ee1e": "c(p,y,t) = \\sum_i b_{ii} \\left( y^{b_{yi}}e^{b_{ti}t} p_i + \\sum_{j\\,:\\,j\\neq i} b_{ij} \\sqrt{p_ip_j} y^{b_y} e^{b_tt}\\right).", "120838cff47154743574e6de33bcf2a5": "\\scriptstyle{X_L}", "12087bb53a2f5ea9352922f6c496c879": "P(R_{NP},\\theta_1)", "1208c67e86ac809b392cd4977bd4a13a": "x_{1}(z) = x_{-1}(z)+a_0\\cdot x_0(z)", "12093ae7210d96911e186b48a93c55ca": "\\scriptstyle\\sigma_2^2", "120972ceaf7ac335b5b73c397e86f2ac": "B(x) = \\int {1 \\over e^{-4x}} e^{-2x} \\cosh{x}\\,dx = \\int e^{2x}\\cosh{x}\\,dx ={1\\over 6}e^{x}(3+e^{2x})+C_2 ", "12098aef5d61b130a68dc519f947bf1f": "C_{D,0}", "1209e36df50bd83b87b011f615da329f": "^{\\;}H(\\xi )", "1209ec2da726bed1b2e675521f2c9619": " I(\\mathbf{q}) = \\int_V\\gamma(\\mathbf{r})e^{-i\\mathbf{q}\\cdot\\mathbf{r}}\\text{d}\\mathbf{r}", "1209fc086f15b78a6c45757aa758636e": "\\sgn(\\omega_n)=\\pm 1", "120a1439c5535d9fa0e985e99b9389d7": "\\mu_0 = 4 \\pi \\times 10^{-7} (\\rm{N / A ^2} ) \\approx 1.2566370614 \\cdots \\times 10 ^{-6} (\\rm{N / A ^2} )", "120a146a8dfbb5f8af851f3f659c7c59": "V_n=2F_{2n-1}-F_{n-1} \\,", "120a29453afe71b97be1434afdc65324": "\nQ = n\\left(n+2\\right)\\sum_{k=1}^h\\frac{\\hat{\\rho}^2_k}{n-k}\n", "120a4b36f3d8bf327b4c84aa8a52678d": "\\epsilon D", "120a527ec2fb6583ec5c438f64ad5c18": "\\mathcal O(1)", "120a9af57ee747e52cba32da79e7c12c": "\\omega_1(t)", "120abe116bf3ca0db9641be23603f06d": "g(\\cdot \\mid x)", "120ac279f23f3c3ae5a6ef9bba6c8a7d": "b_{\\{j,k\\}}=\\begin{vmatrix} a_{1j} & a_{1k} \\\\ a_{2j} & a_{2k} \\end{vmatrix} ", "120aef2dd3e826ad00d6f8d361f71d33": " 2x^3 - 2x^2 -3x + 2 = 0 ", "120af3c5227de4cd6e9a3dd1e45aa9f9": " L_{i_1\\alpha_1}^{i_2\\alpha_2} {\\hat L}_{i_2\\beta_1}^{i_3\\beta_2} S_{\\alpha_2\\beta_2}^{\\alpha_3\\beta_3}= S_{\\alpha_1\\beta_1}^{\\alpha_2\\beta_2} {\\hat L}_{i_1\\beta_2}^{i_2\\beta_3} L_{i_2\\alpha_2}^{i_3\\alpha_3},\\quad 0 0 .", "1212c7db203d48037731de433b2a6b4f": "G = (V\\,, \\Sigma\\,, R\\,, S\\,)", "1212c9f997adce8def4ba2f8f70c41a9": "\\lim_{n\\rightarrow+\\infty}g_n", "121313223ec6d864aab9b8f8e10f2716": "Q_S\\left(n\\right)\\,\\!", "1213266e347e6ebc1748790b5d341018": "\n\\beta_{cr} \\approx (s^2 + 2s)^{\\frac{1}{2}}\n", "12134aa42c1c5f312d1b532963bc5614": "A \\rightarrow (B \\rightarrow A)", "121352ec22d6a433071dbe726f1e8f42": " \\sum_{i=1}^N \\sum_{j=1}^N ", "12135fa5efea91932025817bf1cdd83d": " \\ln 2 = \\frac{1}{1} -\\frac{1}{2} +\\frac{1}{3} -\\frac{1}{4} +\\frac{1}{5} -\\cdots. ", "121413ddbd0b4bb71eeefdf13d444b80": "r_c = -\\frac{2\\gamma}{G_v}", "12143c1b992ca08d35b2cd3f97cb394a": "\\alpha\\approx \\frac{d}{f}", "12146a84ceb263a0c53f1085faafd27d": "{1-e^{-1}}=0.61", "12147ec9106aee686c4b00da28a45e3c": "\\neg \\Diamond", "1214ee19f0c86c16a1ceb0746bef34ee": "u = \\sum_{n=0}^\\infty f_i^{(n)} u_n = \\sum_{n=0}^\\infty e_i^{(n)} v_n,", "1214efd654949f8bc2a73f106b8d0d8d": " r_B(n) > 0 ", "12150c34c7ebacf3ac6d651c9d033d66": "\\vec {f^j} \\in C^1(\\mathbb{R}^s, \\mathbb{R}^s), j = 1, \\ldots, d", "12159869635db63b02c0a0ae6176daeb": "\\nabla\\cdot\\mathbf{D} = \\rho_{\\mathrm{free}}", "12159d69e4fc16397dd1cd7ee3bbdc37": " SubCipher_{n+1}=DEC_{b_{n+1}}(k_{b_{n+1}},C) ", "1215ae164e4d4120e2e91a83843d6f05": "c_{\\text{fil}}", "12166545159517419d339863a7710ac4": " \\Gamma \\cup \\{ \\neg \\varphi \\} ", "12170f55dc268f5a2bf48938dc194a16": "\\mathrm{Ad}: G\\to\\mathrm{Aut}(\\mathfrak g)\\sub\\mathrm{GL}(\\mathfrak g)", "1217388d726a03b92ca781521ce0e03a": "\\textstyle c", "1217546f73d17b94510859ff8d59b705": "\\,\\frac{d\\mathbf{w}}{d t} ~ = ~ \\eta \\, y(t) (\\mathbf{x}(t) - y(t)\\mathbf{w}(t)).", "1217751311083458516371ed22dea7a9": "\\begin{pmatrix} 0 & 1 & 1 & 1 & 0 \\\\ 1 & 0 & 1 & 0 & 1 \\\\ 1 & 1 & 0 & 1 & 1 \\\\ 1 & 0 & 1 & 0 & 1 \\\\ 0 & 1 & 1 & 1 & 0 \\end{pmatrix}", "1218482fc44abff9acb5730296209d10": "\\ D_{heel} ", "1218a2f546e67a8ae154b8cc595c35b8": "(y_1, \\dots, y_k) \\in [m]^k", "1219144e892baf828fb87d05ed2b97d5": "\\gamma = \\frac{5}{3}", "12197d5cd38fa1835f1a8f32828923a5": "k = \\int_{a}^{b} g(x) \\, dx.", "1219eb7d81ba826c407356af7b1cc141": "\\phi\\,\\!", "121a319e4238c37b5ab2d03408442f72": "H(X_{i}| ... )", "121a466bf97e7e552a66009aa8ddb72f": "\\gamma = \\frac{1}{3}", "121a6bda80191ac5d4cdd998566037db": " D_{i} = X_{i} \\oplus Y_{i} \\oplus B_{i}", "121ae3a9ca06e5926057c6b5cce1a6bb": "g(z)=-1/90\\sum_{i=0}^8 z^i", "121b66144d4d915c134836d416e819a4": "{{r}_{O3}}", "121bb53505fa4eb6007832227983b4f8": " |\\psi (t)\\rangle \\equiv \\begin{pmatrix} \\vdots \\\\ \\psi_{j-1}(t) \\\\ \\psi_j(t) \\\\ \\psi_{j+1}(t) \\\\ \\vdots \\end{pmatrix} ", "121c602f6f9c6483569216f5e01b8af6": "k \\in \\{1,2,\\dots\\}\\,", "121ca52489106f28aeeb3c6fc0b2da63": " h_A(x)=\\sup\\{ x\\cdot a: a\\in A\\},", "121cdf43d753c1fe95ac68a7ce679903": "\n\\begin{align}\n0 & = \\frac{d I'}{d \\epsilon} [0] = L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T \\\\[6pt]\n& {} + \\int_{t_1}^{t_2} \\frac{\\partial L}{\\partial \\mathbf{q}} \\left( - \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial \\phi}{\\partial \\epsilon} \\right) + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( - \\frac{\\partial^2 \\phi}{(\\partial \\mathbf{q})^2} {\\dot{\\mathbf{q}}}^2 T + \\frac{\\partial^2 \\phi}{\\partial \\epsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}} -\n\\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} T \\right) \\, dt.\n\\end{align}\n", "121cfdcf4419248f434783a6336936a9": "(1/T) \\,,", "121d499a70c5dc88cf8e65e432afd025": "T=-D^{-1}R", "121d53811f80aaa7b3c66910fe38b54e": "S \\rightarrow A: \\{K_{PB}, B\\}_{K_{SS}}", "121d771b8a0e39fbff64b00c95b7a748": "\\rho^2", "121d9aa5f5eddf45d63336459e783c3a": "c_{pd}", "121db019e3a57df3fe993e8dd1fa2fe6": " H_n^{(0)} = \\frac1n. ", "121dc4263a56311cf360403b36a43e0f": "a \\oplus b", "121e2ecb84f8ba698a3a0595d9aeb902": "\\scriptstyle\\boldsymbol \\nabla V", "121e4918c3a10d79c7cf9d4fe447bb9e": "\n\\tau_{i,j}=A_{ij}+\\frac{B_{ij}}{T}+\\frac{C_{ij}}{T^{2}}+D_{ij}\\ln{\\left ({T}\\right )}+E_{ij}T^{F_{ij}}", "121e49ddecefa1ac21aa339b189d1d97": "\\hat{M}\\approx\\frac{1}{2}\\left (\\frac{1}{F_{ST}} -1 \\right ) ", "121ec24d82aa944c1ef87fa2dd4a4e70": "-100\\pm 10", "121ed2a3ccf49bfb186b15094afafad8": "\\frac{d}{\\log{d}}", "121ed985398c676ff487978e8aff933f": "\n \\cfrac{\\partial W}{\\partial I_1} = -\\cfrac{C_0}{J_m} \n ", "121eef2a0abaeca6998fed61f1e67dc0": "B_{k+1}=B_k+U_k+V_k\\,\\!", "121effd24b0086552605654b52abfc1b": "p_1 = x_1\\,.", "121f1f1345b88d61e8de929c2fba2aae": "h=-a_0", "121f459a3dbf3a52d4e92f9003241485": " \\; {}_2F_1(a+1,b;c;z)- \\, {}_2F_1(a,b+1;c;z) = \\frac{(b-a)z}{c} \\; {}_2F_1(a+1,b+1;c+1;z)", "121f4a4c60a00cc2780613db122af89e": "\nG(u) = - \\frac2{mh^{2}} \\int^{\\frac1u} F(r) \\, dr \n", "121fb37b643d996c919f8e15c44bf226": "L^2=L_x^2+L_y^2+L_z^2=\\ell(\\ell+1)e.", "121ff5f1d1c30418d5d0376357ae0ae7": "\\frac{ax^2+2bx}{2}\\,", "121ff81c8e173410c911f6d7461f6413": " \\boldsymbol{\\omega}' = \\boldsymbol{\\omega} + \\boldsymbol{\\Omega} \\,\\!", "1220328f561803f9df3a1ea222bf1429": "\n\\{f, g\\}_{DB} = \\{f, g\\}_{PB} - \\sum_{a, b}\\{f,\\tilde{\\phi}_a\\}_{PB} M^{-1}_{ab}\\{\\tilde{\\phi}_b,g\\}_{PB} ~,\n", "12204ac9e030bcb065b0d0ca24844451": "10.36 \\%", "12204b6f3a627910dc94d797e198cbb4": " \\langle f^*, w\\rangle = f(w).", "12205bce3fafc529b4e8a2c5b8c4e1a4": "r \\cdot p = p \\cdot r = p", "12211635c7c7c91a4dfed3ecd37652b4": " \\beta(\\phi,\\psi)=(-1)^{\\frac12 m(m+1)}\\beta(\\psi,\\phi),", "1221af30eae8b0aae45f557d7998052e": "\\textbf{F}=-m{d^2\\textbf{C}\\over dt^2}.", "1221c109e5e9e5dd2885d90c2771ccb9": "[A]_p\\subseteq \\operatorname{cl}(A)", "1221d3d1fd9e0cba179df465b66ef57d": "\\lambda_j = c_0+c_{n-1} \\omega_j + c_{n-2} \\omega_j^2 + \\ldots + c_{1} \\omega_j^{n-1}, \\qquad j=0\\ldots n-1.", "1221f85a7b80396c80ab7b3944cd2322": "m_m\\;", "12221d5e47854ad9debadde1920d1287": " \\int f(r) dx dy = 2\\pi \\int f(x) x dx ", "12221fe07fa7fe3c15273412710e0444": " \\partial_t u = \\Delta_x u^m, \\,\\, m > 1, \\,", "1222535e4e93103b092c1eb492d5b777": "e(v)=\\frac{\\int\\limits_{0}^{v}{{}}yf(y|v)dy}{F(v|v)}", "122278ab8aa68e7ad3e9eb9bcc0dcf6e": "a(x,t), b(x,t)", "1222b6bc07bdd5ec8fd316dabb48c05c": "\\mathfrak{so}_{2n+1}", "1222c45a3bba0883fca335a2bfa4928e": "\\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~.", "1222e75665bd66c2233b8d3ef5669676": " L = T - V ", "1223e4fb70d877bba74cf6c4fc19583b": "\\sigma_{ii} > 0", "12241245fb0cea72d91bbe8de86b4fca": "R^r f_* \\to g_* \\circ R^r f'_* \\circ g'^*.", "12243e80df90f120933cb7b35c734faa": "f(\\mathbf{y})", "122477605844386cb335556364107b47": "\\begin{bmatrix} \\sqrt{\\dfrac{\\eta_2}{\\eta_1}} \\\\[15pt] -2\\eta_2 \\end{bmatrix} ", "1224b964b62baf5d6570be748c192a0d": "T_0 = T - \\frac{1}{n-1}\\iota(\\operatorname{tr} \\,T)", "1225673626a600669a2579d1ac4d9715": "\\int\\arctan(x)\\,dx=\n x\\arctan(x)-\n \\frac{\\ln\\left(x^2+1\\right)}{2}+C", "12258850e6551184c4a13195460dc147": " \\nabla X", "1225ad822e57b5a9627250100389e6b1": "(\\widetilde{\\rho}, \\widetilde{V})", "1225d6d6b564c2317d34f22746a61221": "\\mbox{d}T = \\mbox{d}L\\cos\\varphi-\\mbox{d}D\\sin\\varphi = \\mbox{d}L(\\cos\\varphi-\\frac{\\mbox{d}D}{\\mbox{d}L}\\sin\\varphi)", "1225e32c37e9a08903e07b59b0688569": "|00\\rangle, |01\\rangle, |10\\rangle, |11\\rangle", "1225e48212927f828b221d81d9d465e8": "\\mathrm{C}^{\\alpha}", "1225ecade34057f10b5c5b7093e5c798": " \\bar F(x) = o(1/x) .", "12264b412801e03e9aa31ff07c76bc61": "L_{D}\\big[\\rho_{S}(t)\\big] =0", "1226af56e9c19ec8e78b7166632fb140": "\\vec{\\Gamma} = \n\\vec{\\mu}\\times\\vec{B}=\n\\gamma\\vec{J}\\times\\vec{B}", "12277132c4446ec03bdb8b55891b24d4": "100k", "12279c110c35cf254232fcb75f9c4551": "r_5 = 1267.9", "1227a24589c82a235a53865dd8f95b21": "\\mu_\\mathrm{N} = {{e \\hbar} \\over{2 m_\\mathrm{p}c}}", "12280c745065c7236fd62726c50c459f": "\\gamma(x')", "1228f9085b9820aea6ac9da1079d5e4d": " E\\left[ \\Lambda(n+1) \\right] = E\\left[ \\left( \\mathbf{\\delta}(n) - \\frac{\\mu\\, \\left( v(n)+r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right)^H \\left( \\mathbf{\\delta}(n) - \\frac{\\mu\\, \\left( v(n)+r(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} \\right) \\right]", "122926e78aad5d17b565114e475b7a28": "L=-{a_0^2\\over 8\\pi G}f\\left\\lbrack{|\\nabla\\phi|^2\\over a_0^2}\\right\\rbrack-\\rho\\phi\\;", "122963c1d994589b246ccb769ffa284d": " t = \\frac{W(R\\ln p)}{\\ln p} ", "12299fc0f87855188ca4a6c8c4b52689": "h\\boldsymbol{.}v=\\epsilon (h)v", "1229d22508f1f40ae348675979810e14": "\\rho_x^A", "122a602c7504f1711beecbd5be1a1df6": "sim(q_{and},d_j)=1-\\sqrt[p]{\\frac{(1-w_1)^p+(1-w_2)^p+....+(1-w_t)^p}{t}}", "122a647e7714db9742c6e5e3351205dc": "-\\nabla^2 \\vec \\psi = \\nabla \\times (\\nabla \\times \\vec \\psi) = \\nabla \\times \\mathbf v", "122a89af348d8fd8f1e20572419fcd87": "\\Sigma _{XX} ^{-1} \\Sigma _{XY} \\Sigma _{YY} ^{-1} \\Sigma _{YX}", "122a90ef5f11f5ff0f7e0fb42685ba4a": "\\gamma_n^2=c/(2n)", "122a9d74ef2cd7d8c2f4af8b883440ec": "2nm", "122b02fddeff90c6717eb96df33035ef": "E_\\mathrm{p,e} = \\phi q \\,\\!", "122b2c2f74465193a588e41ab16a292e": "\nd_r=\\frac{|x-y|}{\\left(\\frac{|x|+|y|}{2}\\right)}\\, .\n", "122b7298fcd662e04bcd4eb2bc63d32a": "\\mathbf{R}^{1,3}\\rtimes \\operatorname{O}(1,3)", "122b911c4c74cc684e2610c2b5394a15": "\\frac{M_a}{r} = \\frac{1}{r}\\frac{dP(z)}{dz} + P(z).", "122bbad4e2259cbc791139a1aba963e4": " (\\tan x)' = \\sec^2 x = { 1 \\over \\cos^2 x} = 1 + \\tan^2 x \\,", "122c42ae392a1b8225c6b27954b8c778": "\\nu = \\mu /{\\rho})", "122c5bf75de3bea75e6ac3ae9d9c9ed1": "D_{AB}", "122c8661d375cb55a740549ce54a545f": "(x_n,v_n)", "122c9b13feb136f3cb409708edd8d2cc": "(\\mathbf a \\cdot \\nabla) = \\left(\\sum_i a^i e_i\\right) \\cdot \\left(\\sum_i \\frac{\\partial}{\\partial q^i} \\mathbf e^i\\right) = \\left(\\sum_i a^i \\frac{\\partial}{\\partial q^i}\\right)\n", "122cdedc11e700eb0e2b26bd2302ec44": "\\Re s=1", "122d1605ff98ab654b99fb3ced2b7e29": "(a^2+ab+b^2)^2+(c^2+cd+d^2)^2 = ((a+b)^2+(a+b)(c+d)+(c+d)^2)^2", "122d7503d3910ab589be38df4994de48": "\\mathbf{q}^m ", "122d8a8fb9fb25f50f74a03c5c38eb74": "\\exists x\\, \\phi", "122db48b9cefab92ca64236b128959ed": "x\\delta'(x) = -\\delta(x).", "122e1f2a7549858b94d561f3a64e9c68": "\\lVert v_i\\rVert^2 = 1", "122e339be0077372b946d2a9e17c87d0": "\\hat{T} = T/T_m", "122e58646e28a92c19594205fa841430": "r = yield of option", "122e92668cca7de2ab7a4336753e2124": "h\\chi^2|\\Phi|^2", "122eacaf98b0557f59e75a5c20147b18": " \\mathbf{y} = \\begin{pmatrix} y_{1} \\\\ y_{2} \\\\ 1 \\end{pmatrix} ", "122ecd098eef3aec29493300556f4e23": "\\ K({\\sigma}/{E})^n", "122ecfc0c5e1d463b0c835ebf6f02d2c": "\\textstyle S_{1}(r)", "122f35c1eceb0aca505b33e515a1fa0d": "\\mathrm{MA}= \\frac{f_O}{f_E}", "122f3b03fe3a9d6d715b3d87b81f98d1": "\\hat t(s) \\equiv \\frac {\\partial \\vec r(s) }{\\partial s}", "122fb62a42c73bd5c9505c40ea8fcc19": "\\frac{347,373,600}{635,013,559,600}", "122fc242fc2d6bf7d88058e310d2ea39": "1 + max(1 + MD(p), 0) =", "122fcb27b282247b7ffa507dcdd25254": "\\begin{align}\n\\sin (x + iy) &= \\sin x \\cos iy + \\cos x \\sin iy \\\\\n&= \\sin x \\cosh y + i \\cos x \\sinh y.\n\\end{align}", "122fcc4df4015fe4340ce1e50bc3fb53": "\\mathcal{}(H_*(LM), H_*(LM))", "123008b3f7350bfe54c9616559cc267f": "\\mathbf{\\pi}", "12301fc6d254544e5cb648e15b667997": "\\R^N", "1230293f3b27768bb582604a7a04cd37": "\\scriptstyle \\leq8\\times10^{-16}", "12307c23fc78dd0bcb6ec9a9e6d934f0": "(M,g)\\,", "12307fed4e43bdbb9486112b07ec51a2": "\\left\\{ \\begin{matrix} n \\\\ k \\end{matrix} \\right\\} = S(n,k)", "1230dee7c33c22fa088449d30a88f4c5": "\\frac{1}{2}0 ", "1234c59108794113afe9373cdc572041": "\\left(x_0,y_0\\right)=\\left(2,1942\\right);\\left(x_1,y_1\\right)=\\left(4,3402\\right);\\left(x_2,y_2\\right)=\\left(5,4414\\right)\\,\\!", "1234e4cf4710843d42fbb6c9044dab47": " \\vec{v} ", "1235134fa75f7dac02d1d3d6c16021bb": "\\displaystyle \\mu=\\sqrt{\\frac{(ab+cd)(ad+bc)}{(a+c)^2(ac+bd)}} = \\sqrt{\\frac{(ab+cd)(ad+bc)}{(b+d)^2(ac+bd)}}", "12356653e3a5d7857d21840f8562edc7": "\\begin{align}\n L_c &= \\Big(\\frac{Y_w L_{wr}}{Y_{wr} L_w} D + 1-D\\Big)L\\\\\n M_c &=\\Big(\\frac{Y_w M_{wr}}{Y_{wr} M_w} D + 1-D\\Big)M\\\\\n S_c &= \\Big(\\frac{Y_w S_{wr}}{Y_{wr} S_w} D + 1-D\\Big)S\\\\\n\\end{align}", "12357aa804445a44a7a8f7ac3102e32e": "\\mathrm{d}s^2 = 4 \\frac{\\mathrm{d} x^2 + \\mathrm{d} y^2}{(1 - (x^2 + y^2))^2}.", "12358cec9dc75a161dbfd5d16ee21c55": "J\\frac{\\partial}{\\partial z^\\mu} = i\\frac{\\partial}{\\partial z^\\mu} \\qquad J\\frac{\\partial}{\\partial \\bar{z}^\\mu} = -i\\frac{\\partial}{\\partial \\bar{z}^\\mu}.", "12359b5c5fce40bd85efc8bb603e134b": "\\operatorname{dim}R", "1235d6d9ed53384c24a9be1b59de595e": "\n(2j+1)\\sum_{m_1 m_2}\n\\begin{pmatrix}\n j_1 & j_2 & j\\\\\n m_1 & m_2 & m\n\\end{pmatrix}\n\\begin{pmatrix}\n j_1 & j_2 & j'\\\\\n m_1 & m_2 & m'\n\\end{pmatrix}\n=\\delta_{j j'}\\delta_{m m'}.\n", "12360bf6c58ef5cb3fa6137f43ddda52": "x[n] \\cdot y[n] \\!", "12360d64269d04026c458407b7eb77c8": "h, \\, h'", "12360f5d0cd4a103ed69b52d46e4a8b9": "\\|f\\|_p=\\biggl\\|\\prod_{k=1}^n f_k\\biggr\\|_p\\le \\prod_{k=1}^n \\|f_k\\|_{p_k/\\theta_k}=\\prod_{k=1}^n \\|f\\|_{p_k}^{\\theta_k}.", "12369323adef2b6ac037ac7509473605": "ds^2 = \\sum_{k=1}^{d} \\left( h_{k}\\right)^{2} \\left( dq_{k} \\right)^{2} \\ , ", "12369bb81f635b884615d3a06adcd9c1": " M' = M\\Lambda . \\,\\! ", "1236b0ee80dab7abc6c4ed5f3fa884d0": "E(t)", "1236ef6d4c88045281fb35fb7233f30b": "\\frac{ \\tbinom{4}{1}^{13} }{ \\tbinom{52}{13} } = 0.010568\\% = 1:9462", "1236f3661b5883b7d2617fa911e48394": "(0,\\pm\\varphi,\\pm 1)", "123714ae9bbc895bbb005db130c1ea18": "\\text{Per-unit ohms reactance} =\\frac{\\text{ohms reactance * }\\text{kva base}}{kv_{L-L}^2*1000}", "123720fa113ee8a26f845d702bce53b8": "i,j \\neq k", "12374131d5ed6ffcae1aafeda4a5d97e": "c_L(s', x) < c_L(s, x)", "1237e7b869832ee65aee6e1a25da67e8": " \\boldsymbol{M} = \\chi\\boldsymbol{H} = \\frac{C}{T}\\boldsymbol{H}", "1238294932def501af296e852d3a8453": "v_{A|O}= \\frac{v_{A|O'}+ v_{O'|O}}{1+ (v_{A|O\\,'})(v_{O\\,'|O})}", "1238af06827ca828b232b06328e8b157": "\n1 = \\int dx dp \\, |x \\, p\\rangle \\langle x \\, p|.\n", "1238e8c98b5f7025d1ac3c5f4ae2da26": "CS \\sqrt{\\frac{B-\\frac{A^2}{N}}{N-1}}=5.57", "123922b174d948495fe030c5fa3e3170": " H_0: \\theta = \\theta_0 \\text{ vs. } H_1: \\theta > \\theta_0 .", "12393c86c08d0cbd3233ac67404b5f4c": "g = C*a", "12395d6f992052fda8d9e1a84b2af10a": "Pr=Sc=1", "1239f086e7e91edefaecc2f7e5815f9c": "\n q(2\\pi r)(2t)\\Bigr|_{r} \n - q(2\\pi r)(2t)\\Bigr|_{r+\\Delta r}\n - h_c (2) 2\\pi r\\, \\Delta r \\left( T - T_e \\right)\n = 0,\n", "123a12e616746a8b3425c03b01a48908": "\\wedge^3(\\mathbb{R}^6)=20", "123a2a6b5a2242545d496ac741765a0a": "\\text{Var}(\\hat \\theta) = \\frac{\\text{Var}(Y_1) }{2} = \\frac{\\text{Var}(Y_2) }{2}. ", "123abc919f58bb86743c32750f05906c": " \\frac{d}{dt} \\int_S S \\, dS = -\\int_S CB^\\alpha_\\alpha \\, dS", "123aeec2a5af89d9f40e5ecb2228234a": "f\\ ", "123b3a22e1631d6b5ad940988aa3aac9": "\\sum_a p_a c(a, x) \\leq C", "123b5221e62d1bab9ebad987eec7f585": "\\frac {d^2 y} {dw^2} +(a-2q \\cos 2w )y=0.", "123b84ce6042edefb5b79c73453eaa49": "\\varepsilon = \\begin{pmatrix}\n\\varepsilon_{xx}' & \\varepsilon_{xy}' + i g_z & \\varepsilon_{xz}' - i g_y \\\\\n\\varepsilon_{xy}' - i g_z & \\varepsilon_{yy}' & \\varepsilon_{yz}' + i g_x \\\\\n\\varepsilon_{xz}' + i g_y & \\varepsilon_{yz}' - i g_x & \\varepsilon_{zz}' \\\\\n\\end{pmatrix}", "123bafcba93692cc1b377714de1d1fe9": " ACH_{natural}\\,\\!", "123bc18dd032ef9cbbd9c4decf8141fc": "j(-k) =-i\\,", "123bc36c8925dd815c8776ad813f3bc7": "\\ A'_\\mu = G A_\\mu G^{-1} + \\frac{i}{g} (\\partial_\\mu G)G^{-1}", "123bdb08585e858c15dcdbe416a84929": "200/\\pi", "123bdf044cc21fe34e8f3fd079132580": "y=\\begin{cases}a & \\{ft\\} < 0.5 \\\\ -a & \\{ft\\} > 0.5 \\end{cases}", "123bf20abd51db581b1c254bff6488cf": "\\ {(D/L)_{\\alpha}} < tan(\\beta) ", "123c4276e0d91c674e7b16668e1e8381": "p(X^o,x^m,h) = p(X^o,x^m,h,n,b)\\,", "123cb18d654323da0c0b665bc228bb85": "P(i) = \\frac{w_i}{\\sum_j{w_j}}", "123d4135a23024dea37a0a21d4b9ec5e": "\\neg", "123d8ade263cac41d0839caadc24475c": "\n S = 100~ \\mathrm{erf}(3.186\\times10^{-4}~ E^{1/2})\n ", "123dd035d447bd11426bface84055ce3": "R = \\{(a,b), (b,c), (d,c) \\}", "123e54c4a448b6baf5052ea574207e89": "m<2n", "123e8a3aee1ce55cf796ea9786f72c57": "\\textit{occludeon}(t)", "123e8d144d967627071ba9ea3ff9eac3": "{\\rm Homeo}(X).", "123eceb2d71b024607a840b5e2546742": "45 = 101101 \\rightarrow 101111", "123f43d18ac9de268761fa0205b8e831": " \\mathbb Z / k \\mathbb Z ", "123f64184498c92037eea7317a7bbe46": " du = du_0 + du_1 ... + du_n \\,", "123fe8babaa1491298251228ca10c0eb": "O(n^2 2^n)", "123fead50246387983ee340507115ef4": "release", "12401c21f812323d8fe4a5a7d5fa19a0": "\\leftarrow B_{1},\\dots,B_{m},\\hbox{not } C_{1},\\dots,\\hbox{not } C_{n}.", "12406c172f70c0bbb7106bf8d0d8ae86": "g:{\\mathbb R}^n\\rightarrow{\\mathbb R}^m", "12407f685886a1fe269f187dd846e1d2": "\\Delta H=m_2(r_2-r^\\ddagger)+E_a", "12408005ec77bde3747b9b62b6c44013": "g(X_1,X_2,X_3) = f_1(X_1)f_2(X_1,X_2)f_3(X_1,X_2)f_4(X_2,X_3)", "12409d2b253a22166f5edc4f83ee1f5b": "\\alpha \\vDash n", "1240ae0fb72f1a1d67eb40e658ae5e49": "(k_x,k_y,0)", "1240b003124de52825ef062d3f95a099": "\\oplus_{i=1}^nU_i", "1240e5e9e41d04af29a1e5bac903a416": "\\xi=\\frac{1}{2}\\left(\\frac{X_I}{R}-\\frac{R}{X_I}\\right)\\sinh\\beta.", "1240ecbec27152cf0d71adb1ed12c8e9": "\\alpha -\\, ", "1240f923bf13def0103540dca57bef07": "\\scriptstyle d\\leq 2 ", "12416398ed8a94bee1d079fa2d810e33": "\\hat x' = \\hat x \\left(\\cos^2 \\frac{\\theta}{2} - \\frac{1}{3} \\sin^2 \\frac{\\theta}{2}\\right) + \\frac{2}{3} \\hat y \\sin \\frac{\\theta}{2} \\left(\\sin \\frac{\\theta}{2} + \\sqrt{3} \\cos \\frac{\\theta}{2}\\right) + \\frac{2}{3} \\hat z \\sin \\frac{\\theta}{2} \\left(\\sin \\frac{\\theta}{2} - \\sqrt{3} \\cos \\frac{\\theta}{2}\\right) ", "12417f5d28e24bd44da929b2be484104": "\\scriptstyle \\| z \\| \\;=\\; \\sqrt{z z^\\star} \\;=\\; 1.", "124180f6828f80c4e7d303d7bd1aac0c": "\\circ: R \\times R \\rightarrow R", "1241c07e91187ea706ccf6761f87d17c": "r_{i}.", "1241cfe7f164c618d8b6f0fc22c3069e": "Ax + By = C,\\,", "1241d2ebf682fcf1bf4142c43fe208ce": "\\zeta=\\sum_j g_j e^{-E_j/k_B T}", "1241d39477768db0bd74b98eb4c933dc": "\n\\begin{align}\n\\boldsymbol{\\nabla} \\boldsymbol{S} & = \\frac{\\partial S_{rr}}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{rr}}{\\partial \\theta} - (S_{\\theta r}+S_{r\\theta})\\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{rr}}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{r\\theta}}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{r\\theta}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta})\\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{r\\theta}}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{rz}}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{rz}}{\\partial \\theta} -S_{\\theta z}\\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{rz}}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{\\theta r}}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta r}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta})\\right]~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{\\theta r}}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{\\theta\\theta}}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta\\theta}}{\\partial \\theta} + (S_{r\\theta}+S_{\\theta r})\\right]~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{\\theta\\theta}}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{\\theta z}}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta z}}{\\partial \\theta} + S_{rz}\\right]~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{\\theta z}}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{zr}}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{zr}}{\\partial \\theta} - S_{z\\theta}\\right]~\\mathbf{e}_z\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{zr}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{z\\theta}}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{z\\theta}}{\\partial \\theta} + S_{zr}\\right]~\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{z\\theta}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z \\\\[8pt]\n & + \\frac{\\partial S_{zz}}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_r +\n\\cfrac{1}{r}~\\frac{\\partial S_{zz}}{\\partial \\theta}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta +\n\\frac{\\partial S_{zz}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_z\n\\end{align}\n", "1241db460a954e92eda397668c44851e": "x>e.", "1241fb664e80c6d150a1ea2f028e9564": " \\begin{align}\n \\theta_k[n] &= \\frac{2 \\pi}{f_\\mathrm{s}} \\sum_{i=1}^{n} f_k[i] + \\phi_k \\\\\n &= \\theta_k[n-1] + \\frac{2 \\pi}{f_\\mathrm{s}} f_k[n] \\\\\n \\end{align} ", "124285765ef84e388663269b08fc769e": "\\mathrm{2 ROO{^{\\cdot}} \\ \\xrightarrow {} \\ 2 RO{^{\\cdot}} + O_2 \\ \\longrightarrow {} \\ ROH + QO + O_2}", "1242bcaec18bf6267b0f73fe7ba477cc": "x^2 - y^2 \\equiv 0 \\pmod{n} \\hbox{ , } (x + y)(x - y) \\equiv 0 \\pmod{n}", "1242c179234b05ffffd98a49a5ed9539": "x_1+x_2", "1242de1682fb4474d1201906297e595f": "\\int x\\sin^2 {ax}\\;\\mathrm{d}x = \\frac{x^2}{4} - \\frac{x}{4a} \\sin 2ax - \\frac{1}{8a^2} \\cos 2ax +C\\!", "1242e581a0c2efd7027c5d27be1b8661": "A_5/\\{e\\}", "1242ecd57e70f6b94b7613dfb7bf013b": "F^{\\alpha\\beta} = \\left(\\begin{matrix}\n0 & {E_x} & {E_y} & {E_z} \\\\\n-{E_x} & 0 & cB_z & -cB_y \\\\\n-{E_y} & -cB_z & 0 & cB_x \\\\\n-{E_z} & cB_y & -cB_x & 0\n\\end{matrix}\\right).", "1243248f7c2490bc40cd0736c008f547": "=\\frac{1}{2}\\frac{m}{L}\\int_0^L u^2\\,dy", "124367e24296d57e22098b3214a95f38": "E_{b \\lambda}(\\lambda,T)", "1243e8c3af2463f759d6e0936b9a9512": "\\beta_\\text{max}\\,", "1244109828a3a22567f537df52ee9fa2": "{x} = \\sqrt {r^2 + \\alpha^2} \\sin\\theta\\cos\\phi", "124425e8319cb8ce9d2c719154aebb33": "\\scriptstyle{f:\\mathbb{R}^2 -> \\mathbb{R}}", "124429560e9b47daf31f10d235f6e8c4": "\n\\begin{align}\nx& = [a_0;a_1,a_2,\\dots,a_k,a_{k+1},a_{k+2},\\dots,a_{k+m},a_{k+1},a_{k+2},\\dots,a_{k+m},\\dots]\\\\\n& = [a_0;a_1,a_2,\\dots,a_k,\\overline{a_{k+1},a_{k+2},\\dots,a_{k+m}}]\n\\end{align}\n", "124457016256eab931d0737735bdd7ed": "\\phi : U \\times U \\to [0,1]", "1244680ad953edccff29bd4d4d32911e": "f\\colon M\\to N", "1244995069a569a3654e069361470c25": "j(\\tau), \\frac{j^\\prime(\\tau)}{\\pi}, \\frac{j^{\\prime\\prime}(\\tau)}{\\pi^2}", "1244c1bdcebdccb532f8dc75a2f9c090": "\\exp[-x/2]", "1245706d70b0116c2d45bb8cfb88d58c": "x_j = x_m", "1245717a3d05e6f6fe82e2d3334fdbf9": "\\left| h_i(x) \\right| \\in O\\left(\\frac{x}{(\\log x)^2}\\right)", "1245783d4161fda018f51697c88d3b8c": " p(1|0) = 0.5\\, \\operatorname{erfc}\\left(\\frac{A+\\lambda}{\\sqrt{N_o/T}}\\right)", "12458d4526d6ebacc01561035773a72e": " \\boldsymbol{\\omega}_\\mathbf{B} = {1\\over 2}\\mathbf{B} \\times \\mathbf{B'} = -{1\\over 2}\\tau \\mathbf{B} \\times \\mathbf{N} = {1\\over 2}\\tau \\mathbf{T} ", "1245cb3de72982a5c8de03e7a500e8cd": "l_2= a_{00} - \\mathcal{L}(p_6)+p_3p_6, \nl_3= a_{00} - \\mathcal{L}(p_9)+p_3p_9, \nl_{31}, .... .", "1245d412ae9fab2fd7892fb683c6fa61": "O(\\log^2n)", "124664611c787862a0324b191ddaa120": "a_{n+1} = a_{n} + 1", "12470230f048f949d059b336e72b3909": " b \\equiv 8^{2^{2-1-1}} \\equiv 8 \\pmod {13}", "1247271523c63a49a34c35cde5f81c4f": "\\ \\alpha", "12473f8d7960bda6a3531ff5e50dfe59": "F^n", "12474cc97b24718e3063e70fa9666eea": "G=(N, T, P, S)", "12474dbaedded29c0b332c5e45788ade": " r(u,v) > 0", "12475580a93d0ee778183ab1d41a8713": "\\sum_{n=-\\infty}^\\infty m_n \\hat f(n) e^{int}", "12476384f5a379dd4f040efe08a5f740": "\\int_a^b f(x)\\,\\mathrm dx = F(b) - F(a).", "12476d963b3e2219aae926849ed72a6e": " \\int_{\\textbf{R}^{n d}}f(x_1,\\dots,x_n) M^n(dx_1,\\dots,dx_n)=E \\left[ \\sum_{(x_1,\\dots,x_n)\\in {N} } f(x_1,\\dots,x_n) \\right], ", "124793fab7ff1799b76a5f3c32f114a3": "f : A \\to \\mathbb{R}^2 ", "1247cecf9cbbcb26fe51b3182c261d50": "SO(2k+1) \\cong PSO(2k+1) = PO(2k+1),", "1247dfaf13fa0de630671919222ede1b": "q'_\\text{P}", "124817a655ced5c7aac92cde4e35c94d": "E Z = 0", "124855233af08ab80a9b816d884a33c4": " (-1.00, 0.00); ", "12487358679b18e3b307c8c5417d71d7": "\\frac{D(D+2)} {2(k-1)} > n\\begin{pmatrix}r + 1\\\\2\\end{pmatrix}", "1248bb97d27a019202b745b3eb87b54d": " R_\\mathrm{VARIMAX} = \\operatorname{arg}\\max_R \\left(\\sum_{j=1}^k \\sum_{i=1}^p (\\Lambda R)^4_{ij} - \\frac{\\gamma}{p} \\sum_{j=1}^k \\left(\\sum_{i=1}^p (\\Lambda R)^2_{ij}\\right)^2\\right).", "1248cc9b9ae89cd84f3a33977f2aa5f5": "\\phi_{l}\\,= \\frac{\\phi_{N} + \\phi_{L}}{2}", "1248f41cea6c5800827eef338ffe9233": "w =w_N +w_P", "1249255abc34d9746ea2998bcc4e66a0": "A_i \\preceq B_i", "124959c70e14f25a350b4433012e93e8": "\ny = f(x) = \\frac{a + bx}{c + dx}\\,\n", "1249881b4e7472f7500ab50d768f1870": "\\begin{align}\n \\phi_0(x) &= 1, \\\\\n \\phi_k(x) &= x^k = x\\phi_{k-1}(x)\n\\end{align}", "1249d55a013e09bc608e513a47d10794": "p = A \\to w \\in P", "124a121306fde680279af0804a518d4f": "\\phi = \\cos^{-1}{\\sqrt{\\frac{2}{3}}}", "124a4f4d11c05d6ff8f576f98bf74766": "U_0,\\dots,U_{q-1}", "124a76d25daafbd3b64f44bf0f63b410": "1 - 4\\pi r^3/3V", "124a785e7d95506a9a0a87e1fccf6087": "a_{1}*b_{13} ", "124a912c09eef7a570c4c81dd6f021ae": "T_{11}=2 \\eta_0 \\lambda {\\dot \\gamma}^2 \\, ", "124aee59b2cc99d23e863b9c8602a69c": "\\sqrt{r^2+h^2}", "124afdb7adcd1ce02e1f749621999b13": "S(p_x) = - \\int p_x(u) \\log p_x(u) du", "124b0d7b8a012b551fe318b4e3855f7a": " = \\int \\Big[ f(x) \\; \\nabla_{\\theta} \\log\\pi(x \\,|\\, \\theta) \\Big] \\; \\pi(x \\,|\\, \\theta) \\; dx", "124b31831039e075ae68ec8b2d471284": "f(x) = \\frac{k}{\\lambda^k} x^{k-1} \\exp\\left(-\\frac{x^k}{\\lambda^k}\\right)", "124b40f8730a387b59b373453abc35f4": "G(x,y,", "124b55f9d7f201cc5d9d45857cfe5275": "z(\\mathbf{x})z(\\mathbf{x'}) \\approx \\varphi(\\mathbf{x})\\varphi(\\mathbf{x'}) = K(\\mathbf{x}, \\mathbf{x'})", "124b80811383990664fbd06048601c7c": "\\sigma_{xy}", "124b94e13e3ab87f0648002b7d014dff": "\nU = \\int d\\theta \\int \\rho d\\rho \\ \\lambda(\\rho, \\theta) \\Phi(\\rho, \\theta)\n", "124bb64e427a77d713fd3511725a46e4": " y (x, \\ t) = A \\cos \\left( kx - \\omega t \\right) = A \\cos \\left(k(x - v t) \\right) ", "124bbed7f1cc8945b9362a028037eb6a": "{\\mathfrak H}\\,", "124c37a293ec5fd14a840ed9e6b167a1": "\\nabla^2\\mathbf{B} = \\frac{1}{c^2} \\frac{\\partial^2 \\mathbf{B}}{\\partial t^2}", "124c3f1456bcae33c34c3705ee9b97b1": "\\scriptstyle\\mathfrak{C}(\\mathcal{Z})", "124c657d230808bb54c9b80b8d8c7951": "I = \\displaystyle s_wY +(s_c-s_w)P_c", "124cc99f1a965ccda1f1ebc90e27f0ce": "\\alpha: [0,1] \\rightarrow [0,1]", "124d0d3b5a7d9c9d4eafcab3db0a5e17": "\\frac{\\mathrm{d} \\sigma}{\\mathrm{d} (\\cos\\theta)} = \\frac{\\pi \\alpha^2}{s} \\left( u^2 \\left( \\frac{1}{s} + \\frac{1}{t} \\right)^2 + \\left( \\frac{t}{s} \\right)^2 + \\left( \\frac{s}{t} \\right)^2 \\right) \\,", "124d6e4e12cd3587cfc2e894599ceb83": "C_{rr} = \\frac{F}{N_f} ", "124d83121dfc7388b3968154d23df6fb": "(X - X_\\min) / (X_\\mathrm{norm} - X_\\min)", "124daaa9e105e4908792851293009202": "\\tau(M)", "124ddc7a5c21676c228a1889de2423e5": " =\\sum_{i\\neq m}\\text{Tr}\\left\\{ \\mathbb{E}_{X^{n}}\\left\\{ \\Pi\n_{\\rho_{X^{n}\\left( i\\right) },\\delta}\\right\\} \\ \\Pi_{\\rho,\\delta}\n^{n}\\ \\mathbb{E}_{X^{n}}\\left\\{ \\rho_{X^{n}\\left( m\\right) }\\right\\}\n\\ \\Pi_{\\rho,\\delta}^{n}\\right\\} ", "124df6f2540504dc5ea73945b35b9634": "({v_0+v_i})K_w 10^{-b_1E_i + b_0}", "124e0b2d5bbc6c7d1c376e23e5e11b8b": " R = \\sqrt {\\frac{(ab + cd) (ac + bd) (ad + bc)}{(a + b + c - d) (b + c + d - a) (c + d + a - b) (d + a + b - c)}}.", "124e21988bd76b47a0232d0df89107fa": "i=0,1,..,n", "124e35fd39520da2c58603ffc4c5d101": " f_{pm} ", "124e676e83768676656053306aeeffbc": "e^{-At}\\mathbf{y}'-Ae^{-At}\\mathbf{y} = e^{-At}\\mathbf{b}", "124e8e38590efe9ffe05c3fd7fa802ae": "z=\\infty", "124ee9170b68be63c66a6508107ec89f": "b + X", "124ef642b189c7c2450cff56c63e29a3": "\\frac{\\partial u}{\\partial r}", "124f9472da0fcb9f86874a2b0d7d2bac": "\\alpha=\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}\\ = \\frac{1}{V}\\left(\\frac{V}{T}\\right)", "124f987ff771391bb00f37927adf8d51": " X_2 ", "124fd5141bc40b5b468edd12592de2a5": "x-y \\in K_0", "1250211274ac8dbaea6909a65db43e42": "P,Q,R,S", "12503daaa21a370cca9e4c120ad57057": "\\Delta U(V_a,V_b;T^+;V_c,V_d;T^-)\\ ", "12506f2e450d14afb20ccaab244169bc": "S_1\\cdot x_1+S_2\\cdot x_2", "1250b128d30bd4cc75ffced31b5a770a": "\\forall x \\forall X_{\\in A} (Xx \\rightarrow \\exists Y_{\\in B} (Yx \\and \\forall y (Yy \\rightarrow Xy)))", "12510ecc627663872b9a367b7d0a42f9": "(\\overline{C} \\vee A \\vee B) \\wedge \\overline{(\\overline{C} \\wedge (A \\vee B))}", "125172d7b5e75dcb3bb4cea332268e82": " H_3^* \\quad \\longrightarrow \\quad H \\ + \\ H \\ + \\ H ", "1251ef41f88e98de4d6424fac710c96b": "n_i ", "1252156f985856912260dec8d312d04e": "A_x \\hat{\\mathbf x} + A_y \\hat{\\mathbf y} + A_z \\hat{\\mathbf z}", "1252205ef76eda7fb2f3243606f87404": " \\mathbf{F} = \\mathbf{m} \\mathbf{a} ", "1252609ae43efbe1758c91376fe5262b": "\\rho \\otimes \\omega.", "12528bd547d254e55632140682b9adec": "\nt = \\int_{0}^{z_s} { n dz \\over c \\cos \\alpha(z) } \n", "125329315671ce0768bc58235ad51770": "\\left(\\text{P} - \\text{V}\\right)", "12533d0cac5f786c86f7c65d790a9226": "L = T - U.\\quad", "12538201841371db2c2d38d42eebbcd9": "H(\\phi) \\le \\log \\sqrt {2\\pi eV(\\phi)},", "1253967d5a5a67fb641930ab20fea687": "F_{s,n} = gV(\\rho_s - \\rho_a) ", "1253b8522e16c0aef582d10ec00e4fa3": "\\frac{OA}{OQ} = \\cos \\alpha\\,", "12542106b65a8ea67158e83abf8b38fc": "X=2/3.", "1254717bd76e6ef0c39fb10d1de24e7a": "Y=g(X)", "125495e1cd74bc8b3c1e181d3be3ec2b": " a_2(S,H) = \\frac{\\ln(S/H) + (r-\\frac12\\sigma^2)\\tau}{\\sigma\\sqrt{\\tau}} = a_1(S,H) - \\sigma\\sqrt{\\tau}", "12550c1f32bd7c369550848520210d4d": "u'(t) = \\frac{d}{d t} r'(t) = \\frac{d}{d t} r(t) - v = u(t) - v.", "125559e24a2662e0f31ced8149f1ae6a": " \\operatorname{MTF}_s(\\nu)= e^{-3.44 \\cdot (\\lambda f \\nu /r_0)^{5/3} \\cdot [1-b \\cdot (\\lambda f \\nu /D)^{1/3}]} ", "1255c60ed83d144951ed7d372a92fd30": "(X, d_{X})", "125600b11d7831dae7398f10239575f4": "(O(log log n))", "12564d9786930cafbbe0d46ed676a605": "x^2 - 1.786737601482363 x + 2.054360090947453 \\times 10^{-8} = 0", "12565974392c9ea7f9aa41160579b8d8": " f \\in [0, 1] ", "1256e361612e4bf9ff63ba8873202b7b": "c_{k'}^{(1)}=- \\frac{i}{\\hbar} \\int_0^t dt' \\;\\lang k'|H_1(t')|k\\rang \\, e^{-i(E_k - E_{k'})t'/\\hbar} ", "125706542d7f4c6099a06a4f2910fdb4": " y_2(x)=4x-3 ", "12571fdd653be83248f4c367ebe63526": "\\varepsilon / a = \\varepsilon", "12572aeb00945406c0aab1a3db8a243d": "-\\overline{3}= \\overline{-3} = \\overline{1}.", "1257f763703310d4d585c8adfb8c63ce": "\\pm\\sum_{i=-\\infty}^n a_i p^i.", "12580adb261fa0e7dc64dca30ad48490": "O(\\log n/\\log\\log n)", "1258424547d3aa8b9a38c22fd11d64de": "E[F] < 1", "125866e83e92694a1d413a7f07ce6cc8": "\\Sigma U_\\Gamma = F_\\Gamma - A_{\\Gamma 1}A_{11}^{-1}F_1 - A_{\\Gamma 2}A_{22}^{-1}F_2,", "125876e06b2da07ede2d22ea19c4cbee": "{1 \\over \\rho}{\\partial \\over \\partial \\rho}\\left(\\rho {\\partial f \\over \\partial \\rho}\\right)\n+ {1 \\over \\rho^2}{\\partial^2 f \\over \\partial \\phi^2}\n+ {\\partial^2 f \\over \\partial z^2}", "12589b831ded820b78312b4a9e184fd6": " \\Psi", "1258a5e9e73f6d90a2e7f3bbaf6b7858": "1s^2\\,", "1258dbf189da09bf418f8f7198e52383": "\\overline{\\overline{I}}(h) = \\inf\\{I(f) | f\\in C[a,b], h\\le f\\}.", "1258dd3bb09ffd5dd023fb9de14d699b": "[H_k ]", "12590fd370dd0d394f27f2c42e383909": "\n\\begin{align}\nT_A & = [A]+K[A][B]\\\\\nT_B & = [B]+K[A][B]\n\\end{align}\n", "12591df3303c108f4b805aa680b25fdc": " v\n= \\frac{1}{ u} ( {u} \\cdot v) + \\frac{1}{ u} ( {u} \\wedge v )\n= ( {v} \\cdot u) \\frac{1}{ u} + ( v \\wedge u ) \\frac{1}{ u} \n", "1259b792279c498370afd3cf199bda9b": "e_3(\\tau) = \\tfrac{1}{3} \\pi^2(\\vartheta_{10}^4(0;\\tau) - \\vartheta_{01}^4(0;\\tau)).", "1259c9004980830e830c91e7968cdd64": "s \\ \\stackrel{\\mathrm{def}}{=}\\ q/f", "1259d55e1521d7150973e4f0c57b19a3": " (\\nabla_c T)^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_s} = \\frac{\\partial}{\\partial x^c}T^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_s}+\\,\\Gamma ^{a_1}{}_{dc} T ^{d \\ldots a_r}{}_{b_1 \\ldots b_s} + \\cdots + \\Gamma ^{a_r}{}_{dc} T ^{a_1 \\ldots a_{r-1}d}{}_{b_1 \\ldots b_s} ", "125a667f96c903122b119b7d55a7679a": "\n\\tan \\theta\n= \\sqrt{\\mathit l \\over \\mathit l^{\\prime}}\n.", "125ad75d17f3b3cfcdfb0c044c023737": "B_{max}", "125b0511a55b001336a4ac691827b64e": "v \\notin N(T \\setminus S)\\,", "125b3020442c5368e2560aaa34bf6b7d": "\\Pi_0(x)", "125ba9c6dfaa60d16409d0823ae0af2d": "\nm(\\phi)= a(1 - e^2)\\int_0^\\phi \\left (1 - e^2 \\sin^2 \\phi \\right )^{-3/2} d\\phi,\n", "125bc3a895f1cace6b98ec45b9843a6f": " ( a_1 ) + ( a_2 x + a_2) + ( a_3 x^2 + a_3 x + a_3) = x^2 - 1 \\,", "125c1c1e80015d2040a8330045ba5520": "u'''(c)<0", "125c242a0f02072e64f9a0b8ec347ce4": "\\left|\\frac{f(z_1)-f(z_2)}{1-\\overline{f(z_1)}f(z_2)}\\right| \\le \\left|\\frac{z_1-z_2}{1-\\overline{z_1}z_2}\\right|.", "125c44cda569226c190ba8c30f32ecab": "log_g", "125c76b791996b305353eb9ef3267bca": "(G\\times SL(n),V\\otimes \\mathbb F^n)", "125c8a306b5436475841a9ca05531c98": "{\\tau'_{\\rm pb}}", "125c916ea28cb0850dbaeae7deea0981": " n = \\mathrm{JD} - 2451545.0 ", "125d3bebec5585c5d1186740cec6c450": "\\begin{align}\n E_\\text{out} &= \\sum_{i = 1}^\\infty {K_i E_\\text{in}^i \\sin(i \\omega t)} \\\\\n &= K_1 E_\\text{in} \\sin(\\omega t) + K_2 E_\\text{in}^2 \\sin(2\\omega t) + K_3 E_\\text{in}^3 \\sin(3\\omega t) + K_4 E_\\text{in}^4 \\sin(4\\omega t) + K_5 E_\\text{in}^5 \\sin(5\\omega t) + \\cdots\n\\end{align}", "125d3d0b4ab35bf5a521075f21b624f5": "X^+3\\frac{hs}{D}x^2+3\\left(\\frac{hs}{D}\\right)^2x=\\frac{P'}{3}\\frac{h^2}{D^2}", "125d4958d0e1050ee0240844f52260a7": "\\gcd(a,a) = a", "125d61f85d6153a25aee3145e197dd5c": "\\Omega = {\\left(q+N^{\\prime}-1\\right)!\\over q! (N^{\\prime}-1)!}", "125d70c63d089e961501181575d0109c": "N(\\mu,I_k)", "125d83620b063896be9566c9947f4861": "\\alpha_1 = \\frac{4G}{c^2}\\frac{M}{b_1}", "125e33aceadefa2ba7f06cc44ab8b2a3": "E\\, =\\, \\frac12\\, \\rho\\, g\\, a^2\\,", "125e7fd7e93fb8bb74b6c13e295e772d": "[ion]_\\mathrm{out}", "125eaa7cf8392f6df79524d1a2c79ecc": "y = ax^3 + bx^2 + cx + d\\;.", "125ec6280e530d5eb3696c74bdb05b4c": "\\sum_{\\scriptstyle j=1\\atop\\scriptstyle j\\not=i}^n a_{i,j}(\\Phi_{j,1},\\ldots,\\Phi_{j,n}),", "125eee4fbaf566efa60b078555716f27": "W(T,G) := N_G(T)/C_G(T).", "125f3838a0d36b6db2bc4839ec1a3313": "v\\in (F_2)^d", "125f70d920c3b077f993f3c8e55fc7a8": "\nc_{\\mathrm{air}} = 331.3 \\ \\mathrm{\\frac{m}{s}} (1 + \\frac{\\vartheta^{\\circ}\\mathrm{C}}{2 \\cdot 273.15\\;^{\\circ}\\mathrm{C}})\\,\n", "125f86f1bb764f803ffb0ba805dd6e34": "\\dfrac{dR}{dP} < 0 \\!\\ ", "125ff111ae29a09a0d6187c9ad57d47e": " \\Delta_h = hD + \\frac{1}{2} h^2D^2 + \\frac{1}{3!} h^3D^3 + \\cdots = \\mathrm{e}^{hD} - I ~, ", "125ff850e79a4937783d86ec5201e79d": "\\mu = \\left(\\frac{\\gamma-1}{2}\\right)^2 - \\frac{1}{4} \\quad,\\quad \\gamma = 1 \\pm \\sqrt{1 + 4\\mu}.", "126181fb770c0f9cc630588ac25faa51": "p=\\tfrac{1}{6}", "1261e4be5b3b0a32eafdba15427ca7c8": "u(t) = - K ( x - e ) \\, ", "126207c7c197aaac720240a1d6b4e28f": "f(k,i)", "12623305c66087564e25eef11f3a0df6": "z_1,\\dots,z_{k-1},z_{k+1},\\dots,z_n", "126248cb7a44893db99179a8b28cb329": "D(f(d))", "1262780e938a441012b3440423e8680f": "(q_{i,1}, q_{i,2}, q_{i,3}, q_{i,4})", "126282e05e46e5803e9f309388d2d38d": "\\epsilon_{ik} = \\frac{\\mathbf{x}^\\top_{0k}([\\delta K] - \\lambda_{0i}[\\delta M])\\mathbf{x}_{0i}}{\\lambda_{0i}-\\lambda_{0k}}, \\qquad i\\neq k.", "126288d3a8b385054b6ddac5b34c0a96": "\\hat{P}(y\\mid x_1, \\ldots x_n)=\\frac{\\sum_{i:1\\leq i\\leq n \\wedge F(x_i)\\geq m}\n\\hat{P}(y,x_i)\\prod_{j=1}^n\\hat{P}(x_j\\mid y,x_i)}{\\sum_{y^\\prime\\in\nY}\\sum_{i:1\\leq i\\leq n \\wedge F(x_i)\\geq m}\n\\hat{P}(y^\\prime,x_i)\\prod_{j=1}^n\\hat{P}(x_j\\mid y^\\prime,x_i)}", "12629a5d9f1f5f72b4687c383e23d98b": "\\bar{A}^{f}e^{i\\phi_n^f} \\xrightarrow{iFFT} \\bar{A}_n^ke^{i\\phi_n^k}.", "1262b0faeaad9a7ed7a2bff7d9abba81": "\\mathbb{A}^n_k", "1262c8e1c18575a059b5cf91f534a970": "\\scriptstyle \\sqrt{n}", "12631845e3c8705ce99025dc91efae18": "1 - x^2", "126322941f06403fe6622d98ef9d16dd": "g_{(a,k)}(u)=k (1-u)^{-\\frac{1}{a}},", "126327aabeee288cf4009fa57374087d": "X_i \\subseteq M_i", "12633add9295ebce5a88194f6045af38": "s(x) = p(x)\\cdot x^t - s_r(x)\\,.", "1263e48ce658011752e890a21f3cd767": "\\{like\\langle Mary, Sue\\rangle, like\\langle Mary, Bill\\rangle, like\\langle Mary, Lisa\\rangle\\}", "1263f69f7776ee47356ed28dface550e": "n_ib", "12640f1b86c99509ca20c670e0f8357f": "\\ \\alpha_j = A_j \\, e^{-i \\Omega \\,t}, ~~~j=1,2 ,", "1264377932bb51600a94acc2dc4c931f": "\\Delta_+", "12644e32dfd2bae04a4425cc837d092f": "y(t)=Cx(t)+Du(t)\\, ", "126452993412054a722fef0d5b76cc79": "S\\subset P", "1264775e19756e6284ae825567ebe908": "|n m_n\\rangle, n=", "1264d76983fee04d9abff194802e7a91": " \\frac{\\delta J}{J} = \\varepsilon^{2}_S \\frac{\\delta S}{S} ", "1264e7dd7a4c8d655f80511a1896bdf9": "10\\uparrow\\uparrow\\uparrow 7=(10 \\uparrow \\uparrow)^7 1", "126527e7998eb6e56dca42ba00aa3cdd": "V(z) = \\prod_{p \\in P, p < z} \\left( 1 - \\frac{\\omega(p)}{p} \\right).", "12660a9c909b6559c139e463f6ca9b6b": "M_J \\sim T^{3/2} \\rho^{-1/2} \\sim \\rho^{\\frac{3}{2}(\\gamma-1)}\\rho^{-1/2}.", "1266191b557b8e3d371c6e5d8374750e": " \\int_{\\Bbb Z_p} x^k \\, {\\rm d}x = B_k ", "1266217685d9afca644c2ab68efed145": "\\bigcap_{k = 1}^N \\pi_{i_k}^{-1}(A_{i_k}) \\neq \\varnothing.", "12666f427a06cb7ae30b145786f98d33": "P_{move~left} = \\tfrac{1}{2} + \\tfrac{1}{2} \\left( \\tfrac{x}{c+|x|} \\right) ", "12667dc42bf6efaf907508d2989def4e": "\\nabla^2_{norm} L(x, y; t)\\ = t \\nabla^2 L(x, y, t) = t (L_{xx}(x, y, t) + L_{yy}(x, y, t))", "1266ebad88668d1c21cff3bfc59f7419": "\\Delta T(t)= T(t) - T_{\\text{env}} ", "1266ed1bd98c40bf728a91a454894a43": "\\mathbf{e}_i \\mathbf{\\times} \\mathbf{e}_j = \\varepsilon _{ijk} \\mathbf{e}_k, ", "1267408c5f220709bec4fa7d678b36b7": "q(v)=x^\\mathrm{T} Ax,", "12679462f63dc9e6dc59ea6eb6e958aa": "\\delta_F", "12679fa07708083104270037e000e918": "a_E = (D_e - F_e/2)", "1267a22467312d12a24047643ec10f2c": "\\mathbb{Z}_q", "1267c8459f58fc6e452c2cd9809914d2": "\\mathbf{B}(t) = \\sum_{i=0}^n \\mathbf{b}_{i, n}(t)\\mathbf{P}_i,\\quad t \\in [0, 1]", "1267cf6ae1d2ff5f24dd40fb0b2b47a0": "\\rho^{B^R_jC}", "12684fd11b07870ff85ca8c008836b78": " \\displaystyle{H=H^2\\ominus \\varphi H^2,}", "126873d39a4673014dd33b7bdd7994ee": "\\varphi : V\\times W\\to V \\otimes W", "1268788ce463106c00b9b12beb021c17": "\\beta_3 = h = ( s - D )", "1268a97a8e8d3fe14ed2363a47e2f296": "g_i = \\mathrm{gf}_i / \\mathrm{df}_i", "1268c3f34396c75bb802c6d9125981dc": " -\\frac{dT}{dZ} < \\frac{g}{C_p}", "1268e479b5bb6df076517211fbfa5139": "\\displaystyle e^{- i a \\omega} \\hat{f}(\\omega)\\,", "1269001e056be480f8b325e6a5308b1c": "T \\hat{\\mathbf{p}} T^\\dagger = - \\hat{\\mathbf{p}} ", "126919cc76ce8409ab7b152ea124bea7": "d_1(z)= \\frac{1}{z+1}.", "126940ab279fc415187d24bbe9d08241": "C_v = {\\hbar \\alpha \\omega\\over 16\\pi\\varepsilon_0},", "126998c5b3c80066fd81f842da105f9b": "l_1(\\theta) = \\vartheta + \\angle KAP_1 = \\vartheta + (\\pi-\\psi)/2 = \\vartheta + (\\pi - \\vartheta + \\theta)/2 = (\\vartheta+\\theta+\\pi)/2", "1269c1bcb48bde5cf4c600a7a074d9ae": " \\sum_{n=1}^\\infty\\frac{1}{n^s}", "1269f1376e7546db19fbc3a552785bf6": "\\frac{\\text{Var}(Y_1) }{2} = \\frac{\\text{Var}(Y_2) }{2} ", "126a0e232b97bb2f7e45420750cf4dae": "|n_{k_i}\\rangle", "126a3cce63461921d8822a755e6af7a4": " x \\subsetneq y \\, ", "126a5823b36cb789463e52d32837cd37": " \\!\\ S_m^2 = a^2 + mb + 1. ", "126a67027ae68286feac1bf0131407bb": " \\omega = \\frac{\\Omega_c}{2} \\pm\\sqrt{{\\Omega_c}^2/4-q E_r/{m r}} ", "126aa6d864ebacdbeac4e912af9ce4cd": "M_x[a + cx] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&c&0& \\cdots \\\\\na^2&2ac&c^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "126af1cb64b1d7efe4be6aaf77885e73": "-P=(Y_1:X_1:Z_1)", "126b105020a7237b678b44e1697ad8b4": "v^2 = \\eta_{\\mu \\nu} v^\\mu v^\\nu = - (v^0)^2 + (v^1)^2 + (v^2)^2 + (v^3)^2", "126b9e8d2cd37e40a30e3d7b62e4f4ad": "= \\sqrt{2} \\cos(z-\\frac{\\pi}{4})", "126b9f451c423f62ea83ffbc46991a1d": "\\left( \\mu_{k} \\right)", "126bbc117a9e6c77c7174b1820dc904e": "\n\\mathrm {DOF} \\approx \\frac {2 Hs^2}\n{H^2 - s^2} \\text{ for } s < H \\,.\n", "126bf189bee4feac9cb8f4ab526f50ac": "\\left\\{a_i\\right\\}", "126c22d4981a757399b08f1f4ce4b79c": " y = b+r\\,\\sin t\\,\\!", "126c75ce7c9e8e3f9dd9e793863b3bbb": "z = {{x - \\bar x} \\over h}", "126cd512c076e18ccea5109bb3ded14d": " |L\\rangle ", "126ce06c203d6bcbb4aca9807e44de1e": " \\int 1_{\\mathbb{Q}}(t) \\varphi(t) dt = 0 ", "126d60f9ee83b5fd78477bb9b45d3ca9": "H = \\frac{\\dot{a}}{a},", "126d698dd81f8d606d127b703510a9d0": "S = \\frac{\\mu_0LV_A}{\\eta}", "126d7080a8a6b254646be2eaf49237bd": "\\textstyle T_o=1/\\beta _o", "126da68c4e475facabd83fdbc28d1d4e": "(x+y)^n", "126f13dc582261d8923ff3ed7e4d05f1": " \\mathbf{J} \\cdot \\mathbf{\\hat{n}} = \\frac{I}{A} \\,\\!", "126f3b58d2d3cfaa5468124e7981ac4d": "T_y\\pi \\colon T_{y}E \\to T_{\\pi(y)}B", "126f48e90e46c7a1a4e0de492762e893": "L_n(x) = V_n(x,-1).\\,", "126f551514b91594d1b99befd04d4ce2": "\n p(r,\\theta) = \\left[A_\\alpha~J_\\alpha(k~r) + B_\\alpha~J_{-\\alpha}(k~r)\\right]\\left(C_\\alpha~e^{i\\alpha\\theta} + D_\\alpha~e^{-i\\alpha\\theta}\\right)\n ", "126f5d27337e55f461c98dad1807130e": "f(x_0 + h) = f(x_0) + \\frac{f'(x_0)}{1!}h + \\frac{f^{(2)}(x_0)}{2!}h^2 + \\cdots + \\frac{f^{(n)}(x_0)}{n!}h^n + R_n(x),", "126fa22ff26bdcd0757cd665317b7ce7": "k_2 = 1.0617", "126fea4f626ce40601010227c4d520ae": "\\boldsymbol{u}^{(0)}", "12704da6da40361c09c4514b8523ed43": "\\cosh(1) = 1.1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1..._!", "127070cef4ce4b76f4df143571c4f648": "r= K_1 k_2 C_A C_S", "12710456e501635a6b739fb262be8494": "w(k)=\\sum_{k'}P(k,k')=\\frac {2 \\pi} {\\hbar} \\sum_{k'} |\\lang\\ k'|H_1|k\\rang |^2 \\delta(E_{k'}-E_k)", "127145354bb25868eda39d2844e7752c": " \\mathcal{P}_3(-p_2):= a_{30}(-p_2)^3 +a_{21}(- p_2)^2 +\na_{12}(-p_2)+a_{03}=0. \n", "12715dbe04152bffa498eadeab92e182": "\\varphi = f(s)\\!", "1271717094f619bb3eea472dcae38f06": "\\Delta t < \\frac{h^2}{a}", "1271777354724d416ba24bf14928e2d8": "Q^\\pi(s,a) = E[R|s,a,\\pi],\\,", "12718b12e5423663a5e482278ce4afc9": "\\tau_{ind} \\left(\\omega \\rightarrow 1 \\right) = 1", "1271cf2bc4f3b6802ccce3debad8c53e": "\\displaystyle{W(x,y)=(Vx+(I-VV^*)y,-V^*y).}", "127221c0f223a56c51d3b02b24ae9406": "{4 \\choose 2}_q = \\frac{(1-q^4)(1-q^3)}{(1-q)(1-q^2)}=(1+q^2)(1+q+q^2)=1+q+2q^2+q^3+q^4", "12723e4f009caa821e7d8d43ded9905d": "n_1=n_2=0\\,\\!", "1272508ab953e5d45f2129c29d1e37df": "\n\\begin{cases}\nX_1=\\frac{\\alpha^2}{2r}(1+\\frac{r^2}{\\alpha^4}(\\alpha^2+\\vec{x}^2-t^2) ) \\\\\nX_2=\\frac{r}{\\alpha}t \\\\\nX_i=\\frac{r}{\\alpha}x_i \\qquad i\\in \\{3,\\cdots , n\\}\\\\\nX_{n+1}= \\frac{\\alpha^2}{2r}(1-\\frac{r^2}{\\alpha^4}(\\alpha^2-\\vec{x}^2+t^2) ) \n\\end{cases}\n", "127257c2dc29fbeca649a53cfc48327f": "\\mathrm{H_2S_2O_8\\ +\\ 2\\ H_2O\\longrightarrow\\ H_2O_2\\ +\\ 2\\ H_2SO_4}", "127267ed85da0e97d582fe899227ec5a": "k\\geq 3", "1272c7815a1cec170499935656bc980f": " Z_{Fano}'' = x_1.{1 \\over 4} [ x_1^4 + 3 x_2^2 ] = x_1Z_{Klein} ", "1272d1e60873f8f535d5a46f30bc308a": "y^2 - y = x^3 -x", "12731b719e36bfa2dd14969ef6fd8339": "\nc^{2} d\\tau^{2} =\nc^{2} dt^{2}\n- \\frac{\\rho^{2}}{r^{2} + \\alpha^{2}} dr^{2}\n- \\rho^{2} d\\theta^{2}\n- \\left( r^{2} + \\alpha^{2} \\right) \\sin^{2}\\theta d\\phi^{2}\n", "1273334b73dbba47c02a909ec32ffd7a": " \\int_0^\\infty { e^{-x} \\ln^2 x }\\,dx = \\gamma^2 + \\frac{\\pi^2}{6} .", "12735a77f6ec40c2f0d894b3a54c5622": "P=a \\cdot ar \\cdot ar^2 \\cdots ar^{n-1} \\cdot ar^{n}", "12739293e2196eefd1f0e76940a4f001": "E \\subseteq A", "1273da5b5f3ede4282bf6de94a8efe27": "P(X)=P_A(X)+X^A(X-1)R((X-1)^2)", "127417bb3464156474a1d957f94256b7": "\\displaystyle{L(a)=\\frac{1}{2}R(a,1).}", "12746506ea430b0e169517e07c2e6346": "F_1 = F_2 \\,", "1274bdf84ea58e9813fc404ff400f6d9": "\\mathbb{P}\\left(\\textbf{c}=(a,b)\\right)=\\frac{e_\\mu}{e_\\lambda},", "1274e2c92ffc70708ee8e0cd19ee0ef5": "\n\\psi^{(3)}(z)\n", "127513f4e8a24e8f0c202c18bbdfa886": "\n \\Beta(x,y) = \\Beta(y,x).\n\\!", "127546b9fbbc1e5772b0df05d91e4a2e": "\\gcd(|r-s|,n) = q", "12760c648704cf116f46ff0711e79b54": "\nE =\n{a_1 a_2 \\over 4 \\pi r } \\exp \\left ( -k_D r \\right )\n.", "12762cccdeac82da2511403e6f653246": "\\rho_a=\\rho_b=\\rho", "127635e65690d5e566ed1b3dc1af405e": " \\int_{\\Omega} (\\rho \\frac{d u_i}{d t} - \\nabla_j\\sigma_i^j - f_i )\\, dV = 0", "12766cb4974d5305509f554d7f12ee4f": "G^{-}", "127696cb009776c3fe81c0c2eddf4225": "{{\\varepsilon }_{particle}}", "1276a89aa7094a2d46682178f623dce7": "\\phi = - 2\\theta\\,", "1276ac64310337268039c63081976f75": "S_i=Q_i/T_i", "1276af7c25a1c6be3daa6a6143f1913a": "\n{2\\pi\\over T} \\int_0^T dt \\left( {dp \\over dJ}\n{dX\\over d\\theta} -\n{dP \\over d\\theta} {dX\\over dJ}\\right) =1\n\\, .", "1276b659a6173afab77968dd036a7896": "u_0, u_1, u_2, \\ldots, u_n, u_{n+1},\\ldots", "12776187f463071a2f2228cc30abefb4": "n = \\frac{P}{100}(N-1)+1", "12776f31325c1c92a27d5b39832d27c0": " \\Delta \\mu= \\mu_{1} - \\mu_{l} ", "1277b7d580c09bc09e93e884dd7b3707": " \\textstyle n^{th} ", "127872b010fe92685d0579b13e386b02": "F \\subseteq \\Sigma^{\\omega}", "12787a08d245abb645cf851bead3af2c": "\\mathbf{Q} = m \\mathcal{F} = m \\frac{\\nu_{f}}{\\delta\\nu}", "127889bc621f24f455bfadd7d91b4c98": "(n+b, k+b)", "12788b97bbd1ffa842c1637c3d29eb71": "\\bar{M}=(\\bar{X}+\\bar{Y}+\\bar{Z})/3", "1278a3b1f3303891b4151076ef088e7c": "\n\\begin{matrix}\n\\quad 237.5\\\\\n4\\overline{)9^15^30.^20}\\\\\n\\end{matrix}\n", "1278cbd0efac6829e488cbc239bdfb82": "1 \\leq x \\cdot x^l \\qquad 1 \\leq x^r \\cdot x", "1278d3d6e4fbc1dde5c1b3ec24680808": "\\scriptstyle x_0", "1278da82f5feca9e35ec88fbf735dc31": "\\tau_{\\alpha,\\beta} = \\varphi_{\\beta} \\circ \\varphi_{\\alpha}^{-1}.", "127906211e37f84e2fccd019a12ddf56": "s_i=s_{i+1},", "1279165851fcd42eb3b3e24f9f755772": " 0 \\le a_i < b_i \\le 1 ", "12791b5695882095005978122d1afb4d": "X_1,X_2,X_3,...", "127931fc4dcc56591b22abd0fafe0042": "M^{n}+2nH_{2}O+2ne^{-}\\rightarrow nH_{2}+2nOH^{-}+M", "127950221540c41a0df81d590109bbb7": "p = \\frac{R\\,T}{V_m-b} - \\frac{a\\,\\alpha}{V_m\\left(V_m+b\\right)}", "127959b80aa213f9a56e75e52f9d5b71": "S = \\left \\lceil \\log_2 \\frac{P + 1}{\\log_2 17} \\right \\rceil \\,", "12796acdf565c1fd643942f318d9f7f2": "\\partial_x.", "127998f9c8a29713a7100b42b8ba7552": " K(x_i - x) ", "1279a67bd5088266c2b36d4001c06790": "J(n,2)", "127a07c13ce6119c645b731038101c5c": "\nC = {|x|_\\mathrm{peak} \\over x_\\mathrm{rms}}\n", "127a0a4ab20d66195ba7f5825b6ecb75": "w = z^{1-c}u", "127a33d00aa221bf71d830a8ffd23936": "x \\in \\mathbb{R}^m,\\alpha \\in \\mathbb{R}^p", "127a5d5fc17b0afc31b4baacd8e3b4dc": " G = \\sum_{i=0}^\\infty G^i A_i.", "127a7792cc8467e3dbdaa4fac59ae4be": " |\\{ x: f^*(x)>t\\}| = |\\{x: f(x)>t\\}|.", "127ab64bb77b0232593c48234f86d150": "s_1, y_1, y_2", "127b0d77e28c888c1b5c9333439d1216": "\\forall \\phi(\\boldsymbol{x}) \\in L(A,\\boldsymbol{x})", "127b65b4c2c1cd11aa55916245993a33": "\\displaystyle \\Lambda_0(P) = \\min \\sum_{x \\in X, s \\in S}P(x)l(s)", "127b6f24caf5e02a93f9a5d9865dcc45": "\\mathit{alg}(A_1,B_3)", "127b891ef1ca85b0b16821f10084a492": "|\\mathbf{U}|{\\rm tr}\\left(\\mathbf{U}^{-1}\\frac{\\partial \\mathbf{U}}{\\partial x}\\right)", "127ba9d5bb612a2810654de6c4a24fd7": "\\cfrac{1}{3 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{3 + \\cfrac{1}{9+ \\ddots}}}}}", "127bc10827549d7b09c71fcffb9e83cf": "\\mathbf e\\,\\!", "127c423075a3f04249a36367eb845ce9": "f_a([n,n+1)) = a_n", "127c5888c5e3986f030d153f1b7a3f84": "\\text{stick}(T(p,q)) = 2q\\text{, if } 2 \\le p < q \\le 2p. \\, ", "127c60f20391dab1c9c36d8c2002768f": " \\mathbf{L} = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}) + (\\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R}))\\times\\mathbf{V}.", "127cebc613dca053840cf87416f7205f": " \\theta = \\sin^{-1} \\left( \\frac{\\lambda\\ \\Delta \\Phi}{2 \\pi d} \\right) ", "127d24b90d89ed8eeda6504619cd514c": "\\sin 3\\pi", "127d4a62fbb575405cbfc143556b7c04": "\\mathbf{r}_{\\perp}", "127d5a6ee6de8eb7c9cb4fc78abae39f": "\\dot{\\varepsilon}\\,\\!", "127d5cad2a8735f839cbb76bba29fad1": "A_V = K", "127d745b260546a70711a25148fff137": "\\mathcal A \\models_X^+ \\psi \\Leftrightarrow \\mathcal A \\models_X^+ \\psi^I", "127dbacca197947e51b825fa32e38164": " \\sigma_z = \\dfrac{F}{A} = \\dfrac{Pd^2}{(d+2t)^2 - d^2} \\ ", "127dc2bd0312ee9e2f51401b299049db": "\n\\frac{L}{r^{2}} \\frac{d}{d\\theta} \\left( \\frac{L}{mr^{2}} \\frac{dr}{d\\theta} \\right)- \\frac{L^{2}}{mr^{3}} = -\\frac{dV}{dr}\n", "127dcbe8a3b1361825c5b95aecbf05f6": "\\Gamma_{\\varphi}", "127e05514f9d475becf9b799f625ad5e": " a_{ij} = (s_{ij})^\\beta ", "127e2428cc5eee92d8fed461deaa50cf": "\\varphi: E \\rightarrow J(E)", "127e4a477763fb75420c0b93b0040673": "\\scriptstyle{\\sqrt{1-{v^2}/{c^2}}}", "127e5f4f0d1772979ff87dc4d84d6509": "(K+2)/(K+1)", "127f471b60bfba0d2b99176f12cdff0e": "\n \\int_{\\Omega}\\boldsymbol{\\nabla}\\boldsymbol{G}\\,{\\rm d}\\Omega = \\int_{\\Gamma} \\mathbf{n}\\otimes\\boldsymbol{G}\\,{\\rm d}\\Gamma \\,.\n ", "127f8fbd8444a968d9c1afbb8923d25a": "D\\rho -\\bar{\\delta}\\kappa=(\\rho^2+\\sigma\\bar{\\sigma})+(\\varepsilon+\\bar{\\varepsilon})\\rho-\\bar{\\kappa}\\tau-\\kappa(3\\alpha+\\bar{\\beta}-\\pi)+\\Phi_{00}\\,,", "127fe7b019eaff17dc971d3b0ce48039": " ds^2=\\rho^2 g_{00}(dx^{0})^{2}+ 2\\rho g_{0k}dx^{0}dx^{k}+\ng_{kq}dx^{k}dx^{q},", "12802072d74db0d113998a71adedd886": "(T,J)", "12804c191cdc801c96baa00c608b5b60": "k_{\\lambda}.v = c_{\\lambda} q^{(\\lambda,\\nu)} v", "12806f094734430c17ebe2b71965509f": "X = \\varprojlim_{j \\in J} Y_{j} = \\left\\{ \\left. y = (y_{j})_{j \\in J} \\in Y = \\prod_{j \\in J} Y_{j} \\right| i < j \\implies y_{i} = p_{ij} (y_{j}) \\right\\}.", "12809868bff29cd82bd6f5a9307b75cb": "\\log \\frac{k_{t-BuCl, sol}}{k_{t-BuCl, 80% EtOH}} = Y ", "128100f79d6b369d6b5a4dbe7a6019f1": "\n\\frac{\\sigma_\\alpha^2}{\\sigma_\\alpha^2+\\sigma_\\epsilon^2}.\n", "1281940ad676f6185a95c268232a13f9": "a(\\phi_i)\\,", "128272371a72877a2d047697f43e7e21": "{(3/2)}^{12}\\approx 2^7,", "12828aa2d02f8f12551f81ec0f3e6e09": "c = \\begin{pmatrix} a_0 \\\\ a_2 \\\\ a_4 \\\\ \\vdots \\\\ a_N \\end{pmatrix} = D \\begin{pmatrix} y_0 \\\\ y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_{N/2} \\end{pmatrix} = Dy, ", "128292f4faba09cf1b32174bf30e4225": "q = 1 - p = 1 - \\tfrac{1}{2} = \\tfrac{1}{2}", "1282dbcdc9771c92690c92b70668b383": "m > 0", "1282ebdebd93251bba9b2a347a5748bb": "(x_1,\\ldots,x_n)R=A", "12838cc6c9d9f08bd41423b3c18ae865": " {}^\\mathrm{N}\\mathbf{a}^\\mathrm{R} = {}^\\mathrm{N}\\mathbf{a}^\\mathrm{Q} + {}^\\mathrm{B}\\mathbf{a}^\\mathrm{R} + 2 {}^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B} \\times {}^\\mathrm{B}\\mathbf{v}^\\mathrm{R} ", "1283bca13104450a3d644e1f61cc29b9": "\\hat H = -J \\sum_{j =1}^{N} \\sigma_j \\sigma_{j+1} - h \\sum_{j =1}^{N} \\sigma_j ", "1283dfdefd9a083ea421b2128b4d41c2": " \\frac{ | \\Psi(X,a) | ^2 } { \\int | \\Psi(X,a) | ^2 \\, dX } ", "1283edbbf0829fb0c5fac0c5378a3754": "\\overline{X}_n\\pm A\\frac{S_n}{\\sqrt{n}}.", "12842d1001ac44440b1106c77a6352f7": " z : \\mathbb{R}^p \\rightarrow \\mathbb{R} ", "128446a074f73eeed19baa18c8f10e2c": "g_i(x)>0.", "1284501774c40a4e3cd18ff724f171fc": "|\\text{alive}\\rangle", "1284720f4e5ab194021b0ec2614532f6": "\\,_2F_1", "12848b9e61a8a6e0b78591be42a03c0f": "\\overline{\\operatorname{Sp}}(E)=\\overline{\\operatorname{Sp}(E)}", "1284a37300dff1eef415c4671fc4f550": "\\displaystyle{a(z)={1\\over |z|^2 +1}.}", "1284b1184c63d526116f02716eaa0152": "\\overline V", "1285139fcd6053413cf8aafa4469e2f8": "\\begin{align}\nC &= LBA \\div ( SPT \\times HPC )\\\\\nH &= ( LBA \\div SPT ) \\, \\bmod \\, HPC \\\\\nS &= ( LBA \\, \\bmod \\, SPT ) + 1\n\\end{align}", "128523349a618db89d0a57eb780a0fe4": " \n\\begin{bmatrix}\nu & 0 \\\\\n 0 & u^{-1} \\end{bmatrix}", "12852673ea50019ba6e2809f54e64b76": "\\operatorname{ad} (x)(y) = [x, y] . ", "128530b0af27d8a2025932018b18b4d7": "B \\oplus A", "12856e19775c18fd7baba3954d06bbbb": "\\mathrm{Res}_{z=\\infty} \\frac{f(z)}{5-z} = \\exp (\\tfrac{\\pi i}{4}) \\left (5 - \\frac{3}{4} \\right ) = \\exp(\\tfrac{\\pi i}{4})\\frac{17}{4}.", "12859eb919f21acda5de15e886748a40": "(T_s - T_o)", "1285bd102cc096dd43afefeb62566669": "O(A_1:A_2|B) \\triangleq \\frac{P(A_1|B)}{P(A_2|B)}", "1285d4942c89ea2976d708dfa97df851": "X[k]", "1285f0eb201983932c7ca8b8c42480a6": "\nP(k) \\propto k^{-\\gamma}.\n", "12860194205a06179d4444609bdc9aaa": "d = \\frac{{WL}}{{4T}}", "12863bb5fc4524cb660f5dab95b580c9": "|\\Psi^-\\rangle", "128689e74d5fa4c94a5fb7b1c563c756": "\\textstyle(x, y\\pm1, z\\pm1)", "12871792bb70b377f281f1645e374975": " P_e^{(n)} \\ge 1 - \\frac{1}{nR} - \\frac{C}{R} ", "12873aeb286b5c2cca7e9477ec9481c5": "\\left(1+x_{i}\\right)", "128743d2a4c3ecfae83458638d0afc97": "|\\;\\;|", "1287491ec7e30d8610ee89c7fe68f165": "(\\phi \\to \\bot ) \\to \\lnot \\phi ", "12878870fa8bca90746deb93abe62bd7": "\n\\frac{\\sqrt{2}}{Y} dt = \\frac{d\\varphi_{1}}{\\sqrt{E \\chi_{1} - \\omega_{1} + \\gamma_{1}}} = \n\\frac{d\\varphi_{2}}{\\sqrt{E \\chi_{2} - \\omega_{2} + \\gamma_{2}}} = \\cdots =\n\\frac{d\\varphi_{s}}{\\sqrt{E \\chi_{s} - \\omega_{s} + \\gamma_{s}}}.\n", "1287c663689a16554d0912980ce45aed": "e_s", "1287e2f653202e49eb141856077e70a1": "a\\oplus b=T(a,1,b)", "12889c5f93db54fa936d8a765d4eae57": "\\diamond", "1288e7976aeb15fa36c259ee691fa9f6": "\\sigma \\approx 0.45 \\lambda N \\ .", "1288f2505e50d59509893aa9ccacaaab": "\\phi-1=\\left(\\sum_im_i\\right)^{-1}\\left[If'-f + \\sum_i\\sum_j\\left(\\lambda_{ij}+I\\lambda'_{ij} \\right)m_im_j\n+2\\sum_i\\sum_j\\sum_k \\mu_{ijk} m_im_jm_k + \\cdots\\right]", "12893d508020982fb6b2ff7dd7ae1453": "i_{abc}(t) = T^{-1}i_{\\alpha\\beta\\gamma}(t) = \\begin{bmatrix} 1 & 0 & 1\\\\\n-\\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 1\\\\\n-\\frac{1}{2} & -\\frac{\\sqrt{3}}{2} & 1\\end{bmatrix}\n\\begin{bmatrix}i_\\alpha(t)\\\\i_\\beta(t)\\\\i_\\gamma(t)\\end{bmatrix}.", "1289aae3ff46474aba2e05b4d2280e20": "A_\\%(M)", "1289df2c84ee0c3a40a1065037b04e78": "\\mathbf{b}_i, \\mathbf{b}_j", "128a351b89452c4a8d00aaccdb9d768c": "h(x_1,\\ldots,x_m)\\,", "128a3b1f4c29b0e5d8dc4441370d5287": "a_v = \\int_0^\\infty e^{-\\lambda u} \\frac{(\\lambda u)^v}{v!} \\text{d}F(u) ~\\text{ for } v \\geq 0", "128ac735f20a3719bda08037ea22f57e": "\\scriptstyle X \\subseteq \\mathbb{R}", "128aeb60dd0b9b835ea64242ee28689f": " E^{(0)}_{n_1,n_2} = E_{n_1} + E_{n_2} = - \\frac{Z^2}{2} \\Bigg[\\frac{1}{n_1^2} + \\frac{1}{n_2^2} \\Bigg] ", "128afe269fe8494d81c53c597f6884ff": "\\omega_{X|Y}", "128b162f3c9bed28b393d74204dea961": "y=x\\cot\\frac{\\pi x}{2a}.", "128b2315c119aa4eac8fb764208093fe": " \\Lambda ", "128b2c4604b54fdc17544b8bd6ea1c87": " \n \\vec{F}_f = \\int _f \\vec{j} \\times \\vec{B} d^3\\vec{r} \n", "128b3fa96868a2126f747c3f359f5c89": "r=a+b\\theta^{1\\!/\\!x}.", "128b431cae260ba2ecf90420f7788334": "\\frac{{\\rm d}^2 x}{{\\rm d} t^2} = -\\sum_n \\omega_n^2 x ", "128b4b3d701986d8faf7d987d4fc30d6": "A_{192}= 3.1410319509", "128bc044cd07eeaf609bdec5337e028e": "\\lambda_1\\lambda_2\\cdots\\lambda_n vol(K)\\le 2^n vol(R^n/\\Gamma).", "128bed3dc81d6fe9b774969ee1e3f277": "1/e=37%", "128c3c0470addee25dc3e3fb07ba16f9": "\\langle cacao \\rangle", "128c48c1c8125a6befc26888274e95cc": "f'(x) = {\\mathrm{d} \\over \\mathrm{d}x} f(x)", "128c5a8ad52331e3b7d7fbbaec8fcd50": "F_0,F_1,F_2", "128c6a4f7e1a9cb466921f7a7bf54212": "D_{\\mathrm{KL}}(P\\|Q) = \\sum_i \\ln\\left(\\frac{P(i)}{Q(i)}\\right) P(i)\\!", "128c74c8c28c008a33efbd4d8958162e": "Y_s(z)", "128c7ad2d429745ab8a80ae6ba90b4ef": "\\mathrm{E}(X) = \\lambda \\Gamma\\left(1+\\frac{1}{k}\\right)\\,", "128cc12ac724410d689b09aa3cbf6da3": "\n\\begin{bmatrix}\ne_x \\\\ e_y \\\\ e_z \\\\ e_t\n\\end{bmatrix} = \nA^{-1}\n\\begin{bmatrix}\ne_1 \\\\ e_2 \\\\ e_3 \\\\ e_4\n\\end{bmatrix} \\ (2)", "128d68ecea6a7fe76562024915ca9252": "k = k(n)", "128d7a9bd803506a72862602729ff3ba": "\\ (\\phi, \\lambda)", "128e0bbe088ffccc96e270fbb234e941": "\\operatorname{dist}_{\\operatorname{robust}}(T(\\mathcal{M}), \\mathcal{S}) = \\sum_{m \\in T(\\mathcal{M})} \\sum_{s \\in \\mathcal{S}} g((m - s)^2)", "128e249456f495173273bd487e4ff69b": "|r\\rangle", "128e46c906836d75c4f2de51ff9063fc": "P(n)= |\\langle n|\\alpha \\rangle |^2 =e^{-\\langle n \\rangle}\\frac{\\langle n \\rangle^n}{n!}", "128ee115856834e25132858339b49686": "\ng(s) = 2 \\gamma \\int_0^{\\infty} (st)^{\\gamma + \\rho - 1/2} \\; G_{p,\\,q+1}^{\\,q+1,\\,0} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ 0, \\mathbf{b_q} \\end{matrix} \\; \\right| \\, (st)^{2 \\gamma} \\right) f(t) \\; dt,\n", "128f81f3c82c6548b1bff4b7275018e0": "u_2 = \\frac{(x_1^2+ax_2^2+x_3^2+x_4^2)x_6 - 2x_2(x_1 x_5 + bx_2 x_6+ x_3 x_7+ x_4 x_8)}{c}", "128f906f18ac639aa2ee0f1422df9bd0": "a + c", "128fa24c558c3507c9276167b0238205": "B\\rightarrow\\neg C", "12902544313a452e8c8346c520e826e8": "= <\\psi|(A-a)^2\\psi>", "12902fc62eb1ae422beb9b2dfb7f1928": "n_1 (\\lambda)", "1290559cd34e18837285c6e1b47342ff": "\\pi : C(X) \\rightarrow L(K)", "12906248bbe21d18d6b8d96391a922a1": "\\forall p: \\mathcal{BB}p\\to\\mathcal{B}p", "12909992d5d2cbd8a4c871a7228c6621": "\\Gamma(n) = 1 \\cdot 2 \\cdot 3 \\cdots (n-1) = (n-1)!\\,", "1290bbb09acfab4c1ea905f753946b06": "\\mathbf E =\\frac{1}{2}\\left[ (\\nabla_{\\mathbf X}\\mathbf u)^T + \\nabla_{\\mathbf X}\\mathbf u + (\\nabla_{\\mathbf X}\\mathbf u)^T \\cdot\\nabla_{\\mathbf X}\\mathbf u\\right]\\,\\!", "1290c173f86f2eb00adfa9fd8d6cb59b": "\\varepsilon^l_A:A^l\\otimes A\\to I", "1290eac0ed93251cf8fb17d8d3bdf7c6": "p(\\textbf{z}_k|\\textbf{z}_{1:k-1}) = \\int p(\\textbf{z}_k|\\textbf{x}_k) p(\\textbf{x}_k|\\textbf{z}_{1:k-1}) d\\textbf{x}_k", "1290eeffdf3b1e7b492ce720dd10e08d": "(g^a)^b \\bmod{p}", "1290f1ca251579ae051c4cbe263d7a7b": "\n\\int_{-\\pi}^{\\pi} \\sin((2m-n)x)\\sin^n x\\ dx = \\left \\{\n\\begin{array}{cc}\n(-1)^{m+(n+1)/2} \\frac{\\pi}{2^{n-1}} \\binom{n}{m} & n \\text{ odd} \\\\\n0 & \\text{otherwise} \\\\\n\\end{array} \\right.", "12910770b73831d398ee1304962983de": "\\mathcal{Z} \\left\\{ f(t) e^{-a\\, t} \\right\\} = e^{-a\\, m} F(e^{a\\, T} z, m).", "129150c6efea9d3418afcf58346e9b34": "\\mathcal L(D)", "12919629d5d98028f8dacf03e5acaf66": "\n\\mathrm{E_1}(z) = \\int_1^\\infty\n\\frac{e^{-tz}}{t} dt\n", "1291978b40fef6c5d9d4fbba9be0d1c0": "\\mu \\|w\\|_2", "12919dc6b2e16aac8bfa19a25c0a2d4b": "VCA (p_{0 \\frac{1}{2}}, (a, m)) \\cup \\{[m, m]\\} \\cup VCA (p_{\\frac{1}{2}1},(m, b))", "129272275507e29f027b602dc8577020": "\\frac{\\partial T}{\\partial q_j} = \\sum_{i=1}^n m_i \\mathbf{\\dot{r}}_i \\cdot \\frac{\\partial \\mathbf{\\dot{r}}_i}{\\partial q_j}", "12931a7922d292546b5ca8f85c8af3e1": "\n\\langle f|\\mathbf{H}_B|f'\\rangle\\equiv\\langle u,t,s,r|H_B\\otimes\\mathbb{I}\\otimes\\mathbb{I}\\otimes\\mathbb{I}|u',t',s',r'\\rangle\n", "129344bb032523706eabf801ce7ee83b": "f(S)=\\sum_{i\\in S}w_i", "12935094c05d4fe264f2fc36e73f92c3": "\\begin{align} \\phi_1 + \\phi_2 &= \\phi_{\\text{sys}} \\\\ \\frac{\\phi_1}{V_1} + \\frac{\\phi_2}{V_2} &= 0 \\ ,\\end{align}", "12938105a43d5bd52ffd54768a90d4af": "\n\\frac{1}{2}\\rho v^2 + P + \\rho g h = \\mathrm{const.}\n", "1293ec82c0b7832cde09c34855e74c40": "\\perp, \\angle, \\sphericalangle, \\measuredangle, 45^\\circ \\!", "1294693eebe7d20084681aada4103956": "\\vec F \\cdot t = \\Delta m \\vec v", "129469ac0904fdf0ef810058213e7523": "V_1\\otimes V_2\\otimes V_3", "1294788e2618d4229c3e59cb9570d903": "E_\\text{S} = -\\mathbf{d}\\cdot\\mathbf{F}.", "1294c1f68e997ca6d8ecb25082804df9": "\n\\langle r^{2} \\rangle = \\frac{6k_{\\rm B} T \\tau^{2}}{m} \\left( e^{-t/\\tau} - 1 + \\frac{t}{\\tau} \\right).\n", "1295421e1fcaee2a173e113a9d045194": "P(S \\rightarrow S'|E)", "129543f9083a67fe18059cdb5c2e41a6": " \\Delta W = - \\Delta V \\,\\!", "12959e8725dc55f16e20d57052980217": "\n \\cfrac{\\mathrm{d}^2 W}{\\mathrm{d} x^2} - \\left(\\cfrac{1+\\alpha}{\\beta}\\right)~W = \\frac{M}{\\beta D^{\\mathrm{beam}}} - \\cfrac{q}{D^{\\mathrm{face}}}\n", "1295a859b167ac81f4eb51b1c83b9ed4": "C \\in \\mathcal{C}", "1295f7ead87b6432cba321b54eb6a4fe": "phi_i", "12962c2f3464d4162ffa46009a480575": "P=K\\rho^{5/3}", "1296c45635af9dcd2e03f80ad2e7c7e6": "d\\mathbf{X}_t = \\boldsymbol{\\mu}(\\mathbf{X}_t,t)\\,dt + \\boldsymbol{\\sigma}(\\mathbf{X}_t,t)\\,d\\mathbf{W}_t,", "129799ef2fac2e0f9e0e28c16c6d636c": "{\\gamma \\times y_1^2 \\over 2} - {\\gamma \\times y_2^2 \\over 2} - F_d = \\rho q(v_2 - v_1)", "12979e24fa99ffbb7f9dff7add280e9d": "A_{f}(\\infty)", "1297fd52d3a43aef7bb4671d3bdcb0d2": "\\, \\vec{S}_i \\cdot \\vec{D} = \\sum_{A=1}^N \\; M_A \\big(\\vec{f}_i \\times\\vec{R}_A^0\\big) \\cdot \\vec{d}^{\\,A}=\\vec{f}_i \\cdot \\sum_{A=1}^N M_A \\vec{R}_A^0 \\times\\vec{d}^A = \\sum_{A=1}^N M_A \\big( \\mathbf{R}_A^0 \\times \\mathbf{d}_A\\big)_i = 0,\n", "12984b04e8a953c103b5a529a128c99b": "\\frac{dR}{dt} = \\gamma I - \\mu R ", "1298853a34d859cb701ae96805924528": "\\vec{F}\\,(\\vec{q})\\!", "1298a05a415e6456eb1a36ed80aad200": "\\phi \\to \\bot ", "12992240713fa57dea8beac2f9e6b0d7": "\\sin \\alpha = \\frac{\\cos U_1 \\cos U_2 \\sin \\lambda}{\\sin \\sigma} \\,", "129930a70d02c994e22afc8b5858fd3f": "*T^{IJ} = {1 \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} T^{ KL}.", "12994c1a1c3de5f91e4b361ce002e0ba": "\\alpha = [\\alpha _1 , \\ldots ,\\alpha _N ]^T", "1299988740ca877169a14d4791f07d2f": "1 \\,-\\, \\frac{1}{3} \\,+\\, \\frac{1}{5} \\,-\\, \\frac{1}{7} \\,+\\, \\frac{1}{9} \\,-\\, \\cdots \\;=\\; \\frac{\\pi}{4}.\\!", "1299f15555e56cf3ad2a405ffaa33c71": " H_u : V \\to V\\,", "1299f512529d4c7f8a198b278ba90f3c": "\\forall n (n \\in \\mathbf{N} \\iff ([\\forall k \\in n(\\bot) \\or \\exist k \\in n( \\forall j \\in k(j \\in n) \\and \\forall j \\in n(j=k \\lor j \\in k))] \\and", "129a14a0b7f576ef418c269e42a06e0c": "x = \\operatorname{Re}\\,(z) = \\dfrac{z + \\overline{z}}{2}", "129a829c9e91d217377ffc2766a4b4e4": "\\mathit{E_{g,\\mathrm{InPAs}}} = \\mathit{x}\\mathit{E_{g,\\mathrm{InP}}}+(1-\\mathit{x})\\mathit{E_{g,\\mathrm{InAs}}}-\\mathit{bx}(1-\\mathit{x})", "129a8495f1a7863d379c07b4d6823d93": "wp(\\mathbf{if}\\ E_1 \\rightarrow S_1 \\ [\\!] \\ \\ldots\\ [\\!]\\ E_n \\rightarrow S_n\\ \\mathbf{fi}, R)\\ = \\begin{array}[t]{l}\n (E_1 \\vee \\ldots \\vee E_n) \\\\\n \\wedge\\ (E_1 \\Rightarrow wp(S_1,R)) \\\\\n \\ldots\\\\\n \\wedge\\ (E_n \\Rightarrow wp(S_n,R)) \\\\\n \\end{array}", "129ab398b7a82f688c39dbc35665aea9": "w(z)=\n\\left(\\frac{z-a}{z-b}\\right)^\\alpha \n\\left(\\frac{z-c}{z-b}\\right)^\\gamma\n\\;_2F_1 \\left(\n\\alpha+\\beta +\\gamma, \n\\alpha+\\beta'+\\gamma; \n1+\\alpha-\\alpha';\n\\frac{(z-a)(c-b)}{(z-b)(c-a)} \\right) \n.", "129b589447eac570919806a4185e0533": "\\Delta_{\\mathbf{Q}(\\sqrt{d})}", "129bb404ca829f380029ad271169e967": "(X,X)", "129bc753f57cd83e62b01c65567d3da9": "\\exp(z)", "129c1890bcdbfbf0280c06f64917fbec": "\\tilde{S}(\\omega)", "129c62c5d963eb77790ec32a3ba74cee": "P^*=\\frac{1}{2}", "129c9eb5ffb4389eba06a0ef4ad5342a": "(X_{j})_{j\\in S_{i}}", "129cd22e74f096474908e39736962095": "\\mathcal{F}^3(\\hat{f}) = f.", "129ce768e58bd652764f42e4ef302cfe": "\n \\begin{align}\n \\delta K & = \n -\\int_0^T \\left\\{ \\int_{\\Omega^0} \\left[\n J_1\\left(\\ddot{u}^0_{\\alpha}~\\delta u^0_\\alpha \n + \\ddot{w}^0~\\delta w^0\\right) \n - J_3~\\ddot{w}^0_{,\\alpha\\alpha}~\\delta w^0\\right] ~\\mathrm{d}A\n + \\int_{\\Gamma^0} J_3~n_\\alpha~\\ddot{w}^0_{,\\alpha}~\\delta w^0~\\mathrm{d}s\n \\right\\}~\\mathrm{d}t \\\\\n & \\qquad\n - \\left| \\int_{\\Omega^0} J_3~\\dot{w}^0_{,\\alpha\\alpha}~\\delta w^0~\\mathrm{d}A\n - \\int_{\\Gamma^0} J_3~\\dot{w}^0_{,\\alpha}~\\delta w^0~\\mathrm{d}s \\right|_0^T \n \\end{align}\n", "129d83ceec94d5ff850e59267524f9d3": " \\mbox{E} = \\frac{ \\sqrt{30 \\cdot P}}{d}", "129da5e0f6b8bf1b449e80360ba684ed": "\\begin{array}{rcl}p&=&\\frac{\\varphi}{e}\\\\&=&2\\frac{1+\\sqrt{5}}{\\sqrt{10+2\\sqrt{5}}}\\\\&\\approx&1.70130\\end{array}", "129db74edf52b763b3c642533a6e2949": "\\sigma_{(ij)}", "129de37aa714507bdda89a1e20644a24": "\\bar{x}=\\frac{1}{A}\\int_a^b x[f(x) - g(x)]\\;dx", "129e4bd1f992b085d3ec3012930aa1ba": "\\delta(B) = \\frac{\\Pi_{i=1}^n ||b_i||}{\\sqrt{\\det(B^T B)}} = \\frac{\\Pi_{i=1}^n ||b_i||}{d(\\Lambda)}", "129e73ab6e70f84330cfacb9f01f57dd": "\n\n\\mathbf{F}_{12} =\n- G {m_1 m_2 \\over {\\vert \\mathbf{r}_{12} \\vert}^2}\n\\, \\mathbf{\\hat{r}}_{12}\n", "129f008829c016637ce5cd30dbc975f3": " M = {\\mathrm MinN}(L,D,n)", "129f61bb06883ef57d30ffda00b7751b": " \\{ \\gamma^\\mu,\\gamma^\\nu \\} = 2g^{\\mu \\nu} ", "129fe03e48da442355421bad241d5291": "u\\in W^{1,n}(R^n)", "12a02d46fb89297274041849b9f651d6": "D = 2^m - 1", "12a03acd2fc3dc8f1083387fa2ee6a76": "\\bowtie \\!\\,", "12a03c1e0b9d556e1538f637dda40611": "\n C_3 = -\\frac{3125}{24EI}(-1645 + 4 M_c + 64 R_a) \\quad \\text{and} \\quad C_4 = \\frac{25}{12EI}\\left(-40325 + 6 M_c + 120 R_a\\right)\\,.\n ", "12a07588f9667526eec89d4f256fe18a": "\\frac{\\partial^2\\eta}{\\partial \\phi^2} = \\cos\\left(2\\left(\\phi-\\theta\\right)\\right) + h\\cos\\phi > 0. \\,", "12a0ad123bfa94caaea75d2ac1def938": "a \\cdot (0.089490\\dots) ", "12a0da5171ef3a87a5c61d1c5be421c7": "c = \\frac{w}{n}", "12a12fc7bc407a230701ed2849ac22c8": " F_i = A_t u C_i \\, ", "12a190dea50e93cdec7d9e3969af3b83": "C_n\\,\\mathbf{v} = \\mathbf{v}-(\\tfrac{1}{n}\\mathbf{1}'\\mathbf{v})\\mathbf{1}", "12a1961c1fdbc45cfc541d7df4bf4d25": " f_{0} \\frac{\\partial v_g}{\\partial p} = - \\frac{R}{p} \\frac{\\partial T}{\\partial x} ", "12a1b3daf4f9ca64e9ba471d98626bc0": "T\\delta(t) = \\sum_{n=-\\infty}^\\infty m_n e^{int}", "12a1bdc5b304c80049c79fab077ab3d0": "B[\\vec{X}]_{ab} = {{}^\\star R}_{ambn} \\, X^m \\, X^n", "12a20d1603863de6503ddd1777c01212": "\\lVert z \\rVert \\ne 0 ", "12a221a2818cdec906d60bf80b95f925": " k \\equiv \\left( \\frac{\\partial Z}{\\partial p}\\right) \\frac{\\partial}{\\partial p} \\ln\\theta", "12a2708a6820b1e7b153a8cec8842735": "\\Delta = \\frac{\\partial^2}{\\partial x_1^2}+\\cdots+\\frac{\\partial^2}{\\partial x_n^2}.", "12a321efe5c0ca06395e54fba3545253": "\\|\\beta\\|_0", "12a32b6f4b8b0b80a54ec0f9b1da5b0e": "\\Theta= \\Theta_\\mu^a dx^\\mu\\otimes\\vartheta_a", "12a369bca305ce8f34204337129c91d5": "P_a=P_0 a^{D/2-1} \\, ", "12a3d2b96d5ab8447ee9b9bcdb38af8a": "\\scriptstyle p_i", "12a3de3d0a6b9f00973ae83d9f62dfd4": "[A]=\\frac{[A]_0+[B]_0}{1+\\frac{[B]_0}{[A]_0}e^{([A]_0+[B]_0)kt}}", "12a42198be200a3a1aee1299f262bc07": "\\boldsymbol{\\alpha}=(1,0,\\dots,0),", "12a43195bb0211d4707eb2df67ce55fe": "\n {\n \\rho~(\\dot{e} - T~\\dot{\\eta}) - \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} \\le \n - \\cfrac{\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T}{T}.\n }\n ", "12a470bdb6c72e4187daf15d3a379d99": " \\frac{t_r}{t} = \\sqrt{1 - \\frac{r_s}{r}} ", "12a47f4e060848cb0c773c81fa537299": "k^2E[u_1^2]+E[x_2^2]", "12a4b07c5c6410c832dfe6665ec17b24": "t_{m+n} = t_m t_n + D u_m u_n, \\quad u_{m+n} = t_m u_n + t_n u_m \\quad \\mbox{and} \\quad t_n^2 - D u_n^2 = N^n", "12a51479f6ab7bd981c1142867ef2765": "c^1", "12a552b131692d34a36d4127410c4cd9": "\\Omega^0(M, V) \\otimes_{\\Omega^0(M)} \\Omega^p(M) = (V \\otimes_\\mathbb{R} \\Omega^0(M)) \\otimes_{\\Omega^0(M)} \\Omega^p(M) = V \\otimes_\\mathbb{R} (\\Omega^0(M) \\otimes_{\\Omega^0(M)} \\Omega^p(M)) = V \\otimes_\\mathbb{R} \\Omega^p(M).", "12a5aa756afb5ef3c78826e031a261c6": " \\bold{X}(\\bold{u}) + t \\bold{A}(\\bold{u}) = \\bold{X}(\\bold{u} + d\\bold{u}) + t \\bold{A}(\\bold{u} + d\\bold{u}) ", "12a64ebb1365899db65c0e4c4e1109a1": " \\tau_H ", "12a6651c5dface30d2dc51df9c9ec7db": " a,b,c ", "12a6eb8ec8c9a5eadbc0ff06b945661c": "\n\\begin{align}\n(\\gamma a_1)(\\gamma a_2)\\dots(\\gamma a_m) &= \\gamma^{\\frac{\\mathrm{N} \\mathfrak{p} -1}{n}} a_1 a_2\\dots a_m \\\\&\\equiv \\left(\\frac{\\gamma}{\\mathfrak{p} }\\right)_n a_1 a_2\\dots a_m \\pmod{\\mathfrak{p}},\n\\end{align}\n", "12a6f951e73b362e1d2821d112d61d4e": "\\tau_1\\ ", "12a708077f12a14041703e40d277c12f": "={1\\over W}\n\\begin{pmatrix}\nu_2'(x) & -u_2(x) \\\\\n-u_1'(x) & u_1(x) \\end{pmatrix}\n\\begin{pmatrix}\n0\\\\\nf\\end{pmatrix},", "12a72d9ca6bf8ca954b8505d2f8698ae": "c_1 : \\mathrm{Pic}(X) \\to H^2(X,\\mathbb Z).", "12a759005c96067536fecbdf90d9eefc": "K(\\sqrt{d})", "12a816f4089f8b075e6d461f627f11e2": "x = c\\varphi,\\ s = \\ln \\tan \\tfrac{1}{4} (\\pi+2\\varphi).\\,", "12a8262222e759f3bcec22b6a05ac3b9": "g(T)v=0", "12a87db75c2d2db107a04c20062ec14b": "s =r\\theta.", "12a8c4f7411c7748b54c230efb166e7b": "\\langle K, \\prec_K\\rangle", "12a8d229ad0ea9f8fdb27d3e5f39c193": " x \\neq \\omega", "12a9378f2a8d70b0d7d3f185cdd7cf46": "{}_{j \\in {1,2,3}}", "12a966fc3df93e6f1e118da677d74e99": "\\sum_{k=0}^\\infty ar^{2k+1} = \\frac{ar}{1-r^2}", "12a9b0a75fd3f74872afddcb626fd436": "\\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha^2", "12a9b5731d3932fdb0a1f12964b2032d": "\\Delta G = \\epsilon_{surface atom} - \\epsilon_{adatom} \\qquad (3)", "12aa72cea82cb412ecf50f34283a0c6b": "\\operatorname{cont}_F : X \\rightarrow Z", "12aac7e42e7e58d3076235a37300aac7": "(\\gamma=1.4)", "12aaf723354ec1b39f422d06de4c31d1": "(M, L)", "12ab7083e6ae070d8d9438a89631e6eb": "\\sigma^2\\left[1+\\frac{\\alpha\\phi(\\alpha)-\\beta\\phi(\\beta)}{Z}\n-\\left(\\frac{\\phi(\\alpha)-\\phi(\\beta)}{Z}\\right)^2\\right]", "12ab9e63af84b4eb593bb95c514554a6": " \\frac{\\partial r_2}{\\partial t}+(u-\\sqrt{\\rho})\\frac{\\partial r_2}{\\partial x}=0", "12abc05a54c243e633e3f791fd8d5559": " \\hat{x}", "12abe5668eda8e95686ff5a9ff43bcec": "\nW = i \\theta^j \\mathbf{e}_j + \\eta^j \\mathbf{e}_j,\n", "12ac0bd4846b6f11fef6826d03b35037": "\\scriptstyle V_\\mathrm r", "12ac5b96d7e2791264bb4eec4bf2d154": "\\Delta h_{ab}", "12ac71598a569e32691dca97dc431372": "{R_x}=1000\\frac{R^{*}}{M_x} \\, ,", "12ac959723e228ae1db6756a33a731be": "\\pi(\\mathrm{E}\\lambda) = \\pi_{\\mathrm{i}} \\pi_{\\mathrm{f}} = (-1)^{\\lambda}\\,", "12aca7d6645830510fc98827099b6395": " f(n)\\le Cg(Cn+C)+Cn+C", "12acce3b528f43bd40f99793a46b3e06": "C_{lk}", "12ad3df34778052eb6749f0d25faf230": "\\frac{E}{I}=Z_{sc}", "12ad55e6e897c11bf1259b6dca16e5f0": "O(n^2/(\\log_2 n)^{\\log_2\\frac{8}{3}})", "12ad5ff40904a6eb547cee28579be98e": "P=(x^2+bx+1)(x^2-bx+1).", "12ada8d9797e6fdc1708834cde6139a4": "\\frac{\\dot m_0}{\\dot m_{01}} = \\frac{\\epsilon_0}{\\epsilon_{01}} = \\frac{p_{01}}{p_0}", "12add0ec70859801543f9e1ba97e9840": "\\scriptstyle{\\beta}\\,\\!", "12ae08c812c085c4b53741770231b64c": "G'=\\{*n_1, *n_2, \\ldots, *n_k\\}", "12ae0919ab7550811cc121cc4f775965": "H_1\\oplus H_2,", "12ae3b0c6d8a027927a14e470fca74da": "E_r(m) = y^m u^r \\mod n", "12ae637c93e7554e4fa72cb44b296255": "a\\succeq b\\;", "12ae6fa2857922529d49338005d6883e": "-2 \\log(\\Lambda)", "12ae78abff2d05e11b92d824bcfaaaa7": "y_i = \\beta_0 + \\beta_1 x_i + \\beta_2 x_i^2 + \\cdots + \\beta_p x_i^p,", "12ae8713f54cd71b3e51e5713792408b": " \\frac {\\pi}{2 \\log(1+\\sqrt{2})} ", "12aea7af5c39da7b1b97830eb5f73c80": " -\\hbar ", "12aeca6f151fa63be6365ff4b77c5cef": "s^{-1}", "12af61664176d2f4cc1c32b28f591b53": "\\textstyle \\frac{1}{2^n}", "12afbb568b98e61f251766e6a00b966f": " k \\in \\{1,.., m\\}", "12afbf166db5e42dba63d6ebeb3d8cb2": "\\Delta K=K\\otimes K", "12aff02f649bf9e309023654620e3f77": "z_1 , \\ldots, z_n", "12b016b79bf0c182438966edb0577263": " I_1 = \\frac{V_1}{|Z_{total}|}\\angle (-\\theta) ", "12b048cefb0fcb2ca8aa36df5734b4fb": "gx = y", "12b05c41f7a1463f8b1eb9c4e0447ba8": "\\delta W=p{\\rm d}V", "12b08a615453f593e858f645acfe04ed": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) = \\frac {\\partial L}{\\partial q_j} ", "12b0a80feb6c6d42de9618ccea2424c3": "\\displaystyle{W(x_1,y_1)W(x_2,y_2)=e^{i(x_1\\cdot y_2-y_1\\cdot x_2)} W(x_1+x_2,y_1+y_2),}", "12b0c7d1a029e2fb44cccf9bc0f15207": "(i=1,\\ldots,d)", "12b0c7e2a4bbc48e21fdd8593ec16f55": "k_1=3.796866512", "12b0cafaaeaf87d2d182d4e6dbed36b3": "\\tilde I(\\omega)=\\frac{K}{2\\pi c}\\left( \\frac{\\omega_{max}}{\\omega}-1\\right)", "12b0f5f9deec753718bc0b7afb818521": " I_{v}", "12b13fe271fc463930eab38a2efa88ae": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log ((\\frac{\\varepsilon}{3.715D}) + (\\frac{6.943}{Re})^{0.9}))\n", "12b1587adf08c65c0511e068c19061ad": "z^E=z-\\sum_i x_iz^{id}_i.", "12b1be9bd1b2bbe833dc250bf2f43966": "10\\sqrt{\\ell/g }", "12b1dfc0528e77eb301ad31b9bcb08b0": " \\Delta \\nu ", "12b1e0726ac1678cc3029ef8a83b6931": "a_{3,1} x_1 + a_{3,2} x_2 = b_3", "12b1f4ba736cfa59c1849eda9000f37b": "f^{-1}(U )\\cap X", "12b1f4eb936ea0857b8222e3f347292b": "\\left(\\!\\!{\\ \\choose\\ }\\!\\!\\right)", "12b20aa839be216d65d6c9bce2c581ea": "\\iint_D 5 \\ dx\\, dy", "12b2696c2b28b0068d8c41e62f04176a": "w\\rightarrow \\overline{w}^3+w_0", "12b26e4c1c31b3b76558de6bd66702f9": "s \\mapsto s(\\pi)/\\pi", "12b30099bd9645905cc065dc7132dcb2": "\\int L(a-\\theta) f(x-\\theta) d\\theta", "12b36da2f312d7c2499bcf31efc4ebcd": "\\hat{p}_0 \\leftarrow \\hat{r}_0\\,", "12b3f9b15dec4c28fcdc2bbe3da32007": "S = \\frac{h}{2\\pi} \\, \\sqrt{s (s+1)}=\\frac{h}{4\\pi} \\, \\sqrt{n(n+2)},", "12b41d9247a36a3920fbea7cc586ded0": "\\tan{(\\theta)} \\approx \\mu_\\mathrm{s}\\,", "12b451a278920c624e8bfa3974345fa1": "p = x^2 + y^2", "12b45995a97b38006221633f74505dba": "\\Omega \\subseteq \\mathbb{R}^n", "12b4a61685792b17d3859fad5a74e30c": "\\mbox{Dic}_n < \\mbox{Pin}_-(2)", "12b4af413cfb7703002f8bbd3b0b5d6a": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix}\n = \n \\begin{bmatrix} 2\\mu+\\lambda & \\lambda & \\lambda & 0 & 0 & 0 \\\\\n \\lambda & 2\\mu+\\lambda & \\lambda & 0 & 0 & 0 \\\\\n \\lambda & \\lambda & 2\\mu+\\lambda & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\mu & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\mu & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\mu \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ 2\\varepsilon_{23} \\\\ 2\\varepsilon_{31} \\\\ 2\\varepsilon_{12} \\end{bmatrix}\n", "12b4c2b43e16af79101d4796af888bef": "\\mathcal{L}_t\\{f(t)\\}(s) = F(s),\\ \\forall s \\in \\mathbb R", "12b504c59476364d7e750ab7d7eebbff": "I_{D}=\\mathbf{L}\\cdot\\mathbf{N} C I_{L}", "12b50e1ac0a5d717e6bb02eafc9cc559": "{\\scriptscriptstyle\\sqrt[r]{a/b}}", "12b53fb340aa69cb9a729c682d6b202e": "{\\partial f \\over \\partial r} = {1 \\over i r}{\\partial f \\over \\partial \\theta}.", "12b5a8ea98b8ab3e6d0dad02316c1b24": "p_\\textrm{kin} = p - \\frac{qA}{c} \\,\\!", "12b5d858cd06c9470b47c6a3f0dc416e": "\\Phi(s)=\\overline{\\Phi(1-\\overline{s})};", "12b5f7617c346ed49011aa316d494e97": "\n -\\boldsymbol{\\nabla} p \n = \\rho_f \\frac{\\text{D} \\boldsymbol{u}_f}{\\text{D} t} \n - \\mu \\boldsymbol{\\nabla}\\!\\cdot\\!\\boldsymbol{\\nabla} \\boldsymbol{u}_f,\n", "12b60ac23e59a36f0ee3e9559665ee9b": " \\operatorname{I}(S) = \\{f \\in k[x_1,x_2,\\ldots x_n] \\ |\\ f(x)=0 \\mbox{ for all } x\\in S \\}.", "12b63b128dceabbb46222e747274a07d": "L\\!", "12b672be98c714f5bfea96f72e68b3f2": "[x,y] v = x (yv) - y (xv)", "12b6a8e5116dcb1e7bd0773297540bab": " H(\\vec{r},t)", "12b6c396ed5335c16c9931566e01e01c": "\n\\begin{align}\nG_{\\alpha\\beta} &= R_{\\alpha\\beta} - \\frac{1}{2} g_{\\alpha\\beta} R \\\\\n&= R_{\\alpha\\beta} - \\frac{1}{2} g_{\\alpha\\beta} g^{\\gamma\\zeta} R_{\\gamma\\zeta} \\\\\n&= (\\delta^\\gamma_\\alpha \\delta^\\zeta_\\beta - \\frac{1}{2} g_{\\alpha\\beta}g^{\\gamma\\zeta}) R_{\\gamma\\zeta} \\\\\n&= (\\delta^\\gamma_\\alpha \\delta^\\zeta_\\beta - \\frac{1}{2} g_{\\alpha\\beta}g^{\\gamma\\zeta})(\\Gamma^\\epsilon_{\\gamma\\zeta,\\epsilon} - \\Gamma^\\epsilon_{\\gamma\\epsilon,\\zeta} + \\Gamma^\\epsilon_{\\epsilon\\sigma} \\Gamma^\\sigma_{\\gamma\\zeta} - \\Gamma^\\epsilon_{\\zeta\\sigma} \\Gamma^\\sigma_{\\epsilon\\gamma}),\n\\end{align}\n", "12b6e5b1cf0b9416d45705209e7aa455": " (1+s\\text{Ad} \\beta)P(t^p+t^{p-1}s+\\cdots+ts^{p-1})P(s^p)", "12b7c38ef6061bf664b5a90fa79b9cee": " F := \\int d^D x \\ \\left( a(T) + r(T) \\psi^2(x) + s(T) \\psi^4(x) \\ + f(T) (\\nabla \\psi(x))^2 \\ +h(x) \\psi(x)\\ \\ + \\mathcal{O}(\\psi^6 ; (\\nabla \\psi)^4) \\right) ", "12b7f2f3eaf91f10c1e1b83df4a34f3f": "W^\\ast(s) = \\frac{(1-\\rho)s g(s)}{s-\\lambda(1-g(s))}", "12b8a17d82f5e38488446cd9ba19ded4": "V_x,V_y", "12b8b41ff058e83ee14c54b1fde37b9a": "\\Lambda(x,\\lambda)=x^2+\\lambda(x^2-1).", "12b8cb61a9c477d612f909da15d0ae07": "Z'_k", "12b8cb811367cf01d77e980ef4b7f0f1": "m+0.5p(q+|q-n|)", "12b8cd801b7f6604560c36433cfd065b": "\\displaystyle E(k,\\phi)=\\int_0^\\phi\\sqrt{1-k^2\\sin^2\\theta}d\\theta, \\text{ for } \\left|k\\right| \\le 1", "12b8cfd5604adf510aef104e8ed08d71": "Pr_L", "12b8d73616c0ba0b1111564bb04d97c8": "L_{4k},", "12b8eea5eb1be4eae3e9ef1ad4e1f335": "\\,\\! q(\\tau) = q_t (\\tau) + q_s (\\tau).", "12b911ece928743b7fe21ed2d9c9c13e": "\n\\operatorname{rank}(\\widehat D) \\leq r \n\\quad\\iff\\quad\n\\text{there is full row rank } R\\in\\R^{m - r\\times m} \\text{ such that } R \\widehat D = 0\n", "12b91c3bd8e5b26d60407a6ac90a989f": "\\mathbf{E} (Y_{n+1}\\mid X_1,\\ldots,X_n)=Y_n.", "12b9448e1273187047ea33cf8d1d27f1": "\\Lambda=0\\;", "12b96dca22ddf063dd5f998ccaa1b46b": "(F,m):(\\mathcal C,\\otimes,I)\\to (\\mathcal D,\\bullet,J)", "12b9c111bb3ca4ca068a8af30b749dc8": "\\frac{e}{\\sqrt{1-e^2}}", "12b9cea058dec45e846033fcab119d55": "\\varepsilon_{ijk} \\varepsilon^{imn} \\equiv \\sum_{i=1,2,3} \\varepsilon_{ijk} \\varepsilon^{imn}", "12bb10bb09648797e41e349f47192853": " x^3 - x - 1, ", "12bb779c25d4905394e26eceeb8181f3": "m^2 + (a-1)m + b = 0. \\,", "12bbb95e280dac1509f437191f1229c1": " Y ", "12bbe31ba35d7bd3b225be626a2e8920": "H(z,f) = \\limsup_{r\\to 0}\\frac{\\max_{|h|=r}|f(z+h)-f(z)|}{\\min_{|h|=r}|f(z+h)-f(z)|}", "12bbf175cba3329ae6cea2f3b338019a": "(\\land, \\lor)", "12bc20ee215a9e0e2d31f232d33439a0": "\\pi = \\begin{pmatrix}0.885 & 0.071 & 0.044 \\end{pmatrix}.", "12bc3eb8a56d48ca26deac0ca96a629d": "X^{n}\\left( i\\right) ", "12bc96995257c8a723ac0ed50eb06db6": "n\\$=n!^{(4)}n! \\,", "12bca1efb56e3394e1589c9ea2828b7e": "\\mu(gx)=\\mu(x). \\, ", "12bcde8f3a659bac809109dd290bce00": "d=d_{\\mathrm{0}}+D\\cos(\\boldsymbol\\omega t + \\boldsymbol\\varphi) \\;", "12bd2ad68ee2f5396d3f4f6d6dbc81b1": " \\omega_{lab}=\\omega_0\\left(1\\pm\\frac{v}{c}\\right),", "12bd40062aa66bc122b6e84d6784c530": "\\frac{n}{n+1}", "12bd9ec91bb43b70654fdd95c0b75228": "A = 87.7", "12bdb956bf19b9a7d0bd9024cd2a39f8": "dim_{\\Bbb C}M=2", "12bdc80720708ecac900f80afeefb5ec": "\\{(x,1) \\mid x\\in[0,1]\\}", "12bdf1abae9f8c43f5d25ba2c604dfb6": "U - XE", "12be3568f4ded29150c39b4425f69774": "T:\\mathbf{M}\\to\\mathbf{A}", "12be6c7b1ca2a6f21553dda9c2367f6a": "I(t) = |E(t)|^2", "12beacc95a9c786a8e4aedfbdfb10a60": "R''=w^e*R'", "12beb2d82a58c794a4d8f92099b94cbc": "\\mathfrak{P}^{82}", "12bee37145febaa5d96781f1a7515298": "\\omega_1^2,\\omega_2^2,\\cdots\\omega_N^2", "12bf91fb53556a4ab09128332f93e500": " \\mathbf{b}_1,\\mathbf{b}_2, \\dots, \\mathbf{b}_n ", "12bfd56fc9de4bb573aaa882be8a426f": "J_3\\,\\!", "12bfd7162403b1d2a02b8587ce79b253": "M(p) = \\exp\\left( \\frac{1}{(2\\pi)^n} \\int_{0}^{2\\pi} \\int_{0}^{2\\pi} \\cdots \\int_{0}^{2\\pi}\n \\ln\\Bigl(\\bigl|p(e^{i\\theta_1},e^{i\\theta_2},\\ldots,e^{i\\theta_n})\\bigr|\\Bigr)\\, \n d\\theta_1\\, d\\theta_2\\cdots d\\theta_n \\right).", "12c0b16175a1f25afebfc5d195d366bf": " 5(x - 1)\\left(x + \\frac{1 + i\\sqrt{3}}{2}\\right)\\left(x + \\frac{1 - i\\sqrt{3}}{2}\\right)", "12c12c6467be80650810898f91a4ae5f": "\\nu(M) \\,", "12c1974296ca8b37c4c5377ecfd88c94": "dW_t = 0", "12c1c5bc7e48b74a425ca2fcdd28238a": "\\overline{S}_i", "12c23528a899cb3ac176b76655a1f184": "1+\\sqrt{2}", "12c2720d8493b190fa4e41919e992527": "\\{s_m - s_n : m,n \\in \\mathbb{N}, n1", "12cba6f59541487695f7d2c06fb7e60a": "(x + 5)(x - 2) = 0. \\,", "12cbb1e0ff00ce62ce8afe2614ae974a": "\\|\\gamma'(t)\\| = 1 \\mbox{ } (t \\in [a,b])", "12cbc1b787322f173fa108dd979cecff": "c=(1,2)", "12cbc33e5829dce2e2b8ef870b446829": "(a+b)^{n+m-1}=\\sum_{i=0}^{n+m-1}{n+m-1\\choose i}a^ib^{n+m-1-i}.", "12cbf63543c8de9ffae334c3d90bf626": "0\\le ma), ", "12cf746417128179be3623825057202f": "\\bigcup_{i\\in I} V_i=V", "12cf959e944778eb96642e9aaf58718a": "\\lambda^{[l]}_{\\alpha_l}", "12cfe3b552b0704e72cf03bd74b9ad06": " \\ v_{1}", "12d00559154de4707253b83f932533a4": "\\vec{d}=\\vec{d}_{\\text{eg}}|\\text{e}\\rangle\\langle\\text{g}|+\\vec{d}_{\\text{eg}}^*|\\text{g}\\rangle\\langle\\text{e}|", "12d018653b48dffddf983f277f479980": "\\mathfrak{P}^{62}", "12d049c9dab4fbdbe631408a4f12c819": "\\Delta q\\,\\!", "12d05fbc8355d50f9d368ca179fee312": "\\left( \\frac{m}{3}+M \\right) \\ddot{\\bar{x}} = -k\\bar{x}", "12d119262c7d12350b3d5d250c1acace": "P(E)", "12d15fd3f667cf27ad772053b7f9bb61": "\\eta^2+3\\eta+2", "12d1b8d0f7b34da18a98f8719652e363": "\\overline{S}", "12d1e9f979bc038f2b1f67c614b7c77d": "{(e_i)}_{i\\in I}= ( (\\delta_{ij} )_{j \\in I} )_{i \\in I}", "12d1f7f456152efbd318e790d068f8c8": "\\scriptstyle dx\\int{dx\\int{Vdx}}", "12d21cc90bf405d6b9283f2b74d112ea": "\\displaystyle R^+", "12d30f0e910b7cdd8686f952dffdff9b": "f(x_1,x_2,\\ldots,x_n|\\theta)P(\\theta)", "12d31caefcac9be96c3ed4f566c9b38a": "\n\\begin{matrix}\nG(n-1) & = &p^{-1}p^n + &q^{-1}q^n +& r^{-1} r^n\\\\\nG(n) & =& p^n+&q^n+&r^n\\\\\nG(n+1) &=& pp^n +& qq^n +& rr^n\\end{matrix}", "12d35494f5be1c989aa15102988e2ac8": "\\displaystyle T=\\frac{a^2+b^2+c^2}{4\\sqrt{3}}\\quad\\text{(Weizenbock)}", "12d3c78e7f224919086c1e2de9b6ad1e": "{f_{qk}}^{m_{qk}}", "12d3cfa1e8fae898e02015336cc7ed51": "\\gamma \\gamma^* = \\gamma^* \\gamma, ", "12d3d52072cefeefb584c02336ab569b": " \\frac{X_1+\\dots+X_n}{\\sqrt n} ", "12d402d4445f06d63587bdbc2f5b182f": "{\\mathfrak m}_p^{k+1}", "12d420ddbbed595350db3a7a8006c9ff": "P(\\Delta R) = p(\\Delta X) p(\\Delta Y) p(\\Delta Z)=\\frac{1}{\\sqrt{(2\\pi)^{3N} | \\frac{k_B T}{\\gamma} \\Gamma^{-1}|^3}} exp\\left\\{ -\\frac{3}{2} \\left(\\Delta X^T\\left( \\frac{k_B T}{\\gamma} \\Gamma^{-1} \\right)^{-1} \\Delta X \\right) \\right\\}", "12d4a11b43413deccabd9d8ced763ef0": "\nU_{2}=R_{2}I\n", "12d4b4948e84bc8d319e962531ce0802": "{\\textbf{x}}_2", "12d4f613bb8743b635ebae050c82dee6": "\\frac{(p_1+q_1-a)(p_2+q_2-c)}{(p_1+q_1+a)(p_2+q_2+c)}=\\frac{(p_2+q_1-b)(p_1+q_2-d)}{(p_2+q_1+b)(p_1+q_2+d)}", "12d5224355216b72793cbc747cc10e00": "a = 2, \\, b = 2, \\, c = 1, \\, f(n) = 10n", "12d53427a9f8da8e48f0f6fc1e9b20c8": " \\sin a = \\tan(\\pi/2{-}B)\\,\\tan b = \\cos(\\pi/2{-}c)\\, \\cos(\\pi/2{-}A)\n=\\cot B\\,\\tan b = \\sin c\\,\\sin A. ", "12d57689b9da7393795fb7c3bd3fb260": "Y_{4}^{-1}(\\theta,\\varphi)={3\\over 8}\\sqrt{5\\over \\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(7\\cos^{3}\\theta-3\\cos\\theta)\n= \\frac{3}{8} \\sqrt{\\frac{5}{\\pi}} \\cdot \\frac{(x - i y) \\cdot z \\cdot (7 z^2 - 3 r^2)}{r^4}", "12d58b3c1e3be412397ce706eec58187": "q_0 \\le 0", "12d5b8f557eb5e3a965dd3c02f98cbf6": "M(t;s) = E(e^{tX}) = \\frac{1}{\\zeta(s)} \\sum_{k=1}^\\infty \\frac{e^{tk}}{k^s}.", "12d5def95a4093ca6d6dbe53fa67687c": "\\mathcal{L}=\\{\\lambda\\in\\mathbb{C}\\,:\\,e^\\lambda\\in\\overline{\\mathbb{Q}}\\}.", "12d5eef93f7ebd4e4fdfb37121a19705": " \\mathrm{ ID } = ( n - 1 ) a m^{ b - 1 } ", "12d61c7987dfe01a11cb402b838666f7": "\\inf \\theta \\leq 7/22.", "12d6c196f8848a8c536eec200f568b7c": " -0.0905 \\le m \\le 2 ", "12d6cc2a716bba1f2f74b191f8436a63": "y_i \\ell_i(x_i)=y_i", "12d6e7bb8f03bbd9223c3de6ea99611c": "\\varphi = \\operatorname{atan2}(y, x) \\quad", "12d6f502f93f5aa1640f255a5e0fc5d6": " + \\ln\\Gamma_p\\left(-\\Big(\\eta_2 + \\frac{p + 1}{2}\\Big)\\right) =", "12d73e9376972e5f053f0e666b9ce193": " f(k;m,r,p) = \\frac{{k+r-1 \\choose k} p^k}{(1-p)^{-r}-\\sum_{j=0}^{m-1}{j+r-1 \\choose j} p^j}\\quad\\text{for }k\\in{\\mathbb N}\\text{ with }k\\ge m,", "12d7c74e91f24ee4870480bcf601033a": "\\cup_\\xi (\\xi + C_\\xi)", "12d8065aff3cc649a610c26364257d1b": "7x^2y^3 + 4x^1y^0 - 9x^0y^0", "12d815ebe36b7ca49c8a5e1162fa20e6": "-mg(l+a)\\limsup_{z\\to\\zeta, z\\in U} v(z).", "12dc8dc484ebd95d9dbf99ae2e8d8195": "{\\textstyle \\alpha^2}", "12dc8dee8ac8f2d13df4966e760fb8f8": " q_1 ", "12dcb4179e40d7bf9cda1df436787e88": "\\varphi (t) = | t |^{p}", "12dcb60e61d37830185efb04e20b0cec": "m =3", "12dcccc0c4c40664a7d09962a8d0dedf": "K_n x^n + K_{n-1} x^{n-1} + \\cdots + K_1 x + K_0 \\!", "12dd17ecfd7f0fc3e13e33a6f13e4b97": "Y_\\alpha(z)\\sim-i\\frac{\\exp\\left(-i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)}{\\sqrt{2\\pi z}}\\text{ for }0<\\arg z<\\pi", "12dd47e1542dbfcbeb676fa9bfdacf40": "\\theta_E = \\sqrt{\\frac{4GM}{c^2} \\frac{d_S - d_L}{d_S d_L}}", "12ddbd83f5eb39d90d8006c75801341f": "((a, b) \\downarrow F)", "12de7673992b1735e29cdd211851fa05": "a\\,\\!", "12de9a3db97b28001828569f4376c1d1": " \\operatorname{I}(x) = \\{f \\in \\operatorname{C}(X): f(x) = 0 \\}.", "12deb437aef5219835c08ea920fd072f": "L \\cap M", "12ded7db537863e87ed65d5764d663ec": "R_zR_yR_z(x) = R_{zyz}(x)", "12defe87bf1ca924857fe2dc72574735": "w_i = \\frac{n'_i}{n_i} = k - a z_i + b z", "12df2035c45186af8889f458d0e74b33": "\\lim_{z\\to 1^{-}} G_a(z) ", "12df2b4ddcfd3c66a95bd3632128b8ad": "\\mathrm{E}(w(t_1)\\cdot w(t_2))", "12df43ff9277a98334d388b0e8d9fec4": "\nP_{i_1} \\oplus P_{i_2} \\oplus \\cdots \\oplus C_{j_1} \\oplus C_{j_2} \\oplus \\cdots = K_{k_1} \\oplus K_{k_2} \\oplus \\cdots\n", "12df555a4a55830f5fa74afdc2b1adaf": "[S(a,c) + S(c,b) - S(a,b)]/15 \\,", "12dfaf0593af8628c956faa8f43ce9b6": " \\int_0^1 \\sum_{j,k=1} ^q |H_{j,k}(t)|dt < \\infty, ", "12dfcd386801054fee0516dcee4afad3": " Z_{21} = R_c \\qquad Z_{11} = R_c + R_a \\qquad Z_{22} = R_c + R_b \\, ", "12e03421a5114dc69b8d1f5dc5481d99": "10^{n}", "12e08c5918486ddebdac638153b174af": "r_k(n) = |\\{(a_1, a_2,\\dots,a_k):n=a_1^2+a_2^2+\\cdots+a_k^2\\}|\n", "12e09e44f4c0b1f0ced6c36e48e86687": "H_0=1 \\mbox{ and }P_0=0 \\,", "12e0ec62de27894b7f6e0b7bfb6f44a3": "h_0 - h = \\frac{Q}{2\\pi T} \\ln\\left( \\frac{r}{R} \\right) ", "12e13d157bbadd050990bdc63b3a2cb1": "+w=\\frac{Dz}{Dt}", "12e16e4a20d9e78115f974d5df9619f5": "\\rho = n_2/n_1", "12e199b82cc15236dd0fd7c0b3f8230d": "y^p - y = f(x),", "12e1c57c5cc2913e702f823692893b6b": "\\eta(\\phi)", "12e208274a580aa422f9b2ed612800e7": "(S,\\sigma)", "12e210132dceee320960d402cabfbade": " H_p = \\ker\\alpha_p\\subset T_{p}M.", "12e2120355c00c91d80322b211d9aeb9": "Demand_{p,c} + Backlog_{p-1,c}>0", "12e21dc7f9e40df41ebc118bb473d09d": "1-\\text{e}^{-2} \\approx 0.8647", "12e2316629a83f4fbcbbbcf04beefcc9": "{\\epsilon_F}_n", "12e2684254a086311455cbf71bf08a94": "Y^{\\prime}[\\sigma f] \\to Y^{\\prime}[\\sigma]", "12e2c88e6900f70ae80118e68dc0b4a2": "\\beta(g)=-\\left(\\tfrac{11}{3}C_2(G)-\\frac{4}{3}n_fC(R)\\right)\\frac{g^3}{16\\pi^2}~,", "12e2d2e74b4da04b62457157494e4794": "w^{f}(f^{*}) = \\sum_{e \\in E} f_e^{*}(a_e \\cdot f_e + b_e)", "12e317dd0f200b6cf3fcd9f1b4426338": "\n F (\\sigma_{22}-\\sigma_{33})^2 + G (\\sigma_{33}-\\sigma_{11})^2 + H (\\sigma_{11}-\\sigma_{22})^2 + 2 L \\sigma_{23}^2 + 2 M \\sigma_{31}^2 + 2 N\\sigma_{12}^2 + K (\\sigma_{11} + \\sigma_{22} + \\sigma_{33})^2 = 1~.\n ", "12e3701452a420354f5951800b335f1a": "V_o", "12e43c0c1847302d6cb841d015c5e7ad": "\n\\int_{0}^{\\infty}\\cos(2x)\\prod_{n=1}^{\\infty}\\cos\\left(\\frac{x}{n}\\right)dx \\approx \\frac{\\pi}{8}.", "12e4a8c1eea7241a8a6a682eec8b01a5": " \\phi_{va} (r) = \\frac{2 r}{1 + r^2 } \\ ", "12e4f3077c67b562c78b7ad38d95eb65": "V-A = \\{x \\mid x \\not\\in A\\}", "12e50ebc69c490be810bb85a286c598e": "t=\\cfrac{1}{1-\\cfrac{m\\lambda}{i\\hbar^2k}}\\,\\!", "12e5c3f9519de2d2b2515496393afe74": "location_i", "12e6547e74dc186eb07a1bc666b6d315": "\\,\\!d(f,g) = |f(x_0)-g(x_0)|\\;", "12e6c830f554b4f544d3c71edb1b381a": "w / h > 3.3", "12e70da219e570ef638a7cbbdf6fa279": "\n \\mathbf{A} \\oplus \\mathbf{B} =\n \\begin{bmatrix}\n a_{11} & \\cdots & a_{1n} & 0 & \\cdots & 0 \\\\\n \\vdots & \\cdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\\n a_{m 1} & \\cdots & a_{mn} & 0 & \\cdots & 0 \\\\\n 0 & \\cdots & 0 & b_{11} & \\cdots & b_{1q} \\\\\n \\vdots & \\cdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\\n 0 & \\cdots & 0 & b_{p1} & \\cdots & b_{pq} \n \\end{bmatrix}.\n", "12e74ceffdbabef1dc50642de5997544": "\\varepsilon_n\\in\\{-1, +1\\}", "12e78a9a1c063e655f5780d2014e2543": "t\\mapsto S\\,\\mbox{diag}(e^{it\\theta_1},\\dots,e^{it\\theta_n})\\,S^{-1}.", "12e7f8d87ce2beff61056303f2f2739e": "D(t+1) = |V(x,y,t+1)-V(x,y,t)|", "12e81850af31e2928120bb40138ac4ab": "\\scriptstyle \\vec{l} = (l_x, l_y)", "12e821270e647e92be6ff93be4ee05a1": "f(n) = \\sum_{i=1}^k \\left \\lfloor \\frac{n}{5^i} \\right \\rfloor =\n\\left \\lfloor \\frac{n}{5} \\right \\rfloor + \\left \\lfloor \\frac{n}{5^2} \\right \\rfloor + \\left \\lfloor \\frac{n}{5^3} \\right \\rfloor + \\cdots + \\left \\lfloor \\frac{n}{5^k} \\right \\rfloor, \\,", "12e8414a8bccc4c392ddd07cb3783d50": "x'=\\frac{ac(x^2+y^2)+x(ad+bc)+bd}{|cz+d|^2}", "12e84b15e7781c853764804251a11a15": " (M, g) ", "12e859b049da9f0f16ce90703552e392": "MRP_L=MPP_L*P", "12e8816bc42772d56cbf6f39842880ed": "e_1(\\tau) = \\tfrac{1}{3} \\pi^2(\\vartheta^4(0;\\tau) + \\vartheta_{01}^4(0;\\tau)),", "12e881b909aaac1d06011d7196fb1b41": "\\frac{(\\log g)^2}{\\pi g},", "12e8db2338dd0c22bc6e8760d0467124": "g' \\in G", "12e902d3bb1a1e5b89e5ec1e515ab140": "\\int_{y_0}^{y_1} f(x, y) \\,dy", "12e954346c9e336f877391adc7077202": "94906265.625x^2 - 189812534x + 94906268.375", "12e960b12f5df5e10c4c76778194a78c": " h(X) = -\\int_X f(x) \\log f(x) \\,d\\mu(x), ", "12ea069dc94f7fbae7c58b3519ed2283": "|\\mathrm{card} E(K) - (q + 1) | \\le 2\\sqrt{q}", "12ea1f6f35632b89274a1306e1110b63": "1.96x^2 +19.6x ", "12ea4c59e107564a215e0caf5f3098bc": "n \\cdot \\delta p", "12ea9f349f523c8e65466889e361f1f7": "\\frac{\\pi r^2}{4}", "12eb2bc65d74c0d29efc0c590e233d78": " \\hat{X}^n(z^n)=\\left( \\hat{X}_1(z^n), \\ldots ,\n\\hat{X}_n(z^n) \\right) ", "12eb2ef1e21e6656971ea7034aa87b10": "\\lim_{B \\rightarrow x} \\frac{1}{|B|} \\int_{B}f \\, \\mathrm{d}\\lambda,", "12eb651ede12aaa0b4c6a5d089469d8f": " \\chi_n\n\\begin{pmatrix}\na & b\\\\\n0 & a^{-1}\n\\end{pmatrix}=a^n.\n", "12eb8b69f203694b0466180bc2015ca0": "\\textstyle Y", "12eb9bbc4431ae838874afea8be7b85c": "d(x, y) = \\operatorname{arccosh}\\left(\\frac{B(x,y)}{\\sqrt{Q(x)Q(y)}}\\right).", "12ebb833247183cfac8da5b71f0bc3ec": "\\text{im}(A)\\subseteq\\text{im}(B).\\,", "12ebe2c770417549f7646e379c12b1dc": "PDOP = \\sqrt{d_x^2 + d_y^2 + d_z^2}", "12ebee2e4b45140931d45a2d1b1d90ac": "T(x)=g(x) + \\sum_{i=1}^k a_i T(b_i x + h_i(x))\\qquad \\text{for }x \\geq x_0.", "12ec7ff279101059cf73b8573527e420": "B^2 - 4AC > 0 ", "12ec8e137f4da6e137921159720c0344": "N(X)", "12eccac6ea84d2bf1e281e9339543924": "{\\tilde{D}}_6", "12ecf5153e8c8edad5e820479359dfc9": "B=(b_{pq})", "12ed2d0a5bc03c9a56d72b0de4f4af77": "v_{\\it avg}", "12ed44a1c273a0fb33aea3ec863853e1": "(x_1,\\ldots,x_k) \\in R^A_i", "12edae1b896b5cf29c6ec4b0e2bd8789": "1/\\eta_f", "12ee191a47ed137dda9783152b9f0da5": "n=31", "12ee24c9d00f630f859812fede24fdae": " {\\{ f_{1},f_{2} \\}_{M \\times N}}(x,y) = {\\{ {f_{1}}(x,\\cdot),{f_{2}}(x,\\cdot) \\}_{N}}(y) + {\\{ {f_{1}}(\\cdot, y),{f_{2}}(\\cdot,y) \\}_{M}}(x) ", "12ee63ae1f4e1954c6f3b906e1279d6b": " F = a + r\\Psi^2 + s\\Psi^4 + H\\Psi \\,", "12eef979c0d460492ed2e71958a22f0a": "\\Sigma^{-1}(C)", "12efe22166c5612553f8ae0be66380ce": "\\scriptstyle p_{0m} \\,", "12efeae7f866427b231a28461c8e890b": "\n\\vartheta_4^2(q)=\\sum_{k=-\\infty}^\\infty q^{2k^2}\\vartheta_4\\left(\\frac{k\\ln q}{i},q\\right).\n", "12f06183f7b45f8f489d99effe1b2a2a": "\\alpha_1=a+b+e;\\quad \\beta_1=a+b+c+d;", "12f0a2309c99718f3895b58e8e8c6901": "AX + UY = I", "12f0e11d53301ad87b30c9d353966900": " I = I_0 \\left [ \\frac{ \\sin \\left( \\phi/2 \\right ) }{\\left( \\phi/2 \\right )} \\frac{ \\sin \\left( N \\delta/2 \\right ) }{\\sin \\left( \\delta/2 \\right )} \\right ]^2 \\,\\!", "12f0eddd942f836ca3915061b27d62bd": "l(X)", "12f0ee59968eaea8f149568bcbeae97a": "\\qquad \\forall_{i\\neq j}\\, x_{ij} \\geq 0.", "12f11189828a77c8db2ce45b787deba1": "\\left( Z_{i-k}, \\ldots,\nZ_{i-1},Z_{i+1}, \\ldots,Z_{i+k} \\right) ", "12f1734702b5e3c6d8513017e9cf6503": "p_K (\\lambda x) = \\inf \\left\\{r > 0: \\lambda x \\in r K \\right\\} \n= \\inf \\left\\{r > 0: x \\in \\frac{r}{|\\lambda|} K \\right\\}\n= \\inf \\left\\{ | \\lambda | \\frac{r}{ | \\lambda | } > 0: x \\in \\frac{r}{|\\lambda|} K \\right\\}\n= |\\lambda| p_K(x).\n", "12f19ff25bc9100d0900a5aafaea30d7": " \\mathrm{ERH} = a_w \\times 100\\% ", "12f21fd950c570ec93929f7c86bcdff6": "D : \\Gamma(E\\otimes\\Omega^*M) \\rightarrow \\Gamma(E\\otimes\\Omega^*M)", "12f222dd22cac17760b581cda7dfd714": "\\overline \\lambda = \\frac{\\lambda}{\\lambda^{\\text{*}}}", "12f2526ca72325ccb0e5e5b3bb5287cf": "e^{i\\alpha}=\\left\\langle 0|U(\\infty)|0\\right\\rangle^{-1}", "12f27527bbcbebfdebc4ffda139ff725": "\\mathcal{Y}", "12f2cc8365910da12d20f36f89062d94": "\\Phi_b(\\hat p)\\Phi_b(\\hat q)\n=\n\\Phi_b(\\hat q)\n\\Phi_b(\\hat p+ \\hat q)\n\\Phi_b(\\hat p)\n", "12f31853fb6eae539a2a6883f2fca0ea": "Q.", "12f384debded95097d5fe47bf193ae62": "0 = \\hat{x}_1 - x_1", "12f388f68ab8e57d3c532d8c8c1e09b6": "\\sin 18^\\circ = \\cos 72^\\circ = \\frac{\\sqrt5 - 1}{4}", "12f3d8cb2e74ac34a45ec484e937d0f0": "x|U|=y|V|", "12f3e4d62efea37fbd97dcd9c362397a": "\\mathrm{Ir} = \\frac{\\tan \\alpha}{\\sqrt{H/L_0}}", "12f3f32b7110ffc144c00e99d00ee9e2": "B_{n}^{(k)}", "12f429b33a8c7c4f9a5c96ae1991fe25": "r(x+y) = rx + ry", "12f45c4863bb28d2b9d2de7af6741bbf": "a^b = \\left(re^{\\theta i}\\right)^b = \\left(e^{\\ln(r) + \\theta i}\\right)^b = e^{(\\ln(r) + \\theta i)b}", "12f46d0e829afc435aed9f325365f47b": "(F_p)^d", "12f48fbcc1b971f4e41f52fe2794574d": "\\mathbf{N_s}={\\frac {\\mathbf{120}\\mathbf{f}}{\\mathbf{p}}}", "12f4d86c0f31334e102665a4279361be": "A\\models\\forall x\\phi[x,\\bar{a}] \\iff \\phi^{A,x,\\bar{a}}\\in\\forall_A", "12f50006a8e143e325aaac5d37b3b744": " k A\\frac{(T_{m-1}^i - T_{m}^i )}{\\Delta {x}} +k A\\frac{(T_{m+1}^i - T_{m}^i )}{\\Delta {x}}+ {e_{m}}A \\Delta {x}= (\\rho c_{p} \\Delta x A) \\frac{(T_{m}^{i+1} - T_{m}^i )}{\\Delta x} ", "12f517f63b3ddd32c54e8b5232e1033c": " \\frac{Z(u)u^{1/2}}{\\sqrt \\pi }\\sim \\int_{-\\infty}^\\infty e^{i (uV(x)+ \\pi /4 )}\\,dx ", "12f531e442f4d962c2ea9e7aacc89938": "\\gamma = \\frac{3-\\tau}{\\sigma}\\,\\!", "12f59d9ef36d087ca186e02ad958f70b": "\\gamma = \\operatorname{atan2}(X_3 , Y_3).", "12f5c27e369b035e2caf0bf5bfbd6800": "f(x_1,x_2,\\ldots,x_n)", "12f6367e88eb5209ac6c4395c08c6cd9": "\\bigcup\\{\\sigma\\,\n\\colon(\\exists C)(\\langle\\sigma,C\\rangle\\in\n G)\\}", "12f664b3e4f2e51e214768edeb12f0c6": "P(x) \\lor (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\lor Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "12f68af42744fd031ff56476b9c0451e": "\\phi\\vert_K \\in \\operatorname{Gal}(K/\\mathbf Q)", "12f6ecf7fef507aceae8be0eff761f5c": "P^n(x \\smile y) = \\sum_{i+j=n} (P^i x) \\smile (P^j y)", "12f6f461cd49e0808f7ddee474f1d3b9": "L_K=\\left(\\frac{m}{2n_{s}e^2}\\right)\\left(\\frac{l}{A}\\right)", "12f70b4a9cf4455c58a7aee27bbe05cc": " P^{}_t(j,q) ", "12f89e127d7459eecc488aa723db1e43": " \\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\\n 0 & \\cos(\\theta) & 0 & \\sin(\\theta) \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & -\\sin(\\theta) & 0 & \\cos(\\theta) \\end{matrix} \\right] ", "12f8aa6419bdd67dfdb3c40ad076b55b": "\\mu (A) >0,", "12f8c70405aeeb9dacf082090dfb54ff": "\\begin{bmatrix}\nc_2 c_3 & - c_3 s_2 c_1 + s_3 s_1 & c_3 s_2 s_1 + s_3 c_1 \\\\\ns_2 & c_1 c_2 & - c_2 s_1 \\\\\n- s_3 c_2 & s_3 s_2 c_1+c_3 s_1 & -s_3 s_2 s_1 + c_3 c_1\n\\end{bmatrix}", "12f97a3f5226a04def3aa90382c6c340": "\\int_{\\Gamma_1}\\frac{\\mathrm{F}(\\mathrm{X})}{\\mathrm{X}-\\Xi}\\mathrm{d}\\mathrm{X}=2\\pi i\\sum^k_{s=1}\\mathrm{N}^{(s)}u^{(s)}\\mathrm{F}(\\Xi)", "12f98ccba91bd634dfd3ea3b577709f0": "W(\\Tau)", "12f99fe93443d65b450816343da2ca54": "F_{n} = F_{n-1} + F_{n-2},\\, ", "12fa76fe3c2380fde076156344a47f42": " \\alpha_p ", "12fae542fdc986ff628b05c980383420": "v = {d \\over dt}(Li) = L{di \\over dt} \\,", "12faf7b241cc52a383b177bbeee59c6a": " \\nabla \\times \\mathbf{E} = -\\mu_o \\frac{\\partial \\mathbf{H}} {\\partial t}", "12fafa9ac27b00580823b78bf96a95d1": "\\begin{align}\n\\nabla^2 U &= -M_x\\left({\\partial^2 B_x \\over {\\partial x}^2} + {\\partial^2 B_x \\over {\\partial y}^2} + {\\partial^2 B_x \\over {\\partial z}^2}\\right) - M_y\\left({\\partial^2 B_y \\over {\\partial x}^2} + {\\partial^2 B_y \\over {\\partial y}^2} + {\\partial^2 B_y \\over {\\partial z}^2}\\right) - M_z\\left({\\partial^2 B_z \\over {\\partial x}^2} +{\\partial^2 B_z \\over {\\partial y}^2} +{\\partial^2 B_z \\over {\\partial z}^2}\\right)\\\\\n &= -M_x \\nabla^2 B_x - M_y \\nabla^2 B_y - M_z \\nabla^2 B_z\n\\end{align}", "12fb282673e43435f7b5ae1853b69187": "G\\rightarrow 0", "12fc6220753b43847d66009e75120f3c": "\\{\\{q_1\\},\\{q_2,q_3\\}\\}", "12fc9cdd82579fea64aecf1180555792": "\n\\begin{align}\nE(X) & = \\frac{1}{2} \\\\[6pt]\n\\operatorname{Var}(X) & = \\frac{1}{24}\n\\end{align}\n", "12fcc337a262b208240745257f841f17": "a^{-2} + b^{-2} = d^{-2}", "12fce0714a5d61d7685c348011b075ff": "y\\cot\\varphi=\\frac{y}{\\tfrac{dy}{dx}},", "12fd204b34a3f1c84d262945ddc37d29": "H_{0}^{(i)}(x) = xG_{0}^{(i)}[H_{1}^{(1)}(x),...,H_{1}^{(n)}(x)]", "12fd3b4d703b1f89fc2c84ae5a40a4bb": "ab>1", "12fd3cef765bf5411bfc584ecad7b943": "F_mP_m=S_mF_N.\\,", "12fd4f8116f7c244b6d6d3e06cfd0905": "R/p^n", "12fd51e8221d40d266a77b12c3648135": " A \\rightarrow X", "12fda988cc40eb4214181d95e00bbac0": "e^{i(\\alpha + \\beta)} = (\\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta)+i(\\sin \\alpha \\cos \\beta + \\sin \\beta \\cos \\alpha)", "12fdfe327f07b75055766578c5cc5dc1": "2^{15_{dec}}", "12fe2690b6a38adbb1493e2a6f0173c1": " S:X^2 \\rightarrow X^2 ", "12fe535347865177bacade254d11ce9c": "\\scriptstyle x a \\equiv 1 \\pmod p", "12fe535c731a0b7fb49eafe24805e3a2": "y_\\mathrm{atm}", "12fecd3dd54eb8a0e3aa4ebc5a1cbd9d": " d_{1} ", "12fee16699e5bd04103a2932ef58edb1": "\n\\boldsymbol{s}=\\boldsymbol{R_f-R_i}=\\Delta\\boldsymbol{R}\n", "12ff71999646c56fc4a6de1db987e194": " \\rightarrow", "12ff7c40280485def3c4db7c6495791e": "10^{7.6}", "12ffb15026e354690a84a0499fd2f89d": "x_{n+1} = \\frac{x_n^2 + 2x_n}{x_n^2 + 1}.", "13005a3d0853a1970cbcbfeeda952427": "\\oint_C \\vec{H} \\cdot \\mathrm{d}\\vec{l} = I_{\\mathrm{enc}}", "1300cefb3f126d2f188def5f3998da6d": "1_{x,y}", "1300efb50bfdd3e33f4d776a2a64b038": "\\|T^2x\\| \\ge \\|Tx\\|^2 \\,", "1300f7354e7b132ba06e255f8d22d194": "\\quad \\tfrac{t^2+1}{2a}", "1301432ccc745f36b88301dc521459b5": "y_{3}\\left( x\\right)=j_{-4}(x) =\\left( -\\frac{15}{x^{3}}+\\frac{6}{x}\\right) \\frac{\\cos(x)}{x}-\\left( \\frac{15}{x^{2}}-1\\right) \\frac{\\sin(x)}{x}.", "1301973e1d25a624d3f45ec3c45b941c": "_{Mpq}\\!", "13020040a0b735394e11d352cd14ecef": " P(x',t | x,0) ", "13021c358dd5db520006c31a59781395": "e^\\nu_a = e^\\mu_a e^\\nu_\\mu \\,", "1302a21c7f0407dde27fce9c3b35317a": "\\Delta_rG>0", "1302a34d5d567b58f2e52436000dd9f1": " -n(n+1)~r^n~\\cos(n\\theta) \\,", "1302ae880d88b2bdce4868961306bc92": "Z := Z+L_i*w_i", "1302b02a9106f64307e0fa4a85386177": "{x^2 + 2x^3}", "130357f29fa8718819bdff6d0269f48c": "\\frac{d}{dt}{\\bold v}_0(\\varphi_0^{-1}(P(t)))=\\left(\\frac{d}{dt}J_{\\varphi_{01}}(\\varphi_1^{-1}(P(t)))\\right)\\cdot {\\bold v}_1(\\varphi_1^{-1}(P(t))).", "13035b90ba8b496cfc6d04d439eb8d04": "P \\land \\exists x \\, Q(x) \\Leftrightarrow \\exists x \\, (P \\land Q(x)) ", "1303613906a404e9888364cc307ce8bf": "\\lambda_2 \\,", "13037e9c9753cb1f886f429929388f63": "M_G", "13039a55a4720d3c2b91c996311a3612": "\\Delta X_n^1", "1303b0be86ada546ec776915e817c81a": "p_\\pi(\\boldsymbol\\eta|\\boldsymbol\\chi,\\nu) = f(\\boldsymbol\\chi,\\nu) \\exp \\left (\\boldsymbol\\eta^{\\rm T} \\boldsymbol\\chi - \\nu A(\\boldsymbol\\eta) \\right ),", "1303c4d55e7579f5bc020153b1461d5a": "\n \\dot{\\rho} = \\frac{\\partial \\rho}{\\partial t} + \\boldsymbol{\\nabla} \\rho\\cdot\\mathbf{v} ~;~~\n \\dot{\\eta} = \\frac{\\partial \\eta}{\\partial t} + \\boldsymbol{\\nabla} \\eta\\cdot\\mathbf{v}.\n ", "1303c5c75d3eae272b9c84920c5a0466": "\\mathbb C^*", "13040d378cbb9b9ef31d7bdfa27746ae": " x_n=(ax_{n-r}+c_{n-1})\\bmod\\,b\\,,\\ \\ c_n=\\left\\lfloor\\frac{ax_{n-r}+c_{n-1}}{b}\\right\\rfloor,", "1304435c8bc4c0f5b8ecbe7f16727d75": "A^*", "13044b70e7e47b34729d39dd200c14ea": "\\tfrac{t^2}{\\log t}", "13046ae9549d127278b6e26472a79c1b": "\\alpha=\\frac{\\partial u_y}{\\partial x} \\quad , \\qquad \\beta=\\frac{\\partial u_x}{\\partial y}\\,\\!", "1304c9b804161fbf6cd3e1e4f0f3055e": " \\langle \\hat{L} \\rangle_\\textrm{PM} = \\sum_{A}^{\\textrm{atoms}} \\sum_{i}^{\\textrm{orbitals}} |q_i^A|^2 ", "1305054881bb4d43e86f723e845e1d43": "m_k\\ddot {\\vec x}_k(t)=F_k(\\vec x(t))=-\\nabla_{\\vec x_k} V(\\vec x(t))", "130527965708be2b8274e688419cf280": " su(N)", "13058be3d8ceb06273459847a6a3ad30": "\\boldsymbol\\mu_{X,Y}", "1305cf401f3832d052e786f7c2a81982": " \\textstyle {J}_{\\alpha} ", "1305e7910687b7f64827c2808a0ccc91": "6.5\\cdot 10^4", "130600787d4889375d2049fd1806ca7d": "R = \\bigcup_i A(x) \\oplus t_i", "13064eff2904bad4595d093c741db465": "P_\\text{estimated} = \\frac{\\frac{1}{SU} - 1}{\\frac{1}{NP} - 1}", "13067925c30c5b7d4111b06cdfac0313": "u_{1}(\\mathbf{q}) ", "130693682fe4d9d5612c6bc6f7df878f": "3^8", "1306e5d99726c06558d21f4716feed7a": "38 \\equiv 14 \\pmod {12}\\,", "1306eb165f2534c59abd6d11e8367bfb": "a_1\\leq a_2 \\leq \\cdots \\leq a_n", "130738856fc9612571b31e7928e2e614": "\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)", "13073a46df0768c0a6553150f6bf2047": "X_c \\le X_c^0", "1307530bbad17fe83a008987a06fad15": "\\textstyle \\delta", "13077de639cf686dc761c7bf9a5c1c1d": " H_0 = {DF}_T \\int_\\omega H(\\omega)\\, d\\mathbb{P}(\\omega) ", "13078955e1101087e2918cfc6f71e594": "g(\\sigma_1,\\sigma_2)=\\int_X \\frac{d\\sigma_1}{d\\mu}\\frac{d\\sigma_2}{d\\mu}d\\mu", "130810d2c92926f5bb947b917c6fad2a": "D_T^2 z = \\{\\{z,T\\},T\\} = \\{(\\dot{q}, 0),T\\} = (0,0)", "13086b5390a51152ae0cc4dd91df65ff": "\\{a^\\frac{j}{2}\\psi(a^jx-kb)\\}_{j,k\\in Z}", "13091099a7b1fbd221ab526eac2fa699": "P_c \\,", "130923d07b3d567bceaf83fe8f7aa71b": "T^{\\bar k-2}p,\\dots,Tp,p", "13092adb1b5af45587af133463d706b8": "\\mathrm{pOH} = \\mathrm{pK_W} - \\mathrm{pH}", "1309783b0373ce7dbb64485afa366e47": " \\delta_{m+1}=\\frac{T_{m+1}(0)}{T_m(0)} ", "13098eced8194ca1107b909ecaa50cd0": "\\frac{d}{dt} \\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{N}\\\\\n\\mathbf{B}\n\\end{bmatrix}\n= \\|\\mathbf{r}'(t)\\|\n\\begin{bmatrix}\n0&\\kappa&0\\\\\n-\\kappa&0&\\tau\\\\\n0&-\\tau&0\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{N}\\\\\n\\mathbf{B}\n\\end{bmatrix}.\n", "1309a3363d95bf456d2f270e7a4f1683": "\\,O,O_i", "1309bd6f699c1dd81906c4b087c6478c": "u= \\tfrac{1}{6} \\left (x^3 -\\pi^2 x \\right),", "1309bd8fe3cb64bf49bd16aeba59ee1b": "\\text{hom}(-,X):C^\\text{op} \\to C", "1309d31299357ab2306d906f44cce76d": "\\left| \\Gamma_1 \\right| = \\left| \\Gamma_2 \\right| ", "130a0b7752c8cd8ee8802710b1ed97c2": "Z_L=\\infty", "130a14353b67992ed46ed53b9a666fca": "\\color{Melon}\\text{Melon}", "130a3c6d6ca230f74b1e0b2abd20f4b2": "\\nabla f(x^*)", "130a555bc192bac57517e470a9a35afa": "\n \\int\\limits_0^1 \\!\\frac{\\ln\\ln\\frac{1}{x}}{1+2x\\cos\n\\varphi+x^2}\n\\,dx \\,=\\int\\limits_1^{\\infty}\\!\\frac{\\ln\\ln{x}}{1+2x\\cos\\varphi+x^2}\\,dx\n =\n\\frac{\\pi}{2\\sin\\varphi}\\ln \\left\\{\\frac{(2\\pi)^{\\frac{\\scriptstyle\\varphi}{\\scriptstyle\\pi}}\n\\,\\Gamma\\!\\left(\\!\\displaystyle\\frac{1}{\\,2\\,}+\\frac{\\varphi}{\\,2\\pi\\,}\\!\\right)}\n{\\Gamma\\!\\left(\\!\\displaystyle\\frac{1}{\\,2\\,}-\\frac{\\varphi}{\\,2\\pi\\,}\\!\\right)}\\right\\} ,\n\\qquad |\\Re{\\varphi} |<\n\\pi.\n", "130a55d0696be58d22e26eca42afb6ee": "\nt=t_{0}", "130a7ca8d25ee619d31cb1fa29dde94e": "(k+2)(k+1)A_{k+2}+(-2k+1)A_k=0\\;\\!", "130a812f388c01a429719b7beefecaf7": "\nC \\frac{dV}{dt} = I - g(V),\n", "130ab19206799d24bd617e1db01eea1d": "\\Delta \\cong \\operatorname{Hom}(\\operatorname{Gal}(L/K), \\mu_n)", "130acfcea55210419f8a268e091e918c": "pV = nRT\\,\\!", "130af4e066dee5f97a48b2a5008b2520": "\n \\frac{d^2u_\\epsilon}{dx^2} - c^\\epsilon(x)u_\\epsilon = - \\phi^\\epsilon(x), \\quad 0 < x < 2 \\quad (4)\n", "130b2ba6b1fb77823f07397078d9d722": "p^2 > 4\\pi A", "130b67f65c556c8820db1d9554cf439e": "\\vartheta_{00}(z;\\tau) = \\vartheta(z;\\tau)", "130c3b48e8cdf998df86e9f8001569b6": "b\\ge n", "130c4fda13ec679cd4fd8dda55373b41": "\\frac{1}{\\beta}-\\log\\left[\\frac{\\beta}{2\\alpha\\Gamma(1/\\beta)}\\right]", "130c9c02c2e7e2ffe9eb6fc8efa70c2c": "P ::= \\emptyset\\,\\,\\, | \\,\\,\\,a.P_1\\,\\,\\, | \\,\\,\\,A\\,\\,\\, | \\,\\,\\,P_1+P_2\\,\\,\\, | \\,\\,\\,P_1|P_2\\,\\,\\, | \\,\\,\\,P_1[b/a]\\,\\,\\, | \\,\\,\\,P_1{\\backslash}a\\,\\,\\,", "130cf98832a0dc4773147358d3e103d1": "2/101 = 1/101 + 1/202 + 1/303 + 1/606", "130d12788450ada3da40b3d5f89f2e16": "\\operatorname{GL}(V) \\leq \\operatorname{\\Gamma L}(V).", "130d338a47d093b0e5799ab5b99dad49": "{C}_{CF}", "130d60bd2f97bbabe44e82f80080643f": "y=-{g\\sec^2\\theta\\over 2v_0^2}x^2+x\\tan\\theta", "130db251383a3c49afe77ff591563b28": " \\frac{1}{\\sigma^{2} + \\tau^{2}}\\left({\\partial (\\sqrt{\\sigma^2+\\tau^2} A_\\sigma) \\over \\partial \\sigma} + {\\partial (\\sqrt{\\sigma^2+\\tau^2} A_\\tau) \\over \\partial \\tau}\\right) + {\\partial A_z \\over \\partial z}", "130dfb56e1ad751c6a4b5d0f4d3ecb9e": "(Tx,\\mu_x)", "130e2e1609c1b145132ba01cd3dc31de": " W_{KRA}= \\sum_{n=N}^{n=\\infty} 2 \\pi \\omega^{2} p (n-n_\\mathrm{osc})^2 \\int d \\Omega |FT (I_{KAR} \\Psi ( \\mathbf{r}))|^2 J_n^2 (n_f, \\frac{n_\\mathrm{osc}}{2}) ", "130e3d66e7312d085da14666b55d2338": "2(1+\\sqrt{2})s^2\\,\\!", "130e5f8c90c6557bae665228a2cf87bb": " u(t) = \\gamma \\sin(\\omega_0 t + \\varphi_0) ", "130e67d965e74a64298d664f795cf982": "A^{n-1}b", "130ed2b3e1b9d52ab62a058d517715e6": "(X_n, d_n, \\mu_n)", "130f7bcc1a8ab7d37dc972ee3ec3ac4d": "\\neg D\\rightarrow\\neg C", "130fafbaca9391335bd927142d732724": "\\begin{align}\n\\sum_{i=1}^n \\text{E}[ (S_i - S_{i-1})^2] &= \\sum_{i=1}^n \\text{E}[ S_i^2 - 2 S_i S_{i-1} + S_{i-1}^2 ] \\\\\n&= \\sum_{i=1}^n \\text{E}\\left[ S_i^2 - 2 (S_{i-1} + S_{i} - S_{i-1}) S_{i-1} + S_{i-1}^2 \\right] \\\\\n&= \\sum_{i=1}^n \\text{E}\\left[ S_i^2 - S_{i-1}^2 \\right] - 2\\text{E}\\left[ S_{i-1} (S_{i}-S_{i-1})\\right]\\\\\n&= \\text{E}[S_n^2] - \\text{E}[S_0^2] = \\text{E}[S_n^2].\n\\end{align}\n", "130fe413966ba4e4e3071bb45a910e5a": "v_{T-j}", "1310016cef732a1fac36974f1a2792a9": "\\frac{\\left \\Vert v_{Target} \\right \\| }{ \\sin(\\theta_{Deflection}) } = \\frac{\\left \\Vert v_{Torpedo} \\right \\| }{ \\sin(\\theta_{Track}-\\theta_{Deflection})}", "13102c14d5dfbd8248dae67af3e0660f": "k=|f_o-f_e|/f_0,", "131051ea4f96502fce372ee4692631a1": " \\forall x \\, (P(x) \\rightarrow Q(x)) \\rightarrow (\\forall x \\, P(x) \\rightarrow \\forall x \\, Q(x)) ", "1311810f1a89f9ec08638abbe6e4c96e": "dW =Q \\, dV =\\left[ \\int_0^V\\ dV' \\ C(V') \\right] \\ dV \\ . ", "1312028ddc981e4e36cd3b91ef665805": "\\{f, \\mathcal{H}\\} + \\frac{\\partial f}{\\partial t} = 0", "131214d4fdf40a43f0102364f4decbf9": "U=-G\\frac{16}{15} \\pi^2 R^5 \\left(\\frac{M}{\\frac{4}{3}\\pi R^3}\\right)^2= \\frac{-3GM^2}{5R}", "131217925a1a78cecae66513e4864e72": "b \\in \\Gamma", "131220f39668c31132d0c521ec903630": "\\mathbf{P} \\left[ \\sup_{0 \\leq t \\leq T} \\big| \\sqrt{\\varepsilon} B_{t} \\big| \\geq c \\right] \\leq 4 n \\exp \\left( - \\frac{c^{2}}{2 n T \\varepsilon} \\right).", "13127fa065167f088d0a0877e21b3c91": " |x| = p_1^{a_1}\\ldots p_r^{a_r}.", "1312805d1c4c78fbb9dcc5bee3c8a435": "\\alpha ", "1312b06cebbe1e15696198bc426ab9f4": "\n\\sum_x \\psi_+(x)|x,\\uparrow\\rangle + \\psi_-(x)|x,\\downarrow\\rangle\n\\,", "1312cd3f180f13ce045e3ad7c920ebe9": "\\|f_n\\|\\leq M_n", "1312dba4acceffe6ddb18cb7ffa771c0": "t_\\max > \\frac{d}{i_\\max} + \\frac{d}{j_\\max} + O(1)", "1312eba929a970eba0e90b2571773ede": "\n\\begin{align}\n \\delta (\\bar{\\lambda}) &= 2 \\delta_1 (\\bar{\\lambda}) + \\delta_2 (\\bar{\\lambda}) = 2 \\big( \\bar{n}_1 - 1) \\alpha_1 + \\big( \\bar{n}_2 - 1) \\alpha_2 \\ , \\\\\n \\Delta &= 2 \\frac{\\delta_1 (\\bar{\\lambda})}{V_1} + \\frac{\\delta_2 (\\bar{\\lambda})}{V_2} \\ .\n\\end{align}\n", "1312efddb47a74d08895428eb0b19cc9": "w(x^{q}, y^{q})=(x^{q^{2}}, -y^{q^{2}})", "131385d30e2d926d9b2c5ac1e8d3c0e3": "X_j= - {\\rm \\nabla} \\frac{\\mu_j}{T} ", "1313b87e9846aee17064b2ec8b35d6f6": " ta(s)= \\min\\{ t_{si} - t_{ei}| i \\in D\\}. \n", "1313f245c16223eaee992ec499a5f9e7": "\\frac{\\partial ^2\\Pi}{\\partial x_i \\partial x_j}", "1314154d6660eac18e2b083dcf2aa5b7": "f(3)=10.", "131438c56262e9b5f96a3655f3b7919f": "g(z) = \\begin{cases}\n \\frac{f(z)}{z}\\, & \\mbox{if } z \\neq 0 \\\\\n f'(0) & \\mbox{if } z = 0,\n\\end{cases}", "131464ff977d56664e7a983f8ded56be": " [Force \\ on \\ Plate] = [Weight \\ of \\ plate] + [Surface \\ Tension \\ Force] - [Buoyant \\ Force] ", "131472bdcba05ffd00c336914d3cf8f0": "\\mathbb C^n\\otimes \\mathbb C^n", "1314763ced2cedfdd8e32b584235fdf4": " \\frac {2\\pi (x) }{ \\frac{x}{\\epsilon ^{ln(lnx)} } }", "1314c51722f0a48ea12574945e8d223a": "f(-x)=1/f(x)", "1315095d2fedddc2d11f1a8def6c58e7": "\\langle T_C\\rangle = \\frac{1}{\\Delta S} \\int_{Q_{out}} TdS", "13156036dd3fba86bd5f3354e8e62641": "\\frac{d^2y}{dx^2} = (A+B\\weierp(x))y", "1315bd1457554e9533bbc89b53bb90bc": "(T,V )", "1315c1dada771ae80b948d4e2f6d5897": "f(t)=\\frac{1}{2\\pi} \\frac{d\\phi(t)}{dt}", "1315d95a72845a0b5768bc91064530e7": "\\tilde f(x) = \\frac{1}{2\\pi}\\text{ p.v.}\\int_0^{2\\pi}f(t)\\cot\\left(\\frac{x - t}{2}\\right)\\,dt", "1316a1629f7600e21d93f6d417821aef": "S' = S''' = S'''''", "1316de8e1ab2134b3399cc0d78b45c3b": "\\sigma_2\\,\\!", "13173ff34816285ce96238c281b1e72f": "f_c\\,,\\,f_m\\,", "1317c941b60c67b6081284ff6e30e051": " -8.349\\times 10^{-11}\\times 60\\times 60\\times 24\\times 10^9\\approx -7214 \\text{ ns} ", "1318274397f12d88e3920207fb8f460c": " n = \\frac{E}{\\hbar \\cdot \\omega} - \\frac{1}{2} = \\frac{m \\omega A^2}{2\\hbar} -\\frac{1}{2}", "13183ff4a9f45b94417d7823bdda0850": "= O \\sum_{n=0}^{\\infty} \\frac{1}{n!} (-i H t/\\hbar)^{n} |\\psi_0\\rangle", "13185be3e9220ab954bdedcd05c02e4b": " K(x_i - x) = e^{-c||x_i - x||^2} ", "131866f77a89234d6428c42fc06c8d19": "V_c=a_0\\sqrt{5\\Bigg[\\bigg(\\frac{q_c}{P_0}+1\\bigg)^\\frac{2}{7}-1\\Bigg]}", "1318c185f5339fc1a60ca1da067d5cac": "T_m(k) = V_1 \\cdot C_M(k) + V_2 \\cdot (\\alpha \\cdot C_M(k+1) + (1-\\alpha) \\cdot C_m(k+1)) + V_3 \\cdot (\\beta * C_m(k-1) + (1 - \\beta) \\cdot C_M(k - 1))", "1318d541272762b8020f290828504e96": " \\text{length of solar day}=\\frac{\\text{length of sidereal day}}{1- \\tfrac{\\text{length of sidereal day}}{\\text{orbital period}}}. ", "1318dd34e2fea40d32972c9f4002520e": "\\delta = \\arcsin(1/e)\\,", "1318f1898bd8690d1e477b8e6d18e78b": "\\Gamma(s)\\zeta(s) =\\int_0^\\infty\\frac{x^{s-1}}{e^x-1}\\,dx, ", "1319f5d4d78428de32ec985155bb72fa": "\\sigma\\;\\mid\\;\\tau \\equiv_{b}\n\\tau\\;\\mid\\;\\sigma", "131a659fb4e4edd79e2ab569b39b6745": "n(j)", "131a6cc31efd47a5b6bef19568764408": "E=F(\\alpha)", "131a8a5279434571a7c80274988aff10": "\n\\varphi = 2\\pi\\xi\\frac{}{}\n", "131ac7a55fbf77500a953b4eb00d0b84": "G(\\psi) = \\sum_{r=1}^{k_1}\\psi(r)e^{2\\pi i r/k_1}.", "131ae124a06fd38c77be375a4bcd7b94": "\n=\\frac{1}{\\sqrt{2\\pi}} \\sum_m e^{im\\theta_k}\\int_0^\\infty r\\operatorname{d}\\!r\\,\nf_m(r) 2\\pi i^m J_m(kr)\n", "131b25c041e3ea27f4e87ce51a622f71": "H_n(M;\\mathbf{Z})=0", "131b4518a6693db291494c4c8135b1c9": " \\{(U_{\\alpha}, \\varphi_{\\alpha})\\}", "131b68643b1ddb1eeb812048affdecd3": "\\,\\mathcal{R}\\,", "131b7e143eba2be8fbc97f6084d466e3": "\\ pF^n =_{def} \\{x_1...x_n : F^n x_1x_3...x_nx_2\\}.", "131b993f4b6b23c36b6d36e8a36496ec": "\n\\hat{G}(\\boldsymbol{k}) = H \\left( k - k_c \\right), \\qquad k_c = \\frac{ \\pi }{ \\Delta }.\n", "131bd4c2bf2fb2dcd6653f59d2fb8c0c": "\\frac{P(x)}{Q(x)} = \\sum_{i=1}^n \\frac{P(\\alpha_i)}{Q'(\\alpha_i)}\\frac{1}{(x-\\alpha_i)} ", "131c37e2852da6c21088bbafd9c5de32": "0.d_1d_2d_3 \\dots;\\dots d_{\\infty - 1}d_\\infty d_{\\infty + 1}\\dots,", "131c40c8d7f7db51a150c10f8c1a4d68": "(\\Sigma, D, I)", "131c5c32a334854272ed65e40e7b0ca7": "R={\\log_q{|C|} \\over n} \\le 1-H_q(J_q(\\delta))+o(1) ", "131c7d356a34f25bf2583b2beafd84da": "\\partial f_i/\\partial x_j", "131c927249ce3b52d8adadd80644be60": "CCl_4", "131d04366a99f2e9676b62d2c861a9bf": "u'\\frac{d}{dx}\\frac{\\partial L}{\\partial u'}=\\frac{d}{dx}\\left( \\frac{\\partial L}{\\partial u'}u' \\right)-\\frac{\\partial L}{\\partial u'}u'' \\, , ", "131d22e7bca14bf19d7a0855ebb8adad": "|R(z)|\\leq 1", "131d3742ca3d2b210b7d6d10e4f54b1e": "\\vec{f}_j", "131d506244289281ea9fb297f4daa846": "P_{\\pi}^{-1} = P_{\\pi^{-1}} = P_{\\pi}^{T}.", "131d64f3093d2610bdefb597a858c1c3": "P(X_{1},X_{2},\\ldots,X_{n})", "131d9bdbbb43837df3a5e5b7b9775d7f": "\\underbrace{B_j\\land\\dots\\land B_j}_j.", "131db117593e7d6ffae022b771c6e90b": " y(x) = x \\cdot y' + (y')^2. \\,\\!", "131e315d843a666dacdb545dab79d48e": "\\frac{{x}}{{2}}.", "131ea58e218b38e231f2e0ed9619e5cd": "iS = \\int_t \\left[ m \\left(\\frac{dy(it)}{dt}\\right)^2 + V(y(it)) \\right] dt", "131ee96e40a675bd815eded8a6d9c050": "|-4| > |2| + |1|", "131f04ef3dd102f4156afe21bc2f5028": "d'", "131f6304d827bb8c0b31ec47798e3f65": "t\\in\\mathcal{T}", "131f9ab1827a2dcbb3e0f39652f2a961": "Global CO_2 Emissions =(Global Population)\\left ( \\frac{Gross World Product}{Global Population}\\right )\\left ( \\frac{Gross Energy Consumption}{Gross World Product}\\right )\\left ( \\frac{Global CO_2 Emissions}{Gross Energy Consumption}\\right )", "131fcbb7cec215136e10228e0f342aea": "\\mu_{a} \\,", "131fd69e68b644a2b54d36b8d2ae88be": "I^i_{electrode}", "131fdc4636b86ac5f9a3e54a8a7ee4f9": " (e,f),(g,h) \\mapsto \\left(\\frac{eh+fg}{1+ degfh},\\frac{fh-aeg}{1-degfh}\\right)", "131fecb364a8e0856fd04b557de54bd0": "T >0", "131ff6a36cf10e00931d5734549a1395": "\\frac{dF}{dx}=\\frac{e}{\\epsilon\\epsilon_0}\\left (n-n_a\\right )", "13200863b1d4d0620055ef7f9ff9a667": " \\mathcal{L}_X (T(Y_1, \\ldots, Y_n)) = (\\mathcal{L}_X T)(Y_1,\\ldots, Y_n) + T((\\mathcal{L}_X Y_1), \\ldots, Y_n) + \\cdots + T(Y_1, \\ldots, (\\mathcal{L}_X Y_n)) ", "132010180fdb9b50fc20997fe2f115a8": "P = P_s + P_f - P_{sf}.", "1320632218b9ecf3cbffdaac3e3ac696": "\\neg (A\\lor A)\\lor A", "132089cacde27ed8b9020f418d357bff": "\\psi(\\Omega^{\\psi(\\Omega^{\\psi(0)})})", "1320a2664921eb458cfb03364a61db64": " = \\left( \\sum_{i=1}^n \\left| x_i - y_i \\right|^2 \\right)^{1/2}", "13211299daae2d9982b9d048f3937784": " \\bar x_2 ", "1321a455ac4427540303dd5987f5a3c2": "f_j^e[n]", "1321c5dadf11bd90800751315efe739d": "z = \\Phi(w)", "1321f600924e7d690deccf753b1d4b56": "A[\\hat{\\mathbf{k}}]\\,\\!", "13223267f2d1188904b8bf4e5c05efa7": "\\pi \\oplus(\\sigma \\ominus \\tau) \\neq (\\pi \\oplus \\sigma) \\ominus \\tau", "13230b5bfb612251a19e58baeada5e2b": " 0 = - \\Delta\\text{H}_{\\text{f}} + \\text{V} + \\frac{1}{2}\\text{B} + \\text{IE}_{\\text{M}} - \\text{EA}_\\text{X} + \\text{U}_\\text{L}", "132390541f97ed12997bc60618bb285b": "\\mbox{run rate }=\\frac{\\mbox{total runs scored}}{\\mbox{total overs faced}}", "1323b5bbb882153c7fa129a3f55e7913": "\\frac{\\mathbf{B}}{\\left|\\mathbf{B}\\right|}.", "132487d42bb4f40fcfcc85dd0bea4be9": "a_{\\mathit{wf}}=\\left\\{ \\begin{array}{ll}\n\\frac{4\\Delta_{0,50}^{2}}{1+4\\Delta_{0,50}^{2}}, & \\mathrm{for}\\ \\Delta_{0,50}<0\\\\\n\\\\0, & \\mathrm{for}\\ \\Delta_{0,50}\\geq0\\end{array}\\right.", "13249b56109f67473434bff4b692e31d": "\\frac{A(x\\rightarrow x')}{A(x'\\rightarrow x)} = \\frac{P(x')}{P(x)}\\frac{g(x'\\rightarrow x)}{g(x\\rightarrow x')}", "1324d399626dfdf63b91c382533ecd09": " S_{\\text{gauge-fixed}} = \\tfrac{1}{2} \\langle \\Psi | c_0 L_0 |\\Psi\\rangle + \\tfrac{1}{3} \\langle \\Psi,\\Psi,\\Psi \\rangle \\ , ", "132510a47daebc02e46edc2d58b1f513": "\\int_{-\\infty}^\\infty f(t)\\,dt = \\lim_{a\\to-\\infty}\\lim_{b\\to\\infty}\\int_a^b f(t)\\,dt.", "13259605bd89e461ed298c8372a9bbee": "r^2 \\, d\\phi^2 = \\cot(\\phi)^2 \\, dr^2 = \\frac{R^2}{r^2-R^2} \\, dr^2 ", "1325cbab3acd85b62fa10f31c669cd6c": " g_Q(x) = R - 1 \\quad \\text{if } R \\text{ is odd and } x \\ge a^2.", "1325d02cc79ae19ea98ff59226e069de": "\\,x,y \\in M", "132697478da1ebf855774a52fab3b193": "\\chi^2=\\sum^k\\frac{(\\text{observed}-\\text{expected})^2}{\\text{expected}}", "1326bfa9cab2cc91daed2e2054fa085b": "i_s", "132779ec50452e8bd1fe9cf13f9d8834": "=m_1^2 + m_2^2 + m_3^2 + m_4^2 + 2 \\left( m_1^2 - p_1 \\cdot p_1 \\right) \\,", "1327c5a67041c39498c92429828f52bd": " K_i^{*} = \\frac{K_ik_4}{k_3+k_4} ", "1327c68053a3b44efab0e651016e0c60": "E = V - 128", "1327e7eb009322eb5bf8773c2bab7bcb": "g = n+1, \\lambda = \\varphi(n),", "1327fa011c9a108b7e03af4e5e5a6272": "\\displaystyle \\int_Df(\\tau)\\overline{g(\\tau)}E(\\tau,s)y^{k-2}dxdy", "1327fcc8c22e8fa5a446fd71df965e37": "\\frac {dT}{dt} +\\frac {1}{\\tau} T = \\frac{1}{\\tau} T_a, ", "1328288bf2458c7c38005b885b88f1b2": "\\alpha=f_1(d_a), \\beta=f_2(d_a)", "13285769c8cd293bfd541406e0621d87": "d=0", "1328f7ac8e44b744072fe5457b1a787a": "\\rho_I(t) = \\sum_n p_n(t) |\\psi_{n,I}(t)\\rang \\lang \\psi_{n,I}(t)| = \\sum_n p_n(t) e^{i H_{0, S} t / \\hbar}|\\psi_{n,S}(t)\\rang \\lang \\psi_{n,S}(t)|e^{-i H_{0, S} t / \\hbar} = e^{i H_{0, S} t / \\hbar} \\rho_S(t) e^{-i H_{0, S} t / \\hbar}.", "13291972707bbf15282eb48b3a58b85f": "C \\approx \\frac{\\pi a (9 - \\sqrt{35})}{2}", "13293d96229660b4d58f4d598cb00853": "e^{X}Y e^{-X} = e^{\\operatorname{ad} _X} Y =Y+\\left[X,Y\\right]+\\frac{1}{2!}[X,[X,Y]]+\\frac{1}{3!}[X,[X,[X,Y]]]+\\cdots.", "13294ddcd2ae37f16ca81ab399088ecb": "1 \\over 3", "1329561c1be71291edf63fb128558fcf": " \\nabla \\times \\mathbf{H} = \\frac{4}{c^2}\\left( - 4 \\pi G\\mathbf{J} + \\frac{\\partial \\mathbf{g}} {\\partial t} \\right) \\,\\!", "132956250769ae7e923691734e6e9788": " v_e = I_{sp} g_{o} ", "132993a1e61dac8fa6cb8da8d59e465a": "\\min(2,3,2,1,3,6) = 1", "1329a1ea3cf18fa02d323888a46fb846": "\\textrm{E}[\\|\\textbf{x}_{k} - \\hat{\\textbf{x}}_{k|k}\\|^2]", "132a0ec0e90d85720803ceb65ff88b17": "r = c", "132a246a32b3636371660621e977e4ec": "\\therefore", "132a2b9ff7d611c9947b4fb65921f73e": "R(u,v)w=\\Omega(u\\wedge v)w. ", "132a53a090c1a8764a5c3cf45ad356a0": "b \\approx 43 AU", "132b39cdff1c47d9b24723eb36d96a4c": "{k_1 \\over k_2} = \\frac{\\ln (1-F_1)}{\\ln (1-F_2) }", "132b45d748a0dc277832d754f6064583": " \\delta_Y(\\varepsilon) \\ge c \\, \\varepsilon^q, \\quad \\varepsilon \\in [0, 2].", "132b7a718a970d6d5c32f5a3954ce494": "\\mathbb{E}\\left[ \\left(\\int_0^t H_s \\, dB_s\\right)^2\\right]=\\mathbb{E} \\left[ \\int_0^t H_s^2\\,ds\\right ]", "132c1d66533a39a087d903f859f1ad4c": "\\phi_n(\\kappa) = 0.033C_n^2\\kappa^{-11/3},\\quad \\frac{1}{L_0}\\ll\\kappa\\ll\\frac{1}{l_0}", "132c834c23e12064579e7d5727bd243b": "{2,3,1}\\,\\!", "132c8644bf0affdf6857571014cfe3b9": "f(1)=0", "132ca804c2d37103e4a52c9fc645b444": "{\\partial u\\over \\partial x}={u_{i+1}^n-u_{i-1}^n\\over 2h}", "132cc50dbfc565f8ad72c623ed36feeb": "\\sum_{k=0}^\\infty \\frac{1}{k+1}{2k \\choose k} z^k = \\frac{1-\\sqrt{1-4z}}{2z}, |z|<\\frac{1}{4}", "132d340340a15905d4ff1c16aed8f4f2": "2^n \\equiv 2 \\pmod{n}\\,", "132d42e04bdcdb4eac78be6206e7fd6f": "(\\underbrace{-,\\cdots,-}_{k},\\underbrace{+,\\cdots,+}_{n})\\,", "132d47002ae5c3ebf5e37764e5e11921": "V^B", "132d6ca808cf0abfc68b8b26afc104a4": "\\phi(\\tau,z+\\lambda\\tau+\\mu) = e^{-2\\pi i m(\\lambda^2\\tau+2\\lambda z)}\\phi(\\tau,z)", "132d86a0486dd349d6c0fbc5e5aec408": " \\, \\theta \\, ", "132d9204086f03abfd00f2c0f28f3b0f": " F(x_1,\\ldots,x_n,u,u_{x_1},\\ldots u_{x_n}) =0. \\,", "132da3fb6246e107dcf0c89cf4efb65a": "\\sqrt{2/(N-1)}", "132dbb8ebe5d7a782ee5b17a139f28c2": "SP_x (t,f) = |ST_x (t,f)|^2 = ST_x (t,f)\\,ST_x^* (t,f)", "132e0c3e2dfafa2973d0c971b80a6187": "f(x)\\in L_\\omega^2", "132e298c25d7fed88263fa595f83b242": "V[J] = \\int J \\wedge J \\wedge J", "132e2dc6655eefbfee62e01e9b64b571": "\n\\prod_{i=1}^{r}\\prod_{j=1}^c\\frac{1}{1-x_iy_j}.\n", "132e47b9ee867e241feeb4adb582d9a7": "(S, G, P)", "132e76ecc2f0a55d7627df17687f42f8": "\\left[{\\begin{matrix}\n\\sigma'_{11} & \\sigma'_{12} & \\sigma'_{13} \\\\\n\\sigma'_{21} & \\sigma'_{22} & \\sigma'_{23} \\\\\n\\sigma'_{31} & \\sigma'_{32} & \\sigma'_{33} \\\\\n\\end{matrix}}\\right]=\\left[{\\begin{matrix}\na_{11} & a_{12} & a_{13} \\\\\na_{21} & a_{22} & a_{23} \\\\\na_{31} & a_{32} & a_{33} \\\\\n\\end{matrix}}\\right]\\left[{\\begin{matrix}\n\\sigma_{11} & \\sigma_{12} & \\sigma_{13} \\\\\n\\sigma_{21} & \\sigma_{22} & \\sigma_{23} \\\\\n\\sigma_{31} & \\sigma_{32} & \\sigma_{33} \\\\\n\\end{matrix}}\\right]\\left[{\\begin{matrix}\na_{11} & a_{21} & a_{31} \\\\\na_{12} & a_{22} & a_{32} \\\\\na_{13} & a_{23} & a_{33} \\\\\n\\end{matrix}}\\right].", "132e89f895e672f7d768c59b730dea85": "1^2>0", "132e918e08c9e66aca00c78708edbdb1": "m \\rightarrow \\infty", "132ed9975c221bd31a82c24d4d298377": "m_n,\\, 1\\leq n\\leq N", "132f2a7909026ad9d99eb3617e5fa5d2": "P = \\frac {\\pi^2 f^2 I^2 L^2}{2 c^2 h \\sigma} \\,", "132f2e4d607214c35f77a6a8a223976c": "\\langle Tx, y\\rangle = \\int_\\mathbb{R} \\lambda\\,dE_{x,y}(\\lambda)", "132f8ba98a13f25af92dd3489c20d3ed": "B_{ij} \\equiv \\frac{\\partial A_j}{\\partial x_i} - \\frac{\\partial A_i}{\\partial x_j}", "132f9970d5ddbe6e4a617c2ee6fa7592": "(L_{n+1}, R_{n+1}) = (L_{n+1}',R_{n+1}')", "132fbae817d977b2489d77aae8b28b21": "\ne^z = 1 + \\cfrac{z}{1 - \\cfrac{\\frac{1}{2}z}{1 + \\cfrac{\\frac{1}{6}z}{1 - \\cfrac{\\frac{1}{6}z}\n{1 + \\cfrac{\\frac{1}{10}z}{1 - \\cfrac{\\frac{1}{10}z}{1 + - \\ddots}}}}}}.\n", "132fc9d7ae8570da5cae1c756fe647f3": "3 \\sqrt{N}", "1330031b2c0b8f743c415044002c5990": "di_i", "1330295f7365e6b2525765b8b7c5347a": "\\operatorname{exp} \\equiv \\lambda m.\\lambda n. n\\ m", "13306c2f3619496b7e192f85d1fcad35": "F_0 |_A = f_0", "133092a1bf8f87f422416eafe4d02b7f": "\\mathcal{L}_X\\omega=0.", "1330a9073cead0f3f632d228b833e311": " \\textstyle n^2", "1330d7d3a3e04888cb1d564ef967d6ff": "s = \\sqrt{\\frac{(n_1-1)s^2_1 + (n_2-1)s^2_2}{n_1+n_2-2}}", "13310bbf243b195ce5d7a1981a908507": " \\lambda_{mod} = \\frac {\\lambda^2}{\\Delta \\lambda}\\ . ", "13310e071799bf88465bbff9b19d8dc9": " \\mathrm{rk}\\,L=\\tfrac12\\mathrm{rk}\\,E", "13311b0d1e75ba697cd320411dade1eb": "{dx_2 \\over dt} = r_2x_2\\left({K_2-x_2-\\alpha_{21}x_1 \\over K_2}\\right).", "13315da8e16ec5944c65ea78df091456": "\\mbox{eGFR} = \\frac{ {k} \\times {Height} }{Serum\\ Creatinine}", "13319531c7ebbfaa01fcf4eb07a06132": "u(s) = \\left(K_P + K_I \\frac{1}{s} + K_D s\\right) e(s)", "133254c99c18642f9232fd0d29f8935a": "B_0 = 2, \\quad B_i = \\min_{0 \\leq j < i}\\{ B_j + A(n{+}t{+}i{-}j{-}1, 2t{+}2, t{+}i)\\}", "133267ccdfce85bfa2af89d2117a5ab7": "I(x+\\Delta x,y+\\Delta y,t+\\Delta t) = I(x,y,t) + \\frac{\\partial I}{\\partial x}\\Delta x+\\frac{\\partial I}{\\partial y}\\Delta y+\\frac{\\partial I}{\\partial t}\\Delta t+", "1332835298924325e0dceaf965519ef9": "c=u^2+v^2,", "1332edd9216c7f2a410375128ed458ac": "\\scriptstyle \\left( y'(0), y(0) \\right) = (1, 0)\\,", "13331acc64c8ed7feb3098b1672def46": "\\forall\\alpha_1\\dots\\forall\\alpha_n.\\tau", "13333d418ab12f6015bd30339fc516ca": " U '", "1333eb3422ade940edc00a9f109cde4f": "p = B s\\,", "1333f4b108f61e75e8a051ae48b4acd5": "\\frac{\\alpha}{m},\\frac{\\alpha}{m-1},...,\\frac{\\alpha}{1}", "13341109577f21eec23f67e1491daee4": "\\scriptstyle L=\\mathbf{Q}(\\zeta_\\ell)", "13342609c9ad51810130c30eeca6d8c5": "disc(\\mathcal{H}) \\leq 6 \\sqrt{n}", "13342b07689f6b5bcaaa1a0756ad48be": "\\rho_{AC} = x_{13} \\ ", "1334303d61efda2b0c53880edc7145a9": "f \\stackrel{\\leftarrow }{\\partial }_x g = \\frac{\\partial f}{\\partial x} \\cdot g", "133442294cd46df3d13e19548ef29790": "10^{13}", "1334ec1765499b82a9f9016a47fe952e": " ds^2 = -(1-\\omega^2 \\, r^2 ) \\, dt^2 + 2 \\, \\omega \\, r^2 \\, dt \\, d\\phi + dz^2 + dr^2 + r^2 \\, d\\phi^2 ", "13353dd5531be1a5073d752ddb6e3902": "\\scriptstyle(4\\pm8)\\times10^{-12}", "13354afc5556ea6f05b4c3efc3940a82": "y = 20 + 3 \\epsilon_4 + 4 \\epsilon_8", "133553f770ddafddd785a7ee7388b81e": " I(\\mathbf r) = I_1 (\\mathbf r)+ I_2 (\\mathbf r) + 2 \\sqrt{ I_1 (\\mathbf r) I_2 (\\mathbf r)} \\cos {[\\varphi_1 (\\mathbf r)-\\varphi_2 (\\mathbf r)]}", "1335ba9c3fd08cb0f539f168676e4f94": "R(N_2,N_3)", "133603913b08d935fb18f3b315b127dc": "S_1, S_2, S_3, \\dots , S_n\\,\\!", "133620f1e4751470f28e89642031f519": " g\\! ", "133634bd9b338fe7ea0f76d9ece96768": "\\,_nE_x = Pr[G > x + n]v^n = \\,_np_xv^n ", "133706d1605beb1fa881cf4025412167": "\\mathrm{rem}\\left(x_0,m_i\\right) = a_i", "13375c22930246d1b824a15c9a2be130": "\\scriptstyle{R_l^0}", "13379092ec45be1af10deb4c090e3527": "p(x|z)", "1337d9ff34c9a0c6876945f612884af7": "\\vert{\\Psi_{\\mathbf{p}}^{\\circ}}\\rangle", "1337f98310fd31f9d4ddf2cec78347dd": "e=-d\\phi /dt = - SdB/dt", "13383997fff1a7f77ba27c1e9354ab04": "\\lambda=\\int_\\, \\mathcal{E}\\, dt", "13383f6e3ebb812b4bc0c0471be38c48": "\\sum_{k=0}^\\infty k^4 \\frac{z^k}{k!} = (z + 7z^2 + 6z^3 + z^4) e^z\\,\\!", "1338514089e09578d0b1a6700fd6eb29": " U(x,y,z,t) = u(x,y,z) e^{-i(kz-\\omega t)},", "13385cd2626cfc1bfd651830e67e2701": "\n\\begin{bmatrix}\n0.5 & 0 & 0 & 0.5 \\\\\n0 & 0.5 & 0 & 0.5 \\\\\n0 & 0 & 0.5 & 0.5 \\\\\n0 & 0 & 0 & 1 \\end{bmatrix}\n", "1338939c352a99a4780ebcb080dc2468": "A=B^TB=R^TQ^TQR=R^TR", "133919e2f37517dcc77b483f366747b7": "\\left(\\frac{dn_1}{dt}\\right)_{\\mathrm{pos}\\,\\mathrm{absorb}}=-B_{12}n_1 \\rho(\\nu)", "13397149a94379232e8eeb8a4a8af93b": "C = \\dot{V} - C_{osm}", "133976d5ba017d7d872b0e6f6e598d7f": "\\,\\zeta < 1 / \\sqrt{2}", "13398f328ba973b9339f9e03fb7bfba4": "\\scriptstyle \\epsilon_c = p_c / p_{01} \\,", "1339a37da7375f2e51ec474b12db6d12": "\n\\hat{H}_{\\textrm{int}}=\\chi\\sum_{n,\\alpha}\\left[(\\hat{u}_{n+1,\\alpha}-\\hat{u}_{n,\\alpha})\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n,\\alpha}\\right]", "1339d03c679e2fc6efefa28a4ba76646": "1 \\over 7", "133a36d041284fc8ce45d46132a4bd0d": "\\begin{align}\nxy & = \\{ X_L | X_R \\} \\{ Y_L | Y_R \\} \\\\\n & = \\left\\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R | X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \\right\\} \\\\\n\\end{align}", "133a786238a843f2b6f075d701b704ab": " (\\mathbf{b} \\in \\operatorname{Im}(A)) ", "133a8d8d59795d2ef443dec95ffa2cda": "\\sigma_\\mathrm{eq}", "133ae19c283d03c7dcaf8edf823d73ad": "\\lambda^2 d \\mu_{h} = d \\mu_{k}\\,", "133ae4baa2da8a7e24135cc4221f547e": "\\rho_1 \\ge \\cdots \\ge \\rho_n\\, ", "133b491d22f26dcf8dca63855aefe28b": "G_1:\\{0,1\\}^h\\to\\{0,1\\}^n", "133b4acb325c07849cd7b4b06a813ae2": " X(z) = \\sum_{n=0}^{\\infty} x[n] z^{-n} ", "133b7a84b36043b125b9dd0bc5620c69": "\\, {\\hat{C}} = CT", "133bec54d3df720bcda67ce19f16364b": "\\rightarrow (1)", "133c2c358ea4bc9f5734ac6732aea6d5": "\\frac{f(x_0+\\epsilon)-f(x_0)}{\\epsilon} > K/2;", "133c5722f16a7c1d197fabe6edf073dd": "height_{midparent}=\\frac{1}{2}(height_{father}+(1.08\\times height_{mother}))", "133c7f13616bd30d299879c648948055": "\ne_1 = \\frac{P \\int K^2}{n h^D}\n", "133c9151ba673d631a9b7e83ff2b57ba": "\\sum_{j=1}^n w_j x_j = W,", "133cb81b5fb186ec165b1ae5a60265a2": "(\\Bbb{N}, \\cdot, \\uparrow)", "133cc6903f063f92a40f353bcb3091cc": "\\delta X(\\omega)", "133d15a3a6a3956e0a7778ec71d85d39": "\\mathbb{C}^n = \\bigoplus_i Y_i", "133d5a7e53eb1623544ec86487d4a8fb": "\\scriptstyle \\text{range}\\, =\\, \\max_i(Y_i)\\, -\\, \\min_i(Y_i)", "133d5d6776100b4588234f7831489de7": "X = \\coprod_{\\alpha}{X_{\\alpha}}", "133d7f9f31bddd62b6bc8880c8f5d946": "E(\\mathbb{F}_q)", "133dbb123218e2e771e45cea77bd301a": "cr(K_{m,n})", "133dc26295ba5508bdbd353d10f36ade": " \\mathbf{E} ( \\mathbf{r} , t ) = \\mid \\mathbf{E} \\mid \\mathrm{Re} \\left \\{ |\\zeta \\rangle \\exp \\left [ i \\left ( kz-\\omega t \\right ) \\right ] \\right \\} \\equiv \\mid \\mathbf{E} \\mid \\mathrm{Re} \\left \\{ |\\phi \\rangle \\right \\} \n", "133dd4051053de4acd865b646dc1cfcf": "\\mathbf{M}^1,\\mathbf{N}^1", "133dd94b78bcd13c5f8f2aab4623e2c2": "\\lbrace 17, 23, 29\\rbrace", "133df2f858fe61455e9d923f2eedb4fe": "f_e(x_e) = \\sin \\left( \\frac{n \\pi x_e}{L_e} \\right)", "133e121d08e25c65bdf90665b5ec692d": "p(y,x)=\\sum_1^n w_jp_j(y,x), ", "133e75244edb014f03c534d36547ad2a": "|f(x)-(S_Nf)(x)|\\le K {\\ln N \\over N^p}\\omega(2\\pi/N)", "133e76e81033a31efdc0c55fe42c2500": "\\binom{k}{\\alpha} = \\frac{k!}{\\alpha_1! \\alpha_2! \\cdots \\alpha_n! } = \\frac{k!}{\\alpha!} ", "133ebe4e66278209e05d4ef60366950a": " \\lambda f.(\\lambda p.p\\ p)\\ (\\lambda x.f\\ (x\\ x)) ", "133ec32a11ce5196bf065519d1af434a": " z = x + iy = |z| (\\cos \\phi + i\\sin \\phi ) = r e^{i \\phi} \\ ", "133f1eb71de5e7e9f33479bf1ffc09cb": " g_{em} \\ =\\ 2\\left ( {\\theta_\\mathrm{left} - \\theta_\\mathrm{right} \\over \\theta_\\mathrm{left} + \\theta_\\mathrm{right} } \\right ) ", "133f461b9ea2866ad4052fc6f77dde03": "\\mathbf{B},", "133f908a0fe7da6392cf707d7532da28": "0 \\leq l \\leq r \\leq n", "134047c6f2d1271f1f09a7d57f300050": "\\begin{align}\n\\sin y = x \\ \\Leftrightarrow\\ & y = \\arcsin x + 2k\\pi , \\text{ or }\\\\\n & y = \\pi - \\arcsin x + 2k\\pi\n\\end{align}", "134072fb981be1fabaa8d3bc8a157fc9": "x^2+c = x^2-2xt+t^2", "1340912de620d2f69c86b7a8f82d9e4f": "\\scriptstyle TE_{mnl}", "13411e15c34c939e042a26cc78feefee": " \\log {g(\\zeta)-g(\\eta)\\over \\zeta -\\eta} = -\\sum_{m,n\\ge 1} c_{mn} \\zeta^{-m} \\eta^{-n}", "1341914dc76b01b5f48b797ada5e2278": "\\mathbf k", "1341a95f233b84a8046b7951e8d8d1bf": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 1\\\\\n0 & 1 & 1 & 1\n\\end{array}\n\\right] .\n", "1341b662dcad96b90630ab8d84e9f7c2": " B \\ ", "1341d34474d7c9ae8ae6000ae1621705": "y[n] = (x * g)[n] = \\sum\\limits_{k = - \\infty }^\\infty {x[k] g[n - k]}. ", "13425d29c65b33d585c51dde0d20e767": "\\boldsymbol{\\mu}_a", "13425eeed61b9fa166c4f35bd844dfb2": "\\bar{\\mathsf{\\Omega}}(a,x) \\mapsto \\bar{\\mathsf{\\Omega}}'(a,x)=R\\bar{\\mathsf{\\Omega}}(a,x)R^{\\dagger}-2a\\cdot\\nabla RR^{\\dagger}.", "1342da3464211ac0bb6a11410359c008": "O(n^{1/(2k-1)})", "1342de7d0703e1c9abee73fbdd6cb173": " \\langle x(t) \\rangle = \\langle x \\rangle_0 + \\int\\limits_{-\\infty}^{t} \\! f(\\tau) \\chi(t-\\tau)\\,d\\tau, ", "134352ac75a2b6d3ca5766a3a629b8c5": "k[V]^G", "13436c247320284d49c07238c666286d": "c_3 < 0", "1343923278289008b04ad5f121b3ca76": " P(x) ", "1343993b14ffa86c32c3c4d15e72073d": "\\{\\{-2, -1\\}, \\{1, 2\\}\\}", "13439c3307ae1ceba0998aa85991c218": " F_{M} ", "1343e2405a6ece8515c1dbc6740430fd": "\\mathcal{O}^X", "134449c536b1ac37d78b34526fd197d1": "{\\textstyle \\sum}a_kz_0^k = a(z_0) \\, (\\boldsymbol{wB}) ", "1344c33158c30a3d04ede87305193000": "\\scriptstyle \\lfloor\\frac{i+w}{J}\\rfloor", "13451deb165c2bc58d59b2a0673bb9da": "\\begin{align}\n & \\left\\langle j'_n , m'_n , ... , j'_2 , m'_2 , j'_1 , m'_1 |j_1 , m_1 , j_2 , m_2 , ... j_n , m_n \\right\\rangle \\\\ \n= & \\langle j'_n , m'_n | ... \\langle j'_2 , m'_2| \\langle j'_1 , m'_1 | | j_1 , m_1 \\rangle | j_2 , m_2 \\rangle ... | j_n , m_n \\rangle \\\\\n= & \\prod_{k=1}^n \\left\\langle j'_k , m'_k | j_k , m_k \\right\\rangle \n\\end{align}", "13456d2a0b0163e5ed501d708d14f81e": "1922^{12}", "13458606ac0ed75bc3167fb7aa312921": "\\mathcal{N}=(1,1)", "1345ceee29863e5fc5152d2cfa6ea016": "(U,\\varphi)\\,", "1345fadf3b0abf77d1f92de0bd6b9dde": "P[X_1 \\le u_1, \\dots, X_n \\le u_n] \\le P[Y_1 \\le u_1, \\dots, Y_n \\le u_n ] ", "1346280563307dd9139fd4ce45dd2c57": " \\begin{align}\n\\mathcal{L}_K = \\sum_f \\overline{f}(i\\partial\\!\\!\\!/\\!\\;-m_f)f-\\frac14A_{\\mu\\nu}A^{\\mu\\nu}-\\frac12W^+_{\\mu\\nu}W^{-\\mu\\nu}+m_W^2W^+_\\mu W^{-\\mu} \n\\\\\n\\qquad -\\frac14Z_{\\mu\\nu}Z^{\\mu\\nu}+\\frac12m_Z^2Z_\\mu Z^\\mu+\\frac12(\\partial^\\mu H)(\\partial_\\mu H)-\\frac12m_H^2H^2\n\\end{align}", "134630743b6fb4c5cff9042f53149d72": "\\langle E \\rangle^2=m^2c^4+\\langle \\mathbf{p} \\rangle^2c^2.", "1346582c28b50643bd3a23af930310e4": "\\begin{align}\n x^1 &= x \\\\\n x^n &= x^{n-1}x \\quad\\hbox{for }n>1\n\\end{align}", "13465b6d3bf9a84a71ed199ea982c4c8": "\\phi_a (\\omega)", "13467c607ceac041aa451742b18fbaa2": "\\beta_g=", "1346dc43ec12bd9657742c5ae8d0c0a7": " \\frac{1}{x} ", "13470969d10dc4a7bb3884354fda0137": " v_i ", "134766878c4e4b03c0c362b5bfc40f1b": "K^\\text{app}_m=K_m(1+[I]/K_i)", "13477f050c5019c52feea347bdd90ce8": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 \\\\\n-1 & 2 & 0 \\\\\n 0 & 0 & 2 \n\\end{smallmatrix}\\right ]", "134782eba81925e50ae0aff1007edf2c": "F = \\rho SV^2 \\sin^2 (\\theta) ", "1347aed7c5f264dbd5a679a2e94ebbda": "V_-\\,", "1347f9a8c5c697b7fa4992ba182e9b71": "\\frac{k}{(k-1)}", "1347fd20b7d0a229b50d1e432f2a5b68": "\\Lambda = \\frac{h}{\\sqrt{3 mkT}},", "13480623c74d78a30c24aa310d8bb6cd": "\\nu_{Ti} = (ZeKE/m_i)^{1/2} = 1.69 \\times 10^7 Z^{1/2} K^{1/2} E^{1/2} \\mu^{-1/2} \\mbox{s}^{-1} \\,", "13481239c55b3814795d3944a24f171e": "\\mbox{Area} = \\frac{1}{2}a b\\sin C.", "1348c11881b7d1ba814e82f0b60f2c4a": "[\\omega\\wedge\\eta] = \\omega\\wedge\\eta - (-1)^{pq}\\eta\\wedge\\omega,", "1348d4b84f8dd009299afd64b8c1dad5": "\\scriptstyle \\theta", "1348e80284a9ad55a9b73fde790f76fb": "B = \\{R_1, \\dots ,R_K\\}", "134956319b41a7c6ac16713377bdd9a8": "\\frac1{64}\\int_0^1\\frac{x^{16}(1-x)^{16}}{1+x^2}\\,dx= \\pi-\\frac{741\\,269\\,838\\,109}{235\\,953\\,517\\,800}", "13496798b1e2e7bee66667072f59783b": "cr(K_{m,n}) = \\lfloor n/2\\rfloor\\lfloor (n-1)/2\\rfloor\\lfloor m/2\\rfloor\\lfloor (m-1)/2\\rfloor.\\ ", "134995a181146891dcb464a453b0c888": "\nc^{2} d\\tau^{2} = g_{\\mu\\nu} dx^{\\mu} dx^{\\nu} \\,\\!\n", "1349965ed2c5baa35fbd46e88db032ed": "S_n = \\sum_{i=1}^n a_i X_i,", "13499a4e188a19c8e5d1f88037037f17": "\\operatorname{Categorical}(\\boldsymbol\\theta_{x_t})", "1349a8a8d242a4cf4351aa8e8bbe149d": "y=[3,3,5]\\,", "1349aeef7c8ba408bd0c335ef1077f98": "\\displaystyle{(x,y)_0 =\\varphi(f_{x,y})}", "1349bb664f627dbfaab4c6b3ece4350d": "s^{-1} = S\\,", "1349cd4602162f137d131d73ac9881e7": "\\scriptstyle {A+\\sqrt{5}I}", "134a4965cdfb5e02180c5ccb775f1f6d": "\\sum_{n=0}^{N-1}Q_n(x)Q_n(y)\\pi_n=\\frac{1}{\\rho(x)}\\delta_{x,y}", "134a657a7a8e136d733917d789d955a8": "\\mbox{House P/E ratio} = \\frac{\\mbox{House price}}{\\mbox{Rent} - \\mbox{Expenses}}", "134a7ce93d7e65ed846fc1208af4aa21": "D \\approx D' + t", "134b086d86dad362441361c409900502": "C_{n-1}' \\not = C_{n-1}", "134b203250c2c1b0ff0c3918a28057b3": "\n\\langle \\phi(x_1) ... \\phi(x_n) \\rangle = { \\int e^{-S} \\phi(x_1) ... \\phi(x_n) D\\phi \\over \\int e^{-S} D\\phi}", "134b31b03360659b2909ce8be973374f": "\n\\frac{d\\varphi}{dt} = \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{L \\, c^2}{E \\, r^2}\n\\,.", "134ba72d07c49a31137b952b01779e15": "\\operatorname{tr}(x^k) = 0", "134bd8b9d48c083f943399684eba58fb": "k\\propto \\sqrt{\\frac{T}{M}} \\text{ (ideal gas)}. ", "134c0aba8f0f7b4c55a967c6db48ec89": "2^{bh(v)}-1", "134c1f959e7e679d4c31eba0cfd8a5c0": "\\exists x\\,c=x", "134c7737800ed7b1d5738898c7fa5c41": "\\lim_{x \\to \\infty}\\frac{f(x)}{g(x)}=0.", "134c7bf1a1dc1364b03b3c6cea450061": " (u^2 + \\alpha + y)^2 = (\\alpha + 2 y) u^2 - \\beta u + (y^2 + 2 y \\alpha + \\alpha^2 - \\gamma), \\qquad \\qquad (3)\\,", "134ca6c9305ac8ec8f1d36a6278183f1": "2\\theta^\\circ", "134cc2f6b5c1d2301ee7a1c89e922efe": "\\not=", "134ce3f5e3b06d8a6d338bd2e55b484b": "\\begin{matrix}\\operatorname{Ta}(3)&=&87539319&=&167^3 &+& 436^3 \\\\&&&=&228^3 &+& 423^3 \\\\&&&=&255^3 &+& 414^3\\end{matrix}", "134cf0be132297c8ce449fba44833d23": "m=(8/3)E/c^2", "134d1982658be76a6fe7aac6c740136b": "\\iint\\limits_{S}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\subset\\!\\supset \\mathbf D \\cdot \\mathrm{d}\\mathbf A", "134d3af76bbc193f5209e0359b97035a": "\\frac{1}{10} + \\frac{1}{10} = \\frac{1}{5}", "134d3c018786324961c4c581d902b4bf": "\\gamma=P(-z ", "135de5d2a80d9bdfd581c3fb8abd7ed8": "u,v,", "135df1635b0181e64bb76d735a9dd9ab": "\n u_i^{n+1} = \\frac{u_i^n + u_i^{\\overline{n+1}}}{2} - a \\frac{\\Delta t}{2\\Delta x} \\left( u_i^{\\overline{n+1}} - u_{i-1}^{\\overline{n+1}} \\right)\n", "135e615dad444216df9ffb2d06a290c7": " C_{2\\epsilon} = 1.92 ", "135eaaeed8800c89e945ecb35f714f51": "\\sqrt{\\frac{1}{56}}\\!\\,", "135ef7e3a528accf8a7fa14e84c4adb4": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = \\frac{8\\pi G}{c^{4}} T_{\\mu\\nu}", "135f0b21da240c2b181561ef360b1890": " \\sigma^{2}( \\langle A \\rangle ) = \\frac{1}{M} \\sigma^{2}A \\left[ 1 + 2 \\sum_\\mu \\left( 1 - \\frac{\\mu}{M} \\right) \\phi_{\\mu} \\right],", "135f12dabd027f2b6f8b81a0411945dc": "\\frac M m = \\sqrt { \\pi \\frac { \\delta } T \\frac {V^2} {2} } \\mathrm{e}^{ \\left( \\frac { \\delta } T \\frac {V^2} {2} \\right) } \\mathrm{erf} \\left( \\sqrt { \\frac { \\delta } T \\frac { V^2 } {2} } \\right)", "135f1b3b8adb4ccff6b1fa503cfee73a": "TCR = \\left(\\frac{l_{\\mathrm{sum~of~pieces}}}{l_\\mathrm{tot~core~run}}\\right)\\times 100 ", "135f2a9e8499913f9280be68e0547196": "\\Psi(x) = \n\\begin{cases}\ne^{ -\\frac{1}{1 - x^2}} & \\mbox{ for } |x| < 1\\\\\n0 & \\mbox{ otherwise} \n\\end{cases}", "135fbfb66efe89f65aba29ec1f57e088": "\\chi(S,\\mathcal{O}_S)=\\frac{1}{12}(c_1(S)^2+c_2(S))", "135fdeea4e0a32282427ab07523f98a6": "M(t;k,\\lambda)=\\frac{\\exp\\left(\\frac{ \\lambda t}{1-2t }\\right)}{(1-2 t)^{k/2}}.", "136012efaaae697cb42f9977241f0ffb": "(a^p)", "13601c2d567143550b01a3a015505c6b": "K_i=\\frac {y_i}x_i", "13607753dc207af118a6f5906b12be4f": "E_r\\,", "136078e6ebba8718a68add37deb80b44": "L = - \\partial_{x}^2 + v( x, t )", "1360a23956abea8e55de99cfce0f8bff": "b(v) = E(\\max_{j \\neq i} v_j ~ | ~ v_j \\le v ~ \\forall ~ j)", "1360fa40d8a9dbb878b994e8bec3916a": " \\mathrm{H_{R}} = { \\mbox{tube hematocrit} \\over \\mbox{feed reservoir hematocrit}} ", "13610ea260bd4212f416eb1fb90d8fc7": "\\ H^{BM}_i (X)=H^{-i} (X,\\mathbb{D} _X), ", "136134b66637e72916e0caf51d42b2a9": "\\mathrm{SL}(2n,\\mathbb{C})", "136172cde4030f5cdd4c8382afbe12ae": "\n\\tau_{\\max,\\min}= \\pm R", "1361add4a3550f1d8d3b2081a3538c95": "\\mathcal{E}(\\rho) = \\sum_i A_i \\rho A_i^\\dagger", "1361afc324d9da7cdffd4b33b62bc5df": " s = v + u - {v u \\over c} \\, .", "13620a9fb842a4f28288c82d67063545": "\\textstyle \\mathbf{c}_2", "136236fbb4a90ceadaf5a46ddabf6222": "X_1, \\ldots, X_n", "136274abb512b3bbcd47339ef5446eb5": "|\\Psi(t)\\rangle = \\sum_n C_n(t) |\\Phi_n\\rang", "136281cbcb35cce6a6c724b1fdc86c7a": "(S \\subseteq X) \\; \\mapsto \\; \\exists_f S\n= \\{ \\; y \\in Y \\; \\mid \\; \\exists x \\in f^{-1}\\lbrack \\{y\\} \\rbrack, x \\in S \\; \\}\n= f\\lbrack S \\rbrack", "13629935975995fa4da39df7fc0fcd64": "\\frac{1}{H}", "1362a1c16c3cfd959735dd3dd6816fa0": "e^{iz \\cos(\\phi)} = \\sum_{n=-\\infty}^\\infty i^n J_n(z) e^{in\\phi},\\!", "1362b8ef4124561aa318d8102211d65d": "D_\\mu =\\partial_\\mu \\pm ig_s t_a \\mathcal{A}^a_\\mu\\,,", "1362ea3e2bcf8011573390f6075eb3b6": "\\psi_4(x) = (2 \\sqrt{6} \\, \\pi^{1/4})^{-1} \\, (4x^4-12x^2+3) \\, \\mathrm{e}^{-\\frac{1}{2} x^2}", "13630cde0bca1544d12c063c9c44d37a": "R_1 \\simeq R_2 \\simeq R", "13636687d2d40633b6914ad156310399": "\n\\begin{array}\n[c]{ccccccc}\ng_{1} & = & X & Z & Z & X & I\\\\\ng_{2} & = & I & X & Z & Z & X\\\\\ng_{3} & = & X & I & X & Z & Z\\\\\ng_{4} & = & Z & X & I & X & Z\n\\end{array}\n", "136371b757a99c261a0bc00454a8b5dd": "\\lambda_{i} \\ne \\lambda_{j}", "136393b310d4a4d0ce5f4bfab3aed455": "C_2 = \\prod_{p\\ge 3} \\frac{p(p-2)}{(p-1)^2} \\approx 0.66016 18158 46869 57392 78121 10014\\dots", "1363958cf6b0f5ca95aa77e05186bd11": "\\sum _x a^x = \\frac{a^x}{a-1} + C \\,", "1363adc49cab9c2b6c64c76455b4103b": "d\\operatorname{Ad}", "1364196bb8bcabcdf0c8f12048c92fd3": "W_o (t) = \\int\\limits_0^t e^{A^T \\tau} C^T C e^{A \\tau} d\\tau", "13641c4ed4cd6194596e51428ccac0a5": "\n\\sigma _z^2 \\approx \\,\\,\\,\\frac{1}{n}\\,\\,\\left[ {a^2 \\sigma _1^2 \\,\\, + \\,\\,\\,b^2 \\sigma _2^2 \\,\\,\\, + \\,\\,\\,2\\,a\\,b\\,\\sigma _{1,2} } \\right]", "13645114b8c545be75a31d2a5b24646e": "P/", "136452cf44c6429d500587445e911df1": "\\|x\\|_{bv_0} = \\sum_{i=1}^\\infty|x_{i+1}-x_i|", "13645b741cdf628074abe7e46f9a79d7": "0 < t < 1", "1365119e292840ba3dc908b46d070640": "\\mu_\\mathrm{eff} = \\frac{\\mu_r}{1+k(\\mu_r -1)},", "136512749bef4627ba9a55c17811d6a9": "-1/c^2\\mathbf{E}", "136548216702fba97d8e1d193036b91a": "\\bold{g} = -\\nabla \\Phi ", "13656718a70a35e5350ae798900c07f1": "\\bar p_i = \\frac {\\sum_{j=1}^n \\begin{cases} 1 & \\mbox{if }x_{ij}\\mbox{ defective} \\\\ 0 & \\mbox{otherwise} \\end{cases}}{n}", "1365ddb3f8f603942cf11a2b81d39d2b": "m \\ddot x = \\phi(t, x, v)", "1365e88ddd22fc0eebb3cbab27cac319": "B=0.2", "13660530732efac2d04da15353fa88de": "a\\vee\\lnot a=1", "136664d2ec6df8f399903ca112f30f71": "(\\mu/\\rho)_i", "1366bbfc94cea853b23901a19eae3579": " A' = \\frac{1}{2} \\iint \\limits_A | d\\mathbf{A} \\cdot \\mathbf{\\hat{r}}| ", "1366c54e72a2b46ec57dea0cbdcbb6a6": "(q;q)_n=(1-q^n)(1-q^{n-1})\\cdots (1-q)", "1366ed5959e804668ce5f97684910658": "p(x)=M_n(x,\\dots,x)", "136724e5d2ea6f3cd7b9606415b9a21a": "\\left(\\!\\!\\!\\binom{n}{k}\\!\\!\\!\\right)=\\binom{n+k-1}{k}.", "13674d1e116ee2c76b26faa30c1b5c8a": "\\vec r \\cdot \\vec n_0 - d = 0.\\,", "13675276f9bd780371f4a85636cc91a3": "T_0>\\cdots > T_j", "136791245aa2325460447dca2fa71a12": "f_{Y_{[r_1:n]}, \\cdots, Y_{[r_k:n]} \\mid X_{r_1:n} \\cdots X_{r_k:n} }(y_1, \\cdots, y_k | x_1, \\cdots, x_k) = \\prod^k_{ i=1 } f_{Y\\mid X} (y_i|x_i)", "13679fb120b35b229b3e05404f8aa382": "\\begin{align}\n\\forall w_1,\\ldots,w_n \\, \\forall A \\, ( [ \\forall x \\in A &\\, \\exists ! y \\, \\phi(x, y, w_1, \\ldots, w_n, A) ] \\\\\n&\\Rightarrow \\exist B \\, \\forall y \\, [y \\in B \\Leftrightarrow \\exist x \\in A \\, \\phi(x, y, w_1, \\ldots, w_n, A) ] )\n\\end{align}", "1367f6bf32a2668c02fb4a77e76a630c": "\n\\sigma_s = \\frac {n_e\\ e^2}{m_e\\ \\nu}\n", "13681016cf9de2173e0ebcdda7581381": "{e \\over n} \\le {{q-1}\\over q}\\left( {1-\\sqrt{1-{q \\over{q-1}}\\cdot{d \\over n}}}\\, \\right)=J_q({d \\over n})", "13682bd627f2940a3279ea7b0dcdce48": "W_{i-1}", "13683671b6790314712469db219402c9": " d^2\\xi^{\\prime} dz\\rho(\\vec{\\xi}^{\\prime},z) ", "1368b899de5b7e71900fd806dcb8f14c": "|\\Psi_C\\rangle", "1369100fedcec429e3834ca46d847ea1": "P(\\cdot |\\alpha ,\\beta )", "1369120d47bcb570e5c3465b99073a14": "2^5 = 32 > 26", "136995b56f396817203f1f5233ce267d": "d_k=-H_k g_k\\,\\!", "1369bb002c76449871bb2b207bdd91ae": "\\scriptstyle \\vec J = \\vec L + \\vec S", "1369f8fd43395ddd5e49af69beffed02": "\\displaystyle \\hat{f}(-\\xi) = \\overline{\\hat{f}(\\xi)}\\,", "136a1dc73d3479a1482df4e2f293fd4f": "\\xi=0 ", "136a21801e1ae63fab301f3b19d78b7e": "(4)\\qquad F_{ab} = A_{b\\,;a}-A_{a\\,;b}\\;,", "136a2ea462cf67501e39fe49fb602717": " \\mathrm{tr}(\\mathbf{AB}) = \\sum_i \\sum_k A_{ik}B_{ki} = \\sum_k \\sum_i B_{ki} A_{ik} = \\mathrm{tr}(\\mathbf{BA}) ", "136a6c2a06fa87a5b1353466bda0788e": "DR_{p,c} = \\frac{\\sum_{p,c}(DR)}{count_{p,c}(singular cases)}", "136a7d81159f175ef03e659567a7cd63": "g^{r'} ~\\bmod~ p", "136aa4cc2deb07514413c06ba58ca9d4": "\n\\begin{align}\n\\operatorname{ns}(u) & = \\frac{1}{\\operatorname{sn}(u)} \\\\[8pt]\n\\operatorname{nc}(u) & = \\frac{1}{\\operatorname{cn}(u)} \\\\[8pt]\n\\operatorname{nd}(u) & = \\frac{1}{\\operatorname{dn}(u)}\n\\end{align}\n", "136ac4cf53968f2310e80756db6601a5": "c(q_j)", "136acc1e219e7e8465c9bc4dae692e1f": "-(p+1)/r", "136b278e2de250a8682e336712119093": "q \\oplus (a)q", "136b4f8eb8849dc5e87a162e30f1e560": "Q(p)", "136b5d1c2e1a5207a43fa208fac550e5": "\\mathrm H(L, R) = (\\sigma(L),R)", "136bce849d912ba119952d760716f63e": " = \\max_{\\beta} \\left[1 - \\frac{1}{3} \\left[\\exp\\left(-\\frac{1.10 x - 20 \\beta}{10}\\right) + \\exp\\left(-\\frac{1.10 x}{10}\\right) + \\exp\\left(-\\frac{1.10 x + 20 \\beta}{10}\\right)\\right]\\right]", "136c740771455e77934c50250eb85b40": " \nProb(ranking \\; 1, 2, \\ldots , J) = {exp(\\beta z_1) \\over \\sum_{j=1}^J exp(\\beta z_{nj})} {exp(\\beta z_2) \\over \\sum_{j=2}^J exp(\\beta z_{nj})} \\ldots {exp(\\beta z_{J-1}) \\over \\sum_{j=J-1}^J exp(\\beta z_{nj})}\n", "136cc7c76d8b6b6c3561210b71a26d8c": "A, B,C\\subseteq X\\,", "136ce9d75f387c554d1c7a5716f70dd0": " -\\frac{d}{dx}\\left[p(x)\\frac{dy}{ dx}\\right]+q(x)y=\\lambda w(x)y", "136cec92ad00e5750911c5173984995c": " \\varepsilon_{Nd(t)} = \\left[\\frac{\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{sample(t)}}{\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{CHUR(t)}}-1\\right]* 10000 ", "136daf68d2fcdf660449579c7684f588": "R=A / nil(A)", "136e057a89522fda3f2160bf4ae29e6e": "\\scriptstyle x_{\\tau}(u) = \\delta(u-\\tau).", "136e3a7ce0ee1b8181d2b156119b2543": " \n 0 = \\sum_{i=1}^N w_i, \\;\\; 0 = \\sum_{i=1}^N w_i \\, c_{j,i} \\;\\;\\; (j=1,2,...,nx)\n", "136e54ce18481154840ff82bf0921877": "n_{i} = \\sum_{\\sigma}\\hat a^\\dagger_{i\\sigma}\\hat a_{i\\sigma}", "136e7596defa3afe882e06588efceef2": " Z ", "136ece3cbc00399cf85b6790a699b61b": "\\frac {p!}{k! (p-k)!}", "136f2ef5e86dfc3c3946507a080d32c7": " M^*F(t)=QF(t/2).", "136f313de9976a602722f3184c11c17d": " D \\equiv 0\\pmod{4} ", "136f5279da7978992bbb2f26d8d9233c": " E[M_t] = \\int_{0}^{\\infty} m f_{M_t}(m)\\,dm = \\int_{0}^{\\infty} m \\sqrt{\\frac{2}{\\pi t}}e^{-\\frac{m^2}{2t}}\\,dm = \\sqrt{\\frac{2t}{\\pi}} ", "136f6025ea6b5f6bcd989ae94f5977e9": "G = \\begin{bmatrix} I_k | P \\end{bmatrix}", "136f8119128bf5713464b09b91b8acbd": "(C^\\infty)'(M)", "136f8f7b3ff4529f70c24086afff8d8f": "(W-X^T1)r_f + X^Tr = \\mu,", "136f9f7f79f8f3cc054b9b2a650e804f": "W \\theta (0)", "137058ed167a5d2442c69a789769579e": "\n \\sigma_{11} = -p + 2~\\lambda^2~\\cfrac{\\partial W}{\\partial I_1} ~;~~\n \\sigma_{11} = -p + \\cfrac{2}{\\lambda^2}~\\cfrac{\\partial W}{\\partial I_1} ~;~~\n \\sigma_{33} = -p + 2~\\cfrac{\\partial W}{\\partial I_1} ~.\n ", "1370c864eb2f67df67723fa1437f198c": "m\\to 0", "13718bd5f184073cce55400e3c8acc8e": "v_{air} \\,\\!", "1371bdcba6f128e58bca080135f0f683": " \\mathbf{c}_i ", "1371d4f76f50dcbd7710cbf2bc6d3b1e": "{\\tau}_1", "1372369aed4fa7e0b05ac8efac485540": "\\scriptstyle{R_3^3}", "13727862a20b058e29b86f69bb5b4baf": "500truckloads*.10(ECI)=50meatruckloads ", "13728b4491a939a79cbdc12a7141f757": "G(t) = \\sum_{n=0}^N \\pi_n(t) ", "1372a0e75c93f76c8333db2fbd64f7bb": "D_\\mathrm{r}", "1372cce5756a6fb83d8d3f2616cdeb1c": "(x_1,y_1)+\\ldots+(x_n,y_n) = O", "1372eca818fb486c6277ae40b2fa7da4": "\\operatorname{Pr}_Y(x,y) = y", "13730889a73e74c8cab459dcc247aa6c": "\\sum_{n=1}^N x_n", "137321b5e03c14e7c8809c284b484ba4": "R[x]/(x^N-1)", "137332dcf48f1aeb4dec20440b3b10a1": "a,b \\in \\mathbb{Z}_q", "13737d5cb4230ced7d9f3f722336f34c": "F = \\Delta PA", "1373e19ffbe4761848b6b5155cd321a3": "\\Omega_1, \\dots, \\Omega_{n}", "137425aeae8e33d918f4007ec7afe7e3": "s_{k}", "137430895b5795a01ac938af36d78160": "\n\\begin{align}\nL & = 10 \\log_{10} \\frac{x_1^2}{x_2^2} & \\mathrm{dB} \\\\\n & = 10 \\log_{10} {\\left(\\frac{x_1}{x_2}\\right)}^2 & \\mathrm{dB} \\\\\n & = 20 \\log_{10} \\frac{x_1}{x_2} & \\mathrm{dB} \\\\\n & = \\ln \\frac{x_1}{x_2} & \\mathrm{Np}. \\\\\n\\end{align}\n", "1374784edd01ebfc4da0acb387524ddd": "W(\\boldsymbol{F})=\\hat{W}(I_1,I_2,I_3) = \\bar{W}(\\bar{I}_1,\\bar{I}_2,J) = \\tilde{W}(\\lambda_1,\\lambda_2,\\lambda_3)", "1374aa7dbf4849a6383e5c0e4e60e53e": " f(n) = \\sum_{d \\mid n} g(d) ", "1374d1b45ba405e027bd63dcd276b6fe": " x^*", "13752f8569b8fcc49a0bed9f02338545": " \\frac{\\eta^2}{2}", "13754d474526b209002526515675f9a2": "x_1x_2...x_\\ell", "1375836cccc27c1845242446967a079c": "10^r \\equiv 1 \\pmod n", "13758573b562e906f010844c4001f8b1": "B_0 \\subset R^m", "13759f9e38df443bd0155cff248d883f": "\\int_{0}^{\\infty}\\frac{f(t)}{t}\\, dt=\\int_{0}^{\\infty}F(p)\\, dp.", "1375bc8c503cf23cd43dad67cc1e2b8f": " m = k_B = 1", "137620d6eb133dccc06f294763412a5b": "R_{tot}=1/(1/R_1+1/R_2+...+1/R_n)", "137636c30def3f0cdde728451138e0be": " \\lfloor \\log_2(n) \\rfloor = 19 ", "1376485059038fd4e7f3969059bf5c6a": "A-B\\geq 0 ", "1376851cfafa2d01f575745b80c09536": "f^{-1}(y)=\\ln(y)", "137690c4ec2f8ed11ef96b4a882ff7de": "\\phi_i(a v) = a \\phi_i(v)", "137725489a1a2bf3a42dd1c7dc054b03": "C^\\prime = \\frac{2\\pi R}{\\sqrt{1-v^2/c^2}}", "1377e4580b0c29ac513c81832435a18f": "\\textstyle \\sup_{s\\in[0,t]} \\mathbb{E} |M_s| < \\infty ", "1377e4ded193081a4165d3739f0316b8": " v_x ", "13780c8ce9896bf51fbca91eab265e2f": "\\mathrm{crd}\\ 108^\\circ=\\mathrm{crd}(\\angle\\mathrm{ABC})=\\frac{b}{a}=\\frac{1+\\sqrt{5}}{2},", "13782ba3c8f54f06459629a04fa764e0": "V =\\frac{nQ}{M}", "13784c65bea3e22c51e63bc2a76af39d": "\\mu^-", "1378b0ea4f16e22f851bedec7bc2853d": "\\mathbb{Z}/24\\mathbb{Z}", "1378e529acc7dcee18b444de5acc7a8f": " p\\in[\\underline p,\\overline p] ", "1379135365aff8d4271ab4ddccdc3eb1": "ds^2=\\frac{1}{y^2}\\left(dt^2-dy^2-\\sum_idx_i^2\\right),", "13793107b866816ca20ccd668553f8ee": "r_{los}^2 + r_{rg}^2", "1379723a3280a93acdde639dff1c2bfe": "\\mathbf{R}_x = E[\\mathbf{xx}^H] = \\begin{bmatrix}\nR_{xx}(0) & R^*_{xx}(1) & R^*_{xx}(2) & \\cdots & R^*_{xx}(N-1) \\\\\nR_{xx}(1) & R_{xx}(0) & R^*_{xx}(1) & \\cdots & R^*_{xx}(N-2) \\\\\nR_{xx}(2) & R_{xx}(1) & R_{xx}(0) & \\cdots & R^*_{xx}(N-3) \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nR_{xx}(N-1) & R_{xx}(N-2) & R_{xx}(N-3) & \\cdots & R_{xx}(0) \\\\\n\\end{bmatrix}\n", "1379bcc3eac33117fe20475bf7794f77": "U_{ji}", "1379ce43cd5c63a5370d841f49fb19e5": "x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}=\\frac{1\\pm\\sqrt{(-1)^2-4\\cdot 2\\cdot 4}}{2\\cdot 2}=\\frac{1\\pm\\sqrt{-31}}{4}\\,\\!", "1379d37d449fb77abc720e653b2536b6": "RMS_{total} = \\sqrt{{RMS_1^2} + {RMS_2^2} + ... + {RMS_n^2}}", "1379d9368665c4e8a20a437932fbb438": "\\eta=w", "1379e5304064f9b91509ee863b51edf8": "(s + \\vec{v}) (t + \\vec{w}) = (s t - \\vec{v} \\cdot \\vec{w}) + (s \\vec{w} + t \\vec{v} + \\vec{v} \\times \\vec{w})", "137a0dd7db03a19aafa369ca30066012": " C_k = C_i g_{n_1} g_{n_2} \\cdots g_{n_j} ", "137a1de3b187ba130de5cbc87340f310": "\\hat{H} =\\frac{1}{2}\\left[\\hat{p}^2+\\Omega^2(t)\\hat{q}^2\\right].", "137a50086fe42a48a2e6aa6a40acb69a": " y_t = y_0 + \\sum_{j=1}^t \\varepsilon_j", "137a5b787c73bd06df0f88940a46538f": "\\theta (u,\\xi )", "137aaccd9f336df140fc5c56ef994637": "W_C = \\frac{Q_C^2}{2C} = 2\\pi \\alpha W_{LC}. \\ ", "137b4d5df5c1fb3094a4c5d427d238f9": "x_i = \\frac{n_i}{n}", "137c269981b03f3bf744b25af803f5bd": "\\quad W_0=\\frac{\\pi}{2}\\qquad \\text{ and }\\quad W_1=1\\,", "137c48e7106e14d2670ac3c64a67245c": "\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n2 & 4 & 0 & 0 \\\\\n0 & 3 & 3 & 0 \\\\\n0 & 0 & 4 & 3 \\\\\n\\end{pmatrix}.", "137c4c8daac011f2092021bf18f91838": " L = \\lim_{n \\to \\infty} x_n ", "137ca731118c981b9b810850a9f7d762": "p_k,\\ldots,p_n\\ (k A(i)\\}\\right\\vert", "13899053ddf98c757bfe45562e2fce84": "n_1 + n_2+ \\cdots +n_r=n", "13899aa46204b02713a854702142955d": "P_{0}", "13899f65e16e008ce26fa729a050b528": "A_{\\alpha\\beta\\cdots,\\gamma} = \\partial_\\gamma A_{\\alpha\\beta\\cdots} = \\dfrac{\\partial}{\\partial x^\\gamma} A_{\\alpha\\beta\\cdots}", "1389d47d0aa8dc720a497c18ed6481f5": "a \\sin\\left(\\frac{x}{a}\\right)", "1389f24a8338ce719d16d83142d63331": "(x\\pm i0)^\\alpha = x_+^\\alpha + e^{\\pm i\\pi \\alpha}x_-^\\alpha,", "138a2645a4f744c7655fd5a5e6842431": "(X,\\mathcal T)", "138b41111570a898f0329dc1c4e8cb84": "P(t+\\tau) = P(t)P(\\tau)\\ ", "138b5e1b12b3dc93c1b366b1cecf52fb": "-$5.26-3\\cdot$9.99", "138b65ddb20677aed17c4ee7be0bab59": "\\left\\{{4\\atop4/2}\\right\\}", "138b96d357d1dbf023d63302ca4fa236": "\n \\sum_{n=1}^\\infty \\frac{1}{n} = \\prod_{p} \\frac{1}{1-p^{-1}}\n = \\prod_{p} \\left( 1+\\frac{1}{p}+\\frac{1}{p^2}+\\cdots \\right)\n", "138bde2d21ca0ae2352e13c7f4a91a58": "a_r = \\frac{(r + c + \\alpha - 1)(r + c + \\beta - 1)}{(r + c)(r + c + \\gamma - 1)} a_{r - 1}, \\text{ for } r \\geq 1.", "138be8b82adc8730f67c56b92bae0eee": "\\phi^*=-1", "138c6545ca874b5cf3e084cc89a02a63": "\n\\psi(\\mathbf{x},\\tau) = \\mathrm{e}^{K \\tau} \\psi(\\mathbf{x}) \\mathrm{e}^{-K\\tau}\n", "138c8f109e48c641d347f0ca144b0e39": "-U_{A}(\\delta_{B})U_{B}(\\delta_{B})\\leq -U_{A}(\\delta_{A})U_{B}(\\delta_{A})", "138ccf9ccea0f33f6b0c9c6876cb4c16": "\\psi(\\vec r)", "138d127a5421ac7869e068005ccee17e": " \\alpha(t)=\\frac{IR_{P}(t)}{R_S(t)} \\qquad \\qquad (6) ", "138d5f82a5c03aa4eee6d1cd17b696e8": "\\dot{x}=A x", "138d640060375a883fbd74f71590abf4": "E\\supset F", "138d6a6fc53d1b77dc76d881dc22a145": " e^{\\frac{-i\\delta}{\\hbar}H_{k,k+1}}.", "138d9f0cbc8ae0deb9e8348dec2c4c16": "r(x(t),u(t))dt=dR(x(t),u(t))", "138ddb8f654b4ff218add31482d356d6": "Q=\\sqrt{(R^2+a^2)^2-(2 a z)^2}.", "138eede587eb7cb6233a68550ef6eadb": "H=\\sum_i x_i{\\partial\\over\\partial x_i},", "138ef8b49caa515783e0d7e4b3f394dd": "2(\\nabla_X Y,Z)= X\\cdot(Y,Z)+Y\\cdot(X,Z) - Z\\cdot(X,Y) +([X,Y],Z) +([Z,X],Y) + (X,[Z,Y]),", "138f4bf2027bfa7624a45b158b0af615": "M(\\mu,\\sigma)", "138f7dfe3628974a1ca00535b6634cd0": "d = \\gcd(d_1,v_1 + v_2 + h) = \\gcd(x-1,-6x+22) = 1", "138fcbf2237d26bab3c454e45fd87151": "i \\hbar \\frac{\\partial}{\\partial t}\\rho = \\mathcal{H}[\\rho]", "138fee2b86d3c0c4604dc123f34bc454": "\\mathbf{b_{2}}=2 \\pi \\frac{\\mathbf{a_{3}} \\times \\mathbf{a_{1}}}{\\mathbf{a_{2}} \\cdot (\\mathbf{a_{3}} \\times \\mathbf{a_{1}})}", "139005c289391e4dc52df5e595404b5a": "\\Delta(x) = \\mathcal{O}\\left(x^{\\theta+\\epsilon}\\right)", "1390442d61c28d944d7a39a240a26530": "(f,\\hat{O}g) = (\\hat{O}f,g)", "1390d4579c5cc0c926cbe7e66b77d54d": "T=C_1\\cup C_2\\cup \\cdots \\cup C_m", "1390ed3f3f8522eaad1f83b338ddee9c": "\\Delta(a) \\in \\mathcal{C}^\\mathcal{J}", "13912a3ca98677f892f445f5b1300c3d": "\\mu/T", "13913219d68657e24f239b20b865e5e7": "P_c^{(k)}(0) = P_c^{(k+n)}(0)", "13917f94cefdb0d9ff0f84555477bd01": " 2^{2^{\\overset{n}{}}}+1", "139182d52c3c7f28e8ba425ca152c666": "\\frac{dx}{dt} = f(x)", "1391f0d9dfdc1bf756433daef0ea0ce0": "P(A\\ \\mbox{and}\\ B)=g(P(A),P(B|A))", "1392065f3f8c8367cd26d30dbf021df5": "p_{v,w,k}(G)", "13921f3c95c76b3d2f1646c85060d753": " f(q) \\ = f(w + xi + yj + zk) \\ = w f(1) + x f(i) + y f(j) + z f(k) \\ = w - x i - y j - zk \\ = q^*.", "1392370d74da2f451a05f3413430d8c9": " df: T_{x} X \\rightarrow T_{f(x)} Y", "139239d03b317553540c7d23a16223e3": "f(n_i)=\\sum_i (n_i + g_i) \\ln(n_i + g_i) - n_i \\ln(n_i) +\\alpha\\left(N-\\sum n_i\\right)+\\beta\\left(E-\\sum n_i \\varepsilon_i\\right)+K.", "139255cec61f93b2848c497fc367573e": " F_\\theta = a F_A - b F_B = 0. \\,\\!", "1392768521f2c4dd9d791ea425772ac7": "P = \\frac{\\rho A V^3}{2} C_P", "13930b0ba518cd87f3c7aa5294e62619": "\\frac56\\rho", "13937363c58307c2bf9e767cfbc66fde": "\\rho(z_1,z_2)=\\log\\frac{|z_1-\\overline{z_2}|+|z_1-z_2|}{|z_1-\\overline{z_2}|-|z_1-z_2|}", "1393b8b8d829705d728a201b457bfef1": "g_{0es}", "139402e6328f843b622a6bc0069b6942": "\\lambda_{n+1}", "1394092508a282c4515983a71d3cf0da": "\n\\frac{\\lambda}{1+\\lambda} \\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v \\right) = \\chi \\left( C_m \\frac{\\partial v}{\\partial t} + I_\\text{ion} \\right)\n.", "139436d7ba8aa26f22039d940e898a2f": "h{{\\left[ \\frac{\\mu _{v}^{2}}{g{{\\rho }_{v}}\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)k_{v}^{3}} \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{3}\\;}}=0.0020{{\\left[ \\frac{4m}{\\pi {{D}_{v}}{{\\mu }_{v}}} \\right]}^{0.6}}", "139483beaeb74051ce81b3a25e76eca7": "\\mathfrak{gl},", "1394cf45e4685fdf36505e6eb6527bc6": "z\\left(x,y\\right)=\\left(x,y+z\\right)", "1394fcc7a60fcc0ad9769c73b423399f": "f \\in\\sigma", "139580bf570d8ad46ebb7a867d642bd6": "|x-y| \\geq \\bigg||x|-|y|\\bigg|.", "13958edccb8b6efb5b844a25ba92126d": "\\frac{(i\\omega)^2+\\xi^2}{((i\\omega)^2-\\xi^2)^2}", "1395a5b0361707b3c7fb4be218d3690c": " { \\partial \\over { \\partial x^a } } \\equiv \\partial_a \\equiv {}_{,a} \\equiv (\\partial/\\partial ct, \\nabla)", "1395c2c2e805891838cabda81a9ad18d": " {\\rm trig}(M)=(0,2g,g_3) \\,", "13962c1563475279c2fd0dadbc0f9dcc": "\\cos (2 \\sigma_m) = \\cos \\sigma - \\frac{2 \\sin U_1\\sin U_2}{\\cos^2 \\alpha} \\,", "139635b4ef68570a2c388dc2959a05c4": " f(x+) = \\lim_{y \\to x^+} f(y) ", "139650b71b241f871e85a92f3c53e30f": "a \\in \\left\\{0,...,(p-1)(q-1)/4-1\\right\\}", "1396540cbdb53b994df321149be608c6": "\\left\\{\\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3},\\frac{x_1+x_3}{x_2},\\frac{(1+x_2)x_1+x_3}{x_2 x_3} \\right\\},", "13970073048c13fa98f8f25354db4073": "{0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+\\cdots}", "13975e67033638c94eb919a345ebf8f2": "x=918.082", "1397999bd4e9728137aab993fc4f599a": "f: A \\to Q", "13981791e732963ce7efd1774e65e876": "X[f_1 \\dotso f_i f_{i+1} \\dotso f_n] \\to \\alpha Y[f_1 \\dotso f_i] \\beta Z[f_{i+1} \\dotso f_n] \\gamma", "13981c380df46df15370cd0c188cc9f1": "\\operatorname{gr}_m O_{X,x}=\\bigoplus_{i\\geq 0} m^i / m^{i+1}.", "1398253a5cd2be953826b8be16956b58": " {\\mathbf{}}K(t) = P(t)C^\\mathrm T(t)W^{-1}(t) ", "13982f20c005669cfb6c88f7691bf242": "a_0-L", "13983de680535568fb7d4537975b9353": "\\mathbf y=n(\\overline{\\mathbf x}-\\boldsymbol{\\mu})'{\\mathbf \\Sigma}^{-1}(\\overline{\\mathbf x}-\\boldsymbol{\\mathbf\\mu})", "13991f26a97da351e592720ed88a6f3f": "25 \\over 2", "13997636c12724d9f4ce906a479e7b3b": " X_n\\ \\xrightarrow{p}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{d}\\ X,", "1399834dd4c3f5a18ca05f0ec5187593": "\\frac{b}{d}", "1399ab3520ac37ecfe8fffaf77818532": "\n \\frac{\\partial \\eta}{\\partial t}\n + \\frac{\\partial \\Phi}{\\partial x}\\, \\frac{\\partial \\eta}{\\partial x}\n + \\frac{\\partial \\Phi}{\\partial y}\\, \\frac{\\partial \\eta}{\\partial y}\n = \n \\frac{\\partial \\Phi}{\\partial z}\n \\qquad \\text{ at } z=\\eta(x,y,t).\n", "1399afeae9068428fb649b2f58e2d837": "\\|A\\|_F \\le \\|A\\|_{*} \\le \\sqrt{r} \\|A\\|_F", "1399b73092d3c7cc5930fd0d299319ae": "\\boldsymbol\\alpha=(\\alpha_1,\\alpha_2,\\ldots,\\alpha_K)", "1399ee959b9c3e978484e5b38b2de434": "\\hat{X(t|T)} = \\sum^{\\infty}_{i=1} \\hat{X_i}\\Phi_i(t)", "1399f16c0e05287ee03576a9a5a594f2": "Qp_n(x)=np_{n-1}(x). \\,", "139a5f6c5804614df8d2b96c09d599d4": "{\\mathit{He}}_7(x)=x^7-21x^5+105x^3-105x\\,", "139a90c221557c3ed4d82640bb001feb": "1\\!\\!1_{B(k)}", "139aa459357bbc403230f5aeff8c9bd2": " \\hat{\\varepsilon}(\\omega) = \\varepsilon'(\\omega) - i \\varepsilon''(\\omega) ", "139aa6db3d5ffb3c73a79d653af4a26d": "u_8", "139acf60ce8d35cffa79c9e638bda563": " h_{0} = {X_{cross_{1}} \\over 2} \\sqrt{{V_{1}-V_{0} \\over V_{1} + V_{0}}}", "139adac7a4d34459f8200a36670125f4": "{\\text{f}_\\text{P5} \\over \\text{f}_\\text{T}} = 2 ^ {( 7 / 12 )} \\approx 1.49821 \\approx {3 \\over 2}", "139b12acfd62c0c9e58ce3074718e8f2": "I = veN/L", "139b19b7ba394648e575d747ce355c5c": "\n \\overset{\\triangledown}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} - \\boldsymbol{l}\\cdot\\boldsymbol{\\sigma} - \\boldsymbol{\\sigma}\\cdot\\boldsymbol{l}^T\n", "139b2bd3c2a26c61a3ad4f01b039dfae": "f: \\mathbb{N} \\to \\mathbb{N}", "139b31f52f59a6b0baf38be01ce5c286": "C = \\mathrm{d}q/\\mathrm{d}V\\,\\!", "139b8eb65be2ce7670dc5dd97965ea18": "\\dotso", "139bf931df0172d67e975268a0da28cf": "\\theta _0", "139c1c707c03ecdd16ff500bd5d62795": "\\hat \\rho = \\exp\\big(\\tfrac{1}{kT}(A - \\hat H)\\big),", "139c7429b09fd9dcb93be6e1b43445e2": "d_1=\\begin{pmatrix}\n 1&0&0 \\\\ \n1/2 & 1/2&0 \\\\ \n1/2&-1/2&w\n\\end{pmatrix}.", "139cd3b40768e63fcc2477a647912f3e": " \\mathbb{R}_x^n \\times \\mathrm{R}^N_\\xi ", "139ce03bd868c678d141ba0c0694f17e": "\\varphi:\\mathcal{X}\\to\\mathbb{R}", "139ce992f89071078f21c526c73a1129": "\\tilde A_2(R)=2\\pi R", "139d2186190f2751b5744a9c2d2e85c6": "\ndQ = C_{V} dT+P\\,dV.\n", "139d31e416990a77ae77f7707e47c283": "R_n^2(\\xi,-x)=R_n^2(\\xi,x)", "139d5ee757298244ee3fb4a70218cca3": "\\left(\\left(\\frac{b_n}{a_n}\\right)^2\\right)^{N_t} = 10^{N_d}", "139d7a416ec5e0d694e6f8ccb1897598": " \\mathbf{A}_i = \\alpha\\times(\\mathbf{r}_i-\\mathbf{R}) + \\omega\\times\\omega\\times(\\mathbf{r}_i-\\mathbf{R}) + \\mathbf{A}.", "139d8c95b22903818ae3fc6bdbbaf484": "\\arcsec x ", "139da28898a3a8f4a4b540eb4f30197d": "Y=\\beta_{30} +\\beta_{31}X +\\beta_{32}Me + \\varepsilon_3", "139dabbd6d49ca7d4173bfc97e309567": " J_y \\sim \\frac{B_{in}}{\\mu_0\\delta}, ", "139dc70371bc58bd36588eac59273528": "\\Diamond_i", "139de60d60eebe461ce944aed52c5ac1": "\n\\alpha (\\theta )\\,\\, \\equiv \\,\\,\\left[ {\\,1\\,\\,\\, + \\,\\,\\,{1 \\over 4}\\sin ^2 \\left( {{\\theta \\over 2}} \\right)\\,} \\right]^2", "139e4170c21d840b942914ca25ec79e2": "q\\ne 0\\in \\Bbb C", "139e6a0a8ba62b6d91239788708b6c27": "\\Gamma(n+\\tfrac14) = \\Gamma(\\tfrac14) \\frac{(4n-3)!^{(4)}}{4^n}", "139e9bd0bc65b7788031d1f298ade80f": " \\mathcal{F}: L^2_\\mu(G) \\rightarrow L^2_\\nu(\\widehat{G}).", "139f120e78490ad580d52c1c5d7e15cc": "A_{Bq} = nN_A\\frac{\\ln(2)}{t_{1/2}}", "139f1ab475bef4718960f797cd17c360": "D_{f}", "139f93d9d22a9eb7c5378f539d28c52d": "\n s_{n, k}(t) = 2^{1 + n/2} \\int_0^t \\psi_{n, k}(u) \\, d u, \\quad t \\in [0, 1], \\ 0 \\le k < 2^n.", "139f99e3ce496865b4d82434f35be559": "h_a(x) = ", "139fb3f33193989bbc214dade614669d": "\\textstyle G'", "139fc0c23e71138369bd21fe428e217a": "p_k(x_1,\\ldots,x_n)=\\sum\\nolimits_{i=1}^nx_i^k = x_1^k+\\cdots+x_n^k,", "139ff8801186f3cbe1bb867897c42db8": "\\mathrm{verb\\ form} = \\mbox{stem} + \\mbox{thematic vowel} + \\mbox{inflectional suffix}", "13a004c2b20e756d91a85b04eb5e93a4": "(j,k) ", "13a0e3336da790da7f4fe483e4ac4818": "P\\left( y \\right)", "13a13ea0a6972ed94afdfb4256b28b3c": "\\frac x b = \\frac c d", "13a14a36b6b8fe3c5ce9c859bb8e008b": "{}'", "13a1a2181217b0f6f4c471f0e1fd1c1c": "F_2=\\{e\\}\\cup S(a)\\cup S(a^{-1})\\cup S(b)\\cup S(b^{-1})", "13a1d8c76a4ed9746ec064e1ddd05792": "\\sup_{\\alpha} E(G(|X_{\\alpha}|)) < \\infty.", "13a2096381197da1b640b2c00aee9a1b": "f = \\chi_S - \\frac{1}{2}", "13a248d66e721f2b7dbd3caa0161c593": "E_{0}", "13a25ea40c1a73a51817a2e290ad2661": " (a(x_0 - m) + b(y_0 - n))^2 = a^2(x_0 - m)^2 + 2ab(y_0 -n)(x_0 - m) + b^2(y_0 - n)^2 = (a^2 + b^2)((x_0 - m)^2 + (y_0 - n)^2)", "13a2b22efb5eef810e052d13ccf79e18": "x < r", "13a2e790b07a4ff19581d2d1f0e9d71b": "S(q)-1=\\frac{4\\pi\\rho}{ q} \\int_{0}^{\\infty} [g(r)-1]\\sin{(qr)}{d}r", "13a314966acceba14c8e20092d63b8a5": "P = \\exp\\left(20.386-\\frac{5132\\,\\mathrm{K}}{T}\\right)\\,\\mathrm{mmHg}", "13a356c465bf407cd1433e415ff13c11": "\\tilde{4}\\cdot m", "13a3c5bd3187935a20bd65f0cef13e41": "\n\\langle\\mid r \\mid\\rangle \\sim \\langle\\mid r \\mid\\rangle_\\text{free}/n.\t\t\n", "13a498a2e0247f451f6df67ebe7d7131": " n \\times k_2 ", "13a4c31992503616b954f7ca39e65c05": "w\\in V", "13a5d284b4de9e0eca35385d90114d27": " k = \\frac{n\\pi}{L} ", "13a6545a8be7b209501aec5ff4734175": "\n M^{\\mathrm{core}}_{xx} := \\int_{-h}^{h} z~\\sigma^{\\mathrm{core}}_{xx}~\\mathrm{d}z = 0\n ", "13a6b3c899e400191649396c66afec82": "H^{\\sigma\\sigma_j}(\\vec{x},\\vec{x}') = \n\\begin{cases} \n h^{\\sigma\\sigma_j} & \\left | \\vec{x}-\\vec{x}' \\right | \\le c \\\\\n 0 & \\left | \\vec{x}-\\vec{x}' \\right | > c \\\\\n\\end{cases} ", "13a6cd3741dd1e205768fa3c190c623d": "h_{\\text{FE}}", "13a712e2e5e37c4591599e6462a3c539": " P(|| X - \\mu || \\ge k || \\sigma ||) \\le \\frac{ 1 } { k^2 }. ", "13a7980c0175a147ec579c1de932ff7f": " K =\\frac{\\det \\begin{vmatrix} -\\frac{1}{2}E_{vv} + F_{uv} - \\frac{1}{2}G_{uu} & \\frac{1}{2}E_u & F_u-\\frac{1}{2}E_v\\\\F_v-\\frac{1}{2}G_u & E & F\\\\\\frac{1}{2}G_v & F & G \\end{vmatrix}- \\det \\begin{vmatrix} 0 & \\frac{1}{2}E_v & \\frac{1}{2}G_u\\\\\\frac{1}{2}E_v & E & F\\\\\\frac{1}{2}G_u & F & G \\end{vmatrix}}{(EG-F^2)^2} ", "13a7d841fc898c8c02c13f038fe92740": "W=kTN \\ln(V_2/V_1)\\,\\!", "13a806f9b35a643e1fcc46e7004644ca": "\\frac {F}{\\rho AV^2} = f(R_n, \\alpha)", "13a819238aabe85e3290a2ff42fabe4c": "\\frac {d^2r} {d\\theta^2} \\cdot {\\dot {\\theta}}^2 + \\frac {dr} {d\\theta} \\cdot \\ddot {\\theta} - r {\\dot{\\theta}}^2 = - \\frac {\\mu} {r^2}", "13a81e9b5ec3c7289685fe7ff36093be": "[0,1]\\subseteq\\biguplus_k V_k\\subseteq[-1,2]", "13a82b51e585f7d1b00306f43451deda": " SK = \\frac{ F( 1 - \\alpha ) + F( \\alpha ) - 2 Q_2 }{ Q_3 - Q_1 } ", "13a8681d826f0a8ce514209bb3296296": "x^i e^{-a |x|^2} \\in S(\\mathbf{R}^n).", "13a88c5330b738ed79fd6108e89529ca": "K \\wedge P", "13a8ba603816294cce5461895078cfe0": "{\\nabla}^2 \\rightarrow {\\delta_z}^2", "13a8d1c9ab3fdf40db247d818a3923a9": "\\{x|x1", "13b013f5414f688a6d92858b772bd107": "\\stackrel{\\mathbf {E}_{\\bot}}{}", "13b056184eb7ad4941965732413b07b3": "p(x) = x^3+x^2-2x-1", "13b085cc51d035eb76dd0bfa554c839a": "\\dot{\\gamma}_e", "13b09f6e00a4f6ee5820a7e5dd4041e3": " (\\theta^{*}_{(\\alpha)};\\theta^{*}_{(1-\\alpha)})", "13b17d219f5366cd59f9559444470bba": "I_D = \\frac{\\mu_n C_{ox}}{2}\\frac{W}{L}(V_{GS}-V_{th})^2 \\left(1+\\lambda (V_{DS}-V_{DSsat})\\right).", "13b220906096a833240c35ac61bea2f9": "\\Delta v = v_e \\ln \\frac {m_0} {m_1}", "13b2605642a0828f17eaf6b14cc39115": " \\mathrm{Distance}( b^\\mathrm{ideal}, b^{k}_\\mathrm{Inverse~iteration})=O \\left( \\left| \\frac{\\mu -\\lambda_{\\mathrm{closest~ to~ }\\mu} }{\\mu - \\lambda_{\\mathrm{second~ closest~ to~} \\mu} } \\right|^k \\right). ", "13b2617da4326976297e3fda1be47c60": "S(t|\\theta)=1-F(t|\\theta)", "13b2a29edfa63cb4f376f9f462203cff": "\\lnot a\\Rightarrow b", "13b3083b8f2fb1c7b7167c7eb4dac524": "\\scriptstyle H=h(X)", "13b31244048bb600c8df889e693462e8": "\\begin{align}\n P &= 1 \\text{mW} \\cdot 10^{\\frac{x}{10}}\\\\\n P &= 1 \\text{W} \\cdot 10^{ \\frac{ x-30 }{10} }\n\\end{align}", "13b382773db3b0030a79c8c8879acedf": "\\lambda' - \\lambda = \\frac{h}{m_e c}(1-\\cos{\\theta}),", "13b3e1dc8890efe0bca8e279e98536c4": "C^\\prime(a,q,\\xi) = \\frac{\\partial C(a,q,\\xi)}{\\partial \\xi}", "13b42d63d4ed41298de411106db9997b": "\\lim_{x \\to 0} x \\ln(x) = 0", "13b44a5310ad050fd3659fe683480256": " 4p^3+27q^2\\leq 0\\,,", "13b4a21e630d48396eb5b9858a91b508": "\\mathcal{M} = NI", "13b4ac2a4d9f1629bc7d9dbbbdac7953": "f_0\\,", "13b4b01f1960159186f3db3a9635dd81": "\\vec k\\perp\\vec B_0", "13b4dfbf05dcdff8b708363ebea2b131": "\\lim_{i \\to + \\infty} \\lambda_{i} = 0.", "13b4fedf6dd767b15eea81b1a8fdba53": "\\mathrm{SINR}(x) {{=}} \\frac{P}{I+N} ", "13b4ff0f3eb0a5e8c883dea9b78eaa14": " \\left (\\ddot{\\bold{r}} \\right )", "13b51b66631a6b9fd9f3ba5545c49d3a": "d = \\partial + \\bar\\partial", "13b52edc722dfb4460cd039e9fae414e": "1/(1-z)", "13b56608a775d37fe34ef6935a7d89e5": "\\gamma_f(\\theta)) ", "13b5dfc6528f8fd69cfe012ac78bff92": "g^{th}", "13b5e00bc189f331925dcde4c4a240f0": " {\\partial u\\over\\partial x}+{\\partial v\\over\\partial y}=0 ", "13b6f1165f57dec6285dbe3b25ea5fae": "S=kJ^z", "13b6fc21d83d6fcaf39403ccef6ff9d6": "L(f,\\hat f)=\\|f-\\hat f\\|_2^2\\,,", "13b746f1ef5257fb901be21456ef5001": " V_{Atom} ", "13b7b7bb33bf08524eda17f7e13caf9b": "\\scriptstyle - \\frac{1}{7}", "13b7c88f95f9fbbfa455dd466f19d3a1": "V_{oc}", "13b7df217eb78c8c450d688982a5c153": "|\\sigma|", "13b80d2c9594a5f63f41835ef889b7e3": "\\mathbb R\\supset[,]\\ni t\\to x(t)\\in X", "13b8d00f2d98049f91f702cd5dd5bc82": " \\mathbf{\\gamma_1}:I_1 \\to R^n", "13b93faeba2e7c51d1f0a7fe8faa0e3d": " f(x)/f(-x) \\equiv e^{-x} ", "13b9926ff965d20a727bfd2bf5e88e50": "\\mathbb{P}(|X-\\mathbb{E}(X)| \\geq a) \\leq \\frac{\\textrm{Var}(X)}{a^2},", "13b998cf24f1b652e64c68bcd1c2a02e": " \\operatorname{de-let}[\\lambda q.f\\ (q\\ q)] ", "13b9a1a680edce72d6046544b536a690": "u(x,y,z)", "13b9fe7dd1b095da9d0ae037eca47257": " x'_{3} ", "13ba01ba8b7d29e25ee0c6d65ce50b11": "x\\ \\dot{=}\\ 0 \\rightarrow x\\ \\dot{=}\\ 0", "13ba37f62f0b6fe78b1f1fbaff6f951a": "\\mathcal{L}\\left(f\\right)=S_{\\phi}(f)/2", "13ba92980b8067fe29e825bc21b479b9": "x\\le K", "13baac17377b3271ca694e37c98e866e": "\\operatorname{cons} \\equiv \\operatorname{pair} ", "13baad0e3c9fcebffd17a64142690f8a": "\\delta\\in\\Delta", "13baaf3d45bde9201138e629fb4697dc": "R(S)=\\frac{1-\\sum_{i \\in S}{p_i}}{1-\\sum_{i \\in S } \\frac{\\beta_i}{D}}", "13bafdd735f14ed4c4203d8f218c39bc": "\\underline{P}(Cl_t^{\\geq})", "13bb21af2a2261b726e33d0606ec1194": "\\psi(\\vec{x})\\,\\!", "13bb4390518aafa7e2c9fbe68e835962": " \\rho_l = \\mathrm{density \\ of \\ liquid} ", "13bb511b7b20cd22ef71b7c48dc31ba1": "G_{11} = \\frac{- r \\nu'(r) + e^{\\lambda(r)} - 1}{r^2} \\;", "13bbe8151b482046de4ec544a7d63fad": "A \\to \\widehat{A}", "13bc894e6f0a405b62a37563e93fb81b": "\\sqrt{Z}", "13bcac9a6adf62375319ebc69d857e72": "L_\\in", "13bce2e6d2961bc3f71af9fb46195e40": "\\sum r_i.", "13bcf04bfe060069699e0b4b92a53f0f": "\\sin 75^\\circ = \\cos 15^\\circ = \\dfrac{\\sqrt6 + \\sqrt2}{4}.\\,\\!", "13bcf14b62b27c7c459cc7dcd9318a10": "id: X \\to X", "13bcfcd70046d935354b1465d626c583": "M_n(i,j)", "13bd0bf2ac081b9576ebc5d020a4a006": "\\scriptstyle \\dot m_{01} \\,", "13bd1be2356eca4452e4648d4acb4195": " \\beta\\ = \\arcsin\\left(\\frac {b}{c} \\right)\\,", "13bd26cf0060d30f0e487f6a1281d4ab": "a_{ij} = \\left\\{ \\begin{matrix} 1 & \\mathrm{if} ~ v_i \\in e_j \\\\ 0 & \\mathrm{otherwise}. \\end{matrix} \\right.", "13bdee301af9bf655137757ae2926528": "i=i_0 \\frac {nF} {RT} \\Delta E", "13bdffe841517fa58bc5bdf3002ef4b7": "\\mathrm{ob}(\\mathrm{Gr})", "13be0c14eaff4ce2fa73860c7e5931aa": "(n,k)", "13bec184fc576edc414c745fb59404f9": "\\tbinom mr.", "13bef16fb8a1fc44a5e0f6e8cc65becb": "1/2(n-1)", "13bf37566e0f1da297e16fb11ece90a9": "b_i = a_i - 2e_i", "13bf6c3f74e23c1279b9d598699e9cfb": " V_{CB} = V_{CE} - V_{BE}", "13bf800328e595b0e3eeae1957d214ed": "^n\\mathbf{P}_r", "13bfbc7863ef1d8de184392afd3a0baf": "\\displaystyle V_{m}=\\sum\\limits_{n=1}^{K}L_{m,n}\\frac{di_{n}}{dt}.", "13bfdb9ea68fb8f938b2e2c2a2319449": "D_{0} < 0", "13bff50d081ada63b49056f1bf18df84": "p(x_k|y_0,\\ldots,y_k)", "13c01cd3e913665018170fffd3868e66": "\\dfrac{PR}{RB'} = \\dfrac{SQ}{B'S},", "13c0256b854bfa3d7466bf8e465c8554": "(R_g)_{S,[m]}", "13c04abc55b3bf628f2176f9fe41bf85": "P=\\{(a,b)|a,b\\in R\\}\\cup \\{(a)|a \\in R \\}\\cup \\{(\\infty)\\}", "13c07c8f2638661b4745a601cd650c94": " x{d\\over dx}\\psi = x\\psi' \\neq {d\\over dx}x\\psi = \\psi + x\\psi' ", "13c0d15703c030e9aba65821e7cc8c35": "m \\neq 0,1,3", "13c0eba9a3f4908a5af14baad48048b4": " x_i := x(i) \\, ", "13c0f073c5afe7200ac95ca6bbc3bd16": "f_j^{(1/p)} = \\sum_\\beta f_{j\\beta}^{1/p} X^\\beta,", "13c110c72a21001815d1b9ac6d26c69a": "\\ge 0", "13c1431b6249f455f7e9641bcae2470a": "SU(2)_R \\approx S^3", "13c15e19e4ded2841ceba7ced1ce5e42": "\\sin(y) = x \\ \\Leftrightarrow\\ y = (-1)^k\\arcsin(x) + k\\pi", "13c1bffe6821c0a73571a98470ee4338": "\\mathfrak{g}_-", "13c228abf293efad438a0962b1b47859": "f^{-4}", "13c25fb369f84cc7bd181afc909d3b04": "B(x)=B_0\\exp\\left(-\\frac{x}{\\lambda_L}\\right),", "13c2be2d7eed414bc7215717ec722225": "D(\\alpha\\wedge\\beta) = D(\\alpha)\\wedge\\beta + (-1)^{\\ell\\deg(\\alpha)}\\alpha\\wedge D(\\beta).", "13c2db86d73119fa9729125bb82065fd": "i(t) = I e^{-\\frac{R}{L}t}", "13c37739c4d4334a29f8f209a86e01be": "\n L_{x} u + L_{y} u + N u = \\rho(x, y) \\qquad (2) \n", "13c41e19c4c03b414056ea858fbf6a56": "7+5\\sqrt{2}=14.07106\\ldots", "13c5392d8cff46918314ce12261eda68": "\\breve{\\rho} = r \\left[1 + \\frac{1 - r^2}{2\\left(n - 1\\right)}\\right]", "13c581c0c5cfdde1ab31cb97b88f198d": "C_M", "13c5a27dff834d87ddd52aed154e4a14": " A, B \\in {\\mathbb R}^{m \\times n} ", "13c5cd37b56d25c57f12863cfa8b7fa7": "a \\ge b \\iff a = \\operatorname{lcm}(a,b),\\;", "13c61635b1317c01496fe4efef0662f7": "F_n = 2 + \\prod_{i = 0}^{n - 1} F_i.", "13c69d77429db181b5cfbabe0e49072c": "E\\left(t-D_F/c\\right) + E\\left(t-D_V\\left(t\\right)/c\\right)", "13c6cafd953eab410e737cb871dfe891": "\n p(x|\\alpha, \\beta, \\theta)=\\begin{cases}\n\\frac{\\theta \\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \n\n\\frac{(x - a)^{\\alpha-1} (b-x)^{\\beta - 1} }{(b - a)^{\\alpha + \\beta + 1}}\n\n + \\frac{1 - \\theta}{b-a} \n\n & \\mathrm{for}\\ a \\le x \\le b, \\\\[8pt]\n 0 & \\mathrm{for}\\ xb\n \\end{cases} \n", "13c6fc34f3023e69ae5eb5458fa132c8": "(x_j-x_i)", "13c7884743a6887fd6d936627abd680f": "R_X+R_Y\\geq H(X,Y). \\, ", "13c79f61af22d8d64c1cc951e0c785bb": "y\\subseteq d(R)\\,", "13c7a6df336f87699953af24ffd581c8": "A = \\frac{A+A^*}{2} + i\\frac{A-A^*}{2i}", "13c7c3406af6259fef0d1b424749098a": "V_{out} = V_2 - V_G = V_2 - \\frac{R_G}{R_G+R_1}V_1.\\,", "13c84c3fcb77ff93263e5c37c22e9417": "\n\\begin{align}\n\\bar y &=\\frac{1}{T}\\left(\\int_0^{DT}y_{max}\\,dt+\\int_{DT}^T y_{min}\\,dt\\right)\\\\\n\n&= \\frac{D\\cdot T\\cdot y_{max}+ T\\left(1-D\\right)y_{min}}{T}\\\\\n\n&= D\\cdot y_{max}+ \\left(1-D\\right)y_{min}.\n\\end{align}\n", "13c8563ec9ac2e7f9677a569a02c801c": " = \\frac {3} {16^0 \\cdot 1} + \\frac {6} {16^1 \\cdot 3} + \\frac {18} {16^2 \\cdot 5} + \\frac {60} {16^3 \\cdot 7} + \\cdots\\!\n= \\sum_{n=0}^\\infty \\frac {3 \\cdot \\binom {2n} n} {16^n (2n+1)}\n ", "13c88539eb11ea3612be36e0b555b41c": "\\delta_0 > 0", "13c8bbf43b8d850d687d92084164ca6e": "\\tilde f\\colon X\\times [0,1]\\to E", "13c95bd779f41f236b248d209461863d": "{\\delta}_2=\\sqrt{{\\mu}L\\over {\\rho}V}\\,\\!", "13c966d1a315e3f8cfabd6bf30fce881": " (\\frown\\mu) : H^k(C) \\to H_{n-k}(C) ", "13c98e4297d8cd8c4cc4924a0c1aeebc": "g_i / f_i", "13c9bacbd51596eb2e784e95cccda826": "dx = (R+\\chi )\\left ( 1-cos\\Theta \\right )+\\left ( D_{ss} +D_{sp}\\right )U_{x}", "13c9c2059d9486d624292b432f0e8d0e": "\\operatorname{div} \\mathbf{F} = \\nabla \\cdot \\mathbf{F} = \\frac{\\partial F_1}{\\partial x} + \\frac{\\partial F_2}{\\partial y}+\\frac{\\partial F_3}{\\partial z},", "13c9e59705dceb6525ac40e456c4d6cb": "\\frac{0.088\\ \\mathrm{N}}{(5.3\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=0.0017", "13c9fcd74294315a00a0b9fa9d0e169a": "\\|u\\|_E=(u|u)^\\frac{1}{2}_E \\, ", "13ca99b2c58112befc9682db04a97644": "\\bold{r}\\rightarrow \\bold{r} + \\bold{v}t", "13caa4a3bbb94482cc406fdbfcb150ec": "n = r\\times s^{-1}\\pmod{m}", "13cade3cec323560140fe7e7d15babb6": "M = L/\\Delta x", "13caed825f61241d1f0f4ba0433df6c7": " \\theta (L)= 1 + \\sum_{i=1}^q \\theta_i L^i.\\,", "13cb21eeaa1950c5923069e52e9b557e": " g_n = O\\left((\\ln p_n)^2\\right). ", "13cb39597596b588fc8abe4809a88fb3": "(0,1/n,0)", "13cb3993ab6dbfa2363407b76a546d3f": " f(x_1, \\ldots, x_n, \\underbrace{0, \\dots, 0}) = (y_1, \\ldots, y_n, \\underbrace{0, \\ldots , 0})", "13cb8b963522a1dd94ac884ee6be8921": "b=0.1", "13cc0cb81b499dff66d4f0ddd68a2061": "\\tau_{V,W}\\;", "13cc41973e39a7c9f16d908e993272a7": "d(S_1,S_2)=0", "13cc462f4a9b73176bc86f7bdec46744": "[P_i,P_j]=0 \\,\\!", "13cc48d0937dde00e67a6fd4c71bef29": "m:\\mathbb{Z}_p\\rightarrow\\mathbb{Z}_p/p\\mathbb{Z}_p\\cong\\mathbb{F}_p", "13cc4f9d7c953594735c96257da721a6": " \\mu_i ", "13cc6e45fbff19d28450bc66f843f0fe": "\\mbox{linking number}\\,=\\,n_1-n_4\\,=\\,n_2-n_3.", "13cc906a4e683aa486dd36cc891784b2": "\\frac{\\partial v_i}{\\partial x_i} - \\frac{v_i\\, v_j}{a^2} \\frac{\\partial v_i}{\\partial x_j} = 0.", "13cd537318275d481f3fe5fce0df8c4c": "S \\subset V", "13cd6c45ed2ddd2f920cecd0878fb4c2": "C_i\\,", "13cd999fb1ceb40d0d23a4305b9b3d69": "(x - a)^{n} \\equiv (x^{n} - a) \\pmod{(n,x^{r}-1)} \\qquad (2)", "13cdf85ed9179714e663b835a1f9010d": " R_{22} = R_{11} = -\\Delta p.", "13ce1bf9c56de115da720336da04aaae": "\n\\begin{align}\n \\tau_2 & = \\frac {(C_C C_L +C_L C_i+C_i C_C)(R_A//R_i) (R_O//R_L) } {(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)} \\\\\n & \\approx \\frac {C_C C_L +C_L C_i+C_i C_C} {C_M} (R_O//R_L)\\ , \\\\\n\\end{align}\n", "13ce391ffb235b6414114486bbff161f": " 1^\\dagger = 1 ", "13ce48429d23820700301cea0dc74821": " 0 0\\,", "13dcdf1294abd8124f5076faa766fd87": "H=\\,", "13dd50dcd0266c57673a4897f439ce2b": "{\\tilde{B}}_3", "13dd89e569ae31da9a5b4bb8089fe6a8": " D_r", "13ddb885bc1daf16b00d254c350777d4": "x \\sim p(x|\\theta)", "13ddf7a74e9f0aea649eef875366d524": "x^2=ax+b", "13de3ac790a45f1c0d7de7002d985e75": "\\sqrt{\\frac{n e^{2}}{m\\epsilon_0}} =", "13de676e2e8099271de972b5df1970c9": "t_{p-r}=1.5 [s], \\mu=0.7", "13de6980b7616933c24f210f3bd19b6e": "{L \\over D}={{\\Delta s} \\over {\\Delta h}}={v_{\\text{forward}} \\over v_{\\text{down}}}", "13de8df1c9a03f72d57fb7395a13171b": "D_{\\max}", "13ded1c2fd307ba2bbf16e13259999d3": "i : H \\to E", "13df0d3569f262da1f9f4dd32ea612bf": "\n\\mathrm{CR}(x_1,\\dots,x_N) = \\sum_{i_1,\\dots,i_N=-1}^2 f_{i_1\\dots i_N} \\prod_{j=1}^N b_{i_j}(x_j)\n", "13df5b3e4f1ccb9953d13f29fa7bd57d": "\\left(\\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "13df6c5073effc4e7570c7a98fbb7215": "F[r]= -\\frac{r^2}{r'}", "13df9d2a5f69fc01626224bacd519663": " \\alpha , \\ \\ K( \\alpha+1 ) = K ( \\alpha ); ", "13e049f13df403a560f546f07d83514e": "-j \\sqrt \\frac{6}{25}", "13e04d93897c4e0a1bd3710a9a5d0fff": "\n\\Phi(z,s,a)=z^{-a}\\Gamma(1-s)\\sum_{k=-\\infty}^\\infty\n[2k\\pi i-\\log(z)]^{s-1}e^{2k\\pi ai}\n", "13e05336a7e1dd445acd37d8828a265c": "\\epsilon_{\\alpha\\beta\\gamma\\delta} ", "13e06dafb3c3e46440a6610a5df8716d": "m\\leq n\\log b/\\log a+1", "13e07167e2881097861b6091edb70ef6": "t\\left\\{r,{p\\atop q}\\right\\}", "13e076c3da5284fd2ce22a748343c723": "\\boldsymbol{\\mathsf{E}}.", "13e095007c180894278b223b29c42aa0": "\n M_{yy}\\Bigr|_{y=-b/2} = M_{yy}\\Bigr|_{y=b/2}\n", "13e09863e41108f3df357d5b22abdaf5": "v=\\infty", "13e0ca60af320b20ee8888980330fba8": "\\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}", "13e0e9033dd0370e857ab9a9dcd316ad": " [\\lambda]^\\omega ", "13e1203b77a69ca578b9771243e0878d": " x,y ", "13e13c46b4e543e3f349f6b1655edfbe": "\\mathbf{J_1}", "13e141066758b3e71c890019247bc3f3": "\nZ_\\mathrm{in}=Z_L=Z_0 \\,\n", "13e15946bead62b759bb7390782bbdc1": " \\Phi = H J - J H^t \\, ", "13e16d400be638b58fc9139c9c2d12b4": "Md = 0.80 log10(T)^2 + 1.7 log10 (T) - 0.87", "13e17901066f338d9f6fb1196592b4a2": "w(2,2)", "13e19ef4e4201148f8d3b66b69ed6a80": "c_p ", "13e21973282a707ec1f05cc88dcda2ed": "\\phi\\left(x\\right) = \\sum^\\infty_{n=0} \\lambda^n \\phi_n \\left(x\\right)", "13e22d738a6397235374190f6c2a2ea4": "x+p=\\sqrt{(x-p)^2+y^2}. ", "13e251c3e11c15bf285e0bd12a1b701d": "\\left(\\frac{\\partial S}{\\partial E}\\right)_{x} = \\frac{1}{T}\\,", "13e26b7a27c61b9c731e34a12458f944": "{I_1}", "13e2e028dbe3df49c5d9c6571eaa9687": " W = \\int_{V_1}^{V_2}P_1 \\left(\\frac{V_1}{V} \\right)^\\gamma\\, dV ", "13e355e1f421a234afc7fd4486449465": "\nx L_n^{(k) \\prime\\prime}(x) + (k+1-x)L_n^{(k)\\prime}(x) + (n-k) L_n^{(k)}(x)=0,\\,\n", "13e3784aa841558352b6afbc82efbc88": "\\mathcal{S} = \\int_{t_1}^{t_2} L\\,\\mathrm{d}t.", "13e386ec55fefc5ed41079850023dd15": "\\log_a \\lambda", "13e3a9c7c384fa9920781bc3be101b5f": "\\sigma_{ic}, i=1,2,3", "13e3c08c5a53ad234e9848d4a5df09c6": "S \\to C_K \\to S^1", "13e40115f4f5cf791061425ad093ab2d": "T^{\\bar k}p", "13e41043429047515b1a29d607a78c8b": "\\mathbf{n}_i = \\mathbf{R} \\mathbf{N}_i. \\,\\!", "13e44574cd9c6767e3de212754979d99": " \\sqrt{4\\pi\\varepsilon_0}\\left(\\mathbf{E}, \\varphi\\right) ", "13e445f9fc74ad0d5cfde729b163c384": "\\gamma (n) = \\prod_{p \\mid n} p", "13e4811f4614d10e5f46131cdd816539": "t \\mapsto (t^2,t^3)", "13e490055a59bf24eb6f40b101960256": "D_i(\\theta)", "13e490990e435667da0fc7c35e68b700": "T\\colon Q\\times \\Sigma \\to Q.", "13e52c0e33c8d08cae6b94417edfb876": "\\displaystyle T(f)=\\int_{\\mathfrak{a}_+^*} \\tilde{f}(\\lambda) |c(\\lambda)|^{-2} \\, d\\lambda", "13e538ba214c1c43218df090e4c2ab7a": "\nR=|\\langle z \\rangle| = e^{-\\sigma^2/2}\n", "13e54a7f943e30f3e5514db13a79323a": "A(z) = 1 + B(z) + B(z)^{2} + B(z)^{3} + \\cdots = \\frac{1}{1 - B(z)}.", "13e5832f5571902e48f6615c9aa80eb0": "[HG]", "13e6345a299bf399fff856aa3e3f6606": "y'(t) = -A\\, y+ \\mathcal{N}(y), \\qquad\\qquad\\qquad (7)", "13e67957a08cec4bbc4721a87204f97e": "\\Delta_r G = \\Delta_r G^\\circ + R T \\ln Q_r \\,", "13e6aa219798b128c6b99e97bc0da381": "\\mathbf{v} = v^r(r) \\mathbf{Y}_{10} + v^{(1)}(r) \\mathbf{\\Psi}_{10}", "13e6bc6b05b5d9e7e443467900514984": "S_1,S_2,S_3,S_4\\,", "13e6de38f569e23c36495c99ad7fbff9": "\nE[J,z;\\epsilon] = C \\int d \\vec x (I(\\vec x) - J(\\vec x))^2 +\n A \\int d \\vec x z(\\vec x) |\\vec \\nabla J(\\vec x)|^2 + B \\int d \\vec x\n\\{ \\epsilon |\\vec \\nabla \\phi(\\vec x)|^2 + \\epsilon ^{-1} \\phi ^2(z(\\vec\nx))\\}\n", "13e6f378f1e8454839eaf17f4eb1bc55": "\\begin{bmatrix}\\cos(\\theta)&\\sin(\\theta)&0\\\\\n-\\sin(\\theta)& \\cos(\\theta)& 0\\\\\n0& 0& 1\\end{bmatrix}\n", "13e6fb26737d787325e07ef819c94733": "W^* = \\cup_{\\gamma \\in \\Gamma} \\;\\; \\gamma W.", "13e70e36bbdf697429dc57ca79af8d53": "(A,B) \\mapsto Tr(B^{1-p}A^p)", "13e73281966ecd9181d9b8301e044ec5": "\\pi_{i+}=\\sum_{j=1}^c\\pi_{ij}", "13e75326d8bd7b820616c6fd3c1ab3b7": "\\beta>2", "13e75b2d5a262d8e30d67eaa06059361": "[0,0.5]", "13e78439a8c71a3c143a5b3622bfbd9c": "\\mathit{w}_{GC}(\\mathit{q})", "13e7b576b988edc4fae49cac9c78c042": "r_{c}\\le R(q,u)", "13e7c5d053d41b7be434b7cf382327bd": "M w = \\langle w, v_1 \\rangle v_1 + T w = ( \\langle w, v_1 \\rangle + \\alpha ) v_1.", "13e7cb754b1da09c09bebf265b0bddfa": "\\mathbf{d}_j=\\nabla X_j + (X_j-Y_j)\\nabla (\\ln P) + \\mathbf{g}_j\\, ;", "13e81abc6e85e7f9d2f9660381161ad4": "\\mathbf{D} = \\mathbf{E}+4\\pi\\mathbf{P}", "13e8deab1d6ba961f20f865cf76b5d2c": "\\Phi(p) = (\\pi(p), \\varphi(p))", "13e901580cf920b47e6d047b798cce24": "\\sum_{k=0}^\\infty {{\\alpha+k-1} \\choose k} z^k = \\frac{1}{(1-z)^\\alpha}, |z|<1", "13e9553675a4ba9e3a40324a422d8e69": "\\bar{\\psi} \\ \\stackrel{\\mathrm{def}}{=}\\ \\psi^{\\dagger} \\gamma^{0}", "13e9669842c4613654cd545f2fb325ab": "T=C_F\\int [n(\\vec{r})]^{5/3}\\ d^3r \\ .", "13e97b6aed3842977e74d66dee531499": "\\scriptstyle \\boldsymbol{e}_{i, \\text{rec,ECEF}} \\;=\\; -\\frac{\\partial r(\\boldsymbol{r}_i,\\, \\boldsymbol{r}_{\\text{rec}})}{\\partial \\boldsymbol{r}_{\\text{rec,ECEF}}} ", "13e989548e45c4ec185cd72cc621f1e8": "c_B(0,b)=-\\frac{1}{2b}\\mathrm{coth}\\frac{\\beta b}{2}", "13e9908316bc305c9c6f70118da244ab": "M^{(i)}", "13e9b56294ca43dc5ce78517d8b6ede8": "\\prod_{i=1}^N(m_i-1)=M\\prod_{i=1}^N\\left(1-\\frac{1}{m_i}\\right)", "13ea8e9ce4ea2782326060f9dad03d2d": " \\mathrm{FWHM} = 2\\sqrt{2 \\ln 2 } \\; \\sigma \\approx 2.355 \\; \\sigma.", "13eabfc6c56a703fbfa4a181062b8819": "\\ d=\\frac{1}{2}gt^2 ", "13ead68d11118032f72a8f816e85e31c": "\\scriptstyle\\square^2 ,", "13eb0c37628fb783dd162e4739636211": "D(Rf_{\\ast}(\\mathcal{F})) \\cong Rf_!D(\\mathcal{F})", "13eb19f99cede15b8a0d52a7759a6189": "\\frac{dx}{dt}=x-x^2", "13eb38a5c3636f9f57506a3d1e69efec": "g_N = 0", "13ebc40f0650cc9846452a66b3b535c2": "X\\cdot b_k(X)\\mod f(X)=X\\cdot b_k(X)-f(X)", "13ebca8a5244c089e4849bae92e46cee": " \\sum_{j=0}^n (-1)^j\\tbinom n j P(n-j) = n!a_n", "13ebcc7113a6d4928b435f7227f33368": "\\left(traces\\left(P\\right), failures\\left(P\\right)\\right)", "13ebdd01b4ceebb2b033052aa19db14c": "Z_\\alpha = [Z_0:Z_1:\\ldots:Z_n]", "13ec0d6e556beae382494d2a95f78d63": "F= -k X\\,", "13ecb2c067a3c38537f9f283786e69f2": "\\delta=\\sqrt{{2\\rho }\\over{\\omega\\mu_r\\mu_0}}", "13eda1f72c27b90988812d98c8675b39": "\\mbox{DG} =\\frac{a-b}{a}", "13ede354a74f816880b301bce4aea00e": "\\scriptstyle \\phi '", "13ee4a83bf2887706221c37f10d413e5": "\\frac{d[W]}{dt} = -a_1[W][E_1] + d_1[WE_1] + k_2[W'E_2]", "13ee9b2356ceba882c513d31911ab4e9": " r_{1} = \\frac{m_{2}a}{(m_{1}+m_{2})} ", "13ef78a9ffe5c721e87bf22f06afeda8": "\\beta = 0.35", "13ef984bab7f31b28f716dec76e994a1": "A_n(z) = \\sum_{k=0}^n a_k z^k.", "13ef99fef5af39da92ce9ecc1dc9a80d": "\nV(x,T) = D(x),\\,\n", "13efa50c9fc36219f1237a81ed3c1d4b": "\\alpha(m,n) = \\min\\{i \\geq 1 : A(i,\\lfloor m/n \\rfloor) \\geq \\log_2 n\\}.", "13efc496c414effcb2775070a26eb719": "|I| = \\frac{P}{A_\\mathrm{surf}} = \\frac{P}{4\\pi r^2}", "13efd63aac410295eea5fd87df8a1f97": "a(x - \\alpha)(x - \\beta) \\ ", "13f054501b181e44bc8ae6ce4fadf874": "G(\\theta|\\alpha),", "13f0637e8a2c25741aad7ce49b4afe8b": "(g \\circ f)^{-1}(y) = \\tfrac13(y - 5)", "13f0e801c388377344c74c1e979d1003": "N(k,\\epsilon)\\leq\\exp(C(\\epsilon)k).", "13f0f5a748310b9c41469a7a184046f4": "C_{AB} = \\frac{M_B}{M_A} = \\frac{1}{2}", "13f0f7b7f267d9a31e4615ed21bfa332": "\\sqrt{6^2 + 8^2} = 10", "13f15aae861adaae938599523683b922": "ax + by + c = 0\\,", "13f1a511f0ff158c02a9c858b6f72652": "\\gamma_4\\,", "13f1ba4b116ede3f57030c5fd444ddfe": " \\frac{W}{m_\\mathrm{0} c^2} = \\left( 1 + \\frac{\\alpha ^2 Z^2}{(n_\\mathrm{r} + \\sqrt{n_\\mathrm{\\varphi} ^2 - \\alpha ^2 Z^2} )^2} \\right) ^{-1/2} - 1", "13f26e96037bfbcb4d1ee107ef3bd38a": "\\log(*z_1/z_1) = k_{syz} y", "13f29485a675bcd7c648e29cc2b6251f": "q \\psi_0 + \\frac{d\\psi_0}{dq} = 0", "13f2ca0729b20b2d9017a45be64a3f26": "t_p \\equiv t_1^p \\equiv t_1, \\quad u_p \\equiv u_1^p D^{(p-1)/2} \\equiv u_1", "13f2ce330f6afa6357c7b6fa4fcb7d7b": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ -0.25\\end{smallmatrix}", "13f32c617d706cc01d67728e9df4324d": "J_q(n, d, e)", "13f395321385408b8aa03e1460cedaa0": "\\alpha_k \\le \\frac{3k-4}{4k}\\quad(4\\le k\\le 8)\\ ,", "13f3ce4b40dd26c6080d25a54f5333db": "\\ g_{\\phi}= \\left( 9.780327 + 0.0516323\\sin^2(\\phi) + 0.0002269\\sin^4(\\phi) \\right)\\,\\frac{\\mathrm{m}}{\\mathrm{s}^2}", "13f3f9e180ad21a7554ace64550a781e": " F',", "13f41fb0aa05b8c4a7953dd9834e0fb2": "\\mathbf{P}^2 = - (E_\\mathrm{rest}/c)^2 = - (m c)^2 .", "13f41fc1fe5e94692d180cf7d6cd121a": "s, e \\in \\mathbb{Z}_q", "13f4a66e5e47360fb397d6164f8e08e5": "x_{ij} = Q_{ijkl} \\times P_k \\times P_l", "13f4ba65c6d3620a34bbaa1d084a7d7c": " r_i = \\left\\vert \\left \\{x_k:x_k \\geqq x_i \\right \\} \\right \\vert .", "13f560bcbba95daeb7dc06e9407411f2": "R / \\operatorname{ker} f", "13f5621af050d4974be77b6bf13551a5": " \\frac{dN}{dt} = -k N", "13f573d4129ddf7a986e1ec4154936b2": "U(a,b,x)\\sim x^{-a} \\, _2F_0\\left(a,a-b+1;\\, ;-\\frac 1 x\\right),", "13f5cb1ea0c29245c82805824c46c15f": "g \\left ( \\alpha\\, \\right ) = 1 - { \\left ( 1 - {2^{-N}} \\right ) ^ \\alpha\\,} : 0 \\le \\, \\alpha\\,", "13f627f48d45107a1bc58ebb9253e6c7": " \\frac{ \\text{d} [{_{a_j}^{b_{j}}}S_j^{\\beta_{j}} ] }{ \\text{d}t } \\simeq - \\sum_{i=1}^m \\frac{ x_{b_{ji}} \\text{k}_{3(i)} E_0 \\overline{S}_i }{ \\overline{S}_i + K_i \\left( 1+ \\displaystyle\\sum_{p\\neq i} \\dfrac{\\overline{S}_p}{K_p} \\right) } \\qquad \\qquad (9a) ", "13f62c9aa3411c74f55faa6850ff5fb2": "f(c)", "13f637e4f7207a7983076438ea545fb8": "\\rho / \\sigma", "13f6618d09bec656658ffdcce15c36c4": "w_1,\\ldots,w_N", "13f74f1754f3c0e377c3c42c9f890fa6": "\\|x\\|_\\infty\\le\\|x\\|_1\\le n\\|x\\|_\\infty.", "13f7557092bfe957f27b4fce6a6220c5": "\\delta_g", "13f77b2369f8dac673bb67aca059600e": "\\nabla \\vec f = 0", "13f7961a3fd7a41a92786c7c83efe6a3": "fp_{11}/(p_{11}+p_{10})", "13f7f9af207b597ad46a50a11e29bbf5": "\n\\mathbf{x}(t) = \n\\mathbf{r}_{k} e^{\\lambda_{k} t}\n", "13f7fd9341eebda91119595c3a05f048": "\\frac{d}{dt}\\langle\\dot\\gamma(t),\\dot\\gamma(t)\\rangle = 2\\langle\\nabla_{\\dot\\gamma(t)}\\dot\\gamma(t),\\dot\\gamma(t)\\rangle =0.", "13f80ae362303da193277c011c2fbec9": "D+\\sum_{k=0}^\\infty CA^kBz^k", "13f812157a5051a7030462246c9401c6": "\\delta = \\sqrt{(\\Delta H_v - RT) / V_m}", "13f8135b589c6a7384811bd0c5cb8337": "\\int x\\arccos(a\\,x)\\,dx=\n \\frac{x^2\\arccos(a\\,x)}{2}-\n \\frac{\\arccos(a\\,x)}{4\\,a^2}-\n \\frac{x\\sqrt{1-a^2\\,x^2}}{4\\,a}+C", "13f89954f5de407793e2443ab28a541a": "\\log p", "13f8ba4cd54290fc83a7a25cc1a3ca4a": "\\mathbf{s} = 2 \\mathbf{h}/(1 + h^2)", "13f9055fe4e2dcd3cccb57f8999e1b4a": "\\int I(q)q^2\\,dx ", "13f93390f4b4d864816bee8cb63c98f1": "X = (X(t), t \\ge 0)", "13f95b621b1fe44bc70559490145d485": "n,m\\geq N", "13f962b29fda08256826a204bd3fff0a": "\\omega^2+\\omega", "13fa132608cee6c9379d9d712e446c54": "{\\color{white}\\frac{d}{dx} 2^x} = 2^x \\cdot\\text{constant,}", "13fa1e63dda9da93f8f44a795341e7d5": " r_1 ", "13fa8829f2a9f276a410c78ddf113e9a": "a_n^{2n-2}", "13fad4907a80f051079e9768ab244ea6": "\\forall x, y \\geq 0", "13fb41e8d12af3a134c710e571e7d06f": "y=-\\frac{3}{7}x + \\frac{11}{7}\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;z=-\\frac{1}{7}x-\\frac{1}{7}\\text{.}", "13fb42c883d34195c3d9c45b604fd2e2": "\\frac{n(q_1)}{N} \\cdot \\frac{n(q_2)}{N},", "13fb48d034eafb0ea71b2dab96af0d73": "N(d_+)", "13fb4a3eab5c5d693731b14e2aa88cb6": "\\scriptstyle t\\rightarrow \\infty ", "13fbebcbae10fb52668c48da27b4c919": "\\text{d}", "13fbf2741122c6ab7fda130b3a931527": "SO^+_{1,n}\\mathbb{R}", "13fc7b1a064ccade54f847f4dc938a64": "{x_1 = z_1}", "13fc9de7f44bc19ac212dd2ec9d32d81": "CMUAMA = \\frac{\\left ( MUAC - \\left ( \\pi \\times \\frac{TSF}{10} \\right ) \\right )^2 - 6.5}{4 \\pi}", "13fca36f7ef54140559e0d0f6253a895": " \\mathcal{L} [\\varphi (x)] ", "13fcbed8bcac58d95c0cb989f3aca2de": "\\sqrt[n]{x}", "13fcdc132c6487bb6bdc892786083b2e": "a=-\\nabla\\Phi", "13fce668381d93c55c280d0a2f0d3328": "\\operatorname{Var}(X)= \\operatorname{E}[X^2] - (\\operatorname{E}[X])^2.", "13fd0277843a5b1a1409bfe98ba5da89": "\\int x^m\\,\\operatorname{arcoth}(a\\,x)dx=\n \\frac{x^{m+1}\\operatorname{arcoth}(a\\,x)}{m+1}+\n \\frac{a}{m+1}\\int\\frac{x^{m+1}}{a^2\\,x^2-1}\\,dx\\quad(m\\ne-1)", "13fd08bf049e64aa2d3bc65965d909c4": "\\mathrm{Re} = \\frac {\\rho U_\\infty L}{\\mu}.", "13fd1fc00c4c9593b88673c6b494ce6f": "O^{0},\\cdots,O^{T}", "13fdb2f1a3f178de3b5cbf7da8c6c1ad": "t^\\prime = t-v x/c^2 \\,", "13fe16b5f133e0549dafe9ec7bd2095f": "{\\mathcal{A}}", "13fe38ce74a1252e183b1436189178a1": "d(x_n, x_m) \\leq d(x_n, x) + d(x_m, x)<\\varepsilon/2 + \\varepsilon/2 = \\varepsilon", "13fe6dc777724e1cc5872da88f9acf36": "(X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}", "13fe9f44dbf2af655b47d501b25b9916": "(x-\\alpha_i)", "13fea8852297c6f282d61e184a0f69d7": "x^2-2qy^2 = 1\\ ", "13feb4abaef8f5fbd97e61b4e89deb80": "C^{a}_{\\ bc}=0", "13ff24f35935ad57e15ca5647d54f220": "\\Sigma_{SFR} \\propto (\\Sigma_{gas})^n", "13ffcbc1f6b13df1ed8616aac1db77f4": "-P_1=(x_1,-y_1)", "14004376841054065e5ea453af59c836": "\\lim_{\\nu \\to +\\infty} \\operatorname{P} \\left(\\frac{X_{\\tau_\\nu}}{\\sqrt{\\nu}} < x\\right) = \\Phi(x)\n= \\frac{1}{\\sqrt{2\\pi}}\n\\int_{-\\infty}^x\n\\exp\\left(-\\frac{u^2}{2}\\right)\n\\, du, \\quad x\\in\\mathbb{R}.\n", "1400490d2842bcf5fb6ee18640d946f8": "\\mathrm{ad}\\colon\\mathfrak{g}\\to\\mathrm{End}(\\mathfrak{g}):X\\mapsto (\\mathrm{ad}_X\\colon Y\\mapsto [X,Y])", "1400c60e47050cf6bfc0d56896217d6b": "\\sqrt{\\frac{GM_{\\mathrm{planet}}}{R_H^3}} = \\sqrt{\\frac{GM_\\star}{a^3}},", "1400ceb4bf2bd5a7be16d78ffd94ce1e": "Q_{\\mathbf{u}\\mathbf{u}}", "1400db4031668f8c277d89bcebf459b9": "D = \\varepsilon E + \\xi H\\,", "14010200d760ad7b606883fb8960d837": "\\,\\phi(B)", "14010fb34067c69d05ba9d57b3fbca06": "\\rho : \\pi_1B\\rightarrow \\mathrm{Homeo}\\left(F\\right)", "140150dc2edb15cc0f0832e7226a246d": " \\psi = \\psi_{0} e^{iS/\\hbar} ", "140164935f2778d22e2d589db2429c5d": "h(G)", "1401a805930b1bcbc0d4cdfc6c4dae4c": "\\Sigma n_i = N", "1401abc5c8fd2bd9ae45bb067d51e6b9": "\\displaystyle{k(s,t)={1\\over 2\\pi}\\partial_t \\arg (z(s) - z(t)),}", "1401cc82a167526164adad5dcfb7e33a": "\\sigma < 0", "1401d79c7b73bb682d3f68bff8204e04": " C(r,z)=G_1(0,z)S(r)+2\\pi S_0\\int_{0}^{\\infty} G_2(r'',z)I_\\phi (r,r'')r''\\,dr''.\\qquad(14)", "14020d9df85e729771cc8bc0b10fa757": "Z(X,t)", "14022e18fd29b2c531ada9f0293cc773": "(\\lnot z)", "14023d15167d353c0707f48ba70ec2b2": "\n\\begin{align}\n\\int_{x\\in E}|p(x)|^p\\,\\mbox{d}x &\\geq& \\int_{x\\in E\\cap (J\\setminus E_\\lambda)}|p(x)|^p\\,\\mbox{d}x\n\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\\\\n&\\geq&\\lambda^p\\mathrm{mes} E\\cap (J\\setminus E_\\lambda)\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\\\\n&\\geq&\\frac{\\textrm{mes} E}{2}\\left(\\frac{\\textrm{mes} E}{2C \\mathrm{mes} J}\\right)^{p(n-1)}e^{-p\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J}\\max_{x \\in J} |p(x)|^p\\qquad\\qquad\\\\\n&\\geq&\\frac{\\textrm{mes} E}{2\\textrm{mes} J}\\left(\\frac{\\textrm{mes} E}{2C \\mathrm{mes} J}\\right)^{p(n-1)}e^{-p\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J}\\int_{x \\in J} |p(x)|^p\\,\\mbox{d}x\n\\end{align}\n", "14024a9c99802fea090ac45c57e17ecb": "\nm\\Omega>\\kappa_z\\,\n", "14026da1fcd875592978e55743abd618": "-1 = -A + C", "14029858d3a7ae492219886d2f27ae2f": "\\operatorname{Res}(f,i)=\\lim_{z\\to i}(z-i)f(z)\n=\\lim_{z\\to i}\\frac{e^{iz}}{z+i}=\\frac{e^{-1}}{2i}", "1402af796cd0c63255a6d85027341273": "f(n) \\sim g(n).\\,", "1402f3316bdc64712a705e5b8e5bb68e": "r^2 n \\propto r^{-1.5}", "14039829aa64625e99bb06b9c263cdbc": "M=L/R", "1403a9c06353ee7a734082b3b9cb2971": " \\exp \\log \\frac{1}{1-z} = \\frac{1}{1-z}", "1403fa9347c70cfd5b3348a3e0a3fd36": "f(t) = -2t^3 + 3t^2", "14041c3532cd6c4878ea09af53e2f83e": "R[t] \\to S, \\quad f \\mapsto f(T)", "1404dd3f4e27c420c5ab03816552ac8c": "[\\mathbb Z_4^6 \\times (\\mathbb Z_3^7 \\rtimes \\mathrm S_8) \\times (\\mathbb Z_2^{11} \\rtimes \\mathrm S_{12})]^\\frac{1}{2}.", "14053ad84391542856ecfd44ca84861d": "1/\\mbox{trace} \\mathbf{\\left(J^TWJ \\right)^{-1}}", "1405797ea19ff84d492e5d8bab6d163f": " (a_1,b_1,c_1)", "14057ee278db4784c475107da4a770a1": "\\left\\vert h\\right\\vert < \\pi", "14059d451ac251e108ddc5c61bd38c1d": "\\operatorname{Li}_{1}(z) = -\\ln(1-z)", "1405e6e9709a6719f1b48c43be5c2ed5": " \\tau_1 + \\tau_2 = C_2 (R_1+R_2) +C_1 R_1 \\ , ", "14060f1ac8215b6ec3eaad022f03e484": " r = k^2 \\cdot \\frac{1}{d^2}", "14061d083159604d7fcadc603f39ca4a": "\\chi_e\\ = \\varepsilon_r - 1.", "1406f6b20d8beb7d967eb30f17f8c5de": " {{}^3 R}_{2323} = \\frac{-\\omega^2}{2} \\, \\frac{C^\\prime \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right)^2}{C\\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right)^2} ", "1406f9934ee0e7fcdc261228e66f3d17": "F(x,\\theta)", "14070ed17a1fa76a2d1328db994351b6": "P(V_t \\geq 0) = 1 \\text{ and } P(V_t \\neq 0) > 0 \\, ", "1407ab317ac43ec9ecd555dc671cbeb5": "\\omega_{n}", "1407dc401c1bd95b366f793d77da5c7f": " \\text{cat}(X \\times Y) \\le \\text{cat}(X) +\\text{cat}(Y) ", "140835c22e5cd4e2aa12a4238fba7eda": "|{\\psi}\\rangle", "14084093465d434dbdc4d6ae6e7e8eba": "E_4^2 = E_8, \\quad E_4 E_6 = E_{10}, \\quad E_4 E_{10} = E_{14}, \\quad E_6 E_8 = E_{14}. ", "1408598a7ade84d0aa31dffee9c43ecd": "(\\nabla_c\\nabla_d - \\nabla_d\\nabla_c)", "14085c4af1c38b5ac993344c422d2316": " \\log_{10}(F(x)) = m \\log_{10}(x) + b, ", "14088ad333389531f5729ba6a5551b51": "H=max(P1,P2)", "140893524d30a815137e6c0352230082": "K(k)\\,", "1409ea5de7ff862ea96b938784ccec86": "\\lim_{t \\to \\infty} \\mathrm{E}[r_t] = b", "140abeefb251599e23e898ebb923f018": "p\\leq N", "140acc06c3f36fcdd9125c47af3bf7ec": "\\scriptstyle\ni", "140ae05fe8360fc6cf8c02435b06752a": "a_{n+1,k} = - n a_{n-1,k} \\ \\ k =0", "140b1bcf66d4ab868b3b5a7c3fa38fb1": "\\pi_i A", "140b2d7ed48bf632f77f6a89b68ab71f": "u(x(p, w))", "140b9b19460d154360ca9e032e65d727": "V_2 = Q/A_2", "140be2791373d73b906f152de4ddb9f4": "\\Re(s)>\\sigma_a", "140c4da37dec3cb1ae6fda0f71f7a376": "\\bar{\\Phi}", "140c56a63228acff004b72df352f8459": "0 < A \\leq B < \\infty", "140c919096c6c41140f27ca222290f87": "A A", "140ce8d68434e86209c18d118dd6580d": "\\beta_\\nu", "140d166f2fdde666c3b7dad0d9067272": "s=\\frac{k_R}{k_S}=e^{\\Delta\\Delta G^\\ddagger/RT}", "140d229376c9f0ff158878f9931fb57d": "1 \\times 2", "140d479ddd7de08aa1386e951d423b2d": "\\uparrow p", "140dbda52379f7014f8e7ae796531dc0": "|\\mathfrak o|=n+1", "140deece0bd03e960c38e2c1e4d15cf3": "SiO_{2} + C \\rightleftharpoons SiO + CO", "140e114e859a3968201ee3fa818fb2a5": "\nh^2 f_n y_n = 2y_n-y_{n-1} - y_{n+1} + \\frac{h^4}{12}y''''_n + \\mathcal{O} (h^6)\n", "140e9a45d0fceaaeca46489d7edbd731": "K^M_2(\\mathbb{C})", "140eb385391fa1827ad0f8811bcf7354": "\\varphi=Ax", "140eee963d45855a160bb2130b134620": " \\left(\\frac{m_2}{\\rho_2 a_2 D_2^2 Cos(\\alpha_2)}\\right)", "140f1c3ba3a559bde53aee914c1d8442": "\\ x(t) ", "140f5c1979c7f6eeb4cb75bb3e883db8": "n \\ge 400", "140f609f6868ee3e8c53292ce8cb019a": " y^m(\\mathbf{x})+\\delta(\\mathbf{x}) ", "140f61afbd34531b0b3b1611ba0ddec3": "e^t\\geqslant 1+t ", "140f67d345d72883248f1c0d18f450f1": "wp(S,R)", "140f6f80f343d63bda34f5b6de70ae19": "\\frac{3 \\cdot \\pi}{4}", "140f766e56d4637316bcd0680167cf61": "\\ H(z)=\\frac{b_0+b_1z^{-1}+b_2z^{-2}} {1+a_1z^{-1}+a_2z^{-2} }", "140fc56be65d7be882ef1d914336bcfd": "dV_{o} =\\frac{idT}{C}", "140fe34f59b41d4d3493df66917a0191": "\\displaystyle{L(b)Q(a)+R(a,b)L(a)=Q(a)L(b) +L(a)R(b,a)=L(Q(a)b) +2Q(ab,a).}", "140ffb9092c0de46d5479c4d6226826a": "\\mathbf{XC}-\\mathbf{CY}=rs^\\mathrm{T}", "141080726fe73eb629e204f3daa49382": " = 29", "14108ec9fdbd700eeb74da6a1a7ec758": "A_{n}=\\frac{1}{\\pi}\\displaystyle\\int^{2 \\pi}_0\\! f(x) \\cos{nx} \\,dx\\qquad (n=0,1,2,3 \\dots)", "1410b5a09206e20cededee60d0d821d7": "\\theta_x : T_xM \\rightarrow E_x", "14115da1ec59acbd589024ca8e8bbfd9": "n^{O(\\log \\log n)}", "141173d62981a15cbb3d0c15cc52b94a": "\\mathrm{Problem/Theme \\longrightarrow Complication \\longrightarrow Resolution}", "141190230ade937bc794c6556d5f5953": " \\ t' ", "1411a18939c198e9a4d7d2d69104f48d": " A(t) = A + \\frac{it}{\\hbar}[H,A] - \\frac{t^{2}}{2!\\hbar^{2}}[H,[H,A]] - \\frac{it^3}{3!\\hbar^3}[H,[H,[H,A]]] + \\dots ", "1411e672d85c21021050676ccb3441cc": "\\Delta _G U=\\Delta _G H -p\\Delta V_m", "141205e9c60a4b41ff726b65b7685001": "a_1,\\dots,a_k", "1412632206b26eac7c2802b1ee13a77a": "R = \\frac{ \\varepsilon_{\\text{c}} L_{\\text{r}} ^2 } {h^2}", "1412848ec54da3b9f0194a317bf5fd50": "\\mathcal A=(A,\\sigma,I)", "141298e09f0aa6da7eb1a7bc606ddaf5": "C_1 \\leq\\frac{|A|}{B} \\leq C_2.", "1412a7cd8ee572624c8f36da1080cee6": "\n\\frac{\\partial g_{jk}}{\\partial x^i} + \\frac{\\partial g_{ki}}{\\partial x^j} - \\frac{\\partial g_{ij}}{\\partial x^k} = 2\\left\\langle \\frac{\\partial^2 \\vec\\Psi}{\\partial x^i \\, \\partial x^j} ; \\frac{\\partial\\vec \\Psi}{\\partial x^k} \\right\\rangle \n", "1412e4856c55e0e51fdd64b7e611bd21": "4 \\ \\mathrm{m}/\\mathrm{s} = 7 \\ \\mathrm{N}", "1412ff53e7f65e94544c8a81042297c4": "\\mathbb{E}_{E}[\\Pr_{e \\in BSC_p}[D(E(m) + e) \\neq m]]", "14130028d900b9cfb8dbe346c7ea0892": "FVA \\ = \\sum_{m=1}^n C(1+i)^{n-m} \\ = \\sum_{k=0}^{n-1} C(1+i)^k", "141305604e881d529aedb152a89ffa90": "C_K^+ = I_K / P_K^+,", "14131de616d16fbca4559153389e7957": "T_3\\le\\sum_{B} 2^B\\frac{2n}{2^B}\\le 2n\\log^*n.", "1413360a08efa6bfca405b4e49c3aac8": "\\mathbf{V}(\\mathbf{r}) = \\dfrac{1}{2\\pi} \\sum_i \\gamma_i \\cdot \\left[\\mathbf{e}_z \\times \\dfrac{\\mathbf{r}-\\mathbf{r}_i}{(\\mathbf{r}-\\mathbf{r}_i)^2 + \\delta^2}\\right]", "14138bd109e01e8bbb7f8ba328f079cb": "a_0 = \\frac{1}{n}", "14138fb3b370e80a60ee8df852b2b4e2": "\\mu = \\int_{X_{1}} \\mu_{x_{1}} \\, \\mu \\left(\\pi_1^{-1}(\\mathrm d x_1) \\right)= \\int_{X_{1}} \\mu_{x_{1}} \\, \\mathrm{d} (\\pi_{1})_{*} (\\mu) (x_{1}),", "1414c752c2f0fb4cc7b2fec346682238": " M^n \\mathbf{u} = a^n\\, \\mathbf{u},\\qquad M^n \\mathbf{v}=b^n\\,\\mathbf{v}.", "141547df5611319dc2faa75d0d3a7eb7": "2^M", "141588ce49a7f08b8087b543f31745c6": "\\tilde{x}_1 = x(t + 1) \\, \\forall \\, t \\in \\mathbb{R}", "1415972cc497c830c8f538c8ef7d828f": " \\sum_{k=0}^{n} F(n,k).", "1415dce56e764b6285fde94f1d629237": "x \\mapsto \\langle x, y\\rangle", "1415eca759c1e4f6d67a7f87c5cb37e6": " (\\eta K)_X = \\eta_{K(X)}.\\, ", "14162bddab087796c16090eac0eb62a2": " \\textstyle{\\mbox{Illness rate } = \\frac{\\sum{\\mbox{Illness-related Absence Times in Days}}}{\\sum{\\mbox{Planned Working Times in Days}}}} ", "14164269f60422a885306b5c4ef87505": "D_{sc}(P,Q) = \\frac{1}{n}\\sum_{p \\in P} \\arg \\underset{q \\in Q}{\\min} C(p,T(q)) + \\frac{1}{m}\\sum_{q \\in Q} \\arg \\underset{p \\in P}{\\min} C(p,T(q))", "1416787acf04077ebc5cfa925983a228": "\\varphi\\rightarrow\\lambda^{-\\Delta}\\varphi.", "1416b09596a4ab21fddf92d77732bcd2": "|b_0| +|b_1| +\\cdots |b_m| \\le 2^m\\,\\left | \\frac{b_m}{a_n}\\right |\\, M(p)\\,.", "1416b96090c57a43d8b30064349dcd38": "\nf(x)=\\sum_{i=1}^n \\alpha_i \\lambda_i e^{-\\lambda_i x} =\\sum_{i=1}^n\\alpha_i f_{X_i}(x),\n", "1416dfdac57348b793c08f65dac5e508": "\\scriptstyle \\mathcal L_X", "1416f4d87b9def3f0142e334450ce02c": "\\scriptstyle \\vec r ", "1417e86447f83a01b64c278d60cf8a59": "{U}'(x)=f(x)(v-B(x))-F(x){B}'(x)=F(x)\\left(\\frac{f(x)}{F(x)}(v-B(x))-{B}'(x)\\right)=0", "1418062f3f8027d923789c602aa08820": "y=mx=m \\frac{2v^2\\cos^2\\theta}{g} \\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)", "14181df7fa2d6a7e009653f3a272205f": "\\forall \\sigma,\\tau^{\\prime} \\in \\mathcal{X}", "14182749c4e04428deaf4dce25a57fd0": "\\varphi_{X+Y}(t)=\\varphi_X(t) \\varphi_Y(t) =\\exp\\left(it\\mu_X - {\\sigma_X^2 t^2 \\over 2}\\right) \\exp\\left(it\\mu_Y - {\\sigma_Y^2 t^2 \\over 2}\\right) = \\exp \\left( it (\\mu_X +\\mu_Y) - {(\\sigma_X^2 + \\sigma_Y^2) t^2 \\over 2}\\right).", "141962fc4471e7d50a56786f0239016d": "u(k)= -K \\hat{x}(k)", "14196eec4b83188299309b3cc52c11bc": "\\mathrm{Ad}_{a}\\mathfrak{m}\\subset\\mathfrak{m}\\,", "141980babcdd8a0701a2d00bc980dfc2": "\\mu \\frac{\\partial}{\\partial\\mu} y_\\text{t} \\approx \\frac{y_\\text{t}}{16\\pi^2}\\left(\\frac{9}{2}y_\\text{t}^2 - 8 g_3^2- \\frac{9}{4}g_2^2 - \\frac{17}{20} g_1^2 \\right),", "1419c2e60a6b79d5d63a03ae743f34db": " \\textbf{x}_{k} = \\textbf{F}_{k} \\textbf{x}_{k-1} + \\textbf{B}_{k} \\textbf{u}_{k-1} + \\textbf{w}_{k} ", "1419e279235698afbfd7755f6bf5fc88": "m = \\gamma m_0\\,;", "141a17aab8ba4843f521b04affc13200": "F_n(z)=\\frac{Q_n(z)}{P_n(z)}", "141a368dd30157046995da6e8cecfd4c": "\n\\vec n \\cdot \\left( \\mathbf\\Sigma_0 \\nabla v_0 \\right) = 0 \\,\\,\\,\\,\\,\\,\\, \\mathbf x \\in \\partial \\mathbb T\n.", "141a497def9bb5489021f1aa78a35005": "\\left(n_1, n_2, n_3 \\right)\\,\\!", "141a5669a21f36e3af7627ad4deb6e7f": "3.2 U_p > I_p", "141a5d4759e2e45fd42e9e0cf1cd2b24": "|\\Omega\\rangle", "141a8feeed3aa4546bc622e620217295": "\n\\mathrm{NH_{4}^{+} > K^{+} > Na^{+} > Li^{+} > Mg^{2+} > Ca^{2+} > guanidinium}\n", "141b06aefda2202fd9aa03590738e288": "DR_{*}^{S}", "141b0a5b54a6a4ce72c2d574fcd300ec": "\ng' = g{\\rho_1-\\rho_2\\over {\\rho_1}}\n", "141b2459f1671b13ba901e104007b160": "G = H_1 \\times H_2 \\times \\cdots \\times H_l\\,", "141b6116216b2edc432492a52b138294": "\\hat{p} \\psi = -i\\hbar\\frac{\\partial }{\\partial x} \\psi ,", "141b79b30a584e36e304c72504c8c32b": "m_1\\|m_2\\|\\cdots\\|m_x", "141b7fbcbf287b5d86e7a7f850071866": "\\ \\sum_{n=-\\infty}^{\\infty}{\\left|h[n]\\right|} = \\| h \\|_{1} < \\infty", "141bcf4a5cf76ef12ae6f306d336013e": "HK = 139.54 \\frac{P}{d^2}", "141bd2f935f1bce44d710ea6c49bc353": "\\frac{dw/dt}{w}=g_w=\\rho v-\\gamma.", "141c1b932c1a056b00d39687ec9d6d3d": "\n\\omega^4 - (2K+\\kappa^2)\\omega^2 +K(K+Rd\\Omega^2/dR) =0", "141c8dcd33517cce3525436243c7f6e4": " dx = \\frac{x}{\\sqrt{t^2 + a b}} \\, dt", "141c95fd06f83b5427d4d88ca9afa1cb": "\\left(2+\\sqrt{-6}\\right)^2 = 4 + 4\\sqrt{-6} - 6 = -2 + 4 \\sqrt{-6} .", "141caf8f69c7bad12c265e37f2adfa54": "\n0\\to R\\xrightarrow{\\ x\\ }R\\to0. \n", "141cf5f371e76a6d4fe45f877c850b11": "(g^a,g^b,g^c)", "141d368a8b802696c217d15316468993": "\\mathbf{J}_\\mathrm{tot} = \\mathbf{J} + \\frac{\\partial\\mathbf{D}}{\\partial t}", "141d4bcf6119d5d38a60a3beb8cddbdc": " \\{ f,\\{ g,h \\} \\} + \\{ g,\\{ h,f \\} \\} + \\{ h,\\{ f,g \\} \\} = 0 ", "141db70c280e90ffa01d28203151ffb9": "e(v)=\\tfrac{1}{2}v", "141defc7bb8056ad41ef4076ab548e36": "h^2 = x^2 + y^2. \\,", "141e9f0203f7ee6056a19ca2042734d2": " m\\ge n", "141eb513e82b0cb4eb21f3e50ca6513d": "\\begin{cases}\n \\infty & \\text{for }\\alpha\\le 1 \\\\\n \\frac{\\alpha\\,x_\\mathrm{m}}{\\alpha-1} & \\text{for }\\alpha>1\n \\end{cases}", "141eb7cecef1acaf78084308005924e8": "\\alpha_{t_1}", "141f10ee8f1c1df43e4455ffb074a3d2": " \\mathcal{B}(\\mathcal{H})", "141f2d7837d1bbc63598518cb5a224eb": " \\mathcal{S}[\\mathbf{q}(t)] = \\int_{t_1}^{t_2} L(\\mathbf{q}(t),\\mathbf{\\dot{q}}(t), t) dt ", "141f2f6208c6bf606490e0115ad029f0": "\\lim_{n \\rightarrow \\infty} f_n(x) = f(x).", "141f93d115993bc092f008533a014cfa": "\\chi(\\tau,0)", "141fbad5c237cb390746713b73b9bef9": " x_2,x_3,\\ldots,x_s ", "141fc3e984bdb86c22325a3962223a0e": "\n \\mu = C_1\\left(1 + \\tfrac{3}{5\\lambda_m^2} + \\tfrac{99}{175\\lambda_m^4} + \\tfrac{513}{875\\lambda_m^6} + \\tfrac{42039}{67375\\lambda_m^8}\\right) \\,.\n ", "1420991a473b0e136407785761b536a3": "90 < \\theta \\le 180", "1420ff3150fd64b6b4c9725f96af9200": " { \\partial \\over { \\partial x^a } } \\ \\stackrel{\\mathrm{def}}{=}\\ \\partial_a \\ \\stackrel{\\mathrm{def}}{=}\\ {}_{,a} \\ \\stackrel{\\mathrm{def}}{=}\\ (\\partial/\\partial ct, \\nabla)", "14210a2d9fa4821b39a58b859826bbe0": "M\\ddot x + \n\\gamma\\dot x + Kx = F\\,", "142172b9546813e11f78ac92518bbdc4": "\\left(\\frac{p^*}q\\right) = 1.", "1421884db2f413200b9f9272deaebd17": "T:V\\rightarrow V ", "1421fc8049dc113ff83bcb794a74bfeb": "\\hat{f}(\\xi) = \\mathcal{F}(f)(\\xi) = \\int_{\\Re^n} f(x) e^{-2\\pi i \\langle x,\\xi \\rangle} \\, dx", "1422462345eebddf87056618ad9d51e5": "V_R = C_R \\cdot \\exp\\!\\left[{-z\\over\\lambda_R}\\right]", "142252dc7029d4b5698971640730a216": "\\tau_{U} = \\inf \\{ t \\geq 0 \\ : \\ X_{t} \\not \\in U \\}", "142272e402c3424d0c751458f25c7041": "\\frac{c}{c'}\\sim1+\\left(\\beta-\\delta-\\frac{1}{2}\\right)\\frac{v^{2}}{c^{2}}\\sin^{2}\\theta+(\\alpha-\\beta+1)\\frac{v^{2}}{c^{2}}", "142351e9a48de4907251a15fbae7462a": "A_{0,1}\\subseteq A_{0,2}\\subseteq A_{0,3}\\subseteq\\dots", "1423714a046ef79b33d25f61bc7850f6": "\\Delta(G)=n-1", "14237a09b6be33cb266c0a9a467d3627": "i*(TM)", "1423c12947ad7b6a9b8fc0ff70af3691": "\\lim_{x\\to\\infty}\\frac{f(x)}{g(x)} = \\lim_{x\\to\\infty}\\left(1+\\frac{\\sin x}{x}\\right) = 1", "1423f1eca43285573e3e5a981cea044e": "\\begin{matrix} \\frac{9}{7} \\end{matrix}", "1423fe2a603a0984f3b824f4dd88eeaa": "E[h(y)] = \\int_{-\\infty}^{+\\infty} \\frac{1}{\\sigma \\sqrt{2\\pi}} \\exp \\left( -\\frac{(y-\\mu)^2}{2\\sigma^2} \\right) h(y) dy", "1424576de49dc2dcc9ebbe1471441e38": "f(x,y) = \\begin{cases}y & \\text{if }y \\ne x^2 \\\\ 0 & \\text{if }y = x^2\\end{cases}", "142480edc25c3f67ac9b6b587d2bebd1": "\\cosh s = \\cosh b \\cdot \\cosh^2 l - \\sinh^2 l.", "1425709a7e394ffd986d11587dd70b6f": "R_{critical} = \\frac{2 \\cdot \\gamma \\cdot V_{Atom}}{k_B \\cdot T}", "142593160f797362160f0238e043b1a5": "|\\psi(x,t_1)|^2 \\neq |\\psi(x,t_0)|^2", "1425ce4e1db78ed1c2a3a1efe93d253b": "X_i=p_i\\geq 0", "1425eaee8efb8e982d94faa96104dc21": "\\mathrm{Bhattacharyya} = \\sum_{i=1}^{n}\\sqrt{(\\mathbf{\\Sigma a}_i\\cdot\\mathbf{\\Sigma b}_i)}", "142636266e9be0d0c02aee8ee0eac181": "\\zeta(k)", "142648986c0bb09ee0458b22ebea425f": " \\beta_{1}y(b)+\\beta_{2}y'(b)=0\\qquad\\qquad\\qquad(\\beta_{1}^{2}+\\beta_{2}^{2}>0),", "14265577ff8ee9e24a39b0fb2d437dde": "p \\gamma_0 = E + \\mathbf{p}", "14266865cebe9969b708551bfde7e30f": " \\frac{C_V}{Nk} \\sim 9 \\left({T\\over T_D}\\right)^3\\int_0^{\\infty} {x^4 e^x\\over \\left(e^x-1\\right)^2}\\, dx", "142673c1609416cfa9ceb305bf06a449": "E_{x_l,p_k} = \\frac{\\partial x_l(p,w)}{\\partial p_k}\\cdot\\frac{p_k}{x_l(p,w)} = \\frac{\\partial \\log x_l(p,w)}{\\partial \\log p_k}", "142683a703877c13109bc6956a3058d2": "(2-\\eta)^3= 7(\\eta-1)^2.", "1426ac4994ec481343a9455c3ab2e8de": "a_n = a\\,r^{n-1}.", "1426e8fca3932e4e9613cb0f342743ed": "\\mathfrak{P}^{4}", "142767c46cedb3e74f0790a60729f0bb": "a_{p_j}^{+}", "142775222b0df67f72be7e0b1a236361": " \\mu -\\sigma \\sqrt{ \\frac{ 1-q }{ q } } \\le x_q \\le \\mu + \\sigma \\sqrt{ \\frac{ q }{1- q } }", "14278c3e0b76defbfedf2b6e94d9c822": "d > 1/64", "1427d1c3ce6ab5f6510890b718b2bef2": " B_k \\mathbf{p}_k = - \\nabla f(\\mathbf{x}_k)", "1428200c45d995f73c2cc15864b00ac7": "\\scriptstyle D'", "14293d58f5914a1382bf8414e2c1e08f": "\\ \\displaystyle u \\ ", "142953a63755044e215e0366ef46e082": " \\begin{bmatrix}C^{j+1}\\end{bmatrix}=\\begin{bmatrix}AA^{-1}\\end{bmatrix}([BB][C^{j}]+[d]).", "142a0904f51a76d54e43a463353867db": "(-\\infty,a_n) \\cup (b_n, \\infty)", "142a31ba36a0c4226833cc6c92cf346f": "\\int_0^1 f(x)v(x) \\, dx = \\int_0^1 u''(x)v(x) \\, dx.", "142a64ccd835b449d00b24f165185036": "\\overline{\\mathbf{Q}} [[T]]", "142aeca4468bcc28fa1ad7002b67e517": "\\neg K_a \\neg \\varphi", "142b25dcf1eb58c482b4398f2696275d": "P_{\\mathrm{out}} = K_p\\,{e(t)}", "142b37cacd911ba8623fab4dbb59a5bc": "\\mbox{External virtual work} = \\int_{V}\\delta\\boldsymbol{\\epsilon}^T \\boldsymbol{\\sigma} \\, dV \\qquad \\mathrm{(1)}", "142b609233aee1a59ce32a52b7f92cf7": "\\Gamma(W)", "142b7a3a46f09b4078475acffb9656f3": "\\mathcal{E}: D\\times D \\to \\mathbb{R}", "142bdc762b8a4d9528b48131697d07e9": " H_A(\\mu) = \\Phi\\left(\\frac{\\sqrt{n}(\\mu-\\bar{X})}{s}\\right)", "142c195145bbea2637638446a2f082b1": "\n\\Omega^{*}=\\arcsin\\left(n_{1}n_{2}\\right)+\\arcsin\\left(n_{2}n_{3}\\right)+\\arcsin\\left(n_{3}n_{4}\\right)+\\arcsin\\left(n_{4}n_{1}\\right).\n", "142c59268b5a18704f75a79830173662": "n = 1200 \\cdot \\log_2 \\left( \\frac{f_2}{f_1} \\right).", "142c796b930c3645a04c12f50136bc68": " f\\Bigl(\\sum_{k=1}^nA_k^*X_kA_k\\Bigr)\\leq\\sum_{k=1}^n A_k^*f(X_k)A_k, ", "142cf40b2574250a374f7e299ebe03b7": " \\mathbb{R}^n ", "142d2a0290f5aebfa351affb2596f8ff": "U_{Total} = U_{Repel} + U_{Stretch} + U_{Bend} + U_{Trellis} + U_{Gravity}", "142d3eeaab4eed722c8faf528eb611e9": "\\{X(t)", "142d83e2b8f496be11ec117b6e474b48": "S\\ \\stackrel{\\mathrm{def}}{=}\\ M^2_{pl}\\int_M \\epsilon_{abcd}( e^{a} \\wedge e^{b} \\wedge \\Omega^{cd}) = M^2_{pl}\\int_M d^4x \\epsilon^{\\mu \\nu \\rho \\sigma} \\epsilon_{abcd} e^a_{\\mu} e^b_{\\nu} R^{cd}_{\\rho \\sigma}[\\omega] ", "142db804092ae5dfd0153788a41affbc": "c=(p'q'^{-1})^{\\alpha\\beta}=(h^rg^xh^{-s}g^{-y})^{\\alpha\\beta}\n=(h^{r-s}g^{x-y})^{\\alpha\\beta}=", "142e106451174957a0f633988242f918": "\\widehat D^*", "142e86f3ae7a030ceab9be75254dbe22": "V_1 = V", "142ec92f6842e7261b744f2753cb8d13": "\\lim_{n\\rightarrow\\infty} np_n = \\lambda,", "142fbdde23d012ff8976400c3567a8f1": "4.16130x^2 + 9.15933x - 11.4207 = 0", "14311166283b0f182c3234b15228d850": "\n \\underbrace{\n \\begin{bmatrix}\n -1 & 0 & 0 \\\\\n 0 & -1 & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}\n }_{C_{2}} \\times\n \\underbrace{\n \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & -1 & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}\n }_{\\sigma_{v}} = \n \\underbrace{\n \\begin{bmatrix}\n -1 & 0 & 0 \\\\\n 0 & 1 & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}\n }_{\\sigma'_{v}}\n", "14319465a54059760cc6149b457e8e49": " \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0. ", "1431cd84e28cbc734dd7c4e7c464bdff": " f(x) = b ", "1431dceac216e994f13b24e4057314e0": "\\forall x\\ \\exists y\\ \\exists z\\ ((x \\lor z) \\land y)", "1432055a1fc723b7b952ed4cf4156263": "\\mathfrak{D}_{L/K}", "14324d0a9022731fd83e60bdc012e0c3": " \\mathbf{y}", "14325b203add24084e67cd76ea6ca2da": "\\scriptstyle k\\leq n+1", "14328c24ee66f319bcafb22b7ed94340": " \\begin{matrix}\\frac{1}{a}\\end{matrix} ", "1432d0964e3b0179409331bd0c357ee4": "\\eta_{th} \\equiv \\frac{W_{out}}{Q_{in}} \\equiv \\frac{\\text{Electrical Power Output + Heat Output + Cooling Output}}{\\text{Total Heat Input}}", "143330b8f6b24ee34cbc79bcb03855e6": "[I] = \\begin{bmatrix}\nI_{11} & I_{12} & I_{13} \\\\\nI_{21} & I_{22} & I_{23} \\\\\nI_{31} & I_{32} & I_{33}\n\\end{bmatrix}=\\begin{bmatrix}\nI_{xx} & I_{xy} & I_{xz} \\\\\nI_{xy} & I_{yy} & I_{yz} \\\\\nI_{xz} & I_{yz} & I_{zz}\n\\end{bmatrix}.\n", "143330e466809978a62d204ac3e40609": "\\!f(x)=1+e^{-ax}", "143368bf77368c8be688ab66460487e1": " u_{n,i} ", "1433a38f493f455defec103e3a2dae42": "Z=\\sum_{n=0}^{\\infty } \\left (\\frac{2}{27} \\right )^n \\frac{(15n+2)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{3} \\right )_n \\left ( \\frac{2}{3} \\right )_n} {(n!)^3}\\!", "1433cdacaeee0b8cb98fdde716f69018": "\\xi = \\pm\\, \\Delta\\, \\int_0^\\psi \\frac{\\text{d}\\hat{\\psi}}{\\sqrt{1 - m\\, \\sin^2 \\hat{\\psi}}} = \\pm\\, \\Delta\\, F(\\psi|m),", "1433d671dffea5fa94244fa7c2c3ab3c": "u_L^C", "1433d98eac19da5dbbbb05b35277d324": "\\frac{L}{L_{\\odot}} = {\\left ( \\frac{R}{R_{\\odot}} \\right )}^2 {\\left ( \\frac{T}{T_{\\odot}} \\right )}^4", "1433df7c68d31d87f49f4bf730d2a2b9": " \\sum_{i=0}^n \\Delta h_i g_i, ", "1433eca3fb6259959ec22064cea8e510": "\\varphi_\\alpha(\\beta) < \\delta \\,", "1433fa80a3b91ac2bcce6be91fceb9ea": "p_{i_1,\\ldots,i_n}(f_1,\\ldots,f_n)", "143405b7914725e1af955bd2894d8f8d": " Z = A R^b ", "1434101071ef648e8514b2d62094e12f": "\\lbrace \\lambda_i \\rbrace", "1434f4cd0dfeb1fafdb47b468be10cb3": "\\tfrac{2\\pi}{3}", "143501ccd3f55f24062586288a1ccc6a": "\nE\\sum_j (\\hat{Y}_j - E(Y_j\\mid X_j))^2/\\sigma^2,\n", "14356a8f663e67fa16797ad9a6256264": " w'=v_x-iv_y=\\bar{v},", "14362077aac4af29772f1aa0d584934c": "\\mathfrak{X} ", "143680e4361eb2a7c43249848743cb0f": "E^*(\\mathbf{CP}^\\infty) = E^*(\\text{point})[[x]]", "14369e878a206361b693eec2b4075da9": "-[H^+]_{0^{ }}10^{-b_0}/K_w", "143757781b5d8489bdd2ac66b5b7ffeb": "v \\in S", "1437711dd636dc151fe1bc7ef51f9f42": "S^\\circ = 10 \\log S + 1", "14378b6ea28ddb55ea1762203de351b1": "\\ RED=Ra/R_{0}", "1437f56b00ca3fd0c768bba2e7d21d51": "M(t) = M_0 \\mathrm{e}^{-t/T_2}", "143854de9c6845037b2f5438102ea02d": "(x-1)/x", "143855010b900c440e8977c1f1b98639": "a=N_\\mathrm{A}^2 a'", "14399ced3698adf9e88f3585b44fb6ea": "\\left(\\!\\!{n\\choose k}\\!\\!\\right) = {n+k-1 \\choose k} = \\frac{(n+k-1)!}{k!\\,(n-1)!} = {n(n+1)(n+2)\\cdots(n+k-1)\\over k!},", "1439a448cdcbabcb180f357f369ec0f7": "L(p) \\le 2^n M(p) \\le 2^n L(p) ; ", "1439a845ab6be262395878229ffc01a8": "\\left\\|\\sum_{k=1}^m x_k-\\sum_{k=1}^nx_k\\right\\| = \\left\\|\\sum_{k=n+1}^m x_k\\right\\| \\le \\sum_{k=n+1}^m\\|x_k\\|<\\varepsilon,", "1439cb1b3731deb28c4f986a7c4cc301": "[\\Delta(y, E(m)) > (p+\\epsilon)] \\leq 2^{-{\\epsilon^2}n}", "143a0b190ad337896b3083780b42d53f": "S=\\left(\\Omega,\\mathcal{A},\\mu\\right)", "143a75a3c85ba4ff7e34f9dfd9fba0f5": "|{\\mathcal P}|<\\infty", "143a7928d8fb6e86328f39a447ca5cd1": "p \\ll \\rho c^2", "143a8a3672c255f0d031064fe50e67ca": "\\left[{n \\atop r}\\right]_r", "143b831ae565a37b707c5d0b910e2178": "\\arctan(1/\\lambda)", "143bde353c8bcdd3f40722ab93aa6b21": "= \\frac{1 + \\sin(A x) \\cos(B x)}{2}", "143c84eda304ce710729dd74d9280351": "\\phi(\\cdot)", "143cb2c308688f8c25e9d1c0377d53a7": "{100 \\over \\sqrt{2}}", "143cc097f631290d61cf4c959a01f98f": "\\begin{align}&\\textstyle{\\frac{2\\log\\left(\\frac{\\sqrt[3]{27-3\\sqrt{78}}+\\sqrt[3]{27+3\\sqrt{78}}}{3}\\right)}{\\log(2)}},\\\\ &^{\\text{or root of}}\\\\ &2^x-1=2^{(2-x)/2}\\end{align}", "143cc1d4040f5dda5dd38a57fa665d99": "\\sigma'(0) = x.\\,", "143d65ea21768236186b56833b42ae28": "H_n(G,M)=H_n(F\\otimes_{\\mathbf{Z}[G]}M)", "143d8caf7d4f115d46e8038db9d6f2e0": "\\left[M\\frac{\\partial }{\\partial M}+\\beta(g)\\frac{\\partial }{\\partial g}+n\\gamma\\right] G^{(n)}(x_1,x_2,\\ldots,x_n;M,g)=0", "143df6b29b8532a06bc3bdaf9df743bb": "\\rho \\simeq \\rho_0 \\,", "143e11af44f263b2ed499a1b1f5c5f07": "\\ U_0 = \\,", "143e1593f81c07e8b2b45047a73949c1": "x_3 = r\\sin\\psi \\sin\\theta \\sin\\phi\\ ", "143e4e4767ae79e5f11f239a4cfc15a5": " \\left[ \\begin{matrix} 1+\\alpha^2/2 & \\alpha & 0 & -\\alpha^2/2 \\\\\n \\alpha & 1 & 0 & -\\alpha \\\\\n 0 & 0 & 1 & 0 \\\\\n \\alpha^2/2 & \\alpha & 0 & 1-\\alpha^2/2 \\end{matrix} \\right] ", "143e5c40cf53a2e5cd195bb97f2f676d": "\\sqrt{\\lambda_i}V_i=\\mathbf{X}^\\mathrm{T} U_i", "143edc54ae24a633118a04cb763856a7": "E \\{ (g_i^*(y)-x_i) g_j(y) \\} = 0,", "143ef5ed8cf7d3292ee5ecaaabc0bd5c": "F_\\theta", "143f057790f2444eaac137a2ef01e7f0": "\\begin{bmatrix} \\ln p_1+C \\\\ \\vdots \\\\ \\ln p_k+C \\end{bmatrix}", "143f72d458ac43bcb0e1f61fe97946eb": "\\underline{\\mathbf{Z}}", "14402582008e305fb1d857ccb9991cee": "(M,\\Omega)", "14404e74761d07676dbbfe8a67e02b5e": "\\displaystyle \\iint f(x,y) e^{-2\\pi i(\\xi_x x+\\xi_y y)}\\,dx\\,dy", "14409d5a6e9d918d7b8e31ee775ea26f": " U \\mapsto \\pi_U", "1440c93a7208dc2f18c42ee01ea0e10b": "Y = \\frac{\\omega_H}{\\omega}", "1440da304e956c08becad0e9cda005e7": "\\int\\tan^n ax\\;\\mathrm{d}x = \\frac{1}{a(n-1)}\\tan^{n-1} ax-\\int\\tan^{n-2} ax\\;\\mathrm{d}x \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "1440ef44ed08563449987ad3fdb9fd7b": "i_C=C\\frac{d}{dt}v_C \\iff v_L=L\\frac{d}{dt}i_L", "14413d160c862f9212dfe5ca189d99b0": "\\tfrac{dx}{dt}", "14414cb709fc8bade32a579c658d9c6f": "\n \\begin{align}\n \\boldsymbol{\\nabla}\\cdot\\mathbf{v} & \n = \\cfrac{\\partial v_r}{\\partial r} + \n \\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r \\right)\n + \\cfrac{\\partial v_z}{\\partial z}\\\\\n \\boldsymbol{\\nabla}\\cdot\\boldsymbol{S} & \n = \\frac{\\partial S_{rr}}{\\partial r}~\\mathbf{e}_r \n + \\frac{\\partial S_{r\\theta}}{\\partial r}~\\mathbf{e}_\\theta\n + \\frac{\\partial S_{rz}}{\\partial r}~\\mathbf{e}_z \\\\\n & +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta r}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta})\\right]~\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta\\theta}}{\\partial \\theta} + (S_{r\\theta}+S_{\\theta r})\\right]~\\mathbf{e}_\\theta +\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta z}}{\\partial \\theta} + S_{rz}\\right]~\\mathbf{e}_z \\\\\n & +\n\\frac{\\partial S_{zr}}{\\partial z}~\\mathbf{e}_r +\n\\frac{\\partial S_{z\\theta}}{\\partial z}~\\mathbf{e}_\\theta +\n\\frac{\\partial S_{zz}}{\\partial z}~\\mathbf{e}_z\n \\end{align}\n ", "144250497400d99f1ecd90a7602dcd9a": "g(f(k))", "1442b13913e1383c039765fe935eb2c5": " F(b)-F(a) = \\int_0^1 DF(a+(b-a)t)\\cdot (b-a) dt", "1442c814e115d0aca59d389d2ce695c7": "H = {50 \\choose 2}{48 \\choose 2} \\div 2! = 690,900", "144312c1373c86c3a98d1434a706779e": "\\Delta_\\Omega=\\det (I- T_\\Omega^2) =\\prod (1-\\lambda_n^2).", "14432a7c175f6ba320b4b1acff1461b6": "\\langle\\cdot|\\cdot\\rangle", "14435595fd595cdabadb60c07124b653": " \\frac{\\partial W^*}{\\partial t^*} + U^* \\frac{\\partial W^*}{\\partial X^*} + W^* \\frac{\\partial W^*}{\\partial Z^*}\\ = -\\frac{\\partial p_d}{\\partial Z^*} + Pr \\left(\\frac{\\partial^2 W^*}{\\partial X^{*2}} + \\frac{\\partial^2 W^*}{\\partial Z^{*2}}\\right)\\ - {Ra_s Pr_s S} + {Ra_T Pr_T T}", "14435a5e55ac0326df9b871d3b852e91": "p_i=y_i^2", "14435b40fa72ce72b61ea788a1206d14": "4 X^3 - g_2 X - g_3", "14436843492351317959da20d48ae255": "\\left\\{(x, y) \\in R^2: \\sum_{i=1}^n \\sqrt{(x-u_i)^2 + (y-v_i)^2} = d\\right\\}", "144369d646b3419394a676dccc8df430": "F_1 = {x}^{2}+sx+ \\frac{b}{2} + \\frac{s^2}{2} - \\frac{c}{2p}", "14439ad4ab1ed9260c140b560f76f5d7": "\\alpha _V(V,T)\\ = \\frac{\\left.\\cfrac{\\partial p}{\\partial V}\\right|_{(V,T)}}{p(V,T)} ", "1444bcb16f2537b1fbefbc5fcea86b80": "\nH= { p^2 \\over 2 } + {l^2 \\over 2 r^2 } - {1\\over r}.\n", "1444e97b33fb963e49879f931a953b23": "0 < d(x, y) < \\delta ", "1444fa801991e8d988cc449b8281f419": " \nP_{ni}= {exp(\\beta_i s_n) \\over \\sum_{j=1}^J exp(\\beta_j s_n)}, \n", "1445076005a0c08c710016083be07a89": "\\varphi_{F}(e,0,y) \\simeq g(y),\\,", "144574d846374d4901b3ec32c761ab7a": "\\mathcal{G}_{0}", "14458328c2af57aa2782da5e070a561a": "f_1 = (x-2y)\\text{ and }f_2 = \\sqrt{3}y.", "1445b41be33191433c554d8d9f91ab03": "\\zeta_{\\Omega+1}", "1445beec041ec217900b093d16b2b9e6": "r_\\mathrm{max}=\\frac{p}{1-\\varepsilon}.", "144638371e40cb92bc79e7efcd7b21f1": " {\\sigma_S^P}=0 ", "14464ac1dfe6fa8ad8fda94bb6f01571": "k-1", "144660f9c83a4944301f52e95eea4f69": "t^{\\bullet} = \\{ s \\in S \\mid W(t,s) > 0 \\}", "144662c5f1f6ab08e02d5c60fb0bbc45": "S'=S \\cup \\{C\\}", "1446793c9c0ed3142c01fe5ecaef4f8f": " X = [-a,a] \\times [-b,b] \\times [-c,c] = \\{ (x,y,z) \\in \\mathbb{R}^3 \\,:\\, -a\\leq x\\leq a, -b\\leq y\\leq b, -c\\leq z\\leq c \\}\\,.", "14467cfb1fd04a9ca11c589331e46a25": "a_{H^+} = \\exp\\left (\\frac{\\mu_{H^+} - \\mu^{\\ominus}_{H^+}}{RT}\\right )", "1446922b8d015f3fc7b6e854a97bee12": "I(\\rho, \\mathcal{N})", "14483780e738c8868f175e46543e763f": " Z(S,H;s) = \\sum_{x \\in S} H(x)^{-s} . ", "14484357a61bd495ef4cd9907df395d6": " \\tau = \\pm\\frac{h}{r^2+h^2}. ", "14489729ebeff60cdd5ed67bbc0e4ea5": "M' = x^3 M", "14489dbbe75ed0c92a3a12f2ce1c6c6f": "P = 10^{\\frac{-Q}{10}}", "1448a6a0a9de5e5c139f19c6d4832b70": " M_{m}", "1448ca68c4bd5ae9b613ceac0de0e8bd": "\\nabla(f g) = f \\nabla g + g \\nabla f", "1449c17944ea4b2f31c5c417e19e4b6d": "\\displaystyle P(z|d)", "1449d0bd76ba67646e5f8401694f80b0": "a^{\\dagger}(\\phi)\\,", "144a9b252e1102805dd6567a4fe6855f": "\n a_{31} = \\frac{(\\lambda + 2\\mu)(4\\mu + A) + 4\\mu B}{4(\\lambda + \\mu)}\n ", "144acadb0a4d2198041385b5c501a8ae": "\n\\left(\\frac{1}{z} + 1\\right) \\left(1 - \\exp(-z)\\right) =\n\\frac{1}{z} + 1 - \\exp(-z) - \\frac{1}{z} \\exp(-z).", "144ad6c6094e76add7517b072c61c080": "f_{(\\xi,\\mu,\\sigma)}(x) = \\frac{\\sigma^{\\frac{1}{\\xi}}}{\\left(\\sigma + \\xi (x-\\mu)\\right)^{\\frac{1}{\\xi}+1}},", "144b0daffd620f02963fa2f758d051d3": "\n\\frac{1}{\\tilde{Q_s}} = \\langle\\frac{1}{Q_{mnp}}\\rangle_{k\\le k_r \\le k_r+\\Delta k}\n", "144b5833a81e160f270ebe34acb2ccc2": "\\frac{e^{E(1)/\\gamma}}{e^{E(1)/\\gamma} + e^{E(2)/\\gamma}}\\ ", "144b761e4f25f989fbac92f5592a8d10": "\n\\begin{align}\n(x_N * y)[n] & = \\int_{0}^{1} \\frac{1}{N} \\sum_{k=-\\infty}^{\\infty} \\scriptstyle{DFT}\\displaystyle\\{x_N\\}[k]\\cdot \\scriptstyle{DFT}\\displaystyle\\{y_N\\}[k]\\cdot \\delta\\left(f-k/N\\right)\\cdot e^{i 2 \\pi f n} df \\\\\n& = \\frac{1}{N} \\sum_{k=-\\infty}^{\\infty} \\scriptstyle{DFT}\\displaystyle\\{x_N\\}[k]\\cdot \\scriptstyle{DFT}\\displaystyle\\{y_N\\}[k]\\cdot \\int_{0}^{1} \\delta\\left(f-k/N\\right)\\cdot e^{i 2 \\pi f n} df \\\\\n& = \\frac{1}{N} \\sum_{k=0}^{N-1} \\scriptstyle{DFT}\\displaystyle\\{x_N\\}[k]\\cdot \\scriptstyle{DFT}\\displaystyle\\{y_N\\}[k]\\cdot e^{i 2 \\pi \\frac{n}{N} k} \\\\\n& = \\scriptstyle{DFT}^{-1} \\displaystyle \\big[ \\scriptstyle{DFT}\\displaystyle \\{x_N\\}\\cdot \\scriptstyle{DFT}\\displaystyle \\{y_N\\} \\big],\n\\end{align}\n", "144b8c7cb82c965e7bc25dfacfcba334": "d \\in N", "144c25411906427be681acc2216a28ca": "D_{\\mu}=\\partial_\\mu-(e/\\hbar c)A_\\mu ", "144cc1a637818aa02c6ad1c2b96b00da": "\\sqrt[3]{2} ", "144ccf1df809efa97b0563822db6dd08": "0 < k < n", "144d128457d100f7cd931d44990cab2b": "\\frac{t}{-s}", "144d99c40666216b69c1181f94f6f1b8": "W\\in R^{n\\times (n-q)}", "144db4d181e465476aa73f60a23fb909": "\\rho_f\\;", "144df43596a49b779ac3bf9d3250cc02": "P_{\\rm T} = IV \\, .", "144e0701a241fde94bbc33820313b6cd": " -a^{2} ", "144e9660eb462f8a67ff9565c48f1d28": "p_s \\quad", "144ecdf9b9bb7701e5ea15afbc3659e2": "W_{n-2}", "144eddc6a0381db94bc0d2d73fa858be": "f_e,", "144f40fac3f9e476bafd1a17350412a6": "\\lambda_g=\\frac{2l}{n}", "144f5091e84c506b490648b6a4ad00dd": "\\frac{Gr}{Re^2} \\ll 1 ", "144f5982d445d7f98c442f52c84ff0fa": "E_{KL}\\,\\!", "144f6860a0bd87df04a261cf97089a8f": "|A| = \\mbox{ card}(A)", "144f6aedc714321cc8c0721800b6d917": "2\\times \\tau", "144fca33bc06702f96434f6e2c518c82": "x_1 = 2", "144fca730c45a219a060c65eacbe31a9": "v_2 = i_2 Z_2", "1450b45e92af810bba5f71a79da16572": "\\mu([a,b])=b-a", "1451190cb567a5693c5a5d32c2ffe6ea": "a_1a_2\\cdots a_n = d^n {\\left(\\frac{a_1}{d}\\right)}^{\\overline{n}} = d^n \\frac{\\Gamma \\left(a_1/d + n\\right) }{\\Gamma \\left( a_1 / d \\right) },", "145138ee9bcf2622461df9891d215229": "a^{(p-1)/2} \\equiv \\left(\\frac{a}{p}\\right) \\pmod p ", "1451435fe84cd61d9538e3bb7a2119b2": " \\Delta n = \\frac {dn} {dc}* \\frac {\\Gamma} {h} ", "14514b1df8492af0e5a2a3d48a50a825": "\\{8, 4, 11, 9, 8, 11, 5, 1, 13, 9\\}", "14514c2b4f0474a5cbeb41bb3ebaf266": "X_n(C_n)", "1451595ea216c7ae83df6c0351139ab0": " \\equiv Re = \\dfrac{\\rho V l}{\\mu}", "1451a8d40e3f14bbfeddad0ab23464df": "B < \\infty", "1451acef3c9ffe025eb29396a016495a": "x \\in [0; a]", "1451f1e35a7ad34679c1d8b8e2082a4e": " y_{i+1} = y_i + 1", "145281d3abc898cff23ef6c92e783c53": "\\int_A f\\,d\\mu\\leq\\nu(A)", "145283b963cc0de3d65364b18de388da": "(0, t_1)", "1452abf5273ef7ee4d15c812a07a6c5b": " \\mathcal{N}_k", "14532f2f2697576bb8dcc0ef35027fa8": "\\!\\,\\gamma_{2}(t) = \\gamma_{1}(p(t))", "1453a02c852b0015113adc66d1d5c2aa": "\\lfloor kr \\rfloor", "1453a4ec05ff2bc42f036473d8bd0ac9": "\\mathit{d_H}(\\mathcal{B}) \\geq \\mathit{d_{min}}", "1453a5e73801aea0d24d2ab43f40562b": "\\rho \\left( \\dfrac{\\partial \\vec{u}}{\\partial t} + \\vec{u} . \\nabla \\vec{u} \\right) =- \\nabla p + \\mu \\nabla^{2} \\vec{u}", "1453d85c1fc37fdef9f82ded87cc8a34": "L_\\in = \\{\\in\\}", "1453dd13982ccb0d5db67a775c9a7347": "\n A = \\lim_{c \\rightarrow \\infty} A_c.\n", "145410720fa5c8e647679186271f514a": "10^{(10\\uparrow)^{n}x}=(10\\uparrow)^{n}10^x", "145434468b2ddbc4a439f338d0872035": "dV = \\prod_i ds_i = \\prod_i h_i \\, dq^i", "14545a13df05258ce0c55e278a529888": "\\operatorname{erf}(x)", "14547a45931c1dd06aa0b9c456b28c9e": "q(E,a_E,a)=\\frac{M(a)+m(E,a_E,a)}{1+\\frac{1}{2}\\left[ M(a)+m(E,a_E,a)\\right]}", "1454c2d26b5137bc9e9b668717bfb08b": "\\zeta = \\gamma M", "1454fd88afd23bd92cdd2b45cd5ff177": " \\frac{\\partial p}{\\partial t} = \\Delta p. ", "14550f20660cbb8f8001fd753e80a78e": "P_f = \\frac{{r \\choose 3}{4 \\choose 1}^3}{{52 \\choose 3}}.", "145574ab559a521c89de1d83e73b25e4": "f_o = 40Nf_r", "14568204fb0f59f19b49acfba864817d": "a^{L}", "1456958800b4ca3a3c92d54fbe332f57": "(r, s)", "1456b0d0df940945e45b8d98f1b71218": "V_B = V_{be} + I_ER_E \\ ", "14577abec3fedc9c1bba91fabfa0c7d1": "E_n(x^\\mu_0)", "1457833d4549f9d4e7dfecea85dfceab": "\\frac {1}{\\left\\vert B \\right\\vert}", "14578664c7c554b369c821616fb0a548": "\n\\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} \n= p! \\delta^{\\mu_1}_{\\lbrack \\nu_1} \\dots \\delta^{\\mu_p}_{\\nu_p \\rbrack}\n= p! \\delta^{\\lbrack \\mu_1}_{\\nu_1} \\dots \\delta^{\\mu_p \\rbrack}_{\\nu_p}.\n", "1457fe09fac3f54ba553ceda8eec6c3a": " [Z_i] = \\begin{bmatrix}\n \\cos\\theta_i & -\\sin\\theta_i & 0 & 0 \\\\\n \\sin\\theta_i & \\cos\\theta_i & 0 & 0 \\\\\n 0 & 0 & 1 & d_i \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix},\n", "14580a67ce4957176ee7026b89b5880b": "([t,x],s) \\mapsto [s\\cdot t,x]", "145870c5529a5519eae78becd40977a3": "\\epsilon = w/h", "14587bd8bbc1576bf4f79f7a7baa5c3b": "\\Gamma^{[k]}", "1458a437b3c6456f9ebf61d46c9ed13e": "\\Z", "14590907b8fa9305d2f125fe591ac10f": "\\Bbb S^2\\times\\Bbb S^2", "14591ae27b64a2f3a9ba38f6a5e213d1": "Pr(P(x,w)\\leftrightarrow V(x) \\rightarrow 1) =1", "14595148bc7edd382964ab4ac8ab5ff7": "\\mathfrak{p}_2 \\cap A", "1459877961528b9f504a3832b9f318f8": "\\mathrm{EV}_{S} = \\mathrm{EV}_{100} + \\log_2 \\frac {S} {100} \\,.", "145992696c50c2fa65cc7bd97c0cee1a": "\\ln\\mathcal{N}(\\mu,\\,\\sigma^2)", "1459a1fd57bae4bf9e68b4c017a89025": "(E \\in \\mathfrak{E}) \\land (E \\subset E' \\subset Y) \\Rightarrow (E' \\in \\mathfrak{E})", "145a089aba4ba1db57ba7766086d0d09": "BFD=0", "145a2076426515c59e57da4951e75c31": "\\Pr\\{CS\\} \\equiv \\Pr \\left\\{ \\left( \\bigcap_{i \\in S_p} E_i \\right) \\bigcap \\left( \\bigcap_{i \\in \\overline{S}_p} E_i^c \\right) \\right\\}, ", "145a42faff03589c1e2780db64319a5d": "F_{bfo}\\,", "145a6060c986e0e3396b59c0dd6b350c": " \\mathbb{T} ", "145b2f2a5b540ad9b2c0fb3fd2306ec0": "\\bigwedge GFP_{i}", "145b91edffc5b724e1c74074e82160d5": "I am a idiot\n", "145bccda6da126b1707c6c49c3d5ce23": "\\vec{R}_A", "145bd87f3a3bea5578c1ae1be2180f30": " h \\in R ", "145bf4b54cc77c9c24358fde3a0f2aaf": "\\delta\\hat\\mathcal{O} ", "145c07732557a2d8dd7b8fda7bfaa812": " |Z \\rangle ", "145c41c863da0302e73abf7e07cf65c9": "z \\mapsto \\omega z \\ ", "145c5365668b20205d75e80952bd0641": "\\frac {M^d} {P}=\\sqrt {\\frac {tY} {2R}} \\,", "145c64bd70459760c59e8acde633e9d3": "\\pi^4+\\pi^5\\approx e^6", "145c6c1ddac1a590c2c8b3362a7f307c": " E_s(s,t) = E_s(s) \\; \\exp\\left(i \\omega t + i \\phi \\right) ", "145cab29a4f622d580d7ec98561141d4": "\\!\\,T^{*}\\!M", "145d0ccc4bb2baee49c8d06b181c5018": "d(k)\\,\\!", "145d2cb2852c4cf462be4063d491bb71": " m_{pq} = \\sum_{x=1}^M \\sum_{y=1}^N x^p y^q P_{xy} ", "145d45716d4c06854f4089c31806eb56": "W\\left( \\mathbf{p,z}\\right) ", "145d5b55f58daa3ae6140b3772919c47": "\\mathbf{x} = [x_1, x_2, \\cdots, x_{nx}]^T", "145f2596c555364ae719494e52185515": "\\begin{align}\\frac{x^{4n}(1-x)^{4n}}{2^{2n-2}(1+x^2)}\n&=\\sum_{j=0}^{2n-1}\\frac{(-1)^j}{2^{2n-j-2}}x^{4n+j}(1-x)^{4n-2j-2}\\\\\n&\\qquad{}-4\\sum_{j=0}^{3n-1}(-1)^{3n-j}x^{2j}+(-1)^{3n}\\frac4{1+x^2}.\\qquad(*)\n\\end{align}", "145f69f3b8655ea9cadeb82f39e911a8": "V_i/V", "145f8de8904e6d3cde5ae74e2998a25a": "\\hat{\\theta} \\in [0, \\pi]", "145f9562d574a3fb5c688b3181921e74": "\\mathrm{AlCl_3}", "145f9ae47ee87f170d618e9cee6f0332": "\\int d^dx\\, \\sqrt{-g} R^{\\mu\\nu\\rho\\sigma}R_{\\mu\\nu\\rho\\sigma}", "145fc7f265f2720f4783df01bb7f05fd": "2X(s)\\rightarrow X_2(s)+(s)", "14602dd7dd842a44bb97041560abd1da": "H[\\{p_i\\}] = - \\sum p_i \\log p_i \\,", "146041bf39be9d472571b88081bee093": "\\mathbf{B} = B\\mathbf{\\hat n}", "14607278d97d4129e2b66aa6ede21a95": " \\equiv 1", "1460a4047c543d46b285066deadb1a9b": "GDOP=\\frac{\\Delta ( {\\rm Output \\ Location} )}{\\Delta ( {\\rm Measured \\ Data} )}", "1460cd1147b95a64b373ef35a5477420": " G=Aa^{2}+Cc^{2}+Bac-F", "1460d95122e1315e0efa170d119522d7": "d1", "149f569669d1a44647f4d92bc6c93074": "T_pM\\,", "149f7a0280b55b3155e91b98f9b7c3f8": "\\begin{smallmatrix}T_{\\rm eff}\\end{smallmatrix}", "149f90aa5c8c46265a1028dd3ff372a4": "I_q = I_1 \\sin^2 \\left( \\frac {q\\pi g \\sin \\alpha} {\\lambda} \\right) / \\sin^2 \\left( \\frac{ \\pi g \\sin \\alpha}{\\lambda}\\right) \\ , ", "149f97ff8f22132ee28bded5553a3de4": "\\displaystyle{{1\\over 2\\pi}\\iint |F_-(z)|^2 (1-|z|^2)^{-1/2} \\,dx dy}", "149fccab5db3016d91d55f7cd34365dc": " e \\approx 1.602\\ 176\\ 565 \\times 10^{-19}\\;\\; \\mathrm{C}. ", "149ff0831c48cc13d841b48610a256cb": "\\frac{g(\\boldsymbol{x}'\\rightarrow \\boldsymbol{x})}{g(\\boldsymbol{x}\\rightarrow \\boldsymbol{x}')}=1", "14a0038e8b9b837517e3c18942ff1318": "\\delta E/\\delta x", "14a04151804b46d0b3a742a46683cc86": "\\Lambda(x,y,0)", "14a04cf4d8af10b63b005597c21c7f49": "p \\oplus _x n", "14a086e3774e08338d1ec6da0c9b29c1": "\\mbox{If } X \\vdash Y \\mbox{ and }Y \\vdash a \\mbox{, then }X \\vdash a", "14a0ba194ca4874a1766b95b4077450c": "R_\\lambda f = \\int_0^\\infty e^{-\\lambda t}T_t f\\,dt.", "14a105ea781d75edce90d493637386bb": "{1}/\\sqrt{\\omega}", "14a12ff2cdba4702c041047ebd8cdf9a": "\\Pi_k(z_1,\\ldots,z_n):=(z_k-z_1)(z_k-z_2)\\cdots(z_k-z_{k-1})(z_k-z_{k+1})\\cdots(z_k-z_n)\\quad k=1,\\ldots,n.", "14a1989c28e728ff0a674efbf1fbe751": "2 / 7", "14a1b756edad99bb5e85197d5826fc65": "p \\not\\ll p", "14a22e69771467be72f32266adf77c29": "W = \\arg \\min_X ( \\max_Y score(Y, X))", "14a3428e846942683433782b55655f3b": "H = p^{i}_a \\partial_i y^{a} - L", "14a3e8659f24b984dbcdaec48308233b": " c_{t+1} = (1-R^{-1}) \\left[A_{t+1} + \\sum_{j=1}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j-1} E_{t+1} y_{t+j} \\right]", "14a451b226111e0db393ca9afcc90d6d": "f(tx+(1-t)y) \\le t f(x)+(1-t)f(y) - t(1-t) \\phi(\\|x-y\\|), \\,", "14a46dc71d5d7e93bf36cc14931ae283": "P(W_n|[Spam=false])=\\frac{1+a^n_f}{2+a_f}", "14a49175f3782dac4f7d241c68691903": "\\textstyle \\textbf{R}^{ d}", "14a4a69df046e46988db0621397a6773": " p_1 := A^{-1} B p_2 ", "14a4ae2ab11e8c3a5be1f3d56019b2bd": " \\mbox{div}\\,(\\mbox{grad}\\,f ) = \\nabla \\cdot (\\nabla f) = \\nabla^2 f = \\Delta f ", "14a4b0abbb834a812611bf9959b53831": "i,j,x", "14a4bc40b2af8ab2f195803e5ee2f058": "\\begin{bmatrix} -2\\boldsymbol\\eta_1 \\\\[5pt] -(2\\eta_2+p+1) \\end{bmatrix}", "14a4ebfd49a31e699009ccf08552e124": " B=B_{0} ", "14a4ece9e83c329a24d7c343ad313ca7": "h_\\theta = R\\, ", "14a5beaca6422ab203dd7ba36cc8eea1": "\n\\Phi_n(x) =\n\\prod_\\stackrel{1\\le k\\le n}{\\gcd(k,n)=1}\n(x-e^{2i\\pi\\frac{k}{n}})\n", "14a5e1cbf01b176512513d2a81b048cf": " c_1 = 0 ", "14a61ba307e6494ae48d290ce6b42aeb": "i < 0", "14a6dc93d8b195acc00951d4b36f9be4": "K = \\frac{L^2}{t_D U}", "14a6e55d9b5e48cecfb5eb5004754d68": "(M^2,\\partial M) \\subset S^3", "14a74a300d66e724d164da868f36d31b": "w(t,0)=0,", "14a7a87e6d92dc217768e184e793b45f": "dQ = C\\,dV", "14a7fca18e7850a306a2e1a7bd0a3182": "\\mathbf{P} = m\\mathbf{v} + e\\mathbf{A} ", "14a84fa53884614a613c8fc4aa59381b": "\\ \\phi, \\ \\mu, \\ \\kappa", "14a863300a71d05baea19833d441a427": " \\tan\\alpha_2 = \\frac{1}{\\phi}- \\tan\\beta_2\\,", "14a901f13da44d2c8f324a09fb6b1e44": "\\gamma(a)=x", "14a9cb1785a05fe25a4a6b2a335f9486": "\\frac{1}{4\\pi c^2}", "14a9e4d8494ee5ecc7c25c58ea341f5e": "\\text{ESF}_i\\,", "14a9f2fca3e9efd08d709e653b7d10c5": "\\mathbb{X}=(x_1,\\ldots,x_n),", "14aa2258df61c93d6076dfb8eda363b5": "\\frac{\\partial C}{\\partial t} = \\frac{C_{i}^{j + 1} - C_{i}^{j}}{\\Delta t}", "14aa66df6d70aa7bb2ce69faa2b2a55e": " x_{\\perp }", "14aa688ae0424eaa04ede7cff7cddb9b": "\\ell^2_{P}=\\frac{\\hbar\\,G}{c^3}", "14ab3eebd427d1772aec2b22274f32cc": "\\sum_{i=0}^n P_i", "14ab3f17ba9814c384b26da6a881dbff": "\\rho(\\cdot,\\cdot)", "14ab5490ae43841ffa71a17bce172bde": "\\textstyle(1+X)^n=\\sum_{k\\geq0}\\binom nk X^k,", "14ac2b02aac8abf2d2ef6748a78db25a": "\\frac{\\alpha}{\\alpha_G}", "14ac778ed69e05f7a64a2237a643eb09": "\n\\begin{array}{l}\n\\sum\\limits_{n=0}^{\\infty }\\delta \\left( x-\\gamma _{n} \\right) + \\sum\\limits_{n=0}^{\\infty }\\delta \\left( x+\\gamma _{n} \\right) =\\frac{1}{2\\pi } \\frac{\\zeta }{\\zeta } \\left( \\frac{1}{2} +ix\\right) +\\frac{1}{2\\pi } \\frac{\\zeta '}{\\zeta } \\left( \\frac{1}{2} -ix\\right) -\\frac{\\ln \\pi }{2\\pi } \\\\[10pt]\n{} +\\frac{\\Gamma '}{\\Gamma } \\left( \\frac{1}{4} +i\\frac{x}{2} \\right) \\frac{1}{4\\pi } +\\frac{\\Gamma '}{\\Gamma } \\left( \\frac{1}{4} -i\\frac{x}{2} \\right) \\frac{1}{4\\pi } +\\frac{1}{\\pi } \\delta \\left( x-\\frac{i}{2} \\right) +\\frac{1}{\\pi } \\delta \\left( x+\\frac{i}{2} \\right) \\end{array}\n", "14ad0bfa0595277207a88a0ad161da7e": "f \\circ \\gamma : [0, T] \\to \\mathbb{R}", "14ad13e0abab5b454966327be399c54f": "\\lim_{p \\to 0} M_p(x_1,\\dots,x_n) = \\exp{\\left( \\ln{\\left(\\prod_{i=1}^n x_i^{w_i} \\right)} \\right)} = \\prod_{i=1}^n x_i^{w_i} = M_0(x_1,\\dots,x_n)", "14ad464a01c14bac4ba24c17d317efe7": "X=\\frac{Y}{y}x", "14ad82993c8b8e35ed32cc3d0e10a930": "D_f", "14ad8c20e8759cd1e2c26de63cddf745": "\\tilde{A}=\\left\\langle H_{0}\\right\\rangle_{0} - T S_{0} + \\left\\langle\\Delta H\\right\\rangle_{0}=\\left\\langle H\\right\\rangle_{0} - T S_{0}\\,", "14add1279f441439f5c5587163ab3090": "\\sin^5\\theta \\cos^5\\theta = \\frac{10\\sin 2\\theta - 5\\sin 6\\theta + \\sin 10\\theta}{512}\\!", "14addb3443ab67b0b55c7956500c0ec1": "= p(C) \\ p(F_1,\\dots,F_n\\vert C)", "14adf6a1ecbf5150421904ffa40367b1": " \\mathbf{\\hat{n}} = \\mathbf{\\hat{r}} \\times \\boldsymbol{\\hat{\\theta}} \\,\\!", "14ae0cfd4eaffa27e467902d6d622ee1": "\\sigma=\\phi_{1,2}", "14ae2ab8141a09eed79e1ea2b6f24c29": "E = \\frac{1}{2}mU_{\\mathrm{ion}}^2 = neV_1", "14aea17ff026dc1f45913ef79fcf2fb5": "\n\\frac{\\partial \\rho (\\mathbf{r},t)}{\\partial t}+\\nabla \\cdot \\mathbf{j}(\n\\mathbf{r},t)=0, \n", "14aef51bc2b404fe65a900e4c994a91f": "(\\boldsymbol\\beta-\\hat{\\boldsymbol\\beta})", "14af1ffd3776c073f97ae89da0508f55": "v \\mapsto e^{t \\log b}v", "14af742eeca25a23943d80ef558da3be": "k_1=\\tfrac{(a-c)b}{c(c+1)},\nk_2=\\tfrac{(b-c-1)(a+1)}{(c+1)(c+2)}, k_3=\\tfrac{(a-c-1)(b+1)}{(c+2)(c+3)}, k_4=\\tfrac{(b-c-2)(a+2)}{(c+3)(c+4)}", "14af8245a44e5d43831875ac275efc0c": "\\begin{cases} p < \\frac{(d-1)^{d-1}}{d^d} & d > 1\\\\ p = \\tfrac{1}{2} & d = 1 \\end{cases}", "14aff913ec4987e693202110cad3ce9a": "y = Ae^{t} + 2Be^{-5t}. \\,\\!", "14b0191cb071cb958f50f92ddd4753f0": " \\scriptstyle F,", "14b033d9bdc25c33f257c99d09d6845c": "\\lbrack\\mathbf h\\rbrack = \\lbrack\\mathbf h\\rbrack_1 + \\lbrack\\mathbf h\\rbrack_2", "14b05ec3da085dfc96d142f6c04d6409": "\\exp(v) = c(1).\\ ", "14b0d05755a357f0a7a3b7095f0740ee": "O(n + z)", "14b0d0c977d026a668c604c1281c25bb": "p = 2 \\quad \\mbox{and} \\quad d_K \\equiv 1 \\pmod 8,", "14b10b819a86a86e17a8483cffb920bc": "\\rho: \\mathbb{C}^n \\to \\mathbb{R} ", "14b18ed237e269e9f1fb0d0e79aca6a6": "\n{dx \\over dt} = {x(t+\\epsilon) - x(t) \\over \\epsilon}\n\\,", "14b1c0c9185f9fa4ec15e12b90a65e12": "^hp(x,y,z)=z^{\\deg(p)}p(\\tfrac{x}{z},\\tfrac{y}{z})", "14b1d8a94e3282503d370545514f90f5": "[z_{/\\cong_{\\mathcal{B}}}]_X = \\{z\\in z_{/\\cong_{\\mathcal{B}}}\\mid z\\in X\\},", "14b1eba3a7dfa2690a41ebf7bb1f6a6a": " \\empty ", "14b2214382cfdeac478caa51cc4f72ec": "\n\nF (t) = K s (t) = K ((1 cm) sin (wt)) \\!\n\n", "14b2842455aa11b5f613eed22762ed6d": "\\Delta^0_1", "14b34ae33b0ab6685987aa3692e185b7": "\n\\delta^{\\mu_1 \\dots \\mu_n}_{\\nu_1 \\dots \\nu_n} = \\varepsilon^{\\mu_1 \\dots \\mu_n}\\varepsilon_{\\nu_1 \\dots \\nu_n}.\n", "14b3801428a2fe9857893634334d2207": "q\\to0", "14b3822faba0d938a0e9abce7f881fe5": " R = |P-Q|", "14b3ac8a919efae53f6dced1d14fb47c": "E_3=-q", "14b4184aaaff9c86e38f1c6ed2e5d92a": " T_K ", "14b4883589c601fdef221d007a845c5e": "\\frac{ \\text{d}E }{ \\text{d}t } = z \\frac{\\text{d}B}{\\text{d}t} - \\frac{\\text{d}C_1}{\\text{d}t} - \\frac{\\text{d}C_2}{\\text{d}t} - \\frac{\\text{d}C_3}{\\text{d}t}", "14b4a5e4eb6dfd9db07c8a321767272c": "\\mathrm{DF}(t_{\\textrm{tod}},t_{\\textrm{mat}})", "14b4ad51b0f38ab76613a466c7c1b24e": "\\scriptstyle y", "14b4c6ec856b2c53fe76a14bc8660292": "\nr_4 (n)=\n\\pi^2 n\n\\left(\n\\frac{c_1(n)}{1}-\n\\frac{c_4(n)}{4}+\n\\frac{c_3(n)}{9}- \n\\frac{c_8(n)}{16}+\n\\frac{c_5(n)}{25}-\n\\frac{c_{12}(n)}{36}+\n\\frac{c_7(n)}{49}-\n\\frac{c_{16}(n)}{64}+\n\\dots\n\\right)\n", "14b4d7614915d3922532fa7b65c00547": " L_p=10 \\log_{10}\\left(\\frac{{p_{\\mathrm{{rms}}}}^2}{{p_{\\mathrm{ref}}}^2}\\right)", "14b4ffe824a9ea41453b801f7ac7e201": "F=k\\cdot x", "14b54f90bbd7717134c9d927e6965465": "I_1(z)", "14b557f0d69c25a059228a4e4e52c4f0": "\\scriptstyle \\pm 2", "14b559d1492835fa6207d56fb7c5abce": "S(\\alpha ) = S^{max} \\sqrt{1 - \\frac{1- \\sqrt{1 - \\Delta ^2}}{2}}", "14b57afad6adbd260e99275813c87ff2": "b \\mapsto (\\pi_* \\alpha)_b", "14b59f15f19074345ff62f75f5a1aabf": "\\tan(\\theta) \\approx \\theta = \\Delta x/R", "14b5b7073193311e38479f57494dc0cd": "F(t, x, y) = \\frac{F(u, x, y)-F(t, x, y)}{u-t} = 0.", "14b60671587932fbd870acd5571406fc": "a=\\omega-1", "14b60dee94ab94cebe3aaffdf0480025": "x\\in V_\\alpha\\cap V_\\beta", "14b6bcb2240d3eeacabcda0695c0d2ef": "\\mathfrak{R}_k", "14b6ea586dbb753876056f1bc99821cb": "\\scriptstyle \\leq-2\\times10^{-21}", "14b6eb9e096de2f56e9ce0b72fef4edb": "(X,Y,\\langle , \\rangle)", "14b711dbccb916558d43abb0886404da": "\\int_a^b f(x)\\,dx", "14b742b76633e38af9c74ae51f878201": " y^5+p y^3+q y^2+r y+s=0", "14b76ebe44edc9ec2fa73be9de1c55fc": "(M \\geq 0)", "14b78e4112ca310eba1106f1fcc837ce": " \\Delta S_{\\mathrm{overall}} = \\Delta S^\\prime+\\Delta S^{\\prime\\prime}-\\Delta S^\\prime=\\Delta S^{\\prime\\prime} .", "14b79935be74f7e2ac374a544a4d0128": "(x', t') = (x,t) \\begin{pmatrix} 1 & 0 \\\\-v & 1 \\end{pmatrix}.", "14b7e9bfa079c71a1c9b01299bdd5380": "\\frac{x^2-y^2}{(x^2+y^2)^2} = -\\frac{\\partial^2}{\\partial x\\partial y}\\arctan(y/x)", "14b86c80a39556ef49d0ccaef7ffeb0b": "t_2\\,\\!", "14b8c68022b2702f0d1788a8acc04fab": "\\theta_V = Ad(h^{-1})\\theta_U + h^*\\omega_H,\\,", "14b8f97fe567a57911317f7dbbf408aa": "\\scriptstyle p_{21} \\,", "14b931b4b874e8138609a0a5f298e2e0": "A = 2 \\pi r h.", "14b940f2a549b2c92fc3fa5ccf88cb0b": "\n \\boldsymbol{R}\\cdot\\boldsymbol{R}^T = \\boldsymbol{\\mathit{1}} \\quad \\implies \\quad\n \\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T = - \\boldsymbol{R}\\cdot\\dot{\\boldsymbol{R}^T}\n", "14b94255fda9b4cbe904a8a8caa9c460": " \n\\Gamma_n^o= \\langle \\mathrm{Shannon}_n\\rangle^+\n", "14b95950d75d0bcd4fdb0dec9c5447a2": "\\Delta H = - R * (b+2\\frac {c}{T}),", "14b991c77c711171bcad2b04db117190": "v\\in C^2(\\bar{\\Omega})", "14b9b8ea7673699a10967338db764d85": "\\beta(u)=2\\pi \\rho = (4\\pi) u = 4\\pi \\sqrt{2M(r-2M)};", "14b9bda42976d420e1726c2093e446a8": " \\langle \\Gamma_i(t)\\Gamma_j(t') \\rangle \\propto \\delta(t,t') ", "14b9ebeaf82be5c6d696eec0326f6aef": "i:{\\mathrm F}_{\\mathrm O}(E) \\to {\\mathrm F}_{\\mathrm{GL}}(E)", "14b9ef4df305a651cbf1ed7cee3c9f89": "u_{l,k}", "14ba54fbe57cf3c56f09d5efc509b40a": "\\mathbf {e}", "14ba5bcc64da4d7995f081772373cb3b": "i\\hbar\\frac{\\partial}{\\partial t}\\psi=-\\frac{\\hbar^2}{2m}\\nabla^2\\psi + V\\psi", "14ba5f3d519ded167e29e779009d630e": "(\\alpha, \\beta, \\gamma ) = (\\alpha(x,y), \\beta(x,y) , \\gamma(x,y) ) = ( xy , x-y, x+y )", "14ba6d9c3993c5e1e2f5e62d6553d34b": "\n\\frac{{\\partial \\Lambda(T_{ij},\\lambda_i,\\lambda_j) }}\n{{\\partial T_{ij} }} = - \\ln T_{ij} - \\lambda _i - \\lambda _j - \\beta C_{ij} = 0\n", "14ba9598bb7c26b855e212a4554b8ef7": "\\nabla^4", "14baaef38cb2ea4f613ad36072f37c08": "\\forall s,t: W^-[s,t] = W(s,t)", "14bb0cdf38e9a9eb47e7645cd78d39c2": "|w| \\le \\tau.", "14bb23c2bb070e0086f75a30c3ec8109": "s_0(t) = e^{s t}\\left(\\cosh (q t) - s \\frac{\\sinh (q t)}{q}\\right),\\qquad s_1(t) =e^{s t}\\frac{\\sinh(q t)}{q},", "14bb2fe43021ce9e91a4a20e524d8a17": "\\dot{\\varepsilon_{\\rm{p}}}^{*}", "14bb3663551afe19c99efcaab77fca84": " R_n^{(n+1)}(t) = f^{(n+1)}(t) ", "14bb48486188a76c2d7eed9d5465b9f9": "I_{n,m}= I_{m-2,n-1}-a^2I_{m-2,n}\\,\\!", "14bb5a367707365553196bd8d6ff133b": "\n\\left({\\frac{p_i}{p_{\\mathrm{ref}}}}\\right)^2 = 10^{\\frac{L_i}{10}},\\qquad i=1,2,\\cdots,n\n", "14bbabedcf353aee78305b07b6481765": "\\tilde c_i = c_i \\tilde b_{i - 1}\\,", "14bc76512bc1da3166d918135e10e894": "\\forall y \\forall z ( P(y) \\land \\exists x Q(x,z))", "14bc8fa16a5059b9115c6e085d02de0a": "E= \\int_0^L B \\kappa^2(s)ds ", "14bc9de520cd7d3dabc2346d66cdeac5": "z_{\\alpha/2}", "14bcaa9506f743b1f8d6bfb3df9fd549": "H+L(\\alpha)\\ge 0", "14bce8d003e9f64192914e43b4f0d383": "\\frac{1}{\\sqrt{n}}\\sum_{i=1}^n x_i\\varepsilon_i\\ \\xrightarrow{d}\\ \\mathcal{N}\\big(0,\\,V\\big),", "14bce9a3f18b2d8ae5d231a327a477be": "(\\mbox{STr})", "14bd08d2e3938c183ff5ec6081e13305": "\n\\begin{align}\n& {} \\quad \\left[ \\begin{matrix} A & B \\\\ C & D \\end{matrix}\\right]^{-1} =\n\\left[ \\begin{matrix} I_p & 0 \\\\ -D^{-1}C & I_q \\end{matrix}\\right]\n\\left[ \\begin{matrix} (A-BD^{-1}C)^{-1} & 0 \\\\ 0 & D^{-1} \\end{matrix}\\right]\n\\left[ \\begin{matrix} I_p & -BD^{-1} \\\\ 0 & I_q \\end{matrix}\\right] \\\\[12pt]\n& = \\left[ \\begin{matrix} \\left(A-B D^{-1} C \\right)^{-1} & -\\left(A-B D^{-1} C \\right)^{-1} B D^{-1} \\\\ -D^{-1}C\\left(A-B D^{-1} C \\right)^{-1} & D^{-1}+ D^{-1} C \\left(A-B D^{-1} C \\right)^{-1} B D^{-1} \\end{matrix} \\right].\n\\end{align}\n", "14bd28d163393630962ee86e8b212243": " EAC = \\frac{NPV}{A_{t,r}}", "14bd4970ce01b9e821756add9d1e3e53": "\n(1+x)^{-2} = 1-2x+3x^2-4x^3+\\cdots \\quad |x| < 1\n", "14bd66ba5aef2092821bf1261b66f766": "BMO", "14bda14f876b8699dcda7eb93a655790": " \\|x-y\\|^2 + \\|x+y\\|^2 = 2\\|x\\|^2 + 2\\|y\\|^2 ", "14bda6aa361e73c544279855296030e6": "\\varphi_{xx} + \\varphi_{yy} = -\\rho,", "14bdea8555086393973634251f337792": "r=r_0(\\theta)", "14bdf6fd669a50951e3b9f708a744ed7": "\\Phi_1(x) = x - 1\\,", "14be9791e05dd35e5f535a5720573f58": "W = - \\frac{\\mu J_m}{2} \\ln \\left(1 - \\left( \\frac{\\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2 - 3}{J_m} \\right) \\right)", "14bedee5da804bc74223bbcfc5ba89eb": "(A,+,\\cdot,-,0,1,c_\\kappa,d_{\\kappa\\lambda})_{\\kappa,\\lambda<\\alpha}", "14bf287a0a068c1d59dd9e70a602fc86": "-\\frac{1}{2} \\lambda dN_1 = d [(N_2D_1 + N_1D_2) \\frac{dN_1}{d\\lambda}]", "14bf5ff0df232b4b551de7e07ae01643": "P(\\vec R)", "14bf9378dfc74058a4e59e065fccecd2": "\\gamma = \\alpha +j \\beta \\,", "14bf958843dc773e5ad7064be0000af3": "{n \\choose k}_q", "14bfd0a90a0114be4ca5aca2f68d8573": "\n W = C_1(\\bar{I}_1-3) + D_1(J-1)^2\n = C_1\\left[J^{-2/3}(\\lambda_1^2+\\lambda_2^2+\\lambda_3^2)-3\\right] + D_1(J-1)^2\n", "14bfea5f6966e9a4578f22fe82c9f3d7": "\\scriptstyle \\eta(v,x)", "14c01d5f62dbb18e7b5d370b27bcf7d2": " (\\mathbf{b} + \\mathbf{c})\\mathbf{a} = \\mathbf{ba} + \\mathbf{ca} ", "14c029a192de512fc04dec22839e1ac0": " I = \\frac{1}{12}wh^3 ", "14c08a89a28c2265590e3f0cb1d6b809": "T^{00}>0", "14c095a4c2130cb8f51665823633f1b9": "H = \\int \\rho h \\mathrm{d}V,", "14c0c1a69914fe7e8908d13868a014e3": "\\ (U,\\ N,\\ E)", "14c0ff60b4a081e857dbdc419e76dcef": "\\left\\{0\\right\\}= T_0\\subseteq T_1\\subseteq \\dotsb\\subseteq T_k=R", "14c11610e98799a98b74d6973a7f0434": "ROC2 = (1-Price/Price(X2))*100;", "14c13e919506c9cd6022c1a2295f40ca": "\\mathrm{P}", "14c166d8caf03d529673554619ceed96": " \\frac{\\hat p}{\\hat q} ", "14c1fbbbbfd081aaee80274d6468fb9a": "\\sigma_e\\,", "14c2009e893a1535fb198e51a90e4e2d": "I = I_o e^{-Q \\Delta x} = I_o e^{-\\frac{\\Delta x}{\\lambda}} = I_o e^{-\\sigma (\\eta \\Delta x)} = I_o e^{-\\frac{\\rho \\Delta x}{\\tau}} ,", "14c2777e467711300832e9e3c1a2a295": "e^{-1/g}", "14c3136e55c1e556b19a70495143cf86": "\\frac{1}{2} \\Delta v^2", "14c3314ad0051d045b2ff85b02d2dee7": " \\left (z\\frac{d}{dz}+a \\right ) \\left (z\\frac{d}{dz}+b \\right )w =\\left (z\\frac{d}{dz}+c \\right )\\frac{dw}{dz}", "14c3acbe640e1ddbc5220bd84b809171": "|\\mathrm{GHZ}\\rangle = \\frac{|0\\rangle^{\\otimes M} + |1\\rangle^{\\otimes M}}{\\sqrt{2}},", "14c3b3a6539320ccb18e22aa48e888d2": "\\mathbb{N}^\\mathbb{N}", "14c3b932f35ec23203ada6df588d16b2": "S^{d+1}", "14c50c535c224592c570575125c75374": "\n J = 100 \\left( A / A_w \\right)^{c z}\n", "14c517b2613d6d508b61ae507ce2e70e": "f(x) = \\ln x", "14c52509018cc4bbd4348bc3e1b653ab": " \\mathbf{A} \\otimes (\\mathbf{B}+\\mathbf{C}) = \\mathbf{A} \\otimes \\mathbf{B} + \\mathbf{A} \\otimes \\mathbf{C}, ", "14c533e18ef8415bf05d537db2a27732": "x=y+m", "14c561e595ce5709065107017d4d8f09": "\\delta = 1/\\beta-1", "14c56683e30b53b498b7174e28c7bd0b": "L(A) \\subseteq L(B)", "14c577760ebc7810fa5601661d622768": " \\operatorname{max}(a,b) = \\frac{a+b+|a-b|}{2} ", "14c600e00c3967278760bfc2053219fe": "\\forall A [A \\not = \\varnothing \\rightarrow \\exist b (b \\in A \\and \\forall c (c \\in b \\rightarrow c \\not\\in A))].", "14c61e1ce270e52467a13ac075504ce2": " \\frac{(\\mathbf{r}-\\mathbf{p}_i)}{|\\mathbf{r}-\\mathbf{p}_i|^3} = - \\nabla \\left(\\frac{1}{|\\mathbf{r}-\\mathbf{p}_i|}\\right). ", "14c628fcecb9a6431f91f65b56a001d8": "x-\\Delta x \\le \\xi \\le x+\\Delta x", "14c66ffea255e4ee362236443973cb72": "W(\\varepsilon)", "14c672ab3df14adce6d7abae9757cc6b": " \\partial_{ij} ", "14c6873157820ac817c02b4fc3720a89": "\\phi = \\arctan \\left({R \\over L}\\right)", "14c6ca77360582fdb354b9302bfc312d": "{}^{3}x", "14c6e8a587dfdce3609603692f1f9ef9": "1\\leq i \\leq m", "14c70ec8cbb51314c5b6ac584cf2c523": " \\pi P(t) = \\pi ", "14c7452626fa36bf359685372604ae1c": " t_3 ", "14c7a5f699476506c72559b7696a35af": "\\mu : \\mathbb{R}^{n} \\times [0, T] \\to \\mathbb{R}^{n};", "14c87522cd363dad9abe585e94f6d1ef": "\\varpi", "14c8883d2c4b2c396906c46f3a897d9c": "\\varphi_{h(e)} \\simeq \\varphi_{\\varphi_e(e)}", "14c88f20b13d670f33fb09f84517d6c5": "g(\\textbf{a})\\not =0", "14c8ced3a0f6fa9c6d5e8166b3bfcdc9": "\\sqrt{2}^{\\sqrt{2}}.", "14c8d739d8b6fd0db889ce221e884b62": " h \\nu = g_\\mathrm{e} \\mu_\\mathrm{B} B_\\mathrm{0} ", "14c92c0854fed9dd04da1b2d34fbf96a": "46 = 6 + 20 + 20", "14c92e023182e18fd9d0d8071f441b1b": "q^*(\\mathbf{Z}) = \\prod_{n=1}^N \\prod_{k=1}^K r_{nk}^{z_{nk}}", "14c983b8e5474e64fc30679ed0c6d4b5": "\\begin{array}{cl}\n \\underset{\\boldsymbol{\\alpha},\\boldsymbol{\\xi},\\rho}{\\min} & -\\rho + D \\sum_{n=1}^{\\ell} \\xi_n\\\\\n \\textrm{sb.t.} & \\sum_{\\omega \\in \\Omega} y_n \\alpha_{\\omega} h(\\boldsymbol{x}_n ; \\omega) + \\xi_n \\geq \\rho,\\qquad n=1,\\dots,\\ell,\\\\\n & \\sum_{\\omega \\in \\Omega} \\alpha_{\\omega} = 1,\\\\\n & \\xi_n \\geq 0,\\qquad n=1,\\dots,\\ell,\\\\\n & \\alpha_{\\omega} \\geq 0,\\qquad \\omega \\in \\Omega,\\\\\n & \\rho \\in {\\mathbb R}.\n\\end{array}", "14c9d37cf5a78009bc1a75af48c3b6d5": "\\mu =2", "14c9e1137621b47ab94317b664421b45": "\np = \\frac{\\rho}{4} \\left( \\cos 2x + \\cos 2y \\right) F^2(t).\n", "14ca01b4a2151099cb75d1fbeb9b29bd": "\\quad h(\\phi)\\;=\\;\\frac{P'K'}{PK}\\;=\\;\\frac{\\delta y}{R\\delta\\phi\\,}.\n", "14ca3634eb43d7f0b16f867df5ef5ec5": "\\frac{\\partial \\log \\mathcal{L}(\\alpha,\\beta \\,|\\, x_1, \\ldots, x_n)}{\\partial \\beta} = \\frac{\\partial \\log \\mathcal{L}(\\alpha,\\beta \\,|\\, x_1)}{\\partial \\beta} + \\cdots + \\frac{\\partial \\log \\mathcal{L}(\\alpha,\\beta \\,|\\, x_n)}{\\partial \\beta} = \\frac{n \\alpha}{\\beta} - \\sum_{i=1}^n x_i.", "14ca3c9c106d9587d826deee8499c635": "\n \\begin{align}\n \\varepsilon_{rr} & = \\frac{\\partial u_r}{\\partial r}\\\\\n \\varepsilon_{\\theta\\theta}& = \\frac{1}{r}\\left(\\frac{\\partial u_\\theta}{\\partial \\theta} + u_r\\right)\\\\\n \\varepsilon_{\\phi\\phi} & = \\frac{1}{r\\sin\\theta}\\left(\\frac{\\partial u_\\phi}{\\partial \\phi} + u_r\\sin\\theta + u_\\theta\\cos\\theta\\right)\\\\\n \\varepsilon_{r\\theta} & = \\frac{1}{2}\\left(\\frac{1}{r}\\frac{\\partial u_r}{\\partial \\theta} + \\frac{\\partial u_\\theta}{\\partial r}- \\frac{u_\\theta}{r}\\right) \\\\\n \\varepsilon_{\\theta \\phi} & = \\frac{1}{2r}\\left[\\frac{1}{\\sin\\theta}\\frac{\\partial u_\\theta}{\\partial \\phi} +\\left(\\frac{\\partial u_\\phi}{\\partial \\theta}-u_\\phi \\cot\\theta\\right)\\right]\\\\\n \\varepsilon_{r \\phi} & = \\frac{1}{2} \\left(\\frac{1}{r \\sin \\theta} \\frac{\\partial u_r}{\\partial \\phi} + \\frac{\\partial u_\\phi}{\\partial r} - \\frac{u_\\phi}{r}\\right).\n \\end{align}\n", "14ca95091504036f5e660f0cd380b0d0": "0 0", "14de173a8adf1c85c00d0533d848e0f8": "= \\frac {5 - 2.34} {5.09} = .5%", "14de304fb8a6abb19c7d5a9fe5ce6057": "C(u\\otimes v)", "14de5f1f13e662c4c6c6562e3549fced": "A(t) = U(-t)AU(t). \\quad", "14deeddfefcfbacb84366f221cdb02d2": "\ny = \\frac{B-\\sqrt{B^2-4A}}{2}\n", "14df0ff5c4308223b3e2168e251d4447": "\\begin{matrix} \\frac{2}{3} \\end{matrix}", "14df3f5d62fa5eaafb76fabe34dbce62": "\\nabla_X V = \\lim_{h\\to 0}\\frac{\\Gamma(\\gamma)_h^0V_{\\gamma(h)} - V_{\\gamma(0)}}{h} = \\left.\\frac{d}{dt}\\Gamma(\\gamma)_t^0V_{\\gamma(t)}\\right|_{t=0}.", "14df83ae45ba136ce8958f93b0c695e1": "\\displaystyle{\\|u\\|_{(1)}^2 =|(\\Delta u, u)|\\le |(\\Delta_1 u,u)| +|(Xu,u)| \\le \\|\\Delta_1 u\\|_{(-1)}\\|u\\|_{(1)} +C^\\prime \\|u\\|_{(1)}\\|u\\|_{(0)}.}", "14df91d39f7a06a8eaca21bc4de6499a": "\\frac{\\overline{PA} \\cdot \\overline{QA}}{\\overline{CA} \\cdot \\overline{AB}} + \\frac{\\overline{PB} \\cdot \\overline{QB}}{\\overline{AB} \\cdot \\overline{BC}} + \\frac{\\overline{PC} \\cdot \\overline{QC}}{\\overline{BC} \\cdot \\overline{CA}} = 1.", "14df945f4520d87288c1c406b58fee90": "1 +\\lceil p\\log_{10}(2)\\rceil", "14dfc2a5d3a9a165077201fcfde1b0a9": "M_{\\mathrm{u}} = M({}^{12}\\mathrm{C}) / 12 \\,", "14dffd39abb6d74064c1393e199dcffd": "D_{med} = E|X-median|=2Cov(X,I_O) ", "14e02c652e90ad7b729ff84ea33b110f": "C = \\frac{3 \\times \\mbox{number of triangles}}{\\mbox{number of connected triples of vertices}} = \\frac{\\mbox{number of closed triplets}}{\\mbox{number of connected triples of vertices}}.", "14e0406efd339db5f4eaa23f638d8005": "v(p + r,t) = v(p,t) + J(p,t) r", "14e0490b83d7555f9832ebb54a7ca51d": "\\bold{p}=\\frac{\\partial G_2}{\\partial \\bold{q}}=\\frac{\\partial S}{\\partial \\bold{q}} \\ \\rightarrow \\ \nH(\\bold{q},\\bold{p},t) + {\\partial G_2 \\over \\partial t}=0 \\ \\rightarrow \\ \nH\\left(\\bold{q},\\frac{\\partial S}{\\partial \\bold{q}},t\\right) + {\\partial S \\over \\partial t}=0. ", "14e04d5363196d308393629af6bcc333": "\\int_{-\\infty}^{\\infty}e^{- a x^2 + b x + c}\\,dx=\\sqrt{\\frac{\\pi}{a}}\\,e^{\\frac{b^2}{4a}+c},", "14e054604a0e03d668baedc5eb23b93e": "\\|f\\|_{1,w}\\leq \\|f\\|_1.", "14e05bd95dec768c200f5dbb03a20af4": "\\frac{\\partial \\mathbf{D}}{\\partial t} + \\mathbf{J} = \\nabla \\times \\mathbf{H} \\ \\rightarrow \\ \\mathbf{E}\\cdot\\frac{\\partial \\mathbf{D}}{\\partial t} + \\mathbf{E}\\cdot\\mathbf{J} = \\mathbf{E}\\cdot\\nabla \\times \\mathbf{H}.", "14e061011b37262c1d6213a4f276bef7": "L_1\\cdot L_2 = \\{w\\cdot z | w \\in L_1 \\land z \\in L_2\\}", "14e08757093344734a670802306cf705": "\\begin{align}\n e^{i x} &= \\cos x + i \\;\\sin x \\\\\n e^{-i x} &= \\cos x - i \\;\\sin x\n\\end{align}", "14e0df45900873845c67edfe0b8d2ae8": "a_{t}", "14e0f56fb1ecd858b267ea86f7e26e86": "\\gamma_\\text{SG}\\ ", "14e14a8523502755031185532f5d92b4": "\\forall C [\\lnot \\exist W (C \\in W) \\iff \\exist F ( \\forall x [\\exist W (x \\in W) \\Rightarrow \\exist s (s \\in C \\and \\langle s, x \\rangle \\in F)] \\and ", "14e19b344fd2b63f60c7c9d1a8f3a9fd": " U(f) = f(T)\\xi, \\, C[0,1] \\rightarrow H", "14e1e154857d768e838515e68d36ffc2": "P_2 = P_1(1+r)-c", "14e1fbd7b83da4c8661e1a3cd6eb846a": "T_o", "14e2139b9e08d44a4d42c892bc598e13": "\\ x[n] = 0.5^n u[n]", "14e236e27f2f575b8a87b1b44d881856": "(r, \\phi, z)\\in[0,\\infty)\\times[0,2\\pi)\\times(-\\infty,\\infty)", "14e23d2a69b782882a1e6bf5ddbf12d5": "\n\\frac{d^{2}x}{dt^{2}} + \\beta(t) \\frac{dx}{dt} + \\omega^{2}(t) x = E(t).\n", "14e26413a15d7467db850d97b0e6a0e4": "\n\\frac{\\mathbf{s(}n\\mathbf{)} \\,\\,\\mathsf{nat}}{n \\,\\,\\mathsf{nat}}\n", "14e2afb8376dc5a31797418eed321b54": "\\mathcal{L}.", "14e2ca745d0be4bef09547415c48afc8": "10_{137}", "14e2e0c29c6c6d68e209dd61c72b25fa": "\\operatorname{E}(T) = k\\sigma^2 + n\\mu^2 + 2\\mu \\sum_{i=1}^k (n_iT_i) + \\sum_{i=1}^k n_i(T_i)^2", "14e3390d4539cc1370999e6f31da05fc": "\n\\mathcal{P}_3 \\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* =\n m \\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* ,\n", "14e3d87382feaafd9bd43b4357b87016": "877\\,", "14e3e589f905545fa78bc9df02e436f1": "k_BT\\approx0.026\\,eV", "14e3eeb4d53f990467847d3de906187d": "expr \\subseteq \\{expr_{1}, \\dots, expr_{n}\\}", "14e422dfbefe9a24811dd6951618d87d": " \\text{gross margin} = \\frac{0.667}{1 + 0.667} = 0.4 = 40%", "14e463372eceb1e2a386d27b7f58f296": "e^- + H_2O \\longrightarrow H_2^+ + O + 2e^-", "14e4963992aa60057ea937a7979bfca4": "f:\\mathbb{R}_{+} \\to \\mathbb{R}", "14e498b4f9bd48ea2e533c8f4498402b": "\\mathbb{P}\\biggl(\\bigcap_{j\\in J}A_j\\biggr)=\\frac{(m-|J|)!}{m!}.", "14e4c98bfe057d46da3da7fb426c0649": "\n\\begin{align}\nf^{-(n+1)}(x) &= \\int_a^x \\int_a^{\\sigma_1} \\cdots \\int_a^{\\sigma_{n}} f(\\sigma_{n+1}) \\, \\mathrm{d}\\sigma_{n+1} \\cdots \\, \\mathrm{d}\\sigma_2 \\, \\mathrm{d}\\sigma_1 \\\\\n&= \\frac{1}{(n-1)!} \\int_a^x \\int_a^{\\sigma_1}\\left(\\sigma_1-t\\right)^{n-1} f(t)\\,\\mathrm{d}t\\,\\mathrm{d}\\sigma_1 \\\\\n&= \\frac{1}{(n-1)!} \\int_a^x \\int_t^x\\left(\\sigma_1-t\\right)^{n-1} f(t)\\,\\mathrm{d}\\sigma_1\\,\\mathrm{d}t \\\\\n&= \\frac{1}{n!} \\int_a^x \\left(x-t\\right)^n f(t)\\,\\mathrm{d}t\n\\end{align}\n", "14e4dc358686033decc36698b281bdf2": "\\dot{p}(0)=-m\\omega^{2} x_0 ,", "14e4f0aea0e961a68d28dc6b2d387e46": " y = 3x - 3 .", "14e5018d20c891631cae4ae7ba235221": "S(n)=n+1. \\, ", "14e51a4c8a314d559a6647d06ab34e47": "A \\cup (B \\cap C) = (A \\cup B) \\cap (A \\cup C)\\,\\!", "14e5500feea0eced4cd7056759255f38": "C= dE/dT = k \\beta^2\\langle E^2 \\rangle_c = k \\beta^2(\\langle E^2\\rangle - \\langle E\\rangle ^2)", "14e56f4ef5e41d392b474a872b31a68c": "\\{\\to,\\land,\\lor\\}", "14e5c4069fe2064426aa3e4c95c000ef": "N(u) = 1", "14e5dac99630fc6a1bf6df6a3c9bacab": "\\Delta E_A", "14e63f3d1ab0e0df716d0062e050ed9f": "\\nabla_X Y - \\nabla_Y X = [X,Y]\\ ", "14e6981dd7f5eb726461212591524113": "\\gamma^r", "14e6a8efaff1957e45e78449344e0202": "S_1 = \\alpha^{i} + \\alpha^{i'}", "14e6d76fde0d20cf0a85be806816b91d": "i \\neq j ", "14e727541ba6c3c44af619177d541452": "\\{\\psi(x),\\psi(y)\\}=\\{\\psi^*(x),\\psi^*(y)\\}=0 \\, ", "14e728185b3120aff615ef7fc2831b65": "S^{(\\omega)}", "14e7449108e1538f44efcbd44b2eb5f9": "u = \\frac{J_3 P^0_3(\\sin\\theta) }{r^4} = J_3 \\frac{1}{r^4} \\frac{1}{2} \\sin\\theta (5\\sin^2\\theta -3) = J_3 \\frac{1}{r^7} \\frac{1}{2} z (5 z^2 - 3 r^2)", "14e7552457ef87f15eb79aec9f2ff8f8": "\\mbox{Area}\\;=\\;T \\,+\\, 2\\left(\\frac{T}{8}\\right) \\,+\\, 4\\left(\\frac{T}{8^2}\\right) \\,+\\, 8 \\left(\\frac{T}{8^3}\\right) \\,+\\, \\cdots.", "14e7a3651122c733dfee2fa075d7036e": "(x_1, x_2, \\ldots, x_n)", "14e7b6aa827e74a8cb0df9ddc25d580f": "F \\models_\\text{pref} G", "14e7d26a80421f01577ac03673b81fdd": "\\beta=0.280169499023\\dots\\,.", "14e7ff9975624bbc495c2d14d82f4717": "\\prod_{k=1}^4(k+2) = (1+2)(2+2)(3+2)(4+2) ::= 3 \\times 4 \\times 5 \\times 6 = 360", "14e8117c2348050446945d8166ad965e": "c(u,v)-f(u,v) > 0", "14e815a1046597f66ca63a4b4133084d": "C^{++} = (C^+)^+", "14e824db17a01698432596793cb25682": "1944 = [33, 30]_{58}", "14e82900ada7fada0846c3e43a183b0e": " \\Omega_j = \\theta_j+\\phi_j", "14e837d978b467fda5c9915b270fd86d": "m_c", "14e84173ffef7c811198aacca371259f": "{v_C^2 \\over 2}-gh_C+{P_\\mathrm{atm} \\over \\rho}=\\mathrm{constant} ", "14e8421b04c1eda501d764aa0740d9b3": "r = r(s),\\ \\theta = \\theta(s)", "14e853a522a4b902e97bbc63774e5481": "\\deg(cP)=\\deg(P)", "14e8b1f27cf4ec764e59ade3659a8143": " \\phi \\, ", "14e8f43a5bae11336e6f8a330756a580": "\\sum_i \\Big(\\sum_\\alpha a_{i\\alpha} X^\\alpha\\Big) \\otimes b_i = \\sum_i \\sum_\\alpha X^\\alpha \\otimes a_{i\\alpha}^p b_i,", "14e927836ae1392b721ac9aa99652fda": " \\int \\mathrm{Det}({\\partial F\\over \\partial G})e^{iS_{GF}} DA \\,", "14e93652d91ab9b286325e01d678ece0": "\\vec{W} = M \\vec{L} + \\vec{P}\\times\\vec{C}", "14e968b38a8cc064505594710aad830d": "\\mu_2 = \\frac{ 2 }{ \\bar{x} } \\left[ 1 - \\sqrt{ 1 + 2 ( c^2 - 1 ) } \\right]^{-1}", "14e979873497c4bd4af70de5b3b0dee6": "e^{S(x)}\\,", "14e9ec72509a2fedf308bdfd9f2852e8": "2^{803}-2^{402}+1", "14ea01d3a995a921f7ad8b2334a4f1af": "\\frac{1}{p}\\!", "14ea08eaf6b4195da9032a3fccad5b73": " S_{xy} ", "14eab2048d7664df893a5adbdb453601": "\n lu = g(x), x \\in \\partial D \\,\n", "14eac3aadba72699512f712a5324af1e": "\\{ba, abab, aababb, aaababbb, \\dotsc\\}", "14eacef4b222c0df06a463b3a6c7c453": "\\beta =4", "14eafc1151f2f5b3dbee9a115c930330": "\\frac{d^2y}{dx^2} = \\frac{d}{dx}\\left(\\frac{dy}{dx}\\right).", "14eb177540f56701669c15f438deac9b": "e^{i\\pi}, e^{i\\pi\\sqrt{2}}, e^{i\\pi\\sqrt{2}}, e^{2i\\pi}.", "14eba0ad299228f8b098728d0165ba31": "P(x_2) - f(x_2) = - \\varepsilon\\,", "14ec0e00142389334d2b900b57ab4409": " P_u = u u^\\mathrm{T}. \\, ", "14ecadbf797b09b7456d26996ad4ffb0": "\\gamma(0)=x", "14ed1849368bff14300b12c5fed3b49d": "A_j\\,", "14ed61f971f989af85220c5edc55aae2": "H = \\frac{1}{2m}\\left(p - \\frac{e}{c}A\\right)^2 + e\\phi - \\frac{e\\hbar}{2mc}\\sigma\\cdot B.", "14edb01b881a33a8d514f97e9a7c0d95": "x\\cdot (2+x)", "14ee1a75229267df9f2b4f5c258a1875": "\\frac{\\Diamond p\\land\\Diamond\\neg p}\\bot", "14ee368469aa2f9e06e2e90f1cc1854a": "\\int_{-\\pi}^{\\pi}f(e^{i\\theta})\\overline{g(e^{i\\theta})}\\,d\\mu", "14ee48626bfe01caf79e6127438e2ff7": "q = \\mathbf{S}(q) + \\mathbf{V}(q)\\,", "14ee7570caf0393954381bb7cf7f249c": "f( a + x h ) \\approx f(a) + x \\left(\\frac{\\Delta f(a) + \\Delta f(a-h)}{2}\\right) + \\frac{x^2 \\Delta^2 f(a-h)}{2!}.", "14ee907b38cab40339b0f89490279d33": "FMV\\,=\\,{VBAB * TAB_{factor}}", "14eea6eed865ff34a70d0289907bcf8d": "\n\\frac{\\mathrm{d}f}{\\mathrm{d}t} = ROCOF = \\frac{\\Delta P f}{2GH}\n", "14ef32eece00cf2399260cc8441c2fd8": "\\frac{P_2 B}{P_1 P_2}=\\frac{\\sin \\beta_1}{\\sin \\delta}", "14ef5ea590fe892c51d01b02b899e304": "\\frac{1}{n!}\\omega^n = \\operatorname{pf}(A)\\;e^1\\wedge e^2\\wedge\\cdots\\wedge e^{2n},", "14ef66acbd0d15ac6d0a4d12f4599822": "b_i(x)=0", "14ef8444279f153a690863dacd7076dd": "A_{k+1} = A_k \\cup \\{\\sigma(k+1)\\} \\, ; \\quad S_{k+1} = S_k + a_{\\sigma(k+1)}.", "14efd16f77ac0f11999509a4feed1ae2": "\\bar{r}_{i\\cdot} = \\frac{\\sum_{j=1}^{n_i}{r_{ij}}}{n_i}", "14efd330a815e8144983aff5d97ab918": "p(\\sigma[1] \\sigma[2] \\ldots \\sigma[L] \\gamma[1] \\gamma[2] \\ldots \\gamma[L]) = \\prod_{t=1}^{t=L} p(\\sigma[t] \\gamma[t])", "14efdcfb0beb639d80cd0151e0b9a023": "\\begin{align}\nL' & =\\Delta x'-v\\Delta t'\\\\\n & =\\gamma L_{0}-\\gamma v^{2}L_{0}/c^{2}\\\\\n & =L_{0}/\\gamma\n\\end{align}", "14f088991d1ce8b0abb86ef2ae1c1761": "\\sup_{x\\in\\mathbb R}\\left|F_n(x) - \\Phi(x)\\right| \\le {0.33554 (\\rho+0.415\\sigma^3)\\over \\sigma^3\\,\\sqrt{n}},", "14f0e52b57f95fb4e77f1f09b3c7f990": "H^{(m)}_n=\\sum_{k=1}^n \\frac{1}{k^m}", "14f11e2103b92b2f68e484123f74a14d": " W_{4\\to 1} = \\int_{V_4}^{V_1} P \\, dV, \\, \\, \\text{zero work if V4 equal V1} ", "14f12a1313d53610d9f8756932be1cc0": "\\omega_{\\Lambda_1}\\omega_{\\Lambda_2}", "14f1c4eb7f8fdbe942495db69dbc4ad3": "\\scriptstyle {d = {\\left(\\frac{{L_{\\ast}}/{L_{\\odot}}}{S_{eff_{\\ast}}} \\right)^{0.5} AU}}", "14f217fadc773bd00a64218d27e3b66a": "\\delta[f]=|f(0)| < C \\|f\\|_{H^1}.", "14f232890699b03ebdd8e6beb9353e55": "\\vec{\\bold{r}}_{n,n'} = (r_x,r_y,r_z) = d (l,m,n)", "14f23d40e08e1498a64b4aeb5f721523": " T = \\frac{J_T}{r} \\tau= \\frac{J_T}{\\ell} G \\theta", "14f4099da421c9878a59afa621ec8967": "B^a{\\!}_\\mu", "14f4912d41bf5ff96f87cc8b42669d7e": " f_n(z) = \\sum_{k \\ge 0} \\left( \\frac{k!}{|G|} [z^k] g(z)^n \\right) \\frac{z^k}{k!} =\n\\frac{1}{|G|} \\sum_{k \\ge 0} z^k [z^k] g(z)^n = \\frac{g(z)^n}{|G|}.", "14f4c987f49761b29dfd1171e4e2d760": "\\ \\sigma^' = \\sigma + rG(R) = \\sigma + r(\\sigma^+ - \\sigma)", "14f55ae9d0ebd3eb73959ed95484f885": "A=3\\sqrt{7+4\\sqrt{2}}", "14f56fce5fd6843066f94d0cf1b5ab65": "g:TQ\\to T^*Q", "14f649691eccc315695d6820e91d23cb": "(1/2)n^2", "14f64f483ef962ab2971c4fbdc26b645": "\n\\begin{align}\n\\int\\limits_{0}^{2\\pi}{\\left(\\frac{p}{r}\\right)}^2\\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right) \\cos u \\ du \n&=\\ 2\\ e_g\\ \\left(5\\ \\sin^2 i \\ \\int\\limits_{0}^{2\\pi} \\sin^2 u \\cos^2 u \\ du \\ -\n\\int\\limits_{0}^{2\\pi} \\cos^2 u \\ du \\right) \\\\\n&=\\ 2\\pi\\ e_g \\ (\\frac{5}{4} \\sin^2 i-1)\n\\end{align}\n", "14f6a18518c6902aa906eabb60f7b66b": "f(x,y,z)= 2 x + y + 5", "14f6a3821d108d5e53faf8c51b13b3b8": "\\exists x\\, P(x) \\wedge \\forall y\\, \\forall z\\,((P(y) \\wedge P(z)) \\to y = z).", "14f6ccbc8859573f49103a62202e3167": "f'(x) = \\operatorname{st}\\left(\\frac {f(x+h)-f(x)}h\\right).", "14f7895e8ee0209999ff6cf4ea07aa85": "1+z\\le e^z", "14f7bcacf24a58c5902460087ff4f86e": "\\bigg. J = - D \\frac{(S_1 - S_2)}{\\delta} \\bigg. ", "14f7be14fb8bbc9179e4a41cf7690817": " \\delta W = \\sum_{i=1}^n \\mathbf{F}_i \\cdot \\delta\\mathbf{r}_i.", "14f7fa8af01a9b50dce5a51a7fcd099c": "X\\preceq Y\\ ", "14f7fda76e2708132fc2f568c4a9c908": " \\lambda_j = -\\frac{j^2 \\pi^2}{L^2}", "14f81bc3c2ca62c6c63f2632b24e9d1e": "\\tfrac{63}{462}", "14f89cef1062ee7f82d5d964ea375ea5": "Z_C = \\frac{1}{sC} ", "14f8fc1f800491824aaaf4800145c733": "w_{\\alpha \\beta}( n_\\mathbf{p} + 1 \\leftarrow n_\\mathbf{p} ) = (n_\\mathbf{p} + 1) \\left( 1 - {n_\\mathbf{p} \\over \\Omega} \\right) w_{\\alpha \\beta}( 1_\\mathbf{p} \\leftarrow 0), \\quad (16)", "14f910ab18310ff58773fe94892e084c": " J > q_{0.95}^{\\chi^2_{k-\\ell}} ", "14f942c48031a21e13e002e587fec84e": "\\underbrace{a\\diamondsuit\\cdots\\diamondsuit a}_{k\\text{ factors}}.\\,", "14f9963f71cf4cf7c6ea41a98a2f6264": "A_n =(5 \\times 92^\\frac{36-n}{39})^2", "14f9ef1130fcff22dd2d61cb33fab0a0": "y_j = \\sum_i w_{ij} x_i ~~\\textrm{or}~~ \\textbf{y} = w\\textbf{x}", "14fa875aefc7fc270d389366da600ffb": "\\phi\\left[W_u(x)\\cap E_i\\right] \\supset W_u(\\phi x) \\cap E_j ", "14fa8b16e3989bcd739bc66b98862309": "{(x_1 - m')}/{\\sigma},... ,", "14faf2b6cef7cba1384bbc9fc72534db": "\\hat{\\mathbf{R}} = {1 \\over n}\\sum_{i=1}^n x_ix_i^\\mathrm{T}.", "14fb1c6a8602e0cbb50d1a78e2fd9f02": "\nE_{\\ell r} = \\sum_{\\mathbf{k}} \\tilde{\\Phi}_{\\ell r}(\\mathbf{k}) \\left| \\tilde{\\rho}(\\mathbf{k}) \\right|^2\n", "14fb2328bf4cca2d2c0389f7212d5ddb": "f_y(0,0) = p_y(0,0) = a_{01}", "14fbc1fca5dd685bed59a70d040cd9c8": "\n \\begin{align}\n \\boldsymbol{\\nabla} \\mathbf{v} &= \\sum_{i,j = 1}^3 \\frac{\\partial v_i}{\\partial x_j}\\mathbf{e}_i\\otimes\\mathbf{e}_j = \n v_{i,j}\\mathbf{e}_i\\otimes\\mathbf{e}_j \\\\\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} & = \\sum_{i=1}^3 \\frac{\\partial v_i}{\\partial x_i} = v_{i,i} \\\\\n \\boldsymbol{\\nabla} \\cdot \\boldsymbol{S} &= \\sum_{i,j=1}^3 \\frac{\\partial S_{ij}}{\\partial x_j}~\\mathbf{e}_i \n = S_{ij,j}~\\mathbf{e}_i ~.\n \\end{align}\n ", "14fc0108eef4e4abd43be4964b3b4205": "\\mu_{v \\to u}(x_v)", "14fc28c715b45f9fbac4ee662cdcd886": "c_{ii}>0.\\ ", "14fc363aee31b904874a250a5641ce4e": "u > c", "14fc418fd94add531ebafd67f995303c": "\\frac{\\mathrm{d} (T_{h})_{*} (\\gamma)}{\\mathrm{d} \\gamma} (x) = \\exp \\left( \\langle h, x \\rangle^{\\sim} - \\tfrac{1}{2} \\| h \\|_{H}^{2} \\right),", "14fc712ae6153c5256ad88904015ee06": "C_f = \\frac{0.0583}{Re^{0.2} },", "14fcd29ee2244be10098df07fc1e5fda": "PDI=\\frac{M_w}{M_n}", "14fcfeb03a9b6d1ce7060a7484cf09e9": "f(x,y) = x - 2y + 2", "14fd27ac01b79e967a2d348e1110449a": "\n\\begin{align}\nS_{n+2} &= \\int_0^{\\pi/2}S_1 r . S_n R^n\\, d\\theta =\\int_0^{\\pi/2}S_1 . S_n R^n\\cos\\theta\\,d\\theta\\\\\n&=\\int_0^1 S_1 . S_n R^n \\,dR= S_1 \\int_0^1 S_n R^n \\,dR\\\\\n&= 2\\pi V_{n+1}\n\\end{align}\n", "14fd595278208455417e37733d105cde": "P_{1},...,P_{m}", "14fda63bfc585789c4388e47899edf02": "f = a_0 + q + \\sum_{i=2}^\\infty a_i q^i", "14fdae055857594646650f6df422d385": " 1/r_{12} ", "14fdae72d286220a51ca6aab0b568d2c": "a \\equiv \\omega^2 LC \\left[ \\left( \\frac{R}{\\omega L} \\right) \\left( \\frac{G}{\\omega C} \\right) - 1 \\right] ", "14fdd71a84ad2e348e68be392fe131a5": "\n \\frac{z-2}{z+2} = \\left( \\frac{\\zeta-1}{\\zeta+1} \\right)^2. \n", "14fdd8e49e85cb2b687f53ba28e6ecec": "f*V_n = f \\,", "14fde9573cfaf3cfa5b052f0fcb4b08b": "d\\,' = c\\cdot(0-t_1') = -c\\,t_1'", "14fe3060ebf011d3c394abf02d7792f7": "{}^{n}({}^{1/n}a)=\\underbrace{({}^{1/n}a)^{({}^{1/n}a)^{\\cdot^{\\cdot^{\\cdot^{\\cdot^{({}^{1/n}a)}}}}}}}_n\\neq a", "14fe6905fe9c9dce3fe936ec30de29d5": "\\left( \\frac{\\mathrm{d} \\alpha_{1}}{\\mathrm{d} t}, \\dots, \\frac{\\mathrm{d} \\alpha_{n}}{\\mathrm{d} t} \\right),", "14fe9153f6d0831a07952bf376858324": " {} = p^{03} z_0 + p^{13} z_1 + p^{23} z_2 . \\,\\! ", "14fe9938cfd54da7f47f41f9838bf062": "F_{2,n'}", "14ff1c57811659da83f8031d400fc6a7": " | \\langle R_n, g_{\\gamma_n} \\rangle | ", "14ff45db1b78c1fdc0dabb7b41c29e98": " c(r)=e^{-\\beta w(r)}- e^{-\\beta[w(r)-u(r)]}. \\, ", "14ff730f8df66f8c730b44fd84f4fde4": "\nb = x - x_i,\\quad i = 2, 3,\\ldots, n\n", "14ffde9a267388f79d46fbb5af499f56": " \\frac{b^{2n+1}}{n!} I_n\\left(\\frac\\pi2\\right) = P_n\\left(\\frac\\pi2\\right)a^{2n+1}. ", "15004284acb060b9a39420f3c7b2e3b4": " r= R \\frac{h}{H} ", "150086db11283ee59943128676d31ad8": " b^\\alpha ", "1500f358bc15e4cdd7f46f7b298fd765": "\\begin{align} 2\\cdot R_*\n & = \\frac{(180\\cdot 5.60\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 216.8\\cdot R_{\\bigodot}\n\\end{align}", "15010fff3925f351e1bbf8957584fcb8": "\\mu(AB+CD)=\\mu(AB)\\mu(CD).\\,", "15011e5d1c43a1d140a287cd0085e337": "\\nabla_{fX}Y = f\\nabla_X Y", "15014092a8f15a1121bc8170bbb8f379": "\\scriptstyle{(Rt)}", "15015f837616fa6e4b9bf1e892401f3d": "\\ddot{x}_a + \\frac{2}{\\tau_0}\\dot{x}_a + \\omega_a^2 x_a = \\frac{e}{m} E(t,\\mathbf{r}_a)", "150182dd3ab31c1af4c7f6cb4518a55b": "TX_i = X_{i+1}", "15019f6b74fa698963000a977d7e29fc": "\\sigma_{r\\theta}=0", "1501c9b5f71363c53b3856c0fba84f66": "p_{X}", "1501fed59dfe33c518b7f85d9e4ffc79": "f(\\vec{x},\\vec{y},\\vec{z})\\rightarrow g(\\vec{x},\\vec{z})", "1502005c3c9100ff68265f92c5ec021a": "\n\\left.\n\\begin{matrix}\n L_1u\\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_i f_1^i(x)\\frac{\\partial u}{\\partial x^i} &= 0\\\\\n L_2u\\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_i f_2^i(x)\\frac{\\partial u}{\\partial x^i} &= 0\\\\\n \\dots&\\\\\n L_ru\\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_i f_r^i(x)\\frac{\\partial u}{\\partial x^i} &= 0\n\\end{matrix}\\right\\}\n", "150231d2a099a1286f8acdcc2cc9518c": "(a^2+b^2)(q^2+p^2) = (aq+bp)^2 + (ap-bq)^2\\,", "1502556194c82dc003affc7ed12da9b0": "\\chi_i(s) = \\frac{\\langle \\mathbf{e}_i'(s), \\mathbf{e}_{i+1}(s) \\rangle}{\\| \\mathbf{r}'(s) \\|} ", "1502b72b6db27630a1994d2983074858": " n \\approx \\sqrt{1 + \\frac{3 A p}{R T}}", "1502edcb8e318e91227d5aa554047da3": "\\hat{\\alpha} = \\frac{4GM}{c^2b}", "150316342acccb9435deeb75a212f262": " g_{ij} ", "15031fcfb750ea831a00f623b4d9e3af": "q \\prec p", "150327d942f4c55b12c36dffd94535cc": "A = \\begin{bmatrix} 0 & 0 & 0 & -1 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\end{bmatrix}.", "1503409db55a18ec1dd9b8dfa7e4e459": "k \\ge 1", "15035a174ce023dce5f584573f1f19cf": "q_{n+1} = 2 p_n q_n \\,\\!", "1503775f37a96a5cf4b34f2940cbc146": "\\color{Black}\\tfrac{8}{m}\\tfrac{2}{m}\\tfrac{2}{m}", "15038d33b3f2772ebf6b368b9149b4a6": "P(x) \\to (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\to Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "1503ccb0af6e39fa198588a0113d9cc6": "\np = \\hbar k\n", "150427b5fb001afd6ec41d30f1bcdce0": "\n\\sqrt{-g}A^a_{;a} = (\\sqrt{-g}A^a)_{,a} \\;\\mathrm{or}\\;\n\\sqrt{-g}\\nabla_\\mu A^\\mu = \\partial_\\mu\\left(\\sqrt{-g}A^\\mu\\right)\n", "150462ed90f31805e32b937780144b45": "U(\\zeta)", "15049f9a6e066d06efa33ff8af0fabca": "(p_1,\\,p_2,\\dots,\\,p_n)", "15053aadb91c9f5a5efc3ce39367f793": "K_i = \\frac{P'_i}{P}", "15057f605aa5f1b57a03ff73f2f2e7e9": "g(x)=(x^{2t-1}-1)p(x)", "1505c7ec876ec273313aebb7769a5fdf": " p(r_1 , r_2) = P - \\frac {2 \\gamma\\, \\rho\\, _{vapor} } {(\\rho\\,_{liquid} - \\rho\\,_{vapor})}\\left ( \\frac {1}{r_1} + \\frac {1}{r_2}\\right ) ", "1505f59c1c3d31d3fa713ec9440c6ae2": "\nf^\\star\\left(x^{*} \\right)\n= \\begin{cases} x^{*} \\ln x^{*} - x^{*} , & x^{*} > 0\n \\\\ 0 , & x^{*} = 0\n \\\\ \\infty , & x^{*} < 0.\n \\end{cases}\n", "150642ff0a49be7ffd3faffa4fb52762": "t^3 + pt + q = 0,", "15065755f2f20d486095164fef11bb63": " \\textbf{a} = \\textbf{f} \\left(c\\textbf{h} + c\\right) \\pmod q ", "15067153ac49cb6bc4cf7cf472f28523": "C_{dyn} = \\frac{{V_T}}{{PIP-PEEP}}", "15068c177d578c057a0b98d4a8c3b1bd": "\\left(r,s\\right)", "15068c2d533d79587ab59a73fa9d378a": "\\mathbb{E}[2e' + s'] \\le {2 \\over d}\\sum_ie_i < D", "15071208cf21091b3e04d4bb14603d9c": "\\mathbb{Q}\\sqrt{5}", "15078e8cc049f69483b5918ce79ce653": "x' = \\gamma x - \\gamma \\beta c t \\,", "1507ae8ade9566f9e6b996bc31888dac": "K^m", "150820c4df1e69386930ddd44e181551": "l\\propto \\frac{1}{n\\sigma} ", "150833fdbd38dece93356812d407a693": "e^{i\\mathbf{K}\\cdot\\mathbf{(r+R)}}=e^{i\\mathbf{K}\\cdot\\mathbf{r}}", "1508aa9619e3c53a1d56933466e321a3": "\\ln \\sqrt[n]{c} = \\frac{1}{n}\\ln c", "1508b304b2d4c28407e34c00f33a2b07": "f(\\mathbf{v}) = f_1(\\mathbf{v})~ f_2(\\mathbf{v})", "15094c7e1bf7dd22c54932635dfe3490": " \\{ \\cdot,\\cdot \\}_{M \\times N} ", "150a3fd845c2dbf05b9a3659986a5e70": "T_M(\\rho;E)", "150a4aa8b05eadcd26b1fe8bd55527b3": "\\operatorname{Ass}_R(R/I) = \\{P\\}", "150a4aaf80da7ea45c01150e591c8a13": "S\\!\\left(x\\right)", "150a544559f3d554683a1bfaffaced15": "X - Y \\sim \\operatorname{Logistic}(0,b).", "150ad83ebf01ecac380c61775d2aa342": "N(\\alpha) =\\left [ \\mathcal{O}_L: \\alpha\\right ]=|\\mathcal{O}_L/\\alpha|.\\,", "150ae39e08abc996b08d5b1e0f0d6481": "C = \\frac{(1 + 2 + 3 + 4 + 6)^2}{5} = 256/5 = 51.2", "150ae4176b3748297b1d68ff3c228282": " \\varepsilon_{ij} \\varepsilon^{in} = \\delta_i{}^i \\delta_j{}^n - \\delta_i{}^n \\delta_j{}^i = 2 \\delta_j{}^n - \\delta_j{}^n = \\delta_j{}^n \\,.", "150b4470ec420b2cf5eba37b76ea98fb": "\\textstyle S(1)", "150b4514a331722c64b5b83d122ac058": "g^k \\mod q", "150b4983505b125e2d4eff8ab6ac637c": "P \\and P", "150b49b5eb21585e772ae97522446705": " g_{ab,c} = \\frac{\\partial {g_{ab}}}{\\partial {x^c}} ", "150bb819ab8054324855b2370b8f49a1": "\\displaystyle{\\|\\mu_F\\|_\\infty < 1.}", "150c47f986044c824dc942cf025fecaa": "\\scriptstyle OA-BA=1", "150c5598edacfabbb28a9ec6072a7b50": "h : B \\rightarrow B'", "150cc46016b2f4435fcae2e1f7aa30bd": "\\frac{\\partial \\rho}{\\partial \\sigma} = \\frac{\\partial^2 V}{\\partial \\sigma \\, \\partial r}", "150cd263a96f033fa2dc6682c5a8731f": " B \\subseteq A ", "150ce31a4ef9a9b78415da03e8e38b3b": "\\operatorname{MSE}(S^2_{n+1})=\\operatorname{E}((S^2_{n+1}-\\sigma^2)^2)=\\frac{2}{n + 1}\\sigma^4", "150d051c6e4821c4a1dd479cd849bba8": "\\mathbf{r}(\\boldsymbol \\beta)\\approx \\mathbf{r}(\\boldsymbol \\beta^s)+\\mathbf{J_r}(\\boldsymbol \\beta^s)\\Delta", "150d0f570b3eef5c7238379d5d879a25": "\\bar u \\pm 3\\sqrt{\\frac {\\bar u}{n}}", "150d4865acae3768a83fa56116da4865": " = \\operatorname{Tr}\\left( p\\!\\!\\!/' \\gamma_\\mu p\\!\\!\\!/ \\gamma_\\nu \\right) + \\operatorname{Tr}\\left(m \\gamma_\\mu p\\!\\!\\!/ \\gamma_\\nu \\right) \\,", "150d7c7e2dd4107bdb47b66d3877d37c": "\\textstyle [n,k] l", "150df0d71af2b92ab4f8e1c7c35c85f9": "result(a,s)", "150e01e6b022c347c46f3d1322d9bde3": " { {d {u}^{\\mu}} \\over {d\\tau}} =0 ", "150e765a17475a35d6e35834f9762073": "T_{eff} ", "150e8b9dbfcfc60608c9f1dba1850fd1": "\\Gamma:=\\{S:D\\to\\Sigma\\; :\\; D\\subset[0,\\,c],\\, S\\; \\mathrm{ monotone }, \\forall t\\in D\\; (\\mu\\left (S(t)\\right)=t)\\},", "150e9195c9094d721e5a192f7f56fb60": "\\Delta C", "150edf7e9d0efd19d4390b02810351d5": "\\begin{align}\n I_{\\text{C}p}(W) &= I_{\\text{E}p}(0) \\\\\n I_{\\text{C}} &= I_{\\text{C}p}(W) + I_{\\text{C}n}(0') \\\\\n I_{\\text{C}} &= q A \\left[ \\frac{D_\\text{B}}{W} p_{\\text{B}0}\\left(e^{\\frac{q V_\\text{EB}}{kT}} - 1\\right) -\n \\left(\\frac{D_\\text{C} n_{\\text{C}0}}{L_\\text{C}} + \\frac{D_\\text{B} p_{\\text{B}0}}{W}\\right)\n \\left(e^{\\frac{q V_\\text{CB}}{kT}} - 1\\right)\n \\right]\n\\end{align}", "150f0d9abdee95483fd4572a8c7ac008": "\nE_0 = q^2 - pr = 0.00382 \\,\n", "150f66c3c540d4b574aeb17a41d2c382": "\n p = \\cfrac{2C_1}{\\lambda}\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right]~.\n ", "150fc79f870bb19c00b1e1fa395573d3": " y\\mapsto \\int_{\\pi^{-1}(\\{y\\})} f(p)\\,\\lambda_y(dp) ", "150ff1eda68b50306a5a270ca2d0db24": "\\frac{\\pi}{6} \\ (30^\\circ)", "150ffe363439e1f1f6f2429f86bca791": " |\\psi_s^b \\rangle ", "1510679c3b07d27a4670ddc13a8e161c": "a \\sin \\omega t", "151076f35bd13d0ab9e8d8616377de69": "=\n\\frac{\\textrm{Li}_{\\alpha\\!+\\!1}(z)}{\\zeta(\\alpha)}\\,\\tau^\\alpha", "15107b1bc6e82ecbc0b6b9f931380685": "a\\perp b", "1510f277f2474fdff24874670fdba377": "g(x) = \\prod_{j=1}^{n-k} (x - \\alpha^j), ", "1510f60bda74f3cac21a091a35aedece": "F_{[\\alpha\\beta;\\gamma]}=\\frac{1}{3}\\left(F_{\\alpha\\beta;\\gamma} + F_{\\beta\\gamma;\\alpha}+F_{\\gamma\\alpha;\\beta}\\right)=\\frac{1}{3}\\left(F_{\\alpha\\beta,\\gamma} + F_{\\beta\\gamma,\\alpha}+F_{\\gamma\\alpha,\\beta}\\right)= 0. \\!", "15111106d10f28d0059f965773350f68": "b(G, H)", "15111d4f9ec708b6f79b41f3d4a5aa43": " P= \\dot{m} ( h_{02} - h_{01} )= \\dot{m}c_p (T_{02}-T_{01})\\,", "1511728e9d651d34552bb2af8af9ddc0": "f(-1)", "1511baf90a7bc3f076ff8056129474c8": "x^py^q=k", "151216ca439c54d0612b4c743374af87": "\\left(\\begin{array}{ccc} 1 & 5 & 0\\\\ 0 & 9 & 2\\\\ 1 & 7 & 3\\end{array}\\right)", "151216e796580350e1238ad5a4bc2e53": "\\pm \\sqrt { \\left( \\frac {1} {\\tau_1} - \\frac {1} {\\tau_2} \\right) ^2 -\\frac {4 \\beta A_0 } {\\tau_1 \\tau_2 } },", "151217d0d9779b8b6564a5ec3be975e4": "\n \\mathit{JB} = \\frac{n-k}{6} \\left( S^2 + \\frac14 (K-3)^2 \\right)\n ", "15124aa83640bccc56c3246826e1cc7e": "\\bar{\\mathbf{\\gamma}}(s) = \\gamma(t(s))", "15125e137fbe755e4b4ec77d56d4056b": "\n (A + B)^o = A^o \\cap B^o.\\,\n ", "151368e97f0d752291cadd2b01079783": "k = \\theta(log N)", "15136ad82613a162356f1f1aa10627e6": "\\left\\lceil\\frac{\\left\\lceil a/2^{k-1}\\right\\rceil}{2}\\right\\rceil = \\left\\lceil\\frac{a}{2^k}\\right\\rceil", "15137e2dc3b46a1d5472af44f82551d4": "\\mathbb{F}_2[x]/(x^3-1)", "15139ae10521fdb90d95c3d2ef738484": "\\Delta \\Phi\\, =\\, 0 \\qquad \\text{ for } -h(\\boldsymbol{x})\\, <\\, z\\, <\\, \\eta(\\boldsymbol{x},t),", "151419191374b698538f179047ec5bf0": "{\\rm WF}(f) = \\{ (x,\\xi)\\in T^*(X) \\mid \\xi\\in\\Sigma_x(f) \\}", "15143e5a892f05ce9636a1fa864ea05d": "{ll_2}", "1514880b9856b1d62bbf058df1c7afdf": "\n\\sigma^z = \n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "1514bbf7d7e295eaf6b960af77c92921": "m= 100", "151508648895934a41f926f8102093e4": " \\cos y = \\sqrt{1-\\sin^2 y}", "15152205c454443c327089e502ed8510": "2p", "15158307ff0f2a0c1d7d1654fb96b2cd": "\\frac{(-1)^n - 1}{\\pi n^2}", "1515b1c11a95cb1984ed2bd06ab9b26c": "\\textstyle\\frac {Ebm/Gb}{C drop 2}", "1515c4a830a7f60210434975d23b9cd4": "W[1]", "1515efe38a8c56411d967e24dbaa23d0": "f_k(x)\\leq f(x)", "15160f77fd22265e9afa67b9d3e6b0b7": "E = \\sum_{ij} J_{ij} S_i S_j + \\sum J_{ijkl} S_i S_j S_k S_l \\ldots.", "15163700181cddff467e3121e89e9974": "w(n)= 0.5\\; \\left(1 - \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right) \\right)", "1516421e7f1f92e4078e73195701df33": "\\sum x_i = 1", "1516ef6f969afd21d668ce4df621edd1": "v(p+r,t)", "151731dbbdc3f3241411b1c86a52f838": "\nY_{i,j} = \\mu + T_i + \\mathrm{random\\ error}\n", "1517de4431af1c827fef408ac03fcbfc": "X_d", "1517f7076ceb5192d342ae71ad2c324e": "\\int_0^L \\mathrm{d}s", "151845be9c6261c712182bb00fcebfa7": "{x_1,x_2,...,x_j}", "1518ede48e1f7eaf21b2ffcf9f5acd86": "\\scriptstyle \\leq2\\times10^{-7}", "151904e8a1b740fd49a18f7b06b4847b": "\\mathbf{C'}", "15194615d32f041186d6d51a2166e799": "\\operatorname{first} \\equiv \\lambda p.p\\ (\\lambda x.\\lambda y.x) ", "1519ae5f5e8ce3272158240fd5681a95": "f(x,y) = x^2 + y^5.", "1519bde1b34c6a43fa52f03b7b82e2ed": "\\Delta_{uv}=0.003", "1519bfe59cd8b17182e1d261755ba38c": "\n\\psi\\rightarrow{1\\over\\sqrt2}\\left[\n\\begin{array}{cc}1&1\\\\1&-1\n\\end{array}\n\\right]\\psi=\n{1\\over\\sqrt2}\\left(\\begin{array}{c}\\psi_L+\\psi_R\\\\ \\psi_L-\\psi_R\n\\end{array}\\right) .\n", "1519fe86828ff016979d0c3c11430941": "\\alpha_{mk} \\in (-2,-1,0,1)", "151a33700161bc7db6137ae5be20ff6e": "n + k - 1 ", "151a3f9c2230cbe651a43ce3a4202df9": " \\sum_{i>j}^{n}[ \\langle \\phi_i | \\vec{r} | \\phi_i \\rangle - \\langle \\phi_j | \\vec{r} | \\phi_j \\rangle ] ^2 ", "151a45f165ee02c623c89ae6f4d0e1da": "\\phi'(a)=f'(a)-y>y-y=0", "151a9fff5c179a425b234b5ec3f78b21": "\n -40 < \\Re(t) < 40\n", "151ab59fc4d37d024b9971d280067409": "\\mathrm{D} F (\\sigma)\\;", "151ae48ef2fc97b2d10feea4f5a931e4": "W_{m_1}", "151b25977466c263aaea32849e702145": " \\frac{d[B]}{dt} = 0 = k_1 [A] - k_2 [B] \\Rightarrow \\; [B] = \\frac{k_1}{k_2} [A]", "151b2998e20e1c73380c4a22e2ceaa5f": "[x,y] = (-1)^{\\epsilon(\\hbox{deg}\\ x)\\epsilon(\\hbox{deg}\\ y)}[y,x]", "151b42e448c23ac59605891968a4cf2b": "\\mathit{z}", "151b6b5f1caef6e7becd256d823da774": "M_n=\\sum_{k=0}^{n}X_k,\\quad n\\in\\mathbb{N}_0,", "151be16dba694615f34300b5671fc282": "f(x_0) = y_0", "151c60a5653d9496d06d107cccff367e": " [\\textrm{CO}_3^{2-}]_{eq} = \\frac{K_2[\\textrm{HCO}_3^-]_{eq}}{[\\textrm{H}^+]_{eq}} \n= \\frac{K_1K_2[\\textrm{CO}_2]_{eq}}{[\\textrm{H}^+]_{eq}^2}. ", "151c84c4d0ae26ab2fbdc2e2f98410e0": "\n{\\Vert r \\Vert}^2 {\\Vert{ u}\\Vert}^2 = \\sum_{i 0 ", "1523152aead8dabf961fff04942255d4": "\\mathbf{i}= \\sqrt{-1}", "152320d41632044685c51990969bccdd": "f(x,y) = U(x,y,z)\\big|_{z=0} ", "15234f112df70078944cac193de16223": "\n \\begin{cases}\n \\bar{x} = a_1 + 2a_2 \\\\\n \\sigma^2 = a_1 + 4a_2\n \\end{cases}\n ", "1523888e56250d9cb747adbc1486d0cb": "\\omega_{L}=\\sqrt{k^{2}_{z}\\left ( \\frac{C_{s}^{2}C_{A}^{2}}{C_{s}^{2}+C_{A}^{2}} \\right )}", "1523b4bcdca5d98bfe69baeaf407ed32": "|U(T)| > 0\\,", "1523c3fb576603821cbb8ce40a425d5f": "k = \\mathbb{Q}_p", "1523e675d9362bbc17a3bf2b0a7b5c9c": "I_1, I_2, \\ldots, I_n \\subset C_1", "15240e01e99f730acbca49ec179bc689": " \\sqrt{2} = \\frac{10}{7} \\approx 1.429", "1524ca696793b8e9021b13ccce9c7b66": "N_i(c)", "1524da5b0d4c26570167caaf1cebcc78": "\\mathsf{C^*\\!-\\!alg} \\to \\mathsf{KK}", "1524e58e20a937c0060ac07255b78d82": "R = \\frac{\\left(z_2-z_1\\right)}{\\left(z_2+z_1\\right)}\n", "1524ef779a6ad53fa02a94d29b4f9cd8": "C_1 \\otimes C_2 \\in \\mathcal C", "152579ec6455c2e9a01e5f05e6d6ff19": " \\mathcal{A}_2", "1525ea3b0c78924df135d9692fc11964": "\\overline{{{\\beta }_{12}}}", "15260190b3b39932051d61faa6a803f8": "n^3 - n^2", "15261e886be65a84fe205c3977bdc398": "\\scriptstyle \\frac {41}{29}", "15267992b2d9da957e79218e616c9e43": "\\mathbf{F} \\cdot \\Delta \\mathbf{r} = - \\mathbf{\\nabla} E_\\mathrm{p} \\cdot \\Delta \\mathbf{r} = - \\Delta E_\\mathrm{p}\n \\Rightarrow - \\Delta E_\\mathrm{p} = \\Delta E_\\mathrm{k} \\Rightarrow \\Delta (E_\\mathrm{k} + E_\\mathrm{p}) = 0 \\, .", "1526b87bbbadcc1e771ac725d6bdf16e": "\\left(\\frac{x}{1-y},0\\right)", "1526e884b546acfdd0d7c75918b37c53": "g(T)", "1526f04a8949c8498db6337f18c56cf3": "\\int_0^\\infty \\frac{\\log(x)}{(1+x^2)^2} \\, dx", "1526f88328a22f432dd5426202393e6f": "N(\\mu,\\sigma_v^2 )", "15270572da93f4c1e0731271ffe34725": "\\|x'\\|>r", "15272a4ba4fd4758d094293266259776": "\\alpha^A", "152761662c1d7fa527c0243bcbce5068": "B(z^2)", "152773cdcb0db8cbf1b42c4c431be858": "(2n-5)!!=\\frac{(2n-4)!}{(n-2)!2^{n-2}}.", "1527b4869090aa6a4086722308d38855": " G \\subset U ", "152805fc3733c04b94a03a799aef9a71": "U = \\mathbf d_{\\rm e} \\cdot \\mathbf E.", "1528096214598ac2ea2bd31eca0bc8cb": "m = \\rho_f V_\\text{disp}. \\,", "152834aac46fc61d71ac17144ff80e57": "\\forall x\\, (\\exists y\\, A(x) \\vee B(z)) ", "15285354e9a8489a4208c92748d0261d": "\\delta(x) = \\frac{1}{2(2\\pi i)^{n-1}}\\int_{S^{n-1}}\\delta^{(n-1)}(x\\cdot\\xi)\\,d\\omega_\\xi", "1528ea172314511404da128ef88bac03": "L=D(N\\otimes I)", "1528f6665471ce32a32e437bbf1f05c9": "L(G) = \\{ a^*b^*c^*\\}", "1528f9498312a3097a58679ee261ca7c": " |x\\rangle|y\\oplus f(x) \\rangle ", "152913f0b6e9eddbd542450d5e506089": "V_g=V_a", "1529604788d4d5ba54434fa696bdfdef": "E_s = \\frac {1}{\\delta}\\log\\left ( \\frac{k_s}{k_{\\text{CH3}}} \\right )", "1529873aacb4caca884e6cb529112111": "xy \\equiv zw \\rightarrow yx \\equiv zw\\,", "152a259377e0c9586aac82af946bfeec": "\\|A^{-1}\\delta A\\|", "152a3c96fc1f453436715a0ba6158581": "f(z)=f(z+\\omega_1)", "152a4afe03a98f543f34eead9b7c1f91": "\\bold{\\alpha}", "152aaee4411aac160251a27ba4fbde7e": "(d_1,e_1,f_1) + (d_2,e_2,f_2) = ((-1)^{e_1e_2}d_1d_2, e_1+e_2, [d_1,d_2][-d_1d_2,(-1)^{e_1e_2}]f_1f_2) ", "152aece223b8329f8210d1f20a714d90": "x(v)\\cdot Q", "152b13d725619ed50c7ae73cda0398e4": "X_2(z)", "152b1ee0c95d96ab864677eb5e6ffd73": "\\;^+T^{IJ} := {1 \\over 2} \\Big( T^{IJ} - {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} T^{KL} \\Big)", "152b3ff994a2efd470f838ae16c2771a": " \\sqrt[4]{3^4+2^4+\\frac{1}{2+(\\frac{2}{3})^2}} =\\sqrt[4]{\\frac{2143}{22}} = 3.14159\\ 2652^+", "152ba4338d171777433824e1301584a3": "\\gcd(a,b)=1", "152bbb5db0e8b594498189d41f50a517": "W, B, D,", "152bdf8fe8d1f55d0e5e47c9e6bd426b": "Ff \\to X \\to Y", "152c07732064d8afdd0632205ff344ee": "\\Omega (n^2)", "152c4eb645d0aa2601b5b0e9f96be36e": "R_t", "152c979392856a3c360ca78cd801ab30": "\\displaystyle \\frac{1}{\\sqrt{2 a}} \\cos \\left( \\frac{\\omega^2}{4 a} - \\frac{\\pi}{4} \\right) ", "152cbe1462fedaaf0758623fea90a8ed": "(x,y)\\ ", "152cf5ce5c2ca62f1e0f9dc665231ec2": "\n\\Psi_T(x(T))= \\frac{\\partial \\Psi(x)}{\\partial T}|_{x=x(T)} \\,\n", "152d2f4c5aadab30428a16a832f437f7": "\\operatorname{F}(\\dots\\mid z_d)", "152d4e0a650c261d3c1c88d99bee1690": " \\frac{\\partial \\Pi (\\mathbf{\\xi}, t)}{\\partial t} = \\sum_j \\left( \\sum_{ik} -S_{ij} \\frac{\\partial f'_j}{\\partial \\phi_k} \\frac{\\partial (\\xi_k \\Pi (\\mathbf{\\xi}, t) )}{\\partial \\xi_i} + \\frac{1}{2} f'_j \\sum_{ik} S_{ij} S_{kj} \\frac{\\partial^2 \\Pi (\\mathbf{\\xi}, t)}{\\partial \\xi_i \\, \\partial \\xi_k} \\right), ", "152d4e9fe5b2b1546a6ae0a30fa90f35": "\\sum_{n=1}^\\infty a_{\\sigma (n)} = \\infty.", "152dcd4279fa3d4d6ceebbab8b1ee972": "u_0(t) = U_0\\, \\cos\\left( \\Omega\\, t \\right),\\,", "152ddd99c0309af134f533dce4226747": "P_{(k)} > \\frac{\\alpha}{m+1-k}", "152e2e1463822418d92010dfdc8d2687": "J(0) = \\widetilde{J}(0) = 0", "152e4473322f6fa26671e66470368cd3": "m(n) = \\pi(\\phi(e,s(n)))\\, ,", "152e71a5a4dd7787348e88741789653b": "\\mathbf{x}=(n_1,\\dots,n_K)", "152e77da2ce3e36646af590bd58bcb15": " I_{t-1} = 1 ", "152e99ca18c7cc35643c38be9bf9515e": " V[c] = c\\left[ \\iint_D f \\, dx\\,dy + \\int_C g ds \\right].", "152eb48160422b123e39f6381267e9fd": "y \\in P(x)", "152ece20c43b4f01d6729e2d284791d6": "\\pi_j\\ ", "152f34470cd4f7713c1daa9b1b59075b": "p_D=\\bar{D}-D(\\bar{\\theta})", "152fccda9e719a6a6200efde7ac4aaee": "\\cos \\theta = \\frac{\\mathrm{adjacent}}{\\mathrm{hypotenuse}} = \\frac{a}{c} \\ .", "15301b741ac5d2c56353b66f7b994d62": " GDP = f(M, T) \\,\\! ", "15302c4f548df3d14e9b2d09f10cca8d": " Q_b ", "1530478861498456191bf9fa3865d235": "h(x_1,\\ldots,x_m) = f(g_1(x_1,\\ldots,x_m),\\ldots,g_k(x_1,\\ldots,x_m)) \\,", "15304814be9206143c225fb9d1bce059": "\n X\\ \\sim\\ \\text{NB}(r; p)\n ", "15306861686a7216b23ee92133a80fdc": "C \\in \\mathcal{B}_B", "1530a731f2f513d9464f84565e45dec7": " 0\\le r_{i+1}<|r_i|, ", "1530adae49bd73ee1977be18bc2dd09b": "\\begin{align} f*g & = fg +\\tfrac{i\\hbar}{2}\\Pi^{ij}\\partial_i f\\,\\partial_j g -\\tfrac{\\hbar^2}{8}\\Pi^{i_1j_1}\\Pi^{i_2j_2}\\partial_{i_1}\\,\\partial_{i_2}f \\partial_{j_1}\\,\\partial_{j_2}g \\\\\n & {} - \\tfrac{\\hbar^2}{12}\\Pi^{i_1j_1}\\partial_{j_1}\\Pi^{i_2j_2}(\\partial_{i_1}\\partial_{i_2}f \\,\\partial_{j_2}g -\\partial_{i_2}f\\,\\partial_{i_1}\\partial_{j_2}g) +\\mathcal{O}(\\hbar^3)~.\\end{align}", "1530c55e914bd04ae717ffcb80ff3a91": "k=\\frac{1}{3}vl\\frac{C_V}{V_m} ", "1530edd15ae9a59555a0ef06668ee6af": "\\frac{2m^m}{\\Gamma(m)\\Omega^m} x^{2m-1} \\exp\\left(-\\frac{m}{\\Omega}x^2 \\right)", "15313d88d0b6e7ff07ff7d0bdf43a278": "3\\uparrow\\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}} ", "15315b5728bfabbb6b18eff436d64eea": " K(\\omega) = \\frac{\\omega}{c} \\sqrt{\\frac{\\varepsilon_1 \\varepsilon_2 \\mu_1 \\mu_2}{\\varepsilon_1 \\mu_1 + \\varepsilon_2 \\mu_2}} ", "1531810abc9870b35577e941803769ab": "\\{\\ \\Delta P_1\\}", "1531948bc9853e370bd2cf110472e5a2": "\\ \\Rho", "1531c099596cb8b5438e53fbc21550b2": "V_{\\max}", "153253d4a370d4b7ea171bd4153e6da8": "\\vec{a}=a_1a_2a_3a_4 ", "15325c052618a7c29ef4f547f1bc70ff": "\\langle G, R\\rangle", "153266eb50c22e410c33eb0d4fb28514": "\\rho>1", "15333ef0377798fa111a2e02a12dca96": "e^x-1\\approx x + \\frac{x^2}{2}", "1533a14f71e135ab9f0f16e213c5d25f": "\\iota_{X_{H_\\xi}} \\omega = d H_\\xi.", "1533cd3ace9cc3783309dbb2c7c859b1": "b_i < \\max_{j\\neq i} b_j < v_i ", "153459ea620eb2c4a1ad3ca25c80bbed": "n=0,\\frac{1}{2},1,\\frac{3}{2}, \\ldots", "1534b9f9784afde6a2d99605935207b7": "(\\textbf{Q}^0)^\\ast", "1534db3f05fd785536d6c49e25478df3": "\\phi_i^{-1}(x,y) = [(i,x,y)].", "1534f6d0315948dadcec162523acf48b": "E_{\\rm y}, \\nu_{\\rm yz}", "1535bae72561af95bb05465be6f94672": "\\lim_\\omega(M,n d, p)", "15364804ddef909ce9dbd7a067611346": "\n u = \\frac{d x }{d \\tau} = \\frac{d x^0}{d\\tau} + \n \\frac{d}{d\\tau}(x^1 \\mathbf{e}_1 + x^2 \\mathbf{e}_2 + x^3 \\mathbf{e}_3) =\n \\frac{d x^0}{d\\tau}\\left[1 + \\frac{d}{d x^0}(x^1 \\mathbf{e}_1 + x^2 \\mathbf{e}_2 + x^3 \\mathbf{e}_3)\\right].\n", "153694ff307e7e859cc5b250cc78421b": " \\mathrm{nDCG_{6}} = \\frac{DCG_{6}}{IDCG_{6}} = \\frac{8.10}{8.69} = 0.932 ", "153706506053bf3e179b18fb12c9f03d": "\\alpha = \\frac{J}{Mc}", "15370e9107c6f3bf145d1ae13abfa717": "\\varphi(\\beta)(\\xi_1,\\xi_2,\\dots,\\xi_k)=\\beta(\\eta^{-1}(\\xi_1),\\dots,\\eta^{-1}(\\xi_k))", "1537225f8f3d59febf069ea982eec9c6": " v_y=v_0 \\sin(\\theta) - gt ", "1537368a2a3252c7f67cd86c5eb37e4a": "\\; q(q+1)", "1537d0503afaef98c3bf4934094bfdaa": "\\textstyle c_1 = t_1 + at_1^{-1} \\mod n ", "1537d2fc1ba0dde5b92c8f2335de8538": "\\vec{e}_1 = \\sqrt{2} \\omega \\, \\partial_x", "1537edbe9775a05d001eaf14b5863403": "\\alpha \\in\\left(0, 1\\right)\\;", "1538399e2340a8d1b0d303fb0128d85e": "P_{\\rm emt} = \\overline{\\epsilon}\\,P_{\\rm emt\\,bb} \\qquad \\qquad (5)", "15386cb992b107a89d26a7cd42ec044c": "\\mathrm{Re}(\\tilde{k}) = k", "1538becd4f0ffdd4cde7c3d025b70622": "\\gamma_i<\\Omega", "15390ff7aa473475f50967a27a52f5af": "c_n = \\frac{1}{T} \\int_{-T/2}^{T/2} f(x)\\ e^{-2\\pi i(n/T) x} dx.", "1539260a2f29f562d3c54fbf11c56124": " R_0 \\le 1 \\Rightarrow \\lim_{t \\rightarrow +\\infty} \\left(S(t),I(t),R(t)\\right) = \\textrm{DFE} = \\left(N,0,0\\right) ", "15395a7a95f000c4b58d88cb74d6733d": "S_{l}=(h_{l-1}\\; mod\\; N)", "15397cd81cbe40705d9eb757af18f7e2": "\n\\varphi = \\begin{cases}\\arcsin \\left(\\frac{b}{\\sqrt{a^2+b^2}}\\right)\n& \\text{if }a \\ge 0, \\\\\n\\pi-\\arcsin \\left(\\frac{b}{\\sqrt{a^2+b^2}}\\right) & \\text{if }a < 0,\n\\end{cases}\n", "1539cb6795abd233ec6184156fa82fa9": " y_0 = 0 ", "153a79e18e394e9fc40312a6862a4c02": " \\mathbf{b}_{N \\times M} ", "153b846069545b04ac19075b1c671994": "(10\\uparrow\\uparrow)", "153b855950beceb73c1872a4d6862748": "I_\\lambda", "153bf21446cf89c6e5c6738016208f6c": "2(K+3)", "153c51a578068f9b6fd0ca7d5d21fc13": " \\lim_{t \\to \\infty} \\frac{1}{t} Y_t = \\frac{1}{\\mathbb{E}S_1} \\mathbb{E}W_1 ", "153c77131195f087e66acd22181bcd6e": "\\left(1+(1 + \\xi z)^{-1/\\xi}\\right)^{-1} \\,", "153c8216fe0f19a59d835eeef389ded4": "q_{k} = \\frac{p_{k+1}}{\\sum_{j \\geq 1} p_j}", "153c8cfdaf99ba8dc7c56c00854f1285": "u_p e_p = (Q_{pq}(e'_p \\otimes e_q))(v_q e_q)", "153c936b302aa556b81e51fedf234a38": "B=-\\frac{2M }{Q}\\ \\omega ", "153cb5982ebf917524225eb8793d6d9f": "\\pi(x) - \\pi(x/2)", "153cc515f5f927f67faecbba0865e85b": "\n \\begin{align}\n \\lambda_{\\hat{x}} = \\left( \n \\begin{array}{c}\n \\Lambda(a_1,\\hat{x}) \\\\\n \\vdots \\\\\n \\Lambda(a_{|\\mathcal{X}|},\\hat{x})\n \\end{array}\n \\right) \\,.\n \\end{align}\n ", "153ccdf2da686d2402e7e9008a964ee8": " Tv = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad T^2v=0, \\ldots, T^rv=0, \\ldots ", "153cea75645ff7ced81e9e3a4164a482": "C = \\sum_{i=1}^r c_i\\mu_i", "153cfdbdc8fd13865a38a5832d6a81e5": "\n\\begin{align}\nG_0 &= \\sum_{k=1}^\\infty \\pi_{0k}\\delta_{\\theta^*_k}\n\\end{align}\n", "153d1494176b899954e5a9dc83386247": "m=2n-k", "153d21cac185cf38d83ad8e7fc97cf4b": "\\displaystyle{f(e^{i\\theta})=e^{ih(\\theta)},}", "153d4d93f38abf88ba4fbd14069a8dc6": "\\begin{align}\n u^\\prime&= \\frac{u^*}{13L^*} + u^\\prime_n \\\\\n v^\\prime&= \\frac{v^*}{13L^*} + v^\\prime_n \\\\\n Y &= \\begin{cases}\n Y_n \\cdot L^* \\cdot \\left(\\frac{3}{29}\\right)^3,& L^* \\le 8 \\\\\n Y_n \\cdot \\left(\\frac{L^* + 16}{116}\\right)^3,& L^* > 8\n \\end{cases}\\\\\n X &= Y \\cdot \\frac{9u^\\prime}{4v^\\prime} \\\\\n Z &= Y \\cdot \\frac{12 - 3u^\\prime - 20v^\\prime}{4v^\\prime} \\\\\n\\end{align}", "153d86e023b6b7ad827f3b1cb7b3c0c6": "d \\!\\,", "153dae0b85c14068b2bdf5e19521878b": "[0,0.5)", "153e09e49ca01cb423e55a5ed7ac0fbc": "\\text{Had}(x) = \\Big(\\langle x , y \\rangle\\Big)_{y\\in\\{0,1\\}^k}", "153f3747a2ed0464c1b9e8de5375c2a1": "\n\\zeta(2) =\n\\sum_{n=1}^\\infin \\frac{1}{n^2} =\n\\frac{1}{1^2} + \\frac{1}{2^2} + \\frac{1}{3^2} + \\frac{1}{4^2} + \\cdots = \\frac{\\pi^2}{6} \\approx 1.644934.\n", "153f4f9f9158da7fec07cff141d4192a": "\\left\\{\\frac{\\partial}{\\partial x^1},\\dotsc, \\frac{\\partial}{\\partial x^n}\\right\\}", "153f88290a0056f91e66ecd6d041e92f": "\n\\begin{align}\n\\mathcal{H}_1 &:= \\text{Image}\\; P &= \\operatorname{span}\\{|\\omega_k\\rangle \\in B_{\\text{op}} \\;|\\; \\chi(x) = 1\\},\\\\\n\\mathcal{H}_0 &:= \\text{Ker} \\; P &= \\operatorname{span}\\{|\\omega_k\\rangle \\in B_{\\text{op}} \\;|\\; \\chi(x) = 0\\}.\n\\end{align}\n", "153fc2a5a0a49d52dda62d96ae0a293f": "R\\,", "153ff72c134f0f76d039e76e2cd65430": "\\frac{\\displaystyle 2 \\sum_{i=1}^n |X_i-\\mu|}{b} \\sim \\chi^2(2n) \\, ", "15402f6b763418e2dd02a339830723eb": "\\mathrm{M} = \\mathbb{R}^2 - \\{0\\}", "1540c20ea21399dcc5c4471c6a142cf5": " \\frac1n \\sum_{k=1}^n \\mathrm{E} ((M_k-M_{k-1})^2 | M_1,\\dots,M_{k-1}) \\to 1 ", "1540c468af640bcaed6b3f89b22a3312": "A_\\mu A_\\mu + B_{\\mu \\nu}B_{\\mu \\nu} + \\psi_{\\{1}^\\alpha \\psi_{2\\}}^\\alpha", "1540f458c8e0763b13e9390fab6a3bdb": "t(\\vec{r})", "154110dd919695283934c98a88e92d6b": "T_G(2,0) = (-1)^{|V|} \\chi_G(-1)", "15412bf6e129b5dd241dec7b347b5fb0": "\\Omega_\\mathbf{k}", "1541946ac8630b7fbe14f4039c91dea2": "(\\theta,\\psi,\\phi)", "1541a50027e21ce2fee5a27cf76c6f58": " X_0 * ( M_1 + M_2 + \\ldots + M_n ) = X_1 * M_1 + X_2 * M_2 + \\ldots + X_n * M_n ", "15422b5b15231ad4fd8984a3df13a205": "F= F_3(p, Q, t) + qp \\,\\!", "15425dec0e43a568632920f854039cbc": "k\\in \\mathbb{N} ", "154265b2b4d50e678dfb1b46bb1f937c": "C(m,n)", "154275e94de1184fb3838d6971d4ec5d": "2\\rho = 2n\\pi", "15429a42c5db422f056cdf98490f45b0": "\\mu = G M / c^{2}", "1542ba1c3007c0c228c3a0106c590392": "\\{V\\text{ open }:(\\exists{U\\in\\mathcal{O}})\\bar{V}\\subseteq U\\}\\,", "1542c1e2c3eac02cfabefb1d20e09c69": "p(x) = \\frac{\\alpha-1}{x_\\min} \\left(\\frac{x}{x_\\min}\\right)^{-\\alpha},", "1543d6b31be651c36e169d2c8558a387": "R_n'", "1543da840d6f786643629782a4ebb352": "\\kappa = \\sqrt{ \\frac{4\\pi e^2}{\\epsilon} \\frac{\\partial n}{\\partial \\mu} }", "1543dcc83dac32c808bce7474cdea1dc": "\\begin{matrix}\n\\left(\\frac{L/K}{\\cdot}\\right):&I_K^\\Delta&\\longrightarrow&\\mathrm{Gal}(L/K)\\\\\n&\\displaystyle{\\prod_{i=1}^m\\mathfrak{p}_i^{n_i}}&\\mapsto&\\displaystyle{\\prod_{i=1}^m\\left(\\frac{L/K}{\\mathfrak{p}_i}\\right)^{n_i}.}\n\\end{matrix}", "1543f5fcd4b1f5b91bdbafcdd74fe681": " \\mathbb{E} \\left[ g(X_1,\\dots,X_d) \\right] = \\int_{\\mathbb{R}^d} g(x_1,\\dots,x_d) \\, dH(x_1,\\dots,x_d).", "1544123e15697e116718d79b155e96fd": "\\kappa = \\frac{p}{z} \\tan \\phi", "154436025a5a66f99973a4a3466b647a": " (-\\beta)^2 - 4 (2 y + \\alpha) (y^2 + 2 y \\alpha + \\alpha^2 - \\gamma) = 0.\\,", "15443fa24fc052d2556c913a4cd1ffb2": " \\dot{V} \\leq 0 ", "15444eb27adfc21cf2a51f356dc5fa78": "D(x)", "1544582ed1b77bdef9aa36024e53a754": "\\mathbf{S}_k(\\mathbf{p}(t))=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_{k,n}(p_n(t)),", "154474f6c95de68cca407040716ab320": "\\delta_r= c_0/2B", "1545ba8fc8492a3ae4431ff622b4cbde": "~ E =\\frac{I_{\\rm s} G}{I_{\\rm p}A}~", "1545c7f46c4ecfb12b1da8e7f8fb7f93": "I = (a, b) = \\{x \\in \\mathbf R \\,|\\, a < x < b \\}, ", "1545dff4b0c4c9f85ceb877e4df467c5": "k \\in \\mathcal{N}", "1545ee63245cbae99d2d08c44b7d797f": "A.M.A. = \\frac {R} {E_{actual}} ", "1545f48b2b2e2ac9b498e0ba415a6cf8": " Q = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & \\frac{\\sqrt{3}}{2} & \\frac12 \\\\ 0 & -\\frac12 & \\frac{\\sqrt{3}}{2} \\end{bmatrix} ", "1546433c08747cdb968b1b1bbf07dc5f": "\\frac{dP(\\mathbf{x})}{d\\Omega}=\\frac{r^2}{2}\\Re(\\mathbf{n}\\cdot\\mathbf{E}\\times\\mathbf{H})", "1546793d89f6a725b9c17cfaacb0d1fd": "\n\\lambda_k = -\\frac{2}{h^2}(1 - \\cos(\\frac{k \\pi}{n+1})).\n\\,\\!", "15468b529684ae1e10c91109f425e7b0": " \\delta \\int_{t_1}^{t_2} L(\\mathbf{q}, \\mathbf{\\dot{q}},t) dt = 0 ", "1547056eb430ebded6538c1e8ad4e5d2": " \\mathcal{A}\\phi + L = 0 ", "1547a74739c1e2b6be3ebe927017bfbb": "M_{n}", "1547d2572fedf4abb12ce03ff75c8c3e": " \\mathrm{Vol}_{n-1} \\, (K \\cap A) \\leq \\mathrm{Vol}_{n-1} \\, (T \\cap A) ", "15484124ea5cedb553a1be623c7e5ae3": "\\{X_1,X_2,\\cdots\\}", "15484892a80db447c0c1884580f6dd65": "w(u,x_0,\\tfrac{r}{2}) \\leq \\lambda w(u,x_0,r)", "1548517f82c99e391bdca4a51bd7f071": "\n\\dot{V}=(\\dot{e}+\\alpha e)(\\ddot{e}+\\alpha \\dot{e})\n", "154851e976d7eadda0c898eae5bc10ec": "(n-m) \\times n", "1548a2d5199d57b0b2e771f24ca8eb1d": " \\Phi = \\frac {\\rm \\#\\ photons \\ emitted} {\\rm \\#\\ photons \\ absorbed} ", "1548f19723492b717eed6a0bc6c5d38f": "0<\\mathit{MPC} <1", "15493bdbae1015cb61ac1156138b0a66": "\\frac{ dy }{ dt } = -x-x^3-\\alpha y,~ \\alpha \\ne 0", "154a0b74f18ea80cc080b29d407fca8a": "f(\\hat{x}_{n_j},\\theta_{n_j}) > f(x_{n_j},\\theta_{n_j})", "154a10c4cf4093047ee2a384de5d3d24": "{\\rho^2_{XT}} = \\frac{{\\sigma^2_T}}{{\\sigma^2_X}}", "154aeb31b7cde4d54a94604d4e5e076a": "\\displaystyle{T(\\beta)=\\begin{pmatrix}1 & \\beta \\\\ 0 & 1\\end{pmatrix}.}", "154b0b00834c981f1a541c98e30b00a3": "\n\\mu_{\\operatorname{eff}}(\\dot \\gamma, T) = \\exp \\left( A_0 + A_1 \\ln(\\dot \\gamma) + A_2 T \\right) \n", "154b2edabf59f96e7ae5ca18e0bb2e51": "\\sum _x \\cosh ax = \\frac{1}{2} \\coth \\left(\\frac{a}{2}\\right) \\sinh ax -\\frac{1}{2} \\cosh ax + C \\,", "154b35701e243ec3c8bb863e8396063c": " M'= \\rho \\cdot V^2", "154bda1bbf05d3478b67935eced901f0": "g(r,\\theta)", "154be246bdbdb1062d92c5824ca424e4": "\n\\frac{1}{2}\\sum_{i=1}^KQ_i(t+1)^2 \\leq \\frac{1}{2}\\sum_{i=1}^KQ_i(t)^2 + \\frac{1}{2}\\sum_{i=1}^Ky_i(t)^2 + \\sum_{i=1}^KQ_i(t)y_i(t)\n", "154c149de88269008049e6aa6dd5fd59": "T_0=\\frac{abc}{2h}.", "154c4e3c7c89855d5f79f7f3dbf08b51": "\\boldsymbol{\\nabla}\\mathbf{u} = \\boldsymbol{\\varepsilon} + \\boldsymbol{\\omega}", "154c64630c334177ed9650af5fe09166": "D_e ", "154ca2557c060d053770a45b78552adc": "\\left . \\frac{d \\varphi }{d k_1} \\right | _{k_1 = k } = x - t \\left . \\frac{d \\omega}{dk_1}\\right | _{k_1 = k } +\\left . \\frac{d \\alpha}{d k_1}\\right | _{k_1 = k } \\ ,", "154caade0d22c60368adb4679ffc6170": "G_1,~G_2,", "154cc4a27418889f382f820b8fc07f5b": "I_5\\ = \\frac{1}{5} \\cos^4 x \\sin x + \\frac{4}{5}\\left[\\frac{1}{3} \\cos^2 x \\sin x + \\frac{2}{3} \\sin x\\right] + C,\\,", "154ccbfb15b904c565175e8c882f9555": "(x(t_0),y(t_0),z(t_0))", "154d0752c1bac72deba7bc1ecb29457c": " |\\psi\\rang |\\phi\\rang \\rightarrow \\sum_n c_n |\\psi_n\\rang |\\phi_n\\rang \\quad \\text{(measurement of the first kind),} ", "154d1ce852f85f3403286df390e1ada6": "\\ker\\phi", "154d3a2080a981deb50b0b18dadad582": "I[\\vec{g}]=\\sum_{j=1}^{N}\\sum_{k=1}^{N}f_j[\\vec{g}]f_k[\\vec{g}]e^{2\\pi i \\vec{g} \\cdot \\vec{r}_{jk}}.", "154d6d86c025f9b7795b79991754bb0c": "V_\\text{Yukawa}(r)= -g^2\\frac{e^{-kmr}}{r},", "154d734640776c25790af7c2ccd59718": "N \\to M \\to M''", "154d76c244544d1e944cad50ee55e9a6": "A_i \\in N", "154d8b615022832fc79be03f23217b37": "\\phi : \\mathfrak{g}_1 \\to \\mathfrak{g}_2", "154dcc5feeccfe9765608ea046be4894": "N = \\left( -{^{(4)}g_{00}} \\right) ^{-1/2}", "154e62a25c1e6ad0f36057e69326674b": "1/\\alpha \\approx 137", "154e6a4858bb6ab887515a950d028966": "=x\\cdot\\frac{1}{t} \\quad \\mbox{ at }t=1", "154e76ce6e9778985f7e184fb32c9498": " y\\in \\mathbb{F}_{q^n} ", "154ea992cb8a2d25f34eec06d2a1df03": " \\displaystyle{gDg^{-1}=D + A}", "154ec6666949cc4e4ab01dc262a72d38": "\\sigma_0^{ }", "154f0919e57f1e7a8a046c3d58f256f7": "n-p", "154f8988452469bc0d75a0af93ae5772": "Y(1 - (b - bt + t) = I ", "154f9132441870774ff1345ef56b117e": "0 \\equiv z + tf'(r) \\pmod{p^m}.", "154f9e98c064f79f5c0b69bda11f25a6": "R_{sensmin}", "154faf33306e82d5cc078f93d705fcf2": "0\\rightarrow U \\otimes \\Lambda^{k-1}(W) \\rightarrow \\Lambda^k(V)\\rightarrow \\Lambda^k(W)\\rightarrow 0", "1550108d51092e4907cc997e15dc8253": "a^{d\\cdot 2^r}\\equiv -1\\mod n\\quad\\mbox{ for some }0 \\leq r < s .", "15503bbc339d6a9ba4cc35615de9355e": "{k-\\frac{1}{k}} = 2ix", "155041f3477632831ae9fd521399a390": "i \\in \\{0,...,d\\}", "155051be83313e75c95546ff0e0e6996": "\\mathrm{MV(t)}=K_p\\left(\\,{e(t)} + \\frac{1}{T_i}\\int_{0}^{t}{e(\\tau)}\\,{d\\tau} + T_d\\frac{d}{dt}PV(t)\\right)", "15505529a7367412d2a37dc7ae5f3b37": " \\nabla \\times (\\psi\\mathbf{A}) = \\psi\\nabla \\times \\mathbf{A} + \\nabla\\psi \\times \\mathbf{A} ", "15506b335966f5c52eb8809d76a92b28": " W\\left ( J \\right ) =\n -{1\\over 2} \\iint d^4x \\; d^4y \\; J\\left ( x \\right ) D\\left ( x-y \\right ) J\\left ( y \\right )\n", "1551309be9e8e1d4a1f0b11dd126565d": " |{3 \\over 2}, \\pm {3 \\over 2} \\rangle, |{3 \\over 2}, \\pm {1 \\over 2}\\rangle, |{1 \\over 2}, \\pm {1 \\over 2}\\rangle ", "155196fa67465424b3301b2a5139096b": "\\min_i \\sum_j A_{ij} \\; \\le \\; r .", "15519c121f434bd150afd0a174ed43b6": "V_1 \\cap V_2 = \\emptyset", "1551c0517901395ad1ffa5102400ccb3": "\\sum_{n=s}^t C\\cdot f(n) = C\\cdot \\sum_{n=s}^t f(n)", "1551d6e4f95b02691e25f8b633d583d7": "\\mathrm{\\rho}", "1551e928210b66aec5b61d193d9a0c72": "e^{a\\sqrt{-1}}", "15522c4326a1e63c6b63aa0c6d4fc42b": "\\left.\\begin{matrix}\n& {}+\\kappa(\\kappa(X_1,X_2\\mid Y),\\kappa(X_3\\mid Y),\\kappa(X_4\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_1,X_3\\mid Y),\\kappa(X_2\\mid Y),\\kappa(X_4\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_1,X_4\\mid Y),\\kappa(X_2\\mid Y),\\kappa(X_3\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_2,X_3\\mid Y),\\kappa(X_1\\mid Y),\\kappa(X_4\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_2,X_4\\mid Y),\\kappa(X_1\\mid Y),\\kappa(X_3\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_3,X_4\\mid Y),\\kappa(X_1\\mid Y),\\kappa(X_2\\mid Y))\n\\end{matrix}\\right\\}(\\mathrm{partitions}\\ \\mathrm{of}\\ \\mathrm{the}\\ 2+1+1\\ \\mathrm{form})", "1552b8eaa8278eddea644d524240e367": "Weight When Cut - OvendryWeight \\over OvendryWeight", "1552d5aa13fd25b7526cdaf784c9fc49": " h_{g;k_1, \\dots, k_n} ", "1553027480574c1186403d82d0f042bb": "-c_i^*", "155326ae316bdca30ce29027da73080c": "k_g = \\sqrt{1-r^2}/r", "15532c29f79381888e1f3b9d4222d7f2": "\\tau_r = \\lambda_r / \\lambda_2, \\qquad r=3,4, \\dots. ", "155333808941a22eaf3d32e2ca81b67f": "\\varepsilon \\colon C_p(X) \\otimes C^q(X) \\to \\mathbb{Z}", "15533856e5e08c30e6ef6964e09580e1": "u_{i}:\\{1\\ldots m\\}^{d_{i}+1}\\rightarrow\\mathbb{R}", "15533df17d567df5971f41f59b76bcfc": " |\\alpha| = \\sum_{i=1}^n \\alpha_i,\\ ", "15535ea56905b22dd27dbab12c3f6691": "y_t = (1 - \\alpha) x_t + (\\alpha-\\alpha^2) x_{t-T} + \\alpha^2 y_{t - 2T})", "1553ed8fdd8cc175ad91d5419f188f42": " \\tilde W_t =W_t - \\left [ W, X \\right]_t ", "1553efac04c2b29d8bb61a0a4552a34e": "MRS", "15545944ba0670c16344da8125702c19": " \\mathbf{q} = e^{\\tfrac{1}{2}{\\theta(u_x\\mathbf{i} + u_y\\mathbf{j} + u_z\\mathbf{k})}} = \\cos \\tfrac{1}{2}\\theta + (u_x\\mathbf{i} + u_y\\mathbf{j} + u_z\\mathbf{k}) \\sin \\tfrac{1}{2}\\theta", "15546984c88607b38d27b5906ede9321": "\\mathcal{H}^q(X, \\mathcal{F})", "1554706386e567833fa9eeab28987393": "\n K^2 = Z_1(g_1)^2 + Z_2(g_2)^2\\,\n ", "15551fcad0dcba66e39e3d326327d8a1": "\\alpha_F=\\frac{\\mu_\\theta-\\mu_{ref}}{\\mu_\\theta*\\mu_{ref}*(\\theta-\\theta_{ref})}", "15554c0c5178715ca2f55195c7041537": "L_0 = (k+h)^{-1}\\sum_s\\left[ {1\\over 2}X_s(0)^2 + \\sum_{m>0} X_s(-m)X_s(m)\\right], \\,\\,\\, L_{\\pm 1 } =(k+h)^{-1} \\sum_s\\sum_{ m\\ge 0} X_s(-m\\pm 1)X_s(m).", "155577b3e2cfdd9e2ec28122285ba204": "res_{U_i \\cap U_j, U_i} \\colon F(U_i) \\rightarrow F(U_i \\cap U_j)", "1555a930d1270fe8fe23732784d8466d": "J=J_{\\lambda_1,m_1}\\oplus J_{\\lambda_2,m_2} \\oplus\\ldots\\oplus J_{\\lambda_N,m_N}", "1555b19090de5fcdf8e79162bb6940fe": "x_1M_1 = y - x_2M_2", "1555d4776408d6c2e59609e4b3b8b632": "\\pi = \\cup^{\\infty}_{i=1} (\\sigma_i \\cup \\gamma_i)", "1555e4119fef0f664e96cb1979e021f7": "dF_i=\\sum_j \\epsilon_{ij}\\, dA_j\\,", "155662295e5f1b6ef807afa96cac6320": "2 C_6H_6 + 15 O_2 \\rightarrow 12 CO_2 + 6 H_2O", "1556638b996f1898fd9147c1a872003d": "\\dot Q", "15568c0b18bc33bd3f3ca38b7b9ec153": "f, \\; \\; \\theta_{init}", "15568d5930d7c77f5762eda92702b453": "\nN \n", "1556a97e3df85543d499c532ba0c4ba6": " \\{A \\vert A \\subseteq \\kappa \\land \\kappa \\in j (A) \\} \\,.", "1556bfedadd1e72e7c42b82a9496a4e3": " \\left( k_\\perp^2 +\\frac{1}{\\rho^2} \\right)G(\\mathbf{k}_\\perp) = 1 ", "15571b8edc062e222f56cc1eb80e4259": "\\left|A\\right|^2=\\left|B\\right|^2 \\,", "15572737efdba948eb2ca2a6e8de1350": "U+pV-TS", "15576f8043e74f3f26921c77bbdb3db9": "X^\\star", "15580c586bb8d5b0cc15743155ce97d0": "|ab| = |a| |b| \\mbox{ for all } a,b \\in G", "15582d37e233ec3d41b3ae9aa51c97b3": "H=2a+1 \\mbox{ and } P=c ", "1558567a52d52d33add2f0eeb97cf03f": "\\lambda_0\\neq 0,", "1558db63955cc27fdf5cf0c1be4c4ffd": "\\tbinom47", "15592b00063dd40a2051ad58740a5d5d": "A:B:C = \\frac{1} {\\Delta \\Gamma} : \\frac {1} {\\Delta E} : \\frac{1}{\\Delta kT}", "155941981b5bb955103afd98cb722a55": "\\beta_n", "155945b440ff23143412ebeeb7979dcd": "Y \\sim \\chi^2(\\nu_2)\\,", "155973fe048353bc11dab589eabd0be5": "B_{i,k,t}(x) = \\frac{x-t_i}{h} k[0,...,k](. - t_i)^{k-1}_+", "155a113803327e5ceba60ca54025602d": " \\scriptstyle \\tau = Fd ", "155a2141a0435006a0eea4b8c36234d7": "L^{X/Y}_i = 0", "155a69be27e79f5d0573f4a388b95e16": "m \\leq H_\\infty(X) - 2 \\log\\left(\\frac{1}{\\varepsilon}\\right)", "155a738e6a896777be2d95b149cf8c1c": " m = \\frac {\\sqrt{E^2 - (pc)^2}}{c^2}", "155a79ef12c6615b93c9d44ebe9db065": " x(t) w(t-\\tau) = \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} X(\\tau, \\omega) e^{+j \\omega t} \\, d\\omega. ", "155aacb5190b9ddd14f52a060f051df5": "\\mathcal{} f'", "155b28a32f4eb488c543ec39d602926e": " \\langle \\cdot [e], [e] \\rangle_f ", "155b830af7e037c46e586fa71c5d6476": "v(p, w)", "155bd21287ade5e44a71ec14c33d4172": "\\{\\vert n\\rangle\\}_{n \\in \\mathbb N_0}", "155c1f61a4b66c2ee99f12e0cd82c56c": "f_{W}\\upharpoonright N=0\\,", "155c829097296079097025a5ba8421d9": " E = pc \\,.", "155cbda4ef9aad1c2230d475e8b94181": "g_{\\mu \\nu }=\\, 0 ", "155cf475d37727ff852c380674e4eb4d": "\\delta t=6.5\\pm7.4\\ (\\mathrm{stat.}){\\scriptstyle {+8.3\\atop -8.0}}\\ (\\mathrm{sys.})", "155d316d634946d7f446ac19deaf9150": " E_{21} =\\frac{d \\ln (c_2/c_1) }{d \\ln (MRS_{12})}\n =-\\frac{d \\ln (c_2/c_1) }{d \\ln (MRS_{21})}\n =-\\frac{d \\ln (c_2/c_1) }{d \\ln (U_{c_2}/U_{c_1})}\n =-\\frac{\\frac{d (c_2/c_1) }{c_2/c_1}}{\\frac{d (U_{c_2}/U_{c_1})}{U_{c_2}/U_{c_1}}}\n =-\\frac{\\frac{d (c_2/c_1) }{c_2/c_1}}{\\frac{d (p_2/p_1)}{p_2/p_1}}\n", "155d483107e2ee9b932c880779ce140e": " p = \\min \\left( 1, \\frac{ \\exp \\left( -\\frac{E_j}{kT_i} - \\frac{E_i}{kT_j} \\right) }{ \\exp \\left( -\\frac{E_i}{kT_i} - \\frac{E_j}{kT_j} \\right) } \\right) = \\min \\left( 1, e^{(E_i - E_j) \\left( \\frac{1}{kT_i} - \\frac{1}{kT_j} \\right)} \\right) ,", "155dbc0436ba00db8e4a3629082d121a": "H_q(x) = xlog_q(q-1)-xlog_qx-(1-x)log_q(1-x)", "155dcd4f4cf4b9018e087e29a70d3ebb": " \\langle r \\rangle = a \\Gamma(\\frac{4}{3}) = \\frac{a}{3} \\Gamma(\\frac{1}{3}) \\approx 0.893 a\\,.", "155e31404d7c8508854066fffd001772": "-21 - 3\\lambda + 7\\lambda + \\lambda^2 + 16 = 0 \\,\\!", "155e509dc08e54e8bbda792bbc3d255d": "\\hat{e}_j", "155e6bd548873626fba24ec5f415abc6": "\\ell_0", "155e906c6c57b939919e44e46b21ecc0": "\\ang \\theta ", "155e96421d20a886a7a26f2e9d999941": " \\begin{align}\nf'(x_2) = \\ell_0'(x_2) f(x_0) + \\ell_1'(x_2) f(x_1) + \\ell_2'(x_2) f(x_2) + \\ell_3'(x_2) f(x_3) + \\ell_4'(x_2) f(x_4) + O(h^4) \\end{align}", "155ea9abf796ec17d07dd994464287a8": "f=\\frac{a-b}{a}=1 - b:a.", "155eb0542d80a49d7303d6ee86b942df": "G=(\\ast A_v)\\ast F(X)", "155eb8ff9f84f4356ad19ebcdfd92838": "\n \\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\left(\nI-\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\right) \\Pi_{\\rho,\\delta}\n^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\} \\right\\} ", "155ebf3256daa047d3933d06af5fa86a": "R=1", "155ede14a3aec17422ac5779b1545822": "B = \\{x\\}", "155f4cce4dd38781e083ff289f9a2f40": "\\mathrm{Kn} = \\frac {k_B T}{\\sqrt{2}\\pi\\sigma^2 p L}", "155f67ecc3a881fb4ecaaedb59877c01": "B=\\{B_1,...B_s\\}", "155f6e225044fe11ce4fabf81552e036": "X_k^i", "155f894f211e0a351cc118540ed2a44d": "r^n = a^n \\sin(n \\theta).\\,", "155fa186ec31446c9962e24d6891d43b": "U_B", "155fee16a95f9ad34ab979feed4b7506": "\n\\begin{array}{cccc}\nz^{*}= & \\stackrel{DM}{\\mathop{Opt}}&\\stackrel{Nature}{\\mathop{opt}}\\quad & g(d,s)\\\\[-0.05in]\n& d\\in D & s\\in S(d) &\n\\end{array}\n", "156069ac45d280c77ba5cd20960fab8f": " - \\frac{\\partial p}{\\partial z} = \\frac{\\Delta p}{L} ", "156081cfc0c000324719b54f01f6af69": "QB=0", "1560b79a9abbb13e102b75f499e893e1": "3 = 2 + 1", "1560c9dff2d0d96a151f9b6d2e7a5ddb": "\\tfrac{\\beta }{2}", "1560d28f994bdb1c882c28d058b221d7": "\\frac{\\lambda \\operatorname E (S^2)}{2 \\{1-\\lambda \\operatorname E (S) \\}}", "1560d6130a4407023310cb6d8679b3cf": "\\sum^{n}_{i=0}p_{i,j,k}=k", "1560f8483b5e42b9b845ede58dfabfbe": "\\forall\\alpha.\\forall\\beta.\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)", "1561090f4fd4ae85e4fe269d450e0d34": "-1 < \\kappa < 0", "1561629b26389b872170f53e1be54b6d": "V^{\\alpha }", "1561718106fa14629c5d122cfe614413": " D=\\omega(m_1-x_1)+(m_2-x_2)", "15618ab9efcee346b9dd866ea048d87e": " = \\dfrac{5(\\sqrt{3} - 4)}{3 - 16}\\,\\!", "1561be931a03d5c0dea5046b8a6ceda4": "\\tfrac{v}{c}", "1561c3d5b80265e5c980916a453b7cc7": " \\Omega[x^*-c] ", "15620f2bf6b000d170522649a601b2bd": "\\vert{\\Phi_0}\\rangle", "15629e5d956e0d8bd59efb45de72aacf": "T \\in \\R^d", "15630f1d85190e088343788a9b311d78": "\\{y \\mid \\exists x[\\phi(x,y)]\\} ,", "1563162ecebf94912369dfc0f9af1a05": " T = g_m (r_{\\pi}// R_S ) \\ \\frac {R_D//r_O} {R_D//r_O +R_F +r_{\\pi}// R_S}\\ . ", "1563384a0036c665a7531a88f116d80e": "\n\\underbrace{\\underbrace{\\mathrm{root+suffix}}_{\\mathrm{stem}} + \\mathrm{ending}}_{\\mathrm{word}}\n", "156348c36b74f26ca6e7a24235bf3309": " s_k t_{k+1}-t_k s_{k+1} = (-1)^k. ", "15636e811bd1b3596a3284eaf3542050": "G^{-1}", "156371fa68f63dc48e4345abd91d613d": "\\nu_1 \\le \\nu_2, k \\le n, p \\in [0, 1]", "1563a72c6993fe81033e39ae5134ed05": " \\frac{\\partial}{\\partial h} \\operatorname{AMISE}(h) = -\\frac{R(K)}{nh^2} + m_2(K)^2 h^3 R(f'') = 0 ", "1563b8504d179165422dcf11b92fecdf": "c^2=\\gamma(p+p^0)/\\rho", "156410efaa3f06bcb68056331e7b66a4": "v_{D}\\,", "15641e1f5d1554d4b3fe895154bf1d68": "f^n(A) \\cap B \\ne \\varnothing", "15642b77303878d4edb3ac4771dc5aad": "\\mathrm{e}_{i}", "156460d6b5c9a01f27f8e7e92a789f84": "B_2=G^{31}", "1564e81e84079281c809d3c521389fa0": "\\Lambda^1(E)^K", "156507a62f3a70e3f759500c9f05d8ad": " ( \\frac{x}{z} , \\frac{y}{z} ) ", "15650a7541f2a18fd124cb8b7792374d": " h < \\frac {\\lambda}{8 \\cos i} \\;", "1565b0796797071f4f3823eb26127d0e": "K \\hbox{--} \\bar K", "156605ebad8e67f6a389eeec48c4c19e": "p(X)=X^2-5X-2I_2,", "156615bc3ef86e8f898f45390ccc68bb": "e^3", "1566b3941bf0066601d4bcf32286ebf2": "N(x)", "1566db4013b00c30b4a4702d503270c2": " f \\in \\mathcal{M}_{g;k_1, \\dots, k_n} ", "1567070f4dcf5fb26ecb608f3c944e75": "{s_k \\over {n \\choose \\lfloor{n/2}\\rfloor}} \\le {s_k \\over {n \\choose k}}.", "15676f60f1237fd7dce08cf3db9d5c3f": "D_{12}=\\frac{3}{2n(d_1+d_2)^2}\\left[\\frac{kT(m_1+m_2)}{2\\pi m_1m_2}\\right]^{1/2}", "15677511bae0e800d0667059d477a89e": "\\scriptstyle p(bk \\,+\\, a) \\;\\equiv\\; 0 \\pmod{b}", "15678dc8270a263fb1f8c8f1e6bac9d7": "d^{\\dagger}\\omega=i(\\partial-\\bar{\\partial})\\text{ln}||\\Omega ||,", "1567c7fb03847cd101a37fa32db4503a": "\\mathbf{a} = \\mathbf{a}_\\text{x} + \\mathbf{a}_\\text{y} + \\mathbf{a}_\\text{z} = a_\\text{x}{\\mathbf i} + a_\\text{y}{\\mathbf j} + a_\\text{z}{\\mathbf k}.", "15688d3238b42b148eecd42a3278cce1": "\nq_1\\ \\hat{f_1}\\ +\\ q_2\\ \\hat{f_2}\\ +\\ \\ q_3\\ \\hat{f_1}\\ =\\ \\sin \\frac{\\theta}{2}\\quad \\hat{e_3}=\\frac{\\sin \\frac{\\theta}{2}}{\\sin\\theta}\\quad \\bar E\n", "1568dd4ff929e70671a51abdf352c665": "O(n^2\\log n\\log q)", "1568dd8ceade5b85b2ed2f8156a02290": "1^{2^{k-1}}", "1568eff0ee1d358d6a0fa7b08b12e7c0": " GA(m,\\mathbb R) ", "1568fd88c570556f7e078a0e481101e7": "-1/2 \\mu_i \\cdot E_{RF}", "15699da5162c6174d970b58e778138fc": "\\frac{x}{n}", "1569b0652f60d5b4eebaaa2de3178f48": "r_{\\mathcal P}", "1569b73bb8a869da53435f54c19830d0": " = 3 + \\frac{1}{8} + \\frac{9}{640} + \\frac{15}{7168} + \\frac{35}{98304}\n+ \\frac{189}{2883584} + \\cfrac{693}{54525952} + \\frac{429}{167772160} + \\cdots\\! ", "1569ffcfea536239b5328a05e4b4c777": "\\begin{pmatrix} \\overline{\\delta} & -\\overline{\\gamma}\\\\ -\\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix}", "156a349c2cdf2e3ef2ef19e87dc406f0": " T^{ab} {}_{bc} = \\sum_{b}{T^{ab}{}_{bc}} = T^{a1} {}_{1c} + T^{a2} {}_{2c} + \\cdots + T^{an} {}_{nc} = U^a {}_c .", "156aac4cc6dc985f8be0e74c416f6298": "\\overline{w_i}", "156ac61a39a7925e5c69ffcc67224523": "\n \\left|A\\right|_{ij} = a_{ij} - \\sum_{k\\neq i} |A^{kj}|_{il} |A^{ij}|_{kl}^{\\,-1} \\cdot a_{kj}\n", "156ad0d78e964e2ea2a79aee3ea8227a": "\\mathbf{F}_\\mathrm{rad} = { 2 \\over 3} \\frac{ q^2}{ c^3} \\mathbf{\\dot{a}}.", "156af2859026537b52e98582ca4d7255": " \\gamma > 2 ", "156af37236f5daa4d889325fe88baedb": "\n1\\ {\\rm Np} = 20/ \\ln_{} 10\\ {\\rm dB} \\approx 8{.}685889638 \\ {\\rm dB} \\,\n", "156b14c98e5d372d59916e3fea9cb5fa": "dV = \\left(\\frac{\\partial V}{\\partial p}\\right)_{T}dp + \\left(\\frac{\\partial V}{\\partial T}\\right)_{p} dT = V\\left(\\alpha dT-\\beta_{T}dp \\right)\\,\\,\\text{ (2)} \\,", "156b4321c04913ca7f983483b8a101ba": "|Leader_i\\rangle= \\frac{1}{n^{3/2}}\\sum _{a=1}^{n^3}|a,a,\\ldots,a\\rangle", "156b63ebe005a62e831e6d1c301b4be7": "Z_r", "156bb9c17a70f18fc1896deb7f987fa5": "I = C(dV/dT)", "156bc7b9bee44723fa628b738e0e8ee2": "c^2 = 5^2", "156bca7a24d5769e858a6f11c26c826c": " m_2=\\frac{a-a_2}{b_2 (a_2-a_1)}\\!", "156be0d4e3de061d159b7192f263baf6": "P(x_1)", "156bf7552c0780e13b206b571067f74e": "{x'}^j, j=0,1,\\dots", "156c83cb1e70cb24e5799a30e12c5073": "D(fv) = v\\otimes (df) + fDv", "156c8687dbe59f2546579112fdcc5f61": " \nN_j\\left(U^\\left(0\\right)\\right)\n=\\Gamma_{jk}U_k^\\left(0\\right)-U_j^\\left(0\\right)\n==\\left(\\Gamma_{jk}-\\delta_{jk}\\right)U_k^\\left(0\\right)==0\n", "156cae5711114578af2e9398aacf4e86": " r\\frac{\\partial}{\\partial r} = \\frac{\\partial}{\\partial \\rho}", "156cb999e3d3ca190825626a84c48c7b": "x_k = -\\cos\\left(\\frac{k \\pi}{N - 1}\\right) ; \\qquad \\ k = 0, 1, \\dots, N - 1.", "156cbdbf192274123c4951f88bff2a10": "(z - 1)(z + 1) = z^2 - 1 = (z - j)(z + j)", "156cceab26d40dc7ba5af57cc28d5e6a": "I=I_0 e^{i\\omega t}.", "156ce9e375e6cb012dc635b488c01c73": "c_0={1\\over\\sqrt{\\mu_0\\varepsilon_0}}.", "156d008bc155eebecf5e8b8e90df1230": "\nMRF = \\frac {(U - hR_{Fn})(hR_{F1} - L)\\prod^{n-1}_{i=1}(hR_{Fi+1} - hR_{Fi})}\t\t\t\t\t{[(U - L)/(n+1)]^{n+1}}\n", "156d00a9cb2329569c2b4163cac450af": "{\\operatorname{d}\\Gamma\\over\\operatorname{d}y}", "156d3429cd6916eefe2b3b683ef6721e": "\\tfrac{1}{2} (1 \\pm j)", "156dd665999efac8c7ef8ded066d5ed3": " w\\,R\\,u\\land w\\,R\\,v\\Rightarrow u=v", "156e05afcdb99b14dbe2671a850c61bc": "P^{\\lambda}_{ij \\sigma}", "156e0d2b1231ce052224f9d4455c7699": "\\delta = \\theta_0 + \\theta_2 = \\theta_0 + \\text{arcsin} \\Big( n \\, \\sin \\Big[\\alpha - \\text{arcsin} \\Big( \\frac{1}{n} \\, \\sin \\theta_0 \\Big) \\Big] \\Big) - \\alpha", "156f1942b102e583cb7bc12c0f88bf26": "|k_e|, |k_h|", "156f446d2432e8d1f65d1574c09f3f2e": "M \\; [ \\mathbf{s} \\; \\mathbf{t} ] \\; = \\; [ \\mathbf{s} \\; \\mathbf{t} ] \\Lambda_\\theta ", "156f828f9f15b6fdd668d56b86504831": "\\mathbf{j}_r(t)\\in \\mathbb{R}^3", "156f82c61b9ccc4edaad89686de3d321": "\\delta Q = T dS_h\\,", "15704217859c9273fffc6681434d7dae": "\n{4\\choose 2} \\left ( \\frac{1}{2} \\right )^4 = 6 \\cdot \\frac{1}{16} = \\frac{6}{16}\n", "1570627dee1e357f70871ee9315ca87e": "V_{tangential}=V_{wall}", "157075c60304563ebf1397dc9d434ad0": "s_{AB}", "1570bee19fe00d903b33e3bab8f1e4b1": "B=\\dfrac{dN}{dt} = k_1M_T^j(c-c^*)^b", "15712a00763cd12474b0a1ef8ffd3721": "\\{\\lambda_n\\}", "157148969e68174643d663fa61499e64": "\\mbox{SL}(2,\\mathbf{R}) = \\left\\{ \\left( \\begin{matrix}\na & b \\\\\nc & d\n\\end{matrix} \\right) : a,b,c,d\\in\\mathbf{R}\\mbox{ and }ad-bc=1\\right\\}.", "15718ead15aa98daa72cb4f16dd23aff": "H_\\xi = \\langle \\mu, \\xi \\rangle", "1571a63bd42577b33dd46664cf75b6cb": "\n\\begin{align}\nB_{1_1} =& \\begin{pmatrix}0 & 0 & 48 & -16\\\\ 0 & 0 & -8 & 2\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 0 & 1\\end{pmatrix}\\\\\nB_{1_2} =& \\begin{pmatrix}0 & 0 & 4 & -2\\\\ 0 & 0 & -1 & 1/2\\\\ 0 & 0 & 1/4 & -1/8\\\\ 0 & 0 & 1/2 & -1/4 \\end{pmatrix}\\\\\nB_{2_1} =& \\begin{pmatrix}1 & 0 & -48 & 16\\\\ 0 & 1 & 8 & -2\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{pmatrix}\\\\\nB_{2_2} =& \\begin{pmatrix}0 & 1 & 8 & -2\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\end{pmatrix} \n\\end{align}\n", "157210a3a7133f92876dce0788b068ce": "\nI:\\{\\mathbb{X}\\subseteq\\mathbb{R}^n\\}\\rightarrow\\{\\mathbb{Y}\\subseteq\\mathbb{R}^m\\}\n", "1572c4a2036b7316e5687740c9835193": "e\\colon G\\times G\\rightarrow G_T", "15731a3600bd3731157459d5d07a3f30": "Z_P = b_1t_1y_1+b_2t_2y_2\\,", "15733ef796e744976ebf2e9dc5de8942": "B_{2}", "15737fde927f0c945e4d33784063c612": " (R^i f_* \\mathcal F)^\\mathrm{an} \\cong R^i f_*^\\mathrm{an} \\mathcal F^\\mathrm{an} ", "1573f0f7dd2bdd54d3a2e1ae915af0f6": "\\text{inquire}(j)", "157443d24eb255d284b70ff30e8146ec": "\\ I(r,d)=\\delta I_bR\\exp(\\alpha d)", "157471ee6e4a50a9b18cc321f24d3d6e": "\n \\Beta(x,y)=\\dfrac{(x-1)!\\,(y-1)!}{(x+y-1)!}\n\\!", "157482be62239900dd14b755b374d327": " W_{n,k}=\\sum_{v=0}^{k}(-1)^{v+k} \\left(v+1\\right)^{n} \\frac{k!}{v!(k-v)!} \\ . ", "15749dc1f9eaadac752f1c7168a02e27": " (Exa \\leftrightarrow Exb) \\leftrightarrow (a=b),", "1574acf98fadf9844681c5cad9313456": "K_m = \\frac{k_r + k_\\mathrm{cat}}{k_f}", "1574adfffa2f46ff72d8d2aea763ddf0": "6_0", "1574b82063b4508684413e5d70e0992c": "\\mbox{eGFR} = \\mbox{166}\\ \\times \\ \\mbox{(SCr/0.7)}^{-1.209} \\ \\times \\ \\mbox{0.993}^{Age} \\ ", "1574f057e5a4de3b7b53627317967b49": "\\operatorname{Li}_s(z)", "15753780721821850ae30ed7309bbdf5": "s,t\\in V", "15756f2fbcbca2dabaeb57e80e52266f": "\\Delta=1", "157644a52f1c4a0566f2a2d2c7afd239": "\n\\frac{SS_{AB}}{SS_E}.\n", "1576aea5a12637288e73fd0a33103ec8": "\\sigma^2 = \\operatorname{E}( X ^ 2 ) - \\mu^2", "157703d038aa3fa1057bee5a3b799331": " E[X]_{11} = -2m/r^3, \\; E[X]_{22} = E[X]_{33} = m/r^3", "1577506b32c9ed486679b56ca0b2364b": "(p-1)/2\\,", "15776735b8f28a30308748db44fd45cf": "\n\\begin{bmatrix}\n\\begin{vmatrix} 3 & -1 \\\\ -1 & -5 \\end{vmatrix} & \n\\begin{vmatrix} -1 & 2 \\\\ -5 & 8 \\end{vmatrix} \\\\ \\\\ \n\\begin{vmatrix} -1 & -5 \\\\ 1 & 1 \\end{vmatrix} &\n\\begin{vmatrix} -5 & 8 \\\\ 1 & -4 \\end{vmatrix}\n\\end{bmatrix}\n= \n\\begin{bmatrix}\n-16 & 2 \\\\\n4 & 12\n\\end{bmatrix}.\n", "157859fa19dc8e6f0e199d5c967c471e": "t_1[K] = t_2[K]", "157861f05e4e565c764b650e9fb6ceb6": "\\frac{dr}{dt} = \\frac{d\\mathbf{A}}{dt} r_0 = \\frac{d\\mathbf{A}}{dt} \\mathbf{A}^\\mathrm{T}(t) r(t)", "157863ed597755d3bf01136ff4b3e042": "\\Re^2", "15786f316de05ed69ac95e73b97db10b": "\\prod_{k=0}^\\infty \\left ( 1 - \\frac{1}{q^{2^k}} \\right ) = \\left ( 1 - \\frac{1}{q} \\right )^{-1} - 2.", "1578a045d6f0f59bf351e6fbe3db53c3": " \\vec{s}_b ", "1578b5a9b9e3d9b42b963fa877d903f1": "w_{\\boldsymbol{xy}}", "1578c2aa3dfa7a7786dd148e725ffa9a": "f_i(\\alpha)=\\alpha^{q^{i-1}}", "1578e830b9df7b5f2030526e2690d083": "\\tan \\phi = \\frac{\\rm{BC} \\sin \\alpha}{\\rm{AC} \\sin \\beta}.", "157907749afa659528e72aa2093baec8": "\\displaystyle{|\\partial^\\alpha a(z)| \\le C_\\alpha (1+|z|)^{m-|\\alpha|}}", "15790d55a74f61f69c900e5a63bc414a": "{n \\choose \\nu}", "15790eefb3794d0539ab1399e6510c42": "f(x, y)\\!", "15795f0ad37587786eff313052dbf309": "\\Delta\\mathbf{r}_i=\\mathbf{r}_i - \\mathbf{R}", "1579bf75a2ec41a54939df716eed43ff": "\\begin{bmatrix} 1&0&0 \\\\ 0&1&0 \\\\ 0&1&0 \\\\ 0&0&1 \\end{bmatrix} \\begin{bmatrix} a \\\\ b \\\\ d \\end{bmatrix} = \\begin{bmatrix} a \\\\ b \\\\ b \\\\ d \\end{bmatrix}", "157a024493ed769eac263927c17debd5": "L = \\{ a^nb^nc^n : n \\ge 1 \\}", "157a40034a71de5c1b9c41a09a0be9d8": "\\{ H(N) , H(M) \\} = C (\\vec{K})", "157a539de99045b95023087b009870bc": "QE_\\lambda=\\frac{R_\\lambda}{\\lambda}\\times\\frac{h c}{e}\\approx\\frac{R_\\lambda}{\\lambda} {\\times} (1240\\;{\\rm W}\\cdot {\\rm nm/A}) ", "157a68adf787734e9db6707a48502e33": " \\frac{1}{2\\pi i} \\oint_C z^k\\frac{f'(z)}{f(z)}\\, dz = z_1^k+z_2^k+\\dots+z_p^k,", "157a8b4284f5ea75d520c6fb73c366ee": "x\\in\\mu(x,G)\\subseteq G", "157a96185fbdaf3c8c1b01f32bf1f7e7": "sgn(p)", "157b22a03713bdd30c7a0c0425a3d374": "C_{abcd}+C_{acdb}+C_{adbc}^{}=0", "157b565fc02df373b816360feee7ea71": "C = \\frac{Q}{V}", "157bda1e42dd199c05d9a99ebe441725": "\\sqrt{x^2 + b}", "157bde725ce70e551d064dea3b822188": " v \\in D(\\Delta_D) ", "157be742b8b08b79bda1669bf4ae317a": "A\\left(\\frac{h}{t}\\right) + \\frac{A\\left(\\frac{h}{t}\\right) - A(h)}{t^{k_0}-1} \\approx A\\left(\\frac{h}{s}\\right) +\\frac{A\\left(\\frac{h}{s}\\right) - A(h)}{s^{k_0}-1}", "157bf2f09a472e136bc178b5ae2163ef": " p_d = \\left(\\frac{e^{\\sigma\\sqrt {\\Delta t/2}}-e^{(r - q) \\Delta t / 2}}{e^{\\sigma\\sqrt {\\Delta t/2}}- e^{-\\sigma\\sqrt {\\Delta t/2}}}\\right)^2 \\,", "157c051f2c4b2b53df7a0a2d650e4a75": "\n \\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma} = \\mathbf{0} \\qquad \\implies \\qquad \\cfrac{\\partial\\sigma_{jk}}{\\partial x_j} = 0 ~.\n ", "157cdee1e6c1295915d121a016dddecb": "\\Psi_{id}", "157d51eaa8cea4ad0e0c6bd3f58a4f29": "x(t)=(x_1(t), ..., x_N(t)) \\in A", "157d98d2d90cb1539f6c1add75bc4e61": "O(\\log (\\max \\{K(x\\mid y), K(y\\mid x)\\}) )", "157da986ab45e59325519a4877db8ca6": " \\mathbf{L}_{i}^{-1} =\n\\begin{bmatrix}\n 1 & & & & & & & 0 \\\\\n 0 & \\ddots & & & & & & \\\\\n 0 & \\ddots & 1 & & & & & \\\\\n 0 & \\ddots & 0 & 1 & & & & \\\\\n & & 0 & -l_{i+1,i} & 1 & & & \\\\\n\\vdots & & 0 & -l_{i+2,i} & 0 & \\ddots & & \\\\\n & & \\vdots & \\vdots & \\vdots & \\ddots & 1 & \\\\\n 0 & \\dots & 0 & -l_{n,i} & 0 & \\dots & 0 & 1 \\\\\n\\end{bmatrix},\n", "157dc79688333227ac047d010705342d": "|2\\Theta_C|", "157ddf15c35d49507ec68d0e086e92ef": "_k\\mathbf{V}^r=\\mathbf{S}_k\\mathbf{V}^i", "157de57f8e3b05f8f67df6e61e14efd7": "\\ I_A=A_{11}+A_{22}+ \\cdots + A_{nn}=\\mathrm{tr}(\\mathbf{A}) \\, ", "157de6eaeb987616e891407eb92aa2e8": " R \\rightarrow R \\otimes_S R", "157e26d8bbe8d2ffaca5227bf75c1d08": " \\frac{2}{n} = \\frac{1}{n} + \\frac{1}{2n} + \\frac{1}{3n} + \\frac{1}{6n} ", "157e2a2ae28ccabe24227c1625991bd5": "\\omega_c = \\frac{eB}{m^*} - \\ ", "157e5bf8bdfc36bcc4639e3254305012": "R_1R_2 + R_1R_3 + R_2R_3 = \\frac{(R_aR_bR_c)(R_a+R_b+R_c)}{R_T^2}", "157ea4320ab3754080350cc7c47ae59f": "f_k(x_1,\\dots,x_m)", "157ee5ec7f45bd3399c2fd33315b2d40": "\\scriptstyle p_1, p_3", "157f2752be15df83cb72e60c175f350d": "f \\mapsto \\mathbb{P}^0_n = \\dfrac{1}{n} \\sum_{i = 1}^n \\varepsilon_i f(X_i) ", "157f389d05517b0f6162f73c98e1f274": "M(q)=1+240\\sum_{n=1}^\\infty \\frac {n^3q^n}{1-q^n}=E_4(\\tau)", "157f5d953a0086ca508afecb0b131507": "(-1)^i E", "157f6e736149f35649909cc817261596": "T_1 \\neq T_2", "157f994f0ccb5bd0438a2f6c6bcc1606": "{\\color{Blue}~6.5}", "157f9e5c73d648c19a5c58341082853f": "hg\\in V(\\Gamma)", "157fa14781f927c02cb756db273ae81c": "\\Delta = -16(4a^3 + 27b^2)", "157fe76b2767a9665879c415c9260475": " S(r') =\n\\begin{cases}\nS_0, & \\text{if }r'\\leq R \\\\\n\\,0, & \\text{if }r'> R\n\\end{cases}\\qquad(8)", "1580072f8746210f4608ac8b4efac4c4": "\\ N = f/D", "158043874ca5bd5fcda242b43bcaab2c": "\\sharp (L \\cap \\phi_H (L)) \\geq \\sum_{k=0}^n b_k(L; \\mathbb{Z}_2)", "15805d86b511f3ec49655262b4b6c934": "E_\\mathrm v", "158060b729a5d6180bdd61ad215c45f1": " \\rho = \\frac{ N_s }{ N } ", "1580709060bf4c410eb2934cfece4ce0": " u= \\hat u_0 e ^{kt} ", "158082a8651cf87b678bc52e481d5af1": "\\frac{1}{l}", "15809515d71a8f147e8340023a211ad5": " w \\notin x ", "1580ce905b1ec478998b84e85b63c627": "T = 2\\pi\\sqrt{a^3 \\over \\mu}", "1580e4269c3ba3dcf55d55ee44d4f13b": "(\\Omega, \\mathcal{F}, P)", "1581ddbd53bce67eaba8b152c234a1b0": "k: A \\to X", "1581ee7dc8f306dc9a4cb422b6919ed9": "(I < 0)", "158202eb547e8572fecdd9ee36993b11": "i:=i+1;", "158223c871a3023a2b64633a714ca3a2": "\\displaystyle{U^*F(e^{it})=2^{3/4-\\sigma/2} \\pi |1- e^{it}|^{1-2\\sigma} F\\left({1+e^{it}\\over 1-e^{it}}\\right).}", "1582719893f12c430ae4c72b5dc0745f": "(x_\\mu, \\lambda_\\mu)", "15837d3dc0788df45deea5e53d963801": " { \\Gamma(\\frac{1}{n}+1) \\over s^{\\frac{1}{n}+1} } ", "1583cd31c563db6aba62b068cb3f507a": "\\underline{a},\\underline{b} \\in U \\Rightarrow \\underline{a}+\\underline{b} \\in U", "1584194c4b047b0575ede0bdb3a3d614": "Y_1, Y_2 \\subset Y ", "15844fae553a00461f67e6f1ce892907": " \\begin{align}\n\\operatorname{cov}(X,Y) &{} = E(XY) - E(X)E(Y) = E(XY) - 0 = E(E(XY\\mid W)) \\\\\n& {} = E(X^2)\\Pr(W=1) + E(-X^2)\\Pr(W=-1) \\\\\n& {} = 1\\cdot\\frac12 + (-1)\\cdot\\frac12 = 0.\n\\end{align}\n", "1584635f63ff3f4c21d830afafa510e9": "{1 \\over \\sqrt{\\mathit{f}}}= -2.0 \\log_{10} \\left(\\frac{\\frac{\\epsilon}{d}}{3.7} + {\\frac{2.51}{Re \\sqrt{\\mathit{f} } } } \\right) , \\text{turbulent flow}.", "15849d08d4a893d77f574d9793e5b9e6": "\\left[{0\\atop 0}\\right] = 1 \\quad\\mbox{and}\\quad \\left[{n\\atop 0}\\right]=\\left[{0\\atop n}\\right]=0", "15849f78d47c2c541cd79f62ba0af120": "R(x)=a_n{\\prod (x-r)} \\mbox{ for all } r\\in R. \\,\\!", "1584a84028095dfb66e7ab623b8533a0": "\n\\eta_{a\\mu\\rho} \\eta_{b\\mu\\sigma}\n= \\delta_{ab} \\delta_{\\rho\\sigma} + \\epsilon_{abc} \\eta_{c\\rho\\sigma} \\ ,\n", "15853bac5f51b81afdf9d055fb34d560": " G \\ ", "158597e702831d0c1e008828c6a5abfd": "D_\\infty= \\frac{d+1}{2} \\infty_1+\\frac{d-1}{2} \\infty_2", "1585b31846140536e76d45d568e4029d": "(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n},", "15862f920d7221720e4040b4fc9be633": "5! = 120", "15865584e16de0dc617a9cf70e28f479": "432 \\times 204 = 88,128\\,", "15874906dc3b925104b166ba0993e669": "\\varepsilon(x,C)\\equiv\\exp{(-\\int a(C)dx)}", "158794e5928e3e86899499c6ec4d589c": "\\chi_{M}^{I}:\\mathbb{N}\\rightarrow\\mathbb{N}", "1587dc4523849612b4667b8f3bdbf322": "\\frac{\\partial |\\mathbf{U}|}{\\partial x} =", "1587e777c464476c6bfe19b4f65511b0": "\\left\\{ \\rho_{x}\\rho_{x^{\\prime}}\\right\\} =0", "1587eb85745dc11a1a54f97985d06c9b": "\\mathbb{Z}[\\sqrt{n}]", "158810ebe4b50630ad20f9e51652ecb0": "|\\Delta E|", "15882d97815327ba452dfbc531651578": " N_\\varepsilon:=\\max(\\kappa_\\varepsilon, \\lambda_\\varepsilon)", "15882ea00370eb89d1509905e460b940": " m= 0 ", "1588d1b5f367c9ce437155dd0e2182ee": " k\\in\\mathbb{N},\\ t_1,\\dots,t_k\\in\\mathbb{R}", "1588eb607818ab4c59a45141ce345c72": "f : \\mathbb{C}^n \\to \\mathbb{C}", "158904e7db3986c7cfe7de001d824370": "\\prime, \\backprime, f^\\prime, f', f'', f^{(3)} \\!, \\dot y, \\ddot y", "1589102232e24f6a353213849edbdd9e": "{f(x+h)-f(x-h)\\over 2h}.", "15891f39fbba8d83d3976094d9449403": " l \\leq a^T x \\leq u ", "1589238e395f9f7a82e72405cbacc4bb": "\\exists x (\\forall y ((\\pi (y,b) \\land y=a) \\leftrightarrow y = x) \\land a = x)", "158946ad8b0cc12cd9e276f3b8baca2e": "\\binom{p}{n} = \\frac{p!}{n!(p-n)!}.", "15894b92eb08a3a74e300ed34d9a9ca5": "{\\mathbf A}\\sim \\mathcal{W}({\\mathbf\\Sigma},\\nu)", "15895caeec0bde86ecdffa1bf4bef84c": "r_z", "1589b31ea13bf4843d464032a72c3523": "k \\ge 0", "1589be0e2979a112c910175cc8ef46ba": "y^{\\prime} = (y_1^{\\prime} \\ldots y_N^{\\prime})", "1589cfc6c58d5f6b9716a8bd5db181b1": "\\C \\rightarrow \\C, z \\mapsto z^2", "1589daa1872425e0b32116d476beca49": "\\hat{z}= z_1\\hat{I} + z_2\\hat{J} + z_3\\hat{K}", "1589fbea7fa2a30d00e6a516bf1c09dc": "B_\\text{g} = \\frac{2 \\pi G m}{5rc^2 T}", "158a0976973b25e5a3260266a4fa7db0": "(-2)^n\\,n!\\,", "158a20d137d6d44b7494019e81734bd5": "\\mathrm{Hom}(N,\\mathrm{lim} F)\\cong\\mathrm{lim}\\,\\mathrm{Hom}(N,F-)", "158a422d836132d8b373e706cf65bb3c": "M_1 = \\frac{1}{8} \\, S^{ab} \\, S^{cd} \\, \\left( C_{acdb} + i \\, {{}^\\star C}_{acdb} \\right)", "158a598d0471fe0a69e89bf68b6394d8": "\\mathbf{v}_T = \\frac{1}{f} \\mathbf{k} \\times \\nabla_p ( \\Phi_1 - \\Phi_0 )", "158ae6940c6897b5dadb98bc99517f44": "V_\\text{out} = -{I_\\text{p} R_\\text{f}}", "158b04bad8e0515e297a059163d7f372": "2^{s+1}", "158b09cb8321dcc604d309a33db556d0": "xAx', xwx' \\in (N \\cup T)^{*}", "158b2caccc1034af7124e705750c1f33": "\n \\boldsymbol{\\nabla} \\times (\\boldsymbol{\\nabla} \\times\\boldsymbol{\\epsilon}) = \\boldsymbol{0}\n", "158b62ecfee543c353a075eb9d79f755": "\\lambda = \\exp \\left ( \\left |-\\alpha^{(s)} \\right | \\right ) + \\sum_{s=1}^\\infty \\sum_k \\frac{\\exp \\left ( \\left |-\\alpha^{(s)} \\right | \\right )}{k^{(1)^2} k^{(2)^2} \\cdot\\cdot\\cdot k^{(s)^2}}", "158b6aa93f4596f520554719193169a4": "\n\\hat{\\mathbf{x}}_2 = [-\\cos\\beta\\sin\\gamma_2\\,,\\,\\cos\\gamma_2\\,,\\,\\sin\\beta\\sin\\gamma_2]\n", "158bc9745d022ce0120157276894023f": " X \\ ", "158bfa6c128a87092838c40fc12d24de": "E (\\mathbf{r}, t) = E^{(+)}(\\mathbf{r}, t) + E^{(-)}(\\mathbf{r}, t)", "158c03d0c0065bfb38fd1da052a406a4": "P(U)", "158c5cf40a5d67c5fca60cdfac2854f7": "I = -I,", "158c63eda0b4a1aab37091cc0fa52725": " [D \\rightarrow D^{'}]", "158cc82496207c65c4094d7b98c860f8": "\n\\nabla \\cdot \\left( \\mathbf\\Sigma_0 \\nabla v_0 \\right) = 0 \\,\\,\\,\\,\\,\\,\\, \\mathbf x \\in \\mathbb T\n,", "158ce2b2b8e794ce3fc61e12f005dd4a": "\\theta_3,\\ldots,\\theta_k", "158cf43a4455e3772a6ba13f6743e986": " \\ \\mathcal{L}_f, \\mathcal{L}_g, \\mathcal{L}_m ", "158d1bfe08e0474c07dc64238fd950d5": " \\begin{align}\n\\phi(x) &= a x^2-x^3, \\\\\n\\psi(x) &= 1-b x^2. \n\\end{align} ", "158d23aefaecffc74e69e4cc7c9dda48": " \\mathcal{C}(S_\\ast(X),S_\\ast(B))=\\cdots\\xleftarrow{\\delta_2} \\mathcal{C}_{-2}(S_\\ast(X),S_\\ast(B))\\xleftarrow{\\delta_1} \\mathcal{C}_{-1}(S_\\ast(X),S \\ast(B))\\xleftarrow{\\delta_0} S_\\ast(X)\\otimes S_\\ast(B),", "158d485d7cad4f8d9f90c7d2d01b72d6": "\\operatorname{Tr}(\\rho^2) < 1", "158e243a6fd150e10517552ca09a2856": " |f|(x) ", "158e312f0c4baacf72801256e4bbde13": "x = y + 1", "158e5532c5b58379449c30763b22f680": "\n \\boldsymbol{\\sigma} = \\mathsf{c}:\\boldsymbol{\\epsilon}\n ", "158e71cf7fa0b1517568c40bdb9a5cc9": "k' = \\sqrt{1-k^2}", "158e9a39ade58643c2eb42403702faeb": "\\displaystyle f_T(x_n+\\Delta x)=f_T(x)=f(x_n)+f'(x_n)\\Delta x+\\frac 1 2 f'' (x_n) \\Delta x^2", "158ebfae026f4352052046f9b9b54aad": "{52.0 \\mbox{ g }\\mathrm{H_2S}} - {28.4 \\mbox{ g }\\mathrm{H_2S}} = {23.6 \\mbox{ g }\\mathrm{H_2S}}", "158ed842cd5fa65ff039d48af8b1c928": " \\mathbb{Z} ", "158f1955d5de89a3fdb6152e87cd2eb4": "\\color{WildStrawberry}\\text{WildStrawberry}", "158f255aebd5d5af89c72d8fa0268e71": "{\\tilde{C}}_{8}", "158f6b61b11a13b0fa1f1cbecaec80f2": "\\mathbf{C}(A,A)", "15901f103d097397d5f7fdc3313e032f": "(Q \\to P)", "15908194ba88c19c15af20b49a7d1364": "R(t) \\sim (-1)^{N-1}\n\\left( \\Psi(p)u^{-1} \n- \\frac{1}{96 \\pi^2}\\Psi^{(3)}(p)u^{-3}\n+ \\cdots\\right)", "1590b2a8b109a6280423c57230b295ab": "SLG = \\frac{TB} {AB}", "1590b4ecf12b76f32e32ee48fad02602": "\\Psi : L(H_B) \\otimes C(X) \\rightarrow L(H_A).", "1590bdd1ac1de3098ae64ae14bb15bc2": " \\lim_{k \\to \\infty} \\left( \\frac{1}{4} \\right)^k = 0 ", "1590bf11662688c51fe74034fac77c5f": "y = 4 e^{-ln(2) t}= 2^{2-t}", "1590cf5200cbfd405fa15a562aeea4dc": "2+2 < 5", "159108a61907c6a33dca946641595f2a": "\\kappa_\\nu", "1591b39ab84d6f8c5eacef5e7f440e1e": "\\tilde{o} = (T1'-T1-T2'+T2)/2", "1591e9541c3ccdda4d0adf5e0742a4be": "\\text{II}(v,w)=\\langle S(v),w\\rangle", "159200de3062dfbbea4dac89128f4cf1": " {}^\\mathrm{N}\\!\\boldsymbol{\\omega}^\\mathrm{B} = {}^\\mathrm{N}\\!\\boldsymbol{\\omega}^\\mathrm{D} + {}^\\mathrm{D}\\!\\boldsymbol{\\omega}^\\mathrm{B}.", "1592a07a39644dada570b31bb5e35aab": "t_{1/2} = \\frac{[A]_0}{2k}", "1592e428a53437cee99f85c65e741d65": "{\\int}_{S} \\mathbf{F} dS ={\\int}_{\\tilde{S}} \\mathbf{F} d\\tilde{S} ", "1592f21267a349034a3b364022436006": "\\rho(\\theta) = \\rho_\\perp + (\\rho_\\parallel - \\rho_\\perp) \\cos^2 \\varphi ", "1592f7b158da73e482bdccb713ee0402": "k=0, 1,\\cdots, n", "1592f84858cb4edaf8e71ace005b9aca": "a (f * g) = (a f) * g \\,", "159366c42181c8efb408c7139099f3b9": "\n\\rho_{x^{n}}\\equiv\\rho_{x_{1}}\\otimes\\cdots\\otimes\\rho_{x_{n}}.\n", "15939189fdbaddb34870f063c2724a5d": "\n\\sin x \\approx x \\text{ when } x \\approx 0.\n", "1593bbb1848be7f6f0550af1b0ea0567": "\\scriptstyle{\\hat{U}(t_0,t_0) = 1}", "15944619706d6ff55791c24833609ccb": "\\Delta\\left(\\,P-P_e\\right)+\\frac{\\rho}{2}\\,\\left[\\frac{\\,f}{\\,D}+\\frac{f_e}{D_e}\\left(\\frac{F}{F_e}\\right)^2\\right]\\,W^2\\Delta\\,X\n+\\frac{\\rho}{2}\\left[\\left(\\,2-\\beta\\right)\\,-\\left(\\,2-\\beta_e\\right)\\left(\\frac{\\,F}{F_e}\\right)^2\\right]\\Delta\\,W^2\\,=\\,0", "159464c4c1ffdcd5479f1c8f38f6ee72": "(x_2,-x_1,x_4,-x_3,\\dots,x_{2n},-x_{2n-1})", "15946696b88e05d0fdb844f46c31e510": " I = 0.16 Re^{-1/8}. ", "15946b211bc9675a7940db155566ab80": "S(A\\mid BC)\\leq S(A\\mid B)", "159492b89d7997ce974fb61434ca6af8": "\\mathbb{M}_n (\\mathbb{C})", "1594b61cf0b73cbabb74d091025dec41": "nx_i R \\ln(V/V_i) = - nR x_i \\ln x_i\\,", "1594fb94d8791fc03e5b00694c4129a3": "v_{\\alpha,0}", "15951ee46361447e80755c8fafb141de": "(v_{0{^{ }}} [H^+]_0-v_i[OH^-]_0)/(v_0+v_i) = [H^+]_i - K_w/[H^+]_i", "1595357dde7d6179ade6e587c127361e": "\\int\\frac{\\mathrm{d}x}{\\sin ax} = \\frac{1}{a}\\ln \\left|\\tan\\frac{ax}{2}\\right|+C", "15955b7482fa3f61ddc7b9690e620d97": "\nh^\\prime(x)=e^x\\log x+x^{-1}e^x=h(x)+f(x)g(x).\n", "1595ab75534e7e40203ddaf98baeeac1": "s_{ij}^v(x) > s_{ji}^v(x)", "1595c1a44120bc8bf4cdbbfe228a0de1": "ds^2=\\frac{4}{|q|^2 (\\log |q|^2)^2} dq \\, d\\overline{q}", "1595d365f082f21681ddd4ae39de2430": "h(F_1)=A", "1595d708e8b3afca37be7a5a8075ce1c": "n^2 a^n u[n]", "15962be4aac1b52d034a3adbb895bd01": "t^{-n}\\int_{\\mathbb{R}^n} f(y)\\varphi(y/t)\\, dy = t^k \\int_{\\mathbb{R}^n} f(y)\\varphi(y)\\, dy", "1596a7c62a81030cf307d771fefbacbb": "{\\mathbb P}\\biggl(\\bigcup_{i=_1}^{n+1} A_i\\biggr) \\le \\sum_{i=_1}^{n} {\\mathbb P}(A_i) + \\mathbb P(A_{n+1}) = \\sum_{i=_1}^{n+1} {\\mathbb P}(A_i)", "1596a98c1da23daee068026373951cde": "\\langle l_1, r_1 \\rangle_w", "1597122dd9a68542ec0c33d1624f1e2b": " a = \\delta ", "15973cfc8e7ee6c42f9b352459c02c00": "C(h)", "159775a3dd81e203a63c0d55b5c56c2c": " ModVR = 1 - \\frac{ Kf_m - N }{ N( K - 1 ) } = \\frac{ K( N - f_m ) }{ N ( K - 1 ) } = \\frac{ K v }{ K - 1 } ", "159793a4a769bd8edd528c9b73ed9d95": "w = (1 + \\lfloor 2.6 \\cdot 11 - 0.2 \\rfloor + (0 - 1) + \\left\\lfloor\\frac{0 - 1}{4}\\right\\rfloor + \\left\\lfloor\\frac{20}{4}\\right\\rfloor - 2 * 20)\\ \\bmod\\ 7", "1597f846f1853c71c44004c21b980359": "- \\frac{d \\rho}{\\rho} = -\\frac{1}{a^2} \\frac{dp}{\\rho} = -\\frac{1}{a^2} \\frac{-\\rho V d V}{\\rho} = \\frac{V}{a^2}d V", "15983db6289aae8d3a71fc7e15b67153": "{{i}_{B1}}={{i}_{B2}}", "159871b70efc10fbddd951d127560f6a": "B_Z", "15987259d4b76cf46a31c7164f2df634": "y=g(x).", "1598d88f06844008f486c8c51bbcaa14": "\\mathrm{wind}", "15999f6a0ddec237d55fc56843a97cdd": "z \\mapsto z - R \\frac{f}{f'}", "1599a047ffc4a884e936eff8e8572512": "\\{x\\in F : w(x)>0\\}", "159a321fc0413162bbfd5843d689904b": " HRT = \\frac{Volume\\; of\\; aeration\\; tank}{influent\\; flowrate} ", "159a721fa09a49abf0d1fda4f2c4bd5e": "W_0,W_1, ..., W_{N-1}", "159a92b310043eb6dfb1ec235936a100": "V'_{a}", "159aec9bd0b30b8fa9cc3db72fea629c": "\\Gamma_u", "159afee50ea8e265039b582d51a6624f": " 0 < \\lambda < 1 ", "159b0786e476906d332710b8dc91c53a": "{\\tilde{C}}_{5}", "159b5cd78d1990e1596ce704d559b5fd": "B(x;r)\\subseteq B(y;s)", "159bb2c223b17fb81983b683006e7afa": "\\left\\langle\\rho=\\rho_2,Z_2\\right\\rangle", "159bd0a488d278e99d15ef00fd7d89fb": "\\lim_{x\\to a} f(x) = L\\,", "159be9da606742ceb0c03faf7b8bf584": "\\text{gyr}[\\mathbf{u},\\mathbf{v}]\\mathbf{w}=\\ominus(\\mathbf{u} \\oplus \\mathbf{v}) \\oplus (\\mathbf{u} \\oplus (\\mathbf{v} \\oplus \\mathbf{w}))", "159c189938462c8730c82eb8831b6333": " \\lambda = (\\lambda_1, \\lambda_2, \\cdots, \\lambda_k) ", "159c9230e219d55346e8def570a2e524": "\n\\hat{\\Delta} = R_i - \\hat{R} - \\hat{a} \\cos{\\theta_i} - \\hat{b} \\sin{\\theta_i}\n", "159cb9183d9730290ed2f89c463982bd": "\\begin{align}\n\\varepsilon_1(t) &= -\\frac{1}{2}\\sqrt{4a^2 + (\\hbar\\omega_0 - 2\\mu B(t))^2}\\\\\n\\varepsilon_2(t) &= +\\frac{1}{2}\\sqrt{4a^2 + (\\hbar\\omega_0 - 2\\mu B(t))^2}.\\\\\n\\end{align}", "159d02907cc0242abca769b4a6d7a019": "H_{\\nu}( \\omega)", "159d213753d88ac2ba9da6d48f4221ec": "\\mathbf R", "159d32cf7b7e681ee5d5bf88e1057709": "r(t_1,...,t_n)", "159d491622b4ec95a20fa5bcfa9d4725": " \\tfrac{k_0}{p_0}\\,p ", "159d5f60bba230164e90208becdf12a5": "w=z", "159d817837d8424099a67c86ec84b8b3": "\\mathbf{P}_1(R)", "159d8654856a89593e5a6668d67152be": "\\alpha : TM \\rightarrow {\\mathbb{R}},\\quad \\alpha_x = \\alpha|_{T_xM}: T_xM\\rightarrow {\\mathbb{R}}", "159de732021373b213818c1796a9f6fe": " \\text{for } t<\\log_e(2)", "159e0adc038b248158a210e05fa25bce": "A(i,i)", "159e9457856b41152608d43e897a1de1": "A_{ij} ", "159ec4588327ac31c8f201f70e6b20ce": " i\\hbar {\\partial \\psi (x, t) \\over \\partial t} = -\\frac{\\hbar^2}{2 m} \\frac{\\partial^2 \\psi (x, t)}{\\partial x^2} + U(x) \\psi (x, t) ", "159ede73b5141eaf74a84c7714a106d4": "\\scriptstyle J^{\\gamma} ", "159f12ce9148954f5f1c7dedd3def78f": "P\\!", "159f1fd015f2c4a16f360de56c3b80d4": " Bq^ \\ast \\equiv 0 \\bmod \\ (d/z) ", "159f461e95b3364702ab257b8a40d8de": "\ng^{\\mu\\nu} \\frac{\\partial S}{\\partial x^{\\mu}} \\frac{\\partial S}{\\partial x^{\\nu}} = c^{2}.\n", "159f5192acff5e8dc7db30f9384240db": "\\varepsilon = 0.01", "159f69eb8dee26f9c6e864243fc6b120": "\\frac{1}{2\\pi i}\\int_{-i\\infty}^{i\\infty} \\frac{\\Gamma(a+s)\\Gamma(b+s)\\Gamma(-s)}{\\Gamma(c+s)} (-z)^s \\, ds", "159fac319463a70c7ab8f1f6d02442a3": "E_F = \\frac{\\hbar^2}{2m_p} \\left( \\frac{3 \\pi^2 (6 \\times 10^{43})}{1 \\ \\mathrm{m}^3} \\right)^{2/3} \\approx 3 \\times 10^7 \\ \\mathrm{eV} = 30 \\ \\mathrm{MeV} ", "15a044cd3da8d90f7ad8ae7ade5da46d": " S_d [i_r, i_z] = \\frac{S[i_r, i_z]}{N\\Delta V(i_r)}", "15a05b43eaf77133cee28c619e0d7f32": "\\{ y_i \\} ", "15a07fdb25797093c39e7b36fa2b6d7c": "N(E) = \\frac {V}{2\\pi^2} \\left(\\frac {2m}{\\hbar^2}\\right)^{3/2}\\sqrt{E-E_0}", "15a0b2b3b65a87583abfc5864a071b9f": "b < 2^{w-M}", "15a0ced64f2312cfc5e8a290857a1503": " \\ C_{l_\\alpha} ", "15a0cff797c349351ab4f600b5f2df0d": "f_{a}(1) = 4^{1^{0} 2^{0} 1^{1}} = 4^{1} = 4 ", "15a0eae7f7184d6b692478431ab00098": "\\alpha = \\frac{\\sqrt{4\\,b_2\\,b_0 - b_1^2\\,}}{2\\,b_2}. \\!", "15a0edfed7445c8b972b09217adccc65": "d/D \\geq 0.07", "15a0fa890ef5d942c426262efb871dbe": "x^2+2xh+h^2 = (x+h)^2,", "15a152f6e29e18e521cd0187a5e05b03": "\n E=\n \\left( { a_1\\, a_2 \\over 2 \\pi L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 }\n\\mathcal J_0 \\left ( kr_{B1} \\right) \\mathcal J_0 \\left ( kr_{B2} \\right) \\mathcal J_0 \\left ( kr_{12} \\right)\n=\n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_B^2 }\n\\mathcal J_0^2 \\left ( k \\right) \\mathcal J_0 \\left ( k{r_{12}\\over r_B} \\right)\n", "15a19ff7f8b1ed8508009c382fd1c3f7": "T\\;", "15a1beedfb982e81cc06a2cbe4b8d668": "\\mathcal{L}=\\bar\\psi_B\\left[i\\gamma_\\mu (\\partial^\\mu + ie_BA_B^\\mu)-m_B\\right]\\psi_B -\\frac{1}{4}F_{B\\mu\\nu}F_B^{\\mu\\nu}", "15a226a912eed749d24be5f0252e7a55": "\\sigma_{y}", "15a2311476b41023ee77af5d711a61ff": "I(t)=M_at-P_0+P(t) \\,", "15a2c1783c28f988302c23e8b1d17528": "f(\\mathbf{r})=\\sum_{\\mathbf{K}} h(\\mathbf{K}) \\cdot e^{i \\mathbf{K} \\cdot \\mathbf{r}}, ", "15a2cbb963cf3912de9da737cec98841": "\n1+B_t(t,T)+\\alpha(t)B(t,T)-\\frac{1}{2}\\gamma(t)B^2(t,T)=0\n", "15a2d6da795620300ad3e583b255098d": "V_n(\\sqrt{R},Q) = \\frac{a^n+b^n}{a+b}", "15a35368435047a2bcb47b5ab3c42817": "\\beta(\\alpha_s)=-\\left(11-\\frac{2n_f}{3}\\right)\\frac{\\alpha_s^2}{2\\pi}~,", "15a3b2d6b390d100390a7dd5598fc4b7": "\\phi = hf_0\\,\\!", "15a3e569dbae8b65fe55b2e7be9100ac": "\\wp_\\tau(z)", "15a40c10a679907b72ce76117c4535b6": "\\psi_{n_x,n_y,n_z} = \\sqrt{\\frac{8}{L_x L_y L_z}} \\sin \\left( \\frac{n_x \\pi x}{L_x} \\right) \\sin \\left( \\frac{n_y \\pi y}{L_y} \\right) \\sin \\left( \\frac{n_z \\pi z}{L_z} \\right)", "15a418f456760c26776fdef7dc372cca": "\\{p,1-p\\}", "15a432f30c01457fcae6b73310e7de3d": "(x-3) (x-1)^3 (x+1)^3 (x+3) (x^2-3)^2.\\ ", "15a49140db2d4d6b0dce2636b730e2d0": "t \\in \\mathbb{R}", "15a4f2c4f69248dc99cd43b70f28bac4": "\\mathbf{F} = q\\left[-\\nabla \\phi- \\frac{\\partial \\mathbf{A}}{\\partial t}+ \\nabla(\\mathbf{v}\\cdot\\mathbf{A})-(\\mathbf{v}\\cdot\\nabla)\\mathbf{A} \\right]", "15a4f479f3cbc8a5f090a936bb9ddffc": " z_{k+1} = z_k + hf(q_k,p_{k+1}) \\, ", "15a51ca1c3908a562357091e3bb4f285": "\\Box \\phi", "15a5652489e7465d11da99309009febd": "K = \\mathbb{R}^d_+", "15a5d314e912346fbcaf2ca8024a10fe": " S = \\sum_{i=1}^{N-1}(c_{i+1}-c_i)^2 ", "15a5f902bcd0a5374a1b9b80507fa7e9": "\\textstyle n2^{l-1}+1", "15a639ccc4b3e0056a8fc594553642b1": "-\\frac{dN}{dt}= \\lambda N", "15a64955780f6ec48902a71956c120e1": "\\omega_N = e^{-\\frac{2\\pi i}{N}},", "15a668b9bcce8685a0797693dc7e9582": " 2 \\le i \\le m-1 ", "15a67b8b933a6bd77593e3bd4c779317": "\\alpha<2", "15a764354a3f4ceb778a21690fb952cf": "\\mathbf{M}=\\mathbf{F}^\\mathrm{T}", "15a76ff818d343cc5eef7101f97a63ae": "F_0 \\colon M \\to M \\times 0", "15a795d5b4688eef60138943925f3111": "r_a+r_b+r_c=4R+r,", "15a7e3687c8dd090497a498de2f1eca6": " X_n = \\{x \\in X: \\operatorname{dim} H_x = n\\}. ", "15a874ae54da1ba7b9d6c6cc8ac42993": "a_0,\\ldots ,a_N > 0", "15a8bf428849adbfa7472abb73905ab5": "\\alpha \\times Y", "15a8f2c34eb79dfd32cc8f2a0978e299": "|00...0\\rangle", "15a900072b20edb02837602480ea2320": "A^{T}A = I, det(A) = +1 ", "15a96b96777a9f1ad352d52751eaf17c": "\\gamma_n(x)", "15a96f27e285951bde18cf32f5d872bb": "\\cdots\\overset{d_{n+1}}{\\longrightarrow}E_n\\overset{d_n}{\\longrightarrow}\\cdots\\overset{d_3}{\\longrightarrow}E_2\\overset{d_2}{\\longrightarrow}E_1\\overset{d_1}{\\longrightarrow}E_0\\overset{\\epsilon}{\\longrightarrow}M\\longrightarrow0,", "15a9a8e6cca35e1319518b144ac07fd6": " \\begin{align} \\mathbf{r}_\\mathrm{cog} & = \\frac{1}{M \\left | \\mathbf{g} \\left ( \\mathbf{r}_\\mathrm{cog} \\right ) \\right |}\\int \\left | \\mathbf{g} \\left ( \\mathbf{r} \\right ) \\right |\\mathrm{d}\\mathbf{m} \\\\\n & = \\frac{1}{M \\left | \\mathbf{g} \\left ( \\mathbf{r}_\\mathrm{cog} \\right ) \\right |}\\int \\mathbf{r} \\left | \\mathbf{g} \\left ( \\mathbf{r} \\right ) \\right | \\mathrm{d}^n m \\\\\n & = \\frac{1}{M \\left | \\mathbf{g} \\left ( \\mathbf{r}_\\mathrm{cog} \\right ) \\right |}\\int \\mathbf{r} \\rho_n \\left | \\mathbf{g} \\left ( \\mathbf{r} \\right ) \\right | \\mathrm{d}^n x \\end{align} \\,\\!", "15a9b73c37000fa3e83b737099a299c7": "P \\setminus K", "15a9ccb8060b43d920e39e01ff0156d1": "\\mathrm{}\\ \\Pr(X>40 \\mid X > 30)=\\Pr(X>10).\\,", "15a9cedbab792101825311615fd74749": " 0 = \\sum_{m=1}^d b_{m,n} u_m ", "15aa2eaa36aeb7eb649e20eebda1fdf6": "g^{(2)}(\\tau) = 1 ", "15aa3a38fce913805176c7cb29301b64": "F_0(a, b) = a + b", "15aa647c827f671b92a1fd276abac7ed": "V(J)=1", "15aa6eaad71b91fa8df72e39fdf6f744": "\\mathbf p_{m+1}=\\sum_i^m c_i\\mathbf p_i.", "15aab1388fcbe47ff5b9f79235412390": "m=1 0 1", "15aabc8fd5a35230ac695c5186c508a6": "(v,w){:}", "15aacf1aed839305ac36d38bc6167739": "\\liminf_{n \\to \\infty} \\| x_{n} - x_{0} \\| < \\liminf_{n \\to \\infty} \\| x_{n} - x \\|,", "15aae5fc0a6f417399ed38195bee8892": "P_1(y)", "15ab09152ff5b36d264aa5ba472b0c2a": "\\Delta G^\\ominus_{Fe} = -nFE^0_{Fe};\\Delta G^\\ominus_{Ce} = -nFE^0_{Ce} ", "15ab2d2b0b92c13f328635e5c4bdbe64": "i+1", "15ab8efc379b8627b4b644325a4bb0c0": "\\phi(T\\otimes U)=\\phi(T)\\otimes\\phi(U)", "15ab95def80823e45f1a6dfdb8021b77": "\n\\frac{1}{a_{\\overline{n}|i}} - \\frac{1}{s_{\\overline{n}|i}} = i\n", "15abeba2ef8c4882259ac119c4ca3ca2": " \\Delta F_y = \\frac{1}{2} C_L \\rho w^2 ldr \\frac{\\cos(\\beta - \\phi)}{\\cos\\phi}", "15ac0347a6a9075f425d0f9c1e2bfdf5": "V_g f \\in Y", "15ac818c15c8653db90739a08427c531": " l = \\frac{R T} {m_{A}g} ", "15acad78b054830808698dcd07dcba83": " \\tau(w) = w' : (w,w') \\in U ", "15acd4240aa428eeb50998aaee7835c2": "|H\\rangle", "15ad03c50e93cfbbc5b1e515a5b5f785": "{\\zeta_g}", "15ad92e948ea305c298a2867e0f0c6ba": "\\mathbf{I_m} = \\mathbf{I_0} \\sin \\phi_0 ", "15adaceddd1a2ac25858b574e8802da7": "\\,\\hat{\\lambda}_i", "15adbe25adc627c59823907f881c3b1f": "f=\\frac{v}{2l}", "15adcba3bdb0754dd7aedb4540134925": " x(t) = -\\frac{qE_0}{m \\omega^2}\\sin(\\omega t) + \\frac{qE_0}{m \\omega}t - \\frac{d}{2} ", "15adcbc83e8e5da33cd5fc22108ff0b5": "A = 227\\cdot5^2", "15adcf5af2342261722d9cfaf993a7fe": " P(f) = \\frac{1}{f_\\max - f_\\min}\\int_\\Omega \\delta(f - F(\\boldsymbol{r})) P(\\boldsymbol{r})\\,d\\boldsymbol{r} = \\frac{1}{f_\\max - f_\\min}\\frac{1}{V}\\rho(f) \\int_\\Omega \\delta(f - F(\\boldsymbol{r})) \\, d\\boldsymbol{r} = \\frac{1}{f_\\max - f_\\min} = \\text{constant}", "15adef1749db58083539ddd73babdc70": "D_N({\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1})=1", "15ae52d0a5c11ff8a7cd7bffc075762c": "g*", "15ae64b4a18bf00ce833b6dba8114e8f": " \\forall i \\in C^* \\text{, } |\\mathbf{c}_i| \\geq w_\\min ", "15ae75960ffd194d3707fee25eda5dea": "e_j^{N_j} = 1 = \\omega_{jk}^{N_j} = \\omega_{jk}^{N_k} \\,", "15ae88fb367511de5388699de99ffa83": "\\delta(q_1, b, a) = (q_1, \\varepsilon)", "15ae96c14d1517db2bfa3551aaa2c5aa": " \\begin{array}{lcl}\nPrecision ={ \\left \\vert Prelevant \\cap Pretrieved \\right \\vert \\over \\left \\vert Pretrieved \\right \\vert}; \\\\\n\\\\\n\nRecall = {\\left \\vert Prelevant \\cap Pretrieved \\right \\vert \\over \\left \\vert Prelevant \\right \\vert} ;\\\\\n\\\\\nAveragePrecision = \\int_0^1 {Prec(R_{ecall})}dR_{ecall},\\\\\n\n\\end{array}\n", "15aeaca53456b1ef193568e90b9f9141": " N_10 = 32+4+2+1 ", "15aefc48007673476be6000d0a080ee4": " \\ln{p_s \\over p_0} = \\frac{2 \\sigma M}{RT \\rho \\cdot r_p} ", "15af582436a80bdb8c712f7a88315c05": "\\supset", "15af7211b9e69fa7ac8b2ca415229b02": "e^{x_1y_1}, e^{x_1y_2}, e^{x_2y_1}, e^{x_2y_2}.", "15af80a32fb35660a7441d811fa4d063": "\\epsilon^\\mu(q,y) \\,", "15af99f4716e26855fed68cbd82b62ce": "F(\\vec{r},t) = \\varepsilon a e^{i\\vec{k}x-\\omega t} + h.c ", "15b0021e4b3c087ac029030f5336b0cd": "\n \\lim_{n\\to\\infty} \\operatorname{Pr}(X_n\\in A) = \\operatorname{Pr}(X\\in A)\n ", "15b08699ac7e80090714a81e44781870": "\\lambda: A \\to B", "15b088e42dbe089b5302a75efc6b1687": "b=\\det \\begin{pmatrix} z_1w_1 & z_1 & w_1 \\\\ z_2w_2 & z_2 & w_2 \\\\ z_3w_3 & z_3 & w_3 \\end{pmatrix}\\, ", "15b0a8a0c01e5e2f1d901888ed491e00": "n < 214928639999", "15b0c6118b4b283ef9b0e174fef1e40e": "\\pm\\tfrac{1}{2}+G(x-y)|_{\\mathbf{R}^{n-1}}", "15b0d170266cc663d85481ecb3ed4fa6": "\\mathbf{F}^g = \\mathbf{I} + [\\vartheta^{\\perp}-1]\\mathbf{s}_0\\otimes\\mathbf{s}_0", "15b1165ee52e66c76559f28205268924": "\\begin{align}\nL'& = L'_{old}+v\\Delta t'=\\frac{L}{\\gamma}+\\frac{\\gamma v^{2}L}{c^{2}}\\\\\n& = \\gamma L\n\\end{align} ", "15b1359fc78dcd4a855a126c08bb6c32": " \\sup_\\Lambda|\\chi(\\lambda)-1| \\leq \\epsilon, \\quad\n\\chi\\in\\hat{G}.", "15b1cbee5939bc1ea8d8487f639c0691": " {\\rm pH} = -\\log_{10} [ {\\rm H^{+}}] \\,\\!", "15b1d37bb7f5937c66ad4aca512f580e": "\n\\sum_{\\delta\\mid n}\\Lambda(\\delta)=\n\\log n.\n", "15b1e4d6d2feceb255e4acd82cccf6a7": "\\sum_{k=1}^m \\frac{1}{k^2} = \\frac{1}{1^2} + \\frac{1}{2^2} + \\cdots + \\frac{1}{m^2}", "15b1e887fd51a27495a10c72d8f14b1e": "\\scriptstyle \\lfloor k\\rfloor\\,", "15b1e88a9c6cc1a7808db4fa2540a280": "\\sum_j n_j \\text{Reactant}_j\\rightleftharpoons \\sum_k m_k \\text{Product}_k \\equiv \\alpha A+\\beta B...\\rightleftharpoons \\rho R+\\sigma S ... ", "15b201981b91fa3db08249a57516327b": "{v_{esc}}\\,", "15b2710d28a65222c7942cfad48419b8": "\\sum_{k=2}^\\infty (-1)^k \\frac{\\zeta(k)}{k} = \\gamma", "15b2d3d77baa02dff47bf537690a3144": "\\mathrm{O_2 + 4 \\ e^{-} + 4 \\ H^{+} \\longrightarrow H_2O}", "15b2da7e7b99c320b0a07dca941a69b5": " z \\in [x,y] ", "15b2f1c5744e0329f210a15b1278ecb1": "M(j,j) = 1 - \\frac{\\lambda m(j)\\sum_{i=1, i\\neq j}^{20}A(i,j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)}", "15b30a02f62fd433a23d4503a4fca3af": "V_\\mathit{SYNCHSW} = I_\\mathit{SYNCHSW}R_\\mathit{on} = (1-D)I_o R_\\mathit{on} ", "15b330d06e8136d00895dd6b480f8420": "H_3L \\rightleftharpoons H_2L+H:pK_1=-\\log \\left(\\frac{[H_2L][H]} {[H_3L]} \\right)", "15b39bacaa699958e470c8e2d2cad8b3": " (x_i^1, x_i^2) ", "15b3cf12724810c360e9c60e592156da": "P(k,k') = \\frac {2 \\pi} {\\hbar} \\left( D_{ac} \\sum_{q} \\sqrt{ \\frac {\\hbar} {2 M N \\omega_{q} } } | q | \\sqrt { n_{q} + \\frac {1} {2} \\mp \\frac {1} {2} } \\, I(k,k') \\delta_{k' , k \\pm q } \\right)^2 \\delta [ \\varepsilon (k') - \\varepsilon (k) \\mp \\hbar \\omega_{q} ],", "15b41df8b46ce1c1ff53664445c11f73": "-\\frac{\\hbar^2}{2M}\\nabla_i\\cdot\\nabla_j ", "15b47468765e2de05c9e95b5a826d1cb": "\\left({9 \\over 75} \\right)", "15b4a17a9366c9a59085bef8f32de4e5": "x=\\frac{1}{a_1}+\\frac{1}{a_1a_2}+\\frac{1}{a_1a_2a_3}+\\cdots.\\;", "15b4ad58baaa08c32f8c67f18e672c3e": "(x_0,y_0)", "15b4af05e9582885bed95d18844b9e45": "\\Delta R_{0}\\Delta x_{0}=\\Delta r_s\\Delta r\\ge\\ell^2_{P}", "15b4f8f00c591228cb92f88164bdc3a3": "(a_n)", "15b508d2db1619fa3656a5c266274483": " x^n(x^2-x-1) - (x^2-1)\\,", "15b5177220acb64df10fc20902356ff4": "\\displaystyle{Y=\\mathrm{Ad}(X)\\cdot Y= Y + [X,Y] + {1\\over 2} [X,[X,Y]]\\in \\mathfrak{m}_+ \\oplus \\mathfrak{k}_{\\mathbb{C}} \\oplus \\mathfrak{m}_-,}", "15b51ded00bc1cc6fd772d859f151295": "\\mathbb{R}^{2d}", "15b51e0caff1955020c7bfc6d29e3304": "\n \\begin{align}\n &\\nabla^2 \\left(\\frac{\\partial \\varphi_1}{\\partial x_1} + \\frac{\\partial \\varphi_2}{\\partial x_2}\\right) = -\\frac{q}{D} \\\\\n &\\nabla^2 w^0 - \\frac{\\partial \\varphi_1}{\\partial x_1} - \\frac{\\partial \\varphi_2}{\\partial x_2} = -\\frac{q}{\\kappa G h} \\\\\n &\\nabla^2 \\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) = -\\frac{2\\kappa G h}{D(1-\\nu)}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) \\,.\n \\end{align}\n", "15b54adae7e26abe688e1a0bd9dc2da5": "A \\to X", "15b5dc2065bec88d12b000ac4facc905": "{{Tonnage}} = \\frac {{Length}\\times\\ {Beam} \\times \\frac {Beam}{2} \\times \\frac {3}{5}\\times {0.62}} {35} ", "15b5dca93fb69d952a3dd9539cd5f20e": " X + \\mathit{ATP} \\longrightarrow \\mathit{XP} + \\mathit{ADP}", "15b698c308009d5c2d3d2cb432d22510": "\\omega.\\!", "15b6e22b44bb28cb88f1df76cafba3f6": "+ (\\partial_{b_1} X^c) T ^{a_1 \\ldots a_r}{}_{c b_2 \\ldots b_s} + \\ldots + (\\partial_{b_s} X^c) T ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_{s-1} c} + w (\\partial_{c} X^c) T ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_{s}}\n", "15b70d274ddb172cea22c1998c7f68b4": "\\sqrt{2\\pi}\\Big(\\varphi(t)+t\\Phi(t)-\\max\\{t,0\\}\\Big)", "15b77835caca4e41ac84afea64c86869": " \\sigma = {3}\\varepsilon_0\\frac {\\kappa-1}{\\kappa+2} E_{\\infty} \\cos \\theta =\\frac{1}{V} \\bold{ p \\cdot \\hat{R}}\\ . ", "15b77a8e21dd857fa3ee2d0edea24650": "k^{\\beta}_i", "15b7a68d43b66b052d0184693a12b8b2": " \\sum\\limits_{k=a}^{b-1} f(k)=\\int_a^b f(x)\\,dx \\ + \\sum\\limits_{k=1}^m \\frac{B_k}{k!} \\left(f^{(k-1)}(b)-f^{(k-1)}(a)\\right)+R_-(f,m). ", "15b7b1ddb03e6bad0184fcfcd71a8c42": "\n\\begin{align}\nV_\\mathrm{rms} &=\\sqrt{\\frac{1}{T} \\int_0^{T}[{V_{pk}\\sin( \\omega t+\\phi)]^2 dt}}\\\\\n &=V_{pk}\\sqrt{\\frac{1}{2T} \\int_0^{T}[{1-\\cos(2\\omega t+2\\phi)] dt}}\\\\\n &=V_{pk}\\sqrt{\\frac{1}{2T} \\int_0^{T}{ dt}}\\\\\n &=\\frac{V_{pk}}{\\sqrt {2}}\n\\end{align}\n", "15b7b2d36fba832657d78d2045b8f9df": "\\int\\frac{r^7\\;dx}{x} = \\frac{r^7}{7}+\\frac{a^2r^5}{5}+\\frac{a^4r^3}{3}+a^6r-a^7\\ln\\left|\\frac{a+r}{x}\\right|", "15b7f368a251434ef0f694ee96efd6de": "T(y_0) = T_0\\,", "15b817ca48d2c94356b48bdba4f406e5": "a_8\\times (6\\rho^4-6\\rho^2+1)", "15b827ec7905f04f1bf39cb554b08430": "BSC_{p}", "15b83ef1361eb0fb666ef29f9b48c5e2": "f = \\beta y,", "15b841c73f03e6246b7b90dc2167706b": " \\sum_{k=1}^{d+1} H_{B_k} \\geq \\frac{d+1}{2} \\log(\\frac{d+1}{2} )", "15b858c0d0b1c830e8dc63a4466f6e78": "P = \\frac{P_{max}} {2}", "15b8730ba75051c58f866cb2ffcc0a45": "\n \\left( B \\or C \\right) , \\lnot C , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\vdash \\lnot A\n ", "15b88e119ddf26a00567806f64ffee62": "\\begin{align}\n& \\mathbf{T}^2 = \\begin{pmatrix}\n\\frac{1}{16} & \\frac{1}{4} \\\\[4pt]\n0 & \\frac{1}{16}\n\\end{pmatrix}, \\quad \\mathbf{T}^3 = \\begin{pmatrix}\n\\frac{1}{64} & \\frac{3}{32} \\\\[4pt]\n0 & \\frac{1}{64}\n\\end{pmatrix}, \\quad \\mathbf{T}^4 = \\begin{pmatrix}\n\\frac{1}{256} & \\frac{1}{32} \\\\[4pt]\n0 & \\frac{1}{256}\n\\end{pmatrix}, \\quad \\mathbf{T}^5 = \\begin{pmatrix}\n\\frac{1}{1024} & \\frac{5}{512} \\\\[4pt]\n0 & \\frac{1}{1024}\n\\end{pmatrix},\n\\end{align}", "15b8cce139ab68921eb9889be91d6b10": "p_y(x) := \\vert \\langle x , y \\rangle \\vert \\qquad x \\in X ", "15b8fe65a153f3d61e3aa92d7d727d28": "\nmL_{e}^{2}\\ddot{\\theta}=-kL_{e}^{2}\\theta-\\mu B \\sin{\\left(\\frac{\\theta H_k}{B+H_k}\\right)}\\left(\\frac{H_k}{B+H_k}\\right)-2K_uV\\sin{\\left(\\frac{\\theta B}{B+H_k}\\right)}\\cos{\\left(\\frac{\\theta B}{B+H_k}\\right)}\\left(\\frac{B}{B+H_k}\\right)\\Rightarrow\n", "15b9015edadefaca276c4699b4faa442": "\\{j_1j_2j_3\\}", "15b983900b08826c6039f3daea7be214": "P_0=k\\left(V/H\\right)^n", "15b9da7a94585faa52f59490cb35a005": "\\lambda m.\\lambda n.n \\operatorname{succ} m", "15ba550d5555a6d2c09df02798070402": "\\sum_{i=N+1}^{\\infty}|a_{i}| = \\sum_{i=1}^{\\infty}|a_{N+i}|\n< \\sum_{i=1}^{\\infty}r^{i}|a_{N+1}| = |a_{N+1}|\\sum_{i=1}^{\\infty}r^{i}\n= |a_{N+1}|\\frac{r}{1 - r} < \\infty.", "15ba5c2c4de89fbb2a2873e771b895ef": "2\\epsilon", "15ba9457f86ec9a30bc9c1186628cce1": "x \\in A", "15ba9d3b4668ce0a9a087945d9c05734": "y_{tt}=ky_{xxxx}", "15baa5d14f7136e5885fd0a4410b4bce": " \\frac{\\mathrm{ft^3}}{\\mathrm{slug}} ", "15bac40c6a3a27c1ed0594b21fdfa7ca": "y \\left( t \\right) =x(at)", "15bb0105cac825d1941db7ffbd160537": " \\Psi_N(f)=E[\\exp(-N(f))] ", "15bb0e7dcc81e0af498eaadf86e72a34": " y_3=\\frac{y_1^3-a\\cdot x_1^3}{a\\cdot y_1\\cdot x_1^3-y_1} ", "15bb14daeb013fca7424aea2a459a7c6": "k/q", "15bb233b90fe2d2bfced7d9ea2501886": "(-2c , -c^2)", "15bb38bb5739e8eceae8bf32af65c15a": "d = 1", "15bb9113eb813735527dfe406b41e477": "z_c \\le z \\le -z_c", "15bbc42a514b3cf401204ed7695d63d8": "W_A", "15bc21f02f93443df17d3dfbc8880a94": "\\rho =N/V", "15bc2bff3845bdeb9c5a40abdfa8b97c": "x=(RF)a\\lambda", "15bc4951d5b32e4704d864e13be5decf": "f''-2zf'+\\lambda f=0;\\;\\lambda=1", "15bc4b37ea80d103be9b3611ad98f0af": "\\Psi_n = \\sqrt{\\frac{2}{L}} \\sin \\left( \\frac{n\\pi}{L} x \\right)", "15bc6df800ec908fe87f4be767eeafa8": "\n 2C_1 = \\mu\\,\n ", "15bc6e5f4d18ff7409b845d3fbae4544": "T_{1} \\dots T_{n}", "15bc86d481ea9eac6dbd8fa8cbfafedc": " \\hat{f}^n_{i-1/2} = \\frac{1}{2} \\left( f_{i-1} + f_{i} \\right) - \\frac{ \\Delta x}{ 2 \\Delta t } \\left( u^n_{i} - u^n_{i-1} \\right). ", "15bca7119d9e828dead4a9675b32d7d1": "\\psi (\\mathbf{r}, \\ t=0) =A\\ e^{i\\mathbf{k \\cdot r}} \\ , ", "15bcb9807bc17321a713ffc649618a7a": " \\chi_\\lambda (\\exp X) = \\frac{\\sin((2\\lambda + 1)X)}{\\sin X/2} ", "15bcbb94f2d649b10cb7ccd09036c5e7": "=\\lim_{x\\to\\pm\\infty}\\left[\\frac{x^2-1}{x}-x\\right]", "15bcbd06c8084b9a0a4beb424ed4401f": " \\mathcal{C} = \\{ f \\in V^* : \\langle f, e_s \\rangle > 0 \\ \\forall s \\in S \\}", "15bd36046c5714dac49475e884de0db9": "\\pi_i \\colon G \\to G_i\\quad \\mathrm{by} \\quad \\pi_i(g) = g_i", "15bdcda88ab6e25112b1d5a2e10de4fe": " \\frac{e_c^2|\\langle\\varphi_v|e_c\\mathbf{x}|\\varphi_c\\rangle|^2D_{ph-e}[f_e^\\mathrm{o}(E_{e,v})-f_e^\\mathrm{o}(E_{e,c})]}{\\epsilon_\\mathrm{o}^2m_{e,e}^2u_{ph}n_\\omega\\omega} ", "15be3c2519dc3df50beeab4d9eb20dd8": "p_{1}", "15be438baa231c98073b780e3f12a54f": "\\lambda/d", "15be7062c05163f511f39a3ead28b2fc": "(M,\\bar{N})", "15bebe71037e28f744651387ee43421f": "\\displaystyle{|a_n|\\le r^{-n} \\sup_{|z|=r}|f(z)|.}", "15bec5e1a8928d5ef4480b811df43591": "\\sigma_{22} = \\sigma_{33} = 0 ", "15beddf17e3a9b75437d2f2685184871": "U_2\\left(x,y\\right)=\\beta", "15bee9298af680589f79e9b7ec0bd67a": "\\ \\lambda_i=\\lambda_0[1+K_i(\\Delta\\mu/\\mu)],", "15bee9611d495eb5166ba0a51b815b43": "\\sin \\theta = \\frac {\\text{Rise}}{\\text{Length}} \\,", "15bf21e981330cd12c2fa37226efbd36": "T_G(x,y)=(x-1)^{-k(G)} Q_G(x-1,y-1).", "15bf4bfb259c425e6b9389f4817402e5": "\\hat{N} = m + \\frac{m - k}{k}= m + mk^{-1} - 1 = m\\left(1 + k^{-1}\\right) - 1", "15bf62afc7f4ae839219d9deb2e75ba4": "x' = y'^n + r'", "15bf8550544059b33960eb174e210629": "{(\\eta_b)_{max}} = {\\cos^2\\alpha_1}", "15bfc40c1903572994adff2f388e5277": "(\\hat{c}- \\hat{a})", "15bfd1c1468174a48780ca9ea4480488": "\\Pr (y_j = h_i) = (1 - \\pi) \\frac{\\lambda^{h_i} e^{-\\lambda}} {h_i!},\\qquad h_i \\ge 1", "15c09a929654f7dca65a774f4b6fda6b": "\\frac{n(n-1)}{2}", "15c0c070e4b5cc0016f225d630343863": "{p \\choose i} \\equiv 0 \\pmod{p},\\qquad 0 < i < p.", "15c0e11b3ff535ba41419d17c8044630": "F(x,y) := \\frac{1}{2}(x^2 + y^2)", "15c1185ccad287d3246a56ced5f45536": "\\gamma^{*}=(\\gamma(0)+\\gamma(1))e - \\gamma ", "15c1296c292fcd2280427b777de13ec6": "\\frac{w:\\neg \\Box A}{w':\\neg A}", "15c12f702488b8e36dc6975ad739e299": "I_{n,m}= \\int \\frac{dx}{x^m(x^2-a^2)^n}\\,\\!", "15c135a1632eac0ed602fa5e4186fb09": "11=10+1", "15c152914bf894420e9f12cc9ec2a83c": "z \\equiv y^n \\pmod{n^2}. \\, ", "15c1c38363c5aeb6cd3e597a7b6c6990": "\n |((j_1j_2)J_{12}j_3)JM\\rangle = \\sum_{J_{23}} |(j_1,(j_2j_3)J_{23})JM \\rangle\n \\langle (j_1,(j_2j_3)J_{23})J |((j_1j_2)J_{12}j_3)J\\rangle.\n", "15c1db79efffac8cf246350de7f6c62c": "\\frac{(x_i- x)}{R_i}", "15c1f22a7fbd66069ed07d6b1a5e6cce": "\\Delta f = \\frac{f_0}{kA^2}\\langle F_{ts}q'\\rangle \\,", "15c230cf6fc4c47c6d574e7f43641e73": "f*g", "15c26bed66bbec79f2652dbdae1da519": "d \\mod n \\in \\mathbb{Z}^*_n", "15c28ebc86e5ccc23772c423b789caeb": "\\ p = \\rho g H + p_\\mathrm{atm},", "15c2c05f0105d47398bbdccdd6389582": "\\scriptstyle B ", "15c2d1614e1909ce9f49898b9e1e9d14": "\\scriptstyle ({H_n})_{3,2} \\;=\\; (-1)^{3 \\cdot 2} \\;=\\; (-1)^{(1,1) \\cdot (1,0)} \\;=\\; (-1)^{1+0} \\;=\\; (-1)^1 \\;=\\; -1", "15c2f79d2e1fa8e78db1dec66e8f0bbd": "B_{\\mathrm{HT}}:=\\oplus_{i\\in\\mathbf{Z}}\\mathbf{C}_K(i)", "15c32c4c5dfc148f79de581decd7eaca": "d=\\sqrt{(x_0 - m)^2+(y_0 - n)^2}= \\frac{|ax_0+ by_0 +c|}{\\sqrt{a^2+b^2}}.", "15c3786c10ffc750833aad5a8b9d9086": "1-{{{n \\choose {k+1}}{{N-n} \\choose {K-k-1}}}\\over {N \\choose K}} \\,_3F_2\\!\\!\\left[\\begin{array}{c}1,\\ k+1-K,\\ k+1-n \\\\ k+2,\\ N+k+2-K-n\\end{array};1\\right]", "15c3be3382d1d63b98566549b2f9aa9c": "\n\\begin{matrix}\n\\sigma &=& 2/3\\\\\nC_{b1} &=& 0.1355\\\\\nC_{b2} &=& 0.622\\\\\n\\kappa &=& 0.41\\\\\nC_{w1} &=& C_{b1}/\\kappa^2 + (1 + C_{b2})/\\sigma \\\\\nC_{w2} &=& 0.3 \\\\\nC_{w3} &=& 2 \\\\\nC_{v1} &=& 7.1 \\\\\nC_{t1} &=& 1 \\\\\nC_{t2} &=& 2 \\\\\nC_{t3} &=& 1.1 \\\\\nC_{t4} &=& 2\n\\end{matrix}\n", "15c3dc9102a2fee4ae055315a71a7c04": "dk'_x\\,dk'_y\\,dk'_z = dk_x\\,dk_y\\,dk_z", "15c4b56c0ed4c269f908489b5407c8d6": "u_G(x)=A(x)u_1(x)+B(x)u_2(x).\\,", "15c527d694f5d434d0484533a01c77de": "\n\\begin{align}\n\\tau&=\\frac{\\pi K'}{2K}\\\\\n\\lambda&=\\frac{\\pi \\eta}{iK}\\\\\n\\alpha&=\\frac{\\pi u}{iK}\n\\end{align}\n", "15c52fb3edd4cda61eadf9167831b598": "M=\\{M_{1},M_{2}, \\ldots , M_{n}\\}", "15c533c54aab9ce9dd524f99c9e06af8": "\\hat{\\textbf{x}}_{n\\mid m}", "15c541d17491976989227ebd94c5748f": " \\sigma^2 ", "15c569fe3e59fb9f932c306caa7bc8e9": "=(b_{12}-a_{12})", "15c5739e37af3c9188fa2e1561060a6a": "\\mathfrak{X}", "15c5bc0825ddb5684e81a7197df9e8c0": "M_{\\phi}: \\mathcal{L} \\to \\mathbb{R}", "15c5de709e2a590d8a7c0d0363256469": "N'_{pp} ", "15c61ff382ac1faf64bd1990706063b8": "|i-j| \\neq 1", "15c66512756a8421ba470b2577a784e9": "\\textstyle\\frac1{i!}", "15c6ccfab137a7189ad4976991911979": "{\\color{white}.}\\qquad\n\\phi_1 \\le 0, \\quad \\left|\\phi_2\\right| \\le \\left|\\phi_1\\right|,\n\\quad 0 \\le \\lambda_{12} \\le \\pi. ", "15c6d2f0f24e6b0bca4791ecbac2b582": "4(1/2!)\\pi^2 = 2 \\pi^2 ", "15c6e2318abec545bd1f3ead73cd1a01": "(fV)_x = f(x)V_x\\,", "15c70a86798e9389ae36bd8a26ed057e": "\\dot{\\mathbf{x}}(t) = \\left(A - B K \\left(I + D K\\right)^{-1} C \\right) \\mathbf{x}(t) + B \\left(I - K \\left(I + D K\\right)^{-1}D \\right) \\mathbf{r}(t)", "15c7271ffbe1d734471b6e07f268ca9f": "\n\\cfrac\n {\\cfrac\n {\\cfrac\n {\\forall x .\\ [D(x) \\wedge \\neg D(f(x))]\\, }\n {D(d) \\wedge \\neg D(f(d))}\n \\forall_E\n }\n {\\neg D(f(d))}\n \\wedge_E\n \\qquad \n \\cfrac\n {\\cfrac\n {\\forall x .\\ [D(x) \\wedge \\neg D(f(x))]\\, }\n {D(f(d)) \\wedge \\neg D(f(f(d)))}\n \\forall_E\n }\n {D(f(d))}\n \\wedge_E\n }\n {\\bot}\\ \n \\Rightarrow_E\n", "15c87f5f259dd38925d53c40d0e645c7": "Q_i(t)", "15c9609a1f7013765f4bba5762ff2856": "n^{a_2}", "15c9a080077db430a2f3f37f41411556": "C_H", "15c9a8ab6438dc7f591ba7d1cdabc931": "\\frac{{N_W}^2}{N_R} \\left (\\frac{{\\partial}u'}{{\\partial}t'}\\right)+\\left (u'\\frac{{\\partial}u'}{{\\partial}x'}+v'\\frac{{\\partial}u'}{{\\partial}y'}+w'\\frac{{\\partial}u'}{{\\partial}z'}\\right )= -\\frac{{\\partial}P'}{{\\partial}x'}+\\frac{1}{N_R}\\left( \\frac{{\\partial^2}u'}{{\\partial}x'^2}+\\frac{{\\partial^2}v'}{{\\partial}y'^2}+\\frac{{\\partial^2}w'}{{\\partial}z'^2}\\right)\\,\\!", "15c9b3d50ff3c13c737257369a5461de": "D = f_1(F - b)/(F + f_1)", "15c9ccc914a232b6ea4b62e93d759a8e": "Q_v/R \\approx", "15ca08bfb9aa59a71562e6edc64eb9a5": "deg(R(X))", "15ca46370ff3e9ef066b1364171edc68": "\\mathcal{TSL}, or \\mathcal{SRI}", "15ca5aa8985f88aed9c28eb1d636c01c": "\\mu = \\mu ^\\circ + RT\\ln \\frac{f}\n{{P^\\circ }}", "15ca90b09b175c5a4e80f5f06d29ed6b": "\nA = p q \\frac{}{}\n", "15cac3efca8e932720a188c6cbe9aadc": "\\scriptstyle V_n ", "15caf8c7c6b3a06e03a6169b66c35e3c": "\\sum\\limits_{i=1}^n p_i", "15cafe31566fcb5411bf250fa0ab020b": "\\lim_{x \\to \\infty} \\exp(-k \\lambda_i t) = \\left\\{ \\begin{array}{rlr}1 & \\text{if} &\\lambda_i > 0 \\\\ 0 & \\text{if} & \\lambda_i = 0 \\end{array} \\right\\} ", "15cbbcf59b8c3b7a89234a0c521d9679": "3(x+1)=3x+3", "15cbbd53175dd6120a923121ea791e40": "\\mathbb{E}[(R_r - R_\\min)_+]", "15cbc670cc40c103cf14ee7fa0188e7b": "\nP_y(t+dt) = P_y(t) + dt \\sum_x P_x R_{x\\rightarrow y}\n\\,", "15cbe281ef9b7700d442a2e1c5436511": "\\; W \\pi_1 (a) V_1 h = \\pi_2 (a) V_2 h.", "15cbff19fab422b675636e3aca109199": "D = \\{ x^2 + y^2 \\le 9, \\ -5 \\le z \\le 5 \\}", "15cc6fa4e7b4493ea9785dcc0818deb1": "\\left(xzyz^{-1}\\right)\\left(zy^{-1}x^{-1}y\\right) = xzyz^{-1}zy^{-1}x^{-1}y.", "15cc89522a3f7c2dfe6bf3f4279c3cd4": "g_1=0, \\ldots, g_n=0", "15ccab65a0d3d8e92c3cd5b53081a59b": "1.\\ \\mathrm{CrO_4^{2-}+H^+ \\rightleftharpoons HCrO_4^- ; K_1=\\frac{[HCrO_4^-]}{[CrO_4^{2-}][H^+]} } ", "15ccfed3f49be24a67ae0e34109f4939": "f\\circ f^n=f^n\\circ f=f^{n+1}", "15cd2154da295c744400b94a13df90cb": "+\\frac{62}{2835}*x^9-\\frac{1382}{155925}*x^{11}+\\frac{21844}{6081075}*x^{13}+O(x^{15})", "15cd36c81f97890a9ad6f40e9de51c14": " \\| A \\|_{w} = \\sup_{n \\geq 0} (1+n)\\mu(n,A), ", "15cd60143bf155c4f74a5051e7bc3f19": "\\frac {V_\\mathrm i}{V_x} = \\lim_{\\delta x \\to 0} \\left ( 1 + \\frac {\\delta Z}{Z_0} \\right)^{\\frac{x}{\\delta x}}", "15cda45f4d5812906a1d6f3eb52a6e46": "b'=c", "15ce175a6c16fb78e557de6749cce437": "\\ L/D > 10 ", "15ce2b3a61533cbd1bc374db738df222": "ta(s,t_s) = t_s.", "15ce48e12ee10e058c8703b7eaf85f39": "\\log K_0 = \\frac{1}{\\log 2} \\left[\n\\sum_{k=3}^N \\log \\left(\\frac{k-1}{k} \\right) \\log \\left(\\frac{k+1}{k} \\right)\n+ \\sum_{n=1}^\\infty \n\\frac {\\zeta (2n,N)}{n} \\sum_{k=1}^{2n-1} \\frac{(-1)^{k+1}}{k}\n\\right]\n", "15ceb08b14110719751cbc75928ab109": " \\mathbb{}H_*", "15cef79d8979bb44a71c1257a028be0e": "{r\\over a}", "15cf356e0be06f5b0b80f69cf474f397": "S(x)=\\frac{P(x)}{R(x)}\\,\\!", "15cf45b2e38e677f92eec766b59641c8": "\\Psi=\\Phi+b\\;", "15d009971f452f713b4ce9f21694263e": "\\Gamma(\\tfrac14) = (2 \\pi)^{3/4} \\prod_{k=1}^\\infty \\tanh \\left( \\frac{\\pi k}{2} \\right)", "15d0227f4335b85f92d67c39ac2346c0": "{\\rm gcd}(n,q)=1,", "15d0910d6646a361be7642f8eb24b3a9": "\\mu = \\frac{mv_\\perp^2}{2B}", "15d118905e7b54240a45992e64eaf5af": "\\Lambda\\in\\operatorname{Hom}(L^{\\infty}(G),\\mathbf{R})\\,", "15d1204794314b34bcc11ae3aae3bdbc": " \\phi(\\vec{r},t) = \\phi_{0}(\\vec{r}) e^{i\\omega t}. ", "15d12cedba94c857ab59fd9070919861": "z_1,z_2,z_3", "15d170b7c98f459cbaecc0d663e2fa1e": "1 / (N C)", "15d180cdb818fabf450cb5183f8859df": " \\left|\\frac{x}{a}\\right|^{r} + \\left|\\frac{y}{a}\\right|^{r} \\leq 1", "15d189324d42cce986d3274ac59d4628": "V^{\\mathbb C} \\cong V\\oplus iV", "15d1c2bda03172895e59027112c2d5e5": "\\beta=1\\!", "15d1ded340f9aea25761b6325caf4f27": "E[U] = 0\\times1/2 + 1\\times1/4 + (-1)\\times 1/4 = 0", "15d23db95ffa01914bf1fa6fc14da8ec": "qq^* \\neq 0.", "15d25ba877fd68948884a11c80124302": "\\mu = G(M+m)", "15d29f1fec65b19ceed0e6c5024d06fd": "I_{a}", "15d2cafcaafb889fd0ed095fe22e1857": "C_p>0", "15d31fa8dfcae8446857532e9f24ada3": " w(z) = \\left(1-z^2\\right)^{\\alpha-\\frac{1}{2}}.", "15d34c55bb3a669b1eba7a09b9155036": "\\nu >6\\,", "15d34ca776588afc565a7a294f5d2f64": "G(r)\\propto\\frac{1}{r^{D-2+\\eta}},", "15d3c2197ac311e7ffdcaede17dd5728": "\\left[{n\\atop k - 1}\\right]", "15d3f77ec905a77ff67bfe4c1bc4a266": "a^{n-1} = 38^{220} \\equiv 1 \\pmod{221}.", "15d44acfccb3e7ebde38cfb234437fee": "\n D = \\frac{Eh^3}{12(1-\\nu)}.\n", "15d4f829eada1f04155fdf6c9f57e483": "\\scriptstyle A=\\{x\\in Z^d, \\eta(x)=1\\}\\subset Z^d ", "15d5094234d64c66cd150dbbf6b43d67": "T_{lower} = \\frac{F d_m}{2} \\left( \\frac{\\pi \\mu d_m \\sec{\\alpha} - l}{\\pi d_m + \\mu l \\sec{\\alpha}} \\right) = \\frac{F d_m}{2} \\left( \\frac{\\mu \\sec{\\alpha} - \\tan{\\lambda}}{1 + \\mu \\sec{\\alpha} \\tan{\\lambda}} \\right)", "15d52cf8689750029499a0669b415c94": "H_0|n\\rangle=E_n|n\\rangle ", "15d54eb537b515e7be0ef1d73f30ecbd": "S(T,V,N)=N k_{\\rm B}\\left[\\frac{5}{2}+\\ln\\left(\\frac{n_{\\rm Q}}{n}\\right)\\right]", "15d56205fd6de9645bbdb9734a0a64fb": " \\lim_{n \\to \\infty} {H}_{2n} = \n\\textstyle \\left(\\frac{1}{2}\\right)\n^{\\left(\\frac{1}{3}\\right)\n^{\\left(\\frac{1}{4}\\right)\n^{\\cdot^{\\cdot^{\\left(\\frac{1}{2n}\\right)}}}}}\n = {2}^{-3^{-4^{\\cdot^{\\cdot^{{-2n}}}}}} ", "15d596fbe42e42db84dd109b066e5c37": "\\frac{\\mbox{Current Assets}}{\\mbox{Current Liabilities}}", "15d5ac95cdda0dcc3723e8bee0d37015": "\\phi(\\psi)", "15d5b8732f638e752010baaac8090fdf": "\\gamma_{33} = e^\\psi \\sim \\xi^{-(1-s_1^2-s_2^2)}. \\,", "15d5c12bd2d5c56f0d8b4da4168d6e97": "|C|=A_q(n,d).\\,", "15d5c689a52b2ee02abebca3310da138": " {}= \\begin{vmatrix} a^{i} & a^{j} \\\\ b^{i} & b^{j}\\end{vmatrix} ", "15d5c6c9d48c5f1887e99d323c2e9bf8": "[x,x]=0", "15d5d14d342c43ff5236d06f987e9539": "H_m", "15d64454fa0c07c7d0f79e1c975defb5": "m_{rel} = \\gamma m_0 \\!", "15d6446efad9c71737042b3dc5b5bb2a": " K^1(X) ", "15d65a2b0eea0e05a1e334b06c45c6f7": "\\mathbf {X}=\\begin{bmatrix}\nX_{11} & X_{12} & \\cdots & X_{1n} \\\\\nX_{21} & X_{22} & \\cdots & X_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nX_{m1} & X_{m2} & \\cdots & X_{mn}\n\\end{bmatrix} , \n\\qquad \\boldsymbol \\beta = \\begin{bmatrix} \n\\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_n \\end{bmatrix} , \n\\qquad \\mathbf y = \\begin{bmatrix} \ny_1 \\\\ y_2 \\\\ \\vdots \\\\ y_m\n\\end{bmatrix}. ", "15d66ec190cc47fe47e714a2d93604b6": "|+\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle)", "15d6bb440f4ed3fe1f5e16fe6077e727": "(A - 2 I) \\begin{bmatrix}\n0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1\n\\end{bmatrix} = \\begin{bmatrix} \n-1 & 0 & 0 & 0 & 0 \\\\\n3 & -1 & 0 & 0 & 0 \\\\\n6 & 3 & 0 & 0 & 0 \\\\\n10 & 6 & 3 & 0 & 0 \\\\\n15 & 10 & 6 & 3 & 0\n\\end{bmatrix} \\begin{bmatrix}\n0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1\n\\end{bmatrix} = \\begin{bmatrix}\n0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0\n\\end{bmatrix}\n", "15d6c26bf4b3159521919775cd0e72bd": "\\langle r, a \\mid r^3, a^2, arar \\rangle", "15d7195739d5145a299530076a2ef2a1": " {\\Delta}_{\\rho}(f) := \\frac{(-1)^{\\left|f\\right|}}{2}{\\rm div}_{\\rho} X_{f} = \\frac{(-1)^{\\left|x^{i}\\right|}}{2\\rho}\\partial_{i}\\rho \\pi^{ij}\\partial_{j}f.", "15d728bf354d2df98672f1b31eb40189": "0.\\overline{5}\\overline{4}", "15d7d36cf87173aa0eb5303f99d4d0cc": "\n x_{n+1}=f_\\mu(x_n)=\\begin{cases}\n \\mu x_n & \\mathrm{for}~~ x_n < \\frac{1}{2} \\\\ \\\\\n \\mu (1-x_n) & \\mathrm{for}~~ \\frac{1}{2} \\le x_n \n \\end{cases}\n", "15d7da66d761cf7cd60e43d77c270cc4": "P_n(z)", "15d835c33075057ecf09c1184db0a2e3": "{\\textbf R}_K", "15d8420cf5d547366c0636eed59786d1": "h(X)", "15d8ce27fc7984fac063adc6cbf70b2d": "\\bold{g}=-G\\sum_i \\frac{M_i(\\bold{r}-\\bold{r_i})}{|\\bold{r}-\\bold{r}_i|^3},", "15d92c476babe876329314876ecf17a8": " PW_x(t,f) = \\int_{-\\infty}^\\infty w(\\tau/2) w^*(-\\tau/2) x(t+\\tau/2) x^*(t-\\tau/2) e^{-j2\\pi\\tau\\,f} \\, d\\tau", "15d93034172d981b98cc8244c3f840b6": "R^{i+1}\\subseteq G", "15d946399729d25977616ee774c81943": " \\mathcal{L}\\{f(t)\\}", "15d98038949e111deebe7ff282ee8777": " \\frac{dR}{dt} = \\nu I - \\mu R ", "15d9bf81da4b5b53c6609c4c8ceb2cdf": "\\mathcal{A}=E/\\sigma:", "15d9d8f4f078956d8f9f9d5c8e5539ed": "g(p) = [h_1(p), ..., h_k(p)]", "15da0d15d15ef322370f70871d711615": "a=2\\mu\\sqrt{\\rho}", "15da1162a8aa5570ca97019af3768520": "b\\in\\mathbb{Z}", "15da3864b7ae54434184efc742ca5070": "t 0)\n\\end{array}\n", "15e92f6d4a9337d74273af2acf6d83ce": "e = \\sqrt{g(2-g)}.", "15e955e6c0573aa3ecddefe4f88966f0": "n (n-1)!", "15e97c30e5d70fa1ac640b7ebc6492d7": "i \\hbar \\frac{\\partial}{\\partial t}\\Psi = \\hat H \\Psi", "15e9c56e6997f3e3ebe05932377de6fc": "\\Gamma(n/24)", "15e9fbcf229f7869d2f5242fc212d595": "{\\rm MG}_{p}(\\alpha,\\beta,\\boldsymbol\\Sigma)", "15e9fc901189caf056597836e61ffc39": " E_\\mathrm{p} = qU \\,", "15ea0d423ee2c460639733a1db3ae844": "\\tan \\psi = \\left ( \\frac {f} {v' \\cos \\theta - f} + 1 \\right ) \\tan \\theta\n = \\frac {f + v' \\cos \\theta - f} {v' \\cos \\theta -f} \\tan \\theta \\,,", "15eacdf189942184cd35d98a8584d7a7": "\\begin{pmatrix}c & 0 \\\\ 0 & c \\end{pmatrix}", "15eb7e2dd76647080b66c72611df326a": "\\begin{align}\n\\left \\langle \\theta^{G},\\psi^{G} \\right \\rangle &= \\left \\langle \\left(\\theta^{G}\\right)_{K},\\psi \\right \\rangle \\\\\n&= \\sum_{ t \\in T} \\left \\langle \\left([\\theta^{t}]_{t^{-1}Ht \\cap K}\\right)^{K}, \\psi \\right \\rangle \\\\\n&= \\sum_{t \\in T} \\left \\langle \\left(\\theta^{t} \\right)_{t^{-1}Ht \\cap K},\\psi_{t^{-1}Ht \\cap K} \\right \\rangle, \n\\end{align}", "15ebb45bc4af78c49017a7881eb07d6d": "dz(t)", "15ec0a0315d245215dab2d96b3ee7fda": "\\phi_2 \\ ", "15ec6bb05ebf0f9d3856768ea8dd0281": "\\pi_0 := \\text{Pr}[P=1,Q=0]=\\sum_{\\omega \\in S_0} \\Psi^2_\\omega.", "15ec79f3ad4f6cbd00c7a78efb5a6e0d": "T^2(\\Omega)", "15eca165b219f76d826fe5b4dadb789b": "p < p'", "15ecf9ad515de98952af8bf6a1b91a9e": " y(\\lambda) = \\frac{y_0}{1 - 2 \\, x_0 \\, \\lambda + (x_0^2 + y_0^2) \\, \\lambda^2} ", "15ecfb67f3e4dcb5c232c157bd4eae1d": "f(z) = z + a_2z^2 + a_3z^3 + \\cdots", "15ed2beab72d95d9953f645f20da1143": "P \\cong 0.35 \\sqrt{L_\\mathrm w}", "15ed4a8685a73dda8309160bedead685": " \\Phi_E = \\frac{q}{\\varepsilon_0}", "15ed4f4afcdaae445a0f344b78212201": "k\\ .", "15edc95f3e58f30fdc16d1aa34a3a0b5": "\\scriptstyle \\tanh \\ b = w/c", "15ee4696d2ee3cffe69beeca3c2ffc93": "L_{CW} = \\,\\,{{16\\,R} \\over {3\\,\\pi }}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,T_{CW} \\,\\, = \\,\\,{{2R} \\over v}\\,\\,\\,\\, \\ne \\,\\,\\,{{L_{CW} } \\over v}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,T_{RW} = \\,\\,{L \\over v}", "15ee46c29e9315c3de388c04a7ecf7d5": " r_o = r + \\frac{nh}{2}", "15ee4e53672e1e4e6a8d6503c30ee8d5": "\\sigma_\\mathrm{tot}", "15ee5065c28f7afdcb33d26425a3f7d3": "{Action\\over h} = -{Entropy\\over k}", "15ee7be13ac831c107e67ec7db4eea22": "(u'(x,z,t),w'(x,z,t)).\\,", "15ef5de7c23645ae102f968935716de8": "\\chi(K_n)=n", "15ef7183ab62546de1c829b26329a499": "\n\\begin{alignat}{2}\n\\frac{v_{k+1} -2v_k + v_{k-1}}{h^2} & = \\lambda v_{k} \\\\\nv_{k+1} -2v_k + v_{k-1} & = h^2 \\lambda v_{k} \\\\\n(v_{k+1} - v_k) - (v_k - v_{k-1}) & = h^2 \\lambda v_{k} \\\\\nw_k - w_{k-1} & = h^2 \\lambda v_{k} \\\\\n& = h^2 \\lambda w_{k-1} + h^2 \\lambda v_{k-1} \\\\\n& = h^2 \\lambda w_{k-1} + w_{k-1} - w_{k-2} \\\\\nw_{k} & = (2 + h^2 \\lambda) w_{k-1} - w_{k-2} \\\\\nw_{k+1} & = (2 + h^2 \\lambda) w_{k} - w_{k-1} \\\\\n& = 2 \\alpha w_k - w_{k-1}.\n\\end{alignat}\n", "15efdc1fe0c1a7fa1fca7c76b6a34f6f": "[\\omega] \\in H^2(\\mathbb{S}^{2k}, \\mathbb{R}).", "15f04f43fda7aeecc69a346e33e34e10": "\\varphi_z", "15f05029707420fd076c6799cc26b474": "\\,\\Gamma(x)\\,", "15f0dee5881eaf6654bc293de5f47d00": "\\ln(z+1) = z - \\frac{z^2}{2} + \\frac{z^3}{3} - \\cdots, ", "15f105ae7caebe526d8017982d01434d": "\\omega(r)= \\frac{1 }{r} v", "15f1295fc982f9c3c3206243ff909bf0": "F(\\phi,k)=\\sin\\phi R_F\\left(\\cos^2\\phi,1-k^2\\sin^2\\phi,1\\right) ", "15f15a559037df8fc3bb2d079df00c25": "\\frac{\\frac{dx^\\prime}{dt^\\prime} + v}{1+\\frac{v}{c^2}\\frac{dx^\\prime}{dt^\\prime}}.", "15f1985d362a5eb63570e624636f3fce": "\\psi(x+L)=e^{i \\theta}\\psi(x).", "15f1a13c904765463eef368eea432fcd": "\\sigma = -p\\boldsymbol{\\mathsf{I}} + \\boldsymbol{\\mathsf{T}}", "15f1f4c53534021a6c94d6ec96d90638": "\\scriptstyle{{\\lambda\\over 4}}", "15f2222ae692f8a50061050ba1cb489b": " ( a x + b y ) \\wedge b = ( a \\wedge b) x = c \\wedge b", "15f22d78f7c22a53a1b59ab5251bd0f3": "v \\propto \\!\\, r \\omega\\,,", "15f24b46ffaf5056cc4fd0e3ecfac19b": "\\langle\\cdot |\\cdot\\rangle", "15f2552966dce5911dc2cb4f15a1f42c": "N \\left(a+bi \\right) = a^2+b^2 = (a+bi)\\overline{(a+bi)} = (a+bi)(a-bi).", "15f276be5b8559d7ce1e71a9a4faf22a": "U_n\\,", "15f2a6165ea91d2a35bf4c8c4e73ebae": " M_S^2", "15f2f17e220be55393c6d1cf6224267f": "addOne : nat \\to nat", "15f3cf1abe256e7b98dc45b064734df0": "\\ v_i\\ ", "15f4087762549dfab0e4d52e976e59b5": "\\tau_{U}", "15f41f86560631aeeef62ec681c615d1": "v_\\text{out} = F(A_1 \\sin \\omega_1 t + A_2 \\sin \\omega_2 t)\\,", "15f4248566b5e5f0a52b4079bcaf8715": "A = \\frac {f_c} {Q_a/Q_t - 1} = 68.7672", "15f4425e423ccaf2c191123dad358e88": "f(M) = {I(M \\in [-b,b]) \\over 2b}", "15f46126fb9b1e13d18b750195e216de": "\n\\begin{align}\n\\mathbf{B}_{n,m} & = \\frac{K_{n,m}a^{n+2}}{R^{n+m+1}}\\left[\\frac{g_{n,m}\\mathcal{C}_m+h_{n,m}\\mathcal{S}_m}{R}((s_{\\lambda} A_{n,m+1}+(n+m+1)A_{n,m})\\mathbf{\\hat{r}})-A_{n,m+1}\\mathbf{\\hat{e}}_3\\right] \\\\[10pt]\n& {}\\quad {}-mA_{n,m}((g_{n,m}\\mathcal{C}_{m-1}+h_{n,m}\\mathcal{S}_{m-1})\\mathbf{\\hat{e}}_1+(h_{n,m}\\mathcal{C}_{m-1}-g_{n,m}\\mathcal{S}_{m-1})\\mathbf{\\hat{e}}_2))\n\\end{align}\n", "15f47b4f2d1f37111e0eb22dc5e278aa": "\n\\frac{dq^s}{dt}=w^s,\\qquad\\frac{d}{dt}\\left(\\frac{\\partial T}{\\partial w^s}\\right)-\\frac{\\partial T}{\\partial q^s}=Q_s,\\qquad s=1,\\,\\ldots,\\,n", "15f49720e865366cb301c4f794ab0c8a": "[D \\rightarrow D^{'}] ", "15f4e0c846fa6f10e04fa0610ce90090": "\\frac{1}{\\sqrt{1-\\delta}}", "15f4e99345c1a477fe997bd27b108012": "(m - n) \\times (m - n)", "15f4f99c9077a4b63e4d2d597f1420cc": "P(\\alpha,\\alpha^*)=\\delta^2(\\alpha-\\alpha_0).", "15f5392aa1cda774fef26d8819f4c1ee": "\\boldsymbol\\mu\\in\\mathbb{R}^D ; \\boldsymbol\\Sigma \\in\\mathbb{R}^{D\\times D}", "15f55f0770b87d372dc41478715aad6b": "\\pm \\pi/3", "15f5720bab3cbe8d8b33f486a9b03255": "w:\\neg a", "15f6167e3dd7bb170e7c2359784d2e78": "(AA^+)^* = (U\\Sigma V^*V\\Sigma^+U^*)^* = (U\\Sigma\\Sigma^+U^*)^* = U(\\Sigma\\Sigma^+)^*U^* = U(\\Sigma\\Sigma^+)U^* = U\\Sigma V^*V\\Sigma^+U^* = AA^+", "15f64afac10c3369f0c72f86aa54186b": "\\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\\psi_z,-\\psi_x),\\,", "15f65dc39f17bb80472146a13a79ebb1": "V(\\mu) = \\mu^2/n +n", "15f6cea9893b187c80c7874051074360": "f(x,y) = 100\\sqrt{\\left|y - 0.01x^{2}\\right|} + 0.01 \\left|x+10 \\right|.\\quad", "15f772f3a2fae5869536e58235125089": "\\frac{x + x^*}{2} = x_0\\,e_0", "15f79d3522d8582dcf0dd5dd0ac5c8c7": "\\boldsymbol\\beta^{(t+1)} = \\underset{\\boldsymbol\\beta} {\\operatorname{arg\\,min}} \\sum_{i=1}^n w_i (\\boldsymbol\\beta^{(t)}) \\big| y_i - f_i (\\boldsymbol\\beta) \\big|^2. ", "15f7a24e580a3b7e47bd0dc90d633275": "E(r)", "15f7bd58d464de95c5d99171672fc190": "\\frac{\\text{Prerequisite : Justification}_1, \\dots , \\text{Justification}_n}{\\text{Conclusion}}", "15f7eaab3582ac1007cf43f7ec29e0f1": "x < c\\ ", "15f7f62ce09af0d7485d6539c069d605": "t = 1, \\ldots, T", "15f8382f87144f5f87c919ca80cb140b": "|\\mathbf{T}| = \\sqrt{T^{\\alpha}T_{\\alpha}}", "15f846a330669fee1f65d1b4a39f8d99": "P\\Gamma L/PGL \\cong \\operatorname{Gal}(K/k)", "15f8be788083d10e58744f82c8824e01": "f(z) = \\frac {az + b} {cz + d}.", "15f8d41b9e6033dcd95316e5fd9232b6": "3^{ \\lceil\\log_2 n \\rceil} \\leq 3 n^{\\log_2 3}\\,\\!", "15f90b642d4341cf420dfb968b3b0189": "\\sum_i p_i (|\\phi_i\\rangle \\langle \\phi_i|)^{\\otimes t} = \\int_{\\psi}(|\\psi\\rangle \\langle \\psi|)^{\\otimes t}d\\psi", "15f939517325a337c901c544e5ca1981": "a_\\text{i}\\in\\Sigma_\\text{int}", "15f9b90f3b2293bbb124457bfa44dddd": "\\left\\lceil \\frac{n}{m} \\right\\rceil = \\left\\lfloor \\frac{n+m-1}{m} \\right\\rfloor = \\left\\lfloor \\frac{n - 1}{m} \\right\\rfloor + 1, ", "15fa062c7f59a1926685d412eccab9dc": "\\exp\\!\\left( -\\left( (-\\log(u))^\\theta + (-\\log(v))^\\theta \\right)^{1/\\theta} \\right)", "15fa5162e0924d41210a053d34f4efa5": "\n\\begin{align}\n\\sigma_{12} &= s_{12}/R,\\\\\n\\sin\\phi_2 &= \\sin\\phi_1\\cos\\sigma_{12} + \\cos\\phi_1\\sin\\sigma_{12}\\cos\\alpha_1,\\quad\\text{or}\\\\\n\\tan\\phi_2 &= \\frac{\\sin\\phi_1\\cos\\sigma_{12} + \\cos\\phi_1\\sin\\sigma_{12}\\cos\\alpha_1}\n{\\sqrt{ (\\cos\\phi_1\\cos\\sigma_{12} - \\sin\\phi_1\\sin\\sigma_{12}\\cos\\alpha_1)^2 + (\\sin\\sigma_{12}\\sin\\alpha_1)^2 }},\\\\\n\\tan\\lambda_{12} &= \\frac{\\sin\\sigma_{12}\\sin\\alpha_1}\n{\\cos\\phi_1\\cos\\sigma_{12} - \\sin\\phi_1\\sin\\sigma_{12}\\cos\\alpha_1},\\\\\n\\lambda_2 &= \\lambda_1 + \\lambda_{12},\\\\\n\\tan\\alpha_2 &= \\frac{\\sin\\alpha_1}\n{\\cos\\sigma_{12}\\cos\\alpha_1 - \\tan\\phi_1\\sin\\sigma_{12}}.\n\\end{align}\n", "15fa5cd3d15710ca383e68cd96775dd3": "r \\times 1", "15faf806a9b23c7cbffef5d3eebbf22a": "\\frac{a-b}{b}\\,\\!", "15fb06e96fe7bd52d10c9da6992a20d6": "f^*:H^*(X; \\mathbf{Z}/2\\mathbf{Z})) \\to H^*(X'; \\mathbf{Z}/2\\mathbf{Z})", "15fb0debf178116d8d46f2bce152e81d": " = \\frac{GM\\tau}{kR^2}", "15fb5d9f70ec10e5c7912d26d65a9b72": " \\beta(M) = (-1)^{r(M)} \\sum_{X \\subseteq E} (-1)^{|X|} r(X) \\ . ", "15fb843488651759ed1533a70cc957e6": "u=a^2+b^2", "15fb98fb9957c61415120317765b2895": "\n\\frac{d}{dt} = \\frac{ \\partial }{ \\partial t } + \\vec{v} \\cdot \\nabla \\; .\n", "15fba57765ef0639d2beaa6fcaf7edd0": "\\, y_1 ", "16035954482dd051630c8c1fcc2c6223": "\\left(\\frac{-1}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "16035c7ec2bd5cd3dbf8d30735874440": "\\overline{\\mu}_j^{(1)} = \\overline{\\mu}_j^{(2)}", "16037c53e30170ba29e8298dbb695ccd": "X_1X_2=X_2X_1", "1603e1b87bad791f5528a1dbd779cb00": "\\mathbf{QP} = \\mathbf{Q}.", "1603f099b1c079b9b99420b9ef432adf": "\\mathbf{x}'\\mathbf{A}\\mathbf{x} = \\sum_{i,j}a_{ij} x_i x_j", "1604060a637eca70aa8e0de1da9f1434": "\\theta_2 \\,\\!", "16043eb02289b77bc5fce774490eea50": "\\begin{bmatrix} a & b \\\\ b & d \\end{bmatrix}", "1604e99e69e844dbb0a3206c7bfe2c94": "\\tau = RC = \\frac{1}{2\\pi f_H}", "1604f4b7b58ce3600b6a9e057acdee1d": "A=UDV^H", "16059dbdae38a8d786633b935bb732d3": "M_{\\eta}", "1605ebdd8ada06e88406aab800879f56": " M_j(i) = 1 ", "1606168a880f9ceb6218263927db5e96": "x \\cdot y \\leq x^2 + y^{2}/4", "16066b820736bfc0a2702a4c96742db6": "\\scriptstyle\\sqrt{b^2-4ac}", "1606742f064c2c2048309ae60b3029a8": "|a(\\tau, \\zeta)|^2", "1606b0a8f39b9678547ff92c5ef862cb": "|r_1 - r_2|.", "1606bf745931de219e701bbd4053b37c": "(\\phi_1, \\dots, \\phi_n)", "160707b27ae191ee560ff49b9395a8ec": "\\ y[n] = \\sum_{k=-\\infty}^{\\infty} h[n-k] x[k].", "16076ef56e279ae6139dfa0f3638975c": " L = \\sum_{i=1}^n f_i(x)\\frac{d^i}{dx^i} \\,; ", "1607c9d968bb05dd129cea29a24316e8": "\\omega '=\\frac{\\omega r}{2(R+r)}", "16083831472ba45fc050280b08e07960": "d_n(x,y)=\\max\\{d(f^i(x),f^i(y)): 0\\leq i0),", "161baba6c4c3c626ea666ebfc4105591": " \\quad P(D) u (x) = \n\\frac{1}{(2 \\pi)^n} \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} e^{i (x - y) \\xi} P(\\xi) u(y)\\, dy \\, d\\xi ", "161bec377d77e25b12d52e69a701829f": "\\frac{N}{N_0} = e^{-\\frac{\\Delta E^\\ddagger}{kT}}", "161c51f3d771e4dd423e1ad83114dd24": "\\theta_1=90^\\circ", "161c6f1c6482c27c11b7e309de6ee14c": "360^\\circ \\iff 2\\pi r", "161c72a30bb8e124336571042fec1689": " \\nabla ( \\psi + \\phi ) = \\nabla \\psi + \\nabla \\phi ", "161d178a63c6ba1197b179f70d1ce602": "J_\\lambda", "161d99d74cb4c40acb4155d162d8a0ec": "\\operatorname{U}(n_1) \\times \\cdots \\times \\operatorname{U}(n_k).", "161de52026bee27c27aed967b1f880e7": "\\bar g", "161e060938bb662de6074c9db3634ba7": " {dy \\over dx} = {Y \\over X} ", "161e699f709e1ccc5bce07a2a418af13": "\\gamma_\\mathbf{u}=\\frac{1}{\\sqrt{1-\\frac{u_1^2+u_2^2+u_3^2}{c^2}}}", "161e6b2d1dde57ec6896ce4e858e9fa5": "\\langle s,t \\mid s^2, t^3, (st)^5 \\rangle\\,\\!", "161ea786146feb57cd6abc541686c72e": "f^{-1}(V_i) = \\bigcup_j U_{ij}", "161ee5ee127ecf8e27ffde1b3e26f79f": "T_{\\rm E} ", "161f4f6af4de76b1aff051e67e3fd1bc": "Q_n c_n", "161f6f5c4f668245b0918698ae63fbe0": "Fdr(z)", "161fa7b85fdcca5ebd788698081ad8f7": " SUE = \\frac{EPS - Forecast}{\\sigma(EPS - Forecast)} ", "161fddc79b30ddbd8d4d76c1a91b1d63": "n_{e}", "16203be20bacd599b82fea4ec72b194b": "(z,z_2;z_3,z) = (z_1,z;z,z_4) = \\infty.", "16204ccb11f9d08f3dea8134dfa0005f": "\\Gamma_{tot}=\\Gamma_{rad} + \\Gamma_{nrad} ", "16207669b2295a2763f185ae8262a882": "\\,e_x", "1620e76d5789119b7f4db797d2c16290": "\\bar f(x)f(x).", "1620edf3d0bbd27cf2a67e18f5ef26aa": "X_\\alpha", "1620fd53d6f66d0fabf51c67d9f74dbf": "(x,y,z)\\mapsto (-x,-y,-z)", "162120f2b19e12af8a16431dd7608fd2": "\\hat\\phi=Y_D\\circ F", "162153d1df4ffa80997fc4f2dd21d36e": " y = \\cos(6\\pi nt)\\sin(2\\pi nt)-\\cos(6\\pi t)\\sin(2\\pi t), 0 \\leq t \\leq 1 ", "1621ffb153e69579df73c4e2e4e16ffa": "p_1\\,\\!", "162214b565c6fce6fd0a1a966d1e86e6": "\\mathrm{^{244}_{\\ 96}Cm\\ \\xrightarrow[]{(\\alpha,n)} \\ ^{247}_{\\ 98}Cf\\ \\xrightarrow[3.11 \\ h]{\\epsilon} \\ ^{247}_{\\ 97}Bk}", "1622513279631d306a106afca0a9fb5c": "\\frac{1}{2} \\ln \\frac{1+\\alpha v}{1-\\alpha v}=\\mathrm{arctanh}(\\alpha v)", "162271f2afbae19d3d56fc1033ab6676": "\n \\log(a_{\\rm T}) = \\frac{E_a}{R}\\left(\\frac{1}{T} - \\frac{1}{T_0} \\right)\n ", "1622a2eb142201bae06cda9da5993d10": " \\begin{align} & \\left(-\\frac{d^2}{dx^2} + A\\right) \\psi_1^0 = 0,\\qquad \\psi_1^0(0) = 0 \\quad,\\qquad \\frac{d\\psi_1^0}{dx}(0) = 1, \\\\ & -\\frac{d^2}{dx^2}\\psi_2^0 = 0,\\qquad \\psi_2^0(0) = 0,\\qquad \\frac{d\\psi_2^0}{dx}(0) = 1, \\end{align} ", "1622df73b9c5280b05661e8f980bf523": "G_{S_N}(z) = G_N(G_X(z)).", "16231fc31b0ce649586bc7dcdaaebe50": "f_{\\omega + 1}(3) - 2", "16234783ba2ad8a92d11da2927d33519": " [Z_i]=\\begin{bmatrix}\\cos\\theta_i & -\\sin\\theta_i & 0 & 0 \\\\ \\sin\\theta_i & \\cos\\theta_i & 0 & 0 \\\\ 0 & 0 & 1 & d_i \\\\ 0 & 0 & 0 & 1\\end{bmatrix},", "16238fc72724d93eefd77414bc68311f": "\\|x\\|^2 = \\sum_{k\\in B}|\\langle x, e_k\\rangle|^2.", "162391ddafb66ea6a3b803c04b56b9ff": "||(x,y)|| = \\sqrt{x^2+y^2}, \\, ", "1623adeb554952f0ad58fdc5f37b1d41": "\\ln P + n \\ln v = C", "1623b32e64ebd98b825e7b00cdfa1f17": "A_{t,t+1} = A_t \\cap L^p(\\mathcal{F}_{t+1})", "1623d62ce069246a0589ab4354c3e7df": "A\\otimes B", "1623f245369f75fbfdbe221248606faf": "B=A+(360/\\pi)\\times 0.0167\\times \\sin(W\\times (D-2))", "1624827224698be508cdef05fa773167": "z\\in D_R", "1624aae913f625fb636449be53712b76": "d>4", "1624cf5668bfe6f97ee7ac5dd28caaeb": "d\\Gamma_0", "1624dda74d784601344a062ad4043e13": "X=2\\left[\\begin{matrix}\\xi_1\\\\\\xi_2\\end{matrix}\\right]\\left[\\begin{matrix}-\\xi_2&\\xi_1\\end{matrix}\\right].", "162525bf03a300e1303483a4c383f672": "x:S\\to X", "16252d1e952b8d5dbcc652cfa3e1483a": "\\frac{1}{2}\\cdot\\delta", "16252fc0c6a3eb60a27f563e1b7c45f7": "a \\triangleright (b \\triangleright c) = (a\\ \\triangleright b)\\triangleright (a\\ \\triangleright c)", "16253c47c7c3ee7321643f2f8befcc2c": "\\frac{1 + {\\scriptstyle\\frac{1}{2}}z}{1 - {\\scriptstyle\\frac{1}{2}}z}", "1625957bf68fcc1e04bdaa499cbabd80": "(\\forall x \\phi) \\rightarrow \\psi", "1625960b01ba1b6aa2b2dd7957139dd7": " \\operatorname{W}(A)(Ax + ix) = Ax - ix \\quad x \\in \\operatorname{dom}(A). ", "1626041568028f4a773985cbb8abc73b": "\\frac {[D_{ad}]}{p^{1/2}_{D_2}[S]} = K^D_{eq}", "1626d06ac26e06bdf92327a534afca3e": "\\alpha_j(p,q) = P(w_{1(p-1)}, N^j_{pq}, w_{(q+1)m}|G)", "1626f4fa568afb2422c7618b3b319d9c": " ACH_{50} = {Q_{50}*60\\over V_{Building}}\\,\\!", "1627182735a540a231758044b5ba9073": "\n\\mathbf{x}_w^{(k)} = \\underset{\\mathbf{x}_w \\in W(\\mathbf{x}_w^{(k-1)})}{\\operatorname{argmax}} \\,\n\\det(\\mu(\\mathbf{x}_w, \\sigma_I^{k}, \\sigma_D^{(k)})) - \\alpha \\operatorname{trace}^2(\\mu(\\mathbf{x}_w, \\sigma_I^{k}, \\sigma_D^{(k)}))\n", "16276dace2435af3d6698e25188461ab": "g(x)=1", "1627a172c9976b86185ed15f1e74280e": "P_3=(4,4\\sqrt{15},-8\\sqrt{3},16)", "1627e6e02a61a7cbe851b6a5be34cc73": "\\varphi_\\delta(\\alpha)", "16280b2575d6aeaf5ab4c103568e641f": "\\theta = {\\pi \\over q}", "16287aa3baeb06275b1c6d54f20027ad": " f(t) = \\frac{\\lambda^x t^{x-1} e^{-\\lambda t}}{(x-1)!}. ", "1628e53f591f0a4ef4d3f846ed33898d": "\\hat{H}", "1629501ce2f38bfa07421d63a174bf97": "c = 18", "16299b96e4e4d86900f487356828dde9": "M_y = \\int_0^2{\\int_x^{4-x}}{}{}x\\,(2x+3y+2)\\,dy\\,dx", "1629bfcf86cacbab43a76db9cd8917fa": " | e_n | \\le \\frac{\\max_j \\tau_j}{hL} \\left( \\mathrm{e}^{L(t_n-t_0)} - 1 \\right). ", "1629d8150bc0a2e5b4e1cf347cc1707f": "N/{\\sqrt{\\theta}}", "162a6e32f3f45240d9166140d85c14bf": "\\left[\\frac{q}{p}\\right]_3 = 1\n\\mbox{ if and only if }\n\\begin{cases}\nq|LM\\mbox{ or }\\\\\nL\\equiv\\pm \\frac{9r}{2u+1} M\\pmod{q}, \\;\\;\\;\\mbox{ where }\\\\ \\;\\;\\;\\;\\; u\\not\\equiv 0,1,-\\frac12, -\\frac13 \\pmod{q} \\;\\;\\;\\mbox{ and } \\\\\n\\;\\;\\;\\;\\;3u+1 \\equiv r^2 (3u-3)\\pmod{q}\n\\end{cases}\n", "162a78da2f4c128badcaf5d6b8687037": "T_K\\,\\!", "162a7a80205f5cac14df3c47e219ba08": "\\boldsymbol\\alpha+\\sum_{i=1}^n\\mathbf{x}_i\\!", "162a92b0abe7f4a09cb2760e40c1c3a3": "\\sum_{m,n}\\int d^dx_1 \\cdots d^dx_m\\, d^dy_1 \\cdots d^dy_n S_{m+n}(x_1,\\dots,x_m,y_1,\\dots,y_n)f_m(\\bar{x}_1,\\dots,\\bar{x}_m)^* f_n(y_1,\\dots,y_n)\\geq 0", "162acb38e76dc86c88f97b8b0c5464bc": "\\frac{1}{x - 2} = \\frac{3}{x + 2} - \\frac{6x}{(x - 2)(x + 2)}\\,.", "162afdd94e26c656a21e8cddec79c830": "\\zeta_i=dz_i", "162b0d976cdeabd7105ebcf9aa9184cd": "b \\le a ", "162b23614f3de15ba9c77440d9b75780": "F_{2}", "162b2d9697b973899bf85b76991cda0c": "\\mathbb{F}_{q^n}", "162b505a04f7ea940337b689871f22e0": "\\hat{\\mathbf{A}}", "162b6a575da557fe8ed4d3be373390c7": "\\underline{\\mathbf{e}}(\\ell) = \\mathbf{F}\\left[ \\mathbf{0}_{1xN}, e(\\ell N),\\dots,e(\\ell N-N-1) \\right]^T", "162b6ad30dbd1c4133375e92fd5433b5": "\\varphi_K(x)={\\chi_{K_\\delta}\\ast\\varphi_\\delta(x)}=\n\\int_{\\mathbb{R}^n}\\chi_{K_\\delta}(y)\\,\\varphi_\\delta(x-y)\\,\\mathrm{d}y,", "162b98f202cb53bd7014b27f4d97a938": "\n\\frac{\\partial}{\\partial t} \\frac{ 4(1200t + 200) }{ t^2 + 4t - 2t^2 } = 4\\frac{ (4t - t^2)(1200) - (4 - 2t)(1200t + 200) }{ (t^2 + 4t - 2t^2)^2 },\n", "162ba20e77c5ec68cd91627573f08ffd": " \\frac {I}{I_0} = \\frac {1}{2}\\quad .", "162bd8a46babaeb48e7f6955dd130ca1": "| x + iy | = \\sqrt{x^2 + y^2}.", "162be0dba6b9a12a356c0133f5d7ba5b": "G(s)=\\frac{1}{s^2+s+1}", "162bf31bc716b1eedc125e11307f7dc7": "\\mathbf{H}\\ \\equiv \\ \\frac{\\mathbf{B}}{\\mu_0}-\\mathbf{M},", "162bf4aecfe4a8271341df1c23e76deb": " P_k(x) = 1+x+\\frac{x^2}{2!}+\\cdots+\\frac{x^k}{k!}, \\qquad R_k(x)=\\frac{e^\\xi}{(k+1)!}x^{k+1},", "162c565e38457ce882add7bc6486846f": "\\overline{\\rho}=\\frac{1}{\\overline{\\sigma}}", "162c7caa6d566dbf86b5016503631d3a": "(4)\\quad t=v-r-2M\\ln\\Big(\\frac{r}{2M}-1\\Big)\\qquad\\Rightarrow\\quad dt=dv-\\Big( 1-\\frac{2M}{r} \\Big)^{-1}dr\\;,", "162cdead5f8571f34875f3ddb11067a1": "g_p(Y_p,aU_p+bV_p) = ag_p(Y_p,U_p)+bg_p(Y_p,V_p).\\,", "162cf91b72aea9a61cc553dfa22cc4d9": "d \\to z", "162d1cef0825ad9dc2eb09ab220a1cba": "\\exp_{10}^2(5.84259)", "162d4c413f99ae2763b1ced17ed1a14b": "\\Gamma ", "162d616e67e056a8da8a0ffdcd4f4a91": " T(\\psi(x) \\bar{\\psi}(y)) \\ \\stackrel{\\mathrm{def}}{=}\\ \\theta(x^{0}-y^{0}) \\psi(x) \\bar{\\psi}(y) - \\theta(y^{0}-x^{0})\\bar\\psi(y) \\psi(x) .", "162dea7b04a30499ae3eeafd1d424788": "(C)\\int\\, f d\\nu + (C)\\int g\\, d\\nu \\le (C)\\int (f + g)\\, d\\nu.", "162e1e0ce491b1cdb006392bfaacc15f": "\\rho_{i,T_0}", "162e547ee23d422d373ed8b79eb327c1": " \\qquad \\qquad \\mathrm{vibrational }\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ E_{f,v,l} = \\hbar\\omega_{f,v}(1+\\frac{1}{2}) \\ \\ \\mathrm{and} \\ \\ Z_{f,v}\\sum_{j = 0}^\\infty \\mathrm{exp}[-(l+\\frac{1}{2})\\frac{\\hbar\\omega_{f,v}}{k_\\mathrm{B}T}] = \\frac{\\mathrm{exp}(-T_{f,v}/2T)}{1-\\mathrm{exp}(-T_{f,v}/T)},", "162ecc8ae7d6b6e88b0df42d01b3ef45": "\\operatorname{Pin}^\\pm(4) \\to \\operatorname{O}(4)", "162eeb5b8cbe5357686bc3d88f203a34": "d\\alpha=\\sin\\phi\\,d\\lambda.", "162efb1eb8f0378c7d837ea8b30286e7": " \\left[ 1; \\frac{2t}{t^2+1}; \\frac{t^2-1}{t^2+1}\\right].", "162efe6847ed1adcc20122c9f979c1ef": "D_\\mu\\phi=(\\partial_\\mu \\phi - i e A_\\mu \\phi)", "162f2935e6ab84cf7b4d1c8579bfe9ca": "M_0, M_1 \\subset N", "162f439ef32e7eb2d173c00579a1527c": "\\begin{matrix} {11 \\choose 1}{4 \\choose 3}{40 \\choose 2} \\end{matrix}", "162fbd0fb5cfde94641f4910a529232e": "B_0 = \\frac{1}{c}E_0,", "162fc5abe360a63b7d74988459dbce40": "\\|AT-I\\|_F,", "162fcede5603c9d7122deaadcad0b4f4": "E_1=\\frac{v_1^2}{2g}+y_1+\\frac{p_1}{\\gamma}=E_2=\\frac{v_2^2}{2g}+y_2+\\frac{p_2}{\\gamma}", "162ffde5e606b0343377f5bfb48a82d7": "p(x)\\cdot{d \\over dx}", "1630016f05909cf288d9c8c8c12a22e2": "\\Pi^S+\\Pi^I=1", "16303c2c1bc34fff553826356a07558a": "\n f_i = K_{ij}~d_j ~.\n ", "1630787bd562dd16135397a0f7e2e84c": "f,g:{\\mathbb R}^n\\rightarrow {\\mathbb R}", "163098679a068ecd801d34f4a150b98d": "\\lim_{n\\to\\infty} \\frac{\\mathrm{N}(n,S)}{n} = 0.", "1630d5ff7976739ac1a9d3cfd295b54d": "\n\\mathbf{v}_{k} = \\boldsymbol\\omega \\times \\mathbf{r}_{k}\n", "163150331dea672eb07db8bde80ed0bb": "\\begin{pmatrix} p_{n-1} & p_{n} \\\\ q_{n-1} & q_{n} \\end{pmatrix}", "16316bb5c537f4d07cf462dfaf5bfb06": " E = - \\frac{\\Delta\\phi}{d}", "16324897458c1d52a5004148bf814b8f": " \\left( f_1 \\star_\\inf \\cdots \\star_\\inf f_m \\right)^\\star = f_1^\\star + \\cdots + f_m^\\star. ", "16326fbad995488ac0d59a546dd5b5a9": "\\delta=\\frac{\\rho_ap}{k}", "16327e656d07ae22a9a41e498d631640": "\\dot \\gamma = \\frac{V}{H}", "1632ee7fe5d6c343f4479c0968396e31": "\\sigma_n", "16333758fad2598cf8c84b7c5df92264": "A + B \\rightleftharpoons |A \\cdots B|^{\\ddagger} \\rightarrow P ", "16334ad4eb29342eb8f65930e4d44f2f": "K,N:", "16334eff3966c4d239a05520fc557779": " \\begin{align} P(Presentation~WHOIFPI) = P(Presentation~WHOIFPI~by~condition~1) + \\\\\n P(Presentation~WHOIFPI~by~condition~2) + \\\\\n P(Presentation~WHOIFPI~by~condition~3) + etc \\end{align}", "16335d82d31355736892a13801efa0d1": "ku>\\kappa_z", "16337592edce67f98b9ad43c98831c29": "X_w(a,b) ", "163379a708207472f0b48ebd571d58b0": " u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \\Leftrightarrow\n \\Phi(a'_{i},a_{-i})-\\Phi(a''_{i},a_{-i})>0", "1633e609f33108f110e1b223a3dd81ee": "\\begin{align}\n \\mathbf{V} \\mathbf{V}^* &=\n \\begin{bmatrix}\n 0 & 0 & \\sqrt{0.2} & 0 & -\\sqrt{0.8} \\\\\n 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & \\sqrt{0.8} & 0 & \\sqrt{0.2}\n \\end{bmatrix} \\cdot\n\n \\begin{bmatrix}\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n \\sqrt{0.2} & 0 & 0 & 0 & \\sqrt{0.8} \\\\\n 0 & 0 & 0 & 1 & 0 \\\\\n -\\sqrt{0.8} & 0 & 0 & 0 & \\sqrt{0.2}\n \\end{bmatrix} \\\\\n\n &=\n \\begin{bmatrix}\n 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 1\n \\end{bmatrix} \\equiv\n\n \\mathbf{I}_5\n\\end{align}", "163402576a6c172be0e40b729c5c35c0": "{Q_3 - Q_1 \\over Q_3 + Q_1}.", "163418a02552d4134c87bde2b66bedb0": " \\alpha ", "163442fc17db463edca4a31791202bcf": "\\vdash B", "1634a736412bd208c0ecf4d7fb7ed6d4": "x = \\pm \\left(\\sqrt[m]a\\right)^n", "1635515f4a6d6dbe320ce0be8c415af7": " \\csc x = (\\sin x)^{-1} = \\frac{1}{\\sin x}.", "16355ab7f45478b0519aefe1cc9763c3": " -{\\partial p}/{\\partial s}", "163560912eb173d743579a570e2d6033": "(B y +\\beta)^n - B^n y^n", "16357da621cb7d585efb110e0a67808a": "\\Sigma^1_3", "1635b9b702eb57833b992fbc0e321b97": "y(t)=f(x(t))", "163618078d58100a50990cd8aef08d4b": "h(k)", "16364f65af01370b2b21831899ccbf96": " vblood ", "1636a34fb6f146d991358b572651f55d": " e_j y_j + \\sum_{i=1}^m{b_{ij} s_i} \\le d_j", "1637153e8e8cd310f1af6b8b9bf085b9": "n_t^j", "163746cf9c097785b6d2df9d156a8a38": "S^i = \\{z_1 ,...,\\ z_{i-1},\\ z_i',\\ z_{i+1},...,\\ z_m\\}", "1637d979a782c3aedb33f29b5d826736": "y_i \\left[ {w^T \\phi (x_i ) + b} \\right] = 1 - e_{c,i} ,\\quad i = 1, \\ldots ,N .", "1637df3195f37e58a4188ba7d2c17979": "2\\, \\operatorname{Cl}_2(\\theta) +2\\int_{\\pi}^{\\pi-\\theta} \\log\\Bigg| 2 \\sin \\frac{y}{2} \\Bigg| \\,dy= ", "163837903b71c6f56e0256b4ff83e00d": "r = f(x)g(y)", "16385244564dfff6bd0a61fc2bf3d7d4": " \\frac{7}{6} ", "16389d8d2ba844f7e4e27270e357a41a": " \\chi : \\mathbb{Z}/N\\mathbb{Z} \\rightarrow F^*\\ ", "1638a8867c7211973ebbeab3f7e28566": " \\omega = \\frac{\\sigma \\left ( k^2 a^2-1 \\right )}{2a \\mu_A} \\frac{1}{k^2 a^2 + 1 - k^2 a^2 I_0^2 \\left ( ka \\right ) / I_1^2 \\left ( ka \\right )}", "1638b77b7d0b364a035923604795f335": "m \\in P", "1638ca3816cd0772e02b3fc1fff85976": "T(R)=\\frac{T(A_1),\\dots,T(A_n)}{T(B)}.", "1638e8492e71dee9f64c27cc55719866": "p_f > \\frac{1}{2}", "1638f2ed0da101523e2880edaef4fc8b": "p(r) \\ne 0", "1639028839d71cccdeaba64bceff9cd4": "\\displaystyle L = \\mu_0\\mu_rN^2A/l.", "163915d96bf1488f866c78c2183e7378": " \\int_{-\\infty}^\\infty x\\phi(x)\\Phi(bx) \\, dx = \\int_{-\\infty}^\\infty x\\phi(x)\\Phi(bx)^2 \\, dx = \\frac{b}{\\sqrt{2\\pi(1+b^2)}} ", "16394c0c93870bc3f6c9298b5f686b93": "r^{\\prime\\prime} = -\\mathbf{k}\\, r\\, \\mathbf{k}. ", "16394fca0405a7339ada1becb2298ead": "M = 1", "16395f8634039643670729f61a34ab9a": "\\mathbf{P}( X \\le (1-\\delta)\\mu) \\le e^{-\\frac{\\delta^2\\mu}{2}}, \\quad 0 < \\delta < 1", "1639a20f662391c938749087bebac654": "\\phi(x)=\\sum_{k=-N}^N a_k\\phi(2x-k).", "163a88682def3da8260f0b7b7977a349": " t \\in [0.7, \\dots, 1.4] ", "163a98e1eef1667a79813b694b4cf488": "R \\sim \\mathrm{Rayleigh} (1)", "163af789c6bff5f1a86e66190b323255": "(1-(z-d)^2/n)^n", "163b3106450e6b8293d55f9ee242b3fc": "\\Box A\\to\\Diamond A", "163b3b34fd9f9e7c51bcb630c6af12d9": "-\\frac{\\hbar^2}{2m}\\nabla^2 \\psi_n(q) = E_n \\psi_n(q)", "163b7e16ccc676a367d51ddd8354c5a0": "\\sigma = uv = \\prod_{i=1}^k p_i", "163b97624c5346ea00305d1d176fcc9c": "1, 3, 7, 21, 48, 112, \\ldots", "163bdc6c9f31a965fe67eb31f4ef40d3": "(1+z)^n = 1+z^n (\\mbox{mod}\\;n).", "163be9b6f55ef16b0f1259b7146daacc": "\\forall \\mu, \\nu, \\exists \\lambda", "163bfdbfc2c6ab4f516c0465dc08c8e1": "2^{\\mathrm{intermediate\\ result}} = 2^{I+F} = 2^I\\,2^F", "163cde00287e629f33dae509a8414505": "\\phi(x)", "163d42b20ee0642a0e30e9a9bc3a40e9": "SL \\to GL^+", "163d513d322c2502434ea06a393e2404": " \\mathbf{S}X := \\bigcup_{i=0}^{\\infty} \\mathbf{S}_{i}X \\mbox{.} \\! ", "163dadc589bdbbe1b5e0959baedb8d84": "d \\approx 1.23 \\cdot \\sqrt{h}", "163dc45aaa57f2225909859887a7b18d": "(V^*)^{\\mathbb C} \\cong (V^{\\mathbb C})^*.", "163dcd4f1e43a48b97e727112bd591fe": "k \\ne i", "163ddb0e4aa7ad86bb62b4030462e347": "P_n (k) ", "163dea18c142083900d95b7e1fa82f3f": "(\\bar 4,1,2)", "163dfce51229b83841c0d51d0fd21651": "\\mu (A) = \\inf \\{ \\mu (G) | G \\supseteq A, G \\mbox{ open and measurable} \\}", "163e11cbf5ff4357f0bf6207feae7258": "n=\\frac{f}{2-f},\\quad \nA=\\frac{a}{1+n}\\left(1+ \\frac{n^2}{4} +\\frac{n^4}{64} +\\cdots\\right),", "163e1f321990e6d90185dbd06b0cc11b": "\\partial_\\omega(\\omega\\epsilon)", "163e284be69977ee0b7c09456c07f3d1": "l_2(\\theta) = (\\vartheta+\\theta)/2", "163e7f8cc14081491672fb252a14a5b5": " i:=i+1;", "163e8ab46fdc03a113690f3b5b860785": "\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q", "163e9ad8bbc63b2b80c6e2b234419c8d": "EL(\\Gamma)\\ge \\frac{\\log(r_2/r_1)}{2\\pi}.", "163ea35546c9de550a1851bb7a2cbeaf": "\np_{\\mathbf{Y}}(Y)=\\frac{p_{\\mathbf{y}}(\\mathbf{y})}{|\\frac{\\partial\\mathbf{Y}}{\\partial \\mathbf{y}}|}\n", "163eb8a639ffa87bb3d5ec3e7a436dd2": "\\delta(x) = \\frac1{2\\pi} \\sum_{n=-\\infty}^\\infty e^{inx}", "163f41ddc8645ad40cb50b494ff409f1": "\\sum_{i=0}^n \\dim H^{2i}(M,R)\\bmod 2", "163f42ac9a96526aa1add67ec35d36b1": " (L_1 \\times L_2) \\cdot L_3 = \\langle L_1,L_2,L_3 \\rangle = 0. ", "163fbe8cb54f22f59e5cc82b5c729e28": "\\dot S", "16402685a0e4602f2c951ca8df6ade36": " 2 \\le i \\le m - 1 ", "1640503311e070f7e7bfcf49a037d7b1": "\\Theta_{\\mathrm{div}} \\simeq 2\\frac{w_0}{z_R}.", "1640603d3de2bea8fffaddfe7a143c61": "\\tfrac{5}{2}", "1640da851490562e0ca9dae6e8198b64": " (\\mathbf{b_{1}}, \\mathbf{b_{2}}, \\mathbf{b_{3}}) ", "164126a5c9c38f224248aff32ce6e7ad": "\\varphi\\star \\varphi^\\prime", "164209e30f2287f64347bfdb337dc9fb": " v\\, = \\frac {2 \\pi r } {T} = \\omega r ", "1642d7cc623fde3c56a7471079868d7f": " m_{t+1} = m_t - \\overline {\\Delta v}_{t-16} + 1.5 \\left( + {\\Delta p}_D + \\overline {\\Delta q} \\right) - 0.5 {\\Delta x}_{t-1} \\,,", "1642ee17ffab28d42adcacbed7c0fb37": " \\psi_m ", "1642fc54cad195a9f9506852715243d1": " n \\in N\\cup\\{+\\infty\\}", "16430576533683a9271051b3ad4624ae": " d U = \\delta Q + \\delta W \\, ", "16446094d5c28548a6ec20124c0be07f": "\\boldsymbol{n}", "1644b01d18c035656d6e57b206074fd7": "x*g", "16450bd602506f0b37d566de5b28afee": " \\forall P \\in \\text{vbl}(A) ", "1645103e03da3acaba25816554710a35": "\\frac{1}{\\sqrt{1 - \\beta^2}}", "164515215ac18c3392e1c77f6bb7a837": "E_n^{(3)} = \\sum_{k \\neq n} \\sum_{m \\neq n} \\frac{\\langle n^{(0)} | V | m^{(0)} \\rangle \\langle m^{(0)} | V | k^{(0)} \\rangle \\langle k^{(0)} | V | n^{(0)} \\rangle}{\\left( E_m^{(0)} - E_n^{(0)} \\right) \\left( E_k^{(0)} - E_n^{(0)} \\right)} - \\langle n^{(0)} | V | n^{(0)} \\rangle \\sum_{m \\neq n} \\frac{|\\langle n^{(0)} | V | m^{(0)} \\rangle|^2}{\\left( E_m^{(0)} - E_n^{(0)} \\right)^2}.", "1645436a004690d079de9d4b1098a409": "= e^{-i 2 \\pi fT} \\mathrm{sinc}^2(fT) \\ ", "1645ca15abc2c95197ff2f2ca8d6eed7": "=\\int_{P(t_1,t_2)} \\dot Q(t)dt", "16460f0d757edd0a3f7f700687a971ee": "\\mathcal{L}_t\\left\\{ f(t)\\right\\}(s) = F(s)", "16462602ca908d0a89a03d3a156a3d7d": "\\!A", "16463c38031cc857d20f8fb7f4d39bbe": "\\textstyle b_{2} = p(t_x, a_{(-1,2)}, a_{(0,2)}, a_{(1,2)}, a_{(2,2)})", "16467399d678c1f3c1ebf9748719e110": "\\frac{\\partial {{T}_{{}^{1}\\!\\!\\diagup\\!\\!{}_{2}\\;}}}{\\partial P}", "164678f5a92460bc67ae48cd8cc734df": "\\mathbf{p}=(p_1,p_2,...,p_s),", "16467b631d4b7e72d7968123019c1f52": "z^\\mu = x^\\mu + i y^\\mu", "164685fed16995ed1d1cb60398eddfbd": "\\ddot{Z}/Z", "1646e7c616f381772efcb605cba6c83d": "\\vec{Q}", "16474519c7480ba37ef273b9fbcc6ab5": "\\frac{1}{E(\\phi)}", "16477fd3011ff1bbcdacfcd551410fca": "\\eta=1-\\left(\\frac{\\mathit{T}_{1}}{\\mathit{T}_{2}}\\right)", "16479c46a8bec4b2a6377f42a58b4507": "\\omega_B", "1647dd85b1d4d58cae1e001d1dec34b4": "\\frac{\\mathrm{d}\\phi}{\\mathrm{d}t}\\,", "16484169ec44a6340333ce901d9be6f8": "\\frac{3}{4}\\,\\sqrt{\\frac{\\pi}{2}}", "16485e8f988cee3cb1d86eeefc0d2c4b": "x(t)\\approx\\int_{-\\infty}^t dt'\\, \\chi(t-t')h(t')\\,.", "164935bf3f110ca673a798339531a5e0": "c_{k+1}", "1649394019327a77e24d2a4f059541ee": " \\gamma = \\lim_{n \\to \\infty} \\left \\{\\frac{ \\Gamma(\\frac{1}{n}) \\Gamma(n+1)\\, n^{1+1/n}}{\\Gamma(2+n+\\frac{1}{n})} - \\frac{n^2}{n+1} \\right\\} ", "164939437939651e6be0b758f8098a69": "\nz_{i,j} = \\sqrt{(y_{i,j} - y_{i+1,j+1})^2 + (y_{i+1,j} - y_{i, j+1})^2 } \n", "16495c5742548582fff3ab84d15b3aec": " \\sigma^2 = g(r) \\, r \\, d\\theta", "16496926f12dd2010b5744bbd2a69d1f": "\\sum(\\overline{X}-X_i)=0,", "16497c5a9f8ee922bec9f237aed3ce26": "f(\\textbf{x}_{1}) \\leq f(\\textbf{x}_{2}) \\leq \\cdots \\leq f(\\textbf{x}_{n+1})", "164986b728adb47ad55fa7ca3ac2a753": "\ny\\; =\\; y_0\\, \\sin(kx - \\omega t)\\; +\\; y_0\\, \\sin(kx + \\omega t).\\,\n", "1649e712c8b0ee50a07ffdac17318102": "\\displaystyle{gJg^t=J,}", "164a4d74285de53017b4fede37b869c9": "\\ A-\\text{vertex}= -1 : 1 : 1 ", "164a6b7b7e304335bcb246b02ca99655": "\\{k\\}_{k=1}^{M}", "164aab095469b363240d39c73f7459da": "\\frac{\\partial u }{\\partial y}", "164ae789a1915c8b331ea8f8293b3498": "\\Lambda(\\alpha^i)", "164b3923779554606f2f414e4f1db29a": "h(k_3)", "164bb2c5e87cb4246ba5b43dfbbac5cb": "f(n)=O(1)\\,", "164bdb06b8b59a8e9e74a494488ebfd8": " (1+2/e) \\approx 1.736 ", "164c33008d5460f9bbe21283058f42cf": "y_i=g(x_i)+\\sigma\\varepsilon_i", "164c5a364542be3f2784eed0ef2c2f26": " \n \\begin{align}\n \\frac{\\partial \\mathcal{L}}{\\partial f_1} - \\sum_{i=1}^{n} \\frac{\\mathrm{d}}{\\mathrm{d}x_i} \\frac{\\partial \\mathcal{L}}{\\partial f_{1,i}} &= 0 \\\\\n \\frac{\\partial \\mathcal{L}}{\\partial f_2} - \\sum_{i=1}^{n} \\frac{\\mathrm{d}}{\\mathrm{d}x_i} \\frac{\\partial \\mathcal{L}}{\\partial f_{2,i}} &= 0 \\\\\n \\vdots \\qquad \\vdots \\qquad &\\quad \\vdots \\\\\n \\frac{\\partial \\mathcal{L}}{\\partial f_j} - \\sum_{i=1}^{n} \\frac{\\mathrm{d}}{\\mathrm{d}x_i} \\frac{\\partial \\mathcal{L}}{\\partial f_{j,i}} &= 0.\n \\end{align}\n ", "164ced55a7307af6ea4ce518ea4cebe5": "I[0]=0", "164d023a50096f27e480982d74126e86": "J^k_0(x^i\\circ f)=J^k_0(x^i\\circ g)", "164d2d8646ce09e5fc2f8a1ac90806c6": "x(t) = e^{at} \\left(\\cos bt - \\frac{a}{b} \\sin bt \\right) ", "164d3ae4d5a48f402683064c87c34c81": "\\chi_+^a", "164d5a0a2f035a9a6d50b91f2bdda4c7": "f_{\\epsilon_0}\\,\\!", "164d6fdff9a80685d868f5b638f19cf1": "xN_2z", "164d9e2d995ed1383b38a3744490eda4": "\\displaystyle{W=(Z-iI)(Z+iI)^{-1}.}", "164daf5448d54bbcefbde0c8defe3691": "(P^2)", "164dc5156fc56c268db408b4a370921e": "\\nu=(4,3,2)", "164df731e691a1830185268ae22c0f4b": "S(\\mathbf{x}_{\\text{i}},\\,\\omega_{\\text{i}},\\,\\mathbf{x}_{\\text{r}},\\,\\omega_{\\text{r}})", "164e032dee785a1a7f0d4c368c062564": "=\\lambda\\mathbf{r}_{dx}(n-1)+d(n)\\mathbf{x}(n)", "164e31839cda9817457d5bc0e2d68c7d": "\\scriptstyle \\mathcal{N}=\\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\\} ", "164e4d24adf147d550c64683e6c7e780": "F(v_1\\otimes \\cdots \\otimes v_n) = f(v_1,\\ldots,v_n).", "164ebdc38094787328f6096fc110974d": "R_\\mathrm{load}", "164fa3034e56d79832412700b1e82ef4": "f_t(z)", "164fca470391d1420d94c70c3f30b1ae": "K\\subseteq E", "164fe527970fd1e2c6fdb77c0af38250": "\\langle S \\mid R\\rangle.\\,\\!", "16504c6dfbbdfbc46f9b4e5253cf112f": "\\operatorname{Tr}(Q\\rho) \\geq \\epsilon", "1650a49e25841a98079bfd84ae2d5825": "D^k", "1650c60a66df0651e8a0e7d9106a6188": "w[T]y", "1650ef9cdf8ca25428de17cce9de60ee": "\\partial_k", "165131f15d6a3d7c81b6de6c863919cf": "\\left( \\frac{\\pi}{a} \\right)^2 + \\left( \\frac{\\pi}{b} \\right)^2 + \\left( \\frac{\\pi}{c} \\right)^2", "16516fa91b6fefc568800d4afe70994d": "\\langle \\mu, \\xi \\rangle", "1651bc5814e151fa96904c35b425e4f0": "\\Theta_r = \\frac{c_p (T-T_e)}{U_e^2/2}", "1651c8d0d18ccb5537744513e1d21694": "B=f(P,E)", "1651f955ab36f5cc34d4c8ea42f785a4": "F_m^i(x)=\\left \\langle 0 |\\mathcal T\\phi_i(x_1)\\phi_i(x_2)|0\\right \\rangle=\\sum_\\mathrm{pairs}\\overline{\\phi(x_1)\\phi(x_2)}\\cdots\n\\overline{\\phi(x_{m-1})\\phi(x_m})", "1652e05f528a1bfaab2dff61d8e5fdf9": "M,a\\models\\phi\\to\\psi\\iff\\forall b,c((Rabc\\land M,b\\models\\phi)\\Rightarrow M,c\\models\\psi)", "1652e4db0fa082db1030997d869ac5b8": "\n D=\\begin{bmatrix}d_1&0&0&\\ldots&0\\\\0&d_2&0&\\ldots&0\\\\\\vdots&&&d_s&0&\\\\0&0&0&\\ldots&0\\\\\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\end{bmatrix}_{m\\times n}\n", "165342509514b58fe5c69e1b3dcf6ffb": "\n\\eta = \\frac{P_\\mathrm{out}}{P_\\mathrm{in}}\n", "165396c52a4595b2abe2257f713eb3a4": "\\mathbf{e}_1,\\dots,\\mathbf{e}_n", "165402a54d9bd5635bc19abf245ea205": "\\Delta H=\\Delta U + \\Delta(PV)\\,.", "165418bce527a40ce68df045e2fc7d77": "\\frac{{d^2 x(t)}}{{dt^2 }} + \\bar{c}\\frac{{dx(t)}}{{dt}} + \\bar{k}x(t) = 0", "16543ff8b3e39de69197ad3589ef8f84": "0", "166ef4089e5732c873c2526d33e7710c": "\\ell=1,\\quad m=-1,0,+1", "166f06881d2443484d668f428e0edbc7": "r_{C} / r_{A}", "166f4c38702ff86aca210c5e5f5dcc86": "H = c \\sqrt{D}", "166f9e86385fd3afd95705014199864c": "\n\\varphi(\\theta) = \\frac{1}{2}\\sin(\\theta).\n", "166fef34cd9044eae64854011bcb636f": "\\left(s_{1}, s_2\\right)", "167042c9cc7ab2432700baea81ab4551": "NRx_i = 2\\rho \\sqrt{\\frac{D_i}{\\pi t}}", "16705a640ed247a8ac9ccb8b768339fb": "M = \\frac{N \\mu_B g s}{V}", "16708713ee7cafa4138ccb10b0648703": "(\\mu_{(1)}(t), \\dots, \\mu_{(n)}(t)),", "1670dc45bd7bbc920f5f9f31b4f234df": "x_i \\succ x_j\\,\\!", "167113cb54f30e1b2d8cd0ca9f4428e1": "h(t)=\\left( \\frac{\\alpha}{\\pi} \\right)^\\frac{1}{4}e^{\\left( -\\frac{\\alpha}{2}t^2 \\right)}", "16713f4c4e8601fb977cc0d9b6b5237e": "\\bold{E}[H] = \\langle H \\rangle = -\\frac {\\partial \\log(Z(\\beta))} {\\partial \\beta}", "1671d9a186e4d59d6bb343658aed59a0": "\\mathbf{F}(x,y,z)=-x^2\\boldsymbol{\\hat{y}}.", "1672043fe46b2fec6f20afaade81374f": "Y_n = {X_1 + X_2 + \\cdots + X_n \\over n}", "167251a335db6215942049a68dbd32a6": "\nI_1 = \\begin{bmatrix} 1 \\end{bmatrix}\n,\\ \nI_2 = \\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1 \n \\end{bmatrix}\n,\\ \\cdots ,\\ \nI_n = \\begin{bmatrix}\n 1 & 0 & \\cdots & 0 \\\\\n 0 & 1 & \\cdots & 0 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & \\cdots & 1\n \\end{bmatrix}\n", "1672a332ce03149c177157209d513c22": "\\displaystyle{B(z_1,z_2)=(x_1,y_2)(x_2,y_1)^{-1},}", "1672a615ce788a56391fdc6829c01643": "\\sum_{i = 1}^{m} \\frac{1}{p_{i}} = 1", "1672e4681257f3088399ae0e60791814": " \\scriptstyle u(x,0) = e^{-x^2 +ik_0x}", "16730d28aa7355aa4d176c4f742a70de": "n\\equiv 2k \\pmod{4k}.", "16732763560d16f8bb7a9516b4c52008": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \n\\left( \\mathbf{\\hat{k}}\\cdot \\mathbf{\\hat{r}}\\right)^2\n{ \\exp \\left ( i\\mathbf{k} \\cdot \\mathbf{r} \\right ) \\over k^2 +m^2 } = {e^{ - m r } \\over 4 \\pi r } \\left\\{ 1+ {2\\over mr} \n- {2 \\over \\left( mr \\right)^2 } \\left( e^{mr} -1 \\right) \\right \\} ", "16740d7be57aa330dfaeaf568c1ea000": "\\prod_{i=1}^nx_i^{w_i\\cdot q} \\leq \\sum_{i=1}^nw_ix_i^q", "1674704f332635dbe7cd31559d9f6882": "V_{out}=V_{dd}", "16747abb0f9c30cb7b7b535375ab7c42": "[H^+]_{i^{ }}", "1674bd0aeb5c1975dceb4da4afb998b0": " A^{ik} {}_{;\\ell} = A^{ik} {}_{,\\ell} + A^{mk} \\Gamma^i{}_{m\\ell} + A^{im} \\Gamma^k{}_{m\\ell}. \\ ", "1674bf5405d7ec0c7ff38416f63514c5": "\\omega_{\\alpha KI} = {1 \\over 2} e_\\alpha^J (\\Omega_{JKI} + \\Omega_{IJK} - \\Omega_{KIJ})", "1674e892010094e397eb445ebd8a276c": " T(A,D) = \\,\\underset{x \\sim D}{\\operatorname{E}}[T(A,X)]. \\, ", "1676287dc26461863a4a5b591267b6c9": "X \\rightarrow Y, Y \\rightarrow Z \\vdash X \\rightarrow Z", "16762b9416c61b0c19f78a50060a16c1": "\\scriptstyle V_1", "1676b26ab76d2b643e1ecaa470fccd69": "\\nabla \\times ( \\nabla \\phi ) = \\mathbf{0}", "1676ca11a54cd44804f13eba862bf321": "\\Pi^{EXP}_k=\\mathrm{coNEXP}^{\\Sigma^P_{k-1}}", "1676cc5d3594a76d211152c373fc7f30": "\\int_{-a}^{a} (a+x)^{m-1}(a-x)^{n-1}\\ dx=(2a)^{m+n-1}\\frac{\\Gamma(m)\\Gamma(n)}{\\Gamma(m+n)}", "1676f4da60fdf0e2e8d8b657ca9b2f44": " [H_k] = \\frac{1}{C_k^{2n}}\n\\sum\\limits_{i+j=k}{C_i^n C_j^n w_i w_j [H_{ij}^\\ast]}, ", "1677313b2ca799c5023f98104417c7e9": "\\psi(x)=\\left(1-\\frac{\\lambda \\mu}{c}\\right) \\sum_{n=0}^\\infty \\left(\\frac{\\lambda \\mu}{c}\\right)^n (1-F^{\\ast n}_l(x))", "167738efe7dcee510cd9530f7becb675": "2\\cos(\\frac{2k\\pi}{11})", "16773fb299ef418cb45caa2a2e1769a8": "E_{CMI}(\\varrho_{A, B})=S(\\varrho_{A})=S(\\varrho_{B})", "1677631ccc17836e8c579eb52f58716e": "Z[J] =\\int \\mathcal{D}\\phi e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{\\lambda\\over 4!}\\phi^4+J\\phi\\right)} = Z[0] \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\langle\\Omega|\\mathcal{T}\\{{\\phi}(x_1)\\cdots {\\phi}(x_n)\\}|\\Omega\\rangle.", "1677a73721b8a0e1c3e271a16d789274": "i=1,\\dots,k", "1677cc1936c7182653edf7b7bd6e0ece": "Error = \\left (\\frac{(Real Delta) + i (Imaginary Delta)}{(Real Sum) + i (Imaginary Sum)} \\right) \\div \\left (\\frac{(Real Delta Cal) + i (Imaginary Delta Cal)}{(Real Sum Cal) + i (Imaginary Sum Cal)} \\right)", "1677e732300edef0ec6a6198b3f80d1e": "-1.3817", "1677fb4aa8e4bc73c50c9c8947d84b54": "z_\\epsilon = \\frac{1}{\\epsilon \\epsilon_0}\\frac{l}{S}", "1677fc4fc2e375b1a180a50af7982cf3": "N_{m}", "16781f38736f8651413590942dbb7283": "D_tT=\\nabla_{\\dot\\gamma(t)}T.", "1678825430c97f4459e90b6b596867a1": "\\, (1-p+pe^{it})^n", "1678fb690b605f4aa115d64302c45008": "T_{l}-T_{t}=\\frac{2}{c}\\left(\\frac{L}{1-\\frac{v^{2}}{c^{2}}}-\\frac{L}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}\\right)", "1679091c5a880faf6fb5e6087eb1b2dc": "6", "16797a1d4ca01f5e58b63a1b014e788b": "H_{\\leq \\beta} := \\{X \\in 2^\\omega : X\\ \\mathrm{has\\ effective\\ Hausdorff\\ dimension\\ } \\leq \\beta \\}", "16798347341b6d506739a559b107dc3a": " V_t - \\frac{V_1}{2/N^2} = (1-2/N^2)\\left(V_{t-1}-\\frac{V_1}{2/N^2}\\right) = (1-2/N^2)^{t-1}\\left(V_1-\\frac{V_1}{2/N^2}\\right) ", "16799f3c67e13519813f1287274fd6a3": "\n \\begin{align}\n F = & \\cfrac{1}{2}\\left[\\Sigma_2^2 + \\Sigma_3^2 - \\Sigma_1^2\\right] ~;~~\n G = \\cfrac{1}{2}\\left[\\Sigma_3^2 + \\Sigma_1^2 - \\Sigma_2^2\\right] ~;~~\n H = \\cfrac{1}{2}\\left[\\Sigma_1^2 + \\Sigma_2^2 - \\Sigma_3^2\\right] \\\\\n L = & \\cfrac{1}{2(\\sigma_{23}^y)^2} ~;~~\n M = \\cfrac{1}{2(\\sigma_{31}^y)^2} ~;~~\n N = \\cfrac{1}{2(\\sigma_{12}^y)^2} \\\\\n I = & \\cfrac{\\sigma_{1c}-\\sigma_{1t}}{2\\sigma_{1c}\\sigma_{1t}} ~;~~\n J = \\cfrac{\\sigma_{2c}-\\sigma_{2t}}{2\\sigma_{2c}\\sigma_{2t}} ~;~~\n K = \\cfrac{\\sigma_{3c}-\\sigma_{3t}}{2\\sigma_{3c}\\sigma_{3t}} \n \\end{align}\n ", "1679f575a1431c0bdf34817ac3134043": "\\gamma^n \\to Gr_n", "167a43013459122478d65d0486e072f3": "\\frac {d\\theta}{ds} = 2a^2 s", "167a6ca9bf7204de1d67fe059b0bc8c6": " \\|x + y\\|^2 = \\|x\\|^2 + \\|y\\|^2 + 2 \\real \\langle x , y \\rangle. ", "167aafc414fe18093b849c7973cbd52d": "\\sdot \\frac {1+j \\omega C_C R_o/A_v } {1+j \\omega (C_L + C_C ) (R_o//R_L) } \\ . ", "167adf021ddd27f51b332943f9a3eaba": "\\operatorname{pf}(A^{2m+1})= (-1)^{nm}\\operatorname{pf}(A)^{2m+1}.", "167b2c0f5dfdb9c26488887f385114ca": " \\mathbf{\\bar C}_{k} \\sim \\mathbf{C}_{k} \\, \\mathbf{T}^{-1} ", "167b489781f5d6d30157c7da73b38c45": "T=(diag(A-\\lambda_n I))^{-1}.", "167baa44d6110b64adc0d945fdab8764": "|U(T)| \\geq |U(S) \\setminus N(T \\setminus S)| \\geq |U(S)| - |N(T \\setminus S)| > 0\\,", "167bec0c3d393d628af84e4d84b5d304": " p = 1 - e^{ -m } ", "167bec1b4b56516ff646c8d64be75ce6": "a\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}", "167c277f3f6087e57bf32672bfe6aca7": "\\alpha = \\frac{(^{18}O/^{16}O)_{Liquid}}{(^{18}O/^{16}O)_{Vapor}} = 1.0098 ", "167c29112335c2b9314a648f7a212251": "t=t_1\\,", "167c4d78881b5b5cbfee437ca788aa4d": "W_4", "167cc9f062f4d38b1261890524305caa": "mx + a", "167cd4c82c0fa1b3a0035bcf95f8f0fb": "\\scriptstyle \\mathbf{u}", "167d333d2d3be390b69d13e4b2b73bd1": "\nS_u (1\\pm S_u) = \\pm 1 (1 \\pm S_u)\n", "167d4868b1b2c0ae2d4de98781a02181": " w(x) = \\prod_{i=1}^{20} (x - i) = (x-1)(x-2) \\ldots (x-20) ", "167dd4e7be820e8d78b1cd65f032d973": " \\sigma^2_P = n \\frac{1}{n^2} \\sigma^2_i + n(n-1) \\frac{1}{n} \\frac{1}{n} \\bar{\\sigma}_{ij} ", "167ddff2137a87505cb96475e7c2dfb6": "r_t = \\exp{X_t}\\,", "167e29312938b3dbe217b0dc31b21470": "\\overline{\\Omega}", "167e904200ca0672c63498b5c1d7a05b": " \n\\mathbf{a}_{\\mathrm{i}} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\left( \\frac{d^{2}\\mathbf{r}}{dt^{2}}\\right)_{\\mathrm{i}} = \n\\left( \\frac{d\\mathbf{v}}{dt} \\right)_{\\mathrm{i}} = \n\\left[ \\left( \\frac{d}{dt} \\right)_{\\mathrm{r}} + \n\\boldsymbol\\Omega \\times \n\\right]\n\\left[\n\\left( \\frac{d\\mathbf{r}}{dt} \\right)_{\\mathrm{r}} + \n\\boldsymbol\\Omega \\times \\mathbf{r} \n\\right] \\ ,\n", "167ecc84d75e93d135f13984c1cf1e8c": "|\\pi_{0,\\mathrm{c}}|=2\\left(\\gamma_0+Bn\\right).", "167ed8514419d2205f0ff2df3702f235": "\\textstyle [255,223,33]", "167ede0a6f34ccc88cbbedf0f4be1160": "\\phi(f^n(\\phi^{-1}(z)))=e^{2\\pi i\\alpha}z", "167f45cd3a6eee87f6681a44aeacb3ea": " R = df/dn(n=0)", "167f50f8b86df0cc6b109a047bd4569d": "C_{TD} = \\frac{\\epsilon_0 S_{TD}}{a_{TD}} \\approx 6 \\times 10^{-14}\\;\\mathrm{F} = 60\\;\\mathrm{fF}", "167f56648fab33f229d1e190ebe3a3d0": "{\\alpha \\over \\beta}>0", "167f8a50ff76bb9d711ccd2975c771f4": "Z_{i+1} = \\left\\{ \\begin{array}{ll}\nS_{i+1} & \\text{ if } \\displaystyle \\max_{1 \\leq j \\leq i} S_j < \\lambda \\\\ Z_i & \\text{ otherwise}\n\\end{array}\n\\right.\n", "167fe137153670790383cbb917cda28b": "\n\\sum_{k=0}^N a_k =1\n", "167fea3190f7f16b048b48c870d3b509": "h\\in S_A", "167ff787438d34206f480eeb06fadb80": "\\ln\\begin{vmatrix}P\\end{vmatrix}-\\ln\\begin{vmatrix}K-P\\end{vmatrix}=kt+C", "16800f198d2dc25278ad33e4f69593bd": " P=(0,0) ", "1680148e7e7eba23f2f80fbe1ce05b38": "S_n \\to \\operatorname{Aut}(S_n) \\to \\operatorname{Aut}(A_n)", "1680192943391da77061c05a12a56c18": "L_0 a = na", "16802c9cf36be2ec89e817e794ea7789": " V = \\frac{32}{27} d^2\\ h = \\frac{128}{27} r^2\\ h ", "1680cff44ed649c99fb0bb03cb462d9e": "\\vec E,\\vec B", "1680d665d12fc59d7313f1cfd5f4f9e4": "\\, n R = V P / T", "1680e216449d35e28285c3957d58b177": "s^1_1", "16812b18f6c4a43dcba3c5a35f30b398": "G\\circ F", "16813d8ec487fd81bb2935acf8a55203": "\\int_0^T |g(t,0,0)| \\, dt \\in L^2(\\Omega,\\mathcal{F}_T,\\mathbb{P})", "1681a269f207d171e1d6601426869356": "{}_0F_0(;;z) = e^z", "1681b2d3a0b2c8d2426cd46b28e1a7f8": "\\hat{x}(k+1) = A \\hat{x}(k) + L \\left(y(k) - \\hat{y}(k)\\right) - B K \\hat{x}(k)", "1681e73ee89e573a24560a097f72b1fc": "\\frac{d}{dx}\\, \\operatorname{arcosh}\\,x =\\frac{1}{\\sqrt{x^{2}-1}}", "16827544fa24ecd3208034a4ade83805": "\n\\tan \\theta_{2} = \\frac{\\cos \\chi \\cos \\eta \\sin \\lambda + \\sin \\chi \\cos \\lambda - \\cos \\chi \\sin \\eta \\cot(15^{\\circ} \\times t)}{\\sin \\eta \\sin \\lambda + \\cos \\eta \\cot(15^{\\circ} \\times t)}\n", "1682a5545014ce25d84150a004231cb2": "x^\\ast", "16834cd14b9c34e277ef1a3d4756547e": "V_2 = a_{12}V_1 \\,.", "168372b616ea823492ad1dd8eafbf548": " Re_M ", "1683a9be417c472ab92116ccd1b18b55": "=0.58\\overline{3}\\text{ t/kg}", "1683cf4bd6069d993a80978b6a729856": "\\int\\limits_0^{1}\\! \\frac{\\ln\\ln\\frac{1}{x}}{1+x+x^2}\\,dx =\n\\int\\limits_1^{\\infty}\\! \\frac{\\ln\\ln{x}}{1+x+x^2}\\,dx =\n\\frac{\\pi}{\\sqrt{3}}\\ln \\biggl\\{ \\frac{\\Gamma{(2/3)}}{\\Gamma\n{(1/3)}}\\sqrt[3]{2\\pi}\n\\biggr\\}\n", "1683f7fce37ca235a8156e67324774d4": " A^n \\rightarrow A \\otimes_B A ", "16848237c9f6badf633047324f061743": "f(x) = 1/x - b", "1684a6ada91ba7ba5e8ee8a5d3ac6353": "\\mathrm{C^{\\alpha}_{j}}", "1684c9a7919d201d3cda31c30d335c93": " (E_{d/s}) = (E_{u/p} - Z) = (6.04 - 2.0) = 4.04ft \\,\\!", "16852819d6fc5efe295ef21bcb61a228": "\\mathfrak A_P:= (\\mathcal P\\setminus\\overline{P},\\{z\\setminus\\{\\overline{P}\\} \\ | \\ P\\in z\\in\\mathcal Z\\}\n\\cup \\{E\\setminus \\overline{P} \\ | \\ E\\in {\\mathcal E}\\setminus\\{\\overline{P}_+,\\overline{P}_-\\}\\}, \\in)", "16854e1386ed8200a5a8fe7f2ba2a275": "\\left \\{ (1,1), (-1,-1), (0,-1), (-1,0), (i,-i), (-i,i), \\left(j,j^2 \\right), \\left(j^2,j \\right) \\right \\}, \\qquad j = e^{\\frac{2 \\pi i}{3}}.", "1685689005f40a5e5acb81579434af26": "b=mn+n^2,", "168596dbae3631ca956cc83c054eddae": "u_{m,n}^{(0)}", "1685a7a0109d194b75b3f235a35f3a5d": "F_N =( \\dot{m}_{air} + \\dot{m}_{fuel}) v_e - \\dot{m}_{air} v", "1685ba0f9cc9b8ad02cb300309a6baf7": "V(p,T)\\ ", "1685f9dd49c1dbca222d54b1ffeeefa4": " D_n(z)", "16868711d2f80fc5c4b56ed9721a2224": "a_{ik}", "1686cf84ce1b43f43c1ae367d8fac28b": "-jx_C = -j\\frac{1}{\\omega C_M} = \\frac{1}{j\\omega C_M}", "1686d357641f40023539db7d6003a397": "\\displaystyle{J_f(0)=|a_{1}|^2 - |a_{-1}|^2,}", "16879d73e1e2753ff48c3b9447f47c15": "C_1,D_1", "1687c45aeab91edcb6b520726a653881": "m+1 R(x_i), \\forall x_i \\in W(x_c) \\big\\}, \\\\\nR(x_c) > t_{threshold}\n\\end{align}\n", "1699cfa4c2e15c121d1bc27ebdadca58": "D_k\\!", "1699ffa7de42dc727e04687526ebf9de": "y_j = x_N", "169a4f40a5cf867f1b3fc2789aab42b1": "\\frac{\\partial c}{\\partial t} = \\nabla \\cdot \\left[ D \\nabla c - u c + \\frac{Dze}{k_B T}c(\\nabla \\phi+\\frac{\\partial \\mathbf A}{\\partial t}) \\right]", "169a62eb6001e54160963ac839db49f5": "\\cos A = \\frac {\\textrm{adjacent side}}{\\textrm{hypotenuse}} = \\frac {b}{h}\\,.", "169a70df525985ea9e8e998d6317fe86": "G * H = \\langle R_G \\cup R_H \\mid S_G \\cup S_H \\rangle.", "169a79ea33108c3e7f3c69b39e57d9e5": "\\boldsymbol\\tau={{d \\mathbf{L}}\\over {dt}}={{d(I\\boldsymbol\\omega)} \\over {dt}}=I\\boldsymbol\\alpha", "169a85994b624f4ab86ed9a8cda295f3": "I = I_\\mathrm{f} + I_\\mathrm{b} = \\iint_\\Sigma \\left(\\mathbf{J}_\\mathrm{f} + \\mathbf{J}_\\mathrm{b} \\right) \\cdot \\mathrm{d}\\mathbf{S} = \\iint_\\Sigma \\mathbf{J} \\cdot \\mathrm{d}\\mathbf{S} ", "169b2a7f5d167e8ba4c5445dc69bfa73": "\n W = C_1\\left[\\tfrac{1}{2}(I_1-3) + \\tfrac{1}{20\\lambda_m^2}(I_1^2 -9) + \\tfrac{11}{1050\\lambda_m^4}(I_1^3-27) + \\tfrac{19}{7000\\lambda_m^6}(I_1^4-81) + \\tfrac{519}{673750\\lambda_m^8}(I_1^5-243)\\right]\n ", "169b3f1f4947d42cc7dfda0da5ce654d": "~\\displaystyle r \\approx \\exp\\!\\left(-\\sqrt{8\\!~K\\!~L}~\\theta\\right)~", "169b6386685e0a08bbbda63d50f9d4b7": " A[y] = \\int_{x_1}^{x_2} \\sqrt{1 + [ y'(x) ]^2} \\, dx \\, , ", "169b79051910d72cde0b46dac09c127a": "\\theta(x,y)", "169b8c3071d1b8e0db23132d28cf265f": "xdx + ydy = 0.\\,", "169b9f22f8b5f03a6bef4823504c67f1": "\\mbox{Internal virtual work} = \\delta\\ \\mathbf{q}^T \\big( \\mathbf{k}^e \\mathbf{q} + \\mathbf{Q}^{oe} \\big) \\qquad \\mathrm{(13)}", "169c28ffb439d5b2b751a1446380e816": "\\sum_{n=1}^{\\infty}\\frac{1}{n^p},\\!", "169c3d03113bef59c25d02c8cb99afef": " 0 0 ", "169eb94d047bc07b2754f5f8d44d3c4e": "T(n)=O(n^2\\log n)", "169eca1e1b28f5d8137bc45994a46e3b": "|x|_\\ast=|x|_\\infty^\\lambda", "169efdd2169061cf846e8d497305f653": " Q(a)b=\\{a,b,a\\},\\,\\,\\, Q(a,c)b=\\{a,b,c\\},\\,\\,\\, R(a,b)c=\\{a,b,c\\}. \\, ", "169eff002b261e5a69da9c8a9680bc23": "\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx=\\frac{1}{q(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{q(x)p_{n}(x)}{x-x_{i}}dx ", "169f167cab2dff26fb4aab2d311cbceb": "C_n = \\frac{1}{n+1} {2n \\choose n} = {2n \\choose n} -\n {2n \\choose n+1}\\text{ for all }n \\geq 0.", "169f180acdd94d759304288328d61533": "x = 21(4m) + 10 = 84m + 10", "169f4bce463a537e62524b0a1493aecd": " a=\\sqrt{2} ", "169f8521200f5d0c433da5fc19ad2746": "\\scriptstyle g_1, g_2, g_3", "169fccccfac4c03483aee4997242944c": "A_{\\nu ; \\rho \\sigma} - A_{\\nu ; \\sigma \\rho} = A_{\\beta} R^{\\beta}{}_{\\nu \\rho \\sigma} \\, ,", "169fe261b465a1e705f84ae1855e4c33": "CT_{min} = \\begin{matrix}max\\\\j=1,M \\end{matrix}\\lbrace \\tau_j \\rbrace", "169fe46f9e3b26ea660113cc6e77f860": "g_{ac}", "169ff30122149f9639c095347cef88d4": "6^2 = 36 = 10", "16a01ac89474d5630319eb0fb7571e56": "\\Psi(\\mathbf{r}_1,\\mathbf{r}_2 \\cdots \\mathbf{r}_N,t)", "16a06a3cd4e00155267a56d4aa1c340d": "\\ \\frac{\\rho_2}{\\rho_1} = \\frac{1}{1-\\frac{2}{\\gamma + 1}\\left[1 - \\frac{1}{M_x^2}\\right]},", "16a089e396133b12b10421e2456f50fc": "(S \\rarr T) \\times S", "16a0e235f6503acdf412ccabe44c3d54": "d_K^{-p/2}", "16a13757a4124f665161d66e050f425e": "K_\\text{J} =\\frac{e}{\\pi \\hbar} \\,", "16a14abf647efc7e1c4ee6fa1116dc0d": "F = \\frac{A_{\\rm r}M_{\\rm u}It}{m}.", "16a154fe4cca457120af19ac743891d8": "\\chi_{1}\\,", "16a1ffec8671aa049b3d8102ea6835e6": "A=(a_{ij}) \\in \\mathbb{K}^{n \\times n}", "16a29a176abc995c7eb433e1d9295ac1": "p = \\alpha", "16a2a040bc3a10f838621c288e9743bd": "(M\\otimes N)^\\prime = M^\\prime\\otimes N^\\prime,", "16a2ba6e6c88546836698c6ca0667288": "S=\\{A_1 \\lor A_2, B_1 \\lor B_2 \\lor B_3, C_1\\}", "16a2c750fd37f68f3676f0c5d1bfc264": "\\omega_{J,g}(X,Y):=g(JX,Y) \\,", "16a30ca585e448f16cf118208f1e92b1": "\\mathbb{F} = GF(q)", "16a328a0ce8373a0d284c18a264360a6": "g(a_1, a_2)", "16a33dc956d9aec298961523a1dcb25c": "f(z_j) = z_{j+1}\\,", "16a36d421769a95e1bb9308c1fe1e361": "Y_\\alpha(z) \\sim \\begin{cases} \\frac{2}{\\pi} \\left[ \\ln (z/2) + \\gamma \\right] & \\text{if } \\alpha=0 \\\\ \\\\ -\\frac{\\Gamma(\\alpha)}{\\pi} \\left( \\frac{2}{z} \\right) ^\\alpha & \\text{if } \\alpha > 0 \\end{cases} ", "16a3878e41695554a1447e89432d47cb": "f(t,x,y)", "16a38b734abba3099415026dec1f8cc3": "d_n = \\left(\\frac{6 \\times 453.59237~\\mathrm{g}}{n \\times \\pi \\times 11.352~\\mathrm{g/cm}^3}\\right)^{\\frac{1}{3}} = 4.2416~\\mathrm{cm} \\times \\frac{1}{\\sqrt[3]{n}}", "16a3953129d392af21bd5b9da09337b5": "\\textstyle{n=\\frac{1}{2}n(n-1)},", "16a3bcecb5da65488bbab82da108cd0e": "\n\\left( \\frac{ds}{dt} \\right)^{2} = 2T\n", "16a41b243f960f02d199ac609ae53819": "p_k < 0", "16a4212dc6b7f9b139dd8ca1d8157eca": "M' \\rightarrowtail M \\twoheadrightarrow M''", "16a4640ee412133f070022418d0811f7": " \\boldsymbol{B}=\\boldsymbol{F}\\cdot\\boldsymbol{F}^T ", "16a47d7ee62bad534f75e73892c6b43c": "\\sigma = \\sigma_y", "16a48976ff99d162c02117a8970bbc1d": "X = R, G, or B", "16a50da8413e9b786c570ae4d1e9b737": "\\frac{\\partial \\ln|\\mathbf{U}|}{\\partial x} =", "16a58513eecc5e27172b60143e8d6ddb": "I_0=I_{in}\\frac{1-R}{1+R}", "16a59c6c5a0a6afe21215cf9de2f1746": "\n \\Gamma_{ijk} \n = \\tfrac{1}{2}[C_{ik,j} + C_{jk,i} - C_{ij,k}]\n = \\tfrac{1}{2}[(\\mathbf{G}_i\\cdot\\boldsymbol{C}\\cdot\\mathbf{G}_k)_{,j} + (\\mathbf{G}_j\\cdot\\boldsymbol{C}\\cdot\\mathbf{G}_k)_{,i} - (\\mathbf{G}_i\\cdot\\boldsymbol{C}\\cdot\\mathbf{G}_j)_{,k}]\n", "16a5cb2c6cba1f761c791f5a62272fd8": "V(x) > 0 \\quad \\forall x \\in U\\setminus\\{0\\}", "16a5df436ba13bdc1d9e2b63d6f57614": "\\delta \\circ \\psi = \\psi \\otimes \\psi", "16a5fc11b09e10089ab035bae5b5aa18": "\\forall x, y \\,(xRy \\to yRx)", "16a70deb1e982ed3e49f2ee8e304a9bb": "\\alpha_j=-E[S_j-T_j]", "16a73ac697d40736ec5f9ae22a421a88": "[0,\\lambda] = \\{\\alpha \\mid \\alpha \\le \\lambda\\}\\,", "16a743dc2a6ab476e2710703f7169a3e": "U(S,X_1,X_2,\\dots)", "16a79c47a5e344f74fdd77e83e5e116f": "k\\ge 7.", "16a7b8c1735a9904ec407d4bb604ae26": "1+\\sqrt{2(1+\\frac{1}{\\sqrt{5}})}", "16a7c1cc5b3bf22339897a1884bc48fd": "\\frac{d F}{d t}=\\sum_i \\frac{\\partial F(T,V,N)}{\\partial N_i} \\frac{d N_i}{d t}=\\sum_i \\mu_i \\frac{d N_i}{d t} = -VRT \\sum_r (\\ln w_r^+-\\ln w_r^-) (w_r^+-w_r^-) \\leq 0", "16a7fe571c562813c7f001b7ee55539e": " \\mathbf{J}=\n\\begin{bmatrix}\nA & 0 \\\\\nB & C \\\\\n\\end{bmatrix}\n", "16a8292f9c251f63eba17ee3a4ffb0e9": "L_{D}\\big[\\rho_{S}(t)\\big] = \\frac{1}{2}\\sum_{\\alpha,\\beta = 1}^{M}b_{\\alpha\\beta}\\big(\\big[\\mathbf{F}_{\\alpha}, \\rho_{S}(t)\\mathbf{F}^{\\dagger}_{\\beta}\\big] + \\big[\\mathbf{F}_{\\alpha}\\rho_{S}(t), \\mathbf{F}^{\\dagger}_{\\beta}\\big]\\big).", "16a8903fe77d11bb06ee8b0e74c2bf74": "{V^2}/{R}", "16a8985dcd52a6e8e7c8606fbfd999d6": "\\sum_{k=0}^{N-1} \\sin_k(i)\\equiv 0. ", "16a8b4a76f977ee32c0e6b1056793fbb": ". . .", "16a8ccafe26e5d98879c8705900e6106": " H(X|Y) = H(X,Y) - H(Y) .\\,", "16a8df07ccdcdf3110da7b9d03d20bea": "{2 \\over 3} Cr_2O_{3(s)} + {4 \\over 3} Al_{(s)} \\rightarrow {2 \\over 3} Al_2O_3 + {4 \\over 3} Cr", "16a8eca0ea4ba9714d36814c882c5d29": " \\sin(x) \\frac{d^2y}{dx^2} + 4 \\frac{dy}{dx} + y = 0 \\,, ", "16a8f4705030acb2cb92aecd5394f911": "\\eta_c = \\ell m N / L^2", "16a972d00aa8a005ce23e7cff814a943": "\n{\\hat{\\alpha}}(q, {r_{c}}) = \\max \\left \\{ \\alpha: \\ {r_{\\rm c}} \\le \\min_{u \\in \\mathcal{U}(\\alpha, \\tilde{u})} R(q,u) \\right \\} = \\max_{\\alpha \\ge 0} \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})} \\varphi(q,\\alpha,u) \\quad \\quad \\Box\n", "16a9cd46194fa8b7788705f676cf1d74": "f(x)=a + b x+ c x^2+d x^3+\\cdots", "16a9d8aef04c20112bf117547e6d9b75": "TM\\otimes\\cdots\\otimes TM \\to E", "16a9daeee032b8aa28ed65b393dd1482": "D_\\gamma(\\gamma(b)||\\gamma(a))=\\int_a^b(b-s)g_\\gamma(s)ds", "16aa0bc34e6700948541cceae21b5132": "T(*) = A", "16aa0bea3153c70928f6376f5bf2fd64": "X=f(A, B, C, \\dots)", "16aa29af1a2222aa6c4e25b8b424f4e9": "P_n=[n]P=(X_n:Z_n)", "16aac161e34573e60f6477dfbaa20035": "\n\\hat{H} = \\tfrac{1}{2}\\left[ \\frac{\\mathcal{P}^2}{I_1}+ \\mathcal{P}_z^2\\Big(\\frac{1}{I_3}\n-\\frac{1}{I_1} \\Big) \\right],\n", "16aaee0be4310bec4d0a90b4bed35e5e": "b_0 = a_0", "16ab025e32e8e64ef4df917da38fe052": "\\vec S \\cdot \\vec J = \\frac{1}{2}(J^2 + S^2 - L^2) = \\frac{\\hbar^2}{2}[j(j+1) - l(l+1) + s(s+1)],", "16ab1743af9f39327b5022d6f180289d": "f(y)\\ge f(x)", "16ab73b826991d3c50cd61d0770fc0ce": "\nP=I \\cdot V = I^2 \\cdot R = \\frac{V^2}{R} \\,\n", "16abbef98c44386e29d9fdbf34ed42dd": "h_F^{(1)}(z)", "16ac1c6d827ea95108985f89842a4bda": "1/x^2", "16ac2591ca4b5e895b0da5c506267b60": "(\\mathbf{\\lambda} x . x x x) (\\lambda x . x x x)", "16ac3d8d6743d5e4c80689495c2b7104": "r = \\frac{C}{2\\pi}.", "16ac52216bc60f5527ae62d3dc4bf60e": "\n\\ln p(\\mathbf{x}; A)\n=\n-N \\ln \\left(\\sigma \\sqrt{2\\pi}\\right)\n- \\frac{1}{2 \\sigma^2} \\sum_{n=0}^{N-1}(x[n] - A)^2\n", "16ac58e5a01f207741c6bba9fdcbde0e": "\\frac{a^x}{a-1}\\,", "16ac8405f3b110a8a9aba8ae0c9b1853": "A({\\boldsymbol \\eta})", "16acf7c0df028e5213192de2a13e142a": "\n \\begin{matrix}\n a^b & = & \\underbrace{a_{} \\times a \\times\\dots \\times a} \\\\\n & & b\\mbox{ copies of }a\n \\end{matrix}\n ", "16ad3c1d98da48e45bd03f2ff4a95085": "\n\\hat{\\mathbf{T}} = \\frac{d\\hat{\\mathbf{S}}}{dt}= -\\frac{i}{\\hbar}\\left[\\frac{\\hbar}{2}\\boldsymbol{\\sigma},\\hat{H}\\right]\n", "16ad52fbafe38fa0a8cb2104979e119e": "I=\\int_0^a \\frac{1}{z}\\,dz-\\int_0^b \\frac{1}{z}\\,dz=\\ln a-\\ln b-\\ln 0 +\\ln 0", "16ad9a091cbce331ba8b01551f76ae3b": "\\left(a, q, u\\right)\\succsim \\left(c, s, v\\right)", "16adefc4cacd3b68984078ed4156c60e": "\\scriptstyle P(x_t|s_t)", "16ae57e1f40b9d912fe5a835a276d48e": "K_i = \\frac{[A]_i}{p_A\\,[A]_{i-1}}", "16aee0aadc2eea05f6a7b5f5ff540bc9": "(\\tfrac{1}{2}\\pi,\\pi)", "16af36c55c890e104d43c814f6155c21": "O_b:\\mathfrak{H}_b\\rightarrow\\mathfrak{H}_b", "16afcd0d566abc717455f061574e3c41": " \\tau = \\sum_{i=0}^{\\infty} \\frac{t_i}{2^{i+1}} = 0.412454033640 \\ldots ", "16b001475fa4f62cd9c791a6124b1124": "\\Pi (i \\omega_n )=\\frac{1}{\\beta }\\sum _{i \\omega_m } \\frac{1}{i \\omega_m +i \\omega_n -\\epsilon }\\frac{1}{i \\omega_m -\\epsilon '}=-\\frac{n_F(\\epsilon )-n_F\\left(\\epsilon '\\right)}{i \\omega_n -\\epsilon +\\epsilon '}", "16b00f56a247758348fd7a99e067926d": "g_t(z) = \\sqrt{z^2+4t}", "16b0575a7d218647422e64a1c4ba9964": "\\hat{K}\\in L^\\infty(\\mathbf{R}^n)", "16b08e97689b511cadc975ebdf083d7e": "h(A) = h(B)", "16b0a75bfaf7b9b278dd8756ff2ff0e7": "B\\supset A", "16b0e315e31e18b689a57982fecfb00c": "S(t)\\ :=\\ (S_1(t),\\ . . .,\\ S_N(t))", "16b107be450e66526a158d467b243511": "Dep_t", "16b109dc2dec4055ae83fb11e66f7051": "\\scriptstyle{Rc<0.1}", "16b116313055e5c9191d03265ec3ef7d": "= \\frac{a(cf + ed)}{bdf} = \\frac{acf}{bdf} + \\frac{aed}{bdf} = \\frac{ac}{bd} + \\frac{ae}{bf}", "16b137c62f4d3787762c6cc432de5dee": "E\\supseteq U\\supseteq F", "16b167dd37b3ff3684efb40975ae6701": "\n\\hat{\\rho}^{[?]}=\\sum_{m}\\,\\mathcal{P}_{m}\\hat{\\rho}_{m}=\\hat{1}/D,\n", "16b1a0a1182d3bad56f6778e2d143cf8": " r\\ r = ( r\\ r \\to y ) ", "16b1f9bb7d3e8127c682b8758496704f": "A = \\frac{1}{N}\\sum_{n=1}^{N} (G_n)^2 \\times (D_1 - D_2)", "16b21a78d2151d2181f1f90ba6df20f8": "A(\\omega) \\approx \\frac{R_0}{i \\omega L}", "16b21ea5f7fa343162c6f418b5082e5b": "M_{r}=\\frac{(c-\\beta )(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}.", "16b2a3f12c31d6683042b74b8d58e1d2": " \\frac{ | -1000 | }{ | -1 | } = 1000, \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (14) ", "16b2af654956187ce493e9dcc3b2ff0c": "\\pi\\colon E\\to B", "16b335bcc0440032e49f1cfe971dc66e": " J = cl{\\{x \\in M \\mid x \\mbox{ is critical point of } f_t \\}} ", "16b367867c939c67d4bedb0ab2f1ca94": " \\left(\\mathbf{ab}\\right)\\times\\mathbf{c} = \\mathbf{a}\\left(\\mathbf{b}\\times\\mathbf{c}\\right)", "16b3afc7bbec72e53bcb3c7a63a2e809": "H_{\\nu} (\\omega)=-e^{-j {(\\nu-2)} \\frac {\\omega} {2}} P_{\\nu}( \\sin (\\frac {\\omega} {2}))", "16b3b302a249f8ae3d2e20cdc10b7879": "xy''+(c-x)y'-ay=0\\,", "16b3c9a0b5fbcd00decec1cb0da0892a": "\\,d^{(p)}", "16b40ea147efb071bdfb78e052c94881": "i\\hbar\\dfrac{\\partial\\Psi(x,t)}{\\partial t} = i\\hbar\\left(\\psi_{0}\\exp\\left(-i\\dfrac{E_{0}t}{\\hbar}\\right)\\left({c_{0}}'(t) -i\\dfrac{E_{0}}{\\hbar}c_{0}(t)\\right) + \\psi_{1}\\exp\\left(-i\\dfrac{E_{1}t}{\\hbar}\\right)\\left({c_{1}}'(t) -i\\dfrac{E_{1}}{\\hbar}c_{1}(t)\\right)\\right)", "16b410487fc05b66f5df3523c3f08a03": " K_1 = \\frac{d_1 + k_1}{a_1[W_T]} = \\frac{K_{M1}}{[W_T]}, ", "16b416f663e73abc97691dd5600fe620": "Tr(g^a)\\in GF(p^2)", "16b42e4985cdca5c4f7ece787bbae454": "x = (-1)^{\\mathrm{Sign}}(1 + M)2^{E - B}", "16b453556ed1b50bfe1a5cd950ae5d17": "\n\\begin{alignat}{2}\nf(x) & = (a-b)^2 \\\\\n& = a^2-2ab+b^2 \\\\\n\\end{alignat}\n", "16b47caeb49427975a9579db51e50f2e": "S_n \\equiv S_{n-j} \\star S_{n-k} \\pmod{m}, 0 < j < k", "16b4c5517ecbdf1c3828464885a82d50": "- {\\mathfrak{T}_{\\mu}^{\\nu}}_{, \\nu} \\, = \\, - \\Gamma^{\\sigma}_{\\mu \\nu} \\mathfrak{T}_{\\sigma}^{\\nu} \\, + \\, f_{\\mu} \\,", "16b5032415f75e28e64f8586b5d61d0e": "b_8=a_1^2a_6-a_1a_3a_4+4a_2a_6+a_2a_3^2-a_4^2", "16b528beb6f64da8799f67606f8a768b": "A\\subseteq B \\Leftrightarrow \\forall x \\mathbf{1}_{A}(x) \\le \\mathbf{1}_{B}(x). ", "16b52f771e35005c6d579b75ff63a1ad": "sI_{t+n}^{i}", "16b5acdf8eec74920ee94d415cf162cb": "(P(X), \\le^+)", "16b5ca30fac4870070aca8453b01824d": "= \\mathrm{sinc}^2(fT) \\ ", "16b60a8d377cc67d40a7ec26b28c147d": "(C-I) y_n - (C-I) y_m - y_m \\in Y_{n+1},", "16b621b10b40a1d929f7a83c1c973d22": "1 = \\frac{A t_c^2}{a x_c} = \\frac{A}{c x_c} \\Rightarrow x_c = \\frac{A}{c}.", "16b63cb8b46d5eb600da3004c2706262": "z_n= \\cos i\\,", "16b6512af8ffe586e82774107c7af515": "\\sqrt{2}=1", "16b6836995b69af42343bd625e6ee417": "\\varphi(T_i)", "16b6b5217cc0cf8ead4d0a2a4357955a": "g(n)=O(e^n)\\,", "16b6e01e6809948e032f664d1a68f753": "m = \\tfrac{1}{2}a_m + 4", "16b6e2e5771d570a49306496e72f6882": " \\int_0^\\infty f(x)x^{s-1}\\, dx, ", "16b740416a749a2b07aea2da237ceb5b": "P_{3}^{-2}(x)=\\begin{matrix}\\frac{1}{120}\\end{matrix}P_{3}^{2}(x)", "16b753081088754e99bd10dbd96ae89e": "g^{eq}=\\frac{\\rho(\\vec{e}-\\vec{u})^2}{2(2 \\pi RT)^{D/2}}e^{-\\frac{(\\vec{e}-\\vec{u})^2}{2RT}} ", "16b7b8115cf848a7368b3562f2828dd6": " \\min\\left(1;\\exp\\left(-\\beta\\cdot\\Delta E\\right)\\right)\n", "16b7dc0e56b327d68b2f5aa39f9a0d18": "\\{1,2,\\dots n\\}", "16b7f5718038ebec6fc3a8505bf597d9": "x x^{-1} = x^{-1} x = 1.", "16b820b685f5807927ba119037584c1a": "p_i \\approx x_i p_i^{\\star}", "16b8f5098448ccca054a4ed863e34348": "t_i \\cdot t_j", "16b92c61744f571a740eeb0016b63277": "A\\,\\Delta\\,B = (A \\setminus B) \\cup (B \\setminus A).", "16b930b20a815d3e71eb6c7c20defc7e": "\\operatorname{Free}_R(X) \\to M", "16b98cd85e5428c205c3f409bd4dcc63": "f_2(x)/f_3(x)\\,", "16b9e2dd529abfdc9a6cd7b2e011adcd": "\\{A_1, B_1\\} \\times \\dots \\times \\{A_n, B_n\\}", "16ba608e105aced372457c59b96e4456": "D^{\\epsilon}(\\rho^{\\otimes n}||\\sigma^{\\otimes n}) ~\\geq~ D^{\\epsilon}(\\mathcal{E}(\\rho)^{\\otimes n}||\\mathcal{E}(\\sigma)^{\\otimes n}) ~.", "16ba6cadf174457f619abfe75ae7b2a2": "\n\\begin{cases}\nx_{n+1} = 1 - y_n + |x_n|\\\\\ny_{n+1} = x_n\n\\end{cases}\n", "16ba915a0c67a7b3ad1947060149986d": "M_g", "16bab2c58031c640f123ba7a5dc306f7": "\\mbox{male shoe size (Brannock)} = 3\\times\\mbox{foot length in inches}-22", "16bb5b048bce7ede089a385a8e3e502a": "\nf_i = \\left(C_1/L_i\\right)\\left(\\lambda_i^2 - C_2p\\right)\n", "16bb85ec00042758b312da7b058174ce": " \n\\bar{q} (x,y) = (x_{\\bar{q}},y_{\\bar{q}}) = \\left( x - \\frac {\\psi_{\\bar{q}-1} \\psi_{\\bar{q}+1}}{\\psi^{2}_{\\bar{q}}}, \\frac{\\psi_{2\\bar{q}}}{2\\psi^{4}_{\\bar{q}}} \\right)\n", "16bb87dd65ac56545d278d3693f8866c": "{(X-\\mu)}^{-\\tfrac{1}{2}} \\sim\\,\\textrm{FoldedNormal}(0,1/\\sqrt{c})", "16bba2f8367123079307be02ccc7988c": "H[i]=\\operatorname{lcp}(S[A[i-1],n],S[A[i],n])", "16bba52b5b81268a6a2590bf2b9b2c7a": "\\,\\alpha", "16bbc392e34ee783d4406ca06a4b2730": "h\\ge\\lceil\\log_2(n+1)-1\\rceil\\ge \\lfloor\\log_2 n\\rfloor", "16bbea1709e8254ac73ef37344c74b23": "Iz=Ip-In\\,", "16bc082f30b70042ead601a4cf123d17": " \\ \\left(\\textbf{f}_p \\right)", "16bca97ea5fffad3a492554b4918a4b9": "\\psi(x) = \\frac{d}{dx} \\ln \\Gamma(x).", "16bd088352622766e92ca200e79fd028": "\\langle x, y\\rangle \\neq 0,", "16bd4eb8d09f5cacd1ff3ac7cdbd19cd": " \\mathit{I}_{T, v} = \\{ p \\in \\mathbf{F}[t] \\; | \\; p(T)(v) = 0 \\},", "16bd5a6921cbb097b63cc880d3130a10": "H(X)=-\\int p(x)\\log p(x)\\,dx.", "16bd72066d37fac0962c1d7f579f0781": "\n \\begin{matrix}\n a\\uparrow\\uparrow\\uparrow b= &\n \\underbrace{a_{}\\uparrow\\uparrow (a\\uparrow\\uparrow(\\dots\\uparrow\\uparrow a))}\\\\\n & b\\mbox{ multiplied copies of }a\n \\end{matrix}\n ", "16bd8d3bfa5097eb54a5dddeebb9e584": "\\frac{a^2+b^2}{p^2+q^2} = \\left(\\frac{ap+bq}{p^2+q^2}\\right)^2 + \\left(\\frac{aq-bp}{p^2+q^2}\\right)^2", "16be28e76edd78ad6bbd7febd1d6ec72": "1 \\leqslant i,j \\leqslant k", "16be4244f218bcc3f44889854c10ef85": "p(r) = \\frac {2A} {\\pi \\omega_0^2} exp(-2r^2/ \\omega_0^2)", "16be8e97adff6d66dac6cfadc2cf1260": "\n \\begin{align}\n u_1 & = \\cfrac{F_1}{4\\pi\\mu}(\\kappa+1)\\ln |x_1| + \\cfrac{F_2}{8\\mu}(\\kappa-1)\\text{sign}(x_1) \\\\\n u_2 & = \\cfrac{F_2}{4\\pi\\mu}(\\kappa+1)\\ln |x_1| +\\cfrac{F_1}{8\\mu}(\\kappa-1)\\text{sign}(x_1)\n \\end{align}\n ", "16bea4ecf3a7431bcc9a2a313d5f3f63": "w(x_1,x_2)=e^{-s x_1-z x_2}", "16beb1328c11585cf1414a50ec7b3923": "\n\\mathcal{W} = \\alpha \\mathcal{A} \\otimes \\mathcal{A} + \\beta \\, \\mathcal{B} \\otimes \\mathcal{B}\n", "16bec76a7802a76aaa404d069d9c6bff": "f(x)=1+\\frac{4x^2-8x+16}{x^3-4x^2+8x}=1+\\frac{4x^2-8x+16}{x(x^2-4x+8)}", "16bee674c12b129dcffc3f099b8767d3": "z(x_i)", "16bf78207489beac2d2c1b3305e3a46d": "e_0 < e", "16bfaca1528fced3dd873a7f8b199386": "\n\\mathrm{MER (%)} = \\sqrt{ {P_\\mathrm{error} \\over P_\\mathrm{signal}} } * 100%\n", "16bfb97e8eb1bdffc1ab0433d96b9e28": "T1'", "16bfc7f13dbd08cb7ddfd9136ec714d0": "\\,0=\\sum_{j\\ne k}|p_j|u^j-|p_k|u^k,", "16bfd20fc554545f58d30e15aeee40ca": "S\\subseteq Y\\times Z", "16bfe01ce4cffa830ba9bd47ddcde617": "1.645 > \\beta \\ge 1.28", "16c0633d8e14b1ef64266f73e020087e": " \\frac{1}{\\sqrt{f}}= -2 \\log_{10} \\left( \\frac { \\varepsilon}\n{3.7 D_\\mathrm{h}} + \\frac {2.51} {\\mathrm{Re} \\sqrt{f}} \\right)", "16c06809b1abd50dccc9b0b4452fabf9": "z = 0.5", "16c095dc7b230e3acc7413cc95dcb161": " x_i \\rightarrow c ", "16c0ce42e9ce29ac2677a5b3ce87f60c": "I_n^f", "16c0d8acf8b9179918431e74424ca42c": "P(x) \\uparrow (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\uparrow Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "16c10841d7a5a537e6cf660c4e6fb148": "q_k < 0", "16c1f38a909314fbe3ae79c748e64131": " c=\\sqrt{\\frac{f}{\\rho}} ", "16c22f65f46040a73b02c0c586f8c06f": "g^{-1} (0) = \\{ x \\in U \\mid g(x) = 0 \\in Y \\} \\subseteq U.", "16c25137caeae53bb0457c856bdbfc29": "\n\\begin{array}{lcl}\nE(k) & \\propto & k_{\\perp}^{-5/3}; \\\\\nk_{||} & \\propto & k_{\\perp}^{2/3} \n\\end{array}\n", "16c2aef1bc0d558423d3cac2813d18e8": "A=\\frac{1}{2} r^2 \\theta.", "16c2d41dc546deefd4884444e211e0fc": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{5}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "16c2d9b8074678da923688e5bd5ac573": "2^{2^n} + 1", "16c320e56daf3de5e759b678e4d933a2": "\\int_{P_1}^{P_2}\\frac{\\mathrm{d}P}{P} = \\frac {L}{R} \\int \\frac {\\mathrm{d} T}{T^2}", "16c33b2aea95d7a05d44e3953b7b7339": " E_0 = m_0 c^2 \\,\\!", "16c3684e2506c2b41e33e0fbfa64b664": " \\forall x \\exist r Fxr \\,.", "16c38032f3060fd48736eb5585960948": "\\sin \\theta = \\pm \\frac{\\sqrt{a^2 + b^2 + c^2}}{k}", "16c39a87a1212841ba3b7282866e88d7": "\\vec{C}_4", "16c3b2d0ab41db51c39adbdb28a649b5": " an^2+bn", "16c3da2bfdfd7c4ccd7dbc71bc8586b8": "P_1[b/a]", "16c42a5ee061a1af911bca5732810c65": "f : \\bigoplus_{i \\in I} M_i \\rightarrow M", "16c440089ff8c668d9424255742f6189": "d = f + p", "16c4ea3612cb5ea98bacc490e87c400c": " \\varphi \\left ( \\mathbf{x} \\right ) = \\sum_{i=1}^N \\left ( a_i + \\mathbf{b}_i \\cdot \\left ( \\mathbf{x} - \\mathbf{c}_i \\right ) \\right )\\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) ", "16c514d35bebb256433423143ddad593": "\\textstyle{1\\over r^2}", "16c51c36290d4b8798f7a3fb20911d0b": "\\mathfrak{sl}_2(\\mathbb{C}).", "16c530bab59fbeb92f66984ec9e48833": "\n\\left( r^{2} - a^{2} + c^{2} + x^{2} + y^{2} \\right)^{2} = 4r^{2} \\left(x^{2} + c^{2} \\right)\n", "16c5397e29ad3811cfe1e43e07f74da6": "n^{-1} U \\left(z, \\hat\\zeta \\right) = n^{-1} \\sum U \\left(z_j, \\hat\\zeta \\right) = 0", "16c562902f30ec739882c634cd4dc131": "\\scriptstyle \\eta ", "16c5a49a504194bc0e18af85aededdbf": "\\sum_{k=m}^n ar^k=\\frac{a(r^m-r^{n+1})}{1-r}.", "16c5c5ce108d0c7535c7dcf10da2cd84": "g_\\mathrm{rev}\\,", "16c614385de625bdf9326591e0669e95": "G_0^{-1}", "16c632ecacdee7be03cc2217008c3049": "\\omega_1(t):=\\sup_{s\\leq t}\\omega(s)", "16c65d5e3b8286ef18f318bebbc942ff": "(2\\pi/V)^d", "16c6abcd001c113b16487e33f6d7163c": "q\\ ", "16c6bb1e7ff25c5260b286008575c27b": " p = uq + r,\\ ", "16c6bb7496a4e3b1d19d3f0704df42df": "\\begin{cases}x = e^\\rho\\cos\\theta, \\\\ y = e^\\rho\\sin\\theta.\\end{cases}", "16c6bfd6b23c6f45049add0389959756": "T_R=0.9", "16c6c5b94de8822436356c7b733fc01f": "90 = 2 \\times 3^2 \\times 5^1", "16c6eda5a517e1f144a37ccc79ff988e": "-1+Y/\\overline{Y} = c(\\overline{u}-u).", "16c7232f28b44c2de2125326950fa11c": "\\Im(z)-\\Re L_t(w,w)\\in V \\, ", "16c797230fdbc4ea01ae6620b45d0080": "{\\mathbf y}\\sqrt{\\nu/u}={\\mathbf x}-{\\boldsymbol\\mu}", "16c7a057cab1116577b57586002d37e3": "1 < p < \\infty", "16c7d1ac82d31e2b9f4d815f5279a36e": "\\mathbf{i}W \\leftrightarrow W,", "16c7ff98dcdb446c88e9f3d97e6d02f0": "T(\\lambda v) = \\lambda^\\theta T(v),", "16c816575e95a5f2cc028b8f92420f96": " [ a , b ] ", "16c819219f4f98fbc50492d8abfbef42": "\\bar{\\tau}_s = 2.1s, ", "16c81eae0a8e9c12b072c9f5dccbb4ff": " \\frac{V^2}{R} = -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial n}", "16c84a38fa01744df661397d5854e640": "\\mathbf{X} ", "16c84bc40861e9eb993a10e7dc2f3a10": "{\\rm unless}\\quad x = -1.", "16c8af825c013e66ce7a1e627603d508": "d(x,y)>r", "16c8f1c6563a76f46f8afe0fbbf5c843": "1 \\cdot \\mathbf{0}=\\mathbf{0}", "16c94e555cfeb5d52ae72c4432eddb03": "P = V+E+F+C \\,\\!", "16c966bb9d8ce4d1c75a475dd7530a93": "t_3 ", "16ca9e4dd057aab73be1cf835757f8f2": "M_{2}=(\\frac{\\frac{2}{\\gamma -1} + {M_{1}}^{2}}{\\frac{2\\gamma}{\\gamma-1}{M_{1}}^{2} - 1})^{0.5}", "16cad9ed5a1fd86e4fc3a258234e05ad": "\n\\frac{L + 1.25 \\times \\sqrt{S} - 9.8 \\times \\sqrt[3]{DSP}}{0.686} \\leq 24.000 \\, metres\n", "16caf13f5b9a7080370ebef99784919d": "\\mathit{K}_a\\varphi", "16cb48aae3ae2294910eae05f66494ac": "f_\\mathbb{H}", "16cb9bebe1747819ee672898705388b3": "mP = 0", "16cbab002642812cd0809c0d17337c32": "1 \\over 10^6", "16cbad6b5e2cfb980807cef70cdcfd80": "z = -2", "16cbc5bf48660cfe5af2ad29f8b80020": "W^\\ast", "16cbca6493b4098798e6251eb7eb46a2": " O(A|B,C) = \\Lambda(A|B \\cap C) \\cdot \\Lambda(B|A) \\cdot O(A).", "16cc133003f237995ca05ccd1c029bce": "S \\times N", "16cc4117aa11d6fadfe295ec43fda2ec": "k r\\ll 1", "16cc58311e2f72f8c096bc28c6286603": "\\mathbf e_{m+1}=\\sum_i^m\\ c_i\\mathbf e_i.", "16cc6a9cf846e8519da532cde0bdae6a": "\\{X_i(\\omega)\\}_{i\\in\\mathbb{N}}", "16ccf71b13078607a3bc5cd8dc0e7330": "1-e^{-\\hbar\\omega_\\alpha/k_BT} \\approx \\hbar\\omega_\\alpha/k_BT \\, ", "16cd175b544caf24acd4ffb5ee5c0b7a": " S_{M_j} \\subsetneq \\cup_{k \\in S} S_{M_k}", "16cd3f0aef1db407d98b9f8b3be95743": "w[n] = \\ w_0\\left(n-\\frac{N-1}{2}\\right),\\ 0\\le n \\le N-1.", "16cd68cdabaee2bc69fc456e12d83fd6": "BG/BH", "16cd7a00e1ad8c96bffaef367609ee61": "\\mathbf {X} \\boldsymbol {\\beta} = \\mathbf {y},", "16cd8cd85e30de3f2180a0ab1c1048b3": "P(\\mathbf{x},\\mathbf{t}\\mid \\mathbf{x_0})", "16cd97a7898ef945dda8e8e678d652c3": "\\| \\cdots \\|", "16cdbf9053a8e8ccf2cb8edaf7b6083a": "dl^2=t^{2p_1}dx^2+t^{2p_2}dy^2+t^{2p_3}dz^2", "16ce356a35b66ac10b64ac243ab6b4c9": "v_{R_1}=v\\frac{R_1}{R_1 + R_2} \\iff i_{G_1}=i\\frac{G_1}{G_1 + G_2}", "16ce429f1f4fb2b3c8916043cd87c18e": "\\mathbf y(x) = U(x)\\mathbf z(x)", "16ce5d3553abbb977981758212d402a8": "v^{*}:= \\max_{d\\in D,\\,z\\in \\mathbb{R}} \\{z: z \\le f(d,s),\\forall s\\in S(d)\\}", "16ce7039ef7ef21de0aa7aab84fc39bc": "4 \\Delta x", "16ce72b18f657df6f5805128affa3798": "\nC(\\{x\\})=\\{x\\}\\cap X\n", "16ce959d2748e1a3109f216b62ec7b21": "x\\frac{d^2g}{dx^2} + \\Big( (l+\\frac{1}{2}) + 1 - x\\Big) \\frac{dg}{dx} + \\frac{1}{2}(n-l) g(x) = 0", "16ced14ad381d0c55f46a750367db2ed": "f(i)", "16ced8c81e9b88644007c41e5b59e871": "\n\\left.\\begin{align}\n|1, 1\\rangle &=\\; \\uparrow\\uparrow\\\\\n|1, 0\\rangle &=\\; (\\uparrow\\downarrow + \\downarrow\\uparrow)/\\sqrt2\\\\\n|1,-1\\rangle &=\\; \\downarrow\\downarrow\n\\end{align}\\;\\right\\}\\quad s=1\\quad\\mathrm{(triplet)}\n", "16cf0a9a371e7164ea3b6340f91368f5": "\\beta_\\mathrm{Darlington} = \\beta_1 \\cdot \\beta_2 + \\beta_1 + \\beta_2 + 1", "16cf15177829b95493295d847b2b4ff6": "k=1,2,\\dots,b", "16cf1fd6fa42533790803c6a67756d27": "_k\\mathbf{E}_{l,m,n}=_k\\left[E_1,E_2,\\ldots,E_{11},E_{12}\\right]^T_{l,m,n}", "16cf3f93acb1340b11e29743e93d30e5": "\\textstyle P_{X_r} = P", "16cf641f02aa0faa1420b417a3285027": "p(\\tilde{x}|\\mathbf{X},\\alpha) = \\int_{\\theta} p(\\tilde{x}|\\theta) p(\\theta|\\mathbf{X},\\alpha) \\operatorname{d}\\!\\theta", "16cfbfe13e6ad2c3af1781267ebe899d": "e^z = {}_1F_1(1;1;z)", "16cfc8da079c902590e0f61f38c7258e": "M(f(z), 1)", "16cff04968eb771b3f52a6813c1627ea": "20*log_{10}(2)", "16d00a9e0d3adae598b69752f3ff8582": "\n\\{f, g\\}_{DB} = \\{f, g\\}_{PB} + \\frac{c\\epsilon_{ab}}{q B} \\{f, \\phi_a\\}_{PB}\\{\\phi_b, g\\}_{PB}.\n", "16d026ccb78f6c6416731bdd70e8dfe6": "f(\\mathbf{0}_{V}) = f(0 \\cdot \\mathbf{0}_{V}) = 0 \\cdot f(\\mathbf{0}_{V}) = \\mathbf{0}_{W} .", "16d047303bd2cfb65e57195a14cbae77": "(1,2,2)_H", "16d06cc014affbbee42fcb1a9fd6f1a9": " \\sin(\\alpha)\\simeq \\tan(\\alpha)=y/f", "16d08b744ced0e4ca300155818c83561": "(x,a) \\sim (x,b)\\text{ if }x \\neq 0.\\;", "16d09d70ab577692fdd225f531f12bbb": "P_c^p(z_0)", "16d0bbb234cf39043edbf27486b18997": "\\{v_k\\}", "16d0c35deb3ef8acc4e3e5a1f6beab23": "\\hat{A}_{n,\\alpha}^{\\dagger}", "16d1cef20c82a52818efbfb619dc03ba": "B((x_0, x_1, \\ldots, x_n), (y_0, y_1, \\ldots, y_n)) = x_0y_0 - x_1 y_1 - \\ldots - x_n y_n", "16d25daca7eddc4c0d1820e886268c6a": "S^2\\left|s,m_s\\right\\rangle=\\hbar^2 s(s+1) \\left|s,m_s\\right\\rangle,", "16d2b578b3ae98e4a1a5cf6bf8fab1b7": "\\jmath^2 = +1", "16d2d225f3edaf4ebcdcb1beed2792d1": "\nD = \\left(\\frac{r_{ij}}{2a_{ij}}\\right)^{2}, a_{ij} = \\sqrt{a_{i}a_{j}}\n", "16d2f645b9b6da3e8f2f8c1c40a5201d": "\\ln(-\\ln(1-\\hat F(x)))", "16d3189c358fed34f9c25dd51b228f95": " \\mathit{M}_\\mathit{s} \\approx \\frac{\\mathit{m}_\\mathit{p} {\\mathit{t}_\\mathit{rec}}^{3/2}}{\\sqrt{\\mathit{n}_\\mathit{rec} \\sigma^3}}.", "16d3241486079b87b3ddeb0cb9108e03": "h(t,s) = 0\\text{ for }t < s \\, ", "16d3248badead42c6d5054284059e657": " t = cd^2 \\log n ", "16d3592dade3799b8728c37b4e79a510": "D\\sigma-\\delta\\kappa=(\\rho+\\bar{\\rho})\\sigma+(3\\varepsilon-\\bar{\\varepsilon})\\sigma-(\\tau-\\bar{\\pi}+\\bar{\\alpha}+3\\beta)\\kappa+\\Psi_0\\,,", "16d36a5a2201ef89ad7622f5168903a7": "\\left[f\\right]_{\\lambda,p}^p = \\sup_{0 < r< \\operatorname{diam} (\\Omega), x_0 \\in \\Omega} \\frac{1}{r^\\lambda} \\int_{B_r(x_0) \\cap \\Omega} | u(y) |^p dy. ", "16d391f93af773b247c772b0db9e1936": "f(\\alpha x) = \\alpha^k f(x) \\, ", "16d3b59240171479a3689dc146641feb": "\\mathbf{F}\\left(\\mathbf{r}\\right)=\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\delta^3\\left(\\mathbf{r}-\\mathbf{r}'\\right)\\mathrm{d}V'=\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\left(-\\frac{1}{4\\pi}\\nabla^{2}\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\right)\\mathrm{d}V'=-\\frac{1}{4\\pi}\\nabla^{2}\\int_{V}\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'.", "16d3e582572fbaf62853f54006985d94": "{-\\rho_w u_w \\delta y_i\\left(\\frac{\\delta x_i}{2} \\right) \\left( \\frac{\\partial \\phi}{\\partial x}\\right)_{wk}}", "16d4086f98d090eb5916a1f19b61c693": "w(n)=\\operatorname{haversin}\\left(\\frac {2 \\pi n} {N-1} \\right).", "16d46bf95c662acf0f2808cd47d0415d": " e^{-ip q(t)} \\,", "16d48e327ca779bbb517988365846d3f": "\\sin A = {\\cos A \\over \\cot A} ", "16d49477f1f74c07a9f0cbbb92cff43b": " \\sum_{i=1}^{n}\\|\\mathbf{x}_i - L_{k}\\mathbf{z}_i\\|^2 ", "16d49c2cc1f97016bc80ab5bbccfa77b": "{S^a}_m \\, {S^m}_b", "16d4b22a327f67dd9b126ce50247aba3": "\\frac{\\partial^0}{\\partial x_1^0}f(x_1, x_2, \\ldots, x_n) = f(x_1, x_2, \\ldots, x_n)\\,,\\quad \\ldots \\frac{\\partial^0}{\\partial x_n^0}f(x_1, x_2, \\ldots, x_n)=f(x_1, x_2, \\ldots, x_n)\\,. ", "16d4c7f4ae9d09b7c75ded886df2de20": "\\sin^n\\theta = \\frac{1}{2^n} \\binom{n}{\\frac{n}{2}} + \\frac{2}{2^n} \\sum_{k=0}^{\\frac{n}{2}-1} (-1)^{(\\frac{n}{2}-k)} \\binom{n}{k} \\cos{((n-2k)\\theta)}", "16d52aa7c414a1dcad1b1b9d58b94373": " (x_1, x_2, x_3) ", "16d5b899df76ddaa49e8467f34002312": " \\phi_1, \\ \\phi_2, \\ ..., \\ \\phi_n \\vdash \\chi \\rightarrow \\psi ", "16d5d16a26e37974d2ede4292bb447ff": "M\\colon \\mathbb{F}_q^n \\to \\mathbb{F}_q^n", "16d5ff783adb2df2c9f0e7cad49180b0": "\\left\\| I\\right\\| =\\left\\| \n\\begin{array}{ll}\n0 & -E_{n} \\\\ \nE_{n} & 0\n\\end{array}\n\\right\\|,", "16d6290e33f078f6e182829160b5a62c": "|\\lambda|=1", "16d674308ddf8c15053f06bb20b29657": "p_1^2=p_2 p_3", "16d6a13697e6af8dcf74bbc529075a6a": "H_2(X)", "16d6b20cd52ff68fcee0419fba7ca7cb": "(L,\\ast,1)", "16d6d75d3b023c6133ac940926e3e02e": "F_{max} = k \\frac{TLt^{2}}{W}", "16d6eb81aa7417a3cccd1348b5a52ad7": "S^a_0(t)", "16d74a44e587237c2c7e158587bcd5f8": " x = d(R)\\,", "16d7b420f88c417093b1622a3baef0e8": "d = -3(1 - u^2 - v^2).\\ ", "16d7c280105f6458bfa692e85e420992": "a_k \\approx \\frac{2}{N} \\left[ \\frac{f(1)}{2} + \\frac{f(-1)}{2} (-1)^k + \\sum_{n=1}^{N-1} f(\\cos[n\\pi/N]) \\cos(n k \\pi/N) \\right]", "16d7ede6a8d32bcca638bca667fb6f3e": "T^4(\\Omega)<\\Omega", "16d7f69fcfa098fd011c9927ccede0d3": "W = C_X A^T(AC_XA^T + C_Z)^{-1} .", "16d8089ebfd3c8fcc39607cf7aaf34bc": " X_{f} ", "16d812c99b674354df99942714a8deed": " \\textstyle \\|f_n\\|_2 = 1/\\sqrt2 ", "16d86342eae7ebdfbb060ae1fc3a447a": "\n \\begin{pmatrix}\n c t' \\\\\n x'\n \\end{pmatrix}\n =\n \\begin{pmatrix}\n \\cosh \\varphi & - \\sinh \\varphi \\\\\n - \\sinh \\varphi & \\cosh \\varphi\n \\end{pmatrix}\n \\begin{pmatrix}\n ct \\\\\n x\n \\end{pmatrix}\n = \\mathbf \\Lambda (\\varphi) \\mathbf v", "16d8b331df457e39904ae63934398482": "y^T \\bar y = \\hat y^T \\bar y", "16d8b625c3dea1c75661f6bdbbbdfecd": "\n \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + (2\\mu + \\lambda)~\\nabla(\\nabla\\cdot\\mathbf{v})\n ", "16d8d405125edbd84f6beb9fbaff2e8c": "\nv(S) = \n\\begin{cases} \n 1, & \\text{if }S \\in \\left\\{ \\{1,3\\},\\{2,3\\},\\{1,2,3\\} \\right\\}\\\\\n 0, & \\text{otherwise}\\\\\n\\end{cases}\n", "16d8e05e6746c39e1916e622171b12cf": "s_n = \\sum_{k=1}^n (-1)^k k^{1/k}", "16d93961f98af9a8ad087f19f642382f": "(t^c, t^{c+1}, \\dots) A", "16da44421aff5a0a969eb42335a5fc2a": "|(kg)^{-1} (kh)| = |g^{-1} h|", "16da681349f9f28dc3c9f1015ded25fd": "W_0 \\supset W_1 \\supset \\cdots", "16da6d0d6340c3bd7d98b49ef0a6e62d": "\\mathbf{T}(x)", "16db377156b6a727777f391bcbe853c0": "q^2", "16dbdb98a6f3a59f16e0165daba18bc9": "E_\\mu (z)", "16dbf435db489857f73807b1613a58b5": "(X_t)_{t \\ge 0}", "16dc0ab269f1279e3139917e09ac64c3": "a^{a^a}", "16dc1b450aea553f60a08da05b06a5de": "\\varphi=\\chi+\\sum_{j=1}^{3}\\delta_j\\sin\\left(2j\\chi\\right),", "16dc30a0f7fc9d761795ff8e1ce46dec": " r=\\int_c^1\\frac{dt}{\\sqrt{1-t^4}}.", "16dc71817d684313d2808922474e5fc0": "\\alpha_k\\geq0,\\qquad \\lim_{k\\to\\infty} \\alpha_k = 0,\\qquad \\sum_{k=1}^\\infty \\alpha_k = \\infty.", "16dcf6b8256aa0b2cbb7ced79422d780": " H \\neq 0 ", "16dd1fb150b2312c866ea25a1ee939ac": "f = { \\omega \\over 2 \\pi } = {1 \\over {2 \\pi \\sqrt{LC}}}. ", "16dd62904156bb5869755452c2edade7": "\\hat p ", "16ddd25ddf48cc75919754c354efe41e": "(m \\ll p)", "16dea1a61568e2220095d41129d5047c": "\\langle \\hat{B}^\\dagger_\\omega P_\\mathbf{k} \\rangle", "16dea37ca8eb32d1274fbdc55c708aaf": "L^1(G//K)", "16decd582e3cf5e0ffb9ff53d28bd46f": "{\\mathbf{}}\\tau(t)= \\hat{P}(t) \\hat{S}(t) \\left( \\hat{P}(t) \\hat{S}(t) \\right)^*.", "16df41e97ee4fcb81fc3b1d87ab090ab": "\n\\mathbf{g}=\n\\begin{pmatrix}\nI_1 \\sin^2\\beta \\cos^2\\gamma+I_2\\sin^2\\beta\\sin^2\\gamma+I_3\\cos^2\\beta &\n(I_2-I_1) \\sin\\beta\\sin\\gamma\\cos\\gamma &\nI_3\\cos\\beta \\\\\n(I_2-I_1) \\sin\\beta\\sin\\gamma\\cos\\gamma &\nI_1\\sin^2\\gamma+I_2\\cos^2\\gamma & 0 \\\\\nI_3\\cos\\beta & 0 & I_3 \\\\\n\\end{pmatrix}.\n", "16df6eb35437d0639fae2f37a25f7381": " {\\nabla p^{'} } ", "16df91f1e4636f8d9181e17bb3715c41": "\\mathbb{Z}[x]/(f(x)). \\, ", "16dfa009717d98ef88b93627b8191b36": "\\Box\\equiv\\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2}-\\nabla^2,", "16dfa21a75a4faafea5db0b270d156fe": "\\mu(A\\cup B) =\\mu(A) + \\mu (B).", "16dfadadc5a5736ec648f5bf8028f9b8": "b -1 \\le k \\le \\left\\lfloor\\frac{3}{2}b\\right\\rfloor -1.", "16dfb5e1f0da0ba1d0e38e093641c931": "E_n = h \\left( n + {1 \\over 2 } \\right)\\nu=h\\left( n + {1 \\over 2 } \\right) {1\\over {2 \\pi}} \\sqrt{k \\over m} \\!", "16dfb8d8f0531851655e1963ef02ea2c": "\n\\left\\langle\\frac{\\partial f}{\\partial t},\\frac{\\partial f}{\\partial s}\\right\\rangle(0,1) = \\left\\langle\\frac{\\partial f}{\\partial t},\\frac{\\partial f}{\\partial s}\\right\\rangle(0,0) = 0,\n", "16dfe91a7e716214af81a7bcbef1f9af": "\\mu_0,", "16e011e0c27fdc2bbc480fc599f4afca": "f(z_1,z_2, \\ldots, z_n)", "16e02e3f04dc7b857b556698d397cd3c": "\n D_\\mathrm{KL}(p \\parallel q) = \\int p(x)\\ln\\left( \\frac{p(x)}{q(x)}\\right) dx\n ", "16e03d81973cf9ae23f14b23e2f39503": "x'(0) = -A \\sin 0 + B \\cos 0 = B = 0, \\,", "16e03e3b5680c4c26db5976f63c04ba7": "\\R^2 ", "16e047525735026ff8c1cc2f52c49d2d": "\\begin{bmatrix}M\\end{bmatrix},", "16e05d9b06eeb6385b244b7398081b02": "B(v)=\\left(\\frac{n-1}{n}\\right)v", "16e0d1724ff9cfbc25549fa6d34d09d7": "\\begin{align}\n\\mathbf{A}_x &= R\\cos(\\phi) \\\\\n\\mathbf{A}_y &= R\\sin(\\phi) \\\\\n\\mathbf{B}_x &= \\mathbf{A}_x \\\\\n\\mathbf{B}_y &= -\\mathbf{A}_y\n\\end{align}", "16e0e09fcf9ba3c01106d49adc74c19e": "\n\\|y\\|_A=\\sup_{x\\in A}|\\langle x,y\\rangle|,\\qquad A\\in{\\mathcal A},\n", "16e12ca977fa384d8c0a8745aa1e9654": "\n\\left( -\\frac{1}{2} \\nabla^2 + V(\\textbf{x}) \\right) \\psi_{-} = E_{-} \\psi_{-}\n", "16e13e459e99351da5a22f98ef69963b": "\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu\\gamma^\\rho\\gamma^\\sigma\\gamma^5) \\,", "16e17b3f9b70bea48a9e89de7c4a9665": "\\hat{d}=-{3\\over4} \\ln({1-{4\\over3}p})=\\hat{\\nu}", "16e1b9ba3bc0ea1125a2b713016b6922": "f(x) = \\sum_{i=1}^n \\pi_if_i(x), -\\infty < x < \\infty ,", "16e1c4a560a400caf7a4e84a5214e11d": "|z|_p1\\quad \\text{(supercritical flow)}", "16e26bfa366568987cf4db796f8638da": " \\forall A \\in \\mathcal{A} : \\Pr[A] \\leq (1 - \\varepsilon) x(A) \\prod\\nolimits_{B \\in \\Gamma(A)} (1-x(B)) ", "16e2ff425646e5ddc2f1a535ee02344f": " \\sigma(x) := x^p + p\\delta(x) ", "16e310a1ed7e0079ae7f0d1a37fcd43b": "{H}^{(n)}_{ij} := \\langle\\psi_{in}|\\hat{H}_{\\text{JC}}|\\psi_{jn}\\rangle,", "16e35d16063cd69632515f00ed9fe3f1": " \\mathbf{L}^0 = \\mathbf{L}, \\quad \\mathbf{L}^{i+1} = [\\mathbf{L}, \\mathbf{L}^i]\\ ", "16e3790df660a79605a6bdf740d269d0": "x \\leq y \\iff g(x) \\leq g(y)", "16e382ee9c0a96925397b433935c3a01": "\\tau_1 = C (R_1 + R_2)", "16e3959b07a211413a1792253f70d03b": "\\mathcal{H}=\\mathcal{H}_{1}\\otimes\\mathcal{H}_{2}\\otimes\\ldots\\otimes\\mathcal{H}_{n}", "16e3b1c0636921d75c950eb3310952f3": "\\bigcup_n f^{-n}(z)", "16e3c0ac16cd77c7dc8f997e25ea64d0": "H_n = \\sum_{k = 1}^n \\frac{1}{k},\\!", "16e3d07447da62c38c3d77771099b56d": "\n[\\wp'(z)]^2=4[\\wp(z)]^3-g_2\\wp(z)-g_3,", "16e49b4da73008464b35f6203f8942ce": " B=\\begin{pmatrix}\n\\lambda_1 & 0 & \\cdots & 0\\\\\n0 & \\lambda_2 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\ddots & 0\\\\\n0 & 0 & \\cdots & \\lambda_n\n\\end{pmatrix}", "16e49c15023b5a27ed9208a42f6c70a6": "\\boldsymbol{F_{\\dot \\theta}} = -\\boldsymbol {\\omega \\times}\\left(\\boldsymbol{\\omega \\times r }\\right)\\ ,", "16e4c0e1ecb8055b5e485411345bcf11": "D \\zeta = k_B T", "16e4eb6dcceb237abc7b6ca6a4e88ad4": "\\sin(\\alpha \\pm \\beta) = \\sqrt{\\sin^2 \\alpha - (\\sin \\alpha \\sin \\beta)^2} \\pm \\sqrt{\\sin^2 \\beta- (\\sin \\alpha\\sin \\beta)^2}", "16e4f58d4fb62a8a513367812d04dfae": "\nR_\\mathrm{Th} = R_1 + \\left[ \\left( R_2 + R_3 \\right) \\| R_4 \\right]\n", "16e4f97b0940534df8715707fc9812f6": "\n\\left( \\frac{du}{d\\varphi} \\right)^{2} = \\frac{1}{b^{2}} - \\left( 1 - u r_{s} \\right) \\left( \\frac{1}{a^{2}} + u^{2} \\right)\n", "16e52377ecc0439f0b23f5cc76b486b8": "z,x,y,u", "16e53a8a0bbef14e25e7ca0b905e5aef": " \\begin{align}\nk_i^j = \\delta_{ij}+P_{i,i-1}k_{i-1}^j + P_{i,i}k_{i}^j + P_{i,i+1}k_{i+1}^j\n\\end{align}", "16e5df79200ad41c38172c97487f4658": "\\dot{u}_{n+1}=\\dot{u}_{n}+(1 - \\gamma){\\Delta}t~\\ddot{u}_n + \\gamma {\\Delta}t~\\ddot{u}_{n+1}.", "16e5f3f88bbe946adb9b8c8a27ee69f5": "(1 \\; 9 \\; 13 \\; 8 \\; 11)", "16e5f63cd62f3d302473718e0268a1ab": "\\scriptstyle{|n\\rangle}", "16e65f972992070177a4a55062a0a576": "\\frac{\\Gamma(n+k)}{k!\\Gamma(n)} = \\binom{n+k-1}k", "16e68d4ccb1adf12c2bef442253896b9": "\\sigma_{P}^{2}=\\mathbb{E}\\left[\\sum_{i=1}^{n}x_{i}^{2}(R_{i}-\\mathbb{E}[R_{i}])^{2}+\\sum_{i=1}^{n}\\sum_{j=1,i\\neq j}^{n}x_{i}x_{j}(R_{i}-\\mathbb{E}[R_{i}])(R_{j}-\\mathbb{E}[R_{j}])\\right]", "16e6ed33d9ac773f84170a78ddd62195": "2^{4n}-1, 3^{4n}-2^{4n},\\dots,(4n)^{4n}-(4n-1)^{4n}", "16e6f8d793c8cd5ce99a43c14b824d77": "\\hat\\zeta", "16e726c7058178bad641a36d6b3b19b3": "+S_x \\otimes S_x", "16e74e6424378befa9518a89eab147cf": "u(r)\\ \\stackrel{\\mathrm{def}}{=}\\ rR(r)", "16e75f3245ddbed593b5d02616091410": " G/F", "16e785725a7220b7a85526e174790102": "G=(V, E)", "16e79aa2f3c5754429354a0d3ca19f79": "P_i^2 = 1, i=1,2,...,2k", "16e7ace3e532a8caf680ee9be480a708": "\\scriptstyle \\lambda/4", "16e7da158482ea8feb9445179d63c0d8": "\\mathbf{u}(t) = -K \\mathbf{y}(t) + \\mathbf{r}(t)", "16e7e5453849788209d61a0081c7bab1": "\\displaystyle\\partial ^{2}W/\\partial i_{m}\\partial i_{n}=\\partial ^{2}W/\\partial i_{n}\\partial i_{m}", "16e901f7a9cd9c3b2ffb79e2eeac3ce4": "0\\leq \\theta \\leq 2\\pi", "16e90f744846824ef4a53e1c666aa71b": "0< \\alpha_{01} < \\alpha_{02} < \\cdots", "16e9236ce5c9dc1b390c452daaf7817a": "\nC_{1} = - \\frac{m k^{2}}{2 \\hbar^{2}} H^{-1} - I ~,\n", "16e92a4edb6df477c18ba6024decaf45": "p_2 = \\Delta x , q_2 = x_{\\text{max}} - x_0\\,\\!", "16e94be175a986910fc60aa7a2a2ac28": "\\operatorname{var}(\\mathbf{b}^{\\rm T}\\mathbf{X}) = \\mathbf{b}^{\\rm T} \\operatorname{var}(\\mathbf{X}) \\mathbf{b},\\,", "16e9715b466d97e5bb135373cdb9125b": "\\left(3^3\\right)^3=27^3=19683.", "16e9a29195756bdd601e03d359a8f85f": "\n\\begin{align}\n& P_{0}(x) = 1 \\\\\n& P_{n}(x)=\\frac{1}{2^n n!}\\ \\frac{d^n(x^2-1)^n}{dx^n} \\quad n \\ge 1 \\\\\n\\end{align}\n", "16e9c3f8d9aceb5f85b77d80341ce643": "a_{ij} = \\sgn(r_j-r_i) ", "16ea0c3e6859c3d0b32be0ae9e56d506": "T(n) = T(n-1) + (n-1)T(n-2),", "16ea1b135d8b4465523d38135e0f98f2": "A_{i+1} = V_i^R||W_i^L", "16ea29d4ffa526805e62b698e00a3456": "N = E_s-A_s + E_m-A_m + E_x-A_x + E_h-A_h\\,", "16ea3b59c0fd578220c90beba8a2d37b": "\\mathbf{F} = I \\boldsymbol{\\ell} \\times \\mathbf{B} \\,\\!", "16ea4948eb2b1b482baab59de49c6297": "\\sum_i \\rho_i v_i=0", "16ea531aa0702f01b1f02b8be48b8437": " (m_{1}+m_{2})T^2 = \\frac{4\\pi^2 (a''/p'')^3}{G} ", "16eb00fe0ecb2a0c45e9e3f73910cd8e": "\\sum_i dp_i \\wedge dq_i+\\sum_j \\frac{\\varepsilon_j}{2}(d\\xi_j)^2, ", "16eb486d49a62370c99319da4e44b5c2": " k_+(x,y) = J_{x-\\frac{1}{2}}(2\\sqrt{\\theta})J_{y+\\frac{1}{2}}(2\\sqrt{\\theta}) - J_{x+\\frac{1}{2}}(2\\sqrt{\\theta})J_{y-\\frac{1}{2}}(2\\sqrt{\\theta}), ", "16eb95071ac6698f089ac0dd5880a640": "1 < p \\le 2", "16ebb383260b2ccd44340588119b1273": "\\frac{1}{\\theta} \\geq \\frac{1}{\\theta_0} + \\frac{\\tau}{3}", "16ebd29dff3ca3474121f409113e4d03": "W_1^T A W_1", "16ebffdee1f1adf850e48439e4a70f9f": "\\{X_1, \\dots,X_n\\}", "16ec0513f71070478795aa7317d13327": "\\varrho_{A, B}=tr_\\Lambda (\\varrho_{A, B, \\Lambda})", "16ec2d76fed2e2961bb2563b068833b2": "\np(y|f,\\mathbf{x},\\sigma^2) = \\mathcal{N}(f(\\mathbf{x}),\\sigma^2).\n", "16ec3846c09d7c2199beff9c9a085774": "x \\mapsto y \\mapsto x - y = x \\mapsto (y \\mapsto x - y)", "16ec6dd784bf332932334172e2adef93": "T(n) = \\Theta\\left( n^{c} \\log^{k+1} n \\right)", "16ec7e25919e5181fd6f7292481bb2b5": "e_2 = R\\, \\sin{(\\alpha)} d\\theta", "16ec7ff9cb6bacd175147e8ccc587d84": " \\operatorname{tr}(A\\rho_\\psi)=\\left\\langle\\psi\\mid A\\mid\\psi\\right\\rangle", "16ecafd8c5f559e28c19854f89aa1f0d": "\\ B_T = 2(\\Delta f +f_m)\\,", "16ecc7916a0deca54e48e7f968017ed1": "\\chi^2 = {(|121 - 59| - 0.5)^2 \\over {121 + 59}}", "16ed1a1513d96140b77c01f2e7a69065": "\\mathcal E^3", "16ed6b7287a255a769eeaf4decf7886e": "6 \\,", "16ed7dbb721b8a74d0fd23967ee466e1": "\\square = \\frac{1}{c^2}{\\partial^2 \\over \\partial t^2 } - {\\partial^2 \\over \\partial x^2 } - {\\partial^2 \\over \\partial y^2 } - {\\partial^2 \\over \\partial z^2 } ", "16ed8815ba6b3f623fae028b192a440f": "t \\mapsto P(t)", "16ed99ab006a04c38a92a496a72b1925": "R=\\left( \\frac{2mG}{c^{2}}\\right)", "16edd123b4c143e2af769b45442f8702": "\\scriptstyle P_E", "16edf5a2a0abc432770de48913629844": " \\int_a^b K(t,t)\\, dt = \\sum_i \\lambda_i. ", "16edfce07a4f11d7f89383eedd3b1db0": "\\rho^{-1}", "16ee221e8a683dde0f060125840cc065": "M^{}_{}", "16eec1cf23249ccda05a5aa8769827c4": " \\mathcal R ", "16ef1423397fdc62ba2f3c6f0a95a05e": " P(k) \\sim k^{-\\gamma}\\text{ with }\\gamma = 1 + \\frac{\\mu}{a_\\infty}.", "16ef7459b4e668209ce83bf28914c868": "\\mathrm{d}G\n=V \\,\\mathrm{d}p-S \\,\\mathrm{d}T\n+\\sum_{i=1}^I \\mu_i \\, \\mathrm{d}N_i \\,", "16ef8abe15c4c11d7432bf5d128d3360": "\ny_{c} = -\\frac{1}{D} \\begin{vmatrix} A_{xx} & B_{x} \\\\A_{xy} & B_{y} \\end{vmatrix}\n", "16f015a5b0f42fe79edc4606ff5f3c23": "Z = \\sum_{n=0,1} e^{-E_n\\beta} = e^{ \\mu B\\beta} + e^{-\\mu B\\beta} = 2 \\cosh\\left(\\mu B\\beta\\right).", "16f058fd67722c80e6a0109d2fac39e0": "\nP(X,Y)=\\frac{P(Y)}{Q(Y)}Q(X)\n", "16f05d4bb89c5b38182fd66f51fc75d7": "[f(x)]^{k/-h}=f(x)\\cdot f(x-h)\\cdot f(x-2h)\\cdots f(x-(k-1)h),", "16f0d2d48108870420a79fc9c342215d": "(\\phi \\to \\chi ) \\land (\\chi \\to \\phi )", "16f17bdffc8df4a08780f046f8ee6996": "\\log_2(12/3) = 2", "16f1cb15213cef3994ff99b41a74edda": "T^{-3}(|B^A|)", "16f20aacfc64d1965648f62b54d0782a": "\\displaystyle u_x+v_y=0", "16f244d58afede9baf3b0075bcfcb226": " L_1 \\cap L_2 = gL_1 \\times gL_2. \\qquad \\qquad (1)", "16f27c0e6d176466efab8e8e2c4c82e6": "\\ln \\left( {I(z) \\over I_{in}} \\right) = \\gamma_0(\\nu) \\cdot z ", "16f2b860d1c8c447bea413a4a0e53566": "F=GF(q)", "16f33c53f0878c0b1403cacfc85a8808": "\\mathbb{Q}(y)", "16f35b7ba2f0ddbf53f471339c96fb85": " x =t, \\quad y= c, \\qquad z = e^{-{t \\over c}}, \\quad t > 0 ", "16f3662e7b30d40e58b0dca7fbccbba0": "\n\\theta = \\theta_{1} + \\theta_{2}\n", "16f3950f493bb08aab77a9b282fdd8df": "G_0, ..., G_r", "16f3bd88ce08f120182fed45a48d9599": " \\hat{\\rho}_\\mathrm{neq} ", "16f3ff6e7d89d829c66893a0084dd097": "a'(\\varphi_0) \\neq 0", "16f42f4f42aafb415758f210b02fc309": " T_{lm} = \\left(T_{hi}-T_{co}\\right) - \\left(T_{ho}-T_{ci}\\right) \\over \\ln \\frac{\\left(T_{hi}-T_{co}\\right)}{\\left(T_{ho}-T_{ci}\\right)}", "16f43908b65b3f0d9c9e2b84a16fbb7a": "P(t) = P_0 2^t = P_0 exp(ln(2)t)", "16f466ac6bd14bbbc95af323ae71414b": " r=s+a=((b-a)/3)+a=(b+2a)/3. \\,", "16f46f2b997aa981614ad208ca053944": "\\left\\langle {dG}/{dt} \\right\\rangle_\\tau", "16f4fe576ac571fdd3713b2d99d53750": "e_2 := \\sum_{1 \\leq i0", "16f9aa34da1d97384ed54233c900279d": "PER \\,= \\frac{Gain\\ in\\ body\\ mass(g)}{Protein\\ intake (g)}", "16f9b442d02e26ee3ed40f46b46f6887": "z\\bar z = \\|z\\|^2", "16f9ead2c2e45daf6d89916ab375a151": " \\forall \\mbox{ positive } \\delta \\cong 0, \\ (|h| \\leq \\delta \\implies |f(x+h) - f(x)| < \\varepsilon).\\, ", "16f9fc2d60a8ccc3772bd576e5fce6be": "\\mathbf{X}_i", "16fa2ad938f416ac1e50ed325df8cef9": "E=\\epsilon\\sum_{i=1}^N \\sigma_i = \\epsilon \\cdot j", "16fa3ba41bb3c2dab8035a38081dacc5": "n I_n\\ = \\cos^{n-1} (x) \\sin x\\ + (n-1) I_{n-2} , \\,", "16fa878f90dee208c7a65e7638ebbc68": "P(X | Y_k) = \\frac{P(Y_k | X) P(X)}{P(Y_k | X ) P(X) + P(Y_k | \\overline{X}) P(\\overline{X})}", "16fa9edaca4ee1a8b926461817e6b71c": "U^* = \\frac{\\sigma\\epsilon}{2\\rho}\\sqrt\\frac{E}{\\rho}", "16fb33e5ce923b1434adf91f2b4ac2cb": " m \\cdot v=m_{naked eye}-2+2.5 \\cdot \\log_{10}(D \\cdot P \\cdot t) ", "16fb5918eb70d0d6acc3daed889cd363": " \\frac{m+n+2}{4}", "16fb836da2fd9a38138a938d2156502d": "I^-(p) = I^-(q) \\implies p = q ", "16fc00d9c551df704e15cdaa567c0190": "d(m,m') < \\epsilon", "16fc436cacaccd4f1b8f07e26ead3e13": "(\\mathcal{H}, \\langle \\;,\\; \\rangle)", "16fc72f98c9998f9973af1e73be5c81f": "P_{\\alpha, i}", "16fc9fad607ac1b49b5068fd050644ec": "K_{a1}=\\frac{\\rm{[HCO_3^-] [H^+]}}{\\rm{[H_2CO_3]}}", "16fcb2602ba3fd09374ae2137bd519b6": "MPL_{nt,1}=MPL_{nt,2}=1", "16fd5836b763262bfe10b4db1cd3719a": "A\\in\\mathbb{R}^{|V|}", "16fd677cc5b05f7fcb56da20a7788a0a": "\\mathrm{\\tfrac{u\\bar{u} - d\\bar{d}}{\\sqrt{2}}}\\,", "16fd9066d9bef6399cfcaab1384912b7": "\\operatorname{isnil} \\equiv \\lambda l.l (\\lambda h.\\lambda t.\\lambda d.\\operatorname{false}) \\operatorname{true} ", "16fd93e97d6c513d305c9205f1e7e2b6": "a(k,k) \\leftarrow \\sum_{i' \\neq k} \\max(0, r(i',k))", "16fd9a6130542c1032b8e74d6614bc1a": "au+bv-(1+c)w+c=0", "16fdb1ddc8b014d044d9c92c00873a52": "\\operatorname{BG}_p(a_n;x)=\\sum_{n=0}^\\infty a_{p^n}x^n.", "16fdb1e52c7be43e270abe7b1450d431": "n_{\\rm air} < n_{\\rm film}", "16fe0e8d83e4debe0eab3c1e39438b88": "\\sum_{k=m}^\\infty ar^k=\\frac{ar^m}{1-r}", "16fe3682f5be51344c00d749de21f0a6": "\\operatorname{arcosh} \\;u \\pm \\operatorname{arcosh} \\;v = \\operatorname{arcosh} \\left(u v \\pm \\sqrt{(u^2 - 1) (v^2 - 1)}\\right)", "16fe7ffa30c9d157ddca84dbb10caece": "\\mathrm{Homeo}\\left(F\\right)", "16fee590d449f1ca34835ca6bb30557f": "[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0", "16ff15b3bd3276e221eaffed0fb87474": "\\ x-x = 0", "16ff42f525810d45fc8b506f5ef0674a": "3^{(F_n-1)/2}\\equiv-1\\pmod{F_n}.", "16ff5a2504ffa7d2f91279b976fc6ab6": " x(j) = \\sum_{i=1}^p a_i x(j-i) + n(j).\\,", "1700120544789704371a6fa2ed70703e": "\\{X_1, \\ldots,X_m|S=s\\}", "17003e6ac018d3c124275e01537e6ffd": "\\sum_{k=1}^n k^p = {1 \\over p+1} \\sum_{j=0}^p (-1)^j{p+1 \\choose j} B_j n^{p+1-j},\\qquad \\mbox{where}~B_1 = -\\frac{1}{2}.", "17003ebcb5c6ea5663ffafb51f8a63a4": "r(\\theta)=\\frac{a (1-e^{2})}{1 \\pm e\\cos\\theta}", "170051564ed349eb62e02d0cb68996d7": " m_1 \\mbox{cosh}(s_1)+m_2 \\mbox{cosh}(s_2)=m_1 \\mbox{cosh}(s_3)+m_2 \\mbox{cosh}(s_4) ", "170058481dc479049aace6a2a6a4b11b": "\\int x^5 r^{2n+1} \\; dx = \\frac{r^{2n+7}}{2n+7} - \\frac{2a^2r^{2n+5}}{2n+5}+\\frac{a^4 r^{2n+3}}{2n+3} ", "1700629f7e3bcb381fa0b3605097dfbb": "f(x) = a_1 + a_2 2^{-2x} + a_3 3^{-3x} + \\cdots .", "1700a16c07d5cb26b1a8f5290cab743d": "\n Q = \\left(4 / c \\right) \\sqrt{\\textstyle{\\frac{1}{100}} J} \\left(A_w + 4\\right) F_L^{1/4}\n", "170118018f9b80f1d9bf4420a495f70f": "\\{C_n\\mid n<\\omega\\}", "17011906fd2d1e75b78693b709c307fa": "2 m c^2 / \\hbar \\,\\!", "17012a480b0340a9f6e1d443308b1909": "\\omega_L-\\omega_0 ", "17015a191600677337505375c27d4f87": "\\sin x = x \\prod_{n = 1}^\\infty\\left(1 - \\frac{x^2}{\\pi^2 n^2}\\right)", "1701767cb01d555318a0194df6dc58ae": "F \\wedge *F = \\langle F, F \\rangle d\\mathrm{vol}_M.", "17020195fc8e11bf49de6f8e609177b5": "S1\\ \\delta^c\\ S2", "17025c9685770f8491ab762e593e4926": "{\\scriptstyle\\frac{1}{24}} (x^4-16x^3+72x^2-96x+24) \\,", "17027e1e5e0110f0344aee79a61ffa9f": "a_1 = \\frac{F_1}{m_1^\\mathrm{inert}} = a_2 = \\frac{F_2}{m_2^\\mathrm{inert}}", "17028cfb7aa82b62c33f2cd276a969a2": "P_{1}^{-1}(x)=-\\begin{matrix}\\frac{1}{2}\\end{matrix}P_{1}^{1}(x)", "1702d707397cc315266c1bf7cda72359": "dx \\times dy ", "1702df8bd79c3024a19728f32deb3920": "R^{T} = R^{-1}, \\det R = 1\\,", "1703222e96933a9699695d69b9741291": "R(O)=0", "17035143cabdd9496bedb8df17d62b7b": " \\vec{n} = \\partial_v", "170388e14079d913dc130a9ddda089a8": "\nc^{2} d\\tau^{2} = c^{2} dt^{2} - dx^{2} - dy^{2} - dz^{2} \\,\\!\n", "1703f631f4046925b363f51ea1b1300e": "\\frac{1}{\\sqrt{r_\\text{middle}}} = \\frac{1}{\\sqrt{r_\\text{left}}} + \\frac{1}{\\sqrt{r_\\text{right}}}.", "170464305b1e2596eeb96e7aa2a44311": " (f,g)=\\int_a^b f(x) g(x) w(x) \\, dx. ", "1704749c18e9b9dbe9753205d84a5dc0": "s_2 = \\begin{cases} j_2 + 31 + \\left \\lfloor \\frac{j_2}{96} \\right \\rfloor & \\mbox{if } j_1 \\mbox{ is odd }\\\\\n j_2 + 126 & \\mbox{if } j_1 \\mbox{ is even }\n \\end{cases}", "17048bbea88cd4d106163b71bf8bcdf2": "(G, \\mathcal{X}, \\mathcal{Y})", "17049862a7810135a1fb512a22b54a16": "\\neg a \\or b", "1704a5b3719855bf8ab07e3ac5282a30": "p(\\boldsymbol{Y}_v |\\boldsymbol{X}, \\boldsymbol{Y}_w, w \\neq v) = p(\\boldsymbol{Y}_v |\\boldsymbol{X}, \\boldsymbol{Y}_w, w \\sim v)", "1704c55a9996089ebb7088458a60e8c0": " 0 ", "1704c77606125a4ac22fa96bc7a1c41a": "J_v = K_f\\left[ \\left( P_c - P_i\\right)- \\sigma\\left(\\pi_c - \\pi_i\\right)\\right]", "170527704e1eaaa4501530db742d1ec3": "y=f(x) + \\epsilon", "170567754acbad203779ece2210f8c6e": "P_{ij}(s;t) ", "1705d3d1f04a1565c48fab6cf93fd545": "\\{X_i\\} \\sim \\mathrm{Multinom}\\left(k, \\left\\{\\frac{\\lambda_i}{\\sum_{j=1}^n\\lambda_j}\\right\\}\\right)", "170605871c67410da2bc92332d85fe4a": "\\begin{pmatrix}0 & h\\\\h & 0\\end{pmatrix}", "170631e3b82568d82c4087438328c2a6": "J = -E \\frac{dc}{dx}, ", "1706589c297e4560d94f0ce441af4896": "\\tfrac24+\\tfrac34=\\tfrac54=1\\tfrac14", "17066c5b89a0c95072e5efc20e8a05d5": "\\mathbf{A}\\!", "17067162fa1a5c92fbe0c8c792cf8699": "G=Gal(\\mathbf{C}/\\mathbf{R})=\\mathbf{Z}/2 \\mathbf{Z}.", "17068fc99194dcb799dd32c25a6b2ab7": "\\frac{1}{|n|-1}", "1706b282b42f29b469864485a2a352c9": "|\\psi_s\\rangle", "1706f08b44d0f59f6f24ec2caad4dcbe": "dN_{tot}/dt = 0", "170743592bb84a9bda9a2ce7fb4f0cf7": "T(s, t)", "17076994178fcbe0e28e96473339371b": "z+n", "1707b0bec01bcbac82a442b173dc1084": " P^{-1}", "1707f25e0d04882fa406c735dea37c82": "X_k = \\sum_{n=0}^{2N-1} x_n \\cos \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}+\\frac{N}{2}\\right) \\left(k+\\frac{1}{2}\\right) \\right]", "170823da45eb0f429cd8b5758e5414ce": " S(\\rho^{12}||\\rho^1\\otimes \\rho^2)\\leq S(\\rho^{123}||\\rho^1\\otimes\\rho^{23}).", "17084907e1dac20036255d3c01906446": "|\\alpha|^2", "170862221d9fe1f35c61d42b5f5340b9": "B=C+R", "170864677425e84a42ec31e07806bd5e": " {\\left( 1 - {\\left( 1-X \\right) }^{\\tfrac{1}{b}} \\right) }^{ \\tfrac{1}{a} } \\sim \\textrm{Kumaraswamy}(a,b)\\,", "17089e81664ce583e28e592e981c96ca": "D(A)", "1708e3907ab41fbecb6dd5a9b784ed56": " t + x r \\quad \\mapsto \\quad \\exp(a r) (t + x r) = ", "170921844c230efc6e09cb59eecc7904": "\\ r = k", "1709a40a4ffc2b0e27b0a3006953acdc": "\\mathbf{x} = x^i~\\boldsymbol{E}_i", "1709aa4e0b31ef04cdc5cbc8f287687f": " \\langle y_{\\beta} \\rangle_{\\beta \\in B}", "1709b33406ea4ea73e4d9535b0bb7215": " (\\nabla^2+k^2 )\\boldsymbol{u}=0 ", "1709e448e1147f6b2dde0d96c671bf62": "\\chi (G)=\\min\\{ k : P(G, k) > 0 \\}.", "170a021c18372d5cd5f02b5000ca04b0": "\\mathbf{H_{{2}/{1}}}=\\mathbf{r}\\times m_2\\mathbf{v}\\,\\!", "170a3afdaf86e278ca3f41f4516996fd": "B(\\cdot ,\\cdot )", "170a8b2c6c715faa5960e7d9d6c9284e": "G_1 / G_0", "170a9cf30edcf19a6b1316a96bf9aaf4": "{{E^*}_{1/2}}^{red} = {E_{1/2}}^{red} + E_{0,0} + w_r ", "170ab05a5e126c1de8cf36946ebf3936": "L^{q,w}", "170ae1795f2fd8686870d54149200ec4": "\\frac{A_{final}}{A_{initial}}=R.G.e^{{\\frac{-{\\pi}ft}{Q}}}", "170b1ffef2e848ab03962b0b5c8da7f7": "H(z) = \\frac{A(z)}{D(z)}", "170b417a215e4315aaadd0957a34bd06": "\n\\int\\limits_{-a}^{a}dx\\phi _{m}^{\\mathrm{even}}(x)\\phi _{n}^{\\mathrm{even}\n}(x)=\\int\\limits_{-a}^{a}dx\\phi _{m}^{\\mathrm{odd}}(x)\\phi _{n}^{\\mathrm{odd}\n}(x)=\\delta _{mn},\n", "170b6207d8f5fdd990786f522e1a66d0": "dh/dx", "170b68b2f94b5597788d16c07d30e6e3": "r_t^j = \\sum_{i=1}^{n_T}\\alpha_{ij}s_t^i + n_t^j,", "170b8a9c81ffc255884bd6b8e913b204": "|i-j| \\geq d", "170bcf66a8e7ee0c0efa559edc068d7b": "x = P_{2k} + P_{2k-1}, \\quad y = P_{2k};", "170c08c43fe6ad0ff79fdeccf3c64d07": "(\\varepsilon(h_{(1)})h_{(2)})cv=h_{(1)}c\\otimes h_{(2)}v=h(c\\otimes v)=h(cv)=(h_{(1)}\\varepsilon(h_{(2)}))cv.", "170c15fd5e38683aa5a621230785b640": "H(x)=-\\sum_{m=0}^{N-1} Q_m\\ln Q_m", "170c4bb02d619f9e397688d607475f9a": "\\scriptstyle{R_a}", "170c77d8f686791cfdf50da14b2fa91f": "\\mu+\\Psi^\\textrm{T}X", "170c853ac78dbd16b267ff433c931225": "H_2(S(2k+2,n)) \\approx H_2(S(2,n)),", "170cf3f4ef3d2a4c4e4aeedf6e4a95ca": "\\sum_{k=1}^N f_k \\times C_k + G \\times CM = 0", "170d2eecc0f93a4d20868d1b8a87b983": "350 = \\sqrt{50^2 +100c^2}", "170d8791a4ae6face0b12360cfa0ae41": "\\mathrm{STC} = \\tfrac{1}{n_{s}-1}\\sum_{i=1}^T y_i (\\mathbf{x_i}-STA)(\\mathbf{x_i}-STA)^T,", "170d90690d0f47817f97d245ab25bef4": "\\ s = \\sigma + j \\omega", "170dc478c00fce61d2362bfcb05f5ed6": "\\left[W_{\\mu},W_{\\nu}\\right]=-i \\epsilon_{\\mu \\nu \\rho \\sigma} W^{\\rho} P^{\\sigma}. ", "170de09171e462c79a0a467657f7657f": "P = (1 - R_C \\cdot (\\frac{1}{R_B \\cdot 0.96}+\\frac{T_{Sync}}{B_B})) \\cdot 100 \\% ", "170e467f4fd3ffa4f8865f512182be11": "\n\\begin{align}\n\\mathbb{P}(y \\mbox{ received} \\mid x \\mbox{ sent}) & {} = (1-p)^{n-d} \\cdot p^d \\\\\n& {} = (1-p)^n \\cdot \\left( \\frac{p}{1-p}\\right)^d \\\\\n\\end{align}\n", "170eadc653de1095b407b17797e4c8ab": " \n\\cos\\phi = \\frac{c}{a}, \\qquad\nk^2 =\\frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}, \\qquad\na\\ge b \\ge c,\n", "170ed7ef177e15746d13cc285477fc3e": " \\bold g = g_{ab}e^{(a)}e^{(b)}~~~~~~~~~~~\\text{where}~g_{ab} =\\bold g(e_{(a)},e_{(b)}) ", "170edee006367db5753199b060f56efb": "\\forall s_1,s_2\\in S: (s_1{}^\\bullet \\cap s_2{}^\\bullet\\neq \\emptyset) \\to [(s_1{}^\\bullet\\subseteq s_2{}^\\bullet) \\vee (s_2{}^\\bullet\\subseteq s_1{}^\\bullet)]", "170f2ff9222438eff1b1fe52504fd318": "\n\\sum_{i,j=1}^{D+1} x_i Q_{ij} x_j + \\sum_{i=1}^{D+1} P_i x_i + R = 0\n", "170fbd5308fd36ad30c7f6dc718a087f": "\\left [ p_j , p_k \\right ] = \\frac{i\\hbar e}{c} \\epsilon_{jk\\ell } B_\\ell", "171112f2bb94019fcc7f14cf4b455dc2": "n/4", "171223f9e05098de2c6f350469e82847": "v_i \\,\\, (i = 1, \\dots, N),", "17125c99aa83f9857286f13ec27952b1": " \\vec x= \\vec p + s\\vec v + t \\vec w ", "171264d2bf0638a198285613bc47a0cb": "\\begin{smallmatrix}{a}\\end{smallmatrix}", "1712c40a855b15c69ef5b4d83252cd40": "\\begin{align}\n\\int_a^x 0\\,dt &= F(x)-F(a)\\\\\n &= F(x)-C,\n\\end{align}", "17138a41a24d490cc0c203367a68f09e": " L( \\{\\vec X(n)\\}, \\{\\vec M_m( \\vec S_m, n)\\} ) = \\prod_{n=1}^N{l(\\vec X(n))}.", "1713b0b431803bcde713e1f515966200": "3+\\omega = 0+\\omega = \\omega", "1714710c482c335b73bcb11c5c212afb": "n/p_i=n_i", "1714d32d8ad484d79bd943949c8844d8": "T^{\\mathrm{F}}_p", "1714f289f19fb86b2b94f85b8a0257b8": "\\begin{matrix}\\frac{1}{2}\\end{matrix} m v^2", "17153842d71a270755b28a5437b587ad": "a_x = b_y c_z - b_z c_y \\, ", "171542424903a643d769114933ed18af": "J(C_k)", "17156bcef6eef317e4800186b470858e": "W=\\mathbf F\\cdot\\mathbf s.", "171597bdad29d41a5b5271798a4382ef": "0.\\overline{010} = 11.\\overline{001} = 1110.\\overline{100} = \\tfrac15+\\tfrac35\\mathrm i", "1715dce24081993c16ef3b5a7a706e8a": "AU_i", "1715f5c89a324a37473d6cb0fbbb44d3": " ~\\epsilon_{t-1} > 0 ", "1715f71e00ebfb176fb46ac7702fda50": " X \\backslash E", "171604f61e838ee6fed394e7947e75e2": "\\frac{:{\\neg}F}{{\\neg}F}", "1716db8d8c4a9d5d541af07bb0b88ed0": "T'_{0}=L'/v", "1716ed2c573f24825e26a9357a01e344": "\\partial_{[\\alpha} F_{\\beta\\gamma]}= \\nabla_{[\\alpha} F_{\\beta\\gamma]} = 0 ", "1717328f425a00d6595eecabb83d5060": "p,r\\in M", "171792057920e87b953a09f334d8b305": " \\beta_\\rho(\\pi(x_1,x_n)) = \\min \\{ \\alpha_{\\rho}(x_i,x_{i+1}) | i=1,\\ldots,n-1 \\} ", "1717b4fa0d2eb8701e25f6eedc76eda8": "\n\\mathrm{var}(T)\n\\geq\n\\frac{[\\psi'(\\theta)]^2}{I(\\theta)}\n", "1717f46ab3798da532df687c515efa8d": "P(n) = \\int_{N=n}^{\\Omega} P(n \\mid N)P(N) \\,dN =\n\\int_{n}^{\\Omega} \\left[\\frac{1}{c}\\right] \\frac{1}{N \\ln(\\Omega) } \\,dN ", "17181a814248569e5aaf5b4dd08762d2": "F_5(a, b) = (x \\to x^{x^{(a-1)}})^{b-1}(a)", "171823b25b82527cbb9a4fe103f56382": "Q_{xz}Q_{xy}Q = \\begin{bmatrix}1&0&0\\\\0&\\ast&\\ast\\\\0&\\ast&\\ast\\end{bmatrix} . ", "1718285dcfbb4375534b24a6ce4b4de9": "104348/33215", "171842d0fb5a27563dc1e74d31c6a86c": "\\ln(4)/\\lambda.\\,", "1718a8d40af715e6d22c93cbf9aedbd9": "I_z=I_x=I_y = \\frac{m s^2}{6}\\,\\!", "17192c7c7385d28cfb715d8114f2a871": "100\\times\\left[\\sqrt{\\ln({\\rm bills\\ entered})}+\\ln({\\rm hits}+1)\\right]\\times[1-({\\rm days\\ of\\ inactivity}/100)]", "17195d679bbebdf832eea52c4aeeb67a": " a^n + b^n = (a^{2^k} + b^{2^k})(a^{n-2^k} - (b^{2^k} a^{n-2^{k+1}}) + (b^{2^{k+1}} a^{n-2^{k+2}}) - \\ldots - (b^{n-2^{k+1}} a^{2^k}) + b^{n-2^k}).\\!\n", "17196543ea11f4d138e6d964ea599ca3": "\nk_{B} T = \\Bigl\\langle q \\frac{\\partial H_{\\mathrm{pot}}}{\\partial q} \\Bigr\\rangle = \n\\sum_{n=2}^{\\infty} \\langle q \\cdot n C_{n} q^{n-1} \\rangle = \n\\sum_{n=2}^{\\infty} n C_{n} \\langle q^{n} \\rangle.\n", "1719b84b44c5607f08d152e36c1822d6": "F_{66} = 2(F_{11}-F_{12}) ", "1719ce882e9408d8bddf91b0838ed8f7": "\\displaystyle{f_0(z) = f(z).}", "1719dbe616e8e0d7595a881a067f694a": " y[n] = \\sum_{k=1}^{K} r_k[n] \\cos\\left( \\theta_k[n] \\right) ", "1719e500cb70ce6fe5324977af326eae": "\\sqrt{a} = 10^{(\\log_{10} a)/2}.", "1719f1b1de3be5b0a562c541707ad6b3": "\ny=b\\,y_{frac}\n", "171a4a620f686059b754a3e628101f97": "\\langle\\psi|\\phi\\rangle = \\bar{\\psi}\\phi = \\psi^{\\dagger}\\gamma_0\\phi", "171a85425401f3a3f74d635d926ed909": "\\Box(\\Box(A\\to\\Box A)\\to A)\\to A", "171adce7d002f6554d326370d75e1f69": "C = D^2 - \\frac{dD}{dr}.", "171b2995d469895708807c5a205ba7f4": "A \\cap B = \\emptyset,", "171b35751dd1aee0b02241fc538146e0": " \\omega = Uk - \\beta \\frac {k}{k^2+l^2}", "171b840f35653fd7cd771c592ecb12a7": "Num(S)", "171ba604e41847774b6b7660afb06ff2": "S(x) = x \\cup \\{x\\}", "171bbd3dff8eaa3bd8c623e8c31530a1": "E_{c,d}", "171bc67358f9223edc208cc0e7257c8f": "\\begin{align}\n V_1 &\\equiv V_{a,1} = \\alpha V_{b,1} = \\alpha^2 V_{c,1}\\\\\n V_2 &\\equiv V_{a,2} = \\alpha^2 V_{b,2} = \\alpha V_{c,2}\n\\end{align}", "171c0567ea77a56621950fd292f4911f": "\\gamma_{n,\\mathbf{C}}:=(E(\\gamma_{n,\\mathbf{C}})\\to\\mathbf{P}^n(\\mathbf{C})),", "171c6a5ea8091dec84c5174287871f50": "\\mathbf{v}=\\mathbf{v}_{\\parallel}+\\mathbf{v}_\\perp", "171c7af866e00f6506eaaf1e2eb8f63a": "C\\left(F\\right)=F^\\beta", "171d0c2ca1f83e496c822c37e7e0cd81": "A \\times M \\to M", "171d229a4e4ab98e709c3654d5c0daed": "n!=1\\times 2\\times\\ldots \\times n", "171d3aa311f3f29d8e99ac8325c918fc": "M_{ij} = \\sum_x \\sum_y x^i y^j I(x,y)\\,\\!", "171e15ca81b2d98245ff4885ac3a27e2": "(-1)^{k+1} B_k ", "171e64fb2f0d2f8e1497a11eba7f7e71": " \n\\mathbf{A} = \\begin{bmatrix}\n\\mathbf{B}_{1} & \\mathbf{C}_{1} & & & \\cdots & & 0 \\\\\n\\mathbf{A}_{2} & \\mathbf{B}_{2} & \\mathbf{C}_{2} & & & & \\\\\n & \\ddots & \\ddots & \\ddots & & & \\vdots \\\\\n & & \\mathbf{A}_{k} & \\mathbf{B}_{k} & \\mathbf{C}_{k} & & \\\\\n\\vdots & & & \\ddots & \\ddots & \\ddots & \\\\\n & & & & \\mathbf{A}_{n-1} & \\mathbf{B}_{n-1} & \\mathbf{C}_{n-1} \\\\\n0 & & \\cdots & & & \\mathbf{A}_{n} & \\mathbf{B}_{n}\n\\end{bmatrix}\n", "171fd7a6983ab7275e7a8c1e6eed4e63": "c = m^N \\mod N.", "1720049518c54612d768892ca32048c0": "\\frac{X_{b}(t) - X_{b}(0)}{X_{b}(\\infin\\,) - X_{b}(0)} = 1 - \\exp \\left ( \\frac {FDt}{\\beta\\,^2f^2} \\right )", "17204bfde38cd457a51d05c7c95fd7c8": " (\\mu_{i,j}: i\\in I, j\\in J)", "1720789a56ce8752d07f0f918f8a64e1": "\\frac{a-b}{a+b} = \\frac{2\\sin\\tfrac{1}{2}\\left(\\alpha-\\beta\\right)\\cos\\tfrac{1}{2}\\left(\\alpha+\\beta\\right)}{2\\sin\\tfrac{1}{2}\\left(\\alpha+\\beta \\right)\\cos\\tfrac{1}{2}\\left(\\alpha-\\beta\\right)} = \\frac{\\tan[\\frac{1}{2}(\\alpha-\\beta)]}{\\tan[\\frac{1}{2}(\\alpha+\\beta)]}. \\qquad\\blacksquare", "17207dbd5a04c3d90149864006383014": "\\frac{f'(x)}{f(x)}=\\sum_i\\left[\\alpha_i'(x)\\cdot \\ln(f_i(x))+\\alpha_i(x)\\cdot \\frac{f_i'(x)}{f_i(x)}\\right].", "17208bc21c168b8dbe30c84f7f5ad525": " \\!\\ S_m^{11} = S_{(m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m)}. ", "17209e122bf8ceb1729f4dac494bcfc0": "\np_y = \\frac{\\partial L}{\\partial \\dot{y}} = \\frac{q B}{2c}x ~,\n", "1720a41a5a5f58f1bfb25ec584cf44a8": "G_y", "1720bc203c4dddbe2f4e111c59cc6ffd": "\n J_{\\rm IIIc} = G_{\\rm IIIc} = K_{\\rm IIIc}^2 \\left(\\frac{1+\\nu}{E}\\right)\n", "17214be14077a7bb9269ef04f22b99d2": "\nF_{n} = F_{n-1}^2 - 2(F_{n-2}-1)^2\\!", "1721587d08a34ccd759115f0420033c7": "d \\Phi = - U d \\frac {1} {T}+\\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "1721752663bd779c01ca03dce6bd54da": "\\textit{state}(s, 1) \\leftrightarrow \\textit{state}(s \\circ \\textit{open}, 0)", "17219536c279def0b89ac9fddadc20d2": "\n \\rho = \\frac {MP}{RT}, \\,\n", "1721dbc9f60370bfc91ebe3b40bb262a": "\\|\\cdot\\|: G \\to \\mathbb{R}", "1721dc124aaa2e2f5bd410c3aff72717": "\nF_{44} = \\cfrac{1}{S_{\\mathrm{f}4}^2}\\ ,\\ F_{55} = F_{66} = \\cfrac{1}{S_{\\mathrm{f}6}^2}\n", "17227d892ae518eab12eb3f0e596f1a0": "\\mathbf{w}", "17229288b2769adcd181b070b07d59f6": "C''(s) = \\Gamma\\,", "17229b02720250cdd7ab37279edaece1": "\\frac{qu}p = \\left \\lfloor \\frac{qu}p\\right \\rfloor + \\frac{r(u)}p,", "17235296cc0d7139787e99102bf29481": " G_i ", "1723f0367131823396fb44455f3891dd": "{2^{2^{2^{2^{2^{2^{2}}}}}}}-3", "172445a742c9032f03e56b101552d995": "R \\to Wa~~~~~~~~~~~~~\\text{probabilities of generating 4 possible single bases on the right}", "1724964d2765fa9bf14a4e5b07f5dfdb": "x_0=\\left(a+\\omega \\right)^{\\frac{p+1}{2}} \\in \\mathbf{F}_{p^2}", "172498163519cd368a518393e93ba7f1": "|0\\rangle \\otimes |1\\rangle = \\frac{1}{\\sqrt{2}} (|\\Psi^+\\rangle + |\\Psi^-\\rangle),", "1724c619c44897dc6700782b3cf55f59": "\\Delta{V} = -\\int_{m_0}^{m_1} {V_{exh}\\ \\frac{dm}{m}}\\,", "1725021a1fd7adfbd3cba077eb93235d": "x^{ 14 }+x^{ 13 }+x^{ 12 }+x^{ 2 }+1", "172509e7156c407cab071232c7056005": "b(X^*, X)", "17250f10921a8413163fcb1e6e551e58": "\\left(\\frac{b-a}{2}, \\sqrt{ab}, \\frac{a+b}{2}\\right) = \\left(\\frac{b-a}{2}, G(a,b), A(a,b)\\right),", "172552e34863f606b13a9b5d12159573": "\\mathbf{M} = z_0\\mathbf{I} + z_1\\sigma_1 + z_2\\sigma_2 + z_3\\sigma_3", "17257fb51e468cf116b2851ff4fd878a": "\\ell(\\alpha, x_\\mathrm{m})", "1725a9a1f22f0f8f34419e0a68301cf8": "\\displaystyle S =\\int d^{d}x\\left\\{ \\frac{1}{2}\\left( \\nabla \\phi \\right) ^{2}+u\\phi^{4}\\right\\},", "1725df284746a3494c732dc688b5be96": "\\langle \\overline\\psi \\gamma_0\\psi\\rangle", "17269924657084fe6f89b4e5cb37ef46": " (xy)^2 = x^2y^2[y,x][[y,x],y].\\,", "1726cfafae44140b6606fd400f6a4ea4": " \\hat{A} = (A, B) = a_0 + a_1 i + a_2 j + a_3 k + b_0 \\epsilon + b_1 \\epsilon i + b_2 \\epsilon j + b_3 \\epsilon k, ", "1726fd7e38959a179460d2de6236de2d": "O\\left (M \\cdot N \\cdot \\max(M, N) \\right )", "17274951c9305fb389a1735655fc5528": "Q_\\lambda", "172794e66f60a2bd89c30f9058533193": "P^h (S) = \\inf \\left\\{ \\left. \\sum_{j \\in J} P_0^h (S_j) \\right| S \\subseteq \\bigcup_{j \\in J} S_j, J \\text{ countable} \\right\\}.", "1727b9c08154370a391b2e28b8237b03": " \\Gamma(W) ", "1728302e5fa24c84432f55cdd4949cbe": "f_{\\pm}", "172885e5b9d0cde8933ea03c41747df4": "\\mathbf{Z}/4\\mathbf{Z}", "17288667b04c1724e099c79bcc459224": "R[It] = \\oplus_{n \\ge 0} I^n t^n", "1728f3d25e0cccf13ceb8553926b64e1": "V_{GB}", "17294f613b0fd19f088e163d30eacfaf": "\\mathcal{B}", "17296396be5e5c482b8393fbb5ca3058": "\\ \\Delta C_p", "1729b8c9500bfbb58a35601690afb2db": "\\binom xn = \\frac{x^{\\underline n}}{n!} = \\frac{x(x-1)\\cdots(x-n+2)(x-n+1)}{n(n-1)\\cdots2\\cdot1}.", "1729d92b1e53c65e265b69bd586bd7ca": "\nV_R(s) = \\frac{R}{R + 1/ Cs}V_{in}(s) = \\frac{ RCs}{1 + RCs}V_{in}(s)\n", "172a13fefd9f6ef964da471648ef8d0d": "A_m(0,2) = 1,2,1", "172a5c063150e87a4986e5332316bb1a": "R \\in \\Gamma", "172adeae62e42a3f931142a3b6c947cb": " \\mathbf{B} = \\frac{1}{2}\\sum^{N}_{j=1} \\mathbf{Q}^* \\left( z^2_j - z^2_{j-1} \\right) ", "172af6e98ac8d53d4d370e9880b69aa2": "\nT_e \\equiv \\left[ \\frac{S_0(1-\\alpha_p)}{4\\sigma} \\right]^{1/4}\n", "172b188a189ff9160ce3584be3faed94": "-D", "172b3c5b6bfbcaf4e31e2d76c7bbf7fc": " (b) \\subseteq (a) ", "172bb0ef0948f11d57b405877894c7ca": " \\hat{H} = \\sum_{n=1}^{N}\\frac{\\hat{\\mathbf{p}}_n\\cdot\\hat{\\mathbf{p}}_n}{2m_n} + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t)\\,,\\quad \\hat{\\mathbf{p}}_n = -i\\hbar \\nabla_n ", "172bb4f2e3b12c7ad45d389214920339": "\\mathbf{x}_{k+1} = \\mathbf{x}_{k} + {\\Delta t} \\mathbf{v}_{k+1/2},", "172c34e52f2ad58b8aa02d0ea051fd3f": "\\omega_{i,j} = |S_{i} \\cap S_{j}|", "172c3b691d282078cc7565a2dd97eaad": "C_l^m = (-1)^m C^{m\\ast}_l\\ .", "172c400b7a3be1beda6c5d248bebd2f6": "\\scriptstyle \\bigcap U_n", "172c7724d1ac7fdd5ef14cbe1f5b1cab": "\\overline{z}=x-iy", "172c99a29affd5ca98c168e05cb23598": "\\scriptstyle x\\in V \\subset U", "172cae39db5319717355fcf70ebfbc99": " K^{\\Omega(a b c)} \\mathrm{rad}(a b c) \\ ", "172cd15a60d26551b02a67cb1f85bbc9": "\\langle I\\rangle ", "172d321a9936a53cf15b27645872406b": "\\int_a^b\\! h(x) e^{M g(x)}\\, dx\\approx \\sqrt{\\frac{2\\pi}{M|g''(x_0)|}} h(x_0) e^{M g(x_0)} \\text { as } M\\to\\infty \\,", "172d97d4169a28ca8940502f204073a4": "\\prod\n \\limits_{\\scriptscriptstyle 0\\le q 0", "17315dcde1582930cd13b22ffca1b111": "0 \\leq p^{ij} \\leq 1, \\ ", "1731675b9449e566b9a4a3e9461ed119": "\\lambda_i = a_i+ib_i", "17316f6240b76d26354ab4603876984e": "\\psi_n''(x) + (2n + 1 - x^2) \\psi_n(x) = 0\\,.", "173191c2af0ab653e50d4d36c8669ef9": "B_1,\\ldots,B_k", "1731df130fe4bcbf41673ccdf7032cb5": "f (n) = f (n - 1) + f (n - 2)", "1731e3c1b29b2944b20f9fe530039eae": "\\frac{dy}{dx},\\quad\\frac{d f}{dx}(x),\\;\\;\\mathrm{or}\\;\\; \\frac{d}{dx}f(x),", "17327d83679cae8edb03c309813574cb": "\\vec s_{h}", "1732d134ca3ba11a60ff7a05b91ef57b": "\\overline{4}3m\\, ", "173319ec51e0cb83d0aa8fdb9114c57b": "\\hat f_x(x) = \\frac{1}{(2\\pi)^k} \\int_{-C}^{C}\\cdots\\int_{-C}^C e^{-iu'x} \\hat\\varphi_x(u) du.", "1733515c422ad853ae7bff1a46020627": "F[r]= \\int_a^t \\sqrt { r^2 + r'^2 }\\, dt", "17339e49e26410f83248221222de09cb": "S = \\operatorname{logit}(TPR) + \\operatorname{logit}(FPR)", "17341c72e3914addbaf2a060f24f1851": "p_n(z_i) = w_i, \\text{ where }i = 0, 1, \\ldots, n.", "1734251c88262ea01fc9f4a0a499327b": " \\zeta_{H} (s,a) ", "17343dcfd5c053e4fc0541e05b62e0d1": "K = V_{star}\\sin(i)", "1734748cf2157571a75097c150cf1318": " E_{i} = - \\beta_{ij} \\frac{\\partial T}{\\partial x_j} \\,", "17349bd2be4e84440fd0fce9024282d6": "\n\\langle 0|\\varphi(0)|p\\rangle= \\sqrt \\frac{Z}{(2\\pi)^3}\n", "1734dd9a5bd7daaf776bbf2aec444a9f": "\\hbar=m=e=4 \\pi \\varepsilon_0 =1 ", "17356bc0b465dc4f005e9667bd4b74bf": "\\{\\{1,1,0,0\\}, \\{1,0,1,0\\}, \\{1,0,0,1\\}, \\{0,1,1,0\\}, \\{0,1,0,1\\}\\}. ", "173574726983ebaf30966ab11713568d": "\\mathrm{Sh} = f(\\mathrm{Re}, \\mathrm{Sc})", "17364994986119dca6dad2664c6820c3": "R_2=\\sqrt{\\frac{a^2+\\frac{ab^2}{\\sqrt{a^2-b^2}}\\ln{\\left(\\frac{a+\\sqrt{a^2-b^2}}b\\right)}}{2}}=\\sqrt{\\frac{a^2}2+\\frac{b^2}2\\frac{\\tanh^{-1}e}e} =\\sqrt{\\frac{A}{4\\pi}}\\,\\!", "17365806639070f35b9d795ed6d467c1": "\\delta(a+b)=\\delta(a)+\\delta(b)", "17367a5ea3fa4e299e8f743c94d654d2": "\\lim_{r\\to 1^-}\\left|G\\left (re^{i\\theta} \\right)\\right| = \\varphi \\left(e^{i\\theta}\\right )", "1736816c553751b49ff1427556851138": "\\hat K_j (x_1) f(x_1)= \\phi_j(x_1) \\int { \\frac{\\phi_j^*(x_2)f(x_2)}{r_{12}}dx_2}", "1736980c0d884fc7c686aaef2e50cddc": "\nP_y(t_0+t) = \\sum_x P_x(t_0) K_{x\\rightarrow y}(t)\n\\,", "1736c96d8462cc67dc802fe429f285e4": "\\sin^2 A=1-\\cos^2 A", "173729637a572ff803abb41fd008de8c": "E_x^{\\rm HF,SR}(\\omega)", "17373286ba73f225a11f2f63f53153e4": "= \\sum_j Q_j e^{-E_j it/ \\hbar}.", "173732e67770e37f938f5b0b25273def": " 1 = l_B + (1 + r) l_A a_B", "173765c55d3651bf08b53ecfa4360a02": "\nL = \\frac{b^{2}}{a} = a - \\frac{p^{2}}{4a}\n", "173766566526cecae92c644f88999c11": "\\frac{\\partial}{\\partial p}M_p(x_1, \\dots, x_n) \\geq 0", "17378dfaf07e35d67b31fbf2bf3cf9e6": "\\displaystyle{|z_n -w|^2 \\ge (|z_n-\\zeta_n|^2 + |\\zeta_n -w|^2)/2.}", "17379b12d232f584364ddec6589f5c8d": "(\\mathbb{N}^k, \\le)", "173809e93ca6c16931824a8ef54422d2": "\\boldsymbol{\\Omega}=(0,\\ 0,\\ \\Omega)", "17381f2e1b6f22c064711a5031ff4711": "z(1-z)\\frac{d^2w}{dz^2}+(c-(a+b+1)z)\\frac{dw}{dz}-abw = 0.", "173820b37ed0a5a230c612b461393344": "\\langle Ax,y \\rangle = \\langle x, Ay\\rangle \\quad \\forall x,y\\in\\Bbb{R}^n.", "1738c715ce1d2aafe9af90bfe0f2e1d7": "P_2(L)", "173961bd17d03217951462a63890254d": "\n\\begin{array}\n[c]{c}\nX\\\\\nZ\\\\\nI\\\\\nI\n\\end{array}\n\\left\\vert\n\\begin{array}\n[c]{cccc}\nX & I & I & I\\\\\nZ & I & I & I\\\\\nI & Z & I & I\\\\\nI & I & Z & I\n\\end{array}\n\\right.\n", "17396456dd02397800f8119772aa6ced": " \\mathrm{T_{High}}(f) = \\frac {j f/ f_1} {1 + j f/f_1} \\ , ", "17399f7fe04b8f845b32806e6f321047": " \\bigcup_n{\\mathcal{E}^n} \\subseteq RP ", "173a12501c478f5b3ff53c81f2bc834d": "\\det(E+(n-i)\\delta_{ij}) = 0, \\qquad n>1 ", "173a25c28578e35900262628c795b2fc": "\\forall x \\forall y \\forall z \\left( (x,y) \\in R^+ \\wedge (y,z) \\in R^+ \\Rightarrow (x,z) \\in R^+ \\right)", "173a9f5bc66c81d744e870b53ae1debf": "\\mbox{Assets} = \\mbox{Liabilities} + \\mbox{(Shareholders or Owners equity)}", "173b4cbdd2b8cca7b037e5d332c0fef8": "V = -\\bold{\\hat{d}}\\cdot\\bold{E} ", "173b5c7a232c4dcbf8d1dede80d7b3eb": "(I+X)^{-1}", "173b9840ef9471cc14cef7d0ff4bbcd6": "\\displaystyle{[L_{-1},A]=0,\\,\\,\\, [L_{-1},A^*]=A,\\,\\,\\, [L_1,A]=-A^*, \\,\\,\\, [L_1,A^*]=0.}", "173ba88afa3df16251f954483c103981": "\\omega_1\\}", "173bbf5cc516192a8bbccb9f9e0df4bd": "y/a", "173bf0309223a0ac683fbac777c06dc9": "\\bar{m}=n-1", "173bf34eee5800f154eb445ed88673f6": "(\\mathcal{M}, s \\models \\phi)", "173c8073bfd1673a2ab083de3d2cf719": "-12 > v_{g} > -20 \\mbox{km/h}", "173ca4505aa6bf1ee817a4f11ebe7a40": "J_{Ay}=-D_{AB}\\frac{\\partial Ca}{\\partial y}", "173cb3e04e1fa574fa79207939ae4070": "N=\\left(\\frac{16\\pi k^3\\zeta(3)}{c^3 h^3}\\right)\\,VT^3", "173cbfb2d8c11ec7cf732ab3b7bed49d": "k = \\pi/a", "173cc10937fd02da1474c93db88af033": "\n\\begin{align}\n\\omega^2& = 3 - 2\\sqrt{2}, & \\omega^3& = 5\\sqrt{2} - 7, & \\omega^4& = 17 - 12\\sqrt{2}, \\\\\n\\omega^5& = 29\\sqrt{2}-41, & \\omega^6& = 99 - 70\\sqrt{2}, & \\omega^7& = 169\\sqrt{2} - 239, \\,\n\\end{align}\n", "173ce9e659cb9b7dc5327e19db205553": "\\textstyle r \\in \\mathbb{Z}_q^*", "173d3129b6e5ce018b8182501754c9d2": "\\mathfrak{gl}_n(F).", "173d7fcd2d8200e05642ce657be2939f": "P = {2 \\over 27} \\rho_a A C_L G^2 V^3 ", "173d9d565e424853a1738020af455efa": "\\langle x, y \\rangle", "173db3743590aa57d106899b4c34bcb7": "\\,\\tfrac{-1}{2} - i\\tfrac{\\sqrt{3}}{2}\\,", "173df0ffd4eb9c70507ef7554236c433": "[\\cdot,\\cdot]: \\mathfrak{g}\\times\\mathfrak{g}\\to\\mathfrak{g}", "173e28bfa8f53dd008f8555783f9ab4c": "\\mathbf{X}\\boldsymbol{\\beta}=\\ln{\\left(\\frac{\\mu}{1-\\mu}\\right)}\\,\\!", "173e2dfac89205aefbe5fc527f03f518": "(x-y)^2 = x^2-2xy+y^2", "173eb932753a4dc968906d82b84ed4b7": "t_{1} = \\frac{\\frac{1}{2} 9 h}{2 m_e c^2 \\alpha^6 (\\pi^2 - 9)} = 1.386 \\times 10^{-7} \\; \\text{s}", "173eca2b659d6c2c9684731bff40c02e": "(q_1,...,q_{d(t)})", "173eda91467d59e92b43f2a1d21ee177": "N_i", "173f181c4e69501de095ab911751e631": "\n\\Pr(X=x)=P(X_1=x_1,X_2=x_2,\\ldots,X_n=x_n). \\,\n", "173f385fe2356f7930466dcaa3b8e60b": "\n\\begin{align} \nH_2\\left( e\\left(d_{ID}, u\\right) \\right) &= H_2\\left( e\\left(sQ_{ID}, rP\\right) \\right) \\\\\n&= H_2\\left( e\\left(Q_{ID}, P\\right)^{rs} \\right) \\\\\n&= H_2\\left( e\\left(Q_{ID}, sP\\right)^r \\right) \\\\\n&= H_2\\left( e\\left(Q_{ID}, K_{pub}\\right)^r \\right) \\\\\n&= H_2\\left( g_{ID}^r \\right) \\\\\n\\end{align}\n", "173f642a9f40499f67618728f0a7417e": "\\eta_{\\alpha}", "173fc88448988c94a179d9f91deee880": "z/y=x", "173fea468dfcfa2873ea7d9193925d29": "\\frac{dF}{dL} = 2 k_{\\rm A}\\frac{I \\, I^\\prime}{d}", "174020a4fda27b50f8bdd13ca15a4984": "V=\\oplus_{n>0}V^n,\\,", "1740360406a39e1bde26622fe5aa9be6": "\\left\\{ie_1, ie_2, \\dots, ie_n\\right\\},", "1740434d059957cb8d49720e9b8c4a2b": "\\begin{align}\n \\mathrm{Ai}(x) &{}= \\frac1\\pi \\sqrt{\\frac{x}{3}} \\, K_{\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right), \\\\\n \\mathrm{Bi}(x) &{}= \\sqrt{\\frac{x}{3}} \\left(I_{\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right) + I_{-\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right)\\right).\n\\end{align}", "1740488d59aadb04e1399f773a1c81ba": " F = UW_{11} \\cdot UW_{21} \\cdot ... \\cdot UW_{33} = \\sum_{i=1,2,3} \\sum_{j=1,3} UW_{i,j} = \\sum_{j=1,3} \\sum_{i=1,2,3} UW_{i,j}", "17406acdebe7b545b729e831eb48edd6": "p^*(E)=L_1\\oplus L_2\\oplus\\cdots\\oplus L_n.", "1740a8d7884ce281cf8d3886e330006c": "m=r^n -1", "17411bacf6215a744f9ce16a91c3770d": "|p_{k,S}^C-p_{k,U}^C|<\\frac{1}{Q(k)}", "17411c07a625ac4ce3f197f2d3af3688": "\\displaystyle{[v_m,u_n]=m(v,u)\\delta_{m+n,0}I.}", "17419c20cb3bce78262515b00838d114": "\\mathbf{v}\\left(\\mathbf{x},t\\right)", "1741ade17578f33c67660b8d2deb7778": "O(\\log\\left(n\\right))", "1742282a2bd8b3ed6dbf6c1c8c811f0b": "\n p(x,y) \\vdash p(x,y)\n ", "174292d27dc52d87f231ae33de57a704": "\\phi(x^n)\\!", "1742965f5157fa75f883cd81f720681f": "\\left( \\begin{matrix}\n r \\\\\n r^{'} \\\\\n\\end{matrix} \\right)_{4}=\\left( \\begin{matrix}\n r \\\\\n r^{'} \\\\\n\\end{matrix} \\right)_{1}=\\left( M_{1}\\cdot M_{2} \\right)_{4}\\cdot \\left( M_{1}\\cdot M_{2} \\right)_{3}\\cdot \\left( M_{1}\\cdot M_{2} \\right)_{2}\\cdot \\left( M_{1}\\cdot M_{2} \\right)_{1}\\cdot \\left( \\begin{matrix}\n r \\\\\n r^{'} \\\\\n\\end{matrix} \\right)_{1}", "1742f17f7b567d9760a4e56ac957b47a": "L_p(x) = \\frac{\\sum_{k=1}^n x_k^p}{\\sum_{k=1}^n x_k^{p-1}}.", "17433526ae602e81c37ba6aaf3b64607": "\\ \\varphi\\sim \\varphi'", "17436d00389ef3801531d628b9511c65": "X \\sim \\textrm{Exponential}(1)\\,", "174388fa85a9bfa6cf9cb250994ced64": "f=\\left(1+\\frac{\\Delta v}{c}\\right)f_0", "174432b9f4bd922f90d241af67db1dc8": "\n\\begin{bmatrix}\n\\boldsymbol{I}_m & \\boldsymbol{0} & \\boldsymbol{V}_1^{(t)}\\\\\n\\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_1^{(b)} & \\boldsymbol{0}\\\\\n\\boldsymbol{0} & \\boldsymbol{W}_2^{(t)} & \\boldsymbol{I}_m & \\boldsymbol{0}\\\\\n& \\boldsymbol{W}_2^{(b)} & \\boldsymbol{0} & \\boldsymbol{I}_m\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1^{(t)}\\\\\n\\boldsymbol{X}_1^{(b)}\\\\\n\\boldsymbol{X}_2^{(t)}\\\\\n\\boldsymbol{X}_2^{(b)}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{G}_1^{(t)}\\\\\n\\boldsymbol{G}_1^{(b)}\\\\\n\\boldsymbol{G}_2^{(t)}\\\\\n\\boldsymbol{G}_2^{(b)}\n\\end{bmatrix}\\text{.}\n", "17445cc02e4386c0cc3c9f3c395ef829": "c=22", "174468e4c22084ddbd425ccc0ebe19ba": "f: X \\times Y \\rightarrow \\mathbb{R}\\,\\!", "174473646a7721beb9f8452033ac5460": "U(r_0) = \\frac {U_0 e^{ikr_0}}{r_0} ", "174477bd1162f95bbb23a311cde01779": " i_{R}", "1744ede5a55a23cdb01a0953549ad95d": "\\beta = 3+\\sqrt2", "17451fb33613dfc7cbc7c3c12a7a366f": "Q\\;=\\;C\\;A\\;P\\;\\sqrt{\\bigg(\\frac{\\;\\,k\\;M}{Z\\;R\\;T}\\bigg)\\bigg(\\frac{2}{k+1}\\bigg)^{(k+1)/(k-1)}}", "1745a02cc4394ce7b2685c55574b08ab": "F = \\frac{Gm_1 m_2}{r^2}\\ ", "1745b4953c00539ba21e44f9abc8cc97": " \\sin A = \\frac {S} {bc} = \\frac {S} {\\sqrt {S_A^2 + S^2}} \\quad\\quad \\cos A = \\frac {S_A} {bc} = \\frac {S_A} {\\sqrt {S_A^2 + S^2}} \\quad\\quad \\tan A = \\frac {S} {S_A} \\, ", "1745e9d592c2ab0e11cbd8fe427537b5": "v_{\\infty} -\\varepsilon ", "1745ef2bee802b3b9f5f0c18f44cb788": "\\hat{r}_x(\\tau) = \\frac{1}{2T} \\int_{-T}^{T} [x(t+\\tau)-m_x(t+\\tau)] [x(t)-m_x(t)] \\, dt.", "17464456145e52e94d3cd374bcd7d408": "\\alpha = \\pi/2", "174681f99032180611ec2b4174570582": "\\forall x_1\\dots\\forall x_n\\exists !y\\phi(y,x_1,\\dots,x_n)", "1746cbcc187adb28414504c7f97f42af": "zero : 1 \\longrightarrow N", "1746f67f00e854d46fdb8878bde5fa18": "V_{\\rm free}(\\phi)=m^2|\\phi|^2", "17475e30b9d3a31adbf5349b24f842b4": "z\\ .", "17476c882b5f21746e6d9caa0e5a4637": " F_6 = x, S_6 = \\operatorname{false}, A_6 = \\_ ", "174771634f3bff27e9f60fd2b6299004": "\\psi = \\begin{cases} \\psi_1, & \\mbox{if }x<-L/2\\mbox{ (the region outside the box)} \\\\ \\psi_2, & \\mbox{if }-L/2L/2\\mbox{ (the region outside the box)} \\end{cases}", "174776a4d763af020bdc7ba2284d97ee": "\\begin{array}{rcl}\n \\dot x &= &-\\frac{1}{RC}x + \\frac{1}{RC}\nA_cA_r\\sin(\\theta_r(t)) \\cos(\\theta_c(t)),\\\\\n \\dot \\theta_c &= & \\omega_c + g_v (c^{*}x) \\\\\n\\end{array}\n", "1747843e12d3afa55996f7758e7ee45e": "y = 12x - 16. \\,", "1747a47991cb34d6cb09f39dd306afc1": "q_i = \\gamma_i + \\frac{\\beta_i}{p_i} (y - \\sum_j \\gamma_j p_j) ", "1747a5ff1f9339b3296f2d77928addd8": "i^2=j^2=k^2=ijk=-1", "1747f350909032aaed074d59b9ea28e0": "\\displaystyle R^i", "17482df55c7541f5816a0cf1f0528c60": "p_{k-1}(x)", "17483b8b7cbf9673c9b341a5777c575d": "\\,d^2 = b^2 - n^2 = c^2 - m^2.", "17484ba992e6f05ebc523ee77a0606ae": "\\{f_a (x)| 0 \\leq x \\leq 2^{n-1}\\}", "174859f261d1f7c9c0fd4e4cc4c4b312": "M_r = \\frac{c(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r(c + \\gamma)_r}.", "174894e77664dab8849d51fb3b47ece0": "q = \\tfrac12\\, \\gamma\\, p_{s}\\, M^{2},", "1748a220ecd820fb203b06ffb7033737": "\\begin{matrix} \t \n\\varepsilon u^{\\prime \\prime }(x)+u^{\\prime }(x) =-e^{-x},\\ \\ 0 \\in \\mathbb{Z}^+", "174ae2d0892b8c69256466a362894c77": "g = d_1 d_2 - d_1 - d_2 + 1", "174b1cbfc72f642f67b903c405071571": "2^{20}", "174b70c87ce3052f3b3933cce3aee74f": "H_\\Delta (N) = \\sum_\\Delta N(v(\\Delta)) \\epsilon^{ijk} Tr \\big( h_{\\alpha_{ij}} h_{s_k} \\{ h_{s_k}^{-1} , V \\} \\big)", "174c05740fad5257a6361d6857eb10f5": " I_\\Delta=(x_{i_1}\\ldots x_{i_r}: \\{i_1,\\ldots,i_r\\}\\notin\\Delta), \\quad k[\\Delta]=k[x_1,\\ldots,x_n]/I_\\Delta. ", "174c4d96165f80fa7d79b782465b107f": "\\mbox{vec} (\\mathbf{H}_{\\textrm{estimate}}) \\sim \\mathcal{CN}(\\mbox{vec}(\\mathbf{H}),\\,\\mathbf{R}_{\\textrm{error}})", "174c52b4a94e91e2b50026a7592ef8a0": " \\kappa = i k_x = i \\sqrt{\\frac{2m E_x}{\\hbar^2}} ", "174c58df1106208177f43cad26ae0f89": " \\varphi_s(z) =\\int_K f_s(k\\cdot z)\\, dk", "174c9e03e4cd448e4f736b78ad0f2dcb": "\\{X_k\\}_k", "174ca2263fe946ae25c9f97fabcb5821": " \\mathbf{n} = \\begin{pmatrix} -\\mathbf{R}^{-1} \\, \\mathbf{t} \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} \\tilde{\\mathbf{n}} \\\\ 1 \\end{pmatrix}", "174cd94d5d88b59ab8c3452a431b4b7e": "(a_n)_{n \\geq 1}", "174cde27d620166ac1ec839b056b732f": "\n\\sum_{k=1}^\\infty a_k^2\\,\n", "174cde985f3d39e5948ab4b16539001c": "b_{jk}", "174ce8d105c077051ca06b49874f3b4a": "\\chi_\\text{Yates}^2 = \\frac{N( \\max(0, |ad - bc| - N/2) )^2}{N_S N_F N_A N_B}.", "174d360832f8ebb914afc6e652a0bdc3": "\n a = (1 + \\beta^2)~\\sigma_y^2 ~,~~\n b = 0 ~,~~\n c = -\\cfrac{\\beta^2}{3}\n", "174dec5f47173c47d5df2a4716159be4": "\n \\boldsymbol{R} := \\boldsymbol{\\nabla}\\times(\\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon}) ~.\n", "174e1812b3055b16729af7f8c2e7215e": "K_{\\rm w}=[{\\rm{H_3O^+}}][{\\rm{OH^-}}] = K_{\\rm{eq}} \\cdot [{\\rm{H_2O}}]^2 ", "174ecaac70fee70cb474d4460843453c": "\\scriptstyle x_0 \\;=\\; 1", "174ef061d89d32d78550dd2c9750b04a": "D_{KL}(P\\|Q) = D_{KL}(P\\|Q_\\theta)\n + \\int_{\\mathrm{supp}P}\\left(\\log\\frac{\\mathrm dQ_\\theta}{\\mathrm dQ}\\right)\\mathrm dP.", "174ef4d74a2606605b883a3f167a451d": "\\scriptstyle (\\Omega, \\mathcal {F}, \\operatorname {P} )", "174eff7dcd101af0dd39189666f975f9": " A' = - \\ln\\left( \\frac{I_l}{I_0} \\right) = \\alpha' \\ell = \\sigma\\ell N \\,", "174f10edc73d5e91787b56ada00ac773": "(A \\to B) \\to (\\neg B \\to \\neg A)", "174f69398458629f706bdabe0273dc9c": "10^{10^{122}}", "174f7c96898c9a63cff1391c99d00243": "\n\\partial_t u + \\partial_x \\left[\\frac{u^2}{2} + \\frac{G}{2} * \\left(\\frac{b u^2}{2} + \\frac{(3-b) u_x^2}{2} \\right) \\right] = 0.\n", "174fd9ce3a7423e156500a5419fb02b0": "z(r)", "174fe838e8a4cab9a3dd8605464b1193": "\\left \\langle {1\\over r^3} \\right \\rangle = \\frac{2}{a^3 n^3 l(l+1)(2l+1)}", "17504fd21ddcc77ac88048c5efebff64": "\\bar{\\theta}_\\mathrm{BiasCorrected} = N \\bar{\\theta}-(N-1) \\bar{\\theta}_{Jack}", "1750547d58e6cbed636c208e7ed77c46": " Y = X_1 - X_2", "1750af558ad9b2dcd88469fb40956e7a": "\\delta = 2 \\arctan \\left( \\tfrac{d}{2D}\\, \\right),", "1750b042c42f9c594cb18716b276eb0a": "a^{P-1}\\equiv 1\\bmod P", "1750c3b4547dddd44b5a3e54de0a5349": "K=K(1/\\xi)", "1750ef56ae16b301c0937525324c8719": "\\operatorname{Cl}_2(2\\theta)=2\\, \\operatorname{Cl}_2(\\theta)-2\\, \\operatorname{Cl}_2(\\pi-\\theta)\\, . \\, \\Box ", "17510d9922345041fc751157475c1cfe": " N = \\text {notional} ", "1751262e393fabde29e37eea9c80d72a": "2^\\frac{8}{12}x = 2^\\frac{2}{3} \\approx 1.5874 x", "175136b6b4cfd99a3f456fa7583ef49c": "Z(C_4) = \\frac{1}{4}\\left ( a_1^4 + a_2^2 + 2a_4 \\right).", "1751451b6dc35f297a20feaa3ecd4f58": "\\mathbb{H}^{n+1}", "1751a31cc2df3d4594e2a0a4e851125f": "F\\subseteq K\\subseteq E", "1751ce5de61d19323342560c39917f4a": "\\tilde{\\eta} = \\frac{\\eta}{h},", "1751ef48dcc4d8d2ff3b72e77c0ca481": "\\scriptstyle 27", "17520d4d060a3cc2f959e92c458cc7a1": "\\scriptstyle \\hat X_t = X_{T-t}", "17522925d6d23c29bfc7d6055726f041": "{jx,jy,jz}", "175249f0f863294af34f377b2dd61654": "= (1 + sT) \\left( \\frac{1 - e^{-sT}}{sT} \\right)^2 \\ ", "17526fb08909fff52dca340b830143df": " \\frac{c}{\\Gamma} = \\frac{c}{\\Gamma_{max}} + \\frac{1}{K\\Gamma_{max}}", "17527299cb76e2a99e2b35913a405244": "=\\int_{\\Lambda^{m\\mid n}}f\\left( x\\left( y,\\xi\\right) ,\\theta\\left(y,\\xi\\right) \\right) \\varepsilon\\frac{\\det\\left( A-BD^{-1}C\\right) }{\\det D}\\mathrm{d}\\xi\\mathrm{d}y,", "1752755ca5dff8012ab0608cc117721b": "\\int_\\Omega |\\nabla u| = \\int_{-\\infty}^\\infty H_{n-1}(u^{-1}(t))\\,dt,", "17528c55f1ccc8b31fa4fa0b2797faaf": "\n \\boldsymbol{F} = \\begin{bmatrix} 1 & \\gamma & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}.\n ", "1752a3f2513518579a5c5e1e0a2d2a71": " X = \\frac{1}{2}(X + X^{\\top}) + \\frac{1}{2}(X - X^{\\top}) . ", "1752b96aca8a0c7d9c8e8a3573696f76": " \\operatorname{lambda-anon}[\\lambda F.X] = \\operatorname{lambda-free}[X] \\or \\operatorname{lambda-anon}[X] ", "175365a9244a0c319fd719cc7249f5b5": "\\phi_1 = N_1/N\\,", "1753707741af858c26810fff895890e5": "i \\le k", "175382ee410cdc08711fd0275d240df1": "\\displaystyle{Q(a,c)a^{-1}= (L(a)L(c) + L(c)L(a) - L(ac))a^{-1}=c.}", "17539888aee6adb744e9da8ffa591d18": "\\ h_{n,m}", "1754502ab58151debbe9131124689c8c": "e^{\\pi\\sqrt{3}}", "175451350fed58edcb2fabab66691717": "6 n K (N-K)(N-n)(5N-6)\\Big]", "1754694780d6338ae0e49f9574566b34": "n>N", "17548253c5b40740ab4af63ba95b1f2b": "(Y - a)", "1755166fadbc8ff2ded65af9bde12a0b": "N_\\lambda", "17552d8fe194c5daaae6cd4ab21deea1": " \\operatorname{def}[F_5] \\and \\operatorname{ask}[S_5] \\and FV[A_5] \\subset V ", "17553f733ecf3df7a552b73c0e31e04c": "c=(k+m)^2, \\, ", "175547908ad406e937971a72d0ae00a0": " \\left| x - \\frac{p}{q} \\right|> \\frac{A}{q^{n}} ", "17555e08fcaec4fca9e2d2668ba56d92": "\\frac a 1 = \\frac x d.", "1755a890cde845f3d1b32367212adc0b": "\\Phi = \\int{J_N} \\, {\\rm d}t", "1755c1e41132c60a8012914f96c0c12b": "C: P \\rightarrow\\!\\!\\!\\shortmid I\\!N", "1755cc516504aa84bb2999de853350be": "\\mathrm{ cos(Z) = \\frac{sin(dec) - sin(lat) \\cdot sin(Hc)}{cos(lat) \\cdot cos(Hc)}}", "1755ce438b523466ba5e4f4d0783cd95": "\\sum_{n=1}^\\infty \\frac{\\bar{H}_n^{(b)}}{(n+1)^a}=\\zeta(a,\\bar{b}) ", "175690cd0939245d7491c5d3d4aaa8ec": "\\omega_{z}", "1756b3eefaa66a19436acbcee1fb6fde": "(E \\rightarrow (I \\land R))", "1757e499c2621580e2a14d865c590d61": " \\ ,", "17581d9b08b5dbbe1d9a19fa13fb8502": "|xy|_X", "17582f947c63c9d42cbfb41720f1bbb1": "\\alpha,\\beta\\in\\{1,2,3\\}.", "175849e9f7c912307cefeef1b7eb0786": "\\lim\\limits_{x \\to 0} {\\frac{1}{x} =\\frac{\\lim\\limits_{x \\to 0} {1}}{\\lim\\limits_{x \\to 0} {x}}} = \\infty.", "1758745a1d746224c2edc0d9e7c012b9": "C_q(n,d,w)", "175881293316e72678aab82de418cd35": "B\\cdot U(\\$100) + Y\\cdot U(\\$100) + R \\cdot U(\\$0) > R \\cdot U(\\$100) + Y\\cdot U(\\$100) + B \\cdot U(\\$0) ", "175892a3f4a8bf0ab19a58b05e3a0268": "\\scriptstyle z^n+a_{n-1}z^{n-1}+\\cdots+a_1z +a_0", "175897d733004b51f2c1544f684a3f4f": "D_L\\!\\,", "1758da9afa5c8280e7490f43cf60bc2a": "\\mathop{\\textbf{y}}: C \\to \\hat{C}", "17592756f17022bca13a655f532b760e": "E_2 = E_c = 6.00 \\text{ ft}", "175941cf15dface5b5ccc6598e745f1b": "e/(e-1) - \\varepsilon", "1759b64f14aa6a420ddfb0d46aff804c": "\\displaystyle{ \\iint K(x,y) U(x)V(y) \\, dxdy,}", "175a14eff85f19c7717a85eade25564f": " C(t) - P(t) = S(t)- K \\cdot B(t,T) \\, ", "175a1871bf1e4515cdf79890de61bbb7": "x^a\\,", "175a72f96874ebbecef1aff430a212ff": " \\rho_L \\gg \\rho_V ", "175a98399a31bad8588050dd66c51af2": "p=\\frac{\\sum_jw_j}{B}", "175ada960806d9770bca772f36da1fb2": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) \\ \\subseteq \\ \\mathcal{U}(\\alpha^\\prime, {\\tilde{u}})\n", "175adee607d9e5458be2e70d5c779e13": "(q^i,\\dot{q}^i)", "175b01aab4742b658515aa97b5b1f46e": "\\scriptstyle\\mathsf{Boolean}", "175b09eaffbf4ab560223e56eb2b99f1": "S\\setminus\\{0\\}", "175b357719fd5e327f1b35f9341d8993": "\\frac{\\partial}{\\partial{z}}P_c^p(z_0)", "175b43734dd6f6e1bfbf1b2e0807af43": "x_a(\\hat{t})= \\frac{1}{2\\pi^M}\\int_{-\\infty}^{+\\infty} \\! X(\\hat\\Omega)e^{(j\\hat{\\Omega}^{T}t)} \\, \\mathrm{d}\\hat{\\Omega}", "175b77f3130b96d739593010c7e367e1": "\\scriptstyle h\\left(\\varepsilon\\right)\\,=\\,0", "175bc361ec8dd6641b83df29547bd50b": "\\beta\\;", "175be3872bda5b46254c095f0e5a4041": " \\hat{s} \\longleftarrow \\hat{k}z + \\hat{l} ", "175c2a7728d40da4abed6466b5750475": "\\hat{a}^{\\dagger} \\hat{b}^{\\dagger} |0, 0\\rangle_{ab} = | 1, 1\\rangle_{ab}, \\, ", "175c917933eb5001461d8fb66e8a8d6f": " P(y) \\rightarrow Q(y) ", "175cb6e3faeaa0ebffb7242bcb221c66": "\\pi(\\theta)\\,\\!", "175d1fdd05ac104c7e529f3fb33ca0f7": "V = -\\alpha/r", "175d2d37002a91a298e5ef1f04d59760": "\\displaystyle{K(\\mathbf{v}(t)+ \\lambda \\mathbf{n}(t),\\mathbf{v}(t))=-{1\\over 2\\pi \\lambda},}", "175d40d97c68675d5042ddbd1e001446": "S_{z_l}", "175d7cdd44d3f83be8ed5ef305ac03c5": "0<\\int_0^1\\frac{x^8(1-x)^8(25+816x^2)}{3164(1+x^2)}\\,dx=\\frac{355}{113}-\\pi.", "175ec79741187ccb0d653bf0d4496ef5": "f:E\\to R", "175ecc60bca959c190842b52a64fde1b": "\\exists x_1 \\, \\forall x_2 \\, \\exists x_3 \\, \\forall x_4: (x_1 \\or \\neg x_3 \\or x_4) \\and (\\neg x_2 \\or x_3 \\or \\neg x_4)", "175f6b9d8ddc0312e81448a7d7e7a5e4": "K\\subset G", "175faeda2910b65061d640f59c5bc596": "{W_{I}(\\mathbf {r}, t)} =<\\psi \\mid {E^{(-)}(\\mathbf {r}, t)} \\cdot\n{E^{(+)}(\\mathbf\n{r}, t)}\\mid \\psi>", "175fb02dc71571bb97cf42967a26105c": "v_{1}", "175fcebc8deacf3546044cc17448eaba": "\\det\\frac{\\partial}{\\partial (q_0,p_0)}\\Phi_{{\\mathrm{eE}},h}(z_0)\n = \\begin{vmatrix}1&h\\\\-h\\cos q_0&1\\end{vmatrix}\n = 1+h^2\\cos q_0.", "175fd06d917042e11bfaafd50673d327": "\n\\psi_{0} \\left(\\mathbf{r},t\\right) \n= A_{u}e^{i\\left (\\phi_{u} + 2\\pi\\nu t + \\mathbf{k}\\cdot\\mathbf{r} \\right )}\n", "175ff65b914e27276738ba9ffbe19e3d": " \\Delta k = k_1 - k_2 \\,\\!", "176020b4240418af973b5269eb2a6065": " v^i {}_{,j}", "17602a036765eef44433837afdb6ad01": " (\\lambda p.(\\lambda q.q\\ p)\\ \\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ \\lambda f.\\lambda x.f\\ (x\\ x) ", "17605398a1ab4b49b4c0d1921530c956": "(Z,X-\\widehat{X})", "1760b5914bafbb3cf75543953b246a56": "V(y) = \\sum_{i,j} \\sqrt{ |y_{i+1,j} - y_{i,j}|^2 + |y_{i,j+1} - y_{i,j}|^2 }", "1760de4b0e1885f437ef5fe60131ffb0": "d_{k+1} = \\frac{\\pi}{4} \\sum_{j=0}^k \\frac{d_j d_{k-j}}{(j+1)(2j+1)}", "1760e159d6228519538e3dffe9325aed": "\\eta(x,t)\\, =\\, a\\, \\cos\\, \\left( kx\\, -\\, \\omega t\\right)", "17615a2e243f16be6ecef9feb4ef87db": "\\scriptstyle\\ \\hat{p}=0.05 ", "1761717b355ea511f94484ad463a8f52": "L = \\sum_{i=0}^\\infty T^iL^e", "17618083689bf13313acbd196bb685e0": " = {i \\over \\hbar } \\left( H A(t) - A(t) H \\right) + e^{iHt / \\hbar} \\left(\\frac{\\partial A}{\\partial t}\\right)e^{-iHt / \\hbar} .", "1761e24704946375173598549ecd1012": "-\\log_2\\left[ 1-\\frac{1}{(k+1)^2}\\right]", "1761f60bdeaf2d88bdec534f514c9336": "\\gamma = \\frac{1}{\\sqrt{1 - \\frac{v^2}{c^2}}}", "176201a26fe6f1bacb2a114fae9ae83b": "Z_3 = T_1Z_1T_2Z_2 - kX_1Y_1X_2Y_2", "17620ea387c9667fc488f53cbc0913f6": "\n\\sup_{f}\\inf_{g}\\int\\int K\\,df\\,dg = \\inf_{g}\\sup_{f}\\int\\int K\\,df\\,dg\n", "17622817a410c06cb3fda67ad352c896": " \\frac{\\partial \\phi}{\\partial x} = \\frac {2x}{r_0^2} \\big( U + V\\cos \\Omega t \\big) . \\qquad\\qquad (8) \\!", "1762491f4275aa9bea2e2db2b2cdb56a": "f+g = (f \\vee g) \\circ \\Psi", "17627090466ae824e1436544cf7736a3": "(V\\oplus W)_0 = V_0\\oplus W_0", "176279e9ba3c959a7c2ad5ac5e3b31c0": "T \\cong R/P_1^{a_1} \\oplus \\cdots \\oplus R/P_r^{a_r} \\cong R/Q_1^{b_1} \\oplus \\cdots \\oplus R/Q_s^{b_s} ", "17628416b3b5fb3e3a9650b13dc37632": "s_{m}(z) = a_m\\cdot(2\\cdot m + 1)\\cdot(1 + z^{(-1)^m})", "17628862c345dee13e3753bc9404d951": " M = \\sum_{ i = 1 }^K ( f_m - f_i ) ", "176299cad60ea69f96dc88a9a6eb354e": "R(f,\\hat f)=E \\|f-\\hat f\\|^2.\\,", "1762d0e1d148f6f38ba399d6cde28f22": "\\mathbf{\\nabla} \\cdot \\mathbf{D} = \\rho_f", "1762f2b3c74cc1e91ebc4f9e3f3651c1": "E=(1/2) mv^2", "1762f998dd0adbca0ce21062820cebd2": "\n\\begin{align}\n(x_T * h)(t)\\quad &\\stackrel{\\mathrm{def}}{=} \\ \\int_{-\\infty}^\\infty h(\\tau)\\cdot x_T(t - \\tau)\\,d\\tau \\\\\n&\\equiv \\int_{t_o}^{t_o+T} h_T(\\tau)\\cdot x_T(t - \\tau)\\,d\\tau,\n\\end{align}\n", "1762faaa922d868294a5ab1609394fa7": "x \\, \\partial_v + u \\, \\partial_x", "1763106a890a247a6c993683bb6951de": "S_{i+1} = S_i+X_{i+1}-1 = \\sum_{j=1}^{i+1} X_j-i", "17638b2cfef7dad323f09188465ed3bc": "{\\infty \\choose r}_q = \\lim_{m\\rightarrow \\infty} {m \\choose r}_q = \\frac{1}{[r]_q!\\,(1-q)^r}", "17640101c8f06c39d3f1693066b19980": " V_t \\, ", "176404b457074913779b56dc87b5e733": "f(x)= \\sum_{i=0}^\\infty f_i^{(\\alpha)} L_i^{(\\alpha)}(x).", "17644ab31f043b5d89fa42bf3faf3428": "\n \\varepsilon_{xx}^{\\mathrm{face}} \\equiv \\varepsilon_{xx}^{\\mathrm{face}}(z) ~;~~\n \\varepsilon_{zx}^{\\mathrm{core}} = \\mathrm{constant}\n ", "176485ce5fde0a9ef40e9381a49d1b01": "\nSS_{AB} \\equiv \\frac{\\sum_{ij} Y_{ij}(\\bar{Y}_{i\\cdot}-\\bar{Y}_{\\cdot\\cdot})(\\bar{Y}_{\\cdot j}-\\bar{Y}_{\\cdot\\cdot})}{\\sum_{i} (\\bar{Y}_{i \\cdot}-\\bar{Y}_{\\cdot\\cdot})^2 \\sum_{j} (\\bar{Y}_{\\cdot j} - \\bar{Y}_{\\cdot\\cdot})^2}\n", "1764d984660f71b3ca6fc710e423b397": " Q(t_1) ", "1764f63cabdd74d5073884fc55a2439d": "(X:Y:Z)", "17653a5dd5025b5896481bedea79b2da": " {}+2432902008176640000. \\,\\!", "176569038972377570bd31259dd891a7": "\\overline{y}_x=\\frac{\\sum_i y_{xi}}{n_x}", "17656ba78da00c251e7f970e3fde9dd8": " \\Delta_\\sigma(i\\omega_n) = \\sum_{p}\\frac{|V_p^\\sigma|^2}{i\\omega_n-\\epsilon_p}", "17658e936aa274650caa657853f14be7": "L_p=\\sum_{e\\in p} L_e", "1765cd3fb116ffe3eba6be27a5b07311": "\\scriptstyle{L \\ll \\lambda /2}", "1765d6e7f7e6cd1d4e1c6cca31874546": " C_{2n} ", "1765e8f89e1c7c1b5461b43512fad4fd": "h : P \\to N", "17661d4747c70f6cac9b4dcf367593dc": "c_2 < 4.858.", "176644ed9355fa81569a1d04a2feb34b": "(x_0,\\lambda_0)", "17664c8b2270579567ebf180edfa497f": "\n\\frac{\\partial}{\\partial\\theta} \\int_{-\\infty}^{+\\infty}\n f(x; \\theta) \\, dx\n=\n\\frac{\\partial}{\\partial\\theta}1 = 0.\n", "176696e9a382536728fc2404f4975923": "X^{(k+1)} \\gets X^{(k)} - t_k t^{(k)} {p^{(k)}}^T", "1766a950d5bf3e30e9a8d4e3569dc658": " \\mathcal{I}_X, \\mathcal{I}_Y ", "1766cf8974a990000e9268a5232f7efa": "Tavg = 1 * p(1) + 2 * p(2) + 3 * p(3) + . . . + n * p(n).", "176745d0595b87280eb0bd893e3d72c4": "F[y]=\\frac{y''}{(1+y'^2)^{3/2}}", "176758722b72783cd4d234c82901d1c2": "\\operatorname{cov}(X_i,X_j)=-np_i p_j\\,", "1767d92a794bc11bac54f591d2827040": "\\mathbf{F} = I\\int \\mathrm{d}\\boldsymbol{\\ell}\\times \\mathbf{B}", "1767fd7908e742adc3decc1f50a4e30a": "\\theta_E = \\sqrt{\\frac{4GM}{c^2}\\;\\frac{d_{LS}}{d_L d_S}},", "176805ccb06ef2f1aad08fb1fa0eda35": "|00\\rangle", "17681d9f74bcaf4496441d6764fdf9ee": "\\bar{I}", "17688b5953b212ce73fac400bd26a9f3": " \\hbox{GrossEnergyYield} \\div \\hbox{EnergyExpended} = EROEI ", "17689202824b609470b454d76bddd6c1": "\\mathbf{T}^{(\\mathbf{n})} = \\lambda \\mathbf{n}= \\mathbf{\\sigma}_\\mathrm n \\mathbf{n}\\,\\!", "1768e97ca4bf9a473b3887ca2cd169fa": "{\\color{white}-}\\nabla^2\\mathbf{H} = \\sigma \\frac{\\partial \\mathbf{B}}{\\partial t}.", "1769440372afd113afa3dfe3dbb6bd13": "r_i=p_i+q_i", "1769b1634e330cfd2cfb6c79d2a7b1bf": "x_{11} \\ ", "1769bc7411ec457889728713eb45ba1a": "\n\\frac{p}{\\Lambda_{\\chi}}, \\frac{m_{\\pi}}{\\Lambda_{\\chi}}.\n", "1769ce0d7c6ba7e4e3ba4b3484914301": "L_i = S_i / S_n \\, ", "1769d22938331cc5da5beda0790a949e": "1,173,696\\,", "1769efe85750b874aa755898ad7e7832": "\\displaystyle \\frac{1}{\\sqrt{n}} X_n", "1769f59c2429883213f99bde95fc0b31": "P_{m+1}(x) = - \\left( (m+1)xP_m(x)+(1-x^2)P_m^\\prime(x)\\right)", "176a7bf83331ccf31c470c2708c9f27f": "\\cdots \\rightarrow P_2\\otimes_R B \\rightarrow P_1\\otimes_R B \\rightarrow P_0\\otimes_R B \\rightarrow 0", "176aa7194e39eba5f0e5d95556d19286": "\\textstyle \\sigma_k = A_k (\\Delta)", "176ac6c823791503a45e40e4889baf49": "\\equiv \\!\\,", "176af80d345d94230eb34a21552188e3": "\\mathbf{y} = y_1, \\ldots, y_{n-d}", "176b0cdbad28c96c3eb0bb4a353b1b26": "|\\vartheta(x)-x|=O(x^{1/2+\\varepsilon})", "176b2782706ca28a8c15347ec6109d8d": "\\sqrt{1-\\omega^2 \\, r_0^2}", "176b3ce05d8c52640ba016c35513d2aa": "\\mathbf{B}=\\nabla\\phi ", "176b65da777007d9f1de53fb6b53d82c": "[accumulation] = [in] - [out] + [generation]", "176ba9414d317d91a6bd4f84229441af": "\\frac{d\\nu(r)}{dr} = - \\frac{dP(r)}{dr} \\left( \\frac{2}{\\rho(r) c^2 + P(r)} \\right) = \\frac{1}{r} \\left( \\frac{8 \\pi G r^2 e^{\\lambda(r)} P(r)}{c^4} + e^{\\lambda(r)} - 1 \\right) \\;", "176bb9b3ad688f704cbf850f4d565b4d": "Z(t) = t - J_{N(t)}", "176bd80f7f7faa81b018f95a7704109f": " M_{\\mathbf X}(\\mathbf t) := \\mathbb{E}\\!\\left(e^{\\mathbf t^\\mathrm T\\mathbf X}\\right).", "176c105d84355949ba9bf580facdc8a6": "A_i^\\mu", "176c73c858e82a8946004b8a3dbe15b0": "\\ \\mathbf E_J \\cdot \\mathbf e_i = \\delta_{Ji}=\\delta_{iJ}.", "176c85ab780d0c03c7c09283455378d5": " x_{n+1}=rx_n(1-x_n). \\,", "176ca1a09fd104ec061ea0866d48a298": "\\bigg \\}", "176d3416e8d21df33f0a7564f9da5317": "\\Gamma_n ", "176d36a7176bf0c35fafb07b294c0f17": "\\hat m(\\mathbf{s}_0)", "176e1a28d6bba261ab25fc2f03057bda": "E^2 \\propto |\\vec p|^2", "176e2d1e783c286e4332660dbb7b9ab4": "(M\\phi)(v)=\\frac{1}{\\deg v}\\sum_{w:\\,d(w,v)=1}\\phi(w).", "176e3ff49c0c6a81d63c5670fbbc2c36": " (0,1) ", "176e73ec56c6c7bc569799ebb258d956": "\nc_{\\mathrm{fluid}} = \\sqrt {\\frac{K}{\\rho}}\n", "176e90fb1e16a8c389165e9a080d6e74": "mc^2 = hf", "176ed33e2972bb05f049c90b7d3684ae": "var_{ab}(p) = var_{ba}(p) \\ge \\rho_{ab}(p).", "176f2138d490a253db52f22991735fa4": "x = {1 \\over 2}", "176f2b311ff7838e01b3897686a9cf42": "r = \\frac{a(e^2-1)}{1+e\\cos \\theta}", "176f88c4442da55bd38348867589f832": "\\,\n{\\mathbf{U}}_{||} ={\\mathbf{V} \\cdot \\mathbf{U} \\over |\\mathbf{V} |^2 } \\mathbf{V} \\ , \\quad {\\mathbf{U}}_{\\perp} = \\mathbf{U} - {\\mathbf{U}}_{||}\n", "176fcbf217e88e2cd699969dba3f9560": "\\lim_{n \\to \\infty} \\ _{nominal}\\alpha = \\pi", "176fdd253124b72dfe88010ab78a4875": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 1 & 1 & 0\n\\end{array}\n\\right] .\n", "17703819644d613ad6612c213f82ca45": "\nP_A = \\frac{e^{v_A } } {1 + e^{v_A } } \n", "177055d2a03702ee3f24b1c612d51a02": "\\rho_0=z_0=0", "17705fa5092e3c0e5ecb4eff57649048": "\\sigma_1 > (\\sigma_2 = \\sigma_3 = 0)", "17706a5eeda0f3d308d147e4a236e853": "F_{max} = mg + \\sqrt{(mg)^2 + 2mghk} = mg + \\sqrt{(mg)^2 + 2mgEqf}", "1770742ef37d454091b5a8a77715dc1d": "s_1= h_1(g_{\\boldsymbol\\theta} (z_1),\\ldots, g_{\\boldsymbol\\theta} (z_m))= \\rho_1(\\boldsymbol\\theta;z_1,\\ldots,z_m)", "1770c6a3f164b7b8545cdc534c92ed7c": "(\\cos x + i \\sin x)^n = \\cos nx + i \\sin nx.", "1770efac20a3b8c84aa0d011327395ec": "d = L + C_p", "17712dd223c37c2448a5977f4ee8d2da": "\\displaystyle{\\phi_\\lambda=b(\\lambda)^{-1} M_1\\Phi_\\lambda}", "1771922ba3f2f5259277a0da11c7ce71": "I_{\\rm sp}=\\frac{v_{\\rm e}}{g_{\\rm 0}}", "1771b95c53a26c3ef3b49a8db54a6831": "K(u)=-1,", "1771f03abd4e4befe6bafc39ddf672af": "L \\subset K", "17728c4f74c62fed1eb7dde9001a354c": "T/T_{F}", "1772d4458ea641aea33fb110181d6252": "Q \\times (\\Sigma \\cup\\{\\varepsilon\\}) \\times \\Gamma", "1772d85d55fdafb8073bf06e58643fa3": "12^m", "1772f5e3fa387f41c4d370a50f55fb0a": "\\lim_{r\\rightarrow \\infty}\\epsilon_0 r^2 E_r(r,\\theta,\\phi)", "1772fe8c31c29d51809d32973cf44367": "\n\\mathbf{R} = (R_x, R_y, R_z),\\quad \\mathbf{P} = (P_x, P_y, P_z)\\quad\n\\hbox{with}\\quad P_\\alpha \\equiv \\sum_{i=1}^N q_i r_{i\\alpha}, \\quad \\alpha=x,y,z.\n", "1773482855a272c7f6cbb0016fa025c3": "\\, \\mathrm{st}(u_H-u_K)= 0", "177384b1214cbeb26b7ceeae2bf5e1dc": "\\bar{\\rho\\,\\!c}", "17738a7e29a06ff14b41547b42cab0ad": "I_{\\alpha-\\frac 1 2}", "177424a3c885eb3fa306394e3214aba0": "\n \\nabla\\cdot \\mathbf{u} = \\nabla^2 \\phi \\qquad ( \\text{since,} \\; \\nabla\\cdot \\mathbf{u}_{\\text{sol}} = 0 )\n", "177426167a2bcf74f813ef638dac959a": "C_{QL} = e^2\\cdot D_{2D} = \\xi \\cdot C_{Q0}", "17744d29ff700a64df8f9c36cb388e05": "ds^2=-Fdv^2+2dvdr+r^2(d\\theta^2+\\sin^2\\!\\theta\\,d\\phi^2)\\,,\\;\\;\\text{with } F\\,:=\\,\\Big(1-\\frac{M}{r} \\Big)^2\\,,", "17744e18bcb7a165d7853fd826590946": "\nQ\\propto I^2 \\cdot R\n", "177489ca0526a65cfabd1ee858229ede": "\\hat{T}", "1774b4985d74a9bc5610a4a058b8257d": "\\ddot{x}_\\alpha(t) = \\dot{v}_\\alpha(t) = F(v_\\alpha(t), s_\\alpha(t), v_{\\alpha-1}(t))", "17753508725fc4f161632f3e013643fa": "A_1+\\cdots+A_n", "177577158c966ab6acba2ea262931b40": "B_{k}+\\frac {(y_k-B_k \\, \\Delta x_k) (y_k-B_k \\, \\Delta x_k)^T}{(y_k-B_k \\, \\Delta x_k)^T \\, \\Delta x_k}", "1775d28b9d25df4fe462d94a5ebe9b85": " F_{4}^{(1)}(a):= \n\\begin{bmatrix} 1 & 1 & 1 & 1 \\\\ \n 1 & ie^{ia} & -1 & -ie^{ia} \\\\\n 1 & -1 & 1 &-1 \\\\\n 1 & -ie^{ia}& -1 & i e^{ia} \n\\end{bmatrix}\n{\\quad \\rm with \\quad } a\\in [0,\\pi) . ", "1775f7f0ab98e038920e8dea58d951d8": "s_1 \\cap s_2", "1776b3cd9fe220c1aa21653dd7c74534": " \\lim_{x \\to -\\infty}e^x = 0. \\, ", "1776ed614400a592d1878ad01f6c9453": "F_R = dE_R/dz\\,", "177710f052a9e58fefe56b52bb2f69c5": "(A\\cap C)B/(A\\cap D)B", "17775e4524ddb0f6401bfc9880e6fcad": "\n\\begin{bmatrix}\n -1 & 0 & 0 \\\\\n 0 & -1 & 0 \\\\\n 0 & 0 & 1 \n\\end{bmatrix}\n", "1777977b30493e6f7287ea6589b701d3": "\\Psi_A(1,2,\\dots,N_A)", "177797ba021505cd7e5210e726cbc190": "M_1=\\int_0^{r_B} \\frac{4\\pi \\rho(r) r^2}{\\sqrt{1-2GM(r)/rc^2}} \\; dr \\;", "1778714680e63da209e4e6bfeb82315d": "s_\\max \\sim |p-p_c|^{-1/\\sigma}\\,\\!", "1778ee59ad7a3daf9b8b87d899f5f671": " \\mathrm{Dir}(\\alpha) ", "1778f282ca3754aa74a2cc1eecbd52b7": "\\frac{1 + \\mathrm{Interest}}{1 + \\mathrm{Inflation}} = 1 + \\mathrm{Real}", "17790cb45f1c34ae9e690a3ec2eafd04": "\\rho_{He}", "1779a593abb219f5dc33d7bfdc2ca4fd": "\\phi_n(t) = a_{n1}e^{-\\alpha_1 t} + a_{n2}e^{-\\alpha_2 t} + \\cdots + a_{nn}e^{-\\alpha_n t}", "1779f3405a94f64dfe8d80c7721d87be": "\\delta\\hat\\mathcal{O}", "177a075552116b2c5a897ce885ba1955": " \\frac{d}{dx} = i \\frac{1}{i} \\frac{d}{dx} ", "177a11d2ed5d8dd744348a0ef108bcb5": "\n\\sum_{x:x \\neq y} 2^{-D(x,y)} \\leq 1 , \\; \\sum_{y:y \\neq x} 2^{-D(x,y)} \\leq 1,\n", "177a732126b9e041aeaa34f4f67b5f28": "{\\Delta}E=A-I\\,", "177adae3f347234e5b924f9bc07c2744": "h^{-1}:B\\to A", "177b2d5b45d574d3ee2c3f97b4ec7c90": "\\det (b_{i+j-2})_{1 \\le i,j \\le n+1} = \\det (c_{i+j-2})_{1 \\le i,j \\le n+1}.", "177b606113ef85fa923e9f8862e33686": "\\cdots \\xrightarrow{\\delta_n+1} C_n(E)\\xrightarrow{\\delta_n} C_{n-1}(E)\\rightarrow \\cdots \\rightarrow C_1(E)\\xrightarrow{\\delta_1} C_0(E)\\xrightarrow{\\varepsilon} Z\\rightarrow 0,", "177ba492474125cd459f2c1b25aa1d22": "\\ln \\gamma_i^C = (1 - V_i + \\ln V_i) - \\frac{z}{2} q_i \\left( 1 - \\frac{V_i}{F_i} +\n\\ln \\frac{V_i}{F_i}\\right)", "177bacd5d6b71dfe71960cee0973f27c": "e_1+e_2+\\ldots+e_k=d_k", "177bbd1739d89b8478b8cf5f59c8465c": "\\mathbf{S} = \\frac{1}{2}\\left[\n\\begin{array}{cccc}\n0& 0& 1& -1\\\\\n0& 0& -1& 1\\\\\n1& 1& 0& 0\\\\\n1& 1& 0& 0\n\\end{array}\\right]", "177bdabc21ecb8bf905cf74432062a39": "X_{-\\infty}^{-1}", "177c1ae1ba2c169df6b1ea712a78a42e": "\\partial \\circ M \\circ (\\eta \\times \\operatorname{Id}) = \nM \\circ (\\eta \\times \\partial)", "177c221d2d445974750486c15bbcdad5": " (\\sqrt{a}x + \\sqrt{c})^2 ", "177c30c62f58fc8a3ad3ffbad159a6a7": "h_{k-1}, h_k, v_i", "177cb9f826acd274e783e78b93813147": "A=(10+\\sqrt{\\frac{5}{2}(10+\\sqrt{5}+\\sqrt{75+30\\sqrt{5})}})a^2\\approx17.7711...a^2", "177d378a365010537be94217cae9f35d": "|N|=\\frac{E_Y/B_Z}{dT/dx}", "177d3caa3dd1021bb0b0dad1b0c145ad": "\\sin\\beta = \\sin\\delta \\cos\\epsilon - \\cos\\delta \\sin\\epsilon \\sin\\alpha", "177d4cf5ae3516971f738c112d484efc": "\\lambda(\\boldsymbol{r'})", "177d55e40e9e8ed721129cd5248484a9": "\\{\\mu\\}", "177e30cf2f489eb4d125da0d514e1290": "R_{\\mu \\nu} - \\frac{1}{2} g_{\\mu \\nu} R + \\Lambda g_{\\mu \\nu} = \\frac{8 \\pi G}{c^4} T_{\\mu \\nu} \\,.", "177e6387b133e204f24828511a0f6029": "\\scriptstyle{v_i}", "177f3996aeb76e98c1222e5965f549d8": "B\\left(\\omega\\right)", "177fb9f61549d60681a22c28297a536f": "\\Lambda_\\epsilon(A) = \\{\\lambda \\in \\mathbb{C} \\mid \\exists x \\in \\mathbb{C}^n \\setminus \\{0\\}, \\exists E \\in \\mathbb{C}^{n \\times n} \\colon (A+E)x = \\lambda x, \\|E\\| \\leq \\epsilon \\}.", "177fc0526bf0aa11a448148794b5146d": "V \\oplus iV,", "177ff7a0a8afd8ed925d19883df14b09": " \\mathbf{P}^n_A", "17802d1c3257f9eb535d022de318b865": "{n \\choose k-1}+{n \\choose k}={n+1 \\choose k}.", "178124b1758d16949e81aa1f44d98eee": "\\lim_{n\\to\\infty} c a_n = c \\lim_{n\\to\\infty} a_n", "17816666f24174bfe6ad635fcca8c329": "L^{2,h}(D) = L^2(D)\\cap H(D)", "178180505bbefefa1b9e2df946db4b7b": "\\tau_{y\\eta}\\,\\!", "17819b8266769abbc6cfdab5b9bd13ec": "C \\to C[W^{-1}]", "1781bffe49946243e024aa08d1875e83": "\\mathbf{x}^{(k+1)} == \\mathbf{x}^{(k)}", "178205c9bd089ef5c4753e28023874f3": "f\\, '(x_*)=0", "17826de19259fe610a2e507dbde16d9a": "1264460", "178290ff5e89e36da41ad880c703ffdc": "{48 \\choose 5} = 1,712,304", "1782a68bf03aa518c0133b3a0f719fbe": "\\nabla \\cdot \\mathbf{A}", "1782b4feb18f4ff4a707d9b654b9a42c": "x>-1", "1782c20027d6396a72ff230f10a56a13": " E_j\\ ", "1782c7cf7d537d897cfb4f49d114237c": "\\left[ B \\right]=\\left[ A \\right]_{0}\\frac{k_{f}}{k_{f}+k_{b}}\\left( 1-e^{-\\left( k_{f}+k_{b} \\right)t} \\right)+\\left[ B \\right]_{0}\\frac{1}{k_{f}+k_{b}}\\left( k_{f}+k_{b}e^{-\\left( k_{f}+k_{b} \\right)t} \\right)", "1782d655042b1dde95098b237cbc2198": "\\left\\lfloor \\frac{a}{n} \\right\\rfloor \\, ", "17830770eb29367b1aa42df9232cc24c": " c\\left(\\text{largest monomial of }s_1\\right)^{i_n-i_{n-1}}", "1783285049bdf2b33440c1ef59dbb7d4": " y-p(x) = 0.\\,", "17838356b676a558ddab46ff3f3541ec": "\\forall_f", "1783a6cef146a7924ae6050425441f7b": "\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix}:\\mathbf b", "1783af5d653d2c8f657d293124bc9c16": "\\tilde f(x):=\\inf_{u\\in U}f(u)+\\text{Lip}(f)\\cdot|x-u|,", "1783b5c03568f5881cbeab4c314e2f71": "p_i\\leq1+\\sigma(p_1^{\\alpha_1}\\dots p_{i-1}^{\\alpha_{i-1}})=1+\\prod_{j=1}^{i-1}\\frac{p_j^{\\alpha_j+1}-1}{p_j-1},", "1783c04045ff795c6a08356457e01e5f": "T_{20}=A+\\frac{Bk_2}{k}", "1783f0a6148a058095693dd0911a6846": "G / \\omega C", "1783f494cee67967e04e7339f7051cfd": "\\varphi_j^n \\quad ", "1784389e5b4552af6e1f5efc07bcc924": " A=\\frac{G}{3-G}", "1784993a5398a5edf91ac7b5cdeb7838": "M = 0", "17849dc9dc7804ba75421a404bd6584c": " 1/4 ", "1784a093ef579a1471da686ea4f67641": "(11)\\quad {\\hat\\sigma} ^2=\\hat\\sigma_{ab}\\hat{\\bar\\sigma}^{ab} =\\frac{1}{2}\\,g^{ca}\\,g^{db}\\,k_{(a\\,;\\,b)}\\,k_{c\\,;\\,d} - \\Big(\\frac{1}{2}\\, k^a{}_{;\\,a} \\Big)^2 = \\,g^{ca}\\,g^{db}\\frac{1}{2}\\,k_{(a\\,;\\,b)}\\,k_{c\\,;\\,d} - {\\hat\\theta}^2\\;.", "1785004d270166b673ae4b9655e64c5c": "\\boldsymbol{\\mu}_S = \\frac{g_s e}{2m} \\bold{S} ", "178510e30d9d11ffe732c8887f8c3425": " A(t) = 4 \\int_0^t W_s^2 \\, \\mathrm{d} s ", "17852f3942dcbd7b170f99d534d35a7d": " \\mathrm{Gr}_c = \\frac{g \\beta^* (C_{a,s} - C_{a,a} ) L^3}{\\nu^2}", "178555f4aa2e0953c0346f4daa7b7025": "M_2=(0.03000-31.4424x_D+30.0717y_D)/M", "17857fb7e4f853d4ddfedcfa9d2ea7ac": "\\, i > i^{(2)} > i^{(3)} > \\cdots > \\delta > \\cdots > d^{(3)} > d^{(2)} > d", "1785e1d662f804683f08b169ff39baa3": "\\log a = 0.6192290, \\log b = 0.9618637, \\log c = 1.0576927", "17869e223a6460984564ea87da53f358": " \\Psi(w,v) = \\alpha \\left( \\Psi(v,w) \\right) ", "1786a94e78f8b9d3ca65b0d26e3afda4": "\\log \\frac{\\epsilon}{\\epsilon - (1-\\epsilon)\\delta} ~=~ -\\log\\left( 1 - \\frac{(1-\\epsilon)\\delta}{\\epsilon} \\right) ~\\geq~ \\delta \\frac{1-\\epsilon}{\\epsilon} ~.", "178717df2d86b8abe6915c8e34d7329c": "\\mathbf{H}_{\\text{Electric dipole}}", "17873da24045b25d5f2b6312a23b573a": " d \\sin \\theta = m \\lambda \\ , ", "17875ea53adbcfcd53aecc2489ac3296": "L_2>L_1", "1787b1efaef540c5fb7ea8f7c68c9614": "Is=ss^Ts+nn^Ts=s\\langle s,s\\rangle+n\\langle n,s\\rangle=s", "1787b77ce65b27ee0f59df5a9cf57624": "V = i \\gamma^0\\gamma^1\\gamma^2\\gamma^3.\\ ", "1787e2f85c4dfde563002fc2390d555f": "\\psi(x)-x > K\\sqrt{x}", "1788028c10706464be2da741adf591a4": " \\{P_i, y_i\\}_{i=1}^n, \\ y_i \\in \\{+1, -1\\} ", "17886787641252b8a48b1ec11bde58cf": "k_{A^*}", "17888738df4aca4408299efc0441466d": "\\widetilde\\phi_\\alpha(x, v^i\\partial_i) = (\\phi_\\alpha(x), v^1, \\cdots, v^n)", "1788efbde856bef67011f8e7c220da5f": "t = s[r \\sigma]_p", "1788f99cea96fa8638f44a07a0d24426": " \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\sum_i\\frac{\\partial^2 \\mathbf{S}}{\\partial x_i^{2}} + \\mathbf{S}\\wedge J\\mathbf{S}.\\qquad (2)", "1788ff61f604c6d2fd4817d6360b92a2": "dx\\wedge dy=-dy\\wedge dx", "1789285921aaf4d6521e44991781f8bd": "\nD = \\frac{k_{B}T}{f_{tot}}\n", "17896256c1b9e6ef89e3469fc4ad4cae": "^{d_h}M_{P_h}", "1789830e938bff850f904817801f60dd": "\\textbf{r}", "178ae2dac028fcabed5ddc9e65b31a71": " \\begin{matrix}\n0 & = & {}+ p_{12} z_0 & {}- p_{02} z_1 & {}+ p_{01} z_2 & \\\\\n0 & = & {}- p_{31} z_0 & {}- p_{03} z_1 & & {}+ p_{01} z_3 \\\\\n0 & = & {}+p_{23} z_0 & & {}- p_{03} z_2 & {}+ p_{02} z_3 \\\\\n0 & = & & {}+p_{23} z_1 & {}+ p_{31} z_2 & {}+ p_{12} z_3\n\\end{matrix} ", "178b3db30810e8e054a1625de581c2a4": "\\hat{\\beta}_{MSM}=\\operatorname{argmax}\\,\\hat{m}(x,\\beta)'W\\hat{m}(x,\\beta)", "178b4aef597fe388e4ee52a421262826": "Q_r < K_{eq}~", "178b4eb18bbcb8b8f40947df13aeebe5": "K_i=\\frac {P_i^{\\star}}P", "178b631f59a502d7c6e407ab20412ac3": "x_i^-", "178c199267c85aef1ae4270394cb4241": "\\leq j n", "1790b22375d4ec762ff92ccc420a7849": "F_b = -k_{eq} (x_1+x_2) .\\,", "1790bc121dfa00a2596c8ab82b0c700a": "\\frac{d \\theta}{dx} = k .\\,", "17919569f8e772299d2de89183c44ca5": "\\lambda = \\ell(\\ell+1)\\,", "17920c7c691ff7e27dea9c75b8e69535": "\n\\mathbf{P} = -\\frac{\\partial G_{1}}{\\partial \\mathbf{Q}} = -\\mathbf{q}\n", "17927b4bf4452d98ac4b4028ae740c58": "1\\rightarrow K\\rightarrow G\\rightarrow H\\rightarrow 1", "17927d9db511c26d4a938508b33ed9dd": "\\mathbf{r}=(r_1, r_2, \\dots, r_M)", "17928c2d1d3db1ec69e5da3ac7f3ffdf": "\n D \\approx \\frac{2}{3}E^ff^3 + 2E^ffh(f+h) = 2fE^f\\left(\\frac{1}{3}f^2+h(f+h)\\right)\n", "1792a218a0645935327cb8550802aef6": "V = \\frac{-AN\\mu_0}{l} \\frac{dI}{dt}", "17931ef55a26de0f4731092727e66ab9": "b'_i = 1\\,", "179329c50708d1c7519c988b83a75c08": "\nR_n=|m_n|\\,\n", "17932fac05948bbb398143b07c3bbbb0": " H \\left| \\psi (t) \\right\\rangle = i \\hbar {\\partial\\over\\partial t} \\left| \\psi (t) \\right\\rangle.", "1793556bbdd9a64451ce294befbfc1b7": " n = \\infty ", "179392386646efb4d26cf987f20acf02": " \\operatorname{Cov}(X,Y)=\\operatorname{E}[XY]-\\operatorname{E}[X]\\operatorname{E}[Y].", "1793a877238d6da0fe259af3daf8db85": "\\mathrm{COP}_{\\mathrm{heating}} - \\mathrm{COP}_{\\mathrm{cooling}} = 1\\,", "1793c4200a4c56163c6cd7f77714a874": "\n\n\\mathbf{F}(\\mathbf{r}, \\mathbf{m}_1, \\mathbf{m}_2) = \\dfrac{3 \\mu_0}{4 \\pi r^5}\\left[(\\mathbf{m}_1\\cdot\\mathbf{r})\\mathbf{m}_2 + (\\mathbf{m}_2\\cdot\\mathbf{r})\\mathbf{m}_1 + (\\mathbf{m}_1\\cdot\\mathbf{m}_2)\\mathbf{r} - \\dfrac{5(\\mathbf{m}_1\\cdot\\mathbf{r})(\\mathbf{m}_2\\cdot\\mathbf{r})}{r^2}\\mathbf{r}\\right],\n", "1793c47ddcfebb7f8155f46e35e4ceb6": "C \\frac{dV}{dt}=0", "179402e04ecb6a630ac7344a8954555e": "x * y = y * x\\qquad\\mbox{for all }x,y\\in S", "179427f2fbc92ddc7ebac4730f462562": "\\Psi_{id} : \\mathcal{B}_2 \\rightarrow \\mathcal{A}_2", "17942debc698e3def4d4757acf18ca87": "A_{u}", "1794402f3638b827218820aa44994365": " Q_p = \\sum H_p - \\sum H_r = \\Delta H", "179474ed9fdcb4ad56770fe1228e1a75": "c_0", "179537b26ee04a6a1eefe6e8a2ddde69": "\\vec{A}_{||}=0", "17953a9b349b2821519c4d445618cd6e": "R=\\frac{\\eta_j}{c_p} \\frac{C_L}{C_D} ln \\frac{W_1}{W_2}", "17957d878bf4960e34d492340ca6a20d": " a,u,w", "1795d46584feaf59136988d6dba38568": "< \\cdot, \\cdot >", "17960f004fee00eab0fd8dc31fbc48a7": "V(\\phi) -0\n\\end{cases}", "17a04cd02c73f2dd6ad1b42cbf9f9813": "\\R^V", "17a05bd3f52ed80b340510ee96ee2ccd": "\\mathit{q}", "17a05f391c28aea7f21706b9dbdfa424": "\\tau \\sigma", "17a09206d578b43575f7a0c51640dfd2": "|\\alpha' \\rangle", "17a09af768b03dcfa184287545e42097": " {D'} ", "17a0e5b21c35c5084fd512a64b739918": "C_y = -a\\lambda\\ + A_y.\\,", "17a11337560bb7c4b9f3c3589b2d0888": "\\mathrm{Si-OH + OH-Si \\ \\xrightarrow{polymerization} \\ Si-O-Si + H{_2}O}", "17a152fc9de829d1970cff4cce1efb74": "\\operatorname{Spec}", "17a184a43993bddb45ac3ca884a4ac6b": "\\scriptstyle f_s/2,", "17a1d6e8a523d4194783fa994c8fc18c": "\\begin{align}\n &Y + \\sigma(100 - l_1 + \\beta) = X + \\sigma (100 - l_2 + \\beta) \\\\\n \\Rightarrow {} &X - Y = \\sigma(l_2 - l_1)\n\\end{align}", "17a20a60a422a3defa40e901baafdf6b": "\\theta(X)=-X^T", "17a2182ae342971e27f272fea6fc808d": "s(F)=\\infty", "17a2b4d73340f97c1610cee83578c08b": "\\left.\\frac{\\partial}{\\partial r_{\\alpha}}\\bar{\\rho}(r_{\\alpha})\\right|_{r_{\\alpha}=0} = -2Z_{\\alpha}\\bar{\\rho}(0).", "17a2e0b3ddf682d73a80dc7838b2557b": "p_1=1", "17a381949cb658f3d0481ce690136c70": "\\det\\big(\\mathrm{adj}(\\mathbf{A})\\big) = \\det(\\mathbf{A})^{n-1}, ", "17a3ce8d2e0ad6c889d99c68461f6a08": "a_j-a_i", "17a40acb3d14a0f613090927710d3cff": "\\mathbb Z_2", "17a4993817ab6428bb5403cc90569a55": "p(x)=x^3 -7x + 7 ", "17a4a7ba2a8f61ef70c0982be6a125a9": "(I,\\le)", "17a4d45d616eba0eb85a700663753697": "A = \\begin{pmatrix}0&M\\\\M&B\\end{pmatrix} \\text{,}", "17a4f92d86b83d69536e600092b5d32c": "i\\hbar \\frac{\\partial}{\\partial t}\\psi =\\widehat{H}\\psi", "17a5180712427043192a565defdef006": "\n \\sigma_{11} = \\left(\\lambda^2 - \\cfrac{1}{\\lambda}\\right)\\left(\\cfrac{\\mu J_m}{J_m - I_1 + 3}\\right)~.\n ", "17a59ae0a855620d0292bcb1063b200a": "F = \\frac{dW_f}{dt}", "17a5d4967911e7e20345d77861224ec9": "|x\\cdot y|\\leq|x|\\cdot|y|", "17a62842bc7e50de218dd3fd1867f110": "\\Pi'", "17a648e35d5b58384e77d0744f15a2c5": "\\displaystyle{\\sum (1-|\\lambda_i|) < 1}", "17a671a7916725910cd594ad1a71ce23": "\\ \\Phi", "17a689a53f7d8737b57380c3c7245e11": "disc(\\mathcal{H}) := \\min_{\\chi:V\\rightarrow\\{-1,+1\\}} disc(\\mathcal{H}, \\chi).", "17a698156de43e09b45d2228624d26f1": "|a_{i,j}|_{0,\\alpha;\\Omega}, |b_i|^{(1)}_{0,\\alpha;\\Omega}, |c|^{(2)}_{0,\\alpha;\\Omega} \\leq \\Lambda.", "17a6b2f3ab3bae8cacd196c239094d52": " \\mathbf{F}(\\mathbf{x}) = -\\frac{dU}{d\\mathbf{x}} ~. ", "17a6e3786fcee00d26de14049f89ebe1": "\\Phi:\\mathrm{hom}_C(F-,-)\\to\\mathrm{hom}_D(-,G-)", "17a70d8ced682a0a156863b6d2168ccd": "\\ q\\in M", "17a71577171bd97c71e95a5bbcc87083": "x \\not\\ll y", "17a74e3e623011c3c98da41e4c4888aa": "M=4", "17a81c48246c845c6e16d9fef4e07b5e": "\\epsilon^* - \\epsilon_\\infty = \\dfrac{\\epsilon_0 - \\epsilon_\\infty}{1 + (i\\omega\\tau_0)^{1-\\alpha}}", "17a82ce61a79c3c6ac15944a42512eb7": "\nm\\frac{du^a}{ds}=qF^{ab}u_b,\n", "17a836b4bbded5a128f4cffb33077b99": " A = \\sqrt{(s-a)(s-b)(s-c)(s-d)}", "17a8dc0d6d7b993f341d41b392189f51": "|x^{\\rho}|=\\sqrt x", "17a8e639d22844bb61cc5c7b8458ee60": "D = \\{z\\in \\mathbf{C} : |z| < 1\\}.", "17a8f8fa883a0c4dcc09cd8ee62f500f": "\\min_{x \\in X}f(x)", "17a92b25b2e3289c37dc5f44a1ccd28e": "\\sum_{P \\in C}{c_P [P]}", "17a941d7d1417fc0813745695592cda6": "\nQ=\\omega\\frac{\\rm maximum\\; stored\\; energy}{\\rm average\\; power\\; loss} = \\omega \\frac{W_s}{P_l},\n", "17a9bfda729499597088e43c0ea096fa": "\\omega = \\omega_{0} - \\nu", "17a9c2a57b6050eb05a980849218627a": "\\partial_ig_{jk}=(\\Gamma_{ij,k}=0)+\\Gamma_{ik,j}^*=\\partial_i\\partial_j\\partial_k\\psi", "17aa1504fb8fd391c95e17fb2c4081ed": "\\scriptstyle\\delta\\,", "17aa20a885a267996bffe03ac0f92a2e": "g(z)", "17aa75fca3cc0b4ad4ffce5e9538affc": "g^{\\cdot}:(\\mathbb{R},+)\\to(\\widehat{G},\\cdot) :\\lim_{i}q_{i}\\in\\mathbb{Q}\\mapsto \\lim_{i}g^{q_{i}}", "17aa8209a276e9c96addf6c2450b9adc": " K = -(4/9)/J,\\ ", "17aa82a77167a98516267f83b4d068ec": "k_{1H}", "17aa86dc1a0121a9375d5a602db9caae": " N_10 = 39", "17aaa2643df92cff6f25e31f91d5f001": "g_{nm} (k) = s(k-mN) \\cdot e^{j\\Omega nk}", "17ab56afce1cbcde6cf56efab9582d5a": "C^\\omega(X,Y)\\subset \\mbox{Hom}(X,Y)", "17ab83d68ab936d528e1d6c94516b424": " ([A_1,A_2], [A_3,A_4], [A_5,A_6]) =0 ", "17ab9b118b7b926df7ff0cd2c06d8ff0": "E = \\sum_{i=1}^{k} \\sum_{p \\in C_i} (p-m_i)^{2},", "17aba57fa3ba1e52436bde651dd534d1": "\\mu S+\\lambda S", "17abd174add9feacc3e0cc97d1006db1": "\n\\operatorname{E}[\\operatorname{dCov}^2_n(X,Y)] = \\frac{n-1}{n^2} \\left\\{(n-2) \\operatorname{dCov}^2(X,Y)+ \\operatorname{E}[\\|X-X'\\|]\\,\\operatorname{E}[\\|Y-Y'\\|] \\right\\} = \\frac{n-1}{n^2}\\operatorname {E}[\\|X-X'\\|]\\,\\operatorname{E}[\\|Y-Y'\\|].\n ", "17abd966bd70d77a73b47d95d9938945": "V_{a,b}\\subset Y", "17abe57b40fd9d840c7150a52489a424": "\\chi^*", "17ac2f84363af2816dc33f64149a7afe": "\\mathit{KS}(\\mathit{PRIMES}) \\leq 167", "17ac68370640535beb63f1bf83f34b59": "x_i = \\frac{\\rho_i}{\\rho} \\cdot \\frac{M}{M_i}", "17ace7e621c20488079195db5269a5e3": " \\mu = \\sin \\varphi ", "17ad6396d0bc8018ae6f3b0cb6605452": "\\int_{S^2}f^*\\omega=\\langle c_1(TM),f_*[S^2]\\rangle=0", "17adacdc69beb819166653c9eba94116": "\\lambda_i\\,\\!", "17aded73356e21e73d7a4bee820dc865": "\n\\ NEP = \\sqrt{4 k_B T^2 G}\n", "17ae139928090f55ffd98a1aeef2def6": "((A\\to B)\\to C)\\to(D\\to((B\\to(C\\to E))\\to(B\\to E)))", "17ae8a5a7840c436d16bccba8948e7a8": "[h,e]v=hev-ehv=2ev", "17aead184440e2929e84471fa1beb937": "SO(2n) \\supset SU(n)", "17b005d3350f6a036823c05e0a250413": "\\beta=\\frac{2a \\pi ^2(2mE)^{1/2}}{h}", "17b00cad4f9dad1cfae944abdff28ba4": "O(\\sqrt{|S|})", "17b03549ad2fdaec017297e23f7e16c6": "0\\rightarrow B\\rightarrow X''_n\\rightarrow X_{n-1}\\oplus X'_{n-1}\\rightarrow\\cdots\\rightarrow X_2\\oplus X'_2\\rightarrow X''_1\\rightarrow A\\rightarrow0.", "17b08a7e7f246ac9b0dedae92fb24176": " \\mathbb{C}^m ", "17b0ab432dd3922b037c1d0fdf63d83d": "\\operatorname{gr}_I R = \\oplus_{n \\in \\mathbb{N}} I^n/I^{n+1}", "17b0e656c34493ed5b35d97611e857ca": " \\chi_{\\mathrm{top}}(x,y) = x . \\, ", "17b0ef287ca1f56a02f649f56164e0d3": "\\nabla f(x^*) = \\sum_{i=1}^m \\mu_i \\nabla g_i(x^*) + \\sum_{j=1}^l \\lambda_j \\nabla h_j(x^*),", "17b1036d35e80e298e933c6204c02533": "E_{n,\\mathbf{k}} = E_{n,0}+\\frac{\\hbar^2 k^2}{2m} + \\frac{\\hbar^2}{m^2} \\sum_{n'\\neq n} \\frac{|\\langle u_{n,0} | \\mathbf{k}\\cdot\\mathbf{p} | u_{n',0} \\rangle |^2}{E_{n,0}-E_{n',0}}", "17b11950df5099f23d85e7778d2bc537": "a _ {(2)} 0 = 0", "17b1364578f4405182b834ad2d41b20e": " g(t) = \\ln M(t) = \\mu t + \\frac{1}{2} \\sigma^2 t^2", "17b1c1f2a7934842f633b148e5db8c94": "q = \\sgn(y) \\left\\lceil \\left| y \\right| \\right\\rceil = -\\sgn(y) \\left\\lfloor -\\left| y \\right| \\right\\rfloor\\,", "17b1fe3ec31b0960e20f3f520c275e25": "R = \\mathbb{Z}\\left[\\sqrt{-3}\\,\\,\\right],\\quad a = 4 = 2\\cdot 2 = \\left(1+\\sqrt{-3}\\,\\,\\right)\\left(1-\\sqrt{-3}\\,\\,\\right),\\quad b = \\left(1+\\sqrt{-3}\\,\\,\\right)\\cdot 2.", "17b28cacf3d4e06ca759108cd4440a57": "A \\subsetneq \\mathbb{R}^{n}", "17b292ded2032405051317f9eb1cd065": "F = F^{ab} e_a \\wedge e_b \\quad (1 \\le a < b \\le n) ", "17b29a26c1c7f92ed3df3a380276deb7": "\\displaystyle{(F_n,F_n)=(XY^n F_0,Y^{n-1}F_0)=2n (F_{n-1},F_{n-1}).}", "17b29b44e301ca998d62104d1762f0f7": "-2\\,", "17b2b959727f67db65ca4c9b4b5783c4": "\\begin{align}\n\\nu_0' &= \\nu_0 + n \\\\\n\\nu_0'{\\sigma_0^2}' &= \\nu_0 \\sigma_0^2 + \\sum_{i=1}^n (x_i-\\mu)^2\n\\end{align}", "17b2e36d3fde45bfa18cc296c7c037fb": " A = \\sum_{r=0}^{n} \\langle A \\rangle _r ", "17b306cb61dc66cc75bd7094b9b2b385": "\\text{x quus y}= \\begin{cases} \\text{x + y} & \\text{if }x,y <57 \\\\[12pt] 5 & \\text{otherwise} \\end{cases} ", "17b320ff8a083e1cdedf8ff6cf13cdae": "a - bi", "17b33b52d1c4e76ceaa666d8b99c8e10": "0 =\\frac {\\dot Q_L}{T_L} - \\frac {\\dot Q_a}{T_a}+ \\dot S_{i}.", "17b3673d5c28fb72e7cc48549f489dd4": "6/8", "17b379f0674e898b05296b3f08f29ffd": " 0 < C < 1", "17b384259930066e915c92a2b592b8da": "\\text{E}U(W_T) = \\text{E}U(W_0R_1R_2 \\cdots R_T),", "17b3947bb48fbf8c49a7d3c7e04f2d96": "SHA_d(message) = SHA(SHA(0^b || message))", "17b3af4220adc8a4f0087a9f6ccbff1c": " C^1 ", "17b3b672383255c065d75c12dbaf877c": "T_{2A}", "17b3c8af3c291305252228e91d5894c0": " x = \\cos^3\\theta, \\quad y = \\sin^3\\theta.", "17b41ec9f9f1714067a069952816e9f2": "\\int_0^{\\theta}\\log(\\tan x)\\,dx=-\\tfrac{1}{2}\\text{Cl}_2(2\\theta)-\\tfrac{1}{2}\\text{Cl}_2(\\pi-2\\theta)", "17b4248640fd6a50dc5166c8f47656e6": "F_4\\,", "17b4b203de4dd8650b172a4dd671a6b8": "i_c(U_g)=-C_F(dU_g/ dt)", "17b516810a0d5d036805c9a4285baa0e": "N_m", "17b530425cc4459edd0ecab981b190dd": "j\\in\\Lambda", "17b5984ef31861e09ef4e528c72de78a": "g^{(2)}(\\tau)= g^{(2)}(-\\tau) ", "17b59e231ad5afb35f7d4de462e1fefc": " |V|\\geq 2 ", "17b5f85a17684c97647e768f49a97749": "\\mathbf{v} = \\left(\\frac{d x}{d t}, \\frac{d y}{d t}, \\frac{d z}{d t}\\right)", "17b61033810748f99fb169946b443b2a": "b\\Rightarrow \\lnot c", "17b635119b77ff5ca5f140ceaaaf5f75": "P = \\frac{q^2 \\gamma^4}{6 \\pi \\varepsilon_0 c} \n \\left( \\dot{\\beta}^2 + \\frac{(\\vec{\\beta} \\cdot \\dot{\\vec{\\beta}})^2}{1 - \\beta^2}\\right),", "17b649a0b5a7a3a602a95aeebc853ee1": "P(\\vec y)", "17b6a916526da17c8b33e880ff550718": "C' = E(P')", "17b6c7afdd706c907dc8ffc226eb6be8": "|B|", "17b6f761895267dbe18ac18b873cdd26": "P(X=x).", "17b7139ec3e2c52c2e131b4f8102cd90": "x=\\beta^nd_n + \\cdots + \\beta^2d_2 + \\beta d_1 + d_0 + \\beta^{-1}d_{-1} + \\beta^{-2}d_{-2} + \\cdots + \\beta^{-m}d_{-m}.", "17b74d6d6e8aae6fe69598edb594ed19": "S = k[z_2, ..., z_m]", "17b753fbe6c2f5f6e34e8c89a86ccc37": "v=x+\\sum_{k=1}^\\infty\\frac{y^k}{k!}\\left(\\frac\\partial{\\partial x}\\right)^{k-1}\\left(f(x)^k\\right)", "17b7649873a4ffdc8a0c3eaa788b2a18": " G(z) = \\log \\frac{1}{1-g(z)} - \\frac{1}{2} \\log \\frac{1}{1-g(z)^2} =\n\\frac{1}{2} \\log \\frac{1+g(z)}{1-g(z)}.", "17b77e8b5eff8afd676c91f049ccdeb5": " \\begin{align}\n y_1 &= y_0 + hf(t_0, y_0) = 1 + \\tfrac12\\cdot1 = 1.5, \\\\\n y_2 &= y_1 + hf(t_1, y_1) = 1.5 + \\tfrac12\\cdot1.5 = 2.25, \\\\\n y_3 &= y_2 + hf(t_2, y_2) = 2.25 + \\tfrac12\\cdot2.25 = 3.375, \\\\\n y_4 &= y_3 + hf(t_3, y_3) = 3.375 + \\tfrac12\\cdot3.375 = 5.0625.\n\\end{align} ", "17b8185d03cae433573080f13689496e": "\\frac{a}{c} + \\frac{bc - ad}{c} \\left [ \\frac{(cx - a + \\alpha)\\alpha^{n - 1} - (cx - a + \\beta)\\beta^{n - 1}}{(cx - a + \\alpha)\\alpha^{n} - (cx - a + \\beta)\\beta^{n}} \\right ]", "17b84defa2e0dae6a310db2b00ef499d": " e^{-idtk^2}F[e^{idt\\hat N}\\psi(x, t)]", "17b8680fb05cd0387e0b90ee8bc1b519": "( u_i , v_j ) \\!", "17b8af93f0cc289cc17cfa3dc8a637f6": " \\scriptstyle\\mathcal L", "17b8b653163192bdc68546cdaeed51cd": "w\\infty + z\\infty = (w+z)\\infty", "17b915ec7a11610ea3919a4d05da53b9": "\\mathbf{Spec}\\cong \\mathbf{Pries}\\cong \\mathbf{PStone}", "17b998de51c99550d065eb5953b6195f": "\\mathrm{v}", "17b99e166258f650036939b57689bdec": "A_{2}", "17b9c41f280a0a10f1ee92c4ad47d166": "I:\\aleph_0^\\text{op}\\rightarrow L", "17b9d9ac7cab51ca9d5d2a3530a106c3": "I_{\\mathbf{Q}}:[0,1] \\to \\mathbf{R}", "17ba69868c214e00b63b76b944e09aa0": " \\left(\\mathbf{A}_1\\mathbf{A}_2\\cdots\\mathbf{A}_n\\right)_{i_0 i_n} = \\sum_{i_1=1}^{s_1}\\sum_{i_2=1}^{s_2}\\cdots\\sum_{i_{n-1}=1}^{s_{n-1}} \\left(\\mathbf{A}_1\\right)_{i_0 i_1}\\left(\\mathbf{A}_2\\right)_{i_1 i_2}\\left(\\mathbf{A}_3\\right)_{i_2 i_3} \\cdots \\left(\\mathbf{A}_{n-1}\\right)_{i_{n-2}i_{n-1}}\\left(\\mathbf{A}_n\\right)_{i_{n-1}i_n} ", "17ba8482e8d5bc8a3df87219404b7add": "t_{ik}(x) = t_{ij}(x)t_{jk}(x).\\,", "17baa021aa006cc59cf389712ec84848": "F(\\mathbf{x})= \\frac{1}{2}\\mathbf{x}^TA\\mathbf{x}-\\mathbf{x}^T\\mathbf{b}", "17baaa2eb890d9213c96e7603926ba8f": "H_n\\left(S^n\\right)\\cong\\mathbb{Z}", "17bacc08d04b46497c24d78bc7fbd104": " 0 \\leq D_{KL}(f || g) = \\int_{-\\infty}^\\infty f(x) \\log \\left( \\frac{f(x)}{g(x)} \\right) dx = -h(f) - \\int_{-\\infty}^\\infty f(x)\\log(g(x)) dx.", "17baf154e9ec5f7e5a803a6a5a6f73ae": "\\textrm{Cauchy}(\\mu,\\sigma) \\sim \\textrm{t}_{(df=1)}(\\mu,\\sigma)\\,", "17bb84ad5d7806ef7c833a6637a02add": "\\varepsilon \\longrightarrow 0", "17bb9136e71d308ef2a4d36f33feeaa9": "\\frac{\\mathrm{d}y}{\\mathrm{d}X} = y(1-y)\\frac{\\mathrm{d}f}{\\mathrm{d}X}. \\, ", "17bb9a6cb87193db905785ea06ea4d36": " (V_R)^i_j - (V_A)^j_i ", "17bc1dea0303712a9b05ffee5bf9aaa6": "(\\phi \\to \\chi ) \\to ((\\chi \\to \\phi ) \\to (\\phi \\leftrightarrow \\chi ))", "17bc220fb0a441d07dda06b659a47ec5": "g = -G \\cdot M/r^2", "17bc6c9d83379f431daad3c7d94d2cc4": "\\scriptstyle\\mathbf{H}_{\\textrm{estimate}}", "17bc860125f4fa3fa09cbea5fc2ca9d2": "\\widetilde{\\gamma}(0)", "17bcc6e776ce567c1790b7b4c1ba675e": "Y_t = \\mu + \\int_{A_{t}(x)} g(\\eta,t-s,x)\\sigma_{s}(\\eta) L(d\\eta,ds) + \\int_{B_{t}(x)} q(\\eta,t-s,x)a_{s}(\\eta) \\, d\\eta \\, ds", "17bce3012ceb444221a8d9d99789fe7e": "\\widetilde{\\lambda} ", "17bd112a664ba4bc180abbc0e40578a3": "D_{pr}", "17bd3fa0694156b0d2c32ecd1c3f4c37": "x^5 + c_2x^2 + c_1x + c_0 = 0 \\,", "17bd6aaefbcb537765bc764b765eaf81": " \\max_{x \\in J} |p(x)| \\leq \\left( \\frac{4 \\,\\, \\textrm{mes } J}{\\textrm{mes } E} \\right)^n \\sup_{x \\in E} |p(x)| \\qquad\\qquad(*) ", "17bd8244ca43d1c1feee57e8109b92ac": "\\varphi_1=\\varphi_2", "17be220b367ab7eb7164be07306e527b": " \\Diamond P ", "17be2e1cc038e49e99996cd87e0b806d": "\nu = \\Lambda \\Lambda^\\dagger,\n", "17bece06ee7f74a4903ac340f6eb4c84": "\\{ v_1,\\ldots, v_{k+1},w_{k+2},\\ldots,w_n\\}", "17bf0d6faffe3250ad4831afb048a73e": "{CE}_{n}", "17bf1aafd43252d4cc1766f085a6c458": "E\\ ", "17bf2f73fc14052b691b96411f527c3c": "x_1 \\not=x_2", "17bf720bb505942f7ceaa5fa02ef8db7": " -\\frac{ \\beta }{ \\beta + 1 } ", "17bf9966bd5523507bbfc3c38f5c11d2": "\\int\\frac{\\cos ax}{x} \\mathrm{d}x = \\ln|ax|+\\sum_{k=1}^\\infty (-1)^k\\frac{(ax)^{2k}}{2k\\cdot(2k)!}+C\\,\\!", "17bfecdcfb48d00a21b8b869b5df5daf": "m_\\mathrm{tot}", "17bff35e08b8023c479e2dde8c40c9fc": "X_6 = \\$0.50", "17bff462408fcf1a6f676ea989bbf648": "b=\\mathbf{r}_0\\cdot\\mathbf{v}_0", "17c080941341a4449d1ff36fd84eadba": "\\scriptstyle I_\\text{enc}", "17c0ac1344c7f640992ee632e596b75b": "e_b(k,i)\\,\\!", "17c0bbb8bffd337e30b3e6144bdf98ac": "h_{-1}", "17c0c59de18c5c61253ccdcaa90ae2ec": "W = \\tau\\theta. \\!", "17c0e7aa154abbfca74fce94869d8a99": "j((1+\\sqrt{-163})/2)=(-640320)^3", "17c11ef0d8a46e564c37f62f184a3a3d": "\nV(t) = V_0e^{-t/\\tau}+Ae^{-t/\\tau}\\int_0^t \\, dt'\\ e^{t'/\\tau} e^{j\\omega t'} ", "17c137bdbdcc2a7af70caf61b0fb6571": "{\\mathfrak d} ", "17c1631f648ec5c972843d118d821316": "\\phi(\\mathbf{y},\\dot{\\mathbf{y}},\\mathbf{y}^{(\\gamma)}) = \\mathbf{0}", "17c1912bef8757efaa266089a05e0335": " Normal^{}_{}(0, \\sigma^{2} \\delta) ", "17c1c15b827dcd738cb1c0fe211af656": "y(t)\\,\\!", "17c1d359409a0d90975385807111d2e9": "S_{CQB} = \\frac{z}{yz+z+1}", "17c22fd3ecad8dbbc9e4786a58d6aeda": "X(T) = \\{ P_{N-1} , P_{N-2} , ... , P_{N-T} \\}", "17c2b927bc5f87f6be48b69cc7daf38c": " \\left[{n \\atop k}\\right] = \\left|s(n,k)\\right| .", "17c2dfef45b1b363711521464b9209f0": "K_p = \\mathrm{\\frac{p(NO_2)^2}{p(N_2O_4)}}", "17c2e0a209c737d6d6a9c38813b5b601": "d_3", "17c319d4386cfd3be08b4e72658a3d69": "\\mathrm{SO}(2n,\\mathbb C)", "17c33c08a5a3074e881e2c7530c3bff4": " T_{dp:f}\\approx T_{f}-\\frac{9}{25}(100-R\\!H);", "17c34ac7c515bdbad423302548010a51": "\\ \\zeta", "17c35572b75208fd617125290350254c": "\n\\frac{d t}{d \\tau} = \\left\\langle \\frac{p}{m} \\right\\rangle_S < 0\n", "17c391a9b37fa2c1b1406e38646c508c": "D^-f(t) \\triangleq \\limsup_{h \\to {0-}} \\frac{f(t + h) - f(t)}{h}", "17c400798802f0f57618855e38d0dd2d": "\n \\mathcal{P} = \\Big\\{\\ f_\\theta(x) = \\tfrac{1}{\\sqrt{2\\pi}\\sigma} e^{ -\\frac{1}{2\\sigma^2}(x-\\mu)^2 }\\ \\Big|\\ \\mu\\in\\mathbb{R}, \\sigma>0 \\ \\Big\\}.\n ", "17c404b5557ce2ba595af555a3900529": "m\\ddot{x}_j=k(x_{j+1}+x_{j-1}-2x_j)[1+\\alpha(x_{j+1}-x_{j-1})]", "17c4455eee7df04570466b8813f44fd4": "dt = \\frac{\\mathrm{d}v}{g - \\frac{kv^2}{m}}", "17c454d47bc2c11ccffbc4e79285729e": "\n\\left\\vert \\Phi^{+}\\right\\rangle ^{BA}\\left\\vert 00\\right\\rangle\n^{A}\\left\\vert \\psi\\right\\rangle ^{A}.\n", "17c48cb446def2f4705f94a428a39eb8": "\\frac{R_x}{R_g + R_y} = \\frac{R_{B1}}{R_{B2}}", "17c4f6323042441ecc73cc0ad3ea1937": " L_\\text{shd} \\, ", "17c50171dd99868a958e0264ce4810b4": "C_3\\,", "17c525d4a9c29902bec0ba6221908fa5": "\\displaystyle\\frac{(n + 2)(n-1)}{2n}", "17c57b31a421edeefb6e192b2beb0a83": "*2222", "17c5b8360f5cdbf285fa2422a6c8875f": "8r\\le a+b+c+d \\le 8r\\cdot \\frac{R^2+x^2}{R^2-x^2}", "17c5e5068f279fab07a07d35ff3989b1": "\\mathbb{P} (\\vartheta_{s}^{-1} (E)) = \\mathbb{P} (E)", "17c60af8fac70fec9c3e9b8ca1aa7530": "\\tan 2A = \\frac{2 * \\frac{3}{7}}{1 - (\\frac{3}{7})^2}= \\frac{21}{20};", "17c61e39f9cf40cab4a05059a53d8376": "Y = \\text{Output}", "17c67b2bab1aebc4e1f7b5eb6fee0117": "q^{(p-1)/2} \\equiv (-1)^{\\sum_u \\left \\lfloor qu/p \\right \\rfloor} \\text{ (mod }p).", "17c68304c4529a455a48bca26140fca8": "\\frac{N}{C} \\ge 1.0", "17c6af449b4cd6a11cea0ab961ad4748": "\n \\left. \\left[ k_{c}(2H+c_{0})+\\bar{k}k_n\\right]\\right\\vert _{C}=0\n", "17c6d3da4310dd3a50677c17124d7d1b": "\\textstyle p:2q:r = p_1:2q_1:r_1", "17c6e53aae3864376a8f622096aadc59": "\n= \\sum_i |\\lang \\psi |i \\rang|^2|\\lang i|\\varphi \\rang|^2 = \\sum_i|\\psi_i|^2 |\\varphi_i|^2\n", "17c705e5800fcfa64be2b39ebb2260ab": "1/\\mu", "17c73c55ae4d30d27532f2f538a13d9a": "\\operatorname{H}[\\mathbf{X}] = -\\ln \\left (B(\\mathbf{V},n) \\right ) -\\tfrac{1}{2}(n-p-1) \\operatorname{E}[\\ln|\\mathbf{X}|] + \\frac{np}{2}", "17c77fdbc4369d09c0eaa4476f071f50": "\\arccot x = \\frac{\\pi}{2} - \\arctan x ", "17c8012f5ad9fda667a889e3100c98c4": " r_- ", "17c80d41500d6d7a9a9c6b6f98404ae8": "\\int_0^1 \\rho^{D-1}R_n^{(l)}(\\rho)R_{n'}^{(l)}(\\rho)d\\rho = \\delta_{n,n'}", "17c80d92eb4858d09f08dec199b0ef2e": "L = 2\\pi/k ", "17c87d31e54b48968dfbbdc8a9402b47": "PWV = \\dfrac{\\Delta x}{\\Delta t}", "17c8942c9ed21522767ba16348df9fc8": "S = \\int d \\tau \\Big[ {dx \\over d \\tau} p + {dt \\over d \\tau} p_t - \\mathcal{H} (x,t;p,p_t) \\Big]", "17c8a1e9f50d9481d42c2e20562a0ffe": "Z = -20 + \\tfrac{2}{3} x + \\tfrac{11}{3} y + \\tfrac{4}{3} t", "17c8d71b3cb9e7cc90744c303e545195": "\\textstyle\\binom mk", "17c99259a31002d4a2c3f6808be06073": "x\\ne0\\ne y", "17c9a81794db833e07f6f584a6a1f5e5": "Cl_3^{\\geq}", "17c9f5dfb9a82e9274d56673d21f43d6": "f^*=\\prod_{k=1}^n(X-\\alpha_k^*)", "17ca14f8aac63565748d7ddfd947bdd9": "\\nu_m > \\nu_c", "17ca6282072bbed909f3ef091779fc2a": "L_{GD}^*", "17cb493a06665211bf75c645fe365aea": "x+y+z \\leq 1,\\,", "17cb5c634ffb0882870156bfc5c6d026": " \\mathbf{S} ", "17cb5ce8cfac29649bd24e8d99371861": "|r|<\\sigma R", "17cb78d49103e560c1607f407a29fd84": "E_y=E_z=0, E_x", "17cba223b15f99c9d3e05f8d81a31f15": "I=\\sigma_1\\sigma_2\\sigma_3", "17cbad5a6ccad13458f01b9984c3ddf0": "\\,\\{A+iB\\} = e^{i\\theta}\\{A+iB\\}", "17cbdb44f384e6ae465e3a6fad5f0e3b": "VCA(64x^3+576x^2-64x-64,(\\frac{3}{2},2)) ", "17cc2f79a63a0b9472935ee3da247399": "\\text{Var}(X_i) = \\frac{K_i}{N} \\left(1-\\frac{K_i}{N}\\right) n \\frac{N-n}{N-1} ", "17cc4c2a6905b10f8868fabdde4a8d45": "k\\rightarrow\\infty", "17cc679caf1c157c79d1b8f5ee6ad837": "{_4^2}\\text{S}^{\\beta}", "17cc6e28087828386c5c61be16e76e64": "\\sum_{i=1}^{n-1} i^{\\varphi(n)} \\equiv -1 \\pmod n", "17cc7292d05c904e019e499b3c54e4b2": "\\begin{align}\n dy \\, dz &\\hat{\\mathbf x} \\\\\n+ dx \\, dz &\\hat{\\mathbf y} \\\\\n+ dx \\, dy &\\hat{\\mathbf z}\n\\end{align}", "17cd31ca2ad4c993f41a911921171370": "\\alpha = (K_{eq} \\cdot \\Pi \\sigma_{Products}/ \\Pi \\sigma_{Reactants})^{1/n}", "17cd4c40578493bfa013b53eb2a18d1d": "\\hat{\\rho}_k", "17cdaa30a3eb4051f92a1ad6ec3eea8f": "\\sum_{x,y}z(x,y)<\\infty", "17cdc70546189b942bd85708596e5d7d": "S_2 = \\frac{A}{S_1} = \\frac{ \\frac{\\pi ^2}{6} - 1}{\\frac{5}{6}} = \\frac{\\pi ^2 - 6}{5} = 0.77392088021\\cdots ", "17cdf36a23aaa5a7bb966970166c0bc0": "\\frac{d}{dx}\\delta(-x) = \\frac{d}{dx}\\delta(x)", "17ce0156c6dcc475ca036f4501ef3ebf": "\\rightarrow KR_{n-1}^G(\\coprod_{j\\in I_i} G/H_j\\times S^{i-1})\\rightarrow KR_{n-1}^G(\\coprod_{j\\in I_i} G/H_j\\times D^i)\\oplus KR_{n-1}^G(X^{i-1}) ", "17ce27cb1c765a63f8281d55549733ca": "\\gamma_{x}^{-} := \\{\\Phi(t,x) : t \\in (t_x^-,0)\\}", "17ce668a51bb6d0b0444a59cc73462f8": "\\tfrac {K_5}{K_4 + \\text{Reliability}}", "17ce9c5b987418f79e631f97db2d47b5": "Q_0=\\Delta U+W+W_0=W+W_0=W_\\text{total}", "17ceb532aa3b3c830a707488b50fd0f5": "\\frac{d\\beta}{dt}= -r", "17ceb7ba0753f4512e4359bd74969db8": "\\mathrm{d}H = T \\mathrm{d}S + V \\mathrm{d}P.", "17ceca71c3b34dcff1b82d7924668885": "\\psi(\\Omega+\\omega)", "17cf0ad1e9531bb5a463c43fd23f6fde": "\\scriptstyle (\\phi,\\, \\theta,\\, \\psi)", "17cf3b08f51ab9b8b1d3f0e6501721f7": "\n\\alpha = (-i \\tau)^{\\frac{1}{2}} \\exp\\!\\left(\\frac{\\pi}{\\tau} i z^2 \\right).\\,\n", "17cf4602a3e3b1bcbc0e72b28a774fde": "H(s) = \\frac {1}{1+\\alpha s}", "17cf4afaf1b8e94b4fdaabef8ebd31c1": "\\mathbf{r} = \\mathbf{r}_1 - \\mathbf{r}_2 ", "17cfc942478b711d8bf4422277d4778b": " Y(t) = R_n(t) - \\frac{R_n(x)}{W(x)} \\ W(t) ", "17cfd35a502a12aa5a05c07941a1d7df": "\\Lambda^k(V^*) \\otimes \\Lambda^n(V) \\to \\Lambda^{n-k}(V). \\, ", "17cfdb75c84613db2b01f5fc846927fb": "{\\omega {L}} = {{1} \\over {\\omega} {C}}\\,", "17d010993125eb26c35e811ae20210d5": "k \\in \\mathbb{Z}^n", "17d05096790cbdd8958e957958486f35": "\\beta, ", "17d0762a42b1dba98752b272708d079b": "f(x|a,b)= a b e^{-(b e^{-ax} + ax)}\\,", "17d08c915c3a43915454ebfb586ea269": " \\acute{P}_t = \\acute{P}_0 + \\acute{V}_0T_t + \\frac{1}{2}\\acute{A}_0T_t^2 ", "17d0bcc02bc07c0b74d5f33cbc060307": "m=0,\\dots,n", "17d0d1ea51e81fb2acbbb612f626e0c9": "k+k_0", "17d0e11fa183629e20c5d1d6dc67a09e": "\n\\left[ \\begin{array}{ccc|c}\n2 & 1 & -1 & 8 \\\\\n-3 & -1 & 2 & -11 \\\\\n-2 & 1 & 2 & -3\n\\end{array} \\right]\n", "17d1429efeeffd4856a2b694a256b90f": "B_{n+1}=\\sum_{k=0}^{n} \\binom{n}{k} B_k.", "17d146961d6fd0989e8a614104483829": "\\boldsymbol{R}^{-1} = \\boldsymbol{R}^T", "17d16d4daa010d46d058e890010dc3be": "f(z) = \\sum_{k=0}^n f(z_k)\\frac{G_n(z)}{(z-z_k)G'_n(z_k)}", "17d192bcb49d19a39379bd72c95c7302": " \\mathbb Z ", "17d1b5e8b5708d0f5ff55acaa623845b": "\\lim_{\\| \\Delta x_i \\| \\to 0} F(b) - F(a) = \\lim_{\\| \\Delta x_i \\| \\to 0} \\sum_{i=1}^n \\,[f(c_i)(\\Delta x_i)].", "17d1ef843c6f0b7c5aad448a3add820d": " \\left(e'_w\\right) ", "17d21ebc44fa5379d9cb9ad36628829a": "X_k=p(e^{2\\pi i\\tfrac{k}{2^n}})", "17d2207c50da3999aa28e73690766eaa": "g(t,V)=\\bar{g}\\cdot m(t,V)^p \\cdot h(t,V)^q", "17d27ca75f435cede1f55a245f5b17e0": " y^e(\\mathbf{x}) ", "17d28213f49f4ce6885e4fb73e47fb92": "R_{tot}", "17d28b9167937faaa8b7b4c2c535d0c8": "p^{2}/(m\\gamma)", "17d2c51392b33917e88fe1af02247eb6": "g = {\\frac{4\\pi}{3} G \\rho r}", "17d3351614d0cbdb29faca85af012655": "M_{ij} N_{ij}", "17d356e3538088f76021a7460a7aa73c": "k=\\frac{2\\pi}{a}", "17d397110a74f0a092f88e83498f5b97": "D_n = n^2 + 4(n^2 - n)", "17d39d28ebbdd22d46434f959a8ffd20": "|x|=1,", "17d3c881aafd93da5b5f7fc26e4b4634": "\\mathbf{P}(t) = \\int_0^t \\mathbf{V}(t) dt = \\int(\\mathbf{A}t + \\mathbf{V}_0)dt = \\tfrac{1}{2} \\mathbf{A} t^2 + \\mathbf{V}_0 t + \\mathbf{P}_0. ", "17d3f7c9e9890baaecd46cd73efc65f4": "{ -1 , 1 }", "17d40c630ff3797e876aec68840cb050": "f(y,x)=y^2+x^2-r^2", "17d40dcb94593a9fe4c7310f00df1058": "\n\\delta \\varphi \\approx \\frac{3\\pi m^{2} c^{2}}{2L^{2}} \\left( \\frac{4G^{2} M^{2}}{c^{4}} \\right) = \\frac{6\\pi G^{2} M^{2} m^{2}}{c^{2} L^{2}}\n", "17d457e6178abe32066f660033985f3e": "\\mathbb{Z}_1", "17d4e88907d03a20732396e906541e4b": "P_{\\text{x,t}}=\\frac{E_{\\text{x,t}}}{R_{\\text{x,t}}[1-T]}", "17d520aa4f3eafa7af25c085f70ce613": "N> 1", "17d5525e412071fd1062277888a78a18": "\\mathrm{P}(C|AB) = \\frac{\\frac{1}{16}}{\\frac{1}{16} + \\frac{3}{16}} = \\tfrac{1}{4} = \\mathrm{P}(C)", "17d5974b1ec85d536ce08ce952a7a1f2": "f(T)\\leq f(S)", "17d597592b8d90e504ea2213fef9ad7e": "\\scriptstyle{\\tau \\rightarrow \\infty}", "17d59f02b8d46a7461189ef24ebf4176": " (-1)^m n! \\; [z^n] [u^m] g(z, u) = \n\\left[\\begin{matrix} n \\\\ n-m \\end{matrix}\\right]\n", "17d5e5c0307c4829aa24ba82a0d121fb": " [A,BCDE] = [A,B]CDE + B[A,C]DE + BC[A,D]E + BCD[A,E]", "17d648ea50b87e97505a799d87654435": " = \\frac{1}{A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} \\left(\\vec{\\nabla}_{\\vec{r}} \\bullet{} \\frac{\\vec{r}-\\vec{r}'}{|\\vec{r}-\\vec{r}'|^n}\\right)\\vec{F}(\\vec{r}')d\\tau'", "17d6987bc53318b1760eb61d1bafe508": "\\left(\\frac{d}{dq} q- q \\frac{d}{dq} \\right)f(q) = \\frac{d}{dq}(q f(q)) - q \\frac{df(q)}{dq} = f(q) ", "17d6f7f139aaae95eebdbfdfba085d39": "x^2 + y^2 = 0.", "17d79e89fec3c320ef63926fd3862a72": "Y_1=\\frac{1}{T_1}", "17d7f6111f47230fd65ada0707fd1e61": "(x \\cdot f)(v) = x f(v) - f (x v)", "17d7f64b69ae5e67c2ab0b2acd2e2abe": "\\rho: 2^X \\rightarrow 2^Y", "17d808103292821dacc048df5fc5b40a": "\n \\boldsymbol{\\sigma} = -p~\\boldsymbol{\\mathit{1}} + 2C_1\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~(3+\\gamma^2)^{i-1}\\right]~\\boldsymbol{B}\n ", "17d8413d5243ef4f856d002265462720": "S_{a/\\$}", "17d85532452294ee6899ee5305b835b0": "U_1 (\\mathbf r,t) = A_1(\\mathbf r) e^{i [\\varphi_1 (\\mathbf r) - \\omega t]}", "17d86fb528165239ebc895b4b0dc8e23": "\\mu_Y^\\pi", "17d884a358c3b723f0a9e53102c8bfad": "T_O", "17d8bc79fdd575e57161dfa63642a306": "b \\left ( \\alpha\\, \\right ) = \\frac{\\alpha\\,}{{2^N}} : 0 \\le \\, \\alpha\\, \\le \\, {2^N}", "17d94809652ba924fae9a8de54a13295": " e^{\\Lambda} = 1 + \\Lambda + \\tfrac{1}{2} \\Lambda* \\Lambda + \\tfrac{1}{3!} \\Lambda * \\Lambda * \\Lambda + \\ldots ", "17d9c075c38185fc7315e5888f9537e0": "A>0", "17d9f208d6b981f4e7ab37a914cac6a1": "k_{\\lambda}.(v \\otimes w) = k_{\\lambda}.v \\otimes k_{\\lambda}.w", "17da46b46e9e71bdc29ed3087b7f17dd": "\\sup_{x_1,x_2,\\dots,x_n, \\hat x_i} |f(x_1,x_2,\\dots,x_n) - f(x_1,x_2,\\dots,x_{i-1},\\hat x_i, x_{i+1}, \\dots, x_n)| \n\\le c_i \\qquad \\text{for} \\quad 1 \\le i \\le n \\; .\n", "17da6fbed2142f22acfaaca480d2760c": " H[A] = \\int_{- \\infty}^\\infty S(Cr \\lbrace A \\geq t \\rbrace )\\,dt.", "17da84f07ba0cb6220c486de4f46bb6d": "\\Delta Q=Q\\left(\\frac{1}{\\eta}-1\\right)", "17dab9ed9db2de2a595daf012fbe23f6": "36 \\cdot V^2 =\\begin{vmatrix}\n\\mathbf{a^2} & \\mathbf{a} \\cdot \\mathbf{b} & \\mathbf{a} \\cdot \\mathbf{c} \\\\\n\\mathbf{a} \\cdot \\mathbf{b} & \\mathbf{b^2} & \\mathbf{b} \\cdot \\mathbf{c} \\\\\n\\mathbf{a} \\cdot \\mathbf{c} & \\mathbf{b} \\cdot \\mathbf{c} & \\mathbf{c^2}\n\\end{vmatrix}", "17dac61b10d732b54d8450853a3b1804": "\n\\sum_i \\langle (Id \\otimes \\Psi_i)(M_i \\otimes I)(\\rho \\otimes \\omega)(M_i \\otimes I), \\; I \\otimes O \\rangle\n", "17daff7d1f0fba077955fd83d9738f1d": "\n \\alpha_t = \\sup_{\\phi\\in Z:\\,\\|\\phi\\|_\\infty=1} \\| \\mathcal{E}_t\\phi \\|_1.\n ", "17db218b1466f2e08000d399647601ae": "A(x,\\ t)", "17dbe4047de689f1c13508decd4fc23e": "L(5;1)", "17dc50dad6abc4a399913c75bd267f02": "s(n,1/\\epsilon)\\,", "17dc58d253690d82d82ae8d75b50e0c1": "CAT = PL", "17dca03da967dd1094df5beef44a4425": "R_{\\mathrm{ESR}}", "17dcd3c4d1f699c2a07fa3d192e47538": " \\begin{align}\n\\operatorname{E}\\left[ \\operatorname{E}[X|Y] \\right] &= \\sum\\limits_y \\operatorname{E}[X|Y=y] \\cdot \\operatorname{P}(Y=y) \\\\\n&=\\sum\\limits_y \\left( \\sum\\limits_x x \\cdot \\operatorname{P}(X=x|Y=y) \\right) \\cdot \\operatorname{P}(Y=y)\\\\\n&=\\sum\\limits_y \\sum\\limits_x x \\cdot \\operatorname{P}(X=x|Y=y) \\cdot \\operatorname{P}(Y=y)\\\\\n&=\\sum\\limits_y \\sum\\limits_x x \\cdot \\operatorname{P}(Y=y|X=x) \\cdot \\operatorname{P}(X=x) \\\\\n&=\\sum\\limits_x x \\cdot \\operatorname{P}(X=x) \\cdot \\left( \\sum\\limits_y \\operatorname{P}(Y=y|X=x) \\right) \\\\\n&=\\sum\\limits_x x \\cdot \\operatorname{P}(X=x) \\\\\n&=\\operatorname{E}[X]\n\\end{align}", "17dcfdbe98f10e7f081ad8053f904a2a": "\\begin{align}\nV_{n}(\\mathbb R^n) &\\cong \\mathrm O(n)\\\\\nV_{n}(\\mathbb C^n) &\\cong \\mathrm U(n)\\\\\nV_{n}(\\mathbb H^n) &\\cong \\mathrm{Sp}(n)\n\\end{align}", "17dd17f2b91bae92b15063867360844f": "E<0", "17dd7e22f2e19c2490d98e12a8f6a395": "f(t,n) = f(t-1,n-1) + f(t-1,n)", "17ddb805ae48564e4909e568dce535f8": "\\left|\\int_\\Gamma f(z) \\, dz\\right| \\le M\\, l(\\Gamma), ", "17ddf3e3567916ddac68e16b0b2dc3b9": "E[Y(T)| X_t=x] = E \\left [e^{- \\int_t^T V(X_\\tau)\\, d\\tau} u(X_T,T) + \\int_t^T e^{- \\int_t^r V(X_\\tau,\\tau)\\, d\\tau}f(X_r,r)dr \\Bigg| X_t=x \\right ]", "17ddf8e61a6ab211fa78338c327c82a9": " \\begin{cases} \\begin{bmatrix} 1 & 0\\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} a \\\\ c \\end{bmatrix} = x\\begin{bmatrix} a \\\\ c \\end{bmatrix} \\\\ \\begin{bmatrix} 1 & 0\\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} b \\\\ d \\end{bmatrix} = y\\begin{bmatrix} b \\\\ d \\end{bmatrix} \\end{cases} ", "17de3fcadb0e421107f556d9261471e5": " \\langle k \\rangle = \\lim_{\\alpha \\rightarrow \\infty} \\zeta_G ( \\alpha ). ", "17deef526b086afa3169a9a27d4557f1": " \\text{N} \\; = \\; k_0 \\left [ m + \\nu \\theta_2 + \\frac {z \\nu \\omega s}{4} (9 + 4 \\varepsilon c^2 - 11 \\omega ^2 + 20 \\omega ^2 c^2)\\right ] \\, ", "17df1378f55240630e871c62e792c2a7": "\\frac{d}{dt}\\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \\frac{f(t_0 + t)}{S(t_0)}", "17dff42fa1938ffdc5e74c06779b8f81": "2^{m/n}=3\\,", "17e0b3891a3a2f5236fb0167e144365e": "\\textbf{Vect}_K", "17e108b76635be3f4b27ff31aa8bfb0c": "\\bold{A}\\;", "17e12308ccf9654d467c52094f918cb7": " z_k = \\langle X - \\mu, \\mathbf{v_k} \\rangle, k = 1, 2, \\dots, p ", "17e13c57ff1871a28f517738649ad583": " \\vec{F} = -\\ C \\vec{r} ", "17e13ebe4ed9acd37f7e01278383735a": " b(z)= \\frac{\\mu_x }{\\sigma_x^2} z + \\frac{\\mu_y}{\\sigma_y^2} ", "17e1824bb144c81c6d3ac0e6e5a648a0": "Y_{10}^{9}(\\theta,\\varphi)={-1\\over 512}\\sqrt{4849845\\over \\pi}\\cdot e^{9i\\varphi}\\cdot\\sin^{9}\\theta\\cdot\\cos\\theta", "17e193f9695bd4c01f2b52a1a19b8b95": "A(\\rho)=w\\,h", "17e1a23dbbc47a45c6f4dfab217d4b77": "\\mathfrak{k}\\oplus i \\mathfrak{p}", "17e24d6ff3aab946eb9f4f51370ed0b9": " I_{z}' = \\int_{0}^{aL} dz \\int_{x,y} dx dy \\, \\rho'(x, y, z) \\,r^2 ", "17e284288b0fa14e8f43c9ea9d6b268b": "(n-4)2^{n-3}+1", "17e28d9d301cdae584e6e378659377b7": "\\ \\sigma_{rc} = \\sqrt{PDOP^2 \\times \\sigma_R^2 + \\sigma_{num}^2} = \\sqrt{PDOP^2 \\times 2.2^2 + 1^2} \\, \\mathrm{m}", "17e2b8e42b405da0ba0f3bcb72d7fb1d": " b_M = \\infty", "17e2ef344c869b7ea7ef0536ad36ac7e": "\\varepsilon_{\\alpha + 1}", "17e322c32a2e1f442a2373d3a0b702b0": "F_a \\propto \\vec{v}", "17e35b9d873da2bfe8ca3d676364b8d3": "\\gamma:I\\to M", "17e3962169351fef25a32f844841b242": "h \\ll N", "17e3d5f0f7addd53afb1f2019e738996": "X_{(a,0,c,d)}(u) = \\sqrt{d} \\cdot e^{i \\pi cdu^{2}} x(du) \\, ,", "17e3d855d8e4734cdff491abc2c67ceb": "\\Delta l", "17e42881b74c73353df5ae18d8fb8b11": "f_W(t)=\\prod_{d,e\\ge 1}(1-X_d^{[d,e]/d}Y_e^{[d,e]/e} t^{[d,e]})^{d e/[d,e]}=\\sum_{n\\ge 0}D_n t^n", "17e447c4fff19f1dfcee4cf10417360f": " r = 0 ", "17e452bcb87c8a1b8217d6d23e11324b": "O(n-i)", "17e4d31435ffc486ccd366828de4060e": "D_{\\mathrm{KL}}(P\\|Q) = \\sum_j p_j \\log \\frac{p_j}{q_j} \\geq 0,", "17e4f3012bab4a66c45e19c94018062e": "\na_{ij}={\\frac{1}{x_i-y_j}};\\quad x_i-y_j\\neq 0,\\quad 1 \\le i \\le m,\\quad 1 \\le j \\le n\n", "17e503f0700ea0935618b473e566fc9f": "\\{T_r, p_r\\}=const.", "17e52223a61815d0ea2bd345aee6d51d": " q(x,y,z)=d((x,y,z),(0,0,0))^2=\\|(x,y,z)\\|^2=x^2+y^2+z^2. ", "17e522aeebd1905f06c37e69f8664047": " \\mathbf{C}_{ik} = (\\mathbf{A} \\, \\mathbf{B})_{ik} =\\sum_{j=1}^N A_{ij} B_{jk}", "17e54b0e83161c98ed2854b021ad4ecd": "m > \\deg(M(x))", "17e571d37c7867877ed899821c75ecbb": "\\sqrt{\\frac{2}{\\pi \\left ( 2k-1 \\right )}}\\,\\frac{2^{2k-2}\\left ( k-1 \\right )!^{2}}{\\left ( 2k-2 \\right )!}", "17e59ccf437c33faae05999c47d52256": "\\begin{smallmatrix}[\\frac{Fe}{H}]=0.5\\end{smallmatrix}", "17e5b0256f40aef34af8d1cc8d5ff62b": "\\forall s,t: W^+[s,t] = W(t,s).", "17e5ea4c140ad5b30d1faf3a121e954f": "\\sum_{i=0}^n \\mbox{metaball}_i(x,y,z) \\leq \\mbox{threshold}", "17e61d9f31314bbba0b1bbd8fe0bbcd8": "\n\\boldsymbol{\\nabla}\\cdot(\\boldsymbol{A}\\cdot\\mathbf{b}) = (\\boldsymbol{\\nabla}\\cdot\\boldsymbol{A})\\cdot\\mathbf{b}+\n\\tfrac{1}{2}[\\boldsymbol{A}^T:\\boldsymbol{\\nabla}\\mathbf{b}+\n \\boldsymbol{A}:(\\boldsymbol{\\nabla}\\mathbf{b})^T]\n", "17e637885b1c4007253313a5aeec81cf": " Y_i=\\alpha+\\beta X_i+u_i , ", "17e6b2525c9c388d6dd326ee60381750": "D_1(P \\| Q) = \\sum_{i=1}^n p_i \\log \\frac{p_i}{q_i}", "17e6c3efed1d499a57c8409b187c939e": "\\frac{\\text{d} f(x_1^*(c_1, c_2, \\dots), x_2^*(c_1, c_2, \\dots), ...)}{\\text{d} c_k} = \\lambda_k^*.", "17e707a577e6d93dab45da9b7d77c7e0": " \\sec \\theta\\! ", "17e71961946c1870e63767e5d71b4464": "\\ \\vec\\mathrm{M}_{sail/G} = \\vec{F_{forward}} \\times distance_{G - E} ", "17e71eacd30f8cd4e4375cdf8b30ff0a": "\\frac{d^2y}{dx^2}.", "17e7271ffe23a3f4f90c8f8fec96d3c1": " \\left\\{ a_1 + a_2 + \\cdots + a_n \\right\\}_{n=1}^\\infty = O(1) ", "17e84e30aa55a84e3d2e51aca9d2dbae": "\nC(d) = \\sigma^2\\exp(-d^2/2 \\rho^2).\n", "17e89e388328fe6f5b8be54f67510a17": " \\tan\\theta = \\frac{S_1/\\sqrt{n_1}}{S_2/\\sqrt{n_2}} ", "17e8c3dbda94ba325c41c9dc3626a197": "\\chi\\,", "17e8e4449ae48d3c4c33725bf70c768e": "2^{571}", "17e8ec35cedb8027aa7ebe1ca77d0af8": "\\Pi=\\left\\{ I,X,Y,Z\\right\\} ", "17e913e5835be6f896c1a2c573944c21": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = \\frac{f_{2}\\left(x,y\\right)}{0.858 \\exp \\left(-0.541 f_{1}\\left(x,y\\right)\\right)} \\geq 1 \\\\\n g_{1}\\left(x,y\\right) & = \\frac{f_{2}\\left(x,y\\right)}{0.728 \\exp \\left(-0.295 f_{1}\\left(x,y\\right)\\right)} \\geq 1\n\\end{cases}\n", "17e9a133159f21c9c6f879857ef85501": "\\varphi=\\pm \\arccos{(-\\beta/(2\\sqrt{R}))}", "17e9c13d22a77ff2682c7bbc86bb2b25": "\\bold{\\hat{n}}", "17e9ce08b3ecfda0a0c0f6da00595919": "|njm\\rangle", "17e9cf799d6214d684ddcedaef132457": "XZ \\to YZ", "17e9e68ae2640ae3b793de415fecc265": "\\chi_{\\text{e}}", "17ea10df84753d1d5e066f5beaec91ac": "\\left\\|\\cdot\\right\\|", "17ea482e746464da7ade6f19d04a6100": "\\theta_0 = \\omega_r \\quad\\text{ if }\\quad P(\\omega_r\\mid\\xi) = \\max_{s=1,2,\\ldots,R} P(\\omega_s\\mid\\xi) ", "17ea7cf4504f64dbfca9dcb2e873e4b4": "\n\\begin{align}\nd(u\\cdot v) & {} = (u + du)\\cdot (v + dv) - u\\cdot v \\\\\n& {} = u\\cdot dv + v\\cdot du + du\\cdot dv.\n\\end{align}\n", "17ea8c81f955f7ecf305a5a65a492ff3": "~\\mathsf{4BaCrO_4+2Ba(OH)_2\\xrightarrow{NaN_3} \\ 2Ba_3(CrO_4)_2+O_2\\uparrow \\ +2H_2O\\uparrow}", "17eafc5969f5dd6acc87a4b5c0ec3618": " x_n=(x_n, x_n+1 ,\\dots , x_n+s-1) ", "17eb8e2e179b9bc4c11ac3b9c8d9a498": "(i-1)^{th}", "17eb927e31859ce116f5c64cbcc30d75": "\\alpha =", "17ebd8bd5b8095d735ea6687ef82fdc2": "\\mathbf{\\Pi}^1_1", "17ec219d55c8e745978ccb3a839409fe": "H \\; 1000 \\lor \\lnot H \\; 1000", "17ec36c91e738aa8ccc78d949b6ea8cb": "e^{j\\pi/2}.\\,", "17ec52c66977b4ee27cc3cb70b2fe91a": " {81\\over80} ", "17ec6134c938436ab5d31f054fe7c889": "\\cosh(t)\\,", "17ec75419272fa56fe701527393f9d6f": "h_{\\mathrm{FOH}}(t)\\,", "17ec7cd41ca3a8a01a81a7cbf39adc69": "{\\rm pcf}(A)=\\{cf(\\prod A/D):D\\,\\,\\mbox{is an ultrafilter on}\\,\\,A\\}.", "17ec8a053605926c3943b75c1335d79f": " \\kappa^{F(n)}_2-\\gamma_2 = 0\\,,", "17ed423bd00722bfd44f70ac800b9e37": "\\mathcal{F}^W_t.\\,", "17ed59237c2847f885568b7457a7393c": " V(t)=V(0)+\\int_{0}^{t}L_{t}(x^{\\ast }(s),y^{\\ast }\\left( s\\right) ,s)ds. ", "17ed5bc0d01231e7079511b916b3c957": "\\overline{16}_{-1H}", "17ed833de1218186e4c981bd4fb9cace": "|\\langle f,g\\rangle| \\le \\|f\\|_2 \\|g\\|_2\\,,", "17edf411e4e987edb5be66da1089d267": "cov(w_i, z_i) + wz = E(w_i, z_i) = az(1 - z)", "17edfef36bddd2ddf333e14ad7b4f3c5": "0.999^{1000}", "17ee1b1c6e8ebde78abc253ed5d9db76": "\\textstyle -", "17ee82407f59375a680009514e71d0be": "E_n(x^\\mu)", "17ee8b370256539c3ff0c5d37345248d": "F_f = \\mu N_f \\,", "17eee94126b662ee7d4156e34c11f25a": "\\scriptstyle\\otimes", "17eefdd872ea45fa01f79c426ed6902b": "\\widehat{R}(\\theta,\\hat{\\mathbf{n}}) = \\exp\\left(-\\frac{i}{\\hbar} \\theta \\hat{\\mathbf{n}} \\cdot \\mathbf{J}\\right) ", "17ef988738145fa073738ee7a5d90a21": " M \\leq \\left\\lfloor \\frac{2d}{2d-n} - 1 \\right\\rfloor = \\left\\lfloor \\frac{2d}{2d-n} \\right\\rfloor -1 \\leq 2 \\left\\lfloor \\frac{d}{2d-n} \\right\\rfloor. ", "17efccd691a4cec276dd62d69692cdbb": "W_{1} (\\mu, \\nu) = \\sup \\left\\{ \\left. \\int_{M} f(x) \\, \\mathrm{d} (\\mu - \\nu) (x) \\right| \\mbox{continuous } f : M \\to \\mathbb{R}, \\mathrm{Lip} (f) \\leq 1 \\right\\},", "17f08c78c5e3bf040c7b550baf559d5a": "\\frac {1}{T} = \\left ( \\frac{\\partial S}{\\partial U} \\right )_{V, N} \\, .\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(3)", "17f0b3bb7c80669befd5d00ff2f3b4f1": "\\{ x\\}", "17f0bc7fc1b8f99ef9d45fc0d93687ec": "\\ \\ t=\\tan\\alpha\\ ;\\ \\ t\\in \\mathbb{R}\\ )", "17f124bfd50ee0bb5f6cedeb37230021": "\\nu_{ij}", "17f16f9bc981b8ae04f421aa6fc72918": "k(t) = \\sum^{\\infty}_{i=1} \\lambda_i S_i \\Phi_i(t), 00", "17f3977b33d086bcbbb9da8729521343": "A^{\\prime}=A,", "17f3e0655786ef4df373ecbee1773ed2": "\\epsilon_{F}", "17f405fb83dd8bc4825175104470d95a": "\\textstyle \\bar{M}_{\\mathrm e} = \\left( \\begin{array}{cc} \\frac12 & \\frac12 \\\\ -\\frac14 & \\frac14 \\end{array} \\right)", "17f4149c1f2c00ad85e587fc916fd6dc": "(a)\\text{ }\\bigcup\\nolimits_{i=1}^{n}{R_{i}=R.}", "17f42511d5c6b54d2bc99422fca646d2": "\n\\begin{align}\n f&=\\frac{a-b}{a}, \\qquad e^2=f(2-f), \\qquad n=\\frac{a-b}{a+b}=\\frac{f}{2-f}\\\\\nb&=a(1-f)=a(1-e^2)^{1/2},\\qquad e^2=\\frac{4n}{(1+n)^2}.\n\\end{align}\n", "17f426970d265ff3f12b55d280187164": "f_i(\\vec{x}+\\vec{e}_i\\delta_t,t+\\delta_t)-f_i(\\vec{x},t) + F_i=\\Omega(f)", "17f492cdd3a7eb944da6b39b7507ef79": "\ng_{ab} = \\frac{\\partial Z^A}{\\partial E^a}\\frac{\\partial Z^B}{\\partial E^b} G_{AB}\\ .\n", "17f4c9f7d2f67ac6806846b512efb63a": " \\alpha (\\omega ) = \\frac{{4\\pi \\omega }}{{n_b c}}\\chi ''(\\omega )", "17f4d827a7cec674b0d3b8002e546081": "X \\in T_pM", "17f53f0181d25d95449981f1a841cb56": "\\delta n^a-\\Delta m^a=-\\bar{\\nu}l^a+(\\tau-\\bar{\\alpha}-\\beta)n^a+(\\mu-\\gamma+\\bar{\\gamma})m^a+\\bar{\\lambda}\\bar{m}^a\\,,", "17f547bd474a83a51aff9e1b5fbee91a": "\\mathbf{n}_1", "17f58a87bc544f5200da2fc46428254c": "\\min(3-1,4-1,2-0,0-(-1),6-3,9-3) = ", "17f5939bda952cd3ff4a4d34c8f3b408": "O(d)\\,", "17f59a354839efc176a2773aff3f44e0": "\\pi_* W_v = v\\,", "17f59d9ac48df9d60f6fe8999d8b683a": "x^{\\left [ 1 \\right ]} = x \\log(x) - x", "17f5f6cfe7ccbe4d9ca46a9b9da0027f": "\\sin \\left(x-y\\right)=\\sin x \\cos y - \\cos x \\sin y, \\,", "17f60ef594eb982a99682a230d70d57c": "u = (u_{ij})_{i,j = 1,\\dots,n}", "17f665f57c52c9926c2e9855cdc5bff2": "E \\left \\{ \\left| h_{w,ij} \\right |^2 \\right \\} = 1 \\quad \\forall i,j ", "17f66fbdd5bf0d7626acef952d4baea9": " |0\\rangle, \\ldots , |2^n - 1\\rangle. ", "17f67df2f5d9ee3661255e2bec2cb7ba": " h(x) = g(x) -(\\lambda^\\prime-\\lambda) \\int_c^x (\\varphi_\\lambda(x) \\theta_\\lambda(y) - \\theta_\\lambda(x)\\varphi_\\lambda(y))g(y)\\, dy", "17f6942775078d20dc17baeabe217cea": "H_n(0) = \n\\begin{cases} \n 0, & \\mbox{if }n\\mbox{ is odd} \\\\\n (-1)^{n/2} 2^{n/2} (n-1)!! , & \\mbox{if }n\\mbox{ is even} \n\\end{cases}\n", "17f6f18308e191fbac91ee26bb7d7e8a": " u(w) = -w^{\\alpha}", "17f71054ef4cfe4d134500b3d0199d3c": "m = \\mathop{\\mathrm{arg\\ min}}_{p\\in M} \\sum_{i=1}^N w_i d^2(p,x_i)", "17f712769b8bf67b62230eccf552fb7d": "\\mathbf{n} =\\frac{\\mathbf{i}-\\mathbf{r}}{\\|\\mathbf{i}-\\mathbf{r}\\|}", "17f73ca0ee9e3525bbf58451e1d8bf5e": "\\alpha_{\\text{pump off}} (\\omega)", "17f76e61dd4b3788991af8773f778192": "<0.24", "17f7b419875e59455c6f5c6f424f14ee": "c(w) = \\frac{1}{A(w)} \\frac {d}{dw} A(w)\n= \\sum_{n=0}^\\infty c_n w^n.", "17f7cbd51c8256a9a58b5dddad126b5f": "F_{X_{\\gamma}} = -\\frac{\\partial E}{\\partial X_{\\gamma}} = -\\bigg\\langle\\psi\\bigg|\\frac{\\partial\\hat{H}}{\\partial X_{\\gamma}}\\bigg|\\psi\\bigg\\rangle.", "17f7f80674eb042d147a005aeaaac5eb": " 0 < a < b, \\quad a, b \\in \\R", "17f88caaba714ee24023972ba73a2a29": "\\mathbf{A'}", "17f89bb424fdc7b0ba779255907971fa": "n/r+1", "17f8bcd2c38b559ae31aca3101ed43eb": "2x^2+5=3", "17f90493dab3503456118902cd35d216": "{{P}_{X}}\\left( u,\\xi \\right)=E\\left\\{ {{P}_{V}}X\\left( u,\\xi \\right) \\right\\}", "17f94bccc8eae6d01f546eb552c41239": "\\Lambda(s,\\chi)=\\varepsilon\\Lambda(1-s,\\chi^*)", "17f95824364be5599c0feb863bc72e34": "\\psi_{\\mathbf{k}}(\\mathbf{r}) = e^{i\\mathbf{k}\\cdot\\mathbf{r}}u_{\\mathbf{k}}(\\mathbf{r})", "17f9a145cd0169d6029df4f52325dbc1": "J = P^{-1}BP = \\begin{bmatrix}\n4 & 0 & 0 \\\\\n0 & 16 & 1 \\\\\n0 & 0 & 16 \\end{bmatrix},", "17f9a7012baac7cb62742162aab18037": "f(t) ", "17f9c7e2488ccfafed01d6c6908ac0e8": " \\sqrt n ", "17fa1c8ad83d6239a18d7b0206db17f4": "x^2 - y^2 = 1", "17fa5a2827df33beafc249c1a46fa53a": "\\beta=g_m r_{\\pi}", "17fa6b00000ed9b303dd0db4acf18c0c": "{O}(n)", "17fa6b24a5885013b6a9e263545b66a7": "\\left[J^{\\mu \\nu},W^{\\rho}\\right]=i \\left( \\eta^{\\rho \\nu} W^{\\mu} - \\eta^{\\rho \\mu} W^{\\nu}\\right) \\,, ", "17fa7d0703338253c37340eec5a60d6a": " Q = I_3 + \\frac{1}{2}Y,", "17fa98acf41e49bec05845a5fdebfb3e": "a \\neq 0\\, ", "17faafe190cbc8f978fc84f423fbb24c": "\n\\phi_1 = p_x + \\tfrac{q B}{2c} y,\\qquad \\phi_2 = p_y - \\tfrac{q B}{2 c} x.\n", "17fae224ec007d74c9a3cc9bc2a4864c": " d \\log X_i(t) = \\gamma_i(t) \\, dt + \\sum_{\\nu=1}^d \\xi_{i \\nu}(t) \\, dW_{\\nu}(t)", "17faf92b941cc0cb1b978cfbc5c42d73": "\\sin x", "17fb0d8c787ae6675db999d191ac2772": " p_{i1} \\geq p_{i2} \\geq \\ldots \\geq p_{in}", "17fb1e4ff242dd6437a4208b26eb1ad4": " \\mathit{GF(p)}", "17fb39aac58017a7aaad11450f6abe50": "\\text{gcd}_{R[X]}(p_1,p_2)=\\text{gcd}_{R}(\\text{cont}(p_1),\\text{cont}(p_2))\\,\\text{gcd}_{R[X]}(\\text{primpart}(p_1),\\text{primpart}(p_2)),", "17fb78f7bd8d4a1ad2b217e814f7f0f3": "\\cosh(\\mathrm{arsinh}(1/\\varepsilon)/n).", "17fbba8e68fd90588d7e1e5074549619": "E(x)=\\mu,\\,E((x-\\mu)^2)=\\sigma^2", "17fbdcff815277a6e99e2a5f8713b187": "\\psi(\\alpha)^{\\psi(\\alpha)^{\\psi(\\alpha)}}", "17fc1fb2acf04207e6af2fb86727524c": "\\mbox{Pr}(Y\\leq 7.5)\n=\\mbox{P}\\left({Y-6 \\over \\sqrt{2}}\\leq{7.5-6 \\over \\sqrt{2}}\\right)\n=\\mbox{Pr}(Z\\leq 1.0606602\\dots) = 0.85558\\dots", "17fc38b90b8ff5c087c8f62d131d49ba": "f_{\\text{c}} ", "17fc393a100a2f3bbc1293f984ce40b0": "F\\left(n\\right) = {{\\varphi^n-(1-\\varphi)^n} \\over {\\sqrt 5}}.", "17fc6b7c84e59aaafead6eda1bd16605": "\n\\left[ \\frac{\\partial }{\\partial t} + \\frac{p}{m} \\frac{\\partial}{\\partial x} - U'(x) \\frac{\\partial}{\\partial p} \\right] \\langle x \\, p | \\Psi(t) \\rangle = 0.\n", "17fc734cf2d5dd0188f659ae5c9e5e6f": "\\langle \\mathbf{v}, \\mathbf{n} \\rangle \\;dS ", "17fc7a2140a1dded51c24fedd17f8f55": "(C_\\bullet, \\partial_\\bullet)", "17fcd476fd64650175502165a2054a18": "m\\rightarrow\\infty", "17fcf02b3e20f259eee0ed22eeec10d1": "\\psi_1 = Fe^{- \\alpha x}+ Ge^{ \\alpha x} \\,\\!", "17fcfc4ef5fce99a5966fec2a4b5a3d5": "\\operatorname{CAT}(k)", "17fd70474a56d5f564f073db2a959023": "\\{Commit_k(x',U_k)\\}_{k\\in\\N}", "17fde9ac2fedfd385a15ce4e1df11399": "M = C \\sigma + 1 = C \\sqrt{t} + 1", "17fe7713fed4e793724c387e5c0937cb": "\\mathbf{A} = \\begin{bmatrix} 1 & 0 \\\\ 1 & 3 \\\\ \\end{bmatrix}", "17fecbdd82e4642c75405b263f8d588b": "\\sum F_i = \\lim_{\\Delta t \\to 0} \\frac{P_2-P_1}{\\Delta t}", "17ff20034c63f28dddc493a7f840e198": "p \\cdot y_j \\leq p \\cdot y_j^*", "17ff3c89c7bda002dabd7758cc1d93e8": "\\textstyle \\frac{\\partial C}{\\partial t}=\\frac{\\partial}{\\partial x}[D_1 \\frac{\\partial C_1}{\\partial x} -C_1 \\nu]", "17ff802d6881af18c3785b2019ffc168": "|{\\Phi^B_i}\\rangle", "17ff82018a328f7bed6130b2d32f77ef": "A = \\begin{bmatrix} 4 & 3 \\\\ -2 & -3 \\end{bmatrix},", "17ffc7ec88da68caa1e0b2a49f616d6e": "S: x^4+y^4+z^4-1=0", "18002002861c3bf20e361b16a3837555": "G/H_1\\xleftarrow{\\eta} G/(H_1\\cap H_2) \\xrightarrow{\\tau} G/H_2", "1800ba74192dac1a8fc099e7752e6c41": "\\ U(r;R_{1},R_{2})= -\\frac{AR_{1}R_{2}}{(R_{1}+R_{2})6r}", "1801c758f0fe93840b82d798cc9e2643": "MRT = \\left[ \\left(GT+273 \\right)^4 + \\frac{1,1 \\cdot 10^8 \\cdot v_a^{0,6}} {\\varepsilon \\cdot D^{0,4}}(GT - T_a) \\right]^{1/4} - 273", "1801cfc88edd59ca7296ac197514e703": "\\sqrt{7}", "1801f28c921288de06eca4cf4e671350": "i p_0 \\mathbf{\\epsilon^1}(\\mathbf{p})", "18022a974ac7f7544d134b81c1c4b080": "0 < \\delta < T", "18026f514a6f9bb23a6c2a0f2f32a040": "\nf(x) = {a \\over (x + b)^2} + c\n", "1802867ddcb7193daa2d52d00f7ffeaf": "\n \\begin{align}\n u_r &= A~\\cos\\theta + B~\\sin\\theta \\\\\n u_\\theta &= -A~\\sin\\theta + B~\\cos\\theta + C~r\\\\\n \\end{align}\n ", "1802c53e9efe2c1e8c39c0101b6153d3": "\\scriptstyle 5\\times 6", "1802d76a50a819df0635526eadf4b78d": "d^2\\alpha (x\\wedge y\\wedge z) = \\frac{1}{3} d^2\\alpha(x\\wedge y\\wedge z + y\\wedge z\\wedge x + z\\wedge x\\wedge y) = \\frac{1}{3} \\left(d\\alpha([x, y]\\wedge z) + d\\alpha([y, z]\\wedge x) +d\\alpha([z, x]\\wedge y)\\right),", "1802da9f164ef179e4fee2410d0d67ca": "p^* = \\nabla F(p)", "18032c2bc6ad00d41f82aded185b9b00": "r = \\frac{p_F}{\\tilde{p}_P} - 1,", "1803744cb75e4c2b994f9959e4036316": "{\\boldsymbol S}", "180447bebc133a437caa5a46c1e5a594": " \\tilde{\\omega}^i (\\vec e_j) = \\delta^i_j ", "1804f3e945b610e3bfd7a7ffdba1248d": "PCER \\le \\alpha ", "1804fdeac812caa00e039120e2d76250": "\\varepsilon_{tot} = \\varepsilon_1 + \\varepsilon_2", "18050d14a2324b74e4c51241ac49b4a3": "\\mu ={}^{\\left[ \\log \\left( Km'n' \\right) \\right]}\\!\\!\\diagup\\!\\!{}_{\\lambda }\\;", "180545d3bc00563b5f4a619065fc0803": "\\Lambda^n V", "18054b980a948ae82db8ca111f38bc28": "\\tbinom{2n}{n}", "1805799703cfb6852959a04abcf64d18": "\\frac{b}{a} \\cdot \\frac{a}{b} = \\frac{ba}{ab} = 1.", "1805d588198cd97aebab15b4be967dd5": " B \\circ A", "1805d7bca4919079003e4cd024659765": "F(a,q,x) = \\exp(i \\mu \\,x) \\, P(a,q,x)", "18062211d970068bd37d40b5d9b4f9c4": "\\Gamma (X, \\mathcal O(d)) = k_d[X_0, \\ldots, X_n]", "180666cc1eaf31c005972a837bc8e847": "\n Q_x^{\\mathrm{face}} = \\cfrac{\\mathrm{d}M_{xx}^{\\mathrm{face}}}{\\mathrm{d}x}\n ", "180672f922bf4d484a94493a4ee6fb19": "0\\le Re \\ s\\le 1", "18067b9f46e0a1789ffcde1c6814a4da": "(\\alpha_n^2 > 0)", "18069f4398b00eae525578799415d74f": " {\\partial^2 \\mathbf{E} \\over \\partial t^2} \\ - \\ c^2 \\cdot \\nabla^2 \\mathbf{E} \\ \\ = \\ \\ 0", "1806c82a6011d980a7ac56004460bc6c": " A = \\frac{V_o}{V_i} = \\frac {\\left ( \\frac{M}{C} + a \\left ( \\frac {M}{C} \\right ) \\left ( \\beta - 1 \\right ) + \\frac { \\beta + 2 } { 3 \\left ( \\beta + 1 \\right ) } \\right ) } {\\left ( \\frac{N}{C} + \\frac{2 \\beta + 1}{3 \\left ( \\beta + 1 \\right )} \\right ) } ", "1806d217fc8c24b7c1180da5fe84471a": "\\langle\\chi|\\partial^\\mu A_\\mu|\\psi\\rangle=0", "180708f18711731d870976286cc28396": "\\begin{align}\nx&=\\frac{a\\sinh v}{\\cosh v - \\cos u}\\\\\ny&=\\frac{a\\sin u}{\\cosh v - \\cos u}\\\\\nz&=z\n\\end{align}", "18072fd37c78b5b3fd11544442824fe3": "\\bold r_0 = h_1\\bold {n}_1 + h_2\\bold {n}_2", "18075ab9eb580438e37c317057fd7a4a": "p=j^2 \\cdot g", "1807752fa6d342715d0003971f3c0f41": "\\mathbf{NL} \\subsetneq \\mathbf{PSPACE} \\subsetneq \\mathbf{EXPSPACE}", "1807cd6c50789312c1cc88c6c1a8998d": "\\mathbb E(D_u) = \\int_{-\\infty}^\\infty |x'|p(u,x') \\, \\mathrm{d}x'", "1807d8c86d25ce116582a56803c5df0e": "\\Delta m^2_\\odot\\simeq8\\times10^{-5}\\,\\mbox{eV}^2", "1807e33be3dcfdc55408ee2ef33c235a": "1/K_p", "180823239b488c4f827b05ced75230c4": "1 < \\frac{\\theta}{\\sin\\theta} < \\frac{1}{\\cos\\theta} \\implies 1 > \\frac{\\sin\\theta}{\\theta} > \\cos\\theta \\, . ", "180863eaffa00e8b6dff86a9923f8059": "m^2= - I, \\quad z = x I + m \\sqrt{-p}", "18088d589a8d81b9444df964cae50e2b": "\n\\begin{align}\n& {} \\qquad F_{X_n,X_{n+1},\\dots,X_{n+N-1}}(x_n, x_{n+1},\\dots,x_{n+N-1}) \\\\\n& = F_{X_{n+k},X_{n+k+1},\\dots,X_{n+k+N-1}}(x_n, x_{n+1},\\dots,x_{n+N-1}),\n\\end{align}\n", "1809064148dcc02bb7beb9a848be89a9": "\n1 = H' \\int_0^T \\{ x,p \\} dt = H' T\n\\,", "18095107a92891a2e4f4fd2486db5e67": "2N^2", "1809847ce91741ecc9a82882159e044b": "a_q=\\min\\{n\\in\\mathbb{N}\\colon n\\ge(5/2)^{q+1},", "180986a5c166b5d8f40456577d106cc0": "\\frac{\\mathrm{d}F}{\\mathrm{d}t}-P \\leq 0.", "1809998d6fbca11e75ba1344184616c2": " \\mathsf{T}=(0, \\mathbf{v}).", "1809a9122cb55a13bc0af46771724fbb": "P = \\sum_{ij} |i\\rangle \\langle j| \\otimes |j\\rangle \\langle i|", "1809c3e5e783c9346f41afa25b4d6055": "S(x) = \\sum_{i=\\lfloor x \\rfloor - a + 1}^{\\lfloor x \\rfloor + a} s_{i} L(x - i),", "1809c7e7ff9da0b203c92112c6581db2": "\n \\sigma_x = -zE\\cfrac{\\mathrm{d}^2w}{\\mathrm{d}x^2}\n ", "1809d05376474d84d30f35040b8f201f": "S(\\rho^{AB}) = - \\operatorname{Tr} \\rho^{AB} \\log \\rho^{AB}.", "1809e0d2643c824182747d32bf2ce40c": "v = \\frac{3}{2}\\mathbf{e}_1 + 2\\mathbf{e}_2", "1809eeb44977727661f96e4b6976e864": " dP \\propto \\exp{\\left[-\\frac{\\mathcal{H}(\\mathbf{p}_1,\\ldots,\\mathbf{p}_N;\\mathbf{r}_1,\\ldots,\\mathbf{r}_N)}{k_\\text{B}T}\\right]}d\\mathbf{p}_1,\\ldots,d\\mathbf{p}_Nd\\mathbf{r}_1,\\ldots,d\\mathbf{r}_N, ", "180a1dde2077bc80fa44e1339484d091": "-i\\hbar\\boldsymbol{\\nabla}", "180a2870d4cbbac91b35a6c1813f13c0": "HS_K(t)=1\\,.", "180a2a9d56f09498295807239ede8d0a": " |I(t) - I(0)| < \\varepsilon^{1/(2n)} ", "180a33eb34f823e27b4d6f454dbb7e6a": "2\\hbar k ", "180a58c005ad2c2e5dea7450f3f3d1bb": "R(t)=Pr\\{T>t\\}=\\int_{t}^{\\infty} f(x)\\, dx \\ \\!", "180aa8b273e58d0c881edca7d4df39b8": "[S]=[S]_0(1-k)^{t}\\,", "180c59c3a2a8fb21185fe069086eeea3": "\n\\left\\{\\begin{matrix} \\tau_{12}=\\frac{\\Delta g_{12}}{RT}=\\frac{U_{12}-U_{22}}{RT}\n\\\\ \\tau_{21}=\\frac{\\Delta g_{21}}{RT}=\\frac{U_{21}-U_{11}}{RT}\n\\end{matrix}\\right.", "180c8a867d9f782257eafe4e72f56262": " (S_1, \\ldots, S_N) ", "180d3f0cdc3ae1cf35627c95601d898b": " r_1<\\cdots k ", "180eba59ff13816c241ab7b5d6bf1104": "\\mathbf{E}_{l,m}^{(M)} = \\sqrt{l(l+1)} \\left[E_l^{(1)} h_l^{(1)}(kr) + E_l^{(2)} h_l^{(2)}(kr)\\right] \\mathbf{\\Phi}_{l,m}", "180efb12feb4d089b09615f5f21a415c": "\\psi(x) = \\int{\\frac{d^3p}{(2\\pi)^3 \\sqrt{2E} } \\sum_{\\lambda \\pm 1}{\\left(\\hat{a}_p^\\lambda u_\\lambda(p) e^{-i p \\cdot x} + \\hat{b}_p^\\lambda v_\\lambda(p) e^{i p \\cdot x} \\right)} } \\,", "180f42501da3daa476de45218a3534c3": " pb_t ", "180fbe7f9e38c992a5df3eb4797cb8db": "\n \\boldsymbol{\\varphi}_x' (x) \\equiv \\left( \\frac{\\partial \\varphi_i (x)}{\\partial x_j} \\right), \n \\qquad 1 \\leqslant i \\leqslant k, \\quad 1 \\leqslant j \\leqslant n.\n", "180fe72520038e7e96fe41557628018b": "x_t = x_0(1+r)^t", "18101784a9d512383c0dac78f71f0c3b": "\ny^{[n-1]} = \n\\left[\n\\begin{matrix}\ny_1^{[n-1]} \\\\\ny_2^{[n-1]} \\\\\n\\vdots \\\\\ny_r^{[n-1]} \\\\\n\\end{matrix}\n\\right], \\quad\ny^{[n]} = \n\\left[\n\\begin{matrix}\ny_1^{[n]} \\\\\ny_2^{[n]} \\\\\n\\vdots \\\\\ny_r^{[n]} \\\\\n\\end{matrix}\n\\right], \\quad\nY = \n\\left[\n\\begin{matrix}\nY_1 \\\\\nY_2 \\\\\n\\vdots \\\\\nY_s\n\\end{matrix}\n\\right], \\quad\nF = \n\\left[\n\\begin{matrix}\nF_1 \\\\\nF_2 \\\\\n\\vdots \\\\\nF_s\n\\end{matrix}\n\\right].\n", "181043a9bff552b4a3bbb2745ab710ae": "\\mu_X = \\mathbb{E}_X [\\mathbf{e}_x] = \\left(\n\\begin{array}{c}\nP(X=1) \\\\\n\\vdots \\\\\nP(X=K) \\\\\n\\end{array}\n\\right) ", "1810ccc0e3399443049e39f537c570db": " \\Delta W_\\mathrm{ON} = - \\Delta W_\\mathrm{BY} \\,\\!", "1810e57e32c3de41d6778dec879b9ebe": " \\quad (8) \\qquad \\qquad {{\\partial {\\mathbf u}} \\over {\\partial t}} + \\nabla \\cdot {\\mathbf f}\\left( {\\mathbf u } \\right) = {\\mathbf 0} . ", "1811ac550cb73c34cb1df992312e8080": "\\mathbb{T}", "1811be2c89fa4bd425250a9b495af177": " D (d_\\lambda f) = \\lambda \\cdot d_\\lambda f.", "1811de483b0e564d9e631e7f99564c69": "I_X = \\frac{Y_X} {Y_{Total}}I_T = \\frac{\\frac{1}{R_X}} {\\frac{1}{R_X} + \\frac{1}{R_1} + \\frac{1}{R_2} + \\frac{1}{R_3}}I_T", "1811f7eae428ee41f5e8bb015d95695a": "\\eta^{-1}", "181205e7cc6366aaf861ab61194e524c": " MRP_L = MC_L ", "181241214a35d54b2a97ae35a3d34167": "f_j (c_1,c_2,...,c_{N_B}) \\,=\\, [j]_{TOT} - [j] - \\sum^{N_S}_{i=1} \\frac{\\nu_{i,j}}{\\gamma_i}\\, K_i\\, \\prod^{N_B}_{k=1} \\{k\\}^{\\nu_{i,k}} \\,=\\,0 ", "18124e94c021584f29f6e4621673ba79": "s_{kk}=0\\,\\!", "18125e406a3acc3e377df532a4ed09a6": "(M, H, g)", "1812b83ad7e9470a13d9ab8e08fc8843": "j^n(\\kappa)", "1812cfa10c32a5773462a3d2de54216a": " \\sum_{n=1}^{\\infty} x_{n}", "1813496391386430b1fecf04c2db1533": "\\varepsilon \\thicksim N(0, \\sigma^2).\\,", "1813bd3fbc3c2b75ebde60944ee14902": "(X,id_X)", "1813d71ac32a0e0aabc2af49fbf0bf0a": "\\sube", "1813e6b0253367e59eaf37adbcb09190": "\\left(\\lambda+\\mu\\right)\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}\\right)+\\mu\\left(\\frac{\\partial^2 u_x}{\\partial x^2}+\\frac{\\partial^2 u_x}{\\partial y^2}+\\frac{\\partial^2 u_x}{\\partial z^2}\\right)+F_x=0\\,\\!", "18143f739d4c788fdefe9d0f358e83d5": "(A\\to B)\\to((B\\to A)\\to(A\\leftrightarrow B))", "1814c9fb56fbc8b6ae3b1a241ea29aad": "Q=\\mbox{Re}(L)", "1814fcab976450ffcf27d04a82cf5a87": "w(x_1,x_2,\\ldots,x_n)dx_1dx_2\\ldots dx_n", "18154f1a351aed69411dfc1d74afd945": "f(z) = z^m e^{g(z)} \\displaystyle\\prod_{n=1}^\\infty E_p(z/a_n)", "181552e6c8d3178a0e5bcfb93e76999b": "\\mathrm d\\mathbf{r} = \\mathrm d\\rho\\,\\boldsymbol{\\hat \\rho} + \\rho\\,\\mathrm d\\varphi\\,\\boldsymbol{\\hat\\varphi} + \\mathrm dz\\,\\mathbf{\\hat z}.", "18156063a1acd4a642204b6bfa284b2d": "\\frac{4}{3}\\div \\frac{704}{576}={\\color{blue}\\frac{12}{11}}", "1815ac41d05d5259e9e54e83d8c22f92": "P(M>a)", "1815b38a57a4b348b7ff676955a34aa4": "B_\\infty", "1815e5f6c120502d38f2eed8b0b5e2a1": "(X_1 \\wedge Y_1) \\vee (X_2 \\wedge Y_2) \\vee \\dots \\vee (X_n \\wedge Y_n).", "1815efb92ba9df6aa94736e4304db783": "\\log_b (a-c) = \\log_b a + \\log_b (1-b^{\\log_b c - \\log_b a})", "181646f5cc3c6780e968c8b0760781a5": "G^{(\\alpha)}=\\{e\\}", "1816909600ff52b7c8a6fe5da1894e1e": "\\mu(\\emptyset) = 0", "1817b2083540c4801888dd6aebf9b9e6": "\n \\tilde{f}^{*}(\\cdot) = \\sum_{i = 1}^n \\alpha_i k(\\cdot, x_i) + \\sum_{p = 1}^M \\beta_p \\psi_p(\\cdot)\n", "1817fa16f68b0ff291d3fd9ce0bae2d1": "\\frac{2}{3}\\!\\,", "18184e80f69735e66651ff330ed7ec36": "\nH(X,Y)= H(X) + H(Y) - I \\left (X;Y \\right ) = 2\n", "181862a5ec1bff268df5184dd2489f6b": "\\tau(p) = 0 \\bmod p", "18186a80c2c74e2ff966d067f207009c": " \\vec w ", "1818709942d238261498fa72703eaf05": "h_{\\mathcal{D}}(t)", "1818dddfa5de4e4b3f846b29023bd48b": "G:\\mathcal D\\to\\mathcal C", "181902e0f73e7f74a3406034e566e102": "( v, N) ", "18193226b0d4f87b68420151cd720406": "V(\\mu)=\\tau^\\prime[\\tau^{-1}(\\mu)]", "1819c280dd2113ed22ff2311f27df264": "\\mathbf{Top}.", "181a27fff2021c81b6f3fcf14f7d796b": "\n W(I_1)\\biggr|_{I_1=3} = 0 \\quad \\text{and} \\quad \\cfrac{\\partial W}{\\partial I_1}\\biggr|_{I_1=3} = \\frac{\\mu}{2} \\,.\n ", "181a3633f90605c2fc1f907b02924640": "\\Omega = q\\theta - (q-2)\\pi.\\,", "181a3829e04b0262ffd2065a7c1c77ec": "\n|\\psi \\rang = \\sum_i |i\\rang \\psi_i\n", "181a4a90433d2b6ccbc1896dae157bad": "\\sigma=\\frac{\\epsilon\\epsilon_0\\psi_0}{\\lambda_D}", "181a6afe6190dfe00f9471a7e26fd622": "\\csc ^2 x_1 + \\csc ^2 x_2 + \\cdots + \\csc ^2 x_m\n=\\frac{2m(2m-1)}6 + m = \\frac{2m(2m+2)}6.", "181ad54b8fcdb1066fb2288cfdd61d9e": "\\Delta_5", "181b034810968526cce3d1ed24830e5d": "u v^T", "181b14f3c64571384be6c9bf3eaba6bc": "r_{p+i \\cdot k}=r_{p+p \\cdot k}=r_{p(k+1)}=0", "181b8d98d4c1465346d92acde3025ceb": "T^{a}{}_{acd} = 0", "181b944684b8d793134d351dbcaf0638": " H_s(r)=P({N}(b(o,r))=0). ", "181c427b49efda95cd62934c8dfc9935": "B = 0.07780\\frac{P_r}{T_r}", "181cb3dcccd04c0d46c7a6571810ac27": "\\scriptstyle \\hat{s}(t)", "181cb4fa8519e6c0a19fe85dd8d57e2b": "\\mathbf{u} = \\{ u[j] \\}", "181ce4d603bde8eb87d56af0620d4bc9": " \\begin{bmatrix}k_1 & 0\\\\0 & k_2\\end{bmatrix}", "181d10b7e3a35994a766a94799fc3111": "\\|\\mathbf{x}-\\mathbf{x}^*\\|\\leq\\delta;\\,", "181d1319de482194de9ca39c9ff1576d": " \\chi = \\psi_1 + i\\psi_2 \\, ", "181db2fd298f0302c590f1b3babdc612": "\\hat{S}_N \\subset C", "181db821b8b2aac77c3e03c5e67efbb0": "p(\\vec{r})=1/V", "181dcd45678d4499bdbad5e4a29d6d1d": "\n\\lambda_2 = (\\mathrm{E}X_{2:2} - \\mathrm{E}X_{1:2})/2\n", "181ddd0df008d6860b34522ff8b92d58": "\\sigma f(A_1,\\dots,A_n)=f(\\sigma A_1,\\dots,\\sigma A_n)", "181de74002d56eb422c240e61ac0a015": "v_{0^{ }}", "181e05b7f24072b3e0fdf492b5d6a85d": "\\scriptstyle d \\geq 3 ", "181e54f99352a0a0f9f1d1d138f7bd5a": "P(t) = \\begin{cases}\n 0 & tt_o+t_p \\\\ \n\\end{cases}\n", "181e72e1f409b60f5322f9008a8ea469": "\\int_{\\mathbb{R}^{n}} \\prod_{i = 1}^{m} f_{i} (x \\cdot u_{i})^{c_{i}} \\, \\mathrm{d} x \\leq \\prod_{i = 1}^{m} \\left( \\int_{\\mathbb{R}} f_{i} (y) \\, \\mathrm{d} y \\right)^{c_{i}}.", "181edc009ceec906e6db431b342313c0": "G\\, =\\, \\int_{-h}^0 \\left(\\frac{\\text{d}f}{\\text{d}z}\\right)^2\\; \\text{d}z.", "181f3a7a07c9c0f763b1028d304d8c4d": "\\begin{smallmatrix}\\left( \\frac{T_{eq}}{T_{pole}} \\right)^4 = \\left( \\frac{7,600}{10,000} \\right)^4 = 0.33\\end{smallmatrix}", "181f5015bdd98f043258bdff3ccc8bb7": "\\mathbb{R} ", "181f6a46c34ce06a6bd8e1ea5d7ef213": "H_i = V^i(H)", "181fab25d30b8f19c9fc01d32cd59774": "T(x_i,y_i) = M^2 \\iint T(M k^i_x,M k^i_y) ~ e^{j(k^i_x x_i + k^i_y y_i)} dk^i_x \\, dk^i_y", "181fdd3909a3a3a9e3fe179ce6bfe9c5": "|b|^2", "18203051a20839c54417d2891036cbbf": "X_1, \\dots, X_n", "18203483de0584daf8422945759fc3fd": "\\text{pitch}_Z = \\sqrt{6} \\cdot {d\\over 3}\\approx0.81649658 d,", "18203e6e345a3a9b7e570767a7aac3d4": "\\mathbb{E} f", "1820552c3b85931361457e1c3ff6ec3e": "{\\tilde{E}}_n", "182060f212b58d78b2a6d58b79f3496d": "b_1 = B / (1 + (o_1/o_2) + (o_1/o_3) )", "1820726e9191d57202abd54a4d7a39bc": "Q = \\begin{pmatrix}\n{-(x_1 + x_2 + x_3)} & {\\pi_1 x_1 \\over \\pi_2} & {\\pi_1 x_2 \\over \\pi_3} & {\\pi_1 x_3 \\over \\pi_4} \\\\ \n{x_1} & {-({\\pi_1 x_1 \\over \\pi_2} + x_4 + x_5)} & {\\pi_2 x_4 \\over \\pi_3} & {\\pi_2 x_5 \\over \\pi_4} \\\\ \n{x_2} & {x_4} & {-({\\pi_1 x_2 \\over \\pi_3} + {\\pi_2 x_4 \\over \\pi_3} + x_6)} & {\\pi_3 x_6 \\over \\pi_4} \\\\ \n{x_3} & {x_5} & {x_6} & {-({\\pi_1 x_3 \\over \\pi_4} + {\\pi_2 x_5 \\over \\pi_4} + {\\pi_3 x_6 \\over \\pi_4})} \n\\end{pmatrix} ", "18209cace6d67a042a882ca4f3179f21": "2\\beta_E(Q)\\ell(Q)", "1820a23a4a3410b7193215760c1d395d": "\\text{NC}(S)", "1820ae845d6ddabc083fd26f233ac6e9": "a_{14}-a_{13}", "1820bab280edf6a1691cafd2a067ac5e": "(p-p')^2\\approx \\,", "1820e42debc4030e7555fca17b8fe89f": " D = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. \\,", "1821302a913a29379a79c7ddd3758434": "E_4 \\cong A_4", "1821456c71226d0bccd9a057396c8139": " \\frac{ \\mu - \\theta } { \\sigma } ,", "18215de4205af8f9584cde4689e1d404": "\\hat{\\beta} = (1-\\bar{x}) \\left(\\frac{\\bar{x} (1 - \\bar{x})}{\\bar{v}} - 1 \\right),", "18219425e20e5727d544a843f11d26c2": "k \\ ", "1822125c2e9e2ef8c64b4354c50928fa": "H^k_{\\mathrm{DR}}(M) = \\ker d_k / \\mathrm{im} \\, d_{k-1}.", "1822d316572396360302ebd0f07c7c0e": "<24>5_H \\bar{5}_H", "1822e3fbda81bca73026581c2ef5de67": "\\mathsf{PH} \\subseteq \\mathsf{BP} \\cdot \\oplus \\mathsf{P} \\subseteq \\mathsf{P} \\cdot \\oplus \\mathsf{P} \\subseteq \\mathsf{P}^{\\sharp P}", "18230599c03e0cbba169ed1c88e13259": "\\textit{add} \\in F_2", "18236dfdc3f0677bf5145aa77f138cb9": "\\Gamma\\left (\\frac{1}{2}\\right )=\\left (-\\frac{1}{2}\\right )!=\\Pi\\left (-\\frac{1}{2}\\right ) = \\sqrt{\\pi},", "182371b59680b2bb3995d8e8961c17fa": "d_{0,1} = \\log \\left( q_0^{-1} q_1 \\right)", "18237fd134c0886a5e2d0289670e0a10": "\\frac{dP(r)}{dr} = - \\frac{G}{r^2} \\left( \\rho(r) + \\frac{P(r)}{c^2} \\right) \\left(M(r) + 4 \\pi r^3 \\frac{P(r)}{c^2} \\right) \\left( 1 - \\frac{2 G M(r)}{c^2 r} \\right)^{-1} \\;", "1823cf44a3acf6db04f9133807c7897c": "\\mu = GM", "1823f7bf48e8b1b9e556cdf72b9631bf": "L_\\phi=e^\\phi(\\textstyle\\frac{1}{2}e^{-\\phi}\\partial_\\alpha\\phi\\partial_\\alpha\\phi +\\textstyle\\frac{3}{2}e^{\\phi}\\partial_0\\phi\\partial_0\\phi)\\,", "18243f98d62ee0fd1a1f457d44f2da7a": "\\xi_{0}(\\mathbf{q})", "1824892d5451c3075554d64665949840": "\\operatorname{cn}(u)=\\frac{2\\pi}{K\\sqrt{m}}\n\\sum_{n=0}^\\infty \\frac{q^{n+1/2}}{1+q^{2n+1}} \\cos ((2n+1)v),", "1824ac77ba3c5497e869a995d49d0200": "\nx_n = e^{-\\frac{2n\\pi {\\rm{i}}}{N-1}} - \\frac{t}{(N-1)^2}\\sqrt{\\frac{N}{2\\pi(N-1)}}\\sum^{N-2}_{q=0}\\psi_n(q)_{(N+1)}F_N\n\\begin{bmatrix}\n\\frac{qN+N-1}{N(N-1)}, \\ldots, \\frac{q+N-1}{N-1}, 1; \\\\[8pt]\n\n\\frac{q+2}{N-1}, \\ldots, \\frac{q+N}{N-1}, \\frac{q+N-1}{N-1}; \\\\[8pt]\n\n\\left(\\frac{te^{\\frac{2n\\pi {\\rm{i}}}{N-1}} }{N-1}\\right)^{N-1}N^N\n\\end{bmatrix},\\quad n=1,2, 3, \\dots , N-1", "1824d212ba3a9fa8345f51e91d95b859": "\\mu=\\nu_0\\leq\\nu_1\\leq\\ldots\\leq\\nu_k=\\lambda", "182502d59cbc0ac4a2c39722dfa57368": "(-1+1)>0", "18254041af8bbfe82d842bf5668298e0": "1-\\lambda n(n-1)dt", "18256429c0e252c25805a86b1d8e6de0": "k\\to 0", "1825842e493abf125ade3b211e1b6e69": "\\scriptstyle \\frac{k}{\\lambda^2}\\,", "182599009ae897b9003442eb4f24a279": "2^k/2", "1825c9d481bf09bd774ef70ead9b5def": "c_{n}(t)\\,", "1825deb23e17e064408129c0448a3298": "x\\in \\Bbb{R}^n", "1825f7196cf32cb18df1663623af7d88": "y_j < x_i < y_{j+1} < x_{i+1} < y_{j+2}.", "1826490c8f8429d9667762e38807f844": "q \\sim 7 \\times 10^{-4}", "182649aab320ff7d6ef2044500f58712": "a,b,c,d\\geq 0", "18268b8bd7443d16a0b8a0be061c5805": "P(i)=0", "1826a5403bb5e8d3577713f95fe7b673": "x_{43}=-x_{42}\\,", "1826fca994754c3f969b9c2f4e257fad": "S(\\alpha)^3 = \\sum_{n_1, n_2, n_3\\leq N}\\Lambda(n_1)\\Lambda(n_2)\\Lambda(n_3)e(\\alpha(n_1+n_2+n_3))\n = \\sum_{n\\leq 3N} \\tilde{r}(n)e(\\alpha n)", "182718a69a698f2ffd598302d271e014": " E [ y | z ] = \\beta E [ x | z ] + E [ \\varepsilon | z ]. \\, ", "1827d0c969118f3e49c0c9cb9f717505": "\\begin{align}\n\\operatorname{MSE}(\\hat{\\theta})= & (\\operatorname{E}[\\hat{\\theta}]-\\theta)^2 + \\operatorname{E}[\\,(\\hat{\\theta} - \\operatorname{E}[\\,\\hat{\\theta}\\,])^2\\,]\\\\\n= & (\\operatorname{Bias}(\\hat{\\theta},\\theta))^2 + \\operatorname{Var}(\\hat{\\theta})\n\\end{align}", "1827e41102ba5fa95018de456eb74e22": " \\lambda = 0 ", "182815887d67b5c37fdfb692f9741511": "f_X(t)=\\prod_{n\\ge 1}(1-X_n t^n)=\\sum_{n\\ge 0}A_n t^n", "18282c4f4aa63e8675c2214f357e7834": "k_{20}", "18283a5f9881739b7153820471c6092c": "S \\approx R\\, \\zeta - \\tfrac12\\, g\\, \\zeta^2 + \\tfrac12\\, \\frac{Q^2}{\\zeta} - \\tfrac16\\, \\frac{Q^2}{\\zeta}\\, \\left( \\zeta' \\right)^2.", "18284326e895bdc167579e5d6fe8e71f": "\\xi\\,", "18289a96dd8f406635a829de4137cda9": "f\\in k[x_1,\\ldots, x_n],", "1828a9b4f733effa1a056e168bb616f8": "\\textstyle \\frac{\\log m}{\\log r}", "182965c3c19e03a5f4ae0f7557b33707": "\\lambda_i \\to X'", "1829b4ff04228b596b5dbcfb24261756": "\\nabla \\cdot \\mathbf{E} = 0", "1829c110dfcbe7752f1d5601fb606573": "E_{el}", "1829db36a1114b0907f9019bb2584580": " V = m g y_\\mathrm{pend} = - m g \\ell \\cos \\theta . ", "182a047dd965d1c197775f7974c95d06": "T = T_2\\,\\!.", "182a2df236325bec6300cab798e8e02d": "\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)f(x,y)=g(x,y)\\quad \\text{for all } x,y", "182a858f9a691de79896b398babb676b": "f(b)-f(a) <\\infty", "182aaa6221b9f9974ae294fc3750e224": "\\ 1 : 1 : 1.", "182b2736cc68e270e9415f03d8fbfb2e": "X^{(n+1)}=(X^{(n)})'. \\,", "182b60b8d06ba9f4809e7d7638e6a566": "_{q.1=1.p\\,}\\!", "182bcfab235eb3a69949adeed0939be2": "d(a,b) = \\log \\underset{p \\in J}{\\max} (pa/pb) + \\log \\underset{q \\in J}{\\max} (qb/qa) .", "182be08fc613389f0837572c78ecebad": "d_\\mathrm{opt} = \\arg\\max \\limits_{d \\in D} U_D(d) \\,", "182c593ecd33d5c6f8941a54fee7f30d": "p_2/q_2", "182c5bb73ea0a9d09530cc4018fdd58a": "W_9", "182cb7a5e46e32d2d848d1d43f45bbcd": "B - A", "182cf41582a5ab9f160772cded704993": " \\int_0^2 \\! \\int_{0}^{\\pi/2} \\! \\int_0^2 \\! \\bar{f}(r,t,h) \\, dh \\, dt \\, dr = 16 + 10 \\pi", "182d3f463b7bee3e4e1f87f5d1e49d89": " C(\\xi,\\eta)=I(\\xi,\\eta) H^{*}(-\\xi,-\\eta) ", "182d65a0f5a87702d0d1fbaa64e805f8": "V(\\phi) = g\\phi^4", "182d808e6c518dfdf1849a324b77bf06": "\\; (I_{A_1}\\otimes \\Lambda_{A_2\\ldots A_m})[\\varrho_{A_1\\ldots A_m}] \\geq 0 ,", "182deae1aec958933c13359af5afeff4": "V : Y \\times Y \\to \\mathbb{R}", "182e6d6fc99a3d2179a0ffc1945cd3c9": "F^{\\times 2}", "182ee973d8eb209adf21497494e21470": "\nC \\mathbf{E}(\\mathbf{x},t) C^-1 = -\\mathbf{E}(\\mathbf{x},t), ", "182f0c15073f9fed3f9655c1d65d2f69": " \\int g_n \\, d \\mu \\leq \\lim_k \\int f_k \\, d \\mu. ", "182f375fc3b385fd8a5fa35c12174d41": " \\langle 0 | T(\\psi(x)\\bar{\\psi}(0))| 0 \\rangle =iS_F(x) = \\int \\frac{d^4p}{(2\\pi)^4}\\frac{ie^{-ip\\cdot x}}{p\\!\\!\\!/-m+i\\epsilon} ", "182fedbd4c8077dedb3589ec2621a986": "|f(n)| \\le k\\cdot|g(n)|", "18300022313a539444b08eb3d6d856bc": "\\mathbb{P}\\biggl(\\bigcup_{i=1}^n A_i\\biggr) =\\sum_{k=1}^n \\left((-1)^{k-1}\\sum_{\\scriptstyle I\\subset\\{1,\\ldots,n\\}\\atop\\scriptstyle|I|=k} \\mathbb{P}(A_I)\\right),", "183007fc35cfb2d4566f9fdfaf34816b": "A = (a_1,...,a_n,\\acute{a}_1,...,\\acute{a}_v)", "18307c5d5001ced85aa7f10beac023a5": "\ni{ \\partial \\psi \\over \\partial t} = - {\\partial^2 \\psi \\over \\partial x^2} + V(x) \\psi\n", "18309f7c0fdb0615f40cac030d0b1c60": "v: \\mathbb{N}\\rightarrow 2^{P}", "1830c0599ff4366a60437bb6b18374eb": "\\mathrm{D}_{h} (\\delta u) = \\langle u, h \\rangle_{H} + \\delta (\\mathrm{D}_{h} u),", "1830ecf04886ffdba2db365f453d8e50": "q_\\theta(X)", "1831178801c27c1a2f70a9ba0f5ca0d4": "R_i\\;=\\;af", "18311bc088e57f50e7138765a829b098": "P = 1 ", "1831205100a319fc4582f7e1375e61a0": "\nD:(TM\\setminus 0)\\times \\Gamma(TM) \\to TM; \\quad D_XY := (\\kappa\\circ j)(Y_*X),\n", "1831254d1e9f8185cdcd867f654b107d": "\\pi/4=\\sum_{n=0}^\\infty \\, \\frac{(-1)^n}{2n+1}=1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots,", "183136d279a10d602266e75f19e1ce7d": "\\operatorname{lcm}(a,b)=\\frac{|a\\cdot b|}{\\operatorname{gcd}(a,b)}.", "183159f8fda2e9e7e9110b024c8b0f9b": "\\Omega_{j,i}=\\Sigma_{r=0}^{m-1}\\Sigma_{s=0}^{m-1}w_{r,j}w_{s,i}m_{r,s}", "183189b63ec0ba3b0a7adb3a431d9a8c": "(\\gamma,\\alpha)", "18318b67b162f170eb5ae4fea5c5f702": "f: A \\hookrightarrow D", "18319091c56182057a0342558e73da54": "[\\hat{A}] =([A], [DA]),", "1831a3893624edbd6922db91a77e8dc7": "A\\to(B\\to(C\\to A))", "1831b803f26422995da82d10a4f74c0a": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}^2 \\end{matrix}", "1832232229d75037879e2cf0b581b0b3": "\\beta^i", "18327f4705a348b0e245495aaf3938bc": "\\tau_{xz}(x,h+f) = 0", "183282cbe7c574d3e78711c651854469": "\\begin{align} 2\\cdot R_*\n & = \\frac{(56.8\\cdot 0.801\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 9.8\\cdot R_{\\bigodot}\n\\end{align}", "1832aed22ef6c02e89801b869e1ef564": "\\ell(s)", "1832b59045b25ae7f789506efa9a4b97": "\\mathcal{K}_i", "1832bc2c083d9396c00c3fa9e95ba6c1": "(5)~~~~~\n \\left(\\frac{\\partial z}{\\partial u}\\right)_v\n =\n \\left(\\frac{\\partial z}{\\partial x}\\right)_y\n \\left(\\frac{\\partial x}{\\partial u}\\right)_v\n +\n \\left(\\frac{\\partial z}{\\partial y}\\right)_x\n \\left(\\frac{\\partial y}{\\partial u}\\right)_v\n", "183391278b888bd71bb3a10b976151c4": "\\chi_+", "1833b9b3850ba7bb2f00e2c3919adab4": "{\\scriptstyle\\frac{\\sqrt{3}}{2}}+{\\scriptstyle\\frac{1}{2}}i", "1833baa8be00c3f9c6b1c7a51153a9fd": " I + \\epsilon A ", "1834149017904babab5c1383f48bf494": "\\hat{h}(\\xi) = \\overline{\\hat{f}(-\\xi)}.", "18344293880397c6c4efa958d7e05516": "a \\mapsto \\exp(ar)", "18345152ddf5df725c4ecfaf942dc7ed": "\\alpha (s)\\psi (s)+ \\beta (s) \\frac{d\\psi }{dn}(s)=f(s) \\ ", "1834b2eea61af9b42512749e5bc4c7f8": "\\int_V (\\varepsilon_{ijk}\\sigma_{jk}) dV=0\\,\\!", "1834cf8c2e8a99b7f4242098caf1c7c6": "\\frac 1N", "1834d4fb897475d971052e455f525b1b": "C_I = 0", "1834d9be1758cf39f3569ea2cb58f926": "10^{10^{8}}", "1834eea7819f71a5a9fa49116072db37": "u_i: S \\rightarrow \\mathbb{R}", "183501a28322bd68c48a41a01a7cde8a": "\n H = \\cfrac{1}{2}\\left[\\cfrac{1}{(\\sigma_1^y)^2} + \\cfrac{1}{(\\sigma_2^y)^2} - \\cfrac{1}{(\\sigma_3^y)^2}\\right]\n ", "18350ff126006c0cac18f0e773a0b856": "\\displaystyle{\\pi(Z)f(t)=f(-t),}", "18359ddb57435e12f94f4ea62897189f": " V = (8/9)^2 d^2 h = 256/81 r^2 h", "1835be7100b56b8508af1ddf8e1bde96": "i_D=I_R \\left(e^{v_\\mathrm{D}/n V_\\mathrm{th}}-1\\right), ", "1835c5bace2ef9d4fe5a7344f9e53110": "\\operatorname{var} [k_t/K] = p_t(1-p_t)/K", "1835d7f4dd86231983a9ad1c89bf9992": "\\begin{align}\np_{e'}^{\\, 2} &= \\mathbf{p}_{e'}\\cdot\\mathbf{p}_{e'} = (\\mathbf{p}_\\gamma - \\mathbf{p}_{\\gamma'}) \\cdot (\\mathbf{p}_\\gamma - \\mathbf{p}_{\\gamma'}) \\\\\n &= p_{\\gamma}^{\\, 2} + p_{\\gamma'}^{\\, 2} - 2 p_{\\gamma}\\, p_{\\gamma'} \\cos\\theta. \\end{align}", "18361e638512e651e8c23df185725c21": "F((x,y),(a,b,c)) = x^2 + y^5 + ay + by^2 + cy^3", "18362c17c3ee475f77126217be27febd": " M = \\infty ", "1836de901ad6a30e77d360fb3ca4afee": " q_1 = q_1(x,y,z),\\, q_2 = q_2(x,y,z),\\, q_3 = q_3(x,y,z)", "1836f034080a207eed105478c72af25b": " where \\; y' = \\frac{y}{y_c}", "1836f2460e353a21220bd1f362c07722": "s\\geq s'\\rightarrow \\theta\\geq \\theta'\\rightarrow s\\geq s'+\\ell", "183768d73eb7e86c955d1c55eb033bb3": " \\, \\frac{|SC|}{|SD|}=\\frac{|SA|}{|SB|}", "18376f14dace82ffaf0112ca25ea25a1": "Vs_\\mathrm{ideal} ", "1837712d68ef78c5814a9f0d808dc106": "G_m = g_m", "1837881fb32d09869aee4f1571987085": "s_N(x) = \\frac{a_0}{2} + \\sum_{n=1}^N A_n\\cdot \\sin(\\tfrac{2\\pi nx}{P}+\\phi_n).", "1837c55593a059c0e994ee2bb32291cc": "\\hat H = \\frac{{\\hat p}^2}{2m} + \\frac{1}{2} m \\omega^2 {\\hat x}^2 \\, ,", "1837ed9702b1069b77b5e75c0cd7ae6a": " r_\\mathrm{ corr } = r + \\frac{ N - 1 }{ N } \\frac{ m_y - r m_x }{ n - 1 }", "18383f4420cc4cdbff89bed41575530e": "n_T = 2n_0 - 1", "183842eaa5e8b4b40a5a3f8c67454a9a": "\n\\langle z^m \\rangle = \\frac{1}{2\\pi i}\\sum_{n=-\\infty}^\\infty \\phi(-n)\\oint z^{m+n-1}\\,dz.\n", "18386b8a73b75c0fda8f5c9629dba2ca": "p_k= 1- \\sum_{i=1}^{k-1}p_i", "18388e539d1e6bb99c30688cb6ca8c6c": "j=1,\\ldots,n", "1838932f4a6c40a254164ccc211e6b39": "x^{(n)}=x(x+1)(x+2)\\cdots(x+n-1)=\\frac{(x+n-1)!}{(x-1)!}.", "1838c16ae9e555a727d005a9079e6b4e": "\\lambda_\\mathrm{B},", "1838eb6b97e24ae7a5bbd2af56eebfb1": "\\dot{M}=dM/dt=2 \\cdot 10^9 kg/s", "1838f4463a2d45a09bd4810a41de36ed": "T = \\sum_{i=0}^\\infty \\ell_i v_i = ", "18391c75778278e20d13d5755fe04110": "{[x_1, x_2]}^n = [x_2^n, x_1^n]", "1839a0de1867d4b5972a709c41cd84b2": "4x^2 + 49", "1839a9b74c88f400956d248878e1965f": "w^2 + 2x^2 + 5y^2 + 5z^2,\\ ", "1839ae658db9a25c83f4442844ca5dbd": "\\lfloor \\frac{q}{k}\\rfloor \\approx d", "1839da7e98a7cca6bf7f4ae20c4cd214": "G^{\\mathrm{R}}(\\omega)", "1839fc639593bc47f117f595ae611999": "=\\int |f||f + g|^{p-1} \\, \\mathrm{d}\\mu+\\int |g||f + g|^{p-1} \\, \\mathrm{d}\\mu", "183a2919ba17457f6136c18bc8df1ff0": "s_{max}", "183a4943a4487593b6903b239c251831": "u(M) \\subset IM", "183aadeae11df4d7d4cb025424d81af4": "1-4c=0", "183abe8f222f2d2ade7c9afdc3b08915": "\\theta^2/2", "183abfb247b0b35039d65f379665b8f3": "100 / (.50 + .75)= 80 ", "183ae4bcda6933e2cad7ca4fd2e135e3": "O \\left (\\log(q) \\deg(f)^3 \\right )", "183b5717f8f582a0b397696ace269606": "\\omega_1=1", "183b6c3fa732af18876b95be32d1ab8a": "\\frac{4}{\\nu-6}\\sqrt{2(\\nu-4)}\\!", "183b76e2a4c3d295e3a0a81a020abe57": " | f(x) - f_\\epsilon (x) | < \\epsilon ~\\forall~ x\\in M", "183b77c41ab8b4e30c7812f0d1eac4ed": " r^{(g)}=1", "183ba6c8f5b10f0f5ead639e32915971": " \\lim_{x \\to 0} \\frac{Ax}{x} = A . \\! ~~ (5) ", "183bb059e4277a53a853acc75e799771": "\nU = \\begin{bmatrix} \\rho \\\\ u \\end{bmatrix}, \\quad A = \\begin{bmatrix} 0 & \\rho_0 \\\\ \\frac{a^2}{\\rho_0} & 0 \\end{bmatrix}\n", "183bee6b20836c70d8a6c54cd6726e28": "x = {{X+Y} \\over \\sqrt{2}}, y = {{X-Y} \\over \\sqrt{2}}", "183d0e1bcc3962a7bddcb0e98716d236": "e^*=(1/2)(-f+ \\sqrt {f^2+4f})", "183d747a3b2791cd2429af84e99fe397": "\\delta H^3 = 3H \\int_{\\Lambda<|k|<(1+b)\\Lambda} {d^4k \\over (2\\pi)^4} {1\\over (k^2 + t)}", "183dadd6db498a4fbad18a4818171344": "\n\\ln\\left(\\ell^n n\\pi^{n/2} (2E)^{\\frac{n-1}{2}} \\right) = \n\\underbrace{\\ln\\left(\\ell^n E^\\frac{n}{2}\\right)}_{important} + \n\\underbrace{\n\\ln\\left(\n\\frac{n (2\\pi)^\\frac{n}{2} }{\\sqrt {2E}} \n\\right)}_{drop}\n", "183e866071ebaf841e922bed45d3a822": "\\frac{D}{Dt} = \\frac{\\partial}{\\partial t} + \\mathbf v\\cdot\\nabla.", "183ea5d2b4b2e6570c5478c5a1e96e95": "\\text{coNP}\\subseteq\\text{NP/poly}", "183efec275d6b13d632c15f8a65f7bff": "G^n(L/K)", "183f1f1f38cbf05ecaff07e39fc93c54": "\\mbox{NEXPTIME} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NTIME}(2^{n^k})", "183f5ed968f78bd4497ccd28d5db4d5b": " {h_1 \\over h_0} < 1 ", "183f668f1798cdc27eef1992345cd003": "\\mathbf{D}=\\epsilon_0\\mathbf{E}\\,, \\quad \\mathbf{B}=\\mu_0\\mathbf{H}\\,,\\quad c^2=\\frac{1}{\\epsilon_0\\mu_0}\\,, ", "183f8ff2614fa69d652ce4c456fc3557": "\\operatorname{tr}(X \\otimes Y) = \\operatorname{tr}(X)\\operatorname{tr}(Y)", "18401af7241dbe26e4801e6561aa1a08": "\\,F_t", "184050d47e8d9a0a92ae0bf2fd2b8602": "|\\phi\\rangle\\otimes|\\phi\\rangle", "18405e9b1190af7c2c733507980ccaed": "\\displaystyle{|a_{1}|^2 - |a_{-1}|^2={1\\over 2 \\pi^2} \\int_0^{2\\pi}\\int_0^{2\\pi} \\sin (\\theta-\\varphi) \\sin(h(\\theta)-h(\\varphi))\\,d\\theta\\,d\\varphi={1\\over 2 \\pi^2} \\int_0^{2\\pi}\\int_0^{2\\pi} \\sin (\\theta) \\sin(h(\\theta+\\varphi)-h(\\varphi))\\,d\\theta\\,d\\varphi.}", "18406b33f1195399e5eb70bc76d1b1f9": "\\mathbf{S}' = -\\mathsf{V} \\frac{\\partial \\mathbf{D}} {\\partial t}", "1840cb9f974442643de6b7751e969ffd": "\\frac{\\partial \\mathbf{f(g(u))}}{\\partial x} =", "1840cf3e815b0c428155d6eb03946c8d": " I_i = \\text{Inputs}", "1840ff440ea05751174ef653dbd01fe6": "A - B - 2\\sqrt{3} = 0", "18415d1c45cd9731d7ab4ab2fe34162e": "\\left|\n\\begin{array}{cc}\n x_1 & x_2 \\\\\n x_2 & x_1\n\\end{array}\n\\right|=\\left(x_1+x_2\\right)\\left(x_1-x_2\\right).", "184191b81bed4c1b5a9bd4f75598db90": "\n{d\\theta \\over dt} = {\\partial H \\over \\partial J} =H'(J)\n\\,", "1841dca31a5b7a702e317f0023ed2aa1": " R_C || \\left[ (r_{ \\pi}||R_S)(1+g_mr_O) \\right] \n ", "1841eb436e741a9300dd175bf3e41a29": " \\int_{\\Sigma(t_0)} \\left. \\frac{\\partial \\mathbf{B}}{\\partial t}\\right|_{t=t_0} \\cdot d\\mathbf{A} = - \\oint_{\\partial \\Sigma(t_0)} \\mathbf{E}(t_0) \\cdot d\\boldsymbol{\\ell} ", "1842190789f968b9d4a9f64091c8710c": "\\frac{\\delta \\mathcal{S}[\\phi]}{\\delta \\phi(x)}\\approx 0", "18423db170f858c009de471d95495686": "0 = g(y^*). \\,", "184343a8e545da50d752d2b9580f0348": "\\sigma+\\tfrac{\\sigma}{\\sqrt{2\\beta}} H_{-1}\\left(\\tfrac{-1+\\alpha}{\\sqrt{2\\beta}}\\right) ", "1843e0510d9907d4cc4bbed1a14544fd": "\\langle \\mbox{grad} f(x) , v_x \\rangle = df(x)(v_x)", "1843f5db100a3733017b2328aa168c2c": "B \\supseteq A.", "184420a8f05da514fc3eb283a4394acc": "SO(3)\\;", "184464beac37b0d60e9fe0e7691f71c4": "a_1+a_2+\\cdots+a_{24}\\equiv 4a_1\\equiv 4a_2\\equiv\\cdots\\equiv4a_{24}\\pmod{8}", "18448f21ca593c75ec2f34ccf5fc71cd": "\\operatorname{AveP} = \\frac{1}{11} \\sum_{r \\in \\{0, 0.1, \\ldots, 1.0\\}} p_{\\operatorname{interp}}(r)", "1844a2846f1ac12d26a47be0ba087dd8": "\nf(\\mathbf{x}) = arg\\min_j D(\\mathbf{y}(\\mathbf{x}),\\bar{\\mathbf{y}}_j),\n", "184535df04431873a0391b72daf77c68": "gJ", "18453a1953307ec7783f466f30874cbe": "k, n_1, n_2", "1845589f5dfe488a5f15800a409dffb4": "A \\not= S_0", "1845dc1c1cef5c12daa2c302344dff0a": " \\frac{\\sum_i y_i / \\sigma_i^2}{\\sum_i 1/\\sigma_i^2} .", "1845ed3596406b0865f1ccf29931d6ca": "G_{xy} = H(f)G_{xx}(f)", "1846240c415de18b8ce4beb8ba27a0d3": "q_y", "18468b65fb318609c11c8bed98ac686a": "((x \\mapsto (y \\mapsto x \\times x + y \\times y))(5))(2)", "1846a733ccb9acee97c2f742913f500a": "X \\sim {\\beta}_{\\alpha, \\beta}", "1846a8dcd4c79bfdd4115a093b958fd0": "F(\\mathbf u)", "1846f7429e80f6e23211ae4640b20a93": "z(u,v) = u \\cos \\theta + v \\sin \\theta \\,", "184737a7bb3db92bfdb95f77c8b1f603": "f(t)=(f_1(t),f_2(t),f_3(t),\\ldots) \\, ", "184762fd737390b91f5f76d9eee5003b": "\n\\left[ \\Delta - \\lambda^2 \\right] G(\\mathbf{r}) = - \\delta^3(\\mathbf{r}).\n", "18477ed4ed68f37232517fe2deed40e3": "(\\Gamma,S)=(\\mathbb{R},\\mathcal{B})", "18479d8a7b83464aec168bbf151727ec": "| {\\Psi}\\rangle=\\frac{1}{2\\sqrt{2}}\\left( | {00}\\rangle + {\\sqrt{3}} | {01}\\rangle + {\\sqrt{3}} | {10}\\rangle + |{11}\\rangle\\right)", "18483643c8aeeaa7aee767036d28d0c0": " \\left|c_n(t)\\right|^2 = \\left|\\lang n|\\psi(t)\\rang\\right|^2", "1848a6e362d9e3c380e9bc351796a148": "q=\\exp (i\\pi\\tau)", "1848eded3a09c8f8346205a6bb8670a4": "e=\\frac{r_{a}-r_{p}}{r_{a}+r_{p}}=\\frac{r_{a}-r_{p}}{2a}", "18493cd78ab291afd13a2864fbd01566": "Var X_i = 1", "1849e6147f9b9cdf42fea44d24385db1": "\\frac{\\partial \\mathcal{H}}{\\partial q_j} =- \\dot{p}_j \\,, \\quad \\frac{\\partial \\mathcal{H}}{\\partial p_j} = \\dot{q}_j \\,, \\quad \\frac{\\partial \\mathcal{H}}{\\partial t } = - {\\partial \\mathcal{L} \\over \\partial t} \\,.", "1849fabc3398c6a4bf69a754d08a9c2b": "\n\\text{minimize} \\quad \\text{over } \\widehat p \\quad \\|p - \\widehat p\\| \\quad\\text{subject to}\\quad \\operatorname{rank}\\big(\\mathcal{S}(\\widehat p)\\big) \\leq r.\n", "184adbd5c4bfcf858aeb9f218f7ab380": "{}^{238}_{92}\\text{U}\\to{}^{234}_{90}\\text{Th}", "184b35ade951580c0f2f3e24ee8d4f58": "m_i = (i+1)\\cdot m+1", "184b471913188b25ed9177512b8667ae": "t \\mapsto (u+v) \\oplus (u - v) , \\quad s \\mapsto (w + z) \\oplus (w - z).", "184b5881d57b39e794f3d9187ba5545a": "b=2uv(v^2+u^2), \\,", "184bb24318fb76fce070cbebbefb86b7": "v \\propto \\sqrt{E/\\rho}", "184bbcf51935abd38bc62c172292a383": "h(k) = \\frac{1}{k!} \\ ", "184bf62fa08b14a75fc7a4b5310a0e27": "H_*(X;A)", "184c0df2b8a393e6e73f3e310cfd283b": "D_1,\\ldots,D_n", "184c2d1e36fddc5a4ad2bf0f8d73f205": "\\!\\ c_\\mathrm z", "184c3a7d37db7c46b4e66b187529727d": " B-\\text{vertex}= \\sec^2 \\left(\\frac{A}{2}\\right):0:\\sec^2\\left(\\frac{C}{2}\\right)", "184c850f127dcacfe881535591b6d3c9": "Z_n^m(\\rho,\\varphi+2\\pi k/m)=Z_n^m(\\rho,\\varphi),\\quad k= 0, \\pm 1,\\pm 2,\\ldots", "184c88ea75d19e65e665e1af1b6c4be3": "\\boldsymbol{P} = \\varepsilon_0 \\chi_e \\boldsymbol{E} = \\varepsilon_0 (\\varepsilon_r - 1) \\boldsymbol{E}", "184c9454f8f41da2d994ba3ad8678802": "R = \\ell \\cap \\ell '", "184d2b7c778fb230d0911410ed78df60": "\\int ku dx = k \\int u dx.", "184d3d276a6ff583ccf7819bce7ed7fc": "\n\\delta D = \\frac{1}{2}\\rho NW^2 c \\times c_D(\\alpha)\\delta r\n", "184db09385953c5c5c1d4a8c62695e84": "\\bold{p}_\\mathrm{em} = {\\bold{S} \\over {c^2}} ", "184de2f85cdf6b7cacc9ca2086b91e3a": "\\begin{align}\n \\left(\n \\frac{1}{\\sqrt{\\sigma^2 + \\tau^2}} \\frac{\\partial A_z}{\\partial \\tau}\n - \\frac{\\partial A_\\tau}{\\partial z}\n \\right) &\\hat{\\boldsymbol \\sigma} \\\\\n- \\left(\n \\frac{1}{\\sqrt{\\sigma^2 + \\tau^2}} \\frac{\\partial A_z}{\\partial \\sigma}\n - \\frac{\\partial A_\\sigma}{\\partial z}\n \\right) &\\hat{\\boldsymbol \\tau} \\\\\n+ \\frac{1}{\\sqrt{\\sigma^2 + \\tau^2}} \\left(\n \\frac{\\partial \\left(\\sqrt{\\sigma^2 + \\tau^2} A_\\sigma \\right)}{\\partial \\tau}\n - \\frac{\\partial \\left(\\sqrt{\\sigma^2 + \\tau^2} A_\\tau \\right)}{\\partial \\sigma}\n \\right) &\\hat{\\mathbf z}\n\\end{align}", "184e5a7e6e1073c0a3f7243649b2067b": "\\begin{align}\n&\\nabla \\cdot \\mathbf{D} = \\rho_f \\\\\n&\\nabla \\cdot \\mathbf{B} = 0 \\\\\n&\\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B} }{\\partial t} \\\\\n&\\nabla \\times \\mathbf{H} = \\mathbf{J}_f + \\frac{\\partial \\mathbf{D} }{\\partial t}\n\\end{align}", "184e782bf914db1fc2355492bdee9c55": "\n \\begin{align}\n I_1 & = \\text{Tr}(\\boldsymbol{\\sigma}) = \\sigma_1 + \\sigma_2 + \\sigma_3 \\\\\n J_2 & = \\tfrac{1}{2} \\boldsymbol{s}:\\boldsymbol{s} = \n \\tfrac{1}{6}\\left[(\\sigma_1-\\sigma_2)^2+(\\sigma_2-\\sigma_3)^2+(\\sigma_3-\\sigma_1)^2\\right] \\\\\n J_3 & = \\det(\\boldsymbol{s}) = \\tfrac{1}{3} (\\boldsymbol{s}\\cdot\\boldsymbol{s}):\\boldsymbol{s}\n = s_1 s_2 s_3\n \\end{align}\n ", "184e84f97644c4aa7211f644b60809fc": "M_{Y}=\\{Z\\in\\mathbb{R}^k:Z\\preceq Y\\}\\ ", "184f1c169a111c246ae95b090df83096": "{\\mathcal U}^*", "184f1ede41cf2e5e475572c01a83b51e": " f(0) = f_o ", "184fa6b509434f3760b52b6ae75f6a4f": "l(\\theta) = \\alpha+\\theta", "184fc565219eea37cea832d2263e6bde": "f\\left(\\sum_{i\\in I}\\lambda_i a_i\\right)=\\sum_{i\\in I}\\lambda_i f(a_i)\\, .", "1850265a3b39b66d1befbf68eba5b3dc": "A=-x\\sin(\\theta)+y\\cos(\\theta)cos(\\phi)+z\\cos(\\theta)\\sin(\\phi)", "1850f5f395348fa21a7a0909ca424cf5": "a_1, ..., a_n", "185120d886d1312b5765fbcc6ca56b7e": "Prob(f(x_k)-f^*> \\epsilon) \\leq \\rho", "18514abffdf8c0476fc31bbcc67288d9": "\\nabla\\sigma = (\\mathrm d\\sigma^\\alpha + \\omega^\\alpha\\!{}_\\beta \\sigma^\\beta)e_{\\alpha}", "1851c0714960d2de5b1fc4fe90456ab6": "w(E\\oplus F)= w(E) \\smallsmile w(F)", "1851c157eadb0c9b355f1ba8bd7a0ac2": "\\mathfrak{g} \\to \\mathfrak{gl}_V", "1852050689b1c87aafadf313e46821e8": "= \\left \\vert \\frac{1-k(N- \\varphi (N))}{Nd} \\right \\vert ", "185254c23f9d65f50f7cc185b854a1a8": "x^2 + y^2 + z^2 = R^2.", "1852867e59486e21c1b1ef2bb36b7318": "\\mathrm{Ad}_g\\colon \\mathfrak g \\to \\mathfrak g", "18529b6408dd8f7ec05df5cbb594f2eb": "f(x_0+\\Delta x)=f(x_0)+\\nabla f(x_0)^T \\Delta x+\\frac{1}{2} \\Delta x^T {B} \\Delta x ", "1852bd5f3275076ca7061de28284c940": "\n\n\\begin{align}\n-\\gamma^\\mu \\hat{P}_\\mu + mc & = -\\gamma^0 \\frac{\\hat{E}}{c} - \\boldsymbol{\\gamma}\\cdot(-\\hat{\\mathbf{p}}) + mc \\\\\n& = -\\begin{pmatrix}\nI_2 & 0 \\\\\n0 & -I_2 \\\\\n\\end{pmatrix}\\frac{\\hat{E}}{c} \n+\n\\begin{pmatrix}\n0 & \\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}} \\\\\n-\\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}} & 0 \\\\\n\\end{pmatrix} + \\begin{pmatrix}\nI_2 & 0 \\\\\n0 & I_2 \\\\\n\\end{pmatrix}mc \\\\\n& =\n\\begin{pmatrix}\n-\\hat{E}/c+mc & 0 & \\hat{p}_z & \\hat{p}_x - i\\hat{p}_y \\\\\n0 & -\\hat{E}/c+mc & \\hat{p}_x + \\hat{p}_y & -\\hat{p}_z \\\\\n-\\hat{p}_z & -(\\hat{p}_x - i\\hat{p}_y) & \\hat{E}/c+mc & 0 \\\\\n-(\\hat{p}_x + i\\hat{p}_y) & \\hat{p}_z & 0 & \\hat{E}/c+mc \\\\\n\\end{pmatrix}\n\\end{align}\n", "1852de6ec5dfa06cad32a00df9ee261d": "\\mathbf{M} = (\\mathbf{M}_2-\\mathbf{M}_1)(\\mathbf{M}_2-\\mathbf{M}_1)^{\\text{T}}", "185320f6d98bb95ba3588dbe2393fe5f": "P_1,\\ldots,P_4", "18537c930bc137026b2ea3d0813a7b48": "(\\lambda_{\\bold{k}} - \\epsilon)C_{\\bold{k}} = 0", "18538269a292aa77c1d029bc6d0105ea": "|\\varphi \\rang", "18538c9da0cd3169a30025c503983ea7": "\nW_+ = \\sum_{i:h_t(x_i)=y_i} D_t(i)\n", "1853aec857e01032063e88e43ebaec3f": " F_n = \\oplus_{i=0}^n A_i ", "1853fb3bbf5f335001f003dbedccfdcd": "J^\\mu\\,", "1854264739edc59e21798ba131285dd6": "F_\\bold{X}", "185435a919237441cf36f1f63c896e2a": "\\sqrt{\\sqrt{0.1_2} \\cdot \\sqrt{10_2}} = \\sqrt[4]{1} = 1", "185449a1785117afe26a58f039212d04": " \\mathbf \\Pi = \\varepsilon_0 \\frac{ \\partial }{\\partial t} \\mathbf A \\ , ", "18545200d5d476d2bc8427bb8da1e0bf": "\\epsilon\\left(t,z\\right)", "18546e49a068f46d81972bd13a9a67de": "\n(1 - q) \\mathcal{L}[U(x; q) - u_0(x; q)] = c_0 \\, q \\, \\mathcal{N}[U(x;q)],\n", "185475a4564babb661facae2a50a333a": " \\omega = \\pm \\pi / T. \\ ", "1855079f4b10ec37c47495cc078be98f": "b, s, \\beta > 0\\,\\!", "1855385b46bd6c0446dfae06dbcc3684": "\\chi_1(\\omega)", "185555f8187ea28e0eee70a61b00f772": "r \\approx \\pi \\sqrt{N} / 4", "185591a9ae38cd83b5044462a10ddf15": " \\lambda_k(D_1) \\le \\lambda_k(D_2).", "1855972f21eb11bd3b16e231bc9638f9": " W=Fr\\phi=\\tau\\phi,", "1856372dfcb39dd3413daaa4c918f846": "z^* = \\frac{r^2}{\\overline{z - z_0}} + z_0", "185679f33acda4cb07f311b5e5c6c129": "\\mathbf{u}=0", "185680cadfcb6e4f750942dae9a7693c": "\\rho = 0.065*\\mathrm{D}^2 * \\pi / 4", "1856add222a611b85b3d0b0080e48e71": "(3)\\qquad A_a = \\Phi(\\rho,z,\\phi) [dt]_a\\;,", "1856e10b6af995b7af7b8f7513b7a8e2": "\\frac{1}{r^2} P^1_1(\\sin\\theta) \\sin\\varphi= \\frac{1}{r^2} \\cos\\theta \\sin\\varphi", "18570c3481ad123007be7bcd04adab83": "\\dot y = \\mu(1-x^2) y-x.", "1857464d3aea614c6f299ce9bf719c11": "P_3(y)=0", "18574b71aa1fffdc3695c71726b90157": "u \\in L ", "18574eb24482def01156684ff5cafec4": "b \\wedge \\left( \\bigvee a_i\\right) = \\bigvee \\left(a_i \\wedge b\\right)", "1857aa4a5dddf86a1adbf7a7873986b7": "f(s)=\\sum_n a_n e^{-s|\\omega_n|}", "18583cd15b2f403d87db08950a7da0ed": "w(L)", "1858ffa9e5e0e0a36776d25229a263c2": " \\mathbf{\\Lambda} = \\operatorname{diag}(\\lambda_1, \\lambda_2, \\dots)", "1858fff801b6db73d147b01c34fd20e5": "W(r)~", "185945601a09320d7c3e192bec764901": " f+g ", "18595730605f0e917b8b166fda878841": "* \\to G", "18595d5dac54286e2332c93b5ac49817": "\\prod_{i=1}^{p-1} i^{p-1} \\equiv (-1)^{p-1} \\equiv +1 \\pmod p", "1859a071aa5560a33a5fff6ef385231c": "R_{Hp} = \\frac{1}{pq} = \\frac{V_{Hp}t}{IB}", "1859a1f931628a74f6fd532a2e0596ab": "\\bar f:X\\times_Z Y \\rightarrow Y", "1859a405da6a0198f3bd160c94b74887": "III_1", "1859b03afd0fd4ac368db995f8ff41a0": "\\scriptstyle ab", "1859cfba978b10c953c0969e4fff638b": "\\sum_t \\left\\{[(at + b) - y_t]^2\\right\\}", "1859ffa58ca953bc04a9e7b04bfd470e": " J_G(\\mathbf{x}^{(0)}) ", "185a344b8f2b9207948ecfd93a14ccc3": "L(u)=E(u)-\\omega Q(u)+\\Gamma(Q(u)-Q(\\phi_\\omega))^2\\,", "185a6de473e8ea6a0c322c509739a449": " v_i = (0, \\ldots , 0, 1, \\ldots , 0, w_{i_1}, \\ldots , w{i_d})", "185a71b1e8b5ea0f127e8f499994851e": "F(x) = \\begin{cases}\n p^2 & \\mbox{if } x \\equiv p \\pmod{p^3} \\\\ \n p^4 & \\mbox{if } x \\equiv p^2 \\pmod{p^5} \\\\ \n p^6 & \\mbox{if } x \\equiv p^3 \\pmod{p^7} \\\\ \n \\vdots & \\vdots \\\\\n 0 & \\mbox{otherwise}.\\end{cases} ", "185a7a7dbfb9f18344c8a84cf7e87737": "\\sum_{i=1}^{n-1} \\frac{\\partial}{\\partial x_i}\\frac{\\frac{\\partial f}{\\partial x_i}}{\\sqrt{1+\\sum_{j=1}^{n-1}(\\frac{\\partial f}{\\partial x_j})^2}} = 0", "185aea59b36f8863e71df2730ca694aa": "a = b = 1/2", "185b340d0b0f300147c5444b745d5411": " \\operatorname{Tr}(\\bar Q \\rho) ~=~ (\\epsilon - \\mu) + (1-\\epsilon + \\mu)\\operatorname{Tr}(Q\\rho) ~=~ \\epsilon ~.", "185b41b4aa6d23dd8d92382fdb269cdc": "v = \\left|\\boldsymbol v\\right| = \\left|\\dot {\\boldsymbol r}\\right| = \\left|\\frac{d\\boldsymbol r}{dt}\\right|\\,.", "185ba22e9e196d8ad29861510c83f703": "I_A < I_B = I_C ", "185baf079946d5d2e373935d389c30e1": "w(\\mathfrak{D}_{K/Q})", "185bb1669acf9862b0b949bce2ff3b09": " \\sqrt{1 - \\frac{\\lambda_2}{\\lambda_1}}. ", "185bd6d6db66181aa5a52568a5387be6": "\\nabla^2 \\mathbf A - \\frac 1 {c^2} \\frac{\\partial^2 \\mathbf A}{\\partial t^2} = - \\mu_0 \\mathbf J", "185c04daf95b1832a7f47d6a983971ce": "n_{A}", "185c1231f4572167debcaf7e022afefb": "H^{1,1}", "185c342787c08cf38c76a8415c9ddd21": "\\scriptstyle M_0, M_1", "185c3ecfdbed23e47da6e19b61d2f2e1": "\\forall x . (P(x) \\rightarrow P(c))", "185c754c0d9c866c4482aeec6b4229f1": "f(N) \\leq {2N-5 \\choose N-2} + 1 = O\\left(\\frac{4^N}{\\sqrt N}\\right).", "185c75b3ada96b45e7632feea6d58caa": "A=-\\frac{1}{3}\\,U", "185c806477593a3111195edb063e5711": " D = 2R ", "185cfbf64632a8b81deb92c4c4a3205a": "{\\mathit{He}}_{n+1}(x)=x{\\mathit{He}}_n(x)-n{\\mathit{He}}_{n-1}(x),\\,\\!", "185d14a325f74be8ad5bd60ab37f0fb4": "x_i = a_{i,1} s_1 + \\cdots + a_{i,k} s_k + \\cdots + a_{i,n} s_n", "185d38fd286f10138246de48021405e8": "v = v. + a.t + \\frac{1}{2}j.t^2 + \\frac{1}{6}st^3 ", "185d53698528f35e737eb920fe462a13": "\\sqrt{\\frac{5}{18}}\\!\\,", "185d643b00fbefaa137e8b1ad9320ed0": "(0,1/n)", "185d64ac0318fdad5f0877b800fd2f96": "|\\omega|", "185dd26d79796c0c4777836a506da2fa": "\\langle \\exp(i\\theta[\\sigma]) \\rangle_p", "185de6c59b63d0fe203ed8e1f5e0b5b2": " Z_{t} = \\int_{0}^{t} H_{s} \\, \\mathrm{d} s,", "185e85b5f22ddaaafea039631c3b2062": "K \\oplus 0 \\subset \\bar{G}", "185e8c0a3c5efa5e444e5bdc36fd1468": "\\triangledown_Y\\circ\\varphi=\\varphi^\\triangledown\\circ\\triangledown_X", "185eb2ed544b238edd2e7a814b79c89e": "x_1, \\ldots, x_k", "185ed55975ef822b7bb1a658c4b2f2f3": "A^{-1}:y \\mapsto \\left( \\frac{y}{2} \\right)^2 - \\frac{3}{8} ", "185f55ddfbc1363d6514b54225d47da8": "\\propto", "185f5bc2ce2d881bda2bfaca3daeb461": "f(\\boldsymbol{x}) = \\underset{y \\in \\mathcal{Y}}{\\textrm{argmax}} \\quad \\boldsymbol{w}'\\Psi(\\boldsymbol{x},y)", "185fa04ddf788300d0df2961cd76a261": "(ls)P", "185faabe3fdbbe32c06ba4dcbff5e17a": "\\{y_i,x_{ij}\\}_{i=1..n,j=1..p}", "18600f1fc25bd1db4d7707789793dc3a": "H(W)", "18602ef4f9402019a08807e2f5f5b876": "x = \\gamma \\left[ \\gamma \\left( x - v t \\right) + v t' \\right]", "1860493b119ea5605470b4bb32e486a9": "\\nu_{12}\\sigma^2", "18608495abd0dff2716c141fcac60c1c": "V = \\begin{bmatrix} a_1 & a_2 \\\\ b_1 & b_2 \\end{bmatrix} \\in\\R^{2 \\times 2}", "1860d352e43fb9a193707328a8797bd1": "\\theta\\in[0,2\\,\\pi)", "18610afebba9400a76c9222047ef215e": "A \\times B \\cong X", "18616b754bbd47e1aafeef108f889fa6": "\np(z)=a_n z^n+\\dots+a_1 z+a_0.\n", "186198f0889ddd5f9c33978be9fd57d3": "(\\hat{\\boldsymbol{x}},\\hat{\\boldsymbol{y}},\\hat{\\boldsymbol{z}})", "1861a91087cf0697b444cfa539f823d4": "\\frac{h^2}{d}", "1861b28d8f4f3507f88bd58ae86e30f3": "\\scriptstyle R_{sn}", "1861d63716fbe36a863949a5aad1bd77": "\\mathrm{MC}^3\\ ", "1861dba33099bf8f1194dabc69dcf004": "d x^1 \\, d x^2 \\, d x^3 \\, d x^4", "1861e72b8c948af3e22b5ec64ec97980": "y \\in \\{l,r\\} \\setminus \\{x\\}", "1861ee095f63010dc71e2ef5774005b3": "E=-\\frac{\\hbar^2\\kappa^2}{2m}=-\\frac{m\\lambda^2}{2\\hbar^2}.", "18625162087830dc8c5efcf988352c48": "r(x)>r(y)", "18625bfe5e21136c4de45bbffe7d96f9": "{\\eta \\over \\mathit{\\Delta}} = \\cos \\delta \\sin \\alpha", "186278def33580aff74015039c35a86a": "\\Bigg(\\frac{17}{p}\\Bigg)_4\\Bigg(\\frac{p}{17}\\Bigg)_4=\n\\begin{cases}\n +1 \\mbox{ if and only if }\\;\\;p=x^2+17y^2 \\\\\n -1 \\mbox{ if and only if }2p=x^2+17y^2\n\\end{cases}\n", "186301a10c315169618815cfa0f9676d": "F(b) - F(a) = \\int_a^b f(t) \\, dt.", "1863363aafb2bdcffae32ab04b678990": "\\{\\varnothing,\\{1\\},\\{0,1\\}\\}.", "18636d2afe15922099975c8eeb99e766": "T_0\\,\\!", "186386077e6b916e2b10cf6846e5d46a": "e^{ix} = \\cos x + i\\,\\sin x.", "186389b2a6f4a298a3fcf683b501e6a6": "\\chi(t, \\zeta) = \\frac{ |t - \\zeta|^2}{4(1 - |t|^2)(1 - |\\zeta|^2)}", "18639bfc3b6c41225bd852a12786f3a8": " r \\ = \\ (1+i/n)^{n} - 1", "1863b0de8f3dfbeeef0096c03e5d7e13": "L_{\\gamma}(X, Y)", "1863eb80e0e8f209e80feab2b46a4a92": " \\rho_k=k\\frac{U_{MN}}{I_{AB}}", "18641f3cca77893e1c73938c9d01b1e5": "\\tau \\,\\!", "1864472b3e5d7c64a6f1de15a3824c4c": "\\operatorname{Li}_2(z)+\\operatorname{Li}_2(1-z)=\\frac{{\\pi}^2}{6}-\\ln z \\cdot\\ln(1-z) ", "1864ef34a0f34b19a7bb1567e2ffc841": " \\sum\\limits_{i,j=1}^{2n} a_{ij} \\xi_i \\xi_j", "1864f7e6674885847484937d6b4e7a52": "V=\\bigcup {}_{q \\in Q} V_q", "18657a80050f9b39dc140f0c7010defb": "p/m", "1865945d65b5d44c44bc46f94cef8dce": "\\begin{align}\n C &= S_0 N(d_1) - Xe^{-r(T)} N(d_2) \\\\\n\\\\\n d_1 &= \\frac{\\left[\\ln\\left(\\frac{S_0}{X}\\right) + \\left(r + \\frac{\\sigma^2}{2}\\right)(T)\\right]}{\\sigma\\sqrt{T}} \\\\\n\\\\\n d_2 &= d_1 - \\sigma\\sqrt{T}\n\\end{align}", "1865ed583b301d5bb65c70756b1fca7b": "XAX = X", "1865fe9e1b89182e53afbfe3edf37e62": "\n\\begin{array}{cl}\n & P\\left(Searched|Known\\wedge\\delta\\wedge\\pi\\right)\\\\\n= & \\frac{1}{Z}\\sum_{Free}\\left[\\prod_{k=1}^{K}\\left[P\\left(L_{i}|K_{i}\\wedge\\pi\\right)\\right]\\right]\n\\end{array}\n", "18662a1d455ea4bd72218c341f352cb9": "y = \\sqrt{x(x-1)(x-\\lambda)}", "186639c8a2a8b44425af82dc6b84c9fc": "\nv=\\begin{bmatrix}0 & 1 \\\\ -1 & 0 \\end{bmatrix}, \\qquad p=\\begin{bmatrix}0 & 1 \\\\ -1 & 1 \\end{bmatrix}.\n", "18666261315c816a2946cff7d6fe3b26": " \\theta = {[L]^n \\over K_d + [L]^n} = {[L]^n \\over (K_A)^n + [L]^n} = {1 \\over ({K_A \\over [L]})^n+1} ", "18666c11ee0dfbf891d77c2337a0c5b9": "|b| = {\\ln{\\varphi} \\over 90} = 0.0053468\\,", "18668ec6adff0ef97462783657852525": "\\omega = c k.\\,", "1866d848800d1b62d365810bde02c45d": "P_\\mathrm{L}", "1866d9c98df8758a6c0a1964aa6fe14b": "\\omega = e^{\\frac{2 \\pi i}{N}}", "186712eacb73412fd4a48d9729e619c4": "S_M\\;", "1867344aac4679ac6402b8b6d56ec051": "x_0=(q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q)~{mod}~N", "1867ef321d4d3366fd96271ff5ae9d10": "H^*_G(X) = H^*(EG\\times_G X)", "18681430cfcf9b8e7499013e820e18c8": "A \\vdash B \\to A", "1868212d4409c2055f2ee0de17f15273": "y = x \\tan(\\theta)- g(x/v_h)^2/2", "1868aa091794cb79f9771468cd8699d7": "\n\\nu_p\\left(\\frac{a}{b}\\right)=\\nu_p(a)-\\nu_p(b).\n", "1868c96ee5f6376b4c68420d1f66138f": "WTS(O_j) = TS(T_i)", "1868d7b7c25e12ceaa9a407985c5c4b4": "\\begin{align}\n g_{\\mathrm{Alice}} &= \\begin{pmatrix} 1&6&2\\\\6&3&8\\\\2&8&2\\end{pmatrix}\\begin{pmatrix} 3 \\\\ 10 \\\\ 11 \\end{pmatrix} = \\begin{pmatrix} 85\\\\136\\\\108\\end{pmatrix}\\ \\mathrm{mod}\\ 17 = \\begin{pmatrix} 0\\\\0\\\\6\\end{pmatrix}\\ \\\\\n g_{\\mathrm{Bob}} &= \\begin{pmatrix} 1&6&2\\\\6&3&8\\\\2&8&2\\end{pmatrix}\\begin{pmatrix} 1 \\\\ 3 \\\\ 15 \\end{pmatrix} = \\begin{pmatrix} 49\\\\135\\\\56\\end{pmatrix}\\ \\mathrm{mod}\\ 17 = \\begin{pmatrix} 15\\\\16\\\\5\\end{pmatrix}\\ \n\\end{align}", "18691c10200e9ea26a70494882efe103": "\\frac{\\bar{C} G z^{-k}}{1+\\bar{C}G z^{-k}} = z^{-k} \\frac{C G }{1 + C G}", "18691caddabf3374addff8727ba4e557": "\\tilde{\\mu} \\in \\mathbb{R} ", "1869b9ea99aacf415166baed7116e097": "\\mbox{sn}(u; k) = -{\\vartheta \\vartheta_{11}(z;\\tau) \\over \\vartheta_{10} \\vartheta_{01}(z;\\tau)}", "1869d28c749042cfa1c8296c8a4ad012": "f = (f_0, f_1, ..., f_{d-1})", "186a619c3a85cece584b7762e3d763a6": "\\frac{e^{i\\theta} - e^{-i\\theta}}{2i} = x ", "186a69818830137b3bfb5b730e1d5096": " \\frac{dV}{dt} = \\mu N P + \\rho S - \\mu V", "186a9ed8101469fef65efd76d866a250": "\\mathbf{i}=[i_1,\\ldots,i_M]^T", "186ad2176f601254ef09c9f380d51485": " D = \\text{close}_\\text{previous} - \\text{close}_\\text{now}", "186bca604231a6fce5b31dca568a19d9": "|\\alpha_1\\rangle", "186c8e87a13bdf57a2481d8f9b0c2aac": "\\tfrac{4000}{100} + \\tfrac{4000}{100} = 80", "186ca497852157dc0dbd3807900348e0": "\\phi_e \\in F", "186cb8ae024dfc1ecaec185badf296fc": "f''\\;", "186cc8c31c3bc1746689ef8db4909fba": " \\nabla \\times \\mathbf{g} = -\\frac{\\partial \\mathbf{H}} {\\partial t} \\,\\!", "186cfb1f9726d17262aadd50fbbf725a": "\\begin{align}\nx&=\\rho\\cos\\theta\\sin\\phi\\\\\ny&=\\rho\\sin\\theta\\sin\\phi\\\\\nz&=\\rho\\cos\\phi\n\\end{align}\n", "186d22e3d612accd890bde850f52fb5d": "U_{f, P} = \\sum_{i=1}^n (x_{i}-x_{i-1}) M_i . \\,\\!", "186d80b5088cae799f56877e01bc556f": "\\arccos x = 2 \\arctan \\frac{\\sqrt{1-x^2}}{1+x},\\text{ if }-1 < x \\leq +1 ", "186d877666fa2c6f92794b782c19456a": "a\\in A", "186da9423fd39a4e56e886b5bdd78faa": "\\mathbb{Q}_{2^s}=\\mathbb{Q}(i,\\eta_s).", "186dc399d4dee8e5ebf32a0d0e40946f": "i(A)=A^T", "186dce50f4ec2ad9510769c802a6fc5c": "\\mathbf{\\omega_s}", "186e33d462ee2265479857ff43e88aa7": "\\beta \\beta^* = \\beta^* \\beta,", "186e40172a5e5153a4e580c8c50ba462": "\\mathfrak{s} \\oplus \\mathfrak{a}", "186eb136e6fea813dd342852f97c2b49": "\n \\qquad \\qquad u_x^- = \\frac{u_i^{n} - u_{i-1}^{n}}{\\Delta x}\\,, \\qquad u_x^+ = \\frac{u_{i+1}^{n} - u_{i}^{n}}{\\Delta x}\n", "186efbdc699e6a2df50e618c002fbacc": "\\scriptstyle f(n) \\;=\\; 3 \\uparrow^n 3", "186f09a10e0bcea337edcc116633acb5": "e^{-i\\int H(t) dt_{op}}\\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix} \\otimes \\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}", "186f12d43ba7f49da2c9c6d9a60a962f": "~\\or~", "186f5139f589daad12bfdf9b0056a0d7": "\\vec{r}(t)=\\left ( \\rho \\sin\\frac{\\beta c}{\\rho}t, \\rho\\left ( 1-\\cos\\frac{\\beta c}{\\rho}t \\right ), 0 \\right)", "186f550f810feb1fefb24d1498047747": "=\\hat{c}_P NkT\\,", "187004b04626bfbf5a09e242fc0f8e48": "X^*_{1,...,j}=X_{j+1}/(1-X_1-\\cdots -X_j),\\ldots X_k/(1-X_1-\\cdots - X_j).", "1870300c1f670296e1a1b05a66181e1e": "LQ = K + E \\times LP", "1870a034c55f64c041c0333c160e3cc0": "Happens(open, 0)", "18710cfdb66f3ae985aa505d86993db0": "\\mathbf{T}^n = \\underbrace{S^1 \\times \\cdots \\times S^1}_n.", "1871209018150de1315de86e7e4bb3e4": "\n\\begin{align}\n \\gamma &= \\sum_{k=2}^\\infty (-1)^k \\frac{ \\left \\lfloor \\log_2 k \\right \\rfloor}{k}\n = \\tfrac12-\\tfrac13\n + 2\\left(\\tfrac14 - \\tfrac15 + \\tfrac16 - \\tfrac17\\right)\n + 3\\left(\\tfrac18 - \\cdots - \\tfrac1{15}\\right) + \\cdots\\\\\n \\zeta(2) + \\gamma &= \\sum_{k=2}^\\infty\\left(\\frac1{\\lfloor \\sqrt{k} \\rfloor^2} - \\frac1{k}\\right)\n = \\sum_{k=2}^{\\infty} \\frac{k - \\lfloor\\sqrt{k}\\rfloor^2}{k\\lfloor\\sqrt{k}\\rfloor^2} \n = \\tfrac12 + \\tfrac23 + \\tfrac1{2^2}\\left(\\tfrac15 + \\tfrac26 + \\tfrac37 + \\tfrac48\\right)\n + \\tfrac1{3^2}\\left(\\tfrac1{10} + \\cdots + \\tfrac6{15}\\right) + \\cdots\n\\end{align}\n", "187143518bf995c2d20ec8e3043dc4ef": "p-1=m\\lambda", "187143672d650c3b5f6de6c457e7aa51": "[\\![\\mu Z. \\phi]\\!]_i = \\bigcap \\{T \\subseteq S \\mid [\\![\\phi]\\!]_{i[Z := T]} \\subseteq T \\}", "187144cf2baef3a6e11cd1965794b9e4": "\\int \\pi(\\theta) d\\theta = 1.", "187198df91db1bd570cee717cd91264d": "\\hat q = r + d\\varepsilon", "1871b6a81c98445dc262970a77fb5b26": " M_t = F(B_{t \\wedge \\tau}) ", "1871e1372b1682ded35e8e1cd00c3f8f": "\\left|{\\int_\\Omega|\\chi_K|^q\\,\\mathrm{d}x}\\right|^{1/q}=\\left|{\\int_K \\mathrm{d}x}\\right|^{1/q}=|\\mu(K)|^{1/q}<+\\infty,", "1872392ca3e7eb90fe7a4a0d7f05b4e4": "g(x)=\\ln f(x) = \\frac{\\ln x}{x}.", "1872639756b2f931f91b5b9dfe8d16b5": "\\frac{1}{S}=-\\frac{2Gm}{c^2}", "18726499314813a88ecbf87ee2c577fa": " H(z, u) = G(-z, -u) =\n\\left(\\frac{1}{1+z} \\right)^{-u} = (1+z)^u =\n\\sum_{n=0}^\\infty \\sum_{k=0}^n \n\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right] u^k \\, \\frac{z^n}{n!}.", "18729aa6f524eac2da132d98a45da187": "k[t_1, ..., t_n]", "187303928a98b00c0350d8828b07b3f0": "\\tilde{P}_{r}=\\left\\langle r\\left|\\frac{\\exp\\left[-\\beta\\tilde{H}\\right]}{\\tilde{Z}}\\right|r\\right\\rangle=\\frac{\\exp\\left(-\\beta\\tilde{E}_{r}\\right)}{\\tilde{Z}}\\,", "187320c3fbb969b1b393b54e4f32b3f3": " > |V|", "1873d9b7b11aa509eff2fc484ab67eaa": "\\parallel\\cdot\\parallel", "1873dab299ef284bdbbe09125544493a": "(1-f)p_{01}/(p_{01}+p_{00})", "1873f65817f530e6e476612315feeab4": "F_r ", "18740f4295b5f655aeae9efb25bfc461": "\\kappa = \\sqrt{ \\frac{4\\pi e^2}{\\epsilon} \\frac{\\partial n}{\\partial \\mu} } = \\sqrt{ \\frac{6\\pi e^2 n}{\\epsilon E_f} }", "18743e056ddb7395827903e98e8f6c99": "\\begin{matrix}\n a_{11} x_1 + a_{12} x_2 &\\leq b_1 \\\\ \n a_{21} x_1 + a_{22} x_2 &\\leq b_2 \\\\\n a_{31} x_1 + a_{32} x_2 &\\leq b_3 \\\\\n\\end{matrix}", "18744044d44ab57577e37ea2b9960a89": "\\log_{10}(3542) = \\log_{10}(10\\cdot 354.2) = 1 + \\log_{10}(354.2) \\approx 1 + \\log_{10}(354). \\, ", "1874569290841c7e1de30ce6dca99899": "f'(t') = \\frac{\\mathbf{r} - \\mathbf{r}_s(t_r)}{|\\mathbf{r} - \\mathbf{r}_s(t_r)|} \\cdot (-\\mathbf{v}_s(t')) + c \\geq c - \\left|\\frac{\\mathbf{r} - \\mathbf{r}_s(t_r)}{|\\mathbf{r} - \\mathbf{r}_s(t_r)|}\\right| \\, |\\mathbf{v}_s(t')| = c - |\\mathbf{v}_s(t')| \\geq c - v_M > 0", "187493889ad89f38f705454689353c02": " T/3", "1874d0265729cd43d493f155abd96cb6": "q = \\frac{pn}{n-\\alpha p}", "1874d623d539c98b814300c167b581e9": "\\Delta^{n-1} \\twoheadrightarrow P,", "18755c468fb372c4cddba132750e2004": "U,V \\subset \\mathbb{R}^n", "18757e58a5884d3c1b9a74a7e3992a6f": "C^1\\ ", "1875901a60e57216ed47ab4356063ff8": "x\\ge 0", "1875aa150ba914a9608b24632930f58e": " \\delta = \\det\\left(\\begin{bmatrix}A_1 & B_1\\\\B_1 & A_2\\end{bmatrix}\\right) ", "1875ca061052a33c650e9571b6ab4154": "\\frac{dx}{dy} = \\frac{1}{dy / dx} . ", "1875fa35976f0599b3118837e7d8d735": "\\sigma=(i_1 i_2 \\dots i_{r+1})(j_1 j_2 \\dots j_{s+1}) \\dots (l_1 l_2 \\dots l_{u+1})", "1876368e2bac99e6451d747d17c992a3": "\n\\begin{bmatrix}\n0 & 2 \\\\\n4 & 8\\\\\n\\end{bmatrix}\n", "1876a4f175bf5d4f9d04539dc5dc638a": "\\mathbf{a}_{\\text{per}}", "1876b62c5de864e9ce8c9eaaa3a74c77": "\n\\int \\exp\\left( - \\frac 1 2 \\varphi \\hat A \\varphi +J \\varphi \\right) D\\varphi \\; \\propto \\;\n\\exp \\left( {1\\over 2} \\int d^4x \\; d^4y J\\left ( x \\right ) D\\left ( x - y \\right ) J\\left( y \\right ) \\right)\n", "1876c1097bbb344f2a72eebc3f3b76d0": "Z(t) = X(t) + R Y(t)", "1876dc672a0088bf5c1af26e279636c8": " \\bold x^{(m+1)} = (\\bold D - \\bold L)^{-1}\\bold U \\bold x^{(m)} + (\\bold D - \\bold L)^{-1}\\bold k. \\quad (8) ", "18770c7b15d98ad5e1ff0dbd7d01f259": "\n \\mathbf{F} = 0\\,\\mathbf{e}_x - F\\,\\mathbf{e}_y + 0\\,\\mathbf{e}_z\n \\quad \\text{and} \\quad \\mathbf{r} = x\\,\\mathbf{e}_x + 0\\,\\mathbf{e}_y + 0\\,\\mathbf{e}_z \\,.\n ", "18771d41ccabfc4946af8e1af6c804d9": " H^n(G,M)=H^n({\\rm Hom}_{G}(F,M))", "187726061a7b5d0256f0fb0a1b0782fd": "E_{n}^{(1)}=-\\frac{1}{2mc^{2}}\\left(E_{n}^{2}+2E_{n}\\frac{e^{2}}{a_{0}n^{2}} +\\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}}\\right)=-\\frac{E_{n}^{2}}{2mc^{2}}\\left(\\frac{4n}{l+1/2}-3\\right)", "187748e61a55b804310bd7735c5dfcb7": "X_1|X_2=x_2 \\ \\sim\\ \\mathcal{N}\\left(\\mu_1+\\frac{\\sigma_1}{\\sigma_2}\\rho( x_2 - \\mu_2),\\, (1-\\rho^2)\\sigma_1^2\\right). ", "18779b51bce5fbab4541556836f02c78": "G=\\left(V,E\\right)", "1877e79bf17c9bb94711116d27d7e21a": "f(A) = \\sum_{i, j} [A]_{ij} f(e_{ij}) = \\sum_i [A]_{ii} f(e_{11}) = f(e_{11}) \\operatorname{tr}(A)", "1877f35323191ae2c325de0f825b5104": "\nI_i(x|k,t) = \\int_L^x M_i(u|k,t)du,\n", "1877f900717497b9eb9dc884068091e3": "Z^{\\infty}_{in}", "1878228ff796bfddfd0fd54947e0d034": "\\mathrm{E}_{\\mathbf{R}}[\\hat{\\mathbf{R}}]\\ \\stackrel{\\mathrm{def}}{=}\\ \\exp_{\\mathbf{R}}\\mathrm{E}\\left[\\exp_{\\mathbf{R}}^{-1}\\hat{\\mathbf{R}}\\right]", "18783a8efbfafd336e5b5e56fc17778a": "R^{-1}", "18784a361e5f96ca27b114b6f796861d": "V=\\frac{1}{6}\\left(10+8\\sqrt{5}+15\\sqrt{5+2\\sqrt{5}}\\right)a^3\\approx12.3423...a^3", "187898bb1a2a862a0999cf86f8f0e5ae": "d\\, ", "1878e085b76a1a5654fdf769b7de3ab4": "D^{k+1}F(u)\\{v_1,v_2,\\dots,v_{k+1}\\} = \\lim_{\\tau\\rightarrow 0}\\frac{D^kF(u+\\tau v_{k+1})\\{v_1,\\dots,v_k\\}-D^kF(u)\\{v_1,\\dots,v_k\\}}{\\tau}.", "1878f8d8eb6a9f9725df66c327ec4dab": "\nMa = Mg - Kv\\,\n", "18791407f9ee7cc46e94fb7b08edb1fc": "\\partial_1(e) = d_0(e) - d_1(e) = v - v = 0. ", "1879468c7491a981ddb054d01fa46bb0": "\n Q_\\alpha := \\kappa~\\int_{-h}^h \\sigma_{\\alpha 3}~dx_3\n", "1879730781338b7cedca8eadb50cc4a4": "p^\\prime \\equiv p - \\frac{\\Lambda}{8 \\pi G}", "1879a2c3bada456b9205d5ffafb423a5": "y_1 = x_0 - x_1. \\, ", "1879e8731a1184ddc30914ba547720a1": "Bonus = (R_n - R_o - (18 + 2 \\times G));\\ \\ \\ min(Bonus)=0", "187a0d50b45ca04ce2afe3133405d4c0": "g=3", "187a64a9c9654c056a7c99d75714852f": "\\alpha = (\\alpha_1, ..., \\alpha_n) \\in \\R^n", "187ae22789155aea6a3467b9669ed4d4": "d_\\lambda = \\frac{9!}{7\\cdot 5\\cdot 4\\cdot 3\\cdot 2\\cdot 2\\cdot 1\\cdot 1\\cdot 1} = 216.", "187afe26af8552f6f386be9c87f1ccdf": "F(A) = \\{b \\in M \\mid b = f_{\\varphi}(a_1, \\dots, a_n); \\, \\varphi \\in \\sigma ; \\, a_1, \\dots, a_n \\in A \\} ", "187b5a35ea419ef4f511f3bcf4fc6a5d": "U_i \\cap U_j \\neq \\emptyset", "187b8c1e8bb4444f20512f326e1f5c9c": "\\delta_H(d_F) = \\delta_{H1} \\frac{1 - \\exp\\left(-d_F/\\lambda_F\\right)}{1 + d_F/d_0}.", "187bb5014952afaa4369c9979477e9d1": "b_4", "187bb5f36111bc779b615517fdbeda55": " (n-1)(n-2)~r^{-n}~\\cos(n\\theta) ", "187bfe98bdbd2ee8e0b484481dba04a2": "P=(N_\\uparrow-N_\\downarrow)/N", "187cd6236f5eedff9399c4c606ea81af": "I[u] = \\int_D (\\nabla u)^T A (\\nabla u)\\,\\mathrm{d}x", "187cd9936495a2e5cf67dbb7dba8538f": "V = x^2y + y^4 + ax^2 + by^2 + cx + dy \\, ", "187d62995335efabffe91dac7937321c": "X_{i_1},\\dots,X_{i_n} \\text{ and } X_{j_1},\\dots,X_{j_n} \\!", "187e1a4c3d63ccac36ce283d420330a4": " h_{combined}", "187e526b73ab20fc09c2965b87e443ba": "\\operatorname{Cov}(X_i^2, X_j^2) = 2 \\langle X_i X_j \\rangle^2 = 2 M_{ij}^2", "187ecb99c8f8d632222c8040ba7c84ad": "J\\frac{\\partial}{\\partial x^\\mu} = \\frac{\\partial}{\\partial y^\\mu} \\qquad J\\frac{\\partial}{\\partial y^\\mu} = -\\frac{\\partial}{\\partial x^\\mu}", "187ef4436122d1cc2f40dc2b92f0eba0": "ab", "187f23dece76c2c8cbb550ab38bfcfc3": "\\Lambda(E)", "187f3958cab7e623394c2973e801bfe6": "P(t)=\\mathcal{P}_{ab}(c_a^*c_b)exp(i\\nu_1t)+\\mathcal{P}_{ac}(c_a^*c_c)exp(i\\nu_2t)+c.c.", "187fd4fa8be2256a660a7b18fd4b73a5": "p_{0 \\frac{1}{2}} \\leftarrow 64x^3-448x+448", "187fde02d22edfe777d97a947b269419": "\\pi_{1} = 1-\\pi_{0} ", "188079d973ad0be8a1013e48e672258c": "\\sum_{i=1}^k \\mathrm{n_i}^\\mathrm{S}\\,\\mathrm{d}\\mu_i\\, +A\\mathrm{d}\\gamma\\, = 0\\,.", "18808159c9f664bc9a4d04faa92a8638": "\\hat{S}^2", "1881135d49064106948ed208a99690ff": "(\\mathbf{a}\\times \\mathbf{b})\\cdot(\\mathbf{c}\\times\\mathbf{d}), \\quad (\\mathbf{a}\\times \\mathbf{b})\\times(\\mathbf{c}\\times\\mathbf{d}) ,\\ldots", "188156c823ead08aa06494871fec8a63": "\\omega_{S}=\\sqrt{\\frac{k_{z}^{2}B^{2}}{\\mu\\rho_{e}}}", "1881da51ebf6ff6391ae373f1193144c": "\n\\#_{n,m} = {n+m-1 \\choose m-1} = {n+m-1 \\choose n}\\,.\n", "1881e88f33be2eb98dd778054f5c1983": "\\langle x , y \\rangle", "188213b2b0bd3065d7a200c46d874c09": "\\lim_{t\\rightarrow1}f(\\gamma(t))=f_{1}(e^{i\\theta})", "188214f7ce4cff74d5fdcf33c75fca8c": " \\Delta f\n= \\frac{\\partial^2 f}{\\partial r^2}\n+ \\frac{N-1}{r} \\frac{\\partial f}{\\partial r}\n+ \\frac{1}{r^2} \\Delta_{S^{N-1}} f\n", "188231a91ed69207a52b0e59b91d3cf9": "x=1\\ ", "188250b55a4085437d24ba37a2bddfd3": "\\displaystyle a = 1 \\times 5 = 5", "18825fe9b7c871249babd1c757873dd2": "(a_{ij})", "1882745480f80c63870fe2d96e330a5a": "\\dot{G}(t-\\tau)", "188281dd07a8792df31de293dc5a0bda": "\\ \\displaystyle R \\ ", "188286ebc8b3b9c46590c712327d89b9": "c_\\alpha = \\pi^{n/2}2^\\alpha\\frac{\\Gamma(\\alpha/2)}{\\Gamma((n-\\alpha)/2)}.", "1882c64a6cf94a30158ffa7334e95c7c": "10^{\\frac{13-1}{2}} = 10^6 \\equiv 1 \\pmod {13}", "1882c704faa3f93353e9006938a2894e": "\\langle F,R,{\\Vdash}\\rangle", "1882e4ea827fcae43798331cd797ece6": "4\\pi \\epsilon_0", "188341f39fc157faaad493bf5a425cce": " \\textstyle \\int_{0.5}^1 \\mathrm{\\Beta}(49582, 48871) \\approx .983.", "1883d50bb4fbe1033018f925d816a192": "\\hbar = \\frac{h}{2 \\pi}", "1883fa6484e08c82ad8c920ad0008bd7": "S\\subseteq A^\\mathbb N", "1884004e4b00addecc0049d153d2e87d": "E ( f(x) \\cdot g(x) ) = E f(x) + E g(x)", "18843a2371bae8637a5b867636c13986": "x,y\\in A_1\\cup A_2", "188456e85e5258396aa9ebdb7b8178a0": "\\langle f, g\\rangle := \\int_\\Omega (Df(x), Dg(x)) \\, dx ", "18849c52417407de4d69623d07843c20": "n_{i, t}", "18849d4b2ec3504c6aae3f12ce88893b": " Df = -(pf^\\prime)^\\prime + qf", "1884a44435bf6e23a9face1a0bc52c2c": "\\Theta=\\tan^{-1}(m/k)", "1884fd5772c3ca09da0a0e70f4f7d693": "\\sqrt{\\frac{1}{6}}\\!\\,", "18856861ae0142eef654eb99e18259f3": "\\int_{C_1} L(x,y)\\, dx = \\int_a^b L(x,g_1(x))\\, dx.", "188594fb1141ab6ffe1e33ddbbe20c8b": "\\mathrm{VF_5 + H_2O \\ \\xrightarrow{}\\ VOF_3 + 2HF }", "1885dfefcfe67d59db0ce26605e0304e": "\n\\left(\\begin{matrix} {\\bold F} \\\\ {\\boldsymbol \\tau} \\end{matrix}\\right) =\n\\left(\\begin{matrix} m {\\boldsymbol 1} & - m [{\\bold c}]\\\\\nm [{\\bold c}] & {\\bold I}_{\\rm cm} - m [{\\bold c}][{\\bold c}]\\end{matrix}\\right)\n\\left(\\begin{matrix} \\bold a_{\\rm cm} \\\\ {\\boldsymbol \\alpha} \\end{matrix}\\right) +\n\\left(\\begin{matrix} {m \\boldsymbol \\omega} \\times \\left({\\boldsymbol \\omega} \\times {\\bold c}\\right) \\\\\n{\\boldsymbol \\omega} \\times ({\\bold I}_{\\rm cm} - m [{\\bold c}][{\\bold c}])\\, {\\boldsymbol \\omega} \\end{matrix}\\right),\n", "1886817aba72d836d299b56d835b6c9f": "\\mathbf{j}_r=\\int_{t_0}^{t_1}\\mathbf{f}_r dt", "18868d1ecfe0da570f1be63b1d908784": "v_s", "1886acccb9fc762085008aedda7ca779": "Vt=Vg+Vi+Vo", "1886e625b4b5241d722ba825c148a4c8": "s=\\frac{a+b+c+d}{2}.", "1886eda224719f904cfcc2077619b2d7": "C:y^2+h(x)y=f(x)", "1887c9b3a5386c6b83d6711c107dda8e": "P_{D}", "1887d3f69a6d9aaba8eda1127f4cbbfe": "\\frac{vX}{c+vt}\\,\\!", "1887ec7b1748201ee8deb2468e0af960": " {}+1206647803780373360\n x^6-3599979517947607200\n x^5 \\,\\!", "188849597db9198d230f06e96b725773": "2.2\\times 10^{-6} \\ \\mathrm{seconds} \\,", "188858d46d2c549f8a2da42ac3d316f8": "I(q) \\sim S' q^{-(6-d)}", "1888a16042020813c1d29b0954ca215c": "\\boldsymbol{v}\\,", "1888ba03d9fb43a16946a10cb446f645": "\\sigma^{-1}", "1888d1745398648fa07cbccb8926b1cd": "\\langle \\cdot , \\cdot \\rangle ", "188904e7e059c0e5a05da77223ff8281": " S \\circ T := \\{s \\circ t: s \\in S, t \\in T \\}.", "18893eb772c377a31b1746974df39682": " n\\sin \\theta ", "1889a9a8e04e4cf883e27a2f2c6255eb": " \\mathbf{v}_1 + \\boldsymbol{\\omega}_1\\times\\mathbf{r}_1 = \\mathbf{v}_2 + \\boldsymbol{\\omega}_2\\times\\mathbf{r}_2 ", "1889f83b287b0314cc26bae74eea41ec": "s(x_1,x_2,x_3,x_4,\\ldots) = (x_2,x_3,x_4,\\ldots)\\text{.}", "188a03f8de5252635c30495fa4a88c33": "L = \\int_a^b \\sqrt{ \\left|\\sum_{i,j=1}^ng_{ij}(\\gamma(t))\\left({d\\over dt}x^i\\circ\\gamma(t)\\right)\\left({d\\over dt}x^j\\circ\\gamma(t)\\right)\\right|}\\,dt \\ .", "188a1d39c07fa79aa6c51d2152c031cc": "\\hat{f}(r\\alpha) = \\int_{-\\infty}^\\infty Rf(\\alpha,s)e^{-2\\pi i sr}\\, ds.", "188a3d326e5e2280b7d046a8554fc9dc": "\\varinjlim A_i = \\bigsqcup_i A_i\\bigg/\\sim.", "188a58b793667fd45e61a6bb13a42184": "(v_{x},v_{y},v_{z})", "188aa4ab7ba11a91a978f53a101542ba": "F_t - S_t", "188ac3fae1e32f569fc6f1b089cd00fa": "\\mathrm{S_2F_2 + 3H_2SO_4 \\ \\xrightarrow{80^oC}\\ 5SO_2 + 2HF + 2H_2O }", "188b05260f017cedba309d25f64fd5cb": " \\mathbf{F} = \\nabla \\phi.\\,\\!", "188b3ffb15876c4642210083458c5ece": "(\\pm 1, \\pm 1, \\pm 1)", "188b625c6f167af25d9a215f6d6ccebb": "{\\mathit{He}}_n(x)=\\int_{-\\infty}^\\infty (x+iy)^n\\, \\mathrm{e}^{- y^2/2} \\, \\mathrm{d}y/\\sqrt{2\\pi}\\ .", "188b6a454f3adfa535548223ec0929c5": "x<10", "188b6bf67bc2a40ec15a30200802c54d": "a\\delta\\phi", "188b710daba82055cf96cb1d4a59630f": "Vol(M, \\lambda g) =\\lambda^{n/2}Vol(M, g)", "188bcd21b27436ee4a913ac583bd5f41": "A + \\overline{B_{\\delta}} := \\left\\{ a + b \\in \\mathbb{R}^{n} \\left| a \\in A, b \\in \\overline{B_{\\delta}} \\right. \\right\\}", "188bf411bbe9a71c900da19bb53db73b": " z_{12} \\,", "188c151e25bc3cea4641cc722994fbc4": "U\\left(x,y\\right)=x^\\alpha y^{1-\\alpha}", "188c226ff875898861c69c960928378d": "\\int\\operatorname{arcosh}(a\\,x)\\,dx=\n x\\,\\operatorname{arcosh}(a\\,x)-\n \\frac{\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}}{a}+C", "188c886236774d0659c204b94056a07a": "0,\\infty", "188d4ada5f4a8b9309ac3c0a9ce29200": "\\mathbf J", "188dcd29088d10048de9d208b99de1db": " \\vec 0 = \\vec\\mathrm{M}_{sail/G} + \\vec\\mathrm{M}_{hull/G} ", "188de81d3fe7186b3b5d3c48c79dd941": " \\frac{d{\\mathbf r}(s)}{ds}={\\mathbf u}_1({\\mathbf r}(s)) ", "188dfaa1f8283faaa0e5dcd3cee46fd6": "B^{\\ast}\\,", "188e0a27226ea35d2c78cbf5c952ad1d": "\\dot{K}", "188e65fae6efd0d7733ef6bcb00465c0": "f = \\bold{1} * g", "188e8738bcb05c9402ff373fa7c05f15": "\\beta_{min}=\\kappa_3+\\kappa_4 d_2", "188ec85ebad245459752275259b526a7": "dr_t = \\theta (\\mu-r_t)\\,dt + \\sigma\\, \\sqrt r_t dW_t\\,", "188ee2a34be84205542f460fa30736a9": "\\alpha : L(G) \\longrightarrow \\mathbb{F}^n", "188f44cd54ff166a66ac4266fc4892e0": "\\sigma(X, X^*)", "188f5ac1b1ac9c5aa7eec28793155847": " n=2 ", "188f625f814556e9f8ebbca7cfbaa0be": "y^*\\,\\!", "188fafc09550fa0eff705766a7c57d60": "B_3(T)", "188fde19b9ab3ea4bb7ed655bc54a8c0": " (J^\\alpha) (J^\\beta f)(x) = (J^\\beta) (J^\\alpha f)(x) = (J^{\\alpha+\\beta} f)(x) = { 1 \\over \\Gamma ( \\alpha + \\beta) } \\int_0^x (x-t)^{\\alpha+\\beta-1} f(t) \\; dt", "188ffa3164c852925f68f3e80893580e": "P_1=(1,0)", "189054031045ff1382c8129b458d36b0": " \\Theta_d^* = {h | h \\in S_A, j_h = d},", "189062606b55eebd5846d99d9782ffb2": "k=0,1,2,\\ldots,", "18908fff8062dd0029b99bd92ab0a09e": "K(k)=\\frac{2}{1+k1} K(\\frac{1-k1}{1+k1})", "18911f15de48873b70bb53c4b7805a11": "\\frac{2 \\lambda}{R} ", "1891ca4d0ef625a3c4c190c917282422": "H^i( G/B, \\, L_\\lambda )", "1892330c3bf4359ba7d85f5477cd1dc5": "L[u]=f,\\,", "1892449a02312a313a23c49baa2a40cf": "\\langle , \\rangle': (y,x) \\to \\langle x , y\\rangle", "1892b1f129a1c9594967558c3433a408": "\\operatorname{not1}\\ \\operatorname{true} = (\\lambda p.\\lambda a.\\lambda b.p\\ b\\ a) (\\lambda a.\\lambda b.a) = \\lambda a.\\lambda b.(\\lambda a.\\lambda b.a)\\ b\\ a = \\lambda a.\\lambda b.(\\lambda x.b)\\ a = \\lambda a.\\lambda b.b = \\operatorname{false} ", "189316d02d660f0f0478948cb7edb95d": "h=f(x_n)\\ ", "189326fb7b8cd2c93dc280f79a91644a": "\\tilde{h} = H_1(\\tilde{w},\\tilde{s})", "189331fab3bb02edd66a3b5d9c45217b": "-p \\;\\frac{p + \\ln(1-p)}{(1-p)^2\\,\\ln^2(1-p)} \\!", "18937b1c3d76aad31e69c077d76dc204": "\\tau_\\mathrm{n} = -\\frac{1}{2}(\\sigma_1 - \\sigma_2 )\\sin 2\\theta\\,\\!", "1893ff69e5339a235236ba1e6f7dac13": " S:= \\mathcal{M}+\\mathcal{M}^* +\\mathbb{C} 1 ", "18946e4aeec4d299237daa07fa2387b0": "\n\\begin{array}{lcl}\n a & = & \\frac{d^2x}{dt^2} \\\\\n & = & \\frac{d}{dt} \\frac{dx}{dt} \\\\\n & = & \\frac{d}{dt} (\\frac{dx}{dA} \\cdot \\frac{dA}{dt}) \\\\\n & = & \\frac{d}{dt} (\\frac{dx}{dA}) \\cdot \\frac{dA}{dt} + \\frac{dx}{dA} \\cdot \\frac{d}{dt} (\\frac{dA}{dt}) \\\\\n & = & \\frac{d}{dA} (\\frac{dx}{dA}) \\cdot (\\frac{dA}{dt})^2 + \\frac{dx}{dA} \\cdot \\frac{d^2A}{dt^2} \\\\\n & = & \\frac{d^2x}{dA^2} \\cdot (\\frac{dA}{dt})^2 + \\frac{dx}{dA} \\cdot \\frac{d^2A}{dt^2} \\\\\n & = & \\frac{d^2x}{dA^2} \\cdot \\omega^2 + \\frac{dx}{dA} \\cdot 0 \\\\\n & = & x'' \\cdot \\omega^2 \\\\\n\\end{array}\n", "18956d8d721be71de2aed54a7c3bcc89": "\\exp(-\\beta \\varepsilon(\\mbox{state})) = \\prod_\\mbox{vertices} \\exp(-\\beta \\varepsilon_{ij}^{k\\ell})", "1895917fcf2f74fd389a9bc370d5bdf6": "I_{spike} = \\sum_{\\mathbf{s}} P(\\mathbf{s}|spike) log_2 [P(\\mathbf{s}|spike)/P(\\mathbf{s})]", "1895fe06dd85b03071367dd35f2fb74c": "\\coprod_{i=1}^N x_i", "189615efad256e6f603839ab9d05a6dc": "2 U_k", "189640332c952b62d541cc2e00a34d3d": "M_B = -11.569 \\ kN \\cdot m ", "189695234146739ede1f2e9692ae62cb": "(1-\\lambda_p)", "1896d795094586c954ea9b8a46891783": "NID(x,y)", "18973a90cbf22c78e57b5405c6c144e8": "\\langle T, \\Delta \\phi\\rangle = 0", "18975b0a3a106031fc0b45a8659f56a4": "-20\\le x,y \\le 20", "18975f661ac54937ed3ebafd74ae1826": "\\exp(-E/k_B T)\\,", "18978cc2fc5ae6db3b5102353d6a6950": "\\sin \\theta = \\theta - \\frac{\\theta^3}{3!} + \\frac{\\theta^5}{5!} - \\frac{\\theta^7}{7!} + \\quad \\cdots ", "1897be964cf0e1693a2e83aa9d1f7e3d": "D(0, 0) = 0; ~~ D(0, -1) = -1; ~~ D(1, -1) = -1; ~~ D\\left(\\frac{3}{8}, -\\frac{3}{4}\\right) = \\frac{27}{128}. ", "1897dc95f9896bab096535c9def46084": " \\frac {x \\sin \\theta} {\\lambda} = \\pm \\frac {1}{2}, \\pm \\frac {3}{2}, ...", "1897df0ccae432eb35a4dea99a291f7d": " \\boldsymbol{\\beta}(y) \\ge 0, \\mathbb{E}_{Y^{tr}} [ \\boldsymbol{\\beta}(y)] = 1 ", "1897e60fa0f0a08bd59f3a9aff1f0982": "\\phi_2 ", "189823f8b90513488cb6048ddb2fd60f": "\\cos\\varphi", "1898df23a13badc3d56f4de3b76b1b19": "(A,\\phi)", "1899006f6f0679e0964cf6f1fc124588": "(b_1, \\dots, b_n)", "189934d1d98999afd63209dba9fc4e82": "\\breve{\\boldsymbol\\theta}_i=\\mathrm{Inv}(\\boldsymbol s,\\boldsymbol z_i)", "18997d64d3fb1b8b34b50d1d5d0b2036": "k=1\\ldots\\lambda", "1899cddb77666d0e852a4209f2462148": "\\scriptstyle\\frac{-3\\pi}{2}", "1899e9301923b81e4442da9c926ad330": "\\{ a^n : n \\geq 1 \\}", "1899f1559d1d26066caa707897b30d48": " ModD(y) \\equiv - \\frac{1}{V} \\cdot \\frac{\\partial V}{\\partial y} = - \\frac{\\partial ln(V)}{\\partial y} ", "189a3676b44955400e597ca7b79a2aa2": "\n\\operatorname{P}( \\left| \\overline{X}_n-\\mu \\right| < \\varepsilon) = 1 - \\operatorname{P}( \\left| \\overline{X}_n-\\mu \\right| \\geq \\varepsilon) \\geq 1 - \\frac{\\sigma^2}{n \\varepsilon^2 }.\n", "189a562f37b5839c495bb968c1059fb9": "Y|X", "189a62adcb589050d3be98be1f765c0e": "I(\\nu,T) =\\frac{ 2 h\\nu^{3}}{c^2}\\frac{1}{ e^{\\frac{h\\nu}{kT}}-1},", "189a6c69916161df3d709f299a64b337": " (-1)^{\\frac{n-1}{2}} \\cdot C_{\\frac{n-1}{2}} \\equiv 2 \\pmod n,", "189ac807f51cdb474ee0740d9fd0e893": " \\langle Q \\rangle_\\psi = \\int_{-\\infty}^{\\infty} \\, x \\, |\\psi(x)|^2 \\, dx", "189ae40408bef1b9cad90605bb51f47b": " p_1((\\frac{T_2}{T_1})^\\frac{\\gamma}{\\gamma-1}-1)\n\\,", "189b93d19c17dddaccaf7e4fcb4fcaea": "[H^+]_{0^{ }}10^{b_0}", "189bdd68a9c070d450bc218ea2211555": "[[z]]", "189bfb2c2abb79025eb4c6cea0d8cf7b": "\\begin{align}\n G_L &\\to 1 \\\\\n G_R &\\to 0\n\\end{align}", "189c028399d91e0fca282c1955fa491a": "p=0.", "189c4f5febfa6b17825c341346e5a1f7": "\\begin{align}\n \\phi({\\mathbf{X}},z_{n+1}) = \\int p({\\mathbf{X}}-{\\mathbf{X'}}, z_{n+1}-z_{n}) \\phi({\\mathbf{X}},z_{n})\\exp\\left(-i\\sigma\\int\\limits_{z_{n}}^{z_{n+1}}V({\\mathbf{X'}},z')dz'\\right)dX'\n \\end{align}", "189c5675254ab7ef45079aa3c9af7d09": "x[T\\cup S]y", "189c6cbe80dae0f849c56df4e57b8bb1": " R_x(\\tau) = \\int_{-\\infty}^{\\infty}\\left.x(t+\\tau /2)x^*(t-\\tau /2)\\right.\\, dt.", "189cdf573cc9e575ff2f7fe0ded02e44": "\\Phi_i\\rightarrow\\Phi", "189ce067cc5b92d3228402efcffd1ab2": "\\cdots \\to H^1(V, \\mathcal O_V^*)\\to H^2(V, \\mathbb Z)\\to H^2(V,\\mathcal O_V)\\to \\cdots.", "189cf407f19bf0f6fe6ead4392233c19": "x' > y \\sqsubseteq x := y+1", "189d33de47b11995edd6014c9250beaa": " { Q_1 \\over \\ Q_2} = { \\left ( {N_1 \\over N_2} \\right )} ", "189d65ea689ab8abd354c7c4883f9241": "\\|x_i - x_j\\|", "189d673d3b7970a55671dc60f60a1774": " i \\leftarrow I ", "189e47d3077e96af6c3a115fe1bf9f87": "\\varphi=1/(2\\sin(\\pi/10))=1/(2\\sin 18^\\circ)\\,", "189e8c39ea43baed4198f243db129874": "A \\oplus \\! B \\oplus C", "189edb1c7606c629725529fd5210fc9d": "\\ \\hat{x}(t) = g(t)*y(t)", "189f4ae226ce8e2c1c6f6ed5142138fb": "B_\\infty^{p,q}", "189f4c64921adeb80f74c2d4f76e9437": "2 \\ \\operatorname{tr} (\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma) = 8 \\eta^{\\rho \\sigma} \\eta^{\\mu \\nu} - 8 \\eta^{\\nu \\sigma} \\eta^{\\mu \\rho} + 8 \\eta^{\\mu \\sigma} \\eta^{\\nu \\rho} \\,", "189f532fa4d24c07110adbc183c0154f": " m^{2n}\\equiv m^{2n-\\wp(p-1)}\\pmod p \\!", "189f68a2a3a410a2389708559c3f4d4b": "\\delta_y", "189f8851bd75530579c998307e0d67f9": "a\\ne b \\ne c \\ne d, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "189f9d45cdc1369a54d346d6f86dd900": "S(0) = S(0/1) = [1; 2, 3, 4, 5, 6, 7, \\dots].", "189fa3d6ec47083f68846211f25c0d71": "{\\mathbb Q}(x)", "189fad1b0f1de9a3570cda392fdadab5": " \\operatorname{de-lambda}[x\\ f = \\lambda y.f\\ (y\\ y)] \\equiv x\\ f\\ y = f \\ (y\\ y) ", "18a03bd19d746163bac6e05c45218e70": "H = -\\ln\\left(\\frac{\\phi(q)}{2\\pi}\\right)+2\\sum_{k=1}^\\infty \\frac{(-1)^k}{k}\\, \\frac{q^{(k^2+k)/2}}{1-q^k}", "18a0eaae05c4e17c8d499d0bdff02859": " |\\Psi^{(\\pm)}_\\epsilon \\rangle = \\frac{ U_{\\epsilon I} (0,\\pm\\infty) |\\Psi_0 \\rangle}{\\langle \\Psi_0 | U_{\\epsilon I}(0,\\pm\\infty)|\\Psi_0\\rangle} ", "18a10280ff9fec2cb2035def93074577": "_C", "18a157d55c3a810756af692f2a7674c3": " H = \\frac{2}{3}\\cdot\\frac{A+B}{2} + \\frac{1}{3}\\cdot\\sqrt{A B}.", "18a1b53e3e5951aac453e86ddbc284a7": "(R \\to S)", "18a21ad7167c7ea66d80707a3bc51733": "l=\\frac{\\lambda}{4}\\ ,", "18a2422c9c7fa6717043797dd738d3e8": " 0.5 \\ln(1.5 + 5f^{*}) + 0.5 \\ln(0.75 - f^{*}) \\! ", "18a2450249093366c5ae3bccf41225bd": "\\mathit{Z}_p = \\{0, 1, 2, ..., \\mathit{p} - 1\\}", "18a33fea91b4526949a255113206ada0": "= \\sgn( \\tan(\\frac{2\\theta + \\pi}{4})) \\frac{\\tan \\theta}{\\sqrt{1 + \\tan^2 \\theta}} ", "18a3d15e003ab3fde06f39ff59e3bce4": "(N/\\sqrt{T})", "18a3d979d4ad4037702a8021c2524565": "Y \\to Z, S \\to Z, Y \\to S, S \\to A", "18a4ca9a19f345bf6866427902b62e29": "2V/(V + 2)", "18a4fdd9c57ce3f5a140e89d9df15b6f": "\\lambda(s,t_s)=y", "18a5579c6872193e5f1a96343278c52b": "T_p = 1-R_p\\,\\!", "18a57207cfbee0edf6bd2fff690470cc": "\\sum_{i=0}^{\\infty}\\operatorname{P}[\\textrm{first\\ }i\\textrm{\\ rolls\\ are\\ ties,\\ }(i+1)^\\textrm{th}\\textrm{\\ roll\\ is\\ 'the\\ point'}]", "18a59284ce6e3123929431d22a5c3655": "a* = 1 \\cup a;a*\\,\\!", "18a5bd7a2e6026f4e0495bbc5afd600b": "\\theta_R = 1.22\\lambda/\\,\\!d", "18a5e913f49e8dc1fa94296720bd647a": "\n\\frac{\\langle B,s\\rangle \\Rightarrow \\mathbf{true}}{\\langle\\mathbf{while}\\ B\\ \\mathbf{ do }\\ C,s\\rangle\\longrightarrow \\langle C;\\mathbf{while}\\ B\\ \\mathbf{do}\\ C,s\\rangle}\n\\quad\n\\frac{\\langle B,s\\rangle \\Rightarrow \\mathbf{false}}{\\langle\\mathbf{while}\\ B\\ \\mathbf{ do }\\ C,s\\rangle\\longrightarrow s}\n", "18a63a8c4991585eb2858222600ca602": "f(\\mathbf{x})=0", "18a6ee04730b22a17211fe8cad693ac1": "\\hat Y_{j(i)}\\,", "18a71b3d5f7d5b1af3d595a00049595e": "v_{i,m} - \\varepsilon < v_i 1", "18b4729f3b0657e360cb639e7ada72ab": "R^{\\mathfrak T_{\\Phi}} \\overline {t_0} \\ldots \\overline {t_{n-1}}", "18b4c68bfc760bbf3e3624b9466b59a2": "1 \\le p, q \\le \\infty.", "18b4da017912c5e9183df31ac6f8370e": "\\sigma_{i,\\,j}", "18b5349300d054f65d2b26943937c3d1": "2x=a(3x^2-y^2)", "18b540028c08b3564e25f146db9ef2ab": "F[G]\\,", "18b5416b92bc741f5f3dd407738866de": "L= 2S-\\frac{400 + 3.5S}{A}", "18b5c3583e02eb21216625472e25f88d": " \\begin{align}\n\\mathbb{E} ( I - g(X) )^2 & = \\int_0^{1/3} (1-g(3y))^2 \\, \\mathrm{d}y + \\int_{1/3}^{2/3} g^2 (1.5(1-y)) \\, \\mathrm{d}y + \\int_{2/3}^1 g^2 (0.5) \\, \\mathrm{d}y \\\\\n& = \\int_0^1 (1-g(x))^2 \\frac{ \\mathrm{d}x }{ 3 } + \\int_{0.5}^1 g^2(x) \\frac{ \\mathrm{d} x }{ 1.5 } + \\frac13 g^2(0.5) \\\\\n& = \\frac13 \\int_0^{0.5} (1-g(x))^2 \\, \\mathrm{d}x + \\frac13 g^2(0.5) + \\frac13 \\int_{0.5}^1 ( (1-g(x))^2 + 2g^2(x) ) \\, \\mathrm{d}x \\, ;\n\\end{align} ", "18b5c459d9acf84628fa39c81c689b23": "X^*", "18b7828af58814a03317d2ad8d4901ee": "\n W = D_1\\left(\\tfrac{J^2-1}{2} - \\ln J\\right) + C_1~\\sum_{i=1}^5 \\alpha_i~\\beta^{i-1}~(\\overline{I}_1^i-3^i)\n ", "18b7911727a89109b30e9db96ae16dda": "[B_1(t),\\phi(t)]\\Longleftarrow SLR \\Longrightarrow [A_N(z),B_N(z)]", "18b7bb34a2981efb7b59903eeff7bedd": "\\ where ", "18b8533be0dbf96a2eaff6d16df1b38d": " \\dot{V}(x) = \\frac{d}{dt}V(x) \\le 0", "18b8a4aa0f31e06aa7f8a35dd26f88b0": " | \\rho ) ", "18b91870f12e40867d62dda1870a0bf4": "Pmo + Pmf = 1", "18b94090f21af324926114c559ca4b66": "\\sigma_\\mathrm{n} = \\sigma_x \\cos ^2 \\theta + \\sigma_y \\sin ^2 \\theta + 2\\tau_{xy} \\sin \\theta \\cos \\theta", "18b94c714aafaa62984c0b1618036562": "q\\gamma(a,q) = \\frac{q}{d}\\gamma(a/d,q/d)-\\log d.", "18b9f2fdc605563121d97539fc8d5e0c": "\\forall \\vec{y}\\,\\varphi(f(\\vec{y}),\\vec{y})", "18ba02363b7fedc6e301e7d2d930d96f": "\\scriptstyle{p}", "18ba06497be736a00e5d586e7c0eff8a": "f_n(z)=\\frac{a_n}{b_n+z}.", "18ba0877c3b039bcf8af77f16a2cd37a": "\nE(\\phi, k) = \\int_0^\\phi \\sqrt{1 - k^2 \\sin^2 \\theta} \\,\\mathrm{d} \\theta = \\sin \\phi \\,F_1(\\tfrac 1 2, \\tfrac 1 2, -\\tfrac 1 2, \\tfrac 3 2; \\sin^2 \\phi, k^2 \\sin^2 \\phi), \\quad |\\real \\,\\phi| < \\frac \\pi 2 ~,\n", "18ba55b0cd0f940044ec599262597c2c": "\\{ (\\emptyset,\\mathsf{i})\\}", "18ba72e0fe9235a80f0a6ab3e98a49dc": "\\langle |\\gamma(n)|^2 \\rangle = \\frac{1}{c_n} \\sum_{n\\;\\mathrm{step\\; SAW}}|\\gamma(n)|^2 = n^{2\\nu +o(1)}", "18ba8aaaf28689458abcfc03fb6497ae": "\n\\begin{array}{l} \n(\\forall L\\subseteq \\Sigma^*) \\\\ \n\\quad (\\mbox{regular}(L) \\Rightarrow \\\\ \n\\quad ((\\exists p\\geq 1) ( (\\forall w\\in L) ((|w|\\geq p) \\Rightarrow \\\\ \n\\quad\\quad ((\\exists x,y,z \\in \\Sigma^*) (w=xyz \\land (|y|\\geq 1 \\land |xy|\\leq p \\land \n(\\forall i\\geq 0)(xy^iz\\in L)))))))) \n\\end{array} \n", "18ba9473b1f7b96c41e90f5242c156d8": "\\psi_{n}", "18bae66d415041394488d0f810328d9d": "ncp=\\sqrt{n}\\frac{\\mu-\\mu_\\text{baseline}}{\\sigma}", "18bb05897a006e4b06c97d63ed9a74f2": "\\dot{\\theta} = f(\\theta) + g(\\theta)S(t)", "18bb1b76f70276f343d7526478c27f27": "\\mathbb{E}_\\theta \\left[ V(x(\\theta),\\theta) - \\underline{u}(\\theta_0) - \\int^\\theta_{\\theta_0} \\frac{\\partial V}{\\partial \\tilde\\theta} d\\tilde\\theta - c\\left(x(\\theta)\\right) \\right]", "18bb1e8fa7618fb780342b2c803e1637": "\\bar{w}_{1L}(s,2n+\\gamma_{1L};L)", "18bb41f9fbbfa60a43d5e7350d6c9ae4": "\ny\\; =\\; 2\\, y_0\\, \\cos(\\omega t)\\; \\sin(kx).\\,\n", "18bb8cacf20918a41196f9bb562f8869": "xy+ax^3+bx^2+cx=d\\,", "18bb8cd057d57f22e2742ea35ba662d7": "V_3", "18bbae7591c05f96e5a7d44431e2444c": "\\lim_{x\\to0^+}\\log x=-\\infty", "18bbae909b0daf82a2969f5f5ef3a589": "\\,\\kappa_3", "18bc3ca421bd5a60bdfdb2d8c7cf765f": "V_{n+1} = \\int_0^1 S_{n}r^{n}\\,dr", "18bc6acd4040524ea2270b74ca8b389f": " \\begin{pmatrix} y_{1} \\\\ y_{2} \\end{pmatrix} = \\begin{pmatrix} m_{1}(x_{1}, x_{2}, x_{3}) \\\\ m_{2}(x_{1}, x_{2}, x_{3}) \\end{pmatrix} ", "18bc6bc95462fe455140fd689af4f37f": "a = 2r_2, c = r_2, d = r_1\\,\\!", "18bc82f68262b2a5c4408648bd492194": "2\\pi \\varepsilon a\\sum_{n=1}^{\\infty }\\frac{\\sinh \\left( \\ln \\left( D+\\sqrt{D^2-1}\\right) \\right) }{\\sinh \\left( n\\ln \\left( D+\\sqrt{ D^2-1}\\right) \\right) } ", "18bccbfdf90f4137431af6bc072bc5bc": "C_{stat} = \\frac{{V_T}}{{P_{plat}-PEEP}}", "18bd7f90e2be815249783b0021fa338f": " (J^\\alpha f) ( x ) = { 1 \\over \\Gamma ( \\alpha ) } \\int_0^x (x-t)^{\\alpha-1} f(t) \\; dt", "18bd8fc9c1add66574b2c6ad352c1e09": " \\mathbf{r} ", "18bdb74993b655d2d9fc8f2e500a1278": "\\Delta_{\\alpha}", "18be17b6f838e77cb27987a8b4b95fe4": " \\left(\\frac{\\partial T}{\\partial V}\\right)_S = -\\left(\\frac{\\partial P}{\\partial S}\\right)_V = \\frac{\\partial^2 U }{\\partial S \\partial V} ", "18be2b8a85054984b0979facedfeff38": "W_i(x)", "18be2c14dd16f907369b3fb992b42eef": " X_t := W(t + \\tau_a) - a ", "18be7e17673a28353590323423b1cfd3": "\\Rightarrow P = 5.25 \\mbox{ big bets }", "18be99b429a0b6209066c6e48df5d0f4": "\n \\frac{\\partial p}{\\partial t} + \\rho_0~c_0^2~\\nabla\\cdot\\mathbf{v} = 0 ~.\n ", "18bf58ee81d2c796057d130e40e94e3c": " e_s(T)", "18bf9dd809926e4de2b43a9eda126d74": "\\|f\\|_p= \\sum_{n}|a_n|^p \\qquad (0 < p < 1)", "18bfb223f5ce7115f7a4b51094b834ea": "\\frac{x}{\\sqrt{1-x^2}\\arcsin(x)}", "18bfba88133de0dabea899170b5e687b": "O(k)", "18bfc1131d97016f2b5ddb0173cc4077": "\\lnot x \\vee \\lnot\\lnot x = 1 \\mbox{ for all } x \\in H.", "18c038ad087bb019d95934a2bc91ce38": "yt[ab]", "18c08e9499ed484042ace8aa905b506a": ".\\qquad NP/N,\\; N/N,\\; N,\\; (NP\\backslash S)/NP,\\; \\underbrace{NP/N,\\; N}", "18c1a5c895d7377f120febe901bd9bc3": "T(t,\\sigma) = C(t) \\phi(t,\\sigma) B(\\sigma)", "18c1f49e7a9c6685d9a8af46995b1e92": "\\textstyle ", "18c26fbb9faeb672c839801ae8d7ad34": "B^{ij}", "18c2c9e447046792e2c30f4b110ac12d": "2^K", "18c2f4b4d5330d21f199aab54a25ca65": " \\zeta ", "18c36012bc30feba7ea4d3f8c40e9070": "2^{\\frac 1 {12}} \\approx 1.0595", "18c456eb149d0b8085c092f13c390655": "\\langle X,Y\\rangle_A=\\langle Y,X\\rangle_A", "18c5171e237b269287e952f3d3052c6d": "\\begin{align}\n & W^2 = \\sqrt{2\\gamma_2 + 4} - 1, \\\\\n & \\delta = 1 / \\sqrt{\\ln W}, \\\\\n & \\alpha^2 = 2 / (W^2-1), \\\\\n \\end{align}", "18c5bee80a7702628fdb81f660d2d122": " \\gamma = \\int_0^1 \\frac{1}{1+x} \\sum_{n=1}^\\infty x^{2^n-1} \\, dx. ", "18c5f9700e93ac23c1ca12fe82793464": "S_{0}", "18c622a400c959474d52682d250a2545": "\\# X < \\nu(W)", "18c6b428497c5dfc34dc2381fefd2a01": "\\scriptstyle \\mathbb{R}^2 ", "18c6f2a3052cd34e931618a614ee9608": "\\begin{bmatrix}\\Psi\\end{bmatrix}^{T}\\begin{bmatrix}M\\end{bmatrix}\\begin{bmatrix}\\Psi\\end{bmatrix}=\\begin{bmatrix} ^\\diagdown m_{r\\diagdown} \\end{bmatrix},", "18c70df624239185d211541116b4658a": "A_w(\\mathbb{T}) =\\{f:f(t)=\\sum_na_ne^{int},\\,\\|f\\|_w=\\sum_n|a_n|w(n)<\\infty\\} \\,(\\sim\\ell^1_w(\\mathbb{Z}))", "18c783c7483df4341ae61e26e46688e7": "A\\geq\\frac{P^{m}(V)}{2}\\,\\!", "18c79bc7572a8969200765a7e578384e": "\\div \\!\\,", "18c7da0dbd1bf29357ccae6f8d85cae6": " a*(b + c) = a*b + a*c", "18c83df3e8f90750ec5e350a55ec8090": "{e}=\\rho_vR_vT \\,", "18c87d80621e6c261a7575039bf8e7e3": "(\\mathbf{\\hat{x}}, \\mathbf{\\hat{y}}, \\mathbf{\\hat{z}})", "18c8d17bf436f990c16703d01556c1ec": "\\vec{p} \\in P", "18c91a49cd224a3e390dcaf993db2f56": "\\scriptstyle A_0, A_1, B_0, B_1", "18c932f784acdcfb8c7c81591b255ed5": " H_{1-\\alpha}-H_\\alpha = \\pi\\cot{(\\pi\\alpha)}-\\frac{1}{\\alpha}+\\frac{1}{1-\\alpha}\\, .", "18c9879537ecfdb1a719992752a3e4ec": "\\zeta^{\\prime}(-1)=\\frac{1}{12}-\\ln A", "18c9ab7fd3438beaac0f33b87fbf6666": " i \\in D ", "18c9b29a5a484f5e37991318459fcca8": "\\succ_i^p", "18c9ce8c77dc0267a83b6ed5a29c0044": "y = x^m", "18c9df2276ffd7a0ab756bdfa1f04efd": "1 \\leq \\cdots \\leq A_{-1} \\leq A_0\\leq A_1 \\leq \\cdots \\leq G", "18c9ed4dad017ea46713b5309a8c8677": "k_{0} = 1. \\!", "18ca3278a836cd4e54595dd54b77daee": "r_{a}", "18ca374c2173f81f8c19eb161900b95c": "\\Delta g_h = \\left [ G \\, m_\\mathrm{Earth} / \\left( r_\\mathrm{Earth} + h \\right) ^2 \\right ] - \\left[G \\, m_\\mathrm{Earth} / r_\\mathrm{Earth}^2 \\right]", "18cab4412675e6cd3658f0b619095dda": "\\lambda=w\\cdot\\tilde\\lambda", "18cb04d6c0fe962c0924510db4794da0": "\n\\begin{align}\n\\biggl|\\bigcup_{i=1}^n A_i\\biggr| & {} =\\sum_{i=1}^n\\left|A_i\\right|\n-\\sum_{i,j\\,:\\,1 \\le i < j \\le n}\\left|A_i\\cap A_j\\right| \\\\\n& {}\\qquad +\\sum_{i,j,k\\,:\\,1 \\le i < j < k \\le n}\\left|A_i\\cap A_j\\cap A_k\\right|-\\ \\cdots\\ + \\left(-1\\right)^{n-1} \\left|A_1\\cap\\cdots\\cap A_n\\right|.\n\\end{align}", "18cb076d0fe4b615ee592b4b9a56c0fc": "\\underline{\\varphi \\quad \\quad}\\,\\!", "18cb08d66527a64985e2db443c219a67": "\\mathbf{k}^\\mathsf{T} \\mathbf{k}=1", "18cb86e065d27c16648273989c67ab1a": "\\ MU_y=\\partial U/\\partial y ", "18cbb36716dcc00fc63d445446e6b3d1": "\n\\begin{align}\n\\dot{\\mathbf{x}}(t) &= f\\bigl(\\mathbf{x}(t), \\mathbf{u}(t)\\bigr) + \\mathbf{w}(t), &\\mathbf{w}(t) &\\sim N\\bigl(\\mathbf{0},\\mathbf{Q}(t)\\bigr) \\\\\n\\mathbf{z}(t) &= h\\bigl(\\mathbf{x}(t)\\bigr) + \\mathbf{v}(t), &\\mathbf{v}(t) &\\sim N\\bigl(\\mathbf{0},\\mathbf{R}(t)\\bigr)\n\\end{align}\n", "18cc4194f044ecd8d8f8adc24550038a": " c_a = \\sum^a_{ i = 1 } p_j ", "18cc42fd224345464fd8bbe701a4edfc": "C = \\frac{N}{V} = \\frac{p}{kT}", "18cc584457d459aeb3a6277022daacb1": "q = 1/\\log_2(\\varphi) \\simeq 1.44,", "18ccb0e4a67e65f6a84f8eb08236c427": "\\|p_\\sigma\\|\\le\\alpha\\sqrt{n}", "18ccc5f2be3e3400d203c6a74a2f2840": "\nf(x, y)=\n\\begin{cases}\n\\frac{x^3y}{x^6+y^2} & \\mbox{ if } (x, y)\\ne (0, 0)\\\\\n0 & \\mbox{ if } (x, y)=(0, 0)\n\\end{cases}", "18ccc9571afc21e0c83a1ae3976dc19a": "W_CB = \\frac{C_BV_B^2}{2} = \\frac{Q_B^2}{2C_B} = 2\\pi \\alpha W_B, \\ ", "18cccaa1e6cc9c1c23f29598bc9d01a0": "A^{(\\beta)}_X", "18cccd0062230fb90522b99008c96f74": "6x^2-6x+1", "18cd00c8599a379a8854b77b42fce125": "n_3^2", "18cd07f22d73ea3c74863f593399aaec": "x\\ast 0=0 \\implies x=0", "18cd3f61bb42cf1c2db2e54e6e34cd96": "\\alpha(u) = \\sup \\{ \\alpha((Q_E \\otimes Q_F)u; (X/E) \\otimes (Y/F)) : \\dim X/E, \\dim Y/F < \\infty \\}.", "18cd6a26e5fa0a9aecea3362ecd7a164": " \\prod_{k=1}^{n} \\sin\\left(\\frac{\\left(2k-1\\right)\\pi}{4n}\\right) = \\prod_{k=1}^{n} \\cos\\left(\\frac{\\left(2k-1\\right)\\pi}{4n}\\right) = \\frac{\\sqrt{2}}{2^{n}}", "18cd76adf9e419c326dcb8a21d2c4e9b": " \\lim_{n \\to \\infty}\\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1}} \\qquad \\scriptstyle x \\in (3.8284;\\, 3.8495)", "18cdc75d71c5fcc70d6b7c721ccf1ccb": "M_x[cx] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&c&0& \\cdots \\\\\n0&0&c^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "18cdcc46f01170e5181c5fb4f16629d4": "x_2=b(e(x_1,x_2),f(x_1,x_2))", "18ce4578c9438e94ef922dbd60043904": " f_x ", "18cef17f5862cab2886a6e32da6d521a": "d_{2,-1}^{2} = -\\frac{1}{2}\\sin \\theta \\left(1 - \\cos \\theta\\right)", "18cef2dcb65e74d68170aa193d352475": "n = V(+\\infty) + V(-\\infty)\\,", "18cf0be85b837ab17433600fa8ca4edb": "L = \\sum_{n=0}^{\\infty}a_n \\Leftrightarrow L = \\lim_{k \\rightarrow \\infty} S_k.", "18cf2e335b256d116bf5784ff58ffa58": "\\displaystyle\\mathbf r=\\mathbf r(q^1,\\,\\ldots,\\,q^n)\n", "18cf2e8e98f9e04d70f816168f16679f": "R\\ ,", "18cf34aaea29dca8123c5618d3f3d955": "f(\\mathbf{v}) = f_1(\\mathbf{v}) + f_2(\\mathbf{v})", "18cfcc45a6600384d9c42bf89ddaa0b6": "\\left \\langle \\sigma_y^2(N_2, T_2, \\tau_2) \\right \\rangle = \\left ( \\frac{\\tau_2}{\\tau_1} \\right )^\\mu \\left [ \\frac{B_1(N_2, r_2, \\mu)B_2(r_2, \\mu)}{B_1(N_1, r_1, \\mu)B_2(r_1, \\mu)} \\right ] \\left \\langle \\sigma_y^2(N_1, T_1, \\tau_1) \\right \\rangle", "18d003bc91170eee7f18145c5a7b3cee": "\\int e^x \\cos x \\, dx.", "18d047091e14b8004d65a6169f11375c": "F\\rightarrow M", "18d0bbaed8c808511af6d9d78ad4be5e": "\n+_*:T(E\\times E)\\to TE \\quad , \\quad \\lambda_*:TE\\to TE\n", "18d0c516a7f6be3f85f474eccef6dcd7": " x \\in \\mathbb{R}^n", "18d0d9a81efd48585e55e54feffe3bbe": "\\tfrac{1}{2} v^2 - \\mu/r = constant = \\tfrac{1}{2} C_3", "18d0fb3d042601899924692cac9281fb": " {52 \\choose 5} = \\frac{52}1 \\times \\frac{51}2 \\times \\frac{50}3 \\times \\frac{49}4 \\times \\frac{48}5 = 2,598,960.", "18d10943ecf5fa07437faeec929ae5d2": "S(t,u) = p(t) + u r(t)\\ ", "18d14c692224f44ddd06df74f6f0f2e9": " q_e = \\iiint \\rho_e \\mathrm{d}V ", "18d1a51c2657c0f7b4ddb799341ac9de": "\\mu(\\sigma,\\tau)=(-1)^{n-r}(2!)^{r_3}(3!)^{r_4}\\cdots((n-1)!)^{r_n}", "18d1a5b10c18259f453199b2da4705d6": "-1 \\le \\xi < 1", "18d1e62a5cceb8f64cc61fdeeaf2dcf5": "\\tfrac{9KG}{3K+G}", "18d1fff28ff77879fd2a054c3388d9e4": "\\lambda_1, \\lambda-2, \\ldots, \\lambda_k", "18d2d5e9e539970e468fbd0daadf7881": " \\chi_T ", "18d305ca08e3a56a7bf1c2fbe59fa654": "|F_n| = \\frac{1}{2}\\left(3+\\sum_{d=1}^n\\mu(d)\\left\\lfloor\\tfrac{n}{d}\\right\\rfloor^2\\right),", "18d33c0576229a7cdce893831a99aa41": " \\frac{\\partial N_x}{\\partial e}\\frac{1}{X} = \\frac{\\partial X}{\\partial e}\\frac{1}{X} - \\frac{1}{Q}\\frac{\\partial Q}{\\partial e} - \\frac{1}{e} ", "18d37af0cd9f19f6013d9bacfa83aad1": "\\Delta \\left( \\frac{1}{B} \\right) = \\frac{2 \\pi e}{\\hbar c S}", "18d400a78fdb8cafa22d43db883ca9f0": "\n\\max\\{\\alpha: R(q,u) \\in C, \\forall u \\in \\mathcal{U}(\\alpha,\\tilde{u})\\} = \\max_{\\alpha\\ge 0}\\ \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})} I(q,\\alpha,u)\\}\n", "18d40bec0a6e15cc9c6ef691a49874ba": " {\\Psi} = 1 -(1- A{\\Psi}^{\\ast}) ", "18d42ed3943c51df719cd7d12492b572": "\\begin{align}\nx_{12} & = {\\color{BrickRed}a_{11}}{\\color{RedViolet}b_{12}} + {\\color{BrickRed}a_{12}}{\\color{RedViolet}b_{22}} \\\\\nx_{33} & = {\\color{BurntOrange}a_{31}}{\\color{Violet}b_{13}} + {\\color{BurntOrange}a_{32}}{\\color{Violet}b_{23}} \n\\end{align}", "18d4f811912a860a06436cb9fd1c7af4": "\n\\alpha(0,1) = v,\\qquad\n\\frac{\\partial \\alpha}{\\partial t}(s,t) = v+sw_N,\n\\qquad\\frac{\\partial \\alpha}{\\partial s}(0,t) = tw_N.\n", "18d4f9889121377503b8df6613b71b4e": "S = \\{(x_i,t_i): f(x_i) > t_i\\}", "18d57e2afd67380f1b7252ccec01b7f1": "\\boldsymbol{\\nabla \\times}\\left( \\boldsymbol { \\nabla \\times V}\\right ) = \\boldsymbol {\\nabla}\\left(\\boldsymbol{\\nabla \\cdot V}\\right ) - \\nabla^2 \\boldsymbol V \\ , ", "18d59220b9efd10dd2d46bb3e83efc0f": "M_{\\alpha} = \\sum_{i_1 < i_2 < \\cdots < i_k} x_{i_1}^{\\alpha_1} x_{i_2}^{\\alpha_2} \\cdots x_{i_k}^{\\alpha_k}. \\, ", "18d5933a0f0edd7cb24b2318b0a31293": "\\vec r\\,\\!", "18d598bb85551bad919521ddf1779dd9": "\\vec{F} =m\\vec{a}.", "18d5cae1d47696d5b6d9aede698ccf4a": "\\operatorname{Tr} (\\sigma_i) = 0 .", "18d60d4323b6c781b54ffe505afe34d5": "Z_3\\,", "18d615cfeed2397d6b1022303b96f41f": "\\begin{align} \\mathbf{p}(\\mathbf{r}) & = \\sum_{i=1}^{N} \\, \\int\\limits_V q_i [ \\delta (\\mathbf{r_0} - (\\mathbf{r}_i + \\mathbf{d}_i) )- \\delta ( \\mathbf{r_0} - \\mathbf{r}_i ) ]\\, (\\mathbf{r}_0-\\mathbf{r}) \\ d^3 \\mathbf{r}_0 \\\\\n& = \\sum_{i=1}^{N} \\, q_i\\, [ \\mathbf{r}_i +\\mathbf{d}_i - \\mathbf{r} -(\\mathbf{r}_i-\\mathbf{r}) ] \\\\\n& = \\sum_{i=1}^{N} q_i\\mathbf{d}_i = \\sum_{i=1}^{N} \\mathbf{p}_i \\ , \n\\end{align}", "18d6db889b81027f9cdc63e0381dfd0d": "\\left[y^{(1)}\\right] \\supset \\cdots \\supset \\left[y^{(k)}\\right]", "18d6de4bb3829f1be8680bfb8bc10256": "\\rho < 1", "18d6eb73f9f5a3bfcfdb50e69781c92e": "\\sigma_n = \\sigma'", "18d727a8d67fa56e7866f4447b0f9fd7": "\\text{Res}_t(g_1x_1-f_1,g_2x_2-f_2).", "18d72b458cb159d62c23170036801947": "L^{-1}", "18d783995bfcb6d6d17051ef27aabe21": "x_1,x_2\\in \\Omega\\backslash X", "18d7cefd81bdaeaf7163a3f063c440ee": "\\theta_1 = dy - y'dx.", "18d803f3478e06288fdc4b15f3d823bc": "\\mathbf{M}(\\mathbf{q})", "18d8042386b79e2c279fd162df0205c8": "400", "18d80f4bce9c68c24a1efbf2a85dbf52": " \\langle\\bar\\psi \\psi\\rangle = {1\\over Z} {\\partial \\over \\partial \\eta} {\\partial \\over \\partial \\bar\\eta} Z |_{\\eta=\\bar\\eta=0} = M^{-1}", "18d90459172a76697ac496c18c672b0e": "c_1^2>c_2", "18d916a53926bede3895d71f6621da3a": "\\forall i, \\lambda_i \\ge 0; \\lambda_0 = 0", "18d975ee85323d2a1939a43e15666899": "W_s=\\int_{1}^{2} \\sigma_s\\,dx.", "18d9962fdb0adb9b0587bda0c33524a2": "n^2+1", "18d99b829fb348bc558fc223d5c79acf": "(r,\\theta,z)", "18d9be35d247c9ea4926c801c4c199b6": "\\sum\\limits_{k=1}^{\\infty} |x_k\\,y_k| \\le \\biggl( \\sum_{k=1}^{\\infty} |x_k|^p \\biggr)^{\\!1/p\\;} \\biggl( \\sum_{k=1}^{\\infty} |y_k|^q \\biggr)^{\\!1/q}\n\\text{ for all }(x_k)_{k\\in\\mathbb N}, (y_k)_{k\\in\\mathbb N}\\in\\mathbb{R}^{\\mathbb N}\\text{ or }\\mathbb{C}^{\\mathbb N}.", "18d9ddc4fe786f1cd073c3dbc282b5da": "x_1, x_2 \\geq 0", "18d9f795cd0d91c53c3acbb7daad604c": " \\Delta t \\rightarrow 0", "18da665ed77201f6c7402d395e55c21c": "\n\\nabla^{2} \\Phi = \n\\frac{1}{a^{2} \\left( \\sigma^{2} - \\tau^{2} \\right)}\n\\left\\{\n\\frac{\\partial}{\\partial \\sigma} \\left[ \n\\left( \\sigma^{2} - 1 \\right) \\frac{\\partial \\Phi}{\\partial \\sigma}\n\\right] + \n\\frac{\\partial}{\\partial \\tau} \\left[ \n\\left( 1 - \\tau^{2} \\right) \\frac{\\partial \\Phi}{\\partial \\tau}\n\\right]\n\\right\\}\n+ \\frac{1}{a^{2} \\left( \\sigma^{2} - 1 \\right) \\left( 1 - \\tau^{2} \\right)}\n\\frac{\\partial^{2} \\Phi}{\\partial \\phi^{2}}\n", "18da6fd1d28eafe7341e995408a06d00": "D_y\\ f(x)", "18da78172f27d03ad7e9502ee5fb4b79": "g(f(x)) = \\sum_{n=1}^\\infty\n{\\sum_{k=1}^{n} b_k B_{n,k}(a_1,\\dots,a_{n-k+1}) \\over n!} x^n.", "18da8dfa42c5e288b3d8a601179e0a59": "\\mathbf{y}(\\cdot)", "18daef71b5d25ce76b8628a81e4fc76b": "y_{i}", "18db1019fc935b332ccd75838c1a05ae": " 4K^2 = (ad)^2 \\sin^2 \\alpha + (bc)^2 \\sin^2 \\gamma + 2abcd \\sin \\alpha \\sin \\gamma. \\, ", "18db1f7b31478a3839616580ad00ad0d": "C = D \\left[ N(d_+) F - N(d_-) K \\right]", "18db58092a2eba6265a61b7cce5c3ae3": "\n\\begin{matrix}\nD_{\\mathrm{KL}}(P\\|Q) & = & -\\sum_x p(x) \\log q(x)& + & \\sum_x p(x) \\log p(x) \\\\\n& = & H(P,Q) & - & H(P)\\, \\!\n\\end{matrix}", "18db5e44caf2ed2fdde04a9dd6a6c41f": "(\\underbrace{x_n+x_{n+1}-\\alpha}_{=\\,y})\\alpha-x_nx_{n+1}=(x_n-\\alpha)(\\alpha-x_{n+1})>0,", "18db600e9b6993dd9ec8642eb24695dd": "\\pi_k", "18db90767e8ab871f14664034faf96e7": "g(r) dr", "18dbc544e753ec339db6cb5606b70029": "\\frac{\\sqrt{17}-1}{2}", "18dbec4959311ce1ed660197e86282b2": "E = \\frac{\\hbar^2 k^2}{2m} ", "18dbff11522d3c95e274440bdd47deb1": "\\Omega_{-}\\,", "18dc2d7de0b78341c8d30be4ecb7217b": "|N(h)|", "18dc516a4024559888e5f7271fd3f7cf": "T = 4\\sqrt{\\ell\\over g}F\\left( {\\theta_0\\over 2}, \\csc{\\theta_0\\over2}\\right)\\csc {\\theta_0\\over 2}", "18dc5eaca5c946d491477c8aba406a34": "v(\\tau)= \\int \\frac{r(\\tau)}{r(\\tau)-2GM} d\\tau", "18dc8499fdc12a50265acbc280088b5b": "\\mathbb{T}^3", "18dcc10831be4494a9c0fcda3f7bec40": "d_{j}", "18dce07d0b7288f146469052faa6a251": " {u}_{1}^{z} = {g}_{1}^{k z} = h^k \\,", "18dd082ee2e6b606e52cf983f153f3f4": "B=QTZ^H", "18dd11165ce16b83780b7d075019caf7": "d*\\mathbf{F}=*\\mathbf{J}", "18dd30e1971035d87332bb7a9da52125": "ab \\le \\int_0^a f(x)\\,dx + \\int_0^b f^{-1}(x)\\,dx", "18dd6968964f626cb21125eb5a414ad6": " y=\\frac{Y}{Z}", "18dd8a5b501b6e52fab6fdca29986aae": " \\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k+1} - x_{k} \\right) \\left( f(x_{k+1}) + f(x_{k}) \\right).", "18dda4dd3a74b0ad8cba16542145e5bc": "\\vec B\\!", "18ddbcdfc1287d2e4d37590299c4e28d": "R_{mn}(-i c,i \\xi)", "18ddf017f6cd97b31468ebfc3e167b7b": "001100", "18de282afdf186a5a68e15c900084ce7": "E[\\textbf{v}_k\\textbf{v}_k^T] = \\textbf{R}_{k}^a", "18de505a0c1d409e45c8075261763358": "cf \\in \\mathcal{H}", "18deab078d5dde8e2825954f355e7932": "\n \\beta = \\arctan \\left(\\frac{b\\sin \\alpha}{a + b\\cos \\alpha}\\right) + \\begin{cases}\n0 & \\text{if } a + b\\cos \\alpha \\ge 0, \\\\\n\\pi & \\text{if } a + b\\cos \\alpha < 0.\n\\end{cases}\n", "18deda0400bf981d62336050161c54f4": "A^{\\mu } =\\hat{P}^{\\mu }A(r) ", "18df69c7e2afa34bf74da608bb2bd466": "\\mbox{Mortgage Yield: ri such that P} = \\sum_{n=1}^{N} \\frac{C(t)}{(1+ri/1200)^{t-1}}", "18dfdf66424bc93f268b0a52ea1088ef": "\\epsilon_{ij} = d_{ijk}E_{k} \\,", "18e06cc13564d3ae926f420ebf45ebcf": " y^2 = x(x-a^n)(x+b^n). \\, ", "18e08796f10460ab1bc7e5622cd15249": "r^*t' + t^*r' = 0.", "18e0e3c623cb11ea32986af2a0ec35da": " \\scriptstyle \\forall n:\\; \\sum_{j=1}^J P_{nj} = 1 , ", "18e0e8c459b7e584339cbbd03089c555": "m\\in M.", "18e0ff95ddcaa085a20abacb0a6c1ec7": "1-e^{-4.7/4.5}", "18e135ba82fd466fb3d91d10eff214d9": "P(v,u\\in K)=P(v\\in K)\\,P(u\\in K)", "18e151b7cdd4f24b150d6dea6516d4dc": "~y", "18e18adb2e9f5c02ad1b703d0a0528cc": "M_{-1}^{2}", "18e1be1f137eecd5e0d08c063ddf1d21": "{\\partial v \\over \\partial y} = e^x \\sin y", "18e1c440689fbb0b3331a75502d696bf": "(\\forall k\\in\\mathbb{Q}\\setminus\\{0,-1,-2,\\ldots\\}):f_k(x)\\neq0\\text{ and }\\frac{f_{k+1}(x)}{f_k(x)}\\notin\\mathbb{Q}.", "18e1d074eb6ca6bab87c99b95af11ee9": "d_1 \\ldots d_n", "18e1e571ad0dea2afe023c83368239ae": "R_{CF} \\ll R_P", "18e2143a3c1df78caf3a5884732e5fea": "\\frac{{\\rm d}n}{{\\rm d}\\lambda} < 0.", "18e232b29e32721fdfdf28d703e4c490": "a = \\alpha r/2", "18e253d4ce939cf1524819f4bd610674": " \\sigma = [1+\\frac{6.2}{z.n^{2/3}}]^{-1} ", "18e27dd067ddf4ee760b43ba878543d8": "b < c", "18e2dbdb3f4bc32e5b7bf612fdbd0400": "f(x) = \\delta(x) + 3 \\delta(x-2) + \\delta(x-5) + 3 \\delta(x-8) + 5 \\delta(x-10). \\,", "18e2e81fad5c868e323ead730af15019": "\\rho =-\\frac{g_{kq} v^{k} v^{q} }{2v_{0} v^{0}}.", "18e2ea43e61c22335ca1d9200ee50679": "{l_B \\over l_D} = {a_B \\over a_D}", "18e33fe4608b1286390dfd6424135b4f": "\\Psi(\\bold{r},t) = \\frac{1}{\\sqrt{\\Omega_r}} e^{i\\bold{k}\\cdot\\bold{r} - i \\omega t} ", "18e363b1ba578c2b9b706372a409b7d8": "q = 1337", "18e386fab3e73a34d516014b68b079f9": "g=\\det(g_{\\alpha\\beta})", "18e39c093f7c315b0436229b216cc05b": "\\frac{1}{\\sqrt{2}}(|0\\rangle|1\\rangle - |1\\rangle|0\\rangle)", "18e3dd82ef155c4e50f62b98ca54111a": " v_{22}= 1. ", "18e40ab6e49b1477a253210873796f45": "\n\\dot{p}_j = -\\frac{\\partial H}{\\partial q_j} - \\sum_k u_k \\frac{\\partial \\phi_k}{\\partial q_j}\n", "18e4672d88622956c3d2aeb7c75dbafe": "f(r_s)=\\frac{\\tanh(\\sigma (r_s + R))-\\tanh(\\sigma (r_s - R))}{2 \\tanh(\\sigma R)},", "18e49888d88b3813a4d9d80e1c891177": " \\sigma^2(z) = \\sigma_0^2 + M^4 \\left(\\frac{\\lambda}{\\pi\\sigma_0}\\right)^2(z-z_0)^2 ", "18e4c768ee4a496790925ac1c7150b75": "\\mathrm{0.58\\overline{3}}", "18e4da906e92a9599d537d8af3d49ab5": "\\Delta M = -2\\sigma_i ", "18e4eb5c7617cf910330fc5783a51c15": "x_n = \\sum_{i=1}^n N(n,i)\\, k^{i-1} = \\sum_{i=1}^n \\frac{1}{n}{n\\choose i}{n\\choose i-1} k^{i-1}.", "18e4f18165e1482db8d3cd346d864593": "gD=\\{gd:d\\in D\\}", "18e509c71028a2c70b9ab8f482c7671a": "\\left|x- \\frac{p}{q}\\right|= \\left| \\frac{c}{d} - \\frac{p}{q} \\right| = \\frac{|cq - dp|}{dq}", "18e55bc4b98b51fa752496c4b1767be4": " \\sum_{n=0}^\\infty L_n(x) t^n = \\frac{2-xt}{1-xt-t^2}.", "18e5b854e7c6b116a9c12afa63776db7": "\n\\boldsymbol{\\mu}^A = \\boldsymbol{\\mu}^B = \\mu_\\mathrm{HCl} \n\\begin{pmatrix} 0 \\\\ 0 \\\\ -1 \\end{pmatrix}\n\\quad\\hbox{and}\\quad E_{\\mathrm{dip-dip}} = \\frac{-2\\mu^2_\\mathrm{HCl}}{R^{3}_{AB}}.\n", "18e5c6757e1ce7689bde2e941b50101c": "\\widehat{H}_B = - \\mathbf{B} \\cdot \\widehat{\\boldsymbol{\\mu}}_S ", "18e5ee7d665be309e903036512839be6": "F_e = F_m", "18e5f0e76d8125efee320acf3639fa82": "a,b,c,x", "18e62cfdbbba96a6523c26896d72a5d6": "VAG (p, (a, m))\\cup VAG (p,(m, b))", "18e661fcfc00ebfcedde7da052602a82": " \\Delta t ", "18e68dea3c645b212900138827e07bce": "P_n^r= \\frac{3n^2 + n^3(r-2) - n(r-5)}{6}", "18e6d2998d32e036596ac714945c280c": " F_c=ky", "18e7229ebfe4e86fb91d36909d6f2201": "(\\alpha,\\beta)", "18e7a59f42857ecf6181bbc4a00840c3": "\\leq 4/3", "18e7dfefd25afd16c2eb9175fb8dd05d": " K(x,y) = e^{-\\frac{\\|x - y\\|}{\\sigma}}, \\sigma > 0 ", "18e7f09e1ba3ead19d2fc82b4efcd47e": "a^{\\ln x}\\,", "18e833a4416a8cf33f6eb69697cf10f3": "\n \\{\\boldsymbol{m}\\} = \n \\begin{cases}\n \\mathbf{m}_1 = \\mathbf{\\hat{x}}_1 = [1,0,0] & \\| \\, \\textrm{to} \\, \\textrm{applied} \\, \\textrm{tension}\\\\\n \\mathbf{m}_2 = \\mathbf{\\hat{x}}_2 = [0,1,0] & \\perp \\textrm{to} \\, \\textrm{applied} \\, \\textrm{tension}\\\\\n \\mathbf{m}_3 = \\mathbf{\\hat{x}}_3 = [0,0,1] & \\| \\, \\textrm{to} \\, \\mathbf{N}\n \\end{cases}\n ", "18e871e3a2f44664dac2bbe3bdde2356": "n=1.5", "18e8aec7b6bb93acabfcc38b5f352dac": " \\ln(xy) = \\ln(x) + \\ln(y). \\!\\, ", "18e8af50161ff9bb61d83bf509d4ba78": "\\mu = M_{\\mbox{i}} - M_{\\mbox{f}}\\,.", "18e8cd2673a772049abd316473986c6c": "\\left\\vert -h \\right\\vert", "18e91a026e88127189fb8fc07e1b1e4f": "A= \\pi D^2 / 4", "18e965e2958c2b0d27d4026a6b39391c": "\\sigma,\\tau", "18e9981ff27b4c67567ed89fd97460bd": "\\overline q^i_{tt}=0", "18e9ce193cb916c093dc6103d005c35e": "F_r = 1; \\; \\textit{Critical}", "18e9e4628380071621088ec8c296e8db": "a_3b_1", "18e9e4d860da1afcb3b2eb5a2ae6b0f3": "x + x^2", "18ea38550fd57e7c2a64fa2851d7eed5": "\\pi(x) = \\operatorname{li}(x) + O\\bigl(xe^{-\\sqrt{\\ln x}/15}\\bigr)\\!", "18eacf44f99c3a847cf8833245952bd8": "D_{a} = \\frac{313\\Pi}{10800}", "18eb3dc33619724182bacdda52edccf2": "H=H_1\\otimes H_2", "18eb57492e10c8ba84f1907647b1b647": " \\underline{\\hat{\\mathbf{h}}}(\\ell) = \\underline{\\hat{\\mathbf{h}}}(\\ell-1) + \\mu\\mathbf{G}_2\\mathbf{\\Phi}_\\mathbf{xx}^{-1}(\\ell) \\underline{\\mathbf{X}}^H(\\ell) \\underline{\\mathbf{e}}(\\ell) ", "18eb6b29f002e684df90786d417aea35": "U/D>\\pi", "18ec0f8feaca1b6b0dca2651a8c87ed5": "\\scriptstyle A_t ", "18ec6089a8de6865d2a39891d930b03a": "\\displaystyle{X}", "18ec6addef1c5059221a87ace99feaa9": "U_{DC} = n \\frac{\\hbar}{2 e} \\omega, \\ \\ \\ I(t) = I_c J_{-n} (a) \\sin \\phi_0.", "18ec7450b1cade397101e5dd9f496e22": "a(\\overline{y})=1-a(y)", "18ec821f6c25dd91aaeed00426da2ae3": " \\mathbf{x}_j ", "18ec93e899d470f2998ea9875533a148": "\\sqrt{p(R_i)}", "18ecbaf66226ba4ea4e7604b197113dd": " \\frac {\\mu_0 l}{\\pi} \\left( \\ln\\left(\\frac {d}{a}\\right) + \\frac {Y} {2} \\right) ", "18ed0afc43da2399560134b6e8fd05c1": "\\sum F_{\\perp} = F_n - F_w \\cos \\theta = 0 \\,", "18ed25604450022943278b745379b09d": " D^3 ", "18ed92d5aa74d11b26b02495c0225105": "\\{ a^ib^i \\; | \\; i \\geq 0\\}", "18edb17c0fe3647a4edb159d9dbe2f62": "\\mathbf{e}_1 \\wedge \\mathbf{e}_2", "18ee1a620604335fcb97eed24e611d9c": "\nf(T_1,T_3) = \\frac{q_3}{q_1} = \\frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).\n", "18ee261e8fdaf09a093f4fca95284db9": "\\forall a, b \\in X,\\ a R b \\or b R a.", "18ee590341dab70c6f0399729deebe52": " 0 = - \\Delta P2 \\pi rdr + \\eta 2 \\pi dr \\Delta x \\frac{dv}{dr} + \\eta 2 \\pi r dr \\Delta x \\frac{d^2 v}{dr^2} + \\eta 2 \\pi (dr)^2 \\Delta x \\frac{d^2 v}{dr^2}. ", "18ee59b780433789750e641ecc86e92f": "\\int\\cos^2 {ax}\\;\\mathrm{d}x = \\frac{x}{2} + \\frac{1}{4a} \\sin 2ax +C = \\frac{x}{2} + \\frac{1}{2a} \\sin ax\\cos ax +C\\!", "18ee6ecfee0d63526a15095efea248ff": "(\\mathcal{X},\\Sigma)", "18ee7743f4ccf7470f652fd2a1a7238b": "t_2=[v_2, v_3]\\,", "18ee983cb0b3b556bd69afb048eb4e89": " K_t(x) = e^{-tH} \\, ,", "18eebd452f2617cf78ffc2dec6ce8c06": "v^*\\leftarrow v, x^* \\leftarrow x", "18eec6b2df6e8b8bbf3183366172fa07": " \\mathrm{Ref}(\\theta) \\, \\mathrm{Ref}(\\phi) = \\mathrm{Rot}(2(\\theta - \\phi)), \\ ", "18ef107c7fab1ca903a3a8754b7eef0a": "|n(x^\\mu)\\rangle", "18ef2138a9fe6ef69fe7d2f5d7b88fa2": "F = f(W(h_{1}), \\ldots, W(h_{n})),", "18ef2de26637057f0d82d5641dfa5e49": "\\scriptstyle u = u_0 + u_1 ... + u_n ", "18ef4309191f5c9b8813f814787c46fa": "\\displaystyle\\alpha", "18ef54debd2505c76d3ee54354a8d237": "w'_1=1", "18efa680e578ea5c08bd4057de44317b": "ST_x(\\Box_m \\varphi) \\equiv \\forall y ( R_m(x, y) \\rightarrow ST_y(\\varphi))", "18efc5472201a32ac91ddc05d4daf6a6": "\\operatorname{qri}(C) = \\operatorname{ri}(C)", "18f00026f69ba077a7cc4320bb07b329": "\\|x\\|_f \\overset{\\text{def}}{=} |f(x)|", "18f13abd48fe832e3323f2e24c87a82b": "\\mathbb{Z}/2\\mathbb{Z}:=\\lbrace 0,1 \\rbrace", "18f147d88109cddb695b05d78d52ff25": " NPSH_A = \\left( \\frac{p_e}{\\rho g} + \\frac{V_e^2}{2 g} \\right) - \\frac{p_{v}}{\\rho g}", "18f1887869ebade600220dbc16f4562a": "G(A,B)", "18f197b8fa5e4c2ba49cea075cc713ca": "\\mu(x\\wedge y)\\mu(x\\vee y) \\ge \\mu(x)\\mu(y)", "18f1bb52ba5aa4d30fad7bdbfad303fd": "\\cup_{n=1}^\\infty A_n\\in \\Sigma_0", "18f1c04cc33cac73730aea1f1d01e305": "m/n", "18f1e3d02ce97d1ecf4ce1d497979c4d": " \\left(\\frac{p_1}{p_0}\\right)^{R/c_p} = \\frac{T_1}{T_0}, ", "18f1fec2d0d81e43578c0b9957369590": " \\sin^2\\theta_1+\\cos^2\\theta_1=1 \\, ", "18f22aeaffef62020b1b657f172811ec": "\\gamma_0 = \\frac{1}{\\sqrt{1-v_0^2/c^2}},", "18f23635fdd22c2cf89d8e3b668a7a96": "\\Pr(\\tau_{n+1}\\le t, X_{n+1}=j|(X_0, T_0), (X_1, T_1),\\ldots, (X_n=i, T_n)) ", "18f265310240b2e3b1d77eb09b736540": "a \\ne 0,", "18f284a405363e95719439b6fa1b9f8a": "(t_i,s_i)", "18f2c079d88ea8e87f4ed76ebff0fe76": "\\mathfrak{p}=\\mathfrak{g}_-", "18f2c9be3e9dd6426637bea129ddfbf9": "b_1 = 9", "18f2d60a0f21cd6529721272e79c4f00": "\\frac {de} {dt}=0", "18f3017ea924138f657aff2cb1bffbe9": "\\sqrt{\\varphi}", "18f39b8f8bda0dfedbc8fade1db56137": "\\Gamma(n+1)=n!", "18f3b10a7a29947a625031e81b8df176": "\\eta\\colon X \\to G(X),", "18f3c8f2cb09a2eaf55c0b0ac9ae2ada": " m: G\\times G\\to G, (g,h)\\mapsto gh, \\quad i:G\\to G, g\\mapsto g^{-1}, ", "18f3cdd8b0f4cc5e1f8d5439ce61eb2f": "\\operatorname{csch}(z)", "18f411624e9ce3015b527bae9d546a29": "\\scriptstyle \\lfloor x \\rfloor", "18f41580a86278dc12dfd80b1c9b73fa": "\\vert S_j \\vert > t", "18f454cfb034d7a0d0586a1045fc556c": "\\tfrac{1}{p^2}", "18f4550b6bb63d16fad13c06c1db98ef": "\\tfrac{{{f}_{T}}}{10}", "18f46c53d1a1c94e144b56a59b6bbb33": " \\forall^p L := \\left\\{ x \\in \\{0,1\\}^* \\ \\left| \\ \\left( \\forall w \\in \\{0,1\\}^{\\leq p(|x|)} \\right) \\langle x,w \\rangle \\in L \\right. \\right\\} ", "18f5053e0cb6f1a3411f89ae082f21f4": " \\Delta u = O(k)+O(h^2). \\, ", "18f50e3141228f645297a63cc81b571f": " s - \\frac{r^2 - R^2}{s}", "18f584b183ed0444e6daaf778bdfc47f": "N_K(x) = \\left \\{ p \\in V\\;:\\;\\forall x^* \\in K, \\langle p, x - x^* \\rangle \\geq 0 \\right \\}.", "18f59f9764a34a4d7435285ceaf8113f": "(z - \\mu)' ( Az - A\\mu)= \\operatorname{trace}\\left[ {(z - \\mu )'(Az - A\\mu )} \\right] = \\operatorname{trace} \\left[(z - \\mu )'A(z - \\mu ) \\right]", "18f5ade42ad7739d82ef2849344d9c9f": "M_{31}", "18f5e3c31625d4ca5e7af7c0e6c5b43d": "\\lambda\\in\\mathbb R", "18f603cdab6dcb7cdc475029a155b7c6": "\\alpha_i > 0", "18f63800376271ee4b0efe1545744cd6": "f\\,", "18f660f590c9ca42e886d51788fb0467": "\\|f\\|_\\infty=\\|f\\|_{\\infty,S}=\\sup\\left\\{\\,\\left|f(x)\\right|:x\\in S\\,\\right\\}.", "18f6b37b70921418ec756ccf08e2c2db": "e_{f}[n]", "18f6bcb2a23df35bfd6d7b21564b7e65": "10 \\log_{10} r.", "18f6c67ffdaa6186ffcf50b9cddc072f": " G = ", "18f6d5c060dd41efcc31e6d834d6650b": "\\sum_{n=1}^\\infty \\frac{1}{n^p},", "18f6f55e3f5d7e22553835ad93c7c23a": "d = \\begin{pmatrix} 1 \\\\ 2/(1-4) \\\\ 2/(1-16) \\\\ \\vdots \\\\ 2 / (1-[N-2]^2) \\\\ 1 / (1-N^2) \\end{pmatrix}.", "18f71314ac1dc4e2f314e82bcceac49a": "y_1 = f_1(x_1, x_2, \\ldots, x_n)\\,,\\quad y_2 = f_2(x_1, x_2, \\ldots, x_n)\\,,\\ldots, y_m = f_m(x_1, x_2, \\cdots x_n) ", "18f71d2215a6b4a7ec4352c8ebfbdb13": "F = M \\otimes_A A_f = M_f \\qquad G = M \\otimes_A \\hat{A}.", "18f765e1efc6dff1c9fa8864607252bd": "\\langle \\mathcal{M}_{\\rm Tot}^2 \\rangle = \\langle \\mathcal{M}_{\\rm Tot} (t=0) \\mathcal{M}_{\\rm Tot}(t=0) \\rangle", "18f76f7414cba8374e6f6ebcbb5a5009": " \\mathbb{Z}/12\\mathbb{Z}", "18f771d524a5919810aaab8a58eb2e61": " \\mathbf{h}\\cdot\\mathbf{\\hat y} ", "18f78f003aa49ff8df3fefef866a7d90": " \\boldsymbol{y}\\in \\mathbb{R}^{k}, \\nu>0, \\lambda_{j}>0, \\mu_{j}>0", "18f799f827537f60dd46a8d256ba4282": "i =1,\\ldots,m", "18f7d20ecf76be3a747691b7b3e1c8e2": " F_{e}-F_{w}\\,=0", "18f83fb5ad488b078566b61892a9ff40": "n - 2", "18f89f56a678d37acced735a2d961dae": "k'_x = 0.332{D_{AB} \\over x} Re^{1/2}_x Sc^{1/3}", "18f8a08cd35e5fed6565afbfaa1582b0": " L_v ", "18f92f56f72a7d8ce15437db8d79ecc7": "\\theta > \\phi\\,", "18f9af01396d2d5def8b17a47882862e": "\\begin{pmatrix} z & z \\\\ z & z \\end{pmatrix} \\begin{pmatrix} z & -z \\\\ -z & z \\end{pmatrix}\n\\equiv z^2 (1 + j )(1 - j)\n\\equiv z^2 (1 + \\varepsilon )(1 - \\varepsilon) = 0.", "18f9d0bcfb04da11e6954ec61ad78567": "V_{\\sigma, \\sigma'}", "18f9d28289326992546aa45cdf816044": "k = 1/(4\\pi\\epsilon_0)", "18f9e1145b32ce66c8dfe53ba399373f": " \\scriptstyle\\phi ", "18fa9ab0c7793958866516b87ce04e30": "C_S (x) = C_O \\left ( 1 - (1 - k_O) e^{- \\frac{k_O x}{L} } \\right )", "18fafbf8831652b47769858cd64a3235": "\\sum_k\\lambda_k=1", "18fbae7a8d395ca90ea7a0da7fd7333a": "\\pi(.)", "18fbf71efae89c1195cab30f475216d3": "= \\frac{1}{2}\\frac{m}{3}\\dot{x}^2 + \\frac{1}{2}M v^2 - \\frac{1}{2} k x^2 - \\frac{m g x}{2} - M g x", "18fc1da9ea816f51ddb2c56aee3a2430": " r \\in \\mathbb{F}_d ", "18fc2dc5ac82ba13872c570add73a685": "\\displaystyle{P(z) =\\int_0^{2\\pi} {1 + e^{-i\\theta}z\\over 1 -e^{-i\\theta}z} \\, d\\mu(\\theta)}", "18fc7b2c943c858084c4b9086ba49eae": "a_0 = 1\\qquad b_0 = \\frac{1}{\\sqrt{2}}\\qquad t_0 = \\frac{1}{4}\\qquad p_0 = 1.\\!", "18fc9dc6ec01c64f80ea7a7614b5abce": "f\\left(a_0,\\dots, a_{n-1}, s\\right) = \\begin{cases}0 & \\mathrm{if}\\;\\forall i < n \\; \\left(\\beta(s,i) = a_i\\right) \\\\ 1 & \\mathrm{if}\\;\\exists i < n \\; \\left( \\beta(s,i) \\neq a_i \\right)\\end{cases}", "18fd07fb3a62d261e6f788d6ade29d67": "z_2=-1-i.", "18fd0c6be015c9cbf90e423096e916df": " \\mathbf{m}_\\mathrm{i} = \\mathbf{r}_\\mathrm{i} m_i \\,\\!", "18fd2381c30e11080ca90314b63fcc28": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^2 = \\frac{1}{1^2} + \\frac{1}{3^2} + \\frac{1}{5^2} + \\frac{1}{7^2} + \\cdots = \\frac{\\pi^2}{8}\\!", "18fd2cb43fda97771f3604b92b2c8936": "x = e^{-x} ", "18fd607770aa8994093d2ffa91c25fbe": "E_{A,B}={P_{B,1} + P_{B,2} \\over Q_{A,1} + Q_{A,2}}\\times{\\Delta Q_A \\over \\Delta P_B}=\\frac{\\partial Q_{A}}{\\partial P_{B}}\\frac{P_{B}}{Q_{A}}", "18fde71ee19dfc9d4d0e4681878dd0f7": " \\left | {\\epsilon_{n+1}}\\right | \\le M{{\\epsilon}^2}_n \\, ", "18fec27da3bf73cbd2107ecc9dad3d58": "\\tan\\left(\\sum_i \\theta_i\\right) = \\frac{e_1 - e_3 + e_5 -\\cdots}{e_0 - e_2 + e_4 - \\cdots}.\\! ", "18ff0a26db77107bf4ace612e8a25720": "\nX_{(a,b)} = \\frac{1}{n}\\sum_{k=1}^n U_k(a,b).\n", "18ff9a004f917b2dff3623c794c7ab72": "\\Theta(\\vert\\mathbb{D}\\vert L_{ave}+\\vert\\mathbb{C}\\vert\\vert V\\vert)", "18ffa6261de310d729460a3639fcc11e": "H= -J \\sum_{[i,j]} S_i\\cdot S_j", "18ffb35f999ba71c0ea4aa80195f4e22": "k_1 \\approx k_2", "18ffd649733cd86b1089fd6a9ea73130": "\\tfrac{5}{7}", "18ffdc3c2fd1505262af22b3c6a51e57": "x_\\text{int} \\cdot 2^{-23} - 127 \\approx \\log_2(x).", "18ffdcaccdab92703b6be9a8a3310dd9": "\\epsilon_\\sigma", "1900416e47e659c501fb769cdd9c2f5e": "XC \\to YC", "19006a8c50b48247f249cc95f6485e54": "(X, \\mathcal{B}, \\mu)", "19006e5ce93e8986960dc1601ac2a6c0": "| \\phi_{2^i} \\rangle", "19009c9a0ad718ad261bdb70830ebea3": "\\frac{2}{\\Gamma(\\frac{\\nu}{2})}\n\\left(\\frac{-t}{2i}\\right)^{\\!\\!\\frac{\\nu}{4}}\nK_{\\frac{\\nu}{2}}\\!\\left(\\sqrt{-2t}\\right)", "1900be3c950f53113041adf03a6b3b78": "1 = (x-1) - (x-1)^2 + (x-1)^3 - ... + 1 - (x-1) + (x-1)^2 - (x-1)^3 + ...", "1900bfda4a81b6764d6cecd4b02b9103": " \\nabla \\phi \\, ", "190111f541fe3addbf545bf234b023a3": "\n\\frac{\\partial^2 \\Gamma (s,x) }{\\partial s^2} = \\ln^2 x \\Gamma (s,x) + 2 x[\\ln x\\,T(3,s,x) + T(4,s,x) ]\n", "19015826ed7623ddd69314d3f9563baa": "u''/u= v^2 -v'=-S -Rv=-S +Ru'/u\\!", "19017ad3c1a35ddb266fda1fd64947fa": " T_{0}", "19019a087ef2f110932714b92c306688": "d = \\frac{2}{\\omega_c Z_0 C}", "190242186166c1bed8070cf90c11b750": "e^+ e^- \\rightarrow e^+ e^-", "19026037427b4adacd4e88fedaebdb29": "x, y \\in \\textbf{Q}", "190294964ca87a17b674ebe55a090193": "\\pi_\\Sigma(w)\\equiv \\pi_\\Sigma(v)", "1902d2e3dd3dddfac28e1b73765b11b2": "\\mathcal{S}\\left[\\varphi_i, \\frac{\\partial \\varphi_i} {\\partial s}\\right] = \\int{ \\mathcal{L} \\left[\\varphi_i [s], \\frac{\\partial \\varphi_i [s]}{\\partial s^\\alpha}, s^\\alpha\\right] \\, \\mathrm{d}^n s }", "19034fe86ae80ca2a7e403e347cfb7c6": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\ \\left( \\, \\frac{\\partial L}{\\partial \\dot{x}_i} \\, \\right) \\ = \\ m \\, \\ddot{x}_i ", "1903765c0596b8a9a589605ba81dc1b4": "M_Q(\\theta) < \\infty:", "1903964bf699b86f48b07c09420f7298": "\\lambda_1\\,", "1903c9a1dc788d43e8fc2f58d0a06f77": "H^{k+1}_n=\\frac{H^k_1+\\cdots+H^k_n}{n}", "1903fcdf961f87903f9c3a22d491c83e": "x(a + b) + c", "19040849468e88a54777d1889abc6a22": "ax^5+bx^4+cx^3+dx^2+ex+f=0.\\,", "1904c6bbcfcde0fd32f84cf84f23c7ae": "M_\\Sigma > M_N \\,\\!", "1904d27aca1c555d389d975078847239": "A_{ij} = A_{ji}", "1904e29e549219c7d14cf955974e8ea8": "\\pm \\sqrt{\\frac {1} {5}}", "1904ea2370b07a7d979157c028604f2d": "\\frac{\\partial n(x,t)}{\\partial t}=\\frac{1}{2}\\int^x_0K(x-y,y)n(x-y,t)n(y,t)\\,dy - \\int^\\infty_0K(x,y)n(x,t)n(y,t)\\,dy.", "1904ea7d67b03a8d978e8506424b0fc0": "\n J_1 := \\int_{\\Gamma} \\left(W n_1 - n_j\\sigma_{jk}~\\cfrac{\\partial u_k}{\\partial x_1}\\right) d\\Gamma\n ", "19051409c58bc32d1473bb47c5552740": "V(\\phi) = -10|\\phi|^2 + |\\phi|^4 \\,", "1905649b6ea7885554c4419ba187f092": "\\operatorname{Tot}(K)^n =\\bigoplus\\nolimits_{i-j=n} K^{i,j}", "190566692b1effdb3f849a990c73b854": "Au_{xx} + 2Bu_{xy} + Cu_{yy} + \\cdots \\mbox{(lower order terms)} = 0,", "1905ce7705139b9b165d23a6a3dd4e05": " \\mathbf{b} \\succ \\mathbf{a} ", "1905fc641acdb9bea8d26c943679261d": "b_{t+1}=\\eta", "1906336c3f4e493cebc6eb9e18e6a89a": "\\frac{2(2n+5)}{9n (n-1)}", "190651a250b815e68dd858227c1fbc0e": "\\underline{P}(Cl_3^{\\geq}) = \\{x_7,x_9,x_{10}\\} = Cl_3^{\\geq}", "190659faa50c094c619f88465cbd31c4": "Q_\\text{cmb}", "19065fa8b53dac2165c51708f2507fce": " G(\\C) ", "19068b9ba15ee939ab3a1c69306fe055": "[A,A] = 0", "1906aaaaa3e678ee299235fe2d6c9804": "\\scriptstyle\\leq10^{-9}", "1906edafcbaf52ab3d49e826d207d413": " \\frac{120}{90}=\\frac{12}{9}=\\frac{4}{3} \\,.", "19070998a9b4872fdcaf8c07ec04e5ab": "g=h.", "19070d9f38190632818d17aee69b2ec3": "\\ (1/r) ", "19073ae57c6ae431a149f51f528f2004": "\\exists f \\exists g \\forall x_1 \\forall x_2\\phi (x_1,x_2,f(x_1),g(x_2))", "19078e0d21cf083774b72f8bed41493f": "P \\rightarrow \\infty", "1907a6c841851b3db4cff1d67a62cc96": "(E(t)+E(t-\\tau))^2", "1907b3d79cadae2f7bada01c2d62b4f6": "-\\frac{\\eta(n_\\eta(\\xi_1)-n_\\eta(\\xi_2))}{\\xi_1-\\xi_2}", "190822760036f293ff506223b47df974": "s=x+iy", "19088f512b8bcc8ba258062ab4c63300": "$1.4339 \\times \\frac {(1 + 5%)} {(1 + 7%)} = $1.4071", "190919bf666ac59775ab7bf011c5cb98": " = \\frac{15!}{2} \\cdot \\left( \\frac{4!}{2} \\right)^{14} \\cdot 4 \\cdot \\frac{64!}{2} \\cdot 3^{63} \\cdot \\frac{96! \\cdot 2}{2 \\cdot (4!)^{24}} \\cdot \\frac{2^{95} \\cdot 64! \\cdot 2}{2 \\cdot (8!)^8}", "190919d2a19f2a0efff59a4f54a97ee7": "\\mathrm{d}U_{cv} = \\mathrm{\\delta}Q + \\mathrm{d}U_{in} + \\mathrm{d}(p_{in}V_{in}) - \\mathrm{d}U_{out} - \\mathrm{d}(p_{out}V_{out}) - \\mathrm{\\delta}W_{shaft}", "190934e3960941c01c649e915a7fc345": " C(s) = \\frac{rT}{1-e^{-rT}}=\\frac{s}{1-e^{-s}} ", "1909510a27c3009912a5340daa6cc051": "\\zeta\\Bigl(\\tfrac{1}{2}+it\\Bigr);", "19095d5a3ebdf83a709a60458674391b": "Y_{m_n} = \\frac {Z_{m_n}} {k^2 + Z^2 - Z_{m_n}^2}", "1909a078d1c60ee7523a86ecb8342601": "L_{b}", "1909a3f64512211ef691d7b75956252b": "\\sin^2 \\theta + \\cos^2 \\theta = 1\\!", "1909d419aa966ab9ce8853170d3cf71c": "s \\cdot y = 1", "190a24aeaa26edd193cde20ece6f649d": "\\operatorname{E}[L] = \\operatorname{E}[v^{K(x)+1}] - P\\operatorname{E}[\\ddot{a}_{\\overline{K(x)+1|}}]", "190a429490bdd22373bd5ad2cf1b3cf8": "\\cot \\theta = \\frac{i(e^{i\\theta} + e^{-i\\theta})}{e^{i\\theta} - e^{-i\\theta}} \\,", "190a6ae9bde93da07082c39175a31258": "\\scriptstyle P_T", "190a828a81b325eb6cf942ba638cd631": "T: x \\mapsto 1-x", "190a889bd2b8d23c6c72e0e02980111b": "\\simeq \\begin{cases} \\mathbf{R} & \\mbox{if } k = 0,n-1 \\\\ 0 & \\mbox{if } k \\ne 0,n-1 \\end{cases}", "190b448fa5759f21444a5c5cc9146475": "\\displaystyle \\|J^{-1}_{n} - J^{-1}_{n-1}\\|_{F}", "190b622c95192e0c61283b638e832c14": "SL^{\\pm} < GL", "190c06e51be8f8f9b5429a1049c9e5bd": "\n \\left.\\frac{\\partial}{\\partial \\theta}\\right|_{\\theta=0} f(T_{\\theta} \\boldsymbol{x}) = 0 .\n", "190c7c7eccc024910a66603cb74baee7": "a_0^2", "190c8daea9e4dd0245a8d06853261744": "B=\\{f_1,f_2,\\ldots,f_r\\}", "190cafafc7d6824de3a68f053ace1bbe": "\\pi(p) = P(x=1\\mid p)", "190cb555529e76b1dcd46d99a540738a": "\nI_{lm} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{q}{\\left( r^{\\prime} \\right)^{l+1}} \n\\sqrt{\\frac{4\\pi }{2l+1}} \nY_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "190cd091f3f280ff53be552974361331": "\\nu_{it} \\sim \\text{i.i.d.} N(0, \\sigma^2_{\\nu}),", "190cec97991384067a7e054fa870844f": "\\bold{uv}=(u_1v_1,u_2v_2,\\ldots,u_nv_n) \\, ", "190d0b7374f3c6fedf407f2ca7fe35ff": "Z_1 X_2 Y_3 = \\begin{bmatrix}\n c_1 c_3 - s_1 s_2 s_3 & - c_2 s_1 & c_1 s_3 + c_3 s_1 s_2 \\\\\n c_3 s_1 + c_1 s_2 s_3 & c_1 c_2 & s_1 s_3 - c_1 c_3 s_2 \\\\\n - c_2 s_3 & s_2 & c_2 c_3 \n\\end{bmatrix}", "190dc11ab078d0d8c4833a2c5726e63b": "R^+ \\times R^+ \\times S^1 .", "190de919089802132dec0d2ee7657388": "p(x)=p_n\\cdot(x-z^*_1)\\cdot(x-z^*_2)\\cdots(x-z^*_n).", "190e0255aa85683a3bbf33703b8155ea": "\n6.1 \\max(0, x - 13)\n- 3.1 \\max(0, 13 - x)\n", "190e26a6d0f1023ca4a7eac341c1d025": "C_i\\ ", "190e6827fb65b13e18f40e53b484dfdf": "(q,p)\\in\\mathbb R^2", "190e98cf0a6ba7a8b6ed2eafbfb4e8ea": "\\geq 1", "190e99ea84e55217bca401e70b3dc9df": "V_{f1}", "190ee7b4cdc41be86ec22f4e9f1c1861": "x \\vee y = x = y \\vee x", "190f44e1781f224c6f9b36c812ed59c7": " v\\left( L \\right) ", "190f4f2b1ac99dd2775aa1f52a4d3c10": "H_{o'} = gH_og^{-1} \\qquad \\qquad (1)", "190f56a49bfc0a59ce15d58d99ff2027": "X_1, X_2, \\dots, X_p", "190f7b0218407ea4e0f4d1f4d25dcf34": "\ng_{ij} = -\\left ( \\frac{\\partial E_i}{\\partial T_j} \\right )^D\n = \\left ( \\frac{\\partial S_j}{\\partial D_i} \\right )^T\n", "190fcd20efd2b1e077b55aea4304f54f": "(\\overline{A} \\vee \\overline{C}) \\wedge (\\overline{A} \\wedge \\overline{B}) \\wedge (C \\vee A \\vee B))", "190fd64951849ec9e89daf00d496902f": "{\\bold \\ Re}", "19101cff99bee6d94718a3fff84cd621": "N_i =\\int N(t)\\Phi_i(t)dt", "191040d6e194e1b3ab614e0b5bc285d8": "F=\\frac{q_1q_2}{4\\pi\\epsilon_0 r^2}", "19117a1561403de529987a77fea26ef8": "|x|_{\\ast} = |x|^{c} \\text{ for all } x \\in \\mathbf{K}.", "1911893ceb4f5a3fd60ca729ef8c2025": " r:2^E \\rightarrow \\mathbb{Z} ", "19119f28a57fdbd8689c899016d0c1c2": "\\mathcal{F} \\left ( \\mathbf{x} \\right )", "1911d79ef42ee2446c30c8febc9a26c5": "\\left(\\!\\!{4\\choose18}\\!\\!\\right)={21\\choose18}=\\frac{21!}{18!\\,3!}={21\\choose3}=\\left(\\!\\!{19\\choose3}\\!\\!\\right),", "1911e5b18d68d95b6c5576f27ea435bb": "3x+2y=6\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;6x+4y=12", "1911fa78cf29c974650261bff84c5683": " P= A j^{\\star} = A \\varepsilon\\sigma T^{4}.", "1911fb5be175b85f127c81d54f71eca6": "\n\\nabla^{2} \\Phi =\n\\frac{1}{a^{2}} \\left( \\cosh \\tau - \\cos\\sigma \\right)^{2}\n\\left( \n\\frac{\\partial^{2} \\Phi}{\\partial \\sigma^{2}} + \n\\frac{\\partial^{2} \\Phi}{\\partial \\tau^{2}} \n\\right) + \n\\frac{\\partial^{2} \\Phi}{\\partial z^{2}} \n", "1912005d2adff7b55159a85960d6946e": "v_e=\\frac{Ze^2}{n\\hbar 4 \\pi \\epsilon_0} ", "19123f8c65a296836bdfd8788e7a90d9": "\\Phi_\\lambda(g)=(\\pi_\\lambda(g)\\xi_0,\\xi_0).", "19127a96a5a638cfb29f59b692132a41": "S^\\delta_R", "191290e640010866bc8f722797ccb077": "(\\forall u)(\\exists v)(P)\\psi", "1912a09ba11f76693c9f3541536ac4cd": "f_0,\\ldots,f_{n-1}", "1912deebcb89c190eff1799856d0d775": " f'(x)=\\dfrac{d}{dx}f(x)=k x^{k-1}\\;.", "19137c0198719551f249d6a7969d8b06": "\\partial \\mathbf{H}(x) / \\partial t", "19139aac6007de2dd6490497dd01364b": "\\leq d+1", "1913de0238e600441c71cc08694a554d": "(\\mathbf{x}\\times\\mathbf{u})\\cdot(\\mathbf{v}\\times\\mathbf{w})=(\\mathbf{x}\\cdot\\mathbf{v})(\\mathbf{u}\\cdot\\mathbf{w})-(\\mathbf{x}\\cdot\\mathbf{w})(\\mathbf{u}\\cdot\\mathbf{v}),", "1913fd7f2c210d415bba389750aaf504": "VTR=100%*Viewthrough/Impressions", "19141235f0e58be6e45c6ea9eb818c83": " \\ln x \\leq x-1 ", "191414631fb4e51a6e0db753bd90d0e0": "\\rho\\propto r^{-N}", "19151e526b4b8c899a2153220ab1801a": "\\mathbb{X}^{k}", "19154574bbb585eb04dba48fb7303031": " \\mathbf{F} = m\\mathbf{a} = m\\frac{d^2\\mathbf{r}}{dt^2}", "19158ba5333a4646373ecdb03242b974": "k \\in N", "1915a608beda1ef4c49d8c87a0c9d7d8": "\\tilde Y_t = Y_t - \\left[ Y,X \\right]_t", "1915bce0c67cb5da75005eefac4a74c4": "p = (x,0)", "1915dab52874df1248f3b7c5752e8f12": " H_1 C_2 P_n", "1915f6e440e2aa7e4c7249bd5a5c4e89": "\\mu = \\mu^{\\Theta} + RT \\ln \\left( \\frac{f}{bar} \\right) = \\mu^{\\Theta} + RT \\ln \\left( \\frac{p}{bar} \\right) + RT \\ln \\gamma ", "191656e27a7242ada897fb8f2b2a6b01": "c_i + (1 + \\sigma) v_i", "1916a6b578c64574ea211adc91863538": "\\scriptstyle\\phi(t) = t^2\\log\\frac{1}{t}\\log\\log\\log\\frac{1}{t}", "1916bf48ba02a7ec42e37657a545006a": "\\mbox{absolute margin of victory} = \\begin{cases}0; & w \\le \\frac{c}{2} \\\\ w - \\max\\{r, \\frac{c}{2}\\}; & w > \\frac{c}{2} \\end{cases}", "1916c232538093b85e75c7bbef2b9f1a": "E(Nl,t ) = 1 - e2-(aNl,t)", "1916e884f16ba7cd98984c74753f9d93": "y(x,t)\\, = y_0 \\cos(k x - \\omega t +\\varphi)", "1916edb7738d3b4298a20c3b8f9cd8f4": " \\frac{1}{42} ", "19170f32cbe777c9a9cf2aec0a7bca7e": "f^+:X^+= Proj(\\oplus_m f_*(\\mathcal O_X(mK)))\\to Y", "1917627925acb0bfc171301afe5ccaff": "\\liminf_{n\\to\\infty}\\frac{p_{n+1}-p_n}{\\log p_n}=0.", "191801332c4c8db223789478221abe61": "\\frac{\\partial |\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|}{\\partial \\mathbf{X}} =", "19183bff564aea419754540d1e8985e7": "\\frac{2^n}{n!} \\mathrm{vol}(\\mathbb{R}^n/\\Gamma) \\le \\lambda_1\\lambda_2\\cdots\\lambda_n \\mathrm{vol}(K)\\le 2^n \\mathrm{vol}(\\mathbb{R}^n/\\Gamma).", "19188b5d7016d022def74c01e8eab114": "p_{i}=x_{i}/N", "1918a9736f6b21d6b9a70d98a215d27a": "m_{L}=\\frac{m_{0}}{\\left(\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)^{3}},\\quad m_{T}=\\frac{m_{0}}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} ", "1919805b955ddc40ad3a8d6a4a702dc5": "{\\mathit{He}}_n'(x)=n{\\mathit{He}}_{n-1}(x),\\,\\!", "191a0ee133be76689375af2af2dadd8f": "\\Gamma^k_{ij}=(e_i \\cdot De_j) \\cdot e^k", "191a357f585c58425ce783a3329006e5": "C_n(x_1, x_2,\\ldots,x_n) := \\left\\langle \\phi(x_1) \\phi(x_2) \\ldots \\phi(x_n)\\right\\rangle\n=\\frac{\\int D \\phi \\; e^{-S[\\phi]}\\phi(x_1)\\ldots \\phi(x_n)}{\\int D \\phi \\; e^{-S[\\phi]}}\n", "191a92911a45010610972b5f343df714": "df=1", "191ad2ce69d3206c7675b8406b3c1009": "\\iiint_T f(x,y,z) \\ dx\\, dy\\, dz = \\iint_D \\int_{\\alpha (x,y)}^{\\beta (x,y)} f(x,y,z) \\, dz dx dy", "191ad5bc4dec318a1d9486075e1e0bd8": "\\tan(\\gamma+\\pi/2)=1/\\tan\\gamma", "191adff20517fd0a0a85a18812b5e379": "\\lambda d", "191ae49db56239410cd4378e1b545d62": "|\\eta|.", "191bfc8b6e4ae7102318121c0d4a170e": " \nL = \\int_S F_{\\rm rad} \\cdot dS = \\int_S \\frac{c}{\\kappa} \\nabla \\Phi \\cdot dS\\,. \n", "191c64bdae5c35fed2281891274cf4c8": "\\pi_2=\\pi_5=\\mathbb{Z}", "191c7ad983ddfff9ba73849ded89a9d9": "g_{1mm'}", "191cb8b106b45d57bea9f2bd41dda222": "{du}/{dt}=F(u(t))", "191d0544d85e6b3395250a9665f25ebe": "u \\equiv_L v", "191d670b6482c653d233c8d77011a2b3": " \\mathrm{FWHM} \\, = d_{\\mathrm{F}} \\mathrm{ln} (2) \\approx 0.693 \\, d_{\\mathrm{F}}. ..........(22) ", "191d9ce5e8e1141122738cb31a576cc2": "A = \\frac {1 + \\frac {1}{4} (k_1)^2}{1 - k_1}", "191e06be3e8e4dbd08734eaa8834d317": "\\mathbf{r} = \\frac{a}{1+e \\cos\\theta} \\mathbf{\\hat{r}} \\,\\!", "191e49aac76695b80d44c2ad107aad93": " Decrypt_\\varepsilon ", "191e94a2627524f3e4f21ffac612b31d": "\n\n\\begin{array}{lcl}\nminimize: V(\\vec w, \\vec \\xi) = {1 \\over 2} \\vec w \\cdot \\vec w + C_{onstant} \\sum{\\xi_{i,j,k}} \\\\\ns.t. \\\\ \\begin{array}{lcl}\n \\forall \\xi_{i,j,k} \\geqq 0\\\\\n \\forall (c_i, c_j)\\in r_k^*\\\\\n \\vec w (\\Phi(q_1,c_i)-\\Phi(q_1, c_j)) \\geqq 1- \\xi_{i,j,1};\\\\\n ...\\\\\n \\vec w (\\Phi(q_n, c_i)-\\Phi(q_n, c_j)) \\geqq 1-\\xi_{i,j,n};\\\\\n\nwhere\\ k \\in \\left \\{ 1,2,...n \\right \\},\\ i,j \\in \\left \\{ 1,2,... \\right \\}.\\\\\n\n \n \n \\end{array}\n\\end{array}\n\n", "191ead2c11f43c73e7f3970b0b1d201a": "R > r + a", "191f030411bd38b4955c373b2820842b": " \\int_{V}\\boldsymbol{\\epsilon}^T \\boldsymbol{\\sigma}^* dV ", "191f1a8df597bd192c07369a641e91a3": "(A,\\sigma)", "191f7020f5bf3a87b2024dcdd6b90c22": "\nM_{0} \\equiv \\int d\\zeta \\ \\frac{\\lambda(\\zeta)}{\\zeta}\n", "191f7940711bd6730aefc9a3e3ef9fb6": "\\varepsilon_Y = \\mathrm{id}_{TY}.", "192040a481a02c91f4175743c4c1553b": "p=a-b(x+y)", "1920cd19c04008439b17cc72788dcfb0": "\\bold U(x_0,y_0) = \\frac{1}{j\\lambda}\\int\\!\\int \\bold U(x_1,y_1) \\frac{e^{jkr_{01}}}{r_{01}}\\cos\\theta dx_1 dy_1", "19211978ba7e1567773c82d4c6bedcd8": "S(-1)^{\\oplus n+1} \\to S, e_i \\mapsto x_i", "1921660d6bbf2dc4ecd5c78858e3be04": "\\operatorname{E}(\\tilde{Y_k}- \\widehat{Y_k})=0 . ", "192175bb9483bf2588cd1ea6eea6bc13": " \\cos B = -\\cos C \\, \\cos A + \\sin C \\, \\sin A \\, \\cos b ,", "19218cf103615515c67b3c4e989541f3": "\n\\max w\\left( {T_{ij} } \\right) = \\frac{{T!}}\n{{\\prod_{ij} {Tij!} }} \n", "1921bcba2403434dd77df374f622ea16": "\\alpha_k", "1921d0566ae938d2ffd6a9d062956775": "\\{h_n\\}_{n\\ge 0}", "1921fa2bfe99382ee4140f2d7caf0491": "\\alpha_i\\leftrightarrow\\beta_i", "192201d9ff83fdf71c35bf91e9de4f54": "\\dot{\\mathbf{x}} = \\varphi( \\mathbf{x}, t )", "19222ddb0696c213078a8cb0948c63b7": "(i_\\alpha t)^{i_1\\dots i_{r-1}}=r\\sum_{j=0}^n\\alpha_j t^{ji_1\\dots i_{r-1}}.", "19222fec8f1b6be85309fe2382449484": " V \\otimes W ", "19223e2e77817e86e125e9328b46017a": " m l \\ddot{\\theta} = - m g \\sin \\theta - k l \\dot{\\theta} ", "192250ef22b00eabb5ff7b832d5236de": "B^{(n)}", "192264f8d32bb758e69f9279d91e4afc": "F(t')\\!", "19228d0eb1ab46c3dcfd608496f2ea2d": "\\frac{D^2}{dt^2}J(t)+R(J(t),\\dot\\gamma(t))\\dot\\gamma(t)=0,", "192292e35fbe73f6d2b8d96bd1b6697d": "lm", "1922b8e22bae729ffa04ea46d29ac5b8": "L^1", "1922c26b309cf406ae462f001364c213": "L(t_1',\\ldots,t_n') :- B", "1922cd45d2eba9f6c9b03187b517e51c": "\\sigma^2_K=\\sigma^2_{\\beta_{12}}+\\sigma^2_{\\beta_{13}}-2 \\sigma_{\\beta_{12}} \\sigma_{\\beta_{13}}\\rho_{12,13}\\,", "1923029fbd6f645959ca421259dd04d6": "E_{2n}\\left(\\frac{p}{q}\\right) =\n(-1)^n \\frac{4(2n)!}{(2\\pi q)^{2n+1}}\n\\sum_{k=1}^q \\zeta\\left(2n+1,\\frac{2k-1}{2q}\\right)\n\\sin \\frac{(2k-1)\\pi p}{q}", "19233bae16418436417653d8c79a8df2": "F(x_1 + \\Delta x) - F(x_1) = \\int_a^{x_1 + \\Delta x} f(t) \\,dt - \\int_a^{x_1} f(t) \\,dt. \\qquad (1)", "192340153c5bcd6aa2a6d178e3bf2492": " G=SO(3)/U(1) ", "1923533c88582e19945e82caa101e43d": "\\hat{a}_i^{(\\eta)} = \\frac{x_{i+}}{\\sum_j \\hat{b}_j^{(\\eta-1)}},", "19235f840cf32d786c6c2994e209068f": "\\overset{\\cdot }{x}=f(x)+B(x)u+h(x)", "19236c254f82397d4ae5c7cbce41b945": " a_1 = - \\sum_{n=2}^\\infty a_n. ", "19238f4bc02e570e0307d962e220ed03": "H = \\frac{\\dot{\\theta}^2}{2} - \\frac{g}{l}\\cos\\theta.", "19243a0646263bd27c594dca33db048b": " yxyx \\rightarrow x^2y^2 ", "19246823504bc4b4c31674c0fa4b02d6": "R_0 = 0~ \\mbox{since}~0.0078 \\ >= 0.02 ", "192481a8c43bb58d917346eff39e3ad3": "\\scriptstyle\\frac{\\pi}{2}", "19249588a3dcaa56d833265b84881d9e": "\\frac{1}{\\ln(p)} = \\frac{1}{p-1}+\\int_0^{\\infty}\\frac{1}{(x+p)(\\ln^2(x)+\\pi^2)} dx \\qquad \\qquad \\forall p > 1", "1924b6c9f38d8780fa413e43d85312db": "D=R\\sqrt{\\theta^2_1\\;\\boldsymbol{+}\\;\\theta^2_2\\;\\mathbf{-}\\;2\\theta_1\\theta_2\\cos(\\Delta\\lambda)};{\\color{white}\\frac{\\big|}{.}}\\,\\!", "1924d321b3befac1eb6225d40c28a06c": " V_{\\text{out}} = -( V_1 + V_2 + \\cdots + V_n ) \\!\\ ", "19250d87521b8d914daa572fce49e8ba": "\\tilde \\phi = \\phi-\\phi_0", "19253dfa5cc89774dd1d17fc77625115": " \\langle\\mu\\nu|\\lambda\\sigma\\rangle = \\iint \\mathbf{\\chi}_\\mu^A (1) \\mathbf{\\chi}_\\nu^B (1) \\frac{1}{r_{12}} \\mathbf{\\chi}_\\lambda^C (2) \\mathbf{\\chi}_\\sigma^D (2) d\\tau_1\\,d\\tau_2 \\ ", "19255638ed5bbbfc2dd2f00ac6437b4a": "\\left(\\frac{\\partial U}{\\partial \\theta}\\right)-\\frac{d}{dz}\\left(\\frac{\\partial U}{\\partial\\left(\\frac{d\\theta}{dz}\\right)}\\right)=0", "19259731a0d9038d4878b99f240120b6": "e_1, \\ldots, e_k", "1925a6ecea62e67108e05837ad42357c": " T_{c} = \\beta^{-1}", "1925b1bdcaf9d8e96499b51263835338": " \\left \\| A \\right \\| _p = \\max \\limits _{x \\ne 0} \\frac{\\left \\| A x\\right \\| _p}{\\left \\| x\\right \\| _p}. ", "1926312deeeff81b8a4fb89ee30f48e5": "1=1^n=(p+q)^n=\\sum_{k=0}^n {n \\choose k} p^k q^{n-k}.", "19263a9a067a890e402d5cd545860a1a": "V^\\infty", "1926d31e091d7dc30b2c9a3702537101": "U_{\\texttt{name}}\\,", "19272339c815dc47eef0ffe51ab54812": "~|n,\\pm\\rangle~", "19275a041eee84dfb0180bb6eddef687": "\\lambda^nf(a, b, c) + \\mu\\lambda^{n-1}\\Delta_P f(a, b, c) + \\frac{1}{2}\\mu^2\\lambda^{n-2}\\Delta_P^2 f(a, b, c)+\\dots .", "192780821efae6755e9477fdd4a39223": "f:a\\to b", "1927858e73c05099c08b94db09efef41": "P_i = A_i \\oplus B_i", "19278974a2304842ecd802e7cf16bb3b": " |\\psi^T\\rang ", "1928392ac0b4a77dd9a968dc6c69cf32": "\\alpha_R = \\frac{138}{D}", "1928c3c876fb36e5e7c18d9a0e1fb117": " k_\\mathrm{B} = 1 \\ ", "1928ccf03ae5338f733965d62b8770c5": "{d\\theta\\over dt} = \\sqrt{{2g\\over \\ell}\\left(\\cos\\theta-\\cos\\theta_0\\right)}", "1928dc58254957913034062e2de8eb3d": "{n!\\over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\\,du+O(du^2),", "1928efa279cbaedaac2e470d954032c6": "\\frac{\\partial^\\ell f_i}{\\partial x_{i_1}^{\\ell_1}\\partial x_{i_2}^{\\ell_2}\\cdots\\partial x_{i_n}^{\\ell_n}}", "1928f2d87af684d81f758680332b2a40": " d[x,x^* ]=\\max_a |u(a)-u^*(a)| \\,", "1929052caef667d98743df8761d507b9": "f_i(0,0,\\dots,0) = 0", "19298983a27eeab314d1beea9b7d3dde": " \\left[(\\mathbf{AB})^\\dagger\\right]_{ij} = \\left[\\left(\\mathbf{AB}\\right)^\\star\\right]_{ji} = \\sum_k \\left(\\mathbf{A}^\\star\\right)_{jk}\\left(\\mathbf{B}^\\star\\right)_{ki} = \\sum_k \\left(\\mathbf{A}^\\dagger\\right)_{kj}\\left(\\mathbf{B}^\\dagger\\right)_{ik} = \\sum_k \\left(\\mathbf{B}^\\dagger\\right)_{ik}\\left(\\mathbf{A}^\\dagger\\right)_{kj} = \\left[\\left(\\mathbf{A}^\\dagger\\right) \\left(\\mathbf{B}^\\dagger\\right)\\right]_{ij} ", "19298c2af784855a55e7c066f70bf605": " a^2 + b^2 = a^2 - (ib)^2 = (a - ib)(a + ib) ", "19299660df53a56963d814791baf42ed": "F = G \\frac{m_1 m_2}{r^2} ", "192a40b891d9c20925aeb2261dbdea4f": "S_0(P,Q)=\\cdots=S_{d-1}(P,Q) =0.", "192a427f362d1037d7df4509745a8470": "\n\\int D \\rho \\; \\delta \\left[ \\rho - \\hat{\\rho} \\right] F\n\\left[ \\rho \\right] = F \\left[ \\hat{\\rho} \\right], \\qquad (4)\n", "192a762853fa5f186f124bf445aca1ad": "H(A):=-Tr\\rho_A\\log\\rho_A", "192a7808925538a267466b5c6916f851": "y(t) = A \\sin(kx - \\omega t)", "192a842434fcee9247b04972d6c6858c": "\n\\mathcal{L}_\\mathrm{EW} =\n\\sum_\\psi\\bar\\psi\\gamma^\\mu\n\\left(i\\partial_\\mu-g^\\prime{1\\over2}Y_\\mathrm{W}B_\\mu-g{1\\over2}\\boldsymbol{\\tau}\\mathbf{W}_\\mu\\right)\\psi", "192a8ca6c41cfdd77c52d52ab92e3641": "\\sum_{j=0}^p c_{j,k}z^j = [\\operatorname{Res}_{z=\\lambda_k} f(z)] \\sum_{j=0}^p \\frac{1}{\\lambda_k^{j+1}} z^j", "192ad965c4815ef67c86ed1460766ddb": " \\min_{i=1...n}(\\frac{S_i^T}{S_i^0}). ", "192af400fa1b0725c03fbac9e698abdb": "|-\\rangle = \\tfrac{1}{\\sqrt{2}} \\left(1,-1\\right)", "192bbaede1fdef1a7b0145d11040ad4a": "\\begin{matrix}\n \\underbrace{\\begin{matrix}\n \\left[ \\begin{matrix}\n \\frac{\\partial f}{\\partial x_{1}} &\n \\frac{\\partial f}{\\partial x_{2}} &\n . . . &\n \\frac{\\partial f}{\\partial x_{N}} \n\\end{matrix} \\right] \\\\\n {} \\\\\n\\end{matrix}}_{\\nabla f^T} & \\underbrace{\\begin{matrix}\n \\left[ \\begin{matrix}\n v_{x_{1}} \\\\\n v_{x_{2}} \\\\\n \\vdots \\\\\n v_{x_{N}} \\\\\n\\end{matrix} \\right] \\\\\n {} \\\\\n\\end{matrix}}_{v} & =\\,\\,0 \\\\\n\\end{matrix}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{is the same as writing}\\,\\,\\,\\,\\,\\,\\,\\,\\nabla f^T\\,\\,\\,\\centerdot \\,\\,\\,\\,v\\,\\,=\\,\\,\\,0.", "192bcd2ad930494b6485f1216526656a": "f(x_i) \\neq y_i", "192bf38ad826c8b799596aa95acb4195": "A\\frac{dp}{dt}=L_p p.", "192c1b4c97e7cf16034ba365cf8c55a7": "U(t_k+\\delta t)", "192c442ae99c13b6c7440ac3e550cad3": "\\frac{d}{dx} \\log_b(x) = \\frac{1}{x\\ln(b)}. ", "192c572fde9511263584e5b69e4af96e": "\\mathbf{X'}=\\mathbf{wX}, \\mathbf{y'}=\\mathbf{wy}.\\,", "192c5b4983f3b8987a93c4b25b4842ce": "\\mathbf{N} - 1", "192c6d15ee502fdbb42d32464b038da0": "y \\wedge x/y = 0 = x/y \\wedge y", "192c88685e3c0bc9cc206a7752ceffb6": "\\Omega = 2\\pi - 2n \\arctan\\left(\\frac {\\tan{\\pi\\over n}}{\\sqrt{1 + {r^2 \\over h^2}}} \\right) ", "192cb49bf467fb27b0f009d38960a20b": "H_0:\\ m(\\theta_0)=0", "192cdf9f9b1215e11fd8fc45c007e6c2": "| k \\rangle", "192d3b3e09258b9b77a60113b4ec1b73": "\\frac{dx}{dt}(t_n)=\\sum_{k=-q}^q \\frac{-i2\\pi k}{T} X_k e^{\\frac{-i2\\pi k t_n}{T}}, \\quad n=1,\\dots,N.\\,", "192d4f10a3ad648108d7b6dfc9660d26": " \\lambda \\sigma_i + (1-\\lambda) \\sigma'_i \\in r(\\sigma_{-i}) ", "192d5d177f953dbc5942e7947ff67dde": "\\pi F", "192d7d5c5ca46a19f63389a3dd8ee20d": "t=t^\\prime=0", "192d84954d9750750c14b37c8f52a887": "S : f \\rightarrow u", "192db13978a4a8572e8c06886ebfa144": "d_W", "192db2794b64f15ea81817ea9ff30487": "I_E = (\\beta + 1)I_B + (\\beta + 1)I_{CBO}\\,", "192e1f16c897d48f16b1c807064d482a": "\n\\mu _z \\,\\, = \\,\\,\\,{\\rm E}\\,[z]\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sigma _z^2 \\,\\,\\, = \\,\\,\\,{\\rm E}\\,\\left[ {\\left( {z\\,\\, - \\,\\,\\mu _z } \\right)^2 } \\right]", "192e3ceb138349886f6d2f79a846333f": "\\sum F_{\\|} = F_i - F_f - F_w \\sin \\theta = 0 \\,", "192e6e81db824c6662019be0ce399ea5": "\\left( \\frac {x}{\\lambda} - f \\ t \\right) = \\left( \\frac {x+\\Delta x}{\\lambda} - f (t + \\Delta t) \\right)\\ , ", "192ed0507b1fccca48b2aa716d2dc671": "\\textstyle |\\Omega\\rangle", "192eea1bb2e8cf134bcc35715e0c45e7": "\\Delta E = \\frac{\\beta^2}{1-\\beta^2}eE_0l_se^{z/l_s},", "192ef223ed254490b3223bd3f76f791d": "[x^{-1}, y] = [y, x]^{x^{-1}}.", "192efa85fde664088378d0d93bcbf5fb": "DPW = \\frac{\\displaystyle \\pi d^2}{4S} - 0.58^{*} \\frac{\\displaystyle \\pi d}{\\sqrt{S}}", "192f0c0fcb2a45e70b1f15fb29ac9a51": " F\\left(\\dot x, x, y, t\\right) = 0 ", "192f379152753d0f6e087e0eb792c051": "x^2-\\frac12", "192f50ec41d691af8f3c88111d47ae56": "\\phi=+1", "19309f2cc286af9ef849b241ca24622f": "y_k = \\frac{1}{\\sqrt{N}} \\sum_{j=0}^{N-1} x_j \\omega^{jk}.", "1930a0a422d84cad426361a5a881cd04": "\\Rightarrow \\left\\| A v_i \\right\\|^2 = \\lambda_i \\left\\| v_i \\right\\|^2", "1930dfe3b9d903d5f359c763a91870a5": "\\Delta E = E_1 - E_2 = \\left(y_1 + \\frac{q^2}{2g y_1^2}\\right) - \\left(y_2 + \\frac{q^2}{2g y_2^2}\\right) = \\frac{(y_2 - y_1)^3}{4y_1 y_2}", "19310380597f85e281e904af33e77de0": "\\iota: F \\longrightarrow E", "193117293445f89d422795f3e4fdc6b7": "\\textit{mother}: \\textit{animal} \\longrightarrow \\textit{animal}", "19315b8b6eb6ac0c2eb5b79ae702ed70": "\\{(x,y):|x|=|y|\\}", "193161db9089f52be260b4ece5c1cfe7": "\n J_1 := \\int_{-h}^h \\rho~dx_3 = 2~\\rho~h ~;~~\n J_3 := \\int_{-h}^h x_3^2~\\rho~dx_3 = \\frac{2}{3}~\\rho~h^3\n", "1931882a1554796a61e241aae439cce7": "\\mathbf{\\Lambda}(\\varphi_1 + \\varphi_2) = \\mathbf{\\Lambda}(\\varphi_1)\\mathbf{\\Lambda}(\\varphi_2)", "1931c7a7475c79456708bac13d9513e6": "~\\omega_2", "1931d9f57487f77009123860cc6dc57e": "\\sum_{i=0}^{n}\\lambda^{n-i}\\left[d(i)-\\sum_{l=0}^{p}w_{n}(l)x(i-l)\\right]x(i-k)= 0\\qquad k=0,1,\\cdots,p", "193203bdd11eec56406cb25632951832": "Z_{1,t}>Z_{2,t}", "1932680bf99159dc4c21366f34de8091": "x^2-d(y+1)^2= 1", "19329487fb29679f9be6282392a0d113": "SL_3(\\Z)", "1932e200fb070fed86e65d1af49c1d3c": "e^{-\\frac{\\Delta G_F}{k_BT}} = [V_\\mathrm{Mg}'']^{2}", "1932fe953f3822e6b0ca4ac8a5fff2e6": "\\epsilon(\\lambda_{ex})", "19338d854f9df8e21ee89b3484591c89": "\\varphi:V\\otimes V^*\\rightarrow \\varepsilon_H", "1933bc7f5f194c785408953539480ba1": "a^{N-1}_p \\equiv 7^{11350} \\equiv 1 \\pmod{11351}", "1933f6b65b56a3fbe1ffbae527e7ab61": " \\eta=\\frac{9}{2} \\frac{1}{(1-\\frac{a^{3}}{b^{3}}) \\mu} ", "1933fef2f568bd948d257717ac87df5b": "p_1^2 = m_1^2", "19340c3322f86f511a91926308c71a92": "(m + n) r = m r + n r", "19346b8e34b57b09a0bb4f4742e291fb": "s_i = S \\bmod \\ m_i", "1934bd15286c328fb04bd0ad2668a7d7": "(B \\oplus C)", "1934eebebfb7e103a4578459790df54c": "dl^2=r_{0}^2 (d \\theta^2 + \\sin^2 \\theta\\, d \\phi^2)", "193585327f6db8ff0db3628dc7987aee": "\\beta = \\frac{1}{k_B T} = \\frac{1}{\\tau}", "193609b18eb606ed7fde71157f573e0d": "\\int_X \\|f\\|_B\\, d\\mu < \\infty.", "193638893fb777e821eaa1d468ec3caa": "[T_A^1] \\longrightarrow^* T_B^2", "193651e2eb979ea5e5a24d8f84d0f5fe": "Z_\\mathrm{p} = sL_\\mathrm{p} + R_\\mathrm{p} + {1 \\over sC_\\mathrm{p}}", "1936c4ad5c8f3b4a913024eca19e5c2c": "\\mathfrak{e}_{7}(\\mathbf K)", "1936f0f8638c6b51a4687ce25546a6b8": "week(date) = \\left\\lfloor \\frac{ordinal(date) - weekday(date) + 10}{7} \\right\\rfloor", "19376b55f56f17c4cf6bf16b7a7e77b2": "1/28 + 1/14 + 1/7 + 1/4 + 1/2 + 1/1 = 2", "1937807a199f91da25898e2f21180d27": "\\begin{align}\n\\lambda_{12} &= \\omega_{12}\n- f\\sin\\alpha_0\n\\int_{\\sigma_1}^{\\sigma_2}\\frac\n{2-f}{1 + (1-f)\\sqrt{1 + k^2\\sin^2\\sigma'}}\n\\,d\\sigma'\\\\\n&= \\omega_{12}\n- f\\sin\\alpha_0 I(\\sigma_1, \\sigma_2; \\alpha_0).\n\\end{align}\n", "1937b3e880c7debe7b1335b5d669f54e": "\\frac{\\partial (x,y)}{\\partial (\\rho, \\phi)} =\n\\begin{vmatrix}\n\\cos \\phi & - \\rho \\sin \\phi \\\\\n\\sin \\phi & \\rho \\cos \\phi\n\\end{vmatrix} = \\rho", "1937caa6d015eda45795912a693a7967": "\\Delta x^0 = -\\frac{g_{0 \\alpha}\\, dx^\\alpha}{g_{00}} \\equiv g_\\alpha \\,dx^\\alpha.", "1938902477bca0776aa4829c01ae6a81": "\\operatorname{mr} (G)=\\operatorname{mr} (H)", "1938b1b88119b0925c537b7ecd09ed5d": " \\epsilon_i = P(X_i=1) - P(X_i=0),", "19392441d05c74603ef22c51c9fdb551": "B_v = \\log_2", "193a066f6522c2f2057198d112fbf80f": " N = \\frac{\\Delta}{2\\pi}", "193a115c37779fb02577488534b775b2": " 0 < A < K ", "193a414e2c1a2f78be0434b36512f168": " c -c_0 = A(\\beta,t) \\exp \\left[i\\beta x \\right] ", "193a578fa94fa4d6eb18780afc84bfe1": "\nT = -\\frac{\\hbar^2}{2} \\sum_{i=1}^N \\sum_{\\alpha=1}^3 \\frac{\\partial^2}{\\partial \\rho_{i\\alpha}^2}.\n", "193a7f25a92918c36e8b01fcf23cc68d": "J_0~", "193a9724f4210c884c9c0e10aaacba65": "\\mathbf{P} = N \\mathbf{p} = N \\varepsilon_0 \\alpha \\mathbf{E}_\\text{local},", "193b0f87863e178f94f5bd15f6a33eab": "n_{\\rm e} T \\tau_{\\rm E} \\ge \\frac{12k_{\\rm B}}{E_{\\rm ch}}\\,\\frac{T^2}{\\langle\\sigma v\\rangle} ", "193b1cc600400ecd206d9a890a8a98aa": " R = \\int_0^y C e^{\\rho t}\\ dt = \\frac{C}{\\rho} \\left(e^{\\rho y} - 1\\right) ", "193b295b63399cd55397b86a583d9743": "L = [T]_0/[R]_0", "193b791dc926f85d8d56160275fd15c4": "\n\\begin{align}Q(x;\\;y_0,\\;y_1,\\ldots,\\;y_n)&=P\\left(x+1;\\;xy_0,\\;xy_1+y_0, \\;xy_2+2y_1, \\;xy_3(x)+3y_2,\\ldots,\\;xy_n+ny_{(n-1)}\\right)\\\\\n&=R(x)P(x;\\;y_0,\\;y_1,\\ldots,\\;y_n)\n\\end{align}\n", "193b7f54555f257555899890bee2cb3a": "\n\\sin\\theta=\\left(\\frac{\\pi }{2} - \\frac{N}{2} \\phi \\right)\\frac{4}{N \\pi }\n", "193bd097d5140b8a8d4869697c5751a7": "\\xi{\\left(\\theta\\right)}", "193c678c571fcce0c22542f0a2e34a1f": "M_z(t) = M_{z,\\mathrm{eq}} - \\left [ M_{z,\\mathrm{eq}} - M_{z}(0) \\right ] e^{-t/T_1}", "193c7803a0e7c659b80af591f9a213cc": "k_e", "193c7ed4bad69da0e437743b57dabc90": "\\langle\\alpha'|\\alpha\\rangle = e^{(-1/2)(|\\alpha'|^{2}+|\\alpha|^{2}) + \\alpha'^{*}\\alpha}", "193ccc89ae380ee6db4f5b2f29d628a1": "X_j = \\frac{V_{j}}{2^{31}}", "193ce1ed474b4f7d7155a92b25a6cf2e": "D(T) = 1/B(T) = \\exp\\left(-\\int_0^T r(u)\\, du\\right)", "193d1abd31aff3da8944dbf90880aa7e": "\\varepsilon_x", "193d382d84ea96b6ac76731c3335dddd": " a_i < b_i ", "193d38f572568e22431af99b8a8d05d4": "\\beta_{k}\\geq\\frac{1}{4}", "193de046654e45d9d6c124b5bc99bae8": "\\Delta n_{\\text{c}} (0')", "193e4f92c1fd539586d2a5edc2de3c7c": "\n\\begin{align}\n\\operatorname{pmi}(x;y) &=& h(x) + h(y) - h(x,y) \\\\ \n &=& h(x) - h(x|y) \\\\ \n &=& h(y) - h(y|x)\n\\end{align}\n", "193e9c2d67c7aa49e080ca3ddba87daa": "\\dim(\\operatorname{gr}_I(M))-1", "193eb2a7251699912336f11087bb5bde": " \\hat{w}^{(L)}_k = w^{(L)}_{k-1} p(y_k|x^{(L)}_k), ", "193eba137fb2a1e96b77e0131c5c43c9": " + ", "193ec6858a85fc6d257f13852e9dc69c": "\\Delta_{\\psi} \\phi = \\sqrt{\\langle {\\phi}^2\\rangle_\\psi - \\langle {\\phi}\\rangle_\\psi ^2}", "193ece03df83a060b3c0b95c53b2ec0f": "C(i,j), C(i,k)", "193eddd1f8d575471e64333c4430960f": "h^i{}_k = \\cfrac{\\partial x^i}{\\partial q^k}", "193f0a132ba6c6876ff1a339ec393869": "L(p_j)", "193f4b94ce34529cb3c5aefb94af1b4b": "e^{(0)}", "193f6b546cbaeede34d2280d70dea0bc": "\\bigcup_{\\alpha<\\gamma} M_\\alpha\\prec_K N", "193f84589e23e13e08a43ee4db24fef2": "x_1, \\ldots , x_d", "193fc613b993c6ee2ad9fd559b3c4b83": "i = 1, 2, \\cdots, t; j = 1, 2, \\cdots, n", "193ff63c1060169a3e33ac10dcf985f6": " \\iint_{|z|<1}|F(z)|^2 \\, dS<+\\infty,", "193ff898670828ca697d0c5fcdacfba3": "2 + {1\\over2} = {5\\over2}", "194011504a6f100342d8eb8e69290a49": "U(|\\psi_2\\rangle\\otimes|\\mathrm{in}\\rangle)", "19401793ec61890d228748bce42da457": "\\leftrightarrow", "194042655062ba0ea8dbd2d7e9611897": " f_i = a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n \\, ", "194048eabb5cd295e4d1e280fc9b1f44": "A = \\Delta \\cdot k", "194058855e9e750c9f096c84069a5998": "\\scriptstyle z=0", "194059d26c613c4e8b33edade0701207": "F_4 = 4F", "1940a51b704d055f7407e2da3e1ddd2e": "\\overline{Z}-\\overline{M}", "1940ca52c77b631921bd65141f613a99": "[\\mathbf{u}_k,\\mathbf{u}_\\ell] = c_{k\\ell}{}^m \\mathbf{u}_m\\,\\ ", "19410831b5cac5c2d29ea48a83657ed9": "s^{\\ast }(p)", "19413c31389a7ed7a60b410ecb0a745b": "\\textstyle \\min \\sum_i x_i.", "19416526de8c03069196fc3b7e0586e6": "f^{(0)}_n (x) ", "194197ecc394cdc20ba773f400b88f7f": "\n f(\\xi, \\rho, \\theta) = 0 \\, \\quad \\text{or} \\quad\n f := \\rho + \\bar{\\lambda}(\\theta)~\\left(\\xi - \\bar{B}\\right) = 0 \n ", "194216b15719a2a7c5efe027491413fd": "G(V,E)\\!", "194279ec1e5f9373b0f9f97f7effe493": "C_M = \\frac{L_H -L_L}{L_H+L_L}", "1942c65acc6ba509b4b2485bc0f2cd19": "corr(z,z') = corr(\\mu + \\frac{1}{2}g + e, \\mu + \\frac{1}{2}g + e') = \\frac{1}{4}V_g", "1942cfa9c83ccc5897712f712aef3bf9": "\\scriptstyle r\\geq 1+\\sqrt{2}", "1942ed613f55de450ba506b3325b2225": "\\Gamma(U, \\mathcal{F})", "1942f1db6bb2ddd756198eb77d19707b": "\\Omega_{E,\\ell}", "1943126619175eb24198047bce456a5b": " \\mathbf{A}\n\\!\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\times \n\\end{array}\\!\\!\\!\n\\mathbf{A} = 2 \\left[\\left(\\mathbf{a}_1\\times \\mathbf{a}_2\\right)\\left(\\mathbf{b}_1\\times \\mathbf{b}_2\\right)+\\left(\\mathbf{a}_2\\times \\mathbf{a}_3\\right)\\left(\\mathbf{b}_2\\times \\mathbf{b}_3\\right)+\\left(\\mathbf{a}_3\\times \\mathbf{a}_1\\right)\\left(\\mathbf{b}_3\\times \\mathbf{b}_1\\right)\\right] ", "1943805ec95379ac146d709e8a989b7f": "ma=m\\frac{dv}{dt}=-\\frac{GMm}{r^2}\\,", "194393b45be9dca3e129157c785e0015": "\\langle X|R\\rangle ", "1943ca6b4190cf04f061377be986a8ab": "\\frac{2^{3/2}(k+3\\lambda)}{(k+2\\lambda)^{3/2}}", "1943e27e20de250cf4a463893a9b8d86": "\\left|g(x)\\right| \\in O(x^c)", "1943eb832b0958a0863a908f6d8e2254": "R=\\operatorname{C}(R)\\cong \\operatorname{C}(S)=S", "194402baa507fdd77c624b12441d2eb5": "T^{ab}{}_{;b} \\, = 0", "19440afdcfc2b8100fbe1f3b3f0e7ad9": "\\begin{align} 2 \\times b & \\equiv 2b\\\\\nb + b + b & \\equiv 3b\\\\\nb \\times b & \\equiv b^2 \\end{align}", "194464d9c0450a9184ff3815f153e7cb": "uv=ds", "1944b35d928e3bb34fda43ad48a1a11b": "\\cup \\{\\{ z \\in\\C \\mid (z-z_0)\\overline{(z-z_0)}=d \\ \\text{(circle)} \\mid z_0 \\in \\C, d\\in \\R, d>0\\}", "1944c8b606cd7e825933e899045e349b": "\\tilde{a}_{j,k} = a_{j,k}^*", "19452467807fd0cc6c5e1acb5671540b": "P_r =P_t ( {\\frac{\\lambda G}{4\\pi d}} ) ^2 \\times ( 1-e^{-j \\Delta \\phi})^2 ", "1945549fffae6f3522223c949427b214": "{\\Bbb C}", "1945592150dc57ce70dc093104426983": "e^0=1", "19455ac5c29b2010ec728029dd68f587": "E_m \\rho_j E_n^\\dagger = \\sum_{k} B_{m,n,j,k} \\rho_k", "194597a5efd6c389627376ec2fde5cc4": "h(a) = -2b", "19466c1163ddead1f4f9b2441b1cd57d": "\nj_{elec}^{sat} \n= j_{ion}^{sat}\\sqrt{m_i/\\pi m_e} \n= j_{ion}^{sat} \\left( 24.2 * \\sqrt{\\mu_i} \\right)\n", "1946a849341f615050b658e939636ace": "|\\phi_2\\rang", "19474d0e6e5389c1119755dec508681a": "P \\text{(hp)} = {\\tau \\text{(in} {\\cdot} \\text{lbf)} \\times f \\text{(rpm)} \\over 63{,}025}", "1947725787968c88314154082ac2060f": "\\frac{76852}{(1+0.10)^{10}}", "194796193aeeb8e0d4221dd12a3a4c43": "\\Rho_A", "1948050f2614803a4f4404c0b09fd2ed": "\\lim_{N\\to \\infty} \\frac{1}{N} \\ln P(M_N > x) = - I(x) .", "1948598529f6ea51a587de55a057214a": "{\\omega^0}_3 = 0", "19486cb2649ab19b4ef58d56fd1989df": "{Si(\\pi)}", "19486e37c9a977b1170872f33d11984e": "c_{BE}", "19486f2687c77e4a3415727ae5eb1192": "k < l", "194898069dad79ed447b084b96ab6cb4": "B > 0 ", "1948f70653bd15f1433732002bcd31b8": "B_\\lambda(T) =\\frac{2 hc^2}{\\lambda^5}\\frac{1}{ e^{\\frac{hc}{\\lambda k_\\mathrm{B}T}} - 1}", "19491c8f005eda8dfb0274fd15c3f2ca": "\\text{Step 2}", "194953e30316e24fb4420d76b37cafcd": "\\delta/\\pi", "1949656b6054b1d3db99ae022030f65e": "\\pm\\frac{\\sqrt{\\csc^2 \\theta - 1}}{\\csc \\theta}\\! ", "1949dfce6c0997eb45e54c5177fc303b": "Y = 3K(1-2\\nu)", "1949fa95340d1b2ad877c66b6ec79483": " {t=t_0} ", "194a106512985b68e4eac3327944e664": "\\overline{BC}", "194a627a74b816b929fc0317031eb63a": "f: M \\to M.", "194a89c2824f82bb0652f01ba6846eb7": " \nV=\\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\\\a_{21}&a_{22}&a_{23}&a_{24}\\\\a_{31}&a_{32}&a_{33}&a_{34}\\\\a_{41}&a_{42}&a_{43}&a_{44}\\end{bmatrix}=\\begin{bmatrix}A_1&B_1\\\\C_1&D_1\\end{bmatrix}\n", "194a90472811968f7abeb8820178108b": "\\langle A, \\oplus, 0 \\rangle", "194ab452466ff815b5ff05275f83b9aa": " \\zeta(-1),\\zeta(-3),\\zeta(-5),\\ldots, \\zeta(-m) ", "194aba5d80daa7cd28cc1d364e9404b5": "s_a(t)\\,", "194abc078116f951908caa1ccd7c0ac7": "V[e_k] = [e_{k+1}] \\quad \\mbox{for} \\quad k = 0 \\ldots n-1.", "194b273f891dd740717bbf358c22f4af": "H_n(x)=(-1)^n e^{x^2}\\frac{d^n}{dx^n}e^{-x^2}\\,\\!", "194b3067b7171e89f6633b9860173a1a": "x \\in \\mathbb{R}^2", "194ba0016440f9cec26a48a3fe2dbc6b": "{52 \\choose 2}{50 \\choose 2} \\div 2 = 812,175", "194bcfe56baf8ce108926fee5db3e6f4": "A\\parallel_+ P \\parallel_- B", "194c770a6fe09b0e9562c6eba2723cda": "\n\\hat{\\theta}=\\frac{F^{-1}(\\hat{q}_2)\\tau_1-F^{-1}(\\hat{q}_1)\\tau_2}{F^{-1}(\\hat{q}_2)-F^{-1}(\\hat{q}_1)},\\quad\nF(x)=\\frac{1}{\\sqrt{2\\pi}}\\int\\limits_{x}^{\\infty}e^{-v^2/2}dw\n", "194c894c9acf73e811a69934bfa217dc": "\\|f_n(x)\\|_B\\le g(x)", "194d0881f0cc91741a4da51dc881bf3d": "x_{i+1}-x_i=h\\ \\forall i", "194d38ed3c7df173d586ecf3b2a71044": " \\sin\\theta = \\frac{H}{L},", "194d9d1ae7e135a37ece8c0f3b3f2368": "\\mathbb{C}^{n}", "194dcc7caa45b56e1ab889f01de707a4": "\\operatorname{R}(T) = \\operatorname{R}(T_0)(1 + \\alpha\\Delta T)", "194e3a185efd76ee4c356ad1f1abca0d": "X(f)\\ ", "194e41a4599c3d829537b3d0c9c4dc46": "= \\frac{q_{0}}{q_{1}} exp (-\\beta \\Delta U) \\frac{\\int ds^{N}exp(-\\beta U_{0})\\delta(U_{1}-U_{0}-\\Delta U)}{q_{0}} = \\frac{q_{0}}{q_{1}} exp (- \\beta \\Delta U) p_{0}(\\Delta U)\n", "194e447c7b2e57806a89eb161999446a": "X \\succeq 0 \\Leftrightarrow A - B C^{-1} B^T \\succeq 0", "194e544049e383bc2b78c97f145f763f": " \\Delta f(x,y) \\approx \\frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}. ", "194ea13262c0693cded4e399f0ac7681": "=\\ b_{ij} (-1)^{i+j} |M_{ij}|,", "194ecc34fa7f41fbd809e664212047df": "\\|d\\mathbf{X}\\|^2 = dX^\\mu dX_\\mu=c^2d\\tau^2=ds^2 \\,,", "194efeb51dc3cb04870fe56983cc97b4": " \\left [ \\hat{A}, \\hat{B} \\right ] \\psi = 0, ", "194f0320e13b47803e90b2dd736e5737": "\\log^+(x)=\\max\\{0,\\log(x)\\}", "194f57407da821bac52ed80cd55c385c": "\\,^{z_2 = x_2 y_1 + x_1 y_2 + x_4 y_3 - x_3 y_4 + x_6 y_5 - x_5 y_6 - x_8 y_7 + x_7 y_8 + u_2 y_9 + u_1 y_{10} + u_4 y_{11} - u_3 y_{12} + u_6 y_{13} - u_5 y_{14} - u_8 y_{15} + u_7 y_{16}}", "194f7c71cfbe161d5653093f67932e6b": "\\mathcal{H} = \\mathcal{H}(p,q,t) = \\langle p,\\dot{q} \\rangle -L(q,\\dot{q},t)", "194f991f3d55ccd3cb374059e0434a39": "(Out)\\quad m[\\;n[\\;out \\ m.A\\mid A'\\;] \\mid \\overline{out} \\ m.B \\mid B'\\;] \\Rightarrow_{amb} n[\\;A \\mid A'\\;] \\mid m[\\;B \\mid B'\\;]", "194fd5eb705b174f78cea7e27d58dbe8": " A^*A \\,=\\, (QR)^*(QR) \\,=\\, R^*Q^*QR \\,=\\, R^*R", "195003a6255123ad9e0a1941ed6dc0d5": "y = \\sum_{r = 0}^\\infty a_r (x - 1)^{r + c},", "195009052b7cbca0104a915b7bb7563f": "\\gamma=\\gamma", "19500bf5729221abf1c8fad5c45f16bb": " \\dot{V} ( \\mathbf x) ", "19503ee8ccd1f90942f178c4b12131b5": "M_1 (\\vec X,\\vec {\\rm E},Y) = \\left[ {\\begin{array}{*{20}c}\n 0 & 0 & b \\\\\n 0 & 0 & A \\\\\n 0 & 0 & I \\\\\n {A^T } & I & 0 \\\\\n\\end{array}} \\right]\n", "1950bb0c40ceaec16b377f829b96cff1": " r= \\frac {1}{s}", "1950ee158c131d36cf7b83596c240212": "\\beta H_g = - \\beta \\sum_{(i,j)}J_{ij} \\delta(s_i,s_j) - \\sum_i h_i s_i \\,", "1950f10625b96aa4611b31935901fdee": "\\begin{align}\nc(i,k,X) &:= \\left \\{p \\left(X_i|X_{i-k}^{i-1} \\right ) \\right \\} \\\\\nc(i,X) &:= \\left \\{p \\left (X_i|X_{-\\infty}^{i-1} \\right ) \\right \\}\n\\end{align}", "19519dcebc223000df9e7ec676788402": "\\sigma_{yy}\\sigma_{zz} - \\sigma^2_{yz}", "19522d05daea1699923e26e2794c4aa5": "\\theta_{(l)}\\,\\hat{=}\\,0", "195234152f7fb4fdd94ac4b2a3890208": "\\displaystyle R(f)(\\phi)(x) = \\int_G f(y)\\phi(xy) \\,dy = \\int_{\\Gamma\\backslash G}\\sum_{\\gamma\\in \\Gamma}f(x^{-1}\\gamma y)\\phi(y)\\,dy. ", "195246810f9bfc228bca491859062b14": "\\xi", "19524783f77b962726c6a3a32e1d969b": "c(x) \\ge 0~~ x \\in \\mathbb{R}^n, c(x) \\in \\mathbb{R}^m~~~~~~(1)", "19524a01b0bb23fb28170076a61824c4": "\\left\\{{j\\atop k}\\right\\}", "195294d43f24d89a4d84eaf58a604dcf": "\\langle A, \\in\\rangle ", "1952a387272e13895410e766c5ba4dc5": "z\\frac{d^2w}{dz^2}+\\left(C+\\frac D\\sqrt{D^2-4F}z\\right)\\frac{dw}{dz}+\\left(\\frac E\\sqrt{D^2-4F}+\\frac F{D^2-4F}z\\right)w=0", "1952dd2c9b14fde81357c321fccfe516": "7.76 \\times 10^{206544}", "19532796fff88b522980627cbaffa592": "\\mathrm{H=Wb\\ A^{-1}=V\\ A^{-1}s=kg\\ A^{-2}m^2s^{-2}}", "19532812640bd20f43d31023e1071ae3": "\\varphi\\circ\\pi={\\mathrm {pr}_1}\\circ\\psi\\, ,", "195371a0f34f791e88e6eb0743000476": "t(d,n) \\leq \\mathcal{O}(d^2\\log n)", "1953796efcf6b915573c34df836234d5": "\\Gamma =3 \\, ", "1953a4318c4b5edf02c2289be3a0366a": "\\scriptstyle|\\psi(x,y,z,t_0)|^2.", "1953c85dddb1d67afcd41a23688ef8a4": "\\{|r_i\\rangle\\}", "1953f08796bace949f6fdbbdc3f0f6e9": "2t_{d}", "1954b5a14afc4cd694d658a009372c0d": "x_0 = b", "1954e0047441a6c9527ea1cd1955cd6f": "E\\left[e^{t\\log X}\\right] = \\lambda^t\\Gamma\\left(\\frac{t}{k}+1\\right)", "19552212ad592fb4fbb8ac7cf48300b3": " \\inf_{N\\geq 1} \\sum_{n=N}^\\infty \\Pr(E_n) = 0. \\, ", "19552944d265423ce186e9781e0b3411": "\\scriptstyle - \\frac{11+4\\sqrt{5}}{41}", "195532bbf10ce37f6de16665f92d93a5": "p^n\\;", "1955336337eb36003a780cd4a1034492": "\\Delta{s}\\rightarrow0", "19555dbb9b4922bfca024a893f0c1288": "\n\\left(b^{2d} + b^d + 1\\right)^n,\n", "1955aee141ceb183b70fd1f2df16acf4": "\\mathbf{s} = [s_1, s_2, \\ldots, s_{N_s}]^T", "1955b89f7ee3e8a8a8d18615efe03518": "f(a)=0.", "1955ec7f301777ef0a6c6281d15534d4": "a_j-a_i=K", "19561bb22d07427e063434ddbeaac62e": "\\sum_{n\\ge 1} \\frac{d(n)^2}{n^s} = \\frac{\\zeta(s)^4}{\\zeta(2s)}", "19568cba6a2f4d3bb0cd5b4ae05ac110": "A=\\frac{1}{b}\\inf_{|\\gamma|\\in[1,a]}(G_0(\\gamma)-G_1(\\gamma))>0", "1956940fd568edcef6c60b3723bb951b": "m.", "1956d148f8c77df7bf21265fd957e797": "a\\in\\mathbb{R}", "1956e7f4c76d3febd1870fbfbaa9988c": " t^{-1} K(x, t; X_0, X_1) = K(x, t^{-1}; X_1, X_0).", "195716ce17af6b96683e9cfd8c57e49c": "f:X \\rightarrow Z", "19574c28c05ecf3ca919073e93040e30": "\\nabla\\times\\mathbf{H}_\\text{d} = 0", "19578c52c9799a48fb3138dec36c30f6": "\nu = u_{1} + \\left( u_{2} - u_{1} \\right) \\, \\sin^{2}\\left( \\frac{1}{2} \\varphi + \\delta \\right)\n", "1957d47e4158cf35010ac79565813d73": "\\{(A1,A2) | (A1,B1)\\} = \\{ \\{|\\} | \\{|\\} \\}.", "1957d7a6b83de643498aa23bd6949a47": " \\vec q = - k \\vec \\nabla T", "19583fa1a671f0a0810cbe7cd5ee5f75": "\n \\begin{bmatrix}\n L' \\\\\n M' \\\\\n S'\n \\end{bmatrix}\n =\n \\mathbf{M}_H\n \\begin{bmatrix}\n X_c \\\\\n Y_c \\\\\n Z_c\n \\end{bmatrix}\n =\n \\mathbf{M}_H\n \\mathbf{M}_{CAT02}^{-1}\n \\begin{bmatrix}\n L_c \\\\\n M_c \\\\\n S_c\n \\end{bmatrix}\n", "19585c35847bbd6a71e35ae3299dfdd1": "x_{m-1} \\succ x_m", "1958a12f9fac2e513669175780a3641d": "f^{-1}(F(f)) = f(F(f)) = F(f)", "1958d167a60990afb443fca72ea814b7": "f_t : M \\to \\mathbb R", "19591371392c0b82414c07db29ac41f5": "F(A,B) = \\inf_{\\alpha, \\beta}\\,\\,\\max_{t \\in [0,1]} \\,\\, \\Bigg \\{d \\Big ( A(\\alpha(t)), \\, B(\\beta(t)) \\Big ) \\Bigg \\}", "195928339db213e4ca31e731b2fdbd40": "\\boldsymbol{\\Gamma}", "1959334d1a3d88cc15fd96ab15d8e030": " \\int_0^t f(w(s)) \\, \\mathrm{d}s = \\int_{-\\infty}^{+\\infty} f(x) L_t(x) \\, \\mathrm{d}x ", "195973c728080ff7059576a88d4d7ccd": "\\epsilon \\in \\mathbb{C}", "1959cd91477c06cb6bb761ed8169dd66": "x^\\prime [(m - l)\\mod N]", "1959e499d5110d8b4abf857fb94fdb32": "D_{\\mathrm{KL}}(P\\|Q)", "195a0b1b92912ef0c047c789b27dc0dc": "\n J_{(1)} = J_{(2)}\n ", "195a21c626ce9f44e3e2cc6c10f29ca0": " \\hat\\rho = \\sum_i p_i |\\psi_i \\rangle \\langle \\psi_i|.", "195a5b104f0f0d94cc53f4827ff57648": "\\sigma(x,y)", "195a65e0f5f4e2663c174d1fee2a2fe6": "\\textit{true} \\rightarrow \\textit{open}(1) \\wedge \\textit{occludeopen}(1)", "195a98b1b5ba6b44fb1820f3087d3cee": "\\left(1-\\frac a2\\right)\\sqrt{-(a+1)}-a\\left(2+\\frac a2\\right)\\arcsin\\frac1{\\sqrt{-a}}.", "195af63d1c54ae6dad4123723e2a22c3": " \\frac{\\sum_{n=0}^\\infty b_n X^n }{\\sum_{n=0}^\\infty a_n X^n } =\\sum_{n=0}^\\infty c_n X^n, ", "195b2834fd19ddf153b8ee828acde11b": "\\sum_{p|N}\\frac{N}p + 1 = N.", "195b3a7fbc45810c7069800c54e755b8": "\na = \\frac{a^2b^2+b^2c^2+c^2a^2}{\\left(a^2+b^2+c^2\\right)^2}\\Delta\n", "195b51f3ae98c50f1541bb54a442efcd": "\\sigma_f(\\theta)", "195b7f1122557c408cf93fac9770f5c2": "y_c=\\begin{cases}\n-1.1063814 x_c^3 - 1.34811020 x_c^2 + 2.18555832 x_c - 0.20219683 & 1667\\text{K} \\leq T \\leq 2222\\text{K} \\\\\n-0.9549476 x_c^3 - 1.37418593 x_c^2 + 2.09137015 x_c - 0.16748867 & 2222\\text{K} \\leq T \\leq 4000\\text{K} \\\\\n+3.0817580 x_c^3 - 5.87338670 x_c^2 + 3.75112997 x_c - 0.37001483 & 4000\\text{K} \\leq T \\leq 25000\\text{K}\n\\end{cases}", "195bc176a3eecc0e1cb95ab4f7eea1a8": "p_{\\operatorname{interp}}(r)", "195bcb37cd58a7069bf2b8d723364a21": "\\scriptstyle \\delta t_{\\text{clock},i} (t_i) \\;=\\; \\delta t_{\\text{clock,sv},i} (t_i) \\,+\\, \\delta t_{\\text{orbit-relativ},\\, i} (\\boldsymbol{r}_i,\\, \\dot{\\boldsymbol{r}}_i)", "195c48cfb7a585c41f637ab76f4fd590": "\\mathbf{e}_{1}(t) = \\frac{ \\mathbf{\\gamma}'(t) }{ \\| \\mathbf{\\gamma}'(t) \\|}.", "195c599630049736688b0afbe45f87a2": "\\mathcal U^{\\times^r_X} := \\mathcal U \\times_X \\dots \\times_X \\mathcal U.", "195cef8076f3333a5fd5025f74daca4f": "m_\\mathrm{i}", "195d2fdbffb939fe4bc792cadd80bf62": "\\exists f_1,f_2\\in C\\;pf_1=1+f_2", "195e2a44f7c0122d96ac7be568d73207": " 1\\ \\mathrm{V} \\cdot 10^\\frac{-60}{20} = .001\\ \\mathrm{V} = 1\\ \\mathrm{mV} ", "195eaa6f24ce8f721729d5115bae0fd2": "\\theta>0", "195f426da0265c47431bef593a930574": "\\mathbb P(A_1) \\le \\mathbb P(A_1).", "195f4d1cc140d1ca55725b434285ee08": "\\left(10-1.37218 \\frac{\\sqrt{2}}{\\sqrt{11}}, 10+1.37218 \\frac{\\sqrt{2}}{\\sqrt{11}}\\right) = \\left(9.41490, 10.58510\\right). ", "195f4dd692396f9164530339f3c96309": "\\,N(R)=(2\\pi)^{-d}\\omega_d VR^{d/2}+\\frac{1}{4}(2\\pi)^{-d+1}\\omega_{d-1} AR^{(d-1)/2}+o(R^{(d-1)/2}).\\,", "195f6e3e9d12ba6e65b20f2719df81c1": " a=(a_1, \\dots, a_n) ", "195fb9efa4f61e9968b73af9b4aeeef6": "x_0\\!", "195fcf1c043c8af2ad00aaeabfd1912d": "\\textstyle\\epsilon", "1961487550cb1f61914c9800f6957560": "{A}_{18}^{(2)}", "19617304f80da783687c0800e3ac3d7e": "\\theta = \\pi", "1961a97e2b0aa2c4cba24e56315ca139": "E_\\text{exch} = A \\int_V \\left((\\nabla m_x)^2 + (\\nabla m_y)^2 + (\\nabla m_z)^2\\right) \\mathrm{d}V", "1961c355ac953bae13c3727883912638": "P^\\star_{\\mathbf{k}}", "1961d722eb0a0f899070cd88293dc19d": "\n\\begin{align}\n\\sum_{i\\in S}\\sum_{a\\in A(i)}R(i,a)y^*(i,a) \\geq \\sum_{i\\in\nS}\\sum_{a\\in A(i)}R(i,a)y(i,a)\n\\end{align}\n", "1961eeac1fd1559e2cd40a0324eb7411": "\\overline{\\Gamma^k{}_{ij}} =\n\\frac{\\partial x^p}{\\partial y^i}\\,\n\\frac{\\partial x^q}{\\partial y^j}\\,\n\\Gamma^r{}_{pq}\\,\n\\frac{\\partial y^k}{\\partial x^r}\n+ \n\\frac{\\partial y^k}{\\partial x^m}\\, \n\\frac{\\partial^2 x^m}{\\partial y^i \\partial y^j} \n\\ ", "1962304f997d73c67d909135f0a99e1e": "E Z \\mathbf{y} = 0", "19627b9cfe9eb538f4d940082c339f43": "\n\\hat b \n", "1962a11c3dec36526b3a35f65d8c3777": "\\int_E f(x) \\, dx.", "1962aeb01df827ac7650c224e5687417": "H_1, \\dots, H_p", "196331f7ca0856bc1404c81620d5ddee": "A=\\{x \\mid \\phi\\}", "19634dba6597efe959b91b31202c96bd": " \\left \\langle Ax, y^\\star \\right \\rangle = \\left \\langle x, A^\\star y^\\star \\right \\rangle, ", "1963bf9d24e3b9be35691d6026aeed3e": "\\Delta C_t = 0.5 \\Delta Y_t - 0.2 (C_{t-1}-0.9 Y_{t-1}) +\\epsilon_t", "1963f8f15b60547359961bf40f5d83a3": "W^{II}(z,x)=E\\left[ \\beta ^{II}(Y_{1})\\mid X_{1}=x,Y_{1}0\\}, && \\text{the heat equation},\\\\\n-\\frac{\\partial u}{\\partial x}(0, t) &= f(t), && t>0, &&\\text{the Neumann condition at the left end of the domain describing the inlet heat flux}, &&\\\\\nu\\big(s(t),t\\big) &= 0, && t>0, &&\\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature},\\\\\n\\frac{\\mathrm{d}s}{\\mathrm{d}t} &= -\\frac{\\partial u}{\\partial x}\\big(s(t), t\\big), && t>0, &&\\text{Stefan condition},\\\\\nu(x,0) &= 0, && x\\geq 0, &&\\text{initial temperature distribution},\\\\\ns(0) &= 0, && &&\\text{initial depth of the melted ice block}.\n\\end{align}\n", "1987a88fc39f6b7f2fdf571395212e68": "d-1", "1987b6358e69904a4b2620751195dc61": "[0, \\pi] \\,", "1987b8342168ae3d754d4a353832e2b3": "H \\subseteq A", "1987d0c8cb6768a4753bf5df2056dfa0": " V_{i} := \\bigcup_{j 1. ", "19954eb5893dbb80cfd91d6ebcad2680": "O(\\underline{u}:G)", "19954f1ee6438a24f1758b43c4c6e606": "\\Gamma^k {\\mathbf e}_k\\,", "19955180e4f1f7c7c96da365d15b388d": "\\mathbb{P}( \\theta_{up}(X) \\geq \\theta | \\theta)", "1995a69ab16b97dedd15c9ee0abd4ae2": "\\Gamma\\left(\\frac{s}{2}\\right) \\pi^{-s/2} \\zeta(s) = \n\\frac{1}{2}\\int_0^\\infty\\left[\\vartheta(0;it)-1\\right]\nt^{s/2}\\frac{dt}{t}", "19964a2c10afc6d91dc66293130ce42a": "z^N-1 = F_1(z) F_2(z) F_3(z)", "1996a01256151a7de63182612cd5b7e4": "\\alpha(U_{+})", "1996bc29a9052cd839e74c9aba11c599": "\nIM_i = e_i^t \\left( G_i-G \\right)\n", "19974d9ff467ca51c10ed34d6f3bea26": "(q_2, p_2)", "19978c5eae58f5be5c746a2f29a2339f": "\\underline{\\varphi \\vdash \\lnot \\psi}\\,\\!", "1997fbf8e16efe1aeeea49a032454b26": "e^{-x} = \\cosh x - \\sinh x", "19983625ce00458e19cb8d2325fad636": "versin(\\theta)=1-cos(\\theta)", "199841974c59ef53d9ebb97ac335a2fc": "HD = DP.", "199848333e004f3702eb338744d818f2": "H =\\sum_{n=0}^\\infty E_n | n \\rangle \\langle n|", "19989f1685981884934ee90c26d26dcb": "\\mathbf{P}^{-1} \\mathbf{R}_1 \\mathbf{P} = \\mathbf{D}", "1998bc8bcd6e62169bdb4c6f1d826835": "\\displaystyle{G_0=KA_0K.}", "1998bebc8aab93ad11d7cc346f7b9a17": "f\\colon S \\to S", "1998f5b4ce24c28961cec25bcdcebc20": "VCA(64x^3-112x+56,(0,2)) ", "199911eaf439374bc1ca41fe5bfb2dfe": "\\tfrac{1+ \\sqrt{5}}{2}. \\,", "199967db6c19aa7fe20fbd760e2f261f": "{\\tau} = I\\ {\\alpha}", "19996851bbe16e45b824474b587f7d52": "|({\\mathcal F}_af)(at,y)| \\le C e^{-\\varepsilon a},", "19997ce9e1d2be894d4ae5d767dabd76": "\\mathrm{Ad}_g", "1999c50f4e93857f7436bed20696d9e1": "n^{\\underline k}=n\\times(n-1)\\times(n-2)\\times\\cdots\\times(n-k+1),", "1999d0078e9a6e421e00f3b58513d38d": " \\pi / 2", "1999f106954f6ff4669d1f460fab14a0": "\\displaystyle{[\\delta^h,\\sum a_\\alpha \\partial^\\alpha]=(\\delta^h(a_\\alpha)\\circ R_h) \\partial^\\alpha.}", "199a3cef9b2a7c0762fa0a6568fef088": "a_{i,j,k}", "199aa29ac20ea76bed3965887ba7c383": "(\\forall{\\alpha<\\kappa^{+}})A_{\\alpha}\\leq_{f} B\\,", "199ab0fddcf6057c1d12262475af4a8a": "\\Sigma_n", "199b0643221b2d0887c987a1462255ee": "\\mathcal{M}(U) \\to \\mathcal{M}(V)", "199b2da68391c77caa328a519959fdb4": "12.9232102", "199b4394970666ce7528e192cf3b2676": "\\frac{d\\varphi}{d\\alpha} = \\int_a^b f_{\\alpha}(x, \\alpha)\\,dx + f(b, \\alpha) \\frac{db}{d\\alpha} - f(a, \\alpha)\\frac{da}{d\\alpha} ", "199b77219a0793e89f49c7614b42838a": "u_{1}, u_{2}\\in (V\\cup\\Sigma)^{*}", "199ba4358939098d08e196df1a2df739": "A \\circ B", "199bc539a17705dd92c959577d1404a5": "K_i \\varphi \\implies \\varphi", "199be30a7ca6070d6bb1c9e9d325ea94": "C^{-1}_1,C^0_1,C^1_1", "199c2f04393f16410802bee481d0614c": "\\varphi\\left(\\int x\\,d\\mu_n(x) \\right)\\leq \\int \\varphi(x)\\,d\\mu_n(x),", "199c2f6f10dd3cfc94ecc4c05929fd30": "\\frac{\\partial z}{\\partial x}", "199c34ec680fb1368f84dc9a5d564db0": " ~ \\sum_{mnp} ~ \\frac {1}{k^2-\\alpha_m^2-\\beta_n^2-\\gamma_p^2} ~ \\bold G_{mnp} ~ \\bold J(\\alpha_m,\\beta_n,\\gamma_p) ~ e^{j(\\alpha_m x + \\beta_n y + \\gamma_p z)} ~ = ~ \\bold 0 ~~~~~~~~~~~~~~~~(3.3) ", "199c4df01bd6170f7d83145e1189cf56": "\\Delta\\mathbf{v}\\,", "199cb56b173080aeff39332c795afc8a": "\\,{{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}", "199cbda39eae2b91d55c0c1201cb4fe0": "L_n\\left[1/2,1+o(1)\\right]", "199cc2ae9e3f8dac92151ce502826fda": "\\pi = \\frac{F}{l}.", "199d664785a7182bcb78907cc6efd426": "F_5\\;=\\;(\\frac{f}{900})^{-n} \\mbox{ for } 2< n <3", "199d824a6ec719c04d6f349008808846": "\\bar{D_T}", "199dc703138a378f664841de18b43dd9": "\\left\\langle X,Y \\right\\rangle.", "199de46fb7583f7488f5152e492a307c": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{sinc}\\left(\\frac{\\nu}{2\\pi a}\\right)", "199e3e1081482e0787f42d6886e3f6cc": "\\zeta_{a} > \\zeta_{b}", "199e770aefdf9e1969110b945e19e6ef": "R/I", "199e9d4ae76d467d01bf27d9a38d7104": "N \\rightarrow U", "199ea3d974838ca329bcc09f0a3e2449": " x_{k+1} = x_{k} - \\tau \\nabla f(x_k) ", "199eae636d5286e6fa5a6d864976d126": "p(m) = -\\tfrac{1}{8}", "199eb8d4ba1982878efc883c2a8841e5": "\\begin{matrix}\n&& K && \\\\\n& \\eta_A \\swarrow & \\, & \\eta_B \\searrow & \\\\\nA && \\begin{matrix} f \\\\ \\longrightarrow \\end{matrix} && B\n\\end{matrix}", "199eefb242c3eb4ca44ee1dcd612012e": "\\mathcal O = k[[t]]", "199f077e864bd9f00c7533d0c523c2ad": "\\{x_t, y_t\\}", "199f45229705d0c176d17b4b9d58e129": "(x_1,x_2,....,x_n)", "199f70ea7fa4be0b6421b970bda236d2": "NM(X|k_0,\\{p_1, p_2, p_3\\})= 0.00465585119998784 ", "199f78aa20d48a01e124292d955b6357": "w=g_1 g_2\\cdots g_m", "199fa642986296f2fbe02a1258402aa1": "\\int_{\\reals^3} \\rho = N", "199fb1ee6c70970e322bc7f5c9e0d01c": "e_\\alpha^I", "199fb9b43bc6e2b346b45c63c01366ec": "\\delta(X, Y) = \\log \\Bigl( \\inf \\{ \\|T\\| \\|T^{-1}\\| : T \\in \\operatorname{GL}(X, Y) \\} \\Bigr).", "199fdba64b4357d13b0a54611db02c72": "\\mathbf{S} =\\mathbf{E} \\times \\mathbf{H} ,", "199fe0ee55a358eae25d6583a56369a1": "\\frac{\\partial f(g(u))}{\\partial \\mathbf{x}} =", "199fe28b14d7de1f22666e319f0c8ecf": "\\begin{align}\n & \\mu_1(g_2) = - \\frac{6}{n+1}, \\\\\n & \\mu_2(g_2) = \\frac{ 24n(n-2)(n-3) }{ (n+1)^2(n+3)(n+5) }, \\\\\n & \\gamma_1(g_2) \\equiv \\frac{\\mu_3(g_2)}{\\mu_2(g_2)^{3/2}} = \\frac{6(n^2-5n+2)}{(n+7)(n+9)} \\sqrt{\\frac{6(n+3)(n+5)}{n(n-2)(n-3)}}, \\\\\n & \\gamma_2(g_2) \\equiv \\frac{\\mu_4(g_2)}{\\mu_2(g_2)^{2}}-3 = \\frac{ 36(15n^6-36n^5-628n^4+982n^3+5777n^2-6402n+900) }{ n(n-3)(n-2)(n+7)(n+9)(n+11)(n+13) }.\n \\end{align}", "19a029a162a67f2d310b823f89820339": "{\\mathcal P} = \\left \\{ \n\\left(\\frac{74}{511},\\frac{81}{511},\\frac{137}{511} \\right) , \n \\left(\\frac{148}{511},\\frac{162}{511},\\frac{274}{511} \\right) ,\n \\left(\\frac{296}{511},\\frac{324}{511},\\frac{37}{511} \\right) \n\\right \\rbrace", "19a069f087c731d82fbf373afa97bbf8": "(5)\\quad ds^2=-\\Big( 1-\\frac{2M}{r} \\Big) dv^2+2dvdr+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;.", "19a07f42375250eeec2c1f1256cb964d": " \\epsilon = GM \\left( \\frac{2a-r_a}{2ar_a} - \\frac{1}{r_a} \\right) = - \\frac{GM}{2a} ", "19a09e5535341fae17b8a43797a82c6d": "|0\\rangle_B", "19a15e3925d19a6d53e8bf065bce0755": " \\left(1 + \\sum_{i=1}^p \\phi_i L^i\\right) X_t = \\left(1 + \\sum_{i=1}^q \\theta_i L^i\\right) \\varepsilon_t \\, .", "19a1db91013fe6f388f9df252069e3b5": "S_{\\mathrm{f}6}", "19a1f82859cf1cc42f7a174e49a73cf7": "da\\,\\!", "19a236bd33c9d1b25b54078d75438610": "S_1 = {25 \\over 24} \\approx 70.7 \\ \\hbox{cents}", "19a238b9254d274589701eb7f8d6aa94": "\\langle , \\rangle : X \\times Y \\to \\mathbb{F}", "19a25d8b621357c0b7d6bf525e840291": "\\mathbf{r} = \\mathbf{y} - \\mathbf{\\hat{y}} = \\mathbf{y} - H \\mathbf{y} = (I - H) \\mathbf{y}.", "19a279a922e5f4740784161e5a305682": " \\dot{S} = -2\\dot{R}.", "19a2cd3ecad002bbd85764a966f36e03": " \\mathbf{x} = \\{x_{-M}, x_{-M+1},\\ldots,x_M\\} ", "19a31e17d0cf3e104a26a165ba89603c": "\\left(\\omega = \\omega_o\\right)", "19a350289fe7909ac6ae6aaf7fb21cee": "\\left \\langle N,e\\right \\rangle = \\left \\langle 90581,17993\\right \\rangle", "19a3d21a29c9e5afbecd9dd04ddad36d": "\n\\begin{align}\n&D[p||q]=\\frac{1}{2}g_{ij}(q)\\Delta\\xi^i\\Delta\\xi^j+\\frac{1}{6}h_{ijk}\\Delta\\xi^i\\Delta\\xi^j\\Delta\\xi^k +o(||\\Delta\\xi||^3)\\\\\n&h_{ijk}=D[\\partial_i\\partial_j\\partial_k||]\\\\\n&\\partial_ig_{jk}=\\partial_iD[\\partial_j\\partial_k||]=D[\\partial_i\\partial_j\\partial_k||]+D[\\partial_j\\partial_k||\\partial_i]=h_{ijk}-\\Gamma_{jk,i}\\\\\n&h_{ijk}=\\partial_ig_{jk}+\\Gamma_{jk,i} .\n\\end{align}\n", "19a40010111789ab2b002fa427c3f9e1": "\\begin{pmatrix}a & -m\\\\c & n\\end{pmatrix}", "19a4350bbcbe6580b14d35a450257d7a": "L=M", "19a439ed08428eba46fcb31fce6737b9": "m_\\text{S} = \\sqrt{\\frac{e^2}{G (4 \\pi \\epsilon_0)}}", "19a4a2507d68bf2f9d8cef90ed0c85b8": "\\mathbb{N}^{\\mathbb{N}}", "19a569e51bc4fb0f04565b126517215b": "\\psi_{klm}(r,\\theta,\\phi) = N_{kl} r^{l}e^{-\\nu r^2}{L_k}^{(l+{1\\over 2})}(2\\nu r^2) Y_{lm}(\\theta,\\phi)", "19a5d1215959f1bf660ef0836ccb6b69": "\\beta_E(Q)", "19a5dc08814aa09fd5bbeff85538e995": "\\oint_S \\mathbf{E} \\cdot \\mathrm{d}\\mathbf{A} = \\frac{Q}{\\mathcal{E}_0}", "19a5e25d843fc40b787024a57e3c49ba": "q_y=q_y(x,y)", "19a5ec6e8d51a3c90c48aa9548f39c92": "\\ln(L)= -\\frac{1}{2} \\ln (|\\boldsymbol\\Sigma|\\,) -\\frac{1}{2}(\\mathbf{x}-\\boldsymbol\\mu)^{\\rm T}\\boldsymbol\\Sigma^{-1}(\\mathbf{x}-\\boldsymbol\\mu) - \\frac{k}{2}\\ln(2\\pi)", "19a62cc119b1e2575c9316137fdd2b98": "\\scriptstyle L_c", "19a6461776a93d2451ce850c2d8ad598": "{\\color{Blue}~2.15}", "19a73fc3b84f377f6508b7ff40f3e170": "{\\tilde{D}}_8", "19a74ef2d7c2f6370559c559e606fff0": " e^{(a-1)\\Theta}f(x)=f(ax)", "19a7a37a20ea2231cb90ba01577ffe5d": "\\sqrt{E} = (k^2-m^2)^{1\\over 4}", "19a7b0f68176cdfcbcaff8e0001493e4": "U(\\alpha S,\\alpha V,\\alpha N_{1},\\alpha N_{2},\\ldots )\n = \\alpha U(S,V,N_{1},N_{2},\\ldots)\\,", "19a7bdc2300335c921dd2d36f9b8c82c": "\\bar{\\mathbf{e}}{}_j = \\bar{\\mathbf{e}}{}_j\\left(\\mathbf{e}_1,\\mathbf{e}_2\\cdots\\right) \\quad \\rightleftharpoons \\quad \\mathbf{e}{}_j = \\mathbf{e}{}_j \\left(\\bar{\\mathbf{e}}_1,\\bar{\\mathbf{e}}_2\\cdots\\right)", "19a7ed3a8acee2707be22f0831d04827": "\\left( \\frac{\\partial(G/T)} {\\partial T} \\right)_p = \\frac{1}{T}\\left(\\frac{\\partial G}{\\partial T}\\right)_p + G\\frac{\\partial (T^{-1})}{\\partial T} = \\dfrac{T\\left ( \\dfrac{\\partial G}{\\partial T} \\right)_p- G}{T^2} = \\frac{-ST - G}{T^2} = -\\frac{H}{T^2} \\,\\!", "19a852d6b5d898d377fbf3324fd276ca": "\\langle R'(X,Y)Z, W\\rangle = \\langle R(X,Y)Z, W \\rangle + \\sum_{j=1}^k \\alpha_j(X,Z) \\alpha_j(Y,W) - \\alpha_j(Y,Z) \\alpha_j(X,W) ", "19a891cd03ec35200ce75ce8dea8a445": "[\\ ]_+", "19a8b2ea9738dbc9cf366bb300209209": "z_1, z_2, z_3, \\infin", "19a8f3dcb49fdee5dc4a48fb7eb82e4a": "x_{\\omega}", "19a93ebc67feceda8f02609cfcc96847": "\n \\ln\\left[\\frac{1+e}{1+e_0}\\right] = \\ln\\left[\\frac{v}{v_0}\\right] = - \\tilde{\\lambda} \\ln\\left[\\frac{p_c}{p_{c0}}\\right]\n ", "19a93edbfbcb14ce3b9f423896dba1c0": " \\frac{dN}{dt}", "19a94c143c55b7663bcfe27d51a12cf2": "\\{\\,r^{X_n}:n=1,2,3,\\dots\\,\\}", "19a95445d0c61258be64392a9ff1c155": " d/dz Id - E/z ", "19a96b15cf0c9a45e9314e0d76e5f5f4": "\\mathrm{Interest} = \\mathrm{Principal} \\times \\mathrm{CouponRate} \\times \\mathrm{Factor}", "19a9725c7555f62b3c35c410036f9b7e": "\\scriptstyle|\\zeta|^n\\leq\\|a\\|_1\\max\\{|\\zeta|^{n-1},\\cdots,|\\zeta|,1\\} =\\|a\\|_1|\\zeta|^{n-1} ", "19a9d41a1b22e947799c020049da0dfc": " \\begin{align}\n& \\text{maximize} && \\mathbf{c}^\\mathrm{T} \\mathbf{x}\\\\\n& \\text{subject to} && S \\mathbf{v} = \\mathbf{0} \\\\\n& \\text{and} && \\mathbf{lower bound} \\le \\mathbf{x} \\le \\mathbf{upper bound}\n\\end{align} ", "19a9da7015c340be722505f039eee1c4": "\\tfrac{1}{\\sqrt 2}(\\pm 1 \\pm 1i + 0j + 0k)", "19a9dff0655da20749d7605f858ed898": "\n \\varphi(w) = \\exp\\!\\big\\{i\\operatorname{Re}(\\overline{w}'\\mu) - \\tfrac{1}{4}\\big(\\overline{w}'\\Gamma w + \\operatorname{Re}(\\overline{w}'C\\overline{w})\\big)\\big\\},\n ", "19aa9b43a5b37b0e2fb3c76a039e190b": " N \\times H", "19aa9d41967d0ef9a4a149fa8ae6c81c": "P = \\langle \\psi \\vert \\Pi \\vert \\psi \\rangle = \\vert \\langle \\phi \\vert \\psi \\rangle \\vert ^2. ", "19aad067e6c9338626cc507960677fda": "(e,f)", "19aae4a342457e40d2a76175cf81e309": "(x\\cdot y)^l = y^l \\cdot x^l \\qquad (x\\cdot y)^r = y^r \\cdot x^r", "19ab0a90ac537abbb29aebc8b2cf9cd9": "\\frac{{\\rm d}z}{(z+a)^3z^{1/2}}", "19ab32fc4270d0869b1c02223f69cfd3": "U =L^3 \\frac{8\\pi}{h^3 c^3}\\int_0^\\infty \\frac{\\varepsilon^3}{e^{\\beta\\varepsilon}-1}\\,d\\varepsilon. \\qquad \\text{(3)}", "19ab5b91020dd6bff9e9cdd3d0ee0a54": " {d^2 x^\\mu \\over dt^2} =- \\Gamma^\\mu {}_{\\alpha \\beta}{d x^\\alpha \\over dt}{d x^\\beta \\over dt}+ \\Gamma^0 {}_{\\alpha \\beta}{d x^\\alpha \\over dt}{d x^\\beta \\over dt}{d x^\\mu \\over dt}\\ .", "19ab6cb8f6b9c3e72bd8424a447446a8": "s\\cdot\\sqrt[3]{2}", "19abfe4c3cc868998f84e98086dc1540": "m'\\,", "19ac0fa5ab97186b57ea5c6bc38b10cd": "\nP^\\phi =\n\\exp\\left(\n\\int^T_0\\dot{\\phi}(t) \\, dX^\\phi_t\n+ \\int^T_0\\tfrac{1}{2}|\\dot{\\phi}(t)|^2 \\, dt\n\\right) \\, dP.\n", "19ac1c5586c1bf7a7e142abc86f89793": "u,v,x", "19ac43e8c3ca5e3bf53bed85137d249d": "\\scriptstyle A^\\ast", "19acb64bd338ad59c888c5ada9b04cb0": "a_2 - a_1 < a_m - a_{m-1}", "19ace9a4fe006a187f00a311e7fa6450": "\\nu_t^{3/2}", "19acf66f483ecf85b7f8fdab0693d4bd": "\\frac{1}{s^2}", "19ad202815dcdc697be0ca2f8af02ed7": "\\mathbb{R}^d", "19ad212e0e6f4852aaabbffecbd0a598": "\\frac{\\mbox{Moles of Reagent X }}{\\mbox{Coefficient of Reagent X}}", "19ad72400080ee57575ba5e6d12b0656": " f( x ) = \\frac{ 1 }{ \\sqrt{ k \\pi } } \\frac{ \\Gamma\\left( \\frac{ k + 1 }{ 2 } \\right) }{ \\Gamma\\left( \\frac{ k }{ 2 } \\right) } \\frac{ 1 }{ \\left( 1 + \\frac{ x^2 }{ k } \\right)^{ \\frac{ 1 + k }{ 2 } } } .", "19ad87fb64988ee348de0213390271b1": "g(n)\\cdot k_1 \\leq f(n) \\leq g(n)\\cdot k_2", "19adc977676e56a8aca858461697c157": " \\mathrm{U}(n) \\supset \\mathrm{SU}(n) \\supset \\mathrm{O}(n) ", "19adf2d7ec933a5823fa291984bc16a6": "\\frac{cx^2}{(x^2+y^2)^2}+\\frac{dy^2}{(x^2+y^2)^2}=\\frac{x}{x^2+y^2}", "19adf37be719948d41ac2ff84fa61424": "(1-DX_i)", "19ae73bbf8e200f3699d147aff178bbd": "f = \\frac{a_e-a_p}{a} = {5 \\over 4} {\\omega^2 a^3 \\over G M} = {15 \\pi \\over 4} {1 \\over G T^2 \\rho} ", "19aec715af3ed5b3397a9c9d0872f68c": "t\\in K(C)", "19af21bb9b4b023be42204a69025e8bd": " \\times ", "19af5e9e558f60ddf5d1e2b12dcd7cb2": "\\Pr (K=k)=\\frac{1}{n+1}\\sum\\limits_{l=0}^{n}{{{C}^{-lk}}\\prod\\limits_{m=1}^{n}{\\left( 1+({{C}^{l}}-1){{p}_{m}} \\right)}}", "19af94d6e5f3641dde48c0c79bd5d4e4": "\\varphi = (x^i)\\, ,", "19afd72e6a7da26c356dcc81218ce29d": "{\\Delta C}/{\\Delta x}", "19b02d3dd4df1103e1dea10b8d08781a": "\\Psi(\\mathbf{r}) \\leftrightarrow |\\Psi\\rangle ", "19b04680dcf1a693e29153c390ef8bd9": "x\\otimes x\\;\\stackrel{\\tau}{\\longmapsto}\\;q(x\\otimes x)\\;\\;\\Longleftrightarrow\\;\\; g.x=qx", "19b0807947a89c40475ff4fef2b635db": "2-m_0", "19b0aafa05fb4fcc739fefeffd1e293c": "E_B = \\frac 1 {1 + 10^{(R_A - R_B)/400}}.", "19b0c0960cefee0255c780ae26490805": "\\delta_j", "19b13472cc73d015c3cebdabbdb03832": "\\sigma = f(w)", "19b148b116432a1c2ccc40737d41fea3": "\\textstyle x,x'\\in X", "19b15d7ccb975651bbf415e2fa5dade3": "\\alpha_V", "19b17e66ff8c3deb2341fa0d5866a485": "\\nabla A = 0 ", "19b1901285f5c69e73329a670fcc3535": "V_{\\text{opamp}} = V_s \\left( 1 + \\frac{R_2}{R_1} \\right)\\,", "19b19ffc30caef1c9376cd2982992a59": "gh", "19b1e88684c09f1fd97b8fc2fb1059f7": "\\tau_\\mathrm{n} = 0\\,\\!", "19b205c5161b6d426fdeacda1d455bd4": "ABCD=1010", "19b20dfe64a1935dcd440103cd0a36f1": " = \\frac{1}{(2s_{e-} + 1)(2 s_{e+} + 1)} \\sum_{\\mathrm{spins}} |\\mathcal{M}|^2 \\,", "19b214e3bb80e112875045039b8f4101": "\\nabla g ", "19b240a4310ed6d6e48a5fda7fcf9c50": "\nf_n = \\begin{cases}\nf_{n-1}+f_{n-2}, & \\text{ with probability 1/2}; \\\\\nf_{n-1}-f_{n-2}, & \\text{ with probability 1/2}.\n\\end{cases}\n", "19b242a31882dbbe69834a1b36d633c7": "R_{\\text{1}}", "19b26471066c6bd052d95c02f6e6d5d9": " {\\rm h_{total}= \\frac{P_2-P_1}{\\rho g}} ", "19b3358360077bb651612aef0b1408dc": "(P_i)", "19b360f654e577c2abc91424c3433714": "= [1 + M\\cdot \\cos(\\omega_m t + \\phi)] \\cdot A \\cdot \\sin(\\omega_c t)", "19b38bb0573eae6e7e3e5b9baee43006": " \\sim 10^{701} \\,\\!", "19b3c1f925926e79368c8a8951c9a5af": "C = \\frac {1}{2} \\log(1+\\frac{P}{n})", "19b4182291bbe7cd15f295fa37654c27": "R_{\\delta_i}(x) \\geq R.", "19b42157bd087dbc92cd59f6c72286d0": "dis_{Ham}(c,w) \\leq (d-1)/2", "19b4bb879b62a60b4251e4311f5aeb92": "R_{12,34} = \\frac{V_{34}}{I_{12}}", "19b584de7a9bb72964d85dc543ed0977": "a_0=1-a_1-a_2\\,", "19b59cbcbacfc3652fd72592ed0db241": " (32)\\cdot(200) = (2^{5+3})\\cdot(5^2) = ((2^4)\\cdot(5))^2 = 80^2", "19b5bd207aad033c69bd56930ec2e0b3": "p(x; \\theta)", "19b64546284212be1322293c51b8a471": "\n \\mathbf{M}_1 = \\int_A \\mathbf{r} \\times (\\sigma_{11} \\mathbf{e}_1 + \\sigma_{12} \\mathbf{e}_2 + \\sigma_{13} \\mathbf{e}_3)\\, dA \n \\quad \\text{where} \\quad \n \\mathbf{r} = x_2\\,\\mathbf{e}_2 + x_3\\,\\mathbf{e}_3 \\,.\n ", "19b648cf8bee48bc4ca12e97e4df34bc": "\\left(\\frac ab\\right)", "19b661850a75aca92783b06df5b834cc": "\\mathcal{L}\\,", "19b66ec027c9fea880fb1cbf92bd62c9": "X_2 \\sim \\mathrm{Herm}(b_1,b_2)", "19b6d9a7ee1277c9e5eb424b78abc693": "R(\\lambda) = \\left [ \\sum_{i=1}^{16} w_i R_i(\\lambda)^{1/n} \\right ]^n", "19b76c92067e66821878b9e2397bbddd": " u = \\frac{h \\nu}{k T} \\,", "19b78c1a926d6023d4e3897dfbb25b6f": "K_1, K_2, K_3, K_4", "19b79edf6c3a5914d5c1da7fd5e500f7": " g\\ f = \\operatorname{value}\\ v\\ f = f\\ v", "19b7ba0df50b5e36d365dae3bcb57102": "v_y = U \\cos\\theta\\left({1 + {a^3 \\over 2r^3} - {3a \\over 2r}}\\right) ", "19b7e3c2770ca3af80b91bddc3ed8d57": " F = \\log(\\cosh(J (N_+ - N_-))).", "19b81719c021119746a61796df09bad8": "\\alpha=10", "19b844e328cb4ce92bb1b8bda4d5dfaf": "\\mu_j X_j = \\sum_{i=1}^M \\mu_i X_i p_{ij}\\quad\\text{ for }j=1,\\ldots,N.", "19b87ed308ba10ed3803bb45e01f20ef": "\\Delta(y,z) \\geq 0", "19b89ad066560f7b753dc8598a6fc088": "\\ {d \\over dx}\\left( \\ln \\left| x \\right| \\right) = {1 \\over x}.", "19b8c09ff1e7b274e4f90ff858cf8cde": "m_{Moon} = 0.25 + 2.5 \\log_{10}{\\left(\\frac{3}{2} 0.00257^2\\right)} = -12.26\\!\\,", "19b9611e1299acf17e8b4266ffe71e1e": " \\frac{\\mathrm{d} p^1}{\\mathrm{d} \\tau} = q U_\\beta F^{1 \\beta} = q\\left(U_0 F^{10} + U_1 F^{11} + U_2 F^{12} + U_3 F^{13} \\right) .\\,", "19b9653e1742697fe39fb94d2a1aa6da": "e^{-(\\mu_1\\!+\\!\\mu_2)}\n\\left(\\frac{\\mu_1}{\\mu_2}\\right)^{k/2}\\!\\!I_{k}(2\\sqrt{\\mu_1\\mu_2})", "19b96d02a0a50791dd211f8fd4ff657f": "e_q(z) = E_q(z(1-q)).", "19b9ab217e005c45390acf2a12a9a3c4": "N = \\frac{V_s \\times f_c \\times c \\times r_s}{n_r} ", "19b9b21d5a75e5ecd6e1c677a09d140c": "(r_1,\\ \\vec{v}_1) (r_2,\\ \\vec{v}_2) = (r_1 r_2 - \\vec{v}_1\\cdot\\vec{v}_2, r_1\\vec{v}_2+r_2\\vec{v}_1 + \\vec{v}_1\\times\\vec{v}_2)", "19b9e1dac6d1fc5350248c392bd433df": " \\boldsymbol{E} = - \\boldsymbol{\\nabla} \\varphi - \\frac { \\partial \\boldsymbol{A} } { \\partial t } \\ ,", "19b9eaccd850c0226fbdc0e6da1fb8d0": "O(\\log |\\mathcal U|)", "19b9f6300f1a0609ddedc875e6f59b71": "\\omega \\approx \\frac{2 \\mathrm\\pi~\\mathrm{rad}} {86\\,164~\\mathrm{s}} \\approx 7.2921 \\times 10^{-5}~\\mathrm{rad} / \\mathrm{s}", "19ba8fcfdd3a46917058c9ff09a40dfe": "\\scriptstyle\\mathcal S", "19bacc3fd44f26a41335738ae8b52b85": "\\mathbf{ x}(3) = [u(3)\\, u(4)]=[89\\, 85]", "19bb392121a2c5842fa4e386fed7c0a0": " Z = (\\dot{\\epsilon}) \\exp \\left ( \\frac{Q_{HW}}{RT} \\right ) \\,\\! ", "19bb39cc49fb9a0ba02db83dc61e9851": "\\begin{align}\n\\boldsymbol\\chi' &= \\boldsymbol\\chi + \\mathbf{T}(\\mathbf{X}) \\\\\n&= \\boldsymbol\\chi + \\sum_{i=1}^n \\mathbf{T}(x_i) \\\\\n\\nu' &= \\nu + n\n\\end{align} ", "19bb43bf74147106520153729f3fd2b4": "g(r) = \\frac{Gm_1}{r^2}", "19bb737cf15b5e69c5deaeddc8d78a50": "f(x)=\\sum_{n=-\\infty}^\\infty c_n\\ e^{2\\pi i(n/T) x} =\\sum_{n=-\\infty}^\\infty \\hat{f}(\\xi_n)\\ e^{2\\pi i\\xi_n x}\\Delta\\xi,", "19bb74c03097ddc7a101d6793e919f34": "\\Delta_{\\psi} \\hat{X} = \\sqrt{\\langle {\\hat{X}}^2\\rangle_\\psi - \\langle {\\hat{X}}\\rangle_\\psi ^2}.", "19bb8bbe83478f13dfeb105361319621": "\\scriptstyle p_0 \\,", "19bb90e74ce53bb4cedd401c8691f659": "\\scriptstyle U_2\\in\\tau_2", "19bbe31419185078ac46a87025a0148e": "\n\\Delta \\vartheta(\\vec r) = \\frac{1}{\\vec h \\cdot \\cos \\vartheta_B} \\frac{\\partial}{\\partial s_{\\vec h}} \\left[ \\vec h \\cdot \\vec u(\\vec r)\\right]\n", "19bc9c56d34e225de6c5eed015db3b5d": " M(x) \\equiv \\min(e^{-x} , 1) ", "19bd4919d996b1e7f718530f46949a1a": "F_{\\overline{z}}=0.", "19bdb1f98bfdd2fa1faf981f51a80cd0": " \\forall{x}{\\in}X, P(x) \\or Q(x) ", "19bdc220efb66684958f4205e5d07a36": "\\vec{a} = a \\hat{n} ", "19bddd32f384f364df95153dc8fc5775": " \\frac{\\cot(A/2)}{s-a} = \\frac{\\cot(B/2)}{s-b} = \\frac{\\cot(C/2)}{s-c}. ", "19be0b3d90aea5466c70f303c9de324b": " g_{ab} = \\left( \\begin{array}{cc} \\dot{X}^2 & \\dot{X} \\cdot X' \\\\ X' \\cdot \\dot{X} & X'^2 \\end{array} \\right) \\ ", "19bf24d34caa6b55d8c4cc2890b5376e": "\\{0, 1 \\}^{\\ell}", "19bf9442bea375a24abb4c22e9951a92": "b=2", "19c0dfd60bfd23dca9580b9fa54d1df7": "f(n_t(x))", "19c0f84e39b39fb4ba2e3b40baca13b5": "D_{1,0} = 0, \\!", "19c0fc78906a304590d664631e893d55": "2n^2", "19c1ba136e34dcb75add398063b965f3": "f(x) = 6 + 4 (x-1) + 1(x-1)^2", "19c1bdfa004f5787a842819850078c0f": "(a_1,\\ldots,a_n)\\in R^A", "19c23b056930763275c408b1cb7905e8": "z^T P z", "19c2f36a70f1d70231d5a05f5d646661": "A(\\varphi)=-f(\\varphi)", "19c3908ad79a1cce35d53d79f6519e52": "\\mathbb{E}\\|\\theta-\\hat{\\theta}\\|\\leq\\epsilon^2", "19c3a2a42d3bc62185f9332bf9b78b03": " h*(u*v) = (h*u)*v = u*(h*v)", "19c3baa09c717d34fe353b30af53943e": "\\sigma_{13} = \\sigma_{31}", "19c3c82974aa99285d2b1307e488b299": "\\begin{align}\n- \\underset{\\varepsilon \\searrow 0}\\lim \\int _{\\mathbf{R}^d}\\,f(x)\\, n_x \\cdot \\nabla_x I_{\\varepsilon}(x)\\;dx &= \\oint _{\\partial D}\\,\\underset{\\alpha \\to \\beta}\\lim f(\\alpha)\\;d\\beta, \\\\\n\\underset{\\varepsilon \\searrow 0}\\lim\\,\\int _{\\mathbf{R}^d}\\nabla_x^2 I_{\\varepsilon}(x)\\,f(x)\\;dx&= \\oint_{\\partial D}\\,\\underset{\\alpha \\to \\beta}\\lim n_\\beta \\cdot \\nabla_\\alpha f(\\alpha)\\;d\\beta,\n\\end{align}", "19c3da295c28e5a98b62da1c071db4a3": "\n \\begin{align}\n \\tan \\alpha & =\\frac{\\tfrac{\\partial u_y}{\\partial x}dx}{dx+\\tfrac{\\partial u_x}{\\partial x}dx}=\\frac{\\tfrac{\\partial u_y}{\\partial x}}{1+\\tfrac{\\partial u_x}{\\partial x}} \\\\\n \\tan \\beta & =\\frac{\\tfrac{\\partial u_x}{\\partial y}dy}{dy+\\tfrac{\\partial u_y}{\\partial y}dy}=\\frac{\\tfrac{\\partial u_x}{\\partial y}}{1+\\tfrac{\\partial u_y}{\\partial y}}\n \\end{align}\n", "19c4bee3168e813662b3a5c7265e7289": "C_{st}^1", "19c51400c4739ba1d463780561504985": "Exy \\leftrightarrow [Czx \\rightarrow Czy].", "19c524f58d450c09671bbcf56ad9319c": "(u_1 \\sim v_1 \\text{ and } u_2 \\sim v_2)", "19c5364f4eec335e09972d69d7f854ec": "\n \\varepsilon_{\\mathrm{oct}} = \\tfrac{1}{3}(\\varepsilon_1 + \\varepsilon_2 + \\varepsilon_3)\n ", "19c55d1b297410ca4e154f57a7361135": "(a_2, b_2)", "19c5a88cab87d2ab8a9087a3adcc742c": "f^{-1}(-\\infty,c]", "19c5ac5bf58cc1c43a834fb2eeb2dd62": "f(x) = x \\cdot \\psi(x)", "19c5b1c3f20f72791b9c006e2ba7550d": "(I > 0)", "19c5fd810c785d7c02be5caf7155e63e": "\\overline{\\Sigma}_t", "19c6144ed8ef656efd227f1039569f17": "\\Leftrightarrow m\\sum^{N}_{i=1}w_i(x_0) - m=0 \\Leftrightarrow \\sum^{N}_{i=1}w_i(x_0) = 1 \\Leftrightarrow \\mathbf{1}^T \\cdot W = 1", "19c630f91994382664b4d0a43490e00e": " \\left( E_x ,\\, E_y ,\\, E_z \\right) \\propto \\left(\\cos \\frac{2 \\pi}{\\lambda} \\left(c t - z \\right),\\, \\sin \\frac{2 \\pi}{\\lambda} \\left(c t - z \\right),\\, 0 \\right) . ", "19c63bf027a2b276b972c908673e2fcd": "\\ \\mathbf e_i = \\alpha_{iJ}\\mathbf E_J", "19c65dee9dbe1341f19aedb556e4e012": "h(h(h(h(x))))", "19c66269c491582465c0984a7f1badb2": "\\phi_{\\it nk}", "19c68888c8a31f0cdf5bc9c4f2a13502": "\\scriptstyle\\mathbb{R}", "19c6a57b27e96f30d2184428b5cbd1b5": "\\mathbf{F}_\\mathrm{rad} = \\frac{\\mu_0 q^2}{6 \\pi c} \\mathbf{\\dot{a}} = \\frac{ q^2}{6 \\pi \\epsilon_0 c^3} \\mathbf{\\dot{a}}", "19c6ea88d2dd2b91545bf32eaa4c7354": "\\begin{align}\nR_j &=\\int_G \\varphi(g) gH^{(1)}g^{-1} \\, dg, \\\\\nR_{j,\\varepsilon} &=\\int_G \\varphi(g) gH_\\varepsilon^{(1)} g^{-1} \\, dg, \\\\\nR_{j,\\varepsilon,R} &=\\int_G \\varphi(g) gH_{\\varepsilon,R}^{(1)} g^{-1} \\, dg.\n\\end{align}", "19c75c0cadb518b03f76b2ec2c736893": "100^{100^{12}}=10^{2*10^{24}}", "19c7d8a7070d3e0a2848f6ccc987fbf7": "n_\\mathrm{A} = n_\\mathrm{B} \\times \\frac{R_\\mathrm{B}-R_\\mathrm{AB}}{R_\\mathrm{AB}-R_\\mathrm{A}} \\times \\frac{1+R_\\mathrm{A}}{1+R_\\mathrm{B}}", "19c80cc2f677c57189bb16ae5cab2f29": "||p(\\hat{x_0} + \\sigma) - p(\\hat{x_0})||", "19c813e392f357145417b8f60b426696": "g_1(x)=f(x)^e - C_1 \\in \\mathbb{Z}_N[x]", "19c820f1ae9866f44a5b7cef354745af": " P(a,b) = \\int d \\lambda \\cdot \\rho(\\lambda) \\cdot p_A(a, \\lambda) \\cdot p_B(b, \\lambda) ", "19c839db16b077c602f45a2d0dd27959": "\\varepsilon = \\frac{B^2}{4A}", "19c84108184d89f3a1482920d2fae171": "d\\phi = d\\phi' - \\omega \\, dt.", "19c888ba52a35eab3a5af395c6d1faf4": "v_5 \\ge 1", "19c92099d3668b7da1a0cf6ddd1dcee0": "|\\psi(X,t)|^2", "19c992110e5455b198ab03e8d1ad79bf": "\n f(z) = \\int_0^1 \\frac{d}{dt} f(t z_1,\\ldots, t z_n) dt \n = \\sum_{i=1}^n z_i \\int_0^1 \\left. \\frac{\\partial f(z)}{\\partial z_i}\\right|_{z=(t z_1, \\ldots, t z_n)} dt,\n", "19c9bf426005882255abb29721a7150a": " H^{ex}= \\frac {b_1 X_1 b_2 X_2}{b_1 X_1 +b_2 X_2} \\left( \\frac{\\sqrt{a_1}}{b_1}- \\frac{\\sqrt{a_2}}{b_2} \\right)^2", "19c9f4929ee9063820dc84f68ff51e2c": "\\Delta^{\\mathcal{I}}", "19ca14e7ea6328a42e0eb13d585e4c22": "36", "19cba33fcdc47a9f58cec7f2965061e9": " P_1.P_2 \\equiv P_2.P_3 \\equiv P_3.P_1, ", "19cbb2503e262ab7024cc88bcd963e1e": "\nY_i = \n\\sum_{j=1}^s a_{ij} h F_j + \\sum_{j=1}^r u_{ij} y_j^{[n-1]}, \\qquad i=1,2, \\dots, s,\n", "19cbba1a7e1e68a7b0c695568bcc0189": "\\sqrt{2}^{\\sqrt2}", "19cc0601bb5dc8fdd4098eb03f50e0b9": "f(t)\\,", "19cc7cb792134fba45b1afd05d094fd2": "ki = j\\,", "19cc903b5cd8b382913a6a74a4202327": "\\boldsymbol{H}={\\boldsymbol{B}\\over\\mu_0}-\\boldsymbol{M},", "19cca7437823282823f09e331017607d": "\\scriptstyle <10^{-18}", "19ccc747dd43a70a2eb0fe20d641e764": " \\overline{\\Delta M} = - 1.382\\sigma^2", "19ccefb55180e7fdd751bcd8c63cda03": " ab > 1+\\frac{3}{2} \\pi.", "19cd10d86dd5d8353a4df2185f6de2f8": "z'_0 = 1 \\,", "19cd1682b899b2c3d73c7fbe7917b7d0": "\\displaystyle P(x)(v) = v+F(v)x=v+v^qx", "19ce0b99a4d47b4da8c60d07b1b9e15d": " \\dot S_{i} \\ge 0", "19ce3f1ab29759bfa4d3be5f8907ff4f": "E_{i}=B_{ij}^{0}D_{j}+\\tilde\\delta_{ijk}\\frac{\\partial D_{j}}{\\partial x_{k}}=B_{ij}^{0}D_{j}+(ie_{ijl}\\tilde{g}_{lk}k_k)D_{j}\\, ", "19ce4fbf1ad559a3000196a288479889": "r_{\\text{o}}", "19cebccfc5684b769c284fdf48909b14": "CD = \\tan \\theta\\,", "19cebe3bf9410862bf336407424b43fc": " \\langle x, z \\rangle ", "19cf1f5cf01713eb38f2476ea6794f5d": "-0.060607633\\ldots", "19cf6eade436c435282f89a1395a26b2": "\\mathbf{d}_{i+1} = \\frac{(1-t)(1+b)(1-c)}{2}(\\mathbf{p}_{i+1}-\\mathbf{p}_{i}) + \\frac{(1-t)(1-b)(1+c)}{2}(\\mathbf{p}_{i+2}-\\mathbf{p}_{i+1})\n", "19cf8dce89a1372496bfd1be8532d42e": "\\bar{T}", "19cf9e131a73c44f4a4bcc6c6d340b9a": "\\chi^2_\\nu", "19cfcda70d2f5ead370e3b01e6ec3c66": "\\min_{x\\in X}\\sup_{y\\in Y} f(x,y)=\\sup_{y\\in Y}\\min_{x\\in X}f(x,y).", "19d016fa223651f47973cb72766e9527": "\\Delta S = S_{final} - S _{initial}", "19d100d2614e0a93db2379ed36e54876": "1800 \\lessapprox Re_{crit} \\lessapprox 2400", "19d12e5ba4a943ea376b0525ef4a000d": "(a+b-c)ab=24", "19d130cfcb822cf5a8bc39cba9f19aa5": "\n\\begin{align}\nS_\\psi &= I -2|\\psi\\rangle \\langle\\psi|\\quad \\text{and}\\\\\nS_P &= I -2 P.\n\\end{align}\n", "19d16bcb2c1f294750496d3c0f6f7555": "S_{z_U}", "19d16f72700692802d294b69fa873395": "\\textstyle{f(x) = \\sum_{n=0}^\\infty {w^n s(2^{n}x)}}", "19d17e3162ab21978a86db1a21dbae90": "x^2 + bx = a.\\,", "19d18610a6b0c235295aed555c112ba7": "\\psi_n(z)", "19d2515790047bf939dd2a786d1d3cc8": "P_i < G, i=1, \\dots ,n", "19d2ce8beee90458271ab79fd08cf1cf": "\\sum_{n=0}^{N}a_n \\Bigg(\\alpha \\frac{e^{j(\\omega t + \\phi)}+e^{-j(\\omega t + \\phi)}}{2}\\Bigg)^n\n= a_0 + \\sum_{n=1}^{N}\\frac{a_n \\alpha^n}{2^{n-1}}\\frac{(e^{j(\\omega t + \\phi)}+e^{-j(\\omega t + \\phi)})^n}{2}", "19d2f26f196b480e62c56dff95dd107a": "x = \\frac{X}{X^2+Y^2},\\ y=\\frac{Y}{X^2+Y^2}.", "19d329a577b3e139308b3abaab455c89": "\\!l = \\sqrt{3} r", "19d37304f5d5933766c3b54ccf3b7bfd": "\\Pi(z) = z\\Pi(z-1)\\,.", "19d3802e75e05fcce94c9d8d7a94d589": "k = e^{23.1} \\cdot e^{-12,667/T}", "19d3a9a62f475acb99eff3683457f78f": "1 - \\left(1-\\frac{1}{10000}\\right)^{20000} \\approx 86\\%", "19d3b9750b13401d80831e278e9581c6": "MQ \\,", "19d3d79d70f2ea1daca3582cdd52bdd7": "\\|Mf\\|_{L^p} \\leq A \\|f\\|_{L^p},", "19d476d1534b797c69bb94e001f67823": "\\mathbf{E}' = -\\mathbf{B} \\times \\mathbf{v} = \\mathbf{v}\\times \\mathbf{B}.", "19d494636a741f5fe90f02735cdc5250": " N(X) = \\frac{1}{2\\pi e} e^{ \\frac{2}{n} h(X) }.", "19d49fd1539d86c4701153bc4e669b34": "d(x,S) \\leq d(x,y) + d(y,S),", "19d52a1fc0de8a0a9861c2e109eb679f": "= \\oint_{\\partial\\Sigma} \\left( \\mathbf{F} \\times \\mathbf{G}\\right) \\cdot \\mathrm{d} \\mathbf{r}", "19d5467fd2a07ae9225509ec50bcb5d5": "g_{UV} : U\\cap V \\to \\operatorname{GL}(F)", "19d58f2d793f80d2f18fc5edc52d683e": "|1-q|", "19d59c7d61b117f2865b20b59c8b70f7": "x-y", "19d5a905a46d2e3faa1e584d524cdb6a": "\\mathrm{O}_3", "19d5bfd3a00608c306c61445b33f73d6": "\n\\overline{P}(Cl_t^{\\geq}) = \\{x \\in U \\colon D_P^-(x) \\cap Cl_t^{\\geq} \\neq \\emptyset\\}\n", "19d5c6c3002f4dedc7c75e65e416fc13": "W_D(x,s) = \\sum_{i \\in (1\\dots r)} |\\frac{\\part}{\\part s}P_{D,}(d_i,x,s)| ", "19d5e63a41b26deff42a5c627b4e271a": "g_{ik}\\neq g^{ik}\\ ", "19d5e7beaca99d75df5d6e3f2c1df9ff": "L = \\{a^n b^n c^n | n \\ge 1\\}", "19d5f54c1a8fe599c647b41bed0c66f6": "w^\\ast_n", "19d6076fe27eec3683405cd87220e37d": "(\\{T, F\\}, \\lor)", "19d6857016d7c2a1603696bd8a78ce5e": "s_N(x)", "19d698e032aa0c5de7f29b6f0c291cfe": "\\mathfrak{p}'", "19d6b273d7fda9fe614a8b6dcce69814": " Q(\\alpha) \\, = \\, \\{c_1 v_1+c_2 v_2+c_3 v_3 \\mid \n0 \\le c_1 \\le c_2 \\le c_3 \\le 1\\}.\n", "19d704642bc8d8441dc7680d8bd6347c": "E[X] = 0\\times1/2+1\\times1/2 = 1/2", "19d71b55dd67b6d3af7c01c95a25f644": "C = A - B", "19d7691f6d9fc1dd503107094b60ab00": "k_A", "19d80eaa54ef756020d4ba7d67f6e39a": " = n.\\,", "19d8205e9a5e75f4f1b2d7a97f9fffaf": "IF^n \\rightarrow p^{n-2} \\exist p+F^n", "19d83eb051c7ee1bfdb1894b14df4e77": "u,v \\in V, u \\neq v", "19d87dd758fe8a8fd34f4e207c10d878": "v_1, v_2,\\ldots, v_n", "19d8dab4bc8f4433d58fe289d9a1d410": "0 : \\mathbf Z/n \\to \\mathbf Z/n", "19d8dc9f9780eea68129d97f538d9ccf": "d(f(x),f(y)) = r d(x,y).\\, \\,", "19d93df3252419000075075016bdde58": " Merit_{S_{k}} = \\frac{k\\overline{r_{cf}}}{\\sqrt{k+k(k-1)\\overline{r_{ff}}}}.", "19d978e1f7091576893bd7ee5e9f8288": "A\\preceq B", "19d9924ef3d60e751887845dd94cb2c0": "d=2^s (1\\leq s\\leq m)", "19d9a6d3b3f3fa14aebacd9aec364d50": "\\int\\operatorname{arcosh}(a\\,x)^2\\,dx=\n 2\\,x+x\\,\\operatorname{arcosh}(a\\,x)^2-\n \\frac{2\\,\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}\\,\\operatorname{arcosh}(a\\,x)}{a}+C", "19d9ae5c1ab2311eb797d0af294413bd": "Z_{metal} \\, Z_{slot} = \\eta^2 /4,", "19d9c40da08b11d7eb4b9e69e0e834d5": "s_n = \\prod_{i=1}^n\\pi_i", "19da2c42ed41912c144cda0d98e7f8f6": "\n\\sum_{i=1}^{n}a_{i}\\bar{a}_{i}b_{i}\\bar{b}_{i}+\\sum_{i \\lambda\\} =m\\{x: \\, \\omega(f-g)(x)> \\lambda\\} \\le m\\{x: \\, (f-g)^*(x)> \\lambda\\} + m\\{x: \\, |f(x)-g(x)|> \\lambda\\} \\le C\\lambda^{-1}\\|f-g\\|_1.", "19db16814aff96b326320bed21b785cc": "\n\\left(\\part^2+m_0^2\\right)\\varphi(x)=j_0(x)\n", "19db670847403247777e2501e38b2186": "p, q \\vdash (p \\land q)", "19dbbc35c052b7a07c9435928d957f96": "\\sqrt{\\frac{75.625}{BAF}}", "19dccc13f5d90e676c4582fbcb18122a": "W'=-p_{n-1}\\,W,", "19dd05a36a09c3e84a348ecc349229b5": "P_3=0", "19dd0627ce59cfd646f739d6eb1bdedb": "I_2 = \\left( \\frac{Z_1}{Z_1 + Z_2} \\right)I", "19dd2025a9538bd04f531c342b0970c5": "\nh(x) = \\int{\\frac{1}{v(x)}dx} .\n", "19dd5c655a402ccb3d9130e134eb86df": "Z_\\mathrm{S}", "19dd745b55ffbd6ab638d5a9e5c2b72a": "{\\mu_{roll}}", "19dd7be9c3492ddb11062df460ae1d13": "e_\\bar{\\alpha} = L^\\gamma{}_\\bar{\\alpha} e_\\gamma ", "19dd8f409402ed2ebf0a7f7e4bf00cd2": "\\frac{\\sqrt{3} + x}{\\sqrt{3} - x} = 2", "19dd93004dcba98538034e183e1129b0": "u \\in \\mathrm{Im} (\\theta)", "19ddfa387dad84759572266b01b283e5": "[v'\\;\\|\\;N\\;\\|\\;[u\\;\\|\\;v\\;\\|\\;M]_h]_m \\!\\rightarrow\\![w\\;\\|\\;M]_h\\;\\|\\;[w'\\;\\|\\;N]_m", "19de00ba89c28bd16f9e6e4d89e87a2c": "\\mathbb{E}^g[1_A \\mathbb{E}^g[X \\mid \\mathcal{F}_t]] = \\mathbb{E}^g[1_A X]", "19de22b6b6f1eaa2c81473906c2cb0ce": "\\beta^\\prime_n = \\sum_{j\\ge0}\\frac{(\\rho_1;q)_j(\\rho_2;q)_j(aq/\\rho_1\\rho_2;q)_{n-j}(aq/\\rho_1\\rho_2)^j\\beta_j}{(q;q)_{n-j}(aq/\\rho_1;q)_n(aq/\\rho_2;q)_n}.", "19de55e31f7adde48f9d4ff1a7b7cd58": "s_{k+1}=\\pm\\frac{b}{GCD(a,b)}", "19de5995f93d1366db2003d2f2dc9352": "\\beta(Y,X)", "19de5ea00c082bfabab8ed300916e90d": "P(t) = e^{Qt} = \\sum_{n=0}^\\infty Q^n\\frac{t^n}{n!}\\,,", "19de801e778d3555e11f8e65f6cad3d9": "{\\mathfrak d}", "19dec9ddba44f4d334bf71b538bf9935": "j_\\ell(kr')=\\frac{(kr')^\\ell}{(2\\ell+1)!!}+O((kr')^{\\ell+2})", "19df0bd299379fa1b03b3befb0919996": "\\Theta|n\\rangle", "19df16fdd3b97d5e761ed662490747b6": "R_z = \\frac {1} {Y_{21}} \\qquad R_x = \\frac {1} {Y_{11} - Y_{21} } \\qquad R_y = \\frac {1} {Y_{22} - Y_{21} } \\, ", "19df1c2726ed43128440c1157f72a937": "\\psi ", "19df4ca33417c98bcc9aa9bd7dc3e761": "\n\\begin{align}\nN u &= -b \\partial_{x} u^{2} = -b \\partial_{x} (u_{0} + u_{1} + u_{2} + u_{3} + \\cdots)(u_{0} + u_{1} + u_{2} + u_{3} + \\cdots) \\\\\n&= -b \\partial_{x} (u_{0} u_{0} + 2 u_{0} u_{1} + u_{1} u_{1} + 2 u_{0} u_{2} + \\cdots) \\\\\n&= -b \\partial_{x} \\sum_{n=1}^{\\infty} A(n-1),\n\\end{align}\n", "19df61df78d2023969e529969db8666b": " S = \\frac{ 2 }{ \\beta^2 ( 4 + 5 \\alpha^2 ) }", "19dfc03da5a2401a24bcd161e1c7005b": "\\mathbf{E}_{\\text{Electric dipole}}(\\mathbf{x},t)=Z_0(\\mathbf{H}_{\\text{Electric dipole}}\\times\\mathbf{n})", "19dfe1ce8dcf67b391022a230d12ca02": "\n\\begin{pmatrix}\\alpha^{7}+\\alpha^{-3}x&1\\\\ 1&0\\end{pmatrix}\n\\begin{pmatrix}x^6\\\\\n\\alpha^{-7}+\\alpha^{4}x+\\alpha^{-1}x^2+\\alpha^{6}x^3+\\alpha^{-1}x^4+\\alpha^{5}x^5+(\\alpha^{7}+\\alpha^{7})x^6+(\\alpha^{-3}+\\alpha^{-3})x^7\\end{pmatrix}=\n", "19dfe2d759ec6d6869da6d2185fc0294": "\\phi(x,z)", "19dffbdddca00c828918d66ba4dc5c72": "\n{{\\Delta \\hat g} \\over {\\hat g}}\\,\\,\\, = \\,\\,\\,\\,{{\\hat g\\left( {L + \\Delta L,\\,\\,\\,T + \\Delta T,\\,\\,\\,\\theta + \\Delta \\theta } \\right)\\,\\,\\, - \\,\\,\\,\\hat g\\left( {L,\\,\\,T,\\,\\,\\theta } \\right)} \\over {\\hat g\\left( {L,\\,\\,T,\\,\\,\\theta } \\right)}}{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,Eq(4)}}", "19e05ebd8991fb1887041c225c155b85": "\\mathit{EQ_2}", "19e072a673ccaf1a4033f4ad50f09e4f": "cos(\\beta) ", "19e0b5b8efdc14659ec37dcc46c83dc0": " \\sigma_2 = \\left[ \\begin{matrix} 0 & i \\\\ -i & 0 \\end{matrix} \\right]. ", "19e0bdfb900e2453070209e52b1652fd": "\\mathfrak{T}_{\\mu}^{\\nu} = T_{\\mu \\gamma} \\, g^{\\gamma \\nu} \\, \\sqrt{-g}.", "19e0da4efbbbc66c49565ee1ad98bdec": " \\Phi(r) = k_e \\frac{Q_1}{r} ", "19e10bb4977d06dc80bb1a04c601f19f": "\\left(p,q,Tr(g),Tr(g^k)\\right)", "19e11abbbd15592e0b71681078a54b56": "\\langle\\xi_i(t)x_k\\rangle=\\langle x_i\\xi_k(t)\\rangle", "19e16962430ae761a0a084cf561c0fb7": "\\hat{B}^\\dagger_\\mathbf{q}", "19e1d7a54cc56a652bf6cdf9232f1b08": "x_i = w_i \\cdot \\frac {M}{M_i}", "19e1dc236e52a6b58a1b97a3c875f3e3": "e=(e_1~e_2~\\ldots~e_N)^T", "19e1f9ce7932cf347eb1e6c8eb2e9844": "f\\star g = m \\circ e^{\\frac{i\\hbar}{2} \\Pi}(f \\otimes g),", "19e2174c07384de3e01d224eb444dd85": "1/I+1/O = 1/F", "19e2212113b16a50c0b6dca8e2d281f1": "\\sigma(\\mathbf{x}) = \\mathbf{0}", "19e233c942f346c6d3c194b220ec2dcb": "(R \\bowtie S) \\cup ((R - \\pi_{r_1, r_2, \\dots, r_n}(R \\bowtie S)) \\times \\{(\\omega, \\dots \\omega)\\})", "19e2582f3ecf941ec253e61681428bd5": "S_{25} = \\sigma_{put,25} - 2 \\sigma_{atmf} + \\sigma_{call,25} ", "19e25da981a1c93f5b6b1014a8da5d0b": "t_{0.975,11} = 2.20", "19e2adc1d3d62258a2e756cc95311b79": "ku", "19e2c87dfcd499d96746ef27b99584a5": "\\Delta H", "19e2d001c09a4f577d39d167abf246ba": " \\langle u,\\ v \\rangle = \\|u\\| \\|v\\| \\cos \\theta \\ ; \\ (-\\pi < \\theta \\le \\pi)\\ , ", "19e30df859ba67aeef61c8f5d50ff631": "([A,B]:[B,D])\\diamond([A,C]:[C,D])\\,\\!", "19e35b7c71424608ffca4314bad0f968": "R/i-P", "19e363414ccd6e4acbcb5d2871c81440": "i \\in \\{1, \\dotsc, n\\}", "19e371803b9249b55ff650318c5f403f": " G(z) = \\begin{cases} \\exp\\left\\{-\\left( -\\left( \\frac{z-b}{a} \\right) \\right)^\\alpha\\right\\} & z t + s) \\over \\Pr(X > t)} = \\Pr(X > s).", "19e615e86e3d41cab8e2947599fb6087": "[Z_0,\\dots,Z_n] {\\sim} [1,z_1,\\dots,z_n],", "19e65262ebd183549d4ac03426fedc99": "407=4^3+0^3+7^3", "19e682ea3bbfe7bb2e5f7698d09654cb": "\ne = \\sqrt{1 + \\frac{2EL^{2}}{k^{2}m^{3}}}\n", "19e6a1351f4593fa6a5da49794269627": "\\scriptstyle (T-T_\\mathrm{g}) \\,\\le\\, t \\,<\\, T", "19e6a1d69386ac0d16c7dde15eef91da": "z = z_2,", "19e6e1c159523f625c67d4492ed1c447": "f_c'(z_0)\\,", "19e6f6f3a2e7fc4953ef59e0428767b8": "\\eta_\\mathrm{max} = {T_H - T_C \\over T_H} {\\sqrt{1+Z\\bar{T}}-1 \\over \\sqrt{1+Z\\bar{T}} + {T_C \\over T_H}},", "19e7217bfb2e45ed28490e0382fb1b5b": "I_a", "19e72e04be6bd147dff922246c7f44c3": "g_\\mathrm{N}", "19e750dab3777f80f35c1f2a308d91c8": "S_{ff}(\\ell) = O(\\ell^{-s})\\quad\\rm{as\\ }\\ell\\to\\infty", "19e7510288fd3003c4417537f390de4f": "r =\\,\\alpha\\beta;", "19e761868636d00524df5d7c593b12b7": "\\textstyle i=1,\\ldots,n", "19e7d50d4f002dee55277c8e4ef278f2": "\\,p\\in[0,1]\\,", "19e863e91f9e87dfcc34b1a02d54acf8": "AB^+ + h\\nu \\to A + B^+", "19e8654ae6dd5270a6a66ccabff752b2": "C \\frac{dV}{dt}+V \\frac{dC}{dt} = -K \\cdot C + \\dot{m} \\qquad(8)", "19e8fe8dc8c51f66a419e33370a8184d": "\\delta \\psi_{out} = \\left( u + \\frac{\\partial u}{\\partial x}\\delta x \\right) \\delta y + \\left( v + \\frac{\\partial v}{\\partial y}\\delta y \\right) \\delta x.\\,", "19e91a54a86f07291e8fed48b3b39929": "\\frac{k}{2} (1 + \\ln (2\\pi)) + \\frac{1}{2} \\ln |\\boldsymbol\\Sigma |", "19e92905a38d916db89789f0f2ad9ef7": "e^{(2)}_i", "19e97d692138430a6798e7982a6afb4a": " V(t) = Vo(e^{-t/ \\tau}) ", "19e98ae1ed9addc1200ac4848dc319c8": "\\frac{\\partial^2 f}{\\partial x_i\\, \\partial x_j}(a_1, \\dots, a_n) = \\frac{\\partial^2 f}{\\partial x_j\\, \\partial x_i}(a_1, \\dots, a_n).\\,\\!", "19e9b798261f7e946e5f4224ca8bed43": " \\frac{dy}{dx} = \\frac xy.", "19ea073c275437e1989504018be9d469": "\\varphi_0(\\beta) = \\omega^\\beta", "19ea192b116c0c4596b4e4fa425109e2": " \\varphi (m)= 2 \\times 4=8 \\,", "19ea3a21eded0317d0d66c37967a34bd": "\\langle x,y \\mid x^m, y^n, xy=yx \\rangle\\,\\!", "19ea6c34cad73649f5458684f75d4938": "\n x_t = x^*_t + \\eta_t\\,,\n ", "19ea97c73dd38f9fd6de34bbc3643d8e": "\\gamma_1 = 1", "19eacb3d42b4cf08e0cdfd1156aa2a11": "\\{y \\mid y W x\\}", "19eb17812e5818b7937d6aaad1924a31": "\n \\boldsymbol{F} = \\boldsymbol{\\mathit{1}} + \\gamma\\mathbf{e}_1\\otimes\\mathbf{e}_2\n ", "19ebf996e999e8465f8f46aacb4871ba": "\\sigma_y = \\sigma_2\\,\\!", "19ec015a1157fe19cc585137ffc7498c": "\\psi(\\mathbf{r_1},\\mathbf{r_2},\\,t) = \\psi_A(\\mathbf{r_1},\\,t) \\psi_B(\\mathbf{r_2},\\,t) ", "19ec3635ded4bd76dd08dbcf77c70174": "q_{\\mathit{left}}", "19ec3a462af7814b2ec18c24ccc61713": "\\ P_r = S \\frac{\\lambda^2}{4 \\pi}", "19ec53dfabe89169ed7a8bc65cd88b59": "X \\subseteq E", "19ec5e9b380068ccdeb1484dcb25ed2a": "\\kappa=\\pi=\\varepsilon=0\\,,\\quad \\rho=\\bar\\rho\\,,\\quad \\tau=\\bar\\alpha+\\beta\\,.", "19ecaeaf6300fcee068abbd62a647346": "\\sum_{i=1}^k c_i = \\sum_{i=1}^k d_i = 1", "19ece1047cce719d82cc4d85015d7680": "\\mathcal{C}^k", "19ecf8797efa6d5fe048d4cf9eeeb878": " \\tilde{R}_n = \\begin{bmatrix} R_n \\\\ 0 \\end{bmatrix}, ", "19ed6e5bc03178928ac93ba1c3f37fa4": "[[x,y],z]", "19ed86b1d0789172abc1b1d706b65638": "\\mathcal{F}_x = \\varinjlim_{U\\ni x} \\mathcal{F}(U),", "19edf4beb4324355ecacca062fb1591a": " \\operatorname{Var}(X) = \\sum_{k=0}^{n} {n\\choose k}p^k(1-p)^{n-k} (k-np)^2 = np(1-p),", "19ee71762e0afea39366e1117aa8c7d4": "A=CF", "19ee9cac88d24cff99fbd8fb801c3999": "\\mathcal{L} = p_1 \\partial_x + p_2 \\partial_y,", "19eecdc3c1f102237a0373c741b27c67": "\\mathcal{Z}'\\{x[z]\\}=\\frac{\\mathcal{Z}\\{x[z+1]\\}}{z+1}", "19ef01f138f295a96bea25ef79c3ab01": "\\int_C \\mathbf{F}(\\mathbf{r})\\cdot\\,d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t))\\cdot\\mathbf{r}'(t)\\,dt = \\int_a^b \\frac{dG(\\mathbf{r}(t))}{dt}\\,dt = G(\\mathbf{r}(b)) - G(\\mathbf{r}(a)).", "19ef17a8f2028f4e879aff3b3bc287e1": "{ D+\\Delta{P}_t \\over D_i } = \n{\\Delta{P}_t \\over P-\\Delta{P}_t } \\left ( { P-\\Delta{P}_t \\over D_i }\\right ) + {D \\over D_i}", "19ef2509d2a6fa6c934d4610a440c3a4": "|R_2| \\leq \\frac{9}{10}\\frac{1}{2r+1}\\frac{1}{3^{2r+1}};", "19f03da607592325113d5e0ed71242fb": " \\mathbf{g} = \\frac{1}{\\mu_0 c^2}\\mathbf{E}\\times\\mathbf{B}\\,,", "19f044a9f1a4d75e246b101e61f63a95": "\\mid \\bar{q} \\mid< l/2", "19f0b4440903a864457d497731661d2d": " \\mathbb{P} (Y=0|X=x) = \\frac{ \\binom 7 x }{ \\binom{10} x } = \\frac{ 7! (10-x)! }{ (7-x)! 10! } ", "19f0b788b7ce130449e81a92ee66907e": "V_{IN}", "19f152fabc8a6dc415825d82b46c7a3c": "x[n] = \\left \\{ \\cdots, -(0.5)^{-3}, -(0.5)^{-2}, -(0.5)^{-1}, 0, 0, 0, 0, \\cdots \\right \\}.", "19f18ba4e152bad3cdca6ff11780bbef": " \\displaystyle \n \\rho \\left( \\frac{\\partial v_i}{\\partial t} \n + v_j \\frac{\\partial v_i}{\\partial x_j} \\right) = \n - \\frac{\\partial p}{\\partial x_i} \n + \\frac{\\partial}{\\partial x_j} \\left[ \n \\mu \\left( \\frac{\\partial v_i}{\\partial x_j} + \\frac{\\partial v_j}{\\partial x_i} \\right) \n + \\lambda \\frac{\\partial v_k}{\\partial x_k} \n \\right] \n + f_i\n ", "19f2c53db72e572a33e66c2c9a8c9e75": "1000\\times 9.5 = 9,500", "19f37dd848c814d7721337169025fc0b": "\\scriptstyle \\mathbf{b}_2", "19f3b8814470c48c1701ce86eb7fd1ae": "\\alef_0\\rightarrow(\\alef_0)^n_k", "19f3bfaaa153d2f7c8779a9d6bf34acc": "(\\mathbf{a}\\times \\mathbf{b})\\times\\mathbf{c} = (\\mathbf{c}\\cdot\\mathbf{a})\\mathbf{b} - (\\mathbf{c}\\cdot\\mathbf{b})\\mathbf{a} ", "19f3ed0adf8b37229a92669c19f99c5d": "M_1 ", "19f42938f5094baf6e314ab14bf64a73": " \\begin{align} \\mathbf{B} + \\mathbf{C} &= \\mathbf{a} \\wedge \\mathbf{b} + \\mathbf{a} \\wedge \\mathbf{c} \\\\ &= \\mathbf{a} \\wedge (\\mathbf{b} + \\mathbf{c}).\\end{align}", "19f49972ac33aa720457d45522c4e108": "\\mathrm {EV} = \\log_2 {\\frac {E \\cdot S} {C} } \\,.", "19f4ab5a9238057877d5e89024b6d0ed": "g_k", "19f4c529bdb948b6d80e3e0c0cfc1605": "\\sum_{\\sigma}\\sum_{n_1\\geq n_2 \\geq \\cdots \\geq n_k \\geq1} \\frac{1}{{n^{i_1}}_{\\sigma(1)}{n^{i_2}}_{\\sigma(2)} \\cdots {n^{i_k}}_{\\sigma(k)} }", "19f4e43d9772f06bf5e1198d7256e31b": "\\displaystyle \\overline{\\hat{f}(-\\xi)}", "19f4e95b426e10d59deab317f8bfb957": "{E} = {V_1^2}-\\frac{V_{r1}^2}{2}", "19f514e670867a0af1ee7ac228b3b2c2": "(\\forall X (Xx \\leftrightarrow Xy))", "19f5714b750b80517cdc6f9c63fbdc28": "S_8", "19f578931c3af9b77daf85445c2db33e": "3N-3-2 = 3N-5", "19f5cbc206c6c4a4fedb8d5c4e7f19e8": " u_6 =0.69972 ", "19f5fb4486a25886a8d5654f1d31a53a": "N=V\\setminus\\{root\\}", "19f6141bd3a411bb66154601d81e12eb": "c_0^2\\nabla^2\\rho", "19f6296bb1e2e913c9506f28db491ab0": "G=(X, E)", "19f6b7485588607da0a25e5784bd6776": "\\tan \\psi = \\frac {u'} {f} \\sin \\theta \\,.", "19f73c0abf03d7e8cf543f15bda8a808": " \\operatorname{E}_{k-1} e^{\\mathbf{X}_k} = \\operatorname{E} e^{\\mathbf{X}_k}. ", "19f73cdee765afc7c34c68f19e003e83": "\\textstyle S_{11} = -S_{22}", "19f7671aebc0f58827d30f597f582990": "\\Delta G_{w}", "19f79f1f80b8d634a8b43f18d07c2f4b": "\\begin{align}\nw_{HNO_3}&=0.70\\times 0.76=0.532\\\\\nw_{HF}&=0.49\\times 0.04=0.0196\\\\\nw_{H_2O}&=1-w_{HNO_3}-w_{HF}=0.448\\\\\n\\end{align}\n", "19f7ecca7c61bddb1282cefdbcdbe993": "a=-1.25,\\ \\phi^* = 1.2 \\times 10^{-3} h^3 \\mathrm{Mpc}^{-3}", "19f7fbbded647e379da8c9dd976343c3": "\\mathrm{mul_c}", "19f81abe0bafd5776f593d3ddf9a8a76": "h(x), f(x) \\in K", "19f83db071a886771a411d7e436a444b": "1 - \\chi(S), \\, ", "19f8435770c6cf15b75d13df6d3fb29b": "\nk", "19f844d3b28451854c2a97281430c033": "P_D = l_D + (1 + r) l_A a_D = l_D + {a_D (1 - l_B) \\over a_B} ", "19f88407ddcf9043f5fdc39d86b837b0": "\\left(\\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "19f8a960e8fcf709052cd72ad0066101": "(A\\to(B\\to C))\\to((A\\to B)\\to(A\\to C))", "19f8b50de8053e5d7233e892720f0bfb": "d(f_i(x), a) \\leq M \\qquad \\forall i \\in I \\quad \\forall x \\in X.", "19f8e7022cf0c204be2fbd5db3763f7e": "(29)\\quad \\psi(r,\\theta)\\,=-\\sum_{n=0}^\\infty a_n \\frac{P_n(\\cos\\theta)}{r^{n+1}}\\,, ", "19f9046052cad13eeef33b1443203200": "xy \\vee \\bar{x}z \\vee yz", "19f923a0584ed8e2c873a2427fe226bb": "\\omega^{2^{p-2}} = kM_p - \\bar{\\omega}^{2^{p-2}}", "19f92bcef936d797e9fa39a314d8306b": "=A'(x)u_1(x)+A(x)u_1'(x)+B'(x)u_2(x)+B(x)u_2'(x)\\,", "19f93db6f2cd6d7e33c1b6e336095b2a": "\\scriptstyle L,", "19f9619030770233e121c27c86b478e9": "a_0,a_1,a_2,\\dots", "19f96704a10c3249470dfcfba027a021": " Y(z) = -\\partial \\xi e^{-2 \\phi} c(z) ", "19f96cc300776555dc370f33a7447141": "\\mathbf{T}^{(\\mathbf{n})}=T_i^{(\\mathbf{n})}\\mathbf{e}_i\\,\\!", "19f97e2a278f0b500938ded9a6752176": "\n \\begin{align}\n \\boldsymbol{\\sigma} & = -p~\\boldsymbol{\\mathit{1}} + \n 2\\left[\\left(\\cfrac{\\partial\\hat{W}}{\\partial I_1} + I_1~\\cfrac{\\partial\\hat{W}}{\\partial I_2}\\right)\\boldsymbol{B} - \\cfrac{\\partial\\hat{W}}{\\partial I_2}~\\boldsymbol{B} \\cdot\\boldsymbol{B} \\right] \\\\\n & = - p~\\boldsymbol{\\mathit{1}} + 2\\left[\\left(\\cfrac{\\partial W}{\\partial \\bar{I}_1} + \n I_1~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right)~\\bar{\\boldsymbol{B}} - \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\bar{\\boldsymbol{B}}\\cdot\\bar{\\boldsymbol{B}}\\right] \\\\\n & = - p~\\boldsymbol{\\mathit{1}} + \\lambda_1~\\cfrac{\\partial W}{\\partial \\lambda_1}~\\mathbf{n}_1\\otimes\\mathbf{n}_1 +\n \\lambda_2~\\cfrac{\\partial W}{\\partial \\lambda_2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2 + \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3}~\\mathbf{n}_3\\otimes\\mathbf{n}_3\n \n \\end{align}\n ", "19f98b8118a2ab9ce854c05550202ba7": "\\mathbf{R}^{*T} \\mathbf{r} ", "19f9c30fdfe10d1a9a7e5772430d0c87": "b \\in B_n", "19f9d9973aa402aafe31eb3da3a27cb5": " QS_{ij}", "19fa382f6fc93c93b5652cab012b491c": "\\tau = 2 \\pi \\left( \\frac{M + m/3}{k} \\right)^{1/2}", "19fa90906c5c806e36b33d7922263013": "\\begin{bmatrix}4 \\\\ 1 \\\\ 8 \\end{bmatrix}", "19fabf487449c9aa766642f833a7aca0": "\\lbrace\\Phi_j\\rbrace, \\lbrace\\Psi_j\\rbrace \\leftarrow \\text{function DiffusionWaveletTree} ( T , \\epsilon , J ):", "19fad5e8a4cf591dfe8b4f4c97dc1148": "\nk = 20 \\left(\\frac{2}{\\pi}\\right)^{3/2}\\frac{\\left(k_B T \\right)^{5/2}k_B}{m_e^{1/2} e^4 \\ln \\Lambda} \\approx 1.8~10^{-10}~\\frac{T^{5/2}}{\\ln \\Lambda}~ W m^{-1}K^{-1}\n", "19faeace644960898c48dfea7f787ee3": "\\pi_x(o)", "19fb56252ccd12cc469d2c6ea9e54a3d": " \\operatorname{let} x\\ f\\ y = f\\ (y\\ y) \\and q\\ f = f\\ ((x\\ f)\\ (x\\ f)) \\operatorname{in} q ", "19fb782028d1eb2c30190659b5a87af3": "\\{(x_i, r_{im})\\}_{i=1}^n", "19fb87a1f4a0fbddca1534c8429eab8f": "\\omega_{\\Lambda_1}", "19fba609e2095aff720d34f3466f6aff": "X=-\\dfrac{1}{1234567}=-e^{-e^{e^{0.9711308}}}", "19fbae890f9e97ed355a1a663ee8260a": "\\ln\\begin{vmatrix}K-P\\end{vmatrix}-\\ln\\begin{vmatrix}P\\end{vmatrix}=-kt-C", "19fbdfde09ac6929d437c710d305a6ab": "w_i = \\beta_1 + \\beta_2 h_i + \\beta_3 h_i^2 + \\varepsilon_i.", "19fc8699f3b68cddf89a57a71d8b99eb": "\\int_{0}^{110} 82.17 (1.035)^t\\, dt", "19fce25bc926229233026425153a0d1d": "1\\le k\\le r", "19fd15b8143cd57563945293e6d17a46": "y'=f(t,y)", "19fd2b8f45447c22cb4b75d5ae2fa4bf": "z(x, y \\pm 1) \\rightarrow z( x, y \\pm 1 ) + 1.", "19fd34def372b53767d2c9f905f3a1c5": "R_{\\infty}:= 1+\\max\\{|a_0|,\\cdots,|a_{n-1}|\\}. ", "19fdf2beda9bfb86866c30fae14b0964": "L_c \\,", "19feba130d76f3f59699d04acb4524bb": "\\{x_n\\}", "19febf9b289e510f9865fcf7a4e2a47e": "\\scriptstyle \\log\\varepsilon^{-1}", "19ff01d54a6436fd0e5f286343d7e299": "X_o(t) = \\frac{-A+\\sqrt{{A^2}+4(B)(t+\\tau)}}{2}", "19ff0883ea89d2810644e5ebb865c108": " N(z) = C + Dz + \\int_{\\mathbb{R}} \\left(\\frac{1}{\\lambda - z} - \\frac{\\lambda}{1+\\lambda^2} \\right) d\\mu(\\lambda), \\quad z\\in\\mathbb{H},", "19ff267f3441e6595957b46f6e919a56": "\n \\cfrac{\\nu_{\\rm xy}}{E_{\\rm x}} = \\cfrac{\\nu_{\\rm yx}}{E_{\\rm y}} ~,~~ \\nu_{\\rm yz} = \\nu_{\\rm zy} ~.\n ", "19ff298c821ac6f4b7276dced6c75e47": "P_{CO_2} = (1.5 \\times HCO_3^-) + 8 \\pm 2", "19ff72c3dc08564cbd324b0c09e909d9": "\\left(\\frac{\\partial S}{\\partial V}\\right)_{T}", "19ffa9408782a396f46bdaf4ecbdf512": "\\frac{1}{\\omega\\pi} e^{-\\frac{(x-\\xi)^2}{2\\omega^2}} \\int_{-\\infty}^{\\alpha\\left(\\frac{x-\\xi}{\\omega}\\right)} e^{-\\frac{t^2}{2}}\\ dt", "1a002cb8405b5a4c91c62f8cb2689ade": "\\tau_{yz}", "1a004938ef2d7bb94a67f43ec3024229": "\\mathcal{N}_0(\\boldsymbol\\mu_0, \\boldsymbol\\Sigma_0)", "1a00725013aeac96f55d311febcec08c": "6 \\div x", "1a00767c1a5e3ecb1a11f19b585a8a7f": "\nK = K_0 + K'_0 P \\qquad(4)\n", "1a00b0c8872b8e3cdac435e8c6909c3c": "P = p_1p_2...p_m", "1a00cf5959edf106bfdd8814b4c87c98": " \n A_{i,j} = \\phi(||\\mathbf{c}_i - \\mathbf{c}_j||), \\;\\;\\;\n \\mathbf{V} = \n \\begin{bmatrix}\n 1 & 1 & \\cdots & 1 \\\\\n \\mathbf{c}_1 & \\mathbf{c}_2 & \\cdots & \\mathbf{c}_{N}\n \\end{bmatrix}, \\;\\;\\;\n \\mathbf{y} = [y_1, y_2, \\cdots, y_N]^T\n", "1a010300ea95950b79706346ce71bee2": "-v_\\mathrm{rel}\\int_{m_0}^{m_1} \\frac{\\mathrm{d}m}{m} = \\int_{v_0}^{v_1} \\mathrm{d}v", "1a0143e7caa3d8eb2319cbd7025fd68d": "A_1,A_2,\\dots,A_n,\\dots", "1a015980937050432c797d6b80876dca": "\\mu_n(B_n) = 0", "1a016c47c56780b1224308c247ae0b2e": " \\, a ", "1a01c50c3d8d8c852a108796303ed066": " \\scriptstyle \\sin (\\theta + 120^ \\circ)", "1a0260338f8f0b55589202ca0c4ab27c": "\ny(\\tau)=\\frac{w(vt,t)}{w_{st}}\\ ,\\ \\ \\ \\ \\tau\\ =\\ \\frac{vt}{l}\\ ,\n", "1a026315938a57c4f7807d6fb344df25": "T \\!", "1a026cc4a7ef35293437d3ad5473b752": "f \\in \\pi_1(X)", "1a026eb25ff8e253d704c390764ec774": " \\; {}_2F_1(a,b;c-1;z)- \\, {}_2F_1(a+1,b;c;z) = \\frac{(a-c+1)bz}{c(c-1)} \\; {}_2F_1(a+1,b+1;c+1;z)", "1a02742e97ce791495707af1509ec905": "\\mu \\ge 0,", "1a02dc7d64b7114188695db1c10d27ad": "\\rfloor ", "1a02dfe7bdb4514c3c7b3d1d342b7a9e": "O(0)+O(-n).\\ ", "1a030d980d95919f012a49aaa653b5ae": "f(x)=\\frac{P(x)}{Q(x)}.", "1a0322900261cee8f3941e3bdfbcac1a": " q_2 = \\frac{Q}{w_2} = \\frac{150}{3} = 30 \\text{ ft}^2/s", "1a035a132b4401f9d108fa015f112f67": "|(f^n)^\\prime|\\ne 1,", "1a0378e7832a746c0fd960c17cbc554d": "u, v, w, x, y\\in U", "1a0392716db24fee1532cbfd65121b15": "\n\\begin{align}\nI_S & = \\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R}+\\mathbf{d})\\cdot (\\mathbf{r}-\\mathbf{R}+\\mathbf{d}) \\, dV \\\\\n& = \\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R})\\cdot (\\mathbf{r}-\\mathbf{R})dV + 2\\mathbf{d}\\cdot\\left(\\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R}) \\, dV\\right) + \\left(\\int_V \\rho(\\mathbf{r}) \\, dV\\right)\\mathbf{d}\\cdot\\mathbf{d}.\n\\end{align}\n", "1a03d0f7242823c05e0f16ad19f85201": "\\sqrt{x}", "1a04ac8e02a9b4834f79ee7065e1de41": "A_1\\to A_2\\to A_3\\to A_4\\to A_5\\to A_6", "1a04bb5f91f86cd72fc048e74d3b9df6": "z = r \\sin b", "1a04c0ffd8f001c569c03389df28c164": "2^n/2", "1a053b90c12b3e25fdc79d2ba89daaee": "[q_i]=[p_{i+1}], i=1,2,\\dots, n-1", "1a053e758994c115fdb8fb5e235bdf73": " C_{rr}", "1a0540209bdd254e2e2636f18091ce82": "\\bar \\partial_{j, J} f := \\frac{1}{2}(df + J \\circ df \\circ j) = 0.", "1a05491f18531ebcf31233f61ac7e0e8": "\n\\begin{align}\n\\sigma_{0}(12) & = 1^0 + 2^0 + 3^0 + 4^0 + 6^0 + 12^0 \\\\\n& = 1 + 1 + 1 + 1 + 1 + 1 = 6,\n\\end{align}\n", "1a05886ae4ac84a90aa4296a021d60d6": "\\displaystyle \\frac{2\\pi}{\\Gamma\\left(\\alpha\\right)}u\\left(\\pm\\nu\\right)\\left(\\pm\\nu\\right)^{\\alpha-1} ", "1a05898a3e4d0e33d95348e7a515e6d0": "P=\\frac{1}{2}\\rho A C_D \\left(UV^2-2VU^2+U^3\\right)", "1a0589c3278954864fb28597aea5777a": "\\tilde {\\mathbf A}/\\pi_1(\\mathbf A,v)", "1a0620d859f4815107d4dd18453920e8": "\\kappa _T(T,p)\\ ", "1a06a5f619989f11bc9b44f95efcc080": "T^{\\alpha\\beta} = 0", "1a06d47358bed0594e9de5dd597e0a37": "C = b(Y - T) ", "1a0706c07149f219ed4ec1a65c48e52d": "\n \\int P(x-a)~dx = P\\left[\\cfrac{x^2}{2} - ax\\right] + C\n ", "1a07269ff1b568e256ecb4eebc6017d6": "F_b = \\beta g \\rho_o \\Delta T", "1a07a7270b7a5b839fcc97999ab49dec": "E = \\sum_{i=1}^N \\alpha_i X_i^2", "1a07c5253f7174d2c9c7c502cfcabb1d": " x = \\left(\\begin{matrix}\\frac{a-b}{2}\\end{matrix}\\right)\\cos (2 \\omega t) ", "1a07d1497c81d90d553590a4b4feb930": "\\|u\\|_{L^q(R^n)}", "1a08369d90eeff5a19a42315220d0f00": "g\\colon Y\\to X", "1a08660faa1d7631a80cee6f992f1829": "V_{out} = V_{in} \\cdot \\frac {L_2} {L_1 + L_2}", "1a08917c78682ca880ad4468db051a9f": "\\lambda = \\frac{\\sum_{i=1}^t c_i \\mu_i}{\\sqrt{\\sum_{i=1}^t c_i^2 \\sigma_i^2 }}. ", "1a08d5cd8ed9a2e2fec975e219b67626": "P(\\theta|\\mathcal{D}) \\propto P(\\mathcal{D}|\\theta)P(\\theta)", "1a08eba585b41620c22a00c1f3f71961": "\np_{surv} = \\mathbb{E}[ 1_{S_t \\rho_c", "1a0e7c104e11587e09cdb6359fbf9b32": "(4*x^2+644*x)*(4*x^2+644*x+25921)=", "1a0eda170261067891c463698a39840e": "|s_{n+p}-s_n|=|a_{n+1}+a_{n+2}+\\cdots+a_{n+p}|<\\varepsilon.", "1a0f19f5d9bb73a80767125a095830c2": " \\mathbf{E} \\left[ | \\hat{x}_T(\\omega) |^2 \\right] = \\mathbf{E} \\left[ \\frac{1}{T} \\int\\limits_0^T x^*(t) e^{i\\omega t}\\, dt \\int\\limits_0^T x(t') e^{-i\\omega t'}\\, dt' \\right] = \\frac{1}{T} \\int\\limits_0^T \\int\\limits_0^T \\mathbf{E}\\left[x^*(t) x(t')\\right] e^{i\\omega (t-t')}\\, dt\\, dt'.", "1a0f60f3a01da6dc8131bf7902834cd3": "\nZ_\\mathrm{i+1}=Z_\\mathrm{0,i} \\frac{Z_i + jZ_\\mathrm{0,i}\\tan(\\beta_i l_i)}{Z_\\mathrm{0,i} + jZ_i\\tan(\\beta_i l_i)}\n", "1a0f96bf5b18d6c2d897fc713bb757c5": "x_i = \\sum_{j=0}^n a_{ij}x_j, \\;\\;\\; 1\\leq i \\leq n,", "1a0fa318ebc1b3fa6ffac01d86c286ce": "\\mathbf{M}", "1a1076dd909636723e179faa34a243c9": "a^n u[n]", "1a111a571f475971c073db0569639061": "\\vdash A \\Rightarrow B", "1a114ad255499321be443a99fe3223b2": "L_d \\ll L_{nl}", "1a1177a1d8ec7c65c09dc01365e8103d": "\\log z", "1a1211130e9ebcd2bb22893c98729af4": " e", "1a12ad8dd212e7bca659a1069c60eaf7": "\\vec{\\sigma} = \\sigma_1 \\hat{x} + \\sigma_2 \\hat{y} + \\sigma_3 \\hat{z} \\,", "1a12c3f9274f0a122401680b3f987135": "i=\\{1,2\\}", "1a12d22d7ba0eeb246c640888ec4be84": " \\int_a^b f(x) \\, h(x) \\, dx = 0 ", "1a132bc2cbba830a7b68f8e698ed0c92": "\nU(\\rho,z)=2 \\pi a \\int_0^\\infty \\exp { \\left [-\\frac {\\rho'} {\\sigma} \\right]^2} J_0(2 \\pi \\rho' \\rho/\\lambda z) \\rho' \\, d \\rho'\n", "1a1353b888ed0c40c98d7485d51a56c7": " \n\\begin{align} \n\\frac{1}{v_0} &= \\frac{ K_M^A}{v_\\max {[}A{]}}+\\frac{ K_M^B}{v_\\max {[}B{]}}+\\frac{1}{v_\\max}\n\\end{align}\n", "1a136410efbfa73a600f642108604711": "a \\wedge b := \\frac{1}{2}(ab - ba) = -(b \\wedge a)", "1a13ab2a75388a97891f5da138805178": " a_{10} = \\mathcal{L}(p_7)+p_3p_7+p_1p_9,", "1a13d972a2c8473f075ded6eca465169": "\\wp", "1a14960d92068132a0bf2b4af919afa4": " \\mathrm{ T_{Low}} (f) = \\frac {1} {1 + j f/f_1}. \\ ", "1a1531d53a55979d47e156aacec78d8f": " \\nu_2 = p[ \\sigma_1^2 + \\delta_1^2 ] + ( 1 - p )[ \\sigma_2^2 + \\delta_2^2 ]", "1a15595311fac89ba9e35e0705565675": " -y \\, \\partial_x + x \\, \\partial_y", "1a157ce446c4d4303add1ed2743b9c53": " \\theta = 1.220 \\frac{\\lambda}{D}", "1a158c616a63418bcb38ebd16dfb5a25": "\\sum_{S \\in 2^N\\setminus\\{\\emptyset\\}} \\alpha (S) v (S) \\leq v (N).", "1a158e580034a26770fc56570c0bd8b6": " \\bar{E_C} ", "1a159f9c4e423e4bdfc97e70fa40ef43": " 5 = (1 + 2i)(1-2i) ", "1a15a0b20d03015288edec6e2323ff81": "(\\mu,\\sigma^2)", "1a15eba66eae25d869ef4b46d3e95fc1": "(A.2.b)\\quad \\psi_{,\\,\\rho\\rho}+\\frac{1}{\\rho}\\psi_{,\\,\\rho}+\\psi_{,\\,zz} =0 ", "1a160589f3e43c74f4937249d3696eac": " \\frac{\\sigma^2}{\\xi^2}[2\\alpha \\csc(2 \\alpha) - (\\alpha \\csc(\\alpha))^2] ", "1a162ef8231f6511b998943140a294d5": "n=q-1", "1a168d63b24ed54acd26b68abca7bc3d": "\\alpha_t(s, a)", "1a16a59beb0861c9a3e55fc4a5302933": "\\phi-\\beta\\,\\!", "1a16e1d3b1f098c5eac89fad5b0643b1": "z^2+w^2", "1a16fb018d6deada23b0d6f36f4d4ccf": "S[i,n]", "1a172f0c2eec73530f9c8a46483152e9": "H_{(1)} \\ldots H_{(m)}", "1a1753442ad7ebc5ecf4e02c8bb0c2b0": "\\left ( \\frac{\\partial S}{\\partial V} \\right )_T = \\left ( \\frac{\\partial p}{\\partial T} \\right )_V", "1a175da2a613413afd8909856e55f324": "f:S \\to \\mathbb{R}", "1a17f12edaea2b8c11634bfc7e635939": "n_F(z)=(e^{\\beta z}+1)^{-1}", "1a183e02d2dd8ea8a94f071e55b3d237": "P(T) \\approx 10^{-21.85} ~~~~~~~~~~~(10^{4.3} < T < 10^{4.6} K) ", "1a18787cf7f29af14de320c5d5dc3ff3": " g_t g^{-1} = F_t F/2 ", "1a189f3d8c1cfc118988f11d1e5d06d6": "\n\\int x^{b-1} \\gamma(s,x) \\mathrm d x = \\frac{1}{b} \\left( x^b \\gamma(s,x) + \\Gamma(s+b,x) \\right).\n", "1a18ad216b0c9b13b21ffb26f8abda53": "n > \\log_{10} (1/\\epsilon)", "1a18da63cbbfb49cb9616e6bfd35f662": "2.3", "1a197f7095cfb3f9dc8dca1c7f53d11d": "\\frac{v_\\|}{\\omega_c}\\, \\vec{b}\\times\\left[\\frac{\\partial\\vec{b}}{\\partial t} + (\\vec{v}_E\\cdot\\nabla\\vec{b})\n\\right].", "1a19a3958183c94db6c796cf1c7cfce5": "Mod(\\sigma):Mod(\\Sigma')\\to Mod(\\Sigma)", "1a19d562d550955d6f369224aaaefdda": "\\frac{\\partial\\phi}{\\partial t}+\\nabla\\cdot\\mathbf{j}=0", "1a19da4efeaa082b043e74b90523cfa1": " U_0(x,y) = e^{jkn \\Delta} e^{-j \\frac{k}{2f} [x^2 + y ^2]} U_i(x,y) ", "1a19f25c6c41c70f170606d534a70c36": "\\mathbf{P} = \\left(\\frac{E}{c}, \\mathbf{p}\\right)\\,.", "1a1a263d52179f947b00a2d5e60d5c71": "\\operatorname dx", "1a1a7a2af9c4e7a342016c165269a8ab": "\\operatorname{DG}(a_n;s)=\\sum _{n=1}^{\\infty} \\frac{a_n}{n^s}.", "1a1acd502bd38b524fe9a3514edbd68d": "\\mathbf{M} = \\mathbf{U}\\mathbf{D}\\mathbf{U}^{-1}", "1a1af3040befd2ea8a0052c09d5cb367": "p \\in F", "1a1ba3480c81a5596fb27544a2ee115f": "w(t) \\sim \\mathcal{N}(0,\\sigma^2)", "1a1c01c0cafb38c4beaa366617390158": " M(\\vec X) = \\left( {\\begin{array}{*{20}c}\n {\\bar \\mu _1 + \\bar \\mu _2 } \\\\\n {\\bar \\Sigma _1 + \\bar \\Sigma _2 } \\\\\n\\end{array}} \\right)\n", "1a1c70969df916e13c687dc39945b90a": "\\sum_k Y_{ik} a_k[n] + Y_i^{\\text{sh}} a_i[n] = S_i^* b_i^*[n-1] \\qquad (n=0, \\ldots, \\infty)", "1a1cf0c14d19a7c9da326d8c9da1d74e": "\\Bbb{R}^d", "1a1d180acc07655b2faceaa901393763": " \\mathbf {GOP_{\\nu}} ", "1a1d3891d7ea2f2db154c402cd39a885": "(\\overline{Y}-Y)/\\overline{Y} = c(u-\\overline{u})", "1a1d739bbc059614509d400db7ebbaab": "\\beta_i s\\equiv w_i r s\\equiv w_i\\pmod{q}.", "1a1da387d9d61a52ef19911a8f5a968a": "\n=\n\\left[\n\\begin{array} {c c | c c c c | c | c c}\n1 & 2 & 4 & 7 & 8 & 14 & 3 & 12 & 21 \\\\\n4 & 5 & 16 & 28 & 20 & 35 & 6 & 24 & 42 \\\\\n\\hline\n2 & 4 & 5 & 8 & 10 & 16 & 6 & 15 & 24 \\\\\n3 & 6 & 6 & 9 & 12 & 18 & 9 & 18 & 27 \\\\\n8 & 10 & 20 & 32 & 25 & 40 & 12 & 30 & 48 \\\\\n12 & 15 & 24 & 36 & 30 & 45 & 18 & 36 & 54 \\\\\n\\hline\n7 & 8 & 28 & 49 & 32 & 56 & 9 & 36 & 63 \\\\\n\\hline\n14 & 16 & 35 & 56 & 40 & 64 & 18 & 45 & 72 \\\\\n21 & 24 & 42 & 63 & 48 & 72 & 27 & 54 & 81\n\\end{array}\n\\right].\n", "1a1e4db88a701533707cfb28adf26251": "\nV^{AB} = \\sum_{\\ell_A=0}^\\infty \\sum_{\\ell_B=0}^\\infty (-1)^{\\ell_B} \\binom{2\\ell_A+2\\ell_B}{2\\ell_A}^{1/2} \n\\sum_{M=-\\ell_A-\\ell_B}^{\\ell_A+\\ell_B} (-1)^{M} I_{\\ell_A+\\ell_B,-M}(\\mathbf{R}_{AB})\\; \\left[\\mathbf{Q}^{\\ell_A} \\otimes \\mathbf{Q}^{\\ell_B} \\right]^{\\ell_A+\\ell_B}_M\n", "1a1e8192343516e070a02f53c2e12959": "\\mathbb{Z}[G]", "1a1e8c397346e7ddf74dcb73fc212c9a": "\\mathcal{E}= \\omega B A \\sin{\\omega t} ", "1a1ec506a61023dc8691d6704389e898": " \\mathbf{J}_\\alpha = \\sum_\\beta L_{\\alpha\\beta}\\,\\nabla f_\\beta", "1a1f8c23f557524f6f963fd4dccfe6f9": "\\tfrac12 a b \\sin(C)", "1a1f9ccafae1fff64519f639e9c5df06": "2 k_\\text{F}", "1a1fa4952566a429eca578933704d3ef": "\\lim_{n\\to\\infty} diam(C_n)\\rightarrow 0", "1a1fdcd5797f1fb6a1319277cbbafbef": "p=\\frac{h}{2L}\\sqrt{n_x^2+n_y^2+n_z^2} \\qquad \\qquad n_x,n_y,n_z=1,2,3,\\ldots ", "1a202ed3612f0fef7ffc7841b7b82c7c": "p_0 = p + q + \\rho g z\\,", "1a203b4ddb47f654f0f97a816acbae22": " p(\\theta_1, \\cdots, \\theta_m) = C \\prod_{1 \\leq i \\leq m}(1-\\sigma\\cos\\theta_i) \\prod_{1 \\leq k < j \\leq m} (\\cos\\theta_k - \\cos\\theta_j)^2~.", "1a2091196176452b90d8adfee15c1ce0": "i'th", "1a20b84f5f1c061ee1ef068bd1023dd2": "(u_i,h_i)\\in U_i \\times Homeo(F)", "1a20f16e5c64d6b161d7693d1e87c9c1": "S=\\mathbf{r_x^TM_x^{-1}r_x+r_y^TM_y^{-1}r_y}", "1a2158a55cbab875b8ee1db807e62e21": "C_{pq}", "1a21688b940d0b34377aca457c8c8b9c": " \\mathbf{v} = \\mathrm{d} \\mathbf{r}/\\mathrm{d} t \\,\\!", "1a2209a50c840664856ea7720e4f3cdc": " \\mathbf{\\omega} = \\nabla \\times \\mathbf{v} ", "1a221e9991a088d239c4dbfdc4622f62": " \n\\mathcal{L} = \n-\\textstyle{{1}\\over{4}}\\,F_{\\mu\\nu}F^{\\mu\\nu}\n+\\textstyle{{1}\\over{2}}\\,(k_{AF})^\\kappa\\,\\epsilon_{\\kappa\\lambda\\mu\\nu}A^\\lambda F^{\\mu\\nu}\n-\\textstyle{{1}\\over{4}}\\,(k_F)_{\\kappa\\lambda\\mu\\nu}F^{\\kappa\\lambda}F^{\\mu\\nu}.\n", "1a2243b0673bd3de0f98b0e9382a4dc7": "\\ \\epsilon", "1a2274463a7e3927dca69285e57a0313": "XBH = 2B + 3B + HR", "1a2297bc202077a23cc42867fb26e92c": "\nx = 1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\ddots}}}}\\,\n", "1a22d5022456b3fdd82b8fd8be7ae5ae": "J_i = - \\frac{D c_i}{RT} \\frac{\\partial \\mu_i}{\\partial x}", "1a22ef2363a2564fcf6012210ac1c890": "\\mathrm{C^{\\delta}}", "1a2337e565c6c8d8b7248fda95d7f7bf": "z = {1 \\over 2}\\ln{1+r \\over 1-r}", "1a23779360e5d4138a59112d5a4f04f2": "\\lim_{x \\to 0} \\frac{\\sin ax}{x} = a", "1a238b8c1e493810971db1bd3e865a0b": " c^* = \\min_{ w > 0 } \\left[\\frac{1}{w} \\ln \\left( R \\int_{-\\infty}^{\\infty} k(s) e^{w s} ds \\right) \\right] ", "1a23fced934e003b1401139a270c5b7b": "\\scriptstyle x \\;\\in\\; (0,\\, \\infty)", "1a2476517cc63fc8afec181b597a1a38": "F\\star G", "1a2488dcc33fda995f00c0a54d200c51": "\nI = \\frac{1}{T} \\int_{0}^{T}p_\\mathrm{inst}(t) v(t)\\,dt\n", "1a24cab6eed461fd75b52a97d07d762f": "Leader_{j}", "1a2594c0b021589ae05bad3d95f83dc8": "S^2_i, n_i, i=1,2", "1a25aa972d53d71e503d08368a715ad9": "\\begin{matrix} {r \\choose 3}{4 \\choose 1}^3{52 - 4r \\choose 2} \\end{matrix}", "1a25de39bf8617ac6e1a53c26679f691": " \\Delta^o \\overset{S^1}{\\longrightarrow} \\text{Fin}_* \\overset{F}{\\longrightarrow} k\\text{-}\\operatorname{mod}.", "1a25fcfce4d11c72822d702ae83c331f": " V\\subset Y", "1a2643b3e60cdec43207da0d0cbfefc8": "LL(\\alpha,1) = GPD(1,\\alpha,1).\\,", "1a2647b75933b0328eae3f99f425dd3b": " U = N k_B T^2 \\left(\\frac{\\partial \\ln Z}{\\partial T}\\right)_V ~", "1a26aa505912b7e0fb043ef8b3f1733a": "I_{M-1}", "1a26b7963eb20eb0068bd30e0eb8a198": "\\Gamma(\\gamma)_0^t e_\\alpha(\\gamma(0)) = \\sum_\\beta e_\\beta(\\gamma(t))g_\\alpha^\\beta(t) ", "1a2757da1c9f59abaa141d0e59d0cea5": "\n\\omega^2 =\\omega_{pe}^2 +\\frac{3k_BT_{\\mathrm{e}}}{m_e}k^2=\\omega_{pe}^2 + 3 k^2 v_{\\mathrm{e,th}}^2\n", "1a278dbd840884f28111395f12670b18": "\\frac{p(x)}{(x-a)}", "1a279d88b6b11a085b685b8a2c9b5aa5": "\\operatorname{E}[g(X)] = \\int_{-\\infty}^\\infty g(x) f_X(x) \\, dx ", "1a27b65d4a17d6f364230c928295ea64": "G^{33}", "1a286a2615acbff05f1a75b5729813ee": "\nx = b_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{a_i}{b_i}.\\,\n", "1a28a5cc2aa6aace2a45bec4532237a6": "\\varepsilon_{ij} =\\frac{1}{2} (u_{j,i}+u_{i,j})\\,\\!", "1a28f3eca45061acd51ea9e7b4fa92c6": "\\Phi(t)=\\phi\\,(t){\\phi\\,}^{-1}(t_0)", "1a2962da12d6f9ebf6bd4211b30e3137": "\\mathfrak{h} \\subseteq \\mathfrak{g}", "1a297ccc433d980263f70a9318599a89": "p(G(U_\\ell))", "1a2981eb7c3dfb8b3ce7b26baa6f1352": "f: \\textbf{R} \\to Y", "1a29b6c4530e41e3701fa1777ea8de7c": "\\displaystyle{D(x+h) \\ge D(x) + f_\\varepsilon(h) + |h|^2.}", "1a29c002755e05f297c99c693440ea60": " \\log \\left ( \\frac {p(r)} {P} \\right ) \\approx \\frac {R_{critical}} {r} ", "1a29d84310c1c1f5efb6d267e5c724fe": "\\displaystyle s_i = s_{i-1}(s_{i-1}-1)+1,", "1a29e4ae2a9c784e24bf7db9259012c7": " \\text{Area of polygon (on the unit sphere)} \\equiv E_n = \\Sigma -(n-2)\\pi.", "1a29eab4660c57a9984afd9d6316dfaf": "\\sqrt{\\epsilon_0 / 2}", "1a2a377d76bb9d8f65ce131a509c7442": "C_\\bullet\\mapsto H_n(C_\\bullet)=Z_n(C_\\bullet)/B_n(C_\\bullet)", "1a2a50975b5f0f92f11ed99578ace24e": "\\bar V\\,", "1a2abe6648e807b90565e8f674fd916b": "\n\\int_1^\\infty x^{-\\alpha} \\; (x-1)^{\\alpha - \\beta - 1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z x \\right) dx =\n\\Gamma (\\alpha - \\beta) \\; G_{p+1 ,\\, q+1}^{\\,m+1 ,\\, n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p}, \\alpha \\\\ \\beta, \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right).\n", "1a2ae1978c4f7a32d6d8841c1144771c": "p_{f}", "1a2aeea258edc2e2631943480206b928": "D_\\mu D^\\mu\\phi + A F^{\\mu\\nu} D_\\mu \\phi D_\\nu (D_\\alpha D^\\alpha \\phi) =0", "1a2afd5fb3b5d4d3b5be9eb3799ae79a": "\\chi(\\mathbf{S}^n) = 1 + (-1)^n = \\begin{cases}\n2 & n\\text{ even}\\\\\n0 & n\\text{ odd}.\n\\end{cases}", "1a2b216da572f46514b14211db41d931": "\nR = \\left( \\begin{array}{cc} a & 0 \\\\ 0 & d \\end{array}\\right)\n", "1a2b24ba1ab504e66e5da705604e7833": "\nf(r) =\\int_0^\\infty F_\\nu(k)J_\\nu(kr) k\\operatorname{d}\\!k\n", "1a2b5ab4a01b13e4d6f09a48da699851": "KC(\\mathcal{X}) = \\sum_{i\\neq j}KC(x_i, x_j) = 2\\sum_{i \\left \\lceil {\\sqrt{kn - \\frac{1}{2}}} \\right \\rceil > \\left \\lceil {\\sqrt{kn}} \\right \\rceil", "1a55cfd5c620ed82c9e8acd1de5cc345": "f(x)=\\exp(x)", "1a55f2f94e125f450f6dc10fd4efc87e": "\\sin \\alpha_\\mathrm{s} = \\cos h \\cos \\delta \\cos \\varphi + \\sin \\delta \\sin \\varphi", "1a56230b94ffdd34834d107134ff55c3": "\\boldsymbol{F(r)}={q\\over4\\pi\\varepsilon_0}\\sum_{i=1}^Nq_i{\\boldsymbol{r-r_i}\\over|\\boldsymbol{r-r_i}|^3}={q\\over4\\pi\\varepsilon_0}\\sum_{i=1}^Nq_i{\\boldsymbol{\\widehat{R_i}}\\over|\\boldsymbol{R_i}|^2},", "1a568e8c0cad16ab1cb2c9fb56b3d4fc": "\\mathbf{K}_t, \\mathbf{K}_s ", "1a568f568605e054f69eb1345f32b2d7": "S(S-1) > 8m + 20", "1a56e49b4136ae2b142973db3bda0a2e": "(n+1) I_n = x^{n+1}e^{ax} - a I_{n+1} , \\!", "1a57c2d9c9aa76af684beb70ab97851e": "d(x+a) = dx", "1a582a733c885e7a80817518f785f423": "\n\\phi^\\epsilon (x)=\\begin{cases} 2, & 0 < x < 1 \\\\ 2c_0, & 1 < x < 2 \n \\end{cases}\n", "1a586dfcf7603fa9972c39eb9f0bb785": "(u_t, x_{t-1}^{[m]})", "1a5870c4cef6ec993ddc15dc0dcfc29a": "y=(RF)a\\phi", "1a587c09d7b4a5a903053d84402ef3ce": "\\frac{43,416\\ \\mbox{MW·h}}{(366\\ \\mbox{days}) \\times (24\\ \\mbox{hours/day}) \\times (20\\ \\mbox{MW})}=0.2471 \\approx{25%}", "1a588a1e5e2a1fc746d4b034f8f253eb": "r,s\\in\\mathbb{R}", "1a588e03437a2b260b94843e386379bc": "a_1/d", "1a596ce2de6379e55a5468d9d0fc3942": "\\partial_\\beta F^{\\alpha \\beta} = \\mu_0 J^{\\alpha}. \\!", "1a597ed9b45e2d8e4204252322a2734f": "\\mathbf{D} = \\varepsilon \\mathbf{E} = \\varepsilon' \\mathbf{E} + i \\mathbf{E} \\times \\mathbf{g}", "1a59de64e9a037c4fd6ba7aadaf7f81a": "2\\Gamma_\\mathrm{res}", "1a5a0c8af15ca8b225b658c86aa1bb5a": "r' = B^n r + \\alpha - ((B y + \\beta)^n - B^n y ^n)", "1a5a34d2c4ac697f6926ae4bdb0f512e": "\np_w(\\vec\\theta)=\\sum_{k_1=-\\infty}^{\\infty}{p(\\vec\\theta+2\\pi k_1\\mathbf{e}_1+\\dots+2\\pi k_F\\mathbf{e}_F)}\n", "1a5a73d4470168a0ce6ec3ade1e6a242": "(\\mathcal C,\\otimes,I)", "1a5ad276aba92f7db12afdbfaa17e49e": "\\frac{1}{2(2^{100}-1)}", "1a5ad9632ade6008312bbef72539611a": "D = \\begin{bmatrix}\nT1 & T2 & T3 \\\\\nR(A) & & \\\\\n & W(A) & \\\\\n & Com. & \\\\\nW(A) & & \\\\\nCom. & & \\\\\n & & W(A)\\\\\n & & Com.\\\\\n\\end{bmatrix}", "1a5b424a1fb425f7dc28f74af2f35797": "V = (2/3)\\pi r^3", "1a5b539fad975fc5b35be419d2443b55": " r_{i+1}= r_{i-1} - r_i q_i,", "1a5b8c7b268b940e8c9d6755208da9d5": "I(a)=\\int_{-a}^a e^{-x^2}dx.", "1a5c07b678b13e194ab91c2108d83dc1": " 0 = \\sum_{n=1}^e \\sum_{m=1}^d b_{m,n} (u_m w_n)", "1a5c0cab95ce0436eea23929623c4025": "\\phi_k ", "1a5c56588f41d38822cffc99c0cc999a": "\\theta / 3", "1a5c56e93c66d12bfa57b78f6db82040": "a,b \\in \\mathrm{R}", "1a5c6012573eb7661c798d924fd6eefd": "Z_{in}\\left(l\\right)=Z_0 \\frac{Z_L + Z_0 \\tanh\\left(\\gamma l\\right)}{Z_0 + Z_L\\tanh\\left(\\gamma l\\right)}", "1a5c7909892907d3ad966cdbab389fd3": " q = C\\mathcal{E}e^{-t/RC}\\,\\!", "1a5c80a936d80b9897bf4044443bae53": " \\ h", "1a5c88e77842654cdb328a346baa696b": "g'N'=N'", "1a5c8ab2df31a54e52f279254f684e6b": "L_{X\\mid Y=y}(x) = f_{Y\\mid X=x}(y)", "1a5c8f9c140074128cb5c4a23f878e9d": " \\Psi_{1\\ldots N-1}(j^{N-1}\\alpha_1 J_1) ", "1a5c94554cea52bff8e8bb326f30468a": "\n\\int_{\\mathbb{R}^n} e^{- x^T A x + v^T x} \\left( a^T x \\right) \\, dx = (a^T u) \\cdot\n\\mathcal{M}\\;,\\; {\\rm where}\\;\nu = \\frac{1}{2} A^{- 1} v \\;.\n", "1a5cbec8903f64d5834c1e948c5194c8": "\\Bigg[\\frac{\\pi}{\\theta}\\Bigg]\\left[\\frac{\\theta}{\\pi}\\right]^{-1}=\n(-1)^{\\frac{N\\pi - 1}{4}\\frac{N\\theta-1}{4}}.", "1a5ccdbbcd96fc3625389b174c845de2": "\\forall x [\\exists y Animal(y) \\land \\lnot Loves(x, y)] \\lor [\\exists z Loves(z,x)]", "1a5d9645c46f7be2374dc12a26041ab2": "d\\varepsilon = d\\varepsilon_e + d\\varepsilon_p", "1a5dda535de965a6cfdbc2a354ee1ae2": "\\frac{d}{ds}(c_g E) = 0,", "1a5de7979bef6e24c2583689eaccb60d": " {_2^1}\\text{S}^\\gamma + \\text{E} \\underset{\\text{k}_{2(3)}}{\\overset{\\text{k}_{1(3)}}{\\rightleftarrows}} \\text{C}_3 \\overset{\\text{k}_{3(3)}}{\\rightarrow} {_2^1}\\text{P} + \\text{E}, ", "1a5e21d51cd6f450f5ba2a19bae2a14d": " r={n^2\\hbar^2 \\over ke^2m}. ", "1a5e228db5ee8d8f6077498cd40ad9bc": "x^6(x^2-x-1)+(x^2-1)", "1a5e2fc470d7cb10c8d62a10717051be": "g_{ab} = \\phi^2 \\, \\eta_{ab} ", "1a5e50192202479b4d03565ccc2e47f5": "\n\\begin{align}\n|\\boldsymbol{b_1}|^2>&|\\boldsymbol{b_2}|^2+|\\boldsymbol{b_3}|^2 \\text{ (favorable, will decompose)}\\\\ \n|\\boldsymbol{b_1}|^2<&|\\boldsymbol{b_2}|^2+|\\boldsymbol{b_3}|^2 \\text{ (not favorable, will not decompose)}\\\\\n|\\boldsymbol{b_1}|^2=&|\\boldsymbol{b_2}|^2+|\\boldsymbol{b_3}|^2 \\text{ (will remain in original state)}\n\\end{align}", "1a5e7589d1b67b2582217a61abe05b9e": " E_k=\\frac{1}{2}I\\omega^2=\\frac{1}{4}mR^2\\omega^2 ", "1a5ea4a4764a3cd2a7488159260c2cec": "Y_t(u) - T = Y_c(u).", "1a5fa80967ef93cd206235c69222e454": "-\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\nabla\\cdot\\mathbf{j}(\\mathbf{r},t) dV + \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\sigma(\\mathbf{r},t) dV = \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\frac{\\partial \\rho(\\mathbf{r},t)}{\\partial t} dV. ", "1a60202321d13ec160707b458b4843e5": "OC = \\sqrt{OA^2 - AC^2}", "1a60a8a28e78e7f8687c68f3810ca48b": "\n\\begin{bmatrix}\n F_{k_1} \\\\\n \\vdots \\\\\n F_{k_M}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n (e^{\\lambda_1})^{k_1} & \\dots & (e^{\\lambda_M})^{k_1} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n (e^{\\lambda_1})^{k_M} & \\dots & (e^{\\lambda_M})^{k_M}\n\\end{bmatrix}\n\\begin{bmatrix}\n \\Beta_1 \\\\\n \\vdots \\\\\n \\Beta_M\n\\end{bmatrix},\n", "1a60b54e92409d9942ddb745668bde54": "\\hat{q}_{{\\rm w}}", "1a60f2375c09a4f535e258d31ce6f15e": "\\mathbf{w}_{old}=\\mathbf{w}_{new}", "1a60f59a6950089a686df1c9c861db97": "\\ M_{heel_{max}} > D_{heel} \\times F_{forward} \\times \\frac{sin(\\beta)}{cos(\\beta)}", "1a61219ecd70908c2b27d5370742d76e": "f = \\frac{qB}{2\\pi m}", "1a61e3bbdab6197f36a54dca32cc06c5": "\\log_{10} 1000 + 1 = 3 + 1", "1a61fc3a62547c7e7e856e5fedc60324": "f(x_1)", "1a6248c24cd550e0af12637d660f0e27": "\\hat{e}=r_b \\cdot \\hat{r} + \\cos \\alpha \\cdot ( \\hat{b} - r_b \\cdot \\hat{r})+\n \\sin \\alpha \\cdot \\hat{r} \\times \\hat{b}", "1a625564ff18c476c6d14a4ca9aad569": "\\dot\\gamma = \\frac{8v}{d} = \\frac{8\\left(\\frac{Q}{\\pi r^2}\\right)}{2r},", "1a62680a10418c04766f451e8b8fd3d6": "{ds}", "1a62a7f47cbbcdf21d7c4bd92c1ef382": "\\oint_C f(z)\\,dz = \\oint_C \\frac{1}{(z^2+1)^2}\\,dz = \\oint_C \\frac{\\frac{1}{(z+i)^2}}{(z-i)^2}\\,dz = 2\\pi i \\frac{d}{dz} \\left({1 \\over (z+i)^2}\\right)\\Bigg|_{z=i} =2 \\pi i \\left(\\frac{-2}{(z+i)^3}\\right)\\Bigg|_{z = i} =2 \\pi i (\\tfrac{1}{4i})=\\frac{\\pi}{2}", "1a6306e2eff59a0d4ac22658f931d547": "\\ddot u_n={\\left(\\frac{c}{\\Delta x} \\right)}^2 \\left(u_{n-1} \\ -\\ 2u_n\\right)", "1a634659de31ceecf01934f5f856d863": "O \\lnot A", "1a6377c12b6296d58f067ea886528e95": "X = \\{ n_i \\}", "1a63a4784dc7678e17a6b8c2746d7332": "(p',q') \\in R", "1a63ae2b14ef48ddee17fbf5f27ccf57": "\nP_i=\\frac{Y_i}{\\sum_{i=1}^k Y_i}", "1a63e745c9b2a53946eda467620d3c47": " \\zeta = \\exp\\left(\\frac{2\\pi i}{n}\\right). ", "1a6469bf3cc8e2b0bd9f6609934db745": "\\langle\\Gamma(\\gamma)_s^tX,\\Gamma(\\gamma)_s^tY\\rangle_{\\gamma(t)}=\\langle X,Y\\rangle_{\\gamma(s)}.", "1a647bee1de03e394b83433055b43e77": " P = \\mathbf{Tr}_{\\mathbb{Q}(\\zeta) / L} (\\zeta^j) ", "1a647cf8bf64c1b5ba54475e698bc332": "W_{im}(X)", "1a648b693f4a80f009e6cd7d2c04d73a": "x\\frac{B^y -1}{B-1}", "1a64c2793eebda69654632af2b594d7d": "E(Nl,t) = 1 - (1 - p)Nl,t", "1a650918887bebae77d29e2898e366bb": " D_1 \\equiv (\\exists z_1...z_{m+1}) \\phi(z_{a^1_1}...z_{a^1_k}, z_2, z_3...z_{m+1}) \\equiv (\\exists z_1...z_{m+1}) \\phi(z_1...z_1, z_2, z_3...z_{m+1}) ", "1a655149dd84d1ce4c8423604d4b91c0": "= \\{\\gamma^\\mu,\\gamma^\\nu\\} \\gamma^\\rho \\gamma_\\mu - \\gamma^\\nu \\gamma^\\mu \\gamma^\\rho \\gamma_\\mu \\,", "1a655bf339675674ce1d65dac8c4d7a3": "\\mathcal{F}(\\{x_n\\})_k=X_k", "1a655c6a5a5f4c2b16fdac55057f3b15": "\\begin{align}|6x^4 - 2x^3 + 5| &\\le 6x^4 + |2x^3| + 5\\\\\n &\\le 6x^4 + 2x^4 + 5x^4\\\\\n &\\le 13x^4\\\\\n &\\le 13|x^4|\\end{align}", "1a65805ba6780532940834717af63f05": " (\\partial V)_T=-(\\partial T)_V=-\\left(\\frac{\\partial V}{\\partial P}\\right)_T", "1a65c9370bab7419505cbf3054197b60": " - \\hbar^2 c^2 \\mathbf{\\nabla}^2 \\psi + m^2 c^4 \\psi = - \\hbar^2 \\frac{\\partial^2}{(\\partial t)^2} \\psi. ", "1a660d50a5bde5f2a00182b539e5ee45": "\\frac{\\Gamma(2+\\nu-b)}{2\\Gamma(2+\\nu-b+a)}\\left(\\frac k 2\\right)^\\nu e^{-\\frac{k^2}4}\\,_1F_1\\left(a,2+a-b+\\nu,\\frac{k^2}4\\right)", "1a6625b838bfc2f2a625943886df5fc4": "\\left( \\begin{smallmatrix} 9 & 15 \\\\ -10 & -10 \\\\ \\end{smallmatrix} \\right)", "1a6685e8682360033668f96a1e356ee8": "\\mathbf{a} = (a_1, \\dots, a_n)", "1a66874f4d02e4e1f0e1fb1fd5eb0deb": "Z = n_i \\times [Z]_i", "1a66fe7666f8c838a98f68842592af79": "\\scriptstyle [-\\infty,\\, -\\frac{1}{2}] \\,\\cup\\, [\\frac{1}{2}, \\,+\\infty]", "1a672d1ca022697d18a7c264effb2e55": "m_1 = m_2 = m_3 = \\cdots = m_n\\, ", "1a675014f12d411e1f6c3e4bbd8cd29a": "A(z) = 0.5[P(z)+ Q(z)]", "1a67ba8ad67cfa9f00c9acfb8486de3c": "F_s = \\mu_s N \\,", "1a6808b414be52a7e29392d502dd3f3b": "(1-a)(1-b)(1-c)(1-d)+a+b+c+d \\geq 1. \\, ", "1a6870c1806d0999d6f50ff05d9997fc": "x^\\mu_0 = 0", "1a68e1921524f3940b32459a97a922a5": "(\\wedge) \\frac{X \\cup \\{A \\wedge B\\}}{X \\cup \\{A, B\\}}", "1a68ee730acf9a84d5cdeca0f49b3d4b": "\\eta (\\gamma, T)", "1a6966dd780373beb846f896352fd840": "\\frac{r_{01}}{z}", "1a698ec3c23dd6854144511551a1037b": "\\det\\boldsymbol{F}", "1a69c882a15ce6b8e0ff48212cc1d7ca": "T = k[u, z_2, \\dots, z_d]", "1a69e6851da9ee7d69d5853b8c28c298": "\\tau.", "1a69eeca39dc3f620fc2696ee7cd4506": "\\mathbf{x_a}", "1a69f0fe0579f3c6e4481328710bc443": "|\\alpha_i|=\\sqrt p", "1a6a271b54c4eca5c4f826114f8b1153": "c2^{k}.", "1a6a8f5aa0a5827bf2aa808d5457f3b6": " {Gr}(r, \\mathcal E\\otimes_{O_S} k(s))", "1a6a97c01b352176c29df771091402c9": "\\ e_2 = (0,1)", "1a6ae5c0baad0c27ffbe1ba00f30713c": "\\begin{array}{cc}\n \\begin{array}{rrr} \\\\ &1& \\\\ 2&& \\\\ \\\\&&/3 \\\\ \\end{array}\n \\begin{array}{|rrrr} \n 6 & 5 & 0 & \\text{-}7 \\\\\n & & 2 & \\\\\n & 4 & & \\\\\n \\hline\n 6 & & & \\\\ \n 2 & & & \\\\ \n \\end{array}\n\\end{array}", "1a6b1b4f9c5a3d10cb92eba5457a6fca": "(AB+BC) - (AC') = n\\lambda, \\,", "1a6b321258d385e619bc65e5f3f61b81": "x= \\lambda\\,", "1a6b5bc401b71fea09fdb9d86e4e72eb": "\\operatorname{E}[X] = \\frac{1}{k \\theta} + \\frac{1}{k \\theta} + \\cdots + \\frac{1}{k \\theta} = \\frac{1}{\\theta} .", "1a6b8149214fbc8d38ad67c14f0f8ecc": " \\chi = \\frac{x}{x_c} \\Rightarrow x = \\chi x_c.", "1a6bb2427289563c8608a0e17d35c863": "G^\\wedge", "1a6be56064dd18db1a86c91a572d9722": "ds^2=[a(u)(x^2-y^2)+2b(u)xy]du^2+2dudv+dx^2+dy^2", "1a6beb77bddb2f66cf392e0dbd85aa69": "(p-1)/2", "1a6c3467aa7fd759747d4a7fb823f2ec": "\\Phi_D \\equiv \\int_S \\mathbf{D}\\cdot \\mathrm{d}\\mathbf{A}", "1a6c429fe21d3716bdb6e6b304c02e99": "\n\\alpha \\approx \\frac{3\\pi m}{r}\\sqrt{\\beta} = \\frac{3\\pi m}{r}\\sin(\\theta).\n", "1a6c4a7d6e569ed632794aff1e5b1f63": "\\Pi^P_2", "1a6c5ef3aa87a5a6d13d16673a97359b": "\\frac{\\vec{H}\\{ 0.133; \\; 0.65; \\; 0 \\}}{\\|\\vec{H}\\|}=\\frac{\\vec{H}\\{ 0.133; \\; 0.65; \\; 0 \\}}{\\sqrt{0.133^2+0.65^2}}=\\frac{\\vec{H}\\{ 0.133; \\; 0.65; \\; 0 \\}}{0.668}=\\{0.20048; 0.979701 ; 0 \\},", "1a6c7be6c6aaf90360222418445b8eae": "\\mathfrak{sl}_3(\\mathbf C)", "1a6c8740525a46fc102b34d03a1cf2e2": " \\mathbf{v} = (v_x, v_y) ", "1a6c8e8b7623d6584b5d7f5281a56539": "{100 \\over \\sqrt{4}}", "1a6cd631bb5c47f70bd072a4ffc75e9c": "\\displaystyle{\\|\\sum_{i=m}^n T_iv\\|^2 \\le \\sum_{i,j\\ge m} |(T_iv,T_jv)|.}", "1a6d08f3912bb21c2091322a8e239924": "l(t,s)=0", "1a6d6a8904fb0ea0f4e465ab4f426ff4": "\n\\Pr[R(x,y) = 0] > \\frac{1}{2}, \\textrm{if }\\, f(x,y) = 0\n", "1a6dac261da09965540c1a15d7e0eb52": "\n f:= \\cfrac{1+2R}{2(1+R)}|\\sigma_1 - \\sigma_2|^m + \\cfrac{1}{2(1+R)} |\\sigma_1 + \\sigma_2|^m - \\sigma_y^m \\le 0\n", "1a6e4e0f467fc9d39da1758ad2129208": "\n \\begin{align}\n A_{mn} & = \\frac{4}{ab}\\int_0^a \\int_0^b \\varphi(x_1,x_2) \n \\sin\\frac{m\\pi x_1}{a}\\sin\\frac{n\\pi x_2}{b} dx_1 dx_2 \\\\\n B_{mn} & = \\frac{4}{ab\\omega_{mn}}\\int_0^a \\int_0^b \\psi(x_1,x_2) \n \\sin\\frac{m\\pi x_1}{a}\\sin\\frac{n\\pi x_2}{b} dx_1 dx_2\\,.\n \\end{align}\n", "1a6ea6c95ed75bdd123541f9886b3f6c": "A_1^{x}+B_1^{y}=C_1^{z}", "1a6eefaf05c3f65caab7b869ff68e205": "i=1,2,\\dots,m", "1a6fefa5ee8dd24299f086f37a3e090c": "W = 2(\\mathcal{L}(X;\\hat{\\mu},\\hat{d})-\\mathcal{L}(X;\\hat{\\mu},1))", "1a6ff258f201b594f40a5b69b3192fc6": " \\tfrac{360^\\circ} {n} ", "1a703131a00e1bcf0abc991e3982641e": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\left( \\tfrac13 \\mu + \\mu^v) \\nabla (\\nabla \\cdot \\mathbf{v} \\right) + \\mathbf{f} ", "1a70ea4c892391f380b78d9ae28a0a4b": "\\begin{align}\n\\int \\cos^2 x \\, dx \\,&=\\, \\int \\left(\\frac{e^{ix}+e^{-ix}}{2}\\right)^2 dx \\\\[6pt]\n&=\\, \\frac{1}{4}\\int \\left( e^{2ix} + 2 + e^{-2ix} \\right) dx\n\\end{align}", "1a70f94acac1a1cff56a30b320c15e50": "\\int_0^\\infty \\frac{\\sin px}{x}\\, dx=\\begin{cases}\n\\pi/2 & \\text{if } p>0 \\\\\n0 & \\text{if } p=0 \\\\\n -\\pi/2 & \\text {if } p<0\n\\end{cases}", "1a71226f50ca2b457f76a271d8b46749": "y_0\\in\\R^m", "1a7127e19d14f8076463305f808a4fe6": "\\scriptstyle\\{e_{(a)}\\}_{a=1\\dots4}", "1a71788fbb1b36a1b36a07b94700785a": "\\kappa = \\frac{8 \\pi G}{c^4}", "1a71a6001ed87018a99f82028678afb8": "\nC = \\frac{n_c P(c|\\vec y) - 1}{n_c - 1}\n", "1a71a708ba65371d2a498f5d1c48162a": "r\\Pi\\,\\Delta t = \\Delta \\Pi", "1a71c2d916e5a78c3e8684550bc4abc9": "\\bigtriangleup_{SO,dB} = ISOI_{dBm} - P_{in,f,dBm}", "1a71c8ad897631136005ffa374e15e10": "h^2_o=\\frac{2L^2k_o}{rD_cC_o}", "1a71c95dce1de5c5c2ae18ca8819d0e7": "1.44\\log_2(1/\\epsilon)", "1a72048d807c29547d5382b269caba61": "\\hat2", "1a723406116be4bce41d9db031b8d0e1": "p_{ij} \\ge 0", "1a72a02ddd1d46ddf960993b0b964940": "n_e \\propto e^{e\\Phi/k_BT_e}.", "1a72abb3e21b33d0c2371a5a902c9061": "Q = K\\left(\\text{Final}\\right) - K\\left(\\text{Initial}\\right) = \\left(m(\\text{Initial})-m(\\text{Final})\\right) c^2", "1a72e1c52b1e0e6314d233b3ba60b794": "{x e^x \\over e^x - 1}-5=0.", "1a732869a01a7e79c5eb16a68a1d3de1": "\\sigma(\\pi)", "1a7335d95d296135763ee0f305faa9b7": "0.99 \\cdot 100 - 0.98(100 - x) = x", "1a7392f171ee64b2e179badcd8a2337d": "\\displaystyle{ {dg(z)\\over dz} = {1\\over (cz+d)^2}.}", "1a739545d2078c6844af26099315532f": "\n\\frac{1}{\\sqrt{\\pi}}\\int w(z)\\,dz = \\frac{\\mathrm{erf}(z)}{2}\n+\\frac{iz^2}{\\pi}\\,_2F_2\\left(1,1;\\frac{3}{2},2;-z^2\\right)\n", "1a73cbf05c151e03d7c64f466053673d": " \\forall x \\, [\\alpha(x) \\land \\gamma(x)] \\leftrightarrow (\\forall x \\, \\alpha(x) \\land \\forall x \\, \\gamma(x) ).", "1a73da00df44a7180f2edde6598a5fdd": "\\Omega(o\\mid s',a)", "1a73ec89563d5c404c0a50d028565bee": "\\mathrm{St} = \\frac{h}{G c_p} = \\frac{h}{\\rho u c_p}", "1a73f328bad560e4c0fe9f0896a655fb": "x_1 - x_2", "1a7431ece4a4e909c86d5e69dec1d56e": "\\sin t = \\sin(2\\pi k+t) \\,\\!", "1a744e85878a0dde48aa99e18b3e0149": " U = \\frac{1}{2} \\varepsilon \\int_{V} |\\mathbf{E}|^2 \\, \\mathrm{d}V \\, ,", "1a7452e4104f96d3669bf689f18195b9": "t = (r_1 \\cdots r_h)(q_2 \\cdots q_n),", "1a748458bd91e0994006f75c21367da7": "E_2=c/a", "1a74909b6cbaa4532a76d83b72c12de0": "\\bigcirc", "1a74921448294d4a093b21e2b282d879": "\\Delta) = m", "1a74ac7dd63ca38b48896cd693b87f23": "\\hat{G} = \\langle N,\\hat{A}=S_1\\times S_2\\times\\dotsb\\times S_N, \\hat{u} =u \\rangle", "1a75049c643e0fbd930397c02706bc47": "y_n = S \\cdot x_n^2\\, , \\!", "1a7523e278dcbce48c7cf3ee3bb31b1e": "\\chi_U^{-1}(1) = U.", "1a75b8deabcb9580d994fb8f28602355": "\\pi(\\mathcal{C})", "1a75be367908f94bfb46e0b31f9ccac3": " GB(y;a,b,c,p,q) = \\frac{|a|y^{ap-1}(1-(1-c)(y/b)^{a})^{q-1}}{b^{ap}B(p,q)(1+c(y/b)^{a})^{p+q}} \\quad \\quad \\text{ for } 0=k", "1a8797adbc75a653772d2b460945cccd": " \\mathbf{F}(\\mathbf{x}) = \\begin{matrix} \\frac{1}{2} \\end{matrix} \\frac{dC}{d\\mathbf{x}} \\frac{Q^2}{C^2}~. ", "1a87c43deb6b36b27dedc7fa7c8170ef": " S_m=k\\ln \\Omega,", "1a87d3616b3113145971b343c21d4cec": "-\\frac{d\\Phi}{dp}\\left[Ap+Pf(p+\\Phi(p))\\right]\n+A\\Phi(p)+Qf(p+\\Phi(p))=0.", "1a87df7c2c14481aa21ce0dd48430051": "\\tfrac{3M-E+S}{6}", "1a87f4922e4f646da575229a1a6ae1b4": "\\text{mul}_{f(A)}f(\\lambda)=\\sum_{\\mu\\in\\text{spec}A\\cap f^{-1}(f(\\lambda))}~\\text{mul}_A \\mu.\\,", "1a8825691c8f861298868dfc294b7e50": "S_{mk}^{}=\\sigma_{mk}+Rp_{mk}", "1a88a26c7cc184520c0bf97122ce5e68": "f^{(0)}_n(x)", "1a89202c3ef48551b193fdf212df77af": "q=\\gcd{(a,b,c)}", "1a8948f5786cc2a2b27c2738a9b5b27b": "\\Omega = \\Omega_1 \\cup \\Omega_2", "1a8949ae238dd2c3c9285cdbf0eeaec3": "V(r) = \\frac{Z_1 Z_2 e^2}{r} \\phi (r) \\qquad (1)", "1a894b853f9480eef2091046c3b8105c": "\\begin{matrix}1\\le p\\le n\\end{matrix}", "1a897370e99b3bdbd2842a3d6e181096": "b_0 = g_0", "1a898f5e52c455a5cc3698003cb95bdb": "\\tau = K \\left(\\frac{\\partial u}{\\partial y}\\right)^n ", "1a89bfeaf5f474c01cfdc78223a007f0": "12x \\equiv 20 \\pmod {28}\\ ", "1a8a066052a033eed9de1056eba7afde": "\\alpha=(\\alpha_1,\\ldots, \\alpha_n)", "1a8a34a6cf1e4282b31ea247ac3936b1": "\\in \\mathbb F_5 ", "1a8a36d21d63fa0a8c0f1af7d9aac318": "\\alpha = - \\frac{\\mu}{kT}", "1a8a5ae6c2ae1b82c7aa71b1421db69a": "S^2=\\{a\\in E^3\\colon\\|a\\|=1\\}.", "1a8a7397f830da3fa4e6eb0108a95942": "\\big. \\frac{\\partial E_\\mathrm{in}}{\\partial t} - \\frac{\\partial E_\\mathrm{out}}{\\partial t} - \\frac{\\partial E_\\mathrm{accumulated}}{\\partial t} = 0", "1a8ad4c88dd861d0bd473ecfe1cfa18f": "\\eta (x)", "1a8b1f992463adee24d67af3a196deea": "X \\Rightarrow Y.", "1a8ba3d288ebb1c6ba3d0eae81c0fdb7": "\\frac{AH}{AD} + \\frac{BH}{BE} + \\frac{CH}{CF} = 2.", "1a8c26bf5084e8118743b648ecb4b764": "\n\\lambda_{\\pm} = {3\\over 2} \\pm {\\sqrt{ 5} \\over 2}.\n ", "1a8ca35b9618b99de19964b07dd81d2a": "F^\\text{T}", "1a8cd99f0a4c0b00e1478a10285b74be": "\\mathfrak{der}(A)\\oplus\\mathfrak{der}(J_3(B))", "1a8ce6da442aad48c4f9a2c0954e194c": "A_0(x)^2 - B_0(x)^2 = 2m \\left( V(x) - E \\right)", "1a8d1dc4f09158f889f21ddae1829c8d": " {\\mathbf p}={\\mathbf p}_1\\times {\\mathbf p}_2 \\times \\cdots \\times {\\mathbf p}_m. ", "1a8d35fda7748bd4dc345e93bc0c079a": "\\int\\frac{\\mathrm{d}x}{\\sin^n ax\\cos ax} = -\\frac{1}{a(n-1)\\sin^{n-1} ax}+\\int\\frac{\\mathrm{d}x}{\\sin^{n-2} ax\\cos ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "1a8d5719b61cfd85ebd754c952f4f209": " -\\frac{m}{2}\\ln|-\\boldsymbol\\eta_1| + \\ln\\Gamma_p\\left(\\frac{m}{2}\\right) =", "1a8d5f02caa455c7f95570c75992136a": " \\phi(\\omega) = -\\mathcal{H} \\lbrace \\alpha(\\omega) \\rbrace \\ ", "1a8d7519b8f5068cacc05537e2ae2da4": "r_{AB}", "1a8d9cc12436e061ce67f8e35f43284a": "C = {c_{ij}}", "1a8e11552a6809e21105413af53ddcc9": "\\varphi_B = (k_B T/q) \\ln (N_A/n_i) \\ , ", "1a8e6971cffd73e684a6ff22c14f422c": "k^* = S", "1a8ed6dae1864fd2f4eb7f64df03e09f": "\\vec{\\mu}_S = g \\mu_B \\vec{S} = 2 \\mu_B", "1a8f6283ec5209a155dbdf387097ad8e": "\\mathsf{STUVWXYZ} \\!", "1a8f718d39b894547d56008f0501e6ce": "\\,\\Theta(N \\,\\mathrm{slog}\\, N)", "1a8f8945a45db50f58a4d474f56060ca": "k_\\mathrm{spec} = \\left [ \\sin(L, T) \\sin(V, T) - \\cos(L, T) \\cos(V, T) \\right ] ^n=(-\\cos(\\angle(L, T)+\\angle(V, T)))^n,", "1a8fdbf343d292335103d618023ec6c1": "\\forall x: \\neg P(x)", "1a9043f9be4dab0c9bb6fe9c43d1589d": " \\operatorname{de-lambda}[p\\ f = \\lambda x.f\\ (x\\ x)] \\equiv p\\ f\\ x = f \\ (x\\ x) ", "1a90529f34a92813a96700c4294b0afe": "R = \\tfrac{1}{2}[v,v]", "1a90561678516393a75425bf10db083a": " \\mathrm{d}f = f(d_p) \\,\\mathrm{d}d_p", "1a9071e8efdeaceba95466942bcb5366": "\\left(\\beta-\\delta-\\tfrac{1}{2}\\right)=(4\\pm8)\\times10^{-12}\\,", "1a907711a467a9b7a1dcfa7d70af30eb": "\\mathcal{B_A} = (\\mathcal{A}, \\Delta ' , \\varepsilon, \\Phi ')", "1a90808e6fb9d34c17d8e35101921ea8": "\\gamma_m = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L\\left(y_i, F_{m-1}(x_i) + \\gamma h_m(x_i)\\right).", "1a90a6ec2f44d453602421cbfc9e3488": "b_m = (a*1)(m) = \\sum_{n\\mid m} a_n. \\,", "1a90e02e0bdc29efffba77f2b52939c4": "A_1X_1", "1a90e28476d96f6ea05b599082b2bbc9": "\nH_R(s) = { V_R(s) \\over V_{in}(s) } = { RCs \\over 1 + RCs }\n", "1a90f1c5fdae4d81c92c70f2fa5ce580": "f_\\omega(f_1(f_0(3))) - 2", "1a912d74b53523a6cc7810b3fc595e97": "\\Phi_S^R = -m(S\\otimes \\Phi_S^R\\circ P)\\Delta.", "1a918609cb6a66b5316cc7712b123c1e": "f(x)=1/|x-1|", "1a9193d185242c89451cefe024afe9e7": "a\\,\\mathcal{L}\\,b\\Longleftrightarrow a^{-1}a=b^{-1}b,\\quad a\\,\\mathcal{R}\\,b\\Longleftrightarrow \naa^{-1}=bb^{-1}", "1a921f9516272fbf443757f475f7242d": "(\\cos x + i \\sin x)^n = \\cos (nx) + i \\sin (nx), n \\in \\mathbb{Z}.", "1a9250d60dce98ddbeecca72f913cd92": "E = \\gamma(\\mathbf{u})m_0c^2, \\quad \\mathbf{p}=\\gamma(\\mathbf{u})m_0\\mathbf{u} ", "1a9256ddec9b9994ff1d9cec33f734f8": " \\left \\| C y_n - C y_m \\right \\| = \\left \\| (C-I) y_n + y_n - (C-I) y_m - y_m \\right \\|", "1a926eecc71b437898a276c1b780ffb5": "\nJ_{F,\\alpha}(\\theta:\\theta')= \\alpha F(\\theta)+(1-\\alpha) F(\\theta')- F(\\alpha\\theta+(1-\\alpha)\\theta')\n", "1a92a0337571c31ccd3dfab189f0a66c": "\\max_{|z| \\le 1}( |P^{(k)}(z)| ) \\le \\frac{n!}{(n-k)!} \\cdot\\max_{|z| \\le 1}( |P(z)| ). ", "1a92f8e9bf58b924ec5b5c4aa397211e": "U = -\\int\\vec{F}\\cdot d\\vec{x} ", "1a931c2def48d4eea352d9a9fbd59e3b": "q^{n-m}", "1a933888906895bacb397ad221330dc9": "i_n=1,\\dots,r_n", "1a933ede8b514212b3a6bdef389a1dd7": "\nE_{\\mathrm{ground}}=D_{\\alpha }\\left( \\frac{\\pi \\hbar }{2a}\\right) ^{\\alpha\n}. \n", "1a937a42af3ef6ff6e8b13f3d1c49d1f": "\\chi(A)-\\chi(B)+\\chi(C)=0", "1a9381d08198452c6aeef9dc9d039e2d": "\n= {A \\over \\pi} < \\infty.\n", "1a93e86e9842175c41e5468dfdf11b6d": "pE = pN/2", "1a940f5d62a46f0aef587132fb0b28b8": "\\vec P = \\frac{d \\vec \\tau}{dt}", "1a942d6e3e7cf96bc0cf07c5f4c30bcd": "\n\\begin{array}{ccc}\n{1 \\over \\sin A} = \\csc A & \\text{or} & {1 \\over \\csc A} = \\sin A \\\\ \\\\\n{1 \\over \\tan A} = \\cot A & \\text{or} & {1 \\over \\cot A} = \\tan A \\\\ \\\\\n{1 \\over \\sec A} = \\cos A & \\text{or} & {1 \\over \\cos A} = \\sec A\n\\end{array}\n", "1a94e3268c0eaf8f14089e3e84e97827": "\\chi_+^\\alpha = \\frac{x_+^\\alpha}{\\Gamma(1+\\alpha)}.", "1a94ed9fc45e5af6704e271bb651baa6": "A = \\{x \\in \\mathbb{R}^2: x_2 \\geq x_1^2 \\text{ or } x_2 \\leq 0\\}", "1a953cc0fadf728c89ef26fe6c6b091b": "\\text{Quot}(A)", "1a955e5540626e58816bb17f281209f1": "d = r \\operatorname{haversin}^{-1}(h) = 2 r \\arcsin\\left(\\sqrt{h}\\right)", "1a95766f6767c6a343957e18e5a4742b": "\n\\mathrm{SNR} = \\frac{P_\\mathrm{signal}}{P_\\mathrm{noise}} = \\left ( \\frac{A_\\mathrm{signal}}{A_\\mathrm{noise} } \\right )^2,\n", "1a9593007f548b7f31565c4abd411335": "x \\in \\mathbb{R}_+", "1a95b81e6a2cb8cd7300d96e399fb31c": "\\Phi(v) = \\left( \\int_0^\\infty \\bigl( t^{-\\theta} J(v(t), t; X_0, X_1) \\bigr)^q \\, {dt \\over t} \\right)^{1/q} < \\infty.", "1a95e7845d318f5307e1823f9e522f4e": "A(x) = \\prod_{i=1}^n (x-x_i) \\quad\\text{and}\\quad B(x) = \\prod_{i=1}^n (x-y_i). ", "1a9611b7f0c4e4f44aad57a98e867c97": "f = \\frac{c}{\\lambda}, \\quad\\text{or}\\quad f = \\frac{E}{h}, \\quad\\text{or}\\quad E=\\frac{hc}{\\lambda},", "1a9633c244f9c88f511ab4cc2d69d023": "2\\le n\\in\\N", "1a96d24b78a5995681e97e696f2065a8": " e ", "1a96eae5c008381d0e652f607a58557d": " J = 2,1,0 \\, ", "1a97461b7353373189c0be4a1150d99c": "E(\\mathbf k) = E_0 + \\frac{\\hbar^2 \\mathbf k^2}{2 m^*}", "1a97a3e30ffa3d15521abef105847057": " \\sum_{i=1}^N \\Delta t_i = T. ", "1a97f660485876e6e2ea92d858b91d7c": "(\\hbar m_\\ell)", "1a984ae26cdd98f2387b6b61e920ced2": "f^{(n)}(x) = \\lim_{h \\to 0} \\frac{\\sum_{0 \\le m \\le n}(-1)^m {n \\choose m}f(x+(n-m)h)}{h^n}.", "1a987d1cc9164f7818bea6654c4c44f5": "P(x,y,z)", "1a988b93be6d7a771ba072cb98dcb127": " n_{t+1} = \\int_{-\\infty}^{\\infty} k(x-y) R n_t(y) dy ", "1a9896044879d8764e859d413903d919": "\\mathbf{n}_{u}", "1a98b6d80ebdcb60a7721c86f3551f52": " \\frac{1}{\\pi} = 12 \\sum^\\infty_{k=0} \\frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}\\!", "1a99d7d6b1e3c2fcd1882d7737707aca": "\\operatorname{logit}(P) = \\beta_0 + \\beta_1 \\times \\operatorname{logit}(Q) + \\epsilon", "1a9a075f5f14056ae46f7fe18148e131": " \\left(q, \\rho, I, \\mathbf{J},\\mathbf{P}, \\mathbf{p}\\right) ", "1a9a35e3258bdf25c4e4785fe5a1d974": " \\frac{s+\\beta}{(s+\\alpha)^2+\\omega^2} ", "1a9a752aae65f753a327c946d03efb6d": "(B y + \\beta + 1)^n \\le B^n x + \\alpha\\,", "1a9a76745437118fd8c423c3f5b97a36": "\\phi_{u,v}", "1a9a95bd0521864b1f6d46eb3859717d": "G^v = G_{\\psi(v)}", "1a9ab3873881de67769dad4378e52156": "10^k", "1a9abb630f4abe69219824994b7890ad": "\\partial x / \\partial \\theta < 0", "1a9b19229ca3fe5680330a79335ec60a": "(0,1,2,3....,n-1)", "1a9b46b14a1e4ce15b79d99a8f30d4e1": "k_a f_a + k_b f_b + \\cdots + k_N f_N,\\,", "1a9bd525923ce8981e3524656300ba0e": "A=\\frac 1\\sqrt{2k(k-\\gamma)}\\sqrt{\\frac C{n-|k|+\\gamma}\\frac{(n-|k|-1)!}{\\Gamma(n-|k|+2\\gamma+1)}\\frac 1 2\\left(\\left(\\frac{Ek}{\\gamma\\mu c^2}\\right)^2+\\frac{Ek}{\\gamma\\mu c^2}\\right)}", "1a9bd52b2b494ef364bf20824cad2c06": " v_2/c =\\mbox{tanh}(s_2)={\\frac{e^{s_2}-e^{-s_2}} {e^{s_2}+e^{-s_2}}} ", "1a9bdda778ba11f53e2711c7a4a375c6": "I=n\\begin{bmatrix}0.2 & 0 & 0\\\\0 & 10.1 & 0 \\\\ 0 & 0 & 10.1\\end{bmatrix}.", "1a9cf77d65498a343549ea09ba214c00": "Q=-W\\;", "1a9d4831368d128c719d803d1911f984": "c_2 \\in C_2 ", "1a9d650d978ba43940a313836d36ef77": "\\{1, 2, \\cdots n\\}", "1a9da1ef8e4b4bd566b330ff883549aa": "I_{r,r-1}= \\begin{bmatrix} I \\\\ O \\end{bmatrix}", "1a9dd50f4606d0c0057fac6200cd691f": "\\boldsymbol{\\hat\\beta} =( X^\\mathrm T X)^{-1}X^{\\mathrm T}\\mathbf{y}.", "1a9e06fcf8d6146d0e73f567d279a582": "\nG(\\mu) = G_0 \\sum_n T_n (\\mu) \\ ,\n", "1a9e6c60b1b60e7e47a816dcba44c6b2": " \\frac {d \\vec {J_R}}{d t} = \\frac {\\gamma}{\\hbar} \\vec {J} \\times (\\vec {B_0} + \\vec{B_R}) - \\vec {J} \\times \\omega ", "1a9f3f557cfc6f226a63069d9f568925": "\\tfrac{52163}{16604}", "1a9fbde296633dbac2d794dfa832d7e2": "\\forall i\\, [ P_i \\in R_i ]", "1aa03c78298365d8bd5e95ec8f57da89": "\n \\mathcal{M}_{\\rm Tot} = \\mathcal{M}_{\\rm Trans} + \\mathcal{M}_{\\rm Rot}.\n", "1aa081cc2ba4f8142d252c4cec0d1e66": "\\lambda^4", "1aa0b3f8750dea7a4fac798c3d7ed865": "\nX = \\{ 2, 4 , 4 \\}\n", "1aa0b69ace93700ecaee285cd2424ea1": "a = 2^r", "1aa0dcb8bf73775ab5ba718bbed70c70": "\\mathbf {r}", "1aa12dc7c2e0ab1f2dc9a71409e7cd06": "Fd = C \\frac{\\operatorname{d}^2 \\theta}{\\operatorname{d}t^2}", "1aa1303a9ab04ddd702ab3a9fe021e77": "\\sum_{k=0}^n {n \\choose k} k^{n-k}", "1aa16aa11e9490e7f534c4326f4890f8": "y_0=0", "1aa194ce5024713d8a0db656df36b8d7": "\\frac{1}{r^3} P^1_2(\\sin\\theta) \\cos\\varphi = \\frac{1}{r^3} 3 \\sin\\theta \\cos\\theta\\ \\cos\\varphi", "1aa1bda74f474fad29d9654956ae10f5": "\\beta := 2\\rho h/D", "1aa1f6198ae825527ff76de0187af369": "C_{13}=\\frac{1}{N} \\sum_{r=1}^N Q_r(t_1) Q_r(t_3)=1. ", "1aa20d211130ae324333d03bdb26b5dc": "J(n,x) = \\int_0^x t^n \\frac{e^t}{(e^t - 1)^2}\\,dt.", "1aa21c217a57dfafcea07b7fc80e8b23": "\\sigma_1\\,\\!", "1aa2861f962fb86bbbb2d43ff4af92d2": "\n\\frac{\\partial \\overline{\\rho}}{\\partial t} + \\frac{ \\partial \\overline{\\rho} \\tilde{u_i} }{ \\partial x_i } = 0.\n", "1aa2b8822a696953d20dec92d93271e1": " \\frac{1}{2} {\\bold u} \\cdot \\nabla {\\bold u} + \\frac{1}{2} \\nabla ({\\bold u} {\\bold u}) ", "1aa2c1eaa5a11af7eb7311b86b998c4d": "\\mathfrak{so} (n, \\mathbf{R}).", "1aa2d2eddbda71018bd60c149ebd02bb": "\\rho_{\\text{realized }} = \\frac{2}{n(n-1)}\\sum_{i < j}{\\rho_{i,j}}", "1aa30dce386abfc4f75b826fcc5ca26f": " E_\\lambda(n) = E_{\\lambda_n}(n) ", "1aa326395ebe8e7b8d05bc85c4ea7b8e": "R\\left( x \\right) := xH\\left( x \\right)", "1aa329b2c4b0f12ab4f6b1ec1a04da95": "c_{n} = \\sum_{k=1}^{n} a_{k} \\sum_{\\mathbf{\\pi}\\in \\mathcal{P}_{n,k}} \\binom{k}{\\pi_{1},\\pi_{2}, ..., \\pi_{n}} b_{1}^{\\pi_{1}} b_{2}^{\\pi_{2}}\\cdots b_{n}^{\\pi_{n}}, ", "1aa3ecad3243bf1acb3eb09b4986c1db": "A \\to k", "1aa438502c4b923c65a9b72ecbdcd61a": "K_x", "1aa4cb2aefe8cdaaace2c85d4012f72a": "K = 1+ K_1 + K_2 + \\cdots", "1aa4fc6c4c153270c84d61175458ddc7": "f_1, . . . , f_n", "1aa502c8b5ee92630c08a04161cdc580": "\\Phi^TC\\Phi=(XW)^TC(XW)=D_C", "1aa5084befc888be4c182a1db87eb403": "\nn' \\geq sn^{\\frac{s-1}{s}}\n", "1aa51a3622ee847ff6f1a966c6bc7cca": "G_0:=\\bigsqcup_\\alpha U_\\alpha", "1aa57eb79f46dbce421f1712607a021d": "\\left(\\!\\!\\binom{r}{k}\\!\\!\\right)=\\frac{r^{(k)}}{k!}=\\frac{r\\cdot(r+1)\\cdot(r+2)\\cdots(r+k-1)}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}", "1aa5a502b1db0d9750ef007504234a9a": "\\exists x \\in S_{k-1}^{\\perp} \\, \\|x\\| = 1, (Ax, x) \\ge \\lambda_k.", "1aa5d95133f2d7f93ef5f3c1396ae0c8": "(x+1)^{x+1}-x^x\\,", "1aa5fe39fc4d04da53e2d293dcae592a": "\\frac{\\mathrm{d} C_{1} }{\\mathrm{d} t} = g C_1 - 20.8\\alpha r^{3} G C_1^2|", "1aa678d7c56144dfaa26e7a4c44d5068": " 1 > \\beta \\ge 0.75", "1aa678fa785dfbffc14987659e60412a": "\\alpha_1,\\dots, \\alpha_n", "1aa68dd2536c4596fdc9cf97ca5806df": "\\text{Base amperes }=\\frac{\\text{base kva * 1000}}{\\sqrt{3 } * \\text{base volts}}", "1aa6d6592ba6720659b290dd764e1cd7": "{\\pi}/{2N},", "1aa6d783d3063dba6684bf278e7ee136": "\\pi_i \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial {\\mathcal L}}{\\partial(\\partial x_i / \\partial t)}.", "1aa6da72fa4d6bb6f20d91e21e8c1043": "\\log_2(n!) \\approx n (\\log_2 n - \\log_2 e)", "1aa6dda22ff509a4f9f5c73df9683726": "\\left\\{C_i\\right\\}", "1aa6e1ba9fa095f15ad2ce49efa5d744": " \\{ \\mathbf{e}_{23},\\mathbf{e}_{31},\\mathbf{e}_{12} \\}", "1aa70620cb0cd0c1b75bba9b2d801ff1": "\\frac{\\partial E}{ \\partial w_{ji} } = \\frac{ \\partial \\left ( \\frac{1}{2} \\left( t_j-y_j \\right ) ^2 \\right ) }{ \\partial w_{ji} } \\,", "1aa7082d2797393c491580a00c94e6d9": " f_X(x) = \\frac{1}{\\sqrt{2\\pi}}e^{-x^2/2}.", "1aa70e4f96d6b21922eab96291658630": " \\lim_{k \\to \\infty} \\bold T^k \\quad (6) ", "1aa763d7d571d1e8ebc0f2a07b4ed873": "Z_{eff} = Z - S ", "1aa7cf9d18e5fec597b9fa5ed84853dc": "g(a + h) - g(a) = g'(a) h + \\varepsilon(h) h.\\,", "1aa802db0b35a76a2242e7da8504d344": "E(Q) \\leq \\frac{m_0}{m}\\alpha \\leq \\alpha", "1aa83bd9d4d6c1a9578db73c73fa45b6": "P_s=P_r=P=V \\cos\\left(\\frac{\\delta}{2}\\right) \\cdot \\frac{2V\\sin{\\left(\\frac{\\delta}{2}\\right)}}{X}=\\frac{V^2}{X}\\sin(\\delta)", "1aa8545042c8cc55c9bfd042ea68d783": " F(a_1, \\ldots, a_n) = F(a_\\boldsymbol{\\pi(1)}, \\ldots, a_\\boldsymbol{\\pi(n)})", "1aa857e72edc5e805a793c7e3d616602": "|u'(x)-(\\pi u)'(x)|\\le K h \\|u''\\|_{L^2(a, b)}", "1aa85f3d7c5a523cbb53688f030b3774": "\\omega_2^L", "1aa8f07cc43fd4a3b0e2cd7b4b941fbc": "F(y) = \\frac{\\int_{x_0}^{x_1}y(x)\\;\\mathrm{d}x}{\\int_{x_0}^{x_1} (1+ [y(x)]^2)\\;\\mathrm{d}x} ", "1aa8fae67966d47f1298d3fcd8547a57": "\\ \\|x[n]\\|_{\\infty} < \\infty", "1aa923b192fe5854eb638f1229244a50": "(e^{b \\epsilon} e^{-ar}) q (e^{ar} e^{bs\\epsilon}) = e^{b \\epsilon} (e^{-ar} q e^{ar} )e^{bs\\epsilon}) = e^{2b \\epsilon} (e^{-ar} q e^{ar}).", "1aa93e33f7ed6b57746ed181cb547aab": "(\\mathbf{v}_0, \\mathbf{v}_1, \\mathbf{v}_2)", "1aa9518cd94e43191f3ba41046f68b41": "\\Im \\tau >0", "1aa9a41799aa3458645cdf076e47b306": "f(a) + f(b) = f \\left((a+b) \\frac{a}{a+b} \\right) + f \\left((a+b) \\frac{b}{a+b} \\right)\n\\le \\frac{a}{a+b} f(a+b) + \\frac{b}{a+b} f(a+b) = f(a+b)", "1aa9dd34bc41c3db620c1af63d236085": " d=67 ", "1aaa43be82036c17abef5f99eed9ffcd": "\n\\lim_{n\\rightarrow\\infty} a_n = 0\\,\n", "1aaa4ffef6743b01f0eee69e3dd93a78": "\\operatorname {somb} (\\rho) = 2 J_1(\\pi \\rho) / \\pi \\rho.", "1aaa76cbe62e84dcaf2dd492c2f6bf9b": "\\psi_k(x) = e^{i k x} u_k(x)", "1aaa7883ce81785407d702ee0470e1d3": " E = \\sum_i \\left[ \\dot q_i \\frac{ \\partial L}{ \\partial \\dot q_i} \\right] - L ", "1aaa85d122537e89ab6d3c6bd30b8a74": "x^*x = x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2.", "1aaa898a2fcca7ced16a675e9b6ef627": "f(x;\\alpha,\\beta) = f(1-x;\\beta,\\alpha)", "1aaabf98cfc7b0b338ac1e17085c2f55": "IC_p(X) = \\tau_{\\le p(n)-n}Ri_{n*}\\tau_{\\le p(n-1)-n}Ri_{n-1*}\\cdots\\tau_{\\le p(2)-n}Ri_{2*} {\\mathbb C}_{X-X_{n-2}}", "1aab2390bf8013903313e6614f2d6009": "a_n z^n", "1aab3c3aefb40f201b84c9d3f55516d4": "dN", "1aabcbe06016c902922ecb772d9c9717": "\\pi_i (f)", "1aac09608797b7147f80b5305dfc14d3": "K_{\\mathrm b} = \\mathrm{\\frac{[HB^+] [OH^-]}{[B]}}", "1aac0d8ab58b9c1cbfbd805ea6e4e5c7": " \\tau\\;", "1aac384b1eef0ded9c28536e13fb5b07": "d(v) = 1 \\,,", "1aac416b5b8df7462f817f76847c58a8": "\\begin{align} \n\\left| \\frac{f(w) - f(z)}{ w-z } - g(z) \\right|&= \\left| \\int_z^w \\frac{ g(\\zeta) d\\zeta}{w -z} - \\int_z^w \\frac{ g(z) d\\zeta}{w -z} \\right|\\\\\n&\\leq \\int_z^w \\frac{ | g(\\zeta) - g (z) |}{|w -z|} d\\zeta \\\\\n&\\leq \\max_{ \\zeta \\in [w, z]} | g(\\zeta) - g(z) |,\n\\end{align}", "1aac601bf88d00188a0f6f3cb8f94770": "\\frac{1}{2}\\left(1 \\pm |\\vec{a}|\\right)", "1aac73b58786c10d55743184579c5e92": "\\Psi = R e^{i S / \\hbar }", "1aacb8a1f06d3f525c2d52dcde17814f": "\\Omega = 2\\pi\\left(1 - \\cos\\theta\\right)", "1aad288fdab1a2cc333ae1b11fde7171": "\\Gamma = \\frac{1-y_T}{1+y_T}\\,", "1aad2fbcac161490247876059b7b6822": "r_k<\\sqrt m", "1aad642de22f845592b93adcabb1781c": "[1,B^3]", "1aad9eb45b1c24b7c14b395757608a58": "\\scriptstyle\\sigma_1^2", "1aae166bfdcdfb3c41e9474b78098c74": "\\sigma_{1}", "1aae16b5e6fb949662a8cfc55c51da66": "V(\\mathcal{C})", "1aae6b2cfb9b43aa540fb9096a375bc3": "B^I", "1aae8228d2c02a0a841bbbe9f0243b7d": "F_{(\\xi,\\mu,\\sigma)}(x) = \\begin{cases}\n1 - \\left(1+ \\frac{\\xi(x-\\mu)}{\\sigma}\\right)^{-1/\\xi} & \\text{for }\\xi \\neq 0, \\\\\n1 - \\exp \\left(-\\frac{x-\\mu}{\\sigma}\\right) & \\text{for }\\xi = 0.\n\\end{cases}\n", "1aaea31c49d95301fafca03d2e175417": "R_{\\mu }", "1aaf45c4d15258a4669acd07e7c709fd": "\\Lambda^{m\\mid n}", "1aafa2e5b4efd167cb0eb77e0f9eb594": "U_i:A\\longrightarrow\\Bbb{R}", "1aafc2ced604a9974ed7c15f3f8511f4": "x_i = \\tan \\theta_i\\,", "1aafc7bb8e0f10aa0d6074ade5b6a9d4": "2^{-2} \\times 0.100_2", "1aafd4023807be7de8aee982b3f9a4c8": "\n\\frac{BC}{BD} = \\frac{AC}{AD}\n", "1ab023d93061289b8e4dc122b5cb1ace": "\\begin{bmatrix}\n c_2 c_1 \t & -c_2 s_1 & s_2 \\\\\n s_3 s_2 c_1+c_3 s_1 &\t-s_3 s_2 s_1+c_3 c_1 &\t-s_3 c_2 \\\\\n-c_3 s_2 c_1+s_3 s_1 &\t c_3 s_2 s_1+s_3 c_1 &\t c_3 c_2\n\\end{bmatrix}", "1ab02506e653a363608474e8a9dac7de": "\\scriptstyle q \\;=\\; y^2", "1ab039c742bc51077ff5aa4ae8f8b8a5": "P = \\sum_{i=0}^\\infty{l_i}", "1ab0540f1dfcd2c19af04a8d7c838e30": "u_f = dl/dt ", "1ab06556316daa80b830e50aca6310c0": "p_n > {{p_{n - 1} + p_{n + 1}} \\over 2}", "1ab08df0178c231a421a1d866362b7bb": "b(w)", "1ab0e533205cfd46c57f78c4da18b051": "K = \\frac{a + b}{2} \\cdot h", "1ab11c6c85ef77b316d793c7b4f98916": "p=8m+5", "1ab13dc8c122d60b750463dd80c4a7f7": "\\deg(f,\\Omega,p)=\\deg(g,\\Omega,p)", "1ab16a900ad65dd304e17b7940393417": "\\left. \\frac{\\mathrm{d}}{\\mathrm{d}t} I( \\boldsymbol{u} + t \\boldsymbol{v}) \\right\\vert_{t=0} = -\\int_A \\sigma_{ik}(\\boldsymbol{u})\\varepsilon_{ik}(\\boldsymbol{v})\\mathrm{d}x - \\int_A v_i f_i\\mathrm{d}x - \\int_{\\partial A\\setminus\\Sigma}\\!\\!\\!\\!\\! v_i g_i \\mathrm{d}\\sigma \\geq 0 \\qquad \\forall \\boldsymbol{v} \\in \\mathcal{U}_\\Sigma", "1ab2b8ed0b4480f6fe8a50cbe067b350": "N = \\frac {f} {c}\n \\frac { v_{\\mathrm N} - v_{\\mathrm F} }\n{v_{\\mathrm N} + v_{\\mathrm F} }\n\\,.", "1ab2ebb77170c5836399cab7b0e8b8fb": " \\tbinom {n+1} k", "1ab2fb4c84388318d58cce76c8ab9675": " \\displaystyle{AA^*=JA^*AJ=I}", "1ab32c9f25c8affa819513aa0b30a70b": "i, j, l, k", "1ab370c0dfbac1d5c1f9206d83868632": "\\lambda_{x}^{2} \\leq \\lambda_{y}^{2} \\leq \\lambda_{z}^{2}", "1ab3a5bf4068065989c6064b169147e1": "\n\\begin{bmatrix} 1 \\\\ 1 \\\\ 0 \\end{bmatrix}\n\\wedge\n\\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\end{bmatrix}\ny = \n\\begin{bmatrix} 1 \\\\ 1 \\\\ 0 \\end{bmatrix}\n\\wedge\n\\begin{bmatrix} 1 \\\\ 1 \\\\ 2 \\end{bmatrix}.\n", "1ab42883c679fad5383a9cac3f2a4c3a": "\\int_0^\\pi f(x)\\sin(x)\\,dx= \\bigl(F'(x)\\sin x - F(x)\\cos x\\bigr)\\Big|_{0}^{\\pi}.\\!", "1ab45bbf086d8a17c53c13522059d5c2": "\\Delta_I", "1ab49de7222bf270920191a26a9cd052": "f+ \\lambda g", "1ab4e2d1136854fbb042404772121520": "1>0", "1ab5158115ac415d756a00c9cf5dc169": " \\Delta E ", "1ab52d81f116196376dc9f9c549f069b": "i = 1,\\dots, \\min\\{m,n\\}", "1ab5909b949af30693e04c9782baf20f": " \\sum_{i=1}^4 \\Omega_i = 2 \\sum_{i=1}^6 \\phi_i - 4 \\pi ", "1ab61f15198d7ed5c0af334198a0d764": "= \\epsilon/2", "1ab65788298b915edaa589fd7507c5bb": "a_n\\rightarrow1", "1ab67afd638a0b530a9e8c8eb7f147db": " C = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix}. ", "1ab698ea28a21c95680707077e229fa0": "\\alpha = 3\\,", "1ab6c02438070224a4e6e42117db9e98": " \\frac{\\mathrm{d} y}{\\mathrm{d} t} = f(t,y) ", "1ab6dd1ebba23da4c4cf415515c12387": " \\begin{align}\n\\mathbf{u}_k^{(1)} &= \\mathbf{v}_k - \\mathrm{proj}_{\\mathbf{u}_1}\\,(\\mathbf{v}_k), \\\\\n\\mathbf{u}_k^{(2)} &= \\mathbf{u}_k^{(1)} - \\mathrm{proj}_{\\mathbf{u}_2} \\, (\\mathbf{u}_k^{(1)}), \\\\\n& \\,\\,\\, \\vdots \\\\\n\\mathbf{u}_k^{(k-2)} &= \\mathbf{u}_k^{(k-3)} - \\mathrm{proj}_{\\mathbf{u}_{k-2}} \\, (\\mathbf{u}_k^{(k-3)}), \\\\\n\\mathbf{u}_k^{(k-1)} &= \\mathbf{u}_k^{(k-2)} - \\mathrm{proj}_{\\mathbf{u}_{k-1}} \\, (\\mathbf{u}_k^{(k-2)}).\n\\end{align} ", "1ab74872675fa1297f76e8950dc938dd": "\\mathrm{D}_2 \\cong \\mathrm{A}_1 \\times \\mathrm{A}_1,", "1ab76407e86e8a7c491577eb955316b9": "\\nabla \\cdot \\left(\\frac{\\mathbf{s}}{|\\mathbf{s}|^3}\\right) = 4\\pi \\delta(\\mathbf{s})", "1ab77df72ddb65a53b2d82e411c8e206": " \\ C_L = 2 \\pi (A_0 + A_1/2)", "1ab7846b0bfae81bf125c7f8d3cdc9f5": "\\phi^{n+m}=\\left (\\frac{1+\\sqrt{5}}{2} \\right)^{n+m}\\in O\\left(1.62^{n+m}\\right),", "1ab7bfc3b2401aed340f8f631e5556ce": "\\scriptstyle{\\tau = t_1 - t_0}", "1ab7d6c15619557c80e8fcd6b03964aa": " f(X) + E[u(X) - u(B)] ", "1ab8077d9861c4b3e56e863685dc869d": " 0 = MU_x + MU_y\\frac{dy}{dx}", "1ab85836a785e5828c5764871a1b82b3": "\\{p_{1}, p_{2}, p_{5}\\}", "1ab8c4df2b32b6f1921373f0eb03deb6": " \\phi_n : x \\mapsto n^{k-1} B_k(\\langle x \\rangle) ", "1ab8f74662094375be833aa7331a5dec": "DR_{T/D}^{S/V}", "1ab918526661afc3a55e5afe89aa0ed4": " \\text{ }\n\\frac{1}{t}\\sum_{\\tau=0}^{t-1} \\sum_{i=1}^NE[Q_i(\\tau)] \\leq \\frac{B + V(p^* - p_{min})}{\\epsilon} + \\frac{E[L(0)]}{\\epsilon t} \n", "1ab920863900f52cc962a172df58ea63": " d = \\frac{v^2}{g} ", "1ab934a547e2bc5d8afa2b41f85fe0df": "\\mu_2=\\dots=\\mu_k=0", "1ab949e8425c93a96c3625f82709dbc4": "\\left\\langle 0,c\\right\\rangle", "1ab9808a03a46592b2e540f8159c6df1": "P(H_{n})=\\mathbb{C}P^{n-1}", "1ab98f8bbc41e04448d5356018efee8e": "1_{En}", "1ab990ac5e62b96d59ad474058cdad2e": "f_*(u_n) \\in H_n(X)", "1ab9a39ee2d4b6e08dcda32167b23b23": "\\text{Throughput} = (\\text{Sales revenue – Direct Material Costs})", "1ab9cb5b6b65eeb000441c91765659df": "\\lambda_\\max", "1ab9ced7035753d114f561b980d5709b": "\\ge1", "1ab9e9918dcf677575c8aa3ff6123437": "\\hat 8", "1ab9ea8ff2894b022b3f7cd5e6d58e5b": "\\psi \\propto \\sum_{i=1}^{m} A_i e^{i\\bold{k}_i\\cdot\\bold{r}}", "1aba2b3443cfb68c08d0123d11920345": "x''+c_1 x'+c_2=0", "1aba6bc2e8410c9c20c626125896651c": "\\mathrm{Win} = \\frac{\\text{runs scored}^2}{\\text{runs scored}^2 + \\text{runs allowed}^2} = \\frac{1}{1+(\\text{runs allowed}/\\text{runs scored})^2}", "1aba9a316555b2a1f8f6abef61479bfb": "x \\in \\mathcal{N} (X)", "1abac7e5686a1a07099b0bd47416bcd7": "\\binom{p+q}{q} - \\binom{p+q}{q-1} = \\binom{p+q}{q}\\frac{p+1-q}{p+1}.", "1abaca9a286657bcc5335c98b479783b": "\\widehat{\\mathbf{J}} = \\widehat{\\mathbf{L}}+\\widehat{\\mathbf{S}}\\,.", "1abae6323d1a5a6d9ba2d4026013e26b": "\n\\begin{align}\n \\int_0^1 f(x)v(x) \\, dx & = \\int_0^1 u''(x)v(x) \\, dx \\\\\n & = u'(x)v(x)|_0^1-\\int_0^1 u'(x)v'(x) \\, dx \\\\\n & = -\\int_0^1 u'(x)v'(x) \\, dx = -\\phi (u,v),\n\\end{align}\n", "1abb17dc3174c03be1f2fd795a777250": "\\pi<\\widehat{1}", "1abb4c8a30ca619e2395a8d002bbfded": "\\varphi_{\\beta+1}(\\gamma+1) [0] = \\varphi_{\\beta+1}(\\gamma)+1 \\,", "1abb9bd1f9b31964775398d8fae5502f": " g_0(n) = 0 ", "1abbb88228d6436ed45e5741586a747d": "\\Gamma^1_{ik} = \\begin{bmatrix}\n0 & 1/r & 0 & 0\\\\\n1/r & 0 & 0 & 0\\\\\n0 & 0 & -\\sin\\theta\\cos\\theta & 0\\\\\n0 & 0 & 0 & 0 \\end{bmatrix}", "1abbe13a6b9520d438676740cdb15158": "Q_{eq}", "1abc03af71306d9db4a1524d034c2e07": "Vz-=Iz.Zz\\,", "1abc86f1f3b1a7ba8fdaba3bf6ec9729": "\\Delta p=p\\otimes K+1\\otimes p", "1abcc653113c3d1f138ce2feef07bc27": " b = 2i + (h-i-j) = h + i - j \\, ", "1abcec815c3e2c9a6fcac5be41fa230b": "x^\\prime = x - vt", "1abcf05eb36512ac8672493b4a410cd9": "\\operatorname{E}[\\,x_t(y_t - x_t'\\beta)/\\sigma^2(x_t)\\,]=0", "1abcfff16c133734ea77fa457f77aed6": "\\beta^n_{b}", "1abd1944ed6f70fd94104c9db8473d11": "S_n(x)=x j_n(x)=\\sqrt{\\frac{\\pi x}{2}} \\, J_{n+\\frac{1}{2}}(x)", "1abd3d98386280fdf4e48c43896e37f4": "\\ u'w'", "1abd5bc7d61eadc9b29dc764596d11bc": "\\textbf{K}_k \\textbf{S}_k = (\\textbf{H}_k \\textbf{P}_{k\\mid k-1})^\\text{T} = \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T}", "1abdd729b882e1bbb21eabf7f8cdb9b9": "G_\\epsilon(V,E)=D\\cap\\Z_\\epsilon^d", "1abe1e488858561aeec0dce26c295f66": "X \\mathbf{\\operatorname{m}} Y", "1abe21c5f54ab3be3b3a92ce4df7128e": " x=x", "1abe384bf2adb759827fb149cfe66d64": "\nH+L(\\alpha) \\ge 0.\n", "1abe44305ac7e4e6521be709b6d05fcf": "\\Omega^3", "1abeba592078e159bfb4b1c2a9bdcb7b": "H = 2h", "1abed2f9ab62c9c6df7ada4534217ea9": "\\text{(6)} \\qquad \n \\sigma_y(\\varepsilon_{\\rm{p}},\\dot{\\varepsilon_{\\rm{p}}},T) = \n \\begin{cases}\n 2\\left[\\tau_s + \\alpha\\ln\\left[1 - \\varphi\n \\exp\\left(-\\beta-\\cfrac{\\theta\\varepsilon_{\\rm{p}}}{\\alpha\\varphi}\\right)\\right]\\right]\n \\mu(p,T) & \\text{thermal regime} \\\\\n 2\\tau_s\\mu(p,T) & \\text{shock regime}\n \\end{cases}\n", "1abf24343749f2bb74c6f78e7b6bca6b": "m_2=1\\ ,", "1abf2b520ef1a91db8f2a22012de71e4": "\\phi(e_i)=e_j", "1abf3d7c735ea59569e8588ba93482a4": "S_0=\\{0,1,2\\}, S_{i+1}=(S_i\\times S_i)+S_i", "1abf43d94ea93793febf2ceeaebd2c43": "S_n = \\sum_{i=1}^n x_i", "1abf84996f3f707e1b215bbc2b012f2c": "I_R = GV_R", "1abfbd1bb9566bf753a1ba13e4676027": "S\\circ F \\in \\mathrm{D}'(V).", "1abff943e15eaf744beb3885cbf89134": "\\mathcal{G} = \\{A: \\mathbf{1}_{A} \\in \\mathcal{H}\\}", "1ac042b49fb7d8cada7d342e7c4065f2": "\\partial f/\\partial y", "1ac0839b13f4885faded5c2a49afa025": "\\overline{\\mathbb{D}}", "1ac08e0fd0cf461755eef45da79d9c66": "S(p) = \\frac 1 2 + \\frac 1 3 + \\frac 1 5 + \\frac 1 7 + \\cdots + \\frac 1 p.", "1ac0b55f1f613743f7271e10b100ad71": "S^{-1}R=Q(R)", "1ac0d601419879e7f42c90f6698f325e": "h_{\\phi} = a \\sigma \\tau", "1ac10edee329c373b1465dd6491f3595": "P=\\int_0^\\infty d\\nu \\int_0^{\\pi/2} d\\theta \\int_0^{2\\pi}d\\phi \\, B_\\nu(T) \\cos(\\theta)\\sin(\\theta)=\\sigma\\,T^4", "1ac1159bb78e0a96149ecc60295cbb5d": "\\mathcal{L} = \\partial_\\mu\\phi\\partial^\\mu\\phi - (\\phi^2 - 1)^2", "1ac11e8ba2687f97dc0dce38a5da1ce0": " Z_1, Z_2 \\sim \\mathcal{N}(0,1) ", "1ac167a94601d3bd9407cd841b300125": "\\frac{\\partial \\ln |a\\mathbf{X}|}{\\partial \\mathbf{X}} =", "1ac16a82c7efc4d76924cd6e464a3091": "\\{1, 2, \\dots, k^k\\}", "1ac1a4f4fcd5a72da08cc1955d65703c": " Q_i ", "1ac1ed89c9194c9e7779939efaa611d0": " V=\\sqrt{\\frac{(l^2+m^2-n^2)(l^2-m^2+n^2)(-l^2+m^2+n^2)}{72}}. ", "1ac21dcc002aa4bf060ab2c3cba3d029": "-671\\pm 6", "1ac23cbbd1934a654cf3ee328fea1882": "\\phi \\to \\phi \\lor \\chi", "1ac25ad53a1a66592c89613ae74fab24": "\\begin{align}P(Hypercalcemia~WHOIFPI~by~cancer) = \\\\\n P(cancer~WHOIFPI) * r_{cancer \\rightarrow hypercalcemia} = \\\\ \n 0.002 * 0.1 = 0.0002 \\end{align} ", "1ac29daf7266b56354a14880fcf1ad2a": "\\operatorname{sgn}(0+0i)=0", "1ac2a5282e20b69e09578e7ed63c832d": "\\mu_{\\mathrm{B}} = e \\hbar / 2 m_e", "1ac30352595c745bd666db5058d4b43b": "f'(\\alpha) \\neq 0", "1ac3b73863e00a4ee60189f18d0a802b": " \\lim_{n\\to\\infty} \\operatorname{Pr}(X_n \\leq a) = \\operatorname{Pr}(X \\leq a),", "1ac3b7ae1f1053a05eb3af6dfa9efd6d": "-K(T - t)e^{-r(T - t)}N(-d_2)\\,", "1ac42d09412d03d9e4267d2c9534658f": "P(N(t+h)-N(t)=1) = \\lambda(t) h + o(h)", "1ac451f00486b941e4c498570d8073b3": "\\therefore T=\\frac{c}{v}\\left(e^{v/\\alpha}-1\\right)\\,\\!", "1ac48a03d67cd2d89e3dad75529af52a": "s(x)=1", "1ac4b7c8c6e5bc02cda050d6b9eac989": "d_1(z) = a + (1-a)\\overline{z}.", "1ac4fe89f4f267d51da6bcf9a386b49e": "L_{\\mathrm{MV}}(x_0,\\dots,x_n) = \\sqrt[-n]{(-1)^{(n+1)}\\cdot n \\cdot \\ln[x_0,\\dots,x_n]}", "1ac52899339bbbd6dd0c2f01d1e9928c": "(T \\subseteq Y) \\; \\mapsto \\; f^{*}(T)\n= f^{-1}\\lbrack T \\rbrack", "1ac54f12ab5718857144a90269e92464": " \\mathrm{SCL}: r_{a,t} = \\alpha_a + \\beta_a r_{m,t} + \\varepsilon_{a,t}", "1ac59a71348c01c1360c048b7f28c536": " {x_2 \\choose \\theta_2} = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix}{x_1 \\choose \\theta_1}, ", "1ac5da260430003e69811d7015e822d8": "w = \\exp(\\pi i z)", "1ac648ddca94359d2437f0f13aa930d2": "\\beta \\equiv \\frac{1}{k_BT}", "1ac64d00754e43b4abdda0d7b336fd5b": "2^{2n}+1", "1ac66aa19f2577a001c296ee4c3f84fa": "\\begin{align}\n\\sum_{n=1}^{\\infty} \\left|\\gamma_n-\\beta_n \\right| &<\\infty \\\\\n\\sum_{n=1}^{\\infty} n \\left|\\beta_{n+1}-\\beta_n \\right|&<\\infty\n\\end{align}", "1ac67d3a5300ab0c5022cfb9fedf3f61": " (1, 0), (0,1) \\quad ", "1ac722c49b915507c9cff059702cf706": "Z_L\\,", "1ac73d4e1e66919fc33c11419b2c6c2c": " \\sin \\theta \\approx 1.22 \\frac{\\lambda}{d}", "1ac74de873b6051623f6a6666f04e25f": "t\\to\\infty.", "1ac752fd452eea38d6c44306c9522b1e": "x \\in {a,b}", "1ac765d6411a75ad190348f96b70614e": "(M,\\mu,\\eta)", "1ac783c6c43ba3762d481352146e4538": "\\longrightarrow_R^*", "1ac7b0f3dd179c5b5b8c9fa073466d46": " h = 1 - \\frac{2}{3} \\frac{(k+ \\lambda) (k+ 3 \\lambda)}{(k+ 2 \\lambda) ^ 2} \\, ;", "1ac7c1a17a7c88b812debfa6457dba1f": "\n\\begin{align}\ny_1 & = {f_1}^{}(x_1,\\ldots,x_n) \\\\\ny_2 & = f_2^{}(x_1,\\ldots,x_n) \\\\\n& {}\\ \\vdots \\\\\ny_m & = f_m^{}(x_1,\\ldots,x_n)\n\\end{align}\n", "1ac7d34981306d1966c327b4a357ab2b": "\\textstyle ID \\in \\left\\{0,1\\right\\}^*", "1ac8088c44308012141655c125bafac8": "U_{--}=\\bigcup_{n\\ge 0}\\alpha^{-n}(U_{-})", "1ac81b8ad1e2335ce45a23887a7ded18": "\\textstyle (X,\\Sigma). ", "1ac84e66b4aa6d5c44163a47022bd90b": "|a' \\rangle \\to \\alpha = \\frac{|\\alpha' \\rangle}{\\sqrt{\\langle \\alpha'|\\alpha' \\rangle}}", "1ac899ccce2d417ad6fe6e9fa4bbf9ad": "\\omega = exp(\\theta^at^a)", "1ac8cf945037b51f528f5097a2097a51": " |f(x) - f(c)| < \\varepsilon.\\,", "1ac8e3e74660dafa1f6a667bb16ceebc": "D = \\sum_{P \\in C}{n_P P}", "1ac9053477ef2c67dd2ead878f32361e": "R^2 \\equiv 1 - {SS_{\\rm res}\\over SS_{\\rm tot}}.\\,", "1ac93d0427703b0a5ea6255031052d68": "s, e, e_v \\in \\mathbb{Z}_q", "1ac951819db336b3c16b92d393cde79e": " I(X;Y) = \\int_Y \\int_X \n p(x,y) \\log{ \\left(\\frac{p(x,y)}{p(x)\\,p(y)}\n \\right) } \\; dx \\,dy,\n\n ", "1ac970b27d2f76d07d9061a3f8bc0557": " \\Phi(B)", "1aca039fca0fe90ce18c019d54503bfb": "\\hat H ", "1aca1c27d02ba4b9425b4132e0f47f1a": "L^2 Y_{\\ell m}=\\ell(\\ell+1) Y_{\\ell m}", "1aca56b06c43f6675bf5ccaf087b5e62": " a_{i \\pm \\frac{1}{2} } \\left( t \\right) = \\max \\left[ \n\\rho \\left( \\frac{\\partial F \\left( u_{i} \\left( t \\right) \\right)}{\\partial u} \\right) ,\n\\rho \\left( \\frac{\\partial F \\left( u_{i \\pm 1} \\left( t \\right) \\right)}{\\partial u} \\right), \n\\right] ", "1acaac09b7d5de46368f0e9efb4eef75": "\n B_{ijkl}\\frac{\\partial^2 u_k^{(1)}}{\\partial x_j \\partial x_l} = \\rho_0 \\frac{\\partial^2 u_i^{(1)}}{\\partial t^2},\n ", "1acaac888369dfac85fb74c88944f023": "O(2^{O(\\frac{k \\log{k}}{\\epsilon^{2}}))}dn)", "1acac23443d49bc12be4e4c098717dae": "B_n^1", "1acac7a29d4fd172140faaf938e385c9": "\\gamma^5 = i \\gamma^1 \\gamma^2 \\gamma^3 \\gamma^4 = \\gamma^{5+}. ", "1acb598033c89b51148ce7e98c0245b9": "\\,\\gamma = \\arccos\\left(\\frac{a^2+b^2-c^2}{2ab}\\right)\\,;", "1acbaf9dda30a8d0c95687637b60a8f2": "\\sigma(T) = \\bigcup_{i=1}^m F_i.", "1acbb9979bdeda693648b7d31f1c01dd": "A_1,\\ldots,A_k", "1acc02fb7d3da69280926cb4654fbba1": "R_3, \\, M_2, \\, M_3", "1acc77d5bb5a1645b117522235bb23db": "P_{1}^{0}(x)=x", "1acd18f6a4cd5b9931e49823fbb4cf74": "d\\in\\mathcal{D}^n,r\\in R\\,\\!", "1acd2dc7cbc07aee146fe91daa38e943": "\\boldsymbol{\\bar{v}}", "1acd80071923fa3df9af07fae6718a4d": "{\\mathcal H}_{h,v} = [\\eta^2_{h,v} + (\\beta_{h,v}\\eta'_{h,v} - \\frac{1}{2}\\beta'_{h,v}\\eta_h)^2]/\\beta_{h,v}", "1acd988cd6abcf169fd0b14b1e164b70": "\\int \\frac{\\mathrm{d}x}{\\csc{x} + 1} = x - \\frac{2\\sin{\\frac{x}{2}}}{\\cos{\\frac{x}{2}}+\\sin{\\frac{x}{2}}}+C", "1acdbbcba509a96934e167c71ecd73a0": "\\{b_n\\}", "1ace0414b12fa88b25c98bfce55ae3ca": "\\mathbf{A}^{\\mathrm{T}*} \\,\\!", "1ace0a5a3204a493e831125ce4e509e0": "\\langle E(s) \\rangle", "1ace25909136c74fe8d3d9b1effdcd09": " G(v) = \\omega_e (v+{1 \\over 2}) - \\omega_e\\chi_e (v+{1 \\over 2})^2\\,", "1aceea56504628d531c9ab8aa9120c44": "\\Delta_{S^{n-1}} f(t,\\xi) = \\sin^{2-n}t \\frac{\\partial}{\\partial t}\\left(\\sin^{n-2}t\\frac{\\partial f}{\\partial t}\\right) + \\sin^{-2}t\\Delta_\\xi f", "1acf1f8b7a4510e7e56d520b6334f701": "\\lang B,+,\\lnot,1 \\rang", "1acfc4ccaeb8d5d13372dac7f0798adf": "p= \\sqrt{\\frac{ab^2-a^2b-ac^2+bd^2}{b-a}},", "1ad0438311fecd2dda9d26fc710c3f58": "\n\\mathbf{v}\\equiv\n\\begin{pmatrix}\nq_1 \\\\\n\\vdots \\\\\n\\vdots \\\\\nq_{3N-6} \\\\\n0 \\\\\n\\vdots \\\\\n0\\\\\n\\end{pmatrix}\n= \\begin{pmatrix}\n\\mathbf{B}^\\mathrm{int} \\\\\n\\cdots \\\\\n\\mathbf{B}^\\mathrm{ext} \\\\\n\\end{pmatrix}\n\\mathbf{d} \\equiv \\mathbf{B} \\mathbf{d}.\n", "1ad084538eeb6c542fb5188d4237a35e": "z_1,z_2 = \\frac{1}{2}\\left(\\varphi_1 \\pm \\sqrt{\\varphi_1^2 + 4\\varphi_2}\\right)", "1ad10a950d51012a4abdf8af6a94f7f5": "0\\leq y(t)-r \\quad\\perp\\quad N(t)\\geq 0.", "1ad141546f024caa695127aa78b2d96b": "\\zeta(s)=\\frac{s}{s-1}-s \\sum_{n=0}^\\infty (-1)^n {s-1 \\choose n} t_n", "1ad1d1c7d3bfe417d5a1e3e7e49a1263": " \\sqrt{-h} = \\frac{2 \\sqrt{-G}}{h^{cd} G_{cd}} ", "1ad208c737ceea78c03c278168ebf1cd": "\\varepsilon^{\\{i,j\\},\\{p,q\\}}=\\mbox{sgn}\\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ i & j & p & q \\end{bmatrix} ", "1ad21099eee186a10725e09281509ecb": "\\leq \\subseteq S \\times S", "1ad246ac836d81dc93091efbfa00d82c": "m = \\int_0^2{\\int_x^{4-x}}_{}{}\\,2x+3y+2\\,dy\\,dx", "1ad250362e34f47ca7d51569614d2f9a": "\n \\overline{X}_n\\ \\xrightarrow{a.s.}\\ \\mu \\qquad\\textrm{when}\\ n \\to \\infty.\n ", "1ad2e99505fd53dc4849b34b220ad5db": "Y_{p}", "1ad30ef5daf05fcd00b41d70a71773b7": "\nI(2\\omega)=\\left|\\int{|E(\\omega)|^2e^{i\\phi}\\mathrm{d}\\phi}\\right|^2\n", "1ad39da56b90bccfcc436120392f903c": " S(a,b)=(b,a^b) ", "1ad3bd4a3b8e07df0909d4d845c59dd3": "S = F \\cup B", "1ad3de4c753b77af0666a3918886d04c": "(x)_{n}", "1ad3ef5ca8da2adc2a9e7cd396ff0824": "\\supseteq \\!\\,", "1ad40bb74aee47b9e9f178b9a2a01b2d": " F_G = m\\cdot g ", "1ad42097bf0281b3046a04a2af3c44d7": "k^3 \\varepsilon", "1ad458b4f4762e6e73e401659a75d97f": "{\\mathbf{x}}^n", "1ad484422d4e1ff54a1b6dd58fccee96": "\\operatorname{ch}(V) = \\frac{\\sum_{w\\in W} \\varepsilon(w) e^{w(\\lambda+\\rho)}}{e^{\\rho}\\prod_{\\alpha \\in \\Delta^{+}}(1-e^{-\\alpha})}", "1ad587203f5f3126d22a016b85a195fb": "\\ \\frac {D_{pitch} \\times lift \\times ( sin(\\beta) -{(L/D)_{\\alpha}} ^{-1} \\times cos(\\beta)) }{D_{heel} \\times lift \\times (cos(\\beta) +{(L/D)_{\\alpha}} ^{-1} \\times sin(\\beta))} = 10 ", "1ad59df534c1f6f7133fa06ae5bd4aab": "-\\alpha", "1ad5ab4c730c076fe255ba9b87850d18": "\n\\begin{align}\nP\n&= 1 - \\frac{N-1}{N} \\cdot \\frac{N-2}{N - 1} \\cdot \\cdots \\cdot \\frac{N-n}{N - (n - 1)} \\\\\n&\\stackrel{Canceling}{=} 1 - \\frac{N - n}N \\\\\n&= \\frac nN \\\\\n&= \\frac{100}{1000} \\\\\n&= 10\\%\n\\end{align}\n", "1ad610ebd0add956af81456e922ad8ef": "\\, D", "1ad64ff9d9f71a5fe1b6fad1e0a307a7": " \\mathbf{B} ( \\mathbf{r}, t ) = \\frac{1}{r} \\mathbf{B}_0 \\sin( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} + \\phi_0 ). ", "1ad6613e952c2e85fba379464b7eef03": "\\mathcal S = \\langle S, \\rightarrow, \\cdots \\rangle", "1ad6a19f85d3abaa03514e1f391ac1eb": "\\left\\langle v_1\\wedge\\cdots\\wedge v_k, w_1\\wedge\\cdots\\wedge w_k\\right\\rangle = \\det(\\langle v_i,w_j\\rangle),", "1ad7040b9de2599867a04826b5419dc9": "T_p(K) = \\mathrm{Gal}(A^{(p)}/\\hat K) \\ . ", "1ad710cb18694afa8cc43db1f37b8d0a": "\n=\n\\begin{pmatrix}\na&-b&-c&-d\\\\\nb&\\;\\,\\, a&-d&\\;\\,\\, c\\\\\nc&\\;\\,\\, d&\\;\\,\\, a&-b\\\\\nd&-c&\\;\\,\\, b&\\;\\,\\, a\n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\np&-q&-r&-s\\\\\nq&\\;\\,\\, p&\\;\\,\\, s&-r\\\\\nr&-s&\\;\\,\\, p&\\;\\,\\, q\\\\\ns&\\;\\,\\, r&-q&\\;\\,\\, p\n\\end{pmatrix}\n.\n", "1ad71f862b1d76bd199fecbee7a7b198": "T_{t}=", "1ad77b38db007391397f32ad71290205": "\\Delta t' = 0 \\ .", "1ad7b392bd95dfab8e9468840279d17e": "\\overline{B_{\\delta}} := \\left\\{ x = (x_{1}, \\dots, x_{n}) \\in \\mathbb{R}^{n} \\left| | x | := \\sqrt{x_{1}^{2} + \\dots + x_{n}^{2}} \\leq \\delta \\right. \\right\\}", "1ad7d8b9f2f69a879280e302f824d8da": "\\left(\\frac{\\partial T}{\\partial P}\\right)_S = \\left(\\frac{\\partial V}{\\partial S}\\right)_P", "1ad7f005c0cd028353fa6dd2e8c6bf4e": "\\int (Tf)g", "1ad80504847b080a710a00021959ebf4": "\\left[\\Phi(x),\\Phi(y) \\right]:= \\Phi(x) \\Phi(y) - \\Phi(y) \\Phi(x)", "1ad810342a232a36a57e5314053472dc": "n \\ge 1", "1ad827e8c9d18d30dd832cb313109dc9": "m\\dot{r}^2 = 2E - \\frac{L^2}{mr^2} + \\frac{2GmM}{r} = 2E - \\frac{1}{r^2}\\left(\\frac{L^2}{m} - 2GmMr\\right),", "1ad83aa505c260716d973a5d3e5d476e": "a_{4}+b_{4}+c_{4}=2b_{1}", "1ad87ce2f948498f698911a05b157327": "J_2^2|j_2m_2\\rangle=j_2(j_2+1)\\hbar^2|j_2m_2\\rangle", "1ad8fe7cf27c143c862ecc737d61a192": "\\Phi_{\\pm}", "1ad943023f9e4231a37fead0969a5ac3": "(\\mathbb{N}, \\le)", "1ad9459b74efd19572aad78729e56b16": " \\phi, \\ \\phi \\rightarrow \\chi \\vdash \\chi ", "1ad9467ebbcf00fe7fb6fec2167b2ebe": "O(d_{in} \\log p \\log^{1+\\epsilon} q)", "1ad96c376da3011a6f66f75ae0bcb5ab": "P(x) \\in U", "1ad9b76c75f2cb9cfa49956459d83077": "(P\\cap B)+x\\subseteq K", "1ada27fb55c3ab32cb1d63b3ad9b3399": "\\Box \\lnot K", "1ada7aab630bcf6f89e480b4baf646ac": "b(\\lambda)=f_\\lambda(i)=\\int {\\sin \\lambda t \\over \\lambda \\sinh t} \\, dt= {\\pi\\over \\lambda} \\tanh{\\pi\\lambda\\over 2},", "1ada834c5b64222b916a7060cdf8f9e6": " \\delta W = (\\sum_{i=1}^n \\mathbf{F}_i)\\cdot\\dot{\\mathbf{d}}\\delta t + (\\sum_{i=1}^n (\\mathbf{X}_i-\\mathbf{d})\\times\\mathbf{F}_i) \\cdot \\vec{\\omega}\\delta t = (\\mathbf{F}\\cdot\\dot{\\mathbf{d}}+ \\mathbf{T}\\cdot \\vec{\\omega})\\delta t, ", "1ada9851b7773942f2ad049381808b54": "\\pi (\\mu, \\nu) \\le 1", "1adaff66a44a3afe0e927847259e591d": "\\quad \\eta = \\rho \\sin \\psi, \\; ", "1adb033695d2e51e0690730fa5e82a63": "\\delta_{ext}:Q \\times X \\rightarrow S ", "1adb1080e3b2e54604085aa615621ca9": " MC_L = 30 ", "1adb3925456c348576c82a866708f398": "x^4 - y^4 = z^2", "1adb47037809792950943fb40f3a1517": "G(\\chi) := G(\\chi, \\psi)= \\sum \\chi(r)\\cdot \\psi(r)", "1adb4bbcfdce4fa99e3411506ef587df": "\\sum_{J_{z'}}\\langle J,J_z|\\vec \\mu_J|J,J_{z'}\\rangle\\cdot\\langle J,J_{z'}|\\vec J|J,J_z\\rangle = \\sum_{J_{z'}}g_J\\mu_B\\langle J,J_z|\\vec J|J,J_{z'}\\rangle \\cdot\\langle J,J_{z'}|\\vec J|J,J_z\\rangle", "1adb53068c520e0b7b65e6a924b5607e": "IMPL(\\alpha,\\alpha')=s^TL(u\\otimes v)=1-\\alpha(1-\\alpha')", "1adb5d76e6c14cef19c702b1cbadb50a": " \\lambda_D = \\sqrt{\\frac{\\epsilon_0 k T_e}{n_e q_e^2}}", "1adb6a5cd3e5ae167cec17f89a817c77": "dz'=\\frac{dz}{(cz+d)^2}", "1adb8d805ba5e60698e25a35a37691ac": " s_k = \\frac{\\pi k}{2q+1}, k=-q,\\dots,-1,0,1,\\dots,q. ", "1adc163714b0581dce311b5686767089": "\\displaystyle{f=v +\\partial_n u.}", "1adc4d19bb30cabcc86226a8691870be": " e(P,Q) = e(Q,P) ", "1adc5344f42e3e5f0782da68fa3ef848": "= \\frac{-y(x)\\left[p(x)y'(x)\\right]|_a^b + \\int_a^b{y'(x)\\left[p(x)y'(x)\\right]} \\, dx + \\int_a^b{q(x)y(x)^2} \\, dx}{\\int_a^b{w(x)y(x)^2} \\, dx}", "1adc6e3ec105d58d84dd8a7e414600bd": "\n\\begin{align}\ng(Z,w) & = 91125Z^6 \\\\\n& {} \\quad {} + (-133650w^2 + 61560w - 193536)Z^5 \\\\\n& {} \\quad {} + (-66825w^4 + 142560w^3 + 133056w^2 - 61140w + 102400)Z^4 \\\\\n& {} \\quad {} + (5940w^6 + 4752w^5 + 63360w^4 - 140800w^3)Z^3 \\\\\n& {} \\quad {} + (-1485w^8 + 3168w^7 - 10560w^6)Z^2 \\\\\n& {} \\quad {} + (-66w^{10} + 440w^9)Z \\\\\n& {} \\quad {} + w^{12} \\\\[8pt]\nh(Z,w) = & (1215w - 648)Z^4 \\\\\n& {} \\quad {} + (-540w^3 - 216w^2 - 1152w + 640)Z^3 \\\\\n& {} \\quad {} + (378w^5 - 504w^4 + 960w^3)Z^2 \\\\\n& {} \\quad {} + (36w^7 - 168w^6)Z \\\\\n& {} \\quad {} + w^9\n\\end{align}\n", "1adcf2582883c2cadbdca811a6c5391d": "\n u_i^\\mathrm{T} u_j =\n \\begin{cases}\n 1, & \\mbox{if }i = j, \\\\\n 0, & \\mbox{if }ij \\notin E.\n \\end{cases}\n", "1adcf68d8fc50ae80421aeb541eb6970": "\\lambda(T,W) = \\frac{\\int_{-W}^{W} {\\|U(f)\\|}^2 \\,df} {\\int_{-1/2}^{1/2} {\\|U(f)\\|}^2 \\,df}.", "1adcfd50c9e00fbf417ee939945504a1": "|f|", "1add0562d4853ae3e7d487717e311840": "16K^2 = 16(S-p)(S-q)(S-r)(S-s). \\,", "1add12e7dd6cb71bf2db56c642b5537d": "\\operatorname{Pr}(X=x_k) = p_k \\quad\\mbox{ for } k=1,2,\\ldots", "1add3dba0a8521842d4ff1e36249442f": " \\int_{\\gamma} g(\\zeta) d \\zeta = 0,", "1add97377ec9dafbb474c7f8508992fd": " r \\equiv \\mathrm{RAD}~\\bold{cm}", "1add9df4162f037631f347b31eb1383e": " x \\geqq y ", "1add9f2286646ffc9e9923902b69e28b": "d\\boldsymbol{\\varepsilon}_e = 0", "1ade06deda570e2c07559914edbcf850": "K_{s}", "1ade0f8a3afed6749e6ee9c2385386a4": "\\sigma = T ", "1ade18fc80eda153811ffe166cf883c0": "\\vec{e}_1 = \\left( 1 + m/r \\right) \\, \\partial_r ", "1ade2fe2b9ab15923e8f6392e07b09bb": "{M}", "1ade52b43a6888e9f458dc70296bcdc2": "n_1=0,\\,\\,n_2=\\pm\\frac{1}{\\sqrt 2},\\,\\,n_3=\\pm\\frac{1}{\\sqrt 2},\\,\\,\\tau_\\mathrm{n}=\\pm\\frac{\\sigma_2-\\sigma_3}{2}\\,\\!", "1ade53d02d5d0a17297a27f4aecd3282": "B_4 \\bar S", "1ade87eaaa384e7322455afd323580e0": "\\frac{F_A}{F_B} = \\frac{a}{b}", "1ade98012fd87daa808ea7ba82716ee2": "1-e^{-5.0/4.5}", "1adea2b8be39c8108bee671b4cc1a745": "V_D\\gg nV_ T", "1aded4342fe9d330c576dc01a4b4bab4": "\\definecolor{gray}{RGB}{249,249,249}\\pagecolor{gray} g \\mapsto g\\circ h", "1adee0c3551c6f3ec04634a43c34b5a7": " P^2 = P_\\mu\\,P^\\mu ", "1ae04f9254bd1f61562ead77db876ef7": " \\cdot\\left(\\text{largest monomial of }s_n\\right)^{i_1}", "1ae06e1aa73f04423aa9975348ef050c": "f'(x) = (\\ln a)a^x = \\lambda f(x)", "1ae09669bbba392bff06eb2207e2bebc": "\\sum_{n=1}^\\infty \\frac{n+1}{n} \\left(\\frac{1}{2}\\right)^n = \\frac{2}{1} \\cdot \\frac{1}{2} + \\frac{3}{2} \\cdot \\frac{1}{4} + \\frac{4}{3} \\cdot \\frac{1}{8} + \\cdots", "1ae0b142f4e01f6d1f5f86d7c9245cf9": "h_x\\leftarrow (A^TD_v^{-2}A)^{-1}c", "1ae0b23a2363a831199866191d4433ab": "\\,{}^{(-1)}a = 0", "1ae0cde92eed5b79c8556648ec6b680b": "\\mathbb{E}\\left(\\left(\\int_0^t H\\,dM\\right)^2\\right) = \\mathbb{E}\\left(\\int_0^tH^2\\,d[M]\\right).", "1ae1189ceb808bad843e4f5d93100fcc": "\\mathrm{not}~r \\equiv \\mathrm{true}", "1ae13ce616cb08a5c83105ddf5a44817": "\n \\Omega_{d} = \\begin{cases}\n \\frac{1}{ \\left(\\frac{d}{2} - 1 \\right)!} 2\\pi^\\frac{d}{2} & d\\text{ even} \\\\\n \\frac{\\left(\\frac{1}{2}\\left(d - 1\\right)\\right)!}{(d - 1)!} 2^d \\pi^{\\frac{1}{2}(d - 1)} & d\\text{ odd}\n \\end{cases}\n", "1ae1a4d556f4fb861e4235bcdbede541": "S = \\langle U, Q, V, f \\rangle", "1ae261aef27d11634ebcab81d3f8d4e3": "n^2*s^2", "1ae277cfc394d164e4e632b087787cb8": " f(x_1) ", "1ae2cb1e93a77ab84ee40435e84c4313": "v_O", "1ae2d257a35264eeb51c7f6c196fd023": "I[f] = \\mathbb{E}[V(f(x), y)] = \\int V(f(x), y)\\,dp(x, y) \\ .", "1ae2ddebac92c61b4f392152c972c9fe": "A_{1}F_{1 \\rightarrow 2} = A_{2}F_{2 \\rightarrow 1} ", "1ae2f80bd43c7cba007bc6f7781de58a": "(x^2+y^2)^2+18a^2(x^2+y^2)-27a^4 = 8a(x^3-3xy^2)\\,", "1ae334a3278b1eae94b2bf01cff69fdb": "3 \\left\\lceil\\frac\\Delta2\\right\\rceil", "1ae34f534195b92331cccf570d1f286e": "J_2^2", "1ae351b7d3143355015b4c5d0c938716": "{R^2}_2 = (C)-(E)", "1ae38954f6cba2eafda4e9c34df8d944": "C_j", "1ae3bacaa3c96d25914aa2aabd6b6400": "L_x", "1ae3c0d1129eb02888859330b05558bf": " \\sigma_u^2 = \\max(\\sigma_-^2, \\sigma_+^2) , ", "1ae435df7b794701ad49b6c179408458": "\\scriptstyle T_{i+1}", "1ae437dda2d55cf38f3167cf58fe257c": " R^3_3(\\rho) = \\rho^3 \\,", "1ae43bd67873b70aa82b58769422583b": "M=\\mathbb N", "1ae47e75a6b333ab3fa002e906cd54dc": "D' = 2( B'_xC'_y - B'_yC'_x ). \\,", "1ae4abe506d112a3cd2d8bc32e33ca75": "\\mathbf{C} = \\left( \\mathbf{J}^T \\mathbf{J} \\right)^{-1} \\mathbf{J}^T", "1ae5443ca64730ce84e57c692982120f": " 2k_1^2 > 1 - \\rho ", "1ae5926370b8019094da892cbbd7f01f": "T \\sin \\theta = \\frac {mv^2}{r} \\,", "1ae5ef11d69f34d471ab6f9e8bd13751": "\\Sigma^{EXP}_k=\\mathrm{NEXP}^{\\Sigma^P_{k-1}}", "1ae5f88f935f3f6a3260fc08318ea62a": " \\begin{align}\n\\operatorname{E}(X) &= \\mu \\quad \\quad \\quad \\text{for }\\,\\nu > 1 ,\\\\\n\\text{var}(X) &= \\frac{1}{\\lambda}\\frac{\\nu}{\\nu-2}\\, \\quad \\text{for }\\,\\nu > 2 ,\\\\\n\\text{mode}(X) &= \\mu.\n\\end{align} ", "1ae615f85c2c8a72da6c11e3faa2042d": "\\scriptstyle{a=6.1121\\ \\mathrm{millibar};\\quad\\;b= 18.678;\\quad\\;c= 257.14^\\circ \\mathrm{C};\\quad\\;d=234.5^\\circ \\mathrm{C}.}", "1ae6253da2e5447636536685459f4b67": "r_1=\\sqrt{R}", "1ae6c2974417879c11bc464c52e34304": "S\\subseteq Y", "1ae6c67f72f01dbe1006f899e10e09da": "X_{n+1} = \\sum_{j=1}^{X_n} \\xi_j^{(n)}", "1ae6d477182e26a40c2fb736336b7f0d": "\\delta_0=\\left\\{\\begin{matrix}1 & \\text{if}\\ X_1=0, \\\\\n0 & \\text{otherwise,}\\end{matrix}\\right.", "1ae6f2bb35b18be6fc7fcd08b1574326": " \\log_e(Y) = a_0 + \\sum_i{a_i \\log_e(I_i)} ", "1ae7082c8d30dcb49a59f41439bfc911": "\\tilde{H}_d", "1ae7346d959265fa4ed8ec8b5a856305": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ \\frac{\\sqrt{8}}{3} \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ -\\frac{\\sqrt{2}}{3} \\\\ \\sqrt{\\frac{2}{3}} \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ -\\frac{\\sqrt{2}}{3} \\\\ -\\sqrt{\\frac{2}{3}} \\end{pmatrix}", "1ae745ad48763d30ac1f85fada71e162": "a \\cos(x) + b \\sin(x) = \\sqrt{ a^2 + b^2 } \\cos(x - \\operatorname{atan2}\\,(b,a)) \\;", "1ae78e54b1449ba31b989d59863351fd": "A_m, A_e", "1ae7a7f0027ec06f1da999992b2403b8": "A = A_0 \\exp(-z/2H) \\, ", "1ae7ec45b7aec5668bc80a216f29c88d": "\\int_0^1 \\phi(p)dp = 1", "1ae806034c2ba77b4044c196756871ad": "[n:=n+1][n:=n+1]\\ldots \\Phi(n)\\,\\!", "1ae824002506a3f5ce2ca38753bf3fca": "M,a\\models\\lnot\\phi\\iff M,a^*\\not\\models\\phi", "1ae883436e2a70938611aa5d1af0ae9b": "\\textstyle \\mathbf{c}", "1ae8c0f9b51291d9f997641c6d948c66": " F_{i i} = 1 - \\sum_j C_{j i} ", "1ae8c13f1d0b25fce1cbff89f45f9769": "[\\ddot U]", "1ae8cb5ef015bcabe733ef4c8fc38de8": "(2^5/9!!) \\pi^4 = (32/945)\\pi^4 ", "1ae8efcb74e5b61c4d2e52ea85f292df": "\\bar{X}_m", "1ae8ff0eafb50e0559d4a0ef51f7c463": "\\frac{df(v)}{dv} = 0", "1ae919ef6ec1d1f5f9b2a5a219aa1601": "V(x,t)", "1ae98698b8f9f9ff679d95a8c43ac71a": "\\tau: [0,1]\\to D", "1aea2dc12c695e4d9b36b9b5a1d0133f": " \\text{Standard error} = \\sqrt{\\frac{p(1-p)}{n}}=\\sqrt{\\frac{p-p^2}{n}}.", "1aea572dc87627e1d16d240540acd41c": "1 \\le \\mathrm{Ra}_D \\le 10^5", "1aeae92df63ec0a79495589220062dad": "\\mathcal{A}^a_\\mu", "1aeaf3c261f9acaa8ac8e99bf011ea10": "\ng_{ij} (z)\n:=\n\\frac{\\partial^2}{\\partial z_i\\, \\partial \\bar{z}_j}\n\\log K(z,z) ,\n", "1aeb00464964c160b0ea3840de0ccd00": "\\,d_x = l_x - l_{x+1}", "1aeb120f6ea3f5113553d4337931b1ec": "V_{base} = 1 pu", "1aeb1ed45d9f55b1428dad6ca5ed5b44": "P_{D,max}", "1aec2483b0477f359ee72eb858d11ea9": "\\kappa \\ge cf(\\mu)", "1aec5e46df30faddf18262d8160c6572": "BSC = BAC + \\pi/3", "1aec6c8c4e8c7782fa4aae85b0de36e3": "A=(E^{2}-\\omega^{2}L^{2})^{1/2}/\\omega", "1aec8bd65246b75bc289689c744323aa": "\\,x=x_1 x_2 \\in \\Sigma^*", "1aecb5334e90200e1a0a8411633463c2": "2s2p", "1aecf962e2e222b81a3f7cb8716ff5a2": "\\mathbf{y}^{\\prime\\prime} = (y_1^{\\prime\\prime},\\ldots, y_N^{\\prime\\prime})", "1aed01a08b6e79d1dc616180c500ce40": " -\\frac{174611}{330} ", "1aed088131190cd9f70e514814c97e6b": "\\delta(\\mathbf{x}-\\mathbf{x'})", "1aed1b41855e3eee16204d835251eb08": "ds = (1/T)\\,du + (-\\mu/T)\\,d\\rho", "1aed2ee78c0a5c1f09824c55f049b212": "FAP", "1aed45a53e3cd44a6895a5f4b2da1d4b": "{h S}={\\frac{(\\rho_s-\\rho)}{\\rho}(D)}\\left(f \\left(\\mathrm{Re}_p* \\right) \\right)=R D \\left(f \\left(\\mathrm{Re}_p* \\right) \\right)", "1aed7c905c058af93801adf85f46d9f7": "=\\displaystyle{{G_a\\lambda^2\\over 4\\pi Z_\\circ}E_b^2} \\,", "1aed7e6dcb00e784d459219bd9149e76": "F^* \\overset{x^n}{\\to} F^*;", "1aedc54f1f41e3c44632d5c0bdca0542": "T_E = {\\epsilon\\over k} = {h\\nu\\over k} = {h c_s\\over 2k}\\sqrt[3]{N\\over V}\\,,", "1aee05778344022ab93538f353a5d7f7": "S = (X, \\delta)", "1aee131c9e35a67ca3309e83f8ff2c27": "\\sigma_{\\log K}=\\frac{\\sigma_K}{K}", "1aee47dc2cebad185137293a664a4605": " E[\\varepsilon^2_t]=\\sigma^2 \\, ,", "1aee9b8c8a0bf160241e3d3fa152abd4": "\\vec{f}_2 = \\frac{1}{r} \\, \\partial_\\theta ", "1aeebc7bcf55efa7233f61531760078b": " \\mathrm{n_0}\\,=\\, e_{rms}\\frac{ \\pi}{\\sqrt{3}}\\, (2f_0\\tau)^{\\frac{3}{2}} ", "1aeecb0f67090c42293a6fd8f982ce5a": "\n\\langle z \\rangle=e^{i\\mu-\\sigma^2/2}\n", "1aeef03af57d0a3b26888e29399960f7": "A1^{+}", "1aef646d38450e1146d8a855a3311ccf": "\\begin{align}\n &(x_1, x_2, \\dotsc, x_n, x_{n+1},\\dotsc)+(y_1, y_2, \\dotsc, y_n, y_{n+1},\\dotsc) =\\\\\n &(x_1+y_1, x_2+y_2, \\dotsc, x_n+y_n, x_{n+1}+y_{n+1},\\dotsc)\n\\end{align}", "1aef938f991402967c780af0792d9b92": "\\sigma^\\mu \\partial_\\mu \\equiv \\sigma^0 \\partial_0 + \\sigma^1 \\partial_1 + \\sigma^2 \\partial_2 + \\sigma^3 \\partial_3 ", "1aefa448f9b0ed9db73e879cbb71d5d0": "\\mathbf V\\,\\!", "1af0871d9e866330e9ea59cae72d88d2": " v = \\sqrt{ G(M\\!+\\!m) \\over{r}} = \\sqrt{\\mu\\over{r}} ", "1af08bc2d0cd512c8939c6d0570db4f9": "\\Gamma_0 [0] = 0 \\,", "1af08db3081b7836689bf2297ceb6819": " \\nabla \\left( u \\cdot \\nabla f(x)\\right) = u^T H(x) = H(x)u", "1af15663e8ebe3a5d5d01b8a078db690": "\\dfrac{ \\partial u }{ \\partial x } = \\dfrac{ \\partial v }{ \\partial y }", "1af173878deaef50f556e964bb4e2771": "\n \\kappa_2=2Dt,\n", "1af199c4f7213582e2812f300d60ae6b": "\n\\frac{1}{c_{1}} V \\left( \\frac{1}{\\lambda_1} J \\right)^{k} \n\\left( c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right)\n\\rightarrow 0\n", "1af20ac9e418aefa8e36fd283210e1f1": "| \\nu_j \\rang", "1af22538e8a7973f9748d6c5bbca0ebd": "cP = 0\\,", "1af2564dc766de96dd5e6677bd695b65": "k_n |\\mathcal{Z}|^{2K_n}=o\\left( \\frac{n}{\\log n} \\right)", "1af25d11f87465ffdaf6a78f602aed98": "\nx = R_{pullup} = \\sqrt{R_{sensmax}R_{sensmin}}\n", "1af2a5e7f1328ae428d623a8529f0f3c": "\\langle A \\rangle_\\sigma = \\int a \\; \\mathrm{d} \\langle \\psi | P(a) \\psi\\rangle", "1af2bbfa7890a063130883af933d16a6": "\nT^4, SU(2l+1), T^1 \\times SU(2l), T^l \\times SO(2l+1),\n", "1af2d1bcd12380ef8ef0f12b7c837d88": "H(k-1) = \\begin{bmatrix}Y(k) & Y(k+1) & \\cdots & Y(k+p) \\\\ Y(k+1) & \\ddots & & \\vdots \\\\ \\vdots & & & \\\\ Y(k+r) & \\cdots & & Y(k+p+r) \\end{bmatrix}", "1af2de8141afd27e1667da9adeee79b2": "\\forall x ( \\phi \\lor \\psi)", "1af32d4335c674f8b2b658bffc259ff1": "x \\not\\in D", "1af32dad0ac1d436a4489c7fe92e05e0": "< 0.0062", "1af349c66ef07dec75bc7de287758b66": "P_n(x)\\,", "1af3682c0da2ff90a6494748af21f0d2": "\\int_a^b f(x)\\,dx.", "1af386504f5e4171edd57c6ae4bd4f44": "{dn \\over dt} = an^2 - bn,", "1af38af214bc4019e1fae1aaad4140c5": "L/k", "1af3a816afaa0d99b385293f1fcc8273": "P = \n\\begin{pmatrix}\n\\frac{1}{2} & \\frac{1}{2} & 0 & 0 \\\\\n\\frac{1}{2} & \\frac{1}{2} & 0 & 0 \\\\\n0 & 0 & \\frac{1}{2} & \\frac{1}{2} \\\\\n0 & 0 & \\frac{1}{2} & \\frac{1}{2} \\\\\n\\end{pmatrix} + \\epsilon \\begin{pmatrix}\n-\\frac{1}{2} & 0 & \\frac{1}{2} & 0 \\\\\n0 & -\\frac{1}{2} & 0 & \\frac{1}{2} \\\\\n\\frac{1}{2} & 0 & -\\frac{1}{2} & 0 \\\\\n0 & \\frac{1}{2} & 0 & -\\frac{1}{2} \\\\\n\\end{pmatrix}", "1af3cecd0523deda46f04a2404178925": "f_l(X)", "1af40b4606c6faa8813987603da8d6b6": "b' \\in I^j", "1af41556c2fd55bd3afea0318302ae33": "x = -\\frac{b}{2a}", "1af432fe417df6053045ad5ab173a4ab": "{\\alpha} = \\frac{{d\\omega}}{dt} = \\frac{d^2{\\theta}}{dt^2}", "1af455baf8cf53a8d9f71472178234c4": "\\tfrac13 \\left( \\eta' \\right)^2 = f(\\eta)\\,", "1af471feba26b381c155ad93b3614986": "n^2\\,", "1af4ae0db5e582d6f295d4c53389d8ad": "\\displaystyle{[JA,JB] = [A,B].}", "1af4e40ddcbad4cca752eedef2af0fd3": "\\int d^Dx \\left[(A_\\mu^1)^2+(A_\\mu^2)^2\\right]\\,,", "1af4f7d743268fd743f3a5bc4c11da86": "\\mathbf{F}_{p^2}", "1af58d000230b6ec31191d9d26f529ec": "[\\alpha]_D^{20} = +6.2", "1af58fe9c80b6605f2b3aed805bbc079": "g_2(x)=x^e-C_2 \\in \\mathbb{Z}_N[x]", "1af59be9b51bd923c5ada20b43588dce": "\\operatorname{E}(\\theta)", "1af59d01d5e2f7161edbe57591415dd4": "(e_i)", "1af5ea123a721e525ebb311c9ebf901d": "f_i\\in\\mathfrak{g}_{-\\alpha_i}", "1af609ca52ca77c7bbddcef1c9ba62bb": "\\operatorname{length}(E)", "1af651daccfc49790236c9f39855cffa": "\n\\begin{align}\nE^{(1)}_\\mathrm{electrostatic} = & \\sum_{\\ell_A=0}^\\infty \\sum_{\\ell_B=0}^\\infty (-1)^{\\ell_B} \\binom{2\\ell_A+2\\ell_B}{2\\ell_A}^{1/2} \\\\\n&\\sum_{M=-\\ell_A-\\ell_B}^{\\ell_A+\\ell_B} (-1)^{M} I_{\\ell_A+\\ell_B,-M}(\\mathbf{R}_{AB})\\; \\left[\\mathbf{M}^{\\ell_A} \\otimes \\mathbf{M}^{\\ell_B} \\right]^{\\ell_A+\\ell_B}_M,\n\\end{align}\n", "1af6bd199ce2aac9a45c123f97659a0e": "P \\propto \\frac{1}{V}", "1af6c7540c44ac8e8c44a5e3297c6136": "1-\\left(1-\\frac{1}{m}\\right)^{kn}.", "1af7139e781c6797dec3e18ae85b5c7b": "\\scriptstyle n = 1, 2, 3, 4 \\ldots", "1af7393e428e1b9ae5792044d259ecb8": "V_ T", "1af766c105f030c46e281daddcf56e41": "L_\\rho(\\gamma)=\\int_\\gamma \\rho^*\\bigl(f(z)\\bigr)\\,|f\\,'(z)|\\,|dz| = \\int_{\\gamma^*} \\rho(w)\\,|dw|=L_{\\rho^*}(\\gamma^*).", "1af768e593ae489c4917bfe516aa5c78": "2i +2", "1af7884d818f0bad92cf66873f0b12a2": " \\tilde{f}_1(\\lambda)=\\tilde{f}(\\lambda)\\cdot \\varphi_\\lambda(w),", "1af7acc84b33c00ecff4867eb8f32cd2": " p = m \\dot x. \\,", "1af85e561d314cf46f473d4ef8346f53": "x_{(n+1)/2}", "1af8b10681588f6c5cf157b7f6944d73": "L_f + L_c=12 \\,", "1af95adc078fdfccde1eb4b99c6207dc": "\\langle\\mathcal P(F),\\cap,\\cup,-\\rangle", "1af98d15d06890f93e936957e66704d1": " \\Sigma(W,S) = (X \\setminus \\{ 0 \\}) / \\mathbb{R}_{>0}^\\times", "1af992a6aba6cfd7cd4af977c36b829f": "r(t) = g(e(t), c(t)) \\, ", "1af99f41ed5cdd516b3360121fd136eb": "d\\mathcal{O}_s(x_0) = \\mathrm{span}(dh_1(x_0), \\ldots , dh_p(x_0), dL_{v_i}L_{v_{i-1}}, \\ldots , L_{v_1}h_j(x_0)),\\ j\\in p, k=1,2,\\ldots.", "1af9a3f284badda70538449cb633af2a": " c- ab -a_x \\neq 0 \\quad \\text{and/or} \\quad\nc- ab -b_y \\neq 0,", "1af9d4cc61f114b34680210b593cffe2": "\n z = n \\frac{\\left(1+\\frac{1}{\\zeta}\\right)^n+\\left(1-\\frac{1}{\\zeta}\\right)^n}\n {\\left(1+\\frac{1}{\\zeta}\\right)^n-\\left(1-\\frac{1}{\\zeta}\\right)^n},\n", "1af9dcecc465950e25f7153943970180": " r ", "1afa62b4df51ba8156571ba39b48b019": "b=2(1-\\nu)=a+1\\,\\!", "1afa8f5325d8a9235e9a108944ba038b": "1.9761", "1afae345de48dcb83decaa2fb09e98eb": "\\displaystyle \\mathcal Z(z) = \\sum_n z^n g(n,N) = 1+Nz+ \\tfrac{1}{2}N(N-7)z^2+\\cdots", "1afaff502a27cd99fbb7e1dfe2e464de": "\\bar{\\delta}\\delta-\\delta\\bar{\\delta}=(\\bar{\\mu}-\\mu)D+(\\bar{\\rho}-\\rho)\\Delta+(\\alpha-\\bar{\\beta})\\delta-(\\bar{\\alpha}-\\beta)\\bar{\\delta}\\,,", "1afb2ce5e9b16a2d326c9da195e92dc2": " \\begin{bmatrix}\n \\mathbf{N} \\\\\n \\mathbf{M} \\end{bmatrix} = \n\\begin{bmatrix}\n\\mathbf{A} & \\mathbf{B} \\\\\n\\mathbf{B} & \\mathbf{D}\n\\end{bmatrix} \\begin{bmatrix}\n \\varepsilon^0 \\\\\n \\kappa \\end{bmatrix} ", "1afb4b51684ca3abc8d0f4d361258c17": " q <_\\mathcal{O} p ", "1afb6aa26b49a381393f524b4882ce27": "{E}_{3-8}", "1afb87747818374ee07fe60f295dbc7a": "\n\\begin{align}\n\\frac{d m_{12}}{d s_2} &= M_{21},\\\\\n\\frac{d M_{12}}{d s_2} &= -\\frac{1 - M_{12}M_{21}}{m_{12}}.\n\\end{align}\n", "1afbbb386ab798c6b552991f69e18130": "f(\\Theta)", "1afc00238bee17ca5a88497e5c931c63": "\\R^d", "1afc21e08db9a8abb8c34d682609ce27": "\\mathbf{c}\\cdot\\mathbf{\\Delta k}=2\\pi l", "1afc24b08861fbf702476f54795b2a63": "M\\subset E^\\downarrow.", "1afc32f9ecdea6a79b7c763e1065835f": "\\displaystyle{D(x+h) \\le |x+h-z|^2 \\le |z-x|^2 +2h\\cdot (x-z) + |h|^2 =D(x) +f_0(h) +|h|^2}", "1afc42f49b669ad0d6ef5477115a490e": "c\\!+\\!i\\infty", "1afc5128582f9baefeebfde2d9aefb45": " k_x\\in{\\mathbf k}_x, \\ k_y\\in{\\mathbf k}_y ", "1afcfaa63a5afea49f514264538b3a1a": "\\mu^'_2=k+\\lambda^2", "1afcfda102bde92d02d058d3ae82d1dc": "\\Lambda^{n}", "1afd7b9bc7095e2960ce839642923fa0": "\\phi(x) = x^*(x)", "1afd9c8547614dce607f5dba2c1fd0cf": "G = \\frac {\\cos \\theta} {\\sin \\theta + \\cos \\theta - 1}", "1afe0d11723f75be1089d388b38f68b2": "\\qquad i (\\lambda v) = (i \\lambda) v = (\\lambda i) v = \\lambda (i v)\\qquad ", "1afe2cad8fc2cedde53f95b7c1eb90bb": "\\{\\Phi_{00}", "1afe31c4d8ceddf84df3464c592cfd50": "\n\\beta_i = \\frac{\\sin (\\nu \\phi_i)}{\\cosh(\\nu \\eta) - \\cos(\\nu \\phi_i)}\n", "1afe49a9fa63c16926a3feb7e0640f32": "F(t) = \\int_{0}^t E(t)\\, dt \\qquad E(t) = \\frac {dF(t)}{dt}", "1afe83854fb470a16e06fc2c306e175e": "dS/S = \\rho/\\sigma \\cdot ( G \\cdot M/r^2 - \\omega^2 \\cdot r ) \\cdot dr", "1afe83a709a1a412122aade17092fe25": "f^{-1}(n)=m", "1afeeaca605ea59476fe74c104c489d1": "f_i = \\pi_i \\circ f", "1afef125f6afcab921f2dc3748a15ffb": "\\int_0^\\infty e^{-ax^b} dx = \\frac{1}{b}\\ a^{-\\frac{1}{b}} \\, \\Gamma\\left(\\frac{1}{b}\\right).", "1aff20ed758ea7a989f9f154a4f24541": "\nh_t(x,y) = \\sum_{i=0}^\\infty \\exp(-\\lambda_i t) \\phi_i(x) \\phi_i(y).\n", "1aff2c5530b9ce3b9ccc580904206b22": "IV_1", "1aff676d2e4456ca58de4a9549636141": "K(\\sigma) = 1 + 3\\langle JX,Y \\rangle^2", "1aff74ad01db1471c065dfd06c510874": "f(x)=\\delta(x-x_0),", "1affff94ac44e6254d43802aade7dd7a": " h_{jk} \\in C^\\infty(U,\\mathbb C) ", "1b0038ae6e0a54ae4f634238411efc32": "\\frac{1}{n}x + \\frac{1}{n}l", "1b00858e0e33eefa62983374a555f13f": "-a_4=5\\sum_n \\frac{n^3q^n}{1-q^n} = 5q+45q^2+140q^3+\\cdots", "1b00aa69ce82836826f738b97a135e4b": "\\lfloor n/r\\rfloor", "1b00ae279f7b96a86886ca6f15d165b1": "\\frac{B_{in}^2}{2\\mu_0} \\sim \\frac{\\rho V_{out}^2}{2} ", "1b00bc8e537c9b2193aba3c95e56c935": "\\frac{137,200\\ \\mathrm{N}}{(285\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=49.1", "1b00d919da1dc670605fb472e0c6b429": "n_{-}\\,", "1b0116561d756b52af5539658a6a42fc": "\\operatorname{Var}[S-T]>0", "1b018e24dc0b76b2f2b2d1ea33c857bc": "\\mathbf{O_j} = \\mathrm{diag}(b_{*,o_j})", "1b019cb970b7e8eb584a06d91f73c401": "\\mathfrak{g}\\to G", "1b01a8552a79f513b0119c0dd04813ae": "\\boldsymbol{\\varepsilon\\digamma\\varkappa\\varpi} \\!", "1b0216b9681891f44a24a44e28f3d477": "\\|a \\mathbf{v}\\| = |a| \\|\\mathbf{v}\\| ", "1b023c8a1c0f4e6bb39d9c33b59a6d6b": " e^{-S_F} X ", "1b024fce65a4a1b71a19e3f722ce6dbc": "\\ r = \\mathcal{F}^{-1}\\{R\\}", "1b025d92fd92b3e06c33a1875cdc9a70": " {G^{\\prime\\prime}(\\omega) =\\ } [G^{\\prime}(\\omega)G_0 - {G^{\\prime}(\\omega)}^2]^{-{1\\over2}}", "1b0268237ddc2e8d876edc88cd3f8d09": "\\chi ^2 _{0.99}", "1b030a0c8c2c9ce33437b7e8ec3d6c34": " \\cot \\beta_{\\rm 2} ", "1b0333e98862ae2000bb04c4cfcbac58": "T[i]=-1", "1b0430bedb8015c29413b5091ca48eb9": "D_{n,0} = \\left[{n! \\over e}\\right]", "1b04546c2176a0f4bbd2ea67c4b705ea": "1/4-\\varepsilon", "1b045fe6d5677587f838df87afd7b302": "f(\\theta)=f(0)", "1b04817e489e244106a6be4aaacbed01": "\\frac{1-6p(1-p)}{np(1-p)}", "1b04bb9d91cbcfd506c84bcbf3e6cbba": "\\scriptstyle-0.9(2.0)\\times10^{-10}", "1b04cd43f4af6a7904f2a9e2add46a71": "(3/4)^N", "1b04d4b8d95be6e4d0231344c1b80613": "\\left( A^{-1} \\right)^T = \\left( A^{T} \\right)^{-1},", "1b050f0576b87200d049e7e65e341bdd": " u = 0 ", "1b05190451f5afe29e2a16ff0a55eb2d": "\\ C_2^3 (3)=\\frac{16}{49}", "1b052cd3b241da58d06cf7ed003e803e": "\nI_{lm} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int d\\mathbf{r}^{\\prime} \n\\frac{\\rho(\\mathbf{r}^{\\prime})}{\\left( r^{\\prime} \\right)^{l+1}}\n\\sqrt{\\frac{4\\pi}{2l+1}} \nY_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "1b05392e4237526b07a8c891e9225476": "i F = \\nabla\\wedge\\nabla ", "1b054d93d3165ef3eae57a0c1a8bc21b": "\n\\frac{1}{k^{2}} = \\left( \\frac{R}{C} \\right) \\left. \\frac{dC}{dr} \\right|_{r=R}\n", "1b055e10cf989e212d72b842cec4ccd7": "P(x)=1/|S|", "1b0579088363d0cbcbb03faec2fd656b": "p_Z(z) = \\frac{1}{\\pi} \\frac{\\beta}{(z-\\alpha)^2 + \\beta^2},", "1b05822c92b77d143c8d71ab41e8eee2": "\\liminf_{n\\to\\infty}\\frac{g_n}{\\log p_n}=0", "1b058d255444fb05d6146751280865a7": "\\zeta = \\frac{1}{2 \\sqrt{K_p K_v R C}}", "1b058f5ee96d93ddcbec00c3907ece91": "\\; \\varrho_{A_1\\ldots A_m}^{T_{A_{k_1}\\ldots A_{k_l}}}", "1b059ef5acffd2bc1150d3ff659a8140": "p_ie_{k-i}=r(i)+r(i+1)\\quad\\text{for }1\\tfrac 14", "1b0e89ed6ef04d7559592cf9e6c07b3f": "\\exp\\left(t + \\frac{z}{t}\\right) = \\sum_{n=-\\infty}^\\infty t^n {\\mathcal C}_n(z).", "1b0e9a9a0efa483da2349413a7db5ffc": "a_2 = V_2^+", "1b0eb889c60178c7a00bfa6e5a378d0e": "{\\displaystyle\\tilde\\varphi_t(k)}", "1b0f1f8c09bb40a7ad3e8d7777c205aa": "FDR = Q_e = \\mathrm{E}\\!\\left [Q \\right ] = \\mathrm{E}\\!\\left [\\frac{V}{V+S}\\right ] = \\mathrm{E}\\!\\left [\\frac{V}{R}\\right ] ", "1b0f51fd5dddd69b87ba8dd3e2ce584d": "\\mathbb A_k^{n+1}", "1b0f5dc4eccc85b21ee2c35db8c58961": "\\scriptstyle\\bar{x}", "1b0f9a6fc397e23f3f4249156b108dcc": "\\partial\\bar{\\partial}\\omega=i\\text{Tr}F(h)\\wedge F(h)-i\\text{Tr}R^{-}(\\omega)\\wedge R^{-}(\\omega),", "1b0fb89907c90b0cfd071a0cdd7ea0dc": "p<0.1", "1b0fd7c8fbe6c3931bb3d345bb459303": "\\langle e_n, e_m \\rangle = \\delta_{mn}", "1b0fd9efa5279c4203b7c70233f86dbf": "-10", "1b0fe23e62dc07b05a69b73d1bd071fa": "G\\subset {\\mathbb{C}}^n", "1b0fe25146802dc106c54db9bc5fdd13": "F_1(x)=1 \\,", "1b0ff8a1440da24f36b42de8c0012e0a": "\\epsilon(v) > \\epsilon(u)", "1b105b1e9533074584d7bc91d314181d": "\\sigma_B", "1b10710d91eb975193c12955abd2076a": "2.1) \\ \\text{Stock A}\\ -= \\text{Flow}\\ ", "1b108299b9df85ab77ac0ace61b2f6c6": "C_D^{(\\beta)}(\\{x\\})", "1b10a896736a697cd7094aac70a40b46": " \\operatorname{let} M \\operatorname{in} N ", "1b10bad2218174608c7e730723bcf53d": "\\left(0,\\ 0,\\ \\pm1,\\ 0,\\ 0\\right)", "1b10e53be4f6a6819579125899cd553a": " [ L_i , L_j ] = i \\hbar \\varepsilon_{ijk} L_k. ", "1b10e63097337b87541c40654ec099d7": "Dyad: V_2 \\otimes V_2\\to V_2", "1b114a065e8ab5d12728de23768cc065": "1s^2", "1b116c483fcaf1a450b647b7013f9082": "\\sum_{i=0}^n i^3 = \\left(\\frac{n(n+1)}{2}\\right)^2 = \\frac{n^4}{4} + \\frac{n^3}{2} + \\frac{n^2}{4} = \\left[\\sum_{i=1}^n i\\right]^2", "1b119ba70111eab26fa5b4fed8be139e": "m = \\frac{\\Delta y}{\\Delta x}", "1b11aad7da804360e2f924b70f733bd6": "\\frac{d \\rho_{gg}}{dt} = \\gamma \\rho_{ee} + \\frac{i}{2}(\\Omega^* \\bar \\rho_{eg} - \\Omega\\bar \\rho_{ge})", "1b11eb9de4c96790aa8dd236ac790c24": "L-k+1", "1b1209e26e79b8a78f0a945c9abf3533": "v(t) = -v_{\\infty} \\tanh\\left(\\frac{gt}{v_\\infty}\\right),", "1b122c7cf5af5b1e264fa2b2f73f32f8": "\\nabla^2 \\mathbf{E} = \\frac{1}{c^2} \\frac{\\partial^2 \\mathbf{E}}{\\partial t^2}", "1b127b7ac6ed1d2cc682f414fadf8b0d": "A_{{zz}}", "1b128a5e47b8cadcd9d0ff3fdb6bea60": "D\\leq \\frac{\\log{(n-1)}}{\\log(k/\\lambda)}+1", "1b132921f3be55f89eead5bcf306a1c9": "\n\\text{bias}_I(X)\n=\n\\left|\n\\Pr_{x\\sim X} \\left(\\sum_{i\\in I} x_i = 0\\right)\n-\n\\Pr_{x\\sim X} \\left(\\sum_{i\\in I} x_i = 1\\right)\n\\right|\n=\n\\left|\n2 \\cdot \\Pr_{x\\sim X} \\left(\\sum_{i\\in I} x_i = 0\\right)\n-1\n\\right|\n\\,,", "1b133abdb241471bdf2d07238fb6a4c9": " n! [z^n] g_m(z) = \\left[\\begin{matrix} n \\\\ m \\end{matrix}\\right].", "1b1428001688c80e573382c49c3909c6": "f(x,y) = 0 = Ax + By + C", "1b14afd9d15b235d067ee09a9e52dbd2": " V = L^3 ", "1b14c6675b6864645688b221ff428d67": "\n\\begin{array}{lcr}\nA = \\begin{bmatrix} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\ 2 & -2 & 3 \\end{bmatrix} &\nB = \\begin{bmatrix} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\ 2 & 2 & 3 \\end{bmatrix} &\nC = \\begin{bmatrix} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\ -2 & 2 & 3 \\end{bmatrix}\n\\end{array}\n", "1b14ffc1a2dfe5f6bfdaee7845fceb91": "\\mathrm D_{\\mathsf C}=2", "1b15262bb34d66f685defeedea57672a": " I \\propto \\sum_{E_f-eV}^{E_f} |\\psi_n (0)|^2 e^{-2 \\kappa W} ", "1b152b3c29c380b16d50209c76e62c0b": "\\left ( 1 \\right )", "1b1539e8c5d91bc9f00cb6f3dbe1380e": "\\exists x \\in t", "1b155d1e83ad160155b2fb2210c1f77b": "\\alpha = \\frac{k}{c_p\\rho}\\,\\!", "1b1566ddd2bae1fd84f8547c9ff572b1": "\\forall \\underline{y}\\in Y \\forall\\epsilon >0\\exists \\underline{x}\\in X", "1b157ca049809d68e0e5b45d9492f3a5": "\\left \\{ S_1, \\ S_2, \\ S_3, \\dots \\right \\}, \\quad S_n = \\sum_{k=1}^n a_k,", "1b15a6408f9e08a508bca7eaa092c69f": "\\alpha_{\\mathrm{sun}}=\\displaystyle\\frac{\\int_0^\\infty \\alpha_\\lambda I_{\\lambda \\mathrm{sun}} (\\lambda)\\,d\\lambda} {\\int_0^\\infty I_{\\lambda \\mathrm{sun}}(\\lambda)\\,d\\lambda}", "1b15ed4e55d56f85b19efef1526ea0b1": "R^5", "1b165ec0353c6d96260293c1ddef44b1": "\\mathcal{M}_X", "1b17215df910f275ee469afd6b7e757f": "\\tilde{B}_t = \\int_0^t \\sgn \\big( \\hat{B}_s \\big) \\, \\mathrm{d} \\hat{B}_s = \\int_0^t \\sgn \\big( X_s \\big) \\, \\mathrm{d} X_s,", "1b174a16164187f32b5dc73fe2e8d8bd": "\n \\sigma_{11} = 2C_1\\left(\\lambda^2 - \\cfrac{1}{\\lambda}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right]~.\n ", "1b174f1c12b1644b3ecc6993071b66a3": "I_{R2} (= I_{E}) = \\frac{V_{R2}}{R2} = \\frac{V_{Z} - V_{BE}}{R2}.", "1b175bb3eaf1726400f845a802b7b6d4": "1 \\le a < N", "1b1772695902aa5e4239d3c420b46e8a": "\nH = S - \\int \\Pi(x) {d\\over dt}\\psi = \\int {|\\nabla \\psi|^2 \\over 2m} + \\int_{xy} V(x,y)\\psi^\\dagger(x)\\psi(x)\\psi^\\dagger(y)\\psi(y)\n\\,", "1b179aaf55b1e675701b052df628c49d": " (\\boldsymbol\\mu,\\boldsymbol\\Sigma) \\sim \\mathrm{NIW}(\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi,\\nu) .\n", "1b17b8ef9c380ba410f383a006b1334c": "(1,a,a,1,\\varepsilon)", "1b1812ddf9b7652b56a7ddf6a29ccbfe": "\\nabla_{\\mathbf v}\\alpha", "1b1816f393fc2327946eb362c831a7bc": "\\ldots, -\\frac{3}{2}\\hbar, -\\hbar, -\\frac{1}{2}\\hbar, 0, \\frac{1}{2}\\hbar, \\hbar, \\frac{3}{2}\\hbar, \\ldots", "1b1819ddcb8c314306143f1857d25ce3": " \\cdots", "1b18999800000de0768db4cb2e07b171": "(1)\\quad K_\\lambda:=\\sum_{\\alpha\\in S} M_{s_\\alpha\\cdot \\lambda}\\subset M_\\lambda", "1b18a4c4fc578ef4cfd1cc0eb0daa473": "\\leftrightarrow ", "1b18bd12d0d488b511aa808bce7fd2de": "d_1\\ge \\cdots \\ge d_s.", "1b18eaffaaf2bf08ea53a95fea8bc9f9": "\\psi\\left(\\alpha\\right)", "1b18fe3af46e60c4c3fcc4e1674dfcde": " \\tan(\\phi\\,\\!)", "1b190fe270c76b95d333533f4b2613fa": "t_{\\operatorname{ev}} \\;", "1b19118158a8f926a4f12a26947886ee": "\\sum_{k=1}^n A^*_kA_k=1", "1b196fb2855beb459c351c2f9c065c8b": "\\varepsilon(0)", "1b19e0827e3b4f49c5236e9c0b76063d": "\\Delta{Lk=Lk-Lk_o}", "1b19e5ba0224e91619300b1cf3cf12f4": "(\\varepsilon)", "1b1a5d6f6be87c7fe2b52265a3c0d212": "s(t) = S'(t) = \\frac{d}{dt} S(t) = \\frac{d}{dt} \\int_t^{\\infty} f(u)\\,du = \\frac{d}{dt} [1-F(t)] = -f(t).", "1b1aaa12dbcef20f9a460e5997c36b85": "\\bar x = \\displaystyle \\sum_{i=1}^n x_i/n", "1b1ab620e85ab55ea8f0be4b6c8be0bd": "I_0 = \\frac{bh^3}{12}", "1b1bd2d0abe4f4884e8b8a313232120d": "\\lfloor \\frac{1}{2} |V|\\rfloor", "1b1c142ae41eb695982c1e1262e10f80": "\\bar U", "1b1c239c4952b3bfac1797f9d76cddca": "\\frac{\\partial}{\\partial x_{m}}\\left (\\frac{v_{t}}{\\sigma_{k}}\\frac{\\partial R_{ij}}{\\partial x_{m}}\\right )", "1b1c5c0ddd4eb976a57dd171c7413423": " \\frac{c_0}{2} + \\sum_{n=1}^\\infty c_n \\cos \\frac{n\\pi x}{L}", "1b1c5d0bf85b3ffbd399990d095a159e": " \\|v\\|_0 ", "1b1ccf540b427859f88f57d88af45a1e": "\\sgn(\\sigma)", "1b1d238a0a2ee0c76f6bee798a3a2952": "\\alpha(s,t) = \\alpha(t,s)", "1b1d8fd15a63b5e3f0047ba093b92a9a": "0=M_0\\subset M_1\\subset\\cdots\\subset M_{n-1}\\subset M_n=M\\,", "1b1de83a4177bbfd1979f43d1d172ccb": " h'(x) = f'(x) g(x) + f(x) g'(x).\\, ", "1b1deb8ddab79ee00b6f9ce1e02ace5c": "\\mu'_1(P)", "1b1dfcd9b08fcad5eb059292dcf53543": "N=0", "1b1e111e2ba17545f6d7fdde3dc09d90": "K_0^G(C)", "1b1e58b1899ee0b1a94b99e9ae60c3b5": " \\mathbf{\\tau} + \\lambda_1 \\stackrel{\\nabla}{\\mathbf{\\tau}} = 2\\eta_p \\mathbf{D} ", "1b1e9e07d1f5522e9cd9c68c1ef4f3bf": " \\begin{pmatrix} \\varepsilon_{11} & \\varepsilon_{12} \\\\ \\varepsilon_{21} & \\varepsilon_{22} \\end{pmatrix} = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}", "1b1ea0af04dfad48ed5ffca36dd91230": "\\begin{align}\n r:\\ &\\rho \\left(\\frac{\\partial u_r}{\\partial t} + u_r \\frac{\\partial u_r}{\\partial r} +\n \\frac{u_{\\phi}}{r} \\frac{\\partial u_r}{\\partial \\phi} + u_z \\frac{\\partial u_r}{\\partial z} - \\frac{u_{\\phi}^2}{r}\\right) = {}\\\\\n &-\\frac{\\partial p}{\\partial r} + \\mu \\left[\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_r}{\\partial r}\\right) +\n \\frac{1}{r^2}\\frac{\\partial^2 u_r}{\\partial \\phi^2} + \\frac{\\partial^2 u_r}{\\partial z^2} - \\frac{u_r}{r^2} -\n \\frac{2}{r^2}\\frac{\\partial u_\\phi}{\\partial \\phi} \\right] + \\rho g_r \\\\\n \\phi:\\ &\\rho \\left(\\frac{\\partial u_{\\phi}}{\\partial t} + u_r \\frac{\\partial u_{\\phi}}{\\partial r} +\n \\frac{u_{\\phi}}{r} \\frac{\\partial u_{\\phi}}{\\partial \\phi} + u_z \\frac{\\partial u_{\\phi}}{\\partial z} + \\frac{u_r u_{\\phi}}{r}\\right) = {}\\\\\n &-\\frac{1}{r}\\frac{\\partial p}{\\partial \\phi} + \\mu \\left[\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_{\\phi}}{\\partial r}\\right) +\n \\frac{1}{r^2}\\frac{\\partial^2 u_{\\phi}}{\\partial \\phi^2} + \\frac{\\partial^2 u_{\\phi}}{\\partial z^2} + \\frac{2}{r^2}\\frac{\\partial u_r}{\\partial \\phi} -\n \\frac{u_{\\phi}}{r^2}\\right] + \\rho g_{\\phi} \\\\\n z:\\ &\\rho \\left(\\frac{\\partial u_z}{\\partial t} + u_r \\frac{\\partial u_z}{\\partial r} + \\frac{u_{\\phi}}{r} \\frac{\\partial u_z}{\\partial \\phi} +\n u_z \\frac{\\partial u_z}{\\partial z}\\right) = {}\\\\\n &-\\frac{\\partial p}{\\partial z} + \\mu \\left[\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_z}{\\partial r}\\right) +\n \\frac{1}{r^2}\\frac{\\partial^2 u_z}{\\partial \\phi^2} + \\frac{\\partial^2 u_z}{\\partial z^2}\\right] + \\rho g_z.\n\\end{align}", "1b1ec563bf7476aa573a8959735b5819": "\nu^{-1}_{-1}(\\mathbf{p}) = \\sqrt{ {E + p_3} \\over 2 E}\n\\left( \\begin{array}{c}\n{{-p_1 + i p_2} \\over {E + p_3}} \\\\\n1 \\\\\n0 \\\\\n0\n\\end{array} \\right),\n", "1b1ee9800c09587b326af08987bb1097": "n! \\equiv -1 \\mod p", "1b20776490ef05aa236c532c174f93d0": " A(B,B')A' ", "1b2079677975ee6ea03c821a47036eb1": " V(f(x), y)", "1b2091989febeb090b50ee21c832fa3f": "\\mathrm{Dir}(\\vec{\\alpha})\\,\\!", "1b20947216a7d9a21eac6d43d1335bd4": "Q_2=f_a(S_b)", "1b2095aeb23be2435ffc62df21370d07": "\\widehat{T}^{(2)}_{\\pm 2} = \\widehat{a}_{\\pm 1} \\widehat{b}_{\\pm 1}", "1b20c3f8da2a3b530e7bc5d88f2c50d9": "r = r \\times 1 = r \\times 0 = 0. \\,", "1b20e2e7e669b8dd805832d728d2f7e0": " I_P = I_{C, \\text{rod}} + M_\\text{rod}(L/2)^2 + I_{C, \\text{disc}} + M_\\text{disc}(L+R)^2,", "1b21163860d3dd35152d448b936edc06": "F(x',t) > F(x,t) \\implies F(x',t') > F(x,t')", "1b21919945f5543d35a7e0eb3a7a91b7": "(\\alpha', \\beta', f')", "1b21ad3c59bcd4efec61f76a90129960": "U(r+\\bigtriangleup r,w)=U(r,w)\\exp([-bw+ iH(bw)]\\bigtriangleup r ) \\quad (1.5)", "1b21fcb18c994f5454d00136aa5e09cb": "M = \\{ M_{1}, M_{2}, \\dots, M_{m} \\}", "1b21fcdc827eaafe07a3bcae544005d3": "|\\phi(t)|", "1b22953c54e2b44e5f9cce5cbd322258": "\\scriptstyle -\\frac{df(E)}{dE} = \\frac{1}{4kT} \\operatorname{sech}^2 \\tfrac{E-\\mu}{2kT}", "1b22975cf1cd9da405329c03dbe730e8": "l \\cdot w \\cdot h", "1b22aca4fdd9e66bc63bb0af7bc6667b": "2^2 q", "1b231df3e6bc982fad35499758cd755a": "(n,k) = (n, \\delta n) \\Rightarrow k = \\delta n", "1b2348bbf1dd243e00397ab347b3692e": "\\begin{align}\nu_i & = x_i - \\delta_{iJ} X_J = x_i - X_i\\\\\n\\frac{\\partial u_i}{\\partial X_K} & = \\frac{\\partial x_i}{\\partial X_K}-\\delta_{iK} \\\\\n\\end{align}\n", "1b2378c4fe01c9bf699c919383f18b7f": " \\langle x+y,z\\rangle= \\langle x,z\\rangle+ \\langle y,z\\rangle ", "1b23e007aad71555a2966913ddddb764": "D(c,\\rho)", "1b244b225115a4f82e3839b82107f51b": "disc(\\mathcal{H}) \\leq C \\sqrt{t \\log m} \\log n", "1b24866f630c1d4ec0678944aa57a558": "{e}^{-\\Theta /bT}", "1b2491207a778f51461693338fefeaf2": "n = \\left\\{0, 1, 2, \\dots\\right\\}", "1b24b243af9d0f8ab7d830c0ae82dc33": " H(k[\\Delta]; x_1,\\ldots,x_n) = \n\\sum_{\\sigma\\in\\Delta}\\prod_{i\\in\\sigma}\\frac{x_i}{1-x_i}. ", "1b24f7977473be435ca421be58d3cfbe": "F_{\\alpha \\beta} = \\left( \\begin{matrix}\n0 & E_x/c & E_y/c & E_z/c \\\\\n-E_x/c & 0 & -B_z & B_y \\\\\n-E_y/c & B_z & 0 & -B_x \\\\\n-E_z/c & -B_y & B_x & 0\n\\end{matrix} \\right)\\,", "1b25dcf600c55da38f936b7f89980efe": "2^{\\operatorname{LS}(K)}", "1b26002baa591e65e5367c52fab2e293": "\\frac{P}{A} = \\frac{2 \\pi h }{c^2} \\left(\\frac{k T}{h} \\right)^4 \\int_0^\\infty \\frac{u^3}{ e^u - 1} \\, du.", "1b262206c61572286b81d69802c6513b": "\\phi^\\prime", "1b26465fdda19c334798e17d2b12c012": "A \\in 2^\\Omega", "1b26b257e441b257a962f30a4a9a6dcc": "E_A(x) \\geq E_B(x)", "1b26c0e4b5e30055cd3fc6bd3bd77540": "f^{-1}(\\operatorname{int}'(A)) \\subset \\operatorname{int}(f^{-1}(A))", "1b26d21e5198a479a3fee07e069c7cbc": "i(V)", "1b26e3e896b5448109552db32f2be8d3": "\\begin{align}&\\cos \\frac{\\theta}{2} = \\sgn \\!\\! \\left(\\!\\! \\pi \\! + \\! \\theta \\! + \\! 4 \\pi \\! \\left\\lfloor \\! \\frac{\\pi \\! - \\! \\theta}{4\\pi} \\! \\right\\rfloor \\! \\right) \\!\\! \\sqrt{\\frac{1 + \\cos\\theta}{2}} \\\\ \\\\\n&\\left(\\mathrm{or}\\,\\,\\cos^2\\frac{\\theta}{2}=\\frac{1+\\cos\\theta}{2}\\right)\\end{align}", "1b274cbe937c84ed14ed978d8834ffaa": "\\sum^{\\infty}_{m=1} \\frac{y}{m^2+y^2} = -\\frac{1}{2y}+\\frac{\\pi}{2}\\coth(\\pi y)", "1b2765839de3715dbb3d37931c270d8e": "\n\\begin{pmatrix}\n(\\bar{3},1)_{\\frac{1}{3}}\\\\\n(1,2)_{-\\frac{1}{2}}\n\\end{pmatrix}\n", "1b276ac1fe6d4425a75cdb365a367406": "1.25\\sqrt{H}", "1b28486709f46a3ac30f5a4cdc6b3141": "dU=\\delta Q+\\delta W", "1b28aced7e96c3f8b82023a6a2bef154": "\\lambda_i = \\lambda\\text{ for }0 \\leq i < K \\, ", "1b28cdae2c0973205af7e43028048372": "X_1 X_2", "1b28d71dccbf5ff1eeab11fce46f55b8": "Q |\\psi_0\\rangle = -S_\\psi |\\psi_0\\rangle = (2\\cos^2(\\theta)-1)|\\psi_0\\rangle +2 \\sin(\\theta) \\cos(\\theta) |\\psi_1\\rangle", "1b29131ba97d8ea37b117b6d1bb89871": "\\boldsymbol{G}_{Had}", "1b2914a8d8f0f73116c800c8576d2c04": "\\Delta_{\\pi}^{2}=0", "1b291b55c74dc0457647825c89948778": "v = (x_b + k^e) \\mod N", "1b29807fba94568b14f6b7d116556b5f": " \n \\cos(\\text{inner side}) \\cos(\\text{inner angle}) = \\cot(\\text{outer side}) \\sin(\\text{inner side})\\ -\\ \\cot(\\text{outer angle}) \\sin(\\text{inner angle}),\n", "1b29a77805cd22485f0da888771265f2": " a_P T_P = a_W T_W + a_E T_E + {a_P}^0 {T_P}^0 + S_u ", "1b29ab8fdd453f3248f3ac68b609a176": "\\nu(z) = k - \\frac{\\log(\\varepsilon/|z_k - z^*|)}{\\log(\\alpha)}.", "1b29f1325ec37e4cf6033d71494a9346": "F(\\mathbf{z}, \\mathbf{x}) = \\operatorname{Prob} \\lbrace Z(\\mathbf{x}_1) \\leqslant z_1, Z(\\mathbf{x}_2) \\leqslant z_2, ..., Z(\\mathbf{x}_N) \\leqslant z_N \\rbrace .", "1b2a1b945e30d38498cba9146d6043ff": "\\phi (A) = \\frac{KA\\cdot\\operatorname{sock}}{2}\\exp\\left(-\\frac{KA^2}{4}\\cdot\\operatorname{sock}\\right),", "1b2a3668a465885eed04e4509594ba55": " \\boldsymbol\\eta = \\mathbf X \\boldsymbol\\theta ", "1b2a8d80369b97c3086dd8d2361ed876": "XY=\\frac{|a^2+c^2-b^2-d^2|}{2p}.", "1b2aa540bc5a9d35ac3dbd2e36cf1652": "u(c) = \\frac{c^{1-\\theta}-1} {1-\\theta} \\,", "1b2af3b11deb3956ffed253d7bb2cca5": "\\left\\langle M_B\\left(l_B\\right)\\right\\rangle\\sim l_B^{d_B}", "1b2af99c6eb76e81f85b75383bf8e6ee": "\\omega_{pl}^2(q) = \\frac{2 \\pi e^2 n q}{\\epsilon m}", "1b2b32c7df5f76141684bb30d39399e0": "y = \\rho_0 - \\rho \\cos\\theta ", "1b2b45d45abd23bf45bf8b83d83858a1": "Q_{ab}{}^{c}=Q_{(ab)}{}^{c}", "1b2b4cf5107b7a7ad9566dce8dfbad22": "\\omega_f(t):=\\sup\\{ d_Y(f(x),f(x')):x\\in X,x'\\in X,d_X(x,x')=t \\} ,\\quad\\forall t\\geq0.", "1b2b7abdacade6bfee8532c14df40f73": "\nu_2 = \\frac{\\lambda}{2} + \\frac{1}{r^2}\\left(\\frac{p^2}{2m}-\\frac{1}{2}mgz \\right).\n", "1b2b7fcfa3d6ceddab3a6be1df824430": "(R^{-1}P)", "1b2b88b65adbdc568e55a7df404cc854": "x=\\cos y\\,\\!", "1b2bf2869b0b2c84ba7c50ec6198ee12": "\\cos \\left( \\frac {\\pi} {5} \\right) = \\cos 36^\\circ={\\sqrt{5}+1 \\over 4} = \\frac{\\varphi }{2}\n", "1b2c0a0a81fd5346d9ce68b30226bc5d": "f\\in BV_\\varphi([0, T];X)\\iff \\mathop{\\varphi\\mbox{-Var}}_{[0, T]} (f) <+\\infty", "1b2c2a7cee33b6fb0a68211f8500a107": "x^4 - 10x^2 + 1,\\ ", "1b2c8fcf35b342643acb33167d8db4dd": "\n\\frac{\\mu^2}{x}.\n", "1b2c9883e732ab514175f8543c910230": "\\|Tx\\|^2 = \\langle T^*Tx, x \\rangle = \\langle TT^*x, x \\rangle = \\|T^*x\\|^2", "1b2d2a835c5ac47d32c91fee49c9dcb9": "\nF \\left(\\mathbf{x}^{(0)}\\right) = 0.5((-2.5)^2 + (-1)^2 + (10.472)^2) = 58.456", "1b2d6445b054f48a72dd810423cd4e01": "\\tau = I \\alpha\\,", "1b2d8c73ecf1b97144e0f4b489873868": "\\tilde{H}(t)=\\sum_{t_i\\leq t}\\frac{d_i}{n_i},", "1b2de5396f196794b727393419fed9dc": "f(n) > g(n)\\,\\!", "1b2e2cf401b1d188124328c5ea88af01": "y \\ge 0.138 + 0.580 x", "1b2e49c26b804dc9ac8b6343f9fdec8f": " \\hat {\\mathbf{e}}^2", "1b2e5739a139fe87db706788cc8e1e16": "\\mathbf R=\\mathbf N-\\mathbf L", "1b2eb5418feaee2d3f95ec87ee7b7464": "\n\\sum_{i=0}^n p_i(z) f^{(i)} (z) = 0\n", "1b2ecdaf6a6c88fa2bc82edfad84ec6e": "0<\\phi\\left[|m|\\right]<1", "1b2eddf3577dc84a9fa476308eecd83c": " h_{n} = {V_{n} \\over cos(i_{n})} \\left( {T0_{n+1} \\over 2} - \\sum_{j=0}^{n-1}{h_{j}\\sqrt{{1 \\over V_{j}^2} - {1 \\over V_{j+1}^2}}} \\right) ", "1b2f08e4d4e535a01a4040874fc86a5b": "X=Y\\cdot Z", "1b2f0a509545e54f01c59a2bf2a6dfe1": "P=c_s^2\\rho,", "1b2f36e274431ebd48a2baa36c32c1b4": "\\scriptstyle \\frac{V}{R}", "1b2f4f4447872b11bd2b0b2382056302": "\\mathbf{B} = \\mathbf{U} \\mathbf{S} \\mathbf{V}^T", "1b2f688f7c8b1a82bc4585cda2c1bd81": "m^*_\\text{density} = g^\\frac{2}{3} \\sqrt[3]{m_x m_y m_z}", "1b2f9141e3069258db3931fba8ad271f": " u(a)=\\frac{1}{2\\pi}\\int_{\\mathbf{T}}\\frac{1-|a|^2}{|a-e^{i\\theta}|^2}f(e^{i\\theta})\\,\\mathrm{d}\\theta", "1b2fd60cd9ac6b079fd721837a9b8a77": "\\frac{dF}{dx} = f(x).", "1b3019d392e3ee0dc65eee83cec274af": "\\bar{y}=\\frac{M_{01}}{M_{00}}", "1b30706240f0355ce91e70838a143c19": "(s)", "1b30d4cd598edf69006b735efd94fa02": "\\lambda=(\\lambda_1,\\ldots,\\lambda_{m_2})", "1b30ef2a31be91ea72f6f8d7b2d4fb9c": "x+h", "1b3117d018f75ed45c50fdc849645054": "\\Lambda(A_1), \\Lambda(A_2),\\ldots, \\Lambda(A_n)", "1b3134184af0d97cf98e98550f657c7d": "\\displaystyle{\\partial_n u|_{\\partial \\Omega} =0,}", "1b313f382aeab7621a804a829996c451": "E=\\int d^3x\\, \\left[ \\frac{1}{2}\\overrightarrow{D\\varphi}^T \\cdot \\overrightarrow{D\\varphi} +\\frac{1}{2}\\pi^T \\pi + V(\\varphi) + \\frac{1}{2g^2}\\operatorname{Tr}\\left[\\vec{E}\\cdot\\vec{E}+\\vec{B}\\cdot\\vec{B}\\right]\\right]", "1b31d878869a7a2a2243f0a281b4c244": "{K \\choose k}\\cdot{N-K \\choose n-k}.", "1b323c6fb303ace1a6fb6e38d644c47b": "\\frac{\\partial \\mathcal{H}(q,p)}{\\partial p} = \\frac{p(t)}{L} = -\\dot{q}(t) \\ ", "1b32515c2316ef92ea433626a6489a7a": "M_v-N_u=L\\Gamma^1{}_{22} + M(\\Gamma^2{}_{22}-\\Gamma^1{}_{12}) - N\\Gamma^2{}_{12}", "1b329308e0009389e60936f954966404": "\\beta = -\\eta_2,", "1b329aa84296d7d276fa8f082f5bc636": " \\frac{1}{m!}\\sum_{k=0}^m (-1)^{k} \\left[{m+1\\atop k+1}\\right] B_{n+k} = A_{n,m}. ", "1b32b33339b988be580e35465349943e": "F(n+1)", "1b32d078a58b149739c80c76f5c3d1f5": "P_G", "1b3375ba38a555c30a12f04439c7443c": " I(0) < \\delta\\text{ and }I(T) > K \\, ", "1b3385b9d76a7193c4b9bd94d6be5833": "\\mathsf{H}(\\cdot)", "1b33b27a9ff57ef2e67c799b57359f42": "2g", "1b33c4b25aeb798428694efad45ac72a": "e = \\sum_{k=0}^\\infty \\frac{1}{k!}", "1b3402a67b5c39eb34bffd4740445683": "K(x) = h(x) e^{\\boldsymbol\\eta \\cdot \\mathbf{T}(x)}", "1b34543b731e888cb39ff70888596175": "\\scriptstyle \\varphi=\\frac12(1+\\sqrt5)\\,\\!", "1b34680bfad4fd69ba62366786bb78c3": " I_N ", "1b34cea2960b5b23a77f94ec29e5985b": "k\\{ \\tau_p\\}.\\,", "1b34ece2df19d366b8d32e1d039a844c": "\\rho_{Actual}\\,\\!", "1b350f9b23871bccf35b479ae3fb0c5f": "r_i(x_j)", "1b3551633296b23d0445a4219f4cee14": "\\delta+1", "1b355b909deb67616eaa466f2b4e65dc": "(2)\\quad t=u+r+2M\\ln\\Big(\\frac{r}{2M}-1\\Big)\\qquad\\Rightarrow\\quad dt=du+\\Big( 1-\\frac{2M}{r} \\Big)^{-1}dr\\;,", "1b356209d3fef39971420b5bd851f84e": " Z = \\frac{1}{{i} \\omega C_b + 1/R_b}", "1b35c6928b246f99a9b16e5679126109": " h(x) \\rightarrow 0 + h_0 \\delta(x) ", "1b35da8305a59462f6fe75a96bfc485f": "uw=vw", "1b35e567cebf847f3c717a84ab46d37d": "\\textstyle \\mathbf{b}_1", "1b36bb13c11fbd30a693ff6645dd9d11": "\\begin{align}\n\\frac{dy}{dx} &= \\frac{dy}{dz} \\times \\frac{dz}{dx} = -\\frac{dy}{dz} = -y' \\\\ \n\\frac{d^2 y}{dx^2} &= \\frac{d}{dx}\\left( \\frac{dy}{dx} \\right)\n = \\frac{d}{dx}\\left( -\\frac{dy}{dz} \\right)\n = \\frac{d}{dz}\\left( -\\frac{dy}{dz} \\right) \\times \\frac{dz}{dx}\n =\\frac{d^{2}y}{dz^{2}} = y''\n\\end{align}", "1b370d5d59a83a035f2e61c1dabab139": "\\operatorname{club}(\\kappa)", "1b3714cce3f5e6ebffe57748205e733c": "\\textit{open} \\circ \\textit{on}", "1b3723e0b4f1c8a81ad5ab64ae0fa23e": "t=|H|_{x}|\\,\\!", "1b372f792f4d1decf9048f8c4ee6a5dd": "\\tilde{J_n} = -\\frac{J_n}{\\mu\\ R^n}", "1b3731fca4937dbd72f9c627836df1e2": "X \\in C, Y \\in F", "1b373a2a965f31e05343e2b85a556dbe": "m_5(x) = x^2+x+1,\\,", "1b3763b4d746c4ef40c240595fd50baf": "O(2^{2k^2}+n+m)", "1b376fbd71b27090316ce8a7ae5497a1": "T(n) = 2 T\\left(\\frac{n}{2}\\right) + O(1)", "1b377f93f1a3cf825444d7e86eb1cee1": " \\mathrm{Li}(x) = \\int_2^x \\frac1{\\ln t} \\,\\mathrm{d}t = \\mathrm{li}(x) - \\mathrm{li}(2). ", "1b37a63f2416c0e53c0a897c01d232c4": "q = \\left ( \\left [ qx1,qy1 \\right ]; \\left [ qx2;qy2 \\right ] \\right )", "1b3823f10bac3c0a7d8ce5fa8c4129a4": "E_{\\pm} = a \\pm |\\mathbf{r}|", "1b383b961d55892e6b952c3adea3f635": "a_{0j}", "1b388f6e849e75c8a72152dfbb8715e8": "f_\\epsilon(x)=\\mathbb{E}\\big[f_\\epsilon(X_\\tau)\\,|\\, X_0=x\\big]", "1b38aed9648ae9d27b9e27801d10f1ab": "\nP_\\mathrm{L} = {1 \\over 2}{{|V_\\mathrm{S}|^2 R_\\mathrm{L}}\\over{(R_\\mathrm{S} + R_\\mathrm{L})^2}}\\,\\!\n", "1b38f75b0e72c028b4030f714aac5d83": "\n \\dot{\\varepsilon_{\\rm{p}}} = \\left[\\frac{1}{C_1}\\exp\\left[\\frac{2U_k}{k_b~T}\n \\left(1 - \\frac{\\sigma_t}{\\sigma_p}\\right)^2\\right] + \n \\frac{C_2}{\\sigma_t}\\right]^{-1}; \\quad\n \\sigma_t \\le \\sigma_p\n", "1b3934fab43f480b55b91fcd4fc81893": "\\displaystyle{g_a(z)= {z-a\\over 1- \\overline{a}z}.}", "1b398ca551ec519ba826c135203136f8": "k<\\beta,", "1b39ca16b2a0f5f6c6a9f49b017d7d8c": "s = - \\dfrac {q^2-a_2+a_1} {a_1q-a_2+1} \\bmod{\\ell}", "1b3a394794c85d511c66dd960cdfb070": "f(U) \\neq f(V) \\vee P.", "1b3b029a8a42c7c0b8d89a7ff05ae745": "~\\alpha(0)=|\\alpha(0)|\\exp(i\\sigma)", "1b3b0c41edb32817ed869d11402df2f7": " \\Gamma_i ", "1b3b0d39eace66eb6a6e5e6816d21d79": "f(X_t)=f(X_0)+\\int_0^t f'(X_s) \\, dX_s + {1\\over 2}\\int_0^t f''(X_s) \\, d[X]_s.", "1b3b211925112c8a732895669fe4f002": "\\epsilon = \\frac{\\sigma}{E} + \\alpha \\frac{\\sigma_0}{E} \\left(\\frac{\\sigma}{\\sigma_0} \\right)^n", "1b3b3584117fefe62a76ad3395cbdb08": "\\textstyle \\left\\{ O_{i}\\right\\} ", "1b3b5316829b718d24ae9b76afc9b65f": "M=S^3/\\Gamma", "1b3c1a40f9cb094d47e8c6f9b0df773f": "\\circ", "1b3cc0bbd8f0a2f68c3c22a2a0ef865b": "a = -\\omega^2", "1b3cf6b61727d01f863a13b7db408732": "\\{ X \\mid \\exists m \\phi(X,n,m)\\}", "1b3d000397579bb97b3a5e1776e1af53": " X \\,\\sim Bin(n, p) ", "1b3d5826f934b5011047068d49f9cf15": "\\frac {dx} {dt}", "1b3d7a732d58fe6e43a547b840c12641": "y_1,y_2,...,y_t", "1b3d98a676ae1d627fa50d03b8e66857": "(z_1,t_1)(z_2,t_2) \\cdots (z_n,t_n)", "1b3dc69c97f688744bc79c7ed28fd36a": "{s_1}<{s_2}", "1b3e05e131b73cf457ff1731e746c43c": "F_L = L \\otimes_k F", "1b3e3ac75f1b8361df481220890f7585": "s_{Tx}\\,", "1b3ea1017e85ec576d2a82dc6ec24c7f": "x(t) = -1 + e^{-t\\left(x_{0}+1 \\right)}", "1b3eb1b5d7c7f832d9b42c55e5fc86af": "OLD(T_i)", "1b3eec6d8313a155d8a6ff0fe0d7a255": " [\\textrm{CO}_2]_{eq} = \\frac{[\\textrm{H}^+]_{eq}^2}{[\\textrm{H}^+]_{eq}^2 + K_1[\\textrm{H}^+]_{eq} + K_1K_2} \\times \\textrm{DIC}, ", "1b3f3b4ccd035da3eb6fd2afa1f1d423": "O(k/\\log N)", "1b3f4b49ed4628101af2fbf5f092dbdc": "Z_0 = \\sqrt{L/C}", "1b3f905457325b5fe3c22e9666469fe0": "\\hat s(t),", "1b3fc1bac9f0239d5f484cbc612ad43e": " s_j ", "1b3fd21fb65b59e433321883e7a81bf5": "|j_1m_1\\rangle", "1b402923d0746dc6d27a1afa5ff5f03b": "{E'}^2 - (p'c)^2 = (m_0c^2)^2\\,.", "1b40637a8330003a9cd61bc5d0760966": "t:=\\frac{M_1-M_2}{SD_\\text{within}/\\sqrt{\\frac{n_1 n_2}{n_1+n_2}}},", "1b40c5455185ed19cc57b87fffb65193": "|a|:=\\begin{cases}a, & \\text{if }a\\geqslant0,\\\\ -a, & \\text{otherwise}.\\end{cases}", "1b41195424fdfedc75e4caccb6d6932d": "{ S = \\ln \\Omega } \\ ", "1b413646eb2008295c35ec8f603240ea": " P(\\lambda x, \\lambda^2 t) = \\lambda^m P(x,t)\\text{ for }\\lambda > 0.\\, ", "1b4148d654a27793ad826f017779f467": " \\Gamma_a ~,~ \\Gamma_{a_1 a_2}", "1b41cc724e4d1bb15eb70eebf87ca378": "(x_n,0)>(0,1)", "1b41d1c0c4512f8042ca7e985bd521fd": "\\rm{dK = K \\left ( \\frac{BFP-dIBB-dHB}{BFP-IBB-HB} \\right )} \\,", "1b421dd6318ea30ee8682c0f2c59cd3e": " = (z + i\\sqrt5)(z - i\\sqrt5)", "1b4237b50df5b053d3f63da329acc13c": "\\psi(\\mathbf{r})=\\phi_{n}(z)\\frac{1}{\\sqrt{A}}e^{i(k_{x}\\cdot{x}+k_{y}\\cdot{y})}u(\\mathbf{r}).", "1b4279687adfeaa02fdd2bfec5fe4638": "(-\\vec{\\infty},\\vec{\\infty})\\,", "1b429003910d26d98f5568f1d9696d7b": "P_3=(X_3,Y_3,Z_3,ZZ_3)", "1b42b9e56c650c87bf0c7f6e201b6d19": "I \\subset \\R, \\; c \\in I", "1b43008ad84e9f9265fd27f818562daa": " \\operatorname{drop-params}[(\\lambda N.S)\\ L, D, V, R] \\equiv (\\lambda N.\\operatorname{drop-params}[S, D, F, R])\\ \\operatorname{drop-formal}[D[N], L, F] ", "1b432187eac2a64b8eafbf3d484181bc": "{}_pF_q(a_1,\\dots,a_p;b_1,\\dots,b_q;z), \\vartheta\\; {}_pF_q(a_1,\\dots,a_p;b_1,\\dots,b_q;z)", "1b4329fb23abc4a14893d4fd446c6eb1": "R_h \\rightarrow \\frac{b h}{b} = h", "1b432b35835c583d857dea6ba33d82c6": "\\delta^{\\alpha}_{\\beta} \\, A^{\\beta} = A^{\\alpha} \\,", "1b4399ac0fc9fd174a9bded78a61608c": " v_0, v_1, ... ", "1b43c5e851bef33b50a224ae72a362fa": "x \\Rightarrow^{ac}_{m} y", "1b43cbbc95fd5231faf001c79f604285": "T_{\\rm F}", "1b43fafb317533d3532d38613cd54b72": "(n_x,n_y,n_z)=(3,4,7)", "1b4408ddec06de485fef9b8deb30571e": "\\mu((\\tilde X \\cup \\tilde Y) \\cap \\tilde Z).", "1b441e71205bc8c052e778ea96e6faf0": "v = \\begin{bmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix}.", "1b4447b5544fdbb0d0eeb55ec0458be8": "\\omega^{A}_{x\\overline{\\lor} y}=\\omega^{A}_{x}\\;\\overline{\\sqcup}\\;\\omega^{A}_{y}\\,\\!", "1b44f492708bf1379b984f6e5a4d1ab6": "a_i: {\\mathbf S} \\rightarrow {\\mathbf S}", "1b45283d035db8ea8e09a6477dd9ead0": "{\\Omega}/\\sqrt{seconds}={\\Omega}(s^{-1/2})", "1b45644ccd90ebdfafe1e4c6788b3960": " \\textbf{m} = -1 + X^3 - X^4-X^8+X^9+X^{10} ", "1b456ead54371bf51c51ce05b19be298": " \\frac{d\\nu}{d\\lambda}=\\frac{d\\nu}{d\\mu}\\frac{d\\mu}{d\\lambda}\\quad\\lambda\\text{-almost everywhere}.", "1b45d958b7370013a0ebb1a7eccad35f": "O(g(x))", "1b45e859fa657432f287c46240dcd33a": "\\forall t,s \\in [a,b], K_X(s,t)=\\mathrm{E}[X_s X_t].", "1b46097e090cd78100d9e5316dc22cd1": "F(x;s,\\theta) = F(x/s;1,\\theta), \\!", "1b464d960c55c91a4a76daf6c958de82": "z_n=\\left(\\frac{\\pi}{2}+n\\pi\\right)^{-1}", "1b47b53cd6d4ec9cc239fd5b8cb77217": "\\mathcal E_T:=\\mathcal E\\otimes_{O_S} O_T", "1b47baeb7aad9b53d39f85cfa8fd68fe": "G_0 = G", "1b47bc0cd1fc5ae0dbb5cb24777207a1": "\n \\begin{matrix}\n \\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\\\\n 10\\mbox{ multiplied copies of }10\n \\end{matrix}", "1b47f2139ffb5eaff8ca580c27d33b1a": "\\pi_{ij}=\\pi_{i+}\\pi_{+j}", "1b47f56bf2bcc0b15e87f80c4ce571e5": " |z| < 1 ", "1b4807f906c7c3d1d50b86d4a838ee27": "\\mu_{app}", "1b485d1acae11a3b1a9b6738596255e4": "[a_0; a_1, a_2, a_3, \\,\\ldots ] = \\lim_{n \\to \\infty} [a_0; a_1, a_2, \\,\\ldots, a_n]. ", "1b488e5e33d2760243ddc093082c75f6": " \\Delta f = \\frac{\\partial^2 f}{\\partial x^2 } + \\frac{\\partial^2 f}{\\partial y^2 } + \\frac{\\partial^2 f}{\\partial z^2 } = 0.", "1b48ea9ca6e26f197682482ed59aaac6": "\\frac{f(t)}{1-F(t)} = p + q F(t)", "1b490e9a944f7e6e06bf0c5fc13812bc": "\\phi:\\mathbb{D}\\rightarrow D", "1b491b4a89ee669759b0a60d706fc152": "p_s = (1-tanh(s)^2)", "1b4929e68b16fee477efd52fe0ede1de": "\\beta = \\omega \\sqrt {LC}", "1b4977a0055292ac2003ed4d6fef9609": "f(x)=1/x^2", "1b49803cfd0027456248b46bc70ec0c8": "\\alpha \\rightarrow 1", "1b499f2a7a708ad6a2c28505c128c5af": "h \\in H", "1b49adbd471f5cab0dd2d987adbf306e": " \\varphi(ST) = {\\rm f}( \\{ \\langle S e_n , e_n \\rangle \\lambda(n,T) \\}_{n=0}^\\infty)\n= v_{\\varphi,T}( \\{ \\langle S e_n , e_n \\rangle \\}_{n=0}^\\infty ) ", "1b49e6abcbda3ae40e96c9ce0b7690b3": "S \\subseteq T", "1b4a14adec41007ea6e5375a9ddeaa4c": "I_t = K_t - K_{t-1} \\,", "1b4a2a402bba276b957d024ec3420636": "u\\in U_\\alpha\\cap U_\\beta\\cap U_\\gamma.", "1b4a5e9f9368e622cee01761693ae308": "\\Box \\phi = \\frac{\\rho}{\\epsilon_0}", "1b4a6c716ee0fe03a390a9efa14589dd": "\\limsup P(n) / \\log n = 2", "1b4a951adf43a75ec87f6238921b9124": " H_0 + W_{DE}(t) ", "1b4aac870f6357bc8ddff4c729d0b04c": "\\scriptstyle d^{*}_{ij}", "1b4abca54260bd0cc661ea6ea92a042c": "\\rho^{\\mathrm{ent}}_t(X) = \\frac{1}{\\theta}\\log\\left(\\mathbb{E}[e^{-\\theta X} | \\mathcal{F}_t]\\right).", "1b4acfb4fc5569995071577574d5206e": "-\\frac{d^2}{dx^2}", "1b4b77779c688edee9a5ea643ed58935": "\\begin{align}\nred= n - dof = n-(n+m-p) = p-m,\n\\end{align}", "1b4ba3cb2f01720d82e6ff42f5a6ec84": "P(R_t|\\lambda) = \\exp(-\\lambda t)\\,", "1b4bc213275944e02ecbb63dcaca3fd5": "\\mu_1, \\mu_2, \\ldots, \\mu_t ", "1b4be3506b06bc47595b9c28c1c6870b": "r - R = 0", "1b4be3778caa7728d3ab9e49bc8e72c3": "\\frac{d\\ln k}{dT} = \\frac{\\Delta E}{RT^{2}}", "1b4c2e7e05508488a05be6513e9acbe3": " \\dot q = {\\partial H \\over \\partial p} \\,", "1b4c552f30e1067f9215425f8b955d43": "\\overline{A + B}=\\overline{A} \\cdot \\overline{B} \\iff \\overline{A \\cdot B}=\\overline{A} + \\overline{B} \\iff \\overline{AB}=\\overline{A} + \\overline{B} \\iff \\overline{A + B}", "1b4c5d4144aee6c054ed5cd1055a9c19": "\\theta_\\text{c}=\\arcsin\\left(\\frac{n_2}{n_1}\\right)", "1b4c85c07f032f6196cc69fe47162079": " X \\colon(\\Omega, \\mathcal F, \\mathbb P) ", "1b4cc5606b16d3c2f88f20faf90ceca4": "(5)\\qquad D\\sigma=\\sigma(\\rho+\\bar\\rho)+\\Psi_0=-2\\sigma\\theta_{(l)}+\\Psi_0\\,,", "1b4cf0cd01ca9c8d65c2cfb1504b642f": "\n\\mathbf{v} = \\frac{d\\mathbf{r}}{dt} = \\dot{r} (\\cos \\varphi ,\\ \\sin \\varphi) + r \\dot{\\varphi} (-\\sin \\varphi, \\cos \\varphi)\n", "1b4d003146297fb4f3be4fd494526149": "d_M (x_{n_k}, y_{n_k}) < \\frac{1}{n_k} \\wedge d_N ( f (x_{n_k}), f (y_{n_k})) \\ge \\varepsilon_0 .", "1b4d0651ef806ab5744190328623be43": "\\left( \\gamma^0 \\right)^\\dagger = \\gamma^0 \\,", "1b4d6251fd64b973ee623eb795ad2a47": "\\lVert q \\rVert = \\sqrt{qq^*} = \\sqrt{q^*q} = \\sqrt{a^2 + b^2 + c^2 + d^2}", "1b4dad9912359203c05e07e1c5e50728": " U(t) = \\exp\\left({-\\frac{i}{\\hbar} \\int_0^t H(t')\\, dt'}\\right),", "1b4dc54a182a7cb2bfa19f47631ac17b": "\\hbar^2 \\, (1/2) \\, (1/2 + 1) = (3/4) \\, \\hbar^2", "1b4dd6e049bbb7a0f16f52b72a6572c8": "c > d", "1b4dfb5f9d10b23544b634743cfceaa1": "\n\\Lambda(x) = \\frac{ L(\\theta_0|x) }{ L(\\theta_1|x) } = \\frac{ f(x|\\theta_0) }{ f(x|\\theta_1) }\n", "1b4e3e0d4b4d4d08055963947bf7647b": "p_i = P(X = i)", "1b4e63855c02338acdccb214086a8da5": "\\displaystyle U \\sim \\epsilon \\epsilon' a \\omega \\sin \\phi", "1b4e7585bcd85536c0533fd4bf6627b8": " X \\sim \\textrm{GB1}(a, 1, 1, b)\\,", "1b4f61b72b387fd5c9532ba654e379d7": "x_1=1.42>\\sqrt{2}", "1b4f957b738a66b3788d74d02273fec7": "\\{K_{AB}, A,\\mathbf{N_B'}\\}_{K_{BS}}", "1b4fa43f0c88a661a2a15987ee12bff1": "v = V_{max} \\frac{[S]}{K_M+[S]}", "1b4fb5ad8e17e6094bed78c8d9285d66": "2^3(1+x)^{3}p(\\frac{\\frac{3}{2}+2x}{1+x}) = 8x^3+4x^2-4x-1", "1b4fe2190ffbf0dfce4ce9629f72a29a": "Y_x = K x^{\\log_2 (b)}", "1b500c1bb411497a99e4630bd770ab8f": "V(t) =\\ M(q(t)) I(t)", "1b502c3a73a7fdee82caa2ef41704254": "w=1/3", "1b50878e6d4f005a7e6ff25c4d90064a": "t\\mapsto (a\\cos t, a\\sin t, bt), t\\in [0,T]", "1b5097d55f7ad1576b2ad3c383712785": "\n\\langle r^{2} \\rangle \\approx \\frac{3k_{\\rm B} T}{m} t^{2} = \\langle v^{2} \\rangle t^{2}.\n", "1b50bf999080f951478f832d0890ab62": "\\boldsymbol{T}", "1b50e809e15adc21f66b4190985678b5": "\\Re \\left[ \\mathrm{Bi} ( x + iy) \\right] ", "1b5159dad7eb824f669e3bcfbc42b358": "d_{k,n} = a_{k,n} - k/m_n", "1b5161108cfa5abe8069d6bca469b6d5": "\\frac 1 {2}\\left |{\\frac {f^{\\prime\\prime} (x_n)}{f^\\prime(x_n)}}\\right | 0 \\} \\,", "1b735602edd399667deac52c541bd290": "\\mathfrak{M}(d)", "1b735b48cf704e8029cab1e20af3d078": "\n\\int_{0}^{\\pi}\\sin\\theta\\, d\\theta \\int_0^{2\\pi} d\\varphi\\; R^m_{\\ell}(\\mathbf{r})^*\\; R^m_{\\ell}(\\mathbf{r}) \n= \\frac{4\\pi}{2\\ell+1} r^{2\\ell}\n", "1b7368f46bb45dd22ef25c46d6f54619": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = f_1(\\mathbf{x}, z_1) + g_1(\\mathbf{x}, z_1) \\overbrace{\\frac{ 1 }{ g_1( \\mathbf{x}, z_1 ) }\n\\left( u_{a1} - f_1(\\mathbf{x},z_1) \\right)}^{u_1(\\mathbf{x}, z_1)}\n\\end{cases}", "1b7370c73eb8409047acd351389d740c": "\\lim_{p\\to\\infty}{\\left(\\sum_{i=1}^n |x_i-y_i|^p\\right)^\\frac{1}{p}} = \\max_{i=1}^n |x_i-y_i|. \\,", "1b73b46902f9af4b1078f9decf3c9431": "\\psi_1(z) = \\frac{d^2}{dz^2} \\ln\\Gamma(z)", "1b74464353515988215bd5bf2abb0a72": "A = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}", "1b74657b0b8ce073588540d39aed374d": "u/y", "1b7490913900363d51207a4fcb174457": "R=\\frac{(\\rho_s-\\rho)}{\\rho}=1.65", "1b74b6d01d07b242dcbb75969c8ece95": "\\langle n \\rangle", "1b74be694c6066749910c82f1cfdf08b": "0^{(\\omega)}", "1b74dc569e10c78567281a506e9461aa": "C_3 = 36 \\ \\mathrm{pF}\\,", "1b751afbd2349bcb84a75e0942e51086": "K[[x,y]]/(x^2,xy)", "1b7537beff8c524c217c5f1321e00e19": "e^{-\\frac{2 \\pi i}{N}}", "1b756b041148bd177b3712638a3f0af1": "\n\\limsup_{T\\rightarrow\\infty} \\frac{1}{T}\\log\\left(\\frac{Z_\\pi(T)}{Z_\\nu(T)}\\right) \\leq 0\n", "1b7570d1931c89df7bdd6120615e01d9": "\\forall X, Y \\in L^0(\\mathcal{F}_T)", "1b75767a344d24b547a3a740f05863c1": " \\sin E' = \\frac{y}{b} \\ . ", "1b757c17f789d57d1b9d842b06f4eaf8": "{n\\choose k}_2", "1b75b3fabd2029c79b5dec5720e6323a": "(p-1)/2p", "1b75f8b00d53e9fff5826221d62557d4": "(1-x)(1-x^2)(1-x^3) \\dots = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} + \\dots.", "1b75fa0bf829aa8736aa82aa3c12e657": "J \\propto \\Psi^\\delta", "1b762e083205175dcb99111ae0ebdd00": "\\mathit{n_q} = t, q =0, 1, ..., p-1", "1b763d485c59cddde353d9f6cea6834c": " = 1 - \\frac{1}{(1 + e^{\\mathrm{log}(e^{y} - 1)})^{\\theta}} ", "1b764acd730c804b795b63ea90e0b13c": "( w \\ll \\Delta^{1/2})", "1b764ad88cde290846778f96df1b2ed9": " p \\colon E \\to B ", "1b76663e3241b2a3ac0f22f0008ae1f2": "\nQ_{lm} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int d\\mathbf{r}^{\\prime} \\rho(\\mathbf{r}^{\\prime}) \n\\left( r^{\\prime} \\right)^{l} \n\\sqrt{\\frac{4\\pi}{2l+1}} \nY_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "1b767ef177de610ed1d759cf6fe484d8": "B_{z/2}=0", "1b76da0b0550e492d5065c9bf3304fba": "\\, \\varphi=\\ t-\\arctan(\\ t)", "1b76e99344a80d5b7b89d132066e2c9b": "f, g_1 \\ldots g_m : \\mathbb{R}^n \\rightarrow \\mathbb{R}", "1b76f3e5a33306fad957baab52fa5b80": "t\\begin{Bmatrix} 2 , p \\end{Bmatrix}", "1b770f7c8775617e4cb88819c8ff370f": "\n\\begin{align} \n& R\\cos\\left(\\alpha\\right)=R_{0}\\sin\\left(l\\right) \\\\\n& R\\sin\\left(\\alpha\\right)=R_{0}\\cos\\left(l\\right)-d \\\\\n\\end{align}\n", "1b7733fea96aba283e1b6cd6affba3d0": " \n\\lim_{x \\to \\infty} u'(x) = 1.004.\n", "1b7735a9e93908622c9d66fcdb533079": "\n \\Delta m=111,132 -559\\cos 2\\varphi.\n", "1b774e01dcddf51f1f0a9c15f8dc5162": "t_1'<0", "1b776dc3cdd65c6b8d92d255476372de": "V(I) = \\{P \\in \\operatorname{Spec}\\,(A) \\mid I \\subseteq P\\}", "1b77c7997a606e649b4eb4fcb9e89a20": " \\sigma_{12} = \\sigma_{21}", "1b7816047af5232d8bc125c49d4c9883": "e^{i t \\hat{M}_E} = \\lim_{n \\rightarrow \\infty} [e^{i t \\hat{M}_E / n}]^n = \\lim_{n \\rightarrow \\infty} [1 + i t \\hat{M}_E / n]^n.", "1b782558c9fe4f9fbdf04185745495e1": "r_{\\rm max}", "1b7857e0d4b5d61a948fb961d284fff2": " {}_2H_2(a,b;c,d;1)= \\sum_{-\\infty}^\\infty\\frac{(a)_n(b)_n}{(c)_n(d)_n}= \\frac{\\Gamma(d)\\Gamma(c)\\Gamma(1-a)\\Gamma(1-b)\\Gamma(c+d-a-b-1)}{\\Gamma(c-a)\\Gamma(c-b)\\Gamma(d-a)\\Gamma(d-b)} ", "1b78730de93934d43eca8453af483816": "\nM = m_1 + m_2.\\,\n", "1b78a2ba2cf8935925fc806e008eb57b": "E_{1,\\text{thr}} = \\frac{(m_a c^2+ m_b c^2 + m_c c^2)^2 - m_1^2 c^4 - m_2^2 c^4}{2 m_2 c^2}", "1b78c0858e819a50f597dc98350f8034": "\\dot{\\,} \\!\\,", "1b7906ac4c24db91d350e6fb03bdec98": "\\left(0,\\ \\pm1,\\ 0,\\ 0,\\ 0\\right)", "1b79461b3b244d0166e065bd8dfc71a2": "T : K \\to L", "1b79523a0b0bef2ec987bd9bfb9435f6": "\\int Z_n^m(\\rho,\\varphi)Z_{n'}^{m'}(\\rho,\\varphi)d^2r\n=\\frac{\\epsilon_m\\pi}{2n+2}\\delta_{n,n'}\\delta_{m,m'}", "1b79539cc938bf4486effbbd51d2b564": "n= P Q", "1b795a7e3d4498259bd0eaaa4d690fc9": "g \\left(\\sum_{b \\in B} l_b b\\right)\n:= \\sum_{b \\in B}f \\left(l_b^{\\sigma^{-1}} b\\right)\n= \\sum_{b \\in B} l_b f (b)\n", "1b7a15f2c94a6171f41f9e3318261f5a": "\\scriptstyle K_a>K_b>0,", "1b7a1fe49217687bd95fd5b4153543fe": "a \\uparrow b = a^b", "1b7a3f3b4150a0c946e3ebc7311300c6": "\\Omega_{n} R^{n-1} \\sinh^{n-1} \\frac{r}{R} \\,", "1b7a983bd2b06d95dac83700eb5bab7b": "D_{R\\delta}^{2}(\\mathbf{X},\\mathbf{0}) \\left( \\left(\\mathbf{X,0}\\right): \\Omega \\to \\mathbb{R}^2 \\right)", "1b7ab232531f34313df95a803e1e48e3": "\\exp_2^{i-2}(n^{O(1)}))^{\\Sigma_j^{\\rm P}}", "1b7acce0697412647bb92c42729c5c62": " g = G \\frac{m}{r^2} ", "1b7ad3c0ff1ba15b03f7ff013ed02510": "S=\\sum_k \\sum_j r_k W_{kj} r_j\\,", "1b7afc034cd1fa6c19102ed668dd9365": "\\pi(\\tilde{\\gamma}(t)) = \\gamma(t).", "1b7b1eff5898e3b799c58f0c34df4309": "\\eta_{Optics}", "1b7b9f64be4dc475dfb3f269abb52d38": "(m_{i,j})_{i,j=1}^k", "1b7bd384811acae63af2e05877b3655f": "\\sigma_{\\rm e}", "1b7c05daef90262f2ca226e6a3cd8927": " max_{i,j} p_{i,j} \\leq T ", "1b7c22e4097190e5b451fe489abb0806": "G_p^{(n)}=\\left \\langle 0 |\\mathcal T\\mathopen{:}v_i(y_1)\\mathclose{:}\\dots\\mathopen{:}v_i(y_n)\\mathclose{:}\\phi_i(x_1)\\cdots \\phi_i(x_p)|0\\right \\rangle", "1b7c339be6ab6ae47ca404b83f8eb1dc": "\\eta(x,t) = \\eta_2 + H\\, \\operatorname{cn}^2 \\left( \\begin{array}{c|c} \\displaystyle \\frac{x-c\\,t}{\\Delta} & m \\end{array} \\right),", "1b7c82c51c132dc31a920a4b44acdc8e": "\\frac{\\text{supports}}{\\text{supports} + \\text{opposes}} \\times 100", "1b7cede1bebd106339192982aa1c13b6": "S(\\mathbf{q})", "1b7d097cfaccb7213fd2574f25f593d8": "RF = (2W + X - Y)\\sqrt{8}", "1b7d27c3d2842da2b7d340f681586262": " (J^n f) ( x ) = { 1 \\over (n-1) ! } \\int_0^x (x-t)^{n-1} f(t) \\; dt,", "1b7d724abed9413f1199be7002f07a70": " \\displaystyle \\gamma(z)=\\gamma_-(z)^{-1} \\gamma_+(z),", "1b7d955c8da7f8acd3bcf74a0b8c7850": "g_v=0", "1b7e1e089f8753271b38f59863c395e2": "\\tilde r^2 = \\pm 1", "1b7edb89e3ef2335e8520393ad42259b": "m\\!\\left(a\\right)", "1b7f768767c24b69e9847bd5a21605c3": " \\text{h} = E_1-y_2 ", "1b7f7db9071ffe6cd26422c9cc12efa3": "= 2\\eta^{\\mu\\nu}\\gamma^\\rho \\gamma^\\sigma \\gamma_\\mu - 4 \\gamma^\\nu \\eta^{\\rho \\sigma} \\, \\quad", "1b7f8cd1802f3169842aa431387f8375": "d_{xy} = N_2^c \\frac{xy}{r^2} = -\\frac{i}{\\sqrt{2}} \\left(Y_2^2 - Y_2^{-2}\\right)", "1b7fcaa2e80f3cc459ba13babb1338cb": "\\frac{1}{10}", "1b80171f969d96935530a2b127c14332": "p:G_0\\to M", "1b80823027034aa6a1ca8f974c00863f": " = 10^{-23} \\frac{\\mathrm{erg}}{ \\mathrm{s} \\cdot \\mathrm{cm^2} \\cdot \\mathrm{Hz} }", "1b809a07021b54600a36b5a3e4fa1850": "\\displaystyle L = \\mu_0N^2A/l.", "1b80bc8d1448b3cbae3fcf9ef043613d": " (Th)(w)=\\lim_{\\varepsilon\\rightarrow 0} -{1\\over \\pi}\\iint_{|z-w|\\ge \\varepsilon} {h(z)\\over (z-w)^2} \\,dx dy.", "1b80ccb2d4c422d730f4aa47fd783fd5": "B^2-n", "1b80f9b12175158ad3ded9ac0026534a": "N(T)=\\frac{1}{\\pi}\\mathop{\\mathrm{Arg}}(\\xi(s)) = \\frac{1}{\\pi}\\mathop{\\mathrm{Arg}}(\\Gamma(\\tfrac{s}{2})\\pi^{-\\frac{s}{2}}\\zeta(s)s(s-1)/2)", "1b8118e5a29ee40996e1365a66bdb848": "2^w", "1b81437c098fa9628dba16678b880e9e": "g_p < p^{0.499}", "1b815c283d23de967a167378ff2bc6b0": " \\chi ", "1b81a7c0ab6114599c916e9192042672": "\\sum_{p\\leq x}\\frac{|a_p|^2}{p}=n_F\\log\\log x+O(1)", "1b81cca3a59133c4a80eff82a19b074b": "f(x,y)=x^3+Ax^2+x-By^2", "1b81fedbf639bbfce9c2c4d4166a39c1": "m_1 a = m_1 g-T", "1b83391d208cd1e588d2e0ea0d85a212": " \\mathbf{F_1}=q\\left(\\mathbf{E_1}+\\frac{d\\mathbf{x_1}}{dt}\\times\\mathbf{B}\\right). ", "1b83ac77f7ddc4820348a588a87dd9d6": "\\nu = -\\frac{2\\,b_2\\,a + b_1}{2\\,b_2^2\\,\\alpha} \\!", "1b83b6538829913d7e799f40526e541e": "[1, n-1]", "1b8431d9eecbe0f73863a9f647083395": "-\\cos(2U)", "1b8447036c94d44798208764b58b6dcd": "\\text{PAM}_n(i,j) = log \\frac{f(j)M_{n}(i,j)}{f(i)f(j)} = log \\frac{f(j)M^n(i,j)}{f(i)f(j)} = log \\frac{M^n(i,j)}{f(i)}", "1b8495feebe7bdbfe792dc5e6cf1639e": "wt(y)", "1b84d30b8e2115baaf9faad9dc32d79d": "v^2 = Q(v)1\\ \\text{ for all } v\\in V,", "1b84d4db49f0b129d6e98779b0a4fefe": "((R^p f_* \\mathcal{F})_s)^\\wedge \\simeq \\varprojlim H^p(f^{-1}(s), \\mathcal{F}\\otimes_{\\mathcal{O}_S} (\\mathcal{O}_s/\\mathfrak{m}_s^k))", "1b856c211d194ccc5d73c4d16f282980": "MRTS", "1b85ba3c31b65dbd5bafd28d2f552904": " \\alpha > d.\\, ", "1b85f74371eca8f93aca0edac979c194": "\nY_{(j-\\frac{1}{2},\\frac{1}{2})jm}=\\left(\\begin{array}{c}\n\\sqrt{\\frac{j+m}{2j}}Y_{j-\\frac{1}{2},m-\\frac{1}{2}}\\\\\n\\sqrt{\\frac{j-m}{2j}}Y_{j-\\frac{1}{2},m+\\frac{1}{2}}\\end{array}\\right)\n", "1b864433866ef59c2742e830e37ecab9": "\\det(M)", "1b87236b320458c9f43f9264176074b9": "\\textstyle\\frac {2}{2-1}=4", "1b8729c7d13c458fea49216876f07d87": "|\\mathbf{C}|^{p+1}", "1b873842c06a9e4e8f9966b5de2ebd7e": "g(x) = 1", "1b8748df044433752e67451aaf47cc8e": " \\xi = \\sqrt{\\frac{\\hbar^2}{2 m |\\alpha|}}", "1b874b93eafc45ae8405387284583a2b": "x \\times y = \\frac{1}{2}(xy - yx).", "1b87548baf8b937bf17fe8b0028fbedd": "(8)\\qquad Q_1 = C\\;A_2\\;\\sqrt{2\\;\\frac{Z\\;R\\;T_1}{M}\\bigg(\\frac{k}{k-1}\\bigg)\\bigg[(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}\\bigg]}", "1b875b2c0ff5e0e4d60b42ad73afc5d7": "\nJ(\\mathbf{w}) = \\frac{\\mathbf{w}^{\\text{T}}\\mathbf{S}_B\\mathbf{w}}{\\mathbf{w}^{\\text{T}}\\mathbf{S}_W\\mathbf{w}},\n", "1b87d4b994b3b7efd92b9978a7b91fe6": " \\operatorname{Var}(y_{1}) = \\sigma^2 ", "1b8862873c3ba5af1cae7bf97ed24da4": " N(s)= \\frac{1}{\\pi}\\operatorname{Arg}\\xi(1/2+i\\sqrt s)", "1b886b42c6e32bf155d8bd311e9deb9d": " \\mathbb{F}{q^n} \\to \\mathbb{F}_{q^n} ", "1b88cb287e992918a7fe73097849d157": "T_{env}", "1b88f86f59d82a9fb80a5d934d507647": " \\pi(x)={\\rm Li} (x) + O \\left(x \\mathrm{e}^{-a\\sqrt{\\ln x}}\\right) \\quad\\text{as } x \\to \\infty", "1b899005d745ac8a3be24520dc235ee9": "\\begin{array}{rcl}\n y_1'&=&y_2\\\\\n y_2'&=&y_3\\\\\n &\\vdots&\\\\\n y_{n-1}'&=&y_n\\\\\n y_n'&=&F(x,y_1,\\cdots,y_n).\n\\end{array}\n", "1b8991d3a0692e7c568ab956587c289c": "X(\\bold{a})", "1b89aa48b2fe8b2316d378e3a6856897": "f(z)=F(z)\\overline{F(\\overline{z})}", "1b89e3633cfbb7f178ae571f5112b033": "\n \\kappa_n = 2^{n-1}(n-1)!\\,k\n ", "1b89ef0da3fd9a5e3e638c79d2cd8d42": "\\displaystyle r \\approx r_0\\!\\left( \\frac \\ell L C,\\!~K\\sin(\\theta)\\right)", "1b8a33c2ae28ec4042bf8fc01661b131": "G(\\zeta )=\\frac{ \\tfrac{e^{\\zeta}}{4}}{3+\\zeta +\\frac{\\tfrac{e^{\\zeta}}{8}}{3+\\zeta +\\frac{\\tfrac{e^{\\zeta}}{12}}{3+\\zeta +\\ldots}}}", "1b8a9ed8b39f57e8029cf33e1e263147": "\\forall p: \\forall q: \\mathcal{B}(p \\to q) \\to (\\mathcal{B}p \\to \\mathcal{B}q )", "1b8ab6e8102bc2b91763316423e516f1": "\\frac{\\delta L}{\\delta \\psi} = 0", "1b8b1e87d8a0a0190df5f3456d9c3d8d": " - \\sum_{i \\in I} p_i \\ln q_i \\geq - \\sum_{i \\in I} p_i \\ln p_i ", "1b8b252ef3b59b90eda3acccba6157a2": "\n p = \\cfrac{\\mu J_m}{\\lambda(J_m - I_1 + 3)}~.\n ", "1b8b63fdd7dfe788966556a03e6ae788": "U = \\textstyle{\\frac{1}{2}}m\\omega^2 \\langle x^2\\rangle = e^2 E^2/ 4 m \\omega^2 ", "1b8b807e27f114c5da16d417fa95c38a": "\\omega_n=\\frac{2n\\pi}{\\beta}", "1b8bbc1b1ea773af28b9d12b017b8156": "\\sqrt{x^2+c} = -\\frac{t^2-c}{2t}+t = \\frac{t^2+c}{2t}", "1b8bcebdef594b8fade4901e48532218": "\n\\vec{\\nabla}\\cdot\\left[\\epsilon(\\vec{r})\\vec{\\nabla}\\Psi(\\vec{r})\\right] = -\\rho^{f}(\\vec{r}) - \\sum_{i}c_{i}^{\\infty}z_{i} q \\lambda(\\vec{r}) \\exp \\left[{\\frac{-z_{i}q\\Psi(\\vec{r})}{k_B T}}\\right]\n", "1b8bf69b0ecb33dd72ce92a6d1350d51": "2^p", "1b8c0e669f5722f2adef6a9c76a0eaa7": "\n\\tilde{f}(d) = \\left\\{ \\frac{1}{2} \\cdot \\left[ 1 - N \\left( z < \\frac{w \\cdot shared(d)}{\\sqrt{\\theta}} \\right) + N \\left( z < \\frac{- w \\cdot shared(d)}{\\sqrt{\\theta}} \\right) \\right] - \\frac{d}{n} \\right\\}^2\n", "1b8c0fd3c14439e8dc4dddc9c194cb87": "A(x)=\\sum_{n=0}^{\\infty}A_{n}x^{n}", "1b8cd9c110785bf8c22c2e6d4e97d44b": "\\not\\sim", "1b8cfb60d7230535d61d09197f9cf2ab": "\nx\\langle y\\rangle \\cdot P \\; \\vert \\; x(v)\\cdot Q \\longrightarrow P \\; \\vert \\; Q[^y\\!/\\!_v]\n", "1b8d01e89556aa40e3d4989cf8433d90": "\\scriptstyle\\vec x(t)", "1b8d1ca08cb3d0016a33712a5046de57": "\\text{Posterior Probability}(p=x|s,f) = \\frac{x^{s-1}(1-x)^{n-s-1}}{\\Beta(s,n-s)}, \\text{ with mean = }\\frac{s}{n},\\text{ (and mode= }\\frac{s-1}{n-2}\\text{ if } 1 < s < n -1).", "1b8d28173dfc241069b21ddb50c0386f": "\\ddot{x}=g(x)\\cos(\\omega t),", "1b8d35799e359fbd2a762b876773d9cd": "(I\\ nat\\ 3\\ 3) \\to \\bot", "1b8d83dc9f69eec424bb75bda4908ce0": "\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-y| f(x, y) \\, dx\\, dy.", "1b8d8f1392233ef6fe64dfaa0377419e": "n=2^r-1", "1b8db565adf47cb0b40cb813ba8a778a": "H_{\\mathbf{k}}'", "1b8dd6b4295d4358ac7a606667ca9bd7": "X(t) = \\sum^{\\infty}_{i=1} X_i \\Phi_i(t)", "1b8e0fe76f44b5fd447de10b892d5424": " D \\,", "1b8e42e0596fa09a11e98d249c895819": "\\begin{align}\nx &= \\frac{\\lambda\\mu\\nu}{ab}\\\\\ny &= \\frac{\\lambda}{a}\\sqrt{\\frac{(\\mu^2-a^2)(\\nu^2-a^2)}{a^2-b^2}} \\\\\nz &= \\frac{\\lambda}{b}\\sqrt{\\frac{(\\mu^2-b^2)(\\nu^2-b^2)}{a^2-b^2}}\n\\end{align}", "1b8e4ca854088478a8ef2ac8e9b11598": " Z = \\sqrt{ \\left ( \\frac{\\rho_q\\mu_q}{\\rho_f\\mu_f}\\ \\right ) } ", "1b8e6bb9f0ed4ac13c05c9aaaebed760": " F_{net} = \\sum \\, F_{i} = ma ", "1b8e72c7bb5900ba5b0c97855b08e73c": "f(x) = |x|,\\qquad x\\in[-1,1].", "1b8ef0c843318f7dfd122cf88ea440e1": "\\Gamma={B_{\\text{ADS}}\\Gamma_{\\text{max}}\\over (1+B_{\\text{ADSc}})}", "1b8f15a8ff9c5f2abeb637fd49221e01": "\\nu:E \\times\\mathcal F \\rightarrow [0,1],", "1b8f38e9dcc0c262690ff2b79e00c27e": " B=U[\\mu_1, \\mu_2, ... \\mu_n] +P_RV -T_RS=U[\\boldsymbol{\\mu}] +P_RV -T_RS \\qquad \\mbox{(3)} ", "1b8f5d37b59adea244f8fe63d135ebb9": "ax + by = 0, a^2 + b^2 = c^2 (a,b,c \\in \\Q)", "1b8f5e65b073c8497b5d85421ef76f25": "L^x(t)", "1b8f85b2d92e3da91020a26b29427016": "i\\ll J", "1b904339c6b6fdd98c55274c179e49b8": "\\bar{u}_j", "1b905fd92bb184440e5dbd0ea2016490": "4 \\cdot 2^n", "1b908e1d31cb8541c53c718f46cd1258": "a \\in FA", "1b911a296e11ed74633d035379e8e971": "Loans = \\left(1 - \\alpha - \\beta\\right) \\cdot Deposits = \\frac{1 - \\alpha - \\beta}{\\alpha + \\beta + \\gamma}", "1b9146185c2484ec7201403d725168e8": "F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t", "1b915c8a5afbd5b8322c82075457b2d9": "C=\\frac{N_{A}}{3k_{B}}\\mu_{\\mathrm{eff}}^{2}\\text{ where }\\mu_{\\mathrm{eff}} = g_{J}\\mu_{B}\\sqrt{J(J+1)}", "1b91a6b54fd4d4c6308a2f271917b258": "\\nabla\\cdot \\mathbf{v} = 0", "1b91bfd7081a77f5f6a5b897f18dca36": " x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}.", "1b91e7703f745d757b5271b9909e9a12": "G = H - TS", "1b9208a3097a682489d931e9f34e3c1f": "\\textstyle [0,\\infty) \\to \\mathbb{R}, ", "1b927da77a0234060d7d8ded00dfe95e": "x_{t+1}=x_t+\\gamma\\frac{df}{dx}(x_t)", "1b92a923714d2c7b5f0437ba234d6370": "\\mathbb{G}_m", "1b92ce9534b88fe56ca632ba1db49d01": "(n-q)", "1b92e83d8834c55877ab4fc458934fd3": "Q_\\mathrm{lost}", "1b930f09056de56ffe5c0d3c46472881": "f(x)=\\frac{e^{x}}{e^{x}+e^{x_0}}", "1b9377d5257d7a1825263e10f7fa219f": "|\\psi\\rangle\\in\\mathcal{\\tilde{H}}_{S}", "1b93791030d6c8e0fb96989693e8a426": "\\Delta V = 2 V \\sin \\frac{\\Delta i}{2}", "1b938798c34a0778e3660a01a83e23e3": "h_k(x) = \\begin{cases}\n\\frac{f(x) - P(x)}{(x-a)^k} & x\\not=a\\\\\n0&x=a\n\\end{cases}\n", "1b9392d5dbdee5d5a988fbfa96275b47": "\\mathbf{u}=u_1\\mathbf{i}+u_2\\mathbf{j}+u_3\\mathbf{k}", "1b93f28d59e5b3ff2e7712dfbb7ffdaf": "a'-a^2+\\big(2+\\frac{1}{x}\\big)+\\frac{4}{x^2}=0", "1b93f6652ce0eaed2dc2c14585332f2f": "f(x^i)=\\bar{f}(\\bar{x}^j(x^i))", "1b94849513ce2ce5b7b63f0bceb83b01": "H=\\left(\\sqrt{1+\\frac{2}{\\sqrt{5}}}\\right)a\\approx1.37638...a", "1b94987be9517d2b7c4af0c5a9efef17": " \\det(E^{c})=0 ", "1b94a239e09cea4b5ebd0be77f6f06df": " \\mathbf{r}(t) ", "1b950310500e45a509696f8bd9ca2b4e": "f\\colon [0,1] \\to \\mathbf{R}", "1b952078ffc2fe72f8e77d541e804b9f": "\\mathbb{E}[Y_i\\mid \\mathbf{X}_i] = p_i = \\operatorname{logit}^{-1}(\\boldsymbol\\beta \\cdot \\mathbf{X}_i) = \\frac{1}{1+e^{-\\boldsymbol\\beta \\cdot \\mathbf{X}_i}}", "1b95659ce107ea494f031b441b772b32": "\n g = \\det[\\boldsymbol{F}^{\\rm{T}}]\\cdot\\det[\\boldsymbol{F}] = J\\cdot J = J^2\n ", "1b95cf0401693d47d40d2f0a55314989": "h,h' \\in \\mathfrak{h}", "1b96026fb7a76dc571bc4c2fd32257d4": "\\,\\!\\Omega^n = \\omega", "1b967bac203aefb4eaab5d7c0629f6af": "\\frac{d^2\\varphi(x)}{{dx}^2} = \\frac{\\varphi(x{-}h)-2\\varphi(x)+\\varphi(x{+}h)}{h^2}\\,+\\,\\mathcal{O}(h^2)\\,.", "1b96a4967e023afa8087486c336a8645": " |A| = \\sum_{H \\in S} \\varepsilon^{H,H^\\prime}b_{H}c_{H^\\prime}, ", "1b96ec989d9b19140e8bb6adcb7bee41": "\\lim_{n\\to\\infty} \\mu(\\{x \\in F: |f(x)-f_n(x)|\\geq \\varepsilon\\}) = 0", "1b96fbab16a8b545ab0921950366b4d7": "HW=\\text{Headwind}", "1b972691ab16cbbd36a57909ae6af44c": "\\mathbf{F}_{ext} = -\\left(\\mathbf{F}_{gas} + \\mathbf{F}_{other}\\right).", "1b97ad6a5cd67ffdf95714d06f60ed88": " 4 \\pi r^2", "1b97bc41cd3f8991402e81e1db62b3fc": "(u_i, v_j) \\in E", "1b9835840ab3bf0094bf37cc41a02a0c": "x = \\frac{3 \\lambda}{2} \\sqrt{\\frac{1}{3} - \\left( \\frac{\\phi}{\\pi} \\right)^2}", "1b9858a8e501cecbe3fd9a5aee6f3525": "D_2\\left(E\\right) = \\frac {\\pi}{c_k}", "1b9860707fdeb617b0ba9e9cceffa161": " \\mathbf{\\tau} = \\mathbf{\\Omega} \\times \\mathbf{L},", "1b9886ef1423c46156223afeaeda9d10": "z_k\\epsilon_k", "1b98a6f15cbe7e0854c363460c7fecd4": "[P_i,P_j]=0", "1b98be9c5da5c066fc6c5cbd0abcc87c": "k\\sin x", "1b98eab71d7be02eb7a88cc457d2f746": " \\textstyle \\left \\langle {(\\Delta p)}^{2} \\right \\rangle = \\frac{1}{2}K^2n", "1b990f894c176d03669ad2cf4e2f4631": "\\left[\\sigma_a, \\sigma_b \\right] = 2i \\varepsilon_{abc} \\sigma_c \\,, \\quad \\left[\\sigma_a, \\sigma_b \\right]_{+} = 2\\delta_{ab}\\sigma_0", "1b991022245568904f5bcf1d6b7f5cc8": " \n \\nabla\\times\\mathbf{v} = \\frac{1}{h_1h_2h_3} \\sum_{i=1}^n \\mathbf{e}_i\n\\sum_{jk} \\varepsilon_{ijk} h_i \\frac{\\partial (h_k v_k)}{\\partial q^j}\n", "1b995d7895806d0c3819075919b4b27d": " H \\approx A ", "1b99b1954aeea157a00e5c1cada40330": "\\log Z_r", "1b99bc82d606ca025dba1296a231368e": "\\dim A = \\operatorname{tr.deg}_k A", "1b99ee20338e1d716fa65741f0043184": "\\alpha_i \\in \\R", "1b9a139fc73cee28bf64f6cab7dbeeeb": "\\bigcup_{\\xi<\\alpha}f(A_{\\xi})\\,", "1b9a3ef12d1e232f50c053998071e056": " P\\times_G W", "1b9a5dd6ce47cd491ebad9c52d026fec": "\\rho_{F\\pm}=\\frac{2\\rho_F}{1\\pm\\beta},", "1b9a933cb98516e2cc6da0df6ee84d36": "e^{-ix} = \\cos(-x) + i\\sin(-x) = \\cos(x) - i\\sin(x)", "1b9a99af5988f4e53c46820e67915d59": "c \\in C_n", "1b9b7c1c3ef059dbab853c65e57fa026": "\\omega_{A}=\\sqrt{\\frac{k_{z}^{2}B^{2}}{\\mu\\rho_{i}}}", "1b9ba2b5b7c3679d2e484c7c53b8a07d": "\\chi \\equiv H_2 \\left(\\frac{1 + \\sqrt{(1- 2 \\,\\eta\\,p)^2 +4 \\,\\eta\\, |\\gamma|^2}}{2} \\right)-\\sum_k \\xi_k H_2 \\left(\\frac{1 + \\sqrt{(1- 2 \\,\\eta\\,p_k)^2 +4 \\,\\eta\\, |\\gamma_k|^2}}{2} \\right)\\;", "1b9bbcd00bba940854247783357d5c8c": "\\sin 2A \\sin(B - C)x + \\sin 2B \\sin(C - A)y + \\sin 2C \\sin(A - B)z = 0.\\,", "1b9bd824ea866de6319d5938208846a4": "\\{\\mathbf{e}_{k}\\}", "1b9be32604ef1e203c67bb7f940aef4f": "k_i = f\\left(t_n + c_i h, y_n + h \\sum_{j = 1}^{s} a_{ij} k_j\\right).", "1b9c3a09779ab270a183224b3b074e00": "R_2^\\omega(n)", "1b9c5bcc3cad4112ad863c10b029a323": "\\frac{dA}{dt} = {i \\over \\hbar } [ H , A ] + \\frac{\\partial A}{\\partial t}", "1b9d5613fa478270cfa49b6e4c63331d": " \\lambda=u\\pm c ", "1b9d8e99ee03a49de6cd3a346d91634e": "\\beta = \\sqrt{n^2(a) k^2 -(l/a)^2}", "1b9db4b01a27be1df822c286d143ce51": "x_1, x_2, \\ldots, x_n", "1b9e46a0367a244a8adef650d5330dec": "\\sigma_v = \\sqrt{\\sigma_1^2- \\sigma_1\\sigma_2+ \\sigma_2^2}\\!", "1b9e94db68f62af00c677d6340e60333": " T = \\frac{V_{s}}{\\Omega} R_r ", "1b9eb70d341bbdcfb6d9c186128745db": "0=x+f'\\left(\\frac{dy}{dx}\\right).", "1b9ecb45305c08a2b4a33270c07b3046": "\\sin{\\frac{A}{2}}=\\sqrt{\\frac{bc}{ad+bc}}=\\cos{\\frac{C}{2}},", "1b9ed1d63bd083a93909e337efe89ca9": "f:\\R^n \\to \\R", "1b9f0e06a0bd45626e66b1daedf1777b": "T^{\\mu}_{\\mu} = \\operatorname{Tr} \\begin{pmatrix} \\rho_0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0& 0\\\\ 0 & 0 & 0 & 0 \\end{pmatrix} = \\rho_0", "1b9f346553bdd0a0c805bd8949d7aaad": "\\text{Protocol efficiency} = \\frac{\\text{Payload size}}{\\text{Frame size}}", "1b9f4a9c18b7ac57cd8ecacc73278e9d": "\\displaystyle A_i", "1b9f55b06a54bd9b4ee728fb369c5c9a": " \\int_{-h/2}^{h/2} (\\text{area of cross-section at height }y) \\, dy, ", "1ba07dc82bf4c67d3544e5a73184980c": "\\hat{k}", "1ba0e384b7fdbfe20efeb303dc56ac7f": "\\left[\\begin{array}{rrr|r}\n1 & 3 & 1 & 9 \\\\\n1 & 1 & -1 & 1 \\\\\n3 & 11 & 5 & 35\n\\end{array}\\right]\\to\n\\left[\\begin{array}{rrr|r}\n1 & 3 & 1 & 9 \\\\\n0 & -2 & -2 & -8 \\\\\n0 & 2 & 2 & 8\n\\end{array}\\right]\\to\n\\left[\\begin{array}{rrr|r}\n1 & 3 & 1 & 9 \\\\\n0 & -2 & -2 & -8 \\\\\n0 & 0 & 0 & 0\n\\end{array}\\right]\\to\n\\left[\\begin{array}{rrr|r}\n1 & 0 & -2 & -3 \\\\\n0 & 1 & 1 & 4 \\\\\n0 & 0 & 0 & 0\n\\end{array}\\right] ", "1ba0f6b293be009a1ad218973856f3ae": "\\, x_{1}", "1ba0f9382661b3f178b1dfcaedbf877e": "T_{\\mathbf{x'}}", "1ba1113462daa68451a64a5e601c0abd": "Al_2O_3", "1ba17060cc5b59fd6452356c2fd0d9c1": "S \\Rightarrow ABC \\Rightarrow aAbBcC \\Rightarrow aaAbbBccC \\Rightarrow aaabbbccc", "1ba1d7971175b953732933fe254b67b5": " \\max \\;Q_1 (\\alpha )\\; = - \\frac{1}{2}\\sum\\limits_{i,j = 1}^N {\\alpha _i \\alpha _j y_i y_j K(x_i ,x_j )} + \\sum\\limits_{i = 1}^N {\\alpha _i } ", "1ba1da13740e78ddeefe32a93d51f7d0": "ds^2=e^{-2f(\\phi)}dt^2-e^{2f(\\phi)}[dx^2+dy^2+dz^2]\\,", "1ba1e29e2bccf35fb644302af6d930f4": " B_\\mathrm{eff} ", "1ba21232f16b0c10ca7fc33c93f076e3": "\\pi(y_{0})\\in U\\, .", "1ba21947fd9ebb7bb9939b2905f6f696": "q = 1 - p", "1ba2511d68fb6ac2e3d6f9d02d90ed19": "I(t) := \\sin(\\alpha t)", "1ba2871ab9662f8a54c279bdfa29b2d8": "a_{t,j_t}", "1ba34223a90648a3acf3100aa0de7212": "\\qquad\\mathcal{O}_{[ab]}\\oplus \\mathcal{O}_{[b]}", "1ba36185b04c519fcb96e8d04422b573": "f(t)=\\begin{cases}\n\\frac{1}{N^{n-1}} & t=0 \\\\\n\\sum_{x=1}^{N-t}\\left(\n[\\frac{t+1}{N}]^n -2[\\frac{t}{N}]^n +[\\frac{t-1}{N}]^n \n\\right)& t=1,2,3\\ldots,N-1.\\\\\n\\end{cases}", "1ba363cffbc6cb2f4a1e62bc4dac4643": "A_6.", "1ba38ff694c5a152457b9c7f4e45f4cb": "\\lambda = \\frac{AFR}{AFR_{stoich}}", "1ba3af6f4efb002c980e77db30bda5a9": " \\operatorname{de-let}[\\lambda f.\\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x)] ", "1ba3b620f495364c23d6ed4200395931": "n\\ge 0,", "1ba403614259aef1d5b5ee389181a114": "\\tfrac{(0\\cdot\\varepsilon)}{n}", "1ba406d219409513f15af7fb0f530b91": " f^*=f, \\,", "1ba40d0692844bc75deab6532e1fd9b4": " \\Phi_{12}(r, \\theta_1, \\theta_2, \\phi) = 4 \\epsilon \\left[ \\left(\\frac{\\sigma}{r} \\right)^{12}- \\left( \\frac{\\sigma}{r}\\right)^6 \\right] - \\frac{\\mu_1 \\mu_2}{4\\pi \\epsilon_0 r^3} \\left(2 \\cos \\theta_1 \\cos \\theta_2 - \\sin \\theta_1 \\sin \\theta_2 \\cos \\phi\\right) ", "1ba457e50e005b9bdc27898eb9af4318": "\nk(\\mathbf{x},\\mathbf{x}') = \\sum_{i=1}^p \\Phi^i(\\mathbf{x})\\Phi^i(\\mathbf{x}').\n", "1ba45cdbd40481a21931fd3784d80ff9": "u_{i,j} = \\partial u_i /\\partial x_j", "1ba47607aa24a95d008acf806cef3646": "(p_r,p_s)=0", "1ba4c6b10b12d22056579df0e36e8287": "\n(X\\odot Y)\\odot Z\\cong X\\odot (Y\\odot Z),\n", "1ba4f06f68614e5da79a8ebd378d532a": "\\wedge ", "1ba5097697d6035c55ac3e11bc2e1515": "\\begin{bmatrix}\n1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & 1\\\\\n0 & 0 & 0\\end{bmatrix}", "1ba53bd1cb3333c5d204cf3d710cfab5": "\n\\exp\\left( -z + vz + \\frac{1}{u} \\log\\frac{1}{1-uz} \\right) =\n\\exp(-z + vz) \\left(\\frac{1}{1-uz} \\right)^{1/u}.", "1ba564dd72d531dd49b80b21ad320c08": "\\hat{\\Sigma}", "1ba5770634793382494a9aa11d750a43": "\\begin{matrix}x = 0 \\\\ y+z=0\\end{matrix}\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;\\begin{matrix} x=0 \\\\ x+y+z=0.\\end{matrix}", "1ba5ffadbb8590b67dfb9890104f73c9": "\\omega = e_1e_2\\cdots e_n.", "1ba62075ce4788a2352196648b7a0e97": " \\gamma(a) ", "1ba628f0e49ce4d74925f7d99f7504b9": "\\frac {dm} {dt}", "1ba6383f3e9b57430fc008cacf06cce2": "\\scriptstyle \\{(x,y,z,p)=(0,y,U(y),U^\\prime(y));y\\in[a,b]\\}", "1ba679c01367d1a68d16344ac8f85695": "f^{\\mathrm{iv}}\\,\\!", "1ba68e6a28b34be566fdae2231cc4a3e": "M(a,b,z) = e^z\\,M(b-a,b,-z)", "1ba6b6dae533628eceb8dcef93b53d2c": "\\left.\\nabla_{\\partial/\\partial t}\\frac{\\partial}{\\partial x}\\right|_{x=0} = 0.", "1ba6ba9d91fa03b3f2b437047c7cf315": "\\therefore \\ln(L) = \\ln\\left(\\prod_{i=1}^n\\left(\\frac{1}{2\\pi\\sigma_i^2}\\right)^{1/2}\\right) - \\frac{1}{2}\\sum_{i=1}^n \\frac{(y_i-f(x_i))^2}{\\sigma_i^2}", "1ba6bfcae3f0bc6489131695534fea67": "\\frac{a^2b^2}{p^2}+r^2=a^2+b^2", "1ba6e329983b29a9e91bc8b1b9c27ede": " \\left(\\mathbf{\\hat{p}}-\\frac{e}{c}\\mathbf{\\hat{A}}\\right)^2=\\hat{p}^2 -2\\frac{e}{c}\\mathbf{\\hat{A}}\\cdot\\mathbf{\\hat{p}}+\\left(\\frac{e}{c}\\right)^2\\hat{A}^2 ", "1ba6f9e12298898fb7d2531087d83408": "\\tau (\\omega) \\equiv T", "1ba715e46e150f07cd08c78eb06f77cc": "\\begin{matrix}\n \\frac{\\nu+1}{2}\\left[\n \\psi \\left(\\frac{1+\\nu}{2} \\right)\n - \\psi \\left(\\frac{\\nu}{2} \\right)\n \\right] \\\\[0.5em]\n+ \\log{\\left[\\sqrt{\\nu}B \\left(\\frac{\\nu}{2},\\frac{1}{2} \\right)\\right]}\n\\end{matrix}", "1ba78014551b13bed4dcc9e59dd720f9": " e^{s_2}=e^{s_3}{\\frac{m_1 e^{s_3}+m_2 e^{s_4}} {m_1 e^{s_4}+m_2 e^{s_3}}} ", "1ba78f8c7fac4ce56e072aace4faa9fa": "\\int_{[a, b]} f(x)\\,\\mathrm{d}x = \\int_{[a, b]} \\mathrm{d}F = \\int_{\\{a\\}^- \\cup \\{b\\}^+} F = F(b) - F(a).", "1ba7964c15baa9964ee6c980affd3b8a": "2d \\sin \\theta = n \\lambda", "1ba7af6c1c6a47e4c29a4a0790521355": "p^2+1", "1ba7b348cbb77a02c4a4a38a9df0272d": " E_0 = 1,\\quad E_1=1,\\quad E_2=1,\\quad E_3=2,\\quad E_4=5,\\quad E_5=16,\\quad E_6=61,\\quad E_7=272,\\quad \\ldots ", "1ba83a4be9f0b918ee3677be3b1af4b7": "\\int\\frac{\\tan^n ax\\;\\mathrm{d}x}{\\sin^2 ax} = \\frac{1}{a(n-1)}\\tan^{n-1} (ax) +C\\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "1ba8aaab47179b3d3e24b0ccea9f4e30": "x_i", "1ba8b61ad77bd286ec76ea61a78dd372": "v=5\\cdot10^5m/s", "1ba8f20856f67be37e95c776f5e5c872": "\\text{10KC8}", "1ba984497123631be412734f873e665a": "\\operatorname{PSU}_4(2) \\cong \\operatorname{PSp}_4(3),", "1ba9a7cb82e2a8ce69bcf7d135fc5515": "R(z)", "1ba9f0eb891988dae7b4e9eb00bc6b44": "\\langle x,y\\rangle=\\langle x,z\\rangle", "1baa226737e37ee331765dc93c684993": "|\\mathbf{y}-\\rangle", "1baa749130e39c3991dc549ccfc58a80": "\\forall i \\in \\{1,\\ldots,n\\} \\, \\forall j \\in \\{1,\\ldots,n\\}", "1bac4a43c31721780a8d1fde3cb5f923": "i_2(t)\\,", "1bac6a8b865585d3379d347600b797c0": "\\operatorname{EQ} = \\lambda m.\\lambda n.\\operatorname{and}\\ (\\operatorname{LEQ}\\ m\\ n)\\ (\\operatorname{LEQ}\\ n\\ m) ", "1bac7f88668f5508b1304365414a754d": "\\exp(x_i y_j),\\quad (1\\leq i\\leq d, 1\\leq j\\leq l).", "1bacdbb52b14610a6a59ed1fd24019d3": "\n\\cfrac{dN}{dt}=N[r(1-cN)+\\beta M(X+M)]\n", "1bacf1d43bda13ccf528eb8787ee84cd": "z_1+z_2", "1bacf5fb34078b851402046ff077bf88": " \\frown \\colon H_p(X) \\times H^q(X) \\to H_{p-q}(X)", "1bad3fe355eb46c475723240abbc1970": "\\textstyle 0.5 \\frac{\\mbox{dB}}{\\mbox{cm depth}\\cdot\\mbox{MHz}}", "1bad9637f527012609a63d4ca5af92d9": "\\text{Smooth}", "1bada7c3126594146b9ca3da0ff8134d": "\n\\left( Y_{\\ell}^m \\right) ^* = (-1)^m Y_{\\ell}^{-m}.\n", "1badbc31b868244348c80332394ea850": "k_f = \\frac{k_a}{k_{av}}", "1badf4aff1db27ed25e9d37ec9b2dbdd": "\\forall x G(x,y_1\\dots, y_n)", "1bae63129195524ba215be7be4b190ee": "\\lambda_s(3)=3s-2+(s\\, \\bmod \\, 2)", "1bae6cbda605893e4da60577493e40ae": "1.3 \\times 10^5", "1baeb5deea846fce669cb1166c330847": "\n \\begin{align}\n M_{xx} & = -D\\left(\\frac{\\partial^2 w}{\\partial x^2}+\\nu\\,\\frac{\\partial^2 w}{\\partial y^2}\\right) \\\\\n & = \\frac{2M_0(1-\\nu)}{\\pi}\\sum_{m=1}^\\infty\\frac{1}{(2m-1)\\cosh\\alpha_m}\\,\n \\sin\\frac{(2m-1)\\pi x}{a}\n \\left[\n -\\frac{(2m-1)\\pi y}{a}\\sinh\\frac{(2m-1)\\pi y}{a} + \\right. \\\\\n & \\qquad \\qquad \\qquad \\qquad\n \\left. \\left\\{\\frac{2\\nu}{1-\\nu} + \\alpha_m\\tanh\\alpha_m\\right\\}\\cosh\\frac{(2m-1)\\pi y}{a}\n \\right] \\\\\n M_{xy} & = (1-\\nu)D\\frac{\\partial^2 w}{\\partial x \\partial y} \\\\\n & = -\\frac{2M_0(1-\\nu)}{\\pi}\\sum_{m=1}^\\infty\\frac{1}{(2m-1)\n \\cosh\\alpha_m}\\,\\cos\\frac{(2m-1)\\pi x}{a}\n \\left[\\frac{(2m-1)\\pi y}{a}\\cosh\\frac{(2m-1)\\pi y}{a} + \\right. \\\\\n & \\qquad \\qquad \\qquad \\qquad\n \\left. (1-\\alpha_m\\tanh\\alpha_m)\\sinh\\frac{(2m-1)\\pi y}{a}\\right] \\\\\n Q_{zx} & = \\frac{\\partial M_{xx}}{\\partial x}-\\frac{\\partial M_{xy}}{\\partial y} \\\\\n & = \\frac{4M_0}{a}\\sum_{m=1}^\\infty \\frac{1}{\\cosh\\alpha_m}\\,\n \\cos\\frac{(2m-1)\\pi x}{a}\\cosh\\frac{(2m-1)\\pi y}{a}\\,.\n \\end{align}\n", "1baef01203b0e8b736179554de75b6d3": "{\\bar{N}}_3", "1baf6299fb605d5e463f3ba5eedfeb78": "{\\mathbf{}}L(t)", "1baffc5f025c220673370cdf90c0f74b": "\\lceil y/x\\rceil", "1bb00d641a613d384ee177446a060a66": "\n \\cfrac{1}{1+R} (|\\sigma_1|^n + |\\sigma_2|^n) + \\cfrac{R}{1+R} |\\sigma_1-\\sigma_2|^n = \\sigma_y^n \n", "1bb023d595cd42d62a732587e0e9d886": "W(\\mathbf{x},\\mathbf{p};t)=\\frac{1}{(\\pi \\hbar)^3} \\exp{\\left( -\\alpha^2 r^2 - \\frac{p^2}{\\alpha^2 \\hbar^2}\\left(1+\\left(\\frac{t}{\\tau}\\right)^2\\right) + \\frac{2t}{\\hbar \\tau} \\mathbf{x} \\cdot \\mathbf{p}\\right)} ~,", "1bb08c87bcfcea46da21078c228f066f": "{ i \\dfrac{ \\partial f }{ \\partial x } } = \\dfrac{ \\partial f }{ \\partial y } . ", "1bb0dc8712c13daa558af12ec3308ee9": "\n \\begin{align}\n F_2(\\sigma_1 + \\sigma_2) & + F_3\\sigma_3 + F_{22}(\\sigma_1^2 + \\sigma_2^2) + F_{33}\\sigma_3^2 + F_{44}(\\sigma_4^2 + \\sigma_{5}^2) + F_{66}\\sigma_6^2 \\\\\n & \\qquad + 2F_{12}\\sigma_1\\sigma_2 + 2F_{23}(\\sigma_1+\\sigma_2)\\sigma_3 \\le 1\n \\end{align}\n ", "1bb116fb2452850c14d6145d6f2e424a": " P = \\begin{bmatrix} 0 & 0 \\\\ \\alpha & 1 \\end{bmatrix}. ", "1bb1244b2f0943b57515f2c22710f95a": "\\delta(x) = \\Delta_x^{\\frac{n+k}{2}} \\int_{S^{n-1}} g(x\\cdot\\xi)\\,d\\omega_\\xi.", "1bb1d2268c431b8b44cccf99fc992942": "C_g(\\theta)", "1bb237de72baed692b596aaf4f346005": "x\\to\\lambda_i,", "1bb26bd5b612fc5a841fbf5bec114c57": "\\quad (10) \\qquad \\qquad \nv_{i} {{d {\\mathbf {\\bar u} }_{i} } \\over {dt}} + \\oint _{S_{i} } \n {\\mathbf f} \\left( {\\mathbf u } \\right) \\cdot {\\mathbf n }\\ dS = {\\mathbf 0}, ", "1bb273cc23051b7609a8202728bbc779": "C = \\frac{1}{k_B}N \\mu^2", "1bb29b3cd62566ae7d6e857794068fa3": "x=As", "1bb347afdaadf7e80675bbb19286684f": "\n\\begin{bmatrix} 0 & -a & ab - c \\\\ 0 & 0 & -b \\\\ 0 & 0 & 0 \\end{bmatrix}\n\\lrarr\n\\begin{bmatrix} 1 & 2a & 2c \\\\ 0 & 1 & 2b \\\\ 0 & 0 & 1 \\end{bmatrix} .\n", "1bb37a2ddf528c1a19d49ef589f913a4": "\\{ 1, i_1, \\dots, i_7 \\}", "1bb39197af94ea26a5d1f14f6c349a65": "\\mathbb{C}\\setminus\\{0\\}\\ni z\\mapsto \\exp(-1/z)\\in\\mathbb{C},", "1bb3b1843ed8dd44a944e43e754d1a8c": "n_1 \\mathbf{a}_1 + n_2 \\mathbf{a}_2 + n_3 \\mathbf{a}_3", "1bb3de5baed8cda6f806d31c1eff6cb5": "T=-\\frac{1}{r}\\ln\\left(1-\\frac{P_0 r}{M_a}\\right)", "1bb432efe235efc88f4f1b524f08b649": "f=\\sum_{k=1}^\\infty c_k \\psi_k,\\text{ with }\\sum_{k=1}^\\infty c_k^2 < \\infty,", "1bb4357e8dd258c5b718ca0e5b121089": "\\langle T_n f , g \\rangle = \\langle f , T_n g \\rangle", "1bb4e1b4e6f7e4fb7673999b711e6a2e": " u = \\arccos { {\\mathbf{n} \\cdot \\mathbf{r}} \\over { \\mathbf{\\left |n \\right |} \\mathbf{\\left |r \\right |} }}", "1bb5089ed53a341d94b965fe9e7807b4": "\\scriptstyle \\{0,1\\}", "1bb52017967e125651e28bc1c640f3f1": "\nD_N^*(x_1,\\ldots,x_N)\\geq C\\frac{\\log N}{N}\n", "1bb59bc9acd39bd5ee26bf841bdabf19": "\\dot{V}\\,", "1bb64ea58215a3436351a23185596841": "\n\\sqrt{(j+m)!(j-m)!(j_1-m_1)!(j_1+m_1)!(j_2-m_2)!(j_2+m_2)!}\\ \\times\n", "1bb674c08c5f393141a8a02e029223a6": "\\bar{g}", "1bb6a03994b24bf5a99d9445176a8f76": "A~", "1bb6d5d6520e271846aae0e00e0d4309": "f : C \\to X", "1bb6df6108964f54ca8cdcdf37939473": "(E,\\mathcal E)", "1bb7b6bbeb569e94828d982403aa0e02": "\\scriptstyle{t_1}", "1bb7e0a066b32bfebe811a26b0bfad81": "I_{1}(\\sigma_{xx}\\sigma_{zz} - \\sigma^2_{xz}) - I_{3}", "1bb7e5c0c8e2f61792bfc622659c0a8a": "a(u_h, v)=L(v)\\,", "1bb80701b9049281de06538d371bb758": "d ^2", "1bb81124844fcc9de981ca864bb21dac": "\\,d", "1bb84ebd45b81f13c826b785cfc1113e": "\\text{F}", "1bb86e2b18c4c802d545812fe7d876d0": "H(|0\\rangle) = \\frac{1}{\\sqrt{2}}|0\\rangle+\\frac{1}{\\sqrt{2}}|1\\rangle", "1bb88a51685404c7c829f24aa72705b5": "\\hbar/\\Delta E", "1bb8a64d30b1652df26cbbc4affe641f": "\\frac{km}{L^{2}}", "1bb90d11a95c0ce14482ab49b0baf33f": "m\\ddot{\\vec{x}}+\\nabla V=0,", "1bb9c3978699034e38fb91425dc1a246": "\\bigvee_{j=1}^m \\exists x. \\bigwedge_{i=1}^n L_{ij}.", "1bb9c77078bc73dff5ddd7e4125788a7": "M \\ge 1", "1bb9fa53c2fcf62076145a1b39758870": " f : M \\to \\mathbb{R}.", "1bb9fd0cb2a00c8a4bd188d05922bb63": "\\alpha_J = \\Delta_R - \\beta_{iM} \\Delta_M ", "1bba15752ab4b23a19eff6f382dcf88d": "(a_i)^{(\\alpha )}_{\\kappa}", "1bba32b67b426845d8da8c33e3a60d5e": "\\langle n_1,n_2,\\ldots,n_k\\rangle", "1bba6584c573f61525d05e3e60320d76": " \\delta (h\\boldsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)})\n\\otimes h_{(2)}\\boldsymbol{.}v_{(0)}", "1bba65a4ef7ac2537f444674df28bddd": "m_E", "1bbb7fc0876b6f24c96463c2dafc4d10": " f_{pn} = {1 \\over (2.0665 - 1.0665p/100)} = 0.48394 + 0.0024688p + 0.00001561p^2 ", "1bbc0148c454c2c564dfa4455ec6a299": "\\hat{S}^{lm} (f)= \\frac{1}{K} \\sum_{k=0}^{K-1} \\hat{S}_k^{lm}(f).", "1bbc18b02bd479949a94c9c27dba7224": "f(x) = \\cup_{i}f_i(x)", "1bbca599a6a2481d7a4626e8c9cb620f": "\nx_{c} = -\\frac{1}{D} \\begin{vmatrix} B_{x} & A_{xy} \\\\B_{y} & A_{yy} \\end{vmatrix}\n", "1bbcaac5f7e22c56f9002876d427f36f": "(Q_i,\\ \\hat{\\Sigma},\\ \\Gamma_i,\\ \\delta_i, \\ s_{i},\\ Z_i, \\ F_i)", "1bbcf6cd7d3cef1e280486497462488a": "\\binom{t+k}{k}", "1bbcf71c2f5a3b1641d3b4dff2aeac48": "a_{8}+b_{8}+c_{8}=a_{1}", "1bbd2c34e9cbb10eb23fda53e9d99d9d": "=\\widehat{a} + [\\widehat{a},\\delta\\alpha\\widehat{a}^{\\dagger} - \\delta\\alpha^{*}\\widehat{a}] = \\widehat{a} + \\delta\\alpha[\\widehat{a},\\widehat{a}^{\\dagger}] - \\delta\\alpha^{*}[\\widehat{a},\\widehat{a}]", "1bbd3166fdec8ef3e146c8a6412d6323": "\\frac{1}{Z_{TP}} = \\frac{1}{Z_1} + \\frac{1}{Z_2} + \\frac{1}{Z_3} + ... \\,", "1bbd33b9e215654d9f06c53963826d93": " ln(\\gamma_1^\\infty) ", "1bbd5a225030182fe589dbd583012540": "l_q", "1bbd82697edae2e16beaf2582f27c4a2": "P(X_1 = 0)=p_1", "1bbddab710838a5e079c4c4d22f04d98": "b\\geq0", "1bbdec5ae9f4f64d7613e0eb7ce3ef89": "\\mathbf{z}:\\mathbb{R_+}\\rightarrow \\mathcal{R}", "1bbe066e6011ed98c7890e7b9e5233f3": "\\| f \\|_{L^2}^2 := \\int_0^\\infty \\frac{x^\\alpha e^{-x}}{\\Gamma(\\alpha+1)} | f(x)|^2 dx = \\sum_{i=0}^\\infty {i+\\alpha \\choose i} |f_i^{(\\alpha)}|^2 < \\infty. ", "1bbe1541350d803445b2a81323c227d7": "(\\alpha_1-\\alpha2)/2", "1bbe7c704b57c27a2ac6c4783fa7c3a5": "T(x) = \\sum_{n\\le x}\\frac{\\lambda(n)}{n}\\ge 0\\text{ for } x > 0,", "1bbeaa8ed797729d76f0ea23e7ffa3b2": " (\\mathbf{H}_i)_{i \\in I \\cup \\{ 0 \\} }", "1bbec9c72a30ba825f8fdf8712e1dd5a": "\n\\begin{align}\n& {} \\quad \\mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\\\\n& = ac \\langle X_u,X_u \\rangle + (ad+bc) \\langle X_u,X_v \\rangle + bd \\langle X_v,X_v \\rangle \\\\\n& = Eac + F(ad+bc) + Gbd,\n\\end{align}\n", "1bbf1dac71916ec739ce62419036abc3": "a = \\frac{g-2}{2} = 0.00116592080(54)(33)", "1bbfa93467d0b1215d9e91c16bfcca5d": "e^{i\\theta}= \\cos (\\theta) + i \\sin(\\theta)", "1bbff955e91db43588c36b697e816f0d": "\\textrm{Vanna} = \\frac{\\partial\n\\mathcal{V}}{\\partial S}", "1bc0111e74b5890679ae2b76aecc1a89": " G_i E_i= E_i G_i = l^{-1} E_i \\text{ and } E_i G_{i\\pm1} E_i =l E_i. ", "1bc04a77a265ef88dcae82e76ac1175e": " \\rho_n(x_1,\\ldots,x_n) = \\det(K(x_i,x_j)_{1 \\le i,j \\le n}) ", "1bc07fbecdda3200bf8bf1facbeb6342": " q=(s,t_s, t_e) \\in Q_A", "1bc082d30fad094884f7944bf4f545fd": "F = \\frac{Q_1Q_2}{r^2}", "1bc09488f4474b40c8d7034be69237ee": "(a_1, b_1, c_1, d_1)", "1bc095facd7aa3d5f146d1d4e451668c": " e^{j \\omega t} ", "1bc0d5d71244b2fbad5828cd30d084f6": "E_7 \\supset SU(8)", "1bc0de05c2e6e5a09bc5d8eb891a17c9": "15+10+10+10", "1bc119c840bd7ca7b72fcf641a8eda11": "\\frac {\\sqrt 2} 2", "1bc1d245ed07f1edd2d2cce819cf8ce8": "|\\psi(t)\\rangle = \\sum_n c^A_n(t)e^{-iE_nt/\\hbar}|\\phi_n\\rangle", "1bc20f4d7e96716314c87512afc8e4e0": "\\Pi_{(x:A)}B", "1bc23524400091e2ab502d09e8ba0a9c": "s(-2) = s(-1) = 0", "1bc247733faee428bfb487bdac42dbc0": "\\ y ", "1bc259dc28d0fac41eb104afcb51e3a7": " u_g = - {g \\over f} {\\partial Z \\over \\partial y}", "1bc28fe541df9356b27da251e53977b9": " \\mathbf{A} (\\mathbf{r}, t) = \\int { { \\delta \\left ( t' - { { \\left | \\mathbf{r} - \\mathbf{r}' \\right | } \\over c } - t \\right ) } \\over { { \\left | \\mathbf{r} - \\mathbf{r}' \\right | } } } { \\mathbf{J} (\\mathbf{r}', t') \\over c } d^3r' dt' ", "1bc2ae67c035b63e923c0d3aafee04e8": "{1 \\over 4 \\pi} \\int_{\\theta=0}^\\pi\\int_{\\varphi=0}^{2\\pi}Y_\\ell^m \\, Y_{\\ell'}^{m'*} d\\Omega=\\delta_{\\ell\\ell'}\\, \\delta_{mm'}.", "1bc2d021afd8533fb390a0755879a3d2": " 1/\\sqrt{T} ", "1bc3a93d5290eae711b84e134a837a77": "{}_2 Q_3 = 0\n", "1bc3d6195f9df90cf9487debc8e7ec97": "\\ddot{\\mathbf{r}} = a_r \\hat{\\boldsymbol{r}}+a_\\theta\\hat{\\boldsymbol{\\theta}}", "1bc3da55df6b9d3cfd405a27e0277b69": "\\delta[(h(S, X), S), (U, S)] \\leq \\varepsilon", "1bc41df5b6ae524fb5b675215a381ea8": "\\ C_{rr} = \\sqrt {z/d} ", "1bc42f3abe50d807e52bb7172bd0285a": "\n (\\bar{I}_1 := J^{-2/3} I_1 ~;~~ \\bar{I}_2 := J^{-4/3} I_2 ~;~~ J=1) ~.\n \\,\\!", "1bc439f9155578498a3b415bbfc0caf4": "w_m(x) = min\\{w(y)|y \\in \\mathbb{Z}, y \\equiv x \\pmod{m} \\}", "1bc43d4e951cefc8ebe56cc32154ba4a": "\\Rightarrow A(ab\\bullet{}b\\bullet{}abbabb, abbab\\bullet{}b\\bullet{}abb, abbabbab\\bullet{}b\\bullet{}) \\Rightarrow A(\\epsilon, \\epsilon, \\epsilon) \\Rightarrow \\epsilon", "1bc497a86e587fbe9483f246053ec777": " \\text{Re} [{}_1F_1(\\alpha; \\alpha+\\beta; it) ] = \\text{Re} [ {}_1F_1(\\alpha; \\alpha+\\beta; - it)] ", "1bc4ca19c5596cae065d65fdce61ae5b": "\\frac{dM}{dt}=-\\frac{M-M_e}{\\tau}", "1bc50df3342e17d7d9600b97dccfa40e": "\\delta\\vec{A}_{||}=0", "1bc5866b2cb82e351a0f9af0d513b2cc": "f(z_0)=0", "1bc59abd07973e988bba58467b3ed671": "f(q) \\propto q^a (1-q)^b", "1bc5ecc2b5ccbe4899e44f4baa692179": "SCR = \\left(\\frac{l_{\\mathrm{sum~of~solid~core~pieces}}}{l_{\\mathrm{tot~core~run}}}\\right)\\times 100 ", "1bc602e024fb5d8cce515046c5301091": "y \\in colgroups", "1bc681fc533ecae560d765f7cb5ee721": " k = \\begin{matrix}\\frac{\\omega }{c} \\end{matrix} n_1 ", "1bc6948dac8209fc147b46fe7801080b": " |p' \\rangle ", "1bc6cfd8076d0b863b69179d88f7e1eb": " A \\otimes_B A", "1bc6dafd86926517273eb37143b2aa2b": "H(q,u,p,t)= \\langle p,u \\rangle -L(q,u,t)", "1bc6eac9bb4f76beb126b2b0f33504f9": " \\frac{x+z-y}{2y}+\\frac{y+z-x}{2x}+\\frac{x+y-z}{2z}\\geq\\frac{3}{2};", "1bc6f95da3fe9457cdfca20647403589": "\\mathfrak{so}_{16}", "1bc731481c2b613c2a616e48c66ce518": "f_1 = \\frac{1 + \\sin(k_1 x)}{2}", "1bc772367f585f98792bba00c2413ec0": " \n\\frac {x} {\\ln x} < \\pi(x) < 1.25506 \\frac {x} {\\ln x}\n\\!", "1bc7a8eecd75843b5adf929abef0c88a": "t\\mapsto f(t)", "1bc80bba59e7a4e7432d77ada13f7fa2": "\\sum_{n=1}^{\\infty}a_n (e^{-s})^n,", "1bc83122008ede41abea59f585259090": "\n M_{yy} = f_1(x) = \\frac{4M_0}{\\pi}\\sum_{m=1}^\\infty \\frac{1}{2m-1}\\,\\sin\\frac{(2m-1)\\pi x}{a} \\,.\n", "1bc8880c8534e60bab13d6fa7c544573": "(\\Omega, \\mathcal{F}, \\mathbb{P}) := \\left( C_{0} (\\mathbb{R}; \\mathbb{R}^{d}), \\mathcal{B} (C_{0} (\\mathbb{R}; \\mathbb{R}^{d})), \\gamma \\right).", "1bc8a62ae7924077fffe9c4191c83a7c": " [T,b_n] = -nb_{n-1}. ", "1bc8ab8c837a94663c834603776a07da": "\n\\frac{\\lambda + \\alpha\\beta(1+3\\beta+2\\beta^2)}{\\left(\\lambda + \\alpha\\beta(1+\\beta)\\right)^{\\frac{3}{2}}}\n", "1bc8f608669061203b8d5768133e5df2": "[AFO]=[FCO], [AFO]= \\frac{1}{2}ACO=\\frac{1}{2}[ABO]=[ADO]", "1bc94c6d7c3b09999d3eb4081f357914": "(n_x,n_y,n_z)=(3,2,7)", "1bc9ad24e86219fb6be0016d070af7e5": " p(O_{fg} |I, I_t) ", "1bca5cdf5481f06a29a91a4ac16e7aee": "R_{ab} \\, = R^c{}_{abc}", "1bcad372191a195aeadb257a348f6d29": " C \\approx\\ l( b(C_{000}, C_{010}, C_{100}, C_{110}), b(C_{001}, C_{011}, C_{101}, C_{111}))", "1bcaf55db1313514b84b90dc2aad93ce": "\\begin{align}\n E &=\\frac{-A a_1 + a_2}{B} \\\\\n F &=-i \\frac{A^2 a_1 - A a_2 +2 a_1 B}{B \\sqrt{A^2+4B}} \\\\\n\\theta &=a\\cos \\left (\\frac{A}{2 \\sqrt{-B}} \\right )\n\\end{align}", "1bcb0986cca78a9f4d13bb5c6405898d": "\\mathsf{0123456789} \\!", "1bcb13e174b66706c64aff4605ee7f29": "\\Gamma(x,\\alpha+1,1)", "1bcb5fc443a4ef1238c836ec8fb16bec": "\\hat 1", "1bcb6e4e498db7dee7bd4003415f1a1f": "\\ p = \\rho \\cdot g \\cdot z", "1bcb7bf7c91933bcaf4cb1e2aa72988a": "\n\\begin{align}\n\\operatorname{sc}(u) & = \\frac{\\operatorname{sn}(u)}{\\operatorname{cn}(u)} \\\\[8pt]\n\\operatorname{sd}(u) & = \\frac{\\operatorname{sn}(u)}{\\operatorname{dn}(u)} \\\\[8pt]\n\\operatorname{dc}(u) & = \\frac{\\operatorname{dn}(u)}{\\operatorname{cn}(u)} \\\\[8pt]\n\\operatorname{ds}(u) & = \\frac{\\operatorname{dn}(u)}{\\operatorname{sn}(u)} \\\\[8pt]\n\\operatorname{cs}(u) & = \\frac{\\operatorname{cn}(u)}{\\operatorname{sn}(u)} \\\\[8pt]\n\\operatorname{cd}(u) & = \\frac{\\operatorname{cn}(u)}{\\operatorname{dn}(u)}\n\\end{align}\n", "1bcc51e0181cb040d09cf67d0dae83b1": "\\alpha_0 = \\sum_{i=1}^K\\alpha_i.", "1bcc614b6a9911ffd3b21dd15c38bd2b": "\\phi(t_1,\\dots,t_n)", "1bcc6bb41717e20699eefacf42a13b94": "p = mv", "1bcc883a5cfe902e818db11ddcf383aa": "\\mathsf{(CH_2CH_2)O\\ \\xrightarrow{200\\ ^oC,\\ Al_2O_3}\\ CH_3CHO}", "1bcc97a4960578435bfb82efeb18074a": "b,\\beta,s", "1bccc32572670e48d124adf891f7f656": " \\varepsilon \\; = \\; \\frac {2r - 1}{(r - 1)^2} = \\; \\frac{a^2 - b^2}{b^2} \\; ", "1bccd5177620e7bea397b13173c521ce": "\\frac12 = \\nu_2(a) - \\nu_2(b),", "1bcceb1739ade831ff36ab935b1f4a89": " \\frac{d{M}}{dt}=\\gamma{M}\\times{B}-\\frac{M_x\\vec{i}+M_y\\vec{j}}{T_2}-\\frac{(M_z-M_0)\\vec{k}}{T_1} ", "1bcd1d24c31c92fb81001c13784e8450": "k = n - m", "1bcd98c227e8fad247c90cc53757f4fc": "(h,f(x),b \\oplus h(x))", "1bcda3ba30d77e5af56e20a09c22c46e": "\\mathbf{B}(t)=(1-t)\\mathbf{B}_{\\mathbf P_0,\\mathbf P_1,\\mathbf P_2}(t) + t \\mathbf{B}_{\\mathbf P_1,\\mathbf P_2,\\mathbf P_3}(t) \\mbox{ , } t \\in [0,1].", "1bcda555b639307502313373bfa46365": "f_{xx}(a,b)", "1bcdc50c69e3ae5a3df98f77af261ef9": "k\\colon X\\to Z", "1bcdcdfe7fe7c15733da2f9063433d5a": "\\pi_*(\\alpha) = \\pi_*(g \\, dt \\wedge dx_{j_1} \\wedge \\dots \\wedge dx_{j_{k-1}}) = \\left( \\int_0^1 g(\\cdot, t) \\, dt \\right) \\, {dx_{j_1} \\wedge \\dots \\wedge dx_{j_{k-1}}}.", "1bcdfa302fbbfccf2f080a31c84d402f": "\\partial u^j/\\partial x^i(p)", "1bce001a90b4d5fc4782ba842b63508c": "R=\\frac{0.61\\lambda}{\\mathrm{NA}}\\approx\\frac{\\lambda}{2\\mathrm{NA}}", "1bce2786340f97240df47054c15abc60": " P(\\partial_t) := a_m \\partial_t^m + \\cdots + a_1 \\partial_t + a_0,\\; a_m \\neq 0. ", "1bcea7efe500a2118662f1ce388b3179": "\\sin \\theta_m > 1", "1bcec32e763b100d3fc6081331122dac": "Q = I_\\mathrm{RMS}^2 X = \\frac{V_\\mathrm{RMS}^2} {X}", "1bcee5ef91e3358fb98a5e59d1495308": "I_{\\mathcal Q}(0)\\in Q", "1bcf079fcee1fc6eea3d5b191e256ac6": "\\mathbf {X} = [\\mathbf {X_1} \\mathbf {X_2} ]", "1bcf26dde8ccec121c574ffa808bf666": " i = i_L + i_C \\,", "1bcf5a5015afc1e7230f1352b941bbfe": "y(t) = A_c \\cos \\left( 2 \\pi \\int_{0}^{t} f(\\tau) d \\tau \\right)", "1bcf9619feedd85ccda146090dae2b41": "\\tau = \\sqrt{-3502}", "1bcfa8d639c0827f66bc7348405d7069": "\\liminf_{n\\to\\infty}\\frac{p_{n+1}-p_n}{\\log p_n}=0", "1bcfda5bf0cc77ec49adf2ec96177a1f": "v = \\frac{ds}{dt}", "1bcfead2eb189a758dfcb71e8c26e9cb": "\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}, \\quad \\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2},\\quad \\cdots", "1bcfece10d705dc1a01a12d1eaacb181": " \\text{Area of triangle (on the unit sphere)} \\equiv E = E_3 = A+B+C -\\pi,", "1bd010da361d94202bde8e5fc899d075": "b^{-1}", "1bd01cd6ede07179a46ee938af84c0a4": "c_{\\nu_j}", "1bd01e5b7dcb94c8635f13ec4afe6e63": " \\mathbf{a} = \\frac{\\mathrm{d}\\mathbf{u}}{\\mathrm{d}t} \\,\\!", "1bd04dc50382dd83e46aaa28e08c415e": " (0\\mid c_2) \\in C_1\\mid C_2", "1bd086ed753e265ebf843dbf7a3109a8": "\\hat{q} \\psi (q) = q \\psi (q)", "1bd0d928c27718f7d4d1d2783642bd51": "H = H_1 \\otimes \\cdots \\otimes H_n", "1bd0e2fb802e234db043dc71ab762f7d": "\\beta_c = \\frac{1+\\sqrt{1-4c}}{2}", "1bd0e726e5223868d9e94828d998abbd": " \\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\left(\\frac{\\partial f_1}{\\partial \\boldsymbol{S}} + \\frac{\\partial f_2}{\\partial \\boldsymbol{S}}\\right):\\boldsymbol{T} ", "1bd0eaa80ccce6a7a5a74cbf4b357a99": " \\log { \\left ( \\frac{k}{k_0} \\right ) } = mY \\,", "1bd0f17521e98fdd86d15f23ed355e9a": "D(\\theta,\\phi)", "1bd0f2ee48485d88ef3073a09c2f441c": "\\ln{( 1+x^\\alpha )} \\leq \\alpha x \\quad{\\rm for}\\quad x \\ge 0, \\alpha \\ge 1 \\,", "1bd1038286a54f23d9b9b31b939d8b4f": "(2h)^{-1} \\operatorname{sech}(\\pi x/2h)", "1bd1171c2d31c99af30803780fc1ad9a": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = x^{2} - y \\\\\n f_{2}\\left(x,y\\right) & = -0.5x - y - 1 \\\\\n\\end{cases}\n", "1bd13e85d6fef0d794a6b76627e093d5": "C_{OX}", "1bd14d5d9f4779f2b343d3dfbe5021ad": "\\lambda_u/\\gamma", "1bd1533b3f14e293084ada0f9342d52d": "\\mathit{Q}", "1bd17ae62259f77c4c2993e0fef35e40": "d = 36", "1bd1b20f5d71d9ad04975eba148ac0c9": "\ndr=\\left(\\pm 1 - \\sqrt{r_g\\over r}\\right)d\\tau ,\n", "1bd1d7859f18b53a71912db48664941e": "G_1 = 1 - \\sum_{k=1}^n (X_{k} - X_{k-1}) (Y_k + Y_{k-1})", "1bd21153a5cd43e02348425800884962": "x^2=(2\\sqrt{a^2-b})x^0+2a", "1bd27edb6aa60ae131675f69bb4603b2": "r_1(q_2)", "1bd2eacbfeff61915b919add5b9cc378": "\\begin{align}\nP&=\\frac{V^2}{X-Xc}\\sin(\\delta)\\\\\nQ&=\\frac{V^2}{X-Xc}(1-\\cos \\delta)\n\\end{align}", "1bd31d6528f61f805d65e2a1b09ed43f": "K(X_i,X_j) = k_ik_j.", "1bd3381018944fbae7c135405b4b1e4c": "yp_{y}+p-c_{y}=0", "1bd33b4d38275d250db687544f0dcd1a": "\\mathbf{H} = -\\nabla U.", "1bd37145ba1d6e0dacf97875354d9a72": " y = 2 A \\cos \\left ( \\frac{\\phi}{2} \\right ) \\sin \\left ( k x - \\omega t + \\frac{\\phi}{2} \\right ) \\,\\!", "1bd3a0484883ca6deaada8395a8f6e85": "ER", "1bd3a0a036394af6f652a839a394e50d": "\n \\quad \\min \\limits _{D, X} \\sum_{i} \\|x_i\\|_0 \\qquad \\text{subject to } \\quad \\forall i \\;, \\|Y - DX\\|^2_F \\le \\epsilon.\n", "1bd3a67216de1df16947669396c731b8": "\n \\mathbf{u}\\cdot\\mathbf{v} = u^iv_i = u_iv^i = g_{ij}u^iv^j = g^{ij}u_iv_j\n ", "1bd3bc12f14e732ede664b4533bd16f7": "\\alpha(t):=v(t+1)", "1bd3dd08b9e17023952d66db5ed54828": "k_{AXU} AXUHash \\,", "1bd44540e3583032f8a8da76b9596f5d": "\\mu_4=(k)(k+2)\\,", "1bd4917950a5f106cd4d60ae2e453bf5": " E^c_{ij} = x_i p_j", "1bd4a7dd52e10f575b981566b4a9dc42": "T\\left(n \\right) = \\Theta\\left(f(n) \\right)", "1bd4e0a45f9e6595c5c9892a70749e74": " i(v) = enS_z \\frac{1}{4\\pi} \\int\\limits_{\\sqrt{2eV/m}}^\\infty f(v)dv \\int\\limits_0^\\zeta v\\cos \\vartheta 2\\pi \\sin \\vartheta d\\vartheta", "1bd50aca6a19eafdf6b066a5482b5ffc": "\\vec{v} \\times \\vec{w}", "1bd53667c6094ad36b2fefaff9a3a198": "\\scriptstyle f/f_s", "1bd5b24ca53af2660ac41e52d8d89c30": "\\log_b\\colon H \\rightarrow \\mathbf{Z}_n,", "1bd5c321bd2c9f7d11ca953696c871ad": "\\widehat{\\mathbb{C}}", "1bd5fc8bf62f545461553d4db1260ae8": " \\Big( c \\, \\log n / \\sqrt{n} \\Big) \\, 2^n ", "1bd6025bc4c7c1e1d455879c82787003": "{\\mathbf P}^'=[ {\\mathbf A} \\; | \\; {\\mathbf a}_4 ]", "1bd666e2402482fc78b41259f2379f3f": " x_i = x_j = \\frac{1}{n} , \\forall i, j ", "1bd6b82b3156fb27a41e45aaceb5fb62": "\\sum _p 1/(p^s\\log p)", "1bd72a5be0b1aa8bdbe43e863ad2d174": "\\boldsymbol{ a}_C =-2\\boldsymbol{\\Omega \\times v}= 2\\,\\omega\\, \\begin{pmatrix}-v_u \\\\0 \\\\ v_e \\end{pmatrix}\\ .", "1bd732526c90c0bb02936ffafe3e1245": "\\oint \\frac{\\delta Q}{T} \\leq 0,", "1bd7360eabfb28d378d3887e08f8f140": "\\omega_H = 2\\pi f_H = \\frac{B_0 |e|}{m}", "1bd73b6a902126b9426a518b15fe164e": "M_2 = P_1' | P_2 | \\dots", "1bd75746e65a23e91c62fca4a042fead": "L * (x,\\,y) = x ", "1bd77db2d6340fccfcfcdf471b48d92c": "PCAB", "1bd81eefdfdb041fac841ae5aa460083": " \\sigma^2(q_k) ", "1bd826fdb9af2ee7b85c6728d14ed907": "\\xi_1 < \\sqrt{2} \\lambda <\\xi_2", "1bd82d9fef5d54a32abe5c6d00d9222c": "\\tilde{D} \\tilde{D}^{T}/\\left( N-1\\right) ", "1bd8a5072c62d68eb86590ac1b4f302e": "h \\cdot (ab) = (h_{(1)} \\cdot a)(h_{(2)} \\cdot b)", "1bd8ef232b476a728243e7e57a22160c": " cov\\left[\\sum_{jN+1}^{(j+1)N} u_i,\\sum_{k=jN+1}^{(j+1)N} X_k\\right]\\neq 0 .", "1bd938626cd53edc67aba10ed8d58f87": "k^p", "1bd9e32fbaad8de1230a622253b21d71": "\\nabla \\cdot \\mathbf{E} = 4\\pi\\rho ", "1bda13b8625684afc4a1e834834579ef": "p_2=\\frac {q_1-q_2}{2q_4}\\ ,", "1bdab83c0032f4b872140bec5652375e": "\\mathcal{S}=\\mathcal{A}\\boxtimes_{n=1}^N \\mathbf{U}_n", "1bdab9b05e05baef1023dde257789634": " I + A \\, d\\theta ~,", "1bdb2988d0a26f6a36c94786a3180521": "x(t)=e^{A(t_1-t_0)}x_0+\\int_{t_0}^{t_1}e^{A(t_1-\\tau)}Bu(\\tau)d\\tau", "1bdb8d319354b7cfbf552d2f9fc8aa9f": "dU = T\\,dS-PA\\,dx+mg\\,dx", "1bdbc508611dee197a73f9d6c8f6366b": "f(\\textbf{x}_{1}) \\leq f(\\textbf{x}_{r}) < f(\\textbf{x}_{n})", "1bdbccefb882f7ab1f4affeec7919d67": "p^\\mu = -i\\hbar{\\partial\\over\\partial x_\\mu} ", "1bdc21ced07531276a097ef568bb84d8": "V_{Th}^{\\mathrm{low}}", "1bdc78d9a28675f844cb4cdb7d7ce909": "\\alpha_k \\le \\frac{k-2}{k+2}\\quad(12\\le k\\le 25)\\ ,", "1bdcd80e1fc8e946bff20eae9ffca285": "\\mu \\le \\lambda/2", "1bdcea4b6fe26c0c888ac3ef3cde39a4": "M/I M", "1bdd2800bbb25840ad1f6e77e001386d": "P_X:T^*Q\\to \\mathbb{R}", "1bdd47accf53a9472e660de221c506a6": " F_{\\mu \\nu} ", "1bdd5c6cb9161f257e1c9fd5fdb0c955": "(-1)^{ik}[x,[y,z]]+(-1)^{ij}[y,[z,x]]+(-1)^{jk}[z,[x,y]]=0", "1bdd5c99b69267720d83b81faca1d8ed": "s=\\frac{\\omega_{\\rm p}}{\\omega_{\\rm s}}\\frac{P Q}{R^2}", "1bdd74eecdcbc0e5c4104603e2656ce5": "f_n^{(k)}(x)=\\frac{\\alpha_n}{n!\\,\\lambda_n^{n-k}}\\psi_n^{(k)}(\\lambda_n x),\\qquad k,n\\in\\mathbb{N}_0,\\;x\\in\\mathbb{R},", "1bdda4cb9c5e5c4958ea94ccc6f8d399": "\\rho(\\bold{r}) = \\rho_0 + \\delta n \\frac{\\cos(2 k_F|\\bold{r}| + \\delta)}{|\\bold{r}|}", "1bddaa478dfd0e85c2d6bed20147f7c9": "0\\le {1 \\over k} \\le {1 \\over Wk}", "1bddac4d5e2f5d73621270dd4269bc32": "a=b^n", "1bde183c6f329b4292679bcb04478bf9": "\\dot{x} = f(x,u,t)", "1bde67a829c797b4186e0a8c8759664a": "\\mathrm{2VF_5 + 5H_2O \\ \\xrightarrow{}\\ V_2O_5\\downarrow + 10HF }", "1bde741a4b027a6fbb2e83cffed849a6": "AB^{*}=BA^{*}.", "1bde9bc34b6582080f21b032cd6ea52d": "\\Pr(R \\cap B \\mid Y) = \\Pr(R \\mid Y)\\Pr(B \\mid Y)\\,", "1bded72c2382ff9d0ecf45c53678118a": "\\mathrm{AFD} = \\frac{e^{\\rho^2} - 1}{\\rho f_d \\sqrt{2\\pi}}.", "1bdef9c1bc0b1098a622b19c8a0fdc96": "\\mathbf{A}^{-1}=\\frac{1}{\\det(\\mathbf A)}\\begin{bmatrix}\n{(\\mathbf{x_1}\\times\\mathbf{x_2})}^{T} \\\\\n{(\\mathbf{x_2}\\times\\mathbf{x_0})}^{T} \\\\\n{(\\mathbf{x_0}\\times\\mathbf{x_1})}^{T} \\\\\n\\end{bmatrix}.", "1bdf1befa10ca2bbe0fd7e537db5542e": "\\bigcup f(a)", "1bdf3f3fc9f69ece169992c488f7df0c": "\\text{provided that } \\mathcal{M}\\{B\\}>0", "1bdf7884034eda04ad2367a8f0cf9dae": "\n=\\frac{\\operatorname{cov}(\\beta,\\theta)}{\\sqrt{(\\operatorname{var}[\\beta]\\operatorname{var}[\\theta])}}.\\frac{\\sqrt{\\operatorname{var}[\\beta]\\operatorname{var}[\\theta]}}{\\sqrt{(\\operatorname{var}[\\beta]+\\operatorname{var}[\\epsilon_\\beta])(\\operatorname{var}[\\theta]+\\operatorname{var}[\\epsilon_\\theta])}}\n", "1bdf88aa90697f1e367a6834cffadf41": "Q_j\\in \\mathbb{R}^1", "1bdfb537c38c6ca2008cbfe5df779c5c": "n \\choose (n+k)/2", "1bdfe670c1db0224349c9d3cee020cfc": "- \\text{E}^{2}(\\tau|b,s,\\beta), \\quad \\beta \\ne 1", "1be0108445274b64e746144b367919e8": "L=-\\partial_{x}^2+u\\,", "1be03680765d1758574e1a99d121e1bf": "X_{j}", "1be0b4087d78d6c2bffe2f70b8460fb5": "{r_m}", "1be0c936ac35a1237f3d91a5bab13e94": " \\qquad \\qquad b_{\\kappa,\\alpha}^\\dagger = \\frac{1}{N^{1/2}}\\sum_{\\kappa_p,\\alpha} e^{i(\\boldsymbol{\\kappa}_p\\cdot\\mathbf{x})}\\mathbf{s}_\\alpha(\\boldsymbol{\\kappa}_p)\\cdot[(\\frac{m\\omega_{p,\\alpha}}{2\\hbar})^{1/2}\\mathbf{d}(\\mathbf{x})-i(\\frac{1}{2\\hbar m\\omega_{p,\\alpha}})^{1/2}\\mathbf{p}(\\mathbf{x})].", "1be10656885c257a4c98af160b7e8170": "2^{-\\frac{1}{\\alpha}}", "1be112a328f3a4276fcd11289e021d22": "\\left( \\frac{\\partial A}{\\partial y} \\right)_{x,z} \\!\\!\\!= \\left( \\frac{\\partial B}{\\partial x} \\right)_{y,z}", "1be11ba7e5433c85d4f8e1a13572fe8f": "\\lim_{n \\rarr \\infty} p_{jj}^{(n)} = \\frac{C}{M_j}", "1be144c34fd0853d08077d686b5e2088": "M=n+1", "1be155fc754d0ae7841b97a5bbbebd86": "\\int_0^\\delta t^{\\lambda + n} e^{-xt}\\,dt = \\frac{\\Gamma(\\lambda+n+1)}{x^{\\lambda+n+1}} + O\\left(e^{-\\delta x}\\right)", "1be16d00e39d89d873b58a7b77e1d808": " (i,j) \\leq (i',j') \\iff i\\leq i' \\qquad \\textrm{and} \\qquad j\\leq j'", "1be182f7fa88172265721fcf91b8581b": "\\tfrac{3}{2}\\scriptstyle{\\sqrt{2}}", "1be184b6963cbaf5745122e24b7d65c7": "1.\\mu_{8,4}(p_{4}) = \\alpha_{8}(p_{4}) ", "1be1ae1b8153d616f18e7cc7843e1b9f": "c_{d_0}\\;", "1be1b640abdda07d8f3090291f0ca40f": "s,t \\in V(G)", "1be1dfd33d881912b75eb5a462ec2c89": "\\textstyle(x\\pm1,y\\mp1)", "1be21a6bc22b3ad4ec5eb48be6c8fc23": "l_A", "1be30633e9258f7e4604829638482361": " z_1 (x,y) = {x^2 - y^2 \\over 2} ", "1be336e794138c4c30f91c0dd4fe8100": " = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-h} \\left( \\Lambda \\Lambda^{-1} \\right) h^{ab} g_{\\mu \\nu} (X) \\partial_a X^\\mu (\\sigma) \\partial_b X^\\nu(\\sigma) = \\mathcal{S} ", "1be36aeb33833bf58795e6b3f642dbe5": "\\, q", "1be38d91a56c5b60de469bfa43491807": " \\{ C_i \\}_{i \\in I} ", "1be39fccae0fc210477ad8978cd96012": "\n(x^3+Ax+B)((x^3+Ax+B)^{\\frac{q^{2}-1}{2}}-\\theta(x))\n", "1be3dfd47cb6e9cc23030e7854f25ef6": "1/p^2", "1be3e9656794404dce8d35f83552d1ac": "t^a(d,n) ", "1be3fe099395a4ab475678f1602c795c": "\\mathbf{a} = a_1\\vec{r}_u + a_2\\vec{r}_v ", "1be3fed02676c5e01c601bf209a0af3c": "\\mathbf{X}^{1/2}", "1be40b9e096dc090f91585f6114d8dca": " y_i = \\sum_k M_{i k }\\otimes x_k \\in EndPol \\otimes Pol ", "1be44330461d19cbe0f5003716b2f5c8": "u \\in V", "1be46696105293817ab91de9ab176038": "\\mathcal O(n+1)", "1be4ca7b3d5cd777eee7a50db7f74011": "f_{i}(\\theta)", "1be50391e28ecd508f471224325bb583": "f_{t}\\left(\nx,t\\right) ", "1be558f7c415d9e857e4b27511cbe55a": "\\text{ORL}(\\mathrm{dB}) = 10 \\log_{10} {P_\\mathrm i \\over P_\\mathrm r}", "1be56a018a8e1ffcea6e3b2f143b5f79": "\\ (1-\\eta^2) \\frac{d^2 Y_{mn}(c,\\eta)}{d \\eta ^2} -2 (m+1) \\eta \\frac{d Y_{mn}(c,\\eta)}{d \\eta} +\\left(c^2 \\eta^2 +m(m+1)-\\lambda_{mn}(c)\\right) {Y_{mn}(c,\\eta)} = 0, ", "1be59e92cc2818029c7fed4a024b46a1": "\\sin^2(18^\\circ)+\\sin^2(30^\\circ)=\\sin^2(36^\\circ). \\, ", "1be5aa8d4cea80201a31a16b8a9666c7": "\\operatorname{pd}_R M = 1", "1be5b382180319353ccbbda39c4bd7c3": " f(z + \\omega)=f(z) \\ ", "1be614c4e1b209ddfef534ec9247b48c": "\\mathbb{C}\\circledast X\\cong X\\cong X\\circledast \\mathbb{C},\n", "1be64bd809d160c2874421c1083792a7": "\\scriptstyle A \\oplus B", "1be669931bdb641d83b5b4fc38c7bb3c": "\\sum_{i=1}^{k+1}\\delta_i\\!\\int_{z_{i-1}}^{z_i} g_j(z)\\,dz=0\\text{ for }1\\leq j\\leq k.", "1be676dbcb38c53f32cbc0b9f5553026": "F_\\ell", "1be69f76d50b0f2c4edcfcba8ac43fc4": "\\mathbf A = \\|\\mathbf A \\|\\left( \\cos \\alpha \\ \\hat{\\mathbf i} + \\cos \\beta\\ \\hat{\\mathbf j} + \\cos \\gamma \\ \\hat{\\mathbf k} \\right) \\ ,", "1be6f70858bb4e92745f4c48eae7bbcc": " \\Psi = \\sum_{i=1}^N c_i \\Psi_i. ", "1be6fb612b389126b539545989ef67b7": "x = -a", "1be6fc2ce2a6d196c4e363adb61ca5be": "y \\rightarrow y/r", "1be72ec0830c1e6af36794185ef386f6": "2 + 1 \\twoheadrightarrow 3", "1be7511f2ef4e5b483710b80fd3e2b71": "\\Gamma_{12}(l, m, 0) = \\iint U(l, m, P_1) U^*(l, m, P_2) \\, dS", "1be7594e4a1f09b7de7cbfb6605c20bb": "\n\\begin{align}\n\\Theta_i&=\\sphericalangle(\\mathbf{p}_i,\\mathbf{k}),\\\\\n\\Theta_f&=\\sphericalangle(\\mathbf{p}_f,\\mathbf{k}),\\\\\n\\Phi&=\\text{Angle between the planes } (\\mathbf{p}_i,\\mathbf{k}) \\text{ and } (\\mathbf{p}_f,\\mathbf{k}),\n\\end{align}\n", "1be7ad08853563968d35e7b9c5466afd": "n!\n\\; \\; - \\; \\; {n \\choose 1} (n-1)!\n\\; \\; + \\; \\; {n \\choose 2} (n-2)!\n\\; \\; - \\; \\; {n \\choose 3} (n-3)!\n\\; \\; + \\; \\; \\cdots\n\\; \\; \\pm \\; \\; {n \\choose n} (n-n)!\n", "1be7fa233dcda7831ce592bede114223": "P_r = \\left [P_t{{G^2 \\lambda^2 }\\over{{(4\\pi)}^3 R^4}} \\right] \\left[\\frac {c\\tau}{2} \\right] \\left[\\frac {\\pi R^2 \\theta^2}{4} \\right] \\eta = \\left [P_t \\tau G^2 \\lambda^2 \\theta^2 \\right] \\left[\\frac {c}{512(\\pi^2)} \\right] \\frac {\\eta} {R^2} ", "1be8629b9f429ff89175d33603d85241": "\\int_0^1\\ln^{2n}\\left(\\frac{x}{1-x}\\right)\\,dx = (-1)^{n+1}(2^{2n}-2)\\beta_{2n}\\pi^{2n}", "1be8d255bf6b3efa7625c373f92aa19e": " N_0 ", "1be8f15b1c555b7c1ae119f5b616658b": " \\delta/2\\pi = n \\,\\!", "1be921bc3bdeeda4c4e7e2703a027e8e": "PSL(2,\\mathbb{R}).", "1be9d935c62cd1e7b99778baf951744c": "\\operatorname{Gal}(\\mathbf{F}_{q^f} / \\mathbf{F}_q)", "1be9e3fd25a7da54adcd22f67fa89429": " x_n = -\\cos\\left(\\frac{\\pi}{N} (n+\\frac{1}{2})\\right)", "1be9f9f8e460d33db1f1089254e78a39": "e^{ix}=\\cos x + i \\sin x", "1bea86b9e0729f0e7efe21dde34ffba3": "V\\left(x\\right)", "1bea8ba4ceaf923082ab0d43fbe96908": "S[t]", "1bea95bc1fb13d3d185e67411b368d41": " \\sgn\\left(nx-k\\right) = \\begin{cases} \n-1 & nx < k \\\\\n0 & nx = k \\\\\n1 & nx > k. \\end{cases}\n", "1beac5384ab16f9f67515dd08b57b177": "(x)O(\\underline{x}:G)", "1beada934b50b5a8190c03add75c909b": "N^*(x)", "1beadd88d70982f74ea530acda32c043": "p_{n-1}/q_{n-1}", "1beb163b90b89f0ca8e4b0ed3d6ac705": "\\sum_{q\\leq Q}\\max_{y 0.\n", "1bf06980cb4eb314ea1b4bcf3354fc2c": "\n\\frac{\\langle E,s\\rangle \\Rightarrow V}{\\langle L:=E\\,,\\,s\\rangle\\longrightarrow (s\\uplus (L\\mapsto V))}\n", "1bf0b779c9001024285ab7454a249f59": "2^n+1", "1bf12be07abaf625d933facb962b2eb6": "G_N = 1 = c", "1bf15896187ff0f9d7d66b07432c8f1a": "\\left[ B \\right]=\\left\\{ \\begin{matrix}\n \\left[ A \\right]_{0}\\frac{k_{1}}{k_{2}-k_{1}}\\left( e^{-k_{1}t}-e^{-k_{2}t} \\right);\\,\\,k_{1}\\ne k_{2} \\\\\n \\left[ A \\right]_{0}k_{1}te^{-k_{1}t}\\;\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{otherwise} \\\\\n\\end{matrix} \\right.", "1bf198096556483c6f2b1079f9206dac": " \\frac{1}{q}+\\frac{1}{p}=1 ,", "1bf1c13cf84920da719f1860fd9d6d3f": "x \\oplus x \\oplus x = x \\oplus x", "1bf21b2e6f60c278eae7d5e03e39de43": "\\Gamma(N) = \\left\\{\n\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in SL_2(\\mathbf{Z}) :\nc \\equiv b \\equiv 0, a \\equiv d \\equiv 1 \\pmod{N} \\right\\}.", "1bf28f20e2f530cc5813719c6daa75cb": "(m/q)P_p = uq(m/q)P_p = umP_p = 0 \\, ", "1bf292a3e14620ef9316194dab31d354": "\\frac{x^3 - 12x^2 - 42}{x - 3} = x^2 - 9x - 27 - \\frac{123}{x - 3}", "1bf2d677e8320c91e78baa12d1a52721": "\\int \\sec{ax} \\, \\mathrm{d}x = \\frac{1}{a}\\ln{\\left| \\sec{ax} + \\tan{ax}\\right|}+C", "1bf3305525def18f3230ab86d8a9e485": "\\Phi_n(x) = \\prod (x - \\zeta)", "1bf383803d7a344c74729c9494a374d8": "\n\\ln\\ I(R) = \\ln\\ I_{0} - k R^{1/n} ,\n", "1bf38bc5d5af0aa152208f67f2ffef00": "\\gamma=\\sqrt{ZY}", "1bf3b44da47ce3985f33a759a52e8a02": " y_n ", "1bf3ca070beb57c0d5d9c85a3ff0b88f": "A \\in M(m,n;\\mathbb{K})", "1bf417e737d12f0763d9cfea4aec48a8": "B=Y", "1bf41f16dfd33ceec7d09e91e2f90de8": "\n \\lambda_i \\equiv l_i/L_i\n ", "1bf45066aff20080dc142253ad6a5f89": "\\frac{\\partial n(x_i,t)}{\\partial t}=\\frac{1}{2}\\sum^{i-1}_{j=1}\nK(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \\sum^\\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t).", "1bf49a39c7156b5b62ded5cc5e7674dd": "\\displaystyle x(s) = x_0 + \\sum_{t} {s^{|t|}\\over |t|!} \\alpha(t) \\delta_t(0).", "1bf49f5d69bf7910161aefc94f7d2f0b": " i\\hbar \\frac{d}{dt} \\rho_I(t) = \\left[ H_{1,I}(t), \\rho_I(t)\\right].", "1bf4a25fbcef1e257fa670665497dbcf": "{f_2 \\over f_1} = {p_2 \\over p_1}", "1bf4acdcd001c0df98fe6f63f1f2d5fd": "P_0 = (x_0,y_0,z_0)", "1bf4d10c9c7255077d5b7209bb8c3c1a": "Y\\sim HN(\\sigma)", "1bf4df1aff97f5dc6a294225bb309604": "\\mathbf{B}_{\\|}", "1bf4f9175b2e7b4284a8ccf17ab6406c": "\\mathbb{E}[X\\,|\\,\\mathcal G]+c\n\\le\\mathbb{E}\\Bigl[\\liminf_{n\\to\\infty}(X_n+c)^+\\,\\Big|\\,\\mathcal G\\Bigr]\n\\le\\liminf_{n\\to\\infty}\\mathbb{E}[(X_n+c)^+\\,|\\,\\mathcal G]", "1bf51fd0e2af1f4e78ee3124cabccb37": "\\mathbb{C}c.", "1bf53e0cb19c70923b9d853769fa07ac": "\\scriptstyle \\mathbf{R}_R", "1bf55568f54820efd0d8d3d3639c5cda": "P_\\mbox{rej}", "1bf574a0bf5334ddadcacec3c594b642": " clearance\\ ratio\\ of\\ X = \\frac{C_x}{C_{in}}", "1bf5762089537f2be13b54cecf8643a7": "\\vec v_n=\\tfrac{\\vec x_{n+1}-\\vec x_{n-1}}{2\\Delta t}", "1bf58115c16d44e451e2b5f745c4cdeb": "\\dot\\rho=-{i\\over\\hbar}[H,\\rho]+\\sum_{i = 1}^{N^2-1} \\gamma_{i}\\big(A_i\\rho A_i^\\dagger -\\frac{1}{2} \\rho A_i^\\dagger A_i -\\frac{1}{2} A_i^\\dagger A_i \\rho \\big) .", "1bf59e2ed0faf854d0b53249221ba72c": "Q = A. f(K,L)", "1bf5eaea06b6f27fc3f1f5fac4bedeae": "(s x_0, s y_0; s^2 t_0)", "1bf6bd3b2c78a00259dec0f21a719486": "kM_p\\omega^{2^{p-2}} = 0.", "1bf6befa83331a40e84e4ab90663ec7b": "\\neg c \\vee d,", "1bf6fb07acdcac1b6857170e69490930": "\nc^{2} d\\tau^{2} =\n\\left( g_{tt} - \\frac{g_{t\\phi}^{2}}{g_{\\phi\\phi}} \\right) dt^{2}\n+ g_{rr} dr^{2} + g_{\\theta\\theta} d\\theta^{2} +\ng_{\\phi\\phi} \\left( d\\phi + \\frac{g_{t\\phi}}{g_{\\phi\\phi}} dt \\right)^{2}\n", "1bf6fea172e8f53ec71c328f1773e763": "X = \\left\\{ x_1,\\dots x_n \\right\\} \\sub A", "1bf6ffb816890e357bb896d986081897": "G^{(n)} \\neq \\{e\\}", "1bf77084f13b7b646c64e8cb6391a11e": "J^k=\\begin{bmatrix}\nJ_{m_1}^k(\\lambda_1) & 0 & 0 & \\cdots & 0 \\\\\n0 & J_{m_2}^k(\\lambda_2) & 0 & \\cdots & 0 \\\\\n\\vdots & \\cdots & \\ddots & \\cdots & \\vdots \\\\\n0 & \\cdots & 0 & J_{m_{s-1}}^k(\\lambda_{s-1}) & 0 \\\\\n0 & \\cdots & \\cdots & 0 & J_{m_s}^k(\\lambda_s)\n\\end{bmatrix}", "1bf7a75a7e48320092acd6ca45bfd430": "T \\gg T_D", "1bf7eb0f9d5aace4bc3b6189193edaff": "x^{2} + 3x - 10 = 0. \\,", "1bf7f73bece5d5fce7bbb9b7508ec692": " P_2^{-1}A_2P_2=\\begin{bmatrix}0 & B_3 \\\\ 0 & A_3 \\end{bmatrix}", "1bf82d713f0474bcd0e347fad5614540": "N_c = 3", "1bf837fed349552284b3c3c3df8906df": "\\displaystyle{\\partial_{\\overline{z}} f =0,}", "1bf838a198d6e93c0e32da9b1b817459": "\\sum_{i=0}^n a_i x^i", "1bf866429b2a54fd84f0aee2ae662c51": "g_1,\\cdots,g_n", "1bf8d8d6a9a099c337f117ba80365582": "\\varphi(x)\\leq \\varphi(x+y)-(D\\varphi)(x)\\cdot y.\\,", "1bf8dadfbb854d882936ec2879ec1a21": " S(\\theta_0 , \\theta_i) = \\frac{(\\theta_0 - \\theta_i) L_f}{t^{1/2}} ", "1bf8e3390c743e096bf374ffa4ddca25": "\\{f_i\\}_{0}^{N-1}", "1bf94a7242a891b1b716b2be9752f3d8": "\\boldsymbol{m}", "1bf94ab2452a1c00a90ffdf53f4021c1": " \\left| \\begin{matrix} u_x & u_y \\\\ v_x & v_y \\end{matrix} \\right| = u_x v_y - u_y v_x. ", "1bf95797fdcec75939f4d30d01976f96": " \\bold x = (1,1)", "1bf9b0f97c36784c9777cf501bafe8a4": "\\ R_i ", "1bf9cf49aeb5e41fd877ef6a6054046c": "x \\# y \\;\\to\\; y \\# x", "1bfb454cb8ec46ca8b62a9592e71968b": "\\sigma_\\mathrm n", "1bfb4d9609e159a3a98469dd54f7e25f": "\\begin{pmatrix}1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\end{pmatrix}", "1bfb576fcb9f1cfc973c6e67d94d083f": "\\int_{-\\infty}^\\infty 1 \\cdot e^{2\\pi i x\\xi}\\,d\\xi = \\delta(x)", "1bfb8f8268a57cd59f5baee24e661a77": "S\\ =\\ \\mathbf{true}\\ |", "1bfba8f3e6a79196eea583801d10fe00": "\\omega_{i_1\\cdots i_k}", "1bfbc252a8042e91426cc7d65a3c22e2": "1/kT ", "1bfc27d98d7ddc7cee933b74e192bcec": "\\limsup_{n\\to\\infty}1_{X_n>0}>0", "1bfc2f5638af87d5f561ca965d6d8132": "\\frac{dq_2}{d\\sigma} =\\frac{\\partial P}{\\partial u_2} \\quad \\quad \\frac{du_2}{d\\sigma} =-\\frac{\\partial P}{\\partial q_2}", "1bfc95b7db00c43fc91d7b382873d65a": "f_y", "1bfc9a47684e89400f8d71be994d632d": " S = ( \\alpha - 2^{ 1 / \\alpha }[ \\alpha - 1 ] ) ( \\frac{ \\alpha - 2 }{ \\alpha } )^{ 1 / 2 } ,", "1bfcf24be75b3a202003b3a12d80edc4": "\\pm\\sqrt{3/5}", "1bfd0b8e3736ffff7cae934d5053738b": "P_V\\approx 0.64700", "1bfd163ce0059b9a475b6088f420c593": "\\dot{x} = A x+ B u, ", "1bfd1b65a8a1ca265d2e256bf9595e69": "(1~2)", "1bfd6527cc6feb106cc15616f8eb33af": "Gx+Cx=Bu", "1bfd995632a44da79e17f7be8c449bb3": " \\Gamma(s,x) = \\int_x^{\\infty} t^{s-1}\\,e^{-t}\\,{\\rm d}t ,\\,\\! \\qquad \\gamma(s,x) = \\int_0^x t^{s-1}\\,e^{-t}\\,{\\rm d}t .\\,\\!", "1bfdee66bea8333f33d815506c94e5cd": "I = \\int_0^1 \\frac{1}{1+x} \\, \\mathrm{d}x", "1bfe02982bce0776c1fc7570aea7b149": "{\\Gamma,\\;x{:}1{\\to}\\tau_1\\;\\vdash\\;e:\\tau_2{\\to}\\tau_3}\n \\over\n {\\Gamma \\vdash \\kappa x{:}1{\\to}\\tau_1\\,.\\,e\\;:\\;\\tau_1\\times\\tau_2\\to\\tau_3 }\n", "1bfe0e73558c831835b2fd1adddfcbd9": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon r} \\sum_{l=0}^{\\infty}\n\\left( \\frac{r^{\\prime}}{r} \\right)^{l} P_{l}(\\cos \\gamma )\n", "1bfe2d324b29d527a1f275a18fe75284": "\n R_\\alpha = \\sum_{i=1}^N r_{i,\\alpha}.\n", "1bfe3322286a935319deebe894821cf1": "\\mathbf{log_2}", "1bfe3dfdab29a5d4fd77abe31df08416": "\\eta \\in \\Omega^q(M, \\mathfrak g)", "1bfe66ecc330b132953d08d2a66981ec": "f_n\\in \\mathcal{L}^1(\\mu)", "1bfe7dafaf05184e938a2ba95a7eeb1a": "F = \\mathbf Q[\\alpha], \\alpha = \\sqrt[3] m ", "1bfe80701f41249750cf8bd84b6231c6": "\\det\\mathsf{f} = \\underline{\\mathsf{f}}(I) I^{-1}", "1bfecfb756a695b24a50c41856a030db": "\\mathfrak{P}^{6}", "1bfeedaa9ac34f9efee3ae79fbd70271": "s_d", "1bff036e16c55ad582bc9eea9151aa92": "M = (X, d)", "1bff67e994e50e4ab0e9f7186aafefd4": "\n\\begin{align}\ni&\\leftarrow 1 \\\\\nr_1&=r \\\\\nG_1&=(H^HH+\\sigma ^2I_{N_r})^{-1}H^H \\\\\nk_1&=\\arg \\min \\left \\| (G_1)_j \\right \\|^2 \\\\\n\\end{align}\n", "1bff719ee29804967eecf1a1a4b530fe": " \\mu_X ", "1bffc09e811a7e8b411db8af32cd3b80": "g(n) y \\\\\n\\end{cases}", "1c035ebd32780e9598c129db0080f0e1": "|H_{rrc}(f)| = \\sqrt{|H_{rc}(f)|}", "1c03a88506a096ad5e0093f4709bdbf3": " y \\leftarrow \\lambda(s);", "1c0414627a0735be19bd579fbff98b69": "D=R\\sqrt{(\\Delta\\phi)^2+(\\cos(\\phi_m)\\Delta\\lambda)^2}{\\color{white}\\frac{\\big|}{.}}\\,\\!", "1c043ea664f5eeae49a122668816fdc2": " [2](....([2]([2]([2]([2]([2]P+[k_{(l-1)}]P)+[k_{(l-2)}]P)+[k_{(l-3)}]P)+ \\dots ) \\dots +[k_1]P)+[k_0]P= \n [2^l]P+[k_{(l-1)}2^{l-1}]P+ \\dots +[k_12]P+[k_0]P ", "1c04b57af911f4322b635dde5cc7218d": "C_1 \\approx", "1c04dfcb2339a32e464a2dfdd28ee0a8": "\\mu = \\sigma/\\xi,", "1c05096cd8253c71618496a0d6cada8e": "T^{**} = T", "1c057bb27e7d822d124d1eb212b95728": "D_{A} \\subseteq D_{\\mathcal{A}}", "1c059324aad3fbf9c3d0899a20368f79": "w(s(x) - x) \\ge i+1", "1c05b3032149efc238bf8e123466e1a0": "\\scriptstyle y[n] = x[nM].", "1c0619d09229237f572ba6bd8f584a31": "\\hat{T}_f", "1c0659a20edbc8831b3e86ecdf22bab6": "s\\left\\{\\begin{array}{l}5\\\\3\\\\2\\end{array}\\right\\}", "1c0690daf307465960195d4cb564c003": "\\omega = \\sum_{i_1 < \\cdots < i_k} \\omega_{i_1\\cdots i_k}dy_{i_1} \\wedge \\cdots \\wedge dy_{i_k},", "1c0697ae99b4b61084c33016bfcf1495": "\\mathbf F(\\mathbf X,t)=F_{jK}\\mathbf e_j\\otimes\\mathbf I_K\\,\\!", "1c06ba8c53212c805a523132ed7041d6": "r_k", "1c06c425b13b0fb550f2bb2412fc4498": "L(p,d)", "1c071bf89f251a1b30ce6e0cfc33123d": "GF(3)", "1c072128df3708bd9e9c80db1ebdd4a3": "d(id_A) = 0", "1c072de3ea11145b6368a3ab77256594": "\\beta_k = {\\rm rank} (H_k(S))\\,", "1c07496e69c4d19bc9b320876fa72e1d": "pp_\\kappa(\\lambda)", "1c078e4762fd332240e3de05f872113d": " v:T(TM\\setminus 0)\\to T(TM\\setminus 0) \\quad ; \\quad v = \\tfrac{1}{2}\\big( I + \\mathcal L_H J \\big).", "1c081628aae64a513916423c39b48a8a": "\\boldsymbol\\alpha\\!", "1c084216fff1c0d634282612eb0149de": "P_n(L)\\in\\mathbb{Z}[q,q^{-1}]", "1c0881a42bffa6bca5f49e05a56a02b9": " P(X=k) = f(k;N,K,n) = {{{K \\choose k} {{N-K} \\choose {n-k}}}\\over {N \\choose n}}.", "1c08af2a61e4839f29cc9f9d7567e0c3": "\n\\lambda_3 = (\\mathrm{E}X_{3:3} - 2\\mathrm{E}X_{2:3} + \\mathrm{E}X_{1:3})/3\n", "1c08ef73bb15f993223cb1c7ef835c6c": "\\begin{align}\n\\frac{\\partial P}{\\partial z} &= -\\gamma = -\\rho \\, g \\\\[0.5em]\n\\frac{\\partial h}{\\partial z} &= 0\n\\end{align}", "1c0906b70394a5f178dde32d6937b3c2": " \\begin{matrix} \\frac {v_2}{v_1} = \\frac13 \\end{matrix} ", "1c097f373c07c97dcdfa4dad7553f47a": "\\begin{bmatrix} 1 & -\\frac{1}{G} \\\\ 0 & 1 \\end{bmatrix} ", "1c09e49051cf7aa2943815e9e4ef9af1": "g_1(x)=\\sqrt{1-x^2}", "1c0a217aa05c4f4a82274fe515a34a6d": "\\tilde X = \\tilde Y", "1c0a285489acb8aba7fe577769550a6c": "d_m=p_0+p_1d_{m-1}+p_2(d_{m-1})^2+p_3(d_{m-1})^3+\\cdots. \\, ", "1c0a38dcafdbc3d45694f86560d0ec9f": "x_i = -\\sum_{k=1}^P a_k x_{i-k} + y_i", "1c0a699c0d0656ea2793b2587eac4712": " \\frac{V^{1/6}}{T} = const ", "1c0a86c915f66906afd89a7fcd432845": "\n \\frac{\\partial A^{-1}_{ij}}{\\partial A_{kl}} = -\\cfrac{1}{2}\\left(A^{-1}_{ik}~A^{-1}_{jl} + A^{-1}_{il}~A^{-1}_{jk}\\right)\n", "1c0ad6c5bb696ce2f2ab1761495d1b0e": "f(x) = \\begin{cases} e^{-1/x^2} & \\text{if }x\\neq 0 \\\\\n0 & \\text{if }x = 0 \\end{cases} ", "1c0bc1d75a88dac001afe722f7897170": "h_1,\\ldots,h_n \\leftarrow b_1,\\ldots,b_m", "1c0bdd4fe90d28d64b64d0dfb01494e2": "(S,\\mathcal{D})", "1c0be18e4eeb52a84da907ec52261f00": "b_n \\to b", "1c0be6590dc00c2c556e80a363fb9787": "n\\equiv 0 \\pmod{2^{k}}, \\quad n\\equiv 1 \\pmod{5^{k}}\\, ,", "1c0cdad988fe639fb1cdb82a4a0b401e": "R_n^{(b)}=\\frac{1}{b-1}\\prod_{d|n}\\Phi_d(b)", "1c0cded4686ed811b07bf02195fa18c5": "\\Sigma=:\\operatorname{diag}(\\sigma_1,\\ldots,\\sigma_m)", "1c0d574e43b9c4bd3978b5c131405efb": "A_{\\theta}", "1c0dc882ce3fb65a2b90318e8b8f2e1a": "(p_\\theta,q_\\theta)", "1c0de3620ce9b4074eb2c43a6cb9446b": "_{k+1}V^i_4(x,y)=_kV^r_2(x+1,y)", "1c0e1c35c6dfb4c63aca6b95eded4686": "\\sum_{j=k}^{d-1} (-1)^{j} \\binom{j+1}{k+1} f_j = (-1)^{d-1}f_k. ", "1c0e4c84c3d11e378211134e22e5f745": "U = \\sum_{i=1}^{n} \\frac{C_i}{T_i} \\leq n({2}^{1/n} - 1)", "1c0e5ec49df45d37549a8625a5bb24f9": " i_1,i_2, \\ldots i_N ", "1c0e91546f38692e060b264269f16711": "Q_i ", "1c0ee1a9ff010788593027ccb8778b03": "\\! p = \\rho_m RT = \\rho_m C^2", "1c0ef9c8fe53dd6cb90fe834da685688": "n_i=0 ", "1c0f114fe3f5f180fe74607e01a6d90c": "W(L,t)=\\Big\\langle\\frac1L\\int_0^L \\big( h(x,t)-\\bar{h}(t)\\big)^2 dx\\Big\\rangle^{1/2},", "1c0f3a616ebf56542bcac2b5b3a14b3d": "\n\\begin{align}\n& \\sum_{j=-\\infty}^\\infty \\sum_{k=-\\infty}^\\infty 2^{-j} \\left( |a_{j,k}|^2 + |\\tilde{a}_{j,k}|^2 \\right) \\\\\n& {} = \\sum_{k=-\\infty}^\\infty \\left( |a_k|^2 + |\\tilde{a}_k|^2 \\right) + \\sum_{j=0}^\\infty \\sum_{k=-\\infty}^\\infty 2^{-j} \\left( |a_{j,k}|^2 + |\\tilde{a}_{j,k}|^2 \\right) \\\\\n& {} = \\int_{-\\infty}^\\infty |f(x)|^2 \\, dx.\n\\end{align}\n", "1c0fa24c0b1c58166a5ee2c4e9f20a87": "\\left ( 3 \\right )", "1c0fbbbde9215920564d42ae70502b95": "\\begin{align}\n\\Delta^r(\\Delta^s(\\phi_{1,1,1}))\\,&=\\,\\Delta^r(\\phi_{1,2,1} - \\phi_{1,1,1}) \n &=\\,\\Delta^r(\\phi_{1,2,1}) - \\Delta^r(\\phi_{1,1,1})\n &=\\,(\\phi_{2,2,1} - \\phi_{1,2,1}) - (\\phi_{2,1,1} - \\phi_{1,1,1})\n\\end{align}", "1c0fd973428f7e1a7e09b7da9676a592": "N_\\text{a} = N_\\text{s} + 2N_\\text{p}", "1c0fff5e0ca4e6a586809235be89cc79": "X\\mapsto H_n (X)", "1c10245b754c8e6d0698c8cd0f628537": " H_0 = \\hat{h}_1 + \\hat{h}_2 ", "1c10597cc5bc434d706f249e0bfd8ef7": "[X+\\alpha,Y+\\beta]= ([X,Y]_A +\\mathcal{L}^{A^*}_{\\alpha}Y-i_\\beta d_{A^*}X) +([\\alpha,\\beta]_{A^*} +\\mathcal{L}^A_X\\beta-i_Yd_{A}\\alpha)", "1c1087beec7e05baaee96dfdaf96cfe5": "\\overrightarrow{p_1p_2}", "1c10f36ae587282e208fab563b1145e6": "p(1)=3", "1c10f59b2252bc16ddd4de951817ac42": "\\text{sample excess kurtosis} =\\frac{6}{3 + \\hat{\\alpha} + \\hat{\\beta}}\\left(\\frac{(2 + \\hat{\\alpha} + \\hat{\\beta})}{4} (\\text{sample skewness})^2 - 1\\right)\\text{ if (sample skewness)}^2-2< \\text{sample excess kurtosis}< \\tfrac{3}{2}(\\text{sample skewness})^2", "1c113162851f177a8f6944a8c409ee17": "u\\equiv z_1z_2,\\qquad v\\equiv z_3z_4,", "1c114ae1cbff40368ee0bb5189394b52": "\\mathbf{x}(t')", "1c117fdd8fd0c106f4e0decce8983d33": "\\mathrm{Re}(e^{+i\\omega t})", "1c1180e71ea67a1fb5cd129f6fb5fdbc": "\\frac{C_p}{C_V}", "1c11931f15f98d5cf13e6761a9d7fd63": "\\textstyle \\vec{n}", "1c11bede59c9af484633e99262e2e684": "\nC(\\mathbb{C}^m, \\mathbb{C}^n) = C(\\mathbb{C}^{m \\times m}, \\mathbb{C}^{n \\times n}) \n== C( \\mathbb{C}^{m \\times m}, \\mathbb{C}^{n} ) == \\frac{\\log n}{\\log m}.\n", "1c11fe9581c989f7d688665cbc73d90d": "\\textrm{vercosin} (\\theta) := 2\\cos^2\\!\\left(\\frac{\\theta}{2}\\right) = 1 + \\cos (\\theta) \\,", "1c12251cea7debdedf0e62e64c3930ff": "\\int\\limits_0^1\\log\\Gamma(t)\\,\\mathrm dt = \\tfrac12\\log2\\pi.", "1c1226f47a25edc60c9cd653bc4bdebc": "T = {\\hbar \\, c^3 \\over 8 \\pi G M k_\\text{B}} \\;\\quad \\left(\\approx {1.227 \\times 10^{23}\\; \\text{kg} \\over M}\\; \\text{K} \\right),", "1c12376cc505e8a72244aa662621cb54": "\\frac{d^{2}u}{d\\theta^{2}} + u = -\\frac{m}{L^{2}} \\frac{d}{du} V(1/u)", "1c125a2ae5a628c083c9369d242ba13d": "Q(p) = f\\,(L, K),", "1c1263b1646e5bd902079616e3ee13f1": "\\operatorname{cofactor}(X) = |\\mathbf{X}|(\\mathbf{X}^{-1})^{\\rm T}", "1c126fa2fc355e874f34ec98123b3b53": " R_{XY}(\\tau) = CR_X(\\tau), ", "1c12f6a5d232cfc8a2360925180a7f59": "V_T = V_s + V_v = V_s + V_w + V_a", "1c130cac70a0f2aacb737a3e9acc27b9": "g_n(z)=z+\\frac{c_n}{n}\\varphi (z),", "1c132d46fec5ab681325d7b4f19a7f05": "(4)\\; z=\\frac{4.5-1}{22}=0.1591", "1c13371b2c541fb7852977f931330e3c": "z \\mapsto z + a", "1c139968313ae8d5363c6ef1cfc13f35": "(*) \\left\\{\n\\begin{matrix}\n\\mathrm d{p} &= \\theta^1{\\bold e}_1 + \\cdots + \\theta^n{\\bold e}_n & \\\\\n\\mathrm d{\\bold e}_i &= \\omega^1_i{\\bold e}_1 + \\cdots + \\omega^n_i{\\bold e}_n, & \\quad i=1,2,\\ldots,n\n\\end{matrix}\n\\right.", "1c13a5ee6ab71c6b0d7787cb5023019a": "[x,x] = 0", "1c13cc3e4d6e22207c6caac1651250f5": "d(f,g)=\\|f-g\\|_\\infty", "1c13f1a640102992b0703e4028a122c5": "\\scriptstyle T_a(p) \\;=\\; p \\,+\\, a", "1c13f8f8a3dfd13cecfb96bde3fbe101": "\\le k", "1c1404a2fc0d1be70a90da91795c6e30": "centers = \\{ c : P_n(c) = 0 \\}\\,", "1c144b1facb8147be1d364c1f3aa5f83": "\\scriptstyle S_x^{+}(s)", "1c14cd6eac1111393d7f47a9be102249": "(C_n)_{n\\in\\mathbb{N}}", "1c14e50f05f5bfc14d739d3a8a18e3ad": "\\!\\mathcal A \\models_X \\phi \\rightarrow \\psi", "1c15255f6625a8129cb3420ccd2987a9": "a = 2^{-j}", "1c15858c11e32f86b05196bcd40958b8": "\\Phi_{00}", "1c166155a1f76a20a9ef32b0ef6e6bb6": "\\gamma:[0,1)\\rightarrow D^{2}", "1c16cea880f7c1eace3bd11ccf0d4ec8": "\\sum_{n=-\\infty}^\\infty y^{2n}x^{n^2}. ", "1c16ed2adb88b608ce55ac2b22b1e5c3": " f_0(t) = \\frac{1}{T(t)} ", "1c171ac975f69ce91068d4ad59e8e234": " J[y] = \\int_{x_1}^{x_2} L[x,y(x),y'(x)]\\, dx \\, .", "1c17880e229fcbf88dde39a6666438a0": "Q_B |\\Psi_f\\rangle \\neq 0", "1c1843477d61eaa8cee862abd981f66d": "(s,o)", "1c189b8d4ab34a374ea1db5bf8c7a691": "\\overline{T}_2=\\frac{1}{L}\\int_0^LT_2(x)dx.", "1c18ba03a086176ed47863873fb6aa13": "A \\in \\mathcal{X}", "1c195fa11fb6ab2d83133a09f395a515": " \\Pi \\colon (\\text{pairs of pointed spaces}) \\rightarrow (\\text{crossed modules}) ", "1c19aed024aae6bb67ef5941098d7e1c": "OOO\\ldots\\Box\\varphi", "1c19b747b91be9de63ec5e021c355e6b": "g(n) = \\sqrt{n}.", "1c19c002bd5e33b15d6d5398d6c23f85": "\\tan \\theta = \\frac {v'} {S} \\,,", "1c1a33f54e0844557f66fe3e1fc872ab": "R \\le 1- ({q \\over {q-1}}) \\delta + o(1)", "1c1a50ea1152933029c66c6f8854eebc": "M \\rightarrow \\infty", "1c1a6fb40c2f261e7e16ece20dd3da00": " y = b \\sin(\\omega t) ", "1c1a7b95ee3850a9b7d2346d17728bf2": "x(N) = \\frac{P_0\\cdot r}{N(1 - (1 + \\frac{r}{N})^{-NT})}", "1c1b0a06692f0530b7ca3831d3596a48": "p\\circ p = p", "1c1b6c70508edb5d2387db962f3891e7": "\\mathbf{S}_{2}", "1c1b9f928a043719243378b58dfcfcd8": "\\kappa=\\frac{\\alpha_{m}\\alpha_{m-2}}{\\alpha_{m-1}^{2}},", "1c1bb9596aa054e6347b814a0b6e5583": "\\mathbf{Z}_2 \\to S^n \\to \\mathbf{RP}^n.", "1c1bcc2341d2cea15fe70765ca8bf8f9": "B e^{i\\phi}\\,", "1c1be871cfb9fb0a293d5f73288e4ebe": "Dih_4", "1c1cbcfe4d3a1b4d106328daf0875bc0": " \\!\\ 1/S_m = S_m - m ", "1c1cd7af4570f58a2402af7803cb47f1": "u = (a+\\sqrt{a^2-1})^2(b+\\sqrt{b^2-1})^2(c+\\sqrt{c^2-1})(d+\\sqrt{d^2-1})", "1c1d5d5a574ec21053e714ce9a4ce4ae": "\ne^{A t} = B_{1_1} e^{3/4 t} + B_{1_2} t e^{3/4 t} + B_{2_1} e^{1 t} + B_{2_2} t e^{1 t}\n", "1c1d80d4e998316802ce362027bf7aa9": "(f\\circ g)' = (f'\\circ g) \\cdot g'.\\, ", "1c1daf8850c8415a7262cd0dcb19620e": "(n!)", "1c1e0bc9abd49290c928d56275cceb45": "1 \\leq i \\leq n_A", "1c1e453942119efd769f48ad24d4444a": "\\operatorname{Sym}(V)= \\bigoplus_{k=0}^\\infty \\operatorname{Sym}^k(V).", "1c1e8448a47b21b3cd097797832531e8": "PFB = \\frac{(3200)(FC)}{(FW)(MC)} = \\frac{(3200)(200)}{(2.0)(4800)} = \\frac{200}{3} = 66.667", "1c1eccc738a3b8d755e8fc1ac00cd60e": "G_R = | H_R(s) | = \\left|\\frac{V_R(s)}{V_{in}(s)}\\right| = \\frac{R}{\\sqrt{R^2 + \\left(\\omega L\\right)^2}}", "1c1fb988a11789b7d6f121ddb042f451": " \\mu = IA = \\frac{qv}{2\\pi r}\\times\\pi r^2 = \\frac{q}{2m}\\times mvr = \\frac {q}{2m} L ", "1c1fd781089b82b9fe6a42ba88019dc4": "\\mathcal{H}[t] = \\dot{\\vec{x}}[t] \\cdot \\vec{p}\\,[t] +\\frac{mc^2}{\\gamma} + e \\phi [\\vec{x}[t],t]=\\gamma mc^2+ e \\phi [\\vec{x}[t],t]=E+V \\,.", "1c1fdd1de6a32db6ad3aa8a92a51e0c2": "S_-|-\\rangle=0", "1c2038bdcb5b60e6b4f43ea8b1f4e6a8": "222 \\times 60 = 13,320\\,", "1c20f3e7f2ba7598541fc36da8ff6178": "\\textbf I(\\alpha) = \\frac{\\pi^2}{8}-\\frac{\\alpha^2}{2}.", "1c21186725ed1d42410ad321f749053f": "a_{C} \\frac{Z(Z-1)}{A^{1/3}}", "1c213c4726d511f1c1cb6e8ba3c006f9": "\\sum_{i=1}^\\infty a_i^2 K (x_i, x_i) < \\infty", "1c216826199658525800a2ab1e6312db": "K \\to K", "1c21787c738240b28048a0e6e4c5fb08": "\\operatorname{ad}(x)(y) = [x, y]", "1c218969c72c38345eaf0a71e2613712": "\\langle\\widehat{\\mathbf{a}}_j^{\\dagger m}\\widehat{\\mathbf{a}}_k^n\\rangle=\\int P(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)\\alpha_j^n\\alpha_k^{*m} \\, d^{2N}\\mathbf{\\alpha}", "1c21d6f6d066fbfcc7beed0e17c30cee": "K_{xx}", "1c2262d98ed87159978aea0fd2a024f6": " \\{www | w \\in \\{a,b\\}^+ \\}", "1c228984cad817c8b582d458a2b25eba": "=[\\operatorname{sinc}((M\\cdot c) \\cdot \\xi, (N \\cdot d)\\cdot\\eta) *\n\\operatorname{comb}(c \\cdot \\xi, d \\cdot \\eta)] \\cdot\n\\operatorname{sinc}(a \\cdot \\xi, b \\cdot \\eta) ", "1c22c069120c7ebadbbf05810c768983": "e = \\left [ \\sum_{k=0}^\\infty \\frac{1-2k}{(2k)!} \\right ]^{-1}", "1c22d7bec7042246e37d6d46cdce8f23": "P \\lor Q", "1c23625d40d3c6b9107062ef398e5307": "\n\\frac{\\partial}{\\partial t}\\left[\\phi\\left(\\frac{S_o}{B_o}+\\frac{R_VS_g}{B_g}\\right)\\right]\n+\\nabla\\cdot\\left(\n\\frac{1}{B_o}\\vec u_o+\\frac{R_V}{B_g}\\vec u_g\\right)= 0\n", "1c236f8988fd7388ba3a8766342066f8": "s < RN_i[j]", "1c2375bcaedb29772242fb87cb7e1de8": "\\Delta r_j\\Delta c_i\\Delta v_k\\,", "1c237cd3ae701ab11924fbb01e974192": "\\mathbf{e}_3(t) = \\frac{\\overline{\\mathbf{e}_3}(t)} {\\| \\overline{\\mathbf{e}_3}(t) \\|}\n\\mbox{, }\n\\overline{\\mathbf{e}_3}(t) = \\mathbf{\\gamma}'''(t) - \\langle \\mathbf{\\gamma}'''(t), \\mathbf{e}_1(t) \\rangle \\, \\mathbf{e}_1(t)\n- \\langle \\mathbf{\\gamma}'''(t), \\mathbf{e}_2(t) \\rangle \\,\\mathbf{e}_2(t)\n", "1c23d1fbd91890018fda967d45f4b25c": " s_{m} = \\sum_{n=1}^{m} x_{n}", "1c2418649406dc739968814876cf36a1": " z \\to 0 ", "1c24523549562edfa26af03c951bdb8f": "\\gamma=1/\\rho_c^2", "1c24570005d8161946d8f02ea5c9e997": "M=2^{n\\left[ I\\left( X;B\\right) -3\\delta\n\\right] }", "1c2459418dab45e0544cbb3f217cba4b": "\\frac {d \\vec r(t)} {dt} = \\vec{\\omega} \\times\\vec{r}", "1c2466f678c627c3d17525492cc085a3": " Efficiency of Conversion for ingested food(ECI) * Food Calories=Net Production(Calories) ", "1c24898a9a24f0a1010dd5311f4023dd": "{ }^\\dagger:Y^+\\rightarrow Y^+", "1c24ca8017ca81ba734cd54f904d163b": "\n\\begin{align}\nP^{(n)}_e & \\leq P(U) + P(V) + \\sum_{j \\neq i} P(E_j) \\\\\n& \\leq \\epsilon + \\epsilon + \\sum_{j \\neq i} 2^{-n(I(X;Y)-3\\epsilon)} \\\\\n& \\leq 2\\epsilon + (2^{nR}-1)2^{-n(I(X;Y)-3\\epsilon)} \\\\\n& \\leq 2\\epsilon + (2^{3n\\epsilon})2^{-n(I(X;Y)-R)} \\\\\n& \\leq 3\\epsilon\n\\end{align}\n", "1c24ece846319faa3989f0b5e3c53e5e": "(z_{1it},...,z_{kit}\\;)^T", "1c2523069f287e72d2eea598429d616b": "\\scriptstyle \\left(\\alpha_i\\right)_{i \\in I} \\;=\\; \\left(0\\right)_{i \\in I}", "1c2535e439062a1e99a1bdd2e83e9089": "g(x) = \\frac{x - f(x)}{|| x - f(x) ||}. \\, ", "1c256fc01911a73e504ac53ea2f29b50": "\\begin{bmatrix} 1 \\\\ 2 \\\\ 1 \\\\ 1\\end{bmatrix},\\;\\;\n\\begin{bmatrix} 3 \\\\ 7 \\\\ 5 \\\\ 2\\end{bmatrix},\\;\\;\n\\begin{bmatrix} 4 \\\\ 9 \\\\ 1 \\\\ 8\\end{bmatrix}\\text{.}", "1c26802768c8eb6ab905e47dcc74cbd7": "\\left\\{ v_{L} \\right\\}", "1c2681b115ed29f7c42ff60d27bb0e9c": "AB=BA", "1c26a1cab34b9b4427c05450b0df2ee7": "I\\mathcal{Q}_{\\mathrm{Hur}}", "1c26b78904afd9689e1e8a52991d525e": "\\sigma_{a \\theta v}( R )", "1c271a14323dbb2aa3096b22449c34ec": "\\mbox{Scale-inv-}\\chi^2(v_0, s_0^2).", "1c272caebe34b29c949a0c5c1beaef29": "w''(x-) = w''(x+)", "1c275bd042e5387bd4d1858e95835650": " AB^{-1}", "1c27e1ca7f901422b7bb0058893130bc": "Alt^p(V) = (\\wedge^{p+1} V^*)\\otimes V", "1c282d4d08aaf0dc06c6d9db4f8f3551": "\\Gamma(x) = \\begin{cases} \n\\frac{1}{2\\pi} \\log{ | x | } & d=2 \\\\\n\\frac{1}{d(2-d)\\omega_d} | x | ^{2-d} & d \\neq 2.\n\\end{cases} ", "1c28374fa6e239fc1024e8387e8d020f": "\\theta\\in [0,1)", "1c284c091253a8a8923a34720482b691": "\\Pr(A \\mid \\Sigma) := \\operatorname{E}[\\chi_A\\mid\\Sigma].", "1c284e8552b000e5e2a5456b38aea17b": "\\boldsymbol {E=-{v\\times B}}\\ .", "1c2855e948898bc7e18d51953059667a": "\\frac{\\partial^2 \\epsilon_x}{\\partial y^2} + \\frac{\\partial^2 \\epsilon_y}{\\partial x^2} = 2 \\frac{\\partial^2 \\epsilon_{xy}}{\\partial x \\partial y}\\,\\!", "1c2857e8182497818b15b011934c7ca5": "b~", "1c2863f950db4807e3e2e9614cfdca6d": "\n M = R_A x - P(x-a) = Pbx/L - P(x-a)\n ", "1c2884d030f539376a8fa17586ec144e": "\n F_{ii}F_{jj} - F_{ij}^2 \\ge 0\n ", "1c28c6380b87e509f4531b5e9cff17bd": "0\\le Z_{i,t}\\le S_{i,t-1} ", "1c2903397d8833382673bab22aa8b937": "CN", "1c290e9a0eb1c4b2cf2cbcdba32d6b91": "\nPr\\begin{cases}\nDs\\begin{cases}\nSp(\\pi)\\begin{cases}\nVa:\\\\\nS^{0},\\cdots,S^{T},O^{0},\\cdots,O^{T}\\\\\nDc:\\\\\n\\begin{cases}\n & P\\left(S^{0}\\wedge\\cdots\\wedge S^{T}\\wedge O^{0}\\wedge\\cdots\\wedge O^{T}|\\pi\\right)\\\\\n= & P\\left(S^{0}\\wedge O^{0}\\right)\\times\\prod_{t=1}^{T}\\left[P\\left(S^{t}|S^{t-1}\\right)\\times P\\left(O^{t}|S^{t}\\right)\\right]\\end{cases}\\\\\nFo:\\\\\n\\begin{cases}\nP\\left(S^{0}\\wedge O^{0}\\right)\\\\\nP\\left(S^{t}|S^{t-1}\\right)\\\\\nP\\left(O^{t}|S^{t}\\right)\\end{cases}\\end{cases}\\\\\nId\\end{cases}\\\\\nQu:\\\\\n\\begin{cases}\n\\begin{array}{l}\nP\\left(S^{t+k}|O^{0}\\wedge\\cdots\\wedge O^{t}\\right)\\\\\n\\left(k=0\\right)\\equiv Filtering\\\\\n\\left(k>0\\right)\\equiv Prediction\\\\\n\\left(k<0\\right)\\equiv Smoothing\\end{array}\\end{cases}\\end{cases}\n", "1c292d1b5f4509e592cf514474627b37": "\\{0, 1\\}^{\\omega}", "1c297e4b5a0eb679bc0e359b5e8ca21b": "K_{0}-K_{1}=\\frac{E}{c^{2}}\\frac{v^{2}}{2}", "1c298f636304df472f3afe17835056cc": " {\\mathit l \\over \\mathit l^{\\prime}} = 1.", "1c2996b132a5e1a99ab974a7847bb4ac": "[S:T] \\mapsto \\left[\\frac{1}{(a_0S-b_0T)} : \\ldots : \\frac{1}{(a_nS-b_nT)}\\right] ", "1c299de5ca9a68113e73838956143b6e": "\\varepsilon_{d,e}(n) \\ll \\mu_\\infty", "1c29c75443c7353fa8c0e751fe7147e9": "D(E(m_1, r_1)^k\\mod n^2) = k m_1 \\mod n. \\, ", "1c2a30042227ef28e6f80a0357486a17": "\nW = p_{me} V_d\n", "1c2a7a71e9919fa300337686e14c401a": " g_{obs} ", "1c2b18f667fd7890f3cf77fbe3a6b2d3": "\\Lambda_{\\mathit{g}}=\\frac{\\sum_{\\mathit{g}=1}^\\mathit{G} \\overline{\\lambda_{\\epsilon,g}}}{\\mathit{G}}", "1c2b590c64bf541a3776f1f172cc55db": "1/(1-c(1-t)+m)", "1c2b764adae2f300ae3ec5af7955b7ec": " t\\in \\mathbb{N} ", "1c2c2362531608056be5fd4f52d34593": " z^2-z-2", "1c2c24ee2eb5178071df9cffe3fc076b": "X_0 \\rightarrow Hc + X(t)", "1c2c731b59e264d7418cf2e67ff13a80": "[\\mathbf{\\omega}]_\\times = \\left[\\begin{array}{ccc} 0 & -\\omega_z & \\omega_y \\\\ \\omega_z & 0 & -\\omega_x \\\\ -\\omega_y & \\omega_x & 0 \\end{array} \\right] = \\frac{d\\mathbf{A}}{dt}\\mathbf{A}^\\mathrm{T}", "1c2c79bba234d498032a8fa0c99b0249": "\\scriptstyle k \\;\\in\\; \\mathbb{N}", "1c2d1ea3189036f4e7c69fb2aaead855": "TR(Q) = P(Q) \\times Q", "1c2d71e7f7350aa94366105ee126c3ec": "(1-k)\\psi(k) + \\ln \\frac{\\Gamma(k)}{\\lambda} + k", "1c2d73cfb375053f0e8ac0d545939f5c": "A_{11}U_1 = F_1 - A_{1\\Gamma}U_\\Gamma, \\qquad A_{22}U_2 = F_2 - A_{2\\Gamma}U_\\Gamma,", "1c2d7e686e3fbb8df4e3d04d670bbe89": "r_{o1} \\leq r \\leq r_{o2}: B(r) = \\frac{\\mu_o I}{2 \\pi r} \\left( \\frac{r_{o2}^2 - r^2}{r_{o2}^2 - r_{o1}^2} \\right)", "1c2d8136bdf41e80805ad06d618aebb1": " \\lim_{k,\\ n \\to \\infty}\\int_\\Omega |{f}_k (x)-{f}_n (x)|^p \\, dx = 0", "1c2da119093f203b8199820fed981ca5": "m=y/k_0", "1c2e506e6b9cf80dec6b872f2a8df579": "\\nabla \\varphi", "1c2e523fa41945de6b573c3f2bdaf0ac": "\\{\\mathbf{r}_i,i=1\\ldots N\\}", "1c2e6d1e5251188006726e21fb28c9bf": "i_\\Sigma(U_g)", "1c2e99e91c7591f266382696af1cd0ea": "\\dot\\vee", "1c2ecc1e484af46e354c703b9b883320": "Pr[a|x,y] = \\frac{e^{\\frac{S(a)}{T}}}{Z}", "1c2ef73a3ca30e01e84925abc81d20b1": "p = 3 \\sqrt[3]{uv}", "1c2fa9d25e93b49cb5ef04dfd369e034": " \\frac{1}{\\pi} = \\frac{2\\sqrt{2}}{9801} \\sum^\\infty_{k=0} \\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} ", "1c30425f594774f625f07b91e0c74f44": "x^{(m)}_1", "1c30562b13bbd6274b768336b8ad404c": "\\alpha = 2\\beta \\,\\!", "1c3078a6d060a4be8480bc5786c83050": "\\operatorname{Morse}(M) \\subset \\operatorname{Func}(M)\\,", "1c30ef10d44fc702b8ec7b47505d274d": " \\frac{dN(t)}{dt}=((b-aN(t))-(d-cN(t)))N(t) ", "1c3124c4a6828ae5c34e380d1fe2aa6e": "S \\subseteq U", "1c3142900f48dba597d93d2353b416f3": "\n\n{\\partial F\\over\\partial\\tau} = {1\\over 2} {\\partial\\over\\partial\\mu} \\left[(1-\\mu^2) {\\partial F\\over\\partial\\mu}\\right] \n\n", "1c3197309f2484c3acc03cb9cb5e23c8": "N \\ll 1", "1c31c4f07766a89d9ea719eb295714c6": "{\\tilde{A}}_{2+}", "1c322c8b009a068add19f34f14ccbcae": "\\mathcal{I}(\\theta) = \\frac{n}{\\theta(1-\\theta)},", "1c32337809de0fb306e63a187716e426": "(\\Lambda(A) : A \\in \\mathbb{B}_b(S))", "1c323c8ca15a4c35cb36b4da70fe0ff7": "y^2(a+x) = x^2(a-x)", "1c3249898004ed7c0dbf777e27170afd": "\n\\begin{array}{lcl}\n\\#\\mathbb{W}^{k} &=& \\text{number of words generated by topic }k \\\\\nB &=& \\sum_{v=1}^{V} \\beta_v \\\\\n\\end{array}\n", "1c324cad92f5b0e361fb8aabfd970fbb": "n_x=\\cos{\\theta(z)}", "1c3256d49054045ca0fc464c35a5dfba": "\\dim P({\\Bbb M}) = 24", "1c3267b5eeb9bee4f051357c232fa0a9": "f_i\\,", "1c3272b41fbbb0766c15f8a53642fb84": "\\mathrm{argmax}", "1c32745198714b41fac2325dffbda74e": "\\frac{dy_c}{dx} = \\left\\{\\begin{array}{ll}\n\\displaystyle{\\frac{2m}{p^2} \\left(p - \\frac{x}{c} \\right)}, & 0 \\leq x \\leq pc \\\\\n\\\\\n\\displaystyle{\\frac{2m}{(1 - p)^2} \\left(p - \\frac{x}{c}\\right)}, & pc \\leq x \\leq c\n\\end{array} \\right. ", "1c32cc86a864e97cdcbd295d388a031e": "\n\\begin{align}\n \\epsilon_j^n & = e^{at} e^{ik_m x} \\\\\n \\epsilon_j^{n+1} & = e^{a(t+\\Delta t)} e^{ik_m x} \\\\\n \\epsilon_{j+1}^n & = e^{at} e^{ik_m (x+\\Delta x)} \\\\\n \\epsilon_{j-1}^n & = e^{at} e^{ik_m (x-\\Delta x)},\n\\end{align}\n", "1c32da2676256c493882db195b9bb467": "\n\\begin{align} \nProb(choosing \\, 1) \n& = Prob(U_n \\; 1", "1c3e4b496b916e02089dbce96fc9f352": "P_1, \\ldots, P_g", "1c3e5b01cf3fb0746fbdfaef6196bdc5": "\\,p\\,", "1c3ebc7ce19cf99522c07cbfd56cd4b7": "\\langle x,x \\rangle = 0,", "1c3ed3cad133be512c832aa31c66e756": "-\\Delta G_v", "1c3ee470a49884f06eebbcc89cf037db": "\\alpha_F=\\frac{\\mu_\\theta-\\mu_{ref}}{\\mu_{ref}^2*(\\theta-\\theta_{ref})}", "1c3ef2e566aeeff55d60174956f6df09": "\\Gamma(n) = (n-1)! \\,", "1c3f6b03df3e5052c5fa146f834629b9": "\\mathbf{E} (X_{n+1} - X_n \\mid X_1,\\ldots,X_n)=0", "1c40b838dc2d0ea913fbc87f8736ffc9": "(1-\\epsilon)\\int_{\\psi}(|\\psi\\rangle \\langle \\psi|)^{\\otimes t}d\\psi \\leq \\sum_i p_i (|\\phi_i\\rangle \\langle \\phi_i|)^{\\otimes t} \\leq (1+\\epsilon)\\int_{\\psi}(|\\psi\\rangle \\langle \\psi|)^{\\otimes t}d\\psi", "1c40ba79ab3adfea1caefb39bf16beb3": " H^{s} \\left( E\\backslash \\bigcup_{j}U_{j} \\right)=0 \\ \\mbox{ or }\\sum_{j} \\mathrm{diam} (U_{j})^{s}=\\infty.", "1c40d31668eae673d5e93a765605a29b": "e =\n\\pi^2/3", "1c41229e2b74fe6a119548ffc4e2be66": "Ne^{-0.69(\\Delta\\ln\\Delta)^{1/3}}", "1c4143e1a048dcc6bf4042a06c8d4cc7": "\\frac{d^2\\eta}{d\\theta^2} + \\beta^2 \\eta = \\frac{1}{2} \\eta^2 J^{\\prime\\prime}(u_0) + \\frac{1}{6} \\eta^3 J^{\\prime\\prime\\prime}(u_{0}) + \\cdots ", "1c41a54117126c2d3e9fddd89835744d": "\nK \\approx S^2 = (\\Delta \\omega_r/\\Delta_d)^2 > 1 .\n", "1c41c5d4f3efd10f91738280b1ce6404": "(P \\lor \\lnot Q) \\land (\\lnot P \\lor Q)", "1c424ae50281fefe6d4ee5c6fc203d1a": "\n\\frac{d}{d\\tau} \\left[ \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\right] = 0,\n", "1c428fe2154ada309d28ab3fdbf27691": "\\displaystyle{T(t)=e_t(T)}", "1c42fe52256483ab41bb7b17ae8384fe": "S(x,T) = \\sum_{\\rho:|\\Im \\rho| \\le T} \\frac{x^\\rho}{\\rho} \\ . ", "1c43187e6537884cf6df72751ffd3d88": "t_0 = {(E(Y)-u_0)}/{(S/\\sqrt{n})}", "1c43543eaf7dd5e788958ac606719245": "\\text{RR} = \\frac{1}{N^2} \\sum_{i,j=1}^N \\mathbf{R}(i,j).", "1c43a44932f4d872a0822895d8d60002": "\\displaystyle{P=\\bigcup_{\\sigma\\in W_\\lambda} B\\sigma B,}", "1c43e15a6a5759ff701a87602a7e2a68": "e^{-\\phi} = \\gamma(1-\\beta) = \\gamma \\left( 1 - \\frac{v}{c} \\right) = \\sqrt \\frac{1 - \\tfrac{v}{c}}{1 + \\tfrac{v}{c}}.", "1c43f5ea701b38b0911d9f759851eab5": "(u+du,v)", "1c4401173d3ad39aac1763e8af2899d9": "\\rho(x,y)=\\sum_{n=1}^\\infty \\, 2^{-n} \\, \\frac{\\left|\\langle x-y, x_n\\rangle\\right|}{1 + \\left|\\langle x-y, x_n\\rangle\\right|}", "1c448d9a95cb8443ad3aebe570f84a1c": " \\epsilon \\neq 0 ", "1c4538f141eea8fca6edcae46b84952c": "f:\\mathbb{R}^+\\to\\mathbb{R}", "1c45eb5ee0916e325a82f4e65b562e42": "\\hat{X}_i(z^n) = f\\left( z_{i-k},\\ldots,z_{i+k} \\right)", "1c45ebb05173f6e426ded4a1daf8a971": "\\operatorname{gr}_I(R) = \\oplus_0^\\infty I^k/I^{k+1}", "1c462dfa9fb82a30c577e6071ece6058": "w_k = \\prod_{j\\ne k}\\left[\\frac{1}{(x_k-x_j)^2+4\\sigma^2}\\right]\\left[1-\\frac{2\\sigma}{(x_k-x_j)}i\\right].", "1c465b7374b42f36c9793deba30b1424": "\\quad (6) \\qquad D = 3\\pi \\mu d V \\qquad \\qquad \\text{or} \\qquad \\qquad C_d = \\frac{24}{Re} ", "1c46cfa94fcc2df31d3b02d6f005e381": " K\\ge \\delta\\,.", "1c471e07976adc044ea76befbdf83d49": "dN_A", "1c479d86edcf59cac399b3c7ce809692": "P_\\delta = \\{\\,\\epsilon\\tau \\mid 0 \\le \\epsilon \\le \\delta,\\tau \\in \\Sigma\\,\\}", "1c480bfc50af9298b9c1375718c80d5f": "X^2 - X = T^{-1}", "1c4815f29e0ab19d0b2189ecba95b328": "\\left\\langle r_1,r_2 \\mid (r_1)^2=(r_2)^2=(r_ir_j)^3=1\\right\\rangle.", "1c486b8ceb456485b79590f7f74ea4a3": "R(k_{sp})=-m((A + 3B\\phi_{in}^2)k_{sp}^2 + \\kappa k_{sp}^4)=\\frac{m(A+3B\\phi_{in}^2)^2}{4\\kappa} = \\frac{1}{t_{sp}}", "1c487b6309929f01684384c1f6996f4c": "\\mathrm{Spec}(R)\\,", "1c48a42f3cb0847676c6a9037010b507": "\\lambda = n (n+1)", "1c48b8143edc09b9af7372a2f64ed0a8": "n!\\,", "1c48db78ec96cfa89ec7d1be5f76f092": "\\mathbf{\\tilde{W} = WB}", "1c49520f0d8adc0fe1a6a84e6602f009": "\\cdots \\to \nA_{n+1} \\xrightarrow{d_{n+1}} A_n \\xrightarrow{d_n} A_{n-1} \\xrightarrow{d_{n-1}} A_{n-2} \\to\n \\cdots \\xrightarrow{d_2} A_1 \\xrightarrow{d_1}\nA_0 \\xrightarrow{d_0} A_{-1} \\xrightarrow{d_{-1}} A_{-2} \\xrightarrow{d_{-2}} \n\\cdots\n", "1c4988f6d3eb6e846b3e993a74985cc4": "\\langle p? \\rangle q \\equiv p \\land q\\,\\!", "1c499338dc9e5c0944eadda56ce89a0c": " (M,\\{ \\cdot,\\cdot \\}_{M}) ", "1c499c81c22c9e35d499955324741447": " \\textrm{Spec}(K) ", "1c49bd60b17ecb7b07f5421cc0c30f41": "N(q,n)=\\frac{1}{n}\\sum_{d|n} \\mu(d)q^{\\frac{n}{d}},", "1c49ea22300f7dfdc46f257bc0f7cca8": " r = \\frac{ \\bar{ y } }{ \\bar{ x } } = \\frac{ \\sum_{ i = 1 }^n y }{ \\sum_{ i = 1 }^n x }", "1c4a005185dc1442ee5ed80d2cdef37b": "a^d \\not\\equiv 1\\pmod{n} \\text{ and }a^{2^r \\cdot d} \\not\\equiv -1 \\pmod{n} \\text{ for all }r \\in[0,s-1]", "1c4a2e4e903c1e6d7407b5b6b04857bc": "\\mathcal{O}_Y(V)", "1c4a680612d4325954d3598f4f5ccda3": "\\sigma_{zx}\n=-\\frac{\\partial^2\\Phi_{zx}}{\\partial y \\partial y}\n -\\frac{\\partial^2\\Phi_{yy}}{\\partial z \\partial x}\n +\\frac{\\partial^2\\Phi_{xy}}{\\partial z \\partial y}\n +\\frac{\\partial^2\\Phi_{yz}}{\\partial x \\partial y}", "1c4ae5cd5e03ce9525c477cdcc43e697": "a\\geq 1, b\\geq 1, (a,b)\\neq (1,1)", "1c4bcce1e6eaa460149ddee222bcae1b": "\\pi_1(B)", "1c4c2cafb71629c1d53036c979c00745": " A = \\frac{\\theta}{2} (R^2 - r^2) ", "1c4c7e5b289b597cdb93d142c5c314b6": "\\;\n\\sum_k p_k = 1.\n", "1c4c82613dea79522895a92aaf101488": "\\{ x\\}\\;", "1c4c8752888327e81aa3eb6b08f1a267": "\\overset{\\ \\ \\uparrow}{\\leftarrow}", "1c4d15dbd8927e28500ebd372157a666": "2 \\int_M^{\\infty} G(u, t) \\, du = 2 \\int_{\\frac{M}{\\sqrt{t}}}^{\\infty} G(v, 1) \\, dv < \\varepsilon.", "1c4d4639dab2e33c8cfb92a5114f34d9": "\\frac{}{}LC", "1c4d6a41ac38a2b6917ce63f2532368a": "(X_1, \\ldots , X_n)", "1c4d7e42642bd2735940546c805c0d46": "\n\\mathbf{S} \\mathbf{v} = \\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle \\mathbf{e}_{k}\n", "1c4d9c0cedd24d24519f6b736e6fac0d": "\\hat{h}(\\xi)=\\hat{f}(-\\xi).", "1c4dcd3ff8ccfb15e4baeb9a7a4a7445": "\\frac{df}{dx}=-w^2\\frac{df}{dw}", "1c4df245f03e161511e62aaaf07d095e": " = e(p_1, u_0) - e(p_1, u_1)", "1c4e1f5790311980dcc9e27da292725c": "S = S(E(\\boldsymbol{r}))", "1c4e69e5c6d5cac17caf1f5892daa30f": "\\lim_{q\\rightarrow 1}\\frac{1-q^n}{1-q}=n,", "1c4e7194da4aa7c2a1b7cf24a4c9e6c8": "\\lim_{|n|\\rightarrow \\infty}\\hat{f}(n)=0", "1c4e9bfc751d2af604b2c4dd3d25f05a": "\\theta_{p1}", "1c4eda9c7c048b179b530b3d13e6f9cf": "b(1)e^{\\beta(1)}+ b(2)e^{\\beta(2)}+ \\cdots + b(N)e^{\\beta(N)}= 0.", "1c4edb839c44be4902670b37ab7dd9bf": "= 10^{\\left ( \\log_{10}\\left ( \\sqrt{2\\pi n} \\left ( \\frac{n}{e} \\right )^n \\right ) \\right )} ", "1c4ef1790cdaf2a5b7979c108bb5591f": " \\mu_{Y \\mid x} = \\mathbb{E}_{Y \\mid x} [ \\phi(Y) ] = \\int_\\Omega \\phi(y) \\ \\mathrm{d}P(y \\mid x) ", "1c4efd4bab61a242856978b7696d8b31": "\\ln(K)", "1c4f811b3373e9e19a72b4b8f4129d53": "\\tan\\frac{\\pi}{60}=\\tan 3^\\circ=\\tfrac{1}{4} \\left[(2-\\sqrt3)(3+\\sqrt5)-2\\right]\\left[2-\\sqrt{2(5-\\sqrt5)}\\right]\\,", "1c4f818eeadcda33a3dfc8087f7b50b5": "j\\ge p", "1c4f9544f45764cb99c98aa9d1971d26": "\\forall g \\in G \\;\\; \\exists k \\in \\mathbb{N}", "1c4fa11dd36a7a148d168289d83e8ad5": "\\nu_-", "1c4fc8813eab2b90cf9d115c6af601c9": "\n \\mathrm{sf}(n)\n =\\prod_{k=1}^n k! =\\prod_{k=1}^n k^{n-k+1}\n =1^n\\cdot2^{n-1}\\cdot3^{n-2}\\cdots(n-1)^2\\cdot n^1.\n ", "1c4fea42b65c8a992897a3ab09b57be5": " (-\\omega_1,-\\omega_2)", "1c4ff757428239961f07611397dbf2be": "Div^0 (K)", "1c4ff8d63ca74605d1903b21a1c2037e": "\n2 T =\n\\begin{pmatrix} \np_{\\alpha} & p_{\\beta} & p_{\\gamma}\n\\end{pmatrix}\n\\; \\mathbf{g}^{-1} \\;\n\\begin{pmatrix} \np_{\\alpha} \\\\ p_{\\beta} \\\\ p_{\\gamma}\\\\\n\\end{pmatrix},\n", "1c4ffc545b59d3484e489fb0a64051e6": "\\Phi'", "1c500c16e7634d3875f5122c6a7c1e6a": "code_{i-1}", "1c501ad62472068a5a3efdadecdca821": "p_{n}(x)y^{(n)}(x) + p_{n-1}(x) y^{(n-1)}(x) + \\cdots + p_0(x) y(x) = r(x).", "1c50a4a02391a82a7e5052559d914e65": "\\textstyle a:=\\frac{k_f}{k_i}", "1c50b879bdeb42283abf63f666c15c69": "\n\\langle \\psi_2 | \\mu_z | \\psi_1\\rangle =\n\\left ( \\mu_z \\right )_{21} = \\int \\psi_2^*\\mu_z\\psi_1\\, \\mathrm{d}\\tau .\n", "1c50ef63bad5614f30f8222f9855c841": "w = 3", "1c510886acbe8d3685ed0412a5b29643": "\\theta(t) = \\frac{\\log\\Gamma\\left(\\frac{2it+1}{4}\\right)-\\log\\Gamma\\left(\\frac{-2it+1}{4}\\right)}{2i} - \\frac{\\log \\pi}{2} t,", "1c510fe172f7f0d649512631b74d078d": "\\Delta(x_1\\wedge\\dots\\wedge x_k) = \\sum_{p=0}^k \\sum_{\\sigma\\in Sh_{p,k-p}} \\operatorname{sgn}(\\sigma) (x_{\\sigma(1)}\\wedge\\dots\\wedge x_{\\sigma(p)})\\otimes (x_{\\sigma(p+1)}\\wedge\\dots\\wedge x_{\\sigma(k)}).", "1c51329b5519ac641e96d11751ca1703": "\\begin{align}\n \\mathrm{d} \\sigma \n &= \\left( \\sum_{i=1}^2 \\frac{\\partial u}{\\partial x^i} \\mathrm{d}x^i \\wedge \\mathrm{d}x \\right) + \\left( \\sum_{i=1}^2 \\frac{\\partial v}{\\partial x^i} \\mathrm{d}x^i \\wedge \\mathrm{d}y \\right) \\\\\n &= \\left(\\frac{\\partial{u}}{\\partial{x}} \\mathrm{d}x \\wedge \\mathrm{d}x + \\frac{\\partial{u}}{\\partial{y}} \\mathrm{d}y \\wedge \\mathrm{d}x\\right) + \\left(\\frac{\\partial{v}}{\\partial{x}} \\mathrm{d}x \\wedge \\mathrm{d}y + \\frac{\\partial{v}}{\\partial{y}} \\mathrm{d}y \\wedge \\mathrm{d}y\\right) \\\\\n &= 0 - \\frac{\\partial{u}}{\\partial{y}} \\mathrm{d}x \\wedge \\mathrm{d}y + \\frac{\\partial{v}}{\\partial{x}} \\mathrm{d}x \\wedge \\mathrm{d}y + 0 \\\\\n &= \\left(\\frac{\\partial{v}}{\\partial{x}} - \\frac{\\partial{u}}{\\partial{y}}\\right) \\mathrm{d}x \\wedge \\mathrm{d}y\n\\end{align}", "1c5172aeb28d4aeb3cd5087153da0386": "E = \\sum_{i}^N \\varepsilon_i - V_{H}[\\rho] + E_{\\rm xc}[\\rho] - \\int {\\delta E_{\\rm xc}[\\rho]\\over\\delta\\rho(\\mathbf r)} \\rho(\\mathbf{r}) d\\mathbf{r}", "1c51f6cbf795392ea96452762265415a": "\n\\begin{align}\n& p=\\operatorname{prox}_f(x) \\Leftrightarrow x-p \\in \\triangledown f(p) & (\\forall(x,p) \\in \\mathbb{R}^N \\times \\mathbb{R}^N)\n\\end{align}\n", "1c5217190a44289fc4a9eb9e66e80002": " \\frac{\\partial}{\\partial z} = \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x} - i \\frac{\\partial}{\\partial y} \\right) \\quad,\\quad \\frac{\\partial}{\\partial\\bar{z}}= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x} + i \\frac{\\partial}{\\partial y} \\right) \\ .", "1c529cec043f9393c5b9eeff1ee05313": "\n\\begin{bmatrix}\nc t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\gamma&0&-\\beta \\gamma&0\\\\\n0&1&0&0\\\\\n-\\beta \\gamma&0&\\gamma&0\\\\\n0&0&0&1\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nc\\,t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix} ,\n", "1c52eda493d450bd1d581d69967a7440": "{\\frac{\\log 4}{3}}n \\le (\\sqrt{2n}+1)\\log 2n\\; .", "1c53068008783089636dc7dd4c4db0d1": " K\\overset{\\underset{\\mathrm{def}}{}}{=}\\frac{P_d'}{P_a+P_d} ", "1c531023f400b197b6326eee95d0fb57": "\nd(1)+\nd(2)+\n\\dots+\nd(n)\n", "1c5334b07353ad26d30f5ea19c5d13d3": "\\delta n^a=\\mu m^a+\\bar{\\lambda}\\bar{m}^a-(\\bar{\\alpha}+\\beta)n^a\\,,", "1c53a2dc3676eae0a0a1d7bfeaadafec": "A_i\\;=\\;min(w_T, w_R, w)\\;x\\;min(h_T, h_R, h)", "1c53b7f9f523bf2260227916bf59a065": "Y=\\sum_{i=1}^2 X_i.", "1c544dbb59f8edc73f78a75a00058d64": "c_{11} = \\frac{ E (1-\\nu)}{(1-2_\\nu)(1 + \\nu)} ", "1c54740a278e3e6754c6613d37a45dd9": "= \\ln(1.23456) + \\ln(10^2) \\,\\!", "1c5489955c368efe305c6502fa9ef9d1": "F = \\frac{\\text{explained variance}}{\\text{unexplained variance}} ,", "1c54eba980b1dad6b86fd8dcc67a4370": "\n\n\\begin{bmatrix}\n1 & 2 & 3 \\\\\n0 & -6 & 7\n\\end{bmatrix}^\\mathrm{T} =\n\\begin{bmatrix}\n1 & 0 \\\\\n2 & -6 \\\\\n3 & 7\n\\end{bmatrix}\n", "1c54fa1ca18a7f95a9f9a47c415850dd": "\\left( T_{\\rm matter} \\right)_{ab} = \\phi \\, \\rho \\, u_a \\, u_b", "1c5588099e6cfe62a3c92aee15b30623": "W_{1-2} = \\int PdV\\,, \\quad PV = P_1 V_1 = C", "1c55c133bea8aba8e8387e50d98ccace": "W\\stackrel{f}{\\ \\to\\ } X\\stackrel{g}{\\ \\to\\ } Y\\stackrel{h}{\\ \\to\\ } Z", "1c55eb549799ee2b94ae0aff2cc131d1": "[x_a,y_a]", "1c561612f36ddb819edba5c11fb547b4": "\\csc^2x=\\cot^2x+1\\,", "1c56351cf3e3e978c052916892fec228": " \\mathrm{CG_{p}} = \\sum_{i=1}^{p} rel_{i} = 3 + 2 + 3 + 0 + 1 + 2 = 11", "1c563d20d5649f7f8e0bba17756dcc0c": "p\\mapsto \\deg(f,\\Omega,p)", "1c56bea9894c33e806353adcca923dc4": "B_{\\mathrm{cris}}\\otimes_{K_0}H^\\ast_{\\mathrm{dR}}(X/K)\\cong B_{\\mathrm{cris}}\\otimes_{\\mathbf{Q}_p}H^\\ast_{\\mathrm{\\acute{e}t}}(X\\times_K\\overline{K},\\mathbf{Q}_p)", "1c56c39632626797ea585e6ab90cb55f": " \\sum D = d1 + d2 + d3 + ... = W ", "1c5719558eaa6a8d98d3d19967bbb2d9": "(U(t),u(t))", "1c5763b9e8eb7f55dc116b43fea94aa3": "F_{[\\alpha\\beta;\\gamma]} = 0", "1c57ae6cbc9b53c0d31db055651dad64": "S=S_1 \\times S_2 \\times \\dotsb \\times S_n", "1c57b752b116101480666cfbdf066ef6": " \\frac{1}{P_n} = \\frac{2k_{t,d}+k_{t,c}}{{k_p}^2[M]^2}R_p + C_M +C_S \\frac{[S]}{[M]}+C_I \\frac{[I]}{[M]}+C_P \\frac{[P]}{[M]}+C_T \\frac{[T]}{[M]}", "1c57bfff308992577309393944f18144": "\\tfrac{8}{11}=\\tfrac{1}{2}+\\tfrac{1}{22}+\\tfrac{1}{6}+\\tfrac{1}{66}.", "1c581ea484d263fd6e80c4956016cb62": "\n\\frac{d}{dt} \\left( \\frac{\\partial T}{\\partial \\dot{\\varphi}_{r}} \\right) = \n\\frac{d}{dt} \\left( Y \\dot{\\varphi}_{r} \\right) = \\frac{1}{2} F \\frac{\\partial Y}{\\partial \\varphi_{r}} \n-\\frac{\\partial V}{\\partial \\varphi_{r}}.\n", "1c586082f47470b4924982809196cf78": "\\Omega(M, \\mathfrak g)", "1c58710a9ae54b5391bc956b4ad0b739": "w_{12}", "1c58c24c0cace2a7636b63b4a21b2f38": "\\partial \\Omega.", "1c58c49a8c16c94c9dd36d35dc2fefd1": " u(r,t) ", "1c59bab178620da6ded38dab4e9da1e7": "A,H", "1c59f6acef572ff8ecb140f2da10b619": "1 / \\phi^2 \\,", "1c5a0a5eef9692c3c64c564d6f3e7bda": "\\frac{1024}{729}", "1c5a1220efecab7294c9da0decf2e8dd": "H = -\\frac{\\hbar^2}{2m} \\nabla^2 ", "1c5a59d3392d3ae548f2bc5f90d3affe": "d = \\boldsymbol{\\bar{v}}t\\,.", "1c5b7210fa6af8b417c2aaf4029442e2": "Y \\sim N\\left(\\nu \\sin\\theta,\\sigma^2\\right)", "1c5b755b740cc6ec6c6c79ca5fab5faf": "\\lambda_{a'}\\frac{}{}", "1c5b8ad0b38072954978be014250a473": " Q(a)=2L(a)^2 -L(a^2). \\, ", "1c5b9ac51b36f47a2a0133af347d784b": "x_a = x_0 (u_a \\cos \\omega t + v_a \\sin \\omega t)", "1c5bd0d2cefbac52e9efdc1d23eb4d0f": "V^{\\prime }(t)=L_{t}(x^{\\ast }(t),y^{\\ast }\\left( t\\right) ,t)=y^{\\ast\n}\\left( t\\right) ", "1c5be04ffe5ce630ec04bc1e9d4d301e": "\\mathbf{B}_i=\\frac{\\rho_i}{a_i}=\\left \\langle \\exp \\left ( -\\frac{\\psi_i}{k_B T} \\right ) \\right \\rangle", "1c5c38c58a819eec741eaa412529eb61": "(F_T, G_T, \\eta, \\mu_T):C\\to C_T", "1c5c3c3deb192f2386f158c1db09b1c7": "\\nabla^\\perp u(x,y)=(-\\frac{\\partial u}{\\partial y},\\frac{\\partial u}{\\partial x})", "1c5c4d8ad4a109cdb8b02589c504c5e0": "\\scriptstyle f_s/2\\ >\\ |f|,\\,", "1c5c95e86bcc29aa01b84a2fdd0a982f": "\\sum_{i=1}^n \\min(|x_i-y_i|,q-|x_i-y_i|).", "1c5cc9e3cc8428c278542f2db5ae0367": "\\omega_{s(g=2)} = \\frac{eB}{mc\\gamma}", "1c5d311069ad014b707416c9e5b634a4": "\\color{BurntOrange}\\text{BurntOrange}", "1c5d4fb0da7e6cab54ee95e3876644d3": "x_2 = -10^{0.2192318 + 0.2706462} = -3.08943", "1c5df1a76b5213b6806eed9d517febbe": " \\sqrt{\\pi}x e^{x^2}{\\rm erfc}(x) \\sim 1+\\sum_{n=1}^\\infty (-1)^n \\frac{(2n)!}{n!(2x)^{2n}} \\ (x \\rightarrow \\infty).", "1c5e43ea86c386691c991c8393e0a4ab": " \\gamma = \\; F_{\\mathrm{r}} / F_{\\mathrm{M}} = \\beta_{\\mathrm{r}} / \\beta_{\\mathrm{M}}. ..........(39) ", "1c5f0856f920300aaf6c5fa98fa25f8f": "\\beta := 1 / \\alpha", "1c5f435fb535264e196d35a96c4f37bc": "\\frac {v_\\mathrm N - v} {v_\\mathrm N} = \\frac c d", "1c5f96fbaddf8646b1b61f37bee394cc": " \\begin{align} \\mu \\nabla^2 \\mathbf{u} -\\boldsymbol{\\nabla}p + \\mathbf{f} &= 0 \\\\\n \\boldsymbol{\\nabla}\\cdot\\mathbf{u}&=0 \\end{align}", "1c5fab45d7e7ed8040730c315c282c6b": "X \\in L^p", "1c601b274dff34ba5cddd90c78098d2e": "(\\mathrm{Tr}\\;G)^2 = \\left( \\sum_{i=1}^r \\lambda_i \\right)^2 \\leq r \\sum_{i=1}^r \\lambda_i^2 \\leq n \\sum_{i=1}^m \\lambda_i^2 ", "1c60654fc884a3c9f6f68df340a9a3c7": "\n- \\left( r^{2} + \\alpha^{2} + \\frac{r_{s} r \\alpha^{2}}{\\rho^{2}} \\sin^{2} \\theta \\right) \\sin^{2} \\theta \\ d\\phi^{2}\n+ \\frac{2r_{s} r\\alpha c \\sin^{2} \\theta }{\\rho^{2}} d\\phi dt\n", "1c6084fcb37821198a37c9aac724c79d": "\\, \\sim ", "1c611a3c6dcdaa03254da9d0b48a317a": "\\mathbb{R}^n \\to \\mathbb{R}^m", "1c611cb07379069e04f9cb683b9769d0": "H_{max}={v^2\\over 2g}", "1c612194910bba6b902a07432daf99e3": "\\ F_{propulsive}= drag \\times cos (\\beta) ", "1c61331dc90773c0639c886bcd3f0d15": "\n\\mathrm{Eu}=\\frac{p_\\mathrm{upstream} - p_\\mathrm{downstream}}{\\rho V^2}\n", "1c615f65a308eeed3c6141be0884f993": "10 \\log_{10} 2 = 3.010... \\approx 3;", "1c616152acb6c113829bc3d22571a857": "\\int_X \\left[\\sum_{i=1}^n \\chi_{E_i}(x) b_i\\right]\\, d\\mu = \\sum_{i=1}^n \\mu(E_i) b_i", "1c61ba3178870926ac25a4323e2f688d": " n_l ", "1c61bdda2415b95c99a794d20d458cd4": "|P_1(A)| < |P(A)|", "1c6291276f74cc51f5e77a71d39b2703": "[K_r : K_{r-1}] \\cdots [K_2 : K_1][K_1 : F]", "1c62b5a83ad7e7c61dcec7d2170d889d": "\n \\begin{align}\n N_{\\alpha\\beta,\\beta} & = J_1~\\ddot{u}^0_\\alpha \\\\\n M_{\\alpha\\beta,\\alpha\\beta} + q(x,t) & = J_1~\\ddot{w}^0 - J_3~\\ddot{w}^0_{,\\alpha\\alpha}\n \\end{align}\n", "1c64054c43378c4d9ae6023cd7bcd11e": "A + B \\; \\stackrel{k}{\\rightharpoonup} \\;2B", "1c642e2367be346bf1fc473d5bd88995": "\\pm d_0.d_1d_2d_3\\dots\\times b^n,", "1c64cc685525b2e35162727c9479858e": " B(x) =B(x+1)", "1c64d561964fe18cd5da2af686eb6205": "\\mu^\\Delta(A) = \\lambda(\\rho^{-1}(A)),", "1c64e2e5ab36fa13078dd411c0b87c55": "\\frac 1 \\mu", "1c64ff65f0a138b630bc81e5d89a53a7": "\\rho=3.1\\times 10^{-8}\\alpha^{-7/10}\\dot{M}^{11/20}_{16} m_1^{5/8} R^{-15/8}_{10}f^{11/5}{\\rm g\\ cm}^{-3}", "1c652a36cc6d37cfcc80d493acd40005": "r\\left( {\\vec x; q} \\right)", "1c653d402a0c4964779b7add171c5de1": "t_{initial} = 0", "1c6602184b14b54f7c8c77575b9dcd0d": "-K \\leq k \\leq K", "1c66051886d92a7d30efc78da0516de5": "\\nu(\\phi) ", "1c6628f3d3a789a3e0337c89ce034933": "\\pi_2(x)<(2C_2+\\varepsilon)\\frac{x}{(\\log x)^2}", "1c667b5f5eb8b7b1573c85df7942eced": " Pe_{l} \\text{≤} -2", "1c668a16384ae736fa0f458acd2fbeab": "{c\\over\\Gamma}={{1\\over {B_{\\text{ADS}}\\Gamma_{\\text{max}}}} + {c\\over \\Gamma_{\\text{max}}}}", "1c66bdd57870d22b3b88245ca36dc616": "\\partial L'= (\\partial L-\\operatorname{int~im}\\phi)\\; \\cup_{\\phi|_{S^p\\times S^{q-1}}} (D^{p+1} \\!\\times\\! S^{q-1}).", "1c66f77c2b23c82672259e6acf547044": "\\prod_{p^k|24} f(p^k) = f(2) f(3) f(4) f(8).\\ ", "1c670dd6401a5f99e70f67fcf18604f6": "H(x_k)", "1c6738f054f7c3c42610e55869877f3b": "\\mathcal{F}\\{f \\cdot g\\}= \\mathcal{F}\\{f\\}*\\mathcal{F}\\{g\\}", "1c6762424783508ee56aca6114cf840e": "\\beta=1", "1c677b588b3ee01de2e6238a50dbdfa4": "(17)\\quad \\theta_{(\\ell)}=-(\\rho+\\bar\\rho)=\\frac{r-2M(v)}{r^2}\\,,\\quad \\theta_{(n)}=\\mu+\\bar\\mu=-\\frac{2}{r}\\;.", "1c67f492e89efef7536cc587cdd9d729": "\n\\mathbf{N}\\left( \\mathbf{u}\\right) \\equiv N\\left( u_{1}\\right)\n\\otimes\\cdots\\otimes N\\left( u_{n}\\right) .\n", "1c6879bc7895b3c10f3fd741c9794930": "\\sum_{j=1}^{i-1} a_{ij} = c_i\\ \\mathrm{for}\\ i=2, \\ldots, s.", "1c68a22f881267ed0b83ac703bd4de16": " \\angle MHA = \\theta = \\angle ZHA - \\zeta' \\approx \\angle ZHA - \\zeta = 90^\\circ - \\frac{1}{2} (\\varphi_H + \\varphi_A) + \\delta ", "1c68b5c5ce310279c0365d8b30a78399": " c =M_{4,1}\\,", "1c68e8cf9e85a95d36ee5e277c357228": "\\scriptstyle S^{+}", "1c690c7ff32ffce96cf98c0e665ca69f": " l^2 = r^2 + x^2 - 2\\cdot r\\cdot x\\cdot\\cos A ", "1c691959efb790722254fb9c4248fdd2": "\\Delta(v)=1\\otimes v+v\\otimes 1", "1c6972d39651ec7e1dba31e3afe7b583": "\\begin{align}\\pi &\n= \\sum_{k = 0}^{\\infty}\\left[ \\frac{1}{16^k} \\left( \\frac{4}{8k + 1} - \\frac{2}{8k + 4} - \\frac{1}{8k + 5} - \\frac{1}{8k + 6} \\right) \\right] \\\\ &\n= P\\left( 1, 16, 8, (4, 0, 0, -2, -1, -1, 0, 0) \\right)\n\\end{align}", "1c69a4620c0d68697f23fdcd60af8893": " W=F_{2}(\\alpha_{1},\\alpha_{2},\\alpha_{3}.....\\alpha_{k})=0 ", "1c69d7afa8c525697e88b79b2dcd72d9": "\\begin{cases} \\Theta''(\\theta) + \\nu^2\\Theta(\\theta) = 0\\\\ r^2R''(r) + rR'(r)-\\nu^2 R(r) = 0 \\end{cases}", "1c69e1002e35c7f7f902d105198c609b": "\\bar{y}=\\frac{1}{n}\\sum_{i=1}^n y_i ", "1c69e1ec27d9b129f6ef93946e6aca76": "\\mathcal{V}(G)", "1c69ed435f3e2b312a2c89598557892f": "n(\\omega_j)", "1c6a00ccd6683fb3660cf782aa572b51": "\ng_n = p(N_{-},n)\n", "1c6a7adf7c2ec75b937b4cccfed35a91": "\\hat{H}_{\\text{JC}} = \\hbar \\omega \\hat{a}^{\\dagger}\\hat{a}\n+\\hbar \\omega \\frac{\\hat{\\sigma}_z}{2}\n+\\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{\\sigma}_+\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_-\\right),", "1c6ae411d247e441a289a913fc02e2f1": "{\n \\gamma = \\sum_{k=2}^\\infty (-1)^k \\frac{ \\left \\lfloor \\log_2 k \\right \\rfloor}{k}\n = \\frac12-\\frac13\n + 2\\left(\\frac14 - \\frac15 + \\frac16 - \\frac17\\right)\n + 3\\left(\\frac18 - \\frac19 + \\frac1{10} - \\frac1{11} + \\dots - \\frac1{15}\\right) + \\dots\n}", "1c6aef6220c2cf4351c45767effb9026": "\\theta:T^*M\\otimes\\Omega^pM\\rightarrow\\Omega^{p+1}M", "1c6b0ca5e03a886e7a0d1741331e929f": "\nA_v = \\prod_{i \\in v} \\sigma^x_i, \\,\\, B_p = \\prod_{i \\in p} \\sigma^z_i.\n", "1c6b2ba181612dba150a315b6347bdc7": " 1/4 \\leq p_1 \\leq 2/3 ", "1c6ba617904a799bfa9062d749845c7b": " t_i(\\hat\\theta) = \\sum_{j \\in I-i} v_j(x^*_{I-i}(\\theta_{I-i}),\\theta_j) - \\sum_{j \\in I-i} v_j(x^*_{I}(\\hat\\theta_i,\\theta_{I}),\\theta_j) ", "1c6cfb5075461ab5e2a3bc8372013a86": "\n\\bar\\psi(\\mathbf{x},\\tau) = \\mathrm{e}^{K \\tau} \\psi^\\dagger(\\mathbf{x}) \\mathrm{e}^{-K\\tau}.\n", "1c6d0cbab0de2ada403c82149c6a0a2f": "\\operatorname{MSE}(\\overline{X})=\\operatorname{E}((\\overline{X}-\\mu)^2)=\\left(\\frac{\\sigma}{\\sqrt{n}}\\right)^2= \\frac{\\sigma^2}{n}", "1c6d24c6b7f55dd92938207a410900dd": "1 \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{16} \\,+\\, \\frac{1}{64} \\,+\\, \\cdots.", "1c6d53d479dc823b9a5f16a668020cf3": "\nq = 2\\int_0^\\pi \\frac{I(\\alpha)}{I(0)} \\sin \\alpha \\, d\\alpha.\n", "1c6d843e390cc7ec217f046f3fcf1eb6": "\\inf_{Q_{Y|X}(y|x)} I_Q(Y;X)\\ \\mbox{subject to}\\ D_Q \\le D^*.", "1c6dc09098c96cf690cb9beadbb06a0f": "V(r) = \\frac{-Ze^2}{r}", "1c6dc490ee2368603e2732ba7f386055": "f_n(z)=z+\\frac{1}{\\rho n^2}\\sqrt{z}, \\qquad \\rho >\\sqrt{\\frac{\\pi }{6}}", "1c6dcc1aca556dcc6d3d48e509e391fc": "\\frac{H_n}{P_n}", "1c6dffbd1ec7425930842b204a3fea31": "s^2 = \\frac{1}{2N} \\left\\{ \\sum_{n=1}^{N} ( x_{n,1} - \\bar{x})^2 + \\sum_{n=1}^{N} ( x_{n,2} - \\bar{x})^2 \\right\\} ", "1c6e4c7ea9187799fcf4027de228b70e": "m=f_{sc}/f_{ref}", "1c6e55d5abe7c4ef4168fcd59c3db538": "\\ T = \\frac{I_l}{I_0} = e ^ {- \\sigma \\ell N} = e ^ {- \\alpha'\\ell} .", "1c6e747eb9def5c640a74525104ac27a": "S_m(n) = {1\\over{m+1}}\\sum_{k=0}^m (-1)^k {m+1\\choose{k}} B_k\\; n^{m+1-k}. ", "1c6f09f07c06ae3128af4f5bb6557f22": "K_\\alpha(z) \\sim \\begin{cases} - \\ln (z/2) - \\gamma & \\text{if } \\alpha=0 \\\\ \\\\ \\frac{\\Gamma(\\alpha)}{2} \\left( \\frac{2}{z} \\right) ^\\alpha & \\text{if } \\alpha > 0. \\end{cases} ", "1c6f5120a7b01c7bd288c93b64cb3cc5": "\\displaystyle{a^{b+c}=(a^b)^c.}", "1c6f85857795410fcd52bf6f0fe79144": "L_2 = sh/(h+s) = h/(h/s+1) = h", "1c6f91ff6d4fb5c67f510420dd9045d9": "{x} \\ge {0}\\,", "1c6fac357e8f6e53c4673f3db3091f0a": "x^2y,\\,\\!", "1c700ae5dcf9c1305a41e496acf47e3f": "m(x) \\sim 1/x , \\, ", "1c702ab65cf4affa4e895efd8f6c7ece": "F(\\underline{x},-)", "1c70377b92bdfef9317e4b334d30d2a8": "\\frac{dA}{dt}=\\tfrac{1}{2}r^2 \\frac{d\\theta}{dt}.", "1c704bd22fa98873b2a8070a9bf4a041": "\\mbox{Slip} = \\frac{NP-V_a}{NP} = 1-\\frac{J}{p}", "1c705cfecb670563cc60061396f14de4": "\\Omega = -kT \\ln \\Big(\\sum_{\\rm microstates} e^{\\frac{\\mu N - E}{kT}}\\Big)", "1c706071f28fbd72fc368322c4b30488": "\np_k=n\\frac{\\dot{x}_k}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}\n=n\\frac{dx_k}{\\sqrt{dx_1^2+dx_2^2+dx_3^2}}\n=n\\frac{dx_k}{ds}\n", "1c708e1e013f604acc1893dd78feec3b": "x_{n_3} \\geq x_{n_2}.", "1c7109d5368f7c71b0494b40d337ae06": "\n{{\\sin (\\theta )} \\over {4\\left[ {1\\,\\,\\, + \\,\\,\\,{1 \\over 4}\\sin ^2 \\left( {{\\theta \\over 2}} \\right)} \\right]}}\\,\\,\\, \\approx \\,\\,\\,{\\theta \\over 4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\,\\,{\\theta \\over 4}\\,\\,\\Delta \\theta \\,\\,\\, = \\,\\,\\,{{\\theta ^2 } \\over 4}{{\\Delta \\theta } \\over \\theta }", "1c712ddaa8457a7926d7129d93d8c9c0": "dy = a\\,dx", "1c7180361eacf57d714d4399e28bf909": "(\\kappa \\le \\mu) \\rightarrow ((\\kappa + \\nu \\le \\mu + \\nu) \\mbox{ and } (\\nu + \\kappa \\le \\nu + \\mu)).", "1c7186dbfb3f611312b8825f6c1f9a72": " \\varphi(g_1 g_2) = \\varphi(g_1)\\circ \\varphi(g_2) \\quad \\text{for all }g_1,g_2 \\in G \\,\\!", "1c71a3b383e32d4fdaeabb1e43e883b3": "\\chi(M \\cup N) = \\chi(M) + \\chi(N) - \\chi(M \\cap N).", "1c71a523b2398cbd176b796ff63f6feb": "\\textstyle f_C(c) = 0.2", "1c71f2c078e7194c56a9499e600270b3": " y(x) = x \\cdot y' + (y')^2 \\,\\!", "1c724b96ba5973b9023107c7fb44acde": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x)=g(x)", "1c7266bda6399771f14154b48959630b": "S_5=\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 \\\\\n1 & 2 & 3 & 4 & 5 \\\\\n1 & 3 & 6 & 10 & 15 \\\\\n1 & 4 & 10 & 20 & 35 \\\\\n1 & 5 & 15 & 35 & 70\n\\end{pmatrix}.", "1c72b6206da332c036fd47a4a90fbfea": " \\mathbf{I} = \\Delta \\mathbf{p} = \\int_{t_1}^{t_2} \\mathbf{F}\\mathrm{d} t \\,\\!", "1c72b9453cd275f8176a4259a2799580": "3^{F_n-1}\\equiv1\\pmod{F_n}", "1c72c13da4eee1c61daddad8b3828f7b": "M_\\mu\\to M_\\lambda", "1c72c52a434cdb758d0fb79567cf0448": "\\int \\frac{dx}{\\sqrt{x^2+c}} = \\int \\frac{\\frac{t^2+c}{2t^2}dt}{\\frac{t^2+c}{2t}}", "1c72f4b271e661bc0736d3a5dab40b7e": "(\\cdot,\\cdot,\\cdot) : Q\\times Q\\times Q\\to Q", "1c738fcd9733a432602ff6defd139a37": "\\displaystyle{[J_m,J_n]={c\\over 3} m\\delta_{m+n,0}}", "1c739c6ff0405d1d347ea49f7748a29b": "\\scriptstyle\\mathbf{\\hat{n}}", "1c73b5822c23dd7396dfcb36a450fd59": " \\frac{\\mathrm{d}\\mathbf{L}_i}{\\mathrm{d}t} = \\boldsymbol{\\tau}_E + \\sum_{i \\neq j} \\boldsymbol{\\tau}_{ij} \\,\\!", "1c73b6ffebeaf3cbfc5df82ff58cf1fa": "\\bar{G}", "1c73c110dded50675fbcf1565c392b80": " \\mathbf{F} \\cdot \\mathbf{r} = 0 ", "1c73fcf3c87d83e82c59038f41e8c671": "m_{\\mathrm{eff}} \\le m", "1c74234e407004c3b89313435efadc47": "\n\\left(\\widehat{E} - c\\boldsymbol{\\alpha}\\cdot\\widehat{\\mathbf{p}} - \\beta mc^2 \\right)\\left(\\widehat{E} + c\\boldsymbol{\\alpha}\\cdot\\widehat{\\mathbf{p}} + \\beta mc^2 \\right)\\psi=0 \\,,\n", "1c745f2ad0a65bf4634fa2187c63b019": "\\aleph_1 \\times 2 ^ {\\aleph_0}\\, = 2^{\\aleph_0}.\\,", "1c747a0c90dfbc859316f497b8808fba": "C\\subseteq\\kappa", "1c7498d3fd21d52291b85e86451296ff": "k_F = (3\\pi^2 N_e/V)^{1/3}", "1c74df82dd4854589387348363a7a3e0": "S_{\\text{CGHS}} = \\frac{1}{2\\pi} \\int d^2x\\, \\sqrt{-g}\\left\\{ e^{-2\\phi} \\left[ R + 4\\left( \\nabla\\phi \\right)^2 + 4\\lambda^2 \\right] - \\sum^N_{i=1} \\frac{1}{2}\\left( \\nabla f_i \\right)^2 \\right\\}", "1c74f010aea750d8b5099df7e065fdf8": "a - \\sqrt{a^2 - b^2}", "1c74f5def7f7f6d1fd206d987a896857": "\\lambda = B\\left(\\frac{n^2}{n^2-4}\\right) \\qquad\\qquad n = 3,4,5,6", "1c752c7271436df7057aa75f56dc5dd0": " Marginal Cost(MC) = \\frac{\\ dC}{\\ dQ}", "1c758c905658f1dbb33cc6de313998f1": "\\ \\{x : A(x)\\},", "1c761199cc445e9b249285cf6bf5a98f": "\\Psi(x) \\approx C_0 \\frac{ e^{i \\int \\mathrm{d}x \\sqrt{\\frac{2m}{\\hbar^2} \\left( E - V(x) \\right)} + \\theta} }{\\sqrt[4]{\\frac{2m}{\\hbar^2} \\left( E - V(x) \\right)}}.", "1c767451119de5d3c9aae08e244eae9b": " \\gamma_{total}\\left(d\\right) ", "1c76cde5362619ed6d443a8778d7d68f": "E = E_{KIN} + E_{POT}", "1c779a57da1668aa058a883a56e8b6ca": "\\int_{-\\infty}^{\\infty} |f(\\xi+i\\eta)|^2\\,d\\xi < \\infty.", "1c77e6aa877e041ff7d34240fd9ecf4c": "f(a) = \\min_{x \\in R \\setminus \\{0\\}} g(xa)", "1c7840262592bcfeca27aac4b10d4969": "d(k)", "1c784b849e900375636d7cdba245a599": "\\mathcal{F} \\{ f \\}", "1c786d45b40ca94e9a858276bd971f4c": "\nz\\rightarrow \nz^3 - 3 z x^2 + z y^2 + z_0\n", "1c78b486fa89d4f71edbbd0d53d214dc": "ub", "1c78e51631375cfe500f96f321d063a3": "\\omega_{pl}^2 = \\frac{4 \\pi e^2 N}{\\epsilon L^3 m}", "1c78ea100bb0b6822a900c6f5053c40d": "C_{\\mathrm{p}}", "1c7919ee51bf9c43612d9a6aa4d9d68b": "\n \\displaystyle \n S(4,2)\n =\n \\left\\{ \n\t (1111), \n\t (1112), \n\t (1122), \n\t (1222), \n\t (2222)\n \\right\\}\n", "1c795b5da70841d856b4410f4af87290": "T \\vdash \\varphi", "1c795d8bf9c86273efb69c4d580a8457": "s(n)", "1c79ab541f8c525efd74cdb3f18a361e": "\\begin{cases} \\dot{u}(t) = A u (t); \\\\ u(0) = u_{0}. \\end{cases}", "1c79c561f7aa073bb69217a07ffdcffc": "\\,\\alpha ", "1c7a25de0c4ef3098e5a0d92002e9e0b": "\\{ N_{L/K}(x) | x \\in \\mathfrak{b} \\}.", "1c7a7eda90d34f821a7834455648f510": "\\int_{-\\infty}^{\\infty}L(\\Delta f)d\\Delta f = \\frac{f_\\Delta}{\\pi}\\int_{-\\infty}^{\\infty}\\frac{d\\Delta f}{f_\\Delta^2 + \\Delta f^2} = \\left.\\frac{1}{\\pi}\\tan^{-1}(\\frac{\\Delta f}{f_\\Delta})\\right|_{-\\infty}^{\\infty}= 1", "1c7ae0212b67b6dc7104070fe8a2aaba": "M(f)<0\\,\\Rightarrow\\,L(f^{-1})\\leq -{\\frac {1}{M(f)}},", "1c7b1295a3ca164f6f10cb3ab5ae9458": "\\partial /\\partial t", "1c7b5f4e1a5f1092da4dfebdf41af0ee": "\\tfrac{10}{3} \\div 5", "1c7b783ea8c31c2816a193adc996de6c": "b' = \\tau(b,a,o)", "1c7bfccc43787f838cf05e40103a5d91": "\\displaystyle{e_W=\\det (I-W^*W)^{1/4} f_W}", "1c7c0cad838a82350b38d91c1e1d7ad5": "y = b + r \\frac{2t}{1+t^2}.\\,", "1c7c13712c27246ed8b722bf28b7324b": "{1 \\over 1-\\alpha z}=\\prod_{j=1}^\\infty\\left({1 \\over 1-z^j}\\right)^{M(\\alpha,j)}", "1c7c139dfb84d36a62ad85bd61550790": "(7, 4, 3)", "1c7c2038f8c25bd16ace98a12f447c54": "V_{YY}", "1c7c23455301f7b488f798bcc3af5f0c": "B_{\\text{op}} = B", "1c7c6fcce368ca6f327aaa7ade047006": " \\frac{1}{24}\\left(n^6+3n^4 + 12n^3 + 8n^2\\right). ", "1c7c84a53deafa61a815efed32a0c28c": " (\\beta_2 , \\; \\lambda_2)", "1c7c877f919bc8e523de0c79f8482b7d": " 10^{1/30} = 1.079775", "1c7ca8a78509e148b0da0cafce863b28": " h_{\\alpha \\beta,\\gamma} \\eta^{\\beta \\gamma} = \\frac12 h_{\\beta \\gamma,\\alpha} \\eta^{\\beta \\gamma} \\,,", "1c7cab2cde93564526ac136aec9c505f": "N' = \\pi^{-1}(N),", "1c7cb00334a8fe64ac7cf9a120cae5b5": "\\mathrm{^{239}_{\\ 94}Pu\\ \\xrightarrow {2(n,\\gamma)} \\ ^{241}_{\\ 94}Pu\\ \\xrightarrow [14.35 \\ yr]{\\beta^-} \\ ^{241}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{242}_{\\ 95}Am\\ \\xrightarrow [16.02 \\ h]{\\beta^-} \\ ^{242}_{\\ 96}Cm}", "1c7cd00d98876082e5f6d8d8d04ea7f5": "\nf_\\mathbf{v} \\left(v_x, v_y, v_z\\right)\\, dv_x\\, dv_y\\, dv_z.\n", "1c7d013f78c102398b93496d91ad3671": "= \\frac{2}{3}", "1c7da5980eaba46dd7471584b1fb5de9": " \\and ((T_2 = [F_2, S_2, A_2]::[F_1, S_1, A_1]::R ", "1c7dae5409aeeedcb6aa2e4c2d1ada71": "F_j", "1c7e17a14a4bd186001307ac352818d1": " \\sigma ^ 2_t ", "1c7e814a9045da29bc091492b08d32c1": " C =\n \\begin{bmatrix}\n 1/2 & 0 \\\\\n 0 & 1/7 \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n 11 \\\\\n 13 \\\\\n \\end{bmatrix}\n =\n \\begin{bmatrix}\n 11/2 \\\\\n 13/7 \\\\\n \\end{bmatrix}. ", "1c7e889d05243af857153c401b236617": "\\lambda=2\\pi/k", "1c7efd1a42ffd08bae9b2235d2754060": "y:\\Delta^n\\rightarrow Y\\,", "1c7f3f0a8f462f5f400fd91704c75d68": "\n\\left[ \\begin{array}{cc|c}\n5.291 & -6.130 & 46.78 \\\\\n0.00300 & 59.14 & 59.17 \\\\\n\\end{array} \\right].\n", "1c7fa447397b3403dc2f4b810cca18ea": "\\phi_n(0) = 1", "1c7fc52761eb224fa431c5eab519f17b": "\\xrightarrow{C4H}", "1c7ffb424e9c698c8289752ec032e0cf": "\\psi_1(\\varepsilon_{\\Omega_2+1})", "1c801983acd85eb12c45df8969ffcbc8": "E \\{ (\\hat{x}-x) (y-\\bar{y})^T\\} = 0", "1c804e0352c2fd04dd6923b2dd622089": "R_n^2(\\xi,x)>1", "1c807797a1f5e7f6062275fb795f5d50": "n_t(x)", "1c80e7ac4762c7585993604b5e3bb2e9": "P_d(x,y,z)=0", "1c815001366cf21910e7c76ade62bb11": "|0{\\rangle}", "1c81771509cdf9517c7d05eacf348dde": "\\scriptstyle\\lfloor \\lambda \\rfloor", "1c81a3b249a7cd61bcddab88e6de8f6f": " \\overline{AD}^{\\,2} = \\overline{AB}^{\\,2} + \\overline{BC}^{\\,2} + \\overline{CD}^{\\,2} \\ .", "1c820b60ce4540c5339ea64d2d90bd46": "^29", "1c8212910e2d491904e9a5bcf0d80d4e": "\\Big( (\\mathcal{M}, s) \\models AF\\phi \\Big) \\Leftrightarrow \\Big( \\forall \\langle s_1 \\rightarrow s_2 \\rightarrow \\ldots \\rangle (s=s_1) \\exists i \\big( (\\mathcal{M}, s_i) \\models \\phi \\big) \\Big)", "1c8213b9d9bbfd3480b71550dd87e5ed": "\\ L_0=a", "1c827500761de941cba99239aaf76484": "\\phi=(f\\circ\\xi^{-1})^{-1}=\\xi\\circ f^{-1}", "1c828fc0c76726d11ee41c69e2c09dc4": "\\sigma(f(x))=f(x+1)", "1c82b70aacf66befdb20dd9e36393fbf": "M=\\begin{pmatrix} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{pmatrix}, ", "1c82fd822d8f2c860f6fe9a1909b3b71": "(r_ir_j)^k", "1c834430b9871100b53bffd979309472": "\\begin{bmatrix} N\\\\M \\end{bmatrix}", "1c837f53ad1173411db02ee49173a61e": "k_i(k_i-1)", "1c83d2c7ec448e1586fced85203ad83c": "\\Delta \\langle \\hat{B} \\hat{B} \\rangle", "1c8414cf43dc06fb11810afe58fc570e": "\\tfrac{2}{7}\\scriptstyle{\\sqrt{30-3\\sqrt{2}}}", "1c847d4cf6ac267949f1919c24e212c2": "R^{d-1}", "1c848169dba57c91ed187e3f7faa480a": "1 = \\frac{(k_2 - k_1)k_1}{k_2 \\cdot k_1} + \\frac{(k_3 - k_2)k_1}{k_3 \\cdot k_2} + \\dots + \\frac{(k_n - k_{n-1})k_1}{k_n \\cdot k_{n-1}} + \\frac{1 \\cdot k_1}{k_n}", "1c84ca98188b58c3f258c5658e82c6d2": " 1 - [-0.625 \\log_2(0.625) + -0.375 \\log_2(0.375)] \\approx 4.6%.", "1c8518c1bda4a2652bd187c5a1107743": "{B}_{7}^{(1)}", "1c852c5de4573e4e63fc4f853f888362": "0 = (x^{\\alpha} T^{\\mu \\nu} - x^{\\mu} T^{\\alpha \\nu})_{,\\nu} . \\!", "1c85557f3478c8868cf71f227382ea81": "\\sin x = \\sum^{\\infty}_{n=0} \\frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\cdots\\quad\\text{ for all } x\\!", "1c85bd366dda9578bca1bfa13468a4e2": "V(z)=\\frac{2z}{1+\\sqrt{1+4z}}", "1c85f1028949810c6dffc85bb32fd6e6": "B_{k} = B'_{0} = (1 + r_{k})B_{0} - p_{k}", "1c86023efc384cc5c1dd5b88fcf0baec": "\\begin{align}\n I_3 \\cdot R_3 &= I_1 \\cdot R_1 \\\\\n I_x \\cdot R_x &= I_2 \\cdot R_2\n\\end{align}", "1c8604ae4056da7feb0a94276fc4d8f0": "\\Delta f \\approx \\frac{ c}{2nl \\cos\\theta } ", "1c8643fa8c04255cfcafb144796c658a": " b = 2 ", "1c864b26839eacd2aa5df39ded60a9d9": "\\displaystyle\\Omega(k)", "1c86543803e2095a2064ee70cd5776c3": "[* F , * G]^{IJ} = - [F , G]^{IJ} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Eq.7", "1c8703e7c502bf1efa7273435fe78032": "\\theta_{(l)}", "1c8714e724a7f0cc344c30d4c738d4f8": " f_{xx}(x,y) \\approx \\frac{f(x+h ,y) - 2 f(x,y) + f(x-h,y)}{h^2} \\ ", "1c8717bf743dc20b47fd28420f8baaa3": "\\delta \\mathbf{u}(\\lambda r)", "1c87957ed09e68858cc37b66ec7c07a8": "\\left|\\widehat f(n)\\right|\\le {{\\rm var}(f)\\over 2\\pi|n|}", "1c880ddca12fd76766ff725309096805": "\\mathit{a}", "1c880f6fff7af3578e6bd5531378df0d": " \\boldsymbol{v} \\in \\mathbb{R}^n ", "1c8864318c57f63ae29a56c532a2453a": "a^n + b^n", "1c8881c2b72d487bc18d53cd7346a05e": "\\Lambda_{min} \\subseteq \\Lambda \\subseteq \\Lambda_{max}", "1c8896865e21230f8e056c2dae680164": "name_i", "1c88c4abbc6007b0cf5e13958f616db9": "\\mathcal{F}_{t}", "1c8916911546684ecb5b0b598164bfe0": "C^{2+\\delta}", "1c8965dd4db91534cbb67d6973226576": " \\mathbf{E}(z,t) = \\mathrm{Re} (\\mathbf{E}_0 e^{i\\omega((\\tilde{n} z/c) - t)})", "1c89991159992ffc7304ed59ded82efa": "\\lang \\mathbf v , \\mathbf w \\rang", "1c89ce97cc18cd81508173797a6e6264": "q_\\alpha", "1c8a2370746031bd1040ab77818d7a5d": " J(x,t) = \\frac{i \\hbar}{2m} ( \\psi \\frac{\\partial \\psi^*}{\\partial x} - \\frac{\\partial \\psi}{\\partial x} \\psi ) ", "1c8a5bebf295c421d007960e4cf5c258": "1-\\zeta_m", "1c8a7d35ef26d89684fce7ad05bb9e4d": "k = \\frac{2\\pi}{\\lambda}", "1c8b02d6bdd89c337bf1b4781f48e573": " P * V^{\\gamma} = \\operatorname{constant} = 1.58 \\times 10^9 = P * 100^{7/5} ", "1c8b53f09d0c114797831f239ade0e47": " u_{70}(\\mathbf{r}) = \\bar{u}_{lh}(\\mathbf{r}) = \\left | \\frac{3}{2},-\\frac{1}{2} \\right \\rangle = \\frac{1}{\\sqrt 6} |(X-iY)\\uparrow\\rangle + \\sqrt{\\frac{3}{2}} |Z\\downarrow\\rangle ", "1c8b5678830b0f2a6eda1271b3a626a0": "\\Delta v =v_{\\mathrm{g}}\\left(1-\\tfrac{v_{\\mathrm{c}}}{v_{\\mathrm{g}}}\\right)\\approx v_{\\mathrm{g}}", "1c8b5802df8a5f185235bd4446d655c0": "U(y, \\xi)\\,", "1c8bd988c72b27e2355be2bbdbe7e584": "\\mathbf{T}(5) = 5 \\,", "1c8beb3b0b0c1514f36716363c3c2f1b": " (\\lambda x.x\\ x)\\ (\\lambda x.\\lambda f.f\\ (x\\ x\\ f)) ", "1c8c85cde8a69ff8604b10d5fe1a4ca5": "\n\\delta_{2s}(n)=\n\\frac{\\pi^s n^{s-1}}{(s-1)!}\n\\left(\n\\frac{c_1(n)}{1^s}+\n\\frac{c_4(n)}{2^s}-\n\\frac{c_3(n)}{3^s}+ \n\\frac{c_8(n)}{4^s}+\n\\frac{c_5(n)}{5^s}+\n\\frac{c_{12}(n)}{6^s}-\n\\frac{c_7(n)}{7^s}+\n\\frac{c_{16}(n)}{8^s}+\n\\dots\n\\right)\n", "1c8c99796fcd4a0ab2d3c328dc851f9a": "\\biggl(\\frac{1-p}{1 - pz}\\biggr)^{\\!r} \\text{ for }|z|<\\frac1p", "1c8ca73486fe326c54f1559358ddfecf": "X \\cap W_i", "1c8cee48c07a1cdbfdbaa79748cb2e90": "y_{1i}", "1c8d2ef4fc5764198bb6df8299263ee3": "\\alpha = 1 \\mathrm{m}^{-1}", "1c8d401a5790a50d57caa5233aefa6d4": "\\cos (wz) + i \\sin (wz)\\,", "1c8d8ce34f2148797866bbb422292d95": " C(N) \\leftrightarrow \\forall x \\lnot C(x) ", "1c8dd4a654b61c6941618bf1534ba829": "\\int_{\\mathbf{R}} \\frac{\\sin x}{x} dx", "1c8efcab01cf7143097d467c4625e859": "\\tilde{Fr_1}", "1c8f3f00e1925efd56f1d0b6aa4347c3": "\n(c \\delta \\tau)^2 = (c \\delta t)^2 - (\\delta x)^2.\\,\n", "1c8fbeae1557bc0ac8a45d7afc837a93": "P_{dBW/m^2/Hz} = 10 \\log_{10}(P_{Jy}) - 260\n", "1c8fcd356afec21c05421124b8862ad0": "M(n,m+1,p) = M(M(n,1,p),m,p)", "1c902748eb82600ffe44ad79cc82ce7a": "\\frac{\\partial \\mathbf{\\hat{z}}} {\\partial \\varphi} = \\mathbf{0}.", "1c902eb0605242d70cadbe2bc78f2467": " \\begin{align} p(n,k,m) &= 1 - { (m - nk -1)! \\over m^{n-1} (m - n(k+1))!}\\end{align} ", "1c90436cf9c162977e8089732c3440a4": "\\varphi(p)=2^k", "1c9058bd156b8bceef2935bcb68818bf": "n\\in\\mathbb{N}^*", "1c90a40df84e0f1c1e897026fc30e897": "R/n = \\tan \\theta\\,", "1c90e57d7bc9bd980d44fd8965ff5ca3": "k = 8", "1c90e87927def654d6dfda90ea92fda4": "\\varphi(x_1,x_2,x_3,x_4,x_5) =\n\\varphi(x_5,x_2,x_3,x_4,x_1) +\n\\varphi(x_1,x_5,x_3,x_4,x_2) +\n\\varphi(x_1,x_2,x_5,x_4,x_3).", "1c90fba63b8486db2db0ac0f13399e80": "\\mbox{Copper Loss} = I^2 \\cdot R \\cdot t", "1c910bea2d3a2b0796c8508e570a902c": " p_1=1, ", "1c91b2f11123ab24bedf87567864c9b5": "\\displaystyle z=\\frac{G(x)^5}{xH(x)^5}", "1c91b6f7877b71a3d51eebeb61b1d6a1": "e_t^2", "1c91c91f48e191b0c9c7803995d05726": " \\text{NPV} = \\frac{\\text{number of true negatives}}{\\text{number of true negatives}+\\text{number of false negatives}} =\n\\frac{\\text{number of true negatives}}{\\text{number of negative calls}}\n", "1c91e1280eb5b3626c029e2d88ab3b76": "L_\\rho(\\gamma)", "1c9214abf423bd36c4239e71ba55fee8": "P=I/G", "1c924f5e1db39797bea5db87dd7e80b8": "\nf(\\mathbf{X}) \\sim \\mathcal{N}(\\mathbf{m},\\mathbf{K}),\n", "1c9252d5c532fcfa9d50175a938c66c7": " \\ell_2", "1c9261848713b3e7c2c609c5fe39548a": "t^{-n-1}", "1c92a23649ed4c9789b0a04246cd5e5d": " \\frac {d^2 \\mathbf{x}_\\mathrm{A}}{dt^2} = \\mathbf{a}_\\mathrm{AB}+\\frac {d\\mathbf{v}_\\mathrm{B}}{dt} + \\sum_{j=1}^3 \\frac {dx_j}{dt} \\frac{d \\mathbf{u}_j}{dt} + \\sum_{j=1}^3 x_j \\frac{d^2 \\mathbf{u}_j}{dt^2}. ", "1c92b3817f410c97e751dae1e51eb544": "t\\in \\lbrack 0,1]", "1c92c395255065be3acc86f2db8b0f18": "g_m = \\frac {I_C}{V_{T}}\\left(\\textrm{in}\\ \\frac{\\mathrm{mA}}{\\mathrm{mV} }\\right) ", "1c92c7ef37dab7c528d9200dc57951a1": "r_{k+1}=0", "1c92d8ba252e2ff0a6ffcea8ed6f2190": " \\textbf{W}^{-1} ", "1c930d3f99626629cfb7bc70c72a7871": "\ni\\hbar \\epsilon g \\partial_g U_\\epsilon(t_1,t_2) = H_\\epsilon(t_1)U_\\epsilon(t_1,t_2)- U_\\epsilon (t_1,t_2)H_\\epsilon (t_2).\n", "1c9316ac43eba22c820cd8d05dee28d6": "\\begin{align}\nx &= \\frac{a\\sinh v \\cos\\phi}{\\cosh v - \\cos u}\\\\\ny &= \\frac{a\\sinh v \\sin\\phi}{\\cosh v - \\cos u} \\\\\nz &= \\frac{a\\sin u}{\\cosh v - \\cos u}\n\\end{align}", "1c935e25bb7bd4ba1ad8cbf5a67bf173": "k\\mid l \\implies f(n,k)\\mid f(n,l)", "1c93667590f379fe888f55c064ae3c6a": "\\text{var}\\,[Y(\\mu;t)] = a\\mu^pt^{2-d}\\,\\!", "1c93a490ffc51046d58fbd22f9d48256": "I_{cr} = I_p / N_{cr}", "1c93b5ef40f1a929e9f061424bbafc69": "\nc^2 {d \\tau}^{2} =\n\\left(1 - \\frac{r_s}{r} \\right) c^2 dt^2 - \\left(1-\\frac{r_s}{r}\\right)^{-1} dr^2 - r^2 \\left(d\\theta^2 + \\sin^2\\theta \\, d\\varphi^2\\right),\n", "1c93fc7a6403093fdb794859c65cd6af": "\\sum_{n}|a_n|^p < \\infty", "1c93fdbbf342d2d131689f2a5492a9b7": " K_t(x,y) = K_t(x-y) = {1\\over \\sqrt{2\\pi it}} e^{-i(x-y)^2 \\over 2t} \\, .", "1c94057e24b6e3c36d1c799fb6ec5e9b": " CD_\\text{wave} = \\frac {24V} {L^3} ", "1c940d18986659f3a362e3b582d26422": "i \\leq j", "1c944c31217a8d5559c152cafb5f1a4f": "\\frac{1}{\\sigma\\sqrt{2\\pi}}\\, e^{-\\frac{(x - \\mu)^2}{2 \\sigma^2}}", "1c944d3a52acd084050af2ce9ff0e655": " V_A, V_B:\\Sigma\\mapsto\\mathbb R", "1c94e9ee1711656d0dc527c6ef08f12d": "f(x_0+2,y_0+3/2) - f(x_0+1,y_0+1/2) = A+B = \\Delta y - \\Delta x", "1c955d3f150225f23e6d0dcec41478e7": "x^4=a", "1c95700138cb221d5added03a8c70015": "\\sum_{n=1}^{\\infin} 1^n", "1c95a1fad0c57258bf1819bee7212683": "\\beta^*\\theta = \\beta^*(\\sum_i p_i\\, dq^i) = \n\\sum_i \\beta^*p_i\\, dq^i = \\sum_i \\beta_i\\, dq^i = \\beta.", "1c95e241a89bb33fd581bd8610cdde77": " P_{load} \\, ", "1c95e7392296825bddad0404bfe05a07": "\\frac{dx}{ds} = a", "1c95edcecaa3a950fcaf841590eca415": " W_{i_1..i_k j_1...j_k}=V_{(i_1..i_k)(j_1...j_k)}", "1c9605e969bb3f5c8c80d74071bba478": "\\sqrt{I_{L1}^2 + I_{L2}^2 + I_{L3}^2 - I_{L1}*I_{L2} - I_{L1}*I_{L3} - I_{L2}*I_{L3}}", "1c96bf233e7a9087addf7fddd590e073": "\\left(\\frac{am+Nb}{|k|}, \\frac{a+bm}{|k|}, \\frac{m^2-N}{k}\\right)", "1c96f72e1b6a4afd406b04ac8501209e": "\\{ e \\} \\!", "1c9766fc4685abae490b7eadb6bb29d7": "\\left\\{ \\begin{matrix}\n N & \\mbox{if } a = e^{i 2 \\pi k/N} \\\\\n \\frac{1-a^N}{1-a \\, e^{-i 2 \\pi k/N} } & \\mbox{otherwise}\n \\end{matrix} \\right. ", "1c97b0cf1433df7d02b752a961b38f4d": "t = \\frac{\\mu_y z - \\mu_x}{\\sqrt{\\sigma_y^2 z^2 - 2\\rho \\sigma_x \\sigma_y z + \\sigma_x^2}}", "1c985a311fe6854e1bf2225b3e1e479d": "\\frac{kT}{m}", "1c988e8755ae3b64b1bfbfc7ae4347c5": "a . b", "1c98abcebf708f057c7735ddfe3bf252": "\\lambda_3", "1c98f86c5fb58734f117ca79657fd754": "I_{sp} / c", "1c98fb9836626c6feb3a9b692c25bc6b": "\n \\int x^{m+n} \\left(2 a\\,B (m+1)-A\\,b (m+n (2 p+1)+1)+(b\\,B (m+1)-2\\,A\\,c (m+n (2 p+1)+1)) x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}dx\n", "1c99396fe1e7076b433ce55dc1a84fad": " A = \\int_{\\infty}^{-\\infty} y(T) x'(T) dT = \\int_{\\infty}^{-\\infty} TPR(T) FPR'(T) dT = \\int_{-\\infty}^{\\infty} TPR(T) P_0(T) dT = \\langle TPR \\rangle\n", "1c9957f8c25dad33fd3e05db74e039d1": "\\frac{dD}{dr} \\leq 0.", "1c99968db24ffddae0aa89e1ddba57dc": "\\scriptstyle \\mathbf J \\;=\\; 0", "1c99974e6b4cc2b15e75506341ed3b37": "\\overline{D}_{\\hat{\\dot{\\alpha}}}X=0", "1c9a842ee9ad3c637b71018a7a51bf23": "D(y)", "1c9a8f941d8f259af16672f3a21f79e1": "p^\\mu = -i{\\partial\\over\\partial x_\\mu} ", "1c9aa43424adf8d05120f44a1b5b0866": "u(x_i)=u_i", "1c9af354e628d5361597afc6238a7af1": "= 2 p_{T 1} p_{T 2} ( \\cosh(\\eta_1 - \\eta_2) - \\cos (\\phi_1 - \\phi_2) ) .\\,", "1c9b1d75901ea83338820ba3f26b9414": "(\\mathcal F f)(\\xi)", "1c9b28fa6a62e4f679cbdcc68255afbd": "\\mathrm{Hom}(\\mathrm{colim} F, N)\\cong\\mathrm{lim}\\,\\mathrm{Hom}(F-,N)", "1c9b442b89f521da79549d4d571f30a0": "\\left\\vert \\psi\\right\\rangle ^{A}", "1c9b841e06b401ef5d0967e2153df858": "0.\\overline{45}", "1c9baa79f8f2d8e973c9269e0b5ddde3": "\\begin{align}\n&A_{j_0} (x)~,\\\\\n&[A_{j_{0}} (x), A_{j_{1}} (x)]~,\\\\\n&[[A_{j_{0}} (x), A_{j_{1}} (x)], A_{j_{2}} (x)]~,\\\\\n&\\quad\\vdots\\quad\n\\end{align}\n\\qquad 0 \\leq j_{0}, j_{1}, \\ldots, j_{n} \\leq n\n", "1c9bac504188b2152665436fb2ac6dcf": "\\Phi_R(\\mathbf r, t) = \\tilde{\\Phi}_L(\\mathbf r, t)", "1c9bbd54e308aec6a21619347b83639a": " S_{h}^{p}(\\Omega_{h})=\\{v_{|\\Omega_{e_{i}}}\\in P^{p}(\\Omega_{e_{i}}), \\ \\ \\forall\\Omega_{e_{i}}\\in \\Omega_{h}\\}", "1c9bcf491c515eece22a8d7783e1ead2": "g(x)^q", "1c9be3ee8886693a30f34e670e5a43e6": "\\langle \\cdot\\rangle", "1c9c15c0d2a33410277fbf10ab53db9c": "F(b) - F(a) = \\sum_{i=1}^n \\,[F'(c_i)(x_i - x_{i-1})].", "1c9c3fdd1d73cf00b078d94336f6bd6b": "\n\\sigma \\ = \\ 0\n", "1c9c88b480a6054d92374045a1450acc": "\\frac{a}{\\sin A} = \\frac{c}{\\sin C}.", "1c9ca2b01e5e7189cd537ececc519e88": " \\tan E =\\frac{\\sin E}{\\cos E} = \\frac{ \\sqrt{1-e^2} \\sin \\theta }{e + \\cos \\theta} \\ . ", "1c9cb84425b520512f4e6e88a9d56f50": "A^\\text{T} = F^\\text{T}C^\\text{T}", "1c9ccae34809bb9e4997c31189b9f515": "z(t) = t.\\,", "1c9cdcf12b2d951923f539f93de579f8": "\\mu=\\frac{\\exp{(\\mathbf{X}\\boldsymbol{\\beta})}}{1 + \\exp{(\\mathbf{X}\\boldsymbol{\\beta})}} = \\frac{1}{1 + \\exp{(-\\mathbf{X}\\boldsymbol{\\beta})}}\\,\\!", "1c9d246ca12efceb95d9a8dff4ab90f0": "D\\colon K(\\!(X)\\!)\\to K(\\!(X)\\!)", "1c9db336f635ec280e505d27bed0f426": "M_\\lambda=\\{v\\in M;\\,\\, \\forall\\,h\\in\\mathfrak{h}\\,\\,h\\cdot v=\\lambda(h)v\\}.", "1c9db949e178c9969de0565ead550e01": "A + B := \\{\\, a + b \\in \\mathbb{R}^{n} \\mid a \\in A,\\ b \\in B \\,\\}.", "1c9dd5e318b571f3a58075685b58721b": "\\phi(n) = n - \\sum_{i=1}^r \\frac{n}{p_i} + \\sum_{1 \\leq i < j \\leq r} \\frac{n}{p_i p_j} - \\cdots = n \\prod_{i=1}^r \\left (1 - \\frac{1}{p_i} \\right ).", "1c9df30b8c063dbf839f9cc7caf17d36": "\\left( {1 \\over 3}\\times1 \\right) + \\left( {2 \\over 3}\\times{1 \\over 2} \\right) = {2 \\over 3}", "1c9dff5e805a253a78053328522ced0a": " pmax( v_j )", "1c9e0ff5274efffcff70cd4edb80a764": "\\tilde{n}=n+i\\kappa", "1c9e70b8e754419eddbe07880c5512e6": "g_i^{N_i}", "1c9e8e89a21a333db1f4e6b2d5f91bb0": "\\scriptstyle{dy}", "1c9ec4f302f14abdeab0f1fa59ff741b": "|0\\rangle", "1c9ec9accfca7bd71ce8c96189acbdd1": "\\mathcal{B}_r = \\lfloor r+p \\rfloor, \\lfloor 2r+p \\rfloor, \\lfloor 3r+p \\rfloor,\\ldots = ( \\lfloor nr+p \\rfloor)_{n\\geq 1}", "1c9edfbb273dc9269d8e526b1394bef1": "{\\diamondsuit(E^{\\lambda^+}_{cf(\\lambda)}})", "1c9ef6cf3aa261aa1452d8002ad11d5f": "\nM =\n\\begin{bmatrix}\nf^*\\\\\ng^*\n\\end{bmatrix}\n\\begin{bmatrix}\nf & g\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nf^*f & f^* g \\\\\ng^*f & g^*g\n\\end{bmatrix}.\n", "1c9efe46a5bf67fcd84826841738427f": "J_2 ", "1c9f43d4c089779964797f41fc9df06d": "\\gamma=0.739", "1c9f7e0799d4bfc76e07fae718ac0a2a": "K_0,K_1,\\ldots,K_{n}", "1c9fa6ac43ccd9c4f7d5921c17e36f4d": "{x^\\prime}^2\\left(A\\cos ^2\\theta\\ +\\ B\\sin \\theta\\cos \\theta\\ +\\ C\\sin ^2\\theta\\right)\\ +\\ x^\\prime y^\\prime\\left\\{B\\left(\\cos ^2\\theta\\ -\\ \\sin ^2\\theta\\right)\\ - 2\\left(A\\ -\\ C\\right)\\left(\\sin \\theta\\cos \\theta\\right)\\right\\}", "1c9fdd4e0cf4b75d34fab96391f53444": "{}^q\\!D={1 \\over \\sqrt[q-1]{{\\sum_{i=1}^S p_i p_i^{q-1}}}}", "1ca0017996183a41be8cb5a6189379d2": "\\sum_{j=0}^p\\binom{p}{j}\\frac{B_{p-j}}{j+1}=\\delta_p", "1ca01ab7b39875245b4b4f8c244b2dd3": "\\ell(r,p) = \\sum_{i=1}^N \\ln{(\\Gamma(k_i + r))} - \\sum_{i=1}^N \\ln(k_i !) - N\\ln{(\\Gamma(r))} + Nr\\ln{(1-p)} + \\sum_{i=1}^N k_i \\ln(p).", "1ca089289425a0603ffa3f9d1fb11243": "\\partial : k^* \\rightarrow H^1(k,\\mu_m) ", "1ca0d9c40af6e3780dac772fbcf8551e": "P_{W_n}(x)=x((x-2)^{(n-1)}-(-1)^{n}(x-2)).", "1ca125c081b81130b2087bddf9611de5": "X_{2\\pi}(\\omega)\\ \\stackrel{\\mathrm{def}}{=} \\sum_{k=-\\infty}^{\\infty} X(\\omega - 2\\pi k).", "1ca17a0880354f5e21ec3c8a899bb5e0": "A_y", "1ca1ece4374b5be28722d41a326ae7f1": " u_0 ", "1ca1f7f0a199747e3f8a20409c5bd217": "\\lim_{i \\rightarrow \\infty} A_i = \\{ x : \\lim_{i \\rightarrow \\infty} x_i = 1 \\}.", "1ca2603d2e142635b0e03d04ff0d6912": "X^*(s) = \\sum \\bigg[\\text{residues of }X(\\lambda)\\frac{1}{1-e^{-T(s-\\lambda)}}\\bigg]_{\\text{at poles of }X(\\lambda)},", "1ca26c89c1cd3373413036978a190715": "R = {\\lambda\\over\\Delta\\lambda}", "1ca27c90c619a6de32b3863228bdb6de": " 2^{\\aleph_0} = \\aleph_2 ", "1ca2dba6757b23cc0978ba27afcaff08": "\n n \\int\\limits_{-\\infty}^\\infty (F_n(x) - F(x))^2 w(x) dF(x),\n", "1ca338e335cf6b4eb30c9a8f857de31a": "\\tau_d \\simeq \\delta^2 / \\kappa", "1ca34fd8b78d19931cd5c2b4d26ed451": "\\mathcal G^{4,1}", "1ca3877c63f927e1242d20331b83da24": "\\nabla\\boldsymbol{U}", "1ca3889310ad83b6563c7b80eeb57aa4": " a_{t} \\in \\Gamma (x_t), \\; x_{t+1}=T(x_t,a_t), \\; \\forall t = 0, 1, 2, \\dots ", "1ca39ba253deea0b3df7b8a211b7f889": "\\left(2\\sqrt{\\frac{2}{5}},\\ -2\\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm1\\right)", "1ca3a185ef17ba78ac1b3d91f126bc42": "\\mathrm{d}^2 \\Phi \\over \\mathrm{d}A\\ \\mathrm{d}{\\Omega}", "1ca3bf1a6cec9c9bc81455982f5f6cd8": "U'\\cap T(\\Omega)=\\varnothing", "1ca427b2d7eebda250d3cea88aea83ce": "\\C^1", "1ca46a467832ec88aab6627900be4f8d": "\\mathcal{G}(V,E,F) = \\{(V,E+F') | F' \\subseteq F\\}, E \\cap F = \\emptyset", "1ca46dd58a745513acb303e808aa4cda": "a \\uparrow^n b = a \\uparrow^{n-1} \\left(a \\uparrow^n (b-1)\\right)", "1ca4ccab8f963e559324ab49a0ead940": "N_{M_j} = \\max\\nolimits_{m_{j+1}} N_{M_{j+1}}.", "1ca4d3ea7a03b51fba0451cd3bfe01b5": "\\ln 2", "1ca4e36773aae91b7ea23329b0604468": " \\begin{align} \\hat{T} & = \\frac{\\mathbf{\\hat{p}}\\cdot\\mathbf{\\hat{p}}}{2m} \\\\\n & = \\frac{1}{2m}(-i \\hbar \\nabla - q\\bold{A})\\cdot(-i \\hbar \\nabla - q\\bold{A}) \\\\\n & = \\frac{1}{2m}(-i \\hbar \\nabla - q\\bold{A})^2\n\\end{align}\\,\\!", "1ca50e435868e1bd41d56201ac4817af": "[\\mathit{id}_A \\times\\iota_1, \\mathit{id}_A \\times\\iota_2] : A\\times B + A\\times C\\to A\\times(B+C)", "1ca522e0e79cdb32b18b84d0198f7b1e": "\\frac{c^2k^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}", "1ca540f4c3e43c38c1dafbb235f0d405": "z =\\exp(2\\pi i/3).", "1ca57ab59bba961d083e5e529e47c8de": "|\\langle u,v\\rangle| \\leq \\|u\\| \\cdot \\|v\\|.", "1ca5826fa422987c871d4e38394ab94a": " K=\\sqrt{1+t\\over 1-t}", "1ca591d80112a79abcdfc90b3c732d6f": "\\vec A", "1ca5a0b8967b59e89dbb56957c3a5b1a": "\\rho(X)=1", "1ca5c5e806bf5dde85384d5f2d32fcc1": "T={T_1,T_2,\\dots,T_n}", "1ca5d6ace1a97a806845cd92c9bce6dc": "\\psi(\\Omega^{\\psi(0)})", "1ca5e3299c87781bd55e9f001bd5f04d": "\\frac{1}{2}mv_e^2 + \\frac{-GMm}{r} = 0 + 0", "1ca5f874ffd5726fa13c79db690b27a0": "d = (\\sqrt{p} + \\sqrt{x}) (\\sqrt{p} + \\sqrt{y}) (\\sqrt{p} + \\sqrt{z})", "1ca601a62b6ae30bc0953ef1c9925853": "r_E={a\\over\\sqrt{6}}\\,", "1ca617f4ce3795c51622f38fb20df8c4": "f(\\theta,\\varphi)=\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell f_\\ell^m \\, Y_\\ell^m(\\theta,\\varphi).", "1ca61f6043dff952e9a1bc0457d63d66": "h_{g;k_1, \\dots, k_n} = m! \\prod_{i=1}^n \\frac{k_i^{k_i}}{k_i!} \\int_{\\overline{\\mathcal{M}}_{g,n}} \\frac{c(E^*)}{(1-k_1\\psi_1) \\cdots (1-k_n \\psi_n)}.", "1ca645cd01f4d0d2ad4db18338a2b1fe": "r_2 = d", "1ca676a902c7ee21d50d645feaf5007a": "\\omega \\gg \\frac{1}{RC}", "1ca6ac13ab18ab980c9655d93f2e7d03": "\\lceil\\log_2 xy\\rceil \\leq \\lceil\\log_2 x\\rceil + \\lceil\\log_2 y\\rceil", "1ca6b4e9258579f24b12187c5554a729": "1 - \\sum_{i=1}^{N} s_i^2,", "1ca6b7ebbe4d1b1f4abf24340d2598c9": "X = \\begin{bmatrix}A & B \\\\ C & D\\end{bmatrix}", "1ca6fca218312104abb050083a238281": "f_\\text{circ} = \\frac {4 \\pi (2166086)} {39330^2} = 0.0176.", "1ca6fd73832c90423e6a71bbc42aa1cb": " \\sum_{n}^{}{(T_i-T^0)(-\\Delta S^0_i)} ", "1ca7a2b80a80936027dc95b310249e47": "k \\rightarrow \\infty", "1ca7c2e7d77078fe8dc06064bfdf7955": "P=\\frac{\\partial W}{\\partial t}=\\frac{\\partial QV}{\\partial t}", "1ca7f228e45204df5750cbd2b37028ed": "a_b = u^c \\nabla_c u_b ", "1ca7f78a7a2ef95b278c379f88d1420f": " \\tau_i = \\frac{\\beta_i}{\\sum_{j\\in\\Theta}\\beta_j} \\tau,", "1ca7fda31d6b74c1e1b93c13dc25ad01": "\\mathfrak{f}(L/K)=\\eta_{L/K}(s)+1", "1ca809d09d6f0e7f474d8db05ecfc3a3": " \\real\\langle x, Ax\\rangle \\leq \\mu(A)\\cdot \\|x\\|^2", "1ca81a07c3742d220da66dd83a77e649": "Q_f = {Q_w}/\\sqrt {S{p_g}} \\ ", "1ca851b440acbe5ed42a53e3ef91f5b5": "\\frac{\\sqrt{2}}{\\pi} \\,", "1ca8585e8d263b447e4666c928c23d42": "\\sigma \\Vdash_P \\psi", "1ca886e8b4aafb2434404e2f069d948a": "\\{\\neg a, b\\}", "1ca927202171567eb921c14591b1ce0f": "\\frac{6\\text{ failures}}{7502\\text{ hours}} = 0.0007998 \\frac{\\text{failures}}{\\text{hour}} = 799.8 \\times 10^{-6} \\frac{\\text{failures}}{\\text{hour}}, ", "1ca968eca785ff72eefca0ac5dc3c080": " \\mathrm{Tr}\\left( F_i F_j \\right) = \\frac{\\mathrm{Tr}\\left( \\Pi_i \\Pi_j \\right)}{d^2} = \\frac{\\left| \\langle \\psi_i | \\psi_j \\rangle \\right|^2}{d^2} = \\frac{1}{d^2(d+1)} \\quad i \\ne j.", "1ca9dd35e1c40938315ec20605f57174": " D_c: L^2(R)\\rightarrow L^2(R), (D_cf)(x)=\\frac{1}{c^\\frac{1}{2}}f(\\frac{x}{c})", "1caa29949d17b1affd8df5b76a8f15ae": "M_e", "1caa9a1cb1dd3daa0f867d4b9ccb3672": "g \\times g", "1caaa26921f42730131f89f4a2e6f7c8": "m+\\frac{s}{\\sqrt[\\alpha]{\\log_e(2)}}", "1caafc349ee3a1c70d8e312250832c8f": "\\scriptstyle|\\zeta|^q\\leq 1+\\|a\\|_p^q ", "1cab69d594f08ad3327de10c74579623": "\\Delta p\\ =\\ \\gamma \\left( \\frac{1}{R_x} + \\frac{1}{R_y} \\right)", "1cabc3d3496eb21a7a72a8724a08b0d5": "F\\colon \\mathcal{B} \\to \\mathcal{B}; \\, ", "1cabc86b1d544f278b31f698c6fda3a7": " \\gamma = 7/5 \\,", "1cac4f8ce4b344d80393a52ec308420b": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^2 u\\ \\cos^2 u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^3 u\\ \\cos^2 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^2 u\\ \\cos^3 u \\ du\\ = \\\\\n&-\\hat{g}\\ 2\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^4 u\\ \\cos^2 u \\ du \n+\\hat{h}\\ 2\\ e_g\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u\\ \\cos^4 u \\ du =\n-\\hat{g}\\ \\left(2\\pi \\frac{1}{8}\\ e_h\\right) + \\hat{h}\\ \\left(2\\pi \\frac{1}{8}\\ e_g\\right)\n\\end{align}\n", "1cacb3557cd006327eea5bad59c9d5c4": "n!+A = k^2", "1cace7e7d804072a6ceb71623d2ff851": "\\frac{b^2}{\\sqrt{a^2+b^2}}", "1caceefae6966b45241c72a61e56f9fd": "D_l (l \\ge 1)", "1cad3b2baa594483721eddcf4df5845a": "\n {\n \\begin{align}\n \\rho~\\det(\\boldsymbol{F}) - \\rho_0 &= 0 & & \\qquad \\text{Balance of Mass} \\\\\n \\rho_0~\\ddot{\\mathbf{x}} - \\boldsymbol{\\nabla}_{\\circ}\\cdot\\boldsymbol{P}^T -\\rho_0~\\mathbf{b} & = 0 & & \n \\qquad \\text{Balance of Linear Momentum} \\\\\n \\boldsymbol{F}\\cdot\\boldsymbol{P}^T & = \\boldsymbol{P}\\cdot\\boldsymbol{F}^T & & \n \\qquad \\text{Balance of Angular Momentum} \\\\ \n \\rho_0~\\dot{e} - \\boldsymbol{P}^T:\\dot{\\boldsymbol{F}} + \\boldsymbol{\\nabla}_{\\circ}\\cdot\\mathbf{q} - \\rho_0~s & = 0\n & & \\qquad\\text{Balance of Energy.} \n \\end{align}\n }\n ", "1cade8efd5fe62bc3b0416b61bc08ab8": "j^{r}_{p}\\sigma \\in S", "1cadeba717c44a339e0fd6f6e8cb90ff": "\\begin{align}\n\\mathbf{\\hat{J}} & = \\mathbf{\\hat{L}}+\\mathbf{\\hat{S}} \\\\\n& = -i\\hbar \\bold{r}\\times\\nabla + \\frac{\\hbar}{2}\\boldsymbol{\\sigma} \n\\end{align}", "1cadec1bdc0b6ef34dbae3e3cfdc4bd6": " H_2(S) = \\mathrm{ker}(\\partial_2) / \\mathrm{Im}(\\partial_3) \\cong 0", "1cadede0f43f092bada32ca57580f349": "np \\rightarrow \\lambda", "1cadefb52d43df870f32cf0fa772a493": "\\lambda_1 \\ldots \\lambda_n", "1cae0cf7a6cba6196334175ddcb27f4f": "\n \\mathbf{n} = n_1~\\mathbf{e}_1 + n_2~\\mathbf{e}_2 + n_3~\\mathbf{e}_3\n ", "1cae301bbce25bff262b21375b10815b": "\\lambda (x,y)", "1cae352ae6277a58c1b9bc75ad119c99": "\nJ_y = \\frac\\hbar2\n\\begin{pmatrix}\n0&-i\\sqrt{3}&0&0\\\\\ni\\sqrt{3}&0&-2i&0\\\\\n0&2i&0&-i\\sqrt{3}\\\\\n0&0&i\\sqrt{3}&0\n\\end{pmatrix}\n", "1caea9117e343de5bbf11be05f10c4e2": "(a-c)(a+c) = (d-b)(d+b)", "1caecdaaea62e7f17075f96c9e5b286a": " \\sigma = 0 ", "1caed8b1165383841df9f01b2f88ae49": "\\Pi_t = \\frac{\\omega(r_t)*(\\Pi_{t-1}A)}{[\\omega(r_t)*(\\Pi_{t-1}A)]\\textbf{1}'}.", "1caee8942b70b4623129beb343720378": "\\left(\\sqrt{1/15},\\ -2\\sqrt{2/5},\\ 0,\\ 0,\\ 0\\right)", "1caef47ac7fab9442dce44dc427e7a29": "\\Lambda^k[n]\\to C", "1caf033e7eac800a04b16891e875f241": "O(V^2E)", "1caf07ee8252faed628500935d6eb64e": " S(\\rho^{12}) \\leq S(\\rho^1) +S(\\rho^2) ", "1caf51e9e3ca2d379800775f6ac0532d": "r_{m}\\in E(T_{1}(r_{m-1},x_{m} ))\\Rightarrow r_{m}\\in E(T(r_{m-1},x_{m} ))", "1caf526b143335e86eb76a5c283b9b84": "N(\\mu_i,\\sigma_i^2), i=1,2", "1cafa66a707ef4f2520179988f09fd7d": "2\\otimes3", "1cb008539f6d877d8c5a35190ee1fba8": "R=g^{ab} R_{ab}=0", "1cb05d89cc7f7ec8b48570d435d716a3": "q= C_b \\Delta(V_c)", "1cb0635c17f5de308a2d6a31f19b3395": " \\Delta=\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}. ", "1cb0a266f1296117ac4779b1cab36141": "|k|^{-\\frac{3}{4}} e^{-\\sqrt{|k|}}", "1cb0ae7b22fec0bbca487337c204bb1c": " \\left\\{ X(t) \\right\\}", "1cb0beb10a1c28e6cd07f682ca8b1c00": "\\hat{\\Upsilon}_{n} = Y_{n} + a(Y_n) \\delta + b(Y_{n}) \\delta^{1/2}.", "1cb116a276fe5f826ed111ed1a70f67a": "S.A.R = \\frac{{Na^{+}}}{\\sqrt{\\tfrac{1}{2}({Ca^{2+}+Mg^{2+}})}} ", "1cb11b84ca5288ee3350894b1d42a05c": "\\scriptstyle\\vec y", "1cb156453a13a61d0bb1282666e86abe": "\\frac{64}{81} \\sqrt{2}", "1cb24dafc8035d2c720256620066ae73": "\\delta >0", "1cb266746bc89ca135ae5cd90d588cac": "e:\\textit{Eve}", "1cb2b472910ab2f9145e3ce23f9276a8": "f_{max}", "1cb2d3fcd417cc48cba7e3fe824d8c4d": "\\vec{v}_{\\rm inertial} = \\frac{v_\\|}{\\omega_c}\\, \\vec{b}\\times\\frac{d\\vec{b}}{dt},", "1cb2e0fe79c829bb6d883a3a81e684e1": "x_{\\lambda}", "1cb30670494b9a39dd3155194767e368": "|\\mathcal A|\\subseteq |\\mathcal B|", "1cb392867eefe35628f6900f5a3f7aba": " \\langle n|H|n\\rangle= E_0 = E_i - U \\ .", "1cb43ab803bc4fcf0458cdc0220fb1bc": "\\|f\\|^2 = \\sum_{n=0}^{m-1}{|a_n|^2} + \\|R_m\\|^2", "1cb485cfb0886bf602848cf9464ed20d": "\\scriptstyle A \\;\\subseteq\\; \\{ \\lim_n a_n:\\, \\forall n \\;\\ge\\; 0,\\, \\ a_n \\in A \\}", "1cb4a2503f05c07ea4ebe9eb8b283c95": "s(x)=\\min_{n\\in{\\bold Z}}|x-n|", "1cb536c18ef0f7d7957882aea091a61c": "\\leq_c", "1cb539cf31a9bc49ac9a1ccbc12b552f": "E_2=E_1 + \\hbar \\omega", "1cb561e2a7b9a4dcf8e787631e2d543a": "\nM = \\frac{q B}{c} \n\\left(\\begin{matrix}\n 0 & 1\\\\\n-1 & 0\n\\end{matrix}\\right),\n", "1cb6185d789b84280012a1e2c1a4caae": "PA = \\bigcup_{n} PA_{n}", "1cb61f6cac13eef6084f2336b4232846": "\\pi_{i} P_{ij} = \\pi_{j} P_{ji}\\,,", "1cb69307b0779287bd3a69f93b367e70": "\\displaystyle \\delta(\\xi)", "1cb6c310d2c2934b9b91d17ca7b37abf": " \\alpha \\equiv ", "1cb7049065a0f3067920a95ecdde8dac": "\\Delta_k(x)=\\mathcal{O}\\left( x^{\\alpha_k+\\varepsilon}\\right)", "1cb706122d4a127ce184f937c69c4be9": "\n\\theta = \\pm 2 \\ \\mathrm{atan}\\left( \\sqrt{\\frac{1 - C}{1 + C}} \\right).\n", "1cb731ffffcba6391745197b9efe509d": " \\det A_{33} = 0 ", "1cb7ccd2a282e73d862063571c8226e6": "\\cot\\frac{\\gamma}{2} \\sin\\frac{a-b}{2} = \\tan\\frac{\\alpha-\\beta}{2} \\sin\\frac{a+b}{2}.", "1cb8a3872985bd0131ff7d7806f37db4": "-\\alpha-1", "1cb8ae633a26b6181b0ca6e88b3d81d4": "R[e_1, \\ldots, e_k]/\\left\\langle \\{e_ie_i|1\\le e_i\\le e_j\\le l\\}\\right\\rangle,", "1cb8afd87fc11473fe98ebfdfbd306da": "\\scriptstyle \\dot{R}", "1cb8c8bbf8c66366dde4b17c0f835666": "C_{V,m}", "1cb90320c4bbd6ed430ced9542e09e76": "h\\mathbb{Z}", "1cb9372b8daebabd798e972344b47b36": " n_1,\\ldots,n_k", "1cb983f53e57343e2f6370899b4d7282": " E_{act} ", "1cba6786cc8f541030caa7244f3e6430": "r = r_0 \\cos(\\varphi - \\gamma) + \\sqrt{a^2 - r_0^2 \\sin^2(\\varphi - \\gamma)}", "1cba6aeca5622f7ce19ea07e918f9aa8": "\\begin{align}\ne^{\\pi \\sqrt{19}} &\\approx x^{24}-24; x^3-2x-2=0\\\\\ne^{\\pi \\sqrt{43}} &\\approx x^{24}-24; x^3-2x^2-2=0\\\\\ne^{\\pi \\sqrt{67}} &\\approx x^{24}-24; x^3-2x^2-2x-2=0\\\\\ne^{\\pi \\sqrt{163}} &\\approx x^{24}-24; x^3-6x^2+4x-2=0\n\\end{align}\n", "1cba7f343e73b1666ece4c0cb999540d": "\\,CD", "1cba900507099f1394ee8482d9f49f2d": "u^2_i = 48", "1cbab575f605089f277cae2dc59a6d6c": "B_5", "1cbac16f4efe4579cc742613501efb47": "f(\\alpha^{-1}(y))=\\alpha^{-1}(y+1)\\, .", "1cbac314be0c2537a9e5d16882368af0": "\\times\\sum_{r=0}^{\\ell-s} {\\ell-s \\choose r} {\\ell+s \\choose r+s-m} (-1)^{\\ell-r-s} e^{i m \\phi} \\cot^{2r+s-m} \\left( \\frac{\\theta} {2} \\right)\\ .", "1cbaf38e845f301fdf5302a032c8dc98": "f:\\mathcal{Z}^{2k+1}\\to\\mathcal{X}", "1cbb092423afe4c132ebe4567a1221ac": "\\{A, B\\} = AB + BA", "1cbb6a0c917aec4a216cd9d1a7f008d8": "\\psi(\\Omega^2) = \\phi_2(0)", "1cbb6b81beb9b71a9b17ce72a078b7be": "\\Delta m^2<0", "1cbb7e672946849e053ea4a59ba4adf9": "\\mathbf{\\Psi}_{10}= -\\sqrt{\\frac{3}{4\\pi}}\\sin\\theta\\,\\hat{\\mathbf{\\theta}}", "1cbc42372b8f385a270129f40d3ceb41": "x_i=y_i\\cdot z_i", "1cbc4db7d1a3530507323b0f04eb00c5": " v = \\begin{matrix} \\frac12 \\end{matrix} \\cdot (v_1 + v_2) ", "1cbc50fff51d6206e043395083ed818e": "u^{\\alpha}_{I}:U^{k} \\to \\mathbf{R}\\,", "1cbc7c2ae0f92a1a4ea670af843ff776": "\\left[1 + \\left(\\frac{x-\\mu}{\\sigma}\\right)^{1/\\gamma}\\right]^{-\\alpha}", "1cbc9459a63fb82dbea22b1fedb7b5d0": "\\frac{\\dot{A}(t)}{A(t)}", "1cbc99dcc76739a152d2346c0454a903": "z=4.467\\times \\log_{10} [\\alpha2 macroglobulin (g/L)]-1.357\\times\\log_{10} [Haptoglobin (g/L)] + 1.017 \\times \\log_{10} [GGT (IU/L)] + 0.0281 \\times [Age (years)]", "1cbcc67ccff7ece2d4bf172a373ca61f": "\\arg{z}=\\{\\operatorname{Arg} z+2\\pi n:n\\in \\mathbb Z\\}", "1cbcddb8c6d3a87ee51c468ed772f8e0": "\\displaystyle \\pi^{-\\delta}\\Gamma(\\delta+1)\\left|\\frac{\\boldsymbol \\nu}{2\\pi}\\right|^{-n/2-\\delta}", "1cbdf7c5ed9bd6ba0ab007419195da48": "\\dim (R/P) + \\dim (R/Q) = \\dim (R)\\ ", "1cbdf9c177c2d0933865ccb355ced749": " \\langle i,j \\rangle ", "1cbe0fb258b2a216e0accf76ee525a84": "|z|=1/(t-\\epsilon) > R", "1cbe65f54d92fb7a7358e16c40572f51": "\nT_2=\\frac{1}{4}\\sum_{i,j}\\sum_{a,b} t_{ab}^{ij} \\hat{a}^{a}\\hat{a}^{b}\\hat{a}_j\\hat{a}_{i},\n", "1cbe6ec4bf4eb89865a7434d6ff87fab": " A^{-1}[i]", "1cbea6a7320b6cc31e964dbe512128ee": " \\mathrm{St}= {f\\over U}{C^3} ", "1cbef2a2a9ae468d92c38e9de6e33b35": "-S = \\{ -s: s \\in S \\}", "1cbf23be176975393ecabffa2dab73b8": " c + d\\,\\mathrm i = s \\cdot ( \\cos(\\psi) + \\mathrm i \\sin(\\psi) ) = s \\cdot \\mathrm e ^{\\mathrm i \\psi} ", "1cbf4b8617ae5542fdea66c1fb111aac": "\\Pi_1 = P(q_1+q_2(q_1)).q_1 - C_1(q_1)", "1cbf4d434e3bcf063cb73efb4a652edb": "\n n \\equiv {c \\over c'} \\approx \\left(1 - {2 \\Phi \\over c^2} \\right). \n", "1cbfe66b8fa0b227c1a43fd7414c37f1": "J\\subseteq A", "1cc060f045ded3bc5bab0dd42e9cb72b": "V_1 = P_{11}Q_1 + P_{12} Q_2 + P_{13}Q_3, ", "1cc07aeda7870026cd806aa21bae633e": "i_1, i_2, \\ldots, i_n", "1cc0878180d3266d09d8927f7e23a6b1": "{a\\pi\\over 5}\\ {b\\pi\\over 5}\\ {c\\pi\\over 3}", "1cc098e1b55764f2f0b192a23f5ea6ef": "(-\\Delta)^{-\\alpha/2} f(x)", "1cc0c0a24f2576cbe53d268966f8d942": "\\scriptstyle i,j,k", "1cc0efbfef307be7e7b959ddcee5715b": "\nA(\\alpha, \\beta \\,|\\, z) = G_{p+2,\\,q}^{\\,q-m,\\,p-n+1} \\!\\left( \\left. \\begin{matrix} -a_{n+1}, -a_{n+2}, \\dots, -a_p, \\alpha, -a_1, -a_2, \\dots, -a_n, \\beta \\\\ -b_{m+1}, -b_{m+2}, \\dots, -b_q, -b_1, -b_2, \\dots, -b_m \\end{matrix} \\; \\right| \\, z \\right).\n", "1cc1162706b268c0d1033138c4762469": "y_{Q21} = 1.00 + j1.52\\,", "1cc1585ee701f5338f50e49c6383c469": "\\vartheta_{11}", "1cc1896313f1e17e38e59979c1055ddc": "{ { \\frac{1}{\\varrho}}{\\frac{\\partial p}{\\partial z}}} = - g. \\qquad (3)", "1cc18a8a1efe46c7faa9f15b4f04e124": "\\Phi_4", "1cc2076be935cc3893f05d7dd7c8a9a2": "\\frac{X_s}{X_r^'}\\approx\\frac{0.4}{0.6}", "1cc24d7843f23d50d16562d326fa25ec": "r(\\emptyset)=0", "1cc250ab164bfba7b16daaacdc8d7fbb": "-m_1/(m_1+m_2)", "1cc2a1b2e89ad52fa91a8e69ea064176": "k \\varphi (N)<\\varphi (N)d", "1cc2d642ae02fa633ff953fdd4eae851": "\\varphi_{a_1X_1+\\ldots+a_nX_n}(t) = \\varphi_{X_1}(a_1t)\\cdot \\ldots \\cdot \\varphi_{X_n}(a_nt).", "1cc355103aee2cd8a91686dfb9e13305": "(42/9!)\\prod_ p \\left((1-p^{-1})^4(1+4p^{-1}+p^{-2})\\right)", "1cc3ddf38986dbd80fdbb180af426e1a": "I(t) \\to 0", "1cc3ed73d449afce1c25c8344ab61a97": "\n \\mathbf{a}\\cdot (\\mathbf{b}\\times \\mathbf{c}) =\n (\\mathbf{a}\\times \\mathbf{b})\\cdot \\mathbf{c}\n", "1cc3ff30e30d539a35dc81eb33a97281": "f \\, '", "1cc422427571db0aa3bb835d0311c7ea": "D(s)=1+kG(s)", "1cc46c0b4d052113fcc067fdeab855bf": "b_n \\to \\infty", "1cc4cd649adc27e8b546ad8b5fc47770": "TM =\\bigsqcup_{x\\in M}T_xM=\\bigcup_{x\\in M} \\left\\{x\\right\\}\\times T_xM\n=\\bigcup_{x\\in M} \\left\\{(x, y)\\vert\\; y\\in T_xM\\right\\}.", "1cc4dd87545adf85076e1b3fb381f96d": "\\begin{align}R_{F}(x,y,z) & = R_{F}(A (1 - \\Delta x),A (1 - \\Delta y),A (1 - \\Delta z)) \\\\\n & = \\frac{1}{\\sqrt{A}} R_{F}(1 - \\Delta x,1 - \\Delta y,1 - \\Delta z) \\end{align}", "1cc4e93534d5e2271a2c8e3b11d6bdc8": "\\sum_{i=1}^2 \\left(\\frac{1}{2} \\omega^2 Y_i^2 M_i\\right)=\\sum_{i=1}^2 \\left(\\frac{1}{2} K_i Y_i^2\\right)", "1cc5eb97f8145ab4b4bfdb4dcc3ee936": "BWT[i] = S[A[i]-1]", "1cc5fb6d3b10cf0b4029e23d46fa7fc0": "[0,2\\pi]", "1cc77ce43189651af6fecded877ab079": "PGL(n,K) = \\mathbf{P}(GL(n+1,K)),", "1cc7d54323b032d8b8e28e4cb52bc7aa": "+1.0000000000 +1.1913785723 \\cos(x) -0.0793018558 \\cos(2x) -0.2171442026 \\cos(3x) -0.0014526076 \\cos(4x)", "1cc856eaa4864355bc7aa0fb29ae5615": "\\hat\\beta = (X^TX)^{-1}X^Ty\\ . ", "1cc8a9e0e8873dd05558fd7c41b3f0bd": "F_{13}=\\sin{(\\alpha)} \\sin{(\\theta)} d\\alpha\\wedge d\\phi", "1cc8c15587cd0351848b438d37b4d7c9": "a_m", "1cc8cb1735d017609f416303c2001bda": "\\tau_1 = \\frac{K_p K_v}{\\omega_n^2}", "1cc8dbd0c67f648d4cbaa62a06caa3b0": "Q~", "1cc8e279567ce5de5e131f8766ecc402": " \\mathbf{A} \\ast \\mathbf{B} = (\\mathbf{A}_{ij}\\otimes \\mathbf{B}_{ij})_{ij}", "1cc8f926734abc0689153376616c4415": "u = cu_+ + (1-c) u_- \\,", "1cc904f35a8e49ba6d91450f4234369c": "(\\mathbf{a} \\cdot \\mathbf{b})^2 - (\\mathbf{a} \\wedge \\mathbf{b})^2 = \\mathbf{a}^2\\mathbf{b}^2", "1cc93e320814dbaa255fc5169c1a2b90": "\\sigma_{CS}=\\sigma_{SC}=E(\\cos\\theta\\sin\\theta)-E(\\cos\\theta)E(\\sin\\theta)\\,", "1cc9819f58b6f7bc0edc8b0b14f2a17e": "\\mathrm{d}(\\mathbf{p},\\mathbf{q}) = \\mathrm{d}(\\mathbf{q},\\mathbf{p}) = \\sqrt{(q_1-p_1)^2 + (q_2-p_2)^2 + \\cdots + (q_n-p_n)^2} = \\sqrt{\\sum_{i=1}^n (q_i-p_i)^2}.", "1cc99f1904970eee572f4af94eb33df9": "{10}^{\\,\\! 4 \\cdot 2^{80}}", "1cc9a401b56e2abfee9ece6e4ded7c90": "0 \\leq \\varepsilon_{n+2} \\leq \\min \\left\\{\\frac {\\varepsilon_{n+1}^2}{2}, \\frac {\\varepsilon_{n+1}}{2} \\right\\}", "1cc9fe7ce0659e1a37d7da9b0995ef57": "\\mathbf{z}_{k+1}^\\mathrm{T} \\mathbf{r}_{k}=0,", "1ccaa0caff8c07504f2e17efe436eee3": "Z_{22} = {Y_{11} \\over \\Delta_Y} \\,", "1ccae03d4ec9c20db7dd2217110b298d": "\\mu(a_m/a_0) = \\mu(a_M/a_0)", "1ccaf86acc085dac3283262263f5d8e3": " R_a = \\frac {Z_1 Z_2} {R_b} \\, ", "1ccb2fd4de8445c82de205e329c265d5": "N\\,\\!", "1ccb420d7a9e7bfe23bab3ed06f52f86": "f(x) = x^2 + 2", "1ccb56303db4b96362d87746292c87bd": "\\Delta=11-d", "1ccb88e6ee7760657b5deef01fe44993": "\\frac{\\partial \\mathbf{g(u)}}{\\partial x} =", "1ccbe99d6d11036b50d874766f2dc52d": "\\phi(x)=\\int_V G(x,x') \\rho(x')\\ d^3x'+\\int_S \\left[\\phi(x')\\nabla' G(x,x')-G(x,x')\\nabla'\\phi(x')\\right] \\cdot d\\hat\\sigma'.", "1ccc01d3c0998832c2378069d983090b": "2\\pi\\!", "1ccc0a479afff13788f6cb71495bfd6e": "\\gamma = \\alpha + i \\beta\\,", "1ccc4bfd098e47a4765e886f6c93c518": "\\Delta p_{\\text{B}}(W)", "1cccd264fcb55821fc47b052ee6473bf": "\\alpha_p=\\begin{cases}\\frac{1}{\\sqrt{M} },&p=0 \\\\ \\sqrt{\\frac{2}{M} },&1\\le p\\le M-1\\end{cases} \\qquad \\alpha_q=\\begin{cases}\\frac{1}{\\sqrt{N} },&q=0 \\\\ \\sqrt{\\frac{2}{N} },&1\\le q\\le N-1\\end{cases}", "1ccd30acf2008c6f39d121aa6d39a4da": " G(i\\omega_n) = G_{ii}(i\\omega_n) = \\sum_k \\frac {1}{i\\omega_n +\\mu - \\epsilon(k) - \\Sigma(k,i\\omega_n)}", "1ccd391c7a32c629fbd0f1374d5f869f": "\\mathit{n}\\times\\mathit{n}", "1ccd53a382f5e10570851976edcd9dfe": "\\mathrm{bind}: (E \\rarr \\mathrm{M} \\, A) \\rarr (A \\rarr E \\rarr \\mathrm{M}\\,B) \\rarr E \\rarr \\mathrm{M}\\,B = m \\mapsto k \\mapsto e \\mapsto \\mathrm{bind} \\, (m \\, e) \\,( a \\mapsto k \\, a \\, e)", "1ccd8cbc9f5a7e9d2ffc3c999affc700": "[t_0..t_1]", "1ccd900875bba9c7d559acdee0b4fb9d": "t_{1} \\dots t_{k}", "1ccda985f3150fb4bd15f727a19c992e": "1/\\sqrt{2\\pi}", "1ccdd092bfc31bc1293527c0b16cd7be": "\\ln R=B/T + \\ln r_\\infty", "1ccde18c954e788ef8622714daeee0cd": "V^p_n(R) = \\frac{(2\\Gamma(\\frac{1}{p} + 1)R)^n}{\\Gamma(\\frac{n}{p} + 1)}.", "1cce07fd798a0442264d36d53e24ad74": "j = l+1/2", "1cce16180bd8804d26c6edbf9a6e23e2": " \\widehat J_j (1) ", "1cce9bf6cac32b87a3bbbdc7d0d05e72": " A(x,t) = A_0 \\left(1-[1-{\\Lambda}(t)]{x\\over{L}}\\right)^2 ", "1ccea00c097b218c202eaa8e7271fb1c": "M_p", "1cceb86f8c64f5d0a71ea5c16f51c8c8": "\\alpha = 3-\\sqrt2", "1cceec254196e85c5f3f8122ee8d5ef9": "10_3", "1ccf05ad152eb0c5c90f509a2c2e762d": "\\nabla_v w", "1ccf3aafa41fa9e9817790bdb6bbfe21": "\\textstyle i\\neq j", "1ccf4de9a09566ed93bfc633441c2906": "\\Delta E = p \\Delta V \\,\\!", "1ccf6ee6b96969b20880ea6f7d4f6057": "a \\wedge S \\wedge a", "1ccf9d49b2635c61d18bb9d4195ed51e": "\\Phi\\left(x\\right) = \\left(x + \\mathrm{Ker} F_i\\right)_{i \\in I}", "1cd04755bef80e06c4f019e1df39705f": " \\frac{\\sin^2 \\theta}{\\cos^2 \\theta} + \\frac{\\cos^2 \\theta}{\\cos^2 \\theta} = \\frac{1}{\\cos^2 \\theta}\\!", "1cd0542b8a4213c3651d6382bc564e72": "\\alpha\\,=\\,\\sum_{j=1}^{n}\\,\\frac{\\sigma - \\sigma_e}{\\sigma_e + L_j(\\sigma - \\sigma_e)}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(3)", "1cd09da875db2eb61ac45ea99f671405": "\\gamma_{xy}=\\frac{\\partial u_x}{\\partial y}+\\frac{\\partial u_y}{\\partial x}\\,\\!", "1cd0ff1e99d9c944d46c03ef6f61bafd": "\\displaystyle\\dot q^1,\\,\\ldots,\\,\\dot q^n", "1cd11c858b4834a5be54b0f8437a042b": " \\boldsymbol{\\alpha}' = \\boldsymbol{\\alpha} ", "1cd1a221fbeeda54d8a08614860cd1db": "z\\in E-\\{y\\}", "1cd1e2e06eb41e3936b9217aa690b7f2": "\\Box \\pi_1", "1cd1ff934752811a90d7e66379f28338": "\\sigma_1(A)\\ge ...\\ge\\sigma_n(A)", "1cd20690433dfc37b3cbb0f8684ac1c2": "f^{1}(\\theta)", "1cd23684becc903ec5c255244fe6a199": "p_1, \\cdots, p_n", "1cd25a5fc2fb9ebee169900e0e98606a": "p \\leftarrow \\hbox{not } q", "1cd2d9d41e52e99601b1f76c468d1750": " \nu=\\left[\n\\begin{array}{c}\nu_0 \\\\\nu_1 \\\\\n... \\\\\nu_{Ne} \\\\\n\\end{array}\n\\right]\n", "1cd322f30a6aeac3cb47ee07ace5baa2": " \\frac{a_0}{2} + \\sum_{n=1}^\\infty \\left\\{ a_n \\cos (nx) + b_n \\sin(nx) \\right\\}.\\, ", "1cd326ed099fd1fa2a8dfd1040bc0bef": " \\lambda^2 - 2 \\lambda \\cosh(\\sqrt{r}) \\cos(\\sqrt{r}) + 1 = 0 .", "1cd36e87b5c3913d05abfabc7e1b903d": "{\\mathbf r}_0", "1cd3b99a075c7bbe95dff2598110f6e7": "\n\\varphi(t;\\mu,c) = \n\\exp\\left[~it\\mu\\!-\\!|c t|^\\alpha~\\right],\n", "1cd3c693132f4c31b5b5e5f4c5eed6bd": "pk", "1cd3ca12f87a9155303b836c586f501c": "\\forall n \\in {}^*\\mathbb{N}, {}^*\\!\\!\\sin n\\pi=0", "1cd3efe5e1b77b80ad181cb5e4f1ebf3": "\n\\vec{r}(t+dt)= \\vec{r}(t-dt) + \\vec{v}(t+\\frac{dt}{2}) \\, dt\n", "1cd3f31c8b808dd3a48192c5d374370e": "\\dot{\\mathbf{q}} = \\frac{d\\mathbf{q}}{dt} ~.", "1cd424f15e3be6952346fa87e390dbf2": "k=(\\vec{N}\\cdot \\vec{H})^n=(\\vec{N} \\cdot ((\\vec{L} + \\vec{E}) / 2))^n=(\\vec{N} \\cdot ((\\{ -0.6+\\frac{\\sqrt{3}}{2}; \\; 0.8+0.5; \\; 0+0 \\}) / 2))^3=(\\vec{N} \\cdot ((\\{ 0.266; \\; 1.3; \\; 0 \\}) / 2))^3=", "1cd4373a3d973f6625a8be848b1f5202": " \\operatorname{var}(X)= \\mu^2 \\frac{ \\nu+L+1}{L \\nu} .", "1cd46ed590b2194ee3e1f91fc947b976": "\\scriptstyle{D(ab)=D(a)b+(-1)^{|a|}aD(b)}", "1cd47fc412c528599d95b10f1d859e56": "P\\sim |p-p_c|^\\beta\\,\\!", "1cd487965b1ed24da3ec7d8b94f777f5": "X\\subseteq\\mathbb{R}^{n}", "1cd4b5fcad87d31f6b59dd482f5ac8cb": "\n[u^m] g(z, u)|_{u=1/u} |_{z=uz} =\n[u^m] \\left( \\frac{1}{1-z} \\right)^u = \n\\frac{1}{m!} \\left( \\log \\frac{1}{1-z} \\right)^m,", "1cd4d0951222a1c6d57e05ef3c64e449": " m \\ddot{{x}}(t) = - K x(t) - \\Beta \\dot{x}(t)", "1cd60907f570de5a87b8908eeac809a6": "p\\in\\mathbb C", "1cd64c21a3b2e95af070ab72218fa049": "4x^2+16-16x", "1cd65ff01699dd11eb3dfe5b54ead3c7": " f \\,", "1cd7507f4d63278c051fdc6d44cc320c": "\\beta_s", "1cd75e09ec880cf6ee2951b369365b9e": "P(R_{i})", "1cd77b2bb17c91f1dd5cbcde6618fad7": "\\rho(\\tau_s) \\propto \\exp \\left \\{ - \\frac{\\left ( \\tau_s - 2.1s \\right )^2}{4s} \\right \\}.", "1cd7ec5fde8e5c0801bcac12f73c28c8": "\\frac{3}{5}N", "1cd82b14c99e016b291c66efbdd309fa": "(-\\infty, \\infty)", "1cd885cedacc70751adbbef0c940749d": "\\gamma = \\frac{\\lambda_{PN} - \\lambda_{NN}}{(\\lambda_{NP} - \\lambda_{NN}) - (\\lambda_{PP}-\\lambda_{PN})},", "1cd8c453b83f2419b13b01d9265fd7a4": "|[0,1]|=|\\mathcal{P}(\\aleph_{0})|\\,", "1cd8ee2a55797ac390dc1156e40d05bb": "\\hat f(\\nu) = \\mathcal{F}\\{f(x)\\}.", "1cd9138c1db89e13e94da8a5016fafcc": "\\omega_B = \\sqrt{\\frac{1}{L_BC_B}} = \\frac{\\alpha^2}{2}\\cdot \\frac{m_0c^2}{\\hbar} = \\frac{2\\pi c}{\\lambda_B}, \\ ", "1cd91ae4b965b62ef44654d90c159fde": "\\textstyle (E_{1}-E_{2})=0 ", "1cd92fb7962b7d320479e660c6906598": "\n\\sqrt{2} X(0)= \\sqrt{\\frac{h}{2 \\pi}}\\;\n\\begin{bmatrix}\n0 & \\sqrt{1} & 0 & 0 & 0 & \\cdots \\\\\n\\sqrt{1} & 0 & \\sqrt{2} & 0 & 0 & \\cdots \\\\\n0 & \\sqrt{2} & 0 & \\sqrt{3} & 0 & \\cdots \\\\\n0 & 0 & \\sqrt{3} & 0 & \\sqrt{4} & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\end{bmatrix},\n", "1cd94bf033c877c13c30af03158c5785": "B^2X_t=X_{t-2} \\, ,", "1cd99a20e031887eddfbbf0f13dc167f": "A + n \\to B^{*} + \\gamma", "1cd99bd071e595c0f0d56fad91ad578f": "|G|^2", "1cd9dcadee7afefb384b421277f67489": "(\\neg A\\to C)\\to((B\\to C)\\to((A\\to B)\\to C))", "1cd9ddec625f256fa0eb2351577c9e25": " D \\overline{D}", "1cd9e0326aabfbdf163502d4ba0fa2ce": " \\Delta y_{t-k} ", "1cdaad81bd3a9b5bcc8062a2740a125a": "\n \\int_0^a \\sin\\frac{m\\pi x}{a}\\,\\text{d}x = \\frac{a}{m\\pi}(1 - \\cos m\\pi) \\quad\\text{and}\\quad\n \\int_0^b \\sin\\frac{n\\pi y}{b}\\,\\text{d}y = \\frac{b}{n\\pi}(1 - \\cos n\\pi)\\,.\n", "1cdabbbc5aeb71136feb65014a5f23b1": " \\mathcal{H}\\left(\\mathbf{q},\\mathbf{p},t\\right) = \\mathbf{p}\\cdot\\dot{\\mathbf{q}} - \\mathcal{L}\\left(\\mathbf{q},\\dot{\\mathbf{q}},t\\right)\\,,", "1cdabff88cce339aff58dc31d64699fc": "2^\\mathfrak {c} = \\beth_2 > \\mathfrak c ", "1cdac79582f2f6beac46655f65887e55": "\\scriptstyle \\tan (\\alpha/2)=\\sin\\alpha/(1+\\cos\\alpha)", "1cdaef02128c2d5b400d891252f604c9": "\\textstyle \\binom{n}{k} = {n! \\over (n-k)! k!}", "1cdbc922ee64437602c70cdc96536629": "\\times \\!\\,", "1cdbe8a4b1159a7d40b486c20d070fcf": "n+m", "1cdbf789246d6f65618b03de148e4be0": "H(s) = \\frac{P(s)}{Q(s)} = { G \\cdot \\displaystyle\\sum_{m=0}^M {b_m s^m} \\over s^N + \\displaystyle\\sum_{n=0}^{N-1} {a_n s^n } } ", "1cdc33bffa0035bd9e28d912e751902d": "\\mathcal{L}_{V^{r}}(\\theta)", "1cdc759be8124ad9e6a8740f89d90387": "\n [\\mathrm{j}_k,\\mathrm{j}_l] \\equiv \\mathrm{j}_k \\mathrm{j}_l - \\mathrm{j}_l \\mathrm{j}_k = i\\hbar \\sum_m\n\\varepsilon_{kl m}\\mathrm{j}_m, \\quad\\mathrm{where}\\quad k,l,m \\in (x,y,z)\n", "1cdcf69d432207336eb5b5adb0592c43": "K\\in F_n", "1cdd0149b9bb34251d059c4e5377c674": "\\lambda/(1+\\lambda)", "1cdd3c61a09d223e435dcfd791327ecd": "B \\cap C = \\emptyset", "1cdd6d84f016b0d8e0adc923266b2347": "\n\\varphi(x)\\sim\\sqrt Z\\varphi_{\\mathrm{in}}(x)\\quad \\mathrm{as}\\quad x^0\\rightarrow-\\infty\n", "1cddc5b19a6d4bd574a44601c2a9f48d": "\\lambda \\phi(x) - \\int_a^b K(x,y) \\phi(y) \\,dy = f(x).", "1cddcff75aed7e23f1722644f199fc05": "(x,y,z)\\mapsto(x-2(xz+y^2)y-(xz+y^2)^2z,y+(xz+y^2)z,z)", "1cdde264f9d885c0538c0ca09d62c843": " u(t,x,y,z) = \\frac{t}{4\\pi} \\iint_S \\varphi(x +ct\\alpha, y +ct\\beta, z+ct\\gamma) d\\omega, \\,", "1cde93af7086982a4107c892bf2e6830": "\\scriptstyle{\\widetilde{f}(x) = f(-x)}", "1cdf190afc8d25133df9bd083884d909": "2^{r-1}", "1cdf461d8cadd88aa7b166761ea3a149": "(U; z^A,y^a)", "1cdfe650245cf3a6f5fb647a5913f16c": "d(x,z) \\le d(x,y) + d(y,z).", "1ce015ccf21b95ea19abc8fedd03b2e0": "S_N(x)", "1ce030c95233754dc5c8d41c864d6a7f": "\\displaystyle{\\widehat{\\delta_h f}(m,n) = h^{-1}(e^{-ihn}-1)\\widehat{f}(m,n)=-\\int_0^1 in e^{-inht}\\, dt \\,\\,\\widehat{f}(m,n).}", "1ce03cac9d873314495641b5c46d455a": "M = E I \\kappa = E I \\frac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}", "1ce083a8c349416531d476ac432d5ed2": "\\mathbf{k} \\cdot \\mathbf{x} - k_0t", "1ce09f0913db43ca57cccaebc264976f": "\\tan \\delta=\\frac{\\sum_i a_i \\sin\\delta_i}{\\sum_i a_i \\cos\\delta_i}.", "1ce0c055b6c21f755cd037310fd10439": "n_1 = 1/2, n_2 = -1/2, n_3 = 1/2. \\ ", "1ce0cdaf66b9bf4b74d1f118c30080ef": "\\overline{X}", "1ce0fdd7f80fafc38fda2607a030f179": "\\scriptstyle b(k): \\mathbb{N} \\rightarrow \\mathbb{R} ", "1ce1124a62c94540c3fa52466204221f": "E_{\\infty}", "1ce11b33802040e197b34bf394f8ac6e": "\\langle \\psi\\psi^*\\rangle=\\langle \\mathcal{T}_\\tau \\psi(\\tau=0^+) \\psi^*(0)\\rangle\n=-G_\\eta(\\tau=0^+)=-\\frac{1}{\\beta}\\sum_{i\\omega}G(i\\omega)e^{-i\\omega 0^+}", "1ce17f52a3da9f0cac5c05d849bea323": "(\\mathcal{X}, \\mathcal{A})", "1ce18e9eb07124ee80c9fcc310058e3f": "W_c = \\int\\limits_0^\\infty e^{A\\tau} B B^T e^{A^T \\tau} d\\tau", "1ce1c81e90d06b8d11e838885ed1e381": "270^o", "1ce2098b2c9422a038ff3dda1691d035": "\\sum_{\\gamma\\in\\Gamma} H(\\gamma(z)).", "1ce214a6276be7d69ebeb3fb8bc82c94": " p = S_{kl} ", "1ce215ed1f4788e6e001d8966093ba78": "t_{l}=\\bar{t}", "1ce21b9bfb0496c711676169d9203a41": "\ne \\times m = \\psi.\n", "1ce2400a58ddd50f4f1d738e2e9664e8": "H_i = \\{ y \\in ( \\mathbb{F}_2 ) ^d \\mid y_i = 0 \\} ", "1ce240eb4d6eb4e7a06d67f0451f463b": "\\sigma^* = \\left( \\frac{1}{2.48\\rho^*} \\right )\\Bigg[\\log\\left( \\frac{k_s}{k_{\\text{CH3}}} \\right )_B - \\log\\left( \\frac{k_s}{k_{\\text{CH3}}} \\right )_A \\Bigg]", "1ce253abfa7195d2847feed751306a4a": "c=15.2", "1ce2681b830182c7dbf9cb8f2c253ba7": "\n\\mathcal{A}=\\sum_{j+k\\le n}a_{jk}\\partial_x^j\\partial_y^k=\\mathcal{L}\\circ\n\\sum_{j+k\\le (n-1)}p_{jk}\\partial_x^j\\partial_y^k\n", "1ce27fbc45637312633e0d319aa4c219": "c_p=\\frac{\\Omega(k)}{k}\\,", "1ce2957d5c7ae9fec475b8c46433f3f0": "\\ O(k g) = O(g)", "1ce2ab1565da9257186d6d781fa5d2ae": "\\scriptstyle I_{L_{\\text{max}}}", "1ce2af8f7af2409385d6d6a0194375f3": "P=(-1:-1:1)", "1ce2c059f82c35a93a738da9c6384e0d": "M_{n,R}", "1ce2e225eacd40e6f260535b3a26bf7d": " \\tau_{nuc} = \\frac{\\mbox{total mass of fuel available}}{\\mbox{rate of fuel consumption}} \\times \\mbox{fraction of star over which fuel is burned} = \\frac{MX}{\\frac{L}{Q}} \\times F ", "1ce2e339cb54454547e3e7fb1ca12b16": "\\mathcal{B}_t = ( \\lfloor n(t+1) \\rfloor)_{n\\geq 1}", "1ce2fb3c0bc43df43b8fdf58ae46ec05": "f(x) = b^x. \\, ", "1ce3eba5df9f486625aea47d3b77c63e": "\\bar w = {\\sum_{i=1}^n {p_i w_i}} ~~~~~~~~~~(2)", "1ce409f8edd134fc37e504bc10fdf81d": " \\mathrm{Ro} = {f L^{2}\\over \\nu} =\\mathrm{St}\\,\\mathrm{Re} ", "1ce4300a2940e7ecadebdfd13ebd3690": "\\operatorname{Re}(\\lambda_k) = \\operatorname{Re} \\left ( 1+\\alpha_{-2}e^{i2 \\pi k(N-2)/N}+\\alpha_{-1}e^{i2 \\pi k(N-1)/N}+\\alpha_1e^{i2 \\pi k/N}+\\alpha_2e^{i4 \\pi k/N} \\right )", "1ce447f7f87af00c7f626731ce643be9": "\\mathbf{x}=(x,y,z)", "1ce45f59098ba4a7ed6bf15c314b711a": "{\\bar{Q}}_7", "1ce480cea2943b84c93dc05886f859d7": "\\vdash_\\vec{s}", "1ce4d16bb9365ec9bee39910f6435e15": "\\sum_{i=0}^{j-1} \\frac1{s_i} = 1 - \\frac{1}{s_j-1} = \\frac{s_j-2}{s_j-1}.", "1ce4f6cc0332b04d06101ea472baf7c1": "\nu = \\frac{\\gamma}{2} x^2 - \\frac{\\gamma^2}{2}\\left(\\frac{x^5}{5!}\\right) + \\frac{11 \\gamma^{3}}{4}\\left(\\frac{x^{8}}{8!}\\right) - \\frac{375 \\gamma^{4}}{8} \\left(\\frac{x^{11}}{11!}\\right) + \\cdots \n", "1ce5140c3855f1bcf480d6c70594b877": " k=\\frac{t_r-t_M}{t_M}", "1ce5576158df1955460f5f4a96dacee0": "\\eta_1, \\ldots, \\eta_b", "1ce5630bcb590535a9327bcbd3865a51": "S(P,f,g) = \\sum_{i=0}^{n-1} f(c_i)(g(x_{i+1})-g(x_i))", "1ce57d11f9b40e15fff91bec1c6ca43d": "D0", "1ceeef23130c31175b2c9cd34cce205e": "\\mathcal{N}(\\mu_1, \\sigma_1^2)", "1cef9417759ba8bf07cb1050433cebb6": "T_{V,0}", "1cefaeb1fd125eec86f7ad85dde4239b": "{\\pi\\over 3}\\ {\\pi\\over 5}\\ {4\\pi\\over 5}", "1cefb365e65368c8ac7f843f4de4aa4a": "x = \\int_0^\\infty v(t) \\, {dt \\over t},", "1ceff20a10441bedd341aaa90dd541ec": "L/T^2", "1ceff5f76263a1bb77621183fea19c9a": "V_i = (\\bar{n} - 1) / (n_F - n_C)", "1cf01d999318a69d5b7aa91bf285193e": "P_2+P_3=2+5=7", "1cf04dd62248019272dedd8f3a473b34": "C_\\text{sr}", "1cf0d7a19e485e062479c0764f4b172c": "\\pi_{\\mathbf Q}\\colon{\\mathbf Q}\\to M\\,", "1cf115f1bbe885df6427c6c712a48ab0": "(6,10,3)", "1cf135419cd9d294d0eb7d895ff91656": " \n{\\star \\bold{J}} = -\\rho dt + j_x dx + j_y dy + j_z dz \n", "1cf1819449da0dda035ac5a0b98e214a": "[S^2,S_i]=0, \\quad [J^2,J_i]=0", "1cf18829a73835a00db8c2035cf3cb94": "V_D \\approx 0.30 V", "1cf1c03e9683d9d613db148f1f8d6c85": "\\operatorname{Pr}(Y_i=y_i\\mid \\mathbf{X}_i) = {p_i}^{y_i}(1-p_i)^{1-y_i} =\\left(\\frac{1}{1+e^{-\\boldsymbol\\beta \\cdot \\mathbf{X}_i}}\\right)^{y_i} \\left(1-\\frac{1}{1+e^{-\\boldsymbol\\beta \\cdot \\mathbf{X}_i}}\\right)^{1-y_i}", "1cf1c96acf610a8d1b74c56ccc34f670": "\\mathbf{o}", "1cf1d91afafb2698aafb9325fb3093f2": "F_3(p,Q) = \\frac{p}{Q}", "1cf23d2e5df5bb09142dda7c9d61b5c8": "Z_\\Lambda^\\Phi(\\bar\\omega)", "1cf27ed6cc67f92bb14ae5d4259c671d": "\\mathcal S_n(\\mathcal B_{1\\cdots n}, \\mathcal D_{1\\cdots n})", "1cf29347652ea8d52429ea825172febc": "\\beta_k : M_{n_k}(C( \\mathbb{T} )) \\; \\rightarrow \\; M_{n_{k+1}}(C( \\mathbb{T} )).", "1cf2ef7480a1663c36331c9739b7d053": "\\left\\langle\\tilde{H}\\right\\rangle =\\left\\langle H\\right\\rangle\\,", "1cf326e9182840bfece8444098a54864": " = ([2.6m + 7.8 - 2.6m + 0.2])\\ \\bmod\\ 7", "1cf34e5dd31d2479b54e1cb71f245068": "\nd_1 = \\frac{4 \\alpha(2 - \\alpha)}{(1 + \\alpha)(3- \\alpha)}, \\qquad \\alpha=\\frac{MG}{2Rc^2}\n", "1cf36eb340053a174f7e7790b097e17b": "\\Phi(\\tau+16)\\,", "1cf3e126f8b3c6d59a6779a879a2b87a": "g_i(\\mathbf{x},z_1,\\ldots,z_k) \\neq 0", "1cf3fb150f32e804edce72891ef1e672": "f(z)=\\sum_{n=-\\infty}^\\infty a_n(z-c)^n", "1cf4387829ed522a24d38654e5dd5c47": "\\nabla _\\mu T_{M\\;\\nu }^{\\;\\mu }=\\frac 1{8\\pi }\\frac {\\nabla _\\nu \\phi}\\phi \n\\Box \\phi,", "1cf455285d2e7a4b60c283029308a6f0": "\\deg(h^2) \\leq 2g", "1cf516c2f98b957b6da0c5797c929279": "\\mathfrak{q}_1 \\subseteq \\mathfrak{q}_2 \\subseteq \\cdots \\subseteq \\mathfrak{q}_m", "1cf539d6135f019489aa2eea29490b0f": "H^\\bullet(X;R) = \\bigoplus_{k\\in\\mathbb{N}} H^k(X; R).", "1cf56c97d4c0fa4d8bdddb3e2a46f76b": "\\Omega_m = 0.22,\\,", "1cf601be5b401e2e48bc6402d60a60a8": "\\zeta'(-1,x)=\\psi(-2, x) + \\frac{x^2}2 - \\frac{x}2 + \\frac1{12}", "1cf60238995c56da9882fd097c30c0bd": "\\frac{d}{dx} \\int_0^x t^3\\, dt = \\frac{d}{dx} F(x) - \\frac{d}{dx} F(0) = \\frac{d}{dx} \\frac{x^4}{4} = x^3.", "1cf6397b8a31655406688d7725836657": "C_\\mathfrak{st}^\\lambda C_\\mathfrak{uv}^\\lambda \\equiv \\phi_\\lambda(C_\\mathfrak{t},C_\\mathfrak{u}) C_\\mathfrak{sv}^\\lambda \\mod A(<\\lambda)", "1cf65087c4906cbcf20b638409689c9a": "\\lim_{N\\rightarrow \\infty} p = c", "1cf65e8a140011ebf78b8454671cbdd0": "n_0 \\equiv a \\mod m", "1cf6b25a231270f5f28700065c4e274d": "s = \\tfrac12(a + b + c)", "1cf725da6a7a4a5c579f48c3a99815cf": "R_S=R_H \\left(1+\\frac{\\cos(\\theta)\\sin(\\alpha)^2-\\cos(\\alpha)\\sin(\\theta)\\sin(\\alpha)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "1cf731faae065d45ebbc13e2c7c715a0": "\\sum_{n\\geq 1}\\tau(n)q^n=q\\prod_{n\\geq 1}(1-q^n)^{24} = \\eta(z)^{24}=\\Delta(z),", "1cf73e6c36aae5fd4e13eec1213377ef": "O(2k) \\neq SO(2k) \\times \\{\\pm I\\}", "1cf75149b9422cba89f2425a2bcab13a": "f(x_1,x_2,\\dotsc) = x_1", "1cf7e37ff57ca733fc3b734ff6aedd27": "f(a,b) = ab", "1cf908a3f39d126231dc2a0188de1ffb": "f(n)\\le 3n-3", "1cf93f995f6f916485e0070c15f43dec": " x \\not\\in x ", "1cf967455cabaf16644f53878baaf4de": "O(f(k))+n^3", "1cf97c9c847a8b54b24badde907650fc": "2\\log_2(n+1)", "1cf991a9bdf72b1e77edd3e431aa9c1d": "E=0", "1cf9a39837609bbcd177490e800d6658": "v = r\\omega_{rad}", "1cf9af29f60beca43656e9cad0fd964f": "\\nabla_\\beta", "1cfa165c65aa313723b46a93744e107e": "\\langle\\tau^n\\rangle = \\Gamma(n) \\int_0^\\infty d\\tau\\, t^{n}\\, \\rho(\\tau)", "1cfa5cd3d571616b47f464db167449a2": "\\vec{h}_3 = \\frac{\\sqrt{1-2m/r}}{\\sqrt{1-3m/r}} \\, \\partial_\\phi - \\frac{\\sqrt{m/r^3}}{\\sqrt{1-2m/r} \\, \\sqrt{1-3m/r}} \\, \\partial_t ", "1cfa601ba3730bb8056ce65fc98279bb": " d^2(\\ln d)^{O(1)} n^2 \\mathbb{F}_q ", "1cfa671e18be131c7c0f0e60a17972c3": "\nF_{\\phi}(\\mathbf{r}, \\alpha, \\beta) =- \\frac{3 \\mu_0}{4 \\pi}\\frac{m_2 m_1}{r^4}\\sin(2\\phi - \\alpha - \\beta)\n", "1cfab2140efbdd5af94682fc6fc2aa66": "\\int_I \\rho(\\gamma(t))\\,d{\\mathrm{length}}_\\gamma(t)", "1cfabe79d22d83569089441eabba1408": "(\\le) : R \\times R \\rightarrow \\Bbb{B}", "1cfaea59874dead9782c6df0b761ce7f": " u \\left ( x \\right ) = x_1 + \\theta \\left (x_2, ..., x_L \\right )", "1cfb4b62c3fabfea351cba30357798ec": "\n\\lim_{N\\rightarrow\\infty}P_N(\\overline{R})=2N\\overline{R}\\,e^{-N\\overline{R}^2}.\n", "1cfba28a1a5c4897824baaa5a083c448": " \\sigma \\dot{\\sigma} \\leq -\\mu |\\sigma|^{\\alpha} ", "1cfc07378c97b94a5a1786150e83076a": "\\mathrm{Ann}_R(N)\\,", "1cfc0aa03d7aa8ce83a4ea0c373b971d": "\\lim_{n\\to\\infty} \\frac{f(n)}{g(n)} = 1", "1cfc931107decdbed8a865001aba9843": "\\frac{V_{dd}}{2}", "1cfd00088c529e6d258215ef50d7e59f": "\\scriptstyle\\sim", "1cfd4b2437b9e4a2ff0a7f36ba5d578e": "\\scriptstyle A\\,\\!", "1cfd9accc927798abb4d608bbaa80325": " \\ln(-\\eta) -\\ln k", "1cfe28c7956121e2b307e8b1835de656": "J_{ij} > 0 ", "1cfea071e94b76c58b5358e49570bea7": " \\tau = {VQ \\over It},", "1cfea84fb23cd83eeea02873f4f9b9af": "Q_t(W_t)", "1cff150e60dc33dbf91b52a000ce1d34": "\\epsilon_p\\,\\!", "1cff204ad548756e580ee0236dc7b24c": "p(k)", "1cff66606d5aa45c44413e71941c95ca": "D_i = \\frac{ (\\hat \\beta - \\hat {\\beta}^{(-i)})^T(X^TX)(\\hat \\beta - \\hat {\\beta}^{(-i)}) } {(1+p)s^2}.", "1cff6d11b4f6770f15830df4a34a59fb": " R = V S V^{-1}, \\, ", "1cff7f1273ec23bab95cf613aa13f66b": "\\ln \\frac{q_i}{p_i} = \\frac{q_i}{p_i} -1 ", "1cff9b06913ce7a14a720da1e7bdf753": "R=\\frac{r_\\mathrm{o}}{\\sin(\\beta_\\mathrm{o})},", "1d000b5118c83dec82d15e759fe7fe49": "\n\\sum_{p}\\nu_p(n) = \\Omega(n).\\;\n", "1d005c6c722588708966f5fd576a1b7b": "C_{01}=C{11}=0", "1d005cb5fe95e6315c899350ce8635e4": "v= {\\sqrt{rg\\left(\\sin \\theta -\\mu_s \\cos \\theta \\right)\\over \\cos \\theta +\\mu_s \\sin \\theta }}\n={\\sqrt{rg\\left(\\tan\\theta -\\mu_s\\right)\\over 1 +\\mu_s \\tan\\theta}}", "1d0070647c4f8e37197c120d1211e0e3": " E'\\rightarrow \\mbox{Div}^0(E')\\to\\mbox{Div}^0(E)\\rightarrow E\\,", "1d00b3fea8988ffc59fb7df622e2893a": "P(X_1=X_2)", "1d00d7385620b9bab21da659cd57717b": "\n\\cos \\theta = \\frac{g_{ij}U^iV^j}\n{\\sqrt{ \\left| g_{ij}U^iU^j \\right| \\left| g_{ij}V^iV^j \\right|}}.\n", "1d00e7dce692e8dc3f6877f035e3a616": "OR", "1d01486ae6512712dd2e838821363980": "c(c - 1 + \\gamma) = 0.", "1d0189a358bb5596d2094f68f8802358": "\n\\phi ::= p | \\neg p | \\phi \\lor \\phi | \\phi \\land \\phi | \\mathcal{P}_{\\sim\\lambda}(\\phi \\mathcal{U} \\phi) |\n\\mathcal{P}_{\\sim\\lambda}(\\square\\phi)\n", "1d01a98ba02aaa48fcd47b7149ac6222": "\\, C \\bullet D := \\sum_{r,s}\\langle \\langle C\\rangle_r \\langle D \\rangle_{s} \\rangle_{|s-r|} ", "1d02078fc5d3856634ab73a252bfca84": " c_i^{fit} ", "1d021fb48c03fc0011d4a58820b0a984": "T_0 T_p M \\cong T_pM", "1d022b8a2e3fee501dd4e3f6f9d2b779": "m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r} = m \\sum_j \\left[ \\sum_i \\left[ {\\mathrm{d} \\over \\mathrm{d}t} \\left( \\dot{r_i} {\\partial \\dot{r_i} \\over \\partial \\dot{q_j}} \\right) - \\dot{r_i} {\\partial \\dot{r_i} \\over \\partial q_j} \\right] \\right] \\delta q_j ", "1d024df1d507b2f709e2eaa6552e442e": "\\rho(\\mathbf{k},x)", "1d029df1f7a0d87a1f3ea15adb0d001b": "\\{(x_1, y_1), \\dots, (x_n, y_n)\\} ", "1d02c143ba4ec966a7581562630c547f": "B = \\frac{\\ell b^2 + mb + n}{(b-c)(b-a)};", "1d031f5fd5b71ebb7a40e6420eedd760": "\\dot{x} = \\frac{\\mathrm{d}x}{\\mathrm{d}t} = v", "1d039297310d25a38770a0bb01d5273b": " {x}(t+1) \\approx \\varphi(t)=\\varphi [\\varphi(t-1)]", "1d039423b51ea148024ae30fe89fc0eb": "\ny = a \\ \\frac{\\sin \\sigma}{\\cosh \\tau - \\cos \\sigma}\n", "1d03a3efae4a4440309e2476cd68fea0": "\\Delta=e^D-1=\\sum_{k=1}^\\infty \\frac{D^k}{k!}.", "1d03eb69a5b982ca0d68147841c973ed": "\\scriptstyle f_\\text{partial}(2, 3)", "1d041058b22e4094e4be1b2a3e6e4653": "\\zeta^\\prime(a,z)\\ \\stackrel{\\mathrm{def}}{=}\\ \\left[\\frac{d\\zeta(s,z)}{ds}\\right]_{s=a}.", "1d046cdeb1d2a0b59685c72cc5e7cf1e": " \\gamma= \\frac{\\omega F}{\\sqrt {2E_i}} ", "1d0486297fc1f0519981865d661ae52c": "\n\\begin{bmatrix}\n0 & -0.80 & -0.60 \\\\\n0.80 & -0.36 & \\;\\;\\,0.48 \\\\\n0.60 & \\;\\;\\,0.48 & -0.64\n\\end{bmatrix} \\qquad \\left( \\begin{align}&\\text{rotoinversion:} \\\\&\\text{axis }(0,-3/5,4/5),\\text{ angle }90^{\\circ}\\end{align}\\right)", "1d04ab28a442296ed5b91981b3f24e34": "SV = EDV - ESV", "1d04b87647563f698b40f902bf3be219": "\\bold{A} = A \\bold{\\hat{n}}", "1d04bee1e6333486c62cd23320def217": "\\delta V_{xc}[\\rho](\\mathbf{r})=\\frac{\\delta V_{xc}[\\rho]}{\\delta\\rho}\\delta\\rho=\nf_{xc}(\\mathbf{r}t,\\mathbf{r'}t')\\delta\\rho(\\mathbf{r'})", "1d04edc71ddab1778583bcdd223eee54": "\\{P(x),\\neg P(c)\\}", "1d053bb69a72439e60cad0ca3d48fa56": "\\mathit{STOP}", "1d0554c602996b48acdc2601cedceb15": "\n \\hat\\theta_n\\ \\xrightarrow{p}\\ \\theta_0\n ", "1d0562814a2d1571b3917d1b3bd4cbeb": "V_y = V\\cos\\theta", "1d058e0b16481515f9584d013188fe7d": "m_{\\mathrm{p}} \\,", "1d05ad00cd73c54cd23c63c6fb28cf7b": " A(H) ", "1d05c5dab7063bff697b15f0134444dd": "\\mathbb{R}^{mn}", "1d06aa5b77fd46ae43227dca7b73ccc3": "\\mu_5=\\kappa_5+10\\kappa_3\\kappa_2\\,", "1d06ab96b1bc72408795689983f7ad71": "\\gamma_{BY} = \\frac{1}{2\\pi \\alpha} \\ ", "1d06ae8429d08eb73ceb64a7bebd8466": "2+log(w)/log(2)", "1d07081b4d542b7be7aefc5b32035e72": "\\alpha (t_{0}) = p;\\,", "1d072ef3ecb054838e1abc7d9db1e8f7": "I_0(\\lambda)", "1d07af0d67adc3a5d5615df1cb4e188c": "B_{3}= \\langle b_{3}^{*}, b_{3}^{*} \\rangle =\n\\begin{bmatrix}\\frac{-6}{14}\\\\\\frac{9}{14}\\\\\\frac{-3}{14}\\end{bmatrix} \\begin{bmatrix}\\frac{-6}{14}\\\\\\frac{9}{14}\\\\\\frac{-3}{14}\\end{bmatrix}= \\frac{126}{196}= \\frac{9}{14}", "1d087e01ccfd95fd1bb1b1c8988b6200": "\nH^{(\\lambda)}(X)\n =\\sum_{m=1}^n\n \\left[\n \\prod_{\\kappa=0}^{\\lambda-1}(\\alpha_m-s_\\kappa)\n \\right]^{-1}\\,P_m(X)\\ .\n", "1d08abebe95ee454cff00e30ec2ed04a": "F[x,y]=\\int_E xy \\, dt", "1d08ba687006178661bd7a7400347034": "\\vec r(\\theta,\\phi) = (\\cos\\theta \\sin\\phi, \\sin\\theta \\sin \\phi, \\cos\\phi), \\quad 0 \\leq \\theta < 2\\pi, 0 \\leq \\phi \\leq \\pi.", "1d08e9b8066e77b4d3df6437d07ab1dd": "0 = \\frac {\\mathrm{d}^2 \\psi} {\\mathrm{d} \\eta^2}+(\\frac{2 m E l^2} {\\hbar^2}-\\frac{2 m^2 g l^3} {\\hbar^2}-\\frac{2 m^2 g l^3} {\\hbar^2} \\cos(\\eta)) \\psi ", "1d08edf206479de251ec1b3ac29d3970": "\n\\cos\\theta_{xy}=\\frac{\\langle x,y\\rangle}{\\|x\\| \\|y\\|}.\n", "1d08eef584c96ec937cbef62be2b5b18": "\nX_{m,\\delta}(i)={x_i,x_{i+\\delta},x_{i+2\\times\\delta},...,x_{i+(m-1)\\times\\delta} }\n", "1d0903a376aeda2858f95a2b34e21d77": "\\begin{pmatrix}1 & 2 & 3 & \\cdots & n \\\\ 1 & 2 & 3 & \\cdots & n\\end{pmatrix}.", "1d09105b167b9bbf3528f5e7f1fcb6ac": "\\textstyle a \\mapsto \\sum f(a_i) g(b_i).", "1d099d6fa0cc669495c7140ec758a3f0": "\n\\Delta \\tau^{k}_{xy} =\n\\begin{cases}\nQ/L_k & \\mbox{if ant }k\\mbox{ uses curve }xy\\mbox{ in its tour} \\\\\n0 & \\mbox{otherwise}\n\\end{cases}\n", "1d09bb12779c08bace7b3993b11d9a79": "Z_r = \\{ (A, W) \\mid A(k^n) \\subseteq W \\}", "1d09d05816c90dc60b668987aeff61df": " \\lang \\theta , \\phi | l, m \\rang = Y_{l,m}(\\theta,\\phi)", "1d0a1b14ad47f6ae076ca39ae71204b5": "O(P_r+P_s)", "1d0a1e229efe244255ef3a39e7f515bd": "a_1=b_1", "1d0a2997bcaed0081207b7016227b9f4": " MPGe ", "1d0a4578303784dbdcbaccc5ec00389d": "V_A(a) + V_B(a) = 0", "1d0a83572fe667d5ed596a2ec3173019": "\\textstyle\\sum_{i=1}^nx_i\\hat\\varepsilon_i=0.", "1d0aa1ecfacc5ba465979b175c8b0955": "d_\\text{star}", "1d0adc654ce9be12c177436514a0e366": "\\ F = \\frac{N b}{r} ", "1d0ae0aaa70d341c07e688a300930dfc": "\\hat{f}(\\xi) = \\int_{\\mathbf{R}^n} f(x)e^{-2\\pi i \\xi\\cdot x}\\,dx", "1d0b03be98d6579a21d0251bc91f1d03": "A_{2N} < C < A_{2N} + D_{2N}.", "1d0b26da1252fab01230527ea1939f56": "p(\\textbf{x}_k|\\textbf{x}_{k-1},\\textbf{x}_{k-2},\\dots,\\textbf{x}_0) = p(\\textbf{x}_k|\\textbf{x}_{k-1} )", "1d0b8f7ee6c8431323e998f7ed636833": "C = \\underset{x\\in \\mathcal{X}}{\\max}\\ \\bold{E}[c(A,x)]", "1d0c9fb22056bb143996c843419269af": "e[n] = y[n] - x[n] + e[n-1]", "1d0ca11128dc5e6eeead14457ce90c50": "p\\left(\\hat{y}\\right)", "1d0cbeebf87084fca4ecc43242794b45": "E_2:F\\rightarrow F^+", "1d0d7d2f4123fb6a7339dde885243ad7": " D_v(f) = v(f)\\,", "1d0d80333140b5e8356fbc1b45ba662c": "\\mathbb{E} \\big[ \\langle \\nabla_{H} F, V \\rangle \\big] = - \\mathbb{E} \\big[ F \\operatorname{div} V \\big],", "1d0d80fc042cfc960f950c72ac394637": "C_\\beta' = C_\\beta\\begin{bmatrix}0&-k\\\\1&0\\end{bmatrix}", "1d0dc2beb3fad6bcfcf4d3cf02315260": "\\subseteq\\Sigma^*", "1d0dc71f9af4cfce1026bf86c5bac6c1": "\\alpha:= -\\frac{1}{2}\\big(n^a\\bar{\\delta} l_a-\\bar{m}^a\\bar{\\delta}m_a \\big)=-\\frac{1}{2}\\big(n^a\\bar{m}^b\\nabla_b l_a-\\bar{m}^a\\bar{m}^b\\nabla_b m_a \\big)\\,.", "1d0dca57c64680805b725be6d021d47a": "T:= \\{ \\vec x|n: n \\in \\omega, x\\in C\\}", "1d0e3cb6f69c34a2908a9f6d3732e3cd": "I_2(n).", "1d0e7076db037986c574306a8e53404b": "P = (L_0,R_0)", "1d0eaa23a44beef69ff49fe487bbfb94": "\\lambda P(L_1)-\\lambda^{-1}P(L_2)=(q-q^{-1})P(L_3).", "1d0eb07f9c4107f810b036746aa5ee3c": "\\underline{u}_D \\star \\approx \\underline{u}_1", "1d0fc24fae5fa2d847536ab163b29aae": "\\begin{align}\n\\frac{p}{p^*} &= \\frac{\\gamma + 1}{1 + \\gamma M^2} \\\\\n\\frac{\\rho}{\\rho^*} &= \\frac{1 + \\gamma M^2}{\\left(\\gamma + 1\\right)M^2} \\\\\n\\frac{T}{T^*} &= \\frac{\\left(\\gamma + 1\\right)^2M^2}{\\left(1 + \\gamma M^2\\right)^2} \\\\\n\\frac{V}{V^*} &= \\frac{\\left(\\gamma + 1\\right)M^2}{1 + \\gamma M^2} \\\\\n\\frac{p_0}{p_0^*} &= \\frac{\\gamma + 1}{1 + \\gamma M^2}\\left[\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)\\right]^\\frac{\\gamma}{\\gamma - 1}\n\\end{align} ", "1d0fe2abd3b3efc983ee08a9dd749c92": "\nf(x; y_0, A, x_c, w, t_0 )=y_0+\\frac{A}{t_0} \\exp \\left( \\frac {1}{2} \\left( \\frac {w}{t_0} \\right)^2 - \\frac {x-x_c}{t_0} \\right) \\left( \\frac{1}{2} + \\frac{1}{2} \\operatorname{erf} \\left( \\frac {z}{\\sqrt{2}} \\right) \\right) ,\n", "1d0fe3ad28ed27546808efeec5c626a9": "n\\ge 1", "1d10328c121e47d14a02b3c2dd7a6886": "p(x,s)=u_s(x)", "1d10474cbc2fb4c2b12cd9fe48d1595b": "s\\pm t", "1d1087bf50dedbb0044482efbe67c880": "T_0=\\frac{1}{2}m\\left(\\frac{\\partial \\mathbf{r}}{\\partial t}\\right)^2\\,\\!,", "1d10e7d0e9531abc24569eb1b7fa1a6f": "\\frac{1}{2}\\sqrt{0},\\quad \\frac{1}{2}\\sqrt{1},\\quad \\frac{1}{2}\\sqrt{2},\\quad \\frac{1}{2}\\sqrt{3},\\quad \\frac{1}{2}\\sqrt{4},", "1d1101739dd2fa95e6b339a4c4b571ac": "e^{-\\alpha\\epsilon n/4}\\,\\!", "1d1197f94f0f8cd0050d3678e6c84fc3": "V_+ = V_\\mathrm {in} \\frac{Z_\\mathrm {in}+R_1}{Z_\\mathrm {in}+2R_1}", "1d11f439bcbf265215a0a96fe873077d": "\\scriptstyle 2\\pi N r", "1d1201bd70af511725bf6485523f9c11": " \\frac{dS}{dz} = 0 ", "1d120d3d29b1843bf02f03908c40a6cb": "\\gamma_n \\le (4/3)^{(n-1)/2} \\ .", "1d120d743ee9076f8d76ee45d19fe8e6": "\\mbox{NC}^i \\subseteq \\mbox{AC}^i \\subseteq \\mbox{TC}^i \\subseteq \\mbox{NC}^{i+1}.", "1d1216084a835c680ef76e787a537934": "-l\\le m\\le l", "1d1268d5e562f6d2e27c09e6f69321c1": "\\rm{d}(X) = \\aleph_0", "1d12dfb9ea498f0ce8aca799a036c93a": " \\mathbf{m}", "1d1327b738db7b356968b81ceb8a66af": "A,B \\in P", "1d13a92aba7af4d10f9f0e05f3898eea": "\\Phi_t^0\\left(D\\right)\\left(a_0\\right)=\\mathrm{tr}\\left(\\gamma a_0 e^{-tD^2}\\right),", "1d13b555e8fa656c2a08159999c90759": "e^{ar} = \\cos (a) + r\\ \\sin (a) ", "1d1419a1535f715babe897b18e238787": "\\Omega_n^G = \\Omega_n^G(\\text{pt})", "1d14327e03bebee75dea438c86b9ff7b": "\\xi_{\\times +}", "1d14482b496ac79353aed6b0f5d54a69": "u(x)\\!\\,", "1d14569a52587c1586988545b48f305c": "\\! w<-1", "1d1524e69d13cd77795b6d20f2fc2546": "d>0", "1d1582046e8d36fd5ce799a30216337d": " x_0 \\,", "1d15a403c240f27b5dcb1a6a1ccf8c24": "\\mathcal{L}(\\phi) = \\frac{1}{2}(\\partial_t \\phi)^2 - \\frac{1}{2}(\\partial_x \\phi)^2 - \\frac{1}{2} m^2\\phi^2 - \nV(\\phi),", "1d15dbeb4c94fd96cdd91e59d043eae6": "K(x,y) = \\langle \\varphi(x), \\varphi(y) \\rangle", "1d165b280947110da81c36101e38cc3f": "\n\\begin{align}\nK_l & = \\text{lower strike price} \\\\\nK_u & = \\text{upper strike price} \\\\\nC_n & = \\text{net credit per share} \\\\\nN & = \\text{number of shares per options contract}\n\\end{align}\n", "1d16bb9e5363ad5ff192208223914626": "\\begin{align}\n\\operatorname{var}\\left[\\ln \\left (\\frac{1}{X} \\right ) \\right] & =\\operatorname{var}[\\ln(X)] = \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta), \\\\\n\\operatorname{var}\\left[\\ln \\left (\\frac{1}{1-X} \\right ) \\right] &=\\operatorname{var}[\\ln (1-X)] = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta), \\\\\n\\operatorname{cov}\\left[\\ln \\left (\\frac{1}{X} \\right), \\ln \\left (\\frac{1}{1-X}\\right ) \\right] &=\\operatorname{cov}[\\ln(X),\\ln(1-X)]= -\\psi_1(\\alpha + \\beta).\\end{align}", "1d16d8fa7ed9455943cfd7290fa2a99d": "\\mathbf{v_d}", "1d16e6fb0f673014cb357bb9eccbdcc3": "\\delta_X(\\varepsilon / 2) \\le \\tilde \\delta_X(\\varepsilon) \\le \\delta_X(\\varepsilon), \\quad \\varepsilon \\in [0, 2].", "1d16efe78a4740b4259635d2a339490b": "I = \\{1,\\ldots, n\\}", "1d16f95987cd040456c2c25067376ecf": "\\pi / 4\\sqrt{2}", "1d1721a2abc5d2837988edc518ce65a5": "\\scriptstyle A\\otimes_K L", "1d175fdb3ce7c9f111f1c66ef135d7eb": "\\frac{\\sqrt[3]{108 + 12\\sqrt{69}} + \\sqrt[3]{108 - 12\\sqrt{69}}}{6}", "1d17d27be0541975ee753678f711844b": "\\mathcal{L}_X T_{ab}=0", "1d17d685bdc9b4bd3ef4055466d82e3d": "\\omega_k = \\sqrt{2 \\omega^2 (1 - \\cos(ka))}.\\ ", "1d183d57a828a3bb3ca7567a9f163b3f": "P=200000", "1d18a6e569fe5adf17271f3ea5bd8385": " \\left( \\nabla^2 - { 1 \\over {c}^2 } {\\partial^2 \\over \\partial t^2} \\right) \\mathbf{E} \\ \\ = \\ \\ 0", "1d18be8c729ed0c03e30093596f091e6": "\\tau_\\mathrm{bulk}", "1d18d50801ca9cc67ca08cdd963123d3": "U_{kl}^{AB}", "1d18e10b70af19ccf8c4112c49b3336b": "\\sigma_{\\mathrm{T}}", "1d19250a02b3155af1c561e6ff38dc17": "(S,d)", "1d192b41bbfb0fb5c4bff3a91da05d3f": "T-\\lambda", "1d192bb209fa2f0f0cfdbe23eb00310b": "10^{18}", "1d196fe3d16a4493c2e69e69499e4e49": "\\mathbf{B}(t) = \\mathbf{T}(t)\\times\\mathbf{N}(t) = \\frac{\\mathbf{r}'(t)\\times\\mathbf{r}''(t)}{\\|\\mathbf{r}'(t)\\times\\mathbf{r}''(t)\\|}.", "1d1a81c66e1d2c4f2e054e585951fef6": "{\\rm VRR}\\,\\,\\, = \\,\\,{\\alpha \\over {2\\,\\, - \\,\\,\\alpha }}", "1d1adab8af1bec934d272c17ba78f036": "d^3k", "1d1b1e144dcef8174bd5666f38ade04f": "T(\\omega) = 0 ", "1d1b5506d787a1f0f9c4eb9a53a41ddb": "1+0\\sqrt{2}=1.0", "1d1b5c9a5ab264843dd3be53564b121f": "e=\\hat{x}-x", "1d1b67a4315090635fdc1b0ab810c96c": "\\cos\\gamma = 1 - \\frac{c^2}{2a^2}", "1d1b67d7612bb0ca44695a80d17d4753": "v_1 \\le v_2 \\le v_3 \\le \\dots \\le v_N", "1d1c906e20011d74be0c2ee83abaf921": "[0..1]", "1d1d4dc3dd666b887ba8cba6cc215a37": "R/(q_i)", "1d1d9610e0f6cb66cbb43923a733020b": "\\ R_l(r) = r^l \\sum_{p=1,P} c_p A(l,\\alpha_p) \\exp(-\\alpha_p r^2)", "1d1db6bcc58465ede4a89d0d6b82815e": "\\frac{a^x}{\\ln a}\\,", "1d1dc2fafef279934f2a51b4ac4d59dd": "(v,w)", "1d1e2f4b9431bafc7b3471e0dbf0f8a2": "{n+1\\choose k}_2={n\\choose k-1}_2+{n\\choose k}_2+{n\\choose k+1}_2", "1d1e6aaaa43d48bcb97c42991f9dc978": "\\frac{d}{ds}\\Big|_{s=0}\\mathcal S(\\gamma_s)\n= \\Big|_a^b \\alpha_i X^i - \\int_a^b g_{ik}(\\ddot\\gamma^k+2G^k)X^i dt,\n", "1d1ed0671a92ee771eb1c1d9e5277e39": "\\big\\{\\mathbf{F}_{\\alpha}\\big\\}_{\\alpha=1}^{M}", "1d1f36b0adc3a84df596f5e91c9b815e": "\\mathbf{R} = n_{1}\\mathbf{a}_{1} + n_{2}\\mathbf{a}_{2} + n_{3}\\mathbf{a}_{3}", "1d1f4de02833556752efbb3a17e6df36": "\n\\eta(2i)=\\frac{\\Gamma \\left(\\frac{1}{4}\\right)}{2^{{11}/8} \\pi ^{3/4}},\n", "1d1fc588f515ba1ed0950a81b58511dc": "\\eta^R:G\\to \\mathfrak{g} \\otimes \\mathfrak{g}", "1d200ec1a1ca4eac6094ccd7ed425069": "C^\\phi_{MX} =\\left[\\frac{3}{\\sqrt{pq}}\\right]\n\\left(p\\mu_{MMX}+q\\mu_{MXX}\\right).\n", "1d206774fa859376863653284d771a6a": " \\phi(x,y,z) = \\left(\\frac{x}{a}\\right)^2 + \\left(\\frac{y}{b}\\right)^2 + \\left(\\frac{z}{c}\\right)^2 - 1 = 0", "1d2071b227874a89041a480925185758": "X(t)", "1d208ff8689c77f7bfcb9495d1a788fd": " c = \\sqrt{\\frac{N_{rr}^*}{\\rho h}}", "1d20d35a3896ac35d7ed51f16dfe0155": "\\mathcal{ABCDEFGHI} \\!", "1d20e87c46153a58bd076c6e7e02ed55": "\\beta'", "1d20e9a24e4e64512e51afef852e5bc6": "J^k_0({\\mathbb R},M)_p", "1d2118af8c9c571fec6305fff4b35c9b": "T\\cup M", "1d2177b052eaed03c61ace7c1fa54613": "\\theta=2", "1d21d03dadb8329d94da8bb76f7ad49a": "R_f(-\\tau) = R_f^*(\\tau)\\,", "1d223191de06a2371757e9757343129d": "\\pi =T^{a_1}M^{a_2}L^{a_3}g^{a_4} \\, ", "1d223f52e789290f1117815490b53287": "(X \\perp\\!\\!\\!\\perp Y) \\,|\\, W", "1d22b15b9985b38212a8e8a713ced177": " \\mathbf{H} = \\begin{pmatrix} a_1 & c-id\\\\ c+id & a_2\\end{pmatrix}", "1d22bc54639ff8a8581254bf211b0230": "\\vec{q}^A_r", "1d239f79c9f7e82de16b560d96c9e162": "z \\rightarrow z+s, \\ \\ \\infty \\rightarrow \\infty \\quad , ", "1d23cc09d50f87e679b7935ed6459ba0": "Re(p) < 0", "1d23f07abd11ceb4ee03406801920fd0": "\\color{Goldenrod}\\text{Goldenrod}", "1d23fe052b5300f3e02bd97241675abc": "\\scriptstyle z\\, = \\,2\\pi ni\\text{ for } n\\, = \\,\\dots,\\, -1,\\, 0,\\, 1,\\, \\dots.", "1d2402dc131d1a3153c9e9ed520d67f2": "\\mbox{Impairment Cost} = {\\mbox{Recoverable Amount} - \\mbox{Carrying Value}}", "1d2410c062f6dbb0e30df6a396937426": "\\textstyle |S_{11}|^2 + |S_{12}|^2 = 1", "1d243aa59a13db8030d5bd9ce261eb70": "P_2^2 - 4Q_2 = (P+2)^2 - 4(P + Q + 1) = P^2 - 4Q = D", "1d24703d5697cb2cc2a53e21dca00503": " a_n \\ ", "1d247098f8d5b1e465ddb3aea319e8a4": "\n\\begin{bmatrix}\n1 & 2 & 3 & 4 & 5& 6 &input\\\\\n1 & 0 & 0 & 1 & 0 & 1 &flag\\ bits\\\\\n1 & 3 & 6 & 4& 9 & 6 & segmented\\ scan\\ +\\end{bmatrix}\n", "1d2480ccb042a88e9a02225b784344ed": " T_n (y) ", "1d24acc1fd4dfe3aed2e137a0c8fc5b9": "P=\\frac{A_{x}}{\\ddot{a}_{x}}", "1d252e2846757f89d5debf77be43e868": "D\\lambda-\\bar{\\delta}\\pi=(\\rho\\lambda+\\bar{\\sigma}\\mu)+\\pi^2+(\\alpha-\\bar{\\beta})\\pi-\\nu\\bar{\\kappa}-(3\\varepsilon-\\bar{\\varepsilon})\\lambda+\\Phi_{20}\\,,", "1d2559a184122bca9aa0e8a0f42951df": "\\varphi=\\theta-\\theta_k", "1d257c8c6d304f5f82f314447adbdb76": "\\chi(z_1,\\ldots,z_n) = (\\bar z_1,\\ldots,\\bar z_n)", "1d25b957cb614d7ba68f68a392f09ee1": "0/x = 0", "1d2630a70db48afa6a8e27d4e0b7eef1": "M_{2413\\oplus35142} = M_{241379586} = \\begin{bmatrix} &1&&&&&&& \\\\ &&&1&&&&& \\\\ 1&&&&&&&& \\\\ &&1&&&&&& \\\\ &&&&&&1&& \\\\ &&&&&&&&1 \\\\ &&&&1&&&& \\\\ &&&&&&&1& \\\\ &&&&&1&&&\\end{bmatrix} ", "1d2636c91e25cadc0349b96c98ace41e": "\\lim_{n\\to\\infty}\\frac{\\ln(g(n))}{\\sqrt{n \\ln(n)}} = 1", "1d2640cdbd600cbdd72fe6915e8ca26f": "\n \\displaystyle \n \\left\\langle \n \\begin{matrix} \n\t 3 \n\t \\\\ \n\t 4 \n \\end{matrix}\n \\right\\rangle \n = {3 + 4 - 1 \\choose 3-1}\n = {3 + 4 - 1 \\choose 4}\n =\n \\frac\n {6!}\n {4! 2!}\n = 15\n", "1d268cc2ef7f20c744884a9eb3690f00": "\\boldsymbol{\\Omega} = \\begin{pmatrix}\n0 & - \\omega_\\text{z} & \\omega_\\text{y} \\\\\n\\omega_\\text{z} & 0 & - \\omega_\\text{x} \\\\\n- \\omega_\\text{y} & \\omega_\\text{x} & 0 \\\\\n\\end{pmatrix}", "1d26cbe9a42f8d62c5797957ef636062": "d(x,z) \\le d(x,y) + d(y,z)", "1d26f6727e69643cd634718b396fe9f4": "\\begin{array}{rcl}\n\\mathcal{G}[r]\\rho & \\equiv & \\frac{r\\rho r^\\dagger}{\\operatorname{Tr}[r\\rho r^\\dagger]}-\\rho \\\\\n\\mathcal{H}[r]\\rho & \\equiv & r\\rho+\\rho r^\\dagger-\\operatorname{Tr}[r\\rho+\\rho r^\\dagger]\\rho\n\\end{array}\n", "1d27a39a83f24f03b07f92fc1cd1b9d8": " i_1=i_r\\ ", "1d281a0422f9404c3e31b82139063c9c": "b_{T(V)}(T(v),T(w))=b_V(v,w).", "1d28c23070e40220d6e6469a8907bb41": "\\varphi = \\psi\\circ\\iota", "1d28e9e4fde2fdcbc5ddf5e775da4a8e": " \\vec a = {\\vec F \\over m} ", "1d29601709b8f5db6ebfb0a8f499c594": "f(t|\\theta)=\\theta f_0(\\theta t)", "1d29e8837c3c095758938f04277c7b24": " R={3L_{\\text{s}}\\phi_{\\text{d}} \\over \\phi_{\\text{s}}}", "1d29fecebb067f6c48648988d35a9170": " 0 = 2(Q-M) + 2QY , \\,\\!", "1d2a08776d6865edf674c7483274b0ab": "\\partial_{xx} \\left( - \\partial_{xx} \\psi \\right) = - \\partial_{xxx} \\phi \\,", "1d2a75a22aa751c6bd56e7d8a1dee4e2": "e^X = e^{A+N} = e^A e^N. \\,", "1d2ab395697ecff7df4765ecfbbc1a75": "\\forall x, sp(S,P) \\Rightarrow Q", "1d2ad734ad8942abc34f835b99d9f8fd": "\\int_{[A]_{i}}^{[A]_{f}} \\frac{d[A]}{[A]}=\\int_{t=0}^{t}-kdt", "1d2b0795a73e0fe83a86d3dbe925fbc4": "(\\log^{12}(n))", "1d2b0af30d37632489c41b7d100a657d": "{}^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B}", "1d2be4138e36e3f0174a09e105e59ce5": "P_n=P_a(\\sqrt[12]{2})^{(n-a)}", "1d2bfdbb955b25644e406437ace85ec8": "\\mathbf{u} \\otimes \\mathbf{v} = \\mathbf{u} \\mathbf{v}^\\mathrm{T} =\n\\begin{bmatrix}u_1 \\\\ u_2 \\\\ u_3 \\\\ u_4\\end{bmatrix}\n\\begin{bmatrix}v_1 & v_2 & v_3\\end{bmatrix} =\n\\begin{bmatrix}u_1v_1 & u_1v_2 & u_1v_3 \\\\ u_2v_1 & u_2v_2 & u_2v_3 \\\\ u_3v_1 & u_3v_2 & u_3v_3 \\\\ u_4v_1 & u_4v_2 & u_4v_3\\end{bmatrix}.", "1d2c10da76029714f21d877c8bc5a76c": "R_\\text{earth}", "1d2c3603b7999012d70ac5524af2a7c3": "T_{tot} = \\frac{3}{8} + 0.01 + \\frac{1.1 \\times 8}{2} \\left(\\frac{1}{2.5} - \\frac{1}{1} \\right)", "1d2c54c954311d74b323c8e0c7b89271": "ES_1", "1d2c859cd62301bae4a3faa504591742": "H^1(M,{\\mathbb Z_2})", "1d2cc93fbd9cac7967b8fd1d8515b00d": "{Q_f} = {Q_{water}}\\sqrt \\frac{1}{Sg}", "1d2ce798c79c32ec5a818af7916b952b": "F(x,y) = c\\,", "1d2cfe729015cab697f270ff1888209a": "p_i(e^t-1) + 1", "1d2d4195485cd005cfcfd848e68fac35": "\\cos^n\\theta = \\frac{1}{2^n} \\binom{n}{\\frac{n}{2}} + \\frac{2}{2^n} \\sum_{k=0}^{\\frac{n}{2}-1} \\binom{n}{k} \\cos{((n-2k)\\theta)}", "1d2d8ca6b51a04f6e1af12388213a3c4": "I(X;Y|Z) \\ge 0", "1d2dbf930a6b8e7cedc1ab80ef0a0413": "\\varphi(x)=2\\cdot\\sum_{k=0}^{N-1} h_k\\cdot\\varphi(2\\cdot x-k)", "1d2e4a75af5e2ca37f6cb36da7b84503": "\\displaystyle{g(x,y)=\\begin{pmatrix} E & F \\\\ F & G\\end{pmatrix}}", "1d2ea3322e2c8dad27ddcf5b7f9dee18": "\\scriptstyle{\\Delta p}", "1d2ea57ef07652f3d99a8ff58b4cf62c": "k[t]/(t^n)", "1d2eddd7ba651206d34777e23dca96de": "-e({\\boldsymbol{r}}_{\\text{SO}}\\cdot{\\boldsymbol{E}})", "1d2f075daabf3af437d7c9e767beac46": "\\begin{bmatrix}M\\end{bmatrix}=\\begin{bmatrix}1 & 0\\\\ 0 & 1\\end{bmatrix}", "1d2f2a8410f934caccc8a8c490bcde1d": "\\rho_C^2 = 1-D(Y|X)/D(Y)", "1d2f4325ad8a9eb376e910194c05f7ce": " \\operatorname{de-let}[M]\\ \\operatorname{de-let}[N] ", "1d2fd37f34f4719d4d7913085b671de4": "\\mathcal L(x)", "1d30b0ebb0b7bb4c870cc60572f6cee3": "\\frac{F_{\\rm el}}{kT}=\\frac{\\pi^2}{24N^2a^5}\\int_0^\\infty\\left\\{-z^3\\frac{{\\rm d}\\phi(z)}{{\\rm d}z}\\right\\}{\\rm d}z", "1d30efc186ca4fea7fa73770343b0f3b": "\\frac {k_{i-1}}{x_i-x_{i-1}} + \\left(\\frac {1}{x_i-x_{i-1}}+ \\frac {1}{{x_{i+1}-x_i}}\\right)\\ 2k_i+\n\\frac {k_{i+1}}{{x_{i+1}-x_i}} =\n 3\\ \\left(\\frac {y_i - y_{i-1}}{{(x_i-x_{i-1})}^2}+\\frac {y_{i+1} - y_i}{{(x_{i+1}-x_i)}^2}\\right)", "1d30f17eeb1407150f3c23cfd825bc81": "{\\mathbf{f}\\cdot\\mathbf{u}}={d \\over dt} \\left(\\gamma mc^2 \\right)={dE \\over dt}", "1d319b7464fcdeee5495d34cbc509caa": "(f,g)\\!", "1d326adfe9f8565083aab6c5b9228c7a": "K=\\Gamma + \\Theta", "1d327044e41b0e626ea100800e2d3e40": "\ns_1 s_2 = 1 - 2 \\sin(\\pi/18)\n", "1d3290610438676fa6ccb055ceb357ba": "1 < \\frac{1}{p} + \\frac{1}{r}", "1d32a269c403d8433b519ae62633924c": "\n X = \\frac s {y_\\mathrm{atm}}\n = \\frac {R_\\mathrm {E}} {y_\\mathrm{atm}} \\sqrt {\\cos^2 z\n + 2 \\frac {y_\\mathrm{atm}} {R_\\mathrm {E}}\n + \\left ( \\frac {y_\\mathrm{atm}} {R_\\mathrm {E}} \\right )^2 }\n - \\frac {R_\\mathrm {E}} {y_\\mathrm{atm}} \\cos\\, z \\,.\n", "1d32b4f78d324eb6a203be804ff8685e": "X^{\\ast }(t)", "1d32c8887a589d26705465a877b77786": "\\begin{cases} \\pi_r: J^{r}(\\pi) \\to M \\\\ j^{r}_{p}\\sigma \\mapsto p \\end{cases} ", "1d3393a593a8d010c17164c091fc835d": "\\epsilon_N(t)=\\sum_{k=N+1}^\\infty A_k(\\omega) f_k(t)", "1d339c99c60488442eec9d99082b8544": "(A, B) = \\{\\varphi(a) : a \\in A\\} \\cup \\{\\varphi(b) \\cup \\{0\\} : b \\in B \\}.", "1d33b07b5bee96afc9984dcc816496e4": "j = A + B, A + B - 1, ..., |A - B|,", "1d33bce3f353bfc95cbe375bd1cacf54": "\\int_0^\\infty \\frac{\\sin b\\,\\omega}{\\omega}\\,d\\omega = \\int_0^{b\\,\\infty} \\frac{\\sin b\\,\\omega}{b\\,\\omega}\\,d(b\\,\\omega) = \\int_0^{\\sgn b\\times\\infty} \\frac{\\sin x}{x}\\,dx = \\sgn b \\int_0^\\infty \\frac{\\sin x}{x}\\,dx = \\frac{\\pi}{2}\\,\\sgn b", "1d340e6277c8fc93c281886c2168e9ff": "\n\\delta_d (\\lambda) = \\sigma_i \\beta_i^*(\\lambda) \\, \\Delta \\sigma_i\n", "1d3436db5be542b178d90641c39cca00": " \\exp(J) = I + \\frac{ e^n-1}{n} J,", "1d34418628f60dfc415247a9c153ea2e": " \\langle r^k \\rangle = \\int_{0}^{\\infty}P(r) r^k dr = 3 a^k\\int_{0}^{\\infty}x^{k+2}e^{-x^3}dx\\,,", "1d345316ad424b3749d50afe8ddf5986": "l^2=(r_2-r_1)^2+d^2", "1d34b2af147c3fc1da4824e49cee7df6": "S = \\langle A, R \\rangle", "1d34b7bfbd1bb58bca83ceb881a648de": "V=2.357 Y^{0.343}-1.52", "1d34e9b5adf6dcba4e191356dfeb497d": " h_i^\\ast ", "1d3507f9f362d005e060af7db94b5895": "\\sigma = \\left \\langle v^0 , v^1 , \\dots ,v^k\\right \\rangle.", "1d354c8a9e98c3186010e9385b6b7bf9": " \\nabla^2 \\Phi(\\mathbf{r}) =\n\\left(\\sum_{j = 1}^N \\frac{n_j^0 \\, q_j^2}{\\varepsilon_r \\varepsilon_0 \\, k_B T} \\right)\\, \\Phi(\\mathbf{r}) - \\frac{1}{\\varepsilon_r \\varepsilon_0} \\, \\sum_{j = 1}^N n_j^0 q_j\n", "1d3642c30ab2c5484f8350169b4563ce": " X_{t+s} - X_{t} = \\int_t^{t+s} \\mu(X_u,u) \\mathrm{d} u + \\int_t^{t+s} \\sigma(X_u,u)\\, \\mathrm{d} B_u . ", "1d365b3f9f13c0609211a70a86b00f83": "{\\mathbf{w}}_i", "1d36b3a4b0d4a5a9b50d8cd88a9797f9": "\\lambda / r_0", "1d36e954629388450816271e727b296d": " \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} x^2}\\,dx = \\sqrt{2\\pi}. ", "1d3770d156411dd4c88e005d627cd180": "F_2 - 1", "1d37a8eebccbb9658f0445293da44a53": "R \\bowtie R = R", "1d37b6c07075a4c9e96e7aad36b0f9ec": "_{q=p\\ \\Rightarrow\\ q'p=qp'\\,}\\!", "1d37d33fe0039d0a8ec30888ee1a6093": "(1000\\bar 1 000\\bar 1 0)_{\\text{NAF}}", "1d384225a77cf17f948f472379227899": " 2 ^{14}\\text{NO}_3^- \\rightarrow {^{14}}\\text{N}_2\\text{O}, ", "1d387249e9f5f25c7ebe255c27423383": " arg(\\rho_n(c)) \\,", "1d388a6cfe0ec30ae232e93ddf6f498a": "\\frac{\\partial \\mathbf{j}_s}{\\partial t} = \\frac{n_s e^2}{m}\\mathbf{E}, \\qquad \\mathbf{\\nabla}\\times\\mathbf{j}_s =-\\frac{n_s e^2}{m}\\mathbf{B}. ", "1d38c4832f2571b5302710d65162fa8b": "i_1 \\circ p_1 + \\cdots + i_n \\circ p_n", "1d39446ddef5b06cb81be7a7ebe2ccac": "x=16/5", "1d39e8a1e4641edb811735775e09a251": "n_x, n_y, n_z", "1d3a28d9aeb7bbb01e922d3f6c1b5d60": "B\\Z", "1d3a87c5e30b58ca492fec586a514952": " E 2 \\pi rh = \\frac{\\lambda h}{\\varepsilon_0} \\quad \\Rightarrow \\quad E = \\frac{\\lambda}{2 \\pi\\varepsilon_0 r} ", "1d3a8c1c45dcd259e461eb235cc01eb4": "H^\\alpha", "1d3a9bcc849732ca8aa8ebd418fad8c4": " \\mathrm{d.f.} = \\frac{(s_1^2/n_1 + s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)}.\n", "1d3ad27025247c575180b1e2b240759d": "(x_3, y_3)", "1d3b01955fdea74780e1a4edf6c17c1f": "f = \\frac{ 14.227}{Re} ", "1d3b7ac8e4cbd8240a762f35358e57d8": "\\prod _x \\frac{x+1}{x} = C x", "1d3c60d2089d78ba278af2440fd2e193": "Y_i = C + (B_0 + B_1 T_i)\\sin(2\\pi\\omega T_i + \\phi) + E_i", "1d3c782b4cf65afc51aaaacb5e19991d": "S\\otimes T:V\\otimes W\\rightarrow X\\otimes Y", "1d3c85f182100b4c7c6fc89c6be06234": "\\left ( \\frac {^{13}C}{^{12}C} \\right )_{sample} = \\left ( 1 + \\frac {\\delta^{13}C}{1000} \\right ) \\left ( \\frac {^{13}C}{^{12}C} \\right )_{PDB}", "1d3d3f9328743930a4c139944f9c077c": "\\Delta\\mathbf{w}_{n}", "1d3d5a0a1ebc552467c21311862b1f4a": "\n{\\mathbf y} \\sim N({\\boldsymbol \\theta}, \\sigma^2 I).\\,\n", "1d3d85263048ff73d9e59682b0c8bf44": " \\Gamma", "1d3d87beda749a3cfd36e08c70c49689": " f = \\lambda x.x \\to y ", "1d3d942080e4730c027537aac281ee7a": "a_{i}=F(x_i)", "1d3daf0a9c786b400ce07505c8d87344": "\\displaystyle{N_+=N_\\sigma\\cdot M_\\sigma.}", "1d3db74a7915d264c94efcdc1deb7b3b": "\\left(1+\\frac{x}{n}\\right)^n=e^x \\left(1-\\frac{x^2}{2n}+\\frac{x^3(8+3x)}{24n^2}+\\cdots \\right),", "1d3deb12b13167802707b7f44b5f7fb4": " C_v \\ll C_0 , ", "1d3e2e1c3773321d16b9d5209399784a": "[0,1]^2 \\subset \\Re^2", "1d3e96bbf9c719eaa5997ada8c11b4e9": "\\, \\tau", "1d3f02d09cebcfb3ac14e5d79b2550c3": "f(i) = f_i", "1d3f08959ad89a735f7a8bd9cdd66cd8": "\\boldsymbol{v}_T", "1d3fbf50383d9ed5abcd8f1f90482c3f": "\\mathbf{a}\\times\\mathbf{b}=\\det \\begin{bmatrix}\n\\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\\na_1 & a_2 & a_3 \\\\\nb_1 & b_2 & b_3 \\\\\n\\end{bmatrix}", "1d3fd31d4040a5ae75362c68e32d0127": " ~G(k_x,k_y)", "1d4028c806916cd2b17d10a69d4060f0": "\\varphi^*", "1d40d5538760bdfbe2064fcccef3e685": "\\mathcal{O}\\left({A^{m-2}}\\right)", "1d40e87acdff39128077caa9341a519f": "\\min \\{ 2 \\times 2^{d-1-r}, 2^{d-r} \\} = 2^{d-r} .", "1d40f2a8de06149d8c9bf6bdd21d28c5": " D_m ", "1d40f91c0e72a2d0c91d31b3a5e0d3ff": "0<\\varepsilon, \\varepsilon_{1}<1", "1d4163f47a7280ec1bf4458eee4ea5a8": " \\frac{-y_2}{f} = \\frac{x_2}{x_3} ", "1d41c663447da5010acb3c15f1b56a2d": "\\pm\\sqrt{1 - \\sin^2\\theta}\\! ", "1d41e6f55521cdba4fc73febd09d2eb4": "1.", "1d4248f05f5db5ffe09587b0c4f70f8a": "\n= \\frac{\n 0.00198_{TTT} + 0.1584_{TFT}\n}\n{\n 0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0.0_{TFF}\n} =\\frac{891}{2491} \\approx 35.77 %.", "1d427c8367456193613932dda2e3ba5b": "[\\alpha]_\\lambda^T = \\frac{\\alpha}{l \\times \\rho}", "1d43b20acb41616e49414788d25e5cbb": "\\varepsilon \\left[ M \\right]\\le \\frac{\\left\\| f \\right\\|_{\\Beta ,p}^{2}}{\\frac{2}{p}-1}{{M}^{1-\\frac{2}{p}}}", "1d43bb992f4fe5cdf3855c74e50ba30f": " J(g) = R_{emp}(g) + \\lambda C(g).", "1d43cf3e9cc8956a19279855442a7082": "\\bar{V}", "1d4423fdfa639d8e41ae4b5bc7f77930": "\\begin{cases}\n y_t = g(x^*_t) + \\varepsilon_t, \\\\\n x_t = x^*_t + \\eta_t.\n \\end{cases}", "1d443da7a6187301dc4a51d1c2f143a1": "(1-x^2)^{\\alpha+1/2}\\,", "1d4448d69d246bdb3569ccaa133e6d8c": "\\mathbb{C}^* = \\operatorname{GL}_1(\\mathbb{C})", "1d44e82aaa2153d7a6863720f1b2c633": "f(x_0+0)", "1d4511c16a6accc47fadf4d30028e150": " {\\phi} ~ {\\alpha} ~ \\frac { Q } {{N} {D^3}}\\ ", "1d452d075fcc6d09f630b1db8ddcaac7": "\\tilde{Z}[\\tilde{J}]\\sim\\int \\mathcal{D}\\tilde\\phi \\prod_p \\left[e^{-(p^2+m^2)\\tilde\\phi^2/2} e^{-g\\tilde\\phi^4/4!} e^{\\tilde{J}\\tilde\\phi}\\right].", "1d453013429955e91f67836f27a9b9f8": " X ", "1d4563ad8f05a81145fc0dc117173bb3": "\\operatorname{Cov}(X, Y) = \\operatorname{E}(XY) - \\mu_X \\mu_Y.", "1d45a21b927abbd3006925b74888ab7f": "n=1 b_1 b_2 b_3\\dots b_m", "1d45c7dfc919755222e2dda68568ffee": "L_{k+1}(z) = \\frac{(1-z_{k+1}z^{-1})}{(1-z_kz^{-1})}L_k(z),\\quad k=0, 1, ..., N-1", "1d45d439a89e67a6b198fdb8b6078aee": "m_L = 1", "1d45e093a1233f41208d78517154e06c": "x \\times 1 = x \\times S(0) = (x \\times 0) + x = 0 + x = x ", "1d45f1e23075edec79cd564d71e12434": "f(u)= \\left\\{\\begin{array}{ll}\\frac{841}{108}u + \\frac{4}{29}, & u \\le (6/29)^3 \\\\ \\\\\nu^{1/3}, & u > (6/29)^3\\end{array}\\right.", "1d45feee96031bc04aee40ca77147aea": "p-p_\\infty \\approx - \\rho U^2.", "1d464372c2f6fa56541e4fd51e96622a": "dG = \\pi dS \\mathrm{NA}^2", "1d464403210356102cc98246113e82e5": "\\frac{e^{-\\mu n}(\\mu n)^{n-1}}{n!}", "1d464fed4a2d9d927ae50a3f73135e57": "\n\\lim_{t \\to \\infty, F_e \\to 0 }\\frac{1}{t}\\ln \\left( {\\frac{{p(\\beta \\overline J _t = A)}}{{p(\\beta \\overline J _t = - A)}}} \\right) = - \\lim_{t \\to \\infty ,F_e \\to 0}AVF_e,\\quad F_e^2 t = c. \\,\n", "1d465a1b7417fbdae9686f5baaafbc40": "\\ Pxy \\rightarrow Exy.", "1d465eac11c2923d28416d7cf61c23c6": " \\ C2: A(AB)=A(B).", "1d46a97c5cdb3a0f34d54f1abe33a5b7": " {(f (x^{*})- f (x))/ f (x^{*}) < \\epsilon , \\qquad \\forall{x}\\, \\in X} ", "1d46c3a6f3af40613bd1dda44b6dca25": "W_R(K)", "1d470f5ffaea4f02f9110fb88761e3da": "|x_n - 0| \\le x_{N+1} = \\frac{1}{\\lfloor1/\\epsilon\\rfloor + 1} < \\epsilon", "1d4734bc6db8002e08e28eec76ab0b7a": "\\{1,2,3\\}", "1d477f59ca62afb7461d567fee79597f": "v = {\\partial g \\over \\partial y} ~.", "1d4797b7552fe8177b57f559dec9cd75": "\\scriptstyle{b^2 = a^2 (1 - k^2)}", "1d479fa9eda9073d7ca5349b14c702aa": "u^{k+1}=\\overline{u}^k - \\frac{I_x(I_x\\overline{u}^k+I_y\\overline{v}^k+I_t)}{\\alpha^2+I_x^2+I_y^2}", "1d47cea66c20b89dfd6d8405fb88b55c": "I^+(p) = I^+(q) \\implies p = q ", "1d47ee00add2712ecbb662a0761059d9": "\n\\zeta(s) = 2^s\\pi^{s-1}\\ \\sin\\left(\\frac{\\pi s}{2}\\right)\\ \\Gamma(1-s)\\ \\zeta(1-s)\n\\!,", "1d47f6bfb593b3bacb00a81bb09a556c": "p_i(q_i, \\dot q_i, t) = \\frac{\\partial \\mathcal{L}}{\\partial {\\dot q_i}} \\,.", "1d48201eed3a959fa34361f9f3e31adb": "1, 0, 1, 0, \\ldots,\\,", "1d4823040c20fc3503ef2ed71f1c5ef4": "Y[\\mathrm{iso}] = c_{11} + c_{12} -2 (\\frac{c_{12}^2}{c_{11}})", "1d486bef7b21da51b2fd579356bfed3c": "g(x)=x^2+x+1", "1d48918b52a759841acbeb52ba8ffafa": "\\dots,W_1,W_0,W_{-1},\\dots\\dots", "1d48acc6256ae2f04176755fe33bc267": "\\frac{d^2 f}{dx^2}\\ \\cdot\\ \\frac{1}{\\left[\\sqrt{1+[f'(x)]^2}\\ \\right]^3} = 0 \\, , ", "1d48ad52203ce2db0dcff7720a4ab428": "\\pm\\frac{\\sqrt{\\sec^2 \\theta - 1}}{\\sec \\theta}\\! ", "1d48e49bf3bcd6eb886b1ea6e15d1dfd": " S(T\\rho||T\\sigma)\\leq S(\\rho||\\sigma), ", "1d48ec5d0e91220d61ae0a60c712115b": "\\nabla^2 = \\frac{\\partial^2}{{\\partial x}^2} + \\frac{\\partial^2}{{\\partial y}^2} + \\frac{\\partial^2}{{\\partial z}^2}", "1d490f95c5ce690b24b275c6534c5c6b": "y^2 - x^2 = 1 ", "1d492e2555c61e41c66ebe4bd3445d96": "\\mu_{x,\\lambda} \\rightharpoonup \\theta \\; \\mathcal{H}^m \\llcorner P", "1d493fe8c05826374f9f4a7e0796c0e1": "S(T) = C T^A {\\exp\\left(\\frac{-B}{T}\\right)}", "1d494c8741553c6005078f497c4de429": "\\tfrac{9}{64}", "1d49717c355548d0e46ffaca66c79ac1": "E^*(\\mathbb{C}\\mathbf{P}^\\infty) = \\varprojlim E^*(\\mathbb{C}\\mathbf{P}^n) = \\varprojlim R[t]/(t^{n+1}) = R[\\![t]\\!], \\quad R =\\pi_* E = \\oplus \\pi_{2n} E", "1d49888f7d173effb405fd6d82bcad40": "S^3/\\Gamma", "1d49f6b0447a7c24f4d25bbecbd29f8c": "S^{-1} S", "1d4a0d4c0554b9adede9ab4f4770cda9": " \\operatorname{E}(\\theta|y) = \\alpha' \\beta' = \\frac{\\bar{y}+\\alpha}{1+1 / \\beta} = \\frac{\\beta}{1+\\beta}\\bar{y} + \\frac{1}{1+\\beta} (\\alpha \\beta). ", "1d4a71f32d5b082fe2c8d398373ba946": "d(\\mathbf{u},\\mathbf{v}) = \\cosh^{-1}\\left(\\frac{\\mathbf{u}}{\\|\\mathbf{u}\\|} \\cdot \\frac{\\mathbf{v}}{\\|\\mathbf{v}\\|}\\right)", "1d4a807a66bf7a94bed524002019c136": " \\hat{\\pi}^{ij}(t,x^k) \\to -i \\frac{\\delta}{\\delta g_{ij}(t,x^k)} ", "1d4b75ec6ca01707473e5fdea231a12d": "(1 + \\alpha^2)", "1d4c8c8c8dc9be9f621bbe1573bfbaf5": "{\\alpha} = 0", "1d4ca999c879306d9bc62e3d0286bc93": "\\tbinom{5}{5}", "1d4cced60293ce8255ecf5a7c6af8023": "\nT > \\Chi^2_{1-\\alpha,k-1}\n", "1d4d1b1ff131021be62024be4cf7428c": " x_k= P_kx_{k-1}, ", "1d4d8f777bfc9d82f0406139bd10a18c": "h(x) = \\sum_{n=0}^{\\infty} (\\sum_{k=0}^n f_k) x^n", "1d4df73dba8e5102b370c589be328316": "K=\\C", "1d4df8918f52e9e7b47f6352913e1a3e": "a_{65}^{(12)}", "1d4e14a615189406689300061fde8f26": " L(P, t) = \\#(\\{x\\in\\mathbb{Z}^n : Ax \\ge tb\\}). ", "1d4e6a165aec4d835bac5a99e3aaa280": "k > 0\\!", "1d4e742c33c518eb95ba0c7b5eb2f602": "f(a\\mathbf{i} + b\\mathbf{j} + c\\mathbf{k}) = \\frac{1 + \\mathbf{i} + \\mathbf{j} + \\mathbf{k}}{2}(a\\mathbf{i} + b\\mathbf{j} + c\\mathbf{k}) \\frac{1 - \\mathbf{i} - \\mathbf{j} - \\mathbf{k}}{2}", "1d4e77f8edc6910f1ceb23746d41e466": "T_{\\mu \\nu} \\, = \\, - \\frac{1}{\\mu_0} ( F_{\\mu \\alpha} g^{\\alpha \\beta} F_{\\beta \\nu} \\, - \\, \\frac{1}{4} g_{\\mu \\nu} \\, F_{\\sigma \\alpha} g^{\\alpha \\beta} F_{\\beta \\rho} g^{\\rho \\sigma} ) \\,", "1d4edddb3776c42416cc6e868f64d55d": "\\mu_x = \\sum_{y\\neq x}\\mu_{xy}", "1d4eee0ed1882a60f5479952eb627a6e": " log_e(1+x) ", "1d4f4565bf4e38019ef4bd68abbdf54b": "x \\cdot 1 ", "1d4ffec1abf84d616a5fccc0b82ce5ef": " \\Delta V = -\\int_{r_1}^{r_2} \\mathbf{E} \\cdot d\\mathbf{r}\\,\\!", "1d500c527c4bafef0c2509f414a36ea7": "\\delta_\\text{int}:Q \\times \\Sigma_\\text{int} \\to Q", "1d506411839ec4d34d04f651d368fc1b": "G \\triangleright G_2(q)',\\ p^d=q^6\\text{ and }p=2.", "1d506772e9b0365fae9d33937050c36c": "D=i_D\\circ j^k:J^k(E)\\rightarrow F", "1d507a41e72e98114f2899a68e757758": "z = (x-\\mu)/\\sqrt{2}\\sigma", "1d50f2b88311cf645dc2b9b531dfb0c2": "\\phi^1 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\quad \\quad \\phi^2 = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\,", "1d5166eef62bebb51c438fa18f4564f4": "U(\\mathbf{r},t)= A_oe^{i(\\mathbf{k}\\cdot\\mathbf{r} - \\omega t )}e^{i \\varphi}", "1d51c8441aba0bd976918896d08cdf33": "\\mathit{dr}(n)>0 \\Leftrightarrow n>0.", "1d51d95895854e7ce977bb4b4549f40c": "H(A,X) = H(A) + \\mathbb{E}_{a \\sim A} \\big[ H(X| A=a) \\big]", "1d51e47aee3c00362bcd110f0b15da60": "\\scriptstyle Z ", "1d51ee6cbfc93d4c1ef4df9388ab7656": "\\Pr(|\\overline{X} - \\mathrm{E}[\\overline{X}]| \\geq t) \\leq 2\\exp \\left( - \\frac{2t^2n^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right),\\!", "1d522af0612abeb7211c7277a784cc28": " {\\kappa}a >> 1", "1d524d950b3cce35265078a236975954": "\\widehat{E}=i\\hbar\\frac{\\partial}{\\partial t}\\,,\\quad \\widehat{\\mathbf{p}} = -i\\hbar\\nabla\\,,", "1d527a31bc42ee12f6cb40f9cc8119fd": "f(p_1,p_2,\\ldots,p_n) = -\\sum_{j=1}^n p_j\\log_2 p_j.", "1d52b324fdabd94caa8ce7f2932c99b1": "\\mathbf{K}'", "1d52c92dd60ef9f0d2057df6151b1e8d": "e^z = \\cfrac{1}{1 + \\cfrac{-z}{1 + \\cfrac{z}{2 + \\cfrac{-z}{3 + \\cfrac{2z}{4 + \\cfrac{-2z}{5 + {}\\ddots}}}}}}", "1d5329f203d7a63255e2e5db5a4112b1": "\nf(B)=2\\times\\frac{e^{-\\frac{B}{2}}}{2\\pi}\\int_0^{\\frac{\\pi}{2}} \\, dt\n", "1d53d949b390f882462fcbe467410386": "\\tfrac{1}{6}=2^{-1} 3^{-1} ,\\;\\;\\tfrac{3}{4}=2^{-2} 3^1, \\;\\; \\gcd(\\tfrac16, \\tfrac34)= 2^{-2} 3^{-1}, \\;\\;\\operatorname{lcm}(\\tfrac16, \\tfrac34) = 2^{-1} 3^1. \\;\\;", "1d53f3a57a091df3f5ade1f1a82b7a0c": "\\begin{align}\nx &= \\sigma \\tau\\\\\ny &= \\tfrac{1}{2} \\left( \\tau^{2} - \\sigma^{2} \\right) \\\\\nz &= z \\end{align}", "1d54619180cd6a0c14d8e9f4ac419a1e": "\\{x\\}, \\langle x \\rangle", "1d54b43dd99935b88fc2b4d3b088c422": "y^2 = \\frac{1 - \\sqrt{1 - 4 U^2}}{2},", "1d54b5fd261d00aa57786d0e9df9ff09": "\\mathfrak{P}^{18}", "1d54ed8072dd721e45418d5262d4ecef": "\\frac{x}{\\sqrt{1-x^2}\\arccos(x)}", "1d55069e1b3a1643950d574c45ad0255": "\\beta(\\phi)", "1d5515054c9a9e0e7ac990a1eaaf1e87": "\\dot{Q}(x+dx)=\\dot{Q}(x)+d\\dot{Q}_{conv}.", "1d55179592f8269c60366d4d88ab7ae2": "1-\\frac{\\alpha}{m}", "1d55415060896863055888932cf4fe64": "a_{jk}=\\Vert T_j T_k^\\ast\\Vert,\n\\qquad b_{jk}=\\Vert T_j^\\ast T_k\\Vert.", "1d55585e0a170f790fa9cf37a5c7bf88": "J_k(n) = n^k \\prod_{p|n} \\left(1-\\frac{1}{p^k}\\right)\n=n^k \\left(\\frac{p^k_1 - 1}{p^k_1}\\right)\\left(\\frac{p^k_2 - 1}{p^k_2}\\right) \\ldots \\left(\\frac{p^k_{\\omega(n)} - 1}{p^k_{\\omega(n)}}\\right)\n.", "1d557dcab13f9dd21c96f0d41637d596": "\\frac{dK(t)}{dt} = I^g(t) - D(t) = I^n(t)", "1d558400d437662fca851d765c760faf": "\n\\begin{align}\nf(t) & {} = \\frac{d}{dt}\\Pr(T_x \\le t) = \\frac{d}{dt}\\Pr(X_t \\ge x) = \\frac{d}{dt}(1 - \\Pr(X_t \\le x-1)) \\\\ \\\\\n& {} = \\frac{d}{dt}\\left( 1 - \\sum_{u=0}^{x-1} \\Pr(X_t = u)\\right)\n= \\frac{d}{dt}\\left( 1 - \\sum_{u=0}^{x-1} \\frac{(\\lambda t)^u e^{-\\lambda t}}{u!} \\right) \\\\ \\\\\n& {} = \\lambda e^{-\\lambda t} - e^{-\\lambda t} \\sum_{u=1}^{x-1} \\left( \\frac{\\lambda^ut^{u-1}}{(u-1)!} - \\frac{\\lambda^{u+1} t^u}{u!} \\right)\n\\end{align}\n", "1d55ac14cb6a40e57ae7e2b51265adbe": "K_1 = \\oplus H_{\\alpha}", "1d55f0a7b672133457c8cdea68050323": " \\hat{H} = \\hat{H}^{\\dagger}. ", "1d55f10c9df658c98268925d074ca2e0": "\\Delta h\\, =\\, h_1\\, -\\, h_2\\, =\\, -\\, \\frac{1}{g}\\, \\frac{A_1}{A_2} \\left( 1\\, -\\, \\frac{A_1}{A_2}\\right)\\, v_1^2.", "1d5613061b14dc1cb2960aa3db5afdae": "\\scriptstyle M_\\text{A}", "1d56258109956b026d767af1cd300dee": "Vd_F = Vd_{T1} + Vd_{T2} + Vd_{T3} + ... + Vd_{Tn}\\,", "1d5631b985746dafa7e04d0310b95b34": "\\mathbf{H^r}", "1d5675825f91c3465a267c659bd3ea49": "~D(x)=u e^{x}+v~", "1d56f87a4031f10036022423ee209ad7": "\\! E_\\mathrm{h} / k_\\mathrm{B} ", "1d5713b7fac39021d0c83c67243ddcb0": "v_p = \\sqrt{2\\mu/r_p + v_\\infty^2}\\,", "1d5721cf09e47f5f5ed4d9b91b45ac57": "S=\\{s_1, s_2, \\dots\\},", "1d57c1cd49d1bdb80fac5ce2cc88102a": "\\displaystyle{(Rv,v)=|(Rv,v)|=\\sum |(R_iv,R_jv)|.}", "1d581721d86ead4ec5b3356b640ad503": "e^{\\frac{-E_a}{k_BT}}", "1d5875904f4d060f31bef3a06f7b3af7": "\\dot m", "1d58aacbca61067d3a49e219ab45bcc3": " G\\approx 0.8650 {\\rm \\ cm}^3 {\\rm g}^{-1} {\\rm hr}^{-2}.", "1d58f3169c11bb5c1e58cb861ecf17c5": "\\scriptstyle \\psi^\\dagger(x)", "1d58f82b73910c13c8fbc7ff4a224220": "\\alpha_{F}", "1d590407211c73321619d8ff9f69f772": "\n \\lambda_{z,n} = \\frac{2 \\pi \\, H}{\\alpha_n} = \n \\frac{2 \\pi \\, H}{ \\sqrt{\\frac{\\kappa H}{h_n} - \\frac{1}{4}}}.\n", "1d596daea8bb14160d6be0ca21907552": "\\boldsymbol{\\sigma_1}", "1d5996df077e0eb3bc551915ce350b15": "\\sqrt{2} m", "1d5a11b49d0db1090a8b06b07b9e5d0c": "\\scriptstyle \\eta=a\\,\\cos\\, 2\\pi\\left(\\frac{x}{\\lambda}-\\frac{t}{T}\\right)", "1d5a8eed291c76081bc5f3837bf6b03b": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon r^{\\prime}} \\sum_{l=0}^{\\infty}\n\\left( \\frac{r}{r^{\\prime}} \\right)^{l}\n\\left( \\frac{4\\pi}{2l+1} \\right)\n\\sum_{m=-l}^{l} \nY_{lm}(\\theta, \\phi) Y_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "1d5ad01bf19db90e246090e2edd640e2": "B_3(t) = \\sum_{i=0}^{n} z_i b_{i,n}(t) \\mbox{ , } t \\in [0,1]", "1d5ad60e3f3b96e4c163f7c2d3e17079": "\\begin{align}\n a' &= e^{-\\frac{1}{2}{\\pi b}} \\cos{\\frac{\\pi a}{2}} \\\\\n b' &= e^{-\\frac{1}{2}{\\pi b}} \\sin{\\frac{\\pi a}{2}}\n\\end{align}", "1d5add4da2f1cf4a6d634e0364c18eb6": "Fp(1-p)\\!", "1d5aea1a16e1ca0318afebf8afd5dab2": "f(x)-R(x) = c_{m+n+1}x^{m+n+1}+c_{m+n+2}x^{m+n+2}+\\cdots", "1d5b319daed3968632474c3925152ebf": "\\frac{J_{X_t}}{X_t} = \\frac{J_n}{n} = \\frac{1}{n}\\sum_{i=1}^n S_i \\to \\mathbb{E}S_1 ", "1d5bcddd8f72c9fdd5792ecf040b4190": "\\mathbf{H}^n", "1d5bdb4e4f0a2930dbde370fe4329761": " \\rho(\\vec R, \\vec R^') = 1- {6\\sum_{i=1}^N(r_i-r_i^')^2 \\over N(N^2-1)}", "1d5c293884de00260dc3401c9a937af6": "\\mathbf{n} / \\mathbf{N} = (n_1/N_1, \\ldots, n_d/N_d)", "1d5c7a7ceaa34162179ad5899a42355b": "\n\\begin{align}\n \\cos a & = \\cos b \\cos c + \\sin b \\sin c \\cos A \\\\\n & = \\cos b\\ (\\cos a \\cos b + \\sin a \\sin b \\cos C ) + \\sin b \\sin C \\sin a \\cot A \\\\\n \\cos a \\sin^2 b & = \\cos b \\sin a \\sin b \\cos C + \\sin b \\sin C \\sin a \\cot A.\n\\end{align}\n", "1d5c7da00713ff53d2b5572b4a4d3886": " e^{-c x} = a_o \\frac{\\displaystyle \\prod_{i=1}^{\\infty} (x-r_i )}{\\displaystyle \\prod_{i=1}^{\\infty} (x-s_i)} \\qquad \\qquad\\qquad(3)", "1d5cfc8aee4cdfdfae5e607679fc7d46": " \\Pr \\left[ L \\le x \\right] = \\frac{2}{\\pi}\\arcsin\\left(\\sqrt{x}\\right), \\qquad \\forall x \\in [0,1].", "1d5d4b7f9a4e70be1153d2ca59cbb619": "B^\\phi_{MX}=\\lambda_{MX}+I\\lambda'_{MX}\n+\\left(\\frac{p}{2q}\\right)\\left(\\lambda_{MM}+I\\lambda'_{MM}\\right)+\\left(\\frac{q}{2p}\\right)\\left(\\lambda_{XX}+I\\lambda'_{XX}\\right)", "1d5db6f070c64386ff6beb61f5b003d3": " f(z) = {u_1(z)\\over u_2(z)},", "1d5df386b22a85673fb715c105056b35": "\\pm \\sqrt {b^2-4ac}", "1d5e405aaec56a6bd177dc0bc7b4dc03": "\n \\frac{dz}{d\\zeta} = \\frac{4n^2}{\\zeta^2-1} \\frac{\\left(1+\\frac{1}{\\zeta}\\right)^n \\left(1-\\frac{1}{\\zeta}\\right)^n}\n {\\left[ \\left(1+\\frac{1}{\\zeta}\\right)^n - \\left(1-\\frac{1}{\\zeta}\\right)^n \\right]^2}.\n", "1d5e9b163626c6491d87617cc1b5cd9c": " \\frac{ \\partial{L} }{ \\partial y} = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\frac{\\partial{L}}{\\partial{w}} ", "1d5efd5057833752abb9e5b4b4b9c476": "P^{-1}", "1d5f0987dc5ddb9f0e562b71ad1328f3": "\\xi\\to 0", "1d5f13a32cfc9f9dbf439cb713150287": "\\langle (x \\wedge y) \\rangle_1 = 0", "1d5f4ebbc17e65fd05d1f31e27819617": "P_1(x) = f(a) + f'(a)(x-a) \\ ", "1d5fcd4dc23c54451926d7dc44dc3de7": "\\Delta f = h", "1d5fec8f558e051bc01b16d69f8f114b": "2^{519} = 1014 = -5\\pmod{1019}", "1d5ffb0049a570d315d44b167a1691c6": "\\Phi(z=0, t)", "1d6003c5e88533a10ad268d0847541b1": "\\|f\\|_B = \\sup_{x\\in X}|f(x)|", "1d605122a27e4d8dc338b46db6dd0738": " \n\\Sigma _1 =\n\\begin{bmatrix}\n0 & 0 & 0 & \\cdots &0 & 1\\\\\n1 & 0 & 0 & \\cdots & 0 & 0\\\\\n0 & 1 & 0 & \\cdots & 0 & 0\\\\\n0 & 0 & 1 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots &\\vdots &\\vdots \\\\\n0 & 0 &0 & \\cdots & 1 & 0\\\\ \n\\end{bmatrix}\n", "1d6058c3ce9f887f333fb0ded070b74e": " \\tau = \\frac{T_r}{T}, \\delta = \\frac{\\rho}{\\rho_r} ", "1d60842db666ff59bdc22df52c73a05a": "\\bar{\\psi} = \\psi^\\dagger\\gamma^0", "1d60b1442a9d5eacf8dd1eabf79f9c7c": "n_y=\\sin{\\theta(z)}", "1d6116c41063af5b064a7022b2389df8": "H[m]=\\sum_{i=1}^N\\sum_{j=1}^N \\Lambda(x_i-m_j,m_i-m_j,x_i-x_j,i-j)", "1d61b934f3739f8624d0b16e35fca486": "Mq_1 q_2 q_3\\cdots q_{n}", "1d61ee0df5b4f14b76dad026d0ce7237": "(m,n)", "1d624ad71e522bbb7f350285861d662c": "U_{n,n}", "1d6294ec0ec341770a7e42e92f09ebae": "1\\in\\mathbb{H}", "1d62aebfe58e35809090178c7bd470e3": "d=(1+v^2)(u-t)(1+tu)", "1d62bd4ccdce8ac859106a51fdacd1b7": "\\int \\frac{\\operatorname{li}(x)}{x}\\,dx = \\ln x\\, \\operatorname{li}(x) -x ", "1d62bf433344920508bcf728e23a0bba": "\\displaystyle{[a,b^2,c] = 2[a,b,c]b.}", "1d62dac31efa32b7bc1107ac057ad35e": "V_\\mathrm i = Ve^{-\\gamma x'}\\,\\!", "1d631eccc872cf5df8adc79a656c9543": "\\mathbf a\\mathbf b = \\sum_{i=1}^7 f_i(\\mathbf a)g_i(\\mathbf b)w_i", "1d632e41fe2aa21c5e497bc4ea7a5fa3": " N_z ", "1d6334d2b826f324c19a806a337916ac": "|\\psi|^2\\approx 1+2~\\mathrm{Re}\\left(\\frac{f(\\theta)}{z}e^{ik(x^2+y^2)/2z}\\right).", "1d6339c4e9579e94a94eab0ac041c057": "{\\Pr}_{\\theta,\\varphi}(u(X)<\\thetaz,x>y", "1d854f5053c4bf83e5d61e2f492903bd": " \\nabla^2\\mathbf{H} = \\lambda^{-2} \\mathbf{H}\\, ", "1d855628a292b6f30e79cc9a2b97c3e7": " \\Pi = \\mathrm{surface \\ pressure} ", "1d8566f83d829f889db1c6a75fff2f44": " \\frac{| \\nu - \\mu |}{ \\sigma } \\le \\sqrt{ 0.6 } ,", "1d8576b6520a3bdd8529045e2881751b": "-1.0185", "1d85e15e8c42d4c860536e0ed0a1414d": "( \\hat{\\theta}_i,\\theta_i)", "1d85ed6726aabd57598d193483c4d8d8": "^{14}\\text{NO}_3^- + {^{15}}\\text{NO}_3^- \\rightarrow {^{14}}\\text{N}^{15}\\text{N}\\text{O} , ", "1d86318750e84e56e8b35ee8ddb90f44": "\n\\frac{ \\partial \\overline{\\rho} \\tilde{u_i} }{ \\partial t }\n+ \\frac{ \\partial \\overline{\\rho} \\tilde{u_i} \\tilde{u_j} }{ \\partial x_j }\n+ \\frac{ \\partial \\overline{p} }{ \\partial x_i }\n- \\frac{ \\partial \\overline{\\sigma_{ij}} }{ \\partial x_j }\n= - \\frac{ \\partial \\overline{\\rho} \\tau_{ij}^{r} }{ \\partial x_j }\n+ \\frac{ \\partial }{ \\partial x_j } \\left( \\overline{\\sigma}_{ij} - \\tilde{\\sigma}_{ij} \\right)\n", "1d865e3b55875f1ea75c2e130391a393": " Q = (m_\\text{n} - m_\\text{p} - m_\\mathrm{\\overline{\\nu}} - m_\\text{e})c^2", "1d873a0bdc91561d3a2249a8d7bc680b": "\\mathcal{H}^{(0)} = -\\frac{\\hbar^2}{2m}\\left(\\nabla^2_{1} + \\nabla^2_{2}\\right)-\\frac{e^2}{r_{1}}-\\frac{e^2}{r_{2}}", "1d87851171785ace9925fe50a74c6035": " A \\otimes_R B ", "1d8787b3e1d78a2432637c07a70bb4f2": " \\{ (C \\otimes 1) \\Delta(C) \\} ", "1d87e563779e35c308b3e14105676e62": "\\tau_{D}", "1d8801f11813b5ab1cff84d42b3f8248": "\\forall x,y \\in H: \\qquad \\lnot(x \\vee y)=\\lnot x \\wedge \\lnot y.", "1d882c8740ea7acc672e223eb0a5d08b": "K, m", "1d88764db0920f9142118ab7152c8edd": " d^2 F_x = k I I' ds ds' \\left[ \\left( \\frac{1} {r^2} \\left( \\frac{\\partial r}{\\partial s} \\frac{\\partial r}{\\partial s'} - 2r \\frac{\\partial^2 r}{\\partial s \\partial s'}\\right) + r \\frac{\\partial^2 Q}{\\partial s \\partial s'}\\right) \\cos(rx) + \\frac{\\partial Q} {\\partial s'} \\cos(xds) - \\frac{\\partial Q} {\\partial s} \\cos(xds') \\right] ", "1d88b7cfeb6250482de2665829a48d24": "\\displaystyle m_1 = m_2 = 1", "1d891016a1af1c3d2629a64edd89186e": "t=x^0", "1d891b1f7023bfee3bfdc9a37fac6adc": " \\cfrac{\\Gamma \\vdash f = g \\qquad \\Delta \\vdash x = y}\n{\\Gamma \\cup \\Delta \\vdash f(x) = g(y)}\n", "1d892645821c09b9ed596974b6743d3b": "\\widehat{1}", "1d899cab1f6d6425a8953a423247442f": "\\forall a,b \\in F, x,y,z \\in \\mathfrak{g}", "1d89c6558230056446b622fe8270d797": "C_G \\varphi=\\psi\\wedge E_G (C_G \\varphi)", "1d8a8e11d5c23997d3d1f96ac8c8eb16": "B = R_3 C_5 + R_1 C_5 + R_1 R_3 C_5/R_4\\,", "1d8b1160213ecd78f58542e6940e688d": "H_0=\\mathbf{P}^2/(2m) +V(R)", "1d8b1d86e6fac43aa734ce3ff485bb22": "\\Omega < \\Omega_{gp}", "1d8ba0e908e6cf5ad2b9867f2a67e2c9": "P(x_1,\\ldots,x_n)=\\prod_{1\\le ic", "1de5ad497ee53dbb833a1d08983e7454": "\\| X_{t} \\|_{p} = \\| X_{t} \\|_{L^{p} (\\Omega, \\mathcal{F}, \\mathbf{P})} = \\left( \\mathbf{E} \\left[ | X_{t} |^{p} \\right] \\right)^{\\frac{1}{p}}.", "1de5d075d6ef848ba4e376da21c3bdd1": "\\int_{E}\\varphi\\leq M\\mu(A),", "1de5ecffd06da46aa1d53dd092ebae15": "I_{SSP}=|f'_if_if_z\\chi_{iiz}^{(2)}|^2", "1de5f5c3f8ea1d74e467b2556cdd7c7f": "r(t)=0", "1de5fa297e36c4631ad16ab29e156896": "P(M2)/P(M1)=1", "1de64e2f0836654d868a134e60ed8069": "f(x_1, \\ldots, x_n) = \\exp\\Big(\\mathord{-}\\textstyle\\frac{1}{2} \\displaystyle\\sum_{i=1}^n x_i^2\\Big).", "1de664ec707d90bc57f381b34df6493f": "n-1, 2n-1, \\dots, 2^kn - 1", "1de7048a4a0cf6a9727ac843cbdc7d88": "D_2 = \\sum_{P \\in E}{d_P [P]}", "1de7249ae2c2037f27c16de0d8a8e562": "\\alpha d\\ge1\\qquad\\qquad(16)", "1de73570f74e6abebac3149f4bc81a5c": " \n(Eq. 5) \\text { } E[Y_i(\\alpha^*(t), \\omega(t))] \\leq -\\epsilon \\text{ } \\forall i \\in \\{1, ..., K\\} \n", "1de74c386fba4dd379391103c5504b73": "wrap(H(M * share_i))", "1de74cec8715f07eaa1bb70ace22a931": "\\xi\\notin\\Sigma_x(f) \\iff \\exists\\phi\\in\\mathcal D_x,\\ \\exists V\\in\\mathcal V_\\xi: \\widehat{\\phi f}|_V\\in O(V) ", "1de762a65de4dac349ca69a2704cb53a": " a>0 ", "1de7ba39d7d0f39048e0c25d57fb4c43": "h^2", "1de7bba5087b35cd2bd086c543be2a0a": "\\text{DOR}^\\prime = \\frac{26/13}{12/48} = \\frac{2}{1/4} = 8", "1de7d5a40868131086194e5b5cf22fc7": "2^w - 2 + {n \\over w}", "1de8d94eb7eb0f6b16eace0a9e5d29a6": "B(x,r)", "1de8ee568ae3d9835bbc9bb6e0922b84": "N_{E},", "1de8f604fb15e570931224cef7eb781d": "\\log_b \\sqrt[p]{x} = \\frac {\\log_b (x)} p \\, ", "1de902ca15957f704df8261c38059f7e": "\\mu_1+\\mu_2", "1de94aead55527d330f705167ab22c08": "\\pi^{p,q}:E^k\\rightarrow\\Omega^{p,q}.", "1dea274073a62beb01020a77de46f3e9": "\\int_0^T {\\partial f\\over\\partial W}(W_t,t) \\circ \\mathrm{d} W_t + \\int_0^T {\\partial f\\over\\partial t}(W_t,t)\\, \\mathrm{d}t = f(W_T,T)-f(W_0,0).", "1dea2fcd967e24d4f117f5a329bcfbd7": "\\underbrace{a+\\cdots+a}_{n \\text{ summands}} = 0", "1deaac1da29e3d14f16afe72ff798c9d": "|\\psi_{A,B}^\\lambda\\rangle", "1deb083d5023ba7d31e5395102fe07dc": "R_o=\\frac{1}{h_oA}", "1debabc8374a2ef134e215b6a6bd0548": "(x_k\\ ,\\ y_k)", "1debbe54dba95801e3faa292b2ca6be6": "\\mathcal O(-(n+1))", "1debd78014583b732b1f3cf1cc5c09e6": " \\sin(\\phi) \\, \\partial_\\theta + \\cot(\\theta) \\, \\cos(\\phi) \\, \\partial_\\phi", "1dec1a166251074116da41fcdab6a53d": "\\displaystyle{V(z)=\\int_0^z -U_y dx + V_x dy,}", "1dec495631d797b12539f985590dad8f": "\\alpha_i = w_i^0/w_i, \\alpha_j =\nw_j^0/w_j", "1dec6a2bf7b1dcca28934c607dc6231c": "\\alpha : 2^N \\setminus \\{\\emptyset\\} \\to [0,1]", "1dec9b8bb460b92746c53c327fd0714a": "G = \\operatorname{Gal}(L/\\mathbf Q) \\cong (\\Z/p\\Z)^\\times,", "1decb1be6422b2f79cde017fb0d16066": "p(C, F_1, \\dots, F_n)\\,", "1decbdaaa6c63fddefa9b4dec5c0bc67": "\\ LiPE = pIC_{50} - LogP ", "1decc1582611559c6fc0de82d3f8cc28": "\\sin\\frac{\\pi}{5}=\\sin 36^\\circ=\\tfrac{1}{4}\\sqrt{2(5-\\sqrt5)},\\,", "1dece67cba02763af5161628c8319557": "\\mathbf{z}\\in [\\mathbf{x}]", "1decf65a72cb7ac2505dcb1b81c6ba3b": "0 < \\alpha \\le 2", "1ded8e027258a7c5af6b41de2bbbd312": "0 = (C_{\\beta I}^{\\;\\;\\; K} e^\\alpha_K e^\\beta_J - C_{\\beta J}^{\\;\\;\\; K} e^\\beta_I e^\\alpha_K) e^I_\\gamma e^J_\\delta ", "1ded92344130202f59bf76d2654bc6f5": "-\\infty~\\mbox{dB}", "1dedb09e29bee9dbce2484e86436078f": "U_{coul}={{k \\, Z_1 \\, Z_2 \\, e^2} \\over r}", "1dee6dfdf7227bf0dba3870ccf39db7c": "(p_1(x) y^\\prime)^\\prime + q_1(x) y = 0 \\,", "1deeca0bf079b2938a04f3d2320e4710": "\\phi(x,t)", "1def01a15617e8735b7ceac252d4257d": "\\{ -S_\\nu, -S_\\nu +1, \\ldots +S_\\nu -1,+S_\\nu \\}", "1df06e81064e3628e7ea7991f0efdcdf": "\\sum_i b_i \\beta S(c_i) = \\varepsilon(a) \\beta", "1df0bffb6b51ce9b4706f71bf988047f": "Y_{lm}(\\theta, \\phi)\\,", "1df107efd8d34de0c4d18ba3eafe211c": "\\alpha_i\\,", "1df136faa1dc83d592e49a5c6c2af9f7": " +\\beta _3 {\\left(\\frac{{\\left[ {1 - \\exp \\left( { - m/\\tau_2} \\right)} \\right]}}{m/\\tau_2} - \\exp \\left( { - m/\\tau_2}\\right)\\right)} ", "1df17e122bd0e48415de424bca436227": " K = K_{0} \\subset K_{1} \\subset \\cdots \\subset K_{\\infty}, ", "1df181eaa1bb40a0067c06ead197170d": "v_i", "1df183675d8ffd7c88b3cec87a4298fb": "{v_R^3}-\\frac{1}{3}\\left({1+\\frac{8T_R}{p_R}}\\right){v_R^2} +\\frac{3}{p_R}v_R- \\frac{1}{p_R}= 0", "1df190750410b63a67a7363bc4111116": " \\sigma^2\\mathbf{V}", "1df19b202443af9a31a2e550d82b500c": "\n Q_\\alpha := \\kappa~\\int_{-h}^h \\sigma_{\\alpha 3}~dx_3 \\,.\n", "1df19bac7fe0bd2d0b5bcf1135b0b708": " \\infty \\text{ for } 1 < \\alpha \\le 2 ", "1df1ced8e3c3ba4edff179fc7437709a": "|F|^2", "1df20f26154730a6d82bca09dce5224e": "\\aleph_{\\alpha+1}", "1df22c28794d53578c6ca0cd22042322": "e(aP,bP)", "1df24731e63176f329736029af75b458": "y_p+c_1y_1+\\cdots+c_ny_n", "1df2ab3c50118f24be33fba5d440e80c": " n=\\sum_{i,j} n_{i,j} \\, ", "1df2c056cf8e52de12585851b645fe2c": "(a + b)^2 = a^2 + 2ab + b^2\\,", "1df30cf48b5f32023ec8eafbb42fbc5d": "\\mathbb R^n,", "1df3165a9a59617b013ae8d9d11df21e": " T^{ik} \\quad i \\ne k ", "1df31fbb8d73ff712d9ee67af78c8233": "\\phi\\ = \\!\\left({Q\\over{ND^3}}\\right).\\,", "1df32c525d59a2a66ce9adc6c2180489": "\n \\operatorname{E}(X^m) = k (k+2) (k+4) \\cdots (k+2m-2) = 2^m \\frac{\\Gamma(m+\\frac{k}{2})}{\\Gamma(\\frac{k}{2})}.\n ", "1df33a96013c92f9fe241b0a590b84a5": "\n\\begin{align}\nG(0)&=0 \\\\\nG(n)&=n-G(G(n-1)), \\quad n>0.\n\\end{align}\n", "1df35a922fade3b3959e05b9c95ee7af": "C_R =\\sqrt{\\frac{mg h_\\text{after}}{mgh_\\text{before}}}", "1df35ad86dbda1c7e2d633cd454fd549": "\\textstyle X=\\{X^+,X^-\\}", "1df37b4fb42a199fcbfa74d5b53a23bf": "\\ \\sum_{w \\in V} f(u, w) = \\mathrm{excess}(u) \\geq 0", "1df393e4d06908549a854db9a7321736": "f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD. \\ ", "1df39db061257815e95ad24357573b57": "D = \\frac{\\pi}{8}\\lambda^2 \\nu_m", "1df4154727320375999c5308abba1fae": "Q\\to\\mathbb R", "1df41652d340541fb57804fa1eda22b0": "y = mx + b \\,\\!", "1df4347163c1ab391950554ba0080cbb": "\\mathbf{S} = \\mathbf{\\hat{n}}S", "1df4369acaf6c7a982a4816188b7fe2b": "ax^2 + bx +c=0", "1df459705e2c4ee6d34c123e8e7f9e4a": " (\\wp^\\prime)^2=4(\\wp-e_1)(\\wp-e_2)(\\wp-e_3). ", "1df461fcf54bd774d1636134ff74dcc4": " h(x,y) = k + xh_1(x,y) + yh_2(x,y) ", "1df46e7600d3b91caafb3e729fd45157": "\n\\frac{{x'}^2}{a^2} + \\frac{{y'}^2}{b^2} = 1.\n", "1df4757a0d45090031e682efe5d82112": "\n\\sum_{i=1}^n (Y_i - \\hat\\mu(x_i))^2 + \\lambda \\int_{x_1}^{x_n} \\hat\\mu''(x)^2 \\,dx.\n", "1df47b82d28942c9874ab4498956826c": "\\mathit{q_i}", "1df4b45e4433fa83c2fe1a62bd59e9aa": "\nr_{\\mathrm{outer}} = \\frac{a^{2}}{r_{s}} \\left( 1 + \\sqrt{1 - \\frac{3r_{s}^{2}}{a^{2}}} \\right)\n", "1df4c2f1262424302507253c0fb9688a": "b = N/K", "1df505889f0be5b811854368ce12f6ac": " \\mathit{WER} = \\frac{S+D+I}{N} ", "1df5713c50393d5a6d45de18d062efaf": " F^+", "1df599af0ed840284f23532e7dc2a9b2": "\n\\begin{matrix}\nProc & ::= & \\textit{STOP} & \\; \\\\\n&|& \\textit{SKIP} & \\; \\\\\n&|& e \\rightarrow \\textit{Proc} & (\\text{prefixing})\\\\\n&|& \\textit{Proc} \\;\\Box\\; \\textit{Proc} & (\\text{external} \\; \\text{choice})\\\\\n&|& \\textit{Proc} \\;\\sqcap\\; \\textit{Proc} & (\\text{nondeterministic} \\; \\text{choice})\\\\\n&|& \\textit{Proc} \\;\\vert\\vert\\vert\\; \\textit{Proc} & (\\text{interleaving}) \\\\\n&|& \\textit{Proc} \\;|[ \\{ X \\} ]| \\;\\textit{Proc} & (\\text{interface} \\; \\text{parallel})\\\\\n&|& \\textit{Proc} \\setminus X & (\\text{hiding})\\\\\n&|& \\textit{Proc} ; \\textit{Proc} & (\\text{sequential} \\; \\text{composition})\\\\\n&|& \\mathrm{if} \\; b \\; \\mathrm{then} \\; \\textit{Proc}\\; \\mathrm{else}\\; Proc & (\\text{boolean} \\; \\text{conditional})\\\\\n&|& \\textit{Proc} \\;\\triangleright\\; \\textit{Proc} & (\\text{timeout})\\\\\n&|& \\textit{Proc} \\;\\triangle\\; \\textit{Proc} & (\\text{interrupt}) \n\\end{matrix}\n", "1df61e7f2446684a7a300e9420d62011": "[n/2]", "1df620affaeb14fc3171571997bdf3dc": "P\\left(S^{t}|S^{t-1}\\right)", "1df64bfbca56349ab1f424cdf971290a": "f\\left(\\perp\\right) = \\perp", "1df664e1d99be05226de273dea7ef62e": " L_{O} \\; = \\; L_U \\; - \\; 4.78 \\; (\\log_{10} f)^2 \\; + \\; 18.33 \\; \\log_{10} f \\; - \\; 40.94 ", "1df675be3ba9e7f7d83b3d5d80c00f1f": "(s, t)", "1df6dd794032e3223d41e87b5a03b6c7": " 2^\\kappa = \\kappa^+ ", "1df6fe382decdbe5d1e57a57f9781bd9": "n=1,2,\\dots ", "1df76a4943737376230bff97483321f5": "p-q=\\frac{r'}{r}", "1df7965db6cdd504e4fcc0bd5d525600": "X_{\\mu > \\epsilon}", "1df7a88342668cf09c9c7f585750bbe6": " \nI(F_a;C) = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i,c_j) \\log \\frac{p(v_i,c_j)}{p(v_i)\\,p(c_j)}\n", "1df7e018ccee4b10f078eabf749776bb": "\\scriptstyle \\Delta f \\,=\\, \\frac{1}{T_U} \\,\\approx\\, \\frac{B}{N}", "1df7e831297ee6ad1755f34604a7d6ae": "m = e / h^k. \\,", "1df82a65af6404e5ecc02b6996d52e0d": "\n\\begin{matrix}\nS\\\\\nS(4,1,2)_H (\\bar{4},1,2)_H\\\\\nS(1,2,2)_H (1,2,2)_H\\\\\n(6,1,1)_H (4,1,2)_H (4,1,2)_H\\\\\n(6,1,1)_H (\\bar{4},1,2)_H (\\bar{4},1,2)_H\\\\\n(1,2,2)_H (4,2,1)_i (\\bar{4},1,2)_j\\\\\n(4,1,2)_H (\\bar{4},1,2)_i \\phi_j\\\\\n\\end{matrix}\n", "1df83c22566e7ce784d12b63f75e2d7b": "q(\\boldsymbol{x}) = f(\\boldsymbol{x}) / g(\\boldsymbol{x})", "1df891ee336daa1d480e0f4e4bcfe5c1": "\\scriptstyle{A_0 ' \\sim A_0^{k^2}}", "1df8a4a4e0f8a6c31bf31d4c1c55c4b7": "\\begin{align}\n(f * g_N)[n] &= \\sum_{m=0}^{N-1} f[m]\\ g_N[n-m]\\\\\n &= \\sum_{m=0}^n f[m]\\ g[n-m] + \\sum_{m\\,=\\,n+1}^{N-1} f[m]\\ g[N+n-m]\\\\\n &= \\sum_{m=0}^{N-1} f[m]\\ g[(n-m)_{\\bmod{N}}] \\equiv (f *_N g)[n]\n\\end{align}", "1df904db52397b02f7affb126532d43e": "e^{-At}\\mathbf{y}'-e^{-At}A\\mathbf{y} = e^{-At}\\mathbf{b}", "1df9b2496ff2902ab27959eda4522ac5": "C(a,b,\\xi)", "1dfa44dac19fbe129a5c3445c64f935e": "\\frac {ad} {bd} = \\frac {cb} {db}", "1dfa982f9fd6291a0dae769ba5d92300": "f(x)=\\begin{cases}\\sin\\frac{5}{x-1} & \\text{ for } x< 1 \\\\ 0 & \\text{ for } x=1 \\\\ \\frac{0.1}{x-1}& \\text{ for } x>1\\end{cases}", "1dfab721eb29fe296bb2117503cdf833": "\\eta_{IJ} = e_{\\beta I} e^\\beta_J", "1dfac909c5b4f67502bee95250077e2d": "\\frac{\\partial{G}}{\\partial{w_{ij}}} = -\\frac{1}{R}[p_{ij}^{+}-p_{ij}^{-}]", "1dfb693903c04376fbba72707d0c5b51": "\\exists x (\\forall y ((\\phi(y) \\land y=a) \\leftrightarrow y = x) \\land a = x)", "1dfbd026fd0f0c0533329becc24a8f41": "\\int \\operatorname{csch}\\,x \\, dx = \\ln\\left| \\tanh {x \\over2}\\right| + C , \\text{ for } x \\neq 0 ", "1dfbd5f936359002bd59ec4e9495c4b3": "\\! ds_3^2", "1dfc0dc62ab0b72188128f273546f6b4": "O(n!)", "1dfc1ac114077ccfb54c2beaffaa43a9": "\n\\mathbf{Q}_1 =\n\\begin{pmatrix}\n1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\\\\n0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\\\\n0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\\\\n0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\\\\n0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\\\\n0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1\n\\end{pmatrix},\n", "1dfc3aab6c6a1ad0be7d45c4f80c0d15": "(m-1)", "1dfc489688a6d018cf9f3f8d4b36daf7": "\\Pr [ X_{t} \\in A | X_{0} = x ] = P_{t} (x, A).", "1dfc984cd186c14bc2f982051249426c": " C = -2\\log_{10} \\left({\\varepsilon\\over 3.7 D} + {2.51 B \\over \\mbox{Re}}\\right) ", "1dfca7cf06f97da3caffdd92493ae95b": "C_k = 2 \\sin \\left [\\frac {(2k-1)}{2n} \\pi \\right ]\\qquad\\mathrm{k = odd}", "1dfcd0d40e2df0f9246a6a50eec94d07": "\\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \\choose k}},", "1dfd2d833baf110820ff695854938b91": "f(x) = \\frac{p(x)}{q(x)}", "1dfd442197164527a1c17663f87cc910": "\\Delta \\tau =Gb \\epsilon^\\tfrac 3 2 \\sqrt c ", "1dfd84a3a4e325e59d305dae73c9abfa": "=\\sum_{k=1}^{d} \\dot q_k \\ \\boldsymbol{e_k} + \\sum_{k=1}^{d}\\sum_{j=1}^{d}\\sum_{i=1}^{d} q_j \\ {\\Gamma^k}_{ij} \\boldsymbol {e_k} \\dot q_i \\ ", "1dfe00188d41c81852565054e5ffc450": " A\\to I\\otimes A\\xrightarrow{\\eta^l}(A\\otimes A^l)\\otimes A\\to A\\otimes (A^l \\otimes A)\\xrightarrow{\\epsilon^l} A\\otimes I\\to A", "1dfe0367b2af745caabd0c4ecb6227f9": "T_{i_1i_2\\dots i_r} = T_{i_{\\sigma 1}i_{\\sigma 2}\\dots i_{\\sigma r}}.", "1dfe3ddd6a20e3a5597cdd6ac92a8500": " B_k ", "1dfe43ffdcc981625d06c7bcd72d071f": "T_{1} ", "1dfe54c9c1f45d7314aaf7a613e24288": "f(\\chi)=\\chi^\\frac{3}{2}", "1dfecb246f5e3c96b396c40915f3cbcd": "-\\frac{d^2}{dx_e^2} f_e(x_e)=k^2 f_e(x_e).\\,", "1dfed80b15ba628ac3285509c581caf3": "p={1\\over 50}=0.02", "1dff07f05201bd3b498721c16963c2d8": " T_{n} ", "1dff0bbf926bdd57b1cd92e37054479a": "\\begin{align}\n \\sum_{i=1}^nw_i\\log(x_i) &\\leq \\log\\left( \\sum_{i=1}^nw_ix_i \\right) \\\\\n \\log \\left(\\prod_{i=1}^nx_i^{w_i} \\right) &\\leq \\log\\left( \\sum_{i=1}^nw_ix_i \\right)\n\\end{align}", "1dff316f66d4c78c2d42d7cf911c1d76": "\\mathbf{\\Psi}(\\mathbf{x})=\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell f_{\\ell m}(k r)\\mathbf{X}_{\\ell m}(\\theta, \\phi)", "1dff618f0b9e2a63a20fd24a0b366d29": "90^2", "1dff826c9863d64159569d231dcbf583": " \\frac{2a + b}{a} = \\frac{a}{b} \\equiv \\delta_S\\,.", "1dffcc3ca5837e9b45989b58f60d1f3c": "J^1_1Q\\to\nQ", "1e005a5e1667ad463b52df02be54c3d1": "z \\notin L \\implies \\Pr\\nolimits_x[\\phi(x,D_n(x,z),z)] \\leq \\tfrac{1}{3}", "1e0080f795a3f79218c6108fd24e5849": " \\int_0^\\infty x^n e^{-\\alpha x} dx = \\frac{n!}{\\alpha^{n+1}}. ", "1e00b1d5a74ff4fb1dc7f4c860e906cb": "G_\\infty", "1e00d7935e25062f1c069773721987b7": "\\sigma_\\text{avg} = \\tfrac{1}{2}(\\sigma_x+ \\sigma_y)", "1e00f5ee4127339bd403b4427c22ec0f": "\\textstyle \\lambda_{\\diamond N}=\\lambda(a_\\diamond\\mid A^c)", "1e0150a7acdbc56d21b6d3543b44185e": "[0, T]", "1e017ccc1bb70de3e10e0d384224f4d7": " k(x,y) \\geq 0 \\,\\,\\forall x,y", "1e01c0c5257f32a264d1ce400440c13d": " || \\cdot ||_\\mathcal{H} ", "1e020145714349ed65e0141f34fb0ff6": "\\lambda \\ \\stackrel{\\text{def}}{=}\\ \\frac {\\text{stress}} {\\text{strain}}", "1e020b4cef70e46efcfa7eca6e284ce5": "\nv v_i \\delta_{i\\ell} = \\operatorname{trace}D_i D_\\ell = \\sum_{k=0}^n \\mu_i p_i(k) p_\\ell (k), \\quad(13)\n", "1e0229daa8b36edd93b7441685c7a21f": "\\frac{d}{dt} \\left .{\\!\\!\\frac{}{}}\\right|_{t=t_1} (F^*_{t,t_0} \\eta_t)_p = \\left( F^*_{t_1,t_0} \\left( \\mathcal{L}_{X_{t_1}}\\eta_{t_1} + \\frac{d}{dt} \\left .{\\!\\!\\frac{}{}}\\right|_{t=t_1} \\eta_t \\right) \\right)_p", "1e0243c22cf1cea7557e901cc1a648a6": " L=\\frac{d^2}{dx^2}+2\\frac{d}{dx}-3", "1e024eeb367ce692449ac013dcfb4376": " {\\rm {}}_{}^{2}\\Pi_{\\rm g}", "1e02be62dd84e9d143fb56aaf17cff93": "\\sqrt{\\frac{1}{2}}\\!\\,", "1e02c940a2ddb996551d42f499f4a217": "\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow {}_{\\ 90}^{233} \\mathrm{Th} \\xrightarrow{\\beta^-} {}_{\\ 91}^{233}\\mathrm{Pa} \\xrightarrow{\\beta^-} {}_{\\ 92}^{233}\\mathrm{U}+\\mathrm{n}\\rightarrow {}_{\\ 92}^{232} \\mathrm{U}+2\\mathrm{n}", "1e02d3c90ea320fab39d11231b0ae156": " \\frac{\\partial \\rho u_i}{dt} + \\frac{\\partial \\rho u_i u_j}{\\partial x_j} = - \\frac{\\partial P}{\\partial x_i} + \\frac{\\partial \\tau_{ij}}{\\partial x_j} + S_i, ", "1e03062508b649616d73a0ecc6e8455c": "g = \\lambda x.f(x,x_0)", "1e0393e4c92649f72485c1d972890876": " \\sum a_n^2 ", "1e03a25b2cd0c2bb73c7bde586f6c7c9": "\\bigoplus M_i", "1e04093b501bd763008d3a2af51d768e": "\\Psi_{\\sigma}(t)=c_{\\sigma}\\pi^{-\\frac{1}{4}}e^{-\\frac{1}{2}t^{2}}(e^{i\\sigma t}-\\kappa_{\\sigma})", "1e04185298ca24bf4ee6fb138346d74d": "5x^2+8xy+5y^2=\\mathbf{x}^TA\\mathbf{x}= (S^T\\mathbf{x})^TD(S^T\\mathbf{x})=1\\left(\\frac{x-y}{\\sqrt{2}}\\right)^2+9\\left(\\frac{x+y}{\\sqrt{2}}\\right)^2.", "1e042185015b14c1aeebe79b35ede84b": "\\tau = (T * Q) \\div (w * I)", "1e04d386db9d622844a019c53f0f74d1": "p =", "1e04f2bdeaf9585249f17f55593413e1": "\\sum_{n=-\\infty}^{\\infty} x_1[n]x^*_2[n] \\quad = \\quad \\frac{1}{j2\\pi}\\oint_C X_1(v)X^*_2(\\tfrac{1}{v^*})v^{-1}\\mathrm{d}v", "1e059d412a4c83c6a304e8279bd045bd": "\\varphi=2\\arctan t, \\, ", "1e05ae2024bca910ca19d06ed9d6dd70": "= e_\\gamma^I (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_I", "1e06d30fd85493b647f5c0993d7c215e": "\\beta^G_{\\mu\\nu}=\\beta^\\Phi=0,", "1e06f1b5b8b527c142781dcd56a0246f": "\\int x^2\\arccos(a\\,x)\\,dx=\n \\frac{x^3\\arccos(a\\,x)}{3}-\n \\frac{\\left(a^2\\,x^2+2\\right)\\sqrt{1-a^2\\,x^2}}{9\\,a^3}+C", "1e07361523c6c42f2d4e89854088a5b1": "\\left|\\frac{BD}{DC}\\right| = \\left|\\frac{BF}{CK}\\right|,\\,\\left|\\frac{AE}{EC}\\right| = \\left|\\frac{AF}{CK}\\right|", "1e0739add05675a5f8b1eb0ed4648284": "\\sum_{m}\\Lambda_{m} =I.\n", "1e07454f3a79a2e1babcac294b38d943": "n_{i1}", "1e0759765959983006db68634ebe6f1b": " \\cot^2 \\theta = \\csc^2 \\theta - 1\\!", "1e0785f6e45e681b6994e5d2dedf646c": "h([f])([x]) = f(x).", "1e07b95d54ae61ef99004a92f8fe1ee0": " F^{\\,ab}{}_{;b} = {4\\pi \\over c }\\,J^{\\,a}", "1e07f918931f07d8bccbb8fa8d1aed5c": "S^{-1} E \\to QC", "1e07ff3e7fbfaa7a9cce21bc9bd4911a": "\\lim_{k \\to \\infty}\\tilde{A}^k=0.", "1e0810557b7fb9379a2d2da216988d9c": "\\mathit{A}", "1e0829840a8ebcea8ab634477c64baed": "\\begin{align}\n \\operatorname{erfc}(x) & = 1-\\operatorname{erf}(x) \\\\\n & = \\frac{2}{\\sqrt{\\pi}} \\int_x^{\\infty} e^{-t^2}\\,dt.\n \\end{align}", "1e083fdd1771fc156b9671e5e51827ba": "\\displaystyle f(x - a)\\,", "1e08448969430624ea5d6774d8634cc8": " F = \\sqrt{1- \\alpha} ", "1e085d0da474ac3a6750d1dfe8793853": "\\mathcal{C} = \\sum_{n \\ge 1} \\sum_{G\\in \\operatorname{Cl}(S_n)} c_G (X^n/G)", "1e0878d8784d3ac9b439c9a49e759a90": "x^{\\prime}=\\gamma x^{*},\\quad y^{\\prime}=y,\\quad z^{\\prime}=z,\\quad t^{\\prime}=\\frac{t}{\\gamma}-\\gamma x^{*}\\frac{v}{c^{2}}", "1e08cd27033ec496b4afbaeb61a5ba21": "t=\\frac{4 k_0k_1 e^{-i a(k_0-k_1)}}{(k_0+k_1)^2-e^{2ia k_1}(k_0-k_1)^2}", "1e08d45ded57c82199afb08ee697d96b": "\\scriptstyle X \\;=\\; F \\,\\otimes\\, I", "1e090c855460d203f6f724c183b93a14": "{d \\over dt}\\left\\{ X \\right\\} = \\left\\{A \\right\\} + \\left\\{ X \\right\\}^2 \\left\\{Y \\right\\} - \\left\\{B \\right\\} \\left\\{X \\right\\} - \\left\\{X \\right\\} \\,", "1e091611d92c98ad709051b6026a5382": "f(tx) = f(tx+(1-t)\\cdot 0) \\le t f(x)+(1-t)f(0) \\le t f(x)", "1e0936f3c664fdefd77bca2f8160c845": "\\lambda(\\rho, \\theta)", "1e093896a811989d6495e6dceca6b695": " \\eta : k \\to K[G] ~\\text{by}~ \\eta (1) = \\sum_{X \\in G_0} \\mathrm{id}_X ", "1e094a7e906873267a27b015aa06412e": "K_a = \\frac{[Ab-Ag]}{[Ab][Ag]} ", "1e09caee46b6e9e3d6bcda917808afe7": "\n\\zeta = [a_0;a_1,a_2,\\dots,a_k,\\overline{a_{k+1},a_{k+2},\\dots,a_{k+m}}],\\,\n", "1e09d5e72498868ecd01e65f2b13dbf1": " P_\\gamma(u) = \\gamma \\| \\nabla u \\|_0 + \\| u-f\\|_p^p = \\gamma \\# \\{ i : u_i \\neq u_{i+1} \\} + \\sum_{i=1}^n |u_i - f_i|^p", "1e0a4fdd7c0d0022dfe3fa292da0e43e": "\\ k>n", "1e0a66fa5070650229fb4133311a2a27": "1\\!-\\!\\sigma te^{-\\sigma^2t^2/2}\\sqrt{\\frac{\\pi}{2}}\\!\\left(\\textrm{erfi}\\!\\left(\\frac{\\sigma t}{\\sqrt{2}}\\right)\\!-\\!i\\right)", "1e0ae887a8bc1a75eac1e908db71148e": "\\sigma^{\\mu\\nu} = \\frac{i}{2} [\\gamma^\\mu, \\gamma^\\nu].", "1e0bce26efa9df31ee455c350132b69f": "t\\,J_\\nu(ut)", "1e0c472dce5fac8838a1ec2a8339fd8f": " C^m_\\ell", "1e0c515fbbf48571172dd9db1ae76adb": "\\frac{R^2}{4}\\!", "1e0c82468a72255a243355e65210b715": "|f|=1,e^{\\pm 1},e^{\\pm 2},\\ldots", "1e0cdfa223e8363cf8d29d60bba85686": "Y_{9}^{0}(\\theta,\\varphi)={1\\over 256}\\sqrt{19\\over \\pi}\\cdot(12155\\cos^{9}\\theta-25740\\cos^{7}\\theta+18018\\cos^{5}\\theta-4620\\cos^{3}\\theta+315\\cos\\theta)", "1e0d350de3eb8486a13405f1c6a9e2fe": "V_n\\,", "1e0d4bae6b43d66468848fb1b84199fb": " |\\psi_{+}\\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}e^{i\\phi}\\\\1\\end{pmatrix}= \\frac{1}{\\sqrt{2}} (e^{i\\phi}| \\psi_{1}\\rangle +|\\psi_{2}\\rangle) ", "1e0d7f9f483b334c9338bbfa826c6902": "\\lambda=\\int\\limits_S\\, \\vec{B} \\cdot dS", "1e0d98dda5b53d7513c3c04358354a11": "K_c=\\frac{k_+}{k_-}=\\frac{\\{S\\}^\\sigma \\{T\\}^\\tau } {\\{A\\}^\\alpha \\{B\\}^\\beta}", "1e0db62a6c407ed0a9441ed95214d962": "\\zeta = \\sinh \\mu", "1e0df5844efe59994ca77852a7fbe497": "s=-2", "1e0e8cb1678ea775e999fdbd18401dbf": "\\displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx} +2(u^2)_{xt}=0", "1e0ea187e1aea82b76bcca184cb26a68": "a(a+c)=b^2", "1e0eb18249c0be2e7a41d28444a93fdc": " \\frac{1}{\\rho}\\frac{dP}{dr} = -\\frac{Gm}{r^2} ", "1e0f064f1e21869a987dcab0d49af0b5": " \\lim_{\\gamma\\rightarrow 0} \\ln(Y) = \\ln(A) + \\alpha \\ln(K) + (1-\\alpha) \\ln(L).", "1e0f0cb2f38147b79cb6a2f0fbcec918": "f\\colon M \\to \\mathbf{C},", "1e0f6976c889a7e664bbd6a1b016495e": "\\tfrac{12}{5}", "1e0f89617fa4cf0b9e3f39bec7a5b120": "(x+\\sqrt{p^*}) = (x+\\sqrt{p^*})(x-\\sqrt{p^*},q) = (cq, q(x+\\sqrt{p^*}))", "1e105ca1edf10a02f877a447a1eb4657": "N(h)", "1e10c82d417500bb233ebcbff5bf8b48": "b_V\\colon V \\times V \\to K.", "1e10d39f8f01bc372e71f42b8679a619": "\\frac{1}{L_\\mathrm{net}} = \\sum_i \\frac{1}{L_i}\\,\\!", "1e10f7272d7b7efeefc06d7af864565f": "\\{ Y_{ij}: i=1,\\dots,n; j=1,\\dots,n\\}", "1e10f8d6287a173e44f5c08666d326c7": "b^k\\,", "1e11170bdeedd8d7c10a8c33d09805f0": "\\hat{B} \\rightarrow \\langle \\hat{B} \\rangle", "1e113a9765e3544edb5801e4ef809d35": "\\left \\langle \\chi_i, \\chi_j \\right \\rangle = \\begin{cases} 0 & \\mbox{ if } i \\ne j, \\\\ 1 & \\mbox{ if } i = j. \\end{cases}", "1e11406e6bec7febca1885f80ff4150f": " 6 \\frac{2}{3} ", "1e11537148f70bfcd4ae23170eee4bdb": "\\scriptstyle t \\,\\in\\, \\mathbb{R}^1", "1e11b90628e8702455229e8f1ec5a4ea": "J_n = -\\frac{\\cos{ax}}{(n-1)x^{n-1}}-\\frac{a}{n-1}I_{n-1}\\,\\!", "1e1261b0a987624ef3c639af9223629f": "=\\int_{P(t_1,t_2)} C^{(V)}_T(V,T)\\, \\dot V(t)\\, dt\\,+\\,\\int_{P(t_1,t_2)}C^{(T)}_V(V,T)\\,\\dot T(t)\\,dt ", "1e12695db456351073ba964c90ec65b8": "\\mathfrak{m} := \\mathfrak{n} \\cap R", "1e12697a9253051abbab0bd59341b281": "\\frac{d\\eta}{dT_H}(T_\\mathrm{opt}) = 0 ", "1e126d744d10e8cda59bbad027aa1d76": "\\!b", "1e127d1d7caf74f19411c3ad6c748447": "x=\\frac{m}{\\ell} \\cosh{t}; \\quad y = m \\sinh t ; \\quad -\\pi {mv^2\\over r}.", "1e1e0321b4da4f26bccc41b71c61cb26": "\\begin{array}{ll}\n& (\\mathbf{a} \\times \\mathbf{b})_i = {\\mathbf{e}_i \\cdot \\mathbf{a} \\times \\mathbf{b}} = \\varepsilon_{\\ell jk} {(\\mathbf{e}_i)}_\\ell a_j b_k = \\varepsilon_{\\ell jk} \\delta_{i \\ell} a_j b_k = \\varepsilon_{ijk} a_j b_k \\\\\n\\Rightarrow & {\\mathbf{a} \\times \\mathbf{b}} = (\\mathbf{a} \\times \\mathbf{b})_i \\mathbf{e}_i = \\varepsilon_{ijk} a_j b_k \\mathbf{e}_i \n\\end{array}", "1e1e5eccf4a5ea7c3fd1741bdc2a7623": "R_{\\text{E}}", "1e1e62245868f79465d0394d369e4f48": "u(c)=ln(c)", "1e1e69b2b3974ba174f3736736e15421": " S_0 = V_0 U_0 + V_1 U_1 ", "1e1e801aeb11bb244603a70750d4334b": "\n\\frac{\\partial y}{\\partial \\mathbf{x}} =\n\\left[\n\\frac{\\partial y}{\\partial x_1}\n\\frac{\\partial y}{\\partial x_2}\n\\cdots\n\\frac{\\partial y}{\\partial x_n}\n\\right].\n", "1e1e8d5205f8b4bd6bb74bc5b114b14b": "\\ddot x \\to 0", "1e1f05888ba72db97143e254de930780": " A_{ \\rho} = G \\left( R_{C2} // R_{L} \\right) \\ . ", "1e1faa105a807c9426dc22241e9c8569": "(\\bar{3},1)_{\\frac{1}{3}}", "1e1fcaab26b776d30b942ba55db4022b": " h\\nu = 2\\mu_n B \\pm 2d_n E ", "1e1fdaf362331e5dde393bbcf0e588d9": "u_1, u_2, \\dots \\in H^1(\\Omega)", "1e207f198eaa72d1d2a243f8a6dc13dc": "-\\boldsymbol{e}_k\\, a\\, \\frac{\\cosh\\, \\bigl( k\\, (z+h) \\bigr)}{\\sinh\\, (k\\, h)}\\, \\sin\\, \\theta\\,", "1e20a1265e3ff10d80ec18008aadb1ad": "\\frac{R_r^{'}}{s}=\\frac{R_r^{'}(1-s)}{s}+R_r^{'}", "1e20b86c6079c96584c82cb26dacce30": "\\log (\\operatorname{E}(Y|\\mathbf{x}))=\\boldsymbol{\\theta}' \\mathbf{x},\\,", "1e20c98e9a40aec4e34c003313b50d56": "(B_t)_{t\\geq 0}", "1e212f2bd617078f8ece73468e1996b7": "I = F t\\;", "1e215061a23606dce6d47f3eddaf53ac": "f(n,1) = 1", "1e21c820c3d405d35f77578e3a764ae8": " M \\ge 2", "1e22436e93648bf46b139758d94363a9": "\\hat{\\nu} = \\hat{\\alpha} + \\hat{\\beta} = 3\\frac{(\\text{sample excess kurtosis}) - (\\text{sample skewness})^2+2}{\\frac{3}{2} (\\text{sample skewness})^2 - \\text{(sample excess kurtosis)}}\\text{ if (sample skewness)}^2-2< \\text{sample excess kurtosis}< \\tfrac{3}{2} (\\text{sample skewness})^2", "1e22c0a29a0f1efe1152ef209a85f29c": "\\mathbf{r}=(r_1, r_2, \\ldots, r_d)", "1e22cf3958a1e11bd4b597e8b9276584": "y=R2(x)", "1e22e102462d26e31173e264726a8b4b": "(S_1' \\cup S_2')'' = (S_1 '' \\cap S_2 '')' = (S_1 \\cap S_2)' .", "1e22f5cb8e9ca7f4a0cfb2b12c591e59": "\\mathbf{r}_\\bot = \\mathbf{r} - \\mathbf{r}_\\parallel", "1e235263c4f9532c6ff84e5d62b1fd78": "\\emptyset \\neq F \\neq \\mathbf{P}^{(1)}", "1e2378037e79a15246de537d17f45e09": "\\ m_i = a_i M_i ", "1e239f0f11b7ecdc27e614c12cbe461b": "\\log(k) - \\log(k_0) = N^+", "1e23b655e511a3419dba719163cf9155": "\\text{GC}", "1e240e97012e638df064e1ec777733d4": "\\frac{\\zeta(2s)}{\\zeta(s)} = \\sum_{n=1}^\\infty \\frac{\\lambda(n)}{n^s}.", "1e2438a01cd72fa352a6a2220699a7be": "2+\\epsilon", "1e2440eaeae55cbc52c42c6b9bdb8ee5": "a \\ge -1", "1e24756c0182c0072b66095ff4016b79": "e^- + p \\to \\nu_e + n", "1e249431948c84852ec2bf57c9cfc7db": "\nVP(\\alpha(t), \\omega(t)) + \\sum_{i=1}^K Q_i(t)Y_i(\\alpha(t), \\omega(t)) \\leq C + \\inf_{\\alpha\\in A} [VP(\\alpha, \\omega(t)) + \\sum_{i=1}^KQ_i(t)Y_i(\\alpha,\\omega(t))]\n", "1e24ae3e82e399abf54b0db66ea780f3": "\\frac{\\mu_k}{\\sigma^k}", "1e24b57a60fa0c2ee1ec42a041e0b459": " L (\\mathrm {dB}) = -20 \\log | \\tau | \\ ", "1e24d7cbc803f83656cb43dfc048cbf0": "C^* = {d S^* \\over d \\ln T}", "1e24ecdf242587ba77f74f0013b80aeb": " V_{(xu)} \\ \\stackrel{\\mathrm{def}}{=}\\ \\rho^{i}(x,u) \\frac{\\partial}{\\partial x^{i}} + \\phi^{\\alpha}(x,u) \\frac{\\partial}{\\partial u^{\\alpha}}\\,", "1e24f050f6b5031f08488b37a6747ca4": "A = L_x L_y", "1e25927467b6edc075e12ab885b7ca14": "M = T \\oplus P", "1e2598c2436ff79f9b7eda55cb72d6da": "\\scriptstyle(A_1, A_2, \\dots, A_n)", "1e25a28bcc613e69aa14c846f5e20967": "b_\\mu", "1e25b19c863180be3e08a84947d8be4b": "\n y = X\\beta + \\varepsilon, \\,\n ", "1e25c93c2e525e8f77d124c13b5ee4a2": "n=\\underbrace{1+1+\\cdots+1}_{n\\ \\rm times}", "1e25df665765ebc102d573435b6954a2": "\\alpha\\in S_0", "1e262a89a8c9a8c0972fc856219ad291": "2^{n^2}", "1e2760f04e9da338f7c8016af27e2132": " 0 \\le j \\le n ", "1e27b8e81380cae5d9d3f663129d7247": "\\int_a^b f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i).", "1e27d9f62821ae2dc7c972ed836d8a54": "\\gamma_{P}(Q) = 6/10", "1e27e8280ef2841621c6d02b6bc2d30e": "u_{n\\mathbf{k}}(\\mathbf{r})=e^{-i\\mathbf{k}\\cdot\\mathbf{r}}\\psi_{n\\mathbf{k}}(\\mathbf{r})", "1e27f8dff6b94cf7501a78d279c810e0": "f(\\mathbf{r})", "1e27fa611b45d34533cc3f3640e5ff86": " (1+x)^n = \\sum_{r=0}^\\infty {n \\choose r} x^r. \\qquad ", "1e289ef3e74d4da0bf3522a08e3dd560": "Qg", "1e28bf2618ad0b447ac840c5b2a760ec": "X=(0, 2)", "1e28eb7fdb064b732eedfb52ed545db1": " \\delta_{1} ", "1e28f5ccd0a57b77be5be2345d3a4404": "H_2(S(2k+1,n)) \\approx H_2(S(1,n))", "1e294daf7eb32c0569c072418b7ec80c": "{\\mathcal L}_{xy}^4", "1e297312635c9b63a628670bad30ff7d": "\\frac{1 + {\\scriptstyle\\frac{2}{3}}z + {\\scriptstyle\\frac{1}{6}}z^2}\n{1 - {\\scriptstyle\\frac{1}{3}}z}", "1e29a4c9c1cafa1e76cf817422645ae7": " \\ v_{1} = u_{1}", "1e29c0f2be0df7aa866c998b11f15078": "a= -2c, a\\in \\mathbb{R}", "1e29e391c0913065b89ad63d8374d4f1": "\\scriptstyle E_K^T", "1e29f2abcb5c7dd05f5f58e3563205eb": "\\sum I(X_{i};X_{j(i)})", "1e2a5aff951b17199d1b84fcb15911de": "\\psi_1(\\Omega) = \\varepsilon_{\\Omega 2}", "1e2a8d3722ec9ec0bfba6308421eaadf": "\nA_{j, k} := a_{j, k}-\\overline{a}_{j.}-\\overline{a}_{.k} + \\overline{a}_{..}, \\qquad\nB_{j, k} := b_{j, k} - \\overline{b}_{j.} -\\overline{b}_{.k} + \\overline{b}_{..},\n", "1e2aae8ca2d5a3610033c709151ad943": "0<\\lambda_1<\\lambda_2\\leq\\lambda_3\\cdots\\leq\\lambda_k\\leq\\cdots", "1e2b2cce6a896b3f6f54a7f2051b3de4": "\\begin{align} \\tilde G(A,B)\\cap \\tilde G(B,A)=\\emptyset\n\\end{align}", "1e2b810ac17403f923606c61b7b33345": "\\displaystyle \\alpha_0:= \\arg \\min_\\alpha f(x_0+\\alpha \\Delta x_0)", "1e2bf581085e56cb474e467c368bc5b0": "u_5 = \\tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+ax_5^2+x_6^2+x_7^2+x_8^2)x_{13} - 2x_5(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +bx_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "1e2c08643b32f74eed20a4ab0f43eb5d": "\\Lambda^0\\stackrel{d_0}{\\longrightarrow}\\Lambda^1\\stackrel{d_1}{\\longrightarrow}\\cdots\\stackrel{d_{n-1}}{\\longrightarrow}\\Lambda^n", "1e2c45d1e61b80301668f46997ed6108": " \\frac{\\xi(s)}{\\xi(0)}=\\frac{\\det(H+s(s-1)+1/4)}{\\det(H+1/4)}.", "1e2c4f3382d0e45a90e50c4470b05569": " \\Psi(x,t+\\tau)=\\int G(x,x',\\tau) \\Psi(x',t) dx' ", "1e2c648a07414987f099fad84449e0c5": " P = D^{-1}A ", "1e2c7bbfbf004c872ff4dac95436e45e": "F(t) = t^3+t^2+t+1.", "1e2c8e3cb41f7c53eb8acdf16a79a5b4": " \\sigma(\\mathfrak{G}^2 \\oplus \\mathfrak{G}^2 \\oplus \\mathfrak{G}^2) = 5/6", "1e2cbb00091ac276f03054a02f3c5096": "P_1,\\dots,P_q", "1e2cbfe20d37e7a02b0b7b3111920d9e": "\n\\mathbf{HM} =\n\\begin{bmatrix}\nx^1_1 & \\cdots & x^1_n & | & f(\\mathbf{x}^1)\\\\\n\\vdots & \\ddots & \\vdots & | & \\vdots\\\\\nx^{hms}_1 & \\cdots & x^{hms}_n & | & f(\\mathbf{x}^{hms})\\\\\n\\end{bmatrix}.\n", "1e2cedcbd7996f95bce9efb56a675f5b": "\nV = \\begin{pmatrix} \nv_{1,1} & \\cdots & v_{1,n} \\\\ \\vdots & \\ddots & \\vdots \\\\ v_{j,1} & \\cdots & v_{j,n}\n\\end{pmatrix}\n", "1e2d2a07db97dfd80a4a6ea7d9d7f3dd": "S^{1}:=\\{e^{i\\theta}:\\theta\\in[0,2\\pi]\\}=\\{z\\in \\mathbb{C}:|z|=1\\}", "1e2d487248a85db8776b1bd299804741": "\\dot n_k", "1e2d4d3dc451d79c29471f15ae12231e": " \\frac{4}{3}\\pi r^3.", "1e2d529e5f026dbb066eaf336b996cc1": "{x_1, x_2, \\ldots, x_m}", "1e2d6bbe28c18249f5e8f73f6d871709": "\n\\mathcal M=i\\sqrt{\\frac{2\\omega_p}{Z}}\n\\int \\mathrm{d}(x^0)\\part_0\n\\int \\mathrm{d}^3x f_p(x)\\overleftrightarrow\\part_0\n\\eta(x)\n", "1e2d97211b830e72db51d66a5fff5028": " |CK| = R\\sin u = r\\sin\\theta. \\, ", "1e2d99fe6e049f27fc2f36efe005c72b": "\\Gamma^2_{ik} = \\begin{bmatrix}\n0 & 0 & 1/r & 0\\\\\n0 & 0 & \\cot\\theta & 0\\\\\n1/r & \\cot\\theta & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\end{bmatrix}", "1e2db417da24999c108a7b1298feac0c": "E^{(+)}=\\mathcal{E}_1exp(-i\\nu_1t)+\\mathcal{E}_2exp(-i\\nu_2t)", "1e2dd21ff5726cdb39fe594204144f6d": "d\\ell^*(t)=|f\\,'(\\gamma(t))|\\,d\\ell(t)", "1e2e8ac2df3afd1c70b4671ee45973ef": "y_0 = x_0 + x_1 \\omega^k \\, ", "1e2e9beb95534b1b064b66b6f12a53f5": "d(h)", "1e2ead32b82cf738fd51e053419d7b5f": "H_{n}(x) = \n2^n\\,U\\left(-\\frac{n}{2},\\frac{1}{2},x^2\\right)", "1e2ebe6ec2e5271bed3cd936a70753cc": "\\dot{d}(t) = H d(t)", "1e2ed9c70078369aa32adf7258d91d59": "\\bigl( \\begin{smallmatrix}\\\\ ~\\;0&4\\\\ -1&5\\end{smallmatrix} \\bigr)", "1e2f22562bc5e1e31c398ba8dec69ea4": " I_{12} = \\frac{V_{12}}{|Z_\\Delta|} \\angle (30^\\circ-\\theta) ", "1e2fae9a8c75fcdfcc9778a3de1e5b51": "\\ggg", "1e2fbbc6d76a75e54f2f7124366b6e1d": "\\textstyle\\phi\\,\\!", "1e2fd236b0f1f6afe242c308a4f20107": "2 - 2 g - n", "1e2ff085202e0f73bc3a9c1b217ceefe": "\\left[{\\begin{matrix}\nT^{(\\mathbf n)}_1 & T^{(\\mathbf n)}_2 & T^{(\\mathbf n)}_3\\end{matrix}}\\right]=\\left[{\\begin{matrix}\nn_1 & n_2 & n_3\n\\end{matrix}}\\right]\\cdot\n\\left[{\\begin{matrix}\n\\sigma _{11} & \\sigma _{12} & \\sigma _{13} \\\\\n\\sigma _{21} & \\sigma _{22} & \\sigma _{23} \\\\\n\\sigma _{31} & \\sigma _{32} & \\sigma _{33} \\\\\n\\end{matrix}}\\right].", "1e304c87731dfd154242c607c033a040": "c\\,\\! = 2.9979 \\times 10^{8}", "1e30583a668faf2830dde67886eae589": "p \\le \\sqrt{N}", "1e30e4cb4f5a8339aced72cd68e7dab0": " \\left(a y_1 \\right) v'' + \\left( 2 a y_1' + b y_1 \\right) v' + \\left( a y_1'' + b y_1' + c y_1 \\right) v = 0.", "1e310372187f4d8a364537ce9f3dea8e": "\\sum_{k=1}^{\\infty} (-1)^k k^{1/k}.", "1e311d2af722d32aabe125cc7d4d09a0": "\\begin{align}\n\\overline{B}_0 \\pi_0 &= \\pi_0\\\\\n \\quad \\left(\\mathbf e^{\\text{T}} + \\mathbf e^{\\text{T}}\\left(I - \\sum_{i=1}^\\infty \\overline{A}_i\\right)^{-1}\\sum_{i=1}^\\infty \\overline{B}_i\\right) \\pi_0 &= 1\n\\end{align}", "1e3123ca721afdd8faf375e578def3d5": "\\frac{dY}{dt} = \\gamma . \\beta . X . Y - \\delta . Y", "1e313c8fb893ceb573a970f301c28a5d": "\\Delta_0>0", "1e31517a3fadc96248187953035b8d21": "\\mathbf{E}(\\mathbf{x},t)=-\\mathbf{\\nabla}\\phi(\\mathbf{x},t)-\\frac{\\partial\\mathbf{A}(\\mathbf{x},t)}{\\partial t}", "1e315926814501ae561ac55649ca2c2b": "B^\\alpha_\\alpha", "1e3175ad60733025780ecf44a00c3c12": " g(n)+h(n) ", "1e320272dee87c54a0d0c11f37f010c4": "(x_i, y_i)", "1e32120bc32641948a6e1f23bce22a57": "\\,r_\\max - a=a-r_\\min", "1e322e66675741f9385158d461c51179": "\\{1, 2, \\cdots, n\\}", "1e32803809b74ee00277d188d53f31b5": "\nV_{L} = V_{\\{U,A\\}}+DT=E_{U}+DT = E_{L|T=0}+D+DT = E_{L|T>0}+D\n", "1e3283825ac3b269511bcebc810ed162": "E=\\sum_{n=1}^{\\infty}\\frac{1}{2^n-1}", "1e329fd14bfc1dbca6de9d44ef95c8fe": "n(n-1)", "1e32c08b5ab608d74bda7449ae33c40b": " 1 \\to \\Gamma \\to G \\to H \\to 1.", "1e32c7c40639e530a915f8a019184464": "\\bar{W} = \\int_{A}^{B}\\mathbf{F}\\cdot d(\\mathbf{r}+\\epsilon \\mathbf{h})=\\int_{t_0}^{t_1}\\mathbf{F}\\cdot (\\mathbf{v}+\\epsilon \\dot{\\mathbf{h}})dt .", "1e32c88788e71ab442caebe09b5209dd": "c\\|u-u_n\\|^2 \\le a(u-u_n, u-u_n) = a(u-u_n, u-v_n) \\le C \\|u-u_n\\| \\, \\|u-v_n\\|.", "1e33fc59e44f93c771bd7e3571abf63a": "\\frac{T_2}{T_1} ={\\left(\\frac{V_1}{V_2}\\right)^{\\gamma-1}} = r^{\\gamma-1}", "1e34158aea636fcba832e750ea990bd8": "x=z^{\\lambda}", "1e34da68df8b8d139f8715b6adb9210a": "\\frac{\\operatorname{Li}_s(e^{it})}{\\zeta(s)}", "1e34f0b9e24019efa08dcc2aa64d62dd": "Q_{h}\\,\\!", "1e34f6dd45bcfc2c9ca0c6cc148f6e3a": "1,2,3,\\ldots,100\\,.", "1e352d77c0676af89b75de1fe497afd2": "\\zeta_p", "1e35427d0615a090fd1922a361bedaeb": " \\frac{1}{\\lambda} =\n \\frac{E_\\text{i} - E_\\text{f}}{12398.4\\,{\\rm \\AA}\\,\\text{eV}} =\n \\left(\\frac{12398.4}{13.6}\\,{\\rm \\AA}\\right)^{-1} \\left(\\frac{1}{m^2} - \\frac{1}{n^2} \\right) =\n R_\\text{H} \\left(\\frac{1}{m^2} - \\frac{1}{n^2} \\right) ", "1e355379e5cabc37440ddeaee6cf96f3": "f\\cdot \\delta_\\xi = f(\\xi)\\cdot \\delta_\\xi", "1e358be5d3c4865d739eb5e215fe732a": " f(x; \\sigma) = \\frac{1}{2\\pi\\sigma^2} \\int_0^{2\\pi} \\, d\\phi \\int_0^\\infty dr \\, \\delta(r-x) r e^{-r^2/2\\sigma^2}= \\frac{x}{\\sigma^2} e^{-x^2/2\\sigma^2},\n", "1e35af1dd7be0ef842b1465e0d2e18ca": " \\langle X, Y \\rangle = \\int_\\mathcal{T} X(t) Y(t) dt ", "1e35d4b37940830eafff801767ea7878": "\\beta \\left( f \\right)", "1e366ca512b63b7ae4b43a53a09e7ca9": "2R_1 = \\frac{2R_bR_c}{R_T}", "1e3685905926003c67063d929b728f8f": "\\Delta G = R T\\ln{{K_{\\rm d} \\over c^{\\ominus}}}", "1e368b367b562e2b24baf9e7e472fef9": "K_{\\mathrm{max}} = h \\left(f - f_0\\right).", "1e3744b2efc3fa7fd0c96e919c64f59b": "\\displaystyle J_{n-1}", "1e37a3de24daabb18be66a06504edb8d": "k = 0,\\ldots,N/4-1", "1e37d2243ffa0c3e07b9ec3a15de1b3c": "H^{-1} = -H", "1e380e2e57747292097b7c4b1a500a4f": "\\psi(x) = \\sum_{p^n \\le x} \\ln p = \\sum_{n=1}^\\infty \\theta(x^{1/n}) = \\sum_{n\\le x}\\Lambda(n).", "1e381e17eab6ada0fb074d04edba421c": "\\bar t_{i} ", "1e382f269c34a8b0815ff928991765fa": "\nH_{\\text{kin}} = \\tfrac12 m |\\mathbf{v}|^2 = \\tfrac{1}{2} m\\left( v_x^2 + v_y^2 + v_z^2 \\right),\n", "1e38b469b80cfc6dbc4e93eeef8b141e": "\\rho_{SB} = \\rho_{S}\\otimes\\rho_{B}", "1e38bf1e82fb4056e643090ae7c2616d": "\\begin{align}\n \\sigma = \\pi/E\\lambda\n \\end{align}", "1e38c7344e689cd8a9f036f723587855": "\\textstyle \\mathcal Z = e^{-\\Omega/(k T)}", "1e39067b4f7fe5e485ff761c957f6398": " H_{j \\pm 1, j}^{ } \\ne 0 ", "1e390ae7b04ced5cf31a0388edabf6e1": " \\frac{dI}{dt} = \\beta I S - \\nu I ", "1e392c8c9b7fc3bef75bfa231d7f057d": "\\{\\langle A \\rangle\\}", "1e3963276740f08d71278f0cdfca2e96": "\\bar{x} = E\\{x\\}", "1e3a8381a429dfcc8ec660242f159644": " \\left(\\frac{m_2 (t_2)^{0.5}}{p_2}\\right)", "1e3ab99ffdbff3af3048bed72735b709": "\\mathbf{\\hat{\\Omega}}", "1e3b56bc6834a8007fa28f870a3bfe4b": "\\det (I + tA) = 1\\!\\,", "1e3b7caedf11e130b16ffebe3415189b": "P(\\imath \\mu)", "1e3b960faf537820a0d297654df8bfae": "%K = 100 *(C-L5)/(H5-L5)", "1e3bd8cbcbcc9dc95145f2d5934819e6": "(R \\bowtie S) \\cup (\\{(\\omega, \\dots, \\omega)\\} \\times (S - \\pi_{s_1, s_2, \\dots, s_n}(R \\bowtie S)))", "1e3be70ea009f7621f47bf2c2ffa0242": " P(x)=(x^2+ux+v)\\left(\\sum_{i=0}^{n-2} b_i x^i\\right) + (cx+d). ", "1e3bfe87e8f1e4d9bc363f5b21336572": "|z| > |a|", "1e3c62d96bafbcbefcdcf2e76a6d021b": "\n \\frac{\\partial x^i}{\\partial X^\\alpha} = F^i_{~\\alpha} \n", "1e3c830c91f80cde3f1c3ba50bfa1f77": "\n\\mathbf{F}_{jk} = -\\nabla_{\\mathbf{r}_k} V = \n- \\frac{dV}{dr} \\left( \\frac{\\mathbf{r}_k - \\mathbf{r}_j}{r_{jk}} \\right),\n", "1e3c91b3ec49933f2bfc4844769c0e72": "\n\\mathbf{\\nabla =\\partial /\\partial r}\n", "1e3cdb7374d2bebeef8e35d478809387": "a_{ij} = g_i \\ \\log (\\mathrm{tf}_{ij} + 1)", "1e3ce89d36b72bce86120bef0f2a34ee": "{Y_i = \\beta_0 + \\sum_{j=1}^p {\\beta_j X_{i,j}} + \\varepsilon_i},", "1e3d22673f29dd81ee82c4d391c306c1": " \\lim_{x \\to A_x}y(x) = C = \\frac{1}{2}\\ A_x\\left( \\frac{y'(0)+\\sqrt{{y'(0)}^2+1} }{1-\\frac{V_t}{V_d}} - \\frac{1}{ (y'(0)+\\sqrt{{y'(0)}^2+1})\\ (1 + \\frac{V_t}{V_d}) } \\right) ", "1e3d68248046f7494b085aaad411de2d": "\\sum_{k=0}^\\infty (-1)^k k! = \\sum_{k=0}^\\infty (-1)^k \\int_0^\\infty x^k \\exp(-x) \\, dx", "1e3dbd2c487ac922d4bf423893170fb4": "\\cos{5x}=16\\cos^5 x-20\\cos^3 x+5\\cos x\\,", "1e3dd379db944b6a68599c109a8a1ebf": "2\\cdot A_5 \\cong 2I;", "1e3ddc86694466e49b567756823f16c7": "2\\cdot$1\\cdot\\sqrt{100\\cdot18/38\\cdot20/38}\\approx$9.99", "1e3e248876093a873e743eb428d1046a": "\\tau_E=rC", "1e3e4ff16f6baa4cd3550bab5f8a5346": "I = I_0 e^{V/V_T}", "1e3e7ebd591a19b526595b0d33e25e83": "\\boldsymbol{\\xi} \\geq 0", "1e3ebbbb84d762b678c967b4a48e66fc": " = \\sum_{i=0}^\\infty \\sum_{j=0}^\\infty a_i b_j (x-c)^{i+j}", "1e3edc773e96664b21e08f9f57e3909a": "\\varepsilon_t \\,", "1e3efa9f6dbc97637e6c81f6e3e4609c": "\\scriptstyle \\mathbb{P}\\left(\\{\\max(X,Y,Z)\\leq W\\}\\right)=\\mathbb{P}\\left(\\{X\\leq Y\\} \\cap \\{Z\\leq W\\}\\right).", "1e3f45e9978768b2964485a638b34b09": "K_{\\mathrm w} =[\\mathrm{H_3O}^+] [\\mathrm{OH}^-]\\,", "1e3f6b6fb268d6d6846610de88a98517": "H_n(\\widetilde{X},\\widetilde{K})", "1e3fab1f7f9cfbcd9075e8ed644b104c": "10 \\cdot \\log_{10} 0.000 001 = 10 \\cdot (-6) = -60", "1e3faeeeb7bbc004fbb6cddde64c14ad": "a_N.", "1e40088aeb26790910f939c4109b95d0": " \\nabla\\times\\mathbf{E}=-\\frac{\\partial\\mathbf{B}}{\\partial t} ", "1e402d31333e6e9e107ec5c391aa4983": "\nL_n =\n\\begin{pmatrix}\n 1 & & & & & 0 \\\\\n & \\ddots & & & & \\\\\n & & 1 & & & \\\\\n & & l_{n+1,n} & \\ddots & & \\\\\n & & \\vdots & & \\ddots & \\\\\n 0 & & l_{N,n} & & & 1 \\\\\n\\end{pmatrix}.\n", "1e4048c06b5051839aacbeab2292a8b1": "z/2", "1e406e15eb2d26271a2492ecb59fe4cb": "C=\\begin{pmatrix}\n 0& 1& 0& 0& 0& 0& 0& 0 \\\\\n 1& 1& 0& 0& 0& 0& 0& 0 \\\\\n 0& 0& 0& 0& 0& 0& 0&-1 \\\\\n 0& 0& 1& 0& 0& 0& 0&-4 \\\\\n 0& 0& 0& 1& 0& 0& 0&-4 \\\\\n 0& 0& 0& 0& 1& 0& 0& 2 \\\\\n 0& 0& 0& 0& 0& 1& 0& 4 \\\\\n 0& 0& 0& 0& 0& 0& 1& 0 \\end{pmatrix}.", "1e4093c2ec402e5000b88665cc90e8dc": "\nS_\\mathrm{TeVeS}=\\int\\left({\\mathcal L}_g+{\\mathcal L}_s+{\\mathcal L}_v\\right)d^4x.\n", "1e40ce4bdd56981553fb0cf000e5f009": " \\boldsymbol\\epsilon ", "1e4181c897a8e179bdef6c606dfd511c": "\\, =xy ", "1e41a3d7f7a85636818103a782f69b69": "\\langle\\overline\\Psi\\Psi\\rangle=\\int\\frac{m}{(k^2+m^2)^{1/2}}\\frac{2d^3k}{(2\\pi)^3},\\quad 0\\theta > 0", "1e4b2a81708c10a3c0a5ec189e88ce85": "d_\\varrho \\times d_\\varrho\\,", "1e4b34a3445892cc736bdfb854cf71fa": "11^3+12^3+13^3+14^3 = 20^3", "1e4b64ce049b4974dcaaf4a303b65cbb": "\\begin{align}\n\\mbox{E}\\left[T\\right] &= \\begin{cases}\n\\mu\\sqrt{\\frac{\\nu}{2}}\\frac{\\Gamma((\\nu-1)/2)}{\\Gamma(\\nu/2)}, &\\mbox{if }\\nu>1 ;\\\\\n\\mbox{Does not exist}, &\\mbox{if }\\nu\\le1 ,\\\\\n\\end{cases} \\\\\n\\mbox{Var}\\left[T\\right]&= \\begin{cases}\n\\frac{\\nu(1+\\mu^2)}{\\nu-2} -\\frac{\\mu^2\\nu}{2} \\left(\\frac{\\Gamma((\\nu-1)/2)}{\\Gamma(\\nu/2)}\\right)^2 , &\\mbox{if }\\nu>2 ;\\\\\n\\mbox{Does not exist}, &\\mbox{if }\\nu\\le2 .\\\\\n\\end{cases}\n\\end{align}", "1e4b80ad8cb0639e6472d95fcb4334e3": " k_z = \\frac{n\\pi}{D}", "1e4c077dac47dd0ca2aeaf76e91ef292": "\\frac{D_F X}{d s}=\\frac{DX}{d s} - (X,\\frac{DV}{d s}) V + (X,V)\\frac{DV}{d s},", "1e4c3ad17c402f7881a63b4de5119c1e": "\\exist \\mathbf{I} \\, ( \\empty \\in \\mathbf{I} \\, \\and \\, \\forall x \\in \\mathbf{I} \\, ( \\, ( x \\cup \\{x\\} ) \\in \\mathbf{I} ) ) .", "1e4c6ef14d4285258cb4cf256ba54e97": "\\dot{\\gamma}(0)", "1e4c8c2dbd7f101468a3d4d37c713ded": " R_n(x) = f(x) - p_n(x) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!} \\prod_{i=0}^n (x-x_i) ", "1e4ca7bfe6703f29bc91e6fd3d105638": "\\left(T_p,\\Omega_p\\right)", "1e4d249f488e2aed941eeb594249467a": "GBWP_{\\omega > > {\\omega_c}} = {A_1}(\\omega )\\cdot\\omega \\approx const.", "1e4d424194f50c262230c1da7d3fae4a": "\\begin{align}\nA(1,2) & = A(0, A(1, 1)) \\\\\n& = A(0, A(0, A(1, 0))) \\\\\n& = A(0, A(0, A(0, 1))) \\\\\n& = A(0, A(0, 2)) \\\\\n& = A(0, 3) \\\\\n& = 4.\n\\end{align}", "1e4d4d92c1af37c4fdd8f4aa22faaac7": "|\\hat{v}(\\zeta)| \\le C_m(1+|\\zeta|)^Ne^{H(\\text{Im}(\\zeta))}", "1e4d6beda591b5ec1a4faedcf669aeb8": " \\{ e \\} (n) <_\\mathcal{O} 3\\cdot 5^e ", "1e4defbb027a039f2cc6f2d895f3c8cc": " dU = T dS - p dV ", "1e4dff0e53f26b1ebc00a0b6ac2e2dcc": "\\alpha^{-\\infty} = 0", "1e4e0c1917dd0daa3b23b875c152f9a1": "x^{2} - 2x + 1 = 0", "1e4e13fb4446ade6e2b28182eda82a16": "r_a(\\mathfrak{u},\\mathfrak{s})\\in R", "1e4e1c787ac9716ed6d775c9b46f1cfd": "\\sum_{i=1}^\\infty \\omega_f(\\theta^i a) < \\infty", "1e4e247239569ce6ffccc95412afa71d": "\\{\\{1, 2, 3\\}, \\{4, 5\\}\\}", "1e4e53d1de46b0c0385c9f92a7d57486": "L^{p_1}(\\mu_1)", "1e4e63ad1646c182e276da11670ed597": "\\hat\\theta(\\theta) = \\theta", "1e4ea6c5aed1f01961fa5c49ceab2e0e": "\\displaystyle i^n \\frac{d^n \\hat{f}(\\nu)}{d\\nu^n}", "1e4eb1780a2cae2798ee7401b1a49974": "(I - \\alpha A^T)^{-1}", "1e4eecbe642a29d96f004cfdd9b7de77": "\\delta = 2 \\arcsin ( \\tfrac{d_\\mbox{act}}{2 D})", "1e4eef7558b91d7d68e50db23b2a8d5c": "\\Gamma(t;\\gamma,\\lambda)", "1e4efd5777cf6575b29ceb94268c7f75": " \\Lambda_i(x_0) ", "1e4fa5a2f7e75dee62260b2b41e06b92": "|r+k|^2", "1e4fd27ec4108fb1b194a3e68bd3b9fd": "\\frac{1}{(36+n)}", "1e4fe334e1d0dadd55071989f7b4122c": " \\|A\\|_{\\text{max}} = \\max \\{|a_{ij}|\\}. ", "1e4ff68109a469de5724197f2907022a": "\\ell\\,\\!", "1e507617c88afcc15b3ad58f77d93889": " \\Psi \\to e^{-\\Lambda} (\\Psi + Q_B )e^{\\Lambda} ", "1e5091e28a538015bcfa1654ba21ca42": " e^{i \\pi/4}\n\\begin{pmatrix}\n1 & 0 \\\\ 0 & i\n\\end{pmatrix} ", "1e509e2224c03f1491ddbae75742305e": "P\\left(S^{0}\\wedge O^{0}\\right)", "1e50afbd1d095a4b30d39231713df069": "\\delta_{pen}", "1e50f561f62b6d9911310897f42b1b7d": " K_\\text{L}(x,x') = x^T x'", "1e5107910c05c6026d81c9cadcf55080": "\\displaystyle{\\|a^2\\|=\\|a\\|^2,\\,\\,\\, \\|a^2\\| \\le \\|a^2 + b^2\\|.}", "1e5125525e34d539dad2e5826a0855c5": "z(\\mathbf{x})=z_B+\\text{nonnegative terms corresponding to nonbasic variables}", "1e514639f323f456e033fdd8b7a3b961": "\\gamma_k", "1e51567b447d675a61e4d3f346f48e85": "\\Delta\\sigma\n=2\\arcsin\\left(\\sqrt{\\sin^2\\left(\\frac{\\Delta\\phi}{2}\\right)+\\cos{\\phi_1}\\cos{\\phi_2}\\sin^2\\left(\\frac{\\Delta\\lambda}{2}\\right)}\\right).\\;\\!", "1e5176e8dc3ffc13a5b1c575a3ef66c1": "\\begin{matrix}B\\left(\\cos ^2\\theta\\ -\\ \\sin ^2\\theta\\right)\\ -\\ 2\\left(A\\ -\\ C\\right)\\sin \\theta\\cos \\theta\\ &=& 0 \\\\ \\\\\nB\\cos 2\\theta\\ -\\ \\left(A\\ -\\ C\\right)\\sin 2\\theta &=& 0 \\\\ \\\\\nB\\cos 2\\theta &=& \\left(A\\ -\\ C\\right)\\sin 2\\theta \\\\ \\\\\n\\cos 2\\theta &=& \\frac{\\left(A\\ -\\ C\\right)\\sin 2\\theta}{B} \\\\ \\\\\n\\cot 2\\theta &=& \\frac{A\\ -\\ C}{B} \\end{matrix}", "1e517786a0e67442b0e893d616545145": " \\sum_{i,j = 1}^n c_ic_jK(t_i,t_j) \\geq 0 ", "1e519606e956f1a22a01608b9d209ff6": "f^\\prime", "1e51c5bbc73626240b4653e7ce21a196": "F_{A_{CO_{O}}}", "1e51d804270b348c8dc7f76e1ce40781": "\\bold{v}=\\frac{\\mathrm{d}\\bold{r}}{\\mathrm{d}t}", "1e51f69d124674835d343bce4e9317ef": "\n \\varepsilon_{zz}^{\\mathrm{face}} = \\varepsilon_{xz}^{\\mathrm{face}} = 0 ~;~~ \\varepsilon_{zz}^{\\mathrm{core}} = \\varepsilon_{xx}^{\\mathrm{core}} = 0\n ", "1e51f9af3dbb6da3de414a41fe36c0a7": "\n\\omega_p^2 = {4\\pi n e^2 \\over m}\n", "1e524043d819073e5c99baa905aa5ae0": "\\dot Q_A", "1e52e54235a832a1cb5487cc6d6eced3": "\\epsilon\\circ \\eta \\colon K \\to K", "1e53088e4e8de854d6388882d6bfafcc": "\\tilde{Q}", "1e53191b63d73e90b087796538b00d08": " S \\subset \\mathbb{C} ", "1e534c5017df637c225a19d2dab347af": "\\Rightarrow (\\lambda_j - \\lambda_i) v_j ' v_i = 0", "1e53541040158ce823175a992e94b3d1": "\\{p_i\\}", "1e53bf640a7fccf0641120be0fe6b4d5": "\\|f(v)\\| = \\|v\\|", "1e53d393f73a027f66e24444419c8cec": " R(0) = Q(0)-P(0) = 0 ", "1e549d985ee4c7496664c23e8bd864c6": "Z = (-1)^n \\left(|a| \\cdot |2a| \\cdot |3a| \\cdot \\cdots \\cdots \\left|\\frac{p-1}2 a\\right|\\right).", "1e550efc3e079ab4a53032a92120f630": "\\phi_3(0)", "1e559cf7727cb99e66dc2a8895ec93d8": "F[\\rho] = \\int f( \\boldsymbol{r}, \\rho(\\boldsymbol{r}), \\nabla\\rho(\\boldsymbol{r}) )\\, d\\boldsymbol{r},", "1e55c86eb8461e6037b49096e60aaac4": "H=\\frac{|0\\rangle+|1\\rangle}{\\sqrt{2}}\\langle0|+\\frac{|0\\rangle-|1\\rangle}{\\sqrt{2}}\\langle1|", "1e561d65475b286cda62e86aa46b1455": "\\, \\gamma \\, ", "1e56337a9807469ffcbf8cfbb789132e": "\\lambda_2 = \\text{L-scale}.", "1e56ab39ac37d9e52a28ab759d6a5c9d": " \\|f \\|^2 = \\langle f, f\\rangle =\\int \\langle f, x\\rangle \\langle x, f \\rangle \\, dx = \\int f^*(x) f(x) \\, dx ", "1e56e7eb4fdad20212e427be90489579": "c_{p} = \\frac {c_{p0}} {\\sqrt {{M}^2-1}}.", "1e56f07219f471ac26749e4a5a07972d": "\\frac{\\delta \\mathcal{S}}{\\delta x(t)}=0", "1e570eca215ae91746ddf576aabd7506": "\\,g_1 + g_2 + g_3", "1e571e9b12107359edef3b6421538e4f": "[ab, c] = [ba, c]", "1e5746ef63bd8b4d9d5ce55f97a3fcf0": "\\{a_1,\\cdots,a_l\\}", "1e58550590c5e7360a977a8843de23c2": " A = \\begin{bmatrix} 3 & 1/7 \\\\ 4 & -1/7 \\end{bmatrix} \\begin{bmatrix} 5 & 0 \\\\ 0 & -2 \\end{bmatrix} \\begin{bmatrix} 3 & 1/7 \\\\ 4 & -1/7 \\end{bmatrix}^{-1} = \\begin{bmatrix} 3 & 1/7 \\\\ 4 & -1/7 \\end{bmatrix} \\begin{bmatrix} 5 & 0 \\\\ 0 & -2 \\end{bmatrix} \\begin{bmatrix} 1/7 & 1/7 \\\\ 4 & -3 \\end{bmatrix}. ", "1e58846148e5819f8129066addc73cbf": "\n \\frac{S_n}{\\sqrt{n\\log\\log n}} \\ \\xrightarrow{p}\\ 0, \\qquad\n \\frac{S_n}{\\sqrt{n\\log\\log n}} \\ \\stackrel{a.s.}{\\nrightarrow}\\ 0, \\qquad \\text{as}\\ \\ n\\to\\infty.\n ", "1e58e44b4471a36ec24119e084077aad": " (Y_t) ", "1e5905dac89d15fc7561f6e74088f630": "\\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\dot{\\mathbf{q}} - L \\right) T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\epsilon}.", "1e59a5887a7ae4f6a0388d082005bef7": "\\{\\gamma^\\mu = \\frac{1}{{\\gamma_\\mu}}\\}", "1e59be739bec1935e1de3a198e463e2b": "\\begin{align}\n\\int_a^x \\frac{f^{(k+1)} (t)}{k!} (x - t)^k \\, dt = & - \\left[ \\frac{f^{(k+1)} (t)}{(k+1)k!} (x - t)^{k+1} \\right]_a^x + \\int_a^x \\frac{f^{(k+2)} (t)}{(k+1)k!} (x - t)^{k+1} \\, dt \\\\\n= & \\ \\frac{f^{(k+1)} (a)}{(k+1)!} (x - a)^{k+1} + \\int_a^x \\frac{f^{(k+2)} (t)}{(k+1)!} (x - t)^{k+1} \\, dt. \\\\\n\\end{align}", "1e59ca70316288f7068f793f6f65f09b": "x_2=b(x,t)", "1e5a15341a53a855acf9e2ed7ef87c48": "g_k = 2 \\sin \\left [\\frac {(2k-1)}{2n} \\pi \\right ]\\qquad\\mathrm{k = 1,2,3, \\ldots, n}.", "1e5a745950640ec83e2db28fefba5713": "\\mathbf{J}(\\mathbf{x}) e^{i\\omega t}", "1e5a7aa7fa68e5da8e2037688fea874a": " \\bar \\psi \\mapsto \\bar \\psi e^{-i \\Lambda} ", "1e5a99226dcb6668c2ad852cdeaf0b58": "\\scriptstyle A_k\\, \\subseteq\\, A", "1e5ae0790d23df2ab8ef0deeeda01572": " C =\\{\\theta_0 u_0 + \\dots+\\theta_k u_k | \\theta_i \\ge 0, 0 \\le i \\le k, \\sum_{i=0}^{k} \\theta_i=1\\} ", "1e5b90c28bf733fcbc42234d9d2c2e57": "\\displaystyle u^{\\circ}\\ ", "1e5ba4374603d680da6650dccca5efc1": "\\exists x\\colon F(x)", "1e5bbac4af7b8387f6972b60f9451f2e": "f.\\,", "1e5bca6acba54a125db59c97dae06728": "F_{-X}", "1e5c0187d71ff6b16bcb2cf3b3c07266": "v_{max}", "1e5c05a80c9511f22e15fd520d6d62c9": "x=x_0 - \\delta", "1e5c08d41d18ca7ec201a49f599bf1d7": " D^s_{m'm}(\\alpha,\\beta,\\gamma) \\equiv\n\\langle sm' | \\mathcal{R}(\\alpha,\\beta,\\gamma)| sm \\rangle =\n e^{-im'\\alpha } d^s_{m'm}(\\beta)e^{-i m\\gamma},\n", "1e5c312fe8d876897fd931933d471c01": "\\frac{F_1}{F_2} = \\frac{k_1}{k_2} = \\frac{c_2}{c_1} ", "1e5c39d3fa6723a61e41ef9730184404": "\\,\n\\begin{align}\n(n-1)^x(n-1)!&\\leq \\Gamma(n+x)\\leq n^x(n-1)!\\\\\n(n-1)^x(n-1)!&\\leq (x+n-1)(x+n-2)\\cdots(x+1)x\\Gamma(x)\\leq n^x(n-1)!\\\\\n\\frac{(n-1)^x(n-1)!}{(x+n-1)(x+n-2)\\cdots(x+1)x}\\leq \\Gamma(x)&\\leq\\frac{n^x(n-1)!}{(x+n-1)(x+n-2)\\cdots(x+1)x}\\\\\n\\frac{(n-1)^x(n-1)!}{(x+n-1)(x+n-2)\\cdots(x+1)x}&\\leq \\Gamma(x)\\leq\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\left(\\frac{n+x}{n}\\right)\\\\\n\\end{align}\n\\,", "1e5c506eafe162ad5ae6ec998e20de21": "A = \\begin{pmatrix} A_{11} & A_{12} \\\\ A_{21} & A_{22} \\end{pmatrix}.", "1e5cca92761444cf1f0af1d409169ae1": "2\\sum |a_n|", "1e5d15d224c79e93e43cf38cfda3f702": "\\sgn \\cdot R_i", "1e5d33039d6915bc2c092dd3c2c569a5": "N=\\Lambda", "1e5d834efde5c5a6035bb70686966c1c": "\\frac{{4 \\choose d}2^d}{80}", "1e5e187c4dc39fafb616cae64fbd60f4": "M_{i,j} =\n \\begin{cases}\n 1 & (x_i, y_j) \\in R \\\\\n 0 & (x_i, y_j) \\not\\in R \n \\end{cases}\n ", "1e5e452bcbe2e7ab4b4e8a31124f1699": "(b^2 + {{a^2}\\over 2})\\pi", "1e5e98a1d443ce997009f4a934899bb5": "\\lambda = \\lambda^e \\cdot F\\lambda^g\\,", "1e5f4524228e7fbe821213ee3ea49ff4": "\\ \\theta ", "1e5f59030bba1744e19b56d70954ec83": "\\displaystyle ax + by = 1,", "1e5f68e0199b14f2fcf1dd97b4e2e023": "X,\\, P", "1e5f8e371e5f2702dd614118c73a18c4": " \\varphi (\\mathbf{r}, t) = \\int { { \\delta \\left ( t' - { { \\left | \\mathbf{r} - \\mathbf{r}' \\right | } \\over c } - t \\right ) } \\over { { \\left | \\mathbf{r} - \\mathbf{r}' \\right | } } } \\rho (\\mathbf{r}', t') d^3r' dt' ", "1e5fa6bc8b78c30f20e2881cda58c960": "a=E(X)-\\sqrt{3V(X)}", "1e5fcd4e3e504765906bf72c15c72526": " S = 125348 = 1\\;1110\\;1001\\;1010\\;0100_2 = 1.1110\\;1001\\;1010\\;0100_2\\times2^{16}\\, ", "1e5fe5f8e0559984d1b9c83487a88896": "(a+b+c)^3 = a^3 + 3a^2b + 3a^2c + b^3 + 3ab^2 + 3b^2c + c^3 + 3ac^2 + 3bc^2 + 6abc", "1e6090e11edd7143eb8873c5a471f504": " \\zeta(s) = 1 + \\frac{1}{2^s} + \\frac{1}{3^s} + \\frac{1}{4^s} + \\cdots ", "1e60ac5b6a6c68387adbf24423080499": " {A_\\mathrm{v}}= g_{21} = \\begin{matrix} {v_\\mathrm{out} \\over v_\\mathrm{in} }\\end{matrix} \\Big|_{i_{out}=0}", "1e60c5dbe3c4ceadbcdd83c3da14f717": "L\\in\\mathbf{H}_n", "1e60c992dbf0b1058a651aefa5756197": "{D}_{8}^{(1)}", "1e60cbfdbc8e58ee4666f8347c45772d": "\\frac{\\partial}{\\partial t} \\int_V |\\Psi|^2 \\mathrm{d}V +", "1e611d330e969808712c01bfa73373aa": "\\left(x - \\left [ a + \\sqrt b \\right ] \\right) \\left(x - \\left [ a - \\sqrt b \\right ] \\right) = (x - a)^2 - b.", "1e6135086e34aadbecfb6c2db3657684": "\\phi \\in [-\\pi/2, \\pi/2]", "1e6139569ed78daa97e465c4c47f98de": " e^x = \\sum_{n = 0}^{\\infty}\\frac{x^n}{n!}.", "1e614e8c622dcee85c766fcb66f836a2": "b_0\\,", "1e619c7acca2c56de9ed36d8cb38da57": "b>a^k.", "1e6232d9749cf454e8b1e01c3aac263c": "\\gamma^\\mu\\gamma^\\mu=\\eta^{\\mu\\mu}\\,", "1e6239456a8cc478884590c65ca4a49e": "\\sum_{k} \\left| y_k - x_k \\right| < \\delta", "1e625395b869eab77e08995e859ec6c2": " U_\\text{B} - U_\\text{A} \\ll kT ", "1e626c8462d7a9509166e454df2195d9": " \\Delta G_{vap} = \\Delta H_{vap} - T_{vap} \\times \\Delta S_{vap} = 0", "1e628035d8c3452ec5f5956ce7932ee3": "\\Pr(E^{\\tilde P(x)}(x) \\in W(x)) \\geq \\Pr(\\tilde P(x)\\leftrightarrow V(x) \\rightarrow 1) - \\kappa(x).", "1e628e069c87a62f83812cc4d2efdbb2": " \\stackrel{\\nabla}{\\mathbf{T}} = \\frac{\\partial}{\\partial t} \\mathbf{T} + \\mathbf{v} \\cdot \\nabla \\mathbf{T} -( (\\nabla \\mathbf{v})^T \\cdot \\mathbf{T} + \\mathbf{T} \\cdot (\\nabla \\mathbf{v})) ", "1e629c3ae6802b986ae4e40f5a5b78d4": "\\langle F,R,V\\rangle", "1e62b4784321049a13156262c19eb09e": "\\|f\\|_{L^2(\\mathbf{R}^d)}\\leq Ce^{C|S||\\Sigma|} \\bigl(\\|f\\|_{L^2(S^c)} + \\| \\hat{f} \\|_{L^2(\\Sigma^c)} \\bigr) ~.", "1e63062d3a6a7a15572d3dcdd3f1917d": "\\sum_{s} Z_{s} e B \\int d v_{||} d\\mu d\\varphi h_{s} \\left(\\vec{R}\\right) = \\sum_{s} \\frac{Z_{s}^{2} e^{2} n_{s} \\phi}{T_{s}}", "1e632750f338be384b9c593e0b9ba0d5": "\\begin{matrix} \\frac{1}{1} \\end{matrix}", "1e6340e8a3c403489533caa0a82995fc": "y_1(x)=x+1\\,", "1e634bf1ea3b0f28ef7fbbd749f5b062": "(G/H)_H", "1e635cbb83f09f791a47ed0b52ff3f2b": " T_K\\phi(x) = \\sigma_i\\phi_i(x) ", "1e63bdaecd47c11e2ee78040582906c7": "\\vec{S}^\\dagger", "1e6438c55f008afca0ec21157d9fa91c": "r_O = \\frac{V_A + V_{CE}}{I_C} \\approx \\frac{V_A}{I_C}", "1e6470a189d249ba285f55eff7a7b3f1": "{}_3F_2 (a,b,c;1+a-b,1+a-c;1)= \\frac{\\Gamma(1+\\frac{a}{2})\\Gamma(1+\\frac{a}{2}-b-c)\\Gamma(1+a-b)\\Gamma(1+a-c)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+\\frac{a}{2}-b)\\Gamma(1+\\frac{a}{2}-c)}.", "1e64a37925a67754d47b4e9dc3c3cc27": "u''+p(x)u'+q(x)u=0\\,", "1e64eda87b479596e4c27cb7031c73c2": "p_1^{n_1}+1 \\le \\mathfrak{M}(d) \\le d+1. ", "1e653e2c49f14f81b60205a54383a208": " \\mathbf \\zeta=79", "1e655c0d4e30ebee2f566a68094ff520": "V=C_2 \\times C_2", "1e656f2a69ca438488679478c18a08e2": "\\displaystyle{f_{\\overline{z}} = \\mu f_z,}", "1e659c51cfd47a2e9f68e157448c1af7": " f(\\frac{1}{n}\\sum_{i=1}^n f^{-1}(x_i)))=\\exp(\\frac{1}{n}\\sum_{i=1}^n\\log x_i)=\\sqrt[n]{x_1\\cdots x_n}", "1e65aac7dec0a23e038a7178178647d3": "r_b=\\langle\\hat{d}| \\hat{c}\\rangle-\\langle\\hat{f}| \\hat{a}\\rangle", "1e65b4d6d965fcd8d83565cd5f540322": "R_p\\leq0.2", "1e65fb8201c846adcdb178db36b85680": "\\frac{3}{8} \\pi a^2", "1e66110d881ae2168db84d6165f63282": "y= \\pm \\left(\\sqrt{2} + 1 \\right) \\left( x-2 \\right) ", "1e664a6988ea2aeacaf377d679115846": "i_\\mathrm r = -i_\\mathrm i \\,\\!", "1e666f542e40880d8117b2e32c5f58da": " V = \\{I_1, I_2, \\ldots, I_n\\} ", "1e669c437f4f44ed5c3df28aa00728cb": "\\qquad \\sum_{j \\in J} v_j\\,x_j \\ \\ge \\alpha\\,v_i\\,", "1e66a07e9888bda94396dae58a9afab3": " \\varphi_\\lambda(e^X) = {\\chi_\\lambda(e^X)\\over\\chi_\\lambda(1)}.", "1e66f932c5d60285740041dbb7887110": " \\delta= \\alpha_y - \\alpha_x. ", "1e66fb47d18b9cd9db82bd08af3e3fa2": "w_{min} = \\frac{Q}{q_{max}} = \\frac{150}{45.5} = 3.30 \\text{ ft}", "1e671a558c1fd5d1ecd30bc4e907c6c6": " IMM(s)=\\{i \\in D| ta_i(s_i) = ta(s) \\} ", "1e6743a673a02d3aa6df71cffd9e1467": "\\frac{d \\eta}{d t} = -\\eta \\nabla_h \\cdot\\vec{v}_h - \\left( \\frac{\\partial \\omega}{\\partial x} \\frac{\\partial v}{\\partial z} - \\frac{\\partial \\omega}{\\partial y} \\frac{\\partial u}{\\partial z} \\right) - \\frac{1}{\\rho^2} \\vec{k} \\cdot ( \\nabla_h p \\times \\nabla_h \\rho )", "1e67498c9f66d1b7dbf98263daf51d86": "\\omega+2", "1e67824fa95e97cf11ed7e152bcd8f34": " m_c(z) ", "1e67931e0bb08cd4aa009dd0d33a3da5": "\\tfrac{1}{2} \\div \\tfrac{3}{4} = \\tfrac{1}{2} \\times \\tfrac{4}{3} = \\tfrac{1 \\cdot 4}{2 \\cdot 3} = \\tfrac{2}{3}", "1e6796dd3b22411de7aff0d03ed9504d": "y_i=\\sum_{j=1}^2 a_{ij} x_j + \\sum_{j=1}^2 b_{ij}y_j", "1e67eb0e840c08faf67f937b328a70ca": " \\mathcal{L}^{-1} \\left \\{ \\frac{1}{s+a} \\right \\} \\, * \\, \\mathcal{L}^{-1} \\left \\{ \\frac{1}{s+b} \\right \\} = e^{-at} \\, * \\, e^{-bt} = \\int_0^t e^{-ax}e^{-b(t-x)} \\, dx = \\frac{e^{-a t}-e^{-b t}}{b-a}.", "1e6813370e2e3c30585362cc6ea39a66": "(Sg)(z) = 2Q(z)", "1e68393bec7f6c54ac3b093c91eca86d": "\\frac n2 \\langle V_\\mathrm{TOT} \\rangle_\\tau\n= \\left\\langle \\sum_{k=1}^N \\left(\\frac{1 + \\sqrt{1-\\beta_k^2}}{2}\\right) T_k \\right\\rangle_\\tau\n= \\left\\langle \\sum_{k=1}^N \\left(\\frac{\\gamma_k + 1}{2 \\gamma_k}\\right) T_k \\right\\rangle_\\tau\n\\,.", "1e68ce3487dc77148d543eaee8ea98ba": " H^{'}_{j\\gamma} = \\left \\langle u_{j0} \\right | \\frac{\\hbar}{m_0} \\mathbf{k} \\cdot \\left ( \\mathbf{p} + \\frac{\\hbar}{4 m_0 c^2} \\bar{\\sigma} \\times \\nabla V \\right ) \\left | u_{\\gamma 0} \\right \\rangle \\approx \\sum_{\\alpha} \\frac{\\hbar k_{\\alpha}}{m_0}p^{\\alpha}_{j \\gamma}. ", "1e68f3c3bc3a2f57ae71b355e9b97509": "\\sqrt{}", "1e6926bdfaa69e759514d2ffe9a0d913": "P_0(y)=((-a/b)y)P_1(y)+c,", "1e69589c6af42e8428c244ec0fa39fb7": "\n\\begin{align} \nProb(choosing \\, 5) \n& = Prob(U_n > d) \\\\\n&= Prob(\\varepsilon > d - \\beta z_n) \\\\\n& = 1 - {1 \\over 1+exp(-(d - \\beta z_n))}\n\\end{align}\n", "1e69745e09d506d2ef26efedec1555a4": " n^{1/k}", "1e6a270bbc2e1939ea5d07bd54824309": " \\ p\\textbf{r} \\cdot \\textbf{g} + \\textbf{f} \\cdot \\textbf{m} ", "1e6a73d75db921c729655dfd65c4bfb0": "\\frac12 mv^2", "1e6aa8ce268f794b5fb70b85dd5eac6c": "(x_1x_2 - Ny_1y_2 \\,,\\, x_1y_2 - x_2y_1 \\,,\\, k_1k_2).", "1e6add065eaeb84b24416409275cff3d": "g(p_i,q_i,t)", "1e6ae43ee27bc8d9b445f6a4c2cd2785": "\\left| \\langle v| U(t)|\\psi_0\\rangle \\right|^2", "1e6ae83f2fee94e07ae3092f83754ea2": "C \\otimes C", "1e6aeaa9e1123305e73497f7dd3cf29e": " m_1 > m_2 > m_3 > \\cdots > m_n.\\,", "1e6aebdb27572591e955109a44f56085": "A \\approx {360^{\\circ}\\over 2 \\pi} \\cdot {M\\over 1000 } \\approx 0.0573 \\times M ", "1e6af492fc535cd50f9473573526d5d0": " H^q(B^\\bull)/H^q(A^\\bull) \\cong \\ker d^1_{0,q} : H^q(C^\\bull) \\rightarrow H^{q+1}(A^\\bull)", "1e6b0b6ab3635f47cadf5d5a904ae452": "f=m\\circ e", "1e6b498af6527dbdc6dd238b0769ac51": "D \\neq -4", "1e6b5b1b8090ad0dae695e69f873d5d7": "\\begin{pmatrix}w_1\\\\ w_2\\\\ w_3\\\\ \\end{pmatrix}=\\frac{1}{1+\\frac{u_1v_1+u_2v_2+u_3v_3}{c^2}}\\left\\{\\left[1+\\frac{1}{c^2}\\frac{\\gamma_\\mathbf{u}}{1+\\gamma_\\mathbf{u}}(u_1v_1+u_2v_2+u_3v_3)\\right]\\begin{pmatrix}u_1\\\\ u_2\\\\ u_3\\\\ \\end{pmatrix}+\\frac{1}{\\gamma_\\mathbf{u}}\\begin{pmatrix}v_1\\\\ v_2\\\\ v_3\\\\ \\end{pmatrix}\\right\\}", "1e6b962ad166b958db642d1c8f9bb2ce": "\\sqrt{B}", "1e6b9d3c5e221072d0cf678046ccbb3c": "\\operatorname{E}[\\,x_t(y_t - x_t'\\beta)\\,]=0", "1e6bad2d26e82f4ec22c0ec1459693a1": "e^X = \\sum_{k=0}^\\infty{1 \\over k!}X^k.", "1e6c8d4af9f706609017fdf575df3936": "\\gamma = (3\\pi^2)^{2/3} \\frac{\\hbar^2}{2m} ", "1e6c90ab968cdee11dfa4eda42d573b1": " A_\\mu\\,dx^\\mu", "1e6ccffc63598fd20a34d06cbf6cfe7d": "\\textstyle E_{2}", "1e6d676a68b43a677542bfb15c2e3d22": "\\phi(\\rho)", "1e6d69ede2647a75797c95245ce41590": "M_{i,j}^{p,q}", "1e6d8eaee0c0359ab430aa0d60a4755b": "\\gamma_k(a)", "1e6d9ce93f5b5229c3073e1754b38e51": "f^2(\\theta^2(t))", "1e6df665d4baaa1b5e239311c9cc9550": "\\mbox{recall}=\\frac{|\\{\\mbox{relevant documents}\\}\\cap\\{\\mbox{retrieved documents}\\}|}{|\\{\\mbox{relevant documents}\\}|} ", "1e6e0a04d20f50967c64dac2d639a577": "1100", "1e6e3432d141e09032cb6127fb1892ec": "\\{[-\\infty,a[:a\\in\\mathbb{R}\\cup\\{\\pm\\infty\\}\\}", "1e6e63df7e2ac7cf7da2de654081f717": "f(x;h)=\\frac{e^{hx}f(x)}{\\int_{-\\infty}^\\infty e^{hx} f(x) dx}.\\,", "1e6e975c355631d08f98ab1b56bc4c0b": "\\langle \\cdot, \\cdot \\rangle_H = \\langle \\cdot, \\cdot \\rangle_G", "1e6ec2d5200ad13c03cb2f08a59ec4eb": "\\lnot \\;\\exists \\;x", "1e6f02e8fed70befb3f11f21b57feaef": "\\phi-\\xi\\,\\!", "1e6fb6df8558239f4c1e7f5daaa8898e": "F[\\rho] = E[\\rho] + \\frac1{\\beta} S[\\rho],", "1e6fdcc12bc90d0e69cad8f2dc5aff90": "\\sigma\\,", "1e7070c60d0e2863e29d9c2df0690ef7": "f''(x_n) (a - x_n)\\!", "1e707431667377ca0c2d700e0c68e90e": "\\Sigma\\left|a_n\\right|^2 \\ne \\Sigma\\left|b_n\\right|^2\\,", "1e707a6d712ce45fb70d0708804cec98": "0 \\le p \\le \\infty", "1e70c470db2cb9b251c37e6551abd0ae": "\\alpha>k", "1e70d4038de3ca07fb37ecc6cb9dda8c": " \\frac{-b}{2a} + i \\frac{\\sqrt {-\\Delta}}{2a} \\quad\\text{and}\\quad \\frac{-b}{2a} - i \\frac{\\sqrt {-\\Delta}}{2a},", "1e714b5d06631bc8d6d3fc3b94e9dfad": " 0.3333...\\triangleq\\lim_{n\\to \\infty} \\sum_{i=1}^n \\frac{3}{10^i}", "1e714e7dc6b10400f2ca0bb9fa812586": "\n\\nu = \n\\frac{\n \\mathrm{tr}(\\tilde{S}^2) + [\\mathrm{tr}(\\tilde{S})]^2}\n{\n \\frac{1}{n_1} \\left\\{ \\mathrm{tr}(\\tilde{S_{1}}^2) + [\\mathrm{tr}(\\tilde{S_1})]^2\\right \\} + \n \\frac{1}{n_2} \\left\\{ \\mathrm{tr}(\\tilde{S_2}^2) + [\\mathrm{tr}(\\tilde{S_{2}})]^2 \\right \\}\n}.\n", "1e715e612b0cdcd77062a419885a9903": "dI = -k_O C_L dV\\;", "1e71ae53ab23cd5e20eec152e26bdfed": "u_{0} \\equiv 1/r_{0}", "1e720784691b8fa4b97de5b85bd30740": " p_t=+ip^2q-\\frac{i}{2}p_{xx}", "1e7283225ce5499c242a102b3114df13": "A_R = \\{X \\in L^p_d: 0 \\in R(X)\\}", "1e72a2bf7dd3ff3f7d07f61f78fff740": "\\varepsilon_r = 1", "1e72bbfe15ab0ee1d2867751fa481ad1": "(M, 0, +)", "1e72e9ab043b82daed58a3fecdc65530": "y_i^{(\\lambda)} =\n\\begin{cases}\n\\dfrac{y_i^\\lambda-1}{\\lambda(\\operatorname{GM}(y))^{\\lambda -1}} , &\\text{if } \\lambda \\neq 0 \\\\[12pt]\n\\operatorname{GM}(y)\\log{y_i} , &\\text{if } \\lambda = 0\n\\end{cases}\n", "1e7312c49039bcb28334287d601dc600": "(2 f(x))^{-2}", "1e734701cabcb9ef0341bfbcc078099f": "\\mathrm{R{-}COOH\\ +\\ H_2O_2\\longrightarrow\\ R{-}COOOH\\ +\\ H_2O}", "1e736c36a77187bef111184d72efc65c": "S_{xx}\\, =\\, \\left( 2\\, \\frac{c_g}{c_p}\\, -\\, \\frac12 \\right)\\, E\\,", "1e73748946bdea5a3a9598725d447cf7": "f(x,y) = \\left(1+\\left(x+y+1\\right)^{2}\\left(19-14x+3x^{2}-14y+6xy+3y^{2}\\right)\\right)", "1e73851613c1e4ea3349cce62371d06f": "\\mathbf{v}^{\\infty}", "1e7395cf4c6edc4feb9c28cfb7b5605b": " \\alpha \\,\\!", "1e73f1c54aef1f67330ddd4d06f3de9b": "\n \\Delta \\tilde{x}_{j+1,j} \\geqslant \\Bigl[2(\\Delta x_{\\rm meas})^2+\\Bigl(\\frac{\\hbar\\vartheta}{2M\\Delta x_{\\rm meas}}\\Bigr)^2\\Bigr]^{1/2} \\geqslant \\sqrt{\\frac{3\\hbar\\vartheta}{2M}}\\,,\n", "1e73fe9f68e2fffb020194ffc4973230": " t = \\left\\lfloor\\frac{1}{2}(d-1)\\right\\rfloor", "1e7424fb7d3295afda4baff6a394af07": "\\;_j\\psi_k \\left[\\begin{matrix} \na_1 & a_2 & \\ldots & a_j \\\\ \nb_1 & b_2 & \\ldots & b_k \\end{matrix} \n; q,z \\right] = \\sum_{n=-\\infty}^\\infty \n\\frac {(a_1, a_2, \\ldots, a_j;q)_n} {(b_1, b_2, \\ldots, b_k;q)_n} \\left((-1)^nq^{n\\choose 2}\\right)^{k-j}z^n.", "1e743608ddcde5fca35c2ddfbe320b2f": "x=r_k,y=s", "1e745ce7a94e1520a9971a9e632da226": "\\iota: S \\rightarrow M", "1e745e0ad4f266a9731796cfcdaad451": "\\operatorname{pf}(\\lambda A) = \\lambda^n \\operatorname{pf}(A).", "1e74ba795769d7ac9e53d80449a89211": "\\ell=\\log s=\\log h + O(\\log(\\epsilon^{-1}))", "1e7500d608224a00aeddde704d355629": "x(j^{1}_{p}\\sigma) \\,", "1e750e88eb7b47f58efe94f2d6e8ccc9": "\\sum_i n_i\\, {\\rm d}\\mu_i = 0", "1e75163d385267ee084f40ced40434d6": "T=\\frac{n_2 \\cos \\theta_\\text{t}}{n_1 \\cos \\theta_\\text{i}} \\left| t \\right|^2", "1e7545758d4279e834d03fb03793ff56": "y'(t) = f(t, y(t)) \\,", "1e757a9802187d75c3c899bacbec3449": "E^{(2)} = \\sum_{l=1}^{\\infty} E_l^{(2)} = \\sum_{l=1}^{\\infty} - \\frac{Q^2 \\alpha_l}{2 R^{2l + 2}}", "1e75b72b8c5de5758b91c64b2b22d8bc": "V = I.(R'x + L'v) + L'x\\frac{\\text{d}I}{\\text{d}t}", "1e75fd0d939080ce35dbfdbe436a50a8": "H=(h\\nu - \\mu)a^\\dagger a. \\, ", "1e76c16988f7c31f3453c8f97a8973b7": "\n\\int_a^b e^{n f(x) } \\, dx \\ge \\int_{x_0 - \\delta}^{x_0 + \\delta} e^{n f(x)} \\, dx\n\\ge e^{n f(x_0)} \\int_{x_0 - \\delta}^{x_0 + \\delta} e^{\\frac{n}{2} (f''(x_0) - \\varepsilon)(x-x_0)^2} \\, dx\n= e^{n f(x_0)} \\sqrt{\\frac{1}{n (-f''(x_0) + \\varepsilon)}} \\int_{-\\delta \\sqrt{n (-f''(x_0) + \\varepsilon)} }^{\\delta \\sqrt{n (-f''(x_0) + \\varepsilon)} } e^{-\\frac{1}{2}y^2} \\, dy\n", "1e76d3ca183f72dced9912dfecfc0307": "\\displaystyle \\frac{1}{|a|} \\hat{f}\\left( \\frac{\\omega}{a} \\right)\\,", "1e77254ca5fcadd2ce551a5d9a582c0b": "K_M", "1e775635789aa04cfde0076b5309757e": " \\mathbf{F}_{ext} = m_{rocket}(t) \\frac{\\mathrm{d}\\mathbf{V}}{\\mathrm{d}t} + \\mathbf{V}(t) \\frac{\\mathrm{d}m_{rocket}}{\\mathrm{d}t}", "1e777dda72b6d3aa51998a46db2b7705": "\\mathrm{I}", "1e777f6242e02301be04ee07183ef18d": "F_2 = A'B + A'C + A'E.\\,", "1e781fc216d96f315ab12b8fcdda607e": "du/dt=f(u)", "1e788ff9085a1e3b210ea92c78cf3632": " \\mbox{DL} = (\\mbox{C} - \\mbox{DW}) \\bmod 7 ", "1e78a9aba621c1c0d22c97d014a12fe5": "|\\zeta(\\sigma)^3\\zeta(\\sigma+it)^4\\zeta(\\sigma+2it)|\\ge 1", "1e78ae39fce06be746293234e6a76604": "\\xi^d_{b_{min}}(k,0) = \\xi^d_{f_{min}}(k,0) = x^2(k) + \\lambda\\xi^d_{f_{min}}(k-1,0)\\,\\!", "1e78aed60e0426874a9441ac0e49cce9": "x|U|", "1e796a0b3f8b5304b315a469e03d207e": "\\mathcal{Q}_{\\alpha}^t = \\{Q = P\\,\\vert_{\\mathcal{F}_t}: \\frac{dQ}{dP} \\leq \\alpha_t^{-1} \\mathrm{ a.s.}\\}", "1e7a04bf0828dde479ede2a9786cb8cf": "b=gb_1", "1e7a10f3033a4a5115ced31dbc7be2ea": "b^2 + 2b + a", "1e7a3dca7e341ed0a9e192b8e1a41bbd": "F(f,x,min,max)", "1e7a583a2c2f40dc1e0ce0ad44f43fd8": "\\delta(x-\\alpha)=\\frac{1}{2\\pi} \\int_{-\\infty}^\\infty dp\\ \\cos (px-p\\alpha) \\ . ", "1e7a76e54b24d1fd8be98ae26d2aaf02": "f_2(x)>g_2(x)", "1e7a95084764435f336d82520870da56": "c = m^2 - n^2 \\, ", "1e7ade3c8b3e50be8b010c2d3ea788eb": " {d^2 \\mathbf{h} \\over d\\tau^2} + R \\mathbf{h} = 0", "1e7b00feebe46b1c1626493867c4eccf": "\\oint ", "1e7b7d2802cb330b1eeaddab728cfb46": " z' = z\\ ,", "1e7b7e3fd5b47c9570dedbb50136a6cb": "+\\;1\\;2", "1e7bc1632394649623a661545ce1a7cf": "\\mathbf P \\neq \\mathbf{NP}", "1e7bc9a2ab0d5d89a02786140dd211e6": "\\langle E\\rangle = \\frac{1}{V}\\int_\\Omega H(\\boldsymbol{r}) \\frac{e^{-\\beta H(\\boldsymbol{r})}}{Z} \\, d\\boldsymbol{r}", "1e7bebf86d156f7ad96432f008a6e35c": "\\color{Dandelion}\\text{Dandelion}", "1e7c07ccc754819426431c9497d34914": "\\, C \\, ", "1e7c0c3ed3b4738b500cb336f2fb412d": "f(x) = \\sum_{n=0}^N c_n \\phi_n(x)", "1e7c6bbe7dddffd8b881e4d0c8bc33e0": "\\tau_1 \\subseteq \\tau_2", "1e7cd2a0dee459160c9f2bcc1235bc1b": " M_A = \\{A(x) : \\mu_A (x) = \\mu_A (0)\\}. ", "1e7ce94b292b3d89bc8519e65cc3439c": "G(n, t) = \\frac {1}{\\sqrt{2\\pi t}} e^{-\\frac{n^2}{2t}}", "1e7d174b547879e98eb9401a87dedd23": "p \\ \\sim\\ a\\mathcal{N}(0,\\,\\sigma_1^2) +(1- a)\\mathcal{N}(0,\\,\\sigma_2^2)", "1e7d1b003e3176219b2ae5ebbeaf1031": "Z=\\frac{m_1z_1+m_2z_2}{m_1+m_2}", "1e7d5b1120971bec627357369217cb27": "\n\\mathbb{E}_{k-1}\\,\\mathbf{X}_k = \\mathbf{0} \\quad \\text{and} \\quad \\mathbf{X}_k^2 \\preceq \\mathbf{A}_k^2 \n", "1e7d6aeb607c3dede585d9c31e2c0b84": "\\partial_\\alpha", "1e7d7566c106841d6cfe9b1fd747e964": "\\varphi (L) X_t = \\theta (L) \\varepsilon_t\\, .", "1e7da01d39e5161e28caad04c027d09f": "\\alpha_{A,B,C}", "1e7e437b6bda6ebbf2bb803bf0d8b527": "\\mathcal Q", "1e7e5f5a7518cb92b3ab9738cbb66472": "\\Phi_F=", "1e7e737f03162a66cf88ce8d6f914a7f": "q - 1", "1e7ea31ae782c651874ff5025183cbb4": "\\displaystyle-13.26~\\mbox{dB}", "1e7ec45e0a6b809a1c7c568779937091": "y_1, ..., y_m", "1e7f545b52a76f19ae73c16220ecdddb": "{\\mathcal L}^2_i", "1e7f8a9cc31a5c86bea281fcf5469160": "\\surd \\!\\,", "1e800e0087577d6240a8fdc1d55e77b4": "\\mu= -k_{B}T\\ln(Q_{N+1}/Q_{N})=-k_{B}T\\ln\\left(\\frac{V/\\Lambda^{d}}{N+1}\\right) - k_{B}T \\ln{\\frac{\\int ds^{N+1}\\exp[-\\beta U(s^{N+1})]}{\\int ds^{N}\\exp[-\\beta U(s^{N})]}}=\\mu_{id}(\\rho) + \\mu_{ex}", "1e808aad9132c1e4c6a9ae9599a71dd6": "(c-v)", "1e80b4619b3906efb16c182dfed52079": " g = (1 - hi)/2, \\quad g' = (1 + hi)/2 ", "1e8113563f6efbe6fc7582915bd9d196": "B \\subseteq A\\,\\!", "1e81560541aba927c9901960f577c97d": "c(V) := \\sum_{i=0}^n c_i(V),", "1e8174308ccec52a8ed5a958ed5a5286": "E(R_i) = \\alpha+ \\beta_{1}F_{1}+...+\\beta_{N}F_{N}.\\,", "1e8174f459bbbf902d2b5fe5e9d2f9fb": " \\log{\\lambda} \\, \\sqrt {\\frac {n \\, d_2} {4}} ", "1e81c86f8bfe80f31b02c4a26b4b47ef": "PX^rXY = PY^rYX", "1e81f9ac6ddbbaaad4c6546eaa992115": " \\tilde{\\boldsymbol{a}}", "1e822c1b8bc59590eba1cd3cb398345f": "T_{M}", "1e82876a796b9b9c6bf6343c3d250800": " \\sqrt{(x - x_0\\cos\\omega t)^2+y^2} - L=0\\,\\!.", "1e82a85e0675eb78e9b164eed57fd98c": "\\displaystyle x^2 - 7 y^2 = 1.", "1e82dbe1cd5035ffb372877e90e383db": "S_{-}=S_x - i\\cdot S_y", "1e82f1965e874d393899f37a32735133": "(1-|\\xi|^2)^{\\delta}_+", "1e833757db81cc56d508d7e92493ba05": "s\\to \\infty", "1e835d236501c8b76c71ab325a52ce43": " Z = i \\omega L + \\frac {1}{i \\omega C} = i \\left ( \\omega L - \\frac {1}{\\omega C} \\right )", "1e8362535fbd55ce411d3435ae049fa1": "\\frac{\\alpha \\in \\Gamma}{\\Gamma \\vdash \\alpha} \\qquad\\qquad\\text{Assum}", "1e83a0cf3dc85f38e57fcb9ffe0a2e27": "f(t)f(t^{-1})", "1e8412230afd0e7398ffa44be708db25": "C(s) = \\left(K_P + K_I \\frac{1}{s} + K_D s\\right).", "1e8430cd4400644e3be887d094e1f718": "k_1\\lambda/NA.", "1e84bea511a3aad79dba2270cf88357d": "2.9 \\times 10^{-13} \\ \\mathrm{seconds} \\,", "1e84e6be586b9aa4feee7cfa46717109": "\\{z_i\\} = \\{-1, -1, -1, 0, 0, 0, 1, 1, 1\\}", "1e84e7aebef9db2a5b1cf5dfd0412660": "|f(z)-g(z)|< |f(z)|+|g(z)| \\qquad \\left(z\\in \\partial K\\right)", "1e852e0f33b4828c0b6f4c97ad83de37": " P = SD", "1e85405ccdb068dfc9f572ad1e2e99df": "f : \\widehat{\\mathbb{R}} \\to \\widehat{\\mathbb{R}},\\quad p \\in \\widehat{\\mathbb{R}}.", "1e855871a2cd6171c400620ec84b4907": "\\mu_k = \\mu \\star \\cdots \\star \\mu", "1e85aeb8635edcbda752900f27545be5": "\\overrightarrow{Vr} = \\overrightarrow{\\omega}\\times\\overrightarrow{r}", "1e85e17009873544fc0de38506cbc46e": "R(l)= P", "1e85f001be5638f058db9553abc27b4b": "\n{\\rm Var}\\left[ {\\bar x} \\right]\\,\\,\\, = \\,\\,\\,{\\rm E}\\left[ {{{s^2 } \\over {\\gamma _1 }}\\left( {{{\\gamma _2 } \\over n}} \\right)} \\right]\\,\\,\\,\\, = \\,\\,\\,{\\rm E}\\left[ {{{s^2 } \\over n}\\left\\{ {{{n\\,\\, - \\,\\,1} \\over {{n \\over {\\gamma _2 }} - \\,\\,1}}} \\right\\}} \\right]", "1e8645d11348a002175e879941581a47": " \\hat{x} = i\\hbar\\frac{d}{dp} ", "1e866d5f281eac8d5c308adc87f8ff79": "\\omega = {2\\pi f} = \\frac{q B}{\\gamma m_0} = \\frac{\\omega_0}{\\gamma} = {\\omega_0}{\\sqrt{1-\\beta^2}} = {\\omega_0}{\\sqrt{1-\\left(\\frac{v}{c}\\right)^2}}", "1e86942d94320f52ecec0ef9a3c240e6": "a\\in X", "1e869f65393143c1fa4e48406908c913": "\\begin{align}\n \\det P &= h_{\\mbox{e}} \\cdot g_{\\mbox{o}} - h_{\\mbox{o}} \\cdot g_{\\mbox{e}} \\\\\n \\exists A\\ A\\cdot P &= I \\iff \\exists c\\ \\exists k\\ \\det P = c\\cdot \\delta \\rightarrow k\n\\end{align}", "1e86a59aaeacba51ef67c5d782567489": " \\mathrm{Ai}(x) \\sim \\frac{e^{-\\frac{2}{3}x^{3/2}}}{2\\sqrt{\\pi}x^{1/4}}.", "1e86b2387fd7153b65b59f62a0a84da6": "\\operatorname{d}f \\triangleq \\partial_a f \\operatorname{d}x^a", "1e86f20d2b9663939272a08b5fe03347": "\\Diamond\\equiv\\lnot\\Box\\lnot", "1e875930916055e38a1aa149e49e8e0e": "\\sum\\mathbf F", "1e87739c52c29e47366dcf1c2e2ba400": "\\mathrm{Var}[X] = \\frac{1}{\\lambda^2},", "1e878283d8545e84fa5ae0c64ad5c558": "a(x-y)", "1e87afc305aef174158abf0e9ebdef40": "\\beta = (-27-8y-9w+6w^2-18yw-11yw^2)/23.\\ ", "1e87b01155ef0c8a49aa0db71e773285": "\\pi_{jk}", "1e87c89899dbdbb094664e5069042f64": "A(t) = [{\\rm Transmitter Release}(t)/ {\\rm Transmitter Release} (0)] - 1,", "1e8805fea863aeee2e770a679221d945": "[2]P_1", "1e881f1147e5a7d5a0c70f0dd278cc79": "\\mathrm{cov}(\\epsilon_1,\\epsilon_2) = 0", "1e885daabb04b3785157337cd8035b53": " X' ", "1e88d273f7b53f586c2c007fa3eaf27a": "x_n \\to c", "1e88ee1d296e6d1b55a0590a0b6df693": "\\phi[J;k]=\\frac{\\delta W_k}{\\delta J}[J]", "1e88f0c74292063f438ebd9ee1dd3a3c": "\\hat{\\xi}^{i} = (\\hat{x}^1, . . . , \\hat{x}^n, \\hat{p}_1, . . . , \\hat{p}_n) \\in Op(L^2(\\mathbb{R}^n)).", "1e890adfba7893718038ec806bf6f474": "\\frac{v(t)^2}{2}-\\frac{v_0^2}{2} = \\frac{GM}{r(t)}-\\frac{GM}{r_0}.\\,", "1e8926511aacf32a13d69853fccb8b21": "\\textstyle a=\\frac{1}{2}", "1e8935a0ff187de7a0d29e64f8368fe0": "H(V)=-\\sum_{j=1}^C P'(j)\\log P'(j) ", "1e893d796502e82a6397cc05a0b14270": "\\nabla y_t = a_0+a_1t+\\delta y_{t-1}+u_t \\,", "1e898ee02cc457f607e6775a6b41c35a": "\\frac{1}{\\Phi(\\varphi)} \\frac{d^2 \\Phi(\\varphi)}{d\\varphi^2} = -m^2", "1e8a3b0c83f3b1356bb3d0e338f1be1b": "|s\\rang = \\frac{1}{\\sqrt{N}} \\sum_{x=1}^{N} |x\\rang", "1e8a6e39c7d8b2e7cf9244b3fe00206c": "a^2 - b^2 = (B + A) (B - A) \\,", "1e8a6f1eba76f41565744b2b5a60faf1": "[X,O]", "1e8a77e39c48f2a34cbad4b5bde534be": "\\sum_{n=-\\infty}^{\\infty} e^{inx}=", "1e8a9b76ba915e36d15fb02748c98b43": "\\operatorname{tr}(A^k) = \\sum_i \\lambda_i^k", "1e8aa01b05bb3acf8fc94e10f42e2849": "x (x + 1) (x - 3) (x + 2) (x - 2)", "1e8adead525ae74ded132a535815e836": "N=53", "1e8b05e8eb3c10c67aea261922c6a515": " \\int_{t_n}^{t_{n+1}} f(t,y(t)) \\,\\mathrm{d}t \\approx \\tfrac12 h \\Big( f(t_n,y(t_n)) + f(t_{n+1},y(t_{n+1})) \\Big). ", "1e8b25ae6007d18dc5fb9c7d2a327acc": "y_c = (C_1x + C_2)e^{-bx/2}\\,\\!", "1e8b776274ed98a563cb2f33261db947": " \\mathbf{A} \\in \\mathbb{R}^{n \\times d_f}, \\mathbf{B} \\in \\mathbb{R}^{n \\times d_g}", "1e8bb92fd39a8197600aa6a6ab063543": "a=\\epsilon", "1e8bdfb26cee0d20342689d4d42c18a7": "\\Phi \\left( R_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu} R \\right) + \\left(g_{\\mu\\nu}\\Box -\\nabla_\\mu \\nabla_\\nu \\right) \\Phi + \\frac{1}{2} g_{\\mu\\nu}V(\\Phi) = \\kappa T_{\\mu\\nu}", "1e8c46590be47a55c12810c111ad5388": " \\sqrt{1000}", "1e8c9455378cb019a1adc4640bf6b82f": "H^{(4)} = x\\partial_x - y\\partial_y - 2y'\\partial_{y'} - 3y''\\partial_{y''}-4y'''\\partial_{y'''}-5y''''\\partial_{y''''}.", "1e8cb71fdce2f50477e39b5a6193c381": "S(\\forall)", "1e8d5d88133019a244b3b3f7e2ca62ca": "\\rho=|\\psi|^2", "1e8d7c15a990ff1bb7e77997832e2959": " \\frac {T \\left(x \\right)-T_o}{T_L-T_o}=\\frac {e^\\left(Ax \\right)-1}{e^\\left(AL \\right)-1}", "1e8d903de5c6c2c4f790bb10e0ab6c3d": "\n{1 \\over 1-u/c}\n\\, .", "1e8dbf14472f2ff22e6193c24c67e193": "\\frac{10}{\\sqrt{a}}", "1e8de0dca01123264ecf4e9fd548bd0a": " b = 0.26\\ V_c", "1e8e20e14406b61c060c3b0a4549731d": "E_n^{(3)}=\\frac{V_{nk_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}}-V_{nn}\\frac{|V_{nk_3}|^2}{E_{nk_3}^2}", "1e8e33292fa3d52911c13ea72a3ed4e8": "\\dot{\\textbf{S}}(t) = -\\textbf{S}(t)\\textbf{A}-\\textbf{A}^{\\text{T}}\\textbf{S}(t)+\\textbf{S}(t)\\textbf{B}\\textbf{R}^{-1}\\textbf{B}^{\\text{T}}\\textbf{S}(t)-\\textbf{Q}", "1e8e410f33f8835cebf75cbfe47dc6ed": "C_{\\alpha-1}(x) + C_{\\alpha+1}(x) = 2\\frac{dC_\\alpha}{dx}\\!", "1e8e7508cbde21c325411e7ac541a594": "\\dot{r}_1 = \\lambda_1 r_1", "1e8e804a8abb60c2ea93b6e4c2ee346a": "D_q(f(x)) = \\frac{d_q(f(x))}{d_q(x)} = \\frac{f(qx) - f(x)}{(q - 1)x}", "1e8ee50f289d1eb2915c2df6ce794358": " \\frac{1}{\\eta} \\frac{\\Delta P}{\\Delta x} = \\frac{d^2 v}{dr^2} + \\frac{1}{r} \\frac{dv}{dr} ", "1e8fb5fe03bb77c216f73e2da35f99c9": "\\frac{dG(\\mathbf{r}(t))}{dt} = \\nabla G(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) = \\mathbf{F}(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t)", "1e8fecec6574defe9a47a25d5e6ece2f": "f(a)=s", "1e90718d1c8f722d2b1382f2988d200f": "P = \\left(\\frac{8}{47} \\times \\frac{4}{46}\\right) + \\left(\\frac{8}{47} \\times \\frac{8}{46}\\right) = \\frac{96}{2162} \\approx 0.0444", "1e909059821760764bc9a81e277e2b54": "\\beta_1\\le\\alpha", "1e909fdc9a50cd6c3cc08459f767b893": " 02\\,\\!", "1eac97b0bd15177655ce2ac021e5bc78": "\\sum_{k=0}^\\infty \\frac{(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\\arcsin z, |z|\\le1\\,\\!", "1eacbcc0397be0e4a907e3ce1a198110": "\n\\text{ROL}(W)_j^i = W_{j-i \\pmod 4}^{i}\n", "1ead2716db7df7dcc1709a6a33ec4f4e": "p_k^\\prime(t)=\\lambda_{k-1} p_{k-1}(t)+\\mu_{k+1} p_{k+1}(t)-(\\lambda_k +\\mu_k) p_k(t)\\text{ for }k \\leq K \\, ", "1ead44e5ad8e6c3bea2a31f2dc914472": "\\scriptstyle PG(4, q) ", "1eadba1eac75e1625f3b97758c9d387f": "|a_{ab}| = \\sin^2 y \\, ", "1eadcb31337236828c385ff9a720cbd7": "\\tan x + \\cot x = 2", "1eade162f25e730eba454664e8edb94d": "\n\n\\phi(e^{-4\\pi})=\\frac{e^{\\pi/6}\\Gamma\\left(\\frac14\\right)}{2^{{11}/8}\\pi^{3/4}}\n\n", "1eae2486887be456286befc09daedd27": "entry(O_{r})", "1eae3b5da849864d11e3a01f1a639cd3": "\\lambda=0\\;", "1eae8d8bf99a09bf2866cadc7ef1032b": "\\hat{\\beta} = \\frac{\\Gamma(\\frac{n-1}{2})}{\\Gamma(\\frac{n-2}{2})} \\sqrt{\\frac{2}{n-1}} \\frac{\\bar{D}}{s_D}. ", "1eaea362a7da9eab5dfbf953006d0b3d": "\\mathbf X_1 \\boldsymbol\\Theta \\mathbf X_2' ", "1eaeb4e469a7bcd8303e4c5e7bd0d7b1": "\\begin{bmatrix}\na1 & a2 & a3 & a4\\\\\nb1 & b2 & b3 & b4\\\\\nc1 & c2 & c3 & c4\\\\\nd1 & d2 & d3 & d4\\end{bmatrix}", "1eaede3fb376df81307243db8085fb51": "\\Omega = (\\mathbb{R}^n)^T", "1eaf00ae3a22e4ffa8fbf0c72614014d": " \\sigma^2>0", "1eaf1750ac8f09f965f5fa1f4dc78cb3": " x_1, x_2, ...,x_{n+2}", "1eaf2b478b94a8e39838dec596a9678f": "g_1(x)\\ge 0, \\dots, g_k(x)\\ge 0", "1eaf6a8a048a168eebb15c592fc60938": "\\alpha=\\frac{1}{V}\\left(\\frac{\\part V}{\\part T}\\right)_p.", "1eafa4076a4f6cd5f516fb25fd29219d": "\\kappa=0.76/\\sqrt{2}", "1eafb8c762aad9f9e0904f7182cd6602": "f(t) = ae^{it}", "1eafbb3bc6ac35501dac29c6dffa7972": "B : V\\times V\\rightarrow K:(u,v)\\mapsto B(u,v)", "1eaffe0a843e0a227878c0cddff2590b": "d(\\mathbf{X} \\otimes \\mathbf{Y}) =", "1eb0319c222eeeb48f74082365c147bb": "10^\\frac{1}{10}\\,", "1eb0847d06faf9748dfb3de3414c1b9e": "f_v(k,r)\\approx f_0(E,E_{Fp},T_p)", "1eb0ab4694f9f819624bd2c8b267f9ff": " \\overline{u_i u_i} ", "1eb0b99262ae6f45bf59fd05dcfc5bf5": "B \\to bB | a", "1eb0fd27e19297521d5675eab1493b6b": "G = \\xi \\sin y.\\,", "1eb188bbaf4e8189b0e279e691d157c0": "\\omega^2 = \\Omega^2(k).\\,", "1eb18bb9ec47e7ef72520976183ac651": "\\mathbf{M}=\\begin{pmatrix} 1 & 1 \\\\ 0 & -1 \\end{pmatrix};\\quad\\Longrightarrow\\quad\n\\mathbf{M}^2=\\begin{pmatrix}\n1\\times 1+1\\times 0 & 1\\times 1+1\\times -1 \\\\ 0\\times 1-1\\times 0 & 0\\times 1-1\\times -1 \\end{pmatrix}\n=\\begin{pmatrix}\n1 & 0 \\\\ 0 & 1 \\end{pmatrix} = \\mathbf{I}\n", "1eb1a638125a45b12b1920cc39313b7b": " v", "1eb1e2e6fbebe0f85954f10e242e7b94": "y_n = \\Phi(\\ h_{n-1},t_{n-1},y(t_{n-1})\\ )\\ y_{n-1}\\quad", "1eb252f7f2c44e4209ab920309ccd6d0": "A(\\tau) = \\int_{-\\infty}^{+\\infty}I(t)I(t-\\tau)dt", "1eb270cf3d338409d9e60187f92390fa": "\\frac{N_{i+1}}{D_{i+1}} = \\frac{N_i}{D_i}\\frac{F_{i+1}}{F_{i+1}}.", "1eb27c23d4ff90e3f25a86fb74533d6b": "\\vec x(t)", "1eb27d73010feab8ae80aba1c2868504": "\nn_\\gamma = 0.243\\left(\\frac{k_\\text{B} T}{\\hbar c}\\right)^3,\n", "1eb29e3cdf958fac9cc6aadcd673407e": " U'\\left(c - G(l)\\right)\\left(\\frac{dc}{dl} - G'(l) \\right) = 0 ", "1eb2ba37b7483956db17dbc186067fa3": "\\max \\{\\mathrm{Eigenvalues~of~} \\bold{c} \\}>1 ", "1eb2e6e46266c3257075ba1a6f397dac": "P_N = P_0(1+r)^N - c (S)", "1eb30d26c30426eefd6e5a992d5385ac": "p_{xy}(x,y) = \\textstyle \\sum_{i=1}^3 \\sum_{j=1}^3 a_{ij} i x^{i-1} j y^{j-1}", "1eb39a201897580c41fe452696be65e8": " r^2 = k_1^2 + k_2^2 +\\cdots+k_n^2 ", "1eb3ad11f243fc51325908e63c21f4b3": "K(z)=A e^{\\psi(-2,z)+\\frac{z^2-z}{2}}", "1eb3af657da6b3b9ce9a7a20c6bd5548": " m \\,", "1eb3c24d6fe5ad4c15ff82422cbc3fbd": "\\begin{cases}\nx' = \\gamma \\left( x - vt \\right) \\\\\nt' = \\gamma \\left( t - \\frac{v}{c^2} x \\right) \\,\n\\end{cases}", "1eb3e51e34ea37fd500d8b1dc6a094d0": "a_1, \\ldots, a_m\\in {\\mathbb C}^d ", "1eb403d789e2ed4931912221521a01d0": "V_{BE1} = V_T \\ln \\left( \\frac{I_{C1}}{I_{S1}}\\right) \\ . ", "1eb462ffcdc77f555c772f3667aaa5f0": "\\sqrt{S} \\approx A - \\frac{P^2}{2A}", "1eb4747b7534c0d36614bc45dc3a2de4": " = P_1.P_2 ", "1eb47b89c91d71aa2f79919d19291104": "M(n) > (4 - 6^{-1/2})n/5", "1eb4ad517ed799a98fda4d0804e77f64": " \\bold y' = \\bold A \\bold y + \\bold f(x), \\qquad \\qquad \\qquad \\qquad \\quad (5) ", "1eb4b0483cbd69e03b061c51698c5826": "\\gamma < \\alpha", "1eb4d9edfbd8070c1939a09236c9ac84": "p_a=\\frac{2aT}{a^2+b^2-c^2},", "1eb4e4be2223b428703eef6c7124088c": " B = Q \\, \\Delta t", "1eb541989515fb3004afde202e9f96a1": "\\mathfrak{d}", "1eb5523dd5141a3528c44fe436670141": "\\int\\frac{1}{x^2+a^2} \\, dx = \\frac{1}{a}\\arctan\\frac{x}{a}\\,\\! + C", "1eb55750136e4b81e9ca596c930dadbd": "\\textstyle < r", "1eb5cbc1cd53eaa87c75eedd42427d28": " S_{M_j} \\subseteq [t] ", "1eb5e7cf10474125e216d22a7371888b": "Radial \\ Velocity > 0.5 \\left (\\frac {PRF \\times C}{Transmit \\ Frequency} \\right)", "1eb604f4b3d67b9579f88b6101c82d0f": "\\gamma ( x^\\prime + \\beta ct^\\prime )", "1eb63d7bb002ce211e87dcb8fdacfa6c": "Q = \\left(\\frac{V \\times I \\times 60}{S \\times 1000} \\right) \\times \\mathrm{Efficiency}", "1eb67b6313a6bd419e37b9f470230f12": "\\displaystyle \\frac{\\mathrm{d}^2 X}{\\mathrm{d}t^2} = \\frac{c^2}{2} \\varepsilon \\gamma_{00}", "1eb69130f20bcad84623b374650bd4c0": "\\textstyle\\tilde{\\beta}", "1eb6b4dc0e030aa9d9a0fead410245e6": "(x_3,y_3,0)", "1eb6ee0f1d09829a14aa4c231804ebcb": "\n \\overset{\\square}{\\boldsymbol{\\tau}} = \\dot{\\boldsymbol{\\tau}} + \\boldsymbol{\\tau}\\cdot\\boldsymbol{\\Omega}\n - \\boldsymbol{\\Omega}\\cdot\\boldsymbol{\\tau} \n", "1eb713fb2232eb9765474f4445e1f516": "\n\\begin{align}\n \\int \\sec \\theta \\, d\\theta& \n =\\tanh^{-1}\\! \\left(\\sin\\theta\\right)\n =\\sinh^{-1}\\! \\left(\\tan\\theta\\right) \n =\\cosh^{-1}\\! \\left(\\sec\\theta\\right).\n\\end{align}\n", "1eb717a0ad95e460119838cd78cdb560": "E = \\alpha U.", "1eb72583ff9c2e5311bb49e6a74743b2": " g(\\gamma)=\\frac{3}{2\\gamma} (1+\\frac{1}{2\\gamma^{2}}\\sinh^{-1}(\\gamma)-\\frac{\\sqrt{1+\\gamma^{2}}}{2\\gamma})", "1eb73a36533500517ef53038b85bb075": "a = \\sqrt{\\frac{m}{b^3}} = \\sqrt{2^2 \\times 5^2} = 10 \\, ,", "1eb7e11a305ed75dfbdb2e9ec6bd4abd": "\nc^{2} = \\frac{1}{1 - \\frac{r_{s}}{r}} \\, \\frac{E^2}{m^2 c^2} - \n\\frac{1}{1 - \\frac{r_{s}}{r}} \\left( \\frac{dr}{d\\tau} \\right)^{2} -\n\\frac{1}{r^{2}} \\, \\frac{L^2}{m^2}\n\\,,", "1eb7fd7fc72fa6ebc4a1f2c5372dc736": "e(g^x,g^y)=e(g,g^z)", "1eb822c1536ddd256f247f2c18394b57": "\\mathbf{A}\\hat e_1\\ ,\\ \\mathbf{A}\\hat e_2\\ ,\\ \\mathbf{A}\\hat e_3.", "1eb8422215acbc85d7e07cb68b8ebb79": "\\psi^{\\alpha}_{j}(\\mathbf{r}_{j}) = \\psi^{\\beta}_{N_{\\alpha}+j}(\\mathbf{r}_{N_{\\alpha}+j}),\\ \\ \\ 1\\leq j\\leq N_{\\beta}.", "1eb88ef5f847daf8fe938dc577168c22": "z \\leftarrow w", "1eb908b2f3d928c4be9c6a7021d71df4": "x^{\\frac{3}{2}}", "1eb9418dc7a627f01fdb6559de364dae": " c_{i+1,\\sigma} ", "1eb97e787a29b7eba3f3e7952baffa2a": " H(X) = -\\int_X f(x) \\log f(x) \\,d\\nu(x) = -\\sum_{x\\in X} f(x) \\log f(x). ", "1eba2348fa963a9d5633506a55f96a69": "I_J = 0,2 \\mu", "1eba63d2398920deec0b5cb407f6ccdc": " X^\\mu \\rightarrow X^\\mu + \\delta X^\\mu ", "1eba6caacde14b058d866140c4becda5": "V_{\\omega}", "1eba74430ba4c3a6e3c64b2f2a53008c": "\\text{fmap}: (A \\rarr B) \\rarr (M \\rarr A) \\rarr M \\rarr B = f \\mapsto g \\mapsto (f \\circ g)", "1eba79fad5125d7a595e26aed932f2fc": "\\tilde{\\psi} \\in L^2(\\mathbb{R})", "1ebab854b80c84d7bf85081c61e15bb1": "{258 \\cdot {{42}\\over {43}}}\\over {49\\choose 6}", "1ebad413d19b19c7d0993da142e34b53": " \\scriptstyle V(u,\\Omega)<+\\infty ", "1ebaffa1e997c3467a93fb42f6cd5f3f": "\\mbox{SAIDI} = \\frac{\\mbox{sum of all customer interruption durations}}{\\mbox{total number of customers served}}", "1ebb03d778b6d5cc34693b39fcd8521b": "g_{k,n}(z)=z+\\varphi_{k,n}(z)", "1ebb1bee05b6f39e75c62132bda2d9e3": "\\mathfrak{H}(k; \\gamma, \\infty) =\n\\begin{pmatrix}\nk & (1 - k) \\gamma \\\\\n0 & 1\n\\end{pmatrix}.", "1ebb1bf30a0eb0253fccf64abf0340a0": "\\sum_{-m \\leq i\\quad\\, \\atop j \\leq m} \\langle A(i,j) h_i, h_j \\rangle \\geq 0.", "1ebb3364f0e9db61ef1e91d37fd2fe6c": "{\\scriptstyle A}", "1ebb4f51b91b59ad3b759bc796368346": " \\overline{h} = {\\overline{p} C_p \\over \\overline{\\rho} R}.", "1ebba5c39adb8e3bd296222db8bfb921": "F_n\\Bbb R^2", "1ebbac688acaee729f998a9e6366f104": "\n\\begin{align}\n\\mathbf U(\\mathbf x,t) &= \\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad &\\text{or}& \\qquad U_J = \\delta_{Ji}x_i-X_J =x_J-X_J \\\\\n\\nabla_{\\mathbf x}\\mathbf U &= \\mathbf I - \\nabla_{\\mathbf x}\\mathbf X = \\mathbf I -\\mathbf F^{-1} \\qquad &\\text{or}& \\qquad \\frac{\\partial U_J}{\\partial x_k} = \\delta_{Jk}-\\frac{\\partial X_J}{\\partial x_k} = \\delta_{Jk} - F^{-1}_{Jk} \\,.\n\\end{align}\n", "1ebbf6f08e5c59420c4b625fe9c3438b": "d\\tilde{\\rho}(\\rho)=0", "1ebc54845e26c3f8c0e3f3d16d1ba0b2": "\\scriptstyle e' = (v, u)", "1ebc96c7171c6c6e3dd2693c232f979b": "\\int_\\gamma f(z)\\,dz=0", "1ebc9b9ab89e548d244f790a3d49a1ff": "(x-1)-\\frac{1}{2}(x-1)^2+\\frac{1}{3}(x-1)^3-\\frac{1}{4}(x-1)^4+\\cdots,\\!", "1ebdbced25bfd2da459ee8e5e392cb8b": "\\scriptstyle I", "1ebe36cdc84b91a06059e4216dc77cd4": "(r + \\mathrm{d}r, \\,\\theta + \\mathrm{d}\\theta, \\, \\varphi + \\mathrm{d}\\varphi)", "1ebe9460cbcb56b1c2745a1811380bf4": " D_H = D ", "1ebeeeddf6cfbc1739f288b5a1abf54a": " \\langle 0,...,k \\rangle ", "1ebefbc1820dde5c50e0ee06e695c605": "F = -kx \\,", "1ebfb6ba9c25312cf01cb8f62ae61e5c": "I_n \\otimes \\Phi", "1ebfeda33a028acc66c6867405ad96f1": "\nc = \\sqrt{\\frac{C}{\\rho}}\\,\n", "1ec00546f44bc0f23e184ac04e8faa45": "u^c", "1ec0091eff0f72f0b43f941266c5d50a": "i^0 = i^{1-1} = i^1i^{-1} = i^1\\frac{1}{i} = i\\frac{1}{i} = \\frac{i}{i} = 1 \\,", "1ec021f4c3dd9dcb9f443e1681fdf9a8": "\\sum_x \\log d_x", "1ec029cc60e4f40018998a544cb2abda": "\\gamma^{(j)}\\,", "1ec07abf40f29e3bad8c3ed0c7ce13a9": "B^\\prime ", "1ec0a3265956fdcb7000186655bc5179": "d+2w(u)\\geq 2d", "1ec0f4b4a81fd6310c27e26073cae36a": "\\dot{\\sigma} = \\dot{x}_1 + \\dot{x}_2 = \\dot{x} + \\ddot{x} = \\dot{x}\\,+\\,\\overbrace{a(t,x,\\dot{x})+ u}^{\\ddot{x}}", "1ec119c66c955d3dd5b0eeb7427c0014": "(10\\bar 1 1100\\bar 1 10)_s", "1ec1290f8a4a4c061afa10121e93d1ac": "T^{00} = \\rho,", "1ec18a4052c11adfb09f4a75a3975532": "\\mathbf{i}_k", "1ec19cc751ac5590b83da3c41d6b509a": "\\gamma(t)= \\gamma'\\ast\\lambda(t)", "1ec1a0912d95a892b4bab948cd8ce556": "\\Phi_G=\\left\\{ V_1^G,\\dots,V_k^G\\right\\}", "1ec1b0da7355babafa9b7e836e8e7cc4": "\n\\int dq\\rho _{0}\\left( q\\right) f\\left( q\\right) \\delta \\left( A\\left(q\\right) -a\\right) =\\int dq\\rho _{0}\\left( q\\right) F\\left( A\\left(q\\right) \\right) \\delta \\left( A\\left( q\\right) -a\\right) =F\\left( a\\right)\\int dq\\rho _{0}\\left( q\\right) \\delta \\left(A\\left( q\\right)-a\\right).\n", "1ec25ab34fbd021319e99b0e2e78f4ae": "-\\dot q_{\\rm ext} = \\boldsymbol \\nabla \\cdot (\\kappa \\boldsymbol \\nabla T) + \\mathbf J \\cdot \\left(\\sigma^{-1} \\mathbf J\\right) - T \\mathbf J \\cdot\\boldsymbol \\nabla S", "1ec291bf65047f349d3a68b24ddd4713": "0 \\mod q_{\\ell}", "1ec2b08201603b79ba8eaf478495c4ae": "\\scriptstyle \\mathcal F", "1ec2b4e835eb0b5c2f0911b3d98a6fac": "\n\\widetilde{u}(x) = \\frac{u(x)}{u_0}.\n", "1ec3411f57192614871b8895b9063a4c": " R(Q)/R_F(Q) = \\left|\\frac{1}{\\rho _\\infty} {\\int\\limits_{ - \\infty }^\\infty {e^{iQz} \\left( \\frac{d \\rho _e}{dz} \\right) dz} } \\right|^2 ", "1ec37fcc002d088d99d7035fcbf66f5b": "R^i\\,\\!", "1ec39ae7c2acd408004153069552825c": " x \\otimes y \\rightarrow y \\otimes x ", "1ec3da35329c1ddd2e2daeb9a3f39cbd": "\\operatorname{E}[(X)_n]=\\lambda^n.", "1ec3f969df401edf119afc601685afe7": "\\lambda_0 (t) = \\frac{c_0}{y_0 - a_0 t}", "1ec50ef80f5b75f1be0bfb4a313d1999": "\\sigma^a", "1ec525e1383c756df033f0aae02897ea": "w = w_1w_2 \\cdots w_k \\in Y^+", "1ec53c7bc6a4f251d05b39e8658ccd8c": "P_{L1}=V_{L1}I_{L1}=V_P I_P\\sin\\left(\\theta\\right)\\sin\\left(\\theta-\\varphi\\right)", "1ec583308d8b08c6057294571484abd9": " y(t+h) \\approx y(t) + hy'(t) \\qquad\\qquad ", "1ec5b9681a2294e1e67deca9429992ad": "s_{2}(t) =\\overset{\\cdot }{s}_{1}(t)+\\alpha _{2}(t)s_{1}^{\\gamma _{2}}(t)", "1ec5c265d552445b7fb34c244df285d1": "\\tilde{C}(u) = W(u)", "1ec683e05e7921e95e358f72f778dd17": "\\operatorname{E}[X] = \\frac{a}{3}", "1ec7099c3d74c9e4b4e046dbd5aba758": "\\sum_{\\lambda\\vdash n} (t_\\lambda)^2= n!", "1ec77202edb98270c964ba0be7bd7d32": "\\left(A\\partial_x + B\\partial_y + C\\partial_z + \\frac{i}{c}D\\partial_t\\right)\\psi = \\kappa\\psi", "1ec78147bdba402eb81e5d03a81195f7": "A_q(n,d)", "1ec78c301719f36037769309903cb376": " M_{xy}'(t) = M_{xy}'(0) e^{-t / T_2}", "1ec79c69bb84d5219a85d59bf2585e4b": "V_T\\,", "1ec79e91d92df04471972727d4bb9404": "10^{-6}", "1ec801116fc1d9b1f2147a635cadcb7c": "\\mathrm{[H^+]^3} + (K_a + C_b)\\mathrm{[H^+]^2} - (K_w + K_a C_a)\\mathrm{[H^+]} - K_a K_w = 0", "1ec83761b4b7c997bfd5dbed0c07e279": "h^{eff}=h+J z m", "1ec850722475d8e580ae1a17de338859": "A_i \\rightarrow A_j \\gamma", "1ec88b186241729643610e9889038694": "(\\mu_{ab}^*(t))", "1ec8a5108db98ecbe0d0278c0b78e878": "D_{26} = \\langle a, b | a^2 = b^{13} = 1, aba = b^{-1} \\rangle .", "1ec8aa82f6aad4cc49960e44b75c0a16": " \\forall H \\in \\mathbf{H}, \\forall v \\in H ", "1ec8cbe560e8240917a675c75d97b089": "\\pi_1 \\big(PSO(2)\\big) = \\mathbf{Z},", "1ec8d7af63e281a0d5407febbd4416b1": "\\prod_{i\\in I} U_i", "1ec8d7eff29ecd71a82e2a4fb6b63275": "\\|\\tau_{v+h} f - \\tau_v f\\|_p=o(1),", "1ec8f0085668d675f0c970e4bbec9280": "\\delta_{\\vec{\\xi}}h=\\delta_{\\vec{\\xi}}g-\\delta_{\\vec{\\xi}}\\eta=\\mathcal{L}_{\\vec{\\xi}}g=\\mathcal{L}_{\\vec{\\xi}}\\eta+\\mathcal{L}_{\\vec{\\xi}}h= \n\\left[\\xi_{\\nu;\\mu} + \\xi_{\\mu;\\nu} + \\xi^\\alpha h_{\\mu\\nu;\\alpha} + \\xi^\\alpha_{;\\mu} h_{\\alpha\\nu} + \\xi^\\alpha_{;\\nu} h_{\\mu\\alpha}\\right]dx^\\mu \\otimes dx^\\nu", "1ec8f4d6e0bbcc7da8cc8b45bc8753c1": "\\lim_{n\\rightarrow \\infty}f_n(x) = \\begin{cases} 0, & x \\in [0,1) \\\\ 1, & x=1. \\end{cases} ", "1ec911c2b681b9eef78e1a55795a6d9d": "F_{max} = mg + \\sqrt{(mg)^2 + F_0(F_0-2m_0g)\\frac{m}{m_0}\\frac{f}{f_0}} ", "1ec91515d3d64d3f7811f9d115134b0c": "\\aleph", "1ec956b9b4dcc57a48a67a7a81be2255": "(s_i f)(t_0,\\dots,t_{n+1}) = f(t_0,\\dots,t_{i-1},t_i + t_{i+1},t_{i+2},\\dots,t_{n+1})\\,", "1ec9a25a6812b49dbb3712f0a283146f": "\\sum a_{ij}b_{ij} ", "1eca0bffac8452289ff430574fc62790": "\\mathbf u(\\mathbf X, t)\\,\\!", "1eca421f0ee2a202ab9a72b8c31fcb44": "\\kappa_2", "1ecacddd93add60feb6cc079a64f2859": " S_{xx}(\\omega) = \\lim_{T \\rightarrow \\infty} \\mathbf{E} \\left[ | \\hat{x}_T(\\omega) | ^ 2 \\right]. ", "1ecb1b8b55f12b5843e4f10be4ea0242": "I = \\mathbf{J} \\cdot \\mathrm{d} \\mathbf{S}", "1ecb2d11982ea453cf1163ec464a5531": "x_{13}", "1ecba53ee71ad3416181255101a4b593": "\\frac{d}{dk}\\Gamma_k=\\frac{1}{2}\\operatorname{Tr}\\left[\\left(\\frac{\\delta^2 \\Gamma_k}{\\delta \\phi \\delta \\phi}+R_k\\right)^{-1}\\cdot\\frac{d}{dk}R_k\\right]", "1ecbb1dedea742667ad6b1433e573773": ", \\left [s(nT)\\cdot e^{-j 2 \\pi \\frac{B}{2}Tn}\\right ],", "1ecbb1ee49a73ef8de2c078073396750": "\n e^z = 1 + \\cfrac{2z}{2 - z + \\cfrac{z^2}{6 + \\cfrac{z^2}{10 + \\cfrac{z^2}{14 + \\ddots}}}}\n", "1ecbd0049570d70901dae7506e9b5ed7": "\\lim_{a\\to\\infty}\\int_{-2a}^a x f(x)\\,dx, \\!", "1ecc0dc5e22a9589cc0c664537248e3b": "E(y) = \\sigma \\sqrt{2/\\pi} \\exp(-\\mu^2/2\\sigma^2) + \\mu\\left[1-2\\Phi(-\\mu/\\sigma)\\right],", "1ecc0fb85f9c89e1cb406b758fead9b1": "\\mathbf{v} = \\dot{\\mathbf{x}} = \\left[ \\dot{\\mathbf{p}}, \\dot{\\mathbf{r}} \\right]", "1ecc6552a9bdc3dc9d7e5907abcff95d": "\n \\begin{align}\n \\overset{\\circ}{\\boldsymbol{\\sigma}} & = \n J^{-1}~\\boldsymbol{F}\\cdot(\\dot{J}~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T})\\cdot\\boldsymbol{F}^T + \n J^{-1}~\\boldsymbol{F}\\cdot(J~\\dot{\\boldsymbol{F}^{-1}}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T})\\cdot\\boldsymbol{F}^T \\\\\n & + \n J^{-1}~\\boldsymbol{F}\\cdot(J~\\boldsymbol{F}^{-1}\\cdot\\dot{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{F}^{-T})\\cdot\\boldsymbol{F}^T + \n J^{-1}~\\boldsymbol{F}\\cdot(J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{F}^{-T}})\\cdot\\boldsymbol{F}^T \n \\end{align}\n", "1ecc6f8f35c8b73b295f2401531692b0": "{4 \\choose 1}^4 - {3 \\choose 1} = 253", "1ecc96509281ceb9a14ca7752aea73a5": "\\cosh\\delta = \\left| \\frac{a^2 + b^2 - c^2}{2ab} \\right|.", "1eccab7596a75d9efad6b16b8a4ccbdd": " ~1/\\beta = k_B T ", "1ecd27c57b10d1a1d5595f8bc0b753de": "\\mathcal{L} \\left\\{ f^{(n)}(t) \\right\\} = s^n \\cdot \\mathcal{L} \\left\\{ f(t) \\right\\} - s^{n - 1} f(0^-) - \\cdots - f^{(n - 1)}(0^-),", "1ecd4b9ba395716a2e4d946246995f66": "\\lim_{k\\to\\infty}\\frac{G^k}{k!}=0", "1ecd513e7722d5d91e6ca38bf7406ce8": "\nC =\n\\begin{pmatrix}\n0&1&0&0\\\\\n0&0&1&0\\\\\n0&0&0&1\\\\\n1&0&0&0\n\\end{pmatrix}\n", "1ecd9c40df5730b41e9171c3879f1a72": "P = \\rho k_BT-\\frac{2}{3}\\pi\\rho^2\\int_{0}^{\\infty}dr\\frac{du(r)}{dr}r^3g(r)", "1ecda1ec9e56eed9b868b17661d5faaa": "C_n = \\frac{i^n}{2^n n!} m \\circ \\Pi^n.", "1ecddf43eb7e714c44e6c40eda1368ea": ",z", "1ecdedab2122c26ebb4abcc1cd1f6355": "\\sum_{n=1}^\\infty \\lambda(n)\\,\\frac{q^n}{1-q^n} = \n\\sum_{n=1}^\\infty q^{n^2}", "1ece566f374b1e5a56e7b527d662b1a7": "P_d = h^0(X, K_X^d) = \\operatorname{dim}\\ H^0(X, K_X^d).", "1ece9307e8a144b0407bbf299c982297": " \\; \\Omega _ R (s_1) = 2 \\; \\Omega _ R (s_2) ", "1ece9e67f6debc628f87c4940f7a213f": "P_0(z)=1; ~~ Q_0(z) = 1 ; ", "1eced98c2a6b6999908a2ef56fdd1a3c": "D= s\\Delta p_{0}\\cos\\alpha_m", "1ecf17e2f655476074d1a3e8601fbeeb": "\\sum_{n=1}^\\infty \\frac{(-1)^n}{n(4n^2-1)} = \\ln 2 -1.", "1ecf20ba5dc5e9823517fc42523001c5": " \\lim_{t \\to \\infty} T^t ", "1ecf270f0290445050f81ac3b4a5d0e4": "\\int_{-\\infty}^\\infty \\frac{\\sin(\\pi x)}{\\pi x} \\, dx = \\mathrm{rect}(0) = 1\\,\\!", "1ecf547d838b4d85fcded4b9d529ac84": "g(z) = \\exp\\left(\\sum_{d|m} \\frac{z^d}{d}\\right).", "1ecf927a1a35e6df99036eb95154335e": "A\\times A := \\{xy: x,y \\in A\\}", "1ecfb6e44370f0b6b4eb4d2878e5f5e1": "U = -\\int_{r_0}^{r} \\mathbf{F} \\cdot \\, d \\mathbf{r}= -\\int_{r_0}^{r} \\frac{1}{4\\pi\\varepsilon_0}\\frac{q_1q_2}{\\mathbf{r^2}} \\cdot \\, d \\mathbf{r}= \\frac{q_1q_2}{4\\pi\\varepsilon_0}(\\frac{1}{r_0}- \\frac{1}{r})+c", "1ecfd85482f88c8586259a19bbd37772": "z_0\\neq1", "1ed0239331fe8d32570400c9b178a854": "\\{ \\{ M , O \\} , O \\}_{M = 0} = 0", "1ed0ea414ba01502814842052d5de139": "f(w)=w/\\phi(w)", "1ed0f83a527955cdd8d14fb0e648cbec": " \\sigma_{xx} = \\frac {-\\mu b} {2 \\pi (1-\\nu)} \\frac {y(3x^2 +y^2)} {(x^2 +y^2)^2}", "1ed15447914f5bf8ee54dee88ce7bab7": "K = \\tfrac{1}{2} |x_1 y_2 - x_2 y_1|.", "1ed166e6ef6ff34b5afe5f77083cf66c": "g_3=\\tfrac{1}{2}(1+(\\eta^2-2)j+(3-\\eta^2)ij).", "1ed18a8ab3bc29b703218e07354ee43a": "\\zeta(10) = 1 + \\frac{1}{2^{10}} + \\frac{1}{3^{10}} + \\cdots = \\frac{\\pi^{10}}{93555} = 1.000994...\\dots\\!", "1ed196f2325656a83ec23cc61c7e24fc": "A = \\sum_n a_n |e_n\\rangle\\langle e_n|", "1ed1f129f7f85ec8f52eecc00ba8a013": "\\exists x Rx", "1ed208f30969a8a5fd0197a2e4521373": "\n R = I \\cos\\theta + [\\mathbf{k}]_\\times \\sin\\theta + (1 - \\cos\\theta) \\mathbf{k} \\mathbf{k}^\\mathsf{T}", "1ed262cb444f95b8a45106a84b13660d": "c^A_i,c^B_j", "1ed297019b270c06d14408dadd755f0c": "\\Delta U(t)\\Delta V(t)= d[U,V]", "1ed2e6ac8d74840a5cc110e1d760b2dc": "\\mathrm{ K_a\\, =\\, \\frac {[H^+\\,][A^-\\,]}{[HA]} }", "1ed34692f7bdfacfd9f09f6b5c3cdb52": "P(M1)=P(M2)=1/2", "1ed346930917426bc46d41e22cc525ec": "\\phi", "1ed376d481fa7f58d46d052807858828": "C_{Q0} = 8\\pi \\alpha \\cdot C_{QY} \\ ", "1ed389a4c15ffbc7673138ed2131d981": "\\partial_t^k u|_{t=0}=\\phi_k(x), \\qquad 0\\le k\\le m-1,", "1ed39bcd371df8a7ee38ac8351925aae": "\\beta = x_{\\eta}x_{\\xi} + y_{\\xi}y_{\\eta} ", "1ed3dd6f9e7bfd66ed48b6dbf817c918": "M=E\\cdot K^{-1}", "1ed3f3127aa57f10fc1e3c53a03bd840": "\n\\mathbf{H}_{\\alpha -1}(x) + \\mathbf{H}_{\\alpha+1}(x) = \n \\frac{2\\alpha}{x} \\mathbf{H}_\\alpha (x) + \\frac{{(x/2)}^\\alpha}{\\sqrt{\\pi}\\Gamma(\\alpha + \\frac{3}{2})}\n", "1ed442173a51dc0b9c3cf2e7c5352f61": "w=h_1 h_2\\cdots h_n", "1ed4d1b05034e33c314cf915ce670b75": "m_n = E(X^n) = \\frac{1}{\\zeta(s)}\\sum_{k=1}^\\infty \\frac{1}{k^{s-n}}", "1ed52deb0d185243d6eaec665ec3652d": "F_{ij} = G (M_i^{\\beta_1} M_j^{\\beta_2} / D_{ij}^{\\beta_3})\\eta_{ij}^{\\ }", "1ed55474ef550ce4b7b3cf47aa5a87cf": " S2 = \\sum_{j=1}^{k} |High[j] - High[j - 2]| + |Low[j] - Low[j - 2]|", "1ed5867197fe41edbc4177ed4c5fada7": "\\textstyle\\overline{U}", "1ed5ca65c4195b2b1c8bded40794f6bd": "\\theta \\log{\\tan \\theta} - \\int_0^{\\tan \\theta}\\frac{\\log x}{1+x^2}\\,dx", "1ed62ba1ea3ede7d4731810251d61eb6": "|b|_\\ast\\leq 1", "1ed77893fbc1f3afe712f5db696748e1": "\\frac{\\partial H}{\\partial u_t} = p - \\lambda_{t+1} - 2\\frac{u_t}{x_t} = 0", "1ed86f05e38d3a425946046e3ac59b5e": "E(\\delta(X))=\\sum_{x=0}^\\infty \\delta(x) \\frac{\\lambda^x e^{-\\lambda}}{x!}=e^{-2\\lambda},", "1ed872aa38c852cc1e1f37b2b44cc8be": "{ \\partial u \\over \\partial \\lambda } = 8\\pi h c\\left( {hc\\over kT \\lambda^7}{e^{h c/\\lambda kT}\\over \\left(e^{h c/\\lambda kT}-1\\right)^2} - {1\\over\\lambda^6}{5\\over e^{h c/\\lambda kT}-1}\\right)=0,", "1ed894a22a52f869cbb866bc50d6f4a9": "(X^n(j), Y^n)", "1ed8c30a30ec29b18d747f1d6c52c49f": "\\scriptstyle\\widehat{\\varepsilon}_i", "1ed99c2dd202f44ce8a65fc27f48e8c3": "\\frac{\\$10,000}{\\$40,000} = 0.25 = 25\\%,", "1ed9be740d876001d18445bf0f9b7069": " c dt = \\gamma d\\tau ", "1eda154dc6063bfa4c1a9f1d0aa7bffd": "a_{32}", "1eda90079cca7409a9e415de34dbc8db": "v \\otimes w \\in V \\otimes W", "1edaa21a993e6df47a4872f64a335cee": "E=\\mathbb{Z}^2", "1edaaf4a82237b5d111baaec4357d25c": "R[[X]]", "1edada44ebf674c8caaeb29c756caaea": "X_{m}", "1edc06c025db330b27d6b616c9ee2f1c": "f \\not\\in \\sqrt{I}", "1edc146a69b97f24fd3d42b4f099c15d": "W_2 = m_2 g", "1edc4dc654eb77a7ae71e2d2713b2392": "\\nu=\\,", "1edc799361f3e8c3c40286fa9c4bc033": "AF\\phi \\equiv \\phi \\lor AX AF \\phi", "1edccfe03fc23a9364699d8f19d62a07": "f(X)=g(X^{p^n})", "1edd09ab5f2737166b43ccea97bf3845": "= 2\\pi\\cdot I_{max}\\int\\limits_0^{\\pi/2}\\cos(\\theta)\\sin(\\theta)\\,\\operatorname{d}\\theta ", "1edd211800a09dac211e661dfae8eb98": "W.L", "1edd5630ed74cfd4167cd5e6f3511ee6": "-v(A - 0)", "1edd6c1727192ed6cefb1d1e7e397967": "f: {\\mathbb R}\\rightarrow{\\mathbb R}", "1edd7293daa48a3827508d749f619f79": "\\varphi_{\\alpha} (x) = \\inf_{y \\in H} \\frac1{2 \\alpha} \\| y - x \\|^{2} + \\varphi (y).", "1edd8076f9f5d6bb20d01f2aa1293fda": "c = \\sqrt{ \\frac{B}{\\rho_0 }}", "1edd9321f7d0513cdde995c33dfb15d1": "g = g_0(r_0/r)^2\\,", "1edd95e5747d6d9484585ea941426b81": "e(P_{\\infty, i})", "1edd9b126c2fb9b4869a84cf4f0f02e1": "\\kappa_f(k,i) = \\frac{\\delta(k,i)}{\\xi^d_{b_{min}}(k-1,i)}", "1eddfdf60538c4f0d3dd59006ad2a9c4": "\\mathrm{Aff}(n)\\simeq \\mathrm{GL}(n)\\rtimes \\R^n", "1ede65dd57d62e9140674f7bd1ea5662": "\\varepsilon = \\varepsilon_r \\varepsilon_0 = (1+\\chi_e)\\varepsilon_0. ", "1ede801a3909cb17e110112a6ec20a02": "\\chi^1 = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\quad \\quad \\chi^2 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\,", "1edf00e4b6254b847d8e47b2c1cddb2f": "\\mathbf{A} = A_+ \\mathbf{e}_{+} + A_{-} \\mathbf{e}_{-} + A_0 \\mathbf{e}_0 ", "1edf39bdff607c52cec40bb94869f57e": "\\boldsymbol{F}(\\boldsymbol{S}) = \\boldsymbol{F}_1(\\boldsymbol{F}_2(\\boldsymbol{S}))", "1edf6235ff0aa6691866148877ad3395": "\\bar m", "1edf66f6f28accded7d527ba25b85227": "dA = r^2 \\sin\\theta\\, d\\theta\\, d\\phi.", "1edf7a10cb75a5ed6d5b1357e53c88ae": "Y:M\\to\\mathbb{R}^n", "1edf84c6ee3304391923017efd607558": "Y=\\bigoplus_{i=1}^\\infty \\mathbb{R}", "1edfa143a11d4b9bc5e821f7a399814a": "\\left[\\begin{array}{c} R \\\\ G \\\\ B \\end{array}\\right]=\\left[\\begin{array}{ccc}255/R'_w & 0 & 0 \\\\ 0 & 255/G'_w & 0 \\\\ 0 & 0 & 255/B'_w\\end{array}\\right]\\left[\\begin{array}{c}R' \\\\ G' \\\\ B' \\end{array}\\right]", "1edfc958d5a196148a1cb43373a2ccf0": "\\frac{dy}{dx} = -1,", "1edff01f1e3949fc2a9f60b46008155b": "(q^i_t-\\Gamma^i)\\partial_i", "1edffe9debd22ecd5ea5124ce339e908": "\\int_0^{2\\pi} |f(h(re^{i\\theta}))|^p \\, d\\theta \\le \\int_0^{2\\pi} |f(re^{i\\theta})|^p \\, d\\theta.", "1ee0430c5ff75ede5c4fafc29a89bd08": "\\lambda_j=(-1)^j\\binom{n-k-j}{k-j}", "1ee0447fde38a4aa7f88d413c6f0efd3": "SM_3(levol,endo,exo)=RE", "1ee06e198eb28d5cac674daadb9de5f2": "\\operatorname{Ber}(X) = \\det(A)\\det(D)^{-1}", "1ee07d263b25e3dcda7057e9deb3360e": "\\underline{\\underline{\\boldsymbol{A}_2}}", "1ee0ac9be1d0328a58dca551ee263bcc": "\n |t_\\lambda-t_{\\lambda-1}|<\\tfrac12\\,|t_{\\lambda-1}|\n", "1ee109bff381e763045eb7bdc8a4792b": " M_t = p ( W_t, t )", "1ee12efc153c609465e4909f6fdc8a6f": "\n W_1 = 1,\\quad W_2 = b,\\quad W_3 = c,\\quad W_4 = d.\n", "1ee147abfa6420fb97621c10756a699c": " 2g(\\nabla_X Y, Z) - g([X,Y], Z) + g([X,Z],Y) + g([Y,Z],X) ", "1ee150b7ad7e504033b554dc3310db12": "(a+b)'=a'+b'=b'+a'=(b+a)',", "1ee16b1070b825a921e89eb3242a1739": "C_{\\mathrm{Artin}}=\\prod_{q\\ \\mathrm{prime}} \\left(1-\\frac{1}{q(q-1)}\\right) = 0.3739558136\\ldots", "1ee17a051f9512465a3beaba1e26bcee": "\\sqrt{\\pi} = \\left(-\\frac{1}{2}\\right)! = \\int_{-\\infty }^{\\infty } \\frac {1}{e^{x^2}} \\, dx = \\int_{0 }^{1} \\frac {1}{\\sqrt{-\\ln x}} \\, dx ", "1ee1df7fcc344f3c320827ad3bf8507c": "\\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \\cdots = \\frac{9}{1-10} = -1.", "1ee1ee02d11ae3ac38869733bda55607": " J_{n+1} = 2J_n + (-1)^n \\, ,", "1ee252f4f0c5f94fdc1d55ac1da82bd1": " -\\infty", "1ee2e9faa290b6678b2f43a98646ab0e": "\\phi_x(t)", "1ee2ecc57219e2c075c1fa46b77322cf": "10\\log(|H_{NEXT}(f)|^2)=\\begin{cases} -59.2 + 4\\log(f) dB & f < 20 KHz \\\\ -42.2 + 14\\log(f) dB & f >= 20 KHz \\end{cases}", "1ee2fe985288c2aeb44987b8cc68e4e6": "p = 4^a(8b + 7)", "1ee30fe57fb3d6e3a88f99c939dcc7a8": "\\lnot (A \\vee \\lnot C)", "1ee362b96fe261ee6c1b843359900103": " k_B", "1ee3826e912aeb916b230416bda12077": "u \\neq s", "1ee42ca6404c21d6a2e35ba1a4aa8ed4": " \\Phi_{,abe}g^{ef}\\Phi_{,cdf} = \\Phi_{,ade}g^{ef}\\Phi_{,bcf} \\, ", "1ee43acafda81d3e23e7c1747a8e0949": "F = m\\,a", "1ee43e4f7458d3f23849169ce032385f": " P = \\frac{ \\varepsilon_0 }{2} E^2 ", "1ee463a70ffcc5b1473fb0adf84a6c8a": "(a/r)<1", "1ee46d33ce7b7811b8174e3084ddbd66": "\\psi_p(a) = \\frac{\\partial \\log\\Gamma_p(a)}{\\partial a} = \\sum_{i=1}^p \\psi(a+(1-i)/2) ,", "1ee4d32ea5d6ce61083cedc8fb49c1fa": "S \\otimes_R S\\to S", "1ee4e23d128fdc058979a7c272bb6a18": "f=\\frac{i}{g}", "1ee4e6d404e15f7e69516faf0dd74473": "A(\\omega) = \\frac{i \\omega C R_0}{1 + i \\omega C R_0}", "1ee50c13ab1fd6caa44fd1dfe008b00c": "\\nu(i) \\!", "1ee520a293a30663d966f36ddcc76c75": "\\Bigg\\lbrack \\frac{\\rho_2}{\\rho_1} \\Bigg\\rbrack", "1ee525229971f852cd26319e93c691d4": "\\frac{\\partial N_1}{\\partial t}=\\frac{\\partial}{\\partial x}[(N_2 D_1 + N_1 D_2) \\frac{\\partial N_1}{\\partial x}]", "1ee57128bcf4db92a06c0662c7fb6233": "V_{\\text{A}}", "1ee5bed5e674968e100df7851c46a966": "\\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "1ee5c1448db5e8b23058cf86d4aabd2f": "w = \\frac{1}{2}\\left[q + c\\left( e^Q -1 \\right) \\right]", "1ee5df290249b90a32de833a16db6d03": "|a(u,v)| \\le C \\|u\\| \\|v\\|", "1ee5e987f66b3cfc2885688fafc96054": "43.2\\ \\mathrm{MHz} 1", "1ee6d9a5fa503d195d2e3b5bd5ee65ca": " m_{i}\\equiv 0\\pmod {p_{j}}, \\; \\text {for} \\; i\\ne j, ", "1ee6ee63ba880b9c4f012300e626a495": " x_i^*(p_1,p_2,m):= \\arg\\max \\{\\,\\!u(x_1,x_2)\\mbox{ } :\\mbox{ } p_1x_1+p_2x_2=m\\} \\mbox{ for } i=1,2", "1ee7168df412ac4a6756f690c84b2177": " n = m = 0", "1ee758e2534b094fc9d70b930d8dbfef": "F=\\frac{\\pi^2 EI}{(Kl)^2}", "1ee7f61b5986d7f4101df415be88215a": " L_I=10 \\log_{10}\\frac{I_{\\mathrm{rms}} }{I_{\\mathrm{ref}} }", "1ee82323ad342a2205a10be3af98414d": "\n\\begin{bmatrix}\n1 & 0 & 0 & X \\\\\n0 & 1 & 0 & Y \\\\\n0 & 0 & 1 & Z \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "1ee82e35f66fd2b76b7727d5eb7b88bb": "{\\rm Tr}\\, e^{F}e^R \\geq {\\rm Tr}\\, e^{F+R}\\geq e^f.", "1ee87af34e9524c8647fe97472ae25e1": "-F\\cos \\left(\\frac{\\pi y}{b}\\right)", "1ee896764b7b9a94a4065790aafe0ec5": "6:m\\ ", "1ee89e16442cb637bf606cd711160855": " \\hat{x}_U(k) ", "1ee96b94aac8db4b64643e165fd6f4cd": "\\lambda _0", "1ee99ad169b9c53b420a06eadbf9dcd7": "G(k)\\leq k(3\\log k+11). \\,", "1ee9afe1e0c02256726c3cedea348b74": "(\\star)\\qquad(x+2)^2-2(x+1)^2+x^2=2. ", "1ee9d94057e7d7605bfc63415d02976b": "\n J = \\det(\\boldsymbol{F}) = \\det(\\boldsymbol{V})\\det(\\boldsymbol{R}) = \\det(\\boldsymbol{V}) = \\lambda_1\\lambda_2\\lambda_3 ~.\n ", "1ee9de4d3e0a76a925ef6a6e996a4d70": "k_\\mu = \\left(\\frac{\\omega}{c}, -k_x, -k_y, -k_z \\right) . \\,", "1eea1b3e499a0d98b30ce32867363093": "e^k k^{O(\\log k)} \\log |V|", "1eea91b5b846092859e6ff642bd6f07a": "g_{ij}[\\mathbf{f'}]=\\sum_{k,\\ell=1}^n \\frac{\\partial x^k}{\\partial y^i}g_{k\\ell}[\\mathbf{f}]\\frac{\\partial x^\\ell}{\\partial y^j}.", "1eeaed5a852812f46ce2c3ace34b0218": "x_{n+1} = x_{n+2} = \\cdots = x_m = \\alpha.", "1eeb7b99f63813eaeb5f4a8d46a18e1c": "4 \\arctan \\frac{1}{5} - \\frac{\\pi}{4}", "1eec10ca260348b69eba8ad49caff2ee": " \\mathfrak{g}^*", "1eec2c4878a6131fca529f588c64e913": "d=1\\,\\!", "1eec2f60d7698cae0e16c9d44d844992": "V:\\Sigma\\rightarrow\\omega_1.", "1eed6793fbbe439f3999805296cf03ff": "N = \\int_{\\sigma(N)} \\lambda d P(\\lambda). \\,", "1eed99fbfa4589fc221c3ccc318b4adb": " r~\\ln r~\\sin\\theta \\,", "1eedc893a4328896c81612f6ff33547c": "\\sin^2\\theta + \\frac{b}{\\sqrt {ac}} \\sin\\theta \\cos\\theta \\pm \\cos^2\\theta = 0 .", "1eee927fa7d08c21e1326fab2cd1690a": "\\mathrm{im}(f_k) = \\mathrm{ker}(f_{k+1})", "1eeebec4cd4013af45cdf78207df27af": "\\sigma _{a,A}^{2}", "1eeef773e753f748faa0e30b1b7470a0": "f^n(x) = f^m(x)", "1eeefbbedc91b784caee8137d78c8ba8": "\\Omega\\in \\Gamma(\\Omega^2M\\otimes \\text{Hom}(E,E)).", "1eef0c80d2a41e21642c910860dcdd69": "\n\\begin{align}\n i \\sin(2\\delta) &=& -i\\frac{\\hbar^2 \\pi q}{maV}\n \\end{align}\n", "1eef0cefa3d1e95a4b85ae78f149c65d": "\\{l^a, n^a, m^a, \\bar{m}^a\\}", "1eef2ae82273c69e5edfafd0f27cd167": "2^{o(\\sqrt{k})} n^{O(1)}", "1eef41f7a8ad9841878e0db789a753af": "\\mu = \\mu^o + RT \\ln C", "1eef82f7c74c576d995e1aad6e42b4a1": "E_n,F_n,H_n", "1eefa25bdcadaa35cf74332d38fc6b6d": "|X|^2", "1eefbe153094631c685a2f2a3181dbba": "T(\\mathbf{x}) = \\mathcal{F}(\\mathbf{x}^*) / \\sqrt{N}", "1eeff2a905146fd96a38404f20be4b0e": "A \\wedge (B \\vee D) \\wedge (B \\vee E).", "1ef02a90fa9c68748d3491ca5daacf51": "n_P \\in\\Z", "1ef04040bcc33c0e19a494092beac50d": "\nL_N = 40 + 10 \\log_{2}(N)\n", "1ef05e70666c31cf854c01348d923bef": "g(z) = \\exp\\left(z + \\frac{1}{2} z^2\\right).", "1ef07bd84f2b87f2a6259421fb1fdc2c": " g(v_1\\otimes v_2\\otimes\\cdots\\otimes v_k) = gv_1\\otimes gv_2\\otimes\\cdots\\otimes gv_k, \\quad g\\in GL_n. ", "1ef08264fd9d75cbc7eadccf77eca478": "l, d", "1ef09bc7962e10cf53f533c5c150a4b6": "\\tau = F/A", "1ef0a5385296f6aed49d2dad9ff81ed1": "x_0 = -\\frac{r_m}{E} \\quad (11)", "1ef0b5f5cc89a40a75bbef1d6276a37f": "\n[[Category:Canada highway infobox templates|{{PAGENAME}}]]", "1ef0d39a58e0abfa1047fef542c2bcd5": "\\ pH", "1ef0d93e70b9bb2bce782faf50c85441": "\\, \\Re(s) = 1/2 \\,", "1ef11aaf01f76caf91ebb91be3b81453": "A=\\bigcap_{k=0}^\\infty H_k", "1ef131c6360538eb2157797e477b8535": "DX\\ \\stackrel{\\mathrm{def}}{=}\\ \\mathrm{d}X + \\bold{A}X", "1ef161eb7eb9cc7abef753e7a8085d8f": "(\\|f + g\\|_p)", "1ef1662438bc782c8447683b1505ef93": "p(w_i \\vert C)\\,", "1ef17e2c27f55803da91fd23d04fcf9d": "V_\\mathrm{CE}", "1ef227ba79cfbacab970539602c573a2": "\n\\log \\frac{1+z}{1-z} = \\cfrac{2z}{1 - \\cfrac{\\frac{1}{3}z^2}{1 + \\frac{1}{3}z^2 -\n\\cfrac{\\frac{3}{5}z^2}{1 + \\frac{3}{5}z^2 - \\cfrac{\\frac{5}{7}z^2}{1 + \\frac{5}{7}z^2 - \n\\cfrac{\\frac{7}{9}z^2}{1 + \\frac{7}{9}z^2 - \\ddots}}}}}\\,\n", "1ef22f9271cf28bb14c41e743a1d46bb": "x_{32}=0\\,", "1ef241eede58d1e6c65db40b92d979e6": "I = \\int _0^{\\frac{\\pi}{2}}\\frac{1}{\\sqrt{a^2 \\cos^2(\\theta) + b^2 \\sin^2(\\theta)}} \\, d \\theta", "1ef28a62d145a3adf85031bd5b0e0e30": "f(0)^2=f(0)^2+f(0)^2.\\,", "1ef2cac2d529634ee6d3e2d37a783113": "|j_1-j_2|\\le j_3 \\le j_1+j_2. \\, ", "1ef314b17a6445ca6d4fa39931c36b18": "\\mathbb{C}^{p+2}", "1ef3624527a866311b2ec0f5eea3085f": "\n\\psi(\\vec{\\theta}) = \\frac{2 D_{ds}}{D_d D_s c^2} \\int \\Phi(D_d\\vec{\\theta},z) dz\n", "1ef38b6e6615f3349dd0b332928345d3": "{{i}_{B}}={{i}_{B1}}={{i}_{B2}}\\approx {{i}_{B3}}", "1ef3cf04952cb236fc8816e505ea60e6": "v_i^t", "1ef41038f1f51fa9c3aa6f6a3aef0906": "\\nu=\\gamma/(2-\\eta)", "1ef4277ed1d8e21c07dc689651d3f3e6": "\\displaystyle V_\\mathrm{rms}=V_\\mathrm{peak}.", "1ef463d9d90207455224b6e8602d8447": "S\\subset\\Delta", "1ef484e79db2e006dcb9790efe20d7ec": "\\{ G_j , O \\}_{G_j=C_a=H = 0} = \\{ C_a , O \\}_{G_j=C_a=H = 0} = \\{ H , O \\}_{G_j=C_a=H = 0} = 0", "1ef49a79254e216f1bcd6e6054f52568": "L_g(\\psi)(x) = \\psi''(x) + g(x) \\psi(x)", "1ef49ba6e88d1ba9677bf9fa307070eb": "[HG_{eq}]", "1ef4c65d7f3dbbadc1ce6ece9276f5c9": "\\mathbf{D_p h}(\\mathbf{p}, u)", "1ef60ce19ec5dc53a45b70e2bde872a3": "_{2}F_{1}(-n,\\alpha;\\alpha+\\beta;1-e^{t})\\!", "1ef6239e9c13b74e03815653a0a30ea2": "\\sigma(\\pi)=-1", "1ef6497c374eeec5201083094fb7eedb": "Tr(h) \\in GF(p^2)", "1ef66a8c7baefeae1c9b1514657dc7d6": "\\text{Weight density} = \\frac{\\text{weight}}{\\text{volume}}", "1ef6d366eacf543eda392737f58ec070": "\\or \\!\\,", "1ef6eb217fecb5fb96bf076f74c144d6": "T_{1/2}=\\frac{-0.693(6.9\\times 10^{21})}{-3200\\text{ s}^{-1} }=1.5\\times 10^{18}\\text{ s or 47 billion years}", "1ef72658d38a6aabc91b10b883e43707": "\\{f_n(x)\\}", "1ef7dce99947d616b7c793cbf99adbc1": " PE", "1ef8053640bf0ef38885ecfcbdae2a67": "\\pm(\\cosh a + j \\sinh a) ", "1ef8072925c45d1b3981ea0cdd79131c": "x = n\\cdot\\pi", "1ef87c8080a59242121546bea9bc9427": "x_{k-1}=a_{k-1}", "1ef89c48072cb255897aa3f247027eb7": "89^2", "1ef9431a2275300913280d155e0e241c": "\\rho = 0.23 V_p^{0.25}.", "1ef949555dc8084976eb10ff4a6c039c": "m_e\\,", "1ef95eed3cd3ce2a9555803b643ba077": "4\\pi / 10", "1ef963168911972b7276caca19b787b6": "\\sigma_i:\\Delta^n\\to X", "1ef9a04b89fb016dc4a4e359d2b3b627": "p_e = 0.5\\, \\operatorname{erfc}\\left(\\sqrt{\\frac{E}{N_o}}\\right).", "1ef9b98b337e22327a68c68ec75423aa": "{\\rm BW}_Q", "1efa05f20b2a63e1424093803ff4babf": "T_{2^p - 1} = 1 + \\frac{(2^p-2) \\times (2^p+1)}{2} = 1 + 9 \\times T_{(2^p - 2)/3}", "1efa51216eec3754a4c0dced6f32e4cf": "\\oint_{C} f(z)\\, dz = 2\\pi i \\sum_{k=1}^n \\operatorname{Res}(f, z_k)\\,,", "1efa6e1c6e83cd5ec8c7c4994f9cb3e7": " \\pi(G)^\\prime", "1efa8c001160d38b4e98079bdd8612fc": "\\sum_{n=-\\infty}^{\\infty} x[n]\\cdot e^{-i \\omega n},", "1efaa7095258dd303271242a3f6eb977": "\\epsilon{k^2}", "1efad6da2812baf6aec9090867729714": "\\gamma(z) = \\frac{1}{\\pi}\\,\\frac{z^q-z}{(z^q-z)^2-1}", "1efb22fe96de8ff0215a81f08fc77e2c": "\\Delta \\subset \\mathbf{C} \\subset \\widehat{\\mathbf{C}},", "1efb71712325cc3e0a13d0979c38f21c": " V[q] = A_4 ", "1efba7a029eb1fd9ef3880e62005812e": "[\\Gamma_{}e_1 (\\phi_2 - \\phi_1)/ \\delta x) - q_A] + [\\Gamma_{e_2} (\\phi_3 - \\phi_2)/ \\delta x) - \\Gamma_{w_2} (\\phi_2 - \\phi_1)/ \\delta x)] + [\\Gamma_{e_3} (\\phi_4 - \\phi_3)/ \\delta x) - \\Gamma_{w_3} (\\phi_3 - \\phi_2)/ \\delta x)] + [q_B + \\Gamma_{w_4} (\\phi_4 - \\phi_3)/ \\delta x)] = q_B - q_A", "1efbb68627093dd8d725a6990a3a1f92": "E\\to \\mathbb{Z}_2", "1efbb7486d34ae867af95c3e001c7923": "\nz\\,\\, = \\,\\,x^2 \\,y\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{{\\partial z} \\over {\\partial x}}\\,\\, = \\,\\,2x\\,y\\,\\,\\,\\,\\,\\,\\,\\,\\,{{\\partial z} \\over {\\partial y}}\\,\\, = \\,\\,x^2", "1efbcd77389f6824e6f369a4a2582bb8": "\\mathbf{y}(t) = \\left(I + D K\\right)^{-1} C \\mathbf{x}(t) + \\left(I + D K\\right)^{-1} D \\mathbf{r}(t)", "1efbf07a71c9649aa3fafff0c0c1597b": "\\omega_{P}=\\sqrt{\\frac{n e^2}{{\\varepsilon_0}m^*}}", "1efbf11bcac83d29bc87a2252a8b0b34": " p q - q p = { h \\over 2 \\pi i } I ", "1efce36899292a978860d1767ba537c4": "(y - \\beta)^{v}", "1efd17c5334eb10033d6c8d85d7102a1": "Z^{(\\ell)}_{\\mathbf{x}}(\\mathbf{y}) = \\frac{1}{c_{n,\\ell}}C_\\ell^{(\\alpha)}(\\mathbf{x}\\cdot\\mathbf{y})", "1efd1906a27397ce248b5084aa4b6114": "\\scriptstyle{ (TS)^\\prime = T^\\prime\\!S + TS^\\prime }", "1efd2f442f734bd1941a1564a717fd88": "-\\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\beta^2} = \\operatorname{var}[\\ln (1-X)] = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta) ={\\mathcal{I}}_{\\beta, \\beta}= \\operatorname{E} \\left [- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\beta^2} \\right ] = \\ln(\\operatorname{var_{G(1-X)}}) ", "1efd5556e50645e2635406aa5d2efea8": "\\begin{pmatrix} \n1 & 2 & \\ldots & j \\\\\na_1 & a_2 & \\ldots & a_j\\end{pmatrix}.", "1efd67cb3a20180ec9a3acbcc9027e52": "\\bar{d}=k{D_v\\centerdot R \\centerdot \\nu_m \\centerdot \\epsilon \\over D_s \\centerdot N\\centerdot \\nu_l \\centerdot C_s}", "1efd9a635b42115e7c7d97615365932c": "z= \\boldsymbol{\\phi}(y)", "1efddb3de3857e020d408e024131ed06": "\\psi (t) = e^{-2 \\pi i t}", "1efe007ba269e986c5c89e0e0f97c2e4": "f, g\\colon I \\rightarrow \\mathbf R", "1efe05c9929d61163fd26e129d738e8a": "(n,2n,n/2)_2", "1efe5b315515cb3fc28697f8e5387d9a": "\\text{dBSWL} = \\text{dBSPL}+10\\, \\log_{10}\\left(4\\pi r^2\\right)", "1efeafb3273a3444e33378721bee739a": "\\Delta G_{fus} = \\Delta H_{fus} - T \\times \\Delta S_{fus} < 0", "1efedfa53eecc0c70040f7548268358c": "\n\\begin{align}\n\\vec{a} \\cdot \\vec{\\sigma} &= (a_i \\hat{x}_i) \\cdot (\\sigma_j \\hat{x}_j ) \\\\\n&= a_i \\sigma_j \\hat{x}_i \\cdot \\hat{x}_j \\\\\n&= a_i \\sigma_j \\delta_{ij} \\\\\n&= a_i \\sigma_i\n\\end{align}\n", "1eff35924a0a7427b46c41783b29482e": "\\beta=(k_{B}T)^{-1}", "1eff5d5532fb11b974ce63997c6ab09f": "\\scriptstyle\\hat{Q}_n(\\theta)", "1eff6f9ba863f1cc47e7778835eabcc2": "V_{R_2} = R_2 \\cdot I", "1eff9f154dc2deb2631f7684b3aa6aac": "\\mathfrak{m}_K^n", "1effdb4188635e43ed906aa0a042c9cf": "6x^2+x", "1f000654eded2cbd11ca1b30f4587df0": "\\kappa_\\varepsilon,\\lambda_\\varepsilon \\in \\mathbb{N}", "1f003ac607460f29a62af072bc9c09a3": "\\mathbf{\\hat{a}} \\,\\!", "1f0084145af7183c4329d81c4b36887a": "\\overline{\\mu}", "1f00a3726d12ca656512fc80ee56a598": "i \\pm s_1, \\ldots, i \\pm s_k \\mod n", "1f00a874d59f8c770904806aef90c41f": "\\int \\frac{\\delta Q}{T}", "1f00c737b07bd25d2a17f037c778d9ea": "g_B", "1f00f2d25f982aa43bbf74de09d32bdc": "\\mu_4", "1f010219c12d38ad5804fce31ce83588": " \\rm{det}\\left ( \\alpha_{i-j}^{q^j} \\right ) \\neq 0 \\quad (i, j= 0,1,\\ldots,r-1).", "1f0117b50a395695cc3b64c5cbda290f": "\\overline{GM_{L}}", "1f016e2922c4a0c78ee36579a303d5e3": "\\tau = RC = \\frac{1}{2 \\pi f_c}", "1f01c6c162b4191653b4b3d7b322f6ad": "i = \\max\\{{\\lfloor}{\\lceil}\\log_b m{\\rceil}/k_m{\\rfloor}, {\\lfloor}{\\lceil}\\log_b n{\\rceil}/k_n{\\rfloor}\\}.", "1f021fc28ce12eceaed7b63e536ffdf0": "\\mu(x)\\in \\mathbb{Z}_2", "1f023407b09aaee18b9e667bcbe6c32e": " a_1=ta_2,b_1=tb_2,c_1=tc_2 ", "1f024fdb79e82573d2228a17a2f035f1": "(\\mathbf{a} \\cdot (\\mathbf{b}\\times \\mathbf{c})) = \\varepsilon_{ijk} a^i b^j c^k", "1f025434136e300ca3fde75a26f500a8": " \\|y-x\\| = \\|-1(x-y)\\| = |-1|\\|x-y\\| = \\|x-y\\| ", "1f02cb7a20a8040b4334cb20b500371d": "B_\\lambda(T) = B_\\nu(T)\\left|\\frac{d\\nu}{d\\lambda}\\right|.", "1f02d30d6d5256e978b7e9e0b627abad": "(\\hbar^2 n(n+1))", "1f02f33b2d6bb0795cc3fe463002ba11": "T_x M = P", "1f02fefb8b21f8e363974a76ffb7b7a4": "\\sqrt{(x_n + j\\omega) \\cdot (x_n - j\\omega)}= \\sqrt{x_n^2+\\omega^2}", "1f03158808dda5b2a28536079b8b5a8e": " Q_{50}\\,\\!", "1f032b3ce3ca33ff496ce91d5f767fb7": "\\Bigl(\\prod_{j \\ne i}^n n_j\\Bigr) + 1\\quad ?", "1f03375653efd91fc9bb2ee25d25d991": "\\textbf{j}", "1f035115f01c45a324885efc424e7663": "2 (2^{n+1} - 1) + 1 = 2^{n+2} - 1 ", "1f038e50ee8a54cfd39e18fe2a70016a": "T_{o-}^{TE}=H cos(\\frac{m\\pi }{a}y)e^{jk_{xo}(x-w)} \\ \\ \\ \\ \\ \\ \\ \\ (25) ", "1f039f0d70d5de5474a6a025e4b95232": "E(Z_n^2)=1", "1f03c5fb4f7cc4d3211d4621b0babc8a": "\\sec(x)\\tan(x)", "1f03df4826f6535122cf138ccf358189": "\\delta S = 0.", "1f03e39c60ff1a05f4cefad1c188a25a": "\\begin{align}C \\sum_{n=0}^{\\infty} \\frac{2^{n+1}}{\\ln F_{n}} &= \\frac{C}{\\ln 2} \\sum_{n=0}^{\\infty} \\frac{2^{n+1}}{\\log_{2}(2^{2^{n}}+1)} \\\\ &> \\frac{C}{\\ln 2} \\sum_{n=0}^{\\infty} 1 \\\\ &= \\infty,\\end{align}", "1f041e9b64e564523f75291d663a4d70": "L, M, R", "1f0496978b7c523822db641b6eabdf61": "\n\tf(\\boldsymbol{x}) = y_{\\mathcal{P}},\\ \\forall \\boldsymbol{x}\\in \\mathcal{P} ,\n", "1f04afe4d38ceb532641f994e26a421d": "\\sinh k, \\cosh l, \\tanh m, \\coth n \\!", "1f04e69159e3fcc91bb192edd5464858": "v^2 = 1,", "1f04ebf4f97b823896c132dd40a60215": "{}_s\\lceil X \\rceil_N", "1f0500df195dab3ad70d51fb53eef2d2": "g \\Delta \\rho L^2 / \\mu", "1f057274abfd9304c46facb1657190fd": "e = \\lim_{n\\to\\infty} \\frac{n}{\\sqrt[n]{n!}}", "1f05d57705c44d935b0aa3fe94ddde68": "C_i = M X_i - P_i t", "1f05fbde3719e63aef76b645cb302ba5": "\nC^m_\\ell(x,y,z) = \\left[\\frac{(2-\\delta_{m0}) (\\ell-m)!}{(\\ell+m)!}\\right]^{1/2} \\Pi^m_{\\ell}(z)\\;A_m(x,y),\\qquad m=0,1, \\ldots,\\ell\n", "1f060825ab4836bc9fcf50cf260d201a": "L>0", "1f065facd0383670a3191f42592ee725": "p^{-n} H(t)=\\frac{t^n}{n!} H(t).", "1f0696288950daa954db60fd2f824ae5": "y=Ae^{-1/x}", "1f06b2e690571c83ee7560b0a6bdef88": "C = 1/(2\\lambda n)", "1f06e7326edc99094e54d8f40c8b9b9c": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_\\mathrm{tot}", "1f06f4a45dd68ea7380a2365e1ddc441": "\\sqrt[2\n]{E}/\\rho", "1f07489d3b5117f6c843b6e274a3be25": "\\delta \\times \\left( x \\times \\operatorname{p.v.} \\frac{1}{x} \\right) = \\delta", "1f0757ca5effe6dbdd0e3ebe6315c0df": "\n\\omega_{r}^{2} = \\left( \\frac{c^{2} r_{s}}{2 r_{\\mathrm{outer}}^{4}} \\right) \\left( r_{\\mathrm{outer}} - r_{\\mathrm{inner}} \\right) = \n\\omega_{\\varphi}^{2} \\sqrt{1 - \\frac{3r_{s}^{2}}{a^{2}}} \n", "1f07688e9eba01a0b7de63c1aaa97b23": "\\sum_{n=N}^\\infty f(n)", "1f077f11860d444f2540b4a9e38a425c": "{K_1,\\dots,K_8} = {3,1.52183,-0.7607 + 0.8579i,-0.7607 - 0.8579i,1,1,0,0},", "1f07ba9362086bc3911f13dc46801f6f": "Q\\to M", "1f07c918c1596b1661e1e2ed2b5df2d9": "K \\in \\mathcal{K}^k", "1f085faff7a7245811a3abe4465cae4f": "V=iR", "1f088e03d24d8126303e28fae0dd535f": "\\alpha=\\sin^{-1}(e)=\\cos^{-1}\\left(\\frac{b}{a}\\right)=2\\tan^{-1}\\left(\\sqrt{\\frac{a-b}{a+b}}\\right);\\,\\!", "1f08a053f7a87ba537cbaa078b1c3c21": "\\phi-\\phi_0\\,", "1f08cc3575b84272b143d7949ae6ca26": "\\alpha_D = \\frac{116}{D}", "1f090d50661700e511ba18b9ed50b1b9": "\\mathfrak{P}^{94}", "1f0915acdc81986a1afbbc69bf7ae883": " I_r \\, ", "1f093dffb289d91bb99dabc5ac554786": "\\alpha_{i_1} \\ldots \\alpha_{i_K} = \\beta_{i_1} \\ldots \\beta_{i_K}.", "1f094363ce760dff0653e9df2164ca3c": "(x - \\alpha)", "1f09c25c5247c1eaf121df644ca42f8c": "\\theta \\,", "1f0a1363fc89fc5d89439c81643b43ee": "I_2 = {{}^\\star C}_{abcd} \\, C^{abcd}", "1f0a43647c13586708fbe223ef7e4a7b": "\\sum C_i", "1f0a83c5067ef7ce2bad08832aa44515": "[A,B] = -[B,A]", "1f0aa218526653a68a6b8e4bf6cf1ec9": "\\textit{VendingMachine} \\left\\vert\\left[\\left\\{ \\textit{coin} \\right\\}\\right]\\right\\vert \\textit{Person} \\equiv \\left (\\textit{coin} \\rightarrow \\textit{choc} \\rightarrow \\textit{STOP}\\right ) \\Box \\left (\\textit{card} \\rightarrow \\textit{STOP}\\right )", "1f0ab62f04bc58686fa2792d9f54ce37": "\nK(x,y;T) = \\langle y;T|x;0 \\rangle = \\int_{x(0)=x}^{x(T)=y} e^{i S[x]} Dx\n\\,", "1f0ad6b12251675fd1cd347ed4a75d3e": "re^{i\\theta}=\\tfrac{b}{2}(e^{i\\rho}+e^{i\\lambda})=b\\cos(\\tfrac{\\rho-\\lambda}{2})e^{i\\tfrac{\\rho+\\lambda}{2}}.", "1f0af275f0e3eadd85a8079af7554a7e": "y=f \\circ x", "1f0b3a1b46852789ebe48485aa4c1a7d": "\\mathbf{V}_{r}", "1f0b4c0f1e368e8221e44946ef71253c": "a_{i,k}", "1f0b62b42686598457d4e526ba2e422a": "(X \\to UA_i)_I", "1f0c02c6cddbfdec1726d2600e3ba510": "\\pi \\approx \\frac{2l\\cdot n}{t h}.", "1f0c0c5d028cbbc4a091b5c55d91852a": "E_{y}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[-\\frac{k_{z}}{\\omega \\varepsilon _{o}\\varepsilon_{r} }\\frac{m\\pi }{a}(A \\ e^{-jk_{x\\varepsilon }w}+B \\ e^{jk_{x\\varepsilon }w})-jk_{xo}(C \\ e^{-jk_{x\\varepsilon }w}+D \\ e^{jk_{x\\varepsilon }w})]e^{-jk_{xo}(x-w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ (40) ", "1f0c2db70e0a325864589dedd7d4cde0": "x_{n+1} = x_n - \\frac {g(x_n)} {g'(x_n)}", "1f0c3e0e13b0288d850817a4e42f74a8": " C(r,z)=G_1(0,z)\\frac{\\delta (r)}{2\\pi r}+ G_2(r,z),\\qquad(12)", "1f0c46369932f7c599890d7879a9c143": "g_j(x) \\leq 0, j \\in J={1,\\dots,m}", "1f0c47394c86781199452029a96d0941": "E=\\begin{pmatrix}\n0&1&0&\\cdots&0\\\\\n0&0&1&\\ddots&\\vdots\\\\\n\\vdots&\\ddots&\\ddots&\\ddots&0\\\\\n0&\\cdots&0&0&1\\\\\n0&\\cdots&0&0&0\n\\end{pmatrix},\n\\qquad\n\\Delta=\\begin{pmatrix}\n-1&1&0&\\cdots&0\\\\\n0&-1&1&\\ddots&\\vdots\\\\\n\\vdots&\\ddots&\\ddots&\\ddots&0\\\\\n0&\\cdots&0&-1&1\\\\\n0&\\cdots&0&0&-1\n\\end{pmatrix},\n", "1f0c52e43e173d595dafb708be9450b1": " d_f = \\frac {\\lambda} {\\sin \\theta}", "1f0c67f18e607266d583cb4e37c5b67c": "C_{ib}", "1f0c7758c8636b86afe67b24dfeada88": "A\\in R", "1f0cb456a5300604254f0cf4545de981": "(x,y,z) = \\left(r\\cos\\phi,r\\sin\\phi,2\\sqrt{ar}\\cos\\frac{\\phi}{2}\\right),", "1f0cc8ee9c54cbf62b346f2b77546d68": "\\lambda \\ge 5", "1f0cdee6fbc0e83b9d98379de50c1f0c": "y_j \\succsim_j x_j", "1f0cf6d8670b1392997cde57ae1370d0": "\\scriptstyle e^{x} = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}", "1f0d073946036c64527daf08d7005aa2": "\n\\begin{align}\nU^2 V^2 + V^2 W^2 + W^2 U^2 & = z^2 x^2 y^4 + x^2 y^2 z^4 + y^2 z^2 x^4 = (x^2 + y^2 + z^2)(x^2 y^2 z^2) \\\\[8pt]\n& = (1)(x^2 y^2 z^2) = (xy) (yz) (zx) = U V W,\n\\end{align}\n", "1f0d4385d1f315ed430a48e6bb97fc39": "i,\\;j", "1f0d81af885552f90dfc35eb2010b240": "S_G", "1f0dc57b8567c1ff562d4401cd8ad78a": "\\mathrm{N\\,m}\\,", "1f0ddb961f3b8a85c16f52ccb8cac7df": "\\left| m_1 - m_0 \\right| < d_0 + d_1", "1f0ddf157ed9e257c422b65b54da7052": "K(k(N))", "1f0de8c565d46804f8c178c1635626ea": "T^{a,b}_R", "1f0de9257f288579a9ee24056d314f67": "\\| f \\|_{BV_\\varphi} := \\| f \\|_{\\infty} + \\mathop{\\varphi \\mbox{-Var}}_{[0, T]} (f),", "1f0df7cddbbef31af9973db705a689eb": "\\int_a^bf(x)\\;\\mathrm{d}x=(b-a)f(\\xi),\\,", "1f0e0204917b76ee133d43c5286acf75": "R1 = \\frac{V_{S} - V_{Z}}{I_{Z} + K.I_{B}}", "1f0e23c7474f7c3c9d776afa335e78dd": "\\scriptstyle \\leq9\\times10^{-14}", "1f0e3dad99908345f7439f8ffabdffc4": "19", "1f0e593b4d4db6ed469e1a0c0f91e390": "L_S^\\sigma", "1f0e597383d998e392cfa977c5443d1d": " W \\not \\in \\operatorname{FV}[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V, W : E \\and F \\operatorname{in} G] \\equiv \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} \\operatorname{let} W : F \\operatorname{in} G] ", "1f0e9f59f9f518d1263e7ab8562a25e1": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "1f0ea9c12a920292e3aadc3b7463e9e7": "A(x) = - \\int {1\\over W} u_2(x) b(x)\\,dx,\\; B(x) = \\int {1 \\over W} u_1(x)b(x)\\,dx", "1f0ef7b3d5b7ee93e178575aea094e21": "\\gcd(a,b)=\\log_2\\prod_{k=0}^{a-1} (1+e^{-2i\\pi k b/a})", "1f0f06853624b77398207d13bd77f48b": "\\forall c\\in \\mathcal{C}", "1f0f2645bb1bd94f23c80242cadd8299": "\\sum _x a = ax + C \\,", "1f0f47055915e0600173ed65e4ff5ae3": "\\theta = 2\\pi/k", "1f0f61d120eb8257be3ea1f92d6780a1": "r=r'\\left[{1-\\frac{1}{2r'}[2(x_0x'+y_0y')+(x'^2+y'^2)]+ \\frac{1}{2r'}[2(x_0x'+y_0y')+(x'^2+y'^2)]^2+ \\cdots}\\right]", "1f0f6fe5f3151d1d5352dc0f61bd8450": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 3.586434\\log_e(T+273.15) - \\frac {5142.974} {T+273.15} + 39.83149 + 1.342324 \\times 10^{-6} (T+273.15)^2", "1f0fc6b690fec6480519224c92f0f349": "\\hat{H}_{\\mathrm{System}}", "1f10351acf489905516df4452464d535": "\nf(x) = \\sum_{i=1}^k \\lambda_i e^{-x \\lambda_i} \\left(\\prod_{j=1, j \\ne i}^k \\frac{\\lambda_j}{\\lambda_j - \\lambda_i}\\right) = \\sum_{i=1}^k \\ell_i(0) \\lambda_i e^{-x \\lambda_i}\n", "1f10689bdb984f30bda41bbe62dc8d17": "\n \\nabla_j {\\mathbf v} \\ \\stackrel{\\mathrm{def}}{=}\\ v^i {}_{||j} \\;\\;\\;\\;\\;\\;\n", "1f10c7a1d9dd3c508f6d72124f4f7e37": "\\|f\\|_\\infty \\equiv \\inf \\{ C\\ge 0 : |f(x)| \\le C \\mbox{ for almost every } x\\}.", "1f10eb6a61c2ab7de6d3f9631dfb287b": " Z=\\sqrt{(1-u_1^2/c^2) (1-u_2^2/c^2)} ", "1f111d363439acc860f38cea82d191cd": "\\sigma_i=\\sgn(\\sum_{j=1}^{N}w_{ij}x_{ij})", "1f1180d5219a3bf04549683eb077a837": "n! = e^{y} \\sqrt{n} \\left( \\frac{n}{e} \\right)^n \\left( 1 + O \\left( \\frac{1}{n} \\right) \\right).", "1f11d6aa70fb3794664012780e5bda41": "\\begin{cases}\n\nx \\equiv & s_{i_1} \\ \\bmod \\ m_{i_1} \\\\\n& . \\\\\n& . \\\\\n& . \\\\\nx \\equiv & s_{i_k} \\ \\bmod \\ m_{i_k} \\\\\n\n\\end{cases}", "1f11f8b1cd2762f5d6625bce43522778": "-\\frac{1}{n} \\log p(\\tilde{X}_0^\\tau) \\to H(X)", "1f124d182014d68f2f784104a7b2f11c": "\\mathbf J_f = \\sigma \\mathbf E", "1f1274e2c136bc674988bab640846225": "\\tilde{E} (\\mathbf{r},\\omega - \\omega_0) = \\int_{-\\infty}^{\\infty} E (\\mathbf{r}, t ) e^{-i (\\omega - \\omega_0)t} dt", "1f12aa1dbbb754fd8db70d70f2cc6295": "Y_q", "1f12f6f40bac88ff4371bf331110ebb1": "p_{01}/(p_{11}+p_{01})", "1f1374e438c48c98bf7e0242dbee0ef6": " 8y = 2\\sin(\\theta)^2 = 1 - \\cos(2\\theta)\\,", "1f137fdc97c8cc839caddb542aede79a": "exp(-r/r_{TF})/r", "1f13b01d8580038ce3380d8aef6be5f8": "\\zeta\\mapsto\\zeta^2.", "1f13ed0acffce58ece16f13da5b02a96": "\\pi=\\frac{4}{Z}\\!", "1f14109a8e40b633e26f6dd03767b103": "F\\cup\\{1-tf\\}", "1f1444490fb916749ea4b29e75f8e0a7": "\\mathbf{J}^{(e)}", "1f146fa60b94bf3b4edeb7f20a4b459a": "5 + 4 + 2 = 11 ", "1f1484ec8ac6c64e870269f0bab2bc6d": "g^{(2)}(0)", "1f14b5f15a3e86bc7acf4d008f63aaec": "f_k=\\sum_i^n A_{ki} x_i", "1f152b703051276d930f2b9da34ccfcf": " a^m_\\lambda \\, dx^\\lambda\\otimes {\\mathrm e}_m ", "1f154d0b7dda54e136c99a4b0760fec7": "\\eta_p\\;", "1f15818c76372abf27254e2f8377d504": "\\{(\\mathbf{a_i},\\mathbf{b_i})\\}", "1f15fe4e97e978026365a0caa5912baf": "\nL = \\frac{1}{6} m \\ell^2 \\left [ {\\dot \\theta_2}^2 + 4 {\\dot \\theta_1}^2 + 3 {\\dot \\theta_1} {\\dot \\theta_2} \\cos (\\theta_1-\\theta_2) \\right ] + \\frac{1}{2} m g \\ell \\left ( 3 \\cos \\theta_1 + \\cos \\theta_2 \\right ).\n", "1f165425a76b8f4a69177a69e7cd8164": "\\Bbb{Q}(\\beta)", "1f16b45095dba0c7080aa355bdb6e299": "\n \\boldsymbol{1} = \\int |x,p\\rangle\\,\\langle x,p|~\\frac{ dx\\, dp}{h} \n", "1f16d270cf026ff2c513265489755689": "\\langle j m j' m' | J M \\rangle", "1f16f23cbc0e14c99de9c7fd03e09319": "(I)\\,", "1f17013e274b5164a8747e6900759c73": "\\textstyle C_1\\nu", "1f1747bed74e109fd8e5a6b6e196cadb": "\nD x + E y + F = 0\\,\n", "1f17b823935ac964a18c7001a1d2fd66": "E(r)=O\\left(r^{1/2+\\varepsilon}\\right).", "1f17c62ad6fb340a93be97a160549eb5": "S_n = \\{x \\in \\bold{Z}^+ : x|n \\} ", "1f17ee0e526abd2038f2357ee3599c02": "x^2 - Px + Q=0 \\,", "1f180697af0129a82a5435c4d58500e3": "W(k)", "1f18133e171cc6b625890c2f41d22942": "\\langle s,t \\mid s^2, t^3, (st)^3 \\rangle\\,\\!", "1f1814277dbf701617c80e155e9b0251": "\n\\left(\n 1 - \\sum_{i=1}^{p'} \\alpha_i L^i\n\\right)\n=\n\\left(\n 1 - \\sum_{i=1}^{p'-d} \\phi_i L^i\n\\right)\n\\left(\n 1 - L\n\\right)^{d} .\n", "1f189c063ae7f610aff980f643decf40": "d(\\cdot, \\cdot)", "1f193eac3318b935f847edbfa9090807": " [\\sigma] = \\mathbf{Q}[\\varepsilon] ", "1f194e9eb40902cc661a102abeabde72": "\n \\begin{align}\n \\sigma_{\\alpha\\beta} & = C_{\\alpha\\beta\\gamma\\theta}~\\varepsilon_{\\gamma\\theta} \\\\\n \\sigma_{\\alpha 3} & = C_{\\alpha 3\\gamma\\theta}~\\varepsilon_{\\gamma\\theta} \\\\\n \\sigma_{33} & = C_{33\\gamma\\theta}~\\varepsilon_{\\gamma\\theta}\n \\end{align}\n", "1f195e03f74a0f26319f9b3cffe66058": "-2\\, ", "1f196fb83c0d5d3545c15cbc743ef321": "\n \\mathcal J_n \\left ( x \\right)\n", "1f19a1f2c8d3843a9667d9fc9390b681": "({\\forall}R)", "1f19a33bccf24c46afaf9bef75347722": "\\displaystyle{2\\pi\\cdot |K(\\mathbf{u},\\mathbf{v}(t))|={|(\\mathbf{v}(t)-\\mathbf{u})\\cdot \\mathbf{n}(0)|\\over |\\mathbf{v}(t)-\\mathbf{u}|^2} \\le {2|\\lambda| + C_1t^2\\over \\lambda^2 + t^2}\\le {2|\\lambda|\\over \\lambda^2 + t^2} +C_1.}", "1f19d18d6b973e3c1dfbe66d6cfa68f5": "f(X) ", "1f19d4609cc9fd80228ab1fd0182a332": "L([Cicero])", "1f19e94a9f84235543cae494396ebbf9": "x_n \\!", "1f19f7febf2c409bd8638213b934b93a": "\\frac{1}{c^2} \\frac{\\partial^2 \\varphi}{\\partial t^2}-\\nabla^2 \\varphi+g\\varphi^3=0.", "1f1a082be2e2c77fb44d8c4b2778dc25": "L \\times R", "1f1a4b1fbd490641e0c8481dd9450039": "\\delta_{pitman} = \\delta_{ML}=\\frac{\\sum{x_i}}{n}.", "1f1a4d1cf696d07d52a4e8851501c6c1": "[\\text{MOS} - 2\\sigma, \\text{MOS} + 2\\sigma]\\,", "1f1a4db47cd7b281907d0236ac6744fc": " \\#", "1f1ad1bb9067c2bb462426f6b865a200": "\\xi = \\frac{m^*(n_{2D}(T_c))}{m_0} \\approx 3.2677 \\times 10^{-3} \\ ", "1f1b0b7e1aba88c1c70922eef59600d2": "\\frac{\\ln(1+0.026\\times 418)}{0.026} \\approx \\text{95 years}", "1f1b179aa774a59fcdb684f51f58b6e1": "\n\\left( \\frac{\\mathrm{d}S_{\\sigma}}{\\mathrm{d}\\sigma} \\right)^{2} + 2m U_{\\sigma}(\\sigma) + 2m\\sigma^{2} \\left(\\Gamma_{z} - E \\right) = \\Gamma_{\\sigma}\n", "1f1b3bcf1777807f637ab3698cde3bea": "e^{-qV}", "1f1b528620da1fe3e0dd1251f262dc88": "= \\frac{60}{510,260} log_2\\left(\\frac{60*510,260}{260 * 10,060}\\right) + \\frac{200}{510,260} log_2\\left(\\frac{200*510,260}{260 * 500,200}\\right)", "1f1b9cc5042543e5658c3cb7a3815788": "K=\\frac{\\pi}{2} + 2\\pi\\sum_{n=1}^\\infty \\frac{q^n}{1+q^{2n}}.\\,", "1f1bd74b9ab52d5988758510d3149cbc": "F_{SK}(x)", "1f1bf5ab4c65031e32632b5c52c46003": "\\mathfrak{D}_0 \\epsilon \\mathfrak{D}", "1f1bf89c2ea673839ce9f84a8fb08b63": "f : x \\mapsto (1-x)^{-1}", "1f1bffbb3828df45025d5e6bfdd1c4bd": " \\chi_{k\\mid k-1} := [ \\chi_{k\\mid k-1}^T \\quad E[\\textbf{v}_{k}^{T}] \\ ]^{T} \\pm \\sqrt{ (L + \\lambda) \\textbf{R}_{k}^{a} }", "1f1ca0c8c229823a6bddf8d6cc02a98a": "\\eta = \\frac{1}{2}\\left(\\frac{\\partial^2 E}{\\partial N^2}\\right)_Z.", "1f1ccc86d887d5ee0c06629c917f33ab": "\nD(t) \\ge \\inf_{0 \\le \\tau \\le t} \\{A(\\tau) + S(t-\\tau) \\} = (A \\otimes S)(t).\n", "1f1cce2ba0977a0733fd4e181a89b82f": " y_0 = A_0 e^{i k x} ", "1f1d7982665f1416dc241c24048e59d8": "\\mathfrak{so}(3,1)_C \\cong \\mathbf{A}_C \\oplus \\mathbf{B}_C \\cong \\mathfrak{sl}(2,C) \\oplus \\mathfrak{sl}(2,C) \\cong \\mathfrak{sl}(2,C) \\oplus i\\mathfrak{sl}(2,C) = \\mathfrak{sl}(2,C)_C \\,,", "1f1d8b97d26818d0b8027b5535524884": "\\, |n(\\mathbf R(0))\\rangle ", "1f1d8f3d43dd53cf2389c321dd906cc8": "(4n^2,2n^2-n,n^2-n)", "1f1dae7d178d4326ea57b154006d7e55": "\\sinh x = \\frac {e^x - e^{-x}} {2} = \\frac {e^{2x} - 1} {2e^x} = \\frac {1 - e^{-2x}} {2e^{-x}}", "1f1dbf73262215e425e5f37b23540e87": "(AX)^{T} = AX", "1f1e3ccf058cc20821a27ea19702f7f9": "\\mu_r = 1 + 2.5 \\cdot \\phi + 10.05 \\cdot \\phi^2 + A \\cdot e^{B \\cdot \\phi},", "1f1edd60bf1c1ab5fb4c8f45a74a2bbb": "\n\\begin{align}\nf\\left( x^{\\prime },t+\\varepsilon \\right) & = \\int_{-\\infty }^\\infty dx\\int_{-i\\infty }^{i\\infty } \\frac{d\\tilde{x}}{2\\pi i} \\left(1+\\varepsilon \\left[ \\tilde{x}D_{1}\\left( x,t\\right) +\\tilde{x}^{2}D_{2}\\left( x,t\\right) \\right] \\right) e^{\\tilde{x}\\left(x-x^{\\prime }\\right) }f\\left( x,t\\right) +O\\left( \\varepsilon ^{2}\\right) \\\\\n& =\\int_{-\\infty }^\\infty dx\\int_{-i\\infty }^{i\\infty }\\frac{d\\tilde{x}}{2\\pi i}\\exp \\left( \\varepsilon \\left[ -\\tilde{x}\\frac{\\left( x^{\\prime}-x\\right) }{\\varepsilon }+\\tilde{x}D_{1}\\left( x,t\\right) +\\tilde{x}^{2}D_{2}\\left( x,t\\right) \\right] \\right) f\\left( x,t\\right) +O\\left(\\varepsilon ^{2}\\right).\n\\end{align}\n", "1f1f0bcd2db60b8d8cb8b6d8c8ce8a54": " \\operatorname{drop-params-tran}[\\lambda m,p,q.(\\lambda g.\\lambda n.n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n))\\ \\lambda x.\\lambda o.\\lambda y.o\\ x\\ y ", "1f1f8213d69df75c1e9d22d170263e52": " k_{\\rm H,cp}(T) = k_{\\rm H,cp}(T^\\ominus)\\, \\exp{ \\left[ C \\, \\left( \\frac{1}{T}-\\frac{1}{T^\\ominus}\\right)\\right]}\\, ", "1f1fb19d3d65384bb9930e72af26a1f7": "\\sum_{i=1}^k\\sum_{j\\in N_i} p_{ij} x_{ij}", "1f1fe6d334705b9d7f0daf57898482b1": " \\mathbf{k}_{2} = k \\left( \\cos{\\left( \\theta_0 - \\Delta \\theta \\right)} \\mathbf{\\hat{x}} + \n\\sin{\\left( \\theta_0 - \\Delta \\theta \\right)} \\mathbf{\\hat{z}}\n\\right) ", "1f2001af69a455e7065f26cc0d617d2d": "\\psi_1(y),\\cdots,\\psi_n(y)", "1f20bd1635b16fd9da83a0b9765f92e0": "[a,b)=\\{x\\,|\\,a\\,\\leq x \\theta_0", "1f3b0d6372efc310d19d3da4d9598e2c": " R^2_2(\\rho) = \\rho^2 \\,", "1f3b61e91aa24a058a0c44d34329d4e3": "\n\\rho=\\sqrt{x^2+y^2}=\\frac{\\cosh\\tau-\\cos\\sigma}{a\\sinh\\tau}.\n", "1f3b88055acc2cbfee9a93ae871ccef3": "\\scriptstyle [m,\\, 1.35m]", "1f3b9963c18a50a20be0e4f76889e67f": "(t_1,t_0,p) \\in D(X)", "1f3bc08a1906d6e143352b3d35dfb43a": "A\\mathbf{x}", "1f3bd215db6702294038f002385005fc": "\\textstyle \\left \\lfloor \\frac{n}{p^2} \\right \\rfloor", "1f3c2767b1f2e88c7d18a42f45979a6c": "P_c:\\mathbb C\\to\\mathbb C", "1f3c290b8064255881ad4a0aad7979cd": "(\\,s_i\\to s_i\\cdot\\epsilon_i\\quad\\,J_{i,k}\\to \\epsilon_i J_{i,k}\\epsilon_k\\,\\quad s_k\\to s_k\\cdot\\epsilon_k \\,)\\,.", "1f3ca10444992ce792db696a63c1ca05": "|ab|<1.", "1f3cbb67c8d96699e4018225bef4d60a": "\n= \\frac{1}{\\eta} \\; G_{q + \\sigma ,\\, p + \\tau}^{\\,n + \\mu ,\\, m + \\nu} \\!\\left( \\left. \\begin{matrix} - b_1, \\dots, - b_m, \\mathbf{c_{\\sigma}}, - b_{m+1}, \\dots, - b_q \\\\ - a_1, \\dots, -a_n, \\mathbf{d_\\tau} , - a_{n+1}, \\dots, - a_p \\end{matrix} \\; \\right| \\, \\frac{\\omega}{\\eta} \\right) =\n", "1f3cc75b7fa36af13ca0cc3af6b6a35a": "2e^2/c \\epsilon_0 m=8\\pi/137", "1f3cdc34448e6565e762d4ced875d96b": "\\gamma_{\\mathrm{rad},0}", "1f3d4624d792d00e1ab2798c64395cac": "\\operatorname{E} [X] = \\int_\\Omega X \\, \\mathrm{d}P = \\int_\\Omega X(\\omega) P(\\mathrm{d}\\omega) ", "1f3d5637c7b8d794919b3a06cea60e44": "\\cos\\theta_W=\\frac{m_W}{m_Z}", "1f3d71fa4cdc4c4099b532bc1f4b5588": "\\sin^2\\theta \\cos^2\\theta = \\frac{1 - \\cos 4\\theta}{8}", "1f3d72fb5326777ad5cfeb1e5cc90440": "\\displaystyle{V(\\psi)=U(D(\\varphi)+S(\\psi))|_{\\Omega},}", "1f3deeaa18f337548cd5c3c97253ce60": "r_m = a \\cdot \\left(1+\\frac{1}{\\sqrt5{}}\\right) \\approx 1.44721 \\cdot a", "1f3e10978c3c8ca6fa28f2ee456d53b0": "D_{KL}(f_\\theta\\|f_{\\theta'})\\leq \\beta.\\,", "1f3e2a1050d96cf50b27ec001a13c40d": "e_1 ", "1f3e4cf1bd075aef01ebdd178383b837": "\\hat \\rho = \\sum_i P_i |\\psi_i\\rangle \\langle \\psi_i | ", "1f3e52a86740c86d5e1c517c2538a9fd": " Select(IMM(s))", "1f3eae811de99bc27d17d4d84a1b2b6a": "\\mathrm{1\\,Fr = 1\\,statcoulomb = 1\\,esu\\; charge = 1\\,cm\\sqrt{dyne}=1\\,g^{1/2} \\cdot cm^{3/2} \\cdot s^{-1}}", "1f3f1817f5c049fd90ea4a4108dd3fe7": " P_{\\mathbf k} \\bar{P}_{\\mathbf k} = 0. ", "1f3f379e16dab43a0c25b19c455470bc": "Z_{CO}^1", "1f3f6a56f9245d79f9c4e8b49d50547a": "t^{-2}", "1f3f9dafa983e3fdd412115b9628dc9c": "\nTSR={(Price_{end} - Price_{begin} + Dividends)}/{Price_{begin}}", "1f3fb04ef4c6a63473fd68fb3c2ff371": "a_{C}", "1f3fba608646fd938773e3e5d0690858": "{\\rm Ci}(x)= \\frac{\\sin x}{x}\\left(1-\\frac{2!}{x^2}+\\frac{4!}{x^4}-\\frac{6!}{x^6}\\cdots\\right)\n -\\frac{\\cos x}{x}\\left(\\frac{1}{x}-\\frac{3!}{x^{3}}+\\frac{5!}{x^5}-\\frac{7!}{x^7}\\cdots\\right)", "1f3fe37d57b5d75151520b4b34bc7955": "\\mbox{Gross Rental Yield} = \\frac{\\mbox{Monthly Rent x 12}}{\\mbox{House Price}} \\mbox{ x } 100%", "1f4094778137452f8468dae050771d8f": " \\langle \\chi | \\psi \\rangle = \\int\\limits_R d x \\, \\langle \\chi | x \\rangle \\langle x | \\psi \\rangle = \\int\\limits_R d x \\, \\chi(x)^{*} \\psi(x) \\,.", "1f409c9d5128288757505262a36c22f1": "(a\\ b\\ c)", "1f40c763ceeeeeb1983e8b5bdfa161d6": "\\pi_{a_1, \\ldots, a_n}(R \\cup P) = \\pi_{a_1, \\ldots, a_n}(R) \\cup \\pi_{a_1, \\ldots, a_n}(P). \\, ", "1f410e06ccf74c48eede05b6c2e91e1c": "\\scriptstyle{\\varepsilon_\\circ }", "1f4121af8984d0ebd7fa79d32be87b0e": "| \\mathbf V | = \\frac{\\hbar}{2} \\frac{\\nabla \\rho}{m \\rho}", "1f4133da43e7b595d5a9fc119ada2db4": "t \\in \\{1,\\ldots,r\\}", "1f418d379721166253dc07fafea9de67": " \\bold A ", "1f4193490d801c7996a882ca0bb3fc86": "r_1 = \\frac{k_{11}}{k_{12}} \\,", "1f41db851d3307e8af992e525012e296": "\\mathcal{C}_{\\varepsilon}", "1f424e16f292e4f25fec4f3755f6b2c8": "\\lim_{t \\to 0} K(t,x,y) = \\delta(x-y)=\\delta_x(y)", "1f4296b4900a3ea65bb380f9581ab03b": "\\textstyle M^1", "1f42a693986145cd1ff0f482061cac60": "\\Delta \\mathbf{b}", "1f42c93c4f5f3eec4ba70733b656dd31": "E_{internal}=E_{cont}+E_{curv}", "1f42d996e23374db76583a24734ccac8": "L = \\frac{\\mathrm{d}^{3} }{\\mathrm{d} x^{3}}", "1f4326a426629934888b77136e0091ae": "\n0 = \\frac{d}{dq} \\left[ \\ln \\frac{dt}{dq} + \\ln w \\right]\n", "1f43a63a03f9e8fe1beca54e344ae88a": "\\hat{R}_x^\\alpha", "1f43e0ded98cec51b0f5c5e3c70c7de7": "L(\\cdot)", "1f44364091ee843b40ae19679b18fa04": "X = x^{q} _ {\\bar{t}}", "1f444c951dd6595039d509a4c29c3537": "\\begin{bmatrix}\n1&-1&0&0&0&0&0 \\\\\n0&1&-1&0&0&0&0 \\\\\n0&0&1&-1&0&0&0 \\\\\n0&0&0&1&-1&0&0 \\\\\n0&0&0&0&1&1&0 \\\\\n-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&\\frac{\\sqrt{2}}{2}\\\\\n0&0&0&0&1&-1&0 \\\\\n\\end{bmatrix}.", "1f446f46fdec7813d84134cfdb6e473e": "I_2 \\, ", "1f45204f4e062aaf3426ef96ad73fa94": "\\displaystyle \\operatorname{e}^{-a|x|} \\,", "1f45b131da0475894ef4251a14655476": "\\operatorname{gl.dim} R \\le n \\Rightarrow \\operatorname{pd}_R k \\le n \\Rightarrow \\operatorname{Tor}^R_{n+1}(-, k) = 0 \\Rightarrow \\operatorname{pd}_R - \\le n \\Rightarrow \\operatorname{gl.dim} R \\le n,", "1f45dade52b12e5f2940932a6bc48842": "P(S)\\,\\!", "1f4650482e9b45cf1daf841afccf3606": "c_0=\\lambda_J\\omega_p", "1f465f44a79303480395bf7a8c0eecee": "K_c=\\frac{[CO_2]} {[CO]^2}", "1f469cbe9eac6bdcab228b014d4575a2": "x_n=f(x_{n-1}).", "1f46a06bc6bd9d1c680fec7f3d741581": " E_{cont} ", "1f46d7acb595e6ae81bb41b111ff6207": "A \\leftarrow B \\rightarrow C", "1f47e9d92cd2291ea9ee5f59f9cc2519": "\\nabla^2 \\varphi' - \\mu_0 \\varepsilon_0 \\frac{\\partial^2 \\varphi'}{\\partial t^2} = \\Box^2 \\varphi' = - \\frac{\\rho}{\\varepsilon_0}", "1f48aa385a9d223f4976c3ba2a97e691": "e^+e^- \\to e^+e^- \\mu ^+\\mu ^-", "1f48e973d6a9075dbaaf41a9e85f034e": "t = 0", "1f4912bb9695e3a49fe4dd26f27bf7d3": "\\tau_{GL}=\\frac{\\pi}{\\hbar(T-T_{c0})}", "1f492f938b0ebf705bab51e7d3fd4493": "x_k^{(L)}", "1f495127b0b25bf21fa02fe4ca6724ef": "m_f", "1f495ef0b3c1242c76ef57d9089861f2": "X_{i+1} = X_i - {f(X_i)\\over f'(X_i)} = X_i - {1/X_i - D\\over -1/X_i^2} = X_i + X_i(1-DX_i) = X_i(2-DX_i)", "1f49882545df5064b8291982a190dbb4": "\\lambda(V) := \\oplus_{i=1}^{d_\\lambda} V_i \\simeq \\mathbb{C}^{d_\\lambda} \\otimes M_{\\lambda}", "1f498ec83e89f2e24c108ad374add6d6": "U=-\\mu_o\\int\\limits_V {\\vec M \\cdot \\vec H\\, dV}\\,\\!", "1f4996abc4d629767fda76768e8f6dcb": "s = a \\sinh \\tfrac{x_2}{a} - a \\sinh \\tfrac{x_1}{a}.\\,", "1f499d5cf581c80f9cd49e02f19f1c12": "c_{1} \\ne 0", "1f49a248f82d08eacacc767bdd0530d9": "\\scriptstyle \\boldsymbol{r}_{\\text{rec}}", "1f49da4fe243c9c917c6a908b103cb01": "f:X\\to {\\mathbb F}", "1f49f3c06a9c5c0c22ddce381601071e": "r \\neq \\textbf{Q}", "1f49fb8e184a29d979f2f7e6053139dd": " G = \\frac{E}{2(1+\\nu)} ", "1f4a232959727d93b19ca9ba4c05c3de": " \\mathbf {Q_{A}} ", "1f4a4bd67aee269a610747c82c36f2c0": "{C}/{r} = 2\\pi", "1f4aa87ff18be3eb68d92e481da60979": "\\forall a \\in A, L(a) = \\mathit{undec}", "1f4af700e15cc80d29f62724c7f11015": " \\overline u_i< u^{max}_i ", "1f4b251a721693d836d6d83fd18f418e": "\\,{}_tp_x", "1f4b56e8988bc9eb7e3fb118d51984ba": "\n\\begin{align}\n \\hat{C}^{(0)} & \\equiv \\hat{A}\\\\\n \\hat{C}^{(1)} & \\equiv [\\hat{H}, \\hat{A}] = \\hat{H}\\hat{A}-\\hat{A}\\hat{H}\\\\\n \\hat{C}^{(k)} & \\equiv [\\hat{H}, \\hat{C}^{(k-1)}], \\ \\ \\ k=1,2,\\ldots\n\\end{align}\n", "1f4b79c8217ea854e208d8e973a30b13": "H_D(x,s_p)", "1f4b7c8cfd9cb55c24dedefe247d7973": "\\begin{align}\n\\left \\| f_k(T) - f_l(T) \\right \\| &= \\frac{1}{2 \\pi} \\left\\|\\int_{\\Gamma} \\frac{(f_k - f_l)(\\zeta)}{\\zeta - T} d \\zeta \\right\\| \\\\\n&\\leq \\frac{1}{2 \\pi} \\int_{\\Gamma} \\left |(f_k - f_l)(\\zeta) \\right | \\cdot \\left \\| (\\zeta - T)^{-1} \\right \\| d \\zeta\n\\end{align}", "1f4b7da7c6c138503c162e12ab08df14": "\\partial_x \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_x \\psi )} \\right) = \\partial_x \\left( \\frac{1}{2} \\partial_t \\psi + 3 (\\partial_x \\psi)^2 \\right) \\,", "1f4baef7bfc93e82e4be83c7e20715f7": "E(\\ln(\\nu+X^2))", "1f4bc6bee995390170ef32393251b990": " I=\\frac{m_1m_2}{m_1 +m_2}d^2 ", "1f4c594d5033505750c0e076a2ec35a0": "f:\\mathbb{D}\\to\\mathbb{D}", "1f4cdd18c3582fcccc22739423922f99": "\\theta = \\tan^{-1} \\bigg( \\frac {\\text{Rise}}{\\text{Run}} \\bigg) \\,", "1f4d17ed011d3180ad4df3ccf3bd884a": "-a+be^{i\\lambda}=re^{i\\theta}-ce^{i\\psi}\\,", "1f4d56e4466df08245fa11d352cd36a5": "S_F = 0.5 cm^2", "1f4d59f9d9c835cc384886b46530bf82": "\\exp O(\\sqrt{\\log N\\log\\log N}),", "1f4d7287d0029c46e367faae346810a2": "P(E)\\in\\mathbb{R}, P(E)\\geq 0 \\qquad \\forall E\\in F", "1f4d99bc8211046745c8e0ca798c0a80": "\\overline{DE}", "1f4de56f69f3332b2dc5229b447fa701": " \\text{Rebound Rate} = \\dfrac{100 \\times\\text{Rebounds}\\times \\dfrac{\\text{Team Minutes Played}}{5}}{\\text{Minutes Played}\\times \\left (\\text{Team Total Rebounds} + \\text{Opposing Team Total Rebounds}\\right )}", "1f4dede5e09242ee6a3cbe4e193f2470": "G_i = G_{i+1}", "1f4e180942bf80c7ebfc71e6194e9307": "e^{-\\epsilon t}", "1f4e2085ab870d616f775abd87749e8e": "\\textstyle |\\psi_{2} \\rangle ", "1f4e6b621b5b30c2d856258f8c19d63d": " \\lnot Q ", "1f4ef6d70410779f6bfe05d015d0f085": "=O e^{-itH/\\hbar}|\\psi_0\\rangle ", "1f4ef731bc1cd46e96459de132d28fed": " \\dot m = \\int \\rho \\mathbf{V} \\cdot d \\mathbf{A}.", "1f4f235315bad3a7f4380828450c6090": " \\sigma:= -m^a\\delta l_a=-m^a m^b\\nabla_b l_a\\,, \\quad \\rho := -m^a\\bar{\\delta} l_a=-m^a \\bar{m}^b \\nabla_b l_a\\,; ", "1f4f39844cbed8f0a05934d03ae60661": "\\textit{occludeopen}(t)", "1f4f48b79a9baac91524bb305743ada2": "= \\frac{1}{\\sigma\\sqrt{2\\,\\pi}} \\times 1 \\times \\exp\\!\\left[-\\frac12 \\left(\\frac{x-\\lambda}{\\sigma}\\right)^{\\!2\\,} \\right] ", "1f4f9c615ce9efa97f35c271a5689b69": "K=0.606", "1f4fb98f437b18ff59474dca1987de6d": "\\Delta^0_n \\subsetneq \\Sigma^0_n", "1f50ee87dc7103f6b36091ae87fcdaf5": "\\rho_t(X) = \\text{ess}\\sup_{Q \\in \\mathcal{Q}} E^Q[-X|\\mathcal{F}_t]", "1f5109d10a67287eaaf3a5c6d608e171": " \\mathbf{s} = \\{ s[j] \\} = \\{ \\sqrt{C[j,j]} \\} \\qquad \\text{for } j = 1, \\ldots, p ", "1f519684ea0ae730f6acf7bf4adccd1c": "m \\leq 1_WC^TA", "1f51de4d6ffe3fa43e80f6804a3ebd44": " 1 / \\sqrt{ {{|}} C_2 {{|}} } ", "1f52919fea406029c3c456a9b83713ad": " \\left[S 1 \\right] = \\left[S 1 \\right]_0 e^{-\\Gamma t} ", "1f52cbcf2b72398cb1f116c7e2a552bb": "\\hat{s}=\\frac{h_0^*y_0+h_1^*y_1+...+h_{N-1}^*y_{N-1}}{|h_0|^2+|h_1|^2+...+|h_{N-1}|^2},", "1f52e3766731ff08093d8273dd92a1a6": " a_P = {a_P}^0 ", "1f530a9d68b027a16c17ec355edcc18b": " \\langle \\prod_x e^{i h_x \\phi_x}\\rangle ", "1f533f828ab1c804bae93efec26ca986": "k^2 \\pmod{p}", "1f53d6f847ebbf7f1c5d2168318b6db1": "", "1f540901e420b1ac5f7d7ea7e3f8bce6": "\np(0,0,t,y) \\, dy := P(W^+_t \\in dy ) = 2\\sqrt{2 \\pi} \\frac{y}{t}\\phi_t(y)\\Phi_{1-t}(0,y) \\, dy.\n", "1f54379eeba326e08190e38ebcd6afa6": "1RM = \\frac{100 \\cdot w}{101.3 - 2.67123 \\cdot r}", "1f54a6a12d87458b298ca5403b2071cc": "Y=\\beta_{60} +\\beta_{61}X +\\beta_{62}Mo +\\beta_{63}XMo +\\beta_{64}Me +\\beta_{65}MeMo + \\varepsilon_6", "1f54f78a933ec5aaf6d1156971b7f40f": "100\\uparrow\\uparrow n", "1f5572dd81fa4bff2fd57f6e7d51e9fe": "f(z)=\\int_0^z\\frac{\\sin(\\zeta^\\rho)}{\\zeta^\\rho}d\\zeta,", "1f559df95ed5b5d0d01cdbd0dc629275": " AMF = \\frac {1.55L_c + \\frac {80.2} {R} - .012S} {1.55L_c} ", "1f55e5a38b4ce09135b4d303e612648f": "k = \\omega \\sqrt{\\mu \\epsilon}", "1f561e138e1028e3e74607b612bf3ea7": "\\alpha t < D", "1f562013abc2c0a8de95a0d4dabf9a19": "m < n", "1f565c87a1387c552d862e801ec80742": "f(x,y,z)=x^5y^2z^3 \\,", "1f5661bb8dfc8b1370f9155d116c16a4": "\n \\begin{align}\n & bD\\cfrac{d^3 w_x}{d x^3} + q_{x1} = 0 \\quad,\\quad\n \\frac{b^3D}{12}\\cfrac{d^3 \\theta_x}{d x^3} -2bD(1-\\nu)\\cfrac{d \\theta_x}{d x} + q_{x2} = 0 \\\\\n & bD\\cfrac{d^2 w_x}{d x^2} = 0 \\quad,\\quad \\frac{b^3D}{12}\\cfrac{d^2 \\theta_x}{d x^2} = 0 \\,.\n \\end{align}\n", "1f56a102ed351261b47ef180e0ab9f98": "\\lbrace e_i \\rbrace", "1f56d7ef5752dfad2a21ded837164097": "\\nabla(z) = 1-z^2,\\ ", "1f56ffeb9671bebce4998f79cfeeda8e": "Tp", "1f571b6d97464f30552ea6c6cfcbc5fd": "(1) PQ={f}(\\overset{+}M)", "1f576a28f7dc049f0233503f085e9357": "t \\, \\rightarrow t + a ", "1f5792977e29075e5e2228b7a2b90d5e": "Y_T = \\int_0^\\infty Y(\\lambda)I(\\lambda,T)\\,d\\lambda", "1f57a6dbeb471f36a6c6eca1ebbffadf": "\\hbox{DSSIM}(x,y) = \\frac{1 - \\hbox{SSIM}(x, y)}{2}", "1f57e73c701e60ae1cee6e33e047ad7b": "\\displaystyle{(S\\varphi,\\varphi)=\\int \\|\\nabla S(\\phi)\\|^2, \\,\\,\\, (S\\varphi_1,\\varphi_2) = \\int \\nabla S(\\varphi_1) \\cdot \\overline{\\nabla S(\\varphi_2)}.}", "1f57ed7775ad69e449f5075b5a542b42": "\\sum_{n=0}^Nn", "1f57f2f7449ad9c79259e2bb173d483e": "\\approx S_x(at,f)", "1f57fb348443229c000ed7f07a9f4811": "\\mathbf{r_{12}}\\equiv |\\mathbf{r}_{2}-\\mathbf{r}_{1}|", "1f57ff5e4f7a2b3aa0f4d136e973afeb": "\\left[ H,Q \\right] = 0 ", "1f5805695b01f67aca8844e9c5f3da8f": " G_iG_j=G_jG_i, \\mathrm{if} \\left\\vert i-j \\right\\vert \\geqslant 2,", "1f581915882310b23392b2b51858aefc": "\\chi_v^{\\text{SI}}=4\\pi\\chi_v^{\\text{cgs}}", "1f58273c4161e57f62a95804c2ae961a": "\\partial p", "1f5864cee6ca653b8e60ca79ca465fb0": " z_{11} \\,", "1f586f8985bab9775ab6007a59e6677b": "\\, 1-p+pe^{it}", "1f587ed1ad0960440bf3c6f32ec1e40d": "{{v}_{test}}", "1f5893a63559a3a560972a368bf93d80": "v(t) = \\frac{a t}{ \\sqrt{1+ \\left( \\frac{a t}{c} \\right)^2}}.", "1f59153e240c9fd39400d8a645f152cf": "X(\\sigma+2\\pi,\\tau)=g[X(\\sigma,\\tau)]", "1f591a2af2638bcbe66cb85c65badf9e": "A = b^2 \\cdot \\sin \\alpha = b^2 \\cdot \\sin \\beta ,", "1f5989e596ade06c50854512ad10b90d": "H_{formation} \\, = \\, 68.29 + \\sum {H_{form,i}}", "1f598bccb4a94d4d68dda8871b97f95c": "a < 1", "1f59d0a44ed6e37998975a212e4c6055": "H_k", "1f59db1ae2bfcb1bdf242239a99d9f2f": "T^{-1}_{w^{-1}}.", "1f59fb2a6ef28a9605c900503225d4a8": "M \\to M[S^{-1}], \\, m \\mapsto m / 1", "1f5a3bfb39ff53e80ef0aa2ff1fe9c8b": "\\langle\\Delta \\zeta,\\eta\\rangle = \\langle\\zeta,\\Delta \\eta\\rangle", "1f5a42ee292b91dd49e01dbd78f259e7": " (-1)^{\\text{sign}}(1.b_{51}b_{50}...b_{0})_2 \\times 2^{e-1023} ", "1f5a70d10b9c51bb4700059a49d134ce": "\\mathbf{g_i} \\cdot \\mathbf{a_j}=2\\pi\\delta_{ij}", "1f5a98bfe470eef2e3bbc7d15cc153d8": "\\{ F_\\alpha \\}_\\alpha", "1f5aaaadfb44ea0bf4e0921959fd966b": "\n\\begin{align}\ni\\left.\\frac{\\partial}{\\partial t}\\big(\\mathbf j(\\mathbf r,t)-\\mathbf j'(\\mathbf r,t) \\big)\\right|_{t=t_0} &= \\langle\\Psi(t_0)|[\\hat{\\mathbf{j}}(\\mathbf r),\\hat{H}_{v}(t_0)-\\hat{H}_{v'}(t_0)]|\\Psi(t_0)\\rangle,\\\\\n&=\\langle\\Psi(t_0)|[\\hat{\\mathbf{j}}(\\mathbf r),\\hat{V}(t_0)-\\hat{V}'(t_0)]|\\Psi(t_0)\\rangle,\\\\\n&= i\\rho(\\mathbf r,t_0)\\nabla\\big(v(\\mathbf{r},t_0)-v'(\\mathbf{r},t_0)\\big).\n\\end{align}\n", "1f5af69f4cd6ab9670333380d158a3e2": "C(\\varepsilon) = \\frac{1}{N^2} \\sum_{\\stackrel{i,j=1}{i \\neq j}}^N \\Theta(\\varepsilon - || \\vec{x}(i) - \\vec{x}(j)||), \\quad \\vec{x}(i) \\in \\Bbb{R}^m,", "1f5b0b99e708e3a4add7c727d6c94ad9": "|g_2\\rangle", "1f5b28c97db0498173915b7589320a86": "total\\ revolutions/(2400*60)", "1f5b47f72cf8eda5fcb5b9a34f3a7251": "F_X(x) = \\operatorname{P}(X\\leq x),", "1f5b9fb41c0ba54be66cb9b4ada1af58": "q \\mbox{ splits completely in } \\mathbf Q(\\sqrt{p^*}) \\quad \\iff \\quad \\left(\\frac{p^*}q\\right) = 1.", "1f5ba2f6d777f31c877612585d2c1302": "n'_i=\\sum_{j=0}^3 w_{ij}n_j", "1f5bc4998eee5f67169f913df1f0232d": "P = P_{mkt}", "1f5bd84b548f1bb052e69647d162ee12": " (\\alpha^j,S_j) ", "1f5bdf790fae36bb5be86d1315492f84": "1/2^{f(|x|)}", "1f5c31b889f6f35e5a00bbc67eb9cfb2": "f(x) = x^5 - 2x^4-7x^3 + 8x^2 + 12x = x (x + 1) (x - 3) (x + 2) (x - 2)", "1f5c39c031a906f822cbe3468b9ee799": "{1 \\over 2} 2X + {1 \\over 2} X = {3 \\over 2}X", "1f5c5f44a161877aaca866ffc6e477b9": "\\mathfrak{s}", "1f5c6705689b61586ae8b35a7f72fcb6": "V_y = \\sin \\theta \\cdot V_r + \\cos \\theta \\cdot V_t = \\sqrt{\\frac {\\mu}{p}} \\cdot (e +\\cos \\theta)", "1f5c9cb497975dfe20b6d3b6b1237c63": " W_2", "1f5cbb651d2e47ef5a245797a854dee4": "\\varphi(z)", "1f5d2ce620e0483766d7abe4e2f5d524": "\\Delta_*^n = \\left\\{(s_1,\\cdots,s_n)\\in\\mathbb{R}^n\\mid 0 = s_0 \\leq s_1 \\leq s_2 \\leq \\dots \\leq s_n \\leq s_{n+1} = 1 \\right\\}.", "1f5d3e2b0d7cd5a6b81c9d395b27421d": "A_{Bq} = \\frac{m}{m_a}N_A\\frac{\\ln(2)}{t_{1/2}}", "1f5d8ba961eedebfbe6dd9609e360856": "\nX \\sim N(0, P)\n\\,\\!", "1f5d8cc22cdd442cd17b10f785cb6f7d": "\\omega\\wedge\\eta=\\operatorname{Alt}(\\omega\\otimes\\eta).", "1f5dbd3b922ce6357a8d2a71c262eebe": "\\mathcal{F} \\left\\{ e^{i \\frac{2\\pi nx}{P} } \\right\\}", "1f5dec31517a9354b868091d6a5a5d27": "\\begin{align}\nx_n=q_+^{\\;n}&=e^{-\\frac{1}{24}(wh)^2\\,wt_n+\\mathcal O(h^4)}\\,e^{wt_n}\\\\[.3em]\n&=e^{wt_n}\\left(1-\\tfrac{1}{24}(wh)^2\\,wt_n+\\mathcal O(h^4)\\right)\\\\[.3em]\n&=e^{wt_n}+\\mathcal O(h^2t_ne^{wt_n}).\n\\end{align}", "1f5dee5c2208a949084d8232eac8320d": "L_\\phi=\\phi R-2g^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi\\,", "1f5e29794ddc09082ea707bfe10fb199": " \n\\mathbf{A} = \\begin{bmatrix}\n\\mathbf{A}_{(1,1)} & \\mathbf{A}_{(1,2)} & & & \\cdots & \\mathbf{A}_{(1,n-1)} & \\mathbf{A}_{(1,n)} \\\\\n\\mathbf{A}_{(2,1)} & \\mathbf{A}_{(1,1)} & \\mathbf{A}_{(1,2)} & & & & \\mathbf{A}_{(1,n-1)} \\\\\n & \\ddots & \\ddots & \\ddots & & & \\vdots \\\\\n & & \\mathbf{A}_{(2,1)} & \\mathbf{A}_{(1,1)} & \\mathbf{A}_{(1,2)} & & \\\\\n\\vdots & & & \\ddots & \\ddots & \\ddots & \\\\\n\\mathbf{A}_{(n-1,1)} & & & & \\mathbf{A}_{(2,1)} & \\mathbf{A}_{(1,1)} & \\mathbf{A}_{(1,2)} \\\\\n\\mathbf{A}_{(n,1)} & \\mathbf{A}_{(n-1,1)} & \\cdots & & & \\mathbf{A}_{(2,1)} & \\mathbf{A}_{(1,1)}\n\\end{bmatrix}.\n", "1f5e2ae8e9487e9350ef2bc324e92e66": "x \\mapsto f(x) - 1/2", "1f5e3a59b468067e398d838ba15d9fc6": "\\ln(2)=\\sum_{k=1}^{\\infty}\\frac{1}{k2^k}\\, .", "1f5e4a2dfef5715aa564fd5de0f4181d": "\\scriptstyle n F(n x) = f(x) + f(x + \\frac{1}{n}) + f(x + \\frac{2}{n}) + \\ldots f(x + \\frac{n-1}{n}).", "1f5ec808f511b2b96896c94aadfe2642": "M(|\\psi\\rangle) = \\left.\\left\\{\\Big( x(t),\\,y(t) \\Big)\\,\\right|\\,\\forall\\,t \\right\\}", "1f5f10ef607334226c8e8e92f0a2c0ec": "H = 2\\lambda d_p + 2GD_m/\\mu + \\omega(d_p \\mbox{ or } d_c)^2\\mu/D_m +Rd_f^2\\mu/D_s", "1f5f1b879fc74bcfc820fb8c1850f811": "\\hat{\\mathcal{P}}_{S_l^k}\\hat{\\mathcal{H}}\\hat{\\mathcal{P}}_{S_l^k} \\left|\\Phi_l^{-k} \\Psi_{\\mu}^{v+k}\\right\\rangle = E_{l,\\mu}\n\\left|\\Phi_l^{-k} \\Psi_{\\mu}^{v+k}\\right\\rangle", "1f5f1fef02f11fef2f7dd85d59b71e7c": "|D| < C", "1f5f2d59cc7a102a122a0105fa741dbd": "\\pi_2(n) \\sim 2 C_2 \\frac{n}{(\\ln n)^2} \\sim 2 C_2 \\int_2^n {dt \\over (\\ln t)^2}", "1f5f387576250bf959ac8f82fe9e0db6": "-0.150757555\\ldots", "1f5f5a07d2f644f31fa0f288ed4952af": "I_{PSS}=|f'_zf_if_i\\chi_{zii}^{(2)}|^2", "1f5fbd3d6181301b392014d42fe515e6": " x : \\mathcal{A} \\rightarrow (0,1) ", "1f5fccc09c122ed7e7f5fdcfdab0ccd9": "\\lim_{k \\to \\infty}J^k = 0.", "1f604c383f5d67db65a2bbb587d1449a": "b=\\max\\left(|X-Y|,|Y-X|\\right) ", "1f60d152601980549950f63a7be06b9f": "I_0 = \\frac{\\pi}{4} r^4", "1f611be29bbd12600af44ddcc0013721": "\\scriptstyle \\left(\\mathbf{B}^T\\mathbf{A}\\right)^T\\left(\\mathbf{B}^T\\mathbf{A}\\right)", "1f6134d3d72c7e3b5bfd3c44f46045e5": "t_{r,s}", "1f61503a790ac20e06281b5f101f6148": "T \\to T", "1f615fa37e357bf3c02aeef7c1e99eff": "\\begin{matrix}A\\end{matrix}", "1f61d682e465a412200713a54307ed67": "\\hat{H}_0 = \\hat{H}'_0= \\beta m", "1f62149ae34ca2716eed7239bdbb8e22": " \\eta_{ij} =1, G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji} \\,\\!", "1f62ddf4d704367472b521748c951282": "m_w", "1f63231749cb0f86c2fa12e1e16de93b": "\nh_{\\nu} = \\frac{1}{2} \\sqrt{\\frac{\\left( \\lambda - \\nu \\right) \\left( \\mu - \\nu \\right)}{ \\left( A - \\nu \\right) \\left( B - \\nu \\right)}}\n", "1f632fa5629297c093a5d9dbbf1decd3": "\\scriptstyle x^2 + N y^2", "1f639c06103efa0ed2f674d5b96ca127": "q=\\exp(2\\pi iz)", "1f63af51a251d5aac8ca74685b89ffd1": "f(x)=72 \\Rightarrow \\lim_{x \\to \\infty} 72 = 72", "1f63bccf389e2ee83d2b30d556253876": "Z_i \\equiv X_i \\wedge Y_i", "1f63fa9ec75879d0dc2b30d7dc69f75e": "a \\vee b,", "1f648fe51ba8859b35dac59012eb4374": "\\scriptstyle{E_1(t)}", "1f64a24618eb0d7eda60e43a5785ad69": "(-4|V|^2+nJ^2+2\\Delta J)/4\\,", "1f64a2bfd2a4ff9ddac18d83f9a31afe": "f:S^n\\to X", "1f64ae1349c1855bb0f196acdd13611b": " DR_{S}^{D}", "1f64e0499631b1dcecb2bf955b99f18e": "\\mathbf{J}(\\mathbf{x},t) = \\int_{-\\infty}^\\infty \\hat{\\mathbf{J}}(\\mathbf{x},\\omega) e^{-i \\omega t}", "1f652bcfbdfd2b5364feabfdbc0eae45": "\\varphi\\circ\\sigma = \\tau_\\sigma\\circ\\varphi.", "1f653728e97c686b4a02d24492738728": "[G^{[l]},G^{[l']}]=0", "1f6567e853f4d22dd5bf80dd89bc4075": " S^0\\cong\\operatorname{O}(1) ", "1f658e42e61175202e3773d5be52c660": "P = E\\, c_g, \\, \\ ", "1f65919005875ff98b882baf6ba58455": "\n \\begin{align}\n (3) & & \\quad \\frac{\\partial \\varphi}{\\partial x} & = -\\cfrac{m}{\\kappa AG}~\\frac{\\partial^2 w}{\\partial t^2} + \\frac{\\partial^2 w}{\\partial x^2} + \\cfrac{q}{\\kappa AG} \\\\\n (4) & & \\quad \\frac{\\partial^2 q}{\\partial t^2} & = m~\\cfrac{\\partial^4 w}{\\partial t^4} - \\kappa AG~\\left(\\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} - \\cfrac{\\partial^3\\varphi}{\\partial x\\partial t^2}\\right)\n \\end{align}\n", "1f6596ab12e8bc96c0cf2e1cf5b4317a": "(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2 = (x_1x_2 - Ny_1y_2)^2 - N(x_1y_2 - x_2y_1)^2", "1f65a639eec747fe4dd95ce2ff06e62a": "\\dot{\\tilde{\\mu}}=D\\tilde{\\mu}-\\partial_{\\tilde{\\mu}} F(s,\\tilde{\\mu})", "1f66aa69a310c7e9737665ad96ab799b": "\\{\\neg P(a), \\forall x . P(x)\\}", "1f66bf297e556c448c460b2203b9da29": "\\sum_{\\tau \\in S_n\\colon\\tau(i)=j}", "1f66c40bb70317029629ad5fbfde8d64": "\\delta A + (1 - \\delta) B", "1f66c9f9d4101fb01a8bdcaa9b64d432": "\\! e^{it\\mu -\\theta|t|}", "1f66cc86f8cf9fac7390db778ac01629": "A_m(3,2) = 1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690,\\ldots ", "1f66fa94d08d37fd9eb8b263f37ddc90": "[m]_{q_i} = \\frac{q_i^m - q_i^{-m}}{q_i - q_i^{-1}}.", "1f67189450d72f4b7c3ad4b33d4ad3a8": "A=30^\\circ", "1f671f6ac23044e005d8c387522a2e32": "\\scriptstyle \\frac{T}{T^\\prime} \\,=\\, T\\Delta f", "1f67a40837ff4ecebb72d040500d8d81": "\\mathfrak J^2(a)_n=\\sum_{i=0}^n(-2)^{n-i}\\binom{n}{i}a_i.", "1f67b5ee1f06a3961ffec313a7d07e66": "\\,\\Pr(Y(t)=i) = \\sum_{n} \\Pr(Y(t)=i|N(t)=n)\\Pr(N(t)=n) ", "1f67b88348ebe0e62021b0be4e38e1d8": "\\Phi_{Y_1,X_1}(x\\circ f\\circ F(y))\n= G(x)\\circ G(f)\\circ G(F(y))\\circ\\eta_{Y_1}\n= G(x)\\circ G(f)\\circ \\eta_{Y_0}\\circ y\n= G(x)\\circ \\Phi_{Y_0,X_0}(f)\\circ y", "1f6804ae6e67c737ed1542a3e65df89a": "dx/dt=ax-xy\\quad\\text{and}\\quad dy/dt=-y+x^2-2y^2.", "1f6834844c44fbc52880521907cac576": "R_{0}=8", "1f684f8035be836b3978e78a8c5f1dec": "(f+g)(x) = f(x) + g(x)", "1f6893c5c6ccf0232672e13f24b9c9e2": "\\phi=-\\log_2{D/D_0},", "1f68d1a6f6121af6365c3ea277adb542": "\n\\begin{align}\nA_{PT} &= A_P + A_T\\\\\n&= (i_P + b_P/2 - 1) + (i_T + b_T/2 - 1)\\\\\n&= (i_P + i_T) + (b_P + b_T)/2 - 2\\\\\n&= i_{PT} - (c - 2) + (b_{PT} + 2(c - 2) + 2)/2 - 2\\\\\n&= i_{PT} + b_{PT}/2 - 1.\n\\end{align}\n", "1f68eba0366bb98e17905aff8d072bef": "\\mathbf x_0", "1f691ed1d29fb3b2c3f983b2d6b37eb5": "4\\cdot153-9=603", "1f694bd0db6da6e22a7f9ebb1fe22d0d": "\\frac{1}{x^2 - x + 2} = \\sum_{k=0}^{\\infty} a_k x^k.", "1f69686b65fc4b95be1c0994e4648133": "\n\\delta^{i}_{j} = \n\\begin{cases}\n 0 & (i \\ne j), \\\\\n 1 & (i = j). \n\\end{cases}\n", "1f696c97024a2aad21065062130c3da7": "\\sum (\\cdots) \\rightarrow \\int(\\cdots)\\rho\\mathrm d^3r", "1f69905b2b6369f94fdb871b35854461": "I={\\Big\\langle\\Big\\langle}\\partial_{xx}-\\frac{1}{x}\\partial_x,\\partial_y{\\Big\\rangle\\Big\\rangle}", "1f69b8f8775417ad1dab192b3400cb22": "y, y'", "1f69ba1af577c7c8cf5f4389f16c19b4": "4\\pi r^3/3", "1f6a17edb5481b7f9891f481d1b1de71": "\\scriptstyle |\\langle a|\\phi\\rangle|^2", "1f6a3103ea914a36e60577953bc23937": " = \\pi- \\frac{\\pi^3}{3\\cdot3!} + \\frac{\\pi^5}{5\\cdot5!} - \\frac{\\pi^7}{7\\cdot7!} + \\cdots ", "1f6b4ceedcec1d18fed928901d828e36": " J^\\mu A_\\mu ", "1f6bcc26c4ba221bfd48aa9895ee8af7": "\\begin{bmatrix}X_{00} & X_{01} \\\\ X_{10} & X_{11}\\end{bmatrix}\n\\begin{bmatrix}Y_{00} & Y_{01} \\\\ Y_{10} & Y_{11}\\end{bmatrix} =\n\\begin{bmatrix}X_{00}Y_{00} + X_{01}Y_{10} & X_{00}Y_{01} + X_{01}Y_{11} \\\\\nX_{10}Y_{00} + X_{11}Y_{10} & X_{10}Y_{01} + X_{11}Y_{11}\\end{bmatrix}.\n", "1f6c2d98d504bac823837e914717b980": "(n,k) \\mapsto n^{\\underline{k}}", "1f6c4256d82215aac198ef1012bef4f5": "\\theta^2=1", "1f6c529f6f1dd7e5dfc6610d6884056b": "u = u*\\chi_r = u*\\chi_r*\\cdots*\\chi_r\\,,\\qquad x\\in\\Omega_{mr},", "1f6c90e12f2fe95be75dfbb40f4d420d": "d^{'}(k)", "1f6cbb91631fa3884ee19639f69b7ddd": "\\left(\\frac{1}{\\sqrt{10}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "1f6cea662d4b24b78f363b58a706e616": "(A_i)", "1f6d97ce7983b0730688d98f5ce81609": "\\Delta^n=(E-I)^n=\\sum_{j=0}^n\\binom nj (-1)^{n-j}E^j,\\qquad n\\in\\mathbb{N}_0.", "1f6de4f12d7200e32790fb5fa21c3099": "\\begin{align}\n C_1 &= L^\\prime_a - M^\\prime_a \\\\\n C_2 &= M^\\prime_a - S^\\prime_a \\\\\n C_3 &= S^\\prime_a - L^\\prime_a\n\\end{align}", "1f6defb5ec75972758578819da1ba5d3": "w\\; R\\; u", "1f6df871e77794605a75559e947eb360": " A : B :C", "1f6e16b33208a64ab75cde9ed14d2c38": "K_{eq}=e^{- \\frac{\\Delta G^\\circ}{RT}}", "1f6e32059561360f584142d5c2979670": "\\big[\\frac{n-m}{n+m}\\big]^{0.5}K_{n-1,m}", "1f6e8ac0727646bb23053df73216611c": "\\tanh x = \\sum^{\\infty}_{n=1} \\frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1} = x-\\frac{1}{3}x^3+\\frac{2}{15}x^5-\\frac{17}{315}x^7+\\cdots \\quad\\text{ for }|x| < \\frac{\\pi}{2}\\!", "1f6ef380e2c528e4caf2ada353c7219b": "\n\\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^p}{e(m+1)}\\,-\\,\n \\frac{p (d+e\\,x)^{m+2}(b+2\\,c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^{p-1}}{e^2(m+1)(m+2)}\\,+\\,\n \\frac{2\\,c\\,p\\,(2\\,p-1)}{e^2(m+1)(m+2)} \\int (d+e\\,x)^{m+2} \\left(a+b\\,x+c\\,x^2\\right)^{p-1}dx\n", "1f6f012e96e510adbb3165d18ffb8d6f": "J\\,", "1f6f0fc533d1f6bf3b2142625a3c339e": "41^2 - 67\\cdot(5)^2 = 6.", "1f6f2ad921baec0407e382f852dbdc6e": "ax - qm = 1,\\,", "1f6f799eb71e0d92d87190de13262109": "x \\succ^p_W y", "1f6f91dfc3b786abd708f2b4149f8732": "\\mathbb{Z}(p^\\infty)", "1f6fc8ab13e60a608aa266b823879e34": " \\mathbf{p} = m\\mathbf{v} \\rightarrow \\mathbf{p} = \\gamma m_0 \\mathbf{v} ", "1f7003f4a797732f4c20dffee01914fe": "(d+b) = hl'", "1f70089acb31c62a886da430252b42c9": "G(\\chi)=\\sum_{a=1}^N\\chi(a)e^{2\\pi ia/N}.", "1f702d6dda61c3231972e4e25a8c7558": " \\int_x^y (Df) g - f (Dg) \\, dt = W(f,g)(y) - W(f,g)(x).", "1f705928a431ff7f17a24e309baf4c39": "A = {h \\over{8\\pi^2cI_{\\parallel}}}", "1f70bec3523e3ee69dc4764426dfe3c2": "e_v=e", "1f70c55be68f446a25c680a65b3658e7": " \\epsilon_{t}\\sim^{iid}WN(0;\\sigma^{2})\\, ", "1f70c6f98f55e5085f98172d9392429b": "\\scriptstyle \\mu \\;=\\; \\mu_0 \\mu_r ", "1f7121a146049708198991dca5fa2ee5": "\\textstyle \\mathbb{F}_{2^m}", "1f71ab960efa0d97fb962750910d425c": " \\psi_i = |\\psi_i| e^{i \\phi_i}, i=1,2 ", "1f71b1572a414b1109dbdce15c4d9965": " \\mathrm{Throughput} \\le \\frac {\\mathrm{MSS}} {\\mathrm{RTT} \\sqrt{ P_{\\mathrm{loss} }}}", "1f71cf4490419b8754a837bbd403d9b3": "\\mathfrak{m}", "1f7220a656ad996e5f778161cab9d333": " \\scriptstyle \\omega ", "1f72463940128d359a3675b36004e675": "\\begin{bmatrix} \\dfrac{-y_{11}}{y_{12}} & \\dfrac{1}{y_{12}} \\\\ \\dfrac{\\Delta \\mathbf{[y]}}{y_{12}} & \\dfrac{-y_{22}}{y_{12}} \\end{bmatrix}", "1f72473e713b197b30f6c5a0a9494e2c": "=(r+1)\\left | c(u)-c(w) \\right | \\geq r +1 ", "1f726d2624e29f3f2c7ffb19365f08da": "h^n \\colon A^n \\to B^{n - 1}", "1f726fc2d0d4f4135fe39bca8064cf2e": "\\, (|k|)", "1f727b5e71e794004e7d5f64c59dea48": "\\phi(r) = r^2 \\ln(r)\\;", "1f72b91cac9f4d09148f276f15d7fc3b": "n^{\\Omega (1/\\varepsilon^2)}", "1f72c0be7b04b5215f1dc7907d16fcaa": "\\Pr[A_j] = p = 2^{-k}.", "1f72df1d70a864ab58cc9ab7bed106ab": " Fr^2 = \\frac{v^2}{gy} = \\frac{q^2}{gy^3} = \\frac{gy_c^3}{gy^3} = (\\frac{y_c}{y})^3 \\; or ", "1f730da8a71d9268f54cd05703f9c050": "p=r\\sin \\psi\\ = r \\cos \\theta = a \\cos^2 \\theta = a \\cos^2 {\\alpha \\over 2}.", "1f737616ccf8c6eb1678bdf3ac1f9d6b": "Q = S_o \\frac{R_o^2}{R_E^2}\\cos(\\Theta)\\text{ when }\\cos(\\Theta)>0", "1f739322941c6c0a3988da8753d9e1fe": "\n\\frac{\\partial c}{\\partial \\zeta} + c = 0\n", "1f7433251704ab78402297095d1b75a2": "\\ell=\\sqrt{\\frac{\\hbar}{2m\\omega}}", "1f7435e186e1674a8ec46647bc7e071b": "0 \\le B_k \\le 1", "1f7454b690cfd499a92b31e637154250": "(1337; 12, 3)", "1f74b35153d4b8589d59a970040474fc": "\\frac{\\mathbf{r}-\\mathbf{r}'}{|\\mathbf{r}-\\mathbf{r}'|^3} = -\\nabla\\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right)", "1f75068ed26ed32ec9c58d7286beadb4": "b'_{\\nu, n}(x) = n \\left( b_{\\nu - 1, n - 1}(x) - b_{\\nu, n - 1}(x) \\right).", "1f750d5ef3c153fc96d9bc2535f039b8": " \\Psi_0 = \\frac{1}{4} \\, \\left( \\left( H_{xx}-H_{yy} \\right) + 2i \\, H_{xy} \\right).", "1f75711854a34370646d90bfe893eae2": "l_{t,j}", "1f75b46c230d976c7b4dbdf5b0e97eae": " f : \\Omega \\to \\mathbb{R}^n \\,", "1f7604f572d60d019fc02094aa89752b": "\\mathfrak{P}^{77}", "1f76101821e0fdcc44c8005b074a06c8": " E_{u/p} = y_1 + \\frac{v^2}{2g} \\,\\!", "1f7687e0136b2939ff35ca53d68d9059": "\\{a,b,c,d,e\\}", "1f7690b98656f2c8c186057d9144cea6": "\\pi_a", "1f76a005b92071a645846a3e0d47f29c": "-T \\left(\\frac{\\partial p}{\\partial V}\\right)_{T} \\left(\\frac{\\partial V}{\\partial T}\\right)_{p}^2 = -T\\left(\\frac{-p}{V }\\right) \\left(\\frac{R^2}{p^2}\\right)=R", "1f76e2404b9fa24edbfbcbea997d0658": " \\phi(t,x,y,z) = C tz e^{tx-yz} + A \\sin(3\\omega t) \\left(x^2z - B y^6\\right) = 0", "1f76f920c90927cc350a43475ff62403": "\\left ( \\frac{\\hbox{Total points}}{\\hbox{Total rides}} \\right ) \\times4", "1f76fb19d415dc27bc9b2c96e86d4275": "Z_2", "1f7759a6d875cc94b8a4a01bd3cdbb2c": "\\to_G", "1f77f89be58905adacbd429ac2e90cd8": "\\vec{J}(\\vec{r},t)=-D\\nabla \\Phi(\\vec{r},t)", "1f780e7515b4875df8af45756b64d680": "\\left\\langle q(x)[F] \\right\\rangle +i\\left\\langle F q(x)[S]\\right\\rangle=\\left\\langle q(x)[F]\\right\\rangle +i\\left\\langle F\\partial_\\mu j^\\mu(x)\\right\\rangle=0.", "1f784ec5434f8f9533d0f3113207e37d": "\\exists a ( \\text{Phil}(a))", "1f78b4a727501b38facda7f4ba43de6e": "I_\\Delta=\\frac {I_0}2\\left[\\left(1-\\sin\\left(\\delta_0\\sin \\omega t\\right)\\right)10^{-A_-}+\\left(1+\\sin\\left(\\delta_0\\sin\\omega t\\right)\\right)10^{-A_+}\\right]", "1f7980eefdde26f3d798185efbf24822": "\\pi_5 = z_0", "1f79816dc0aed43748bab3dbe522ab92": "a=\\frac{5 l (3 l^5-4 m)}{m^2+l^{10}}\\qquad b=\\frac{4(11 l^5+2 m)}{m^2+l^{10}}", "1f79919fa4cd374d5fbbd38cb1f91f7f": "F=\\mathbb Q(\\sqrt m)", "1f79bd220adebedde86d8f480387ab23": "x(t) = a\\cos(t),\\,", "1f79cff37d49f5ecfcd7a2ed0515b658": "Z = S J_{z} S^{-1} \\ ,", "1f7a01f320e61049f9caab900de6c8c5": "\\ a \\,", "1f7a03a30225fa2a6f4c7a30a01aba3c": "\n\\lim\\sup \\frac{\\varphi(n)}{n}= 1,\n", "1f7a0475def51abcfc24d4bae3c59e65": "[W_t, W_t] = t", "1f7a0af9cc469b34a2ecb0900b054c4e": "AUC_{k,k}", "1f7a2076398a463bf7b804eae94d14c2": "H_n=\\frac{(1+\\sqrt2)^n+(1-\\sqrt2)^n}{2}.", "1f7a2282c1b87361b78d0e482dfb3a6b": "h_{UV}(x) = s_V\\circ s_U^{-1}(x),\\quad x \\in U \\cap V.", "1f7a8242f444344f1d38b9bf8ec4f2bf": "\nm(x, y) = g \\{ m(x, y') \\mid y' \\in f(x) \\} .\n", "1f7ab1be359da8cec58ac3fa12ea7380": " \\ C_{min}", "1f7aebd96c654d4801987febb871dc94": " F_d = -6 \\pi r_p \\mu V_{r} .", "1f7b25d25ec49d743c2a2874fcf6829c": " \\mathbb{T}=[0,\\infty)", "1f7b3522f3b3bed9e913fdd88d46589b": " [Hf](x) = - \\frac12 \\frac{\\mbox{d}^2}{\\mbox{d}x^2} f(x) + \\frac12 x^2 f(x). ", "1f7b4a736a7b7602983079aee9570a62": "\\Sigma^{0}_n", "1f7b599a15d834071d22b23c416307b9": "\na_2 = \\frac{a_1 \\cos\\beta }{1-\\sin^2\\beta\\,\\cos^2\\gamma_1}-\\frac{\\omega_1^2\\cos\\beta\\sin^2\\beta\\sin 2\\gamma_1}{(1-\\sin^2\\beta\\cos^2\\gamma_1)^2}\n", "1f7b9318648e593fadf6354cad799415": "p_{dyn}", "1f7cce85dac307ff99fa7256bbfcacfe": "\\mu_{\\alpha}", "1f7cd82ae9a9d3216218fb12dab4cfa9": "\\frac{1-G(tx)}{1-G(t)}\\xrightarrow[t\\to +\\infty]{} x^{-\\theta}, \\quad x>0", "1f7cf91278a3e855c94c6522e53d061a": "\\lambda_\\delta(n)", "1f7d08374c1a84af1d31c26bd6dd107e": "L_U \\; = \\; 69.55 \\; + \\; 26.16 \\; \\log_{10} f \\; - \\; 13.82 \\; \\log_{10} h_B \\; - \\; C_H \\; + \\; [44.9 \\; - \\; 6.55 \\; \\log_{10} h_B] \\; \\log_{10} d", "1f7d961980f72a9b177c1e1d18684079": "[T]_\\beta^\\gamma", "1f7dd5ddf86f117be08408867eeb7b22": "S_\\mathrm{total} = \\frac{\\left ( v_\\mathrm{m} N s \\right )}{V}, \\qquad (5)", "1f7dde3d771e96b3f67fc88478c03c75": "\\theta^{*} = g\\left(\\theta^{*}\\right)", "1f7e1d0f45f1ca1997d2a0570b01fd5d": "\\textstyle < 1", "1f7e34bed65711fcdf7f01154c81ff52": "\\sigma_x \\sigma_p = \\hbar \\left(n+\\frac{1}{2}\\right) \\ge \\frac{\\hbar}{2}~ .", "1f7e7016cbf09526b85723fc5dfa2bc3": "x \\wedge (y \\vee z) = (x \\wedge y) \\vee (x \\wedge z)", "1f7e9f621d27f705126be23079eab4bf": "\\begin{align}\n\\alpha_i&=\\frac{\\boldsymbol{p}_i^\\mathrm{T}\\boldsymbol{r}_i}{\\boldsymbol{p}_i^\\mathrm{T}\\boldsymbol{Ap}_i}\\text{,}\\\\\n\\boldsymbol{x}_{i+1}&=\\boldsymbol{x}_i+\\alpha_i\\boldsymbol{p}_i\\text{,}\\\\\n\\boldsymbol{r}_{i+1}&=\\boldsymbol{r}_i-\\alpha_i\\boldsymbol{Ap}_i\n\\end{align}", "1f7ee3ae4faccfd528c14e6afaa11370": "= \\gamma^0 \\gamma^{\\mu n} \\gamma^0 \\dots \\gamma^0 \\gamma^{\\mu 2} \\gamma^0 \\gamma^0 \\gamma^{\\mu 1} \\gamma^0", "1f7efb11a5062e35c4c103407d504274": "sp(S_1;S_2\\ ,\\ R)\\ =\\ sp(S_2,sp(S_1,R))", "1f7f1d6e89b60f2be92eeb4072c300cb": "a(n+4) \\equiv a(n) \\pmod{10},", "1f7f73a7d98c9557ebb18edcfe4c51ce": " f(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_2 x^2 + a_1 x + a_0 \\, ", "1f7f73c7e8ed6da874ec067f99c09cd5": "A_1 \\lor\\cdots\\lor A_k \\lor C", "1f7fd21193f25f0c5c04f5a4f5eeb128": "\\mathbf{E}^{a} \\left[ f \\big( B_{\\sigma_{k}} \\big) \\right]", "1f800f5a94616fd62bd0bd168c6af7d2": "1 \\ \\mathrm{Gy} = 1\\ \\frac{\\mathrm{J}}{\\mathrm{kg}}= 1\\ \\frac{\\mathrm{m}^2}{\\mathrm{s}^{2}}", "1f801258c264b85e8e0bb3eea2ebda9b": "\\ell ' = \\alpha(\\ell)", "1f802ac05865de81ee4ebea7e9a923c4": " |x\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} ", "1f80925efd94cb3fc32d5e819f21b731": "\\omega_{F0n} = \\frac{\\omega_{F0}}{n}. \\ ", "1f80a18c00065c19dbcf57589a0358aa": "|P(V_{\\omega + \\alpha})|", "1f80bbe06d1bc10023d855d9751c0f25": "X^* = X \\cup \\{\\infty \\}", "1f80cd05ff7868722abdc53bd90fef4c": "T_{f^{-1}P}(X)", "1f8101037e3afb9c21a8883823bf88f4": " \\textbf{a} = p\\textbf{r} \\cdot \\textbf{g} + \\textbf{f} \\cdot \\textbf{m} \\pmod q ", "1f815e0e42cb81964f6e99531cd64e46": " \\overline{\\rho_f} ", "1f819b6c431402316e13dfda49cdf745": "\\{ [M(OH)]^{(z-1)+} \\} = K_{1,-1}\\{ M^{z+}\\} \\{H^+\\} ^{-1}", "1f81d5eb0fe2f45b2f8d8c2ad4c63d12": "0\\varepsilon} K(x-y)f(y) \\, dy. ", "1f91795b5c570049cdcaf73add8215d1": "\\forall x (x \\in a \\leftrightarrow x \\in A)", "1f917e19e02c2345838ab694e3ee6dd9": "\\displaystyle \\sqrt{\\frac{\\pi}{a}} \\cos \\left( \\frac{\\nu^2}{4 a} - \\frac{\\pi}{4} \\right) ", "1f91c52f1a198771670ab18fed186555": "N=(V,E)", "1f91f6ce62d1ce9ba7447981981447e6": "x\\wedge\\bigvee Y = \\bigvee \\{x\\wedge y \\mid y \\in Y\\}", "1f9211e8cf33d0a4fa93afe3a059243e": " \n \\begin{bmatrix}\n {1\\over \\eta}\\\\ - \\left( { a - \\lambda_{-} \\over c\\eta } \\right)\n\\end{bmatrix} \n ", "1f92448d4e6d4ad647b427349e4480a9": "U = ", "1f926e96e3354f4a635403dc1fca133c": "(p + 1) \\times (p + 1)", "1f9271778b3f16601ff4b29d26b2b610": "I=m s^2 ", "1f9296350287268701819ffbeef8105a": "\\displaystyle x = x_0", "1f92f4e75e6b9ccd79395801a27b4cee": "PG_a = \\tfrac{4}{3}PG_v.", "1f9351fbf04b3aef582230fec73e233e": "N\\sim\\operatorname{Poisson}(\\lambda),", "1f93bad7415d79432555fcd6a857ac2b": "\\displaystyle{\\sigma(a,T,b)=(b,-T^t,a).}", "1f93d5afff4b28e0f0c52056368781b2": "F_{eachAnchor}=F_{load}\\frac{Sin(\\alpha )}{Sin(2\\alpha)} \\,", "1f93db021b5b139c283ca168bbf204d0": "\\oint_\\gamma (v\\,dx+u\\,dy) = \\iint_D \\left( \\frac{\\partial u}{\\partial x}-\\frac{\\partial v}{\\partial y} \\right )\\,dx\\,dy", "1f941ff8f996e4eb9e49d2b87ceef78f": "\\scriptstyle 0 \\;\\leq\\; \\alpha \\;\\leq\\; 1", "1f94fb38335d4138271b1794010e4750": "\\displaystyle A=B=C=60^\\circ", "1f95691b519aef3c48c1663e47d963a4": "[\\widehat{a},\\widehat{a}^{\\dagger}] = 1", "1f95d9c308e1cbee4861bf63edd51bd2": "td(E) = \\prod Q(\\alpha_i)", "1f95de6207444b162046eac435c087d1": " \\mathfrak{sl}(3,\\mathbb C)", "1f95fc1ca06160b80e6d2a3734882aa6": "\\ln(f(x))=\\ln(g(x)h(x))=\\ln(g(x))+\\ln(h(x))\\,\\!", "1f9640b2dc1da48b14e9028afaac496f": "y = g_2(x)", "1f964de0e7de97915329fad8482efbb9": "\\cos \\frac{\\pi}{5} = \\cos 36^\\circ = \\frac{\\sqrt 5+1}{4} = \\frac{\\varphi}{2}", "1f966d3a5742d7653b2b5d2cb80d7ec3": "r( \\text{in}, \\text{out})", "1f967e815b21989b041a773b59242310": "\\frac{[A]}{[B]} = \\frac{[A]_0}{[B]_0} e^{([A]_0 - [B]_0)kt}", "1f96b648c52c26eefd587a1ffb8325a5": "\n\\mathcal{H} |0\\rangle = (-Js^2 -g \\mu_B H s)N|0\\rangle\n", "1f9719b14c24ef318f1bf7f1e2cf3a24": "\n\\left(\n\\begin{smallmatrix}\n X & X & X & \\cdot & \\cdot & \\cdot & \\cdot & \\\\\n X & X & \\cdot & X & X & \\cdot & \\cdot & \\\\\n X & \\cdot & X & \\cdot & X & \\cdot & \\cdot & \\\\\n\\cdot & X & \\cdot & X & \\cdot & X & \\cdot & \\\\\n\\cdot & X & X & \\cdot & X & X & X & \\\\\n\\cdot & \\cdot & \\cdot & X & X & X & \\cdot & \\\\ \n\\cdot & \\cdot & \\cdot & \\cdot & X & \\cdot & X & \\\\ \n\\end{smallmatrix}\n\\right)\n", "1f974c9f11aaf2340f59ee57960191bc": "\\Delta(n)=O\\left(\\frac{1}{n^2}\\right).", "1f978226deaa49b61ad98f59ec684582": "\\bar{I}_1 = I_1/J", "1f97ded0269589e6de9986986596b429": "CO_2", "1f97e6939ad24f53f41347befef01b3b": " \\left\\{{n \\atop k}\\right\\} \\sim \\frac{\\sqrt{n-k}}{\\sqrt{n (1-G)}\\ G^k\\ (v-G)^{n-k}} \\left(\\frac{n-k}{e}\\right)^{n-k} \\left({n \\atop k}\\right) \\quad\\forall k, 11 \\\\\n 0 & \\text{if}\\ r\\leq 1\\end{cases}", "1fa926410c869d167ee022e2f5dd7520": "M_{m}", "1fa92a141bd7196d5a899706bca57b7b": "V_{OUT} = V_S \\cdot \\frac{R_{L}}{R_{L} + R_S}", "1fa964bd3d7810755e38cb5ac8bdc92c": "~\\langle n \\rangle =\\langle \\hat a^\\dagger \\hat a \\rangle =|\\alpha|^2~", "1fa98d52609723bc930372993cdaa8fc": "T\\colon V \\to W", "1fa9b26f7365a8ded43a38c4ee11b156": "\\frac{2 + 2\\!\\times\\!0 + 3\\!\\times\\!0 + 4\\!\\times\\!0 + 5\\!\\times\\!0 + 6\\!\\times\\!3}{11} = \\frac{20}{11} = 1 + \\frac{9}{11}", "1fa9e78917b3d3493ca593691fc15c27": "SDecode\\,", "1faa2c93a3ec023bb2c9223048f3f494": " E_v / hc = \\omega_e (v+1/2) - \\omega_e\\chi_e (v+1/2)^2\\,", "1faa6aa0bd93ed5bbcefc7841caa78b2": " \\mathcal{I} = \\int \\rho \\, (\\partial_x \\ln \\rho)^2 dx = - \\int \\rho \\,\\partial^2_x \\ln \\rho \\, dx", "1faa858ca2abe2b3aa9f96d65ee58641": " \\nabla_a G^{ab} = G^{ab} {}_{;a} = 0. \\ ", "1fab2c9874da1b07df366b9da5d7b4d0": "\\theta = n p", "1fab35819992d248f017e9a25a932ada": "hv=fg^{-1}v=\\sum_{b \\in B} l_b^\\sigma b", "1fab5d7c97b9c2c85134e1f0a7de8e26": "\\lambda=\\lambda_0\\sqrt{\\frac{c-v}{c+v}}\\,\\!", "1fac0a296ee042dd60c7d010fbe2c43d": "X_\\alpha = \\frac{c-b}{a-b}", "1fac195ae37518af8c49062eabf906d9": "\\mathbf{v} = (v[1],v[2],...,v[n])", "1fac2717165a3b978ff96b0dca2daa57": "\\Omega =\n\\begin{pmatrix}\n0 & I_n \\\\\n-I_n & 0 \\\\\n\\end{pmatrix}.\n", "1fac55fda14af9b0f523d440f6b3e67f": "\\,(x_2,\\;f(x_2))\\,", "1fac7dadd89836f0f6067e29eb8b9dad": "\\scriptstyle \\left(0\\right)_{i \\in I}", "1fac9c3e8d14989d754cccf7959a65c6": "\\scriptstyle p_{2m} \\,", "1facc2c2ff757ec505f683cb05e83a83": "(3)\\qquad \\kappa\\,\\hat{=}\\,0\\,,\\quad\\text{Im}(\\rho)\\,\\hat{=}\\,0\\,,\\quad \\text{Re}(\\rho)\\,\\hat{=}\\,0\\,,\\quad\\sigma\\,\\hat{=}\\,0\\,,\\quad R_{ab}l^a l^b\\,\\hat{=}\\,0.", "1fad09f744892580d9e2a7011552f358": "\\phi '", "1fad52e9ab3aaca935a52e04f39c3998": " I(z) = I_{in}e^{\\gamma_0(\\nu) z} ", "1fad6fb55ae33616839fd6efaff3d65f": " \\ \\textbf{f} \\cdot \\textbf{f}_p =1 \\pmod p ", "1fad78a534b0f4bce77cc199ea12e8d6": " \\min \\{\\, d(x,y) : x \\in \\mathcal{A},\\, y \\in \\mathcal{B} \\,\\}. ", "1fada30449a302274c8154bca665b89a": "|AE|=|BD|,\\,\\alpha=\\beta,\\,\n\\gamma=\\delta ", "1fadb14ce7b0c4d3ff72761ac56602e6": "\\mathrm d \\varphi_x\\left(\\frac{ \\partial }{\\partial u^a}\\right) = \\frac{\\partial \\widehat{\\varphi}^b}{\\partial u^a} \\frac{ \\partial }{\\partial v^b},", "1fadba73d16884e8eb9d0d5759c2a3ce": "\\hat{\\mathbf{J}}= J(r)\\mathbf{\\Phi}_{lm}", "1fae8ae4cd24bdf1d07032bfa1aec3a1": "R = \\frac{\\sum_{\\mathrm{all\\ reflections}} \\left|F_{o} - F_{c} \\right|}{\\sum_{\\mathrm{all\\ reflections}} \\left|F_{o} \\right|}", "1faeb5a85963286c305ea018ec8f80cf": "u : \\mathbb{R}^L_+ \\rightarrow \\mathbb{R}_+ \\ .", "1faecc4dab1b59b0ffb508741106b414": "\\frac{d}{d t}\\left(\\vec r(t)\\right) = \\nabla g", "1faee40204cf10774ef0e6e5a36f674f": " S_1S_2S_1 ", "1faef6f25aaf90eb0b18b5660c863d70": "10 \\uparrow ^{10 \\uparrow ^{3 \\times 10^5} 10} 10 \\!", "1faf146dfd7cb3cfdaa429a2e1180c31": "\\boldsymbol{\\nu}", "1faf2087d8c8e3b01e9b74327d36cbcc": "\\iint _{ D }^{ }{ \\left( \\frac { \\partial (\\varphi f) }{ \\partial x } +\\frac { \\partial (\\varphi g) }{ \\partial y } \\right) dxdy } =\\oint _{ C }^{ }{ -\\varphi gdx+\\varphi fdy } ", "1faf9a5495803e0f69c960caf661a4ac": "Z \\to Z'", "1fafdbccfb571d7aa2644d01c407a521": "\n E_2 - E_1 = \\tfrac{1}{2}\\,(p_2 + p_1)\\,\\left(\\tfrac{1}{\\rho_1}-\\tfrac{1}{\\rho_2}\\right) = \\tfrac{1}{2}\\,(p_2 + p_1)\\,(v_1-v_2) \n ", "1fb00bdb877b46c5733e41cb124cae04": "f(z)=z+1+\\exp(-z)", "1fb0266c45408f54306c7de03baa455d": "f_! : \\text{Mod}_R \\leftrightarrows \\text{Mod}_S : f^*", "1fb035acb226f98a6db5e42ac9b1f597": "V_m", "1fb07bcb9c06a1aebd89952d9feaf4c2": "X_1 = \\sqrt{\\frac{R_2 R_1^2}{R_1 - R_2}} ", "1fb08ef1a34b9faa675756e053c6dde1": "\\delta(x) \\,\\!", "1fb0b2f1bc03d2d85ddf720fbab3510c": "\\int ds \\sqrt{v}", "1fb0c65e1513d515f6c134c4311db6d8": "|\\mathbf{r} - \\mathbf{r}_s(t_2)| = c(t - t_2)", "1fb0d028e0c7b8573681a7f42c2d4efa": " -\\frac{1}{2} (\\mathbf{a}-\\mathbf{x})^2 + \\frac{1}{2} \\rho^2 = 0 ", "1fb0d9e574d4a8dfa17b15c0a4ed38b4": "Y_s = Y_{N_t} + \\alpha [ P_t - E\\left(P_t | \\Omega_{t-1} \\right) ] ", "1fb1796c7637dee6e153e4443b16aeb0": "{\\mathbf{X}}_{t_1, \\ldots, t_k} = (\\mathbf{X}_{t_1}, \\ldots, \\mathbf{X}_{t_k}) ", "1fb1919af4234e6802dec9dc716a6ac9": "S: X \\to W", "1fb1c3dd21b6d638e029aabcab98f5ee": "S^{\\mathrm WZ}(\\gamma) = \\int_{B^3} \\gamma^{*} c.", "1fb1ecd403869dc2743712b600e83e80": "E(t) dt", "1fb2751e36aa7ee7a58a656b96882674": "b^n + a_{n-1} b^{n-1} + \\cdots + a_1 b + a_0 = 0.", "1fb27adeb836505973906ae13194b614": "Y^{c}=\\max(0,Y)", "1fb28683a5575f143e9315835c154793": " D = \\sum_{k=1}^{M} d_k ", "1fb29d291d9073c4adbc5091ac0ea9fd": "\n\\frac{d}{dz} \\left[ z^{-b_1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) \\right] =\n- z^{-1-b_1} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ b_1 + 1, b_2, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right), \\quad m \\geq 1,\n", "1fb2aafe1caf3f07796d6bdb8d2f21f8": "\\sigma: \\mathbb{R}^n \\mapsto \\mathbb{R}^m", "1fb326aec8788f6df25829f701154fa0": "\\textstyle{x<0}", "1fb33067cb5fa847c62d7b356bd8d099": "2[x,x]=0.", "1fb3315e27fbbad8db2c3146f4910096": "i = 1\\ldots n", "1fb35632d73d869b3f21afb0dc1256a8": "\\,\\sigma_{j, k}", "1fb382a46b80e13ce933747251179910": "\\operatorname{Riesz}(x) = O(x^e)\\qquad (\\text{as }x\\to\\infty)", "1fb38ac2aa147f6aca29ee7ffaa9675e": " S_{uvw} ", "1fb39d7cebf6a4dbd7490988e3f93c57": "\\bold{r}' = x' \\bold{\\hat{x}} + y' \\bold{\\hat{y}}", "1fb3b304980d61ce94691724464e7737": "N^\\pm(X)\\sim\\frac{A_\\pm}{12\\zeta(3)}X+\\frac{4\\zeta(\\frac{1}{3})B_\\pm}{5\\Gamma(\\frac{2}{3})^3\\zeta(\\frac{5}{3})}X^{\\frac{5}{6}}", "1fb3c3f4c5b1d00cda1e19f92e455d84": " \\varphi_n \\ ", "1fb4649654c08362e432d4122f0d9e10": "\\lambda_{c} = \\sqrt{\\frac{\\gamma}{\\rho g}}", "1fb4a6fbea22c7ff968bd40ae9e3b158": "\\int_{0}^{+\\infty} x^\\alpha e^{-x} f(x)\\,dx.", "1fb4aa001bc075df9e1572b9849aa430": "\\textrm{VD}=\\frac{\\textrm{DS}}{\\sqrt{\\left(\\frac{\\textrm{NHR}}{\\textrm{NVR}}\\right)^2+1} \\cdot \\textrm{CVR} \\cdot \\tan{\\frac{1}{60}}}", "1fb528d8993fc05cfb59d882c12b24e4": " \\frac{\\det \\left(-\\frac{d^2}{dx^2} + V_1(x) - m\\right)}{\\det \\left(-\\frac{d^2}{dx^2} + V_2(x) - m\\right)} = \\frac{\\psi_1^m(L)}{\\psi_2^m(L)} ", "1fb540e9c9a2f2e282f38c6ab3b51a2c": "\\ A^* \\vee B^*", "1fb5520dcd1b01e2bfab33e2ff16093b": "\\lim_{\\underset{h\\in \\mathbf{R}}{h\\to 0}} \\frac{f(z_0+ih)-f(z_0)}{ih} =\\frac{1}{i}\\frac{\\partial f}{\\partial y}(z_0).", "1fb5a8da55877e07a8d0d5a5efc8bc4e": "\\mathcal{O}_s", "1fb6035d496acc3a2cc1a66f2a1743b6": "w\\colon E\\rightarrow \\mathbf R^+", "1fb623471d5eab2b9ad3de8b3549dfd0": "\\chi_{\\text{2}}(2\\omega)=\\frac{\\varepsilon_0^2 m \\zeta_2}{N^2q^3} \\chi_{\\text{1}}(\\omega)\\chi_{\\text{1}}(\\omega)\\chi_{\\text{1}}(2\\omega)", "1fb62380d61606d120fcda6f8c418a78": "= V_\\text{d} \\cdot k_\\text{e} = \\frac{D}{AUC}", "1fb63781a11b5a567349fba3a81d7e6e": " \\frac{k(t)}{k} = s. A - n", "1fb64da16298cade5031507606c4d8f7": "\\psi(x,y,z)=C \\cdot Ai({{-eEx-\\epsilon+\\hbar^2k_y^2/2\\mu_y + \\hbar^2k_z^2/2\\mu_z} \\over {\\hbar\\theta_x}})", "1fb66244dedcb6d5448434977ea56f76": "X_i Y_i", "1fb68aa8ed8484819dde02aab6e1f900": "s(N-1)", "1fb69f73c9269e2b8b2cd0bfe4be00a5": "f^{(k)}_{0},\\ldots,f^{(k)}_{N^{(k)}-1},\\,", "1fb6ff3edc72f426caac14ac4e3a4554": "\\supseteqq, \\nsupseteqq, \\supsetneqq, \\varsupsetneqq \\!", "1fb7194838c79a33be3bffd36b785baa": "\\cfrac{A+T}{G+C}", "1fb727587954bb88da496f93270c3b9f": "K_p =\\mathrm{ \\frac{(p_T)^2(x(NO_2))^2}{(p_T)(x(N_2O_4))} = \\frac{(p_T)(x(NO_2))^2}{(x(N_2O_4))}}", "1fb729604087c74338e709f171ee54bd": "\\frac{D}{\\sqrt{2E}}", "1fb76e2ebc0eae759b6a4a36f544c646": "p(b) \\leftarrow", "1fb79f6304c9b4d8127c513fed28f459": "\\langle O(n), O(n) \\rangle ", "1fb7e280baed28452bb0a22572bb067e": "\\left\\{ x_1,x_2,x_3 \\right\\},", "1fb7e803f0886d2bc4e45844e7ea2fef": "\\int\\frac{x}{R}\\;dx = \\frac{R}{a}-\\frac{b}{2a}\\int\\frac{dx}{R}", "1fb83c04d48fa6fbebce4ac9e113390f": "F_2=F_{load}\\frac{Sin(\\alpha )}{Sin(\\alpha+\\beta)} \\,", "1fb85b7836468f8b53e9515000f7468d": "G ", "1fb8edf718f6e25b6e0c2a8fcbc8fb6c": " a_{11} = \\mathcal{L}(p_5)+p_3p_5+p_2p_7+p_1p_8,", "1fb914ba5bd1e318e3c585e0449ad6aa": "\n\\begin{align}\nj & = \\sum_{i=1}^3\\frac{\\partial L}{\\partial \\dot{x}_i}Q[x_i]-f \\\\\n& = m \\sum_i\\dot{x}_i^2 -\\left[\\frac{m}{2}\\sum_i\\dot{x}_i^2 -V(x)\\right] \\\\\n& = \\frac{m}{2}\\sum_i\\dot{x}_i^2+V(x).\n\\end{align}\n", "1fb94faa966d0d6666e65f6ac14d0a13": "\\mu_n = (-nq)\\mu_n(-\\frac{1}{nq}) = -\\sigma_n R_{Hn}", "1fb9d1774ad1d205d4638da8187490a3": "c \\equiv z^Q", "1fba55d0040df49a890f8b8a3a05cf6f": "\\vec{D_{\\beta}}=|\\vec{C_2}.\\vec{X_{\\beta}}-\\vec{X}|\n", "1fba648f51eefc37c7f8ad89472f9562": "A = \\frac{M}{\\rho} \\frac{n^2 - 1}{n^2 + 2},", "1fba8a94ec81415fbeaefc7ab437b827": "\ng=\\frac{Q^*}{L \\rho}\n", "1fbaae231d01ec1cafacff645b3fce46": "\\mathbf{C}_f", "1fbab1bd9145501b6784a0f9f6480a83": "\\Sigma_{n} = \\left\\{ (x, y, u_{n}(x, y)) \\in \\mathbb{R}^{3} \\left| - \\frac{\\pi}{2} < x, y < + \\frac{\\pi}{2} \\right. \\right\\}.", "1fbac4603bc6152029bc2598f76868d3": "\\left[A\\rightarrow a,B\\rightarrow b,C\\rightarrow c\\right]", "1fbaceeb6f72e4127e8692227ba2f028": "UC,", "1fbafd7495b60c9332f7887a3ae2c07e": "x^0=1", "1fbb0379d1725759cfcc9d51dfdc027e": "\nV(\\xi, \\eta) = \\frac{-\\mu_{1}}{a\\left( \\cosh \\xi - \\cos \\eta \\right)} - \\frac{\\mu_{2}}{a\\left( \\cosh \\xi + \\cos \\eta \\right)}\n= \\frac{-\\mu_{1} \\left( \\cosh \\xi + \\cos \\eta \\right) - \\mu_{2} \\left( \\cosh \\xi - \\cos \\eta \\right)}{a\\left( \\cosh^{2} \\xi - \\cos^{2} \\eta \\right)},", "1fbb0f6eac80b34d73fa1c11735360f5": "\\frac{dE}{dy} = \\frac{d}{dy}\\biggl(\\frac{q^2}{2gy^2}+y\\biggr) = 0", "1fbb65cea0e8b9208ab3ea1e08f26636": "\\left(\\frac{a}{p}\\right)=-1", "1fbb95b2c41aad34b4b5f0fb2e1ad29b": "\\forall n_1 \\forall n_2\\cdots \\forall n_k \\psi", "1fbba07c0bd22a305bb115381d0b6e10": "\n\\boldsymbol{\\mathsf{S}} = \\frac{10}{3} \\pi \\mu a^3 \\left[ -2\\boldsymbol{\\mathsf{E}}^\\infty + \\left(1 + \\frac{1}{10} a^2 \\nabla^2\\right) \\left(\\boldsymbol{\\nabla} \\mathbf{u}' + (\\boldsymbol{\\nabla} \\mathbf{u}')^\\mathrm{T}\\right)\\right],\n", "1fbc5fb6d366e5ceb577ab61a5f3fde0": "\\rtimes", "1fbcc0261c41a1aef94a58a83beae2e0": "{\\it{D}}", "1fbcce8e1bd64cc4ea326cc2b7bdbbbc": " \\rho = | \\mathit{before} \\rang \\lang \\mathit{before} | = |\\psi \\rang \\lang \\psi | \\otimes | \\epsilon \\rang \\lang \\epsilon |", "1fbcd9af825730402ca191ba60bd308a": "\\frac{\\partial}{\\partial t}w(t,\\xi)=-\\frac{\\partial}{\\partial\\xi}w(t,\\xi)+u(t),", "1fbd30b0ebfba626656e448399b0e742": "\n\\beta_\\text{max} = 0.072 \\left(\\frac{1+2^2}{2}\\right)\\frac{1}{1.25} = 0.14.\n", "1fbd3975d499eebef25a2b2d2b8fd1fd": "R'_{pb} = R_{pb} - R_{mb}", "1fbd6d93cf88841f1aa016abae1c7828": "\\langle s_{c'},s_{c'}\\rangle (t) = \\int_{-\\infty}^{+\\infty}s_{c'}^\\star(-t')s_{c'}(t-t')dt'", "1fbd992fde967e84fd7a6e40d8d7658c": " RA=\\frac{\\sqrt{(\\lambda_1-\\mathbb E[\\lambda])^2+(\\lambda_2-\\mathbb E[\\lambda])^2+(\\lambda_3-\\mathbb E[\\lambda])^2}}{\\sqrt{3\\mathbb E[\\lambda]}} ", "1fbe10d9ae2ac2b69b1d213a3d251ea3": "\\Rightarrow -D(p(x)\\|q(x))=\\int p(x) \\log \\frac{q(x)}{p(x)} \\, dx \\le \\log \\int p(x) \\frac{q(x)}{p(x)}\\,dx\\,=\\log \\int q(x)\\,dx=0 ", "1fbe4e209e458735fa42abbac15916ae": "a(x+y)=ax+ay", "1fbe5e34f4b68f9b863566ed10dea598": "\\Delta c = 0{\\,}", "1fbe81dab0a48aced10bbcc4d9be023b": "\\sum_i \\log\\left( 1 + e^{-y_i f(x_i)}\\right)", "1fbe94943b146359cce2a52e1c68362d": "\nH\\;\\Psi(\\mathbf{R},\\mathbf{r}) = E \\; \\Psi(\\mathbf{R},\\mathbf{r}) \n", "1fbebc8e81db2bd1b2c1a794daff1483": "mn = a,\\ pq = c\\,\\!", "1fbec501807d7c386de2b308c0a2a12f": "\\sum_{n=1}^\\infty a_n x^n = \n\\sum_{n=1}^\\infty b_n \\frac{x^n}{1-x^n}", "1fbecc4a7298ea2729d912cd53047e17": "k \\sim \\Delta \\nu_\\circ \\sim 2(10 \\mathrm{cm}^{-1}) (300 \\cdot 10^8 \\mathrm{cm/s}) \\sim 6 \\times 10^{11} \\mathrm{s}^{-1} \\cdot", "1fbf9482b98e7ebee9e8b4325defc2a3": "X[x,y]=\\frac{y'}{yx'-xy'}", "1fbfa88cb01eaf0471b05b9c96c4e8ec": "1 \\leq L \\leq p.", "1fbfb28b89e54b0fc955e7e3226b2530": "A,B\\subseteq S", "1fbfca3317a52b97193eb0cc263d99fa": " \\rho= \\tfrac{1}{\\alpha+\\beta+1} \\!", "1fbfd42692c6f74f1367978e014f7c92": " q = \\iiint F \\mathrm{d} A \\mathrm{d} t ", "1fbfe6e45680007dda3e916da01c7548": " 1 \\leq j \\leq n ", "1fc0206ab1e5f828e9ab40cc21951b08": "\\eta_j", "1fc05ead76f2d6b9c4b9d68f855e010f": "\\mu^{-1}", "1fc080587c795c986e36244dd7ad9a69": "X>2", "1fc0821f7fe8bd240fa2bf466df4e057": "\\frac{\\pi}{3\\sqrt 2} \\simeq 0.74048.", "1fc0940dc26a177aab0d82003f11ab03": "(S\\,y)'' = S\\,y'' + 2\\,S'\\,y' + S''\\,y", "1fc0bc211ee47dee1ec609f450ad9b4a": "a_i \\in I^i", "1fc0dd3998a2de27dd5dc13fe32ac9ab": "\\cos \\pi z = \\prod_{q \\in \\mathbb{Z}, \\, q \\; \\text{odd} } \\left(1-\\frac{2z}{q}\\right)e^{2z/q} = \\prod_{n=0}^\\infty \\left( 1 - \\frac{4z^2}{(2n+1)^2} \\right) ", "1fc1523c7e6de14066b43a40f1c05c8d": "\\scriptstyle \\{|0\\rangle_A, |1\\rangle_A\\}", "1fc164dc85d156f67eb71706d0d982cb": "\\mathbf{n} = \\frac{\\mathbf{r}_1\\times\\mathbf{r}_2}{|\\mathbf{r}_1\\times\\mathbf{r}_2|}.", "1fc19afb794c4b2368ae082be5dd92c5": " \\omega = \\omega^r \\, \\cap \\, \\omega^l ", "1fc1ae3cd4e91f49a4cf9ff4c2ba014f": "\\mathbf{v} \\in V", "1fc1d1474faa1718ebfa13afd7e0a213": "\\nabla^2V = -\\frac{\\rho}{\\varepsilon_0}", "1fc1f7b000b1faa032f4aba0365049f7": "-g^{(k)}", "1fc26eae19dc1a77f822d6cfd408177f": "H = - \\frac{\\hbar^2}{2m} \\int d^3\\!r \\ \\phi^\\dagger(\\mathbf{r}) \\nabla^2 \\phi(\\mathbf{r}) + \\frac{1}{2}\\int\\!d^3\\!r \\int\\!d^3\\!r' \\; \\phi^\\dagger(\\mathbf{r}) \\phi^\\dagger(\\mathbf{r}') U(|\\mathbf{r} - \\mathbf{r}'|) \\phi(\\mathbf{r'}) \\phi(\\mathbf{r}). ", "1fc2e24908f22c141e31d784ba6c981e": "E\\supseteq K\\supseteq F", "1fc2eae4b74acbd862dd18c59301ade4": "\\frac{z^3 - \\tfrac13 a}{z^2 - z^3}.", "1fc33df1a678c1b88ebf1296f44e6706": "\\displaystyle{Q_r(\\theta)={2r\\sin \\theta\\over 1 -2r \\cos\\theta + r^2}.}", "1fc3846a47807c5850a84319a75e9b16": "L(T)", "1fc3a72bbaae4e22c94971ef3ef2b87f": "C_i \\left( \\mathbf{r}, t \\right)", "1fc3cc59c43a5922b75132c2c4bdd247": "\\,\\,f = \\frac {\\omega} {2 \\pi}\\text{.}\\,\\!", "1fc3f9dcad3b87a2966b40f5fd357b94": " r_i = r - \\frac{nh}{2}", "1fc4629200f55e2110fb4c571263cebf": "\\bold{j} = \\rho \\frac{\\bold{\\nabla} S}{m} ", "1fc4e4f82497357b11205a031af26f0e": " X_{f}[g] := (f,g) .", "1fc4e7e27edb61a19b11a83a971d82f7": "F_1 = (z^{N/2}-1)", "1fc500923d2353f29a0bac9484bcbbfb": "a_{m,n} = \\frac{2n+2}{\\epsilon_m\\pi}\\langle G(\\rho,\\varphi),Z^{m}_n(\\rho,\\varphi) \\rangle,\\quad b_{m,n} = \\frac{2n+2}{\\epsilon_m\\pi}\\langle G(\\rho,\\varphi),Z^{-m}_n(\\rho,\\varphi) \\rangle.", "1fc504ff79d40d107a9462846763a793": "\n2E = \\dot{\\mathbf{s}}^\\mathrm{T} \\mathbf{G}^{-1}\\dot{\\mathbf{s}}+\n\\mathbf{s}^\\mathrm{T}\\mathbf{F}\\mathbf{s}\n", "1fc51e2b44083e5b687bf632543991c6": "\\lambda_k=0", "1fc5ef9f424597d11cc4df7b51cb4b40": "\\scriptstyle i (\\hat x \\hat v \\,-\\, \\hat v \\hat x) \\;=\\; 2i (\\hat x \\,\\wedge\\, \\hat v)", "1fc5faa3e5665e1e49335257346e88f1": "G = \\frac{E}{2(1+\\nu)}\\,\\!", "1fc60e48b417ee760896887baecb5cfe": "\nL \\propto \\sigma^{3.1}\n", "1fc6739b4b3eda38d449d4fdb9eb29f3": "\\mbox{Quick (Acid Test) Ratio} = {\\mbox{Cash and Cash Equivalent} + \\mbox{Marketable Securities} + \\mbox{Accounts Receivable}\\over \\mbox{Current Liabilities}}", "1fc6a5a8a0be46bdd5cdde820e524e43": "u^3=a^3(\\tfrac{u}{a})^3=a^3(\\tfrac{u}{a})(\\tfrac{v}{u})(\\tfrac{b}{v})=a^3(\\tfrac{b}{a})=2a^3", "1fc6b6f3884a3c5f19269335c1f5fafa": " a(u,u) \\geq c \\cdot \\|u\\|^2 ", "1fc6d8b994af7bde5fadfb8e78e06d0e": "\n\\left(\\mathbf{A} \\mathbf{B}\\right)^{-1} = \\mathbf{B}^{-1} \\mathbf{A}^{-1} .\n", "1fc6e6f313532186a8d96cc6910bf25e": "M(\\hat{x})", "1fc75db554dc991d1f8cf5668c582b79": "(c,b) \\in R", "1fc7d6d5b5d7964c051d68d889978731": "\\cos E\n=\\varepsilon+\\frac{1-\\varepsilon^2}{1+\\varepsilon\\cdot\\cos \\theta}\\cdot\\cos \\theta\n", "1fc80270fd84bfd02eece8812c33b2c9": "\n\\mathbb{E}_{X^{n}}\\left\\{ \\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}},\\delta}\n\\ \\rho_{X^{n}}\\right\\} \\right\\} \\geq1-\\epsilon.\n", "1fc839208f85d5dce5778ed901c3139f": " y = f(k)", "1fc86ac1032be25eca633b7544f777eb": "Z = \\int Dx\\, e^{{\\rm i}\\mathcal{S}[x]/\\hbar}", "1fc8819fc98b84e1deacffebf4bf4b59": "\\textbf{M}_{k} = \n [\\textbf{F}_{k}^{-1}]^{\\text{T}} \\textbf{Y}_{k-1\\mid k-1} \\textbf{F}_{k}^{-1} ", "1fc89784c87362b7983a165bb98a7fb8": "\\frac{4}{5} - \\frac{2}{5} = 0.4 = 40%", "1fc8a40e4104f198e87e696887ea80e4": " \\mathbf{a} ", "1fc8d41944f3f64afdae3b54c2d2ecaf": " \\mathrm{M}^{+z} + z \\mathrm{e}^{-} \\rightleftharpoons \\mathrm{M} . ", "1fc99ac427a2fc26dd414a641b9fcc51": "\\operatorname{MIAE} (\\bold{H}) = \\operatorname{E}\\, \\int |\\hat{f}_\\bold{H} (\\bold{x}) - f(\\bold{x})| \\, d\\bold{x}.", "1fc9daa3ef27c43d9726b587a03c04d6": "\\mathit{H}( \\mathit{n}, \\mathbb{C}_p)", "1fca04efa8462cf833826246202ebe71": " e^3(e + 2)(a + 1)^2 + 1 - o^2 ", "1fca1ed477c05ac9ba4fa12a7a5e07ae": "a + b \\sqrt{-1}, a, b \\in \\mathbf{Z}", "1fca2047eb5ddba83d5a487b46762bf6": " -\\frac{\\mathrm{d}^2\\mathbf{r}(t)}{\\mathrm{d}t^2}m=\\frac{\\partial V[\\mathbf{r}(t)]}{\\partial x}\\mathbf{\\hat{x}}+\\frac{\\partial V[\\mathbf{r}(t)]}{\\partial y}\\mathbf{\\hat{y}}+\\frac{\\partial V[\\mathbf{r}(t)]}{\\partial z}\\mathbf{\\hat{z}}, ", "1fcadb08bff81d8ce0420b1afbf65ac0": " S_{i + \\nu} + \\Lambda_1 S_{i+\\nu-1} + \\cdots + \\Lambda_{\\nu-1} S_{i+1} + \\Lambda_{\\nu} S_i = 0. ", "1fcaeaab429d24583c738c40b4ad5a94": "\np_{k,i}^{\\mathcal M}<\\frac{1}{2}+\\frac{1}{Q(k)}\n", "1fcbd3c93e923b362d00abe9173f37ca": "k[x_0, ..., x_n]", "1fcc7d66dd0358306585155cca8dd467": " \\sum_{n\\le x}d(n)=x\\log x+(2\\gamma-1)x+o(x)", "1fccb59f752c8d14c6c45b5bfa6e6987": "r * e", "1fccc31fd9fb43d716f47f1a5e980228": "\\operatorname{coth}(z)", "1fcd5af5f402845603fcd016135f1ba7": "-\\frac{d[A]}{dt} = k [A]^n", "1fcde9c7cb11774abe82bfaacd66d365": "\\langle O(1), poly(n) \\rangle ", "1fce27f4bdb1769982216f30a0e34430": "\\mathrm{E}\\left[\\hat{A}_1\\right] = \\mathrm{E}\\left[ x[0] \\right] = A", "1fce774d8fba992688a16f1e59db178c": "\\mathbf{a} = (a_1, a_2, a_3).", "1fcebc486bde79742426965170807f63": "y = 0\\,", "1fcec14a18d4361fc19b6029a400f985": "F = [S - PV(Div)] \\cdot (1 + r)^{(T-t)} \\ ", "1fcee716e2996ef968f74e4f38db6427": "\\scriptstyle E_\\vec{p}", "1fcef9f6803f17930b733c743e4c50c6": " \\frac{d \\theta_i}{d t} = \\omega_i + \\frac{K}{N} \\sum_{j=1}^{N} \\sin(\\theta_j - \\theta_i), \\qquad i = 1 \\ldots N", "1fcf001307506fb97c7d1b64fb64191b": "x_k \\ne 0", "1fcf115fe35ee54ac21ed724e675b078": "O(2^km)", "1fcf22b3d5d987dca010507dc8b5d278": "\\scriptstyle{E/c^2}", "1fcf323a90975ddd11f9a2e5f497ffe2": "x(t) = x_1(t)\\begin{bmatrix} 1 & e^{-j\\omega\\Delta t} & \\cdots & e^{-j\\omega(M-1)\\Delta t} \\end{bmatrix}^T ", "1fcf70ef1a0772e4085892530938a676": "L_{x}(\\mathbf{x},\\sigma_{D})", "1fcf841ba6cb2c8ae44b2ca642079582": "q_k", "1fcf8b73b74b29e57619233527678428": "\\,i = 0.12", "1fcfaa7dc3845b6f3ea2e9da6ea7872a": "V_n^2= kT/2C", "1fcfd80ca2ff94600509ff85ea85f28a": "k = A e^{-E_a/(R T)}", "1fcfe57fca607ad295918e02809dc58c": "n_1 = 0", "1fd0040e4536f962319411ad4a97ad39": "0=A_{21}n_2+B_{21}n_2\\rho(\\nu)-B_{12}n_1 \\rho(\\nu)\\,", "1fd02ff915a31059b6326fdee5c760e7": "n\\in{\\mathbb{N}},\\;n>1", "1fd04858035a621bd3c4d4f9ac6e3f84": "X^2+Y^2=1\\,", "1fd089f0f4cece02f59759415980140d": "R(t) = P(\\{T > t\\}) = \\int_t^{\\infty} f(u)\\,du = 1-F(t).", "1fd08f6f5c641d939dc5d558f161bca9": "m_2 \\ ", "1fd0f4ce52d05dcefaed9324d2f95569": "33^8+1549034^2=15613^3\\;", "1fd164fcc71fcbb2514cbfb400eb21dc": "u_{xx} + u_{yy} = 0", "1fd17283506e24d39281068cc05afbf2": "\\kappa = \\Omega", "1fd1ba4abf5af186a5fd797f9450974b": "z_1,z_2\\in\\mathbb{H}", "1fd1c35df6b07d9227eba1f95e975e49": "\\epsilon_{0}", "1fd1d5ccd8873172d395d866d529bc6b": "e_i e_j = \\Bigg\\{ \\begin{matrix} -1 & i=j, \\\\\n - e_j e_i & i \\not = j \\end{matrix} ", "1fd1dc9845ddbbed5201319bc00d29f7": "\\delta J(y)(h) = \\lim_{\\varepsilon\\to 0} \\frac{J(y + \\varepsilon h)-J(y)}{\\varepsilon} = \\left.\\frac{d}{d\\varepsilon} J(y + \\varepsilon h)\\right|_{\\varepsilon = 0},", "1fd20e5a539570a179ce3d65f7a2daac": "d(1-2\\varepsilon) > d/2\\,", "1fd21fe376a3332e8fd67ed25d32f4bb": "\\deg(F)", "1fd244f6feab5985d8b3f8bc348441da": "\\begin{bmatrix} A & B \\\\ B & C \\\\ \\end{bmatrix}, ", "1fd2a1293dd3850017ed2b95a3957f2c": "\\pm \\sqrt{1/3}", "1fd2c5228de66642b72d9a2e747dfdb2": "V = k^n", "1fd2e0b1caa9af835977c93b0c9f9a42": " \\boldsymbol{\\beta}^{(s+1)} = \\boldsymbol\\beta^{(s)}+\\Delta;\\quad \\Delta = -\\left( \\mathbf{J_r}^\\top \\mathbf{J_r} \\right)^{-1} \\mathbf{J_r}^\\top \\mathbf{r}. ", "1fd32fc57b08616399f84f6546a4dcf1": "\\eta_{II} = \\frac{\\eta_{th}}{\\eta_{th,rev}}", "1fd38986e807462ae37dad3b58dd6795": "{\\;\\,dS^\\alpha\\over d\\tau}={e\\over m}\\bigg[{g\\over2}F^{\\alpha\\beta}S_\\beta+\\left({g\\over2}-1\\right)u^\\alpha\\left(S_\\lambda F^{\\lambda\\mu}U_\\mu\\right)\\bigg]\\;,", "1fd399a2880edd5fcf798968ffb57e92": "Q_{\\mathrm{max}} = P", "1fd3b9dc3137a01f24a070e90e55d483": " 0.0235 \\times W^{0.51456} \\times H^{0.42246} ", "1fd41058e9fad1ba04b393195d340fce": "{\\tilde{C}}_1", "1fd45e3467029291005ada5ed15dbc46": "A\\left(t, z+h\\right)", "1fd5410dc84ea247a8bfc1c2e3c2f632": "d_f(\\alpha) = \\mu(\\{x \\in X : |f(x)| > \\alpha\\}).", "1fd5e94b586ef6429747b5914d475480": "\\hat{H}'_0 = (\\alpha \\cdot p + \\beta m) (\\cos \\theta - \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta )^{2} = (\\alpha \\cdot p + \\beta m) e^{-2\\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\theta} = (\\alpha \\cdot p + \\beta m) (\\cos 2\\theta - \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin 2\\theta ) ", "1fd60c9a78e8faa71fc216a281d63384": "\\lambda_1=e^{\\beta J} \\cosh \\beta h+ \\sqrt{e^{2\\beta J} (\\sinh\\beta h)^2 +e^{-2\\beta J}} ", "1fd630296b1f93f433df34dc7c273ade": "\\sigma_{i0}", "1fd69d9a090ba0001b0e5394959b494f": " a_m =\\frac{p_m}{N} \\sum_{n=0}^{N-1} u(x_n) (-1)^m\\cos\\left(\\frac{m\\pi}{N}(n+\\frac{1}{2}) \\right)", "1fd6c6fa8a7e525affc2c3317ab3491c": "\\textstyle v(x) = x^ia(x) + x^{i + g(2l-1) + r}", "1fd7302c96142ceef8c7ee57cdfaae6b": "\\scriptstyle\\mathcal{Z}", "1fd73412b32c4c19c2c03c8a74d7499e": "s_{2}=(1 - u_{\\min})^{-\\frac{1}{a}} k", "1fd73ffdce10723eb2c8b591234164f5": "\\frac{\\partial L}{\\partial q_k}=0 \\quad \\Rightarrow \\quad \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{q}_k} = 0 \\quad \\Rightarrow \\quad \\frac{d p_k}{dt} = 0 \\,,", "1fd744d93512f9948b4c5f6d47013971": "\\mathit{i}^2 =(\\mathbf{e_1e_2e_3})^2 =\\mathbf{e_1e_2e_3e_1e_2e_3}= \\mathbf{e_1e_2e_3e_3e_1e_2} = \\mathbf{e_1e_2e_1e_2} = -1 \\ . ", "1fd8110c514c213b06482dd214af81e2": "\n\\begin{matrix}\n\\sin 0 & = & \\sin 0^\\circ & = & \\sqrt{0}/2 & = & \\cos 90^\\circ & = & \\cos \\left( \\frac {\\pi} {2} \\right) \\\\ \\\\\n\\sin \\left( \\frac {\\pi} {6} \\right) & = & \\sin 30^\\circ & = & \\sqrt{1}/2 & = & \\cos 60^\\circ & = & \\cos \\left( \\frac {\\pi} {3} \\right) \\\\ \\\\\n\\sin \\left( \\frac {\\pi} {4} \\right) & = & \\sin 45^\\circ & = & \\sqrt{2}/2 & = & \\cos 45^\\circ & = & \\cos \\left( \\frac {\\pi} {4} \\right) \\\\ \\\\\n\\sin \\left( \\frac {\\pi} {3} \\right) & = & \\sin 60^\\circ & = & \\sqrt{3}/2 & = & \\cos 30^\\circ & = & \\cos \\left( \\frac {\\pi} {6} \\right)\\\\ \\\\\n\\sin \\left( \\frac {\\pi} {2} \\right) & = & \\sin 90^\\circ & = & \\sqrt{4}/2 & = & \\cos 0^\\circ & = & \\cos 0 \n\\end{matrix}\n", "1fd82f5b59dc829eaeda90a9084ebd81": "\\; (m-1)", "1fd8397e614bf2466baea635d1c07767": "(\\mathbf{F}_2)^n", "1fd9103246379056b637fea02a7446bc": "P(X) = {Q(X) \\over E(X)}", "1fd91969442deaa96407f8f6fe20185a": "l= \\frac{1}{2} \\lambda_d = \\frac{1}{2} k \\lambda_0 = \\frac{1}{2} k \\frac{c}{f}", "1fd9bb5b485ab782ebaa65b3c1aeb01e": "H(x,p;V)=K(p)+\\varphi(x;V)", "1fd9e04643bdb6e5eb06c88fa8a13c0d": "M_A \\text{ and } M_B ", "1fda275a1dd0f7d854b5fdb6cbb47944": "0 \\le r < m ", "1fda459eb5e9d6260cdb28e474c2bf21": "\\sum_{k}{p_k}=1", "1fda5454dd56f797a8d5d705febddf73": "C_{N_c}=\\sum_{n=0}^{N_c-1} \\log_2 \\left(1+\\frac{P_n^* |\\bar{h}_n|^2}{N_0} \\right),", "1fda650214c16fad58a13d74b54dc7a3": "\\mathit{QthD} = \\frac{S \\times 17 \\times 3600}{T_s} \\times \\mathit{SF}", "1fdad8dd6929df0a2c4bcef48413791d": "v_r=(1-s)v_s", "1fdae70739ffc2d807b8e987e28485b8": "\n \\phi(x)(g_1,g_2,\\dots ,g_m) =\n \\begin{cases}\n (a,g_1,g_2,\\cdots ,g_m) & \\text{if } g_1\\in B. \\\\\n (ag_1,g_2,\\dots ,g_n) & \\text{if } g_1\\in A \\text{ and }ag_1\\neq 1. \\\\\n (g_2,g_3,\\dots ,g_n) & \\text{if } ag_1=1.\n \\end{cases}\n", "1fdb1c9b08b0d8ad3c79e79641a46451": "\\mathcal C \\times \\mathcal C", "1fdb6f5253b3d1a276fbefd596cdddf7": " s \\equiv s' \\cdot r^{-1}\\ (\\mathrm{mod}\\ N) ", "1fdb7691f8fc8c5d4f6dad96d3577c6d": "\\phi=\\sum_i\\phi_i; P=|\\phi|^2=|\\sum_i\\phi_i|^2", "1fdba7cf1a337c4ec8fefe92f9a55155": "\\forall a \\in A \\backslash E, \\exists b \\in S", "1fdbbb394d37d7b01a244836739e06eb": " \\Pr \\left \\{ \\lambda_\\max \\left ( \\sum_k \\mathbf{X}_k \\right ) \\geq t \\right \\}", "1fdbea72e3e8352547153089ebc6012c": "\\tfrac{2x}{x^2-1}", "1fdc65aca27b87a04b413b5bf979d227": " m < 1,", "1fdc73bfc2587735eb9eb40bba6dd46e": "K(iw) = H(a(w))+ i a(w) \\quad (1.4)", "1fdc88fc69eba3b8d69741c81a0591eb": "f \\times \\textrm{id}: \\mathbb{A}^1 \\times \\mathbb{A}^1 \\to \\{x\\} \\times \\mathbb{A}^1", "1fdc9843acea0cf3a5668fba4d0039dd": "\\mu_k=\\mu", "1fdccf0aac89e32b19fb6e6a1beb8e52": " \\neg \\varphi ", "1fdce0811a837fa680ca47033e33e47d": "\\begin{array}{ccl}\n\\mathbf{T}& = & T_\\text{xx}\\mathbf{e}_\\text{xx} + T_\\text{xy}\\mathbf{e}_\\text{xy} + T_\\text{xz}\\mathbf{e}_\\text{xz} \\\\\n& & {} + T_\\text{yx}\\mathbf{e}_\\text{yx} + T_\\text{yy}\\mathbf{e}_\\text{yy} + T_\\text{yz}\\mathbf{e}_\\text{yz} \\\\\n& & {} + T_\\text{zx}\\mathbf{e}_\\text{zx} + T_\\text{zy}\\mathbf{e}_\\text{zy} + T_\\text{zz}\\mathbf{e}_\\text{zz} \n\\end{array}", "1fdd0ac0fd53e56f0bdb7ee77bb3b7b2": "\\;2d\\sin\\theta =n\\lambda", "1fdd23a72e0b99289cb976d33c5728a9": " { \\frac{\\partial{(\\rho T)}}{\\partial t}} + { div\\, (\\rho u T )} ={div\\, (k\\, grad\\, T )}+ {S_{T}} \\, ", "1fdd6c3742b12671150b2adefd77c73b": "X:k-\\mathrm{Alg}\\to\\mathrm{Set}", "1fddddb98e7ab8973937ac93c8c58cf1": "\\eta = \\frac {R_R}{R_C + R_D + R_L + R_G + R_R} \\, ", "1fdde5e85b1f6bc4eefcf55168df45ce": "8.7 \\times 10^{-9} \\frac{\\text{Sv}}{\\text{Bq}}", "1fde746b4db28890e9cb609b459c3fd9": " \\left[{n+1\\atop k}\\right] = n \\left[{n\\atop k}\\right] + \\left[{n\\atop k-1}\\right]", "1fde8e8f4bafc19cde9dcac3618a2da2": "u_t = \\varepsilon u_{xx} + uf(u) - vg(u), \\,", "1fdf26ffc2b0016f84a5c2970c929487": "\\{X(t)\\}_{t\\in[0,T]}", "1fdf278c3fd9e34664b7721ba7f38755": "\\frac{\\text{d} [{^1_2}S^\\beta]}{\\text{d}t} = \\text{k}_{2(2)} C_2 -\\text{k}_{1(2)} {^1_2}S^\\beta E", "1fdf66be41f4d4e9bc457f2c57e0d493": "\\gamma = 90^\\circ", "1fdf794476ee9feb1cdfd32d540865a4": "C_{AS}", "1fdfc38196157c0cd4249899084d68bc": "\\Theta_\\Lambda(\\tau)", "1fdfd776456a2f3c83631d40807aa3e8": "6(1/3!)\\pi^3 = \\pi^3 ", "1fe038cf7229c92229444ff795f12250": "\\frac{d^2y}{dt^2} = 6 y^2 + t ", "1fe08dfddc5c307ce3d7455cad606651": "G_{i,j} = B(\\alpha_i,\\alpha_j) .", "1fe0c3440fed9bf7410917d637001ef3": "X: \\Omega \\subset \\Bbb{R} \\times M \\longrightarrow TM", "1fe1f69527c131c3bf0ad93bb46ab2c5": "da\\,", "1fe200e6fe318e7bea5a811371a89597": "x=(x_1,\\ldots,x_m)^T", "1fe22abaf09c45e7d36af37a3bc94c8e": "H^q(X, \\textstyle\\bigwedge^p\\Omega_X|_Y)", "1fe27dbceb53d79c07a6c7c3543ca9c0": " f(z) = \\frac{1}{2\\pi i} \\oint_{|\\zeta| = 1} \\frac{\\zeta + z}{\\zeta - z} \\text{Re}(f(\\zeta)) \\, \\frac{d\\zeta}{\\zeta}\n+ i\\text{Im}(f(0))", "1fe2b1f34376c890533c94f63df043d6": "\nf(k)=\\lim_{\\varepsilon\\rightarrow 0}~\\text{Im}~\\int_\\varepsilon^\\infty \\frac{\\exp\\left(ikx\\right)}{\\exp\\left(x\\right)-1} \\, dx.\n", "1fe30ffb70ccc4d59431919c15642226": "(\\tfrac{a}{p}) = -(\\tfrac{a}{q}) = 1", "1fe32316bf4df8e56d60dbe2a4236433": "k \\in \\Bbb N", "1fe3365f852fb174b334f0dade219a55": " \\frac{ Y }{ Y + Z } = beta( m / 2, n / 2 )", "1fe351dcf3fa4c54529e77f889566dc9": "\\sigma(E) = \\frac{\\pi g}{k^2}\\frac{\\Gamma_{ab}\\Gamma_c}{(E-E_0)^2+\\Gamma^2/4}", "1fe36481d77488844835e27ba6eda719": "\\mathbf{H_{{2}/{1}}}\\,\\!", "1fe3f262315374b6493a8ac8dad4220c": "\\sigma_{a \\theta b}( R ) = \\{\\ t : t \\in R,\\ t(a) \\ \\theta \\ t(b) \\ \\}", "1fe449dcfe6f94c964d0556ba1553615": "\\Bbb{E}\\{\\varphi(Y)\\} \\ge \\varphi(\\Bbb{E}\\{Y\\})", "1fe45a313f537433f21619644034a400": "Y-u", "1fe47c1924089f1c8a589b5a4f1c7835": "\n \\left(\\hat{C}^{(k)}\\right)^\\dagger = (-1)^k \\hat{C}^{(k)}\n", "1fe4a617adba9425700a7a455d59a545": "N_\\mathbf{P} = \\sum_{j=0}^N \\sum_{i=0}^M P^{\\text{old}}(i|s_j) \\leq N", "1fe4d59cfde43cfe08f2661ad32ece18": "4x+5-2x-3", "1fe5363abe0dca20456da627469cc2ba": "S_N(f;t)=\\sum_{n=-N}^N \\widehat{f}(n)e^{int}.", "1fe53bff45fe95a922acd453e1472ad9": "S = \\frac{1}{2}\\sqrt{-\\frac23\\ p+\\frac{1}{3a}\\left(Q + \\frac{\\Delta_0}{Q}\\right)} \\quad\\qquad\\ {\\color{white}.}", "1fe54a783c128a0b01689a84f3704d73": "p=\\rho g z\\,", "1fe58b069e98dc6324bec7bb834fcb24": "T_1=\\sum_i\\ m\\frac{\\partial \\mathbf{r}}{\\partial t}\\cdot \\frac{\\partial \\mathbf{r}}{\\partial q_i}\\dot{q}_i\\,\\!,", "1fe5c66f198808a55e24a553de763741": "A\\cdot B = 0", "1fe5ebae3456ee91e408cc2b6dddc9fe": " X_t = \\mu + \\varepsilon_t + \\theta_1 \\varepsilon_{t-1} + \\cdots + \\theta_q \\varepsilon_{t-q} \\,", "1fe5f60432801f80a5d75e91fbc02922": " \\sum_{i=1}^{n}{\\alpha_{i} \\cdot x_{i}} = \\alpha_{1} x_{1} + \\alpha_{2} x_{2} + \\cdots +\\alpha_{n} x_{n}, ", "1fe69fda9d1d29b016bc860404d5281d": "s_a^*(t)\\cdot e^{j2\\pi f_0 t} = s_{lsb}(t) +j\\cdot \\widehat s_{lsb}(t)\\,", "1fe6cb9b12a48b50c508c3c442d1b805": "|G| = \\sum_i d_{\\varrho_i}^2", "1fe76c3c063af050248c6bd83a021dd4": "\\nabla_X \\xi= (X^*\\otimes I)\\xi.", "1fe7935f9f3c5a4250696e30201f3459": "n_i=n_j", "1fe7f6f1f2f7d261e9472cd1148ab069": "\\begin{align}\n\\frac{2^{1000}}{10^{300}}\n&= \\exp \\left( \\ln \\left( \\frac{2^{1000}}{10^{300}} \\right) \\right) \\\\\n&= \\exp \\left( \\ln \\left( 2^{1000}\\right) - \\ln\\left(10^{300}\\right)\\right)\\\\\n&\\approx \\exp\\left(693.147-690.776\\right)\\\\\n&\\approx \\exp\\left(2.372\\right)\\\\\n&\\approx 10.72\n\\end{align}", "1fe8441aeabe880beeabdb317bb84658": "\\delta = \\frac{3}{4}", "1fe877d97ed9eb57ecc3e3272e1eee19": "\n\\begin{vmatrix} E_{11}+n-1 & \\cdots &E_{1,n-1}& E_{1n} \\\\ \\vdots& \\ddots & \\vdots&\\vdots\\\\ E_{n-1,1} & \\cdots & E_{n-1,n-1}+1&E_{n-1,n} \\\\ E_{n1} & \\cdots & E_{n,n-1}& E_{nn} +0\\end{vmatrix} =\n\\begin{vmatrix} x_{11} & \\cdots & x_{1n} \\\\ \\vdots& \\ddots & \\vdots\\\\ x_{n1} & \\cdots & x_{nn} \\end{vmatrix}\n\\begin{vmatrix} \\frac{\\partial}{\\partial x_{11}} & \\cdots &\\frac{\\partial}{\\partial x_{1n}} \\\\ \\vdots& \\ddots & \\vdots\\\\ \\frac{\\partial}{\\partial x_{n1}} & \\cdots &\\frac{\\partial}{\\partial x_{nn}} \\end{vmatrix}.\n", "1fe8a2c0601e8441247eb26e61a07562": "\n\\mathbf{S}=\\frac12\\left[\n\\begin{array}{cccc}\n1& 1& 1& -1\\\\\n1& 1& -1& 1\\\\\n1& -1& 1& 1\\\\\n-1& 1& 1& 1\n\\end{array}\n\\right]\n", "1fe8b0f77a38eebe4ceead55e25c1662": "\\displaystyle{a^b= (a^{-1}-b)^{-1}.}", "1fe8e8333f3e29dd673bc64ca4a9c8f7": "K(k) = \\frac{\\pi}{2}\\left\\{1 + \\left(\\frac{1}{2}\\right)^2 k^{2} + \\left(\\frac{1 \\cdot 3}{2 \\cdot 4}\\right)^2 k^{4} + \\cdots + \\left[\\frac{\\left(2n - 1\\right)!!}{\\left(2n\\right)!!}\\right]^2 k^{2n} + \\cdots \\right\\},", "1fe93cf7d7ccb38f56a2396634b337e3": "g_{X+k}(t)=g_X(t)+kt.", "1fe95a1184ac851bec647a471302ec8d": "T_v", "1fe9ba770548aa8b5c6bfc53b70071b4": "a_k = \\frac{{- a_2 a_{k-1} + \\sum_{j=2}^{k-1} (-1)^j \\, \\zeta(j) \\, a_{k-j}}}{1-k}", "1fe9d5a35ab4e906fb846044b9385ae2": "S(\\rho||\\sigma)={\\rm Tr}(\\rho\\log\\rho-\\rho\\log\\sigma)\\geq 0 ", "1fe9fed64721279e599a5d849ee5afda": " v = \\frac{2\\pi r}{T} ", "1fea0827be59497c40a8f0c278c9bb4a": "[\\mathfrak{g},\\mathfrak{g}]", "1feab5bee339e5cd4d3f647fdba74c35": "v_1\\otimes v_2\\otimes \\cdots \\otimes v_k \\mapsto v_k\\otimes \\cdots \\otimes v_2\\otimes v_1.", "1fead980aff0e8d1f18ea12452a99f7d": "|x|_P = c^{-\\operatorname{ord}_P(x)}.", "1feb910149c4e2adfd6d0565c589b389": "m^2 =~m", "1febc9a94e0affc527427c0d8acf5455": "f(q) = - \\tfrac 1 2 (q + iqi + jqj + kqk)", "1fec0ad8e0aa654366fb116b9b5a1827": "100 \\,", "1fec93c23c3f7232a231393d55fbac99": "V_{\\alpha+1} = P(V_{\\alpha})", "1fec9532f5222e915512151a2da5b169": "\\lambda\\left(\\Sigma+\\frac{1}{n+1}\\cdot K\\right)-\\lambda\\left(\\Sigma+\\frac{1}{n}\\cdot K\\right)", "1fecf198d6b6dac579a6c361c298a4b2": "d[\\cdot,\\cdot]", "1fed7fac5e1cb1f73935ee396a82aea4": "\\sigma_1^{(t+1)} = \\frac{\\sum_{i=1}^n T_{1,i}^{(t)} (\\mathbf{x}_i - \\boldsymbol{\\mu}_1^{(t+1)}) (\\mathbf{x}_i - \\boldsymbol{\\mu}_1^{(t+1)})^\\top }{\\sum_{i=1}^n T_{1,i}^{(t)}} ", "1fed909e37a65d3cf8c0b0567e76118c": "z \\notin L \\implies \\Pr\\nolimits_x[\\exists y. \\phi(x,y,z)] \\leq \\tfrac{1}{3}", "1fedbf2fbf3f1f3ee948cfa093235060": "\\varepsilon(\\alpha,-\\alpha)=\\varepsilon(\\alpha,0)=\\varepsilon(0,\\alpha)=1.", "1feddb68fda8705a2945e916ce1c61d1": "\\mathbf{C_1}", "1fede4d60bf2984ef5d9a3052a1fe7ba": " c > 0 ", "1fee71f3df550cd0f911ddda8dc3e5f0": "G=\\epsilon D", "1fee89eb9f6defa9a339bba48af66e89": "y \\sinh R \\,.", "1feed622f38bffb5bafc73abc37ce300": "\\lim_{x \\to 0} \\frac{1-\\cos x}{x} = 0", "1fef1620c1f394615594661919b65820": "N_{\\cdot}(\\omega) = \\sum\\limits_{i=1}^{Z(\\omega)} \\delta_{X_i(\\omega)}(\\cdot)", "1fef4ccc8a592606893fc1d5ac15a50a": "R \\approx 0.17 \\text{ mm}", "1fef59cc5cb36eb2912ac0b60f9b74ce": "\n\\frac{d\\Omega}{dt} = \\frac{2GS}{c^2a^3(1-e^2)^{3/2}} = \\frac{2G^2M^2\\chi}{c^3a^3(1-e^2)^{3/2}}\n", "1fef637cd581bc96ca8b90d9540b65a6": "C_{H_2O} = V - C_{osm}", "1fef648eb6b8c5df6940415461b7bd5b": "[\\![[a] \\phi]\\!]_i = \\{s \\in S \\mid \\forall t \\in S, (s, t) \\in R_a \\rightarrow t \\in [\\![\\phi]\\!]_i\\}", "1fef666ee59b638703edd5761c1e3234": "d \\le wt(\\boldsymbol{c'}) ", "1fef8ad8da4ddb479acc614014398798": " {\\kappa}a <1 ", "1fefab0afb5f96cf49a2a297c10ad0af": " \\psi^{(\\operatorname{Sha})}(t)=2 \\cdot \\operatorname{sinc}(2t - 1)-\\operatorname{sinc}(t), ", "1fefb51f912faf4d7b5492909002bebd": "\\Delta p_i \\, \\Delta q^i ", "1fefef4e73e239627356fc5754d0e513": "x+y+z", "1ff0149dcb540c9f2aa8d5c2c896830a": "\\bold\\lambda", "1ff0495fb838d78a99b66c4eebd8d063": "(L + R) + (L - R) = 2L", "1ff0c7beaca6fe0b96e9aae622d7bb39": "\\langle A \\rangle_\\sigma = \\sigma(A)", "1ff0fcba6806d7847a8c41a0de3230b1": "z =\\frac{S}{\\sqrt{\\operatorname{VAR}(S)}}=\\frac{40}{\\sqrt{185.212}} = 2.939", "1ff1259634a6ac4d0b5a3c1eef269d21": "\\Psi^\\dagger", "1ff12a5acda775c5bab92daa74008547": "h,x\\,", "1ff14003d9a42832730c63e2c0a57762": "c_\\text{man}", "1ff18d25b2eb29d2363ede64da7925a0": "{2 \\pi}", "1ff1cdb5504d68ae31a9b49260bab342": "A\\models R (a_1,\\ldots,a_n)", "1ff1d97415ecb181916778609427f577": "\\delta\\boldsymbol B\\ ,", "1ff1de774005f8da13f42943881c655f": "24", "1ff1f5d3d9f35a3ea9919c13026866b0": " \\mathbf{} A ", "1ff202b3c9d393e51fabe205889065f6": "E_\\mathrm{st}= \\sigma \\int dx\\, dy\\, \\left[\\sqrt{1+\\left( \\frac{dh}{dx}\\right)^2+\\left(\\frac{dh}{dy}\\right)^2}-1\\right]\\approx \\frac{\\sigma}{2} \\int dx\\, dy\\, \\left[ \\left( \\frac{dh}{dx}\\right)^2+\\left(\\frac{dh}{dy}\\right)^2 \\right], ", "1ff22ff94ade4fbf7b49c3ee44b36ea2": "M_{cycles}", "1ff232ea2e2dcdb4f6d57e7d3bf956ff": "x \\approx y", "1ff255db0229ad6fbc10596d4fa61116": " IC = \\sum \\frac{ f_i ( f_i - 1 ) }{ n ( n - 1 ) } ", "1ff2ec97b1eef19f42854107729dcffc": " x = x_1 2^{n-1} + x_2 2^{n-2} +\\cdots + x_n 2^0.\\quad ", "1ff31fa359f487e13b71f5ac7d32f216": "\\mathcal{P} = \\frac{\\Phi}{NI}", "1ff3538d6e1da90eeb7dc7eb5773060b": "T_{ij} = \\cfrac{1}{2}(S_i^+ S_j^- + S_i^- S_j^+) ", "1ff3572ed7b400d7b8d032be1bf0527f": "\\Sigma_j p_j f(r_j), ", "1ff37ed118824272399e65808644ec23": "\\mathfrak{gl}(V).", "1ff3840c1a8a45266a65ac3c4e7bee6f": "\\sin\\theta = \\left (\\frac{ m\\lambda_0}{n\\Lambda} \\right)", "1ff3bf454807c8dc19037254634ba246": "r=r_0A^{1/3} \\,\\!", "1ff3f4bb5aa85f4a91c784c1db0346de": "\n\\ell \\ddot \\theta - g \\sin \\theta = \\ddot x \\cos \\theta\n", "1ff413f5d8bed6830140433b661c2d43": "\\Delta \\vec{p}_\\mathrm{avg} = \\langle \\Delta \\vec{p} \\rangle_\\Omega", "1ff446a62cd2afe18c54b1769177a85f": " Q = (\\sqrt{F + T} -\\sqrt{F})^2 ", "1ff499587e86790b65b0446f871850a5": "\\text{s.g.}=\\frac{140}{130+\\text{degrees Baumé}}", "1ff4aa19f042337a731b322c618ae897": "\\Delta t_{i=1}=\\frac {\\Delta S_{i=1}}{\\left(\\frac{\\Delta S}{\\Delta t} \\right)_{i=1}} = \\frac{130.5\\text{ ft}^3}{2.9\\text { ft}^3/\\text{ s}}= 44.9 \\text{ s}", "1ff4df1af6cc3ec92ea0790fd0597fb4": "\\Phi(e)\\,\\!", "1ff4e1b0e340bfdb4666f52eefdfce87": "\\left(\\pm\\frac{(\\varphi+1)}{2}, \\pm\\frac{1}{2}, 0\\right)", "1ff4e7c4ea49e4f89fcea2a90968d87f": "A^{-1}", "1ff501e87fcaa0c9540d2579cc518901": " \\alpha=-k \\cot(k L/2)", "1ff5162360bdfc6fa7e0e8d160a87919": "\\mathbf{R} = \\left( d \\mathcal{M} + \\frac{1-d}{N} \\mathbf{E} \\right)\\mathbf{R} =: \\widehat{ \\mathcal{M}} \\mathbf{R}", "1ff530e35d29a87a18afb6b3e08eb4a8": "\\mathcal{S}\\subset\\mathcal{S}_{drs}", "1ff60dc22ce0d4fde64f6d5b9f5f0f1b": " S_{xyz}", "1ff6384a373ab9fcaba03f902b643b4a": "f^{(k)}", "1ff63c25d28601b1dc2fefde50759425": "\\psi\\rightarrow\\gamma_5\\psi", "1ff66da1a15e819564d488926c87eb35": "ds^2 = g_{ij}(q) dq^i dq^j\\,", "1ff69a9028511a2cbf16f4a10ec64c5b": "b=40", "1ff6a3df32b4eb55ddf93e67f25bd1ed": "r_{443}", "1ff70241875e2a9e46eac25ae203a35c": "\\int \\csc{ax} \\, \\mathrm{d}x = -\\frac{1}{a}\\ln{\\left| \\csc{ax}+\\cot{ax}\\right|}+C", "1ff763ea8c3c373f046f13cd9bb77620": " \\sum_{j=1}^n{g_j y_j} + \\sum_{i=1}^m{h_i s_i} ", "1ff7703188f686aa337b6b9225a066c5": "N(|fg|) \\le \\bigl(N(|f|^p)\\bigr)^{1/p} \\bigl(N(|g|^q)\\bigr)^{1/q}.", "1ff7af9a810bd160a7334c3de425ed9d": "\\nu_2", "1ff7b0f5809662e4d8c79d69b6501f74": "\\mathfrak{P}^{34}", "1ff7d920c350dc17d36721e9b0afef9b": "\n\\mathcal{F} \\left \\{ \\mathbf{x\\cdot y} \\right \\}_k \\ \\stackrel{\\mathrm{def}}{=}\n\\sum_{n=0}^{N-1} x_n \\cdot y_n \\cdot e^{-\\frac{2\\pi i}{N} k n}\n", "1ff8d33a7184bc39ba3e430744d7fccf": " S_v = \n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & \\frac{1}{s} \n\\end{bmatrix}.\n", "1ff8db085ecbde5d955cdce9ac55bf54": " S = \\begin{bmatrix}1 & 1 & 1 \\end{bmatrix} ", "1ff91f1c77c49a8eece699a459554f49": "\\{f_1 \\circ \\iota, f_2 \\circ \\iota \\} = \n-\\{f_1, f_2\\} \\circ \\iota", "1ff9d567c2cb356621879748e6d73740": "\n\\lambda_r = r^{-1}{\\tbinom{n}{r}}^{-1} \\sum_{x_1 < \\cdots < x_j < \\cdots < x_r} {(-1)^{r-j} \\binom{r-1}{j} x_j}.\n", "1ffa2fefbe4226fd2add379163878eb9": " f(r)=0 ", "1ffa4b01ff47ce9d0a926de5ec57b9e5": "\\frac{2r}{\\pi}", "1ffa5ff8cfea16c7baf501eee962dab9": "(\\lnot x\\lor\\lnot x)", "1ffabbfae67d7d9501beaade83e265a2": "\n \\begin{align}\n \\sigma_{xx}^{\\mathrm{f}} & = \\cfrac{z E^{\\mathrm{f}} M_x}{D} ~;~~ &\n \\sigma_{xx}^{\\mathrm{c}} & = \\cfrac{z E^{\\mathrm{c}} M_x}{D} \\\\\n \\tau_{xz}^{\\mathrm{f}} & = \\cfrac{Q_x E^{\\mathrm{f}}}{2D}\\left[(h+f)^2-z^2\\right] ~;~~ &\n \\tau_{xz}^{\\mathrm{c}} & = \\cfrac{Q_x}{2D}\\left[ E^{\\mathrm{c}}\\left(h^2-z^2\\right) + E^{\\mathrm{f}} f(f+2h)\\right]\n \\end{align}\n", "1ffacc3a50a1ad437de9ccc3934a4fdb": "\n S = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}\n", "1ffad26dcb702890f25a814f8f0cefa6": "\\Omicron \\, \\omicron \\,", "1ffb0d011e1344221b0df274d4d24ea6": "F_n(s_0)=F_n(s_1)", "1ffb9bcf71d24055c35021b6c5204e13": " -\\infty < \\omega_a < +\\infty \\ ", "1ffbb3f1764b8667cdea7ce3e31d2e78": " \\bar \\psi := \\psi^\\dagger \\gamma^0 ", "1ffbc8110f352d18b3d751c8c541d401": "\nH + \\gamma \\longleftrightarrow p + e^{-}.\n", "1ffc5ab1f049fd3bc8b25bc5dd96fc07": "f^{\\,liq}(T_s,P_s)", "1ffcb55f54fe598265bab98e908baac7": "\n\\begin{bmatrix}\n1 & 0 & 0 \\\\\n2 & 8 & 0 \\\\\n4 & 9 & 7 \\\\\n\\end{bmatrix}\n", "1ffccf54dc3d227758f594374940b188": "G(N_{i,j})", "1ffd00c904f05bb5043048d1d5e218b5": "W_{\\alpha,B} = aW_\\alpha", "1ffd117c4f731e104e47644cf852b9f3": "L^{\\infty}(\\Omega)", "1ffd23601da637b0c1b8361916cac987": "L^p(M)", "1ffd939ccc161d01c93a89cf4a7e878b": "[\\hat{A},\\hat{B}]=\\hat{A}\\hat{B}-\\hat{B}\\hat{A}.", "1ffda2ff0033acb551bee16545116d8b": " \\left\\vert u-v\\right\\vert _{W_p^k(\\Omega)}\\leq Cd^{m-k}\\left\\vert u\\right\\vert _{W_p^m(\\Omega)}. ", "1ffdb12fd32ae3a02f0933bf6a3a7690": "\\kappa_p(g) \\colon f \\in L^p(\\mu) \\mapsto \\int f g \\, \\mathrm{d}\\mu", "1ffe0fa175f9731368f378e4136bb52e": "(\\pi_1(u\\!:\\!\\sigma\\times\\tau) , \\pi_2(u\\!:\\!\\sigma\\times\\tau)) =u\\!:\\!\\sigma\\times\\tau", "1ffe55d7847a72ec5631b1d2835bdc63": "\\big. Z =Z_0 \\ Q", "1ffe77c486b70da9015e47a5d6fba4df": "Z_\\mathrm{eq} = \\frac{Z_1Z_2}{Z_1 + Z_2} .", "1ffeae0c82829e1da8bd3bfc85d23177": " z = r \\cos \\theta\\, ", "1ffecdc55d3ee76bc9a6d30d86dd9ffb": "\\boldsymbol{\\sigma}", "1fff047e447cf6aaaaca58a02593d80a": "n\\prod_{p|n}\\left(1-\\frac{1}{p}\\right)", "1fff0d37ae014f2bfc5841d86b5c8eb0": "\\ln K= \\sum_k \\ln {a_k}^{m_k}-\\sum_j \\ln {a_j}^{n_j};\nK=\\frac{\\prod_k {a_k}^{m_k}}{\\prod_j {a_j}^{n_j}}\n", "1fff0ea56e91ec01bba42de58d0e11f0": "\\qquad\\mathrm{first\\ form:}\\qquad t_{i+1} = \\frac{\\sqrt{t_i^2+1}-1}{t_i}\\qquad\\mathrm{second\\ form:}\\qquad t_{i+1} = \\frac{t_i}{\\sqrt{t_i^2+1}+1}", "1fff27dfb1db5357e7e6bbcbfae9a769": "A_1, A_2, A_3, \\ldots", "1fff2a1469a9e92e36e35c3b37239f1f": "\\ \\displaystyle R\\ ", "1fff8fc7b08c987172c55e7ee34db023": "\\begin{align}\nW_{A\\to B} & = \\int_{V_A}^{V_B} p dV = \\int_{V_A}^{V_B} \\frac{nRT}{V} dV = nRT\\int_{V_A}^{V_B} \\frac{1}{V} dV \\\\\n & = nRT(\\ln{V_B}-\\ln{V_A}) = nRT\\ln{\\frac{V_B}{V_A}} = nRT\\ln{\\frac{p_A}{p_B}} = p_A V_A\\ln{\\frac{p_A}{p_B}} \\\\\n\\end{align}", "200001b9bdcbbd4866d2d9967ff0f58a": "A_\\text{tri} = \\frac{1}{2} A \\times H. \\,", "20003eca9da435f8b370f3c0bb475e0b": "\\omega_0 \\,", "20005e0e39ea4898273a810c4375c10e": "| \\psi \\rangle = a | 0 \\rangle + b | 1 \\rangle =\\! \\begin{bmatrix} a \\\\ b \\end{bmatrix}", "200074c3ec406eab19c1cee19b0b4812": "H_\\infty(X) \\doteq \\min_{i=1}^n (-\\log p_i) = -(\\max_i \\log p_i) = -\\log \\max_i p_i\\,.", "2000a0aaaf1de40248b2803d9ef4dac1": "\\mathcal{Z} \\left\\{ t^y f(t) \\right\\} = \\left(-T z \\frac{d}{dz} + m \\right)^y F(z, m).", "2000d0474e4fb713257d7f790c2da5d9": "F_S^{-1} : S \\to S", "20010cd052c678363ef065402abfd43a": "f_{xx}(a,b)<0", "20014589ec4e1bacd2d73dae53898cb6": "\\bar c \\pm 3\\sqrt{\\bar c}", "200178f43ae034fecf8ef3789d8871e4": "\\frac{e^{\\frac{(u-t)^2}{4}}}{i\\sqrt{4\\pi}}", "2001bbf17b3c2e1271f02649d586fea7": "h_{XY}(\\pi_1(M_X,P))\\ ", "2001bfed9ab477365d726c5a2f6f2496": "a_1, \\ldots, a_k", "2001c7dc227dae6c2ed150256c2869c6": "D_{ep} = \\frac {D}{M} = \\frac {254}{36} \\approx 7", "2001fd8c3454841df58dbae8c18dfc82": " A_i^2 = B_j^2 = \\mathbb{I} ", "200212880cc59736ee6cfadcf59d049f": " A_{ij} = B[\\varphi_j,\\varphi_i].", "2002246e223667d1db74269e78636caa": "p=\\cfrac{1}{3}({\\sigma}_{11}+{\\sigma}_{22}+{\\sigma}_{33})", "200229231e19a635680d8aac80957faf": "L_n W(z)=-z^{n+1} \\frac{\\partial}{\\partial z} W(z) - (n+1)\\Delta z^n W(z)", "200246851af1c0cbdc39bb613bdaceae": "\\arg\\min_\\mathbf{x} \\|\\mathbf{Ax} - \\mathbf{y}\\|_2", "200279c0e076531f5be4d73b97e8c5ac": "x' = x\\cos\\left(-\\Omega t\\right) - y\\sin\\left( -\\Omega t \\right)", "2002b4b1ccbe5187fc71806e658913e2": "\\,L_v", "20031d09f41aceb6b72723ae7f870904": "E_K(P)=C,", "2003287af2e1e2f115f68560671bad83": " Y_\\ell^m( \\theta , \\varphi ) = (-1)^m \\sqrt{{(2\\ell+1)\\over 4\\pi}{(\\ell-m)!\\over (\\ell+m)!}} \\, P_\\ell^m ( \\cos{\\theta} ) \\, e^{i m \\varphi } ", "20032c63135ce4f83ceb8dbefe2fcecb": "y_x \\approx \\frac {Nc} {f} \\frac {u'} {\\tan \\theta} \\,.", "20034de3c0ff8f16f6a853579097d756": "x_{\\mathrm{FOH}}(t)\\,= \\sum_{n=-\\infty}^{\\infty} x(nT) \\mathrm{tri} \\left(\\frac{t - T - nT}{T} \\right) \\ ", "2003df41a2bcc9d99734f2c3497ee8fd": "p_k = { \\sum_{i=1}^n ({p_{k - i} + p_{k + i})} \\over 2n}.", "20040c0de9b36df41894aed76f23e2fe": "\n[P+iD,X]=0\n\\,", "2004296813796b103db06c62c6cced5a": "s:X\\to Y", "200433c08f424f73a31ea5f11bbb5035": "~A \\cap B", "2004a8c979dc795cb77e75a9e888ca1e": "K=\\mathbf{Q}(\\sqrt{d})", "2004ca542a715b2d1a9bb8ad177358ba": "\n\\mathbf{W} = \\alpha \\mathbf{A} \\otimes \\mathbf{A} + \\beta \\, \\mathbf{B} \\otimes \\mathbf{B} ~.\n", "2004cbb2d77588c0e44982c73d552f31": "f(U) \\subseteq Y", "200520c4284d4a8bdecc000a08e9e1f0": "(\\cdot \\cdot)_\\infty", "20054ac83fe5975c381297d700d32016": "t_c", "20056eb66b0ad7af659f809a45a8ab02": "j=1,\\ldots,k", "20059cb173a964f7bbd60a448d7c4b0c": " x_\\mathrm{IQM} = {2 \\over n} \\sum_{i=\\frac{n}{4}+1}^{\\frac{3n}{4}}{x_i} ", "2005ac65ac7b86ed720fed3c553e78ee": "\n\\sum_{i=1}^n x_i^2 \\sum_{i=1}^n y_i^2 - \\left( \\sum_{i=1}^n x_i y_i \\right)^2 \\geq 0. \n", "2005ed949d8f3b999573de9f6b0e456e": "2\\sqrt{p \\over n}", "200607d6c3f4855c41feb4c6e09c3f70": "\\exists a \\in S :\\forall a' \\in S :\\exists b \\in S :\\forall b' \\in S :\\exists c \\in S :\\forall c' \\in S ... :(a,a',b,b',c,c'...) \\notin G ", "20066410add2796dab7db09aad75a195": "u^\\prime", "2006a17fbc28713a02706a711872ce41": " x_k,y_l ", "2006e35780708ef5f30cdef8a93cd2fe": "A = \\frac{M_{ssd}-M_{dry}}{M_{dry}}", "20070bd19b85c926c31c69acc6c5313f": "\\mathfrak{sl}_2", "200717b4466746b1e97568996c9fdeb4": "G/Z(G)", "20072032f6fc6b99e24565d3ffc96a0a": "\\mathcal{S}_{E}=\\left\\{\n\\bar{Z}_{s+1},\\ldots,\\bar{Z}_{s+c},\\bar{X}_{s+1},\\ldots,\\bar{X}_{s+c}\\right\\}\n", "20075572813d4890608f0b322b483394": "x\\mapsto \\sigma(x)-x", "20086a80157ca571ef45bd1ef2270cf7": "xy^q-yx^q=1", "20088b03c8c2cf94649d95c70c5f50f7": " \\tau_1 + \\tau_2 \\approx \\tau_1 \\ , ", "20088e4bcc1760d7ecf44c8a264b4331": "X=\\mathbb{R}^n,", "2008adc200eb02bc70d1e841ce1fde76": "\\textstyle m \\in \\mathcal{M}", "2008d8d73fc675f5252f776230782558": "\\tilde f\\colon X \\to M_f", "200947dc4ef7dda1ee74851bea2cdcc4": "(|R\\rangle-|L\\rangle)/\\sqrt{2}", "20097029ea3d7331d5efd358070658d4": " x = \\sgn(x) \\cdot |x|\\,.", "2009b07f736feb7c45558c64a9f8653f": " j^k=\\frac{B_{k+1}(j+1)-B_{k+1}(j)}{k+1}. \\!", "200a61aa87f727274107256f38462b7c": " R_{critical} = \\frac {2 \\gamma V_{molecule}} {k_B T} ", "200a9e41549895521891194051239438": " E \\,", "200abf4544218b2f2f2233d76990ae47": " f_X(\\mathbf{x}|\\boldsymbol\\theta) = h(\\mathbf{x})\\ \\exp\\Big(\\boldsymbol\\theta^{\\rm T} \\mathbf{x} - A(\\boldsymbol\\theta)\\ \\Big) \\,\\! ,", "200adb9e4b61f49dde8acb7b22ea72f9": "i=k-1, k-2, \\ldots, k-m", "200af22a98e6e075b04f1d4d87e7e9f1": "(x_1, y_1), (x_2, y_2), \\ldots", "200b41992bec33d9022291ea8d535c8a": "x_1,\\cdots,x_n", "200b4200a7979af8b180932e8b3ef811": " \\sum_M\\frac{|W_0^M|}{|W_0^G|} \\sum_{\\gamma\\in (M(Q))}a^M(\\gamma)I_M(\\gamma,f)\n= \\sum_M\\frac{|W_0^M|}{|W_0^G|} \\int_{\\Pi(M)}a^M(\\pi)I_M(\\pi,f) \\, d\\pi", "200b4b6b2a3237c3a1fd84c261e18895": "\\omega = \\frac{2\\pi}{T},", "200b5d17eafdfbfe67224a16b8bae32e": "E(a, b) = \\int \\underline {A}(a, \\lambda)\\underline {B}(b, \\lambda)\\rho(\\lambda)d\\lambda \\qquad (4)", "200b9de0c47b57c2581b5bc8665cc2fe": "f(\\alpha, \\beta) = (a \\alpha + b \\beta, c \\alpha + d \\beta) = \\begin{pmatrix} \\alpha & \\beta \\end{pmatrix} \\begin{pmatrix} a & c \\\\ b & d \\end{pmatrix}.", "200ba05d1c4235fa5f8a3abe04e5b6d1": "R_2 =(S-T_1) \\times g ", "200bae0b0dc6739ee745be2adb6a50a1": "C = \\frac{1}{m}\\sum_{i=1}^m{\\mathbf{x}_i\\mathbf{x}_i^\\mathsf{T}}.", "200bbcb3c1bb3e74abe993e9e0c1fbb0": "x(k+1) = A x(k) + B u(k)", "200bf984ac4351fec8c13c7c21d1a4e3": "a + \\sqrt{a^2 - b^2}", "200bfa983d89313e94be35ccfc651d16": "f ^ t", "200c01ac5ee3419ca49b1f03fd2d3f59": "10^5-10^6", "200c4901f0bd0ab7c3ef59e8e3e28c8c": "\\mathbf{v}(\\mathbf{x}, t)", "200c5f003b49eb70d6ac3632e24107a6": "\\scriptstyle A_t\\subset Z^d ", "200c7d61e89a469e16a576820c6000f7": "(\\Omega_2,F_2,P_2)", "200cdc8a4fec2104b8123268ae665096": " 1, \\, \\left(1+\\frac{ix}{1}\\right)^1, \\, \\left(1+\\frac{ix}{2}\\right)^2, \\ldots, \\, \\left(1+\\frac{ix}{n}\\right)^n ", "200df5d23573c5a9af71a862a9914b78": " I = U_r ", "200e2230da70130a9c1604ea44428524": "v=v_{O'|O}", "200e49aa2c62d9acacbdf012387a8df4": "\n \\big(\\,\n \\overbrace{1\\, \\square\\, 1\\, \\square\\, \\ldots\\, \\square\\, 1\\,\n \\square\\, 1}^n\\,\n \\big)\n", "200e6497f8108e82ea3445d3b85de192": "| SA | : | AB | =| SC | : | CD | ", "200e8a9925cd2653b6a2514a36b82ccb": "\\displaystyle -i\\pi\\sgn(\\nu)", "200f1e2349d73818120a48e680d93268": "\\alpha > 2.", "200f32e7a4295b416a6483b18cfcaa4a": "Re(\\alpha)=Re(\\beta)=p^{11/2}.", "200f3e833bde508a7fc92bc788de32d5": "k/2 - 1", "200f97de129bc399309c550f6befc5fc": "\nF_{11} = \\cfrac{1}{X_{\\mathrm{f}}X^\\prime_{\\mathrm{f}}}\\ ,\\ F_{22} = F_{33} = \\cfrac{1}{Y_{\\mathrm{f}}Y^\\prime_{\\mathrm{f}}}\n", "200fb59701b9bd0a8b763bf6abffd673": "I_h = \\frac{1}{12} m\\left(w^2+d^2\\right)", "200fe503a74540f60f6c8ef9b63c41be": "S = -\\frac{V_{left}-V_{right}}{T_{left}-T_{right}}", "201022e258ff15130972d727002647a3": "\\,\\phi(0,\\bar{c}) = 0 ~~ \\textrm{for} ~ \\textrm{all} ~ \\bar{c},", "20102365216dff092a1e49ad64ab1560": "\\theta_1 = \\begin{bmatrix}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n\\end{bmatrix}\\qquad \\theta_2 = \\begin{bmatrix}\n0&0&0&0\\\\\n0&0&0&0\\\\\n1&0&0&0\\\\\n0&-1&0&0\\\\\n\\end{bmatrix}\\qquad \\theta_1\\theta_2 = -\\theta_2\\theta_1 = \\begin{bmatrix}\n0&0&0&0\\\\\n0&0&0&0\\\\\n0&0&0&0\\\\\n1&0&0&0\\\\\n\\end{bmatrix}.\n", "201033ebe336e21b40e85354b56eb722": "\\mathbf{abcdefghijklm} \\!", "20103d49e434ed6da124ad94f13d10fd": "\\sin(\\theta) = |P |\\psi\\rangle|= \\sqrt{G/N}", "20106470c24912c405cb864f34729a09": " \\acute{R} = \\acute{R}_{\\alpha}^{ \\alpha} = 8\\pi { G \\over { c^4 } } \\left ( {A\\over 2}\\acute{T}_{\\alpha}^{ \\alpha} + {B \\over 2} \\acute{T} \\delta_{\\alpha}^{ \\alpha} \\right ) = 8\\pi { G \\over { c^4 } } \\left ( {A\\over 2} + 2B \\right ) \\acute{T } ", "20111eed844ef5aaca21baaf6764cec7": "y_s(x)=0", "2011d4d37e456bacc31fa7ae7e4d8939": "f = {v \\over \\sqrt{\\|v\\|^2_2+e^2}}", "2011e57b4da471e895fd0b082258456e": "T(G) = \\frac{3\\delta (G)}{\\tau (G)}", "20122d059ed1a20a6f75460236c6004b": "r_q = y^{((q+1)/4)^{L}}~mod~q", "20123137a0c82b01eda55afa45e2d7aa": "\\omega=\\frac{d\\theta(t)}{dt}.\\,", "2012b89ffaa36af607737b3a2fdaac3d": "\n p(\\eta | y) = \\int p(\\eta | \\theta) p(\\theta | y) \\; d \\theta .\n", "2012d3be38fe508842908bf182923326": "\n\\begin{align}\n\\mathcal{F}\\{\\operatorname{tri}(t)\\} \n&= \\mathcal{F}\\{\\operatorname{rect}(t) * \\operatorname{rect}(t)\\}\\\\\n&= \\mathcal{F}\\{\\operatorname{rect}(t)\\}\\cdot \\mathcal{F}\\{\\operatorname{rect}(t)\\}\\\\\n&= \\mathcal{F}\\{\\operatorname{rect}(t)\\}^2\\\\\n&= \\mathrm{sinc}^2(f) .\n\\end{align}\n", "2012db2e145e2ba9a0281677d78333fd": " \\bar{\\psi}_{\\rm D} ", "2013318ff8c29061e4b3f55fb741685a": " Y(s) = e^{- \\int_t^s V(X_\\tau)\\, d\\tau} u(X_s,s)+ \\int_t^s e^{- \\int_t^r V(X_\\tau,\\tau)\\, d\\tau}f(X_r,r)dr", "201333d839f4c5c86404ce011dd177ef": " \\vec{H} = -\\nabla U, ", "2013375db1e9bcfafec7e8e15a0ab741": "\\left(\\sqrt{1/55},\\ \\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ -\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "2013e7f910b3ad7a8ca424c6c595f49f": " C = 1/B \\,\\!", "201400d073866463c42632818d600ec1": "K_i", "20140cad4d60856d0523dc485a53491a": "\\Omega + x ", "20146a71790cb22bf57a9dc7a6635d42": "\\bold{u} \\in U", "20147c33cd3f60bf25a075a94a30e831": "\\forall b\\in B", "2014ae35f262614f8c25f371f442733b": "\\big(\\sigma_H(\\omega)\\big)^2 = e^{\\pm i\\pi} = -1 \\qquad \\text{for } \\omega \\neq 0", "2014fdc1cbe9447487bdf23d005fd891": " x^4 + a x^3 + b x^2 + c x + d = 0, ", "20155d866107391be25917b0e0f51c03": "V_0\\subset L^2(\\R)", "20156b59cc794e6fffaecbdafdca48ce": " G(z) = 1 + \\sum_{n\\ge 1} \\left(\\frac{1}{|S_n|}\\right) g(z)^n = \n\\sum_{n\\ge 0} \\frac{g(z)^n}{n!} = \\exp g(z).", "20158603ea6b9c587746cb1690128eac": "\n\\sum_y K_{x\\rightarrow y} = 1\n\\,", "20158a4ffa5d497f0bc3ae9c02e50af6": "\\begin{matrix} {5 \\choose 3} = 10 \\end{matrix}", "2015cf0fc2bf47fdce8ecba5e5d0dc4e": "0 \\to W \\to V \\to V/W \\to 0", "2015d9bf0c6335d7be587d7aaf11c963": " a_t ", "2015e468b5d574c5d4461b86714c6e15": " \\left(D^2 + \\frac{b}{m} D + \\omega_0^2\\right) y = 0, ", "2016356256defe42c3cc28fb8395f1f9": "{z} \\ge {0}\\,", "20168cc271fd1cc842852737e2eb3c4c": "d = \\partial + \\bar{\\partial}", "2016a704d85bb3fd22de1c1e8c84c932": "D_\\Gamma:J^1Y\\to_Y T^*X\\otimes_Y VY, \\qquad\n D_\\Gamma =(y^i_\\lambda -\\Gamma^i_\\lambda)dx^\\lambda\\otimes\\partial_i, ", "2016c67940ecaf7dd07f0a9ff50cdba7": "\nm(\\varphi)=a\\big(1-e^2\\big)\\,\\Pi(\\varphi,e^2,e).\n", "2016dcded84431e5dcffe74b504077f7": "\\displaystyle{|T_rf|\\le f^*.}", "201758c0875f4fd664aaff970658e00f": "{\\mathcal L}^2_2", "20178e73e38b59447c1df6e7e591191b": "\\mathcal{O}_{X,x}", "2017e5f52020adfe2439467b8edd154e": "|\\phi|_p = (\\int|\\phi|^p)^{1/p} < \\infty", "2017ea3a8c7580ee97b5c3b4d613d343": "A \\rtimes M", "20184613b9b2e6b97ec1f92477b708ea": "s(\\theta_i)", "20188c84a6d5f200f682b89222904584": "V_{An}", "201897cfea5e1475b846b4b43c71d40b": "{\\ \\atop {\\ }} {{\\underbrace{a^{a^{\\cdot^{\\cdot^{a}}}}}} \\atop n}", "2018aa53f275c90824a0645ec65c0d67": "S_{\\mathrm{FN}} = \\; - b {\\phi}^{3/2} / \\beta. ..........(47) ", "2018d4c3d1d6bf445544371983da3319": "H_2(\\mathrm{A}_n,\\mathbf{Z})=\\mathbf{Z}/2", "2018ee47b429e6afa4a5aa175dde10c5": "U\\ ", "20196b4cb80ce92f9bfb43f9512c85c6": " \\delta = \\frac{\\sigma_\\varepsilon^2}{\\sigma_\\eta^2}. ", "20197ba1d2398678eac31c9b6140e8ea": "\\mathbf{u}_n(s) = \\left[ y'(s),\\ -x'(s) \\right] \\ , ", "2019ff2e43966b2b4e29ce21465ce09f": " A_4 = \\frac{ \\Gamma (-1+\\frac12(k_1+k_2)^2) \\Gamma (-1+\\frac12(k_2+k_3)^2) } { \\Gamma (-2+\\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } ", "201a3b8a9fbfecf4786a2fbe17a0b453": "Q_i = {\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'\\otimes Q \\otimes {\\bold 1}\\otimes\\dots\\otimes{\\bold 1}", "201a43085a9f350e2ab654ba5b1c6296": "\\vartriangle \\!\\,", "201a95ae50fc9c25d2a745afedc7c44d": "q^{-1} + 20q - 62q^3 + \\dots", "201a977cc7cc9c097ccacb28d3a5423f": "[\\mathrm{AB}]^{\\ddagger} = K^{\\ddagger\\ominus}[\\mathrm{A}][\\mathrm{B}] ", "201ab2acc2532c51366486edb54cb680": "=\\iiint_{V}\\left(\\psi\\nabla^{2}\\varphi-\\varphi\\nabla^{2}\\psi\\right)dV\\,\\!", "201ae8beaa5802350b0b9b40d154d4c3": "\\scriptstyle (2\\pi)^{-1}", "201b8de1971547855998ee521b5eab17": " u(w) = \\log(w)", "201bc2e3922b02dc235b91b60c756a0f": "\\theta_i \\theta_j = -\\theta_j \\theta_i\\qquad\\theta_i x = x \\theta_i.", "201be10430645ea2b9dd61d8778a75b6": "\nf(k)=\\boldsymbol{\\tau}{T}^{k-1}\\mathbf{T^{0}},\n", "201c1fde84fd061bb99a5884a945a854": "8\\,\\bmod\\,5 = 3", "201c3c93f5d5ea6871ff56b38c1b214a": "\nF(x)=\\sum_{i=1}^k G_i(x)'G_i(x) .\n", "201c5344a19c395a763e135ef02f2408": "\\text{Mean time between failures}\n= \\text{MTBF} =\\frac{\\sum{(\\text{start of downtime} - \\text{start of uptime})}}\\text{number of failures}. \\!", "201c7e241ef26d91eca7d251c1a792e9": "\\|\\mathbf{v}\\| \\ge 0 ", "201c85f01b16ec4217870763f4256d6d": "\\scriptstyle \\hat v", "201c9062d348ec6953a2500e2ece47c8": "z \\in I^*", "201ca9a59e64c984b4e8eee84d9e69aa": "\\vec{e}_0 = \\partial_t, \\; \\vec{e}_1 = f(r) \\, \\partial_z, \\; \\vec{e}_2 = f(r) \\, \\partial_r, \\; \\vec{e}_3 = \\frac{1}{r} \\, \\partial_\\phi - h(r) \\, \\partial_t", "201d2179e81e6eabf33675ccb4b71768": "\\textstyle a_\\diamond", "201d31d0d9f83384accff0b4e0d69ca1": "Oxy \\rightarrow Cxy.", "201d423d29f6ec052d70ed3cf7782bbe": "\nr_{xy}=\\frac{\\sum\\limits_{i=1}^n (x_i-\\bar{x})(y_i-\\bar{y})}{(n-1) s_x s_y}\n =\\frac{\\sum\\limits_{i=1}^n (x_i-\\bar{x})(y_i-\\bar{y})}\n {\\sqrt{\\sum\\limits_{i=1}^n (x_i-\\bar{x})^2 \\sum\\limits_{i=1}^n (y_i-\\bar{y})^2}},\n", "201d544169dfc667c3e1215c648949b4": "\\mu_\\infty", "201d9f9cca290b4fafd5c4b1e30cc642": "\\Psi : L(H_B) \\rightarrow L(H_A)", "201db00595dbe04b7342f371d3b59af6": "{\\lambda }_{L}=\\frac{1}{3V}\\sum _{q,j}v\\left(q,j\\right)\\Lambda \\left(q,j\\right)\\frac{\\partial }{\\partial T}\\epsilon \\left(\\omega \\left(q,j\\right),T\\right),", "201dbb2fde10937103359420fe8d8b84": "\n\\begin{array}{|rlr|rlr|rlr|}\n\\hline\n\\alpha & \\mathrm{alpha} & 1 & \\iota & \\mathrm{iota} & 10 & \\varrho & \\mathrm{rho} & 100 \\\\ \\beta & \\mathrm{beta} & 2 & \\kappa & \\mathrm{kappa} & 20 & & & \\\\ \\gamma & \\mathrm{gamma} & 3 & \\lambda & \\mathrm{lambda} & 30 & & & \\\\ \\delta & \\mathrm{delta} & 4 & \\mu & \\mathrm{mu} & 40 & & & \\\\ \\varepsilon & \\mathrm{epsilon} & 5 & \\nu & \\mathrm{nu} & 50 & & & \\\\ \\stigma & \\mathrm{stigma\\ (archaic)} & 6 & \\xi & \\mathrm{xi} & 60 & & & \\\\ \\zeta & \\mathrm{zeta} & 7 & \\omicron & \\mathrm{omicron} & 70 & & & \\\\ \\eta & \\mathrm{eta} & 8 & \\pi & \\mathrm{pi} & 80 & & & \\\\ \\vartheta & \\mathrm{theta} & 9 & \\koppa & \\mathrm{koppa\\ (archaic)} & 90 & & & \\\\ \\hline\n\\end{array}\n", "201dbc14304a12449ea74847bb74f4cc": "\\mu_{A}(x)=1", "201df202f2c20ac904dd728e83149379": " f_P = 2u_P + t_P + \\delta_P , \\, ", "201df2c079672eecd6fb5f7bbc470ce8": "Q_j = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial T}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial T}{\\partial q_j} = -\\frac{\\delta T}{\\delta q_j}=\\sum_{i=1}^n \\mathbf{F}_i\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j},", "201e449b5c0ad4f72e4c75be4e871ba9": "\\sigma_{Age = Weight}( Person )", "201e4c97bb7492d7a1486234cf3b7951": "(xy \\equiv zu \\and xy \\equiv vw) \\rightarrow zu \\equiv vw.", "201e5d97363081be8d7d9ba3950979b7": "\\scriptstyle P(G, t) = P(G_1, t)P(G_2,t) \\cdots P(G_c,t)", "201ec3d1014d1b23827e94d1f12f1628": " \\cos(a)\\sin(b) + \\cos(b)\\sin(a) = \\sin(a + b)", "201edf47f9bc92b0d80b898f3a0f3325": "y_I = B\\left( {1 - e^{ - \\tau } } \\right)= B\\left( {1 - e^{ - t/\\epsilon } } \\right).\\,", "201fafdb2155f6d76c5c1b07165ea248": "H(0) = PDQ^T", "201fef0a1b6e41c52bbb3f413281698c": "\\boldsymbol{\\gamma}_A = n", "20206a1cf0576e5a53dcf98e1b224f4e": "\\|f(x+h)-f(x)\\|=\\left\\|\\int_0^1 (Df(x+th)\\cdot h)\\,dt\\right\\| \\leq \\int_0^1 \\|Df(x+th)\\| \\cdot \\|h\\|\\, dt \\leq M\\| h\\|.", "2020fc9835e7d2c3452e01c81324cc88": " \\mu - \\frac{ 1 }{ 2q } \\operatorname{ E }| X - \\mu | \\le x_q \\le \\mu + \\frac{ 1 }{ ( 2 - 2q ) } \\operatorname{ E }| X - \\mu |", "202132bc21a6efe9877e3c1b66c1c4c2": "L(u) = (a_1, a_2)", "20213a5d9d8c020f03389a3f258f6dce": "n(n+1)/2", "20213ccbbfc03eec6a9af8f5f8a7eacc": "S_B(1) > S_L(1)", "2021807fc420a850ee0c8c970872304c": " \\overline{\\rho_f} \\cong \\rho ", "2021be6537e8cc3008de7f024495dbcd": "\\mathfrak H\\,", "2021c518b2aeba3432e062299defff7f": "x(w).w(y_1).\\cdots.w(y_n).[P]", "2021ecb257f4e84d3a7f7ae10302330f": "\\frac{a+b^n}{n}=x", "2022087cbb38390ede0e909a2d83d004": " \\qquad \\qquad S_\\mathrm{vib} = -\\frac{\\partial F_\\mathrm{mix}}{\\partial T} = 3Nk_\\mathrm{B}T\\int_0^\\omega\\{\\frac{\\hbar\\omega}{2k_\\mathrm{B}T}\\mathrm{coth}(\\frac{\\hbar\\omega}{2k_\\mathrm{B}T}) - \\mathrm{ln}[2\\mathrm{sinh}(\\frac{\\hbar\\omega}{2k_\\mathrm{B}T})]\\}D_p(\\omega)d\\omega,", "20221eb0dfd040cecfb1adf15012b96a": "F_X", "2022b5245c9db75248b3e6d92b875b8e": "P_{ij} = \\frac{\\partial V_{i}}{\\partial Q_{j}}", "2023780fd7b69a828d64e91f54dba30b": "R_{WM}", "20237af8291aa5af64eaca03645a2b81": "\\kappa(P) = \\lim_{Q\\to P}\\sqrt{\\frac{24\\left(s(P,Q)-d(P,Q)\\right)}{s(P,Q)^3}}", "2023b8707ce7b23a12d7c93feb59c3a8": "\n\\Delta i\\ =\\ -2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\cos i\\ e_g\\ (1-\\frac{5}{4}\\ \\sin^2 i)\n", "2023d76b1604e1566a570f76915d16a6": " E_2 = \\frac{ I }{ I_{ max } } ", "202410d9b87c4632cc661c7040dfcbf0": "_{ordinal}\\alpha \\ge \\rho", "20247f49f235be7930c80fa6ff8dccb7": "F'(x) = f(x).\\ ", "202481cace524e514625905a6bd4c310": "~n", "2024877e575c02cce04cd613b0493aa9": "Natural increase_t = Births_t - Deaths_t", "2024a6c2fed6ca688781b0e72a19b8c3": "\\lVert L_n\\rVert_\\infty = \\overline{\\Lambda}_n(T) = \\max_{-1 \\le x \\le 1} \\lambda_n(T; x),", "2024cb82c08f7f013d91d0045d60d531": "\\mu \\in {\\mathbb R}^d", "202512f55d8eb4bd14e2216cc032fc34": "\\mathcal B = \\{ U_1 \\times \\cdots \\times U_n\\ |\\ U_i\\ \\mathrm{open\\ in}\\ X_i \\}.", "20254ab2405c961827d66348819dfdea": "\\Gamma = K K^T", "202568a675719177e4ea5ca5dc10ab24": "\\begin{align}N&=q^{\\left\\lfloor\\frac n2\\right\\rfloor}A_n\\left(\\frac\\pi2\\right)\\\\&=q^{\\left\\lfloor\\frac n2\\right\\rfloor}\\frac{\\left(\\frac pq\\right)^{n+\\frac 12}}{2^nn!}\\int_0^1(1-z^2)\\cos\\left(\\frac\\pi2z\\right)\\,dz.\\end{align}", "20257d7bd7864e94ecd986c9d56fc5b8": "\\tbinom nc", "202590a2bb0742584ca0c87d0d1b034a": " a_2 = \\sum x_ix_j = \\frac{t_1^2 - t_2}{2} \\qquad \\text{ where } i < j ", "2025b313372c877b7f158c1ba24e27fd": " \\frac {\\left ( \\frac {MW} {N_0} \\right )} {\\rho\\, _{liquid}} = ", "2025d3b478bc8746b52e8d2f7ebde7fa": "z_\\mathrm{max}", "2025eeb10c0332ceea31d0c3c86c2746": "\n\\bar{\\kappa}_{es} = 0.2 (1+X) {\\rm \\, cm^2 \\, g^{-1}}\n", "20262b3067a9c34a38847f09a31b89b0": "\\varphi = \\frac{1+\\sqrt{5}}{2} = 1.61803\\,39887\\ldots.", "2026564560a754f9a0ce49cfbff28440": "\\tau \\in \\{0, 1, \\ldots, t-1\\}", "202695d8a5b1d6fd98e4d71bb9102862": "\\ln T_{max} \\le \\ln S \\le \\ln T_{max}+\\ln M. \\ ", "2026a7659cee3a094ba8c7a6deb2670b": "(14)\\quad\\quad u_s\\left(\\rho_2 E_2 - \\rho_1 E_1 \\right) = \\left[ \\rho_ 2 u_ 2 \\left( e_ 2 + \\frac{1}{2} u_2^2 +p_2/\\rho_2 \\right)\\right] - \\left[\\rho_1 u_1 \\left( e_1 + \\frac{1}{2} u_1^2 + p_1/\\rho_1 \\right)\\right].", "2026e0999bcbbc95ec019745d9743365": "\\operatorname{E}[|V^S - V^B|] \\approx \\alpha \\mu", "202712e66d20009d5d0d4fba6a56a367": " \\ddot{\\theta} + \\frac{g}{L}\\sin\\theta=0.", "2027262d72c81053fa2532297951103e": "\\displaystyle{m(gh,z)=m(g,hz)m(h,z).}", "20276d4423912c82016bcef747ac3a7b": "[\\mathbf{S\\cdot}\\hat{\\mathbf{n}}]_{\\mathrm{retarded}} = \\frac{q^2}{16\\pi^2\\varepsilon_0 c}\\left\\{\\frac{1}{R^2}\\left|\\frac{\\hat{\\mathbf{n}}\\times[(\\hat{\\mathbf{n}}-\\vec{\\beta})\\times\\dot{\\vec{\\beta}}]}{(1-\\vec{\\beta}\\mathbf{\\cdot}\\hat{\\mathbf{n}})^3}\\right|^2\\right\\}_{\\text{not retarded}} \\qquad \\qquad (3) ", "202787eaf6f029f94cd7b354a33ab027": "| i_1,i_2,..,i_{N-1},i_N \\rangle", "2027c36f1375765d9aa0da699c5e0b31": "\\mathrm{sum}_k(n)", "20280007df9667a74b09ee1b3f62fadd": "r=\\frac{|\\sec(\\theta)|}{(1+\\tan^{2/3}(\\theta))^{3/2}}", "20280e36b2fbda17e56490e7c6d6130d": "\\scriptstyle{\\varkappa}", "20287cbf7ffbe07a540d466290148ed8": "x=X(t)c+X(t)\\int_a^t X^{-1}(s)g(s)ds.\\,", "2028b90d01c714dd7616f5f540c02ce4": "\\textstyle \\{(1, \\alpha), (1, 0)\\}", "2028f7c78214b8dc370fd88c95000f4e": "\nr_8(n)=\n\\frac{\\pi^4 n^3}{6}\n\\left(\n\\frac{c_1(n)}{1}+\n\\frac{c_4(n)}{16}+ \n\\frac{c_3(n)}{81}+ \n\\frac{c_8(n)}{256}+\n\\frac{c_5(n)}{625}+\n\\frac{c_{12}(n)}{1296}+\n\\frac{c_7(n)}{2401}+\n\\frac{c_{16}(n)}{4096}+\n\\dots\n\\right)\n", "20291f16b1504b18156891854f8c7d11": "f^{123} = 1 \\ , \\quad f^{147} = f^{165} = f^{246} = f^{257} = f^{345} = f^{376} = \\frac{1}{2} \\ , \\quad f^{458} = f^{678} = \\frac{\\sqrt{3}}{2} \\ . ", "202948e3bb6111f1ad9638fec26abc78": "N_i\\geq N_{i+1}", "202954ce8c9346ac395af2d57afaaf40": "{}^{n}a", "20298f409797be80636e5bafb8aa22bb": " MP = \\text{Real wages} = AB = \\text{Constant institutional wages (CIW)} \\,", "2029b7138507452307c0e9cbb3fe4273": "v\\in\\mathcal{D}_{L^p}", "202a0ee5b47227bd3a526a5e731d26aa": "\\frac{dy}{dx} = a", "202a4b9d16e86af5dcd8c09d2fa5fca5": "\\langle x,f(x)\\rangle", "202a86d8abfa9829b553274a99894c74": "\\theta_I", "202aaa3a17fa98f2f303f7c62098816a": "\\int_0^T \\left(\\partial_t \\nabla_H F(\\sigma)\\right) \\cdot \\partial_t h := \\langle \\nabla_{H} F (\\sigma), h \\rangle_{H} = \\left( \\mathrm{D}_{H} F \\right) (\\sigma) (h) = \\lim_{t \\to 0} \\frac{F (\\sigma + t i(h)) - F(\\sigma)}{t}", "202ab9eba63a75ab31902732f98de991": "\\psi(n_1,\\ldots,n_k)", "202ad7254f4d72227e10deb23f3b4f3d": "E_i", "202ae15749171ac0ae694b2a3dc01cbc": " D = \\log(2)/\\log(3) ", "202b54f177c9a01bd0d0517e4730e2de": "\\wedge^2_n = \\vartriangle^1_n", "202bc50f0f9343720b378d2bc37ae380": " D^* = L.v_{blood}/6 + D_{blood} \\,", "202bf91a04dcc5cfb0018c947234387f": "\\sigma_k", "202c4a9ccef929426a4b204e404db902": "R =(1- \\frac{\\delta}{H^{-1}(1-r) - \\varepsilon})", "202c53880a0ee33a70ea86d12dda7e1c": "\n \\operatorname{E}[B_t] = \\operatorname{E}[S_t] - \\frac{\\mu \\alpha_t\\delta_t}{\\epsilon + \\mu \\alpha_t\\delta_t} (\\operatorname{E}[S_t] - S_B) \\;.\n ", "202c93d72704df80c9fc4732d9244152": "S = \\sum_{i=1}^n (r_i-s_i)^2 = 2 \\sum r_i^2 - 2\\sum r_is_i ", "202ca8d0bfd5b20907bf47b7504add2d": "\\mu_{s_1}(X)=1\\iff\\mu_{s_2}(\\{t\\mid t\\upharpoonright lh(s_1)\\in X\\})=1", "202cb962ac59075b964b07152d234b70": "123", "202cc8e72185ef42c80b5f2d3f1bc83e": "u[n]", "202d373173698a48ce63b03f341376d4": "\\mu_L(\\bar{x}; \\Sigma_t, \\Sigma_s) = g(\\bar{x} - \\bar{\\xi}; \\Sigma_s) \\, \\left( \\nabla_L(\\bar{\\xi}; \\Sigma_t) \\nabla_L^T(\\bar{\\xi}; \\Sigma_t) \\right)", "202d3a8163f2af315be21682002cecd1": "a = b = c = d, \\alpha = \\gamma = \\zeta \\ne 90 ^\\circ, \\beta = \\epsilon = 90 ^\\circ, \\delta = 180 ^\\circ - \\alpha", "202d7bfd6e62d356d5585ec66924d718": " D(\\alpha)", "202d8c6d892fe574b0b2943377dcc3ef": "\\operatorname{per}\\,T(A) = \\det A", "202d9889278de31a993cf61ecc82845a": "\\mu'_n=\\sum_\\pi \\prod_{B\\in\\pi}\\kappa_{\\left|B\\right|}", "202db8b67b532cbd03115f29f83d03e8": "\n \\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\underline{\\underline{\\mathsf{S}}}~\\underline{\\underline{\\boldsymbol{\\sigma}}}\n ", "202ddca09c74f52cd6072c5055c70648": "\n\n\\mathbf{F} =\n- G {m M \\over {r}^2}\n\\, \\mathbf{\\hat{r}} =\nm \\mathbf{g} \\left ( \\mathbf{r} \\right ),\n", "202df56bb401c0d06150a381b6b1a451": "{4 \\choose 2} = 6", "202e38de5603f797119fa0d1796ba145": "Sq^i(w_j)=\\sum_{t=0}^i {j+t-i-1 \\choose t} w_{i-t}w_{j+t}.", "202e442482c5ac8b73dcf0222ebaa92b": " \\vec u =\\vec u (\\vec x,t)", "202e49f7001aee774d56164622b80f4d": "p_{ij}^k", "202e8b1b0bf69ce12a37ebb4cfa88276": "(x-a)^2 + y^2 + z^2 = r^2.", "202e8da35d1b37e2d5c53dddf0b394d1": "n^2=\\frac{\\rho}{\\beta} ", "202e98b3e90ad6d5cda1d66b0d0ae05c": "p^q", "202eac52ce417589722cbd46023360cb": "BM(X)", "202ef06779ba1b6046699183c2118b7f": "\\frac{L^{(r)}(E,1)}{r!} = \\frac{\\#\\mathrm{Sha}(E)\\Omega_E R_E \\prod_{p|N}c_p}{(\\#E_{\\mathrm{Tor}})^2}", "202f247c10e898c4b1ae7c8f3c04e643": "L_n = (1-\\exp\\left({-{5 \\over {15 \\times 60}}}\\right)) \\times Q_n + e^{-{5 \\over {15 \\times 60}}}\n\\times L_{n-1} = (1-\\exp\\left({-{1 \\over {180}}}\\right)) \\times Q_n + e^{-1/180} \\times L_{n-1} = Q_n + e^{-1/180} \\times ( L_{n-1} - Q_n )", "202f51e6ffd7f1783368b6580f52645f": "m_0,\\dots m_{n-1}", "202f6b106a95a75a36c315fe79728d36": "t\\le s", "202fcbc50eb4db2ed93c872d85832129": " R = VSV^H + \\sigma^2I ......(4) ", "202fdc1df1806c90f286c4e38105e942": " + a_2 (x-c)^2 + a_3 (x-c)^3 + \\cdots", "20302a5c0df2a50af4b5f3099eee8e8a": "C_{qr},*(R_r),C_{rs}", "2030b1b9b05f892724def93ccf44bd46": "\\tfrac{\\mbox{total number of fission neutrons}}{\\mbox{number of fission neutrons from just thermal fissions}}", "2030c6a6b1ef8928b0c1a1d546029fa3": "V_g^f = R_g^f I_g^f + L_g^f I_g^f", "203139f56ac2b701abb63ae53746fea9": " A^{(e)} \\in [\\underline A^{(e)},\\overline A^{(e)}] ", "203160b2a2774c4744a130b8a9ba41d5": " x + \\frac{ab}{x}.", "20317b4d18199ac8894527671494030b": " f(R_1) \\approx \\frac{x_2-x}{x_2-x_1} f(Q_{11}) + \\frac{x-x_1}{x_2-x_1} f(Q_{21})", "2032191e40dc1ba8a6d7bacff95970d2": "\\alpha^{d-1}", "203224857eb0a3bfa91ac942870c35e8": "q,\\ \\frac{1}{q},\\ 1-q, \\frac{1}{1-q},\\ \\frac{q-1}{q},\\ \\frac{q}{q-1}", "2032391e480df6fafd7a32d99a38a5a6": "R(\\hat{n},360^\\circ) = +1", "2032634e35ef8c2f6d3fc6a4bfd432ae": " \\mathbf{W} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} ", "203271c4bf22121879e145b992860770": "\\cos (\\alpha - \\beta) = \\cos \\alpha \\cos - \\beta\\ - \\sin \\alpha \\sin - \\beta\\,", "2032e226763d55bb71ed594dcd15ca9f": "n = \\frac{V_V}{V_T} = \\frac{V_V}{V_S + V_V} = \\frac{e}{1 + e}", "203333db73a0febc74eabc674f2bdad8": " \\frac{d}{dx} \\frac{hi}{lo} =\\frac{ lo\\cdot hi' -hi\\cdot lo' }{lo^2 } ", "203349f4e7c71abedeba8b93dc6104f4": "D = \\sqrt{knr(r-1)} \\, ", "20336f16076197c1e0e7abe117ef10b4": "\n\\begin{matrix}\n\\frac{}{\\mathbf{0} \\,\\,\\mathsf{nat}} &\n\\frac{n \\,\\,\\mathsf{nat}}{\\mathbf{s(}n\\mathbf{)} \\,\\,\\mathsf{nat}} \\\\\n\\end{matrix}\n", "203383ab9ea787a9f3cd3faaea530331": "\\mathfrak{U}=(-\\infty,\\infty),\\ ", "2033c6683f330920a0e81fa189f9c008": "\\omega_{\\alpha+1}", "2033efd99874b50125b50baa49ab11d7": " G(0)", "20341452a68c4d743f860411e67aa0aa": "c(c-1)", "203482d5afecfefcc1fc3b7ff09a6866": "S(e_n) = e_{n+1}, \\quad n \\ge 0. \\,", "20352ae779618f8c3c952c3eb8762931": "f(x)=3x^3-x", "203535e0462000fa0aff937bbab1a963": "\\mu:T^2\\Rightarrow T", "2035415b27876f488040794abae50208": "n^2\\hat{f}(n)", "2035b02f339caba3a93a603a2b108b8d": " \\wp=[\\frac{2n}{p-1}] \\!", "2035c716411ae05a6225ce7cb735080e": "B^* A^{-1}", "2035ccea99a1d349aff3959f1ee60fbe": "\\phi_Y(u)=E[e^{iuY}]", "2035d847e8bf5b899ace29c254bdaebe": "q \\sim p^{2}", "20360ed2f3459bccc2eb1972b4984565": "\\boldsymbol\\theta=\\tan^{-1}\\frac{Y}{X}", "203636049812a9a2e2bcbdabd83fd528": "\n a \\preceq b := \\begin{cases}\n1 &, \\ \\ a\\le b \\\\\n0 &,\\ \\ a>b\n\\end{cases}\n", "20364188d7f21997427137fea2884c4a": "M(t)=\\frac{2}{\\pi}\\int_0^\\pi e^{Rt\\cos(\\theta)}\\sin^2(\\theta)\\,d\\theta", "20364c0769e8d234355c9c55028ca7eb": "b_i=\\frac{a_{n-1}\\times{a_{n-2i}}-a_n\\times{a_{n-2i-1}}}{a_{n-1}}.", "203672b557ed05d928d335b344dfcc53": "(N, M, d, \\gamma, \\alpha)\\,", "203691d3dbce41e8b3844a9f6c11ada6": " f(x) = \\frac{1}{2}\\int_{\\mathbb{R}}\\left(\\frac{x-t}{|x-t|}+\\frac{t}{|t|}\\right)d\\mu(t).", "2036a808ee21bbcf2c59ee19ef445c5f": "T^a_a=0", "2036e0c98c776a5749c1fc8710897765": "x \\in \\mathbb{R}^+", "203747583974154343acc18e4c2344af": "\\begin{align}\nP_{0}(s) &=-\\alpha \\beta, \\\\\n P_{1}(s) &= (2-\\gamma )s^2+(\\alpha +\\beta -1)s, \\\\\n P_{2}(s) &= s^3-s^2.\n\\end{align}", "20376defb6278d75a6b8164aa0678f36": "S_C - \\ ", "2037d7c874e96e2590141ddfa40006dd": "P(R_{NP},\\theta_1) \\geq P(R_A,\\theta_1) \\iff\nP(R_{NP} \\cap R_A^c, \\theta_1) \\geq P(R_{NP}^c \\cap R_A, \\theta_1). ", "2037e6d50299db3ae33cc7163565a0f2": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} Q_{1lm}\n\\left( \\frac{1}{r^{l+1}} \\right)\n\\sqrt{\\frac{4\\pi}{2l+1}} Y_{lm}(\\theta, \\phi) \n", "2037e9d9cd494a1c0f1d41b0ad55d3d6": "n_T=n_1+n_2+ \\cdots ", "20380359266cad80d4ffc3471f9aac31": "K_n(R)\\oplus K_{n-1}(R)\\hookrightarrow K_n(R)\\oplus K_{n-1}(R)\\oplus NK_n(R)\\oplus NK_n(R)", "2038417a58ce7c85ce4d285f97276f2b": "E(x,y,z,t)=\\psi(x,y,z)exp(-j\\omega t)", "203882e7318bff85cd4be8de76e9c597": "m_r\\rightarrow m", "20388f2486a5e230651bd183756ca3cf": "\\scriptstyle \\alpha=\\alpha_j.\\,", "2038b6513d55bf09ecdc4fee846133fb": "9x+1 =\\frac{9n^2 + 9n + 2}{2} ={(3n+2) \\choose 2}", "2038b79d7cd36c70ce0391dac4eb5d75": " \\mathbf{z}_i = \\mathbf{x}_{i}^{k} = V_{k}^{T}\\mathbf{x}_i \\rightarrow ", "2038dc3228f50bbaa92add3ad546ab6a": "\\mathrm{colim}F=\\mathrm{Lan}_E F", "2038dfb91e03138996b554d71d12f3ca": "\\Big [ 1 + \\int d^3x \\lambda^j (x) \\hat{G}_j \\Big] \\Psi (A) = \\Psi [A + D \\lambda] = \\Psi [A],", "20393c3c1b996c28b22be809f0c6174f": " I(y)=I_{0}\\exp(-\\mu y) \\,", "20393ca9ddc076dcff0304ea82d86d71": "AFC=\\frac{FC}{Q}.", "20399d4c636d9f981f8aa5514893fe58": "\\mu=\\sqrt{2 + \\sqrt{2}}", "2039ed44e4cb974ccd148ac7a0217ed3": "-\\Delta f = \\lambda f.", "203a010a29f65f259129ceace16bf7c4": "P(k,k')=\\frac {2 \\pi} {\\hbar} |\\lang\\ k'|H_1|k\\rang |^2 \\delta(E_{k'}-E_k) ", "203a2ddbdfd61eb9b495bd814bbc1e9e": "\\cos(2i+1)\\frac{\\pi y}{2}", "203a5514c05960964289d7ed500356a9": "(g, gs),", "203ad18d956014e6b38e110b924d9d4e": "2 \\pi r h \\,\\!", "203af86679d37b77967d2f6b69eca672": "S\\,\\cap \\,U = T \\,\\cap\\, U.", "203b72e01aead089a31554ab7f19520e": "K,", "203bc9f691ad42bb8a61de285930a8ec": "\\mathrm{Re}_D\\,\\!", "203be82fb8a0190c92a25c7d25e66956": "\\mu_1 \\neq 0", "203c0af8a744562d4fbd246a0c51abfc": "y_{it} = X_{it}\\mathbf{\\beta}+\\alpha_{i}+u_{it}", "203c6c0ed52fc22493566593542f1b10": "\\nabla^2\\phi(x)=0", "203c9d98329c8d8813a748ed7a6ccf51": "m = [P_1 \\to Q_1, \\ldots, P_r \\to Q_r].", "203cd67e50bbe55a394401476ba32ad0": "\n\\bar{T}_\\text{fixed} = \\frac{-4N_e(1-p) \\ln (1-p)}{p}\n", "203d1c981d2f1a9e483ecc4670f963cc": "\\int\\frac{dx}{x^2 - a^2}", "203d3156a6ab6b2e0cb9434710695dbf": "nK_{X'}", "203d42e4ae0cdda20ed9898e0f46ecb3": "\\tau = 1 / ar", "203d4a8f1e080479c8212983c98b4eaa": "s^2 = \\sigma^2 + \\nu^2 ", "203d50c3bd0305720f25937edb8333ed": "y_k = \\Delta\\cdot\\left(k+\\tfrac1{2}\\right)", "203d592ad1127f6fafd9086e8f8048e5": "\nV_{nm}(x,y) = R_{nm}(x,y)e^{jm\\arctan(\\frac{y}{x})}\n", "203da0c9ff18113bb2d218578f6d208e": "pK_w = -\\log ( [\\mathrm{H}^+][\\mathrm{OH}^-] )", "203dbf08881b4384d10c4d75caa9c949": "\\cos (wz) + i \\sin (wz) \\text{ is one value of } \\left(\\cos z + i\\sin z\\right)^w.\\,", "203dceaa71c9b022d3a263e5323a9202": "E_1 = E_2=5.06ft", "203df8fbc04f32d79455b4dfede33d96": "\n\\begin{align} \n \\epsilon(f) & = \\Big[ 1-G(f)H(f) \\Big] \\Big[ 1-G(f)H(f) \\Big]^*\\, \\mathbb{E}|X(f)|^2 \\\\\n & {} - \\Big[ 1-G(f)H(f) \\Big] G^*(f)\\, \\mathbb{E}\\Big\\{X(f)V^*(f)\\Big\\} \\\\\n & {} - G(f) \\Big[ 1-G(f)H(f) \\Big]^*\\, \\mathbb{E}\\Big\\{V(f)X^*(f)\\Big\\} \\\\\n & {} + G(f) G^*(f)\\, \\mathbb{E}|V(f)|^2\n\\end{align}\n", "203e17ba75294f65b427c448cd80dac9": "\\left|x- \\frac{p}{q}\\right|= \\frac{|cq - dp|}{dq} \\ge \\frac{1}{dq}", "203e512d62eb42752be4d5705f8b183f": "S^n \\Rightarrow_{f} ... \\Rightarrow_{f} A^{2n} \\Rightarrow{g} A^{2n} \\Rightarrow{h} ... \\Rightarrow{h} S^{2n} \\Rightarrow{k} S^{2n}", "203e5226dbc89b1e10a193132149d1d2": " \\mathbf{j}_1 \\cdot \\mathbf{S}_1 = \\mathbf{j}_2 \\cdot \\mathbf{S}_2 ", "203ebca743b5d98f495a7486f5d8f578": "x + 0 = x \\and x \\cdot 0 = 0", "203ec63042dd375a3b29ea01a9c78c78": " \\tfrac{D\\rho}{Dt} = \\tfrac{\\partial \\rho}{\\partial t} + \\mathbf u \\cdot \\nabla \\rho = 0", "203ecab60dda5c2f66a4f4d1929e3784": "[M+H]^+\\,", "203ef25b3c7ebcf377cf9efa0daecd75": "\\scriptstyle V \\times V \\times V \\to \\mathbf{R},", "203f25bf91b5f66a9445f085ca5470e0": "{\\frac {|AB|} {|BD|}}={\\frac {|AC|}{|DC|}} ", "203f50efaf747583be4a2fe1573fb274": "i \\in W", "203f5a19123d1fc58ec1322c177738d6": "f \\colon X \\to Y", "203fe3f061619ecd7f4145314a1294e9": "\\mbox{Current ratio} = \\frac {\\mbox{Current Assets}} {\\mbox{Current Liabilities}}", "203ff16ffef753b59319494d8d37d701": "\\hat{x}_2=x_2(1+\\delta_2)", "2040284354875e6ee18b159421143ff7": "\n\\begin{array}{rrrrl}\n&&4B_0&&=1\\\\\n2A_0 &&& + 2B_1 &= 0 \\\\\n-4A_0 &&&& = 0 \\\\\n&-2A_1 &+ 2B_0 && = 0 \\\\\n\\end{array}\n", "204059374481b7692bcc7df238f2a9ff": "\\varphi_\\ast(v)=w", "2040a4632a008550f80d79fa9d6e745e": "z^2 \\frac{d^2y}{dz^2} + z \\frac{dy}{dz} + (z^2 - \\nu^2)y = z^{\\mu+1}.", "2040ae2a83e7d05572da70dfd15d1641": "I_{{(Q)}_{[\\epsilon]}} \\varpropto \\epsilon^{\\tau_{(Q)}}", "2040bb4b583f25fcec49f290f50fc306": "\\mathbf{p}_{k+1} := \\mathbf{z}_{k+1} + \\beta_k \\mathbf{p}_k", "2040cef9f1ee0e832073b7d2d8f74af2": "\\Psi=24\\pi^3\\frac {a^2} {\\tau^2 c^2(1-\\epsilon^2)}", "2040f215e04ca3a38f2415924a2ef7cb": "\\Box \\phi - \\frac{\\partial }{\\partial t} \\left(\\frac{1}{c^2}\\frac{\\partial \\phi}{\\partial t} + \\nabla\\cdot\\mathbf{A}\\right) =-\\left(\\frac{mc}{\\hbar}\\right)^2\\phi \\!", "204207c8d7d1f183d2f17e098b467a8f": "H = H_0 + \\sum_{\\mathbf{k}\\sigma} E_{\\mathbf{k}\\sigma}\\gamma_{\\mathbf{k}\\sigma}^\\dagger \\gamma_{\\mathbf{k}\\sigma}", "2042c57a8594fd481a86ed09150e7940": "a(i,k) \\leftarrow \\min \\left( 0, r(k,k) + \\sum_{i' \\not\\in \\{i,k\\}} \\max(0, r(i',k)) \\right)", "2042cb759e460d4f7e1a7b64636695e1": "|G:H| = \\frac{|G|}{|H|}", "2042dc926cdcb316e1ca51acac530ff2": "k^3+k^2+k-1=0", "20430349f06c24e96da01a2d87d4e1eb": "\n \\quad \\min \\limits _{D, X} \\{ \\|Y - DX\\|^2_F\\} \\qquad \\text{subject to }\\quad \\forall i , \\|x_i\\|_0 = 1.\n", "20430f2a60485445bbe717043289b53d": "\\Delta(y, E(m))", "2043ae3195102845d4282184b4ce309d": "(x_1, x_2, \\ldots, x_n) : \\mathsf{T}_1 \\times \\mathsf{T}_2 \\times \\ldots \\times \\mathsf{T}_n", "2043c650fe9b7badf7a84526c59003ad": "f_0 = \\frac{v_p}{p}", "204461f3a954991bd6a23e6cbca5bc3f": "d^{O(n^2)}", "2044be5b4190df0f3a5329d67cd81b2d": " C_q", "2044ef98cee1ccb5a305af5666b15315": "\\lambda(L(B)) \\geq \\gamma(n).d", "20451f27f52a9eeecfa03f72097c964e": " \\int \\cdots \\int_T\\;f(x_1,x_2,\\ldots,x_n) \\;dx_1 \\!\\cdots dx_n ", "204525c222ecda99e85600541180aa96": "n_r = 4", "204561e6ba8a7e1b6f6ba708e381f6a6": "\\Vert T f\\Vert_{L^2}\\le\\sqrt{\\alpha\\beta}\\Vert f\\Vert_{L^2}", "20457b2d9f82e369552c95dd95a02f2b": "K=K_c \\cdot [H_2O]\\,", "20458d655602f1000157cdccaa8cf6c2": "T \\wedge ", "2045ae15f2dd4be5335c1a311bf8133e": "\\zeta(1+iy)=0", "2045f467a8140abc9cbc9aee0ef38b3c": "\\lim_{x\\to c}{\\frac{f(x)}{g(x)}}=L.", "2045fac6dfd26e6847d378cfcc725653": "I(v)", "2046329e7f7efc6218449be7d6b228cd": "\\mathbf{w}(\\mathbf{X}_A)", "2046418f021f57a61b365b767031423d": "\\ \\ ", "20468fd7ea8d02f9e6325ee347fc593b": "\\mathbb{C} \\setminus \\{0\\}", "20474e637aea89a205b98de37327ce55": "\\bar{3}", "20477a5c69344d9880d5646859d2e942": "\nz = a\\ \\sigma\\ \\tau\n", "20478c9ffc87050bb7c3ef6777c5231b": "\\textstyle{\\frac{1}{2}}", "2047cd0dae5279273852c51490304e2b": "P_{avg} = I_{rms}V_{rms}\\cos\\phi = I_{rms}^2 \\operatorname{Re}(Z) = V_{rms}^2 \\operatorname{Re}(Y^*)", "2047ead45ca6f21887ab2103559ee6cd": "\\ddots\\,\\!", "2048368ed761bd149a142bada7b69b4c": "\\iiint_T \\rho^4 \\sin \\theta \\, d\\rho\\, d\\theta\\, d\\phi = \\int_0^{\\pi} \\sin \\phi \\,d\\phi \\int_0^4 \\rho^4 d \\rho \\int_0^{2 \\pi} d\\theta = 2 \\pi \\int_0^{\\pi} \\sin \\phi \\left[ \\frac{\\rho^5}{5} \\right]_0^4 \\, d \\phi = 2 \\pi \\left[ \\frac{\\rho^5}{5} \\right]_0^4 \\left[- \\cos \\phi \\right]_0^{\\pi} = \\frac{4096 \\pi}{5}.", "20483ccf1d525f4e7244f45e44beff88": "C\\ell^{\\,i}(V,Q)C\\ell^{\\,j}(V,Q) = C\\ell^{\\,i+j}(V,Q)", "20487f3fc4870604823b57d84585d521": "(a+b\\,x)^m (c+d\\,x)^n (e+f\\,x)^p", "2048f874b7f49cc705aa00f16fd02a5b": "y = y_0 + u (y_1 - y_0) = y_0 + u \\Delta y\\,\\!", "2049d730c41156679db0639b8996a25d": "\\frac{\\sum_{i=1}^np_i f(x_i)}{\\sum_{i=1}^np_i}-f\\left(\\frac{\\sum_{i=1}^np_ix_i}{\\sum_{i=1}^np_i}\\right)\\le\\frac{\\sum_{i=1}^np_if(2a-x_i)}{\\sum_{i=1}^np_i}-f\\left(\\frac{\\sum_{i=1}^np_i(2a-x_i)}{\\sum_{i=1}^np_i}\\right).", "2049f39b58752335ac7596cd4103ac5e": "\n\\langle [u],[v],[w]\\rangle = \\{[\\bar s w + \\bar u t] \\mid ds=\\bar u v, dt=\\bar v w\\}.\n", "2049fa62070c05f0282a98e6bae5511c": "0\\le\\lambda_0\\le\\lambda_1\\le\\cdots\\rightarrow\\infty.", "204a117c43be2990f56f146e6e36a10b": "\\tilde{\\mathbf{x}}_0 = (0,0,0)", "204a2ecdcd5df7e5397090158d38688b": "\\boldsymbol{r_j }= \\frac{1}{m_{0j}} \\sum_{k=1}^j m_k\\boldsymbol {x_k} \\ - \\ \\boldsymbol{x_{j+1}}\\ , ", "204aa5563c205da00017aed8ee13bff3": "q_{0}", "204ab3355e47207871f7e205000a5b8b": " y_{k+1} ", "204abe0fe4062fc3f086e6e2bfe6e8c0": "\\mathrm{\\frac{1\\,statvolt}{1\\,abvolt}}=\n\\mathrm{\\frac{1\\,stattesla}{1\\,gauss}}=c", "204ad303624cc8d3a95b287fdb591759": "s_{i+128}=s_i+s_{i+7}+s_{i+38}+s_{i+70}+s_{i+81}+s_{i+96}", "204ae41c9f2d97a460fe17fb5ea3bd30": "\\alpha_s=g^2/4\\pi", "204af65a59a3302947657da139d1d531": "U > 0", "204b4a555f494b07495342b6dd29439e": "\\mathcal D.", "204baab08e71d8a8906d6189187d6974": "RD'=\\sqrt{\\left(\\frac{1}{RD^2}+\\frac{1}{d^2}\\right)^{-1}}", "204bce9220cd8af6db2d5a1b2d0eb9be": "x > 2.5", "204c67cda0406debb25bfbf373a76395": " 1 + 2 + 3+ ... + 34 + 35 + 36 = 666", "204c9aaba2398fcf04c876537c08cb07": "[-2U,2U]", "204d3d55e29ce977037d3582d193a451": "D_{F_1 + \\lambda F_2}^q(p, q) = D_{F_1}^q(p, q) + \\lambda D_{F_2}^q(p, q)", "204d66232acb3b331a33ed086427ae11": "p=d+1", "204d752083b4f8cf5e72f022da431f2a": "M^{\\alpha\\beta\\mu}_y(x)=M^{\\alpha\\beta\\mu}_0(x)+y^\\alpha T^{\\beta\\mu}(x)-y^\\beta T^{\\alpha\\mu}(x)\\,,", "204dba7f4676afc3860775970826a607": "\\mathrm{af}(n) = \\sum_{i = 1}^n (-1)^{n - i}i!", "204dd132ab2f7058d6f5d4c2db308d76": "\n\\frac{1}{2m} \\left( \\frac{\\mathrm{d}S_{z}}{\\mathrm{d}z} \\right)^{2} + U_{z}(z) + \n\\frac{1}{2m \\left( \\sigma^{2} + \\tau^{2} \\right)} \\left[ \\left( \\frac{\\mathrm{d}S_{\\sigma}}{\\mathrm{d}\\sigma} \\right)^{2} + \\left( \\frac{\\mathrm{d}S_{\\tau}}{\\mathrm{d}\\tau} \\right)^{2} + 2m U_{\\sigma}(\\sigma) + 2m U_{\\tau}(\\tau)\\right] = E.\n", "204e0c6532cd435b9ed7472232525770": "\\nabla^2 \\mathbf A - \\frac 1 {c^2} \\frac{\\partial^2 \\mathbf A}{\\partial t^2} = \\mu_0 e \\psi^{\\dagger} \\boldsymbol{\\alpha} \\psi ", "204ebb982944bbbe234738f0f182b210": "\\frac{f_{t}-f}{f_{t}}\\cong \\frac{1}{2}(\\frac{f_{L}}{f_{t}})^{2}\\propto \\frac{1}{L^{4}}", "204ed65a6066a366af330efbb1b6aa54": "\n \\varepsilon_c < \\varepsilon_3 < \\varepsilon_1 < \\varepsilon_t \\,\n ", "204ee47f6b4f19e278483ba4368661ed": "g(n, 1) = (X_1)^n \\text{ for }n=0,1,\\cdots,N.", "204eff4ca49030cd6f1699fa62ea15d3": "k=1\\,", "204f229f1526ee197915660c00172d38": "\\csc A = \\cot A \\cdot \\sec A \\ ", "204f3129d219241b43702105b4a4132d": "(-1,1,0,\\ldots),", "204f9e3a6df76f200dc59fe06bde7e03": "E(R_{i})", "204fc3dff9532da29132e59a6ec38452": "y=e^{-3x}\\left(\\int 2 e^{3x}\\, dx + \\kappa\\right). \\,", "204ffe9e6ef2e939447ffa8c47b31117": " T_\\textrm{inv} = \\frac{2a}{b k_B} = \\frac{27}{4} T_c ", "205004edee9808a164afb3df3700f192": "a_n \\frac{d^n x(t)}{dt^n} + a_{n-1} \\frac{d^{n-1} x(t)}{dt^{n-1}} + \\ldots + a_1 \\frac{dx(t)}{dt} + a_0 x(t) = \\sum_{k = 0}^n a_k \\frac{d^k x(t)}{dt^k} = Af(t). ", "20505ab4007e928f63cf5ca43ad33782": "\\deg^+(v).", "20507138f83081613cb2501cb625cea8": "K=\\frac{W_{i}-W_{f}}{24^{2}}", "20507620d3551f1aaf80633f91297c47": "Z_\\rho", "2050a14457b7612ba7933b13eaea90e2": "\\left(1-\\frac{2}{n}\\right)180", "2050d0571d5055fca960ce1f83039126": "F(x',t) \\ge F(x,t) \\implies F(x',t') \\ge F(x,t')", "2050fd4181d055173dfd65ec559c1777": "f_0(z) = z^2", "20513588752806e9bfb2ed1bc964b694": "G=Z/2", "20514493f67e911591b7b3bc7fac78a6": "j\\left(e^{\\frac{2}{3}\\pi i}\\right) = 0", "2051bd927e73bb3db18273fdc7a64a60": "\\; \\langle N, v \\rangle \\;", "2052810d60550741aab58c3fc2edf189": "\\textstyle{\\int_0^1 dx = 1},", "2052bb19c4653bc524568888f7899081": "u_s=0", "2052c78f76daae16a03d205108038d25": " \\psi\\ ", "2052dbb2e849f42de8efe914a950b565": "L(n) = \\sum_{k=1}^n \\lambda(k), ", "20533123b3fcf98d6fac64bc80c56124": "\\scriptstyle X \\,+\\, Y \\;\\sim\\; \\mathrm{Erlang}(k_1 \\,+\\, k_2,\\, \\lambda)\\,", "20536c3ed8571e0c5126720acd8a4706": "f(\\mathbf{x}_i)", "20537467225a9a19279f80961c13128b": "\n\\begin{align}\nd g(t) & = \\left( \\frac{\\partial g}{\\partial t}+\\mu \\frac{\\partial g}{\\partial S}+\\frac{1}{2} \\sigma^2 \\frac{\\partial^2 g}{\\partial S^2}+h(t)\\int_{\\Delta g} (\\Delta g \\eta_g(\\cdot) \\, d{\\Delta}g) \\, \\right) dt + \\frac{\\partial g}{\\partial S} \\sigma \\, d W(t) + d J_g(t).\n\\end{align}\n", "2053a0ea36275c2f63fa6a919e6a173b": "C_1, C_2 \\in \\mathcal C", "2053b64241fe97c69cff5ad5b63c4a39": "p_n=\\frac{100}{S_N}\\left(S_n-\\frac{w_n}{2}\\right)", "2053d93b55d0eb582645dffb756c1e74": " \\bar V_t = r_{t} + \\sum_{i=0}^{\\infty} \\gamma^{i+1} r_{t+i+1} ", "20542f4e8343028344b17de6e34792fa": "(A.1)\\quad \\theta_{(\\ell)}:=h^{ab}\\nabla_a l_b\\;,", "2054cd77dfaaf3a71e4ec3772945e6d5": "H_n = f^n(S_0)", "2054ddef5ff1df5a5e8c1d50a812fefe": "L(x)", "20551ac845e9324b14111cda5898b060": "S_{fg}(\\ell) = \\frac{1}{2\\ell+1}\\sum_{m=-\\ell}^\\ell f_{\\ell m} g^\\ast_{\\ell m} ", "20554bde52fe1dec5de93aba0fb4f06a": "\\mathrm{d} U = \\frac{\\partial U}{\\partial S} \\mathrm{d} S + \\frac{\\partial U}{\\partial V} \\mathrm{d} V + \\sum_i\\ \\frac{\\partial U}{\\partial N_i} \\mathrm{d} N_i\\ = T \\,\\mathrm{d} S - p \\,\\mathrm{d} V + \\sum_i\\mu_i \\mathrm{d} N_i\\,", "2055a02c19f5f22260c8171133f9040b": "\\tan y=x\\,\\!", "2055ce02dfb28c4f27f3380f091ddc6b": "z\\in\\mathbb{R}^n", "2055facda651ef041290496532fe596c": "\\scriptstyle\\bar c \\not = 0", "20564c16f30faa276c1b566a7b2526b9": "\\left|\\frac{f(0)}{f(1)}\\right| = \\left|\\frac{e\\alpha -1}{\\alpha - e}\\right| = 1.", "205658bd0aac797daddf05c5faf88368": "n^2 + \\frac{7}{4} T \\left( \\left\\lfloor \\frac{1}{2} n \\right\\rfloor \\right) + T \\left( \\left\\lceil \\frac{3}{4} n \\right\\rceil \\right)", "205666098a1de1940139b86d6e249010": "E = \\gamma(\\mathbf{u}) mc^2 ", "2056e378a89734e28902601172873c26": "E\\approx \\frac{\\pi Q}{R}+\\frac{\\pi}{3}\\lambda\\sigma_0^4 R^3", "2056fe3cfd812cca3ef8e6d81dfe7821": "h(x,0)=x", "2057444b6785795febd5c824279015a0": "S_0=\\epsilon, S_k = V_kS_{k-1},\\, k\\ge 1,", "20574c2c244f53789533eebcc7d722ed": "\\rho':\\bar{V}\\otimes V\\rightarrow L", "2057fb04ead3ad75b331a38b5dc60710": "\\deg(h) \\leq g", "205815fb0908beec0e4fdd62939363f3": "\\forall \\varepsilon >0 \\, \\, \\exists \\delta >0 ", "20582dc932a1afff55ccb94246448d18": "L_k(m-1) \\approx \\frac{m-1}{m} L_k(m)", "2058a496d4d2dc7fcb88a886e222012b": "\nC : y^2 + h(x) y = f(x) \\in K[x,y]\n", "2059266a7c166658db3899e6a76c7231": "\\begin{matrix}\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{f(0+h) - f(0-h)}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{f(h) - f(-h)}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{\\left\\vert h \\right\\vert - \\left\\vert -h \\right\\vert}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{h-(-(-h))}{2h} \\\\\n\\\\ f_s(0)= 0 \\\\\n\\end{matrix}", "2059b60122713e7c81b0928af2e20cd1": "(x_n)_{n\\geq 0}", "205a145f1e14c8745b9be077b7563952": "{{i}_{c1}}", "205a23b7bcfb4fb4267f9380ac41ab79": "y = x^{32} + x^{26} + x^{23} + x^{22} + x^{16} + x^{12} + x^{11} + x^{10} + x^{8} + x^{7} + x^{5} + x^{4} + x^{2} + x^{1} + x^{0}", "205a2fe5e15d3e5f38376ea01cb06d34": "(f) = nP + n\\overline{P} - 2n O", "205a45bcbb1927c5deb31e5f7dbd8406": "\\mathbb{Q}(\\sqrt[3]{2})=\\{a+b\\sqrt[3]{2}+c\\sqrt[3]{4}\\in\\mathbb{A}\\,|\\,a,b,c\\in\\mathbb{Q}\\}", "205aad06bd09fcb5edec890d9ea3be56": "\\mathbf z = \\boldsymbol\\eta + \\mathbf W_\\delta^{-1}(\\mathbf y - \\boldsymbol\\mu)", "205aaf7c16957725177f1f6b6c46806e": "[\\omega] \\in H_{dR}^p(M)", "205ad2f6a0ff4d1930feb1ac26e1fcbc": "\\textstyle n = \\mathrm{LCM}(9,31) = 279", "205b7801d2d1c683bee12560d59679e6": "x + y", "205b9653225df73515b0f66c3344dc3e": "V = \\left( \\frac{kba}{3} \\right) \\left( \\frac{Tn}{P} \\right)", "205ba2211474d3b0c7180d3e29f082f7": "q^{\\mathrm{II}}", "205ba9dea3392f4ff5f3395f5d0849e9": "(P \\and (Q \\or R)) \\leftrightarrow ((P \\and Q) \\or (P \\and R))", "205bd15e7103b85e0697fc4a787783f0": " \\mu l_idu_i =dr_i ", "205bdef5c5df475e646a6087adffe3b9": "a + (+ \\infty) = + \\infty", "205c152a628418c40f42cc6d4a2de52b": "e^{[-a_1+a_2]} \\sum_{i=0}^{\\lfloor x\\rfloor} \\sum_{j=0}^{[i/2]} \\frac{a_1^{i-2j}a_2^j}{(i-2j)!j!}", "205c805f50846e03136e47c04e5b7638": "{R_B}", "205cbe965c3f779f59983c5b2f1b5e08": "\\text{Accuracy}=\\frac{tp+tn}{tp+tn+fp+fn} \\, ", "205d1dedd9aa680e4933f92b47b24421": "x_p(t) = \\frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}", "205d2a2609ddb5dd05e8daa2dbcbfaa1": "\n \\boldsymbol{\\Omega} = \\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T\n", "205d4b8ec12621104046523548638d3b": "P\\in E(K).", "205d562b0cd80d24b9b8c008c634358f": "\\bar{c}_{k_0}(s;L)", "205d61cdc5551457dace656369b87957": "\\ln \\left ( \\frac{4}{\\pi} \\right )", "205d832ce327e2acad75558a3c308914": " \\mathit{V}", "205d8911e249c5139ec76303508bcbc3": "GG^T=G^T G=I_2.", "205dd32b23d692d77d5aeb8386bacb78": "\\frac{49}{64}", "205dd432cb13231391c269fc1d666dc6": "Z_0 = \\frac{E}{H} = \\mu_0 c_0 = \\sqrt{\\frac{\\mu_0}{\\varepsilon_0}} = \\frac{1}{\\varepsilon_0 c_0}", "205dff55cfcd6394bcb95b10d2fafea3": "\\frac{a+\\sqrt{b}}{c}", "205e707d9306d90601465607cb15130b": "\\pi(x \\,|\\, \\theta)", "205e7babefed275b957af261449bd284": "g_{\\mu\\nu} = \\eta_{\\mu\\nu}+h_{\\mu\\nu}\\;", "205ecdabb6e07c16fdb01089230a0ae5": "a=a_\\mathrm{then}", "205ed822adf8d4270adf45b6b06a936f": "\nk_0^2 = \\omega_p^2 +\\vec k^2\n,", "205ee73e08578d5214f3277847bdf752": " f(\\tau) \\propto \\tau^k, \\quad \\tau\\approx 0 \\text{.}", "205eea09c4c50d88ea1a0d64d968350d": " \\mathbf{F}=\\mathbf{F}_1+\\mathbf{F}_2 = 2(\\frac{\\mathbf{B}+\\mathbf{D}}{2}-\\mathbf{A})=2(\\mathbf{E}-\\mathbf{A}),", "205f482d85d3197002a4090d6397252f": "g(z) = z +b_1z^{-1} + b_2 z^{-2} + \\cdots", "205f62a21471a1ee45a17a8dd774a1c4": "f' = f\\left(\\frac{u'}{u} + \\frac{v'}{v}\\right).", "205f6b95f6eb654f95a8a1b765f4d9e9": "\\mathcal L\\{f*g\\}=(\\mathcal L\\{f\\})(\\mathcal L\\{g\\})", "205f7c4cb76a8d7a3ffed74fceab3f56": "JMJ=M',", "205fab0af12124d9aee0b08d8fb2bb6d": "v_j^{T}\\sum_{i} v_i^{T} x v_i", "205fac105ae50467cd7a88f130566933": "F = -C U^2 \\left(\\frac{2k}{rd^2}\\right) \\theta", "205fd7dbb011fd50e2f18e8ed22e205c": "w\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right)=\\exp\\left(-\\epsilon\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right)/k_{B}T\\right).", "20602149bc620308c95b7f1bf547f553": "RC = \\left ( \\frac{(2.4C+A)\\;(3C+B)}{9C} \\right ) - .9C", "20606089822f65139f3b4fb5d159d596": "S_0 = v^2\\, h + \\tfrac{1}{2}\\, g\\, h^2.", "20607b924915f2ad0f21bed0a36543b3": "cos(\\theta_E)=e^{-(d1+at)/T_1}", "206090b3503a2efab7d30e7a8edc7e8f": "(\\eta,\\sigma) \\in P_1 \\times \\Sigma", "2060b80d3753bdb5ebb8f103d73bea70": "\n \\cfrac{\\partial^2 \\varepsilon_{11}}{\\partial x_2^2}\n - 2\\cfrac{\\partial^2 \\varepsilon_{12}}{\\partial x_1 \\partial x_2}\n + \\cfrac{\\partial^2 \\varepsilon_{22}}{\\partial x_1^2} = 0\n", "2060fea633b9f238d10e6e0256e59cd6": "\\qquad \\qquad m_j\\frac{d^2\\mathbf{d}(jl,t)}{dt^2} = -\\sum_{j'l'} \\boldsymbol{\\Gamma} \\binom{j \\ j^\\prime}{l \\ l^\\prime}\\cdot \\mathbf{d} (j^\\prime l^\\prime, T), ", "2061500006bc9c20d7feeeb35b006b77": "\\mathrm{CaCO_3 + H_2CO_3 \\longrightarrow Ca^{2+} + 2 \\ HCO_3^{-}}", "20617328aa6593d5b7a83981483de541": "\n \\delta K = \\int_0^T \\int_{\\Omega^0} \\int_{-h}^h \\cfrac{\\rho}{2}\\left[\n 2\\left(\\frac{\\partial u_1}{\\partial t}\\right)\\left(\\frac{\\partial \\delta u_1}{\\partial t}\\right) +\n 2\\left(\\frac{\\partial u_2}{\\partial t}\\right)\\left(\\frac{\\partial \\delta u_2}{\\partial t}\\right) +\n 2\\left(\\frac{\\partial u_3}{\\partial t}\\right)\\left(\\frac{\\partial \\delta u_3}{\\partial t}\\right) \\right]\n ~\\mathrm{d}x_3~\\mathrm{d}A~\\mathrm{d}t\n", "2061b5417b4830010ba7230945d07c2b": "\\mathfrak{sl}_2 \\cong \\mathfrak{so}_3 \\cong \\mathfrak{sp}_1 ", "20632063e7966f0d8be28872d9e7ec41": "[0.16, 0.88]", "20636c8f6cb428ec06e12eb768b6b5a6": "\\mu_{0} \\vec{J} =\\frac{1}{r} \\frac{d}{d \\theta}B_z \\hat{r} - \\frac{d}{dr}B_z \\hat{\\theta} ", "2063afb4ceb2f7be9e69d670e24c35a3": " \\min_{\\hat x} \\max_{(\\Delta ,x) \\in G} \\left\\{ \\left\\| {\\hat x} \\right\\|^2 - 2{\\hat x}^T x + \\operatorname{Tr}(\\Delta ) \\right\\} ", "2063b0e291fb57c35bfcaffc3512f791": "w(i,j)= {1 \\over Z(i)}e^{-{{\\lVert v(\\mathcal{N}_i)-v(\\mathcal{N}_j)\\rVert}_{2,a}^2\\over h^2}},", "2063c1608d6e0baf80249c42e2be5804": "value", "2063d0affeef5d6910cbbcf34e7aaa03": " (a,b) = \\omega((-1)^{\\operatorname{ord}(a)\\operatorname{ord}(b)}b^{\\operatorname{ord}(a)}/a^{\\operatorname{ord}(b)})^{(q-1)/n}", "2064304e1ccd34ba21a833d7c0e53cac": " c_s = \\sqrt{k_B(ZT_e+\\gamma_iT_i)/m_i}", "2064ab0ee8c30521856289fb49b77303": "\\tau\\in(0,1)", "2064bee67c6253602fa1641939864f98": "\n R = I \\cos\\theta + [\\mathbf{k}]_\\times \\sin\\theta + (1 - \\cos\\theta) \\mathbf{k} \\mathbf{k}^\\mathsf{T}\n", "2064decc737bdae581967090428344e1": "(-1)^{\\text{signbit}} \\times 2^{-16382} \\times 0.\\text{significandbits}_2", "20651f27bdf0de3917790d052683c999": "\\ Q_T P_W Q_T \\psi_n=\\lambda_n\\psi_n,", "2065257cf6e0dcc272dcdd78ed42d498": " \\mathcal{F}\\left[\\frac{dW(t)}{dt}\\right](\\omega) = i \\omega \\mathcal{F}[W(t)](\\omega) ", "2065300734dd5a1601e0aa3f77b6a5e1": "\\begin{align}\n \\rho\\frac{\\partial^2 u_i}{\\partial t^2}\n &= \\partial_i\\lambda\\partial_k u_k + \\partial_j\\mu\\left(\\partial_i u_j + \\partial_j u_i \\right) \\\\\n &= \\lambda\\partial_i\\partial_k u_k + \\mu\\partial_i\\partial_j u_j + \\mu\\partial_j\\partial_j u_i\n\\end{align}", "206531fd76f97729d7a2fafb7c89b8fd": " A_1,A_2,\\dots. \\,", "20659aa667a2bb32f3f657ebb9ac94ae": "\\frac{\\operatorname{Li}_s(e^t)}{\\zeta(s)}", "2065a40804b013b6991324c6e537ee43": "\\scriptstyle M_\\text{B}(H)", "2065ccddc9d2e5a5775dba303175c899": "S_{-1}(x,y)", "2065dc32759240079154c80c68197d3b": " \\lfloor \\, \\rfloor ", "20662f05b5c034e0b13f26ebfb8ec126": "\\lim_{k \\to \\infty} c_2(k)= - 2.157782996659\\ldots.", "20663615ea3c66d0f3574986dd4d451a": "E_\\mathrm{h} = {\\hbar^2 \\over {m_\\mathrm{e} a^2_0}} = m_\\mathrm{e}\\left(\\frac{e^2}{4\\pi\\epsilon_0\\hbar}\\right)^2 = m_\\mathrm{e}c^2\\alpha^2 = {\\hbar c \\alpha \\over {a_0}} ", "206685a8b2ab441b326a9ae36f4b6a55": " {{documentation}}\n", "2066b304fa942ebd4a361706d666449f": "Poss(a,s)", "2066c1f1a8900c8221e8af51f739270e": "\\text{Pressure} = \\frac{\\text{force}}{\\text{area}} = \\frac{\\text{weight}}{\\text{area}} = \\frac{\\text{weight density} \\times \\!\\, \\text{volume}}{\\text{area}}", "20673b94d2fc45249d711e9537a5e199": "n\\ge k.", "20678e52a5a38e7cac895a95032560c8": "\\tau_b=\\rho g h \\sin{\\left(\\alpha\\right)} \\,", "2067ada9a78d139b52366b7d1fcde7f2": "f(\\theta): \\Theta \\rightarrow X", "2068415ef79ba0ebccd64a69eb4e39c8": "\\left( \\frac{2}{3} \\right) ^6 \\times 2^4", "2068f32facfa7b1aa7c05c8ebaca5598": "\\mathbb{R}^s", "206933123ea016e7a32f62a8228bed70": "2\\cos(\\theta) \\in [-2,2]", "2069341942491d969789bebf869bb515": "\nQ = \\frac{1}{2m} \\sum_{ij} \\sum_r \\left[ A_{ij} - \\frac{k_i k_j}{2m} \\right] S_{ir} S_{jr}\n = \\frac{1}{2m} \\mathrm{Tr}(\\mathbf{S}^\\mathrm{T}\\mathbf{BS}),\n", "206961687cca6a6d73589655b9c519cc": " \\acute{x}^{\\mu} ", "2069624b1edb6f4f49638ecba24b62bb": "\\begin{matrix}LU\\end{matrix}", "206967bde31f59e3738700c424e6aa60": "\\vec{A} \\cdot \\vec{p}", "206985becba81fae5c16d0a337205f03": "\\vert\\psi\\rangle", "206994b52dd2efafb56f5c7331505dc5": "p^2-p+1=r^2+2rkq+k^2q^2-r-kq+1=r^2-r+1+q(2rk+k^2q-k)=q(1+2rk+k^2q-k)", "2069b2da1dd33fee288586b45874989a": "\nz^{\\rho} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} + \\rho \\\\ \\mathbf{b_q} + \\rho \\end{matrix} \\; \\right| \\, z \\right),\n", "206a5ce347a6319110f01671ec55eda4": "1 \\le n \\le m^{\\varepsilon}", "206a879d3092a3a69c6333762fa43017": "\\overline M \\not= 0", "206ac5421ae2d42d7eec63d13e565774": "(pq)^* = q^* p^*, \\quad (pq)^{\\star} = p^{\\star} q^{\\star} , \\quad (q^*)^{\\star} = (q^{\\star})^*.", "206ad198647ed7b157cf69f8ddae5815": "U_sU_\\omega", "206b07dd0662a648b57cf38257dad50b": " \\left( \\sqrt{\\frac{\\nu}{\\nu+1}}\\mu,\\,\\sqrt{\\frac{2\\nu}{2\\nu+5}}\\mu \\right)", "206b0d88fdb7f78ddb24fa76ed96d514": "\\textstyle A>0", "206b5818102fa109c532a00771827df5": "\\left( \\frac{1}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "206b8e2aaa06147783d6b2b1a74fe5e7": "Y(\\mathbf{x}) = \\int_{S^{n-1}} Z^{(\\ell)}_{\\mathbf{x}}(\\mathbf{y})Y(\\mathbf{y})\\,d\\Omega(y)", "206bbd1a2a64e8b7fe2b83e7be3ddf0a": "\\nabla \\boldsymbol{v}(\\boldsymbol{x},t)", "206bc708461fb2d643b849a51e05891d": "\\forall a, a \\nrightarrow a", "206c14f3c74acb981b8a97bd2f9c6778": "a_{\\ell}\\geq 0", "206c3e312d7eca5ffef26210af670ba2": "(B y + \\beta)^n - B^n y^n \\le B^n r + \\alpha.", "206c4038a3708ae85c43a6908d95a0b4": "(A \\and \\neg B \\and \\neg C) \\or (\\neg D \\and E \\and F)", "206c439059aa2f3b67e467160bd4fd32": "\\| u \\|_{BV} := \\| u \\|_{L^1} + V(u,\\Omega)", "206c6ac133a0bbdea3c46da831e54a1e": " \\sum{c(S_i) \\cdot x_i} \\leq B ", "206c78301070322bb1a720756bb9233a": "\n\\{a^\\dagger(\\mathbf{k}),a^\\dagger(\\mathbf{l})\\} = 0, ", "206d5d2d5d9e396b2bb2d4cb7adcb550": "1_{5}\\rightarrow (1,1)_1", "206d6bab922b99c880f70866794b2981": "\\bar{6}_H", "206d6fd394df9d600c8d4f2543bc7b2b": "(\\operatorname{artanh}\\,x)' = { 1 \\over 1 - x^2}", "206dec5cf451ab5a8246de91d8423db2": "L_{q} = 0, L_{qq} \\geq 0, |L_{qq}| \\geq |L_{pp}|.", "206df40ba048f9db44de55a465eb0558": "\\mathrm{SL}(n,\\mathbb C) \\times \\mathrm{SL}(n,\\mathbb C)", "206e195a0f979781f522f00ccfd20dc9": "2^{f(k)} \\cdot \\text{poly}(n)", "206e5823b03c7d1e7e410c9bbdf37a3c": "V_1,V_2,\\ldots,V_k", "206e7271cd45995cabb08ce22fcfa34b": "\\mathrm{OSR}\\,=\\,\\frac{f_s}{2f_0}\\,=\\,\\frac{1}{2f_0\\tau}", "206ead868b0cca3b1a4361b7fdb1ab7c": "n = \\frac{\\sqrt{8x+1}-1}{2}.", "206f456fce9dbe75a05e706c2209fc19": "\\vec{r} = (\\sigma_1,\\sigma_2,...,\\sigma_N)", "206f631c216cbe73637e2c316dd93e9e": "x^4+360*x^3-270000*x^2+20736000*x+1866240000=0", "206f852748d691594e2c56c687a1a019": "-\\left | y \\right \\vert \\leq \\frac{x^2 y}{x^2+y^2} \\leq \\left | y \\right \\vert ", "206f9a4c66837577bcd76cb05206aa92": " |p \\rangle ", "206fb2258d45d00879d85a9adcc03109": "\\Delta p_{\\text{B}} (x) = D_1 x + D_2", "206ff7f7b3a92ebbe4e47954fe123ea9": "n=(n_1,\\cdots,n_k)", "207037226fa58e4d57d7a77bc9e0302b": "A B C A^{-1} B^{-1} C^{-1}", "20707bf68f4fb4abb0e76ad879d53443": "\\theta = 45^\\circ\\,\\!", "2070bc588b47e00a8993466a73acbe87": "T = \\frac{e^{-2\\int_{x_1}^{x_2} dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}}}{ \\left( 1 + \\frac{1}{4} e^{-2\\int_{x_1}^{x_2} dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}} \\right)^2}", "2070bd63aa1c00592882b4552d3c6259": "\\overline{P}(Cl_3^{\\geq})=Cl_3^{\\geq}", "2070c9d3875256af2db4f9dd7714abd8": " \\frac{dS(t)}{S(t)} = \\mu\\, dt + \\sigma(t)\\,dz_1(t)", "2070cb4e8fa2d2ff1f81c2dc25c99357": " \\pi: N \\to N ", "2070df98a5a32adb6a58c24fe0711daa": "\\tau_{ind}\\left(\\omega\\right) = -\\Gamma_{cap}(\\omega)", "20714ab30e31ded7a10b95dda6e5e641": "\\pi_n", "207152a8f129786e2397aaf321a1300e": "\\mathcal{Q}_{\\mathrm{Hur}}=\\mathbb{Z}[\\eta][i,j,j'].", "2071617678698e26842fc35370cbf2b0": " G = \\oplus G_n ", "207169934204c3b2fe19369e648ea9f8": "\\lambda_3 = 1,", "20716d61b83df5b4491e8af347e446ae": "d[x,y] = xdy-ydx~", "20719dba0b11ea8c7110f78a229023ad": "J_{X_t}", "2071e4ac842502f8495b6a9295aa9890": "\\frac{(1 - z^3/6)}{(z - z^{2}/2)^2} + c", "20721b29492ad3e86ee8c4c252994e60": "\\beth_0", "207289a61ac1a0b28ac396b19be58e3d": "\n\\mathrm{tr}(\\Lambda) = \\lambda_1 + \\lambda_2 + \\lambda_3\n", "2072a3bd5ae2ccadfe1c5823abf8cea1": "D(t)= \\begin{cases}\n1/m & \\text{if } |t| \\le 1 \\\\\n0 & \\text{otherwise}\n\\end{cases}\n", "2073054630cb7e10828f84bd2e27b7a7": "Q_n=\\frac{\\pi}{n \\tan \\tfrac{\\pi}{n}}.", "2073266755beab8dcc86a74a2cf0815f": "[p_1] - [p_0]", "207382741d990214940a479473be1455": " \\mathrm{Force} = \\mu \\frac{d^2Q}{dt^2}", "207387eaeefe20c9b2db223df88ecfe5": " g_{ij}(q^i,q^j) = \\cfrac{\\partial x_k}{\\partial q^i}\\cfrac{\\partial x_k}{\\partial q^j} = \\mathbf{b}_i\\cdot\\mathbf{b}_j ", "2073e1a8b5f6bbfd69ef66bc1378e427": "M(q)", "2073eb7da1bbb5c8a78332c8cb5adb8b": "\\mathbf{P} = \\hat{\\mathbf{a}} + \\alpha \\mathbf{e}_\\infty ", "2073f901fe08a2b05a0e7a4bd5ad6850": "\\langle\\rho F,\\le\\rangle=\\langle F,R\\rangle/{\\sim}", "207414d039b69d01f9cac3feea2fa810": "f(x) \\rightarrow u^T\\nabla f(x)", "20744fde0fdbe04fdf0431ac73fb66e1": "V_{nk}", "2074550af4223c91d90914db05822ee6": "\\phi_0(q) = \\sum_{n\\ge 0} {q^{n^2}(-q;q^2)_{n}}", "20745c1d00357ecc3fc2bd9c036ad3e6": "g(r) = h(r) = \\sin(r)", "20746c5bb5a448a879fef06531fd81d8": "\n\\begin{pmatrix} \n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\\\\\n\\cdots & M_{ij} &\\cdots & M_{il} & \\cdots \\\\\n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\\\\\n\\cdots & M_{kj} &\\cdots & M_{kl}& \\cdots \\\\\n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\n\\end{pmatrix} =\n\\begin{pmatrix} \n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\\\\\n\\cdots & a &\\cdots & b& \\cdots \\\\\n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\\\\\n\\cdots & c &\\cdots & d& \\cdots \\\\\n\\cdots & \\cdots& \\cdots&\\cdots&\\cdots\n\\end{pmatrix}\n", "20748d644d61c6b2e159d34b7bb8602c": "qSC_{p}", "2074d39ef94b3dd4c74e082594324705": "[a_1\\cdots a_m,b_1\\cdots b_n]=\\sum_{i,j}(-1)^{i+j}[a_i,b_j]a_1\\cdots a_{i-1}a_{i+1}\\cdots a_mb_1\\cdots b_{j-1}b_{j+1}\\cdots b_n", "2074eba49c67c835d4a3f9d123543181": "h_\\lambda (X_0)", "2074f85dbbc2bc49e580c90bb4faa6f1": " \\dot{A} = (\\sigma - \\sigma_0) A - |A|^2\nA", "2075641805a5a19e5148a44be83ceb16": "(\\mathfrak{k}, \\mathfrak{p})", "207566963d710aa1e1649d5b692647cc": "\\hat{\\gamma}_3", "207598acc692f8e1ffca65d23bbbeb9e": " \\frac{1}{2}[(\\kappa+1) \\theta~\\sin\\theta + \\{1 + (\\kappa-1) \\ln r\\} ~\\cos\\theta] \\,", "20759e21ff650169d9316fa62c5a6415": "\n \\begin{align}\n &\\frac{\\partial^2 M_{11}}{\\partial x_1^2} + \\frac{\\partial^2 M_{22}}{\\partial x_2^2} + 2\\frac{\\partial^2 M_{12}}{\\partial x_1\\partial x_2} +\n \\frac{\\partial}{\\partial x_1}\\left(N_{11}\\,\\frac{\\partial w}{\\partial x_1} + N_{12}\\,\\frac{\\partial w}{\\partial x_2}\\right) +\n \\frac{\\partial}{\\partial x_2}\\left(N_{12}\\,\\frac{\\partial w}{\\partial x_1} + N_{22}\\,\\frac{\\partial w}{\\partial x_2}\\right) = P \\\\\n & \\frac{\\partial N_{\\alpha\\beta}}{\\partial x_\\beta} = 0 \\,.\n \\end{align}\n ", "2075b9dae17795ff2f98b7b33aeb8c8a": "P^2", "2075d27e08213919ca36d7e20259c2c4": "\\it{n}", "2075d5ae7df6b71f3af7407be8430e25": "\\varphi:U\\to \\mathbf{R}^n", "2075ddd04285d19e8a3c75c4512126c5": "(s^2 - s_1^2)(s^2-s_2^2)(s^2-s_3^2).\\qquad (2)", "2076531fb3d1ad20a62189f5a7cc599c": "\\, e^{t^\\mathrm{T} \\mu + \\frac{1}{2} t^\\mathrm{T} \\Sigma t}", "2076a8c0fd580b12f5d4e122293a0672": "Z \\to Y", "2076d0174cd9eeab2f1214c44ebfe3b9": "\\{x|x<2\\}", "20771de5872107097a8b16ef61dc7c84": "\\mathrm{L\\Pi f}", "2077245a1790b5d220de388d5fdc3a18": "\n\\begin{array}{lll}\n\\omega(0) &= W_0(1) &\\approx 0.56714 \\\\\n\\omega(1) &= 1 & \\\\\n\\omega(-1 \\pm i \\pi) &= -1 & \\\\\n\\omega(-\\frac{1}{3} + \\ln \\left ( \\frac{1}{3} \\right ) + i \\pi ) &= -\\frac{1}{3} & \\\\\n\\omega(-\\frac{1}{3} + \\ln \\left ( \\frac{1}{3} \\right ) - i \\pi ) &= W_{-1} \\left ( -\\frac{1}{3} e^{-\\frac{1}{3}} \\right ) &\\approx -2.237147028 \\\\\n\\end{array}\n", "207775be4940d5b794e994c9886ceabd": "T_f x", "2077a4df35530fa21c75805d81eb3822": "(u,v)\\in E", "20781fbb3c21bd05b096c1f105d7b2c1": "T_B = \\frac{K V}{k_B \\ln \\left(\\frac{\\tau_m}{\\tau_0}\\right)}", "20782b093c43ae61ddfa354b15d14a61": "P_\\pm", "20782b8c73d7645b74a76f7d240ad4c5": "\\sinh x = - {\\rm{i}} \\sin {\\rm{i}}x \\!", "2078432ee5ce779cf504b669e32e6c83": "\\scriptstyle\\mathbf J = 0", "20784f49791eb12824dffcbbe64d1879": "f[x_0,\\dots,x_n]x^n", "20785b1fd4010c141d303552d0d56b4c": "V= \\{a,b\\}", "20787d0db6fe6ce4e643f6163582b8b1": "E^{(1)}_{lm}", "20788323e35116f5b75c24b873f87182": "H(s)=\\frac{K \\omega^{2}_{0}}{s^{2}+\\frac{\\omega_{0}}{Q}s+\\omega^{2}_{0}}.", "20788529a74289c9aa5830c206e5e389": "\\phi(t)\\rightarrow x_1\\quad \\mathrm{as}\\quad t\\rightarrow+\\infty", "20788fc6e749c259bd34532804fa3931": "p_{X} (x) = \\mathbb{P} (x \\in X)", "207901533929b2bb847c5bec50369024": "\\left\\{ \n\\begin{pmatrix}\na&b\\\\ 0 & c\n\\end{pmatrix}\n\\ :\\ a,b,c\\in\\mathbb{C}\\right\\}", "207911b636e27a8100352b0a5865afea": "\\tfrac{3(K-\\lambda)}{2}", "207958891181c4c70f6dc50f54589c42": "\\mathrm{-C(=O)-\\left(CH_{2}\\right)_{12}-CH_{3}}", "20797bbaeafe6f03dc99b2407c05ff82": "x^{q^{i-1}}\\mod f^* ", "2079ce51aaff670dee5c28e7b6f2b617": "\\textstyle p", "2079d20e2f6c414ecca721f4e32a19bf": "C^c_{\\ ab}", "207a144a3a9c97669246f0c5b0775f38": "x_{l}", "207a2029feaca2d5b411a99129e09fe8": "\\begin{cases}\n\\overbrace{ \\begin{bmatrix} \\dot{\\mathbf{x}}\\\\ \\dot{z}_1\\\\ \\dot{z}_2 \\end{bmatrix} }^{\\triangleq \\, \\dot{\\mathbf{x}}_2}\n= \n\\overbrace{ \\begin{bmatrix} f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_2 \\\\ z_2 \\\\ 0\\end{bmatrix} }^{\\triangleq \\, f_2(\\mathbf{x}_2)}\n+\n\\overbrace{ \\begin{bmatrix} \\mathbf{0}\\\\ 0\\\\ 1\\end{bmatrix} }^{\\triangleq \\, g_2(\\mathbf{x}_2)} z_3 &\\qquad \\text{ ( by Lyapunov function } V_2, \\text{ subsystem stabilized by } u_2(\\textbf{x}_2) \\text{ )}\\\\\n\\dot{z}_3 = u_3\n\\end{cases}", "207a62a838ea9503c8526235c3f260c4": "I_c= K_L a^2 \\pi \\frac{U_{str}}{L}", "207b37f465033f6f26a6586c48d143e7": " I_5 = \\int \\cos^5 x dx . \\,\\!", "207b9ac161ce3de9b4eb309be8b9ebd5": "p_H(x|\\alpha) = {\\displaystyle \\int\\limits_\\theta p_F(x|\\theta)\\,p_G(\\theta|\\alpha) \\operatorname{d}\\!\\theta}", "207bdc22da9732fe0057ade0c4ff2912": "R^\\rho_{\\sigma\\mu\\nu} = \\partial_\\mu\\Gamma^\\rho{}_{\\sigma\\nu}\n - \\partial_\\sigma\\Gamma^\\rho{}_{\\mu\\nu}\n + \\Gamma^\\alpha{}_{\\sigma\\nu}\\Gamma^\\rho{}_{\\alpha\\mu}\n - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma} ", "207bea04dc6e1fb45324059a327c5413": "\\scriptstyle \\boldsymbol\\omega_s", "207c36c980f7436de6f3cbda26ff14b4": "T_G= T_H \\cdot T_{H'}", "207c95d1571012ca5f2b969c28c854b3": "quot = \\frac{V}{2s+1}", "207cb816461e0107ac882ae191f7b577": " m_{i+1} = \\frac{m_i}{2} = \\frac{m}{2^{i+1}} ,\n", "207cbde45d6daf5462111dc4909fbea9": "\\text{Per-unit volts}=\\frac{\\text{volts}}{\\text{base volts}}", "207cce2fa09af07a07d4b07a18fd18bb": "[a \\cdot D, \\, b \\cdot D]F=-(\\mathsf{S}(a) \\times \\mathsf{S}(b)) \\times F.", "207cee98846317c6585e709825debb95": "F_\\omega = \\underset{1 \\leq i}{\\bigcup} F_i", "207d00ce1e082e33e804f9581229e848": " q(x) ", "207d04ab5fc1d23d0005098e2c08be74": " \\gamma_p ~ = ~ k_0 ~ \\cos \\theta_0 ~ + ~ \\frac{2p\\pi}{l_z} ~~~~~~~~~~~~~~~~~~~~~~(2.2c) ", "207d9f091422a706b30b61fbb183dd43": "f(n,\\mathrm{lcm}(a,b)) = \\mathrm{lcm}(f(n,a),f(n,b))", "207da16cc8f9d9fa927af4cceb570577": "T_i^i", "207db55304371765030f0620bac2e291": "E_6, E_7, E_8, F_4,", "207df3eb785318293fed2c850b270633": "P(b)", "207e5272447a09e113d098df65d1220b": "X^-", "207e746526e5222ecbf49a6452561d50": "\\langle \\partial_{t} u , v \\rangle = \\langle \\partial_x \\left(-\\frac{1}{2} u^2 + \\rho \\partial_{x} u\\right) , v \\rangle + \\langle f, v \\rangle \\quad \\forall v\\in \\mathcal{V}, \\forall t>0", "207e7b7bce2323d6b9ac928479641dc5": "c_2=1.4380 \\times 10^{-2} \\text{m·K}", "207eb0107485f9dcae3e744c56b4b740": "\n\\begin{bmatrix}\n0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0\n\\end{bmatrix} \\qquad (\\text{permutation of coordinate axes})", "207ec0782f5d52153e5c67480a137c58": "a_{0,i}=H(X_i)", "207eca66b1e44d27964af57d63bd77c4": "x (2x + 3)", "207ed1b47adb503f5641942bbd58345a": "g: X' \\to P", "207ed940dad9d4051b5db5f65856fd2d": "\\mathfrak{gl}_n(\\mathbb{C})", "207eea647354ee56fa464951d421977c": "\\langle x,y\\rangle := y^*M x", "207eea7ca4cab14e6ba251be11386ae4": "r = \\frac{t}{\\sqrt{n - 2 + t^2}}.", "207f0eb0a76c4d6e97c08be9a4bb364c": "p\\in M\\, ,", "207f357a51ff79289b5baec16d8d1e9d": "\\varepsilon -A", "207f56123e92280bae7a2c0074908549": "u,v,\\dots", "207f576a883d318726577787d6486ea0": "f(X\\cup \\{x\\})-f(X)\\geq f(Y\\cup \\{x\\})-f(Y)", "207f5a50db1cfe9e29ded884d412b99f": " \\langle(1 \\; 2) \\rangle = \\{id,\\; (1 \\; 2)\\} ", "207f8f62269e90c1360d8f78e99262b2": "(i,a,j)*(k,b,n)=(i,a p_{jk} b,n)", "207f90ce614be95ba66ee1e78ff7044c": "\\frac{dx^{\\mu}}{d\\tau}", "207fcfc23446f43162e637c77dde949a": "n^{(\\lambda)} = 4\\, .", "207fd2354f3cb9caaba67edf2a151876": " \\{ \\mathbf{e}_1,\\mathbf{e}_2,\\mathbf{e}_3 \\} ", "207fff85ab28770d611c5b760dc1cce1": "\\displaystyle{f^2 -(Hf)^2=(f_+ + f_-)^2 + (f_+-f_-)^2 =2(f_+^2 + f_-^2)=-2iH(f_+^2 -f_-^2)=-2H(f(Hf)).}", "2080029f4d2e6a034da928677cb4bf69": "\\scriptstyle n\\log_2n", "208080d6d26cc44588610c941efa5a2e": " (\\mathbb{Z} S)_n = \\mathbb{Z} \\langle S_n \\rangle, ", "20809753072d8216d2d267a6975d3f59": "\\begin{bmatrix}\n1&0&0&0&1&1&1&1 \\\\\n1&1&0&0&0&1&1&1 \\\\\n1&1&1&0&0&0&1&1 \\\\\n1&1&1&1&0&0&0&1 \\\\\n1&1&1&1&1&0&0&0 \\\\\n0&1&1&1&1&1&0&0 \\\\\n0&0&1&1&1&1&1&0 \\\\\n0&0&0&1&1&1&1&1\\end{bmatrix}\n\\begin{bmatrix}x_0\\\\x_1\\\\x_2\\\\x_3\\\\x_4\\\\x_5\\\\x_6\\\\x_7\\end{bmatrix}\n+\n\\begin{bmatrix}1\\\\1\\\\0\\\\0\\\\0\\\\1\\\\1\\\\0\\end{bmatrix}", "2080e5fb6a39b4b63f56e0ae19e42b59": " A \\ ", "20810b9d5d8241560758acb05ec26a10": "\\ {a} \\sin \\theta = n \\lambda ", "2081129cb58749c4c1fd729f2e9c0316": " \\textstyle P+B=k \\cdot (W+M) \\,\\ ", "20815a319b71e93ce06725170fb27b5a": "\\mathrm{S} = \\left\\{ \\mathbf{x} \\in \\mathbb{Z}_{0+}^c \\, : \\, \\sum_{i=1}^{c} x_i = n \\right\\}", "20816b42df35c3c31c997e2cc29298f9": "\\lim_{x\\rightarrow0} {}^{n}x = \\begin{cases} 1, & n \\text{ even} \\\\ 0, & n \\text{ odd} \\end{cases} ", "2081ad12a23f4b0338e234e9ccc335d2": "\\frac{1}{2}m\\dot{r}^2 = E - U_\\text{eff}(r),", "2081b8a37cccaf4c1fccb8cc98fb8181": "\\int_1^e \\frac{1}{t} \\, dt = 1.", "2081c4b1f0106ed4730bc6c98d3534fa": "\\sum_{j\\in J} r_{ij}x_j = m_i", "2083002807c96abfd625edbbf1d8aa8b": "[F]:=\\big\\{([X],[F(X)]):[X]\\in[\\mathbf{T}]\\big\\},", "20832663121267cf1e870a93012b852d": "R=1+\\frac1{|a_n|}\\max\\{|a_0|,|a_1|,\\dots, |a_{n-1}|\\}", "20832717ce694af3178f64db70967e96": "{(x_{i},y_{i})_{i=1}^n} ", "2083ff843480dc3cf763b67d9c1e84e6": "\\mu_n = \\{x\\in K \\mid x^n =1 \\} ", "20845714172d35a29ea1354b1cd28128": " ta(s) ", "20847b3e450b7099c538aa18704dcd3d": "\\; \\Psi (f) = \\sum_i f_i F_i.", "208483e1b22f0faac3568c16f4d0a830": "f_{pegtop}(a,b) = (1-2b)a^2 + 2ba", "2084ee7ae2ffd8c3b03beec4c26148fb": "A(c)=\\alpha", "208506a77740ceb65deabe81b8830464": " \\sqrt{\\pi/2}", "20856da3bd9d0da62cc5ed62b53c5915": " |6x^4 - 2x^3 + 5| \\le 13 \\,|x^4 |.", "2085774286f7871235c13a7175e0f500": "W(\\mathbf{x}_w^{(k-1)})", "2085d15bd12658fa6b669fd344bb4d3e": " \\beta(S) = U S U^* \\, ", "20860728c901636fb62b8c24a9fae162": "\\Delta W > 0", "20862345fd71eea49f2f0c77b4cd9022": "\\varphi(\\cdot,t)", "20865956f9e89cba1d9a2d3e93a071c9": "O(\\log^5 n)", "20867301d8255889fb1e2d828c4040a1": "|E_{+}\\rangle", "20868fa29dfc38ac154b8ef762766b41": "P_1", "208694c8fef4503dc4b4bb316d72aa36": "P\\in \\mathcal P", "20872ae883c008735d4ba2f01caa2645": "N(E) = 8\\sqrt{2}\\pi m^{3/2}E^{1/2}/h^3\\,\\!", "20874875b73599d964f317c9131b7c9d": "T_\\text{S} = \\sqrt{\\frac{c^4 e^2}{G (4 \\pi \\epsilon_0) {k_\\text{B}}^2}}", "2087c4d19d3950cdf80c1bd41e55d685": "E=\\int_{\\R^n}[\\vert\\nabla\\theta\\vert^2+f(\\theta)]\\,d^n x", "2087c90f4a5ea9520f71a5adf6a10671": "(w_{j_1}, v_{i2})", "2087dee0ddeb74165914a0235a1dedc0": "q = || \\mathbf{u} ||", "2088415d40f5bb6d98f073b2adc5d0f8": " D_r = 0% ", "2088a4950220c4f835240ee5f8f2ae39": "A =\\frac{\\ell a^2 + ma + n}{(a-b)(a-c)};", "2088ce368027c19e7bbcc881bc17096b": "4(n/4)^{2^d}", "2088ee0289517c52ce0909b2eb04aeb1": "\\,_2F_1(a,b;c-1;z)-\\,_2F_1(a,b+1;c;z) = \\frac{(b-c+1)az}{c(c-1)}\\,_2F_1(a+1,b+1;c+1;z)", "2088f5efc9e499ec688808ab22345f66": "|x - y| \\cdot |u - v| = 0 \\!", "20893a79ab3703cc4306a163d28457f3": "C_{t} = C_{0} + (C_{ss} - C_{0})*(1 - e^{-k_{e}t})\\,", "208962ff8e78316fb4bfac7fb1477315": " |e_d| = \\alpha ", "208977d2d0f39115cfb566943cc7bae1": " \\Pi(h) = - {dW \\over dh},", "2089d8bf9f740c545594ad0084769d20": "R_{\\text{b}}", "208a2e1e25a1d2471a9ec43d12904f0a": "\\nu(H)", "208aa0b1bf7a5c9bdc99942a5a975c8e": "\\overrightarrow{C_i^\\alpha C_{i+2}^\\alpha}", "208aac8531e8b12f465db7687ba06536": " v_t = (1/m) \\sum_{i=1}^m v_i", "208ab7ff1846bd299e38eafff2074ab5": "v_{3}'=-q_{2}^{*}\\,v_{1}+i\\,\\xi\\,v_{3}.", "208abb92643ee22b27ebe8e0c41cfe96": " \\operatorname{de-lambda}[M\\ N] \\equiv \\operatorname{de-lambda}[M]\\ \\operatorname{de-lambda}[N] ", "208ae34312e0b8353e975021e5c0c117": "\\scriptstyle a/0 = \\infty", "208afe57c5c87b32248a9ba3ff7a902f": "\\ell^1({\\mathbf Z})", "208b1a2faf3b3ce6d1771f818d553f79": "2^{5/12} = \\sqrt[12]{32}", "208c0ca2c177cc62481f57d1ea37e8b3": "\\mathrm{GF}(p^{m_i})", "208c1a15c01e03fbaf20b24adb9ec893": "(\\xi_i,\\zeta_i)", "208c205b3a6a8dcdded0c9965d794513": "\\zeta_T(s)= \\operatorname{Tr}(T^s)= \\sum{\\mu_i^s}", "208c463516de35fd37c9c05690481754": "\\mathbf{x}\\,\\!", "208c8871e5341088d747ebfa33200ccb": "I = 1.1 \\times I_\\mathrm{o} \\times 0.56^{(AM^{0.715})} \\,", "208cf1f183f6deb761a946d3f9f54d83": "\\hat{\\Gamma}(G,H) ", "208d043b83da5b0f7a5e20c0d8365066": "{}_{\\ 81}^{208}\\mathrm{Tl} \\xrightarrow{\\beta^-\\ } {}_{\\ 82}^{208}\\mathrm{Pb}\\ \\mathrm{(3\\ m,\\ 2.6\\ MeV)}", "208d20f4755e1076b7887452984c035d": " c_{s} ", "208d41fe62ce2fd1c59a919be8cdc5da": "V_t = V_d", "208d62271248374409db31e13a8236ab": "(X_{i},Y_{j})", "208d7198d8fe154f9818bf70544cd742": "\\|f\\|_{p,w} = \\sup_{t > 0} ~ t \\lambda_f^{\\frac{1}{p}}(t)", "208dabbf44304ced4c124225fc8dcec0": "\nx = \\cfrac{1}{1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cfrac{a_4}{b_4 + \\ddots}}}} =\n\\cfrac{1}{1 - \\cfrac{r_1}{1 + r_1 - \\cfrac{r_2}{1 + r_2 - \\cfrac{r_3}{1 + r_3 - \\ddots}}}}\\,\n", "208db2e0f67134f3139d31e6d42a2d32": "(b_{14}-a_{14})+(b_{15}-a_{15})", "208de4a9da7c299c8297125594a9abcc": " p^* = \\frac{p L}{\\mu U} ", "208dfb484df3a17a2283ddf04dd68fef": " \\lambda_n ", "208e54fbbe29178500b9cff00644664b": "\\mathbf{x} = \\begin{pmatrix} x_1 \\\\ \\vdots \\\\ x_n \\end{pmatrix} \\mathbf{r} = \\begin{pmatrix} r_1 \\\\ \\vdots \\\\ r_t \\end{pmatrix}", "208e74d334ffb4457c1d81e6d19665b7": " ({X} + i{P})", "208eceaa97e484a14d2ead1ef0d5c170": "u_z(\\boldsymbol{x},z,t)\\,", "208f14f63c4e5e316cd158fc16021a3a": "z=C_1-C_2x, w=2C_2y", "208f606cd2ac0c09099a90673fba102e": "\nK_H(x') = [x' - x_0(T)][1 - \\exp(-\\beta u_0)]\n", "208f9b6d94b8c1ba746970adebfe72be": "\\alpha_{-\\sqrt{s}}\\beta_{\\sqrt{s}} \\alpha_{\\sqrt{s}}(x)", "208fa5fb0c777e3607d8aaae87f04e46": "r_{a,t} = \\alpha + \\beta r_{b,t} + \\varepsilon_t", "208faa0748518dd4a15ec03052c3ef40": "ds^2 = \\left( \\frac{2}{1 + x^2 + y^2} \\right)^2 \\, \\left( dx^2 + dy^2 \\right) ", "208faf50c7d0d6ff541a5acb72dfe9a0": "x\\,,", "208fe5ba27bf9b429f3e9038eb37f418": " P_g ", "208fef0aa47c42303fc43171e9208bd6": "G^{-1} = G, G^0 = G_0", "209014186ceb22097909ea08902ad592": "\\log P = A-\\frac{B}{T}", "2090ccc580c1018be58d0e475731ed4b": "I_{N} = \\begin{bmatrix} 1 & 0 & \\dots & 0 \\\\ 0 & 1 & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & 1 \\end{bmatrix}", "2090dfd4ebff94e15276443d70bce100": "\\frac {\\Delta Y} {\\Delta L}.", "2091197a46f36821862906517d736b23": "f(x_i,\\boldsymbol \\beta+\\boldsymbol \\delta) \\approx f(x _i,\\boldsymbol \\beta) + J_i \\boldsymbol \\delta \\!", "20913802caafee6a96a2bd74a09d4404": "\\Delta H^*_{uv}", "20914a8b61d19586f9299be423c38444": " \\beta = \\frac {2 \\pi}{\\lambda} ", "20916dabf96955e868230d6e1bcf2256": "\\arcsin x ", "20917dd7710034fbb4b8225d13f919d0": "\\lambda_n = {2L\\over n}\\,,", "2091903a94ea18141c35943959df7409": "R_{2}", "209190b60ca73a4441c11ae1414a845c": "\\hat\\beta = \\frac{\\tfrac{1}{T}\\sum_{t=1}^T (z_t-\\bar z)(y_t-\\bar y)}\n {\\tfrac{1}{T}\\sum_{t=1}^T (z_t-\\bar z)(x_t-\\bar x)}\\ .", "20919f1863f9c6cb0e51c24baba6adc6": "s-c", "2091a08f9ec1dc2d7c9a5439c4ce0ba0": "a^{N-1}_p \\equiv 2^{11350} \\equiv 1 \\pmod{11351}", "2091a58e31371b3fb52ca32c254a2d1e": "x^k_R", "2091dd23b300678ba522157d608e8b56": "{\\overline{b}}", "20922aeebb8b2897957287c81f9eecbf": "A(\\lambda_i)", "209260bc81cdac9309a866a1bd753b29": "orb_G(f).", "2092ba9d89943bba987adcf439d63ae2": "1) \\ \\text{Flow}= \\sin( 5t ) ", "2092bc5439fcc9101275015c68cfe80b": "x \\leftarrow w \\rightarrow y", "2092db726c76d96d6498bbafd938b1b3": "T \\colon X \\to Y", "2092fb7abc34b184470968282565e956": "w=e^{\\pi {\\mathrm{i}}z}\\,", "209300826abe7aaadad6f689b8e3a3cc": "(\\epsilon \\times c \\times h)", "20935a593906878a804568b25caa17c5": "\nR_\\mathrm{eq} = R_1 + R_2 + \\cdots + R_n.\n", "2093fa8bfe7f315d11a92ca192012fb6": "d\\vec S", "209413243fe2fab39f5a44771dcc5eb1": "\\scriptstyle \\Delta G \\ < \\ 0", "2094deafa65af634815619d9a44c1d61": "\ng_{ij^*}=K_{ij^*}=\\frac{\\partial^{2}}{\\partial z^{i}\\partial \\bar{z}^{j^*}}K\n", "20950550c4f35405aa7e23ce1326af79": "x_1 ... x_n", "2095511883bc4525333f9c17a5859189": "{A_{final}}", "20955fbbb092dbb1b32bcc23dcb8eb5a": "h(t - \\tau ) = \\frac{1}{{m\\omega _d }}e^{ - \\varsigma \\omega _n (t - \\tau )} \\sin [\\omega _d (t - \\tau )]", "2095c87259ad731c60cc913b472d1632": " \\operatorname{fact}(n) =\n \\begin{cases}\n 1 & \\mbox{if } n = 0 \\\\\n n \\cdot \\operatorname{fact}(n-1) & \\mbox{if } n > 0 \\\\\n \\end{cases}\n", "2095dde5a62bdcf100ad3708be0e0846": "x_1, x_2, \\dots, x_m\\,", "2095e0c85dee4edec84f5de072e87cb8": "c \\leq a+b", "20961389e3333aee40f8c9be05205c36": " = - x_0 y_0 + x_1 y_1 + \\cdots + x_n y_n + x_{n+1} y_{n+2} + x_{n+2} y_{n+1}.", "20964233f5adf84ae8fbfc6caa650166": "P_{\\,e\\,CO_2}", "20965d489ba2a92f1914aadab977f2ce": " -\\ln R_\\text{OC}", "20967b3bbdf971d29f86509ca172cf0b": "i>k>\\ell>j", "2096db7ddc796bf5e596b3b5cc5b0bec": "||x|| < \\delta", "2097e0fe2c8c767a6a4f5e125b261346": "\\frac{\\partial h}{\\partial t} + h \\left ( \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} \\right ) - K_E T = 0,", "209868cdaac76ca50991af2141e79dbd": "\\, \\operatorname{Cl}_2(\\theta) -2\\, \\operatorname{Cl}_2(\\pi-\\theta) + 2\\, \\operatorname{Cl}_2(\\pi)", "2098be7861f7fa2e2776049e3034990f": "\\rho=2 \\pi a \\frac{V}{I}", "2098e6223965e8601f9d526c38ca6d55": "E = nhf,\\quad \\text{where}\\quad n = 1,2,3,\\ldots", "2099c0bb59279cbba0ab1911d587e7c8": "(4)\\qquad F_{EW}^2 M_{\\pi T}^2 \\cong \\frac{g_{ETC}^2 \\langle \\bar{T}T \\bar{T}T\\rangle_{ETC}}{M_{ETC}^2} \\cong \\frac{16\\pi^2 F _{EW}^6}{\\Lambda_{ETC}^2}\\,.", "209ab57524e199179d5a350e411bbd0c": "0\\leq\\pi_j\\leq1.", "209ad8d77bacb41213ac2bd354b6fde6": "\\left| M_{1} \\right|=\\left| M_{2} \\right|=1", "209ad9e23cfcdfbb4fe713bbde018816": " G(1+z)=\\exp \\left[ \\frac{z}{2}\\log 2\\pi -\\left( \\frac{z+(1+\\gamma)z^2}{2} \\right) + \\sum_{k=2}^{\\infty}(-1)^k\\frac{\\zeta(k)}{k+1}z^{k+1} \\right]=", "209ae5901db49699801a059981e07f58": " C_1|s-t|\\le |f(s)-f(t)| \\le C_2 |s-t|", "209af6ce94b98a80e92fb1c100604c90": "A(T)\\propto (T-T_c)^\\alpha", "209b69c1589a664d3dbfecfed453d313": "\\{ e \\} (n) <_{\\mathcal{O}} \\{ e \\} (n + 1) ", "209b6e1499b9f66a28c7999b4a632eb2": "(l/2)\\sin\\theta", "209b778dbd6169b713f78105b1fd63e2": "\\Delta B_k=B'-B=(\\lambda_k rB-\\lambda_k p)=\\lambda_k \\Delta B \\;", "209c3a11e14f5cd7681e9cee9a804c4d": "\\tfrac{1}{2}+it", "209c8c791cf564ee2481069eaf517270": "m_\\text{H}", "209cd04713b086a562ae4089ac7c3e02": "161 + 77 \\over 2", "209cd1db751ab1d3b7e3fd49f86ed040": "\n|JKM \\rangle = (D^J_{MK})^* \\quad\\mathrm{with}\\quad M,K= -J,-J+1,\\dots,J\n", "209cdfd9890c7cb6448afe3d5175ea7f": "N^b a\\mathrm{inf}_b = m/r^2\\,", "209d3d43d72ce1cf85533c545f3b502a": "t = B \\left( \\frac{V}{A} \\right)^n", "209d42169b8361e22d2a77f90e7cbd41": "\\|f\\|:=[f,f]^{1/2},\\quad f\\in V", "209d77700b3e63f2670d8e070361c295": "G \\colon D \\to C", "209db172a2921ab26bd2470781425de4": "\\left(2\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "209dc3fa4d7c07f0d500ae8fd4129de4": "|Gx| = |G:G_x|.\\!", "209ddfc6b9725cf89cc12a22fd2505ad": "\\min\\left\\{ D + \\lambda \\cdot R \\right\\}", "209e0435487b47557f289debb4f40904": "3 \\rightarrow \\infty", "209e08a8880a47d34233388fdf123285": "\\frac{\\partial}{\\partial t}\\Bigl( \\rho\\, h\\, \\tilde{U} \\Bigr)\\, +\\, \\frac{\\partial}{\\partial x} \\left( \\rho\\, h\\, \\tilde{U}^2\\, +\\, \\frac12\\, \\rho\\, g\\, h^2\\, +\\, S_{xx} \\right)\\, =\\, \\rho\\, g\\, h\\, \\frac{\\partial d}{\\partial x}\\,", "209e7cba885bd45ad022868b7e0068ab": "x \\,\\bmod\\, y = x-y\\left\\lfloor \\frac{x}{y}\\right\\rfloor.", "209e7f62d93a6b720b53c8238cfe33a7": " \\mathbf{\\hat{n}}\\,\\!", "209e989a13dc3c8a4905a664732d1b59": "\\Gamma\\vdash B", "209eea4efe796cd199c34e40e2b3ad0a": "\\frac{\\partial}{\\partial t}\\left(\\nabla^2 \\psi\\right) + \\frac{\\partial \\psi}{\\partial y} \\frac{\\partial}{\\partial x}\\left(\\nabla^2 \\psi\\right) - \\frac{\\partial \\psi}{\\partial x} \\frac{\\partial}{\\partial y}\\left(\\nabla^2 \\psi\\right) = \\nu \\nabla^4 \\psi", "209efd9f67a854ace01461d5fbb434fc": "g\\left(\\tfrac{\\pi}{2k},s\\right)=0", "209f8221aa83586d6cbefa8904fa375e": "u \\rightarrow \\langle Wu, u \\rangle ", "209f8c323e8c9814e29704d7069a64fc": "g_{\\lambda, \\mu}", "209f913967396f03eee9fe65b329e5a6": "\nL(\\mathbf{x},s) = G(s) \\otimes I(\\mathbf{x})\n", "209fe01340e6ff7f8c5101179362a7f0": " \\frac{d \\theta_i}{d t} = \\omega_i + K r \\sin(\\psi-\\theta_i) ", "20a0032877c145184fefabfc60f11bd0": "\\Gamma\\left(a,x\\right)", "20a043d0cdb93b03d7bad2aeb8a8ee36": " (2/T) \\tan(\\omega T/2) \\ ", "20a053769e404267a2ae12156ac8dafa": "a\\in P(A)", "20a0638015bc9fdf42a79dd9eb3e4ba3": "c = 12k, n_0 = 1", "20a07672786cc81927503b82d25bcad0": "\nd = \\frac{L}{1 - e \\cos \\theta}\n", "20a0904afcde68e76cca984559fbe70e": "g:\\mathbb{R}\\to\\mathbb{R}", "20a0b741e487b744ca07836a9fe38d81": "f(z)=az^2+bz+c", "20a0b7e630f0a1cf0af6344ac11f62df": "\n\\sigma = \\sigma_s \\begin{Vmatrix} \\dfrac{1}{1+\\beta^2} & \\dfrac{-\\beta}{1+\\beta^2} \\\\ \\dfrac{\\beta}{1+\\beta^2} & \\dfrac{1}{1+\\beta^2} \\end{Vmatrix}\n", "20a119fabb991c8890a168765b20da50": "\\omega_{\\mu} \\xi^{\\mu} = 0", "20a12d82dbaaf5e08ad45b32441a8dd9": "\\ d[x, x^*]", "20a157c4bbe3d768becd7f7a7a0293da": "a_{n+1} = \\frac{a_n + \\frac{2}{a_n}}{2}. ", "20a1669b2a01139280a8d3587e683ae4": "\n\\begin{align}\n&F_r = J_3\\ \\frac{1}{r^5}\\ 2\\ \\sin\\lambda\\ \\left(5\\sin^2\\lambda\\ -\\ 3\\right) \\\\\n&F_\\lambda = -J_3\\ \\frac{1}{r^5}\\ \\frac{3}{2}\\ \\cos\\lambda\\ \\left(5\\ \\sin^2\\lambda\\ -1\\right)\n\\end{align}\n", "20a1909eeafdc138c7fa99b12684226f": "f(\\mathbf{x}) = f(x_1, x_2, \\dots, x_N) = \\sum_{i=1}^{N/2} \\left[100(x_{2i-1}^2 - x_{2i})^2\n+ (x_{2i-1} - 1)^2 \\right].", "20a1a63a1b2852836a29bc68db220567": "2^{\\mathfrak{c}}", "20a1b334b1266b6e4506a69af1020ea1": "D_{\\mathrm{KL}}(P\\|Q) \\geq 0, \\,", "20a1d58ffcc7687155924937659432c4": " Ld = (Nlex/N) * 100 ", "20a1e1d8cfad3f526ad7c27d2ed4e240": "J_2\\,\\!", "20a227b6d92677d03653bb6114b03213": "5 \\cdot 2 \\quad\\text{or}\\quad 5\\,.\\,2", "20a22dc9d88cd27ea9374b693ff0379d": "q^{n-1}+1", "20a230756ee08c0172213ee817f5ad80": "0 \\to (I^n M \\cap M')/I^n M' \\to M'/I^n M' \\to M/I^n M \\to M''/I^n M'' \\to 0,", "20a272e26348790364d474801503037c": "\\alpha = (\\alpha_1,\\dots,\\alpha_N) \\in \\mathbb{N}^N,", "20a28def80a071ab78c7f5adceb3dec3": "\\textstyle k_f", "20a2c2b5c075bcd541ccbc5cd98e3b9f": "M \\le \\frac{5 N_2}{6 N} \\log_2 N_2", "20a30b85a6d73094e7182ac2da584d71": "\\left(\\frac{\\partial U}{\\partial x}\\right)_y = T\\left(\\frac{\\partial S}{\\partial x}\\right)_y - P\\left(\\frac{\\partial V}{\\partial x}\\right)_y", "20a310c2d1b01188e45a7cfcbb4015a5": "m_a \\mathbf u_a + m_b \\mathbf u_b = \\left( m_a + m_b \\right) \\mathbf v \\,", "20a32afb81ddb586733e34d97b4102d9": "H^\\dagger W^{\\mu\\nu}W_{\\mu\\nu}H/\\Lambda^2", "20a33039235de3fbcafec0890c0e1f23": "A(P) = \\text{true} \\,", "20a33076c14333cca00dae49ab668c7e": "X_k = \\frac{(-1)^k}{2} x_{N-1} +\n \\sum_{n=0}^{N-2} x_n \\sin \\left[\\frac{\\pi}{N} (n+1) \\left(k+\\frac{1}{2}\\right) \\right] \\quad \\quad k = 0, \\dots, N-1", "20a34d575190cd2f81fbb0fceac31ca2": "(\\sqrt{2}/2,\\sqrt{2}/2)", "20a35deccc43bc3649a30db2a05459c5": "\\bar{\\mu}_i=\\mu_i + z_iF\\Phi", "20a3ea17c77f9cc20f66e0fd5947ab39": "l_2 \\, ", "20a41d766611e10616be2d515661c6ca": " \\forall{x}{\\in}\\mathbf{X}\\, P(x) \\to\\ P(c)", "20a42a5dde88287861c70c71083e04c7": "\\begin{align}\n\\det(I + A) = \\sum_{k=0}^{\\infty} \\frac{1}{k!} \\left( - \\sum_{j=1}^{\\infty} \\frac{(-1)^j}{j}\\mathrm{tr}(A^j) \\right) ^k\\, ,\n\\end{align}\n", "20a42ee1a5e60ad864ff5b9ff00d70ee": "10^{-15}", "20a44d27f648ac287bb99a4fac421306": "\n\\left( \\frac{du}{d\\varphi} \\right)^{2} = r_{s} \\left( u - u_{1} \\right) \\left( u - u_{2} \\right) \\left( u - u_{3} \\right) \n", "20a457360e28737ac4b5e8ad4da5a04e": "x_2=l_2\\|r_2", "20a4691fdac93a7c2752215d7583fbd8": "P_i=\\frac{\\partial L}{\\partial \\dot{Q}_i}.", "20a46a3e73e1b7f12946865fac62c6b4": " \\zeta \\,", "20a46d50bf5985db93fed1b6d8132d8a": "x^2=x+1\\, .", "20a4e88e0dcf3ec01fde74300eee2c17": "\\varphi(y,x_{1}, \\ldots, x_{n}) \\,,", "20a5584ee629eebc4e7cc21cf7e10e76": "argmax_{x_i}U_i(x_i)-w_i", "20a559c816dd998ed51efc6eeafdcf6f": "\n\\frac{\\partial^\\alpha u}{\\partial t^\\alpha}=K (-\\triangle)^\\beta u.\n", "20a579522374bfc5abecb9104f95fa4f": "\\beta(3)\\;=\\;\\frac{\\pi^3}{32},", "20a5814ca881712da464550aa9f1e4b0": "\\gamma_{\\mathrm n}", "20a5843ff628cb806e9cbaa6e4564f72": "\\displaystyle d= |W|^{-1}\\prod_{\\alpha>0} \\alpha.", "20a593338e6c44cb42fc416c9595fb9c": "f(x) = a^x \\bmod N,", "20a6a6d5c2b2a5d46d69823465819be1": "\\theta(t)t\\mathrm e^{-\\gamma t}", "20a750aa7e547a07ca92d82beaab7762": " H^{s,p}(\\mathbb{R}^n) := \\left \\{f \\in L^p(\\mathbb{R}^n) : \\mathcal{F}^{-1}\\left (1+ |\\xi|^2 \\right )^{\\frac{s}{2}}\\mathcal{F}f \\in L^p(\\mathbb{R}^n) \\right \\} ", "20a7bcc8c96e0dd950a5b609684a01cf": "w \\mapsto R(w+1)/(w-1)", "20a7d3666e8b3881d5f647d2cbb68960": "S(t) = \\Pr(T > t) = \\int_t^{\\infty} f(u)\\,du = 1-F(t).", "20a7f792c1d57dbffad5b060d69dd636": " p(t) = p_0 + p_1 t + p_2 t^2 + \\cdots + p_{n} t^{n}. ", "20a7fa65c56a8e19071add5fab21d4d0": "(\\hat{b})", "20a828cfd0116783d64e4ba6b53a1093": "h(n)=\\sum_{all tiles}distance(tile, correct position)", "20a88ac3888a899d399311da6d255677": "\nP = \\frac{\\sigma}{\\rho_0}\n", "20a897e52793d8a753747acc6a49ccae": "\\pi(s') P(s',s) = \\pi(s) P(s,s')\\,.", "20a8a0d25273d0aa5c6cd9f9a593150d": " \\left\\vert t_0 \\right\\vert > t_{a/2,n-1}", "20a8b67b292fffe9c5c0497dd4c88d4f": "\\frac{1}{\\sqrt{2}}\\left( \\left|\\uparrow \\downarrow \\right\\rangle - \\left|\\downarrow \\uparrow \\right\\rangle\\right)", "20a8c5af26c8467c25578b3c290f7d3c": "c(e)", "20a8f1d6bfce22843ae66bc937214aff": "0 \\le \\left\\| x - \\sum_{k=1}^n \\langle x, e_k \\rangle e_k\\right\\|^2 = \\|x\\|^2 - 2 \\sum_{k=1}^n |\\langle x, e_k \\rangle |^2 + \\sum_{k=1}^n | \\langle x, e_k \\rangle |^2 = \\|x\\|^2 - \\sum_{k=1}^n | \\langle x, e_k \\rangle |^2,", "20a900fd9e7b00c046e5c3383cc6043b": " \\sum_{i,j,k} n_{i,j,k}=n", "20aabe43c0d85ccfa2e0569c14a4f2fa": " r(\\chi)", "20ab11ea5dac33395e8bcd91b5139a35": "t\\!:\\!1 = ()", "20ab5788949fb57a8079d320118b75fc": "Y_{1,0} = \\omega_e", "20aba0e07a6973414bbb80a9bc38ec44": "Z_\\mathrm{L} = Z_\\mathrm{S}^*.", "20abe4fe92c6ff63ce8662baa381b4e8": "\nRS_i = h_i^t \\left( g_i-G_i \\right)\n", "20abf4a461caa1aaf33cdb4ba7f2f817": " T \\!", "20ac1091b574264f19428c5e8a16cf18": "\\scriptstyle\\{ E_n \\}_{n \\in \\mathbb{N}}", "20ac2604c5f01012e0ae891f821889e2": "K_i = \\frac{K_p}{T_i}", "20ac642354edd3b8c533da348a2a7d4f": "\\tau_{0}", "20aca1d0e1d213f44a364112c73b3733": "\\mathrm{Var}(A)/\\mathrm{Var}(P)", "20acb8dc532569a72c5300bb12c2a64b": "\\Omega_e", "20ad0beb992b77b965467a7ce969fe7b": "\\dot{Q}/T,", "20ad1a1315fb26f3855b0fa1ecf448f8": "\\mathbb{S} \\subset \\mathcal{P}(X)", "20ad3def259e01a1988d612f560b9261": "(2\\pi)^{-p/2}\\det(\\Sigma)^{-1/2}", "20ad3ee341af55b0f7e85294b37858cf": "\\pi_1 > \\frac{1}{2}(\\pi_0+\\pi_1)", "20ad5e53e4d83d825c1b82028c029bf9": "S_2 \\wr S_4", "20ad69def1d8850663d3f7ff3c6bad87": "\\tilde{u}(x,z,t)", "20ae02bcb7f2aca2f71c7f3222859cf5": " f''(x) > 0 \\,\\!", "20ae0ca384c2b6db18f2015758f20e44": "\\Delta_1^0", "20ae67cc67cece0d01cc618deda067c4": "\n\\mathrm{NaCl}_{(s)} \\leftrightarrow \\mathrm{Na}_{(aq)}^+ + \\mathrm{Cl}_{(aq)}^- \n", "20ae81b3ec202d2f45a2b086df3ad10e": "K\\otimes_\\mathbf{Q}\\mathbf{R}", "20aedc820017e944714601d421c802bc": "F _{n+1} (x, y+1) = F _n (F_{n+1} (x, y), F_{n+1} (x, y) + y + 1), \\ n\\ge 0.\\,", "20aef2db5978094f6a4f342cf37bb8ff": " \\nabla\\times\\mathbf{F} (\\mathbf{r}) = \\mathbf{0}\\text{.}", "20af07b3b4880fac6c9ad98a5f167fc5": "\\ell_2=\\frac{x-x_0}{x_2-x_0}\\cdot\\frac{x-x_1}{x_2-x_1}=\\frac{x-2}{5-2}\\cdot\\frac{x-4}{5-4}=\\frac{1}{3}x^2-2x+\\frac{8}{3}\\,\\!", "20af3400ab1736816d0f9182f38bdb4f": "\\prod_{r=1}^m (k_r^+)^{\\lambda_r}=\\prod_{r=1}^m (k_r^-)^{\\lambda_r} \\, .", "20af5b95dee8298982ba48d50301b8d7": "\\sum_i p_i=1", "20af69574ebf361b959ad65f73e21c5e": "K,D,\\epsilon", "20af8d60b7ef72d5e54813776f6ec9e6": " n_{i,\\downarrow, \\uparrow} ", "20af94dc731f882792321b75bcaa5f40": "\\ln\\left(\\frac{1}{3}\\right)=\\frac{-1}{RC}\\frac{T}{2}", "20b0a85c57598e8a84c691b40efab1c2": "\\textstyle P(K) \\ge 1-\\varepsilon. ", "20b0af8cc41c4a218b9df8a52c44ae63": " \\sum_{m=1,3,5,\\ldots\\leq n}\\frac{2}{m^{2}}S_{n,m}=\\sum_{m=2,4,6,\\ldots\\leq n} \\frac{2}{m^{2}} S_{n,m} \\quad \\left(n>2\\ \\text{is even}\\right).\\ ", "20b0f2beecf44c57d7f48b8f7425cea7": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}} \\left(\\boldsymbol{A}^{-T}\\right) : \\boldsymbol{T} = - \\boldsymbol{A}^{-T}\\cdot\\boldsymbol{T}^{T}\\cdot\\boldsymbol{A}^{-T}\n", "20b1029d22d53d8c55b969ae3fac4ab1": "^{99m}Tc", "20b139859ee6ebb1959a029f89b72c69": "\\omega_3^0 \\approx 2.018236", "20b199e0fa0e7ccf734ae90645c6274b": "E(x,v)=\\tfrac12|v|^2+V(x)", "20b1d149bb56e5a0a98a6daf5eb85d0a": "x^2 = a^2", "20b2922f9886239ebd293c2c3aaf72ad": "\\Delta_3", "20b2ea8f6d11644de7777aaec3ddfab4": "S \\circ \\theta", "20b31398cd6cb929f79cd77b4fc4dc8e": " \\frac{d^2y}{dx^2}+[a-2q\\cos (2x) ]y=0. ", "20b31b2a6b9cc5155ed8be0a34b2267a": "v_e \\approx \\sqrt{k_BT_e/m_e}", "20b32f3e1f9c08ff1061937a3bc6f9a4": " \\sigma_e ", "20b339b1d01b344d8425414a7361b189": "I_{O}", "20b3434c2b08d6836f709fac78cb5535": "\\delta W_b", "20b38cad687f3a42223a6b76ee618584": "\n\\mathrm{P}= V_\\mathrm{rms}^{2} = \\frac{A^{2}}{2}\n", "20b3b43fa0a5232467c736032ae5872f": "\\frac{x-x_0}{x_1-x_0}", "20b3c312e73d36724d0da0236da891e0": "|O_{Jac(C)}(2\\Theta_C)|\\cong\\mathbb{P}^{2^2-1}", "20b5a430be9d21dbf6426a473694c8e3": "Q(z,u,v) = \\exp\\left(\\frac{v}{u}\n\\left(\n\\frac{zu}{1} \n+ \\frac{z^2 u^2}{2} \n+ \\frac{z^3 u^3}{3} \n+ \\frac{z^4 u^4}{4} \n+ \\frac{z^5 u^5}{5} \n+ \\cdots\n\\right)\\right)", "20b5b39134571ced128a48a17c99239b": "\\log n=H_0\\geq H_1 \\geq H_2 \\geq H_\\infty", "20b5bc423a9465ea9e92b3b548de636e": "\\log_e(4)=1.386\\ldots", "20b5d88e4ae28e345fdc41a7981795be": "(x-y)^2 < 0", "20b620923ab918a6f2b7a0eb419f8fc4": "c_3", "20b651ac31475acd10f2960db5a7b906": "D_\\mathrm{EO}", "20b681f83c861cda77a086148916d0c8": "\nc = c(r)\\,=\\,\\frac{1- (r-1)^2}{4}\n", "20b695481bc167da71c5cd2e1f37798a": "\\ t^* ", "20b6cb72922664013f0cdccae982f913": "{\\mathcal F}", "20b6d9304641233858e5a753de073b9d": "\\mathcal{D} = \\left\\{ (\\mathbf{x}_i, y_i)\\mid\\mathbf{x}_i \\in \\mathbb{R}^p,\\, y_i \\in \\{-1,1\\}\\right\\}_{i=1}^n", "20b6db30395adbefd322cfcdfbe75568": "L=\\{[a,b]|a,b \\in R\\}\\cup\\{[a]|a \\in R \\}\\cup \\{[\\infty]\\}", "20b6e64a7fa44c24f451be4d922f083e": "((a), [c,d])\\in I \\Longleftrightarrow a=c", "20b749114df2b1c07fbc352b74f080e9": "f^{-1}[0]", "20b7a52f8f715aaef45597e9c69389aa": "(p(x) - \\beta)", "20b805bd59e258906141ccdfb2b14da5": "f(\\mathbf{x}), \\quad \\mathbf{x}=(x_1,x_2,...,x_d) ", "20b822946db0592437bac132e7c8c2e1": "i_0, i_1, ..., i_{n-1}", "20b892eb9ba2d63443af1b8b0cea3a42": " B = \\frac{u^2}{1024} \\left\\{ 256 + u^2 \\left[ -128 + u^2 (74-47 u^2) \\right] \\right\\} ", "20b8a05d21a9e5375159943d884cdc3e": "a_{W}=D_{w} + \\max(F_{w},0)", "20b8c531784bd9b94acd3847a085f5d9": "t.i \\in T_r ", "20b8cd5baaf102c114df2b6d230219a7": "\\tau(y)", "20b8d3b7b3a7299ab26c0faea4c1c6dc": "(\\bold{x},t) \\mapsto (G\\bold{x},t)", "20b94052ecc1f9710fd7cb8860090a52": "\\lambda_j = -\\frac{4}{h^2} \\sin(\\frac{\\pi (j-\\frac{1}{2})}{2n + 1})^2", "20b9a5134bfb3eff46e780f9be0aee43": "\\liminf P(n)/\\log n=1", "20b9c4f885244e3fb95721fe0cede08d": "Q>0", "20ba34dd04da9010b0f8a9e48a0ead5d": "H^{i}(j_x^!C)\\ne 0", "20ba8e458cd798886cdf940d0c03f0fe": " \\kappa < \\operatorname{cf}(2^\\kappa)\\,", "20babae90780fc6b00ddff0308d30750": "v_\\text{out} = \\alpha_1 (A_1 \\sin \\omega_1 t + A_2 \\sin \\omega_2 t) + \\alpha_2(A_1^2 \\sin^2 \\omega_1 t + 2 A_1 A_2 \\sin \\omega_1 t \\sin \\omega_2 t + A_2^2 \\sin^2 \\omega_2 t) + \\ldots \\,", "20bb0f2f1c51c05f8d7fe46d1cfe15fa": "V_w=f^{-1}(w)", "20bb162142fabbf69624ad8edf5f4909": "\\mathbf{E(r)} = \\frac{1}{ 4 \\pi \\varepsilon_0 } \\int \\frac{\\rho(\\mathbf{r'}) \\left( \\mathbf{r} - \\mathbf{r'} \\right)} {\\left| \\mathbf{r} - \\mathbf{r'} \\right|^3} \\mathrm{d^3}\\mathbf{r'}", "20bb45adbec97d06c584996d95e61f5b": "v=\\left(\\left(g^{u_{1}}y^{u_{2}}\\right)\\bmod\\,p\\right)\\bmod\\,q", "20bbe5d77ead81de0f42a83de89e1551": " \\mathcal{B} = A_0 B_0 + A_0 B_1 + A_1 B_0 - A_1 B_1 ", "20bc064261b7a7ef94a243c2d3055af1": " \\frac{Z}{r}", "20bc4de03db1afd161ee457adbbe77d5": "g=\\tfrac{1}{2}m w_i w_i", "20bc7929074736fce1ec19d3c4014dd8": "k_2=\\gamma/J_2", "20bc9b46b67fd66d7dbaa60eeb4211a1": "G_1=\\langle U,D,L^2,R^2,F^2,B^2\\rangle", "20bca72a4078dda835a09d04a42b6ac7": " \\psi \\rightarrow e^{-i \\sigma_z \\omega_r t/2}\\psi ", "20bcb091288350de78be686f978e0af7": " P_A O_2 = P_I O_2 - \\frac{V_T}{V_T-V_D}(P_I O_2 - P_E O_2)", "20bd0cde5d9b9ca34b7eac666c36f489": " j = (3/2)N - 2. \\!", "20bd73670af509549323edec0ecc2908": "\\left \\lfloor 2^{2^{2^{\\cdot^{\\cdot^{2^{\\omega}}}}}} \\!\\right \\rfloor \\scriptstyle \\text{= primes:} \\displaystyle\\left\\lfloor 2^\\omega\\right\\rfloor \\scriptstyle \\text{=3,} \n\\displaystyle\\left\\lfloor 2^{2^\\omega} \\right\\rfloor \\scriptstyle \\text{=13,} \n\\displaystyle \\left\\lfloor 2^{2^{2^\\omega}} \\right\\rfloor \\scriptstyle =16381, \\ldots ", "20bd79c7e8fc0332f739c4b23024405d": "\\pi_1(\\mathbb{R}^3 \\backslash K).", "20bda037fbe16c7c412958bd780c75b2": "vs^{2} =(\\mathbf{y}- \\mathbf{X} \\hat{\\boldsymbol\\beta})^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\hat{\\boldsymbol\\beta}), \\text{and}\\; v = n-k,", "20bdca70df2cb0f14692db9a49459d13": " dz=dx+idy=ds(\\cos\\phi+i\\sin\\phi)=ds\\,e^{i\\phi} \\qquad \\Rightarrow \\qquad d\\bar{z}=e^{-i\\phi}ds.", "20be63df70211320461d7c28274fa158": "\\mathcal{B}(X^*_{\\sigma}, Y^*_{\\sigma})", "20bece015bc0b4532a910251c85d944d": " K ", "20bed1a202baffaf13072a180c841a75": "Y = c_1 X_1 + \\cdots + c_N X_N,", "20bed5031a50dde7a2a01a4527c8a532": "\\textstyle 2", "20bf01a92dbb20e410671da659b63f42": "\\mathcal{S} = \\int_{t_1}^{t_2}\\int \\mathcal{L}(\\mathbf{r},t) \\mathrm{d}^3\\mathbf{r} \\mathrm{d}t.", "20bf4ae8960323bcc6729f73a0f12e4f": "k(\\cdot,\\cdot)", "20bf643f86666317010a2c125c415722": "\\int\\frac{mx+n}{ax^2+bx+c} \\, dx = \\begin{cases}\n\\displaystyle \\frac{m}{2a}\\ln\\left|ax^2+bx+c\\right|+\\frac{2an-bm}{a\\sqrt{4ac-b^2}}\\arctan\\frac{2ax+b}{\\sqrt{4ac-b^2}} + C &\\text{(for }4ac-b^2>0\\mbox{)} \\\\[12pt] \\displaystyle \\frac{m}{2a}\\ln\\left|ax^2+bx+c\\right|-\\frac{2an-bm}{a\\sqrt{b^2-4ac}}\\,\\mathrm{arctanh}\\frac{2ax+b}{\\sqrt{b^2-4ac}} + C &\\text{(for }4ac-b^2<0\\mbox{)} \\\\[12pt] \\displaystyle \\frac{m}{2a}\\ln\\left|ax^2+bx+c\\right|-\\frac{2an-bm}{a(2ax+b)} + C &\\text{(for }4ac-b^2=0\\mbox{)}\\end{cases}", "20bf7d15202b1aee628da0448e04249e": "\nID(x,y) = \\min \\{|p|: p(x)=y \\; \\& \\;p(y) =x \\},\n", "20bfa168ce610d00f699745357feac58": "U(a,L)^{\\dagger}A(x)U(a,L)=S(L)A(L^{-1}(x-a)).", "20bfa47aa60668c81d7b30f2bed6f111": "x_{i+1}=x_{j+1}", "20bfeaa797f4eea5ac0f3b4e63251947": "\\sigma_{\\hat{X}}^2", "20c0096d84b3cc07999e442af53c51bc": "\\sum_{n=1}^{\\infty} f_n (x)", "20c0315b91a6c38d14a7027727ea99df": "{d \\over dt}\\left\\{ X_1 \\right\\} = \\left\\{A \\right\\} + \\left\\{ X _1\\right\\}^2 \\left\\{Y_1 \\right\\} - \\left\\{B \\right\\} \\left\\{X_1 \\right\\} - \\left\\{X_1 \\right\\} + D_x\\left( X_2 - X_1 \\right)\\,", "20c035aa42b5810ac4b8697a376b1ffa": " \\hat{S}", "20c0a109edb71ec82aa27d1ab30d2b10": "f_i : X_{i+1} \\to X_i", "20c0c2a43c0526784fd7990f0cace32e": "\\min J_2 (w,b,e) = \\frac{\\mu }{2}w^T w + \\frac{\\zeta }{2}\\sum\\limits_{i = 1}^N {e_{c,i}^2 } ,", "20c14e48ea891a030419ad510a091f3d": "\nJ_G = \\begin{bmatrix}\n 3 & \\sin(x_2x_3)x_3 & \\sin(x_2x_3)x_2 \\\\\n 8x_1 & -1250x_2+2 & 0 \\\\\n -x_2\\exp{(-x_1x_2)} & -x_1\\exp(-x_1x_2) & 20\\\\\n\\end{bmatrix}\n", "20c1b341b3e5f08cd3aabce814a68fb2": "C(r,z)=\\int_0^\\infty G(r'',z)r'' \\left [ \\int_{0}^{2\\pi} S \\left ( \\sqrt{r^2+r''\\,^2-2rr''cos\\phi''} \\right ) \\, d\\phi'' \\right ]dr''\\qquad(4)", "20c2a133fc1344dcbe39ac49205bc6f0": "{ 3\\ln\\left(3\\right)}-{\\pi\\sqrt{3}\\over3 }", "20c313bc356ccfa9910fe693f4ad7e90": "E=\\frac{(\\hbar k)^2}{2m}", "20c337d75e048c578e64dd7edf0b9c52": " Z(S_n) = \\sum_{j_1+2 j_2 + 3 j_3 + \\cdots + n j_n = n} \\frac{1}{\\prod_{k=1}^n k^{j_k} j_k!} \\prod_{k=1}^n a_k^{j_k}", "20c3436d03583003315ea675d5937609": "S_N f", "20c408bbe4c0287d4208f9021733e784": "w(m+n)\\leq w(m)w(n),\\quad w(0)=1", "20c4506ca1a74fd02ee0db3bce103947": " K(u,v) = {-4 \\over a^2 b^2 \\left(1 + {4 u^2 \\over a^4} + {4 v^2 \\over b^4}\\right)^2} ", "20c49e9494e4d9ac6129823e949bb499": "|c_{11}| \\ge |c_{12}| + |c_{13}|", "20c4d77dedb38bf0037b4541b7678bfb": "[abababbca]_D", "20c536c76ec00fc5629ff6706225106f": "r_i\\in R", "20c57af8d1b0993791bf95f30dd376e3": "{x_1 + x_2 +\\cdots + x_n \\over n} \\ge \\sqrt[n]{x_1 x_2\\cdots x_n}, ", "20c57b8eaa12808def61c7485cc6f513": "\\mathrm{d}E = -\\mu_0 M_s \\int_V (\\mathrm{d}\\mathbf{m})\\cdot\\mathbf{H}_\\text{eff}\\,\\mathrm{d}V", "20c5919ba51759e1facf7e26c315eeeb": "\\frac{1}{(1 - P) + \\frac{P}{S}} = \\frac{1}{(1 - 0.3) + \\frac{0.3}{2}} = 1.1765", "20c5a5694e53f4089d67ceabe80d4185": "\\varepsilon\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{hc}{2L}n,", "20c5b2c7c9c1212a92cbe33244a0ba3c": " \\alpha(u)e^{\\pi H(z,u)+H(u,u)\\pi/2}\\ ", "20c5cc83f43c6278e02526dc2b68d7d9": "\\frac{\\mathrm{d}\\mathbf{M}}{\\mathrm{d}t}=-\\gamma \\mu_0 \\mathbf{M} \\times \\mathbf{H_{eff}} + \\frac{\\alpha}{M} \\left( \\mathbf{M} \\times \\frac{\\mathrm{d}\\mathbf{M}}{\\mathrm{d}t}\\right)", "20c67f1c30d70d528f7f47c6379cfd46": " T \\Delta S_{int} - \\Delta H \\ge 0\\, ", "20c67f6f4616c675e0c79631c6758535": "A_v =\\begin{matrix}\\frac {v_{out}}{v_{S}} = \\frac {R_L}{R_S} \\end{matrix} ", "20c6ddd38f53084219eb44ebb5db226b": "p,\\, q \\geqslant 0,\\, p + q = 1,\\,p = q ", "20c72299ca7a678df883523be23130ef": "\n\\ln 2 = 3 \\ln \\left( 1+\\frac{1}{4} \\right) + \\ln \\left( 1+\\frac{3}{125} \\right)\n= \\cfrac{6} {9-\\cfrac{1^2} {27-\\cfrac{2^2} {45-\\cfrac{3^2} {63-\\ddots}}}}\n+ \\cfrac{6} {253-\\cfrac{3^2} {759-\\cfrac{6^2} {1265-\\cfrac{9^2} {1771-\\ddots}}}}.\n", "20c7804ad15bd45ec87e4f57cfd0455d": " r = n_1 / n_2 ", "20c79511c10764bf5b3c841563c1ad54": "\\vec{p}_{i}", "20c7e667ae456bb7f99e6432344c57d8": "\\varphi(t) = \\int_{\\mathbf{R}} g(t+\\theta)\\overline{g(\\theta)} d\\theta .", "20c8017f0ec6b1f09ec5d3ada1d387fb": "S=F", "20c855fce927bc79b3005831c5e396b6": "e^{-\\alpha t}", "20c8ce4143446376fb48d1766507c0b4": "I \\mapsto \\Gamma_I", "20c8f80b4052c67d08f2fde427397642": "T'(t) = - \\lambda \\alpha T(t),", "20c90d0049e2805935fa6624a7851ff7": "K = \\frac{\\det II}{\\det I} = \\frac{LN-M^2}{EG-F^2}.", "20c92a5711dc3cfc4cbb239400989e45": "440 \\rm{ Hz}\\cdot (\\sqrt[12]{2})^{-19} \\approx ", "20c935729081116936e1028c9a0ac22f": "\\textstyle\\left(\\frac{m_{k+2}-m_{k+1}}{\\sigma_{k+1}}\\right)^T\\! C_k^{-1} \\frac{m_{k+1}-m_{k}}{\\sigma_k} \\approx 0", "20c980b08233e5e56e3618389d84c401": "\\mathcal F", "20c9ae9cc379ee8869915a7a59cbac3a": "a/b \\ge 0.3", "20ca18f64d855fafba1388f9754fa9ff": "f(i)=v_i", "20ca2061bd29f4acd103fbb51febee03": "(-1)^n \\, Z\\left(g_{n}\\right) > 0", "20ca601153d0257d8301cf041dcc75b8": "\\int \\sec^2{x} \\, \\mathrm{d}x = \\tan{x}+C", "20ca9a3d4b18cda9f2435c5ef0db734e": " R_N^k(n)", "20caa0c4e7dc6781e52cea13ab24c431": "\\scriptstyle BV_{loc}(\\Omega)", "20caa2174db477f7e70f1e0521411b1d": "H=T", "20cb0bd46f91db8d4035ea9410549614": " \\alpha \\hat{x}-i\\beta\\frac{\\partial}{\\partial x}", "20cb2b8185895d37704f276ca7ffa4c9": " \\widehat{\\mathbf{S}} = \\frac{\\hbar}{2} \\boldsymbol{\\sigma} ", "20cba01b64876ff7813889d00f872ef3": "\\operatorname{logit}^{-1}(\\alpha) = \\frac{1}{1 + \\operatorname{exp}(-\\alpha)} = \\frac{\\operatorname{exp}(\\alpha)}{ \\operatorname{exp}(\\alpha) + 1}", "20cbabe7b7b66592f9bb126bc3bba84c": "2s = 2^{1} + 2^{2} + 2^{3} + \\cdots + 2^{63} + 2^{64}.", "20cc7e4af4ff3c37a73652b3daffe414": "A = \\sum_{k=0}^r a_k D_x^k", "20cc86b07fda93e7f5beaf4d818cb916": "K\\ge 4r^2", "20cc95ecc649d54bfac200a9de4c9d3a": "\n\\begin{align}\nI(x,y)\n&=\n\\frac{1}{e}\\cdot\n\\lim_{(\\xi,\\eta)\\to(x,y)}\n\\sqrt[\\xi-\\eta]{\\frac{\\xi^\\xi}{\\eta^\\eta}}\n\\\\[8pt]\n&=\n\\lim_{(\\xi,\\eta)\\to(x,y)}\n\\exp\\left(\\frac{\\xi\\cdot\\ln\\xi-\\eta\\cdot\\ln\\eta}{\\xi-\\eta}-1\\right)\n\\\\[8pt]\n&=\n\\begin{cases}\nx & \\text{if }x=y \\\\[8pt]\n\\frac{1}{e} \\sqrt[x-y]{\\frac{x^x}{y^y}} & \\text{else}\n\\end{cases}\n\\end{align}\n", "20ccb3f2890a1717c1a82836c908f6ed": "[-1,1] \\subseteq \\mathrm{Supp}(\\rho)", "20cceca1ea9ebf5e96033ac7c935ef44": "\\Gamma_L=\\left(Z_L - Z_0\\right)/\\left(Z_L + Z_0\\right)", "20cd0007c70777b8a741bfcf078ddd0c": " |\\mathbf{a}| = |\\mathbf{r}(t)| \\left ( \\frac {\\mathrm{d} \\theta}{\\mathrm{d}t} \\right) ^2 = r {\\omega}^2\\ ", "20cdaf79975da27f6d7805988de90cc4": "D^\\mathrm{BW} = \\bigotimes_{r=1}^{2j} \\left[ D_r^{(1/2,0)}\\oplus D_r^{(0,1/2)}\\right]\\,.", "20cdbb1f9187cacde999eaf2e22872ed": "n, f = 0, 1", "20cde1c4bdf2968a7179cef9df71c509": "f(x)=\\Omega_-(g(x))", "20ce7fa8448ea84ee6b413a6cee13a39": "\\ell_f", "20ce92eb8aa8c082a577c8bdceff718e": "\n\\lim_{t \\to \\infty} \\sup_{x \\geq t} f(x)^2 ~-~ f(1)^2 = \\limsup_{t \\to \\infty} \\int_{1}^{t} (f(x)^2)' \\,\\mathrm{d}x\n", "20ce97c91c124e7cd5c89325fe05d341": "\\scriptstyle\\{x_1,\\,x_2,\\,\\ldots,\\,x_N\\}", "20ceb395355c2b74b998a7ce3c3ebdee": "\n \\sigma_{xx} = 0 ~,~~ \\sigma_{yy} = 0 ~,~~ \\sigma_{zz} = 0 ~, ~~ \\tau_{zx} = 0 \n ~,~~ \\tau_{yz} = \\mu(z)\\,\\frac{dV}{dz}\\,\\exp[i(k x - \\omega t)]\n ~,~~ \\tau_{xy} = i k \\mu(z) V(k, z, \\omega) \\,\\exp[i(k x - \\omega t)] \\,.\n ", "20ced19c091d56592b7672ffd717c229": " J_\\mu = -v^2 \\partial_\\mu \\theta ~.", "20cf05f6d27c0e78cd8fcebea0c361ee": "\\angle(U' M V')", "20cf129cd0789619a8de032f0a11be24": "C = Q/V", "20cf2bfd7a8e8ae19dd02f2b94f71fe0": "\n\\mathcal{G}^{(n)}_{\\alpha_1\\ldots\\alpha_n|\\beta_1\\ldots\\beta_n}(\\tau_1 \\ldots \\tau_n | \\tau_1' \\ldots \\tau_n')\n= \\langle T\\psi_{\\alpha_1}(\\tau_1)\\ldots\\psi_{\\alpha_n}(\\tau_n)\\bar\\psi_{\\beta_n}(\\tau_n')\\ldots\\bar\\psi_{\\beta_1}(\\tau_1')\\rangle\n", "20cf53a41ee43600b05be1321ab55c83": "H_{\\frac{1}{4},2}=16-8G-\\tfrac{5}{6}\\pi^2", "20cf7f599291741f768ab0a868bbd6b0": " \\rho= \\frac{s}{1+c} , ", "20cf8f453b412b5182e12b4d296219b6": " \\textbf{E} = k\\frac{q}{r^2} \\hat{r} ", "20cf8fbe230129cb7649e46075914b6e": "0 \\le\\alpha, \\gamma \\le 2\\pi", "20cf940f82b8c291365cc57df94a8855": "a_m = \\frac{1}{(n^2-4\\cdot m^2)\\cdot a_{m-1}}", "20cf9e4d0af1a4d443c4b7557e3a4801": "\\beta=-1/2", "20cfadf5d2ce6a1eb1905b9a5680ef87": "\\cos x/x", "20d030ef843b639bb3abfc4fd2c0aaac": "\\{0,1\\}^d", "20d03ff568918a64596dcebf9b67e243": "1+r= \\frac{M_2 - C_2}{M_1}", "20d0481557295b669184875b9819bedb": "a \\in \\mathbb{A}", "20d06d7aa143953677590f94d3c1f746": "{\\rm tr}\\left( \\left(\\frac{\\partial g(\\mathbf{U})}{\\partial \\mathbf{U}}\\right)^{\\rm T} \\frac{\\partial \\mathbf{U}}{\\partial X_{ij}}\\right)", "20d077c69f6963ac0ba5da366ce2a923": " a + a r + a r^2 + a r^3 \\cdots ", "20d0851abebbc3844348605389aebf6a": "\\sum^{40}_{k=0} -\\frac{1}{2k+1} \\frac{\\sin (2k+1)x}{\\left|\\sin (2k+1)x\\right|}", "20d095e37702a33ea5751dee71ab3c5d": "U[] \\to \\epsilon", "20d0cb9db0e0158dc37a127e6dc5ecc6": " \\frac{d\\mathbf E_{1s}}{d\\zeta}=\\zeta-\\mathbf Z=0", "20d0e274357ecad4683762bd23f2af91": "\\kappa \\le \\beth_{\\alpha}(\\mu).", "20d159120deba8433025b88ad0412f2f": "\\tilde{K}", "20d15e139938616554a8b836dc0bf312": "l_j(x) := \\prod_{\\begin{smallmatrix}i=0\\\\ j\\neq i\\end{smallmatrix}}^{n} \\frac{x-x_i}{x_j-x_i} ", "20d172080f660ac3d58312127d2d6a3f": "i,j,k", "20d177acdb4cbbe32c7efbb92200bafc": " log_e(1+x) = x - \\frac{1}{3}x^3 + \\frac{1}{5}x^5 - \\frac{1}{7}x^7 + ........ ", "20d17e960003bc8d6f7e253c9cc9cb5d": " F(x)+R(x)", "20d19303a5605bb10b28d7ff01de48df": "N_{c} = \\frac{30}{\\pi} * \\sqrt{\\frac{g}{\\delta_{st}}}", "20d193762effc4b034c3a40cac7fe965": "\\{p_1,\\ldots,p_n\\}", "20d19aced1cf6b50e7396ea98b01ec4b": "2+\\frac{3}{4}=2\\tfrac{3}{4}", "20d268de25571167bcaf59c0e1620db8": "e \\cdot p_{cv} = {1 \\over V} \\int_v e^{i(k_p +k-k') \\cdot r} {u^*}_{ck'}(r)e \\cdot (p+ \\hbar k)u_{vk} (r) d^3r", "20d293a9d465c6550229b114a3167474": "A^{k}", "20d2c204a22cf4fcea332d24ced5da12": "-\\pi\\le\\theta\\le\\pi", "20d2d4f4ae963686eeab302a75c4f6d0": "H : V \\longrightarrow G", "20d2da23cedade95311c2bbad8a20081": "(xy)z = e", "20d2ead06a023e5482b1d98175f488d9": "\\sqrt{\\sum_{x\\in R}\\left [F(x+h)-G(x)\\right ]^{2}}", "20d2f0395dde46c301322dc09a22c049": "A(Z)", "20d2f715850fc500d5539e505954c2d0": "\\hat 1, \\hat 2, \\hat 3...", "20d2ffc9f78da23d6e917cf339b84f95": "(\\lambda b.(\\lambda v.b(v)(b)))", "20d393454bfe0f00c745866e9d92a118": "{q_v}", "20d43b66a2e94d95b5f023e9766e48b8": " (\\mathbf{A} \\or \\mathbf{B}) \\and \\mathbf{C} = 0, ", "20d454fa9095f9e40799379ee6767546": "|h(x) - h^*(x)| = O(\\log h^*(x))", "20d4a7120fbc6352ee55ca710b0ffef4": "(x^{15}+3x^{14}-16x^{13}-50x^{12}+94x^{11}+310x^{10}-257x^9-893x^8+366x^7+1218x^6-347x^5-717x^4+236x^3+128x^2-56x+4).", "20d4c9addfa527f384b56406b8050346": "\\mathrm{d}s^2 = -(1+2\\Phi)\\mathrm{d}t^2 +\\alpha^2 (1-2\\Psi)\\delta_{ij}\\mathrm{d}x^i \\mathrm{d}x^j \\,", "20d509482e96c8775462390fd8b82213": "\\mathbf{A}\\mathbf{B} - \\mathbf{B}\\mathbf{A} \\neq \\mathbf{0}.", "20d52abc09173f86219fb55221cbe607": "\\varepsilon = \\omega^\\varepsilon, \\, ", "20d53966e8be0803570e7a0ea0cfbf7e": "\\left( T^{(\\mathbf n)} \\right)^2 = T_i^{(\\mathbf n)} T_i^{(\\mathbf n)} = \\left( \\sigma_{ij} n_j \\right) \\left(\\sigma_{ik} n_k \\right) = \\sigma_{ij} \\sigma_{ik} n_j n_k.", "20d541b011ff78bc440d2cfbf8ad887b": " R = V/I. \\,\\!", "20d54e562a8b26ae2d07db119a34415d": " r^n~\\sin(n\\theta) \\,", "20d5560f2544cbe795e33e94283944e7": "\\Omega^{p,q} \\xrightarrow{\\overline{\\partial}} \\mathcal{F}^{p,q+1}\\xrightarrow{\\overline{\\partial}} \\mathcal{F}^{p,q+2} \\xrightarrow{\\overline{\\partial}} \\cdots \\, ", "20d560b50a6e9a9a6c9a295e42e00a29": "\\displaystyle iu_t + L_1u = \\phi u", "20d5613f94bff9d298999ad004515694": "\n \\begin{matrix}\n \\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\\\\n \\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\\\\n 10\\mbox{ multiplied copies of }10\n \\end{matrix}", "20d5c45fa36239071fc39f9da0b64b56": "\\langle \\sigma v \\rangle", "20d5e793fe2bae035c61ebc175ca100f": " P = (2w + t + x)/(2g)", "20d6404b6699189eaace5e299743a287": "X = \\{x_1, ..., x_n\\}", "20d64e44f690a99cab045a75f4945c4e": " H_{n}=\\sum_{1\\leq k\\leq n}k^{-1} ", "20d66aff35587e970f1594bf04d9ab2a": "r= k_2 C_S", "20d6ce2bfe89316407fe33f37f5f4057": "\\kappa_0= \\left( 1-e^2 \\right)^{-1} .", "20d6d9712e7752c095c696b1ae8546d3": "\\log_2(n)", "20d6ec8acf2f9cb3251c8627b07cf434": "z(2,\\sqrt{3}) = \\frac{1}{2 \\cdot 2 \\cdot \\sqrt{3}}\\sqrt{(2)^2 + (\\sqrt{3})^2} = \\frac{1}{4\\sqrt{3}}\\sqrt{7} \\,, ", "20d6fa8706af2f0b4d35728d4f246a59": "[A,B]_D=[A,B]+d\\langle A,B\\rangle", "20d75a4511a598975f5b08db2d51a778": "E = \\frac{\\hbar^2 \\pi^2 n^2}{2mL^2} = \\frac{n^2h^2}{8mL^2}.", "20d767482f2b6d8a9ba0ae2795986dfd": " K = 2000 ", "20d79abe87b4fe200a9f4fb8c99809e5": " f_X(x|\\theta) = \\exp \\left (\\eta(\\theta) \\cdot T(x) - A(\\theta) + B(x) \\right )", "20d7a7d2b7b46e18837cec0b98de23a1": "p(x)\\in \\Bbb{R}_+", "20d7d7fb0c9eb438a0c0231c6f03c370": "\\operatorname{Br}(k) = \\mathbb{Q}/\\mathbb{Z}", "20d7e92c66c971b964b9d516481faa72": "N = \\sum_{i=1}^c m_i", "20d81c5cd67e4e6607548b46f849d5c4": "\\operatorname{corr}(U,V)", "20d832093010c3145e67518f2d373212": "\\scriptstyle - \\frac{2}{5}", "20d867527ce0138b640d18c18998a01f": "d\\alpha|S|\\,", "20d8b96be2eb751c30ee5f07ef79b9da": "\\frac{\\partial V}{\\partial t}", "20d8fc0c8edfc3f982a3a87a913b9193": "(a,b,c,d,\\ldots,y,z) = (a,b)\\cdot (b,c,d,\\ldots y,z)", "20d902e1c18ed79cb5014a9cffc3b7af": "y - p(x)", "20d9062d72216393b3f4e3f4712d8794": "\\Bbb{R} \\times M", "20d911b659d6fe0e2ff505a55afa4a41": "\\mathbf{a} \\cdot \\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4 + a_5b_5 + a_6b_6.", "20d9395ee2c26340c3472ea724441d14": "l_{i} ", "20d99ad067aa972f3caa30d66ad0571e": "r \\leftarrow min(L(e)-i,m+16)-16", "20d9db3f2cde18e3bad04d5f0e0041b9": "\\omega > 5", "20d9f8e44e17b9fffc6c1d5c2699d050": "\\ell(s)=m", "20da2ba13fd330de7fd25dbc076eba9d": "\\underline{F}=F_{z}\\underline{z_{o}}=L^{TM}(z)T^{TM}(x,y)\\underline{z_{o}} \\ \\ \\ \\ \\ \\ (11) ", "20da3f2895d89a7b6d88a22ff2c862d5": "{\\mathfrak c} ", "20da8f5e5ea8d8a3919479d21d7e600e": "\\cos(y) = x \\ \\Leftrightarrow\\ y = \\arccos(x) + 2k\\pi \\text{ or } y = 2\\pi - \\arccos(x) + 2k\\pi", "20dab6b69cfe9451cb4270bbf30e917a": "a = q^{h+k-1} \\frac{\\sin h \\alpha}{\\sin \\alpha} = q^k \\cdot\\sum_{0 \\leq i \\leq \\frac{h-1}{2}}(-1)^{i}\\binom{h}{2i+1}p^{h-2i-1}(q^2-p^2)^i,", "20db790a20fa81c5222dbd2535251d58": "", "20dbc545ed110cf28853e2cffff404fa": " \\operatorname{Var}(X) = \\operatorname{E}[(X - \\operatorname{E}(X) )^2]. ", "20dbd027f37eb9ae2ae420388f05976c": "S_2(x,y)", "20dc355b5839b6667f4f68bfda9e7f3b": "|f|:=f^++f^-", "20dc50a57ecb7832263b29328a648e47": "\\chi^2_k\\!", "20dca66d2ad86a50c3dd4e109943a45a": "\\psi_i\\circ\\psi_j^{-1}\\in {\\rm Aff}({\\Bbb R}^n)", "20dcbf7b35b5b91f95d171ebd7c15668": "\\begin{align}\nd\\mathbf{x}^2 - d\\mathbf{X}^2 &= d\\mathbf X\\cdot\\mathbf C\\cdot d\\mathbf X-d\\mathbf X\\cdot d\\mathbf X \\\\\n&=d\\mathbf X\\cdot (\\mathbf C - \\mathbf I)\\cdot d\\mathbf X \\\\\n&= d\\mathbf X \\cdot 2\\mathbf E \\cdot d\\mathbf X \\\\\n\\end{align}\\,\\!", "20dcf6e9c4706a97d4dcc5740057631f": " (\\lambda_k)^{n/2}\\sim\\frac{(2\\pi)^nk}{\\omega_n Vol(M)}.", "20dd02ee0c556fe3161f2c8cd88b6b6b": "\\begin{smallmatrix}\\alpha=\\arccos\\left(\\frac{c}{a}\\right)\\,\\,\\!\\end{smallmatrix}", "20dd07091086404ca1ad129bbdc56ff0": "\n2m K_\\mathrm{NR}(p) = {i \\over (p_0-m) - {\\vec{p}^2\\over 2m} }\n", "20dd67fbd6baf6ab5f921defa3d1fab5": "\\pi_1*u_i(x_{1i})+\\pi_2*u_i(x_{2i})", "20ddb8f45d952b2b89f6e83b42f6e106": "b_2 = 0.51465", "20dde788ca784c85072583be4f1c1bc2": " G/H ", "20de10b3bb137e5e4a140200a998eab0": " SH = {{0.622 p_{(H_2O)}}\\over {p-0.378*p_{(H_2O)}}}. ", "20de642efff12648d361ef057e0e6f22": "\\begin{align}\n0 & \\le (x-y)^2 \\\\\n& = x^2-2xy+y^2 \\\\\n& = x^2+2xy+y^2 - 4xy \\\\\n& = (x+y)^2 - 4xy.\n\\end{align}", "20dec4a500cd77c0439ab9a7e9d0d593": "P_{hs}= \\frac{R\\, T}{V_m}\\, \\frac{1 + \\eta + \\eta^2 - \\eta^3}{(1 - \\eta)^3}", "20ded0edb3b9ea15104fafcb11aa63e1": "\n \\lambda_1 = \\lambda ~;~~ \\lambda_2 = \\cfrac{1}{\\lambda} ~;~~ \\lambda_3 = 1\n ", "20df44ef93547d846dff479b1e50221c": "O(N^{K+1} \\, K \\, T)", "20df632a6718c25c2ba1d19116c506f4": " r_O = \\frac{v_{ce}}{i_{c}}\\Bigg |_{v_{be}=0} ", "20e0297ca80d4aec3ae36dc9ca05f984": "\\frac{\\partial^2 u}{\\partial t^2} - c^2\\frac{\\partial^2 u}{\\partial x^2} = 0", "20e031c66d65a43ca472dd567f875046": "H^i(K,V)\\times H^{2-i}(K,V^\\prime)\\rightarrow H^2(K,\\mathbf{Q}_p(1))=\\mathbf{Q}_p", "20e068b26e271fa140694d1fd26e47fe": "P_k(x) = f(a) + f'(a)(x-a) + \\frac{f''(a)}{2!}(x-a)^2 + \\cdots + \\frac{f^{(k)}(a)}{k!}(x-a)^k ", "20e081d6fb099f1f8a5ba23fbb4d02a1": "\n D_{\\alpha\\beta} := \\int_{-h}^h x_3^2~C_{\\alpha\\beta}~dx_3 \\,.\n", "20e0b52e6d3e923b93e49d6e76ae95ef": "s \\models_K f_1 \\land f_2", "20e0bf7c2a3e16c86607cfbc5936c365": "\\mathrm{*}", "20e0dcdef543077c4f381bdc541f60c8": "\\tau_y", "20e0df549f6996e9f1b5a006e70e74b0": "(\\exists x (x^2 = 1)) \\land (0 = x)", "20e0ec70a6a972a34857e889c7107fbf": "a^{\\dagger}(\\phi_i)", "20e18000db1a74a9ad1701f6525c66d8": "\\left\\{A_i : i \\in I\\right\\}", "20e203719b50e9c60a5beb721a2abcd3": "p_X(x)", "20e3b3320ec256387c12e3a417c8e15b": "a^2=(c-d)^2+h^2,", "20e3d8da75328aac4f408a5453232f12": "\\varphi\\left(n\\right).", "20e454e6e64e161875b69f7f7d4167bf": "2 \\cdot 3 \\cdot 5 \\cdot p + 1", "20e4771cda51db739d2329c563d59a2e": "\\Pi^1(f_1,f_2)=\\{f_1,f_2\\}= \n\\frac{\\partial f_1}{\\partial q}\n\\frac{\\partial f_2}{\\partial p} -\n\\frac{\\partial f_1}{\\partial p}\n\\frac{\\partial f_2}{\\partial q} ~,\n", "20e4b1ce4cef9b637a7966fda43fdef5": "\\mathbb F_5[x]/(f)", "20e4bd895f17797637f5fd8a9283346a": "\n a <^d (A_i) a' \\iff (a <_i a')\n", "20e4ddef904130f6591f68da27414d00": "\n F_n(x) = \\frac1n \\sum_{i=1}^n \\mathbf{1}_{\\{X_i\\leq x\\}},\\qquad x\\in\\mathbb{R}.\n ", "20e4e0380e23aa4f093495700facc65d": "y_i = \\frac {\\sin(2^{i }x)}{2 i} \\text{ where } 0 \\leq i \\leq n-2 \\text{ and } i \\in \\mathbb{N}. ", "20e4fbc46786f1717f5bdb2ddbfd2bde": "bc < ac", "20e50ea98f1917b17e293afe55d8d6c5": "a \\in A.", "20e56303799661b54915694e9eb7e154": "\n{\\mathbf B} = \\left ( \\begin{matrix}\nw & 1 & w^2 & 1 & w^2 & w^2 \\\\ \nw & 1 & w & 1 & 1 & w \\\\ \nw & w & w^2 & w^2 & 1 & 0 \\\\ \n0 & 0 & 0 & 0 & 1 & 1 \\\\ \nw^2 & 1 & w^2 & w^2 & w & w^2 \\\\ \nw^2 & 1 & w^2 & w & w^2 & w \\end{matrix} \\right ).\n", "20e5886995251c389345ea216e9fdd38": "\\tfrac{11}{2}N_c", "20e5c67db9158cb5b79fae195918ee7f": "= (f_1'/f_1) + (f_2'/f_2) + (f_3'/f_3) + \\cdots + (f_n'/f_n).", "20e5d7490cc18392f6fbc28b77bb84b0": "P_x=(1-p)p^{x-1} \\,", "20e5df8a2f763c7895545137b4eddc20": "T_{p_j}M", "20e5e6fffaa85ba30963acb79d00d9a8": "\\displaystyle\\Delta \\phi+h(x)\\phi = \\lambda f(x)\\phi^{(n+2)/(n-2)}", "20e60181b180db8417eb68128394ae63": "{{\\alpha }_{i}}", "20e61a57965d7e65f46f4f6802bd7882": "\\hbar \\omega = W_{\\rm e} ", "20e631924672dc1155dca72bf7334ea8": "f^n(\\bot) \\le f^{n+1}(\\bot), n \\in \\mathbb{N}_0", "20e6491eb3efdfe277b7912efb2b6323": "\\mathcal{FS}", "20e68c2b18a35e29522616fabfe529af": "x = a\\lambda,", "20e69b05cff64a158fe7bea8babd9fae": "F^gX", "20e6ec0547e0d8d13187d11f6fbe1633": "\n{\\prod_{i} \\sigma_{i}} = \\Big|{\\prod_{i} \\lambda_{i}}\\Big|.\n", "20e70aceb66796005abdce2113badc12": " g_j = 1 \\pm \\frac{g_S-1}{2l+1} ", "20e71051eb695c0d296bc068390c9f64": "K(k) = F(\\tfrac{\\pi}{2},k) = F(\\tfrac{\\pi}{2} \\,|\\, k^2) = F(1;k).", "20e73cacd234c7f03921b09b65ad177c": "\\frac{\\pi}{4} = 44 \\arctan \\frac{74684}{14967113} + 139 \\arctan \\frac{1}{239} - 12 \\arctan \\frac{20138}{15351991}", "20e7a8eb368f098d27e340400a81a1e0": " arg(\\Phi_M(c)) \\,", "20e7e0a26b48eb8a18e3bca64ce74460": "n' = M_l^{-1T} n", "20e82ee1c41177d7e6b44df075d442e1": " p(x)=(x^2 - 2\\operatorname{Re}z\\,x + |z|^2)^k \\, ", "20e84f400a48a40903f9e786b066f127": "\nf(x) = \\sum_{\\alpha \\in \\mathbb{N}^n} a_{\\alpha} \\left(x - c \\right)^{\\alpha}.\n", "20e87ebc8b74accf0dc58661d7acabc1": "\\Delta t = -\\frac{R_s}{c} \\log(1-\\mathbf{R}\\cdot\\mathbf{x})", "20e8899e1d861fec592fab7ddab0ca5a": "d\\nu =-\\lambda (\\nu - \\overline{\\nu })dt+\\eta \\sqrt{\\nu }dZ_{2}", "20e89a5313e00566076a030ae2344daf": "\\sum_{k=1}^5 k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55.", "20e8ba263c4eafcb327737e5f4575875": "V: \\mathcal{E} \\rightarrow \\mathcal{F}", "20e8d40c191a93c61d5b04e6196de385": "\\frac{\\mathrm{d}K(k)}{\\mathrm{d}k} = \\frac{E(k)}{k(1-k^2)}-\\frac{K(k)}{k}", "20e916b6f9b40f4c8f63eabebf8c68a3": "+ \\,\\! ", "20e9c629c343d4968c0ae4cf598667f4": "\n\\mathbf{F} = q\\mathbf{E} + q\\mathbf{v} \\times \\mathbf{B}\n", "20ea29fc22a090d4463058a5cfae9e54": "x \\succ^p_v y", "20ea66eeabeb580bf90c4f1bee0557be": "\\frac{|A_\\epsilon^{(n)}|}{|\\mathcal{X}^{(n)}|} \\equiv \\frac{2^{nH(X)}}{2^{n\\log|\\mathcal{X}|}} = 2^{-n(\\log|\\mathcal{X}|-H(X))} \\rightarrow 0 ", "20ea9dc754ea9280c0f9ce51f3b3b663": " \\lambda = \\tfrac{1}{2} \\left (-\\frac{b}{m} \\pm \\sqrt{\\frac{b^2}{m^2} - 4 \\omega_0^2} \\right ). ", "20eaf0e6bfa304272088d48daeda9d0d": " (b^c)^d = b^{cd} \\!\\, ", "20eaf88c894f1913cf5117c411ed00f2": "M(\\vec X,Y)", "20eb09e06351effcdfdc88e2ce6a556a": "\n\\mathcal{G}(\\mathbf{k},\\omega_n) = \\frac{1}{\\mathcal{Z}} \\sum_{\\alpha,\\alpha'} \\mathrm{e}^{-\\beta E_{\\alpha'}}\n\\frac{1-\\zeta \\mathrm{e}^{\\beta(E_{\\alpha'} - E_\\alpha)}}{-\\mathrm{i}\\omega_n + E_\\alpha - E_{\\alpha'}} \\int_{\\mathbf{k}'} d\\mathbf{k}' \\langle\\alpha |\\psi(\\mathbf{k})|\\alpha' \\rangle\\langle\\alpha' |\\psi^\\dagger(\\mathbf{k}')|\\alpha \\rangle.\n", "20eb4e80b3443a2fc460b8cdfc4b66f5": "\\varphi\\colon \\Omega^k(P,V)\\cong \\Omega^0(P,V\\otimes\\bigwedge\\nolimits^k\\mathfrak g^*)", "20eb649ce82b7ec3a48f047d37d4ad53": "\n R_A + R_C = P,~~ L R_C = P a\n ", "20eb6f1040b81c05187a66039258b57c": " h=\\sqrt{H}", "20ebc192a9c432fd749a25e7987d2619": "DPW = \\frac{\\displaystyle \\pi d^2}{4S} \\left(1 - \\frac{\\displaystyle 1.16^{*} \\sqrt{S}}{d} \\right)^2", "20ebced512876bf282311e46808f575c": "p\\in X", "20ec368a325f740974924da740630500": "B^\\phi_{MX}", "20ec3ef44ff93672ef0f91965de31c7e": "F\\cap E \\ne \\varnothing,\\,", "20ec5d048278f71758b7cc98de10468a": "W(y_1,y_2)(x)\n=\\begin{vmatrix}y_1(x)&y_2(x)\\\\y'_1(x)&y'_2(x)\\end{vmatrix}\n=y_1(x)\\,y'_2(x) - y'_1(x)\\,y_2(x),\\qquad x\\in I,", "20ec656f4b690644781ff41ae15a9f5b": "e_3=-\\Omega(\\alpha^2)/\\Xi'(\\alpha^2)=(\\alpha^{7}+\\alpha^{2})/(\\alpha^{-7}+\\alpha^{-6})=\\alpha^{-3}/\\alpha^{-3}=1.", "20ece8409268c1c97991c72df7f7b24f": "E_0 = 0", "20ecfc4fb3963dd37f92f5f956ce24fb": "10log(0.693) = -1.59 dB", "20edf5c413f6c13d73d2cd279dc8dc27": " P(s) ", "20ee19f18f99994d1deed1cede2c76b1": "F(x+h) \\approx F(x)+hF'(x),", "20ee299fde926034cd775132421602ed": " L_{2'}(L_{2'}(C_a)\\cap C_b) = L_{2'}(L_{2'}(C_b)\\cap C_a)", "20ee3c4fda4df3b7df6a26eb0a98b95c": "u_{a1}", "20ee45d48f5b31f28323f3d9dcec07bf": "O = \\{X \\cup Y\\}", "20ee4bd3c57c752e5af4df1af2da7eeb": "\\textrm{Gal}(K_n/K)\\simeq \\mathbb{Z}/p^n\\mathbb{Z}", "20ee85afe621c5b693d8dfb6dd5cd115": "ds_{IX}^2 = \\tfrac{1}{4}c^2 \\left ( d\\xi^2 - dz^2 \\right ) - ab \\left \\{ \\sin^2{y} \\left ( 1 - \\chi \\cos{z} \\right ) dx^2 + \\left ( 1 + \\chi \\cos{z} \\right ) + 2\\chi \\sin{z} \\sin{y}\\ dx\\ dy \\right \\}.", "20ef6ea10ff05af1298a87f7d29f102e": "p\\left(z=\\eta\\right)=-\\sigma\\kappa,\\,", "20efb61e7aaa1c24419921a78ff18937": "\\lambda\\geq\\mu", "20efdda3d77cb43b821a90cba7c806a5": "\\tau_{MAP}", "20f011826070657b0f0092278d65ec74": "-i", "20f013db38a481ecb70acf8e41b8ab59": "\\eta_{\\mathrm{PAR}}(T) = \\frac{\\int_{\\lambda_1}^{\\lambda_2} B(\\lambda, T)\\,d\\lambda}{\\int_0^{\\infty} B(\\lambda, T)\\,d\\lambda},", "20f039dd90bf8c27d3bad3d9ac6daae4": "S_y", "20f040b12cbd7144edfec9f56525f106": "\\scriptstyle \\,s_{0,12} = 0.8046006... (+2.4\\%)", "20f056c4b95492acae5a715bd738cc27": "D(a,b)>0", "20f0c094631ad021ea89a7c1d38a4c7f": "An = m", "20f0c768a98d8b235bf0632aab83776e": " \\|g\\| = \\left(\\int_a^b g^*(x)g(x) \\, dx \\right)^{1/2} ", "20f1393be4efac6e7e12e5bead09c36d": "f(U) \\,", "20f1857ecb28af03362514f373883caf": "\\scriptstyle \\frac{p}{q}", "20f1a09a3c069fdf0fe3e5bfab14068a": " K_\\text{SE}(x,x') = \\exp \\Big(-\\frac{|d|^2}{2l^2} \\Big)", "20f1a2931818d3d66250da25dad395b6": "\\tan 2\\theta_0 = \\frac{K\\sum_{k=0}^{K-1}\\sin 2\\omega t_k - 2\\left(\\sum_{k=0}^{K-1}\\cos\\omega t_k\\right)\\left(\\sum_{k=0}^{K-1}\\sin\\omega t_k\\right)}{K\\sum_{k=0}^{K-1}\\cos 2\\omega t_k - \\big(\\sum_{k=0}^{K-1}\\cos\\omega t_k\\big)^2 + \\big(\\sum_{k=0}^{K-1}\\sin\\omega t_k\\big)^2},", "20f1a322fdf387ae8d55344af2420f8e": "M^{+\\bullet}", "20f1e3ed4c495dc14fab17ea99c757db": "\\gamma_\\mathrm{la}\\ >\\ 0\\ >\\ \\gamma_\\mathrm{ls} - \\gamma_\\mathrm{sa}", "20f1fd21cc52a413261f5785db47d395": "K^A_{eq} = K^{A,0}_{eq} \\mathrm{e}^{-(\\Delta H^0_{ad}\\,\\alpha_T \\,\\theta / k\\,T)}", "20f20e28d803a40b26b36c30a091f9e0": "(a*(a+1))*(a+2)", "20f215e7f6d0ce2da52b45114602170b": "\\langle p \\rangle", "20f21ddfebdc5fe8545d810077835346": "s_{pm}=i\\,\\mathrm{cd}(w,1/\\xi)\\,", "20f22d577843609e922439f5cc25199e": "\\varphi (x) = k_0 n (x) L = k_0 L [n + n_2 I(x)]", "20f257992ca15946e44f09e6a1c75a70": "\\begin{align}\n\\iint_D &\\sqrt{(\\vec{r}_u\\cdot\\vec{r}_u)(\\vec{r}_v\\cdot\\vec{r}_v)-(\\vec{r}_u\\cdot\\vec{r}_v)^2}\\,du\\,dv\\\\\n&\\quad=\\iint_D\\sqrt{EG-F^2}\\,du\\,dv\\\\\n&\\quad=\\iint_D\\sqrt{\\operatorname{det}\\begin{bmatrix}E&F\\\\ F&G\\end{bmatrix}}\n\\, du\\, dv\\end{align}\n", "20f27b2c7fc9ea3ef1929069fec0c678": "G\\not=H.", "20f28e5eab4848360edb8d12ed7e10af": "\\cos{(\\omega t+\\phi)} \\ ", "20f2b5df65abf7eabf1a33e774da9b1e": "V^2 - U^2 = \\left(1-\\frac{r}{2GM}\\right)e^{r/2GM}", "20f312ee72f7c11b18aa61f8bec45901": "{c}_{1},{c}_{2}\\ =0\\ or\\ 1", "20f318bcdff4f6671c214a5879c6f2e5": "\\omega:TM\\to T^*M, ", "20f3289d8ea77e7fa9e383bca697fe8f": "{\\color{Red}\\tfrac{3}{m}}", "20f32ebdcb84b5c527f71f06f65e47b3": "\\frac{6(\\alpha^3+\\alpha^2-6\\alpha-2)}{\\alpha(\\alpha-3)(\\alpha-4)}\\text{ for }\\alpha>4\\,", "20f337d4f696c19aef8092a04a3d2bcd": "-10 \\le x,y \\le 10", "20f345c02d0d0cfe94e015beea799a87": "\\scriptstyle kE_b", "20f389c6c2e5772002e158267812736f": "L_{\\infty} = 0", "20f3a2bf12102a1b2bea413f120bc892": " T_0 = T_1 \\left(\\frac{p_0}{p_1}\\right)^{R/c_p} \\equiv \\theta. ", "20f3b0ad27d8efefa70322e562bed881": "{~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~}", "20f3b928b89c294fd4d77e5207cdb2e5": "C_m = \\lambda_p \\alpha \\beta t", "20f3c6f714eca082ebe481740c044aaa": "\\Delta\\mathbf{B}=\\mu_{\\Delta} \\Delta\\mathbf{H}.", "20f40b40f35d779133c4ce09a9b04a33": " 0.0003207\n \\times weight \\mbox{(g)}^{(0.7285 - 0.0188 \\log_{10}{weight \\mathrm{ (g)}})}\n \\times H^{0.3} ", "20f462cf01b3cd1486e2f778344f188b": "n_{ij}", "20f48d7a2997abd00ab3ce9150044f8e": "g\\approx Gm_s/R^2", "20f4f84a176d1a289dfe885b7329230d": "\\frac{AF}{FB} = \\frac{AF'}{F'B}.", "20f4fc9845a6d15afc7325d0c8de7100": "\\delta_D(k,i) = \\lambda\\delta_D(k-1,i) + \\frac{e(k,i)e_b(k,i)}{\\gamma(k,i)}", "20f565b0e375e4530e9e06572d540815": "W_{i}(x_j) = e^{- \\frac{(r_i(x_j)+s)^2}{c}}", "20f582adbd88049d88a7e3fc28b77abd": "\nS_{i+1}(c) = \\begin{cases}T(N_i(c)), & c \\in I\\\\\n S_i(c), & c \\in \\Z^k \\setminus I \\end{cases}\n", "20f5bb7768a2835572ec8afd83908156": " \\scriptstyle \\mathcal{P} =\\left\\{P=\\{ x_0, \\dots , x_{n_P}\\}|P\\text{ is a partition of } [a,b] \\right\\} ", "20f63e25158b1a65da45158667578138": "x (\\theta) = r (k + 1) \\cos \\theta - r \\cos \\left( (k + 1) \\theta \\right) \\,", "20f65f5ce12aba395d6bde91fad3fbcc": "2n-3j-3=0", "20f68045237ddb0c784f763d2dfcbcc7": "\\Pr(X=x) = \\sum_{y} \\Pr(X=x,Y=y) = \\sum_{y} \\Pr(X=x|Y=y) \\Pr(Y=y),", "20f69ee8b55dfa42fbcd834a08e51e6d": "\\displaystyle{E=E_0(e)\\oplus E_{1/2}(e) \\oplus E_1(e),}", "20f6c3cf5ba601cb6b721de9000c9fcf": "-\\omega_{p}", "20f708c6b908a93c04614f84ec9755f8": "\n A = \\begin{bmatrix}\n 0 & 1 \\\\\n 0 & 0 \\\\\n \\end{bmatrix},\n", "20f72e3a694f33b819326f512e033e8a": "\\mu = \\frac{1}{n} \\sum_{v \\in V} f(v)", "20f733ab3007d15c80c69b09ba23cb7a": "\n\\sum_{j=1}^p a_j + \\sum_{j=1}^q b_j = \\sum_{j=1}^m c_j + \\sum_{j=1}^n d_j,\n", "20f75d80400b9fad74bf07698f1f233c": "\\scriptstyle a^2 \\,+\\, 2", "20f794b4581f8eb8790f9de4506a79be": "\\!v_1", "20f7a05bbd8892a2883a8187a92a8a59": "\\Gamma(\\tfrac18) \\approx 7.5339415987976119047", "20f7d74d908ea49f9ae41a4fe8f99407": "\\mu = \\cos(\\theta)", "20f7e12a8a2fde1243dba86c544f4a16": " \\nu_e, \\nu_{\\mu}, \\nu_{\\tau} ", "20f7e76830a65a841b88e32259f08c02": "A^{-1}[j]", "20f7ee512c044c85911c9c71c1abd4c1": " \\leq\\sum_{a^{n},b^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}},\\ b^{n}\\neq a^{n}}\n\\Pr\\left\\{ E_{a^{n}}\\right\\} 2^{-\\left( n-k\\right) }", "20f865f3d198abbeedbde0c70f886dcb": " f^{-1}(I) = \\{x\\in X \\,|\\, f(x)\\in I \\}. ", "20f86b6a2b331ecfbcc72e9da3cfbb78": " |\\lambda - a_{ii}| = \\left|\\frac{\\sum_{j\\ne i} a_{ij} x_j}{x_i}\\right| \\le \\sum_{j\\ne i} \\left| \\frac{a_{ij} x_j}{x_i} \\right| \\le \\sum_{j\\ne i} |a_{ij}| = R_i", "20f88010d58f526bc934d1be41e9f6a2": "\\begin{array}{rcl}\n \\dot x &= &Ax + b \\varphi(\\theta_e),\\\\\n \\dot \\theta_e &= & \\omega_e + g_v (c^{*}x). \\\\\n\\end{array}\n\\quad\nx(0) = x_0, \\quad \\theta_e(0) = \\varphi_0.\n", "20f8927b47392d78410d77c1f9fea521": "\\mathbf{B}(\\mathbf{r}, t) = \\frac{\\mu_0}{4 \\pi} \\left(\\frac{q c(\\boldsymbol{\\beta} \\times \\mathbf{n})}{\\gamma^2 (1-\\mathbf{n} \\cdot \\boldsymbol{\\beta})^3 |\\mathbf{r} - \\mathbf{r}_s|^2} + \\frac{q \\mathbf{n} \\times \\Big(\\mathbf{n} \\times \\big((\\mathbf{n} - \\boldsymbol{\\beta}) \\times \\dot{\\boldsymbol{\\beta}}\\big) \\Big)}{(1 - \\mathbf{n} \\cdot \\boldsymbol{\\beta})^3 |\\mathbf{r} - \\mathbf{r}_s|} \\right)_{t_r} = \\frac{\\mathbf{n}(t_r)}{c} \\times \\mathbf{E}(\\mathbf{r}, t)", "20f8c201a44fe1396fdaf93ca731c6a6": "e^{2\\pi i ( \\pm\\alpha + k ) / N}", "20f955c07e2b78ce401cb5957f562829": "\n\\begin{align}\n\\lambda &= \\frac{3x_p^2 + a}{2y_p}\\\\ \nx_r &= \\lambda^2 - 2x_p\\\\ \ny_r &= \\lambda(x_p - x_r) - y_p\n\\end{align}", "20f9785ef496bbc909e3bdb4cbb672a5": "\\beta>0.", "20f9cc840114df83f38f208b082a24f9": "\\delta\\rightarrow0", "20fa156064ac8e04951081b1b13a24de": "\\rho = \\mathrm{Diag}(1-T, T/(d-1),\\dots,T/(d-1)) \\,", "20fa6aa9e7878378c2d71004716b15f7": " a+ L s_1 +L s_2... + L s_{D-k}", "20fafdce54af27770f7b3ef6278b5812": "\n \\begin{align}\n \\sum f_1 & = F_1 + \\int_{\\alpha}^{\\beta} \\left[\\sigma_{rr}(a,\\theta)~\\cos\\theta\n - \\sigma_{r\\theta}(a,\\theta)~\\sin\\theta\\right]~a~d\\theta = 0 \\\\\n \\sum f_2 & = F_2 + \\int_{\\alpha}^{\\beta} \\left[\\sigma_{rr}(a,\\theta)~\\sin\\theta\n + \\sigma_{r\\theta}(a,\\theta)~\\cos\\theta\\right]~a~d\\theta = 0 \\\\\n \\sum m_3 & = \\int_{\\alpha}^{\\beta} \\left[a~\\sigma_{r\\theta}(a,\\theta)\\right]~a~d\\theta = 0\n \\end{align}\n ", "20fb4d8ede3411cc607eda7243facd2d": "\\bar{t} \\phi(P) = 2\\bar{q} P", "20fb4da8a6c902f16c462cdc9262eab6": "A \\Vert c_{jk} \\Vert^2_{l^2} \\leq \n\\bigg\\Vert \\sum_{jk=-\\infty}^\\infty c_{jk}\\psi_{jk}\\bigg\\Vert^2_{L^2} \\leq \nB \\Vert c_{jk} \\Vert^2_{l^2}\\,", "20fb90202322cf9249d954b2c9dfe49c": "\nT_j =\n\\begin{cases}\n0, & \\text{if } f'(j) \\text{ is an integer}; \\\\\n\\min\\left(\\frac{1}{||f'(j)||}, \\sqrt{U}\\right), &\n\\text{if } ||f'(j)|| \\ne 0; \\\\\n\\end{cases}\n", "20fba7355d3343ae5512ac81dbc86711": "\\textstyle B \\subset \\mathbb{R} ", "20fbf4588c21b11d252391ab537df1f8": "\n\\operatorname{dCov}^2_n(X,Y) := \\frac{1}{n^2}\\sum_{j, k = 1}^n A_{j, k}\\,B_{j, k}.\n", "20fc23b3c8c153837ab352af17737d64": "2^{2^k}", "20fc9bd90179cc1e0efed843ad24baaa": " \\mu_0 < \\frac{ x + m }{ 2 } \\pm k | x - m |", "20fcafe0b441c06f880bebb0bf710f03": "as_k+bt_k=r_k, ", "20fcc2b0b0db33ad9173adf8c846b2b6": " i,j\\in[n]", "20fcfc90e076ffd4c3383e1cf53e54ed": "f_j(x) = x^{m_j}", "20fd09ec99f9adf3093962f9014bd228": " \\forall a \\forall b \\;a \\vee (a \\wedge b) = a ", "20fdd72c3baf5f669923c415149b7bca": "{1 \\over 2} \\ln(a/b) + \\ln 2 = \\ln\\bigl( 2 \\sqrt{a/b} \\bigr).", "20feaada7405d1e5d23377966f803dd9": "v\\in C", "20fed57f6034f18c49ae6be9d1b5f2d5": "\n I_1 = \\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2 = \\lambda^2 + \\cfrac{1}{\\lambda^2} + 1 ~;~~\n I_2 = \\cfrac{1}{\\lambda_1^2} + \\cfrac{1}{\\lambda_2^2} + \\cfrac{1}{\\lambda_3^2} = \\cfrac{1}{\\lambda^2} + \\lambda^2 + 1\n ", "20fedcb1ee65fa5b298b9f81ff297d9d": "e(n) = \\|x(n) - \\widehat{x}(n)\\|\\,", "20ff348f7148c598a3116aab8a6efbc7": "\\frac{36}{12}=3", "20ff553787d7e49312e41b129a86394f": "1.00U($1\\text{ M}) < 0.89U($1\\text{ M}) + 0.01U($0\\text{ M}) + 0.1U($5\\text{ M}),\\,", "20ff593c848a70c0dabb59b9ad133748": "p(n) \\approx {n^2 \\over 2m}", "20ff69eab81f54bda6a85e7a571105ae": "\\delta(w,C)>\\delta", "20ffc407a7df1e1a33a10467ebab7083": "\\frac{\\mathrm{d} P}{\\mathrm{d} T} = \\frac {P L}{T^2 R}.", "20ffc5333da43150cce288425448048e": "\\frac{1}{\\lambda} = RZ^2 \\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)", "21002a065f857b236bdeb31eeb5903f9": "N,T", "21005d7e8daa3cbad69503aa1690b27b": "\\scriptstyle\\mathbf{F}_\\mathrm{B}", "21008212015dff44f37d38b5323be66c": "A(G(U_\\ell))", "210085fa8d344c8d675a5a95c14160b0": "\\mu(E)=1\\,", "2100a14a01f7575769649a657dc173e8": " \\lambda = \\frac{4 (\\beta^2 - 1)}{h^2}. ", "2100cba1a2d20db5e948a8ee6b375efc": "m=30, s_\\Lambda=72.82", "2101799bfb33d9ecb0ddfb6d9a945c04": "U \\in SU(2)", "2101a0dae11357eadf8218353debbb13": " \\sum_{i=1}^N", "2101c52ab5c2165a9ab05e436e5c1225": "\\ F_{propulsive}= - drag \\times cos (\\beta) ", "2101e755479497032bdb2be4540a733d": " \\forall n : \\sum_{i_n=1}^{I_n} w_{n,i_n}(x_n) = 1 ", "2101faf5535187deafb25b1f29820a5b": "N - 1", "210213e0226d4a0a771d39cad2f8be36": "\\int_{-\\infty}^\\infty \\frac{e^{ix}}{1+x^2}\\,dx=2\\pi i\\,\\operatorname{Res}(f,i)\\,.", "210232a31406fa33daaa96c8891b893d": "f'(r)^{-1}", "21024fda36099e6e51162b85632c4050": "f(\\partial X) = f(X) \\cap \\partial Y", "21025654afc9c9d35b1c6c001c0ab0c2": "\\ v_i = \\sqrt {\\frac{2GMd}{r^2}}\\ ", "21025746c635a505f874340700583b32": "f \\in \\Lambda^0(M)", "21029821e28fff21e7e513a03cd7935b": "\\mathrm{Var}[X_i] = \\frac{\\alpha_i (\\alpha_0-\\alpha_i)}{\\alpha_0^2 (\\alpha_0+1)},", "2102dd1b2044d44f90fa82e962aedbb4": "A=\\Gamma +\\sigma ", "21031b3f34f0ba738d4b950e92c371fd": "B(\\mbox{G})= \\mu_r H(\\mbox{Oe})", "21033766da7e8eb0f8fec8d49a59366e": "h(p, u) = x(p, e(p, u)), \\ ", "21033b6683ac17eff06663bdc84ad8b4": "E = \\langle H \\rangle = \n\\int_{\\Omega_r}\\psi_{\\bold{k}}^*(\\bold{r})[T + V]\\psi_{\\bold{k}}(\\bold{r}) d\\bold{r}\n", "2103c5d935de1b88534eba03240c2f86": "[x] E [y]", "2103fa777b761eb5f96147eb13b9ea1c": "\\psi(\\Omega^\\Omega \\psi(0))", "210406315148712b30c14738a90c7912": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log ((\\frac{\\varepsilon}{3.71D}) + (\\frac{7}{Re})^{0.9})\n", "21043b06a7c2486a7f60484d7b6ed109": " \\Delta_3 = \\begin{vmatrix} I_1 \\left ( ka \\right ) & ka I_0 \\left ( ka \\right ) - I_1 \\left ( ka \\right ) & -ka K_0 \\left ( ka \\right ) - K_1 \\left ( ka \\right ) \\\\ I_0 \\left ( ka \\right ) & I_0 \\left ( ka \\right ) + kaI_1 \\left ( ka \\right ) & -K_0\\left ( ka \\right ) + ka K_1 \\left ( ka \\right ) \\\\\n\\frac{\\mu_A}{\\mu_B} I_1 \\left ( ka \\right ) & \\frac{\\mu_A}{\\mu_B}kaI_0 \\left ( ka \\right ) & -ka K_0 \\left ( ka \\right ) \\end{vmatrix} ", "2104539ef0681ef8b631148125964af5": "A\\ \\mathsf{Type}", "21046019a83639b34b1a4554f090dae6": "m := (k (T \\bmod R))\\bmod {R}", "2104a6c38162f12e8ee445ee880a95fd": "\\int\\mathrm{versin}(x) \\,\\mathrm{d}x = x - \\sin{x} + C", "2104c757c287b58721a5eadd66dd6e74": "AT = T_{\\rm a} + 0.33e - 0.70ws - 4.00", "2104cef2f88129d565f97bce6dc33772": "\\dot{\\mathbf{x}}(t) = A \\mathbf{x}(t) + B K \\mathbf{y}(t)", "21055bff5955e4cc9c27d48e936cab73": "XX'\\,", "21055e9940bad315fe988f32326b4532": "A = \\frac{I^{right}-I^{left}}{I^{right}+I^{left}}", "21056027de74fcbd53667fce78c80309": "G = P^{(\\pm)} G + P^{(\\mp)} G", "210575ab02fd679c06facac66677c8c9": "\\Box A^\\alpha = \\mu_0 J^\\alpha", "210581cd06642c70a8e15d8b62a96ed1": " \\mathbf{b}^i\\cdot\\mathbf{b}^j = \n \\begin{cases} g^{ii} & \\text{if } i = j \\\\\n 0 & \\text{if } i \\ne j,\n \\end{cases}\n ", "21060f2b7d0257a90489be69c28de392": "\\kappa_0", "210614a1117e89a255b9d0b39516b604": "L_s", "21063e45d6cd23e1a9bd54efe94fefaf": "\\left\\langle\\mu_z\\right\\rangle = \\mu L(\\mu B\\beta), ", "2106559595e7e9ff37bce9ae5a4e9dfd": "G(K_1, K_2, \\ldots, K_n)=\\sqrt[n]{K_1K_2\\cdots K_n}=\\left(\\prod_{i=1}^n K_i\\right)^{1/n}", "210698eaa5ec36e1aa91641fed266fcc": "\\rho = \\frac{1}{h^n C} \\frac{1}{W} f(\\tfrac{H-E}{\\omega}),", "2106e21acdeca804d32fce51647b5337": "a_2b_2a_2^{-1}b_2^{-1}", "2106e48ad038fad267b57b1dd6ea9bce": "|\\mathbf{W}|", "2106f45529bc655f7837b9cfc58b5e52": " n_{i,j} \\ge n_{i,j+1} \\quad\\mbox{and}\\quad n_{i,j} \\ge n_{i+1,j} \\,", "21071ec7d32c3de41502a24fdad955f6": "\\nu(\\empty) = 0, \\nu(\\Omega)=1).", "210738af164bceaefde27ec1c8849854": "\n\\begin{align}\nU(\\rho,\\omega,z)\n&\\propto \\int_0^\\infty \\int_0^{2 \\pi} A(\\rho') e^{-i \\frac{2\\pi}{\\lambda z}(\\rho \\rho' \\cos \\omega \\cos \\omega' + \\rho \\rho' \\sin \\omega \\sin \\omega' )} \\rho' d \\rho' d \\omega'\\\\\n&\\propto \\int_0^{2 \\pi} \\int_0^{\\infty} A(\\rho') e^{-i \\frac{2\\pi}{\\lambda z}\\rho \\rho' \\cos (\\omega - \\omega') } d \\omega' \\rho' d \\rho'\n\\end{align}\n", "2107779565348eb246bd0cd86956f98a": "q = e^{2i\\pi\\tau}", "210791c540797d0752f4374e61a3dfc2": "\\begin{align} 2\\cdot R_*\n & = \\frac{(28.6\\cdot 0.70\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 4.31\\cdot R_{\\bigodot}\n\\end{align}", "2107ac18348daad232de0dcbff57ec35": "\nNSE_i = h_i^t \\times G \n", "2107bb7fb91501a501bc1a39117fdb0a": "X^2\\backslash\\left\\{ y~\\backepsilon~y\\succ x\\right\\}=\\left\\{ y~\\backepsilon~x\\succcurlyeq y\\right\\}", "21082734df9c3874201e90b00aaf7d32": "\\dot{x}(t) = Ax(t) + Bu(t)", "21084b46aa1ea7f5ed5684b74d102a84": "a_0+a_1x+a_2x^2+\\cdots+a_{n-1}x^{n-1}", "210852a558482b5b517bed4854f026ee": "TE = \\omega =\\sqrt{\\operatorname{Var}(r_p - r_b)} = \\sqrt{{E}[(r_p-r_b)^2]-({E}[r_p - r_b])^2}", "21086ffb2a96c5ce628d0bab4a880c04": "\\mathit{prob}_{\\mathit{before}}(\\psi \\rightarrow \\phi) = \\lang \\phi | \\rho_{\\mathit{sys}} | \\phi \\rang = \\lang \\phi | \\psi \\rang \\lang \\psi | \\phi \\rang = {| \\lang \\psi | \\phi \\rang |}^2 = \\sum_{i} |\\psi_i^*\\phi_i|^2 + \\sum_{ij;i \\ne j} \\psi^*_i \\psi_j \\phi^*_j\\phi_i", "2108ca3cc3e0f456e3d02d406e218e9d": "\\|A\\|", "2108e0e0a702e6a6e2c10721c6525a78": " E(X,U_D) ", "2108f821a8320e949b80836c467fa065": "Y_i=I(\\boldsymbol{X}=i),", "210908c31a22a5686e1bf1d6927ee932": " f(x)= {\\mathcal F}^2 f(-x)", "210912d7ada6f25132d0b99ece462deb": "{\\Bbb Z} + \\alpha\\cdot {\\Bbb Z}", "210999d61bacf4ccb392353212aaa396": "P_{k+1}= P_k+ \\left( 1-P_k\\right) u_k \\otimes {v_k^* A\\left(1-P_k \\right) \\over v_k^* A\\left(1-P_k \\right) u_k}.", "210a2008c014aa4224a8879073870164": "|S_N|=\\sum_{i=1}^\\infty|S_i|\\,1_{\\{N=i\\}}.", "210a252876d793288f2fa87d4584d801": "\\epsilon = \\hbar\\omega = h\\nu", "210a64e0047f7efdd5b2e2fed364a783": "M(p)\\le \\sqrt{|a_0|^2 +|a_1|^2 +\\cdots |a_n|^2}\\,.", "210a92bee716e5e7ce6793cd30d98945": "\\mathrm{P}(A_m \\cap A_n) = \\mathrm{P}(A_m)\\mathrm{P}(A_n)", "210adbc7aca929cfe25208c13de793bc": "\\sigma_T = \\sigma \\sqrt{T}.\\,", "210afcc5dabc5d571a560937efc7bbdf": "\\operatorname{str}(A^{-1} T A)=(-1)^{|i'|} (A^{-1})^{i'}_j T^j_k A^k_{i'}=(-1)^{|i'|}(-1)^{(1+|j|+|k|)(|i'|+|j|)}T^j_k (A^{-1})^{i'}_j A^k_{i'} =(-1)^{|j|} T^j_j\n=\\operatorname{str}(T).", "210b858ee9912f329ee30b89ee11377e": "\\delta ^{13}C = \\Biggl( \\frac{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{sample}}{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{standard}} -1 \\Biggr) * 1000\\ ^{o}\\!/\\!_{oo}", "210be343b12fd63b62fa20d1bae83423": "X_g \\in T_g G", "210c2bad6a6071e6d3d354d214810bc2": "x_{n}=\\sum_{k=1}^{L-1} a_k x_{n-k}", "210c50e20fd80389a61bb47811bc02c4": "\\mathbf{\\mathit{n}}<\\mathbf{\\mathit{d}}", "210cb4d4b54d55fc8e199f2a9325180a": "v = v_0, v_1, ...., v_{n-1}", "210d19383a60789aee5cd4279ac40f7c": " I(x)= \\frac{k}{t_1-t_0} \\int^{t_1}_{t_0} \\Psi\\Psi^{\\mathrm{*}} \\, dt", "210d3b1eff8734130cbbc88f5f8a7781": "Retic Index = Retic Count * {Hematocrit \\over Normal Hematocrit}", "210d440b51e4a0d583dda554afca940c": " C = \\left[\\sum_{i=1}^n a_{i}^{\\frac{1}{s}}c_{i}^{\\frac{(s-1)}{s}}\\ \\right]^{\\frac{s}{(s-1)}}", "210d8ee6f28e6db6da836357fb8b3b23": " T(z) = \\sum L_n z^{-n-2}.", "210dabdf79eaa352193ae8c434ca4999": " \\ g_{\\phi}= \\left(9.7803267714 ~ \\frac {1 + 0.00193185138639\\sin^2\\phi}{\\sqrt{1 - 0.00669437999013\\sin^2\\phi}} \\right)\\,\\frac{\\mathrm{m}}{\\mathrm{s}^2} ", "210dce79f6446af4a23630f02d0c89ea": " |T_j \\cap T_k| > \\frac{2t}{d^2}] \\leq n^2 \\cdot n^{-4} = n^{-2}", "210dcf7c0826f72842e034eefcef60e4": " f_n(x) = \\sin(n \\pi x) \\quad n= 1,2, \\ldots ", "210e25391dc4d1fa712937abc3502fc3": "\\begin{align}\n t &= \\frac{c}{g} \\operatorname{arctanh}\\left (\\frac{c T}{X}\\right) \\;\\overset {X \\,\\gg\\, cT}\\approx\\; \\frac{c^2 T}{g X}\\\\\n X &\\approx \\frac{c^2 T}{g t} \\;\\overset{T \\,\\approx\\, t}\\approx\\; \\frac{c^2 }{g}\n\\end{align}", "210e7fbb7f07dc95138df019481df201": "y = b+r\\,\\sin t\\,", "210efea0d341988ec7e5c6d3427dba91": "\\lim_{t \\to \\infty} \\frac{1}{t}m(t) = 1/\\mathbb{E}[S_1].", "210f450adcd4b85d828fceec2646ca18": "\\left(\\bigcup A_\\alpha\\right)^C", "210f4634b0ea12fe19ad659528cd57a0": "p_1: H\\rightarrow G", "210f5e8a28ec8ccadfc74756c048e84c": "f' \\circ g = T(h) \\circ f", "210f9a4f8aa2e5d730f8d7b30f74af11": "\\lim_{r \\to 0} \\frac1{\\mu \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\mu (y) = f(x)", "210fa56cfae1e6d423a7029786c411c4": "\\mathrm{Gamma}\\left(\\alpha=3, \\theta= 2/3 \\right)", "210fb18336e9db0a0c4f43ecec3af608": "\\zeta(\\theta,\\tau)", "210fb4ef99cd876a142d5be659f19551": "c_{1}(t') = \\dfrac{\\mu_{01}\\epsilon_0}{i\\hbar} \\int_{0}^{t'} \\mathrm{d}t \\exp\\left(-i\\frac{E_{0}- E_{1}}{\\hbar}t\\right)\n\\exp({i\\omega t})=\\int_{-\\infty}^{+\\infty} \\mathrm{d}t \\exp\\left(-i\\frac{E_{0}-E_{1}}{\\hbar} t\\right)H\\left(\\frac{t}{t'}-\\frac{1}{2}\\right) \\exp\\left(i\\omega t\\right)", "2110031e1734e005d86fcd041e2227b4": "(X_\\infty, d_\\infty)=\\lim_\\omega(X_n,d_n)", "21108df1fdaa3d9d4d6537ddf33a8935": "x^2 + \\frac{b}{a} x= -\\frac{c}{a}.", "2110c431de90c55f41f318401085fd33": "ck\\frac{x^{c-1}}{(1+x^c)^{k+1}}\\!", "2110eb159e2837a4c5ba8607afd39842": "\\eta = (d-a+r)/2c", "21111d16b98301c1a7689d454b01d926": "R=q_1+\\left(\\frac{q_2'}{q_2}\\right)", "211173d48327977749523fd0cf1cd1db": "\\delta=\\omega", "2111a4ca6b798d17e1e24a71e2133417": "\\nu_{2,1}( \\mathbb{R}_- , \\mathbb{R}_+)", "2111debd4daefa01aa7fd538468aff13": "\\sqrt{\\frac{4\\alpha}{\\pi}} \\,", "2111f5cb30941292d805e3f23c37b59d": " \\beta_0^{(P)} = \\frac{-1}{V_0^{(P)}} \\left ( \\frac{\\partial V}{\\partial P} \\right )_T = \\frac{0.4343 C}{V_0^{(P)}(B+P)}", "21127b7c198b4b23cffaf6e165d5811d": "\\left\\{K_t(\\omega)\\right\\}_{t=0}^T", "211280d6ac65cbea1979a6605899eb13": "\n(H_\\mathrm{sat} - H_0) \\cdot \\lambda \\cdot k' = (T_0 - T_\\mathrm{wb}) \\cdot h_\\mathrm{c}\n", "211281577c395023b46990d7232b99ff": " s = - \\alpha \\pm \\sqrt{\\alpha^2 - {\\omega_0}^2} ", "21128eb55b23661e0bd31443607b7008": "\\rho_A", "2112d62ca10eb93d2f30104298fbefab": "\\scriptstyle 0 \\le i \\le n", "2112d8b4f21da3344c70855f72d46566": "\\beta\\left(t\\right)", "21132b004174eb9f6d2f8270c7840198": "B_H", "21137d59846cb3308e7ca2b1df56613c": "B_p=\\left \\langle s_p(t),h_p(t) \\right \\rangle", "21138ddf8bfc90594a31b76e7613f041": "x(t) = \\sum_{n=-\\infty}^{\\infty} x(nT)\\cdot \\mathrm{sinc} \\left( \\frac{t - nT}{T}\\right),", "2113bf8814c95c6b906bd14e90814d2b": "\\textstyle \\prod", "2113de175e87a01dbc1a3e9a2de21f24": "\\epsilon_{\\rm XC}(n_\\uparrow,n_\\downarrow)", "211417d5fc075d7dd97ce20375a1c7d2": "\\mathbf{b} \\in [\\mathbf{b}]", "21148ab8c96c67f17ae1a377d3d25804": "\\ \\beta \\approx 90 \\deg", "21149001bffe394775cc14d867c6a0d5": "S_1=\\{v_{k+1},v_{k+2},\\ldots,v_n\\}", "21149c37d56deec56d426d926e6bf21a": " \\bold{x}_i = (\\underbrace{0,\\ldots,0}_{i},1,\\underbrace{0,\\ldots,0}_{k-i}) \\ \\ \\text{for all} \\ 0 \\le i \\le k .", "21149f5c3e297243c120e0b5ba67a9cb": "\\displaystyle{|f(z)-f(w)|\\le C |z-w|^d,}", "2114b0474c49a7b51407aefb6bd0fa0c": "c_4", "2114ced7573391d9d333a7445ed4e84b": "\\scriptstyle \\mathrm{N}", "21151a45612d9ca246743b9ed723d972": "\\frac{\\Gamma(\\frac{\\nu+1}{2})} {\\sqrt{\\nu\\pi}\\,\\Gamma(\\frac{\\nu}{2})} = \\frac{(\\nu -1)(\\nu -3)\\cdots 5 \\cdot 3} {2\\sqrt{\\nu}(\\nu -2)(\\nu -4)\\cdots 4 \\cdot 2\\,}. ", "21154b597796752cfca7b95438a0a101": "g>1", "21155319d91de83485dcb50ed8921daf": "x^2 = H(mU) \\mod n", "211583217c1b986c221fbba8ff28a722": "k_1 \\rightarrow k_2", "2115c611d7d0b89269dbac416a156588": "n=n'", "2115e4cdb2479fe88369518753500fcb": " r_1 \\approx -\\frac{b}{a}, ", "2116b38bafca8f9ed4a322933759117e": " E_{snake}^*=\\int\\limits_0^1E_{snake}(\\mathbf{v}(s))\\,ds=\\int\\limits_0^1 ( E_{internal}(\\mathbf{v}(s))+E_{image}(\\mathbf{v}(s))+E_{con}(\\mathbf{v}(s)) )\\,ds", "2116d3d06ec5a6087b916394b5b3fdb2": "\n C \\circ C([x]) = C([x]).\n", "21170fe882e70504bd1ab64a6c66ead1": " \\mathcal Q", "2117792ab55e0c69b9083d88bbd58990": "D_{\\mathrm{KL}}(Q || P) = \\sum_\\mathbf{Z} Q(\\mathbf{Z}) \\log \\frac{Q(\\mathbf{Z})}{P(\\mathbf{Z}\\mid \\mathbf{X})}.", "2117b6f5085f8ddd319b323b304d263c": " P_i = \\Pr( \\varepsilon_i - \\varepsilon_a > V_a - V_i ) ", "2117c2b079a80ae8c736299613fbef07": "U \\rightarrow X", "2117c36a7d4ffb499a7711f2f728394a": "b + 1", "21181731231c92f29b583c210be2c22b": "U_p= -\\frac{1} {4 \\pi} \\iiint\\limits_V\\left(\\frac{\\nabla^2\\cdot\\mathbf{U}}{R}\\right) dV_Q", "21189b731b9643e7c074925dec2b0982": " 2 \\mathbf{A} = 2 \\begin{pmatrix}\na & b \\\\\nc & d \\\\\n\\end{pmatrix} = \\begin{pmatrix}\n2 \\!\\cdot\\! a & 2 \\!\\cdot\\! b \\\\\n2 \\!\\cdot\\! c & 2 \\!\\cdot\\! d \\\\\n\\end{pmatrix} = \\begin{pmatrix}\na \\!\\cdot\\! 2 & b \\!\\cdot\\! 2 \\\\\nc \\!\\cdot\\! 2 & d \\!\\cdot\\! 2 \\\\\n\\end{pmatrix} = \\begin{pmatrix}\na & b \\\\\nc & d \\\\\n\\end{pmatrix}2= \\mathbf{A}2.", "2119519f8c668c4f3d888a7652803e04": "\\gamma_1^1", "2119ded08945b62cc65af8ac978d99ac": " \\operatorname{build-param-lists}[q\\ q, D, V, T_6] ", "2119e2a2916bb0681bb1c78e628d87a4": "(p_{\\alpha})", "211a0fcc6ae7e91141dea389445255dc": "(1+\\sqrt{-163})/2", "211a5f1e2be060cf6061b48659e97e7f": "p_1^2 = S \\cdot q_1^2 \\pm 1", "211a8d29e41f913abbaea0f6fa595518": "\\sigma \\colon \\{1, \\cdots ,n\\} \\longrightarrow \\{1, \\cdots ,n\\}", "211a91193c9bb9f557c997884842f72c": "\\scriptstyle k\\left[M\\right] / \\mathrm{Ker} F_i \\,\\to\\, F_i\\left(k\\left[M\\right]\\right) \\;=\\; k", "211a9403e14d8709c63a3914bc3598d8": "p(\\theta|M)", "211aa45515c41464c690f98b0693d281": "u_\\nu(T) = {8\\pi h\\nu^3\\over c^3}{1\\over e^{h\\nu/k_\\mathrm{B}T} - 1}.", "211adc5c7e0543cc8ee753574846e3c7": " \\mu_s (\\sigma)=-kT \\ln \\int p_s (\\sigma') e^{- \\frac{E_{int}(\\sigma,\\sigma')-\\mu_s(\\sigma')}{kT}} d\\sigma'", "211ade7224e0efc1d390159956191b56": "2^b-M = 16-10 = 6", "211b1d1d7e5573449c1f9ad4d1af78f6": " q \\leq n ", "211b390281a508a10972c1154a2b54b3": "d(x,y)=d(z,x)", "211b77134cd9624c3fb9f750f5a8ceac": "O(f(x))", "211b7ac1f9be504c00a742ec71212848": "\\hat{S_{i}}\\otimes\\hat{B_{i}}", "211ba1f30e0c0f90dc20e2dcd3fee427": "\\begin{align}\n \\left(\\frac{\\partial A_z}{\\partial y} - \\frac{\\partial A_y}{\\partial z}\\right) &\\hat{\\mathbf x} + \\\\\n+ \\left(\\frac{\\partial A_x}{\\partial z} - \\frac{\\partial A_z}{\\partial x}\\right) &\\hat{\\mathbf y} + \\\\\n+ \\left(\\frac{\\partial A_y}{\\partial x} - \\frac{\\partial A_x}{\\partial y}\\right) &\\hat{\\mathbf z}\n\\end{align}", "211c0c63f4ec05cb32817be102b178a3": "\\textbf{R} = \\textrm{E}[\\textbf{v}_k \\textbf{v}_k^{\\text{T}}] = \\begin{bmatrix} \\sigma_z^2 \\end{bmatrix} ", "211c11414e7f4f449cabdc9f8698f5c2": "\\,{}^{x}a \\approx a^x", "211c18dc3fcdc6b621185ddef385bf17": "n>A", "211c23f12fd466e20ea54fa89fd4b660": "f(\\zeta,\\bar{\\zeta}) = \\frac{1}{2\\pi i}\\iint_D \\varphi(z,\\bar{z})\\frac{dz\\wedge d\\bar{z}}{z-\\zeta}", "211c39650123980b1f1139685d464aa5": "2\\frac{\\ddot a}{a} + \\left(\\frac{\\dot a}{a}\\right)^{2} + \\frac{kc^{2}}{a^2} - \\Lambda c^{2} = -\\frac{8\\pi G}{c^{2}} p.", "211c4899f0dd793bba64a02586f50b49": "\n\\{ \\alpha(f_n), \\alpha(f_m) \\} = 0 \\quad \\mbox{and} \\quad \\alpha(f_n)^*\\alpha(f_m) + \\alpha(f_m)\\alpha(f_n)^* = \n\\langle f_m, f_n \\rangle I.\n", "211cef4edee330569392fe4dc2bd34a7": "b\\cap a", "211d0301a34fe1b26ab49b576989dd94": "\\chi\\,\\!", "211d88963272883183a29c021a4473c4": " T(0,a) = \\frac{1}{2\\pi} \\arctan(a) ", "211e1ef71689b2d0a2f96e0341a13dd7": "\\mathrm{error}\\bigl(x(t_0 + 3\\Delta t)\\bigl) = 6\\,O(\\Delta t^4)", "211e2de732456fdc7d133c2592822b49": " n_0 ", "211e6a6dfe4c5d6a30825a9f727eedcb": "\\text{excess kurtosis} =\\frac{6}{3 + \\nu}\\left(\\frac{(2 + \\nu)}{4} (\\text{skewness})^2 - 1\\right)\\text{ if (skewness)}^2-2< \\text{excess kurtosis}< \\tfrac{3}{2} (\\text{skewness})^2", "211e7ecf33ea40db9f17c31c40dbb71b": " \\begin{align}\nDR(n) &{}=\n\\begin{cases}\n 9, & \\mbox{if } SOD(n) \\mod 9 = 0\\\\\n SOD(n) \\mod 9, & \\mbox{ otherwise}\n\\end{cases} \\\\\n&{}= (n - 1) \\mod 9 + 1\n\\end{align}", "211e8e03586e3ce8f3111a2d989c8c80": " {D^{\\mathrm{eff}}} = f", "211e9d86525447163ec551304527acb9": "\\mathbf{G_1}", "211e9e1b9a1deaebe40b5327408325a6": "\\frac{\\pi}{4} \\approx 1 - \\frac{1}{3}+ \\frac{1}{5} - \\cdots + (-1)^{(n-1)/2}\\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)", "211ee22d6bb06cc02eeb32323149c639": " t_\\text{score} = \\frac{(\\widehat\\beta - \\beta_0)\\sqrt{n-2}}{ \\sqrt{\\text{SSR}/\\sum_{i=1}^n \\left(x_i - \\overline{x}\\right)^2} }. ", "211ef1c36db15621421a0bb8966542f9": "T[g \\sigma] \\to b T[\\sigma]", "211f75c471bdb4d01bf0e20fdcafa052": "\\pi^- + C\\to \\bar\\Sigma^- + K^0 + \\bar K^0 + K^- + p^+ + \\pi^+ + \\pi^- + \\hbox{nucleus recoil}", "211f75e4e049ce88f909ed2d6121741f": "M^{i}", "211f76b374349d65654ac466c2151c65": "R=L-\\mathbf{p}\\cdot\\mathbf{\\dot{q}}\\,,", "211f795a4e11243aa46499fbb4fa3148": " a_{02}-\\mathcal{L}(a_{30}\\omega^2+a_{21}\\omega+a_{12})= p_3 (a_{30}\\omega^2+a_{21}\\omega+a_{12})-\\omega p_8.", "211f8ad93197385ab3022f269af0182b": "|(Ax,y)|\\le \\|x\\|\\cdot\\|y\\|.", "2120b9eb2815b7508011a866620a5090": "=\\widehat{a} + (\\delta\\alpha^{*}\\widehat{a} - \\delta\\alpha\\widehat{a}^{\\dagger})\\widehat{a} + \\widehat{a}(\\delta\\alpha\\widehat{a}^{\\dagger} - \\delta\\alpha^{*}\\widehat{a}) + O(\\delta\\alpha^{2},(\\delta\\alpha^{*})^{2})", "2120d770928c7c5926578c3ef8bea274": "\\textbf{G}(s)=\\frac{\\textbf{P}(s)}{\\textbf{Q}(s)}=K\\frac{(s-a_1)(s-a_2)\\cdots(s-a_n)}{(s-b_1)(s-b_2)\\cdots(s-b_m)}", "2121151af0ca2705fa39bc3024216e5b": "v,v' \\in \\mathbb{R}^d", "2121e7a6dc8ed64fc93b6c002a6e20b8": "f_\\pm(z)", "2121ecf663aa5ebb7e4e7346588c451d": "E = \\alpha - \\pm 1 \\times \\beta", "21220b38dfb7eda7c4e2e2c5bf1ba2ee": "\\lambda_B = \\frac{4\\pi a_B}{\\alpha }- \\ ", "212249d4697805c6577cce97b2441033": "\\cos\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\tfrac{1}{2}(1 + \\cos x)}", "21225d4856fbd75e5bf47e907766ee94": " g_{DS} = \\frac{\\partial I_{DS}}{\\partial V_{DS}}", "2122cec3f6e5240bb424e4f99d9ec79a": "\\bigcup\\nolimits_{n,m} \\left \\{ x \\in X \\ : \\ \\sup\\nolimits_{T \\in F_n} \\|Tx\\|_Y \\le m \\right \\}", "2122f738725ca53a7844f2be811c500e": "\\sum_{n=k}^\\infty \n\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right] \n\\frac{z^n}{n!} = \\frac {\\left(\\log (1+z)\\right)^k}{k!}\n", "21235a0884c91abd5366b2f39d43c170": "W=\n\\frac{1}{\\sqrt{8}}\n\\begin{bmatrix}\n \\omega^0 & \\omega^0 &\\omega^0 & \\ldots & \\omega^0 \\\\\n \\omega^0 & \\omega^1 &\\omega^2 & \\ldots & \\omega^7 \\\\\n \\omega^0 & \\omega^2 &\\omega^4 & \\ldots & \\omega^{14} \\\\\n \\omega^0 & \\omega^3 &\\omega^6 & \\ldots & \\omega^{21} \\\\\n \\omega^0 & \\omega^4 &\\omega^8 & \\ldots & \\omega^{28} \\\\\n \\omega^0 & \\omega^5 &\\omega^{10} & \\ldots & \\omega^{35} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\omega^0 & \\omega^7 &\\omega^{14} & \\ldots & \\omega^{49} \\\\\n\\end{bmatrix}\n", "212369dee6600287df0b32fbbc41526f": "x_n \\in C^*(\\theta_n)", "21237eea68173ccbd84d3e49a3a4b533": " \\varphi:(\\mathbb R^n\\times\\mathbb R)\\times\\mathbb R \\to \\mathbb R^n\\times\\mathbb R; \\qquad\n\\varphi(\\boldsymbol{x}_0, t_0, t)=(\\varphi^{t,t_0}(\\boldsymbol{x}_0),t+t_0)", "21239a5b1e10cd0db525d4bbaf54495f": "\\scriptstyle \\cos\\theta = \\cos\\left(\\theta + 2\\pi k \\right).", "2123cdd736e0847396860bd9ab07fb23": " \\rho_{L^*}=\\frac{\\sum_{j=1}^l L_j (\\phi_j\\rho_w + (1-\\phi_j)\\rho_g)}{L^*} ", "2123d2f52593debfd7975686ee23ebce": "\\textstyle f(X)=\\sum_k f_k X^k \\in R[[X]]", "212446327682b0b29d2bf3d2aa9f40fb": "s, h \\models P -\\!\\!\\ast\\, Q", "21244b0dc08c3ec734a7f3a4fc809137": "\\displaystyle{Tf(w)={1\\over 2\\pi}\\int_{\\partial\\Omega}\\partial_n (\\log|z-w|) f(z)={1\\over 2}\\Re (Hf)(w).}", "2124732b29a8cf7e1c7b8a0f10ff7513": "=-\\left( A_1 \\mathbf{e_1} + A_2 \\mathbf{e_2} + A_3 \\mathbf{e_3} \\right) = - (\\star \\mathbf A )\\ . ", "21247db6e87679d507efabe47d0fe495": "\\psi'_W(t)+t\\psi_W(t)\\approx 0", "2124d8e264ed693811271631c7fa8114": "\\frac{1}{2}+(\\pi_{1}-\\pi_{0})/(2^{1+B(n)+2B(n)G(n)}),", "21253cd82d899d67c25399e1441426bb": "S_q = -\\lim_{x\\rightarrow 1}D_q \\sum_i p_i^x ", "212540b9663b79dae6a1756f12ca8f53": " \\begin{align}\n(\\mathcal{L}_X T)^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} = X^\\gamma T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s , \\gamma} & - \\, X^{\\alpha_1}{}_{, \\gamma} T^{\\gamma \\alpha_2 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} - \\cdots - X^{\\alpha_r}{}_{, \\gamma} T^{\\alpha_1 \\cdots \\alpha_{r-1} \\gamma}{}_{\\beta_1 \\cdots \\beta_s} \\\\\n& + \\, X^{\\gamma}{}_{, \\beta_1} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\gamma \\beta_2 \\cdots \\beta_s} + \\cdots + X^{\\gamma}{}_{, \\beta_s} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_{s-1} \\gamma} \\,.\n\\end{align}", "2125bf68acdb12d921a6f98810f19499": "B_\\mathrm {v} = \\log_2 \\frac {B} {NK} \\,.", "2125ffc6864ab39b3c34a712115a945f": "\\bold{j}_{{\\rm n}, \\, i} = n_i \\mathbf{u}_i ", "21266c7a829c230ef5cdc467bbfcdd49": " { 1 \\over q(z) } = { 1 \\over z + iz_\\mathrm{R} } = { z \\over z^2 + z_\\mathrm{R}^2 } - i { z_\\mathrm{R} \\over z^2 + z_\\mathrm{R}^2 } = {1 \\over R(z) } - i { \\lambda \\over \\pi w^2(z) }.", "2126bbe09c2f409b8fbd517a757ae841": "s = r - \\sqrt{r^2 - \\ell^2}", "2126c397d14354598fea1232133f900d": "\\frac{d\\varphi}{ds} = \\frac{\\cos^2\\varphi}{a}\\,", "2126c71158ed6f42ed70125278a13808": "f \\sim \\operatorname{GP}(m(\\cdot),C(\\cdot,\\cdot)),", "2126d1446d273b5be35e0826cabc727d": "\\displaystyle [a_0; a_1, a_2, \\ldots, a_k-1,2].", "212763f9db7d33a473fcc1e6bf513ea1": "a < \\Im(s) < b", "2127dd385957196f1ebd4560b9041d9d": "\\begin{align}\n\\operatorname{Pr}(X_n\\leq a) &\\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(|X_n-X|>\\varepsilon) \\\\\n\\operatorname{Pr}(X\\leq a-\\varepsilon)&\\leq \\operatorname{Pr}(X_n\\leq a) + \\operatorname{Pr}(|X_n-X|>\\varepsilon)\n\\end{align}", "21283c2d43001596bd3772536a86c10b": "12^2 + 33^2", "2128831dd136ed12ac08450dc1eb8ed6": "\\pi_2(x)\\sim2C_2\\frac{x}{(\\log x)^2}.", "2128932afe1ea5d1a376cd1e04cdaeeb": "\\langle F,\\phi\\rangle =\\int\\int f(x+iy)\\phi(x,y)\\, dx\\, dy,", "2128e3cbaf09a616dfa6d6213cb75e7d": "\\tilde R = e^{-2\\varphi}\\left(R + 2(n-1)\\triangle\\varphi - (n-2)(n-1)\\|\\nabla\\varphi\\|^2\\right) ", "212920f6b2f02ec197cf6c6dbdce97ee": "\\psi(\\bold{r}) = N_x N_y N_z e^{i(k_x x + k_y y + k_z z)}", "212998b2ff69c216de369f89adf41533": "A\\in\\Sigma_0", "2129d0a9ce8d45ced11f68400c0b65be": " \\sqrt{6} ", "212a06a728f5a87715d96892dec70902": "K_{ijk}= \\frac{1}{2} (T_{ijk} - T_{jki} + T_{kij} ),", "212a3567df6fe6fe11caf388a08cd95a": "\\varepsilon>0 ", "212a5f55a2cd108a969af1ca8d6e602c": "I_D=\\Pi - \\Pi_0 - T \\,", "212a83d39f05a6c6e8398d561bffa3e1": "\\mathbf{J}:P\\to\\mathfrak{g}^*", "212abcc00be63b38e4cd48603ee59f7d": "m_{i}", "212ad034d8f5a3267775b134271474ce": "A \\oplus B = (A^{c} \\ominus B^{s})^{c}", "212b4ac94f46affa0d3f4e4c79366ccf": "\\tan[\\arccos (x)]=\\frac{\\sqrt{1 - x^2}}{x}", "212b5b6eb6cfc4b0566d4170b619e313": " {3\\over2} \\cdot {1\\over2} \\cdot {3\\over2} \\cdot {3\\over2} \\cdot {1\\over2} \\cdot {3\\over2} \\cdot {1\\over2} \\cdot {8\\over5} = {81\\over80}", "212b7c3587327c94815ef6e6308e30ed": "\\beta\\mapsto\\lfloor \\beta \\rfloor", "212bb69afb2b28c05db2f77492c35827": "E[|\\eta|^q] < \\infty", "212bf522bb938b1b30a08776ca7c5702": "F = -kx", "212c21acd2153d472c5d4d7cea240804": "W \\triangleq 2 \\sqrt{V}", "212c5bc6aec0d7d0ef21a9eb168dc147": "h \\left( (1 - \\lambda) x + \\lambda y \\right) \\geq M_{p} \\left( f(x), g(y), \\lambda \\right),", "212c65e23fb48bbd1117d6a196787694": "x^2 + y^2 = 1. \\, ", "212c97b4c8663c693e903ad8397db815": "x_0 + tx \\in A", "212ce081962da99ceaf0f174e6dea9e4": "\\lbrace \\left \\langle x, y, z \\right \\rangle \\mid \\phi_x(y)=z \\rbrace", "212d15fb7bf1e166c56ef4d659cf0e3a": "\\mathfrak{H}(\\beta; \\gamma) =\n\\begin{pmatrix}\n1+\\gamma\\beta & - \\beta \\gamma^2 \\\\\n\\beta & 1-\\gamma \\beta\n\\end{pmatrix}", "212d2883c38dfe911872d14954ae3922": "\\scriptstyle {}^{n}z", "212d4cd8917d593fbb5f373e0f120cd0": "\\eta(2) = {\\pi^2 \\over 12} ", "212d511ae81118fe6ed289359ad8fca9": "Q = NX_{S}", "212d920902289b751d63f2f09f473c01": "\n \\eta_{11} = \\langle \\mathbf{e}_1 \\bar{\\mathbf{e}}_1 \\rangle =\n \\langle \\mathbf{e}_1 (-\\mathbf{e}_1) \\rangle_S = - 1,\n", "212dcafb83b41e3df531dfccb65bfb3e": " p \\in K ", "212dce1da2f0bd80366b5e1de813ac04": "g^{abc}", "212ddf6d0ae36a842fc3e390da024d9d": "\\frac{ \\phi \\left( r \\right)}{r} = \\phi \\left( \\frac{1}{r} \\right) ", "212df6022ee1798525c5b59d475949dc": "\\lim_{t\\to\\infty}f(t) = \\lim_{s\\to 0}{sF(s)}", "212e0f9b1a3b926410e12b0857d9d1d3": "j_0(x)=\\frac{\\sin x} {x}", "212e28a0426d14377c80f95b032e2d82": "1, 2, \\dots, d", "212e2ff8a0bea5f2efa58d0e6c1c5b3d": "Rate \\approx 4 mm/s", "212ef7c5e874ed5afbabb177adb1b599": "e_f(k,i)\\,\\!", "212f038037d2d4b09431a7cd618abe5a": "d^3", "212f453914b387160ac98d737b9c7005": " I_1x^2 + I_2y^2 + I_3z^2 =1,", "212f4dbe3ff9046736bf2a9996af85d5": "I_{ii} \\cdot I_{ij} = I_{ji} \\cdot I_{jj} \\ \\ (i 0, \\alpha > 0, \\beta > 0 \\,\\! .", "213e590462f19fe6ea0b3f2a89ebd9e6": "w(\\cdot)", "213e5fc1c033082d04358907bd04252c": "e>1\\,\\!", "213f1f4415a4710e51d06683d3b6d534": "\nG = \\frac{1}{a} + \\frac{n - 1}{b}\n", "213f4fb5a8e8c7dbd4eea72703ad72b6": "F_n = \\frac{\\varphi^n-(-\\varphi)^{-n}}{\\sqrt 5}", "21401031c7b4952c7eaf66699024c74c": "[T] = [Z_1][X_1][Z_2][X_2]\\ldots[X_{n-1}][Z_n],\\!", "21401732cf37f63c4e284e8ee1803c08": "V_{bi}", "21403f41e6740242b98efdf83a6c9728": "\\psi^\\ast", "21404f4fc13796efcc7c147e9e18daf6": "P_1\\in\\{0,1\\}=\\{\\mbox{AB},\\mbox{BA}\\}\\,", "21409bc14c5e30c7c3c772a2c6ef9c70": "\\operatorname{Aut}(A_n)=\\operatorname{Aut}(S_n)=S_n", "2140f74715e50750afb1b1fc403c43f1": "\\frac{1}{n^{i_1}_{1}n^{i_2}_{2} \\cdots n^{i_k}_{k}}", "2140f981496f532d911daf78ec6cb3ac": "\\mathbf{x}(\\cdot)", "214127a972fef1ba141d369d3e3e8e59": "\\Pi_{1}\\cdots\\Pi_{N}", "21416146b1d79fe6c1e3d8f75db8ff6e": "\\rho(x_1,x_2)\\geq 0", "21416eb08876f081761ab76cc5c3bc86": "{b}_{eq}\\,", "21417e9440ad5a08913a33d94e698d0c": "\\scriptstyle r(S - \\{k\\}) = \\sum_{i,j \\in \\{S-\\{k\\}\\}, i < j} r_{ij}", "2141cd9db263cdb933616b882673f29c": "\\varepsilon = k_\\mathrm{B}Tx,", "2142142fa6cafc36b0c8c4ce67b969c9": "B'=UB", "2142bc59bcb2fec37901eb33b8263509": "\\left(\\tfrac {3} {4} \\right)^{168} \\approx \\tfrac {1} {9.77 \\times 10^{20}}", "2142d836a54911776e63096e517795be": "9 \\times 4", "21435193036e4d97f9a30f6377dcdbe8": "\nA(t) = X(t) + i P(t) = \\sqrt{2E}\\,e^{it}, \\quad A^\\dagger(t) = X(t) - i P(t) = \\sqrt{2E}\\,e^{-it} \n", "21436fccc6643c7fa421a022a713fdc4": "\\chi_{\\text{1}}(\\omega)=\\frac{Nq^2}{\\varepsilon_0 m} \\frac{1}{D(\\omega)} ", "2143dbb8db825c0562dc26c2541e4afc": "0 \\le x_1 \\le x_2 \\le \\cdots \\le x_r \\le \\log (w)", "214415d46f96d2c27060ce171cf914a5": "F(b) - F(a) = \\sum_{i=1}^n \\,[F(x_i) - F(x_{i-1})]. \\qquad (1)", "2144561e37b5ea500ff468e8577bdbcc": "e_{\\alpha}^i(\\mathbf{x})", "21446fa5e08910556c296fe47a171aae": " \\mathbf{L}_M ", "21447660b9d614b4246c10b3677a23a0": "(a+bi) (c+di) = ac + bci + adi + bidi \\ ", "214533ee348e26130277770d9f467127": " \\Delta \\omega =\\star \\mathrm{d}{\\star \\mathrm{d}\\omega}= \\frac{\\partial^2 f}{\\partial x^2} + \\frac{\\partial^2 f}{\\partial y^2} + \\frac{\\partial^2 f}{\\partial z^2}", "214569d5487ac8f0f5a318dd96b493c4": " \\mathrm{MA} = \\frac{W}{F} = 5,", "21459703919901473642b097d4e3ab66": "\\begin{matrix}\\left({\\left\\lfloor{\\frac{66}{12}}\\right\\rfloor+66 \\bmod 12+\\left\\lfloor{\\frac{66 \\bmod 12}{4}}\\right\\rfloor}\\right) \\bmod 7+\\rm{Wednesday} & = & \\left(5+6+1\\right) \\bmod 7+\\rm{Wednesday} \\\\\n\\ & = & \\rm{Monday}\\end{matrix}", "214597b9dfe9eea9a81539303dcbc806": " g_n(z) = z\\prod_{j=0}^{n-1} F(f^j(z)).", "21460991870f3fa71b854c42967d34be": "\\mathbf{U}_1 \\mathbf{D}^\\frac{1}{2} \\mathbf{V}_1^* = \\mathbf{M} \\mathbf{V}_1 \\mathbf{D}^{-\\frac{1}{2}} \\mathbf{D}^\\frac{1}{2} \\mathbf{V}_1^* = \\mathbf{M}", "2146100929297ad02137b5c01d808c44": "\\text{Var}(\\hat \\theta) = \\frac{\\text{Var}(Y_1) + \\text{Var}(Y_2) + 2\\text{Cov}(Y_1,Y_2)}{4} ", "2146302266622d205c94666a852deae5": "x_{\\mathrm{eq}} = -\\frac{1}{k}\\left( \\frac{mg}{2} + Mg \\right)", "21463443378344a1bd10634f1f71ff12": "v_f^2 = v_i^2 + 2av_it + a^2\\left(2\\frac{\\Delta d - v_it}{a}\\right)", "2146490e3b600609c0ad92ddb0fde68b": "1 \\to Z(G) \\to G \\to \\operatorname{Aut}(G) \\to \\operatorname{Out}(G) \\to 1.", "21464dff80eaceff5627a77eb3f1ef21": "L:V\\to \\mathbb R", "2146973702ce4cc98344ece108b16e3e": "\\cos\\theta = \\frac{\\operatorname{Re}(\\mathbf{a}\\cdot\\mathbf{b})}{\\|\\mathbf{a}\\|\\,\\|\\mathbf{b}\\|}.", "2146a0c8dbec6eb128ebf28ef4ac30dc": "e^{i\\mathbf{R}\\cdot\\mathbf{K}}=1", "2146aad0f1f8a72368a0f6700e25f0d5": "H_n(X) = \\ker \\partial_n / \\mbox{im } \\partial_{n+1}.", "2146c47d037bdb3fb7830dec1741995e": "\\textstyle A_0", "2146d3556af3d87d7f3846c47b5a8a97": "\\nabla E_{snake} (\\bar v_i) = w_{internal} \\nabla E_{internal} (\\bar v_i)+w_{external} \\nabla E_{external} (\\bar v_i) ", "2146ebb2f9e54cbcbc86dc77923318f6": "\\boldsymbol{Y} = (\\boldsymbol{Y}_v)_{v\\in V}", "2146ef82a4a85ec1afe9bdeff66ccc28": "\\nabla \\times \\mathbf{u} =0,", "2146f5c4c2929461f095a23112f0b29a": "\\begin{align}\n -\\frac{\\hbar^2}{2m} \\frac{\\partial^2\\Psi(x,t)}{\\partial x^2} + V(x,t)\\Psi(x,t)\n &=E\\Psi(x,t)\n\\end{align}", "214712ef5d28ca313fd2381cdb21bb3d": " \\mathit{G} ", "214713912c2a22c2384d268530019500": "\\Sigma s_i \\otimes t_i \\mapsto \\Sigma s_i \\otimes t_i-\\Sigma s_i.t_i\\otimes 1", "21476d7de9bad5eddf6b90650f36d535": "A^i{}_j", "214780684f6e3ee77f860fc1c31e90e3": "E_1(x)=x^2+2", "2147d56ef27b84387e646f1fda5e8ce9": "x_2 \\ ", "21482a212e53f75a0228cbddc5679c64": "\\neg(p \\to q) \\to (p \\land \\neg q)", "21484c2341fc71109329858f72b1f96e": "m_1,m_2", "21489198e75f2bc0fff6d0df80127ca3": "\ns = A \\cos kt \\,\n", "2148b9bf5363f383f0a832fc187cec6d": " \\frac{2}{3} T^{4} dV = 4 V T^{3} dT ", "2148ffb00f544dedb8024f6530ba7f0f": "x^5-x^4-x^3+x^2-1", "2149632d7ed77deb84a78589dda4035c": "G = (V=V^0\\cup V^1\\,, \\Sigma\\,, R\\,, S\\,)", "21497996e331495f03b63c43249905d5": "\\Omega (n)", "2149b12d12b120845482ef0cf1675189": "\\Omega_\\Lambda", "2149bc67e8e8fce67b6e7ba56c013aa0": "\\textstyle \\frac{h}{\\lambda}", "2149beedadeb8f78d902718214ac4bf7": "\\mu \\circ T \\eta = \\mu \\circ \\eta T = 1_{T}", "2149ca955e33dd9262d0586c330a54d8": "\\varphi=|\\det M|\\cdot D_{M^{-1}} (h * \\varphi)", "2149feaff53ea25e2eeae1d52f72ede6": "\\frac{d{\\vec{x}}}{dt} = \\vec f(\\vec{x})", "214a0eb3efeb6236fb87ae2f50081f7b": "x_0 \\approx 5.8", "214a50b0454241e7aa07d99012ff8616": "c(r)", "214a58e80cf58837f696d5cf6184fee0": "\n\\begin{bmatrix}\n\\boldsymbol{I}_m\\\\\n& \\boldsymbol{I}_m & \\boldsymbol{V}_1^{(b)}\\\\\n& \\boldsymbol{W}_2^{(t)} & \\boldsymbol{I}_m\\\\\n& & & \\boldsymbol{I}_m & \\boldsymbol{V}_2^{(b)}\\\\\n& & & \\ddots & \\ddots & \\ddots\\\\\n& & & & \\boldsymbol{W}_{p-1}^{(t)} & \\boldsymbol{I}_m\\\\\n& & & & & & \\boldsymbol{I}_m & \\boldsymbol{V}_{p-1}^{(b)}\\\\\n& & & & & & \\boldsymbol{W}_p^{(t)} & \\boldsymbol{I}_m\\\\\n& & & & & & & & \\boldsymbol{I}_m\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1^{(t)}\\\\\n\\boldsymbol{X}_1^{(b)}\\\\\n\\boldsymbol{X}_2^{(t)}\\\\\n\\boldsymbol{X}_2^{(b)}\\\\\n\\vdots\\\\\n\\boldsymbol{X}_{p-1}^{(t)}\\\\\n\\boldsymbol{X}_{p-1}^{(b)}\\\\\n\\boldsymbol{X}_p^{(t)}\\\\\n\\boldsymbol{X}_p^{(b)}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{G}_1^{(t)}\\\\\n\\boldsymbol{G}_1^{(b)}\\\\\n\\boldsymbol{G}_2^{(t)}\\\\\n\\boldsymbol{G}_2^{(b)}\\\\\n\\vdots\\\\\n\\boldsymbol{G}_{p-1}^{(t)}\\\\\n\\boldsymbol{G}_{p-1}^{(b)}\\\\\n\\boldsymbol{G}_p^{(t)}\\\\\n\\boldsymbol{G}_p^{(b)}\n\\end{bmatrix}\n", "214a5a020528dbb02664821aed9af539": "L_{[\\omega]}", "214a5cd0a911ea05513a2e310ce7f1f1": "P_R = \\frac { e^{-\\beta E_R} }\n { \\displaystyle \\sum_{R'} e^{-\\beta E_{R'}} } ", "214a7a7c4276b143c12d700fa81fce32": "D_i^*", "214a7c41c34216123fa7e631722543fb": "V_G", "214aa0dff8979450561373b30cb5d7a3": "R_1(x)=\\frac{x-1}{x+1}\\,", "214ae64b54d6a83f0c13491afcbc9e7e": "x^5-20 x^3 -80 x^2 -150 x -656 ", "214b1b5f6e3184bd56a225d27e0a2dc4": "\ny_i = \\mathbf{w}_i^{\\text{T}}\\phi(\\mathbf{x}) \\qquad i= 1,\\ldots,c-1.\n", "214b9755c12d6e477f995a96fa658f9b": "a \\to b \\to p \\to 2 = a \\uparrow^{a \\to b \\to (p-1) \\to 2} b = f^p(1)", "214bd1778c0f28627be4bfbb57f98618": "Q_i=\\sum_{j=1}^n\\sum_{k=1}^n U_j B_{j,k}^{(i)} U_k .", "214be445ff875afb0e6a3df35eaab91c": "\\mathbf{D}^2_{x_1, x_2, \\dots , x_n},", "214c1eb0236ed7b85651ff25b28de30c": "C_n = \\sum_{i=0}^na_{n-i}(B_i-B)+A_nB\\,.", "214c43f6a6207470c10de4cee9793f01": "q=\\gamma^s", "214c7f683e396afe067ef8a2e7b20423": "{\\mathcal S}({\\mathbb R}^n)", "214c8c18c2a9f6dcd68dd9a9e9ba617a": "S_t(z)= e^{\\alpha t} ( 1+ t (z-\\alpha )) ~, ", "214cc996737585817a7f60112e8f7002": "\\sigma_{A}(R\\cap P)=\\sigma_{A}(R)\\cap\\sigma_{A}(P)=\\sigma_{A}(R)\\cap P=R\\cap \\sigma_{A}(P)", "214cdbe3dfd8c1df793b6b1e935d36d5": "v_{0.5} = 1.", "214d2e46c693d4bf19e7684d6707ee20": "B = I", "214d60a9f18f1d58599b251e15f7cd54": "X \\cup \\{B\\}", "214d698dfe1f26913479438406822e7c": "\\boldsymbol{p}_i", "214d72717319ae697dd66c4491dd40ea": " XY=2\\cdot X\\cdot Y ", "214d8943f83f71fbe732ffb8ba443dce": "a,\\ a+d,\\ a+2d,\\ a+3d,\\ \\dots,\\ ", "214dad1f45ee676a45098461036e445b": "\\mathbb{P}(A\\mid E) = \\frac{10}{19.2}=50.25% ", "214dd5dd93f127065a74cc10ab28646a": "{\\tilde x}^{\\mu}(\\tau), Y^{\\mu}(\\tau), R^{\\mu}(\\tau) ", "214dead024f155b0ef339e46c00467a6": "E_{\\rm XC}^{\\rm LSDA}[n_\\uparrow,n_\\downarrow]=\\int\\epsilon_{\\rm XC}(n_\\uparrow,n_\\downarrow)n (\\vec{r}){\\rm d}^3r.", "214e28ac05a2d73ead8853760d7dc525": "S(\\sigma)", "214e5554cf0932ea8f6acbf25ae63df7": "G_\\mathrm{dB}", "214e7c07b9bf6ad0f7346c1207da6036": " \\sum_i C^{S_n}_i \\varepsilon^i_{S_m} = 0 \\quad n \\neq m ", "214e86a0ee9e470f4ad615659b1f4e76": "{f(a+h)\\over h} = {f(a)\\over h} + f'(a)+{R_1(x)\\over h} ", "214ee9d3d5cbebc9ed1c146936602b89": "Z = \\left[ \\frac{Y_{O,0} \\cdot W_F \\cdot v_F}{1+Y_{F,0} \\cdot W_O \\cdot v_O} \\right ]", "214f4143615634764cffd340bc425e7e": "q^a", "214fb3c22126224ce7013bb3e7f27cc7": "E_{SOFC}", "214fb6278befd0b5e631b59c9b9edda7": " (q,\\omega,q') \\in \\Delta ", "214fdc70ce236d5a3c0b6a2b2cfe630f": "P_{local}^{'}(k_{i})=P^{'}(i\\in Local-World)\\frac{k_{j}}{\\sum_{j\\in Local}k_{j}^{}}", "2150a564609399b7595ebdd1be33076b": "a \\in \\mathfrak{a}", "2150ac0ad364a4f70b4a9df632f8c830": "\\sum_{i=1}^{n} F_{2i} = F_{2n+1}-1.", "2150d53556a7bd70b63e80e6751c7916": " D < \\sqrt{ m }", "215131d069570a1a4a64a413da4cbe7e": "u_\\nu(T)", "2151820d79e6871b41aa1aac67ccb886": "\\emptyset = M_{-1} \\subset M_0 \\subset M_1 \\subset M_2 \\subset \\dots \\subset M_{m-1} \\subset M_m = M", "2151a2bc77807b81113febbf50c4bc95": "yz", "2151aa9a29a14b359b85d09d75ce6834": " \\phi_{u_n, u}( \\gamma )", "2151c6fe4352df3b95933852fa645490": "1-\\frac{k'}{n}\\le \\{x\\}<1-\\frac{k'-1}{n} ,", "2152245fd17771d88e2fd4626b2cc3b0": "c = q^{h+k-1} \\frac{\\sin (h+k) \\alpha}{\\sin \\alpha} = \\sum_{0 \\leq i \\leq \\frac{h+k-1}{2}}(-1)^{i}\\binom{h+k}{2i+1}p^{h+k-2i-1}(q^2-p^2)^i,", "21525b0c4ff7b63585d27fbc808ba96e": "\\frac{c}{2\\pi} \\log \\frac{t}{2\\pi}", "2152bdba25027b304ef1a31c2c774136": "\\mu_G(x,y) := \\sum_{k\\geq0} m_k x^k y^{n-2k}.", "2152ca0da322c3cc36a8a4ee84f07568": "b \\cdot ((a \\cdot b)^{-1} \\cdot a) \\stackrel{R17}{\\rightsquigarrow} b \\cdot ((b^{-1} \\cdot a^{-1}) \\cdot a) \\stackrel{R3}{\\rightsquigarrow} b \\cdot (b^{-1} \\cdot (a^{-1} \\cdot a)) \\stackrel{R2}{\\rightsquigarrow} b \\cdot (b^{-1} \\cdot 1) \\stackrel{R11}{\\rightsquigarrow} b \\cdot b^{-1} \\stackrel{R13}{\\rightsquigarrow} 1", "2152ec57e055d160ee671d9f7db9fabe": "\\mu_0\\colon \\Sigma_0 \\to[0,\\infty]", "21532074d288652c348e1cd545288e1a": "u(x(t), t)=f(x_0)=f(x-at)\\,", "215322861d6e59cf9616e76b925e88c8": "l^a\\partial_a=\\partial_v +U\\partial_r +X^3\\partial_y+X^4 \\partial_{ z } \\,,", "21537be5949464fdc880864ebc8f714a": "S = \\{ y|wt(y)", "2153a2c40271946e05ae4c7153c2df05": "x=v\\cos u, y=v \\sin u, z=c\\sqrt{a^2-b^2\\cos^2u}.\\,", "2153a7841decb406fd100761e790e45d": "5 Pmf = 3", "21547892a9e76b766f5fd9fc83c7f488": "\\bar{S}_{j_1j_2\\cdots j_p} = \\det(\\boldsymbol{\\mathsf{L}}) \\mathsf{L}_{i_1 j_1} \\mathsf{L}_{i_2 j_2}\\cdots \\mathsf{L}_{i_p j_p} S_{i_1 i_2\\cdots i_p}\\,.", "2154a8c5eb28da41cc1db2d020e08952": "a_{ij}=\\overline{a_{ji}}", "215534191ca42201fd9f91bac83ab0ad": "\\textit{gra}(x,z) \\leftarrow \\textit{fem}(x) \\land \\textit{par}(x,y) \\land \\textit{par}(y,z)", "2155665235e3ad4085c5072d17173e46": "= \\sum_j Q_j e^{-E_j it/ \\hbar}\\langle j|j\\rangle", "21557fdc5f825fb68550ff9d5443217a": "\\textstyle \\mathbb{F}_{28}", "2155886635c73c180e3aec175547d562": "\\widehat{E}_{i}=u_{i}^{*}(\\mathbf{x},t)\\widehat{a}^{\\dagger} + u_{i}(\\mathbf{x},t)\\widehat{a}", "215635940f98d215f371cdf9c4affe49": " 0 = g^{(n)}(\\xi) = f^{(n)}(\\xi) - f[x_0,\\dots,x_n] n!", "21565c85ad0642126d5063783a6c6fd2": "\\frac{x-\\lambda-a_2}{a_2-a_1}\\!", "2156e65e3415be9fca943c4ad94d191b": "\\lambda _x", "2156f2d337ff249c1c6112762c28a87c": "\\sum_{i=1}^n y_i(x) \\, \\int \\frac{W_i(x)}{W(x)} dx.", "2156f4fbb9478ce0bd6e7a6f2a8a5c02": "\\le\\epsilon", "215708ac5db22cc820aff07e0f81a461": "\ni \\rightarrow (\\mathbf{k}, \\mu)\n", "21571258e12fe31c4de2962dc5195ef3": "x_0 \\in [0,1)", "21576fbf20483dd8b931db7ec2cafce9": " \\sqrt{\\frac{4\\pi}{\\varepsilon_0}}\\mathbf{D} ", "2157d36aca58dd16a87d63d5384f644d": "{\\hat{\\alpha}}(q)", "21581c365855f157f6998301ff9dbab1": "\n\\mathcal{S}_{\\alpha}\\left(\\left\\{x_i\\right\\}_{i=1}^{n}\\right) = \\frac{\\sum_{i=1}^{n}x_i e^{\\alpha x_i}}{\\sum_{i=1}^{n}e^{\\alpha x_i}}\n", "21582826f799e12abf977b04a279b276": "\\sum_\\gamma \\frac{1}{1 + e^{l(\\gamma)}}=\\frac{1}{2}", "21583d57ace96b1e8d372909fe88f904": "\\epsilon_{\\mathcal{O}}", "2158de7a462de49f9721534811cbc1dd": "C_{1},C_{2},\\ldots", "2159606287f8923df61d4a7175c217c6": "U_{QFT} \\left|x\\right\\rangle\n= Q^{-1/2} \\sum_y \\omega^{x y} \\left|y\\right\\rangle.", "2159685fccd0df371b20d4bb9be92bdc": "R_{ff}(\\tau)", "21599e78d7d6b04b5d835278d17116ad": " \\dot{r}^2 = (E^2 - V) \\; ( 1 + m/r )^4", "215a20d86144826378bfb74029c1a7ad": "\\text{sign} = 0 ", "215a63ff6b692faadc4216e5676370d4": "\\ R(r) = B(l,\\alpha) r^l e^{-\\alpha r^2},", "215adc564082c7d57b9e1e7377f10077": "T_{mn}(\\tau) = A_5\\,y_1(n^2/2m,i\\tau \\sqrt{2m}) + A_6\\,y_2(n^2/2m,i\\tau \\sqrt{2m})", "215ae5e740cc3604526138eacec5b40c": "\nF(x)=\\left(x^{\\{m\\}}\\otimes I_r\\right)'\\left(H+L(\\alpha)\\right)\\left(x^{\\{m\\}}\\otimes I_r\\right)\n", "215b4edb5604346787e69c27e6d4e973": "\\,\\!\\beta_n = \\frac{\\pi n}{M+1}", "215b8176572cb6de40e1ab4b221c0a80": "H_\\alpha = \\int_0^1\\frac{1-x^\\alpha}{1-x}\\,dx\\, .", "215b856a837275173a1050a7716e9788": "\\!\\,\\text{if }p, q \\equiv 3, 7 \\pmod{ 20 }\\text{ then } pq=x^2+5y^2.", "215c045aa2c05a65ed553651e88ff182": "M_\\alpha<\\frac{Z_\\alpha}{mU}M_q", "215c13b2f0452d81ee3f116dd951b90b": "r(n)", "215c145ada675357d5835168b9cc8617": "\n\\tfrac12\\, \\rho\\, v^2\\, +\\, \\rho\\, g\\, z\\, +\\, p\\, =\\, \\text{constant}\\,\n", "215c34c171eeaacbe88a8ab26e3822df": " \\int_X^\\oplus H_x \\ d \\mu(x). ", "215c351dad7df03a7da9b0bebcf16113": "\\partial_b F^{ab}=\\mu_0j^a.", "215c667339152210f91f6a3a11bd3038": "\\mathbf{(2)}", "215c7ade61c26147f021e4a13f289e62": "V:\\mathbb{R}^n \\to \\mathbb{R}", "215cba6438874cb00b3ed39293c6c6c0": "\\alpha = e^2", "215cdc6429994d168d8e5cbbf4fc7796": "\\textstyle\\frac{17\\cdot(17+1)}{2}=153", "215cf6963bffc6b1fcba4ce7495e5dfe": "{ f(t,i) : 0 \\leq i \\leq n }", "215cfb36e4660ac13f127efa973b66ba": " E( n_r ) = \\alpha \\frac{ X^r }{ r } ", "215d11c7da8017d2a3e26a1c21ddef79": "\\mathfrak{P}^{19}", "215d6650983658131ce7d3a57d95679c": "\n\\Psi(x)= \n\\begin{cases} \nx, & 0\\le |x| \\le a \\text{ (central segment)}\\\\\na\\, \\operatorname{sign}(x), & a\\le |x| \\le b \\text{ (high and low flat segments)}\\\\\n\\frac{a(r-|x|)}{r-b}\\,\\operatorname{sign}(x),& b\\le |x| \\le r \\text{ (end slopes)}\\\\\n0,& r\\le |x| \\qquad\\, \\text{(left and right tails)}\n\\end{cases}\n", "215d673b52bc6e812160a31d85d890a4": "VAS(x^3+x^2-2x-1,\\frac{2x+3}{x+2}) \\cup VAS(x^3+2x^2-x-1,\\frac{x+3}{x+2})", "215d73433947c21d994745f62903f4f2": " \\alpha_2 = \\arctan \\left( \\frac{\\sin \\alpha}{-\\sin U_1 \\sin \\sigma + \\cos U_1 \\cos \\sigma \\cos \\alpha_1} \\right) \\, ", "215d91de917e1a257151e6aaba5bb169": "{1 \\over \\sqrt{2}} \\bigg( |0\\rangle_A \\otimes |1\\rangle_B - |1\\rangle_A \\otimes |0\\rangle_B \\bigg)", "215da4440856b257b54ff09ac087e7c0": "\\displaystyle J_{j}", "215e4f09b06550eafda3261bb3e75325": "E[\\vec{X}]_{\\hat{m} \\hat{n}} = q^2 \\, \\sin(\\omega u)^2 \\, \\operatorname{diag} (0,1,1)", "215f0bb243fd7434a6824c7179a7a7f1": "\\sum_{n=-\\infty}^\\infty a_n e^{in\\varphi}", "215f118a76dd58593a3e98b82506e5e2": "p_0= \\sqrt{2m|E|}", "215f208cfd7ebeec8c4c41ac6b08fa29": " R = \\begin{bmatrix} a_{11} & a_{12} & a_{13} \n \\\\ a_{21} & a_{22} & a_{23}\n \\\\ a_{31} & a_{32} & a_{33} \\end{bmatrix} ", "215f49fa25256a68ca705cb603afc4a4": " \n\\lim_{n\\to \\infty} \\mu_n(A-A_k)=\\mu(A-A_k),\n", "215fae126b3edbdd749d19bd90a2ab8c": "\\overline{\\rho} \\widetilde{\\phi \\psi}", "215fcbd7e09c4924ff9775d47e708c9f": " \\begin{align}\nP_{0,0}&=1\\\\\nP_{i,i-1} &= \\frac{N-i}{N} \\frac{i}{N}\\\\\nP_{i,i} &= 1- P_{i,i-1} - P_{i,i+1}\\\\\nP_{i,i+1} &= \\frac{i}{N} \\frac{N-i}{N}\\\\\nP_{N,N}&=1.\n\\end{align}", "21603be00aeaeb57c0a8c86fc5845fa7": "U_n=X\\setminus C_n,\\forall n", "21607520759867ecd7103c24497dde80": "O_1^{(\\alpha)}(t)=2\\frac {\\alpha+1}{t^2},", "216097380339a780506cea20eba32558": "Spectral \\; purity = \\frac{\\triangle v}{v}", "2160a951d5f560e879012cdb0567d4b5": " x_{n+1} = 1", "2160caf02cec9d8a820fa3318b325c9d": "\n(Sv)(ds) = \\int_0^1 (v(tI + ds)- v(tI))dt.\n", "216114e8b20b59521349b8751caf91c5": "||h'||_{\\infty}", "216133a5837497a9396013022c52702d": "u(i,j)", "21616361dd54d54056f2e78dde513117": "|\\psi\\rangle = \\sum_{i,j,k,...} c_{i,j,k,...} |a_i, b_j, c_k,...\\rangle", "21617aa5947cb08be2150d6213b106e0": "\\{ n_{\\alpha} \\}", "2161bb833eeef43bd9c006051757d1f2": "\n H(P, Q) = \\frac{1}{\\sqrt{2}} \\; \\sqrt{\\sum_{i=1}^{k} (\\sqrt{p_i} - \\sqrt{q_i})^2},\n", "2162262b2486de3eb68bd84b056b2bc0": "e^{-\\frac{2\\pi i}{N}k}", "21624d046368043e5228246d820f08e4": " \\epsilon = \\frac{v^2}{2} - \\frac{ \\mu }{ x } ", "21624e3acbd65bbabdcde4364ba33cd4": "\\cos \\alpha \\approx 1", "21629e666003ef380f80138933e18f86": "_kV^i_1", "2163c6a09c338534fe90ca3d0dd60d17": "\\begin{align}\n I_2 &= -\\frac{Z_{21}}{Z_L + Z_{22}} I_1 \\\\\n V_1 &= Z_{11} I_1 - \\frac{Z_{12} Z_{21}}{Z_L + Z_{22}} I_1 \\\\\n &= \\left(Z_{11} - \\frac{Z_{12} Z_{21}}{Z_L + Z_{22}}\\right) I_1 = Z_\\mathrm{in} I_1\n\\end{align}", "2163f86c92ad4d27859706de5e0eb057": "\\Delta{t}=t_P-t_Q=\\frac{(x_M-x_U)}{v_{QP}}", "216402bddea7e5b7830e6352487e5a12": "q=(s,t_s,t_e)", "216432f71bd4a75f32da5a6356e65746": "\\frac{1}{p}", "21645b9223ed94ad3f13aa0ffafc7cd8": "A,D,F,G", "216467ad1fa7b02c4336b2e478796413": "0\\le\\theta\\le\\pi", "216527b58339e266e24299269c19f604": "cr\\geq ax+by", "216554093aa007ab9947ed316b9c44a1": "\\sqrt{10}", "2165a3d695d6bb48e546fbccf56cb7f1": "\\textstyle P(w_j\\mid[x])", "2165c70c5937cbc066661173c6a805bf": "b_r=\\sum_{p+q=r}h^{p,q},\\ \\ \\ \\ h^{p,q}=h^{q,p},\\ \\ \\ \\ h^{p,q}=h^{n-p,n-q}.", "21660f25f2b7918a42f2e31416b9be35": " d(E,F) = \\inf\\{d(x,y): x \\in E, y \\in F\\} > 0, ", "21660fbc2c6c800dffad4c713ac6bb2c": "P(\\bar{W}) = 1 - p = q", "21667337d35b3088f0886e094a0735ae": "q(\\mathbf{s})", "216688c25dc0048065a97a4bf41a3468": "\\geq\\gamma", "21669de2ac24f04eaf1e71bf4fc0a816": "r_{k}=\\frac{\\pi^{2}kRn_{k}^{2}}{16p^{2}\\left( 1-\\frac{p^{2}}{6n^{2}}\\right)}", "2166e53e332c832ba939cf34ea2066ba": "\\sigma_\\infty = \\sigma_f = \\sigma_m", "2166e5c72213000c38110dc49d281755": "(n_c-n_b)(n_a-n_b)", "2166eb389329df913db3c4a0da8ad491": " a=\\gamma^x\\in\\langle\\gamma\\rangle\\text{ with }0\\leq x<\\omega", "2166f84f98663924d0bbc97bb9a74a28": "\\scriptstyle t,", "21670535a673053399a24a3c90dac59b": "A=A_1\\oplus A_2", "2167964f4466bfa4db38a8751fb64270": "\n\\begin{bmatrix}\n1 & 1\\\\\n0 & 1\n\\end{bmatrix}.\n", "2167d4ef0d12b26b1fc3849840467956": "d_f = \\frac {2 \\lambda z}{W}", "2167e8074c05c593950e63e729adaf1e": "\\ln\\left(F/K\\right) \\Big/ \\sqrt{\\tau}.", "2167e809dd41963976e7f077c6def3c5": "e^{\\theta\\mathbf{e}_{12}} = e^{i\\theta} = \\cos{\\theta} + i\\sin{\\theta},", "2168f809c48d845c008727a64b82317d": "\\mathcal{L}=\\frac{m}{2}\\mathbf{\\dot{r}}\\cdot\\mathbf{\\dot{r}}+e\\mathbf{A}\\cdot\\mathbf{\\dot{r}}-e\\phi\\,\\!", "21692a6d1653c6178f6ed80a91c20417": " I_n = \\int \\frac{\\sin{ax}}{x^n} dx\\,\\!", "216962e12dcab274b3b0442a17e13ac2": "\\partial \\Omega_D \\cup \\partial \\Omega_N=\\partial \\Omega ", "216981343002548ccd2ca27ad4c78afb": "\\mathbf{z}(0)=\\mathbf{z}_0,", "21698d76abdd91c9c3642cceeca0159d": "\\mathbf{PSPACE} \\subseteq \\mathbf{EXPTIME} \\subseteq \\mathbf{EXPSPACE}", "2169974babd649b56bc24b639c16650b": "{\\partial}\\!\\!\\!/", "216a0253bf5eb89f199bb229c4443461": "x_{i}=x_{i}\\left( y,\\xi\\right) ,\\ i=1,...,m;\\ \\theta_{j}=\\theta_{j}\\left( y,\\xi\\right) ,j=1,...,n", "216a1d6f59fbbf8fd59a8ce9d5856626": "\\Delta \\epsilon = \\int v\\, d (\\Delta v) = \\int v\\, a dt", "216a24e1d9bafa2ab9a7bf67f3ce58d9": "|B'|\\leq w(X)\\,", "216a73eaac72e61b567454572f8e37b1": " \\mathbf{A}\\times\\mathbf{B}=\\mathbf{-B}\\times\\mathbf{A} ", "216af02b7bd7129ef08358c9fb766254": "\\Theta(\\sqrt{n})", "216c2c3e54e824efc60ad36ac56e81fa": "E_{\\mu\\nu}", "216c34d4761fbb92ab296f45178aae1b": "x_i+\\sum \\bar a_{i,j}x_j=\\bar b_i", "216d158cc28f3a40bcf1351169cb14aa": "|A_{ij}X'-a_{ij}T'|=0", "216d35e5170732b9f0554582122689af": "q < N^k,\\,", "216d86e1e73c439696f68b15478549a4": "\\displaystyle \\operatorname{sech}(a x) \\,", "216dc846a5621702ec11827e1be19184": "x^2 + 1 \\times 10^{-5} x - 1 \\times 10^{-6} = 0", "216dd24b72ce9f9225f1c3bc6c2a4360": "F_{n-2} = F_n - F_{n-1},", "216e07da96820fa4612b11103459c30d": "\\rho_{2}", "216e42b0775214caaa5deb6fc86e0710": "\nv_1 - v_0 = 0, \\ v_{n+1} - v_n = 0.\n", "216e557d82d0d8ba73382256bf4e7c71": "x,y,u,v", "216e574eb39ef684ce611f1ce09274dd": "\\alpha^*=\\dot{\\alpha}", "216e6f7a0adc7dbbb16a307c30dc384d": "A=\\frac{t^{k_0}A\\left(\\frac{h}{t}\\right) - A(h)}{t^{k_0}-1} + O(h^{k_1}) = \\frac{s^{k_0}A\\left(\\frac{h}{s}\\right) - A(h)}{s^{k_0}-1} + O(h^{k_1})", "216e7bbdc97a94791215c19d6df9e1d0": "\\scriptstyle \\frac{1}{2\\left|I_o\\right|}D\\left(1-D\\right)=1", "216eaedde9b004436fc1d6cf5f80ee71": "f(z)=z+a_2z^2+a_3z^3+\\cdots", "216ec65feacd80553dd8f7124e5c4f49": "A_v", "216ed087408910451a172f71cb1f06c2": "L(M^n) \\,\\!", "216ee234caf09b6e31528b617db709f6": "M_{L/2} = \\tfrac{PL}{4}", "216f0150835f93248a8f37e28b80274b": "\\left(\\nabla^2 - { \\mu\\epsilon } {\\partial^2 \\over \\partial t^2} \\right) \\mathbf{B}\\ \\ = \\ \\ \\mathbf{0}", "216f10db4cd96e855662a14e32f5dda8": "b_{n}=\\sum_{k=0}^{n}\\binom{n}{k}^{2}\\binom{n+k}{k}^{2}.", "216f35086b904ef8c0c8336c57a16f04": "T\\rightarrow \\infty", "216f55ef93dd090ba0d9980d5044d8fe": "z^n = e^{sT n}", "216f5a97587b24ecaf1b78dbca6fae98": " \\frac{\\partial^2}{\\partial x^2} \\left( EI \\frac{\\partial^2 w}{\\partial x^2} \\right) = -\\mu \\frac{\\partial^2 w}{\\partial t^2}.", "216f7b026626e2d7f965f190b9d25d45": "_k\\mathbf{H}_{l,m,n}=_k\\left[H_1,H_2,\\ldots,H_{11},H_{12}\\right]^T_{l,m,n}", "216f8fcff0aaf2636efa2f85a562d2ca": " M_2 = \\left\\{ \\frac{| 0 \\rangle+i | 1 \\rangle}{\\sqrt{2}},\\frac{| 0 \\rangle-i| 1 \\rangle}{\\sqrt{2}} \\right\\} ", "216fb32d1fbe9ec88d234123e1c66d2a": "S_k(\\Gamma(1))", "216fd508338caa344e883065a6e00e73": "\\le \\sum_{k=-\\infty}^{\\infty}{\\left|h[n-k]\\right| \\| x \\|_{\\infty}}", "216ff7c17ae1adfd008adbb47e911e0f": "w^K", "217001962de8bfc5c6093154f2ef943f": "\nH(x,y, p_x, p_y) = \\dot{x}p_x + \\dot{y} p_y - L = V(x, y).\n", "2170191a93318450a9046ee5b62fdee3": "i=1,\\ldots, n\\!", "21707804fb8dd4b4169f358a8fd8cd24": "\\mathcal U\\subset\\mathcal Q", "21707a21977e5fcc5c5ba87c169e4c90": "-3q_p(3) \\equiv 2\\sum_{k=1}^{\\lfloor\\frac{p}{3}\\rfloor} \\frac{1}{k} \\pmod{p}.", "217080a8b9250f8c0e78401f816a34ca": "X_{n+1}=X_n-[F^{\\prime}(X_n)]^{-1}F(X_n),\\,", "2171554e6aaf295b93ae39f390cdcd29": "A = a_{i_1}^{\\varepsilon_1} \\ldots a_{i_L}^{\\varepsilon_L}", "217169df3e25e6e9f40cb72757e33adc": "v = {\\ell}{d\\theta\\over dt} = \\sqrt{2gh}", "2171772eb618af31c5c56217cb21983f": "\\varepsilon= \\frac{|\\partial\\Omega_a|}{|\\partial\\Omega|} \\ll 1", "2171b66d13f14bd0f80141075607d451": "\\left(q_1\\left(\\sigma\\right),q_2\\left(\\sigma\\right),q_3\\left(\\sigma\\right)\\right) \\ ", "2171b7e602e0a148cfd7422862e85cfb": "f(L)=\\bigcup_{s\\in L} f(s)", "2171d9170d8e821e65233aa7fdc200cc": " k = O (q^{2}) ", "2171da95668913b97e2fc405ba7fb381": "b(x,r) = \\langle x, r \\rangle", "2171dd31597818e4437fbdd01e37080f": "\\int \\left| \\cot{ax} \\right|\\,dx = \\frac{1}{a}\\sgn(\\cot{ax}) \\ln(\\left|\\sin{ax}\\right|) + C ", "21721fadbf56fc5ab7075c8ca94ee308": "Nu_c=3.657", "217234240e7d3b86f10ee6f450cd370a": "\\rho_{y}", "217251e2d9f732a6af34ebc9f1e4478e": "S_L(1) = 0\\%", "217257c15f3cfff45a94db0d3ac9d732": " \\nabla\\varphi = \\cfrac{1}{h_i}{\\partial\\varphi \\over \\partial q^i} \\mathbf{b}^i ", "21728774737384bd69b2e7144471bf90": " \\operatorname{F}_{u, t} (\\operatorname{F}_{t, s} (x)) = \\operatorname{F}_{u, s}(x). ", "217296ed3e90eb644bb9af93b0165e78": "|\\psi(x)|^2", "2172d88b2ca6bed5858269ba90642ac4": "\\mu = \\nu = \\rho", "21731f47a47e24cbffdf187bb61f3e7c": "E = \\frac{\\mathbf{p}\\cdot\\mathbf{p}}{2m}+V(\\mathbf{r},t)=H", "21734b9552c3c72ab4c9647e12c2a451": "217 = 6^3 + 1^3 = 9^3 - 8^3", "2173525bb2bfb2692f153b7eda96f765": "y^2=4ax \\,", "2173a8fa0f870294d0ee1413242294c6": "qN_Aw_P \\approx qN_Dw_N \\,", "2173dae8fb10c121262fc191a0deeb38": "\\sum_{A \\in 2^X} m(A) = 1 \\,\\!", "2173f2c15f23ade85d6f3ea166c513ff": " \\delta \\int_{t1}^{t2} \\left[ T - (U + V) \\right]dt = 0.", "217401af5341068d6ad73f9222981888": " \\mathrm{E}[X_i] = \\frac{\\alpha_i}{\\alpha_0},", "217488d6ef0aef2c480a151b1bfa7207": "\\{\\iota^*(c_{i,j})\\otimes b_k\\}", "2174a98efb5a9bb1bdbca996c55f5a1f": "m(\\text{brain tumor}) = \\operatorname{Bel}(\\text{brain tumor}) = 1. \\, ", "2174b470b7ca41a29312ff3e064e6e49": "L_y(x, y)=-1/2\\cdot L(x, y-1) + 0 \\cdot L(x, y) + 1/2 \\cdot L(x, y+1),\\,", "2174f7d89f00fef810dfe35d9f97015d": "T^* = L^{*2} \\,", "2175082e9d5c0e42173b7bcb17e6962d": " \\widehat{S}(\\theta , \\hat{\\mathbf{n}}) = \\exp\\left( - \\frac{i}{\\hbar}\\theta \\hat{\\mathbf{n}} \\cdot \\widehat{\\mathbf{S}}\\right) ", "2175116772a76f991347b206f139ef40": " F_{C} ", "21754135855442e39bb42f5ea3a663cc": " \\psi(x) = \\delta(x - x_0) ", "21759ccac7f9f0d7e448b7fc2156180c": "\\mathbf{r}(t)=\\langle f(t), g(t)\\rangle", "2175a31c3c28a40d9f186080697e9642": " B = False", "2175a3e04eb3560a467b7d5f85ebc77b": "(a, b, c) = (672, 153, 104).\\,", "2175da22a04cb9157833d32eefed9641": "\\mathbb{M}^c", "21761720a1ab42cbaff13db5f0bb79c7": "\\mu'_4=\\kappa_4+4\\kappa_3\\kappa_1+3\\kappa_2^2+6\\kappa_2\\kappa_1^2+\\kappa_1^4\\,", "217646c734966e740b12aed89a227d51": "z^n\\overline{p(\\bar{z}^{-1})}", "2176735cb8168585a77f1071d706456e": "\\mu = \\sum_{i} a_i \\delta_{s_i}", "21768190c4610b071e183cfecfa419f8": " V(x_1,x_2) = \\frac{g}{l} (1 - \\cos x_1) + \\frac{1}{2} x_2^2 ", "2176a581c35515a6c8065ea1bfe59b80": " w = 1 + {a_v\\over 100}", "2176b1376b8d3aa1049ed7a89a24e0a6": "\\log n! = \\sum_{x=1}^n \\log x.", "21771eaddf05f9e1fb9f24d22f64196c": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}{10 \\choose 1}{4 \\choose 2}{36 \\choose 1} \\end{matrix}", "21772425eaf34a5cd088130f3db9bcd8": "\\textstyle N \\ge 2 ", "2177392e865e0c9492369739704a3dd8": "\\Pi_1 = \\bigg(a - b(q_1+q_2(q_1))\\bigg) \\cdot q_1 - C_1(q_1).", "217748bdd5565eebb0f90f50462edbc0": " U \\subset 2^{(\\Gamma^* \\times \\Sigma^*)}", "21779345d862b73b208fe559b1b51d45": "\\nabla \\times \\mathbf a = \\sum_{i, j, k} \\mathbf e_k \\epsilon^{ijk} \\frac{\\partial a_i}{\\partial q^i} = ", "2177ccab356d5cac318ef51a78236fbf": "\\frac{3}{\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{4}} = \\frac{1}{\\frac{1}{3}(\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{4})} = \\frac{12}{7}\\,.", "2177dc8d8b89c99797fc39e27c242a78": " P = \\{ p_1 , \\ldots , p_n \\} ", "2177ec6c5347e1ee0e225fcc6deaeefa": "C_{e_1}", "217802005caaa90d4653aeb33e357ec2": "v_1,v_2,v_3\\,", "21781fc46dfcbf5dee2d12d5a584d29c": "\\frac{\\operatorname{d}V}{\\operatorname{d}t} \\leq -\\mu (\\sqrt{V})^{\\alpha} \\leq -\\mu \\sqrt{V}.", "217821161c83a9d7726e89408bf31a5f": " {3 \\choose 2} = 3.", "21783d1a926860e14547aa96be59023e": "m_x(y)", "21785b9b960ba6ef639cb6db55a9f043": " \\hat {\\textbf{Q}}(t) =\n\\sum\\limits_{i = 0}^n {N_{i,p}(t) \\hat {\\textbf{Q}}_i } =\n\\sum\\limits_{i = 0}^n {N_{i,p}(t) \\hat {w}_i \\hat {\\textbf{q}}_i }\n", "2178702fcc7ded8ceabe1d28b312653e": "R=\\sqrt{x^2+y^2+z^2}", "21787168223d7e78d73447712712cfce": "\\mathcal P_n", "21787a050b12ff7becad8d6f0a0f3c89": "\\rho_{00} \\partial_t u = - \\partial_x p", "21789ac7219ef88f42f5ae355f02872b": " H_{i,j(k)}=\\dot{W}_{i,j(k)}+W_{i,j(k)}^2 ", "2178c68ef3c29094186802c6387267ec": "\\{ a_n \\}", "2178df1bebb97c3aa437190ae553148b": "X_{(1)},\\dots,X_{(n)} \\, ", "21791350b90bf4ad79c8d68ba7ff6ee1": "\\frac{{\\rm d}^2 \\theta}{{\\rm d} t^2} = -\\omega^2 \\theta ", "21792d9d3b7032e95d5af386b0c6c375": " H_0 : \\mu_1 = \\mu_2 \\ \\ \\text{vs} \\ \\ H_1 : \\mu_1 \\neq \\mu_2.\n ", "21796de18cb46aafa3f3a363a096d776": "c_j = c_{o} \\cdot r^j", "21799a4a0c32cda0e4175a5615a9328a": "r_{m}^{(i)}", "2179f48c5eff5bf09a78c23ffc652887": "\n\\frac{n_{\\rm u}}{n_{\\rm l}} = \\frac{g_{\\rm u}}{g_{\\rm l}} \\exp{(-\\frac{\\Delta E}{k T_{\\rm ex}})},\n", "217a1998a837516871dfc76b24fe994d": "\\nabla^2 \\Phi_N = 4 \\pi G \\rho", "217a350a86fd20f90a3e0bcd057af6b8": "\\mathbf{\\hat{f}_{0:5}}", "217a6185f155a31cb281cbb0756148bd": "\\mathcal L=|\\partial_\\mu\\Phi|^2+\\frac{1}{2}(\\partial_\\mu\\sigma)^2-g|\\Phi|^2\\sigma^2-\\frac{\\lambda}{4}(\\sigma^2-\\sigma_0^2)^2\n", "217a98860bfd1007140bfb1aded08c33": "e_{i_1...i_p}", "217aadf0144b75ed5f2170868b5c5d2e": "t \\mapsto \\Phi(t) + c", "217adf31610fd8213234b4ccf0e741a5": "\\gcd(p, qr)=\\gcd(p, q)\\,\\gcd(p, r)", "217b449377c3cfdcc93fa28b212ce62e": " r = \\operatorname{rank}\\left[H_1(G,\\Z)\\right]. ", "217b8a95c021c25c824c81ee40b2b28c": "\nP(E_{n}) = \\frac{e^{-n\\beta h\\nu}}{Z},\n", "217baf66c6e4061605ba139a1caed99b": "n > 4", "217bc58072358947de65f93d2a3d6512": "\\mathcal{J}_{i,i}=J_{i,i}=a_{i-1},\\, i=1,\\ldots,n", "217bd2cdd730c9e010ee477318dfa2cf": "\\begin{matrix} {12 \\choose 1}{4 \\choose 2}{11 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "217be54fadc95f6fe4a637f6689a1c39": "\\psi (X)= \\left\\{\\begin{matrix}\nX & \\mathrm{if} \\quad 0 \\leq X<1 \\\\\n1+ \\psi (\\ln X) & \\mathrm{if} \\quad X \\geq 1\n\\end{matrix} \\right.", "217c294495dd5c85768b52127fd0ef52": " \\lambda^*=\\frac{\\frac{\\partial u(x^*_1,x^*_2)}{\\partial x_1}}{p_1}= \\frac{\\frac{\\partial u(x^*_1,x^*_2)}{\\partial x_2}}{p_2}", "217c7f0433ae5ea45963384a9802b834": "\\begin{pmatrix} 1 \\\\ f_1(t,x_1,x_2,\\ldots,x_n) \\\\ f_2(t,x_1,x_2,\\ldots,x_n) \\\\ \\vdots \\\\ f_n(t,x_1,x_2,\\ldots,x_n) \\end{pmatrix}", "217ca48c96862ca304bb6e11932fd305": "\n\\begin{align}\n\\Pr(C_i = c|C_1,\\ldots,C_{i-1})\n& {} = \\begin{cases}\n\\dfrac{\\theta + |B| \\alpha }{\\theta + i -1} & \\text{if }c \\in \\text{new block}, \\\\ \\\\\n\\dfrac{|b| - \\alpha }{\\theta + i - 1} & \\text{if }c\\in b;\n\\end{cases}\n\\end{align}\n", "217ceedd1a815bcbf02f345358158df3": "n^{*}", "217cf6883d5315ae356bdc232176ba89": "m(m-1)\\cdots(m-k+1)\\frac{x^m}{x^k}", "217e064d9a83d54005f3059c663c8150": " b_n = \\sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2}.", "217e367c9c26e887ca561010e123ced5": "C_n = \\frac1{n+1} \\binom{2n}{n}", "217e6ecadea770b3576db741285e9f30": "= (P^+ \\Omega_{\\alpha \\beta})^{IJ}.", "217e725ccc2519a857f93afb871d805c": "a = \\frac{u^2}{v}+v, \\ \\ b=\\frac{u^2}{w}+w, \\ \\ c=\\frac{u^2}{v}+\\frac{u^2}{w} - (v+w) ", "217eb5768e3734fc04b42e6f9e6056a9": "\\pi\\approx\\left(9^2+\\frac{19^2}{22}\\right)^{1/4},", "217eda8f9466e8bd352d2721bc6f56a3": "\n\\tilde v_i = \\left\\{ \\begin{array}{lcl}\nv_1+v_n-v_0 & &i=0 \\\\\nv_{i+1}+v_{i-1}-v_i&\\qquad\\qquad&0\\\\\nv_{n-1}+v_0-v_n & &i=n \\\\ \\end{array}\\right.\n", "217f0daf17ff768519e1719db0e9991a": "(\\varphi,\\partial_\\mu\\varphi)", "217f791a974496a62399f2af77740933": " \\ N(r)\\ = a_0 + a_1*r + a_2*r^2 + ... + a_t*r^t .", "217f871736e88ae84650c9a14575aa1e": " |0 \\rangle \\ ", "217fcbfee83c5f2161c4648b0e3a2892": "u_\\epsilon(x)", "217fceb9e055913453cdc5df16f098bc": "\\scriptstyle f\\, \\circ \\,\\phi^{-1}:\\; \\mathbb{C}\\, \\rightarrow \\,\\mathbb{C}", "218002f80c2146d160f0a971b56299c5": "\\mathbf{a} \\times \\mathbf{b} = [\\mathbf{a}]_{\\times} \\mathbf{b} = \\begin{bmatrix}\\,0&\\!-a_3&\\,\\,a_2\\\\ \\,\\,a_3&0&\\!-a_1\\\\-a_2&\\,\\,a_1&\\,0\\end{bmatrix}\\begin{bmatrix}b_1\\\\b_2\\\\b_3\\end{bmatrix}", "21800b32a1b608fb5d507ce2d302a460": "\\lambda/4", "2180271cc6dced8cef8da9e9c2083ef9": "D_Y(f_*M) \\cong f_!(D_X(M)).", "21808daa1d65eea33b42649cd40f749d": "F(U) \\rightarrow \\prod_{i} F(U_i) {{{} \\atop \\longrightarrow}\\atop{\\longrightarrow \\atop {}}} \\prod_{i, j} F(U_i \\cap U_j).", "2180a228cca6f7bf19cd3e895bc5a29e": "\\bar{e}\\,", "2180b0d941e4eb8fddc3a3ff67f1070b": "X_\\mathbf{k} = \\sum_{\\mathbf{n}=0}^{\\mathbf{N}-1} e^{-2\\pi i \\mathbf{k} \\cdot (\\mathbf{n} / \\mathbf{N})} x_\\mathbf{n}", "2180b76803a964de63e337b4627ca6bc": "(a \\times b) \\times c = a \\times (b \\times c)", "2180bb9c32e453402e673ddd5eef141c": "\\mathbf{W}_\\mu", "2180c5b619fac6dd8af3882d1f447dd3": "C^1([a, b])", "2180c88e88976ba8f197f7463f2e71e5": "x \\in L \\Leftrightarrow \\#\\{r \\in \\{0,1\\}^T\\mid f(x, r)=1\\} \\ge 2^{T-1}", "2180f11cad5ad387710f6b6570c32909": "p\\in S'", "218154caad3c5613d38c37bd42d6abc4": "If = \\lim_{n \\to \\infty} Ih_n", "218239dc837f5b391c72492f17a6c453": "\np(x) = \\sum_{i=0}^{\\infty} f^2(\\lambda_i) \\phi_i^2(x)\n", "2182d241246465cc9703604be20703fd": "\\scriptstyle T^1_1(V) \\rightarrow \\mathrm{End}(V) ", "2183053d0fa6fb9108a5a4683dacf8b7": " \\text{Observe: } \\omega(t), Q_1(t), ..., Q_K(t)\n", "21833b669d3b5876b6b5f6f9363165ce": " A^+ = A^{-1} ", "21838ae802ab1ae5b2df6aadd31a76d9": "R_{\\mathrm{ads},i-1} = k_i P \\Theta_{i-1}", "2183d8f3770be285177b65cb8269f615": "\\langle F \\rangle", "21846d0110acae240f582d2d5d0e758e": "V_b\\,", "21846dc6f689a02ac3eda9491286f24f": "a+R < r", "21848b5a0ed3e6aaa486459c9a676195": " 1 - 1 = 0 ", "21849574c2f0088ab6c9303151b4b35b": "\\mathbb{Z} = \\sum_\\mbox{states} \\exp(-\\beta \\varepsilon(\\mbox{state})) ", "21852413d464fc53e8e363e8d00d6369": "(F^m \\and G^n) \\leftrightarrow (F^m \\times G^n) \\leftrightarrow (F^m \\exist^m G^n); m0,\n\\end{align} ", "21905692161563b3412cba070633fa5d": "|z|^2< {1\\over 2} (1 + (z\\cdot z)^2) < 1 ", "2190729126e01a6e0bca46ba317aaf95": "E \\in \\mathcal{F}", "2190772e7cee0d1c38929df30a209acc": "c = M_{2,2} = i \\,", "2190f11ad8d6c85164aa477d9af4a153": "Z=\\sum_{n=0}^{\\infty } \\left ( \\frac{4}{125} \\right )^n \\frac{(33n+4)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{3} \\right )_n \\left ( \\frac{2}{3} \\right )_n} {(n!)^3}\\!", "21914e895a78d22fdfc8697ea6767aa2": "2\\pi \\int_{a}^{b} yh(y)\\,dy", "21914f5c5cb29d69e4b5a09057f8d88f": "A=0.7\\,162\\,162\\,162\\,\\cdots .", "21916701de34e75d8627a7f2d04451b5": "b(v) \\sim 1 + \\beta v^{2}/c^2 \\,", "2191b48b16a1ac22f0b98df02d86e23c": " \\!\\ S_{m}^n = K_{n}S_{m} + K_{(n-1)} ", "2191ec67979ff4d85c86941d9c099727": "\\,\\lambda_i\\ge0", "21926fbdb516b0d13b6babd728522e50": "\\nu = \\mu/\\rho", "2192a006517c647900bd7a805a778976": "\\pi_k \\circ \\delta_a = id_a", "2193148b619ac10a6f31cb8ec9dc9cfc": "r = \\frac{\\hbar}{\\sqrt{2m(V_0-E)}}", "21931fdd29a1b3cf9fb338534b73d30a": "N = \\begin{pmatrix}0 & 1 \\\\ 0 & 1\\end{pmatrix}", "219323bb72cf06f8fbfc5060a07b4ab1": "\\overrightarrow{\\mu} = g^{(l)}\\overrightarrow{l} + g^{(s)}\\overrightarrow{s} ", "21932895058c780ca922d1c4cae29a90": "S(\\phi) = \\int \\mathcal{L}(\\phi) dx dt = \\int L(\\phi, \\partial_t\\phi) dt", "219332220478cdc3913ce2b6ab9160cc": "{29^2\\equiv 2^1\\cdot 11\\pmod{91}}", "21936b63da64f2186685a392e5bb4da2": " f_1 = S_1(Y-f_2), f_2 = S_2(Y-f_1) ", "21938e8ebc7552abdaa866f2d9bd1b28": "d(u\\cdot v) = v\\cdot du + u\\cdot dv \\,\\! ", "2193c0ffd6e3180ea5062649c074b8aa": "0 \\leq a_n \\leq 1/n", "2193dc8df6dab6ef728ddaf403ae568f": "\\sqrt{2GM/r}", "2193e0c0223c798179faec9d4f1dc2c8": "S(\\mathbf{q}) = \\frac{1}{N} \\left | \\sum_{j=1}^{N} \\mathrm{e}^{-i \\mathbf{q} \\mathbf{R}_{j}} \\right | ^2", "21941ad54f38291b6dcd99693d60311d": "f_2 = f_2(z) =\n\\sum_{j=0}^{\\infty}\\frac{a(j)}{b(j)}\\frac{z^j}{j!} , ", "2194555f48693165cc98db004f19c8f6": "\\simeq H_{\\mathrm{dR}}^{k}(S^{n-1}).", "219468df07c99ba2b66370ee4901b586": "C_{\\Psi}", "21947a886f4cc608dcffca2281029477": "D_{MLD}(y) = \\arg \\min_{c \\in C}\\Delta(c, y)", "21948c8acb91cdb55fa17cb8f63e172e": "T_q^*Q", "21949897e63154d0901b52494ec5cf44": "Z(T) := \\sum_{n=1}^\\infty \\exp \\left(\\frac{-E(n)}{k_B T}\\right) = \\sum_{n=1}^\\infty \\exp \\left(\\frac{-E_0 \\log n}{k_B T}\\right) = \\sum_{n=1}^\\infty \\frac{1}{n^s} = \\zeta (s) ", "2194c107cf0eca36caa33fdb998cf399": " (A_3, E', F', G', A_0) = (A_3, H'', I'', J'', A_0) ", "2194fd3f7ed676cabc87a0ef7592c060": "P( | X - m | \\ge ks ) \\le \\frac{ 1 }{ [ N( N + 1 ) ]^{ 1 / 2 } }\\left[ \\left( \\frac{ N - 1 }{ k^2 } + 1 \\right) \\right]", "2195605d639e96a5729795e7a72a9da9": "\\, r=a\\cdot\\sec\\alpha=\\frac{a}{\\cos\\alpha}", "2195654fc4d972af24f6bf4d8fadec86": "c \\leftarrow g_1^x rem P", "21956742389d9d22c22317c84fc9cf95": "\\mathbf{L_z} = m\\hbar.", "2195ad440ca8575c79d0ba37346586d5": "4K^2 = (pq + rs)^2 - \\frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2", "2195b3f6edffdf08bcb55afe484c7855": "m\\ddot{x}+ q\\left(\\frac{\\partial A_x}{\\partial t}+\\frac{\\partial A_x}{\\partial x}\\dot{x}+\\frac{\\partial A_x}{\\partial y}\\dot{y}+\\frac{\\partial A_x}{\\partial z}\\dot{z}\\right)= -q\\frac{\\partial \\phi}{\\partial x}+ q\\left(\\frac{\\partial A_x}{\\partial x}\\dot{x}+\\frac{\\partial A_y}{\\partial x}\\dot{y}+\\frac{\\partial A_z}{\\partial x}\\dot{z}\\right)", "2195cc57caeddee2548ec4f9445860a1": "\\det(\\exp(A)) = \\exp(\\mathrm{tr}(A))\\, ", "21960fdf21e7764f37e80137585d270f": "\\mu = m", "219612d73ea46ad66f68d40ac42608d2": "2^n\\,n!\\,\\sqrt{\\pi}", "21969ea72c67bd9f8fcc0517eab6fb92": "m_\\mathrm{Earth}= 5.9736 \\times 10^{24}\\,\\mathrm{kg} ", "21970cb4c9f7e16003fb4f6bb5c5e2c6": " y^2=x^3+ax^2+16ax ", "2197457d64b9da6dbfeac1ee45355738": "\\begin{align}\n\\operatorname{Cov}(z',z'A') &= E\\left[\\left(z' - E(z') \\right)\\left(z'A' - E\\left(z'A'\\right) \\right)' \\right] \\\\\n&= E\\left[ (z' - \\mu') (z'A' - \\mu' A' )' \\right]\\\\\n&= E\\left[ (z - \\mu)' (Az - A\\mu) \\right].\n\\end{align}", "219769a87487526f8865e0e39a5273fb": "0\\le p_i \\le 1", "219777140f6fe25f388e0b9ded74a35c": "s_p: M \\to M, h'K \\mapsto h \\sigma(h^{-1}h')K", "2197831c783e534c8a0f59a63ae6d248": "\\varepsilon_n = \\frac {x_n}{\\sqrt{S}} - 1", "2197c8ec7856b0314cbf2a728655065c": "\n[a,bc] = abc - bca = abc - bac + bac - bca = [a,b]c + b[a,c]\n\\,", "2197d5bceba5fe732952fc70e386f45b": "\\frac{AE}{EC} = \\frac{DE}{EB} = \\frac{AD}{BC}.", "2197f06949db5d28e82bee14a5915942": " \\boldsymbol{v} =\\frac{d}{dt}\\boldsymbol {r} =\\sum_{k=1}^{d} \\dot q_k \\ \\boldsymbol{e_k} + \\sum_{k=1}^{d} q_k \\ \\dot{\\boldsymbol{e_k}} ", "2197f09034ad29709154d05d4693ca6b": "n_k = \\lfloor k \\phi \\rfloor = \\lfloor m_k \\phi \\rfloor -m_k \\,", "219802318bd299f9ad24b80b40198274": " \\psi(f) = \\int_X f(x) d \\mu(x) \\quad ", "21982bd70bd6e0cd6f10a64ad3df1ac7": "J^k_p(M,N)", "219848d8701d5ed6d5812669f35b7359": "\\displaystyle z(\\alpha_1,\\beta_1,\\gamma_1,\\delta_1)+(1-z)(\\alpha_2,\\beta_2,\\gamma_2,\\delta_2)", "2198520b18bf1d54d1776639d54efe00": " \\frac{\\sin A}{a} \\,=\\, \\frac{\\sin B}{b} \\,=\\, \\frac{\\sin C}{c} \\!", "219882225a16b72a8cd923b5a26cd66b": "\\frac{(a_1+a_2+\\cdots+a_n)!}{a_1!a_2!\\cdots a_n!}.", "2198c38fed3d7958b10f26dd2b997702": " \\lim_{\\theta \\to 0} \\left(\\frac{\\cos\\theta - 1}{\\theta}\\right) = \\lim_{\\theta \\to 0} \\left( \\frac{-\\sin^2\\theta}{\\theta(\\cos\\theta+1)} \\right) = \\lim_{\\theta \\to 0} \\left( \\frac{-\\sin\\theta}{\\theta}\\right) \\times \\lim_{\\theta \\to 0} \\left( \\frac{\\sin\\theta}{\\cos\\theta + 1} \\right) = (-1) \\times \\frac{0}{2} = 0 \\, . ", "2198db912e7df085b4fbdff0c7f8f01c": "\\cap", "21994c70f5a75ee3e495452ebb848f43": "B'=SA'\\cap OB.", "2199e77fe8985fdc76b2e0f353c06e6a": "\\gamma'=-\\frac{1}{r}\\mathbf{r}\\cdot\\mathbf{Q}\\alpha_jsc_1+\\alpha_j'\\big[-ac_0+2b\\alpha_js^3\\bar{c}_3+\\frac{1}{2}\\gamma \\alpha_j s^4c^2_2\\big]", "219a2a169da4f5d9c0a48dc5cd7e155e": "\\displaystyle{H^*=JUH U^*J}", "219a5d496f8fbe1fccb81110f6439f08": "\nY = \\mathrm{constant} + \\mathrm{error }\n", "219ae189251bb9b8fae5e14581243ba6": "1.09", "219b1a6aa471120bbcaec3177d90d70b": "\n\\left(\n \\left(\n \\left(\n 1 \\times 2\n \\right)\n \\times 3\n \\right)\n \\times 4\n\\right)\n\\times 5\n= 120\n", "219b9e4cdcc5810b06cbfe5c1a466094": "(X, Y) = \\left(\\sqrt{\\frac{2}{1 - z}} x, \\sqrt{\\frac{2}{1 - z}} y\\right),", "219bf11299b2d7ba9c98962f38670996": " A(x) := \\sum_{n \\le x} a(n) .", "219bf5e4ef25edbc50ce770c1ae9c5db": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {H(A)} \\right) =\\mathrm{Re}\\ \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A} \\right)\n ", "219c3a03696e437dab545b6bc8824a60": "u=\\frac{x}{B}", "219c3a44c547a6d2478694ab5ce49a4b": "\n\\left\\{\\frac{\\varphi(n)}{n},\\;\\;n = 1,2,\\cdots\\right\\}\n", "219c3b87d8bbdf44aa782b6a0dcc0e28": "k_{a ;b} = 0", "219c446833ddcc2b4adbd7892ee711ee": "b(t) = b1 \\times e ^ {\\frac{-t}{\\tau}}", "219c44b0752226bd7fa1d9d976370f07": "\\displaystyle{C=\\begin{pmatrix} {1\\over 2}I + T_K & S\\\\ H & {1\\over 2}I - T_K^*\\end{pmatrix}.}", "219cb6630f157dd3cc997f173aa4c405": "\\mathrm{sys\\pi}_1 \\leq 6\\; \\mathrm{FillRad}(X),", "219cdd6a0fbfb8200eb7d345cf471291": "\\{B_\\theta:\\theta\\in {\\rm pcf}(A)\\}", "219d1394a5750927b6b96d842caff51e": "\\bar{G_i}", "219d311774dda5f7d41dee43795919de": "f_\\max", "219d32caa42c38ae364187299de27674": "\\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon)", "219d4767458bdcb419b19aa038473c1d": "\n\\nu = \\sum_{j = 1}^q b_j - \\sum_{j = 1}^p a_j .\n", "219da01a766fa0c263ae5be8a84311bf": "\\int_V \\Delta\\varphi\\,dV = \\int_V q\\, dV,", "219db81e380381a0af3354fd274187b4": "\\mathbf c=\\mathbf F^{-T}\\mathbf F^{-1}\\,\\!", "219de227675053bc12f6dca9936d0113": "\\delta(t/a)=|a|\\delta(t)\\,", "219e2f3e8418557860664004661edf98": " \\mathbf{E}' = \\mathbf{U} \\, \\mathbf{S}' \\, \\mathbf{V}^{T} ", "219e43d163fc31f6e3862a10a7b22c11": "U(1)^k", "219e624547fd0f23ce41fecdd2913e14": "\\Phi \\vdash \\lnot\\phi", "219eb343cc35d8e05e64f66df1e8058c": "C(\\alpha)_n", "219eb7ec52f9d5f67c38cca1c985cd64": "E_\\text{fusion}", "219eddc2f3dbb165221efe47feed094b": "\\scriptstyle\\mathcal{P}", "219ee16176deb0d6415dc361d70959a2": "(a^p)^q", "219f03cc5d2577d07cd7548fcd7f588b": "\\mbox{mex}(\\left \\{1, 2, 3\\right \\}) = 0", "219f06b96c9ac36bc7b15218cdc766c0": "\n\\mathbf{a \\times b} = \\mathbf{c}\\Leftrightarrow\\ c^m = \\sum_{i=1}^3 \\sum_{j=1}^3 \\sum_{k=1}^3 \\eta^{mi} \\varepsilon_{ijk} a^j b^k\n", "219f5122ac51e237892a779e9e604abe": "\\ G(\\tau)=G(0) \\exp(-\\tau/\\tau_B) +G(\\infty)", "219f775cb652d64218be939388e2b4e5": "2\\cos^2(A/2)=1+\\cos A", "219f7e9eeab9f437b53a8dc3488dc52b": " \\mathbf{x} = \\left ( \\begin{array}{c|c} \\mathbf{R} & \\mathbf{t} \\\\ \\hline \\mathbf{0} & 1 \\end{array} \\right ) \\mathbf{x}' ", "219fa2b389b3fbaccde6adc972cb6dc5": "2^7\\ln(2) =2^7\\sum_{k=1}^{\\infty}\\frac{1}{k2^k}\\, .", "219fac52e3d4591ff08ec9f9022e11b8": "\\exists{}HO^i", "219fad12ff3320f7e420d483f8a46940": "| 1 - \\omega \\lambda_j |< 1", "21a0217e626766f8b415278366a7b6ce": "\\scriptstyle BV_\\varphi([0, T];X)", "21a05a87645983bef4aacd9878b63e8a": "F_X(x) = \\int_{-\\infty}^x f_X(u) \\, du ,", "21a05bde42ad77a1873d79e0a4e08f3e": "\\eta \\left\\{\\eta_bEv/x + P\\right\\} = \\left\\{W C_{rr1} v + \\frac{1}{2}\\rho C_d A v^3\\right\\} ", "21a06a1a7de8ecfebd96b4581f51acdf": "V = \\{w\\in W : \\chi(w) = w\\}.", "21a0733e2b73149cea1c3986ea696462": "g_{uc}(\\langle 0 \\rangle) = \\varepsilon", "21a0b82edd1eedf49dd4f3eb4e6f51cc": "\\tilde{H}", "21a0bcaa65ef95104444f8cdae75d880": "\\langle u, Au\\rangle \\leq -\\pi^2 \\|u\\|^2", "21a0c8bfcbd982d292d737bead64ef3b": "M \\times \\{i\\}", "21a0d87d4f3d97af5fd2f61706d82be5": " \\vec x = \\vec x(s,t)", "21a10423e8b4d72dfd61f42f29584eb6": "{}_2F_1(0,b;c;z)=1", "21a11e1dfff2741aa9860f3f8e3b4ba8": " dV_t = r\\left(V_t-\\frac{\\partial f}{\\partial S}S_t\\right)\\,dt + \\frac{\\partial f}{\\partial S}\\,dS_t.", "21a18c9aff0f8639bcdc41560aaf6676": "B(\\mathbf x_1,\\|\\mathbf h_0\\|)", "21a20f6e1cf26e09ca079e3637c781c8": "\\dot{Q}^\\mathrm{T} = -\\dot{Q} .", "21a22d84d38af9a58deb1813f6f4f223": "A_\\alpha, B_\\alpha", "21a254f696171178f3710aaa5520bfb3": "\\phi_{\\bar z} \\neq 0", "21a2716caab13db9a186b8d89913b682": "\\mathbf{J} = \\mathbf{J}_\\mathrm{f}+\\mathbf{J}_\\mathrm{b} ", "21a273d78ad462e392de05819a8346f3": " P_4 = \\left[ \\begin{matrix} 1 & \\alpha \\\\ 0 & 1 \\end{matrix} \\right] ", "21a27a014af615af8ca081ddca28d907": "r_n=a_0.a_1a_2\\cdots a_n", "21a2b2e5e1cfc105ff09d51c3762050b": " v(\\vec{p},s) = \\sqrt{E+m} \n\\begin{bmatrix} \n\\frac{\\vec{\\sigma} \\cdot \\vec{p} }{E+m} \\chi^{(s)}\\\\\n\\chi^{(s)}\n\\end{bmatrix} \\,", "21a2cb0a67a5b3c2d27e28e6085589b8": "2+3X=a+bX \\,", "21a391b8e38d3d324bb6f9fd1d9d38b2": "K(x,t;x',t') = \\langle x | \\hat{U}(t,t') | x'\\rangle ", "21a3cad64734cbdf9a67fb16cdcc72b1": " e_t ", "21a3e817665d005a1846fd4e26541527": "[f]_x = \\{g:X\\to Y \\mid g \\sim_x f\\}.", "21a4bf60d4e339e630e37d9559b3cfbc": "m_{x,0}", "21a5234ddafd9efc17eb04b8941c9f4f": "q = 0, 1, ..., 7", "21a53a20a0a6172c51ad8a45ad7ad09d": "P_{D}(d_i,x,R)", "21a584010929a6f7bc32e9add73348ee": "10^n = 2^n \\cdot 5^n \\equiv 0 \\pmod{2^n \\mathrm{\\ or\\ } 5^n}", "21a5928563662602aeed5efc90682ef6": "\\operatorname{Ti}_2(\\tan \\theta) = \\int_0^{\\tan \\theta}\\frac{\\tan^{-1}x}{x}\\,dx", "21a598deb8b0ac865d65dfec564006e2": "\\kappa(V)\\ge\\kappa(V_w)+\\kappa(W).", "21a5bedfeb46006a7a45ffe902459694": "\n\\nabla \\times (\\nabla \\times \\vec \\psi) = \\nabla(\\nabla \\cdot \\vec \\psi) - \\nabla^2 \\vec \\psi = -\\nabla^2 \\vec \\psi\n", "21a5c8a26ec2aae23ed1a3cd86c8a853": " \\Pi_{xy}^{(0)} = \\sum_{i}\\vec{e}_{ix}\\vec{e}_{iy} f_i^{eq} =p\\delta_{xy}+\\rho u_x u_y \\,\\!", "21a5e27ef44f94e5bf0ae1ff9ed8e7ef": "T=\\frac{C D}{V}", "21a5ef591ceb9f3ee4345203d42b121b": "\\int_{-1}^1 \\sqrt{1-x^2}\\,dx = \\frac{\\pi}{2}", "21a616bf4c681017567121c3b38b79ea": " \\frac{N + \\Xi}{2} = 1128.5~\\mathrm{MeV}/c^2 ", "21a642a5a7a88dca190ea846eed208b1": "\n\\begin{align}\n |f(z)|-|f(0)|\n &\\leq |f(z)-f(0)|\n \\leq \\frac{2r}{R-r} \\sup_{|w| \\leq R} \\operatorname{Re}(f(w) - f(0)) \\\\\n &\\leq \\frac{2r}{R-r} \\left(\\sup_{|w| \\leq R} \\operatorname{Re} f(w) + |f(0)|\\right),\n\\end{align}", "21a666f271e6798ed49f5c9769cf9059": "\\eta=\\frac{\\epsilon^2}{54}+O(\\epsilon^3)", "21a699fba656bff051113323238e0471": "s = \\frac{((y_1 - y_3) (x_2 - x_3) - (y_2 - y_3) (x_1 - x_3))^2}{((y_1 - y_3)^2 + (x_1 - x_3)^2)((y_2 - y_3)^2 + (x_2 - x_3)^2)}.\\,", "21a70d47bb5899bc506c5b9868c0d3d5": "\\begin{align}\n {\\partial^2 \\mathbf{E} \\over \\partial t^2} - {c_0}^2 \\cdot \\nabla^2 \\mathbf{E} \\;&=\\; 0\\\\\n {\\partial^2 \\mathbf{B} \\over \\partial t^2} - {c_0}^2 \\cdot \\nabla^2 \\mathbf{B} \\;&=\\; 0\n\\end{align}", "21a72f9348451ecde99da6d28f886ea6": "\\frac{CN}{\\log^A N}\\int_0^1|S(\\alpha)|^2\\;d\\alpha \\ll \\frac{N^2}{\\log^{A-1} N}", "21a77a08a8b5fc8c27eee72bd195b1b3": "\ndL_j(t) =\n\\begin{cases}\nL_j(t){\\sigma}_j(t)dW^{Q_{T_{p}}}(t) - L_j(t)\\sum\\limits_{k=j+1}^{p} \\frac{\\delta\nL_k(t)}{1 + \\delta L_k(t)} {\\sigma}_j(t){\\sigma}_k(t){\\rho}_{jk}dt \\qquad j p\\\\\n\\end{cases}\n", "21a7a1c172b43bd471c7d51fcf171250": "\n \\int_{\\Gamma} f_j~n_j~d\\Gamma = \\int_A \\cfrac{\\partial f_j}{\\partial x_j}~dA\n ", "21a7bcd8dc635202ea92dfbc60189d5a": "\\Rightarrow^{ac}_{l} aSSS \\Rightarrow^{ac}_{l} aaSS \\Rightarrow^{ac}_{l} aaaS \\Rightarrow^{ac}_{l} aaaa", "21a7c1e709f78e098c19f4103aedaed6": "U(1)=e^{iq\\theta}", "21a8159f7a1ed14d87666f82b1249bc7": "P_{\\mathbf{v}^K}(\\mathbf{x}) = \\langle \\prod_{i=1}^K \\delta(x_i - \\mathbf{s} \\cdot \\mathbf{v}_i) \\rangle_{\\mathbf{s}}", "21a82c1170f096a900f5968cb974e946": "f_\\tau(x) = e^{\\frac{1}{2}i\\tau x^2}", "21a870c7f3e21a18cbd4e17d50307cba": " \\lambda x . x ", "21a89f2090efe32537fab73de5fe6576": "q'^0=q'^0(q^0,q^k), \\quad q'^i=q'^i(q^0,q^k), \\quad\n{q'}^i_0 = \\left(\\frac{\\partial q'^i}{\\partial q^j} q^j_0 + \\frac{\\partial q'^i}{\\partial\nq^0} \\right) \\left(\\frac{\\partial q'^0}{\\partial q^j} q^j_0 + \\frac{\\partial q'^0}{\\partial q^0}\n\\right)^{-1}.", "21a8a09f4db0fe4cd2937c3cde905084": "E(M,K)", "21a8ac2e9b9c2bbb2d33c99fd8004136": "\\hat \\mu=\\bar X = \\frac{1}{n}\\sum_{i=1}^n X_i.", "21a8b82e294ace0af8d6dfc9b3c106bc": "K_O(x) = \\sup_{\\xi\\not=0}\\frac{\\langle G(x)\\xi,\\xi\\rangle^{n/2}}{|\\xi|^n},\\quad K_O(x) = \\sup_{\\xi\\not=0}\\frac{\\langle G^{-1}(x)\\xi,\\xi\\rangle^{n/2}}{|\\xi|^n}.", "21a92feea1ac714a8879fbb7571af88d": "A = \\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}", "21a961d2422c57fda332346d822d982e": "(A(\\mathbb{R}), \\circ, +)", "21a9a8c563a40cfa8232139660aaa54f": "E=E^0 + {RT \\over F}\\ln {a_{\\mathrm{H}^+} \\over (p_{\\mathrm{H}2})^{1/2}}", "21a9ba04fdb60d769a6fbadfb0ef6d43": "K_n=\\frac{2\\lambda}{d}", "21a9eb087eddd40f26e4d9ea1fd52abb": " \\frac{\\partial\\psi}{\\partial t}+u_x \\frac{\\partial\\psi}{\\partial x}=0 ", "21aa65a2cc27e6986d2463788197ef3d": "\\operatorname{\\widehat{MSPE}}(L)=\\sum_{i=1}^n\\left(y_i-\\widehat{g}(x_i)\\right)^2-\\widehat{\\sigma}^2\\left(n-2\\operatorname{tr}\\left[L\\right]\\right).", "21aa98ce5e5122192256ca834e3b16df": " \\begin{align}\n\n\\nabla_\\mathbf{a}\\det(A) &= \\mathbf{b} \\times \\mathbf{c} \\\\\n\n\\nabla_\\mathbf{b}\\det(A) &= \\mathbf{c} \\times \\mathbf{a} \\\\\n\n\\nabla_\\mathbf{c}\\det(A) &= \\mathbf{a} \\times \\mathbf{b}.\n\n\\end{align} ", "21aaa585089a8af662a881765d53eaaf": "Z=\\infin \\,\\!", "21aacba5b2d5a4edb5f71e36df15e37b": "\\Lambda_{\\chi'}{}^{\\psi} \\,", "21ab045de04ee078ecaed9e2cf771bff": " \\mathbf{\\nabla}^2\\psi-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}\\psi = \\left(\\frac{m c}{\\hbar} \\right)^2 \\psi ", "21ab116f87935a8ab62a5822eff38fa1": "Q = U A \\Delta T.", "21ab432bee0d12ab6048fbcf82534fd0": "\\operatorname{ad}(xy) = \\operatorname{ad}(x)\\operatorname{ad}(y) ", "21ab989245a35ab4aac4d0b8347b5729": "E=E^0 + s\\log_{10}[A]", "21abe8380a871d73c31889902896ff5a": " (\\theta_{xi},\\theta_{yi} ) ", "21abf108f5ac054b37011bc2d56119f7": "\\hat {\\textbf{Q}}= \\textbf{Q} +\n\\varepsilon \\textbf{Q}^0", "21abf34a02364751b5cdf2a937985fd9": "D_0(L \\ 1s \\rightarrow \\psi^*) = const \\ \\vert \\langle L \\ 1s \\vert \\mathbf{r} \\vert \\psi^* \\rangle \\vert^2 \n= \\alpha^2 \\ const \\ \\vert \\langle L \\ 1s \\vert \\mathbf{r} \\vert L \\ np \\rangle \\vert^2 ", "21ac784dadad918e06ec61ecac9100dc": "\\begin{array} {ll}\n\\nabla^2 f &= \\frac{\\partial ^2 f}{\\partial x^2}+\\frac{\\partial ^2 f}{\\partial y^2}\\\\\n\\nabla^2 f &= \\frac{f\\left(x + h,y\\right) + f\\left(x - h,y\\right) + f\\left(x, y + h\\right) + f\\left(x, y - h\\right) - 4f(x,y)}{h^2} - 4\\frac{f^{(4)}(x,y)}{4!}h^2 + \\cdots\\\\\n\\nabla^2 f &= \\frac{f\\left(x + h,y\\right) + f\\left(x - h,y\\right) + f\\left(x, y + h\\right) + f\\left(x, y - h\\right) - 4f(x,y)}{h^2} + O\\left(h^2\\right)\\\\\n\\end{array}", "21ac7c6ccca0262e51fe5f90ee13f8b0": "\\begin{align}\n P(S, t) &= Ke^{-r(T - t)} - S + C(S, t) \\\\\n &= N(-d_2) Ke^{-r(T - t)} - N(-d_1) S\n\\end{align}\\,", "21acc3c91f0859755da1a796bbd2b59f": "\\xi \\stackrel{q} \\longrightarrow \\acute{\\xi}", "21acdb4b4a82954016549e00c3b258fe": "\\forall x \\exists y \\; Rxy", "21acec853474dc91e8a731c4802a3ff6": "\\displaystyle \\frac{P(I|c_f)}{P(I|c_b)} > \\frac{\\lambda_{Ac_b} - \\lambda_{Rc_b}}{\\lambda_{Rc_f} - \\lambda_{Ac_f}}\\frac{P(c_b)}{P(c_f)} ", "21acf1cb957157441bc6225b7a3b618d": "R(t) = s t,", "21ad0bd836b90d08f4cf640b4c298e7c": "bb", "21ad1e7bcddc9ac0a73707b2c67fa180": "\\ h(t) = \\Pi\\left( \\frac{t}{T} \\right).", "21ad46960bd03ad6a13bd66fb151dd20": "\\mathbf{G}^{\\circ \\frac{1}{2}}", "21ad58214bf92fc8248308f411dfe06b": "\\mathcal{R}_m", "21ad6ffe12ed760ae44e0a2f022fcf5d": " D = \\frac{ n ( m - a ) }{ s } ", "21adb1e2fa95ca7917e3501971e9d33d": "\\ Ax^2+Bx+C=0", "21aded5bc02cb46da611b6d06728cfd1": " \\partial{X}/\\partial{{n}} ", "21adf2555d341667416ddafb147528e9": "J=\\frac {Lf \\cdot \\sqrt{S}} {\\sqrt[3]{D}} \\le 3,2", "21ae44554c8a9abd918b378db07bfc26": "R_3=\\sqrt[3]{a^2b}\\,\\!", "21ae4b9ec57cd4f9466ee9d227cd7dec": "f\\circ h=h", "21ae5be16635dceac0663f61c8166ed2": "\n C(x) = \\cfrac{Q_x E^f f(f+2h)}{2D}\n", "21aec3f46816de8f2cf79cd830fa9355": "\\boldsymbol{r_2}", "21aec79060493aefb649c1f18a089364": "\\gamma = \\frac{\\mu\\,\\mu_N}{hI}\\,", "21aeea5542d054ef5319ba5408b22919": "Pxy \\leftrightarrow \\forall z[Ozx \\rightarrow Ozy].", "21aef3b32ae97d3e6e8ff4d095fb3d56": "O(E)", "21aef423aa6dcc39d9de66e65dd84fd3": "\n I = \\cfrac{ah^2}{4} ~;~~ S = 0.5 a h\n ", "21af4c38641ef6027157fda1aa11ebeb": "\\frac12(1+Z)^2\\,\\left(-\\frac12 + 2\\,Z - \\frac12\\,Z^2\\right)", "21af72b899e10fe2b905482164c5521f": "c_c=1", "21afbc1472f1730a357f8fb094fec734": "|Q\\mathbf{x}| = |\\mathbf{x}|.", "21afc3b9dd9406c8d0805dd2134f7ac5": "V^{M}_{N \\setminus \\{b_i\\}}-V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_i\\}}", "21afe9fbbff33112073d4503058c61de": "\\ \\lambda_{\\max} = s^\\mathrm{H} R_v^{-1} s, ", "21b04446dad94958c45d30dd528a0992": "\n U_A=\\{x\\in V:\\quad ||\\varphi||_A<1\\},\\qquad A\\in {\\mathcal A}, \n ", "21b07a37faaf6965b72f89cbd8251d94": "\\Delta T_s =~ \\lambda~\\Delta F", "21b0de2803e0673a3130518b97dd4b6b": "\\tilde{\\boldsymbol\\theta}", "21b104594c0d5568d9366efe574c413f": "X\\rightarrow \\mathbf{S}^3\\rightarrow K(\\mathbf{Z},3).\\ ", "21b17b93b982731158ed26218d82f304": "z = \\sqrt{\\frac{n-3}{1.06}}F(r)", "21b17cb041aaa50a306879929f6fff25": "N/ \\Gamma ", "21b18ded62ac8435bfe537a692b177ac": "{dy \\over dx}=\\frac{-1}{1+x^2}", "21b1c45277875087ae9ce842c35abbe4": "T_1 \\times \\ldots \\times T_n", "21b242289444ad94ca1dc6922d859997": "\\Sigma\\subset\\mathbb R^3", "21b26028bbb53216ae5978a061c255e6": "\\mathrm{0.91\\overline{6}}", "21b322eb587ea005d92ee7cbf31b6953": "\\mbox{(3) }\\bigcap_{n=0}^\\infty T^{-n} \\mathcal{K} = \\{X,\\varnothing\\}", "21b3570a016496d218587902e82b01ae": "\\alpha = \\beta{(\\beta +1)^{-1}}", "21b373d65f9f091033b97d828e58c582": "A, A\\mid(B\\mid C)\\vdash C.", "21b3864e99cdc2b18e22c2034108ae77": "x_{min,i} < x_{max, i}", "21b387ff69652b64238a378f7b57f552": "\\mathrm{Sim_2}(\\sigma,\\tau, y)", "21b3951c14556e2a9fac1220836edae3": " f_1(x)", "21b399ddb43f8c61fbea617b595cba1b": "A \\in C", "21b40848d984c4a50c94583e6c7871ee": "\\langle X, Y \\rangle \\triangleq \\operatorname{E}(X Y),", "21b410c2307373fa06388d06f9970d7e": "\\textstyle 3", "21b4ffe598d9ff594fb8ad4198457584": "\\Pr(\\text{exact})=\\sum_{\\mathbf{y}\\,:\\,T(\\mathbf{y})\\ge T(\\mathbf{x)}} \\Pr(\\mathbf{y})", "21b5414cd8f577e94cbf3072de25a10b": "-3\\le y \\le 4", "21b549fd24899eb3aedb092a6e9e00c4": "C(\\varepsilon) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N^2} \\sum_{\\stackrel{i,j=1}{i \\neq j}}^N \\Theta(\\varepsilon - || \\vec{x}(i) - \\vec{x}(j)||), \\quad \\vec{x}(i) \\in \\Bbb{R}^m,", "21b54d0d64993cdc81871210b39ddd9c": "r(t) = (x(t), y(t), z(t)) = (a \\cos(t), a \\sin(t), b t).\\,", "21b59a98e61e687ae42cc03575069635": "R_\\text{green}", "21b5b6ea6a518c72a1f8460520c3289e": "B = \\mathbb{Z}\\left\\lbrack i \\right\\rbrack", "21b5c609141cd6336ea77401613d9e02": "x,y\\in X", "21b6301d57103a6328873f3b82c2e7f7": " \\frac{T_1}{K_1} = \\frac{T_2}{K_2}", "21b65a4c2822c5ee260bd244b3533c22": "\\chi_M^I (n)=\\sum_{i=0}^n H(\\operatorname{gr}_I(M),i),", "21b7a36819ff40089ce611c0fe777f11": "\\mathfrak p^\\perp", "21b81bd42db26f41a3b8a8705cae901e": "8 e > n^2 / 8", "21b83bda2f7d7ceeb49a469f14dd308e": "U(\\phi^*\\phi)", "21b8576ef395661df94261210e7bd739": " u''_i =\\frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2} ", "21b859709111590884742608e81ec288": "\\frac{D \\Gamma}{Dt} = 0", "21b87a1b4f67cdd4c7b08ac8f9ca88eb": "\n\\mathrm{pH} = 14 + \\log \\frac {C_b V_b - C_a V_a} {(V_a + V_b )}\n", "21b87b2162f412e1d531342e20e9a70f": " |1/2,-1/2\\rangle\\;|1/2,-1/2\\rangle\\ (\\downarrow\\downarrow)", "21b8fce671acf5fa4690193ad7ef3461": "[0,t]", "21b90015ce3b6040c6cd306292f6f5a5": "\\ v_z = v_g \\cdot \\left( \\frac {z} {z_g} \\right)^ \\frac {1} {\\alpha}, 0 < z < z_g\n", "21b970c03b3da263c29fed5869cea5da": "T_0 = id_X :X \\rightarrow X", "21b9e734083b65195474f11a70f7c66d": "\\tfrac{Z}{\\beta}", "21b9fe3f5ea62eab70d5fbd8d9fc0321": "t < k < n", "21ba3d5eb71a31978ba48e8455dc56da": "\\vec{x} = \\begin{bmatrix}x_1\\\\ \\vdots\\\\ x_n\\end{bmatrix}.", "21ba933b0bc877b5ad5e2c8806d29404": "\\frac{\\operatorname dV}{\\operatorname dh} = \\overbrace{\\frac{\\pi r^2}{3}}^\\frac{ \\partial V}{\\partial h} + \\overbrace{\\frac{2 \\pi r h}{3}}^\\frac{ \\partial V}{\\partial r}\\frac{\\operatorname d r}{\\operatorname d h}", "21ba937cd92bb61c444b999878069fc4": "S ^2", "21bb1b15b8f2b05d17222e8bc221ca4e": "\\begin{align}\n16_1&\\rightarrow 10_1 \\oplus \\bar{5}_2 \\oplus 1_0 \\\\\n10_{-2}&\\rightarrow 5_{-2} \\oplus \\bar{5}_{-3} \\\\\n1_4 &\\rightarrow 1_5\n\\end{align}", "21bba5b4b8e8d6b47060140c145b451c": "p>2", "21bbc98e29201c146415a0bfa1e9e375": "b \\ne 0 ", "21bc13fe5056aab09be15cf1f0d74d26": "(499^2/113)x^7-212x+3^4", "21bc59c4af87c0e60e8a9934d3248eb4": "\\ e=\\frac{\\Delta L}{L}=\\frac{\\ell -L}{L}", "21bc83cf4a53b207df3036528243c16c": "\\Delta x = \\Delta y = \\Delta z", "21bca4849a6c32af37f07134830fa8f2": "\\Sigma A", "21bd12bf50083f0383866bad81d7cd00": "\\displaystyle \\cos{A}+\\cos{B}+\\cos{C}=\\frac{3}{2}", "21bd3325ed1c6660b473dd500b8d64b6": " N_v (\\mu_2 - \\mu_1) = \\frac {df}{dc} ", "21bd4ac9ae8f37ea9dfd40fde2480f4e": " \\frac{\\Delta_h[f](x)}{h} - f'(x) = O(h) \\quad (h \\to 0). ", "21bd6f944f2d2c8f0be7887d57d8924c": "{\\rm ATIME}(f,j)", "21bd7c80ad49a3a667d0b949a3cdb2f1": "\n EI w(L/2) = \\dfrac{PbL^2}{48} -\\cfrac{Pb}{12}(L^2-b^2) = -\\frac{Pb}{12}\\left[\\frac{3L^2}{4} -b^2\\right]\n ", "21be2b16da3ee028399208a558ea6cfd": "\\lambda(\\mathrm{lcm}(a,b)) = \\mathrm{lcm}( \\lambda(a), \\lambda(b) )", "21be923a2b2132284994c5bd8b97120c": "\\mathrm{NAG}_v", "21be99bf30cd4c16e58b13b348ec83f3": "\\mu \\leqslant x \\leqslant \\mu-\\sigma/\\xi\\,\\;(\\xi < 0)", "21bec8a1b7392f36d5ba2eb7cab517be": "\\scriptstyle C_{-1} \\;=\\; E_A(I)", "21bf2d7fcc1241739f999554ef066363": "\\operatorname{tr}\\boldsymbol{E}^q", "21bf37951cc8e00646b1d53b0508924b": "\\lim_{z \\rightarrow a}(z-a)^{m+1}f(z)=0", "21bf8cc923343da81207d352ea5d376b": "\\langle \\exp [-i \\mathbf{q} (\\mathbf{R}_j - \\mathbf{R}_k)]\\rangle = \\langle \\exp (-i \\mathbf{q} \\mathbf{R}_j) \\rangle \\langle \\exp (i \\mathbf{q} \\mathbf{R}_k) \\rangle = 0", "21bfa6203e00d318d5f83337e2565f99": " -\\,\\Gamma ^d {}_{b_1 c} T ^{a_1 \\ldots a_r}{}_{d \\ldots b_s} - \\cdots - \\Gamma ^d {}_{b_s c} T ^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_{s-1} d}.\n", "21bfaca9f12f961ce907de3b07b92119": "\\vec{\\sigma}\\vec{p} = \\sigma_1 p_1 + \\sigma_2 p_2 + \\sigma_3 p_3 =\n\\begin{bmatrix} \n p_3 & p_1 - i p_2 \\\\\n p_1 + i p_2 & - p_3\n\\end{bmatrix}", "21bfc0f40564c8e7a0a0d326169fe067": "\\ddot{\\delta \\mathbf{r}} = \\mathbf{\\ddot{r}} - \\boldsymbol{\\ddot{\\rho}}.", "21bfdc09a752a2146a235c75c1e6d1be": "p_\\text{H}", "21bfe20e958d7e594a1f7bdf86d904bd": "C_{XX}(t,s) = C_{XX}(s-t) = C_{XX}(\\tau)\\,", "21bff0735aa0f83cf7b2e82cbc71d956": "\\psi(\\omega)=\\frac{K}{2\\pi c}(\\hbar\\omega_{max}-\\hbar\\omega)", "21c06467164694b0c2371a1dcbe47877": "\\mathbf{F}_{jk} = 0", "21c0841b6ad2cf73baf36e63225c9f02": "\\lceil \\log_2(r) \\rceil", "21c0b34ea58c1f739cb7a3d3e0ea31d1": "\n f(k;\\mu,\\mu)\\sim\n {1\\over\\sqrt{4\\pi\\mu}}\\left[1+\\sum_{n=1}^\\infty\n (-1)^n{\\{4k^2-1^2\\}\\{4k^2-3^2\\}\\cdots\\{4k^2-(2n-1)^2\\}\n \\over n!\\,2^{3n}\\,(2\\mu)^n}\\right]\n ", "21c0d1834bc2e077338ee7cc5e756b90": "a=\\frac{p}{1-\\varepsilon^2}.", "21c1192485826411f9832876565d9ac4": "\\mathbf{\\hat b}", "21c1369312cbb505f1c5c9ccd1a31245": "\\cup_{i=1}^RU_i=\\cup_{j=1}^C V_j=S", "21c13fc1d626e728e3b9e43133ca9fd0": "\\frac{1}{2 \\alpha (25812.807) (299792458)} \\ ", "21c15de1c01ac39cccbd399f0f67060c": "\\mathcal{P}_{\\geq 1}(S) \\,.", "21c1c06705b8dbf9a41f88b96fd53efb": "\\bold{F}_{12} = - \\bold{F}_{21}.\\!\\,", "21c1d7ad02e551a3577c6bc909aed682": "\\mu\\!", "21c1de103aa1aa8d7206d41bc13effd6": "\\textstyle\\frac{1}{2}(-\\textstyle \\sum_{i=1}^je_i+\\textstyle \\sum_{i=j+1}^8e_i)", "21c1e0489fb47e3c0fc1afe02d937719": "\\alpha \\to 0", "21c1e7526266604679965449f6671ee9": " K_y = K_2 = i \\left.\\frac{\\partial \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_y)}{\\partial \\varphi}\\right|_{\\varphi=0} = i \\begin{pmatrix}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{pmatrix} \\,,", "21c1e76b34325a824daf403598ab3ca2": "K(\\frac{\\sqrt{(a^2 - b^2)}}{a})=\\frac{2a}{a+b}K(\\frac{a-b}{a+b}) ", "21c22b677573f3d63c5869a242da7c3c": "\\left|\\Gamma(w)\\right|^2 + \\left|\\tau(\\omega)\\right|^2 = 1", "21c24539e3545e49a9442443cd8ffff7": "\\hat{V}(t)", "21c254aa9d46621286aba6e057dc00ae": "h(\\lambda) = e_{\\lambda}", "21c271f7447e2223eba7afc8ae421604": "\\eta (a,b,c):F(a,b,b)\\rightarrow G(a,c,c)", "21c2e59531c8710156d34a3c30ac81d5": "Z", "21c31e49b1198d4503c0f45dd58bcb94": "a_m^T", "21c320134d8391a110980e7d36b5ece7": "\\Phi (0 \\leqslant \\Phi \\leqslant 2 \\pi)", "21c3cb737a47ce4168b5c0969839eab2": "G(f : X \\to TY) = \\mu_Y \\circ Tf\\;", "21c3f076a0f7021d825a5e331bfc42f7": "1. \\ \\frac{\\partial}{\\partial \\theta} \\frac{\\partial u / \\partial x_k}{\\left|\\partial u / \\partial t\\right|} > 0 \\ \\forall k", "21c3fbf8e4d82135c471f8b84222bc3d": " H_{5,s}=H_{f}+x_{5,s}H_{fg} \\,", "21c46aa00609bbf6c4f04e3f48016d4f": "d_{LS}\\ne d_S-d_L", "21c496a5c0465f6ed18fea3adbd8edce": " K = -{\\ddot{\\varphi}\\over \\varphi},\\,\\, K_m = {-\\dot{\\psi} +\\varphi(\\dot{\\psi}\\ddot{\\phi} -\\ddot{\\psi}\\dot{\\varphi})\\over 2 \\varphi}. ", "21c51974d9f075258474465849fa0b78": "y^2=x^3-mx", "21c5405de8f9fdcf289788ff27c3a3b9": "Z=N+P", "21c54cc667bb7e76babb6d01c610d6ad": "P\\phi_y = y\\phi_y.\\;", "21c5a20a62d0add53666d1c6b1ce0570": " S_n = a_i b_i + \\cdots + a_n b_n ", "21c628c474b8a15e12145bf86cbb7ea8": " -\\tfrac1{ 24} (b-a)^3 f''(a) + O((b-a)^4) \\quad\\text{and}\\quad \\tfrac1{12} (b-a)^3 f''(a) + O((b-a)^4), ", "21c640ff51ce2908e16105e6a6a6d49c": "\n\\mathbf{M}_{\\text{in}} = \\mathbf{Q}\\mathbf{M},\n", "21c64c32fa2bfe330b51871455345f58": "(t_1,t_2,\\ldots,t_n)", "21c67960cb7b2b08abc7b835db8948f9": "\\frac{1}{\\rho}\\frac{\\,d\\,P}{\\,d\\,X}-\\tfrac{\\,f}{2\\,D}\\,W^2+\\left(\\frac {2-\\beta}{2}\\right)\\frac{\\,d\\,W^2}{\\,d\\,X} \\,=\\,0", "21c685c73142fd23b47dac7f5fea115e": "-1 < \\beta \\le -0.75", "21c68f31bf6db00547183f0985d533b4": "(\\boldsymbol\\mu,\\boldsymbol\\Sigma^{-1}) \\sim \\mathrm{NW}(\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi^{-1},\\nu)", "21c701d4913b4baa6a9d259a4c28897c": " D = \\begin{vmatrix} h_3 & h_4 & h_5 & h_6 \\\\ h_1 & h_2 & h_3 & h_4 \\\\ 1 & h_1 & h_2 & h_3 \\\\ 0 & 0 & 1 & h_1 \\end{vmatrix}.", "21c755a92838b5266375443f943e66c8": "\\hat{H}_{\\textrm{ph}}", "21c7571812e736d745ba0a095483c464": "\n\\textrm{efficiency} = \\frac {w_{cy}}{q_H} = \\frac{q_H-q_C}{q_H} = 1 - \\frac{q_C}{q_H} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(4)", "21c77b9a7835730e5cd63cbbdce5bb09": "K = \\sup_{z\\in D} |K(z)| = \\frac{1+\\|\\mu\\|_\\infty}{1-\\|\\mu\\|_\\infty}", "21c785ced9aa6216ba4f7da5898f647a": "\\mathbf{y}_1=\\mathbf{x}_1-\\boldsymbol\\Sigma_{12}\\boldsymbol\\Sigma_{22}^{-1}\\mathbf{x}_2", "21c7be7dfd22e403ead3ddc3fcc71b9a": "\\mathbf{Q}(i,\\sqrt[4]{1+2i})/\\mathbf{Q}(i)", "21c7d89f1ea9018a256a242de3f9cbe8": "\n\\sum_{\\delta\\mid n}\\delta^sJ_r(\\delta)J_s\\left(\\frac{n}{\\delta}\\right) = J_{r+s}(n)\n", "21c7e04ab228d557038b1c86f1a7d3f4": "\\Gamma^a_{bc}=0", "21c83c4b80485ef168fb5580c9c1048b": "\\langle f(z)\\rangle= \\int_0^{2\\pi} p_w(\\theta')f(e^{i\\theta'})d\\theta'", "21c84aeeb81773fdedf0a0921c333eda": "(k-k')^2= \\,", "21c85dd4693a3bb121e7f5a07ab95440": "O(\\theta_{ex})", "21c86e6a06ad1fb5314eef5b829b2d3a": "_{s.3.right\\,}\\!", "21c8e45ae2a20305a3879255bb250aff": "h^b", "21c95d4c8936a39bb904563a8fcdbfdd": "i=1,\\ldots, k", "21ca490699c289310b46d98921017cad": " R_A^\\infty = \\inf \\{ r \\geq 1 \\mid \\exists n \\in \\mathbb{Z}^+, R_A(x) \\leq r, \\forall x, |x| \\geq n\\}. ", "21ca5d5ab1034997251573bd3cb96022": "z \\mapsto z + \\lambda_q,", "21ca69461b1164668edff077d8e6c879": " {N \\hbar \\omega \\over V} = \\mathcal{E}_c = \\frac{\\mid \\mathbf{E} \\mid^2}{8\\pi} ", "21ca7f31773c8f7cbae3c19fa5a26659": "\\, \\mu_h", "21cac1c56e3ca5d185185164169844b0": " \\mbox{P-to-W} = \\frac{|\\mathbf{a}(t)||\\mathbf{v}(t)|}{|\\mathbf{g}|}\\;", "21cacdf2a61d99a9e876d6ea5eef4c98": "\\nabla\\cdot{\\mathbf A}(\\mathbf{r},t)=0\\,.", "21cb16b322901cf5a7ea71c15e02570f": "B(T) = \\exp\\left(\\int_0^T r(u)\\, du\\right)", "21cb21c7c1de5b4c61da90794c62bc1a": "X\\,", "21cb349c4eb81423924e4725169705ad": "\n\\begin{align}\n & \\int\\limits_{I_x \\subset \\Omega_x} f(x) e^{\\lambda S(x)} dx \\equiv \n \\sum_{k=1}^K \\int\\limits_{I_x \\subset \\Omega_x} \\rho_k(x) f(x) e^{\\lambda S(x)} dx \\\\\n & \\xrightarrow{\\lambda \\to +\\infty} \\sum_{k=1}^K \\int\\limits_{\\text{a neighborhood of }x^{(k)}} f(x) e^{\\lambda S(x)} dx \n = \\left( \\frac{2\\pi}{\\lambda}\\right)^{n/2} \\sum_{k=1}^K e^{\\lambda S(x^{(k)})} \\left[ \\det \\left(-S_{xx}''(x^{(k)})\\right) \\right]^{-1/2} f(x^{(k)}),\n\\end{align}\n", "21cb523daa8742d2e3c2d1616a82f40e": "f'(x) = \\begin{cases}\n1 & \\text{if } x = 0,\\\\\n1 + 2\\,x\\,\\sin\\left(\\frac{2}{x}\\right) - 2\\,\\cos\\left(\\frac{2}{x}\\right) & \\text{if } x \\neq 0.\n\\end{cases}", "21cbbf3f2fec9b685f5f4f9d0080018c": "k=2^r-r-1", "21cbdbb49c9593eda7f8a5b8b7f91beb": "(F_1)", "21cbe545595a3a3bb4abd093cae5c23d": "\\scriptstyle\\prec", "21cbf60b5cf640edeab2e133d099672f": "w(a) := 1", "21cc02743caf32a7473a553a60deaeb8": "g'(x)\\neq 0", "21cc0296c4ffdcda0def415a578628f0": "\\frac{1 + \\tfrac{1}{x}}{1 - \\tfrac{1}{x}}", "21cc35251dc254d546249bec179e1c18": "[a]_E", "21cc5121dfaa1c274dad0b91604c058b": "\\sigma_\\mathrm{theoretical} = \\sqrt{ \\frac{E \\gamma}{r_o} }", "21cc77c693e86268948a6bdf84a4ac15": "\\pi(x) = \\sum_{p\\le x}1\n", "21cc879ea266df2c7eb756c8e89e19a8": "(X, \\mu)", "21ccb026eca7fcd7ef9199d311cc5971": "\\frac{\\alpha}{\\beta}=\\,{\\alpha}\\times\\frac{1}{\\beta}", "21ccca36f04c9d7feb2a6144829511c8": "\\scriptstyle 230 V \\times\\sqrt{2}", "21cd291945e68dde6823d75a7bb11263": "x=v\\cos u, y=v\\sin u, z=cu.\\,", "21cd68a66073cb64978d45ac19a7436d": " \\textbf{G}(s) = \\frac{\\textbf{N}(s)}{\\textbf{D}(s)} = \\frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}", "21cd74b937cc73b4f874e866b30a6a53": "x=t-\\tfrac{b}{3}", "21cd74ef05532869557ee641b0fdb453": " \\boldsymbol{D} = \\varepsilon \\boldsymbol{E} \\ , ", "21cdb0fe2644150136453d08f15b691f": "(x_i-\\bar{x})^2", "21cdd99a0d0275336bfd1f0b40bfcd37": "\\eta^T\\textbf{x}", "21ce4100c74705354653e363cdbbef62": "(k-p')^2= \\,", "21ce51f55520bb24e2e8921a42df2263": " \\mathbf{C}_{N} ", "21ce96636b51870a329ae7ee42a54301": "\\textstyle N", "21cea3548576bea6c96918ed857a23df": "\\int_a^bf(x)\\,dx=\\lim_{c\\to b^-}\\int_a^cf(x)\\,dx", "21ceb0d41549ee8b5c7a87c2a28913d7": "df_p = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x^i}(p) (dx^i)_p.", "21cecf7312747f554c9f27205170bd9a": "|\\mathbf{a}| = \\frac{GM}{x^2}.", "21cf170e4f4944aab8f399d58ce7ba2c": "[N_i,p_0]=Dp_i", "21cf46d9d1ff93c724548378ad3d67b9": "f(\\mathbf{x}) = \\psi^i(\\mathbf{x})", "21cf7234f16ef41bfe752433f61842e7": "\n\\ddot{\\mathbf{R}} = 0\n", "21cfd9d4dcb76da6be683980afe33c22": "(\\ell+s)<(P+q)", "21d02a4967bc8e159ffbf819214540b4": "W_k(m) = \\frac{L_k\\left(m-1\\right)+1}{\\mu_k}.", "21d088cac9c105200619b0e25e0ddb26": " c = \\frac{2}{\\alpha} \\ ", "21d0a88940bcd2d935ef0e7bd357931a": "2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{\\ddots}}}}", "21d0aafe1f8d3a118e9982abc0ccf3ce": " X_n (x) ", "21d0bad3ad4c2b0ed56a639a6a2efe7c": "U(\\mathfrak{h})", "21d0e50b69b614b110a6c83f6a6b4ae8": "h_{0}", "21d0f74cc012c3c9c601e6a1688b3735": "\\sim p(X,Y)\\leftarrow\\hbox{not }p(X,Y)", "21d12e9bed5387438c53b1d46cedc77a": " \\rho(\\mathbf{r},t) = \\psi^*(\\mathbf{r},t)\\psi(\\mathbf{r},t) = |\\psi(\\mathbf{r},t)|^2", "21d194eed3d57fc32553dc8caae15eb0": "\\tau_q(r) = -\\pi^{s_p(r)}\\prod_{0\\leq i \\nu_{\\rm yx}", "21e20f5b08399f2bfbb34f42da1e3e23": "\\bar{y}_i = \\bar{y}(Ti, \\tau) \\, ", "21e2a83e2af3b9027ce11e1f3f0cff5d": "\\bold{j} \\times \\bold{B} = j_z (\\hat{\\bold{z}} \\times \\bold{B_\\perp}) +\\bold{j_\\perp} \\times \\hat{\\bold{z}}B_z ", "21e2c0c0472b331622877accbe29b91b": "2n", "21e2ff6dafe11cd84a15ab5ca6e87d35": "b = \\nu\\sigma^2/2", "21e32273ce601b8acabf7dc65ec8b185": " n_r ", "21e3547c407da5c382577451a75758f0": "a_k n^k+a_{k-1} n^{k-1}+\\cdots+a_2 n^2+a_1 n+a_0", "21e36780a20412f14d86f6464c026c2c": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r},t) = -\\frac{\\hbar^2}{2m}\\nabla^2\\Psi(\\mathbf{r},t) + V(\\mathbf{r},t)\\Psi(\\mathbf{r},t) ", "21e37710fe723c7024f42ad99689ac46": " F^{-1} \\dot{F} = {{\\rm id} - \\exp - {\\rm ad} g(t)\\over {\\rm ad} g(t)} \\cdot \\dot{g}(t),", "21e3a1b406f440431e62942027c10444": "\\gamma_{xy} = \\theta", "21e3a548c696935211633e68c9ee3ee9": "\\int x^m\\arcsec(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arcsec(a\\,x)}{m+1}\\,-\\,\n \\frac{1}{a\\,(m+1)}\\int \\frac{x^{m-1}}{\\sqrt{1-\\frac{1}{a^2\\,x^2}}}\\,dx\\quad(m\\ne-1)", "21e3f1aabb00f4a43571fa5183f39987": " Ext_{\\mathbb Q[\\mathbb Z]}(H_1(X;\\mathbb Q),\\mathbb Q[\\mathbb Z]) \\simeq Hom_{\\mathbb Q[\\mathbb Z]}(H_1(X;\\mathbb Q),[\\mathbb Q[\\mathbb Z]]/\\mathbb Q[\\mathbb Z] )", "21e3fa339c256536b5b8f4e3c5dbe5dd": "x_t|x_{t-1}", "21e43c09d2e82c67da128f8fae7f4665": "t=1,...,T", "21e4b1c6304d92c1868e92aa6249b2f7": "\\tilde\\omega=\\sqrt{\\frac{I_{1}\\sin\\delta}{I_{2}}}\\,\\sqrt{|\\dot{\\psi}|\\Omega}\\,,", "21e4e65f1864235e4de1d73732a866d9": "\\mathcal V_i", "21e50d603bf0437d8d015a9f0daa3a42": "g[m_1:b_1:1]_L . g[m_2:b_2:1]_L = g^{-1}(g[m_1:b_1:1]_L \\times g[m_2:b_2:1]_L) ", "21e58db0cc0dd23a1c12ad1d9b78bbb5": "Hom(A,B) \\otimes Hom(B,C) \\rightarrow Hom(A,C)", "21e625c55747befa134186bf825f041a": "\\pi(a,b)=\\displaystyle\\sum_{k=1}^qP_{k}(a,b).w_{k}", "21e63896d757575b7e8e3e375c98041f": "\\mathcal{L}_{X} g = 0 \\,.", "21e6474fe1fe6adfd37a5f88b706b39b": "DGS_{\\sqrt{n}\\gamma(n)/\\lambda(L^*)}", "21e65807162540f4366a63c2ea32e567": "f(2,1) = (2) - 2(1) + 2 = 2 - 2 + 2 = 2", "21e65da76bbea608758e4b3975bc9d3e": " x^{p/q} = \\sqrt[q]{x^p} ", "21e690c7e12f19b7a4f6a9af031dcaa5": "Q = \\left(\\frac{V \\times I \\times 60}{S \\times 1000} \\right) \\times\n\\mathit{Efficiency}", "21e70074dc992a23b1ba28c5521ed3a1": "k_{\\rm C} \\epsilon_0=1", "21e70e478a8de65a8e6f5a84e0c60036": " \\log |\\{x\\}| =0 ", "21e71656ba372de0586dc86d483bf2c9": "\\{ X_1,\\ldots,X_n \\}.", "21e7dc01138446b187a289582101b3fd": "\\left( \\Pi,\\Lambda \\right)", "21e81452b36b188644972ae10823d164": " G^{(k,k+1)}(t) := \\log(\\mu_{(k)}(t)/\\mu_{(k+1)}(t)),", "21e851bdae85c7399c902cd949acaffd": "rE_g", "21e897c34f94928b35c74795069a310b": "(w,x,y,z)", "21e8e5bc8dc4cd709a6e0883c4492a1f": "X(\\cdot)", "21e8f0cc7489f6353d3c8f470355704e": "\\Gamma,R", "21e912297d672900424fb5399eafd195": "\\{p_{\\perp }^{2}+\\Phi (x_{\\perp })\\}\\Psi\n=b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\\Psi \\,,", "21e9481059b59d0f3dcd27d2566e79aa": "\\mathbf{y}^*_j", "21e95751c1bda66db33f3eda534b443c": "K(z,w) = A(w)\\Psi(zg(w)) = \\sum_{n=0}^\\infty p_n(z) w^n", "21e99750d94034f933e392918f6bae4a": "\\{ M , M \\} = 0", "21e9b4cfdaa86e2a2f8312f7e83c8026": "g^2/(4 \\pi)", "21e9c8bcd59a3047981a64534fe99c7d": "\\textstyle{\\frac {\\log(32)} {\\log(4)} = \\frac{5}{2}}", "21e9cd269752010b61d2d071fde00d38": "\\frac {p(x)} {q(x)}\\,", "21e9cd9d46715d1282c7a6d9eae99f0b": "\\lambda_{\\epsilon,g} = (CV_{\\epsilon,g})^2 = \\left( \\frac {\\sigma_{\\epsilon,g}} {\\mu_{\\epsilon,g}} \\right)^2 ", "21ea0ff38e34b3ebaa31cf86e3bc5862": "\\mathfrak{so}(3,3)\\cong \\mathfrak{sl}(4,\\mathbb R)", "21ea68c33e83f2a59e03195928ae4a8a": "17\\zeta(7)-10\\zeta(2)\\zeta(5)", "21ea8b26e1aaadd3be289c4ba772ca67": "\\left.\\frac{d}{dt}f(\\gamma(t))\\right|_{t=0}.", "21eaa0bd4da3fa13459974c651c9dfb6": "\\Sigma_{I,b}", "21eaf78fb8776f3439a573a383143150": "\\delta_{t}", "21eb418a27252c36a64aa5a98e271c33": "\n \\Omega^2(k) = \\frac{g\\, k (\\rho - \\rho')}{\\rho\\, \\coth( k h ) + \\rho'\\, \\coth( k h')},\n", "21eb58e47aabfdc803791bd20ca80db3": "U = b", "21eb6d4d75eddcd1f00cb5d41baf1cf8": "{\\sigma^2_T}", "21eb7c7821fbeae4ed45c4e11d1302af": " y = f(x_1,\\dots,x_n), \\, ", "21ebcef62c355619930aff670065aa44": "\\mathbf{R}_{x}(n)\\,\\mathbf{w}_{n}=\\mathbf{r}_{dx}(n)", "21ebd8909c12a33d9af1122b7c9b2a83": "\\hbar / E_\\mathrm{h}", "21ebf7cef931ca55aa2c54896d5c8fee": "\\hat{T}_{1}", "21ebffcd9bf01b4bd9333cb5d1ac506e": "|{\\Psi}\\rangle=\\sum_{\\alpha_k}\\lambda^{[k]}_{{\\alpha}_k}|{\\Phi^{[1..k]}_{\\alpha_k}}\\rangle|{\\Phi^{[k+1..N]}_{\\alpha_k}}\\rangle", "21ec2250bb4ac62aa9cf39947574abb1": "[N_i,N_j]=-i\\epsilon_{ijk} M_k,", "21ecb996a8c58670fc837e8947a4738e": "|p'_{R_0}(x,y) - p(x,y)| < 0.1", "21ecd620a26c35f2131d96e1352664c0": "\\ R = \\frac{ \\mathbf{G}_a \\circ \\mathbf{G}_b^*}{|\\mathbf{G}_a \\mathbf{G}_b^*|}", "21ecf75eff332c6b984b25654374e82b": " \\det \\left({\\begin{array}{*{20}c} a & b \\\\ c & d \\end{array}}\\right) = \n\\left\\lbrace{\\begin{array}{*{20}c} -cb & \\text{if } a = 0 \\\\ ad - aca^{-1}b & \\text{if } a \\ne 0 \\end{array}}\\right. . ", "21ed281c3c1a46b17604d7a2a334f0e3": "R=2\\sqrt{\\frac{\\nu}{\\alpha}}.", "21ed6b7e4066a1a82734fa6c9d82de37": "{10 \\choose 1}{4 \\choose 1} = 40", "21ed9ed557da313b63ec7f6540423c25": "\\nabla \\times \\mathbf{B} = \\frac{1}{c^2} \\left(\\frac{1}{\\epsilon_0} \\mathbf{J} + \\frac{\\partial \\mathbf{E}} {\\partial t} \\right)", "21edd6a56a792599782c03b790062358": " \\beta_2, \\beta_3,\\ldots ", "21ee199ddc4d248a43f348ef621f31ab": " e^{s_3}=\\sqrt{\\frac{c+u_1} {c-u_1}} ", "21ee2bbe52ab65fa45a793b13a19fa1c": " \\exp^{[x]}_a ", "21ee3b0904fd08b42a4f0d1f526663c3": " S_x(t,f) = \\int_{-\\infty}^{\\infty} X(f+\\alpha)\\,e^{-\\pi\\alpha^2 /f^2}\\,e^{j2\\pi\\alpha t}\\, d\\alpha ", "21ee592e825a4e1dc861d3305dd7c1b6": "\\{O_{3}, O_{7}, O_{10}\\}", "21ee6c0ce8b50c639460f4363d64fc6a": "S_i +", "21eeb7f27fafff850e2879be42c2812e": "m = \\{p_{i_1}, p_{i_2}, ..., p_{i_n}\\}", "21eeeaaae03e5f62b56f4b234c535337": "\\Psi(\\bold{r},t) = \\psi(\\bold{r}) e^{\\alpha t} ", "21eeeafe03259c10670306d25432863d": "F(y)=2\\int_y^\\infty \\frac{f(r)r\\,dr}{\\sqrt{r^2-y^2}}.", "21ef120a20ad19cdbe6df748e4095a90": "\\int x^2\\arcsin(a\\,x)\\,dx=\n \\frac{x^3\\arcsin(a\\,x)}{3}+\n \\frac{\\left(a^2\\,x^2+2\\right)\\sqrt{1-a^2\\,x^2}}{9\\,a^3}+C", "21ef28ecab8df9c1ef7c69512f5c63a9": "\\Phi(x) = 1- \\Phi(-x)", "21ef6015c37d59a5b99052fe71db2704": "\\stackrel{\\mathbf{v\\times B}}{}", "21f00a5a3bb19d3c08967e8c8d656c5b": "F\\left(x, y, y', y'',\\ \\cdots,\\ y^{(n-1)}\\right) = y^{(n)}", "21f00c1ae631f17e60fec725359bf0ae": " f(t)=\\frac{x_1 + \\cdots + x_n + t}{n+1} - ({x_1 \\cdots x_n t})^{\\frac{1}{n+1}},\\qquad t>0.", "21f051bda58ab781c99d9e78c9d6f369": " S_z = \\frac{\\hbar}{2} \\sigma _z = \\frac{\\hbar}{2} \\begin{bmatrix}\n1&0\\\\ 0&-1 \\end{bmatrix} ", "21f058f0e0d49f66a0d61db52288b53e": "\n\\tfrac{ 1}{20},\\,\\tfrac{ 1}{10},\\,\\tfrac{ 3}{20},\\,\\tfrac{ 1}{ 5},\\,\n\\tfrac{ 1}{ 4},\\,\\tfrac{ 3}{10},\\,\\tfrac{ 7}{20},\\,\\tfrac{ 2}{ 5},\\,\n\\tfrac{ 9}{20},\\,\\tfrac{ 1}{ 2},\\,\\tfrac{11}{20},\\,\\tfrac{ 3}{ 5},\\,\n\\tfrac{13}{20},\\,\\tfrac{ 7}{10},\\,\\tfrac{ 3}{ 4},\\,\\tfrac{ 4}{ 5},\\,\n\\tfrac{17}{20},\\,\\tfrac{ 9}{10},\\,\\tfrac{19}{20},\\,\\tfrac{ 1}{ 1}\n", "21f0ba6062febcb2d1f7bb42495fb01b": "e_\\lambda = \\sum_{\\mu\\uparrow\\lambda}e_\\mu,", "21f0bf2686f22020a22b90198be2891d": "\\mathcal{C}_x^k\\,", "21f0e8f4b03d775ed256a52422ec5143": "\\Sigma = \\{0,1\\}", "21f1210cdf6c001738c78fc2496f8a3e": "\\inf \\theta \\le 27/82.", "21f14abb6be09d790a6cf7f4c9445f08": "|n_{{\\mathbf{k}}_l}\\rangle ", "21f1a519c9345d5b1d6829b0cbaf6435": "m_k=\\frac{1}{k+1}\\sum_{i=0}^k a^ib^{k-i}. \\,\\!", "21f1af307837d7582360f4dc4c83d343": "s_\\lambda s_\\mu = s_{\\lambda \\mu}", "21f1b0b3c18ef8a4fb2875d937c89f5f": "a_n = \\sum_{k=0}^n k! \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}", "21f1f173facad48671acbe78a72df3e1": "VC(x)_{z'} < VC(y)_{z'}", "21f259bd908801edab1c75bcf661154c": "\nI=I_0 \\exp \\left [ \\sum_i \\left (\\beta_i^* + \\alpha_i \\right ) \\sigma_i \\right ]\n", "21f282307ffee0cc16fde83c743b3c9b": "(3-z)^{\\frac{1}{4}} = \\exp \\left (\\frac{1}{4} \\log(3-z) \\right ) \\quad \\mbox{where} \\quad 0 \\le \\arg(3-z) < 2\\pi. ", "21f292b3e60ea6999d815f2bf73c1a6e": "v_2\\in V\\,", "21f2fb441d848219516a92d9de8824a0": "\\lim_{(x_0,\\dots,x_n)\\to(z,\\dots,z)} f[x_0] + f[x_0,x_1]\\cdot(\\xi-x_0) + \\dots + f[x_0,\\dots,x_n]\\cdot(\\xi-x_0)\\cdot\\dots\\cdot(\\xi-x_{n-1}) = ", "21f306eb4ff5d34801fc729e60af3a1f": "k =\\frac{mg}{v_t} = \\frac{(0.145 \\mbox{ kg})(-9.81 \\ \\mathrm{m}/\\mathrm{s}^2)}{-33.0 \\ \\mathrm{m}/\\mathrm{s}} = 0.0431 \\mbox{ kg}/\\mbox{s} , \\ \\theta = 45^o", "21f342182cbaf44d1434bd61f3014e3b": "\\frac{d}{dt}\\hat{A}(t)=\\frac{i}{\\hbar}[\\hat{H},\\hat{A}(t)]+\\frac{\\partial \\hat{A}(t)}{\\partial t},", "21f34e1a45602b5cf3f8e747e51cfaa4": "\\text{Cl}_2(m\\pi) =0, \\quad m= 0,\\, \\pm 1,\\, \\pm 2,\\, \\pm 3,\\, \\cdots ", "21f3687d464060a935d843275f140576": "\n\\int_a^b e^{n f(x) } \\, dx\n\\le \\int_a^{x_0-\\delta} e^{n f(x) } \\, dx + \\int_{x_0-\\delta}^{x_0 + \\delta} e^{n f(x) } \\, dx + \\int_{x_0 + \\delta}^b e^{n f(x) } \\, dx\n\\le (b-a)e^{n (f(x_0) - \\eta)} + \\int_{x_0-\\delta}^{x_0 + \\delta} e^{n f(x) } \\, dx\n", "21f381bcd4d5313b0d86a4c5432a2cbf": "M/(x_1, \\dots, x_{i-1})M", "21f3b94eb2a302b8005c814eb45f72fc": "x^{22} + x^{21} + 1", "21f3c8070405ff3631fb7d0e90322aea": "\n f(\\phi, \\psi) \\propto \\exp [ \\kappa_1 \\cos(\\phi - \\mu) + \\kappa_2 \\cos(\\psi - \\nu) + (\\cos(\\phi-\\mu), \\sin(\\phi-\\mu)) \\mathbf{A} (\\cos(\\psi - \\nu), \\sin(\\psi - \\nu))^T ],\n ", "21f3cc554e07dbf6af72c84721ae6605": "1.5553", "21f3d0371601ec2ac4c983b477df3b43": "X=A+B+AB+E \\,", "21f3f5609dc3721743a7db91000a63c0": "M_x = \\int_0^2 (-2x^3+4x^2-40x+80\\,dx", "21f3fc234d9bb4d5d89d6340d987f518": "f^{-1}(y) = \\left\\{ x\\in X : f(x) = y \\right\\} . ", "21f4bcaa35f84fbf8d3d357372cd7004": "{Y_i}", "21f4c2c2eac51023aa0a26d98dd5a7a4": "\\vec{e}_0 = \\frac{1}{\\sqrt{2}} \\, \\left( \\partial_u + \\partial_v \\right) + \\frac{x^2+y^2}{\\sqrt{2}} \\, \\left( -q^2 \\sin(\\omega u)^2 + \\frac{\\omega^2}{2} \\, \\frac{C^\\prime \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u \\right)^2}{C \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u \\right)^2} \\right) \\, \\partial_v ", "21f4d5ba6ebd04c2e0afc9490e9f2cc6": "\\frac{3\\sqrt{3}}{4} \\left(1- \\sum_{n=0}^\\infty \\frac{1}{(3n+2)^2}+ \\sum_{n=1}^\\infty\\frac{1}{(3n+1)^2} \\right)= ", "21f4dd91224e9aea8bb290669be19bd8": "\\sum_{i=0}^L q_i M^i x = 0", "21f5301d0f52b1d88600a5f3301bae3c": "\\varepsilon_{\\alpha\\beta}", "21f572bc150c614b4da9098dfcf72348": "\\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{u}}{\\partial x}", "21f57a353001ba468ee4c37744c6646c": "p \\le \\frac{1-\\epsilon}{n}", "21f57d8b1feed863be9fbe769f3af631": " \\, 0 0:~~~~~~~~ R = \\frac{M}{2 E} (\\cosh\\eta - 1)~,~~~~~~~~ (\\sinh\\eta - \\eta) = \\frac{(2 E)^{3/2} (t - t_B)}{M}~;", "22022562a563391c724ddde602fa2086": " ITGAP = \\frac{(r-g) (b_t - \\sum_{i=1} ^ {\\infin} (\\frac{1+g}{1+r})^{i} pb_{t+i})}{1+g}", "22025301a52cbe9f4e26b2098c4a7a4c": "\\mathit{d}_H^{RC}(\\mathit{p},\\mathit{q}) = \\mathit{d}_H(\\mathit{p},\\mathit{q}^{RC})", "2202ca535b0da497c61f58deb49dc13b": "\\tau_{(12)} (v \\otimes w) = w \\otimes v", "2203061a5961bb1c8930c42039ce7967": "E_{\\mathrm{int}} = - \\sum_{i=1}^3 F_i \\int \\rho(\\mathbf{r}) r_i d\\mathbf{r} \\equiv\n- \\sum_{i=1}^3 F_i \\mu_i = - \\mathbf{F}\\cdot \\boldsymbol{\\mu}", "220311a8754754d5b9cea9d4d0a6e4d6": "\\frac{2\\times S\\uparrow \\times\\ S\\downarrow}{S\\uparrow +\\ S\\downarrow}", "220315c41c00e02c99b4a1db2b752916": " E_{av} = E_0 + \\frac{3}{5} E_F ", "2203413082fcdf541b0f2a2e2ae2e676": "p_L(y) = (y-1)(y-m-1)\\cdots(y-[n-1]m-1) ,", "22034964c1ebe6228ad0ef8ecbe55798": "|T| < 2(1-\\varepsilon)\\gamma n\\,", "220357cdfce1c39f1f6ae23a3ffb324c": " U_9(x) = 512x^9 - 1024 x^7 + 672 x^5 - 160 x^3 + 10 x. \\,", "22036bf382317be0cf3f0fe97438311b": "\\Delta=\\frac{D-2}{2},", "2203afc382f0fd3dbf33b1e3c4f829be": " g(r) = -\\frac {GM}{r^2}.", "2203b25fe01105d229917494c693c44e": "y^{i}", "2203ccab854940fbb4a69a75fafb7c0c": " |\\bar{\\psi} \\rangle = U(R)|\\psi \\rangle ", "22044d076d49f14e3a53df7106374ddd": "I_{m,n} = \\int \\frac{\\sin^m{ax}}{\\cos^n{ax}}dx\\,\\!", "220493f3b19884c7e297e2239c96e51b": "( \\mathbf{A}^\\mathrm{T} ) ^\\mathrm{T} = \\mathbf{A} \\quad \\,", "2204a224e95c42057b6d7daa1982b25b": "d=\\frac{-(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c})) \\pm \\sqrt{(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))^2-\\mathbf{l}^2((\\mathbf{o}-\\mathbf{c})^2-r^2)}}{\\mathbf{l}^2}", "2204a3e2080fc9c09eff788f9fb870ac": " \\mathrm{Gr}_D = \\frac{g \\beta (T_s - T_\\infty ) D^3}{\\nu ^2}\\, ", "2204c2b4d60ae2fd5279ec54cfaed2a4": "2x", "22050a8034e3b7aa0b6c1c0d7e8282bc": "\\cdot:S\\times S\\rightarrow S", "22052c5ec2eb75c3ed60007dc407074a": " \\lim_{h\\to 0}{(6 + h)} = 6 + 0 = 6. ", "22059e96225d83824a489c15e16df795": "x'_{v_{i1}j}= x_{ij} ", "2205a844de8489299dfcdc94854ffcd3": "\n\\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ \\left(3\\ \\sin^2 i\\ \\sin^2 u\\ -\\ 1\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\right)du \n", "2205eb6d84c9e4fdb31bffeb3985caff": "V=\\{{{v}_{1}},...,{{v}_{n}}\\}", "2206201a5e45eef63325349495157b49": "\\scriptstyle 7 \\frac34", "22062280a22f5bb83e61c8c505c99730": "m_\\mathrm{solvent}", "220683701f0349cee2f1486988bc81e5": "t.i", "22069d1b5b6cfd8c40c5a36a09ff9e93": "\\Delta\\phi=\\phi_2-\\phi_1", "2206a5f55d5ca3673011e8950515c717": "1 \\;\\xrightarrow{}\\; A \\;\\xrightarrow{f}\\; B \\;\\xrightarrow{g}\\; C \\;\\xrightarrow{}\\; 1", "2206ada777980db3afedddfc359a6804": " \\varepsilon(t) = \\int_{u(0)}^{u(t)} \\frac{F^{h}(u)}{m} \\mathrm{d} u = (1-a)\\frac{k_i}{m} \\int_0^t z(\\tau) \\dot{u}(\\tau) \\mathrm{d}\\tau ", "2206d80622823ef284615384c2571f9a": "\\sum_{i=0}^n a_i p^i", "220706312b4b2735e34e8976ade8e853": "\\Delta_{\\pi}", "220735b2831beee988dbf3e81cc873b3": "p=3", "22073a4470ab419111518edb4ff18e62": "O(n \\log\\log n)", "2208041b7c34dcf146ce994216b2e83f": " \\mathrm{tr}(\\mathbf{AB}) = \\mathrm{tr}(\\mathbf{BA}) ", "22084ea77de533cc1ca20d79d438022d": "X=A\\sum_{i=0}^{m-1} a_i.", "220876ed22fec26384299279d7c0c462": "\\theta^* \\approx \\theta_{0} + \\mathcal{J}^{-1}(\\theta_{0})V(\\theta_{0}). \\,", "22088400ede476760e73ab87d3e80d48": " {\\mathbf A}_{11}^{-1} {\\mathbf A}_{12}| {\\mathbf A}_{22\\cdot 1} \\sim MN_{p_{1}\\times p_{2}}\n( {\\mathbf \\Psi}_{11}^{-1} {\\mathbf \\Psi}_{12}, {\\mathbf A}_{22\\cdot 1} \\otimes {\\mathbf \\Psi}_{11}^{-1}) ", "2208dd816285a4d3689ca38bb789159a": "t' = t{\\omega}\\,\\!", "2208f21c6385673f7f5757329a5ac544": " \\mathbf{r} \\cdot \\mathbf{\\hat{n}} = \\left | \\mathbf{r} \\right | \\cos \\alpha \\,\\!", "22091444e3da60d0a52fcc4b2cee6d70": " \\chi^{\\prime \\prime}(\\mathbf{Q},\\omega)", "220984254487f8707d685479a3761c25": "h(x_0+\\Delta x) = f(x_0+\\Delta x) g(x_0+\\Delta x) = f(x_0) g(x_0) + \\Delta f g(x_0) + f(x_0) \\Delta g + \\Delta f \\Delta g", "2209cdd73bf13391bbd4ceb8bf75b1a0": "\\int_0^\\infty \\frac{x^3}{e^x-1}\\,dx = \\frac{\\pi^4}{15}", "2209e21d3d3c62419b29c1bcf5fca29f": " C^T_{(-)}=C_{(-)};~~~C^2_{(-)}=1 ", "220a0c5ce505a28b7d9dfb9fa53d69ed": "F_n(z)=f\\circ f\\circ \\cdots \\circ f(z)\\to \\alpha,", "220a97a21b885785e62eadc7a4c78236": "\\frac{\\partial \\chi}{\\partial t}=\\nabla\\cdot( D\\,\\nabla \\chi+ D_{T}\\, \\chi(1-\\chi)\\,\\nabla T)", "220abf6bc46dcc1c63937fc38f315032": " 2 \\Pi_2 \\left(\\prod_{p|n; p \\geq 3} \\frac{p-1}{p-2}\\right) \\int_2^n \\frac{dx}{\\ln^2 x}\n\\approx 2 \\Pi_2 \\left(\\prod_{p|n; p \\geq 3} \\frac{p-1}{p-2}\\right) \\frac{n}{\\ln^2 n}\n", "220ac14f14c1a1efed582072f8d77770": "10/81 = 0.\\overline{123456790}", "220b19101c8bbdf0a36f62a9c641c913": "\n \\mathbf{e}^i(c_1 \\mathbf{e}_1+\\cdots+c_n\\mathbf{e}_n) = c_i, \\quad i=1,\\ldots,n\n ", "220b5ed3b6746762a49bf602e16d9d26": "\\cot \\frac{\\pi}{10} = \\cot 18^\\circ = \\sqrt{5 + 2 \\sqrt 5} ", "220b812198ad2aec784cbe90738b880c": "b_0 \\centerdot 1 + b_1 \\centerdot 1 + b_2 \\centerdot 1 + b_3 \\centerdot 1 = 0", "220ba13e9bec9ee194f6ab5c635eb034": "\\mathcal{L}=\\mathcal{L}_{1}+\\frac{1}{2}\\, I_{2}\\Omega^{2}\\cos^{2}\\delta+\\frac{d}{dt}(I_{2}\\alpha\\Omega\\cos\\delta)\\,,", "220bb215b8cc6dbe2c8c8244941d3fb1": "x_n = \\tfrac{2}{\\pi}sin^{-1}(y_{n}^{1/2})", "220c0ff010dbd6b4eaccd8c260f9ffe5": "x_{i-\\frac{1}{2}} \\ ", "220c2d191a1f2377194b2c2bdd0619d2": "K(z)=\\exp\\left(\\psi^{(-2)}(z)+\\frac{z^2-z}{2}-\\frac z2 \\ln (2\\pi)\\right)", "220cc04c478ff42bc4a41bcd9f586e27": "\\hat {U}(t,t_0)=1-i\\int_{t_0}^t dt'\\hat V(t')", "220cf0b6aa780d764d26d6292f1de38c": "a = -b", "220d0eaba897c0f6628aae07cafb3076": "L = \\int_a^b \\sqrt{ \\left[r(\\varphi)\\right]^2 + \\left[ {{dr(\\varphi) } \\over { d\\varphi }} \\right] ^2 } d\\varphi", "220d1237794470250148ee37a0af3ef1": "0 < \\Im{z} < \\beta", "220d21a517172a2a589add2f409355d4": "g_N > 0 ", "220d6b2c0d44810aaab1e548204997f6": "\\scriptstyle{i=1,\\ldots,m}", "220d7a1030e205f9570c702431c103a2": "\n\\pi=\\cfrac{4}{1+\\cfrac{1^2}{2+\\cfrac{3^2}{2+\\cfrac{5^2}{2+\\cfrac{7^2}{2+\\cfrac{9^2}{2+\\ddots}}}}}}\n=\\cfrac{4}{1+\\cfrac{1^2}{3+\\cfrac{2^2}{5+\\cfrac{3^2}{7+\\cfrac{4^2}{9+\\ddots}}}}}\n=3+\\cfrac{1^2}{6+\\cfrac{3^2}{6+\\cfrac{5^2}{6+\\cfrac{7^2}{6+\\cfrac{9^2}{6+\\ddots}}}}}\n", "220d8023f37720b8da0754d367daf08b": "P(A_1, \\ldots, A_n) \\iff P(*A_1, \\ldots, *A_n) ", "220d918ef21cbfe62ad0d92710d44c97": "\\sigma^{2}_x\\sigma^{2}_y/[\\sigma^{2}_x+\\sigma^{2}_y]", "220d927e869c2390c1fda7eea6361023": "a_k := | L \\ \\cap \\Sigma^k |", "220dbe8a0972557222940bbbf80d7103": "\n\\hat{R} = \\frac{1}{N}\\sum\\limits_{i=1}^N R_i\n", "220deedeea68edced24ebf77f4e3efd3": "X:=\\bigcup_{k=1}^m 3\\,B_{j_k}", "220ec1fd6364da4809a5bbd4d695f0bc": "\\!\\mathcal A \\models_{X[F/x]}^+ \\phi", "220eebf8f8c8c5b7044a42816ab5f56c": "condition_j", "220f251b894c9f00202cad338d1be09a": "\\theta_S", "220f4e8081048e24f3aa6b2114d1174b": "s(x) = \\sum_{i=1}^n \\chi_{E_i}(x) b_i", "220f5f2dab41dbddf09457b5491942dc": "\\textstyle c_n = n+1", "220f6cae28ee5cb53474ce99bac13c05": "\\beta_{F0}", "220f9cd35d0f2f9b80d13d73bcda009b": "\n\\frac{\\partial}{\\partial t}\\left(\\nabla^2\\phi-\\phi\\right)+\\left[\\phi,\\nabla^2\\phi\\right]-\\left[\\phi,\\ln\\left(\\frac{n_0}{\\omega_{ci}}\\right)\\right]=0.\n", "220fbc0f092419ba112d734a5369e9d8": "\\ \\begin{array}{rrcl} & P^*(F^*)^T & = & J^*\\sigma ^* \\\\\n \\Rightarrow & P^*(QF)^T & = & J Q \\sigma Q^T \\\\\n \\Rightarrow & P^*F^T Q^T & = & Q J \\sigma Q^T \\\\\n \\Rightarrow & P^*F^T Q^T & = & Q P F^T Q^T \\\\\n \\Rightarrow & P^* & = & Q P. \\end{array}", "220fc943a0224aa48cda73ea86e998f6": "p \\colon C \\to X\\,", "220fe190634db161b0a7d244646394a8": "[p] \\mapsto [m]", "2210248fff97dbc40ca58394b52d9036": "\\int_{-N}^{N} f(x)\\, dx", "221056de6a39bf400060292db53a4fba": " \\bigcup_n B_n = B.", "2210862e586d661e35708384d9c3c957": "\\displaystyle{[L_m,\\,J_n]=-nJ_{m+n}}", "22109003a1884d96d9f69b7f7e5735c2": "w_2w_{4k-1}.", "2210cc7648fa8173829e59639e012641": "L(s_r)=aL(s), \\qquad L(s_\\ell)=a^{-1}L(s).", "2210e5a25f7eecce23c67b527913b105": "{\\mathbf z}", "221120ba3536e487a32e92676b14f911": "\\begin{align}\n dA_{\\bold{x}}\\,dA_{\\bold{y}} &{}= \\begin{bmatrix} 1 & 0 & d\\phi \\\\ d\\theta\\,d\\phi & 1 & -d\\theta \\\\ -d\\phi & d\\theta & 1 \\end{bmatrix} \\\\\n dA_{\\bold{y}}\\,dA_{\\bold{x}} &{}= \\begin{bmatrix} 1 & d\\theta\\,d\\phi & d\\phi \\\\ 0 & 1 & -d\\theta \\\\ -d\\phi & d\\theta & 1 \\end{bmatrix}. \\\\\n\\end{align}", "221127e1633dc0a4863bb235db9bd976": "\\frac{\\partial \\rho}{\\partial t}+\\nabla\\cdot \\left(\\rho\\mathbf{u}\\right)=0", "221139977c206ec99f734c5b6d17818e": "\\displaystyle{\\omega(g,h) =\\Omega(g,h) \\beta(gh)^{-1}\\beta(g)\\beta(h),}", "22113d887e01888316f5755413ee79d1": "\\tilde{U} = \\text{constant}", "22116896dfcf2ea54f842e9a9e489a83": "|w| \\le \\tau", "221193c150f7ebad479701e16265e561": "\\|S_1\\| \\cdot \\|S_2\\| < 1", "2211d6f3b3b84f1ffc9d5bb45a3f6827": "(x_i)_{i \\in I}", "221208aa5328f227392e786def848f87": "(\\neg A\\to A)\\to A", "221208e49f32879e4044303eb88d09fe": "f_c = \\frac{1}{2 \\pi R C} = \\frac{1}{2 \\pi \\tau}", "2212099388254949ff008d0ec8a0dd91": "[T^i , T^j] = 2 i \\epsilon^{ijk} T^k", "2212512dabfa2a117c1943024ce6e01b": " {[G(x)]}^n = G{(a_n x + b_n)} \\, ", "221299193f96c540ae9f38cf64ac7dfa": " 3 \\mid a_1^2+b_1^2 \\, ", "2212bfd884f6df6f8a643fbd2a8219d9": "\n\\min\\ ||\\mathbf{v_w} - \\mathbf{v_d}||^2 \\qquad s. t.\\quad \\mathbf{S}\\cdot\\mathbf{v_d}=0\n", "2212d0c6199e14737dcacf7a66394234": "\\dot{\\mathbf{x}}(t) = \\mathbf{A}(t) \\mathbf{x}(t) + \\mathbf{B}(t) \\mathbf{u}(t)", "221334506ba2dcb828d221e71ed0f507": "x+t", "2213a573866cc3a01015b285a0497dec": "Sf=\\sum_{i\\in J}\\langle f, \\phi_i \\rangle\\phi_i", "2213acc02bfb76c74b0e9a37d13509dc": "y_j \\not = c_j", "2213b5ce4b9b042f46a3d6cdd88a7584": "2 \\times \\sqrt{6}", "2213be3a40642d54539be537adf83cd0": "\n\\mathrm{Ein}(z)\n= \\int_0^z (1-e^{-t})\\frac{dt}{t}\n= \\sum_{k=1}^\\infty \\frac{(-1)^{k+1}z^k}{k\\; k!}\n", "2213e0878e8ac143520a0dcdd18c04f8": " \\mathcal{M} \\subseteq \\mathcal{A} ", "2213e1b4cbd940603a6702ebe6ba6ae4": "3r", "2214485fc65a1dcd92b453e7c67d3795": " n = \\frac{pAl} {RT} ", "2214d4a7928dcb6ea83559bd8705ef2b": "\\,\\frac{e^{h-\\lambda e^h}\\lambda^k}{k!}", "2215026cf1f70449f7e2952be00fe7a4": "\\scriptstyle(x_1-\\overline{x},\\; \\dots,\\; x_n-\\overline{x}).", "2215093743db6194cf6ada00469b56ee": "\\begin{align} \\frac {d M_{xy}'(t)} {d t} = i \\gamma B_{xy}' M_z (t)\n\\end{align}\n", "221515b5aa06aceed08af2d973b2821d": "\\mu(a_m/a_0) m a_m = GmM/r^2 = GMm/r^2 = \\mu(a_M/a_0) M a_M", "2215b59c88015fb83a27d020c7876665": "\\mathrm{\\Lambda}(A \\otimes B) = \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A \\otimes B} \\right).\n", "2215ceecf3e5d33ad40a470d13d10143": "u_z = \\frac{u_*}{\\kappa} \\left[\\ln \\left(\\frac{z-d}{z_0} \\right) + \\psi(z,z_0,L)\\right]", "2215d77738a7148e3887d4852e22fcdf": "D \\in \\Delta", "221609d2bf909d45d661979c396c4f16": "Q=\\frac{\\text{gap}}{\\text{range}} = \\frac{0.177-0.167}{0.189-0.167}=0.455.", "221624388bab536d75aa96e7edd5931a": "\nI_o = I_e \\left(\\frac{q^4}{m^2c^4}\\right)\\frac{1+\\cos^22\\theta}{2} = I_e7.94.10^{-26}\\frac{1+\\cos^22\\theta}{2} = I_ef\n", "2216667551c62a6524a5d2127500a86f": "\\mu(x)", "221671d8a139ff2b0c8d6c3cd2e80dac": "\\frac{dv}{dz}", "22169b1420143c14ead6e48bfcb9c955": " \\mathbf{u}_i=-L_i\\hat{\\mathbf{x}}_i. \\, ", "2216c9f8ff7ef6e6e0757d44d224f7a2": "E_{1}(\\mathbf{R})", "22171e2d949bacb18801616495bf984c": "s_{ij}>\\tau ", "22172fff96e170b3c10015784f31f1c2": "\\mathrm{MA}_{compound} = \\mathrm{MA}_1 \\mathrm{MA}_2 \\ldots\\mathrm{MA}_n \\,", "22174ae7212cdb08f779c3f416f611c0": "{{\\Bbb R}^{2n+1}}", "221814f0e8a26cc03151e44c36139876": "f(\\mu)=|{\\partial \\mu^{(1)} \\over \\partial \\mu}| f(\\mu^{(1)}) = f(\\mu+b)", "221825347b108bed93476c58a740404c": "(k-1)", "221839f750ef50666ab1b7fabf2bcd71": "C_{diffmap} = ( m |F_{obs}| - D |F_{calc}| ) exp( 2\\pi i \\phi_{calc} ) ", "22185b2c60d34219fadd22e4825f2055": "|\\ \\rangle \\!\\,", "221871dd6f60317e2b09fa62f6cfc24c": "\\begin{align}\n x_{n+1} &= x_n + f(t_n, v_n) \\, \\Delta t\\\\[0.3em]\n v_{n+1} &= v_n + g(t_n, x_{n+1}) \\, \\Delta t\n\\end{align}", "2218befd39fdaf83cbf2992a51196237": "p \\le \\gcd \\left( x-y,n \\right) \\le n", "2218c392302ec9a2b0bd03bb94556e90": "\\mathcal{L}_{V^{2}}(\\theta_{1})\\,", "2218d976a5d4315bc4d5ee15cab7b526": " = y + b (\\alpha - a) - \\delta (\\beta - b) \\ ", "2219018b5eb6a0a9eee1ba29c4257304": "\\hat \\eta = \\left ( \\theta_1, \\theta_2, \\ldots, \\theta_k \\right )", "221945ee4f93c359c2e5ed2fffea0053": " f \\left( \\frac{x}{n} \\right) = \\frac{1}{n} f(x) \\ ", "22194ac0a68811d159a39093a6c7b5a9": "\\pi_1(S)", "221975c3903b34e463f15f43a56ba925": " \\phi_U = -V_\\infty \\cdot \\mathbf{n} ", "22198b86edb5e7de28851ce9f8b8283e": "\n \\sigma_{i} = 2C_1 J^{-5/3} \\left[ \\lambda_i^2 -\\cfrac{I_1}{3} \\right] + 2D_1(J-1) ~;~~ i=1,2,3\n", "221993df96d492ff7f5903c9283b3914": "\\scriptstyle T_\\max", "2219a9f7865e881a74b4ee585db08b16": "\n\\frac{{\\Delta z}}{z}\\,\\,\\, \\approx \\,\\,\\,\\frac{{a\\,\\Delta x_1 + \\,\\,\\,b\\,\\Delta x_2 }}{{a\\mu _1 + \\,\\,b\\mu _2 }}", "2219beee0b45b55be029df4d479009f1": "\\omega_J = \\frac{q}{\\hbar}\\cdot V. \\ ", "2219ef8a39b20e9eddde42a61835e452": "w_{i} = \\mathrm{weight}\\left(a_{i}\\right), 1\\leq i \\leq n", "221a116083f87d4f06615bb781ff8d0f": "\\psi\\mathcal{U}\\phi", "221ade70cc2786150d39b2f8db658d17": "\\text{Cov}(X,Y)\\le\\text{Cov}(X^*,Y^*)", "221af9da4d5dffa6d63bc209d7fbcc9e": "(ab)(a c_1 c_2 \\dots c_r)(b d_1 d_2 \\dots d_s) = (a c_1 c_2 \\dots c_r b d_1 d_2 \\dots d_s)", "221b95feacf04f368578e81b7dced556": "m 1_R = m", "221beccb58823ee171a9b857d5d2d82e": "\ni\\hbar | d \\Psi(t)/dt \\rangle = \\hat{H} | \\Psi(t) \\rangle,\n", "221befdf6eae02b1e4d49c3132467ad6": "\\mu_1=\\mu_1^+-\\mu_1^-", "221c0543f6730384203a254d367997d3": "P(x) = \\langle \\mathbf{x}, W \\mathbf{x} \\rangle ", "221c58ef9b428b01ffbf6f2452941bbb": "x^{\\alpha+1}\\,e^{-x}\\,", "221c5c2b55a969010749ab3a78145b76": "\n\\nabla_\\mathbf{X} y(\\mathbf{X}) = \\frac{\\partial y(\\mathbf{X})}{\\partial \\mathbf{X}} ", "221c6947ebece069ce42d39104fde809": "I_r^'", "221c786b331969eef91620ccf284b91a": "\\left(\\int_{\\mathbb R} |g(y)|^{2\\beta}\\,dy\\right)^{1/2\\beta}\n \\le \\frac{(2\\alpha)^{1/4\\alpha}}{(2\\beta)^{1/4\\beta}}\n \\left(\\int_{\\mathbb R} |f(x)|^{2\\alpha}\\,dx\\right)^{1/2\\alpha}.\n", "221c7fb8e8f49035f169fb9e3b18715f": "\\Pi ", "221c830ead58aba253bcc8c0c6ca8315": "d_H(S,T) = \\max \\{ \\sup\\{d(s,T) : s \\in S \\} , \\sup\\{ d(t,S) : t \\in T \\} \\} ", "221c8e862c3cc667a9665abfc938c0b1": " \\binom{m}{k} ", "221ca05ac39e48b3f5bb48470ca50cce": "\\sigma_m^{2}=E_\\pi[\\sigma_f^{2}(\\theta)]+E_\\pi[\\mu_f(\\theta)-\\mu_m],", "221ccf451d1e86a2ada2b1d27fc2da3c": "\\mathrm{return}: A \\rarr E \\rarr \\mathrm{M} \\, A = a \\mapsto e \\mapsto \\mathrm{return} \\, a", "221cd0cf08ec47534f9ec4a2f85fa263": " \\tau_{Wall} = \\frac {D \\Delta P} {4 L} ", "221d16fe871541dfe85913346464e16b": "I_\\mathrm{NMDA}(t,V) = \\bar{g}_\\mathrm{NMDA} \\cdot B(V) \\cdot [O] \\cdot (V(t)-E_\\mathrm{NMDA})", "221d34767145014b134a6473c37783cf": "z = n - c\\,", "221db8ccb6aab742201b2882f3f99bb0": "\\displaystyle{f_-=D(\\varphi)|_{\\Omega} + S(\\psi)|_\\Omega,\\,\\,\\,\\,\\, f_+=D(\\varphi)|_{\\Omega^c} + S(\\psi)|_{\\Omega^c}.}", "221df38694e658cd5dbaed60e0ce0f39": "\n \\operatorname{E}[\\,y_t|x_t\\,] = \\int g(x^*_t,\\beta) f_{x^*|x}(x^*_t|x_t)dx^*_t ,\n ", "221e173faf90c030933b36de017f8c59": "\n\\int_N^{M+1}f(x)\\,dx=\\sum_{n=N}^M\\underbrace{\\int_n^{n+1}f(x)\\,dx}_{\\le\\,f(n)}\\le\\sum_{n=N}^Mf(n)\n", "221e1cdcadaf602988e259936603aba6": "\\mathfrak{B}(V_q)=k[x]/(x^n)", "221e883a0f7f6f1fb3a57690de684c90": "x^{1/n}", "221e8aff5b8660c28babe46e4013e440": "\\{t,r,\\theta,\\phi\\}", "221e8b5819c99d5fa3748d8244f1d9da": "sW(s,\\xi)=-\\frac{d}{d\\xi}W(s,\\xi)+U(s),", "221ea1d11882ec6de661d2cfa95dec6c": " {} = x_{i}y_{j}-x_{j}y_{i} . \\,\\!", "221ecb1d2c96d59c24f22beca73b517d": "(X^2, \\mu \\otimes \\mu, T \\times T)", "221eecf8c8a6f69ec2eefc1cb14c9444": "\n|\\delta \\vec x| \\approx |\\delta \\vec x_0| e^{\\lambda_1 t}.\n", "221efb49aa5d6137bd51717b3d1f7900": "\\Gamma(s+1)=\\int_0^\\infty x^s e^{-x} \\, dx", "221f2fe2d639a951158966101b301a96": "x \\rightarrow \\infty,", "221f772f52e35588c3cc6de43af981af": "\n \\mathbf{b}_i = \\cfrac{\\partial\\mathbf{x}}{\\partial q^i} = \\cfrac{\\partial\\mathbf{x}}{\\partial x_j}~\\cfrac{\\partial x_j}{\\partial q^i} = \\mathbf{e}_j~\\cfrac{\\partial x_j}{\\partial q^i} \n ", "221fa9fd4a0f43e4289e27fa49206858": "w_\\mathrm{eff}", "221fca41660e44a19daae729f90480ea": "J = \\det(\\boldsymbol{F}) = 1", "221fcae732f0b7d83af3b140a6b84d0e": "\\lceil \\text{slog}_e(-x)\\rceil = -1", "221fe2b1a2a50054c7daa69e1efead29": "\nr_{\\mathrm{min}} = \\frac{1}{u_{2}} = A (1 - e) \n", "221ff9000d1eef9d1f6152d888048616": "\\forall \\alpha. (\\alpha \\rightarrow \\alpha) \\rightarrow \\alpha \\rightarrow \\alpha", "2220759a127017880918765ba56d6a9a": "\\liminf_{n\\to\\infty}x_n := \\sup_{n\\geq 0}\\,\\inf_{m\\geq n}x_m=\\sup\\{\\,\\inf\\{\\,x_m:m\\geq n\\,\\}:n\\geq 0\\,\\}.", "2220883d6d94e5a745ed87c456f28fb5": "p(t) = \\frac{v^2(t)}{R}", "2220a969492b1e2236a06c268e8691c9": "Cl(p+8,q)", "2220bca035dc1ca4b3c43a0465cf7de4": "\\,\\!T[n]", "2220db6065a8fbbafcdddbe44f43c16e": "\\frac{ \\left( \\frac{ K^- + \\bar{K}^0 }{2} \\right)^2 + \\left( \\frac{ K^+ + K^0}{2} \\right)^2}2 = \\frac{3\\eta^2 + \\pi^2}{4}", "2220e37f566a5e0ec392c53d1be00d22": "[\\bold{k},\\bold{k} + d\\bold{k}]", "22212958906cc4ee62de0b991b64c031": "M_\\lambda\\to M_{\\mathfrak{p}}(\\lambda)", "2221a16c5d2f0e18546ddc0d6908f91d": "Y=f(\\beta_1^\\top X,\\ldots,\\beta_k^\\top X,\\varepsilon)\\quad\\quad\\quad\\quad\\quad(1)", "22221b657ee7ad9ecd2bfe6b9571d60e": "R_a = 1.22 \\, w f,", "22221d7f2e51c00d35e8eefee65090ef": "ds^2 = -\\left(\\alpha^2- \\beta_i \\beta^i\\right)\\,dt^2+2 \\beta_i \\,dx^i\\, dt+ \\gamma_{ij}\\,dx^i\\,dx^j", "22230e27d6cea1a1def99a921859833b": "\\displaystyle{\\mathrm{Tr}_{\\mathbf{R}}\\,XY=\\mathrm{Tr}_{\\mathbf{R}}\\,YX,\\,\\,\\, \\mathrm{Tr}_{\\mathbf{R}}\\,(XY)Z=\\mathrm{Tr}_{\\mathbf{R}}\\, X(YZ).}", "222396d6b6003dfca2c070ef98dfeb07": "\\rho(x)=\\frac{e^{-\\frac{x^2}{2}}}{\\sqrt{2\\pi}}", "2223e0180eae55db3545442c9e49689a": "x^{\\ast }\\left( p\\right) ", "22249858a0839ff24ae59955c4af7686": "(x+y)_n=\\sum_{k=0}^n {n \\choose k} (x)_{n-k} ~(y)_k ~,", "2224ddcd6c0dc550e8b1a3fdc924d474": "f(x) = c \\, ,\\,\\, c \\neq 0", "2224e658ccf23b1b58dd91759f1a29db": "\\omega^{2^{p-2}} + \\bar{\\omega}^{2^{p-2}} = kM_p", "2225254da29c2022cc60e8ad66d647d4": "f(\\phi/c^2)=\\exp(-\\phi/c^2-(\\phi/c^2)^2/2)\\,", "2225583ba281dbe31d86257406b1155b": "\\sigma_{xy}\n=-\\frac{\\partial^2C}{\\partial x \\partial y}", "222562ff9046bc9b7c04c5686fe36932": " c = 1 \\ ", "2225a02cdd4daf072d2a9ef5b91e1383": "I_{x,A}:F_{x,A}\\to F", "2225a33c170f19586f1ee1de06e36962": "p\\!", "2226aa2c8cb2fae9564e61b68d62da1d": "\\begin{align}\nx(t) & = e^{-\\alpha t} \\mathcal{L}^{-1} \\left\\{ {s \\over s^2 + \\omega^2} + { \\beta - \\alpha \\over s^2 + \\omega^2 } \\right\\} \\\\[8pt]\n& = e^{-\\alpha t} \\mathcal{L}^{-1} \\left\\{ {s \\over s^2 + \\omega^2} + \\left( { \\beta - \\alpha \\over \\omega } \\right) \\left( { \\omega \\over s^2 + \\omega^2 } \\right) \\right\\} \\\\[8pt]\n& = e^{-\\alpha t} \\left[\\mathcal{L}^{-1} \\left\\{ {s \\over s^2 + \\omega^2} \\right\\} + \\left( { \\beta - \\alpha \\over \\omega } \\right) \\mathcal{L}^{-1} \\left\\{ { \\omega \\over s^2 + \\omega^2 } \\right\\} \\right].\n\\end{align}", "2226d8be45d510354555f60a9dabbcf3": "T_d \\subseteq T", "2226e9cfddef0a4c2521be342f24b47d": "\\begin{matrix}{r \\choose 2}\\end{matrix}", "2226efbbc3e8bd017487e77e040141f2": "R(u,v)w=\\nabla_u\\nabla_v w - \\nabla_v \\nabla_u w -\\nabla_{[u,v]} w .", "2226f830d7912cb8ecb96a2eeaaa3aba": "\n\\partial_t (\\rho e) +\\partial_i(\\rho e u_i) + p\\partial_i u_i=0\\,\n", "2226f8bb8106ac95fb2db9b191fe6deb": " \\prod_{i=1}^{s} x_{i}^{\\nu_i} = K", "2227111c676c40339debd4a010708a7a": "\\theta (f,b)=0", "22275308eaa32ab19c032fa7594bbc09": "E(X^n) = \\sum_{k=1}^m \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}.", "22279224d3434f44f995197df53c43c9": "{\\tau}", "2227d091e1a568ba04be669214b32cf7": "a, b \\in H_1(S)", "2227d8afc43d38f338fadea18383d84b": "R_X(t,s) = E\\{X[t]X[s]\\}", "2227fc6b1daac62daf62a1253b870030": "DF : U\\times X\\to Y", "222809caa99a6ac65909bdb8821ccc74": "x_0,y_0,z_0", "22281defad6e2d58653198aea6fce8fd": "y,t", "2228562089c383a77ce74771d157b5eb": " T(g) \\cdot T(h) = f_g \\cdot f_h = f_{g*h} = T(g*h) .", "2228ac3354ce4b278faae0fcf1ae05e9": "y = 1.80 - j3.90\\,", "22290fe6a2e26be2ed37aa263a664fa4": "ds^2 = -dt^2 + \\alpha^2 \\sinh^2(t/\\alpha) dH_{n-1}^2,", "222924df98b78878ce8dd47d80d7711d": "\\int_\\gamma f(z) \\,dz = F(\\gamma(b)) - F(\\gamma(a)).", "22292f1f0079317e05c9463ddec0d10b": "a_{2,1} x_1 + a_{2,2} x_2 = b_2", "222940f2e774445e0755a45550549e2e": "n = (-1+ \\sqrt{-163} ) / 2", "2229498613991f866abcc16b06bbdadf": "m(\\mathbf{x})", "2229819e8cb2bd103c6c16257adde5d4": "\\frac{r}{2}\\frac{\\cos (\\pi /n)}{n}", "222a2db67ea0ee77a516fc42bcd41be2": "symb(L)\\equiv\\sum_{i+j=n}r_{i,j}(x,y)X^iY^j", "222a3fa8402831cba5b5b30101950ad6": "A(\\lambda) = {SRM\\over 12.7}(0.018747e^{-{(\\lambda - 430)\\over 13.374}} + 0.98226e^{-{(\\lambda - 430)\\over 80.514}})", "222a5c4abd0a41e794c06da5fbed1f1f": "\\frac{2}{\\pi} \\arctan\\!\\left[\\exp\\!\\left(\\frac{\\pi}{2}\\,x\\right)\\right]\\!", "222a689be9f378cc408cefbe9098e2e7": "\\hat\\sigma^2 = \\frac{ \\|\\hat{r}\\|^2}{ \\hbox{tr}\\left( (I-H)'(I-H) \\right) },", "222aa6b0ef9632880a3e99be90e3a960": " \\operatorname{d} U \\propto \\operatorname{d}T ", "222ac3de6b1502e116388d9614079d7b": " \\mathbf{\\pi}^{(k)} = \\mathbf{x}(\\mathbf{U\\Sigma U}^{-1})(\\mathbf{U\\Sigma U}^{-1})...(\\mathbf{U\\Sigma U}^{-1}) ", "222ae801471de7c2fb00b40944059e26": "v=|\\Delta\\lambda|c/\\lambda_0\\,\\!", "222aec8b2aa2268f9a7f9b27a1ccfc0a": "\\mathrm{Yield} = 2 \\left[ \\left(\\frac{a-b}{cd}+1\\right)^6-1\\right].", "222b1c1713b0aaea1eac6ce4e6012064": "\\hat x_{free} ", "222b2d694042ec4ca03110a9fe562534": "\\left( h \\right)", "222b869619a9316a8a4cd6d277e71fd0": "\\frac{d}{{dx}}\\log \\left( \\kappa \\right) = \\frac{{\\kappa'}}{\\kappa}", "222bec494dbefef35a81f2f7f509bb14": "\\tan \\psi = \\frac {u'} f \\sin \\theta.", "222c4b94c1ea472a9bb798958f79783b": "\\pi 12.5^2", "222c712cd0c12d2369bf9d4f1f0f17e9": "(\\log f)' = f'/f", "222ca92183d49f526b200cb797382583": "\\delta(a)", "222cc9d39d988b23f191ea9b230f1364": "\\frac{1}{y} \\frac{dy}{dx} = \\frac{f'(x)}{f(x)}", "222cd21492d718a11bb6b44d70e22ef0": "537^2 \\mod 84923 = 33600 = 2^6 \\cdot 3 \\cdot 5^2 \\cdot 7", "222cfdeee227a3bdb45b5f8c24f9f80c": "D \\subset \\Omega", "222d574a209171c87ef83aafdc6015f0": "\\chi_\\lambda", "222d8343c991fefb0ad3d04e509dbcdf": "Target \\ Height > \\frac{ \\left( Target \\ Range - \\sqrt{2 \\times H \\times Re } \\right)^2 }{2 \\times Re}", "222e2caf9c7b49d3432466e360eceba6": "\\bar{b}", "222e3759bff6266c3657d2605dd0d3c3": "M_i = \\sum_{j=1}^n a_{ij}\\left( \nv \\frac{\\partial u}{\\partial x_j} -u \\frac{\\partial v}{\\partial x_j} \n\\right ) + uv \\left( \nb_i - \\sum_{j=1}^{n} \\frac{\\partial a_{ij}}{\\partial x_j} \\right ), ", "222e8d8a0206971bcfef33d5df0cd5b3": "K=\\frac{[\\text{M}(\\text{OH})] } {[\\text{M}] K_\\text{w} [\\text{H}]^{-1} }", "222e928942f6b98b9735c3507a5f1218": " \\Psi(1,2)- \\Psi(2,1) \\ne 0 ", "222ead7e5a28ba3bfa54297e8cec7848": "R_{\\text{C}}\\,", "222f9c69d670649db60ab306d8067676": " \\frac{\\partial}{\\partial t}p_{\\varepsilon}(x,t)=D \\Delta p_{\\varepsilon}(x,t)-\\frac{1}{\\gamma}\\nabla ( p_\\varepsilon (x,t) F(x))", "222fb9cd78c40c4159dfe7037be035ea": " A = A \\left ( t \\right ) ", "2230281fd061be3809247518cc5475c3": "A^* = A^T", "22304c1694a706284cd8fa43a39d21d6": "e^{i \\pi / 2}", "2230aa3e24e99e12d7e98baa3fcaf1cb": "k = \\sqrt{n}, \\, ", "223129f0f6dee1a483eea56e6c1772ab": "G=U+pV-TS,", "2231d00269d93bf209149ab3e4d7d954": "\\mathbf{P}(X \\ge a) = \\mathbf{P}\\left (e^{tX} \\ge e^{ta}\\right ) \\le \\frac{ E[e^{tX}]}{e^{ta}} = \\frac{E[\\prod_i e^{tX_i}]}{e^{ta}}.", "2232150810ebf9eafff772a344ee2afe": "f(x_1,a,g(z_1),y_1) \\{ x_1 \\mapsto x_2, y_2 \\mapsto y_2, z_1 \\mapsto z_2\\} = f(x_2,a,g(z_2),y_2)", "2232207c8c8080c643581dddad302d3e": "{I}^{2}=I\\times I", "223266d636e7de07caa11d1932c67960": "B\\Sigma_\\infty^+\\simeq \\Omega_0^\\infty S^\\infty", "22329c7bcc696978d61a533c5e731931": "\\theta_e = T_e \\left( \\frac{p_0}{p} \\right)^\\frac{R_d}{c_{pd}} \\approx \\left( T + \\frac {L_v}{c_{pd}} r \\right) \\left( \\frac{p_0}{p} \\right)^\\frac{R_d}{c_{pd}} ", "2233034547204d23736f9d678ccc29cc": "U_{n+m} = U_n U_{m+1} - Q U_m U_{n-1}=\\frac{U_nV_m+U_mV_n}{2} \\,", "22330c6e44f0b4606fe2c256ba4dc504": "_{P}(f)", "223331028c1cd2de584aed1c07802329": " \\mathbf E_{1s}=", "2233e573678ec41f964e2c7a84713e27": "c = \\hbar = 1", "2233e9aa2a260a799da251119f19a591": "a_1=\\frac{1}{2}\\textrm{ln}\\frac{W_+(p\\wedge c)+1}{W_-(p \\wedge c)+1}", "22340c794846330532f7134892854875": "\\begin{align}\n\\mathbf{L}^2Y &= \\lambda Y\\\\\nL_zY &= mY\n\\end{align}\n", "2234183167c70d3c9d29467fff0a5f82": "\\,q_{x+t}", "22344b37955faff24a5f0d8738f6bd79": "d\\theta.", "22345e775d02d325fb3868cd8605d792": "\\log_{10}(\\frac{N}{S}) = -k \\cdot \\log_{10}(r) + m", "22347b673da12b1762a8be43132a7457": " \\hat{f}: \\hat{M}\\to\\hat{N},\\quad ", "2234d561f3d703ad89ea8fbf38149f5b": "\\mathfrak{H}(G)", "2234d9213618f3a9d0ad9562eb2689b1": "r_0 \\in R_n", "22350f234515f924b1d780028a105000": "\\psi(x_1, x_2, ...,x_N)", "2235e2a6524268c02c325a2b5f421847": "\\succcurlyeq~\\subseteq~X^2", "2235ea211c48c8890b062a889642da43": "\n\\mathfrak{P}(\\mathfrak{C}_\\operatorname{odd}(\\mathcal{Z}))\n\\left(\n\\mathfrak{P}_{\\ge 1}(\\mathfrak{C}_2(\\mathcal{Z})) +\n\\mathfrak{P}_{\\ge 1}(\\mathfrak{C}_4(\\mathcal{Z})) +\n\\mathfrak{P}_{\\ge 1}(\\mathfrak{C}_6(\\mathcal{Z})) + \\cdots\n\\right)", "223635019e1f1f9f08d57ceb7ba1f0e9": "\\frac{M \\mu_0}{B+\\lambda M} =\\frac{C}{T}", "2236418f3f6e3ce8823154e3a2cb642d": "g(x)=f(x)\\cdot h(x)", "223663e6160588369676df1c64a0da13": "\nJ_{\\mu \\nu} = \\mbox{tr}\\left[ \\rho \\frac{L_{\\mu} L_{\\nu} + L_{\\nu} L_{\\mu}}{2}\\right].\n", "2236fa96dbe0616c465504d8304e6695": "pa_i", "22374d9016e74189ca3c8d07e708791a": "{\\mathrm G\\mathrm L}(n,\\mathbb R)", "22376fa0b6c15326ef87486a34a0c3f5": "\\mu_{\\operatorname{eff}} = K \\left( \\frac {\\partial u} {\\partial y} \\right)^{n-1} ", "22378879221d36a8a7cb55080aa89325": "(A.1.a)\\quad \\psi_{,\\,\\rho\\rho}+\\frac{1}{\\rho}\\psi_{,\\,\\rho}+\\psi_{,\\,zz}=\\,(\\psi_{,\\,\\rho})^2+(\\psi_{,\\,z})^2 +\\gamma_{,\\,\\rho\\rho}+\\gamma_{,\\,zz}", "2237b7669af7f66884a4633a7d148a3a": "\\scriptstyle i,j ", "2238fa09b11be0d1ec1ef9566cd71e63": "\\displaystyle\\mathbf F", "2239b1aa4eafd9521e10ab1c772c88e7": "\\alpha t = \\pi", "223a2ab31fa3378e0704f97a28a23490": "{3,2,1}\\,\\!", "223a32b4cae9183f2aa207f44678539f": "\n \\boldsymbol{N} = J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\n", "223a44e9ac64f3b34b57a3fd52c33ac6": "z=y/|c|", "223a4c7fe5a38eeb7c156f661da89fe5": "I(0)", "223a4da3924e1261bb886c1cebb6a2b4": "p \\cdot (x_i + \\Sigma_k x_k^*) \\geq r = p \\cdot (x_i^* + \\Sigma_k x_k^*)", "223a6c161200973fb2499c7eb4cb404a": "\nU_0 [\\mathbf{R}] = \\frac{k_B T}{4 R_{g0}^2} \\int_0^1 ds\n\\left| \\frac{d \\mathbf{R} (s)}{d s} \\right|^2.\n", "223a9a209f36205a1a3d5ed5109dfb09": "\\frac{d [ES]}{d t} = k_f[E][S] - k_r[ES] - k_\\mathrm{cat}[ES]", "223ad665207f596361afa1cbbc7c2b30": "D_u = \\frac{1}{m_u (m_u -1)}\\sum_{i=1,j=1}^{m}{_{metric}} \\delta_{c_{iu} k_{ju}}^2", "223aef8d300309527c240820d1f79c06": "Q(A)", "223b26bb5784686cb86485c52065d727": "\\{\\{\\emptyset\\},\\emptyset\\}", "223b87a72aff030cdb1b3f7c469b1a48": "\\mathbf C", "223b9082743bc4ca199916e7da490ee4": "Con := \\{ \\empty \\} \\cup \\{ \\{ n \\} \\mid n \\in \\mathbb{N} \\}", "223bac8cfcb457c56e07e83ffaadb980": "n_1 = 2 n_{11} + n_{12}", "223c2862459819ff4c33649c11ef8c98": "{\\tilde{E}}_{6}", "223c39c524bdf2bf4c8bbf904ab164b1": "(\\alpha+1)_n", "223c5f932d52e55a9006250b9ef75e79": " p_{ij} ", "223c63cbec2d9e24b2fe53154b3ea80e": "h_f = \\frac{8 f L Q^2}{g \\pi^2 d^5} ", "223cfbcb711b638187c911a76985e998": "\\frac{t}{V_L} = \\frac{\\eta\\cdot\\alpha_\\text{c}\\cdot c\\cdot\\frac{\\left(E_\\text{crit}-E\\right)}{E_\\text{crit}}}{2\\cdot\\left(\\Delta P_H+P_e\\right)\\cdot A^2}\\cdot V_L", "223d0261369101390bbd7bd804a9249e": "v(0) = 0", "223d2a7a0dd96168b21b6359e9e19e39": "\\int_{a}^{b} f(t) \\, \\mathrm{d} t := \\lim_{n \\to \\infty} \\int_{a}^{b} \\varphi_{n} (t) \\, \\mathrm{d} t,", "223d38f172bcfb9fa351581d51d00178": "F^{-1}(\\alpha/2; k,1) \\le \\mu \\le F^{-1}(1-\\alpha/2; k+1,1),", "223d4771230e69468777a3c416955562": "r_t = .95 r_c", "223d62b222458cfb4428eb7071ab68bb": " P_\\infty= p'[1-\\exp(-\\langle k \\rangle P_\\infty)]. \\, ", "223d7ffd94124c09d7297fea2a547837": "J^\\mu = c \\bar\\psi \\gamma^\\mu \\psi", "223dcfe974611928ed0043b1b06f6e50": "m_\\mathrm{H_2O} = \\left(\\frac{120. \\mbox{ g }\\mathrm{C_3H_8}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{C_3H_8}}{44.09 \\mbox{ g }\\mathrm{C_3H_8}}\\right)\\left(\\frac{4 \\mbox{ mol }\\mathrm{H_2O}}{1 \\mbox{ mol }\\mathrm{C_3H_8}}\\right)\\left(\\frac{18.02 \\mbox{ g }\\mathrm{H_2O}}{1 \\mbox{ mol }\\mathrm{H_2O}}\\right) = 196 \\mbox{ g}", "223de349ec8fb021a9a62be621abcb01": " T_{\\mathbf{v}} \\mathbf{p} =\n\\begin{bmatrix}\n1 & 0 & 0 & v_x \\\\\n0 & 1 & 0 & v_y\\\\\n0 & 0 & 1 & v_z\\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\np_x \\\\ p_y \\\\ p_z \\\\ 1\n\\end{bmatrix}\n=\n\\begin{bmatrix}\np_x + v_x \\\\ p_y + v_y \\\\ p_z + v_z \\\\ 1\n\\end{bmatrix}\n= \\mathbf{p} + \\mathbf{v} ", "223e2733e86a820e720bbf1e576fa8f7": "\n\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = 0\n", "223e60e37db7f0ff6159878c4fbda615": " {\\mathcal G}", "223eac722791eb38fab27c68e9e818c7": "A_{\\mu\\nu}^{}", "223ec0ca02ec3f40feda5b84931cfbac": " (W\\mathbf{r}) \\cdot \\mathbf{s} = * ( *L^* \\wedge \\mathbf{r} \\wedge \\mathbf{s}) = * (\\omega \\wedge \\mathbf{r} \\wedge \\mathbf{s}) = *(\\omega \\wedge \\mathbf{r}) \\cdot \\mathbf{s} = (\\omega \\times \\mathbf{r}) \\cdot \\mathbf{s} ", "223ec47cc11dec206b411abe0d3438d4": "y \\in F_q^{2k}", "223f66531b1616f96e05d245abd53226": "\\sum_k a_k m_k.\\,\\!", "223f9097f15d3718a97e71f249d1156c": "0\\le P(x_1,x_2,\\dots)\\le 1", "223fbfbe052c81e79519eae973e5031f": "(b-a)^2+4\\frac{ab}{2} = (b-a)^2+2ab = a^2+b^2. \\, ", "22402084bcd1328735d8b7f88915b3d2": "\n \\begin{align}\n I[f] & = \\int_{\\Omega} \\mathcal{L}(x_1, x_2, f, f_{,1}, f_{,2}, f_{,11}, f_{,12}, f_{,22},\n \\dots, f_{,22\\dots 2})\\, \\mathrm{d}\\mathbf{x} \\\\\n & \\qquad \\quad\n f_{,i} := \\cfrac{\\partial f}{\\partial x_i} \\; , \\quad\n f_{,ij} := \\cfrac{\\partial^2 f}{\\partial x_i\\partial x_j} \\; , \\;\\; \\dots\n \\end{align}\n", "22405e2588aeae52551d82e3357cd205": "\nV_W = \\frac{1}{3}(RA+LA+LL)\n", "22409d008af3f0f56c6bf526efb1a7ac": "\\,X(s)", "2240a3a2f59af5a5ca1daaa0b5609aa7": "\\begin{array}{lcl}\nb'_i x_i + c'_i x_{i + 1} = d'_i \\qquad &;& \\ i = 1, \\ldots, n - 1 \\\\\nb'_n x_n = d'_n \\qquad &;& \\ i = n. \\\\\n\\end{array}\n\\,", "2240ce32f133de4aea4a324fae607b6f": "\\text{SSMD}= \\frac{X_i - \\bar{X}_N}{s_N \\sqrt{2(n_N-1)/K}},", "224113b8b6965461c6778a9ac583ac49": "C^\\infty_N(\\mathbb{R}^n)", "2241516c7bc19c2347a15f953aa0ccbb": "H \\ge T^{\\frac{27}{82}+\\varepsilon}", "224159ef585a875eb3fe1e83992d7b54": "g\\otimes t^n", "22417f146ced89939510e270d4201b28": "\\frac{1}{5}", "22419ac61847201a249ca1f20b856584": "\\lambda_o\\,", "2241a76302b85300767fddee76c569ff": "c = 55000", "2241b0b2f93a790ac4f2be9c3817d86b": "x = 1.61633... ", "2241d424233d02badfaf8419e5b7cc99": " m\\ll M \\ll m_{0}+t ", "2241e369c36cd20ae3c19afcd2046949": "h(t) = \\frac{f(t)}{R(t)} = \\frac{\\lambda e^{-\\lambda t}}{e^{-\\lambda t}} = \\lambda .", "2242813bbb729d8c0d26394043414885": "\\pi_{\\ast}^S=\\bigoplus_{k\\ge 0}\\pi_k^S", "2242b62a9f3cc176a289724bf656bb71": "\n2 c^{2} \\frac{d\\tau}{dq} \\delta \\frac{d\\tau}{dq} = \n- 2 r^{2} \\frac{d\\varphi}{dq} \\delta \\frac{d\\varphi}{dq} \n\\,.", "2242bbc8a52f22e15b530681bf33cc5e": "\\langle x_i \\rangle = \\operatorname{E}(x_i)", "2242e31e940d691621bc4180904b3fa0": "^{\\;}H(\\xi )=H(\\star q(\\xi ,\\tau ))", "2242e3a216462f3e09b1b7ab646fb358": "\\varphi(y)=a_0\\cos\\frac{\\pi y}{2}+a_1\\cos 3\\frac{\\pi y}{2}+a_2\\cos5\\frac{\\pi y}{2}+\\cdots.", "2242f747e29686a564b1d02dad55a501": "(s,0)", "224329cbd2e73a03613eae039e07ca19": "\\Delta U = Q - W \\, . ", "22438d0bd5a3515a3417a4f1bbc49a33": " \\omega(P) ", "2243b8e5a4963aea64fe054e030a37bc": "(\\pi,V)", "2244052c5e8b25a58c0fd1624c90e3c5": "\\begin{align}\n \\mathrm{Area}(r) &{}= \\int_0^{r} 2 \\pi t \\, dt \\\\\n &{}= \\left[ (2\\pi) \\frac{t^2}{2} \\right]_{t=0}^{r}\\\\\n &{}= \\pi r^2.\n\\end{align} ", "22440e3b3c7bb02fa840ffc896c293f9": "u(x)\\ge u(y)", "22441de56f68f79e82d5fd259aa0be1a": "U^{(k-1)}\\mathbf{x}_w^{(k-1)} = \\mathbf{x}^{(k-1)}", "224461e6e20eeee4e93682a56ca9ed1f": "\\alpha(x_1,\\ldots,x_n):=(\\alpha x_1, \\ldots, \\alpha x_n)", "224497d30d971b29ae596ffb2eb66c32": " \\left(-\\sqrt{2}, \\sqrt{2}\\right) \\cap \\mathbf{Q} = \\left\\{ x \\in \\mathbf{Q} : x^2 \\le 2 \\right\\} \\, ", "22449d6ad757b0c2d9357ef8dfb82aa1": "u= \\,", "2244af6c06adde0b8f2bbabf8fbbd7dd": "\n(1)\\cfrac{\n (2)\\cfrac{\n (1)\\cfrac{C_1 (1,3)\\qquad {\\color{red} {C_8}^*}}{C_3 (2,3,5)}\n \\qquad\n C_4 (1,-2)\n }\n {C_7 (1,3,5)}\n \\qquad\n (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{C_8 (-1)}\n}\n{\nC_9 (3,5)\n}\n", "2244f86344bb8e0e36e2d71ddda5dc1d": "\\frac{1}{4 \\pi} \\,", "22450705f389f291b9067609a71e524c": "\\chi(M \\# N) = \\chi(M) + \\chi(N) - 2.\\,", "22451bea04a5f3aa2064a2332894c757": "\n\\mathbf{p} \\times \\mathbf{\\epsilon^1}(\\mathbf{p})=-i p_0\n\\mathbf{\\epsilon^1}(\\mathbf{p}), ", "22457e91c28c2b794eddafacfef67e6b": "\\operatorname{Var}(\\theta) =\\sigma^2 =\\frac{n-1}{n} \\sum_{i=1}^n (\\bar{\\theta}_i - \\bar{\\theta}_\\mathrm{Jack})^2", "2245afcd303302e719ebacfba93f012a": "\\left\\{ y~\\backepsilon~x\\succ y\\right\\}", "22463d9c1bc15c257b2f293199a5d33f": "\\hat{y} = {\\arg\\min}_{l \\in \\mathbf{Y}} \\|\\vec{\\mu}_l - \\vec{x}\\|", "224699c6511b7daa5bef4e8592df21e8": "b_2 = V_2^+", "2246f6c1e31a1a1d205ca85f2865980c": "\\displaystyle{\\|F_n\\|^2_2=2^n n!\\sqrt{\\pi},}", "22474115ab11dc153654afa1ebf9470a": "\\cos(\\theta-\\alpha)=\\cos(\\theta)\\cos(\\alpha)+\\sin(\\theta)\\sin(\\alpha)", "2247880f9af2adc7d58c0347b06f0e1e": "\\lambda_B \\,", "22478ae47519f744dc6486756c4f0fc6": " \\,Z ", "2247dcb235fa744046fbb98801b82f69": "\n\\text{average queue (}Q_\\text{avg}\\text{)} = \\frac{\\text{total delay experienced by }m\\text{ vehicles}} {\\text{duration of congestion}} = \\frac{TD} {(t_2-t_1)}\n\n", "2247f35d207dd0f781fe9ab1e72aebeb": "\nx = \\frac{A_k \\zeta_{k+1} + A_{k-1}}{B_k \\zeta_{k+1} + B_{k-1}}\\,\n", "2247fa6106d68a9de4fc57932e875f0c": "\\frac{d^2 x^\\alpha}{{d \\tau}^2} = - \\Gamma^\\alpha_{\\beta \\gamma} \\frac{d x^\\beta}{d \\tau} \\frac{d x^\\gamma}{d \\tau} \\,.", "2248836a69373c04ae26d5ffed2f3c04": "\\mathfrak{a}_0=i\\mathfrak{a}", "224897321fbf64a276e08322476804a7": "\\Delta f = f_x \\Delta x + f_y \\Delta y + \\cdots", "224902b3cdbcd914cbe6eedcbfcdc53a": "\nF(\\rho_1,\\rho_2) = \\left[ \\mbox{tr}( \\sqrt{ \\sqrt{\\rho_1}\\rho_2\\sqrt{\\rho_1}})\\right]^2\n", "224974d4a0bc74dffaa3b99e64dd2968": "\\left\\langle\\mu_z\\right\\rangle = {1 \\over Z B} \\partial_\\beta Z.", "224988b42bde97f9049e354e4730d05e": "\\mathrm{Ass}_R(R/J)\\,", "224a052c1b83e5445af51c14a293aba6": "3 \\times 4", "224a371dff3f0e6e0893041fb23ded0c": "D_1 - D_2", "224a6a92aa2c5776682a8555a7ca3901": "\\alpha S \\subseteq S", "224a9787a32d8d94a16b1cce4f9f1c62": "\nf(x) = \\begin{cases}\ne^{-1/x^2}&\\mathrm{if}\\ x\\not=0\\\\\n0&\\mathrm{if}\\ x=0\n\\end{cases}\n", "224ab7ca239f7f9f114abb12aa3f898f": "\\frac{\\partial \\left( \\overline{u_i} + u_i' \\right)}{\\partial x_i} = 0,", "224add98dbb4102205b7cc52696c238c": " \\mathbf{J}_{\\mathrm{P}} = \\frac{\\partial \\mathbf{P}}{\\partial t}\\ ,", "224ae3da68337fd43644c9708b5a7acb": "M < N", "224b228211378c9cbb2344a81a87c5ea": "C_{i,m} = \\left(\\frac{\\partial C}{\\partial n}\\right)", "224b7987439a3ad6bc0abb106ec15852": "\\text{id}_C \\colon C\\nrightarrow C", "224ba64bbd16cef44085c714ff69b794": "\\theta_1=0", "224ba99b3fb5b904b04eb512a3d52e03": "T = D (D^T D)^{-\\frac{1}{2}},", "224bb16ec7e44b24f3dc762016e822c6": "R_{12}(w) = \\phi_{12}(R(w))", "224bc68d103acb944486dd1c8702cea7": "T \\simeq 70 / r", "224c150f515fa3d1d21ca37d82d46b0c": "y = a_0 \\sum_{r = 0}^\\infty \\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r^2} x^{r + c}.", "224c1aa2b458ad9eede00fba40a12134": "=\\left(\\frac{1}{2} + \\frac{1}{3}\\right)", "224c588642db365ab53301bb2ee04dda": "f:D^{n}\\times {\\mathbb R}\\rightarrow{\\mathbb R}", "224c9a51d32b92072e4093dd32ee9cf8": "Y \\to S", "224ce0466f008fcbad03857213dc2c11": "\\sqrt{2m V_0}a/\\hbar=7", "224cf105fd3bf7e30d03121de57dab1b": "x(i)=y(idx(i))", "224cfcb593539a162be37acc415ec8e4": " \\sum_{n=1}^\\infty \\frac{1}{n} z^n,", "224d5a7b9cb5a86c63024620d6ac8277": "H_\\nu=\\frac{1}{2}\\int^1_{-1}\\mu I_\\nu d\\mu = \\frac{b}{3}", "224d6adb76f909cef9aaa12c03cb0dbd": " \\mathbf{n} \\cdot (\\mathbf{r} - \\mathbf{r}_0) = 0 ", "224d9526d1f440f6ddb470df2eccf341": "L(s,a)=\\prod_p\\biggl(1-\\frac{a(p)}{p^s}\\biggr)^{-1}.", "224e576f5ed099b14809be4616955a7c": "O(l^2) = O(\\log^2q)", "224e892053e28d2967da28f94d54eee0": "V_{\\Delta} = \\dfrac {V_{ref}}{R C} \\dfrac {1}{f_{clk}}", "224ebd77737b731bf34d0b36a76c0249": "2^{\\frac{n}{2}}.", "224ef0d69ea56ed4dc5505b15ff790bd": "\n q(x,y) = \\sum_{m=1}^{\\infty} \\sum_{n=1}^\\infty a_{mn}\\sin\\frac{m \\pi x}{a}\\sin\\frac{n \\pi y}{b} \n", "224f234655d55a9a2091d13bcc581c50": "Q_2 = (13)\\mathbf Z[i] + (i - 5)\\mathbf Z[i] = \\cdots = (2-3i)\\mathbf Z[i].", "224f2c3316f24cf76167e7fae02054b3": "\\mathbf{x}+ \\Delta \\mathbf{x}\\,\\!", "224f85acfae409c455e7fe1cf1c13721": "Td_1 = c_1/2", "224f8bf8f60107d9e89597c5afbcf8f3": "\\mathbf F=\\langle F,R,V\\rangle", "224fc36d0645f715ffd53f036eab84f2": "R = 1 - P(Z)", "224fe56c808f87dc2fb5a8dec91d2b97": " \\lambda = \\lambda^*+\\nu\\,\\!", "2250a0e39f8812c9ec22990659f46ee9": "\\mu\\in\\mathcal{P}(\\mathbb{R}^k)", "2250c692208e619106a587c2b8f9d11e": "\\phi-\\psi\\,\\!", "2250f6776f153eca535f89a5a0a42846": "w_i^{(t)} = \\frac{1}{\\big|y_i - X_i \\boldsymbol \\beta ^{(t)} \\big|}.", "2251256882201348283b1eb49dd7afaa": "w/r", "22512f6fe88d2a74f016c63a1674e4d2": "\\left(\\Gamma A \\frac{\\partial \\phi}{\\partial x}\\right)_{r} - \\left(\\Gamma A \\frac{\\partial \\phi}{\\partial x}\\right)_{l}", "2251d571f667db67d95733219e7b860b": "s_i>0", "2252270c2bcb6ac92270bdad39dcd3bd": "P(G|T) = \\frac{P(T|G) P(G)}{P(T)} = \\frac{0.5 \\times 0.4}{0.8} = 0.25.", "22526c8adc68ebc06217902d4953d382": "q=e^{i \\lambda}", "2252921b4e948fa1887c763a38683576": "(5)\\; h_j=\\frac{y_1 \\sqrt{1+8 F r_1^2} - 3 y_1}{2}", "2252b7e7efc31bb535d1a0d6a2410b52": "\\left( \\frac{2}{3} \\cdot 3\\right) + \\left(\\frac{1}{3} \\cdot 5\\right) = \\frac{11}{3}", "225307b866e46c96a58bd7cf8180a612": "N_R\\equiv\\frac{r}{\\sqrt{L\\cdot{l_p/3}}}<1", "2253167e685877214053c59fdd4d7382": "P_r=\\frac{\\eta C_p}{M\\kappa}.", "22533fd98da46c0a90133768edfd4492": "A = B = 2000", "22538fec3670ec24af4efef1cc5bffae": " \\lim_{\\varepsilon \\to 0}\\int_\\varepsilon^1 x^{-1} \\cos\\left(x^{-1} \\log x\\right)\\,dx", "2253a8014cfc390789d836412bcb5475": "\\mathrm{\\Lambda}(A \\otimes B) = \\mathrm{Co}(\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A \\otimes B}\\right) ).\n", "2253caead7a72c8416ce0f8161c14145": "\\scriptstyle \\mathcal{N} ", "2253ce8932d0ba3c7946661746d7d699": "P _f ", "22540e32376525bceef75d2743196324": "*,", "22542cbebe63b6383f14ae041a9a7866": "C{out}", "22544ac13fbd4046fc510862b025da5b": "q^n/(1 - q^n ) = \\mathrm{Li}_0(q^{n})", "22545c621dbc9000b8d1f449365ace27": " L(\\theta) = \\prod_{T_i\\in unc.} \\Pr(T = T_i|\\theta)\n \\prod_{i\\in l.c.} \\Pr(T < T_i|\\theta)\n \\prod_{i\\in r.c.} \\Pr(T > T_i|\\theta)\n \\prod_{i\\in i.c.} \\Pr(T_{i,l} < T < T_{i,r}|\\theta) .", "2254fe0371bfd7074b154ddde44ba6c1": "\\epsilon = 0 ", "2255a1c588d3b537bb8d35e7811b1632": "v_s = \\sqrt{\\frac{\\gamma_{e}ZK_{B}T_e+\\gamma_{i}K_{B}T_i}{M}}", "2255ba3951bc9fae71a480cc3a1bb66c": "\\scriptstyle\\mathbf J", "2255cf3008297d509206eaaecdc9a1da": " x, y \\in \\Omega ", "2255eb650798c675d09495ec6b616e33": "100k = \\frac{v_{eff}}{d} + \\frac{9}{d} \\sqrt{\\frac{v_{eff}}{d}} =\\frac{v_{eff}}{d}\\big( 1+\\frac{9}{v_{eff}} \\sqrt{\\frac{v_{eff}}{d}} \\big)", "2256043da6e4d9df4b48915eabb39337": "k = \\frac{h}{r} = \\frac{\\operatorname d h}{\\operatorname d r}.", "225630f24ffd3ca9be1dd23456778558": "E_{el}=\\frac{24\\pi\\,\\Kappa\\,\\mu\\,_0 r_0 (r_1-r_0)^2}{3\\Kappa\\,+4\\mu\\,_0}", "22564b919d5fcdd4896d5437ee1d43b4": "f''(N) < 0 \\,", "2256bb17f67bc890ddfec8a7fd270535": " \\displaystyle{\\Phi_\\lambda = M_1 \\phi_\\lambda.}", "225756910055d7867b2d45b47a481caa": "\\pi_\\Sigma (L)=\\{\\pi_\\Sigma(s) \\vert s\\in L \\}", "225792d88332053454a4a10c96dcae61": "\\frac{a}{0} = \\infty", "2257e4e8eb68c4ddf72db66dfe52658b": "\\theta\\circ(1,\\ldots,1)=\\theta=1\\circ\\theta", "2257ff2235ca04785756d0137d13cf5f": "\nw \\equiv z \\sqrt{e_{1} - e_{3}}.\n", "2258c437b0b70a2925ca396b03d5010f": "(V, E-e, A)", "225912916785fe763e001ff218d0f63d": "i^i = \\left( 1^0 e^{-\\frac{1}{2}\\pi} \\right) e^{i \\left[1 \\cdot \\log 1 + 0 \\cdot \\frac{1}{2}\\pi \\right]} = e^{-\\frac{1}{2}\\pi} \\approx 0.2079", "225926ed24143d50534540d2d2cdee44": "R_{NP}=\\left\\{ x: \\frac{L(\\theta_{0}|x)}{L(\\theta_{1}|x)} \\leq \\eta\\right\\} .", "22594c3b127adc1cfe9c82c889df929f": "f\\colon D \\to \\mathbb{R}", "2259b43b2ec201307b5f231bd3258b68": "\\begin{align}\n& p = \\hbar k\\\\\n& E = \\hbar \\omega\\\\\n\\end{align}", "2259c52ba1edc33f7e7349889e40b857": "\\; |\\Psi_{A_1\\ldots A_m}\\rangle = \\sum_{i=1}^{min\\{d_{A_1},\\ldots,d_{A_m}\\}}a_i |e_{A_1}^i\\rangle \\otimes \\ldots \\otimes |e_{A_m}^i\\rangle", "2259d58c43891bb24335eb4f73bfd05a": "\\Phi(x,t)=\\frac{1}{\\sqrt{4\\pi kt}}\\exp\\left(-\\frac{x^2}{4kt}\\right).", "225a1c27174d3bfd2631cdbbf3e65efe": "f_{*}^{}\\colon T_x M \\to T_{f(x)} N", "225a4f2d192ccb7c1215d7a1072d1719": " { \\mathit l \\ne \\mathit l^{\\prime} } ", "225a709ac745169be9dbc703ccd33e65": "\\beta_k = - \\frac{\\mathbf{r}_{k+1}^\\mathrm{T} A \\mathbf{p}_k}{\\mathbf{p}_k^\\mathrm{T} A \\mathbf{p}_k}", "225a811a16e48bf7fb883f64202ec7d2": " \\delta u_{i + \\frac{3}{2} } = \\left( u_{i+2} - u_{i+1} \\right) , \n \\delta u_{i - \\frac{3}{2} } = \\left( u_{i-1} - u_{i-2} \\right),", "225a8724d43101211ddb2f25a1ed7f73": "\nT = \\frac{1}{2} \\boldsymbol\\omega \\cdot \\mathbf{L}.\n", "225abb6e83bbf56ed833bab4d11be026": " { (12L - 3kn(n+1)^2)^2 \\over kn^2(n^2 - 1)(n + 1) } ", "225acff7e527bcebbdcf305fae426d0f": "x\\in [x_i,x_{i+1}]", "225b215449daf89f2bc292812f3703a6": "\\psi\\rightarrow e^{iq\\theta}\\psi", "225b9a82c2424f66c3db040f14e92ec6": "\\frac{a}{c}", "225bf11d6a263581c98de34642c8d1fd": "(d,q)", "225c563bd7f7a9644aaa9021873c8d45": "(200M+200M)/100M = 4", "225cd05bd866eff662fa8a97f4ba70da": "\\begin{align}\nx_{\\mathrm{square}}(t) & {} = \\frac{4}{\\pi} \\sum_{k=1}^\\infty {\\sin{\\left (2\\pi (2k-1) ft \\right )}\\over(2k-1)} \\\\ \n & {} = \\frac{4}{\\pi}\\left (\\sin(2\\pi ft) + {1\\over3}\\sin(6\\pi ft) + {1\\over5}\\sin(10\\pi ft) + \\cdots\\right )\n\\end{align}", "225cdb526fd9f98f5660dcf440b03565": "\\beta^n_{f}", "225cddcb96716c76a48be11d17377772": "p_i = \\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}- bc, \\;i=1,2,...,n", "225d63108bc58fbf3320d4c7b9471488": "\\mathfrak{n} = [\\mathfrak{b},\\mathfrak{b}].", "225d8a6795640a069a67052964fd7d36": " \\lambda(s,t_s)=y", "225da8f9a9e045e43fb37e3402a54b1a": "U_{k,n}", "225e47a9fafc21d09c8113152a250c09": "\\frac{\\mathrm{d}U}{\\mathrm{d}t} = \\Sigma_k \\dot Q_k + \\Sigma_k \\dot H_k - \\Sigma_k p_k\\frac{\\mathrm{d}V_k}{\\mathrm{d}t}-P,", "225e5224aeab194b26597c3c71f63a8e": "I^\\pm(x)", "225e786eb1ece17cb97f9bb4d6f99559": "g=\\prod_{i=1}^k p_i^{n_i}.", "225edcd0c2eac275c6296fe6b3a8687d": "g_1(\\mathbf{x},z_1) \\neq 0", "225f3a649cf01bb9595127c3d63d14ed": "R = \\frac {\\pi \\times 12.5} {250} \\approx 15.7%", "225fa9b9bb06d25a08e4f28d5ca2e796": " p^n = 257^{64}", "225fd69130a867ea4c239e1d6784b62b": "1 \\over 81", "22600f64559c7f8bac880cf2e515a510": "Nu = \\left[Nu_0^\\frac{1}{2} + Ra^ \\frac{1}{6} \\left(\\frac {f_4\\left(Pr\\right)}{300}\\right)^\\frac{1}{6} \\right]^2 ", "226032b68ca57cab15ad578d0d829b13": " A \\ast B = A + B + \\tfrac12 [A,B] + \\tfrac{1}{12} [A,[A,B]] - \\tfrac{1}{12} [B,[A,B]] + \\cdots ~.", "22605460414d12fd3dc7e55651e47a7c": "\\sqrt{N} \\sigma", "226087a515dcb143035c31c67c62f839": "f(t)=\\sin(\\ln(t)),\\; t>0", "22608ba7a68432367466a1be451b81bb": "g_1,\\,g_2\\ldots\\,g_n", "226154735e81aae3518edb0f0325adaf": "\\bigoplus^k_{i=1} A=\\N.", "2261952b6cfe72cac2524f034717f079": "\\pi\\,\\hat{=}\\,\\alpha+\\bar{\\beta} \\,,\\quad \\varepsilon\\,\\hat{=}\\,\\bar{\\varepsilon} \\,,\\quad \\bar{\\mu}\\,\\hat{=}\\,\\mu\\,. ", "22619593c0fbc1e140dcb640797c3864": " I_{SN} ", "2261c7733c26e6528897e54351fba28f": "\\displaystyle{L_n=-\\pi\\left(i e^{in\\theta}{d\\over d\\theta}\\right)}", "2261f7f97154a49e8f3cb88b3fe36a1f": " \\int_A f(x) \\, d\\mu \\,\\!", "22622153c9b67a263ee317eff42fdc66": "\\mathcal{C}^\\mathbf{2}", "2262516b6e53309bd21d5ef437e42add": "v=v_N", "22626a8fcb1e214bb21713aaddfce9d3": "B_{n-1}", "22629ab7e8c584608cf26d2d7d4f1bd1": " \\widehat{\\Omega} = \\widehat{\\Omega}^\\dagger ", "2262c3350a6b3901ba193a5df2002184": " \\mu (A) >0\\, ", "2262d6762e030e102c2ae0d8069ea924": "n^2 / 4", "2262dad4d16e5fe0c9acb45b3ad0a340": "\\displaystyle{f_i(x) = e^{-x^2/2},}", "2263734592a53171cd4c3698ec6861d9": "F(t) = E_t\\left\\{S(T)\\right\\} ", "226392170cb6694015de6e52b4439687": "\\frac{dy}{dx}=y(1-y).", "2263993744d371b7781cb26a879af2f5": " P(P(X_1^n(i')) > P(X_1^n(i))) \\, ", "226410410a2cf571745c0ceee7e2db66": "f(x,n) \\geq k", "2264124205ac059db59fdef14bd6b4ad": "W_{21}", "22643d03eebf368665338b9135e0b354": "\\frac{\\mu_i}{k_0}p_0=p_i", "226485b2a4d52c54299bb4d81807acff": "E[X;P] = \\int_{a=-\\infty}^{+\\infty} a\\,P(X\\in[a;a+da]) ", "2264869f1f8f3e104112b5733112fce1": "\\tanh \\frac{\\hbar \\Omega(n) \\beta} {2}= 4 \\frac{\\omega}{\\Omega}\\sqrt{n+1}", "2264abf871c877cb8ff7fbc20badfbf1": "y \\in [0,1/2), x \\in [1/2,1]", "2264cea93c5894cfe2028f03999ff4a5": "\\alpha(L_n)=L_n +{1\\over 2} J_n + {c\\over 24}\\delta_{n,0}", "2264e8fc1e460956e73d941e58a03bf5": "f_n(z)", "2265081cfabbbdfb0fe32b2470788b3a": "\\frac{c(a_1+n)\\dots(a_p+n)}{d(b_1+n)\\dots(b_q+n)(1+n)}", "22653600bd600eeaea111eec9e3dbea9": "B+B'", "2265ea919196a64603cce82ab29cb0dc": "f = \\frac{P(z)}{Q(z)}", "2265f94210d0bb3ed52a92bf469753a9": " e^Te = (My)^T(My) = y^TM^TMy = y^TMMy = y^TMy. \\, ", "2266859dfa2a82c894311a81fa37c0a5": "\\sum_{n=0}^{\\infty}\\tbinom{n+k}k x^n={1\\over(1-x)^{k+1}}.", "22669228e8b758bacf747d8edccb2cb9": "=\\frac{y''(s)}{x'(s)} = -\\frac{x''(s)}{y'(s)} \\ ,", "2266b19beb55aedf4eeabc06ecff01d6": "\\textstyle \\nu(\\varepsilon)", "2266d496aee9eeafa9fac38168432991": " g_0 = \\frac {-K_P K_V K_C K_M } {s^3 L_M M} \\, ", "2266f958d5b6d1a5e70c49bc6cbfd749": "\\Leftrightarrow \\operatorname{gl.dim}R < \\infty \\Leftrightarrow \\operatorname{gl.dim}R = \\dim R.", "2267285faeaaab31fdef7c2076a7a2fa": "C_2 = 138 \\ \\mathrm{pF}\\,", "22673c3a677cc01c9657579ad724d73c": "\\sum_{j \\in \\mathbf{S}}p_{ij}V(j) \\leq V(i) - 1", "22674641d6c678893d803618d7c44f7f": "Q[\\mathcal{L}(x)]=\\partial_\\mu f^\\mu (x)", "2267d584dc5a275b5e2759d0e2ee6ec7": "\nz \\le \\frac{\\mu}{\\mu+x}\n", "22681addb9f61e033b57a4e9a241f269": "((\\and_{\\epsilon < \\delta}{(A_{\\delta} \\implies A_{\\epsilon})}) \\implies (A_{\\delta} \\implies \\and_{\\epsilon < \\delta}{A_{\\epsilon}}))", "226823a43503af76deed62f3dae9aef3": " \\overline{x} = \\frac{x_1+\\cdots+x_n}{n} ", "22683886b351a6d6e86df531e6f92511": "p\\gamma = \\ell\\pi\\,", "226849c3c1225387d930ec1d7f68843c": "\nx = a \\sqrt{\\left( \\sigma^{2} - 1 \\right) \\left(1 - \\tau^{2} \\right)} \\cos \\phi\n", "226882a5358b2995c80cc5c52caf5cb9": "\\displaystyle{A=(T+I)(T-I)^{-1}.}", "2268a4ebb90af21d37c61d23669b55bb": " \\sum_{ij;i \\ne j} \\psi^*_i \\psi_j \\phi^*_j\\phi_i ", "2268b534827bf910c7adf0adc132dfcd": "=\\Pr[B]\\Pr[A|B]+\\Pr[B^c]\\Pr[A|B^c]", "226901b0a97afaa088dc0d1b03c8544b": "T_{(0,0)} M_1 = \\mathbb{R} \\times \\{0\\}", "22694bd165f4573e4de384414870ba22": "D=\\sum_{i=1}^r P_i-r\\infty_2", "22697bea86be9f5fa4bd1c7a01057a3b": " \\lim_{y_0 \\to 0} \\arccos \\sqrt{ \\frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \\frac {\\pi} {4} ", "22698d55e75769bbf77c6a4740784ace": "F_l(x) = \\frac{1}{\\mu} \\int_0^x \\left(1-F(u)\\right) \\text{d}u.", "2269b7011477c8198ccce0be6831621b": "\\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix}", "2269ce8576a07bf1200c38d76b9583b3": "i_1 \\leq i_2 \\leq ... \\leq i_n", "2269df51c696d2c936a8f39d437a8b6e": "\\begin{matrix}abc \\\\ a/bc \\\\ b/ac \\\\ c/ab \\\\ a/b/c \\end{matrix}", "226a3611c04c0195c3eeefeea9874979": "\\lim_{x \\to -\\infty} a^x = 0", "226a3a8ab5a08cf7e07a32f7934e295a": "y \\gg x", "226a6fa4e14c89a07f20590279625b22": "\\hat p \\pm z^* \\sqrt{ \\frac{\\hat p \\left ( 1- \\hat p \\right )}{n}}", "226a80326ba7b969338188a6eec43084": "x=\\pm 1.\\,", "226aae9407d7e4e227264578fee61f04": "1_X", "226b1835b5bc6785ca974568a3ebcaf1": "- \\left(\\frac{d t}{d x}\\right)^{-1} \\frac{d^2 t}{d x^2} = f(x)", "226b330635b5795c5a4c35d7adafb340": "H^*(M,\\mathbb K)", "226b4102c3a771a9eea412a5b0b2a4b7": "S : X \\rightarrow D", "226b4a6dd11c3e3a357ba20363ff2eca": "g_{n,k}(r)\\text{ and }f_{n,k}(r)", "226b7338fedfbeffa9c3f1704ad8ca1e": "\\text{Level 1:} \\ \\ 266 = 2 + 2 + \\dots + 2 \\ \\ \\text{(with 133 2s)}", "226ba5c7020271363249ed79de8b28aa": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{1-n})}", "226bf0c0a9ff13ad1ba07d51f7146ad6": " \\int_{-\\infty}^\\infty p dp, ", "226bf15cf2f1805a0e75266274ca7bb4": "\\mathbf{a}\\otimes\\mathbf{b}=a_i\\mathbf{e}_i\\otimes b_j\\mathbf{e}_j=a_i b_j\\mathbf{e}_i\\otimes\\mathbf{e}_j", "226bf73809883928549b784b38b05ab1": "det \\; q^{(2)} = {\\epsilon^{3ab} \\epsilon^{3cd} q_{ac} q_{bc} \\over 2}", "226c04efecfa7d5b71fd9f7e577074ed": "\\sum v(t) \\times \\Delta t", "226c282e3406665df3099cc7846c4e4d": "{\\tilde{A}}_{2n}", "226c5c2d4f3900230fe868998f914776": "\\langle a b \\rangle_2 = a\\wedge b\\,", "226c82c7c67baf94f37b8665e6e44855": "i = \\{1 \\dots k\\}", "226ca979c6ec4b377cf65fbbc9cfcb98": "s=\\sin\\theta", "226ced0d1f1d6a992ab8d73a50553199": "{}^q\\!D={1 \\over \\sqrt[q-1]{{\\sum_{i=1}^R p_i p_i^{q-1}}}}", "226d14875dbf4bd2bc5ad4e186a9ea8f": "B = \\mu H + \\chi \\sqrt{\\varepsilon \\mu} E", "226d6149db2d22ff1f3fe7f8de4335d5": "\\rho:\\pi_1B\\rightarrow Homeo\\left(F\\right)", "226d81cd7e156aa12d7cf723ec95b732": "\n\\begin{align}\n \\mathbf{F} & = \\left(\\mathbf{p}\\cdot\\nabla\\right)\\mathbf{E}+\\frac{d\\mathbf{p}}{dt}\\times\\mathbf{B} \\\\\n & = \\alpha\\left[\\left(\\mathbf{E}\\cdot\\nabla\\right)\\mathbf{E}+\\frac{d\\mathbf{E}}{dt}\\times\\mathbf{B}\\right], \\\\\n\\end{align}\n", "226dad79459a4965dc74922b2c8806f9": "\\psi_1(\\alpha) = \\frac{d^2\\ln\\Gamma(\\alpha)}{\\partial\\alpha^2}=\\, \\frac{\\partial \\psi(\\alpha)}{\\partial\\alpha}", "226dc2c78e429964a36315c25003b36b": "\\Delta p = 0", "226ddae13c247a83e462106a4b91a3f2": "\\text{IE}_{\\text{M}}", "226e4e424a35e70a3b73010c8332f8b2": "(-\\infty,+\\infty)", "226e4e7ddbaeaaa707ecec449b3643df": " K = \\sqrt {(s-a)(s-b)(s-c)(s-d) - abcd \\cdot \\cos^2 \\left(\\frac{\\alpha + \\gamma}{2}\\right)},", "226e72f3c8524592b874e5608bfbd47e": "f(1) = e", "226e7afd0e1a288e6dc4629fd3ca157a": " \\left\\vert C \\cup D \\right\\vert + \\left\\vert C \\cap D \\right\\vert = \\left\\vert C \\right\\vert + \\left\\vert D \\right\\vert \\,.", "226e9249ecc997c2861fa3fcd71276fe": "\\,\\overline{A}_x\\! = E[v^T] = \\int_0^\\infty v^t f_T(t)\\,dt = \\int_0^\\infty v^t\\,_tp_x\\mu_{x+t}\\,dt,", "226eade4534836ddc41cfa424762e203": "\n\\overline{C}_1,\\dots,\\overline{C}_{m}\n", "226f41bebdc17c042b30b9f30a446172": "s \\mid_p", "226f88dfa19b49c359e6a8b05a4b95b5": "\\mathcal{I}(\\alpha)=\\operatorname{E} \\left [\\left (\\frac{\\partial}{\\partial\\alpha} \\ln \\mathcal{L}(\\alpha|X) \\right )^2 \\right],", "226f8c7d3fd00a687c4898cea554f1f0": "|p_z,i+1_{x,y};\\downarrow\\rangle", "226facaa77bdc83dd7a591b81eab77c7": "P \\Lambda = \\frac { \\sum_{g=1}^G { \\left ( {\\frac{A_g}{\\overline A}}-1 \\right ) ^2}}{G} ", "226fde4c9564579c4397f1b5b9d101f1": "l=gm,\\quad k=nf,\\quad h=j[1]i,\\quad ig=u[1]n,\\quad fj=mv.", "226ff57c061c7c2d84c5c79cc3b5800c": "Var(\\epsilon) = \\sigma^2", "227063a0613b40bee6f0fdc8000e546f": "\\mathbf{a}\\odot\\mathbf{b}", "2270a2c8191552ee95b50f15ce1c8521": "\n \\begin{align}\n u_r = u_1 & = \\cfrac{F_1}{4\\pi\\mu}\\left[1 - (\\kappa+1)\\ln |x_1|\\right] \\\\\n u_\\theta = u_2 & = \\cfrac{F_2}{4\\pi\\mu}\\left[1 + (\\kappa+1)\\ln |x_1|\\right]\n \\end{align}\n ", "2270a978930a622d7f7f71ab6d8328e2": "\\text{Im}(b_\\lambda) \\cong\n\\bigwedge^{\\mu_1} V \\otimes \n\\bigwedge^{\\mu_2} V \\otimes \\cdots \\otimes\n\\bigwedge^{\\mu_k} V \n", "2270abb8d03f3f3dfd8d156106ffca77": "\\lambda^*(A) = \\inf \\Bigl\\{\\sum_{B\\in \\mathcal{C}}\\operatorname{vol}(B) : \\mathcal{C}\\text{ is a countable collection of boxes whose union covers }A\\Bigr\\} .", "2271023011b790d9e76063ed69f3d310": "\\sum_{i=1}^K N_i = N", "2271087912c89cf4a1598cf92424a0c4": "x_{marker} = {x_{marker\\_orig} \\over x_{orig}} \\cdot x_{scaled} - {marker\\_size \\over 2} + x_{adjust}", "227121311da87d0e3c70dbe4de52c297": " s = \\sqrt[3]{1880} \\approx 12+1/3 ", "2271277f219f0c9d043d20c52b347204": " \\mathbf{P}\\, = \\,(\\cos(s),\\, \\sin(s)) \\, \\Rightarrow \\, \\mathbf{F}\\, = \\,2\\sin(s)\\mathbf{i} + 5\\cos(s)\\mathbf{j} \\, .", "22717e1e758fd2673ed8df3c42fb1898": "(\\mathbb{C}P^n,\\Sigma,\\{U_{\\sigma_1},U_{\\sigma_2},\\dotsc,U_{\\sigma_p},\\})", "2271b96386d7bae7ff90ed544fbded4d": "nym", "2271e6168d7774017270b743b36505a6": "\n\\left(\\frac{2D^{\\mathrm{face}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} - \\left(1+\\frac{2D^{\\mathrm{face}}}{D^{\\mathrm{beam}}}\\right)\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} = \\frac{M}{D^{\\mathrm{beam}}}-\\cfrac{q}{S^{\\mathrm{core}}}\n", "227257860c77fa00f2f99cbd3394882c": "A = {1 \\over 2} A_0 e [e^{i(k_p \\cdot r - \\omega t)}+e^{-i(k_p \\cdot r - \\omega t)}]", "2272f2e6b87d10518445ebdd9145108c": "D = \\varepsilon E - i \\kappa \\sqrt{\\varepsilon \\mu} H", "227318fc4085a330ead6c7306364680e": "\\rho=2 \\pi a \\frac {V}{I} ", "2273248e0465f9d3b0ae75cc6309c3a2": "(Sx)_n=x_{n+1}", "227369ab78112212e471d8d14bcd256e": " \\Pr\\{(M_n-b_n)/a_n \\leq z\\} \\rightarrow G(z) \\propto \\exp \\left[-(1+\\zeta z)^{-1/\\zeta} \\right] ", "22737ab7c5a33e22fae42a02622594b7": "\\sum_{p=0}^{q-1}\\zeta(s,a+p/q)=q^s\\,\\zeta(s,qa) \\ .", "22739d947baf1fe8db6cd0af5b975188": "\\Gamma=+1", "2273a299c2452e3a37bb1bf05f9b544c": "\\mathrm{error}\\bigl(x(t_0 + n\\Delta t)\\bigr) = \\frac{n(n+1)}{2}\\,O(\\Delta t^4)", "2273cb4d87e3bb44fbd63a81fe86bd13": "O(b^{3d/4})", "227430bed634bfed57d72512c96579df": "X \\times X^*_{\\mathcal{G}} \\to \\mathcal{F}", "22743bf12516741749f26f3c5f4de552": "B=\\mu_0\\mu_r((1+\\chi_0)(H_{ext} + H_{exc}) + N_e (H_{ext} + H_{exc})^3)", "22745f534bd8eefc03d911ac4d0eaef2": " M_{ij} = \\frac{\\mu_0}{4\\pi} \\oint_{C_i}\\oint_{C_j} \\frac{\\mathbf{ds}_i\\cdot\\mathbf{ds}_j}{|\\mathbf{R}_{ij}|} ", "227467a6e2e9997bd58cac601460f305": "-m g \\ell \\theta=I \\alpha", "227518080d8aa94cba952ad8d8993a99": "D_n=\\langle x, y \\mid x^n = y^2 = (xy)^2 = 1 \\rangle.", "2275313af06aac1aa13e170975b089f9": "\\beta=-2 e^{i\\theta}-e^{-2 i\\theta}", "22754160dbb35f8af7a58dfa77019184": "r \\times n/w", "22759181021b36d7f0d67d9e51c255d3": "C\\subseteq N", "2276514e5ab82e283c11603081bbb011": "H_q", "2276c1144ab01cd3f25f4431378801da": "k_{o}^{2}=\\left ( \\frac{2\\pi }{\\lambda_{o}} \\right )^{2}=k_{xo}^{2}+k_{y}^{2}+k_{z}^{2}=- \\left | k_{xo} \\right | ^{2} +k_{y}^{2}+\\beta ^{2} \\ \\ \\ \\ (1) ", "227728bdb14346248f49f78a229f9a92": " FA=\\frac{\\sqrt{3( (\\lambda_1-\\mathbb E[\\lambda])^2+(\\lambda_2-\\mathbb E[\\lambda])^2+(\\lambda_3-\\mathbb E[\\lambda])^2 )}}{\\sqrt{2( \\lambda_1^2+\\lambda_2^2+\\lambda_3^2 )}} ", "2277301ed8f09dec845731b78c8de40e": "\n p_{\\mathrm{av}} =\\frac{F}{A}\\approx\\frac{1}{2}E^*h'\n ", "227734aacbe93673bc7690d5db079ffb": "\n\\begin{align}\n\\frac{\\Delta^2F(P_0)}{\\Delta_1P^2} & =\\frac{\\Delta F'(P_0)}{\\Delta_1P}=\\frac{\\frac{\\Delta F(P_1)}{\\Delta_1P}-\\frac{\\Delta F(P_0)}{\\Delta_1P}}{\\Delta_1P}, \\\\[10pt]\n& =\\frac{\\frac{F(P_2)-F(P_1)}{\\Delta_1P}-\\frac{F(P_1)-F(P_0)}{\\Delta_1P}}{\\Delta_1P}, \\\\[10pt]\n& =\\frac{F(P_2)-2F(P_1)+F(P_0)}{\\Delta_1P^2};\n\\end{align}\n", "22776e4755070319688ad67d2ba31792": "\\displaystyle f(x) = \\begin{cases}\n 0.5 x &\\text{for } x \\in Z, \\\\\n 0.5 + 0.5 x &\\text{for } x \\in (0,1) \\setminus Z\n\\end{cases} ", "2277d2411368b9b4f49069698fb3d2ca": "|ix|_B = B(ix,ix) = i^{2}B(x,x) = -|x|_B", "2277f1a42853d774c23fdbfe2b9c77c0": "\\partial^2 w/\\partial x^2 = 0", "227859f1a2a285796c8205901726c7b6": "y_i'(t)=f_i(t, y_1(t), y_2(t), ...)", "227943546052d74a24059ca3d69a49f2": "\\left[ \\nabla^2 - k_0^2 \\right] \\phi(r) = - \\frac{Q}{\\epsilon_0} \\delta(r)", "22794cb2b733cc8ebb9d611c61d681c1": "\\nabla\\times\\left(f(r)\\mathbf{Y}_{lm}\\right) =-\\frac{1}{r}f\\mathbf{\\Phi}_{lm}", "2279689f6a085fbee4812ab8d1d59f02": "x_{21}=p_2q_1-D", "2279a6aa7461805a50f4a48e95e98c50": "B_{\\text{op}} := \\{|\\omega_k\\rangle \\}_{k=0}^{N-1}", "2279b1c42b4353ba60defb0f1a3099e7": "[\\cdot]: A \\longrightarrow B", "2279ccee0f36d7513f6d5377fa63fc41": " a= 0~ mod~ p_1", "2279f3c6247904961abd5a25d1b935af": " \\lfloor x \\rfloor=\\max\\, \\{m\\in\\mathbb{Z}\\mid m\\le x\\},", "227a357c6842c0dcff5a79898f1919d7": "\\mu_{2,1}= \\frac{\\langle b_{2}, b_{1}^{*} \\rangle}{B_{1}}=\n\\frac{\\begin{bmatrix}4\\\\5\\\\4\\end{bmatrix} \\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}}{3}=\\frac{13}{3}(>\\frac{1}{2})", "227a37229b074c000759054e905d3373": "u_{n\\boldsymbol{k}}(\\boldsymbol{r})", "227a3b89d3ccabbeea3231109e2120df": " P(X < (1 - \\delta) \\mu) \\le e^{ \\frac{ -\\delta^2 \\mu }{ 2 + \\delta } }. ", "227a596608db91cda8b7badb9d00e318": "\\lim\\inf_n \\frac{f(x_n) - f(x)}{x_n - x} > \\lim\\sup_n \\frac{f(x'_n) - f(x)}{x'_n - x}.", "227a8acaef192b432d8a9c025dba9806": "\\mathcal{}BP_*/I_n", "227af28dda4008a745cdf602ff14759f": " i = p_1 ", "227b22b0b6222ff30e6a91a78770436a": "\\begin{array}{lcl}s_n & = & s_{n-1}^2 - 2 \\\\\n & = & \\left(\\omega^{2^{n-1}} + \\bar{\\omega}^{2^{n-1}}\\right)^2 - 2 \\\\\n & = & \\omega^{2^n} + \\bar{\\omega}^{2^n} + 2(\\omega\\bar{\\omega})^{2^{n-1}} - 2 \\\\\n & = & \\omega^{2^n} + \\bar{\\omega}^{2^n}, \\\\\n \\end{array}", "227ba3fd670437fe111640f3f14ad637": "\\Omega \\subseteq^d", "227c25f6f71eb0791cd1709f7d0ea089": "\\operatorname{dist}(t) = \\infty", "227cbbb1283b7e00e04c126a76868c41": "\\psi=\\begin{pmatrix}\\psi_{+} \\\\ \\psi_{-} \\end{pmatrix} = \\begin{pmatrix}\\psi_{+\\uparrow} \\\\ \\psi_{+\\downarrow} \\\\ \\psi_{-\\uparrow} \\\\ \\psi_{-\\downarrow} \\end{pmatrix} ", "227cc397230b4109597580cb85b55db8": "f(x)\\le_T f(y)", "227cca83ba68bff68e953b5e70fb8106": "S(Y)", "227ce333769088e212af5de5bbf00220": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}} \\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{f(g)}}{\\partial \\mathbf{g}}", "227d83c49d2731dfd04a990c284358a5": "G(\\mathbf M) ", "227dc4637de62614a5fe6ced410526af": "X' = \\frac{\\partial X}{\\partial \\sigma},", "227df9cde82b3f98fac8ed3d583aa95a": "(\\mathbb{Z}_N)^\\times", "227e005edb52415812f6f527a829ae19": "u_2(x)", "227e2321f71de511a85dd64c13a90dec": "\\begin{align}\np(\\mu,\\sigma^2|\\mathbf{X}) & \\propto p(\\mu,\\sigma^2) \\, p(\\mathbf{X}|\\mu,\\sigma^2) \\\\\n& \\propto (\\sigma^2)^{-(\\nu_0+3)/2} \\exp\\left[-\\frac{1}{2\\sigma^2}\\left(\\nu_0\\sigma_0^2 + n_0(\\mu-\\mu_0)^2\\right)\\right] {\\sigma^2}^{-n/2} \\exp\\left[-\\frac{1}{2\\sigma^2} \\left(S + n(\\bar{x} -\\mu)^2\\right)\\right] \\\\\n&= (\\sigma^2)^{-(\\nu_0+n+3)/2} \\exp\\left[-\\frac{1}{2\\sigma^2}\\left(\\nu_0\\sigma_0^2 + S + n_0(\\mu-\\mu_0)^2 + n(\\bar{x} -\\mu)^2\\right)\\right] \\\\\n&= (\\sigma^2)^{-(\\nu_0+n+3)/2} \\exp\\left[-\\frac{1}{2\\sigma^2}\\left(\\nu_0\\sigma_0^2 + S + \\frac{n_0 n}{n_0+n}(\\mu_0-\\bar{x})^2 + (n_0+n)\\left(\\mu-\\frac{n_0\\mu_0 + n\\bar{x}}{n_0 + n}\\right)^2\\right)\\right] \\\\\n& \\propto (\\sigma^2)^{-1/2} \\exp\\left[-\\frac{n_0+n}{2\\sigma^2}\\left(\\mu-\\frac{n_0\\mu_0 + n\\bar{x}}{n_0 + n}\\right)^2\\right] \\\\\n& \\quad\\times (\\sigma^2)^{-(\\nu_0/2+n/2+1)} \\exp\\left[-\\frac{1}{2\\sigma^2}\\left(\\nu_0\\sigma_0^2 + S + \\frac{n_0 n}{n_0+n}(\\mu_0-\\bar{x})^2\\right)\\right] \\\\\n& = \\mathcal{N}_{\\mu|\\sigma^2}\\left(\\frac{n_0\\mu_0 + n\\bar{x}}{n_0 + n}, \\frac{\\sigma^2}{n_0+n}\\right) \\cdot {\\rm IG}_{\\sigma^2}\\left(\\frac12(\\nu_0+n), \\frac12\\left(\\nu_0\\sigma_0^2 + S + \\frac{n_0 n}{n_0+n}(\\mu_0-\\bar{x})^2\\right)\\right).\n\\end{align}", "227e31dc9b5fba5b17e457149c943543": "\\sum_{k=1}^n k = \\frac{n(n+1)}{2}", "227e569fcfd998e36f71dc6add69e4f4": "\\Pr[A] \\cdot \\Pr\\left[\\overline{A_t}\\right] \\le e^{-t^2/4} \\, ,", "227e5999bd72ae167f9df0a54e2dae93": "\\boldsymbol{\\chi} + \\mathbf{T}(x){\\chi}", "227e71e02805ad1a3af1a39dcfb114de": "S_{\\mathrm{sat}} = \\frac{78}{H_{\\mathrm{sat}}},", "227e9e6ea96659f752771b4ec095b788": "\\frac{\\sqrt{3}}{3}", "227ec2f6235ca10a60b3dbdcf2d6a2d6": "d\\nu(x) = \\frac{1}{2\\pi \\sigma^2 } \\frac{\\sqrt{(\\lambda_{+} - x)(x - \\lambda_{-})}}{\\lambda x} \\,\\mathbf{1}_{[\\lambda_{-}, \\lambda_{+}]}\\, dx", "227eefcde3831848c9f061c203a54a8e": "(m-n)\\sigma^2", "227f0a2b17854530d758163f006c4c17": "\\displaystyle{\\nabla f=(\\partial_xf,\\partial_yf).}", "227f51a03add5697607dd2e43d252d10": "\\alpha = \\beta = 1/2", "227f9708f96cc5a8b10b059eae7fae23": "\\int^{\\overline\\theta}_{\\underline\\theta} \\left( \\frac{\\partial V}{\\partial x}(x,\\theta) - \\frac{1-P(\\theta)}{p(\\theta)} \\frac{\\partial^2 V}{\\partial \\theta \\partial x}(x,\\theta) - \\frac{\\partial c}{\\partial x}(x) \\right) p(\\theta) d\\theta = 0 ", "227fc08ff898b88a9d6372b0b4119f14": "PAI= \\frac {Y_2-Y_1} {T_2-T_1} ", "227fe3b6d887ce0652cf6b31c79be01f": "\\mathbf{h}=\\mathbf{r}\\times\\mathbf{\\dot r}", "22802944eafccd9c716ba44dc2b72404": "l, j, m_\\text{l}, m_s", "2280564bd195de2b6fa87c324114f7b3": "H= - k A\\frac{\\mathrm{d}T}{\\mathrm{d}x}.", "2280a32694681e20f981c0ac7da9e8e8": "\\mathbf 1", "22810e81f0ece5fca221b4646654ca62": "B=\\{b_{ij}\\}", "228110fca3070ee8a51a4cfe19af18d2": "\\mathcal{A} \\times \\mathcal{B}", "22814532056645082db9285e61829350": "\\gamma(1,p,v)=\\exp_p(v)\\ ", "22814b0c0ef80d4859ad64adf4df1cee": "\\chi=(1/3)\\chi_{||}+(2/3)\\chi_{\\perp}", "22817cdefe557180f3bfc1d7f26e2a5d": "x \\mapsto Ax+ b", "2281ab85abb157e602df0522a63947c5": "{\\Delta T}", "2281f9451273e0946363b14f9d515097": " 0 \\longrightarrow L_\\bullet \\stackrel{f}{\\longrightarrow}\nM_\\bullet \\stackrel{g}{\\longrightarrow}\nN_\\bullet \\longrightarrow 0,", "22822211393ae7ba4f3ba47a4aaf16d7": "P\\cdot p\\approx 0,", "2282547b896d6479da1e7395d0228350": "\\begin{smallmatrix}\\rho_{\\odot}\\end{smallmatrix}", "2282d786e48974120f13d5e110c9af75": "\\begin{align}\nP(\\{1,2\\} \\mid 2) &= \\frac{P(\\{1,2\\})/2}{P(\\{1,2\\})/2+P(\\{2,4\\})/2} \\\\\n&= \\frac{P(\\{1,2\\})}{P(\\{1,2\\})+P(\\{2,4\\})} \\\\\n&= \\frac{1/3}{1/3 + 2/9} = 3/5,\n\\end{align}", "2282fe81a8a5b19e5ed8e55d872e9e37": "\\mathrm{RED} \\subseteq P", "228309397e41304acbe86da5cf74c117": "R=\\emptyset", "22831c74c8457fc7914fdd0ed1ae3a7d": "\\displaystyle F_2(q) = \\sum_{n\\ge 0}{q^{n(n+1)}\\over (q^{n+1};q)_{n+1}}", "228384324495e0007a20814a0ac67d58": "R\\not\\equiv 0\\pmod{2\\pi}", "2284588ce1230ed395de263a7a11e3bd": "x=t", "22847c5fc0529b8aa1499e73649153ed": "+\\omega_{p}", "228496ce41925056aed635886c4801d4": "U|a\\rang", "2284b1ec3ab27148142b9eb254e7c65d": " \\textstyle\\ \\mbox{M}^- + HX \\overset{k_{term}} \\longrightarrow \\mbox{M-H} + \\mbox{X}^- ", "2284dfb06618756726306db417c3b1ba": "H^i(X, \\mathcal{L}^{-1}) = 0", "2284fb75e2414865ccbe78ad02e7b38a": "L(k, \\theta) = \\prod_{i=1}^N f(x_i;k,\\theta)", "228506be5ba2d72337b0cb82e61c4878": "\\begin{bmatrix}\n\\langle \\hat{d}| \\hat{a}\\rangle & \\langle\\hat{e}| \\hat{a}\\rangle & \\langle\\hat{f}| \\hat{a} \\rangle \\\\\n\\langle\\hat{d}| \\hat{b}\\rangle & \\langle\\hat{e}| \\hat{b}\\rangle & \\langle\\hat{f}| \\hat{b}\\rangle \\\\\n\\langle\\hat{d}| \\hat{c}\\rangle & \\langle\\hat{e}| \\hat{c}\\rangle & \\langle\\hat{f}| \\hat{c}\\rangle\n\\end{bmatrix}\n", "228522fb5ddacb55fbfee30aa764859e": "\\delta_i \\equiv \\mu - x_i", "22852db1843bb875b101dba86b3014be": "\\tfrac{n}{\\sigma^2} \\hat\\sigma^2|X = (\\varepsilon/\\sigma)'M(\\varepsilon/\\sigma)\\ \\sim\\ \\chi^2_{n-p}", "2285337eabb02aaf78fdfb0f0604125f": "Q \\propto I^2 \\cdot R ", "2285a65da4c02fbc98f8fff91cc9171d": "\n\\frac{\\partial(x, y)}{\\partial(r, \\theta)} =\n\\begin{pmatrix}\n\\cos\\theta & -r\\,\\sin\\theta \\\\\n\\sin\\theta & r\\,\\cos\\theta\n\\end{pmatrix}\n", "2285ac68ad27595ee79cb73e480038e2": " Y_3 = T_1Y_2Y_1T_2 - Z_1X_2X_1Z_2 = 4\\sqrt{15}", "2285baaa031d4539b75da963410965e0": "B = {V \\over 3000}", "2285d625ec7ca05677d53bd0b7a5f0d0": "\\displaystyle \\Delta_G= H^{-1}\\circ \\Delta_A\\circ H - \\|\\rho\\|^2,", "2286627ca88b2665bc202d5f909cd00f": "u_1^2+u_2^2 = (b_1^2+b_2^2)^2(b_3^2+b_4^2)", "2286693e822d82d351a9d673f3507d0e": "A=\\sum\\frac1{a_i}", "228690d6e971e8a5254bbff3e0eed6d1": "\\wp(z;\\Lambda)= -\\zeta'(z;\\Lambda), \\mbox{ for any } z \\in \\Complex ", "22875afd90ad1c2b92df9ff60e8de80b": "\\textstyle g'(\\alpha) - \\frac{k}{\\alpha} g(\\alpha) = 0", "22879e773b76f063d272c50e129955e5": " 0.75 > \\lambda > 0.5", "2287fc14c08365ade485c15ecd6fe355": "syn(x_v)=s'-s", "22882ac2d2f6949a9bdcdae163004c07": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = 1 \\times \\frac{1 + \\tan^2\\theta}{1 - 0} = 1 + \\tan^2\\theta .\n", "228847ace1dcd7bc400a6cbd6fd8efbd": "\\psi(\\beta) < \\delta", "22884d574b92407e456ac7f99fdefa0a": "x_{t+2} = Ax_{t+1} + Bx_{t}", "228881f61fb647f466f0136bc32ac2d0": "x_{n+1} = x_n^2 \\bmod M", "22890114671a8e131fb24cbdf9dd8c4e": "i[\\hat{H}'_0,\\hat{v}_i']=i[\\beta p_0,\\beta v_i]=0", "2289ba90f290ac31612326dc5a85d6d7": "a = g{m_1-m_2 \\over m_1+m_2}", "2289c700810a20d629484f7348490277": " |y\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} ", "2289cdb87c8d9405b62f9e5b05f78893": "\\det\\begin{bmatrix}\n|\\mathbf{v}|^2 & v_x & v_y & 1 \\\\\n|\\mathbf{A}|^2 & A_x & A_y & 1 \\\\\n|\\mathbf{B}|^2 & B_x & B_y & 1 \\\\\n|\\mathbf{C}|^2 & C_x & C_y & 1\n\\end{bmatrix}=0.", "2289f25989e23ff385128410a3ea839d": "\n\\begin{align}\nd\\log(Y) &= \\frac{1}{Y}\\,dY -\\frac{1}{2Y^2}\\,d[Y] \\\\\n&= dX - \\frac{1}{2}\\,d[X].\n\\end{align}\n", "228a0e2d73ee6203c784a64e07329d11": "\\scriptstyle{\\mathbf{n} = (n_1, n_2, \\ldots, n_m)}", "228a4e11b53e500007ff622b8f453f29": "L = rp \\sin\\theta = rmv \\sin\\theta \\, ,", "228ad9eaef032864e1c7edfea2a83ad1": "2^{- D^{\\epsilon}(\\rho||\\sigma)}\\geq \\frac{\\epsilon - \\delta}{\\epsilon} ~.", "228b208e2ac025be451a6553086f76ce": "(2,n)", "228b66e923d2392c41bc9cfdccbd710f": "10\\uparrow\\uparrow (n+1)", "228bb5b0c2039c8bd09b435b92a94674": "\\boldsymbol{a} = \\frac{d}{dt} \\boldsymbol{v} = \\sum_{k=1}^{d} \\dot v_k \\ \\boldsymbol{e_k} + \\sum_{k=1}^{d} v_k \\ \\dot{\\boldsymbol{e_k}} \\ . ", "228bcf5a2258bb4fed05b6fbc5f44ea7": "\\underline{P}(Cl_1^{\\leq}) = \\{x_1,x_5\\}", "228c16e5f1d9dd19534ab8b18fa869e9": " \\mathfrak{sl_2} ", "228c1d23540424a876a8aa7ecd264c71": "\\mathbf{A \\cdot B} = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 ", "228c4d4b19a627aec27b93846e1f8627": "\\mathbf{a} = \\begin{pmatrix}\n10 & 8 & 9 & 6 & 3 & 5 \\\\ \n\\end{pmatrix}", "228c4e84b7cb4d5c726aae1067ff8eb9": " \\forall j \\in [n] \\text{, } |S_{M_j}| \\geq w_\\min ", "228cb04c8c17ffd5817982ff9f06021b": "\\mathfrak{D} = \\{d\\}", "228cc1099bb99471c9e663dc3179923a": "q\\in {\\Bbb C}", "228cd14961ec9120661c673a1ceeeda3": "C_{in}^{\\alpha_1}", "228ce10760234445865375e58ea6518a": "X_{A}=\\frac{n_{A}(t=0)-n_A(t)}{n_{A}(t=0)}=1-\\frac{n_A(t)}{n_{A}(t=0)}=\\frac{100-10}{100}=0.9=90%", "228d1770f0d3e778245ad74459489fc9": "{f_{st}=\\frac{m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}", "228d2c8af9e8c6eb744b7eac6902d632": "R[\\Delta^n] := R[x_1,...,x_{n+1}]/(\\sum x_i -1)", "228d7376d3b7de0ae7e87873f2fe5b49": "\\begin{align}\n& (\\lambda, \\mu, \\nu)\\\\\n& \\lambda < c^2 < b^2 < a^2,\\\\\n& c^2 < \\mu < b^2 < a^2,\\\\\n& c^2 < b^2 < \\nu < a^2,\n\\end{align}", "228d786c7bda8abfb547ca15bd735ab2": "\\mathbf{\\sigma}(t) = -p \\mathbf{I} + \\int_{-\\infty}^{t} M(t-t')h(I_1,I_2)\\mathbf{B}(t')\\, dt'", "228d8c6685bfd3a85a02a50b9f7a467d": "F_0,F_1,\\dots,F_n", "228dc618ebe67c35c2409bbb110e8f44": " \\begin{cases} B^0(G,M) = {0} \\\\ B^n(G,M)= \\operatorname{im}(d^{n-1}), \\ n \\geq 1 \\end{cases} ", "228ddb83fd0d7dfe8ff991fc77e52e15": "\\phi_1(x,z,t) = A e^{-kz} \\cos(kx - \\omega t)", "228e51411e36e647b1c8990eeed58d51": " \\lim_{x\\to 0} f(x) = \\frac{\\varphi(0)}{2} = \\frac{1}{2\\sqrt{2\\pi}} ", "228e5f04182ca99a3a8baff2afd59ae7": "\\lim_{\\lambda\\to\\infty}\\int_0^\\lambda\\left(1-\\frac{x}{\\lambda}\\right)^\\alpha f(x)\\, \\mathrm{d}x ", "228ee4cd41d869f0272d22618352be63": " \\Delta_rG_{T,p} = ( \\sigma \\mu_{S}^{\\ominus} + \\tau \\mu_{T}^{\\ominus} ) - ( \\alpha \\mu_{A}^{\\ominus} + \\beta \\mu_{B}^{\\ominus} ) + ( \\sigma RT \\ln\\{S\\} + \\tau RT \\ln\\{T\\} ) - ( \\alpha RT \\ln\\{A\\} + \\beta RT \\ln \\{B\\} ) ", "228f29f708f1152c8d9dedf4dbc28bf4": "q_{S}", "228f2f944e5ec4e7768bb80f7b6e73cb": "\n \\boldsymbol{\\sigma} = \n \\cfrac{2}{J}~\\boldsymbol{V}\\cdot\n \\left[\\cfrac{1}{2\\lambda_1}~\n \\cfrac{\\partial W}{\\partial \\lambda_1}~\\mathbf{n}_1\\otimes\\mathbf{n}_1 +\n \\cfrac{1}{2\\lambda_2}~\n \\cfrac{\\partial W}{\\partial \\lambda_2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2 +\n \\cfrac{1}{2\\lambda_3}~\n \\cfrac{\\partial W}{\\partial \\lambda_3}~\\mathbf{n}_3\\otimes\\mathbf{n}_3\\right]\n \\cdot\\boldsymbol{V}\n ", "228f59fa279fc92ba37cad3ddc6d8706": "f_{V}(\\boldsymbol{v};\\boldsymbol{x},t) d\\boldsymbol{v}", "228f811498304aabee538781d1634cc5": " \\ m_1\\left(v_1^2-u_1^2\\right)=m_2\\left(u_2^2-v_2^2\\right)", "228fb002018b7f761aafb546c9b4d791": "\\sum_{i} ( \\mathbf {F}_{i} - m_i \\mathbf{a}_i )\\cdot \\delta \\mathbf r_i = 0,", "228ff921a9925bae33fb70737b864968": "d\\sigma(H\\boxtimes H)=H", "22900789cbf6adedfa9d145707d553e9": "e^i \\left (p,u^i \\right )", "229050832f937c4bb1680b9ca4a2adee": "=4 \\left( {p'}_\\mu p_\\nu - \\mathbf{p' \\cdot p}\\eta_{\\mu\\nu} + p'_\\nu p_\\mu \\right) + 4 m^2 \\eta_{\\mu\\nu} \\,", "2290572e74cfad695e4c2b7da592faae": "\\tfrac{1}{2}|z^n|<|p(z)|<\\tfrac{3}{2}|z^n|", "229081090a81261cba501fbaf91121f3": "\\left({x \\beta^k}\\right)_{k=0}^\\infty", "2290e1273b42b44a08bcac75b17091ea": "\\delta W = - p \\mathrm{d}V\\,", "22914b951c175ba068c843c13653ab4b": "\\Delta m_e v_\\perp \\approx \\frac{Ze^2}{4\\pi\\epsilon_0} \\, \\frac{1}{vb}", "229158eddd2ad6c22c44a34af85713ce": " d^{2}= 0 ", "229162b12758995ed36e5785c4acad5a": "\\nabla^2_{u,v} w = \\nabla_u\\nabla_v w - \\nabla_{\\nabla_u v} w ", "22920791f3a280b61ea4c2776559be5a": "\\tfrac{8}{11}=\\tfrac{6}{11}+\\tfrac{2}{11}.", "229213e513572a144a98a568448ef5d8": "|\\Psi|e^{i\\phi}", "229218c7c9f68a250ee71a1b24c432ea": "\nI =\n\\begin{bmatrix}\n \\frac{1}{5} m (b^2+c^2) & 0 & 0 \\\\\n 0 & \\frac{1}{5} m (a^2+c^2) & 0 \\\\ \n 0 & 0 & \\frac{1}{5} m (a^2+b^2)\n\\end{bmatrix}\n", "229244d9f09dfdf10569b8071d74362d": "(**) \\quad \\frac{\\partial u}{\\partial t}\n + \\sum_{j=1}^d \\frac{\\partial}{\\partial x_j}\n {f^j} (u) = 0,\n", "229282de2b3745ba2bef6548a25d41a6": "{\\mathcal E}(M)", "22930a506af72953a249fdd645041ae1": "a \\in X", "229335fdf754f5a62f27d4c3751c0654": "\n \\cfrac{1}{\\sqrt{3}}~\\sigma_b - A + 2B\\sigma_b - 4C\\sigma_b^2 = 0 ~.\n ", "229352b92899350cb603fef8536617f2": "n = \\frac{40}{100}(5-1)+1=2.6", "22935ffe733ec15477ad6920cb8d7bb6": "c=\\frac{\\log_d a}{\\log_d b}", "2294126e00363abf7787d7a5cf807f11": "\ne(t)=A\\sin\\left(\\frac{t}{\\sqrt{\\lambda}}\\right)+B\\cos\\left(\\frac{t}{\\sqrt{\\lambda}}\\right)\n", "229418db26e1710f8219dcafa5deca3b": "f * (g + h) = (f * g) + (f * h) \\,", "22942360063a4b9991511ce68a0461b8": "\\vartriangleleft \\ntriangleleft \\vartriangleright \\ntriangleright \\!", "22948cf6e922063902010c3768f9de34": "c_{empty}", "229498fe11f48d939518685fda992b3a": " \\bold{k} ~ = ~ k_x \\bold{\\hat{x}} + k_y \\bold{\\hat{y}} + k_z \\bold{\\hat{z}} ", "2294a377b91d385af3475fe1ccfd846f": "\\sqrt{x^2+y^2} \\le r.\\!", "2294fc246b8132598fdbda4da0aba4c5": "\\sqrt{2 \\pi n}", "22950efd1440048172275183213174f4": "\nds^2 = \\frac{1}{V(\\mathbf{x})} ( d \\tau + \\boldsymbol{\\omega} \\cdot d \\mathbf{x})^2 + V(\\mathbf{x}) d \\mathbf{x} \\cdot d \\mathbf{x},\n", "22953ff277b71ad0cb92286f15395053": "(V\\oplus W)(a)", "22955bf34cd182e2280a20fb7cb9659f": " \\phi,\\psi\\vdash\\alpha,\\beta", "2295d400661c4e91b783149de8c8a597": "- q_j \\, \\nabla \\Phi(\\mathbf{r})", "2296a7a9533a95821c4479777847e096": " \\mu(A) = \\lim_k \\mu(B_k) \\leq \\lim_k (1 - \\varepsilon)^{-1} \\int f_k \\, d \\mu. ", "2296e726c488eb1fdeee1117d0a55f33": " \\mathbf{J} = \\left( \\rho c, \\mathbf{j} \\right) ", "22970b9f7fed3c90cdd8f5abfcdc6761": "P \\subset {{\\mathbf{k}}}[\\mathbf{x}]", "22973d0baca7f9fdc1523bab4cd87c51": "x^{\\prime} \\leftarrow (y^{\\prime})^{r^{\\prime}}h^{m_h} rem N", "229753e26b7b603dc78eadf516da3bed": " M_2 = \\left\\{\\frac{1}{2}(1,-1,-i,-i),\\frac{1}{2}(1,-1,i,i),\\frac{1}{2}(1,1,i,-i),\\frac{1}{2}(1,1,-i,i)\\right\\} ", "2297a004b93e13216f88d60eb6d11d56": "\\hat{r}\\,", "22980760eab1a35798b611e9640feb8f": " x^{(i)}\\in R^{N_i}", "22980799dae7ef030e0148eaf0d1bd01": "a = b \\approx \\frac12c", "22982b06ad10d66a74f2270bb9c7a0b5": " {\\partial p\\over\\partial x}=0 ", "229859d04721676e6526be1a18bd673a": "(x_n)_n, x_n\\in X_n", "2298a6266bf5cbf0679a1f225de035e9": "S^{*} = \\{ (o_{i}, o_{j}) | o_{i}, o_{j} \\in X_{k}, o_{i} \\in Y_{l_{1}}, o_{j} \\in Y_{l_{2}}\\}", "2298e3d98d2926f07869cda28a26485d": "(5, 12, 13)", "2298e531e032c1fd9b735b5161ad178c": " \\left(f_w\\right) ", "2299996b80e5cbb9676731c645fcbe9f": "0\\leq t\\leq m", "2299c58aac41b30e2e0e7feabbc0dbc2": "\\mathrm{st}(C)", "2299d85e1e8103b2a062bf94d1f1ef87": "f(x_i)=\\bar{y}", "2299decf3743ac391990a30b61e63ec1": "\\sigma \\in \\mathcal{K}", "229a39b090f7f93582415ad6faa9b711": "\\scriptstyle \\nabla_a", "229a481a0e0872210cf2e9ab449bd970": " ds^2~=~dx^2~+~dy^2~+~dz^2~-~C^2~dt^2 .", "229a7e8a338ac8ebeeebfe394c6e7a8e": "t \\in V", "229a9e812343e543064dd2be9b5362e5": "10\\uparrow\\uparrow\\uparrow 6=(10 \\uparrow \\uparrow)^6 1", "229aa93417de537ff635e51826e67c99": "\\Phi: B(G) \\rightarrow B(H)", "229adde30c61804a35efe6dd1f3ba73e": "N_x' = m' x' - p_x' t' = \\gamma(V)\\left(m-\\frac{V p_x}{c^2}\\right)\\gamma(V)(x - vt) - \\gamma(V)\\left(p_x - \\frac{VE}{c^2}\\right) \\gamma(V)\\left(t - \\frac{Vx}{c^2}\\right) = N_x ", "229ae01424652497290d14605294665d": "\n \\left[\\boldsymbol{S}\\cdot\\mathbf{u}, \\boldsymbol{S}\\cdot\\mathbf{v}, \\boldsymbol{S}\\cdot\\mathbf{w}\\right] = \\det\\boldsymbol{S}\\left[\\mathbf{u}, \\mathbf{v}, \\mathbf{w}\\right]\n", "229b01315fc3ee50634005122c82a466": "x_k\\to x^*", "229b387c4e672e84b228590b2a74c00b": "\\varphi(f)\\psi(f)^{-1}=1", "229b8e4c6cbfc6e7968a6b7056341975": "\\ P_{ij \\ldots}=P_{ij\\ldots}(\\mathbf X,t)=P_{ij\\ldots}[\\chi^{-1}(\\mathbf x,t),t]=p_{ij\\ldots}(\\mathbf x,t)", "229bc369f62d28d8a253073adb48ade5": "A,B \\subset X", "229c413ad59c64a6b6f2f86c042481fc": "~ G = N_2\\sigma_{\\rm se} -N_1\\sigma_{\\rm sa} ~", "229c79cadfc584ff4225845e292d72eb": "R(p)", "229c7dd944b8fbf0184d360de6e97984": "c_{13} = 1.97483 \\times 10^{-7},\\,\\!", "229ce98b41975d20b509835b29d9268b": "d_i\\!", "229d1f6a5914c10c145ae01ea9d61b7a": "N(z\\cdot w) = N(z)\\cdot N(w).", "229d72351c39e1b958065df1e7f03222": "{x}_i = \\frac{1}{2} (a+b) - \\frac{1}{2} (b-a) \\cos\\left(\\frac{2i-1}{2n}\\pi\\right). ", "229d9eca632b2acc7dedd37b40b17673": "a=f'(z_0)", "229dce70d3f32f93af25ae8a61f18f82": " \\sigma=\\frac{1}{2}\\left(\\sigma_x - i\\sigma_y\\right) ", "229dd9459eef1ca0f064fc56af46fbf6": "s_i (\\Delta)", "229f263a5350c3d7548dfac4157cc2ab": "V \\in [0,1]", "229f61e0bc8ab1cf2094e3b0eeac6a1c": "D^n\\,,", "229fa3258a3e2167b5d2c4d0bffefa09": "\\stackrel{\\mathrm{def}}{=}\\sum_{m=-\\infty}^{\\infty} f[m]\\cdot g[n - m]\\,", "229fbae624540adc8105c92a1d8fe74e": "c\\in \\mathbb Z^{*}_{n^{s+1}} ", "229fdf8e5d83002ca54e6d8f31fb4f5c": "\\mu = 1/\\lambda", "229fe9eeb7bafbd24069261c0eef947e": "R_1R_2C=CR_3-O-R_4", "22a015a4ac284614fa5e5aebae92fbee": "\nn_i = \\frac{g_i}{e^{\\alpha+\\beta \\varepsilon_i}-1}.\n", "22a05cbec068688a70a10a2073d68164": "\\Gamma_X: \\mathcal F \\mapsto \\mathcal F(X).", "22a06d30ccf3c7e8ab9fe0f6c3a1fb06": "\\bar{T}T", "22a16ef42e0e15a2363b3e6deb90bb0a": "u=F_X(x)", "22a1b2d902069340618a2cdff221cad6": "i > \\operatorname{rk}E", "22a2218d6501ef08e25255ecb6ef7cc6": "H1=0.541266 * P", "22a2368d74c955d340f9d5c19933a038": "D = D_{max}\\cdot (P_{max} - P)", "22a24d0a8fee23437196d4726128ec2d": "Q_C = \\frac{X_C}{R_C}=\\frac{1}{\\omega_0 C R_C}", "22a270b99d811f74be44e37fce3d4075": "[x_0,\\ldots,x_n;f],", "22a2c0be1acdb616c4ab144870516fad": "Z_\\mathrm{in}=Z_0 \\frac{Z_L + iZ_0\\tan({\\pi \\over 2})}{Z_0 + iZ_L\\tan({\\pi \\over 2})}=Z_0 \\frac{iZ_0\\tan({\\pi \\over 2})}{iZ_L\\tan({\\pi \\over 2})}=\\frac{{Z_0}^2}{Z_L}", "22a30be03491a5692ec41f1e349a146c": " \\lambda_{tot} (t_{sc}) = \\sum_i \\lambda_i. ", "22a37cb64bafddb3d7949e57a1823655": "mkT", "22a40e0d7a0856ffb1f6a8ebdbe5aa08": "V = \\frac{4p}{4 - (p-2)(q-2)},\\quad E = \\frac{2pq}{4 - (p-2)(q-2)},\\quad F = \\frac{4q}{4 - (p-2)(q-2)}.", "22a4709ac96af95e41cb78d6a82bc438": " \\frac{1}{\\sqrt{2}} ", "22a47cbe71bd434887b9a5829eda6400": "n_1 \\times n_2", "22a48ba1c4f204f3d96be834d354f8e1": "P \\,", "22a50bf3e8f2ebe23a22c5120c9dbd11": "\\hat{R}", "22a53da2ba6bc1f0d5993c9204d1dfb4": " |f(x) - f(y)| \\le M \\, |x - y|^\\alpha, \\quad x, y \\in [a, b]", "22a59ab9300a67e7239c0808e90e0c32": "\\sigma = \\sigma_R+\\sigma_I", "22a5e4487011b3fb5a47053cb7595504": "\\lambda^{(n)}", "22a62e7c3050e9c2e75e65b185f67d37": "X, Y \\sim \\textrm{N}(0,1)\\, X, Y", "22a64f1493b9a4f61c3db7745a7a2c51": "f(x) - p_n(x) = f[x_0,\\ldots,x_n,x] \\prod_{i=0}^n (x-x_i) ", "22a68463505ebd43886cb03513291958": "\\eta_{\\mu\\nu}", "22a69378ea2160345fdf1d500bb4e6b1": "\\Gamma_{ij,k}^{(\\alpha)}+\\Gamma_{ik,j}^{(-\\alpha)}=\\Gamma_{ij,k}^{(0)}+\\alpha T_{ijk}+\\Gamma_{ik,j}^{(0)}-\\alpha T_{ijk}=\\partial_ig_{jk}", "22a6b88432bbca1acd71e971b0d4371a": "B^\\alpha_\\alpha = -1/R", "22a6f6afd068c21597507178eeea9fad": "A = (2\\pi r)(2\\pi R) = 4\\pi^2 R r.\\,", "22a7042b12b52418b61e4dc5b41bcc0d": "\\operatorname{perm} A \\leq \\prod_{i=1}^n (r_i)!^{1/r_i}.", "22a755a82d173d75f4e52d622665c138": "\\psi_{2,m}", "22a761d68fa745d8a4cc71ca8dfbcb2d": "a^\\dagger a \\psi_0 = 0 = \\left(\\frac{\\hat H}{\\hbar \\omega} - \\frac{1}{2} \\right) \\,\\psi_0 = \\left(\\frac{E_0}{\\hbar \\omega} - \\frac{1}{2} \\right) \\,\\psi_0.", "22a769b60d470fc6c37f1f9cd6dc22e8": "\\exp_{10}^2(8.56784)", "22a7c7266fe187fc829c235da96ead3d": "\nterm(x,j) = \n\\begin{cases} \n \\left \\{ \\right \\}, & j \\geq \\#input\\\\\n \\left \\{ j+1 \\right \\}, & j^{th} \\mbox{ element of } input=x\\\\\n \\left \\{ \\right \\}, & \\mbox{otherwise}\n\\end{cases}\n", "22a7d94e48f27443084943725603ecb7": " G = \\frac { G_1 \\Delta_1 } { \\Delta } = \\frac { -y_{21} R_L } {1 + R_{in}y_{11} + R_L y_{22} - y_{21} R_L y_{12} R_{in} + R_{in}y_{11} R_L y_{22} } \\, ", "22a8234e4fdd3bc701326a4bef77f4c5": "1 - t + t^2 - \\cdots + (-t)^{n-1} = \\frac{1 - (-t)^n}{1+t}", "22a8244308bcb651c842bd0a205748ba": "-\\mathbf{X}^{-1}\\mathbf{A}^{\\rm T}\\mathbf{X}^{-1}", "22a8a2128364a8ea862aa20f13cd15a2": "{2x^2-x+4}=0.\\,\\!", "22a9b1494679181dc52202d0ba1f31ec": "w-M", "22a9c6dcbd0208a13a6c3d91d2a157c5": "\\bot=I^*", "22aa2bff8aa681ee2891df8ce6a31c63": "\n C_{14} = C_{15} = C_{24} = C_{25} = C_{34} = C_{35} = C_{46} = C_{56} = 0 ~.\n ", "22aa50b172e16f7e74061d1dd4f356c3": "\\begin{align} \\gamma = \\lim_{n \\to \\infty} \\left( \\ln n - \\sum_{p \\le n} \\frac{ \\ln p }{ p-1 } \\right)\\end{align}.", "22aac8d36955638219f72323821d785e": "\\begin{align}\n & (\\varphi + \\psi)(x) = \\varphi(x) + \\psi(x) \\\\\n & (a \\varphi)(x) = a \\left(\\varphi(x)\\right)\n \\end{align}", "22ab80d98be6ba3a9c4a490313739aaf": "aq_n - bp_n", "22abccf2370a96166c5f2d79376223fa": "D<\\frac {10^k} n - 1,", "22ac1b7a739599100f9f72a046c8f90d": " p(\\boldsymbol{\\theta},\\boldsymbol{\\phi}|\\rm{data}) ", "22ac23194a1cabb43ad2478cbaa41224": "z^{}_{}", "22ac714ffb9af4d2a92b5b3dd9691da4": "r\\in V\\,\\!", "22accd3047daa3dab24e04f158c97c46": "\\int_{-\\infty}^\\infty\\!dA\\, e^{-\\frac{1}{2} A^2 + A f} = \\sqrt{2\\pi}e^{-\\frac{f^2}{2}}", "22acdfc0b7b024b89a91859e542dcde3": " \\displaystyle{W(z,w)=e^{-izw/2} U(z)V(w)}", "22acedc7ba733d3aeb517e79de401799": "XXZ", "22acf4791c3a74b310d3605ecd18bcce": "\\scriptstyle h(\\tau)", "22acfffa5abb07d762bc57ab5ac340b1": "g(x)=e^{-\\lambda_0-1-\\lambda(x-\\mu)^2}", "22ad456fb0d74b9f48f29688f43aec1c": "k_e q_1 q_2/m", "22ad47f404ea32dd988a4fbdaad50d50": "\\begin{cases}\n2ab = 1\\! \\\\\na^2 - b^2 = 0\\!\n\\end{cases}", "22ad78c2ee68841d4520da3fbf7bf1da": "f(x) = \\int_{-\\infty}^{\\infty} \\hat{f}(\\xi)\\ e^{2 \\pi i \\xi x}\\,\\mathrm{d}\\xi, ", "22ae54f18ab71732cb2a3530a4901017": "\\operatorname{lift}_\\tau(e)", "22ae666a3cd3a651f61269ca0506defd": "t=F_{\\hat\\theta_n}(x)", "22aed3510591eeac0f958dc9bebea90f": "p_i(m)=m(i)", "22aeed46d7274e42d16f9e3e251512b7": "w\\sqrt{\\theta}/{\\delta}", "22af4663c8b81b46f33b115b45ce76a9": "T = 1.41 ", "22af6cbed482a4c8f6de941833c0ee91": "T_{[\\alpha\\beta\\mu]}=0", "22af7d1f2e55c8916a0850fa940b0947": "k_{\\rm on}\\,[{\\rm R}]\\,[{\\rm L}] = k_{\\rm off}\\,[{\\rm RL}]", "22af8ec879e356c2ac01d4c4d5dcd9d6": "\\exp(\\gamma l)\\,", "22b0085b5fd016929fb8cb89ab9d549a": "\\{(a b^n)^n | n \\geq 0 \\}", "22b022754506b390a3b5414fcebfad9b": "(x,a) \\sim (x,b)\\text{ if }x < 0.\\;", "22b04348dc60a3508c31118ba163533c": "\\overline {OB}", "22b04a52da7d54bcb03d1c2182adbcff": "\\frac{d}{dx}\\chi_2(x) = \\frac{{\\rm arctanh\\,} x}{x}.", "22b0787b820d7b94ac706a6749339ac9": "t_1 < t_2 < \\ldots < t_n ", "22b08324b2d8a0e8a81c4a41b91d117c": "M_a=\\lim_{N\\to\\infty}N\\cdot x(N) \\,", "22b0c363a8d77a3b783519d52f3092bf": "2^{n+1}", "22b168be7c5b4cbb08e7fff21a7e6043": "\\{ 0, 1, 2, \\ldots\\}", "22b1a317fbcafd82d03c4165eb3663d0": "L^{<->}", "22b1b7527bd6b270d0da9f1796dce2aa": "~\\Phi_n(x) = 1+x+x^2+\\cdots+x^{n-1}=\\sum_{i=0}^{n-1} x^i.", "22b1bb1a8179fdeae8d7b9c2c1467671": "open(door,result(opens,s))", "22b1c2e66eb01489c2c231414c8a6884": "(f * g) + \\varepsilon = h \\, ", "22b1f7d85314571dc1e1dcfd40154e8e": " p(\\pm x_1, \\pm x_2,\\ldots,\\pm x_n)=0. \\,", "22b1fc7b4f996437e16a0b7b318a2dc6": "m_\\text{red}", "22b218e8955210ea5e10226c2b5d9e06": "\\,NA", "22b21e43056af1cc82aa19c1b4ae2b52": "\\hat{\\beta} - \\beta ", "22b24e1d84ff18af22bb23323caf1731": "\\lim_{\\Delta x\\to 0} \\frac{\\Delta h}{\\Delta x}", "22b297e10a9746252315abecfd3ff4d0": "(\\lambda\\leq\\lambda')\\leftrightarrow(s_{\\lambda'}\\leq s_\\lambda)", "22b2f497188e759b7e75d8ad879a1c30": " D = R - r ,\\,", "22b3028f41872d929a5faadc56f66a1b": "c\\subseteq \\left(c'\\right)^+", "22b31c9ebdf0c1ee047224a011a1cd8c": "Q \\,\\!", "22b369532f78351aaf1e4e642e6e584d": "V(x) = V_0*\\Theta(x)=\\left\\{ \\begin{align} 0 & \\quad x<0 \\\\\nV_0 & \\quad x>0\n\\end{align}\\right.", "22b36edb7824baece8731d66b6ec6eec": "\\frac{\\text{d}[{^{b_j}_{a_j}}S^{\\beta_j}_j]}{\\text{d}t} = \\sum_i x_{b_{ji}} [\\text{k}_{2(i)} C_i -\\text{k}_{1(i)} E \\overline{S}_i] \\qquad \\qquad (8a) ", "22b46f9b460bd11b80a712f9c181a096": "\\hat{S}_N", "22b474e48c46f805f571fc9bdff89efe": "\\; \\Lambda^k[\\psi] = sup_{\\phi \\in S_k}|\\langle \\psi|\\phi\\rangle|^2", "22b4c68332930d02a2f874460da513ea": "E_{r}", "22b4f59cb8348c7d6373fa0d0e3ed055": "(\\mathrm{C_6H_4(CH_3)_2})", "22b505bc46bc4a9efa6c680256705f04": "V_\\beta^I", "22b59ed7a72604f34413115e7f046960": " \\scriptstyle {\\mathbf r}_1(t) ", "22b5abefcd0a9eb941640b1947621dd9": "\\alpha^2u_{j,iij} = 0\\,\\!", "22b5b5cecc8d2260f24543201e571c8b": "\\Big( (\\mathcal{M}, s) \\models \\phi_1 \\land \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big((\\mathcal{M}, s) \\models \\phi_1 \\big) \\land \\big((\\mathcal{M}, s) \\models \\phi_2 \\big) \\Big)", "22b60909b0f6cfc7b72d6f1b63f670c3": " \\hat{\\textbf{x}}_{t-i\\mid t} ", "22b63c0948778288c4c9f3860b411df5": "{\\rm tr}((\\mathbf{B} - \\hat{\\mathbf{B}})^{\\rm T}\\mathbf{X}^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1} \\mathbf{X}(\\mathbf{B} - \\hat{\\mathbf{B}})) = {\\rm vec}(\\mathbf{B} - \\hat{\\mathbf{B}})^{\\rm T}{\\rm vec}(\\mathbf{X}^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1} \\mathbf{X}(\\mathbf{B} - \\hat{\\mathbf{B}}))", "22b641fa5ac4714d0d111eec5786aa0d": "\\ell_0(x)={x - x_1 \\over x_0 - x_1}\\cdot{x - x_2 \\over x_0 - x_2}\\cdot{x - x_3 \\over x_0 - x_3}\\cdot{x - x_4 \\over x_0 - x_4}\n ={1\\over 243} x (2x-3)(4x-3)(4x+3)", "22b647628f488d35fc7a07e821cbcff6": "V:=L(\\underline{a}_1,\\ldots,\\underline{a}_r)", "22b68bd688762b5f6a2136e8c974e779": " g = -G \\cdot M/r^2 + \\omega^2 \\cdot r", "22b6e3dd03cb9a8688980c1a8fbcca0d": "\\pi(g)f(x)=f(g^{-1}x),", "22b6f54eaa543e0735135e60d5b13cab": "-c\\frac{df}{dX}+\\frac{d^3f}{dX^3}+6f\\frac{df}{dX} = 0,", "22b71634db2e7a17be047da9b0301089": "\\sqrt{ax^2+bx+c}\\,\\!", "22b78845d32790f6b041bee362863a7c": " V_{S} ", "22b7bb0c699f5c82b02fd8b51e3812ec": "X_r=\\{(x,t):0\\le t\\le r(x),x\\in X\\}/(x,r(x))\\sim(fx,0). ", "22b7d404620033d69e1c4929f3fd672e": " \\hat{A}\\hat{A}^* = (A, B)(A^*, B^*) = (AA^*, AB^* + BA^*) = (1, 0).\\!", "22b7dc4a2e7161229f7fa237d1689d4b": "\\frac{\\mathrm{d}\\mathbf{A}}{\\mathrm{d}t} = \\frac{\\partial\\mathbf{A}}{\\partial t}+(\\mathbf{v}\\cdot\\nabla)\\mathbf{A} ", "22b7e63eac22c6b06c4070e776f628ba": " f(c_1) = (1.5)^3 - (1.5) - 2 = -0.125 ", "22b80d566556e632e7ee5467dcab4c44": "(+\\alpha, -\\alpha)", "22b812eac0dc95c05573f232e51a8355": " Y_{i+1, 2} = 40,692 \\times Y_{i,2}\\pmod {2,147,483,399}", "22b867147f4fe11a2d41a9ef13f6ca09": "W \\in \\alpha", "22b888cb5dbe74050a5416436a29bb57": "-\\sqrt{\\frac{1}{6}}\\!\\,", "22b8959dca36615c493537b1bc760a05": " \\| Tx \\| = \\| T^* x \\| ", "22b8ae2f554cb5cd4dd83d8b79db3107": "V_{\\,t}", "22b8c33fed8bd7cf6a22988a9d172de2": "{}_j \\bar{P}^l_{L} (\\theta) = \\sqrt{\\frac{2L+j-1}{2} \\frac{(L+l+j-2)!}{(L-l)!}} \\sin^{\\frac{2-j}{2}} (\\theta) P^{-(l + \\frac{j-2}{2})}_{L+\\frac{j-2}{2}} (\\cos \\theta)", "22b8eefb59e47bfc8edc7277baa9fe1c": "\\begin{array}{rccrcrcrcr}\n{\\color{BrickRed}P}{\\color{RoyalBlue}Q}&{{=}}&&({\\color{BrickRed}2x}\\cdot{\\color{RoyalBlue}2x})\n&+&({\\color{BrickRed}2x}\\cdot{\\color{RoyalBlue}5y})&+&({\\color{BrickRed}2x}\\cdot {\\color{RoyalBlue}xy})&+&({\\color{BrickRed}2x}\\cdot{\\color{RoyalBlue}1})\n\\\\&&+&({\\color{BrickRed}3y}\\cdot{\\color{RoyalBlue}2x})&+&({\\color{BrickRed}3y}\\cdot{\\color{RoyalBlue}5y})&+&({\\color{BrickRed}3y}\\cdot {\\color{RoyalBlue}xy})&+&\n({\\color{BrickRed}3y}\\cdot{\\color{RoyalBlue}1})\n\\\\&&+&({\\color{BrickRed}5}\\cdot{\\color{RoyalBlue}2x})&+&({\\color{BrickRed}5}\\cdot{\\color{RoyalBlue}5y})&+&\n({\\color{BrickRed}5}\\cdot {\\color{RoyalBlue}xy})&+&({\\color{BrickRed}5}\\cdot{\\color{RoyalBlue}1})\n\\end{array}", "22b93da3890cf46eb0b150f7878b9071": "\\sqrt{n/2-3/8}", "22b95bcb1de3e047408bc7a83934005d": "\\log \\frac{k_{x,sol}}{k_{x,80% EtOH}} = mY ", "22ba0642369aad29f9fb30cbb3ee20f5": "T_{se}", "22ba10e403d2fbb63084c676966d5f63": "x_\\mathrm{a}(t) = \n\\begin{cases}\n\\ \\ e^{j |\\omega| t}\\cdot e^{j\\theta} , & \\mbox{if} \\ \\omega > 0, \\\\\n\\ \\ e^{j |\\omega| t}\\cdot e^{-j\\theta} , & \\mbox{if} \\ \\omega < 0.\n\\end{cases}\n", "22ba3df1ffa88853a725f9aa70271d54": " S_m=C_V\\ln\\frac{T}{T_0}+R\\ln\\frac{V_m}{V_0}.", "22ba7f3c9b64390cc173abcb7b2bba40": " 0\\leq deg(D_x )+v_1(D)\\leq g", "22ba8403045255f335e95f0a6991e146": "\\phi=\\pm\\varphi", "22badfb1e212c287a4db5c70237313f3": "\\scriptstyle \\theta<\\frac{1}{4} ", "22baeba5a637ed9bc00f744931dee7c7": " \\frac{a+b}{a} = \\frac{a}{b} \\equiv \\varphi", "22bafd6a4d2255a6c8b96ac73b92ea79": "t\\in[0,1]", "22bb7366806b4ad3012332ef01c5577e": "h(x)\\leq 0", "22bba63018a0bb86116fe1da65478529": "f_\\ell = \\frac{S_\\ell-1}{2ik} = \\frac{e^{2i\\delta_\\ell}-1}{2ik} = \\frac{e^{i\\delta_\\ell} \\sin\\delta_\\ell}{k} = \\frac{1}{k\\cot\\delta_\\ell-ik} \\;.", "22bbc5c7044ea97ef7f07eff75d01f27": "z \\ ", "22bbd54d13c77f28dc92a2401e67eab0": "\\, e^{it\\mu - \\frac{1}{2}\\sigma^2t^2}", "22bbede97f45426877830300a702e52d": "I = 30/300 = 0.1", "22bc40bf712ae7f4908381369fdea7db": "x_1,x_2\\in R^d ", "22bc4cbf3b3f4b737baac878df8336bd": "d \\Omega_0 = \\frac{d S_0}{|\\vec{r} - \\vec{r_0}|^2} \\frac{\\hat n_0^s . (\\vec{r} - \\vec{r_0})}{|\\vec{r} - \\vec{r_0}|}. ", "22bc4e390a8a9014b293f0826d32cfcd": " Q_D (l_A a_B + l_B) l_D ", "22bc9d624fbca98ab53fef969b68e2d4": "\\frac{T}{P} = \\frac{x_{p} - x_{f}}{x_{f} - x_{t}}", "22bcaeefcaa1b7e712df20e3f8c556f7": "\\mathfrak{sp}(2n,\\mathbb C)", "22bd077637a7a9fd6c42e5cd7339a59c": "f'(x)\\,=\\,Df(x) = n a_n x^{n - 1} + \\cdots + 2 a_2 x + a_1", "22bd0e466382096a4d54bc510ac07c7a": "\\tilde{M} = |M_{1}|e^{i\\theta_{1}}e^{-i\\phi_{1}} + |M_{2}|e^{i\\theta_{2}}e^{-i\\phi_{2}}", "22bd11870acfeecb092e5489a752655f": "\\mathbb{C}\\left(\\textbf{g}\\otimes\\textbf{n}\\right) \\textbf{n}=0.\\qquad {(4)}", "22bd69f86c3300dd9fae5fd90260c5d6": "a_i,_J u_i,_J = \\sum a_nb u_nb - \\Delta V_u \\frac {P_I,_J - P_{I-1},_J}{\\partial x_u} + S \\Delta V_u ", "22bd6b9028fd636ae7c7645c70f381f7": "Select:2^D \\rightarrow D", "22bd7f9b5e766d9bf85195216db60d95": "\\begin{align}\n\\sup_{y\\in [-\\varepsilon,\\varepsilon]} |Q_{1-\\varepsilon}(y)| &\\le \\varepsilon^{-1}.\\\\\n\\sup_{y\\notin (-\\varepsilon,\\varepsilon)} |Q_1(y)-Q_{1-\\varepsilon}(y)| &\\to 0.\n\\end{align}", "22bdb2e60be4a709b885eed926c4f3fd": " \\|a-b \\|_1 = \\sum_i |a_i-b_i| ", "22be5ef5aa61a0c1449c19deadf13b71": "f(x_1)=y_1", "22be842f9527b4b0ef3b7085b5486b7b": "\\left[a,b\\right]", "22bf33e6cde821d4237c35cc7713611f": "\\mathbb{R}_{--}", "22bf9624469967c4fafbc0759ca05313": "A_1 A_2 \\ldots A_n", "22bfe044e763ff1cbb0ee1d54b4d4cd2": "Q_b^{(i)}", "22c02fcb3c26b705ce285fab39fe0698": "G \\hookrightarrow U", "22c062b79ff14c547953fc25d7aff1af": "\\scriptstyle |\\zeta| \\;\\leq\\; 1", "22c073e1631e10ad78dddf1c6681f398": " Y^{(n+1)}(\\xi) = f^{(n+1)}(\\xi) - \\frac{R_n(x)}{W(x)} \\ (n+1)! = 0 ", "22c092b03d5cf415fecd20a7832f513b": "\\pi: Y\\longrightarrow X\\ ", "22c0dea095c6efd1d8603fa8dda8456a": "R_{d}/100", "22c0fa0bbf2a22a2a32c4a16b7acb329": "\\mathbf r(0) + \\left(\\frac{s^2\\kappa(0)}{2}+\\frac{s^3\\kappa'(0)}{6}\\right)\\mathbf N(0) + \\left(\\frac{s^3\\kappa(0)\\tau(0)}{6}\\right)\\mathbf B(0)+ o(s^3)", "22c1331bd91a23f7e9ed38e5aa35e4be": "{x_1}^{e_1}\\cdots {x_n}^{e_n}", "22c13fd01ba06a4d700000f2495eef6a": "-\\sqrt{\\frac{4}{15}}\\!\\,", "22c17ed039c41f337db4b60659671c13": "f_\\mathrm{{\\omega}}=N{\\omega}^2r.", "22c198f267e4b618677adcd0bdb675f0": "\\int_{A} dA \\nabla f = \\oint_{\\partial A} dx f", "22c1ad678d3c6547e0ec61b6d3b8efc3": "r=\\frac{l}{{1+e\\cos\\theta}},", "22c1b95d0b51bf3b5d08fc1f455aa2e2": " vm_{pq\\mu\\gamma} = \\sum_{i=2}^{images} \\sum_{x=1}^M \\sum_{y=1}^N U(i,\\mu,\\gamma) C(i,p,g) P_{i_{xy}}", "22c1c5fe8131c74196a508aaab6eab9e": "\\mu_{k} = \\frac{b_{k}^{*}Ab_{k}}{b_{k}^{*}b_{k}}", "22c1d034b2416df93506e1b66b4d05f2": "\\xi\\mapsto\\theta=\\theta\\left(y,\\xi\\right) ", "22c1e12cf94b7c16ca4d04c3403bd807": " \n\\hat{E}\\left\\{\\mathbf{x}(n) \\, e^{*}(n)\\right\\}=\\frac{1}{N}\\sum_{i=0}^{N-1}\\mathbf{x}(n-i) \\, e^{*}(n-i)\n", "22c1e4c2633d9b563f2c5c1f3ac2f52d": "\\hbar \\ ", "22c204a666cbbe332d51b68b15d2b2d8": " T(t,r) = t - \\int \\frac{\\sqrt{2m/r}}{1-2m/r} \\, dr = t + 2 \\sqrt{2mr} + 2m \\log \\left( \\frac{\\sqrt{r}-\\sqrt{2m}}{\\sqrt{r}+\\sqrt{2m}} \\right) ", "22c274b8e98bdb49fdb95f8bd9e352bf": "q_\\mathrm{1}, q_\\mathrm{2}", "22c2afaa9d18cdbc249dd99607477df0": " f(x)g(y) = f(x')g(y') ", "22c32dce6b76f3d06f2aa5eecdbaaf63": "S^{\\sigma}(1853)=dr(1853)=8.\\,", "22c3717ba1dfac337b183f3b9949b9a3": "S(t)=(r_1, r_2, \\cdots ,r_N)", "22c3bbaf6078188688c09d8fd47f39d2": "C_n\\left(Y\\right)", "22c3fda0bab16352dbf2135dd00710c9": "m_{(3,1,1)}(X_1,X_2,X_3)=X_1^3X_2X_3+X_1X_2^3X_3+X_1X_2X_3^3", "22c41e4cb82486019b10e43932785f0a": "dY = de^{- \\int_t^s V(X_\\tau)\\, d\\tau} u(X_s,s) + e^{- \\int_t^s V(X_\\tau)\\, d\\tau}\\,du(X_s,s) +de^{- \\int_t^s V(X_\\tau)\\, d\\tau}du(X_s,s) + d\\int_t^s e^{- \\int_t^r V(X_\\tau)\\, d\\tau} f(X_r,r)dr", "22c45d34d43506fde9b57bc2fecfa313": " \\operatorname{ var }( r ) = \\frac{ 1 }{ s_x^2 + m_x^2 } \\left[ ( s_y^2 - s_{ x^2 [ y^2 / x^2 ] } ) - ( s_{ x [ y / x ] } )^2 +2 m_y s_{ x[ y / x ] } - \\frac{ s_x^2 }{ m_x^2 }( m_y - s_{ x[ y / x ] }^2) \\right] ", "22c471f4d0c633bc09ca7b5da07eba7c": "F^*(H)=O_p(H)", "22c4b22afee2c15de89a176f0f3be2d7": "Q^{-}(2n+1,q)(n\\geq 2)", "22c50e2263d50cc11c9b087099987602": "\\theta = n(\\lambda - \\lambda_0) ", "22c523fa8e1c05573e7d47deacb79ef8": "\n\\frac{1}{\\zeta(s)} = \\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^s}\n\\!", "22c59af425ae2fc25b9ce9fa8522347a": "\\mathbf{A}^{*}", "22c5c97c10c6ce627cf54510bc15d9ce": "\\frac{P \\to Q, P}{\\therefore Q}", "22c60cff0fe91b977f01efe3015c845d": "\\boldsymbol{U} = \\boldsymbol{S}\\cdot\\boldsymbol{T}", "22c774ffaf3b9d43995af95f849ee0cf": "\\mathfrak{a}^e", "22c7ad124dcee12145c7d3eb3bccffab": "A_v = \\log_2", "22c7e18f2ea59e4c2564d09f31d2f96a": "P(\\text{ill}) = 1% = 0.01\\text{ and }P(\\text{well})=99%=0.99.", "22c7e32877d4d61963168e0846fa0f66": "\\frac{d^2 \\theta}{d t^2} + 1 = 0.", "22c826d0f93b82d4487b4d1c7287d5bd": "\\hat{\\boldsymbol{x}}", "22c85dce207d7deb6e0c71bd5cd20983": "\n \\mathbf{y} = \\begin{pmatrix} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{pmatrix}, \\quad\n \\mathbf{X} = \\begin{pmatrix} \\mathbf{x}^{\\rm T}_1 \\\\ \\mathbf{x}^{\\rm T}_2 \\\\ \\vdots \\\\ \\mathbf{x}^{\\rm T}_n \\end{pmatrix}\n = \\begin{pmatrix} x_{11} & \\cdots & x_{1p} \\\\\n x_{21} & \\cdots & x_{2p} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n x_{n1} & \\cdots & x_{np}\n \\end{pmatrix}, \\quad\n \\boldsymbol\\beta = \\begin{pmatrix} \\beta_1 \\\\ \\beta_2 \\\\ \\vdots \\\\ \\beta_p \\end{pmatrix}, \\quad\n \\boldsymbol\\varepsilon = \\begin{pmatrix} \\varepsilon_1 \\\\ \\varepsilon_2 \\\\ \\vdots \\\\ \\varepsilon_n \\end{pmatrix}.\n ", "22c8881ed41502f4c10ff52e7a77be8b": "j(p)-j(q) = \\left({1 \\over p} - {1 \\over q}\\right) \\prod_{n,m=1}^\\infty (1-p^n q^m)^{c_{nm}}", "22c8b6d45aa503fe11828c07d8f72070": "\n\\left(\\frac{\\partial T}{\\partial N_j}\\right)_{V,S,\\{N_{i\\ne j}\\}} =\n\\left(\\frac{\\partial \\mu_j}{\\partial S}\\right)_{V,\\{N_i\\}}\n", "22c8e0ca89de0a991eb80ba8e4909788": "\\scriptstyle B'(t)", "22c91e66de1ec0110c6e56792d6bd354": "\\scriptstyle f(b)", "22c93f1db08e62f9be885995800a23f7": "\\begin{align}\nx&= a_1 + n_1\\,x_1\\\\\n&\\cdots\\\\\nx&=a_k+n_k\\,x_k\n\\end{align}", "22c9cadd7be326b089a02a690bcfae93": "(AC_XA^T + C_Z)^{-1}", "22ca0bf4f8a252846e1230ab2317ee27": "\\ell^2(\\Z,\\R)", "22ca1e4bf74cbeaf4a3e975fd61d2a00": " \\chi_j ", "22ca4f25f895d9eff1b217a46c51e235": "72\\,\\text{Hz}\\,", "22ca8242cf7a3701611fcdab4e8dfa64": "{v_A^2 \\over 2}-gd+{P_A \\over \\rho}=\\mathrm{constant} ", "22ca90cf38b94a0cacdcfcbb81888d65": "w(n) = \\alpha - \\beta\\; \\cos\\left( \\frac{2 \\pi n}{N - 1} \\right),", "22caaf8a6d3b83a60147cf1aeebd2a2c": "[x_{k-1},x_{k+1}]", "22cacd68e86fdaf2c4e89036b65cf7fa": "R E", "22cb1d066965711886f7f32d7e339f5d": "R = {-1 \\over \\beta - \\alpha} = - P", "22cb5b460e26ec3f84212b29890c8444": "\\mathbf{Z}[\\sqrt{-1}]", "22cb7353becd1d379c0665375bf3ed08": "\\int_{-\\infty}^{\\infty} t^m\\,\\psi (t)\\, dt = 0.", "22cb79a41a510659669eda0a6beb14ae": "\\sigma_t: V\\to P(\\lbrace 1,2,\\dots,n\\rbrace)", "22cc17e596c8a7f56c0a0cf91e52da14": "||f|| = \\int_I \\frac{|f(x)|}{1+|f(x)|}dx.", "22cca3c2763975bc466d7c093f75ccdf": "(M,x) \\to (N,y).", "22cce6485be59b549cfa47d345cb4658": "i=1,2,3", "22ccfd23882d7b133ea545e86ec9b379": "[B(t,t_0),B^\\dagger(t,t_0)]=t-t_0", "22cd026d16ab1560ebc7ea257a07119e": "E(Y) = f(x)", "22cd966c78cf0bdc86e33a6f65e2cd3d": "P_{n+1}P_{n-1}-P_n^2 = (-1)^n,", "22cda15fa012b0aa16b56e2b782b5a8e": "\\sigma_{it}, i=1,2,3", "22cdab75fa4f8f4fa943c380088c7e83": "(18)\\quad k^c\\nabla_c \\hat\\sigma_{ab}=-\\hat\\theta\\hat\\sigma_{ab}+\\widehat{C_{cbad}k^c k^d}\\;,", "22cdfba0a1ebeb4a28e9923cae52f3da": "\\tilde\\lambda", "22ce083a837c03f6ac00fa32f090faa4": "1,\\ldots,m", "22ce395e620847f4bb68a699a4e3b054": "A-BD^{-1}C.\\,", "22ce56f9641b574163434fc04ae313e9": "\\sqrt{2} = 1.41421356237 \\dots\\,", "22ce89a67ea7c0606f279b68246bdfdb": " \\vartheta \\,", "22ceb3021cef9c0c6440cd66db26518f": "0\\leq i\\leq2^k", "22cec9fb06f2632dae2e18f9dc341597": "{52 \\choose 5} \\times 5 = 12,994,800", "22cee76ced886e62267b5ccdf7c6944c": "\\langle W,\\subseteq\\rangle", "22ceeec583e2a37b64d3b902df5b159f": "p \\mid m_i - \\left(i+1\\right)\\cdot m", "22cf2a620dbbf9604a6987d3f92cd549": "\\iota_{\\Omega_\\alpha} \\omega = \\alpha ", "22cf60311c48f84042ec15829ca57e33": "[m_1:b_1:1]_L \\cap [m_2:b_2:1]_L = g([m_1:b_1:1]_L \\times [m_2:b_2:1]_L)", "22cf6d90476d6b87655fb83d74ee08f5": "x_i\\ ", "22cf8f2060388a102b3b19f20eff55fe": "f_d", "22cfba13b9e3941f5d42ba93aeafc2f0": "a = \\sqrt{\\gamma\\, R\\, T \\over m_m}", "22cfcaee6fdd930609e7ef0912a8b2bd": "SV = \\bigcup_{x \\in V}S_xV \\subset T^*V", "22d0625861255a03788c33b708a968bb": " \\theta(y) ", "22d06c72798fbff71fd04ff95a0b4847": "a_n, b_n \\,", "22d0a945eea27eb58d7dc0db5269e8a7": "\\varepsilon = \\varepsilon_r \\times 8.854\\times 10^{-12}", "22d0b62b6c49ba34a65562e2c95468a4": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = J^{-1}~\\phi_{*}[\\dot{\\boldsymbol{S}}]\n", "22d0f55ea4fe03fcd11235288cfd7192": "\\eta^a_{\\mu\\nu} = \\begin{cases} \\epsilon^{a\\mu\\nu} & \\mu,\\nu=1,2,3 \\\\ -\\delta^{a\\nu} & \\mu=4 \\\\ \\delta^{a\\mu} & \\nu=4 \\\\ 0 & \\mu=\\nu=4 \\end{cases} . ", "22d10b0fa35414f1135574f773e217f3": "G(x,y)=\\sum^n_{i=1}{w_if(x_i,y_i)}", "22d130a1e88fa413d5125b364911a41e": "\\scriptstyle (m|k) \\;=\\; \\Pr(M\\;=\\;m\\mid K\\;=\\;k)", "22d220cf1140901d1acfea1561ce5bb8": "(\\frac{1 + 2\\gamma}{3})G_M, ", "22d263f15b992f2d921d0f468f32d67c": "\\mathbb{D}_8\\times\\mathbb{Z}_2", "22d28c8a344b73518482174da583d4b0": "\\times_{Ke}", "22d2bd888aca987a88ca290ec2ce84d2": "\\textstyle g^{(k)}", "22d2c1a2560de0cb328f9e1b5ab3fa93": " L=v+\\sum_{x,y \\in P \\cap \\mathbb{Z}} \\mathbb{Z}(x-y)\\subseteq \\mathbb{Z}^d", "22d347d27a47be575a278c8b08ae610e": "|jk> = e^{ik \\cdot r} u_{jk}(r)", "22d3661276d7c08d8dfcaad397db53f0": "\n Y\\Gamma = X\\Beta + U.\\,\n ", "22d389a801fe4e36114bf4d7be3253e9": "\\displaystyle{f(D_r) \\subseteq g(D_r).}", "22d3b80f23edf0208833afb47a9664e1": "P(x) = x^{10} + x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1,", "22d3e1c778f6f0b7d7999b957f076456": "\\psi(x)=\\int_a^b T(x,y) \\phi(y) \\,dy", "22d3f3e6a790b98c678c3737cb89dcf2": "VO_2 = \\frac {FR \\cdot (F_{in}O_2 - F_{ex}O_2)} {1 - F_{ex}O_2}\n", "22d418f8a8b2c6f1f211dc9dc86facab": " (\\mathbf{a \\times b})\\mathbf {\\cdot}(\\mathbf{c}\\times \\mathbf{d}) \\ ,", "22d45f795da3d935ba8f360cc95caa2b": "V_{T-j+1}(k)", "22d522d3cb8903ab87de21df83c457f9": "1 \\rightarrow {\\rm Homeo}_0(X) \\rightarrow {\\rm Homeo}(X) \\rightarrow {\\rm MCG}(X) \\rightarrow 1.", "22d5692a36f28033482dfbb336031aef": "\n\\Phi =2m_{w}S+S^{2}+2\\varepsilon _{w}A-A^{2}. ", "22d57784214e1849ca39755700509021": "f : Y \\to X", "22d5787ce98c757f51265c4189835526": "H_{n,m}=\\sum_{k=1}^n \\frac{1}{k^m}.", "22d58f986a91a64a65c36b719ec53b51": "R^{\\mathrm{irr}}(M)", "22d5bab3c27751760ca6706d617bc025": "\\mathrm{H m^{-1}=kg\\ A^{-2}m\\ s^{-2}}", "22d5c218958f1c20cbb1ad0935b7da31": "\\scriptstyle L^{+}", "22d5e330c821503795a8c584386b2435": "e[n] = \\mathbf{g}(\\hat{s}[n])-\\hat{s}[n]", "22d62ff86c45846a9f3aad19d1d93496": " x= \\pm {\\textstyle \\frac{R}{2}} ", "22d6b321d799f9745fb385204412a88a": "\\eta = \\frac{P_m}{P_e}", "22d7398bdcfabffac1dfb2667f2a0327": "2/2^M=2/m", "22d769999ddd33eaac5fd3b22662715c": "f(t) = \\pi^{-1/4}e^{(-t^2/2)}", "22d7910dba3761b7279fee9afb9e1882": "L(G) = \\{ww^R:w\\in\\{a,b\\}^*\\}", "22d7a1e557c0ca842e1b75ba6d2c7531": " t_g = \\sum_{i=1}^n a_i \\otimes_B g(b_i)", "22d7e9cd9df3c67c815d73aaba7142ed": "\n 2,\\ 4,\\ 4,\\ 4,\\ 5,\\ 5,\\ 7,\\ 9.\n ", "22d80b96c76d1e7a52bb57b85c1d6a0e": "\\Psi_m", "22d816cdaffe473e9e39f69e723f64a8": "\nF_{g} = sin\\theta gM\n", "22d8730cbb2430d3d6fb7d0cfac07eaf": "\\bot \\to P", "22d89488220b08661427036e3fff70b2": " \\text { Enrichment ratio} =\\left ( \\frac{\\text {Protein concentration in the foam}} {\\text {Protein concentration in the initial feed}} \\right )", "22d8f9e35eab2b8be32667c4bf0da1f4": "n_j = 1", "22d9bb2875d7a70aeb68696096f3b9b2": "k=0", "22da033f3890bc136cba5580f84b96c9": "\\scriptstyle \\| z \\| \\;=\\; 1.", "22da2023f8f367eeb6206a82ca1ca4ad": "f_!", "22daa808fe379d6d394a9dbb0c249e2b": "\n\\vec{w}= \\begin{pmatrix} w_{ATM} \\\\\nw_{RR} \\\\ w_{BF}\n\\end{pmatrix}\n", "22dae327fda5db8fec6b131d011aabc2": "(\\ln(R))^2", "22daf8bc7f5ca04eb9e9ce418cf5912f": "x \\stackrel{*}{\\leftrightarrow} y", "22db1031dcb36bf530e4f9b3b2c35c36": "d\\varepsilon_p > 0", "22db457b2fae003aa8dea3e5abc5f98b": "\\sqrt{ \\frac{3}{7} }", "22db8860461b91290d24c916adad2638": " \n(p,q') \\overset{\\alpha}{\\rightarrow} (p',q')\n ", "22dc00aa64c364f678244faa2f73df38": "E_0 = m c^2 \\,", "22dcb10dfd8956fcfdbc2778bfc020df": "a(r) = \\frac{F(r)}{m_2} = -g(r)", "22dcd2539d1d56be8a89fa0435b91851": "\\hat{X}_t(\\omega)=\\sum_{k=1}^N A_k(\\omega) f_k(t)", "22dd3045a0e13f527d39bbca5e01e3bf": "V_1, \\ldots, V_e \\subset {{\\mathbf{K}}}^n", "22dd4413280a578900d2d0dfb281e1f2": "E_k(t=0)=0\\frac{}{}", "22dd54aa750673b490ed6f91c78015ea": "[*:*:*:\\dots:*].", "22ddbf62f98628ea786257aff5b7a79b": "isopen", "22de1dad9c0e2ca378424e69bd58ca84": "\\mathbf{R}^{n-1}", "22de1f94595109c91ceec4427839ebab": "D_{mn}(x,\\alpha) = D_m(D_n(x,\\alpha),\\alpha^n) \\, . ", "22de2f6b95a4696194cab0645cfb0f4f": "N(x + iy)=x^2+y^2", "22de5f19c5b9a3350a930bd1769b8a78": "[\\hbar] = M^1L^2T^{-1} \\ ", "22dee644ebbdcf03082ee5a318d5ef1a": "C_Z^{-1}", "22def76b0c5b26c0fe458372886e3b5a": "\\sum_{a \\bmod n} \\chi(a) = 0 \\ ", "22df3bfa159427841071fc5bde856d8d": "\\mu : \\mathcal{R}^1 \\times \\Omega \\to \\mathbb{R}", "22df64cfced466dc281f190c8b118999": "\nP(q | \\vec y) = \\frac{1}{\\sqrt{2 \\pi} \\sigma_q}\n\t\\exp \\left \\lbrace - \\frac{\\left [q - \\bar q (\\vec y)\\right ]^2}{2 \\sigma_q} \\right \\rbrace\n", "22dfafcd62c3550fd5a4a1a72b89cc19": "r = \\sqrt{\\frac{(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}}; ", "22dffc78863472e96fc21f12a59ad781": "e/n", "22e01032d5fcaa51fd1709f479866b20": "S^u", "22e031cfce65a2cb553499b57e7ec119": "\\mathrm{1\\ Square\\ of\\ Land} =(\\frac{\\mathrm{77\\ acres}}{\\mathrm{3\\ Squares\\ of\\ Land}}) \\cdot 1 = 25.41\\ acres ", "22e03bff9f647ce776eab446cf8e6905": "\\int_{-\\pi}^{\\omega} X(e^{i \\vartheta}) d \\vartheta \\!", "22e0638796dbbb922223fef51aaaa483": "S = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\n & R(A) \\\\\n & W(A)\\\\\n & Com. \\\\\nR(A) & \\\\\nW(A) & \\\\\nCom. & \\end{bmatrix}", "22e0920e1aff68d5f69d8eb76cf5a7b4": "\\rho^{AB^L_j}", "22e0ac2f652a61a657ac17bf856c7c27": " n = 1+2\\pi \\frac{Nf(0)}{k^2}, ", "22e0cff7e3d0aae59f6778f95c3fa552": "\\langle C,X^*\\rangle_{\\mathbb{S}^n} = \\langle b,y^*\\rangle_{\\R^m}.", "22e0e32d8bbb4f7340a07d4501a3b134": "\\log_{10}", "22e12092f9902162c1c6582e543c92f3": " q_{ult} = \\frac{2}{3} c' N '_c + \\sigma '_{zD} N '_q + 0.5 \\gamma ' B N '_\\gamma \\ ", "22e192ed29440b29adf4c7ab579b8beb": "\n f^*(\\varphi) = \\varphi \\circ f \\,\n ", "22e1d7251d0b26b9f07d387c23012d60": "\\eta.\\,", "22e1f708b0c0894783ee62418f70f5ea": "C=\\frac{1}{n}R^{1/6}", "22e1f9a10a44975974b865c1a64f2eb7": "y(t)=h(t)=\\sum_{n=0}^{N-1}{\\rho_n e^{j\\phi_n} \\delta(t-\\tau_n)}", "22e200fea1113dbe93603000cc7532b2": "\n\\frac{\\partial(\\rho, \\theta, \\phi)}{\\partial(x, y, z)} =\n\\begin{pmatrix}\n\\frac{x}{\\rho} & \\frac{y}{\\rho} & \\frac{z}{\\rho} \\\\\n\\frac{xz}{\\rho^2\\sqrt{x^2+y^2}} & \\frac{yz}{\\rho^2\\sqrt{x^2+y^2}} & -\\frac{\\sqrt{x^2+y^2}}{\\rho^2}\\\\\n\\frac{-y}{x^2+y^2} & \\frac{x}{x^2+y^2} & 0\\\\\n\\end{pmatrix}\n", "22e2539e7481867fecb346fa2d0d069b": " P(n) = \\int_{N=n}^{N=\\infty} P(n\\mid N) P(N) \\,dN = \\int_{n}^{\\infty}\\frac{k}{N^2} \\,dN ", "22e2744501d61a9e6d970545cf6564f7": "f(x_1)=-0.931596", "22e286e3641bc6ec4931d833727b2e10": "\nP_\\mathbf{k} = P_\\mathbf{k} P_\\mathbf{k} = P_\\mathbf{k}P_\\mathbf{k}P_\\mathbf{k}=...\n", "22e28ce56139a3f95d4451b40101eba1": "ST_{i+1}", "22e32525978de2afcb523c07dc5392cf": "y(a) = A", "22e34fbd2ad7b4e2a51eb56bb09fbe73": " \\mathcal{P} = \\mathfrak{P}(\\mathfrak{C}(\\mathcal{Z})).", "22e3e0b04c69568c816aee6360c6556a": "\\hat{\\beta}_{FE}", "22e4725b1acb9670542fdb7c895dcd7b": "i=1,2,\\ldots,r", "22e47b98a0c45aa63d843362b64b9542": "\\!\\mathcal A \\models_X \\phi", "22e489002c01094d900991d247371f2e": "r_1,r_2 \\in \\{pino,exo,phago\\}", "22e491b93131c5b703d9f91a45f37aed": "f_2(z)=\\frac{i}{\\sqrt{3}}z + \\lambda", "22e4e3bb019bbf06a2a17e7c7e70c1ee": "F_i = \\frac12 \\times \\rho \\times S \\times C_i \\times V^2", "22e570efeb540a531b7e0db25305935f": " H(x)=H_0(x) + x f_0 ", "22e5a6f72a75b47314dd93e2d36e433c": "\nR_j = \\sum_{i=1}^b R(X_{ij})\n", "22e5cf28a04f45dee59f0181a1d9d8a2": "(S,O,P)", "22e61eb97def5e6b43a38244a4cb761e": "\\begin{align}\n y &= \\frac{E}{(\\gamma)_{-\\gamma}} \\sum_{r = 1 - \\gamma}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r (1)_{r + \\gamma - 1}} x^r + F x^{1 - \\gamma} \\sum_{r = 0}^\\infty \\frac{(1 - \\gamma) (\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(2 - \\gamma)_r (1)_r}\\Biggl(\\ln(x) + \\\\\n&\\qquad \\qquad + \\frac{1}{1 - \\gamma} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{\\alpha + k + 1 - \\gamma} + \\frac{1}{\\beta + k + 1 - \\gamma}- \\frac{1}{2 + k - \\gamma} - \\frac{1}{1 + k} \\right) \\Biggr) x^r.\n\\end{align}", "22e6c3f0ee1f8905e041d6c21c92cc26": "g_\\textrm{eff}", "22e6d5746e6ab192c386725cf9ed26fc": "\\nabla^2 \\omega_\\varphi = \\frac{1}{r}\\frac{\\partial}{\\partial r}\\left( r\\, \\frac{\\partial\\omega_\\varphi}{\\partial r} \\right) + \\frac{\\partial^2 \\omega_\\varphi}{\\partial z^2} - \\frac{\\omega_\\varphi}{r^{2}} = 0.", "22e6fb33c18e1bf8c36d363cb929a188": "[\\tfrac{\\mathrm g}{\\mathrm m^3}]", "22e790427895c1318b592dfb13a0415a": "E_\\gamma'", "22e798ea25efd6ba1b134cc5d658d0e9": "0\\times\\infty", "22e7b39854553cc52020035ba6860ca5": "\\frac{{d^2 Y}}{{dx^2 }} = \\frac{1}{{h^2 }}\\left( {2a_2 + 6a_3 z} \\right) ", "22e80712e1ef6ee60d30f2781c99625f": "\\mathbf{B} = \\nabla \\times \\mathbf{A}", "22e997671c8b8d0fc6e123375bd85df0": " \\sin \\delta' = \\frac{y_1'}{d\\,'}=\\frac{y_1}{-c\\,t_1'}\n=\\frac{y_1}{-\\gamma \\cdot (c\\,t_1-\\beta \\cdot x_1)}\n=\\frac{y_1}{-c\\,t_1 \\cdot \\gamma\\cdot \\left(1-\\beta\\cdot\\frac{x_1}{c\\,t_1}\\right)}=\\frac{\\frac{y_1}{-c\\,t_1}}{\\gamma\\cdot\\left(1+\\beta\\cdot\\frac{x_1}{-c\\,t_1}\\right)}\n=\\frac{\\sin \\delta}{\\gamma\\cdot(1+\\beta\\cdot\\cos\\delta)}", "22e999a52dc7430687a131e54ae312d2": "\\min|\\nabla^2 \\lambda_s | ", "22e99b2bf4aa1f5139945247d9d691cf": "\\rho: \\mathcal{L} \\to \\mathbb{R} \\cup \\{+\\infty\\}", "22e9a1fd370ba9921a4acf82ac96f684": "\\liminf_{n\\to\\infty} J(u_n) \\geq J(u_0)", "22e9aa730e42e918d65058d10d04c5c9": "C^\\infty(M) \\to \\Omega^1(M)", "22ea539b47a9cd3bd00b4f15fc100b60": "D = \\{ (x,y) \\in \\mathbf{R}^2 \\ : \\ x \\ge 0, y \\le 1, y \\ge x^2 \\}", "22eaaf4d1af87378b86b3b3f68ad6230": "K_s=\\frac{G\\cdot T\\cdot d}{a}", "22eb067ded0491350b556a4c3d47339c": "v_t={-\\operatorname{d}[M\\cdot]/\\operatorname{d}t}=2k_t[M\\cdot]^2", "22eb11bb1f6326c98bb47a97490c0b7a": "a^2-b^2 = (a+b)(a-b)\\,\\!", "22eb193dd71821ac481fd5a7b3c35b28": "\\mathit{b}_1\\mathit{b}_3\\mathit{b}_5...\\mathit{b}_{2n-1}", "22eb33dabe332a8d56b2e8c59d922217": "i\\colon X \\hookrightarrow \\mathbf{R}^m \\cong \\mathbf{R}^m \\times \\left\\{(0,\\dots,0)\\right\\} \\subset \\mathbf{R}^m \\times \\mathbf{R}^{N-m} \\cong \\mathbf{R}^N.", "22eb3b7a8e4ca76647ba5ad73118f58b": "C_r=t", "22eb527a1eaa4ce9a4005f4ad53aefcb": "({256-Priority)}/{256}", "22eb5f099b94df3465627a6c6d43ac17": "U(\\mathbf r)\\propto { \\int_\\text{Aperture} A(\\mathbf {r'}) e^{-i\\mathbf{k} \\cdot (\\mathbf{r'} -\\mathbf r)} dr'}= { \\int_\\text{Aperture} a_0 (\\mathbf{r'}) e^{i\\mathbf{(k_0-k)} \\cdot (\\mathbf{r'}-\\mathbf r)} dr' }", "22eb666e418ec71e5684e65f968e1ca7": "\n\\text{Corr}_r(X,Y) = \\frac{E[XY]}{\\sqrt{EX^2\\cdot EY^2}}.\n", "22ebcc0f00cf6301feaced0cfd484ea0": "G/(1+GH)", "22ebffaa068f8d9e635a7f1e485a6703": "J^\\alpha = \\rho_0 U^\\alpha = \\rho\\sqrt{1-\\frac{u^2}{c^2}} U^\\alpha ", "22ecaedf4651bf6d3fb6fa434968946b": "M'/IM'", "22ecb84d5c753f26be914c51a31ff86c": " f(x) = \\begin{cases} n & \\text{if }x \\in \\left[n, n+\\frac{1}{n^4}\\right], \\\\ 0 & \\text{else.} \\end{cases} ", "22ece8703114e57f45d62da7bf13ff36": "H_d^L", "22ed815ec63bca38eee6db9b2ee0f6ec": "\\langle X, ({\\mathbf e}\\cdot g)^*\\omega\\rangle = \\langle [d(\\mathbf e\\cdot g)](X), \\omega\\rangle", "22ed847edd18a81b80c3740b474d419c": "\\displaystyle{\\Phi_-=\\partial_z D(\\varphi)|_\\Omega \\in A^2(\\Omega),\\,\\,\\, \\Phi_+=\\partial_z D(\\varphi)|_{\\Omega^c}\\in A^2(\\Omega^c).}", "22ed949936c7631f7ba8bb3284e52b2d": "b_{8}=b_{9}", "22ee096edc2ef98efbed688e1ebf59f7": "MV_i", "22ee438dd5b1b796564a32eaa1945728": "(x^2-4\\alpha)y'' + 3xy' - n(n+2)y=0. \\, ", "22ee57f531d782c4879a12d9d2f9e637": " \\text{area}(\\Delta)=v_\\mathrm{interior}+{v_\\mathrm{boundary}\\over 2}-1 \n", "22ee87e2422ab33407db87ed858c7865": "\\begin{matrix} {1 \\choose 1}{3 \\choose 1}{44 \\choose 1} \\end{matrix}", "22eeba46963601c5e3d47f2f76a0e5c3": "52!/(13!)^4", "22eee2a7796e208e35c837e0f267fb9b": "\\begin{bmatrix} \\dfrac{a_{22}}{\\Delta \\mathbf{[a]}} & \\dfrac{-a_{12}}{\\Delta \\mathbf{[a]}} \\\\ \\dfrac{-a_{21}}{\\Delta \\mathbf{[a]}} & \\dfrac{a_{11}}{\\Delta \\mathbf{[a]}} \\end{bmatrix}", "22eee3168be4b62b963096d4fcbbd920": "c_i u(x_i)", "22ef00d9b773564414497cd5ed0b8f34": "G_{Newton}", "22ef086fb5099e38cce2c95badc3ca05": "\\mathrm{On}(\\mathrm{box},\\mathrm{table})", "22ef0ab2ecfc80f99751536f12372062": "K_\\text{joint}=\\frac{\\Delta M}{\\Delta \\theta}", "22ef3e19c49e6f387bd23164fbd94bff": "{\\mathbf{}}+P(t)C'(t)W^{-1}(t)C(t)P(t),", "22ef3fa14d2c529866b606a2ce8d0f2b": "P_K(p^0,p^1,q)=\\frac{C(f(q),p^1)}{ C(f(q),p^0)}", "22ef68e8e6c6ac5245c71f4a59132a76": "\nx_n\n=x_{n-1}-f(x_{n-1})\\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}\n=\\frac{x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}\n", "22f1243f5a04fe18c6bdf57e74e423fa": "\\begin{matrix} {48 \\choose 5} = 1,712,304 \\end{matrix}", "22f125c2773a73ac8d13eda027963b48": "a_1,\\ldots,a_n", "22f18ffb924fac1950c89b1acb04bbba": "xu\\sim yv\\,", "22f19570a3c98d06252ea2d62ef7dd70": "\n\\begin{matrix}\nV &=& C + i\\theta\\chi - i \\overline{\\theta}\\overline{\\chi} + \\frac{i}{2}\\theta^2(M+iN)-\\frac{i}{2}\\overline{\\theta^2}(M-iN) - \\theta \\sigma^\\mu \\overline{\\theta} v_\\mu \\\\\n&&+i\\theta^2 \\overline{\\theta} \\left( \\overline{\\lambda} + \\frac{1}{2}\\overline{\\sigma}^\\mu \\partial_\\mu \\chi \\right) -i\\overline{\\theta}^2 \\theta \\left( \\lambda + \\frac{i}{2}\\sigma^\\mu \\partial_\\mu \\overline{\\chi} \\right) + \\frac{1}{2}\\theta^2 \\overline{\\theta}^2 \\left( D+ \\frac{1}{2}\\Box C\\right)\n\\end{matrix}\n", "22f199833017f92f64fb5db5064f58ef": "\\int x^2 e^{cx}\\;\\mathrm{d}x = e^{cx}\\left(\\frac{x^2}{c}-\\frac{2x}{c^2}+\\frac{2}{c^3}\\right)", "22f1acd1a4fbf71dc20b7ec840ba10a1": " B_5", "22f26060bb20519f5b0e5903562ad2e0": "\\scriptstyle D(E)", "22f264899eb897c3c2b86c1841d2748b": "h_V(F)", "22f288dff7f7fd02841a6f9faf866615": "\\sqrt{re^{i\\theta}} \\,=\\, \\sqrt{r}\\,e^{i\\theta/2}", "22f315c35d1b1baae66ac8ea06b4cb62": "H=-\\boldsymbol{\\sigma_1} \\cdot \\bold B-\\boldsymbol{\\sigma_2} \\cdot \\bold B-\\boldsymbol{\\sigma_1} \\cdot \\boldsymbol{\\sigma_2}", "22f36059ec66f6fccbbf213adcf799ce": "M\\setminus X=(M^\\ast/x)^\\ast", "22f36e0df2b5695fd40510ba95e2ff11": "i \\in [0,n]", "22f401e22c11f23ecc63c154e4242f8f": "\\frac{1}{c}\\sqrt{(hf - hf' + m_{e}c^2)^2-m_{e}^2c^4} > \\frac{hf - hf'}{c}. ", "22f44c8d97bb37e28b0fd9862e733667": "b = |S^{*}|", "22f45b29e686e354046350faedc38e23": "\\sum_{a=1}^f\\chi(a)\\frac{te^{at}}{e^{ft}-1}=\\sum_{k=0}^\\infty B_{k,\\chi}\\frac{t^k}{k!}.", "22f46facc109a4f629ea7810679b6d37": " \\begin{align}& f( \\boldsymbol{x} ) = \\sum_{|\\alpha|\\leq k} \\frac{D^\\alpha f(\\boldsymbol{a})}{\\alpha!} (\\boldsymbol{x}-\\boldsymbol{a})^\\alpha + \\sum_{|\\beta|=k+1} R_\\beta(\\boldsymbol{x})(\\boldsymbol{x}-\\boldsymbol{a})^\\beta, \\\\&\nR_\\beta( \\boldsymbol{x} ) = \\frac{|\\beta|}{\\beta!} \\int_0^1 (1-t)^{|\\beta|-1}D^\\beta f \\big(\\boldsymbol{a}+t( \\boldsymbol{x}-\\boldsymbol{a} )\\big) \\, dt. \\end{align}\n", "22f4c43698aaffeb960a7431f56e7f2b": "\\gamma(t) = (t, f(t))", "22f5ad893189686c8162b5fe4a52f4a1": "\\left( \\begin{smallmatrix} 1 & -2 \\\\ -1 & -2 \\\\ \\end{smallmatrix} \\right)", "22f65ff3cfbd5e2581bde6550f97ed04": "\\theta_2=\\theta_4\\;", "22f6672d27ebd439ab873803bd5eb74f": "x \\div 2", "22f6e8f374dba554beb0f2253b3522d0": "K^*_p(s;\\theta,\\lambda) = \\begin{cases} \\lambda\\kappa_p(\\theta)[(1+s/\\theta)^\\alpha-1]\n & \\quad p \\ne 1,2 \\\\ -\\lambda \\log(1+s/\\theta) & \\quad p = 2 \\\\ \\lambda e^\\theta (e^s -1) & \\quad p = 1 \\end{cases}\n", "22f6fa494752b44996195ef3e7460c7f": "\\cdots\\rightarrow\\mathrm{Tor}_2^R(M,B)\\rightarrow\\mathrm{Tor}_1^R(K,B)\\rightarrow\\mathrm{Tor}_1^R(L,B)\\rightarrow\\mathrm{Tor}_1^R(M,B)\\rightarrow K\\otimes B\\rightarrow L\\otimes B\\rightarrow M\\otimes B\\rightarrow 0", "22f705717a70e84ebedcab10d55f3a48": "\\omega_1=R_1", "22f80facb2dca684c48d3ecf5b5e830a": "\\Delta\\ R(n)", "22f852dd1a0d559fad4e7285d87dbbe9": "RQ = \\sin \\alpha \\sin \\beta\\,", "22f865b7424640766aa415fd04678c44": "{V^2 \\over 2} \\left({g \\over g+2}\\right)", "22f890ed84298aa3ed42a31142816ef9": " x \\neq 0", "22f8c6d9223b2aec1ca4596fcea846fd": "\\widehat{H} = \\frac{\\widehat{\\mathbf{p}}\\cdot\\widehat{\\mathbf{p}}}{2m} + V(\\mathbf{r},t) ", "22f8da1d27bb6ded691a104bfe83278d": "r_n = \\sqrt{n \\lambda f + \\frac{n^2\\lambda^2}{4}}", "22f938662ef8f0191ea1e88788b80bea": "\\eta = \\zeta + f", "22f9aec3b1a0b58958335d25543ca153": "M'|_{\\sigma}\n\\models_{\\Sigma} \\varphi", "22fa0023b1cce6f0cc59dcf34f49c978": "G^{(n)} := [G^{(n-1)},G^{(n-1)}]", "22fa92a4f599d9d4e6e8f647af631f65": "B=T_{b}P_{b}^{'}+E_{b} \\,", "22fad7b1c3f32ec9d4b77f3c3e39a055": "V_\\mathit{Au}", "22fbb20a50589c272943586195928d7c": "\\frac{-b + \\sqrt {\\Delta}}{2a} \\quad\\text{and}\\quad \\frac{-b - \\sqrt {\\Delta}}{2a},", "22fbfe2dd949cade4ee4367d509a541c": "\\operatorname{Cov}(X_i Y_i, X_j Y_j) = \\langle X_i X_j \\rangle \\langle Y_i Y_j \\rangle = M_{ij} N_{ij}", "22fc0f000d103985b67453ee5ea6de9e": " \nP(A)\\geq P(t \\in \\cup_{k=1}^\\infty [U_k,V_k])\\to 1 \\quad\\text{as}\\quad t\\to\\infty\n", "22fc2a9064fdf57fe13242c7953dbbf3": "\\zeta(a,a)=\\tfrac{1}{2}\\Big[(\\zeta(a))^{2}-\\zeta(2a)\\Big]", "22fc3804bc17c2e60fc92057d0c8f681": "\\mathrm{E}(e_t e_t') = \\Omega\\,", "22fc5185530c55ddeade7b62048fdd85": "\\langle f,g\\rangle=\\int_0^1 f(t) \\overline{g(t)} w(t) \\, dt.", "22fc9da9357cf962536f136d35d1d0fe": "K:=\\mathsf{Quot}(D)", "22fc9ee0ce97e758226989f934a0b2ce": "(u_1, u_2, e, v)", "22fcb20dddcf288e461ba1a0f6ee14ae": " I(\\cdot,t): \\Omega \\rightarrow \\mathbb{R} ", "22fdbb398361c988e1426aa3b8003667": "(\\forall x\\,D(x)\\Rightarrow M(x))\\land \\neg(\\exists y\\,M(y)\\land B(y)) \\Rightarrow \\neg(\\exists z\\,D(z)\\land B(z))", "22fdbbe8801aef3fc41b45b6e50ecc61": "\\mathbf{x}=\\mathbf{0}", "22fdd2644056adc3082414529a73cb51": " F=\\sum_i X_i X_i^T ", "22fe010f477e0c8fc8e0387ea420851c": "x^n(x^3-x-1) = -(x^3+x^2-1) ", "22fe10ce60d8fdc529a8871cc21852ff": " \\exists{n}{\\in}\\mathbb{N}\\, \\big(Q(n)\\;\\!\\;\\! {\\wedge}\\;\\!\\;\\! P(n,n,25)\\big) ", "22fe172a4e348397ad14e33e28f3c42b": "\\sin \\frac{\\pi}{10} = \\sin 18^\\circ = \\frac{\\sqrt 5 - 1}{4}=\\frac{\\varphi-1}{2}=\\frac{1}{2\\varphi}", "22fe3cdddae451c56ce053bf01b5f677": "{Z_0}^2 - Z_0 \\delta Z = \\frac {\\delta Z}{\\delta Y}", "22fe80e01877c307adcb1259a6be52c5": "\\mathfrak{P}^{79}", "22feba68f74656cc2728f2680f70653d": "\\Phi(\\mathbf{u})=R_{l}(u)Y_{lm}", "22ff393a1512745b7e8047514e231915": " \\phi (p) ", "22ff5fab92f1b68f3929d883c7ccc134": "r_1 e_1 + r_2 e_2 + \\cdots + r_n e_n = 0_M", "22ff82baf2e3c2d7457ae7c44a243bcf": "\\mathrm{p}I=\\frac{\\mathrm{p}K_1+\\mathrm{p}K_2}{2}", "22ff96923684c09029dd90763f969478": "\\scriptstyle f_n(x) = \\frac1n \\sin(nx)", "22ff9ce6ad7f1d3fc56099e93bf0883b": "(1+n)^x=\\sum_{k=0}^n {n \\choose k}x^k = 1+nx+{x \\choose 2}n^2 + higher\\ powers\\ of\\ n", "22ffb27c9be319dbdbecd3f4e05fa7cb": "sk_n(K) := i^* i_* K.", "230012c6f99aca97efed4e1bcef70310": "\n\\begin{align}\n\\operatorname{var}(X) & = \\operatorname{E}(\\operatorname{var}(X\\mid Y)) + \n \\operatorname{var}(\\operatorname{E}(X\\mid Y)) \\\\\n & = \\frac{1}{9}\\operatorname{var}(X) + \n \\operatorname{var}\n \\left\\{\n \\begin{matrix} 1/6 & \\mbox{with probability}\\ 1/2 \\\\ \n 5/6 & \\mbox{with probability}\\ 1/2\n \\end{matrix}\n \\right\\} \\\\\n & = \\frac{1}{9}\\operatorname{var}(X) + \\frac{1}{9}\n\\end{align}\n", "23004d0620341a3d9071989768839a2c": "\\langle R(u,v)w,z\\rangle =\\langle \\mathrm I\\!\\mathrm I(u,z),\\mathrm I\\!\\mathrm I(v,w)\\rangle-\\langle \\mathrm I\\!\\mathrm I(u,w),\\mathrm I\\!\\mathrm I(v,z)\\rangle.", "230061e941713013a0db9fc929beba0e": "y^k(0)", "2301009e66e55e01b984dc8c6f5fbba0": " v_i = \\left\\{\\begin{matrix} \n-(\\sqrt{5} -1)/2 & \\mbox {with prob. } (\\sqrt{5} +1)/(2\\sqrt{5}) \\\\\n(\\sqrt{5} +1)/2 & \\mbox {with prob. } (\\sqrt{5} -1)/(2\\sqrt{5})\n\\end{matrix}\\right.", "230177a498dcf374b3b53957fa8e05b1": "x \\mapsto \\lceil x \\rceil ", "23019b2c16f2bbf38ea3614b93e0cc7d": " \\operatorname{E}[X]=\\int_0^\\infty P(X \\ge x)\\; \\mathrm{d}x", "2301abbe24cd34957bd2da20069e26dd": "\nz_{pj}^{new} = z_{pj}^{old} + \\eta (x_{ij}^p-z_{pj}^{old})\n", "2301bb6eca121fd14c03eb848b9131e7": "\\begin{pmatrix} z & \\pm z \\\\ \\pm z & z \\end{pmatrix} \\equiv z (1 \\pm j) \\equiv z (1 \\pm \\varepsilon)", "2301d21430babc6749805421a309085d": "\\begin{bmatrix}-1&0\\\\1&0\\end{bmatrix}:\\mathbf b", "23021d7deda8dd985feea3361ddff3c5": "\\begin{cases}\n \\frac{ b^2 \\left( -(n-2){\\zeta(n-1)}^2+(n-1)\\zeta(n-2)\\zeta(n) \\right)}{(n-2) {(n-1)}^2 {\\zeta(n)}^2} & \\text{if}\\ n>3 \\\\\n \\text{Indeterminate} & \\text{otherwise}\\ \\end{cases}", "23026a02c604089bf625625c47df7c4d": "\\mathcal{L}(\\theta | x)/\\mathcal{L}(\\hat \\theta | x).", "2302867fe83a7e4e3e837097afcaa2d7": "\\lim_{x\\to a} f(x)\\,", "23029a3f3167126c932fae635bf27942": "\n\\begin{align}\n\\mathbb{A}:\\ \\sum_{i\\in\\mathbb{A}} 2^i &= n-k,\\\\\n\\mathbb{B}:\\ \\sum_{j\\in\\mathbb{B}} 2^j &= \\left\\lfloor\\dfrac{k - 1}{2}\\right\\rfloor,\\\\\n\\end{align}\n", "2302a1ebf61ffd94d4f2b54fbfc8fc0d": "E(x) = x^i + x^k = x^k \\cdot (x^{i-k} + 1), \\; i > k", "2302aad96d1bb3751d6830750da33d80": "Hf := p.v. \\frac{1}{\\pi} \\int_{-\\infty}^\\infty \\frac{f(y)}{x-y} dy", "2302ac9939a0aed0905c827aad7f6fc3": " C \\vee D ", "2302f615a22721d010362b96785d9224": "\\text{dBW} = 10 \\log \\left(\\frac{\\text{power out}}{1\\,\\mathrm{W}} \\right)", "23047fe13afc543e209a67d1cb8c234e": "\\delta\\mathit{u}=({\\mathit{c}_{v}})({\\delta{T}})", "230511aef09db5df73f8cee151aedd5d": "Y_{10}^{4}(\\theta,\\varphi)={3\\over 256}\\sqrt{5005\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(323\\cos^{6}\\theta-255\\cos^{4}\\theta+45\\cos^{2}\\theta-1)", "230524ed32654bb0474613f749cd1160": "R_{\\text{C}}", "23054e4b7f917c799ae6e50126cdfe3a": "(\\mathbb{Z}/n\\mathbb{Z})^\\times", "2305a1f45a161a61d14a96715fb63c2e": "\\frac{dx}{ds} = \\frac{1}{\\frac{ds}{dx}} = \\frac{a}{\\sqrt{a^2+s^2}}\\,", "2305c5f77d759d096f6baa1eb0e63f36": "x=10^7", "2305e2ea419df87e959238e900e1c0de": "M_i = \\frac{v_i}{a} = \\frac{1}{a} \\frac{\\partial \\Phi}{\\partial x_i}.", "2305e50866edb8b78d64ad4d722deda3": "\\ P_t", "2305e7121098d4e0453f0e966a5e2efb": "\\frac{1}{1828}", "23060848695a4e86616b04628c690052": "f(x) = a(x - x_1)(x - x_2)\\,\\!", "230683d7d3c44607b309d1398a9bfb3d": "k=\\alpha+\\beta", "230690b6009633aef7f3952d13f5bbe5": "\\phi(1,0,\\alpha)", "2306afe17e5a963a6db8f0c5bfef940a": "x\\mapsto P(x)", "2307257f77f2f6f7b529dbd0b716bc8c": "2\\theta", "230725c8a902fece069278555e44b20e": " f(E;\\beta)=\\frac{e^{-\\beta E}}{\\mathcal{Z}(\\beta)}\\Omega(E), ", "2307b3524d2b98ad095260b2a07d26dd": "\n-\\operatorname{dn}^2(u)+m_1= -m\\;\\operatorname{cn}^2(u) = m\\;\\operatorname{sn}^2(u)-m\n", "2307c3b04fae9951bd06cb895aa5596d": "dist(a,F_l) = min_{x:W^{\\prime}x = \\gamma}\\|x-a \\|_F^2 = \\|W(W'W)^{-1}(W^{\\prime}x - \\gamma) \\|_F^2 = \\|W^{\\prime}x-\\gamma \\|_F^2. ", "2307d7a384963a0d4ba7d43c108a25ee": "SiO_{2}(s) + H_{2}(g) \\rightleftharpoons SiO (g) + H_{2}O (g)", "2307fd207601cbdd65c9657ac82d3af7": " E_{\\textrm{out}}=A\\,(E^+-E^-)=A\\,(E_{\\textrm i}-E_{\\textrm r}) ", "23080dd8dabe6f5f3f1257f3afa1f431": "(\\nabla\\cdot \\vec E(x) - \\rho(x)) |\\psi\\rangle = 0.", "230870d60568d3489294d8a442c03a50": " \\alpha = \\frac{ 648000 }{ \\pi } \\frac{ D }{d } ", "23087ec4b16e9e22b7a74a9ee85cbf3f": "\\overline{u}(x,y)", "2308ce4093a9f53fb7c79dcc57b69421": "\\mathbf{J}_u ", "2308f08f54c2aa0e02fa012e8ba23e69": "u(x)=\\int_0^\\ell f(s) G(x,s) \\, ds,", "230907278d4536f7774c88169d4cebfa": "X^2 \\sim \\chi^2_k", "23098ff0969bf5f743a51dc144bb14b6": " \\begin{bmatrix} a & b \\\\ b & a \\end{bmatrix}. ", "2309dcac9d120bf6dace40b5da2f8657": "|P_c^n(0)|", "2309f5d7da7451c2ebb83f0492a53060": "\\mu_1= \\frac{ 2}{ \\bar{x} } \\left[ 1 + \\sqrt{ 1 + 2 ( c^2 - 1 ) } \\right]^{-1}", "230a19c4f2a98b247a96bf2bad958849": " B_0", "230a2f63233afc6a69916e7b48655457": "\n\\beta_{i}=\\frac{cov(r_{i},r_{M})}{\\sigma^{2}(r_{M})}\n\\qquad (2)", "230a92030ee28292aecceec8b8e648c7": "\\dot{x}=n/R", "230a96b33c823b586f9a118fda5589a0": "\\frac1k+\\frac1k=\\frac1k+\\frac1{k+1}+\\frac1{k(k+1)}.", "230aab57abce22285fb273b1b2f1b0a4": "t > 0", "230ac4f533ceebb6a21f5cf31d45d8d0": " \\{ (1 \\otimes C) \\Delta(C)\\} ", "230af5f8816bfde0a5fbfb0da04aca77": "l(E^2)", "230afda05fea500e61a98b5bf6ed7d31": "\\,t_i", "230b05027bba27a1a7e0b39522c9fae7": " I_{\\mathrm{xy}}= I_{\\mathrm{yz}} = I_{\\mathrm{zx}} =0.\\,\\!", "230b0a094fed614a9837462f91313e9b": "\\!f(k; \\lambda)= \\Pr(X=k)= \\frac{\\lambda^k e^{-\\lambda}}{k!},", "230b0c186f7270d21df0a906be2e708a": "x_{\\mathrm{per}}(t)", "230bba38bf8502715eab8532e6161075": "\\psi: G \\to \\mathbb{C}^\\times", "230c5be17836808745534341c2b013e4": " a = \\omega_a + \\ln(\\omega_a) ", "230c78e77f33a4c022acbfcf22deafe3": " t=0 ", "230c934da5d4b5b490ed13d934ce7dcf": "X_t = \\int_0^t Y_s\\, d W_s.", "230ca2b209963f5f766cdac6016305fb": "GDGT ratio-1=\\left(\\tfrac{[GDGT-2]}{[GDGT-1]+[GDGT-2]+[GDGT-3]}\\right)", "230cbd34cd6d77898ee190e915cae0af": "V_m^2=2 \\frac{C^2}{C^2-1} \\frac{P_{settl}-p_m}{\\rho} \\approx 2 \\frac{\\Delta p}{\\rho}", "230cf541b93aad9f46dcd2f8768b5697": "c=\\cos\\theta", "230d00fbfda6756e68062a7308f3c74c": "K(p\\mid x)\\approx 0", "230d0baf751e4d052a3ca8c2fa716bc0": "\\begin{matrix} {13 \\choose 4}{4 \\choose 2} \\end{matrix}", "230d459bbe09ecc9b72c1649b749dfe9": "3f \\le 2e", "230d862bc744aa2593ff6284d77fbb66": "\n\\eta_{a\\mu\\nu} \\eta_{a\\mu\\nu} = 12 \\ ,\\quad\n\\eta_{a\\mu\\nu} \\eta_{b\\mu\\nu} = 4 \\delta_{ab} \\ ,\\quad\n\\eta_{a\\mu\\rho} \\eta_{a\\mu\\sigma} = 3 \\delta_{\\rho\\sigma} \\ .\n", "230e0fa157fca83b162e2aa9ea804a1c": "x+x^5\\,", "230e7414692f4ac8812c78fe0021a2b4": "\\{ \\mathbb{I}_{ \\{ x \\} } \\mid x \\in X \\} ", "230f0c736c8c262e3b210008b4e2b84b": "\\underline{\\underline{\\mathsf{S}_k}}", "230f1a588f3ad7335bd21a06826c0a89": "\\mathbf{S}= n\\mathbf{U}", "230f3ee5eeb9715975662d97dbe3834d": "3.04 = y_2 + \\frac{q_2^2}{2gy_2^2}", "230f86cde757defd574865449bd4cccb": "n=\\lim_{x\\rightarrow+\\infty}(f(x)-mx)=\\lim_{x\\rightarrow+\\infty}\\left(\\frac{2x^2+3x+1}{x}-2x\\right)=3", "230fe39110734f96f8a1339f996e96c1": "\\dot n_3", "231014c4b811712d23ae0065656fa636": "d \\mathbf{S}= d\\mathbf{r} \\mathbf{ \\times} d\\mathbf{l}", "23107cbfd471282bfcd256c03b4c0a5f": "|\\mathbb F|^m", "2310b7efec383853fd5ebf4450fb32e4": "\nt_{1/2} = \\frac{\\ln (2)}{\\lambda_e}\n", "2310bf4db2d4df990d3968cf24341b61": "A\\star B\\simeq \\Sigma(A\\wedge B)", "2310f4d1db9b79c9823ef2921fdfbe89": "\\vdots \\!\\,", "2310fcb903a834976a8cb874163cea36": "Term \\rightarrow Term\\,*\\,Factor\\,|\\,Factor", "231139902b600f0a2667f05e171178a5": "\\begin{matrix}\n \\underbrace{3_{}^{3^{{}^{.\\,^{.\\,^{.\\,^3}}}}}}\\\\\n 7{,}625{,}597{,}484{,}987\\mbox{ multiplied copies of }3\n \\end{matrix}", "23114540a0d686317aec37435cc4fabe": " G = 2\\pi / d .", "23115a683378406a49f2ecc06c3cb215": "\\lambda(I-P)=\\gamma \\, , ", "23118956d0ef63833d33f87979517f00": "S=\\lim_{\\delta \\to 0} \\sum_{k=1}^m f(P_k)\\, \\operatorname{m}\\, (C_k)", "231193c9a440a576e96d31fcb7d3a840": "22 x 34 x 16.5 cm^{3}", "23119775abd0f5e44d5d6d464dc9c5b5": "\\sqrt{8}", "2311bc09829951d345e0e2134f73d92a": "\\underline{P}(Cl^{\\le}_t)", "2312318a3d6f0523ddeb199d81b979c2": "g:U\\to V", "231235977c18c870b95ad93f2d891c19": "b_r = a \\cdot p^n", "2312387d03bbc97d0603638ba7aea397": "I_z = I_x + Ar^2,", "23128b8cb7328e6e876d2b58cc568f80": "g^{xy}", "23128e728469b331b523c975833e6b26": " (\\boldsymbol\\mu,\\boldsymbol\\Lambda) \\sim \\mathrm{NW}(\\boldsymbol\\mu_0,\\lambda,\\mathbf{W},\\nu)", "2312c4994315c260141cb9f1566ad9aa": "d(df)=d^2f=0", "2312e2593eb0306fbfc855ff56a3a3fc": "\\lnot (P \\lor Q)", "23135c83bec9892a858f5df36b570e15": "\\scriptstyle F_n", "23135fe7433de2cb77eb245c58b94009": " \\hat{A} ", "2313b8be1c82dc40fc40b928a8ae5af4": "\\,\\,\\sigma_{ij} = -P\\,\\delta_{ij} + 2\\mu\\dot{\\varepsilon}_{ij} + \\lambda\\dot{\\varepsilon}_{kk}\\delta_{ij}", "2313c6bd3548a7c09f99089c9d819e8c": "\\lambda_{1} ,\\dots ,\\lambda_{n}", "2313ec33c5a56d1926036bf6ccaf8496": "\n\\begin{align}\n\\lambda = \\frac{h}{\\sqrt{2mE}}, \\qquad \\lambda[\\textrm{nm}]=\\sqrt{\\frac{1.5}{E[\\textrm{eV}]}}\n\\end{align}\n", "23145d60e7030c6dcbe596d2b2bc3b46": "0_K \\, ", "2314f3fd7a034e8881916792e583d3ad": "y_1 = \\ell\\cos\\theta\\,", "2314f8332cc30a42b15c5c68c040b3c2": "\\ni_X", "231519a78c94ecac2754031288439eea": "\\begin{bmatrix}A & B/2 & D/2\\\\B/2 & C & E/2\\\\D/2&E/2&F\\end{bmatrix}", "2315b140e19d399613f103c678f54c43": "r_1, \\ p_2", "2315cbb6356aacb11611c3589a61a27e": "S=-(n_\\mathrm{s}-n_\\mathrm{\\bar{s}}),", "2315d8baf06210e5c406fb036f8a36b7": "A_1~r~\\cos\\theta\\,", "2316258b9a321d49e079a1f92b7c220e": "\nR(t) = \\frac{R_{+} - gR_{-}}{1 - g} \n", "23165550808dc05ba62fd9793ba25b08": " \\lim_{N \\rightarrow \\infty} \\left(I + \\frac{i}{\\hbar}\\frac{\\Delta\\mathbf{r}}{N} \\cdot \\widehat{\\mathbf{p}}\\right)^N = \\exp\\left(\\frac{i}{\\hbar}\\Delta\\mathbf{r} \\cdot \\widehat{\\mathbf{p}}\\right) = \\widehat{T}(\\Delta\\mathbf{r})", "2316b6ae497378cbbde1e9c6a3d62223": "(a_1 + \\cdots + a_n) p = a_1 x_1 + \\cdots + a_n x_n", "2316d6de164515ad7cc41181d980b13a": "[x, z y] = [x, y]\\cdot [x, z]^y", "23170ce210a776a1b748591e7bea019d": " \\ f_c", "231713988facdd83b119e7d2199f52bf": "c^2\\ = a^2 + b^2 - 2ab\\cos(\\gamma)", "231757a21070724a0c33ceaecc78f31a": "a(\\sigma+\\tau)", "2317793a8de61ab32c0f17adff9ea8d4": "x, y", "2317978c7fba5d4b31c782cd418ac89d": "\\scriptstyle\\angle PCP'\\ =\\ 60^{\\circ}", "2317a1ccc9c873b1b1fe3ab54f532ccc": "x^3-tx^2+(t-3)x+1", "23180b39cb9ea45e15f5c4c5b297a328": " \\{\\mu_\\ell\\}_{1 \\leq \\ell \\leq \\omega} ", "23180d3bfa1b6e008cc678c31c92829c": "\\frac{d^{2}\\eta}{d\\theta^{2}} + \\eta = \\eta J^{\\prime}(u_{0}) + \\frac{1}{2} \\eta^{2} J^{\\prime\\prime}(u_{0}) + \\frac{1}{6} \\eta^{3} J^{\\prime\\prime\\prime}(u_{0}) + \\cdots", "2318a1b670d8d9d3932392055f1b5815": "\\ v_z", "2318e923074452066aefd71555d84015": "\\left\\langle \\sum_{j=1}^n b_j K_{y_j}, \\sum_{i=1}^m a_i K_{x_i} \\right \\rangle = \\sum_{i=1}^m \\sum_{j=1}^n \\overline{a_i} b_j K(y_j, x_i).\n", "2318eee43508b4fc2c38bbfae4f83830": "\\left(c, t, y\\right)\\succsim \\left(d, q, y\\right)", "2318fd28280e98162d1997aef2c47266": "\\Phi(x,z,t) = \\beta x - \\gamma t + \\varphi(x-ct,z),", "231913f7b50c942bea5dcba554f7136e": "\\mathbb{G}(k, n)", "23195da5a5101c716bdf110938e63e3f": " \\forall \\epsilon >0 \\, \\underbrace{\\forall x \\in \\mathbb{R} \\, \\, \\exists \\delta > 0} \\, \\forall h \\in \\mathbb{R} \\, \\left( \\, |h| < \\delta \\, \\to \\, |f(x) - f(x+h)| < \\epsilon \\, \\right)", "23195e81cb3f118faed4391f01e55ef6": "\\mathbb Z^k", "2319741eb6cbaadad952a3a5f8eaf937": "(\\hbar^2 j(j+1))", "23198942fb12e0346d3c3a0c7a568a29": "\nx_n = \\sum_k [a_k \\cos (2\\pi \\nu_k n) + b_k \\sin (2\\pi \\nu_k n)].\n", "231a5e59501c03138fc7b97fa1d64f99": "\n \\dot{\\boldsymbol{U}}\\cdot\\boldsymbol{U}^{-1} = \\left[\\begin{array}{ccc}\n\\dot{\\lambda}_{X}/\\lambda_{X}\\\\\n & \\dot{\\lambda}_{Y}/\\lambda_{Y}\\\\\n & & \\dot{\\lambda}_{Z}/\\lambda_{Z}\\end{array}\\right]=U^{-1}\\dot{U}\n", "231a7d9c3ebb0b43d27e1ef688dbf41a": "\\beta_j^+(r_m^+-r_f )_t", "231aa2bff58513c68b5ae63332d4b5f6": " \\beta_{FB} = \\frac {1} {G_{\\infin}} \\ , ", "231aad2be32ac2295b4e74ebf73435e8": "E_g(T)=E_g(0)-\\frac{\\alpha T^2}{T+\\beta}", "231abf4361067da96e3fa12983b3590f": "\\scriptstyle{ \\mathrm{R} =_{\\mathrm{def}} \\{ x\\,|\\,x\\notin x\\} }", "231ad25673de638082c402391d702ed5": "\\begin{align}\n\\ln (n!) - \\tfrac{1}{2}\\ln(n) & = \\tfrac{1}{2}\\ln(1) + \\ln(2) + \\ln(3) + \\cdots + \\ln(n-1) + \\tfrac{1}{2}\\ln(n)\\\\\n& = n \\ln(n) - n + 1 + \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)} \\left( \\frac{1}{n^{k-1}} - 1 \\right) + R_{m,n},\n\\end{align}", "231b61a8f020db4bc50dfff33143ada9": "\\frac{dy}{dx}=\\frac{r'(\\varphi)\\sin\\varphi+r(\\varphi)\\cos\\varphi}{r'(\\varphi)\\cos\\varphi-r(\\varphi)\\sin\\varphi}.", "231b8bbeb4fa45885c97d90d3d2637be": "t>0", "231ba258c315092ab9e8413697d3cf6b": "h(w;\\rho) = \\rho \\, \\exp(-\\rho\\,w)\\, .", "231be161dcd193963413b740ed6b93f3": "\\pi \\, Q = 0\\,.", "231c01def816b7f8d44ef229646388a3": "\\mathcal{N}= \\frac{\\left\\langle \\mathrm{MFT}\\right|\\left.\\mathrm{RPA}\\right\\rangle}{\\left\\langle \\mathrm{MFT}\\right|\\left.\\mathrm{MFT}\\right\\rangle}", "231c1815bcf79ba04aef591f3b115baf": "\\mathbf{Z}/n\\mathbf{Z}", "231c7156745b50c7d196cbd4452c836b": "\\int \\! x^{-1}\\, dx= \\ln |x|+C,", "231caefb182582246d50c67eaebe9f85": "y^2 = x^3 + ax + b \\pmod{N}", "231cc60965f06073a43ef55ab620830f": "\\scriptstyle O(\\frac{y\\log^2 y}{\\log\\log y})", "231cee4f7f919f94f548fd81679d9bbe": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\n = \\cfrac{E}{1-\\nu^2}\n \\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \n ", "231cf45da6abaad8b308bc5789e33732": " y=\\frac{x-\\mu'}{|c|} ", "231d18b7b72dcf05cf206d6e3faa4746": "f(x) > -\\infty", "231d1a2d41cf6dfdd4908db8e68db7a9": "\\dot\\alpha,\\dot\\beta,\\dots", "231d368ae058d2a789709f907f5bd2b1": "d := 1 - \\frac{H_M - H(X)}{H_M} = \\frac{H(X)}{H_M}", "231d53659de6e3ce9aff3cfc3fe599e7": "E\\to{\\mbox{Pic}}^0(E).", "231d94a8c0bf34e69b4bb12314f0ef2e": "\n\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}} =\n\\begin{bmatrix}\n\\frac{\\partial y_1}{\\partial x_1} & \\frac{\\partial y_2}{\\partial x_1} & \\cdots & \\frac{\\partial y_m}{\\partial x_1}\\\\\n\\frac{\\partial y_1}{\\partial x_2} & \\frac{\\partial y_2}{\\partial x_2} & \\cdots & \\frac{\\partial y_m}{\\partial x_2}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial y_1}{\\partial x_n} & \\frac{\\partial y_2}{\\partial x_n} & \\cdots & \\frac{\\partial y_m}{\\partial x_n}\\\\\n\\end{bmatrix}.\n", "231e1cafb9ca5b0db1ff4680a8ebf6c4": "\\mbox{E}(x) = (-x\\mod{m})+1 ", "231e4262985d8d8725ee11abe226833d": "d_{2} \\sigma= 2\\sigma/\\sqrt \\pi .", "231e70c2ef91a82daf1e552748f86400": "\\bar{e}^{ch}=\\left[ \\bar{g}_{F}+\\left( a+\\frac{b}{4}-\\frac{c}{2} \n\\right)\\bar{g}_{O_{2}}-a\\bar{g}_{CO_{2}}-\\, \\frac{b}{2}\\bar{g}_{H_{2}O(g)} \n\\right]\\, \\left( T_{0,}p_{0} \\right)+a\\bar{e}_{CO_{2}}^{ch}+\\, \\left( \n\\frac{b}{2} \\right)\\bar{e}_{H_{2}O(l)}^{ch}-\\, \\left( a+\\, \\frac{b}{4} \n\\right)\\bar{e}_{O_{2}}^{ch}", "231e8e41e749b6b7c9d92b5162bdb26b": "U=TS-PV+\\sum_i \\mu_i N_i\\,", "231f33bb0acc36647826b75a3af1289a": "R_0^0 = T_0^0 - \\frac{1}{2}T,\\ R_{\\alpha}^{\\beta} = T_{\\alpha}^{\\beta}- \\frac{1}{2}\\delta_{\\alpha}^{\\beta}T,", "231f3e688aee8ae9a16497a902b4ed93": "q=\\int_{t_1}^{t_2}\\iint_S \\bold{J}\\cdot\\bold{\\hat{n}}{\\rm d}A{\\rm d}t ", "231fec80038de71293915cb76e01f49e": " j", "2320809b6aad6116b4d965b8ebe426dc": "\nr^\\ell\\,\n\\begin{pmatrix}\n Y_{\\ell m} \\\\\n Y_{\\ell -m}\n\\end{pmatrix}\n=\n\\left[\\frac{2\\ell+1}{4\\pi}\\right]^{1/2} \\bar{\\Pi}^m_\\ell(z) \n\\begin{pmatrix}\n A_m \\\\\n B_m \\\\\n\\end{pmatrix} ,\n\\qquad m > 0.\n", "232095999b139073422402e7a7d99180": "d_{pore}", "2320b34c7e52ecd5fecf070c57c721d4": "A_0 \\leq B \\leq A_{\\infty}", "2320f8341b90e8fc3d126f0d9589ce16": " \\sin \\alpha = \\cos U_1 \\sin \\alpha_1; \\,\\,\\,\\, \\cos^2 \\alpha = (1 - \\sin \\alpha)(1 + \\sin \\alpha) ", "232106f2ffc5cb3ab802c5db61abfda6": " { \\Gamma(q+1) \\over s^{q+1} } ", "23210a6f7a6269e7143a1479fb4d5bf9": " 2\\; \\mathrm{H_3 O^+} \\; + \\; 2\\; \\mathrm{e^-} \\quad \\overrightarrow{\\leftarrow} \\quad \\mathrm{H_2} \\; + \\; 2\\; \\mathrm{H_2 O}", "232154ce43270546a2f51a381610508f": " V_C = I_C X_C\\,\\!", "23219e49231e79f7a2038c40eacd101b": "C^{(k)}(f)\\leq\\bigg\\lfloor{n\\over k}\\bigg\\rfloor+1.", "2321a22c502ea202bfbb0899f9c6db8a": "\\mathbf{Q}_{M \\times 1} = \\mathbf{B}_R \\mathbf{R}_{N \\times 1} + \\mathbf{B}_X \\mathbf{X}_{I \\times 1} + \\mathbf{Q}_{v \\cdot M \\times 1} \\qquad \\qquad \\qquad \\mathrm{(5)} ", "2321d873510c23f952b659d98d72b3be": " \\frac{1}{\\exp(\\psi(x))} = \\frac{1}{x}+\\frac{1}{2\\cdot x^2}+\\frac{5}{4\\cdot3!\\cdot x^3}+\\frac{3}{2\\cdot4!\\cdot x^4}+\\frac{47}{48\\cdot5!\\cdot x^5} - \\frac{5}{16\\cdot6!\\cdot x^6} + \\dots\n", "2321e4a6beb8f7f8e2fc3db2cda337b3": "W \\sin \\theta = n \\lambda, n=0, \\pm 1, \\pm 2, .....", "23221d115f2d2dfdb561ebb8d0381213": "\\rho({\\vec x})", "2322b5d74b4c6531231b44907c63e19d": "\\textstyle f : \\mathbb{R} \\to \\mathbb{R} ", "2322bed50680b12610e1de3ed1bb6e49": "e^{1 + 4 \\pi i n - 4 \\pi^{2} n^{2}} = e", "2322c35a6584d4de80833c06cbf5e660": "t^{p_j}\\Delta x^j", "2322e378cc10be5269f3aa394173f611": "L=\\frac{\\left[T_0\\right]}{\\left[R_0\\right]}", "23231d7551563a696689ebcf26c3ad89": "R_S=\\frac{v_{Bullet}^2 \\sin(2\\theta)}{g} \\, (1-\\cot(\\theta)\\tan(\\alpha))\\sec(\\alpha)\\,", "2323224adc9eba3042ff8bc091e13d5a": "A \\leq X", "23233b069fbf87cc459239ac7a9a3d12": "X\\times[0,1]", "23238f7945b0132ba8d0a626f430f231": "x^{\\rho} = x\\backslash e \\qquad xx^{\\rho} = e", "2323e6383dae6e770d2a04cfd26efb62": " u_i(s^\\prime_i,s_{-i})\\geq u_i(s_i,s_{-i})", "232401d163456315f4e43684c23a90ad": "p^2 + 2pq + q^2=1 \\,", "232410dff222adec1a01b5622968330d": "\\sqrt{1+u} = (1+u)^{1/2} = 1 + \\frac{u}{2} - \\frac{u^2}{8} + \\cdots", "2324874785f036ca1c0544ed044b93b9": "\n \\langle j_1\\,m_1 | j_2\\,m_2 \\rangle = \\delta_{j_1,j_2}\\delta_{m_1,m_2}.\n", "2324e37190effb1ac8ec120eb30e66c8": "\\frac{\\partial}{\\partial \\overline{z}} w(z) = 0.", "2324e442204d6804a4f366eaf7c0bf61": "\\frac{\\pi^2}{16}\\cong0.61685.", "232541ce34c356ff4c4a89d3d9c02a54": "\\mathcal{D}_{L^p}", "23254e1da8500aa9f42f028a48c31059": " \\hat{\\gamma}_{ij}(t,x^k) \\to \\gamma_{ij}(t,x^k)", "2325c4299726b2882245ed19b4a729f9": "\\mathrm{^{249}_{\\ 98}Cf\\ +\\ ^{2}_{1}D\\ \\longrightarrow\\ ^{248}_{\\ 99}Es\\ +\\ 3\\ ^{1}_{0}n \\quad (^{248}_{\\ 99}Es\\ \\xrightarrow[27 \\ min]{\\epsilon} \\ ^{248}_{\\ 98}Cf)}", "2325e7f41bebcbc36375d5ad894c49df": "S^n, n\\ge 1", "232610533a5221611a29fcab57a94949": "\\beta=\\nu B d\\,", "2326261f97ce716dcaa54130a44800fd": "\\pi_{1} \\ge 2\\pi_0", "232626674bf22554c40ddae795431c9c": "M=B\\times F", "2326280d96e91557660e34dfae7d3741": " E_7^{\\mathbb C}", "2326293cdaefca46aa48a0b9e2019063": "\n\\widehat{\\theta} \\; = \\;\n\\frac{\\log(1-P_a) - \\log(1-P_b)}\n{\\log(b) - \\log(a)}.\n", "232644f3a5cdd6c6c291ef64e680bdc0": "J_{mB - sV}", "232679d7b0b4e10c1d7036944b9bb054": "A\\mathbf{x}=\\mathbf{b},", "2326c4ef49073348a7bfc3e4664dc43f": "C_c\\,", "2326f5a7d8d423517cc9e11687f759a4": "x, y \\mapsto \\Omega(A(x), y)", "232713107d94d28733937b2e8a4640d4": "dz = h(x,t) \\, dt + \\eta dv", "232752419ea9a788f263f491c73d5ebf": "f(r) = r^2 - a \\equiv 0 \\,\\bmod{p}", "23276174fbe6520075ed81f47fd75424": "f(x) = - \\int_{D} L_{X} f (y) \\, G(x, \\mathrm{d} y).", "232842acbb0ea982cb93ef74b5ab9bbd": "\\sum_{w\\in N^{(t)}(u)\\cup\\{u\\}} \\frac{1}{d(w)+1} \\le (d'(u)+1) \\frac{1}{d'(u)+1} = 1 ", "23289f855511140daf15c53e505e12a7": "\\lim_{x\\rightarrow\\infty}\\pi(x) / \\operatorname{li}(x)=1\\!", "2328b9a25b41ea4f5a2cb4c7165b86ed": "\\Psi_g", "23293447e60df54b74844d7a2a8fea02": "\\bot^*", "23293d8e1539bc098ed6697f3f787376": "m_{\\pi}^2=\\lambda m_q F", "2329b355331544ba5d62ec7c483fa962": "\\omega_{-}", "2329b6d5250a30af4fa11ec2cfd30405": "L(I,E)", "2329d83d3aabf668ed1dbfbdd68e1583": " [[en:Category:Little Egypt", "232a239af70024857e58402597bddb62": "u_1 = \\Re\\left\\{ F(z)\\; \\text{e}^{-i\\, \\Omega\\, t} \\right\\},", "232a257a6a66bb71c16f041814f97f5e": "\\frac{\\gamma N}{2}", "232a5f881029483ce28d22701afc69d3": "R(z;\\tau) = \\sum_{\\nu\\in Z + \\frac{1}{2}}(-1)^{\\nu - \\frac{1}{2}}\\left({\\rm sign}(\\nu) - E\\left[\\left(\\nu + \\frac{\\Im(z)}{y}\\right)\\sqrt{2y}\\right]\\right)e^{-2\\pi i \\nu z}q^{-\\frac{1}{2}\\nu^2}", "232aacc2dea01263736e030917b4f7fd": "p \\in \\partial G", "232aee5d5ef373c4a399cf849c7af49e": "\\ \\frac{dM^2}{M^2} = \\frac{\\gamma M^2}{1 - M^2}\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)\\frac{4f}{D_h}dx ", "232ba1b1b52154ea1f7ada0cd59236c3": "F(x,y,z) = \\text{constant}", "232bb26c60bdf9c1588703dfe41bea96": "\\rho\\rightarrow r^{-5}", "232c11ad705f64366c244a5c14d00d0f": " \\cfrac{f_{i+1/2}^{n+1/2} - \\cfrac{f_i^n+f_{i+1}^n}{2}}{(1/2) * \\Delta t}=\\cfrac{g_{i+1}^n - g_i^n}{\\Delta x}.\\,", "232c15632fc5873535b7cec2ab1cc6a6": "X_1\\,\\!", "232cd3d624c8c047a1ed85c10f781d4e": "L_{pq}", "232d74eef31de47e4f6fc05ed827c066": "\\Gamma=\\frac{Z_2/Z_1-1}{Z_2/Z_1+1}", "232d799240386ad2cef225c9658cb41f": "|2 b^2 - a^2| \\geq 1", "232e25eac629a6caf56137617e048dc5": " \\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{M} f(-\\ln x) \\right\\}(s)", "232e692208ba68e806e728df049b3426": "\\sum_{j}(h_i - \\lambda \\gamma_j) \\delta O_j = 0\\,\\!", "232eb7cc7a84426b9c8f9018f864488f": "-u''=f\\,", "232ec1e29d38b3e23c7069719e5a683d": "\\ I(r,d)=\\frac{1}{2} \\eta I_bpS \\int\\limits_{0}^{d} \\ln \\left[ 1+\\left( \\frac{r}{h} \\right)^2 \\right] \\exp\\{\\alpha (d-h)\\}\\,dh", "232ee56e6a8db37f94fb4a1092584774": "\\lambda \\neq \\nu", "232f3642e7d22258a083c3201f5d89ed": " \\bar{x} = {Cm + \\sum_{i=1}^n{x_i} \\over C + n} ", "232f86038a9523597db834a2e07d0d49": "a\\uparrow b", "232fe9c5351bbdd433e6372d99059bb5": "S=\\sqrt[3]{K_{sp}\\over4}", "232ff693e1d0c7801103fbcd774244f0": "R = \\frac{ 8 \\eta \\Delta x}{\\pi r^4}.", "2330121f5b60d4059618793cd46a36de": "J=J_\\mathrm{S} \\,e^{-{d/\\delta }}", "23304e6f4636f8928748ef37e228ac3a": "A_1\\cup A_2", "23305b935bc7ae09773e2fdda5e9e903": "\\nabla_\\ell A^{ik}=\\frac{\\partial A^{ik}}{\\partial x^\\ell} + \\Gamma^i{}_{m\\ell} A^{mk} + \\Gamma^k{}_{m\\ell} A^{im}, \\ ", "2330f832f570abb9f1e74b16c1db8dc2": "\\sum_{s\\ni e} x_s \\ge 1.", "2331169b1529cf60b773c4e32b559307": "\n \\lambda_i\\frac{\\partial J}{\\partial \\lambda_i} = \\lambda_1\\lambda_2\\lambda_3 = J\n", "23317a24fe445dee548eb78d67331c37": "A \\land (B \\lor \\lnot B)", "2331ac7cad0ef331aad5a86b6fcd97ec": " [\\mathcal{L}_X, d] = 0 ", "2331b73e435e1e964b2a624977d6f02a": "R(n_1,\\ldots,n_k)", "2332013279f13f9456d4c5f762874fee": "\\hbox{ }^tP_i\\rightarrow \\,^tS(R)^{-1}\\,^tP_i\\,^tS(R) = (CS(R)C^{-1})\\,^tP_i(CS(R)C^{-1})^{-1}", "23323329fc56bd5f967123243ce410c1": "\n {\\mathbf e}'_i = \\frac{\\partial}{\\partial {x'}^i} =\n \\frac{\\partial x^j}{\\partial {x'}^i}\n \\frac{\\partial}{\\partial x^j} =\n \\frac{\\partial x^j}{\\partial {x'}^i} \n {\\mathbf e}_j\n", "233265f2755d63ebe4c53956ce590b57": " \\neg q ", "2333045d4b886c7f7a0e88e015b50404": "\\oplus,\\lnot,", "2333143cc15e1f0f4a5c90dc5c1128ad": "\\pi_A = \\pi_T = {(1-\\pi_{GC})\\over 2}", "23332375d2d6411d73209f584ea0d1cb": "b=\\sqrt[2]{r_{a}\\cdot r_{p}}", "23337dd98563f83f8f813676de53f783": "\\Lambda^{1,0}M", "2333ad596f03a73f3ae4e7c3bcd7deb1": "\\beta (t)", "23341d52b446cf32ee4690f0ca940433": "i\\ll_\\equiv j", "233480c8487464adeb1ece918e1efa26": "g(a, b)=\\begin{cases}\n\\frac{a^3}{a^2+b^2}& \\mbox{ if } (a, b)\\ne (0, 0)\\\\\n0 & \\mbox{ if } (a, b)=(0, 0).\n\\end{cases}", "2334c8dbde2064f1233818452783f9e7": "\n\\begin{matrix}\nl_{1,1} x_1 & & & & & = & b_1 \\\\\nl_{2,1} x_1 & + & l_{2,2} x_2 & & & = & b_2 \\\\\n \\vdots & & \\vdots & \\ddots & & & \\vdots \\\\\nl_{m,1} x_1 & + & l_{m,2} x_2 & + \\dotsb + & l_{m,m} x_m & = & b_m \\\\\n\\end{matrix}\n", "2334cd758d520a39a1145f8e772b88eb": "\\scriptstyle I'_2 = -I_2", "233506ca9bbed051e0c7b94610c61272": "\\mu(t) = \\sigma(t) -t.", "2335460affc1028c6adcf25fd95bfa3f": "0\\le\\{x\\}<1.\\;", "23356e38dd2b2a9abee9b9bba5194521": "\\gamma_\\mathrm{SG} = \\gamma_\\mathrm{SL} - \\gamma_\\mathrm{LG} \\cos \\theta_\\,", "23357f61f717162a44f9298b93024e63": "\\hat{g}=\\hat{a}\\,", "2335b9effd79e7fd6f4d72d92fc70cdf": "F = m\\frac{dv}{d t} = m a,", "2335c44ebf82aafa079bfa8d44ad5a9d": "\\inf_{x \\in X} [F(x,0)] - \\sup_{y^* \\in Y^*} [-F^*(0,y^*)]", "23360ad55340de0bf869970c25c30920": " = {1 \\over p+1} T\\left(\\sum_{j=0}^p {p+1 \\choose j} b^j n^{p+1-j} \\right)\n= T\\left({(b+n)^{p+1} - b^{p+1} \\over p+1}\\right). ", "23362b6a9cf8baeda38a0c42f5546aa9": "\\mathbf{J} = \\frac{\\nabla\\times\\mathbf{B}}{\\mu_0}", "2336381d9797c76cd95c41cd1058dbda": "\\tfrac{1}{4}x+\\tfrac{1}{12}y=1", "2336428ac0c3fd74dfb8a24194470967": "H_\\alpha^{(2)} (x) = \\frac{J_{-\\alpha} (x) - e^{\\alpha \\pi i} J_\\alpha (x)}{- i \\sin (\\alpha \\pi)}.", "2336615428eab2afccc2ac25e8f6fa82": "\\mathrm{If}\\; X,Y \\in \\mathbf{L} ,\\; \\mathrm{then}\\; \\rho(X + Y) \\leq \\rho(X) + \\rho(Y).", "23366b866af9998f0e0f9d71e384afd8": "|n,\\pm\\rangle~", "2336b323371eeb24c77850ebdd813597": "\\ x^2 - 92y^2=1 ", "233724c5adf28da47784390134db3c66": "LP", "233739a45217375ed25dff7f40981967": "\\text{maximum speedup } \\le \\frac{2}{1 + 0.25 \\cdot (2 - 1)} = 1.60", "233748e19aa603307b32a1ecb5f5c5e3": "\\ \\Delta S", "233751764b03943325d8c8105c80103e": "E = 1 - \\left(\\frac{\\alpha}{P} + \\frac{1-\\alpha}{R}\\right)^{-1}", "2337afb449890b668b909a691aff05ce": "\\epsilon\\left| \\frac{dQ}{dx} \\right| \\ll Q^2 ,", "2337c4458facd9d85d41861fe6cfa24f": "G_{ij}=\\int_{t_0}^{t_f} \\ell_i(\\tau)\\bar{\\ell_j}(\\tau)\\, d\\tau. ", "2337f8f5bc8c054982e4095745934447": "\\vec J = \\frac {\\vec E}{\\rho}", "23388d3e952d73c823bd73d2e36c193f": "c_i(0)", "2338b165cb0c6d8e81795427c696178a": "\\mathbf{A}^{-1} = \\begin{bmatrix}\na & b \\\\ c & d \\\\ \n\\end{bmatrix}^{-1} =\n\\frac{1}{\\det(\\mathbf{A})} \\begin{bmatrix}\n\\,\\,\\,d & \\!\\!-b \\\\ -c & \\,a \\\\ \n\\end{bmatrix} =\n\\frac{1}{ad - bc} \\begin{bmatrix}\n\\,\\,\\,d & \\!\\!-b \\\\ -c & \\,a \\\\ \n\\end{bmatrix}.", "2338b4622b5f09bb0224354019972fe8": "P_A = A (A^\\mathrm{T} A)^+ A^\\mathrm{T}", "2338bde451058e64de2ffb3e0f6ffaed": "x=(x_1,\\dots,x_k)\\in F^k", "2338cf290c8adf956d23f809731161a3": " \\mathbf{a} = \\mathbf{E}_\\text{g} + \\mathbf{v} \\times \\mathbf{B}_\\text{g} - \\frac{ ( \\mathbf{E}_\\text{g} \\cdot \\mathbf{v} ) \\mathbf{v} }{c^2} \\,,", "23390b320c5e0587c6c69e017578349c": "q < 1 ", "23390e8880564338ee3d0000189447bc": "\\approx 2.3632718012073547031\\,,", "23394a6d0cacaaf92fd60c71c6ec6684": "P_{4}=1", "23399feab2e14145b433823229fc2f47": "b(z)", "2339a20c078567400faaed5e64ca3862": "H = \\dot{x}_1 p_1+\\dot{x}_2 p_2 - L", "2339af4300318a95889ce984b5e0da2c": "v=\\text{constant}", "233a45460285415f4db07c8cd4db0a64": "\\sum_j n_j(\\mu_j^\\ominus +RT\\ln a_j)=\\sum_k m_k(\\mu_k^\\ominus +RT\\ln a_k) ", "233a4b9703803a852db5d6e79a74712f": "M^k=f^{-1}(1,0,\\dots,0) \\subset S^{n+k}", "233a9ef42f2d3a8ad026c7e6e0b8abf4": " \\square_x = \\frac{\\partial^2}{\\partial t^2} - \\nabla^2 ", "233b30a7af1fd107aa48e7c51002ef25": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}~\\boldsymbol{F}\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{F}^T\n = \\cfrac{2}{J}~\\boldsymbol{F}\\cdot\\cfrac{\\partial W}{\\partial \\boldsymbol{C}}\\cdot\\boldsymbol{F}^T\n ", "233b34e613ad045674818ff0463b7bfe": " [n=2^m, n-1 -\\lceil {d-2}/2\\rceil m, d]_2", "233b7351292b6bdf37f49680619f2abd": "w^0\\in\\mathbb{R}^d", "233b93e9880422468f3de127f5d35650": "\\Pr(W=n)=\\frac{k}{n}\\frac{e^{-\\mu n}(\\mu n)^{n-k}}{(n-k)!}", "233ba484d99cff1c9c5bf1f5c2d3f1ac": "Q=\\begin{pmatrix}\n-\\lambda & \\lambda \\\\\n\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&&\\mu & -(\\mu+\\lambda) & \\lambda &\\\\\n&&&&\\ddots\n\\end{pmatrix}", "233c0ffc8923672a3fbb1b236ff4b2bf": "R(t)", "233cb03365463a02b3dc2e27888e1c34": "m'_0, m'_1", "233cd4d0d439b88f954ec5c1386eb31d": "\\textstyle{{n \\choose m} = {n \\choose n-m}}", "233ce6c358743a7a3cf222891e69d780": "\\sum_{k=1}^n k^p = {1 \\over p+1} \\sum_{j=0}^p {p+1 \\choose j} B_j n^{p+1-j}\n= {1 \\over p+1} \\sum_{j=0}^p {p+1 \\choose j} T(b^j) n^{p+1-j} ", "233cf01fbd0ad3dfe5f6cad2df5ba448": "\\neg P \\rightarrow Q ", "233cf5fa593c8789337a5f5acf10fe03": "=\\,\\!", "233cf870a12a7529768a7652736dd420": "Z=\\int_{p^2\\leq \\Lambda^2} \\mathcal{D}\\phi \\exp\\left[-S_\\Lambda[\\phi]\\right].", "233cf9693d17043e811ffd6b68a7e14c": "f_{\\mathrm{BE}}(E) = \\frac{1}{\\exp\\left(\\frac{E-\\mu}{k_BT}\\right)-1}", "233d12c1b1057ed9f9190079f9f81ff7": "\\Psi(\\omega)", "233d46fef141ec6815e57014337d4872": "FAD", "233de11b2feb965d4b6a724f67ff8412": "\\sin(\\arcsin x) = x\\quad\\text{for} \\quad |x| \\leq 1 ", "233de1dc6a21d2b5d5c78e19305b8de5": "\\theta_a=\\theta(x)|_{x=ja}\\,", "233e358a243eb6ea560c49ff206c227c": "kx^{\\prime 2}/z", "233e480656c9a5c6a42ab6ce57062b00": "\\tau_{ax}", "233e7af80710717c2f009f9fb5573a93": "z_T = \\frac{Z_T}{Z_0} = 0.35 + j0.65\\,", "233ec80ab66c64260d74c1ab87efb94d": "G \\circ G' : \\mathcal{C} \\to \\mathcal{E}.", "233ed2fb96370017307e35365abb1c5a": " R_1^{-1} Q_1^{-1}XQ_1R_1", "233ed40fba1ece94870ad675746ed120": "s(10011)=00111", "233f33c80532b1878f4ba200a82e6c88": "\\begin{align} \n\\ln \\,\\operatorname{var_{G(1-X)}} &= \\operatorname{E}[(\\ln (1-X) - \\ln G_{(1-X)})^2] \\\\\n&= \\operatorname{E}[(\\ln (1-X) - \\operatorname{E}[\\ln (1-X)])^2] \\\\\n&= \\operatorname{E}[(\\ln (1-X))^2] - (\\operatorname{E}[\\ln (1-X)])^2\\\\\n&= \\operatorname{var}[\\ln (1-X)] \\\\\n& \\\\\n\\operatorname{var_{G(1-X)}} &= e^{\\operatorname{var}[\\ln (1-X)]} \\\\\n& \\\\\n\\ln \\,\\operatorname{cov_{G{X,(1-X)}}} &= \\operatorname{E}[(\\ln X - \\ln G_X)(\\ln (1-X) - \\ln G_{(1-X)})] \\\\\n&= \\operatorname{E}[(\\ln X - \\operatorname{E}[\\ln X])(\\ln (1-X) - \\operatorname{E}[\\ln (1-X)])] \\\\\n&= \\operatorname{E}\\left[\\ln X \\ln(1-X)\\right] - \\operatorname{E}[\\ln X]\\operatorname{E}[\\ln(1-X)]\\\\\n&= \\operatorname{cov}[\\ln X, \\ln(1-X)] \\\\\n& \\\\\n\\operatorname{cov}_{G{X,(1-X)}} &= e^{\\operatorname{cov}[\\ln X, \\ln(1-X)]}\n\\end{align}", "233f44fd7bc7220ba7e705f6430c2947": "D_{192}", "233f4ea7ce4ce7eeb467ecb1c7c7a65a": "Z^2_4", "233f509eb47fd05032c61682f1b722f2": "\\mathfrak{sp}_n,", "233f7b2492df8fce12e24c77d47c0ff7": "h:S^{d+1}\\rightarrow S^{d+1}", "233fab408950ad3f1f4c8aa993c8a09a": "c \\equiv b^{e} \\equiv d^{\\left|e\\right|} \\pmod{m}", "233fb88eb9865a2ea95f18e5475c8190": "(p,q) \\mapsto q f(p/q)", "233fdc21188d5d2d0c7a751119ee20dd": "-ds^2 = -(c \\, dt)^2 + dx^2 + dy^2 + dz^2, \\ ", "23406d7586dd6f3fab7f26cdb8bc56e5": "\ni{d \\psi_n \\over dt} = c^* \\psi_{n+1} + c \\psi_{n-1}\n", "23409caafc0a7acd1c1a5e1bfdc61fd3": "\\begin{matrix}\n\\lim\\limits_{x \\to p} & (f(x) + g(x)) & = & \\lim\\limits_{x \\to p} f(x) + \\lim\\limits_{x \\to p} g(x) \\\\\n\\lim\\limits_{x \\to p} & (f(x) - g(x)) & = & \\lim\\limits_{x \\to p} f(x) - \\lim\\limits_{x \\to p} g(x) \\\\\n\\lim\\limits_{x \\to p} & (f(x)\\cdot g(x)) & = & \\lim\\limits_{x \\to p} f(x) \\cdot \\lim\\limits_{x \\to p} g(x) \\\\\n\\lim\\limits_{x \\to p} & (f(x)/g(x)) & = & {\\lim\\limits_{x \\to p} f(x) / \\lim\\limits_{x \\to p} g(x)}\n\\end{matrix}", "2340ab547795f525e876ea2aa43afd35": "\\tau*_c", "2340c61cc12792eabc36261b46a564ee": "w^1 = ~w", "23410b34b4eed832e5dc2e74999989ce": "N({1 \\over 2}\\ln{{1+\\rho}\\over{1-\\rho}}, {1 \\over n-3})", "23412e26eaf7dbe3d75bf54113c30660": " D_{\\nu} ", "2341bef88c41a7ddd194710c806d18c3": "\\displaystyle{\\int_{\\partial\\Omega} gk =0,}", "2341ec116e8c14cf06d6242f8dba1fde": " P\\subset TM", "2342172b3413abe4d07f038e9b364d96": "c=2, \\phi=-20^\\circ", "2342592b6bf34e67ab5c919d09811d24": "k = 7l + 3", "2342b48312bd302535c591a695c604ce": "\\psi_{5,7}=1", "2342b4aef789ab838dc8e31f0e7bc6a3": "l_{l}\\, ", "2342c04efb9091b1882f8d86de1ec83e": " q_{jk}=\\frac{\\sum_{i=1}^{N}w_i}{\\left(\\sum_{i=1}^{N}w_i\\right)^2-\\sum_{i=1}^{N}w_i^2}\n\\sum_{i=1}^N w_i \\left( x_{ij}-\\bar{x}_j \\right) \\left( x_{ik}-\\bar{x}_k \\right) . ", "2342e2937aa9c7a2bede7aaf54f1bdf0": "L(P) = \\sum_{i=0}^n |a_i|.\\,", "2342f6a1e9a6ec09cb6666eb10c65cfa": "\\nabla_\\mu \\phi\\;", "2342f863ee74d3b15b6a9b6d441d0d15": " \\bar{\\mathcal{M}} ", "23432b43afaed9c5e08413111781379d": "\\begin{align} 2\\cdot R_*\n & = \\frac{(91\\cdot 2.06\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 40\\cdot R_{\\bigodot}\n\\end{align}", "2343ae36a4aef6c9c78fd7893c6a1ff3": "\\scriptstyle n=9", "2344b5cdce19cc822937215901c0474d": "\\textstyle R_k = A_{k+1} A_k^{-1}", "23451b9fd03698800326e3e3d9528c48": "\\boldsymbol{\\nabla}P", "2345397ce9a108564892fc0ad8ad2e50": "F = 1 - {138 \\over 141.2} = 0.023", "2345ad2ff3f824cf10dce7fea2dee3d6": "\\scriptstyle P \\left ( { {a}{|}{A,\\lambda} } \\right )", "2345b6bcd2bb6331a0e16a0d6e76c195": "\nH_\\alpha =D_\\alpha (-\\hbar ^2\\Delta )^{\\alpha /2}+V(\\mathbf{r}), \n", "23467fa733d75a9d3a6368e6aef0d216": "\\mathbf{H} := \\begin{pmatrix}\n1 & 1 & 0 & 1 & 1 & 0 & 0 \\\\\n1 & 0 & 1 & 1 & 0 & 1 & 0 \\\\\n0 & 1 & 1 & 1 & 0 & 0 & 1 \\\\\n\\end{pmatrix}_{3,7}.", "2346f1c4cf0c69cf0b3bb07ef2a32e2f": "H|\\Psi_E\\rangle = E|\\Psi_E\\rangle", "234701287ef75a9fc06b34fed1c10647": " N=2^{40},~N=2^{42}, ~N=2^{44}", "23471ed40851377bdb5e5dc7e1d79465": " \\lim_{n\\to \\infty} F_{1,n}(z)", "2347290b3b5b09e01a1a948ea5856d2a": "|c_n| \\ge \\sum_{k=0}^n \\frac{1}{n+1} = 1", "23473bb1c2488766e85dec0082631dcc": "|-3| \\ge |1| + |2|", "234747638eede29658077df51b44a1bd": "\\lambda(t) = \\frac{R(t_1)-R(t_2)}{(t_2-t_1) \\cdot R(t_1)}\n = \\frac{R(t)-R(t+\\triangle t)}{\\triangle t \\cdot R(t)} \\!", "23475d68badb023184833bdb56865946": " u = \\sum_{ i = 1 }^n{ \\frac{ 1 }{ k_i^2 } } + 2 \\sum_{ i = 1 }^n \\sum_{ j < i } \\frac{ \\rho_{ ij } } { k_i k_j } ", "2347dc43179793e67df48a725dc6f305": "\\lambda = \\frac{h}{p},", "2347eb890bc072736464ea007f0ce7ed": "~x=0~", "23482ee9cd54af3d10d2585f4f993780": "\\frac{\\partial E}{\\partial w_i} = (y - t) y (1 - y) x_i", "2348471303684feb2284eef20d132170": "ED_A>ED_B", "2348591ff3dca719242cee8fdb317635": "+S_z \\otimes I", "2348b63eaeb088d0664d5c8bf5e0e90f": "c_3 = 5.37941,\\,\\!", "23491e0f43a649c1d36334cec1b4e16a": " \\frac{\\ln \\zeta(s)}{s}=(1-e^{\\Theta(s)})^{-1}g(s)", "23493f0c83e580864617f5f0539440e5": "\\partial H/\\partial x_3=-\\partial L/\\partial x_3", "2349669b34f3919642db317737d5b83a": "{\\varepsilon_0}^{\\omega^\\omega}", "2349a21aa47e05e5357b5477a9be0c28": " \\delta^h_{n+k} = O(h^{p+1}) \\quad\\mbox{as } h\\to0. ", "2349baad5a455edaf15b953f1109edd0": "\\,\\varepsilon\\in(0,1)", "234a01199629059ae39994975f3fc672": "ID(x,y)", "234a30d800ce9fcb08c900c0af6d8c24": " u \\smile (v_1 + v_2) = u \\smile v_1 + u \\smile v_2. ", "234a7c2fc724e523fd7e5316bf668eae": "a-x_1 = a_0", "234aa4d9049f32d3d45fb4f881b03de0": "f(t)+(p-r(t)){f'(t)\\over r'(t)}\n=t-i+{p-\\sinh(t)+i(1+p\\sinh(t))\\over\\cosh(t)}\n=t-i+(p+i){1+i\\sinh(t)\\over\\cosh(t)}.", "234ab41f40d521af361c69e06036c78d": "1\\over\\sqrt{7}", "234ac57e05424fa1be970a1e6fd3d12a": "[Gf](x) = \\sum_{n=1}^\\infty \\frac {1}{(x+n)^2} f \\left(\\frac {1}{x+n}\\right).", "234afa2f2f61de3e3046b5fb45f1e80a": "EMA(m_0,n)=\\frac{2}{n+1}\\left[m_0-EMA(m_1,n)\\right]+EMA(m_1,n)", "234b184ecc674deea1dec0186a96087f": "\\mathbf{G}_b(u,v) = \\mathbf{G}_a(u,v) e^{-2 \\pi i (\\frac{u \\Delta x}{M} + \\frac{v \\Delta y}{N}) }", "234b5a65a80b5eba5a6c669cc69793b5": "s \\in S, r_1, r_2 \\in R", "234ba6bcc30b4055ba58372d588bb60d": " =\\frac{2\\pi}{\\hbar} Z_{DP}^{2}\\frac{\\hbar \\omega _{q}}{2V\\rho c^{2}} (\\frac{kT}{\\hbar \\omega _{q}}) \\sum_{k} \\delta _{k', k \\pm q}\\delta [E(k')-E(k) \\pm \\hbar \\omega _{q}] ", "234baa95c6797c787cccadc28b777ed2": "Y_{5}^{-5}(\\theta,\\varphi)={3\\over 32}\\sqrt{77\\over \\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta", "234bb2baf02732d5dc6a5c99b2c63e64": "\\lambda_3^2 = \\lambda_2", "234be622bf8a2780f3af13636d96cace": "\\tilde{u}(x)=\\prod_{i=1}^L x_i^{\\lambda_{i}}", "234c401016cac18ecf029eac4f77bccf": "\\sum_{i=1}^m r^{-\\ell_i}", "234c9253117921d5c6e9596f3d052e99": " |F(a)| \\leq \\|F_a\\| \\|F\\| = \\exp(|a|^2/2)\\|F\\|. ", "234ca5cbf0da3ed2871f75588e6c5a61": "B QC", "234cd21913a2b229d4b6762cc206ca16": "p + 2a^2", "234ce3a3a5f25164b277fd3c0358c6e4": "\\varphi(y)", "234d00f67b1e6f07c05b396f93052b63": "\n\\left( \\mathbb{Z}_{2}\\right) ^{2n}=\\left\\{ \\left( \\mathbf{z,x}\\right)\n:\\mathbf{z},\\mathbf{x}\\in\\left( \\mathbb{Z}_{2}\\right) ^{n}\\right\\} .\n", "234d045b51cb63e629f76bf64dfcf160": "{\\Bbb Q}(q)", "234d1483579df1cf3580be575c75b6a4": " \\mathbf{J_r} = \\frac{\\partial r_i (\\boldsymbol \\beta^{(s)})}{\\partial \\beta_j}", "234d36aa7a269d91e4145eb15886a48a": "Q\\mathbf{x} = (\\alpha, 0, \\cdots, 0)^T.\\,", "234dd936588c26b061e493bd9893eb5c": "y = vt \\sin \\theta - \\frac{1}{2} g t^2", "234e24b907b6b4f01f5f92d85bdbf3e2": "fg(s) = \\begin{cases}f(s) & 0\\leq s \\leq |f| \\\\ g(s-|f|) & |f| \\leq s \\leq |f|+|g|\\end{cases}", "234e3086c911173bf299e0acb1425e43": "p_0, \\dots, p_{s-1}", "234e499e88b87ec1734605deefcb49b9": "g\\colon S^l \\to X, \\, ", "234e8a186be07ff7d9345a83c3db0dcc": "K=\\mathbf{C}, V=\\mathbf{C}^n,", "234ed327f2b88768ab89e1d82ba6e4f8": "x\\asymp y", "234eed3258ea9f33d4e5e309ee30ecd5": "{k_n}", "234f1d51dadb82902864301f413e2a69": "M = M_r\\left[\\sin(k\\phi)\\hat{\\rho}-\\cos(k\\phi)\\hat{\\phi}\\right]", "234f3ca5f473d2bfc6fbec3010a3884a": "\\mu'_2=\\kappa_2+\\kappa_1^2\\,", "234f6d5069f37733fc2f0c34d3d279da": "a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2", "234f8093911ead8601518be7507bcbda": "\\eta_Y(y)(t) = y \\otimes t \\quad \\text{for } t \\in X.", "2350366b7c4ec7cc0dfe7223007a2385": "\\ell=\\ell^{(1)}=\\log p ( X^{(e)}=\\frac{1}{p}X^{(m)}", "235041d5e2608942c2c6b197647f4b12": "IV = \\int_0^t \\sigma_s^2 ds,", "235045dbfe14a1a9536141c7de5fbf21": " E = KV \\left(1-\\gamma^2 \\right) = KV \\sin^2\\theta,", "2350602ebf365f745426f31e86dbc8cd": "I \\approx \\frac{n V_T}{R} W \\left(\\frac{I_S R}{n V_T} e^{V_s/(n V_T)}\\right)", "2350748743ebdcc9885503739fa90e4c": "\\varphi_1, \\ldots, \\varphi_p", "23508d585f85400287f68b56a2588d40": "\\frac{\\varphi^n}{\\sqrt 5}\\, .", "2350a843c8f43af69be3f86a933d7b7c": "y(t + \\delta)", "2350c68d803e3cbaf529df86bc075ffe": "\\{92, 19, \\mathbf{101}, 58, \\mathbf{101}, 91, 26, 78, 10, 13, \\mathbf{2}, \\mathbf{101}, 86, 85, 15, 89, 89, 25, \\mathbf{2}, 41\\} \\qquad (N = 20)", "2350e285ff44ad0829ad9ffba1477913": " = \\frac{1}{\\rho} x'(s)\\ . ", "2350fcc56d715c9f0009672d0af44a87": " \\int_c^d f(x) \\, dx \\leq \\int_a^b f(x) \\, dx. ", "23510dcc19d85fe5480876cd69bd24d9": " \\langle v,w \\rangle_\\Phi = (\\Phi (v))(w) = [\\Phi (v),w].\\,", "2351172b4e1c20a5423cfb36f18fa0c1": "C = \\alpha_1 \\alpha_2 \\beta_1 + \\alpha_1 \\alpha_2 \\beta_2 + \\alpha_1 \\beta_1 \\beta_2 + \\alpha_2 \\beta_1 \\beta_2\\,", "235164d2cf3431287af070ce1cf2a341": "i^{\\ast}_{\\mathrm F_{SO}(M)}(T\\mathrm FM)=T\\mathrm F_{SO}(M)\\oplus \\mathcal M(\\mathrm F_{SO}(M))\\,,", "23518477a30967adff51b953abfc869d": " U = e^{i H \\Delta t}", "2351c247b6ca644c057234899c9ab831": "\\begin{align}\nJ_Y=\\sum \\limits_{l} \\sum \\limits_{u} \\frac{{Y_u}^2}{r}\n\\end{align}\n", "2351e0244ac705b29b075f0da3cf1273": "k = 2\\pi / \\lambda", "2352042fcd550f1c30df3ad3a162ff9c": "\\theta_1(x)=x+1\\,", "235220b37b7ef719f379a94c2fd29b76": "j:A\\to A", "235223d6c51d418e4a2f025021fc5a27": " {(-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu) {d \\over d\\tau} (g_{\\lambda \\nu} \\dot x^\\nu + g_{\\mu \\lambda} \\dot x^\\mu) + {1 \\over 2} (g_{\\lambda \\nu} \\dot x^\\nu + g_{\\mu \\lambda} \\dot x^\\mu) {d \\over d\\tau} (g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu) \\over -g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} = g_{\\mu \\nu ,\\lambda} \\dot x^\\mu \\dot x^\\nu \\qquad \\qquad (5) ", "235229cf71f83fb6d1cdd43e507d4756": "\\mathbb R^G", "23522e6a4fd4e6a2396a9d78edcd943b": "\\textstyle \\{(1, \\alpha), (0, 1)\\}", "2352370de300e377d936eebeb6a19fca": "s_\\lambda=-\\frac{1}{\\lambda}\\sum_{i=1}^m \\log u_i.", "23523f57588abc942a605b3d8189349b": "=\\pi\\,r^2\\,h\\,", "2352440e134e788a22913b9413152c51": "k \\approx \\frac{\\Phi}{\\tau}\\left(\\frac{V}{S}\\right)^2", "2352a1c6ea85d3da98e16dc1454014a7": "\\Delta f_\\text{Lichte}=-0.75 C_s(u_\\max\\lambda)^2,", "2352bc178fd5518eafe59b217cb70038": " \\phi = \\frac{n_{\\rm C_2H_6}/n_{\\rm O_6}}{(n_{\\rm C_3H_12}/n_{\\rm O_56})_{st}} = \\tfrac{1}{0.286} = 3.5 ", "2352df0abb89e8e75697e39058eb45c6": "F(x_i) - F(x_{i-1}) = F'(c_i)(x_i - x_{i-1}). \\ ", "23530a5ed42f5b3e5c368d8552aa4047": "\\textit{open}(t)", "2353580fd98524d7ac4503f04bf0fb6e": "\\alpha_T", "2353a1995f42a9d31d92b156c912116b": "H(X|Y) \\le H(X) \\, ", "2353b33d05b46ebd075cb80fa0b85df2": " \\nabla \\times \\mathbf{B} = \\frac{1}{c} \\frac{ \\partial \\mathbf{E}} {\\partial t} + \\frac{1}{c} \\mathbf{J} \\,", "2353e0b2c403513744a1b4a9f0b4e846": " \\tfrac{223}{71} ", "2353fee5e449a2af9c07990fd016f096": "1\\le i\\le M, \\ 1\\le j \\le N, \\ 1\\le k \\le P ", "23540abc37a11f15ce6f9ea51c0317cc": " a = (m^2-n^2) \\,,\\ b =2mn \\,,\\ c = (m^2 + n^2)", "23541a8b0eb5625d82c596e27e8d9092": "\\hat{\\beta}_i", "23549fa4187a90ea50a65765eb129e0d": "e_n = (0, 0, \\ldots, 1) \\,", "235516b7c7eeb9656ab251af3da5d9f0": "\\forall y ( R(x, y) \\rightarrow ST_y(\\Box p))", "23553a3efed3e14076b7d2711c12c59f": " TE = PE + KE", "23553a42f3592ec43606472c3a0b69a4": "k = \\frac{2\\pi}{\\lambda} = \\frac{2\\pi\\nu}{v_\\mathrm{p}}=\\frac{\\omega}{v_\\mathrm{p}}", "23555dbf85c1ea8daecbb06576281135": "\n \\hat\\sigma = \\sqrt{ \\frac{1}{n-1.5} \\sum_{i=1}^n(x_i - \\bar{x})^2}\n ", "2355710c7695014f70afed018c833287": " \\hat{z}\\,", "2355c7d2486944f160451a088f0c0093": "a\\uparrow\\uparrow b", "2355cb8b5650d0dae2b9ed76d4fb816b": "c^2\\!\\left(1 - \\frac{3}{\\chi^2} + \\frac{\\chi\\phi(\\chi)}{\\Psi(\\chi)}\\right) - \\mu^2", "2355cf1586fe371e5e9ba793b8f4cc1d": "\\forall X \\left[ \\emptyset \\notin X \\implies \\exists f \\colon X \\rarr \\bigcup X \\quad \\forall A \\in X \\, ( f(A) \\in A ) \\right] \\,.", "2355df5ba6e2d5374a59e708dd274537": "\\, B_n \\equiv B_n(0)\\, ", "23563cc5158ad32029640e4654069d46": "\n\\begin{align}\nR(1)&=1~ ;\\ S(1)=2 \\\\\nR(n)&=R(n-1)+S(n-1), \\quad n>1.\n\\end{align}\n", "2356526b18d2bf2b63f7aae655b3e6eb": "p(Y_i|X_i; \\theta)", "23566566f5c6e1bbe000952182c6a765": "S_{n+1} = 4S_n \\left( 8S_n + 1\\right)", "235713ba0e6e557501874ebb6bd5ed98": "\\sum_{k=0}^\\infty \\frac{z^{2k+1}}{2k+1}=\\operatorname{arctanh} z, |z|<1\\,\\!", "23576f84cf7a8eb541fb1d375f83e843": "~c~", "235787cb80f6c57b68fcad12be72c73a": "\\subseteq [D \\rightarrow D^{'}]", "2357906cde9be9fbeeeddad94ead2195": "\n\\Psi \\bar{\\Psi} = 1\n", "2357937b260fcb196b683fc955f0d85d": "\\operatorname{str}[M^2]=\\sum_s(-1)^{2s} (2s+1)\\operatorname{tr}[m_s^2].", "2357e0797ceb7b6442606d72127f77a6": "\nR = Y\\left( {P - c} \\right) - Ytd\n", "2357eb40d94bb16a2924d28d49878ff4": "A,A\\to B\\vdash B.", "235857b6eea7dee4025536f1c3b70606": " M_2 = 2^m M + r_2", "235862adb1aee17962054e1f56979856": " U = c N T, ", "23587ed20409fa2d2991cdb38c2e369a": "p_G(t), \\ p_C(t), \\ \\mathrm{and} \\ p_T(t)", "2358af47b78c047fc06818896464288e": "\\frac{\\partial V}{\\partial p_{1}}=-\\lambda x_{1}^{m} ", "23592977b41d94ac274b7994df5f73f1": "V(x) \\approx \\frac{1}{2} x^2 V^{(2)}(0) = \\frac{1}{2} k x^2", "23594482424dd1ca0fa546997a50edcf": "3r_s", "235951f6537cab516ef388c374ecaf85": "H_{t} = c(t, Y_{t}) + \\gamma (t, Y_{t}) \\cdot \\mbox{noise}.", "2359530b0cfdb29b89ccad184e6a7f84": "\\kappa(z) = \\lim_{N\\rightarrow\\infty} \\mathcal Z(z)^{1/N} = 1+z-3z^2+\\cdots", "23595c5b094be7af9ee0cb827749cd8c": "{AE}_{8}", "2359b305d824ee6e76797bc5893cd2dc": "g^{(k+1)} \\in \\mathbb{R}^n", "2359e542ad7475044ada953081f084c2": "f \\circ m = g \\circ m", "2359fb39d00cab953824a9b705ca394d": "\\tau \\in \\mathbb{H}", "235a07e42d1a1df80cd847a0995ecf01": "\nf_n=\\frac{d\\Delta x}{4\\rho_i Lc} \\frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\\Delta x^2}\n", "235a317a7ea55294b48a6628e1a866d3": "\nr(\\alpha,\\beta)=\\frac{\\sum_i (j^\\alpha_i-\\bar{j^\\alpha})(k^\\beta_i-\\bar{k^\\beta})}{ \\sqrt{\\sum_i (j^\\alpha_i-\\bar{j^\\alpha})^2} \\sqrt{\\sum_i (k^\\beta_i-\\bar{k^\\beta})^2} }.\n", "235aa52652b0ea705fcb82b459b713b4": "\\nabla\\cdot{\\mathbf A} + \\frac{1}{c^2}\\frac{\\partial\\varphi}{\\partial t}=0", "235ae6b195bb2da952f164bfab68f5af": "\\hat{i}", "235afc8e5e8f2df2733e9f963c692f9d": "\\sin\\frac{\\pi}{3}=\\sin 60^\\circ=\\tfrac{1}{2}\\sqrt3\\,", "235b22b61b91378858950314fc1aeb24": "\\Gamma^{\\alpha}{}_{\\beta\\gamma} \\,", "235b5d5ac9cc93b581a03a111099e3a6": " H_{ max } = \\sum ( \\frac{ x_{ ij } }{ X + Y } log \\frac{ X + Y }{ x_{ ij } } + \\frac{ x_{ kj }}{ X + Y } log \\frac{ X + Y }{ x_{ kj } } )", "235b6050567deb4869e17c67a67d8fa0": "E_{bw}", "235c00252b00294922d25475ea14b751": "\\mathbb{D}^qf", "235c1346c13200a614d3de059559eb3e": "V_o = NkT_o/P_o", "235c5146ab110558897640c34dad7d97": "\\textstyle j", "235c6c3907e3665a528ff5e2eef98828": "\\begin{align}\n\\mathcal{H} & = \\mathbf{P}\\cdot\\dot{\\mathbf{r}} - L \\\\\n& = {mc^2\\over \\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2}} + e \\phi \\\\\n& = \\sqrt{c^2(\\mathbf{P} -e\\mathbf{A})^2 + (mc^2)^2} + e \\phi \n\\end{align}", "235c96b10c31fbc9850f675b3b40c1ac": "m = (n'-n)", "235cb7d36b84f7025dd7368a261c66c1": "\\displaystyle w(3,3)", "235ce462121ca365eeebb50a07dae474": "\\frac 1\\beta \\log\\left(\\int_{\\mathbb R} |g(y)|^{2\\beta}\\,dy\\right)\n \\le \\frac 1 2 \\log\\frac{(2\\alpha)^{1/\\alpha}}{(2\\beta)^{1/\\beta}}\n + \\frac 1\\alpha \\log \\left(\\int_{\\mathbb R} |f(x)|^{2\\alpha}\\,dx\\right).\n", "235cfa13c27f1cf0d8c4f73f9e26d6e1": "\\mathbf{\\bar{A}}_{l}|j\\rangle = g_{l}\\mathbf{\\tilde{U}}|j\\rangle,\\quad \\forall{l}.", "235d06e215c331069f42afdf1e358a81": "Q(t_2) = U \\left( t_2 - t_1 \\right) Q(t_1) U^{-1} \\left( t_2 - t_1 \\right)", "235d4fc18a47f1c0af61502e8a77ab12": "\\overline{\\mathcal{H}}", "235d6de4f1342359794904a98d7b075a": "f(x)=- \\omega \\phi(x) + \\int K(x,y) \\phi(y)\\,dy", "235e1d04c4d21aaa94da65ab69723ee1": "{{MU^L}\\over{MU^Y}} = {{dY}\\over{dL}}", "235e56112ae68064b3e62bb0568189de": "\\scriptstyle M_V = m_V - 5 \\log_{10} \\left( \\frac{100}{\\mathrm{parallax\\ in\\ milliarcseconds}} \\right)", "235e775d3ea79387b5672b378cb376b8": "n_e = N\\nu_A - M -z", "235eabe6f46f60ca11d8531057325b48": "\nR_2 = \\frac{R_1 X_1^2}{R_1^2 + X_1^2}\n", "235ee020747b9898945419fb2013d380": "\\Omega=W^TMW", "235fc4f0cc15c1d797335d94366b5af3": "x_1 = (1.786737601482363 + 1.786737578486707) / 2 = 1.786737589984535", "235ffe7e77710e69a63177379eb46e70": "0.10266547\\ldots", "23604a79b07ed670fbbe5ea69e85a21e": " \\mathrm{area}(g) - \\tfrac{\\sqrt{3}}{2} \\mathrm{sys}(g)^2 \\geq 0. ", "2360d41faa0467daa4123eb94e627830": "\\omega\\,=\\,\\frac{2\\pi}{T}\\,=\\,2\\pi\\,f.\\,", "23615a2a3cae58cfc4abf1efe72c9d70": "\\langle Pu,u \\rangle \\geq 0, \\forall u \\in V", "23617856b7344c2616ae5b177e91be6b": "\n \\square = \\langle \\bar{\\partial} \\partial \\rangle_S = \\langle \\partial \\bar{\\partial} \\rangle_S\n", "23619bdeaf491eae74c8c0b8ee2d96b0": "\\pmod{\\mathfrak{p}},", "2361b568349137ab9f6f9e9b755b60b8": "c = \\sum_{i = 1}^n \\alpha_i \\beta_i.", "236202cab115a5c3f9f16a3fbc54d8fa": " cov\\left[u_i,\\sum_{k=jN+1}^{(j+1)N} X_k\\right]= 0 \\quad \\text{ for all } i. ", "23620a0d966ef0a180663e677f989219": "\\tau'_{ij} \\equiv \\rho\\, \\overline{u'_i\\, u'_j},\\,", "2362325508d65682119cd54a8300f765": "\\frac{2\\pi^{n/2}}{\\Gamma(n/2)}", "23624eb2b836a2ebb457a2066df37a28": "\\textstyle{\\bigcap_{k=1}^\\infty I^k M = 0.}", "2362f200efa80e8e069234c06e176812": "(p_1 + \\cdots + p_n)^c", "2362f28c5a8a2b22e2e10ac974286df0": "\\left(\\sqrt{1/45},\\ -4/3,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "23631343b1457b33689f556b7fa29516": "\\varphi_\\alpha(\\beta) < \\varphi_\\gamma(\\delta) \\,", "23633e7e4f60dcc9b42883e83148f2de": "\\tilde{\\mathsf{T}}", "236385978fcc1f9676435034e18da0d8": "\\phi\\in(0,\\pi)", "2363cf525f777cda62557af32c64b075": "\\varnothing=\\mathrm{cl}(\\varnothing)", "2363d8d614e9d4e921a7554ce1f0146e": "\\Phi(-x) = 1 - \\Phi(x)", "23647e2b2afb00b78d0df9577ef46854": "\\int_0^\\infty t^a e^{-xt}\\,dt = \\frac{\\Gamma(a+1)}{x^{a+1}}", "23648d3f1d8baedfc570e43faab317bb": "T_\\text{cool}", "236536186ff53fbf31de6fa822e385fa": "2^{2^{h-2}}", "23664b4fba104db53e7d3eaefdf74c2c": "\\text{Gal}(\\overline{\\mathbf{Q}}_{p}/\\mathbf{Q}_{p})", "2366728f0a3f337be4645682c8984bc9": "I_{L1} + I_{L2} * \\cos\\frac{2}{3}\\pi + j * I_{L2} * \\sin\\frac{2}{3}\\pi + I_{L3} * \\cos\\frac{4}{3}\\pi + j * I_{L3} * \\sin\\frac{4}{3}\\pi", "2366947f34186f36e13ae124aad86b09": "\\sigma_k = \\gamma_k \\gamma_0", "2366bd5446dc3b6a239076a083472e37": "\\mathcal{E}(u) = \\int (A\\nabla u,\\nabla u)\\; \\mathrm{d} x, ", "2366d570e9dfccfa473963f9c6ed4c74": "\\widehat\\delta_{ab}=\\delta_a \\circ \\delta_b", "2366ea6eef15986d41d31b72266ef715": "\\tan{\\theta\\over 2} = \\frac{\\cos(\\pi/q)}{\\sin(\\pi/h)}.", "2367104f17127148a69711afc06422dc": " e^{-b {\\mathrm{Log}[x]}^2}x^{-2-a} \\left( -\\tfrac{2b}{a} + \\left(1 + a + 2b \\mathrm{Log}[x]\\right)\\left(1+\\tfrac{2b \\mathrm{Log}[x]}{a}\\right)\\right) ", "2367138ffe7e282e2a6ae2cd8685e4d8": "n_z^e", "236779514b8ad71053b6ca13af512fb3": " f( x ) = \\frac{ 1 }{ 2 \\sqrt{ 3 } } \\quad \\text{if} \\quad | x | < \\sqrt{ 3 } ", "2367926ea7250ce863c566930ac3bd3d": " (\\mathbf{A} \\mathbf{B})\\lambda=\\mathbf{A}(\\mathbf{B}\\lambda )", "2367dc82da759521a88eaf98587d8e45": "\\mathbf{p}=\\gamma(\\mathbf{u})m_0\\mathbf{u}", "236819c437d8b1cd0383b13798b2535b": " \\operatorname{build-param-lists}[g\\ q\\ p\\ n, D, V, K_1] ", "23689dc4a677a22115dac83b43008b11": "\n \\frac{\\partial A^{-1}_{ij}}{\\partial A_{kl}}~T_{kl} = - A^{-1}_{ik}~T_{kl}~A^{-1}_{lj} \\implies \\frac{\\partial A^{-1}_{ij}}{\\partial A_{kl}} = - A^{-1}_{ik}~A^{-1}_{lj}\n ", "23691b0a9421c76fd0952811483c3cf2": "\\mathrm x \\approx \\sqrt{.20 \\times (1.8 \\times 10^{-9})} = 1.9 \\times 10^{-5}", "2369665ca29a60404bd292319f39bd8c": "\\sum_{i=1}^n \\mathrm{NegativeBinomial}(n_i,p) \\sim \\mathrm{NegativeBinomial}\\left(\\sum_{i=1}^n n_i,p\\right) \\qquad 00 ", "23756c5030d08070ac91df03b7fdf5c7": "x \\in \\mathbb{R}^N", "237575586a56b316e56ee84559734faf": " \\left (\\frac{p_1}{p_2} \\right )^\\frac {1}{\\gamma}", "237579ab9744ae9f71b788e67b3124ac": "p_{1},p_{2}", "23758582a8255b75590b7cb4b6e2d6a0": "[01\\overline{1}2]", "23758dd51f91d093c8627e71bea89b1d": "(\\mathbf F^+)_+", "237591bc7e6ea76fe0d65161a4060cc0": "\\frac{n-5\\pm\\Delta_n}{2-2n}", "2375a5deb49069f214314679f307be3a": "\\alpha_i\\nmid\\alpha_{i+1}", "2375a71d49a71c8a4f4f53000318c51c": " A_0 = |0\\rangle\\langle 0| +\\sqrt{\\eta}|1\\rangle \\langle 1| ", "2375b0c83441ed229b84914eb93c9420": "B=\\begin{bmatrix} 0 & a & b \\\\ -a & 0 & c \\\\ -b & -c & 0 \\end{bmatrix}.\\qquad\\operatorname{pf(B)}=0.", "2375ed3e6c9ca2eb8b716aecf32c52a3": "m \\frac{\\partial^2 x(t)}{\\partial t^2}", "2376435019508a9eefbf970a93e540e6": "P_{\\alpha}\\,", "2376946ea2c96e1271fe1c701383d7e6": "\\hat{Z}(x)", "23769a8d77dd49f082b6b3dfe5f0cc8d": "P \\or Q", "2377355ca640f8a52026b876b620faa0": " \\frac{t}{x} = \\frac{S}{S-L} \\Rightarrow S = \\frac{L}{1-x} ", "2377543493a57fd7b5022e6b2d2b447a": " s_1(T) \\ge s_2(T) \\ge \\cdots s_n(T) \\ge \\cdots \\ge 0", "23776b94da51114c679f9f0cb970350c": "\\{x_1, \\ldots, x_k\\},\\{x_{k+1},\\ldots,x_n\\}.", "23778b1022fc8bb9bcd84fafc7159666": "T_{B}", "2377c945680222d8ac6084fc0501055d": "b\\in e_2", "2377d1b826c82fe2a508d2544d0960d8": " \\tilde{\\psi} (\\eta, \\tau) = \\frac{1}{\\sqrt{2 \\pi}} \\int{\\psi (\\xi, \\tau) e^{i \\eta \\xi} d\\xi} = \\sqrt{2 \\pi} e^{i \\tau} \\left[ \\frac{1+2 i \\tau}{\\sqrt{1+4 \\tau^2}} \\exp \\left( -\\frac{|\\eta|}{2} \\sqrt{1+4 \\tau^2} \\right) - \\delta(\\eta) \\right] ", "2377e8b25b3edd80cd8d32c69588aae1": "{}^\\perp", "2377f2b7f6e882d65432133c1674bd3f": " J= \\sum_i | X_i - L M_i R^T| ^2 ", "2378182c67dfb9efc306ea55080dbebc": "Z=\\Phi(\\beta)-\\Phi(\\alpha)", "23782db3097d425aad3a7de0bcfb39c6": " 1 = |\\psi_R|^2 + |\\psi_L|^2 . ", "23785c83f96ce16d3ef39c928607b10a": "y' = x\\sin\\left( -\\Omega t \\right) + y\\cos\\left( -\\Omega t \\right)", "23786eb10c4d9583aa62e2be47214bff": "\\rho: G \\rightarrow \\ker\\, f", "237897ffeb8ab2371f1fb350cc3de54c": "\\ w_r > \\frac{l_e}{l_{us}} w_u ", "2378df30463c6263f6c597191efd3b24": "\\sigma_i^2 = h(z_i'\\gamma)", "2379f2d9fcb8804f8f4abb661817aeb2": "(a \\pm \\sqrt{a^2 - 4})/2", "2379ffdee680960ff908060d756e6db5": "\\Delta F = F(B) - F(A)", "237a2dfc9d74b20f86a58afa5f636cad": " \\operatorname{let-combine}[\\operatorname{let} p : \\operatorname{de-lambda}[p\\ f] = \\operatorname{let-combine}[\\operatorname{let} x : \\operatorname{de-lambda}[x = \\lambda x.f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x)] \\operatorname{in} p] ", "237a3fc7110eaf54a6abda52d0700abe": "H_x = - \\int |\\psi(x)|^2 \\ln (|\\psi(x)|^2 \\cdot \\ell ) \\,dx =-\\left\\langle \\ln (|\\psi(x)|^2 \\cdot \\ell ) \\right\\rangle", "237a57b8d8eecd8d201e9608c89e4f02": "I \\otimes_R M \\to R \\otimes_R M \\cong M", "237af5f18d650c377d23ebd2837d6567": " R^{\\lambda \\sigma}_{\\mu \\nu}= e^{\\lambda}_a e^{\\sigma}_b R^{ab}_{\\mu \\nu} ", "237b1e5150b595330a18876a348aec8a": "\\delta(A,0)=\\arg\\max_{\\delta\\in{NS}}U_{A}(\\delta)", "237b49ed1be171cae222e45e319c353a": "\\frac{ 4 }{ \\tbinom{52}{13} } = 0.000000000629908\\% = 1:158753389900", "237bbb45046cc6f091ce32c38f1394e5": " _0 = -2.90", "237bc5f379ba03d325cbd7e68dc587d1": "\\mu^{\\otimes n}(A_n(s,t))\\le\\frac{\\bigl(\\mu(I_{s,t})\\bigr)^n}{n!}", "237be621b09d243f5efd72fdd9d8dc4c": "\\pi \\cong \\pi_1(M) \\cong \\pi_1(W) \\cong \\pi_1(N).", "237bf0471d8776da81fee9ce62a99972": "\\begin{align}\n\\frac{dH}{dt} & = rH - cHP \\\\\n\\frac{dP}{dt} & = acHP - mP \\\\\n\\end{align}", "237c0cd81f3b0024201d79f3d9ecd726": "c=1-\\zeta_n^2+x_i^2\\zeta_n^2/\\xi^2", "237c6c591bfd76cb00fd29d266dc33e5": "\\tfrac{DGxDF}{FH-DG}", "237c8d751559b361bc60246dcf6a700e": "L(u\\otimes v)=D(N\\otimes I)(u\\otimes v)=D(Nu\\otimes v)=D(Nu\\otimes NNv)=", "237ccaf4172f5eee73b9914a166293ed": "a_2b_2", "237cd679ed94572b222ae88fe5d3027a": "V=\\{V_\\gamma | \\gamma< \\delta < \\beta \\}", "237d08720d6bd10135017a72d0fab0fb": "\\Omega^0(M,E) = \\Gamma(E).\\,", "237d817de718a94e4e290a836db315e5": "u=1/r", "237dccd3133f35c013d7673ca2a706e8": "f(U)", "237dd051b49fa19fb73e8afe6f3beb07": "arg(z(z-a)) = const.", "237e74255e59d6d0e534c2d26942fb38": "C(v)=\\int\\limits_{0}^{v}{{}}xf(x)dx", "237e8cbfd0e5837c9b0b05422147a950": "\\mu(\\bar{S}_{2t})\\,\\!", "237ea2504d2bb6bea643d21912ae9164": "\\dfrac{\\mbox{b.hp.}}{\\mbox{b.hp.} + \\mbox{avg. f.hp.}} \\times 100", "237f0134f94505e054a0e4afcfe5c2d7": "{\\rm E}[z]\\,\\,\\, = \\,\\,\\,\\mu _z \\,\\, \\approx \\,\\,\\,\\,a\\mu _1 + \\,\\,b\\mu _2", "237f1ba24374434f8078fb3f372497d4": "R_U=\\mathrm{C}_E(\\mathrm{C}_E(R_U))\\,", "237f5daac464adafffb85d6975de4fc3": "h/b \\ge 1", "237f6174f1e520ad0dc3f11738a2b7a5": " o(dtdu) ", "237f94b37825bf506d3a1dff916aeddb": "\n0 = \\nabla_{\\vec{\\theta}} \\tau = \\vec{\\theta} - \\vec{\\beta} - \\nabla_{\\vec{\\theta}} \\psi(\\vec{\\theta}) \n", "23801f0b589df9a8994604fa47a70629": "F \\times_G B \\to B", "2380987640578a06dbcaace09c71d566": "\\psi(s+1)=-\\gamma-\\sum_{n=1}^\\infty \\frac{(-1)^n}{n} {s \\choose n}", "2380c1c885fee165e5b39b18a2b0fd02": "i^n = i^{n \\bmod 4}\\,", "238116692f4ee6f75942fdbe61ba1ec5": "\\displaystyle{Cf(w)=T_\\Omega F(w),}", "238126cb40b224b4aef6af5e51f29227": "\\mathrm{Hom}(U\\otimes V, W) \\cong \\mathrm{Hom}(U,\\mathbf{Hom}(V,W)).", "2381484424253e9d6c57aa5192e74a3d": " \\lim_{n \\to \\infty} a_n = \\operatorname{st}(a_H) ", "238180514e48be9bf0f85ba9a7ec5df0": " \\int^\\oplus_X H_x \\ d \\nu(x) \\rightarrow \\int^\\oplus_X H_x \\ d \\mu(x). ", "23819be0599b381a8c0e87d3b0626ed2": "K(x_i ,x_j ) = \\left\\langle {\\phi (x_i ),\\phi (x_j )} \\right\\rangle", "2382172c93f1ac204fa7c10f2f9b683d": "\\sqrt{\\gamma_n}", "2382846a7a5785b46866da2e1ca4dfec": "\\tfrac {d(\\hbox{Victims})}{d(\\hbox{Time})}", "2382b6abb273218ca5d4671ded8fe172": "p(x_i) = y_i \\qquad\\mbox{for all } i \\in \\left\\{ 0, 1, \\dots, n\\right\\}.", "2382c4917af5ed37cb90605c3766a491": "h = h_1 \\cup h_2", "2382ed508ccfd013f65c11775a121625": "\nh_{rf}=\\frac{14.2(1-\\nu^2)}{\\rho_wg}\\frac{\\sigma_t^2}{Y}\n", "23834e353307c767c28711b41823b8d9": "\\star \\mathrm{d}t \\wedge\\mathrm{d}x = - \\mathrm{d}y\\wedge \\mathrm{d}z", "238351f95190b407b5d9fb859e20452a": " fg = f\\circ g = (1\\ 2\\ 4)(3\\ 5)=\\begin{pmatrix} 1 & 2 &3 & 4 & 5 \\\\ 2 & 4 & 5 & 1 & 3\\end{pmatrix}.", "2383bd43d68a54dcbd5502106997681e": "2\\pi \\,", "2383ce0ca57d0ff95d4a67f8914e19a5": "\\lambda_\\mathrm{impurity}", "2383f2fa68a5acd665dceb30749c31c0": "X \\cup E", "2384172053ad501905269dc99e8f1d7f": "x_{Ai}", "23841a994404a10914aef93d4dc2bd74": "\\mathcal{A} \\setminus (\\{A \\} \\cup \\Gamma(A))", "23844370a14cda68defd9dceb8e35c08": "\\vec{x}_A", "2384515c6f2b85927f5384e3051df992": "x^{(n)}=\\frac{\\Gamma(x+n)}{\\Gamma(x)},", "23846db5d05a3b92d77a743039357fcf": "x' = \\sum_{i=1}^k e_ip_1^{e_1}\\cdots p_{i-1}^{e_{i-1}}p_i^{e_i-1}p_{i+1}^{e_{i+1}}\\cdots p_k^{e_k} = \\sum_{i=1}^k e_i\\frac{x}{p_i}.", "23848cef89ce4cbeb31a6eb8b3910be0": " T^2 \\sim \\nu pF_{p,\\nu-p+1}/(\\nu-p+1), ", "2384b8df829cd86139562bcae6ac6912": " \\pi \\rho_S(x)^2 = 2\\pi x - \\pi x^2.\\,", "2384d2709593fcabb11aa6609035d169": "\\begin{pmatrix}a_1 \\\\ b_1 \\end{pmatrix} = \\begin{pmatrix} T_{11} & T_{12} \\\\ T_{21} & T_{22} \\end{pmatrix}\\begin{pmatrix} b_2 \\\\ a_2 \\end{pmatrix}\\, ", "2384d8e539390199adca8dec11680d5c": " \\int_G f(g) \\,dg = \\int_S\\int_K f(x\\cdot k) \\, dx\\, dk = \\int_S\\int_K f(k\\cdot x) \\Delta_S(x)\\,dx\\, dk.", "238526af1095d1d92c03cb5704755132": "\\alpha,\\beta", "2385cbeaa2abe7575d167ac4075664de": "\\textstyle{-\\frac{\\partial}{\\partial x}\\left(EI\\frac{\\partial^2 u}{\\partial x^2}\\right)}\\,", "2385fde6b300ad088e0c2711c9da8a74": "f: X \\to Y", "238615d0ac8432bb816b02baa36ea904": "f:X\\to{\\mathbb F}", "238645f3fe21752b7ae115372af76d48": "(x_1,y_1,z_1)", "23864fdbf6ba66eaa4a2f5101797b807": "w_p \\cdots w_q", "23868a17ccbec489b6bee19f0efdc750": "\\mathbf{AB} = \\begin{pmatrix} \na & b \\\\\nc & d \\\\\n\\end{pmatrix} \\begin{pmatrix} \n\\alpha & \\beta \\\\\n\\gamma & \\delta \\\\\n\\end{pmatrix} =\\begin{pmatrix} \na \\alpha + b \\gamma & a \\beta + b \\delta \\\\\nc \\alpha + d \\gamma & c \\beta + d \\delta \\\\\n\\end{pmatrix}\\,,\n", "2386a28918264fbc7e3276cf6dbc8ce8": "\\textstyle\\ I_{max}=\\underset{f}{max}\\underset{C}{max}I(C, F_{f}) ", "2386a3fde7097a0f9550ae4cacd201ca": "k \\,", "2386ae014238cad8ddde254421155123": "... \\Rightarrow SS \\Rightarrow^{ac}_{f} AAS \\Rightarrow^{ac}_{f} AAAA \\Rightarrow^{ac}_{g} AAAA", "2386f69294fb579d19a8f23a7326a80b": "T_T", "23874fcd9a755f0cb8c441c0b638723a": "{\\sqrt{z}\\over \\tanh(\\sqrt z)} = \\sum_{k\\ge 0} {2^{2k}B_{2k}z^k\\over (2k)!}\n = 1 + {z \\over 3} - {z^2 \\over 45} +\\cdots ", "23876585065cf839a9d9c54b0fac8fa9": "\\min \\left\\{ d_i \\right\\} = \\left| m_q - M_n \\right| - d_q", "23876a350e1516e8360aee7a14a88b79": "\\rm HCO_3^- + H^+ \\rightarrow H_2CO_3 \\rightarrow CO_2 + H_2O", "2387946cfc90e40a61f381d6509241ef": "\\ge \\tfrac{1TeV}{c^{2}}", "238853b38d39941942795591e260d2c6": " S = \\{ (x,y,z) \\in \\mathbf{R}^3 | x^2 + y^2 + z^2 = 1 \\}. ", "238855f99bc46041d67124d358a109dc": "\\frac{d T_n}{d x} = n U_{n - 1}\\,", "238864cdb6c51a514c83eba92384d9ee": " C_{ijkl}\n= K \\, \\delta_{ij}\\, \\delta_{kl}\n+\\mu\\, (\\delta_{ik}\\delta_{jl}+\\delta_{il}\\delta_{jk}-\\textstyle{\\frac{2}{3}}\\, \\delta_{ij}\\,\\delta_{kl})\n\\,\\!", "2388672dc3566e4f4633f7c4c3de12a6": " P(S_i) = P(X_1^n(i)) \\,. ", "2388d4d89b206844a3a394ed42d7e7bf": "\\coprod f_\\alpha : \\coprod X_\\alpha \\to Y", "2388de8f5034cdb4e6cab155850aa63d": "\\mathbf{e}_\\text{y}\\times\\mathbf{e}_\\text{x} = - \\mathbf{e}_\\text{z}\\,\\quad \\mathbf{e}_\\text{z}\\times\\mathbf{e}_\\text{y} = -\\mathbf{e}_\\text{x}\\,\\quad \\mathbf{e}_\\text{x}\\times\\mathbf{e}_\\text{z} = -\\mathbf{e}_\\text{y}", "238957a918afc3365e5e323c018e81e5": "g_{ab}=-g_{tt}dt^2+g_{rr}dr^2+g_{\\theta\\theta}d\\theta^2+g_{\\phi\\phi}d\\phi^2\\,,", "2389b49c96b5d85c40421b6f9822025f": "\\textbf{z}_{n}", "2389b5a8925c6beffa1b119c2bb770e0": "\n\\omega_{k} = \\pm \\sqrt{\\beta_{k} - \\frac{1}{4}}\n", "2389c100b5dda2fd31576fac04f31556": "\\psi(\\Omega^\\omega) = \\phi_\\omega(0)", "2389ddfd028c6cd7ff3adb88a080a44b": "\\sqrt [39]{92}", "238a3262a795dd7faf2d5d8ea056982b": "{} = - 20\\log \\left|1+j{\\omega \\over {{\\omega_\\mathrm{c}}}}\\right| = -10\\log{\\left(1 + \\frac{\\omega^2}{\\omega_\\mathrm{c}^2}\\right)}\n", "238a3a42dbfa6d81eebb6f7a95c52a3c": "\n\\alpha^{\\prime} = 1 + \\frac{[I]}{K_{i}^{\\prime}}.\n", "238a85eb4bda03d1a04693c67aea5fe3": "R(\\theta,d') = R(\\theta,d) - (n-2)^2\\mathbb{E}_\\theta\\left[\\frac{1}{|\\mathbf{X}|^2} \\right]", "238aaffb94a970fdcbb4af65c56bea91": "\\scriptstyle \\psi x \\,= \\,x+ \\frac{x^2}{2^2}+ \\frac{x^3}{3^2}+ \\cdots+ \\frac{x^n}{n^2}+ \\cdots", "238ac359e88c840a09058d93876324bf": "\nH_{R}=\\alpha(\\boldsymbol{\\sigma}\\times\\bold{p})\\cdot \\hat{z}\n", "238aca5bcf24c08bae38ce4c34911a54": "\\begin{smallmatrix}R_R = {\\left ( {\\frac {d_R}{2}} \\right )} = {\\left ( {\\frac {3.740}{2}} \\right )} = 1.870 AU \\end{smallmatrix}", "238b200b5118a4cc0861756b00125757": " d^4 p \\to \\mu^{4-d} d^d p", "238b920eb9099971ebcd2818f464368c": "x \\equiv_{qc} y", "238bdc3e65bb66614ddacc39edbd8b9d": "Y = \\{Y_1, \\ldots, Y_s\\}", "238be49f72833929caf8ae9920d83178": "n(\\lfloor\\frac{rB}{r+1}\\rfloor - \\lfloor\\frac{B}{r+1}\\rfloor)", "238beb6701b94915256eccc0a727af78": "S(x) = - \\int p_x(u) \\log p_x(u) du", "238c250f1aa3d0266edb98bf95910304": "|R_i| \\geq \\sqrt{N-1}", "238c251f2e4a9d0d6d67bd0a05939e84": "O(k^2)", "238c25fc2c22e400d1a424d62c116d49": "a = 1 - \\gamma", "238c2ae511378acdf3d80257b24aa2c5": "P \\underline{\\lor} Q", "238c4ca8ae138e02783454e6728a0d4c": "\\ e_t", "238c9ad82f55cc6ac411519d3cf05738": " \\cos(\\theta_2+\\theta_4)+\\sin\\theta_2\\sin\\theta_4=\\cos\\theta_2\\cos\\theta_4 \\, ", "238d30203e50860341d9c8edf5b4266c": "T > 0", "238d4d00dd73cbde448fcce08074ac68": "\\mathcal{P} \\, \\exp", "238d4f6abb27ad17e50c8c742730d8a7": "\\sin \\alpha = \\left( \\frac{w}{l} \\right)", "238d6fea37f08fcff762402d8d887de2": "\nk(z,y) = G_{p+2,\\,q}^{\\,m,\\,n+2} \\!\\left( \\left. \\begin{matrix} 1 - \\nu + i z, 1 - \\nu - i z, \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\; y \\right),\n", "238d7aed8f22662e832d56ca109c0669": "A_{\\mathrm T}=A^n \\ ", "238d7c3eb38f1f63a6956416977dd17f": "(n, k) = (255,223)", "238daed21dea66f05db354ce045740c9": "\\scriptstyle w_1, \\dotsc, w_n", "238dd3e454c07523ddacafe5918cbe5c": "S_1(A_n)=S(A_n).", "238e1e63379bc49db327f9386c2da30e": "\\mu_{1} \\equiv E[W^1]=g_{1}(\\theta_{1}, \\theta_{2}, \\dots, \\theta_{k}) , ", "238e215539bc986852cd4cfe25aca0c8": "\\mathbf{a} \\cdot \\mathbf{b} .", "238e280f09ea50bfb37c6c03c4967374": "\\delta = \\frac{V_i-V_o}{V_o}D", "238e58962004992c54f3db69d239e45b": "\n\\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & -i & -1 & i\\\\\n1 & -1 & 1 & -1\\\\\n1 & i & -1 & -i\n\\end{bmatrix}\n", "238e60693cccdab5e4844673d0b6565a": "(2)\\ \\ H_n(x)=(-1)^n e^{x^2}\\frac{d^n}{dx^n}e^{-x^2}=\\bigg (2x-\\frac{d}{dx} \\bigg )^n\\cdot 1 ,", "238e6ae69670fa2008d54c4e37bded3b": "M_G(x) := \\sum_{k\\geq0} (-1)^k m_k x^{n-2k}.", "238e8cc76ba7adfb1211da47d36f1db5": "{k'}^2=m_1\\,\\!", "238e984bfd53f92e48fe40526c14ba2f": "\\ v = \\frac{V}{m}\\ = {\\rho}^{-1} ", "238ea6b940d3a3d5abdc962e4ca0bdaa": "x^2+y^2+z^2=0", "238ecf0920fda7c5cc8326b6d2f6a48f": "\\Gamma \\subset V \\times W", "238f05d594779801bc7f1a7cfdb0cd7f": "\\phi=\\phi_0 e^{\\frac{-x}{\\lambda_D}}", "238f6e7476de18fd893637ef736be09e": "a=1-2\\nu\\,\\!", "238f70cbb76eb9971a22df56c7253e52": " A\\mapsto {\\rm Tr} f(A)=\\sum_j f(\\lambda_j),", "23908618399eeea28692428caa459379": "x_k^*:= x_j^*+\\left(b^*- x_j^* A \\right) P_k A^{-1},", "239098b65fded7e8d3ac5307c68e5336": "\\scriptstyle \\sum_n |c_n|^2 |\\psi_n\\rang\\lang \\psi_n|", "23909d5f27eefc3f6b24a7f82507b07f": "y_{k+1}=n-\\sum_{i=1}^ky_i", "2390aa392ed52d52147ba520bdec82e5": " \n\\bold{J} = -\\rho dt + j_x dx + j_y dy + j_z dz \n", "2390b4a08d441ad9adb1b76b742a9b57": "\\Rightarrow_{S \\to XX}\\ XXS \\ \\Rightarrow_{S \\to XX}\\ XXXX", "239103f41782fb2d7d5e43f9d27a22f0": "\\left( \\left| {{H}_{WCM}}\\left( s \\right) \\right|=2.4 \\right)", "23915299bcbc02470f6d56e5050cf5ae": "x^2z^2 > xy^2z > x^3 > z^2", "239158be952fd5b200781d73d04c1e14": "F_{\\rm fluid}=-k_BT_e\\nabla n_e,", "2391a32feb98c643e952e6404103389a": "p'_G-p'_L=g\\eta\\left(\\rho_G-\\rho_L\\right)+\\sigma\\eta_{xx},\\qquad\\text{on }z=0.\\,", "2391bbeeb5b0869378b98bd1b0e4c141": "\\frac{\\partial \\mathbf{y}}{\\partial x} ", "2391c76ff1b8acbf14b99a46a439afa4": "\n[\\wp'(z)]^2|_{z=0}\\sim \\frac{4}{z^6}-\\frac{24}{z^2}\\sum \\frac{1}{(m\\omega_1+n\\omega_2)^4}-80\\sum \\frac{1}{(m\\omega_1+n\\omega_2)^6} ", "2391cf7a07ccaf0c37f4a7e691755e5d": "t \\to \\infty", "2391ef400234f70fd83e0eac6a57a2c0": "\\left|\\frac{\\sqrt{x^2 + y^2}}{r}\\right|^p + \\left|\\frac{z}{h}\\right|^p \\leq 1", "2391f65e83f43fc79933e459137a88c4": "\\,h_1 = 0", "2392948d1f505aaa5ad677305cc81019": "e = \\sum_{k=1}^\\infty \\frac{k^n}{B_n(k!)}", "2392c3be9fd69cf5e77eccfc8f16fda3": "= \\frac{T(x,t)-T_\\infty}{T(0,t)-T_\\infty}", "2392ec97fe6e7f855d788933c202c36d": " \\displaystyle \\alpha = -\\operatorname{sgn}(a^k_{k+1,k})\\sqrt{\\sum_{j=k+1}^{n}(a^k_{jk})^2} ", "2392f0dce629068d9184a545432c7c52": "\\varphi^N", "239335c749997f502334b305a7771f63": " IG(T,a) = H(T) - H(T|a) ", "23936a48ad55b7d6583f95ee5ec3b66c": "\\partial f_{i_m}/\\partial x_{j_n}", "2393b884af08b2044143851934cedaff": "X \\times_S P \\to P", "2393e9ae74251d903ac59710283e6df0": "r = \\sqrt{\\frac{A}{\\pi}}", "239417505db3134d8402553ded71e953": "\\gamma_c(A) = 1", "23941c0e5146df2c92402295ed7ad5df": " -1+j0 ", "23942fdca9dba114b4644be5dab1edea": "v_{s}(\\mathbf{r},t)=v_{\\rm ext}(\\mathbf{r},t)+v_{J}(\\mathbf{r},t)+v_{\\rm xc}(\\mathbf{r},t).\\,", "23943ab5eddecf19de0515a8c9d3736a": "\\left(\\mathbb{R},\\mathfrak{B}\\right)", "239463091cd28199463a3ed2f4a3590a": "\nZ \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{k=1}^{N} m_{k} \\left| \\frac{d^{2} \\mathbf{r}_{k}}{dt^{2}} - \\frac{\\mathbf{F}_{k}}{m_{k}} \\right|^{2}\n", "2394716fa39131e443b4267df83bec7e": "\\mathcal U\\leftarrow\\mathcal U - s'", "23948fdc171ad44f7311e78f0cc253d8": "k(\\mathbf{x_i}, \\mathbf{x_j}) = \\varphi(\\mathbf{x_i})\\cdot \\varphi(\\mathbf{x_j})", "2394a5883d598fd4f2f11222b5617cc0": "\n\\max_{\\mu\\in{\\mathcal M}_X}\\,\\inf_{\\lambda\\in A_Y}k(\\mu,\\lambda)=\n\\inf_{\\lambda\\in A_Y}\\,\\max_{\\mu\\in{\\mathcal M}_X} k(\\mu,\\lambda).\n", "2394af39a0188d6347ac0e8ebe3bab16": "\\textrm{Beta}\\left(\\alpha_1, \\sum_{i=2}^K \\alpha_i \\right)", "2394cf96e6b78ac1050249265c29c7ab": " E[A] = \\int_0^\\infty Cr \\lbrace A \\geq t \\rbrace \\,dt - \\int_{-\\infty}^0 Cr \\lbrace A \\leq t \\rbrace \\,dt.", "2394dba51aa9e08f4b4632f2d4bf59f1": "Q^{n}(c) = 0, n = 1, 2, 3, ...", "23951ca44c73dfbd0ddc9928c64ce935": "Q(x'\\mid x_t) \\approx P(x') \\,\\!", "239520e7273cec8c3f84cf32958223bd": "\\chi_E", "239524b42c6cd49b5a89d78d3034a69c": "log BCF=1.119 log Koc-1.579", "2395a0e2858e528bc86da62d4975bb4a": "\\begin{cases}\n\nx \\equiv & s_{i_1} \\ \\bmod \\ m_{i_1} \\\\\n & \\vdots \\\\\nx \\equiv & s_{i_k} \\ \\bmod \\ m_{i_k} \\\\\n\n\\end{cases}", "2395d95eb0d33660a2e4633fe768c60d": "2g/R = 0.3086", "2395fec912343299bf6ca0cbcf2406f6": "-\\frac{1}{\\lambda^k}", "2396067fdff30ce1346a1ac7e0229601": "m^a\\partial_a=\\Omega\\partial_r+\\xi^3\\partial_y +\\xi^4\\partial_{ z } \\, := \\,\\delta \\,,", "23960e7bd640dd34bec78ed78df96b74": "2^{32}-1", "239693968f7b490432a60823494e2381": "\\gamma_\\alpha = \\lim_{n \\to \\infty} \\left[ \\sum_{k=1}^n \\frac{1}{k^\\alpha} - \\int_1^n \\frac{1}{x^\\alpha} \\, dx \\right],", "2396eaa516b5621ecd34523c0ccaef23": " \\frac{ ( \\frac{ f_m }{ N } ) - \\frac{ 1 }{ K } }{ \\frac{ N }{ K }\\frac{ ( K - 1 )} { N } } = \\frac{ M }{ N( K - 1 ) }", "23974652112a6f1319fa3c24e0f1cbea": "e^{i \\pi} = -1.\\!", "2397510aaaee5caa79c250cc987ac54e": "\\mu(x,G)", "239826a0a6a1c947d34924ba94b7aef8": "\\displaystyle{\\gamma = \\gamma_1\\gamma_2\\left(1 +{\\beta_1\\overline{\\beta_2}\\over \\alpha_1\\alpha_2}\\right)^{1/2},}", "239859719908f5b818416fb06d6b3018": "p_2,\\ldots,p_n", "23986d9b2a622a418cb84ad397a8df80": "h_{t}", "239891422d5b3bb67af33d93c9b76bb9": " (\\partial S)_V=-(\\partial V)_S=\\frac{C_P}{T}\\left(\\frac{\\partial V}{\\partial P}\\right)_T+\\left(\\frac{\\partial V}{\\partial T}\\right)_P^2", "23996d75b8ea74deaa0f05c057b2ed08": " \\begin{align} \n|\\mathbf{A}| & = \\begin{vmatrix} A^0 + A^3 & A^1 -i A^2 \\\\ A^1 + i A^2 & A^0 - A^3 \\end{vmatrix} \\\\\n& = (A^0 + A^3)(A^0 - A^3) - (A^1 -i A^2)(A^1 + i A^2) \\\\\n& = (A^0)^2 - (A^1)^2 - (A^2)^2 - (A^3)^2\n\\end{align}", "239a0d67ec7c87ba303b64d0f2f6102b": "\\delta U + \\delta V_{\\mathrm{ext}} = \\delta K ", "239a330b376765b9a6539f2f958e753e": "\\overline{\\Omega}_{\\theta\\phi}=\\Omega_{\\theta\\phi}/\\sin\\theta=1/2", "239a649ca1f3611c6756dae314fffc6b": "\\dot V(x) = V'(x) f(x) = \\mathrm{sgn}(x)\\cdot (-x) = -|x|<0.", "239a945679982606cb34e373d12f39cf": "(-1,0),", "239bda9e973ed934b8b9190b21e5ab40": "[W_T] = [W] + [W'] + [WE_1] +[W'E_2]", "239c179c407a5968cd05a75cc50ff292": "T_{Chorok} = T_{Dhuhr} - T(0.833)", "239c244fb3678b77ad4946105b79dd2f": "PK\\{(x): y_1= g_1^{x} \\land y_2={(g_2^a)}^{x} g_2^b \\},", "239c42d35e7c4548370c82ddf2cbce99": "\\scriptstyle \\theta ", "239c89327a02417de78b0f12f9dbbe66": "\\mathbf{1}_{A \\times B}(x,y) = \\mathbf{1}_A(x) \\cdot\\mathbf{1}_B(y). ", "239c97996db1251ae54ba9641466897d": "\\frac{q}{\\omega} \\ll 1", "239ce1c5e25a42de1d9befab1a20d68f": "2x(m+\\lambda)c_{mk}^\\lambda(x;k) = (m+2\\lambda)c_{mk+1}^\\lambda(x;k) + mc_{mk-1}^\\lambda(x;k)", "239cf037248aa80419b280e7247a9951": "U(-n,-2n,z)", "239d03df6eba1d0826665bafded86cd5": " Y_i = \\alpha + \\sum_{j=1}^p f_j(X_{ij}) + \\epsilon_i ", "239d0a7df79a33d003515d8154e55a93": " K_\\text{Matern}(x,x') = \\frac{2^{1-\\nu}}{\\Gamma(\\nu)} \\Big(\\frac{\\sqrt{2\\nu}|d|}{l} \\Big)^\\nu K_{\\nu}\\Big(\\frac{\\sqrt{2\\nu}|d|}{l} \\Big)", "239d80b28cf56b7451ad4a8215e694fd": "|r|", "239dd9a0fb8f15d164c9276b6b9abcf2": "\\mu_{ij}", "239e598b0761292677c59856acffd419": "S = S_{\\sigma}(\\sigma) + S_{\\tau}(\\tau) + S_{z}(z) - Et", "239e6aabd39cc7d05cbfefd79f33acaa": "H_b = 5L / 9", "239e8301daacc33676597ad1076edd54": "\\text{max}(X)", "239eef8d93c4dbbdcf7288f3b518928e": "y=r\\sin\\phi.", "239f29cf2cb510a6a635a1f26dde7a2a": "\\frac{a}{b} < \\frac{c}{d}", "239f582295e0212effea57f1d78fb944": "\\mathrm{SR}(K) \\leq \\frac{\\pi}{\\sqrt{8}},", "239fd97453551fa7922d8e122a6ae591": "-\\nu", "239fe35f187fc4e957f54b892987f260": "f(x + b)", "23a04f8a07f13d42a8482b7079861ac8": "\\{ n+1, n \\stackrel{.}{-} m, \\lfloor n/m \\rfloor, n^{m} \\}", "23a086f0edf8d1ad3deffad53608e8f1": "f = \\frac{1}{2 \\pi \\sqrt{LC}}.", "23a0a79f2d63407554c2be91a50a2172": "z_i = \\lambda \\bar x_i + \\left ( 1 - \\lambda \\right)z_{i - 1}", "23a1310d1120d016f48912b033305226": "{1 \\over \\rho}{\\partial \\left( \\rho A_\\rho \\right) \\over \\partial \\rho}\n+ {1 \\over \\rho}{\\partial A_\\phi \\over \\partial \\phi}\n+ {\\partial A_z \\over \\partial z}", "23a151eebcbede0cb7c833bbbe1449a3": "\nP\\left(Searched|Known\\wedge\\delta\\wedge\\pi\\right)\n", "23a153d91b7dd0bc7f3b7a7c602d0c1f": "t=-0.5,", "23a1967d654f959dd43f22bb255d330b": "\\scriptstyle{\\bar k \\sim \\ln N}", "23a1a539958ae0123a278e164ab77f30": "\\frac{\\left|{\\mathbf\\Psi}\\right|^{\\frac{\\nu}{2}}}{2^{\\frac{\\nu p}{2}}\\Gamma_p(\\frac{\\nu}{2})} \\left|\\mathbf{X}\\right|^{-\\frac{\\nu+p+1}{2}}e^{-\\frac{1}{2}\\operatorname{tr}({\\mathbf\\Psi}\\mathbf{X}^{-1})}", "23a1daa7cd08229dfcf67f642ab33607": "E_t", "23a25c3fced257b61da4af1bd5eaf276": " C_k = H g_{n_1} g_{n_2} \\cdots g_{n_j} ", "23a298bf814fdb25f587cbb703ddc7c2": "\n -\\sqrt{2}~\\xi~\\sin\\phi + \\rho[\\cos\\theta - \\cos(\\theta+2\\pi/3)] - \\rho\\sin\\phi[\\cos\\theta+\\cos(\\theta+2\\pi/3)] = \\sqrt{6}~c~\\cos\\phi\n ", "23a2b356551c0b040af48b08dc08ae11": "I_{\\mathrm{Alice}}, I_{\\mathrm{Bob}} \\in GF(p)", "23a2f3546bcfc25675fcefce51056465": "PC_x = C_x = QC_x.", "23a31c52414c99221ef910fce5b36399": "\\vec n_i\\cdot\\vec p_0=d_i, \\ i=1,2,3 , ", "23a343321e5dd03c5fcfeae6cdeb5299": "S(K)=(P,B,I)", "23a3441ddcb52730f26d485b7599d638": "Ty = y. \\, ", "23a3605478ef02cbbb98871a555610d3": "\n-K_{22}^{-1}K_{21}x_{1}=x_{2}\n", "23a3650f37741dfa7394f4711ea081c3": "{\\Bbb Z}", "23a36a1dc6ce08dbf01396df6cdc1217": "\\left\\{{B_n : n = 1, 2, 3, \\ldots}\\right\\}", "23a375a1ae51d74da9ee54c67369a2ce": "H_2A \\rightleftharpoons A^{2-} + 2H^+ :\\beta_D = \\frac{[A^{2-}][H^+]^2} {[H_2A]}=K_1K_2", "23a3897e299ad31ee0a49fef114f7b89": " \\vec B ={\\rm curl\\,\\,}\\vec A =\\left\\{ \\frac{\\partial A_3}{\\partial x_2}-\\frac{\\partial A_2}{\\partial x_3} , \\frac{\\partial A_1}{\\partial x_3}-\\frac{\\partial A_3}{\\partial x_1} ,\\frac{\\partial A_2}{\\partial x_1}-\\frac{\\partial A_1}{\\partial x_2}\\right\\},\\text{ or }\\Phi_B={\\rm d}\\mathbf A.", "23a3afef797162db2d239fca85e0ad42": "^A_Z E", "23a3c81abab62099dcfb6c08284b8331": "= \n{1\\over (j+1)}\\left(g^{(l)}{1\\over 2} \\left(j(j+1) + l(l+1) - s(s+1)\\right) + g^{(s)}{1\\over 2} \\left(j(j+1) - l(l+1) + s(s+1)\\right)\\right)", "23a4493c718a73272da4e3ba1248ed60": "(a+b\\sigma_1\\sigma_2)^* = a+b\\sigma_2\\sigma_1 = a-b\\sigma_1\\sigma_2\\,", "23a45ad10669944a029db1d16895b8dc": " \\omega_{i} ", "23a48eb292114d7fabcc83c653e46cfb": "g(r,r^\\prime)=g(r^\\prime,r),\\,", "23a4c65d87d1725a311b299b7f458550": " x = \\forall +", "23a4ce747750e06ece8d17d70a2f1c78": "\\varphi\\rightarrow\\lambda^{-1}\\varphi.", "23a51b36d5d18a407d6b0dd347f07e12": " \\lambda = -\\sigma =1 \\,", "23a51ca43e95162f2d61f3b929d4ec7a": "a\\otimes(1-a) \\, ", "23a53888f9da74144627bc2466a21718": "(-1)^k \\frac{\\partial}{\\partial u} \\left[ {\\left( \\frac{d}{dt} \\right)}^{2k} H_u \\right] \\ge 0 ,\\, k=0,1,\\cdots", "23a559d727323acfb796d87142eef2aa": "\\begin{Bmatrix} x \\end{Bmatrix}= \\begin{bmatrix} \\Psi \\end{bmatrix} \\begin{Bmatrix} q \\end{Bmatrix} ", "23a57408645e14fdc1ad9521daabfae0": "\\mathbf{B} = e^{-i \\omega t} \\sum_{l,m} \\sqrt{l(l+1)} \\left[ a_E(l,m) \\mathbf{B}_{l,m}^{(E)} + a_M(l,m) \\mathbf{B}_{l,m}^{(M)} \\right]", "23a63f4431ad1392176c6b0dfc6936d0": "L_{-1}^2 \\Psi = 0,", "23a673f00eeac011bf6fa07a729444c1": "B_2(r, \\mu) = \\frac{1 + \\frac{1}{2}\\left [ 2r^{\\mu+2} - \\left (r+1\\right )^{\\mu+2}-\\left |r-1\\right |^{\\mu+2}\\right ]}{2\\left ( 1-2^{\\mu}\\right )}. ", "23a6a55c2e1f7574ea763601a3a53b77": "(\\text{extend} \\, f) \\circ (\\text{extend} \\, g) = \\text{extend} \\, (f \\circ (\\text{extend} \\, g))", "23a6bbb53a41138cdbec7eb5a61b1e19": "\\mathfrak J(a)_n=b_n.", "23a6e0f6cd66756fdfb28d181da13ce2": "\\Pi_{\\rho,\\delta}^{n}", "23a747a1bb2060b23c65a7d930dd6a5e": " \\displaystyle{P=i{d\\over dx},\\,\\,\\, Q=x}", "23a76a73c8fcc330078c8915ad481662": "\\begin{array}{cc} P_{j}(d_{j})=\\left\\{\n \\begin{array}{lll}\n \\frac{|d_{j}|}{p_{j}} & \\text{if} & |d_{j}| \\leq p_{j} \\\\\n\\\\\n 1 & \\text{if} & |d_{j}| > p_{j}\\\\\n \\end{array}\n \\right.\n\\end{array}", "23a76f078afcdf1abac5e68b57be2cd0": "A(f)=2.0+20\\log_{10}\\left(R_A(f)\\right)", "23a79b700e61c75945a8ca545c360d39": "\\bar{R}_P", "23a7dd4a99d3dab7ff18ddab1590f332": "p:P\\to X", "23a867e8ff309993734c69271e021ae3": "\n\\mathrm{CR}(x) = \\sum_{i=-1}^2 f_i b_i(x)\n", "23a8b60c55364e140f2bf980986ff433": " T_i = \\frac{\\mu_i - \\bar{X}_i}{S_i/\\sqrt{n_i}}\\text{ for }i=1,2 \\, ", "23a8f63f4604e8422942a3c194f245dd": "S:R^m \\to R^m", "23a94169ff080fcf979759b5152be9fe": "\\{1,\\ldots,n\\}", "23a98e177f47b98e0e69c4dd9c5b073a": "\\sigma_{xx} + \\sigma_{xz} - \\sigma_{xy}", "23a9d6498ebd2e2516325125b95f5487": "\\mathfrak{P}^{117}", "23a9e65fcebd8b93cb2f34f72e470751": "\\displaystyle{\\mathfrak{h}_i=\\mathfrak{h}\\cap\\mathfrak{g}_i,} ", "23a9f81480e531b259106bb768b50d8a": "|c| = b", "23aa193272e880d05e5edb1bf28c9a43": "\n\\Phi_{pre} = \\frac{\\frac{1}{2}m_ic_s^2}{Ze} = k_B(T_e+Z\\gamma_iT_i)/(2Ze)\n", "23aa2b30caa48a902b5f1b098c90f2c3": "\nS^2=c^2\n\\left.\n\\frac{\\partial p}{\\partial e}\n\\right|_{\\rm adiabatic}.", "23aa406cd9eac1a3eabf5e352cc1ab1e": "\\theta_a\\,", "23aaea169d1d875537281c7b2c857d8e": "\\log(C_N)/N", "23aafbb9fed33af6b165eb862ffefcf3": "(\\neg P \\or Q)", "23aaffb330da7c647a08f30396a229cf": "16y^5-20y^3+5y-1=0\\,", "23abd9baea9c20c3bbfcc3399c734a1b": "f_i g_j=0", "23abf0416e59bee199a8119efffae661": " GJ \\frac{d^2 \\theta}{dy^2} = -M'", "23ac45d769a17e4b37a2bf2ca94b4b4c": "\\Delta L = Ed_{31} = \\left ( \\frac{d_{31}L}{t} \\right )U_x", "23ac91a8667937c895c3d6cdc2b8f33b": "\\beta_j+\\delta \\beta_j\\,", "23ac9895c8117cde10077d87166f9d67": "\\mathbb P^n \\to \\mathbb P^N", "23ac989c5916fa0f1327d642ed8edf63": "z = x + iy\\ ", "23ac996bad415c332fbb59ed77c514c2": "(\\text{ })", "23aca7bc64ac6120c5f43348a53d4d50": "\\langle X-\\alpha_i\\rangle", "23acab630dbfc6af384824c9ee7fe984": "O\\left (\\exp \\left ( \\left (\\tfrac{64n}{9} \\log(2) \\right )^{\\frac{1}{3}} \\left ( \\log(n\\log(2)) \\right )^{\\frac{2}{3}} \\right) \\right )", "23acaedd883a80eb5bed45d6d23267ee": " A(K\\backslash G /K).", "23acd870f145b90a0347c61dc7a0ddf5": "|S(P,f,g) - A| < \\varepsilon \\, ", "23acf87fa1a3e56dce02aef0ac7b9399": "U=(Y,Z)", "23ad0ee06b2c26cf10aad272dc8b5229": "end\\, while", "23ad95ed53baf9237f7365e441e73623": " T_{ant}", "23adb4af8f148e6bf490477f73c76f36": "d = \\frac {D}{m} = \\frac {130}{18.6} \\approx 35", "23adf8420e4622106eb134378be472de": "\\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = R_a,", "23aea4b7d12cd13a89aeb3e58dd5de73": "2^i\\operatorname{value}(x_i)", "23aef064683dda7b6efc577b136c2646": "\\alpha = 2 \\,\\!", "23af098b65939cb9b1a8d5f9f7d2893f": "\\Delta S_{vap} = \\frac{\\Delta H_{vap}}{T_b}", "23af122fc1d9de834d75d743f71af921": "H=(V, E)", "23af3eb99906c8ae50c67bd1522bc519": "\\land,\\lor,\\lnot", "23af853bd144b951de6349d131bd5057": "m_1=\\left\\lceil m/2 \\right\\rceil", "23afc5da73974ff83aa0207d14fb141d": "=\\mathbf{R}_{x}^{-1}(n)", "23afec566cf78d7a392b9c8b88d85570": "Q(2n,q)", "23b0cad8bd941a5f2b9119a05d70ca75": "k_h", "23b0e2a288ff19edf3eb766ae9837f7e": "SU(N_f)_L^3", "23b10a65a6c1b6dee97a31cd4f03436d": "\\sigma_{gt}", "23b111ab578d31773fad406694483bcd": "\\omega_1\\times\\omega", "23b1b2dd06a03b694c8a9d2daea99758": " \\; {}_1F_1(a+1;b;z)- \\, {}_1F_1(a;b;z) = \\frac{z}{b} \\; {}_1F_1(a+1;b+1;z)", "23b20ef3b27490f589843505e708242a": "\\ker(L) = \\left\\{ v\\in V : L(v)=0 \\right\\}\\text{,}", "23b212925dda1686cdc401f390653ce2": "\\{n, l, m\\}", "23b22ecd3125f01a95e03114d1fe3515": "\\mathcal A_k,\\mathcal B_k,\\mathcal C_k", "23b271603807fb72ed74507183d7faea": " \\mathbb{Z}_{p_{i}}", "23b2d65e7203d77e0c82610240710588": "3 \\rightarrow 3", "23b31d3f31d2735be679aa3b5f7a9c27": "\\ddot{x} + 2 \\zeta \\omega_0 \\dot{x} + \\omega_0^2 x = 0.\\,", "23b3421a91ee73f84e5dd44ce2743d6d": "\\begin{pmatrix} 3 & -4\\\\4 & -7 \\end{pmatrix}\\begin{pmatrix} x\\\\y \\end{pmatrix}.", "23b353ef9d1a4c4c6485b898e7f82590": "\\begin{align}\\oint_{\\partial V} \\mathsf{L} (dS;x) &= \\oint_{\\partial V} \\langle F(x) dS I^{-1} \\rangle \\\\\n&= \\oint_{\\partial V} \\langle F(x) \\hat{n} |dS| \\rangle \\\\\n&= \\oint_{\\partial V} F(x) \\cdot \\hat{n} |dS| \\end{align}", "23b374d014fd7adc2e0c6f3d74d05e9d": "\nR_c \\simeq \\ell \\sqrt{s} \\simeq \\frac{\\sqrt{s}}{n \\sigma}\n", "23b380b4c4720500069c5178f7341f19": "=\\frac{2v^2\\cos^2\\theta}{g} \\sqrt{\\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)^2+m^2 \\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)^2}", "23b3d324b0c3346906af9f52a9777b33": "L_x \\times L_y", "23b402ff95ceb6034741c161086af6c2": "R/P_i^{a_i}", "23b4091f193cd663f97501b87d0bd379": " \\frac{1}{\\mu_0\\varepsilon_0}=c_0^2. ", "23b42c80589a9a4843f6ff06f5ea079f": "Q = \\frac{M \\omega}{\\Gamma} \\,", "23b4ab61064190df13af7b7a3dc32681": "\\left\\vert \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right\\vert = \\sqrt{-{g}}\\,,", "23b4cf6ce31c03eb8ad55ac3b9a12702": "{\\mathbb N} = \\{0, 1, 2, \\ldots \\}", "23b4e9baf4059782e0fe4bcca4624132": "\\begin{align} q_0(x) &{}\n= \\int_\\mathbb{R} \\! \\frac{t^3 - x^3}{t - x} \\rho(t)\\,dt \\\\\n&{}\n= \\int_\\mathbb{R} \\! \\frac{(t - x)(t^2+tx+x^2)}{t - x} \\rho(t)\\,dt \\\\\n&{}\n= \\int_\\mathbb{R} \\! (t^2+tx+x^2)\\rho(t)\\,dt \\\\\n&{}\n= \\int_\\mathbb{R} \\! t^2\\rho(t)\\,dt\n+ x\\int_\\mathbb{R} \\! t\\rho(t)\\,dt\n+ x^2\\int_\\mathbb{R} \\! \\rho(t)\\,dt\n\\end{align}", "23b55fe0f610bd2c1f56cf3d0dd9e5b5": "\\int_{K} f(x + c y) \\, \\mathrm{d} x \\geq \\int_{K} f(x + y) \\, \\mathrm{d} x.", "23b583065f5b7336c728011ccd7375b2": "s=2", "23b58def11b45727d3351702515f86af": " ", "23b6559edfe9ac80f3036952ef4475ef": "\\mathrm{Factor} = \\frac{\\mathrm{Days}(\\mathrm{Date1}, \\mathrm{Date2})}{\\mathrm{DiY}}", "23b6882b77ee4423111844725fea3211": "\\varepsilon >0 ", "23b6cf6cb393df40b34b28d11a0ae402": "s = 1-s\\rho", "23b6ded51212dde5ce0120d1b04f8ee4": "D^{2n}", "23b6fd17d8b54b9d376cd838bdc9a667": "n \\in \\mathbb{Z}_{>0}", "23b70914f040c18f76b9ce8e79b26b7a": "\\hat H = - \\frac{\\hbar^2}{2\\mu} \\nabla^2", "23b75d836de111c58a9222754d56232a": "B_i \\rightarrow A_{ij}", "23b7c8d325e72db6c0f77bba87c6f7c3": " f^{-1}(x)= \\sqrt \\pi \\frac{d^{1/2}}{dx^{1/2}}N(x) ", "23b7e6f3b6ef6687fad60e3ebcf0e538": "\\scriptstyle [m,\\, 4.5m]", "23b82211d1500bafb08c9592e48fb022": "x=DL(a)", "23b8bb554bc3f9301d0ab92a9d70ce98": "\\Re (s+a)<0.", "23b8d4d489e0d6f9cbe2019ca2970b45": "\n\\mathrm{TAS} = 39M\\sqrt{T}\n", "23b8f234501b668d0a2b631279f8cd7c": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2} + \\frac{m\\omega^2}{2}x^2 ", "23b8fba3208ac35982d80854b03836b8": " \\ F_p(z,f) ", "23b90b58292a238cba5b94ce50a920a9": "\n\\frac{d\\Sigma(t)}{dt} = a\\Sigma(t) + \\Sigma(t) a^\\top + \\sigma^\\top \\sigma-\\Sigma(t) c^\\top\\eta^{-\\top} \\eta^{-1} c \\Sigma(t).\n", "23b9145e0b96b523fb4fb0ebad89862a": "\\tan \\theta \\simeq \\theta, \\quad \\theta \\ll 1", "23b935a9e8ba7b787a2fed9e032731c7": "\\rho v_i v_j ", "23b99a350c9539dbf7f07a6d88b303a4": "\\sqrt{b^2-4ac} = b \\ \\sqrt{1-\\frac{4ac}{b^2}} \\approx b \\left( 1 -\\frac{2ac}{b^2} + \\frac{2 a^2 c^2 }{b^4} + \\cdots \\right ). ", "23ba1317325b796879752bebaddf491f": "(\\cosh x + \\sinh x)^n", "23ba1d2ca255fc8b583d89e3438d96d3": "\\Gamma(n) = (n-1)!", "23ba4a06c16b5af59c7ea899eb018d74": "\\ {F_{p}} ", "23ba9e54564986150c054cc03ce19851": "o_0, o_1, \\dots", "23baf1c8533a27d7e0b71354215df8be": "~ \\sigma_{\\rm e}~", "23bb3977222abdc31ab56dc750dcd81a": "\\frac{j_\\odot}{j_c}=\\left(\\frac{R_\\odot}{r_c}\\right)^2 \\approx 10%", "23bb5ace61d39fe73bb071c7b681b0c8": "\\left ( I_1=\\int_0^\\infty {\\frac{1}{s^2+1}}\\, ds = \\frac{\\pi}{2}\\right ) = \\left ( I_2=\\int_0^\\infty \\sin t\\, \\frac{1}{t} \\, dt\\right ) \\text{, provided } s>0.", "23bb7f57f852b9ad4a15d4172ca7f51c": "\\forall x \\in \\mathbb{N}. x \\div 0 = 0", "23bb838830357a14004d0a3d4c5ee9cb": "y_2(t).\\,", "23bbb5f94412bba42ad2d9146c557128": "i\\in {1,\\dots,p}", "23bbc6767ef583c5c110f0d8600fc225": "\\rho = F \\cot^{n} (\\frac14 \\pi + \\frac12 \\phi)", "23bbda368ef2588119b20bfa2f0e7862": " (R \\rightarrow S) ", "23bc026f54dc6c8dcdb51cdec3484dd4": "\\overline{\\psi}=\\psi^*\\gamma^0", "23bc4ea7524a65c4bd221c7a5fe3060a": "\\{ 1, 1/\\ln 2, 1.38\\times 10^{-23}\\}", "23bc67683a24998f3a433faa44e375aa": "(p, q) (r, s)\n = (p r - \\gamma s^* q, s p + q r^*)\\,", "23bc91d4d8a0e8019afc06d91b1d5a32": "\\operatorname{core}(A) \\neq \\operatorname{core}(\\operatorname{core}(A))", "23bcc90039d834be67867ed3c50c3014": "=\\mathbf{J}_{\\mathrm{M}}+\\mathbf{J}_{\\mathrm{P}}\\ , ", "23bcf0fcbfbbbcb548cef20f0d777ce1": "Z=P(D|M)", "23bd5b653b621f2e3ec719b3f714137b": " W_t^2 - t = V_{A(t)} ", "23bdbca5262c205034b42c05238e547a": "\\scriptstyle \\mathbf{A}_2", "23bdc504737ab44ddc76420a05a0dc51": "g \\in \\operatorname{GL}(V).", "23be1cf7d1cce64e15bf457a6799241d": " U =\n \\begin{bmatrix}\n 0 & 3 \\\\\n 0 & 0 \\\\\n \\end{bmatrix}.", "23be41a6881417ca0ae511ad34a155d2": "[\\mathcal{L}_X,i_Y]\\alpha=[i_X,\\mathcal{L}_Y]\\alpha=i_{[X,Y]}\\alpha,", "23be5ec342503d75ec343bc8ab875dcb": "T;", "23be6d112e3978dd56e59b7b77d32cf7": "1-\\frac{1}{2}\\left(\\frac{c}{R}\\right)^2 + O\\left(\\frac{1}{R^4}\\right) = \\left[1-\\frac{1}{2}\\left(\\frac{a}{R}\\right)^2 + O\\left(\\frac{1}{R^4}\\right) \\right]\\left[1-\\frac{1}{2}\\left(\\frac{b}{R}\\right)^2 + O\\left(\\frac{1}{R^4}\\right) \\right] \\text{ as }R\\to\\infty\\ .", "23be8d9c45c361293949eaaa4f0fc589": "ds=\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}dx", "23be92cc1bd7e5281672bfd49ba0fce5": "\\phi,\\psi\\in{\\rm Fm}", "23becc99d590ed7e80a82f683279d9e5": "C_{\\rho}(x,y)\\ge \\lambda", "23bed8a531cf6e2aa30e4edb506de660": " |\\psi_{s'}^{b'} \\rangle ", "23bf0b74c7792113e1315e4b0c327f55": " |S-S(D)|\\leq kL,\\quad |S-S(\\gamma)|\\leq kL, ", "23bf0b74e80cefeaaf6e21373d906360": " c = A {|S_2 - S_1| \\over S_2} {f \\over S_1 - f} \\,.", "23bf3694cdeeb4e0e6fd9a33fbeb863e": "V = V_1", "23bf484ade61fac15f9ba5ef09e7f6d1": " \\bold E(\\alpha_m,\\beta_n, \\gamma_p) ~ = ~ \\frac {jk\\eta}{k^2-\\alpha_m^2-\\beta_n^2-\\gamma_p^2} ~ \\bold G_{mnp} ~ \\bold J(\\alpha_m,\\beta_n, \\gamma_p) ~~~~~~~~~~~~~~~~~~~~~~~~~(3.1) ", "23bf904c0087862e9e96c68a2e0574a1": "A=\\frac{1}{4}(60+\\sqrt{10(190+49\\sqrt{5}+21\\sqrt{75+30\\sqrt{5}}}))a^2\\approx33.5385...a^2", "23bf95b1e11954528cbf52e4b91e5f85": "b(f)b^*(g)+b^*(g)b(f)=\\langle f,g\\rangle, \\,", "23bf99ccc3bbd46e0635c7260bc979c3": "\n \\frac{\\partial \\boldsymbol{\\mathit{1}}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\boldsymbol{\\mathsf{0}}:\\boldsymbol{T} = \\boldsymbol{\\mathit{0}}\n", "23bfb08773af94a0fc5d4cd6a3fe49d6": "\\quad\\text{if}\\quad A\\subseteq B\\quad\\text{then}\\quad P(A)\\leq P(B).", "23bfe92930440fd633290fd303ad4dca": "\\mathcal{GW}^{-1}(\\mathbf{\\Psi},\\nu,\\mathbf{S}) = \\mathcal{W}^{-1}(\\mathbf{\\Psi},\\nu)", "23bfef6efe19f38f7b263d0b05451f90": "PV = nRT", "23c059d3aff1ac5ac14f9e2fce20dedc": "g = h^{-1}\\circ f\\circ h", "23c165292d248a6a2cc4b6e2e0a15a51": "(\\Lambda^n A)(v_1 \\wedge \\dots \\wedge v_n) = \\det(A) \\cdot v_1 \\wedge \\dots \\wedge v_n.", "23c1966daecfca8fa2e3356303d4c332": "\\psi(\\Omega^{\\Omega^2+\\Omega 3})", "23c1a48ad9fbe3ad64d7aa5385aa1e2e": " \\Sigma_b = \\frac{1}{C} \\sum_{i=1}^C (\\mu_i-\\mu) (\\mu_i-\\mu)^T ", "23c1d6e3d23eff609b8d40053726f979": "c(E)c(F)=1.", "23c1eebec72f1d26b3301755984c99af": " \n\\lambda = \\pi_1^{\\alpha_1}\\pi_2^{\\alpha_2}\\pi_3^{\\alpha_3} \\dots", "23c2219ceec497e49ae22eb2293ade74": "2\\pi\\ a\\sqrt{\\frac{a}{\\mu}}\\,", "23c23f37519259cc0488f451412cf7b8": "T_{1L}=A-\\frac{Bk_1}{k}e^{-kL}", "23c24870014daf855f7f8b128123fda2": "{\\operatorname{d} \\over \\operatorname{d}x} e^{f(x)} = f'(x)e^{f(x)}", "23c24d0093409e12701e17c328ef0021": "-\\frac{p\\dot r}{r^2} = -\\varepsilon \\sin \\theta \\,\\dot \\theta ", "23c264b6f6e6a07ae77dd2361052edbf": "\\epsilon = \\lambda_{\\bold{k}} \\pm |U_{\\bold{k}}|", "23c279316b512b1b58b126b8f6ee4478": "\\begin{align}\n\\int_{\\gamma} |\\mathbf{x}|^{\\alpha - 1} \\mathbf{x} \\cdot d\\mathbf{x} &= \\frac{1}{\\alpha + 1} \\int_{\\gamma} (\\alpha + 1)|\\mathbf{x}|^{(\\alpha+1)-2} \\mathbf{x} \\cdot d\\mathbf{x} \\\\\n&=\\frac{1}{\\alpha + 1} \\int_{\\gamma} \\nabla(|\\mathbf{x}|^{\\alpha + 1}) \\cdot d\\mathbf{x}= \\frac{|\\mathbf{q}|^{\\alpha + 1} - |\\mathbf{p}|^{\\alpha + 1}}{\\alpha + 1} \n\\end{align}", "23c283d7c7b2825417f5c1a14e1a326a": "D_F^y(x,y)=\\tfrac{1}{2}(x-y)^T Q (x-y)", "23c28afed27aef38002f04431081d759": "f^{-1}\\left(Z\\right)", "23c2a750e8fa642f81831a37a6e29cc4": "\\nabla (\\rho_i v_i)", "23c2bf51098c5a3eaf30b5cf67a08690": "\\text{End}(X)", "23c32f7b4e87e1501a1919b688a597b1": " \\tan \\theta = \\sqrt{\\mathit l \\over \\mathit l^{\\prime}}", "23c384b02f4434e913e60342303544e4": "\n\\theta_i\\sim N(\\varphi, \\tau^2)\n", "23c39258ce7abbc878ee06c9032b700e": "S^+ = U^+", "23c39f47c90d0ff8fdc98599843b3f1b": "F = F_3(p,Q).", "23c468dd1827fd95bc5896543f9ea09c": " \\lim_{k \\rightarrow \\infty} \\phi(x,k) = f(x) ", "23c4a4cdc7ec77a3051417cba3f0adad": "\\epsilon\\,\\!", "23c4f7f4f6c497d66282ecead1c3fe89": "[2]P=(0,a_3)=-P", "23c4fdd03f5ce7d27dccf925a15650bf": "J_k(n) \\star 1 = n^k\\,", "23c548c956f11a965860a2d604a44a01": "L=\\frac{\\alpha\\pi r}{180}.\\,\\!", "23c562f36d97050019d1ad9ebf8b598f": "\\left[\\begin{matrix} a & b \\\\ b & c \\end{matrix}\\right] ", "23c592dc2d084b509b4b478ba107308b": "\\log P", "23c597174ff34699ce82b9efd1667324": "P_\\ell( \\mathbf{x}\\cdot\\mathbf{y} ) = \\frac{4\\pi}{2\\ell+1}\\sum_{m=-\\ell}^\\ell Y_{\\ell m}^*(\\theta',\\varphi') \\, Y_{\\ell m}(\\theta,\\varphi). ", "23c5e9916fa1537ddd64310b9711af84": " H(f) ", "23c615a2a27a4fa59b2dafbf99d2f587": "600 \\,", "23c63758b6ac4633019e26628b9a045a": "\\scriptstyle f \\;=\\; t_B / (t_A \\,+\\, t_B) \\;=\\; 0.25", "23c6690a32ac89060a7d6db76c7b9e00": "\\log_R \\colon R \\to \\mathbb{C}", "23c66a1c57b4b7dcabe64dcf376931ef": "r=f_2(\\theta)-f_1(\\theta+2\\pi),\\ r=f_2(\\theta)-f_1(\\theta-2\\pi),\\ \\dots", "23c6a6b2aa19db9a65ed2a6b127b85f7": "\\scriptstyle \\beta>0", "23c6afa5185a0ce8befd144048f9cff7": "S\\subseteq \\Omega", "23c6e52ef0e6dd1b810d94294b8814d0": "T '= 0", "23c7619f9591397e0693cb905481b0f5": "\\langle 1,2 \\rangle", "23c7b7066a6682d5b74dba6c61d9353a": "\\lim_{(x,y) \\to (0, 0)} \\frac{x^2 y}{x^2+y^2} = 0", "23c7e6f1043b05ab8e990b65072c70e1": "- \\frac{\\hbar^2}{2m} \\frac{\\partial^2\\psi_n (z)}{\\partial z^2} + U(z) \\psi_n (z) = E\\psi_n (z) ", "23c83f25c5adef8eba735a863fb3b260": "\\overline{\\mathsf{f}}", "23c8bbdcc111df29633ca3286fdac77e": "v_{1}=30 Hz", "23c8cf99d0de441831078d97d1767974": "\\;ord_P(G)=-\\deg(G)", "23c8e0c37863d9787a6eef8ec8b36595": "\n\\phi={1\\over\\sqrt{2}}\n\\left(\n\\begin{array}{c}\n\\phi^+ \\\\ \\phi^0\n\\end{array}\n\\right)\\;,\n", "23c921d15b1eed7f54fd608bd050f2af": " \\varphi_\\delta(E) = \\inf \\biggl\\{ \\sum_{i=0}^\\infty p(A_i)\\,\\bigg|\\,E\\subseteq\\bigcup_{i=0}^\\infty A_i,\\forall i\\in\\mathbb N , A_i\\in C_\\delta\\biggr\\}.", "23c93efb376fe6e04cc6679ea7334630": "\n\\begin{align}\n\\frac{\\delta F[\\varphi(x)]}{\\delta \\varphi(y)} \n& {} = \\lim_{\\varepsilon\\to 0}\\frac{F[\\varphi(x)+\\varepsilon\\delta(x-y)]-F[\\varphi(x)]}{\\varepsilon}\\\\\n& {} = \\lim_{\\varepsilon\\to 0}\\frac{e^{\\int (\\varphi(x)+\\varepsilon\\delta(x-y)) g(x)dx}-e^{\\int \\varphi(x) g(x)dx}}{\\varepsilon}\\\\\n& {} = e^{\\int \\varphi(x) g(x)dx}\\lim_{\\varepsilon\\to 0}\\frac{e^{\\varepsilon \\int \\delta(x-y) g(x)dx}-1}{\\varepsilon}\\\\\n& {} = e^{\\int \\varphi(x) g(x)dx}\\lim_{\\varepsilon\\to 0}\\frac{e^{\\varepsilon g(y)}-1}{\\varepsilon}\\\\\n& {} = e^{\\int \\varphi(x) g(x)dx}g(y).\n\\end{align}\n", "23c93f662a57841a6fa19e6541235a3b": "\\scriptstyle\\hat\\beta_2", "23c99dd46138f40f5a11fa6a8393bace": "\\sigma_c=\\limsup_{n\\to\\infty}\\frac{\\log|a_{n+1}+a_{n+2}+\\cdots|}{\\lambda_n}.", "23c9b37fc12b486b33472f6b4b3ee67a": "(\\psi * \\mu) * \\alpha \\equiv \\psi * \\alpha", "23c9dd77a36cd09b6b9cc1652764e5b1": "\\forall x\\exists y\\exists z ((\\lnot x = y) \\land x R y ) \\land ( (\\lnot x = z) \\land z R x ) ", "23ca036df52a0f463097dbd900fe2db9": "\\rho' = {M_i \\rho M_i^\\dagger \\over {\\rm tr}(M_i \\rho M_i^\\dagger)}", "23ca2799f321c43c6040f19abe563a70": "I(\\theta) = I_0 \\left ( \\frac{2 J_1(ka \\sin \\theta)}{ka \\sin \\theta} \\right )^2 = I_0 \\left ( \\frac{2 J_1(x)}{x} \\right )^2", "23ca45b083741524ef60d5bba26241cd": "G = (\\frac{X}{E})*R", "23ca8327f2391b57a0efc9af4d6d2dd5": "\\begin{matrix}\n x = a\\sec t + h \\\\\n y = b\\tan t + k \\\\\n\\end{matrix}\n\\qquad \\mathrm{or} \\qquad\\begin{matrix}\n x = \\pm a\\cosh t + h \\\\\n y = b\\sinh t + k \\\\\n\\end{matrix}\n", "23ca8707a3713f7ecdce63873914898c": " c_1 = \\cos \\phi ' \\qquad s_1 = \\sin \\phi ' ", "23cb6ca4c2ccdf8d7ea726f2145aadde": "\\psi_{jk}(x) = 2^\\frac{j}{2} \\psi(2^jx - k)\\,", "23cb7c2267ceb3b88c6590e0afb6363c": "x = \\begin{bmatrix} k_{1} \\\\ k_{2} \\end{bmatrix} c_{1}e^{\\lambda_1 t} + \\begin{bmatrix} k_{3} \\\\ k_{4} \\end{bmatrix} c_{2}e^{\\lambda_2 t}. ", "23cbb311f5bdd17fa67cfe566f03971b": "K_\\infty", "23cc1d479117eb4f2b7595c429a6838a": "\\{1\\}=G_0\\leq G_1\\leq\\cdots\\leq G_k=G", "23cc3455946bd067502400723ba748d6": "\\ MRS_{xy}=MU_x/MU_y.\\, ", "23cc3cf75618fe3af7094ff8fbde2fd7": "\\textstyle _2", "23cc7fb43c8e28748be8b74e934a7405": "i < j", "23cca4487de814c9653bac4bcd1c2ede": "P(n)\\sim \\frac{1}{4n\\sqrt{3}} e^{\\pi\\sqrt{2n/3}}.", "23ccc190cc397e76117a197d40915e28": "\\delta p_i", "23cd4bb78b9d8af4025b7b4139026b4a": "A_{n}= (A_{n-1}^{yz+1}) (B_{n-1}^{yz }) (C_{n-1}^{yz }) ", "23cd50af2a8db2f13e7dd2eb756f9215": " exp(-x)-1= \\int_{0}^{\\infty}dt \\frac{f(t)}{t}\\rho ( \\sqrt{x/t}) ", "23cd7d1d22255d61f92b8e9375013e67": "k, b, x_b", "23cd849b312b27b78f370683e76a14b2": " P_{i} ", "23cdb7f8b858a2be5858c86f70940e1b": "|\\mathbf{x} \\times \\mathbf{y}|^2 = |\\mathbf{x}|^2 |\\mathbf{y}|^2 - (\\mathbf{x} \\cdot \\mathbf{y})^2 ", "23ce0012b58994bde9553529275aa0ff": "H_i, i=1,\\dots, n", "23ce2e0cfebbe60cbeb97d66b5352aaf": "(c,x)", "23ce4b77e453a9c5e9ae38459e05d50a": "\n\\rho^{XA} = \\sum_x p_X(x) \\vert x \\rangle \\langle x \\vert^X \\otimes \\rho_x^A ,\n", "23ce545ff720d48680e012f822471658": "\\Delta z_i = z'_i - z_i", "23cf864ad158d8c0e6867f49b922a534": "\\tau_0=0", "23d006765c734bab47a3416820ba8492": "\\rho(g_1 g_2)[x]=\\rho(g_1)[\\rho(g_2)[x]].", "23d04054562c5f97feaa798f56ef3bc8": " \\!\\ \\sum_{n=1}^{\\infty}{1 \\over {n(n+1)}} = 1 .", "23d0614564e2ab88b0d26962559c5ce2": "\\phi (B)X_t= c + \\varepsilon_t \\, .", "23d06ab0a7c428fe2042720a23c91b5f": "{\\rm non}({\\mathcal K})", "23d0d3165b30750f6752333e454dfe3a": "G_{ab} = R_{ab} - \\frac{R}{2} \\, g_{ab}, \\; \\; R_{ab} = G_{ab} - \\frac{G}{2} \\, g_{ab}", "23d134331b3b3a5db078495097ecefb7": "v_{2}= \\begin{pmatrix}0.7073 \\\\ 0.07278 + 0.7032i \\\\ 0.0042 + 0.0007i \\\\\\end{pmatrix}", "23d14b29fce7dadc236a53410ba506a0": "\\text{s.g.}=\\frac{144}{134 + \\text{degrees Baumé}}", "23d1701680d100aa0b056552d10a5653": "Velocity = Velocity + (Action) *0.001+\\cos(3*Position)*(-0.0025)", "23d1a4026c80eab276fada4bf77ec77a": "H(X,Y) = \\mu(\\tilde X \\cup \\tilde Y),", "23d1be32c9a29cae71db43443bb7f609": "= 2\\ \\eta^{\\mu\\nu} \\gamma^\\rho \\gamma_\\mu - \\gamma^\\nu \\{\\gamma^\\mu,\\gamma^\\rho\\} \\gamma_\\mu + \\gamma^\\nu \\gamma^\\rho \\gamma^\\mu \\gamma_\\mu . \\,", "23d2091e9e838ed37bb32acbbb618faa": "z_{cv} = f_c(z_{cr}) \\,", "23d2bf907a02e6f725981012c9ac3768": "\\frac{1}{2}\\left(-p_1 + p_2\\right) = \\frac{1}{2}\\left(-p_1 + p_1 + (2i, 2j)\\right) = (i, j)", "23d2ce2828a555f14827f879787f441e": "\\sqrt{\\frac{1}{m} q^2 \\rho^2}.", "23d3341f1f74172efeea5b8199af219e": "M= ", "23d348b1bb1082c3b8f29399e4ea57d1": "\\xi(B)", "23d373c00ae4d31d07131cd67bb1cb1e": "\\gamma_x(0) = x\\,", "23d37ee4c7752085ce47f0e08867d86e": "A^*A = (UP)^* UP = PU (PU)^* = AA^*.\\,", "23d3951ceb825305e9cdb59f6db21820": "D=3", "23d429e839a9f1e759eb789e303b9777": "\\tau_\\mathrm{n}\\,\\!", "23d436f386aa1c3879c3e6b928ed5a2f": "c_{ab}^\\mathrm{opt}(t)", "23d4677b4f913df994eb085f2af76f9a": "\\frac{kg \\cdot m^{2}}{s}=J \\cdot s.", "23d489baff18c52e301316fe66e0399f": "2\\cdot n", "23d4c274f1466e7a2aa4b978566e9fc1": "(F_n-1)/2", "23d4edadc63deda482784bcbe3f219b2": "{d\\over dz} V(a,z) = [L_{-1},V(a,z)]= V(L_{-1}a,z),\\,\\, [L_0,V(a,z)]=(z^{-1} {d\\over dz} + \\alpha)V(a,z)", "23d4efad9cb40c1e5b4cf17bcffcd1c6": " R + \\delta \\le 1+\\frac{1}{n}", "23d549d108ac186706cfd25e7bd3fd99": "x^4-2s(t^2+1)x^2+s^2t^2(t^2+1)", "23d5798ed0ba2cab790a634bfe5d8ff7": "f_w(z)= \\frac{1}{1-w z}", "23d5b4ec9c8ed6d5fabbedc26af854e7": "\\left(V_i-V_o\\right)DT - V_o\\delta T = 0", "23d5d35448c76214b072a4574b261c6a": "\nR_{x\\rightarrow y} = { K_{x\\rightarrow y}(dt) - \\delta_{xy} \\over dt}\n\\,", "23d5d5902487a8ef1d4d4ce0cc26b6d6": " G^{(i)} \\neq G^{(j)} ", "23d628ba6e7d5d6e226f573c92b65df5": "x=s+n\\ ", "23d65004d7f63b6cb23e01d2c0ffa366": "x \\in L \\iff \\exists t_1,t_2,\\dots,t_{|r|}\\, \\forall r \\in R \\bigvee_{ 1 \\le i \\le |r|} (M(r \\oplus t_i) \\text{ accepts}).", "23d6a3dd8d83e20ff7f35d3ab53b8256": " c_7 = 0.000687678, \\,\\!", "23d6b9df3f2e80bc5ffc16bb97e02a71": "\\scriptstyle \\frac{U}{V} \\;\\sim\\; \\mathrm{Pareto}(1,\\, n)", "23d6eefab6856ed4358d2b9441680a8e": "|\\det(A)|=|\\det(R)|=\\Big|\\prod_{i} r_{ii}\\Big|,", "23d71a6b56b5391a141bb8d367ffbf48": "\n\\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} \n= \\sum_{\\sigma \\in \\mathfrak{S}_p} \\sgn(\\sigma)\\, \\delta^{\\mu_1}_{\\nu_{\\sigma(1)}}\\cdots\\delta^{\\mu_p}_{\\nu_{\\sigma(p)}}\n= \\sum_{\\sigma \\in \\mathfrak{S}_p} \\sgn(\\sigma)\\, \\delta^{\\mu_{\\sigma(1)}}_{\\nu_1}\\cdots\\delta^{\\mu_{\\sigma(p)}}_{\\nu_p}. \n", "23d739962942fc054c55a89cd67e9d6b": "\\{(-,+,+,+); l^a n_a=-1\\,,m^a \\bar{m}_a=1\\}", "23d82421eca87bfbc476709b38271e5d": " h = S + 1 \\mathrm{\\tfrac{BTU}{lb}\\;} W t' ", "23d866926e56d1a284f29e1a6190294e": "A \\cdot \\neg B \\cdot \\neg C + A \\cdot B \\cdot C \\,", "23d8a7ee54a72bb853d21a58b7d1f591": "\nS(r_{ij}) = \n\\begin{cases} \n 0, & \\mbox{if }r_{ij} \\le R_L \\\\\n \\frac{(r_{ij} - R_L)^2(3R_U - R_L - 2r_{ij})}{(R_U - R_L)^2}, & \\mbox{if }R_L \\le r_{ij} \\le R_U \\\\\n 1, & \\mbox{if }R_U \\le r_{ij}\n\\end{cases}\n", "23d8b8d110bde0cd6d2bd8ae7e32607c": " x_1 = 0.0 + 106/1121 = 0.09455842997324", "23d8c75253607767632b097c2e5ffb97": "(1,2)_{\\frac{1}{2}}", "23d8ccc41660f6590d657c10229bf389": "\\Gamma(s) = \\int_0^\\infty t^{s-1}\\,e^{-t}\\,{\\rm d}t = \\lim_{x \\rightarrow \\infty} \\gamma(s, x)", "23d8eaae7fdb5e514a3937b4974d508e": "\nE = - \\sum_{ij} J_{ij} S_i S_j\n\\,", "23d90a6f0d8da5d137980ba2a3110305": "3^6 (1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6) \\equiv (1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6) \\pmod 7.\\,\\!", "23d919c5c2b9f223dcdad8e4e6bd8d34": "R=\\rho_1-\\rho_2= \\frac{Q(x)Q(x^5)}{Q(x^3)^2}.", "23d9609b985033ce76fec3fffc3633d8": "r_s < 1", "23d973454eb4411090d7caa2e562715e": "-j1.52 = \\frac{-j}{\\omega L_m Y_0}= \\frac{-jZ_0}{2\\pi f L_m}\\,", "23d997d4996e0b119cdbb7ef6531051c": "V=(\\frac{\\sqrt{3}}{2})a^3\\approx0.866025...a^3", "23da1c930bd68cabcda4a09b5794ad24": "{}^{n}i", "23da2548db5f0c6be9a7f74924bb6bc0": "\\overline{x} = \\left( x_1 + \\cdots + x_n \\right) / n", "23da5aa14c1980e5e97d34b2f13df592": "\\Phi=0", "23dab08094c604888829a5ec46d6f668": " C_0 = I", "23dae17f89257c3a092a043dc7dd1be4": "n\\in N", "23db1b7eda16b0d2612259db4a694f74": "R = \\sqrt{(E^2-P^2) \\, s^2 + 2 \\, s \\, \\sqrt{ (E^2-P^2) \\; R_0^2 - L^2 + R_0^2}} ", "23db274b4bbbabc08f31b9010282d9b5": "\\displaystyle H_{k+1}=B_{k+1}^{-1}", "23db6b295827d9e0550a2e70a566bc23": " \\frac{1}{1-x} = \\sum_{n=0}^\\infty x^n = 1 + x + x^2 + x^3 + \\cdots,", "23dc0259bef08d9fe92fec01cd597a30": "r_{min}", "23dc2c615dc734f7d54bff5766375c1f": "q \\in [0,\\infty)", "23dc4c955c65ff558a575af49e4fe16d": "\\varprojlim_{i\\in I} A_i = \\Big\\{\\vec a \\in \\prod_{i\\in I}A_i \\;\\Big|\\; a_i = f_{ij}(a_j) \\mbox{ for all } i \\leq j \\mbox{ in } I\\Big\\}.", "23dc89fb40cf563663c82871b6047fa0": "\\int_0^1n(x)dx", "23dcd751d57c61c044b858a78891b740": " E_{hh} ", "23dd20f26e81a98748a108e77865a369": "S_R f = \\int_{-R}^{R}\\hat{f}({\\xi})e^{2\\pi i x\\xi}\\,d\\xi", "23dd262e42fcb11ceb6d5d047ac7c9e3": "\\frac{\\sin{\\theta}}{v}=\\frac{1}{v}\\frac{dx}{ds}=\\frac{1}{v_m}", "23dd2f81505a1251726e530eda0f6b7c": " \\mbox{rank}_+(A) = \\min\\{ q \\mid \\sum_{j=1}^q R_j = A, \\; \\mbox{rank}\\,R_1=\\dots=\\mbox{rank}\\,R_q =1, \\; R_1,\\dots,R_q \\ge 0\\},", "23dd2f927191e44f73f4d781ed3dcf4c": "\\omega(t)", "23dd469c4426142d58c4042044a3d620": " f(\\theta_m) ", "23dd9612439c9135b55719bba1c80165": "P \\lor (Q \\land R)", "23ddbc15d0e477a2e23e2208974b5d24": "\nN_0 = \\oplus_{i \\geq 1} ( M_i \\ominus N_i ) \\quad \\oplus \\quad \\oplus_{j \\geq 0} ( N_j \\ominus M_{j+1}) \\quad \\oplus R.\n", "23ded4c3f6dae981aea9b1ad7949f3e3": "D=0", "23df15e889065fea6de753e93a33600d": "S_m=\\int d^4 x\\sqrt{g}G_N L_m\\;", "23df1c2774dc2fcdc6d8929f0ab1998c": "T(p) = Ap + o\\!", "23df1f25c482a1f9f005d6d28900efcc": "1 \\,+\\, 2\\left(\\frac{1}{8}\\right) \\,+\\, 4\\left(\\frac{1}{8}\\right)^2 \\,+\\, 8\\left(\\frac{1}{8}\\right)^3 \\,+\\, \\cdots.", "23df3a6fff1a01d6ec00692b0deab731": "\\mathbb{Z}_N^*", "23df6d2c6361af6323c3ff0636fe535d": "(X, A)", "23dfaa15305ed565e20cd0db432e9357": "N=\\prod_{l\\in S} l > 4\\sqrt{q}.", "23e0399205ca41b972912e899bf07e02": "\\begin{align}\nc^2 & {} = (a - b \\cos\\gamma)^2 + (- b \\sin\\gamma)^2 \\\\\nc^2 & {} = a^2 - 2 a b \\cos\\gamma + b^2 \\cos^2 \\gamma + b^2 \\sin^2 \\gamma \\\\\nc^2 & {} = a^2 + b^2 (\\sin^2 \\gamma + \\cos^2 \\gamma) - 2 a b \\cos\\gamma \\\\\nc^2 & {} = a^2 + b^2 - 2 a b \\cos\\gamma\\,.\n\\end{align}", "23e089b13385e1e642c5ee3a4bdfd271": "4 \\pi^2 a^3/T^2 = \\mu", "23e0a1aad2df9118e737b91371ef9b49": "\n(f \\ast g)(a) = \\sum_{b \\in G} f(ab^{-1}) g(b).\n", "23e0d1abd0f4c120a04dd9fbec8c343d": "Int1\\,", "23e10a53e00dddee62ab390541c84a23": "\ny^{2} +\n\\left( x - a \\coth \\tau \\right)^{2} = \\frac{a^{2}}{\\sinh^{2} \\tau}\n", "23e19a5e67f634103bab55a89e40af40": " \\cdot\\left(\\text{largest monomial of }s_{n-1}\\right)^{i_2-i_1}", "23e1a7473140f07ec726d8c2a56be341": "L+R", "23e1b428b0fc5151e814bb5aecd1cbf4": "d={{d}_{p}}+{{d}_{n}}=\\sqrt{\\frac{2\\varepsilon }{q}\\frac{{{N}_{A}}+{{N}_{D}}}{{{N}_{A}}{{N}_{D}}}\\left( \\underbrace{{{V}_{bi}}}_{\\text{ built-in voltage}}-\\underbrace{V}_{\\begin{smallmatrix}\n \\text{external applied} \\\\\n \\text{voltage}\n\\end{smallmatrix}} \\right)}", "23e1d14f7ed944578835806021118038": " I = 2 \\pi i \\frac{\\exp(\\tfrac{\\pi i}{4})}{-1+i} \\left(\\frac{17}{4} - 5^{\\frac{3}{4}} 2^{\\frac{1}{4}} \\right) = 2 \\pi 2^{-\\frac{1}{2}} \\left(\\frac{17}{4} - 5^{\\frac{3}{4}} 2^{\\frac{1}{4}} \\right)", "23e235bf17f43bad9cbba7d643410429": "(\\alpha \\mathbf{I} - A) R_{\\alpha} g = g.", "23e2421d9e55c3e325e80695c3febcb8": "\\pi^{ab}/tor", "23e305f5c59cfde43b557424c099f31f": "A \\rightarrow B: \\{N_A, A\\}_{K_{PB}}", "23e34142844417943efef1fc7b90b4e6": "(X, Y)", "23e39697bf142d80db47f28147d96238": "\\scriptstyle \\,s_{8,4} = 0.7853982... (+4.7 \\times 10^{-8})", "23e39b6ca29aa8919650c25d9c13c784": "\\bold{v}_1 = \\left(-\\frac{5}{9}\\right) \\bold{v}_2 + \\left(-\\frac{4}{9}\\right) \\bold{v}_3 + \\frac{1}{9} \\bold{v}_4 . ", "23e3dca93afb4313a95dddb2168027d9": "w=g_1g_2\\cdots g_n", "23e409125d0dec1b5e8de58c0471a991": "L_1,\\ldots,L_m", "23e4466cef9c96eeb0957ac8bdc22166": " A(\\theta,|\\varepsilon_1|) = \\frac{|\\varepsilon_1|+1}{|\\varepsilon_1|-1} \\frac{4}{1+\\tan{\\theta}/| \\varepsilon_1|}", "23e48292968e5312bc26785270f6dfd8": "\\Delta f = \\mathrm{tr}(H(f))\\,\\!", "23e488a0e115510162ebf0de68f86fe5": "T(X_1^n)= \\left( \\prod_{i=1}^n{x_i} , \\sum_{i=1}^n{x_i} \\right)", "23e48f687e5e6d8396b088bc24a62aa3": "F_3(x)=-\\frac{7}{12}x-\\frac{5}{12}x^3", "23e49ae1e02da2126813c8a2b7883a62": "TAT^{-1}", "23e4cb5ae1fa2cc6a7392eba7f8fa43a": "B=X^TNX=\\int_{-1}^{1}\\int_{-1}^{1}[x_1^j x_2^i\\delta(x_1,x_2)]_{i,j=0}^{i,j=m-1}dx_1dx_2", "23e4cef02c3a564e22687b8dd764d166": "G(n)=\\begin{cases} 0&\\text{if }n=0,-1,-2,\\dots\\\\ \\prod_{i=0}^{n-2} i!&\\text{if }n=1,2,\\dots\\end{cases}", "23e56d5fa289f90d21c09761263ff3fe": "E(x) = x + e_0", "23e583672624c06585e951e656d20cfe": " D = (\\frac{16 T_\\max}{\\pi {\\tau}_\\max})^{1/3}", "23e63a25f7824e2986f9f43ed10cf207": " \\mathrm{Ref}(\\theta) = \\begin{bmatrix} \\cos 2 \\theta & \\sin 2 \\theta \\\\\n\\sin 2 \\theta & - \\cos 2 \\theta \\end{bmatrix}. ", "23e6510a92f69fd27b75330c38cdc75c": "\\varepsilon_{\\mu \\nu \\rho \\sigma}", "23e6586a97e32cca35d8ab3252372f3a": " V(x) \\rightarrow - V(x) ", "23e669d6ad693609e0e8f00a802d066d": "10\\uparrow\\uparrow(7.21\\times 10^8)", "23e680d0444b05718f4ef44806aa398d": "\\left[\\frac{d}{d\\nu}K_\\nu\\left(\\sqrt{a b}\\right)\\right]_{\\nu=p}", "23e693f316b16953ff7b32c0edd2374f": "F(i)=\\sum_{j=1}^i f(j).", "23e6fe08f9235acfffcc2f35b7323d4f": "\\vec b\\cdot \\vec c = \\Vert \\vec b\\Vert\\Vert\\vec c\\Vert\\cos \\theta", "23e7447a67cc1c81672b56dd0124e598": "\\left\\langle\\tfrac{-1+\\mathrm i\\sqrt7}2,Z_2\\right\\rangle", "23e77967ec81ef64052d2a3a5e7c3941": "\\min_g \\sum_{i=1}^n w_i (g(x_i) - f(x_i))^2", "23e7b16af5dc92102362735e2ca02351": "f(0)=f(1)", "23e7cd22bbd57dad770596c468508ecd": "\\mathbf{r} = \\mathbf{i} -2 \\left ( \\mathbf{i} \\cdot \\mathbf{n} \\right) \\mathbf{n} ", "23e7f62effdf1421d013678a23dd6659": "\\sum^\\infty_{k = 0} |a_k|^2r^{2k} \\le {M_r}^2", "23e84d859854635723fb981e89a57b9d": "\\left. +\\frac{|V_{nk_1}|^2}{E_{k_1 n}^2}\\left(\\frac{3|V_{nk_2}|^2}{4E_{k_2 n}^2}-\\frac{2|V_{nn}|^2}{E_{k_1 n}^2}\\right)-\\frac{V_{k_2 k_3}V_{k_3k_1}|V_{nk_1}|^2}{E_{nk_3}^2E_{nk_1}E_{nk_2}}\\right]|n^{(0)}\\rangle", "23e8b9d5ba66e70b9908c75c554fcdc8": "NI = B \\left(\\frac{L_{\\mathrm{core}}}{\\mu} + \\frac{L_{\\mathrm{gap}}}{\\mu_0} \\right) \\qquad \\qquad \\qquad \\qquad (1) \\,", "23e8e5e378c9b93c40d2764e452093e4": "{\\tilde{D}}_5", "23e92ab7b76eb6aac23963e0878f58a1": "\\forall i \\; \\alpha_i + c_i > 1", "23e94a5b63b9b7094898ecb49f0863b9": " (x , \\sqrt {1 + \\langle x,x \\rangle}) \\in R^{n+1}", "23e9629c607ede03f40f01c4cea9d240": "M + X^+ + A \\to MX^+ + A", "23e97042c4556020f81aac26ef95e508": "\\mathbf{a}_A=\\mathbf{a}_B + 2\\boldsymbol{\\Omega} \\times\\mathbf{v}_\\mathrm{B} + \\frac{d\\boldsymbol{\\Omega}}{dt} \\times \\mathbf{x}_\\mathrm{B} + \\boldsymbol{\\Omega} \\times \\left(\\boldsymbol{\\Omega} \\times \\mathbf{x}_B \\right)\\ .", "23e972dc5b9e5bf3522364d5e3f768d8": "\\mathbf{M} \\vec{v}_1 = \\sigma_1 \\vec{u}_1", "23e9977507f20de084570a97152b3af8": "y(n_1,n_2)=\\sum_{r_1=0}^{N_1-1}\\sum_{r_2=0}^{N_2 -1}{a(r_1,r_2) x(n_1 -r_1,n_2 -r_2) }-\\sum_{l_1=0}^{M_1-1}\\sum_{l_2=0}^{M_2-1}{b(l_1,l_2) y(l_1,l_2) }", "23ea0a754ade8b53333d854ad775b358": "\\mathbb{C}P^\\infty", "23ea4cbdd18453356a82bfb9216bf0c9": "\\frac{\\partial (\\mathbf{a} \\cdot \\mathbf{u})}{\\partial \\mathbf{x}} = \\frac{\\partial \\mathbf{a}^{\\rm T}\\mathbf{u}}{\\partial \\mathbf{x}} =", "23ea65cd896afe45430888242b5e61d7": "N = \\frac{n+\\Delta}{2}\\,", "23ea6dea311ba937d7450552d8beaaf2": "\np_{N} = \\frac{1}{1 + e^{-\\Delta G/RT}}\n", "23ea7d9b55a3bdb471880901cc5b7758": "\\ d\\theta / dt", "23ea985f6cd0a19f97cc6e3f1f4b64f2": "\\displaystyle{\\iint S(\\varphi_1)_x S(\\varphi_2)_x + S(\\varphi_1)_yS(\\varphi_2)_y = \\int_{\\partial\\Omega} S(\\varphi_1)\\varphi_2.}", "23eb297c3a6ee517ff8f26e031907e69": "\\tilde{R}", "23eb8bffcb179a47fe2ea4c5bf3dc0b3": "w^{f}(f^{*}) < w^{f}(f)", "23eb9d5715769559c1e7624647b2cdcd": "p_n\\in X_n", "23ebc230da36435d5f4600385ddb5165": "xp_x+p-c_q+s=0", "23ebe6fa56d557cddcdae0dc4436ca22": "P(\\mu)\\,", "23ebe87e3f9d8119ea281e47375a9968": " G = (V, E) ", "23ebe90e3be361a6e2936507c05baee3": "(n_d", "23ebf6197fdb1f854a3efc5dc86a4e3b": " 1 \\le i \\le N-m+1 ", "23ec5047df7cbb88e2c1e827d22c616d": "\n\\nabla^2 p = f(\\nu,V)\n", "23ece1c2f96360b74bcb1c6eedb50ec4": "T_{(i)}", "23ece7ea6a98c019f654cbd9fa4cadf0": "q_p(a/b) \\equiv q_p(a) - q_p(b) \\pmod{p}", "23ed0075ab50afbefe6a8a8bfe0e1d40": "\\nabla^\\Gamma s = (\\partial_\\lambda s^i - \\Gamma_\\lambda^i\\circ s) dx^\\lambda\\otimes \\partial_i, ", "23ed065f2c1ba7780b18d16a0eeb47c0": " \\ \\gamma ", "23ed36e6b86dcbe40ca30bddf36e81a5": "\\Pi_e+\\Pi_f+\\frac{1}{\\Gamma}=1", "23ed3efc2b9106d0307318a41086dd8a": "\\int_{-\\infty}^\\infty {e^{itz} \\over z^2+1}\\,dz=\\pi e^{-t}.", "23ed5f7a98e50fd920dfaeeadd5360c6": "\\ Qx[\\beta \\and \\alpha (x)] \\leftrightarrow (\\beta \\and Qx \\alpha (x)).", "23edcb9327d85120654714b936cc0f19": " T = - \n \\begin{bmatrix}\n 0.500 & 0.000 \\\\\n -0.357 & 0.143 \\\\\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0 & 3 \\\\\n 0 & 0 \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.000 & -1.500 \\\\\n 0.000 & 1.071 \\\\\n \\end{bmatrix}, ", "23ee29ef83f60d3f32c0b1680d59d90e": "y|_x \\leftarrow x", "23ee3f452c3aad6bc12d81231bf8fb2e": "T_a f(x) = \\lim_{N\\to\\infty} T_{a/N} \\cdots T_{a/N} f(x)", "23ee904e8caec48159c5bba168ae56e1": "\\boldsymbol{\\mathsf{T}} = 2 \\mu \\boldsymbol{\\mathsf{E}} + \\mu'' \\Delta \\boldsymbol{\\mathsf{I}},", "23eec04e7f0be4698551f961314f897f": "( \\lambda x . x ) s \\to x[ x := s ] = s ", "23eec0b29e5467085dd1fc8b26c201b9": "x_2 = (1.786737601482363 - 1.786737578486707) / 2 = 0.000000011497828", "23ef39fdab1df82328d69a275a52cab5": "p_i\\,\\!", "23ef51f78990ef2c65191e9ad5909324": "E(\\hat x,u,y)", "23ef929c866f2f0dc1fec0c895da6947": "O(bd)", "23ef938f84d93700d8017a592b678550": "K = \\frac{I\\omega^2}{2} \\,\\!", "23efe8a2b97c6e1f2df67446053750f6": "R[\\epsilon] \\otimes_{R[t]} R \\simeq R[\\epsilon]/(\\epsilon^2)", "23efef1c9c261682f57bf11576c89249": "\\partial U_0/\\partial\\boldsymbol{\\epsilon}", "23f005d6d75b38b132994257b7f76e7e": "\\mu'(N,T) = F(N+1,T) - F(N,T),", "23f06d1b4292d6bb9c9d2dbb9960fbe9": "y(x) = \\frac{y_0}{(x_0-x)y_0+1}", "23f09b1d20e5e2371d05fc0ac99b3398": "5040, 45360, 453600, 3326400, 39916800, 363242880, 3874590720, 34767532800,\\ldots", "23f0a994552a72ad850fb67d20273008": "\\min\\left(1,\\frac{P_s(i)P_c(i \\rightarrow j)}{P_s(j)P_c(j \\rightarrow i)}\\right)", "23f1209db01e2bd973dd54ed575d8a96": " \\bigl(h_{i\\bar{j}}\\bigr) = \\frac{1}{(1+|\\mathbf{z}|^2)^2} \n\\left[\n\\begin{array}{cccc} \n1+|\\mathbf{z}|^2 - |z_1|^2 & -\\bar{z}_1 z_2 & \\cdots & -\\bar{z}_1 z_n \\\\ \n-\\bar{z}_2 z_1 & 1 + |\\mathbf{z}|^2 - |z_2|^2 & \\cdots & -\\bar{z}_2 z_n \\\\ \n\\vdots & \\vdots & \\ddots & \\vdots \\\\ \n-\\bar{z}_n z_1 & -\\bar{z}_n z_2 & \\cdots & 1 + |\\mathbf{z}|^2 - |z_n|^2 \n\\end{array} \n\\right] \n", "23f141db8c7c538e4eb658fab6fcdcf5": "\\tfrac{M-E+S}{4}", "23f17c3ebe2939624eeda5f2f92d3861": "T_{\\mathrm{h}}", "23f18ae80e0c68bd6871650213d8b1a6": "D_\\text{eff}", "23f19c3d0489a1e01402484a3462e486": "X_i \\wedge Y_i", "23f25076e201b54bcb68c47cae6aa475": "\\int\\mathrm{vercosin}(x) \\,\\mathrm{d}x = x + \\sin{x} + C", "23f25d9544e98f1c27466c16982f3d9a": "\\psi(\\theta) = b(\\theta) + \\theta", "23f268583e97d29d7b1ab373c0cb145d": "u_2(\\mathbf{x},z_1,z_2)", "23f2c057b3f9db616a28c49cee1e4f22": "(S^\\cdot M) \\check{~} = Hom_A(S^\\cdot M, A)", "23f3a795eca7cd9de4464b8e7d06e5a8": "\\operatorname{card}", "23f3f8ea3be572e93e6acae8acf5620e": " \\mathbb{R}^{n}", "23f4c88a3c2b4632a53259d69e62ff13": " x = {{u+v} \\over {\\sqrt{2}}},\\, y = {{u-v} \\over {\\sqrt{2}}} ", "23f51e4e301cdd9c436e1506b08560a6": "P(f|f_1, ..., f_N)", "23f53de3a9456226d9b066e2fb00f83b": " \\hbar \\omega = ", "23f5480b72f3c09d45504cb82dfc54b6": " \\mathbf{Y}_{n \\times 1} = \\left(y_1,...,y_n\\right)^T ", "23f54a107747f29da7cbcbb8dcfd89d9": " g_{(a,k)}(u)=k u^{-1/a}. ", "23f55577f9c25c7e41e27c87b63b9d57": "a^k a^m x = a^{k+m}x", "23f559b4005e88ca7c2eeb5b24ac831b": "\\sum\\frac{f_{xx}f_y^2-2f_{xy}f_xf_y+f_{yy}f_x^2}{f_y^3}=0", "23f567a1eb2cab6bd4fb673b147459b3": "v_1,v_2", "23f57708c806f20830e26b5cb7b0c3e4": "S=f_1(S)\\cup f_2(S) \\cup\\cdots\\cup f_N(S)", "23f5caa7f761e8950c46167c3c713ffe": "vec(A)", "23f5ff9e226365a71944bd9655808b04": "Q= \\frac{V_\\mathrm{A}^N}{N! \\Lambda^{3N}}", "23f651668c79c691ec7a499b30772421": "\\boldsymbol{y}", "23f66ed116e42c054176e22722f019aa": "\\phi_R \\to 0", "23f67728366e547d92590bc017f83b94": "a\\cdot b", "23f6938bf27e396eabd46cbfc0ae358e": "\\frac{(i\\omega)^2}{(i\\omega)^2-\\xi^2}", "23f6c9e1206becd729f5069e7d71910d": "X_i \\geq E(X_i)-a_i-M", "23f71271a144ce25039e2bd83a080726": "V(\\cdot, \\cdot)", "23f71825e2a150405fd063df2ea73fe9": " (12,17);", "23f72d55cda1e90ca8207fee140d367a": " \\mu(\\{x \\in E: f(x) \\neq g(x)\\}) = 0. ", "23f742eac5875f476efe7762b68792ce": "x^5 - x + 1 = 0", "23f7a639b1b7a0349721ba2f6f02597e": " \\Delta \\subseteq Q \\times \\Omega_{Z,[t_l,t_u]}\n\\times Q", "23f7c0d1068a53889c1ac136454653e9": "J = \\left [ \\begin{smallmatrix}0 & I_n \\\\ -I_n & 0\\end{smallmatrix} \\right ]", "23f8009443f21fda045bc1018d18d6d1": "F_{BG} \\; = \\; \\frac{G_B}{4}", "23f816496d58b555eeb80304e15142f7": "F\\,'_{Lorentz}=q\\gamma vB", "23f81dc07682f6c4c9ef558ea9c82499": " w e^w = \\frac{I_SR}{nV_T} e^{V_D/nV_T} e^{\\frac{I_SR}{nV_T} \\left(\\frac {I}{I_S}+1 \\right )} ", "23f8679ceaf79cfb1c393c266784dfb6": "\\ L_{u\\rho}", "23f8909045f964e1af29d9ac8710abce": "\\epsilon_M", "23f8d0c45b63968e89c6aedda61828f8": "\\{Q^\\dagger,Q\\}=2H", "23f8da485fc099bd262a70e5f99934e4": "f(i) = a + (b-a)(i-1)", "23f94f8b9478854fae24896be7727f6a": " \\left\\vert u\\right\\vert _{W_p^m(\\Omega)}=\\left( \\sum_{\\left\\vert \\alpha\\right\\vert =m}\\left\\Vert D^\\alpha u\\right\\Vert_{L^p(\\Omega)}^p\\right) ^{1/p}\\text{ if }1\\leq p<\\infty ", "23f9bd823a6e14842166adb43939c83b": "c_k = {{\\sum_x {w_k(x)} ^ {m} x} \\over {\\sum_x {w_k(x)} ^ {m}}}.", "23f9d6f6dfe27a1717841fef43a43fd9": "\\scriptstyle\\boldsymbol{r}_A", "23f9e37a0b6430f531c7e3733e7a6fc1": "SS(w)=s", "23f9e5af37a1756372c5998e6bd84388": "I(Y;Z|X)>I(Y;Z)", "23fae11cdff623c92b7990de4ea689c4": "y\\cdot a = b", "23fb5e1f04d1bbf245f9436fa5b401af": "OLD(T_i).\\mathrm{add}(O_j, WTS(O_j))", "23fbdee7ae3251bc2a9a4387e84c1730": "X_1 \\oplus X_2 \\oplus \\cdots \\oplus X_r \\cong Y_1 \\oplus Y_2 \\oplus \\cdots \\oplus Y_s", "23fc24180ed54848815b5f089cb85e77": "m,E", "23fc4dab9dcc62c777316418c76498f0": " \n\\mathbf{A} = \\begin{bmatrix} \n\\mathbf{A}_{1} & 0 & \\cdots & 0 \\\\ 0 & \\mathbf{A}_{2} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\mathbf{A}_{n} \n\\end{bmatrix}\n", "23fca4b3ca08964ff17e0ccda7ec165e": "\\frac{\\partial u}{\\partial \\varphi} = - \\frac{\\partial u}{\\partial x} r \\sin \\varphi + \\frac{\\partial u}{\\partial y} r \\cos \\varphi = -y \\frac{\\partial u}{\\partial x} + x \\frac{\\partial u}{\\partial y}.", "23fd65060978db561c7263597635bb4f": " \\mathbb{Z}[x]/\\langle f \\rangle ", "23fdd249f8a14a37ce304f95c445d21b": "W_d = \\frac{\\pi\\; d^2\\; \\sigma_f^2\\; l_d}{24\\; E_f}", "23fe60ab88b9d0143b905bfe4312ab76": "P_{F_{4}}(x) = (1+x^3)(1+x^{11})(1+x^{15})(1+x^{23})", "23fec5a7dd51d840c9c9001300fc9815": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V\\left(\\mathbf{\\nabla}\\cdot\\mathbf{F}\\right)dV=", "23fed1bf4ae66d6f433e2976253a6c36": " f_j (\\mathbf{x}) = f_j (\\mathbf{\\phi} + \\Omega^{-1/2} \\mathbf{\\xi}) = f_j( \\mathbf{\\phi} ) + \\Omega^{-1/2} \\sum_{i = 1}^N \\frac{\\partial f'_j(\\mathbf{\\phi})}{\\partial \\phi_i} \\xi_i + O(\\Omega^{-1}). ", "23ff1f724367fb5e7e84d4a4119fb0e2": "\\theta_1=q\\theta+\\theta_0=q\\varphi+\\theta_0", "23ff39fb1f877cc41825a04a56ff796b": "\n\\begin{align}\n\\xi & = \\int f_3(x)E^2~dx, \\\\[6pt]\nu & = \\left(y+\\dfrac{f_2(x)}{3f_3(x)}\\right)E^{-1}, \\\\[6pt]\nE & = \\exp\\left(\\int\\left(f_1(x)-\\frac{f_2^2(x)}{3f_3(x)}\\right)~dx\\right)\n\\end{align}\n", "23ff4f88d24b2ab879022e08470371d1": "(2n+1)\\times{(2n^2+2n+3)\\over 3}", "23ff52fec673aefb0fe63a8a5a515835": "K_s+(K_1\\cup K_t) (s,t\\geq 1) ", "23ffddee7308c3074e598b33ea734cb9": "\\int_{B_r(x)} |u(y) - u_{x,r}|^2 dy \\leq C r^{n+2\\alpha},", "2400327ee0a9ab5f6ac42bc5d9f68fa3": "\\psi_1^* \\mu \\psi_2", "240134b8dfc597471cebc9632476d8b5": " g(x) = 2x^2+x ", "24014aad976921599876393ccb4505b2": "N(T+H)-N(T) \\geq cH\\log T", "24017038961c2e19b400d9d48161f2fc": "\n\\psi(x) |N; x_1 ... ,x_N \\rangle = \\delta(x-x_1) |N-1;x_2 ...,x_N\\rangle + \\delta(x-x_2)|N-1;x_1,x_3...,x_N \\rangle + \\ldots \n\\,", "240173ccb9bbf446b512851fa10562cf": "\\frac{\\mbox{Net Profit}}{\\mbox{Net Sales}}", "24018524cadd986ddc0c4a5ac8ae1bc3": "\\|x-y\\|^2 =\\langle x-y, x-y\\rangle= \\langle x, x\\rangle - \\langle x, y\\rangle -\\langle y, x\\rangle +\\langle y, y\\rangle. \\, ", "2401a92796de212edebb573a3f9bb570": " 2 {_1^1}\\text{S} + \\text{E} \\overset{\\xrightarrow{\\text{k}_{1(3)}}}{ \\xleftarrow[\\text{k}_{2(3)}]{} } \\text{C}_3 \\xrightarrow{\\text{k}_{3(3)}} {_2^2}\\text{P} + \\text{E}, ", "2401f2f849d12bf941087bce37b05e47": "h^*Y\\subset Y", "24022c769eecbaedaafc669396c108f8": "\\scriptstyle r(\\boldsymbol{r}_i,\\, \\boldsymbol{r}_{\\text{rec}}) / c \\,+\\, (t_i \\,-\\, t_{\\text{rec}}) \\,+\\, \\delta t_{\\text{atmos},i} \\,-\\, \\delta t_{\\text{meas-err},i} \\;=\\; 0 ", "24025cc13aa5ee724bda1e2ee758faac": "R_1 = \\frac{1}{2!}f^{\\prime\\prime}(\\xi_n)(\\alpha - x_n)^{2} \\,,", "2402a79d9775fe2647dc4c3f7ff5d08e": "\nW_{ij}(\\tau;L)=\\sum_{n=0}^\\infty w_{ij}(\\tau,2n+\\gamma_{ij};L).\n", "2402a9d2129977cf71ca984aace5945c": " \\in [0,1]", "24032c923c6a222208b06b4fcbc32f4e": "{\\epsilon_F}_p", "24033353660ebc9cdcd08b42a4b48053": "\n\\Delta x = S \\cdot \\frac{d}{D} = \\frac{S}{D} \\cdot d\n", "24034ed2a38db3e976df517686f2ca99": "F_{\\varphi} = mr \\ddot \\varphi +2m \\dot r \\dot {\\varphi} \\ . ", "24036210a715b56461455780d4328a8e": "Q_p - p\\Delta V\\;", "2403a5949bf4e2a162f178426657a24e": "K_0(kz)\\,", "2403c7d6aadc2e805495cee7a9b4a01b": "h(\\gamma) = \\int_0^1 \\log\\,(Q'(p; \\gamma))\\,\\mathrm dp = \\log(\\gamma)\\,+\\,\\log(4\\,\\pi).\\!", "2403c99af623e6d14539dcd1322f91d6": "C_V = 3Nk\\left({\\epsilon\\over k T}\\right)^2{e^{\\epsilon/kT}\\over \\left(e^{\\epsilon/kT}-1\\right)^2}", "2403e2473d422a58a37da1e30257a255": "\\left|\\psi(t+s)\\right\\rangle=U(t)\\left|\\psi(s)\\right\\rangle", "240410838ce7ff09ca6f36b495ce9419": " \\mathbb K(\\mathfrak g^*)", "240462904ee0fb4460a569d0bfb64833": "y = A' y_1 + B' y_2.", "2404d6607ee5fd25ec09349c0b8d79a4": "(x^2+y^2)x=2ay^2", "2404e5f41f5b9d922c6b25ad9adc2d79": "y_1 \\cdot x_1 = b", "24057184845d7591124f3a972dd8fc9a": "\\Im Z > 0", "24059f8d3f14fbb9a073f7c7695a972c": "H(z) = \\frac{1}{3} + \\frac{1}{3}z^{-1} + \\frac{1}{3}z^{-2} = \\frac{1}{3}\\frac{z^{2} + z + 1}{z^{2}}", "2405cafa6b9395667a87d54422165fdc": "\\operatorname{Li}_2(1)=\\frac{{\\pi}^2}{6}", "2405d1c907533cea98024ef7160926f3": "u(D) = u(E)\\ ", "2405d1e7aeea61f12b8e810e9c2eca12": " = SP - PV ", "2405e4da3da2906003cd5a1d3c2e0adf": "\\left( a^2(-a^2 + b^2 + c^2), \\;b^2(a^2 - b^2 + c^2), \\;c^2(a^2 + b^2 - c^2)\\right), \\,", "2406296eb2187f4e21db33e97bd039b4": "q\\le\\sigma(n/p_k^{\\alpha_k})", "24067d95feb5392a997c89d626a35b1a": "\n\\begin{align}\nx & = r ( \\sin \\phi + \\phi ) + C_x \\\\\ny & = r ( \\cos \\phi ) + C_y \n\\end{align}\n", "24068064c988504afa7fe67630781aa5": "\nx_{2}^{1}+x_{2}^{2} \\leq \\omega_{2}^{tot}\n", "24071a7dd978247f26f5e21ebfb26a3f": "(\\psi_j)_{j\\in J}", "24074b0aded9c4d45dd5f0f65d027202": "\\mathrm{ZnO + CO \\rarr Zn} (vapour) + \\mathrm{CO}_2", "2407a35c78f62f9a191c3e908b5a57af": "\\tfrac{a}{c} - \\tfrac{b}{d} \\cdot \\tfrac{a}{c} = \\tfrac{a(d-b)}{cd}", "2407c25545c0b1d8bb34048625ae4d8e": "\\bigl( L^{p_0}(R, \\Sigma, \\mu), L^{p_1}(R, \\Sigma, \\mu) \\bigr)_\\theta = L^p(R, \\Sigma, \\mu) \\ \\ \\text{if} \\ \\ \\frac 1 p = \\frac{1 - \\theta}{p_0} + \\frac{\\theta}{p_1},", "24084c207c0e3e676fa8096de415fbe7": "\\scriptstyle\\psi", "2408886c8483c573f783bc106b63664f": " He^{- \\alpha L/2} = B \\cos(k L/2)", "2408cb4d8ee3e0022e9cc798ff9cdc0c": "A^n_m", "2408ea4b733afbff3c9c0d237012453d": "\\theta_{i,1 \\dots V};", "2408ee21a1fa3d4c111be443bc5995ab": "e^{\\operatorname{Log}(z)} = z", "2409490d9a3ebf3bdd6c260322ef4beb": " \\xi \\rightarrow 0 ", "240995c3564979561dc6c553378a32ed": " \\textstyle v_{\\infty } ", "2409e0cc6c54f2521b9e2662fe74abbb": "\n\\rho\n\\frac{d{u}}{d{t}}\n= \\mu \\, \\Delta u - \\nabla p + \\Lambda[\\Upsilon(V - \\Gamma{u})] + \\lambda + f_{thm}(x,t)\n", "2409f4ea11c6c5105d5142778466be1a": "H\\psi = E\\psi", "240a0160550f4bbeb9314b34bdaac144": "= \\delta \\int d^4 x \\; e \\; e_M^{[\\gamma} e_N^{\\beta]} C_\\gamma^{\\;\\;\\; MK} C_{\\beta K}^{\\;\\;\\;\\; N}", "240ac9bf44a953a5c69f54afe5c26e4b": "c_\\mathrm d\\, ,", "240b014a3e2b839d3fde8e6f2771b15c": " \\lambda^\\dagger_A=\\lambda^{-1}_A: A \\rightarrow I \\otimes A", "240b9031d3ea0805ea08027ce0da089b": "P(G, \\mathcal{X}, \\mathcal{Y}) = \\{ (A,g) \\in \\mathcal{Y}\\times G: g^{-1}A \\in \\mathcal{Y} \\}", "240bbf26af031154a783e1e50f18a824": "\\frac{ \\text{d}E }{ \\text{d}t } = - \\sum_i \\frac{\\text{d}C_i}{\\text{d}t} \\qquad \\qquad (8d) ", "240bc0e838354eaa0340663feba3372e": "\\text{DSPACE}\\left(\\left(\\log n\\right)^2\\right)", "240bcde4ee17a54e39df97f90d3a91c7": "{\\alpha \\over 3} \\left ( (x - \\beta)^3 + (\\beta - a)^3 \\right )", "240be3b61f4a2fd8112c881bced19f8c": "B[t]", "240bee42bcd5d5bf642f9d695d73273a": "m > n", "240bf90ae5c4e2821bb0cc785672f2a0": "\ndA = h_{\\mu} h_{\\nu} d\\mu d\\nu \n = a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right) d\\mu d\\nu\n = a^{2} \\left( \\cosh^{2}\\mu - \\cos^{2}\\nu \\right) d\\mu d\\nu\n = \\frac{a^{2}}{2} \\left( \\cosh 2 \\mu - \\cos 2\\nu \\right) d\\mu d\\nu\n", "240bfee86a84fd31cb1c595d292afce9": "|x_{2,i} - x_{1,i}|", "240c542a2de8ecff5d3dc6e6930a2e1b": "\n{dR \\over dx} = g = - {dV\\over dx}\n\\,", "240ca5d9ac67864959486d6164bb291a": "\\vec{x} = (x_1,\\ \\ldots,\\ x_j,\\ \\ldots,\\ x_k)", "240cb752542bdd47e05557913bc1a2b1": "\\frac{\\partial \\bigg(a - (q_1+q_2)\\bigg) }{\\partial q_i} \\cdot q_i + a - (q_1+q_2) - \\frac{\\partial C_i (q_i)}{\\partial q_i}=0", "240cca0f4ba92b82a32460272d802110": "\\hat{\\lambda}= median(Y_1^2,Y_2^2,\\ldots Y_L^2)/0.456\n", "240cd6157888502d48efd27cad607796": " 1 - \\lambda = 1 - \\sum_{i=1}^R p_i^2 = 1 - 1/{}^2D", "240d21a3e55502570dea21946efdccbc": "\\lim_{\\Delta v \\rightarrow (\\Delta)} h_{\\alpha_{ij}} = I + {1 \\over 2} F_{ab} s_i^a s_j^b", "240d6ba63629fe52b91247e6495e9879": "\\alpha\\in \\mathbb{C}", "240dfc8139c07b05ccdd3a5214b30c23": "p=\\sum_{i=0}^{n}10^ia_i", "240e9eb92207d1cc19b88ac26ae02e89": "\\delta_n(\\varepsilon) = \\varepsilon^n", "240ed1caf4bfaead0e0a7ed2ac2c163c": "\\textstyle \\mathcal{F} ", "240f20dd1ffdf90209159cbfd6a00eb1": "\\scriptstyle \\frac{T}{4}\\,", "240f35b706a7942e1d4bbaa9a69dcc45": "~\\beta+\\theta~", "241011517ef934025e8046adb0d12150": "\\cosh\\tau", "24102b8810e709c22e5fd469aeb382e4": "\n \\boldsymbol{\\omega} := \\frac{1}{2} [\\boldsymbol{\\nabla}\\mathbf{u} - (\\boldsymbol{\\nabla}\\mathbf{u})^T]\n", "24104125c56ad21393cf61aef16679cb": "(\\sigma(\\mathbf x))(n)=x_{n+1}", "2410a721b635895e0ac56e581383749e": "\\alpha^{k} \\ne 1", "2411211a4a69c1448a58cb0dd61b4cda": "b+q\\sqrt{ - 1}", "24113a23216377402ea721cf190d51dc": "\\exists k \\in \\mathbb{N}", "241160933a4a8cf96721871427e08599": "\\operatorname{sink}[(\\lambda p.p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x)), X] ", "2411864754bbb6e5022835d9845e615e": "\\int_Y\\left(\\int_X |f(x,y)|\\,\\text{d}x\\right)\\,\\text{d}y", "24118873fb2c93ac1bd1c2f28a8a403c": "Happens(a,t_1)", "2411b1411112cde09ae1a1955e0bd016": " f(t,z) = \\left(2t, \\tfrac{1}{4}z + \\tfrac{1}{2}e^{it}\\right).", "2411ca58d417a38ae2cf8b2aa8a68ef0": "d(f(x),f(y)) < 1/n", "24121a01794db7941870290c800103ae": "A(x)=\\sum_{n=-\\infty}^\\infty a_ne^{inx}", "24129487a0710fea4d16212dad58d52a": "\\langle\\rangle", "2412be3c6a0e9b799c10c8518f456cd8": "\\left [\\begin{smallmatrix}\n\\;\\,\\, 2&-3\\\\\n-1&\\;\\,\\, 2\n\\end{smallmatrix}\\right ]", "241379628a0d7176065b5a90525513ad": " \\left\\lfloor ~ \\right\\rfloor", "2413cc8e1087319b5805fc163f9dcb8d": "O(d)", "24146412282a6f4d924c894dfa0c78c8": " \\operatorname{E}[H(X)] = \\sum_{i = 1}^n w_i \\operatorname{E}[H(X_i)],", "24146b4863f968ca22d68077bd962523": "\\! w=p/\\rho", "24146fae61cef46dcc98b85d1511eee2": "\\tau(X^*, X)", "24147dce52ab7a70fab349e1189a1fd3": "\n\\begin{align}\n\\sin x & = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots \\\\\n& = \\sum_{n=0}^\\infty \\frac{(-1)^n x^{2n+1}}{(2n+1)!}, \\\\\n\\cos x & = 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + \\cdots \\\\\n& = \\sum_{n=0}^\\infty \\frac{(-1)^n x^{2n}}{(2n)!}.\n\\end{align}\n", "24149d372a375f03fdc210b4c5b73f2f": "288 \\cdot V^2 =\n\\begin{vmatrix}\n 0 & 1 & 1 & 1 & 1 \\\\\n 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\\\\n 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\\\\n 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\\\\n 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0\n\\end{vmatrix}", "2414a7564965f430f73abda3413f8691": "\\alpha \\in \\mathbb{R}", "2414a91c52961be6980ea4413cad3a1d": "\\sin(\\alpha \\pm \\beta) = \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta", "24150779bbb609f0a576e9ac392c13a6": "H[\\vec{\\sigma}]", "24155d52f61ea4f32cc8773ec814c18f": "n^2 = 1 - \\frac{X}{1 - \\frac{\\frac{1}{2}Y^2\\sin^2\\theta}{1 - X} \\pm \\frac{1}{1 - X}\\left(\\frac{1}{4}Y^4\\sin^4\\theta + Y^2\\cos^2\\theta\\left(1 - X\\right)^2\\right)^{1/2}}", "2415617337a0c3166705cd79ae82b48c": "e_1 e_2 = e_{12}.", "24158b912b74355de1137f8c382faa43": " K[2(1-p)-\\Delta]=(N-K)(2p+\\Delta) \\! ", "2415ebc2189743f85b6261915877c515": "P = -P", "24164ff9fec86e44f7fc76b93f9225e2": "2A = 1.\\,", "2416671ff27d8feb73fb4a9c9a84e204": "1 \\mbox{ year} =10^0 \\mbox{ year} = 10^{0+7.50} \\mbox{ seconds } = 10^{0.50 + 7} s = 3.16 * 10^7 s", "2416ea9ae300408b9a44ab8290980e4f": " P(i)={\\rm tr}(\\rho F_i),\\; ", "2417104e69f3f67f7b49cd5bc9a47edf": "(A,B,C,D,E,F);", "2417ac592cb91d9bbf701d7c6fecb4ed": "F_{\\mu\\nu}=K_{\\nu;\\mu}-K_{\\mu;\\nu}\\;", "2417bbdb4cf4a028325e4672ada77d5c": "H^{H^{}}", "24181f9b52fb80ea5ae1328be6c21ee1": " \\{ \\Gamma_a ~,~ \\Gamma_b \\} = 2 \\eta_{a b} I_N ", "24184a0d99632378ab37c1380d5f4cec": "ax + by = \\gcd(a, b).", "2418b341ccfaf67090455c4d096a3c5d": "(ax+b)", "2418c6d324f6bf7a6fba298607222305": " \\frac{d\\omega}{dt}\\propto \\frac{2\\pi}{\\hbar}|\\langle \\psi_{f}|\\hat{H}'|\\psi_{i} \\rangle |^2 \\rho_{f}=|M_{fi}|^2 \\rho_{f} ", "2418df3231cefa6b45e218a949f8be46": "\\rho,\\phi,z\\,\\!", "2419b93ef3604c5a3264af97a499c7d0": "L\\in \\mathbb{R}\\;", "2419f6ec37e18bef68fe519fed46bb58": "\\displaystyle \\left\\{ 2, 3 \\right\\}", "241aa15ca419f2b7a55a3d9768e491a7": "H(\\sigma) = - J\\sum_{}\\sigma_i \\sigma_j -h\\sum_{j}\\sigma_j.", "241aa9aa5d4399e2b72b1d2905628e03": "\\text{Hom}_{\\mathcal{H}_s(T)}(\\mathcal{Y}, \\mathcal{Z}) \\to \\text{Hom}_{\\mathcal{H}_s(T)}(\\mathcal{X}, \\mathcal{Z})", "241acd54d1978b594b634c2a178c525d": " | \\beta A_\\text{OL} ( f_\\text{0 db} ) | = 1. \\ ", "241adb70191c5fdcf8696c8440c80a59": "1, 2, 6, 24, 120, 600, 4200, 28560, 257040, 2207520, 24282720, 258128640, \\ldots", "241b0850aaf82d4a063421daaf8930e7": " u(x(\\theta),t(\\theta),\\theta) \\geq \\underline{u}(\\theta) \\ \\forall \\theta ", "241b274a394e50ea293729c4f6c7573c": "(\\hat{A} \\cdot \\hat{B})", "241b2b804deca23af2cd4e5bee12d971": "\\xi\\colon E\\rightarrow X", "241b326f29c91a4985c3c29e11defa23": "\\vartheta^*", "241b6843cbc504c15aa56bcd7ada3c4c": "q'_x(a,b)=q'_y(a,b)=p_{d-1}(a,b)=0,", "241b6ae687af83d9105af1d39fe65871": "L = \\frac{\\kappa}{\\sigma T} = \\frac{\\pi^2}{3}\\left(\\frac{k_B}{e}\\right)^2=2.44\\times 10^{-8}\\,\\mathrm{W\\,\\Omega\\,K^{-2}}.", "241b93e940b673843d90cd22ff5769ae": "T_n \\left(e^x \\right) = e^{-e^x} \\frac{d^n}{dx^n}\\left(e^{e^x}\\right)", "241bc0e67997a62f6f01e5ffaee2ca94": "(-0) \\cdot (-0) = +0\\,\\!", "241c08a511cdb97665c315fea0ca70b7": "\\Delta x_0=-\\nabla_x f (x_0) ", "241cc27e21b71612d68e06bd075e7c3b": "\n \\begin{matrix} 4\\times 3 & = & \\underbrace{4+4+4} & = & 12\\\\\n & & 3\\mbox{ copies of }4\n \\end{matrix} \n ", "241cd16f723e93837a908a7ab5cc744f": "x-x_0=-\\lambda (x_P-x_0)", "241ceb844d4f74769f81f803fd5019c2": "\\frac{F(n,k+1)}{F(n,k)} ", "241cfb1fae34d452e8a2c3f516b0ba09": "\\Lambda(f)\\geq 0\\,", "241d1e24afc5d646b2b85d67180c0877": "\n \\Beta(x,y) =\n 2\\int_0^{\\pi/2}(\\sin\\theta)^{2x-1}(\\cos\\theta)^{2y-1}\\,\\mathrm{d}\\theta,\n \\qquad \\mathrm{Re}(x)>0,\\ \\mathrm{Re}(y)>0\n\\!", "241d5eac62cecf2ddf71c3c1309b7a7d": " \\frac{\\Gamma(s,x)}{x^s} \\rightarrow -\\frac 1 s", "241d5fe1c7b6ae59665e690935915df3": "\\lim_{n\\to\\infty} x_n = T\\left(\\lim_{n\\to\\infty} x_{n-1} \\right). ", "241d8be2f6acb2b26a4c527ddda29912": "2^m,2^m-m-1,4", "241d970020e5a85935611e1a21ca2492": "R = \\epsilon \\frac{\\gamma l v_{eff}}{2g \\omega d} ", "241db4d9abc9b28fa107f33829dbaa91": "(K_i\\varphi \\land K_i(\\varphi \\implies \\psi)) \\implies K_i\\psi", "241dfe447bdcc6a628b26b03b373f89a": "\\frac{1}{\\sqrt{2^{n+1}}}\\sum_{x=0}^{2^n-1} |x\\rangle (|f(x)\\rangle - |1\\oplus f(x)\\rangle )", "241e3d7f47575ba4f78c407e52624196": "\\alpha=\\pi/2", "241eaa1fc195495d2413cd6340253152": "I(X;Y) = \\mathbb{E}_{X,Y} [SI(x,y)] = \\sum_{x,y} p(x,y) \\log \\frac{p(x,y)}{p(x)\\, p(y)}", "241ee7772718f1dbba6132708283bb46": "\\rho_g", "241f07d6ab890aaddbe1be15afa3ee31": "\n\\sigma(\\theta)\n= \\epsilon_0 \\frac{\\partial V}{\\partial r} \\Bigg|_{r=R}\n=\\frac{-q(R^2-p^2)}{4\\pi R(R^2+p^2-2pR\\cos\\theta)^{3/2}}\n", "241f1f977405d69c414b305dd1231605": "\\textit{ADJ}", "241f2c2ece90e3aaea33cb68974338ae": "SU(3)_L\\times SU(3)_R", "241f4d180074e6128af716201733bfe6": "C=\\frac{D(N+R)}{D+T}-R", "241f58916abd8d14ee53a4e93ac1523f": "N=\\langle X, < , (h_i)_{i<\\omega} \\rangle", "241f63ea68af23a32f4dfd57235903da": "| e^{+} e^{-} \\rangle \\to \\frac14 \\left(-3|c^{+} \\rangle |c^{-} \\rangle +i |c^{+} \\rangle |d^{-} \\rangle + i |d^{+} \\rangle |c^{-} \\rangle - |d^{+} \\rangle |d^{-} \\rangle - 2| \\gamma \\rangle| \\gamma \\rangle \\right).", "241f8c44e56ef5aad194ce849daa7b8b": "\\,(e^z)^n = e^{nz}, n \\in \\mathbb{Z}", "241fc0dbf8a2026a91d27f1ff562c508": "\\langle j_1 m_1 j_2 m_2|J J\\rangle", "241fd430d256b028d54827ce01d696b5": "\\phi_{sl,v}=\\frac{1}{1+SG_{s}(\\frac{1}{\\phi_{sl,m}}-1)}", "241fe717afac381d2f1c2bb2e8cc47bd": "v_x=\\frac{-K }{\\mu}\\frac{d u_e}{d x}", "24201fa264f16618654527c9397e61c7": "\\displaystyle{K_r(e^{i\\theta})=\\sum_{n\\in \\mathbf{Z}} r^{|n|}e^{in\\theta} ={1-r^2\\over 1 - 2r\\cos\\theta + r^2}.}", "2420985db3b2cf97ef88f5afeeda328f": "\\{ |\\phi_i\\rangle \\}", "2420a564c10620d51603591e2e229c68": "Y_{AC} = Y_{ref} + c{dC_l\\over dC_z} + c{dC_n\\over dC_x}", "24212f98da731bb5089af174b48eae8d": "\\boldsymbol{F} = m((\\ddot r - r \\dot \\theta^2) \\boldsymbol{\\hat r} + (r \\ddot\\theta + 2 \\dot r \\dot\\theta) \\boldsymbol{\\hat \\theta})", "242207bb1e4afe948d672b22fa2f266a": "\\forall \\theta\\in\\Theta, T(F_\\theta)=\\theta", "24221c77f2c8e840a8feeede5c9c46c6": "Ln = 10\\, Laborers \\cdot 0.1\\,\\frac{Ph}{Laborer} = 1\\,Ph", "24222b71f652dede388e6f6a2f4d8598": " \\omega. \\ ", "242264d45a6f335037e99fd55e9270c0": " F = hA_s \\left( T(t) - T_a\\right ), ", "2422a40f372d5458139b57d1c17d7996": "\\textstyle\\R^2", "2422e38ab4ab81b9ffc5b1719c200094": "e_1 = \\{e_2\\}", "242333ca9ea668feee4365b2a5801f2e": " H' ", "242382e94d383195c580445532bf54bf": "f(z)=\\frac{z+i}{iz+1}", "242424100ca47448be646d0a449fb59f": "v = \\sum_{i =1} ^m \\alpha _i u_i \\otimes v_i", "242453842814c9f382714eaed0796558": "\\left(\\frac{p^{2}}{2m}+V\\right)\\vert\\psi^{0}\\rangle=E_{n}\\vert\\psi^{0}\\rangle ", "2424748e5ec008cbe69b8435d94acc86": "\\lambda \\Delta t", "242481206124a6391cd827193a9a6666": "r = a \\frac {\\sin [(q-1) \\theta + \\theta_0]}{\\sin [q \\theta + \\theta_0]}\n= a \\frac {\\sin [(1-q) \\theta - \\theta_0]}{\\sin [((1-q)-1) \\theta - \\theta_0]}", "24249186e96870782319d236ac5ab372": "= \\int \\underline {A}(a, \\lambda)\\underline {B}(b, \\lambda)[1 \\pm \\underline {A}(a^\\prime, \\lambda)\\underline {B}(b^\\prime, \\lambda)]\\rho(\\lambda)d\\lambda", "2424a5d08456b0aeddc3dd59e07345e0": "L_\\theta", "2424dfe11a036b8bfcfe32e55bcb5e13": "\\operatorname{Ext}^1_{\\mathbb Z}(Q,N)", "2424fd19d256658f663a44eb19f4ca37": "\\ \\mathcal{L} = h^\\mathrm{H} s s^\\mathrm{H} h + \\lambda (1 - h^\\mathrm{H} R_v h ) ", "242510caefe1265ff8011beef0009775": "\\scriptstyle C r \\sigma(2r)", "24254816db548f97c3f720d80e4e21ac": "\\ \\hat{X}(f)", "24256265347d650c27922ed7e60354d2": "\n\\begin{align}\n0 & \\le \\operatorname{var}(X_1 + \\cdots + X_n) \\\\\n& = \\operatorname{var}(X_1) + \\cdots + \\operatorname{var}(X_n) + \\underbrace{\\operatorname{cov}(X_1,X_2) + \\cdots\\quad{}}_\\text{all ordered pairs} \\\\\n& = n\\sigma^2 + n(n-1)\\operatorname{cov}(X_1,X_2).\n\\end{align}\n", "2425e19dbb6f681de91e8d855649d8a3": "A_{i+1}:=A_i+A_i\\left(I- A A_i\\right);", "2425ffea73f5aa3fe13b298ec0f2d548": "(1 + x)(1 + qx) \\cdots (1 + q^{m + n - 1}x)", "24265d43c429b6f3e99087e635bd8322": "\\frac{S(t)}{M(t)} = E_Q\\left[\\left.\\frac{S(T)}{M(T)} \\right| \\mathcal{F}(t)\\right]\\qquad \\forall\\, t \\leq T.", "24266eddfd108c33abf8e9683c99d8f5": "|S|\\geq\\gamma'p", "24269bbebfe003c4f208928fbb15bd9a": "\\mathfrak e_8\\cong \\mathfrak{so}_{16}\\oplus \\Delta_+^{128}.", "2426a99211d9cfab74f51ebbb25d9bcf": "n^k", "2426e1d85a8a79af4cace140015a067b": " \\alpha_t(E) = U^*_t E U_t. ", "2426e73b8bff0fbb8226d97deeadc28a": "0 \\leq i < m", "2426ecc7cd74dab649502dbaee279a33": "\\mathcal O_F^\\times \\rightarrow \\mathbf R^{r_1 + r_2}, \\ \\ x \\mapsto (\\log |\\sigma (x)|)_\\sigma ", "2427274af162168a637094d331d15a18": "\\frac{e^{it}-1}{i} = t\\mbox{ , }i = 0\\,", "24279e2ff50be56de0b160aa2608865d": "H(s)=\\frac{V_o(s)}{V_i(s)}=\\frac{1}{1+2s+2s^2+s^3}.", "2427beb7b9371824d1d6a45638115709": "\n\\begin{align}\n\\sin\\phi'&=\\sin\\lambda\\cos\\phi,\\\\\n\\tan\\lambda'&=\\sec\\lambda\\tan\\phi.\n\\end{align}\n", "2428163deef899905962f79890768d95": "g=E \\cdot d", "24282da3fe52079e6fdd7f7a902b5021": "a_1 \\ldots a_n", "24283c9944de9400ac2e4a8ad796936a": " p = \\Pr( X > \\operatorname{ E }( X ) ) ", "242852df083c52259fe2b6c1aaffff1e": "\\mathbf{R}_i=\\mathbf{R}+\\mathbf{r}_i", "2428892f474a249918e07e46ac641219": "s_g = g\\sin\\zeta", "2428ad6532e7ae9342ea8a3cb9cd1561": "\\Gamma=(V,E,s,t)", "2428cf94fb92da6c2fe5f558d428e71f": "P(O_j | A)", "24291822e17b06250ca5c5d6a1b9839e": "i_{V_s}", "242923e989687b37952059bf2d304ad5": "\n d\\boldsymbol{\\sigma}: d\\boldsymbol{\\varepsilon}_p \\ge 0 \\,. \n ", "242949e33aa6cbdad3f260b738f3fb1e": "\n |(j_1,(j_2j_3)J_{23})JM \\rangle = \\sum_{m_1=-j_1}^{j_1} \\sum_{M_{23}=-J_{23}}^{J_{23}}\n |j_1m_1\\rangle |(j_2j_3)J_{23}M_{23}\\rangle \\langle j_1m_1J_{23}M_{23}|JM\\rangle\n", "2429662bc6bd92759a97555705a475bf": "\\int_0^\\infty \\frac{dx}{1+e^{nx}} = \\frac{1}{n}\\ln 2;\n\\int_0^\\infty \\frac{dx}{3+e^{nx}} = \\frac{2}{3n}\\ln 2.", "24297a1d3877eb702c4fb9738668c724": "\\sum_{n=N}^{M-1} |a_{n+1}-a_n|", "24297bfad56825c23b85fe0b3d840f35": "T_{\\delta}^{X^{n}}", "24299185be4aeb562884f784e8a2f1f2": "u\\equiv v", "2429979ed7dcea52dc8ec15d47ef8419": "\\bold{j}", "2429b8369df78c15000b25fff4e2b2fa": "\\ln\\lambda=-a_1\\frac{Z}{\\sqrt{E}}+a_2", "2429c530ad389f58ca8dfa8c14029ce4": "s_1=3u", "242a07516b50fdfd0a849361e768ea96": "Y = a_0 + a_1 z + a_2 z^2 \\cdots + a_k z^k.", "242a323c8cf39cd383a6d6e43d223116": "||n(t)||=1", "242a5a0dbf1d010d7cba18256aee9e13": " \\Big[ \\big [\\mbox{nuclear} \\big] \\big [\\mbox{physic(s)} \\big] \\Big] \\Big [\\mbox{-ist} \\Big] ", "242a806131579109a40ea247f4738568": "Z = \\frac{Z_{0}}{\\sqrt{1 - \\left( \\frac{f_{c}}{f}\\right)^{2}}} \\qquad \\mbox{(TE modes)},", "242a899af774fd774f98dc2244ccbbbe": "\\{h^\\lambda\\}", "242a90af18bf216720c75144e27e0fa0": "\\omega 3", "242aa8dd622ddba1ed9f92e3124fd7c5": "E_k = \\alpha + 2\\beta \\cos \\frac{k\\pi}{(n+1)}", "242b0ca05590373248074786044d9ca9": "V_\\theta(r,t) = \\frac{\\Gamma}{2\\pi r} \\left(1-\\exp\\left(\\frac{-r^2}{r_c^2(t)}\\right)\\right), ", "242b144c97e372558f9fe4a63224107f": "{^h\\!P}(x_0,x_1,\\cdots x_n) = x_0^d P \\left (\\frac{x_1}{x_0},\\cdots \\frac{x_n}{x_0} \\right ),", "242b18b1a08f6f3ce757c6d01374f206": "\n\\mathfrak{P}(\\mathfrak{C}_\\operatorname{odd}(\\mathcal{Z}))\n\\mathfrak{P}_\\operatorname{even}(\\mathfrak{C}_2(\\mathcal{Z}))\n\\mathfrak{P}_\\operatorname{even}(\\mathfrak{C}_4(\\mathcal{Z}))\n\\mathfrak{P}_\\operatorname{even}(\\mathfrak{C}_6(\\mathcal{Z}))\n\\cdots\n", "242b21c0dbb30219f25255908b1f82d0": "e^{\\int^x P(\\lambda)\\,d\\lambda}", "242b72f236e229cd9fbec5120ab5f383": "w \\not \\in L \\Rightarrow \\Pr[V \\leftrightarrow Q\\text{ accepts }w] \\le \\tfrac{1}{3}", "242ba00b2d535fe0336bea8b63b44bf6": "\\lim_{t \\to \\infty} \\frac{G(t)}{t} = \\infty", "242bc13cbbb878c6eaa143e6fd8766fc": "\\frac{dx}{d\\varphi} = \\frac{dx}{ds}\\frac{ds}{d\\varphi}=\\cos \\varphi \\cdot a \\sec^2 \\varphi= a \\sec \\varphi\\,", "242bf38b6700d710f1652ff2e0e3f956": "l=ct", "242c223bfeb2a43c7d19f46aefcc8f51": " \\text{shortcut distance} = m,\\,", "242c5c46ef10c431acd3863dbf9f88b9": "\\pi_1(S^1 \\times D^2) \\cong \\pi_1(S^1) \\cong \\mathbb{Z},", "242ca47e01a4730b1bd7d9359053424d": " tr([\\sigma(X),\\sigma(Y)])=tr(1)", "242cbccc536df14351262be70c5e0b2b": "(p \\vdash q) \\vdash (p \\to q)", "242cfab32f6406e5f29f186fd60fa8cf": "rR=\\frac{abc}{2(a+b+c)}.", "242d0b2fed4787636bd8580b15ffd33b": " x^n ", "242d1e34520df9ea564ed1e96f605c2e": "b(z)=\\sum_{n=-N}^Nb_nz^{-n}", "242d3aa7363acea77309a43bcd66ce71": "\\delta\\rightarrow 0", "242d9831e6119ccd5638c7d736e0314c": "F(t) = \\Pr(T \\le t) = 1 - S(t).", "242e2965f0cdcdc179a3e89cc7f15ed2": "\\left [ -\\pi < \\theta < \\pi \\right ]", "242e2d9ab287449110b3042bd205d04f": " W + E_1 \\overset{a_1}\\underset{d_1}\\rightleftharpoons WE_1 \\overset{k_1}\\rightarrow W' + E_1 ", "242e78a1c09de76f4a8640840c940419": "e^{s*2k\\pi i}", "242e9a5304d0556cfa19bdbf1c9351c6": "h^2 = \\frac{b}{r} = \\frac{t^2}{r}", "242ecc1e1cbb9a9abd76d500128ad62d": "\\sum_{n=0}^{\\infty}U_n(x) \\frac{t^n}{n!} = e^{tx}\n\\left( \\cosh(t \\sqrt{x^2-1}) + \\frac{x}{\\sqrt{x^2-1}} \\sinh(t \\sqrt{x^2-1}) \\right). \\,\\!", "242f17df8ba2a9a9ffab6072b45d1914": "N_2 + 3H_2\\leftrightharpoons 2NH_3; K=\\frac{{f_{NH_3}}^2}{f_{N_2}{f_{H_2}}^3}", "242f38c7cf1bae1a1aa52a4d4ec434a2": "s \\ge t", "242f4ec3930d49d5b4fb6f24840cc7df": "= \\frac {1}{2} \\sqrt{\\pi}\\,", "242f9e8954acf29d57f61cfdf72d6ba0": "\\kappa^{+}\\leq\\kappa\\,", "242fb4eabf10782385559744a34e6ed9": "f(x)={2 \\over \\pi R^2}\\sqrt{R^2-x^2\\,}\\, ", "242fcbd26fc724c44d99b7c0aa3d6255": "(i\\omega-\\xi)^{-3}", "242fd799bd3cf89b2e6559cd759161ff": "\\delta = {\\rho \\over \\rho_{SATP}}", "242fec0f6f4d66625e876bae4fd5986b": " \\ PV \\ = \\ { 1 - e^{(-rt)} \\over r } ", "242ff8a89b48fd3dea9c0cbe7f325217": " c=2.\\xi.\\omega.m, \\; k=\\omega^2.m", "243025d4bd16bb2d631c4c237c9e2188": " C_K' = C_T' \\setminus \\{U\\}, ", "2430bcde069033ed91e8c1f08b45af47": "w = f(z)", "2430e8cac753cd80ea633c4620547221": "U(S,V,N)=\\hat{c}_V N k \\left(\\frac{N\\Phi}{V}\\,e^{S/Nk}\\right)^{1/\\hat{c}_V}", "2430fe39e225818bd957ed4cb53eb770": "loaded", "2431182f37baf8fe883c2b2d878adcc4": "u(\\mathbf{x},t) = \\int_{\\mathbf{R}^n}\\Phi(\\mathbf{x}-\\mathbf{y},t)g(\\mathbf{y})d\\mathbf{y}.", "24314abb2689c5f3ec005d7a28f38df5": "\\ AFG(p)", "243162513db9068187d17df7eb3f5c30": "\\lambda \\in \\mathfrak{h}^*", "24319e2ecb7a53b3423e5757b71299cd": "t \\mapsto Q(t)", "2431b9f60c225fda83dc6a505db3d52f": "x^2 + y^2 + z^2 = 3xyz", "2431c81bde187d7a90cad3006728bfd3": " \\mathbf{F}= \\mathbf{B}-\\mathbf{A} = (B_x-A_x, B_y-A_y, B_z-A_z). ", "2431d2c7cab36c00c3365201ceecc7f8": "\\mathrm{[Fe(H_2O)_6]^{2+}} +\\mathrm{[[Co(H_2O)_6]]^{3+}} \\rightleftharpoons \\mathrm{[Fe(H_2O)_6]^{3+}} + \\mathrm{[Co(H_2O)_6]^{2+}}", "24322ff156ea659506ba7cd79f95367e": "V^2=\\frac{2kb(a+b)}{Ma}", "24324be2b5fc384366011d9e66b09385": "2g\\,\\!", "2432510585320e8ef5261eef8e3e2230": "(t',R')", "2432845effe541006ad4a21224531332": " \\alpha (s) ", "24328c00062b34932a316adbb4cf8fe4": " \\frac{}{}|I| < I_0", "243321a04ab592c414530a7aa6d25e6d": "I_1 \\times I_2 \\times I_3=[a_1,b_1] \\times [a_2,b_2] \\times [a_3,b_3]", "24332b4dad061e8406d25172ae82cd72": "\\Psi''_\\theta(0) = \\frac{d\\mu_\\theta}{d\\theta} \\lim_{h \\rightarrow 0} \\frac h \\tau.", "2434102e9be337298e17c9e419643480": "g'", "2434292fb6f8a5f474e4dad2017c5d34": "G_{V_1, E_1} \\boxtimes H_{V_2, E_2} \\rightarrow J_{(V_1 V_2), (V_1 E_2 + V_2E_1 + 2 E_1 E_2)}", "24346689eb6ecd29c005d3b922e2b664": "p_H(\\tilde{x}|\\alpha) = \\int_{\\theta} p_F(\\tilde{x}|\\theta) \\, p_G(\\theta|\\alpha) \\operatorname{d}\\!\\theta", "24346c32d3cc2d481f1172172d2d152b": "PB + D > C,", "24347c396fd889e7296136560780553e": "E = 3 h \\nu - \\frac{3}{4} \\frac{h \\nu \\alpha^2}{R^6}", "2434909c1229f430c7f314f6408a3806": " \n V(\\alpha) = \\frac{V_0 V_{90}}{V_0 \\sin^2\\alpha + V_{90} \\cos^2\\alpha}\n", "24349f7f2ed7850d4a617488de293dd3": "F_\\text{avg}", "2434f9a3ed45bcb781a653613282e848": "\\epsilon_{i+1} = - {\\epsilon_i}^2 \\,.", "243598ddf46dd59abef780e6e59a335b": "\\int\\mathbf{\\Phi}_{lm}\\cdot \\mathbf{\\Phi}^*_{l'm'}\\,\\mathrm{d}\\Omega = l(l+1)\\delta_{ll'}\\delta_{mm'}", "24359c25dbbae53bc7aa028591b1301f": "\n c(\\psi) = \\int_G \\vert\\langle\\psi | U(g)\\psi\\rangle\\vert^2\\; d\\mu (g)\n", "2435b4ecaa7db0d307d6e9d7ed6d860d": "\\mathbf{r} =[x^1,\\ x^2,\\ \\dots\\ , x^n] \\ .", "2436035702ebfa512c151e3db6ce3818": "\\phi_{c_1,c_2,t}=\\exists m_1(\\phi_{c_1,m_1,\\lceil t/2\\rceil}\\wedge\\phi_{m_1,c_2,\\lceil t/2\\rceil}).", "24360894c15f3d07fa45ca0df04bd422": "\\mathcal{L}\\left\\{ f^{(n)} \\right\\}\n = s^n \\mathcal{L}\\{f\\} - s^{n - 1} f(0) - \\cdots - f^{(n - 1)}(0).", "24361916e44e4a5b85bd83722de77ee5": "\\gcd(n, q - 1)", "24362966659f2ef47654f464801f8753": "x^{\\mu}(\\tau)", "2436cdeee6481d15d6d69b56478984c2": "{}- 2 \\boldsymbol\\Omega \\times \\mathbf{v}_{B} ", "243709dfcaf238130d8a5c741921d12a": "\\left(\\frac{\\gamma}{\\delta},\\frac{\\alpha}{\\beta}\\right) ", "24371d52e3e877b36285ec4a768a173e": "\\sinh^{-1}\\left(\\frac{L}{2a}\\right)= \\frac{k}{2a} ", "24373250a2769355d4267adb69c0b86b": " x_0 \\in M", "2437591c91057ac8301b055d43ea5268": "g(n,k)", "2437637a4dd5be8d46a156da6f10977f": "\\sum_{j=1}^n \\ln(j) \\approx \\int_1^n \\ln(x) \\,{\\rm d}x = n\\ln(n) - n + 1.", "24378fecc92de5e4ab5e0159d6fc6617": "u + v = \\int \\left(\\frac{du}{dx} + \\frac{dv}{dx}\\right) \\,dx \\quad \\mbox{(2)}", "24379644b3a1cd8f206e34bff3a13513": "\\frac{d [X_i]}{dt}=0", "2437c9c3f8c71cbf3b8abb2040095788": "BRET = \\frac{{SV}_{dose}}{{SV}_{background}} \\cdot 365 \\,", "24380e87f075e06d8b4e2e8b7405daf6": "A = \\begin{bmatrix} 19 & 3 \\\\ -2 & 26 \\end{bmatrix} ", "2438952cdb0ccf66aa8ddac532a4448e": " PR = 0.829PIMP + 25SOIL + 0.078UCWI - 20.7 ", "2439719d95fa43252410c291ce1c1382": "\n\\begin{align}\n\\cos(nx) & = \\mathrm{Re} \\{\\ e^{inx}\\ \\}\n= \\mathrm{Re} \\{\\ e^{i(n-1)x}\\cdot e^{ix}\\ \\} \\\\\n& = \\mathrm{Re} \\{\\ e^{i(n-1)x}\\cdot (\\underbrace{e^{ix} + e^{-ix}}_{2\\cos(x)} - e^{-ix})\\ \\} \\\\\n& = \\mathrm{Re} \\{\\ e^{i(n-1)x}\\cdot 2\\cos(x) - e^{i(n-2)x}\\ \\} \\\\\n& = \\cos[(n-1)x]\\cdot 2 \\cos(x) - \\cos[(n-2)x] \\ \n\\end{align}", "24398ad1035a912f50ef81f8ec23b571": "TCPI_{BAC} = { BAC - EV \\over BAC - AC }", "2439aadd2bb46509e581c407202a1036": "\nz(\\tau) = \\exp[\\tau (D_T + D_V)]z(0). \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (4)\n", "243a755f44e23db09ff326871c09868f": "C_3H_8 + 5O_2 + 18.55N_2 \\to 3CO_2 + 4H_2O + 18.55N_2", "243a993106f5075714c0660b46722a63": "A\\otimes B(L^2(G))", "243aabb75efbee694c96c1e8e8959901": "A:=\\begin{bmatrix}\na & b \\\\\nc & d \\end{bmatrix}.", "243abdafc8064e8d50323d6df0b853d5": " P,Q ", "243af2c054114524134b019fecfa6e56": "B^2 - 4AC", "243b601b937fe5a0435dbc7c4ce1464b": "\\sqrt{(x^2 + a^2) (x^2 + b^2)} = 2x \\sqrt{t^2 + \\left( \\frac{a + b}{2}\\right)^2}", "243bc6e21cbf3cf3325ce698d9963049": "f\\colon PG(V) \\to PG(W).", "243c1697d6a4e8a0a7cf8e401254b394": " V = \n\\begin{bmatrix} T & D_{T^*}\\\\ \n\\ D_T & -T^* \n\\end{bmatrix}. ", "243c23e2bae2da6ee1821bf6425a64ff": "\\operatorname{head} \\equiv \\operatorname{first} ", "243c6d50e0d4cede5228158592982342": "\\Phi(m,n)=\\sum_{k=0}^{+\\infty}P_k(m,n)", "243c912695da5a4d625b19b4efc92e57": "{\\mathbf{x}}_r(0)=H(0)E({\\mathbf{x}}(0)).", "243cd43194b9adc20721f3042fd7866e": "\nR_\\text{score} = \\frac{C(A,B)}{\\min(n_A, n_B)}\n", "243d004a44b4d0fb0416f544abc9fcca": "x_i = (d'_i - c'_i x_{i + 1})/b'_i \\qquad ; \\ i = n - 1, n - 2, \\ldots, 1.", "243d966e78d91d39f375d05c64724211": "\\left [\\begin{smallmatrix}2&-1\\\\-2&2\\end{smallmatrix}\\right ]", "243dbfc2d884c043fcb510c397236781": "\n\\sum_{i, j = 1}^{n} Q_{i,j} x_i x_j + \\sum_{i = 1}^{n} P_i x_i + R\n", "243dfd8ea41852d0851558c81abc7498": "\n\\arctan z = z - \\frac {z^3} {3} +\\frac {z^5} {5} -\\frac {z^7} {7} +\\cdots\n", "243e49aa369f9b49c8dd0e42018d8f8f": "P_{\\mathrm{max}} = 0.36\\, \\mathrm{kg\\, m^{-3}} \\cdot h \\cdot r \\cdot v^3", "243f0a3d4598abbc7ca89beb29c137df": "\\frac{\\mathrm{d}^2 y}{\\mathrm{d}x^2} - (2 - k^2) y + 2 y^3 = 0", "243f3ecbe9fc68ee30ddb3b195e47b0a": "\n\\begin{align}\n\\Pr(Y_i=1) &= \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\, \\\\\n\\Pr(Y_i=2) &= \\frac{e^{\\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\, \\\\\n\\cdots & \\cdots \\\\\n\\Pr(Y_i=K) &= \\frac{e^{\\boldsymbol\\beta_K \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \n\\cdot \\mathbf{X}_i}} \\, \\\\\n\\end{align}\n", "243f680e6aec259838dd2bfdf4ffb03f": "0 \\ne I \\subset B", "2440130bca4cf4e5d5fb084a38542e03": "z \\in \\text{int}(\\Pi_A)", "244047292af4d3e23737741e96b33aa4": "f=F(x)", "2440b202faa88ee2121a65cc6f1078e0": "\\delta=\\arg\\max_{\\delta'\\in{NS}}\\{\\pi(\\delta')\\},", "2440bd96fa11f3157389f599e61af0ab": "\\mathcal{B}_{X,D}", "2440c0e92da317521c66c34c334f72d7": "\\displaystyle [a_0; a_1, a_2, \\ldots, a_k+1]", "2440c76f978106801754f5704eac7677": "\\scriptstyle f_\\mathrm{image}(0) = f\\,", "2440e52818e53754435521bd0a671098": "t^2 g_{ij}(y)\\,dy^i\\,dy^j+2\\rho \\, dt^2+2t\\,dt\\,d\\rho,\\, ", "2441a1576fb6f0154cc9bfb54969425f": " D_\\text{wave} = - \\frac {1}{2 \\pi} \\rho U^2 \\int_0^\\ell S''(x) \\mathrm{d}x \\int_0^x S''(x_1) \\ln (x-x_1) \\mathrm{d}x_1 ", "244267c4ed7833da3e0931c4b73bd1ee": " f(t) = \\int _0^\\infty {\\hat f}^c \\cos (2\\pi \\nu t) d\\nu + \\int _0^\\infty {\\hat f}^s \\sin (2\\pi \\nu t) d\\nu,", "2442843ca6d60f18bb03e41622b6bd6d": "X \\sim \\mathrm{GH}(1, \\alpha, \\beta, \\delta, \\mu)\\,", "24428d786877a897d17450ee310d0e97": "t\\mapsto f(s,t)", "2444992556a3cbce4e2c8a165c19de8e": "B\\in U", "2444f161d3357ef9adb644f5fb640e19": "\\sin\\alpha_1\\cos\\beta_1 = \\sin\\alpha_2\\cos\\beta_2.", "244587ed3606ffd53995be138c333f51": "x\\in \\mathbb{R}^*", "2445b10c68a38304f4659bcf751ca109": "SD \\approx \\frac{\\mathrm{ES}}{1^{\\prime\\prime}} = \\frac{1 \\, \\mbox{AU}}{(\\tfrac{1}{60 \\times 60} \\times \\tfrac{\\pi}{180})} = \\frac{648\\,000}{\\pi} \\, \\mbox{AU} \\approx 206\\,264.81 \\mbox{ AU} .", "2445c3058a8435ec01d1d9afcba8b807": "\\; \\pi_{i.} = {\\sum_j O_{ij} \\over N} \\; ", "2445d101cfdc5a52dbd9bcbfa52f0e92": " \\begin{bmatrix} V_2 \\\\ I'_2 \\end{bmatrix} = \\begin{bmatrix} 1 & -R \\\\ -sC & 1 + sCR \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix}", "2445e21e060c798f60b2267a28a84779": " \\text{(1)} \\qquad \\Delta U = \\alpha R nT_2 - \\alpha R nT_1 = \\alpha Rn \\Delta T ", "2445eb98d73933994ded7abd95eb2744": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x)=f(x)", "2445f64e7b38fe3e2a55f729b70da344": " \n\\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & -1 & 0 \\\\ \n0 & 0 & 0 & -1\n\\end{pmatrix}\n\\quad\n", "244652dd7c78a2a5ba1e017f1551b888": "\n\\left(b^{d\\left(n+1\\right)}+b^d+1\\right)^n,\n", "24467b3b75a804536b3763ada09f5cdc": " \\left [ \\hat{A}, \\hat{B} \\right ] = \\hat{A} \\hat{B} - \\hat{B} \\hat{A} ", "244693c6e2c5dab007b514b422d741cd": "\\lambda x.~e", "2446f40c44d0c2c2d5160c9da98a62a7": "\\langle a_n \\rangle_{n=0}^\\infty \\subset \\mathbb{N}", "2447296319d6618f48dc970147c79613": "[\\overline{C},\\overline{S}]", "2447543af6d49189acf0dd0e4aa290c0": "a(f(x))", "244778145f264d691c2cea0c39bc3578": " \\dot{x} = v(x). \\, ", "2447f5eca0cfbcc8d5bf5f083fa0a427": " T^{\\hat{j} \\hat{k}} = \\frac{q^2 \\sin(\\omega u)^2}{4 \\pi} \\, \\left[ \\begin{matrix} 1&1&0&0\\\\1&1&0&0\\\\0&0&0&0\\\\0&0&0&0 \\end{matrix} \\right] ", "244801ab19702a917aec76fc6185555f": "T = \\{x \\in L_\\beta : x \\in S \\wedge \\Phi(x, z_i)\\} = \\{x \\in L_\\gamma : x \\in S \\wedge \\Phi(x, w_i)\\} ", "24483c9ef7c2d7e4dffa8631de4dc5b7": "a_{rel} = a_0 \\, \\sqrt{1-v^2/c^2}", "2448700b9f3718a00ce54a2ba83c097c": "\\langle X+\\xi,Y+\\eta\\rangle=\\frac{1}{2}(\\xi(Y)+\\eta(X))=\\frac{1}{2}(\\epsilon(Y,X)+\\epsilon(X,Y))=0", "2448dcb128af25515be8da1c63e06b79": "\\sin \\frac{\\pi}{6} = \\sin 30^\\circ = \\cos \\frac{\\pi}{3} = \\cos 60^\\circ = {1 \\over 2}\\,,", "2449100368528312c338b371fe47282e": "\\begin{bmatrix} 0 & -1 \\\\ 1 & 1 \\end{bmatrix}", "24491083c736320c8cc2eaf94a35423c": "Tail^+(X) \\gtrdot Head^*(Y)", "244933cfba8c3170f0b78c358a308214": " Q_{H} \\ ", "24497f94a02864da45e42b4d0aa18105": " y' = 0 ", "2449b2b12012224dbe9f384565e9d9a6": "\\Theta(n)", "2449d7a4ddc3683a4ea3e26113008cac": "\\! r", "2449e763ba410d15e19d157dca2b84fb": "E_2-E_1=h\\nu", "244a4edba2e64de6d63c462e112457ea": " f(\\mathbf{x}) = \\frac12 \\mathbf{x}^\\mathrm{T} \\mathbf{A}\\mathbf{x} - \\mathbf{x}^\\mathrm{T} \\mathbf{b} , \\quad \\mathbf{x}\\in\\mathbf{R}^n. ", "244a5e6267e626cf954c9c5f07ff11a7": " |\\eta|<=1 ", "244a828ce9d2a1be8337b80ed6d7ac53": "y_{i,j}\\in Y", "244a9c5de5fbf9edb1fd963d048235b9": "\\sum_{i=0}^{n-1} i 2^i = 2+(n-2)2^{n}", "244aa9fde966071f8506747a10e3bbc4": "\\scriptstyle \\left(1 \\,-\\, \\frac{t}{\\lambda}\\right)^{-k}\\,", "244af61415aef8b1b0e09b7be8edca0a": "x_r \\geq b_i-\\sum_{k=1}^{r-1} a_{ik} x_k", "244b514554bd62c441f1c7c3f5c22489": "\\dot{\\mathbf{f}}", "244b9167481853192f86172ee1e494bb": "P_\\mu (\\tau )", "244bbdc27565643bc21136fe5dc90d3e": "D[x_0,\\ldots,x_n]f", "244bfdd19c862ff8a1def77e44ff0938": "A = L U_3 U_1 P", "244c3860eecf01974210e62004dc6c1c": "MRRT = A(B-C) - D - E", "244c4c56d63caa6e162fc0f66b2cc2b4": "E_{0,1}= 510,260 * \\frac {510,000}{510,260} * \\frac {10.060}{510,260}", "244c9b45fd4b6d1ec882ba14017ed53f": "\\mu(t)=1", "244cc30fc4411b8d6f3726697f7e496f": "child_i", "244cfde86976377e8abddef73f8ea3fd": " S_k = \\{1, 2, \\ldots, n_k\\}. ", "244d0e81de5d4d54781156359a0f7196": "n=\\left\\lceil\\frac{n}{m}\\right\\rceil + \\left\\lceil\\frac{n-1}{m}\\right\\rceil +\\dots+\\left\\lceil\\frac{n-m+1}{m}\\right\\rceil,\n", "244d469c6900dc9c105dccd7052a886b": "\\textstyle \\mathbf{b}_2", "244d7aa2b96b5914f99b5e42b7a4bf15": "expr = A_{i}", "244e37af0beebc59ffe40ccc01c155ce": "\\operatorname{E}((\\delta_1(X)-\\theta)^2)\\leq \n \\operatorname{E}((\\delta(X)-\\theta)^2).\\,\\!", "244e49456af1cc5706c9d6fcb4ab2472": "HV=0.0018544\\times\\tfrac{L}{d^2}", "244eef0c57f18ff9556a81e16c13b47f": " Z = \\int e^{\\bar\\psi M \\psi + \\bar\\eta \\psi + \\bar\\psi \\eta} D\\bar\\psi D\\psi = \\int e^{(\\bar\\psi+\\bar\\eta M^{-1})M (\\psi+ M^{-1}\\eta) - \\bar\\eta M^{-1}\\eta} D\\bar\\psi D\\psi = \\mathrm{Det}(M) e^{-\\bar\\eta M^{-1}\\eta}", "244f54ae45e7cee8bf4f6232400dbc65": "\\tan\\alpha_0 = \\frac\n{\\sin\\alpha_1 \\cos\\phi_1}{\\sqrt{\\cos^2\\alpha_1 + \\sin^2\\alpha_1\\sin^2\\phi_1}}.", "244f80b6384ad586f65b49e353452512": " \\sum_{i,j} de_i de_j' u_x u_r = -dq dq' (w_x w'_r+w'_x w_r) ", "244f8a8950b08319ca83df2b979ef548": " \\frac{p_2}{p_1} = \\frac{2\\gamma M_1^2}{\\gamma + 1} - \\frac{\\gamma - 1}{\\gamma + 1}", "244f9eb65594e5b633f7824843bc823d": "X_{i}=\\frac{n_{i}(t=0)-n_i(t)}{n_{i}(t=0)}=1-\\frac{n_i(t)}{n_{i}(t=0)}", "244fed5b7e7ce4f9789c0e721d13e353": "\\textstyle K ", "24503f7a99422feb310621692bc83d45": "\\,^{238}_{92}\\mathrm{U}\\ + \\,^{24}_{12}\\mathrm{Mg}\\to \\,^{259}_{104}\\mathrm{Rf}\\ + 3 \\,^{1}_{0}\\mathrm{n}", "245046eb150ed2c547e7d7fd2e76d785": "\\int_{-1}^1\\frac{\\mathrm{d}x}{x}{\\ }\n\\left(\\mbox{which}\\ \\mbox{gives}\\ -\\infty+\\infty\\right).", "24505b9fa4f63b41dee0e32a2f76bae4": " {\\rm E}_1(z) = \\int_1^\\infty \\frac{\\exp(-zt)}{t}\\,{\\rm d} t \\qquad({\\rm Re}(z) \\ge 0) ", "24508b7adec3c6b74d857690f2d0f155": "\\sum a_{i,j}(x) \\xi_i \\xi_j \\geq \\lambda |\\xi|^2", "245095537e35690bc3ab52834617f330": "x(t) \\rightarrow X^*(s) \\rightarrow x^*(t)", "24509e79859838cd4f18abf9c4b8b24f": "50i+8", "2450dbe33f813053f861d027e07433d5": " \\frac{p}{q} = \\Omega ", "2450eb54d756a3c3aa5f1047f05ae24f": " \\mu =\\phi_U - \\phi_L ", "2450ff67454e08a2f200b0fa45e46ad1": "e > 1", "24510848c8c770ba84dc1b49662d8abf": "L(A) = \\bigcup_{\\alpha} L_{\\alpha}(A) \\! ", "24512306ec1d0e218647f19389ba8ca7": "\\Delta W = \\sqrt{e^3 F \\over 4\\pi \\epsilon_0},", "245137fd8f8adedf3cfcb35f6007a9a0": " \\Delta_g u = 0 ", "2451a2737600e06be25d25ea911745f0": "\\frac{a}{b} \\cdot \\frac{c}{d} = \\frac{ac}{bd}.", "2451cdf7638b8a5ee073f85cec264a0d": " s_0 \\in S ", "2451d7c7b1b3db49966f2e7cb7705fb1": "\\begin{matrix} {4 \\choose 2}{3 \\choose 1}^2{9 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "2452185351ff69c60b0d1d1c91fe369d": " V = \\frac{ H - E( H ) }{ SD( H ) } ", "24522dcf5c8109082a791413ee40dfb6": "\\rho = \\sum_i \\rho_i \\frac{V_i}{V}.\\,", "2452dd6fd20532beca558815574e25bf": " 2-m^2 =\\sqrt{2+(2-M^2)}\\, ,", "2452fee413f58bb9509e88d80d4b9f8d": "T_1", "245322caae7a2d3c852e000db88953f8": "\\lambda_k=\\left(\\frac{1}{(k-\\frac{1}{2})\\pi}\\right)^2,\\qquad k\\geq 1", "2453932be4f298a60996cb10e00df32a": " A = F_1^{-1} F_2 ", "2453b7e9c144fd28cedf8891935348e8": "~x_1 \\leftrightarrow ... \\leftrightarrow x_n", "2453bacd77eac08e9d8b8b385f62e9f6": " \\begin{align}\n\\lim_{\\beta \\to 0} \\text{median}= \\lim_{\\alpha \\to \\infty} \\text{median} = 1,\\\\\n\\lim_{\\alpha\\to 0} \\text{median}= \\lim_{\\beta \\to \\infty} \\text{median} = 0.\n\\end{align}", "2454070f8b93297ec10fe81246271641": "\\pi = \\frac {P_o - P_e}{1-P_e} ", "2454075020a3066631a1428a19ad9b53": "\\{U_\\lambda : \\lambda \\in\n\\Lambda\\}", "24547d2a00606bb3e907ac0eb0479ba3": "[n]\\times[n-1]\\times\\cdots\\times[2]\\times[1]", "2454db539b568a96c228d8da5d8118f3": "e^2(t) = s^2(t + \\alpha) - 2s(t + \\alpha)\\hat{s}(t) + \\hat{s}^2(t),", "2454e6f9d47ed8a8828b5c41faff111e": " \\mathbb{E}[\\langle \\varphi(x), \\varphi(x') \\rangle] = \\langle x, x' \\rangle", "2454eb77baff6695c637a9e13543de62": "\n\\begin{align}\n& {} \\qquad D(X_1,\\ldots,X_n) \\\\[10pt]\n& \\equiv \\left[ \\sum_{i=1}^n H(X_1, \\ldots, X_{i-1}, X_{i+1}, \\ldots, X_n ) \\right] - (n-1) \\; H(X_1, \\ldots, X_n) \\; .\n\\end{align}\n", "24551f19411355b0536faf9cbefd0b30": "\n{\\mathfrak{T}}^\\alpha_\\beta =\n\\left( \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right)^{W} \\, \\frac{\\partial {x}^{\\alpha}}{\\partial \\bar{x}^{\\delta}} \\, \\frac{\\partial \\bar{x}^{\\epsilon}}{\\partial {x}^{\\beta}} \\, \\bar{\\mathfrak{T}}^{\\delta}_{\\epsilon}\n\\,,", "2455a8a7d97ec8b95a1a30edd357d41c": " p_k (x_1, x_2, \\dots,x_n) = \\sum_{i=1}^n x_i^k \\, .", "2456b7e422105920fa68d4bc3f26dc7c": "= \\mu_0rN^2 \\left[ \\left( 1 + \\frac{1}{32w^2} + O\\left(\\frac{1}{w^4}\\right) \\right) \\ln(8w) - 1/2 + \\frac{1}{128w^2} + O\\left(\\frac{1}{w^4}\\right) \\right] ", "2456d4989e097c16eaf2840d4ed575be": "x^{10} + x^9 + x^5 + x^4 + x + 1", "2456e3c6455cf5f0ef7699d8402c593c": "\\breve{L}", "2457137e54d3deb8f40af46c2b91d0f6": "\\Delta S=2", "245718652e75023ed42583f4aa9eae43": "-1.6326", "24576f92e39e9f1414a0a3274f54dfd1": "F_i\\,", "245788b7873f368fccc0e60527b506aa": " \\mathbf{b}: G \\rightarrow \\mathbf{Bohr}(G). ", "24578a31f08ed1c309dc19d26ac2ff56": " v \\; : \\; 2^N \\to \\mathbb{R} ", "2457912d4aedc72628414f6e3af49956": " \\displaystyle A(s,\\lambda)\\xi_0 =c_s(\\lambda)\\xi_0,", "2457beb505c56430bb2766f1e6560319": "u(0,t)= 0", "2457de0e8d9c6ff5bdee4d8b16d22a52": "\\mathbf{F}=d\\mathbf{A}", "2457eb40a864b7ec5938b8529a840a30": "D^*=\\frac{D_n D_p(n+p)}{p D_p+nD_n}", "24581b6da3ebcd3b67aedf87eb3064ed": "A = {1 \\over 2} \\int r^2 \\, d\\theta ", "24586132f0c6282a02427606e8edc229": "f(x_1^{|q|}, x_2^{|q|}, x_3^{|q|}, \\ldots).", "24587072e08dad9a498c6368faf629d3": "P = 2 L / \\pi D", "2458ada6c84b426854fc3cba2b496658": "\\mathrm{cd}\\left((2m-1)K\\left(1/z\\right),\\frac{1}{z}\\right)=0\\,", "2459347edf2a3aa48d415d1ab6dbefeb": "v_{\\perp}", "2459559fef7bb6a2ba28485295b759d0": "f(f^{-1}(L))\\ne L", "24595899053c8b3129863642a3af6fd1": "F_{1},F_{2},\\dots,F_{b}", "245a0d7f5a309d7fba90767f2954da9a": " \\frac{\\part^3 f}{\\part x^3}(0, r_{o})\n\\left\\{\n \\begin{matrix}\n < 0, & \\mathrm{supercritical} \\\\\n > 0, & \\mathrm{subcritical} \n \\end{matrix}\n\\right.\\,\\,\n", "245a11fe34daba1bd187173380903e51": "\\bold{S}=\\frac{c}{4\\pi}\\bold{E}\\times\\bold{B}. ", "245abd5ab96370bdebeb42ea6ae350ff": "\\operatorname{st}(x) \\le \\operatorname{st}(y)", "245ac8a4c70fda0d388b7a28f87e812b": "0^o \\le \\theta \\le 180^o", "245ace04106ef045600bd5e5aff6088f": " X_{i} ", "245ad35e07ae8c1bf8735e60e89b903e": "\\Omega^{\\infty - n}", "245ae94ccc26532e160fa6595447b49a": "l=2 \\pi R", "245affc8fbd8b1fdf069c46737a42317": "S_{21}=S_{31} = -3\\,\\text{dB} =10 \\log_{10} (\\frac{1}{2})", "245b4065edc742bbee4211427948e0a6": "M,g", "245b4df5eddfa06a01b6271cc6b24e49": "A = \\tfrac{1}{2} \\cdot p \\cdot r.", "245b73f1cf7bd93ee112f210c45f709b": "F = eE \\, \\exp(-i\\omega t)", "245d1a3a3b91ccec6df63ecda744d4dc": "R=\\sqrt{E^2+9\\lambda^2 + 2E\\lambda}", "245d621658e06a1539bfa71238254da7": "p_1(w)=1/w", "245d6e9f87ec4aedcf399c1c593b0643": "\\nabla_{[v, w]}(m) = [\\nabla_{v}, \\nabla_{w}](m)", "245deed2b19c7bc7eee605f65d4778e3": "R_m", "245e1186ebbe95f8c367e7c53cc219ee": "W_{1-2} = \\int PdV = 0", "245e30ffc38fd76a0e66469c65fedab8": " \\Delta E_C ", "245e49e03c1aff7dfa8f672583ef0787": " \\operatorname{Corr}[dz_1(t),dz_2(t)] = \\rho \\, dt ", "245e87fef88092abbcee43101a49ec1a": "\\Delta u = u_{xx}+u_{yy}", "245ec15c7f91ba3eac42fa7c359b4593": "\n\\begin{align}\nX_i &= \\bar{M} + (\\bar{X}-\\bar{M}) + (X_i-\\bar{X})\\\\\nY_i &= \\bar{M} + (\\bar{Y}-\\bar{M}) + (Y_i-\\bar{Y})\\\\\nZ_i &= \\bar{M} + (\\bar{Z}-\\bar{M}) + (Z_i-\\bar{Z})\n\\end{align}\n", "245fdf0a63da0b7785a2053075468879": "v= {\\sqrt{rg\\left(\\sin \\theta +\\mu_s \\cos \\theta \\right)\\over \\cos \\theta -\\mu_s \\sin \\theta }}\n={\\sqrt{rg\\left(\\tan\\theta +\\mu_s\\right)\\over 1 -\\mu_s \\tan\\theta}}", "245fe2514d99ad59089374bedfe13d63": " (-a e , 0) \\,\\!", "24605685c7a717e67bd6cba22d58c86b": "\\gamma(1-1/k)+\\ln(\\lambda/k)+1 \\,", "2460d413afee164e62afdabf88dea1a1": "p_{1\\infty} \\leftarrow p(x + 1), M_{1\\infty}\\leftarrow M(x + 1)", "24614cce729e2ce03e2cc2616ebc7833": "{\\color{Blue}~2.10}", "246151b993bb0d33f3ce10c84bb3ce42": "k = k_2 K_1", "246163c5d68d70388f2ad8c9de1d6574": "\\cup_{n=1}^{\\infty}A_n\\in D", "2461a7355d77e296c24b1ea113d92f50": "\\scriptstyle H_\\mathrm{norm}=0", "2461b178936988bc7debd4270f341ac0": "H(f) = \\begin{bmatrix}\n\\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{\\partial x\\,\\partial y} \\\\[10pt]\n\\frac{\\partial^2 f}{\\partial y\\,\\partial x} & \\frac{\\partial^2 f}{\\partial y^2}\n\\end{bmatrix}.", "2461b7f5d1624dcb655db12b9c5844cc": "(a, 0)", "24627148c3db6e7db25302e606d45201": "G^{\\alpha\\gamma} + \\Lambda \\mathrm{g}^{\\alpha\\gamma} = (\\mathrm{const}) T^{\\alpha\\gamma}~", "246275f41e3f6d3a6f1ca6f426a56eb0": " THD = \\frac{\\sqrt{{a_2}^2+{a_3}^2+..}}{a_1}", "2462874e9db0669fa8628425e3ba8aa4": "\\Box \\bar{h}^{\\alpha \\beta} = -16\\pi T^{\\alpha \\beta} \\,", "24629c2297fda97bb844dcb324dbae4b": " \\mathrm{Br} = \\frac {\\mu U^2}{\\kappa(T_w-T_0)} = \\mathrm{Pr} \\, \\mathrm{Ec}", "2462af80e4e5654179d9901d27e86570": "\n[L_z , X] = iY\n\\,", "2462bd0db7e8e91137d13dc7222e5bc6": "J: X \\times Y \\to \\mathbb{R} \\cup \\{+\\infty\\}", "2462e861bff7cdbccf0bac8a204b035a": "G \\times H", "2462ea1b25d1a7d2fc4cd5554dbf07d3": "\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix}", "2462ef886428ee7d949936d41281f003": "\\bot ", "246314289bb4d3cae42bd012459358dd": "\\begin{align}\n\\operatorname{E} \\left [\\ln^2(X) \\right ] &= (\\psi(\\alpha) - \\psi(\\alpha + \\beta))^2+\\psi_1(\\alpha)-\\psi_1(\\alpha+\\beta), \\\\\n\\operatorname{E} \\left [\\ln^2(1-X) \\right ] &= (\\psi(\\beta) - \\psi(\\alpha + \\beta))^2+\\psi_1(\\beta)-\\psi_1(\\alpha+\\beta), \\\\\n\\operatorname{E} \\left [\\ln (X)\\ln(1-X) \\right ] &=(\\psi(\\alpha) - \\psi(\\alpha + \\beta))(\\psi(\\beta) - \\psi(\\alpha + \\beta)) -\\psi_1(\\alpha+\\beta).\n\\end{align}", "24634e1eb7a9a49d8175fb7d4720f78b": " \\Delta Y ", "246352fe547c6354c96b1fb0cbd30bb1": " D(1-\\epsilon)\\leq deg(x)\\leq D(1+\\epsilon)", "24637ead895f29c197c74d17a7250db7": "\\begin{cases}\n0 & \\text{if } q > p\\\\\n0, 1 & \\text{if } q=p\\\\\n1 & \\text{if } q < p\n\\end{cases}", "24637f35534c0bdd12f2e890bb840ad7": "x \\le y", "24638d5ad5f80002af26b8f159381ac4": "\n\\exp\\begin{pmatrix}\n. & . & . & . & . \\\\\n1 & . & . & . & . \\\\\n. & 2 & . & . & . \\\\\n. & . & 3 & . & . \\\\\n. & . & . & 4 & .\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & . & . & . & . \\\\\n1 & 1 & . & . & . \\\\\n1 & 2 & 1 & . & . \\\\\n1 & 3 & 3 & 1 & . \\\\\n1 & 4 & 6 & 4 & 1\n\\end{pmatrix}, ", "2463f86b8a747be74b42f55656cfebfc": "B_{123}= C_{12} C_{23} C_{13}^{*}", "2464f59d8fcbbad4435bb4978f8f80cf": "P = D\\cdot\\frac{1+g_1}{1+k} + D\\cdot(\\frac{1+g_2}{1+k})^2 +...+ D\\cdot(\\frac{1+g_n}{1+k})^n+ D\\cdot\\sum_{i=n+1}^{\\infty}\\left(\\frac{1+g_\\infty}{1+k}\\right)^{i}", "246529b44d956a3d1739d23d7c3918f8": "BE = 0.02786 \\times pCO_2 \\times 10^{(pH - 6.1)} + 13.77 \\times pH - 124.58", "24652fa9146a070e2e38904fef6bb146": "a_0=\\frac{4\\pi\\varepsilon_0\\hbar^2}{e^2m_{\\text{e}}}", "24652fba5172d3fd6435bcab46ff6997": "\\phi,\\,\\psi\\in L^1(\\mathbb{R})", "246537dfe54b530bd2bddf4ab215af04": "T_\\theta", "246540b85cef944e2aefcf0987d11045": "\\epsilon_{ij}^{\\text{v}} = \\tfrac{1}{3} \\delta_{ij} \\sum_k \\epsilon_{kk}", "24657ad5835f99223af9ac24f7d22f90": "\\tilde{g}(\\alpha,\\beta) = \\alpha[\\mathbf{f}]G[\\mathbf{f}]^{-1}\\beta[\\mathbf{f}]^\\mathrm{T}.", "246595d0c2db3416e0b1c7175fb397ff": "a\\nmid b", "2465cced4f707b58ab37aa4c85a0b365": "\\scriptstyle a x^2 + b y^2 + c z^2 = 0", "24664c09161c78b37f67bad4fd44578a": "(X_1,\\ldots,X_n)=_\\text{d}(F_{X_1}^{-1}(U),\\ldots,F_{X_n}^{-1}(U)), \\, ", "24664e398ad211282fb044af6dbe4e2e": "f_\\alpha(x) = f_\\alpha(y)", "24669d87b55b58ecb660c6c44a624bf4": "-\\sqrt{\\frac{1}{3}}\\!\\,", "2466dedb6adef8c35a739aee75db366b": "y = e^{rx} \\, ", "2466e83a69bc5a8d5d79f0bd8496ef42": "F_{\\rm D}", "246704587935ce9dcf8bc8ef106b94ed": "B(\\mathbf{r},\\mathbf{s}) = (W\\mathbf{r}) \\cdot \\mathbf{s} ", "24670f0c65670221482cf73cc6ccb62b": " \\frac{x}{x^2 + ab}. ", "246723f2e31d7d02bd8aca78a7b80daf": "M_r", "24673c56a685473485a252d1ef8c11ba": "A_ {\\infty}", "24676386ecd05d355fd4eb574e9314aa": "J \\propto E \\exp \\left ( \\frac{-q \\left ( \\phi_B-\\sqrt{qE/(\\pi \\epsilon)} \\ \\right ) }{k_BT} \\right ) ", "2467a6f23482a3998ece4c914a59ab11": "\\mathrm{height}(u)=k", "2467d1392f5267532e6bb77af613e63e": "w_t = y_t -d/c", "2467d79a942f59706793caa941da8097": "Y_{t} - (1 + b )cY_{t-1} + b cY_{t-2} = (C_{0} + I_{0})", "24684994d7738b95e35337b8b794f313": " f '(x)", "24684da853eb9bfe9551a77af8735e3d": " \\partial_{\\overline{z}} f(z) = \\mu(z) \\partial_z f(z)", "24688c2998c234c1fb1832a93e9543df": "\\left| S_k - L \\right| \\leq a_{k+1}\\!", "2468abfd6f9b307b299480117c9357d9": "\\ C_S = kC_o(1-f_S)^{k - 1}", "2468be4097b6976b886c237da395ffc2": "\n\\Omega =\n\\begin{pmatrix}\n{\\mathbf 0} & - {\\mathbf l} \\\\\n{\\mathbf l} & {\\mathbf 0}\n\\end{pmatrix}\n\\,, \\qquad \n\\beta = \n\\begin{pmatrix}\n{\\mathbf l} & {\\mathbf 0} \\\\\n{\\mathbf 0} & - {\\mathbf l} \n\\end{pmatrix}\n\\,, \\qquad \n{\\mathbf l} = \n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1\n\\end{pmatrix}\n\\,.", "2468c6a240f4bca092e6b72fab1655a2": "\\displaystyle u'(x)>0\\mbox{ }\\forall\\mbox{ }x\\in \\left(x_0,x_1\\right]", "2468e2042f49e1dbaa66c9d3c000614c": "\\lim_{h\\to 0}\\frac{f(a+h)-f(a) - f'(a)\\cdot h}{h} = 0,", "2468f34f7bcfc04b8dc4e8b02e82d3d8": "x \\vee y = y \\wedge x", "246a2ad6ed979571de053aaa546b0bc9": " f(x,\\lambda)=0 \\, ", "246a45d39169e57be898a4f7c0266da8": "P(A_n^\\epsilon)>1-\\epsilon", "246a5a9681223e6f86c751d2ccc75e8d": "\\sum_{n=-\\infty}^{+\\infty} {E_n}e^{j({\\omega}t-{k_n}z)}", "246a7af4b0002703c1e2f2893c4f0749": "(R_i)_{i \\in I}", "246ac82fd099529153b2f77dcbb34aa0": "\\textstyle \\text{Var}[{N}(B)]", "246ad60b77c59cc8d50ba80f069880b6": " f \\left(K_\\alpha \\right) = \\left(3.29 \\times 10^{15}\\right) \\times 3/4 \\times \\left(Z - 1\\right)^2", "246b422897f30671287899e9b10bae05": "\nK_{x\\rightarrow y}(T) = \\sum_{x(t)} \\prod_t K_{x(t)x(t+1)}\n\\,", "246b56c160a1cf80eac88068e2c80052": "T_J > 3", "246b9d3adbbf81c7a7e6de83e021f1f8": " \\{x_1, \\dots, x_n\\} ", "246baba6b98e66c88c532355f0b442ca": " (-0.50, 1.25);", "246bd5f89d1e5369386ad10faa3a7514": "f=v/\\lambda", "246bf7fa6f68f91c8fbacbaafd71292a": "\\frac{\\partial p}{\\partial r}=\\frac{\\rho V^2}{L}, ", "246bfa1954a3377f0870eca10268dc94": "f_\\bullet", "246c34c550a7977ad4c582df9b57f0b6": "Q_k", "246ccc14b1b11a00fb8d4970dd1cad22": "\\mathbf{j}_s", "246cd0290a6d2a5035216c453a5736ce": "R(X, L) = \\bigoplus_{d=0}^\\infty H^0(X, L^d).", "246cda3e2d79544b723caf66a11c140c": "P_2=(2:\\sqrt{17}:1)", "246d69873133ce38e4f388cfe0366489": "\\scriptstyle\\sqrt n", "246db1bb1d0e97de1a270dd9ad493c22": "(-\\alpha, +\\alpha)", "246e0e4d80939d910c75df8b16e682ec": "\\mathbb{P}^x\\{X_{\\tau-}-X_\\tau 0 ", "248fac604a7b2866118d8ad60aa44cb5": "V_\\mathrm{rms}=\\sqrt{\\frac{1}{T} \\int_0^{T}{v^2(t) dt}}.", "248fe2b4c32922f410d2c4254b78ab9e": "A_1(\\omega)", "248ff36395cbf1144dcd1f862261999d": "\n \\|\\varphi\\|_A = \\sup_{x\\in A} |\\varphi(x)|, \n ", "24907c8f307e5dee7921fac642e00f3b": "\nP(V) = \\frac{K_0}{K_0'} \\left[\\left(\\frac{V}{V_0}\\right)^{-K_0'} - 1\\right] \\,.\n", "24908481f44c18214b59a7a808b67842": " m \\in (-\\infty,\\infty) ", "2491378d33154a96a2c9af449897f8b5": "(m)", "24913b81a5bfa4e88180f16b257b19f1": " Q = k_f - k_i ", "24916ff2fc7f6d6ab6b27242c06383ff": "(2x + 7i)(2x - 7i)", "249192a9e27a59ea5fa7eaaa62d2430b": "\n\\begin{matrix}\n\\sum_{i=0}^t \\binom{n}{i} < |C| \\\\\n\\end{matrix}\n", "2491ae3fc417527afa536d03fe9d89eb": " \\begin{align}\n\\mathbf{A} & = A_x \\mathbf{e}_x + A_y \\mathbf{e}_y + A_z \\mathbf{e}_z \\\\\n& = A_x \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix} +\nA_y \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix} +\nA_z \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} \\\\\n& = \\begin{pmatrix} A_x \\\\ 0 \\\\ 0 \\end{pmatrix} +\n\\begin{pmatrix} 0 \\\\ A_y \\\\ 0 \\end{pmatrix} +\n\\begin{pmatrix} 0 \\\\ 0 \\\\ A_z \\end{pmatrix} \\\\\n& = \\begin{pmatrix}\nA_x \\\\\nA_y \\\\\nA_z \\\\\n\\end{pmatrix}\n\\end{align}", "2491bc398e209b15f0fdb9e2c5ba9b5d": "A=(1-\\kappa)\\left[\\begin{array}{ c c } 1 & 0 \\\\ 0 & 1 \\end{array}\\right]-\\gamma\\left[\\begin{array}{ c c } \\cos 2\\phi & \\sin 2\\phi \\\\ \\sin 2\\phi & -\\cos 2\\phi \\end{array}\\right]", "2491dcb2cb7af403d6e2919067ee5b67": "x \\ \\stackrel{\\mathrm{def}}{=}\\ q e^{-D(t)}", "2491edac6a2e000320344a81ed495041": "G'/N'", "2491f248be5dc4ce53c7e4c71ef5f2c5": "G_Q", "24920da9de402da836513cd42905fd2c": "1 / 0.02 = 0.02x / 0.02", "249242eb3951489052d4817a186bc3d7": "i_c", "2492a2642e37b60a57d624cd2a7b0481": "\\frac{\\mathbf{v}_{k+1/2} - \\mathbf{v}_{k-1/2}}{\\Delta t} = \\frac{q}{m} \\left( \\mathbf{E}_k + \\frac{\\mathbf{v}_{k+1/2} + \\mathbf{v}_{k-1/2}}{2} \\times \\mathbf{B}_{k} \\right),", "2492d92018bbf90beebdad43cfd94db0": "C^{SL}", "2493b6785c2b61271276d9d95c922e45": "\\lambda_m \\ne \\lambda_n", "2493de0b8b5a0cafed15dec56532e0e5": "g(t)=\\int_a^b K(t,s)f(s)\\,\\mathrm{d}s", "24945af137fce862851d363c5b3f815e": "\\langle Y(m), m\\rangle = 0, \\qquad \\forall m\\in \\mathbf{S}^2.", "249475e5352e7697fed5f70afe08b16c": "\\phi^{A,\\bar{x},\\bar{a}}=\\{(x_1,\\dots,x_n)\\in A^n\\colon A\\models\\phi[\\bar{x},\\bar{a}]\\}", "24953a2ff725454733f859ab05e22e89": "x> 1", "24953cb957250615795fde69e9eaa485": "w(n)=w(n-i_1)w(n-i_2)...w(n-i_k) \\text{ for } n \\ge \\max(i_1, ... i_k) \\, .", "24955ccde15784b0553f7b017688f522": "\\% \\mbox{ change in } y = \\frac{y_2 - y_1}{(y_2 + y_1)/2}.", "2495b620f9fe1e74ee63a12dbd0505af": "\\rho=\\frac{M(G)}{M(G\\setminus e)}", "249618d56fde8a84831bcba20d0335d1": "\\mathbf{y}_i^{\\rm T}", "24965242343cfdc15070483df7dd5ba0": "(I_2-I_3)/3", "24967e4c685321c2de26d045eafe6eac": "I_n[w] = \\frac{1}{n}\\sum_{i = 1}^nV(\\langle w,x_i \\rangle, y_i) \\ .", "249686a76141115e53bddd9d3b343cac": "\nh_t(x,x) = \\frac{1}{4\\pi t} + \\frac{s(x)}{12\\pi} + O(t).\n", "2496ad327ba72d3eba715f38ad0a1c9f": "T_{F}", "2496e19090045bb9934209147bc15ef7": "q \\to 1", "2496ecc23ba1b56e9e24d721c68edab4": "\\frac {c_{metal}}{c_{air}} = \\frac {f \\lambda_{metal}}{f \\lambda_{air}} = \\frac {\\lambda_{metal}}{\\lambda_{air}} = \\frac {L}{d}\\,", "2496f5b9e347f307d1b0968e0814e7f8": "\nP_{D-} + \\frac{1}{2}\\rho v_{D,~z}^2 = P_{\\infty} + \\frac{1}{2}\\rho v_{w,~z}^2 = P_{D-} + \\frac{1}{2}\\rho (v_{\\infty}(1 - a))^2\n", "24971138d67f6c0ea3e617ee86c3615f": "\\cos(a)\\cos(b) = [\\cos(a + b) + \\cos(a - b)]/2", "249716552568c05f7088d49aeccc2076": "\\left|\\frac{a}{b}\\right| = \\frac{|a|}{|b|}\\ ", "2497400a9ba7423587c3556b3f98913a": "e^{-\\frac{\\Omega}{k T}} = \\sum_{N_1 = 0}^{\\infty} \\ldots \\sum_{N_s = 0}^{\\infty} \\int \\ldots \\int \\frac{1}{h^n C} e^{\\frac{\\mu_1 N_1 + \\ldots + \\mu_s N_s - E}{k T}} \\, dp_1 \\ldots dq_n ", "2497a006e5d4ffa2c1f8cd28d6d95968": "L \\in \\Sigma_2 - \\mathsf{SIZE}(n^k)", "249808a73c8d68bfebb903d6671d9f23": " A =\n \\begin{bmatrix} 6 & 5 & 0 \\\\\n 5 & 1 & 4 \\\\\n 0 & 4 & 3 \\\\\n \\end{bmatrix}", "2498173b86ccb56a03f008bcaeccb27d": " \\delta_{ext}(q, x)=((\\ldots,(s_i', t_{si}', t_{ei}'), \\ldots),b) ", "249885905d0e164b83a0e8e50faf3d72": "\\mathrm{A + B \\longrightarrow AB}", "24989ee522c37a12fb2cf2fbf5b21d6f": "m_1...n", "24991f14f1c5bb692e681b48cf98c3b5": "z^{\\star} = a - bh", "24998f188b7fe55cfc8e1a74b15f4491": "P = I^2R \\,", "2499930d8c19771316386635407f0685": "\\sqrt{\\Delta} = 1.786737578486707 ", "2499ea491442016d7e606662980fd27f": " y = Ce^{0.85t} ", "2499f36a8dfdc34b857a32c5a89bff7d": "\\phi^{+}=0", "249a19de06ae2179da1a1edbd07c3585": "17.93^{-n}", "249a6f7a124c5e629a59e212ecf9fc3e": "\n\\varphi _{k-p}^{n + 1} - \\varphi _{k-p-1}^{n + 1} = {\\gamma _p \\left( {\\varphi _{k}^n - \\varphi _{k - 1}^n } \\right)} < 0 , \\quad \\quad ( 7)", "249accf4fffcd97ed6fad6da05ec2a6a": "\\dim \\operatorname{ker}(AB) \\le \\dim \\operatorname{ker}(A) + \\dim \\operatorname{ker}(B)", "249acd69b59226e13718e66b07941082": "\\tau_{23},\\tau_{12},\\tau_{31}", "249b0568ea087a98d6ea67843bea5b1f": "\\omega_r = \\sqrt { \\alpha^2 - {\\omega_0}^2 }", "249b09615be83fe41f1928ffdc946984": "A + B + A... \\rightarrow ABA ...", "249b4c584ee566c0dbe934080d723381": "\\left\\{E_i\\right\\}_{i \\in \\mathbf{N}}", "249c1ed19cc4cfd4387c799673b5dd60": "i=1,\\cdots,n\\,\\!", "249c2d6e4dec0a904576fde16bf34822": "e_i \\in B", "249c4f25cdc5525e3ad7de912960294b": "\n\\frac{\\mbox{d}}{\\mbox{d}t}\\left[\\delta(x-vt)m\\frac{\\mbox{d}w(vt,t)}{\\mbox{d}t}\\right]=-\\delta^\\prime(x-vt)mv\\frac{\\mbox{d}w(vt,t)}{\\mbox{d}t}+\\delta(x-vt)m\\frac{\\mbox{d}^2w(vt,t)}{\\mbox{d}t^2}\\ .\n", "249c7e060ed36f85f4c29c575da881df": "f_{U}(x)=1\\,", "249cdbfbd582d76f1cae60545bb42535": "6/\\pi^2", "249d61a1c9812ef9aa72fccafe2c3c5c": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 1}{4 \\choose 3} \\end{matrix}", "249da23ffe87a313ab1ed0fb22cbabae": "\\vec{\\Omega}", "249da942355f2a9352ba5ea6e9c5f1a7": " \\operatorname{cl}(\\varnothing) = \\varnothing \\! ", "249e18c2d3d0f24ed8c3d9369218bf2f": "\\frac{dG(s)H(s)}{ds} = 0\\text{ or }\\frac{d\\overline{GH}(z)}{dz} = 0", "249e6aa25aac245d991375fa480857d0": "\\int_a^b f(x)\\, dx", "249e78ff656ca5fac54b5641b976abd2": "\\tan^2 + 1", "249e7bc44bca05c93afdd803e9e30736": "E\\in V^n", "249eee7e901b96a04c07fe271c64419e": " \\overline \\varepsilon = \\varepsilon(\\underline E,\\overline P) = \\frac{\\overline P}{\\underline EA} ", "249f05209600c740604e68878055057f": "(x,u,w) \\ \\stackrel{\\mathrm{def}}{=}\\ (x^{i},u^{\\alpha},w_{i}^{\\alpha})\\,", "249f4c688e26cebec24dda77c94052ca": "r\\left\\|A\\right\\|_\\alpha\\leq\\left\\|A\\right\\|_\\beta\\leq s\\left\\|A\\right\\|_\\alpha", "249f7f11c19dfd71a4ac13915b1b1a07": "\\mathbb{E}\\bigl[|X|^p1_{\\{U=0\\}}\\bigr]\n= \\mathbb{E}\\bigl[1_{\\{U=0\\}}\\underbrace{\\mathbb{E}\\bigl[|X|^p\\big|\\,\\mathcal{G}\\bigr]}_{=\\,U^p}\\bigr]=0,", "24a11bee8257982e773660bdb81269f2": "k^h", "24a1366c88a506ebe02c15977d9087aa": "\\displaystyle{|f(x)|\\le C \\left(\\int |\\widehat{f}(t)|^2 (1+t^2)^s\\, dt\\right)^{1\\over 2}= C((I+Q^2)^s\\widehat{f},\\widehat{f})^{1\\over 2}\\le C^\\prime \\|\\widehat{f}\\|_{(s)} =C^\\prime \\|f\\|_{(s)}.}", "24a15b1db2548d1b0fcb5b4635f9c4f2": " K = {e^{-\\Delta G/RT}}", "24a15eedee98a63ecd810b75a2485668": "A(\\boldsymbol\\theta)", "24a17d2e2b000c187e42eafb8faf1059": "\\rho = \\rho(x)", "24a1dd1f7257a670e1ade13ed3b5911f": "\\ell = I-\\frac{1}{k} A", "24a1e5c13bb0e805cdd787d0d9b273fb": "V^* : H \\rightarrow K", "24a30534e1fe74568cb9e8e02833e17d": "R_{ijk}^l", "24a30ae32a63fdf7ef771bba6df75b18": "\\lim_{P\\to\\infty}\\frac{1}{\\frac{1-\\alpha}{P}+\\alpha} = \\frac{1}{\\alpha}", "24a357ee515b42f211bac808553e0691": "\n\\rho_0 = \\frac{v_3^2}{\\sqrt{v_1^2 + v_3^2} \\sqrt{v_2^2 + v_3^2}}.\n", "24a3e44064471d8b817d8cd5e82bc327": "X_1^n=(X_1,\\ldots,X_n)", "24a443043ad5cfe44f3e8bf46d900075": "{\\Bbb P} H^0(L^N)", "24a472a6b170218c5ec1d2c4104784ee": "\\log_2(x^2) = 2 \\log_2 (x). \\,", "24a4a511e6dd27921fa4b30560e20ad5": "D = \\frac{|(\\mathbf{r}_1 - \\mathbf{r}_0) \\cdot \\mathbf{n}|}{|\\mathbf{n}|} = \\frac{|\\mathbf{r}_1\\cdot \\mathbf{n} - \\mathbf{r}_0 \\cdot \\mathbf{n}|}{|\\mathbf{n}|} = \\frac{|\\mathbf{r}_1\\cdot \\mathbf{n} + a_0|}{|\\mathbf{n}|} = \\frac{|a_1x_{11} + a_2x_{21} + \\dots + a_Nx_{N1} + a_0|}{\\sqrt{a_1^2 + a_2^2 + \\dots + a_N^2}}", "24a4b0b887e1b2dbd2331de88598a6d3": "\\sqrt{(350^2 - 50^2)/100} = c", "24a4d5e671be5bb7026ad86a68d14220": "l\\,\\!", "24a50daac817f05c417d95eb1594d01c": " q_k \\leftarrow Aq_{k-1} \\,", "24a515992482e3b61870c98e7a3d11c7": " \\{ b^n a^m b^{2n} : n \\ge 0, m \\ge 0 \\} ", "24a5b2f2acd079f75981e6d679984669": "\\scriptstyle{AB=Rc-Rt=h(Y,X)}", "24a5b831b2a036aae4037979b2684412": "a \\times b \\times c", "24a5d8fdaf168d10d65281b502d81b77": "(E,\\rho)", "24a616dd6d3f439d19fc73f8c5730c8b": " E_{1ss} = y_{1ss} + \\frac{q_1^2}{2gy_{1ss}^2} ", "24a6423c7547bf00b2965d8006744abb": "f^{(n)}(c)", "24a715c68f0e7aa01f37ea50d5f5097e": "\\,\\mbox{R}(z, dt) = \\mbox{T}_x (-y dt) \\mbox{T}_y(x dt)", "24a78ab07f629474b1adfa6d8a28c2b6": "y_{tt}=c^2y_{xx}", "24a79614c9dea1f695109be16950b771": "p^k_{ij}", "24a79bd440c8de8936305ba1d446559b": "\\frac{\\text{d}C_1}{\\text{d}t} = \\text{k}_{1(1)} {^0_2}S E - (\\text{k}_{2(1)} + \\text{k}_{3(1)}) C_1", "24a7a95f596acb7970f244c548f9243d": " \\mathbf{e} \\,", "24a8042b1facc2ce188626f45ea0c332": "E_\\mathrm{nonbonded} = \\sum_{i>j} f_{ij} \\left(\n \\frac {A_{ij}}{r_{ij}^{12}} - \\frac {C_{ij}}{r_{ij}^6}\n + \\frac {q_iq_j e^2}{4\\pi\\epsilon_0 r_{ij}} \\right) ", "24a8315094adbb2a7669d46774e04007": "\\varepsilon_x = \\frac{\\partial u_x}{\\partial x}\\,\\!", "24a855e468f4680ca98638a05ae8405c": "\n \\mathrm{E}[\\,G_F(t_1)G_F(t_2)\\,] = F(t_1\\wedge t_2) - F(t_1)F(t_2).\n ", "24a88d099bf4d379d88f05eefe00577a": "T_{cold}", "24a8a9e4491a57822186f76fea9d33f0": "F_{total} = F_1 + \\frac{F_2-1}{G_1} + \\frac{F_3-1}{G_1 G_2} + \\frac{F_4-1}{G_1 G_2 G_3} + ... + \\frac{F_n - 1}{G_1 G_2 ... G_{n-1}}", "24a8fec9119c2ada2e64a219eaa84200": "e^{i a(\\hat{n} \\cdot \\vec{\\sigma})} = I\\cos{a} + i (\\hat{n} \\cdot \\vec{\\sigma}) \\sin{a} \\,", "24a9cba1145c9db7a6ff6ddb6d04f4a7": "\\mbox{AC} = \\bigcup_{i \\geq 0} \\mbox{AC}^i", "24a9eb07c65a72ffea1be24501ee8ee0": "(C \\otimes y)_j\\ = Y_j= \\sum _{i=-(m-1)/2} ^{i=(m-1)/2}C_i\\, y_{j+i}\\qquad \\frac{m+1}{2} \\le j \\le n-\\frac{m-1}{2}", "24aa249550e0e3ff769e970ee6935284": "\\kappa_{\\mathit{ri}} = \\kappa_i / \\kappa", "24aa669497bfafeeefd673619c4092b3": "u \\colon C \\to J(C), u(p) = \\left( \\int_{p_0}^p \\omega_1, \\dots, \\int_{p_0}^p \\omega_g\\right) \\bmod \\Lambda.", "24aa964ae15fb4eb6571e1409ccd553c": "(x-az)^2+(y-bz)^2 - r^2z^2 = 0,", "24aa966d5cc7017e292a62f7346b33b8": "H(\\varepsilon)", "24aaaef7c95258440ec788ae63e3a9f5": "L_c", "24aacc78f1b57600cf0ce46ef027d918": "\n\\mathbf{j=}\\frac{\\mathbf{v}}{\\mathrm{v}},\\qquad \\mathbf{v}=\\alpha D_{\\alpha\n}|\\mathbf{p}^{2}|^{\\frac{\\alpha }{2}-1}\\mathbf{p,} \n", "24aafc6683975951198e6f23019e9487": " n(n-1)f[n+1] + 3nf[n+2] -4f[n+3] -3nf[n+1] -f[n+2]+ 2f[n] = 0", "24aafe7bd251499482422dfe3e244486": " \\mathrm{nDCG_{p}} = \\frac{DCG_{p}}{IDCG_{p}} ", "24ab0b3c20d6ea64b5c6c9233b77142a": "S(q) = 1", "24ab5ae3904669f68999ec38eacafe27": "P=\\left[\\frac{u_i^* u_i - v_j^* v_j}{1-f_i^* f_j}\\right]^{n-1}_{i,j=0}", "24abb4502a39fc5003509e54f618ff18": "I^{*}", "24abdbbf72ebeef64a1626b7c9a1e27e": "1.34 \\times \\sqrt{\\mbox{LWL}}", "24ac30a2296609d4fcf5c5d6d4e9f2d1": " 1-s \\rho = s \\, ", "24ac87f341319a196ff395579e2ffb8d": "{{V}_{bi}}=\\frac{kT}{q}\\ln \\left( \\frac{{{N}_{A}}{{N}_{D}}}{{{p}_{0}}{{n}_{0}}} \\right)", "24ad323f18c8a9f7f4949cb23d9a2c58": "\\sigma=\\nabla \\times \\nabla \\times \\Phi ", "24ad65ff63b2e8e09ec3f313600e1b55": "\n\\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & 1 & -1 & -1\\\\\n1 & -1 & 0 & 0\\\\\n0 & 0 & 1 & -1\n\\end{bmatrix}\n", "24ad896dccf6fd94eb18478f521558ff": "\n S^{ij} = g^{ik}~S_k^{~j} = g^{jk}~S^i_{~k} = g^{ik}~g^{jl}~S_{kl}\n ", "24adaa6bd51b1b184e6ae5f5a7b57496": "\\displaystyle{F_m(t,x)={t^m\\over m!} \\cdot \\psi\\left({t\\over \\varepsilon_m}\\right)\\cdot f_m(x),}", "24adbe30df874788f8cb8a05201eff45": "H=H_0 + gV", "24adf935c53fd37ad66fd6f27cd7d5ad": " \\mathcal{O}_k / \\mathfrak{p}", "24ae1144aee8e5a8f3c7178e5b037e29": "\\scriptstyle =(1.2\\pm2.2)\\times10^{-43}", "24ae25188fa7a1403baf9d625166cf6f": " | f |_{C^{0,\\alpha}} = \\sup_{x \\neq y \\in \\Omega} \\frac{| f(x) - f(y) |}{|x-y|^\\alpha}, ", "24ae4b8be5bb0b0fbbd227549427d439": "n\\times n\\,", "24ae7848ddf8493dff50fcd63721a21d": "\\begin{bmatrix}1&B_{0;1,2} \\\\ B_{0;2,1}&1\\end{bmatrix}\\begin{bmatrix}y_{1,t} \\\\ y_{2,t}\\end{bmatrix} = \\begin{bmatrix}c_{0;1} \\\\ c_{0;2}\\end{bmatrix} + \\begin{bmatrix}B_{1;1,1}&B_{1;1,2} \\\\ B_{1;2,1}&B_{1;2,2}\\end{bmatrix}\\begin{bmatrix}y_{1,t-1} \\\\ y_{2,t-1}\\end{bmatrix} + \\begin{bmatrix}\\epsilon_{1,t} \\\\ \\epsilon_{2,t}\\end{bmatrix},", "24aeb1bbe0a85f8165835190b05c716f": "\\gamma=\\frac{1}{ \\sqrt[]{1 -\\frac{v^2}{c^2}} } ", "24aeb2a72a3293beb348aa59ebaac536": "\\mathrm{Re} \\to \\infty", "24aef4304dfaee6a7a5eaac7491e9b6f": "[y_j,y_{j+1}]", "24af5322ad0079f3db48eef6e8190130": "P_AO_2=F_iO_2(P_{atm}-P_{H_2O})-\\frac{P_aCO_2}{0.8}", "24aff3c26c898ef818e63c20cbe93d65": "\\bar{k}_{\\alpha} (s)=\\frac{s\\bar{\\psi}_{\\alpha}(s)}{1-\\bar{\\psi}_{\\alpha}(s)}", "24b0c35c03f00eeb7945be343bd8827e": "\\alpha \\in K ", "24b0c67ec78db0f84243cfef40f3431b": "\\mathbf{F_L}", "24b0c99d5a15453372ae301d7d172706": " A = \\mathrm{diag}(z_1,\\dots,z_n)\n +\\begin{pmatrix}1\\\\\\vdots\\\\1\\end{pmatrix}\\cdot\\left(w_1,\\dots,w_n\\right).\n", "24b0cbe10c7ac27d956405e412924d9f": " 0=\\lambda_0\\le\\lambda_1\\le\\lambda_2\\cdots, ", "24b11757e7c581a4eb0a9b26789bb0d7": "w e^w =\\frac{I_SR}{nV_T} e^{(V_s+I_sR)/(nV_T)} ", "24b12535ad9f3951b69180dba8ed91a0": "\\Theta(n+z)", "24b137769c59638a8b6dc6e8a3387e93": "\\textrm{LWL}", "24b14c8e1f1a9d5509d94061b588ef03": "\n1 = {1 \\over \\Gamma(s)} \\int_0^\\infty e^{-t} \\,t^{s-1} \\,\\mathrm{d}t \\qquad (\\textrm{Re}(s) > 0) \\,,\n", "24b1b2e995a24cdff8e24baf095affa5": " \\operatorname*{\\arg\\min}_{\\hat{x}} \\max_{x \\in Q} \\left\\| x - \\hat x \\right\\|^2. ", "24b1d4672b02ee071d7754ef81b072b0": "R(y)", "24b246725cf6911efcb5021c22a08667": "\\gamma_\\nu", "24b28756001eed7b00fb3b1b103a6972": " \\frac{\\partial u}{\\partial t} + \\nabla \\cdot \\left( \\boldsymbol{v} u - D\\nabla u \\right) = f, ", "24b2aa616e53ba1f76754338d671eb7e": "E(|x-\\mu|)=b\\,", "24b2fdde8920609c4fd71f9885e4eb5e": "T = T_0 - \\alpha y \\,,", "24b31fb2a6e90ebc37ce5fc98c425733": " C^*_{(-)}= C_{(-)} ", "24b32ecb6235989bf18164742203fe2b": "\\pi_3 E = \\pi_5 E = \\cdots", "24b396886f16889928536fa6e4656bd8": " \\mathbb E_{{Z_1,\\cdots,Z_{k-1}}}{\\lVert x_k-x \\rVert^2} \\leq \\left(1-\\mathbb E_{{Z_1,\\cdots,Z_{k-1}}}\\left|\\left\\langle\\frac{x_{k-1}-x}{\\lVert x_{k-1}-x \\rVert},Z_k\\right\\rangle\\right|^2\\right){\\lVert x_{k-1}-x \\rVert^2}.", "24b39e635daa9bd6215d812a91fee642": " p^{k} > (p^{k}-1) = (p-1)(p^{0} + p^{1} + ... + p^{k-1}) \\ge \\sigma_{1}(w)(p^{0} + p^{1} + ... + p^{k-1}) ", "24b3a8d369da4efdac42fdb0b50059a2": "d(X,Y) = \\mu(X\\,\\triangle\\,Y)", "24b3bfc531bf7328c942fbfa7c7a9e3c": "Y \\sim \\Gamma(\\beta,1)\\,", "24b3f3219178e0c85f4835adfe4b76ba": "G_s(L/K)=\\{\\sigma\\in G\\,:\\,v_L(\\sigma a-a)\\geq s+1 \\text{ for all }a\\in\\mathcal{O}\\}.", "24b4140897713f6edb65a504b638e318": "{\\mathbb R}^3", "24b425f1edb88d255ba074dbb28b9f56": "\\pi_1, \\pi_2, \\ldots, \\pi_n", "24b44f491a25097b86f06fd8bb94af27": "\\pi_3 = \\frac{a k_b T \\varepsilon_0}{q^2}", "24b493c54f79ba1b99db14a4c0df7da2": "\\Delta\\lambda'=\\lambda_2'-\\lambda_1'", "24b4b07899c733c26e5d57f69ef9407d": "f(X) = \\sum_{n\\ge 0} a_n X^n.", "24b4b53fc3b78062cec9c41c82ad37f2": "O=\\{ X\\in 2^\\omega | \\exists n (X(n)=1) \\} ", "24b4fa5e6ee7a1f6bb5083ed874a0ef5": "\\frac{d [P]}{d t} = k_\\mathrm{cat}[ES]", "24b505136a199a9f9d6089be89f2db23": "(a_1, b_1) \\circ (a_2, b_2) = (a_1 \\ast a_2, b_1 \\cdot b_2)", "24b50a5d8711145c1b7e851201a4f3d9": "x(t-T)", "24b5247734788329276b614ba95a1612": "\\widehat{T}", "24b57c1f4b2ca24a4902229f7002a7cc": "\\operatorname{CAT}(\\ell)", "24b58c4a77c90cb84f2ff9bdaec29d84": "| a_\\max | \\Delta t < \\Delta x/2 \\, ", "24b5be5ad64e63aa50bca7fa046f399a": "\\omega(r_t)=\\left[f(r_t|M_t=m^1);\\dots;f(r_t|M_t=m^d)\\right].", "24b63acac7a9337befca09cfe12774c6": " \\|x+y\\| \\geq \\|x\\| + \\|y\\| \\; \\forall x, y \\in V", "24b6c570798307c10a1c4555649ab6a5": "\\frac{d\\mathbf{y}}{dx} = \\left(\\frac{dy_1}{dx}, \\frac{dy_2}{dx}, \\ldots, \\frac{dy_n}{dx}\\right) ", "24b6e9480fb7028e53033cfea3ff4682": "\\dbinom{n}{m}(n-m)!", "24b7953c35aa07650844de907ec18c0c": "J(u):=\\int_\\Omega |\\nabla u|^2\\, dx", "24b7c69dff795ac056de9fa8a759894b": "\\mathrm{proj}_0\\,(\\mathbf{v}) := 0", "24b7e5d884c263515f87bb2a994193d9": "\\textbf{y}(t) = \\textbf{G}(\\infty)\\textbf{u}(t)", "24b8179390cbe81d12c66d96cf39f2c5": "_2\\text{H}_2", "24b821dcf1efc2ea1794eb8069959b71": "n(r)= br^{-3/2}\\exp{(-cr)}/V_\\xi", "24b8a39ea1e03e97811c6d4a1bb665b8": "\\lim_{s\\rightarrow 1+}P(X\\in A)\\,", "24b8cbe5a7bf12b2db849f10244222f1": "\\textstyle (A_1,A_2,\\dots) ", "24b93cff3eb3737ebabe2e2633b1553a": "\\deg(u') \\leq g", "24b946056d8467e2d0af4355560e30ee": " u^*=Qu ", "24b96ccdadbe944372ee4820efef5178": " \\begin{align}\nx_i = \\frac{\n {\\displaystyle\n 1 + \\sum\\limits_{j=1}^{i-1}\\prod\\limits_{k=1}^{j}\\gamma_k } }\n { {\\displaystyle\n 1 + \\sum\\limits_{j=1}^{N-1}\\prod\\limits_{k=1}^{j}\\gamma_k } }\n\\qquad \\text{(1)}\n\\end{align}", "24b970e0f23fe457bd12aa39c1c9654c": "\\frac{P_{Rx}}{P_{Tx}}\\;=\\;\\frac{c_0F_g}{d^{\\gamma}} ", "24b9b87b9227d78edc8830883a21fc2a": "\\Big( (\\mathcal{M}, s) \\models EG\\phi \\Big) \\Leftrightarrow \\Big( \\exists \\langle s_1 \\rightarrow s_2 \\rightarrow \\ldots \\rangle (s=s_1) \\forall i \\big( (\\mathcal{M}, s_i) \\models \\phi \\big) \\Big)", "24b9f46a106a80650c22f6207104dd48": " B = [b_{ij}] ", "24b9f5ff78481c5490044dd3fd1cc20d": "y_t = \\alpha + X_t \\beta+\\varepsilon_t,\\,", "24ba8ff1842c2cab6cb6a26be956d77f": "E_{x}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\\frac{-jk_{xo}k_{z}}{\\omega \\varepsilon _{o}\\varepsilon_{r} }(A \\ e^{jk_{x\\varepsilon }w}+B \\ e^{-jk_{x\\varepsilon }w})+\\frac{m\\pi }{a}(C \\ e^{jk_{x\\varepsilon }w}+D \\ e^{-jk_{x\\varepsilon }w})]e^{jk_{xo}(x+w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ \\ (45) ", "24baa138ce2df06c511010f55149feba": "L_1=\\ln(-x)", "24bad83487ba662d25dc953467c2907d": "r_{SOI}", "24bb0621fafd46721e3e4a7c5eced3ee": "f(x,y) = \\zeta(\\alpha(x,y),\\beta(x,y),\\gamma(x,y)) = \\zeta(\\alpha,\\beta,\\gamma) = e^\\alpha[\\sin (3\\beta) - \\cos (2\\gamma)] \\,.", "24bb2719a34aa1245dcc03508501a913": "u(x,t)\\,", "24bb4f0638626718b281d971771cf7a3": "Y\\stackrel{\\mathrm{d}}{=}\\sum_{k=1}^n Y_{n,k}, \\, ", "24bb7c5899d7b92fba748392f46e3d98": "H^2 \\times R", "24bb978698041e0766e5222ec0ea2d4f": "\\left(1 - \\frac{t}{\\lambda}\\right)^{-1}\\,\\exp \\{ \\mu t + \\frac{1}{2}\\sigma^2 t^2 \\}", "24bbbf66a1261ac1164939a97a6dc6a8": "Forming time = [L+n(d)]/V", "24bbd0b3e4d56d260993cba2f4754f06": "~A=A_0\\frac{U+s}{1+p+s}~", "24bc04db033f74eb5e39c7b3e93d0fea": "\\mathcal{G}_n", "24bc05d66a2dfe8527271f192ed842ca": " U_{nit} = \\bar{\\beta} X_{nit} + (\\sigma \\eta_{n} X_{nit} + \\varepsilon_{nit}) ", "24bc3c8968c24096637441060dd9e1db": "\\exp\\left({-i\\omega t}\\right) = \\cos(\\omega t) - i \\sin(\\omega t),", "24bc4eab5e7c9cd9a0f2729889dd04b1": "\\mathfrak{gl}_n(F)", "24bc8101c171957a7afa41b517f20d63": "U_\\nu~d\\nu = \\frac{8\\pi h\\nu^3 }{c^3}~\\frac{1}{e^{h\\nu/kT}-1}~d\\nu ", "24bc8b45cc9515c2d88487e9f0b11be0": "y \\in \\{0, 1\\}^k", "24bcfd8fc713e9ef5503a2f39bcb070b": "N_n = \\frac{q^n}n + O\\left(\\frac{q^{n/2}}{n}\\right).", "24bd13ca8544708ecf12e110c2f59713": "\\mathrm{VIF}= \\frac{1}{1-R^2_i}", "24bd55218e20f5d425c2514396c9475c": "m > \\sqrt[4]{q}", "24be0ace00b62ac1d1a34be3a56c95ca": "O(mn \\cdot \\log(mn))", "24be585e8cf4999121c4b9a8596cc1fa": " L*P*(1-R+R*R-R*R*R+\\cdots) = 1", "24be9b8f1355f2bf5bfecc5e34a81517": "(m v^2/2)", "24bece2ffba23c79e598e90b8b2b4a5a": "\\begin{align}\n (Q_1 Q_2)^T (Q_1 Q_2) &{}= Q_2^T (Q_1^T Q_1) Q_2 = I \\\\\n \\det (Q_1 Q_2) &{}= (\\det Q_1) (\\det Q_2) = +1.\n\\end{align}", "24bed801f103c9657f661c9409cc923b": "-x\\,", "24bf0202fbf5e59462d2be6a194bbd7f": " c_i = \\frac{\\left(p^2+\\left(1-e^2\\right) z^2 \\kappa_i ^2\\right)^{3/2}}{a e^2} .", "24bf8825218e19ce3174a9b53bb255fc": "\\frac{\\pi}{4} = 4 \\arctan \\frac{1}{5} - \\arctan \\frac{1}{239}", "24c009e3c295ddbb6dd02c8315c0a30f": "(\\mathfrak{k},\\mathfrak{p})", "24c0619297000e3bae5daa73eb0b9fef": "\\mathrm{ENOB} = \\frac{\\mathrm{SINAD}-1.76}{6.02}", "24c0659911c578501609d7fc0c76f71d": "\\nabla \\cdot \\boldsymbol{\\mathsf{T}}", "24c06e851945e06335d239267288c10e": "K_{TE}=\\frac{X_m}{X_s+X_m}", "24c078a13d168b16b0e02a719bae63f9": "\\pi_0(X),", "24c1362cd48718caf806bc95ea379b0e": "\\frac{d \\tau}{d t} = \\frac{1}{\\gamma(\\mathbf{v})}", "24c13af59c9b0b68aae83135bb470957": " 6 \\pi R \\eta \\frac{\\partial x(t)}{\\partial t} ", "24c15e4235314183c77878ad115266c3": "s_n=\\sum_{k=1}^r a_k s_{n-k}", "24c1ad5da6e1aed092d9f790085f97e0": "\\mu_A^*", "24c29d944e732b4b942998e88f8f5da0": "\n (a,b) <^d (A_i \\times B) (a',b') \\iff\n a <^d (A_i) a' \\lor ( a=a' \\ \\land \\ b <^d (B) b')\n", "24c2a81e85ef85fa2b705bb745a7df20": "\\mu:A\\otimes A\\to A,\\qquad \\eta:I\\to A,\\qquad\\delta:A\\to A\\otimes A\\qquad\\mathrm{and}\\qquad\\varepsilon:A\\to I", "24c2b25c22df82cbb06ec8151d0afe1b": " R_{;l} - R^n {}_{l;n} - R^m {}_{l;m} = 0.\\,\\!", "24c32aeec391b2003e3bba31372424e7": "\\sum_{i=0}^{n-1} F_{2i+1} = F_{2n}", "24c3eb73d7da000365eee0af15b9c545": "\\beta \\ = G\\,M/R\\,{c}^{2}", "24c42af13a3834012b17b85ae0494863": "y \\in L", "24c43ce76343d2934a0c50725c63b5de": "X_{i}=\\frac{n_{i}(t=0)+\\int_0^t\\dot{n}_{i,\\text{in}}(\\tau)d\\tau-n_i(t)}\n {n_{A}(t=0)+\\int_0^{t_{\\text{end}}}\\dot{n}_{i,\\text{in}}(\\tau)d\\tau}", "24c471c83f0366ad7e69f2e1b9f3a9b3": " \\textbf{a} = a_0 + a_1 X + a_2 X^2 + \\cdots + a_{N-2} X^{N-2} + a_{N-1} X^{N-1} ", "24c49a4deebaf691030a91d8d841206e": "\\mathcal{H} = 2 - e_1 - e_2 ", "24c501341139788bc6c224ad419c3fa5": "G(x) = x K_{\\frac{2}{3}}(x)", "24c50bed2164421ecd6c68ad30e4ac3b": "\nK_3 = \\frac{1.25 \\times ( GT + 10000 )}{10000}\n", "24c538e715b1412d067a2e8b6aa886b1": " f : M \\to \\mathbb{R} ", "24c5982c7bb706ae7fd100f7b121ff68": "x = (\\textbf{A}^{\\mathrm{T}}\\textbf{A})^{-1}\\textbf{A}^{\\mathrm{T}}\\phi", "24c5e3cc2ad9098da423c0128b22d96b": "1 \\Rightarrow (3, 1, 1)", "24c65104eff33337589e8fb013b1a9ba": "\n J_n^{-1}=J_{n-1}^{-1}+\\frac{\\Delta \\vec{x}_n-J^{-1}_{n-1} \\Delta \\vec{F}_n}{\\Delta \\vec{F}_n^T \\Delta \\vec{F}_n} \\Delta \\vec{F}_n^T\n", "24c65bb14b5c3fb29ab36e460036f9b0": "\\mathcal{N}=3", "24c6916d4af0fa7318550fd45840efba": "\\binom{\\alpha}{\\beta} = \\binom{\\alpha_1}{\\beta_1}\\binom{\\alpha_2}{\\beta_2}\\cdots\\binom{\\alpha_n}{\\beta_n} = \\frac{\\alpha!}{\\beta!(\\alpha-\\beta)!}", "24c696b8e54e0f2fd066ff1a07671784": "f_{k-q} - f_k = f_k - \\vec{q}\\cdot\\nabla_k f_k + \\cdots - f_k \\simeq - \\vec{q}\\cdot\\nabla_k f_k", "24c6d63c42123501905bacb333fb4cfe": "LL(\\alpha, \\beta) \\to\n L(\\alpha,\\alpha/\\beta). ", "24c71577c69bc60531e52b93bca4d205": "s_x = 0", "24c77c7193a63959ff5559007f44c3f7": "S = \\cfrac{\\pi r^3}{4} = \\cfrac{\\pi d^3}{32}", "24c795eb1ea4acd2196677895b464e80": "\\begin{align}S(\\beta_1, \\beta_2) =&\n \\left[6-(\\beta_1+1\\beta_2)\\right]^2\n+\\left[5-(\\beta_1+2\\beta_2) \\right]^2 \\\\\n&+\\left[7-(\\beta_1 + 3\\beta_2)\\right]^2\n+\\left[10-(\\beta_1 + 4\\beta_2)\\right]^2 \\\\\n&= 4\\beta_1^2 + 30\\beta_2^2 + 20\\beta_1\\beta_2 - 56\\beta_1 - 154\\beta_2 + 210 .\\end{align}", "24c7be430ce43ffd7f8a68f81fa60c1c": "R^{2n}", "24c846bc486b0866b3fe3baa11a8336c": "\\varepsilon_i: \\ \\vec n_i\\cdot\\vec x=d_i, \\ i=1,2", "24c8538e32f233a9a540389ab367b02e": "\\|x\\|", "24c856463af06ddf255f0a562a204930": "\\displaystyle{C_k=\\sup_n \\sum_m {(1+n^2)^{k-1/2}\\over (1+m^2 +n^2)^k} <\\infty,\\,\\,\\, c_k=\\inf_n \\sum_m {(1+n^2)^{k-1/2}\\over (1+m^2 +n^2)^k}>0.}", "24c8594555b16056e42e5ff17e999a78": " C = \\Sigma X_i ", "24c865567dce8c0be1ed3c757d851551": "t_{prt}=1.5", "24c881539adac6252bc3a25fe9874c42": "b = a \\sqrt{e^2-1}.", "24c8a5615d9c6cad23d79d448c7b848c": "E_n(x)", "24c8b4ccba82809cd0b19f9ed4d9e04a": "x \\mid y", "24c8b71eb3c8121becf7796ea3150bfd": "\\Re", "24c91c9077e792ce00390be740e2ac86": "{\\bold u} \\cdot {\\bold v} = \\left(Q {\\bold u}\\right) \\cdot \\left(Q {\\bold v}\\right) \\, ", "24c96602e63ed1085ee85286841c7812": "s \\in\\mathcal S", "24c98e0a64ae0022152f49e86de95e8d": "Y = C + I + G ", "24c9d04f199a1b2dfc9231d2b3b28971": "N < n/2", "24ca7e4cd97fe3179cf94dd7479de673": "\n\\begin{align}\n4\\Phi_5(z)\n&=4(z^4+z^3+z^2+z+1)\\\\ \n&= (2z^2+z+2)^2 - 5z^2\n\\end{align}\n", "24ca7e6d3b3c475b7ee1d6593c9ad23f": "\\mathbf{\\omega'}_2 = \\mathbf{\\omega}_2 + j_r \\mathbf{I}_2^{-1} ( \\mathbf{r}_2 \\times \\mathbf{\\hat{n}} )", "24ca850ee659319371c5fc9ed11fbbd3": "(f\\circ g)'(c) = \\nabla f(a)\\cdot g'(c),", "24ca8e9872cd9cb7d99e86da685df2f8": " \\int_x K_t(x) = 1 \\, ,", "24caa403875e7b84795fdae480248496": "B \\subset \\mathbb{N} ", "24caa4fffe8840052e159ec8f888d42f": "\\Gamma \\vdash \\phi", "24caa6db3531dac6a32a694eac451bb9": " \\langle \\Psi , \\Psi \\rangle = \\sum_{s_{z\\,N}} \\cdots \\sum_{s_{z\\,2}} \\sum_{s_{z\\,1}} \\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r}_1 \\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r}_2\\cdots \\int\\limits_{\\mathrm{ all \\, space}} d ^3 \\mathbf{r}_N \\left | \\Psi \\left (\\mathbf{r}_1 \\cdots \\mathbf{r}_N,s_{z\\,1}\\cdots s_{z\\,N},t \\right ) \\right |^2 = 1", "24cad463a4ca8bcf93b49b9b2f29d6d1": "-i_Z", "24cae4c88d2f629223c647cc1e8c0ea3": "\\scriptstyle y_0,\\ldots,y_m", "24caf49a99714d4c466e6bd759949beb": "\n\\begin{cases}\n\nx \\equiv & b_1 \\ \\bmod \\ m_1 \\\\\n& . \\\\\n& . \\\\\n& . \\\\\nx \\equiv & b_k \\ \\bmod \\ m_k \\\\\n\n\\end{cases}\n", "24cb10700246bb693160cc460fddfde1": "Z(E(K), T) \\equiv \\exp \\left(\\sum_{n=1}^{\\infty} \\mathrm{card} \\left[E(K_n)\\right] {T^n\\over n} \\right)", "24cbb88823412cc196e156c0255ab214": "\\Delta\\arg f(iy)", "24cc010f8117221a235a4f8fc9d3992e": "C_G(Q)", "24cd002826cb3d4aa26d5285314034eb": "\\mathcal{H}_0:\\theta=\\theta_0", "24cd0ae95f7f25806759e27f463707d0": "\\mu = rv^2 = r^3\\omega^2 = 4\\pi^2r^3/T^2 \\ ", "24cd14ea331934a2be851546d0e351b7": "\\liminf m(n) < -1.009", "24cd2cf83946eda8aac994156a74f5a6": " \\gamma_{ws} \\,", "24cd6419b1fa8abc606e6b2978c7c955": "r = {1 \\over {b + a \\cos \\theta}}", "24cd8b9aabdbd91db4ce9b18389974fc": "\\vert n\\rangle \\rightarrow \\vert n^\\prime\\rangle", "24cdba721bbf4ea463fb641108a245b7": "V_{\\gamma}", "24ce1658aafe200ec43851c32646cf63": "K_{\\rm w}=[{\\rm{D_3O^+}}][{\\rm{OD^-}}] = K_{\\rm{eq}} \\times [{\\rm{D_2O}}]^2 ", "24ce7e2fec80b8ed6142240d4b6c1208": "h = r_p v_p,\\,", "24ce8953bdb297cc751ee115cafb7813": "\\log{P}", "24ce8d751b27aafd08dca496d00a3fe3": "e_c", "24cefb3e7f1d7c8c48c6a1a320e00c9c": "v = \\sum_i w_i\\cdot \\bar{v_i} = \\frac {1}{\\rho}", "24cf222b30c46230d11e207b48de49f4": " \\left\\{K_{G}^{(a)} \\right\\}_{a\\in V} ", "24cf4e3a231d4fe8869ac75e5f21c4cb": "Ci = {{Cz^2} \\over {\\pi \\times \\lambda \\times e}} ", "24cf93dbb1409a51849f4964a5794191": "R^m_n", "24d0184a70a56c5a8a532b2770625258": "\n\\varphi = C_1 r \\theta\\sin\\theta + C_2 r\\ln r \\cos\\theta + \nC_3 r \\theta\\cos\\theta + C_4 r\\ln r \\sin\\theta \n", "24d02fee1e0c89439e444d9d8bbd8639": "f(w) = z\\,", "24d03b9eb698c8b0d7975c37b6ad84b4": "\\zeta_{L}", "24d03e5b6bdd5656de3927f3c92a0dd1": "H_{ii}\\,", "24d0797edf7d9f9a8a25b044b2cd85e4": "\\alpha = \\frac{\\sum_P - \\sum_Z}{P - Z}", "24d10f0d1c1927757f556ac0965ee9bf": "(a-b)\\sum_{k=0}^{n-1}a^kb^{n-1-k}", "24d1112839dc9c111a5b91bf4b6c8cb7": "A\\leq\\tilde{A}", "24d121f049e9f3ba281bd8f5d7ac87af": "\n\n\\begin{align}\nL_n^{(\\alpha)}(x) & = L_n^{(\\alpha+1)}(x) - L_{n-1}^{(\\alpha+1)}(x) = \\sum_{j=0}^k {k \\choose j} L_{n-j}^{(\\alpha-k+j)}(x), \\\\[10pt]\nn L_n^{(\\alpha)}(x) & = (n + \\alpha )L_{n-1}^{(\\alpha)}(x) - x L_{n-1}^{(\\alpha+1)}(x), \\\\[10pt]\n& \\text{or } \\frac{x^k}{k!}L_n^{(\\alpha)}(x) = \\sum_{i=0}^k (-1)^i {n+i \\choose i} {n+\\alpha \\choose k-i} L_{n+i}^{(\\alpha-k)}(x), \\\\[10pt]\nn L_n^{(\\alpha+1)}(x) & =(n-x) L_{n-1}^{(\\alpha+1)}(x) + (n+\\alpha)L_{n-1}^{(\\alpha)}(x) \\\\[10pt]\nx L_n^{(\\alpha+1)}(x) & = (n+\\alpha)L_{n-1}^{(\\alpha)}(x)-(n-x)L_n^{(\\alpha)}(x);\n\\end{align}\n", "24d14a05a0021ef5bf253753557eda12": "\\Delta G^*", "24d1fc8b14e421e90840cf7dad3fe609": "\\tan 2\\chi =-2\\kappa \\sin \\boldsymbol\\Gamma\\, ", "24d1ff1cbf41d5cf671dd4cf8576e647": "M_r = \\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r^2}.", "24d2097e9b54ed90562bdca5038bf828": "\\langle K,\\prec_K\\rangle", "24d23668025b73afeea8e0de0ace1dc5": " z^{\\mu}_W(\\tau, \\vec \\sigma) = Y^{\\mu}(\\tau) + \\sum_{r=1}^3 \\epsilon^{\\mu}_r(\\vec h) \\sigma^r, ", "24d23c2ab814daae0f2ec8279680c271": "P_{G_{2}}(x) = (1+x^3)(1+x^{11})", "24d258ad87747387267eb8951b57cff6": "M\\left(|\\downarrow \\rangle \\otimes |O_{\\downarrow} \\rangle \\right) = |\\downarrow \\rangle \\otimes |O_{\\downarrow} \\rangle", "24d2b71f9352a6ff379861d0b05b30dc": "p(x,\\theta)", "24d313e99333ff40d29b2cb55a3a81da": "u-u_h", "24d31477f93f2aed080842416266b3b5": "(a,b)=\\begin{cases}1,&\\mbox{ if }z^2=ax^2+by^2\\mbox{ has a non-zero solution }(x,y,z)\\in F^3;\\\\-1,&\\mbox{ if not.}\\end{cases}", "24d380e4719ca7b8872f5a6e36baecc7": "\\frac{\\partial k}{\\partial t}\\, +\\, \\frac{\\partial \\omega}{\\partial x}\\, =\\, 0\\,", "24d3dbb0cb57274687125cb3902a889b": "\\gamma_0=\\ln(3) / \\sqrt{8}\\pi,", "24d407c5414fce0d5f17c10309bd44a3": "G(f) = G_1 \\left( \\frac {f}{f_1}\\right)^{ge} ", "24d45a895b7e7f3a5ba85faed04f07c3": "n_\\text{jets}", "24d4820eb06c70ec7f32a145c843bcce": "\\mathbf{A}^{mn} \\times \\mathbf{Gr}(r,m)", "24d4be2a338af5d4eed84ad950291db1": " X \\cdot F_i \\subseteq F_{(h(i))} ", "24d501a358d23a029e34f893f916569c": "[A] = \\frac{N_A}{N_0V}", "24d580b23ee359be4b45e306f4fb20e8": "d_n = 2\\lambda Kp_nq_n + Kq_n^2 = Kq_n[2\\lambda + (1 - 2\\lambda)q_n]", "24d5869acf270765e226263ba8035c56": "\\frac{\\partial^2 \\psi}{\\partial x \\partial y} - \\frac{\\partial^2 \\psi}{\\partial y \\partial x} = 0.", "24d5a2783af3f9bbf1778a7261b4e88c": "E = \\sum_{i=2,0} \\sum_{j=2,0} TL_{i,j} = \\sum_{j=2,0} \\sum_{i=2,0} TL_{i,j}", "24d5ba4cf67d644edf16fbed287b099a": " \\mathbf{w} = w_1 \\mathbf{e}_1 + w_2 \\mathbf{e}_2 + w_3 \\mathbf{e}_3, ", "24d5e1589f6997e8c0ed006359e375f5": "e^{\\prime} \\circ f = f = f \\circ e", "24d60a338c142d1e211e3469c5fa05ce": "a_{ij}=\\frac{1}{(k_{i}-j)!}\\lim_{x\\to x_i}\\frac{d^{k_{i}-j}}{dx^{k_{i}-j}}\\left((x-x_{i})^{k_{i}}f(x)\\right),", "24d6127a7bfcbc7831cc1f1380e43ea3": "FNR", "24d628f6733e02250117fd77afb5f5a1": "[p_i,F(\\vec{x})] = -i\\hbar\\frac{\\partial F(\\vec{x})}{\\partial x_i}; \\qquad [x_i, F(\\vec{p})] = i\\hbar\\frac{\\partial F(\\vec{p})}{\\partial p_i}.", "24d64c911117e16a01b64898ce50d531": "\\nabla_\\tau^\\Gamma s=\\tau\\rfloor\\nabla^\\Gamma s", "24d68bd59c164ccd0cb412c789535140": "V_k(x)", "24d69ffb3854e91cade6249b8079b855": "a_{13}", "24d6c8cea3738c83c1b5876704807cfb": "\n \\hat\\beta = \\frac{ \\sum{x_iy_i} - \\frac{1}{n}\\sum{x_i}\\sum{y_i} }\n { \\sum{x_i^2} - \\frac{1}{n}(\\sum{x_i})^2 } = \\frac{ \\mathrm{Cov}[x,y] }{ \\mathrm{Var}[x] } , \\quad\n \\hat\\alpha = \\overline{y} - \\hat\\beta\\,\\overline{x}\\ .\n ", "24d6ca4f3307f5bdc876b6e38d315eb2": "\\Rightarrow \\frac{f(y) - f(x)}{y-x} = f'(x) + f''(x)\\cdot\\frac{y-x}{2!} + f'''(x)\\cdot\\frac{(y-x)^2}{3!} + \\dots ", "24d6cab6ee1f5b4df02b47020c22fb7a": "i\\eta_3=\\sqrt{-2}", "24d6dcc292561c878fff58dfc91b6aee": " \\begin{align}\np(\\lambda) &\\propto L(\\lambda) \\times \\mathrm{Gamma}(\\lambda; \\alpha, \\beta) \\\\\n&= \\lambda^n \\exp\\left (-\\lambda n\\overline{x} \\right) \\times \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\lambda^{\\alpha-1} \\exp(-\\lambda \\beta) \\\\\n&\\propto \\lambda^{(\\alpha+n)-1} \\exp(-\\lambda \\left (\\beta + n\\overline{x} \\right)).\n\\end{align}", "24d6f4ea258e1be0e085dc535c5c5866": "e-\\operatorname{cr}(G)", "24d71f273eaf176c80fbb8c63ff96914": " -2 p_1 \\cdot p_3 \\approx \\,", "24d7836c99c57de6c4f1109acb35dd94": "R = 3\\sum_q e_q^2,", "24d78f205a1372d8c1ed894dc1964ea5": "( \\Delta \\mathbf{v})", "24d7914700d54485b5ca76b6ce7f2f36": "[-,-] : E\\otimes_k E\\rightarrow E", "24d7af823e21715ebdc3ddabf978c610": "\\phi_{0}", "24d83a37508452537d66cff51c78d168": "u_{i}", "24d8d6484c80cde7b102d46b14189ce0": "\\scriptstyle{\\pm \\frac{1}{2}}", "24d8eabcdfb4c6dc65e6b1049bceddda": "y = f(x,u)", "24d909921059b46dd6639d02daea61a2": "Z_k^{-1} = X_n", "24d945dd059c2ed6e8ee6031dcaf3b4f": "AUC_1 = {U_1 \\over n_1n_2}", "24d94bf56b70ea7875bfa7a592508d6c": "\n[\\xi( \\mathbf{p}^{\\prime}),\\eta^\\dagger( \\mathbf{p})]\n= 0 \\quad\\quad\\quad\\quad (14)\n", "24d9f4572924ea53e97901880434d754": "\\left(1 - e^{-bx}\\right)e^{-\\eta e^{-bx}}", "24da7705413aebfc6ee4267545304945": " \\sigma_{\\epsilon} = 1.30 ", "24dac387533624890b1f009c376b4976": "\nQ_{r,s}(n) =\n\\begin{cases}\n1 , \\quad 1 \\le n \\le s, \\\\\nQ_{r,s}(n-Q_{r,s}(n-r))+Q_{r,s}(n-Q_{r,s}(n-s)), \\quad n > s,\n\\end{cases}\n", "24daf38045457445fdfdf14ea6f6d8f2": "[k]b\\,\\!", "24db79e02d60a103be48a8741621a148": "\\mathbb{X}^{I}", "24dba8162dd15a07f4fc686acddd0364": "y_1(x)=c_1\\ln x+c_2,\\qquad x\\in I,", "24dbca283bf50aac7c83a5ff91dcff5c": "S(\\rho)N", "24dbcdc7e16a64c1ca05b49a723b3fbb": "FDR \\le \\frac{{{m_0}}}{m}q", "24dc004b2a162849cc94a209a4cc8179": "S_1 , S_2 , S_3 , S_4 , S_5, \\ldots ", "24dc658c7d1a81ddde3724364a495837": "H = \\alpha H_\\text{norm}", "24dc6fc32961a2176df0aa3a52ab18d9": "\\omega \\in S_0", "24dc7610f253c0163a9c87083a8e01fe": "\n(\\mathbf{M}_{*})_j = \\frac{1}{l}\\sum_{k=1}^{l}k(\\mathbf{x}_j,\\mathbf{x}_k).\n", "24dcaae077391f9d6a58d9d523f66f18": "D_1=6P_1+ 4P_2- 5D_{\\infty_1} -5D_{\\infty_2} ", "24dcc5d4dcd3ca613a2f9a3ca80e6c45": " \\langle \\psi | F(A) | \\psi \\rangle\\, . ", "24dd5a270bfda8301ed421642ad8c0b0": "\\Sigma_s(\\mathbf{r},E'\\rightarrow E,\\mathbf{\\hat{\\Omega}}'\\rightarrow \\mathbf{\\hat{\\Omega}},t)dE^\\prime d\\Omega^\\prime", "24de032c4429b1e400821dd8d2d09fdc": " 0 \\leq \\phi < 2\\pi", "24de0f931509f7b5e1dab5e2772f7ba1": "P\\longrightarrow M", "24de86cb6f4854a91c3241f87455ed83": "T\\approx\\frac{M}{M_{contact}}", "24de89f6fe6f897e7e9937d0b0c0968d": "\\int_0^{\\frac{\\pi}{2}} \\sin^{n}(x)\\,dx = \\int_0^{\\frac{\\pi}{2}} \\sin^{n-2}(x) \\left[1-\\cos^2(x)\\right]\\,dx", "24deb65c9483db51a191d620be5b85d4": "\n\\text{ or if } a_i \\neq b_j\n", "24ded52ad667983821193712d754cf03": "\\mathcal{H}[t] = \\dot{\\vec{x}}[t] \\cdot \\vec{P}\\,[t] - \\mathcal{L}[t] = c \\sqrt {m^2 c^2 + {\\left( \\vec{P}\\,[t] - e \\vec{A} [\\vec{x}[t],t] \\right) }^2} + e \\phi [\\vec{x}[t],t] \\,.", "24defe516f8f7de3587d5c420a828a15": "K_i(X) \\otimes \\mathbf Q = 0 \\ \\, i > 0.", "24df0e3e4d4467501d969f8e52cab983": "z = 1000", "24df22f27c601f112dd3e91419d580e8": "{\\mathbf e}_i = \\partial/\\partial x^i", "24df36c3111d445a3efa5b459f96911a": "s_1=m\\log k+1/a \\sum_{i=1}^m \\log u_i", "24df482628dbea25d86d464caf520193": "f(k)\\cdot |x|^c > |x|^{c+1}", "24df948cfd7a531ac26e5bb882ea659d": "t_i \\cdot t_j = SR((t_i, t_j), (s_i, s_j), O)", "24e01848941ab242ea0770788cf1b8f5": " u \\in C^{\\infty}(\\mathbb{H}) ", "24e05b3963753071a519405aef2b57a5": "0 < x < \\infty", "24e05f8fc4f86d7dfa4511480aa7deb7": "\\kappa(X,K_X)", "24e077c398a3c0f32dc476002ccecf57": "n = \\frac{A}{\\begin{matrix} \\frac{4}{3} \\end{matrix} \\pi R^3 } \\approx 1.2 \\times 10^{44} \\ \\mathrm{m}^{-3} ", "24e087a68378b5454b9912739381aa08": "x \\geq 0,\\ f(x) \\geq 0 \\text{ and } x^{T}f(x)=0 \\,", "24e09ceb345febc761db3067a145569c": "V_\\mu^\\pi = \\mathbf{E} \\left ( \\sum_{i=1}^\\infty r_i \\right )", "24e0e8a4b8f5b55874686dbf7319d2ad": "\\phi_1=88.24^\\circ", "24e102fdc99c52cf7acc40d5d3243978": "\\scriptstyle \\sin \\theta = \\sin 2 \\pi U_2", "24e1070f3253f350f1bd95e235b6260f": "n_1\\ge n_2\\ge \\cdots \\ge n_r\\ge 1", "24e15b2d0e85b238c2e6e028973792e7": "\\frac{L}{S} = \\frac{\\ell}{s}", "24e180e49cd734ee1b9b6ffa13c961e2": "\\int_0^t H \\,d B =\\lim_{n\\rightarrow\\infty} \\sum_{[t_{i-1},t_i]\\in\\pi_n}H_{t_{i-1}}(B_{t_i}-B_{t_{i-1}}).", "24e1b4e9fad7307762ae9f9408da30ce": "\\delta(t' - t_r')", "24e1b6b34b130315b88255b525b3ad6d": "\\mathfrak{n}\\subset S", "24e2201d64371523d267c1187c036c06": "r\\times r", "24e23ac6b43d18965a0d812b6e631fb1": "\n\\Xi(\\xi,V,\\beta) = \\gamma_{\\bar{\\Phi}} \\int D w\n\\exp \\left[ - S [ w ] \\right],\n", "24e25702302dfc5fa09485b5a8bfd570": "A \\leq_T B", "24e2af9b6775cfdb4e7c279881f3707f": "F\\dfrac{dS}{dx}=ixFS", "24e2e9dc10aa6e3f0018058ea533c7d8": "\\textstyle l_1 \\le l", "24e2ecf61343989356d3d8ed704c90f7": "L= -\\rho.", "24e2f656ce8abf5759a9a3f27e4dfc25": "V_T = kT/q", "24e317400e08a014fc48d4c42cfb715f": "(\\operatorname{log}) = \\ln \\frac{b_{min}}{b_{max}} = \\ln \\frac{2}{\\theta_{min}}", "24e3a997781f0c1fe7850265ab0cf3c9": "\\mathbf{x} = \\sum_{j=1}^k \\lambda_j \\mathbf{x}_j-\\alpha\\sum_{j=1}^k \\mu_j \\mathbf{x}_j = \\sum_{j=1}^k (\\lambda_j-\\alpha\\mu_j) \\mathbf{x}_j", "24e463e3b91c97dddecec5945e4c4c42": "\\gamma_{\\mathrm{LV}}", "24e46b3565b1fa76ffcce5250c39f32d": " = (\\lambda p.p\\ (\\lambda x.\\lambda y.x))\\ ((\\lambda x.\\lambda y.\\lambda z.z\\ x\\ y)\\ a\\ b) ", "24e53fce1121e435eba6724ab6e9f1e9": "Y_w", "24e593163da78d9456ef4e44b5fae7c3": " \\frac{1}{\\gamma } \\approx 1-\\frac{v^2}{2 c^2} ", "24e59aea0e6cdf1c0747f0b9de73979f": "U_g = mgh", "24e5d395ee1cac1a3dceee5d552dc9ee": "L=0", "24e5f6e272c6b4aab25731740ac6db38": "y(x_*) = y_{k-1} + \\frac{x_* - x_{k-1}}{x_{k}-x_{k-1}}(y_{k} - y_{k-1}).", "24e6364f68b3aacc7e359f30fbe270b3": " M(x) = 2\\pi x. \\,", "24e68e2f4be55a94363ab329b7d1a187": "\\mathfrak X=\\{x_1,x_2,x_3\\}", "24e694cb9945254e0b35e259bf58edc9": "d=2t+1", "24e69b051ffb2f8569e99e8682a070bf": "\\delta_1>0", "24e6a539fecbe62603a707f51a16e864": "e^{i\\pi/4}", "24e70836845f5642d842c4f13772a410": "c \\in (a, b)", "24e779109890ff62e67e281d145c6ef0": "(a_2, b_2, c_2, d_2)", "24e7cba57e66fed366d96c2b17eaf2f8": "a 2) \\Rightarrow a \\le 2", "24ecc899bd14b3aa417c00485013e94a": "\\mu \\, ", "24ecda7ae483893d9b5f7067c0762f5e": "\\nu_i + \\rho_n \\le \\mu_i \\le \\nu_i + \\rho_1\\, ", "24ed0825e3e1581782a9e2f9eec52704": "\\scriptstyle n_i\\,", "24ed39c417b81b294aeb7f4254d8064d": "\\zeta(s_1,s_2,\\ldots,s_n) = \\sum_{k_1>k_2>\\cdots>k_n>0} k_1^{-s_1}k_2^{-s_2}\\cdots k_n^{-s_n}.\\!", "24ed45b7698c9f29f3784e1be345b643": "\n \\begin{matrix}\n a\\uparrow b= a^b = & \\underbrace{a\\times a\\times\\dots\\times a}\\\\\n & b\\mbox{ multiplied copies of }a\n \\end{matrix} \n ", "24ed48f676744bfa8e9ea9c6392b24df": "\\exist b \\in B", "24ed4bfc9a0a0b6dbd672bc9e95c153a": "\\mathrm{Var}(Z(x_0))=\\mathrm{Var}(\\hat{Z}(x_0))+\\mathrm{Var}\\left(\\hat{Z}(x_0)-Z(x_0)\\right).", "24ed528ee2191bdc8bb3484035b1e8c5": "83^\\mathrm{o}", "24edb2ea71428b08548c56890a88df9b": "A[i,j]", "24edce33157cd57ec63d505a4cfa4e7b": "\\delta_S = 2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\ddots}}}\\, .", "24ee2d6f1184b980ffbc9c4a9e2683b5": "(\\forall x \\phi) \\lor \\psi", "24eeb3d50c0aa4c9ac79ff1e040f748e": "dM=\\frac{R^2/2 d\\theta sin^2{\\theta}}{\\pi R^2}M=\\frac{ sin^2{\\theta}}{2 \\pi}M d\\theta ", "24ef13e9fd9fbae1b5a1a824f2148ab6": "E( B | A =a , A > B) = E( A/2 | A =a , A > B) = a/2.", "24ef65f019cdeff3dd168dbbc1cffda4": "\\textstyle {\\{e_1,\\ldots,e_n\\}}", "24ef74c7fc988f1595bd0dcf26fad3ae": "x = x_0 + x_1 \\epsilon_1 + x_2 \\epsilon_2 + {}", "24ef86dc89c61f8eefb663cc113a45f5": "\\tfrac{\\partial I}{\\partial t}", "24efb8b3d47995de686703e4a61ca4b6": "8911 = 7 \\cdot 19 \\cdot 67 \\qquad (6 \\mid 8910;\\quad 18 \\mid 8910;\\quad 66 \\mid 8910).", "24efbc381c64afe6caab140825745c2c": "B_\\lambda", "24efc5444fdc314cad0efbd7a2b32136": "\\text{Geom}(p) = \\text{NB}(1,\\, 1-p).\\,", "24f011e35eb6f32491e5dade56a07292": "\\mathbf{E} [W(h)] = 0,", "24f064825d181debea866c76cc0a8dff": "\\scriptstyle 1,000,000\\sqrt{N}", "24f068c5eda18182c4e987e30361dde5": "\\tau_c", "24f09494d50a573dc0b88e5606762fee": "A(t) = \\mathbf{U}_n^T \\mathbf{A}", "24f0ded95131e41d741b2305a48b89b1": "=M*V*\\ddot sE+B*V*\\dot sE+pE ", "24f1319f9fcdeb0929fcfb97c311efad": "\\pi-\\pi", "24f152d852ee24e9eda729027991a6d4": "\\chi(n) = \\left(\\frac{n}{p}\\right),\\ ", "24f15e0ed871f58aa67cdf8b404d88cd": " r = \\frac{\\frac1{n-1} \\sum_{i=1}^n (X_i - \\overline{X})(Y_i - \\overline{Y})}{s_X s_Y} ", "24f15fe91fe77af4a20095de542bd9b8": "\\ f_{tuning}(t) = K_o \\cdot \\ v_{in}(t) ", "24f16ae6ff5964def1d42c687bea8fdd": " R = \\lceil \\log_2 M \\rceil ", "24f20d0a0d4685f592bc3f345b494f43": " A(t) := e^{iHt / \\hbar} A e^{-iHt / \\hbar} .", "24f2409d3cea6f8b2b12eaff194b2f58": "\\text{return}\\colon A \\to A^{?} = a \\mapsto \\text{Just} \\, a", "24f25cdcd2eb36e86f10afc40fb322c9": "a\\wedge I=0", "24f28446139ad8828f346d0732af0f6b": "\\Delta \\mathbf{A} \\equiv \\nabla^2 \\mathbf{A}", "24f2851d3a81304b4217edeeeac4604d": "R_{\\infty} = \\frac{m_{\\rm e}c\\alpha^2}{2h} \\Rightarrow m_{\\rm e} = \\frac{2R_{\\infty}h}{c\\alpha^2}", "24f28becfa0d33c285b226a2013cd1a0": "b>a>c", "24f30e7175d822fc09d4649525bb8af5": "\\mathbf{Q}_v", "24f32180dfcdc43e3a1a30438ae32832": "D\\!\\!\\!\\!/ \\psi_i = -\\lambda_i\\psi_i.", "24f35934b2810ea6bde280fbe6c63277": "\n k_r^2=k_x^2+k_y^2+k_z^2,\\, k_x=\\frac{m\\pi}{l},\\, k_y=\\frac{n\\pi}{w},\\, k_z= \\frac{p\\pi}{h}\\, k_{xy}^2=k_x^2+k_y^2\n ", "24f38fb2e400574684c81ad625e782ba": " bh + l(a + b + c) ", "24f3a172a9966bdeb92d7d2f277d9ca3": "\\hat{a}, \\hat{c}", "24f3a38f4b3207edb37da23fad6e21da": " \\mathbb{P}\\left[ A \\cap B \\right] = \\mathbb{P}\\left[ A \\right] \\mathbb{P}\\left[ B \\right], \\qquad \\forall A \\in \\mathcal{I}_X, \\, B \\in \\mathcal{I}_Y, ", "24f401c1e4aab19ec1a825ef2715cadf": "\\operatorname{ad}_g", "24f4209d94383ad4b096b27533faa5bd": "C_d = \\left( \\frac{s} {c}\\right) \\left(\\frac{\\Delta p_0} {\\rho W_{1}^2 /2}\\right) ", "24f469f44fc90664c7c15a6979a6e265": "\\mathbf{x}(t_i)", "24f4b31421e34fe2f8e4866cd53d18d5": "\\overrightarrow{\\operatorname{div}}\\,(\\mathbf{\\underline{\\underline{\\epsilon}}}) = \n\\begin{bmatrix}\n\\frac{\\partial \\epsilon_{xx}}{\\partial x} +\\frac{\\partial \\epsilon_{xy}}{\\partial y} +\\frac{\\partial \\epsilon_{xz}}{\\partial z} \\\\[6pt]\n\\frac{\\partial \\epsilon_{yx}}{\\partial x} +\\frac{\\partial \\epsilon_{yy}}{\\partial y} +\\frac{\\partial \\epsilon_{yz}}{\\partial z} \\\\[6pt]\n\\frac{\\partial \\epsilon_{zx}}{\\partial x} +\\frac{\\partial \\epsilon_{zy}}{\\partial y} +\\frac{\\partial \\epsilon_{zz}}{\\partial z}\n\\end{bmatrix}\n", "24f4f0adfd92d6b4868ef9af99277f5a": "\n\\boldsymbol\\mu_{Y|X}\n=\n\\boldsymbol\\mu_Y + \\boldsymbol\\Sigma_{YX} \\boldsymbol\\Sigma_{XX}^{-1}\n\\left(\n \\mathbf{x} - \\boldsymbol\\mu_X\n\\right)\n", "24f50554a2dfde596785f645117caf0f": "9801\\sqrt{2}/4412", "24f564060c02ae068cf53f47d4424bb3": "{V}={I}{R}. \\ ", "24f571c9812028d15a47c7fbc0d75d6f": "J(v) = \\frac12\\int_{\\Omega}{|\\nabla v|^2\\,d\\mu}-\\frac1{p+1}\\int_{\\Omega}{|v|^{p+1}\\,d\\mu}", "24f5a8d494fa9048e5efbd42f26f53ce": " t_e ", "24f5da9722db93967591a4d0dd3b17b8": "BP*(BP)", "24f64fabf10e083a750b85667f6b2b72": "\\ O[s(t)] = G s(t) - D_3 s^3(t) + \\ldots", "24f68a56ed9f3b6ff81063d2e272d703": " x^{(k+1)}=D^{-1}(b - Rx^{(k)})", "24f6af6b5d380b3a471cab99fbde1c35": "d = 2^{w-1}, 2^{w-1}+1, \\dots, 2^w - 1", "24f6f76b6b7b8d44d6a1d90ab63e5db0": "\\mu \\approx \\frac{-2c}{b - \\sqrt{b^2-4ac}} \\,", "24f73ee335ad46e8862dc2b1320ea152": "\n \\mathbf{e}_i \\mathbf{e}_j + \\mathbf{e}_j \\mathbf{e}_i = 2 \\delta_{ij} \n", "24f7973de403049504bf2409dba2715e": "9^\\frac{1}{2} = 3", "24f7f47d0ec0de18bb9511ddbe1633a1": " \\mathcal{N}(np,\\, np(1-p)),", "24f8ccfcc865ccebd9890eaa4fc90b13": "f(\\theta) = \\sum_n a_n e_n(\\theta),\\quad a_n = \\langle f,e_n\\rangle", "24f914094f00026d223831b9e4eb870d": "\\frac{1}{3}=e^{\\frac{-1}{RC}\\frac{T}{2}}", "24f939bf6084ab078b3aa9365a4f17bf": "_{lex} b", "24f98c870c2a31493ef530b1fea423bf": "d_i\\colon G_{i-1}\\rightarrow G_i", "24f9d0cdc32eb430e6849f098541b043": "m_{rel}", "24fa9b05afcf0704f480a3da2de0301f": "s=\\log_{32}31\\approx 0.991", "24fadd8e528f493243017a74f972c80a": "P(f) (z) = \\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\mathrm{Re} \\frac{e^{i t} + z}{e^{i t} - z} f(e^{i t}) \\, \\mathrm{d} t.", "24fb1491d717328be73a8231f64817fb": "2^{101}", "24fb18ba5fd008eb2f8ea515343988f2": "x \\# y \\;\\to\\; (x \\# z \\;\\vee\\; y \\# z)", "24fb5c2ede9d8cae2f20a109b11dd155": "\nZ_\\text{lower}(x)=Z(x)-\\frac{1}{2}T(x)\n", "24fb71cf064857488a308f649e7ab6b2": " \\delta s_i \\equiv s_i - m_i ", "24fba1c9c992fe9b15a43c056089b5c9": "a_i=\\mathrm{sin}\\frac{(2i-1)\\pi}{2n},", "24fbf281eb28e940e9ec7ddaadc8cca3": "f_c(z) = z*z + c", "24fc27b22b1e2d7bd039a9af4c9d4dd3": "\\hat{\\boldsymbol\\beta}", "24fc8725e617037c6e3951fc0de87745": " \n\\int_{L} \\phi_A^2 ~dV = \\int_{R} \\phi_B^2 ~dV \n", "24fc8fd5506c0aaa6d4ccef215528a5d": "d=8", "24fc904db28c965c8d15ab65a2a9cea0": "\\pi_n(t) ", "24fc98a1b981211033b41407bdbbbcbc": "E = 0.0050\\, ", "24fd1ff7d4b6cd13a58a1af078614fe2": "T^{-1}(E)=E\\,", "24fd4ddd4f9975086c50b5b9271b2493": "\n(Sv)(ds) = \\int_0^1 f'_{t\\mu(I)}(\\mu(ds))dt\n", "24fd8820615f690fb770fbb9e301ad19": "\\Diamond\\Box p \\rightarrow \\Box\\Diamond p", "24fd9276cf3c7dbaeebe5812da978f4f": "\\begin{array}{lllll}\\langle & a,b,c,d,e,p,q,r,t,k & | & &\\\\ \n&p^{10}a = ap, &pacqr = rpcaq, &ra=ar, &\\\\\n&p^{10}b = bp, &p^2adq^2r = rp^2daq^2, &rb=br, &\\\\\n&p^{10}c = cp, &p^3bcq^3r = rp^3cbq^3, &rc=cr, &\\\\\n&p^{10}d = dp, &p^4bdq^4r = rp^4dbq^4, &rd=dr, &\\\\\n&p^{10}e = ep, &p^5ceq^5r = rp^5ecaq^5, &re=er, &\\\\\n&aq^{10} = qa, &p^6deq^6r = rp^6edbq^6, &pt=tp, &\\\\\n&bq^{10} = qb, &p^7cdcq^7r = rp^7cdceq^7, &qt=tq, &\\\\\n&cq^{10} = qc, &p^8ca^3q^8r = rp^8a^3q^8, &&\\\\\n&dq^{10} = qd, &p^9da^3q^9r = rp^9a^3q^9, &&\\\\\n&eq^{10} = qe, &a^{-3}ta^3k = ka^{-3}ta^3 &&\\rangle \\end{array}", "24fdb18c78112712708683fe92101c65": "a, b, c,", "24fdeee659f6f28a73c00424cfd299c5": "H_{\\gamma_i}", "24fe0020dbf0309ca4c054343282d962": "G(E/F)", "24fe1566a087e4efbff1404c153c356b": "u \\in H^r(E, E \\setminus E_0; \\mathbf{Z})", "24fed37420cd2363018dfdd6a5c4ff89": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{B(d+e\\,x)^m\\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{c(m+2 p+2)}\\,+\\,\n \\frac{1}{c(m+2 p+2)}\\,\\cdot\n", "24ff01c6ea93f337e37a627c8682b82c": " U_P = U_R ", "24ff36361545cd7977d15fdf9a2f3126": "p(v;\\theta)=exp[C(v)+\\theta^iF_i(v)-\\psi(\\theta)]", "24ff5b6517fdee6e49b054eddb2adc93": " T^* = \\frac{T - T_B}{T_T - T_B}", "24ff69661169374ae12d9075e8d5086c": "v = \\sqrt{\\frac{GM}{r-r_S}}", "24ff77d01658ab0096e85c40a6949343": "|\\lambda_1| < d", "24ffa2914dd7757a34fc8e3e9bf48d17": "S_n = \\alpha(t_{n}-t_{n-1}) \\times Y_n + (1-\\alpha(t_n-t_{n-1})) \\times\nS_{n-1}.", "24ffd2bb3264edebb496fca1b057e4ca": "\n\\Delta S = \\alpha k_{B} \\ln \\det \\mathbf{W}\n", "24ffe8a6cc14594c1f78360715b2fb81": "\n \\cfrac{\\partial^2 }{\\partial x^2}\\left(EI\\cfrac{\\partial^2 w}{\\partial x^2}\\right) = - \\mu\\cfrac{\\partial^2 w}{\\partial t^2} + q\n ", "25001837e317336d725d0c79dcb8a885": "{A}_{6}^{(2)}", "25005310cdfced3d10c1814de5f4d8fa": "\n\\begin{align}\n\\sum_{i=1}^n a_i\\log\\frac{a_i}{b_i} & {} = \\sum_{i=1}^n b_i f\\left(\\frac{a_i}{b_i}\\right)\n = b\\sum_{i=1}^n \\frac{b_i}{b} f\\left(\\frac{a_i}{b_i}\\right) \\\\\n& {} \\geq b f\\left(\\sum_{i=1}^n \\frac{b_i}{b}\\frac{a_i}{b_i}\\right) = b f\\left(\\frac{1}{b}\\sum_{i=1}^n a_i\\right)\n= b f\\left(\\frac{a}{b}\\right) \\\\\n& {} = a\\log\\frac{a}{b},\n\\end{align}\n", "250092b1aee75dd12ad3ab30bb3a5f33": " P_2 \\rightarrow P_3\\,", "2501079dc34945b14dd3ef2efbca873f": "\\nabla \\times \\mathbf{H} = \\frac{4\\pi \\sigma}{c} \\mathbf{E} + \\frac{\\epsilon}{c}\\frac{d\\mathbf{E}}{dt}", "2501300be0653d4a19dbc7a308c516f4": "\\mathbf{K} ", "2501a785a905ab34c41240ef25722923": "n \\wr", "2501bf53c8754506bfe166566130eb64": "\\displaystyle G_{ab}=R_{ab}-\\frac{1}{2}g_{ab}R", "2501f54cbc730ea4c8385ddc249b339a": "\\mathcal{F}\\subset L^2_P(S)", "2502027c3b2bf300076c0d86979cbef9": "E^2=p^2c^2+m^2c^4", "250223629129304eed51579eafa142f1": "S(q) = 1 + \\rho \\int_V \\mathrm{d} \\mathbf{r} \\, \\mathrm{e}^{-i \\mathbf{q}\\mathbf{r}} [g(r) - 1] = 1 + 4\\pi\\rho\\frac{1}{q} \\int \\mathrm{d} r \\, r\\, \\mathrm{sin}(qr) [g(r) - 1]", "250276e886ac8fdb99802946a1bc7a64": "\\sum_{n=1}^\\infty n", "25028623388915fa056a40abfdb2a54a": "_1F_1\\left(\\begin{array}{c}\\frac{m+1}{n}\\\\1+\\frac{m+1}{n}\\end{array}\\mid ix^n\\right)\\sim \\frac{m+1}{n}\\Gamma(\\frac{m+1}{n})\ne^{i\\pi(m+1)/(2n)} x^{-m+1}", "2502af3ce636a93fe9f08924df12374a": "\\vartheta(p_k)\\ge k\\left( \\ln k+\\ln\\ln k-1+\\frac{\\ln\\ln k-2.050735}{\\ln k}\\right)", "2502c16af86b2c570537c8cbead5d7c0": "\\frac{1}{n} \\displaystyle \\sum_{i=1}^n V(f(\\vec{x}_i,y_i)) + \\gamma\n\\|f\\|_{\\mathcal{H}}^2", "2502e50bf173bdadbff755449fb318e6": "\\Omega(k^{1/2}/\\log^2 k)", "2502f59595863bf6198fe00c1609c475": "\\mathbf D", "250349024bb4f1b3a8e4d5bfffc15f8f": "E_F = \\frac{\\hbar^2}{2m} \\left( \\frac{3 \\pi^2 N_e}{V} \\right)^{2/3} \\,", "25037518947b23ae741d613e46bd1506": " Z_k = - \\frac{a_o}{2n(\\mathbf{r})} \\frac{dn(\\mathbf{r})}{dr} |_{r \\rightarrow \\mathbf{R_k}} ", "2503a55d9c0e5e2def266d4eae140f5d": "x_i^+", "2503a69b2ec6da70d95d2e796feaa6b4": " \\{0, 1, \\infty\\} ", "2503cdd81b3f91dee2c4472f17dd6101": "A_k=\\{0^k,1^k,2^k,3^k,\\ldots\\}", "250433a385243cc6841384ab83e9bb71": "\n\\Delta \\tau' = \\Delta t'^2 - \\Delta x'^2 =\\Delta t{\\sqrt{1-u^2/c^2}\\over 1-u/c}.\n\\,", "250433bcab763f6da124c899a08d970b": "\n s.t. \\sum_{i=1}^n \\pi_{i} = 1, \\sum_{i=1}^n \\pi_{i} h(y_{i};\\theta) = 0\n ", "25043eae2e95be3fb0bc5c47c6b97ea7": "f(\\alpha \\mathbf{v}_1,\\ldots,\\alpha \\mathbf{v}_n)=\\alpha^n f(\\mathbf{v}_1,\\ldots, \\mathbf{v}_n)", "25046031aa2f0f8e5a3483fe8df14cde": "\\tilde{\\rho}:K[G]\\rightarrow \\mbox{End}(V),", "2504a2dd05ddcd94e1510843858ed14d": "f_X(x;k,\\lambda)={{\\rm e}^{-\\lambda/2}} _0F_1(;k/2;\\lambda x/4)\\frac{1}{2^{k/2}\\Gamma(k/2)} {\\rm e}^{-x/2} x^{k/2-1}.", "2504b3aea1643d683868a8a3beb20951": "\nV_{mn} = \\frac{1}{n^m} \\sum_{i_1=1}^n \\cdots \\sum_{i_m=1}^n h(x_{i_1}, x_{i_2}, \\dots, x_{i_m}),\n", "2504b7add1502ac03d266726c65e0baa": "\\rho(\\boldsymbol\\beta,\\sigma^{2})", "2505e244b079a4e45d821ebb9bceb3c5": " \\bar{X}_2 - \\bar{X_1} \\pm A \\cdot S_\\mathrm{pooled} \\sqrt{\\frac{1}{n_1} +\\frac{1}{n_2}}, ", "2505e4b3d0a17579920b744d4e6c8941": " \\nabla \\cdot \\mathbf{J} = - {\\partial \\rho \\over \\partial t} ", "250617cd46e7199b6377aac2199e3850": "\\Phi(h) = \\int_0^h g\\,dz\\ ", "2506362d2734981a5a79b208e83adaee": "\\scriptstyle a^b", "250643f368176509d18910e74d6c34fb": "\\Omega_{ab} = -g_{ac}{J^c}_b", "250645f735eac34ed9b70f57013aaa72": "\\cdots K_nG^{m+1}\\to K_nG^{m}\\to \\cdots \\to K_n", "2506844ffb4817d0dc5f811914d92e65": "\\frac{\\pi}{4} = \\sum_{n=0}^{\\infty} \\bigg(\\frac{1}{4n+1}-\\frac{1}{4n+3}\\bigg) = \\sum_{n=0}^{\\infty} \\frac{2}{(4n+1)(4n+3)}", "250706cfc4e74822c5c3a64c3be07d7a": "M^{\\mu\\nu} = (X^\\mu - Y^\\mu) P^\\nu - (X^\\nu - Y^\\nu) P^\\mu ", "250743a96dbc77709d34ff298a2fca74": "f^L(\\bold{x})=\\mathbb{E}(f(\\{i|x_i\\geq \\lambda\\}))", "2507722f94e690af47605fd0038bdd43": "\\liminf_{x\\to a} f(x) = \\sup \\{ \\inf \\{ f(x) : x \\in E \\cap U - \\{a\\} \\} : U\\ \\mathrm{open}, a \\in U, E \\cap U - \\{a\\} \\neq \\emptyset \\}", "2507c0c4c67b3e82d939c01164b5124b": "\\sum_{k=0}^\\infty (-1)^k {k+\\nu+1 \\choose k} \\left[\\zeta(k+\\nu+2)-1\\right] \n= \\zeta(\\nu+2)-1 - 2^{-(\\nu+2)}", "25080516aa649a94adf1272bca3495a3": " (w,r,x): set P = (SS(w;r),x),R = Ext(w;x), ", "250843c02104f066eb091cdb6a9531fe": " = \\pi \\! \\left(-\\!\\ln (\\tfrac14 !)+\\tfrac34 \\ln \\pi -\\tfrac32 \\ln 2+\\tfrac12 \\, \\gamma \\right) ", "250868fe6d5b913c49b7077f98d7d120": " \\mathbf{a}\\cdot\\nabla ", "2508c7f68f4adb30a8d00ef5269ecf9c": "\n[L_z,Y] = -iX\n\\,", "2508d6b205244eb66c03c5da2b19701c": "T_xX\\,\\!", "250920779989dbbc1e761116f7cad0be": "u_i \\in GF(q)", "250931ebbbd8ce051ba6d9dc6475effe": "(\\mathbf{A}-\\lambda \\mathbf{I})", "2509813c930232c52108f932ac90de3b": "a \\approx 1", "250993074bb293eafa15e449212bab92": "= \\int u \\,dx + \\left(-\\int v\\,dx\\right)", "2509e7b34d293f3091c965f8071570ea": "\\mathbf u(\\mathbf X,t) = \\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad \\text{or}\\qquad u_i = x_i - \\delta_{iJ}X_J\\,\\!", "2509fed2d1d14bccce1f54eb20fbf788": "(Z_1 \\cdots Z_n)_x := \\dim_k \\mathcal{O}_{X, x} / (f_1, \\dots, f_n)", "250a267e76a8212681235107190c2d1b": "J(t,t')", "250a39f269f2877cd09ded30de0e7705": "a(v) \\sim 1 + \\alpha v^{2}/c^2 \\,", "250a5407d0e75163c2a91fcfb472776c": "T_{alt} = 1 / f_{alt} = 10\\; \\mathrm{s}", "250a7e1919495a6d0f1a3c4c1f2a6f83": "a \\to bd", "250ac95d8b29e0fe4e5024d71b235aaf": "\\textstyle\\int_S g\\,d\\mu<\\infty", "250b19a37bb2c39db2e37b39fbdcf015": "n^2(n^2+1)/2", "250cd749207c0e6566c197ff0ef4a9ba": "t\\mapsto at+b", "250cf9f988c9ade64f38a406ae530184": " A_{FB} = \\frac {A_{OL} } {1 + \\frac {R_2} {R_2+R_f} A_{OL} } \\ , ", "250d37579fa36a6cd114ffe70a15b0d4": "\\mu (p)\\ ", "250d4a91f5420c509d73ed27693ce251": "\\mathrm{Var}[X_i] = \\frac{\\alpha_i (\\alpha_0-\\alpha_i)}{\\alpha_0^2 (\\alpha_0+1)}.", "250de32696532bcb35869b1d0f7813c9": "y_k", "250e2e52e6646bef19e94714220275de": "{\\rm Beta}(1-\\alpha,\\alpha)", "250e31b28f3c9e9eccd1cf679bbddb89": "\n \\mathbf{M}_1 =: -M_{12}\\mathbf{e}_1 + M_{11}\\mathbf{e}_2\\quad \\text{and} \\quad\n \\mathbf{M}_2 =: -M_{22}\\mathbf{e}_1 + M_{12}\\mathbf{e}_2 \\,.\n ", "250e4d0ee0cccb782a54597d98738744": " \\mbox{Im}(\\Phi) \\pitchfork \\mbox{orb}(f) ", "250e7f4233c56f8ed0908674003c1d43": " h_{\\alpha \\beta , \\gamma} \\, g^{\\beta \\gamma} = \\tfrac12 h_{\\beta \\gamma , \\alpha} \\, g^{\\beta \\gamma} \\,;", "250f27733e487733ad53b0e8a1bed351": " \n\\mbox{cas} (a+b) = \\cos (a) \\mbox{cas} (b) + \\sin (a) \\mbox{cas} (-b) = \\cos (b) \\mbox{cas} (a) + \\sin (b) \\mbox{cas}(-a) \\,\n", "250f496c3d3f6d514bb640c3338e428d": "\\lim_{n \\to \\infty} \\sum_{i=s}^n f(i) ", "250fc3779f8c74dab8b480e70de0bbe3": "(A, \\mathfrak{p})", "250fc7216664fd47f29d2ca1e8eee57b": "|d(p,y) - d(p,z)| > d(y,z) - \\epsilon.", "250fdfa0018cc5ef4fb3852670864921": "\\langle \\mathrm{axis}, \\mathrm{angle} \\rangle = \\left( \\begin{bmatrix} a_x \\\\ a_y \\\\ a_z \\end{bmatrix},\\theta \\right) = \\left( \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix},\\frac{\\pi}{2}\\right)", "251049edc52e2dd12c45479e16e7de42": "P = K\\rho^{1 + 1/n}", "25104ae3e0ab8b0531e1b65f3a82bb8f": "t \\mapsto e^{tX}, \\qquad t \\in \\mathbb R", "251055b97d941e8309198c918c4fb527": "h \\; ", "2510c39011c5be704182423e3a695e91": "h", "2510cc52d1d6da1c57a9b461ebaa595f": " \\delta_i f = {f-f\\circ s_i\\over \\alpha_i}.", "2510ef50e6f55ad28e7bd866550b2fcf": " \\Delta v = \\pm 1, \\Delta J = 0 ", "2510f269804c9225fd2fb7438c2ea4a0": "\\det(M) =\\begin{vmatrix} a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{vmatrix} = a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.", "25119edbd6a4e77842e2bdd4b8373f59": "\\text{Power (MW)} = 0.85\\times 62.5 \\times 2800 \\times 480 \\times 1.356 ", "2512191aec19c0c4d31150b3a29fc730": "G(k) = (\\frac {B_1^+ + B_1^-} {B_1^- - B_1^+}) \\frac {1} {\\gamma_1}", "2512cf8fe0b277c46ac2f9cbfd873ab4": "\\mathrm{d} U = \\delta Q -\\delta W.", "2512f106c50874d9df041ec7287a5fc0": "= adx + b\\sigma'(x)dx + c\\sigma''(x)dx + e\\sigma'''(x)dx\\,", "2512f2c5c25acddc510d885fb78549e2": "\nV = a b c \\sqrt{1+2\\cos(\\alpha)\\cos(\\beta)\\cos(\\gamma)-\\cos^2(\\alpha)-\\cos^2(\\beta)-\\cos^2(\\gamma)}.\n", "251312d089d129e7601aa491ca46f8a8": " \\geq 1 - n^{-d} - n^{-2} \\geq 1 - \\frac{1}{n} ", "2513161aff07686e16706b6e53bc78cf": "u* = 1- \\frac {(\\delta+\\alpha+\\beta) \\sigma} {s}", "2513941340243f23c94bbb5611fa7d4e": "\\textstyle P(\\theta > 0.5 \\mid k, n)", "2513a9d5a0c040c4a27e1d8b41cfccec": " x_j = \\sum_k X_k \\cdot z^{k \\cdot j} = X_1 z^{1\\cdot j} + \\cdots + X_n \\cdot z^{n \\cdot j}", "2513bfa346dd311530deaebc461e40c4": "{1.17741 \\times 2^{N/2}} = {1.17741 \\times 2^{64}}", "2513c01addac92d01146cc8ac1d15aa0": "\\pi_{k}^{S}", "2513d98b7890af62450c5bfa0c650b66": "L_{4k+2},", "2513e49a55a44df7f603976a3d1a0f50": "A A^+ = I_m\\,\\!", "2513e546a51dc571bda3d6b600f65685": "T_d", "2513fd76ca54388d2af360a50e1b482f": "\\frac{L}{D} \\gtrsim 10", "2514049e919f1aafa67f512ee5ab402d": "k_s", "251410f3e334a087035bb64ee6aeb148": "V(G)\\setminus S", "25142b8236261abe3815a9a221a0dd8d": "m\\{x:\\, |Tf(x)| > a\\} \\le m \\left \\{x:\\, |Tf_a(x)| > \\tfrac{a}{2} \\right \\}+ m \\left \\{x:\\, |Tf^a(x)| > \\tfrac{a}{2} \\right \\}\\le 4a^{-2}\\|T\\|^2 \\|f_a\\|_2^2 +C a^{-1}\\|f^a\\|_1.", "2514443c4335be6fe0739dbe636b9a3f": "[SU(3)\\times SU(2)\\times U(1)]/\\mathbb{Z}_6", "25144a95720d66e8b8b37a94f7c0ea52": "c_0(-\\infty)^{n-2}\\,", "2514e999798ed17212f9b7e877a9bb0f": "u = \\bigvee D", "2514ee206e2d9b11c81e37e721b959fd": "\n\\frac{d}{dt} \\left( \\mathbf{p} \\times \\mathbf{L} \\right) = \n-m f(r) r^{2} \\left[ \\frac{1}{r} \\frac{d\\mathbf{r}}{dt} - \\frac{\\mathbf{r}}{r^{2}} \\frac{dr}{dt}\\right] = \n-m f(r) r^{2} \\frac{d}{dt} \\left( \\frac{\\mathbf{r}}{r}\\right)\n", "25159730f437291c74260dde3aa49f1e": "H(X) \\gets \\lg(S)", "2515b51a8baa4ba411e7daba3ebf8ebf": "\\ell_{OB}\\cdot \\ell_{OD}=y(y+2x)=y^2+2xy ", "25160192b44f50bc7d38f3f7508e9a1d": "\\left[P\\left(\\eta\\right)+p'\\left(0\\right)\\right]=-\\sigma\\eta_{xx}.", "25162f2714050ac3a4030f1de2eedeb8": "\\begin{align}\nQ_0 &= 0\\\\\nQ_k &= Q_{k-1}+\\frac{k-1}{k} (x_k-A_{k-1})^2 = Q_{k-1}+ (x_k-A_{k-1})(x_k-A_k)\\\\\n\\end{align}", "2516755bb6b32de0e005e9a8a9c70940": "[a;\\sigma,\\tau]=\\frac{\\theta_1(\\pi\\sigma a,e^{\\pi i \\tau})}{\\theta_1(\\pi\\sigma ,e^{\\pi i \\tau})}", "2516dd3c734e1795c809446c2ec0eb19": "\\chi(\\Sigma)=-26", "25172e21f15f3198e0699f00d35f206a": " \\frac{1}{4}\\left( \\ddot{h}_{\\hat{\\theta}\\hat{\\theta}} - \\ddot{h}_{\\hat{\\phi}\\hat{\\phi}} \\right) = -R_{\\hat{t}\\hat{\\theta}\\hat{t}\\hat{\\theta}} = -R_{\\hat{t}\\hat{\\phi}\\hat{r}\\hat{\\phi}} = -R_{\\hat{r}\\hat{\\theta}\\hat{r}\\hat{\\theta}} = R_{\\hat{t}\\hat{\\phi}\\hat{t}\\hat{\\phi}} = R_{\\hat{t}\\hat{\\theta}\\hat{r}\\hat{\\theta}} = R_{\\hat{r}\\hat{\\phi}\\hat{r}\\hat{\\phi}}\\ ,", "2517369c29f4d583fb115f4cdc3c7b46": "H_{jk}=2\\sum_{i=1}^m \\left(\\frac{\\partial r_i}{\\partial \\beta_j}\\frac{\\partial r_i}{\\partial \\beta_k}+r_i\\frac{\\partial^2 r_i}{\\partial \\beta_j \\partial \\beta_k} \\right).", "25179ea592df9c7eb7b55172ec0d0e03": "\\forall g_1,g_2 \\in G\\;\\; f(g_1 g_2)=f(g_1)f(g_2)", "2517a9579d887eb51869981920f99e44": "TE = \\omega =\\sqrt{\\operatorname{E}[(r_p - r_b)^2]}", "2518177c1e1a44b330e3671a3abeaa7d": "l \\in L", "2518301a3461fe4cf1555427ec80a424": "E_{\\gamma'} = hf'\\!", "25184a2d8fc43075e4fa3e72198f6f6c": "d J_S(t)", "2518d30b2602ca9fde4a416076c761c8": "(2 D)^2 = d^2 (1 + \\cos \\alpha)^2 + p^2", "2518db83877cc10c4ae9d8ba96918bae": "F_{p,n-p}", "2518fe5277d1dd3ebf3508810ed3c5a4": "x(t+1)=A x(t)", "25197ab000bc8117d287a84dc51fbfb2": "V ( e(u, P), P) = u ", "25198a9d78486aa67e265053095fa762": "\\displaystyle \\beta_n", "2519b299f4a20a25a371fafeb9d0cb44": "\\textstyle (b_n)_{n\\geq0}", "2519d32fe90b632c473ca2c1e1e49e8b": " K(BG) \\cong R(G)^{\\wedge}.", "2519ef98bec9314266fc7d111d9fce0a": " 2.44 \\lambda \\cdot (f/\\#)", "251a0977e76f3fced3f85b7c6e9d0134": " \\dot{x}=f(x,r)\\,", "251a367597bb5bfbf269db7214ec1e0d": "E_\\nu = J_\\nu \\cdot W \\cdot \\rho", "251a3de35a5b50f3356dd7e4454a7f50": " \\epsilon = \\hbar \\omega ", "251aaa1e171349ddc454ac5134bb63b3": "\\hat{H} \\rightarrow -\\hat{H}", "251ad222abc2421b9e7cf63f4e605e12": "X\\to \\mathcal{M}_{fg}", "251b0166793505d39e34da77651ea565": "p \\to p ", "251b2cc19ca3c4aeaafc1250df847c40": "V_a =\\,", "251b811a6dd69d7d9faa4bd7ce5d9b8a": "n = \\sum_{i = 1}^{c + 1} \\varphi^i(n),", "251bd15dbe5045f451baa2c4a04c7b7c": " \\omega = \\frac{E_+ - E_-}{\\hbar},", "251bf594f207311dd2152b890c1184e3": "\\{1,\\dots,n\\}", "251c074f11a2adec72228d7b4bf822d1": "\\overline{r}m:=rm\\,", "251c78ad41e62b76bb30af46e86baa31": "\\mathbf{W} ", "251c88f7760037e7b10c274ce0989501": "\\{\\to,\\bot,\\Box\\}", "251cb0150094f9a642fc36a608b3b3c7": "\\Omega^{^+}", "251cc7df3616c65649d4791db844f58c": "A \\times A \\le 2", "251ccaf6ff19a7f1e7444ba012c8be7c": " r''(x) = -3R_{max}\\{[4x(1-x)]^{-5/4} (1-2x)^2 + 2[4x(1-x)]^{-1/4}\\}", "251d2762a233ed83330de44b18b14722": "\n\\mathbf{\\hat \\beta }_\\mathtt{GLS} = \\left( \\mathbf{q}^\\mathbf{T} \\cdot\n\\mathbf{C}^{ - \\mathbf{1}} \\cdot \\mathbf{q} \\right)^{ - \\mathbf{1}} \\cdot\n\\mathbf{q}^\\mathbf{T} \\cdot \\mathbf{C}^{ - \\mathbf{1}} \\cdot \\mathbf{z}\n", "251d357c35814dcd39e7769210076a06": "\n f(k;\\rho)\n \\approx \\frac{\\rho\\,\\Gamma(\\rho+1)}{k^{\\rho+1}}\n \\propto \\frac{1}{k^{\\rho+1}}\n .\n\\,", "251d66bf3dae93705a252e5337edf362": "x\\subseteq y\\subseteq d(R)\\,", "251dfe163b89f528e1dd5e1a20e060a8": "a := b + c", "251e00f8d165c93f69778ead938a36e9": "\\, (1-p+pe^t)^n", "251e354587e0c5d29579d3830444b949": " L = \\sqrt{D \\tau_\\mathrm{bulk}}", "251e5e78943b0ddcdcd4993e102d9104": "{Y \\over R}", "251eaea5b19d866d0d2953a0ee0cf341": "2^{n + m}", "251edaa02ee7eb6557a2c2b952093c2f": "\\lim_{\\lambda \\to \\lambda_{0}} \\frac{B(\\lambda) - B(\\lambda_{0})}{\\lambda - \\lambda_{0}}", "251f430c3024b6864ccf68e3a975a8a5": "m_9", "251f439513749e51bb87010ca526a4cf": "J_\\alpha(z)\\sim\\frac{\\exp\\left(-i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)}{\\sqrt{2\\pi z}}\\text{ for }0<\\arg z<\\pi", "251f43dff90d27c060706c621e2f837e": "2^L", "251f460eca2ce7f8d63bacceb3dca659": "\\frac{\\partial^2 \\Psi}{\\partial t^2} = c^2\\nabla^2\\Psi - \\left(\\frac{mc^2}{\\hbar}\\right)^2\\Psi \\,\\!", "251f5260cb3388faaaa7b0054a810b4a": "U^2 V^2 = 0\\,", "251fc379065ae9668fa68b1c9cd61675": "\\int x\\,\\operatorname{arcoth}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{arcoth}(a\\,x)}{2}-\n \\frac{\\operatorname{arcoth}(a\\,x)}{2\\,a^2}+\\frac{x}{2\\,a}+C", "252038d1a6b80e3c0afc8ece6f19c9b3": "\\operatorname{var}[X] = \\frac{\\alpha\\beta}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}\\!", "252053821c08ccbdcebf8936c5bdd7a3": " s-1 ", "2520978a04c536e26f52d4044dc0bef8": "\\sum_x f(x)\\Delta g(x)=f(x)g(x)-\\sum_x (g(x)+\\Delta g(x)) \\Delta f(x) \\,", "252098e684a8b6077cac50a27f937add": "|Y|<|X|", "2520ed8a38f75d2bdeb14869af2fb45c": "K_6\\,", "2521125ef5a8f0cf0d1a5263f62b0b89": " Q^{-1} \\sum_x \\sum_y \\omega^{x y} \\left|y\\right\\rangle \\left|f(x)\\right\\rangle.", "25211c4704deb25a33a7772ca8695418": "\\omega,\\omega^{\\omega},\\omega^{\\omega^{\\omega}},\\dots", "2521514c197f9e53fb3d7e6ec51eef63": "-\\mathbf{j}_r", "252169e59f5c3823a3f0bf1ac0f88753": "P \\le q^{H_q(H_q^{-1}(\\frac{1}{2}-\\varepsilon)) \\cdot 2k} = q^{(\\frac{1}{2}-\\varepsilon) \\cdot 2k} = \\frac{q^k}{q^{2\\varepsilon k}} ", "2521a31207feeebd95b31e1d72fa1696": " \\theta = 2\\arccos\\frac{d}{R} = 2\\arcsin\\frac{c}{2R}", "2521c07da56242d9c2647134dfeb1a75": "G_{IC}=\\frac{3E^* \\delta^2 t^3}{16 L^4}", "2521fdab5ed578f00c634ab23460357b": "\\gamma '' = \\gamma_1\\ast \\cdots \\ast \\gamma_{j-1}'\\ast\\gamma_{j+1} \\cdots \\gamma_m", "25223af2ca995f8f4201f0ebcaf9a3af": "I_L = {A \\over \\omega L} = {A \\over 2\\pi f L}.", "252244098f0ff5efcc6bdb8c17bc4a28": "e_i + \\omega_i \\ge d", "2522569284e3b4ef914a360cd45d95bc": " \\chi_\\mathrm{red}^2 = \\frac{\\chi^2}{\\nu} = \\frac{1}{\\nu} \\sum {\\frac{(O - E)^2}{\\sigma^2}}", "25227b50ca3cad66143c4ead83307970": " \\hat{n}_{\\nu_j}|n_{\\nu_j} \\rang=n_{\\nu_j}|n_{\\nu_j} \\rang", "2522a2600e1ec72ca00a8ef6ee6a6888": "\\frac{\\partial {\\rm tr}(\\mathbf{AXB})}{\\partial \\mathbf{X}} = \\frac{\\partial {\\rm tr}(\\mathbf{BAX})}{\\partial \\mathbf{X}} =", "252307d8397acf15be49b316720e622f": "g(x)=x", "2523358aa2e2d98a86baf120d342b69c": "\\mathcal{V}^{\\mathcal{A}^{op}}", "252367bed4ec751f1969f4f038b004d5": " x_1 ", "25236f1df7abf414351793e89b557190": "f(x) \\to g(y)", "252387f90c875205b3da15bb1845d285": "H = - \\zeta {\\epsilon_{ijk} F_{ab}^k \\tilde{E}_i^a \\tilde{E}_j^b \\over \\sqrt{det (q)}} + 2 {\\zeta \\beta^2 - 1 \\over \\beta^2} {(\\tilde{E}_i^a \\tilde{E}_j^b - \\tilde{E}_j^a \\tilde{E}_i^b) \\over \\sqrt{det (q)}} (A_a^i - \\Gamma_a^i) (A_b^j - \\Gamma_b^j) = H_E + H'", "2523ae54c47852a8cc196f81711645d1": " \\lambda_n = {n \\choose 2} \\frac{1}{N_e\\tau}", "2523cdb202c2dee9ce97c9f68096fd27": " C(t) - P(t) + D(t) = S(t) - K \\cdot B(t,T)\\, ", "2523d9d37898495692dfcb3b20f07f1e": "A^{2} S^{2} \\left|F(\\mathbf{q}) \\right|^{2}", "2523fd01e090cb397b54b00e16e7dda8": "\\frac{dL}{dt}=[P,L]", "252409d7bc136e9c7263ca5fd1565704": "\\{\\to,\\neg\\}", "25241c06b7b309ae269afc7db958e1aa": " I = \\iint z^2\\; dy\\; dz,", "252457dad3d625c6a447b665098dcdf1": "\\left|{\\alpha \\choose k} \\right|^2=\\prod_{j=1}^k \\left|1-\\frac{1+\\alpha}{j}\\right|^2 \n\\leq \\left( \\frac{1}{k}\\sum_{j=1}^{k} \\left|1-\\frac{1+\\alpha}{j}\\right|^2 \\right)^k. ", "252489a766af284a4a602a98084c9eae": "a \\ ", "25248e6b08e40a1e228e9241d20f109b": "L_\\rho(\\gamma)=L_{\\rho^*}(\\gamma^*)", "2524b581e7629fb9cb4a427c1c287eeb": "X_t^\\tau=X_0+\\sum_{s=0}^{\\tau \\and t-1}(X_{s+1}-X_s),\\quad t\\in{\\mathbb N}_0,", "2524d282675f03f34a85fc9dbfdcb75c": "p_0\\subset p_1\\subset \\ldots \\subset p_d", "2524e5a4284c8f3c0baf719ae69cbef6": " {\\mathcal L}^2_2: L=Lclm(l^{(1)}_2,l^{(1)}_1); ", "25250b2c61331aca9522bc76895d7bdc": "b = \\frac {fm_\\mathrm s} {N} \\frac { x_\\mathrm d } { D }\\,.", "25252c6d01e06067af9005756c27c31b": "\\kappa\\neq\\lambda\\mu", "25255508ab4a7f16f62de203ccbe96f9": " \\boldsymbol{\\sigma}=\n\\left[{\\begin{matrix}\n \\sigma_{11} & \\sigma_{12} & \\sigma_{13} \\\\\n \\sigma_{21} & \\sigma_{22} & \\sigma_{23} \\\\\n \\sigma_{31} & \\sigma_{32} & \\sigma_{33}\n\\end{matrix}}\\right]\n", "2525893fd2e0f6506aa355bd57863b1c": "(r\\bar{r}+b\\bar{b}+g\\bar{g})/\\sqrt{3}.", "2525e1b5206f5300e3fb8aeea46590b7": "{\\bar{N}}_4", "2526147fdba451085a02683a7fd5f2b4": "\n V_\\mathrm{st}\n = \\sigma \\iint dx\\, dy\\; \n \\left[ \n \\sqrt{ 1 + \\left( \\frac{\\partial \\eta}{\\partial x} \\right)^2\n + \\left( \\frac{\\partial \\eta}{\\partial y} \\right)^2} \n - 1\n \\right]\n \\approx \\frac{1}{2} \\sigma \\iint dx\\, dy\\; \n \\left[ \n \\left( \\frac{\\partial \\eta}{\\partial x} \\right)^2\n +\n \\left( \\frac{\\partial \\eta}{\\partial y} \\right)^2 \n \\right],\n", "2526a5c6396d902e770a177c0b18fab9": "\\left[\\Pi_i(\\mathbf{r},\\ t),\\ A_j(\\mathbf{r'},\\ t)\\right]=-i\\hbar \\delta_{ij}\\delta (\\mathbf{r-r'})\\ , ", "2526b82881f1a421d7934fcf01b3b787": " C = e^{-r_{DOM} T}\\Phi(d_2) \\,", "2526c157039ee718758651ec0fce38b2": "\\Box\\phi\\to\\phi", "2526d419fa11a3531cecdf869096acb4": "\\frac{\\mathrm{d} \\sin \\theta }{\\mathrm{d}s} = \\frac{\\mathrm{d}}{\\mathrm{d}s} \\frac{y'(s)}{\\sqrt{x'(s)^2 + y'(s)^2}}", "2526e28b43e7c658a133d9da7c209352": "\\Psi (x) = \\langle x| \\phi\\rangle", "2526f4f31e4527c5b0e174883525089e": "P(\\Omega)=1.\\,", "25271175c4285c948207d70b9797233f": "[\\![\\neg \\phi]\\!]_i = S \\smallsetminus [\\![\\phi]\\!]_i", "252724839704aed8ab35a4916428087a": "r_D", "25272eeea61c6f971332fa0406d01314": "4n+1", "25273a18977833db3dc0f0fea7963624": " \\{ x_k \\} ", "2527e6b3f4bcfb1164781f59df730464": "\n\\alpha = \\frac{n_i}{n_n}\n", "25280223de731b21e5d287cbd3e36648": "a=n-1+2/n", "25282363f65988c29a4a04c9fff0e950": "Q_P = \\left( \\frac{mP}{2 \\pi \\beta \\hbar^2} \\right)^{P/2} \\int \\cdots \\int {\\mathrm d}x_1 \\cdots {\\mathrm d}x_P e^{-\\beta \\Phi_P (x_1\\cdots x_P;\\beta)}", "25282978c6dd325321c01dfd5a45a6e0": "6 \\cdot m\\ ", "2528442474fe24e441e26de82836d125": "z_{t+k-1}", "25285c6a18af2c7fde66decb5a9db7dd": "\\sqrt{4 k_B\\cdot T\\cdot B\\cdot R}", "25289c2360eb3a15723cf04c639a786c": "\\sigma_i = 1^{\\otimes i-1} \\otimes \\check{R} \\otimes 1^{\\otimes n-i-1}", "2528a2c43c74a683cbe8ad03f948c2e7": "\\Omega\\approx \\int_0^\\infty \\ln\\left(1-ze^{-\\beta E}\\right)\\,dg.", "2528d54c4e828896ebc6a74a813d38a5": "[(\\mathbb Z_3^7 \\rtimes \\mathrm S_8) \\times (\\mathbb Z_2^{11} \\rtimes \\mathrm{S}_{12})]^\\frac{1}{2}", "2529216eb18546a71fb3f90a82aea019": "C \\dot = D", "252944da9a860bd83d6af5b5d7c4ee52": "\\mathfrak{gl}_n \\times \\mathfrak{gl}_m", "252954a8a456f85e7692878f53bbeb82": "\n\\begin{bmatrix}\n0 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix}\n", "252969cde74f8930674f9d397242126f": " G_\\theta \\subset \\mathrm{proj}_\\theta F ", "2529801c16412170af4539dfbb463ac5": "(\\bar x - 2\\sigma/\\sqrt{n},\\bar x + 2\\sigma/\\sqrt{n}).", "252994c9859d4faddb09a4b2f7728a89": "y_i=wx_i+b", "2529b01e560826ba25853b5702096a96": "y_1(x), \\ldots, y_n(x)", "2529c353780da79136fea0d5d0d77391": "O(\\log^6 q)", "252a0b496f9e3fe4a85f134165d47db6": " r = k[sucrose]\\, ", "252a1000ca3cef3841e2e4cb377e855b": " a_1 = - \\sum x_i = -t_1 ", "252a28040fd777ba9cbf861d21291f18": "\\frac{\\Gamma(2n+2\\alpha)\\Gamma(1/2+\\alpha)}{n!\\,2^n\\,\\Gamma(2\\alpha)\\Gamma(n+1/2+\\alpha)}\\,", "252a7bb801067ae1570b5756e38f57f4": "\\scriptstyle\\mathcal{S}", "252aef9c3bdf27e8b38a235263134762": "v'_p=Q_{pq}u_q", "252b44b53debbdfe35ab1c20631f92ce": "y' = \\frac{y}{y_c}", "252b57b47e5b225f3fb591572d84809b": "I(f)", "252b9745d226c4589cdaa6f40d5a8532": "\\Box n = -\\nabla^2 (|u|^2_{})", "252babdcb94e84237a256fc257a2713d": "\\mathbf{x}^\\prime", "252bbc8b495c965ffeecea51ae4c2711": "\\scriptstyle X \\,\\sim \\,\\text{Frechet}(1.7,1)", "252bd6db39ce21c2ec843cd88eb7c749": "\\operatorname{dim}B/\\mathfrak{m}_A B \\ge \\operatorname{dim}B - \\operatorname{dim} A.", "252c1ccab5087fbab854a56b73ac0c6e": " \\int^x \\frac{P_1(\\lambda)}{P_2(\\lambda)}\\,d\\lambda + \\int^y \\frac{Q_2(\\lambda)}{Q_1(\\lambda)}\\,d\\lambda = C \\,\\!", "252c4fcd10a6f3cf1d88d43fe76b3d78": "\\mathbf F = m\\mathbf a = m\\mathbf A + m\\mathbf a'", "252c61ff0ddc7290061310f2549a7105": " \\operatorname{let} x : \\operatorname{get-lambda}[x, x\\ f = f\\ (x\\ f)][x:=x\\ x] \\operatorname{in} x[x:=x\\ x] ", "252c702ee4cd7552bb08de67d2c190ed": "\\frac {d}{dx} \\, x^{\\left [ n \\right ]} = n \\, x^{\\left [ n-1 \\right ]}", "252c72fc62729e136eaee52b384afcee": "f(xy)", "252c81ff3ea78fe65494aa34b37b9151": " \\epsilon_i\\epsilon_j = \\delta_{ij}\\epsilon_i ", "252c8d879fa2f4f34de5d38408180875": "-1<\\sin x<0", "252c8f52673beded4e9bf339c40f4fa4": "\\tilde \\nu /cm^{-1}= \\frac{1}{\\lambda /cm} = \\frac{\\nu /s^{-1}}{c /cm \\ s^{-1}} = \n\\frac{\\nu /s^{-1}}{2.99792458 \\times 10^{10}}", "252c902f64d80aaa0464b5ed3393d042": "w''\\in W", "252cae61fa653fa4cd005710a212e658": " d = {1 \\over 2} at^2 ", "252cb2aa03a67003c67dc927b2433d33": "E={mv^2 \\over 2}", "252cbdbb1d3ff4ed38eb6476a70902b8": "B(x;r)\\cap B(y;s)", "252cc0500ec93eb00a839062e0def483": "E=\\sum_{x}\\left [F(Ax+h)-(\\alpha G(x)+\\beta)\\right ]^2", "252cd2d871b205f894cc37e338395ebf": "x^2 - y^2 = 1 ", "252d3754c0db62a55b9e25c870a524a5": "n\\times m", "252d3991d2d5d6f74785c3beda3f81cd": "\nP(E_\\gamma,\\theta) = \\frac{1}{1 + (E_\\gamma/m_e c^2)(1-\\cos\\theta)}\n", "252d7a61a0b0ade9656f37755ebcfb95": "D = 2( A_x(B_y - C_y) + B_x(C_y - A_y) + C_x(A_y - B_y)).\\,", "252dfadab7ba2f60caa5966ec6892679": "\\mathbf{f} = - \\nabla P ", "252e10c14628d682991fa536a296b58b": "I_{\\mathcal R}", "252e248c96703881aaae9b0b53ce1618": " \\frac{\\partial^2 U}{\\partial s\\,\\partial p}<0,", "252e2ab7c953c90e6d04370b2574119e": " ds^2= g_{\\alpha\\beta}dx^\\alpha dx^\\beta ", "252e2de7f1f06ca485765d556aaf36e5": "\\varepsilon = 0", "252e45c62b0e80987c3c01c7f09cf41f": "V(t) = \\dot{X} (t). ", "252e80401de4d6d945f335fae289b773": "\\textit{dog} \\subseteq \\textit{mammal}", "252ea420a8ac99da86b1cf61ceaab992": "H^1(k^{\\text{al}}/k)", "252ed1ace626b1b0da066475c5e53670": "\nh_z'(:,:,-1) = \n\\begin{bmatrix} \n+1 & +2 & +1 \\\\\n+2 & +4 & +2 \\\\\n+1 & +2 & +1 \n\\end{bmatrix}\n\\quad\nh_z'(:,:,0) = \n\\begin{bmatrix} \n0 & 0 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0 \n\\end{bmatrix}\n\\quad\nh_z'(:,:,1) = \n\\begin{bmatrix} \n-1 & -2 & -1 \\\\\n-2 & -4 & -2 \\\\\n-1 & -2 & -1 \n\\end{bmatrix}\n\n", "252ee2a99a895f36d84ac0c53c192c96": "\\Bigg[\\frac{\\pi}{\\theta}\\Bigg] =\\left[\\frac{\\theta}{\\pi}\\right], ", "252f14ed53109d707e1aed7496c91d21": "\\min(c(A,B)-f(A,B) ,c(B,C)-f(B,C), c(C,D)-f(C,D))=", "252f1c48f3ddfa3167b1e2c47b0f2d2b": "P(recalling~m_{ab})~=~P(similarity(a,m_{ab})>criterion)", "252f5df1a5605c0cd21e9e9e47314ee8": "D = X \\oplus Y \\oplus Z", "252f643d7d66d08b72c830f7d62254ce": "l = \\rho V \\Gamma\\!", "252f9846458ef401bb19ec13240228ae": "x \\mapsto x e^m", "252faf0f410727ffa46828abaa601cfe": "\\!\\,p=x^2+3y^2\\text{ if and only if } p=3 \\text{ or } p\\equiv 1 \\pmod3.", "2530008dfead9e203878cbe7a5f8ae38": "\\kappa\\ge\\aleph_1", "25307939f61f821cbaced6f49840afda": "\\Gamma(5 - 3i) \\approx 0.0160418827 + 9.4332932897 i.", "2530d677b034a008acdda4faf8d81cbd": "\\begin{pmatrix} 1 & 0 \\\\ \\frac{n_2-n_1}{R_2n_1} & \\frac{n_2}{n_1} \\end{pmatrix} \\begin{pmatrix} 1 & t \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ \\frac{n_1-n_2}{R_1n_2} & \\frac{n_1}{n_2} \\end{pmatrix}", "2530ef679c88e7e49285b0ae412a9032": "{\\,\\!256^{256^{256^{256^{257}}}}}", "25311b98169e49fadc9ff483131e3515": "\\nabla^2 \\varphi - \\frac 1 {c^2} \\frac{\\partial^2 \\varphi}{\\partial t^2} = \\frac{1}{\\varepsilon_0} e \\psi^{\\dagger} \\psi", "253173b63bd4bbc42e1afa5ca6e82331": "m,n\\in\\mathcal{H}_L", "2531c96acddc48a80b8dbdff154162b2": " \\sum_{n=0}^{\\infty}\\log(n+a) ", "2531cabd3d99df8dbcfb1750b13cfcdd": "2\\theta_0", "2531cdfb8e31efa0a6a36a283c0e820f": "\\eta^{\\alpha\\beta}", "2531e2dcb98cf5efa58a6f3a80da9564": "g=2", "2531f34408f46344aac5c58a14b3f1e9": "P \\vee (\\neg P)", "253220f1715e47c49285ad1a29950104": "\\lVert f - L_n(f)\\rVert_\\infty \\le (1 + \\lVert L_n\\rVert_\\infty) \\inf_{p \\in P_n} \\lVert f - p\\rVert", "2532964bdf9169751be561fdf6860288": " (x^2 + 3x -4)y^{[3]} -(3x+1)y^{[2]} + 2y = 0", "2532accc186b203e263542dbf707955b": "\\textstyle \\Gamma_{\\mathrm f}(V)", "2532b315a0da83e2c33f97856f49e435": " \\int dx | x + \\epsilon \\rangle \\langle x | \\psi \\rangle = \\int dx | x \\rangle \\langle x - \\epsilon | \\psi \\rangle = \\int dx | x \\rangle \\psi(x - \\epsilon) ", "25334350eac8dc810ec1af74079de9a2": "F(t)=Kt+\\psi \\, \\Delta\\theta \\ln \\left[1+{F(t)\\over \\psi \\, \\Delta\\theta}\\right].", "253375e09f2c8034e3460d54c4255e5b": "H_{\\mathrm{top}}(u) = \\lim_{n \\rightarrow \\infty} \\frac{\\log p_u(n)}{n \\log k} \\ . ", "2534671c2fca22d204122bc754559a6d": "\\int_{-1}^1 (1-x)^{\\alpha} (1+x)^{\\beta} P_m^{(\\alpha,\\beta)} (x)P_n^{(\\alpha,\\beta)} (x) \\; dx =\\frac{2^{\\alpha+\\beta+1}}{2n+\\alpha+\\beta+1} \\frac{\\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)}{\\Gamma(n+\\alpha+\\beta+1)n!} \\delta_{nm}", "25347dc20efb4a3c4c12f1138af37b68": "\\psi_{0} = 0 ", "2534bd1325eff468c3a34599f00936fd": "\\,F_m", "2534dd5abe10b08435a57f96cd95e5d1": "\\ \\sigma_1 \\le \\sigma_y \\,\\!", "253501fd7bf03d0cf3ccbff7396a900a": "\\langle n! \\rangle", "2535035f5e606db6255481e0dde7f8d6": "\\beta = 1 \\,\\!", "25350f3497f973fbd51840327267016c": "N+\n1", "2535276ba191907b194e8a3ae0927c85": "x_k = g(x_{k-1}) + w_k \\,", "2535bc8bb3cc7d5f8b5b7443f9bf7d51": "E(t) = E_{0} \\sin \\omega_{0} t", "2535c595455c9896782f36c405813169": "K = \\int_0^{\\frac{\\pi}{2}} \\frac{d\\varphi}{\\sqrt{1-k^2 \\sin^2\\varphi}}", "25362dc7326f52f03903c7e90ef5d097": "T_f + K\\;", "25362f222a54592f07bb2e5f696ce2eb": "\\Lambda(x) = \\prod_{k=1}^\\nu (1 - x X_k ) = 1 + \\Lambda_1 x^1 + \\Lambda_2 x^2 + \\cdots + \\Lambda_\\nu x^\\nu", "25366d983b0ad35728b8754d889c425c": " F_{tot} = 0 = F_{el} + F_{f} + F_{ret}", "25367509c8f05f84cfaccebdf7afd2e7": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} = \n \\left\\{\n \\int_{-h}^h \\begin{bmatrix} C_{11} & C_{12} & 0 \\\\ C_{12} & C_{22} & 0 \\\\\n 0 & 0 & C_{66} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix}\n", "253681d8085cb847dd94aaa4eec929cb": "Profit=Py", "2536aabb4f39977e0779588f6f1829d4": "\\textbf{Y}_{k\\mid k} = \\textbf{Y}_{k\\mid k-1} + \\sum_{j=1}^N \\textbf{I}_{k,j}", "25370ad2289ae803d06fcae9391cd014": "\\begin{align}\n \\mu_{X \\cup Y} &= \\frac{1}{N_{X \\cup Y}}\\left(N_X\\mu_X + N_Y\\mu_Y - N_{X\\cap Y}\\mu_{X\\cap Y}\\right)\\\\\n \\sigma_{X \\cup Y} &= \\sqrt{ \\frac{[N_X - 1]\\sigma_X^2 + N_X\\mu_X^2 + [N_Y - 1]\\sigma_Y^2 + N_Y\\mu _Y^2 - [N_{X \\cap Y}-1]\\sigma_{X \\cap Y}^2 - N_{X \\cap Y}\\mu_{X \\cap Y}^2 - [N_X + N_Y - N_{X \\cap Y}]\\mu_{X \\cup Y}^2}{N_{X \\cup Y} - 1} }\n\\end{align}", "25371db8886a432545653efa3890f916": "club ", "253743d2bc60c0a9403d1e12ed243a44": "\\alpha = 4", "2537730551a67b48e154fe9b0ccfbd80": "\\gamma=\\sinh^{-1}{\\sqrt{ZY}}", "2537760f4407b5b66ef4ab8e35a5d554": "\n\\operatorname{Li}_2(\\rho^6) = 4 \\operatorname{Li}_2(\\rho^3) + 3 \\operatorname{Li}_2(\\rho^2) - 6 \\operatorname{Li}_2(\\rho) + \\tfrac {7}{30} \\pi^2\n", "25377e3dda4936694799023001ad3dec": " \\Pi^{\\pm} ", "2537a7dd4b3151e22d524b26f76a0a1e": " \\rho = \\rho_\\text{f} -\\nabla\\cdot\\mathbf{P}. ", "2537b7d80d015adb0c918ed22a7facd6": "2^{n^2}(1-1/2)(1-1/2^2)\\cdots(1-1/2^n).", "25380848e46b356b9f45ce651454e882": "P_2=(x_2,y_2)", "25388fba62f375b638bb63a32976f4df": " \\int f^+ \\, d\\mu ", "253891b0d9cbe442b08b712402e79d4e": "\\widehat{\\lambda M}= 00\\widehat{M}", "25389b2aabac718ce6248d3461e2823f": "S^2\\times S^1", "2538a035e8c3b9705615e139ff07c510": "\\begin{align}V_1^2 &= V_a^2(1+a)^2+4\\pi^2r^2(1-a')^2\\\\\n \\mbox{d}D &= \\frac{1}{2}\\rho V_1^2C_D\\mbox{d}A = \\frac{1}{2}\\rho C_D[V_a^2(1+a)^2+4\\pi^2r^2(1-a')^2]b\\mbox{d}r\\end{align}", "2538a9c77d7e8d43edecef4f12aa39d0": "S_{21} = {2 Z_0 Z_{21} \\over \\Delta} \\,", "2538bf118cd2548f1a6daf2a9949b594": "a_\\lambda=\\sum_{g\\in P_\\lambda} e_g", "2538cca9cd259bc00089dacddcdc8025": "\\mathfrak{H}^2", "2538d233fcde1fb7cbb81704f0934eba": "\\psi_{i_1\\dots i_\\ell}", "2538d5218e0536176ced4dbfd005a600": " \\bar{\\sigma} ", "253918fe42259276aea497527ea0bb7e": "U_i \\cap U_j \\neq \\phi", "25392f574efe9e01337a78abbd44e9f5": "D_{ep}", "25395b9e7b08ab3941b209acd96d0c52": "M^{\\prime \\prime}", "2539b0eaef09c39961bdae3e3217b6f9": "\\bar{\\nabla}", "253a0b21d77247ef84ff9a7b14a76702": " T \\Delta S_{SA} \\,", "253a169f1411d70ded3b77fdcd427fb2": "G_\\infty, G_i, \\tau_i", "253a20d5692a7f9b580ebc19794d7d48": "\\textstyle{\\frac {\\log(4)} {\\log(3)}+\\frac {\\log(2)} {\\log(3)}=\\frac {\\log(8)} {\\log(3)}}", "253a6f7c3fb04b3a6c6c8b561a330349": "(q/m)\\vec E=\\part \\vec v_s/\\part t+(1/2) \\vec \\nabla v_s^2.", "253b44ec81de66df3fce13fb21ba782e": " C : V^* \\otimes V \\rightarrow k ", "253c9a2f87701bb0871fc4a79c63a908": "f={O(k)}", "253cb43a575f663ff78d74eb7de7ae0e": "\\N_0", "253d57f637ccdc30b513f8e6c0d99a48": "2^{j+1}-1", "253d6f7cacdc7af9cb80d8fd2708ac61": " \\mathbf{\\hat T}(\\lambda) ", "253d6f830af505a128e8bcd73a930d94": "\\varphi_\\alpha \\colon U_\\alpha \\to {\\mathbf R}^n", "253d8fc2cee6ead59eec545ca7baedc7": "\\nabla_{\\mathbf X}\\mathbf u\\,\\!", "253db92a514fff4d73f63034ea46635a": "\\int_{\\mu^\\circ }^\\mu {d\\mu } = \\int_{P^\\circ }^P {\\bar VdP}", "253de0d24465abd2294d7aa234b4bfaf": " \\int \\cdots \\int_\\mathbf{D}\\;f(x_1,x_2,\\ldots,x_n) \\;dx_1 \\!\\cdots dx_n ", "253df31451013693e7659b47131a5fc5": "D_1+ D_2=7P_1+ 9P_2", "253e23bb5e72f125b0954f85fc47908b": "A^e", "253e6c1496f2c5124c07897e9fed7677": " \\Delta = 1 - ( L_1 + L_2 ) \\, ", "253e93545d645b46e10a771f9c2eefa0": "d_\\text{f}\\,\\!", "253f1ac21faf86ab8306d4dc4826b893": "3:4:5\\ ", "253f30871a50c26d798cdb6ce6ff13a2": " v_i = \\mathbb{I}_{ H_i } ", "253f4cecf653a35d4de16a4ac62b6509": "100_2", "253f6102780979e9b25919ecd38271a4": "s = 60 \\times m", "253f8cc4950445407a2a5296e9fa1c3b": " g= \\frac {G}{\\gcd(K,G)} ", "25400f376be4435bd631d544653a1b60": " \\psi^{(\\pm)}_g(t)=\\int dE\\, e^{-iEt} g(E)\\psi^{(\\pm)}", "2540416e5de096e5d1ccce004eaae434": "\\mu := G", "254049e4635d1a954b2007cf1b082ab7": " e^{(a)} (e_{(b)}) = e^{(a)}_\\mu e^\\mu_{(b)} = \\delta^{(a)}_{(b)},", "25408015549da6c15b025ef8dcfc1b79": "g \\,", "25408898feac32ce94ffe011afc1262d": "\\scriptstyle \\approx R_0", "25417c76d43dea5b5bc5c8cefe5c8463": "\\, m_\\mathrm{e} c^2 ", "2541c5915ac83d5626e7931ecb438d1b": "\\hat{c}_p", "25421de4e7036ed8c1d1a2aa8091e506": " L y(t) = f(t)", "25427e5e912393af18f27e6988bbb6ca": "\n\\begin{bmatrix} x & y & 0 & 0 & 1 & 0 \\\\ 0 & 0 & x & y & 0 & 1 \\\\ ....\\\\ ....\\end{bmatrix} \\begin{bmatrix}m1 \\\\ m2 \\\\ m3 \\\\ m4 \\\\ tx \\\\ ty \\end{bmatrix} = \\begin{bmatrix} u \\\\ v \\\\ . \\\\ . \\end{bmatrix}\n", "2542877c9bb97829cc57b7d798c0db2a": "\\lim_{N\\to\\infty}\\frac{\\#\\{p\\leq N:\\alpha\\leq \\theta_p \\leq \\beta\\}}\n{\\#\\{p\\leq N\\}}=\\frac{2}{\\pi} \\int_{\\alpha}^{\\beta} \\sin^2 \\theta \\, d\\theta. ", "2542b9eddf5af4037bc0d427d1e0a407": "{W_{I}(x, t)} =\n\\frac{\\hbar c}{2}\\mid \\sum_{k}\\sqrt{k}c_{k}e^{i(kx - \\omega t)} \\mid\n^{2}", "2542d7e6654f326ee8c97bc7230bc506": "\\varepsilon_{ijk} \\varepsilon^{imn}=\\delta_j{}^{m}\\delta_k{}^n - \\delta_j{}^n\\delta_k{}^m ", "2542f19f3bf1b32558f05c5436c85b9b": "\n \\sum_{n=1}^N\\sum_{c=1}^NQ_n^{(c)}(t)\\left[ \\sum_{b=1}^N\\mu_{nb}^{(c)}(t) - \\sum_{a=1}^N\\mu_{an}^{(c)}(t) \\right] \n", "2543246ad1e6be543dabc46a11e6f599": "F=(F_1,\\dots,F_n):\\R^{2n+1}\\to\\R^n", "2543559ca21eef709226ff0896efbb03": "\ndV = a^{3} \\sinh\\mu \\ \\sin\\nu \\ \n\\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right) d\\mu d\\nu d\\phi\n", "2543b9d7533c3a9bc39460920aa4cca1": "(g, h) \\circ (g', h')", "2543f79eff7ecf0087b750b553da8aa5": "C_{P}", "254437e58ff9923caab3e866110e2043": "(u,\\xi )", "254440538938893a89ed3c138951ae29": "P_i; P_j", "2544489079fd4bd046ff34b4754be610": " J_{x,y}\\sim |x-y|^{-\\alpha} ", "2544b5206106cf08739b1acc6498003f": "0=\\frac{\\partial}{\\partial x}[D_1 \\frac{\\partial C_1}{\\partial x}+D_2\\frac{\\partial C_2}{\\partial x} -C \\nu]", "2544ba8a3d6678eefbba525723e6f776": "|\\gamma'^2(t)|\\,\\!", "2544bcf8ddaed0661c85307a03ec8610": "d=2\\,", "2544c3e5a330516efba8b6641f41d349": "\\textstyle(x, y\\pm1, z\\mp1)", "2544ecc769e36b14357e27bf30e0a7c2": "M(u_1, u_2, \\dots, u_n)", "2544fc061f344ca9b9d7ef1c3e03044a": "G = \\int^T_0 k(t)x(t)dt", "25456431c0c15f8f14d28aa3c9f19a46": "y=r\\cos\\theta +\\frac1{32}\\varepsilon r^3\\cos3\\theta +\\frac1{1024}\\varepsilon^2r^5(-21\\cos3\\theta+\\cos5\\theta)+\\mathcal O(\\varepsilon^3)", "254592628f751cea5ccb8819aab5bea5": "=\\frac{1}{N} (\\mathbf{X * Y_N})_k, \\,", "25463b2b8e495ed00f3f4823d3fc03f3": "AF = \\operatorname{min}\\{100+4N_W+2N_D+N_R , 150\\}", "25468819f701e41188307ca0f02f0fa7": " \\left [ \\frac {\\partial f} {\\partial \\left (\\nabla^{(i)}\\rho \\right ) } \\right ]_{\\alpha_1 \\alpha_2 \\cdots \\alpha_i} = \\frac {\\partial f} {\\partial \\rho_{\\alpha_1 \\alpha_2 \\cdots \\alpha_i} } \\qquad \\qquad \\text{where} \\quad \\rho_{\\alpha_1 \\alpha_2 \\cdots \\alpha_i} \\equiv \\frac {\\partial^{\\, i}\\rho} {\\partial r_{\\alpha_1} \\, \\partial r_{\\alpha_2} \\cdots \\partial r_{\\alpha_i} } \\ , ", "25468eb68b750545294cdc2a53c02f38": "x=(x_1,...,x_n)", "2546c0adbf3dd8999fcd67606f809ae4": "(x_{1},x_{2}) \\succeq (y_{1},y_{2})", "2546c8e6b888eb37c53df2004700566c": "\\frac{d}{dt} (pe^{(1-q)t} + (1-p)e^{-qt}) = (1-q)pe^{(1-q)t}-q(1-p)e^{-qt}", "254742ebce7ea7ac552bd164a3583a63": "A - B \\in \\mathcal{R}", "25474f22e3282ef1d9d9f4ba593f6952": "M_x = \\int_0^2 (xy^2+y^3+y^2)|_x^{4-x}\\,dx", "2547777c940eaf010b8607935ce496eb": "2 \\leq k \\leq n", "25477b2c139abf95020e3c2ffc584e08": "u_1(\\mathbf{x},z_1) = \\frac{1}{ g_1(\\mathbf{x},z_1) } \\left( \\overbrace{-\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1(z_1-u_x(\\mathbf{x})) + \\frac{\\partial u_x}{\\partial \\mathbf{x}}(f_x(\\mathbf{x})+g_x(\\mathbf{x})z_1)}^{u_{a1}(\\mathbf{x},z_1)} \\, - \\, f_1(\\mathbf{x}, z_1) \\right)", "2547ada68717d8a20610002a185f0a9f": "\\left| \\alpha - \\frac{p}{q}\\right| \\le \\frac{A}{q^n} \\le A< \\min\\left(1, \\frac{1}{M}, \\left| \\alpha - \\alpha_1 \\right|, \\left|\\alpha - \\alpha_2 \\right|, \\ldots , \\left| \\alpha-\\alpha_m \\right| \\right) ", "2547b213cb02f569a046f088414c0024": "(x_{ij})", "2547e4ad6c49d3508def1b1e6cdc420c": "B^{-1}A", "254819d2ba4db43e243b6af833bfccb1": " w/(\\delta * \\sqrt{\\theta})", "254842ca03be8af623146ed7b5685b1f": "1\\leq m \\leq n ", "25485a3effeaa7f5c6316aab41dd447b": "\\displaystyle{g(a,b)=(ga,(g^t)^{-1}b).}", "2548efd01db09ab2a4d253d3819743e4": "?\\left(\\frac{\\sqrt3-1}{2}\\right)=\\frac{2}{7}.", "25490cd0e5045f30483952aeaf911f0d": "x+a_0", "254914c90b584b9f3eb12cc91a989301": "t\\in\\mathbb{T}", "25492e0dc919fcf95c86690ebd5d7a40": " 0 \\leqslant r \\leqslant 1", "25494f91159b4900496f57cfda71141f": "\n\\begin{align}\nLOD = Z & = \\log_{10} \\frac{ \n\\mbox{probability of birth sequence with a given linkage value}\n}{\n\\mbox{probability of birth sequence with no linkage}\n} = \\log_{10} \\frac{(1-\\theta)^{NR} \\times \\theta^R}{ 0.5^{(NR + R)} }\n\\end{align}\n", "25495af5f12e144da18a998fa075774b": "\\mathrm e^{tA}x(0)", "2549b9b685b6b5e5eb1887f856b760d0": "R_{AB}=0\\,", "2549f9df53fa89a44d956dc199327c4a": " \\phi = \\sin^{-1} \\frac {4A}{\\pi \\Delta T_s \\Delta d}", "254a3de2cb6b1b0a54111edab7211819": "\n\\begin{align}\ndf(t,X_t) =\\left(\\frac{\\partial f}{\\partial t} + \\mu_t \\frac{\\partial f}{\\partial x} + \\frac{1}{2}\\sigma_t^2\\frac{\\partial^2f}{\\partial x^2}\\right)dt+ \\sigma_t \\frac{\\partial f}{\\partial x}\\,dB_t.\n\\end{align}\n", "254a999a0efc83bd121c16d43123f81a": "\\nabla\\cdot\\mathbf{B} = \\mu_0 \\rho_m", "254ab56f8b99c2eb239fcc3ff44ab0a8": "\\vec{1} = (1,\\ldots,1)", "254b152ced5c3e1b201c711d5248d339": "V_\\beta/V", "254b5d753d392780c7959ef9449100af": "\\cos \\theta + j \\sin \\theta = e^{j\\theta},\\,", "254b78de4a280030e89ddb70e179baf3": "c_{BE} \\simeq 1.18.", "254bbbb0979358af22e17ce8aa103b92": "\\mathcal{H}(p,x,t)= \\frac{1}{2}p^2 + K \\cos(x) \\sum_{n=-\\infty}^\\infty \\delta(t-n)", "254bd4d1093d85cfef105d13b3e65327": "\\int \\frac{1}{x}\\,dx,", "254bff34b7fed0f8cc2e6fba9ca77d74": "N(n) = {n \\choose 2} p", "254c91ec208b3eb6a8e11c6f2245658c": "d(S)=y\\,", "254cdd7028c03c9ef71cca068827617e": "\\displaystyle{\\mathrm{Ad}(K)\\cdot X \\rightarrow \\mathfrak{t},}", "254dad8d7cd9fe82604b9004162cf653": " S = \\sqrt {1 -x (1 - \\frac {\\rho_L} {\\rho_G} }) ", "254dae1eda9bf5d7eecfe42817525457": "d_\\Gamma(\\alpha,\\beta)=\\sup_t\\ d_X(\\alpha(t),\\beta(t))", "254de19311d9fbea26cf072f8cbbe131": " \\frac{1}{|B|} \\int_B f(y) \\, \\mathrm{d}y - f(x) = \\Bigl(\\frac{1}{|B|} \\int_B \\bigl(f(y) - g(y)\\bigr) \\, \\mathrm{d}y \\Bigr) + \\Bigl(\\frac{1}{|B|}\\int_B g(y) \\, \\mathrm{d}y - g(x) \\Bigr)+ \\bigl(g(x) - f(x)\\bigr).", "254e238e7727536b116f56e0eefa5535": "R(u,v)w=\\nabla_u\\nabla_v w - \\nabla_v \\nabla_u w ", "254e2b1b88631d117378bb2bfff605a4": "\\omega(a'_1)p", "254e48412f5342d5be659e4bf68f72f5": "g(y) = \\int_0^\\infty f(x) K_{iy}(x) \\, dx ", "254e56bfbe171aaaa265920bd0c7d0d6": "\\alpha_2(t_1,t_2) = \\left(\\beta_1(t_1,t_2),t_2\\right) = \\left(\\frac{1}{(1+t_1^2)(1+t_2^2)},t_1 - \\frac{2t_1}{(1+t_1^2)(1+t_2^2)},\\frac{t_1t_2}{(1+t_1^2)(1+t_2^2)}, t_2 \\right).", "254e5d8374714d8f581c8a7b78e41854": "\\mathbf{S}_B\\mathbf{w}", "254e6411045bcd2fbcdad9098a15a91b": "\\ F(K,L)=AK^{b}L^{1-b}", "254e6b185f8b00bbaa6b21612328cdfe": "\\phi_0=1", "254e8ef8081a17541120c662cf9a7e1f": " H \\left| a \\right\\rangle = E_a \\left| a \\right\\rangle.", "254e9fb4e38dd529fc08d47f26abc522": "x[T]z", "254eac4804ea2ee88171e6f51368dc4d": "4\\arctan\\frac{1}{5} - \\arctan\\frac{1}{239} = \\frac{\\pi}{4}", "254efe90f4986539485f5b6363a6dc9f": "\\tilde{6}", "254f24909aebee6dfabd35153cbd9bbc": "\\scriptstyle T=2 ", "254f2f4f5954954aee0b8569bd94f4ec": "\\sin(\\theta) = 0", "254f3f7199dae27be7c539fd9807cd7b": "x\\mapsto (x,1)", "254f7b28c63701c72e615aeff70c2e33": "\\int_0^x dx = \\int_{I_O}^I \\frac{dI}{C_O - \\frac{k_O I}{L}}", "254f8bd4bb5cc2b69c99253578351797": "ik = -j\\,", "254fcdef530d12899f858adf6da29c6a": "a = mq+r", "255016e591501ca3eac1fab91e82f8c3": "V_{br}", "255112d5251e3baaf3ad04e674ef3fad": "f(x)=\\Omega(g(x))\\ (x\\rightarrow a),", "25517d097f49f0086fb6956ef48268df": " [2n + p + q + z - e]^2 - ", "25518944cc9299566187e85cc2e0d29e": "(Tf)(x) = \\int_x^{x+1} f(u)\\,du", "25519360adc624d0e5f4fd4fe4d69e8a": "a_3 = b_2-b_3", "2551a96d9646aa3bfe0145656d72887f": "H(t,\\xi) = G(t,\\xi) \\chi_{[0,\\infty)}(t) ", "2551ae8977d9c721256076dd6d20ee9f": " i \\in \\{1, \\dots, n \\} ", "2551c422d8cb2efa9555d6da4dd6215a": "\\Lambda \\alpha . \\lambda x^\\alpha . \\lambda f^{\\alpha\\to\\alpha} . f x", "25526fec68a6dce76bbe531ddae0eabb": "q \\ \\stackrel{\\mathrm{def}}{=}\\ -\\frac{\\ddot{a} a }{\\dot{a}^2}", "255270250b7d852a6257712264503781": "\\mathrm{I}_A= \\mathrm{tr}(\\mathbf{A}) = A_{11}+A_{22}+A_{33} = A_1+A_2+A_3 \\, ", "2552bd7d1b25f4f54e7667c59e720d2a": "\\gamma \\in \\{0.1, 0.2, 0.5, 1.0\\}", "2552c49d20963bcd22f1e9a856e67016": "H(X_{1,2})", "255368a486b0e2019a696af262cc2c67": "\nL(x,v) = \\frac{1}{2}\\left(\\frac{v - b(x)}{\\sigma}\\right)^2\n", "2553744408c159f00229055366efeeaa": " \n\\mathrm{HFC} = \\sum_{i=0}^{N-1} i|X(i)|\n", "25537f472b26e867b4cf339b1bfd31f4": "\\mathrm{sgn}(\\rho_n) = (-1)^{\\lfloor n/2 \\rfloor} =(-1)^{n(n-1)/2} = \\begin{cases}\n+1 & n \\equiv 0,1 \\pmod{4}\\\\\n-1 & n \\equiv 2,3 \\pmod{4}\n\\end{cases}", "2553ada18a2b780dc5768c89c651e7d6": " D_{nr} ", "2553b244e537f9fa55460b6fb989a422": "\\Gamma\\left({1 \\over 2}\\right)=\\sqrt{\\pi}\\!", "2553b2f5761b4f1305d909c354f387ed": "v_v", "2553eea9c775f908412b5132f7fdbe44": " \\begin{alignat}{1}\n\\mathbf{c}_3 &= -3\\mathbf{c}_1 + 5\\mathbf{c}_2 \\\\\n\\mathbf{c}_5 &= 2\\mathbf{c}_1 - \\mathbf{c}_2 + 7\\mathbf{c}_3 \\\\\n\\mathbf{c}_6 &= 4\\mathbf{c}_2 - 9\\mathbf{c}_3.\n\\end{alignat}", "25540be1099e6835d8987bb62ab43404": " L=\n\\begin{bmatrix}\nl_{1,1} & & & & 0 \\\\\nl_{2,1} & l_{2,2} & & & \\\\\nl_{3,1} & l_{3,2} & \\ddots & & \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\\\\nl_{n,1} & l_{n,2} & \\ldots & l_{n,n-1} & l_{n,n}\n\\end{bmatrix}\n", "25540cf68f86fc419ba56aa6b748859c": "p_{k} = \\frac{p}{r} r_{k}", "2554545d7be77a533307c317b3c99ae5": "\\mathfrak{g}^{\\mathrm{ss}}", "25547b0b89ce9564e6b4b2ea2a6eeba6": "\\hat{b}\\, \\hat{b}^\\dagger = \\hat{b}^\\dagger\\, \\hat{b} + 1.", "2554a2bb846cffd697389e5dc8912759": "\\theta", "2554bebde0a0860a1c27a6ed145d4ab0": "\\varphi\\left(\\int_{-\\infty}^\\infty x\\, f(x)\\, dx\\right) \\le \\int_{-\\infty}^\\infty \\varphi(x)\\,f(x)\\, dx.", "2554c54efee3e60ea7632340791da4f7": "\\mathcal{N}(\\boldsymbol\\mu,\\,\\boldsymbol\\Sigma)", "2554dba2c9241485a946f34144842493": "R = B_e R_f\\,", "2554e13fe0a10be188bddee5c632d7ea": "\\phi(E_i)\\subseteq F_{1-i}.", "255519ed046428549e0f8063e4a97fb7": "\\widehat{\\theta}(X)=\\frac{(a+n)\\max{(\\theta_0,x_1,...,x_n)}}{a+n-1}.", "25554e391c2a86f22a6d715ccc5cb7c5": "\\mathcal{F}_i\\,\\!", "25554ed2e5e6b0730ea58b85bc6e4b0e": " s^{unsigned}_{ij}", "25557dbb7481e76da40b6abcba01c3de": "\\mathrm{im}(f) = \\ker(\\mathrm{coker} f)", "2555af831cf60e7901e5fd3b604032ae": "\\sigma\\leq\\tau", "2555c81396a078d56243496affce6023": " \\cos \\Theta = \\cos \\varphi \\cos \\varphi' + \\sin\\varphi \\sin\\varphi'\\cos(\\theta -\\theta').", "2555d74a0a10e4a19d92045a28da8771": " \\sigma_{ij} ", "2555e7525be0e42250212b462c30a58a": "{\\underbrace{\\partial \\overline{hu} \\over \\partial t}}_{\n\\begin{smallmatrix}\n \\text{Change in}\\\\\n \\text{x mass flux}\\\\\n \\text{over time}\n\\end{smallmatrix}}\n+ \\underbrace{{\\partial \\over \\partial x} \\left( \\overline{hu^2}+{1 \\over 2}{k_{ap}g_zh^2}\\right) + {\\partial \\overline{huv} \\over \\partial y}}_{\n\\begin{smallmatrix}\n \\text{Total spatial variation}\\\\\n \\text{of x,y momentum fluxes}\\\\\n \\text{in x-direction}\n\\end{smallmatrix}}\n= \\underbrace{-hk_{ap} \\sgn \\left({\\partial u \\over \\partial y}\\right){\\partial hg_z \\over \\partial y}\\sin \\phi_{int}}_{\n\\begin{smallmatrix}\n \\text{Dissipative internal}\\\\\n \\text{friction force}\\\\\n \\text{in x-direction}\n\\end{smallmatrix}}\n- \\underbrace{{u \\over \\sqrt{u^2+v^2}}\\left[ g_zh \\left(1+{u \\over r_xg_x}\\right) \\right]\\tan \\phi_{bed}}_{\n\\begin{smallmatrix}\n \\text{Dissipative basal}\\\\\n \\text{friction force}\\\\\n \\text{in x-direction}\n\\end{smallmatrix}}\n + \\underbrace{g_xh}_{\n\\begin{smallmatrix}\n \\text{Driving}\\\\\n \\text{gravitational}\\\\\n \\text{force in}\\\\\n \\text{x-direction}\n\\end{smallmatrix}}\n", "2555fb92144d39b35afea3c41e3ec718": "U = \\sum_{i=1}^{n} w(p_i)v(x_i)", "2556271457018ddab6679e152f1fc70a": "\\left[ A\\right] ", "25562ca8b050ce10d8e57352978f764e": "\\mathbf{X}(s)", "25566ae6804f5deb1adfabccee5a5ec8": "D \\in \\mathrm{Div}^0(C)", "25567ad13be8225351a8d1c5bf12667d": "\\tfrac{1}{24}", "25569259ba6b7958edc520f623bb8617": "\\left(\\frac{\\partial\\Omega}{\\partial x}\\right)_{E} = -\\sum_{Y}Y\\left(\\frac{\\partial\\Omega_{Y}}{\\partial E}\\right)_{x}= \\left(\\frac{\\partial\\left(\\Omega X\\right)}{\\partial E}\\right)_{x}\\,", "255696048a5e1a0ae5ae75f2f86729ef": "\\mathrm{I}_{\\mathrm{O},\\mathrm{P}}\\,", "2556caf5470671f5f3a77453ede1dcf2": "h\\ =\\ 2 \\sqrt{\\frac{\\gamma} {g\\rho}}", "2557031a57d67ebb7bb70042942550e0": "v_e = \\sqrt{\\frac{2GM}{r}} = \\sqrt{\\frac{2\\mu}{r}} = \\sqrt{2gr\\,}", "25576aa54ce14b81a1c15b50f261d135": "\n\\mu(x; t, s) =\n\\int_{\\xi \\in \\mathbb{R}^k} \n(\\nabla I)(x-\\xi; t) \\, (\\nabla I)^T(x-\\xi; t) \\, \nw(\\xi; s) \\, d\\xi \n", "255795cd9acac1e89b36fdec81d7c7a2": "\\gamma_n = -\\Omega_n (1 - \\delta_n). \\,", "2557b6328a9fb8ce2995639a657b853d": "p = \\frac{8ac-3b^2}{8a^2}\\qquad\\qquad {\\color{white}.}", "255830b5efaaf6264a7a2022281bbd0f": "\\displaystyle \\left (\\frac{i}{2\\pi}\\right)^n \\frac{d^n \\hat{f}(\\xi)}{d\\xi^n}\\,", "255854475f333de326c5de5d49900238": "{\\rho_{air}} ", "25588471e26e64b53bd43a86904c30b2": "K:\\mathbf{M}\\to\\mathbf{C}", "255895d445fba419e08eca0b027c9a07": "\\operatorname{div}(\\rho \\phi \\upsilon)", "255914d86a24ad1606afa3a62df33683": "\n F(t) = A e^{i\\omega t} + B e^{-i\\omega t} \\,.\n", "2559422cb58b669fc8c56a566bc13b09": "\\Delta Q(P(t_1,t_2))\\ ", "255a1aa8e3493f351e72862cda7f3ba8": "\\mathbf{Z} = \\{Z_1 \\dots Z_n\\}", "255a63c38928e6f9575c86526ee81be8": " \\mathcal{C}_{XY}^\\pi = \\mathbb{E}_{XY} [\\phi(X) \\otimes \\phi(Y) ] ", "255a8b6e871fa8defd03be7befc32ffd": "s=k \\ln(1 / p)", "255a99ac60ad3c5f6e4e178a7b12e6c6": "\\ldots,", "255aa235e6845bb5e8334d4a77248b96": "DR_{D}^{S}", "255aa5ce12bea936f4447c696a34332b": "|b|", "255b10c452c2af014953b528fe9eb352": "\\mathbf{r}_{1}", "255b4cd0abc731563ce510e10b05c040": " G_{S_N}(z) = \\operatorname{E}(z^{S_N}) = \\operatorname{E}(z^{\\sum_{i=1}^N X_i}) = \\operatorname{E}\\big(\\operatorname{E}(z^{\\sum_{i=1}^N X_i}| N) \\big) = \\operatorname{E}\\big( (G_X(z))^N\\big) =G_N(G_X(z)).", "255b6ebfa66372ab0b7935076936f442": "\\rho w_E = \\hat{ \\mathbf{k}} \\cdot (\\nabla\\times\\tau)/f \\ ", "255bca5d39927a33cd568c4258f7d713": "q^{2n+1}+1", "255bcd90a789eeac8077592a3198277e": "\\frac{\\sqrt{i}+i \\sqrt{i}}{i}\\text{ and }\\frac{\\sqrt{-i}-i \\sqrt{-i}}{-i}.", "255c0c87232256a1c7b19a7616e2981b": "q=e^{\\pi i\\tau}", "255c1fae8c9cfdb62d40554320e2be1f": "\\begin{align}\n\\bold{A}+\\bold{B} & = \\begin{bmatrix}\n a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\\n\\end{bmatrix} + \n\n\\begin{bmatrix}\n b_{11} & b_{12} & \\cdots & b_{1n} \\\\\n b_{21} & b_{22} & \\cdots & b_{2n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n b_{m1} & b_{m2} & \\cdots & b_{mn} \\\\\n\\end{bmatrix} \\\\\n& = \\begin{bmatrix}\n a_{11} + b_{11} & a_{12} + b_{12} & \\cdots & a_{1n} + b_{1n} \\\\\n a_{21} + b_{21} & a_{22} + b_{22} & \\cdots & a_{2n} + b_{2n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m1} + b_{m1} & a_{m2} + b_{m2} & \\cdots & a_{mn} + b_{mn} \\\\\n\\end{bmatrix} \\\\\n\n\\end{align}\\,\\!", "255c4330d01b0b960275e3fab7899c15": "a = \\left( \\frac{3}{4 \\pi n} \\right)^{1/3}.", "255c478b3e67ab21cc2282bf6b14cbf4": "J_k(n)=n^k \\prod_{p|n}\\left(1-\\frac{1}{p^k}\\right) .\\,", "255c7dceecbc1fd5827f7bdba0944d49": "\\sigma_\\mathrm{min}\\,\\!", "255c82804707a93fc92e0a055994b178": "\n\\left|k_1,\\ldots,k_n\\ \\mathrm{in}\\right\\rangle=\\sqrt{2\\omega_{k_1}}a_{\\mathrm{in}}^\\dagger(\\mathbf{k}_1)\\ldots \\sqrt{2\\omega_{k_n}}a_{\\mathrm{in}}^\\dagger(\\mathbf{k}_n)|0\\rangle\n", "255c9428c27429532fd1ba66bd8945d9": "{{{({{\\partial \\over \\partial t} + {\\overrightarrow{V_g} \\cdot \\nabla}})({-\\partial \\Phi \\over \\partial p})}-\\sigma \\omega}={kJ \\over p}}", "255ceb11ed822fdd54a4980da3934cc5": "\\scriptstyle V_\\mathrm{P-P}", "255d141fd458c93122200ca8e8be9030": "\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=0.", "255d19f3f02489ecd08f33c414a91645": "A_{ij}=\\left\\{k\\in\\{1,2,\\dots,p-2\\}: \\left(\\frac{k}{p}\\right)=(-1)^i\\land\\left(\\frac{k+1}{p}\\right)=(-1)^j\\right\\},", "255d35062a53d3a1845483ebd237b15c": "d N_R", "255e14f48ee5491cdd51a252c9eda89c": "U_{ii}", "255e8abf7d6b2e6efa26d2757f821e2b": " \\mathbf{E}_8 \\supset \\mathrm{O}(16) ", "255e8ca75b82f396f815953a05f9831b": "g_Y={F_A}A/{Y}*g_A+\\alpha*g_K+(1-\\alpha)*g_L", "255e9f3db3e2810fcf2ed16d8a6a4807": " [L_{u,v},L_{w,x}]:= L_{u,v}\\circ L_{w,x} - L_{w,x} \\circ L_{u,v} = L_{w,\\{u,v,x\\}}-L_{\\{v,u,w\\},x} ", "255ec49ebf3d6ad790a9336d4f19a8dd": "\n\\begin{align}\n\\left[\\frac{\\alpha}{\\pi}\\right]_2 &= \n\\pm 1 \\equiv \\alpha^\\frac{\\mathrm{N} \\pi - 1}{2}\\pmod{\\pi} \\\\&=\n\n\\begin{cases}\n+1 \\text{ if }\\gcd(\\alpha, \\pi) = 1 \\text{ and there is a Gaussian integer }\\eta \\text{ such that } \\alpha \\equiv \\eta^2 \\pmod{\\pi} \\\\\n-1 \\text{ if } \\gcd(\\alpha, \\pi) = 1 \\text{ and there is no such }\\eta.\n\\end{cases}\n\\end{align}\n", "255f401be8c1801f07cf6b7e315edbb4": "A^i = 0", "255f4ce023d93837dc2790ae2eeaad81": "\\mathbf{X}_2", "255fd2c1bedf5334a708a63a2027c18c": "\\left(\\frac{2mn}{n^2+m^2}, \\frac{n^2-m^2}{n^2+m^2}\\right)", "2560232f9ad271eefce568640ec41b55": "\\sup_{f \\in \\mathcal{F}}\\vert f(x) - Pf \\vert < \\infty. \\forall x", "25607e2945a13ef784b465c4349695b5": "{\\mathrm{i}}", "2561086530f0c6483d1b3dc415ff3753": "\\pi\\alpha", "2561131dc7d4c7c80cf5aad09544bf19": "Y(s)=U(s)(e^{-s}+s-1)/s^2", "25613c08c6b68a9704ffc406287fdbd2": "\\kappa = 0", "25616bc4ff7a0e11ba9fb588b8f82f3a": "H(P) = \\underset{i}{\\max} \\,|a_i| \\,", "2561826a22ce9e570b348d2cc7ac9cd0": "{r \\over \\sin 3\\theta} = {a \\over \\sin 2\\theta}\\!", "2561b2044908d7496b015ccbbc7c7ee2": " g_n < (\\log p_{n})^2 - \\log p_{n}", "2561e2121d99e378b780f48a2e74234f": "BC_n.", "2561f35154361734b7646bba2a94a916": "\n\\int_{-\\infty}^{+\\infty} f(x) dx = \\int_0^1 {dt\\over t^2} \\left(f\\left(\\frac{1-t}{t}\\right)\n+ f\\left(-\\frac{1-t}{t}\\right)\\right) \\;.\n", "25626ddc70ac76d8b42d4dd908f48145": "w_1=w_2=1", "25628bde18547f3964d5413e189daf9e": "du \\cdot dv = u' v' (dx)^2 = 0\\,\\!", "2563198fdfafae0dbdc71cbec9000b60": "q_{1}", "256352aa9a66cdcd602cc204c46ad920": "{t}", "25639e2275027414b46b9188790c4936": "\\scriptstyle \\langle ", "2563e3348b3e6389fb2522e752e0fa5e": "g_{00} > 0.\\,", "256413eb906f3a97cd3e605f5ee7700e": "\\pi_{n-1}(A) \\to \\pi_{n-1}(X)", "25649c3a234023c1126b3c20a5093e7d": "X_1,X_2,X_3", "25652162bba3af6ecd25aa64a9878ce0": "\n\\prod_{k = 1 \\atop \\gcd(k,m)=1}^{m} \\!\\!k \\ \\equiv\n\\begin{cases}\n-1 \\pmod{m} & \\text{if } m=4,\\;p^\\alpha,\\;2p^\\alpha \\\\\n\\;\\;\\,1 \\pmod{m} & \\text{otherwise} \n\\end{cases} \n", "25655081f5b649195d0d605aadf84b7c": "e_q^x = \\sum_{n=0}^\\infty \\frac{x^n}{[n]_q!}.", "2565530a65b6f96f25845fe829642a29": "[F^{[l]},F^{[l']}]=0", "25655bfb0b3ddecad59e2941e83a723f": "\\phi(x) \\equiv x^q \\text{ (mod }\\beta) \\,\\!", "25656b59a2a1984d6e8afb9bf7d588f0": "\nT^{i}_{j;i}=0.\n", "25656d52d660036362e969afd45ea4d1": "\\Box \\Diamond A", "2565f0a9b6ad9c5ad3bc7d2224acb9ab": "\\{ \\mathbf{\\tilde{e}}_{k} \\}", "256603ffe6f86b3a2b13aacc67873492": "e=1\\,\\!", "25661fd7ddbd2cb22ced5d996d36c536": "\n f(p,q,p_c) = q + M\\,p\\,\\ln\\left[\\frac{p}{p_c}\\right] \\le 0\n ", "25664e3edaaa20446da9dcd4aab2bd94": "N\\choose n_1", "2566a5265afa21d30c33510c4b7b1193": "(\\textbf{x},y) = (x_1, x_2, x_3, ..., x_k, y)", "2566e01a66863fa49cad3c96850c652c": " \n\\begin{cases}\n \\displaystyle \\frac{d\\vec{x}_P}{dt} = \\vec{u}_P(\\vec{x}_P,t) \\\\[1.2ex]\n \\vec{x}_P(t_0) = \\vec{x}_{P0}\n\\end{cases}\n", "256725d40a9342b793c90b69d18cc13b": " \\lim_{x \\to 0^{+}}\\frac 1 x = +\\infty", "25678557805d88847a0e87ad678ad2d3": "\\frac{L_{vap}}{T_{boiling}} \\approx 87-88 \\frac{J}{K mol}", "256791f454ba8ad9aaf418612da842a1": "V_{\\text{out}} = V_{\\text{in}} \\left( \\frac{1}{\\beta + 1/A_{OL}} \\right)", "2567b60415bd43aaa741dc8b9b638e14": "\n J_1 = \\int_{\\Gamma} \\left(W \\delta_{1j} - \\sigma_{jk}~\\cfrac{\\partial u_k}{\\partial x_1}\\right)n_j d\\Gamma\n ", "2567e41466de3a8d4495b674b2a20550": " \\sum_{i=1}^\\infty \\lambda_i |e_i(t) e_i(s)| \\leq \\sup_{x \\in [a,b]} |K(x,x)|^2, ", "2567f61cd0cf997933a4498197d14e26": "\\sigma_3^{ }", "25680201bc240752fb5165b1efbff89f": "\\mathbf{\\Pi}", "256836f64a563df684913be7c29b2ccd": "g^{(2)}(0) = 0 ", "25686f4b7adb7313fc37bfdb01478272": "(u_r,u_\\phi)", "25687cf457ad00866c89a51f3bf828ee": "\\mathbf{C}\\sim W_p^{-1}(\\mathbf{V}^{-1},n)", "2568a96eec6be9f969228757abcb44d8": "\n\\bar \\epsilon_{sh}(t,t_0) = -\\epsilon_{sh\\infty}\\ k_h\\ S(t),~~~k_h =\n 1-{h_e}^3~~~~~\n", "2568dab807674550805ec161ca308527": "C(\\mathcal{X})", "2568f2cf5a729cfa7c05be91d0c423c0": " 0 = \\iiint\\limits_V \\left( \\frac{\\partial \\rho} {\\partial t} + \\nabla \\cdot \\mathbf{J} \\right)dV.", "2568f6507d084b139cbbc1c2c47d6079": "{-dE \\over dt}={\\sigma_t B^2 V^2 \\over c \\mu_o} ", "25694fb3fed429a87fc65b0571eff26d": " k \\in \\{1,...,(p-1)\\} ", "25698c7b0873e2903575dce42d8ec0ea": "R \\to R, x \\mapsto uxu^{-1}", "256a3c5fec7439e324970d15eb002fc6": "{\\pi\\over p}\\ {\\pi\\over q}\\ {\\pi\\over r}", "256a52017aefc4b3e12e8b1fb59da9b1": " \\boldsymbol{\\rm S}=\\{A,B\\}", "256a5a2319ac8c8c0feb1d649b9af51a": "x^4+x^3+x^2,\\quad x^4+x^3+x+1,\\quad x^4+x,\\quad x^4+x^2+1.", "256a5a99ef7e9fb789684a153a523388": "\n0 \\rightarrow \\operatorname{Tor}(K^*, K^*)^{\\sim} \\rightarrow \\operatorname{K}_3(K)_{ind} \\rightarrow \\operatorname{B}_2(K) \\rightarrow 0\n", "256a5db732a59e61ca5a7221059db921": " v = \\frac{P_F + P_C}{2} - P_P ", "256a9a3e9e20f0593bfe25dc4d095b43": " |SA| ", "256aad94d33dcce3e113e24009b69bd5": "I_D", "256b2a04d5d2068cd7e07d1944fb7799": "\\gamma(0)=\\gamma(1)", "256b588196ff499cef9d989070b1538d": "n a + m b = 0\\;", "256b87c9dd69b58688ee5fe48b812611": "e^{-|x|}", "256bda0aeae7109348d945b6d2b2ab18": "\\sqrt{(2^{AV})}", "256bed78c559f82e94a4589e0e8a15cc": "N_n", "256c0c5dc46ac8604a68c4581884823b": "\\frac{d\\mu}{dx}=(e^{\\int p(x)dx})\\frac{d}{dx} (\\int p(x)dx)", "256c30e11797273e7f7f047426ad4c41": "\\frac{a_i+b_i}{2}.", "256c3f5f9db8f8161b596d669c0a751a": " q_m = (C_d)(E_v)(Y)\\left [ \\frac{\\pi}{4} \\right ](d)^2 \\sqrt {2 \\rho \\Delta P} ", "256c58751184deac8ca43869b6263bee": " M= ", "256cd2fd9f0a2dd10189796a403581bc": " \\tau = N_{AO} \\int \\frac{1}{(-r_{A})V_{R}(1-\\delta_{A}f_{A})}\\,df_{A} ", "256cf2559101d6fa422cd63ed6960933": "v_A \\gg c", "256dfd235839c8fe602f1d61b422e2bc": "F(u) = \\frac{1}{2} \\int_a^b\\! u'(x)^2\\,dx - \\int_a^b\\! u(x)f(x)\\,dx.", "256e3cfb9430a14b6f8d5cf745b62eb8": "\\delta[n]", "256e72fb9339d35b8d2900be31070c83": "a^2+b^2=1", "256eb7d8fd6cd53b89c086b170135da2": "n\\mathrm{LiCoO_2}\\leftrightarrows\\mathrm{Li}_{1-n}\\mathrm{CoO_2}+n\\mathrm{Li^+}+n\\mathrm{e^-}", "256ef35a588d032cd55ec94e1b174297": "P = p(x,D) = \\sum_{|\\alpha|\\le k} a_\\alpha(x) D^\\alpha.", "256f38fab870148921b032022d16a3a3": "g(t|x) =\\frac{f(x|t)g(t)}{f(x)} = \\frac{1}{t(\\ln(T) - \\ln(x))} \\quad \\text{for all } t > x .", "256f7794ae0474224f1fabc58859a65b": "p(\\textbf{x}_k\\mid \\textbf{x}_0,\\dots,\\textbf{x}_{k-1}) = p(\\textbf{x}_k\\mid \\textbf{x}_{k-1})", "256f794a5147cc866416157f9bcbeb0a": "y_0=y", "256fa059915945369657d31b52f2b751": " q_2 = 30.0 \\text{ ft}^2/s", "256fb976024a94d6a34e0458d71a7059": "s_{ij}\\,\\!", "256ff1583b33a0b93e4bb748d8bfa199": "V=V^1 + V^2", "2570601a00fb7bed607190dd8d0b41f9": "\\operatorname{Var}(\\overline{x}) = \\sigma^2/n", "257099ff91578e85efd8ccdc7dc36c11": " \\sum \\left(\\Phi\\left(r_i(x_j) + \\alpha h(x_j) y_j + s - t\\right) - \\Phi\\left( r_i(x_j) + s \\right) \\right) = 0 ", "25709aa3d8d3531cc691b771c7b5773f": "f \\colon Y \\rightarrow X", "2570c2f79c151da6f194fc1c2cc483fa": "\\sqrt{-1} \\cdot \\sqrt{-1}=(-i) \\cdot (-i)=-1", "2570e6cf9a01f0b85b85f7cc39a8e7e7": "V_{out} = V_2\\,", "2570fb2ddc32a7655a586d096fcc2601": "\\sum_{n=-\\infty}^\\infty \\left | a_n \\right \\vert^2 < \\infty,", "25716c8d0b9dc065c9a1f600c4796897": "L=\\int d^3x\\,N\\gamma^{1/2}(K_{ij}K^{ij}-K^2+{}^{(3)}R)", "2571a85a1d3d0b8b26f8f16f6581ddc4": "a^{p-1} \\equiv 1 \\pmod{p}.", "2571c5d1f72bf84de9f44f6e70ca2895": " \\vec{e}_0", "2572112c0c0545e1265d7aa59f0e804f": "\nA(\\alpha)=\\frac{1}{2}\\int_0^{\\alpha^2}dx\\frac{x}{(x+\\theta_E^2{(1-\\beta^2))}^2}\n", "257246acb84acb9b2ad3473fc093fd3b": "g^{ij}A_j = \\cancel{g}^{i \\cancel{j}}A_\\cancel{j} = A^i\\,,", "257267ec594513755e6479afc6a1f1df": "|r_1 - r_2| < d < r_1 + r_2", "2572897b44d3fb4f042c10f38324a7bb": "A_{HP} = A_{LP} = \\frac{R_1}{R_G}", "2572a1a4a0a66277510e68dd703aaf42": "L(Q)", "2572e48fc25e5f3ec7e19fb7a0cbfe39": "\\Pi^1_n \\subset \\Sigma^1_{n+1}", "2572f7680cbe17e12e94dd7170e2b0ea": "s_i=\\sigma^i(s) \\,", "2573a79b19f261a77dacbbaee7a0428a": "P^{+}(V)", "2573ce98b8cff4467fc7aa0d99156f92": " \\mathbb{E} (X_t | \\mathcal{F}_{t-1}) = 0, a.s. ", "2573dc40996c7a8140d8c3a1658b8d87": "- \\leftarrow \\rightarrow -", "2573eb0da83247ba07e595db9c8a6a47": "\\hat{r}=\\cos u\\ \\hat{g}\\ +\\ \\sin u\\ \\hat{h}", "257425eb46517e99a9208e9ac3403c4d": "\\stackrel{*}{\\rightarrow} \\circ \\stackrel{*}{\\leftarrow}", "2574f531ad3c0f4a19879fb7246d87e9": "x_1, x_2, \\dots, x_n\\,", "257537180f35862819c6f914a2485de6": "y^{v}", "25756253f2319029cb01ceae5d43bf0a": "\n \\begin{align}\n S_{xx} &= \\overline{ \\int_{-h}^\\eta \\left( p + \\rho \\tilde{u}^2 \\right)\\; \\text{d}z } \n - \\frac12 \\rho g \\left( h + \\overline{\\eta} \\right)^2, \\\\\n S_{xy} &= \\overline{ \\int_{-h}^\\eta \\left( \\rho \\tilde{u} \\tilde{v} \\right)\\; \\text{d}z } = S_{yx}, \\\\\n S_{yy} &= \\overline{ \\int_{-h}^\\eta \\left( p + \\rho \\tilde{v}^2 \\right)\\; \\text{d}z } \n - \\frac12 \\rho g \\left( h + \\overline{\\eta} \\right)^2,\n \\end{align}\n", "25756d9d744fcbf92add41fa20ef8739": " |\\mathbf{E}| = E = \\frac{1}{4\\pi\\varepsilon_0}\\frac{Q}{s^2} ", "25759c593653405551a6ddf8c148f91a": "x \\wedge z = y \\wedge z", "2576072a1f8dbe146ebffe18ffbf9a90": "R = A_1(t-\\tau)B_2(t) - A_2(t - \\tau)B_1(t)", "25760b3a378d8f50a7355582814665d4": "\n\\frac{1}{1-z}\n\\exp \\left( (u-1) \\left( \\frac{z^{n+1}}{n+1} + \\frac{z^{n+2}}{n+2} + \\cdots \\right) \\right),", "25763d53b952045a3c24251af8f7c094": "\\textstyle v_i=\\frac{1}{n}q(\\alpha^{n-i})", "257641b23b54de45e6998598333eba95": "J^1Y\\to Y", "2576a16a78f87590c8df90de805929f7": " i \\not \\in \\bar{K}", "2576d6c691e2d64bbdc7316beb2e77fc": "\\operatorname{de-lambda}[\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))] ", "25772e7487dfbac95945412f9e37377a": "a = 2 \\beta", "25774fde9a9d341c9fdee60bcd7e7d52": "\\delta\\! = \\bar x_B - \\bar x_A", "2577802aa24edbd7c80a4a4a4b3bf89c": "h_n^{(2)}(x) = j_n(x) - i y_n(x). \\, ", "2577a937df249ed54ed3d506285a025e": "I^{i + j + 1}", "2577f7966a52063a5c9e7951b6e83c8e": "\\frac{|c|}{-c}\\sqrt{a^2 + b^2}.", "25782ec0f0d1ee522fb8f6f902abb64d": "\\Pr(\\chi(v) = -1) = \\Pr(\\chi(v) = 1) = \\frac{1}{2}", "257841f659f34ef98c7943f318a4c401": "\\R^2", "25788a3ab91d914e9c8bdb3996c3e1cb": " =q_j |\\psi_t\\rangle", "2578d1052a034fe4fe6c777934ed9675": " x \\in S ", "25791d06413f53b809dff035d60880b9": "V_1 - V_2", "25791fa85505f270f5729009c9e9d5c0": " \\left(\\frac{dy}{dx}\\right)^2 \\!\\! - 4x\\frac{dy}{dx} + 4y = 0. ", "257934e7d3c32029a49e5933753e2fc5": "A = \\Gamma(F,X)", "257994e5507f0412d3c9fd343c40a9f4": "P\\times P", "25799a384e6bd27bbbfd43e2d704de97": "\\scriptstyle \\hat{\\mathbf{w}}", "2579b5ebd928bafd0ad4d21ccebe5bee": "v_\\text{out} = \\alpha_1 (A_1 \\sin \\omega_1 t + A_2 \\sin \\omega_2 t) + \\alpha_2(A_1 \\sin \\omega_1 t + A_2 \\sin \\omega_2 t)^2 + \\ldots \\,", "257a1ac0c5506b1fc1f6bc6e58b3db8c": "\\tau_v=\\omega_{xy}/v", "257a5ed0d30f578090194c63d0d59ce1": "u+v = \\int \\frac{du}{dx} \\,dx + \\int \\frac{dv}{dx}\\,dx", "257aaf8fe257416ba7339e1f7cb3ffbc": "{{{V^a}_{bc}}^d}_e.", "257ab0e3515807355247827cfd5132b0": "G^{\\alpha\\gamma} + \\Lambda g^{\\alpha\\gamma} = -\\frac{8\\pi G}{c^2} T^{\\alpha\\gamma}~", "257abe933651e1149e8a8009accb1d9f": "\\{Ax_n\\}", "257b18739ac1a2f3bbca21eda271388c": "A_{[\\alpha\\beta]\\gamma\\cdots} = \\dfrac{1}{2!} \\left(A_{\\alpha\\beta\\gamma\\cdots} - A_{\\beta\\alpha\\gamma\\cdots} \\right)", "257b398c4e6f14efca8077858e6c8f16": "\n \\lim_{n\\to\\infty} \\operatorname{E}\\left( |X_n-X|^r \\right) = 0,\n ", "257c543ac161e854604c310870394afb": "k:=2", "257c9247bfe53afce4c0bfc781ba7784": "\\begin{align}\n\\cos a &= \\cos b\\cos c + \\sin b\\sin c\\cos A\\\\\n\\cos A &= -\\cos B\\cos C + \\sin B\\sin C\\cos a.\n\\end{align}", "257cde2020af155846bd9d5f931a28e4": "\\lbrace \\psi_p \\colon \\mathcal{X} \\to \\R \\mid 1 \\le p \\le M \\rbrace", "257cdee0f8255d54f70f7fae0dd0110c": "U_0(q) = \\sum_{n\\ge 0} {q^{n^2} (-q;q^2)_n \\over (-q^4;q^4)_n}", "257ce9b841eabc55b761885feccdf83a": "\\phi(x) = Ax + x_0", "257d1669965cf81bf085833eddf383ac": "L = \\frac{1}{\\sqrt{2}}(5.9(Y_0^{1/3} - \\frac{2}{3}) + 0.042(Y_0 - 30)^{1/3}) - 14.3993)", "257d588c74db79798235f434bea5d55c": "\\exp(i \\pi) = -1 \\,", "257d83f9c15cd5249e0d58e5b620715e": "\\lim_{N\\to\\infty}S_N = \\sum_{n=1}^{\\infty} a_n.", "257ddacbbbbe3d530069e06ef4241e0d": "|P-Q|_\\pi = |P\\smallsetminus P'| + |Q\\smallsetminus Q'|", "257e392aa90c4fd1a0697238ff7bb191": "\n \\boldsymbol{M}\\, =\\, \n \\overline{\\int_{-h}^\\eta \\rho\\, \\left( \\boldsymbol{U}+\\boldsymbol{u}_x\\right)\\; \\text{d}z}\\, \n -\\, \\int_{-h}^0 \\rho\\, \\boldsymbol{U}\\; \\text{d}z\\, \n =\\, \\frac{E}{c_p}\\, \\boldsymbol{e}_k,\n", "257e5cbc0b279a609e843eea7da3a203": "A_{,i}[\\phi] \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\delta}{\\delta \\phi^\\alpha(x)}A[\\phi]", "257e5f24aabe934202e6b8e7a6950e6e": " \\textstyle S=I\\,\\ ", "257e7a0aca03f1b15a5dbe294a11fbf1": "\\scriptstyle \\delta(k-n)", "257f1964a1492d0ee1a38962b3a2d829": "r^2+h^2=x^2", "257f1e981682ff14a1c1bf1171a1e87a": "\nG_{p,q}^{\\,1,p} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_h, b_1, \\dots, b_{h-1}, b_{h+1}, \\dots, b_q \\end{matrix} \\; \\right| \\, (-1)^{p-m-n+1} \\;z \\right), \\quad h = 1,2,\\dots,q,\n", "257f62b563c40aa8c585f007509f8883": "\\scriptstyle x=\\exp\\, {-\\frac{u}{n+1}}", "257fb25ccb6118b81921b0c6cbba728d": "\\Delta E\\ge \\frac{\\hbar^2k_F}{2M^*\\Delta x}=0.47\\frac{E_F-E_0}{rk_F}\\ ,\\qquad\\qquad (10)", "257fcec61e49c97ded2ea179ed04c966": "\\mu \\rho = e", "257fd12d96f8ade79b1e0f881ab0e0be": "(0 \\le \\rho \\le 4, \\ 0 \\le \\phi \\le \\pi, \\ 0 \\le \\theta \\le 2 \\pi).", "2580806736615fcf600ac7d3301775a4": "\\hat{\\Pi}_{t,n} = \\Pi_tA^n.\\,", "2580a01d337c6b459107c6813f01af58": "\\displaystyle f'(x) = 0", "2580b4e476dd015aafcc9280a285e7c2": "V_{\\text{CB}}=V_{\\text{CE}}-V_{\\text{BE}}", "2580ea3d7f3cc39ea5113ea9f4789171": "\\frac{1}{pq} = \\frac{1}{N}\\times\\frac{N}{pq} ", "2580fd4c70200976cc955171378d6c29": "E = \\frac{1}{2} n J_{ex} S_F S_{AF} + M_F t_F H", "25811bee3b5571486abe010104ab848f": "\\frac{\\pi}{\\sqrt{k}}", "258123156d0a58164bec582bbfc3d19f": "\\rm votes \\over \\rm {seats+1}", "25812331d4832328c0f976fbe4c102c9": "\\psi_{xy} = \\psi_{yx},", "2581267ae634803df536da7f5b7b18a9": "\n\\begin{align}\n(h_\\text{eff})_{AB}&=E\\begin{pmatrix} \\delta_{ab}&0\\\\\\\\0&\\delta_{\\bar a\\bar b}\\end{pmatrix} \n+\\frac{1}{2E}\\begin{pmatrix} (\\tilde m^2)_{ab}&0\\\\\\\\0&(\\tilde m^2)_{\\bar a\\bar b}^*\\end{pmatrix} \\\\\\\\\n&\\quad+\\frac{1}{E}\\begin{pmatrix}[(a_L)^\\alpha p_\\alpha-(c_L)^{\\alpha\\beta} p_\\alpha p_\\beta]_{ab}&\n -i\\sqrt2p_\\alpha(\\epsilon_+)_\\beta[(g^{\\alpha\\beta\\gamma}p_\\gamma-H^{\\alpha\\beta})]_{a\\bar b}\\\\\\\\\n i\\sqrt2p_\\alpha(\\epsilon_+)_\\beta^*[(g^{\\alpha\\beta\\gamma}p_\\gamma-H^{\\alpha\\beta})]_{\\bar ab}^*&\n [(a_R)^\\alpha p_\\alpha-(c_R)^{\\alpha\\beta} p_\\alpha p_\\beta]_{\\bar a\\bar b}\\end{pmatrix} .\n\\end{align}\n", "25815092a30eaf8405481ec7178b9f71": "dV = \\left(\\frac{\\partial V}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial V}{\\partial y}\\right)_x\\!dy", "2581a53ab6ebb625d39a4a19292b97fc": "\\operatorname{E}(Y|\\mathbf{x})=e^{\\boldsymbol{\\theta}' \\mathbf{x}}.\\,", "2581f12a1cc8c14fdb184cc4d34bfa30": "\n\\mathrm{p}K_{a} = -\\log_{10}{K_{a}}\n", "2582692ac5c840b75927b4dd7e31ae01": " \\ln 2 = \\left[ 0;1,2,3,1,5,\\frac{2}{3},7,\\frac{1}{2},9,\\frac{2}{5},...,2k-1,\\frac{2}{k},...\\right] ", "25826b403062325e2f82e8f5b6abf380": "\\int_0^{2\\pi} \\cos(m\\varphi)\\sin(m'\\varphi)d\\varphi=0", "2582863653817674e01910af23103340": "\\alpha A +\\beta B ... \\rightleftharpoons \\sigma S+\\tau T ...", "2582932c993f722f12dd3114a2b87946": "f:D\\to D^*", "25829ebd858b6d059936eb1425013093": "t=\\left\\lfloor \\frac{(2t+1)-1}{2} \\right\\rfloor", "258333c489503551871413cb17502b1a": "f < O(g)", "2583711f97116ed7878443b1159fce50": "\\color{blue} (x_1=\\dots=x_n\\neq none)\\Rightarrow[kill(x_i)](x_1=\\dots=x_n= none)\\;\\color{black}", "2583e4421e35e9ce588f0f78aa2c4e54": " u_L = (\\rho_V/\\rho_L)dR/dt ", "2583ed91a3377b0f3acae0ecbf95bf7e": "\\Box (p \\rightarrow q)", "2583f6ecb237d15352da01087a766011": "\\ell \\cap \\ell' = \\varnothing.", "258402d8f64241a9f62afd358a124bb8": "\\mathcal{E}(u) = \\int_{\\mathbb{R}^n} |\\nabla u|^2\\;dx", "2584810974eb4d3c3108d7e02aefcb3d": "q_p(a+np^2)\\equiv q_p(a) \\pmod{p}.", "2584aad55268a51c23929ea75efcafe6": "\\alpha^{jk_i} = \\sum_{s=0}^{m_i-1}a_{ijs}\\beta_{i,s}", "258519b1e7f0f0b9f724ffdd1ff539cb": "x = 2. \\,", "258564fc35a86e3c30fc7356958f2d52": "\\sqrt{\\frac{3}{14}}\\!\\,", "2585ce32da4a19c18854ba2a8d030696": "u=\\left( 0|1\\right) ", "25866d00f4b3c0941c24ed29f077e6a6": "f_t(z)=h^{-1}(e^{-t}h(z)), \\, ", "25869812f51439d3e0e89a14aeff5632": " \\dot{p}_j = - \\partial H/\\partial x_j ", "25869af2b802e5a1122b6483889cc9a9": "\\mathbf{P}( X \\le \\mu-a) \\le e^{\\frac{-2a^2}{n}}, \\qquad 0 < a < \\mu", "2586bf5ae603661fccaacfe27368ace7": " K_d = K_p T_d \\,", "2586dcfd3f9b4613e466ec717aba1f61": "\n\\frac{m}{n} + \\frac{p}{q} = \\frac{1}{2}.\n", "2586dfc6f6a75276b39268390b5f9d6e": "A=4 \\, x_m \\, y_m \\, z_m \\times \\sqrt{\\frac{1}{x_m^2}+\\frac{1}{y_m^2}+\\frac{1}{z_m^2}}", "25870d71035565dda0a4f645868d6a90": "\\mathrm{UT}_x (M) = \\left\\{ v \\in \\mathrm{T}_{x} (M) \\left| F(v) = 1 \\right. \\right\\}.", "258789eb58f49a79691b6ce2b9f5bf57": " \\alpha \\cap \\beta = \\partial \\alpha = \\partial \\beta ", "2587f7f2149b0f7ffed3faee2362bd00": "\\text{x quus y}= \\begin{cases} \\text{x + y} & \\text{for }x,y <57 \\\\[12pt] 5 & \\text{for } x,y \\ge 57 \\end{cases} ", "258859c29e15b1bedc30517026ae834f": "\\alpha\\in\\big[\\,0.76,\\,0.96\\,\\big], \\quad \\beta\\in\\big[\\,{-2.06},\\,{-1.58}\\,\\big]", "258873cfff2d05d64a1d47d10e3f27bf": "\\phi = \\sum^{A,B}_{n} a_{n} \\phi^{(0)}_{i}", "2588fa381f599c3fb00498ce9f851539": "\nH(2^k) = \\begin{bmatrix}\nH(2^{k-1}) & H(2^{k-1})\\\\\nH(2^{k-1}) & -H(2^{k-1})\\end{bmatrix} = H(2)\\otimes H(2^{k-1}),\n", "25890ec447e116c782cdb50fd7ca7798": "E \\subseteq \\mathcal{E}", "25890ef29d44270bb1106b9216c3028f": " \\mathbf{y}'_{1} ", "25896464e4ef7cd7c42cc5db248a7de0": " j^{\\underline{m}} \\!", "258a009aa30ced303fdd9d81385dcda5": "f(x_1,\\ldots,x_k)\\mapsto\n\\begin{pmatrix}\n\\partial_{\\underline{x_1}}f\\\\\n\\partial_{\\underline{x_2}}f\\\\\n\\ldots\\\\\n\\partial_{\\underline{x_k}}f\\\\\n\\end{pmatrix}", "258a34ea18714e89ead04b2c37788161": "P(x) = \\frac{1-\\sqrt{1-4x}}{2}", "258a46eccd6e12c7b4b7308931ac1bab": "I_n \\ + (n-1) I_n\\ = \\cos^{n-1} x \\sin x\\ + \\ (n-1) I_{n-2} , \\,", "258a5641ac372949139a3688d89a8249": "\\ Kc ", "258aa5f3a1f6e2726ae0d99f51a6f388": "\nS = \\begin{pmatrix} 0 & I_{r} \\\\ I_{w - r} & 0 \\end{pmatrix} A\n", "258acbb837f2358dcb73ce46b2e7a89e": " y - \\hat{f_+} ", "258acca4ae65744ac82900e887856e2e": "f_\\textrm{sim} = \\frac{f_\\textrm{p}NH} {(f_\\textrm{p}NH)+H}", "258addb8f80a122e66f637aca19ee248": "L={1\\over 32\\pi G}\\Omega_\\nu^\\mu g^{\\nu\\xi}x^\\eta x^\\zeta\\eta_{\\xi\\mu\\eta\\zeta}\\;", "258aeef2529e15e28e5b6cba9651f632": " {\\nabla\\times\\vec{B}}={\\nabla\\times(\\nabla\\times\\vec{A})}={\\nabla(\\nabla\\cdot\\vec{A})-\\nabla^2\\vec{A}}={\\mu_0\\vec{J}+\\frac{1}{c^2}\\frac{\\partial\\vec{E}}{\\partial t}}={\\mu_0\\vec{J}-\\frac{1}{c^2}\\nabla\\frac{\\partial\\varphi}{\\partial t}-\\frac{1}{c^2}\\frac{\\partial^2\\vec{A}}{\\partial t^2}}", "258b2250c36a6cb8af0b390bd8c6d1ba": "|H_{\\nu}(\\omega)|=| \\frac {P_{\\nu} ( \\cos { \\frac {\\omega} {2})}} {P_{\\nu} \\cos (0)}|", "258b3c7e254132d3753e9078ea883d7c": "c = \\frac{|\\mathbf{r}-\\mathbf{r}'|}{t - t_r}", "258b8a3c97e6b038ec7e2fb1f72d329b": "R_\\text{F}C_\\text{F}", "258b8f53fd3c3174ecfe53e86a756aef": "d_0(L_\\text{target})=1.24\\sqrt[3]{L_\\text{target}-15}-1.8", "258b9727fd33da7b429acab7c0207c92": "\\mathbf S1:\n\\begin{cases}\nX^2+Y^2-T^2=0\\\\\n4X^2+Z^2-T^2=0\n\\end{cases}", "258bdddda3cf4ef2f318069c16b81cb0": " \\sum_{i=1}^n S_i S_i^* = I.", "258be85bd0628d273fb9cf28abb10b45": "\\Delta p_{0} =\\frac{p_{01}-p_{02}}{d\\dot{m}}", "258c19e59f9111a76fed292d3b727aec": " \\vec{P}", "258c36a1807b1f8704ea862de2663394": "C(\\alpha) = C(\\beta)", "258c4a33b193be16ff6a73dccd7647a5": "\\textstyle \\sum a_n", "258c9d5db9574c0a47afe7ffd045618a": "a,b,c\\in\\R^2", "258d5f9354f634312078c8333c7a3ac8": "a \\approx (bc)^2 f", "258e1086316d884ddff3ce2d0d9ec6b9": "\\alpha = \\frac{\\lambda_{PN} - \\lambda_{BN}}{(\\lambda_{BP} - \\lambda_{BN}) - (\\lambda_{PP}-\\lambda_{PN})},", "258e1ef86c30e66c3ef463123cca8535": " y , k , l ", "258e30594939efbddf16ee82eb474bbc": "\n\\mu _z = \\mu ^2 + \\,\\,\\sigma ^2 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sigma _z^2 \\,\\, = \\,\\,2\\,\\sigma ^2 \\left( {2\\mu ^2 + \\,\\,\\sigma ^2 } \\right)", "258e34240a67ad7d2d8f86027d54bb4a": " = \nf*\\mathrm{d}*\\mathrm{d}h + \n*(\\mathrm{d}f \\wedge *\\mathrm{d}h + \n\\mathrm{d}h \\wedge *\\mathrm{d}f) + \nh*\\mathrm{d}*\\mathrm{d}f", "258e41e5f3438bf96cdaf0b3d6169cb1": "ZFC+\\lnot \\operatorname{Con}(ZFC+H)\\vdash\\exists T(\\operatorname{Fin}(T)\\land T\\subset ZFC\\land(T\\vdash\\lnot H)).", "258f24906256b2db348fe874b109a043": "w(z)=U(0,0,z)=1", "258f9da82a5a9e3acd4607d103aee9b6": " a^{2}=b^{2}=c^{2}=1 ", "258fe12c6fa8331589203e803ff78293": "\\mathbf{[A]}= s^2 \\mathbf{[L]} + s \\mathbf{[R]} + \\mathbf{[D]} = s \\mathbf{[Z]}", "259028bceae44a55f13a92539082435a": "u(x) = \\frac{1}{n-1} \\sin^{n-1}(x)", "25903137f80eba9feeddc41f7da2cdbb": " y(t) \\approx \\cos \\left( -4 t \\pi \\Beta \\sin ( 2 \\pi f_{m} (2t + \\delta t) \\sin ( \\pi f_{m} \\delta t) + 2 \\delta t \\pi \\Beta \\cos (2 \\pi f_{m} ( t + \\delta t) ) \\right)\\,", "25904b18fd41f277e5b74fba36d7cd25": "Y_{right}", "2590587061477b689828d99d3a409bf4": "\\bar{e} + \\Delta \\bar{e}", "2590d0da3e123f1fdc695d8832526bef": "\\lambda \\in \\mathbb{K}", "25911990ec899da76dc3f7e916a9542c": " I_x(a,b) = \\dfrac{\\Beta(x;\\,a,b)}{\\Beta(a,b)}. \\!", "259132779c57997f83cc4e797de1822b": "\\vec{X}", "259166b7d93afd495c2724ad42a2d2ca": "\n\\mathbf u \\times \\mathbf v = \\sum_{i 0\\,", "259473a2166ee1a54585f08d75d8facf": "f(u_{n}) \\to f(u_{0}) \\mbox{ as } n \\to \\infty.", "2594be7a2c65de6fab78cf240239f11a": "\\frac{\\mathrm{d}U}{\\mathrm{d}t} = \\dot Q - p\\frac{\\mathrm{d}V}{\\mathrm{d}t}+P.", "2594c427d5b4fb0cede2634481fcdd22": "[0,0,1,10000]", "2594df56ff847e95bcf5384264beffa0": "X+\\xi\\mapsto X+\\xi+i(X)\\alpha", "259501d5dad9632809330ccec6e6c173": "\ndy/dt= a y^{2/3}\n", "25952b317143375572bf0d5fc08f8c1f": "D(p||q)+D(q||r)-D(p||r)=\\sum(p-q)(\\log;p-\\log;q)", "2595623bce5282d9baa8d6de13409b07": " (\\text{vector length})^2 = B_X^2 + B_Y^2 + B_Z^2 \\, ", "25957117245b2aa70ff2fa7f609cfc29": "r/s, r \\in R, s \\in R^\\times", "2595736a37536ee2c322154c85c34dc7": "(x_4,y_4)\\,", "25958882b8538051b2a78425045f6073": "S^n = D^n/S^{n-1}", "259599a5651e0579b8be314eb1db9c2b": "2^{cn}", "2595bb9d21e06c0196d21c95f805aa71": "\\delta^3\\left(\\mathbf{r}-\\mathbf{r}'\\right)=-\\frac{1}{4\\pi}\\nabla^{2}\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|},", "2595ce4b8a99b4006355d425b39f55bb": "\\!\\tau(x)", "2595cfb91a0f53a8d0956953fc7e6b8c": "y:\\mathcal{P}(S)\\rightarrow2^S", "2596565c1b36cc258bbd08037575219b": "(v_0,\\ldots,v_{n-1})", "2596a2d2fcef3007c2e0783aca180b56": "Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0,", "2596ca7c3fa57c89d937c89ff6237ce9": "g(\\boldsymbol\\eta)", "2596f457f57060553cf0737fe2ed27c6": "\\forall x: P(x) \\Rightarrow Q(x)", "25972673cf0f310e877ec635b4cce313": "{\\bar{L}}_5", "259755d00ed7b4d2323961431d2ae35e": "\\vec F_L", "2597b32b36935a132d199355d58ca58a": "f_n = \\frac{n}{2L}\\sqrt{\\frac{T}{\\mu}}", "2597c0b0a1464ed51ad1ec27ea6bc046": "81.9\\pm 0.5", "25985b0c0d9cb7007f71e16153a20040": "\\lambda r^2 - (\\lambda + \\mu + s)r + \\mu = 0.", "25986f008505072f83781287db5a60a0": "IR = \\frac{E[R_p-R_b]}{\\sigma} = \\frac{\\alpha}{\\omega} = \\frac{E[R_p-R_b]}{\\sqrt{\\mathrm{var}[R_p-R_b]}}", "259881d3201258f509f2425ac5eef713": "deg(p_i)", "259896284211e022f1dc9ba91ca59cb4": "|\\mathrm{O}(2n+1,q)|=2q^n\\prod_{i=0}^{n-1}(q^{2n}-q^{2i}).", "2599298e21387bb12606afe0bd537318": " N_2(k) = 2 \\pi k ", "25994f51b2443f84f9ce1bfe0f66e943": " \\hat \\Sigma = \\frac{1}{T} \\sum_{t=1}^T \\hat \\epsilon_t\\hat \\epsilon_t^{'}", "25995ec689b2617916c0d9f65d4df25b": "a_1 n_2 [n_2^{-1}]_{n_1} + a_2 n_1 [n_1^{-1}]_{n_2} \\equiv a_1 \\times 1 + a_2 \\times 0 \\times [n_1^{-1}]_{n_2} \\equiv a_1 \\pmod {n_1}", "2599af27f8acd90072f1e8b8d13121a5": " y_i^* = \\beta x_i + u_i, u_i \\sim N(0,\\sigma^2) \\, ", "2599b488fe46b0904e8847aa2b41e787": " \\Delta G_{ad} = \\Delta G_{p} + \\Delta G_{c} ", "259a4d8d9adb4bf649336c4e11857c79": "\\frac {C_{13}^2 C_4^2 C_4^2 \\cdot C_{11}^1 C_{4}^1} {C_{52}^5} = \\frac {78 \\cdot 6 \\cdot 6 \\cdot 11 \\cdot 4} {2{,}598{,}960} = \\frac {123{,}552} {2{,}598{,}960} \\approx 4.75\\% ", "259a67c52216b72aabbfe1cce1ed03f4": "\\{H, H, H, \\dots\\}", "259a76a6eb182367a3af4cdbaf5ca4d5": "x_z^2=\\xi^2/x_p^2", "259a7b00a69fca141f4d6b5684d7d2c6": "~n_{s,j}~", "259b3c532c7ebdf9eeff05b7c2b14a9d": " q_n(x) = \\int_\\mathbb{R}\\! \\frac{p_n(t) - p_n(x)}{t - x} \\rho(t)\\,dt. ", "259b51965d63fcde3aa92b8123d10824": "\\lambda (X, f, \\alpha)=(X, f, \\lambda\\alpha)", "259b5fc1b83ad7a2701579e6cefcd07c": "\\alpha = \\frac{\\mbox{number of de-excitations via electron emission}}{\\mbox{number of de-excitations via gamma-ray emission}}", "259bf5c87105bb02ca50410840ce94cc": "f(x) \\,\\!", "259c16c17d1c524fb919fb29b0d19e2a": "j=1,\\ldots, n", "259cb9e09299038e2c6b59b59b4f342c": "{q}={F(k)}", "259cd84c7e5abdc44922d2a5a8d611cd": "r \\in \\mathbb{R}_+", "259d73361aa7312ce9c1d7e113cbc9ad": "W(x) = \n\\begin{cases}\n (a+2)|x|^3-(a+3)|x|^2+1 & \\text{for } |x| \\leq 1 \\\\\n a|x|^3-5a|x|^2+8a|x|-4a & \\text{for } 1 < |x| < 2 \\\\\n 0 & \\text{otherwise}\n\\end{cases}\n", "259df63ece171d871ac28eec06cd2bff": "\\tau H_{n-i-1} M", "259df87b8ef012696e77cf59e26f8e84": "\\mathbf{O} = \\begin{bmatrix}\n\n1 & x_{F} & y_{F} \\\\\n1 & x_{G} & y_{G} \\\\\n1 & x_{H} & y_{H}\\end{bmatrix}.", "259e1dd16454b6c16cb820bf203cb73e": "\\hat{M},\\hat{N}", "259e38b65a86aea4501c0a3c02ff098a": "\\vec j_s = {\\hbar \\over {4 \\pi}} g^{\\uparrow \\downarrow} {1 \\over M_S^2} \\langle \\vec M(t) \\times {{d \\vec M(t))} \\over {dt}} \\rangle ", "259e554e785b3d12cc363922b941c294": "u,v,x,y", "259ef820d1596b7dd83b706e83629595": "t\\text{ and }t - \\Delta t", "259f0e46103d040d01d9e2bf94ecaebf": " \\land ", "259f25e0fe4d9d8cdd9bde31b6677d42": "C_v\\,\\!", "259f35031eb7561091713eca2a6847e9": "\\scriptstyle O(N\\log N)", "259f9b91dbcdc1567f079bef86ce0526": "\\ \\displaystyle \\ \\mathcal{U}(\\alpha^{*},\\tilde{u}) \\ ", "259f9d6cd8e591971a46d5ee64224772": "P = \\frac{P_{max}} {3}", "259fa4d96872e723316c3536350c69b2": " 1 - \\epsilon ", "259fabdff41a6272665d1b9090335447": "\\mathbf{C}V = \\ker c \\cap \\ker b.", "259fae9a12508a84f35cea2f6ad649ee": "P_E~dE = \\frac{1}{N}\\,\\left(\\frac{f}{(\\hbar\\omega\\beta)^3}\\right)~\\frac{1}{2} \\frac{\\beta^3E^2}{\\Phi}\\,dE", "25a0089b5d82307945c55a0842526127": "55_{11} \\ ", "25a02cd84fc2db9c0a26a146751cfdba": "F[x,y] = \\int_a^t\\sqrt[3]{x'y''-x''y'}\\, dt ", "25a07727e3646a693e1e3a4f5de16a28": "E(x)=k/\\lambda,\\,E(\\ln(x))=\\psi(k)-\\ln(\\lambda)", "25a0903e8caa68c62daecfc0e6884635": "g(t) - G(t) \\approx 0", "25a090dad425532d05bb79b2ee8b4818": "L_{4k+2}(\\mathbb{Z})=\\mathbb{Z}_2", "25a0c9a6afcc3945e60901ae32ffecdd": "x(u) = r \\cos u;\\qquad z(u) = r \\sqrt{2} \\left[ E(u,\\frac{1}{\\sqrt{2}})-\\frac{1}{2}F(u, \\frac{1}{\\sqrt{2}})\\right]\\text{ for }u \\in [0, \\frac{\\pi}{2}] \\, ", "25a0ef5fb7e73104fe0c2220cd15d721": "P(G,4) \\neq 0", "25a21b9f575914a01deedf0e4a927e19": "M_n = E \\bigl( f | \\Sigma_n \\bigr)", "25a22f0a3134167043d5c720b8bc2f71": "\\sum_{j=1}^n \\ M_{ij}(\\boldsymbol q ) \\ddot q_j + \\sum_ {j,k=1}^n \\Gamma_{ijk}\\dot q_j \\dot q_k +\\frac{\\partial V}{\\partial q_i} =\\Upsilon_i \\ ; i= 1, ... , n \\ , ", "25a25ae65beac79c88c741b3a9212108": "\\mathbb P_{\\mathbf k}^n", "25a2766c207b0ec2811dc40d4434bfaa": "\\mathbf{y}(k) = \\mathbf{C}(k) \\mathbf{x}(k) + \\mathbf{D}(k) \\mathbf{u}(k)", "25a2af99383a033c55be403b39aa184a": " M^{(n)}(B\\times,\\dots,\\times B)=E [{N}(B)({N}(B)-1)\\dots ({N}(B)-n+1)], ", "25a2b59487fd039c61aac9ca3b74dd69": "\\operatorname{E}[|S_N|]\\le\\sum_{n=1}^\\infty\\sum_{i=n}^\\infty\\operatorname{E}[|X_n|\\,1_{\\{N=i\\}}]\\le\\sum_{n=1}^\\infty\\operatorname{E}[|X_n|\\,1_{\\{N\\ge n\\}}],", "25a2f45bf56d3271d707630a550b688e": "\\ \\epsilon_d", "25a373da14ac41b26a34db638e790242": "Z=V\\rho \\ ", "25a3b4b1848928f25e0b66aad10aa919": "\\left( a \\right)", "25a3cb6cac366e81f9da57a8f35e6b03": " L(n,2) = (n-1)n!/2", "25a3d207d5eb113a3cf19ce8303db051": "\np(\\mathbf{X}|\\mathbf{M}, \\mathbf{U}, \\mathbf{V}) = \\frac{\\exp\\left( -\\frac{1}{2} \\, \\mathrm{tr}\\left[ \\mathbf{V}^{-1} (\\mathbf{X} - \\mathbf{M})^{T} \\mathbf{U}^{-1} (\\mathbf{X} - \\mathbf{M}) \\right] \\right)}{(2\\pi)^{np/2} |\\mathbf{V}|^{n/2} |\\mathbf{U}|^{p/2}}\n", "25a44a250490727df635042c6bd98e36": "H^p(Y,{\\rm R}^q f_*\\mathcal{F})\\implies H^{p+q}(X,\\mathcal{F})", "25a4666732ff7c5e4c83a326afbbe8a4": "\\hat{\\beta} = 1", "25a4c77c966b4817588312d9ed956450": "\\sigma_1 > 0 ", "25a5d03acecdbe195b198608346808da": "d_\\Lambda=d_{\\lambda_1}\\cdots d_{\\lambda_k}, \\qquad\nd_\\lambda=\\partial_\\lambda + y^i_\\lambda\\partial_i +\\cdots,", "25a5d9103a6eae8cc70d729dccaa49fd": "\\lceil mx \\rceil =\\left\\lceil x\\right\\rceil + \\left\\lceil x-\\frac{1}{m}\\right\\rceil +\\dots+\\left\\lceil x-\\frac{m-1}{m}\\right\\rceil,\n", "25a5e1be3b89b1fb9b69c905c958944e": "Initiates(a,f,t) \\equiv\n[ a=open \\wedge f=isopen \\wedge HoldsAt(haskey, t)] \\vee \\cdots\n", "25a6377f043661c6808cd71d5bda40c7": "q=3", "25a66bf1d5decd9de746a4c83fba2b0e": "\\mathbf{v}(\\mathbf{x},t) = \\dot{\\mathbf{u}}(\\mathbf{x},t)", "25a6b584eb0f72084bf0d6b8192691f0": "\\prod_{i \\in I} A_i = \\bigcap_{i \\in I} \\pi_i^{-1}(A_i) ", "25a6ecc95e99fc4750aba27923364d60": "i=2,\\dots,99", "25a6f763ddf2c60ddf79f03b7633fde0": "1-365!/((365-n)!\\cdot365^n)", "25a7149bee46c56eec772bc7da8c210d": "O\\left(e^{(\\ln N)^{1/3}(\\ln\\ln N)^{2/3}(C+o(1))}\\right)", "25a77e87b658f3a8afef8ac3e8842a5b": "a_i \\in \\mathbb{C}", "25a7851359f1ba2c2ad013c8a6794b02": "\\exist C_1 > 0,\\ \\exist C_2 > 0,\\ \\forall \\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n:\n C_1 d(\\mathbf{x}, \\mathbf{y}) \\le \\sqrt{q(\\mathbf{x} - \\mathbf{y})} \\le\n C_2 d(\\mathbf{x}, \\mathbf{y}). ", "25a7e04a96ee3fac9b7479ffdf429eac": "\\overline{\\omega^2} = \\omega", "25a7eaeea04d826bff41777a9b460936": "E_n = -me^4/8\\epsilon_0^2h^2n^2 = 13.61eV/n^2\\,\\!", "25a7f2d95acf15663de89dcb69d82567": "(P(n))_{n\\in\\mathbb{N}}", "25a7fbd008c5b05bc741a24750bffbb1": "\\omega_E \\,", "25a81dde30d0e639428aaaafdc6c3e25": "\\begin{align}P(no~disease~in~population) = 1 - P(PH~in~population) - \\\\\n P(cancer~in~population) - P(other~conditions~in~population = 0.997\n\\end{align}", "25a886b1ed0f1ab38d66ceef1c0e0168": "\n\\left[ L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^{\\mu} \\right) - \nL \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) \\right] \n= \\frac{\\partial}{\\partial x^{\\sigma}} \\left( \\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\right) \\bar{\\delta} \\phi^A + \n\\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\bar{\\delta} {\\phi^A}_{,\\sigma}\n= \\frac{\\partial}{\\partial x^{\\sigma}} \n\\left( \\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\bar{\\delta} \\phi^A \\right)\n\\,.", "25a8afe7b836ac8b794dd22503cef68c": " \\beta\\ = \\arccos\\left(\\frac {a}{c} \\right)\\,", "25a8b802657bed9d27681ab643e52118": "A_8", "25a8b8693d7a146f37a2273437a74c7c": " \\mu_{c}", "25a8cd9626fd02f8fdf65de5d151c5b0": "1 \\le r < n", "25a9047b7b9fd4514fb91d53cbb18909": "\\hat R_P", "25a946fc81fb5268989250ee0dde7717": "Pr(H|X),", "25a94ee98b25f34f8678526199b347bf": "(f\\wedge g)(x)=f(x)\\wedge g(x)", "25a9b7812e7b63552a8971ade951455e": "D^{j}\\left( \\mathcal{G}_{0}\\right) ", "25a9ced74b3233d87c3a0e3355b3a895": "L^*=\\frac{X}{\\mathbf{E}[X;P]}\\geq 0", "25a9deaa3f10d93f42974f327cd8f5ed": "\\mu=rv^2", "25aa862e63b4872d5bd8e4298fef3a4a": " c_i ", "25aa9fbe1cb037c8da62f854a252bd42": "=\\mu \\sin \\theta\\ .", "25aaaba799f1bd6cd76d6844869a6f4b": "\\hat{C} = \\Big( i {d \\over dx} - k \\Big)", "25aad42addaadd752382df2753dc5b66": " -2, \\frac{2}{3}, 1.21\\,\\!", "25aaedbefb7d7d1c7c65e850f463894f": "5 \\}", "25ab742e51736ee707ed42302c1d8c1b": "w\\Vdash A\\land B", "25ab8398f6bb5db8e238193e4b8ea629": "\\mathbf{P}( X \\ge (1+\\delta)\\mu) \\le e^{-\\frac{\\delta^2\\mu}{3}}, \\quad 0 < \\delta < 1", "25ab87160d5626187b1ff609ae471aae": "> 4.6 \\times 10^{26} \\ \\mathrm{years} \\,", "25ac029a9f81a19f308f15ceb12fa475": "AB = r^n-1", "25ac580f519817e6fcaa31d349888b61": "X' = (X_1, \\cdots, X_i + X_j, \\cdots, X_K)\\sim\\operatorname{Dir} \\left(\\alpha_1,\\cdots,\\alpha_i+\\alpha_j,\\cdots,\\alpha_K \\right).", "25ac910d0eda99893ee18a71e7b91e55": "j\\in\\mathbb Z", "25acce8af3450e1843178ad127aa28ba": "\n E_1 > 0 , E_2 > 0, E_3 > 0, G_{12} > 0 , G_{23} > 0, G_{13} > 0\n ", "25acff9ab7e7616274ae2b993ef7aca1": "\\int \\left|\\cos {ax}\\right|\\,dx = {2 \\over a} \\left\\lfloor \\frac{ax}{\\pi} + \\frac12 \\right\\rfloor + {1 \\over a} \\sin{\\left( ax - \\left\\lfloor \\frac{ax}{\\pi} + \\frac12 \\right\\rfloor \\pi \\right)} + C\\;", "25ad100b8ab14e7403ad00ff0dc29d70": "\\textstyle f^{-1} ", "25ad10834aaa3f14dd15ad5d0671de19": "\\vec{F} = \\frac{q_{1} q_{2} \\vec{r}}{4 \\pi \\epsilon_{0} r^3}", "25ad588dfdb4871d864287e633cfe807": " \\mathbf{E}( \\mathbf{r}, t ) = g(\\phi( \\mathbf{r}, t )) = g( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} ) ", "25ad8b2cef744ba6d169c5a779761c50": "g^{(1)}(\\tau)=e^{-i\\omega_0\\tau-(|\\tau|/\\tau_c)}", "25ad90f64c93d166bc730e898cb9d4ef": "{M}=\\frac{q^2}{gy}+ \\frac{y^2}{2}", "25adbfaa950f162fd40dc3c6b7c63d43": "T_{\\mp1}^{(1)}", "25ae7ed1c28cc50b492a0d7f905e9ae4": " \\dot{x_i} = x_i [ f_i(x) - \\phi(x)], \\quad \\phi(x) = \\sum_{j=1}^{n}{x_j f_j(x)}", "25ae8cac059f96485f346a2be5805212": "n_\\text{G}", "25aec4ef2ab12a03780de97f74b04f87": "\\omega^{2}\\equiv\\left( \\omega_o^2 +\\frac{\\mu BH_k}{mL_{e}^{2}(B+H_k)}\\right)=\\omega_o^2\\left( 1 +\\frac{\\mu BH_k}{kL_{e}^{2}(B+H_k)}\\right)", "25af01576ff96df1f8f74cc55a1cc913": "p_a(z)", "25af54de5611f36f0c598577ee8daafd": "A^T X + X A - X B R^{-1} B^T X + Q = 0 \\, ", "25af810aa748842731df94a4b0e9aa06": " z ", "25afb5b3bbcf1be6e9f7a7bc1ff5e95e": "(a\\cdot b)\\cdot c = a\\cdot(b\\cdot c)", "25afb7203c03c65b50e979a3c2ea46ff": "a_{ji}", "25aff1a98fbb6453f0d2c81f32ad70b8": "N_2\\,", "25b00365e84b70670f681371316fa8cb": " \\Delta f = \\frac{1}{\\sqrt{|g|}} \\frac{\\partial}{\\partial \\xi^i}\\!\\left(\\sqrt{|g|}g^{ij} \\frac{\\partial f}{\\partial \\xi^j}\\right) =0, \\qquad (g=\\mathrm{det}\\{g_{ij}\\}).", "25b011dcc8e1f70c761b4244c135fae0": "\\frac{1}{\\sqrt{2\\pi}}e^{-x^2/2}", "25b075e7e914b9dbfb20c590f9b924a3": " H = H_0 + \\frac{\\hbar}{4m_{0}^{2}c^{2}}\\bar{\\sigma}\\cdot\\nabla V \\times \\mathbf{p} ", "25b0be86daad501d2ee3200059e54061": "0=\\sum\\limits_{n=0}^{\\infty }\\frac{(\\sqrt{\\mu }\\varsigma ^{\\mu\n})^{n}}{\\sqrt{n!}}(E_{\\mu }^{(n)}(-\\mu |\\varsigma |^{2\\mu }))^{1/2}|n>,\n", "25deee057750ce13907359f2f0996b84": "\n\\mathbf{q} = -\\frac{\\partial G_{3}}{\\partial \\mathbf{p}}\n", "25df18ba239d20c062d5bebc102b51dc": " HP_{R_n}(k) = {{k+n}\\choose{n}} = \\frac{(k+1)\\cdots(k+n)}{n!}\\,.", "25df4e8bbac576f1bff9a90fa85291f4": "(d_\\mathfrak{g} \\alpha)(X) = d(\\alpha(X)) - i_{X^\\#}(\\alpha(X))", "25df5b4729a1f1fc283169879cd78e95": "(d_1, d_2, p_{1}^r, p_{2}^r)", "25df7a91dac1fae2c27b5a4049a28b03": " H(s) = \\frac { R_1 / (1 + sC_1 R_1) } {R_1 / (1 + sC_1 R_1) + R_2 + 1/(sC_2)} ", "25dfd15c1ddf32a42872aecb4a84bdff": "g(X_{i},X_{j})=\\eta_{ij}\\,", "25dff6299ac80287ec301ad3d9162861": "\\frac{Z'}{Z_0}=\\frac{Z_0}{Z}", "25dff90858f5546dcac1139bc72b02fc": "\\mathbf{p} \\times \\mathbf{\\epsilon^1}(\\mathbf{p})", "25e06f0578d77de682fcb6ccabc09e57": "\\begin{matrix} {r \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "25e0742bdeb6a95431a067175c08b5a6": "|s_2| = 8", "25e093c59df3f7799a0fc2ac4bc7d125": "g_{\\varepsilon} (x) = f (x) + \\varepsilon \\eta (x)", "25e09869f741efbb23117fd65a997f55": "GFR = \\frac{U_{[\\text{creatinine}]}\\times \\dot{V}}{P_{[\\text{creatinine}]}}", "25e09c195d10cf43c35d6a81bf90605a": "E_B(v,V)", "25e0b3939da142f651dbe6fa41bbd9e8": "0\\le x_{i} \\le 1", "25e10282ddfae9363a4585b081a755e4": "x = {a\\over 2} + b \\cos \\theta + {a\\over 2} \\cos 2\\theta,\\, y = b \\sin \\theta + {a\\over 2} \\sin 2\\theta.", "25e1137601c1c685384cd511fd22cc79": "\\frac{2\\pi}{45}", "25e15fb1ba6947f77b7844252441ba50": "t\\neq 1 \\Longrightarrow s\\leq t^2", "25e160cae17e639db0cab486502321f1": " \\frac{\\Delta P}{P} = \\frac{\\Delta \\mathit B}{\\mathit B} + \\frac{\\Delta \\mathit A}{\\mathit A} \\left ( \\mathit 1 + \\frac{\\Delta \\mathit B}{\\mathit B} \\right )\n", "25e1d3ef3ef39a75ae15060c220658ad": " T_{eff} ", "25e1f43e4cfc39568d0044de10c3ed7f": "- L {dI \\over dt} = R_2 \\cdot I", "25e1fd7d8507cd11d1ddd6fefdb26970": "\\phi^{-1}(S)=\\{x\\mid \\phi(x)\\in S\\}\\in\\mathrm{REC}(M) ", "25e21916c510a451a3d032c664d4ce0f": " \\mathbf T^n \\begin{bmatrix} 0 \\\\ \\vec b^{n-1} \\\\ \\end{bmatrix} =\n\n\\begin{bmatrix}\n t_0 & \\dots & t_{-n+2} & t_{-n+1} \\\\\n \\vdots & \\ & \\ & \\ \\\\\n t_{n-2} & \\ & \\mathbf T^{n-1} & \\ \\\\\n t_{n-1} & \\ & \\ & \\ \\\\\n \\end{bmatrix} \n \\begin{bmatrix} \\ \\\\\n 0 \\\\\n \\ \\\\\n \\vec b^{n-1} \\\\\n \\ \\\\\n \\end{bmatrix} = \n \\begin{bmatrix} \\epsilon_b^n \\\\\n 0 \\\\ \n \\vdots \\\\\n 0 \\\\\n 1 \\\\\n \\end{bmatrix}. ", "25e224e926b0f85e79372bbbc8fb09c8": " \\xi \\,", "25e2c0c6aa8227f65923457925d531fe": " \n\\vec{R}_1 \\times \\vec{R}_2 = A \\left( \\vec{r}_1 \\times \\vec{r}_2 \\right)\n", "25e2c7173ddf3bc45afaf3232cf31e6a": "t^{(p-1)(q-1)/2}\\frac{1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^2}.", "25e3f33d7f862bcbc49eaf854761adea": "X_\\mathrm L = 2 \\pi f L ", "25e4162bfc616a3af471c655d32c7a13": "0.3 rad / s \\sqrt{Hz}", "25e429d17ebcf0858fb256609818c180": "\n \\left(\\frac{\\partial w}{\\partial x_1}\\right)^2 ~,~~ \\left(\\frac{\\partial w}{\\partial x_2}\\right)^2 ~,~~\n \\frac{\\partial w}{\\partial x_1}\\,\\frac{\\partial w}{\\partial x_2} \\,.\n ", "25e4acb15a3c12e809c61253c2399441": "\\langle I \\rangle_e", "25e56358cfeee42568285723578129b9": "Q(\\eta)=\\int_{-\\pi}^{\\pi} \\frac{I(w)}{f(w;(1,\\eta))}\\, dw", "25e583ffbc6eefd5c6ea68fdba535f44": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = x \\\\\n f_{2}\\left(x,y\\right) & = \\left(1 + y\\right) \\exp \\left(-\\frac{x}{1+y} \\right)\n\\end{cases}\n", "25e59010356db818689bf1c0b08876b0": " \\lambda f.\\operatorname{let} x = \\lambda x.f\\ (x\\ x) \\operatorname{in} x\\ x ", "25e5c301c7c78b7ea0199980a287cd79": " {P}_\\alpha ", "25e61144ef0afb77d87baec5e56e065f": "\\widetilde\\phi_\\alpha\\colon \\pi^{-1}(U_\\alpha) \\to \\mathbf R^{2n}", "25e64b24ddeb89dac4cd34dc76dd08e8": "\\underline{\\land}", "25e6574b331aa9685cf6b73ab6b727ef": "J^rY", "25e72971f523d97359091f9b69447429": "\\mu_p, \\mu_q, \\nu", "25e76f75d8c479e282a6a6a3ef18da10": "\\mathbf{V} \\setminus \\{X_i,X_j\\}", "25e7be4ee0124b0dc6c281c7851a4a4b": "dW \\propto dQ", "25e7d150e6eb867b0f59ecb10c4370a0": "E_C^{\\rm PBE}(\\omega)", "25e7da7705c2c086bee78f9df9e37d33": " \\phi(x) = \\frac{e^{- \\frac{x^2}{2}}}{\\sqrt{2 \\pi}} ", "25e80ff375bc1c922cd5ab6e8bf68cdc": "\\eta=\\frac{R T_c}{P_c}\\Phi\\frac{1}{V_m}.", "25e84141a9e268f0ee8a38106cc343b0": "R_{\\alpha\\beta}=0", "25e882a5ed62c729c512f1783e022bce": "\\left\\| \\frac{f + g}{2} \\right\\|_{L^p}^p + \\left\\| \\frac{f - g}{2} \\right\\|_{L^p}^p \\le \\frac{1}{2} \\left( \\| f \\|_{L^p}^p + \\| g \\|_{L^p}^p \\right).", "25e8e0e034f606b7047a5b6cf1d5eff7": "0 0", "25f6b9f54baafde2b1693692627eaff8": "H_d(z) \\ ", "25f6c0db710c5ccf0780858f0ce230a8": "A - B C^{-1} B^T", "25f6c98ca8211955bf8e9ceec6055ae5": "\n \\begin{align} \\pi(\\theta|\\mu,M) & = \\operatorname{Beta}(M\\mu,M(1-\\mu)) \\\\\n & = \\frac{\\Gamma(M)}{\\Gamma(M\\mu )\\Gamma(M(1-\\mu))} \n \\theta^{M\\mu-1}(1-\\theta)^{M(1-\\mu)-1}\n \\end{align}\n", "25f6fb3d9c4cb2dfb7dc0229cdb5fb56": "f(qm)=f(q)m", "25f77b14bfc643155b563641ac27bb50": "f_1(d)", "25f7874321d6a78f6b5ba112fb5953b9": "\\displaystyle (1-e^{2\\pi i\\alpha})(1-e^{2\\pi i\\beta})\\Beta(\\alpha,\\beta) =\\int_C t^{\\alpha-1}(1-t)^{\\beta-1} \\, dt.", "25f792cf700d49fe24efa94280c2bbf9": " \\mu_X^\\pi ", "25f7c8647f0efc88bd753bbbc6ac2b6c": "(1/2)b \\pm (c^2 - a^2)/b", "25f7e414d76106f3f7c99b439bbdcb6e": "\\chi_e(\\Delta t) = 0", "25f7f3448a7d08e290e55ed407b4793f": " V_2 ", "25f83ceb5ce70937f0f0a51c2296a2b6": "h: H^n (C; G) \\rightarrow \\text{Hom}(H_n(C),G).", "25f8763c5af5fa164c23046c9d98e909": "f(x;\\alpha,\\beta,c,\\mu)=\\frac{1}{\\pi}\\Re\\left[ \\sum_{n=1}^\\infty\\frac{(-q)^n}{n!}\\left(\\frac{i}{x-\\mu}\\right)^{\\alpha n+1}\\Gamma(\\alpha n+1)\\right]", "25f8b8807f99b07cc3d94b8a24f4399e": "v_{0.5} \\neq 0 ", "25f91780f1181b22a9342596a0c53a52": "(\\mathbf{a} \\cdot \\mathbf{b})^2 - (\\mathbf{a} \\wedge \\mathbf{b})^2", "25f93d53f7a72c019d91a984c072d432": "\\Pr\\{h_{a,b}(x) = h_{a,b}(y)\\}\\le 1/m", "25f966a5374dc451d5c386d0778e2f78": "\\widehat{p} = \\left(1 + \\frac1n \\sum_{i=1}^n k_i\\right)^{-1}. \\!", "25f995ab96a6af776e7d445a51f784ec": " u(0) = u^{1,0} \\mbox{ in } \\Omega", "25f99b25d676f4240ab633d2c6ad2085": "\\frac{AP}{BP} = \\frac{AC}{BC}.", "25f9ab91477595c9e9b5d6658e662cf6": "R_{4,2} = 36 r^4-56 r^3+21 r^2", "25f9ae96a5021937aa517e982dc0e192": "\\limsup_{n\\to\\infty} d(S \\upharpoonright n) = \\infty,", "25fa2375c2a54ff028d553e6e083db68": "\\mu=2.394", "25fa44563151151076269d2eab7bbdbd": "\\prod_{d|n} f(d)\\;", "25fa5f13570947da66b0d20e90b33724": "q \\approx kd", "25fab1e2a6439f91ff9f1d1c67fb446e": "v_\\mathrm{rms} = \\sqrt {{3RT}\\over{M_m}}", "25fb0beeab6bbc745ef65576317fbf6d": "++", "25fb1fed93190afb86d6e7f2b056e11e": "\n\\mathbf{j} = \\sigma \\mathbf{E}\n", "25fb3646e2c1c65a58906e818c7adb1a": "p_mf_m", "25fb896cad16acce5dc92116490fcd55": "\\frac{ \\partial V [e(u,P),P]}{\\partial Y} \\frac{\\partial e(u,P)}{\\partial p_i} + \\frac{\\partial V [e(u,P),P]}{\\partial p_i} = 0", "25fbb65d93d398451ffbca8cd824dd16": "fg = \\Lambda(f, g) + \\Lambda(g, f).", "25fc133c9da87a7b1fe4c7dfadc67ed9": "A_z(\\mathbf{r},t) = \\frac{\\mu_0}{4\\pi} \\int_\\Omega\\frac{j_z(\\mathbf{r}',t_r)}{|\\mathbf r - \\mathbf r'|}\\,{\\rm d}^3\\mathbf{r}'", "25fc2f91140cc449aed7efdc615fb73f": "F(t_{11},\\ldots,t_{1n}) \\vee \\ldots \\vee F(t_{k1},\\ldots,t_{kn})", "25fc45816f5e341d2ac808d19d264aa7": "{e^x \\over \\cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \\cdots\\!", "25fc99a8798c1d6b6dfa0fb518cad15b": " H = H^*, H^{-1} = H^{T}, \\text{i.e. } HH^{T} = I ", "25fd034ca05e19ef700a47a747d3dac1": "\\frac{1}{m_\\text{red}} = \\frac{1}{m_1} + \\frac{1}{m_2} \\,\\!", "25fd5591f13dd0a7d509ad4227844d0d": "\\mu(Av_1,\\ldots,Av_n)=|\\det A|\\mu(v_1,\\ldots,v_n), \\quad A\\in GL(V).", "25fd677a8aae44daac8afba5c182dbc3": " \\widehat{\\beta}_\\mathrm{IV} = (Z^\\mathrm{T} X)^{-1}Z^\\mathrm{T} y \\, ", "25fd6e4e3f7f67f9753102bfc1ad1d95": "(z_1,z_2,z_3) \\in \\mathbf{C}^3,\\qquad (z_1,z_2,z_3)\\neq (0,0,0)", "25fda1a4b7dee5ff36b4ece1f48e4b3d": "\\Re\\{W\\}", "25fda6be1e958dd7c93ac388b9718558": " \\text{PPI} = \\frac{\\text{savings} \\times \\text{probability of success}}{\\text{cost} \\times \\text{time of completion}}", "25fdaadb54fd40b61bda97535323b838": "\nd^j_{m'm}(\\beta)= \\langle jm' |e^{-i\\beta J_y} | jm \\rangle\n", "25fe2f4be9536e006ae2efea18e84a0d": "K>K_c\\approx0.971635\\dots", "25fe940e08e8c622c0b69d7db24bfd1b": " U_{in}=cos2\\theta+90 ", "25ff2038fbecfef7255539b176b731c5": " W_E(2) = \\frac{\\sigma^2 (c_{11} + 2 c_{22})}{2c_{11}}", "25ff8153bc9215f993d5cb26530e8225": "D_Q = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty\n P_{X,Y}(x,y) (x-y)^2\\, dx\\, dy = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty\n Q_{Y|X}(y|x)P_{X}(x) (x-y)^2\\, dx\\, dy. ", "25ffdb224a66d8a81311e294590b3237": " b_{ii}=0.0778 \\cdot \\frac{R \\cdot T_{c,i}}{P_{c,i}}", "2600a1802f049e1ae2a1129a11a11ca6": "F_t|_A = f_t", "2600e9829be252823634ff453498a654": " z^{\\mathrm{T}}N z = \\begin{bmatrix} 1 & -1\\end{bmatrix} \\begin{bmatrix} 1 & 2 \\\\ 2 & 1\\end{bmatrix} \\begin{bmatrix} 1 \\\\ -1\\end{bmatrix}=\\begin{bmatrix} -1 & 1\\end{bmatrix} \\begin{bmatrix} 1 \\\\ -1\\end{bmatrix}=-2 \\not > 0.", "2601322cc2051cbf1a2b7966b39a83cc": "(4q^{2n}(q^{2n}-1)/(q-1),q^{2n-1}(1+2(q^{2n}-1)/(q+1)),q^{2n-1}(q^{2n-1}+1)(q-1)/(q+1))", "2601335cf248e1f383c00415cbad2b59": "(x + 1)^{\\deg(p)}p(\\tfrac{1}{x+1})", "26014d72fb73347ccc6522901634bb80": " m_j\\omega_j = m_k\\omega_k. ", "2601d8ffb7521c0457a2f4938716e9c2": "E_{\\pm}(n) = \\hbar\\omega \\left(n+\\frac{1}{2}\\right) \\pm \\frac{1}{2} \\hbar\\Omega(n),", "2601e22d221336f196364e7992bc8504": "\\displaystyle \\hat{f}(x)\\,", "2601e35d8c91429bee12effc91c4eaa8": "F_{c,v}(p) = p - 2tv.\\,", "26022e62f04ca7b7cd7f30bbb1a693b0": "0 \\to D \\overset{x}\\to D \\to D/xD \\to 0", "26024536a6ab8cb7a119b1f2e415f2da": "NEXP \\subseteq EXP", "2602b62af20a5c97cc109e074b2c6c9a": "dH \\neq 0", "2602da02f1e0dd5d90a6066fca6a4d1b": "f_t:\\Omega_0\\to \\Omega_t, \\mbox{ for } 0\\le t\\le t_0.", "2603471ca85409a34b02eeacafd6168f": "\n\\mathrm{area}(D_r) = -\\frac 12\\, \\Re\\int_0^{2\\pi} \\sum_{n=-1}^\\infty\n\\sum_{m=-1}^\\infty\nm\\,r^{n+m}\\,a_n\\,\\overline{a_m}\\,e^{i\\,(m-n)\\,\\theta}\\,d\\theta\\,.\n", "2603a0e8267c601830d36280f7707fdc": "{\\Bbb C} \\times H/\\Gamma_n", "2603a57935aa077f2c7de0149542e2fe": "\\dot{\\mathbf{p}_i} = \\dot{m}_i \\mathbf{v}_i + m_i \\dot{\\mathbf{v}}_i", "2603ac16346b83991a9ffab96d858f22": "D=4", "2603ce233f95dbba95fde3c6f63a261e": "I_{x} = \\int_A y^2\\,\\mathrm dA = \\int^{b/2}_{-b/2} \\int^{h/2}_{-h/2} y^2 \\,\\mathrm dy \\,\\mathrm dx = \\int^{b/2}_{-b/2} \\frac{1}{3}\\frac{h^3}{4}\\,\\mathrm dx = \\frac{b h^3}{12}", "2603cf4793d36a2d46544facc8d04582": "\\ \\ t\\ (\\ ", "2604223c49bde495937d595925c317f7": "P \\, = \\, f(G_{ij})", "2604293556409495ef9a034e3aa4401c": "\\chi(S') = N\\cdot\\chi(S) - \\sum_{P\\in S'} (e_P -1) ", "26043f7118c8a0956bf41843f40d0b98": "\n \\nabla^2\\nabla^2 w = 0\n ", "2604974757011ea2c086cc62cebba4b7": "\\ \\Delta G(T)=\\Delta H -T \\Delta S", "26051e1e5bb31458e6da232b0e8b3af7": "X(t) = ((k + \\cos(t))\\cos(t), (j + \\cos(t))\\sin(t), \\sin(t)) \\, ", "260545a96c730fc4c721b4007c45594f": " (U, z_i) ", "2605a29607e49db6bfd6c0a476467d7b": "\n\\beta \\,\\,\\,\\, \\approx \\,\\,\\,\\,\\frac{a}{2n}\\left[ \\begin{array}{l}\n \\left( {\\alpha \\left( {\\alpha - 1} \\right)\\mu _1^{\\alpha - 2} \\mu _2^\\beta } \\right)\\sigma _1^2 + \\\\\n \\left( {\\beta \\left( {\\beta - 1} \\right)\\mu _1^\\alpha \\mu _2^{\\beta - 2} } \\right)\\sigma _2^2 + \\\\\n \\left( {2\\,\\alpha \\,\\beta \\,\\mu _1^{\\alpha - 1} \\,\\mu _2^{\\beta - 1} } \\right)\\sigma _{1,2} \\\\\n \\end{array} \\right]", "2605f657072d8970af099eb548b31503": " \\int_{-\\infty}^{\\infty} |q\\rangle \\langle q| dq = 1 ", "2605fb05b0ab77a103c004518d8cac08": "d: Hom_n(A,B) \\rightarrow Hom_{n+1}(A,B)", "26061eb021150a379495d3897bc1754e": "Tr\\,60 \\times 18 (P9) LH", "260627e8347f059ca3f373fb2becdbd1": "\\scriptstyle \\leq1\\times10^{-6}", "2606b581c7a5d5d4b40cff188c6a9a9d": "\\pi_0 (B^+C)", "260733f816b891e2e4c3782d4e2d7317": "\\left (\\frac{{\\partial}u}{{\\partial}t}+u\\frac{{\\partial}u}{{\\partial}x}+v\\frac{{\\partial}u}{{\\partial}y}+w\\frac{{\\partial}u}{{\\partial}z}\\right )= -\\frac{1}{\\rho}\\frac{{\\partial}P}{{\\partial}x}+{\\nu}\\left( \\frac{{\\partial^2}u}{{\\partial}x^2}+\\frac{{\\partial^2}v}{{\\partial}y^2}+\\frac{{\\partial^2}w}{{\\partial}z^2}\\right)\\,\\!", "26073a43b1a44a43666c01a26c2ef6a9": "\\Phi(z,s+1,a)=-\\,\\frac{1}{s}\\frac{\\partial}{\\partial a} \\Phi(z,s,a).", "260771353e750e04adbff9d451eddcaf": "\\begin{align} h\n& = \\frac{1}{4}\\sum_i p_i(\\theta) \\; d(\\log p_i(\\theta))\\; d(\\log p_i(\\theta)) \\\\\n&= \\frac{1}{4}\\sum_{jk} \\sum_i p_i(\\theta) \\;\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_j}\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_k}\nd\\theta_j d\\theta_k\n\\end{align}", "2607a0783fe83a1332bfd46825a2e267": "\\displaystyle f * g\\,", "2607a9cdd0478c6e3e74210dfecfdf40": "\n \\omega^2\\, =\\, g\\, k.\n", "2607b4e8c424fe06d9d8758ba9011ed3": " \\theta, \\sigma, \\nu, \\mu_p, \\mu_q ", "26083d438c762cdbfd9549b1369322c4": "\\alpha_l = \\left[ \\frac{q^2}{\\mu \\omega^2} \\right] \\left[ \\frac{(2 l - 1)!!}{l} \\right] \\left( \\frac{\\hbar}{2 \\mu \\omega} \\right)^{l-1}", "260855aa0bd47eb2299090579cf71f7f": " n=0 ", "26087b0b676d811f5163ec265e83fc53": "\\scriptstyle{I\\ge 1}", "2608a6214ce69434fb63e7b047040415": "F_A \\; = \\; F_{BH} \\; F_{BG} \\; F_{MH} \\; F_{MG} \\; F_{F} ", "2608df8a7b610c08d47d6114f9e7fc7f": "A.a = f(\\alpha_{j1}.a_1, \\ldots ,\\alpha_{jm}.a_m)", "2609040706534c6fe5afc2aa4f167b25": "\\mu \\propto \\lambda^{-2}", "2609163838730964b8cb9690068c2c6e": " \\frac{(\\Pi_1)^{ 0.5}}{(\\Pi_2)^{ 0.75}} = ", "2609419f2c6acbb9b25149979e163172": "D_{s,t}", "2609b57414c8d5582997504db19617ec": "\\scriptstyle -V_{CC}", "260aba52b702c01d89bc65ec0e76c3f0": "\\mathcal {E}_p/e", "260ad813b782b3619de8882732d1ad7a": "A/\\Phi", "260af3e5b7bd8509be9d318a8401a9a6": "2 \\circ 4 = F_{3+4} + F_{3+2} = 13 + 5 = 18.", "260b07d287366cf70efe68becbc02457": "f(\\sup(\\mathbb{M})) = \\sup(f(\\mathbb{M}))", "260b0ec1ed0067264f62c96965442a9a": "\n\\eta(1) = \\lim_{n\\to\\infty} \\eta_{2n}(1) = \\lim_{n\\to\\infty} R_n(\\frac{1}{1+x},0,1) = \\int_0^1 \\frac{dx}{1+x} = \\log 2 \\ne 0.\n", "260b1c8924c2baf353aafc35b52abfab": "\\boldsymbol{r}(0) = \\boldsymbol{r}_0\\in\\mathbb{R}^{2N}", "260b2f81e867503921875ad7a8a53eed": " \\lim_{i \\to \\infty}", "260b307f4d300644a2affb7698d385e7": "[Q=R^2] \\sim \\chi^2(N)\\ .", "260b57b4fdee8c5a001c09b555ccd28d": "\\omega", "260b6099fd117588a809e4fb924e1d63": "n=", "260b628da799118dd8118462c0e00d29": "q = ae(-1) + af(+k) + ag(-j) + be(-k) + bf(-1) + bg(+i) + ce(+j) + cf(-i) + cg(-1)\\,", "260b6e3c94e05305f5643bcec244259c": "\\rho (\\vartheta) = \\rho_0 + \\Delta \\rho \\cdot cos^2 \\vartheta", "260b8cee97f646cec505bd3d6380db1e": "f(\\Phi, I)=0", "260bfa6adbe8062be00d63d7ac624873": "x_2^T \\omega x_2= 0", "260c74541d883220ee7ae41939c49721": "\\gamma: Y\\longrightarrow Z\\ ", "260c8863416e86ac180747e2d0d1f332": "c\\cdot(1 - \\sum_j h_j) = \\sum_j h_j \\cdot \\Phi(-j)", "260ca71cafb4af9e777f33f4c99d25b6": "R = ar^2 + br + c", "260d0632e390b1284ef8fb500a64340a": "0\\sum_{i = 1}^n w_i", "26195a0f50e1cae3446ac73d5c0a1634": "\\mathbf{X}(s) = (s\\mathbf{I} - A)^{-1}B\\mathbf{U}(s). \\,", "2619724bf426aa59f4c41b26983a5120": "F_{E} = -e|E(r)|", "2619783d56716cc9c5f41857a499613d": "g \\frac{\\partial \\eta}{\\partial x} = f v", "2619a66d5263460e0277381a556d5327": " L^2(G)", "261a15fc7febb966e4c1d4bde12f00da": "Z ", "261a22e15384daf71dd5b8abc33391cd": "r(\\cdot)", "261a70c47bce4988014b555fd3e33d49": "\\hat{P}^{\\mu })", "261aad7982a137a6759c3e3b58245bfe": " D_{\\mathrm{JS}} = \\tfrac{1}{2} D_{\\mathrm{KL}} \\left (P \\| M \\right ) + \\tfrac{1}{2} D_{\\mathrm{KL}}\\left (Q \\| M \\right )\\, \\!", "261ab7b4ab65de1afdc9126decea58bc": "\\Delta = 0", "261ad7ab7aa923709ac9a0878237945a": "\nL(\\mu, \\lambda)=\n\\left( \\frac{\\lambda}{2\\pi} \\right)^\\frac n 2 \n\\left( \\prod^n_{i=1} \\frac{w_i}{X_i^3} \\right)^{\\frac{1}{2}} \n\\exp\\left(\\frac{\\lambda}{\\mu}\\sum_{i=1}^n w_i -\\frac{\\lambda}{2\\mu^2}\\sum_{i=1}^n w_i X_i - \\frac\\lambda 2 \\sum_{i=1}^n w_i \\frac1{X_i} \\right).\n", "261adf4ac0a0c093ad6040cb2b82f1d9": "\\mathbf{\\nabla}\\cdot(\\epsilon \\mathbf{\\nabla}\\phi)= 0", "261aea3796669e93390b54299964c7ab": "x_{1}^{1} = x_{1}^{0} + u_{1,1}^{0} + u_{2,1}^{0} = 8 + 2 + 3 = 2 \\in Z_{11}", "261b250b1ce246d05e89e1dd906b8a91": "\n\\mathrm {DOF} \\approx \\frac {2 H s^2}\n{H^2 - s^2} \\text{ for } s < H \\,.\n", "261b5152b766f8d99b1d62233cc4e3ce": "{}^{+}", "261b9ec2ab2f2564411255e00dac15e7": "256^n", "261bde4b35a681350ff36b01dd74ba1c": "I_i\\to I_i+1", "261be5b691700e02c5450406e86ceaf9": "G_{dBd} = 10 \\, \\log_{10}(1.698/1.64) = 0.15 \\, \\mathrm{dBd}", "261c19dfc27dc09f892a460947bf4940": "100111", "261c4fdddb3542b9f7f215c92e423962": "T_n^t", "261cbc98a50f5120121cb4c3be03959c": "\\Delta^{-1} f(x)\\,", "261cc8ade0910f423d718458a2ceb7c3": "\\Phi = \\frac{\\sqrt{5} - 1}{2}.", "261d050b7eb298b222ba4220f15f86b3": "g(k;\\lambda) = P(X = k \\mid k > 0) = \n\\frac{f(k;\\lambda)}{1-F(0)} = \n\\frac{\\lambda ^ k e^{- \\lambda} }{k ! \\left ( 1 - e^{- \\lambda} \\right )}. ", "261d0f23d71a972fe61c98929fae3e55": "U(1)_Y", "261d0f36f095e319dda2395a80a063a7": "\\sum_{k=0}^n a_k \\sim n", "261d260048a0aa6acc1a5c7018c911e9": "a \\land 1 = a", "261d33fc846bac78f451c85ab0a9f360": "\\bar\\theta_i^{}(t_n^{})", "261d91a209af577aca43e0f506c6ba14": "\\geq 7", "261dec347e7132f55de8d744b2c7815e": "\\bmod\\,n", "261eae0c170eebc56303490a063f656f": "y =x \\tan \\theta - \\frac{gx^2}{2v^2}(1+ \\tan^2 \\theta)", "261eef8e83eaa0ea15ebee22c8aee0a1": "pt \\sqcup_{S^{n-1}} pt", "261f07894c7a50fc332e0664b7689927": "P(e, e', T)", "261f0cd12c623cdd8bdb63fd4dabbb54": " H_\\mathrm{e}(\\mathbf{r,R} )\\; \\chi(\\mathbf{r,R}) = E_\\mathrm{e} \\; \\chi(\\mathbf{r,R}) ", "261f512c31a17e6242b18461daa988e7": "\\langle j_1j_2;m_1m_2|j_1j_2;jm\\rangle=(-1)^{j-j_1-j_2}\\langle j_1j_2;-m_1,-m_2|j_1j_2;j,-m\\rangle", "261f55e79db500135590dc6661da947c": "\\bold{v}=(v_1,v_2,\\ldots,v_n) \\, ", "261f5ccd650465e45d7a1d7d06154ae9": " p_k = P[x \\in I_k] = \\int_{b_{k-1}}^{b_k} f(x)dx ", "261f71bd601d4ee8032c97caedb44dd0": "\\begin{matrix}\n x_1 \\geq 0 \\\\\n x_2 \\geq 0\n\\end{matrix}", "261f99f678a722399b2b755f55e5a87a": "\\Phi^{ext}(\\mathbf{q}_i)", "261ff01d39006cacc0a93c032ab4be63": " \\hat{p} = -i\\hbar \\frac{\\partial }{\\partial x} \\,\\!", "26201e0d230b32c6986f07b30057524b": "\\frac{3h}{10}", "262069c0b77d451cffdd65004c1799e7": "\\det(A-\\lambda B)=0", "262069fe1b77c6d95206e1de6003586d": "q = \\beta: \\alpha = OB:OA \\ ", "2620872d7d8ba4fa9124131fbcdd9fd1": "y(x) = 1", "2621084503b1e4c7ae6f480c10d568d1": "\\mathcal{L}_X f =\\sum_{\\{i\\}} \\lambda_i e^*_i(f) \\otimes e_i. \\, ", "2621709eccc6edfbc2f88d3447b1ebaa": "\\frac{1}{2} \\frac{\\partial^2 a}{\\partial \\xi^2} + i\\frac{\\partial a}{\\partial \\zeta} + N^2 |a|^2 a = 0 ", "2621724eb0dde08e3a0419662eb90e70": "\n\\begin{align}\nm'' & = m' r^e\\pmod n \\\\\n & = (m^e\\pmod n \\cdot r^e)\\pmod n \\\\\n & = (mr)^e \\pmod n \\\\\n\\end{align}\n", "2621893e5ebaf67fef801abaa0ae36e6": "x = r_1 \\cos(\\omega_1 t) + r_2 \\cos(\\omega_2 t), \ny = r_1 \\sin(\\omega_1 t) + r_2 \\sin(\\omega_2 t),\\,", "2621901c2308423e78207b0b84618da9": "\\Psi_n(t)=\\psi_n(t) e^{i\\theta_n(t)}e^{i\\gamma_n(t)}.", "2622ad12ceda7ef263d5b0950ffe8514": " \\displaystyle A(s_1s_2,\\lambda)=A(s_1,s_2\\lambda)A(s_2,\\lambda),", "2622d7cfb442fea585ec9c8f93cff16b": "p\\Psi=0", "2622e9b348f05f74e3e4bb9b7d213dc8": "RM", "262332df709a278045b9ed154c9c2588": "=\\mu N-NkT\\,", "26237469281bb04986256179d98d39e3": "\\sup_{z\\in K} |f(z)| \\le C_K\\|f\\|_{L^p(D)}.", "26238485c5ad19f2bb0c2c9e9323a8d7": "\\left\\langle \\cdot \\right\\rangle_t ", "2623d2f1c321db000ac891e074436d4b": "(14.d)\\quad 2\\psi_{,\\,\\rho}\\psi_{,\\,z}= 2e^{-2\\psi}\\Phi_{,\\,\\rho}\\Phi_{,\\,z} ", "2623d49f9338a3601a1f941d40e03431": "\n{\\rm Pr}\\Big(\\hat{f}(x)-w(x) \\le f(x) \\le \\hat{f}(x)+w(x)\\Big) = 1-\\alpha,\n", "2623efb05a8a39e42ccb4ef058c71923": "\\scriptstyle i \\,=\\, 1,\\dots ,n", "26240362df5500d91bf11b5e9e4e67ee": "\\pi_n(x) \\sim 2 C_n \\frac{x}{(\\ln x)^2} \\sim 2 C_n \\int_2^x {dt \\over (\\ln t)^2}", "26242bea98bb2f8a596fbe05601de6bf": " n \\approx 2.5 \\sqrt{ \\frac{ 1 }{ \\sum_{ i = 1 }^k { \\frac{ 1 }{ c_i } } } } ", "2624346167ec52a931cf45725f44a872": "\\scriptstyle \\sqrt{20} \\ = \\ \\sqrt{4}\\sqrt{5} \\ = \\ 2\\sqrt{5}", "262449ee5fb5d7363c6767b028300fd9": "\\mathrm{Power}_{ext}", "2624af45ed1dc4c051ed5f7bb72db550": " \\tau_N = \\tau_0 e^{\\frac{K V}{k_B T}}", "2624b76ef060439d628c7c1ca57cfb2d": "b \\equiv x^2 \\mod n", "2624da2470d717aca6614b1f7c0ec6bf": "((P \\leftrightarrow Q) \\leftrightarrow R) \\leftrightarrow (P \\leftrightarrow (Q \\leftrightarrow R))", "2624f7c891830f07ef43b705864ad7af": "(\\mathbf{S})\\,\\mathrm{d}(ab)=a\\,\\mathrm{d}b+\\mathrm{d}a\\,b\\,.", "262795eaf1590876a9ebce8f96753e2c": " \\cdot : V \\times V \\rightarrow \\R ", "2627afcd51f49d9a2c776f3e3432f3b3": "g_{\\star} \\left( x^{*} \\right) := \\inf \\left \\{ \\left. \\left\\langle x^{*} , x \\right\\rangle - g \\left( x \\right) \\right| x \\in \\mathbb{R}^n \\right\\}", "2627d0e830c07ca2e61b306fc82c8f1d": " \\chi(\\mathcal{F}) := \\sum_{i \\in \\mathbf{Z}_0^+} (-1)^i \\,{\\rm rank}\\, (H^{i}(X, \\mathcal F)). ", "2627e3d7d3c3ce3b0630af60c312fe86": "\\Box\\, \\mathbf{E} = 0,", "2627fbc92df012f8c513367733f89aad": "|f(z)| \\leq \\frac{2Ar}{R-r}", "26282668fe59eb6d3739b73f225ac791": "S_N f(0) = 0 = \\frac{-\\frac{\\pi}{4} + \\frac{\\pi}{4}}{2} = \\frac{f(0^-) + f(0^+)}{2}", "2628895fa7e68bc1ac8f5dd2fb380ea8": " \\bigg| { \\partial A \\over \\partial z } \\bigg| \\ll | kA | ", "2629298f1a7855098111d2ef159611de": "L = \\phi^2 \\, \\eta_{ab} \\, \\dot{u}^a \\, \\dot{u}^b", "262957b9975ec187a6116a788e78f7f1": "(A + \\lambda \\operatorname{diag}(A) ) \\Delta x = b", "26296222ee64bf84e3820fd15687c38b": "\\tau\\mathbb Z", "262a3bd0318d5034272a8f904d6fad24": "\\mathbf{x}_1", "262a4584304c4546e261209782f1c4f3": "\\operatorname{Ext}_R^n(A,B)=(R^nG)(A).", "262a772700d605b6e749091a80ccee25": "\\delta(\\rho \\mathbf{x}) = \\delta(\\mathbf{x}).", "262a7cb37bd4bfdde88b03e865f1b918": " \\beta_p = (1 - p_\\infty)/\\tau_p", "262ac0d0a5152e06d017d837891c2e15": "\\begin{align}\n\\mathcal{F}^{-1}\\{S(f)\\} &= \\int_{-\\infty}^\\infty \\left( \\sum_{n=-\\infty}^\\infty S[n]\\cdot \\delta \\left(f-\\frac{n}{P}\\right)\\right) e^{i 2 \\pi f x}\\,df, \\\\\n&= \\sum_{n=-\\infty}^\\infty S[n]\\cdot \\int_{-\\infty}^\\infty \\delta\\left(f-\\frac{n}{P}\\right) e^{i 2 \\pi f x}\\,df, \\\\\n&= \\sum_{n=-\\infty}^\\infty S[n]\\cdot e^{i\\tfrac{2\\pi nx}{P}} \\ \\ \\stackrel{\\mathrm{def}}{=} \\ s_{\\infty}(x).\n\\end{align}", "262af2e0176f997b94e4e82076618023": "I_x(M)/I_x^n(M)", "262b11e130004bdaf776aa9984e2b2dc": "C = \\frac{1}{2 \\pi f R} = \\frac{1}{6.28 * 5000 * 10} = 3.18 * 10^{-6} ", "262c7dd89f0c5792a84310bf193cf11f": " \\langle P \\rangle =\\mathcal{E}I_\\mathrm{rms}\\cos\\phi\\,\\!", "262c95dfaa6fee4f7f2fe531098eebd2": "\\mathrm{li}(x^{1/2})/2", "262ce00955eccdce97c2cc50caca7291": "\\frac{1}{1-\\frac{a}{p}}H(t)=\\sum_{n=0}^\\infty a^n p^{-n} H(t)=\\sum_{n=0}^\\infty \\frac{a^n t^n}{n!} H(t)=e^{at} H(t) .", "262d5ec61d4727236470a56c2e8433ef": "\\mu_i", "262d8fd63e656ffa6e41f9907f2bc635": " \\mathrm{Cdim} X = \\inf_{Y \\in \\mathcal{G}} \\dim_H Y", "262dfb07bd84ad8f918c1a4035f3232b": "w\\Vdash A", "262e016a156bc7286277d35342224383": "F=N\\varepsilon_0+k_BT\\sum_\\alpha \\log\\left(1-e^{-\\hbar\\omega_{\\alpha}/k_BT}\\right)", "262e042acef0bb91653712ba0609f5dd": "E_{\\mathrm tcS} (\\varepsilon, t, T)", "262e0afc75c8a9fc536a7dce57e6ebe1": "B_1", "262e75aaa58a61e45bb7097b830f478d": "\\scriptstyle d(x,\\, A) \\;=\\; \\inf\\{d(x,\\, a) \\;|\\; a \\,\\in\\, A\\}", "262ec9ad165d37027c5a31654c4ffdaa": "\\int_a^\\infty f(t) \\; dt", "262eebaa13a0c4361a4c30b6d5aa3608": "R_{\\theta}", "262f1e8b6eb40a3e90f7dd4288437008": "P(x\\rightarrow x') = g(x\\rightarrow x') A(x\\rightarrow x')", "262f21c02905d92fd7cde701deb1b086": "s^2(\\vartheta)", "262f265eb5cfb857e8baa0c83a050d28": " \\mathrm{lift}(X\\Rightarrow Y) = \\frac{ \\mathrm{supp}(X \\cup Y)}{ \\mathrm{supp}(X) \\times \\mathrm{supp}(Y) } ", "262f4266de2347543f3e5c86ee5c77b3": "\\begin{align}\nU_{f,P_n} &= \\sum_{k = 1}^{n} f(x_{k})(x_{k} - x_{k-1})\\\\\n &= \\sum_{k = 1}^{n} \\frac{k}{n} \\cdot \\frac{1}{n}\\\\\n &= \\frac{1}{n^2} \\sum_{k = 1}^{n} k\\\\ \n &= \\frac{1}{n^2}\\left[ \\frac{(n+1)n}{2} \\right]\n\\end{align}", "262f57469ad842e72459d47449608b04": "-1.5913", "262f5d08b5ea34d4a82039f839bd7a67": "\np_i := p_{i+1} q_{i+1} - p_{i+2} \\text{ for } i \\geq 0.\n", "262fedc550e771be210750af0ac3e3c4": "1 + x^{-1} + x^{-2} + \\cdots", "26302f0e1c87ba0db4847c24ed9cea8a": "\\frac{\\partial r_i}{\\partial \\beta_j}", "26306baf9f1d96b3b320c15f63275c92": "\nZ = \\sum_i g_i e^{-\\varepsilon_i/kT}\n", "2630807dbdafa8aaedeb4d2c13f49c40": "\\nabla \\times \\left( \\mathbf{v \\times F} \\right) = \\left[ \\left( \\mathbf{ \\nabla \\cdot F } \\right) + \\mathbf{F \\cdot \\nabla} \\right] \\mathbf{v}- \\left[ \\left( \\mathbf{ \\nabla \\cdot v } \\right) + \\mathbf{v \\cdot \\nabla} \\right] \\mathbf{F} \\ . ", "263092b85467e0396b858eb86b921f46": "m_p\\,", "26309d9c15677ba25b9737386ab12d11": "\\mathfrak d", "2630bdd130b8f08da440faac635c86e2": "\\dim X \\leq h^{1,0}", "2630d50cc3b82b44a8e74f59778aed27": "\\Box A^\\alpha = \\mu_0 J^\\alpha ", "2631162c92ff1b1e31a061124a8c1907": "h_{ij}(z)", "26311c5eaa94a8a9e9e710c21788c541": "\\sum_{k \\mathop =a}^b f(k) = \\int_{[a,b]} f\\,d\\mu", "263136042d889b43a694d24249234674": "G_F/G_N= 10^{-32}", "26316c9f3590bf95b86da67699382e0d": "S(x)\\Lambda(x)=\\sum_{j=0}^{\\infty}\\sum_{i=0}^j s_{j-i+1}\\lambda_i x^j.", "2631c53c624694323b2b5a61e078f6ab": "\\delta_n(\\varepsilon)", "26323adf064e40aed7415437058397a6": " e^x = 1 + x + \\frac{e^\\xi}{2}x^2 < 1 + x + \\frac{e^x}{2}x^2, \\qquad 0 < x\\leq 1 ", "26324692b10ffcd9737cfa0af0fd9c4b": "C = (m_1+1)(n+1)\\omega", "263266b9b2c767aaa5c6a89a2f1d7c81": "\\mathcal{G} \\rightarrow f_*f^{-1}\\mathcal{G}", "26326b9f9c16843fb47648df5528d4b5": " [C] ", "263278a2cac5d60dcd8fe92559019a1d": "\\|v\\|_V := \\|\\nabla v\\|,", "26332a74ec5c45dac0efa5a380dcc05a": "\\scriptstyle\\ P_\\text{Q}\\,", "263356f3a46d00372b0e40490452b11b": "\\rho'(X)=\\lim \\frac{|X\\cap I_n|}{|I_n|}", "2633f1d228fcc4115cfead5a4bdfb7b2": "{|B_4|\\over 8}{|B_2|\\over 4}{|B_4|\\over 8}{|B_6|\\over 12} = {1/30\\over 8}\\;{1/6\\over 4}\\;{1/30\\over 8}\\;{1/42\\over 12} = {1\\over 696729600}.", "2634026dcd44e888e4ad2e2c83c2a0fc": " \\frac{\\partial u}{\\partial t} - fv = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x}", "26340b855541235759b517c194748c0b": "E_c = (1.5)\\biggl(\\frac{q^2}{g}\\biggr)^\\frac{1}{3} = (1.5)\\biggl(\\frac{10^2}{32.2}\\biggr)^\\frac{1}{3} = 2.20\\text{ ft}", "2634534aa793904448042140b29cf862": "\\sin\\lambda=a/M", "2634c9f2397a835e622928523310d614": "\\sum_{k=0}^\\infty (-1)^{k} k! = \\int_0^\\infty \\frac{\\exp(-x)}{1+x} \\, dx = e E_1 (1) \\approx 0.596347362323194074341078499369\\ldots", "2634ce4ba4df2a17f5aaccc1a7db3cd1": "\\mathbf{u}=\\nabla\\varphi+\\nabla \\times \\mathbf{A}", "26353212a9319cbac77949ee5c4c359c": "1 + \\sqrt{2}", "26353882829d58dd2dddd1c17cd4e59e": "\\langle (gG)^2\\rangle\\ \\stackrel{\\mathrm{def}}{=}\\ \\langle g^2 G_{\\mu\\nu}G^{\\mu\\nu}\\rangle \\simeq 0.5 {\\rm\\ GeV}^4", "263546e6e75c4be45bd3c04362d412d0": "\\frac{1 + \\sqrt{5}}{2}", "2635819030654633cb5379be075adb75": "f \\equiv g", "2635fc6f4059a4bfc51bcda132e314f5": "\\Xi_k = \\sum_{i=0}^s N_i (\\xi_i - k_B ln (x_i))", "263617b0a5607d4b33daeac75915cd3b": "V_\\emptyset", "2636774b4f85a16a103b1484c0aa3359": " \\varphi \\in C^p(X;R). ", "26368e705157c34231fd9465321c6857": "\\mathbf{j} = e\\mathbf{v}", "2636dc24e3c1030a50ae9371c5589302": "\\Delta{H} = 235\\mbox{ km} + \\left ( 1000 C_R \\frac{A}{m} \\right )\\mbox{ km}", "26375114f173e76250bc31591cdcb9be": "\\scriptstyle c\\,t_2=0", "26377132a7c925ad5dc79f2a6eedc248": "\\scriptstyle{\\tau \\rightarrow 0}", "2637b6d6c5770cf208348f499bd4ac13": "U \\approx\\int_0^{\\sqrt[3]{N}}\\int_0^{\\sqrt[3]{N}}\\int_0^{\\sqrt[3]{N}} E(n)\\,\\bar{N}\\left(E(n)\\right)\\,dn_x\\, dn_y\\, dn_z\\,.", "2637edefe0d300d9dedf07b31e598246": "\\mathcal{L}_{V^{1}}(\\theta^{\\alpha}) ", "2637ffd918ed61bfc5c449980ac2ed41": "\n\n \\frac {v_i} {v_a} = \\frac {R_i} {R_i+R_A} \\frac {1} {1+j \\omega (C_M+C_i) (R_A//R_i)} \\ ,", "263821672bab5de331ab902480ed2782": "\\ m = \\frac{e \\cdot r^2 \\cdot B^2}{2 \\, U} \\approx 9{.}1094 \\cdot 10^{-31} \\, \\mathrm{kg} ", "26383bc14a05f7c12c81799d29d907ff": "\\frac{\\text{FC}}{\\text{TC}}=\\frac{\\text{FC}}{\\text{FC}+\\text{VC}}", "2638b65751352da859f2492b2df9bdfb": " K^{2n+1}(X) \\cong K^1(X) ", "2638cd9001fb42d560ee963978b2cfca": "C_{P}, l, \\alpha, \\sigma, [s]_{E}", "2638d67c8761dfd6ca75aaff63b40daf": "Cl(S)=S", "2639219572a9210946f1416d5f89c5f8": "\n\\alpha_{m_{i+1}-j}(i+1) = \\alpha_{m_i-2j}(i) +\n\\alpha_{m_i-(2j+1)}(i)\\frac{(m-1-2^{i+1}j)!}{(m-1-2^i-2^{i+1}j)!}\n,", "26394d899c429d9196ba2c62f9be4ac9": "|i m_i\\rangle", "26394f2b832072282f5f36b24ff8e861": "\\scriptstyle \\hat x' \\;=\\; R_\\alpha \\hat x R_\\alpha^\\dagger", "26399633eebe207586e7c719b7f3fa83": "0\\leq\\operatorname{dCor}(X,Y)\\leq1", "26399fbfbb7b3a019cbb4bad94aa5d9e": "(1-\\beta\\cos\\theta)", "263a32929dbc5881c5c3f55ea4b21769": "F_{\\nu}^{-1} \\circ F_{\\mu} : \\mathbf{R} \\to \\mathbf{R}", "263a4cf26f3e8fdcc9c15129387fa6c1": "\n \\frac{ \\sum\\limits_{i=1}^n (X_i - \\bar{X})^2 }{\\sigma^2}\n", "263a50f81c0f916a991f882c8a314a60": "\\mathrm{W = \\frac{V^2}{\\Omega} = A^2\\cdot\\Omega}", "263a82430b039c75fe6e09312913f125": "t = 1- \\frac{r_m}{r_c}.", "263add1e3e7da7a49ec17f8f3ac63945": "d\\mathbf X_2\\,\\!", "263b994ecf7b18ab2218aca6ca0a4641": " E_\\textrm{kin} = E_\\textrm{pot} \\,", "263b9cba83ff3dbdf04188c0d9d613a1": "matrix_{multiply} : matrix(k, m) \\times matrix(m, n) \\to matrix(k, n)", "263bb3ad87361809a071a8ae76e9ef92": "g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\\,", "263bf5fc07d9887733d132eaa13e869c": "C_x(t, f)=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}W_x(\\theta,\\nu) \\Pi(t - \\theta,f - \\nu)\\, d\\theta\\, d\\nu \\quad = [W_x\\,\\ast\\,\\Pi] (t,f)", "263c0975f0f6699c28d00d9b02dea161": " f(x) = \\frac{x}{\\sqrt{x^4 + 10 x^2 - 96 x - 71}},", "263c33d5ba58e2fa0cc0fa476824ef23": "\\varepsilon(a,b,c,d)", "263c66f67842bd5f2968aafbd96d2ed1": "\\forall x\\, Q(x)", "263c7f3ccc82208a0852875c1cdba230": "A'(m)+ B'(m-e) = A(m) + B(m-e) \\ ", "263cef3c88c2f2f444082ea6b633fba6": "x J'_\\alpha(x)+cJ_\\alpha(x)", "263d03917d95633f5c4e8cc583d1c223": "i^{m}", "263d2b717407a53ce54ba54a016efb13": "\\mathbf{Q} \\!\\,", "263d3f34131d1d8de87627c66e4824f5": "z=z", "263d5301461977dfde807f58f9ee6988": "f_i(g_1/g_0, \\ldots, g_n/g_0)=0. ", "263d76ed838e079263ce5e21689adcce": "T^*_2", "263d849350fe0aa04498865a1543695c": "\\begin{matrix}\nA = (ei-fh) & D = -(bi-ch) & G = (bf-ce) \\\\\nB = -(di-fg) & E = (ai-cg) & H = -(af-cd) \\\\\nC = (dh-eg) & F = -(ah-bg) & I = (ae-bd) \\\\\n\\end{matrix}", "263e0bd08e9a6ed00a76281259a82dcb": "x\\left(t \\right)", "263e8c1b703bcdd26104bf57030d1c54": "\\operatorname{fnchypg}(x;n,m_1,N,\\omega) = \\operatorname{fnchypg}(x-1;n,m_1,N,\\omega) \\frac{(m_1-x+1)(n-x+1)}{x(m_2-n+x)}\\omega\\,.", "263efcdb40455b0effe547ed409adeb3": "\\scriptstyle j+k-n \\,\\ge\\, i \\,\\ge\\, r+s-1,\\dots ,n", "263f58b888dc7181dd75f75d666d3e38": "h_{T} = \\dfrac{ \\mbox{diffusion time} }{ \\mbox{reaction time} }", "263f8d44dae203bcb942e38e5b17ff40": "\n\\mathbf{\\hat{b}_{4:5}} = \\alpha\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix}1.0000 \\\\ 1.0000 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.6900 \\\\ 0.4100\\end{pmatrix}=\\begin{pmatrix}0.6273 \\\\ 0.3727 \\end{pmatrix}\n", "263f906ac9f708532898828a81c61efc": "\\begin{cases}\n \\frac{n\\beta}{\\alpha-1} & \\text{if}\\ \\alpha>1 \\\\\n \\infty & \\text{otherwise}\\ \\end{cases}", "263fb2c3864020d3ac31681d43dd580f": " \\frac{dm_L}{dt} = \\rho_LAu_L = \\rho_L(4\\pi R^2)u_L ", "26403ec6d537fa31f63e294b44831734": "lb", "2640a34096ef4392606817e6cc9f8443": "\\Delta_{H^{n-1}} f = \\Box f\\left(x/q(x)^{1/2}\\right)|_{H^{n-1}}", "2640c8a3fe3838d64f95cdd31b786f65": "\\tfrac12ab", "2640e33d330298a4cb9a2ccc5602c87f": "(x,y)\\in\\mathbb R^2", "2641beafbf73b933e19a94c776189cc7": " \\rho_{X,Y}={\\mathrm{cov}(X,Y) \\over \\sigma_X \\sigma_Y} ={E[(X-\\mu_X)(Y-\\mu_Y)] \\over \\sigma_X\\sigma_Y} ", "2641ceaf2d4487af403228a918690909": "a_{ij} = a_{ji}. \\,", "2641d746823af32a05f9739287ed0a92": "\\mathrm{ind}(D) = \\prod_{i=1}^r p_i^{m_i} \\ ", "26423f225b030a35616ff13f74676f87": "S(T) = \\frac{C}{\\exp\\left(\\frac{c_2}{AT + B}\\right)-1}", "26424abc355235f5fca9ee7c22f4ba6e": " 3^3 - 1 = 2 \\cdot 3^2 + 2 \\cdot 3 + 2 ", "26425187ea71acf17552a351519e6bf9": "f_c : \\mathbb{\\C} \\to \\mathbb{\\C} \\,", "26427cff6cfbbd418c7c9942403790f1": " \\nu_1 = \\nu_2 \\left(1+\\frac{GM}{r c^2}\\right). ", "2642967a858d4825c7531a3ebd993403": "\n \\det(L_fR_g)=\\sum_{S\\in\\tbinom{[n]}m} \\det((L_f)_{[m],S})\\det((R_g)_{S,[m]}), \n", "2642a4d37f8a0996f38f2417d545815a": "j^{-i}(\\alpha)", "2642d19b27180c52c08c15902f43a744": "[\\hat{\\xi}^{k},\\hat{\\xi}^{l}]=-i\\hbar I^{kl}.", "2642e4307a0773c5dfbacac7870a9691": "e^{-i \\omega t}", "26437ff269e7eb50c8c7ae8029657e24": "A_i\\rightarrow A_j\\rightarrow A_k", "26439c75afe493f6dc75e044c39be13a": "\\int \\frac{\\delta Q}{T} \\ge 0", "26447a666b41f929a983ec99f48ceb77": " K(x,y) = K(\\|x - y\\|) ", "2644bb116b8be92f1f26619adcd2ce30": "O(\\sqrt{n^{2/3}}) = O(n^{1/3})", "2645273ab5c9c2a3dbf0331bb908f733": "{52 \\choose 4} \\times 48 = 12,994,800", "264536e8fdb3dc30cd764c727b8fecbc": "\\Gamma(b-1)/\\Gamma(a)", "26459813a522c7d3ebaf532ab41117c8": " \\sigma \\ ", "2645c5865ba60b5386a9ba0c0145ee77": " \\Delta t/T ", "26461eef607ef249a04022df09903825": "\n\\begin{align}\n \\Psi_{n\\boldsymbol{k}} &=\\mathrm{e}^{i\\boldsymbol{k}\\cdot\\boldsymbol{r}}u_{n\\boldsymbol{k}}(\\boldsymbol{r}).\n\\end{align}\n", "2646482daced2a9e6658246ece6c0efe": "\\lim_{n \\rightarrow \\infty} \\prod_{i=1}^n {a_i \\over 2}=\\frac2\\pi", "26467577c5b093d9fda14fa7a3b81502": "\\displaystyle c_g= \\frac{\\partial\\Omega}{\\partial k}", "264688b4fc0958a9fe78e931ed5653ed": "2 \\, z", "26469e99f0610d934cab50182188d647": "\\ R = |Z| \\cos{\\theta} \\quad", "26470d3baf589e9b274e302e20219f9a": "\\begin{align}\n\\Pr \\left[\\left|-\\frac{1}{n} \\log p(X_1, X_2, ..., X_n) -\\overline{H}(X)\\right|> \\epsilon\\right] &\\leq \\frac{1}{n^2 \\epsilon^2} \\mathrm{E} \\left [\\sum_{i=1}^n \\left(\\log(p(X_i) \\right)^2 \\right ]\\\\\n&\\leq \\frac{M}{n \\epsilon^2} \\to 0 \\ \\mbox{as} \\ n\\to \\infty\n\\end{align}", "264731e0e5881b171c226914acb8dfd6": "\\mathfrak{P}^{17}", "264732d643f446fa4b258b456ef247b6": "L_k = \\pi_{k+1}(X)", "26473cd19fca54957c900301221b0cf0": " |\\downarrow\\rangle ", "26475ca5ca4e4cf82e0cff1083c1183d": " 1.5 \\ \\mbox{mol}\\,C_6H_6 \\times \\frac{15 \\ \\mbox{mol}\\,O_2}{2 \\ \\mbox{mol}\\,C_6H_6} = 11.25 \\ \\mbox{mol}\\,O_2\\ ", "2647e2a1163e35a351dab718086a95c2": "\\mathbf x\\,\\!", "264884439b70ab09a86bc848421c6de6": "[0, 1]", "2648e871cc14f061a0ce481f7d98c9df": "\\phi_x = \\frac {q x} {6 E I}(3L^2 - 3L x + x^2)", "264933c95573c19829593f8cb905ad80": "\\phi(x)e^{-i\\omega t},", "2649a41cfa0b81e904a431f5b7a2172e": "\\widehat{f}(a/p)", "2649b01c12a3e7387d7247870e07d7f8": "e^i \\cdot e_j = \\delta^i{}_j,", "264ad77a8edf4f221cc5662094c9bfa9": "\\langle A(a) B(b) \\rangle + \\langle A(a') B(b') \\rangle + \\langle A(a') B(b) \\rangle - \\langle A(a) B(b') \\rangle = \\frac{4}{\\sqrt{2}} = 2 \\sqrt{2} > 2", "264b188f5344ce4571c749312061f4f9": "a^2 + ba - ab - b^2\\,\\!", "264b5033da304d28fb4c2cd9c78fbe02": "\\sum_{i+j=r+s} a_i b_j.\\,", "264b81e24425cc061bb466634ae925f5": "z^{z^{z^{\\cdot^{\\cdot^{\\cdot}}}}} \\!", "264b9592a542352eced142a5045369b2": "\\Phi_{3}\\left(\\mathrm{R}_{i+1}\\right)", "264bd75a6e18dc8d7822bc5b5dc45da5": "\\eta=2\\cos(\\tfrac{2\\pi}{7})", "264bded76d581e781f4e6955792284c2": "T f", "264bef9ea70f0ec07479b13c30755100": "\\mathbf{l}^2=1", "264c2c065d309877d5e44f939386dafd": "q = 1 - p = 0.046\\,", "264c71cbb3812112f80a6db21b8e54a3": "B = \\lbrace g \\ \\bmod\\ f : g \\in I \\rbrace ", "264c761f560646414bce67f9cc678398": "0< p \\leq 1", "264c778b552d99ddc271b57d6d29a402": "\\mathcal A_n", "264ca23960f7bf433d7970ec8c2adf50": " f(z)=0 ", "264ca5067821a872493bb515cc9ace9f": "\\Phi: \\{(x,u)\\in M\\times EG\\mid\\,f(x)=\\pi(u)\\} \\longrightarrow P", "264d19d7d8d488d45e9a1be1c3650e50": "\\frac{a}{b}\\,\\bmod\\,n = ((a\\,\\bmod\\,n)(b^{-1}\\,\\bmod\\,n))\\,\\bmod\\,n", "264d2f726f3e7b242a1c367faa19d7a6": "w_t^{[i]}", "264d3f5bdfa06c27ddf29993db9b49c3": "R_t^i", "264d68c4baddac5e0dd31caaf043c48b": "V \\in \\mathbb{C}^{n \\times n}", "264d88e0b241adef0f8d7d369fc43c75": "S_s = \\gamma_w (\\beta_p + n \\cdot \\beta_w)", "264d8e0a9d831ea1fd774761d0112065": "\\gamma_{Tot}=\\gamma_1 + \\gamma_2 + \\gamma_3 + \\cdots + \\gamma_n", "264dafeaaf8c98459e52389479cf8684": "\\mathbb{T}^3 = S^1 \\times S^1 \\times S^1.", "264dda783a8dc7bfe0c272d0809301fb": "(\\bar{r}\\ ,\\ \\bar{v})", "264de3afcbde1ce9fcf7517f20256d8b": "\\overline{\\mathrm{Nu}}_D \\ = 0.3 + \\frac{0.62\\mathrm{Re}_D^{1/2}\\Pr^{1/3}}{\\left[1 + (0.4/\\Pr)^{2/3} \\, \\right]^{1/4} \\,}\\bigg[1 + \\bigg(\\frac{\\mathrm{Re}_D}{282000} \\bigg)^{5/8}\\bigg]^{4/5} \\quad\n\\Pr\\mathrm{Re}_D \\ge 0.2 ", "264dfa4e7ec16b812bfe058eb771b599": "\\frac {V}{V_e} = \\sqrt{ \\frac {\\rho_0 }{ \\rho}}", "264e0770b23c418f9aa24bfc7d6bcc97": "\\frac{\\theta \\wedge \\phi \\vdash \\psi}{\\theta \\vdash \\left(\\phi \\rightarrow \\psi \\right)}", "264e0b6ad2c0819b2c5fe489aed02d6c": "X_2 = x_2", "264e12e974d155530dc1691ece2dc6b7": "(x_n) \\mapsto \\sum_n{2^{-n} x_n/(1+x_n)}", "264e4175f5f6cd463aa955d1eb3a0086": "V\\!", "264e561e58330908c02fd8c750ad22c1": "N_n \\sim \\frac{q^n}{n}.", "264e5f7b51323a639fcbc51a3ee86845": "\\lambda=\\lambda_1+\\lambda_2+...+\\lambda_L", "264e7a8a215eb5cde64b0725b158ca98": "\\int_{a}^{b} x^2 dx= \\frac{b^3 - a^3}{3}", "264eacca6a8a8468dc0003e8660cea9b": " T\\colon S\\times X \\to X", "264eb0f968992fc72e6d441ea14e3410": "\\mathbf{A}_{ij} = (-1)^{i+j} \\mathbf{M}_{ij}", "264eda38e4aaa29ac324b7527478190e": "\\! a", "264efcb413bb468f06394b52576cdad6": " w_1 \\,", "264f153cac098318fde5859880c52408": "\nec(K_n) = 2^{(n+1)/2}\\pi^{1/2} e^{-n^2/2+11/12} n^{(n-2)(n+1)/2} \\bigl(1+O(n^{-1/2+\\epsilon})\\bigr).\n", "265002bd74a4357bd8387e900c60fac0": "\\mathbf{f} = \\rho\\left(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}\\right)\\,\\!", "265014f31c4f52be7619286cd05155d2": "\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} + m~\\frac{\\partial^2 w}{\\partial t^2} - \\left(J+\\cfrac{mEI}{\\kappa AG}\\right)~\\cfrac{\\partial^4 w}{\\partial x^2 \\partial t^2} + \\cfrac{mJ}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial t^4} = q + \\cfrac{J}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial t^2} - \\cfrac{EI}{\\kappa A G}~\\frac{\\partial^2 q}{\\partial x^2}\\quad\\square\n", "265016bc9b2cf90dfe5851018fef6135": "[\\alpha]", "2650348b2ed07ffce211b1c4aa1a84c1": "VCA (p_{0 \\tfrac{1}{2}}, (a, m)) \\cup VCA (p_{\\tfrac{1}{2}1},(m, b))", "265078bc2c81b55a9ad4c3e16f5f7319": " y(0)=y_0\\ne 0 ", "265098ada3cb350a7ab5c2bec7ea16e8": "\n\\Xi(\\mu,V,\\beta) = \\sum_{n=0}^{\\infty} e^{\\beta \\mu n} Z(n,V,\\beta),\n", "26509e58e58e005e519fb76a6db4ccfa": "\\pi+3=\\sum_{n=1}^\\infty \\frac{n2^nn!^2}{(2n)!}", "26511cc37094caad6cdc705b860cd134": "2 r_s", "26513b7ba05d6a3a49f7033a6a5f2572": "\\Delta H < 0", "2651a9ab8589e600aa57a0f46f4799a0": "p_\\text{P} = \\frac{F_\\text{P}}{l_\\text{P}^2} = \\frac{c^7}{\\hbar G^2} \\approx", "2651bb9dfd0204a9863ac97f5d2b64b8": "\\psi = n\\theta \\pm \\pi/2", "2651ca440061b25e5d1f51868b228b21": "A^i B_i \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_M \\sum_\\alpha A^\\alpha(x) B_\\alpha(x) d^dx", "26523e563611c8dc0828b22e049a9f80": "P_{sl}", "2652b9b69d77a9956173dbcac7901759": " \\mathbf{x} \\times (\\mathbf{y} \\times \\mathbf{z}) = (\\mathbf{x} \\cdot \\mathbf{z}) \\mathbf{y} - (\\mathbf{x} \\cdot \\mathbf{y}) \\mathbf{z} ", "2652dab209b459b619d2979bd2edda86": "( z - T)^{-1} = \\frac{1}{z} \\sum_{n \\geq 0} \\left( \\frac{T}{z} \\right)^n.", "2652ff71a58663124b3e0bebaf9dcf01": "\\scriptstyle N \\;=\\; n_1 n_2 \\cdots n_k ", "2653e83097e88df0d8dbd77624360bb6": "\\displaystyle{Q(y)R(c,Q(y)d)Q(y)^{-1}=R(Q(y)c,d),}", "26546b95c061ace3cb97c6cb20f1dcc8": "P_{\\rm{80}} = \\left(\\frac{-\\ln(0.2)}{e^{intercept}}\\right)^\\frac{1}{m}", "265545e66c9dcb3e3ee7020c15103872": "cX \\sim \\mathrm{Gamma}( k, c\\theta).", "26554dc594cfc84fd8f5efb1b38623dc": "\\csc^2 A - \\cot^2 A = 1 \\ ", "2655c4c8d25f7b5b71cc898c5ef00623": "\\le ( n^2 + n )T_6 + ( n^2 + 3n )T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7", "2655e5101f5b5640a5b3e381ff6bd3a4": "\\,\\!\\Psi", "265625dd99f23a0d0be29e9aa8a487d7": "d\\omega + \\frac{1}{2}[\\omega,\\omega]=0.", "26562ad33c426b3c7514db8973498f05": "F = D \\wedge A", "26564bf1fed0e4063993de5a6d6733f8": " x=\\begin{bmatrix}\n7.111\\\\\n-3.222\n\\end{bmatrix}\n.", "26564ccf6f1d7ee5a8fa1609552905a2": " p(x) = a_1 x + a_0 \\,", "26568eca2cba1ee98d6c7df95b7aa8c4": "6.14*7.95=48.813", "265692078486662cd135d5a965e5c3e6": "\\frac{\\delta T}{\\partial t} = \\frac{\\partial T}{\\partial t} + u \\frac{\\partial T}{\\partial x} + v \\frac{\\partial T}{\\partial y} + w \\frac{\\partial T}{\\partial z}", "2656cf8aea13f96db826dcd83e476e7d": "{N_p}^{2/3}", "2656e23b0b8f0cb0fbf698b87ac48edb": "E(M) = N - N(1 - \\frac{1}{N})^{K}", "26571360ac198dc526defd387a6cb751": "\n\\mathbb{P}(y \\mbox{ sent}\\mid x \\mbox{ received} ) \n", "2657a4caa93f2f6c043153ef76f0502c": "\nC_{XY}=\\frac{I(X;Y)}{H(Y)} ~~~~\\mbox{and}~~~~ C_{YX}=\\frac{I(X;Y)}{H(X)}.\n", "26587c81978bec5d4f878cb698008669": "X_{n+1} = 2X_n - X_n B X_n.", "26588d4e847a87d5e735fa9970676588": " \\mathbf{} d ", "2659046dd4afc3ba5d7d63ac5429aff9": "(p, q)", "265913405f8e08ed59d2275374cda700": " K(k)=K^{+}(k)K^{-}(k), ", "2659581698ffe90fdbe59981f2d4d120": "(f*S)(x) = \\left \\langle S, \\tau_x\\widetilde{f} \\right \\rangle.", "26595f6f8f88813a302a67aaa7c3eafe": "\\scriptstyle\\lesssim10^{-15}", "26596d2c9050ff432e847ee172a069d5": "J = \\frac{1}{2} x^T(t_1)F(t_1)x(t_1) + \\int\\limits_{t_0}^{t_1} \\left( x^T Q x + u^T R u \\right) dt", "26598d1886e00e87d1fd90657e508ae4": "(x_2, y_2, k_2)", "2659e70945fa71f89e7c3e3f4ca4fdf3": "\\mathbf{r}_s(t')", "265a20e261d576ec430d0c1795ce148a": " n^{1/h} \\leq |B \\cap [1,n]| ", "265a3284d6976734f87d6cf458320a4e": "CBF = CPP/CVR ", "265a3336a14e63d3dc7daab9a3f940d5": "\\color{blue}\\mathcal{S} \\color{blue}\\rightarrow \\color{blue}\\mathcal{E} \\color{blue}\\rightarrow \\color{blue}\\mathcal{I} \\color{blue}\\rightarrow \\color{blue}\\mathcal{S}", "265a5201651753542e92e3000a7741e6": "(p-1)(q-1)/2", "265a91067a73d87fbfa561c7f6e63f4a": "b:[0,1]", "265ac578d68687206e8a4f9c57a46c93": " Aw=\\lambda Bw ", "265ad18e859b061889fcb38033f01746": " f_i: A \\rightarrow B ", "265b14899ebf2f3b8310acf6c65c4a16": "Y \\approx f (\\mathbf {X}, \\boldsymbol{\\beta} )", "265b14d373a858899ae5a4ac805c84f5": "i\\circ f\\circ r:X\\to X", "265b2449220f7a288344a30eba016761": " J = \\frac{\\Delta P}{(R_m + R_c) \\mu} ", "265b32ec6074370744b732b8c0001327": "\\;P_t=P+q_c", "265bc1b608507839cca7f4c76118b8a7": "\n\\frac{1}{\\tau} = \\frac{1}{2}\\left(\\frac{1}{\\tau_+} + \\frac{1}{\\tau_-}\\right)\n", "265bd60bcc56dad14b99cf8cdf734b0e": "X\\in \\mathbb{D}", "265bfb61096cd435683fea67761f7f15": "(n,t_2)", "265bfcafeb15b89a6917e6c853885d3d": "J_{z, circle} = \\iint r^2\\,dA = \\int_0^{2\\pi}\\int_0^r r^2\\left(r\\,dr\\,d\\theta\\right) = \\int_0^{2\\pi}\\int_0^r r^3\\,dr\\,d\\theta = \\int_0^{2\\pi} \\frac{r^4}{4}\\,d\\theta = \\frac{\\pi}{2}r^4", "265c328a8656663e00bb3eed325bddb6": "S = -k_B\\sum_i p_i\\ln p_i", "265c7030896dc7b56d6d4c21daf337e0": "Target \\ Power > \\left( \\tfrac{Clutter \\ Power}{Subclutter \\ Visibility} \\right) ", "265c78ef9b5fef418ad6f502fc67b673": "l \\varpropto \\langle\\rho\\rangle^{-1}", "265ce0fb455bf1fefcf9779fd71e728d": "anJ_n = \\sin{ax}\\cos^{n-1}{ax}+a(n-1)J_{n-2}\\,\\!", "265d027ce8920c351a1c593242b6ede3": " \\phi_{\\infty} = - E_{\\infty}z = -E_{\\infty} r \\cos \\theta \\ . ", "265d1a1c5f0575be7bc8ecce88a59122": "L^2(\\R)", "265d1eb8abe09c91a472927365f15542": "g=\\sum\\limits_{\\mu,\\nu}g_{\\mu\\nu}\\mathrm{d}x^\\mu\\otimes \\mathrm{d}x^\\nu\\quad\\text{with}\\quad\ng_{\\mu\\nu} = \\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{pmatrix}\n", "265d2b550f8eccfb61a9aee79d7aa516": "\\pi_{n}", "265ed59ea6f3b2aaacbe46d7bef6adcb": "\\left(\\frac{a}{n}\\right),", "265f4d60ea930f18f3091783ced65602": "f(Y)\\equiv \\frac{Y}{1-e^{-Y}}-\\frac{Y}{2}", "265f61cddda429d01585ec31dae9d026": "m_r=m-\\rho\\mathcal{V}", "265f75dbc4bf5fff33ad62292a6024ac": "1+6\\left({1\\over 2}n(n-1)\\right)", "265f99a1d3b62521ae509705c7446998": "\\mathbf{M}^+ = \\mathbf{V} \\boldsymbol{\\Sigma}^+ \\mathbf{U}^*", "265fc62311e3b50eafa66b2231f0649d": "\\frac{1}{2\\pi i}\\frac{d\\alpha}{\\alpha}.", "266000dfca90694e4b3b6ad9b79f7779": " \\psi\\left(\\frac{1}{2}\\right) = -2\\ln{2} - \\gamma", "26600e1f68a9766ada7cc0d6d20f6915": "\\{0,1\\}^k \\rightarrow \\{0,1\\}^n", "26604eb2daf4750222b52b54fdb37205": "R_E \\frac{r_E + R_E}{r_\\pi + 2R_E}", "2660689d4124d12dcdecff670868c221": "S_K=-\\int d^4x\\,\\sqrt{-g}\\omega \\left( \\frac14 B_{\\mu\\nu} B^{\\mu\\nu} + V(K) \\right)\\;", "2660b7bb33c8e129fd15e9bfb98f2efa": "\\kappa(X_1,\\dots,X_n)\n=\\sum_\\pi (|\\pi|-1)!(-1)^{|\\pi|-1}\\prod_{B\\in\\pi}E\\left(\\prod_{i\\in B}X_i\\right)", "26613b523d46878898a60f99574a0050": "100\\uparrow\\uparrow 2=10^ {200} ", "2661c3970881e9fbbaa2cc0df080bf78": "x = 379 ; 239", "2661f6d687884cf90fc3b10845bba5dd": "(x_3,y_3)", "26620f2598189af535ed9aec5f74a924": "m\\to\\infty", "266213c3e961aa81bbde92c9c27a3fb2": "r + s = (z_0+w_0) 1_G + (z_1+w_1) a + (z_2+w_2) a^2\\,", "26621d0e2dce9e56c4bb28e4476d906b": "\\displaystyle \\zeta = m_p / m_0", "26626b19b0afed24c614ae0f3313c426": "\\Delta x' = \\frac{\\Delta x}{\\gamma} ", "26629485120e6c54325171a23a7c0830": " \\int_X^\\oplus T_x d\\mu(x) ", "26629951d34eb84641f4d20ad0c96166": "|\\pi(x) - \\operatorname{Li}(x)| < \\frac{1}{8\\pi} \\sqrt{x} \\log(x), \\qquad \\text{for all } x \\ge 2657. ", "266381f1d1b849f6fe37e8ba7afd8d64": "W = \\frac{N!}{n_1! \\, n_2! \\, \\dotsb \\, n_m!}", "26639cf045bb53b25e0029a7c654f721": "f(y)>f(x)", "2663c4b7f1ad1f76970ad1e0932490a0": "S=\\int dt \\, d^dx \\left[\\psi^*(i\\hbar \\frac{\\partial}{\\partial t}+\\mu)\\psi-\\frac{\\hbar^2}{2m}\\nabla \\psi^*\\cdot \\nabla \\psi\\right]-\\frac{1}{2}\\int dt \\, d^dx \\, d^dy \\, V(\\vec{y}-\\vec{x})\\psi^*(\\vec{x})\\psi(\\vec{x})\\psi^*(\\vec{y})\\psi(\\vec{y})", "2663e06beeafd394c3965843c577c57f": "g(x) = \\ln \\frac{\\pi(x)}{1 - \\pi(x)} = \\beta_0 + \\beta_1 x ,", "2663f0e5efb0800f0ed77a11d111e585": "B=A", "2664d768d3d5f2961c39e421b7a7b33d": "x = (x_1, \\dots, x_n)", "26650c11c3812f14ae80aa98a730bc63": "d(x,y) = \\sum_n |x_n-y_n|^p.\\,", "26652e370814596003842fd41d7242ea": " \\tau ", "26653965d428ac175a444bc5d62795af": "(aX^2+Y^2)Z^2= Z^4+dX^2Y^2", "26657c41115dddb2eaf517e8c3d44dfc": "\\frac{a}{r} = 1 + (e -e^3/8) \\cos M + e^2 \\cos 2 M + \\frac{9}{8} e^3 \\cos 3 M + ...", "2665a800ffc06ad0e9d266eeff2ce3b4": "(\\mathbf u,\\lambda)", "266617d5d5cfe4ca48db29497b66f8cd": "S_r(m) \\approx S_{r_1,r_2}(m)", "26664eaf6f30370a448568daf55900d4": "W_{1-2} = \\frac{P_2 V_2 - P_1 V_1}{n-1}", "266664f562edd776813555382bcd56c7": " \\frac {a_z}{a_x} = \\frac{\\pi \\nu}{(1 - \\nu)} . \\frac{d}{ \\lambda}", "26669912de04c3c98d23a7fdd3d623ad": "E_k \\varpropto \\frac{\\sigma_t}{\\rho}m ", "2666b1788e867a84c7c7392b63b1f598": "q = e", "266714d3e6cceed45a7bc03b80bbf882": " \\pi_k = \\frac{(\\lambda/\\mu)^k e^{-\\lambda/\\mu}}{k!} \\quad k \\geq 0", "26672648aff2a51db4e0eb78b9c7f9fb": "w \\in\\Sigma_{\\epsilon}", "266727659f714ec57bb630d4422a8fdc": "\\left(x_0,y_0\\right)", "26679b2cee4234caebb63a48ddc78bc7": " C(h)\\frac{\\partial h}{\\partial t}= \\nabla \\cdot K(h) \\nabla h ", "2667da0d0b9152a85eb33aebc4a129d3": "\\psi_1(0)", "2668486d116e526ad9ba5ad18a93ecc3": "\\mathfrak g,", "2668507a9c768c2e7d0cc92e11596c26": "\\mathbf{r}_{i}", "266853e925b3410402dded85501a25d5": "\\begin{align}\n T &= 5 \\; months = .4167 \\; years \\\\\n\\end{align}", "26685b0a9379ff4005a398579768f596": "[D_1,D_2] = D_1\\circ D_2 - D_2\\circ D_1.", "2668be1101347b8b2866b7cd9dc6fa5b": "S_n,", "2668c3c4f7f719c3957f1b19be8b287e": " \\overline{ \\mu } = m - s \\left( \\frac{\\gamma_1} {2} \\right)^{ 1 / 3 } ", "2668f106c4340d1e398fdc6c1c7c4dae": "\\textstyle c > 0", "2669345227314a151c2995fd5774d9ef": "\\mathit{(p - 1)}\\mathit{p}^{k-1}", "2669541c1094a7ebe9f3a0661b4be109": "\\vert A\\vert", "2669d6f77e06eb6142928af9990bc9cf": "\\gg", "266a00226eac234cc217d7b94054e217": "\\sum_{i=1}^K q_i = P", "266a143e770ffca8cadd1c3324fd1634": "n=3,", "266a261ef188444921d723b8551b9173": "(ax+by) v = a (xv) + b(yv)", "266a9ba52dac10cfed2b1a9b42770dfb": "S_{12} = {-2 Z_0 Y_{12} \\over \\Delta} \\,", "266abe80af4999d44cf72ce102be83e1": "\\mathrm{exp}(iz) = e^{i z} = \\cos(z) + i \\sin(z), \\, ", "266ad3fe947dbf1f84901b7162ca6079": "\\Delta p_{\\text{B}} (x)", "266aefa39e04bac79ce325f426bba103": "T_\\mathrm w\\,", "266b581d5be1b2d1d3fa8aa6f1f9a2cf": "\\max_N \\left|\\sum_{n=1}^{N}\\left(\\frac{d}{n}\\right)\\right|>\\frac{1}{7}\\sqrt d \\log \\log d", "266b72702bc4fc011d9891c69b94ee1f": "x \\pm t", "266b87a0c07c8fed3b8bf891dd5466b4": "\\alpha_c = 1.02056", "266be3a8836b65a8a27c91346b7e8790": " |\\alpha\\rangle", "266c0273df772915e068ff8d1e5927da": "\n\\theta = \\exp {(- \\beta u)}\n", "266c494ec5fd7f48718fd94a39b0dedd": "\\int_a^b \\frac{f(t)}{t-x} \\, dt", "266c5d7fcf32909c05bbd7d24f77ad5c": "\\omega(a_n) = \\lim\\sup a_n - \\lim\\inf a_n.", "266cae595f409e0e2bfb0c329d821806": " -D_2\\frac{dC_2}{dx}", "266cf824cbaa72b4fa91e6387799e939": " {\\hat f}^s ", "266cf8b127ca08bbb10f8e6b9e658485": "aaSbb", "266d730251e5d573139d8f350429a831": " \\sum_{n=1}^\\infty c_n \\sin \\frac{n\\pi x}{L}", "266d8a30639b362857b3aa51fd1f36da": "\n \\begin{bmatrix}\n 1 & 3 \\\\\n 1 & 0 \\\\\n 1 & 2\n \\end{bmatrix}\n+\n \\begin{bmatrix}\n 0 & 0 \\\\\n 7 & 5 \\\\\n 2 & 1\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1+0 & 3+0 \\\\\n 1+7 & 0+5 \\\\\n 1+2 & 2+1\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1 & 3 \\\\\n 8 & 5 \\\\\n 3 & 3\n \\end{bmatrix}\n", "266da83adb524171b27816969121dea7": "\\forall x (p(x) \\rightarrow P(x)) \\wedge \n\\neg \\forall x (P(x) \\rightarrow p(x))", "266e1c7527186ea3b4463792f3566073": "\\lim_{(x,y) \\to (0, 0)} \\left |y \\right \\vert = 0", "266e6027841ee7c4464563f356ebb3c2": "~\\mathrm{2AmF_3+3Ba \\ \\xrightarrow{1150-1350^\\circ C} \\ 3BaF_2+2Am}", "266e71cb2f4d9f16b67f47774e5fe387": "\\underline{E\\Gamma}", "266e84bb62d9bee123e5f5176133a5b6": "(\\mathcal{U},\\mathcal{S})", "266e96d8a9bface34d5fbc576a05f051": "\\max \\sum_{(i,j) \\in E} \\frac{1-\\langle v_{i}, v_{j}\\rangle}{2},", "266eb6a0c2c2cec33e943ebe07b6c660": "\n\\frac{d\\mathbf{L}_{\\text{in}}}{dt} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{d}{dt} \\left( \\mathbf{I}_{\\text{in}} \\cdot \\boldsymbol\\omega \\right) = \\mathbf{M}_{\\text{in}}\n", "266f46c1c2685eb93ec3077c7cf9d174": "D_k=C_k\\cap C^1_k\\cap C^2_k\\cap \\dots", "266f52938f84a1b3df9287ac61011078": "S_0(\\lambda), S_1(\\lambda), S_2(\\lambda)", "266fe0709f0fbabd96ac39549fa95787": "12n - 1", "266fff00a1066271ccb1d448ff16c281": "\\frac{\\beta - 1}{r}", "26704c52602491a16218a00bccc218a9": "\\frac{8388613}{25165824}\\,", "2670a221ce80dcaadfc3fd6499513163": "r=\\frac{1}{4(1+\\sqrt{n})^2} \\, ", "2670bd1be44e0717f23d36652647d704": "O(N \\log{N} )", "267110e216b0283e2bdb8a5b0e3fa875": "9 \\times 5", "26712969fd3cbb868b40550717f7ab28": "(t_2, p_2)", "2671410613b2065a35c0d4dbbb518c1b": "C_n = C_2 \\prod_{q|n} \\frac{q-1}{q-2}.", "267148ca6923ee5341b8226de6341734": " \\bold{F} = \\bold{E} + Ic\\bold{B} = E^k\\sigma_k + IcB^k\\sigma_k", "26714b3711a9cb13f2bd270c64371d2d": "\ne^z = \\cfrac{1}{1 - \\cfrac{z}{1 + z - \\cfrac{\\frac{1}{2}z}{1 + \\frac{1}{2}z - \\cfrac{\\frac{1}{3}z}\n{1 + \\frac{1}{3}z - \\cfrac{\\frac{1}{4}z}{1 + \\frac{1}{4}z - \\ddots}}}}}.\\,\n", "2671ae1eb456013380c59ef9d1cc4344": "\\frac{F_m-F_0}{F_m}", "2671d32b1d3ecedbd9787b2aa1c6c0bf": "\\theta_{r_i}(x)\\big|_{x=-j\\infty} = \\angle(-\\mathfrak{Re}[r_i],-\\infty) = \\lim_{\\phi \\to \\infty}\\tan^{-1}\\phi=\\frac{\\pi}{2} \\quad (10)\\,", "26725ec9a70a1e106f3c61f019935deb": " N^b a_b = \\left( \\frac {\\partial g_{tt}} {\\partial r} c^2 \\right) / \\left( 2 g_{tt} \\sqrt{g_{rr}} \\right) = \\frac{m}{r^2 \\sqrt{1-\\frac{2m}{r c^2}}}", "26726077c418e810d31060589b2e9662": "c = \\log_b a", "26729eae84c73103f076624bafd107cd": "\n\\mathbf{C}^0_{xx} = \\int_{\\Delta} x^2 \\, dA = \\int_{x=0}^1 x^2 \\int_{y=0}^{1-x} \\, dy \\, dx = \\int_0^1 x^2 (1-x) \\, dx = \\frac{1}{12}\n", "2672c2e2fa771f55e21b493f17200615": "\\theta_n = \\frac{2\\pi n}{q},", "2672e2763e9dd40c08fcddcdd675bcc9": "\\alpha_{12}=-1", "26733e216c33c8a7691c09a8333c04c3": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ \\frac{\\sqrt{8}}{3} \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ -\\frac{\\sqrt{2}}{3} \\\\ z_2 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ -\\frac{\\sqrt{2}}{3} \\\\ z_3 \\end{pmatrix}", "26737c7a2bcbdd49ac7c22fcad0b08ff": "\\omega_{x} = (b,d,u,a)\\,\\!", "2673935bf54d9ab04129f8fd316037dc": "Y\\stackrel{\\pi}{\\to} X", "2673df4a49c1e899b298259778795b29": " |\\psi\\rangle\\rightarrow \\operatorname{e}^{-i\\omega_l t\\left(a^\\dagger a + \\sigma^\\dagger\\sigma\\right)}|\\psi\\rangle", "2674663231c00e5720fe175729d4e705": "B_\\nu(T) = \\frac{2h\\nu^3/c^2}{e^\\frac{h\\nu}{kT} - 1} \\approx \\frac{2h\\nu^3}{c^2} \\cdot \\frac{kT}{h\\nu} = \\frac{2 \\nu^2 kT}{c^2}.", "2674f3010db1eb5588dde7e6975f60e6": "P_c = \\,", "26752ed6517fdb33649bfedbeecdf05a": "\\scriptstyle{7({1/3})^s+6({1/3\\sqrt{3}})^s=1}", "267572b8963bb529092a22aaf9b536b8": "\\mathfrak g_P:=P\\times_G\\mathfrak g.", "2675b3728b908ca397d9e5fae3c22af5": " 2 x ( x^2 + y^2 ) + x^2 ( 2x + 2 y \\frac{dy}{dx} ) = 2 a^2 y \\frac{dy}{dx} ", "2675ce6fcff505e0923703eafff80b43": "\\left( \\frac{1-w_i \\overline{w_j}}{1-z_i \\overline{z_j}} \\right)_{i,j=1}^N", "2675ff27c7ea1646962c44ba00e126e2": " z < 0 ", "26766491ee80d0b6e3a24da8acbdfb0e": "i^* i:\\Phi\\subset H=H^*\\to\\Phi^*.", "26766df637276caed72d0a589c0b3fad": " \nu_{n,i+1/2} = \\frac{u_{n,i}+u_{n,i+1}}{2}, \\quad u_{n+1/2,i} = \\frac{u_{n,i}+u_{n+1,i}}{2}, \nu_{n+1/2,i+1/2} = \\frac{u_{n,i}+u_{n,i+1}+u_{n+1,i}+u_{n+1,i+1}}{4}. \n ", "2676768089581d43d30da54d02355ec7": "-Q_{Coble}", "26768ad9076d1fd679d255852be7a2d7": "\\theta = \\arcsin \\left( \\frac{\\text{opposite}}{\\text{hypotenuse}} \\right).", "2676a74097e6c3471b165eb3abae0598": "\\cos^4\\theta = \\frac{3 + 4 \\cos 2\\theta + \\cos 4\\theta}{8}\\!", "2676adec50bb7794c1f00cbb607d6069": "\\tau: \\ z^{\\tau}", "2676b06af354fb3b834b0ba0b4a2e13d": " {\\partial \\overline{u}\\over\\partial x}+{\\partial \\overline{v}\\over\\partial y}=0 ", "2677086aefc732eab0acaac7bde0698b": "\\frac{\\partial h_{s}}{\\partial t} + \\left( v_{||} \\hat{b} + \\vec{V}_{d s} + \\left\\langle \\vec{V}_{\\phi} \\right\\rangle_{\\varphi} \\right) \\cdot \\vec{\\nabla}_{\\vec{R}} h_{s} - \\sum_{s'} \\left\\langle C \\left[ h_{s}, h_{s'} \\right] \\right\\rangle_{\\varphi} = \\frac{Z_{s} e f_{s 0}}{T_{s}} \\frac{\\partial \\left\\langle \\phi \\right\\rangle_{\\varphi}}{\\partial t} - \\frac{\\partial f_{s 0}}{\\partial \\psi} \\left\\langle \\vec{V}_{\\phi} \\right\\rangle_{\\varphi} \\cdot \\vec{\\nabla} \\psi ", "2677a78da638bd86011fd4bced2d20a7": "g(C) := dim_k \\Gamma(C, \\Omega^1_C)", "2677be747b73393907eb13c451af030a": "\\rho=T_{\\mu\\nu}\\delta^{\\mu \\nu}", "2677c183da96e57fd50fe8a3156c72e2": "R^2-r^2", "267835087891678e9f8ffe46bd69783b": "x(u)=a \\mathrm{sn(u,k)}+(a/k)((1-k^2)u - E(u,k))", "26785473d6d245a0e7def889c530101f": "\n\\left( \\mathbf{H}(z_1,\\ldots,z_k,\\ldots,z_n) - \\mathbf{H}(z_1,\\ldots,z'_k,\\ldots,z_n) \\right)^2 \\preceq \\mathbf{A}_k^2,\n", "267905407294192856d94d0fb7a3b2b0": "\\varphi_{\\beta}(\\gamma+1)", "26794630b04f565641b4c9576677fa61": "2k", "267957c68d0ff420619cdb14dcf37dc9": " \\langle u \\bar{v} \\rangle_S. ", "267960fd5b43b833707186f15a03ac9d": "\\begin{array}{cccccccccccccccccc}\n& & & & & & & & & 1 & & & & & & & &\\\\\n& & & & & & & & \\frac{1}{2} & & \\frac{1}{2} & & & & & & &\\\\\n& & & & & & & \\frac{1}{3} & & \\frac{1}{6} & & \\frac{1}{3} & & & & & &\\\\\n& & & & & & \\frac{1}{4} & & \\frac{1}{12} & & \\frac{1}{12} & & \\frac{1}{4} & & & & &\\\\\n& & & & & \\frac{1}{5} & & \\frac{1}{20} & & \\frac{1}{30} & & \\frac{1}{20} & & \\frac{1}{5} & & & &\\\\\n& & & & \\frac{1}{6} & & \\frac{1}{30} & & \\frac{1}{60} & & \\frac{1}{60} & & \\frac{1}{30} & & \\frac{1}{6} & & &\\\\\n& & & \\frac{1}{7} & & \\frac{1}{42} & & \\frac{1}{105} & & \\frac{1}{140} & & \\frac{1}{105} & & \\frac{1}{42} & & \\frac{1}{7} & &\\\\\n& & \\frac{1}{8} & & \\frac{1}{56} & & \\frac{1}{168} & & \\frac{1}{280} & & \\frac{1}{280} & & \\frac{1}{168} & & \\frac{1}{56} & & \\frac{1}{8} &\\\\\n& & & & &\\vdots & & & & \\vdots & & & & \\vdots& & & & \\\\\n\\end{array}", "267969f90e236db9822e2ba2082b4c37": "A_{21} \\in \\mathbb{R}^{(n-1)}", "26797144bdcd6c58ce21907642dc9aaf": "0=t_0 < t_1<\\cdots< t_n=T.", "26799cca5fb2e6da8d2df24d7ab11c74": "E_{(X, d, \\mu)} \\colon [0, +\\infty) \\to [0, +\\infty]", "2679a36a4e30221335d81cdf3d511035": " \\widetilde{o}^1, \\dots, \\widetilde{o}^T", "2679a744724159aed5707c14cd119aee": " \nGM_\\bullet = f R_\\mathrm{BLR} (\\Delta V)^2 .\n", "2679c067f9ee00806541be74aecb7828": " V_n = \\sum_{k=0}^n a_k e_k", "2679d813a6deb1012a035bf5bfdd0a5d": "\\lim_{(x,y)\\to(0,0)}\\left|\\frac{x^2y}{x^4+y^2}\\right|", "2679d84d952b1514604055568c4e4b2e": "\\hat{T}T\\rho=\\rho.", "2679e5921fa81fa9c59509ae328ce507": "\\mathbf x + \\mathbf y = (x_1 + y_1, x_2 + y_2, \\cdots, x_n + y_n)", "267a08cc0d6db246e07bee055a3f80e0": "=\\int \\left( \\frac{1}{2}\\partial^\\mu \\phi \\partial_\\mu \\phi -\\lambda \\phi^4\\right ) \\, \\mathrm{d}^4x ", "267a170160d48fd600cd1f6ec9f383ef": "t \\ne 0", "267a1e6227a70c619d49d2a2765cfe5d": "y''(t)=f(t,y(t),y'(t))", "267a68d847d7eea649d77d14aaa34b4a": "=\n\\begin{vmatrix}\n \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\[5pt]\n \\cfrac{\\partial}{\\partial x} & \\cfrac{\\partial}{\\partial y} & \\cfrac{\\partial}{\\partial z} \\\\[12pt]\n A_x & A_y & A_z\n\\end{vmatrix}\n", "267a6cfc60c0a8a95a3dc4978903be24": "SU(m/n)=OSp(2m/2n)\\cap OSp(2n/2m)", "267a775855ad3b6cef34c019d69ed7ba": "\\mathbf{v}' = (\\det R)(R\\mathbf{v})", "267a84a79826997cbb4efb9053aed28d": "\\dot{n}_A", "267aaea22c674e86d8beade2270d0c61": "W(h) =\\textstyle \\int_{h}^{\\infty} f(z)dz", "267ab9dd0cda374f693e5ee2193869ae": "W \\ ", "267b24e94b7463de5ea99abdab392596": "\\sigma_{xx}<0", "267b8f5dede5b8e491a778b873d73f8a": "\\tan(\\delta'-90^\\circ) = \\frac{\\sin(\\delta-90^\\circ)}{\\gamma\\cdot\\left(\\cos(\\delta - 90 ^\\circ)+\\beta\\right)}", "267bff78ef51ee21aa59189098e319cc": "\\ \\alpha ", "267c3ed6f2673eaa8e6d53d8abd8b728": "\n\\left(\\frac{2}{n}\\right) \n= (-1)^\\tfrac{n^2-1}{8} \n= \\begin{cases} \\;\\;\\,1 & \\text{if }n \\equiv 1,7 \\pmod 8\\\\ -1 &\\text{if }n \\equiv 3,5\\pmod 8\\end{cases}\n", "267c4a3a7956f2c7c49cc9e1d36a624a": "O(nL)", "267c6a7e8db15c950261aae6d2ac8d42": "\n\\Pi_{k} = {1\\over\\sqrt{N}} \\sum_{l} e^{-ikal} p_l.\n", "267c8703324f01130b67924cfc864127": "\\scriptstyle P_k", "267cfa6d4f2285b7770325512f62def5": "\\Phi^{-1}(x)", "267d8e54e2ee4a05960f12833311c005": "\nX_n(t) = \\sum_{k=-\\infty}^{\\infty} e^{ik\\omega t} X_{n;k}\n", "267de8efb9e9e5311cbb8f3928827a70": "E_{gate}(t-\\tau)", "267df38c240591e3efbd3d50b6f04883": "\\scriptstyle \\omega.", "267e0e1db9f82a38a1ff31012744a613": " \\sum _j M_{i,j} v_j = \\lambda v_i^{}", "267e1317d688e899206ea3866f642c55": "2\\pi\\sqrt{\\alpha'.}", "267e3321b35709212e21b874b3f88ea0": " \\frac {\\tau_1} {\\tau_2} \\approx \\frac {(\\tau_1 +\\tau_2 ) ^2} {\\tau_1 \\tau_2} \\approx A_v \\frac {R_i} {R_i+R_A}\\sdot \\frac {R_L} {R_L+R_o} \\ ,", "267e40536216faff418a75a2c5b3f381": "\\displaystyle \\int {(|y|^2)^{-u}\\over |y|^2 +q_\\mu^2} \\, d^Dy = \\pi^{D/2} (q_\\mu^2)^{-z-u} {\\Gamma(-u +D/2)\\Gamma(1+u-D/2)\\over \\Gamma(D/2)}.", "267e6e234b7a828dd5bdfb0ece74418d": "\\frac{1}{1+x} = 1 - x + x^2 - x^3 + \\cdots", "267ec5e201499dd05bbf74bd1601732d": " \\|x - z\\| = \\|y - z\\|.\\, ", "267f23144ed0a72e1010a6d5338f14ef": "\\C \\setminus -\\N_0", "267f3a2f19fbb496aabc04317b723f61": "(g^a,g^b,z)", "267fa0cd57b5d52930b58ed9794fed29": "m = m_1 + m_2", "26801500b1945d7171a00f3af4b9d9c1": "", "26803535adfc0ccb2717ea148d80a1f1": "\\ ax^2+bx+c = 0", "268040e2f4f91da62ada109f02b8690e": "N_{R_3}", "26804af7f94f530583605ba8f6f34298": "a_0x_0+\\ldots +a_nx_n=0", "268053fda3c06027a1727ed3bad40129": "H_n(M, M\\setminus\\{p\\}; \\mathbb{Z})", "268066e471c5bbf7fc3eee31d5284f4f": "u>0", "2680b78fa14ef53189341c7647c8aca8": "\\hat\\mu", "2680f716949e39bccf78df6bf345f68e": "x(t)\\in\\R^n", "2681e7877ba15c04280d0d864386e16f": "\\sigma\\in\\Gamma_{m},\\,\\!", "268218361b4c7c68e71482d88f745745": "p_i = q_i ", "26822f6702c369ee454256fe4b9e3515": " \\lambda: B \\rightarrow \\mbox{End}\\, A_B, \\ \\lambda(b)(a) =ba ", "268260d6ac67f93e8d9562524b86d16f": "f : X_1 \\rightarrow X_2 \\,", "26826df017dfdfb620e9f946b73599ff": "r\\to \\infty", "26828d204611db6ec7b39b07f84ebd81": "\\lnot (A \\vee B)", "26829e29c2f429b82e3d73743bf4f162": " a,b \\ge 0. \\, ", "2682beb0f7eb9b3f8bed0612a1868a3e": "d \\phi: \\mathfrak g \\to \\mathfrak{gl}(V)", "2682ea9b359d6d66b32b9b4f3820c548": "n \\sin \\alpha", "268354384b548acee2f4aa185c9e0709": "w_{2^i}", "26842c97d9db3b5d24e7e649f3da4182": "\\lambda = 2\\pi d", "2684306b928eb1de2297a464b6ce64d8": "L_{ij} = S_{ij} + X_i P_j - X_j P_i", "268439c1c4bb5b17015bd276de464a11": "p \\rightarrow q", "26846903055b0502be5302302e622f98": "\\varepsilon = \\omega^{\\varepsilon}", "2684e73f5b0f1c54849b4853cbf8ce79": " \\boldsymbol{lb} \\leq \\boldsymbol{\\beta} \\leq \\boldsymbol{ub}", "2684f315468882817a189207c4b7d653": "n = \\frac{ck}{\\omega}", "26852d19627b23cfe7c23d5844e50cc3": "T = \\{x_i, y_i\\}", "26857864600298db48bd1c8beb318aa1": "A, C, n", "268582d85d37b7b852bbde44dd4f42c3": "\\alpha \\in \\mathbb{F}_{q^k } - \\{ 0\\}", "26860f75d6e16b71372774d5ddaf0b4f": "\\mu(\\phi)={\\displaystyle \\frac{\\pi}{2}\\frac{m(\\phi)}{m_p}}\\,\\!", "26867ea512c4bcf9d97fbda362dd3233": "1\\leq i 0, \\quad a x > 0\\mbox{)} \\\\\n -\\frac{2}{\\sqrt{b}} \\mathrm{artanh}\\left( \\frac{S}{\\sqrt{b}}\\right) & \\mbox{(for }b > 0, \\quad a x < 0\\mbox{)} \\\\\n \\frac{2}{\\sqrt{-b}} \\arctan\\left( \\frac{S}{\\sqrt{-b}}\\right) & \\mbox{(for }b < 0\\mbox{)} \\\\ \n\\end{cases}", "26a192124d2b772107e72bf7deb069a3": "P \\oint_{\\mathrm{surface}} \\mathbf{q} \\cdot d\\mathbf{S} = P \\int_{\\mathrm{volume}} \\left( \\nabla \\cdot \\mathbf{q} \\right) dV = 3PV,\n", "26a1f4dc9c611cfad9a1234fb1dca3d3": "{A}_{8}^{(2)}", "26a23e2b73be519406b32d5c398d4fc6": "\\mathcal{I}\\left( \\beta ,\\theta \\right)=\\text{diag}\\left( \\mathcal{I}\\left( \\beta \\right),\\mathcal{I}\\left( \\theta \\right) \\right)", "26a26e5369baf000281c72fc6ddaa045": "\\mathcal{O}(m),\\,", "26a28b8255a8329189dd4c4467c0f5a9": "y^2=x^2(a^2-x^2)", "26a2a2b0f4e538fd17fbaa7b169c11d2": "R \\rightarrow \\infty", "26a2a4f695952e6bf6fbc1cc60a45490": " Sc = \\nu/D_{AB} = 1", "26a2df4e86602be51adbbe82ac3b1a4a": "C \\subset \\langle F \\rangle", "26a3121b682435a5e492c652ce4223ab": "x_{\\mathrm{FOH}}(t)\\,", "26a364d355ba4c16768a318817d53feb": "d \\le r < m+d ", "26a3764002b6f72d847ea720ab36bfcd": "Q_j = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial \\mathcal (L+V)}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial \\mathcal (L+V)}{\\partial q_j} = \\left[\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) +0\\right] - \\left[ \\frac {\\partial L}{\\partial q_j}+\\frac {\\partial V}{\\partial q_j}\\right] = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial L}{\\partial q_j} + Q_j. ", "26a396c9347f3b470ffa7db5758552ea": "\\Delta(y, (P (\\alpha_i))^N _{i=1})", "26a3cedf49f0783e34f2fd5972c52bdd": "{\\bold \\ He}", "26a3fa558efe47d73d6de0ef21a3bc8e": " \\delta \\mathbf{u} = \\mathbf{N} \\delta \\mathbf{q} \\qquad \\qquad \\qquad \\mathrm{(6b)}", "26a46d228717df196e811048ec5e856a": "W_{I}(\\mathbf {r}, t)d{\\it t}", "26a4ce2c8be90690193e6efad2703610": "\\det\\Phi(x)=\\det\\Phi(x_0)\\,\\exp\\biggl(\\int_{x_0}^x \\mathrm{tr}\\,A(\\xi) \\,\\textrm{d}\\xi\\biggr)", "26a4e4fe7b02bc5bfd2b163bfd6da622": "R=\\tfrac15", "26a4e9060ef202525a8d9737c82a84fb": "\\begin{align}\np_0(x) & := p(x),\\\\\np_1(x) & := p'(x),\\\\\np_2(x) & := -{\\rm rem}(p_0, p_1) = p_1(x) q_0(x)- p_0(x),\\\\\np_3(x) & := -{\\rm rem}(p_1,p_2) = p_2(x) q_1(x) - p_1(x),\\\\\n&{}\\ \\ \\vdots\\\\\n0 & = -\\text{rem}(p_{m-1}, p_m),\n\\end{align} ", "26a53bd57f336cd11f055cd793e6baaf": "\\forall w\\,[B(0,w)\\land\\forall x\\,(B(x,w)\\to B(x+1,w))\\to\\forall x\\,B(x,w)].", "26a5ed7299e0f4b22987cf1f21d24d72": "G = \\sqrt[4]{32}e^{-\\frac{\\pi}{3}}\\left (\\sum_{n = -\\infty}^\\infty (-1)^n e^{-2n\\pi(3n+1)} \\right )^2.", "26a5f8db9fb39c2b3410344d7812398b": "\\mathfrak{gl}_n,", "26a6a58d80b7cd2aca16d95bbdf75f04": "P(H2)", "26a6a8d43dbb81c2ea968033e74765eb": " l \\geq 2n ", "26a6ba41e06b80e6107857730fac2b35": " v_j(x) = \\begin{cases}\nL^{-\\frac{1}{2}} & \\mbox{if } j = 1.\\\\\n\\sqrt{\\frac{2}{L}} \\sin(\\frac{j \\pi x}{L}) & \\mbox{ if j is even.}\\\\\n\\sqrt{\\frac{2}{L}} \\cos(\\frac{(j+1) \\pi x}{L}) & \\mbox{ otherwise if j is odd.}\n\\end{cases} ", "26a6dfb3f123d46777f275b1be3fce66": "\\mathcal L = -\\frac14F_{\\mu\\nu}^a F_{\\mu\\nu}^a", "26a7be5c5517fd580bd94f7d28e0afa5": "T_c = \\frac{F \\mu_c d_c}{2}", "26a7dff9037a1534a2d3db55930c5fd2": "\\frac{P_1(x)}{Q_1(x)}", "26a7e99432c7a243de8d727784b56453": "\\displaystyle \\exp(x_0\\,i\\,t-\\gamma\\,|t|)\\!", "26a80432776d78349d18abf507254fc2": "\\scriptstyle \\frac{r_1}{2} = \\frac{-p}{2}=\\frac{-b}{2a}\\!", "26a81e9c8f4240727a4c9c8e4d4c796a": "|\\alpha_{\\psi}\\rangle\\equiv\\hat{D}_{\\psi}(\\alpha)|0\\rangle", "26a8216db94bc2766055185d5b144a65": "\\bold E ( \\bold{r} ) =-\\frac {1}{4 \\pi \\varepsilon_0} \\nabla_{\\bold {r}}\\int \\frac {1}{|\\bold r - \\bold{r}_0|}\\ \\bold{p} \\cdot d\\bold{A_0} \\ , ", "26a851c1da05cb7344dab62add5b2856": "\nV(x)=\\bar\\lambda x+ \\frac{\\bar\\lambda^2 r^2}{4\n\\rho},\n", "26a8bc4b89d263c9facecdd4e5e746aa": " \\frac{\\partial (\\mu M)}{\\partial x} = \\frac{\\partial (\\mu N)}{\\partial y} \\, \\! ", "26a923ceac0346a3bfacac9cbb039773": "T(f) = \\int f(x)\\, d\\mu(x).", "26a943b15a2c2a13f8a2ea9e947d9ded": "r \\geq 1", "26a99814d1f3ff6e359647e1d4d586d4": "L = \\{a^nb^n:n\\geq1\\}", "26a99f9b2f21ca3ba6f222efa40c7958": "^{3}", "26a9c9553e2cc94706b6745138b3a70a": "= u_m(t) \\cdot \\cos(\\omega t + \\phi) + i\\cdot u_m(t) \\cdot \\sin(\\omega t + \\phi)", "26a9e7eb8ac630c1261e34fd57e7ca39": "g_1, \\cdots, g_m", "26ab460270eded2601f403e9054621d7": "\\nu >8\\!", "26ab4749b4dafd87fd0902093e560a65": "\\scriptstyle \\vec{F}", "26ab61fa8024f78da62a405a1e574d8f": "\\Delta^n", "26ab7b049445e8aa150a013929849c58": "(3,\\bar{3},1)", "26abbd3b085205a290dadc2d700b5e18": "\\operatorname{recc}(A) \\cap \\operatorname{recc}(B)", "26abc361ab9c3bf557e489b770982d35": "M(t)", "26abedbeb8af48df427a153933f6498e": "\\exp[ik(x-v_{ph}t)-\\gamma t]", "26ac10be0d4e70ad04c8951000575476": "a_{12}=\\frac{1}{x_1-x_0}", "26ac99baf405f378717596eecc8ac783": " P", "26ac9eece1cf4b16d04117ed135de65e": "K \\subseteq \\alpha / A \\leftrightarrow \\exists H: \\exists J:[\\langle H,J,K \\rangle \\in R_1 \\wedge H \\subseteq B \\wedge J \\subseteq \\alpha / B ].", "26acb6a2c3ee11b18e0fb4f2926790d8": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-2t})(1-p^{-t})}", "26acd95bf33dfe594ddf1e16892519f9": "{SU(2)_L\\times SU(2)_R\\times U(1)_{B-L}\\over \\mathbb{Z}_2}.", "26ad5a81cae7043365de5f4906a7be8e": "K_{p,q}", "26adbc57d90f6c048126a8c06dd7b3e6": "\\!\\mathcal A \\models_X^- \\phi", "26ae260615c61bbe6e0783851926a0c5": "\\frac{1-6pq}{pq}", "26ae4241f36eeeea983277b3e9c91535": "\n\\phi(\\{\\mathbf{r}_i\\},t+T) = \\phi(\\{\\mathbf{r}_i\\},t).\n", "26aeb065deae5a41ef3bf16a6681e88e": "P( A_K / A_L ) = \\prod_{j=1}^n ( a_{Kj} / a_{Lj} ) ^{w_j}, \\text{ for }K, L = 1, 2, 3,\\dots, m. ", "26aec6374089dff58d82ee0b3bd97a64": "\\operatorname{gr}_I R = \\oplus_0^\\infty I^n / I^{n+1}", "26af2e336c085d2dbf29088557efc483": "\n\\mathrm{action\\_delta} \\times \\mathrm{mk\\_max\\_speed} \\times \\left(\n \\frac{ i } { \\mathrm{mk\\_time\\_to\\_max} } \\right)\n^{\\frac{ 1000 + \\mathrm{mk\\_curve} } { 1000 }}\n", "26af694d60b9f98179dd9b880d8c34a3": "\\ (2f_a - f_b), (2f_a - f_c), (2f_b - f_a), (2f_b - f_c), (2f_c - f_a), (2f_c - f_b)", "26afca5530aad74bd80a8f10ab64fda7": " \\psi=\\frac{1}{\\sqrt{2}} (1,1), ", "26aff2a6e27e70966ab6951a4191f2f3": " u(t,r) = \\frac{1}{r} \\left[F(r-ct) + G(r+ct) \\right],", "26b018af791aa77b9a7ff0c645ab1d48": " \\Gamma,\\mathcal A,\\mathcal B,\\Delta\\vdash\\mathcal C", "26b056081d50fb5e264ced364aa29190": "g (q_i , p_j)", "26b09a1294c21a17d6315956e25fa39c": "f_p \\le 2B \\, ", "26b0eba40be66dbcf3bd0f6e6928dfde": "\\int \\csc^2 x \\, dx = -\\cot x + C", "26b118de71be211ee4d6d1c1041c0d1b": "s = \\frac{(a_1 b_2 - a_2 b_1)^2}{(a_1^2 + b_1^2)(a_2^2 + b_2^2)}.\\,", "26b171eca8dfdbeb0e9bc9ec52779160": " \\mathbf{[T]}=\\begin{bmatrix} 1 & 0 \\cdots 0 \\\\ T_{21} & T_{22} \\cdots T_{2n} \\\\ \\cdot & \\cdots \\\\ T_{n1} & T_{n2} \\cdots T_{nn}\\end{bmatrix}", "26b17225b626fb9238849fd60eabdf60": "+", "26b18a8e53fae3d6181ef230c56c1896": "ns^n", "26b19c07bd6bb27371bdfdd34f78a24b": "\\mathbf{e}_1(t), \\ldots, \\mathbf{e}_n(t)", "26b1bd36dc827f561251174c04d194ee": "|x-y|^2=(x-y)\\cdot(\\overline{x}-\\overline{y}) = (x-y)\\cdot\\left({1\\over x}-{1\\over y}\\right) = -{(x-y)^2\\over{xy}} \\ .", "26b21b408b4a25089933eb08c700a95b": " 0 \\le y \\le \\pi, y \\ne \\frac{\\pi}{2} \\, ", "26b259fb0497216f29e0eacc0d84900f": "f(x_1, \\ldots, x_n)/\\alpha", "26b262ed7e40e133b188b43e298f0a22": "\np(t) = \\int_0^\\infty h(\\tau) q (t - \\tau) \\, d \\tau\n", "26b267d0c8457ca611c2ad1479263e9d": "f_{X,Y}(x,y)", "26b27098b70ac6082624063de3b23859": "y_i = R C \\, \\left( \\frac{x_i - x_{i-1}}{\\Delta_T} - \\frac{y_i - y_{i-1}}{\\Delta_T} \\right)", "26b2846a2e8c4c2b2a1d464f08df13db": "\\frac{dQ^D/Q^D}{dP^D/P^D}", "26b2883c80bf87e565acc1c9830cf00d": "\n\\frac{d}{dx}u(x) + \\int_{x_0}^x f(t,u(t))\\,dt = g(x,u(x)), \\qquad u(x_0) = u_0, \\qquad x_0 \\ge 0.\n", "26b33544488bd972dd8a3eb29f0f3b24": "g_{o}", "26b365d36b8467351ea01f96fac79cbe": " T \\rightarrow 0, \\ \\ e^{-K}, e^{-L} \\rightarrow 0 ", "26b3a09059d77ae3e8307d7c536e14f4": "\\Delta{v_i}", "26b43a02bd1f0533f8442312cf7010ea": "(X_i, Y_i,)", "26b450466710ed7cb561c21e9f334c44": "\\gamma_i(t)", "26b478285a574ae2d540e73a99358d20": " P = \n\\begin{bmatrix}\n 0 & 0 & 1/2 & 0 & 1/2 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n 1/4 & 1/4 & 0 & 1/4 & 1/4 \\\\\n 0 & 0 & 1/2 & 0 & 1/2 \\\\\n 0 & 0 & 0 & 0 & 1\n\\end{bmatrix}.", "26b48baa6b7e307ff5452508c507be82": "h = \\left(q + \\left\\lfloor\\frac{13(m+1)}{5}\\right\\rfloor + K + \\left\\lfloor\\frac{K}{4}\\right\\rfloor + \\left\\lfloor\\frac{J}{4}\\right\\rfloor + 5J\\right) \\mod 7,", "26b520ac7006c593761e51d463332ae9": "(\\xi_\\nu)_{\\nu=1}^k", "26b543f629b812602f41558e8d2530d6": "Var(X) = V(\\mu) = \\mu", "26b5566703d20e9c19598f1cdc1a8841": "|\\psi_{E}\\rangle = l|0_{E}\\rangle + m|1_{E}\\rangle,\\quad l,m\\in\\mathbb{C}.", "26b568e4192a164d5b3eacdbd632bc2e": "kp", "26b57ccb4491dcbaf736d40cc5e859be": "{\\gamma_{\\alpha\\beta}}^\\chi = -{\\gamma_{\\beta\\alpha}}^\\chi", "26b5e39b6db5357b35357301250234d1": " A\\otimes_B \\cdots \\otimes_B A", "26b5e64f4e0e9123340a1e229e22ef65": "\\mathcal{L}_j", "26b62d6c9f347c44b7bf44aac3da8695": "x \\in \\mathcal O_L", "26b66a702fedc65f0c3454b7687c20d3": " 3^{3^3} + 3 ", "26b69c66794bc0cdaba6a53212fb7d13": "\n\\frac{1}{4}S_0(1-\\alpha_p)+\\epsilon \\sigma T_a^4 - \\sigma T_s^4 = 0 \n", "26b6b931b6c258428e315810692e400f": "\\delta-1", "26b6d603363104a2d0ebc1f11bc2f845": "\\quad\\partial_{\\eta} = t^{\\gamma/2} \\partial_y.", "26b6f64c665c370db1ddfb73093401dc": "( x_2 - x_1 )", "26b7b741c50febab49ae56981f53546a": " \\Omega Y", "26b7c6ee3e38c6d0e153032ca062358d": "\\displaystyle E_A=K_1\\sin^2\\theta+K_2\\sin^4\\theta+K_t\\sin^3\\theta\\cos\\theta\\sin3\\phi ", "26b7c897c6ffe20104b64da201ab6344": "TK_R'^{}", "26b7d778161ca7bf181776a786f50b71": "a_i^Tx \\leq b_i", "26b7ff56473b53ec24426e699af78654": "x_1.\\exists x_2. R(x_2)", "26b8131e3c3b5e4ed9afd0e3bf1aedb0": "\\tau_i^2 = 1", "26b82bc76d3d4fd7a9213b88d9be7033": "\\scriptstyle [m,\\, 1.82m]", "26b87bf5f94e3f4fee01d77f111d86cb": "\\pi_1(X) = \\Z^n", "26b88c4ab183ea019a6adde46ead0335": "M_{KL} ", "26b8b493123418f571776ca5070ccd02": " p(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_2 x^2 + a_1 x + a_0 ", "26b993b6f84a9de4db810dc30b912578": " \\mathrm{T} ", "26b99d3cb8a610400af9f885724c959d": "(\\mathbf{v}\\cdot\\nabla)\\,\\mathbf{v}", "26b9eb8c2784c194bb8c61fb925183e9": "f(x)=\\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} e^{2\\pi i(x-y)\\cdot\\xi} \\, f(y)\\,dy\\,d\\xi.", "26b9f13e24da67ea84cbb44c13336024": "\\cap W = \\cap_{S \\in W} S = \\emptyset", "26ba0cd4dd68d1717bdeb32504c0a3b2": "\\langle \\mathbf{No}, \\mathrm{<}, b \\rangle", "26ba123829294a9d299e0939621ba0f5": " p_{\\alpha , \\beta} (\\varphi) = \\sup_{x \\in \\mathbf{R}^n} | x^\\alpha D^\\beta \\varphi(x)| ", "26ba39b2f3aaead20e454e3629e8c97a": "H=-c_0-2\\sum_{n=1}^\\infty c_n \\phi_n", "26ba4ce0c3c27bb14b5edf62e1c8b2cc": "D_{E}+H_{E}\\approx H_{E};", "26baa2666f336904159f50f42c0a8f52": " p = \\frac{2\\pi r_A}{N_A}.", "26bc16dd3f32eb059837c61c14ff24e4": " p_{\\mathrm{tot}} = p_{\\mathrm{rad}} + p_{\\mathrm{gas}} = \\rho c_{\\rm s}^2", "26bc1d47df866aaadd42a8e018c9730f": "-2\\sum_i r_i\\frac{\\partial f(x_i,\\boldsymbol \\beta)}{\\partial \\beta_j}=0,\\ j=1,\\ldots,m", "26bc32c6863d0276dcd5fb12fd678378": "\\sum_{p}\\frac{1}{p-1}= {\\frac{1}{3} + \\frac{1}{7} + \\frac{1}{8}+ \\frac{1}{15} + \\frac{1}{24} + \\frac{1}{26}+ \\frac{1}{31}}+ \\cdots = 1.", "26bc46534ac19146f9fe821a78434e36": "\n\\frac{d\\theta}{dt} = -\\alpha_{\\mathrm{max}} \n\\left[\\sin 2\\theta - \\sin 2\\theta_{\\mathrm{eq}} \\right]\n", "26bc4f3bbbcfbe4272ef19f160f3ce62": "\\sqrt{\\frac{\\gamma \\pi }{2}}", "26bc5477295cb67ac2ac6ac7debe3a0b": "\\Omega_1 = \\{ \\lnot \\},", "26bc7511d65f1a966d2f435c51b4aa70": " \\dot{y}=\\frac{\\partial y}{\\partial x_j}\\dot{x}_j+\\tfrac{1}{2}\\frac{\\partial^2 y}{\\partial x_k \\, \\partial x_l} g_{km}g_{ml}. ", "26bc8b6679140e0c85541070555f194c": "Q_{yz}Q_{xz}Q_{xy}Q = \\begin{bmatrix}1&0&0\\\\0&1&0\\\\0&0&1\\end{bmatrix} , ", "26bcdb5fa720fc4c43f0e2737252631a": "\\left\\{{n \\atop K_n}\\right\\} \\geq \\left\\{{n \\atop K_n+1}\\right\\} > \\cdots > \\left\\{{n \\atop n}\\right\\}.", "26bd14b18ea311b141277f8f308a2c54": " v_{n1} = v_{n2} =0", "26bd2a2b77f032d99c0ee4da15cc17ae": "\\,y = \\sum_j w_j x_j, ", "26bd2a9995658194c0b0ff8697fd7abd": " f_y(x) = f(x, y)", "26bd4c6ff77b543f7b6334eb700114e3": "\\frac{6(90-\\pi^4)}{5(\\pi^2-6)^2}\\,", "26bd563a67f2fa50d5d5e1771cf6e730": "\\arccos(-\\frac{7}{11})", "26bd8a80ec13a31bffa6f99ad69dec36": "\\frac{\\theta \\vdash \\phi \\quad \\theta \\vdash \\psi}{\\theta \\vdash \\phi \\wedge \\psi}", "26bdd98bd26985cabb09e265fc66654a": " T_{\\parallel} ", "26be0f3b464716780a48233cff7f468e": " \\Phi(\\operatorname{probit}(p))=p", "26be3afe504e0e33ee08871209b348a9": " 2 \\alpha = 2 \\zeta \\omega_0 = \\frac{ \\omega_0 }{ Q } = \\frac{ C_1 + C_2 }{ R_2 C_1 C_2 }. ", "26be536dfe382a1923e9dd120d99e314": "x_{i,0}", "26beac9e817f135acec7c603ddbeb111": "\\psi=\\begin{pmatrix}\\psi_{\\uparrow} \\\\ \\psi_{\\downarrow} \\end{pmatrix}", "26bf421cf1549e80620dc23654201bf0": "V_{2n} = V_n^2 - 2Q^n \\,", "26bf4a980b9ba1af8415680853c3e09f": "(I_r,G)=FP", "26bf63c8e5ff3fff07ed47a54dc54b55": "H_5(a, b) = a\\uparrow\\uparrow\\uparrow{b}\\,\\!,", "26bf6ea4b6a9fc3a70db9a887f75b205": "\\pi/4=\\left(\\prod_{p\\equiv 1\\pmod 4}\\frac{p}{p-1}\\right)\\cdot\\left( \\prod_{p\\equiv 3\\pmod 4}\\frac{p}{p+1}\\right)=\\frac{3}{4} \\cdot \\frac{5}{4} \\cdot \\frac{7}{8} \\cdot \\frac{11}{12} \\cdot \\frac{13}{12}\\cdot\\frac{17}{16} \\cdots", "26bf879162c8f396c9f96602acd7c584": "f : V \\to S^3", "26bfb06404127074c87aed13b12d5dbb": "\nL^{2} = \\Delta \\xi^{2} + \\Delta \\eta^{2} + \\Delta \\zeta^{2}.\n", "26bfb88e25141378608867135b6736af": "i\\,=\\,1,2,\\dots ,n.", "26bff523176aaff5a81cfbe4a79a9dd0": "n/(n + 1)", "26c02addc6319fd035ca4416674da534": "\\varphi=\\frac{1+\\sqrt{5}}{2} ", "26c04c4f8decaf9cfc60bcdee12a8a3b": "[D(d) \\wedge \\neg D(f(d))]", "26c1c2fa95c6ba5eb3b7085295c084c0": " Q= I_{3L}+I_{3R}+\\frac{B-L}{2}", "26c1d2bcd06cb14b59f06851520c3f00": "(X, d, \\mu) ", "26c1e30789bc8e8dd563c56f177de79b": "(a\\uparrow^n)^k b", "26c1f98a352eab1db56ec6532992421a": "dM = \\frac {2\\pi R^2\\sin\\theta }{4\\pi R^2} M\\,d\\theta = \\textstyle\\frac{1}{2} M\\sin\\theta \\,d\\theta", "26c2338890453f4d7a04a8353fc88bbb": "\\mathbf{B'}", "26c2454298e583986270da432b139c8e": "\\mathbf f \\ ", "26c25d4ae008288393f7a9264c554dc2": "\n\\frac{1}{2}\\rho W^2NcC_x = 4\\pi\\rho\\left[(a'\\Omega r)^2 + U^2_{\\infty}a(1 - a)\\right]r\n", "26c273d3a925e42cfcb4f11157f63f88": "\\mathrm{tr}(A) = \\log(\\det(\\exp(A))). \\,", "26c2be9e04e04af3e0e4cc804f9c9a9d": "g(x)=\\sum_k c_{n,k} f^{(0)}_{n,k}(x)", "26c2d23bef79e0e9febe6a4854522327": "\\,\n\\begin{align}\n\\Gamma(x+1)&=\n\\lim_{n\\rightarrow\\infty}x\\cdot\\left(\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\right)\\frac{n}{n+x+1}\\\\\n\\Gamma(x)&=\\left(\\frac{1}{x}\\right)\\Gamma(x+1)\n\\end{align}\n\\,", "26c2f26fda59f51040cbdf34c5ac0279": "Y_{CPE}=Q_0(\\omega i)^n", "26c30fb1f5f05ea0ec7e530e1633c014": "r={{\\ell^2}\\over{m^2\\gamma}}{{1}\\over{1-e}}", "26c3160169bcaf62013f30ff9977aefd": "\\frac{w}{(w - w')}", "26c329ef8376fca1edddae81491cce2d": "-w\\ ", "26c36aab2464d1ef4f7436c3d7254c0e": " = z^2 - (i\\sqrt5)^2", "26c373554caa28c5cd630ef6e528bc38": " \\frac{V_{in}}{V_A} \\sim \\frac{1}{S^{1/2}} ", "26c3c623197f1fd0789a626dbdc02730": "Q = Q_{xy}^{-1}Q_{xz}^{-1}Q_{yz}^{-1} . ", "26c3fa9aaea37b9938d6dc02286bccf5": " \\geq1-\\epsilon,", "26c4e812364ebac03a62da6aea219421": "A_{2n-1} \\to C_n", "26c4f5da45981859f4b0ba9f29b999cb": " \\langle \\phi(k_1) \\phi(k_2) ... \\phi(k_n)\\rangle", "26c4fb5180ecff0561c5bb3d2f1034a1": "\\ MRS_{xy} \\ge 0 ", "26c51f87b6a18b74e3c2841d550dfa3d": " |V_{cb}|", "26c58301a4757c7072b1df5f508573f6": "\\mathbf{\\hat{f}_{0:t-1}}", "26c5b3ed8e42984f507a6220bd975d31": "{\\mathbf{A}}({\\mathbf{r}})=\\frac{\\mu_{0}}{4\\pi}\\frac{{\\mathbf{m}}\\times{\\mathbf{r}}}{r^{3}},", "26c5c8d27426796c1f4fd7757c8b30ff": " \\,I_k =1", "26c5e28d191a17bf1884c7d2a53e9d8a": "E(v)=0", "26c5ebf6dcfc09ec6b90519d2019a7b9": "F(2) + F(5)", "26c660287a296abc741bcf6f516c0f5e": "\\log{C_t} = \\log{C_w} + m \\log{\\phi}\\,\\!", "26c6873692b59f40cac54c1cb920b74d": "[\\partial_\\nu[\\partial_\\sigma V_\\nu] - \\Gamma^\\rho{}_{\\sigma\\nu} \\partial_\\sigma V_\\rho - \\Gamma^\\rho{}_{\\mu\\nu}\\partial_\\mu V_\\rho - \\Gamma^\\rho{}_{\\mu\\sigma}\\partial_\\rho V_\\nu] - [\\partial_\\mu [\\Gamma^\\rho{}_{\\sigma\\nu} V_\\rho] - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma}V_\\rho - \\Gamma^\\alpha{}_{\\mu\\sigma}\\Gamma^\\rho{}_{\\alpha\\nu}V_\\rho] ", "26c69c8d54faf7a5e80f28c019bbd8dd": "\\,A+iB = [i\\phi_1-\\phi_2] + [A_1 +iB_1]\\mathbf{i}+ [A_2+iB_2]\\mathbf{j}+ [A_3+iB_3]\\mathbf{k}\\quad ", "26c7e7dab645083b9bd006831a24c706": "\\text{failed}", "26c7ef913dfaa7cd923dbe81b7fe310b": "C_{xy}\\le 1", "26c82ace22ee147093712760c9b8cc82": "\\phi^-_j", "26c833bc04fb96df970d8559a5772984": "k_{\\rm C}=k_1=k_{\\rm E}", "26c8461f3396e3b5d7e975b1cdb66063": "\n\\beta \\,\\,\\, \\approx \\,\\,\\,{{3\\,k} \\over {\\mu _T^2 }}\\,\\left( {{{\\sigma _T } \\over {\\mu _T }}} \\right)^2 \\,\\,\\, \\approx \\,\\,\\,30\\,\\,\\left( {{{\\sigma _T } \\over {\\mu _T }}} \\right)^2\n{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(19)}}", "26c84cdf3743e8221c058e68be6d64eb": "\\int\\!\\int_D g(x,y)\\, \\mathrm{d}x\\, \\mathrm{d}y = \\int_0^R z \\left(\\int_0^{2\\pi} g(\\mathit{xc}+z \\cos t ,\\mathit{yc}+z \\sin t ) \\,\\mathrm{d}t\\right)\\,\\mathrm{d}z", "26c86b4a0b5f3990941fde4859a2d25a": "\\triangle ADC ", "26c8abd4645b3f81caf2153c293f3a5a": "f(x_1,a,g(z_1),y_1)", "26c924f41e6624aeae0d090149e67db5": " I_R \\le RM_R\\int_0^\\pi e^{-aR\\sin\\theta}\\,d\\theta = 2RM_R\\int_0^{\\pi/2} e^{-aR\\sin\\theta}\\,d\\theta\\,.", "26c96c0e80f62091460174baee9f7e6f": "f^n - g^n = d_B^{n - 1} h^n + h^{n + 1} d_A^n,", "26c98462beeb585d0b3f172a6590a218": "c_{ijk}", "26c9b915984f83831ec787c7c58b8c48": "M \\succeq 0", "26c9ccafebf2eddd3146fd187d7269ee": "X^{L,R}", "26c9d04dbe5b72fae1f62238cde54372": " B_n = - \\sum_{k=1}^{n+1} \\frac{(-1)^k}{k} \\binom{n+1}{k} \\sum_{j=1}^{k} j^n ", "26c9d13539db28ab2728ed8411e1b1fb": " \\Delta A \\, \\Delta B \\geq \\frac{1}{2} \\sqrt{ \\left|\\left\\langle\\left[{A},{B}\\right]\\right\\rangle \\right|^2 + \\left|\\left\\langle\\left\\{ A-\\langle A\\rangle ,B-\\langle B\\rangle \\right\\} \\right\\rangle \\right|^2} ,", "26ca11ec1b23655f704c7d7e416aec98": "G = 1-2 \\left (\\int_0^1L(F)dF \\right ) = \\frac{1}{2\\alpha-1}", "26ca6d2e27d6a9257723c40e719d666c": "\\begin{align}\n V_n(R) &= (2\\pi) V_{n-2}(R) \\cdot \\left(-\\frac{R^2}{n}(1 - (r/R)^2)^{n/2}\\right)\\bigg|_{r=0}^{r=R} \\\\\n &= \\frac{2\\pi R^2}{n} V_{n-2}(R),\n\\end{align}\n", "26ca710c9dfcd9f91f257f0461f3a91b": "\\{1,x^{s_1},x^{s_2},x^{s_3},\\dots\\} \\,", "26cab0e72cc514a66a509d0adbad194c": "c_i = \\frac {{c_{i,T_0}}}{{(1 + \\alpha \\cdot \\Delta T)}}", "26cab7402a83eb4790ba2b807b1879ce": "\\scriptstyle S^1\\times I", "26cac5fc2feaa56340efdfa429cc4cb1": "b \\in \\mathbb{R}^m", "26cacd725a4e95b6e717929764a27e50": "A = sd.\\,", "26cb488b84bccd0e8277e1b66456c0f6": "\\frac{e^{\\mu z}\\gamma^\\lambda}{(\\sqrt{\\alpha^2 -(\\beta +z)^2})^\\lambda} \\frac{K_\\lambda(\\delta \\sqrt{\\alpha^2 -(\\beta +z)^2})}{K_\\lambda (\\delta \\gamma)}", "26cbc9fec2e36b71cf6b0cda07107566": "\\kappa^{-1}R^2", "26cbcef7525376da91e891f75ee6357b": "\\eta(z) = \\sum_{n=1}^\\infty \\chi(n) \\exp(\\tfrac{1}{12} \\pi i n^2 z),", "26cbf6ff44380f11896307ab788d49ed": "\\frac {p}{1 + e}", "26cc56dfd078a74cf504b62440f829d0": "J_\\nu(z)=\\sum_{k=0} (-1)^k \\frac{z^k}{k!} I_{\\nu+k}(z);", "26cc67f35fa2131b6387e660c283a103": " \\Omega^1={\\Bbb C}.{\\rm d}x,\\quad ({\\rm d}x)f(x)=f(x+\\lambda)({\\rm d}x),\\quad {\\rm d}f={f(x+\\lambda)-f(x)\\over\\lambda}{\\rm d}x", "26cc860955c9d73a7e0b449459fa79ae": " k = \\frac{2 \\pi}{\\lambda} = \\frac{2 \\pi f}{v} = \\frac{\\omega}{v},", "26cc864f03d3833569b0ffeeee578ef0": "\\bigvee\\varnothing=0", "26cc89ff187186bcfacb69de349e397b": "w (\\mathbf{R})", "26ccb135f4ce36c2013e6a6ee083bb37": "-0.7,", "26cd3d4129c87f3cabb842b12d3be904": "W(L,t)", "26cd7d5932f7bf04540ea95da3877be4": "\\Phi_3(x)=x^2+x+1", "26cd813845709cc1954aef23d6038bbb": "n_\\mathrm{A} = n_\\mathrm{A*} \\frac{R_\\mathrm{A*}-R_\\mathrm{A*B}}{R_\\mathrm{A*B}-R_\\mathrm{B}} \\times \\frac{R_\\mathrm{B}-R_\\mathrm{AB}}{R_\\mathrm{AB}-R_\\mathrm{A}}", "26cdb8feaa3c5ba862ba03289b2571fc": "\n \\begin{align}\n x_1(t) &= \\log\\frac{(\\lambda_1-\\lambda_2)^2 (\\lambda_1-\\lambda_3)^2 (\\lambda_2-\\lambda_3)^2 a_1 a_2 a_3}{\\sum_{j 0\\!", "26d27c7fadca9cbd95de96a0880130e4": "\n\\bar{\\phi}_{\\bar{x}\\bar{x}} + \\bar{\\phi}_{\\bar{y}\\bar{y}} + \\bar{\\phi}_{\\bar{z}\\bar{z}} = 0\n", "26d2be6c9e8375aea57694a9cdf89831": "\\lambda_f\\otimes\\cdots\\otimes 1 + \\sum_{i=2}^n 1\\otimes\\cdots \\otimes\\triangle_i\\otimes\\cdots\\otimes 1,", "26d2f491715729c3072cc8a7fd245173": " \\mathbf{C} = \\frac{\\mathbf{b}\\times\\mathbf{d} - \\mathbf{b}\\times(\\mathbf{b}\\times\\mathbf{d})}{2\\mathbf{b}\\cdot\\mathbf{b}}.", "26d30a2131b371f4322b9bd6723145e3": " H(f)=(x_{i+1} \\partial f/\\partial x_{i+1} - x_{i-1} \\partial f/\\partial x_{i-1})x_i \\partial/\\partial x_i +\n(y_{i-1} \\partial f/\\partial y_{i-1} - y_{i+1} \\partial f/\\partial y_{i+1}) y_i \\partial/\\partial y_i ", "26d34f122c7d90a3b9b52a8fe00c7c92": "X \\times I", "26d3619925e090880f2e99d7da74f696": " A \\subseteq B \\implies B \\subseteq B^* \\subseteq A^* ", "26d37a8a294d396b98d9f1cfec827623": "\\begin{align}\n\\underline{\\int_{a}^{b}} cf(x) &= c\\overline{\\int_{a}^{b}} f(x)\\\\\n\\overline{\\int_{a}^{b}} cf(x) &= c\\underline{\\int_{a}^{b}} f(x)\n\\end{align}", "26d3c40a1487faf12fe2b7fae709a426": "\\sqrt 6", "26d3db983b439aa3b433b487a41999b8": " \\mathrm{E}(T) = -\\frac{1}{\\theta},\\quad \\mathrm{var}(T) = \\frac{1}{\\theta^{2}} ", "26d448d6cc5261500f4ae877a1fd27c9": "y = -x-1", "26d46389ba7ae04c0fa330e7dd7f88fc": " \\sigma_{y} = \\sigma_{y,0} + {k \\over {d^x}} ", "26d4795ee1f02913f0dd9c0f4a0a68f5": "(h=2\\,r)", "26d47f8115ac4e9c767b6961ad3fdad0": "g_{j+1}[n]", "26d4848460d3d1e96ed7052c1ae9b421": "=\n\\zeta(s+1)\n\\left(\nn+\\tfrac12+\n\\frac{U_2(n)}{2^{s+1}}+\n\\frac{U_3(n)}{3^{s+1}}+\n\\frac{U_4(n)}{4^{s+1}}\n+\\dots\n\\right)-\n\\tfrac12\\zeta(s)\n,\n", "26d49e1d13f868ac8cb28218942077a7": " \\phi(x) ", "26d4ce98ec92e12a3f71a50aa6b3978e": "d(u, v) \\leq 1,", "26d4dd4dcc849e48edffbc71442271de": "\\ G_f(V,E_f)", "26d4e0bfb74c9e35e3da2bd2c88b8235": "P(1)", "26d5074361e2c4b2188df4fa71f51625": "\\lim_{n\\rightarrow \\pm\\infty}f^n(x)=p.", "26d5579325275b5591a648f133dc2805": "Q\\;=\\;C\\;A\\;\\sqrt{\\;2\\;\\rho\\;P\\;\\bigg(\\frac{k}{k-1}\\bigg)\\Bigg[\\,\\bigg(\\frac{\\;P_A}{P}\\bigg)^{2/k}-\\;\\,\\bigg(\\frac{\\;P_A}{P}\\bigg)^{(k+1)/k}\\;\\Bigg]}", "26d55e289e10e467792f12ed2ece04ad": "p(1-p)\\,", "26d57c4ab1ccef5f55f53712d50ae45f": "rm = 0", "26d5cf1de3b457150077701d163bcaca": "x,t", "26d60f2701dae40fe225d043c1197f3d": "k_{2(i)}", "26d6109507c3f0ab30dfe509458ad8ec": "T_xf(C)\\subseteq T_xX", "26d615700ddecfc9c12815300cf35b5a": "\\Phi.", "26d63478e01213a317db1123c14d2759": "y=f(u)", "26d727f0c595159b3cbe27eaf91ee461": "\n\\mathbf{P}(t) = [A(t)]\\mathbf{p} + \\mathbf{d}(t).\n", "26d736f90c5c8777f42c5ef00164ab33": "B_S = \\{U\\cap S : U \\in B\\}", "26d7886bece13f655ba16f1b580ccee4": "\n\\hat{\\mathbf{x}}_1 \\cdot \\hat{\\mathbf{x}}_2 = 0\n", "26d7a2da120e72d0bef2f7ee4ffed5a8": "K(X_i\\mid\\pi_i)", "26d82089ceaf9fdcae25102efdbe2ec8": "(1 - B^{-P}) (B^{U + 1})", "26d8388f3b948b6afd966d17039c60d6": "\\mathcal{E}(\\rho_{AB})= \\mathcal{S}(\\rho_A)= \\mathcal{S}(\\rho_B)", "26d8432ab80efedc6705a4391a640bd4": "r_n\\leq x < r_n+\\frac{1}{10^n}.\\,", "26d87e0b93f554d1157369a93d82deb1": "v_{total} = H_0 d + v_{pec}", "26d8e7f18b1587460a6128c30498b0a8": "\\gamma_c(A) = {(2^\\rho - 1)d^2_\\min (A) \\over 6E_s}.", "26d8f34da292c6bb1aeea8ae6760f4a6": "\\sigma(A+B)\\geq \\alpha+ \\frac{\\alpha(1-\\alpha)}{2k}\\,,", "26d9169164af04d3c5ab84c723de98c3": "m_{k+1} = \\tau_k \\beta_k \\Delta_k", "26d943309f5ca3185a957a4a631a17d8": "{[f(x)]}^{g(\\theta)} = e^{g(\\theta) \\ln f(x)}", "26d9442dae1586280a0e17c6cd2ac7ac": "a < b \\Leftrightarrow \\ln(a) < \\ln(b).", "26d94cc65e4e14b14f09113c488e0eb1": " p_{\\mu} = \\left(E, -p_x, -p_y, -p_z \\right) \\,", "26d9b164547b8f417e4561efe013d42b": "\\int_{C_{0}} \\mathrm{D} \\varphi(x) h(x) \\, \\mathrm{d} \\gamma(x) = \\int_{C_{0}} \\varphi(x) (A h) (x) \\, \\mathrm{d} \\gamma(x),", "26d9c38cba4348db478387e6ba4df358": " \\frac{S(x+t)}{S(x)} = S(t). ", "26d9c9b3e0acaef0e8f973dbcc055046": "\\Rightarrow v_j' M v_i = \\lambda _i v_j' v_i", "26d9d09111fcd16eabfe2f553b840d48": "n r(t)", "26d9dd2238fde5789c22fbe102d3262e": "2^{9/12} = \\sqrt[4]{8}", "26da0832f0a39ad00d5befb392a81764": "\\tilde{\\mathbf{x}}_{3,4} = \\left( \\pm q_+, - \\frac{\\alpha}{q_+}, 1- \\left(1-\\frac{\\gamma}{\\alpha}\\right)q_+^2 \\right)", "26da3320a69e7f68eb6c4061fd13cb21": "PV\\,=\\,\\frac{C}{i}. \\qquad (2)", "26da3a9a79128264eabaabf257bb7cd3": "f(n) \\not = \\phi_n(n)", "26da9e2accb5c9a9be65d1ff036f715b": "f(x,y,z)= \\ 2x+3y^2-\\sin(z)", "26daa7dea8a6c6a8543e2c964eaa7a6c": "2^{(\\log n)^{1-\\epsilon}}", "26daabd2963ff57c48744a708bef63a2": "x = 5 \\pm \\sqrt{7}.\\,", "26db24f52105acab7b07d94e0ef1e319": "S= K\\cap T = T^\\tau.", "26dbf4eb93bf6a7c84e4d2c5c22ef524": "|U(k)-I|=0.\\,", "26dbfe6ca9b4590eef5c40f1ac6f265c": "Q_v = 12400, - 2,100d^2", "26dc40286f3faad495be165e1a40bd43": "m_1=\\sin^2\\left(\\frac{\\pi}{2}-\\alpha\\right)=\\cos^2 \\alpha.\\,\\!", "26dceb9d876229ed4560097da466bcc2": "\\text{Crosswind} = \\sin[60^\\circ] \\cdot 15 \\mathsf{knots} \\approx 13 \\mathsf{knots}", "26dd141ec4710aa4acc87428c8f0e613": "\\det(\\mathsf{A}-\\lambda\\mathsf{I}) = 0", "26dd27be09deef4793b9b8b0571e1632": "2 \\times 3 = 6\\,", "26dd69250fbb5d298abf8614c9a22a6d": "\n\\theta_\\mu=\\mathrm{Arg}\\langle z \\rangle = \\mu\n", "26ddb74b05bfc0894ff60f81ce6e046e": "\\det(\\Delta_n) > 0\\ \\mathrm{and}\\ \\det\\left(\\Delta_n^{(1)}\\right) > 0", "26ddda519f97d1507efe14ad5c810730": "\\scriptstyle G_{ab}", "26de084b9f19f609a5d7f3886820a922": "\\mathbf{x}^k", "26de7de5687111d53bd53ed4d710e1f1": " r \\to \\infty\\,\\!", "26de8b574a447838373c78574c0ab451": "x_{n/2}", "26decdd7aea1e4e23bbfa64044fc831a": "\\frac{\\partial u}{\\partial t} = \\alpha \\left({\\partial^2 u\\over \\partial x^2 } + {\\partial^2 u\\over \\partial y^2 } + {\\partial^2 u\\over \\partial z^2} \\right) + \\frac{1}{c_p\\rho}q.", "26df04bf3fef7ea19d441d2dc9b48e0f": "R_p = k_p \\cdot [M] \\left( \\frac{f \\cdot k_d \\cdot [I]}{k_t} \\right)^{1/2}", "26df06ede6ad4afb7342e5814dece2de": "H = \\int d^3x { N \\over \\sqrt{det (q)} } \\Big( \\tilde{\\pi}^2 + \\tilde{E}_i^a \\tilde{E}^{bi} \\partial_a \\varphi \\partial_b \\varphi + det (q) V (\\varphi) \\Big) + N^a \\tilde{\\pi} \\partial_a \\varphi", "26df5203238f27fc19ee78dec8fd77bb": "\\frac{d}{dt} \\begin{bmatrix}\nu \\\\\nv \\\\\nw \\\\\n\\end{bmatrix} = \n\\begin{bmatrix}\n0 & -\\delta & 0 \\\\\n\\delta & 0 & \\kappa E \\\\\n0 & -\\kappa E & 0\n\\end{bmatrix}\n\\begin{bmatrix}\nu \\\\\nv \\\\\nw \\\\\n\\end{bmatrix}\n", "26df63a08a30deb38870e7ea7c000f55": "\\exists \\bar{x}. \\forall y. y \\prec \\bar{x} \\leftrightarrow F(y)", "26df6ea796628ea196fb25f8d9869493": "V_{O_2}", "26dfca3201528440a91f205ec1e13c99": "\\frac{3^3 2^0 + 3^2 2^3 + 3^1 2^4 + 3^0 2^6}{2^7 - 3^4} = \\frac{211}{47}", "26dfdc5e5326c37114784ed8d6951f8c": "n = n_1 N_2 + n_2 N_1 \\mod N,", "26dfe60a54a40b3dd014993eefcd4c0e": "s \\in \\prod_{i=1}^l \\Z^k", "26e01c7bb3dc63aad40ef5438cecaef1": "A_i^\\lambda", "26e0435e3126a2b9e3701bd9c2bfbea4": "y^2-az^2=P(x), \\, ", "26e043923c7b7a8f24d0b0125f3f623a": "B\\rightarrow A:\\{N_B\\}_{PK(A)}", "26e045cfb77418927feb2515d43e1887": "A_{T}", "26e05bf63d1897a5effec88e8d98b303": "t < \\frac{n}{4}", "26e07223e303dcde222bc539155d0e21": "A^{c}", "26e0e053f40e15adf2d2d1f0d1c00ccf": "\\left\\{ {A^i} \\right\\}_{i =1}^n ", "26e10b64500f0121685a2b920221c96c": "q=\\frac{1}{2}(1+3w)", "26e1370d041e272b870b0e96bb172882": "\\triangleleft \\!\\,", "26e1fccd2f6bdd7343c15dc6ee7b1e6c": " I_p = 0.5 ( \\frac { I_d - 1 } { M_u - 1 } ) ", "26e209bc935a766e62f7ceca48094f01": "S\\propto\\nu^{-\\alpha}.", "26e26fea3e978ec91940d8c7ca593edf": "x = a \\frac {\\sin [m p + \\theta_0] \\cos n p}{\\sin [(m - n) p + \\theta_0]}\n = a + a \\frac {\\cos [m p + \\theta_0] \\sin n p}{\\sin [(m - n) p + \\theta_0]}\n = {a \\over 2} + {a \\over 2} \\frac {\\sin [(m + n) p + \\theta_0]}{\\sin [(m - n) p + \\theta_0]}\\!\n", "26e28bece6aace617f417e0b6ec5cad3": "(A\\mid(B\\mid C))\\mid[([(B\\mid D)\\mid(A\\mid D)]\\mid(D\\mid B))\\mid((C\\mid B)\\mid A)]", "26e291b3c44b02ce7303d5e08227534f": "\\mathrm{Offensive Rating} = \\frac{\\mathrm{Points Scored*100}}{\\mathrm{Possessions}}", "26e2bcd4bfc1c7b297b775e6389cb94d": " \\frac{\\pi}{4} = 12 \\arctan\\frac{1}{49} + 32 \\arctan\\frac{1}{57} - 5 \\arctan\\frac{1}{239} + 12 \\arctan\\frac{1}{110443}", "26e3d8bac39f9313d584a6025bd7544d": "_2", "26e3dcf458e2d4832999973519646098": "x_2=\\frac{1-\\sqrt{-31}}{4}.\\,\\!", "26e5003e405d5a92b49cb064a2b2b783": " a(u,v) = f(v) \\quad \\forall v\\in V,", "26e523249449fcf19edb3f4eccc6bd38": "z = \\min \\{c(x) : x \\in X \\subseteq \\mathbf{R}^{n}\\}", "26e52f01a8e6c1d3a243c6d26ee449f7": "\\chi(\\mathbf{R}) = e^{-\\sqrt{k}R^{2}}.", "26e59f612b72670f328973e0245e0645": "\\frac{I}{4\\omega^2}.", "26e5f521c0a935db49eb5c7312aebd6b": "\\ Y=AK^\\alpha L^{1-\\alpha}", "26e5f67a2917800ec75553fa3fee7d22": "\\Delta Q = 0", "26e616fbee356831cc434555b6a3dd00": "R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\\cdots+b_1x+b_0", "26e634477c7a1285bb21c5df84371894": "k_{i}", "26e658af79b935eeadbe4520e7b4ca16": " P(X=4) = f(4;50,5,10) = {{{5 \\choose 4} {{45} \\choose {6}}}\\over {50 \\choose 10}} = {5\\cdot 8145060\\over 10272278170} = 0.003964583\\dots. ", "26e6e1759b0f8ac1c98af0938476eb50": "\\lim_{r \\rightarrow 1-} (1-r) \\sum_{n=1}^\\infty \\frac{n a_n r^n}{1-r^n} = A . \\, ", "26e6e8e39fb2a664c38774cbea8a5885": "q_z", "26e6ebbb242690b03732a7dced6967bb": "\\dot{v}_4 = {1 \\over C_4} ({{v 1 - v 4} \\over R_6} - {{v 4 - v 3} \\over R_2} )", "26e78e5fcf969e68fec94e6f5d5a8a01": "\\prod_{i \\in I} A_i", "26e7bf580eac82cf856068189f3fafa0": "\\scriptstyle \\hat{\\mathbf u},\\, \\hat{\\mathbf v},\\, \\hat{\\mathbf w}", "26e7da6f5d5897f027a0d18f0965d342": "opens", "26e858b6e02518cfb1d28704e85bdf56": "Xf(y)={d\\over dt} f(y+tX)|_{t=0}.", "26e8b3a8726777e943bef7f3ebc86dbc": "S \\circ op \\circ \\overline{op} = S", "26e8eda7095a007887bdedb5c6a5ff62": "v(x) = -4x+12", "26e8f04ccedbe8469265e977baca5e67": "\\rho_{X_iX_j\\cdot \\mathbf{V} \\setminus \\{X_i,X_j\\}} = -\\frac{p_{ij}}{\\sqrt{p_{ii}p_{jj}}}.", "26e9a323631bf4ba50594e52da23544c": "= 1", "26e9b2bbf149fd536bd0cd06ed99dcc6": "\\tau_{W}(m,\\mu;\\nu)= -\\mathrm{sgn}\\langle y,m\\rangle s(\\langle x,m\\rangle,\\langle y,m\\rangle)+\\mathrm{sgn}\\langle y,\\mu\\rangle s(\\langle x,\\mu\\rangle,\\langle y,\\mu\\rangle)+\\frac{(\\delta^2-1)\\langle m,\\mu\\rangle}{12\\langle m,\\nu\\rangle\\langle \\mu,\\nu\\rangle}", "26e9fe8b4f05888550a5fb7d50a7fbea": "\\overline{V}=V", "26ea01bef16e9643cc3c62de2bfb40bc": "\nR = \\frac {\\log M}{n}\n\\,\\!", "26ea1866bf5112e2e317ebc1faedf882": "n_1=1", "26ea89aab768e8af4da1a7d15c73ca32": " E[\\delta_i^2] = \\sigma^2 ", "26ea9f584282d16897a67849d52c7e1b": "Q_t \\cdot Cv_{O_2}", "26eac4311a8a8fd7faf2867e6dfc1379": "\\chi _{m}=\\frac{\\partial M}{\\partial H}=\\frac{N_{\\text{A }}}{3k_{B}T}\\mu _{\\mathrm{eff}}^{2}\\text{ ; and }\\mu _{\\mathrm{eff}}=g_{J}\\sqrt{J(J+1)}\\mu _{B}", "26eb916c153b5db8077c9661a02db25b": "\\left(A_z\\right)_{m'n',mn} = \\delta_{n'n} \\left(J_z^{(m)}\\right)_{m'm}\\,\\quad \\left(B_z\\right)_{m'n',mn} = \\delta_{m'm} \\left(J_z^{(n)}\\right)_{n'n}", "26ebcd8b76999b03bb99289a7b6442ac": "\\begin{align}\nM_x \\beta = - \\beta M_x\\,, \\\\\nM_y \\beta = - \\beta M_y\\,, \\\\\nM_x^2 = M_y^2 = M_z^2 = I\\,, \\\\\nM_x M_y = - M_y M_x = {\\rm i} M_z\\,, \\\\\nM_y M_z = - M_z M_y = {\\rm i} M_x\\,, \\\\\nM_z M_x = - M_x M_z = {\\rm i} M_y\\,.\n\\end{align}", "26ebf6664153aeafc5dce5630c0a7c04": "\\begin{align}\n \\tan x &= 1\\frac{x}{1!} + 2\\frac{x^3}{3!} + 16\\frac{x^5}{5!} + 272\\frac{x^7}{7!} + 7936\\frac{x^9}{9!} + \\cdots\\\\\n \\sec x &= 1 + 1\\frac{x^2}{2!} + 5\\frac{x^4}{4!} + 61\\frac{x^6}{6!} + 1385\\frac{x^8}{8!} + 50521\\frac{x^{10}}{10!} + \\cdots\n\\end{align}", "26ec68acfc259413ea1f513bfce572f9": "n(\\omega_{k,s}) = \\frac{1}{\\exp(\\hbar\\omega_{k,s}/k_BT) - 1}", "26ec6b68521a53951c7f1c0028226757": "\n\\rho(q) = \\sum_{n\\ge 0} {q^{2n(n+1)}\\over \\prod_{1\\le i\\le n}(1+q^{2i-1}+q^{4i-2})} \n", "26ed352f3dcaf6fc8e653ab91a56cb1d": "A_{n}(x) = \\sum_{m=0}^{n} A(n,m)\\ x^{m}.", "26ed85588662bf409ab009d1d739368b": "\\mu : A \\otimes A \\to A", "26edac09bc64d341c7f2b0e3652e3bc8": "\\alpha_1", "26edbf29b7347287ca128d904d033598": "\\phi^2 = \\frac{\\chi^2}{n}", "26ee0751225c6ba17dedccc89db80797": "\n\\begin{align}\nG_2 & = \\frac{k_4}{k_{2}^2} \\\\\n& = \\frac{n^2\\,((n+1)\\,m_4 - 3\\,(n-1)\\,m_{2}^2)}{(n-1)\\,(n-2)\\,(n-3)} \\; \\frac{(n-1)^2}{n^2\\,m_{2}^2} \\\\\n& = \\frac{n-1}{(n-2)\\,(n-3)} \\left( (n+1)\\,\\frac{m_4}{m_{2}^2} - 3\\,(n-1) \\right) \\\\\n& = \\frac{n-1}{(n-2) (n-3)} \\left( (n+1)\\,g_2 + 6 \\right) \\\\\n& = \\frac{(n+1)\\,n\\,(n-1)}{(n-2)\\,(n-3)} \\; \\frac{\\sum_{i=1}^n (x_i - \\bar{x})^4}{\\left(\\sum_{i=1}^n (x_i - \\bar{x})^2\\right)^2} - 3\\,\\frac{(n-1)^2}{(n-2)\\,(n-3)} \\\\\n& = \\frac{(n+1)\\,n}{(n-1)\\,(n-2)\\,(n-3)} \\; \\frac{\\sum_{i=1}^n (x_i - \\bar{x})^4}{k_{2}^2} - 3\\,\\frac{(n-1)^2}{(n-2) (n-3)}\n\\end{align}\n", "26ee4310fbee994ad4d3751e2b8bd3f8": " U_n(R) = 0 = A_nJ_0 \\left( \\alpha n^{1/2}i^{3/2} \\right) + \\frac{iC_n}{\\rho n \\omega}\\, .", "26ee4c3a61a5d968fb1d07411d44c764": "t_1, t_2, \\dots, t_n,", "26ee688d727ea0c771dbdf3f456895bd": "d_{2}", "26ee7195db770c6621a9090b7adc6c83": "f\\left(\\mathbb{N}_1\\right)\\subseteq A", "26ee7f33cb176694db35573ad5220b3b": "z = \\frac{Z}{X+Y+Z} = 1 - x - y", "26eeda31559ef28700564b8fb5bca59b": "P_{TNL}", "26ef0f8a1ea18d2fe5d2e63e5df2f01a": "\\bold{M} = \\chi_m \\bold{H} \\,", "26ef35925b56766a1a91018a9a3e7a2d": "{n^2} / 8", "26ef9a600660c8eff36adcc3e88179f0": "\\partial M ", "26efc5bfcfb2548faba05e445f18d5c3": "{}_2F_1 (a,b;c;z) = (1-z)^{-b} {}_2F_1 \\left (b,c-a;c;\\tfrac{z}{z-1} \\right )", "26efd2da36397fbf91a9d93c63c3e00c": "\\{a_1, \\ldots, a_k\\}", "26effacf274c4c68c2ac4cb31f3a2827": "k_0\\neq 1", "26f08bbd852c1de2a68309f4e67a29c5": "\\epsilon \\sigma T", "26f0c21bb5156021bdb7ada4ded246f2": "\\int\\arcsin(x)\\,dx=\n x\\arcsin(x)+\n {\\sqrt{1-x^2}}+C", "26f0c4532677d807b3fd5e263f9bb071": "=\\frac{\\textrm{Li}_{\\alpha}(z)}{\\zeta(\\alpha)}\n\\,\\tau^\\alpha", "26f0dba3deea96431da624e45d46ecf1": "\\dot{u}_a \\ll \\omega u_a", "26f10ff3baf0882fc83c7354aa92ff46": "\\mathbf{*3\\cdot24}. \\ \\ \\vdash. \\thicksim(p.\\thicksim p)", "26f14a67fda683a9425a3b4f481a501d": "\\ M_y = \\frac{W - u_1 + u_2}{a_2}.", "26f167f212bd8f42ae4b8ee42e3ddf14": "\n \\begin{bmatrix}\n a_{11} & a_{12} & a_{13} \\\\\n a_{21} & a_{22} & a_{23} \\\\\n a_{31} & a_{32} & a_{33} \\\\\n \\end{bmatrix} =\n \\begin{bmatrix}\n l_{11} & 0 & 0 \\\\\n l_{21} & l_{22} & 0 \\\\\n l_{31} & l_{32} & l_{33} \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n u_{11} & u_{12} & u_{13} \\\\\n 0 & u_{22} & u_{23} \\\\\n 0 & 0 & u_{33} \\\\\n \\end{bmatrix}.\n", "26f179bae6dfee8b90dd5ef2e3b808b1": "\\left.p_0(\\vec{r})=\\int_{\\Omega_0} \\frac{d \\Omega_0}{\\Omega_0} \\left [2 p(\\vec{r_0},v_s t) - 2 v_s t \\frac{\\partial p(\\vec{r_0},v_s t)}{\\partial (v_s t)} \\right]\\right|_{t=|\\vec{r} - \\vec{r_0}|/v_s},\\qquad \\quad(4), ", "26f1993b461107644b85c8f4d28b5f4b": "\\log^{-1}(\\delta^{-1})", "26f19ca14f39a3efc5c38f244ec3f59e": "\\epsilon_i=-\\frac{1}{\\beta}\\ln{\\eta_i}", "26f1c92918f36841630308e55602e2d8": "\\langle c(\\mathbf{x},\\mathit{t})\\rangle = \\sum_{i=1}^{m}\\boldsymbol{\\psi}_i(\\mathbf{x},\\mathit{t}) ", "26f1e7f13d94b8ab0fdee1e67877e845": "\\scriptstyle\\tau_p^n(x)", "26f27b8985b5109ed7b21507986e5502": "a^*_b = \\varepsilon_r\\left(\\frac{m}{\\mu}\\right) a_b", "26f292d0f3f53c5577abe946d98ff641": "\\textit{closedoor}(t)", "26f2ab25e693b6175bd49632afaadb9b": "|G_{\\nu}( \\pi)|=1", "26f32361e1d541b933f6f2c2f5c8ba7f": "E=\\angle zcx", "26f33cb7f2916ef99e7f93e87e853d5f": "\\frac{d\\beta}{dt}=\\frac{Y_\\beta}{mU}\\beta-r", "26f35c46cb8057f1f68f51cfd52bdd8e": "u_1, \\cdots, u_n", "26f36f9158a9dccd6efb839b110f1253": "C(u,v)=-C(v,u)^{}_{}", "26f38028f22c5673f7d44b957f23d7ed": "(-\\infty,\\theta].", "26f3ca65cbccdbec877f1b8557519832": "\\mathbf{n}\\,", "26f3ce86edceef0706a016012d3ee38c": "{4}a_{A}\\frac{((A/2)-Z)^{2}}{A}", "26f3f8c54e57639ba8e1f347da7d53bb": "[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0.\\,", "26f40fbd8dc75261c300e264c47175a8": "\\left[\\frac{L}{p}\\right]_3 = \\left[\\frac{M}{p}\\right]_3 =1", "26f417d9ce1533efeac796cfb03a5782": "\\Pr(\\chi^2_{g-1} \\ge K)", "26f439f8d5cf82407894e3c041b59c3f": "f(x;r,\\lambda)= \\lambda/\\Gamma(r) (\\lambda x)^{r-1} \\mathrm e^{-\\lambda x} I_{[0,\\infty]}(x)", "26f44b68e6ccc6644cc407025bf3f6a8": "\\Upsilon_i", "26f46941f67564404a18a8481513ff29": "\\{P(x_1, \\ldots, x_l)\\}", "26f4c8246f090e860ce12422f4ab12da": "X \\in \\mathcal{X}", "26f516d40acc1aa13d7eaabefd8f0362": "\\scriptstyle 2 \\pi f t \\,+\\, \\varphi", "26f54cdc052769bd3b084de1248f6807": "\\aleph = 2^{\\aleph_0}", "26f55b1ee3c8ccfeaa610c1279a94414": "\\mathcal T \\left\\{A(x) B(y)\\right\\} := \\theta (x_0 - y_0) A(x) B(y) \\pm \\theta (y_0 - x_0) B(y) A(x), ", "26f5bfd7253d740dc120503f0593f5c4": " \\left(\\frac{a}{2}\\right) = \n\\begin{cases}\n 0 & \\mbox{if }a\\mbox{ is even,} \\\\\n 1 & \\mbox{if } a \\equiv \\pm1 \\pmod{8}, \\\\\n-1 & \\mbox{if } a \\equiv \\pm3 \\pmod{8}.\n\\end{cases}", "26f5dc5cc334cfdc99bf2b0dab015b54": "2eV=\\hbar\\frac{\\partial \\phi}{\\partial t}", "26f5de97c92b0973d3e1f71e08892164": "h(x; t)", "26f629916ff48f187f646ab41287e184": "x_{n+1} = f(x_n)", "26f67b2c0c8c620904a393aea234c7f2": "\\kappa(y)", "26f6b925a99ff4e977a40898337267bc": "n=1, \\, 2, \\, 3\\, ...", "26f6c03682a499770d1fe17ce4e5bced": "\\mathbf{R} = \\sum_{i=1}^{N} \\mathbf r_i", "26f6e4f6e647998ec07a536f0f1c2767": "K^*=K", "26f6f8c5261675473f6d17fbcade95c8": "\\dot{x} = f(x) + \\sum_{j=1}^mg_j(x)u_j ", "26f740429f9b45dc49585a246d5e496a": "Cl_X(S)", "26f74836981a83726edf260d34c20733": "\\Phi : M \\to \\operatorname{Hom}_{R} (N, L) ", "26f78c73c39a57184aeb02a8c1c3b5bf": "P(in)", "26f7a6df39a1f4fb15523a6acecf34ee": " [\\Gamma L,\\Gamma L]\\subset \\Gamma L", "26f7b570ed83fedde09930e84d1e8824": "71 \\cdot 36 \\cdot 990 = 2,530,440\\,", "26f83ce8b1b8d64fcbb6f763641a0392": "\\mathcal{H}_{C}", "26f859a24717c23cbac7f287d83584e1": "F_{i} = 1+x^{2^i}", "26f8ce20c7d8193d0f4473f5a987fe0e": "\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\end{pmatrix}\n", "26f90c37a9bdc0c73da3e844a9e052ce": "B^2 - 4AC,\\,", "26f917583380478b1ed798e53434a501": "Q(X)", "26f996c53f99c3b39f3a2906a7837a06": "R_D", "26f9bf91d862ce576b3a38a3079f796b": "KS(x x) \\leq \\ell(x) + 66", "26fa1a688d94abb4aae215d3fce67334": "\\frac{\\lambda}{4!} \\phi^4", "26fa309b0588dc6624bcf44ee18e3fc3": " \\frac{dN}{dt} = r N \\left( 1- \\frac{N}{K} \\right)", "26fa6cd1d8fd5c5972bf80acf55081d2": "\\; S_R = k \\ln \\Omega _R", "26faa26d869d709e90181d129777daee": " \\lambda m,p,q.(\\lambda g.\\lambda n.(n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n)))\\ \\lambda x.\\lambda o.\\lambda y.o\\ x\\ y ", "26fad89e746b1074b40ac4b76d536f25": "E=hc G(v)", "26fafdbd7bbecc8e496fe34d922344dc": "C_\\mathrm{linear}", "26fb94687b8c2b9295c76f0cecd07084": "\\textstyle \\ell\\left( v\\right) =0", "26fbaba6b7470aeec676e1205227f869": " \\frac{ \\alpha - 1 }{ \\alpha + \\beta - 2 } \\le \\text{median} \\le \\frac{ \\alpha }{ \\alpha + \\beta } ,", "26fbb924fe45c49baea75a6d7b6652c5": "SO(10)\\supset SU(5)\\supset SU(3)\\times SU(2)\\times U(1)", "26fbef2fe823a0516645b0ac6846d8be": "m_\\mathrm{PbO} = \\left(\\frac{200.0 \\mbox{ g }\\mathrm{O_2}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{O_2}}{32.00 \\mbox{ g }\\mathrm{O_2}}\\right)\\left(\\frac{2 \\mbox{ mol }\\mathrm{PbO}}{3 \\mbox{ mol }\\mathrm{O_2}}\\right)\\left(\\frac{223.2 \\mbox{ g }\\mathrm{PbO}}{1 \\mbox{ mol }\\mathrm{PbO}}\\right) = 930.0 \\mbox{ g}", "26fc3f4f62cd5c0be10374b1dd7cf722": "p_M", "26fc41e0b44218380d2dad640c61a1b1": "(\\mathbf{0},0)", "26fc457eb3db576befaa90506e411a1b": "\\mu_1 = k_1 = a_1 +2a_2", "26fc6254e962f329dafd5cd72d2329d5": " \\sigma = \\frac{NPSH}{H} ", "26fc75e6623d39201877a23b2e314909": "V_f = \\frac{4}{3}\\pi p_{f}^3(\\vec{r}) .", "26fcc76c3fe0c945f54524665c04f685": "\n F^D = -2\\sigma_0 m^2a^2\\left[\\cos^{-1}\\left(\\cfrac{1}{m}\\right) + \\frac{1}{m^2}\\sqrt{m^2 - 1}\\right]\n ", "26fcdc804ab481605a2ee739330b9995": "B' \\subset B", "26fd32bc1691648785b96930792288da": " M_K = \\sqrt{|D|} \\left(\\frac{4}{\\pi}\\right)^{r_2} \\frac{n!}{n^n} \\ . ", "26fd5ec5b0177251689a2e5810aa18c1": "G = \\langle N, E \\rangle", "26fd608e1a013135e7286babdb72de9c": "\\sigma_{xy} = \\nu e^2/h", "26fd738f1230b209710db1faea32e532": "\n\\begin{align}\n\\left| \\mu-m\\right| = \\left|\\mathrm{E}(X-m)\\right| & \\leq \\mathrm{E}\\left(\\left|X-m\\right|\\right) \\\\\n& \\leq \\mathrm{E}\\left(\\left|X-\\mu\\right|\\right) \\\\\n& \\leq \\sqrt{\\mathrm{E}((X-\\mu)^2)} = \\sigma.\n\\end{align}\n", "26fd892b6929e8f16e658824838a12c1": "H = {50 \\choose 2n} \\times (2n-1)!!,", "26fda9fbf4b01770e9b045ce809b28ee": " \\dot m = \\rho AC ", "26fdae23a1790cbefc06c0bdfe8cdd8c": " v = \\sqrt[4]{ G M a_0 } ", "26fdcb0d3d378a325ce844bbd442d978": "J_0 \\supset J_1 \\supset \\cdots", "26fdd7d135ba9bcc508fb06ae1100cc7": " K_x \\;=\\; 0 \\quad \\text{ if and only if } \\quad f(x) = 0 \\quad \\forall \\; f\\in H. ", "26ff1202aeba011f3471243966982224": "z_1 = 0", "26ff2690c198b417816ba1316f38d1f7": "1.096854 \\pm 0.000004", "27000e61f0442c6ca3efc00d32188f5a": "A = - \\log_{10}\\mathcal{T}\\ = - \\log_{10}\\left({I\\over I_{0}}\\right)", "27005d51c739fde9326e42ca13589d00": "x_n x_m^{-1} \\in U", "27007f402ce8752fe81fd8151efbd8ef": " \\scriptstyle EAC = \\frac{NPV}{A_{t,r}}", "2700ac479255997949615e5b2dfc4fc8": " \\Delta g \\ne x_B - x_A ", "2700d7a0980474ab18cdaf8979f1f93d": "\\frac {33.33% + 50.00%} {2} = 0.417", "2700e8b3a4ec9965e6e7bc147689dcae": "\n\\left[ {\\begin{array}{*{20}{c}}\n \\bullet & x \\\\\n y & \\bullet\n\\end{array}} \\right],\\left[ {\\begin{array}{*{20}{c}}\n x & y \\\\\n \\bullet & \\bullet\n\\end{array}} \\right],\\left[ {\\begin{array}{*{20}{c}}\n y & x \\\\\n \\bullet & \\bullet\n\\end{array}} \\right]\n", "2700fc75fc6ed607525f27b462082ba0": "f:A\\rightarrow B ", "27011ac77ba433e0a02943702c0f5ae2": "x_i x_j = x_j x_i", "2701603777d27808f7b28936e2233991": "\\stackrel{\\mathfrak{p}}{}", "2701bdca83dc42ed0ced6e5975b65f43": "U'", "2701d73149129f48b690985a997d5d5a": "V_i \\rightarrow u_i", "27020fcb2803efe218bbb26fb4cb1424": "\n p^D(r) = \\begin{cases}\n -\\cfrac{\\sigma_0}{\\pi}\\cos^{-1}\\left[\\cfrac{2-m^2-\\cfrac{r^2}{a^2}}{m^2\\left(1-\\cfrac{r^2}{m^2a^2}\\right)}\\right] & \\quad \\mathrm{for} \\quad r \\le a\\\\\n -\\sigma_0 & \\quad \\mathrm{for} \\quad a \\le r \\le c\n \\end{cases}\n ", "27027264e544fa6cc41b3dd2d1d6bba7": "s_0 \\in \\Sigma^*", "270284e9d8219ea073f422bae443d856": " (s_k, t_k) ", "2702b4c643237cf1f8273fd1eddce998": "\\cos(\\pi/7)", "2702c04394874331ffacefa0ed3a6be5": " {s} = {M} {\\lambda} + {q} \\,", "2702d78abd1b0c358616113da1d117d3": "X|\\mathfrak{D}_0", "2702f9f9109bec7289a08bddd4952c8a": "\n r^2~\\cfrac{d^2R}{dr^2} + r~\\cfrac{dR}{dr} + \\cfrac{r^2\\omega^2\\rho_0}{\\kappa}~R = \\alpha^2~R ~;~~ \\cfrac{d^2Q}{d\\theta^2} = -\\alpha^2~Q\n ", "270314605c8f4802e903259100eaecae": "(1-L) X_t = X_t - X_{t-1} = \\Delta X. ", "2703290fbe8c589b13561a55a1cd8b97": "\\sigma (\\Omega) = -\\Omega", "27036c71ba699259df2d3c560f647675": "= \\frac{31!}{2}\\cdot 60^{31}", "27043cf2b1341d557147b4cd7024bb45": "Na|n\\rangle=(n-1)a|n\\rangle.", "270475d41d9194a65a211c637f6dd729": "r \\in I^{+}(q)", "2704a81a01e96b81a9dd3471ea90577d": " u_t+uu_x+(c^2/\\rho)\\rho_x=0 ", "270555012642b902e1336f99a32d1c25": "2x^2+3x+1 = 0", "27056a1579a7bb8ac91056d20f21ab60": "D(E(m) + e)", "27057a6556fecc0df357910852969a6f": "r^\\ddagger=\\frac{E_a}{m_1}+r_1", "2705d975e49eb93a40c99af99f893c94": "f(g(a + h)) - f(g(a)) = f(g(a) + g'(a) h + \\varepsilon(h) h) - f(g(a)).", "2705fdd6d92f123055bf01749569f715": "k(\\bar{y})", "2706a002ceff469953fef0e33cf59ef6": " |x| \\ll 1 \\ ", "2706bf6fba5d6d10a55e0e1e23ce8ab7": "S_{\\rm semi} = \\frac{\\sigma_{\\rm C} S_{\\rm C} + \\sigma_{\\rm V} S_{\\rm V}}{\\sigma_{\\rm C} + \\sigma_{\\rm V}}, \\quad \\sigma_{\\rm semi} = \\sigma_{\\rm C} + \\sigma_{\\rm V} ", "2706e644566c335e00fab859772f896e": "=\\mathbf{a}_{AB}\\ +\\mathbf{a}_B\\ , ", "27071423b9a403c9ffa1895ef3fefb6e": "c=1", "270723cdccc0f3dc59350031a1f16d2c": "((a \\vee n) \\wedge (\\neg n \\vee b))", "2707787db23e3a0c6c8e85807a6dd229": "\\langle a_1,a_2,a_3\\ldots\\rangle", "2707962508b28f176355391b6b8ccf59": "a^2 + |D|b^2 = 4N \\, ", "2707c95949326626c91b956e135e33ce": "\\psi_R=\\begin{pmatrix} I_2 & 0 \\\\0 & 0 \\end{pmatrix}\\psi,\\quad \\psi_L=\\begin{pmatrix} 0 & 0 \\\\0 & I_2 \\end{pmatrix}\\psi.", "2707cdce01d26f2442db43ae75a8a086": " {XX' \\over YY''} = {AX \\over CY}, ", "2708a7ce5aa8bd216396727399ff0e4c": "C^b", "2708deeea96bf410e230562632495ef7": "\\epsilon = 1+\\chi_\\text{e}", "2708e78d7629165e264d14628dd8a3fb": "\\displaystyle{[\\mathfrak{g}_p,\\mathfrak{g}_q]\\subseteq \\mathfrak{g}_{p+q}}.", "2708eef135e71e2e7226a20a3c04cdeb": "\\varepsilon_3''' = \\frac{1}{E}\\sigma_3", "2708f90a9e3c4dd1fb7c8a71dbfc9cbc": "i=(k^2+4ik-5)c_3+(-k^2+4ik+5)c_4", "27095c895734ad8643adc90f0d7ac56f": "S_i^+ \\longrightarrow S_i^+e^{i \\theta_i}", "2709665e0a487a0e385e332b0a7f27a7": "\\frac{1}{2}\\log(1+P/N)", "27099b4b2a33741eb51b64121b947485": "J=(1,1,1,...,1)^T", "2709b9a1bbb18623580ffc86a4bd452c": " F_B \\big ( u^* + \\frac{ \\partial u^*}{\\partial x} dx \\big ) - F_A u^* \\approx \\frac{ \\partial u^* }{\\partial x}\n\\sigma dV + u^* \\frac{ \\partial \\sigma }{\\partial x} dV = \\epsilon^* \\sigma dV - u^* f dV ", "270a5de65dedb213ba1ea386512f791c": " L[t] = T[t] - V[t] = {1 \\over 2} m \\dot{\\vec{x}}[t] \\cdot \\dot{\\vec{x}}[t] - m \\zeta [\\vec{x} [t],t] .", "270a67b48931b2e948a36c185afa99fc": "[I_C^B]=[Q][\\Lambda][Q^T],", "270a8f32a847e623f200eb1092806bf1": "\\varepsilon_{\\color{Orange}{\\color{Orange}{2}}\\color{BrickRed}{1}\\color{Violet}{3}\\color{RedViolet}{4}} = -\\varepsilon_{\\color{BrickRed}{1}\\color{Orange}{\\color{Orange}{2}}\\color{Violet}{3}\\color{RedViolet}{4}} = -1", "270aa4b97e71defc9ce4b8b761704a65": "A'_\\mu(x) = \\partial_\\mu f(x)", "270aae64e82be75653345dd6955a621a": " S(A,P,z) = X \\cdot W(z) \\cdot \\left({1 + O\\left((\\log z)^{-b \\log b}\\right)}\\right) + O\\left(z^{b \\log\\log z}\\right) .", "270ac04ef8471368d16d23db66e65ae8": "\\sigma = 1/\\sqrt{2 \\mu \\omega}", "270ada5119a0137c94b83acac6b5f5c5": "\\displaystyle{\\psi=T\\psi +{1\\over 2}\\psi -(\\lambda- {1\\over 2})S\\varphi=-(T\\psi -{1\\over 2}\\psi) +(\\lambda+{1\\over 2})S\\varphi}", "270b45447668908d2c43583dea36cf38": "\\displaystyle{H=A+iB.}", "270b567948a0aa5966f0202415c0c807": "\n\\varphi(x) = \\langle x_n^2, \\ldots, x_1^2, \\sqrt{2} x_n x_{n-1}, \\ldots, \\sqrt{2} x_n x_1, \\sqrt{2} x_{n-1} x_{n-2}, \\ldots, \\sqrt{2} x_{n-1} x_{1}, \\ldots, \\sqrt{2} x_{2} x_{1}, \\sqrt{2c} x_n, \\ldots, \\sqrt{2c} x_1, c \\rangle\n", "270b6bc72975b59f6a764e887522bc0a": "C = \\frac{f}{16} \\cos^2 \\alpha \\big[4 + f(4-3 \\cos^2 \\alpha) \\big] \\,", "270c10868af5da5d4ccd445df7379e42": "\\delta(X_1, X_2, \\ldots, X_n)", "270c2bdda0bc4086d603e6985fc854c5": "X_{k+1} = 2X_k - X_k A X_k.", "270c3ec8bb0d2394a93a44d22e2d9619": "G = S_0\\cup S_1\\cup S_2", "270c46da70a7caa1e4128dac9c67dd49": "a b^3 c^{-1} c a^{-1} c\\;\\;\\longrightarrow\\;\\;a b^3 \\, a^{-1} c.", "270c5326bef04fc8380220384f593f41": "\\hat{\\textrm{t}}_i", "270c64a980bc0e8443c55d3ccdfd652f": "A_{o}^{FI} = \\left( \\frac{ Total \\ Time }{Total \\ Time + Fault \\ Isolation \\ Down \\ Time } \\right) ", "270c6934da2438b1807df18f7f9981c1": "k\\propto\\exp\\left(\\frac{-\\Delta^\\ddagger G^\\ominus}{RT}\\right)", "270d34e5fd341708d731b9dcffdd465d": "\\,= \\mbox{R}(z, t) [\\mbox{R}(z, dt) - 1]/dt", "270d9abdf2eaff03751d5c8838706ec4": "S(t^-)", "270e333d86cd24092f053b605a5eb786": "\\Sigma_k^*", "270ed143b3fca7cdbaf9b684af626abc": "I = -qnv_xtW", "270edb587a53fe341c3fc3ee6ec6b089": "\\psi(x)=0", "270ee9c6bed722c22a54bd5d4025f1ec": "x*y = x+y-k", "270ef91f3c83cc0967b567d93b9e6e61": " \\operatorname{P}[E_1\\cap(E_1\\cup E_2)]=\\operatorname{P}\\left[E_1\\mid E_1\\cup E_2\\right]\\operatorname{P}\\left[E_1\\cup E_2\\right]", "270f2f3792ee9a744b0098929fdd1a50": "\\mathbf{x}= (x_1, x_2, ..., x_n)", "270f388654d61c59bcbe2339d97bfd31": "( \\sinh x )'= \\cosh x = \\frac{e^x +\n e^{-x}}{2}", "270f3adf243ab096a429fd145065be5d": "\\int_G F(g) \\, dg = {1\\over |W|} \\int_{\\mathfrak{a}} F(e^X)\\, |\\delta(e^X)|^2 \\, dX.", "270f4a678da5c90d5ab807ce09807149": "E = {\\Delta h_f+\\Delta h_m}", "270f7659761ea30f892569bde98806a8": "W:\\mathbb R \\rightarrow \\mathbb R^+ ", "270f8016114715c9801b2896d2378c24": "W = C_e A^T C_Z^{-1}", "270fe5f6ef0acb3e3f811e1423e023cd": "\\dot f", "271022ddb7e70d6a9628ee9bf8c20d42": "(A{\\mathbf x})_k = \\sum\\limits_{l=1}^n A_{kl} x_l", "2710235820156c8b766a3d96c015d5a4": "(p\\lor\\neg p)\\land(q\\lor\\neg q)\\land(r\\lor\\neg r).", "27104399a18330853a1eb19a96eff9da": "\\mu > 0", "27104f404ed2fb511d82f2fe8d6e2187": "\\sum_{n=0}^m 1_{\\{N=n\\}}E^n=\\prod_{j=1}^m(1_{A_j^{\\mathrm c}}I +1_{A_j}E)", "271088698edc1bb87a961f3e8695be43": "\\ F_t", "2710905829de4b69ace1e10ae40a753f": "\\scriptstyle(k-1)", "2710dc64d680e5f96cfb56d025717fe8": "T=\\frac{\\beta A}{\\rho c d^3}", "2710ed90aa9d1a6303e17239ce32c113": "\\mathcal{V}(x)", "271103eb78d216d814c50e82e661cc4c": "~T~", "27110e52d1e55c68aba818eb2e4bc5df": "c=\\nu \\lambda", "2711367841d9e106c018d446887951e3": "[x,y]=z,\\quad [x,z]=0, \\quad [y,z]=0", "2711ba43786e7fc2c890529be11dc6d9": "n_{\\mathrm{dB}}", "27122d9947ad2b554dcca97527e860dc": " = \\frac{4}{3} \\pi \\rho_p r_p^3 \\frac{V_t^2 }{r} .", "2712c102eb5f6ec3919e12f4bfd131e0": "\n{} - p^6 + 28p^4r- 16p^3q^2- 176p^2r^2- 80p^2sq + 224prq^2- 64q^4\n", "2712c552ef453cfd40b586fabbed0da2": "M_{(i,j),k_1} = M_{(i,j),k_2} = 1", "2712d70c435029c9b96cf7fe0b3a5af7": "\\hbox{ad}_g(R_g^*\\omega)=\\omega", "271333d7aa2bed9564a9281a698ab316": "z\\in\\mathcal{Z}", "27133668b08c8e94e71f4beddb4ab6e1": "|g_1\\rangle ", "27138029e6edfe27251aa78b745d042a": "Y=Z\\,\\!", "27141c6f42f5d19abb786e3a39c7f839": " N = N_0\\,e^{-t/ \\tau}, ", "2714380d2a3edbacf1f64854bc8992c5": "O(p*s)", "271476fdd7a1572b71f3b161ee4184c3": " F_\\mathrm b = \\frac {\\Delta \\omega}{\\omega_0} ", "27152207b5aed9d9f94787930812cdb1": "p = m v. ", "2715eef5eb01f99922e7aaa18fd7c6f5": "f^{(i)}, i \\in \\{0, ..., n-1\\}", "2715f1e2f6baa87dcdb8e860fe0ce260": "\\ln y = \\ln y_0 + rS + \\beta_1 X + \\beta_2 X^2", "271638047b21bec11fe5f12c96abea1a": "G(k)\\le k\\log k+k\\log\\log k+Ck.", "2716d3bb8e097bc71156dea958a479b3": "\\gamma_{\\text{m}} ", "27172809b72d72bfd29ecb9547047834": "\\mathbf{F}_l=-\\boldsymbol{\\nabla}\\Phi=-\\frac{1}{4\\pi}\\boldsymbol{\\nabla}\\int_V\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'", "27178175a444521f96e0ba9b070f3275": "\nm_\\mathrm{em} = \\int {1\\over 2}E^2 \\, dV = \\int\\limits_{r_e}^\\infty \\frac{1}{2} \\left( {q\\over 4\\pi r^2} \\right)^2 4\\pi r^2 \\, dr = {q^2 \\over 8\\pi r_e},\n", "271831f34e8f14fe4b59c4e82cce24e3": "T_{\\pi,\\lambda}:[0,1]\\rightarrow [0,1],", "271856303c9419b6ec323bc5529e6ac9": "at+b", "27188a376951ffc4d0134815805884ef": "\\beta_2^{(0)} = \\beta_2", "27190e3f0498615ef381de55b46a8056": "M_{1}{}^{2} =m_{1}^{2}+2m_{w}S+S^{2},", "2719ae0fd018fefdb833bbb6a2138286": "(d \\sigma / d \\Omega )_{\\rm Th}", "2719baa2e157e11e1e115ec6492bdd87": " S = \\int_k {1\\over 2} k^2 |\\phi(k)|^2", "2719ee323ad10e2e12b1c6519db9b4ac": "X(z^{-1})", "271a3fc65144601a177368f8b0121522": "\\Sigma\\to X", "271afe7ed7fb34aabfa67e5e09956295": "\\mathbf{J} = \\mathbf{J}_\\mathrm{b} + \\mathbf{J}_\\mathrm{f},", "271b43fbc04895ae627d2e48f35b6b62": "f_\\omega(3) - 2", "271b604dbc38adcb1913438fb7e9c1a2": "\n\\sigma^*_i(a) = \\left(\\frac{1}{C-1}\\right)\\text{Gain}_i(\\sigma^*, a) = 0\n", "271bbce94f3e39f7f951d6c77c24a7e2": "H_n (x) = (H+2x)^n\\,\\!", "271c55dec1713a74b129cd9c00ab5ed6": "\n{\\frac{\\sqrt{n+1}}{n!\\sqrt{2^n}}}\n", "271cabfb4c4b5a83d963deb152a4eee9": "\\kappa=\\frac{a}{s^2+a^2}", "271d1f67d105cacfeb98a901b002e46e": " x[n] = \\frac{1}{2 \\pi} \\int_{-\\pi}^{+\\pi} X(e^{j \\omega}) e^{j \\omega n} d \\omega.", "271d27f52ec2722277cacb0941c9b61b": "\n 0 = \\frac1n \\sum_{i=1}^n \\nabla_{\\!\\theta}\\ln f(x_i|\\theta_0) + \\Bigg[\\, \\frac1n \\sum_{i=1}^n \\nabla_{\\!\\theta\\theta}\\ln f(x_i|\\tilde\\theta) \\,\\Bigg] (\\hat\\theta - \\theta_0),\n ", "271d7049d6dae75a87206d5c99e22561": "\\delta_\\odot = - 23.44^\\circ \\cdot \\cos \\left [ \\frac{360^\\circ}{365} \\cdot \\left ( N + 10 \\right ) \\right ]", "271d7a9fcf6a6eaa16f1c6f9684d9bc5": "n \\to \\infty.", "271d90ed2edaa04fba4126811f274c0f": "= V", "271dc98c738720b23af59fcf8afa089c": "A=(10+\\sqrt{\\frac{5}{2}(10+\\sqrt{5}+\\sqrt{75+30\\sqrt{5}})})a^2\\approx17.7711...a^2", "271dec5b34962227e8e94175c57db4e0": "Z_r^{p,q}", "271e0e2865ff6a72a19b550043f8d9b2": "p_{n-i}(z)\\,\n", "271e78c8b2c5ff19e7402d9af42d95e9": "\\lfloor x \\rceil", "271ec3320b67630e0396ebe32a417967": "(e^{ar} e^{-br \\epsilon} )^{-1} = ", "271edd37909014ccb24a5277d5fb497b": "\\textstyle \\lbrace{n\\atop k}\\rbrace", "271ee017284df0fa305a66e0d9d3c351": " \\mathbf{\\hat Q \\hat T}(\\lambda)|q\\rangle = \\mathbf{\\hat T}(\\lambda) (\\mathbf{\\hat Q} + \\lambda)|q\\rangle = \\mathbf{\\hat T}(\\lambda) (q + \\lambda)|q\\rangle = (q + \\lambda)\\mathbf{\\hat T}(\\lambda)|q\\rangle ", "271efef3c48c27e20228b06587fd8a71": "dy \\wedge dz", "271f152f11da730c7c6a0d139484e370": "f(g) = \\sum_{\\sigma\\in\\Sigma} d_\\sigma \\operatorname{tr}(\\hat{f}(\\sigma)U^{(\\sigma)}_g)", "271f34ea27b67c987755acb5056ca8c1": "(L/2\\pi)^{d}", "271f3a5312853160bc812ca05a707243": " \n\\text{volume} = \\frac{6\\cdot L^2 \\cdot \\frac{L}{2}}{3} = L^3,\n", "271f5aee3cb1e9825f52e96c86c81f1e": "f\\colon X \\rightarrow T", "27202c197a32aa14c1cc0fab50393a82": " dE = F \\cdot dx ", "272046b75ab4366ee647e77a8f0c0952": "G_1\\ast G_2", "272090c2bf557823e164225c97327437": "(a,b,c,d)=(xz, yz, z^2, xy). \\, ", "2720b99a1d016ed36f887f52cbbe063d": " \\frac{GM_\\text{vir}}{R_\\text{vir}} \\approx \\sigma_\\text{max}^2 ", "272193a81c5c72bbf2a89d8f7d475f7c": "\\!\\, I=I_{o}e^{\\frac{-t}{\\tau}}", "27219bcaad63d305920bcb58027e779d": "S = \\frac{m(m^3+1)}{2}.", "2721a535d23241f23e10717ae00154cd": "\\gamma=2-2\\alpha", "2721cd572bfad45985a6e055fc2c5ecb": "\\nabla_{\\bold u} {\\bold v} = D_{\\bold u} {\\bold v} + \\Gamma(\\varphi)\\{{\\bold u},{\\bold v}\\}", "2722507f492b9f44ae9c9abc0188b28e": "\\rho(a,b) = d(p^{-1}(a),p^{-1}(b))", "2722a27f161295a14d8d129a7aadad9b": "\\tan (\\arcsin x) = \\frac{x}{\\sqrt{1-x^2}}", "272309cb7aa31f4ac8ffb4f013657f64": " \\tau_{xy} ", "27232eb3a713273f57e5859d98768d0b": "f(\\mathcal{Z}(M))\\subseteq \\mathcal{Z}(N)\\,", "27238506c1549f5079b6c7b31a03c708": "\\left(J_z^{(m)}\\right)_{m'm} = m\\delta_{m'm} \\,\\quad \\left(J_x^{(m)} \\pm i J_y^{(m)}\\right)_{m'm} = m\\delta_{a',a\\pm 1}\\sqrt{(a \\mp m)(a \\pm m + 1)}", "2723ae0b4c42b64fe8fb495322da10b2": " f(x)g(y) ", "2723fe64199bf849b28f7b6e7e5fbd67": "f=f(E)", "27246e52868d656102bf905ef9aab6e3": "\\frac{\\partial y}{\\partial \\mathbf{x}} ", "27247403fb09ec1edee645ad9e07e454": " G = H - T S_{int} \\,", "2724a23b16ca720546476383e9bc8610": "\\phi_x", "27250c680d6309c7511d97267f74686f": "X'_\\beta", "272564c8cdaba01a36380357c60dea2c": "D(\\cdot)", "2725725bf31da5361bed83ac9888b95b": "g^{\\alpha \\beta}{}_{, \\beta} = k \\, g^{\\mu \\nu}{}_{, \\gamma} \\eta_{\\mu \\nu} \\eta^{\\alpha \\gamma} \\,.", "272594d6858105a512ee9b10d837b7d4": "Af(x) = \\sum_{i} b_{i} (x) \\frac{\\partial f}{\\partial x_i} (x) + \\tfrac{1}{2} \\sum_{i, j} \\left( \\sigma (x) \\sigma (x)^{\\top} \\right)_{i, j} \\frac{\\partial^{2} f}{\\partial x_i \\, \\partial x_{j}} (x),", "27259969b444266306de543d25651403": "C(v)=\n\\begin{cases}\nm, \\mbox{ if } v\\equiv 0\\;\\; (\\mbox{mod}N)\\\\ \n\\\\\nmc, \\mbox{ otherwise }\n\\end{cases}", "2725a6b17d894abf6c4b6ea71c596f5b": "S = \\{o_1, \\ldots, o_n\\}", "2725bcf5aa3be2c12ee559ab48f468b3": "\\ d[\\mathbf{x}(1), \\mathbf{x}(1)]=\\max_a |u(a)-u^*(a)|=00) ", "272dfe092bae4beafab716f38be6aa67": "\\det\\begin{pmatrix}A& 0\\\\ C& D\\end{pmatrix} = \\det\\begin{pmatrix}A& B\\\\ 0& D\\end{pmatrix} = \\det(A) \\det(D) .", "272e2b19c6c85a1876428fa4e340d7da": "d=n-k+1", "272e458d3dca9d8c009a89f5645c59f2": "T_{ij}=\\rho v_i v_j - \\sigma_{ij} + (p- c^2_0\\rho)\\delta_{ij},", "272e51dcc3dd38589dab55a249bc1426": "x=-\\sqrt{1-y^2}", "272e96042e6302c6415cf814f1be2600": "\\delta(B) \\le c", "272f259c6011fb0b291e266456dda3a9": "\\langle E_1(t) E_2^*(t - \\tau) \\rangle = \\langle ( E_{a1}(t) + E_{b1}(t) ) ( E_{a2}^*(t - \\tau) + E_{b2}^*(t - \\tau)) \\rangle", "272f9580a3fcb461e00aa20343002b99": "T_m = T_1 + (T_2-T_1){e_2 \\over (e_2+e_1)}", "272fbe3e4089c58e62a042322f7e555e": "H^*_G(M) = H^*_{\\text{dr}}(M_G)", "272ff88e896857905b1d384ab3f07072": "\\langle x,\\, p |\\Psi(t) \\rangle", "27301936ef3fd7c774fe94213d1effcf": "l(\\theta) = \\alpha", "27302d8fd3387cca567f2067f4b4f2f9": "q_1''(q^c)=0", "27305e00f744bed2c72b98c86bde971d": "{\\partial u\\over \\partial t} =\n \\left({\\partial^2 u\\over \\partial x^2 } +\n{\\partial^2 u\\over \\partial y^2 }\n\\right)\n = ( u_{xx} + u_{yy} )\n= \\Delta u ", "273068d9988f024a06942438c8e99be0": "N(u) \\setminus \\{v\\}", "273068e6b8868e1b95b9176187588792": "a_1x_1 + \\cdots + a_kx_k=0.", "27308c709962c69aa6fd62d24482a381": "K \\big( \\tfrac{1}{4}(\\sqrt{6} - \\sqrt{2})\\big) = 2^{-\\frac 7 3} 3^{\\frac 1 4} \\pi^{-1} \\Gamma \\big(\\tfrac 1 3\\big)^3 ", "2730980eadd097914e11cf8a762bf03f": " \\star du =dv,", "2730a4212af0788b28e3ea2b2eeeb574": "(J^2 - J_z^2) = (J_x^2 + J_y^2)", "273142f91a8bfde342d386ddfa02c8d2": "(19)\\quad ds^2=-\\Big( 1-\\frac{2M(v)}{r}+\\frac{Q(v)}{r^2} \\Big) dv^2+2dvdr+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;.", "2731456537f2255917a445dceb1e9fdb": "\\gamma_\\tau", "27315dfd3971aab09d6a8e52151253d8": "A_{eff} = \\frac {P_o}{PFD} \\,", "2731a8db5befb60c2d7825282eacd372": "S \\in \\mathcal{A}", "273224ac9b0fbb35a97d89c586171e8f": "\n\\begin{align}\n\\operatorname{Re}\\left\\{\\frac{d}{d t}(A e^{i\\theta} \\cdot e^{i\\omega t})\\right\\}\n&= \\operatorname{Re}\\{A e^{i\\theta} \\cdot i\\omega e^{i\\omega t}\\} \\\\[8pt]\n&= \\operatorname{Re}\\{A e^{i\\theta} \\cdot e^{i\\pi/2} \\omega e^{i\\omega t}\\} \\\\[8pt]\n&= \\operatorname{Re}\\{\\omega A e^{i(\\theta + \\pi/2)} \\cdot e^{i\\omega t}\\} \\\\[8pt]\n&= \\omega A\\cdot \\cos(\\omega t + \\theta + \\pi/2)\n\\end{align}\n", "27326b1b2e401285ac8f33cce12aded5": "V(t)^2/Z", "27326c064bb748e01d8469e2eee6b94b": "\\mu_1,\\mu_2", "27326eaea95493aedd18697142882df0": "{DF}_T", "2732ccc1df31bcec9df77697931274fa": "\\displaystyle{(f,g)_S=(Sf,g).}", "273376e6e51dab1a756368e033fe92b3": "\\sum_{n=0}^\\infty(an+b).", "2733a5527c58793650f09341d82595ce": "R_s = \\frac{1}{\\int_0^{x_j} \\mu q N(x) dx}", "2733f6cd7314219f0318616ed8166651": "E[h(y)] = \\int_{-\\infty}^{+\\infty} \\frac{1}{\\sqrt{\\pi}} \\exp(-x^2) h(\\sqrt{2} \\sigma x + \\mu) dx", "273409400dc642485c4a2be9d987681f": "f \\rightarrow \\langle h, f(T) h \\rangle", "27342c9a10f65aab8a44d6ff2301e363": "C_{abcd}=R_{abcd}-\\frac{2}{n-2}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+\\frac{2}{(n-1)(n-2)}R~g_{a[c}g_{d]b}", "27348e00eee50bc77c7521cffe028c51": "b(z)=b_0+\\tbinom{n}{1}b_1 z+\\tbinom{n}{2}b_2 z^2+\\dots+b_n z^n", "27349e3432e336d849b069608d2909bf": "g \\left ( \\alpha\\, \\right ) = \\mbox{isolated guessing; address space is re-randomized after each attempt}\\,", "2734ce623bc12ed8eb6eafec22ab46d4": "\\overline{\\theta}", "2734fd6cfb087caf20b85c37f0ef1b0c": "f(n) \\in \\Omega(g(n))", "273513b017764885c376cff21b5ecd82": "(N-1)(N-2)", "27354ffafa010686959ef57bf0510760": " \\varphi_n = \\sup \\{|f(x)| : x \\in F_n, \\; \\|x\\| \\le 1 \\}", "273595a80ccb2f9dfc683b33a2f8fcd4": "(A+B+C)x^2 - (3A+2B+C)x + 2A = 1.\\,", "2735c1f8e2b446903d0489951800dcaf": "\n\\Pr_{\\mathcal{S}}\\left\\{ E_{a^{n}}^{\\dagger}E_{b^{n}}\\in N\\left( \\mathcal{S}\n\\right) \\right\\} =\\frac{2^{n+k}-1}{2^{2n}-1}\\leq2^{-\\left( n-k\\right) }.\n", "2735f5bfd8a9ed0062df80a0c0960db1": "\\int f \\, dV= \\int_{-\\infty}^{\\infty} f(t) \\, \\sinh^2 t \\, dt.", "27361797574a55a7a543e6dce7f7ad2e": "A\\rightarrow \\alpha", "2736996057a533311c4b64ab342bdc49": "\\gamma[L+1]", "2736c5b9264fd174b425f09803c70c04": "e(Z-1)", "2736d327efadfbd0e9d3d5fe41cfd660": "g_1, g_2", "2736e11c017eec93b2735d395fff92e5": "I<10^{-3}", "2736fa5d4d16619198139bda55a6df0b": "Velocity = 0.0", "27370f4a0d4637879cfbfb3c5573d2f5": "\\left( 1 + \\beta\\sin(\\Omega t)\\right) Ae^{i\\omega t} = Ae^{i\\omega t} + \\frac{A\\beta}{2i}\\left( e^{i(\\omega+\\Omega) t} - e^{i(\\omega-\\Omega)t} \\right) . ", "2737364a2bd27c93d28d95b771bfb090": "\\mathrm F(E)\\times_{\\rho}V", "2737725072e41660df24fc564a017d57": "n! = (n)(n-1)...(2)(1)", "273774476b615c4956b4fc8fdd9a47be": "\\log(xy) = \\log x + \\log y", "2737b027901e1b6811cc275c4db39e97": "\\{ \\gamma\\,_m \\}", "2737c4d15de5fe830975d5795cafdb56": "\\color{Black}\\tfrac{\\infty}{m}\\infty", "2737ce399344c73e17c42185c211b3f8": "A_{(\\alpha}B_{|\\beta|}{}_{\\gamma)} = \\dfrac{1}{2!} \\left(A_{\\alpha}B_{\\beta \\gamma} + A_{\\gamma}B_{\\beta \\alpha} \\right)", "273802aae2355f16e5f5f43ca18074ad": "n_a(\\mathbf{k}) ", "2738439e6db96030205f02509efc0547": "a_2' = a_0 \\oplus a_1 \\oplus a_2 \\oplus a_6 \\oplus a_7 \\oplus 0 = 0 \\oplus 1 \\oplus 0 \\oplus 1 \\oplus 1 \\oplus 0 = 1", "2738695e271ec60d8ca1fe16da705bc4": "\n\\ddot{r} = \\frac{d^{2}r}{dt^{2}}\n", "2739107d20f9b46d9169acd038d6e59e": "A^\\mathfrak{n}", "2739214e3cb8b4e6ce30b84a42cdcbe3": "\\begin{align}\n \\bar{I_L} &= \\left(\\frac{1}{2}I_{L_{max}}DT + \\frac{1}{2}I_{L_{max}}\\delta T\\right)\\frac{1}{T}\\\\\n &= \\frac{I_{L_{max}}\\left(D + \\delta\\right)}{2}\\\\\n &= I_o\n\\end{align}", "27395b7ab42c3383bd8ba4d3976a6646": "\n{\\rm i}{\\partial \\over \\partial t} \\psi_t = - {\\nabla^2\\over 2} \\psi_t\n\\,", "273a383345e167ee1791232c40eaf917": "v(t)", "273ad1a55db5eb251418161c4734fe02": "Height(t) > k", "273b0dfa8d54fbb6c3ed1d2bc26ee2d7": "\\therefore, \\because, \\And \\!", "273b1b32341db7227180543aa12f06d5": "\\begin{matrix} \\frac{3}{51} = \\frac{1}{17} \\end{matrix}.", "273b41404f9e076c3264efb39726825a": "W_w", "273b55d642a822a5bf1bd829e454fcf4": "\\Rightarrow\\dfrac{7385}{33} ", "273b947d255f328e21163da6e3af9da3": "d.f. \\cong {e^\\left(\\ln \\frac{N-1}{2n} \\ln \\frac{(2n+1)(N-1)}{4}\\right)}^{- \\frac{1}{2}}", "273bceda822e194e1b1eccdb6f67e815": "\\text{true}", "273c8b2ab7bb3acf9ec2d790c88ee1ef": "lb_{computed} ", "273c9d6047355050192260495b413941": "\\gamma_2 = 1", "273ca24b1aa43b0a863a2b80415a2c21": "x \\prec \\bar{y}", "273d4557819e10ea0dd94ba8e91bf251": "\\mathrm {DOF} = 4 N\nc", "273d89150f20417c299b6d41216a2511": "\nF_t =\\ -J_3\\ \\frac{1}{r^5}\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right)\\ \\sin i \\ \\cos u\n", "273da6412bb76f3f49287e4db7c8f5f9": "\\lim_{n \\rightarrow \\infty} \\mu_n(X) = \\mu(X)", "273dff67eba5fe343aac9946c70d1a83": "\\varphi'(a_i) = -\\frac{A_i e^{-\\kappa a_i} (1 + \\kappa a_i)}{a_i^2} = - {1 \\over 4 \\pi \\varepsilon_r \\varepsilon_0}{z_i q \\over a_i^2} = \\varphi_{sp}'(a_i)", "273e187a61dd341f2b2d96cef3d9ecdb": "\\psi_n(x)=x^n\\,h(x),\\qquad x\\in\\mathbb{R},", "273e3bd7999abb12af0c0bdaf3ec1bf7": " mU\\frac{d\\gamma}{dt}=-Z_u u", "273e3f766f178f55f3108f429c02a840": "A_{(\\alpha\\beta)\\gamma\\cdots}", "273ed8b39a916ed779b7d0f616a9192c": " \\lambda>0", "273ef2b4be84c3ab034dbed66af5e48b": "\\chi^2 = {(|b-c|-1)^2 \\over b+c}.", "273efb966e0a9faf69e0cf69d21eaea4": "\\begin{bmatrix} X \\\\ Y \\\\ Z \\end{bmatrix}^B = \\begin{bmatrix} c_x \\\\ c_y \\\\ c_z \\end{bmatrix} + (1 + s\\times10^{-6}) \\cdot \\begin{bmatrix} 1&-r_z&r_y \\\\ r_z&1&-r_x \\\\ -r_y & r_x & 1 \\end{bmatrix} \\cdot \\begin{bmatrix} X \\\\ Y \\\\ Z \\end{bmatrix}^A ", "273f38d10aeb8f535c3bad342e1e4252": "RMR^{-1}.", "273f75a47297087306deba5501531c59": "A(i) > A(j)", "273f765deba06af4b937fedff2c2fb3e": "\\sup_V u \\le C \\inf_V u", "273ff250d6ed9ff94e2ea774603f8a01": "Y^k\\ ", "27403c2daa55b104ef80b186167ca301": "x_i = y_i\\ ", "27406d836990b1d394906916c59c8028": "z_t = y_t - y_{t-1}", "2740b4ab0224dba921f390db5d3a7f5b": "\n\\int_{\\mathbb{R}^n} e^{-x^T A x+v^Tx} \\, dx = \\sqrt{\\frac{\\pi^n}{\\det{A}}} \\exp(\\frac{1}{4}v^T A^{-1}v)\\equiv \\mathcal{M}\\;.\n", "2741d556b96698c7ec7672166f996a65": "p \\gg \\Lambda", "2741f9b0051c0b0418fb36daa0559c26": "m_\\ell=-2,-1,0,1,2", "27422b7a94e97b3fecaf9f1f9195bfcf": "\\{S\\mid T\\subseteq S\\subseteq A\\mbox{ and S totally ordered}\\}", "274235d3c8c0f0991d1f6a54d34065e4": "\\begin{matrix}(\\mathcal{A}B)(\\psi)= \\\\ (\\forall \\phi:\\phi_0=\\psi\\to B(\\phi))\\end{matrix}", "2742f5baabd3c896d37d1d18124405cc": "bA_4 = A_1 \\cup A_2 \\cup A_4", "2743611decaf752d841c21b28fc798e4": "\\tilde{Y_k}", "274372371f2edd16f3b4d357c074d957": "i_{V_s} + i_R = 0", "274392175c4682d9c471ce92ef255f65": "\\mathbb{D}(A,B) ", "2743922d0685f249a012ab2ffe852c6b": "\\bar{r} = \\frac{1}{nk}\\sum_{i=1}^n \\sum_{j=1}^k r_{ij}", "2743e38f46e4b83c5e2516babff4eb20": "\\digamma = \\frac{3FL}{2d}", "27440a41978b7d37992788a5f4c26f05": "\\displaystyle{\\mathrm{dim} \\,\\mathfrak{k} - \\mathrm{dim} \\,\\mathfrak{k}_a = \\mathrm{dim} \\,\\mathfrak{m} - \\mathrm{dim} \\,\\mathfrak{m}_a,}", "2744584198cf6b5801d281dd3e75e745": "\\mathbf{r}_j", "274464ffdd99a069f6d52e7f017e90cc": "\n\\begin{align}\n \\sigma(x, a) &= 0 \\\\\n \\sigma(x, x) &= \\sigma^2(x) \\\\\n \\sigma(x, y) &= \\sigma(y, x) \\\\\n \\sigma(ax, by) &= ab\\, \\sigma(x, y) \\\\\n \\sigma(x+a, y+b) &= \\sigma(x, y) \\\\ \n \\sigma(ax+by, cw+dv) &= ac\\,\\sigma(x,w)+ad\\,\\sigma(x,v)+bc\\,\\sigma(y,w)+bd\\,\\sigma(y,v)\n\\end{align}\n", "2744931aa0c18c50218f0195578f4c14": "\\Delta \\overline{u}", "2744b8272939c5e4f88f6dc77c12c90d": "\n\\bar{w}_{1L}(s, 2n+\\gamma_{1L};L) = \\delta_{n0}\\bar{\\Gamma}_{1L}(s) + \\sum_{c=1}^{[L/2]} (-1)^{c+1}\\bar{w}_{1L}(s, 2(n-c)+\\gamma_{1L};L)\\bar{h}(s,c;L). \n", "27451f893b3b7d536f0c01862879971e": "\\mathbf{u}_1, \\ldots, \\mathbf{u}_n", "27457b029b382d6371057fe7f2a27801": " T_3 = (T_1Y_2)^2 + (Z_1X_2)^2 = 16", "2745a719cb8d9bb76b107d35fcd04b0a": "(f_j f_k)(g)= f_j(g) f_k(g)", "2745b7de130e087b3d5a0e765dfe4c87": "\\frac{L^\\alpha}{1 - \\left(\\frac{L}{H}\\right)^\\alpha} \\cdot \\left(\\frac{\\alpha}{\\alpha-2}\\right) \\cdot \\left(\\frac{1}{L^{\\alpha-2}} - \\frac{1}{H^{\\alpha-2}}\\right), \\alpha\\neq 2", "2745e451109e98fe2cfc022ec0c5bb06": " J^{\\mu,\\nu}(z) = \\sum_{k\\ge 0} \\frac{(-z)^k}{\\Gamma(k\\mu+\\nu+1)k!}.", "2745ef042f8cd42fa6f627c5f0e62cbd": "{}_2F_1(a,b;c;z) = 1 + \\frac{ab}{c\\,1!}z + \\frac{a(a+1)b(b+1)}{c(c+1)\\,2!}z^2 + \\frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)\\,3!}z^3 + \\dots\\,", "2745f778411586078eb847365808f476": "\\mu : T^2 \\to T\\,", "2746271b2ccb4ac7ec7409546bdcd631": "|z|=a>1", "2746325ce40ce297977ef99cf076bedc": "\\, R \\,", "2746f6ad41814755ddf43e9bff638fdd": "v_{xy}=0", "274701d8c17341ed11d68088a3da3895": "P_i = (x_i, v(x_i))", "27470832e77b40587fcdba53c5009d50": "f(\\sigma^*) = \\sigma^*", "274708f04fe3333db664eb753745c82d": "\n \\boldsymbol{\\nabla}\\times(\\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon}) = \\boldsymbol{0}\n ", "2747deb4bda32d52514dbcb1fcbc98fa": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(\\boldsymbol{x}\\right) & = x_{1} + x_{2} - 2 \\geq 0 \\\\\n g_{2}\\left(\\boldsymbol{x}\\right) & = 6 - x_{1} - x_{2} \\geq 0 \\\\\n g_{3}\\left(\\boldsymbol{x}\\right) & = 2 - x_{2} + x_{1} \\geq 0 \\\\\n g_{4}\\left(\\boldsymbol{x}\\right) & = 2 - x_{1} + 3x_{2} \\geq 0 \\\\\n g_{5}\\left(\\boldsymbol{x}\\right) & = 4 - \\left(x_{3}-3\\right)^{2} - x_{4} \\geq 0 \\\\\n g_{6}\\left(\\boldsymbol{x}\\right) & = \\left(x_{5} - 3\\right)^{2} + x_{6} - 4 \\geq 0\n\\end{cases}\n", "2748bb44615a7859b19a958469200f1b": "3.\\overline{3}", "2748be1ecae5019bb08f353c3230702b": "\\mathbf{3}\\otimes \\mathbf{\\overline{3}} = \\mathbf{8} \\oplus \\mathbf{1}", "2749264821871f76e74e56829e430476": "\\alpha = i", "2749c4111d84fb1911141689ca95c933": "f_{x} : E_{x} \\to E'_{f_{0} (x)}", "2749e8ff2fc48c656451643749bc72bc": "IMA = \\frac {F_{out}} {F_{in}}. ", "2749fe8e710a06b7301c082e27dbd803": "\\mathrm{spt}(13n+6) \\equiv 0 \\mod(13)", "274a210151565a0b89a12fba5470b55a": "L(j)", "274a4de4dba5deab523aad21aeb4e6aa": " \\frac{\\partial C}{\\partial t} + \\mathbf{v}\\cdot \\nabla C =0.", "274a5f0569d1e219981d6147ec062425": "t \\rightarrow \\pm \\infty", "274a6367f06056ff9ef9597c14588c44": " f(500)=500(\\sqrt{501}-\\sqrt{500})=500(22.3830-22.3607)=500(0.0223)=11.1500", "274a90691bb0359c7d12ef37a056faef": "DN_c = 1 + (DN_1 - 1) + (DN_2 - 1) + ... + (DN_n - 1), \\ ", "274ab8e8dc930b3609d8248a47a5d111": "\\big. U = \\frac{k A} {\\Delta x}, \\quad", "274ac86b27a6521c4b0c711bd91514c5": "~\\sigma_{\\rm ep} ~", "274b4810b6ea396de824bcdea4c5a773": "\\omega = 2\\pi/86164", "274b5d0c8b33506551742b2564f9c661": "\\det T_h = (-1)^{\\lfloor\\frac{b-a+1}{4}\\rfloor}\\cdot h_a\\cdot h_b\\cdot\\mathrm{res}(h_{\\mathrm{e}},h_{\\mathrm{o}})", "274b905ffd111cb886d605eb67de88c2": "G_{V_1, E_1} \\square H_{V_2, E_2} \\rightarrow J_{(V_1 V_2), (E_2 V_1 + E_1 V_2)}", "274bb8be45054e2093b55dee1569bc81": "\\mathbf{\\hat{n}} \\cdot \\mathbf{\\hat{v}} = \\cos(\\theta)\\,", "274c7beee2ccab7eefceb36038ce4f12": " y_k = \\mathcal{P}_D( x_k + p_k ) ", "274c96707e6dd23c702e2cc6d2c3a942": "D=v_1f_1+v_2f_2", "274d1181b531f1fa4be4a7eb87c03d40": "(x + 1)^{3}p(\\frac{1}{x+1}) = 512x^3+512x^2-128x-64", "274d451e072af48bd0a82bd5dd0a47f2": "\\exists x\\forall y (\\phi(y) \\iff y=x \\and \\psi(y))", "274d4cc0ebee8feefb24798cb04dfb55": "\\mathbf{\\hat{e}}_i \\cdot (\\mathbf{\\hat{e}}_j \\times \\mathbf{\\hat{e}}_k) = \\varepsilon_{ijk} ", "274d95b0d5faeb0e766fc5486c6acb38": "-1, 1\\,", "274dc5087b22c2c1b19fb3e2ee80f32e": "k[x_1, \\ldots , x_d]", "274dc9c31e7eabf01ae5d1cc4f618a3c": "y=S/N", "274dd0431919a6dd17e19820a505db13": "x_1 (t)=\\begin{cases}\n\\cos( \\pi t); & t <10 \\\\\n\\cos(3 \\pi t); & 10 \\le t < 20 \\\\\n\\cos(2 \\pi t); & t > 20\n\\end{cases}", "274dfdba46ac7ef6561fca7ec765267f": "\\ max(m,n)", "274e072c81900310c789143e8ec677b1": " J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\, \\Gamma(m+\\alpha+1)} {\\left(\\frac{x}{2}\\right)}^{2m+\\alpha} ", "274e3682289056f732f253205550354f": "\\Delta E_{00}^* = \\sqrt{ \\left(\\frac{\\Delta L'}{k_L S_L}\\right)^2 + \\left(\\frac{\\Delta C'}{k_C S_C}\\right)^2 + \\left(\\frac{\\Delta H'}{k_H S_H}\\right)^2 + R_T \\frac{\\Delta C'}{k_C S_C}\\frac{\\Delta H'}{k_H S_H} }", "274e9493a3a437dc2fcae487471409c9": "w_i = \\frac {m_i}{m_{tot}}", "274ea7dd52614b381a01dc942fd2e7d8": "Q(\\omega)\\,", "274ed9633cd0696c350a505ae4eb3039": "x (\\theta) = (R - r) \\cos \\theta + r \\cos \\left( \\frac{R - r}{r} \\theta \\right)", "274f97d6fd93dd1926bf998e332bcb4e": " \\mathbf{E} = \\mathbf{K}'^{\\top} \\; \\mathbf{F} \\; \\mathbf{K} ", "27507d53b6f4eaa2c507cdc12e17b4d2": "E_d > 1 \\!\\ ", "2750fc7ed3ddda1d2dcca7e74183c879": "\\ldots f(x,q_k) \\geq r_k\\,\\!", "27516f7d1bc941a1810087cf9796f8df": " W = -\\int_{1}^{2} \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{s}", "27518edcf5038ba208f50e61859b125d": "u\\,J_\\nu(ut)", "275201561e2fcc42c147997417eb7625": "\\exp \\left ( -\\sum d_i\\right)", "27520466c0dcb2fdd0acd6bd77193739": "16*x^4+5152*x^3+518420*x^2+16693124", "27526e1672d2f4f9db8ca0a5d04959dc": "\\eta : I\\to M", "2752c313e4c3583391429ea73e5882c6": "u \\cdot d", "2752fc873e24e06963596584eb15e9a7": "\\mathcal F(U) \\to \\mathcal G(U)", "2752fe31fd48c9ad40cd6781534c3ad8": "P_2(0) = P_2 (1)=0.", "27536adacd8ee817df66722e0483d870": " \\operatorname{Var} (X_i)=\\sigma^2 ", "27537cd086a8e9f62bba71321bd87403": "X_B", "27539b46a71ca477e033401acbe8e22e": "\\frac{\\delta l}{\\delta t}=\\frac{\\sum P}{8 r^2 \\eta l}(r^4 +4 \\epsilon r^3)", "2753a06f3f52c90fd3e8fc243fada346": "A = \\frac{p \\cdot q}{2} ,", "2753a7be181ccde924a436eba34ef1f0": "A(n,m)=\\sum_{k=0}^{m}(-1)^k \\binom{n+1}{k} (m+1-k)^n.", "2753c1390f7d25bb15a1b61f29e832c6": "x_2=KI\\oplus x_1", "27540f6921a0646a06570dde652bb017": "= 9000 \\times 1.1304 \\times 1.1304 \\times 1.1304 = 13000", "275488a53dfd39c8fa0f7d365a6820b2": "b / a", "275496a85f248b8d094a345332fd4266": "P(M,S) = P(M,S_0) + P(M,S_1) + \\dots + P(M,S_n)", "2754ef5a26aa5b053d28cf609ec2f48d": "B_0=3.12\\times10^{-5}\\ \\textrm{T}", "2754fa3e07b2d98d2c58c79f20512b58": "\\;p(t) = P r^t - A \\frac{r^t-1}{r-1}", "2754fadb9f556c033589c1e5ce85e695": " \\displaystyle L_p(1-n, \\chi) = (1-\\chi(p)p^{n-1})L(1-n, \\chi)", "275517a7bfa1c9f113d2e1f197fa8700": "\\textstyle (c_n)_{n\\geq0}", "2755714f8654906e9829a74c412e3464": " \\Gamma^\\mu {}_{\\alpha \\beta}\\ ", "2755735b2dd7b15a651370c88d3a8d29": "\\chi = B \\rightarrow A", "2755b22abce254d35f89293ff1ba6247": " t^{*}_{(1-\\alpha)} ", "2755f5717fbcdcf457d033da65def3f2": "\\epsilon = I_\\mbox{linear} + I_\\mbox{nonlinear} = 0", "27560cccbd150a50a2cf3be11f61c062": "(s^2+0.5176s+1)(s^2+1.4142s+1)(s^2+1.9319s+1)", "2756438f419aab48ab30ac35a1e0090b": "m^{e - 1}\\text{ (mod }p\\text{)} = 1", "275667c283fa17dfe2e5b03c1c9875eb": "-\\csc^2(x)", "2756ae341c28ecc5ac73293da66c326e": "\\mu_B B / k_B T", "27571223560bb0486eddf5e2b505179f": "\\tfrac{BF}{AF} ", "27571c4b401b45a4691ccbcce0b83cb6": "P(X=x)", "27577bb7e608b77faf24300e302085bd": "D=\\frac{1}{\\ell \\nu}", "2757f10aaa345ef4178642164895d8d8": "\\frac{\\rm d}{{\\rm d}z}\\,\\mathrm{erf}(z)=\\frac{2}{\\sqrt{\\pi}}\\,e^{-z^2}.", "275828025cde6a5e68b33960e1e9481d": "b=\\frac{I}{AM}.", "275850a434439f4c91561fc949b9b4e0": "\\mathbf{H}", "2758a232ea11474a55b32d55425abc4b": "\\varphi\\Big(\\varphi(x)\\Big)={\\rm e}^x", "275927a68c49aa1e97f8897df962caf3": "F(t_1, t_2, \\dots)/(p, q, \\dots).", "27592bcdffd7f2e1bc027f89075caf46": "\\left(3\\sqrt{\\frac{2}{5}},\\ \\pm\\sqrt{6},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "27596623be4332515c975fc35488a568": " R = \\frac{ \\sum_{ i = 1 }^K \\sum_{ j = 1 }^L R_{ ij } }{ \\sum_{ i = 1 }^K n_i } ", "2759f89dbe6f3b7665df5cf686c9adb1": "( G1 + G2 ) / 2", "275a1b7d253c6554ef51fd2cc2b53492": "ELR=\\sum_{a=a_E}^ \\infty \\left[M(a)+m(E,a_E,a)\\right]S(E,a_E,a)-\\sum_{a=a_E}^ \\infty M(a)S(0,a_E,a)", "275adc3d2dd05b8a995f3cc55f7d3c9a": "\\mathbf \\nu (x) = \\int \\theta (x) = -\\frac{5}{12} x^4 + \\frac{75}{6} x^3 -1406.25x (m)", "275b2f471d49a7f76745136620f4e43b": "x^7+x^6+x^5+x^3+x", "275b9b1a4ed62e142f27b4565df3cab5": "A(t) = A(0)e^{gt}", "275ba6890cc5db6dc8db920a1caf425d": " \\Delta v V_\\text{esc} ", "275bb6ebbe2a815ef757d9cfe4d0e006": " r_t = \\mu + \\bar{\\sigma}(M_{1,t}M_{2,t}...M_{\\bar{k},t})^{1/2}\\epsilon_t, ", "275beccbaa7ee56fbe17692df9c282f0": "(A,d)", "275c0fa76f168671c9728ee6b225506c": " Z_t = 1 + \\int_0^t Z_s\\, d X_s.\\, ", "275c9a28afb8e92b2f3d0085f6c45224": "B = \\sqcup_{\\lambda} B_{\\lambda}", "275c9b0124c604ab7bc5603feccfb615": "p\\leq 2k-2", "275cf1574a89936fbebf4d8d827b9082": "\nP^{\\prime\n}(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6})=P(X_{6}|X_{5})P(X_{5}|X_{2})P(X_{4}|X_{2})P(X_{3}|X_{2})P(X_{2}|X_{1})P(X_{1})\n", "275d1db172f7bad9992a841b1c5d0100": "\\beta(i)", "275d2ad3cc8d2ac9f7c6372fbb672fbe": "-2\\leq k\\leq 2 \\pmod 8", "275d33d92153da2dd486a049d5a1d776": "{{\\hat{\\mathcal{H}}}_{\\text{KS}}}=-\\sum\\limits_{\\mathit{i}}^{{N}}{{{\\gamma }_{\\mathit{i}}}\\cdot {{{\\hat{\\vec{I}}}}_{\\mathit{i}}}\\cdot {{{\\hat{\\mathbf{K}}}}_{\\mathit{i}}}\\cdot \\vec{B}}", "275d3f445c1cbdb2efb83ac8406c9d27": "\\mathbf{V}_i = \\mathbf{V}+\\frac{d\\mathcal{R}}{dt}\\mathcal{R}^T\\mathcal{R}\\mathbf{r}_{io}", "275d5864e98f807904d403068dfc2a57": "\\widehat{\\boldsymbol \\theta}_{LS}", "275d93aed63268d7f17314861156435b": "\\theta = \\frac{\\pi}{2}", "275daf7d20ada2812241b6fbe246aae9": "d\\omega(X,\\mathbb{D})/dH^1", "275dc431f9270317c68595220d6b8730": "v_{j}", "275de3118360796a2e523dbed9f4c044": "\\mathbf{v'}_r \\cdot \\mathbf{\\hat{n}} = -e \\mathbf{v}_r \\cdot \\mathbf{\\hat{n}}", "275dfd3a7281b1ba81bdd8ab1be29465": "V_D", "275e5785cd9aa8c6141e54d6378fe0de": "\\displaystyle \\operatorname{Tr}(R(f)) = \\int_{\\Gamma\\backslash G}K_f(x,x) \\,dx.", "275e68530f43b1233e17955c496f6be3": "\\neg A \\or B \\iff \\neg A \\or (\\neg \\neg B)", "275effecf23e17e575a1e5ef0b480385": "S:\\{0,1\\}^8\\to\\{0,1\\}^8", "275f1d59d7ef4ee5d1a381e9b83ccce7": "\\scriptstyle f/f_s,", "275f3d8e81cd44f0fc70179f4852e725": "R_i\\,=\\,\\sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}", "275f48850d587181d5e397125b162a14": "\\Pr(X \\geq a) = \\Pr(\\Phi (X) \\geq \\Phi (a)) \\leq \\frac{\\textrm{E}(\\Phi(X))}{\\Phi (a)}.", "275f6b0cbab448a7a6f8e52da71aa7bf": "\\check{R} = T \\circ R", "275fa8e2c161ed4a15cf65b6fb619f5a": "2(2n+1)XP_n(X)=-P_{n+1}(X)+(2n+1)P_n(X)-n^2P_{n-1}(X).", "27600dac5d54ba3e39941d5cc6cfa3da": "n=\\pi\\left(\\sqrt[3]{m}\\right)", "2760555d69e9fc19df02ab85f2440a3b": "R_\\alpha[f](s)= R[f](\\alpha,s)", "2760aa4b069c475548ea095563ab2a96": "\n\\begin{align}\n \\varphi \\colon \\mathcal{X} &\\to \\R^{\\mathcal{X}} \\\\\n\\varphi(x) &= k(\\cdot, x)\n\\end{align}\n", "2760c769ad326e9af6a83ddf5a51131b": " \\frac{dS}{dt} = \\mu N - \\mu S - \\beta(t) \\frac{I}{N} S ", "2760ceb1f2f894726e40ce3387b99a0f": "\\begin{cases}\n \\;[~]_{\\mathcal{S}(\\sigma)} & \\mbox{if } P=\\sigma(~)\\\\\n \\;[\\mathcal{T}(R)]_{\\mathcal{S}(\\sigma)} & \\mbox{if } P=\\sigma(R)\\\\\n \\mathcal{T}(Q)\\;\\|\\;\\mathcal{T}(R) & \\mbox{if } P = Q\\,\\mid\\,R\\\\\n\\end{cases}\n", "27614dca15e222a355daab25528a0eab": "\n \\{ <_1, <_2, \\ldots, <_n \\}\n", "27614e270bda489e4eee7038e76f74bf": "p(c+\\rho z).", "27618de2d512b4918e032c24b793ab71": "\\sigma\\cdot(x_1,\\ldots,x_n) = (x_{\\sigma^{-1}(1)},\\ldots,x_{\\sigma^{-1}(n)}).", "2761f9529f1ee0b20b58ce5ecddff9e1": "C e^{a x}\\!", "2762084c8ab6713405b97010c9bd8c4d": "\\overline{X_n}:=\\frac{X_1+\\cdots+X_n}{n}", "27625e2398b270762ec395c1fcec2d30": "\\hat{G},", "276269c9fbc9a3c07e05ab12ae4600cb": "\\frac{\\partial}{\\partial t}(\\rho\\phi)+\\nabla (\\rho \\mathbf{u} \\phi)\\,= \\nabla (\\Gamma \\cdot \\operatorname{grad} \\phi)+S_{\\phi} ", "276279c8d1c2bd8f187c15b0834ec189": "\\Delta{\\vec{p}_{1,2}} = - \\Delta{\\vec{p}_{2,1}}", "27628bb77f94f52fc32b0ab3b77d8bdd": "\\mathbf{d}_z", "2762993a69d1b1c0c0a90fc7c63ac7e4": "x ^ {13}\\,", "2763295b50ce1bdac5f3a61b469d2def": "\nV \\approx \\frac{1}{2} \\sum_{s=1}^{3N-6} f_s q_s^2.\n", "276367f3edb021ea5e9463ed509ffbc3": "\\text{PCSA}_2 = {\\text{muscle volume} \\cdot cos \\Phi \\over \\text{fiber length}} = \n {\\text{muscle mass} \\cdot cos \\Phi \\over {\\rho \\cdot \\text{fiber length}}},", "276427364d6f1b07ed0a19d3c673e174": "rR''(r)+R'(r)-KrR(r)=0.\\,", "27643d2c5e4581732750efc2253347ec": "f_L (x;\\mu, \\sigma) = \\prod_{i=1}^n \\left(\\frac 1 x_i\\right) \\, f_N (\\ln x; \\mu, \\sigma)", "276457088f5bcc11f3ab6724fa1ed6a5": " \nI_1 \\left( mr \\right)K_1 \\left( mr \\right)\n\\rightarrow\n{1\\over 2 }\\left[ 1- {1\\over 8}\\left( mr \\right)^2 \\right]\n . ", "276466d4d0dfb02721775d26bde78090": "{\\mathfrak M}(K)=({\\mathcal P},{\\mathcal Z};\\parallel_+,\\parallel_-,\\in)", "276487f23a58ca0e6e32a6ebd539494a": "f(i, Y_{i-1}, Y_{i}, X)", "2764b53c84193b755ab4ca2decc87d74": "H=\\cup_{g\\in\\Gamma}\\, g\\overline{F}", "2764c01f9eb9624e8d01af0c07988c64": "\\mathfrak{B}(V_1)=k[x]/(x^2,y^2,xyxy+yxyx)", "27653afcc6a53702237297f96ebc8153": "Y(L_{-1} a, z) = \\frac{d}{dz} Y(a, z) = [Y(a,z),T]", "27657fab3a9b5731ceb5db440fb1914a": " S = Y \\log( p ) + ( Y - 1 ) ( \\log( 1 - p ) ) ", "2765802181072b3aa2be59dae8c72b0d": "b'", "2765ccf3a286be18c4c6f6ea271d9e83": "u_{0,k} = k \\pi", "2765fbff363c1babf12f12163b9485c3": " \\,\\,\\, = -k_B\\,\\sum_i dp_i (-E_i/k_BT -\\ln Z)", "27660d020719153b246cb01e754e1b10": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-2t})(1+p^{-t})}", "27661d60c1191af5e5b8f3cb22618dc3": "\\{(\\mathbf{a_i},\\mathbf{b_i}) \\}", "27665faa8edd1df61e6d1c1e6766e57c": " \\sigma_{ph,\\omega}[I_{ph,\\omega}(\\omega_{ph},T)-I_{ph}(\\omega_{ph},\\mathbf{s})]+ \\dot{s}_{ph,i}. ", "27666f9e2305c1fb80c6489f31036d2d": " \\mathbf{R}^{T} \\, \\mathbf{R} = \\mathbf{I} ", "2766aede88f76ff1a4159b47b28620bb": "x^2 y", "2766eb0d7b795c0563f89611547f28cc": " \\epsilon \\sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 ", "27671e790706777bc1031f3d8903da9a": " f\\left( \\frac{1}{\\frac{1}{r_n}} \\right) = 0 ", "276790d304068427af9e377fe64da34a": "\\lim_{x \\to \\infty}{\\frac{A}{x - 1} + \\frac{Bx + C}{x^2 + x + 1}} = \\frac{A}{x} + \\frac{Bx}{x^2} = \\frac{A + B}{x}.", "27680532e871ad86f92138e3feb018eb": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ -\\sqrt{\\frac{3}{2}},\\ 0,\\ \\pm2\\right)", "27681508c78678529ab81bc43ca12e97": "\\phi \\land \\chi \\to \\chi", "27681b7d5d9808e45b033b9734fff00e": "\\{\\varnothing, \\{\\varnothing \\}\\}", "276822ed44ee8d235b176d650d8e1ecb": "f \\longmapsto \\mathrm{Hom}(A,f) = [\\![ \\mathrm{Hom}(A,X) \\ni g \\mapsto f \\circ g \\in \\mathrm{Hom}(A,Y) ]\\!] ", "276824969ee0b642db5425f2a835ccde": "\\mathbf{v}_0", "27686af236aa56f334c6af2aef5b53eb": " L \\approx 0.318131505204764135312654 + 1.33723570143068940890116{\\!~\\rm i} ", "27688d8a5c35fe922e19ec29eb4bd21b": "\\tau >1", "2768ad4f672cd88cee064df228ac2a6f": "\\mathcal{O}(\\sqrt{\\omega})", "2768cfbdcda8d3f953a301764a286d9b": "\\gamma_i = 0", "276925d9f494e4ae53bea5184a3ebba4": " \\theta|_{y=0} = \\frac{d\\theta}{dy}\\biggl|_{y=L} = 0", "27695ef13053008baedb4fde723531d0": "Q = C L H^n", "2769617597ca4aac74079cf7e76d6685": "\\lambda = \\frac{g}{\\pi}T^2tan \\beta,", "2769fa447dac799fd4d54d734fd5cfd3": "\n\\begin{matrix}\n\\\\\n \\begin{bmatrix}1\\\\4\\\\2\\\\-3\\end{bmatrix},\n \\begin{bmatrix}7\\\\10\\\\-4\\\\-1\\end{bmatrix} \\mathrm{and}\n \\begin{bmatrix}-2\\\\1\\\\5\\\\-4\\end{bmatrix}\n\\\\\n\\\\\n\\end{matrix}\n", "276a36a8de51acc04265b78e5055945b": " \\mathbf{F}_c = 2\\omega \\frac{{\\rm d}r}{{\\rm d}t} \\bold{\\hat{e}}_\\theta = 2\\omega v \\bold{\\hat{e}}_\\theta \\,\\!", "276a99a28ea953338346c1131112576a": "M_A = 0 \\ kN \\cdot m ", "276b2aff8dfe17e21e0a27ae2ee8ea2d": "\\langle \\alpha', j'm'|T_{q\\pm 1}^{(k)}|\\alpha, jm\\rangle", "276b4a6542c0ae3cce06cc323f4150e9": "\\pi_2 = \\varepsilon_r", "276b641f3c57e4dfe79b3378fb70a2ef": "\n\\approx\n0.95\\times 10^{10} \\!\\left({\\sigma\\over 200\\,\\mathrm{km\\,s}^{-1}}\\right)^{\\!3} \\!\\!\\left({\\rho\\over 10^6\\,M_\\odot\\,\\mathrm{pc}^{-3}}\\right)^{\\!-1} \\!\\!\\left({m_\\star\\over M_\\odot}\\right)^{\\!-1} \\!\\!\\left({\\ln\\Lambda\\over 15}\\right)^{\\!-1}\\!\\mathrm{yr}\n", "276b647a7050a689e389a2658420a5e9": "\\sum_{n=1}^\\infty |z_n|^2", "276b6c4692e78d4799c12ada515bc3e4": "ana", "276b882cb9f2ecca73a495e1f0da925b": "\\overline{xy} = \\tfrac{1}{n}\\textstyle\\sum_{i=1}^n x_iy_i\\ .", "276b954f2a31deade9d3b18082bff936": "321 = (2)(13)", "276ba7f587cc2e4cdea0a452c2e6e0f4": "(I_3)", "276bc0f6849abfd539fbb2906a1287a0": "\\frac{{}_2F_1(a+1,b;c+1;z)}{{}_2F_1(a,b;c;z)} = \\cfrac{1}{1 + \\cfrac{\\frac{(a-c)b}{c(c+1)} z}{1 + \\cfrac{\\frac{(b-c-1)(a+1)}{(c+1)(c+2)} z}{1 + \\cfrac{\\frac{(a-c-1)(b+1)}{(c+2)(c+3)} z}{1 + \\cfrac{\\frac{(b-c-2)(a+2)}{(c+3)(c+4)} z}{1 + {}\\ddots}}}}}", "276bcdef15805a4aedbae90eb2849b26": "(0,1/n,1)", "276be6df76fb6dcd2a37ea9d10f2d98c": "p_{l}+\\rho_0gz_{l}-p_{fl}=p_{r}+ \\rho_0gz_{r}-p_{fr}", "276bf0578d572d7ad8f089c47c6f7216": "\\ \\Pi", "276bf6d36b19e5e075886d31e5421f34": "(Q_2, P_2)", "276c36dc91f0227dd1dd6de28156b820": " z = {{\\ln ( x ) - \\ln ( \\mu_g )} \\over \\ln \\sigma_g } = {\\log _{\\sigma_g} (x / \\mu_g)}.\\, ", "276c524b8fbdc720ad7a3b06fc310ea7": "PostCaP~ICO_{all} = 0.00875 + 0.0002 + 0.0005 + 0.0014 = 0.01085", "276c82a5468967167eeceacd6137cdbd": "\n\\operatorname{Li}_s(1) = \\zeta(s) \\qquad (\\textrm{Re}(s)>1) \\,.\n", "276d1346964c30b18876e6cbed120ca6": "\\displaystyle{T(x)=T_1x_1 + \\cdots + T_Nx_N,}", "276d15c5c74a9631e81dd6f31b47c3d1": "M(S)= M \\ / \\ \\sim_S.", "276d5c4039b4f274b98e1cb3c0b0484e": " x(t) = v t \\cos \\theta ", "276f3e7717e238a776cff0e656955855": "s_4(x)=\\frac{3}{2}x^2+\\frac{1}{16}x^4;", "277071d79ee0b5cc4dfa8846c52c613a": " \\ C_M(1/4c) = - \\pi /4 (A_1 - A_2) ", "2770865b917b1e47b31e439571cf2267": "\\mathit{S}", "2770f92b9acd33b84fe536a84419d9ae": " \\mathcal T ", "27715c32915f62d045488e5b2a0517d7": "\\lim_{x\\rightarrow c}\\frac{f(x)}{g(x)}", "27718aa0995792066ccdaf545221f3d2": "\\,K_i,K_j", "2771ba70fa13f13a15cd82a20b2c4108": " \n \\nabla^2 \\mathbf{A} - {1 \\over c^2} {\\partial^2 \\mathbf{A} \\over \\partial t^2} = - \\mu_0 \\mathbf{J} ", "2771fe1b76e8424110362d809d10cbc3": "f:X'\\to X", "27723048a786c1ccd2c59a484b7612cf": "w_j ( k + 1 )", "27726364d29e7c390d4e2cc48e696941": "[H,P_i]=0 \\,\\!", "27735626145ee7efdfb52259f1eb3745": "\\phi_1,\\lambda_1", "2773873bc183412daaf58b90a80f4c7b": "\\mathbf{M} = \\sum_i \\mathbf{A}_i = \\sum_i \\sigma_i U_i \\otimes V_i^\\dagger", "2773d3f3207e769483bf759171d05c90": "\\langle a,b\\rangle=(-1)^{w(a)w(b)}\\frac{a_0^{w(b)}\\prod_{i,j>0}(1-a_i^{j/(i,j)}b_{-j}^{i/(i,j)})^{(i,j)}}\n{b_0^{w(a)}\\prod_{i,j>0}(1-b_i^{j/(i,j)}a_{-j}^{i/(i,j)})^{(i,j)}}", "2773f41894cd947b9f5937684719d2f2": "~G(x,y,z)~", "2774038c84cfa89a32ac99ec96a8e725": "X_\\mu=(ct,-x,-y,-z)", "27740890e14f3b355b2a0ede6b0afa4a": "[x_\\mu,x_\\nu]= i \\theta_{\\mu\\nu}", "277409f676982d122fabc2809b14321f": "[f(x)]'\\,\\!", "2774990487f5973714ce57f25f278727": "H_p(B(S^{-1}S)^0) \\subset H_p(B(S^{-1}S)) = H_p(BS)[\\pi_0(BS)^{-1}] = H_p(BS)[e^{-1}].", "2775086897bc112fbcedbde39eeaeec3": "\\frac {(z_i-z)}{R_i}", "277551180bc42abbcec1a6dee5435059": "\\textit{Person} = (\\textit{coin} \\rightarrow \\textit{STOP}) \\Box (\\textit{card} \\rightarrow \\textit{STOP})", "277553df7f8b3fca553fffc9b493c849": "(U-c)^2\\left({d^2\\tilde\\phi \\over d z^2} - \\alpha^2\\tilde\\phi\\right) +\\left[N^2-(U-c){d^2 U \\over d z^2}\\right]\\tilde\\phi = 0,", "27756ecb2a661a0d820fab830157c678": "Q_c = Q_t - Q_s", "27757d2c07ae8bb420b7f07a22390d88": "P_\\text{fusion} = n_A n_B \\langle \\sigma v_{A,B} \\rangle E_\\text{fusion}", "2775a5f8640069e88d98163fc5215fc4": "\\langle E\\rangle = k_B T^2 \\frac{\\partial \\ln Z}{\\partial T}.", "27766270a8a2b056aafcbef3d5291e8c": "f(r) \\equiv 0 \\pmod{p^k}", "2776d992d78fb95984a7dbbd9ff024a2": "\\Diamond p \\leftrightarrow \\lnot \\Box \\lnot p", "27771121ff8c496a29efb698016f20ac": "C_n^{(\\alpha+m)}(x) = \\frac{\\Gamma(\\alpha)}{2^m\\Gamma(\\alpha+m)}\\! \\ C_{n+m}^{(\\alpha)[m]}(x).", "27778c9dc8a5ab5164523800c0e511ff": "x^{p-1}-1=0", "2777a9c48348dc2e0c8d818023ac6873": "\\mu_{p^{\\infty}} = \\left\\{ \\exp\\left(\\frac{2\\pi im}{p^{k}}\\right) : m,k\\in\\mathbb{Z}\\right\\}", "2777cf1cf8193b9df55bd36f4b118448": "w_{ni}^* = \\frac 1 {k^*} \\left[1 + \\frac d 2 - \\frac d {2{k^*}^{2/d}} \\{ i ^{1+2/d} - (i-1)^{1+2/d}\\}\\right] ", "2778282001f60831df0f501e37e0542c": "\\frac{d}{dr}\\sum_{k=0}^\\infty r^k = \\sum_{k=0}^\\infty kr^{k-1}=\n\\frac{1}{(1-r)^2}", "277867cc0b3b9c2c5a405749f9303cda": "\\ w_r < \\frac{l_e}{l_{us}} w_u ", "2778b188ef3f90ab1ed493da5f51979e": " \\alpha_h: \\mathrm{M}(a,b,c) \\rightarrow \\mathrm{M}(-h^{-1} b,h a, c -a b) ", "2778d9a8e2d5fed1ffb2445b446fd76d": "\\widehat{H}_\\alpha ", "2778f7b14a855985ad597a65abc2a579": "b = f - g", "2779269f045f8bbda797e5afbb852bef": "T_{n} = N_{n-1} = 3 \\cdot 4^{n-1} = \\frac{3}{4} \\cdot 4^n\\, .", "27793261375ecdf7dd47d7a25fe07a16": "r = r(t)", "27799505fdd23814dedab3b11e8345d6": "R = \\mu_1 + \\sqrt{S_1}X.", "2779ce5eec21976ed46a207ca9f9e334": "\nf_\\mathbf{v} d^3v = f_\\mathbf{p} \\left(\\frac{dp}{dv}\\right)^3 d^3v\n", "277a24d5c56626eeef4a229908a01905": "u^2 = \\cos^2 \\alpha \\frac{a^2 - b^2}{b^2} \\,", "277a860ea9ad635a55eddbc0f6c61eec": "p_{d-1}(a,b)\\neq 0,", "277a8c4a4283993d6c5fd438cce284ae": "\\Pi\\left(V\\right)", "277a9622da034f38ec117ed3e8b3ec63": "\\kappa(q_f) = \\kappa(q_r)", "277ab670bf31c63bb078b21122494a64": "y={{a-{1\\over T}}\\over c}", "277af8b5c5c2e7ca75dee13cef3758ab": "Q = q^k", "277b9d9777b3bccf36df1f678d7b4548": "\nP = \n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "277ba5faec4f25552f01dab091ba87d8": "\\scriptstyle \\rho=\\mu(\\{\\eta:\\eta(x)=1\\}) ", "277bb4c832f6f6fdc7d60c554bd2100b": " Y = {1 \\over 3} (n_u + n_d - 2 n_s).", "277bec74e1b0363a35a7d0ffa25f101e": "M^{\\bullet +}", "277c01cef39ce9b909029c3fdf5b76d9": "c_n \\Delta^{\\frac{n+1}{2}}_x\\int\\int_{S^{n-1}} \\varphi(y)|(y-x)\\cdot\\xi|\\,d\\omega_\\xi\\,dy = c_n\\Delta^{\\frac{n+1}{2}}_x\\int_{S^{n-1}} \\, d\\omega_\\xi \\int_{-\\infty}^\\infty |p|R\\varphi(\\xi,p+x\\cdot\\xi)\\,dp", "277c363162bd39dad34a80f8c3bbe897": "h:H\\rightarrow\\mathbb{C}", "277c77db3b85305869ed6ae26ddf123e": "\\binom{p+q}{q}-2\\binom{p+q-1}{q-1}=\\binom{p+q}{q}\\frac{p-q}{p+q}", "277c8cf393b13ebd0a34ae9d57495f86": "\\{x \\mapsto h(a,y), z \\mapsto b\\}", "277d9451dc0a15827eb130f81600a1bd": " M = 6n - \\sum_{i=1}^j\\ (6 - f_i) = 6(N-1 - j) + \\sum_{i=1}^j\\ f_i ", "277dd201ac2206d5ee43956fdb2ad69b": "j_1 = 1", "277dd68fe14717dcff20e76ca03772e4": "Gm", "277e5fee4b7a2c3f392e0caf6d455e1b": "\\begin{array}[b]{r}\\text{motor}\\\\SE\\end{array}\\;\n\\overset{\\textstyle\\tau}{\\underset{\\textstyle\\omega}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup\\!\\!\\!|}}\\;\\text{wheel}", "277ef04e7cde4aaf291aab9c523674e3": "P \\rightarrow P(T)", "277ef62e7aa17e0a4f5e3060a56e4ae6": "K \\le \\sqrt{(s-a)(s-b)(s-c)(s-d)}", "277f78988007fb392744f8b41daee855": "\\pi^{+}\\rightarrow\\mu^{+}\\rightarrow e^{+}", "277fa65207488c36e26a21e963dc7f5e": "\\ 0 \\leq x < 8", "277fccceea698fb9a3678a88a008fe38": "E_\\sigma", "278009087b515eb9e9effda62ab3a86b": "\\prod_{n>0}(1+q^{n-\\frac{1}{2}}z)(1+q^{n-\\frac{1}{2}}z^{-1})=\\left(\\sum_{l\\in\\mathbb{Z}}q^{l^2/2}z^l\\right)\\left(\\prod_{n>0}(1-q^n)^{-1}\\right).", "278042aa3dc61abd7e4acf4b0d7421ea": "\\omega=\\eta=\\epsilon=0\\;", "27806f29a817b174524f75eaea5b4e31": "1 \\to V \\to V \\rtimes_\\rho G \\to G \\to 1.", "278096b54b53e37123816a46e70ed0a8": "F[\\mu_j]=U-TS-\\mu_jN_j\\,", "2780a1eb8c1cfc7839c8f8f32f7fd3d8": "\n\\begin{align}\n\\Pr(Y_i=1) &= \\frac{1}{Z} e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i} \\, \\\\\n\\Pr(Y_i=2) &= \\frac{1}{Z} e^{\\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i} \\, \\\\\n\\cdots & \\cdots \\\\\n\\Pr(Y_i=K) &= \\frac{1}{Z} e^{\\boldsymbol\\beta_K \\cdot \\mathbf{X}_i} \\, \\\\\n\\end{align}\n", "2781100a20d1601ba6de2b39322653cc": "\\|fg\\|_1 \\le \\frac{1}p + \\frac{1}q = 1,", "27814e7a1c31a1afc1f321fce695bd37": "\\frac{2}{\\sigma^3 \\lambda^3} \\left( 1 + \\frac{1}{\\sigma^2 \\lambda^2} \\right)^{-3/2}", "2781afb6c92c8b845208de487f380d0c": "N_{s}=10^{-28/2.5}\\times\\frac{S_{0}(V) \\times B}{h\\nu}", "2781eefcce7b29c76f0ecd9107ae518a": "y/x \\leq 1. ", "278230924d309fdc65f33cec3b68c0f0": "p(A_k)", "278277eb98a0e3006b46980c5ac55547": "\\lambda_i = \\lambda\\text{ for all }i. \\, ", "2782a65fb858cea722d94af15bb6055f": "\\exists x (\\forall y (F(y) \\leftrightarrow y = x) \\land G(x))", "2782d8137a4a79615ffdc479e097d40f": " [\\mathrm{O}/\\mathrm{Fe}] = \\log_{10}{\\left(\\frac{N_{\\mathrm{O}}}{N_{\\mathrm{Fe}}}\\right)_\\mathrm{star}} - \\log_{10}{\\left(\\frac{N_{\\mathrm{O}}}{N_{\\mathrm{Fe}}}\\right)_\\mathrm{sun}}\n", "2782dc80168e214f664b0cd83bdaba27": "\\scriptstyle{\\theta=0}", "2782f462365c766ced3f3d5c4780f38f": "C_1\\ ", "27830add9dbbf2042e6da9411b974160": "x = a\\,", "27830e6e8d3eff22e76251e95a043b72": "\\operatorname{H}^2(\\mathbb{C}\\mathbf{P}^\\infty; R) \\simeq \\operatorname{H}^2(\\mathbb{C}\\mathbf{P}^1;R)", "278319e7836052d3cae89c209faf3ff1": "\\Phi_E(Q_d,k) \\geq k(d-\\log_2 k)", "2783403db5224fa06fe0a182ba793308": "L(\\mathbf{q},\\mathbf{\\dot{q}},t) = T(\\mathbf{q},\\mathbf{\\dot{q}},t) - V(\\mathbf{q},\\mathbf{\\dot{q}},t)", "27845a4e11b6879d8122a0a5b4abbe56": "I_N:\\{\\mathbb{X}\\subseteq\\mathbb{R}^n\\}\\rightarrow\\{\\text{newMin},..,\\text{newMax}\\}", "278496314a4cb5c510a96a538033909d": "\\phi_2,\\lambda_2", "278499dda22b50cdb53ffe795d2c472d": "{(21\\cdot 29)^2\\equiv2^1\\cdot7^1\\cdot11^2\\pmod{91}}", "2785026c4ed4574b65009dcb2485167a": " \\log \\left( |H(j \\omega)| \\right) = \\log \\left( |H(j \\infty)| \\right) + \\mathcal{H} \\lbrace \\arg \\left[H(j \\omega) \\right] \\rbrace \\ ", "2785288d99838cbe6a9d5d92a1021f8f": "\\Gamma=-1", "2785467333afc9ec461a225a5b5f94fd": "Q_1 = I - {2 \\over \\sqrt{14} \\sqrt{14}} \\begin{pmatrix} -1 \\\\ 3 \\\\ -2 \\end{pmatrix}\\begin{pmatrix} -1 & 3 & -2 \\end{pmatrix}", "27855269317242574206360c52adc400": "\\rho \\Phi + (8 \\pi G)^{-1} a_0^2 F(|\\nabla\\Phi|^2/a_0^2)", "27861a049634da5c376d4f0ee717a2cf": " (x+y)^2=x^2+2xy+y^2.", "278624858ec5139c481d5eeab6b033a1": "log_a(x) = b", "2786498c7e07749eb4a7022d71fee2a4": " n_0 + n_1 \\zeta + n_2 \\zeta^2 + ... + n_{m-1} \\zeta^{m-1}\\ ", "27865f16aa96098e5f083f8dd77a0858": "\n\\frac{\\partial c}{\\partial t} = \nD \\frac{\\partial^{2}c}{\\partial z^{2}} + \nsg \\frac{\\partial c}{\\partial z}.\n", "2786b220fb78cf6e30e21f26bf0bb48e": "h_i = \\left|\\frac{\\partial\\mathbf{r}}{\\partial q_i}\\right|", "2786b62736b5a7859ed30c0d06008d50": "z_1, z_2, z_3, z_4", "2786d392fc756a183f2a4a347b160331": "z=we^w", "2786d63a3193f49bbd1d588b0671d7d7": "\\mu=\\sqrt{2}\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}", "2786f63b0af6caa8cf83113c4b654b30": "\\mathtt{rec}\\ v = e_1\\ \\mathtt{in}\\ e_2\\ ::=\\mathtt{let}\\ v = \\mathit{fix}(\\lambda v.e_1)\\ \\mathtt{in}\\ e_2", "278714589a64c20353b8acb7f99afc38": "\\pi/2 ", "278724a1c6095bdbffc5ba8b8e28c8ad": "y = f(a) + M(x - a)", "278766b6672e787ba80de2ded169b523": " P( {ax,by}{|}{AX,BY} ) = \n\\begin{cases}\n\\frac{1}{2}, & \\mbox{if } x \\oplus y = XY \\\\\n0, & \\mbox{otherwise}\n\\end{cases} ", "27877e43e6809e202f73605d1ae6300d": " \n\\begin{align}\nf(x,y) \\approx & \\, \\frac{f(Q_{11})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y_2-y) \\, + \\\\\n & \\, \\frac{f(Q_{21})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y_2-y) \\, + \\\\\n & \\, \\frac{f(Q_{12})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y-y_1) \\, + \\\\\n & \\, \\frac{f(Q_{22})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y-y_1) \\\\\n = & \\, \\frac{1}{(x_2-x_1)(y_2-y_1)} \\Big( f(Q_{11})(x_2-x)(y_2-y) \\, + \\\\\n & \\, \\qquad \\qquad \\qquad \\qquad \\; \\; f(Q_{21})(x-x_1)(y_2-y) \\, + \\\\\n & \\, \\qquad \\qquad \\qquad \\qquad \\; \\; f(Q_{12})(x_2-x)(y-y_1) \\, + \\\\\n & \\, \\qquad \\qquad \\qquad \\qquad \\; \\; f(Q_{22})(x-x_1)(y-y_1) \\quad \\Big)\n\\end{align}\n", "278787e660237506b2acd60564befb65": "0.00303993", "278793f3bb0863943f1e3c0c1f124505": "\\mid \\uparrow \\rangle \\to \\mid \\downarrow \\rangle", "27879488452eee2a1b5ff65c794acfc1": "\\varepsilon \\cdot\\bigl((M-m)+(b-a)\\bigr) = K\\varepsilon,", "2787a64c0db89fee93b7696cb340035d": " j_{\\mathrm{F}} [ \\,= z_{\\mathrm{S}} d_{\\mathrm{F}} D_{\\mathrm{F}} ] ", "2787d3aa73e809b30000da5417abc2bd": "x_n=\\ell+a^n+b^n", "2787dc05d00aa076f0f534bd7c26a415": "x = g(z,u)", "2787f66b0a4fe60163c5dec33ccca6a7": "\\begin{bmatrix} 1 & 0 \\\\ -sC & 1 \\end{bmatrix} ", "278842ba20f0911f76d9c3e3bf1addfe": "\\{F,G\\}=\\frac{\\partial_rF}{\\partial z^i}\\omega^{ij}(z)\\frac{\\partial_lG}{\\partial z^j}.", "2788914065c0778587e2c22849860121": "\\operatorname{Spec} (B \\otimes_A R) \\to \\operatorname{Spec} R", "278891d667d681e9d0fba2c786ef0749": "PA_{n}", "2788b6a89313f71ec28a36b3c3e7898d": "q(t)f(t)", "2788e071b64971b7e4edd78982ce711b": "\\|T\\|", "2788fceede6fb2f41860d100e2fde661": "\\in D", "27896d8fa05b9ade38dd6f1566a38e0b": "\\left\\{ \\mathbf{k} \\in \\mathbb{Z}_{0+}^c \\, : \\, \\forall i\\ k_i \\le K_i , \\sum_{i=1}^{c} k_i = n \\right\\}", "2789ae5a6db7b750b3329c848e7d1677": "J_i=-\\Gamma_{,i}. \\, ", "2789be610f2235c8d9bf2a408d619011": "|w|<1", "2789c618b572916756f8a7efde45ef76": "Y=U", "2789d39a74487af568da6d2bc6440058": " \\frac{d[C]}{dt} = k_1 [A]", "2789f9cc1afbe55db73eec0a7dde3588": " \\Delta(x)=\\left\\vert \\psi(x)-x \\right\\vert < \\sqrt{x} \\log^{2}(x)/(8 \\pi)", "278a3e63434453b1c123458fd8ca8558": "\\mathbf{e}\\,\\!", "278a5c41893b234bc7ce50f94ddaa587": "p_\\mathrm{\\varphi} = m r^2 \\dot{\\varphi}", "278ab54f4842959cabca9f63296aa637": "{dx \\over d \\tau} = \\lambda {p \\over m}, \\;\\;\\;\\; {dp \\over d \\tau} = - \\lambda m \\omega^2 x ; \\;\\;\\;\\;\\;\\; {dt \\over d \\tau} = \\lambda, \\;\\;\\;\\; {dp_t \\over d \\tau} = 0,", "278ae7eb6a62d45166f28b78b0db9b61": "\\begin{align} \\Phi(p,t) & = \\frac{1}{\\sqrt{2\\pi\\hbar}}\\int\\limits_{-\\infty}^\\infty d x \\, e^{-ipx/\\hbar} \\Psi(x,t)\\\\ \n&\\upharpoonleft \\downharpoonright\\\\\n\\Psi(x,t) & = \\frac{1}{\\sqrt{2\\pi\\hbar}}\\int\\limits_{-\\infty}^\\infty d p \\, e^{ipx/\\hbar} \\Phi(p,t).\n\\end{align}", "278b02c71884316642d17c880d397ab5": " \\theta_0 = \\varphi,\\,", "278b04bace20474a4fb53ed31bcdfa11": "mP \\neq 0", "278bc796fc1eb57e965e35bad1636342": "\\frac{1}{0} = \\infty", "278bcacaa64e0c1de4f54215f17885c0": "\\operatorname{Id} *1 =\\sigma_1 =\\sigma", "278c413fc153007da628afabde022232": "D(a) = \\begin{pmatrix}\nD(a)_{11} & D(a)_{12} & \\cdots & D(a)_{1n} \\\\\nD(a)_{21} & D(a)_{22} & \\cdots & D(a)_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nD(a)_{n1} & D(a)_{n2} & \\cdots & D(a)_{nn} \\\\\n\\end{pmatrix}", "278c480989bb273d9b1928e3c6e5a5be": "\\qquad \\qquad \\langle\\varphi\\rangle = \\langle\\varphi\\rangle_\\mathrm{o} + \\sum_i\\sum_\\alpha\\frac{\\partial\\langle\\varphi\\rangle}{\\partial d_{i\\alpha}}|_\\mathrm{o}d_{i\\alpha} + \\frac{1}{2}\\sum_{i,j}\\sum_{\\alpha,\\beta}\\frac{\\partial^2\\langle\\varphi\\rangle}{\\partial d_{i\\alpha}\\partial d_{j\\beta}}|_\\mathrm{o}d_{i\\alpha}d_{j\\beta}+ \\frac{1}{6}\\sum_{i,j,k}\\sum_{\\alpha,\\beta,\\gamma}\\frac{\\partial^3\\langle\\varphi\\rangle}{\\partial d_{i\\alpha}\\partial d_{j\\beta}\\partial d_{k\\gamma}}|_\\mathrm{o} d_{i\\alpha}d_{j\\beta}d_{k\\gamma}+... + ", "278c4c35ddf978e5ed9edd1a076ad7d8": "g: \\pi^{ab} \\to \\pi^{ab}/tor", "278c64c28fc40fc2da7a8c54a01d7d4e": "\\to\\,", "278c93854dce3f4aaa2062d37ed7db84": "\\Xi_i(t)", "278d16f919cbecd1f344de14c397f03a": "p = 4^a(8b + 7) = (2^a)^2((8b + 6) + 1)", "278d1bef0e46890e914e6480716141af": "\\lambda_D = (kT/4\\pi ne^2)^{1/2} = 7.43\\times10^2\\,T^{1/2}n^{-1/2}\\,\\mbox{cm}", "278d4af61479db825fcdc1232439828b": "R(N_1,N_3)", "278d5464a3ad83f50cd6a14afe7b268c": "\\lambda_k \\rightarrow 0", "278d8c3103ca7e8384e0843e88214a5b": "\\scriptstyle\\hat{s}", "278da0421a6ba0a7cd3acb612cda1c94": "X_1 \\vee \\cdots \\vee X_{k-1} \\vee Z", "278da48d6363aef31fcd284eb3230389": " TPR(T)= \\int_{T}^\\infty P_1(T) dT ", "278dc0c0edbd70a79deaf77fb4117bc8": " \\alpha(\\gamma)= 2(\\sinh^{-1}(\\gamma)-\\frac{\\gamma}{\\sqrt{1+\\gamma^{2}}})", "278dd91adf3e671184af92c29ca567d5": "Vol_q(pn,n) \\le q^{H_q(p )n}", "278dfef6de4292a2fb7f26d718968870": "\\frac{\\mathbf{\\Psi}}{\\nu - p - 1}", "278e097ec5efd59ec0ef3ad16f9f4bfc": "R \\times S", "278e10d973aff035eaf74fa0fd2421be": "[x]_Q = \\{Q_1, Q_2, Q_3, \\dots, Q_N \\}", "278e36c188bf833338cacfae1c15a6f7": "{}^5_3", "278e82059adbbd5921a129414f5323da": "\\overline{d}(A) = \\limsup_{n \\rightarrow \\infty} \\frac{n}{a_n}", "278e8ebb29da7d1dfd79d4a3e49d7d56": "T_{TF}[n]= C_F \\int n(\\vec{r}) n^{2/3}(\\vec{r}) d^3r =C_F\\int n^{5/3}(\\vec{r}) d^3r\\ ", "278e9c7cf97ff5bca9c568f16c8b58dd": "W_i^L || W_i^R = W_i", "278f07fc5f5be794993672bbc5cfc167": "[x_1, x_1]", "278f49c68eaf3c21246738a88f60963a": "\n\\frac{dG}{dt} = 2 T + \n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = 2 T - n V_\\text{TOT}.\n", "279032e265d6f19c137eea07601d0b62": "\n F_2\\sigma_2 + F_3\\sigma_3 + F_{22}\\sigma_2^2 + F_{33}\\sigma_3^2 + F_{44}\\sigma_4^2 \n + 2F_{23}\\sigma_2\\sigma_3 \\le 1\n ", "27903ea4c673d6da9760181ab3565712": " \\omega \\in L(\\mathcal{G},\\infty)", "279044e5d399eac4972990945c1bc199": "|t - 1| \\,", "27906170850e5338f2680b069d89ed94": "\\begin{matrix}\n\\times & d & e & f & g \\\\\na & ad & ae & af & ag \\\\\nb & bd & be & bf & bg \\\\\nc & cd & ce & cf & cg\n\\end{matrix}", "27906689b24df409980c177214bbb54f": "x^3 = y^2", "27907c3735f3cb3a7554dbc87516cbba": "\n\\begin{array}{c|ccc}\n0 & 0 & 0 & 0 \\\\\n1/2 & 5/24& 1/3 & -1/24\\\\\n1 & 1/6 & 2/3 & 1/6 \\\\\n\\hline\n & 1/6 & 2/3 & 1/6 \\\\\n\\end{array}\n", "279082712e06a4e84a1ef17f047d7268": " I = \\int_0^{eV}\\rho_S\\left(r,E\\right)\\rho_T\\left(r,E-eV\\right)T\\left(E,eV,r\\right)\\,dE\\ ,\\qquad\\qquad (5)", "2790a140944185518dda5f808f6714dc": "\\nabla G(\\theta,\\phi)=\n\\left({{\\partial}G\\over {\\partial}\\theta},{{\\partial}G\\over {\\partial}\\phi}\\right)\\!\\left(\\theta,\\phi\\right)=(0,0),\\,", "2790e3e7089f798657f49a9d0d4c1e92": "d(f,g) = \\sup_{x \\in X} d(f(x),g(x))", "27910c515720cbf2ebeecb4807bd18b7": "\\mathit{k_t}", "2791535868f5a81e79af88fe6cf17736": " \\ A \\vee B := (A \\rightarrow B) \\rightarrow B. ", "27916dc0137bebe168302a3f4c8b27cf": "\\phi^1=Re{\\phi}", "2791938e0e42cfd7f491c8dc89472b04": " h(i+1) \\geq \\text{max }(i,h(i)) \\text{ for all} 1 \\leq i \\leq n-1. ", "27925d253a84ba533da9712b3adb140d": "b + \\frac{aW}{1-\\gamma} > 0.", "27927abcdc1b5f2a5532be3975233415": "\\Omega_z", "2792ca8f343c1af025648623daa2f3e3": "\\lambda_1,\\, \\lambda_2, \\,\\dots,\\, \\lambda_M", "2792f3248e6fab6e7a8243919f8b9a79": "\\Gamma^{\\lambda}{}_{\\alpha\\beta}=\\frac{1}{2}g^{\\lambda \\tau} \\left(\\frac{\\partial g_{\\tau\\alpha}}{\\partial x^\\beta} + \\frac{\\partial g_{\\tau\\beta}}{\\partial x^{\\alpha}} - \\frac{\\partial g_{\\alpha\\beta}}{\\partial x^{\\tau}} \\right) ", "2793202d75b2f767f816f29bd32baea0": "n \\equiv 3 \\pmod{6}", "279354cf67df39c6278075b103118bfb": "P_0 = \\int_R^\\infty g\\rho dr", "2793a6a7a73f4aac4c4e432f9546353e": "(X_{t_{i+1}} + X_{t_{i}}) / 2.", "2793d8f16e891373a4006cb39d821be5": "\\sigma(p_{S_i}) = \\alpha_i(p_{S_i}) \\prod_{v_k \\operatorname{adj} v_i} \\mu_{k,j}(p_{S_k\\cap S_i})(2).", "2793e6b7b72be69fe3e1bc2e29b526c3": "b \\mapsto (F^pH^k(X_b, \\mathbf{C}))_p.", "27940520375a3749ad77c0219c017d79": "M_{PAW} = \\frac{(R)(T_i)(P_I)+[60-(R)(T_i)](PEEP)}{60}", "27942f80a0bf2a74e414b70181805efd": "a_{ji} \\in R", "27951c1b59c5c18b99439d61b7f90765": "\\sum_k n_k^{(-n)} + \\alpha_k = A + \\sum_k n_k^{(-n)} = A + N - 1", "2795375d3c455eb85839237dec9f5d24": "\\textrm{B}(n,\\lambda/n)", "27956635477bf9342ec0a93147ba1b8d": " \\begin{align}\n&\\lim_{\\alpha = \\beta \\to 0} G_{(1-X)} =0 \\\\\n&\\lim_{\\alpha = \\beta \\to \\infty} G_{(1-X)} =\\tfrac{1}{2}\n\\end{align}", "279581f5551f8c88b73f46a2e171bf1d": "\\mathbf{E_T}=\\mathbf{E_0}e^{i(\\mathbf{k_T}\\cdot\\mathbf{r}-\\omega t)}=\\mathbf{E_0}e^{i(xk_T\\sin(\\theta_T)+zk_T\\cos(\\theta_T)-\\omega t)}", "2795b6c64e68214badc28fe466884913": "d(a\\mathbf{X}) =", "2795ff0ee8570ad8abccbb8b4adc7370": "2^{n^k}", "27961e6dadfdaa121bcc6f1266c8eae0": "\\; U_R (s_i) ", "279632db6704008436fb7729f03962ef": "\\bold j = \\frac{1}{m} \\mathrm{Re}\\left ( \\Psi^*\\bold{\\hat{p}}\\Psi \\right ) ", "2796384a6eb9df635b956bfcf17ea87d": "\\mathbb{C}^{n+1}", "27964916d0a9d03db97ccda5fa6f89cd": "\\ AB = BA", "279669cfce7b50380c69c2f7726b61af": "f_* : P(X_1) \\rightarrow P(X_2) \\,", "279678fbb2ebca8a5941b2929e0a6bc7": "\\scriptstyle { x \\in S_*: x < 0 }", "279688d4413ecbfda9518757ed89969a": "\\sigma_{i}", "2796ae9874431344b7f0393b612ff4aa": "\\left(\\frac{x}{a}\\right)^2 - \\left(\\frac{y}{b}\\right)^2 = 1", "2796af5074a7f27ecccd3cd17e165d53": "c^2", "2796c393553c1ae31239afc9217421fa": "\\Pr[p_i = 0] \\leq \\frac{1}{2}", "2796ce5b2e690868654778e01c06caf2": "\\sqrt{n(n+1)}", "2796f963954b8fa957de06a9ebe0b57e": "{(-\\Delta)^{\\frac{1}{2}} u = f},", "279701cc8fc997aa92967ed4a7efc10d": "\\tilde Z(s) = \\left( \\frac{\\lambda}{\\lambda+s} \\right)^n.", "27970285948903787962b5a5dd620273": "\\begin{align}\n (r + c - 1)(r + c + \\alpha + \\beta - 1) + \\alpha\\beta &= (r + c - 1)(r + c + \\alpha - 1) + (r + c - 1)\\beta + \\alpha\\beta \\\\ \n&= (r + c - 1)(r + c + \\alpha - 1) + \\beta(r + c + \\alpha - 1)\n\\end{align}", "27973c6f638ca1138e089c0c808513f1": "a_\\alpha", "27973d540e8c9f23fc9e9c5c5655c206": "\\ a^* = \\dot{v}^* = \\dot{Q}v + Qa + \\ddot{c} + \\dot{\\Omega}(x^*-c) + \\Omega(v-\\dot{c}), ", "279745c4603d4a2b7326742e9e5ad7e5": "a \\cdot W_0^a \\cdot \\text{E} [R_1^a \\cdot R_2^a \\cdots R_T^a],", "279774a05fb144a502945a240531c90e": "\\delta W_s", "279799552be553615e434ab17245584a": "f(a, b) = b", "2797eb44363ca064e6727a6dbb4ac9b2": "\n\\begin{align}\n\\sum_x \\frac{\\delta H}{\\delta p(x)} \\, \\phi(x) \n& {} = \\left[ \\frac{d}{d\\epsilon} H[p(x) + \\epsilon\\phi(x)] \\right]_{\\epsilon=0}\\\\\n& {} = \\left [- \\, \\frac{d}{d\\varepsilon} \\sum_x \\, [p(x) + \\varepsilon\\phi(x)] \\ \\log [p(x) + \\varepsilon\\phi(x)] \\right]_{\\varepsilon=0} \\\\\n& {} = \\displaystyle -\\sum_x \\, [1+\\log p(x)] \\ \\phi(x) \\, .\n\\end{align}\n", "27981a0c2bd554cba6e7035c359028da": "\\Delta u = f", "2798492849fd554d789c1cb2f9338bb9": "Z = n_i \\times (c_{ij} \\times [Z]_j) = (n_i \\times c_{ij}) \\times [Z]_j", "279852f50a749fd08b9548a8474ed695": " \\Pr(| x \\le m - k \\sigma |) \\le \\frac { 1 } { k^2 } \\frac { \\sigma_u^2 } { \\sigma^2 } .", "27985b3610e8c739e119ac305044aff4": " f(S_{x,i}, t) ", "27991710535b65e6025686c1e8670158": "\nM_i(x|1,t) = \\frac{1}{t_{i+1}-t_i}\n", "27996dffc97f69742e5f5ae1ac57c6d2": "f_{n-1} \\circ d_n = e_{n} \\circ f_n ", "2799ab38e367ec92f11784aa84f0cf54": "\\Delta S = R * (a-\\frac {c}{T^2}).", "2799d3215bc1b5dc7f4cdd1a8c7a044a": "mc^2", "2799e0879627f8907c4f98efe9abdf88": "T = \\{ x_2^2-x_1^2, x_2(x_3-x_1)\\}", "279a252ee34ee8ee2f8473a55587090d": "\\Gamma_0(N) = \\left\\{\n\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in SL_2(\\mathbf{Z}) :\nc \\equiv 0 \\pmod{N} \\right\\}", "279a3c1a9cb094e1deb57d50bc499e9f": "m+S(n)=S(m)+n", "279a541fc719ff2c042c75e6718e47b8": "\\Gamma = \\frac{V_R}{V_F} = \\frac{B \\exp(-\\gamma l)}{A \\exp(\\gamma l)} =C \\exp(-2 \\gamma l)\\,", "279aa2547e65651c954067b66e2ce868": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 1}{4 \\choose 2}{11 \\choose 2}{4 \\choose 1}^2 = 3,294,720", "279aa49cb2e8d29aeb5181bc3c80e9e0": "\n \\frac {52!}{(52-5)!} = \\frac {52!}{47!} = 52\\times51\\times50\\times49\\times48 = 311,875,200\n", "279aa94e1cc07bc98744a6fe5cac570b": "k = \\left(\\sqrt{\\left|\\real\\{Z_{p}\\}\\right|}\\right)^{-1}\\,", "279ae870aeb864a102f0ee59af7ef490": "\n\\mu = \\frac{n\\alpha}{\\alpha+\\beta}=n\\pi\n\\!", "279aff2fa8a7a986117a23e93ffc621c": "f(x)=x^2+3x+4+\\frac{2x^6-4x^5+5x^4-3x^3+x^2+3x}{(x-1)^3(x^2+1)^2}", "279b6672e4cfb15e41a1e4deebfc9f22": "(x_n^{(1)})_{n\\geq 0} = (0,1,2,3,4,\\dots)", "279c1af2d4950cf38a2364984484afaa": "\\lambda_k\\rightarrow\\infty", "279c52d08b4fb24e43173940c137a3c7": " = \\mathbf{u}_{\\rho} \\left[ -\\frac{v^2}{r}\\right] + \\mathbf{u}_{\\theta}\\left[ \\frac {\\mathrm{d} v} {\\mathrm{d}t}\\right] \\ ", "279cd731931c0e2318aca0de26c7cc54": " x = \\frac{1}{k_1 + \\frac{1}{k_2 + \\cdots}} ", "279cf7547a58b656db0e05bc2c2934c4": "a(n,k,X(\\Omega))", "279d1a6893a98cd638d8a8c4a7ce632d": "p = p^{\\star}_{\\rm A} x_{\\rm A} + p^{\\star}_{\\rm B} x_{\\rm B} + \\cdots", "279d1d55f61d7f5854542f7df1aa20a7": "A \\times B ", "279d520ebed641c0cd1854227a3e304d": "F(X,Y) = P(X) + Y*Q(X)", "279d91e3a85738e1a99827cc2fb1e478": "x_{i_j} \\in X_{i_j}", "279d9621fdedfe6f13cb322ea107e862": "e^{-x^2},", "279dac11169af97009e127f834b6b195": "9 \\times x", "279db70eb0feab5767eae729a98ce527": " \\Delta(\\tau^k)=\\sum_{i+j=k} \\tau^i\\otimes \\tau^j ", "279dc6c6dc1a7d5362f6ecb4e0ed828b": " a_n := \\frac{2}{L} \\int_0^L f(x) \\cos\\left(\\frac{2\\pi nx}{L}\\right)\\, dx", "279de18dce0d9a45538934526c4ac45b": "V_{\\text{cc}}", "279ebb36693495fef0683bb02f9c05c5": "s\\rightarrow t", "279f0b9e99f9c8b30282f11ded4b7136": "\\dot{\\theta} = \\omega^2 - \\sigma^2 - \\frac{\\theta^2}{3} - {E[\\vec{X}]^m}_m", "279f3aca7e1f966251a3f205d6da9de7": "\\sigma_x^2(-1) = \\lambda\\sigma_d^2(-1) = \\epsilon\\,\\!", "279f5cd10832a8d1f30ec7b3b9ea961a": "r_{FOR}", "279f9912912de25162dfeb20b4c1f2c0": "\\tfrac{3\\pi}{2}", "279fa09409eafe6144cb3ca00d16f2a3": "\\varepsilon_1 \\neq 0", "27a021b65d522849d7d2753577fa01b3": "\\sum_{n=-\\infty}^{\\infty} x(n\\cdot MT)\\ e^{-i 2\\pi f n(MT)} = \\frac{1}{MT}\\sum_{k=-\\infty}^{\\infty} X\\left(f-\\tfrac{k}{(MT)}\\right).", "27a050edfb4cd6af1c12ff2084077909": "\\mu_{\\mathrm{JT}} = \\left( {\\partial T \\over \\partial P} \\right)_H = \\frac{V}{C_{\\mathrm{p}}}\\left(\\alpha T - 1\\right)\\,", "27a15290a1f6bc2149e7ebd8d435d3a6": "f \\sim g \\quad (\\text{as } n\\to\\infty)", "27a17f736f9b10ea1f9fe2a00c82dfa9": "\\displaystyle x - y", "27a19f5b299251143745fc275925313c": " h = {v_0^2 \\sin^2(\\theta) \\over {2g}} ", "27a1b185ddb89e5586285d9373526702": "\\Phi_o", "27a1ca1e814fd66c7b8c00777f0698fb": "n \\ge \\ell+1", "27a1f5a6c607a8978b1a7847d7af1b6d": "\\tilde{e}_i B \\subset B \\cup \\{0\\}", "27a247e7b6344dd7a5441367b899777f": " \\overline{X} ", "27a28ee9a7c46f034a67670a9d8f7684": "S \\rightarrow ab", "27a2fa3776d5c9dc1265f32ea8494057": "m^2 + 1", "27a3299f3e86235eb2d3447a0111c5dd": "I(X:Y) \\leq S(\\rho) - \\sum_i p_i S(\\rho_i)", "27a3388c165427c902bd205be3df6845": "\\iota = (-8-16y-18w+12w^2+10yw+yw^2)/23", "27a356297ad9e4bac3d1c40d9f827b63": "R_n = R_o + (W-L) \\times 16 + D * 0.04\\ ", "27a37c4fe93e5908db2dc03e0f1d6f18": "H_n\\left(p_1, p_2, \\ldots \\right) = H_n\\left(p_2, p_1, \\ldots \\right)", "27a3872b281ce0ac5eb15f4719960df1": "M_{rot}=\\left( \\begin{matrix}\n \\cos \\theta & \\sin \\theta \\\\\n -\\sin \\theta & \\cos \\theta \\\\\n\\end{matrix} \\right)", "27a399a8ccbc92b409c763abf8f35f22": "|U|=|V|", "27a3ac0f49e13bf96c7de04cc47a5012": "D\\approx P \\mathrm{e}^{-G} \\approx \\mathrm{e}^{-G}. \\qquad\\qquad (7)", "27a3bd0109bf5414d13052cf84b5db8a": " \\epsilon_i \\in \\{\\pm 1\\}, t \\in [-1,1]", "27a3ffaeb0cc60ab591414f32ba29abb": "2^\\mathfrak c = \\beth_2", "27a4634bc931edbca23e3c16df7d2f56": "\\boldsymbol{\\sigma}*", "27a4741dc69513bd2fdaf2bce8de5355": "\\Delta \\subseteq K^{\\times}/(K^{\\times})^n, \\,\\!", "27a47eb93c74bbe3df1c6e3413d51ed0": "|\\sigma_{ef}|^2", "27a4bafa06a43269b6741fa3bcf63ca3": "\\Delta G^0 = -287kJ ", "27a510e920511aee7430a0f676899b6f": "L\\geq 3", "27a57511a2df4d705df5b9c7ef28e2d8": "\n\\mbox{If } \\alpha\\equiv\\eta^n\\pmod{\\mathfrak{a}}\\mbox{ then }\n\\left(\\frac{\\alpha}{\\mathfrak{a} }\\right)_n =1.\n", "27a5892346ed9715d2852b2309022bf5": "\\Phi(z, s, q) = \\sum_{k=0}^\\infty\n\\frac { z^k} {(k+q)^s}\\!", "27a595ea4c3e3e0397ceccd58a465601": "X_b", "27a5d4e6447556cd8c23dbd067ea4092": "6 \\rightarrow \\infty", "27a6341c2863995eef4fc01feed13597": " t \\equiv \\frac{T-T_c}{T_c}", "27a65ac1b4863f3eb9a56d95157dbd01": "V(\\rho,\\varphi,z)=\\sum_{n=0}^\\infty \\sum_{r=0}^\\infty\\, A_{nr} J_n(k_{nr}\\rho)\\cos(n(\\varphi-\\varphi_0))e^{-k_{nr}|z-z_0|}", "27a6e74689cbdd9dcec0b4832f918b71": "\nM_{k} \\equiv \\int d\\zeta \\ \\lambda(\\zeta) \\zeta^{k}\n", "27a7579d0309d06241abeb0bccb4ac99": "A\\times B", "27a79e7b529d1c2537b65ad971f64ac6": " G = (\\{S, A, X\\}, \\{a\\}, S, \\{f,g,h,k,l\\})", "27a7ca65090df9d37e1a58a92d2a120b": "\\sigma_f \\approx a \\frac{\\sigma_A}{A \\ln(10)}", "27a80f01d2e612e23e2906929bab23f6": " P_\\ell(\\cos\\theta)", "27a869566bd3162b0e5815cb9c79fec8": "\\sum_{n=1}^\\infty \\frac{(-1)^n}{n(9n^2-1)} = 2\\ln 2 -\\frac{3}{2}.", "27a88b9153e137a48135d5a0734f9639": "{mv^2\\over r}= {mg\\tan \\theta}", "27a89e2e3385601726b145dbb57c330b": "t = (N+1)\\tau", "27a8a828cafff3320cc5c71bac2d9d82": "\\Rightarrow \\int \\frac{dx}{x} = \\int k \\, dt", "27a8ebad4e6b9550dd062968667d229c": "f(x)=\\frac{a_0}{2} + \\sum_{n=1}^\\infty \\, [a_n \\cos(nx) + b_n \\sin(nx)].", "27a91cd57a2c707d8973bdbd5a22f4a6": "f\\colon \\coprod_i Y_i \\to X", "27a97831c0ed4004ddd04c3b1a962c7f": "\ni\\bar{\\partial} \\Psi \\mathbf{e}_3 \\rightarrow \n(i \\bar{\\partial} \\Psi) R \\mathbf{e}_3 + (e\\bar{\\partial}\\chi) \\Psi R \n", "27a9a02a29335319f5104184ef93789d": "(\\mathbf{F}, \\mathbf{G}) = \\int \\mathbf{F}^* \\cdot \\mathbf{G} \\, dV", "27a9ff394d53cc33da67bc3b592df069": "\\nabla^{LC} g = 0 ", "27aa222bd2e56b43d428a769bb91f451": "\n q - \\mu\\ddot{w} - (EI w_{xx})_{xx} = 0\n ", "27aa289889bd651838d87a1863679bd6": "\\mathbf{B} = (B_x,B_y)", "27aa4d0b8c7f3bfbd86f04c285c7fab9": " \\mu = \\frac{mM}{m+M}", "27aac050c82759a25e222b09b89b6c9a": "R_E \\,", "27aac0e35e6d150d4155bed236c5d77b": " \\begin{align}\nx_i = \\frac{1-r^{-i}} { 1-r^{-N} }\n \\quad \\Rightarrow \\quad\nx_1 = \\rho = \\frac{1-r^{-1}} { 1-r^{-N} }\n\\qquad \\text{(2)}\n\\end{align}", "27aacc49b6ded542c74e53dbb60b7a96": " -ln(1-X) \\sim \\textrm{Exponential}(b)\\,", "27aaf22833c0b52e1bee9fa3d0852baa": "\\sqrt{1-v^{2}/c^{2}}", "27aaf79646b6395a3f0fcce815f008e2": "(m,k+1)", "27aafacee7dca7a5b0f0e046f1b53389": " \\phi \\not\\in Y", "27ab591d01b4a69a03332d2d6dd8a87e": "(\\mathbf u \\times \\mathbf v) \\times \\mathbf w \\neq \\mathbf u \\times (\\mathbf v \\times \\mathbf w)", "27ab84052867f0f4b180754747d25304": "K=mv^2/2", "27aba3533827bf2818737f694b1705cd": "\\mathfrak{p}_1 \\subset \\cdots \\subset \\mathfrak{p}_n = \\mathfrak{p}'_n \\cap A", "27ac0cb30064f5d1bfaac43025994aef": "m = \\left\\lfloor \\left( \\left\\lfloor \\frac{m}{r} \\right\\rfloor + 1 \\right) r \\right\\rfloor ", "27ac38419a218269236e224b213187bf": " n\\sigma^2. ", "27ac61084e65eb3b8fa1b990d77038ab": "I= \\{\\mathrm{milk, bread, butter, beer}\\}", "27acac66e6d10fb206e6f72b749111a1": "L = 10 \\cdot 0.5 = 5 ", "27acc1746d27841b372a18c0f53e8726": "\\mathrm{M} = \\frac {{v}}{{v_\\text{sound}}}", "27ad202dffe45e9624f67be7142747d8": "v_{LZ} = {\\frac{\\partial}{\\partial t}|E_2 - E_1| \\over \\frac{\\partial}{\\partial q}|E_2 - E_1|} \\approx \\frac{dq}{dt}", "27ad663f56e987dc8af4ffa51f23be46": "\\langle a | b\\rangle", "27ae082b3a967141f2b263a0b3651fa0": "\\{(i,x)~|~x\\in r(i)\\}\\subseteq I\\times X", "27ae76f1088e9028683775e574a5c224": "0 \\lambda.", "27af462209ce360802ebefefa87ed397": "g^{-1}([y-\\varepsilon,y+\\varepsilon])", "27afa964a6ce77d50cc84c699e5aa9dd": "\\nu_{t_{1} \\dots t_{k}} \\left( F_{1} \\times \\dots \\times F_{k} \\right) = \\nu_{t_{1} \\dots t_{k} t_{k + 1}, \\dots , t_{k+m}} \\left( F_{1} \\times \\dots \\times F_{k} \\times \\mathbb{R}^{n} \\times \\dots \\times \\mathbb{R}^{n} \\right).", "27afaf7478927436d0aa59f999497118": "\\mu_w \\approx \\lambda/4", "27afc85f9af832ef6e93301afefda923": "n = 0, \\ldots \\infty", "27b00dc5eac3d437f3662585c2411cd8": "\\mathrm{Hom}_{\\mathcal C/\\mathcal R}(X,Y) = \\mathrm{Hom}_{\\mathcal C}(X,Y)/R_{X,Y}.", "27b019a630a74005270466f918da8f83": "L_\\text{aligned}", "27b0a40f069250c87998d1f3f5fc4773": " G_{i,j}=\\begin{cases}\n1 & \\text{if } S_i\\leftrightarrow S_j \\\\\n0 & \\text{otherwise}\n\\end{cases} \n", "27b0bf02fa820b274ba3902e5e9066f1": "c=Tr(g)", "27b0d4f0319b09a84eea89c2dac88a33": "\\left(\\begin{matrix} \\frac{4}{3} \\end{matrix} / {{\\begin{matrix} (\\frac{9}{8}) \\end{matrix}}^2} = \\begin{matrix} \\frac{256}{243} \\end{matrix}\\right)", "27b10b370ee57be9f3511514dec3603d": "{\\bold \\ g}", "27b10e9039ed092d9f5a246a596e4c34": "\\hat{f} \\,\\hat{f}^\\dagger = 1 - \\hat{f}^\\dagger \\,\\hat{f} .", "27b14bd802425593dd76d6620fd0631a": "2^4\\cdot 3^2\\cdot 5\\cdot 7", "27b158888ad28de51445a67fdcc16bbe": "c\\in\\mathbb{R}", "27b17637d3243fffe576793c03e342ce": "\n\\partial_t^2\\psi=\\Delta\\psi-\\frac{1}{2}f'(\\psi),\\qquad \\psi=\\psi(x,t),\\quad x\\in\\R^n.\n", "27b1a66802f5f1b25164cf0288fff5fe": "S_{\\sqrt{3}}= \\{x = -\\alpha+i\\beta \\ \\ |\\ \\ \\alpha>0 \\ \\text{ and }\\ |\\beta| \\le \\sqrt{3}\\,|\\alpha|\\}", "27b1e8d69b3976a62776f4f5739f3d5b": "= -2\\left[ -\\omega v \\left( \\sin\\alpha - \\omega t \\cos\\alpha\\right),\\right.", "27b2100657a3e547ccfe19854dd5f256": "0 = \\frac {\\mathrm{d}^2 \\psi} {\\mathrm{d} \\eta^2}+(\\frac{2 m E^2} {\\hbar^2}-\\frac{2 m^2 g l^3} {\\hbar^2}-\\frac{2 m^2 g l^3} {\\hbar^2} \\cos(\\eta)) \\psi ", "27b24b1c6e9b103638dd8d17202f7e67": "Fc", "27b27afcd86c4bd2dff0134d4e09597d": "V\\textbf{y}=\\textbf{x}", "27b2ba25830e40e1de7776305d00b107": "\ne_i^2=\\gamma_1+\\gamma_2z_{2i}+\\dots+\\gamma_pz_{pi}+\\eta_i.\n", "27b3048beb4ac9f5f207912a6671b06b": "f\\colon X \\rightarrow \\mathbb{P}^{n},\\ x \\mapsto [a_0(x): \\dotsb : a_n(x)],", "27b340836a051ddc222e5ba0c78e9e9d": "\\mathbf{\\hat{e}}_i ", "27b3ac694e515eb0a2bcfabf7798619d": "\\frac{1}{l}+\\frac{1}{m}+\\frac{1}{n}=1.", "27b3c87564726cedbf9fce2a239ef7a4": "\\begin{align}\ny' &= y \\\\\nz' &= z\n\\end{align}", "27b43ece74a399fdb21cfe0d3e893e2d": " \\frac{\\partial \\sigma}{\\partial\\varepsilon}=\\frac{\\partial }{\\partial\\varepsilon}(E\\varepsilon)=E> 0", "27b43efaa927397aaf229b2b2f454c03": "f(x) = \\begin{cases}\n0 & \\text{if } x = 0,\\\\\nx + x^2\\sin\\left(\\frac{2}{x}\\right) & \\text{if } x \\neq 0.\n\\end{cases}", "27b44237a08f43c6f622236f07f2fc93": "|E(\\mathbb{F}_q)| = q", "27b4e6152e680c2e851eefd84b3d8992": "\\frac{d}{dt}A=(i\\hbar)^{-1}[A,H]+\\left(\\frac{\\partial A}{\\partial t}\\right)_\\mathrm{classical}.", "27b5a39e249a52de631a969a229dd1f4": "\n00\\mbox{)}\\,\\!", "27d65eb589a4cebc66d7890fcc934454": "{\\mathrm {Sp}}(n,{\\mathbb R})", "27d693676731525613f731e4a6d556b8": "\\varepsilon_{tot} = \\varepsilon_1 = \\varepsilon_2", "27d72e9bc3bf7027e606906e49099c57": "z = T(x) = \\begin{bmatrix}z_1(x) \\\\\nz_2(x) \\\\\n\\vdots \\\\\nz_n(x)\n\\end{bmatrix}\n= \\begin{bmatrix}y\\\\\n\\dot{y}\\\\\n\\vdots\\\\\ny^{(n-1)}\n\\end{bmatrix}\n= \\begin{bmatrix}h(x) \\\\\nL_{f}h(x) \\\\\n\\vdots \\\\\nL_{f}^{n-1}h(x)\n\\end{bmatrix}", "27d79633bd5cb642f1d263d044919d93": "EU(n)=\\left \\{e_1,\\ldots,e_n \\ : \\ (e_i,e_j)=\\delta_{ij}, e_i\\in \\mathcal{H} \\right \\}.", "27d7a7c2c02b7f3740a2989881e7bf0f": "\n \\boldsymbol{\\sigma} = \n \\begin{bmatrix} \\tfrac{4C_1}{3}\\gamma^2 & 2C_1\\gamma & 0 \\\\ 2C_1\\gamma & -\\tfrac{2C_1}{3}\\gamma^2 & 0 \\\\ 0 & 0 & -\\tfrac{2C_1}{3}\\gamma^2 \\end{bmatrix}\n ", "27d80789574cb352c76cf0551a1ccc8f": "\\scriptstyle{A= B +} \\tfrac{PL}{2}", "27d8079815fe6ec88887e8fca83e2bae": "nRT\\ln\\frac{V_2}{V_1}\\;", "27d80fc352b727c9aed82973c3131d79": "\\approx 15.04", "27d81d5a29a86b950d70802eccc48801": "\nH_\\mu(x) = \\sum_\\mathbf{p} {\\sqrt{p_0} \\over \\sqrt{2 V }}\\left\\{\n\\left[Q_R(\\mathbf{p}) \\epsilon_\\mu^1(\\mathbf{p})\n- Q_L(\\mathbf{p}) \\epsilon_\\mu^2(\\mathbf{p}) \\right]e^{i p x}\n\\right.\n", "27d82a35d65ae2ca34825d1fb937b701": "e_i\\in\\mathfrak{g}_{\\alpha_i}", "27d83137d6d827b497bbc7a7b62d3bc5": "S_{e} = Lx * (AdjFactor * UFP)^{\\frac{Entropy}{1.2}}", "27d8918145e7dc16d7bf4adc7bfbbb92": "\\lim_{x\\rightarrow a}f'(x)", "27d89261c2b7ea56a1232aaaef86faee": "\\gamma=0.8", "27d8c3268a69c7b04a6c09c634ff8785": "Q=\\sqrt{\\frac{x_1^2+x_2^2+ \\cdots + x_n^2}{n}}", "27d93479cad69a6f99cbfa3fbeb482f8": "\\{it+ju + kv :t, u, v\\in\\mathbb R\\}.", "27d966046f5d5711a230b1a7429c4d82": "|p - p_c|\\,\\!", "27d9d9ea0ebbe84b5d40dc11977bcf4f": "{\\rm REACH}_{\\rm out}[S] = {\\rm GEN}[S] \\cup ({\\rm REACH}_{\\rm in}[S] - {\\rm KILL}[S])", "27d9f6043ebe6e6260e926fef15ce43f": "Q(z-x)", "27d9fb1ffc621b4f9fd6645320932159": "E[|\\xi|^p]< \\infty", "27da571b27422598b7d08b39dad06ced": "p(A)=A^n+c_{n-1}A^{n-1}+\\cdots+c_1A+(-1)^n\\det(A)I_n =0 ~,", "27da58578c907649da175ac03922661f": "A = 270^\\circ + \\arctan \\left ({\\frac{\\left ({\\frac{\\partial z}{\\partial x}}\\right )}{\\left ({\\frac{\\partial z}{\\partial y}}\\right )}}\\right ) - 90^\\circ \\left ({\\frac {\\left ({\\frac {\\partial z}{\\partial y}}\\right )}{\\left |{\\frac {\\partial z}{\\partial y}}\\right |}}\\right )", "27da6ada9031d408cb884cfc6a16690e": "\\hat{x} = \\frac{x}{|x|}", "27da8afa442611c608dd049d764da04b": "L \\in \\text{NSPACE}\\left(f\\left(n\\right)\\right)", "27daa58a04d1b7f298a11f5dd2184195": " U ", "27dab1f9acacac39a41902c1122fb6db": "\\!\\forall x \\phi", "27dac40d6a46ef32506cd3a86b133e9d": "\\gamma_\\text{gw}", "27daf37517119ac992837c5a64776027": "v_v^2/2g", "27dafa448bd8ce45a1724284101a2321": "\\boldsymbol{\\Alpha\\Beta\\Gamma\\Delta\\Epsilon\\Zeta\\Eta\\Theta} \\!", "27db16b136dd4e0e121965c38b3ffe33": "+1/3", "27db75a3b79d095b2898588229b1da1d": "v^2 = Q(v)\\,", "27db9d8bb8ce04ed17ff82b58ec4f8e9": "(\\phi, \\theta) = \\left(2 \\arccos(R / 2), \\Theta\\right).", "27dba00e378b6fdc3bf50453c5033406": "T = \\frac{1}{2} \\left| \\det\\begin{pmatrix}x_A & x_B & x_C \\\\ y_A & y_B & y_C \\\\ 1 & 1 & 1\\end{pmatrix} \\right| = \\frac{1}{2} \\big| x_A y_B - x_A y_C + x_B y_C - x_B y_A + x_C y_A - x_C y_B \\big|,", "27dba611aef816d0ef0d6c63a51c1a22": "\\tau = \\int_R^\\infty k\\rho dr", "27dc57829c9dac76b5e1c07fe5fe27fa": "P^{(L)}=L\\, P", "27dcb3702426feb7fde72b3fc0008972": "(\\mathcal{F}g)(y) = \\frac{1}{\\varepsilon^n}e^{-\\frac{\\pi}{\\varepsilon^2}|x - y|^2}.", "27dcc1404a9f6470a7c3beaadbdc0e0c": "V(r)", "27dd4cff71772ef8f006b36c14615f60": "\\ -\\frac{\\mu}{2a}+\\frac{\\mu}{R} = \\frac{\\mu (2a-R)}{2aR}.", "27dd5e8368a931dc6195ff9ec55921e4": "a(x,z)", "27dd630c8cb6aefd909f276b2ee737ec": " U_t = e^{i t A} \\quad t \\in \\mathbb{R} ", "27dda408c0dbb6e027a2e684c536cbf1": "\\sum_{i=1}^k \\mathrm{n_i}^\\alpha\\,\\mathrm{d}\\mu_i\\,= \\sum_{i=1}^k \\mathrm{n_i}^\\beta\\,\\mathrm{d}\\mu_i\\, = 0\\,.", "27ddabb6aa6a2f3efb914342507355f5": "\\int_0^{2\\pi} \\! I^2\\,d\\omega t = ... = \\pi", "27ddda85ea020143b8119c8e8b398df3": "I\\cong A_5", "27dddef5a301d4de6df69ddfa5026fd0": "\\sigma = \\sigma_f + \\sigma_b\\,.", "27de3b8c9e7b4df1706c75523b53443f": "Y=\\{y_i|i=1,2,\\cdots,k\\}\\,\\!", "27deb2a762eab7f9105efa39c8931066": "Z(G) \\hookrightarrow G \\overset{\\sigma}{\\to} \\operatorname{Aut}(G) \\twoheadrightarrow \\operatorname{Out}(G).", "27dec493c1d9c2f38faca1a145fe9554": "a_S", "27ded278381e8c673da9be04c9ff99d1": "\\lambda\\ = \\frac{ hm^2 }{ m^2 - n^2 }", "27e03e3bd9f598ce965c88b553764f60": "\\zeta^2+\\zeta+1=0", "27e056cfdcbe531778630615bff26fa0": "m= (-15, -26) U^{-1} = (3, -7).\\,", "27e0af5dd1a2311dd4e2758c96f45b6c": "\\boldsymbol{\\nabla}\\boldsymbol{T}", "27e0b277aa8b4c6143c169fbb88147d8": " \\langle (1 \\; 2 \\; 3) \\rangle = \\{id, \\; (1 \\; 2 \\; 3),(1 \\; 2 \\; 3))\\} ", "27e0da99896208032d6c68cc7bae46e3": "x=\\frac{a\\sin t\\cos t}{t}", "27e0ea8fc4b5a17d7fa08ed089249b2e": "\\begin{align}\n {\\rm Pr}_{y_1,\\dots,y_m}(\\exists z& A(x,y_1 \\oplus z)=\\dots=A(x,y_m \\oplus z)=0)\\\\\n &\\le \\sum_{z \\in \\{0,1\\}^m} {\\rm Pr}_{y_1,\\dots,y_m}(A(x,y_1 \\oplus z) = \\dots = A( x, y_m \\oplus z) = 0)\\\\\n &\\le 2^m \\frac{1}{(3m)^m}\\\\\n &< 1.\n\\end{align}", "27e0f96563277444ec54a666352ff058": "\\models_{\\mathrm P}\\phi", "27e14658c241aa9f117b9dbb33e297a5": "S=\\frac{a}{\\sqrt{2}}+a+\\frac{a}{\\sqrt{2}}=(1+\\sqrt{2})a", "27e18af31e9b5a97a4f7aa4e19515f22": "\\cot\\frac{\\pi}{8}=\\cot 22.5^\\circ=\\sqrt{2}+1\\,", "27e1b955e80670f9116b6bb439df7869": "\\mathbf{a} = \\mathbf{a}_x + \\mathbf{a}_y + \\mathbf{a}_z = a_x \\mathbf{i} + a_y \\mathbf{j} + a_z \\mathbf{k},", "27e22067ccdacf738a731e51b933bfd5": "Q = \\left( \\frac{1}{Q_c}+\\frac{1}{Q_d}\\right) ^{-1}\\,", "27e289d40ec0a3b00255dbb0148c4f6e": "f_\\alpha(g) = \\lim_{n\\rightarrow \\infty} F_{\\alpha,z}(g^n)/n", "27e29960e16256defede6f6e0697ccf7": "P_n", "27e3579f027677c27cf69744d5204be6": "a^2 \\neq 1", "27e38b6d01778665c5311345519fb5ac": "\\mathrm{kT/q}", "27e3bceec491b2e7e1b2aebc45825852": "v_s^2 = {\\gamma_eT_{e0} \\over 1+\\gamma_e(k\\lambda_{De})^2} \\sum_i {Z_i^2f_i \\over \\bar Z m_i}", "27e3d18eaa7acc19d4cf6158dedb8fcb": " \\langle x , y \\rangle = \\delta (x-y), ", "27e4529acb3935a8f800239c133d7e48": "A_{ij}~", "27e499647c0f21f38a54b1462733ad4f": "\\Delta^{\\prime }(x)", "27e4aab1a54227b6174b6142d888ac8b": "g(0)=g(1)=1", "27e4e19e734012cbd24593a8c8c5ce29": "A_{11}", "27e4e39e413da1a2bda38dc183f8fa68": "(\\mathcal{O} _X)_p", "27e5a88671ca0fe1314d45683376fad2": "\n \\dfrac{Pbx^2}{2L} -\\cfrac{Pb}{6L}(L^2-b^2) = 0\n ", "27e5b72facc4de10d5a8a0151d8e3e66": "1 \\leq \\sum_{k=1}^\\infty \\lambda(V) \\leq 3.", "27e5ea63e4485fb63d807072b21b6214": "~{\\rm Re} (x) \\gg 1~", "27e641be47bab4e97f54b08c11780d9b": "\na=\\sqrt{\\frac{\\gamma p}{\\rho}} = \\sqrt{(\\gamma-1)\\left[H-\\frac{1}{2}\\left(u^2+v^2+w^2\\right)\\right]}.\n", "27e6c4f8d7629a0970b30449bca4cd67": "g_\\text{I}", "27e6dfad6fec76276e25e2c61682faa6": "\\mathbf{J} = \\mathbf{L} + \\mathbf{S}", "27e736508c1666c43d43dde5d3199e7f": "\\frac{\\Delta \\left( \\omega /2\\right) }{-\\Delta f}\\approx \\eta \\omega\nJ_F^{\\,\\prime }", "27e743a371802c6f5ef6abdfe71a02f8": "\\Omega=\\frac{\\pi}{12 hours}", "27e7b1346e011cf2896a04df87832534": "n_{2}", "27e7c9600e0c4edda297d058215417a7": "\n\\begin{align}\n \\mbox{Capital account} & = \\mbox{Foreign direct investment} \\\\\n & + \\mbox{Portfolio investment} \\\\\n & + \\mbox{Other investment} \\\\\n & + \\mbox{Reserve account} \\\\\n\\end{align}\n", "27e7e01f1bd6b7f9b9b0f2d33368d08d": "D(X) = R(X - \\mathbb{E}[X])", "27e8314f8bde178770636924dae3e9a8": " \\delta=\\frac{2\\pi \\omega}{\\omega_\\text{res} (\\theta)}", "27e864e7c006cdc151f1c46e1e0d4131": "r = \\frac{w\\cos \\left (\\theta \\right )}{\\delta \\cos \\left (\\phi \\right )}", "27e870c56cdb8c8f4097b7f8c4d76d7d": " \\{\\cdot,\\cdot\\} ", "27e89e226ca1bc86dcbd5ae1b613ce5d": " \n\\frac{\\mathrm{d}^{3} u}{\\mathrm{d} x^{3}} + \\frac{1}{2} u \\frac{\\mathrm{d}^{2} u}{\\mathrm{d}x^{2}} = 0 \n", "27e8ba661396ac3961350c5f512b93cb": "\\Phi, \\Psi", "27ea1040b1e7109e708e9cd74bdbd001": " f(x) = f(a) + \\int_a^x g(t) \\, dt ", "27ea1dd7d65f5777ba487de4b56a90d6": "\\displaystyle{U_x=-V_y,\\,\\, U_y=V_x.}", "27ea225d5a2c2504e3a9c6f4fd037e57": "\\mathbf{v} \\cdot \\nabla \\mathbf{v}", "27ea3b2a24beff39ae9e38b8bca5f5c0": "S=\\sum_{i=1}^{m}r_i^2", "27ea673590abb25964c0d59178086f25": " t= \\rho \\sinh\\sigma.", "27eaba7588be9ecdc8a11fb052f660ef": "P_{\\sigma\\,\\circ\\,\\pi}", "27eac782422adb62c41a6f3c2c99a5d1": "2^4", "27eb99a77cab658667371cf42f73a8a9": "\\begin{align}\n|C_n - AB| &= \\biggl|\\sum_{i=0}^n a_{n-i}(B_i-B)+(A_n-A)B\\biggr| \\\\\n &\\le \\sum_{i=0}^{N-1}\\underbrace{|a_{\\underbrace{\\scriptstyle n-i}_{\\scriptscriptstyle \\ge M}}|\\,|B_i-B|}_{\\le\\,\\varepsilon/(3N)\\text{ by (3)}}+{}\\underbrace{\\sum_{i=N}^n |a_{n-i}|\\,|B_i-B|}_{\\le\\,\\varepsilon/3\\text{ by (2)}}+{}\\underbrace{|A_n-A|\\,|B|}_{\\le\\,\\varepsilon/3\\text{ by (4)}}\\le\\varepsilon\\,. \n\\end{align}", "27ebe01457588b35c369065a897d327f": "R = \\sqrt{ax^2+bx+c}", "27ebf14ee602621ce4c15e2b2b1b814f": "\\Delta \\varphi", "27ec49f37d75272912635455af6a321e": "i_\\alpha\\circ i_\\beta = -i_\\beta\\circ i_\\alpha.", "27ec60fd5c6d10afa2b5ac64a18ec247": " p_\\theta(x) = \\frac{ \\theta e^{-x} }{(1 + e^{-x})^{\\theta + 1} } ", "27ec9719ca96a0a1cacce9f19e39a900": "\\vartheta(z; \\tau) = \\prod_{m=1}^\\infty \n\\left( 1 - \\exp(2m \\pi i \\tau)\\right)\n\\left( 1 + \\exp((2m-1) \\pi i \\tau + 2 \\pi i z)\\right)\n\\left( 1 + \\exp((2m-1) \\pi i \\tau -2 \\pi i z)\\right).\n", "27ecccaed73a596c0a8b168171d13de7": "\\langle A \\rangle=\\frac{1}{n}\\int A f\\,d^3p", "27ed54230b571ff8c3d655e7fa719ef8": "\n \\boldsymbol{\\epsilon} = \\tfrac{1}{2}\\left[\\boldsymbol{\\nabla}\\mathbf{u}+(\\boldsymbol{\\nabla}\\mathbf{u})^T\\right] \n \\qquad \\implies \\qquad\n \\epsilon_{jk} = \\tfrac{1}{2}\\left(\\cfrac{\\partial u_k}{\\partial x_j} + \\cfrac{\\partial u_j}{\\partial x_k}\\right) ~.\n ", "27eda47b4f20d504e8f8e200bf9071bd": " w''/w'=-2u'/u", "27eda7899fa7635139ec9cbd013cfa01": " u_{tyt} = u_{tty} = u_{xxy} + u_{yyy} ", "27edb1441562c5d0490e8e6d7d512869": " g=G=|g\\rangle ", "27edeec0c3a8ede087b8f5ce2d0227fd": "t/X_t", "27ee4f0f4a2fd47728d5d2dfa8f9067f": "r_1,\\dotsc,r_n", "27ee690567bc40e124b828243f559f29": "\\frac{\\partial^2 S}{\\partial \\beta_j \\partial\\beta_k}", "27eea3d72f23bcb97741b5d178a264bd": "\\scriptstyle \\vec{r}", "27ef2dc21d7d62304916b3fcc6bbff70": "((b^{-1}\\,\\bmod\\,n) \\, (b\\,\\bmod\\,n))\\,\\bmod\\,n =1", "27ef2fb1e6019af1d64369ef4ce13121": "\\Gamma(E\\otimes\\Lambda^pT^*M) = \\Gamma(E) \\otimes_{\\Omega^0(M)} \\Gamma(\\Lambda^pT^*M) = \\Gamma(E) \\otimes_{\\Omega^0(M)} \\Omega^p(M),", "27ef639d78050675826ab112ed73f033": " \\operatorname{build-param-lists}[g\\ m\\ p\\ n, D, V, K_2] \\and \\operatorname{build-param-lists}[g\\ q\\ p\\ n, D, V, K_1] ", "27efcfd6bdb886eef71ae47a819dcb2e": "\\gamma\\!_{polymer}", "27f00b1280b4f7c0716159a1eaa39998": " 1 \\leq i \\leq r ", "27f02682fb2cf882c5e16c7acb2fee08": "V \\otimes_k F", "27f05b13ae778bcb82ca97a51cfa61db": "s_{iw} = 1", "27f0aeb7a9260d882114347e0bee9eaf": "g:\\R^{n+m+1}\\to\\R^m.", "27f0e5db21cd9bc45d5b87992f27d9a8": "\\scriptstyle{\\pi(B)}", "27f0eb52024c5a525b8d3bc3f82e9a09": "\\exp(o(n))m^{O(1)}", "27f11357c767502e831ee0c0babf89aa": "f^2(x) = (f(x))^2", "27f13a5616ca23f0308f9b3044083826": "\\left(\\pi_A = \\pi_G = \\pi_C = \\pi_T = {1\\over4}\\right)", "27f13e0ce509291bc3d8580a201e44a4": "n_s \\sim s^{-\\tau}\\,\\!", "27f19a540bce340378a4383bbcbe734f": "\\widehat{O1QP1}=\\widehat{O2QP2}", "27f29139b38518f98e8ddc2a258d7ae7": "\\lambda>\\xi", "27f314f4819f7261b9313dd748cd9d27": "2\\tfrac{2}{5}", "27f31ebb9d94ab3f0fa68afe9ef3eba7": "r=3", "27f364891b77402222dfab50ed3727d6": "\\mathcal{A} (\\omega) = \\overline{\\bigcup_{B} \\Omega_{B} (\\omega)},", "27f3815f127e8e63b069cd528c0c46f7": "IV_0=0", "27f39869a1b44022e8da8f307a844189": "\\Pr (X_1 \\le x_1,X_2 \\le x_2,...,X_n \\le x_n) = \\int \\prod_{i=1}^n F(x_i|\\theta)\\,dP(F).", "27f3dd3c963b4ce4817ac47c78998fcd": "a \\lor(b \\lor c) = (a \\lor b)\\lor c", "27f3f92ce9a006f9140ae1d5c367be78": "\\begin{matrix} (13 - r) \\times 4 = 52 - 4r \\end{matrix}", "27f4057b5217c010d2b43c5c15599952": "\\gamma=\\sqrt{1-\\beta^{2}}", "27f425e03aedae73d4dc8bf19b29f35d": "P_i=d_{ijk}\\sigma_{jk}+\\mu_{ijkl}\\frac{\\partial\\epsilon_{jk}}{\\partial x_l}", "27f4292ee91d9061c0f99ea5c45e42d1": "\\xi_1=\\frac{1}{\\sqrt{2\\alpha_1}},\\xi_2=\\frac{1}{\\sqrt{2\\alpha_2}}", "27f43bb53abf108238c535f0b978f2a7": "n(A)", "27f49cf94e949a7f823f42ef57d6a38d": "q_{k}\\sim (k+1)^{1-\\tau}\\mathrm{e}^{-(k+1)/\\kappa}", "27f4a8bbc9e658d3b154d82c12f6a799": "r_5 = (A \\to a, \\emptyset, \\{S\\})", "27f4ac54592545affda7458f1003269b": "\\hat g(n) = { (-1)^{n+1} \\over \\pi n }", "27f4baf5f01bcd4d7b8a7603bd9b743d": "Lu=f\\left( x,y \\right),\\ \\ \\left( x,y \\right)\\in \\Omega ", "27f4be53df95a1871a12c3f3c7615c26": "\\scriptstyle \\boldsymbol \\omega dt", "27f50c0a9859482a7eecbc889f8baa73": "\nE_\\text{red} = E^{\\ominus}_\\text{red} - \\frac{RT}{zF} \\ln\\frac{a_\\text{Red}}{a_\\text{Ox}}\n", "27f560ca74d15c4ce23ce0e099e7b949": "\\Delta_\\text{2D}", "27f609ba5015ce0cef1995a4f3532398": "36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2", "27f60fc38013d2fd533cae68931a0e5d": "\\varrho\\mu\\gamma\\angle'", "27f618831cd285cf86733674f0cacd55": "S(E) = \\sum_i \\oint p_i\\,dq^i", "27f63557b67cf9474676551a40758386": "A_{q}(n,d)", "27f68a9083d1c31cade081485e6f6f4a": "z = \\pm\\frac{j}{2}", "27f6a12cf284e9947becc82315602b29": " \\iint_D f \\, dx\\,dy + \\int_C g \\, ds =0.\\,", "27f6afd553845ddc99d9db733d24cf8c": " 1 \\le i \\le n", "27f831c4337484a3eec9f4a9ee24ca10": " x_c", "27f83baf58075463929b3bc4d4f7c71b": "\\frac{4^n}{2n+1} \\leq {2n \\choose n} \\leq 4^n\\text{ for all }n \\geq 1", "27f8b21afdb7fbabae2ce2b29b412930": "\\text{Break-even(in Sales)} = \\frac{\\text{Fixed Costs}}{\\text{C}/\\text{P}}.", "27f8d74467a8158a29c133af6d172af4": "\\lim_{x \\rarr 0^-}{1 \\over 1 + 2^{-1/x}} = 0.", "27f8dcfaa08c45d85f41c9c269c3f0f0": "\\mathbf{\\Omega}^{-1}=\\mathbf{G}'\\mathbf{G} ", "27f9020ab606dc5e945bbe73ff3a5917": "f_c(z). \\,", "27f93195e00b293a5d48892a6723a7dd": "X(0)", "27f999f4adeb24f85e4a2a5c3fb4b41f": "\\left\\{t'_{i,m+j}, i=1\\ldots m,\\; j=1\\ldots n\\right\\}", "27f9a52862060d3c27e5813d25601d96": " \\begin{align}\ng_{jk}(\\theta)\n = 4h_{jk}^\\mathrm{fisher}\n&= 4 h\\left(\\tfrac{\\partial}{\\partial \\theta_j},\n\\tfrac{\\partial}{\\partial \\theta_k}\\right) \\\\\n& = \\sum_i p_i(\\theta) \\;\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_j} \\;\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_k} \\\\\n& = \\mathrm{E}\\left[\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_j} \\;\n\\frac{\\partial \\log p_i(\\theta)} {\\partial \\theta_k}\n\\right]\n\\end{align}", "27f9a87d164b6d49cc7780547b4dcf66": "(a_i, b_i, c_i, d_i, e_i)", "27f9e340f9ad3918d61ab9980d52c130": "\\mathcal{D} = \\{(x; a; r)\\}", "27fa384a85afbde1b89c109635f880fa": "\\ k_b", "27fa6cac97d0ab80f03c90de3961c045": "\n T^{*}_{11} := \\cfrac{T_{11}^{\\mathrm{eng}}}{\\alpha - \\alpha^{-2}} ~;~~ \\beta := \\cfrac{1}{\\alpha}\n ", "27fa6dee32669f9bcbe7b1a7aa15e711": " \\scriptstyle \\Vert\\;\\Vert_{L^\\infty(\\Omega)}", "27fad19a03a94b6acf1e4b65d560fafd": "\\displaystyle \\int_{\\mathbf{R}^n}f(\\mathbf x) e^{-i \\mathbf x \\cdot \\boldsymbol \\nu }\\, d^n \\mathbf x ", "27fada207a1c27e876075a912465983c": "\\displaystyle{\\mathcal{K}=\\mathcal{H} \\oplus \\mathcal{H}.}", "27faed787a905e1a574b7141762e307f": "\\sqrt{n(P+N)}", "27fb09eba9204db1db850bc094f1063d": "\n \\begin{align}\n \\frac{1}{\\lambda}\\, \\int_0^\\lambda \\eta^2\\; \\text{d}x \n &= \\frac{1}{\\lambda} \\int_0^\\lambda \n \\left\\{ \\eta_2 + H\\, \\operatorname{cn}^2 \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m\\end{array} \\right) \\right\\}^2\\; \n \\text{d}\\xi\n = \\frac{H^2}{\\lambda} \\int_0^\\lambda \n \\operatorname{cn}^4 \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m\\end{array} \\right)\\; \\text{d}\\xi\n - \\eta_2^2\n \\\\\n &= \\frac{\\Delta\\, H^2}{\\lambda} \\int_0^{\\pi} \\cos^4\\, \\psi\\, \\frac{\\text{d}\\xi}{\\text{d}\\psi}\\; \\text{d}\\psi - \\eta_2^2\n = \\frac{H^2}{2\\, K(m)} \\int_0^{\\pi} \\frac{\\cos^4\\, \\psi}{\\sqrt{1 - m\\, \\sin^2\\, \\psi}}\\; \\text{d}\\psi - \\eta_2^2\n \\\\\n &= \\frac13\\, \\frac{H^2}{m^2}\\, \\left[ \\left( 2 - 5\\, m + 3\\, m^2 \\right) + \\left( 4\\, m - 2 \\right)\\, \\frac{E(m)}{K(m)} \\right] \n - \\frac{H^2}{m^2}\\, \\left( 1 - m - \\frac{E(m)}{K(m)} \\right)^2\n \\end{align}\n", "27fb6f05171808ff6304a72ab273eaa6": "12\\,", "27fba3fd905517efb91ddf247cb5ec32": " S(\\Psi) = \\tfrac{1}{2} \\langle \\Psi |Y(i) Y(-i) Q_B |\\Psi \\rangle\n +\\tfrac{1}{3} \\langle \\Psi | Y(i) Y(-i) |\\Psi * \\Psi\\rangle \\ ,", "27fbad4d812e76b47303a6b00a3ffbf5": "(a^2 - b^2)^2 = (A+B)^3(A+B-4h) \\,", "27fbce8c68ff35a097081bc9cc25c5db": "\\left\\lceil \\log_2 \\frac 1 \\frac 1 4 \\right\\rceil + 1", "27fbece4c8c5d1d75d1bf153ef43a950": "i,j,\\dots", "27fc1509fcb9340ee056a0a82115156f": "|I_i(\\alpha_k)|\\le|\\alpha_k|e^{|\\alpha_k|}F_i(|\\alpha_k|)", "27fc2a6e19a02f734cbe2f8bd4bc2672": "\\gamma\\in\\Gamma", "27fc3399be3d91097ffd0952c1e71a61": "T\\vdash_{\\mathcal{S}}\\alpha(\\phi,\\vec{\\chi})", "27fc3746bbf605c6aba5e54ebb4a5a42": " f_s = {R \\over N}. ", "27fc75a8139f2b6842b2349f0128da47": "\\vdash \\Gamma\\ \\mathsf{Context}", "27fca5bab3d70edcfb77ef603dd0d78c": "[H, H]", "27fca613ea0dfaaf9e48fc5acb29f52e": "\\frac{f_{o}}{f_{s}}=\\frac{c}{c\\pm v},", "27fcb267ccc84d61483bf7e04a68ad63": "C_j = P_j \\oplus O_j", "27fcb9d89098ecff906ff3aa5f3348da": "A.a", "27fd3c9bc0d7e3d18322268862237740": "E(T^k)= \\nu^{\\frac{k}{2}} \\, \\prod_{i=1}^{\\frac{k}{2}} \\frac{2i-1}{\\nu - 2i} \\qquad k\\text{ even},\\quad 0m+n \\mid X > m)=\\Pr(X>n).", "2800ad59002cdf687b646b5e335a3032": "M_{\\phi}(X)=-\\delta\\sum_{s=1}^S\\phi_sX_{s:S}", "2800fc565c33ed23870778ca086b81aa": " \nA=\\{x\\in E |\\varphi(x)>a\\}\n", "28013e3a784e4bfc0682f7980f772d63": "\\frac{a}{b}\\,\\bmod\\,n = \\frac{a\\,\\bmod\\,n}{b\\,\\bmod\\,n}", "28015def75a790a459ce4df1732371ea": "\n\\frac{\\partial }{\\partial t} \\rho(x,p;t) = \\langle \\Psi(t) | x,\\, p \\rangle \\frac{\\partial }{\\partial t} \\langle x,\\, p |\\Psi(t) \\rangle \n+ \\langle x,\\, p |\\Psi(t) \\rangle \\left( \\frac{\\partial }{\\partial t} \\langle x,\\, p |\\Psi(t) \\rangle \\right)^*\n", "28018a31d4880aab81125eaa55970b96": "\\langle\\langle\\tau_e\\rangle\\rangle", "2802066d41a4e848f94121b26e06a178": "10 \\cdot \\log_{10}", "28023398c861b18571c4114666826084": "\\scriptstyle p^\\mu \\;=\\; \\{\\pm\\sqrt{m^2+\\vec{p}^2},\\, \\vec{p}\\}", "28024c884e786bd0e2974e7e9f044c6e": "x \\in \\{1,\\dots,k\\}", "28029c5f1a8512ea7efb18cabbb404f6": "S = \\{ (x,y) \\in \\R^2; -N-\\frac{1}{2} \\leq x \\leq N+\\frac{1}{2}, \\vert \\alpha x - y \\vert \\leq \\frac{1}{N} \\} ", "2802f947a473c21551e968b94746b06e": "\\Omega=[a_1,b_1]\\times[a_2,b_2]\\times ... \\times[a_N,b_N]\\subset R^N", "2803016d9871a3f1a263456a1afd84fa": "\\mathrm{Aut}(\\mathrm{S}_6)=\\mathrm{S}_6 \\rtimes \\mathrm{C}_2.\\ ", "28033ba00f7438ce839471781027a772": " \\{0\\} \\subset F_0 \\subset F_1 \\subset \\cdots \\subset F_i \\subset \\cdots \\subset A ", "28034a11a3de1ca566826ad371113968": "\\scriptstyle\\mathrm{tr}(\\cdot)", "2803934d96b376a6b582474cd3468897": " P = P^*\\,\\!", "2804140db39734ac81b83dcff4e714b8": "P(z)|_{z = \\beta_m} = 0 ", "2804aaca3f5dced813f6f0a3723e4b66": "\\rho \\circ \\kappa = \\operatorname{id}_{\\ker\\, f}", "2804ae2d240ced2f3f613120c674da50": " \\frac{1}{c^2}\\frac{\\partial^2 u}{\\partial t^2}(t,x) = \\Delta u(t,x) \\quad \\mathrm{for} \\quad (t,x) \\in \\mathbb{R}^+ \\times \\mathbb{R}^n,", "28051e3719a3cc90d5c4cb6104456b47": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-n})(1-p^{-n/2})}", "280632de07af20a4f93f65ee3186abfd": "\\theta \\rightarrow - \\theta", "28063499b41a40a52f1fe38faf5a14e6": "\nH_{\\mathbf{k}}=e^{i\\mathbf{k}\\cdot\\mathbf{r}}H e^{-i\\mathbf{k}\\cdot\\mathbf{r}}\n", "2806854ac8d69074d5a019bed52140ed": "MRS_{xy}=MU_x/MU_y \\,", "280695c08e2577b6f3688f92ceeaa3a5": "A\\wedge B \\equiv A \\mathbin{\\And} (A \\rightarrow B).", "2806c14e63b316fdab07605959cd87df": " I = S \\sqrt{t}\\ + A t ", "2806c80bdf23a1b0ffd3b40aec05049e": "X^{\\text{op}}", "2806cf41b068979bad724cd437ca3ac5": "O_j = \\ E_K (I_{j})", "2806dd52c577a8cb88eee4dc76e86dba": "P(R_{NP},\\theta) = P(R_{NP} \\cap R_A, \\theta) + P(R_{NP} \\cap R_A^c, \\theta),", "2806f8e8724b5f7fc1ffaf7db56c7d0a": " \\tilde f (\\vec v) = f_\\gamma v^\\gamma ", "2807046d163521aa66cff3da9b423dbc": "G \\cong \\mathrm{SO}(2, \\mathbb{Q}).", "280713008b5a86c4303bb3445ad33a51": "\\mathbf y = \\textbf{vec}(\\mathbf Y)", "2807182680f4e5e92c78c39a5778f4e9": "\n\\begin{align}\n0.95 & = 1-\\alpha=P(-z \\le Z \\le z)=P \\left(-1.96 \\le \\frac {\\bar X-\\mu}{\\sigma/\\sqrt{n}} \\le 1.96 \\right) \\\\[6pt]\n& = P \\left( \\bar X - 1.96 \\frac{\\sigma}{\\sqrt{n}} \\le \\mu \\le \\bar X + 1.96 \\frac{\\sigma}{\\sqrt{n}}\\right)\n\\end{align}.\n", "28076aa4a2eec33aa5cd4bb8d54342d4": "\\sum_{i=0}^L q_i S_y[{i+r}]=0 \\forall r", "280788e5a5322a52e66ccb30ebe86eaa": "E_m = \\log_2 \\left (l \\right ) : \\beta\\, = 1, l \\ge \\, 1", "2807c995216e38f050312743227225c8": "L + R \\ \\leftrightarrow \\ L\\! \\cdot \\!R ", "2807d68c7b2e036120e44086db1c62f0": "C=v^i N_i \\, ", "2807decebf4d8e2037ea37f0461b6f7f": "n(d) = n_0 e^{-d/\\langle d \\rangle} dD", "280850314e7623dbb55528c4a6eb1159": "[D(f(d)) \\wedge \\neg D(f(f(d)))]", "28086ba41ce50eaa4f08c378b1624075": "\\max\\nolimits_{m_{j+1}} * \\Pr[V\\text{ accepts }w\\text{ starting at }M_{j+1}].", "28087a724bd01324e5a162431b68789f": "\\mathbf x = \\begin{bmatrix} x_1 & x_2 & \\dots & x_m \\end{bmatrix}.", "28087c504e209ae70ca4f6370e0c5713": "Chow^{eff}(k) := Split(Corr(k))", "2808fe6abd57d1f3cfdbff5cc347001a": " \\eta = k \\left | \\frac{du}{dy} \\right | ^{n-1} ", "28091d1b5d3294c1f4463a0a94a55cba": " K = -\\frac{1}{2e^\\sigma}\\Delta \\sigma,", "28093612c48b29eeebf0c895b2463f18": "(a \\wedge b) \\cdot (c \\wedge d) = (a \\cdot c)(b \\cdot d) - (a \\cdot d)(b \\cdot c).\\,", "280969f675fa9cfde8c0ef9e45b33aa1": "S_{\\beta \\gamma}^{\\;\\;\\;\\; \\beta} = 0", "280977f5eb0a373a76facb445b5166ee": "X(s)", "2809a048691bc27c0cac99e5ab15be2f": "H = (V,E)", "2809acf39536b82b5470530e6aed78bc": "\\alpha\\ominus\\beta", "280a3d3635b953300c3590c60697534f": "\n\\nu(t) = \\arg\\max_{p \\in \\mathbb{R}^n} (p' \\alpha(t) - \\tfrac{1}{2}p' \\alpha(t) p)\n\\qquad \\text{ for all } 0 \\leq t < \\infty\n", "280a525278fde87ef845c2877fa5d1a7": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{j} \\ ", "280a84120fd9cd5508ac31ba7149eed3": "\\scriptstyle{L}", "280ab03a3d7fae4506ec3159faae0cb9": "C_{ab}^* = \\sqrt{a^{*2} + b^{*2}}", "280acd0b7dc27e1c15be6d96208c483b": "y = \\frac{Y}{Z}", "280b9d871a2779ae274c9fdf2f92a7d9": "\n\\int d\\mathbf{Q} d\\mathbf{P} = \\int J d\\mathbf{q} d\\mathbf{p} \n", "280bdbe1d2750a51b97892b7e0b6f36c": " \\begin{bmatrix} x \\\\ y \\end{bmatrix} ", "280c0a986f03b7daa88d99c832d8d6af": "\\left[1 + \\frac{x-\\mu}{\\sigma}\\right]^{-\\alpha}", "280c83b9d178ee52cdcb96fe32305d76": "B_\\theta = -B_0\\left(\\frac{R_E}{r}\\right)^3\\sin\\theta", "280cab3bc6f8ecda6ea25059c0a91bd1": "\n(\\leftarrow \\backslash) \\quad\n{Z \\leftarrow \\Delta Y \\Delta' \\qquad X\\leftarrow\\Gamma\n \\over\n Z \\leftarrow \\Delta \\Gamma(X\\backslash Y) \\Delta'}\n", "280cff47b4bf53ba29b780aed56d20f3": "\\mathsf{f}(I) \\propto I", "280d0bd4d8017c71a48ebbd88ddde576": "x \\mapsto x^q", "280d1021809f3eb41545fead32394ed6": "\\frac{C}{i} + FVA = \\frac{C}{i} (1+i)^n", "280d520fa6238d7473aded92a1cab757": "N \\to M", "280d6a71e6cda81773a33482fe7d3438": " \\mathbf{p}_{\\rm j} = m_j \\mathbf{v}_{\\rm j} \\,\\!", "280d890533c1eee90fa363d000bac2b9": "\\phi\\ ", "280d8afbba1dae2ced2afe18b3c0e9e4": "U = m V,", "280dcb9822efc6175734bbb9733a3e3f": "P={E(1-b) \\over {k_e-br}} \\,", "280de26ecaae81e6af2475a0aed4d71f": "| \\gamma' | (t) := \\lim_{s \\to 0} \\frac{d (\\gamma(t + s), \\gamma (t))}{| s |},", "280df504cc6af5335768fc2291c7afb0": "\n\\begin{bmatrix}\nR(i+1,j) \\\\\nS(i,j+1) \\\\\ny(i,j) \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nA_1 & A_2 & B_1 \\\\\nA_3 & A_4 & B_2 \\\\\nC_1 & C_2 & D \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nR(i,j) \\\\\nS(i,j) \\\\\nu(i,j) \\\\\n\\end{bmatrix}\n", "280e6ccedb10278d5a745ab3dc8c469b": "\n(H-\\mu N)\\psi_\\alpha^\\dagger|n \\rangle = (E_n + \\xi_\\alpha) \\psi_\\alpha^\\dagger |n \\rangle.\n", "280edd3c20bcc641aba370599b4623b9": "f(\\prod x_i) = \\sum f(x_i)", "280ee16d36bd690683ee2315b3d5819c": "d_1, \\ldots, d_h", "280ee3a23290913b9b4830f41e2094e9": "\\left|\\begin{pmatrix}\\lang\\omega|\\omega\\rang & \\lang\\omega|s\\rang\\end{pmatrix}(U_sU_\\omega)^r \\begin{pmatrix}0 \\\\ 1\\end{pmatrix} \\right|^2 = \\sin^2\\left( (2r+1)t\\right)", "280f50172ffd5f0a0856dbaa895b0b90": "\\Phi(\\vec{r},t)", "280f59d83993f00bdac5721ff792b0ab": "T = t, M = m, L = \\ell, g = \\ell/t^2. \\, ", "280f5c08423ff11b7f24e9cdf23398e2": "Z=X \\cup Y^\\phi", "280f77ac6f4a90318854c2fa58da0222": "\\mathrm{AmCl_3\\ +\\ \\ H_2O\\ \\longrightarrow \\ AmOCl\\ +\\ 2\\ HCl}", "280f797410c8cc0234c2eed13dc4039f": " \\cos\\theta\\hat{x}-i \\sin \\theta \\frac{\\partial}{\\partial x}", "280f95b2e4395bebecb1daf3eba09a19": "h_{x}(\\alpha) = \\min_S \\{\\log |S| : x \\in S , K(S) \\leq \\alpha \\} ", "280fb4db74a540a8e6e5b2dc3c448254": "g_0 = W_N(f_0)\\,", "280fba0f49355074d59c01012952b506": "A,A' \\in \\mathcal{A}", "2810eedc2569564b48c1e808d7004834": "\\zeta /r = \\epsilon^\\theta", "28112b4dfed2fd9257085836248439c1": "P_n(k\\rho)", "28113eddf896e0ade3f150896800ebef": "\\scriptstyle A \\;\\Rightarrow\\; B", "281171842bfc5a9c47475ebd89f33976": "uv \\in L \\Leftrightarrow vu \\in L", "281177396b89e2b58f7f3f097afc0a19": "E = \\hbar\\omega = \\frac{\\hbar^2 k^2}{2m},", "281187fe095a9609cf54b2b3e17d3b1e": "C_p = \\left(\\frac{\\partial H}{\\partial T}\\right)_p", "28119b1609ab96728f02dd7d1d747bb9": "t - 0 = {1 \\over g}\\left[{\\ln(1 + \\alpha v^\\prime) \\over 2\\alpha} - \\frac{\\ln(1 - \\alpha v^\\prime)}{2\\alpha} + C \\right]_{v^\\prime=0}^{v^\\prime=v}={1 \\over g} \\left[{\\ln \\frac{1 + \\alpha v^\\prime}{1 - \\alpha v^\\prime} \\over 2\\alpha} + C \\right]_{v^\\prime=0}^{v^\\prime=v}", "28125304c0b88c2636dd9c38a4400f0a": " f^{-1}\\left( \\, f(C) \\, \\right) = f^{-1}\\left( \\, \\tfrac95 C + 32 \\, \\right) = \\tfrac59 \\left( \\left( \\, \\tfrac95 C + 32 \\, \\right) - 32 \\right) = C\\text{, for every }C\\text{.} ", "281262e8067bda6e4fe9319a97c1ed4a": "\\frac{u \\cdot a}{2} + \\frac{s \\cdot a}{2} + \\frac{t \\cdot a}{2} = \\frac{h \\cdot a}{2}", "2812630fa96c0cd3644f04b61a6dd230": "(E_{11}+E_{22})", "28127894d229b31def8883d8fe770849": "\n p_0 = \\cfrac{3F}{2\\pi a^2} = \\cfrac{1}{\\pi}\\left(\\cfrac{6F{E^*}^2}{R^2}\\right)^{1/3}\n ", "2812862617caab2c46d3fe75e6037132": "\\frac{2}{\\frac{1}{60} + \\frac{1}{40}} = 48", "2812aca5dcca0561b9bde7b0e7a255e1": "\\sim \\frac{34}{9} N \\log_2 N", "2812c143d1f0b96be9fb3f08e3539777": "\\ln(2)/\\lambda", "2812d5753ee5836c2e89f0285de173b8": "\\int_0^{\\pi/2}\\sin^{2m} x\\, dx=\\int_0^{\\pi/2}\\cos^{2m} x\\, dx = \\frac{1\\times3\\times5\\times\\cdots\\times(2m-1)}{2\\times4\\times6\\times\\cdots\\times2m}\\frac{\\pi}{2} \\ \\ m=1,2,3,\\ldots", "28134c72b74d855489e3cae58c8096d6": " 100 \\frac{\\text{rise}}{\\text{run}}", "2813724c8a547017eff1575e9f11d674": "\\scriptstyle \\tau=\\inf\\{t>0 \\,:\\, X(t)<0\\}", "28138b68372ed79835d80a83dab014b1": " 1 - \\alpha \\approx 0.15", "28139d800f1e0a7d2e3d3002eda9bb49": "\\sum_{k=d}^n {n\\choose k}{k\\choose d} = 2^{n-d}{n\\choose d}", "2813e55405aa218b929fed2903e5a5c3": "i\\,\\partial_tu + \\nabla^2 u= V(u)u", "28142ba52dba61cfd7f87558fe61be34": "(1-p_x)p_y", "28145a278d17e03c5a0b34a9e532b9e3": "S\\rightarrow\\mathbb R", "281467171fdeab86696bac11b8ca89ab": "i < n", "281497ca85f54f83090313623abdecce": " p = \\frac{k+ 2 \\lambda}{(k+ \\lambda) ^ 2} ;", "2814c0af3a187ac8e598c03b181b1092": "t' \\leq t", "2814dc0cfa1c0964a7360e67f4efdffb": "T^{\\mu\\nu} =\\begin{bmatrix} \\frac{1}{2}\\left(\\epsilon_0 E^2+\\frac{1}{\\mu_0}B^2\\right) & S_x/c & S_y/c & S_z/c \\\\\nS_x/c & -\\sigma_{xx} & -\\sigma_{xy} & -\\sigma_{xz} \\\\\nS_y/c & -\\sigma_{yx} & -\\sigma_{yy} & -\\sigma_{yz} \\\\\nS_z/c & -\\sigma_{zx} & -\\sigma_{zy} & -\\sigma_{zz} \\end{bmatrix},", "2814fd4ef9f2bcc12b109671ad50c859": "u=v_{A|O}", "281585943fb4d94d7617a818ac5e0d08": "\\lnot(\\lnot x \\wedge \\lnot y) = x \\vee y \\mbox{ for all regular } x, y \\in H,", "2815e87cb5066a9fd753acdeb2b2017b": "\\left(x_1-x_2\\right)^2+\\left(y_1-y_2\\right)^2=\\left(r_1 + r_2\\right)^2.\\,", "2815f0437f4639f673f704f7d6d75ed1": "\n\\nabla \\times \\mathbf{E}(x) = - \\partial \\mathbf{H}(x) / \\partial t, ", "28163f588c1e7560acfe548e3f307dd7": "\\rho = \\cot \\varphi_1 + \\varphi_1 - \\varphi\\,", "2816a8e150a2830e8d27bbbbb2aa67c7": "\np(\\mbox{foot size} | \\mbox{female}) = 2.8669e-1\n", "2816d033e6d7dc13b961661f8d673c25": " \\frac{d^2y}{dx^2} = - \\frac{d^2x}{dy^2}\\,\\cdot\\,\\left(\\frac{dy}{dx}\\right)^3 ", "2816dbff8b5ee1479103db02de16b6d1": "\\int_{0}^{\\infty }\\frac{\\sin px\\cos qx}{x}\\ dx=\\begin{cases}\n0 & \\text{ if } p>q>0 \\\\ \n \\pi/2& \\text{ if } 00 \n\\end{cases}", "2816ead63939515d1c49d5b9ad1e2bbf": " \\Delta_0=1, \\Delta_1, \\ldots, \\Delta_n=\\det A. ", "28176e6a04d460a132e3e7a0680401a6": " \\sin \\frac{\\pi}{n} = \\frac{16(n-1)}{5n^2-4n+4}.", "2817e279c78754a12fc2aef82ce2c670": "T_p(N)", "28183a54575d473f0cb5982ed500f01d": "\\mathrm{STA}_w = \\tfrac{T}{n_{sp}} \\left(X^TX\\right)^{-1}X^T \\mathbf{y}. ", "2818bc0f1e8388585ba001a01f78168a": "n4700 m_e", "2818de80d899be761fb6deed6bbf8e23": "r^n = a^n \\sin(n \\theta)\\,", "2818e99ff4a319fbab58c1c5747dabc9": "V/m_0", "2819143673ac99247451cda13d1d531e": "|P(s)C(s)| \\gg 1", "28196e5c13728253f927973617493ebb": "M = {d_i \\over d_o} = {h_i \\over h_o} = {f \\over d_o-f} = {d_i-f \\over f}", "2819ce0013cc2efdd2c40a35f5d416fb": "\\scriptstyle{c \\, = \\, 3\\times 10^8 \\, m/S}", "2819e818f5c92e1c2e51a1e43dc212b1": "\\cos \\theta=\\frac{A_{11}+A_{22}+A_{33}-1}{2}", "281a0e5efb84ad78a412138b9fc11857": "\\Gamma \\vdash_{\\mathrm FS} A", "281a35a65047ce911106efe80d16e4f5": " m - n ", "281a58ae1cd9d848b0c4f876a3c33b15": " F_k = \\rho \\int_A \\sum_i (\\tfrac12 u_i^2 - u_i v_i) n_k \\, \\mathrm{d} S. \\qquad (3) ", "281a70c20b16a38d7781189936e1ac9f": "E = mc^2", "281a89b63c44bf403aaf2e3fde944034": "(n_1, \\ldots, n_{d/2})", "281b75a4775396656b19a7f00debb026": "\\mathbb{Z}[i]=\\{a+bi \\mid a,b\\in \\mathbb{Z} \\},\\ \\mbox{where}\\ i^2 = -1.", "281b767d43b900a7f3965b1d572eee16": " E= {1\\over 2} m_\\mathrm{e} v^2 - {Z k_\\mathrm{e} e^2 \\over r} = - {Z k_\\mathrm{e} e^2 \\over 2r}. ", "281bee3735e89cd59262dc1a5e4a44c1": "f(x) = \\mathbf{E}^{x} \\big[ f \\big( X_{\\tau_{D}} \\big) \\big] - \\int_{D} L_{X} f (y) \\, G(x, \\mathrm{d} y).", "281c22885742aa5f9bc649b2fc6f13a4": "\\sigma_{xx}-\\frac{\\sigma^2_{xz}}{\\sigma_{zz}} +|\\sigma_{xy}-\\frac{\\sigma_{yz}\\sigma_{xz}}{\\sigma_{zz}}|", "281c7dd75248c5070ef01fd0eb54140b": "2 f_\\Delta\\,", "281c9aaad475ee64e68124d68b47f010": "t _4[\\beta] = t _2 [\\beta]", "281cd41514f1c24fa2dbe44c3e3f3f4b": "3 - 5 \\equiv -2,\\ -2 + 6 \\equiv 4\\pmod 6 \\,", "281d033605797b20a134f4e803525aa7": "D_n = \\{(X_1, Y_1), \\ldots, (X_n,Y_m)\\}", "281d1818fcfa34029f148efe673ea40e": " K_f ", "281d692def23e015c2b2190656763759": "\\mathbf{F}_q = q \\left(\\mathbf{E} + \\frac{\\mathbf{v}_q}{c} \\times \\mathbf{B}\\right) \\,", "281d6e900867a49b21e3394f8312541c": "\\begin{align}\n E &= \\sum_{k=0}^\\infty \\frac{1}{2^{k+1}}\\cdot \\min(2^k, W) \\\\\n &= \\sum_{k=0}^{L-1} \\frac{1}{2^{k+1}}\\cdot 2^k ~+~ \\sum_{k=L}^\\infty \\frac{1}{2^{k+1}}\\cdot W \\\\\n &= \\frac{L}{2} ~+~ \\frac{W}{2^L}\\,\\,.\n\\end{align}", "281d7e8ef9853b8080dcbe41aad538cf": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^3 u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^4 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^3 u\\ \\cos u \\ du\\ = \\\\\n&-\\hat{g}\\ \\left(\\int\\limits_{0}^{2\\pi}\\ \\sin^4 u \\ du\\ +\\ \n3\\ {e_g}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^4 u \\ du\\ \\ +\\ \n3\\ {e_h}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^6 u \\ du\\ \\right) \\\\\n&+\\hat{h}\\ 6\\ e_g\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^4 u \\ du = \\\\\n&-\\hat{g}\\ \\left(2\\pi \\left(\\frac{3}{8}\\ +\\ \\frac{3}{16}\\ {e_g}^2\\ +\\ \\frac{15}{16}\\ {e_h}^2\\right)\\right)\n+\\hat{h}\\ \\left(2\\pi \\left(\\frac{3}{8}\\ e_g\\ e_h\\right)\\right)\n\\end{align}\n", "281db53d244447933f726620ac6fd7c0": "x\\in(-\\infty,\\infty)", "281dc3696d54353a521f22da74918f90": " X'^n = B_n(X)\\, ", "281de83336672b5c72dc53e363d25dbd": "x \\to 4(x - \\tfrac{1}{2})^{2}", "281dfd461ed86b23764c77df036f238c": "s = v_0 t + \\frac {1} {2} a t^2 ", "281e189f0280eb27b50110850f6adc17": "f(3) = 3 + 1", "281e67bdb4474965b8e7df2234a16467": " f_i(x) \\le 0 , i = 1,\\ldots,m", "281e7d736c5392618178650dc68a3cbf": "v = v'", "281e9dd962f4605c477a3f4578ca80d3": "(\\mathrm{d}G)_{T,p} = \\sum_i \\mu_i \\,\\mathrm{d}N_i\\,", "281f727db8f6c1ae35e00d43f64d8d26": "\n K_{\\rm I} = 2\\sigma\\sqrt{\\frac{a}{\\pi}} \\,.\n ", "281f89c0b9969c84003dfc12da6d393a": "Q, L: \\R \\to \\R", "281fa7dba612f89cc36bfcd6504909f1": " \\partial_\\phi, \\; \\; \\sin(\\phi) \\, \\partial_\\theta + \\cot(\\theta) \\, \\cos(\\phi) \\partial_\\phi, \\; \\; \\cos(\\phi) \\, \\partial_\\theta - \\cot(\\theta) \\, \\sin(\\phi) \\partial_\\phi", "281fb4623f1eea000f2093f7f9b6c1b9": "x_{k+2}=(2^{32}+6 \\cdot2^{16} +9 )x_{k}=[6 \\cdot (2^{16}+3)-9]x_{k}\\,", "281fc83a911feffe51f18c681ca881c8": "r_3=r_1+2^{-k}\\,\\log{|s|}", "281ffc88e975c71953e5139af5d36453": "\\mu_k^{'}=\\sigma^k2^{k/2}\\,\\Gamma(1\\!+\\!k/2)\\,L_{k/2}(-\\nu^2/2\\sigma^2). \\,", "28204b91e2d6b472484979781042c627": "(r_1 + r_2) \\otimes s = (r_1 \\otimes s) \\oplus (r_2 \\otimes s)", "28208ac9405fd04bfe26f04a82ec387e": "\\{w_1,w_2,\\ldots,w_n\\}", "28209e455abe9ea5e8dcaa0d421e14da": "z^{-\\alpha} f(z)", "2820c85eb5f3b978ce16452386181120": "x^2+y^4", "2820e508856bce1a5550abf04b58625b": " 2 \\cos \\left(\\frac{2\\pi}{17}\\right) = \\zeta + \\zeta^{16} \\,", "2820eccbb4bdc8b1a33750b29ba9e912": " p(a/2) < 2p(a) ", "2821130ecb31647a5f5a06ff6dff0ec1": "s_{0}^{2}", "2821732e30912d6e8b5d0f65a8c2d00f": "x = \\pm \\sqrt[m]{a^n}", "2821992cbb4373430866e32ce07a9405": "M \\simeq 10^{26}", "2821b251f6115ff2e06973617d0ce4ba": "R_\\mathrm{ab} = R_aR_b(\\frac 1 R_a+\\frac 1 R_b) = \\frac{R_aR_b(R_a+R_b)}{R_aR_b} = R_a+R_b", "2821cb8255fd755e0cc0360058c76bcf": "d = \\frac{\\sin\\alpha\\,\\sin\\beta}{\\sin(\\alpha+\\beta)} \\,l = \\frac{\\tan\\alpha\\,\\tan\\beta}{\\tan\\alpha+\\tan\\beta} \\,l", "28220cdd2ae5553d11f151d1b6215939": "y^{calc} = c y^{obs}", "282225ee9806cb0783a6fc8f100b385a": "\\tilde\\psi\\upharpoonright_\\alpha", "28223ff05a6ea1a63fdec9528aceb64c": "f_n(x)=\\frac{\\sin (n^2 x)}{n} ", "28223ffe334074fc595ed8d6caf08baa": "Beam = LOA^{2/3} + 1", "282253d80209604ef46ae039d29cf5fb": "\n p = a_0 + a_1 x + \\cdots + a_n x^n,\\quad a_n\\not = 0,\n", "28228fcdbd1463e5e5a16cd0c1025eca": "\\mathfrak{e}_{8}(\\mathbf K)", "2822bd0cb10b0f399659da47b50dc162": "\n\\begin{align}\n\\tilde{I}_N [f] = A_1\\sum_{i=1}^6 f(a_i^1)& + A_2\\sum_{i=1}^{12} f(a_i^2) + A_3 \\sum_{i=1}^{8} f(a_i^3) \\\\\n& + \\sum_{k=1}^{N_1} B_k \\sum_{i=1}^{24} f(b_i^k) + \\sum_{k=1}^{N_2} C_k \\sum_{i=1}^{24} f(c_i^k) + \\sum_{k=1}^{N_3} D_k \\sum_{i=1}^{48} f(d_i^k),\n\\end{align}\n", "28237807fc4d6f2305b63a8f0e57f1b8": "\\scriptstyle \\leq5\\times10^{-23}", "28239bacd4ca052f4ead09b06a8801e9": "\\int_0^{\\theta} \\operatorname{Cl}_{2m}(x)\\,dx=\\zeta(2m+1)-\\operatorname{Cl}_{2m+1}(\\theta)", "28240e1d02b0db1a4040189ebfef3196": "\\ C ", "28241226029bf2f89e0c1b139f1bb56f": "a x^2 + b x + c\\, ,", "28246c22d62808930b5f473c188dd456": "\\; - \\log q_j.", "2824a348a786343ddc38182c6cc693f1": " H_{t-1} = K +A'H_tA - A'H_tC(C'H_tC+R)^{-1}C'H_tA, \\,", "2824ca4de4430067aa4c914f9768f3e1": "{{V}_{TH}}", "2824f21de871d1118a3efac23d4ca430": "\\frac{1}{2}\\!\\,", "2825151dd7b28833ca68f978f5757e5d": "f = \\frac{2 r_{N} \\Delta r_{N}}{\\lambda} ", "28252280e4a6a2bbe22c9d2f568dd923": "\\frac{\\partial S}{\\partial x_k}= \\int \\frac{\\partial L}{\\partial x_k} \\, dx_3 = \\int \\frac{dp_k}{dx_3} \\, dx_3=p_k", "28252bc6e93be53331ee998ac1e426a5": "E = D^2 = 9", "282540820ff74ff47d353f3f72714902": "\\left (\\frac{\\Delta\\;h}{T}\\right) = \\left(\\frac{u}{\\sqrt{T}}\\right)\\cdot\\left(\\frac{\\Delta\\;v_w}{\\sqrt{T}}\\right)", "2825b5dd089ac4f5817f8bdcb807e7b9": "{\\mathbf{x}}^\\mathrm T(T)F{\\mathbf{x}}(T)", "2825b5f578c8fd7c338fc3d80329b10c": "f(x) = \\frac{x^{k - 1} \\exp(-\\frac{x}{\\theta})}{\\theta^k \\Gamma(k)}", "2825c4004b22329b10f9cc0210977591": "|d_0|>0", "2825c4d31e015e354eb3b409fdbe9333": "\\sum_{l=1}^M S_l (1 + \\mathrm{APR}/100)^{-t_l} = \\sum_{k=1}^N A_k (1 + \\mathrm{APR}/100)^{-t_k}", "2825dd5825664ca99c78d9c5c5aafc59": "\\alpha=\\frac{d\\log(\\lambda F_\\lambda)}{d\\log(\\lambda)}", "2825fda4cda7c4b4283b921167a002fb": "\n\\Omega_{n,\\mu\\nu}(\\mathbf R)=i\\sum_{n'\\neq n}{\\langle n|(\\partial H/\\partial R_\\mu) |n'\\rangle\\langle n'|(\\partial H/\\partial R_\\nu) | n\\rangle-(\\nu\\leftrightarrow\\mu)\\over(\\varepsilon_n-\\varepsilon_{n'})^2}.\n", "282642668875973e68fac1bfa91e1227": "C_1, C_2 \\mapsto C_1 \\otimes C_2", "282654f74c924ef27bfa39f49305af61": "\\left\\langle {dG}/{dt} \\right\\rangle_{\\tau} = 0", "28265d45ded171762eef09cf0766d542": "y(u)=-\\mathrm{cn(u,k)}+(a/k) \\mathrm{dn(u,k)}", "2826620b910812fe94d8a009e4ef4bd6": " f_1 \\, ", "282667de5cb89611fb86409738a84bb7": "P_{\\mathrm{VWAP}}", "28269d24a5e789d9173414362d012617": "\\mathrm{Diff}_k", "2826d392bcafcf2fb02e4eeaa7b90603": "Y_{21} = {-Z_{21} \\over \\Delta_Z} \\,", "28279454da0fc28ffa53194de06f99ab": "Z(\\cdot)", "2827b73f9cedd981dc3a0cb74f1d5fb4": "\\left [ 1 - {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y \\right ] dz = \\left [ {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z + {\\left ( \\frac{\\partial z}{\\partial y} \\right )}_x \\right ] d y.", "2827c0460c2a0f427899d74046d67100": "r=C_S^2 K_1K_2C_AC_B", "2827ef02f32f9ad88b2cdb012cf505d1": " \\frac{\\partial \\Psi}{\\partial t}(x,t) = \\mu (1-x)\\frac{\\partial{\\Psi}}{\\partial x}(x,t);\\qquad \\Psi(x,0)=x^i, \\quad \\Psi(1,t)=1.", "2827f2c33fe59d31c37b7f8822c4d446": "\\left\\{ 1,...,P-1 \\right\\}", "282805a4b296689136c18281442b235e": "\n\\cos \\sigma = -\\frac{4a^{2} - d_{1}^{2} - d_{2}^{2}}{2 d_{1} d_{2}}\n", "2828183aab2fe2abcb6066a505d8cc71": "(a_{21} w \\cdot x_1 + K a_{22} \\cdot w x_2) (1+g) = K x_2", "28283eb26f3b84805a1bcbec1e475d46": "\\ell_A\\colon A\\rightarrow\\Sigma_A", "2828d562ec89efba33b07eb7412af5d0": "\\left[\\hat{f}_i^\\dagger, \\hat{f}_j^\\dagger \\right]_+ = 0 ", "28292cb917e5527a549d86fe36011484": "\\text{2. }\\omega \\notin B : P(\\omega|B) = 0", "28297a01714f309f41f03b92c934281e": "\\vec{\\pi}_E = {\\tilde{r}_E}^{-1}", "282a0fa350c861e987b2d054c1a59169": "AJ = kJ", "282a24cc78a7f580f0f0598dd4622b22": "r=\\frac{mv_ern\\hbar 4 \\pi \\epsilon_0}{mZe^2} ", "282aa95583fb772109d3b9d9c59354c5": "L_y' = z' p_x' - x' p_z' = \\gamma(V)[ (zp_x - x p_z) + V(p_z t - z E/c^2) ] = \\gamma(V) [ L_y - V (m z - p_z t) ]", "282b182ad629456543a036fff3d7daf1": "z_{11} = -z_{22}", "282b581d27f8bb09fa048f5f22f96db9": "\\Delta H_p", "282b72fc2e97581553379bb1aee2aba9": "\\frac{(i\\omega)^2}{((i\\omega)^2-\\xi^2)^2}", "282b9c5a5e060cc36ae36b9f6b3921b3": "P_n(-x) = (-1)^n P_n(x). \\,", "282be2b28622fdff2108ba09a009c856": "x_{n+1} = x_n - \\frac{f(x_n)}{g(x_n)}", "282c7e565887bd16e68d098dd12ca91f": "y_i = f(x_i ;\\beta ) \\cdot TE_i \\cdot \\exp \\left\\{ {v_i } \\right\\}", "282c8b894d92b2f571ce5ff246465abd": "\\{x \\mid x \\in B \\wedge \\phi\\}", "282ce3e0be0ea2c075b358e722938c17": "\\frac{X-np}{\\sqrt{np(1-p)}}", "282cfca5cf7b4b5ad8376ff6c7562cf3": "\n\\mathbb{E}[f(x)] = \\sum_{j=0}^\\infty f(j) \\frac{\\lambda^j}{(j!)^\\nu Z(\\lambda, \\nu)}.\n", "282d1b3b6b7c775fb8571bad71d88fb8": "5+5+3+7=20", "282d223b7195128ecbb0623e783b053f": "\n\\begin{align}\nm = s' \\cdot r^{-1} \\pmod{n}\n\\end{align}\n", "282d4fca6d0c63bc6fd7078452c1a037": " \\int x^n \\Phi(x) \\, dx = \\frac{1}{n+1}\\left( \\left (x^{n+1}-nx^{n-1} \\right )\\Phi(x) + x^n\\phi(x) + n(n-1)\\int x^{n-2}\\Phi(x)\\,dx \\right) + C ", "282d7b02daf8ca7b4818dd13afb16172": "T_{12} = \\frac{S_{11}}{S_{21}}\\,", "282e2c4eb6b2a897cfc5562516edfadd": "3-n", "282e460e3eaa80d5c744970759cdd4f0": "P(event) = \\frac{\\text{number of outcomes in event}}{\\text{number of outcomes in sample space}}", "282e48e0982c2a2b03b868e6e19c2882": " E\\psi = -\\hbar^2\\left[\\frac{1}{2\\mu}\\left(\\nabla_1^2 +\\nabla_2^2 \\right) + \\frac{1}{M}\\nabla_1\\cdot\\nabla_2\\right] \\psi + \\frac{e^2}{4\\pi\\epsilon_0}\\left[ \\frac{1}{r_{12}} -Z\\left( \\frac{1}{r_1}+\\frac{1}{r_2} \\right) \\right] \\psi ", "282eb9aecddbf4051be94a714b45d867": "y R x", "282f45a4eacb40f0f3098ee43999346e": "2 \\sigma \\over \\bar{\\delta}", "282fba55f27ef23de22362dacbf5e00d": " \\ \\tau = Fk ", "282feeb6714ee76ff0a1f7b6f5f91551": "0, 1, .., m-1", "283004b30f9f239a582bbe4bb6cf4b7e": "\\dfrac{\\partial \\vec{u}}{\\partial t} = \\nu \\partial^{2}_{y} \\vec{u}", "28308a1c6a71a8b874a17a516d0db48e": "\\sum_{n=0}^\\infty {a_n \\over n!}t^n", "2830c53827f4b9ecaaf52120547a7c2f": "\n\\Rightarrow y[n] + \\frac{1}{4} y[n-1] - \\frac{3}{8} y[n-2] = x[n] + 2x[n-1] + x[n-2]\n", "2830e9afdffdb79bd779e67fc8ed3a88": "n_A=n_B=10", "28310a55ffd1c04b3fa6842a2ba62cf7": "2\\lim\\limits_{\\tau\\rightarrow+\\infty}\\langle T\\rangle_\\tau = \\lim\\limits_{\\tau\\rightarrow+\\infty}\\langle U\\rangle_\\tau", "28311e4d513f284d05fb138d09f18c07": "p^2 = m^2", "28312ac12818caa3729bba0dc7b6ff3f": "g^{x_1}", "28313d45fa5075ba2c52991d6219d47a": "i \\equiv g^j \\bmod p", "2831a72437535d7e3b161f09debd7bea": "L = \\oint p\\mathbf{n} \\cdot\\mathbf{k} \\; \\mathrm{d}A, ", "2831ad611236f92b164627e9b1679687": "\\displaystyle \\frac{(\\alpha+2)_n}{n!}{}_3F_2(-n,n+\\alpha+2,(\\alpha+1)/2;(\\alpha+3)/2,\\alpha+1;t)", "2831b3626b729f14ffc0871fbc3dd5b6": "u=u_{0}, v=v_{0}", "2831dae3dfe4647c94b5572d1eed7fe3": "M+i\\Gamma", "2831e20b5046e457e343e70fbd5d1f3e": " TSS = (y - \\bar y)^T(y - \\bar y) = y^T y - 2y^T \\bar y + \\bar y ^T \\bar y.", "283215c0662bd9d94369fcab9f1f16cf": "q_1 < q_2 < \\cdots < q_m", "28322245845b9159a31cb6aeff9e609f": "\\Pr[y_i^{\\prime\\prime} = ?] = {2\\omega_i \\over d}", "28322ccb357834b3566eab2757c02870": "\n{\\dot p_{\\theta_1}} = \\frac{\\partial L}{\\partial \\theta_1} = -\\frac{1}{2} m \\ell^2 \\left [ {\\dot \\theta_1} {\\dot \\theta_2} \\sin (\\theta_1-\\theta_2) + 3 \\frac{g}{\\ell} \\sin \\theta_1 \\right ]\n", "2832c1520ce1c6bfb82632eed6bf42ae": " V_t = W_1 - W_{1-t} ", "28331130743bbe540de433906fe9d884": "(q,1,A,q,\\epsilon)", "28333e889c633d00db89278c0c19688a": "\\subset P \\subset \\mathcal x : A(x)", "2833eb4ae9add4054b7962c3ffbda650": " t << x ", "2833ffbc34408e77079fbbb9e3f9dabb": "w_0(n) = \\mathrm{sinc}\\left(\\frac{2n}{N-1}\\right)\\,", "28346139543e02efa9ba4d2f6bb88f65": "F=U-TS,", "28353a4275331b17e2185ed0d8d182fb": "2(A\\xi)(A\\xi)^T = A X A^T\\,", "28353b30466e447c99a511a9c60386a6": " A_{kl}[\\mathbf{k}]=\\frac{1}{\\rho} \\, k_i \\, C_{iklj} \\, k_j.\\,\\!", "28353cbd8feba30e9befe9319ffe3dd3": "EER=(389-(41.2*age))+PA*((15.0*wt)+(701.6*ht))", "2835588df15e7eb8447d62beacd61bee": "\\displaystyle \\hat{f}(\\omega)", "28355a8bb3f33425034c26c9f95b07d4": "\\scriptstyle F:\\boldsymbol{K}\\rightarrow\\boldsymbol{E}^\\ast", "28359a8cf0593326d4ba560ee4a629b4": "\\min_{S_{k-1}} \\max_{x \\in S_{k-1}^{\\perp}, \\|x\\|=1} (Ax, x) = \\lambda_k.", "2835d71691cab553feb74dd8dcaa424a": "E_\\text{B} = -\\dfrac{Ry}{(n-\\delta_l)^2}", "2835f9d62898b04657b96ca9043f29b1": " P \\Bigl( \\Bigl | \\frac{ \\sum_{ i = 1 }^n X_i } { \\sqrt n } \\Bigr| \\le 1 ) \\ge 0.5. ", "28366d39ca3a174375e3b89627744103": "\\forall t. \\textit{changeopen}(t) \\leftrightarrow (\\neg \\textit{open}(t) \\leftrightarrow \\textit{open}(t+1))", "283674d0aa19610406934c17c2a5606f": " \\mu_m = \\mathrm{P}(Y \\leq m) \\,", "28367ca898b5132d5efb1a3be1b79dd5": "\\sum_{n=1}^\\infty \\left (\\frac{a_1+a_2+\\cdots +a_n}{n}\\right )^p<\\left (\\frac{p}{p-1}\\right )^p\\sum_{n=1}^\\infty a_n^p.", "2836c245b6a1eb0d9a64eaba3e9a35f1": "x = \\frac{P_0\\cdot i}{1 - (1 + i)^{-n}}", "2837003d4f12f26ebe679737aa30c873": "e^{-it}", "28373481dba8f851a32610296a82c9ff": "B_n(x) = \\sum_{k=0}^n {n \\choose k} b_{n-k} x^k,", "283745fe8d5e49b5531e927366bff5aa": "\\mathbf{E} \\cdot \\hat{\\mathbf{k}} = 0", "28384c2e06619c584b4f4954218e0726": "N=688,060", "28386902191279712af28ffe91791525": "(P^{(\\pm)} [F, G])^{IJ} = [P^{(\\pm)} F, G]^{IJ} = [P^{(\\pm)} F, P^{(\\pm)} G + P^{(\\mp)} G]^{IJ} = [P^{(\\pm)} F, P^{(\\pm)} G]^{IJ} ", "2838861045afb689a6c9552ab0149779": " \\frac{m}{r_0^2} \\, \\sin(\\theta) \\approx \\frac{m}{r_0^2} \\, \\frac{h}{r_0} = \\frac{m}{r_0^3} \\, h", "28388c9e646885f90376b62c98a1c223": " E = F^0{E} \\supset F^1{E} \\supset F^2{E} \\supset \\cdots \\, ", "2838a866669cdb3aef82a675ba5f9d9a": "g_{3}", "2838b1e7a3fd5dd2188bdab21274d5de": "\\Pr(N=n) = \\begin{cases}\n 0 &\\text{if } n < m \\\\\n \\frac {k - 1}{k}\\frac{\\binom{m - 1}{k - 1}}{\\binom n k} &\\text{if } n \\ge m\n\\end{cases}", "2838fab43c60b7db5b43d2cc70366878": " d\\int_t^s e^{- \\int_t^r V(X_\\tau,\\tau)\\, d\\tau}f(X_r,r)dr = e^{- \\int_t^s V(X_\\tau)\\, d\\tau} f(X_s,s) ds.", "2838fdb4549123936e97c0bfde0cb88a": "g, h\\colon X \\to \\mathbb{R}", "283944737f68e93d74319be731225697": "\\Delta=\\frac{1}{\\pi}\\theta(x)\\Big|_{-j\\infty}^{j\\infty} \\quad (15)\\,", "28398e4e56aff8d1c88bda7683cf432a": "d \\approx 4.12 \\cdot \\sqrt{h} ", "283994a33db70a97db400262fb392b64": " r_\\text{mirror} = \\frac{B_\\text{max}}{B_\\text{min}} ", "2839a5fbb217ac2043c6cfea1c9595a1": "\\begin{align} \\langle\\psi_{\\varepsilon}|\\mathbf{\\hat X}|\\psi_\\varepsilon\\rangle & = \\int_{-\\infty}^{\\infty} \\, x \\, |\\psi_{\\varepsilon}(x)|^2 \\, dx \\\\\n& = \\int_{-\\infty}^{\\infty} \\, x \\, |\\psi(x-\\varepsilon)|^2 \\, dx \\end{align} ", "2839afec110adf595d04617851439665": "S(f)\\colon S(A)\\to S(B).", "283a3a1dd559434b48426981d664a643": "\\left \\lfloor 1000/(2^{\\frac{4n-3}{8}})+0.2 \\right \\rfloor", "283a738c6960f95e25b8cab026251cfb": "\\omega^2 = {1 \\over R_1 R_2 C_1 C_2}", "283a8de164b77e42be45c2c8f0559a9b": "\\Pr(z_n=k\\mid\\mathbb{Z}^{(-n)},\\boldsymbol{\\alpha}) \\propto n_k^{(-n)} + \\alpha_k", "283aaf753c398118bac56510008bfd7f": "qy", "283ad67eb7f09f67e95dc924d06acdaf": "\\mu(p,T)", "283b116ac9928ebb0259590bce2317a5": "\\omega_0 = 2 \\pi f_0", "283b1b45009463b55aa6759c8315ad4b": "x^2+a_{11}x-2b_{2}a_{11}=0", "283b7fdd5f496fac26869d32f656773c": "\n\\bar{P}_{\\mathbf k} = \\frac{1}{2}( 1 - \\hat{\\mathbf k} ),\n", "283bac12893eb53da88e32574552904d": "\\mathbf{R}_2^{-1} \\mathbf{R}_1", "283bbdfa0b5621ab5ad695dab29e7ce8": "\\sec^2(\\theta)-1 = \\tan^2(\\theta).", "283bd7bab980fe084385632bf685ff83": "\\mbox{SO}_2 \\mbox{Cl}_2 (l) \\rightarrow \\; \\mbox{SO}_2 (g) + \\mbox{Cl}_2 (g)", "283bdc4cc5f737a4b4cbe924c95b7e1c": " \\mathbf{F} = -\\nabla P = - \\left(\n\\frac{\\partial P}{\\partial x},\n\\frac{\\partial P}{\\partial y},\n\\frac{\\partial P}{\\partial z}\n\\right), ", "283c174b0ffe96619d1b639d5aa31c76": "(\\phi,\\psi) = \\sum_{s=1}^d \\phi_s^* \\psi_s.", "283c65e79cf48f4214a5ee96f90e74d2": " \\frac{\\partial V}{\\partial y_k} = - \\frac{1}{(1+y_k/k)} \\cdot \\sum_{i=1}^{n} t_i \\cdot \\frac {CF_i} {(1+y_k/k)^{k \\cdot t_i}} = - \\frac{MacD \\cdot V(y_k)} { (1+y_k/k)} ", "283cec22b7687d5c477afa0b0bcfe167": "x^2+xy+y^2=1", "283d2d72ce966de46a22d805d4744e4d": "\\tfrac{\\pi}{3 \\sqrt{3}}", "283d2e52fe9675dd7fe38f83c0b85949": "\n \\tau_{xz}(x,z) = \\cfrac{Q_x}{D}\\int z~E(z)~\\mathrm{d}z + C(x)\n", "283d3d617ca8d9a7a7903a003f49ee2d": "h H ={\\color{Blue}R}\\tan\\!\\left(\\frac{\\pi}{2 n}\\right)\\!", "283da786336dced22605c7369db5ba8f": "[C] = \\frac{k_2}{k_1+k_2}[A]_0 (1-e^{-(k_1+k_2)t})", "283dc1d74cc8a4e586f2fca8a97c12b8": "g=\\bar{g}=\\tilde{g}=\\{g_c:g_c(x)=x+c, c\\in \\Bbb{R}\\}", "283ddbb64c54a4533de6411ad1aea3ae": "f\\left(0\\right)=0", "283dde22697949bb776f707e12938a0b": " =, \\ \\in , \\ \\le,\\ <, \\ \\sub,... ", "283df1870c199c9d9963765b3cf4cc26": "\\tan(11\\pi/16)", "283df78fd3a0f5f5d94605e4b6f0abf9": "\\ \\displaystyle \\psi \\ ", "283e03c9c57ef43f18f1dd4a5355f4ac": "\\text{Prim}\\subseteq \\text{Tp}(\\text{Prim})", "283e0dd0461b844c560d1c61b47aceab": " M_2 = q, N_2 = q ", "283ef1ad8264bb8eb737b9892b880674": "\\sum_i X_i \\beta S(Y_i) \\alpha Z_i = \\mathbb{I},", "283efbbb374b2f6587b3fb05fbde4f5a": "X = Y = C^{0} ([0, 1]; \\mathbf{R}),", "2840069ef8c635cbfac9e6cb0d5b2869": "\\psi = \\vartheta - \\theta", "2840097376437134d9d999b25abf67df": "\\tilde E_6 \\to \\tilde F_4", "28408fe8fe1e0203f74ab4daf12e3c82": "2^{3 \\times 8} + 2^{5 \\times 8} = 1099528404992", "28409009aa222427023ffd47ba801c85": "Z_{i,t}=0 ", "284096a5d219766f720d37d83f9b741f": " \\ddot{x}=a(t,x,\\dot{x}) + u", "2840b397c7ebdaba003ea20281ad9908": "\\Gamma>0\\,", "2840faa55b733382f6d059c6a96dd9ee": "T = \\frac{\\epsilon}{\\delta S} = \\frac{2 \\times 10^{-23}J}{70 \\times 10^{-23}J/K} = \\frac{1}{35} K", "284126970e43f7ffb46f9127dd30e421": "P = {1, 2, 3,\\ldots, n}", "284163c26c8a616741409b8060472d13": "\\mathbb{E}[r|\\theta^\\ast,a^\\ast,x]", "2841aef7a5af7a0c9430c0f088fe9c47": "RR^{-1} = kN + 1", "2841b11a2b2bd36a84fb102cc80cb037": "\\frac{DS^{\\mu\\nu}}{ds} + u^\\mu u_\\sigma \\frac{DS^{\\nu\\sigma}}{ds} - u^\\nu u_\\sigma \\frac{DS^{\\mu\\sigma}}{ds} = 0", "2841d1f57bcad856ca8c9dff4edaed7f": "G= \\frac{a(a+1)}{2}. ", "2841d7351508128398154618b9804ce4": " \\frac nm = \\frac{m}{n-m}.\\qquad (*) ", "284228c433b4a2388e4fe048388a2c34": "\\xi = A_x f_x(z) e^{i(\\omega t - kx)} \\quad \\quad (1) ", "28424d5404816c8b7eb01a9c2cb8b8a6": "|m_I,m_J \\rangle ", "28425fa768271bdf6b3e27bf8d94ebbe": "\n\\sum_i p_i (\\log p_i - \\sum_j (\\log q_j )P_{ij}) \\geq \\sum_i p_i (\\log p_i - \\log (\\sum_j q_j P_{ij})\n", "284276368e293ce1d801f6ac0ce25589": " U_\\text{eff}(\\mathbf{r}) = \\frac{L^2}{2mr^2} + U(\\mathbf{r}) ", "28429bb4a67f9fdd116462b10fede3f3": " \n\\begin{bmatrix}\n a & b \\\\\n c & d \n\\end{bmatrix}\n= \n\\begin{bmatrix}\n 1 & \\lambda z \\\\\n 0 & 1 \n\\end{bmatrix}.\n", "2842de00abe300fbaff649e74c25f3e3": "IPxy \\leftrightarrow (Pxy \\and (Czx \\rightarrow Ozy)).", "284305b44f9b2c4f57193fe8bd17706d": "C = \\lbrace U_\\alpha: \\alpha \\in A\\rbrace", "284307c2182414dc93d37f673e36933c": "G_2\\setminus G_1", "28432a9919394851ab7cfe2d9fbc9765": "RR=\\tfrac15", "284395965384ad307f8acf12c5017a9d": "\\operatorname{Li}_2(e^{i \\theta}) = \\zeta(2) - \\theta(2\\pi-\\theta)/4 + i\\operatorname{Cl}_2(\\theta)", "2844051ec28da950fe01fcbe59e38cc4": "\\ln(c_{T-j}) + b V_{T-j+1}(Ak^a-c_{T-j})", "28441f65f29e9cd84256c4e41ea72e9e": "\\mathbf{n}\\,\\!", "28445999aa4b87bb4fde9c67cd92eab2": "\\begin{array}{cc}\n \\begin{array}{rr} \\\\ &3 \\\\ \\text{-}1& \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & \\text{-}12 & 0 & \\text{-}42 \\\\\n & & & \\\\\n & & & \\\\\n \\hline \n 1 & & & \\\\ \n \\end{array}\n\\end{array}", "284499281807875f8f358e48ac647bfd": "r\\in(0,1)", "2844aba8b65a4c2b47378788082a892a": "x:G\\to Y", "2844d2ebc068ccf99556f5454e23c8d8": "h(t)=h_0 \\sin(\\omega t)", "2844d30f147ee7c01c5e607ee4fae014": "L_{\\text{i}}(\\mathbf x,\\, \\omega_{\\text{i}},\\, \\lambda,\\, t)", "28450326cc4d39ed0a4b0804f77b4323": "A',S',T'", "28454c2dfb3ca842e861debef3ab4ea0": "f(x) = \\sin(2x)", "284555eabf8f5a0b52d48b5e7eb73293": "\nP = \\left\\langle \\psi'\\right|\\boldsymbol{\\mu} \\left| \\psi \\right\\rangle =\\int {\\psi'^*} \\boldsymbol{\\mu} \\psi \\,d\\tau,\n", "2845650f3467218190805286ce04287c": "\\frac{\\pi_1}{\\pi_2}", "2845710b9b89463d603c6ef9e94a0617": "\\succ^p_v", "28458041c206e033109a9fb1baec9ca3": "Cl_t^{\\geq}, t \\in T", "2845860934de4f42775606cc78d3c176": "\\triangle\\delta = - \\beta\\cdot\\sin(\\delta-\\alpha)", "284593eee75e69b2d6421edc5382ee69": "\\mathbf{B}(t) = (1-t)\\mathbf{B}(t) + t\\mathbf{B}(t)", "28459a7631fbd31468e0dab7c52c0751": "(\\log 5 - \\log 3).", "2845c55255ebc9076875c397054b9356": "\\scriptstyle \\mathbf{F}", "2845e91b1773d0d63ac5c8bdf58417de": "H_{p+q}(LM\\times LM)", "28462d62b8f34681c3102a8225af4ebe": "x_i=\\frac{g_i}{g_0}(u_1,\\ldots,u_d)", "28463ca1f2f2ae194927fe2c395fdf5e": "\\left | S_k - L \\right \\vert \\le \\left | S_k - S_{k+1} \\right \\vert = a_{k+1}.\\!", "2846601d9a14d87b9f4eaec14b9b7bae": "\n (\\lambda~\\boldsymbol{\\mathit{1}}+\\boldsymbol{A})^T\\cdot\\left[\n \\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^2 + \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda + \n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}}\\right] = \n \\det(\\lambda~\\boldsymbol{\\mathit{1}} + \\boldsymbol{A})~\\boldsymbol{\\mathit{1}} ~.\n", "284661569ec69cb58d73827c0bf7185b": " h(X,Y) = -\\int_Y \\int_X f(x,y) \\log f(x,y) \\,dx \\,dy", "2846711728a14c311e4e0903f534b01c": "\\lim_{x\\searrow0} \\frac{e^{-1/x}}{x^m} = 0,", "2846a528b93c0234575ced2eafe3cd8b": "\n\\Delta \\hat{z}\\ =\\ \\frac{r^2}{\\mu}\\left[\\ 2\\ F\\int\\limits_{0}^{\\pi}\\sin u \\ du \\right]\\quad \\hat{h}\\times \\hat{z} =\n\\ \\frac{r^2}{\\mu}\\ 4\\ F\\ \\quad \\hat{g}\n", "2846d20f5eaaafcd08d5d6adb30a17ff": "\\hat{a}_{j}\\hat{a}_{j}^\\dagger", "28472aed5e1239b54f3d16b3bc3ab3ba": "1/distance", "284780e49866c5b42d7eae735c87400b": " 1/4\\left(1/4-1/4e^{-4/3t}\\right)\\ ", "284790fad2154fb9332b066aff36bdb3": "\\partial_i := \\frac {\\partial}{\\partial x^i}", "2847a52983f7ba1fd700d124cff6de4b": " G := \\left\\{ \\left. \\begin{pmatrix} a & b \\\\ 0 & 1 \\end{pmatrix}\\ \\right|\\ a > 0,\\ b \\in \\mathbf{R} \\right\\}.", "2847f314b7826dcee416abfb3f02fb0c": "\\frac{\\mathrm{s}}{\\mathrm{m}} \\times \\frac{\\mathrm{m}}{\\mathrm{s}}", "28481f773b834e7f8a4a0ea3ccace4d7": "\\frac{s}{1-e^{-s}}=1+s_e ", "284890c881bd6f2c0f92f4c1444fe6ab": "R(0,m)", "2848d1c0a2914f6cc2b7bbef19aea479": "{\\textbf{x}_{t+1}} = A_{i_t}\\textbf{x}_t,\\quad A_{i_t} \\in \\{A_1, \\dots, A_m\\}", "28490a38a6e9cf83fb4ebdd34f96b564": "\\rho = A a^{-3} + B a^{-4} + C a^{0} \\,", "28491b99d2da0d0b821b768753d41a9d": "B < \\tfrac{1}{M}\\cdot\\tfrac{1}{2T},", "284926c45f7465a039b1d4fd5bedc9d0": "S_{RBB}(n)= S_{RRB}(n) = \\tfrac{1}{2} 3^n - \\tfrac{1}{2} ", "28492fde94aabf7f86a39ee66b6092cf": "\\omega^k{}_{ij} := {{\\mathbf{u}}}^k \\cdot \\left( \\nabla_j {{\\mathbf{u}}}_i \\right)\\, ,", "2849437afdc4314e32f366a312729b5c": " \\sum_\\text{cyclic} \\sin A = \\frac {S} {2Rr} = \\frac {s}{R} \\quad\\quad \\sum_\\text{cyclic} \\cos A = \\frac {r+R} {R} \\quad\\quad \\sum_\\text{cyclic} \\tan A = \\frac {S}{S_\\omega-4R^2}=\\tan A \\tan B \\tan C \\, ", "28498eeabbf20326ae32c85db600ad6b": "496 = 2^4(2^5-1) = 1^3+3^3+5^3+7^3", "284a3b4595c993374b7c0da777bc13b0": "\\Pi(x) = \\sum_{p^k\\le x}\\frac{1}{k}.\n", "284a3c2f37ea20a2318ca10ccb62dde5": "\\mathbf{e}_{ij} = \\mathbf{e}_i \\otimes \\mathbf{e}_j", "284a984fc35ba2a8642aa7e19347be59": "O(\\log |V|)", "284b3567801d1127ea5db52b3fe71319": "F_{i}", "284b3b4a1dfd5b757a138ae6ce43c3e9": "A \\rtimes G", "284b4ca8fc22248876abb4330bf9149b": "(\\ddot{r},\\ \\ddot{\\theta})", "284b6f4c95696bd001a0efadce74e8f0": "\n\\begin{align}\n\\biggl|\\bigcup_{i=1}^n A_i\\biggr| &= \\sum_{k = 1}^{n} (-1)^{k+1} \\left( \\sum_{1 \\leq i_{1} < \\cdots < i_{k} \\leq n} \\left| A_{i_{1}} \\cap \\cdots \\cap A_{i_{k}} \\right| \\right).\n\\end{align}", "284bad3c753dc88f43867ca3801f7267": "\\alpha=\\frac{E}{RT}", "284bbbe7eb3ce45b2f31dd1c0e447f87": "\\mathfrak{so}_{2n}(\\mathbf{R}),", "284be86c439db7e4606bc748fe786fd6": " f^{-1}(y) =\n\\displaystyle \\sum_{n=1}^{\\infty}\n {\\frac{y^{\\frac{n}{3}}}{n!}} \\lim_{ \\theta \\to 0} \\left(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d} \\theta^{\\,n-1}} \\left(\n \\frac{ \\theta }{ \\sqrt[3]{ \\theta - \\sin( \\theta )} } ^n \\right)\n\\right)\n", "284c1cc2e58d21121bc4aa0ca47a531a": "\\delta S = S_{n+1} - S_n = 0 \\,,", "284c2b7ba8e57fe5880e4b5ac5ade92e": "\\rho_c = \\frac{3H_0^2}{8 \\pi G}", "284ca24d7c4086d2020f52ca40f536e5": "\n\\frac{d\\phi}{d\\zeta} = \\omega\n", "284d2d366d20f826cd96671d1c069b6e": "\\begin{align}\nf_X(x_1,\\ldots,x_n)\n &= \\frac{1}{\\theta}\\mathbf{1}_{\\{0\\leq x_1\\leq\\theta\\}} \\cdots\n \\frac{1}{\\theta}\\mathbf{1}_{\\{0\\leq x_n\\leq\\theta\\}} \\\\\n &= \\frac{1}{\\theta^n}\\mathbf{1}_{\\{0\\leq\\min\\{x_i\\}\\}}\\mathbf{1}_{\\{\\max\\{x_i\\}\\leq\\theta\\}}\n\\end{align}", "284d3b3c3cf2193c002edd719f7a1ea5": "\\mathbf{u} \\otimes \\mathbf{v} = \\mathbf{u} \\mathbf{v}^\\mathrm{H}.", "284d4192c04968481460de3c9b1f8054": "\\chi^{d}_{ij}", "284d4bd3a21dd5f68ebc2e3ef5925804": "W_{j} = \\frac{N_j}{R_j^2}.", "284eac35128962f7a0e15301e6a253c2": "\\det( \\overrightarrow{P_1 P} , \\overrightarrow{P_1 P_2} ) = 0. ", "284eb525fec959505ccd8ecf768d5e2d": "n_1,\\,n_2,\\,\\!", "284ec89120a4d0e922e4a27759aa0e4d": "Q_B (\\Psi_1 * \\Psi_2) = (Q_B \\Psi_1)*\\Psi_2 + (-1)^{gn(\\Psi_1)} \\Psi_1 * (Q_B \\Psi_2)", "284f02874ad8387074091513d7010c0b": " s_{iw}= d_{ib} / (d_{iw} + d_{ib}), 0 \\le s_{iw} \\le 1, i = 1, 2, . . ., m ", "284f0b27f6c69d25662e79a6148321ab": "\\alpha =0\\, ", "284f4e15d1292910d3499d7cfea214b4": "S_{ab}", "284f8c4bb92f6bdff536d4a25fa4b8bb": " \\mathbf{p} = m \\mathbf{v} \\,\\!", "284f9bb31edb1a603422fc732d93858b": "E \\left\\{ \\| {\\boldsymbol{\\beta}} - \\hat{\\boldsymbol{\\beta}} \\|^2 \\right\\} ", "284fb172370810f4927aec4a471f4800": " \\pi: T \\to R^2 ", "284fcad0e4e2a59d3e3ddaa6f3487fbe": "x^2 + y^2 + z^2 - w^2 = 0.", "28502db5b98be2e3ff86657d57773f4c": "||dE[A]||^2=\\langle Y^{(e)},Y^{(e)}\\rangle=E[(A-E[A])^2]=V[A]", "2850464b46f985b381df8eec76a48af6": "\\mathbf{M}^{T} \\vec{u}_{1} = 0 + 2 \\lambda_{2} \\vec{v}_{1}", "285059880dc839618907823875294369": "M u", "2850608512d9db3ba1b078e7d0ea5b37": "\\langle C,\\alpha\\rangle", "28507c2f96bccbb9a2f50a604d9affb0": " \\theta = \\frac {\\pi}{2} ", "28509b2b87e09999f905aa0d6cba6b24": "\n\\mathbf{J}(\\mathbf{r}', t') = q\\mathbf{v}_s(t') \\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t'))\n", "2851016dfbe2f2f91f5eb310a6c7ac2a": "\\displaystyle q=y,\\quad p=y^{\\prime}+y^2+t/2", "285107b48b7602928a3913566106422e": "\\textstyle Y_{3}", "28512d18146b015b1e834ebef493dea5": " y_c'(-2 \\cdot c) = c \\,\\!", "285131b84de641b2991f54f3251be358": "(-)^X", "285191bb78e2df8601c182fa2c6ebf9d": "\\mathsf{M}=(0, \\mathbf{M}),", "2851ca436d11c05fe4fc20dc2a868bc1": "\\frac{D} {Q} K", "285238ef08d50f2c5b3e973f6a3bdf16": "\\mathbf{\\hat{w}}", "285250af6bbfc57aeedc8ab017a600cd": "A \\to A", "2852b4e49e74ac1c75b5feded6edc522": "\\ Q = K ( [P_c - P_i]S - [P_c - P_i] )", "2852bb14c03e2436f250558e2e9727fc": "\n\\sigma_2 = \\cos \\psi \\, d \\theta + \\sin \\psi \\sin \\theta \\, d \\phi\n", "2852d4de8777766a328a2d17a719364a": "a_{1,0}=H(a_{0,0}||a_{0,1})", "285360598f96719e61e0714cb3904b5a": "\\frac{\\Gamma(\\alpha,\\beta/x)}{\\Gamma(\\alpha)} \\!", "2853f37cbe96d57d836c6d5bbf30b1da": "\nq_{xx} = a^2 \\cos^2 \\theta + b^2 \\sin^2 \\theta\\,\n", "28541066f69e3cc66f50d2957696e69a": " \\mu_t ", "285413e0971dbffbf5e42a0e5d34a18d": "0\\le t\\le 1", "285428b50439091fc36866df1eea2dd4": "\\mathbf{u}^k=\\eta^{k\\ell}\\mathbf{u}_{\\ell}\\,", "2854574b1815c1e66b50dbc1f38f3df9": "d=-4,-8,-12,-16,-28.\\ ", "2854a95e886f6ab6127abe8234074e88": "\\dot\\theta^{1,2}(t)", "2854f6c09b8473d4a3215b7c2d694197": "0\\rightarrow B\\rightarrow X'_n\\rightarrow\\cdots\\rightarrow X'_1\\rightarrow A\\rightarrow0", "2854fb9c44b9e63cbfbda5be4116ddc5": "0 \\le \\zeta < \\infty", "2855b266acef94849411ad909328a5a5": "\\sum_{i \\in \\mathbb{N}} | e_i \\rangle \\langle e_i | = \\hat{1}", "2855ba314b861ae3675a3a41151f5db8": "[d^{-1}]", "2855ccd7047e7890860686946b92fb6c": "I_\\text{channel}", "2855e79eacd46cbc5fd77f2befbbfce1": "\\frac{mean\\ distance}{\\frac{1}{2}\\sqrt{\\text{density}}}", "2855fe0c1b7cbbf96c1facd4751372b2": "\\frac{u^3}{a^3}=\\frac{b}{a},\\, u^3=a^2b", "28561096c911938c41f71b8bb65927f6": "\\bigvee_{n=0}^N T^{-n}Q = \\left \\{Q_{i_0} \\cap T^{-1}Q_{i_1} \\cap \\cdots \\cap T^{-N}Q_{i_N} \\text{ where } i_\\ell = 1,\\ldots,k ,\\ \\ell=0,\\ldots,N,\\ \\mu \\left (Q_{i_0} \\cap T^{-1}Q_{i_1} \\cap \\cdots \\cap T^{-N}Q_{i_N} \\right )>0 \\right \\}", "285657e37d61def36f931193630183cb": "\\textstyle g ", "2856693c073922f59799e321862e3ad1": "a^{\\prime\\dagger}_i", "2856afbc1abd54fd0121d1dd08547784": "\\displaystyle dE = - \\iota_H d\\alpha", "2856bd6d0e04d44a64c851ef98def6c9": "(13)\\quad\\quad\\; u_s\\left(\\rho_2 u_2 - \\rho_1 u_1 \\right) = \\left( \\rho_2 u_2^2 +p_ 2 \\right) - \\left(\\rho_1 u_1^2 +p_1 \\right)", "2856d04d586c50a1f56e66edd3e30b9d": "z \\in \\Omega", "28570ecdcc326c36c408fc7cb4468f78": "\\mu_{u \\to v} (x_v) = f_u(x_v)", "28572ab1864ef462bc504258cefe27eb": "\\Lambda_{Roy} = max_p(\\lambda_p)", "2857a1840370c99b8e0ecfb0db210e6b": "\\rho(\\mathbf r,t)=N\\sum_{s_1} \\cdots \\sum_{s_N} \\int \\ \\mathrm d\\mathbf r_2 \\ \\cdots \\int\\ \\mathrm d\\mathbf r_N \\ |\\Psi(\\mathbf r_1,s_1,\\mathbf r_2,s_2,...,\\mathbf r_N,s_N,t)|^2.", "2857b4fd34e4e5eb74c88b872110da5f": "o_1", "28580724e256f4ed9298cf6418346c73": "\\tau, \\tau'", "28580757bd34c4107f68598d5a55ee02": "k = ap", "28584ee9397b91dd00ece5d7c9eaf1ad": "\\mathbf{a} \\wedge \\mathbf{b} .", "28584f74407126559e4b4fc19fc72bdd": "\\frac{DV}{Dt} = -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial s} - K V", "2858bba5687ac80ed5391eb909efdd32": " T(n,1)=1,\\;T(1,k)=1,\\;n \\geq k:T(n,k) = -\\sum\\limits_{i=1}^{k-1} T(n-i,k),\\;n\n\n==Notable categorization oddities==\n*'''Category:The League of Extraordinary Gentlemen''' under:\n**'''Category:Alternate history comics'''\n**'''Category:Steampunk comics'''\n*'''Category:100 Bullets''' under:\n**'''Category:Crime comics'''\n*'''Category:Sin City''' under:\n**'''Category:Crime comics'''\n*'''Category:Bone (comics)''' under:\n**'''Category:Fantasy comics'''\n*'''Category:Conan the Barbarian comics''' under:\n**'''Category:Fantasy comics'''\n*'''Category:The Dark Tower comics''' under:\n**'''Category:Fantasy comics'''\n**'''Category:Post-apocalyptic comics '''\n**'''Category:Science fiction comics'''\n**'''Category:Western comics'''\n*'''Category:Elfquest''' under:\n**'''Category:Fantasy comics'''\n*'''Category:Oz comics''' under:\n**'''Category:Fantasy comics'''\n*'''Category:The Sandman''' under:\n**'''Category:Fantasy comics'''\n**'''Category:Horror comics'''\n*'''Category:Warhammer Fantasy comics''' under:\n**'''Category:Fantasy comics'''\n**'''Category:War comics'''\n*'''Category:Army of Darkness comics''' under:\n**'''Category:Horror comics'''\n**'''Category:Zombies in comics'''\n*'''Category:Cthulhu Mythos comics''' under:\n**'''Category:Horror comics'''\n*'''Category:Dan Dare''' under:\n**'''Category:Science fiction comics'''\n*'''Category:RoboCop comics''' under:\n**'''Category:Science fiction comics'''\n*'''Category:The Stand comics''' under:\n**'''Category:Post-apocalyptic comics '''\n*'''Category:James Bond comics''' under:\n**'''Category:Spy comics '''\n*'''Category:Warhammer 40,000 comics''' under:\n**'''Category:War comics'''", "2865c613c99aaa3d69551405dfc218ba": "\\ \\dot{c} + \\Omega(x^*-c) = 0, ", "2865caa8a46c6717434c908272c2f810": "x \\div 9", "2865d444b3b35c5bcc305c75e3fd29c8": "p_{mk}^{}", "2865eda9ce5e110adea2b60a14de4b71": "Q^{\\ell_A}_{m_A}", "2865efcf42bdde4488c4e4e4ef817385": "\n\\begin{align}\n1 & = \\binom{t}{1} - \\binom{t}{2} + \\cdots + (-1)^{t+1}\\binom{t}{t}\\\\\n & = |\\{A_i \\mid 1 \\leq i \\leq t\\}| - |\\{A_i \\cap A_j \\mid 1 \\leq i < j \\leq t\\}| + \\cdots + (-1)^{t+1}|\\{A_1 \\cap A_2 \\cap \\cdots \\cap A_t\\}|,\n\\end{align}\n", "286625a4b61be71f298d1241b0698b10": "\\mathfrak{G}", "28668c0c0d61ec32a1ab13d4a6823999": " g(C)=g(N)+\\sum_{\\text{infinitely near points }x}m_x(m_x-1)/2", "2867bc363a7253bff561ffd037ffeb26": "f(g)= 1\\cdot 1_G + \\sum_{g\\not= 1_G}0 \\cdot g= \\mathbf{1}_{\\{1_G\\}}(g)=\\begin{cases}\n1 & g = 1_G \\\\\n0 & g \\ne 1_G\n\\end{cases},", "2867f4af706d38569867485889cd8c96": "\\bold{a}", "28680de838f8f339669ac059ac5005c2": "K_\\nu(z)\\,", "28680f88314d249a8c170ac1691fab97": "[-1,\\ 1]", "2868388fa9c06d5194c139457ec00d93": " \\frac{d \\varphi}{d a} = \\varphi_a(x,y,u,A,w(A)) + w'(A)\\varphi_b(x,y,u,A,w(A)) =0. \\,", "286854f1af247f9cec36acf237f2c493": "S_P^2 = 2.765 \\, ", "2868af0bc6eab6aa1e3e05e22a7d3493": "\\rho_h(M(a,b,c))\\;\\psi(x)= \\exp (ibx+ihc) \\psi(x+ha)", "2868bdeb19c2fbc67d4c0c422b1b8973": "\\mathrm{J}=\\frac{\\partial\\left( x,\\theta\\right) }{\\partial\\left(y,\\xi\\right) }=\\left(\\begin{array}[c]{cc}\nA & B\\\\ C & D\\end{array}\\right) ,", "28693f36da6b00f80c8bc58b196ef7a1": "\nH(\\omega) = \\left(\\delta(\\omega) - {i \\over \\pi \\omega}\\right) * G(\\omega) =\nG(\\omega) - i\\cdot \\widehat G(\\omega) \\,\n", "286953ca19adb87687468dd5f943c3ce": " \\sigma_p^2 = w_A^2 \\sigma_A^2 + w_B^2 \\sigma_B^2 + 2w_Aw_B \\sigma_{A} \\sigma_{B} \\rho_{AB}", "2869b734143e805a46e0bd4348ede6eb": "P(|X_n/a_n| > M) < \\varepsilon,\\ \\forall n.", "2869dd2bcec667c9306f171c159630b1": "C_2 = \\{U^3 = - I\\},", "286a08da142b39864018779678bb07de": "x \\geq [x]", "286a0fe07f8fc1f50ec995d93ddda362": "M_{57,885,161} \\approx 5.81 \\times 10^{17,425,169} \\approx 10^{10^{7.2}} = (10 \\uparrow)^2 7.2", "286a12120a9296205c7ce4da3f05003f": "\\,pL + (1-p)N\\,", "286a1398fcc0759a3ed7f98cb6e6b8de": "f^{-1}([c,c'])", "286a1f471e752a8643dbe53f15a46946": "C_{in}^\\alpha (x) + C_{in}^\\alpha (y) = (x,\\alpha x)+(y,\\alpha y) = (x + y,\\alpha (x + y)) = C_{in}^\\alpha (x + y)", "286a232b5107c68f7cb2c78d233a3f8e": "k_v", "286a2ff13eafdc3542dfedef94320557": "\\rangle", "286a42733f3ff6c6a156d7ec4c9aa10b": "\\hat F(x)", "286a7c43e4c18da745808373aa193819": "f:k\\to j", "286adb3a7c288faf021ea7ae95aecefe": "(f(x))^j = \\sum_{k=0}^{\\infty} M[f]_{jk} x^k.", "286b02707b1a36ca99a961b53fae4ae9": "N! = \\Gamma(N+1) ", "286b4e6e67ee000352458410bcd9b96a": " \\int_E f \\, d \\mu = \\int_E f\\left(x\\right)\\, \\mu\\left(dx\\right)", "286b89fffa11ef2b83c2eb8068f9807e": "T_{1}, T_{2}", "286c097ff48cad0fb038a7ac9918956a": "A_\\mathrm{vdB}", "286c57016f11e68e131e21ead158f853": "\\liminf_{x\\rightarrow c}\\frac{f(x)}{g(x)}=\\limsup_{x\\rightarrow c}\\frac{f(x)}{g(x)}=L", "286c7c031bc5e164dd0cf8fcc4ca1dee": "\\not=\\varnothing", "286d1e0b3e88ca0c9b4e019e9ca79cd3": " \n (\\lambda \\lambda \\lambda 6 6 2 (\\lambda \\lambda 6 (\\lambda 1 (2 6) 3) (\\underline{15} (5 1 (\\lambda 1)) (5 (\\lambda 1) 1)))) (1 2 (\\lambda \\lambda \\lambda 3 1 2))) 1 (\\lambda \\lambda 2))))) (3 (1 (\\lambda \\lambda \\lambda \\lambda 9 (1 (\\lambda 5 1 (\\lambda 1 5 4)))\n", "286d27874401cea6ac9301ac3d53ce91": " G = \\frac{2}{\\pi}\\int_0^1\\frac{dx}{\\sqrt{1 - x^4}} ", "286d2cf24aa5a675fa729356990d08d9": "r_1=r, r_2=-r\\,\\!", "286d3bbc9892c343673303a6f66d5bf7": "A\\bullet B = (A\\oplus B)\\ominus B, \\, ", "286d6570ea75b46d39d8c4239542542f": " w = w(r,t)", "286d8cdac8faa121e4c3c846a117c00f": "v = \\frac{\\kappa\\,\\Delta P}{\\mu\\,\\Delta x}", "286e0e80e1c2db9f4dde113eeb5f5bf4": " \\mathbf{v} = \\frac {\\mathrm{d} \\rho }{\\mathrm{d}t} \\mathbf{u}_{\\rho} + \\rho \\mathbf{u}_{\\theta} \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t} = v_{\\rho} \\mathbf{u}_{\\rho} + v_{\\theta} \\mathbf{u}_{\\theta} = \\mathbf{v}_{\\rho} + \\mathbf{v}_{\\theta} \\ . ", "286f19a46fc199ab18ee0a7421f09438": "\\mathcal{F}_n", "286f40e05e4f9a74be01f2a63bd5da45": "2^in - 1", "286f4a171dbe56dd41892b3fc4badd13": "\\mathbf T^n \\vec b^n = \\hat e_n.", "287078a3da80f9b3a39007425724e4b9": "\\left(a \\csc\\left(A + \\frac{\\pi}{6}\\right), b \\csc\\left(B +\\frac{\\pi}{6}\\right), c \\csc\\left(C + \\frac{\\pi}{6}\\right)\\right)", "2870da910b1ad6292810b1c3a7181ba8": "[Z, \\Omega X]", "287128bf557db73ad6efaa80e7577070": "{}^{1}i = i", "2871896333dbd34312a604b2625c8f9c": "\\hat{H}_{s}", "28718d182c2578ea2c81a53baec06e79": "\\frac{\\epsilon}{k} =\\frac{1.000+0.945(c-1)+0.134(c-1)^2}{1.023+2.225(c-1)+0.478(c-1)^2}", "28720155915560e4635f8e9edcf4ad97": "P(W|L) = \\frac{0.75\\cdot0.50}{0.75\\cdot0.50 + 0.15\\cdot0.50} = \\frac56\\approx 0.83,", "28726f6b66ad687fa43aab0083e2dbe3": "(N,D,\\lambda)", "2872a8a59bedb00b19f0f4ec8a4c1cc1": "X^{\\beta_{\\gamma+1}}_n", "2872b50c8f6dab67e5f70c4d75d9a13d": " (a_1, \\ldots, a_n) = ((a_1, \\ldots, a_{n-1}), a_n).\\!", "2872ec6e85009daa0ee74e6240a1fc6c": "\\vec{W}_\\mu", "2873411454166a7fdc9c5a23220e267f": "\\begin{smallmatrix}M_v\\ =\\ m + 5 (\\log_{10} {\\pi} + 1)\\ =\\ -1.47 + 5 (\\log_{10}{0.37921} + 1)\\ =\\ 1.42\\end{smallmatrix}", "2873a7bc7615707d281da7ce3e828fa3": "f\\wedge g", "2873edfc8eb06138b5cff10e375d037e": "ce^{i\\psi}-a=\\tfrac{b}{2}(e^{i\\rho}-e^{i\\lambda})=i b\\sin\\alpha e^{i\\theta}.", "287435c44c7b5aaae57c91c0c8aa41bd": "d^* = d(d-1)-2\\delta-3\\kappa.\\,", "287457257b63a341593b1b58c8f859bf": "\\;=F_{id}(N,V,T)+F_{ex}(N,V,T)", "287493795d19a0e23be24874d365df98": "L(i)", "2874bb2c3a9cc6217705eead6273d30e": "{\\tilde{C}}_n", "2874c284675cd977c0144210092eadde": "\\sigma_y^2 = \\frac 1n \\sum_{i=1}^n \\left(y_i - \\overline{y} \\right)^2 ", "28751ae47c2cb13551948f3880aa0e2d": "F(\\mathbf{x}+\\mathbf{h}) \\approx F(\\mathbf{x})+\\mathbf{h}\\left(\\dfrac{\\partial}{\\partial \\mathbf{x}}F(\\mathbf{x})\\right)^{T}.", "287534998454db10659f41e200838084": " \\varepsilon_{ijk} A_j \\nabla_k ", "2875548d7438f9df84bdcbefe7012b19": "G(t)- x_f(t)", "287563016c5fcf69aea236ee94452d9d": "\\frac{\\partial T}{\\partial t} + u \\frac{\\partial T}{\\partial x} + v \\frac{\\partial T}{\\partial y} + w \\frac{\\partial T}{\\partial z} = Q", "28757c54c5480274b2cdf06ddd955555": "\\sin A = \\frac{a}{c} \\text{; } \\sin B = \\frac{b}{c}", "287581670c3a4e8adf7c9990901ece45": "\\mathcal{E}_\\leftrightarrow", "2875c41487d51965c29a1ad097af15af": "\\scriptstyle D^2\\times S^1", "2875e17f4a5214f15921cb23c6b9e69a": "\\begin{align}\n\\text{E}(e^{-tx})=\n\\begin{cases}\\displaystyle\n\\beta^s \\frac{sb}{t+sb}{\\ } {_2\\text{F}_1}(s+1,(t/b)+s;(t/b)+s+1;1-\\beta), & \\beta \\ne 1; \\\\\n\\displaystyle\n\\frac{sb}{t+sb},& \\beta =1.\n\\end{cases}\n\\end{align}", "28760e9d47256f96866f1180fca8de57": "\\operatorname E[X^2_{ik}]=\\operatorname E[\\operatorname E[X^2_{ik}|\\vartheta]]=\\operatorname E[s^2(\\vartheta)+(m(\\vartheta))^2]=\\sigma^2+v^2+\\mu^2", "28761eacdc2d801a4e73edeb52d9eb2b": "vRPM=50/[0.5/SSD IOPS (Iwrite)] + [0.5/SSD IOPS (Iread)]", "287642cba9efe66a80d442a1eb9e6c29": "\n\\cfrac{\\cfrac{pq \\qquad p \\overline{q}}{p}\\, q \\qquad \\cfrac{qr \\qquad \\overline{p} \\overline{q}}{\\overline{p} r}\\, q}{r}\\, p\n", "2876539253e404436c3f51e5c24b4bc2": " \\sigma = 1-\\frac{1.98}{z(1-\\phi_2\\cot\\beta_2)} ", "2876c7f02f54b79183bd1396a64fdf07": "Q(I=7/2)=8\\left( \\frac{4(1+7P^2+7P^4+P^6)}{11+35P^2+17P^4+P^6} \\right)^{-1}", "2876ce6c6f72aa36a8fc5485a58d270c": "\\mathcal{F}(t)", "28772a35288c945522e5b411b3fbcdd9": "\\mathbf F_i=\\langle F_i,R_i,V_i\\rangle", "28772f6544a39018e7e4179e09e46f55": "\\rho=x", "28783a7f113b32800178fc5b84d34f6f": " d ( \\mathbf{ p }, \\mathbf{ q } ) = \\sum_{ i = 1 }^n \\frac{ |p_i - q_i |}{ | p_i| + |q_i | } ", "287891642756800c9447d46cad10015d": "\\varphi _j^n \\ ", "2878dece2fdc3e5a688901cc90a50de6": "\\mathbf{l}_a + (\\mathbf{l}_b - \\mathbf{l}_a)t, \\quad t\\in \\mathbb{R}", "28790af6471dd9c6d4ff451d660b51d5": "A = \\frac{1}{2}b \\times \\frac{\\sqrt{3}}{2}b = \\frac{\\sqrt{3}}{4}b^2", "28790c5ba309dfe563a9a6e3a4593620": "r=\\frac{1} {s}", "28793b4addb5b2c1014c5eea8b41434f": "P(X_N - X_0 \\leq -t) \\leq \\exp\\left ({-t^2 \\over 2 \\sum_{k=1}^N c_k^2} \\right). ", "28793c1c9f01c1428df2753ec7fbd5c1": " c_0 + c_1x +\\cdots+c_{n-1} x^{n-1} ", "287945a8879d85a7bc550633107adfec": "x^n = 1", "2879a1dbfba7a1d8e079430911020af8": "c_1,\\ldots,c_N", "2879afd2d0e6ca3e4c7183af7961ec9c": "\\sin(A + \\pi/3) : \\sin(B + \\pi/3) : \\sin(C + \\pi/3).", "2879bf34712c8bf592ec1b9098c14655": "\\{\\mathrm X_1, \\dots,\\mathrm X_4\\}", "2879c0e4b4982f1d10802d7bb6558b79": "(x_\\mathrm{u},\\ y_\\mathrm{u})", "2879cb2b74dc2006d295272465b1cae8": "M/P=L(i,Y)", "287a0a75b8ec132318b77f7b907b2dcb": "Z_2\\rightarrow 1", "287ac9701a263b0f7ff6a5400c610d90": " F(z) = \\sum_{n=0}^{+\\infty} c_n z^n, \\quad |z| < 1", "287ae932718e3fb63658825213c805e4": "X^\\omega", "287b2535fff4e23932fb7ff7b74af6e0": " \\widetilde X^{(n)}(t) = \\begin{cases} \\int_a^x \\widetilde X^{(n-1)}(t)y_0(t)^{2} w(t)\\,\\mathrm{d}t &n \\text{ odd}, \\\\ \n- \\int_a^x \\widetilde X^{(n-1)}(t) p(t)^{-1} y_0(t)^{-2} \\,\\mathrm{d}t & n \\text{ even}\\end{cases}", "287b81f1878fc90975a0f1c67e1e8225": "S(q) = 1 + \\rho \\int_V \\mathrm{d} \\mathbf{r} \\, \\mathrm{e}^{-i \\mathbf{q}\\mathbf{r}} g(r)", "287bf1714c331d2110554f7c2f83fb8e": "\\Gamma(1) = 1", "287c46af37ba6987b96a433dbfe0df3e": " L_1 \\cap L_2 = L_2 \\cap L_3 = L_3 \\cap L_1. ", "287c47b0bfc7d89fff7e9a36d23c73d9": "d_H", "287c95eedb857a122992b2c177c311d7": "\\sqrt[n]{x_1 \\cdot x_2 \\cdots x_n}.", "287ccb6e6bb4d2453a18e2fe8956879d": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) = \\left \\{ u : \\ \n|u - {\\tilde{u}} | \\le \\alpha \\tilde u \\right \\} , \\qquad \\alpha \\ge 0\n", "287cdbd2f3922c9d49db9202ed01e741": "W_\\lambda = \\int (1 - F_\\lambda / F_0) d\\lambda", "287d10d5e6a413c68afea0683409befe": "\\tan\\frac{7\\pi}{60}=\\tan 21^\\circ=\\tfrac{1}{4}\\left[2-(2+\\sqrt3)(3-\\sqrt5)\\right]\\left[2-\\sqrt{2(5+\\sqrt5)}\\right]\\,", "287d116e3e48f6a7ae2ae0494217ecd0": "B_{ji}", "287d178fcc2d4bc7b592d31a905d2fee": "\\int (u \\pm v) \\,dx = \\int u\\, dx \\pm \\int v\\, dx", "287d5b49c0427ceb28c77529d91eae84": "S\\subseteq R", "287dad0e675fe3db165ee6ad14037314": "F \\to G", "287e27875bd6e45f1ede90cb8edd4976": "\\frac{\\pi}{C\\, \\sqrt{2}}\\, \\frac{1}{\\sqrt{p}} \\sim \\frac{\\sqrt{\\pi}}{2}\\, \\frac{1}{\\sqrt{p}}", "287e3e2e88971dcd86f8850aeaeaf986": "\\mathbf{q}=(q_1,q_2,...,q_s),", "287e41fa5a3ed61b5ce2617cd7c3b335": "T_{\\mathrm {2e}}", "287e79f54ba68a25aff64db6ec948a19": "B = \\frac{NI\\mu}{L} \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (3) \\,", "287eb3cd1b2cfea33cbbdf1297783be9": "\\gamma(t) = \\begin{pmatrix} t\\\\t^2 \\end{pmatrix}", "287f560acfb8d0684a5c3b1f903986f2": "\\begin{Bmatrix} p, q , r \\end{Bmatrix}", "287fd91e561e08718940e001af5cbc58": "x^2-680x+96000", "2880ef97acc3f8e59e3374553a4b5231": "[L_{ij},P_k]=i\\hbar[\\delta_{ik}P_j-\\delta_{jk}P_i]", "2880f439a7504c5a5527a5683cc4f87a": " r=3.5,3.51,...,3.9 ", "288149dbad49e8a6122590d773c0d8fc": "1 + \\alpha(\\epsilon)", "288188bba1c9a32874e1bbafabb8bc2d": " g:X\\times \\left[ 0,1\\right] \\rightarrow \\mathbb{R}^{K}. ", "28818b01362a90b8e6d119e6f19e1c5b": " G_x, ", "28818e35f17db8587af52589136b6964": "R=k/n=1-r / (2^r-1)", "2881e21e9442022a9608e748e56d58b3": "E(N_B)/ |B| = \\lambda|B|/|B| = \\lambda", "28827087c40d708a235bd41e90d17d28": "s_\\lambda s_{\\mu/\\nu} = \\sum_{\\lambda+\\omega(T_{\\ge j})\\in P}s_{\\lambda+\\omega(T)}", "28829c28370832e24c294853abfd38fe": " dB_0/dt ", "2882a4d4e8e34bca0fd367e54412e896": "c_e \\in K", "2882aa6c39644f6f83701de027213cdd": "\\overline{\\Gamma(z)} = \\Gamma(\\overline{z}) \\; \\Rightarrow \\; \\Gamma(z)\\Gamma(\\overline{z}) \\in \\mathbf{R} .", "2882e7d04732ae5350ea1b0474c4c7ed": "1 - \\frac{c}{h}", "2882efc8086632f12f05fe87fb4c0b13": "H \\sim {p^2 \\over 2m} + V(r) + {e \\over m} A \\cdot p", "2882fbf1372dd8c0221dc92cc8e1f3b0": "\n\\gamma(v')\\gamma(v)-v\\delta(v')\\gamma(v)=-v'\\gamma(v')\\delta(v)+\\gamma(v')\\gamma(v)\\,\n", "288309381a9450b6cedad50d0d712178": "P = \\{ (a, b) \\in \\mathbb{Z}^2 : 1 \\leq a \\leq N; 1 \\leq b \\leq 2N^2 \\}", "288387f5a34cb9438abd9ad648c5fb7d": "r^2\\theta = k \\,", "2883aebe263b70b65aa5d5e3cd14c338": "\\delta(x-x')=\\sum_{n=0}^\\infty \\Psi_n^\\dagger(x) \\Psi_n(x').", "2883c2ff6d6fd372e7fc5aef08e64210": "V(p_0)=0", "2884257bc7637641cffdd6e2f3e7c4ca": " \\displaystyle \\int_{G/U} (Mf) \\cdot F = \\int_{G_0/K_0} f\\cdot (M^*F).", "288432b2f7a0392546e5b7621aae7101": "\\scriptstyle{2 -\\log(a)/\\log(b)}", "28845640c351ed877261edaff3e8335e": " z = \\rm{tanh}^{-1} r = \\frac12 \\ln \\frac{1+r}{1-r}", "2884733f7282d5d05a97c5b04cb27cc3": " { \\mathit l \\over \\mathit l^{\\prime} } = 1 ", "28848bda21344cbc03dbbb15eba37313": " \\textbf{f}_p = 1 + 2X + 2X^3 +2X^4 + X^5 +2X^7 + X^8+2X^9 \\pmod 3 ", "288493872e7c220bc24f478aece573de": "p_4(x) = -\\tfrac{3}{16}", "2885368c979dd4105be0cb074f1d8a32": "\\!\\{v_1, v_2, v_3\\}", "28854d5d90290a869c6b9322af6a62ee": "g \\prec h\\quad \\Longleftrightarrow \\quad eg \\,\\,\\omega^r\\,\\, eh\\,,\\,\\,\\, gf\\,\\, \\omega^l \\,\\,hf\\, ", "28858a6b49b224e96d8d6ddc645236da": "S_\\nu", "2885d29322b1439be5500ebfd072e43a": " Spin(11,\\mathbb C)", "2885df70387032b488fb87b0c9e53980": " \\sqrt[y]{x} = x^{1/y} ", "2885fbb8b21dad521828d020c70b5568": "x^{\\ast }", "2885ff17a17e8ff9429703cbb779f425": "\\,t_{Mn, O}", "28862e88d23e036188940923acde8e44": " T_m (0)=0 ", "2886b17d111307ca168c5354c3f55eb1": "\\|(L+I)u\\|_{(-1)} =\\sup_{\\|v\\|_{(1)}=1} |((L+I)u,v)|= \\sup_{\\|v\\|_{(1)}=1} |(u,v)_{(1)}|=\\|u\\|_{(1)}.", "28872c3df4882de21b1dae06aaa2c4a9": "A_2=\\{29.89,\\ 29.93,\\ 29.72,\\ 29.98,\\ 30.02,\\ 29.98\\}", "2887bb21b20a9521a53cde222f5459e4": "~a|\\alpha(t)\\rangle=\\alpha(t)|\\alpha(t)\\rangle", "2887ffea1da7d78c22a9353e4ee64476": " u_i< u^{max}_i ", "2888a2c04bd775a01962b60f35e56aed": "k \\in \\{0,1\\}\\,", "2888e0877e06d153ec298f8313af0f09": "M\\to X", "2888f0f19c726ee0df442d6172211523": "c_2 \\in C_2", "288932d97753a2bbfe4106e70d1d79c2": "\\mathfrak{P}^{12}", "288935b35f18b8bf5e36e6585234d12f": "\\scriptstyle P(B)\\,=\\;0 ", "28893eac20d725ae9788ab4aa0ff9878": " \\frac{y_n}{10} ", "28895045a0a6648c67bb6afc22b5ef1a": "\n\\begin{align}\nm & = d_8\\times 1000 + d_7\\times 100 + d_6 \\times 10 +d_5\\\\\ns & = d_4\\times 10 + d_3\\\\\nt & = d_2\\times 10 + d_1\n\\end{align}\n", "28899584a5706a4d79bedc452ac9f669": "E_k = \\frac{1}{2} mv^2", "2889a970cf281c7f877c6163d2769498": "F_0 = S_0 e^{(r+u)T}", "2889b225ad2f2d6ad5f4addebb42860f": "\n{dV \\over dt} + \\frac{1}{\\tau} V = f(t) = A u(t), \n", "288a0c0b6c1bf0221ada34b9e65462c2": "L_n^{(\\alpha)}(x) = \\frac{n^{\\frac{\\alpha}{2}-\\frac{1}{4}}}{\\sqrt{\\pi}} \\frac{e^{\\frac{x}{2}}}{x^{\\frac{\\alpha}{2}+\\frac{1}{4}}} \\cos\\left(2 \\sqrt{nx}- \\frac{\\pi}{2}\\left(\\alpha+\\frac{1}{2} \\right) \\right)+O\\left(n^{\\frac{\\alpha}{2}-\\frac{3}{4}}\\right),", "288a1bac616cc778e7774e995e5defe5": "\\partial_\\mu E_n", "288a5a3d491f7df4a2d1a09e02946b3a": "f(x)=q(x)g(x) + r(x)\\,,", "288ab004b00d40fabd293d74d63b9bdb": "u = G(n-1), v = G(n), w = G(n+1)", "288abde878eff3d9ac32367ece7eb372": "|\\psi\\rangle^{\\otimes m} \\rightarrow |\\phi_{m}\\rangle", "288ac28b4d84ca52c85920f93b5d4341": "a\\Box_\\eta B^{\\mu\\nu}+f\\eta^{\\mu\\nu}\\Box_\\eta B=-4\\pi G\\sqrt{g/\\eta}T^{\\alpha\\beta} (\\partial g_{\\alpha\\beta}/\\partial B_\\mu\\nu)", "288ad3973173982f646ce3813092aec9": "I_{k}", "288add767e5dceaae47719ae4c20abec": "M(x) = O(x^\\frac12)", "288af5d49137922510261b691ba33a01": "\\ \\displaystyle \\varphi \\ ", "288b488920b3ebfcaf7e401d25998968": "F \\subset {{\\mathbf{k}}}[x_1,\\cdots,x_n]", "288baff759925f414cd8444bdea0df42": "S_x(f) = \\frac{1}{(2\\pi)^2}h_2", "288c1cbea90bab7634879ad06ec9a621": "H^k_{\\mathrm{dR}}(M)\\cong \\check{H}^k(M,\\mathbf{R})", "288c6746eb0a69c5c342ea997526695b": "T_n(-1) = (-1)^n", "288c9943baf2cf5a88de4051d71a2fb6": "a,b \\in \\mathbf B", "288cbf128aa6dda28a698cf69a8f3789": " R_m ", "288cc7b5bab21bbb0f51908201b6537e": "\\gamma = 7/8", "288d1352e8e0d035a4609a52f974a804": "h_1 = c \\sin A = a \\sin C \\,", "288d1ec8fa987b23cd8baa59fd092999": "\\,\\! (I_z^+I_u-I_z^-I_u)=(I_x^+-I_x^-)(I_y)", "288d385f0412f4f1915c7f2da0a66214": "S = B + \\overline{A} + 1", "288dc66834f40e0b74998ffb82b590d8": "|\\det(Q)|=1", "288e276e74c939c6e44647639d043466": " p=\\frac{1}{2}\\rho U^2\\left(2\\frac{R^2}{r^2}\\cos(2\\theta)-\\frac{R^4}{r^4}\\right) + p_\\infty. ", "288f148e96efbfce47e28407377f7410": "\n \\int (d+e\\,x)^m (A\\,c\\,e (b\\,d-2 a\\,e) (m+2 p+2)+B (a\\,e (b\\,e-2 c\\,d\\,m+b\\,e\\,m)+b\\,d (b\\,e\\,p-c\\,d-2 c\\,d\\,p))+\n", "288f3f2eb542924007d7dc4a95606ff5": " \\frac {v_\\mathrm{D}} {v_\\mathrm{G}} \\approx -g_\\mathrm{m} (r_\\mathrm{O} \\| R_\\mathrm{L})", "288fd826b4373862f54dadaa0dab63c2": "\\rho_A = \\sup_X \\frac{\\mathrm{cost}_A(X)}{\\mathrm{cost}_{\\mathrm{opt}}(X)}.", "28900227fa5352643ad8ee0542ff6422": "(0.30-1)^2 = 0.49", "2890231ea2bda2f43eda5cc58413b5bb": "\\mathbf{\\overline{y}}", "28904f05ea9158d65500d170855d5d81": " \\begin{align}\n\\dot{\\mathbf{p}}_i &= \\partial \\mathcal{H} / \\partial \\mathbf{r}_i\\\\\n\\dot{\\mathbf{r}}_i &= -\\partial \\mathcal{H} / \\partial \\mathbf{p}_i.\n\\end{align}", "28909f4b0cdeb1a0256d756df5d25f47": "\\phi(\\mathbf{k})=\\int_{\\mathbf{r}{\\rm-space}} \\psi(\\mathbf{r}) \\phi_{\\mathbf{r}}(\\mathbf{k}) {\\rm d}^3\\mathbf{r}", "2890e29bcbc63e340c3dd5657b5a1b1f": "p\\gtrsim2", "2890fbb92c3cb3b5566db682c96c9d29": "z\\in D", "289129caa08adb66cca7985657882450": "\\Delta k = k_{SP}- k_{x, \\text{photon}}", "2891a9a034cb8b430fc455a2f5099461": " a_b ", "2891ab2d7f91bee86ba81b443227c10e": "P=\\frac{F}{(1+r)^t}", "2891d1edc17669b6f2745e9c25333591": "a = x", "28922f65b91e662f6587a8dba22d4980": "\\{ 1, 2, 3, \\ldots, 100 \\}", "28923709d8cd4ffd1fe87031ff8ce621": "\\limsup_{\\varepsilon \\to 0} \\varepsilon \\log \\mathbf{E} \\big[ \\exp \\big( \\gamma \\phi(Z_{\\varepsilon}) / \\varepsilon \\big) \\big] < + \\infty.", "28923a47de92f0b763a0945cf14ed1bc": "i = I_o (e^{\\frac{v}{V_T}}-1)", "28924ee779f9f22969064ef0c2b159e8": "\\frac{1}{\\tau_B}=\\frac{V}{D}(1-p)", "2892a2cc5c3d9951133ea87ec7ed0ecd": " \\mathrm{ACC} = 0 ", "2892d99ba2b88f86db34951dc3f090f5": "q:=[q_1,q_2,\\ldots,q_n]^T.", "2892fc978549ce2f1c0482c98ee988e2": "\\Phi(t,x) = \\Phi(t+p,x)\\,", "28931db859b77b575072e958cd4d9a33": "A = \\frac{n}{c} \\cdot \\frac{l}{2.303} \\cdot \\left ( \\frac{1}{\\tau} - \\frac{1}{\\tau_0} \\right) ", "2893529574157e1c24cd118842cdc1dc": "\\cos(\\alpha') = \\cos(180 \\text{º}-\\alpha) = -\\cos(\\alpha)", "289355a8315a1edeafc1d3659c35e64d": "\\vec{x}(t + \\Delta t) = 2\\vec{x}(t) - \\vec{x}(t - \\Delta t) + \\vec{a}(t) \\Delta t^2 + \\mathcal{O}(\\Delta t^4).\\,", "28936e1c94850b61a3a0922dde94cb3b": "ds^2\\simeq \\frac{1+\\cos^2\\theta}{2}\\,\\Big(-\\frac{r^2}{2M^2}\\,dt^2+\\frac{2M^2}{r^2}\\,dr^2+2M^2d\\theta^2 \\Big)+\\frac{4M^2\\sin^2\\theta}{1+\\cos^2\\theta}\\,\\Big(d\\phi +\\frac{rdt}{2M^2}\\Big)^2\\,.", "2893707f0fc096f5ec70459b081bc828": "T \\mapsto \n\\begin{bmatrix}\n0 & \\; & & & & \\; & & T_z \\\\\n\\frac{1}{2}I & \\ddots & & & & & & 0 \\\\\n\\; & \\ddots & \\ddots & & & & & \\vdots \\\\\n\\; & \\; & \\frac{1}{2}I & 0 & & \\; & & \\\\\n & \\; & & I & 0 & & & \\\\\n & & & \\; &\\frac{1}{2}I& \\ddots & & \\; \\\\\n\\; & & & &\\; & \\ddots & \\ddots & \\vdots \\\\\n\\; & \\; & & &\\; & \\; & \\frac{1}{2}I & 0 \n\\end{bmatrix} \n.\n", "2893cf7896cde9c555a127223d44807a": "k\\in I", "2893d46a7114fd1985da48c3e16ee02f": "(v_i,v_j) = w", "28941eb500933a0f8740c945a9cd7a19": " g=\\sum_{i,j}g_{ij}\\mathrm d x^i\\otimes \\mathrm d x^j.", "2894791f6a85352ac2f9728370b034f0": "\\sum_{n=1}^\\infty\\frac{1}{n^s} = \\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}}", "2894dece574ee0372b5997b362a36c78": " \\int \\mathcal{D}\\phi F[\\phi(x)]F[\\phi(\\bar{x})]^* e^{-S[\\phi]}=\\int \\mathcal{D}\\phi_0 \\int_{\\phi_+(\\tau=0)=\\phi_0} \\mathcal{D}\\phi_+ F[\\phi_+]e^{-S_+[\\phi_+]}\\int_{\\phi_-(\\tau=0)=\\phi_0} \\mathcal{D}\\phi_- F[\\bar{\\phi}_-]^* e^{-S_-[\\phi_-]}. ", "289543d6534de2ea31f3b327023df566": "V \\not \\in \\operatorname{vars}[\\operatorname{let} F \\operatorname{in} L] ", "2895823d307a780249c8a7f0934d6ed7": "a^{(r)}, \\cdots, z^{(r)}", "28959f1365688edc4b8b61df75218dcb": "~A \\leftrightarrow B~~\\Leftrightarrow~~\\neg(A \\oplus B)", "2895cb088f8d9fe6f4c4b74ca091be2e": "r=\\frac{y^2}{8x}+ \\frac{x}{2}.", "2895e34b66f8d3191502b1b3507fd4a5": "P(x,y) = P_1(x)P_2(y)", "2895e60475a5936279ed1d338930f45c": "1 + \\tfrac{1}{3} + \\dots + \\tfrac{1}{p-2} \\equiv 0 \\pmod{p^2}", "28964e072d8e798ada23b4a47fad3faf": "\\omega_G=g^{-1}dg,", "2896772b220e0c487ed5482c0fc26edb": "I \\theta_f=h\\int_0^{t_f} m_b V_b\\,dt = h m_b L", "289716bdbdb65cf8a03455cb846566b4": "L = \\frac{1}{5} l \\left[\\ln\\left(\\frac{4l}{d}\\right) - 1\\right]", "2897285c7182e0d774632cecf61efd9d": "\\operatorname{\\Gamma L}(V)", "289798b13023f95b9b11c18937ef8dd7": "\\langle X_i Y_i X_j Y_j \\rangle = \\langle X_i X_j \\rangle \\langle Y_i Y_j \\rangle + \\langle X_i Y_i \\rangle \\langle X_i Y_j \\rangle + \\langle X_i Y_j \\rangle \\langle X_j Y_i \\rangle", "2897a722c63b90240557ecf4c4b8b0f5": "\\alpha = (\\beta+1)/2", "28986330e1e8af69c78b3528fa7adb2b": "B_n(q)", "2898cafb6713b2142d6131a331e8af59": "f\\colon\\mathbb{R}^n \\rightarrow \\mathbb{R}", "2898e489f8c6fa73df7537192087b5d0": " i \\in [n] ", "2898f26b70f3c7ccfef65fc39863bac2": " E_a = \\sqrt{(m_k c^2)^2 + (m_t c^2)^2 + 2 E_k E_t} ", "28992df1a62b8e873d95ccc061595055": "X=\\left[\\begin{matrix} A & B \\\\ B^T & C \\end{matrix}\\right].", "28993b2c21f66477742e3e01c3e4c2cb": "\\mathbf{V}_0", "28994f8239cbb760853418fa46b7a608": "\\mathbf{\\tilde{H}}", "2899710f6c0273355128cec8f8a83482": "\nN^{i}_{j}=\\frac{1}{2}\\gamma^{\\alpha \\beta}(g^{hi} g_{hj /\\alpha})/ \\beta\n", "2899ba385b979d87039058da2ed5c181": "E_{photon} - E_{ionization}", "2899cc8afc5d2f39af9a604ae14a1267": "\nm \\ddot x_2 = - k x_2 + k (x_1 - x_2) = - 2 k x_2 + k x_1 \\,\\!\n", "2899d0657e82afa924bf15823c143373": "\\sum_x f(x)\\Delta g(x)+\\sum_x g(x)\\Delta f(x)=f(x)g(x)-\\sum_x \\Delta f(x)\\Delta g(x) \\,", "289a2c3e20d92575e3ce661b52547e7e": "e = \\sum_{k=1}^\\infty \\frac{k^4}{15(k!)}", "289a88a6c249b2c7a807878f336fe043": "(-\\varphi)^{-n}", "289a8a7cbb8032e06e6b0f565b4d6b76": "i = p \\mod 2", "289a9040746d7efe1bc5a1775d402e73": "\\,y_i", "289a9448e93a15e4ba326fab1dba89a5": "P_{max} =P_m +H", "289a99765e815a3359e283783f2149ae": "\\mathbf{M}_{7} := (\\mathbf{A}_{1,2} - \\mathbf{A}_{2,2}) (\\mathbf{B}_{2,1} + \\mathbf{B}_{2,2})", "289b48d1c964bba9c630c39cec36497a": "n=\\sum_i n_i", "289b8bfaef332a03c02eae86342b974e": "\\cos \\phi", "289b8ffc7fc5fe0f17c2e2d9f2b637ae": "f(v,i) = 0 \\,", "289b941dd74244f3ee86cf293d1ec7c3": "a^2 - Nb^2 = k", "289ba239b0da5007c2461e5d005e4019": "x_{n+1} = x_n - \\frac {2 f(x_n) f'(x_n)} {2 {[f'(x_n)]}^2 - f(x_n) f''(x_n)} ", "289ba6de6d0826d36b2f24b4214d298e": " \\rho (\\beta \\nabla S - \\alpha \\nabla \\theta) ", "289bfa1a7e0fd299f9c8cc636b63d2d4": "dA = a(t_e)r\\ d\\theta a(t_e) r \\sin(\\theta) d \\phi = a_e^2 r^2 d\\Omega = \\frac{a_0^2 r^2 d\\Omega}{(1+z)^2} ", "289c0ba43efecd7bfd0d897b56d7085b": "\\leq |V|", "289c2493bc1206737ca146b3934e5d44": " \\log_2 10=1/\\log_{10} 2 \\approx 3.321928095", "289c4ffd657f270802df314dd9c33fbf": "\\alpha_G=\\frac{Gm_e^2}{\\hbar c}", "289c6a7640313b0b7a44d5cfa6530b27": "\n \\mathcal H = -\\frac{\\hbar^2}{2m}\\nabla^2+ V(\\mathbf{r},t) \n", "289ce2a0908ea96c0e51d5c7bcdba7bf": "\\nabla \\times \\nabla \\phi = 0", "289cea743b4a53bc952c141919466642": "\n\\langle \\mathbf{S} \\mathbf{v} \\mid \\mathbf{v} \\rangle =\n\\sum_{k} |\\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle|^{2}\n", "289d158b6e01895b3e171f89714cf8e6": "(x,0) < V:", "289d35466c92174cff9e1fb4f7236bd9": "\\lim_{t\\to\\infty}\\frac{L(tx)}{L(t)}=1", "289d40f9cc876c96ebb84b57141fad65": "A\\subseteq G(A,B)\\subseteq G(A,B)^{-}\\subseteq X\\backslash B", "289d776ca8fcbaca0a532bc9bddfc8b4": "\\frac{\\partial y}{\\partial \\mathbf{x}} = \\nabla_\\mathbf{x} y", "289da00c6e02b6f73599a0cbda720de8": "r_{nk} = \\frac{\\rho_{nk}}{\\sum_{j=1}^K \\rho_{nj}}", "289da3b2848c7b54f09dd9c14aa0d7c2": "(e_R)^c", "289daab18c3e7b53a043814c4ca413ce": " A = \\frac{R T}{p} \\frac{n^2 - 1}{n^2 + 2} ", "289df0c489d4d8ace0c87723e4cf278d": "[-,-]", "289dfb0cb6af9192d6fce1a73bd37a4f": "m_0 =3", "289e4961dd78b14f96f369df3fc3e42d": " \\int_0^\\infty x\\Phi(a+bx)\\phi(x) \\, dx =\\tfrac{b}{t}\\phi(\\tfrac{a}{t})\\Phi(-\\tfrac{ab}{t}) + (2\\pi)^{-1/2}\\Phi(a), \\qquad t = \\sqrt{1+b^2} ", "289e8a6fb8fb4895482b85055904aab7": " \\exp \\left[ \\log \\left( \\frac{ t_a } { t_b } \\right) - z_\\alpha \\left( var \\left[ \\log \\left( \\frac{ t_a } { t_b } \\right) \\right] \\right)^{ 0.5 } \\right] ", "289ea787231ca246b0d7f8d131bbb249": "\n \\boldsymbol{\\sigma} = \\cfrac{2}{J}\\left[\\cfrac{1}{J^{2/3}}\\left(\\cfrac{\\partial{W}}{\\partial \\bar{I}_1} + \\bar{I}_1~\\cfrac{\\partial{W}}{\\partial \\bar{I}_2}\\right)\\boldsymbol{B} -\n \\cfrac{1}{J^{4/3}}~\\cfrac{\\partial{W}}{\\partial \\bar{I}_2}~\\boldsymbol{B} \\cdot\\boldsymbol{B} \\right] + \\left[\\cfrac{\\partial{W}}{\\partial J} -\n\\cfrac{2}{3J}\\left(\\bar{I}_1~\\cfrac{\\partial{W}}{\\partial \\bar{I}_1} + 2~\\bar{I}_2~\\cfrac{\\partial{W}}{\\partial \\bar{I}_2}\\right)\\right]~\\boldsymbol{\\mathit{1}}\n ", "289ece2095668fab620ff14b2fa6799f": "F(x) = \\int_a^x f(t) \\,dt.", "289ee6fd84327a367e221436229240a0": "U + W = \\left\\{ \\mathbf{u} + \\mathbf{w} \\colon \\mathbf{u}\\in U, \\mathbf{w}\\in W \\right\\}.", "289f401fb375087b4746ed0ca23abfd0": "\\max U=U(S, \\Pi - \\Pi_0 - T) \\,", "289f6fe586c93d42b90a4104a98a377e": "17=2^4+2^0=(-2)^4+(-2)^0", "289f74d6700ce41a84671c44654203b6": " r-s=a \\,", "289f90d6e3bbd92b6c4fe5768f92c01f": " \n\\sum_{n=1}^{\\infty}(-1)^{n-1}\n\\frac{\\sin a n \\cdot\\ln{n}}{n} \\,=\\,\\pi\\ln\\left\\{\\frac{\\pi^{\\frac{1}{2}-\\frac{a}{2\\pi}}}{\\Gamma\\left(\\displaystyle\\frac{1}{2}+\\frac{a}{2\\pi}\\right)}\\right\\} - \\frac{a}{2}\\big(\\gamma+\\ln2 \\big) -\\frac{\\pi}{2}\\ln\\cos\\frac{a}{2}\\,,\n\\qquad -\\pi 0) \\\\\n&= \\Pr(\\boldsymbol\\beta \\cdot \\mathbf{s_n} - e_n > 0) \\\\\n&= \\Pr(-e_n > -\\boldsymbol\\beta \\cdot \\mathbf{s_n}) \\\\\n&= \\Pr(e_n \\le \\boldsymbol\\beta \\cdot \\mathbf{s_n}) \\\\\n&= F_e(\\boldsymbol\\beta \\cdot \\mathbf{s_n})\n\\end{align}\n", "28a0aea70335430ff7c1d59952eb06f3": " \\frac{\\partial \\varepsilon}{\\partial P} = \\frac{1}{L} \\frac{\\partial u}{\\partial P} = \\frac{1}{EA} > 0 ", "28a108ddeb980b5839a72626307cd6b1": "\\displaystyle{Hf_\\pm = \\pm i f_\\pm.}", "28a12aaebc464bf7f355e7d9f1d432e1": "\\frac{A(A - 1)}{2}", "28a2240d45c97ef3c8d305fd39d9611f": "V_\\alpha := \\bigcup_{\\beta < \\alpha} \\mathcal{P} (V_\\beta)", "28a22b14eb433428ede245820690485d": "2^{1-s}\\,\\Gamma(s+1)\\,\\eta(s) = 2 \\int_0^\\infty \\frac{x^{2s+1}}{\\cosh^2(x^2)} \\, dx\n= \\int_0^\\infty \\frac{t^s}{\\cosh^2(t)} \\, dt.\n", "28a244e16b11086aec5b5f5b02cce52e": "y_t = a + w_0x_t + w_1x_{t-1} + w_2x_{t-2} + ... + \\text{error term}", "28a25713723148648277ae60cd2ae70d": "\\mathrm{ROOH + ROO{^{\\cdot}} \\ \\longrightarrow {} \\ ROOH + Q{^{\\cdot}}OOH \\ \\longrightarrow {} \\ ROOH + QO + ^{\\cdot}OH }", "28a28f32687c353c47c1a1fa98ecb731": "f(t)=\\sum_{n=1}^\\infty {p_n\\,'(0) \\over n!}t^n.", "28a2cc66a7b1dd190d8739118ec2456f": "\\Delta S \\ge 0", "28a36c36a7b6cf70b3c27f4c6498eedd": "\\frac{T_{2}}{T_{1}}=\\frac{1+\\frac{\\gamma -1}{2} M_{1}^{2}}{{1+\\frac{\\gamma -1}{2} M_{2}^{2}}} = \\frac{(1+\\frac{\\gamma -1}{2} M_{1}^{2})(\\frac{2\\gamma}{\\gamma - 1}M_{1}^{2}-1)}{\\frac{(\\gamma+1)^2M_{1}^2}{2(\\gamma-1)}} ", "28a400d8a32f368b2370ed8dbe3e5742": " f(R,t) = \\frac {4}{9} \\left(\\frac {3}{3+\\rho}\\right)^\\frac {7}{3} \\left(\\frac {1.5} {1.5 - \\rho}\\right)^\\frac {11}{3} \\exp \\left(- \\frac {1.5}{1.5 - \\rho}\\right) \\rho < 1.5 ", "28a416b0591ee084dd6ad42d9ae281db": "k>m>0", "28a4358e9cc4d30d1d7d176d03fc0485": "uv^ixy^iz", "28a44724238fb2d5e107a9ca13327dac": "p^{2}", "28a48dd058f5b93a6fd51ac867a963b9": "\\displaystyle{Q(a)R(b,a)=R(a,b)Q(a) = 2Q(Q(a)b,a),}", "28a4acef15cb47fcb71b6c4b5f6a087b": "\\bigcup \\mathcal{S}", "28a511bd345ff77ee207e6267d0f60a4": "\\sigma(E_1, E_2, \\ldots)", "28a558672bf11580d700153b4dbdcbe3": "\\sum_{k=- \\infty}^{k=+ \\infty} {h_k =-1}", "28a58ce99c998ec80377b0084ae39bb6": "|x|_{\\ast}=|x|_p^c", "28a5909a166165424cf1deb5523ce4ab": "\\begin{matrix} T(\\phi) = T_{hold}, & \\phi\\in[\\phi_{hold},\\phi_{intf}] \\\\\n T(\\phi) = T_{load} \\exp(-\\mu\\phi), & \\phi\\in[\\phi_{intf},\\phi_{load}] \\\\\n \\phi_{intf} = \\log(T_{load}/T_{hold}) / \\mu &\n \\end{matrix}", "28a5cb229fc2b2d80847cf580a5334a5": "Y_{lm}(\\theta,\\phi)", "28a5d263ab37c04aff26e10cef131159": "\\pi_4 = a^3 I", "28a63a49a1531b2004bcb609132b6fd6": "D(n) = \\bigl(1+o(1)\\bigr)\\exp\\left(\\frac{\\pi\\sqrt{8\\log n}}{\\sqrt{3}\\log\\log n}\\right).", "28a6c9c7fafbb8cd16e2053a1b71e1ac": "\n\\begin{smallmatrix} 3 & 1 \\end{smallmatrix} \\quad \n\\begin{smallmatrix} 3 \\\\ 1 \\end{smallmatrix} \\quad\n\\begin{smallmatrix} 2 & 1 & 1\\end{smallmatrix} \\quad\n\\begin{smallmatrix} 2 \\\\ 1 \\\\ 1 \\end{smallmatrix} \\quad \n\\begin{smallmatrix} 1 & 1 & 1 \\\\ 1 \\end{smallmatrix} \\quad\n\\begin{smallmatrix} 1 & 1 \\\\ 1 \\\\ 1 \\end{smallmatrix}\n", "28a6eed312909b540d6dc9948fbf8043": " \\Delta/\\lambda\\to \\mathbf{J}^T \\mathbf{r}", "28a7192dfd2f4221c2280bbce03b823a": "\n \\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} = \\cfrac{C^{\\mu\\gamma}}{2}\\left(\\frac{\\partial C_{\\alpha\\gamma}}{\\partial X^\\beta} + \\frac{\\partial C_{\\beta\\gamma}}{\\partial X^\\alpha} - \\frac{\\partial C_{\\alpha\\beta}}{\\partial X^\\gamma}\\right) \n", "28a77ce91cd222adee7b246d7e6961f7": " \\omega _0 = \\sqrt {\\frac {k_c}{m+m_A}}", "28a7caed5ad631f4829c68ff128d4d67": " ds^2 = d\\bold{x}\\cdot d\\bold{x} = g(d\\bold{x},d\\bold{x}) ", "28a80487f015493e17f128a4d8315303": "U_3=\\begin{bmatrix}\n\\lambda_1 & z_{1\\,2} & z_{1\\,3} & \\cdots & z_{1\\,n} \\\\\n0 & \\lambda_2 & z_{2\\,3} & \\cdots & z_{2\\,n} \\\\\n0 & 0 & \\lambda_3 & \\cdots & z_{3\\,n} \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n0 & 0 & 0 & \\cdots & \\lambda_n\n\\end{bmatrix}", "28a828b6c2a018913febb2f8608c0833": "\\text{period of }\\tfrac{1}{p^{m}} \\ne \\text{ period of } \\tfrac {1}{p^{m+1}}", "28a85f41b7e440fc40265f6665a032c8": " [w]^{-1} \\begin{bmatrix}s \\\\ t \\end{bmatrix} [w] = \\begin{bmatrix}s \\\\ - w s + t \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ -w & 1 \\end{bmatrix} \\begin{bmatrix} s \\\\ t \\end{bmatrix} ", "28a85f612fe1ceb158a78659e547adff": "(W(tf))_{t\\in\\mathbb{R}}", "28a875195e159317e4607813969ffa7d": "\\left\\{6,{3\\atop3}\\right\\}", "28a89c9700a0b80c5ca1d1c909c27f90": "\\left(-r, 0, \\pm\\sqrt{R^2 - (r+a)^2}\\right);\\quad\n \\left(0, \\pm r, \\pm\\sqrt{R^2 - (r-a)(r+a)}\\right);\\quad\n \\left(+r, 0, \\pm\\sqrt{R^2 - (r-a)^2}\\right).", "28a8bd68171f25888dd7398d05b47e89": "X^\\sharp = g^{jk}X_{ij} \\, dx^i \\otimes \\partial _k. ", "28aa42bc74779623be657b1a2a98d6af": "\\theta_{r_i}(x)\\Big|_{x=-j\\infty}^{x=j\\infty} = -\\pi\\,", "28aa893b2b548c4e6878c6254adb21ef": "\\mathsf{(CH_2CH_2)O+HNO_3}\\rightarrow\\mathsf{HO\\!\\!-\\!\\!CH_2CH_2\\!\\!-\\!\\!ONO_2\\ \\xrightarrow[-H_2O]{+\\ HNO_3}\\ O_2NO\\!\\!-\\!\\!CH_2CH_2\\!\\!-\\!\\!ONO_2}", "28ab1f0d75625a70a7a1ce8dbcc3441e": " \\Gamma \\vdash A,\\Delta", "28ab282fa60e420606e527dd145df0ef": "p_{mk}", "28ab58046e070814096863faf21afc62": "(\\nabla \\cdot \\mathbf{T})_j = \\nabla_i T_{ij} ", "28ab691ed97e590459baed5de7ecf76a": "-\\frac{2}{3}\\!\\,", "28abc630b7646482bb321849e1f9c34b": "\\scriptstyle \\beta l = \\pi ", "28ac4b7cd5d2866dadb8739de7672d8b": "\n I(\\lambda) \\equiv \\int\\limits_{I_x} f(x) e^{\\lambda S(x)} dx \n = \\left( \\frac{2\\pi}{\\lambda}\\right)^{n/2} e^{\\lambda S(x^0)} \\prod_{j=1}^n (-\\mu_j)^{-1/2} \\left[f(x^0) \n + O\\left(\\lambda^{-1}\\right) \\right], \\qquad \\lambda \\to + \\infty,\n", "28ac5dc56ae513b58701fa54760224b3": "(p \\times 1)", "28ac7f7c3a6e9dc7b4d23ae4157419b1": " \\Delta Q(t) ", "28acfa532fdc51ed7ea9db4f107298db": " (\\mathbf{\\lambda} x . z) ((\\lambda w. w w w) (\\lambda w. w w w)) ", "28ad4f627695b5a7c563a43ae9bdf1bc": "\\Phi_x(k)", "28ad7ffa2d3ade85771e5200774d94be": "\\theta_B", "28ad90166337a5ef4cc3541db97cd01f": "p,q>0", "28ad9f6878f4499d8a155ffafd85376a": "\\cfrac{\\partial \\mathbf{b}_i}{\\partial q^j} = \\Gamma_{ij}^k~\\mathbf{b}_k\n ", "28adef701946f6edd7a2823653e18098": "\\int |D_N(t)|\\,dt > c\\log N+O(1)", "28ae049b4b3acd7eb817e3b59f718fb6": "\\sum_k |V_{ik}|^2 = \\sum_i |V_{ik}|^2 = 1", "28ae1a5231e18296b559d2e3efe4e5d7": "(\\theta_2=0)", "28ae691713048c742a88452c090aa904": "\\phi(q) = \\sum_{n\\ge 0} {(-1)^nq^{n^2}(q;q^2)_n\\over (-q;q)_{2n}}", "28aeb5f9d1709c5f74f2e2589681d7b6": "v \\in \\mathbb{R}^J", "28aed79e47133025e96e21ea75724f76": "A=-27q", "28af2c24e43ad3464a98d876d1fd9440": "e(g^x PK, p_{SK}(x))=e(g,g)", "28af5813a0714e012dfabc885f6a107c": "\\frac{\\operatorname{E}[|V^S - V^B|]}{V}", "28af5913d08b318946eff48978e34d74": "\\mu \\alpha.[\\alpha]u \\; \\triangleright_c \\; u", "28af99ca5113abb8ccd5b9fefc2e78c3": "J_\\mathrm{drift}(x) = \\mu \\, F(x) \\, \\rho(x) = -\\rho(x) \\mu \\frac{dU}{dx}", "28b0036ec2336ddd2250f488b1a7dac3": "\\begin{align}\n \\boldsymbol{\\nabla}\\cdot\\mathbf{v} &= \\cfrac{\\partial v_i}{\\partial x_i} \\\\\n \\boldsymbol{\\nabla}\\cdot\\boldsymbol{S} &= \\cfrac{\\partial S_{ki}}{\\partial x_i}~\\mathbf{e}_k\n \\end{align}", "28b045ff2150137538d36419232ec822": " \\int_0^\\infty x^{s-1} f(x) \\, dx = \\Gamma(s)\\phi(-s) \\!", "28b0b02364dda204ea262886e24c1310": "\\operatorname{succ}\\ n\\ f\\ x = f\\ (n\\ f\\ x) ", "28b11043a4d6a27cf18b864d246e377c": "R_{ab} = \\frac{2\\Lambda}{n-2}\\,g_{ab}.", "28b12f3cad9f805ee1b9ffb9c64aa0c4": "C_3\\,\\!", "28b13d8e7cab68f952838e9da42d0f6f": "H(\\epsilon)=-\\epsilon\\cdot \\log \\epsilon - (1-\\epsilon)\\cdot \\log (1-\\epsilon)", "28b16e43e42d74406e65d1b0865a92f6": "\\lceil \\chi_c(G) \\rceil = \\chi(G)", "28b2142b164673a57523c1ec04ac70f2": "1-2x+3x^2-4x^3+\\cdots = \\frac{1}{(1+x)^2}.", "28b271012f2a23f67747425e6761498f": " R' = \\langle R \\rangle (1 + 1 / e_{\\langle R \\rangle})", "28b2b45ca95d89009acc6da6a84fdc5a": "S_3 = {v}(\\alpha^3)", "28b2f4552d914e2d199eed87b15d41f8": "C_k = \\left\\{\n\\begin{array}{lr}\n\\mathbb Z^2 & 0 \\leq k \\leq n \\\\\n0 & \\text{otherwise}\n\\end{array}\n\\right.", "28b3644f18f7b4b8f4a585a5db6132e8": "A \\times P", "28b3822cba12d64ec1e8b28f05769110": "X^*_{1,2}=X_3/(1-X_1-X_2),X_4/(1-X_1-X_2),\\ldots,X_k/(1-X_1-X_2).", "28b3e72ae6a190dc1972734e9012d660": "\\Delta f = 0\\,", "28b42858df9b90cc3c3fe31d01a2f1e5": "a_i \\equiv a_j \\pmod{\\gcd(n_i,n_j)} \\qquad \\text{for all }i\\text{ and }j", "28b42f8d23471cfee2622d8dcaa21457": "\n\\mathbf{P} = \\bigoplus_i{D_i} = \\mathbf{D}_0 \\;\\oplus\\; \\mathbf{D}_1 \\;\\oplus\\; \\mathbf{D}_2 \\;\\oplus\\; ... \\;\\oplus\\; \\mathbf{D}_{n-1}", "28b4392ea6c6cfb147dcf03c5287aac3": "\\gamma_3 \\approx 0.9952717", "28b4945770d6eb2452ccb987631e5dfa": "\\int_0^\\pi f(\\cos \\theta) \\sin(\\theta)\\, d\\theta = a_0 + \\sum_{k=1}^\\infty \\frac{2 a_{2k}}{1 - (2k)^2} .", "28b4fb80b2cb7ac657433a4533a8f791": "T_{k,n} = T_{k,n-1} + T_{k-1,k-n} \\quad \\text{for } k \\ge n > 0. ", "28b53c55e4b4a8aa0e3185dff146c629": "B = \\frac{1}{\\omega_\\mathrm{d}}(\\zeta\\omega_0x(0)+\\dot{x}(0)).\\,", "28b5430d9bc55ec25af7a9a7adbc9422": "\\scriptstyle \\Rightarrow ", "28b59d0bacbcc089cf5c618301ca1a06": "\n{S}=\\left[\\begin{matrix}\n-\\beta_1&\\beta_1&0&0&0&0\\\\\n0&-\\beta_1&\\beta_1&0&0&0\\\\\n0&0&-\\beta_1&0&0&0\\\\\n0&0&0&-\\beta_2&\\beta_2&0\\\\\n0&0&0&0&-\\beta_2&\\beta_2\\\\\n0&0&0&0&0&-\\beta_2\\\\\n\\end{matrix}\\right].\n", "28b5a997f757c9cdd323087ae2d4f789": "Ar/3", "28b61b964a071d9810f7c155153bb395": " \\left[\\frac{ \\mathbf{E}\\left[e^{tX_i} \\right] }{e^{tq}}\\right]^m = \\left[\\frac{p e^t + (1-p)}{e^{tq} }\\right]^m = [pe^{(1-q)t} + (1-p)e^{-qt}]^m.", "28b620c932cd787f1c778280868054d3": "(T_{x} f)(y) := f(y - x),", "28b62e9202264ceed8bdf728986de1fb": "k_{cat} \\approx k_2", "28b68db57fef06260fe0897e8b97a8b4": "\n\\begin{align}\n& \\lim_{x\\to 0} \\frac{\\sin x}{x} =1, \\\\[10pt]\n& \\lim_{x\\to 0} \\frac{1 - \\cos x}{x} = 0.\n\\end{align}\n", "28b690ae93159f62319293fad317a164": "y=(t+1)\\inf \\mathrm{supp}X-\\mu(X),", "28b6a1dab720d2d83ee6ddda3a3d27c6": "\\ (u,v) \\in E", "28b6b6d603667291e4e4081bb3bd689e": "M _{CB} ^f = \\frac{qL^2}{12} = \\frac{1 \\times 10^2}{12} = + 8.333 \\ kN\\cdot m", "28b6d5807e83fc7affde05e0b2cb1c06": "s_p = \\sqrt{ \\frac {p \\, (1-p) } {n} }", "28b7539dc5966c0342a7b1a63244f5a1": " R[\\varphi] = \\iiint_D r(X) \\varphi(X)^2 \\, dx \\, dy \\, dz.\\,", "28b7707502b10fc2a6df36cf73d2f0a4": "\\textstyle \\mathbf{r}", "28b787f1bbf117ee55e45b572084b840": "\nU = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} Q_{1lm}\n\\int d\\mathbf{r} \\ \n\\rho_{2}(\\mathbf{r})\n\\left( \\frac{1}{r^{l+1}} \\right) \n\\sqrt{\\frac{4\\pi}{2l+1}} Y_{lm}(\\theta, \\phi)\n", "28b7b29fd6b4fd86838fb7150f2509e9": "\n \\omega^2 = |k| \\left( \\frac{\\rho-\\rho'}{\\rho+\\rho'}\\, g + \\frac{\\sigma}{\\rho+\\rho'}\\, k^2 \\right),\n", "28b7c2f8fe999b0996c0bfe0201ad649": "1 +\\sum_{k=1}^{\\infty} \\left( \\prod_{r=0}^{k-1} \\frac{\\alpha+r}{\\alpha+\\beta+r} \\right) \\frac{t^k}{k!}", "28b80a684dc23eb94544adb278ceae87": "\\phi(\\alpha, \\beta) = \\alpha - \\frac{(2 \\alpha + \\beta - 3)^2}{3(\\alpha + \\beta - 2)}", "28b879780ab0472f166462988d485a6e": "\\frac{P(t)}{P_0}=\\frac{1}{2}=\\frac{1-e^{-r(T-t)}}{1-e^{-rT}}", "28b87b9ac8e02a5ae7946b03463eeb0f": "e^{it} = \\cos t + i \\sin t,", "28b895637058979e494eeb1b21607c5a": " \\mathrm{succ} : \\mathbb{N} \\to \\mathbb{N}", "28b8a26d7f0397d613226b9b5f91bf26": "X_{(m+1)}", "28b8c557df53bc4583a5fd427e843509": "E=E^0 + \\frac{RT}{z_iF} \\ln \\left [ a_i + \\sum_{j} \\left ( k_{ij}a_j^{z_i/z_j} \\right ) \\right ]", "28b90931b04d7d48fa55a43c85b51ffb": " m \\sqrt{\\langle v^2 \\rangle} \\,\\!", "28b90e5875b7e6ead67900601a63ddb3": "\nm \\overline{\\Psi^\\dagger} \\rightarrow \n m \\overline{(\\Psi R_0)^\\dagger} = m \\overline{ \\Psi^\\dagger }R_0,\n", "28b93b4e37f39ca8eb5420f88cb09029": "\\ y=(d/2) \\sqrt{(\\xi^2+1)(1-\\eta^2)} \\cos \\phi, ", "28b9a066c594489b73e63c644652a468": "=Q\\left(\\frac{\\nu}{2},\\frac{\\tau^2\\nu}{2x}\\right)", "28b9c3643584bbd557ee327db6ca45f6": "M_\\mathrm{left}", "28ba4b1c47dc94db458fddb4a406ed19": "\\Gamma_{\\mu\\nu\\alpha}=\\{_{\\mu\\nu\\alpha}\\} +S_{\\mu\\nu\\alpha} +\\frac12 C_{\\mu\\nu\\alpha}", "28baf1b07c1c21b18f00d07764929aca": "x = L\\cot(\\theta)", "28baf2bf0eda6cd1e2df7d17b7eb457b": "(\\omega^2+m^2)^{-1}", "28bb59a115934bed928b08a7e655a150": "X \\times_S T \\to T", "28bb8c90a2ff86f10508973b70b84f50": "\\frac{dN}{d\\theta}={\\frac{1}{2 \\pi \\sigma^2}} e^{ -\\frac{\\theta^2}{2 \\theta_0^2}}", "28bbb18ab6a0ab92a1db71071fac6bdd": "M_{PAW} = \\frac{(P_{high} * T_{high})\\, + (P_{low} * T_{low})} {T_{high} + T_{low}}", "28bbf08592798bf0f19d3d0ae99c9144": "e_x = E[K(x)] = \\sum_{k=0}^\\infty k\\, Pr(K(x)=k) = \\sum_{k=0}^{\\infty}k\\, \\,_kp_x \\,\\, q_{x+k}.", "28bc61b908d05c94c2429f4c1a908ef0": "d=2mn(m^2-n^2), \\,", "28bd34e95174fe595cb535b5d4484c5f": " \\Psi(x,y) = x\\cdot \\rho(u) + O\\left(\\frac{x}{\\log y}\\right)", "28bdc367b47d1e6c434f115aca7eb86e": "\\lim_{n\\to\\infty} \\int_S f_n\\,d\\mu = \\int_S f\\,d\\mu", "28bdcf12dce597687c5b9211402b3b27": "\\nabla \\mathbf{v}", "28be0b2ff0ab1254fe24437ea657174e": "P 0", "28c4b79c00453a1edeac1c47cbfb66b5": "\\mathrm{v}=\\alpha D_{\\alpha\n}p^{\\alpha -1}", "28c563c8d2a6c96d45e652210435c7f4": " u_n(x) := (x-1)^{-2n} P_{n}(x) ", "28c56e33539bec9a846149fe3d4090ed": " v_x > 0\\ ", "28c5bf7f78b9219ef528407815ad9b1b": " \\le r ", "28c5eac946471f68eefb01f7a53b1844": "x_3", "28c602df45800225c950a013ddc09202": "L_{n+1} (\\pi_1 (X))", "28c619cc0d686353b6fd04ba4c1d6408": "g = q \\circ h", "28c633da6754fcbb4ae86fe4aedddf3a": "\\xi = \\prod_{j=2}^\\infty \\left({1 - \\frac{1}{d_j}}\\right) \\ . ", "28c655e9cdc8468134db0d6198f229d3": " \\beta_F = \\beta_{F0}\\left(1 + \\frac{V_{\\text{CB}}}{V_{\\text{A}}}\\right)", "28c68b70ab9de8b616fb4cecb80217d0": "\\sum_{n=1}^{\\infty} \\frac{\\varphi(n) q^n}{1-q^n}= \\frac{q}{(1-q)^2}", "28c6e908f5f6724899dd4ab9850b2ac7": " \\bigsqcup", "28c71c0152449f92252035c3be84621d": " n = ( t / d_m )^2 a m^{( b - 2 )} ", "28c7595dd3198b1ca9e0aa9f61eab6bc": "{e}^{-\\frac{1}{e}} \\color{white}..........\\color{black}", "28c7c1aff5b5edc11e28efeb0787fd39": "\\displaystyle{A_\\varepsilon = (R+\\varepsilon I)^{-1}SK(R+\\varepsilon I)^{-1}.}", "28c7c64780580ce4287069efca6fa2e4": " F_{\\alpha\\beta}=\\eta^{\\gamma\\mu}F_{\\alpha\\beta\\gamma\\mu}", "28c7cd83ad9cee2297835637032cc83a": " \\left( \\frac{\\mathrm{d}S_{\\phi}}{\\mathrm{d}\\phi} \\right)^{2} + 2m U_{\\phi}(\\phi) = \\Gamma_{\\phi} ", "28c7ead189650c5e6eff0fb302d4c178": "| \\vec n |", "28c8505bd610ce17dabf08f7016e970a": "2e + k + 1", "28c85e81e97bdc597fa00c11c56b41b4": "G = G' + iG''", "28c89a0cd693a550a97357ba14a00ac9": " \\Phi_{V} = \\frac{\\pi r^{4}}{8 \\eta} \\frac{\\Delta p^{\\star}}{\\ell}", "28c89aa2ca416b00536c8118f1125852": "\n\\begin{align}\nf_m^{(k)}(n) & = \\frac{m(m-1)\\cdots(m-k+1)}{n(n+1)\\cdots(n+k-1)} f_m(n) \\\\\n& = m(m-1)\\cdots(m-k+1) \\frac{f_m(n)}{f_k(n)},\n\\end{align}\n", "28c89ca9fb324c3ff8eef65eb8444439": "\\operatorname{Log} z := \\text{ln } r + i \\theta = \\ln |z| + i \\operatorname{Arg} z = \\operatorname{ln}\\sqrt{x^2+y^2} + i \\operatorname{atan2}(y,x).", "28c8c22f82b5ace4474f888372f7366b": "\n\\Pr \\left\\{ E[f(X_1, X_2, \\dots, X_n)] - f(X_1, X_2, \\dots, X_n) \\ge \\varepsilon \\right\\} \n\\le \n\\exp \\left( - \\frac{2 \\varepsilon^2}{\\sum_{i=1}^n c_i^2} \\right)\n", "28c8f13981b150bcab7900a156d0a1bb": "\\lambda_B=\\hbar/(m_e k_B T_e)^{1/2}", "28c8fdb55cbbeb686581d8f9b4e83c65": "x \\vee y", "28c93829ef1a3dc9e402cf5430fe5bf1": "Q^{\\ell_B}_{m_B}", "28c9943ff90d9822baf1b68d23009cd1": "(b_n<0)", "28c9a9c918b323e3d97bff7726d09d78": "k(\\mathbf{x}_i,\\mathbf{x}_t)", "28c9cb88abe6073de58c67c1de224203": "15^2+21^2=225+441=666.", "28c9fd978710a3aa6bb7f70eded23ab1": " \\langle s+\\sqrt{\\lambda_i}p_i | s+\\sqrt{\\lambda_j}p_j \\rangle = \\langle s | s \\rangle + \\sqrt{\\lambda_i} \\langle s | p_i \\rangle + \\sqrt{\\lambda_j} \\langle s | p_j \\rangle + \\sqrt{\\lambda_i \\lambda_j} \\langle p_i | p_j \\rangle = 1+0+0+ \\sqrt{\\lambda_i \\lambda_j} \\cos{\\omega_{ij}} = 1 +\\cos{\\omega_{ij}} ", "28ca45c80daf202a121cea4ffc764ccf": " \\cos(k t) = \\frac{\\exp(i k t) + \\exp(- i k t)}{2} = \\frac{z^k + z^{-k}}{2}", "28ca4990e68c146a6b71da9a78abebc4": "\\Theta((|E| + |V|) \\log |V|)", "28ca90f36d10138610a089bec6c49603": "\\frac{1}{n!}\\sum_{k=0}^{\\lfloor x\\rfloor}(-1)^k\\binom{n}{k}(x-k)^n", "28cab36c24bc9a734e2d113daf524385": "a\\left(\\sum_{j=1}^n u_je_j, e_i\\right) = \\sum_{j=1}^n u_j a(e_j, e_i) = f(e_i) \\quad i=1,\\ldots,n.", "28cb2324757fca72a867d4e12ff9fdf3": "\\tan \\psi = \\frac {m + 1} {m} \\tan \\theta \\,.", "28cb43a74d53b47717fa0e0cd5302fd1": "\\mathcal{F}(\\mathbf{x}) \\approx \\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(x_n),", "28cb4b0766fb8b454a52b03f54477d9b": "e = 4", "28cb57b0978a8dbe7b0daf72ae4400bb": "\n A_1 \\times A_2 \\times \\cdots \\times A_n\n", "28cb67cb9650d61b3ab336e684e6db48": "\\sin(\\delta_1 +\\delta_2 )\\sin(\\delta_2 +\\delta_3 ) = \\sin(\\delta_1)\\sin(\\delta_3) + \\sin(\\delta_1+\\delta_2+\\delta_3)\\sin(\\delta_2). \\,", "28cba20a0c17aa20794963b3c834b062": "G = G_0^{} + G_0 \\Sigma G.", "28cba6de8441f20c37eb88b1a4da9f9b": "\\frac{n_x+n_y-p-1}{(n_x+n_y-2)p}t^2 \\sim F(p,n_x+n_y-1-p;\\delta),", "28cbcb3bb68f4932682359edaa98e7d0": "\\vec{P}_{1,4}", "28cbcc4aab9c2a147e76cbc0c8ddb500": "f(0) = (1, 0, \\dots, 0).", "28cbec68a6b311f39d8fa027d71ea7f7": "\\Pi(k_i) = \\frac{\\eta_i k_i}{\\displaystyle\\sum_j \\eta_j k_j},", "28cc3797f616b7542a97179c74438356": "u_1 = \\frac{(ax_1^2+x_2^2+x_3^2+x_4^2)x_5 - 2x_1(bx_1 x_5 + x_2 x_6+ x_3 x_7+ x_4 x_8)}{c}", "28ccb80471f2fe36a2ba1cb2fc78b45f": "\\scriptstyle R_{ws}[-i] \\,=\\, R_{sw}[i]", "28cd1257ee5de94d6fbc0f0ea89829dc": "\\gamma_0=\\gamma", "28cd90cc994d482cbbab9f256d7bac4e": "f(t)g(t)", "28cdb69bf439f116e22c57bb36c0f32f": "\\begin{align} & \\mathbf{\\hat{A}} \\psi = \\mathbf{\\hat{A}} \\psi ( \\mathbf{r} ) = \\mathbf{\\hat{A}} \\left\\langle \\mathbf{r} \\mid \\psi \\right\\rangle = \\left\\langle \\mathbf{r} \\mid \\mathbf{\\hat{A}} \\mid \\psi \\right\\rangle \\\\\n\n& \\left ( \\sum_{j=1}^n \\mathbf{e}_j \\hat{A}_j \\right ) \\psi = \\left ( \\sum_{j=1}^n \\mathbf{e}_j \\hat{A}_j \\right ) \\psi ( \\mathbf{r} ) = \\left ( \\sum_{j=1}^n \\mathbf{e}_j \\hat{A}_j \\right ) \\left\\langle \\mathbf{r} \\mid \\psi \\right\\rangle = \\left\\langle \\mathbf{r} \\mid \\sum_{j=1}^n \\mathbf{e}_j \\hat{A}_j \\mid \\psi \\right\\rangle \\\\\n\n\\end{align} \\,\\!", "28cdecab88496bf35ee30677dc26a20f": "\\lnot \\alpha", "28ce006791acc547ef12975e97d03440": "H_A(s)= {k_A \\cdot s^4\\over(s+129.4)^2\\quad(s+676.7)\\quad (s+4636)\\quad (s+76655)^2}", "28ce3f4dbe770799c2295b0366ed26a5": "i_\\mathrm i = \\frac{v_\\mathrm i}{Z_0} = Iu(\\kappa t-x)", "28ce75dc16591d4b552dc37c94ecea89": "\\frac{100-95}{95} = 5.26\\%", "28cedf53540543e78462fdc49019fb82": "\\operatorname{Var}(X|Y)", "28cf0975824c61a963ba357235ea6ddc": "Q_{u,v} (\\alpha, \\beta) x^u y^v ", "28cf3ccd501f08c436bcf478d18b60dd": "\\mathrm{H_2O + CO_2 \\longrightarrow H^{+} + HCO_3^{-} \\longrightarrow H_2CO_3}", "28d0002cc23364df1a800fbbfd02e58e": "[P,Q]=PQ-QP=-i\\hbar,\\,", "28d02db8f9cf7c0537fac9666ff53223": " Q \\equiv \\max_{p\\in[0,1]} \\; \\Big\\{ \\; H_2 (\\eta\\, p) - H_2((1-\\eta)\\, p)\\; \\Big\\}\\; ", "28d03866159d72ada6e5afcc82194a1a": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}\\left[-p~\\boldsymbol{\\mathit{1}} + 2\\left(C_1 + \\bar{I}_1~C_2\\right)\\bar{\\boldsymbol{B}} -\n 2~C_2~\\bar{\\boldsymbol{B}}\\cdot\\bar{\\boldsymbol{B}} -\\cfrac{2}{3}\\left(C_1\\,\\bar{I}_1 + 2C_2\\,\\bar{I}_2\\right)\\boldsymbol{\\mathit{1}}\\right] \\quad \\text{where} \\quad\n \\bar{\\boldsymbol{B}} = J^{-2/3}\\,\\boldsymbol{B} \\,.\n ", "28d0444cbac239d581aa0b475ab95c80": "\\tan(z) = \\sum_{k=0}^{\\infty} \\left[\\left(\\frac{-1}{z - (k + \\frac{1}{2})\\pi} - \\frac{1}{(k + \\frac{1}{2})\\pi}\\right) + \\left(\\frac{-1}{z + (k + \\frac{1}{2})\\pi} + \\frac{1}{(k + \\frac{1}{2})\\pi}\\right)\\right]", "28d07eb76cebafadbcddbece0388494d": " \\mathbf{x} \\in \\mathbf{R}^m, \\mathbf{y} \\in \\mathbf{R}^n \\ : \\qquad \\left\\langle \\mathbf{x} |A| \\mathbf{y} \\right\\rangle ={}^{t}\\mathbf{x}A\\mathbf{y} ", "28d0b784ce50f818da13317af4642156": "\\pi r^2 h\\;", "28d18665fabd87e3247a75de13a106a7": "\\mu = m_i/m_p", "28d1cbd88f29f0b63c26c224e909e8bf": "\\begin{align}\n U &{}= (A - \\bold{i}I) (A + \\bold{i}I)^{-1} \\\\\n A &{}= \\bold{i}(I + U) (I - U)^{-1}\n\\end{align}", "28d1f06fd4b27dd7c51107aad497ab86": "\\mathrm{Hom}(\\mathrm{Gr},\\mathrm{Gr})", "28d243b04dd6708200122c614a114d7b": "\\frac{\\partial V}{\\partial S}", "28d2647a12e43c3e8cb5d7dec8e77356": "\\scriptstyle\\sqrt 10", "28d2844e9467acfcc62ccf5e37a7add4": "f\\approx f^0+\\frac{\\partial f}{\\partial a}a+\\frac{\\partial f}{\\partial b}b", "28d28669a693b48e36824e48e032d3b5": "\\mathfrak{a} = \\mathfrak{p}_1 \\mathfrak{p}_2 \\dots\\mathfrak{p}_g.\n", "28d292a93ad6b61b67b324228df0628c": " \\nabla \\cdot \\mathbf{J} + {\\partial \\rho \\over \\partial t} = 0.\\,", "28d299cd29fdd90d84afd49e80a40289": " T(f) = Pm(f)P, ", "28d2e245d62b9aaeb77521a2d6a42c37": " 0 = \\frac{dS}{db'}(\\hat\\beta) = \\frac{d}{db'}\\bigg(y'y - b'X'y - y'Xb + b'X'Xb\\bigg)\\bigg|_{b=\\hat\\beta} = -2X'y + 2X'X\\hat\\beta", "28d386998e75e7820dc75a998c825c0c": "\nB_\\mu B^\\mu \\mp b^2 = 0 .\n", "28d3984d5d86f6c759ce3472ee14e223": "\\ d[\\mathbf{x}(1), \\mathbf{x}(3)]=|u(2)-u(4) |=5>r ", "28d3f513fbc1ffc400e1ac62ce40e7c8": "A \\in \\mathcal{F}", "28d3fa169349d84254463df586e69fa4": "X^\\nu = (X^0, X^1, X^2, X^3)= (ct, x, y, z).", "28d3ff7a00a2a0093577c6231e26050a": "F = true ", "28d41a5033c281880a8435443133f857": " P(d\\sigma | \\eta) = \\otimes_{k \\in G} p_k(d\\sigma_k | \\eta) ", "28d43ec89932150e1c5291dae277c928": " v(1), v(2),\\ldots,v(d) ", "28d476a23968b2ab20414ea6c44bfe55": "X' \\widehat \\otimes_\\varepsilon X ", "28d5013b0a47deea517d5736b5880d31": "\n \\det S''_{ww} (\\boldsymbol{\\varphi}(0)) = \\left[\\det \\boldsymbol{\\varphi}'_w(0) \\right]^2 \\det S''_{zz}(0) \n \\Longrightarrow \\det \\boldsymbol{\\varphi}'_w(0) = \\pm 1.\n", "28d5256eb61c0054aac4d6f733e0b6f5": "|L|", "28d526a033db0b7760b660efbe0d06b8": "\\displaystyle \\mathfrak{f}(L/K)=\\prod_\\mathfrak{p}\\mathfrak{p}^{\\mathfrak{f}(L_\\mathfrak{p}/K_\\mathfrak{p})}.", "28d537631badbd24f1061bb566250c5e": "\n\\beta = 180^{\\circ} - (\\lambda + \\iota)\n", "28d549a92404c71e6b0150f24cfd488d": "\\rm 4\\; Cyt\\,c_{red} + O_{2} + 8\\; H^+_{matrix} \\rightarrow 4\\; Cyt\\,c_{ox} + 2\\; H_2O + 4\\; H^+_{intermembrane} \\! ", "28d5bb1760b3b528ad3799b57b587ac5": "\\mathbb{Z}_n \\!\\,", "28d5c441661cab6e884957e65ce8f69e": "f(z)f(1-z)={\\pi \\over \\sin(\\pi z)}\\,\\!\\,\\,\\,", "28d6486f8767c909c034381ea8c3ab48": "h\\ :=\\ f(h,\\ m_i)", "28d667fb566e56b2ac3989b622eb52fc": "|\\bar{h}_n|^2", "28d71df4347bc4f693941d665e372687": "\\frac{\\cos^2\\left(\\frac{\\pi}{q}\\right)}{\\sin^2\\left(\\frac{\\pi}{p}\\right)} + \\frac{\\cos^2\\left(\\frac{\\pi}{r}\\right)}{\\sin^2\\left(\\frac{\\pi}{s}\\right)}", "28d72a8a977bc6a683245b46a02e9a05": "\n\\tau =\\ r_{m} c_{m} \\,\n", "28d72fc1c22e76644ac880b80ef85c83": "G_{k+1} = G_k", "28d75f06d7a7ff000953a2e67ed7e280": " i_R = \\frac {v_E} {R_2} + ( \\beta +1 ) i_B \\ . ", "28d7772add19980f3717650b6a1717c5": "d \\,\\ ", "28d7c6372fcd5994049e0df3a9f0a490": "\\begin{align}\nb_1 x_1 + c_1 x_2 & = d_1;& i & = 1 \\\\\na_i x_{i - 1} + b_i x_i + c_i x_{i + 1} & = d_i;& i & = 2, \\ldots, n - 1 \\\\\na_n x_{n - 1} + b_n x_n & = d_n;& i & = n.\n\\end{align}\n", "28d7e6c627af51ae5e3dc7d13e42d38b": "\\left\\{\\overline{D}_{\\dot{\\alpha}}, \\overline{D}_{\\dot{\\beta}} \\right\\}=F_{\\dot{\\alpha}\\dot{\\beta}}=0", "28d7fa312ba6b57eae3c374dd4a47f4e": " \\sim 10^{89} \\,\\!", "28d8c698f4c891c8e9f498c08c99489a": "C(1,1) = 1\\,", "28d92b6f3f7c50cdc6095a33c4f29514": "\n f(t)=\\frac{\\Delta x}{\\sqrt{4\\pi Dt^3}}\\sim t^{-3/2},\n", "28d92c7e2cd156877779149fed889a84": "\n\\nabla_y\\cdot\\left(A(\\vec y)\\nabla w_j\\right)=\n-\\nabla_y\\cdot\\left(A(\\vec y)\\vec e_j\\right). \n", "28d9d4b3fde95c6b94ceb397e5398326": " \\psi\\nabla^2\\phi-\\phi\\nabla^2\\psi= \\nabla\\cdot\\left(\\psi\\nabla\\phi-\\phi\\nabla\\psi\\right)", "28d9e034e50e7f293df13ae8273dc668": "\\mathbf{f}(\\mathbf{v}) = \\mathbf{f}_1(\\mathbf{v})\\times\\mathbf{f}_2(\\mathbf{v})", "28da01697074a1dcb4c44691d7d46a3f": "x_1x_2x_5=213", "28da0b2a60d1d778b7254ee9b0c59c36": "(a,b)\\,", "28da5d2b6211555eb46aca2138f79d68": " y' ", "28dad983f9e23004ea831ed535ec9a8f": "E^* = E", "28db353127e9ec3071296e2a93243a49": "I_1(\\mathbf r) = {\\sum\\limits_i \\gamma_i \\cdot \\exp(-\\left|\\mathbf{r}-\\mathbf{r}_i\\right|/\\varepsilon)}", "28db55b59688d9fa1e76b39c4685b856": "\\overline{x_i-a_i}", "28dc04dbd6e0d5356c224522dfb3dcba": "\n\\mu_t = \\rho \\tilde{\\nu} f_{v1}\n", "28dc0d8885edab31b148abbc16107ed3": "1_{\\{\\tau_X\\le T\\}}", "28dc2326da0088bef4fc8ba7c3b54bec": "\nJ(A,B) = \\frac {|A \\cap B| } {|A \\cup B|} = \\frac{TP}{TP + FP + FN}\n", "28dc628c16e626812035292a35656cb3": "\\alpha = 0.54,\\; \\beta = 1 - \\alpha = 0.46,", "28dc8bc4541ce9bff87f7237a83b94fb": " \\Gamma' \\vdash C:D ", "28dc930e7c69157c9b0876863eee407c": "q_1", "28dcc18b5802f5df1ac6f8fc6a26470e": "\\mathbf{E}(\\mathbf{r}, t) = \\frac{1}{4 \\pi \\epsilon_0} \\left(\\frac{q(\\mathbf{n} - \\boldsymbol{\\beta})}{\\gamma^2 (1 - \\mathbf{n} \\cdot \\boldsymbol{\\beta})^3 |\\mathbf{r} - \\mathbf{r}_s|^2} + \\frac{q \\mathbf{n} \\times \\big((\\mathbf{n} - \\boldsymbol{\\beta}) \\times \\dot{\\boldsymbol{\\beta}}\\big)}{c(1 - \\mathbf{n} \\cdot \\boldsymbol{\\beta})^3 |\\mathbf{r} - \\mathbf{r}_s|} \\right)_{t_r}", "28dd800d51e26dea97c620a5e4859c5c": " S_k = \\frac{\\displaystyle \\sum_{ 1\\leq i_1 < \\cdots < i_k \\leq n}a_{i_1} a_{i_2} \\cdots a_{i_k}}{\\displaystyle {n \\choose k}}.", "28de4fc9f3f5e5321333e9b12d846d0e": "p_1=v_1", "28de699f2b6d7e9e5bb64d5866ea23f2": "\\mathbf{p'} = \\mathbf{q} \\mathbf{p} \\mathbf{q}^{-1}", "28de8242135b71682fca7f1d80675d22": "f_\\text{e}(x) = \\tfrac12[f(x)+f(-x)]", "28de8bbac4a8fcfb92684a87c153316a": "U_i^'(x)=p", "28de8f3fc552c8735b27b487062f0d1e": "2 \\equiv 2^{p+1} = \\left(2^{\\frac{p+1}{2}}\\right)^2 \\pmod{M_p}", "28df2e8ac1def8027b9afc053d7bbfc8": "\n\\sigma _{\\hat g}^2 \\,\\,\\, \\approx \\,\\,\\,\\,\n\\begin{pmatrix}\n {{\\partial \\hat g} \\over {\\partial L}} & {{\\partial \\hat g} \\over {\\partial T}} & {{\\partial \\hat g} \\over {\\partial \\theta }}\n\\end{pmatrix}\n\\begin{pmatrix}\n {\\sigma _L^2 } & 0 & 0 \\\\\n 0 & {\\sigma _T^2 } & 0 \\\\\n 0 & 0 & {\\sigma _\\theta ^2 }\n\\end{pmatrix}\n\\begin{pmatrix}\n {{{\\partial \\hat g} \\over {\\partial L}}} \\\\\n {{{\\partial \\hat g} \\over {\\partial T}}} \\\\\n {{{\\partial \\hat g} \\over {\\partial \\theta }}}\n\\end{pmatrix}\\,=\n\\,\\left( {{{\\partial \\hat g} \\over {\\partial L}}} \\right)^2 \\sigma _L^2 \\,\\,\\, + \\,\\,\\,\\left( {{{\\partial \\hat g} \\over {\\partial T}}} \\right)^2 \\sigma _T^2 \\,\\,\\, + \\,\\,\\,\\left( {{{\\partial \\hat g} \\over {\\partial \\theta }}} \\right)^2 \\sigma _\\theta ^2\n{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,Eq(17)}}", "28df68cc4a1dd080e6cf499a9e917fa9": "(\\phi \\to \\psi ) \\to ((\\chi \\to \\psi ) \\to (\\phi \\lor \\chi \\to \\psi ))", "28dfb7d8bd8bb68eb38e2476b84712ba": "\n\\begin{align}\nF(0)&=1~ ;\\ M(0)=0 \\\\\nF(n)&=n-M(F(n-1)), \\quad n>0 \\\\\nM(n)&=n-F(M(n-1)), \\quad n>0.\n\\end{align}\n", "28e00059d4c25537a714d7be81b6bf1f": "\\mathrm{0.41\\overline{6}}", "28e076a6ab479db26b01109c13a29df9": "\\mathbf{\\mathit{C}}", "28e087909eef108bd6c9cb7060297370": "\\mathrm{T}(v_i)=w_i", "28e0aca9c25422ad78047c497f82627e": "\nP(s) = \\frac{1}{Z} e^{- E(s) / kT},\n", "28e0ff7958c2f125d9998fe9eae0cc83": "V_{TN} = V_{TO} + \\gamma ( \\sqrt{ | {V_{SB} + 2\\phi_{F} | } } - \\sqrt{ | 2\\phi_{F} | } )", "28e12dd467effc871ee03f0e6fcf564a": "\\!\\mathcal A \\models_X^- \\phi \\vee \\psi", "28e1558cf428043d0daaade25923438b": "\\|Ww\\|_{(1)} \\le C(\\|VWw\\|_{(0)} + \\|W^2w\\|_{(0)})\n\\le C\\|(\\Delta - V^2 -A)w\\|_{(0)} + C \\|(WV+B)w\\|_{(0)} \\le C_1 \\|L w\\|_{(0)} + C_1 \\|w\\|_{(1)}.", "28e162a6418dd5f7edc5895d55ba436b": "z, w", "28e166a43cf0012124de5a19bc0a99c9": "\\textstyle 3.\\ Check\\ which\\ a(\\theta)\\ \\epsilon span\\{\\bold E_{s}\\}\\ or\\ \\bold P_{A}a(\\theta)\\ or\\ P_{\\bold A}^{\\perp}a(\\theta),\\ where\\ \\bold P_{A}\\ is\\ a\\ projection\\ matrix.", "28e185494a796b8af017271e5893494b": "k\\in\\Bbb N", "28e1bb8f85d27e2fcc0af0ef9ad1a74d": "d(x, y) = d(y, x)", "28e1f27d1f838dbe5700d79acb5f6aad": "i^2=-1.", "28e20014de881c3696bec844c941c7a2": "\\textstyle G_1", "28e2121ee734ce066f7799b15d6be9ab": " \\chi_V ", "28e218199ff1d5b2bfc52b11029c6898": "\\underset{\\alpha > 0}{\\lim_{\\alpha\\to 0}} \\|I^\\alpha f-f\\|_p = 0", "28e281b189e72449fcbc1a707f63a878": "H^* (M; o(M))", "28e2b9aee164a85fc8add169fb38fb28": " k = \\left(\\frac{E-E_0}{c_k}\\right)^{1/p} \\ , ", "28e31dcb3e39ce0c0b29d60a2156d88f": "- 8 y^3 - 42 y^2 + 72 y + 378", "28e4058b15e9acc845ad842139379729": "\\mathbb{P}^1 \\times \\mathbb{P}^1", "28e41309882ddb824db739780d2ecba1": "\\mu_\\mathrm{N} = {{e \\hbar} \\over {2 m_\\mathrm{p}}}", "28e44d5c96883b6bc6e9814567e87def": "\\epsilon / D", "28e512a882025a421ca855d03924af28": "-J, -(J-1) \\cdots 0 \\cdots +(J-1), +J", "28e59f3d592d74cc4452eb26f653d9b8": "\n u_s^{\\mathrm{topface}}(x,z) = -\\left(z - h - \\tfrac{f}{2}\\right)~\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x}\n ", "28e5d910a6734e9209c07a7a37ae4785": " \\left(\\begin{array}{cc} +&-\\\\+&-\\end{array}\\right),\n\\quad \\left(\\begin{array}{cc} +&+\\\\-&-\\end{array}\\right), \\quad\n\\left(\\begin{array}{cc} -&+\\\\-&+\\end{array}\\right), \\quad\n\\left(\\begin{array}{cc} -&-\\\\+&+\\end{array}\\right). ", "28e624a508df728e0273abb1e6394210": " \\bold y(x) ", "28e6504280202e340435c4409bd5fa3d": "f(0)=a", "28e71e5fd2f8e19d7a0e518814e83c16": "\\Delta(w,C)=\\min_x\\{w,C(x)\\}", "28e77ed4e9ed74c2034ffe46a8b72018": "\\prod _{1\\le i\\ne j\\le n}(1-t_i/t_j)^{a_i}", "28e78032188ba926bb889e202bd108e8": "ab\\le a", "28e7d11582dc85c5b92920e705bb93f0": "E_{m}", "28e7e81229f7a91e1d8f8f053510cdee": "S_{mk}^{}=\\sigma_{mk}+p_{mk}/R", "28e82c22b9fa89388865172311e899a1": " E_3=2 q_4 q_3", "28e90ba64dc301c559ce754e1211a38e": "\\scriptstyle\\sum_{p|N}\\frac{1}{p}+\\frac{1}{N}=1", "28e92e148f2d5d3214fffc67cd7a4772": "H_0 = {\\rm p}K_{\\rm a} + \\log {{c_{\\rm B}}\\over{c_{\\rm BH^+}}}", "28e93d307001615f55e1706a2c7f4795": "\\Rightarrow D_s ^2 = \\frac {C}{C_s} D^2 ", "28e9634b77bc25a1cca9783a5b4e0acb": "\\text{Gain} = \\ln\\left( {\\frac{P_{\\mathrm{out}}}{P_{\\mathrm{in}}}}\\right)\\, \\mathrm{Np}", "28e9d67574a03b995e5f2bf1954096ec": "e_k(t):=e^{2\\pi i k t}\\,", "28e9e7985215109604037e2159faf2d8": "\\sin\\left(\\frac{x}{2}\\right) = \\pm \\sqrt{\\tfrac{1}{2}(1 - \\cos x)}", "28e9e8aad305db2dd028ac959505e30c": "~\\alpha", "28e9fd68184ad0aee9c687964f68ed92": " c = (64/9)^{1/3}", "28e9fe25681eb18184ad9f1010968a1e": "\\lambda_i^{(m)}", "28ea2617536b390d269bd319e07e06f6": " T_e G ", "28ea300596fd7db0e48bc310ac89717c": "\\scriptstyle(x_0,y_0)", "28ea323dad136b15aae0eef007970f6a": "P = |I| \\cdot A_\\mathrm{surf} = |I| \\cdot 4\\pi r^2 \\,", "28eaaf87fe74d9a5f1d491724ec01d65": "-\\cot\\phi = \\frac{a}{b} \\tan t = \\frac{\\tan \\theta}{(1 - g)^2} = \\frac{\\tan \\theta}{1 - e^2},", "28eab5b5254aad71ceee7be78678935d": "\\mathcal{L}_{V^{r}}\\,", "28eac0884358a673e9b31ab6cadaa6e6": "\\rho_i = \\frac{\\lambda^+_i}{\\mu_i + \\lambda^-_i}", "28eae72477860d091f533dd5d9717d18": "\\mathcal{E} ", "28eb2e70cd20b1c82bc81c0859e1f785": "0 \\le \\beta \\le \\alpha < k", "28eb993320ac2fee076c7ca63bf31796": "v= {\\sqrt{rg\\tan \\theta}}", "28ec74c953db003ff184e94c7aa46117": "x_n \\in X", "28eca1fe7a62d3288c078ed2ccd8a7e4": "A \\cap B = \\{x \\mid x \\in A \\wedge x \\in B\\}", "28ecf4836380cd43da5a6b1a2be4dddf": "avg\\_tch\\_req\\_success\\_city = (tch\\_req\\_success\\_bts1 + tch\\_req\\_success\\_bts2 + tch\\_req\\_success\\_bts3) / 3", "28ed014c67f0bba1685500e7709dc293": "u: \\mathbb{N}^2 \\to \\mathbb{N}", "28ed0660b7f8243ca3a8ef7011fdd7ea": "p_{\\perp }^{\\mu }", "28ed430ac09499f732df9a53871a662e": "\\alpha(x;\\theta)=0", "28ed6846e09b07a07ab99e4a4df21ced": "T = \\{ 2 \\le \\rho \\le 3, \\ 0 \\le \\phi \\le \\pi \\}.", "28ed7236433b740be445180b4eac21ee": "10\\uparrow\\uparrow n", "28edaec959e7ec9d24fb8b8273e8d9a9": " \\psi_1(2) = \\frac{\\pi^2}{6} - 1", "28edc170bf5b805254f982fe62ed2646": "2 Y \\dot{\\varphi}_{r}", "28edc29e66d58cb2355f86712ca8da87": "\\omega^{2^{p}-1}", "28edc666106e3612db1bc618064e0819": "\\mathbf{y^{\\prime\\prime}}", "28edcc9038ea0b4e7ab41617a9e823d9": "x^2 + y^2 = 1.\\,", "28ede9ce4a63b564c597990d412cd059": "GA(n,\\mathbb R)", "28ee1683fcca1a6808007a66e57d8379": "\\lim_{x\\to 0}\\sin(\\tfrac{1}{x})", "28ee1d2ebda817c34abe041493100eb2": "\\frac{{\\mu_0}} {8 \\pi} I^2 (a) +\\frac{1}{2}G\\overline{m}^2 N^2 (a) = \\Delta W_{B_z} + \\Delta W_k", "28ee56e712680cd09523c2ddeb7cd2f4": "\\scriptstyle \\frac{2}{a}\\left(1-\\frac{x}{a}\\right)", "28ee83aeadbf12de0a94aa42a915e4d8": "\\psi_3 = \\psi", "28ee8c700522510cfbad8d2bcdc63594": " k_i=\\lceil \\sum_{j} x_{ij} \\rceil", "28eed60c40749d3dc0d432a6e5fdb757": "\\mathbf{M}_{1} := (\\mathbf{A}_{1,1} + \\mathbf{A}_{2,2}) (\\mathbf{B}_{1,1} + \\mathbf{B}_{2,2})", "28ef2977b6c39e3e571038f42de19d02": "\\lambda(T_i)", "28efe43d6cbe7a5150aa80a9e7856281": "p_{i}\\leq q_{i}", "28f0218eb59518fb74c57271d8ecb62e": "L[u] = u'' - x u' = -\\lambda u", "28f0aa23a0aff301451ac11805463b60": "\\lVert \\cdot \\rVert", "28f0cab27a4a49c82ab309c92e43771e": "E_\\mathrm{angles} = \\sum_\\mathrm{angles} k_\\theta (\\theta-\\theta_0)^2 \\, ", "28f0cc7ce7adf060c6a9f1846cd892de": " \\gamma_{a} ~,~ \\gamma_{a_1 a_2} ", "28f0d79c8435e5815f8cd4a02072ba20": "V_g=\\sqrt{ V_a^2 + V_w^2 - 2V_aV_w\\cos\\left(\\frac{\\pi(d-w+\\Delta a)}{180\\deg}\\right) }", "28f11947369bc8682f056b05237fc518": "\\log_{10} (xy) = \\log_{10}x + \\log_{10}y", "28f1943da0271c21d1120b6e98c5e2d6": "x\\geq\\beta", "28f1a7be6d4ff6ffb865c0f5bc1e9f6d": "\\chi: A \\rightarrow C ", "28f2106ec2f8e7d0e461cb0c989a0d07": "3 \\tfrac{1}{2}", "28f2288b685cd0c342be58526e03caef": "\\mathbb{Z}[\\varphi]/(2\\mathbb{Z}[\\varphi])=\\mathbb{Z}[T]/\\langle 2,T^2+T-1\\rangle", "28f2b23f2fb8712ed30f468c6236b91a": "\\{P^iT\\}", "28f2ba976907b53b47ad27a1ae8b4154": " \\pi(x y) = \\pi(x) \\pi(y) \\,", "28f3886c4ee47e8fd5b150c3264710bb": "\np'_n = \\sqrt{\\rho_n}\\;p_n\n\\,", "28f3921768a66a2ad6ad0e77217db6b1": "(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2", "28f4048ba8fe8d689b47a6925164d731": "U(t) = \\frac{1}{2} k x^2(t) = \\frac{1}{2} k A^2 \\cos^2(\\omega t - \\varphi).", "28f43469d14782636b04121fbdc8dbfe": "\n\\begin{align}\n\\sum_{n=1}^N \\sin\\left(n\\right) & {} = \\sum_{n=1}^N \\frac{1}{2} \\csc\\left(\\frac{1}{2}\\right) \\left(2\\sin\\left(\\frac{1}{2}\\right)\\sin\\left(n\\right)\\right) \\\\\n& {} =\\frac{1}{2} \\csc\\left(\\frac{1}{2}\\right) \\sum_{n=1}^N \\left(\\cos\\left(\\frac{2n-1}{2}\\right) -\\cos\\left(\\frac{2n+1}{2}\\right)\\right) \\\\\n& {} =\\frac{1}{2} \\csc\\left(\\frac{1}{2}\\right) \\left(\\cos\\left(\\frac{1}{2}\\right) -\\cos\\left(\\frac{2N+1}{2}\\right)\\right).\n\\end{align}\n", "28f43ba0b9c347828fbd9909a52e34e4": "m\\pm1=m'", "28f44e0c82412884b8a0c374e9fadb2c": "R(\\hat{n},\\phi_1+\\phi_2)=R(\\hat{n},\\phi_1)R(\\hat{n},\\phi_2)", "28f44f15a5f7bd629bdc032d89a08a50": " \\bold x(0) = \\left( \\begin{array}{c}\nx_0 \\\\\n0\n\\end{array} \\right), \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (13) ", "28f46e5bc903dad43ee1c6e7bcd26972": "i \\omega = (e^{i\\pi/2} \\cdot \\omega).\\,", "28f4b17cf30f2620b191f4a542378de1": " E_\\mathrm{total} = E_\\mathrm{electronic} + E_\\mathrm{vibrational} + E_\\mathrm{rotational}+ E_\\mathrm{nuclear}", "28f4c35d8ed5c6473ca59d6c08788da8": "(6)\\,", "28f4dd4346de6f064cefb0173dde37b2": " |z| = 1 \\ ", "28f4fc72136524f41d1225d8ab8b2d7c": " \\langle O \\cup L \\rangle = (-A^2 - A^{-2}) \\langle L \\rangle ", "28f50c5fb82a5e81ebd62157a9d901bb": "( B'_x(C^{'2}_x + C^{'2}_y) - C'_x(B^{'2}_x + B^{'2}_y) )/ D' \\,", "28f54f6989d177bffb9d9341ccf06c4a": "S^{-1} M = 0", "28f6277fb4f8d3628a679047042a5ba9": " a_m =\\frac{p_m}{N} \\sum_{n=0}^{N-1} u(x_n) \\cos\\left(\\frac{m\\pi}{N}(N+n+\\frac{1}{2}) \\right)", "28f6328b41dd4c95823346eecc26bf0c": "\\Pi_{2}=\\left\\vert 0\\right\\rangle \\left\\langle\n0\\right\\vert ", "28f67b1cb502b3704835c7b23d4100a8": "\\mathbb{Z} \\!\\,", "28f68e4aa00d7407136b32d08cc7b3e3": "x=q^{3/2}", "28f68e995791310bfc8297a65fa9842a": "\\frac{\\mathrm{d}^2 q}{\\mathrm{d}\\tau^2} + 2 \\zeta \\frac{\\mathrm{d}q}{\\mathrm{d}\\tau} + q = \\cos(\\omega \\tau) + \\mathrm{i}\\sin(\\omega \\tau) = \\mathrm{e}^{ \\mathrm{i} \\omega \\tau} .", "28f6c67856f887851562701b206676b5": "\\Delta H=-\\boldsymbol{\\mu}\\cdot\\boldsymbol{B},", "28f6e4c81d98e0b1ad27145f3b826faa": "H(s)=\\frac{1}{1+\\underbrace{RC(m+1)}_{\\frac{2 \\zeta}{\\omega_0} = \\frac{1}{\\omega_0 Q} }s+\\underbrace{mnR^2C^2}_{\\frac{1}{{\\omega_0}^2}}s^2}", "28f71594c297178d6480ccb556c96572": "p\\trianglelefteq q ", "28f75b88aa53997504a653236eee75a4": " \\underline \\sigma = \\sigma (\\underline P) = \\frac{\\underline P}{A} ", "28f76236d5f213947e8079c1dfc5aac5": "p(x,y,z)", "28f79c4f20ed7d6d31313f3ad24f4c73": "\\displaystyle{C(s,t) ={1\\over \\pi i}\\left({\\dot{z}(t)\\over z(t) - z(s)} -{\\overline{\\dot{z}(s)}\\over \\overline{z(t)} -\\overline{z(s)}} \\right),}", "28f7ac08c1263de9b78d81e8d0465000": "M = \\int d^3x {[H (x)]^2 \\over \\sqrt{\\mathrm{det}\\; q (x)}}", "28f7f73fd1749cde4c0367e9bf75228c": "cd=s(s-a)-(s-b)(s-c).", "28f823d23197b5ad6768107a2034c246": "\\exists n_1 \\cdots \\exists n_k \\psi(n_1,\\ldots,n_k,m)", "28f82935e633b12de2a503c0357343bd": "\\mathbb{C}^n = \\bigoplus_i \\; \\mathrm{Ran}\\; e_i (T) = \\bigoplus_i \\; \\mathrm{Ker}(T - \\lambda_i)^{\\nu_i}", "28f852a5a605dc5e63462ef3fc38d333": "(4)\\quad ds^2=-\\frac{L-M}{L+M}dt^2+\\frac{(L+M)^2}{l_+ l_-}(d\\rho^2+dz^2)+\\frac{L+M}{L-M}\\,\\rho^2 d\\phi^2\\,.", "28f8ec7af7bd5cc33eefe2218d4f2d0c": "p(\\sigma^2|D, I^\\prime, \\mu) \\propto \\frac{1}{\\sigma^{n+n_0+2}} \\; \\exp \\left[ -\\frac{\\sum{ns^2 + n_0 s_0^2}}{2\\sigma^2} \\right]", "28f8efd60742eb828cd7faa92c34ba60": "\\mathbf{c}_{x,y,z}=\\langle 0,0,0\\rangle,", "28f966bca4bd3971bf40a9ee90bdf4af": "\\phi_2", "28f9e74262ae703001e42b07c4f61d67": "s_{i+1}:=s_{i-1}-qs_{i};", "28fa009873095ad623e23b5b109787a4": "v_k:(\\rho_1,\\ldots,\\rho_k)\\mapsto \\rho_1\\wedge\\ldots\\wedge\\rho_k", "28fa05bc6074b36c3c03608ef9f119ec": "A \\otimes B = \\begin{bmatrix} a_{11}B & \\dots & a_{1n}B \\\\ \\vdots & \\ddots & \\vdots \\\\ a_{m1}B & \\dots & a_{mn}B\\end{bmatrix}", "28fa27bfe21fa458e6185cf303ca3705": " [\\delta(f)](x) = {(\\mathrm{d} f)_{x}}((X_{\\delta})_{x}) ", "28fa288240f2b1c00145ae6d2b05e250": "\\cos^2(x/2) = \\frac{1+\\cos(x)}{2}", "28fa557a364e86b9d8f6d0430793f368": "a_0=1,\\ a_1=2=1+1,\\ a_2=4=2+2,\\ a_3=8=4+4,\\ a_4=16=8+8,\\ a_5=32=16+16,\\ a_6=31=32-1.", "28fa72bb9f9c5ceec1d5cc949eedeee3": "u,v in R^k", "28fa9917956bacafe5b752f19b1a074b": "VX - C^{-1}Y = 0", "28facf7a629d7685da022f368da94ce2": "\\epsilon_0 = \\mu_0 = 1", "28fb87b225ee25fd7fe60b9997e256e1": "\\mathbf{B}'(t) = 3(1-t)^2(\\mathbf{P}_1 - \\mathbf{P}_0) + 6(1-t)t(\\mathbf{P}_2 - \\mathbf{P}_1) + 3t^2(\\mathbf{P}_3 - \\mathbf{P}_2) \\,.", "28fbbb8b927081f966c9aa722c1413d7": " \\frac{m \\left | \\mathbf{v} \\right | }{\\left | \\mathbf{r} \\right |} = q \\left | \\mathbf{B} \\right | \\sin \\theta, \\,\\!", "28fbd49d9efa7ea54fccbf2e47c1ad5b": "\nf(B)=2\\times\\frac{e^{-\\frac{B}{2}}}{2\\pi}\\int_0^{\\frac{B^2}{4}}A^{-\\frac{1}{2}}(B^2-4A)^{-\\frac{1}{2}}dA\n", "28fc6f0bca428d27502ee1d54a7833ae": "\\textstyle{(\\frac{5}{9})^n}", "28fc7dbb6673cd5fdcb67fa63f8dd424": "s_p(n) = \\sum_{j = 1}^{\\lfloor \\log_p(n) \\rfloor} \\left\\lfloor\\frac{n}{p^j}\\right\\rfloor ", "28fccbcd2fe248ae51194fb57fae6886": "\n r\n= \\frac{1}{ u} ( u \\wedge v )\n= \\frac{ u}{ u^2} ( u \\wedge v ) \n= \\frac{1}{{\\Vert u \\Vert}^2} u ( u \\wedge v ) \n", "28fcd91e20e83f59493c69cdf7e6e340": "\\scriptstyle \\mathbf{R}^{n+1}", "28fd03973b851e682d9415f5cbd367c1": "x_1^{a_1}x_2^{a_2}\\dots x_n^{a_n}", "28fd60fe633b7021c96410efd584b6f6": " d(z_1,z_2)=\\tanh^{-1} \\left|\\frac{z_1-z_2}{1-\\overline{z_1}z_2}\\right| ", "28fd6baea99c414b78390fd08885a3a4": "P^* (T ')", "28fd742caa23117b8d33d57ac38a58f1": "\\chi = 2 - 2g.\\ ", "28fd7eda58830e95e369792ed81a0758": " \\sum \\frac{(\\text{observed}-\\text{expected})^2}{\\text{expected}}\n= \\sum_{k=1}^6 \\frac{(X_k - n/6)^2}{n/6}, ", "28fdc6259f42bdce714db92ea49a985b": "S_{mk}^{}=\\alpha_{mk}-\\beta_{mk}+p_{m,k-1}+p_{m-1,k-1}", "28fdcdda6450230131c2cb4c91d5df14": "\\omega \\; = \\; \\theta_4 - \\frac {\\varepsilon}{60} x^3 c_1 \\left ( 10 - \\frac {4 x^2}{{c_1}^2} + x^2 {c_1}^2 \\right )", "28fde301ba978f13a3ccc6b839cb2118": "\n\\left( x(1-x) \\frac {\\partial^2} {\\partial x^2} + y(1-x) \\frac {\\partial^2} \n{\\partial x \\partial y} + [c - (a+b_1+1) x] \\frac {\\partial} {\\partial x} - b_1 y \n\\frac {\\partial} {\\partial y} - a b_1 \\right) F_1(x,y) = 0 ~,\n", "28fdec14c225cece3ed4879585dbbb1f": "\\log_{10}(\\log_{10}(10^{10^9})) = 9", "28fe12aaa7ed3135d7bf3a1edfa3fe53": "\n \\int_0^b \\int_0^a q(x,y)\\sin\\frac{k\\pi x}{a}\\sin\\frac{\\ell\\pi y}{b}\\,\\text{d}x\\text{d}y = \n \\frac{b}{2}\\sum_{m=1}^{\\infty} a_{m\\ell}\n \\int_0^a \\sin\\frac{m \\pi x}{a} \\sin\\frac{k\\pi x}{a}\\,\\text{d}x = \n \\frac{ab}{4} a_{k\\ell} \\,.\n", "28fe1c151db5f2f852b9ffcf550c55c8": "\\binom{n}{n/2}=\\Omega\\left(\\frac{2^n}{\\sqrt n}\\right)", "28fe2ac17babc5405c0390d16461b5f9": "^*K_{A \\to 0}", "28fe84a342a0b1a74a6744de9a8ccff4": "\\frac{\\operatorname{d}Z_{j}}{\\operatorname{d}t}", "28fecab2c5e2385aa75cf152455e73ce": "p<\\tfrac{(1-\\epsilon)\\ln n}{n}", "28fef8130dd377e7a181e3d2b90a4ec5": "\\tau_{n+1}^{(1)}=c\\left(\\frac{h}{2}\\right)^2+c\\left(\\frac{h}{2}\\right)^2=2c\\left(\\frac{h}{2}\\right)^2=\\frac{1}{2}ch^2=\\frac{1}{2}\\tau_{n+1}^{(0)}", "28ff1a3cdc836acbc092849a4bf4a535": "\\boldsymbol{Q}(t) = (Q_n^{(c)}(t))", "28ff8bd39c71bd246c0ebef0acfef8be": " X^{1:n} ", "28ff9fe8c57180919f5ab847ceef9be1": "L = E_{KIN} - E_{POT}", "28ffda8bdd2baf111789aedbcebf0272": "\\displaystyle \\Phi(r, \\theta, z) = (r \\cos(\\theta), r \\sin(\\theta), z)", "28fff4fc22ac0fa842df951288402097": "\n-\\frac{\\nabla p}{\\rho} = \\frac{\\kappa}{c} F_{\\rm rad}\\,.\n", "290037d965e1cb8d1cce9e4b099644cf": " f(x) = f(a) + f'(a)(x-a) + h_1(x)(x-a), \\qquad \\lim_{x\\to a}h_1(x)=0.", "29007c9e743c9bc42b4865bf52b46007": "\\begin{matrix} {2 \\choose 2}{2 \\choose 2} \\end{matrix}", "2900aa55cb6cc90f7ed954768c9cb343": " f_o = \\frac{f_s}{\\gamma\\left(1+\\frac{v\\cos\\theta_o}{c}\\right)}.", "2900ecebca7d907b4f7754e71a08733d": "| A \\rang", "290114a63c19ae9f72372e29d23bbec7": "-\\theta_i", "2901b827dc852b884fcd242cbde643f7": "\\{Q_i,P_i\\} \\rightarrow \\frac{1}{i\\hbar}[\\hat{Q}_i,\\hat{P}_i]\\,.", "2901bd2d29bc9121dcd576fccaed7b0c": "X,X_0 \\in \\mathbb{R}^p", "29020a5efd0fecbd2fd01bbf560e0760": "A \\in R^{k \\times k}", "29022d77c051a575bae656de6aeccded": "Q_t \\cdot Ca_{O_2} = Qs \\cdot Cv_{O_2} + Q_t \\cdot Cc_{O_2} - Qs \\cdot Cc_{O_2}", "290241a1862233137a1edbbbaad37eed": "\\tfrac yx", "29024cfb1dd2f82110371a19d38b87e0": "\\log|b|_{\\ast}/\\log b\\leq\\log|a|_{\\ast}/\\log a ", "290277e4d8d2066cce6baa43d34af02d": "P \\to M", "29027ca64be86f248958ec0336862513": "s_k=s_0", "290288955945f5ff6b45c326f3e7c5a9": "\\frac{p}{1-p}", "2902ad34307ffe5aa26954a495c2c900": "\\! \\text{MAP} \\approxeq P_{\\text{dias}} + \\frac{1}{3} (P_{\\text{sys}} - P_{\\text{dias}}).", "2902f2ed2a20d140e29522acd29bf5d1": "H(x)= 1 + W[(H(0) -1) \\exp((H(0)-1)-\\frac{x}{L})],", "2903229d4a2768399bbd3f8b9bcb4768": "\\dot{\\gamma} : E \\to V^{*}", "29035f0a53d33b13ffb203014aa6b933": "T(A\\land B)=T(A)\\land T(B),", "290372d03fa54d8b10736a9537511467": "C = (\\Delta x) b", "2903a7a7c1f3e0cd14114932ec514c2a": "DU \\ ", "2903e29e304f5528e0c3ac05f746bfd3": " \\operatorname{lambda-drop-tran}[L, X] = \\operatorname{drop-params-tran}[\\operatorname{sink-tran}[\\operatorname{de-let}[L, X]]] ", "290402c497027ff7d6ccd6fbf61323e4": "\\phi(front)=\\theta-\\psi-\\frac{a}{V}\\frac{d\\theta}{dt}", "29041aa00bbe602271134596450e1897": " \\frac{ 2 ( p q )^{ 1 / 2 } } { ( p + q )^{ 1 / 2 } } ", "29041be3d0df5cc2b3a6ff203217f94b": "\\phi=\\phi_0+\\delta\\phi\\,", "2905597a31e2102bf55a72077ef7a946": " p^0 = \\sqrt{m^2 + {|p|}^2} ", "29058a92e64c5b0e37da8bdf1e4deec3": " \\delta\\ \\mathbf {u} \\equiv \\mathbf{u}^* ", "2905e54494d41b6488baf25d662ed4a0": "\\int_{\\mathbb{R}^n} e^{2\\pi i x\\cdot\\xi}(\\mathcal{F}f)(\\xi)\\,d\\xi = \\lim_{\\varepsilon \\to 0}\\int_{\\mathbb{R}^n} e^{-\\pi\\varepsilon^2|\\xi|^2 + 2\\pi i x\\cdot\\xi}(\\mathcal{F}f)(\\xi)\\,d\\xi.", "2905e857750c715b249af28ad7dafde6": "I = I_J\\cdot \\sin \\phi, \\ ", "290662d447eda832874732eccd70a74d": "\n\\begin{pmatrix}\n \\operatorname{Re}(w) & -\\operatorname{Im}(w) \\\\\n \\operatorname{Im}(w) & \\;\\; \\operatorname{Re}(w)\n\\end{pmatrix}\n", "2906a62916765fc4d9a3c35de940ee8a": "Y_{6}^{-6}(\\theta,\\varphi)={1\\over 64}\\sqrt{3003\\over \\pi}\\cdot e^{-6i\\varphi}\\cdot\\sin^{6}\\theta", "2906ca704342939ba89988f920fac0a9": "\\lim_{n\\to\\infty}\\left(1+\\frac{X}{n}\\right)^n", "2906ecc796232280ee5c0c6ac63f8ddd": "\\chi^2_{1}", "29075e7b72a05964a885814829572a49": " \n \\partial = \\partial_0 - \\nabla,\n", "290762a998bc55f95ae48c40360bdb36": "\\epsilon < 1", "2907640e75de5b483aa5916d28753ec7": "\\mathcal{N}(\\theta,\\sigma_\\theta).", "29078a02fb98cba9e964212f805cade4": " \\left(\\mathbf{ab}\\right)\n\\!\\!\\!\\begin{array}{c}\n _\\cdot \\\\\n ^\\times \n\\end{array}\\!\\!\\!\n\\left(\\mathbf{c}\\mathbf{d}\\right)=\\left(\\mathbf{a}\\cdot\\mathbf{c}\\right)\\left(\\mathbf{b}\\times\\mathbf{d}\\right)", "2907a2d67e9dc801624ae8f70cfc658a": "\\begin{align}\n\\tilde{s} & =\\tilde{{g}}^1 (\\tilde{x}^1,\\tilde{u}^{(1)})+\\tilde{\\omega}_s^{(1)} \\\\ \n\\dot{\\tilde{x}}^{(1)} & =\\tilde{f}^{(1)}(\\tilde{x}^{(1)},\\tilde{u}^{(1)})+\\tilde{\\omega}_{x}^{(1)} \\\\ \n \\vdots \\\\\n\\tilde{u}^{(i-1)} & =\\tilde{g}^{(i)}(\\tilde{x}^{(i)},\\tilde{u}^{(i)})+\\tilde{\\omega}_u^{(i)} \\\\ \n\\dot{\\tilde{x}}^{(i)} & =\\tilde{f}^{(i)}(\\tilde{x}^{(i)},\\tilde{u}^{(i)})+\\tilde{\\omega}_x^{(i)} \\\\ \n \\vdots\n \\end{align}", "2907cdc65704c9f89131ed925e5a6cd4": "\\neg (A \\wedge B)", "2907ec3a52b7f97ecbba196e1fb022df": "0.725 < 0.77976\\ldots", "2907edddbc3765242c631377deb356ab": "F(v)=kT\\log \\left(\\frac{hv}{kT}\\right)", "2908b8818a1b11cef72c94008cceb0c1": "{\\Delta f} = \\frac {2{f}{v}} c \\,", "2908b9b4667a3ef9dae096668dd44d77": "\\tfrac{1}{2}\\le\\sigma_{0}\\le 1", "2908f437f50c6a7d1399ab537e2a4021": "(\\lambda x.(\\lambda y.y))", "29090b9222536680337ebf5e11301160": "\\frac{\\partial f_{\\alpha}}{\\partial t} + \\vec{v} \\cdot \\frac{\\partial f_{\\alpha}}{\\partial \\vec{x}} + \\frac{q_{\\alpha}\\vec{E}}{m_{\\alpha}} \\cdot \\frac{\\partial f_{\\alpha}}{\\partial \\vec{v}} = 0,", "29092d9678eb67b961dade48d9c93acd": "F, G : \\mathbb{R} \\to [0, 1]", "290a49434765c0e03c05965effcd3cb6": "\\partial\\!\\!\\!/^2=\\partial^2.", "290ab090399fd56ac95530cfaa56dcb8": "x(t)=e^{i2\\pi kt^2}", "290ac05ef93439fa17bd5005ad115e26": " u_0^n ", "290aca50a9c85002684bf7d67282c71e": "\\mathbf{x}_1, \\dots, \\mathbf{x}_N", "290b5855249618ae5a3d3ee809968b4b": "H_{a}", "290b695d692dbba19ef43d32f4e34eb8": "(P^{2}+w^{2})\\Psi =0\\,,", "290c9f6d22b0e6c3a88032ea4548e688": "x_{i}\\;\\text{,}\\qquad i = 1, \\dots, n", "290cd2abbc719aeb7d2ee6c9cd920474": "\\lambda \\ll z", "290cd5316b0d23768b5cb6c23c1e2dd2": "D = H - \\tfrac{H}{4} - 2 ", "290cd72bb261fc3ced3dd60972bcc5f4": "\\frac{\\Re[w(z)]}{\\sigma\\sqrt{2\\pi}},\n ~~~z=\\frac{x+i\\gamma}{\\sigma\\sqrt{2}}", "290cf855517c551fc582fa17a90764ee": "y'(a)=\\beta \\ . ", "290d16b8b7dbc30417ee77805d512bf2": "[X_1\\times X_2]=[X_1]\\cdot[X_2]", "290d4555b111998297eaeb53b1073163": "B = 55.8", "290d5dcbf53445025284b94d51b8bcd5": "R=\\tfrac{1}{2}\\sqrt{a^2+b^2}", "290db3323f5e11c084a925bef63b0ba4": "pV=Nk_{B}T\\qquad\\qquad(7)", "290e135ea26231ef4898005be78acb93": "\\|x\\|_\\infty = \\sup_n |x_n|,", "290e4d7cb84560d9d5c282c4d31899ab": "R_K =\\frac{h}{e^2} \\,", "290e6d3c1afaf6a3ed3764cf1965f7e7": "\\epsilon\\rightarrow 0", "290e7792308c633104fe32c3f34e3d80": "\\ O(E)", "290e96c7c6034c54e48dd90340ebd3d9": "H^*(M^{2n}; \\mathbb{Z})", "290ea13bae7754c0cdf632e5f34596ea": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}\\mathbf{A}^{\\rm T}", "290f3d7fed00ccc3690ae1ebd2298fa4": "\\vec{F}_{bolt} = P_{max} \\cdot A_{internal}. ", "29100c5e7efd13e5f215c8339a40ca07": "\n \\mathrm{i}\\hbar \\frac{\\partial}{\\partial t} \\langle\\hat{N}\\rangle = \\mathrm{T}\\left[ \\langle\\hat{N}\\rangle \\right] + \\mathrm{Hi}\\left[ \\langle\\hat{N}+1\\rangle \\right]\n", "29101366d72a0df17fbc9a1c56b16af9": "\\beta_a", "291035dc596dafc01052a5eff7b4e0d7": " \\lambda = \\int_0^\\infty \\beta(a) i(a,t) \\, da .", "291038f13f394897101a67d56975965f": "X \\, \\mathrm{type}^y \\ ", "29106773b33f800246a47d322e4ae4e5": "E_n(\\bold{k}) = \\frac{\\hbar^2 (\\bold{k} + \\bold{G_n})^2}{2m}", "2910729892423c4785ea51c85dd6d6ea": "{\\mu}_d \\in \\mathbb{R}", "29107a22babc009a5580c6a90d7f80e9": "D^{2} = -\\Delta_{n}", "29109768cbe7b3d2a8f428241656008a": "\nI_R = \\frac{V_{in}}{R}\n", "2910bde44a000cb621f68bebe84a326f": "e_\\alpha' = \\sum_\\beta e_\\beta g_\\alpha^\\beta.", "291165d6f276a3bd95963c4dbcc26aa2": "\n2A\\mathbf{x}-2\\mathbf{b}=0 \\Rightarrow A\\mathbf{x}=\\mathbf{b}\n", "291179e9265de26ffaa13b7cae42b15a": "(X_t, \\mathcal F_t^*, P^\\mu)", "2911b81431e8a591c18e02bb3f87cedf": "= {0.1945 \\over 6}", "2911cc75e8a7eda1f415c41c2d1da539": "T^1_0(V) \\cong L(V^*;\\mathbb{R}) \\cong V", "291207e41008f2bf64eef9e88a3b09e4": "S_{h,k} = S_{g,k} T S_{h,f}, \\,", "291224aac1db94b777cc6c3e03e12717": "4. \\; \\; \\mathrm{NO} + \\mathrm{O}_3 \\; \\xrightarrow{} \\; \\mathrm{NO}_2 + \\mathrm{O}_2 \\; ", "29123cdd05229a3fa9964229611063c1": "\\text{HOMA-IR} = \\frac{\\text{Glucose} \\times \\text{Insulin}}{405}", "291245bf2ff1371e5105e468c0cc17bc": " \\phi_1, \\ ... , \\ \\phi_n \\vdash \\chi \\rightarrow \\psi ", "29134719bd06263575a2b9ede1cc2cec": " \\operatorname{de-let}[f]\\ \\operatorname{de-let}[q\\ q] ", "29135233303170d386427eb1390786b8": "MSL = \\frac{mmmm + \\tfrac{3}{2} mrrr + 2 rmmr + \\tfrac{1}{2} rmrm + \\tfrac{1}{2} rmrr}{\\tfrac{1}{2} mmmr + rmmr + \\tfrac{1}{2}rmrm + \\tfrac{1}{2}rmrr}", "2913ac8eee12db0a8b32f687f8bced10": "T \\subseteq N", "2914345b9c144285a76b6cee8e8a84b1": " E = K' + \\frac{RT}{nF} \\ln(c)", "291438a9be7741bee699625e2861912e": "u(x) = c,", "29145924fc62e4a37b9044c2b58daaeb": "Q = \\lambda ,", "2914812848d79da2ad19bdf97cf4b3d0": " a_i^\\dagger", "29153292855d779359626258b812b78b": " k= \\frac{n\\pi}{L},", "29153db24797008d5d27076ca7b758b0": "\\beta = 2\\left(2 - \\alpha\\right) - 4\\sqrt{1 - \\alpha}", "29156fd93eef86589d1b0afab41c0494": "\\frac{\\gamma}{2}", "2915a2b17659e1d5b84be14961ad9c59": "E_{ij} = ", "2915c8e169c6ada74f4547008c7924c5": "R_1^{(b)}={b-1\\over{b-1}}= 1 \\qquad \\text{and} \\qquad R_2^{(b)}={b^2-1\\over{b-1}}= b+1\\qquad\\text{for}\\ b\\ge2.", "2915fccb3bd02748022823d7c05c5ae1": "\\begin{matrix} {1 \\choose 1}{3 \\choose 2}{44 \\choose 2} \\end{matrix}", "2915fcd0874ebe4ed6326df127279f5e": "C \\to cC", "29162236f55887d72a52a8327b8bd9d4": "\\pi_p(\\mathbf{S}^{2k-1})", "291624f656831254c80fc9a190e42ab2": "\\Delta V = V(\\vec{r}+\\delta \\vec{r})-V(\\vec{r})=\\delta \\vec{r} \\cdot \\nabla V + \\frac{1}{2} (\\delta \\vec{r} \\cdot \\nabla)^2V(\\vec{r})+...", "2916855ac433b6f6a938b5d57f44c0f0": "E=M^\\uparrow", "2916c6773f16381cca8ff1f5bb4b35d7": "T[\\cdots]", "2916eb1ad9ef8adf480eb2afcd067e99": "D_{\\mathrm{KL}}(P\\|Q) = \\infty", "2916f14909f9ed216a13530e2b72b953": "x_1=1,\\ldots,x_n=n", "29170126b0c1a8d01a3bfa64c969d94a": "f : Q \\times R \\rightarrow S", "29171786faa6b27dd589ce042b86ec34": "\\frac{{10\\sqrt[3]{b}^2}}{{b}}", "2917189c4df5eaed43f7249cb3bf5814": "\\forall u\\forall v\\,\\partial (u + v) = \\partial u + \\partial v\\ .", "29172e74a6705ff2a4386ce250b4f1e3": "(X_1,X_2\\ldots, X_n)", "291737e6f460cd039690dab7255da155": "\n \\limsup_{n\\to\\infty} \\frac{\\sqrt{n}}{\\ln n} \\big\\| \\alpha_{X,n} - G_{F,n} \\big\\|_\\infty < \\infty, \n ", "2917443132756479e850e6c876206408": "n > N", "291792e8f76fbcb9e8d29ace6b730643": "a^*=\\left(-1\\right)^\\frac{a-1}{2}a.", "29183bddab1919643bed88cca776cc00": "\\Delta\\otimes\\Delta^* \\cong \\bigoplus_{p=0}^k \\Gamma_{2p}.", "29183e69afc20b9728933f28cc19fe7e": "S = k_B \\ln W \\! ", "29184a014c10f35fc201af66ccd3e7c2": "g_2 = -4(e_1e_2+e_2e_3+e_3e_1)", "2918887cc0d46f6afe29dfbf97a7d447": "(x/w \\cdot y) \\backslash z = y \\backslash (w/x \\cdot z)", "2918909c75eb97848d7a7f2258b26448": "b_{ij} = (x_j - y_i) A_j(y_i) B_i(x_j) \\,", "291890b22d98c6955542fa29af6446ed": "\\tan (\\arccos x) = \\frac{\\sqrt{1-x^2}}{x}", "2918a18f29fe7fbc7b60f549fe476c3c": "{{S}_2=\\varphi(5 + 3\\varphi) = 5\\varphi + 3\\varphi^2 = 5\\varphi + 3(1 - \\varphi) = 3 + 2\\varphi }", "29191da0cb92f259d688539792c4f80f": "\\varphi=45", "29195bc86c096d4d8498c9cdb6d57af3": "K_M^{N + 1}", "2919644b6749bc78c9e450c9eb34340d": "\n\\sum_{\\stackrel{1\\le k\\le n}{ \\gcd(k,n)=1}} \\gcd(k-1,n)\n=\\varphi(n)d(n),\n", "2919fc5ff9456db0606e9ddae9431498": "\\ E_z(k_z)=\\frac{\\Delta}{2}(1-\\cos(k_z d))", "291a14abc0caca27f61efc7d4d7fb249": " \\max_x \\sum_{i=1}^L {\\alpha_{i}}\\ln x_i \\text{ s.t. } \\sum_{i=1}^L p_i x_i= w ", "291a4ddb86419421a5f566fb0d7de29b": " n-Tm ", "291a6193e0b6283e1b5f839e68d61a09": "_{metric} \\delta_{ck}^2 = _{metric} \\delta_{kc}^2", "291a70545c58742062ae8e0ff4c9f655": "\\forall X_i \\exists Y_j: X_i \\leftrightarrow Y_j", "291a83de0d29f5071a5042ad101ed16e": "T_b=\\frac{I c^2}{2k\\nu^2\\Delta\\nu}", "291aa2eb092693df0941c2c9edc0dce5": "\n\\Gamma_{\\sigma,K\\sigma}(x,y)\n=\nI*(\\frac{1}{2\\pi \\sigma^2} e^{-(x^2 + y^2)/(2 \\sigma^2)} - \\frac{1}{2\\pi K^2 \\sigma^2} e^{-(x^2 + y^2)/(2 K^2 \\sigma^2)})\n", "291b25b6531fca9f96afd4b4d26e3426": "PVA \\,=\\,A\\cdot\\frac{1-\\frac{1}{\\left(1+i\\right)^n}}{i} \\ = \\ 1000\\cdot\\frac{1-\\frac{1}{\\left(1+.07\\right)^{20}}}{.07} \\ = \\ 1000\\cdot {1- 0.258 \\over .07} \\ = \\ 1000 \\times 10.594 \\ \\approx \\ \\$10,594", "291b84328b4e61c02ee348cd74580d21": " -\\tan t = \\frac{b}{a} \\cot\\phi = \\sqrt{(1 - e^2)} \\cot\\phi = (1 - g) \\cot\\phi = \\frac{-\\tan\\theta}{\\sqrt{(1 - e^2)}} = -\\frac{a}{b} \\tan\\theta.", "291ba2dacaa33653b062ed432db5fa57": "z_k'=z_k-\\frac{F(z_k)}{F'(z_k)}=z_k-\\frac1{\\frac{p'(z_k)}{p(z_k)}-\\sum_{j=0;\\,j\\ne k}^n\\frac1{z_k-z_j}}", "291c02c2774b8c7fa2befe6db00b023e": "\\boldsymbol\\Sigma", "291c455ae03646abfc80a97c512bd96f": "R>0", "291c4f7a3117a2744cd981b9df679563": "A = C", "291ccaa5b8fb4d075679baa3139b6bd2": "p + \\rho v^2/2 + \\rho gy = p_\\mathrm{constant}\\,\\!", "291cfb38f92413d5596c642023287da3": "\\ -\\frac{d[A]}{dt} = 2k[A]^2", "291d6229001a92633bd6e7ca3bdbe434": "J\\equiv\\langle g_1,\\ldots,g_q\\rangle, ", "291d7b2ae4db902bea996c4de1a084ee": "150 \\frac{ml}{kg}", "291d7d48c952d0f238538ef1e2c93697": "\\langle \\cdot |\\cdot \\rangle_E", "291e167621422fcc8c46acd4cfd1edcd": "n\\in\\omega, f(n)\\leq g(n)", "291eb52763bc2636c46ded92c760d422": "\n(n-1)\\frac{s^2}{\\sigma^2}\\sim\\chi^2_{n-1}.\n", "291f0ce6b6884782c8ce3e33548ed9de": "\\mathbf{\\gamma_5}", "291f14cc88d426dbf70889d95d379b3d": "L(v,q)=\\tfrac{1}2\\langle v,Mv\\rangle-V(q)", "291f3c6dc729008ef269839f2e2f5ac6": "\\mu_{M_{J}}", "291f4bb5ed11b6767c2c66152be73284": "\n\\frac{\\sqrt{\\pi}}{2} e^{z^2} \\operatorname{erf}(z) = \\cfrac{z}{1 - \\cfrac{z^2}{\\frac{3}{2} +\n\\cfrac{z^2}{\\frac{5}{2} - \\cfrac{\\frac{3}{2}z^2}{\\frac{7}{2} + \\cfrac{2z^2}{\\frac{9}{2} -\n\\cfrac{\\frac{5}{2}z^2}{\\frac{11}{2} + \\cfrac{3z^2}{\\frac{13}{2} -\n\\cfrac{\\frac{7}{2}z^2}{\\frac{15}{2} + - \\ddots}}}}}}}}.\n", "291f847b9ce3439d881e0b0fdb89fd3e": "|\\hat{f}(a)|=2^{n/2}", "291f88cb6b44aede0762227b53eb4aa0": "x_{i+1}^p \\equiv x_i \\pmod p", "291fc2d6174cc1faa23a3eb29bce16ac": "\\nabla_{x,y,\\lambda} \\Lambda(x , y, \\lambda)=0", "291fe8013e5a5df54ae0a3e599c72f48": "\\log(n/w)", "292040bd0b93ae1610955a6bc08a0c70": " \\mathrm{sinc}\\, x = \\frac{\\sin \\pi x}{\\pi x}.", "2920631e1a42471196209d2b3298e0d4": "\\lfloor x \\rfloor + \\lfloor -x \\rfloor = \\begin{cases}\n0&\\mbox{ if } x\\in \\mathbb{Z}\\\\\n-1&\\mbox{ if } x\\not\\in \\mathbb{Z},\n\\end{cases}", "2920c2d255bf53bc6d66507415c4270f": "\\mathcal{C}^i", "2920d23b3055384057ce288be7ffea59": "\nU_{t+1} (N_{i,j}) = U_t(N_{i,j}) + 2 - \\sum_{N \\in G(N_{i,j})} V_t(N)\n", "2921806082f591b3304e32c2d8cdca65": "2\\omega_i \\over d", "292195c0c5b9f1bf51e736135819a43f": " \\mathfrak g\\otimes_{\\mathbb F}\\overline{\\mathbb F}", "2921a820a9cd16690efd0dccbbf27f8d": "\\mathcal{O}_{X,x}\\otimes \\kappa(f(x))", "2921d0631d6f88e3143486e8fd577639": "S_{i,t}=0", "29224bf896f66036b2262b17a4a0bfe1": "\\mathbf{n}(\\alpha)= (\\cos \\alpha,\\sin\\alpha).", "2922758edbc30cc3fff9fdc1acfa4deb": " x(N_i)\\rightarrow y. \\,", "29229a6ae7e7cabbe087fa75f585ebeb": "y(1)=1", "292330170a1b29a428cb1a599fcffa05": "\\mathbf{U} \\mathbf{I} \\mathbf{V}^*", "29233df7e8990bbb7330db3388142952": " G^{-1} = \\frac1d \\left[ \\left(d+1\\right)I - \\mathcal{I} \\right]", "29237f68f3e06b1e743edd4e18221fbc": "\\mathrm{AgOCN + 2HNO_3 + H_2O \\longrightarrow}", "2923beed5e42cd6c2ca1a820cdf0edbd": "E_7\\,", "292416f0bae8c1ee6758eeb4058b4c84": "\\lambda x + (1-\\lambda )s", "292449054898c976f1e1f40986f47c5b": "\\mu = {G}(m_1 + m_2)\\,\\!", "29245b2a513768a2e41a9280a1d1235b": " \\sigma = \\uparrow , \\downarrow ", "292471e8d2c018f25050961c03b90c9b": " -\\frac{\\hbar^2}{2m}(\\frac{d^2}{d x^2}\\psi +\\frac{d^2}{d y^2}\\psi) =E\\psi", "2924a494c04f46fc75b69e8aa4f69484": " \\theta_k[0] = \\phi_k. \\,", "29251f74c9dbe0812baf49c5cacd2acd": " \\frac{\\mathrm{d} x_i}{\\mathrm{d}t}= F_i (x)-V_i(x)", "292541333210954cdea9a3db307b47ee": "a = b = L/2", "292546742ecb79a7a09526f138e40efb": "\\mathbf{\\Delta}^1_{n}", "2925779ecee11918465920d51fcd3b64": "\\displaystyle \\times J_{n/2+\\delta}(|\\boldsymbol \\omega|)", "2925867c0ec7b5f66b5d214f43e6ec13": "x\\in (a,b)", "292592fd60abc577bf03f55ada5a555b": " \\rho \\sqrt{\\pi} \\, \\mathrm{e}^{-\\rho^2 x^2 / 4} = \\int \\mathrm{e}^{isx- s^2/\\rho^2}\\, \\mathrm{d}s, \\quad \\rho > 0. ", "2925bbac3f6291dc9db9cce6904c3bb2": "\\sigma_{yy}", "29262b3fbb8cd0eb7cb6e71126877d03": "50^2", "29268545bda778f7b461c44e25c25e7d": "\\frac{d u_i}{d t} = \n- \\frac{1}{\\Delta x_i} \\left[ F^*_{i + \\frac{1}{2}} - F^*_{i - \\frac{1}{2}} \\right] \n+ \\frac{1}{\\Delta x_i} \\left[ P_{i + \\frac{1}{2}} - P_{i - \\frac{1}{2}} \\right]. ", "2927545276cfd81445e1ce3b04aea858": "\\ EI {dy \\over dx}\\ = \\int_{0}^{x} M(x) dx + C_1", "292764cf6c9366b6633b42eaa8c9ecb2": " \\sin x^\\circ - \\frac{4 x (180-x)}{40500 - x(180-x)}", "29277be4418837b26e2ff9f2c651e7ce": "\n\\int_{0}^{\\infty} e^{- \\omega x} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\eta x^2 \\right) dx =\n\\frac{1}{\\sqrt{\\pi} \\omega} \\; G_{p+2,\\,q}^{\\,m,\\,n+2} \\!\\left( \\left. \\begin{matrix} 0, \\frac{1}{2}, \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{4 \\eta}{\\omega^2} \\right) ,\n", "29278cf709c8e5f1acd01b2e6b00a90a": " S=\\int^{\\mu_a}_{\\mu_b}L d\\mu=-\\int^{\\mu_a}_{\\mu_b}\\rho d\\mu, ", "2927a3ff535ff07d901538c6a88f4d3a": " f_0 \\ ", "2927aeeb198e7f8a99a04a8df1e44d5e": "V_\\pm=kx", "2927ca2f2959c0b840c6720c10a25e75": "\\langle B,F\\rangle", "2928318549a2643477988da5667f5225": "q = \\frac12 \\cdot\\left(1 + \\frac{\\sqrt{\\sum_{i=1}^M N_i}}{\\sum_{i=1}^M \\sqrt{N_i}}\\right)", "2928b402f6b86569b4d363f685c6e726": " \\begin{align}\nY &= \\sqrt{y} (1_{\\!N} - S) (1_{\\!N} + S)^{-1} \\sqrt{y} \\\\\n &= \\sqrt{y} (1_{\\!N} + S)^{-1} (1_{\\!N} - S) \\sqrt{y} \\\\\n\\end{align} ", "2928e82409da933a00f98c7f598a0164": "G(n^2;x)=\\sum_{n=0}^{\\infty}n^2x^n={2\\over(1-x)^3}-{3\\over(1-x)^2}+{1\\over1-x}=\\frac{x(x+1)}{(1-x)^3}.", "2929392adb0e92275cb920bddde62f66": "x \\mapsto [x]_\\sim", "2929747478093b1e66eae476a683144e": " \\mathbf{v} = \\bold{\\hat{e}}_r \\frac{\\mathrm{d} r}{\\mathrm{d}t} + r \\omega \\bold{\\hat{e}}_\\theta ", "2929ec8b803840e5180435aef7529190": "G = pe + PE - kT[\\ln(n!N!) - \\ln(n - p)!p!(N - P)!P!]", "2929f07151df351cefe0f74fc9bfd0f7": " \\alpha \\in [0, 1] ", "292a3606dc937039e115a232428e574c": "\\phi\\lor\\psi", "292a884afb7f0decbe3298d8ea524373": "\n \\mathbf{F}_1 = \\int_A (\\sigma_{11} \\mathbf{e}_1 + \\sigma_{12} \\mathbf{e}_2 + \\sigma_{13} \\mathbf{e}_3)\\, dA\n ", "292b04e199c59262149feb6231df25b5": "\\omega_1,\\omega_2 \\in \\Complex", "292b752275ba39830c3b4f79bedabebe": "\\mathbf{I}=\\int_V \\rho(\\mathbf{r}) \\left( \\left( \\mathbf{r} \\cdot \\mathbf{r} \\right) \\mathbf{E} - \\mathbf{r}\\otimes \\mathbf{r}\\right)\\, dV,", "292b899320b1048641e3ecbea5fc0c76": "v'=\\sum_{i=1}^d p_iP_iv", "292bdbed75aef7772141d0c068934bdb": "\\mathrm{Cliff}(V,q)", "292beeee007be9bd949047435c0e4c89": "x \\mapsto \\alpha \\cdot f(x)", "292c27ca08795e01cd3ff092eefce474": "s = \\frac{\\sigma + \\tau}{\\sqrt 2},\\qquad\\qquad t = \\frac{\\sigma - \\tau}{\\sqrt 2}.", "292c6cae649bcee60dba1806ffdf1a5f": "CAS=a_{0}\\left[\\left(\\frac{q_c}{P_{0}}+1\\right)\\times\\left(7\\left(\\frac{CAS}{a_{0}}\\right)^2-1\\right)^{2.5} / \\left(6^{2.5} \\times 1.2^{3.5} \\right) \\right]^{(1/7)} ", "292c85ea2f9c2db0624de19cb762988f": "\\begin{align}\n\\int_0^T e^{-xt}\\phi(t)\\,dt &= \\sum_{n=0}^{N} \\frac{g^{(n)}(0) \\ \\Gamma(\\lambda+n+1)}{n! \\ x^{\\lambda+n+1}} + O\\left(x^{-\\lambda-N-2}\\right) + O\\left(x^{-1} e^{-\\delta x}\\right) \\\\\n&= \\sum_{n=0}^{N} \\frac{g^{(n)}(0) \\ \\Gamma(\\lambda+n+1)}{n! \\ x^{\\lambda+n+1}} + O\\left(x^{-\\lambda-N-2}\\right)\n\\end{align}", "292c8b98ea452964edada5557c5a03b6": " x\\mapsto x+b ", "292c9448644c38f64d5fb8a87201b5d6": "c = 1+\\frac{A}{I}.\\qquad (4)", "292cd856afcdd36c910d6e7e58b65416": "\\Delta(t) = -2t+5-2t^{-1}, \\,", "292d75eed6ccba9c98646d0e1346575f": "n \\neq 1\\,\\!", "292dacc09337aa51119d053c2a3f8f5f": "\\sigma = \\sigma_0\\,", "292de4c9ed4553d1b09583c87c3b3315": " V_r = \\frac{2}{9} \\frac{r_p^2}{\\mu} \\frac{V_t^2}{r} (\\rho _p - \\rho _f)", "292e7322fad40ca84cdb01d4189e6543": "G^a_{\\alpha \\beta} = \\partial_\\alpha \\mathcal{A}^a_{\\beta} - \\partial_\\beta \\mathcal{A}^a_\\alpha \\mp g_s f^{abc} \\mathcal{A}^b_\\alpha \\mathcal{A}^c_\\beta \\,,", "292e7e012700b8e5573c4b8b2f99024f": "d \\geq n", "292e81656c9df1ed02ec70d77eb4ace2": "f'(x_*)=0", "292e983f978119219ed398ffea9b0e78": " f(x) \\sim \\sum_{n=0}^\\infty a_n \\varphi_n(x) \\ (x \\rightarrow L)", "292eb625322c318c6c7a920dd21f6fef": "ee_\\text{product}=ee_\\text{max} ee_\\text{auxiliary} (1+\\beta)/(1+g\\,\\beta)", "292ecd51b122246edd2a81cca44a4623": "s_{j,t}^{*}=\\frac{1}{2}(s_{j,t}+s_{j,t-1})", "292eebd5b0cb0c7e4c602c751d96665e": " t\\in [0,\\infty)", "292efef686b2da669c0ac372b703804a": " A_{0} ", "292f23aa44a37df3400c79c80cb70b40": "\\kappa = \\frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}", "292f413b7616f72f863d57fb9a82a8c3": " \\frac{S_2-S_1}{n} = C_p \\ln\\left(\\frac{T_2}{T_1}\\right) - R\\ln\\left (\\frac{T_2V_1}{T_1V_2} \\right ) = C_v\\ln\\left(\\frac{T_2}{T_1}\\right)+R \\ln\\left(\\frac{V_2}{V_1}\\right)", "292fbf846a8feab13426b4beeaaef38b": "r_s = \\frac{2 G M(r)}{c^2} \\;", "292feac8b55587c7dad121f05b52c69c": " \\text{(1)} \\qquad\n bD \\frac{\\mathrm{d}^4w_x}{\\mathrm{d}x^4}\n = q_1(x) - n_1(x)\\cfrac{d^2 w_x}{d x^2} - \\cfrac{d n_1}{d x}\\,\\cfrac{d w_x}{d x}\n - \\frac{1}{2}\\cfrac{d n_2}{d x}\\,\\cfrac{d \\theta_x}{d x} - \\frac{n_2(x)}{2}\\cfrac{d^2 \\theta_x}{d x^2}\n", "2930008b32df801c90a18bb8c32d5845": "\\tan \\alpha=\\frac{\\dfrac{\\partial u_y}{\\partial x}dx}{dx+\\dfrac{\\partial u_x}{\\partial x}dx}=\\frac{\\dfrac{\\partial u_y}{\\partial x}}{1+\\dfrac{\\partial u_x}{\\partial x}} \\quad , \\qquad \\tan \\beta=\\frac{\\dfrac{\\partial u_x}{\\partial y}dy}{dy+\\dfrac{\\partial u_y}{\\partial y}dy}=\\frac{\\dfrac{\\partial u_x}{\\partial y}}{1+\\dfrac{\\partial u_y}{\\partial y}}\\,\\!", "29306b904bfcd45a546bc9781d805ae0": "\\forall n < t\\, \\phi \\Leftrightarrow \\forall n ( n < t \\rightarrow \\phi)", "293081255cf59b8a6a4cf4694bc3365c": "-2C", "293089b84708ceec3debafb55e35eebf": "O(\\log{n})", "2930d7b186bb7cab0c797501aa054e89": "\\mathbf{v} = (v_x,v_y,v_z).", "29310baa780f54ccad3090fafb1bbcb4": " \\sum_{n=0}^\\infty F_n(x) t^n = \\frac{t}{1-xt-t^2}", "293129ca00e30931254ba1b00b881d39": "0 \\le i \\le n \\,\\!", "293141d017cbcac980f871834c7ada87": "(J_1)^2, (J_2)^2, J^2, J_z", "293172d1aa4f2c34dd1fcc190fe8bc90": " V_{Dmech} = V_T - V_{Dphys} - \\frac{PaCO2(V_T - V_D - V_{Dmech})}{P_{ACO_{2}}}", "2931788c29787fa2f7d0ca5b125199f3": "f(x) \\le f(x+y) - f(y)", "29319c6efb81d3f7d45d5f4d474def90": "{1 \\choose 2} = 0", "2931c5fea1bebf45bff90f775fc80766": "{{{9}}}", "2931f6b9797e9b39e598194161e55f7e": "\n\\int_1^M\\frac1x\\,dx=\\ln x\\Bigr|_1^M=\\ln M\\to\\infty\n\\quad\\text{for }M\\to\\infty.\n", "2931f86f7e9301f9e7153edb4aaae133": "t^2 = \\frac{n_x n_y}{n_x+n_y}(\\overline{\\mathbf x}-\\overline{\\mathbf y})'{\\mathbf W}^{-1}(\\overline{\\mathbf x}-\\overline{\\mathbf y})\n\\sim T^2(p, n_x+n_y-2)", "29329f45178d8f738ced1886d12d58a2": "t > 0\\,", "2932ed5765ad13137ae767f1adf9789a": " w = w(r)", "293313058418b73a68e81127b1e145ee": "(a+b\\mathbf{i})(c+d\\mathbf{i}) = ac + ad\\mathbf{i} + b\\mathbf{i}c + b\\mathbf{i}d\\mathbf{i} = ac + ad\\mathbf{i} + bc\\mathbf{i} + bd\\mathbf{i}^2 = (ac - bd) + (bc + ad) \\mathbf{i}", "293396ec32af55d813b7bcbb49c4cb2d": "\\tilde{C}^+\\rightarrow \\tilde{\\ell}^+ \\nu", "2933dbea20923654d3f9acc2f7094509": "\n{\\varphi}_{{\\lambda}_{1}}[Int({I}^{2})]\\cap{\\varphi}_{{\\lambda}_{2}}[Int({I}^{2})]=\\varnothing", "2933e2e70df46551b8d6dc0a9f41eb14": "\\tfrac{1}{2}+\\tfrac{1}{3}+\\tfrac{1}{16}", "29342c18a7d38bef495a97641e4ed166": " \\big( X_1, \\dots, X_i, Y_{i+1}, \\dots Y_r \\big) \\, ", "2934473ecbee019919b5e153d22c945e": "n_{th}", "293454042c53cc0c8389d194c51005a7": "x= \\gamma\\left(1 + v/c\\right) x'. ", "29348a459966fb8609b1d9e6f402750a": "\\tfrac{1-\\rho}{\\rho}", "29348f35a86422145870950cb6367435": "\\Delta x_i =x_{i,1}-x_{i,0}", "2934adcb166ff11727d04057cd5e1833": "(x+5)^2 \\,=\\, x^2 + 10x + 25.\\,\\!", "2934dca8e0b8b5d09b97bd4e8e5f5492": "X_1,...,X_n\\,", "2935119c2f415dd3cc9562cd11a4f6ec": " k_\\text{surf}", "293557b587cbef2634b0f217ac6b64fe": "\\sum_{i,j=1}^5 a_{ij} x_i x_j =0", "29356aaa101db61bda55d37de74cb839": "(p,\\omega_2, q') \\in \\Delta ", "29359d3c3855e09b30cfac86a7049be1": " \\|x\\|_{K(X_0, X_1)} = \\sup_{m \\ge 1} \\Bigl\\| \\sum_{n=1}^m a_n K(x, b_n / a_n; X_0, X_1) \\, y_n\\Bigr\\|_Y < \\infty,", "2935fdc45bc7bb1275a6a5912dfece94": "\\operatorname{excosec}(\\theta)", "293659b48715d04088158dcfc276ae79": "\\frac{\\sin(z)}{z}", "29367371c91a10923b32d1a8368bd98a": "\n\\frac{4+0}{10},\\frac{4+3}{10},\\frac{4+6}{10},\\frac{4+12}{10},\\frac{4+24}{10},\\frac{4+48}{10},.....", "2936a850adb90c22c60649f23860a8ee": "d(x) = \\mathrm{GCD}(p(x),(x^{2l-1}+1))", "2936dec4a9eeced481fad24368cda712": "M/N", "29373cca99c2680ad8d8c7ca709a4209": "x = \\cos(a t) - \\cos(b t)^j", "293781c753971e39e67fa655672737f8": "(0.008 \\cdot t)", "29378edd6c1829a96c0f8a0bff7a625e": "C_V=\\left(\\frac{\\partial E}{\\partial T}\\right)_V=Mk_B,", "2937e3057aa2a4701e78e4b9ce43e79c": " e^+e^- \\to \\eta^\\prime,~~ a_0,~~ f_0,~~ a_2,~~ f_2 ", "293800d9d0a269bc58ba7372ca7ccac0": "E= \\, h \\, \\nu", "293808a6148faf157159bd0c927dd2b6": " V_ {\\omega C} \\, ", "293835a349723dd4872cbc522c002d3f": "\\nabla \\cdot (\\nabla \\times \\mathbf{A}) = 0", "29384bd416e67af13fc145dd6e7afad4": "\\frac{\\pi}{8}", "2938c35a5fb79f44224bcefe6ef10447": "\n\\frac{\\partial X_i}{\\partial P_Y} >0\n", "2938e5d8a1353ddb36bb4471d83e9abe": "\\scriptstyle \\left \\lceil \\log_2(n) \\right \\rceil", "29390f651578055ea5089667bb0373eb": "\\frac{1}{18} = \\frac{1}{2} - \\frac{1}{3} - \\frac{1}{3^2}. ", "2939419410b658835778371c60ec1a96": "\\frac{dQ^N}{dQ} = \\frac{M(0)}{M(T)}\\frac{N(T)}{N(0)}.", "293963d987a6836b57663debb90039ab": "K-L", "2939afe3f2265d04bbe1069815c8e8bc": "EQUI(\\alpha,\\alpha')=1-XOR(\\alpha,\\alpha')", "2939b545d04021022641daf80f9060a8": "F = \\frac {P} {\\frac {1} {2} v} = \\frac {2 P} v", "293b01aa1e7b100d17f18474ff77facb": "u(t,z)", "293b098f4d185a5687c93e0a92386ac3": " k_1 = hf(t_n, y_n), \\, ", "293b20bbe2eb2e5807164601025d81bf": " u(0,x,y)=0, \\quad u_t(0,x,y) = \\phi(x,y), \\,", "293b227a3f3489c0f003726872d08340": "\\operatorname{E}(Y)= \\operatorname{E}(N)\\operatorname{E}(X) ,", "293b481a43ca82f0083b9024f89d2290": " T_{CHUR}=(\\frac{1}{\\lambda})ln \\left[1+ \\frac{\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{sample}-\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{CHUR}}{\\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{sample}-\\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{CHUR}}\\right]", "293ba9537bd511c0af996c508214445b": "Q_1=\\frac{1}{2}\\left[(p-iW)b+(p+iW^\\dagger)b^\\dagger\\right]", "293bd1c8ca8f04858eb6b84292abd331": "P(A > O_j)", "293bd5976bd48dcdcc71b06592ddca25": "\\vdash (p \\lor \\neg p)", "293bda3b898ecf872fa26c966ad39b77": " m_{ox} = \\frac{f_{st} - f }{f_{st}}\\left(m_{ox, 0} \\right) , 0< f_{st} < f , m_{fu} = 0 ", "293bf7ce9d2103cfb0977c40fe1bdc72": " C_{2\\epsilon} ", "293c3da0163f9a32bdd1dd93009dcde7": "x_1,\\ldots,x_n\\,", "293c457cbaf0b0275b607bf58ea35937": " P^{\\mu } ", "293cc250030a54fb148429a2a1054fb8": "\nG^{III} = (d\\Phi- \\delta_{ab}I^a d E^b)^2 + \\Lambda\\, ( E_a I_a)^{2k+1} \\left( dE^a d I^a\\right)\\ ,\n\\quad\n E_a= \\delta_{ab} E^b \\ , \n\\quad I_a = \\delta_{ab} I^b \\ .\n", "293d0536b3311ee8663909034780258d": "X_{ij}", "293d07f855b09ae54e12c89080779dd8": "v_j = \\frac{1}{n}\\sum_{k=0}^{n-1} f_k\\alpha^{-jk}.\\qquad (3)", "293d167e62c2bf69ea1fcd65e6e1d3b6": " P^{\\mu },J^{\\mu \\nu } ", "293d26a2b54b3333912c11111df7beb3": "alive(2)", "293da1aa27c4e9d46ffefecbfb585449": "v_F\\, \\vec \\sigma \\cdot \\nabla \\psi(\\mathbf{r})\\,=\\,E\\psi(\\mathbf{r}).", "293dc92d85c835d67ff9693c4e1a0b92": "X_{c} = \\{ (x,y) \\in L(c) : y \\notin \\mathbb{Q} \\}", "293dd6f4b54b6d6d6b0901edb002988d": "\\sum_{n=1}^\\infty \\varphi(n)\\,\\frac{q^n}{1-q^n} = \\frac{q}{(1-q)^2}.", "293df7c93d10ef4f6a5949097534c1f0": "L^*=116 f(Y/Y_n)-16", "293e02377644a6e2c3ed981dc89ef903": "q = \\frac{\\text{range}}{s},", "293e3cdeb4ed5826e98afe5f57487344": "(\\epsilon nGS_{q})\\,\\!", "293e68c3d6b5a0325da46e61341e5e3c": " n_{k}", "293ea3c591aaf0890c3498611ee3ed37": "\\angle PC_1C_2\\,\\!", "293eeb018eb901b92484ca9c90cf1482": "\\psi': G \\to G^{op}", "293ef95667a77e5797cf23d1cc43cdc7": " \\mu=(\\mu)_{j=1}^J ", "293f59aec7d40ff7a64ad74faba20aaa": " \\sigma_{\\mathbf{A},i}, \\qquad i = 1, \\ldots, r_\\mathbf{A}. ", "293f6c033ec925a24ee943c0f3d579aa": " A \\cong \\bigoplus_{e \\in \\operatorname{min}(A)} A e, ", "293ff78cbcfcef9394b8d9ead90aae32": " \\tau_{P} ", "2940731b88273c3fafffff8a9777a1c6": "\\sum_{n=1}^{\\infin} 1", "2940d4b63e5ada5afac1a7b0ede451b1": " \\lim\\limits_{k \\to \\infty} \\psi_k = 2", "2941bf463add9a364ff28bdefa16ca66": "TS(T_i)", "2941ceac1c752b6380fb9aaaeed008e4": "\\zeta < 1", "2941de190de4872390dcd288afeaea40": "\\zeta_2=\\frac{2}{(1+t)\\sqrt{1+\\epsilon^2}+\\sqrt{(1-t)^2+\\epsilon^2(1+t)^2}}", "29421a09add0305bf1723de034c682f7": "T \\mbox{ is a set of tokens (the basic units of information)} ", "294238f05bae433c3b50100e9cdf2dd1": "\\Pi_{h}(m)\\,\\!", "29426f7d49d82fdc0b3c7496f6095db9": "m=\\int_0^2 (2xy+\\frac{3y^2}{2}+2y)|_x^{4-x}\\,dx", "294317a83b7a370e059c8e237d3c3f65": "{100 \\over \\sqrt{5}}", "294317b4a541e4ab2ab5e0da65e4ab0e": "{K_{BS}}", "2943213061be1bf708f215b56eb70682": "\n\\begin{align}\n& {} \\quad \\Phi(1)+\\Phi(0)-\\sum_\\rho\\Phi(\\rho) \\\\\n& = \\sum_{p,m} \\frac{\\log(p)}{p^{m/2}} (F(\\log(p^m)) + F(-\\log(p^m))) - \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty\\varphi(t)\\Psi(t)\\,dt\n\\end{align}\n", "294385740e2e49aac4542846c9b61dc5": "\\frac{d\\theta_1}{dt}=\\omega_1, \\quad \\frac{d\\theta_2}{dt}=\\omega_2,\\quad \\cdots, \\quad \\frac{d\\theta_n}{dt}=\\omega_n.", "2943b2b51283445c7d54d5915aafa9c8": "\nH(X) = - \\sum_{i=1}^nf(x_i)\\log \\left(\\frac{f(x_i)}{w(x_i)} \\right)\n", "294403d277e3d8d34f97c4206d282c7d": "s=\\frac{\\pi r \\theta}{200},", "294432a50c93fb54ad208ba99b7f7b5f": "\\Pr'(S|W)", "29443d22fa60d7da2ca366ee11a6b38e": " c+\\Delta c", "29444293846c37a05a7098a134be17da": "\\scriptstyle S=\\{0,1\\}^{Z^d} ", "2944624f004b21a81ba5ee35599de050": "(0,n+1)", "2944b8b2e128f5b75c421d3557abe699": "\\left| 6 \\int_{\\mathbb{C}} | \\mathcal{C}_{\\varepsilon} (\\mu) (z) |^{2} \\, \\mathrm{d} \\mu (z) - c_{\\varepsilon}^{2} (\\mu) \\right| \\leq C \\| \\mu \\|", "2944cd1baadf8820eec530a4463ee990": "T_n(x) = \\{nx\\} < 1, \\,\\!", "294540a3dfb032f1b174144b3a0a8f70": "\\Pr(Y\\ge y)\\le {n \\choose y}(1-p)^{y(y-1)/2} \\le n^y e^{-py(y-1)/2} =e^{-y/2\\cdot (py -2\\ln n - p)}=o(1)", "294546a500131f8ac293ad4aaf26d255": "B_\\delta(\\mathbf{WZ}) = |\\mathbf{W}|^{-\\delta} \\int_{\\mathbf{S}>0} \\exp\\left({\\rm tr}(-\\mathbf{SW}-\\mathbf{S^{-1}Z})\\right)|\\mathbf{S}|^{-\\delta-\\frac12(p+1)}d\\mathbf{S},", "294570c6004ae1a20dca25fcd67eff63": "(u_0,\\lambda_0)\\,", "29459ee97e9c09ec5dd9f29f691115b0": " \\mathbf{\\hat T}(\\varepsilon)|\\psi\\rangle = |\\psi_\\varepsilon\\rangle ", "29465d167bea75dee5c7daec109de55c": "\\delta^\\dagger \\circ (\\psi\\otimes 1_A)", "29469ac4fbd9fcfeec9390cef4026c35": "RTI_{20}=\\frac{h}{b} \\times 2924", "29472298d50a0ac838c1c7e187db05d4": "1/\\sqrt{8\\pi}", "29476b753ea559cf052cad95e916c541": "\\Sigma=\\{a_1,a_2,\\ldots,a_n\\}", "2947c6e4d0f4f9adea4c9afea63ccbff": "\\mathrm{u}(t)=\\mathrm{MV}(t)=K_p{e(t)} + K_{i}\\int_{0}^{t}{e(\\tau)}\\,{d\\tau} + K_{d}\\frac{d}{dt}e(t)", "2947f2fcefb34d657bffe9ca17a2f4f6": "x = 2^5 - 2^1 - 2^0", "2948442ee0119a40952e6a90b6b40b01": "\\scriptstyle \\mathcal{R}", "2948453c2caad99ed613daf451f989ed": " g_k(X_k) = \\sum_{X_{\\bar{k}}} g(X_1,X_2,\\dots,X_n)", "29489e883805e0e8831c822a25d120fd": "2+\\left\\lfloor { {n} \\,\\varphi} \\right\\rfloor - \\left\\lfloor {\\left( {n + 1} \\right)\\,\\varphi } \\right\\rfloor", "29491fab17df6efb1955549b53655bb2": "-x^4 +15245x^2-6262506.25=0", "294938c000fdaf9ed70ded86fdd73585": "|H_{jk}|=1 {\\quad \\rm for \\quad} j,k=1,2,\\dots,N ", "294946ab4c45219f74c629d5acfa9d28": "\\hat{H}_{\\textrm{qp}}", "29496677a4f449513f71015be966c41e": "e^{-2\\pi}", "2949b7f0b10306de3524dfb5634d6a55": "\\{y_i\\}_{i=1}^m", "2949cd7538ff2fb58368ad1a4fa94f31": "(z_1,z_2,z_3) \\equiv (\\lambda z_1,\\lambda z_2, \\lambda z_3);\\quad \\lambda\\in \\mathbf{C},\\qquad \\lambda \\neq 0.", "294a08b6d51d7ec0c28e4371dc0b3232": "z_i:=Pz_{i-1}", "294a0f1f0425e0ff841e884db4b552bc": "\\scriptstyle Q_{ext}", "294a37b6e652cff58db912851813b5cd": "\\mathsf{G}(b,c)", "294a4794d7549882922c15b9c774d191": "\n\\tan \\theta = \\sin(\\lambda + \\iota) \\tan(15^{\\circ} \\times t)\n", "294a7f82eb911ddf59387507a3e0d761": "\\operatorname{ad}_XY=[X,Y]", "294b0425012747bd6c131a798f0cf1ff": "J = -\\frac{i\\hbar}{2m}(\\phi^*\\nabla\\phi - \\phi\\nabla\\phi^*)", "294b19c767ebb747f42cb9e128902c03": "\\scriptstyle n \\;>\\; m", "294b3d1285766e6ba5fa89bb5a2ee7ce": "\\pi_3\\colon{\\mathbb{R}}^3\\to{\\mathbb{R}}", "294b7f3aefbd7a1172e1469d8fd37801": "(m+1)(l+1)+d \\begin{pmatrix}l + 1\\\\2\\end{pmatrix} > n", "294b8a72d22cd0e577b3ba80fa3d69c7": "\\dot{x}_1 = f_1(x_1,x_2), \\,", "294b8e7dc7d8731d91e833d536452198": "\\frac{\\sin A}{\\sin_K a} = \\frac{\\sin B}{\\sin_K b} = \\frac{\\sin C}{\\sin_K c} \\,.", "294b8e86731883d20832410b3c947bcb": "v(t)=\\dot x(t)", "294bc70d90db955ef14b799dcc50a3f9": "\\left\\vert \\text{supp}\\left( \\mathbf{A}\\right)\n\\right\\vert ", "294c0d388c64bacf88e7c8a3630b6823": "\n\\begin{align}\n\\chi_{k\\mid k-1}^{0} & = \\textbf{x}_{k\\mid k-1}^{a} \\\\[6pt]\n\\chi_{k\\mid k-1}^{i} & = \\textbf{x}_{k\\mid k-1}^{a} + \\left ( \\sqrt{ (L + \\lambda) \\textbf{P}_{k\\mid k-1}^{a} } \\right )_{i}, \\qquad i = 1,\\dots,L \\\\[6pt]\n\\chi_{k\\mid k-1}^{i} & = \\textbf{x}_{k\\mid k-1}^{a} - \\left ( \\sqrt{ (L + \\lambda) \\textbf{P}_{k\\mid k-1}^{a} } \\right )_{i-L}, \\qquad i = L+1,\\dots,2L\n\\end{align}\n", "294c339936e52cae9fcbf151825bdc5d": "y_{i+1} = y_i + \\frac{h}{2}(f(t_i, y_i) + f(t_{i+1},\\tilde{y}_{i+1})).", "294c35d6ea23aae50e3de6c3426c51c1": "m(\\theta) = C_0\\,\\theta + b_1(\\theta)\\sin \\theta.", "294cd75e54a893eb61d83a8246b47cec": " r \\, ", "294ce890c4a3b9529d80d402420ecd46": "r^2", "294ceda3a27dc78c24a1bb45d9efb479": " |z-b|^2 - |b|^2 + c , \\,\\!", "294db25a677c9b8b563e179477d9385b": "q!", "294ddf436ecf167bdc980d9511dd8d43": "S_{11} = \\frac{b_1}{a_1} = \\frac{V_1^-}{V_1^+}", "294e0631e0d7a600549b185b99d57e98": "\\scriptstyle \\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\\leq 1 ", "294e0e67e3482c95b9ed2e52e05bde93": "R_{\\mathrm{g}}^{2} = \\frac{I}{A},", "294e3e2c69994dc359eeddb5a11dc359": "dC=\\lim_{\\delta x \\to 0}(C\\delta x)=Cdx", "294e96e04778ca3569d0520f93ecedb6": "A\\hookrightarrow A\\oplus B\\twoheadrightarrow B", "294ebac37bab6950ad26d870d57046de": "\nc_1 = 1,\\qquad c_2 =-\\tfrac{2}{3},\\qquad c_3 = \\tfrac{2}{3}\n", "294eec6a8177af125ddede54d40a2937": "\\vec{S}_{t}=\\vec{S}\\wedge \\vec{S}_{xx}. ", "294f09cf3c59786e3399a916e1a14c18": "S_i^\\ell", "294f1053bc456b1b7207f9d2efd8eede": "\\dot{\\psi}", "294f390713e0756aa86134e59db62a19": " \\{ |f_{k_0}^i \\rangle, |f_{k_1}^i \\rangle \\} ", "294f4a83a3a842a9e9929f4fd67c8610": "m = -\\infty", "294f56b2d15043c283f1754b2d825976": "y = t(t^2 - 3) = t^3 - 3t.\\,", "294f64140e6f3cf0e161ee432dc3ce59": "\n\\text{for all process } i:\\sum_{j=1}^{n} a_{ij}\\dot{\\rho_j} = 0 \\;,\n", "294fc7ac225a8db4604137240961bfc6": "s = \\sqrt{\\frac{Ns_2-s_1^2}{N(N-1)}}.", "294fcfe0b57e5322891a3f209537fdf1": " \\widetilde P\\to X", "2950463d03c66e1fa373b5ea101d77dc": "S^{n-1} \\to S^{n-1}", "29504708ed38e907c2cad2a24354a291": "\\omega_3=1/\\omega_2", "2950688a80fc216750d6fff5a31c6247": "K \\colon \\Omega \\to 2^{M}", "2950a03ca2476416f98a235bf40917af": "p(x_1,\\ldots,x_k)=0\\,", "2950b19264901201b8adc0948993265c": "\\text{API gravity} = \\frac{141.5}{\\text{SG}} - 131.5", "2950bfdce098187358e6a52b274d2790": " \n\\frac{1}{\\pi}\\sum_{n=1}^{\\infty}\\frac{\\sin 2\\pi n x \\cdot\\ln{n}}{n} =\n\\ln\\Gamma(x) - \\frac{1}{2}\\ln\\pi + \\frac{1}{2}\\ln\\sin\\pi x - (\\gamma+\\ln2\\pi)(1-2x)\\,, \\qquad 0a+\\varepsilon) \\\\\n &\\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(Y-X\\leq a-X,\\ a-X<-\\varepsilon) \\\\\n &\\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(Y-X<-\\varepsilon) \\\\\n &\\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(Y-X<-\\varepsilon) + \\operatorname{Pr}(Y-X>\\varepsilon)\\\\\n &= \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(|Y-X|>\\varepsilon)\n \\end{align}", "295df0ec140b4cdc8cf61bd54c48105f": "\\eta_\\mathrm{Carnot} = 1 - \\frac{T^0}{T_H} ", "295dfbc881d2a48c5c49cea6beed3dc8": "\\hat{b} \\, \\hat{b}^\\dagger ", "295e03c55d097de3e1ccbfdd619854ad": "\\frac{d^2 L(F)}{d F^2} = \\frac{1}{\\mu\\,f(x(F))}\\,", "295e0c5e7f0c44f3793d2260322e32db": "B_{k+1} (\\mathbf{x}_{k+1}-\\mathbf{x}_k ) = \\nabla f(\\mathbf{x}_{k+1}) -\\nabla f(\\mathbf{x}_k ).", "295e204c52e638376dc2db1d4cbcc9f1": "\\epsilon_n = \\frac{1}{p-1}\\sum_{a=1}^{p-1} \\omega(a)^n \\sigma_a^{-1}.", "295e2ec4a59763f1a58397d118b975b8": "\\frac{\\partial k_{i} }{\\partial t}=\\frac{\\delta k_{i}+2(1-\\delta)m}{2t}", "295e3dcccad50cf2440a4ff0f39f0279": "\\scriptstyle a_1b_1,a_2b_2,\\dots a_d b_d", "295e71e017bdd5e1fe188812c7f70db8": "g_{\\alpha\\beta} = g_{\\beta\\alpha}", "295ec05211ca7f6177d387e788f268a8": "\\big( \\Big( \\bigg( \\Bigg( \\dots \\Bigg] \\bigg] \\Big] \\big]", "295edb9fd0d6c5468a69dadcc17f25da": "c\\in \\R^n ", "295f17a6289edd9b7d4eab641f60d944": "\nM_n=\\frac{\\sum M_i N_i} {\\sum N_i},\\quad \nM_w=\\frac{\\sum M_i^2 N_i} {\\sum M_i N_i},\\quad \nM_z=\\frac{\\sum M_i^3 N_i} {\\sum M_i^2 N_i},\\quad \nM_v=\\left[\\frac{\\sum M_i^{1+a} N_i} {\\sum M_i N_i}\\right]^\\frac{1} {a} \n", "295f4a7c5bde861af018e4f626a50438": "f'(x) = 3x^2.\\!", "295f614533305b7f150f6b290a2af846": "\\displaystyle{\\int_{\\partial \\Omega} |K(z,w)|\\, |dw| \\le C}", "295f6c62cb1b54de8f3fb7ca42b88a06": "a \\triangleright b = ta + (1-t)b ", "295fd4c5adffdee4925878733143937d": "aX_n+bY_n\\ \\xrightarrow{p}\\ aX+bY", "295ff21f17ea0559297bd7754bc4a01f": "\\langle X, \\phi_i\\rangle", "29600f9b306a2ea0b9ece242d13d8d80": "\n \\oint_{\\partial\\Omega} \\boldsymbol{A}~ds = 0 \\quad \\implies \\quad\n \\int_{AB} \\boldsymbol{A}\\cdot d\\mathbf{X} + \\int_{BA} \\boldsymbol{A}\\cdot d\\mathbf{X} = 0\n", "29602944c0c89012ec41166b72856b53": "u_{tt} - u_{xx} = 0.\\,", "2960a358606b6aaf12f4e71a38b9002f": "TRR=(Viewthroughs + Clicks)/Impressions", "2960c5e9428ac317772fae258a28128f": "\\left| a \\right|", "2960ce2a06d503771d547d45edf8da10": "\ng_b(n)=\\sum_{k=0}^{L-1}d_k(n)b^{-k-1}.\n", "2960d4135902c2432b7ae490e7f60d65": "\\mathbf{k}", "2961246ae270353b46fc19632ce63e47": "dS=\\delta Q/T", "296150779bdd21a883cfb284bce270aa": "2 \\times (6+6+9+9) = 60", "29615af881af39504a58085cd32cb5a3": " \\lambda_i(x_0) = \\log \\Lambda_i(x_0).\\,", "2961713c49fcd407a54dea778690502d": "X_{t_2}-X_{t_1}, X_{t_3}-X_{t_2},\\dots,X_{t_n}-X_{t_{n-1}}", "29619a9a3bf838c9caf7560af0b451df": "\\textbf{k}-\\textbf{k}_0 = \\textbf{G}_\\textrm{hkl}", "2961d26b07d61318b281bc7d776182c5": " {u}_{1} (\\mathbf{q})", "2961f649a7553fc301387f44307ede76": "\\alpha = 0 ", "296205b780755bbf462ddbe87aac8a00": "\\mathcal{H}=\\mathcal{H}(\\boldsymbol{q},\\boldsymbol{p},t)", "29622e4f38cbdc7d0f0a9e79176ce0a3": "a+bS(\\mathbf{r},i)", "2962302aedbf0aa2240ddc7b8b216780": "\\int_0^1 x^{\\alpha-1}(1-x)^{\\beta-1} dx = \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)} ", "29629a5fcb9dcaf4a2573fee5ef9651d": "I = \\frac{V}{R}", "29629d0d283e07d875be97e5206761cf": "I[w_t] - I[w^\\ast]", "2962a03148088e0580bdcb1f63566944": "0 < q,m < n", "2962bc6755475f5e37245e97c739bb30": "f(x)=\\frac{x^9-2x^6+2x^5-7x^4+13x^3-11x^2+12x-4}{x^7-3x^6+5x^5-7x^4+7x^3-5x^2+3x-1}", "2962d2eca2aa4228339d79bdf77d46a2": "e^{i(k x + m z - \\omega t)}", "2962f8e8d84adb65cef3ecb04e384b1e": "D_t^{D_e}", "296345a6017d2522cb51803bfb93ffbf": "\\scriptstyle{\\theta_n(t) = -\\frac{1}{\\hbar}\\int\\limits_{0}^{t}E_n(t')dt'}", "296364ac797301a6e3ec0efa998e818d": "H_{D}(x,s) = -\\sum_{i \\in (1\\dots r)} P_{D}(d_i,x,s) \\log P_{D}(d_i,x,s)/10", "2963d09a88619ddffb869aeb61c8f74e": "\\Delta_v = \\operatorname{diag} \\{v_0,v_1,\\ldots,v_n\\},\\qquad \\Delta_\\mu = \\operatorname{diag} \\{\\mu_0,\\mu_1,\\ldots,\\mu_n\\}.", "2963d8d9354b7d51a2067250db358590": "{n \\choose 2} / {I(t) \\choose 2} \\approx 2{n \\choose 2} / {I(t)^2}", "2963f7025ef030071de4503d38942620": "j=0, 1, ..., n-1, n", "29644bd5cf7905a218c59650cfe1d041": "\\begin{align}\\pi &= D^{-1}T^1V^1\\\\\n &= TV/D\\end{align}", "2964828e18620992af14de12f730dc35": "\\mathbf{w}(n)=0", "2964897bebdc3ffcf5f1094dd37d48f5": "dA = -S dT - p dV.\\,", "2964a3e04ad61c1bfd08894ee38793c0": "\\|c'(t)\\|=1", "2964e74ed9a29c634b639b8cf6146372": "(s_i,z_i) \\rightarrow (s, 0)", "29651cf014cb087cd0238009156e113c": "\\operatorname{sgn}\\colon S_n \\rightarrow \\{+1, -1\\}\\ ", "296576aaafc516ac3e53175026bc4ad8": "\n\\begin{align}\n-\\mathbf{q}^2&=-|\\mathbf{p}_i|^2-|\\mathbf{p}_f|^2-\\left(\\frac{\\hbar}{c}\\omega\\right)^2+2|\\mathbf{p}_i|\\frac{\\hbar}{c}\n\\omega\\cos\\Theta_i-2|\\mathbf{p}_f|\\frac{\\hbar}{c} \\omega\\cos\\Theta_f\\\\\n&+2|\\mathbf{p}_i||\\mathbf{p}_f|(\\cos\\Theta_f\\cos\\Theta_i+\\sin\\Theta_f\\sin\\Theta_i\\cos\\Phi).\n\\end{align}\n", "29658d7da409d99b6ff34fbfb5c04262": "\\scriptstyle \\tilde{N}", "29659f3125a9c8cba9c2366fa4ca5352": " |1, ix, 0, 0, 0, \\cdots \\rangle,", "2965f522bafff03e3e5a5851a7b12a17": "\\sigma.", "2966120af2e579172b786c88760407e6": "^{\\rm st}", "296624a25e61e5880c7d863d86af01e8": "\\bold{\\hat{r}}\\psi=\\bold{r}\\psi", "29663bf743d0120f6df29c379710bbfa": "G^{ab}", "2966cccc5ca067c06240bb9527001340": "m_i - m_j = (i-j) \\cdot m", "2966d9c66c6b9471e38eb6ab614ed27b": "M^+_\\infty = \\limsup_{t\\to\\infty} M_t. ", "296726f958d14bcfaa3e0c8e9491e685": "\n\tZ_{CO} = \\sum_{i=1}^m e^{-y_ig_1(\\boldsymbol{x_{1,i}})}\n\t \t + \\sum_{i=1}^m e^{-y_ig_2(\\boldsymbol{x_{2,i}})}\n\t \t + \\sum_{i=m+1}^n e^{-f_2(\\boldsymbol{x_{2,i}})g_1(\\boldsymbol{x_{1,i}})}\n\t \t + \\sum_{i=m+1}^n e^{-f_1(\\boldsymbol{x_{1,i}})g_2(\\boldsymbol{x_{2,i}})}\n", "296776858327899d31c3fe8761e68efd": "a_i/b_i", "29679deb33430dc9a256e529f826f377": "\\hat{\\Psi}(\\omega)=\\frac{2}{\\sqrt{5}}\\pi^{-\\frac{1}{4}}\\omega(1+\\omega)e^{-\\frac{1}{2}\\omega^{2}}.", "2968820dbe977cf77877fc39a7c0139a": "\\begin{matrix}\n1 & 2 & 1234 & 4\\\\\n\\end{matrix}", "29688fee361c308285d244df2d7873d3": "\\Delta z_i", "29689a53a72bacfbd231962aa4d1dce8": "\\langle 0 \\rangle", "29689afb3178687fe391b5dc97fa3107": "\\left(b, z\\right) > \\left(c, y\\right)", "2969230a7f3bf08cf0aeef095ce34023": "\\sum_v d(v) = 2e\\,", "29692464bf2cf6a91fe12b7d66ce42fb": "\\{k ~|~ (\\exists n \\in \\mathbf{N}) \\land (k = 2n)\\}", "2969479e73423a8bedaf3bf99748b288": "t \\mapsto (t^2-1,t^3-t)", "2969866d4a448913d6117dc576806e3f": " \\frac{P(X_1^n(i'))}{P(X_1^n(i))} \\geq 1 \\, ", "2969b560aa1c2010b4df11a53a624e5d": "\\Big( \\pi \\models [\\phi_1U\\phi_2] \\Big) \\Leftrightarrow \\Big( \\exists n\\geqslant 0: \\big(\\pi[n] \\models \\phi_2 \\land \\forall 0\\leqslant k < n:~ \\pi[k] \\models \\phi_1 \\big)\\Big)", "2969e381b654f643bd55a479b84b43a4": "E(x, S, C, \\lambda)", "296a159c3bdc95761203cc8ccd9ae079": "\\mbox{LOP2}=340", "296a19492ee706a3a513f346bc74eea9": "\\mathbf{y}=\\mathbf{W}_L^T\\mathbf{x}", "296ae38d7f7ce705ec9fdd87abe0c36d": "\\sum_{i=0}^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2.", "296b4a5eff095ad9848b4dd6076b0ae7": "x=x_k", "296b5cf05184adc3da637199cecbac70": "\\int_M \\mathrm{d}(\\eta \\wedge \\star \\zeta)=0 =\\int_M (\\mathrm{d}\\eta \\wedge \\star \\zeta - \\eta\\wedge \\star (-1)^{k+1}\\,{\\star^{-1}\\mathrm{d}{\\star \\zeta}})=\\langle \\mathrm{d}\\eta,\\zeta\\rangle -\\langle\\eta,\\delta\\zeta\\rangle", "296b5f592d88a35d5e79653f1655d88c": "p_k = d_k + MCF(G - e_k) - MCF(G)", "296b6e9b3987d1908bfa862757b7d157": " g \\colon [0,1] \\rightarrow [0,1]", "296b7b1398947a89a3bd839693d3b699": "1280 Y' + 1024", "296ba7d1266a91ea6c8dca8791ba101c": "10^{6}", "296ba7e42f3b7817913f4d121029af59": "P_{wc}(\\theta;\\psi)=P_{cc}(\\theta,e^{i\\psi})\\,", "296bb613539c536ebfa973014866b3b2": "r_m = \\frac{2\\sqrt{2}}{3}a \\approx 0.94280904158a,", "296bd3e406e832542d6913b2faf034e8": "(p,q,a)\\,", "296ced34b83e7d0d41d6179160423a4f": " W<\\left(1-\\frac{T_2}{T_1}\\right)Q_1.", "296cee5c68ce118752c96b2d207caeb4": "\\mathbb{V}", "296e687300f6070aaf4e1ea3cb8d25ca": " x^2 + (b/a)x +c/a = 0.", "296e94f6e6a24140febfded4ba9e4cb7": "\n \\sigma_{xx}(x,z) = \\cfrac{z~E(z)~M_x(x)}{D}\n", "296f6add78d1c46dcbf74ad8eb8814a2": "Va", "296f6df26958087f79f9c7f1752a3fc7": "e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\\frac{t^2}{2} [X,Y]} ~\ne^{\\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~\ne^{\\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \\cdots", "296fc6947fdadcd9433988a8f23679bd": "(Q \\and P)", "29701250d25211e5345764edd53464be": "\\mathcal{M}_g", "29703b162051cc1974f463440c8bbe96": "\\frac{d}{s} < \\frac{d}{u} < \\frac{d}{v} < \\frac{1}{3}.", "29704eb8b2b3bf8eedce708a7f8adefd": "b_2=-k_2(x_2-x_1)+(y_2 - y_1)=-1.6875", "297058034a97da0a16cfa618e153905b": "\\mathbf{E}(t) = E_1e^{-i\\omega_1t}+E_2e^{-i\\omega_2t} + c.c.", "297078e591e35dc0b3047de701ed0da0": "\\mathrm{diag}\\left(J_{0,3},J_{i,2},J_{i,2},J_{7,3}\\right)", "29708b7d31697fcdb3aaf6b06da73ab8": "\\Pi_f = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}, \\qquad \\mathrm{det} \\Pi_f = 1.", "297091fb98c36f672bad33e260750f3d": "\\mathbf{r(x)}", "2970a23051a13b00f547d0bbb34342a4": "\\displaystyle BE+BF=DE+DF", "2970ad322a806d37cdf19e08a6841c5d": "\\langle \\epsilon \\rangle = \\langle \\gamma \\rangle", "2970e0f6c05ff4bf363abb4b4b307eec": "E < NJ\\sqrt{\\log 2}", "2971580d912a5c63e602666b10071620": "I(x) = (t_x^- , t_x^+)", "2972656449f9b40f32e671f594cc4c3d": "\\{a, b\\} = ab + ba", "2972669f9a225c61b38d62fe12e3be28": "\\forall p \\in X, \\exists N_{p} \\in T \\mbox{ s.t. } p \\in N_{p} \\mbox{ and } \\left| \\mu (N_{p}) \\right| < + \\infty.", "29727d5c98dfc27ae846755349b9d1db": "V\\operatorname{d}t", "2972826e0b066abbbe3d95ae5d1884a4": "q_0 \\in Q", "2972abe93c94121b50477a599eb65013": "h(x) \\ne h(y)", "2972d1f39d5913dbeaa4b9bf2ea3294e": "m_0 < ... < m_n", "2972edaa6307591100bdb30e62d9642c": "|f(x) - I_C(x)f(x)| < \\varepsilon", "2972edb9764d1d452bc585ccef1c4195": "(ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1.", "2973003e53d8fd0c32eb7d96e2d2b02c": " + \\ln\\left[ (2\\pi m_{\\mathrm{u}} k T / h^2)^{3/2} k T / p \\right]", "2973061327c63f7191c1b5ff5e920151": "\\displaystyle \\exists x_1 \\forall x_2 \\exists x_3 \\cdots Q_n x_n \\phi(x_1, x_2, x_3, \\dots, x_n)", "297377cd1188a7c93eb76886fd449346": "z = f(x,y) = \\,\\! x^2 + xy + y^2.\\,", "2973bcc2e4b57a4bcd9369a7d1813a24": "\\sum_{p \\in P} 2^{-|p|}", "297416e547ece92814101625f9b333e7": "\\textstyle P(E)", "29745043dec70132656654e5d00c2114": "k\\geq n", "29751f19f462bbf7a21c955230bcf142": " \\alpha^{-1}(\\alpha(f(x)))=\\alpha^{-1}(\\alpha(x)+1)\\, .", "297525a24713407e165a29ed8f5f5f2d": "min/cm^2", "297588c39e6efbce5876a33072dd4291": "\\Delta = h\\, \\sqrt{\\frac{4\\,h}{3\\,H}\\, \\frac{c}{\\sqrt{g\\,h}}}", "2975920ac0a53b847ab299688c8ba536": "\\mathcal{L}_f = \\overline{Q}_i iD\\!\\!\\!\\!/\\; Q_i+ \\overline{u}_i^c iD\\!\\!\\!\\!/\\; u^c_i+ \\overline{d}_i^c iD\\!\\!\\!\\!/\\; d^c_i+ \\overline{L}_i iD\\!\\!\\!\\!/\\; L_i+ \\overline{e}^c_i iD\\!\\!\\!\\!/\\; e^c_i \\,\\!", "2975aebdfadcd99a4b739442fcaf79bb": "\n0 < \\varepsilon \\ll k.\n", "29762c4809a999b30e4c0236bc6bbf24": "\\tilde{x}, \\theta, \\alpha", "2976546b98bc210cc9c643931b914018": " \\vec J_2 ", "29766cc8b08684a95a1785e1f0a3efd0": "\\pi(a,x) := x", "2976c80de3897c0e48a30c825198b500": " H = H(q_1,q_2\\cdots q_{k-1}, q_{k+1}\\cdots q_N; p_1,p_2\\cdots p_{k-1}, p_{k+1}\\cdots p_N; \\psi; t). ", "2976f29654737a6ec246bafe2d96c569": "\\nu_{k}", "29770e8288ca16148a22c3249a81a064": "f,g \\in C(X,M)", "29773950662c8d694fa44b196e54dbce": "p_\\varepsilon(s)=s", "29774052aad38b16577833aaa5a9de35": "IL = 10\\log_{10}\\frac{\\left|S_{21}\\right|^2}{1-\\left|S_{11}\\right|^2}\\,", "297760ce3c132bccfb7b17ba414cb415": "S_C(n)=2^n", "29778c3ee3d8f97762d726a8226baad9": "P(E)=1", "29779c59200ee297f14aeb6d3935ac60": "C = \\frac{1}{m}\\sum_{i=1}^m{\\Phi(\\mathbf{x}_i)\\Phi(\\mathbf{x}_i)^\\mathsf{T}}.", "2977f76e14673b9dc1f341443d8a58e1": "2^{2^{n}} +1", "2978021c998785a6f01489c5b0eec871": "P + Q = R", "29789122a44da415985a4fefd2ae254a": "\n J_1\\left( x\\right)\n=\n{a_1 \\over L_b} {1 \\over 2 \\pi r} \\delta^2\\left( r - r_{B1}\\right)\n", "29789caf944216ece56657595ca9145d": "(t,x)\\,", "29792f533a8e651788aa39737229f921": "D_{crit}", "297972eb3f8a9bfb1c3a64f095c524ae": "C(\\varepsilon)=\\lim_{N \\rightarrow \\infty} \\frac{g}{N^2}", "29798b65d20a2705a0f636fd3d0fee9b": "\\Box A^\\alpha = -\\mu_0 J^\\beta", "297995ae95c8d815fffa507a555baa4a": "\n \\left\\{ \\theta \\Big| P \\left[ \\mathrm{Bin}\\left( n; \\theta \\right) \\le X \\right] > \\frac{\\alpha}{2} \\right\\} \\bigcap\n \\left\\{ \\theta \\Big| P \\left[ \\mathrm{Bin}\\left( n; \\theta \\right) \\ge X \\right] > \\frac{\\alpha}{2} \\right\\}\n", "2979c79d42096abf62f6e0889e2107e2": " (\\beta_1 , \\; \\lambda_1)", "297a15d7926199106390913dff3a73ca": "\\Psi _{BETA}(\\omega )", "297abf4d30459b64c62e0be4b042f62c": "\\chi(\\mathbf{x})", "297b0f0b06db01aa54e2ceb195e11ab0": "\\left(\\frac{a}{n}\\right)\\left(\\frac{a_i}{n}\\right)=\\left(\\frac{a\\cdot a_i}{n}\\right),", "297b20fe497e7cb284f25e5766c15d31": "F_0=1", "297b726923b944213b62a0b4130a60f5": "\\overrightarrow{\\mathrm{OA}}, \\overrightarrow{\\mathrm{OB}}, \\overrightarrow{\\mathrm{OC}}, \\overrightarrow{\\mathrm{OX}}", "297b7d57565992652023428b852dd926": "4 \\pi \\,", "297b7e83ef1555425016ca004bf9e124": "d = {\\sum_{t=2}^T (e_t - e_{t-1})^2 \\over {\\sum_{t=1}^T e_t^2}},", "297bbeaa8cfdc35046ffc213b4ebb490": "N \\approx 1", "297be72dc3830cc9f687aef239b8901f": "\\textstyle v(z)", "297befa90860fe809026918e45d1d54b": "h \\leq \\eta,\\,", "297c3872a104e93b38fc118af395adac": "L^{-1} \\xi (x) = \\int dx \\int \\mathrm{d}x \\int \\mathrm{d}x \\;\\; \\xi(x) ", "297c57d3ea9627dcac8a0a12305d084f": "\\Omega = \\sum_{i=1}^{n}\\left(A d\\lambda_i - g_i D \\left( \\sum_{j=1}^{r_i}T^{(i)}_j \\right)g_i^{-1} \\right)", "297cafa3373d074f69ec9ceb2c790956": "(1+2^{-f(n)}2^{B})^{G}-1 < \\frac{1}{6}2^{-n^c},", "297ce6e10c399f2c9c9f5b3c8656c231": "\\hat\\rho", "297cefd50fdb0c3102c89b79f92847c2": "b_i\\ ", "297d33f1e34cd9d9733cf0f22763149d": "s = \\tfrac{1}{2}(a + b + 2c)", "297d6c92233468dda0b24805a6467086": "2^{340} \\equiv 1\\pmod{341}", "297e170b111298ab1a67ee0991969ce6": "X = T^{-1/2} + \\frac{1}{2} + \\frac{1}{8}T^{1/2} - \\frac{1}{128}T^{3/2} + \\cdots", "297e50c28b654cdd93bbb047f8d7c636": "\\Phi_D = Q", "297e5b8e87867f9f851c1373cd360125": " \\operatorname{ker} f := \\{(a,a') \\in A \\times A : f(a) = f(a')\\}\\mbox{.} \\! ", "297e6c5ec950a2c5b5703847ef69f6ff": "\\left(\\nabla_X Y\\right)(m) = d_mY(X) + \\langle X(m),Y(m)\\rangle m", "297ec4016762982ea16ad7b60a49c121": "N_1~", "297f1fa5c65487468338851aa8b5b0aa": "\\exist", "297f281068a6eaeb5cd75cda5fa838ba": "x,y \\in A^{\\ast}", "297f5e350d4ebcd69990829993bcba07": " \\, dS ", "297fdeac90a89bfb26d9e46ce4d9f59e": "\\mathbb{Q}^{\\omega}", "298046673e8ba6ef3112885a578d4ae6": " \\Phi(z/\\sqrt{2})", "2980695d161e7cbba71edff2ffe5f41d": "d(x, a)< \\delta.", "2980732210cf4ca66f48fc94ff3fb888": "(2~4)(1~2~3)(4~5)(2~4)=(1~4~3)(2~5).\\ ", "29809adc8854c5935ee265f13888294c": "\n\\frac{\\partial f}{\\partial A} = - 2A \\sum_i \\sum_{j \\in C_i} p_{ij} \\left ( x_{ij} x_{ij}^T - \\sum_k p_{ik} x_{ik} x_{ik}^T \\right )\n", "2981128a7d0da9a031bec84fb1279241": "f(z) = z^m e^{\\phi(z)} \\prod_{n=1}^{\\infty} \\left(1 - \\frac{z}{u_n}\\right).", "2981c9cdfa92cefc4fc5bb30f1fefb8b": "\\textstyle v\\in P_{k-1}", "29820b9e4fe9e84ecb5ae65facf83700": "\\begin{align}\n\\lim_{n\\to\\infty}\\operatorname{Pr}\\left( \\left |X_n-c \\right |\\geq\\varepsilon\\right) &\\leq \\limsup_{n\\to\\infty}\\operatorname{Pr}\\left( \\left |X_n-c \\right | \\geq \\varepsilon \\right) \\\\\n&= \\limsup_{n\\to\\infty}\\operatorname{Pr}\\left(X_n\\in B_\\varepsilon^c(c)\\right) \\\\\n&\\leq \\operatorname{Pr}\\left(c\\in B_\\varepsilon^c(c)\\right) = 0\n\\end{align}", "2982129bf60db5a7a818fcc3cffcfe7e": "M_\\alpha\\prec_K N", "298225341f913dc3131efd66de412501": "\\begin{align}\n\\{P_X,P_Y\\}(q,p) &= \\sum_i \\sum_j \\left \\{X^i(q) \\;p_i, Y^j(q)\\;p_j \\right \\} \\\\\n&=\\sum_{ij} p_i Y^j(q) \\frac {\\partial X^i}{\\partial q^j} - p_j X^i(q) \\frac {\\partial Y^j}{\\partial q^i} \\\\\n&= - \\sum_i p_i \\; [X,Y]^i(q) \\\\\n&= - P_{[X,Y]}(q,p). \n\\end{align}", "29827a050ec2ec492f8a284ee4b37ad8": "B_I M", "2982c190ff52ec994e56f740e79cfd2c": " \\mathbf{P}={\\begin{bmatrix} p_1, \\, p_2, \\, \\ldots,\\, p_{20}, \\, p_{20+1}, \\, \\ldots, \\, p_{20+\\lambda} \\end{bmatrix}}^{\\mathbf{T}}, \\,\\,\\, (\\lambda < L ) \\qquad \\text{(3)} ", "298317dd5db29fab529d3dd54bba9eb2": "r_a= \\cos u\\,", "298367f9408e82ed582ca32619ca1b26": "a_{1}-a_{14}", "298373800e77a265d14dcdb55fed0d46": "\\left\\{ y~\\backepsilon~y\\succcurlyeq x\\right\\}", "29838baa3f247d1f671f329051d822cb": "\\gamma'|S||T|", "2983da7f4f85c66ca69d709c9e2ecee6": "\\phi_n=\\phi_n(t)", "298427cba26ebae4025a0c53060f4b02": "r = 0, 3k \\equiv 140 \\pmod{167}", "29847eaed22e1c5644b0da0f7cbd7733": "P_RV", "2984993605adaf595ac01918fbc98424": " H_{2^n} ", "29851af092cf7d66e2af5f689f3153d4": "n_{52}", "29858177fc5ffbc23ce8d7d4c4175280": "R(x)=1(x-1)(x+1)(x+2).\\,\\!", "2985d6d700f30d4eb8685ac1b4d8dc70": " Var( m ) = m^2 ( c_1 + c_2 - c_3 + MSE ) ", "298663ba018c0ab7bb6efd3043df2146": "70 = 2mn ", "29866ca37330e7f21b3c289589c491c8": " x(0)=y(0)=1. \\,\\!", "2986d90f74ce5822f7a1d94151ce0c5c": "A\\subseteq\\mathbb{R}", "2986dacb70217d910b0aa9a42bd0e4c7": " = \\dfrac{5(\\sqrt{3} - 4)}{\\sqrt{3}^2 - 4^2}\\,\\!", "2986f363f7b4a9d8ccfac9344c3c5f13": "a \\in \\mathrm{RED}", "2987246605c4546ef569af5d0ff8c0f8": "d = - \\log_{10} \\frac{I}{I_0}", "2987a169a8ccd3aa9993270e8cd15bcf": "\\nu_{k}(\\mathbf{J})", "2987d22eb88b33040c96ea96d3d9add4": "E_\\mathrm{p,m} = -\\mathbf{m}\\cdot \\mathbf{B}", "29885132518bd780052e3f5bb059d714": "\\sigma (a) = b", "2988d1d05f2542f7505a13ee0a0c8033": " \\kappa^{-1} = \\sqrt{\\frac{\\varepsilon_r \\varepsilon_0 R T}{2 F^2 C_0}}", "2988dc1f6920f5700f03118208d3d853": "\\omega3", "298913824d76f9a3a28bc2e0d684058c": "\\frac{1}{\\tau_c} = \\lambda_c = \\lambda_1 + \\lambda_2 = \\frac{1}{\\tau_1} + \\frac{1}{\\tau_2}", "29892aa63b7927620ea20034898b7c40": " \\displaystyle{T(f)T(f^{-1}) -I,\\qquad T(f^{-1})T(f) -I}", "29894549c71105f4244004cce44a59d9": "SU(n) \\supset SU(p)\\times SU(n-p) \\times U(1)", "29895f617f417861b3534caf7a9ef469": "I_A \\otimes \\mathcal{E}^{\\dagger}(|\\phi^{+}\\rangle\\langle\\phi^{+}|) = E_{AB}", "2989924e83c655c3be63497bcb27c24c": "\\sum_{\\beta = 0}^{3} \\gamma_{00|\\beta|\\beta} = \\sum_{i=1}^3 \\frac{\\partial {}^2 \\gamma_{00}}{\\partial {x_{\\beta}}^2} = \\frac{\\partial {}^2 \\gamma_{00}}{\\partial {x_1}^2} + \\frac{\\partial {}^2 \\gamma_{00}}{\\partial {x_2}^2} + \\frac{\\partial {}^2 \\gamma_{00}}{\\partial {x_3}^2} = -\\kappa \\, \\rho_0", "298993e4971b900f8f7e1ae6c229582f": "\\int \\frac{dQ}{T} \\ge 0", "2989957ef3678c4026149a18f934a6b0": "\\mathbf{G}=\\frac{8\\pi G}{c^4}\\mathbf{T},", "2989d6d8753930b47f8d70189d133b4d": "\\tau = R_pC_p", "298a8a9b05d06d7c5547b37b18949914": "\\scriptstyle \\mathbf{E}[\\ln X] = \\psi(k) +\\ln(\\theta)", "298ae558fe1a3bb5aab7fab0bad18e42": "R0", "298b3f79c6077c8db6143a4a2a11e25d": " \\nu = \n\\frac{p+p^2}\n{\n\\frac{1}{n_1}\\{\\mathrm{tr}[(\\tilde{S}_1 \\tilde{S}^{-1})^2]+[\\mathrm{tr}(\\tilde{S}_1 \\tilde{S}^{-1})]^2\\} + \n\\frac{1}{n_2} \\{\\mathrm{tr}[(\\tilde{S}_2 \\tilde{S}^{-1})^2]+[\\mathrm{tr}(\\tilde{S}_2 \\tilde{S}^{-1})]^{2}\\}\n}.\n", "298b47a023ff26625e0069ccdde5cade": " T_{ij}^{(5)}=w_{ik}s_{kl}s_{lj}-s_{ik}s_{kl}w_{lj}", "298b589996732da8f294a8c1a60598ec": "\\alpha=\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial T}\\right)_P\n\\quad = \\frac{1}{V}\\,\\frac{\\partial^2 G}{\\partial P\\partial T}", "298b59b61311971800b0c847574f7d34": "\\begin{align}\nj((1+\\sqrt{-19})/2) &= 96^3 =(2^5 \\cdot 3)^3\\\\\nj((1+\\sqrt{-43})/2) &= 960^3=(2^6 \\cdot 3 \\cdot 5)^3\\\\\nj((1+\\sqrt{-67})/2) & =5280^3=(2^5 \\cdot 3 \\cdot 5 \\cdot 11)^3\\\\\nj((1+\\sqrt{-163})/2) &=640320^3=(2^6 \\cdot 3 \\cdot 5 \\cdot 23 \\cdot 29)^3.\n\\end{align}\n", "298b73296106a81d8f7981b2c3bdb2cd": "\\frac{|SA|}{|SB|}=\\frac{|DG|}{|BD|}", "298ba4c9cfaa832a763e6bdd973cb1bb": "\\Pr\\left(\\bigcap_{n=N}^{\\infty}E_n^{c}\\right) = 0", "298bbced3b43082b191e452807035b64": " \\vec{p}_0 = \\frac{1}{\\sqrt{1-\\omega^2 \\, r^2}} \\, \\partial_t", "298bbed3936772cf15ec81410d52649e": "N_{(i)} + N_{(\\Delta)} = N_{(f)}\\,", "298be7be3882b958a97e6018418deb52": "\\frac{|C|}{|C_v|}", "298c3324e64d39e025ca2a8501f57b93": "[0,L]", "298ca849cc1722cb7e712336d253d9d0": "\\ i = 1,2,\\ldots,n ", "298cb1f6d4ea68a49e0cc3d49b9f9d56": " 1.4 ", "298cbdf9a3bad96f96ea95ce5205d0ed": "E_f (\\%) = \\frac{SV}{EDV}\\times100", "298ce5c9cd110deafa638714a13ec22a": "c_{p} = \\frac {c_{p0}} {\\sqrt {|1-{M_{\\infty}}^2|}}", "298dbd8022c8a94b08ffca2dad57bcd0": " \\sum_{i=1}^\\infty \\lambda_i e_i(t) e_i(s) ", "298ddd9620bc8ee584fff6005ae3519f": " N! = a_1 \\prod_{k=2}^{\\text{length}(N)} a_k!. ", "298e218f93cbe3258f5c954bff99981b": "\\nu = r_p v_p \\int_{t_p}^{t} \\frac{1}{r^2} dt", "298e4329f0931022ac461d9d844b9c23": "\\delta c\\approx c\\,(\\ell^2_P/r^2)", "298e4b31e00e8750f03fe10e778dfca9": "\\iota : A \\hookrightarrow X", "298e50f619f08f384c3f22aa30785695": "\\hat{h}_E(Q)=0", "298eb5586a42b888fe6f33e34689cec6": " \\omega \\mapsto \\omega + \\alpha ", "298eec79cc2cfdcbfc87b6f78c745995": " h'(x) = f'(g(x)) g'(x).\\, ", "298f5cc6a5fca3d46d15df417785258e": "u=0 \\,", "298f7a96990adf6e751c20eb62b93d25": " \\frac{dy}{dx} = 2x \n\\mbox{ }\\mbox{ }\\mbox{ }\\mbox{ };\n\\mbox{ }\\mbox{ }\\mbox{ }\\mbox{ }\n\\frac{dx}{dy} = \\frac{1}{2\\sqrt{y}}=\\frac{1}{2x} ", "298fd78cabaaef1550a22f1775fdbd89": "\nR_1 = 100,\\ R_2 = 200,\\ R_3 = 300\\text{ (ohms)};\\ \\epsilon_1 = 3,\\ \\epsilon_2 = 4\\text{ (volts)}\n", "298ff7ceaaa5eae6814d7d5bcdf0548a": "N_J=0", "299018c8b41e10c1a09e4c56e024ec5e": "\n \\tau^{\\mathrm{face}}_{xz}(x,z) = \\cfrac{Q_xE^f}{D}\\int_z^{h+f} z~\\mathrm{d}z + C(x)\n = \\cfrac{Q_x E^f}{2D}\\left[(h+f)^2-z^2\\right] + C(x)\n", "29902ee5273592b03d5e83c3c9dc36fe": "ax^2+bx+c,\\,\\!", "2990486afbc5fb591045fed63bfd72d3": "\\left( x_i,y_i \\right)", "299063602f3daef9ff88817c4ad20b50": "\\ S(t) ", "29906bd102406dc514b5a81c9bd12d3c": "f, g: \\mathbb{N} \\to \\mathbb{N} \\,\\!", "29909324984f0a2fdf5085b4b2ca8ed5": "x^2 - 92y^2 = 1.", "29909761a60c518c60785d71cc5737c6": "\nf_{tot} = f_{sphere} \\ f_{P}\n", "2990d808ee46a0256b09f8bd28b983ca": "{}^q\\!D_{\\alpha}=\\dfrac{1}{\\sqrt[q-1]{\\sum_{j=1}^N{\\sum_{i=1}^S p_{ij} p_{i|j}^{q-1}}}}", "29910f315f7244d6618e2725797352f1": "b_0=\\left\\lfloor\\frac{\\lfloor\\sqrt{kN}\\rfloor-P_{i-1}}{\\sqrt{Q_i}}\\right\\rfloor,P_0=b_0\\sqrt{Q_i}+P_{i-1},Q_0=\\sqrt{Q_i},Q_1=\\frac{kN-P_0^2}{Q_0}", "29914cbfb0f71dbbcc7170cc3f2266ab": "2^{At(x)}", "299177a1a4abe3d8bedfc1f09b6740e3": "\\{\\mathcal{L}^*g\\}(s) = \\{\\mathcal{L}g'\\}(s),", "299188db95c1a26289783da5ed2970da": " \\left(\\frac{a}{p}\\right) \\equiv a^{(p-1)/2}\\ \\pmod{ p}\\;\\;\\text{ and } \\left(\\frac{a}{p}\\right) \\in \\{-1,0,1\\}.", "29918f00ef8b3d89ae2f853d58e79094": "2^{a-1} 3^{b+1}", "29919f35d6826d4fa46fab3971246e6d": "\\tau'=L_{0}\\left(f'_{x}-f'_{y}\\right)=0", "2991bce87b9ee2a41bdca3c0f11ffcbd": "f^{abc}=f_{abc}", "2991c9c54c48e7057dab1d02c90af8c1": "-L_p N_\\beta", "29925d0cb6e1dffa0590bb7d159aac31": "v^{\\alpha}_{i}, v^{\\alpha}_{i_{1}i_{2}},\\ldots,v^{\\alpha}_{i_{1}i_{2} \\cdots i_{r}}\\,", "2992710439b8ce7967ab3f516b657a07": "N\\cdot x(N) = \\frac{P_0\\cdot r}{1 - (1 + \\frac{r}{N})^{-NT}}", "2992bf5ad6727356d72cafdc4370854b": "\\log S^2(k)", "2992bf9e899d0696ba7407ad0174e2d1": " P_0 = \\sum_{t=1}^T\\frac{C_t}{(1+r_t)^t}", "29930240e9946d58623b1fa895ebbc41": "H^1(X, \\mathcal O_X^*)", "29937bf2b6678cbc3e76e36a83aab94d": "x^{\\mu} \\to x^{\\mu} + \\varepsilon h^{\\mu}(x)", "2993afd280828c962cd55fcbcbf12531": "\\frac {1} {R} =\\frac {2ky}{rd}", "2993bc3c54a317b815d6d99b25d73197": "v(z) = N(z). \\,", "2993fcdd8fcc3198172f7c382e2a9b48": "u(r)", "299494f13c235522fbd14aa1abafe93c": "z_i q \\varphi", "29949651f83255ad642efc5ec04c8ff4": "\\eta = 1 - \\frac{q_C}{q_H} = 1 - \\frac{T_C}{T_H}\\qquad (4).", "2994dcde440045a2a1f332b874d69ab3": "\nK_{d} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{k_{-1}}{k_{1}} = \\frac{[\\mathrm{R}]_{eq} [\\mathrm{L}]_{eq}}{[\\mathrm{C}]_{eq}}\n", "29953f7b76db0e74b9cda05b5ad31538": "z = 6.42", "29956ab11bb3dbd86b1aa75b4d7cb3a0": "c_{3,1}(\\widehat{a}, ab\\widehat{b}ccd, \\widehat{d})", "29960d62748463e432ecc2b937556282": "(w,d)\\,", "29960eaebe5e76d86b2b87502661f0af": "T(t)", "29967fad2a6ac6ddb152d8a085690416": " k1={\\sqrt{1 - k^2}} ", "29969933bdca1dae68ff29ec11dadde3": "Y^m_{\\ell}", "2996c6576aad984b1b74816b98d1e397": "\\lambda(t|X)", "2996c7367c9915d98e97ea7367429ea5": "x = p_i(e^t-1)", "2996cd3f9bf9789c646bcb08fc5d5033": "\n\\mathcal{D}_\\alpha \\mathcal{D}_\\beta V_I = \\mathcal{D}_\\alpha (\\nabla_\\beta V_I + C_{\\beta I}^{\\;\\;\\; J} V_J)", "29972cb0cfd6e8f825bff6896ad7eb75": "R(u,v)=\\nabla^2_{u,v} - \\nabla^2_{v,u} ", "2997331e7df473b3a692f32600079154": "A^{\\circ\\circ}=A", "29973971113c0e221480c373918ea9b0": "(a,b) = 1", "29977c623f57bc0a9c17e32727d473f1": "m+i\\,\\!", "29979e3ff34488093d96105326e8cfa4": "\\mathrm{OR} = \\lambda x^{\\mathsf{Boolean}} \\lambda y^{\\mathsf{Boolean}}{.} x\\, \\mathsf{Boolean}\\, \\mathbf{T}\\, y", "2997c7fe3b027a6f21a661b6f5faaefb": "P_1, \\ldots, P_8", "2997eae62dd1cafd5645f0a84c526f89": "X = \\{ ([X_0:X_1:X_2],[L_0:L_1:L_2]) \\mid P_0L_0 + P_1L_1 + P_2L_2 = 0,\\, X_0L_0 + X_1L_1 + X_2L_2 = 0 \\} \\subseteq \\mathbf{P}^2 \\times \\mathbf{P}^2.", "29982789904d6284dcdd90859b8a9513": "t^a(d,n) \\leq t(d,n)", "29985df4e5dda29b2e573c95f746e659": "\\vec{Z} = \\vec{h}_0", "29988f0c67f0c7e2b78aa5f12b013286": "\nx = \\sigma \\tau \\cos \\varphi\n", "299899a5f409e5b6b87c2f91d277bc55": "B\\in K", "2998aaabf72fb563909a308d4c8292c6": "{ L }_{ \\bigodot }=\\left( \\sigma { { T }_{ \\bigodot } }^{ 4 } \\right) \\left( 4\\pi { { R }_{ \\bigodot } }^{ 2 } \\right) ", "2998d3dc61c6707f3f87cada4bf28ba4": "\\approx 1.57080", "2998e2124b58268125618f4aa3790514": "m[w]", "2998e698de522752f9dc89571f015ab9": "P_{r}", "2998f634e7026c1b4e12b18f2a232112": "\\alpha(1)", "29991949962575015aed17d0d5baaf96": "(a/g)", "2999be8892bd78085ebae093daba73dc": "\\mathfrak{q} \\cap A", "2999d8caab51ca57ca0a23b8d3fcc613": "2\\sqrt{2} \\operatorname{erf}^{-1}(1/2) \\approx 1.349", "2999db4bb032cac2a3704ce1f04335e4": "(21)\\qquad \\pi\\,\\hat{=}\\,\\alpha+\\bar{\\beta} \\,,\\quad \\varepsilon\\,\\hat{=}\\,\\bar{\\varepsilon} \\,.", "2999ebe407dcccc1555047d201d0a31a": "\\int\\sin b_1x\\sin b_2x\\;\\mathrm{d}x = \\frac{\\sin((b_2-b_1)x)}{2(b_2-b_1)}-\\frac{\\sin((b_1+b_2)x)}{2(b_1+b_2)}+C \\qquad\\mbox{(for }|b_1|\\neq|b_2|\\mbox{)}\\,\\!", "299a0abf26659d83281db31e235e2af8": " \\theta \\rightarrow \\theta + e\\alpha\\,", "299a0f6681b038b8f69e6334c86d275d": " Z_C = {1 \\over Cs} \\iff Y_L = {1 \\over Ls} ", "299a2efcaec96dd6cebcde7a3b363a51": "\\Delta M_J = 0, \\pm 1", "299a2fbeb41a10abc78e033df0dd7f14": "V_+ = \\frac{-e^2}{4\\pi \\epsilon_0} \\frac{1}{r} + \\frac{h^2 (l+1) (l+2)} {2m} \\frac{1}{r^2} + \\frac{e^4 m}{32 \\pi^2 h^2 \\epsilon_0^2 (l+1)^2}", "299a84602ef520e0ae4ad6a9a632b4d7": "f (\\psi(x)) = \\langle y, y \\rangle - \\langle z, z \\rangle", "299a9a20e70b23605025bf0b9f7c74d5": "z = r \\tanh(m (\\lambda-\\lambda_0)).\\,", "299aa6bcce7f8267cabedd29730adbde": "l = \\sqrt{ d^2 + {r_1}^2 + {r_2}^2 - 2 \\, r_1 r_2 \\cos(a)} - r_3", "299b3fb5d5dc1b172c568fd8ec707b71": "\\bar{\\theta}_\\mathrm{Jack}", "299b51d9edaa1b5d6512ad1981a6a9db": "r_i = x^\\prime_i/x_i", "299bf85af317fec3af49aab857fa5e94": "\\ 4\\frac{fL^*}{D_h} = \\left(\\frac{1 - M^2}{\\gamma M^2}\\right) + \\left(\\frac{\\gamma + 1}{2\\gamma}\\right)\\ln\\left[\\frac{M^2}{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)}\\right]", "299c15f2a7b3c8e9a4ffc0b95340b494": "x(N-1)", "299c7b425e4c7fb7004a30bbc4781497": "\\bold{A}:\\bold{B} = \\sum_i\\sum_j A_{ij}B_{ij} = \\mathrm{tr}(\\mathbf{A}^\\mathrm{T} \\mathbf{B}) = \\mathrm{tr}(\\mathbf{A} \\mathbf{B}^\\mathrm{T}).", "299c7c06835c26bee41471e5a0ea879e": " H_n^{(r)} = \\frac{1}{n!}\\left[{n+r \\atop r+1}\\right]_r, ", "299c7d21eae3146ea87100792d21f062": "q^{i}(\\xi,\\tau)\\wedge q^{j}(\\xi,\\tau)=\\xi ^{i}\\wedge \\xi ^{j}=- {I}^{ij}.", "299d2619c5a66147e6c61f44c6b567dd": "s = |\\mathbf{a}|\\cos\\theta = \\mathbf{a}\\cdot\\mathbf{\\hat b},", "299d3c6b3db3c43642bcb7d3f395e418": " O_n =x_1\\cdots x_n ", "299d4dfe938dffc1429b4e41f83cbcfe": "\\scriptstyle\\{e_{(a)} = e_{(a)}^{\\mu} \\partial_\\mu\\}_{a=1\\dots4}", "299d7398d257123ebccd0d15f015cb2a": "p_{i|j}", "299d7c8336e4904432ccb3375874ea64": "\\log_{10} (2) = 0.301", "299daa9eec27df5a815f8039f94ddcaf": "E_0 = E[n_0] = \\left\\langle \\Psi[n_0] \\left| \\hat T + \\hat V + \\hat U \\right| \\Psi[n_0] \\right\\rangle", "299e341fb4ec0ce161a3a37fb241c6b9": "\\mathbb{R}\\cap\\mathbb{\\overline Q}", "299e79cf84e4abe380758de8b81a368f": " f(r)=1+\\frac{r^2}{\\alpha^2} ", "299ebc7c637d7a9fd4a530aeff87781b": "\\varepsilon_{ijk} \\varepsilon^{ijk}=1", "299ef052a6ef4ec69b07a26a5f4d3b7f": "A\\times -", "299f5f1ac13f0bf406e8de36c47f9cc8": "\\overline{x + iy} := x - iy", "299fc36b761e3ffd8d046b81060bfd84": "\\scriptstyle s[n]", "299fd858030052a55e2a4dd3f69e5642": "\\alpha\\leq \\beta", "29a033adab5f6d9dabb8d5473269d093": "n_\\eta(-\\xi)=-\\eta-n_\\eta(\\xi)", "29a0375271063dc45f57c5cc195f8343": "\\tilde X = X \\cup e^1 \\cup e^2", "29a068f3ebe093b6492b2adae8e380e2": "\\displaystyle u(x)", "29a071467e7df9285af0c818b08855c9": "2n-k", "29a0f81bc3224d1695da750e454cb974": "\\frac{x_1 + x_2 + \\cdots + x_n}{n}.", "29a12a627158d1bfc5dc129d2495afd1": " [(a_{32}-a_{23}),(a_{13}-a_{31}),(a_{21}-a_{12})] \\,\\! ", "29a131ce6dd3486c64f9ea2390abae10": "\\psi(\\Omega+\\psi(\\Omega+\\psi(0)))", "29a1786c623c1e298d472daa300d6bf1": "\\frac{e}{m} = \\frac{4 \\pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\\frac{\\delta D}{D\\Delta D} \\ .", "29a19284620dd44aa204dfe301f8b1b3": "\\varphi(\\mathbf{r}, t) = \\dfrac{1}{4\\pi\\epsilon_0}\\int \\dfrac{\\rho(\\mathbf{r}',t)}{|\\mathbf r - \\mathbf r'|}\\mathrm{d}^3\\mathbf{r}'", "29a1c855139579289d7a2f367fa7f995": " x = -\\infty ", "29a2292ccf7e40beb0f9e4b26d5bd67e": " \\ c\\textbf{g}+c\\textbf{f}-qK \\pmod p ", "29a2a1a62b21a2efc5b059f4fe2f6d71": "\\forall i,j,k,l \\in \\left\\{1,2,...,m\\right\\}, a_i-a_j=a_k-a_l \\iff i=k \\and j=l.", "29a2ab6c0c1c8550e5f64caf074da3fc": "G(F)", "29a2d11fe0468f2cb52489ab392882da": "g^\\prime \\sim g", "29a352056e8d69f1ae5192cd42085a0e": "f:\\sigma\\rightarrow X", "29a3560e02cc48c5b870763114dacf35": "\\ln(\\phi(q))=\\sum_{n=1}^\\infty b_n q^n", "29a3c08d0f009fe4e336562ed946de7e": " k_i ", "29a415b2dce5f2ec448a8854e926eb1e": "\\scriptstyle W_p^l", "29a42eee21608e6c4e85903b70051b60": "FH", "29a4364cc27c582bf3425312fd6354d0": "r_B(n) = n^{1/2 + o(1)} ", "29a4a24abd0cb4abac843bd001b19a70": " Pipe Diameters = 4.4D \\left [R_e \\right ]^{1/6}", "29a50004d3fb30b61ca09d509bd397f5": "c (S,T)", "29a51c2fc6311a4b965763dbfe8d37d0": "\n\\begin{bmatrix}\nx_0+x_1 & x_2+ix_3\\\\\nx_2-ix_3 & x_0-x_1\n\\end{bmatrix} =\n2\\begin{bmatrix}\nz\\\\ w\n\\end{bmatrix}\n\\begin{bmatrix}\n\\bar{z}&\\bar{w}\n\\end{bmatrix}.\n", "29a597831a7b137abc0a20f138a18156": "\\chi_{sp} = \\tfrac{n \\mu_0 \\mu^2}{3k_BT}", "29a5d6e72e8070afa3b474d732b0d39e": "P(n\\mid N) = \\frac{1}{N}", "29a6e03dc9c2e02efe4f8c895bc4e32f": "\\!\\alpha^2 \\approx 5.32\\times10^{-5}", "29a70930761c10f2056193a2607a7e60": "O(n k^\\alpha\\,\\log(n)^\\beta)", "29a70e8dea30c564974fd107e672b9c8": "\\Delta \\tfrac{W}{L}\\equiv \\frac{{{W}_{2}}}{{{L}_{2}}}-\\frac{{{W}_{1}}}{{{L}_{1}}}", "29a7389faceedd983be4746a15a41775": "h_\\eta^{(2)}(z)", "29a76e800808fed5b30af14c69cbf558": "S = a x_+^\\alpha + b x_-^\\alpha", "29a821766dc3f48a94e4c91320522c58": "\\lambda\\vdash n", "29a849ca01f3300b8e3501d49366af07": "\nR_D = \\Bigg[(n+1)^{(n+1)} \\prod^n_{i=0}{(R_{F(i+1)}-R_{Fi})\\Bigg]^{\\frac{1}{n}}} \n", "29a8aa94e6eeb42b2de9d207b80f5941": "|A| =_{\\mathrm{def}} \\{B \\mid B \\sim A\\}", "29a8e3efc3134c43017bebbd432a4ccf": "V(r) = \\frac {2}{3} \\pi G \\rho (r^2-3R^2),\\qquad r\\leq R,", "29a9050945ce2654a41ed3c58bd81dd1": "x^3 + px = q \\,", "29a90c3466dd60e655f0c7c75791a01e": "\\pm \\hbar", "29a9207db3e097c7beec06fb7bab133e": "(pq|rs)\\in\\mathbb{R}", "29a92d03b6876ec9e7f0e082aa0c7302": "\\omega_{\\alpha \\beta}^{\\;\\;\\; IJ}", "29a9a3a3be62e5b4f7d1fef0adfb3097": "Q_z=\\frac{4\\pi}{\\lambda}\\sin\\theta=2k_z", "29a9a6b9753d4ee46a0570b2bec65f56": "n \\sim \\text{Poiss}(s+b)", "29aa22ec3d3bf3609e8faba7166b8321": "\\Phi_B", "29aa5fbcc440a321158aaf9af49d405c": "\\textstyle\\mathbf{IPC}+\\bigvee_{i=0}^n\\bigl(\\bigwedge_{j0", "29abc36f3a21655bedcf9866fb753dfb": "S[\\gamma] \\ge r\\int_a^b|\\theta'(t)|\\,dt \\ge r|\\theta(b)-\\theta(a)|.", "29abd0026e992df3eb99d32327f05047": "t_i = s_j", "29ac0a962cb46af35d366942a64b9592": " Q \\bold{v} = \\lambda \\bold{v}, \\,\\!", "29ac0da90074e77d80f59d3abbe0f647": "\\rho (L) = \\{ \\lambda \\in \\mathbb{C} | \\lambda \\mbox{ is a regular value of } L \\}.", "29ac5e34ee9208af7d5778a4c2f76cc7": "P_r = {{P_t G_t}\\over{4 \\pi r^2}} \\sigma {{1}\\over{4 \\pi r^2}} A_{eff}", "29acc5c30288c32a4df628d6890e617f": "P,aP,bP,cP", "29ad2e125e9c54c6a0886b0128f5446a": "\\vec x \\in \\mathbb{R}^d", "29ad3b5cbd560ab226939c72ef8898a1": " f(N+\\tfrac{1}{N}) - f(N) = N^2 + 2 + \\tfrac{1}{N^2} - N^2 = 2 + \\tfrac{1}{N^2}", "29adc6f138008786753940782d7767e4": "\\operatorname{Spec}A", "29adf134fcdf1ac67eb7e7f1e9bf856c": "\\displaystyle \\hat{f}(\\nu_x,\\nu_y)=", "29ae56505caed64e4a633ac1f2c103d8": "\\frac{r_1(t) - a_1}{dr_1(t)/dt} = \\frac{r_2(t) - a_2}{dr_2(t)/dt} = \\cdots = \\frac{r_n(t) - a_n}{dr_n(t)/dt} ", "29ae5c157afa814e817986d828310e7e": "e_1(t)=\\dot\\gamma(t)/|\\dot\\gamma(t)|", "29aeda5edc922fd393d4491f984a6029": "\\sigma(\\mathbf{x})", "29aee2f5a89c54a9c930558cc3ed0048": "\\lfloor x\\rfloor=\\left\\lfloor x+\\frac{k'-1}{n}\\right\\rfloor\\le x<\\left\\lfloor x+\\frac{k'}{n}\\right\\rfloor=\\lfloor x\\rfloor+1.", "29af18978a7f9242822cf79cd30ee44e": "\\Delta\\tau_v = - \\frac{1}{2c^2} \\sum_{i=1}^{k}v_i^2 \\Delta t_i", "29af5ac141fd817074b23b97edefb0df": " c(r)=e^{-\\beta w(r)}-1+\\beta[w(r)-u(r)] \\, \n= g(r)-1-\\ln y(r) \\,\n= f(r)y(r)+[y(r)-1-\\ln y(r)] \\,\\, (\\text{HNC}). ", "29af847b0ebfc95fb0195320e799f4a3": "\n A = \\cfrac{6~c~\\cos\\phi}{\\sqrt{3}(3+\\sin\\phi)} ~;~~\n B = \\cfrac{2~\\sin\\phi}{\\sqrt{3}(3+\\sin\\phi)}\n ", "29afbc9975444371e8c48fb436cb0830": "F[y]=\\prod y", "29afc5ae50efa09eaf732a35a9f0bcd7": "player \\leq score", "29affef422d55808197d8a961e9598d8": "N:=\\bigcup\\nolimits _n \\left\\{p\\in \\mathrm{Supp}(\\mu): (T_0U_n)(p) \\sigma_{i+1} \\}", "29b6518844fd31c574e5bfbf2684e2ca": "V(A \\lor B,0) \\Leftrightarrow V(A,0) \\ and \\ V(B,0)", "29b675d2ebb243e9c32090e18b031bac": "\\frac{1}{\\tau^*}=\\frac{p\\mu_p\\tau_p+n\\mu_n\\tau_n}{\\tau_p\\tau_n(p\\mu_p+n\\mu_n)}.", "29b6925e7a5686a742ea70fe47a8688d": "\\mathbf{P}(\\omega)=\\varepsilon_0 \\chi(\\omega) \\mathbf{E}(\\omega).", "29b6ab9cc222880da4dec87b34a046be": "O(A_1:A_2) = \\frac{P(A_1)}{P(A_2)}", "29b6dc638161d180edfb11032e5573d4": "\n\\frac{\\partial^2 F^m_{~\\alpha}}{\\partial X^\\beta \\partial X^\\rho} = \\frac{\\partial^2 F^m_{~\\alpha}}{\\partial X^\\rho \\partial X^\\beta} \n \\implies \n \\frac{\\partial F^m_{~\\mu}}{\\partial X^\\rho}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} + \n F^m_{~\\mu}~\\frac{\\partial }{\\partial X^\\rho}[\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta}] = \n \\frac{\\partial F^m_{~\\mu}}{\\partial X^\\beta}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\rho} + \n F^m_{~\\mu}~\\frac{\\partial }{\\partial X^\\beta}[\\,_{(X)}\\Gamma^\\mu_{\\alpha\\rho}]\n", "29b6e122577d3441925b26874f745868": "A_\\sigma \\hat{\\boldsymbol\\sigma} + A_\\tau \\hat{\\boldsymbol\\tau} + A_\\phi \\hat{\\mathbf z}", "29b7342da0e216a9d99a222895d1620b": "Z_K\\,", "29b754ae9147c0635423b21816875c1b": "(\\beta_1^\\top X,\\ldots,\\beta_k^\\top X)", "29b777239effd2bc18a36620d27c7252": "\\pi_{k+1,k}:J^{k+1}(\\pi)\\to J^k(\\pi)", "29b81101892c51a370a761e93b5026e2": "\\Sigma^x =\n\\begin{pmatrix}\n \\sigma^2_1 & \\text{cov}_{12} & \\text{cov}_{13} & \\cdots \\\\\n \\text{cov}_{12} & \\sigma^2_2 & \\text{cov}_{23} & \\cdots\\\\\n \\text{cov}_{13} & \\text{cov}_{23} & \\sigma^2_3 & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\end{pmatrix}\n", "29b82d2bfe884b230ecba64782f336a9": "\\int_0^\\infty e^{-st} f(t) \\, dt,\\quad \\int_{-\\infty}^0 e^{-st} f(t) \\, dt", "29b88c27df6aea000d29635d7f0192ce": "\\beta_0^{(1)} = \\beta_0^{(0)} (1-t_0) + \\beta_1^{(0)}t_0 = \\beta_0(1-t_0) + \\beta_1 t_0", "29b9c3b31d0a78aa123273f664b8710f": "l = \\sqrt{(d-r_3)^2 + (r_2 - r_1)^2} .", "29ba2e66ff6d8038690f7fc9cbbfd052": "\\binom{x}{k}", "29ba32847ba5f357f4dc7e880316dc1e": "\\mathbf{P}(0)", "29ba48a400b99bc1af1a042ef2271a3e": "\\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}} = \\frac{1}{1-2^{-s}}\\cdot\\frac{1}{1-3^{-s}}\\cdot\\frac{1}{1-5^{-s}}\\cdot\\frac{1}{1-7^{-s}} \\cdots \\frac{1}{1-p^{-s}} \\cdots", "29ba6d08667038089a1d5b7759bdfe80": "\n\\frac{\\partial}{\\partial t}\\left(u_e + u_m\\right) + \\nabla\\cdot \\left( \\mathbf{S}_e + \\mathbf{S}_m\\right) = 0,\n", "29ba837f6917c895e85d2ac0530a3712": " L = \\frac{\\lambda}{i}", "29bad86b3d1e30ce154cfdf6edfbc7cd": "Q^{n+1}", "29bae56585ca839094d976ab6e20a8f8": " \\sigma_{\\mathrm{discr}}(T) = \\sigma(T) \\setminus \\sigma_{\\mathrm{ess}}(T). ", "29bb6dbc797a1d4733c1a383bdc43ec0": "\\|f\\|_{H^\\infty} = \\sup_{z\\in\\mathbf{H}}|f(z)|.", "29bc15db3f59e793965861197514c7c6": " \\frac{\\partial \\bar{u_i}}{\\partial x_i} = 0", "29bc1e76ad879b2010bf9e616feb8d62": "\\vec{Y}", "29bc1f2c10458af64198bfbfddb4db12": "{}^{4}i = i^{\\left({}^{3}i\\right)}", "29bc535dc12c70c70d2634b0469e51bd": "(\\overline{A} \\vee \\overline{B} \\vee \\overline{C}) \\wedge (A \\vee C) \\wedge (B \\vee C)", "29bccee4c2e0ccdeeebd9cbf0dc3efa6": "\n \\boldsymbol{\\sigma} \\approx 4C_1\\left(\\boldsymbol{\\varepsilon} - \\tfrac{1}{3}\\mathrm{tr}(\\boldsymbol{\\varepsilon})\\boldsymbol{\\mathit{1}}\\right) + 2D_1\\mathrm{tr}(\\boldsymbol{\\varepsilon})\\boldsymbol{\\mathit{1}}\n ", "29bce09baa6903ac6347ae366bdbaf08": "|\\Im(z)| >2 ", "29bd1155882b2200f0e4c900bf0013e2": " \\kappa < \\lambda\\,", "29bd1c430e6310b139175821685db160": "{BC}^{2}+{AC}^{2}={AB}^{2} \\ .\\,\\!", "29bd1dff25998fe3eb500c626b782925": "X(y+d,t) - X(y,t)", "29bd2535fe70780a9b40325ea03cbc20": "\\mathrm{C^{\\beta}}", "29bd68a263458b967201104f57cf62aa": "|D\\rangle", "29bdda7a25b4b34bc63f2504fcdcd45f": "[n]_q=\\frac{1-q^n}{1-q}.", "29bddacb40c90e334700868b889400aa": "C_n^2", "29be3503303f17e56d238129eacab0ba": "Z_i \\sim \\mathcal{N}(0, 1)", "29becc720fb9ef8335a19e17843eb9ce": "\\sum_{\\Omega(n)=2} \\frac{1}{n^2} \\approx 0.1407604", "29bece9915c73278a45289e922cd4f1b": "x_q^{(q-1)/2} \\equiv 1\\pmod{q}", "29bed172e783845d4feb8f1d116d628b": "\n \\|\\varphi\\| = \\sup_{\\|x\\| \\le 1 } |\\varphi(x)|.\n ", "29bf0599e94766343ee7d0fff43780d4": "r_{p}=(1-e)a", "29bf28029d7ea6bab8e8063369d72e5c": "\\Phi(x,t):=\\frac{1}{\\sqrt{4kt}}\\exp\\left(-\\frac{x^2}{4kt}\\right)", "29bf432841549d61bd87ac91336fa451": "C^i{}_{jkl}", "29bf6831dbf3e375977ec61bb3130224": "\\coprod", "29bfbac3406724739d5d5a41832ee372": "f(T) = \\frac{1}{2 \\pi i} \\int_{\\Gamma} \\left( \\sum_{n \\geq 0} \\frac{T^n}{\\zeta^{n+1-k}} \\right) d \\zeta ", "29bfe6c6b36c1ce31ca3fbcfb6f7a021": "\\det\\begin{pmatrix}\nu + kv\\\\\nw\\\\\n\\end{pmatrix}\n=\\det\\begin{pmatrix}\nu\\\\\nw\\\\\n\\end{pmatrix}\n+ k\\det\\begin{pmatrix}\nv\\\\\nw\\\\\n\\end{pmatrix}", "29c0c3443dde5e4b3d2f1c1de9f67ec3": " \\{Z_T \\mid \\text{T a finite valued stopping time} \\} ", "29c0d7c30f806c4c3b52dc3418158680": "A\\approx1.2824271291\\dots", "29c103f223ec9867b9db77b4842cfd20": " [t]_q=\\frac{q^t-1}{q-1}. ", "29c11585248d2b9ba404b7356b81d7ac": "^\\top", "29c1590bcfe1c907116a45a9f499bdb4": "\\displaystyle{gZ=(AW+B)(\\overline{B}W+\\overline{A})^{-1}.}", "29c15a6652271b30a3f23e05303dd6ac": "-V_{YY}^{-1}", "29c1db4ec652f87c2cbe138ff66bd9e3": "w\\in\\{0,1\\}^*", "29c27250401a3bd055667e64bc4fe4d3": "G(\\mathbf{k},z)", "29c30dda15ca3ad33ff32c1cde78d049": "H^3 = \\{ q \\in M: q(q^*)=1 \\} \\!", "29c3746d395b0d422cc0e02bc412523f": "\\Delta v/c=10^{-5}", "29c4690d7aefe8e74c25c0efd2926f8e": "e^{-i\\omega t}\\phi_\\omega(x)\\,", "29c4c8e3126ea02838b76dac205f8d77": " x + x^3/3!+...", "29c4e6a51a97c34fa1ec0719e0bfe3fe": "\\left(1/b\\right)\\ln\\{\\beta\\left[\\left(1/2\\right)^{-1/s}-1\\right]+1\\}", "29c4e7235bbbd18b6b9ea346959a18e7": "y = x^2.\\ ", "29c4fc430ed5674f0534ba379e026551": "\nK_\\max = \\underset{(i,j)\\in S}{\\max} K_{ij} \\,\n", "29c537605144f52e39729ef506b75531": "\\frac{W_{Sieve}}{W_{Total}}", "29c5578e78c66cece596aa101d366b32": "\\ Pxx.", "29c5c9eea7260b1be7aac58b135097c2": " E = \\frac{Z}{2 \\, \\sqrt{n}} \\, = \\frac{4.4172}{2 \\, \\sqrt{12000}} \\, = 0.0202 ", "29c605e88b9e17013e81fd0222cff374": "*\\exp(1)*x^{11}+\\frac{150349}{6227020800}*\\exp(1)*x^{13}+", "29c62cec19b0ffa0852baf4677d211d5": "\\sum_{i} u_i", "29c6553e3657ba1d462d5f9f83480167": "\\lambda(\\mathbf{x},\\mathbf{y})", "29c662f78660c43b2d1ff32c3919e0ae": "f(aa)", "29c6777fae17c1fe92b9001c10b41a54": "s_a^*(t),", "29c6b5f8c89dc12061c9159c6fb941de": "\\scriptstyle \\mathbf{X}\\sim\\mathcal{W}_p({\\mathbf V},m)", "29c6f3841f966081742c048709bee263": "p(r,\\theta) = \\ln (I/I_0) = -\\int\\mu(x,y)\\,ds", "29c6f3f470ffd452cdda2fa6b1efd83c": "\\scriptstyle \\log_e(\\frac {760} {101.325}) - 12.4379 \\log_e(T+273.15) - \\frac {6340.514} {T+273.15} + 95.14704 + 1.412918 \\times 10^{-05}(T+273.15)^2", "29c6f6209e92255f1d58bf3a7aa957af": "\\partial R(w)", "29c7075d62fc2d8ceeed999e838205dc": "\\bold{P} = \\epsilon_0 \\chi_e \\bold{E} \\,", "29c71d1792b60186fa5f1ef207a52d98": "\\mathbb{F}^{\\binom{l+d}{d}} \\rightarrow \\mathbb{F}^{|\\mathbb{F}|^l}", "29c759b95a7a4d013fdf52a0062b0ef4": " - \\frac{\\partial \\phi}{\\partial x}", "29c75bae1175f1b21628102213b1b229": "\\ln \\left ( \\frac{ \\varepsilon_k }{ \\varepsilon_0 } \\right ) \\equiv \\ln \\left ( \\frac{ \\varepsilon_0 ' }{ \\varepsilon_0 } \\right ) = 2 (k - 1 + x ) \\delta_0 \\Omega_0.", "29c7bfcb43d2800dc72a6507a663efe6": " W(K) = W(k) \\oplus \\langle \\pi \\rangle \\cdot W(k) ", "29c7ceffd70e0ddeea0d8208c7e7376a": "100+x", "29c8b7410bc311393d83d22792a3ba33": "a_1, a_2, a_3,\\ldots ", "29c8cffcddbd71cce6f4912a2270a9cf": "P_{S/A}(n_1, \\cdots n_N \\rightarrow m_1, \\cdots m_N) \\equiv \\bigg|\\lang m_1 \\cdots m_N; S/A \\,|\\, n_1 \\cdots n_N; S/A \\rang \\bigg|^2 ", "29c8fecb3f6d12a471adaac1de2901a5": "\\prod_{i=n-k+2}^n m_i < S < \\prod_{i=1}^k m_i", "29c904ea86cb70b375762506737725e3": "\n\\vartheta(z; \\tau) = \\sum_{n=-\\infty}^\\infty \\exp (\\pi i n^2 \\tau + 2 \\pi i n z)\n", "29c94efc19d61f59ec6252f22c01808f": "t - 1", "29c97105e48b88a35ff51a080b603d4c": "\\mathfrak{P}^{89}", "29c986d7edab1b30417165b17a02cff7": "\\frac{5}{12}+\\frac{11}{18}\\;=\\;\\frac{15}{36}+\\frac{22}{36}\\;=\\;\\frac{37}{36}", "29ca071de840e72c9495903d8508985f": "~\\alpha,\\beta,\\gamma", "29ca59850511726af8b6349453601a5c": "\\phi(x)\\!", "29ca5bb060f7ccad71fa42d45b7e1e83": "\\Pi^k", "29ca70a48706c29e463a214c43b0cfdf": " a{{\\partial}Q\\over{\\partial}t} + bQ^2 = {\\Delta}p ", "29cb1e025b54781dff1c2421136949be": "\\Rightarrow I(2n+1) \\le I(2n) \\le I(2n-1)", "29cb620c3e516075507b5caebe32df34": "[\\alpha]_D^{20} = +36.5", "29cbb9f7466208eb226969a3b1a60ff1": " r(1,1)+s(-1,2)=(a,b). \\,", "29cbec1ce86435f1d044129f46909212": "C_2 = (pq + Tr[P\\circ Q])^2 + p Tr[Q\\circ \\tilde{Q}]+q Tr[P\\circ \\tilde{P}]+Tr[\\tilde{P}\\circ \\tilde{Q}] ", "29cbfe5753fcc6faa8920c0b046c4d53": "\\sum_{i=1}^N G_i=1", "29cc22d1ff65896638ae89b4481963fa": "L(n)=2F(n-1)+F(n)", "29cc2e9a9d7a1be41a537618e2963c54": "k_{et} = \\frac{2\\pi}{\\hbar}|H_{AB}|^2 \\frac{1}{\\sqrt{4\\pi \\lambda k_bT}}\\exp \\left ( -\\frac{(\\lambda +\\Delta G^\\circ)^2}{4\\lambda k_bT} \\right )", "29cc3ebdd9a4c31bec0c1e26acbc47f6": "\\Delta E \\Delta T \\ge \\frac{\\hbar}{2} ", "29cd5953cbf2de60bac83aafe885b7ca": " \\langle f|g\\rangle ", "29ce5c3f9f7c63d8232a5557d2f7d312": "\n\\Bigg(\\frac{q}{p}\\Bigg)_4 \\equiv\\Bigg(\\frac{a/b - c/d}{a/b+c/d}\\Bigg)^\\frac{q-1}{4}\\pmod{q}.\n", "29ce5dcae557481ff4afc13e272d3de3": "\\mathcal{C}(I_{a}(t_0),B_b(y_0))", "29ce976e3c36431d295a36a70c66b4a8": "\n\\begin{array}{rcl}\nmass_{ingredient} & = & formula\\ mass \\times true\\ percentage_{ingredient} \\\\\ntrue\\ percentage_{ingredient} & = & \\frac{baker's\\ percentage_{ingredient}}{formula\\ percentage} \\times 100% \\\\\nmass_{ingredient} & = & formula\\ mass \\times \\frac{baker's\\ percentage_{ingredient}}{formula\\ percentage} \\\\\n& = & \\frac{formula\\ mass \\ \\times\\ baker's\\ percentage_{ingredient}}{formula\\ percentage}\n\\end{array}\n", "29cf5f7ff4ebd351eae3d74e512cb752": "=-\\frac{{\\pi}^2}{10}-\\operatorname{arcsch}^2 2", "29cf67e7fb88230d002be176d0ac7342": "\\sum_{i=1}^0 i=0.", "29cf99d655ca38f2e909ce42e181a775": "\\Omega_{+}(k)^2={{\\omega^2+\\Omega^2+\\sqrt{{(\\omega^2-\\Omega^2)}\n^2+4{g}\\omega^2\\Omega^2}\\over 2}}", "29cff640523ca25516e89dbdfaf6aa01": "C^\\gamma\\ge b_\\gamma(M).", "29cff6c7a1e2e9ac793b50606336c76f": "A \\mathbf x", "29d069673697bf68758a507c39dd2a61": "\\Phi(f)=\\frac{10}{11}BOC(1,1)+\\frac{1}{11}BOC(6,1)", "29d074f506b26a8868c1318e14d73b03": " R= 1 + \\frac{\\Delta W}{2 U^2} - \\frac{C_{y2}}{U}", "29d182ae089419e46d7a6220f2d0601b": "\n\\lim_{\\tau \\rightarrow \\infty} \\left| \\left\\langle \\frac{dG^{\\mathrm{bound}}}{dt} \\right\\rangle_\\tau \\right| = \n\\lim_{\\tau \\rightarrow \\infty} \\left| \\frac{G(\\tau) - G(0)}{\\tau} \\right| \\le \n\\lim_{\\tau \\rightarrow \\infty} \\frac{G_\\max - G_\\min}{\\tau} = 0.\n", "29d2059e1c907cd3b9a4a089afecef59": "\n \\overline{\\rho}{\\partial_{t} \\overline{u}_i}+\\overline{\\rho} \\overline{u}_j {\\partial_{j} \\overline{u}_i}=-{\\partial \\overline{p}_{i}}+\\eta {\\partial^2_{jj} \\overline{u}_i}-{\\partial_j}(\\overline{\\rho v_i v_j} ).\n", "29d23935be7090adcdc5bb6fa1b5d598": "10^{-123}", "29d291aaf81253196e7f70c40b1beaec": "\n\\ D_{ \\omega } =(1+j \\omega { \\tau}_1 )(1+j \\omega { \\tau}_2 ) ", "29d2931354df4ec1b8ba642ed3e29910": "0= \\Delta M_1 - \\Delta M_2 = \\rho_1 A_1 v_1 \\, \\Delta t - \\rho_2 A_2 v_2 \\, \\Delta t", "29d2d7b729d69a3917a2f9915a4e1f26": " \\phi\\ ", "29d2df3db77911bf72b6a2ca43b5dbbf": "\\mbox{1. } \\quad c_n \\rightarrow 0, \\quad a_n \\rightarrow 0 \\quad \\mbox{ as } \\quad n \\rightarrow \\infty ", "29d31c8fe0a5ffd57a09fe1fba6cc6b1": " a=0. \\,", "29d37cfb78e3b190a1cae677c5b85928": "\n\\sigma_1 = \n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix},\n\\quad\n\\sigma_2 = \n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix},\n\\quad\n\\sigma_3 = \n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "29d44b6361939e7460bd079565b164f2": "\\langle \\phi | \\psi \\rangle = 0", "29d457ef8414ab5dec309b57860dbef6": "\\{a_1^n \\dotso a_{2^{k+1}}^n|n\\geq0\\}", "29d4e50b7f5db1666f939ae4e1ad9131": " \\frac{PV}{T} = k_{\\text{arb}} \\,\\!", "29d4e7f8177fb2597fb5abf98a7f2627": "H_\\ast(\\mathbf{\\mathcal{C}}_{\\bullet {\\text{ }\\Box_{S_\\ast(B)}}}S_\\ast(E))", "29d559dd13e630678dd70f55ab317c44": "V \\le 3", "29d5aadcea06f01866a79cd64ecc19dd": "\\mathbf{u}= (u_i)", "29d5c2029015c549ba4ed75a0f310ca7": "\\mu_{11} = M_{11} - \\bar{x} M_{01} = M_{11} - \\bar{y} M_{10},", "29d5ee60e498dd6f298f21dff84b89a0": "\\sigma_m-\\tau_m", "29d6a0ae56b8e7d91f62084927efbcd8": "\\tau_{ij} = 2 \\mu \\left( e_{ij} - \\frac13 \\Delta \\delta_{ij} \\right)", "29d6a903ce3acf8eb3036561a292ef10": "L_0 - \\{e\\}", "29d6ed80ff6e69fc257acc7521bdf14f": " \\ell", "29d7c953b003600e5665eb8b0645fe03": "= \\sum_{i,j} (v_i^{T} x) v_i)^{T})(v_j^{T} x) v_j\\lambda_j)", "29d7cfafc3aabbcc328271c9b6d09436": " \\begin{align} \n&\\lim_{\\nu \\to 0}\\text{excess kurtosis} = (\\text{skewness})^2 - 2\\\\\n&\\lim_{\\nu \\to \\infty}\\text{excess kurtosis} = \\tfrac{3}{2} (\\text{skewness})^2\n\\end{align}", "29d7efd2b036dd9c716892b04d3e15e0": "Q_{f} \\subseteq Q", "29d824e3f566790c1cb6ddbaaa06c36d": "\\underbrace{111\\cdots111}_N 0", "29d82737f05ab6714af4e5df5e980a37": "w_1,\\dots,w_n\\in\\mathbb C", "29d84560c50051ac3d57d55903558b5c": "\\frac{\\pi}{2}t^2", "29d87c0ead9455d1a5f7a45f8551f3d0": "x \\mapsto |x|", "29d93cdaa0274c194cb24f004a7c898d": "x_2=3\\,", "29d983a804a4a60956f54ef6051c40a9": "f_{\\rm{Frechet}}(x;k,\\lambda)=\\frac{k}{\\lambda} \\left(\\frac{x}{\\lambda}\\right)^{-1-k} e^{-(x/\\lambda)^{-k}} = *f_{\\rm{Weibull}}(x;-k,\\lambda).", "29d9f341b6ba248329cbdad34e4877b9": "x \\neq 0", "29da154f6b2e8515a7866e328374e8bf": " k_i = hf\\bigl( t_n + c_i h, y_n + \\sum_{j=1}^s a_{ij} k_j \\bigr), \\quad i = 1, \\ldots, s.", "29da1b8202f623fceddc7e4ff8f00d1b": "\\forall x \\forall y [ \\forall z (z \\in x \\Leftrightarrow z \\in y) \\Rightarrow \\forall z (x \\in z \\Leftrightarrow y \\in z) ]", "29da83ccb966e893ca639f16c1f7c383": "\\mathcal{O}_V", "29daa54ac11d4660aa9fadd7bddd0596": "Lclm(l_2,l_1)={\\langle\\langle}L_1,L_2{\\rangle\\rangle} ", "29db107276ab833df5af8e1f0cfc2260": "Q = C T", "29db38d3a38f3d0804c77102aa8ccd14": "(a,b,k) = (8,1,-3)", "29db3bbcc1faaa8110bdbf2fcb725448": "K(S|x^*)=O(1)", "29db40d10de66a2415411205df239f62": "\\lbrace \\cosh t + \\jmath \\ \\sinh t : t \\in \\mathbb R \\rbrace", "29db6ed5b31232b157d965ff3649def7": "\\langle \\Psi , \\Psi \\rangle = \\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r}_1 \\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r}_2\\cdots \\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r}_N |\\Psi(\\mathbf{r}_1 \\cdots \\mathbf{r}_N,t)|^2 = 1", "29dbc185c5777aa449c9d0abc3bd616d": "\\tilde b_1 = b_1\\,", "29dc52f66496df3476527c29936b3bfa": "y(u,v) = -\\cos \\theta \\,\\sinh v \\,\\cos u + \\sin \\theta \\,\\cosh v \\,\\sin u", "29dc61e4ce400b3396fdfa03800bf2ff": "\n\\left[1-\\exp\\left(-2\\pi k\\right) \\right]\\int_\\varepsilon^\\infty \\frac{\\exp\\left(ikx\\right)}{\\exp\\left(x\\right)-1} \\, dx = i \\int_\\varepsilon^{2\\pi-\\varepsilon} \\frac{\\exp\\left(-ky\\right)}{\\exp\\left(iy\\right)-1} \\, dy + i\\frac{\\pi}{2}\\left[1 + \\exp \\left(-2\\pi k\\right)\\right] + \\mathcal{O} \\left(\\varepsilon\\right) \\qquad \\text{ (1)}\n", "29dc87d4b2ba3fd62e73521327159ea8": "\\rho_i^A", "29dc8b6768728e7bf6ed14d7c99910fb": "T=a\\otimes b\\otimes\\cdots\\otimes d", "29dca2be408d8b006bfad868822343a9": "\\Box x", "29dca634d30b1e8b7eaa21a272f168cd": "\\tilde Z_0", "29dd17282cc62fc9f41c8a54e02343f5": "(X_i,Y_i)'", "29dd38655dba9c3daa346901edcabd2f": "P(D|H)P(H)", "29dd40002500444b7c5db68a388d3f0e": "A=\\begin{pmatrix}\n0&0&0&0&0&0&0\\\\\n0&0&0&1&0&1&0\\\\\n0&0&0&1&0&0&0\\\\\n0&0&0&0&0&0&1\\\\\n0&0&0&0&1&0&0\\\\\n0&0&1&1&0&0&0\\\\\n0&0&0&0&0&0&0\\\\\n1&0&0&0&0&0&0\\\\\n\\end{pmatrix}", "29dd4cb1ba58aa32db38a479652718ac": "\\sin^2\\theta+\\cos^2\\theta = \\sin^2\\left(t+\\frac{1}{2}\\pi\\right) + \\cos^2\\left(t+\\frac{1}{2}\\pi\\right) = \\cos^2t+\\sin^2t = 1.", "29dd6f057b6ddd9245524abcbfb7c230": "V_{(xuw)} \\ \\stackrel{\\mathrm{def}}{=}\\ \\,", "29dd87dcf0f45df8f3f8101552f3fc91": "\\gcd(N_i , N_j ) = 1", "29ddb0e2a623d8420ea562b5d58b3cf1": "\\frac{d\\phi(x)}{dx}|_{x=1} = \\frac{d\\phi(e^x)}{dx}|_{x=0}", "29de0f50964264190f64597eaaa7a04c": "\\Lambda = \\frac{(n-1)(n-2)}{2\\alpha^2}.", "29de2d7d4b2ca9599108a2d736933d76": "i=j ", "29de49e2281462ef86a5bce8bec6ba9d": "\\rho = \\frac{A_{observed}}{Z_{calculated}}", "29de4f64a9f58531fba0b7206d13dec9": "\\tfrac{m-(k-1)}k", "29de5ef00b215282c7052ad4eabee7c6": "\n \\begin{align}\n \\bar{\\Pi}^0_0 & = 1 &\n \\bar{\\Pi}^1_3 & = \\frac{1}{4}\\sqrt{6}(5z^2-r^2) &\n \\bar{\\Pi}^4_4 & = \\frac{1}{8}\\sqrt{35} \\\\\n \\bar{\\Pi}^0_1 & = z &\n \\bar{\\Pi}^2_3 & = \\frac{1}{2}\\sqrt{15}\\; z &\n \\bar{\\Pi}^0_5 & = \\frac{1}{8}z(63z^4-70z^2r^2+15r^4) \\\\\n \\bar{\\Pi}^1_1 & = 1 &\n \\bar{\\Pi}^3_3 & = \\frac{1}{4}\\sqrt{10} &\n \\bar{\\Pi}^1_5 & = \\frac{1}{8}\\sqrt{15} (21z^4-14z^2r^2+r^4) \\\\\n \\bar{\\Pi}^0_2 & = \\frac{1}{2}(3z^2-r^2) &\n \\bar{\\Pi}^0_4 & = \\frac{1}{8}(35 z^4-30 r^2 z^2 +3r^4 ) &\n \\bar{\\Pi}^2_5 & = \\frac{1}{4}\\sqrt{105}(3z^2-r^2)z \\\\\n \\bar{\\Pi}^1_2 & = \\sqrt{3}z &\n \\bar{\\Pi}^1_4 & = \\frac{\\sqrt{10}}{4} z(7z^2-3r^2) &\n \\bar{\\Pi}^3_5 & = \\frac{1}{16}\\sqrt{70} (9z^2-r^2) \\\\\n \\bar{\\Pi}^2_2 & = \\frac{1}{2}\\sqrt{3} &\n \\bar{\\Pi}^2_4 & = \\frac{1}{4}\\sqrt{5}(7z^2-r^2) &\n \\bar{\\Pi}^4_5 & = \\frac{3}{8}\\sqrt{35} z \\\\\n \\bar{\\Pi}^0_3 & = \\frac{1}{2} z(5z^2-3r^2) &\n \\bar{\\Pi}^3_4 & = \\frac{1}{4}\\sqrt{70}\\;z &\n \\bar{\\Pi}^5_5 & = \\frac{3}{16}\\sqrt{14} \\\\\n \\end{align}\n", "29df36b59e13961d5098b25c5eb8b89f": "\\int x^a\\,dx = \\frac{x^{a+1}}{a+1} + C \\qquad\\text{(for } a\\neq -1\\text{)}\\,\\!", "29df8eed0edf792dc32b95b1794a0db7": "\\scriptstyle{\\mathrm{SO}(1,3)}\\,", "29e028a2ee894d51729754483dc8d289": "t^*\\ = \\frac{t}{L/U}\\,", "29e0e948001a820afcea89d23befd3d4": "q \\in \\mathbb{Z}", "29e10ddf7e411df5e3591b4f816bb3bb": "\\vec F=\\mu_o (\\vec m \\cdot \\nabla ) \\vec H \\,\\!", "29e10e212593beee479afd73eaa98499": "\n \\rho G + G \\rho = d \\rho^{\\,}\n", "29e1b04f802250ef9108c1eda51e6eab": "f(u)\\cos v", "29e1bab60c2317fd7fc2cb2b906023de": "S= S_0 + \\frac {\\hbar}{i} S_1 + ...", "29e1c632a4285bb340a9de65065fd101": "E_f", "29e202060222d0e68299da5a1f596e14": "Q(R)", "29e27b8fcc390cf38efb2e3993247d65": "R\\left( t,s \\right)=E\\left\\{ X\\left( t \\right)X\\left( s \\right) \\right\\}", "29e29d26ebb3b16a9d1f498361b6b410": "\\phi(w) \\geq \\psi(w)", "29e2c895934ead4079192ea9087560e2": "r_k=\\lambda_k r\\;", "29e2e280da908915b0eaab0c5e4a0bb0": "G_{(1-X)} \\approx \\frac{\\beta - \\frac{1}{2}}{\\alpha+\\beta-\\frac{1}{2}}\\text{ if } \\alpha, \\beta > 1.", "29e301a547559e8104024bc29b9704aa": "\\rho_{X,Y}=\\frac{E(XY)-E(X)E(Y)}{\\sqrt{E(X^2)-(E(X))^2}~\\sqrt{E(Y^2)- (E(Y))^2}}.", "29e30e5fcafe92ead661b3a8c1718512": "(x-3)(x-1)^3(x^3+2x^2-3x-5)(x^3+2x^2-x-1)(x^4+x^3-7x^2-6x+7)(x^4+x^3-5x^2-4x+3).\\ ", "29e3814be957e0db0bafd72c9113ecb3": "( R", "29e3d59df5bd9b0b118c858976c8b292": " \\dot{z}(t) = \\dot{u}(t) \\left\\{A - \\left[\\beta\\operatorname{sign}(z(t)\\dot{u}(t)) + \\gamma \\right]|z(t)|^n \\right\\} ", "29e4096d282c57bc0556fab0b6b30c77": "r_\\pm := M \\pm \\sqrt{M^2-Q^2-J^2/M^2}", "29e4b109debcb015655b1f118536c7b9": "\\left( \\frac{\\pi}{a+2\\delta} \\right)^2 + \\left( \\frac{\\pi}{b+2\\delta} \\right)^2 + \\left( \\frac{\\pi}{c+2\\delta} \\right)^2", "29e4d5355935120a7024e3beacc0d37e": "\\bigl( \\begin{smallmatrix}\\\\ ~\\;2/3&4/3\\\\ -1/3&7/3\\end{smallmatrix} \\bigr)", "29e52023719b292d6b1616d9d6f4a311": "P^{1-\\gamma}T^{\\gamma} = C", "29e52ae7e29e7c0e78e6bc15861ff6cd": "\n\\mathbf{A}=\n{\\displaystyle\\bigotimes\\limits_{i=0}^{\\infty}}\n\\ A_{i},\n", "29e55ecd8c15b5c71f9df7d98343a7ab": "\\frac{dy}{dx} = \\tan \\varphi.", "29e5c1eaba7e88efad0bc3ee3529ff7d": "Q(x_i,f(x_i)) ", "29e5ce9ea76409d81bdf3c1013f44ca6": "0 \\leq \\alpha \\leq 1", "29e5f410bbec6dcedd52fa5398862888": "\\alpha = 2 - \\nu d\\,\\!", "29e64c4b30ff6900f33237a4d950a8ad": "pB_{p-1} \\equiv -1 \\pmod p.", "29e6a86a081a3345883b7b36e2e9834d": " \\lim_{|z| \\to \\infty} f(z) = 0", "29e6ac2362597ecf6518e6630a3d993d": "\\forall p((p \\land \\Diamond Kxp) \\Rightarrow Kxp)", "29e6ed858202b4a55808817f4d35383f": "\nf(x) = \\left\\{\\begin{matrix}\n\\beta x & x\\in S_0\\\\\nx^2 & x\\in S_1\\\\\n\\alpha x & x\\in S_2\n\\end{matrix}\\right.\n", "29e740b43dad747a1610bfc1d290feb1": " \\sigma_{x'} = \\sigma_{y'} = \\sigma_{z'} ", "29e79ab629d8b9458ccaaad5cc653317": "e^{[-a_1+a_2]} \\sum_{j=0}^{[n/2]} \\frac{a_1^{n-2j}a_2^j}{(n-2j!)j!}", "29e7a116098ff9d1ba835cdb66b5e45c": "\\lVert e \\rVert = \\lVert e^* \\rVert = e^* e = 0 .", "29e83a118576ae33c3193037d219223a": " (V, cl) ", "29e843aff61b1b0b5420077eaf68c3e6": "\\|f\\|_{p,w}\\le \\|f\\|_p", "29e9228cc13c4a6251ed718578e025ee": "p(\\vec{r})", "29e94c844ca81ebcbde16976c1b58ed3": "Z^T Q Z", "29e9588dd63153a7e7449d95415b0e2b": "\\rho(X) = \\mathbb{E}[-X]", "29e9dcb6982fac0cf29b881c8dc498a9": "a {}_{(n)} b\\,\\!", "29ea29c98ee95a8a0ffac04837f513a1": "\\cos(\\mathbf{X})", "29ea3d1a2f25cfe972b4944f3498577d": "J(x_1,y_1)", "29ea80fa1357ae0fbf430b56279bace6": "\\nabla(FG) = \\nabla FG+\\dot{\\nabla}F\\dot{G}", "29ea887347d76cb0159ec2a543021615": "S(A_n)", "29eaa4edcdafcb3b75e273fdc32fb4f4": " \\hat{H} = \\frac{\\hat{p}^2}{2m} + V(x,t) \\,,\\quad \\hat{p} = -i\\hbar \\frac{\\partial}{\\partial x} ", "29eabb735634a8ea6c082aad32092c9c": "g_H(v,w) = g(v,w),\\quad v,w\\in H", "29eabc8b6ffc6b3ee625a20621b8a461": "\n u_b^{\\mathrm{face}}(x,z) = -z~\\cfrac{\\mathrm{d} w_b}{\\mathrm{d} x}\n ", "29eb38de1ea5fee2d669d4bb4eab405d": "d(i,j)", "29eb74c7f0eb73017d3cec973e83bb49": " i \\frac{\\partial \\psi}{\\partial t} + \\Delta \\psi + \\psi \\ln |\\psi|^2 = 0. ", "29ec0d1a9c528185adb7f7b4ca06048f": "\\displaystyle{\\pi(R)f(t) =Q(t)^{-1} f(t).}", "29ec7059829975ba85edee0812b82d90": "\\gamma \\approx 1", "29ec7f83e0e2721cd9e0ede57a1d74b8": "\\zeta_{\\pm}=z\\pm \\frac{L}{2},", "29ec85a7369f34aa0041f1afb29c85c5": "Y=\\{ x_1=\\cdots=x_n \\}", "29ec92bd04c53faa4008de00fea3d507": "A = L \\times W", "29ed926ca57bede248f975b0c6e19bfe": " b_1 = \\frac{h_1\\rho_c}{\\rho_m-\\rho_c} ", "29edd219e48e5b86eaed08a48a57bf35": "\\frown\\ : H_p(X;R)\\times H^q(X;R) \\rightarrow H_{p-q}(X;R)", "29ee1c2488d3abd1fb84c6245b2b8347": "c = h / y_\\mathrm{atm}", "29ee530a4b63960bcf57dfa0ba882aa3": "Tt", "29ee537b48661644b126d39a44ddc264": "\n\\| u \\|_{L^{2r}} \\leq C r \\| u \\|_{L^r}^{1/2} \\| \\nabla u \\|_{L^2}^{1/2}.\n", "29ee58e857450851cfdd2daee17f5120": "\\omega = y_n", "29ee8d099b18a26660fdaea5e220d87f": "r_1(\\theta_1)+r_2(\\theta_2)=a\\,", "29ef1da92b3f32be82a51653ab10fb3a": "\\left(\\frac{u}{v}(p_1 u' v - p_2 u v')\\right)' = \\left(q_2 - q_1\\right) u^2 + \\left(p_1 - p_2\\right)u'^2 + p_2\\left(u'-v'\\frac{u}{v}\\right)^2.", "29ef2276ca4617121e4f8a31f19a9c37": " \\Omega^1={\\Bbb C}.{\\rm d}t,\\quad ({\\rm d}t)f(t)=f(qt)({\\rm d}t),\\quad {\\rm d}f={f(qt)-f(t)\\over q(t-1)}\\,{\\rm dt}", "29ef6869c9df08aaf7e28c34657a31ca": "S^3(A_n)=S(S(S(A_n))),", "29efa36295b617dcbe4d98deec8a0048": "d = gh + p = (5/3)(1) + 2 = 11/3", "29efa716c94f7676c6cc56ea3df29588": "\\,0 < p < 1", "29f0787ff1b077c79b7cf5b73fb5c210": "c\\in\\mathbf{R}^{+}", "29f08b4745bf8856b6b3204a68118ed3": "\\gamma < \\left(\\dfrac{1-(\\dfrac{\\lambda}{d})}{\\delta-(\\dfrac{\\lambda}{d})}\\right)", "29f0a873369d56efb43b9fa3b695c69c": "\\hbar\\vec I", "29f0b4d764228acdac68e4608cf136d2": "w_i^+=w_i+\\sigma_ix_i\\Theta(\\sigma_i\\tau)\\Theta(\\tau^A\\tau^B)", "29f0b888d6759e1c0db6e75dc0193c4b": "\\nabla \\times \\mathbf{V}=v_x - u_y =0.", "29f0d0349488ec62260de03ce44192cf": "\\mathrm{d}U_{cv}=\\mathrm{d}H_{in}-\\mathrm{d}H_{out}+\\delta Q-\\delta W_{shaft}\\,", "29f11cdc9db81f220388e3b935301dfc": "S_2 = S_2''\\,", "29f168ebd31816ff8777eca9478f26d9": "\\eta=\\frac{1}{8\\pi^2}", "29f1f37c25a1c2f824abd1d451aa5a15": "\n\\binom\\alpha\\beta=\\binom{(\\alpha_1,\\alpha_2)}{(\\beta_1,\\beta_2)}=\\binom{\\alpha_1}{\\beta_1}\\,\\binom{\\alpha_2}{\\beta_2}.\n", "29f22bbf90c8c9e9794a553216b50ed9": "\\sqrt{n}D_n>K_\\alpha,\\,", "29f2347aadd8ded83428ae17017d77c0": "10^{10^{10^{1000}}}", "29f24a19117e7ef8f03fdbd7599953ab": "A = \\frac{n}{2} ( \\cot{\\frac{\\pi}{n}} + \\sqrt{3}) a^2.", "29f26741d36e1ee12e6e7f748820cbc8": " r_{ij} = \\frac {x_{ij}} {\\sqrt{\\sum_{i=1}^{m} x_{ij}^2 }}, i = 1, 2, . . ., m, j = 1, 2, . . ., n", "29f26cf9fc8aa164f8880b9a1c202a31": "a_1,\\dots,a_n", "29f2a960eeb3d9620740cda8349a656a": "\\dot{\\sigma}", "29f2b5ecf05fbd7c5099a693390a85f8": "f(r + tp^{k}) \\equiv 0\\,\\bmod{p^{k+1}}\\, ", "29f2c8c1632f8a5ac61637aeef1243f8": "n_1/n_2", "29f2f1d0c0c5ad8f1b185ce5375cfa81": "\\int_a^b \\psi(x) \\overline{K(x,y)} \\, dx = \\overline {\\lambda}\\psi(y).", "29f356e1c2b9207a9dd6515b5e7a3896": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\mathbf{f} +(\\mu + \\mu^v) \\nabla (\\nabla \\cdot \\mathbf{v}) ", "29f3953a4b44af7d68d5aa6fa89cbd55": "\n P(d|q) = \\frac{P(q|d) P(d)}{P(q)}\n", "29f3d17687737b2a44357e7738cf17bb": "\\scriptstyle\\sum^s_{j=1}(1/x_j)+(1/(x_1\\cdots x_s))=1", "29f405a6ec408716f9ccafba985baeec": " K(a,b;m) = \\sum_{d\\mid\\gcd(a,b,m)} d\\cdot K\\left(\\frac{ab}{d^2},1;\\frac{m}{d}\\right).", "29f42142e0855daebbcf74bf58faae83": "D \\left( \\frac{\\mathrm{D} u_x}{\\mathrm{D} t} + u_z^2 \\right) = -D \\frac{\\partial p}{\\partial x} + \\nabla^2 u_x ", "29f46a5548103f58714aa1bcd07b896c": "\\scriptstyle\\sqrt{m}", "29f46f717e18759f92191dd657a050cd": "\\nabla_X : \\Gamma(E) \\to \\Gamma(E)", "29f487be7e9393c3c17a8d182febe245": "A[[t]]", "29f4e47bed6b7b7d90221f60a67493c6": "w^k_j=v^i\\frac{\\partial^2 y^k}{\\partial x^i\\partial x^j}+v_j^i\\frac{\\partial y^k}{\\partial x^i}.", "29f510d8efa68f06bc0f66f50d2243ec": "A,B_{1},\\dots,B_{m},C_{1},\\dots,C_{n}", "29f59b948eac0bfef780d13305f02639": "\\operatorname{X}(u,v) = \\langle \\operatorname{x}(u), \\operatorname{y}(u) \\cos(v), \\operatorname{y}(u) \\sin(v)\\rangle \\, ", "29f5f916cffe93380adb55d45a611f48": "b = \\frac{1}{2}\\, \\frac{(V - Z_{0}^{*} I)}{\\sqrt{\\left|\\real\\{Z_{0}\\}\\right|}}\\,", "29f64c0cb2484b3afe1f4b13c6333a41": "0\\rightarrow\\varprojlim A_i\\rightarrow\\varprojlim B_i\\rightarrow\\varprojlim C_i\\rightarrow\\varprojlim{}^1A_i", "29f6697d8f34b3077722bdaf8ef13edf": "\\Sigma^{0,Y}_1", "29f6936b753d2f1dbef865e00931c202": "\\text{size of effect}=\\left| \\frac{\\text{second value − first value}}{\\text{first value}} \\right|", "29f6a421b93ebad35c234a806eec931c": "\n \\Theta(G)\n = \\sup_k \\sqrt[k]{\\alpha(G^k)}\n = \\lim_{k \\rightarrow \\infty} \\sqrt[k]{\\alpha(G^k)},\n", "29f756aaa65d9704f53667fead1686f0": "B_n(x)=-\\frac{n!}{2^{n-1}\\pi^n}\\sum_{k=1}^\\infty \\frac{1}{k^n}\\cos\\left(2\\pi kx-\\frac{\\pi n}{2}\\right), 0 1", "29fdafb9742bde3e49fbb1f6a287120f": "v_S", "29fdcea51732a605db59c3a3a642066a": "2x=-4", "29fe3dc612102ac9b5b745d6326fb757": "n(\\omega)", "29fe6ceff060711e7b783b5640996757": "P_i = E_K (C_{i-1}) \\oplus C_i", "29fe9336063076aff6b3e12571cdc281": " (x+y)^{n+1} = x(x+y)^n + y(x+y)^n, \\, ", "29fea4557dabf715e003f641b49f8800": "\\forall x\\in\\mathbb R:\\;g(x)=f(2^{l-k}x)", "29fedeef1251f2ee6e7443e71a2b37c1": "\\int_0^{\\frac{\\pi}{2}} \\sin^{n-2}(x) \\cos^2(x)\\,dx = \\left[ \\frac{1}{n-1} \\sin^{n-1}(x) \\cos(x)\\right]_0^{\\frac{\\pi}{2}} + \\int_0^{\\frac{\\pi}{2}} \\ \\frac{1}{n-1} \\sin^{n-1}(x) \\sin(x)\\,dx = 0 + {1\\over {n-1}}\\,W_{n} ", "29ff551e344017c0a375768961d3adf5": "\\mathrm{PSL}_n(F)", "29ff87757199b5b5ff2a96ff1467d848": "t\\neq 0", "29ffe080edd3f48e675f644b337b9ded": " [A \\cup B]_p = [A]_p \\cup [B]_p", "2a000f708d9c68124181b57b24579703": "I_2", "2a0014bfbd42b0cbc00b44004bb5ec29": "\\Omega(k)\\, =\\, \\sqrt{g\\, k\\, \\tanh\\, (k\\, h)}\\,", "2a0016a91adcc889e1bb8313948322c1": "a=n^2, \\, ", "2a005daec2fddc366257d2f83e3628ab": "\\scriptstyle x_1=4\\,Ly\\;,\\;y_1= 3\\,Ly\\;,\\;z_1=0 ", "2a0078f00633ef475cbd5e928d7f3391": "\\mathrm{\\frac{(SNR)_{O,FM}}{(SNR)_{C,FM}}}=\\frac{3k_f^2P}{W^2}\n", "2a00a3e6aec8c3ebd18e27bf0632fb84": "W,", "2a00dffad3f44d4d8b8c3910ec014811": "p \\left( {\\rm X}_i \\right ) = x_i p\\left( {\\rm mix} \\right ) ", "2a010e537a656830f0f2f385e4c13d50": "\\gamma_{\\alpha\\theta}+\\gamma_{\\theta\\beta}\\cos{\\theta}+\\gamma_{\\alpha\\beta}\\cos{\\alpha}\\ = 0", "2a01323589e14227e619f7df3ffbcd25": "\\exists f \\in F \\mbox{ . } \\forall x \\in \\mathbb{N} \\mbox{ . } x \\in A \\Leftrightarrow f(x) \\in B", "2a013a997bf1473a7e0ebac9b502d414": "1 \\to \\{\\pm 1\\} \\to \\mbox{Pin}_\\pm(V) \\to O(V) \\to 1.", "2a01c8c3c350d94e803db761713fe3a3": "\\displaystyle{\\varphi(z) = c B(z) e^{-P(z)},}", "2a01cfe8b174f71fc0fde16ba67a382b": "q(t_2 - t_1) = \\int_{t_1}^{t_2} \\iint_S \\mathbf{j}(\\mathbf{r},t) \\cdot d\\mathbf{S} \\,dt\\, ", "2a0269919787a4eb3a07572a4e179d4f": "\\scriptstyle Z_{\\text{eq}}", "2a02798a2abd0a7fa130a6c9ea4ad68c": "\\mathrm{Nu}_D = 0.023\\, \\mathrm{Re}_D^{4/5}\\, \\mathrm{Pr}^{n}", "2a02af11a33d4456267d5eff148e5e4e": "cf(\\prod A/D)<\\lambda", "2a02cdc5ab2a494dec8e9d490865115c": "GapCVP_\\beta", "2a02e5f00edfa63cabe932f2d87de2b5": "+ 7 \\cdot 8^{(7 \\cdot 8^7 + 7 \\cdot 8^6 + 7 \\cdot 8^5 + 7 \\cdot 8^4 + 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 6)} + \\cdots", "2a02f40ad4ebb983d44ca3ed8213c77e": "{\\tilde{B}}_{8}", "2a0303bea10d24fa75fc238681b8b623": "\n\\operatorname{Ran}(J)=\\operatorname{Ker}(J)=VTM, \\qquad \\mathcal L_VJ= -J, \\qquad J[X,Y]=J[JX,Y]+J[X,JY],\n", "2a032b7b7dd2d7eaeb0a0c62f7659342": "N(X)= N_\\text{max}\\left(\\frac{X-X_0}{X_\\text{max}-X_0}\\right)^{\\frac{X_\\text{max}-X_{0}}{\\lambda}}\\exp\\left(\\frac{X_\\text{max}-X}{\\lambda}\\right),", "2a033e8ff79d2de1e28c690fec881585": "f^2 = {\\eta^2 \\over 1 - \\eta^2}", "2a0342333d835bc74042e424fa5f9190": "\\frac{\\partial \\phi}{\\partial t} + c^2 \\frac{\\partial u}{\\partial x} = 0", "2a0345cc4af71271e05e9e1ebe2a852a": "{dx \\over d \\tau} = {\\partial \\mathcal{H} \\over \\partial p}, \\;\\;\\;\\; {dp \\over d \\tau} = - {\\partial \\mathcal{H} \\over \\partial x} ; \\;\\;\\;\\;\\;\\; {dt \\over d \\tau} = {\\partial \\mathcal{H} \\over \\partial p_t}, \\;\\;\\;\\; {dp_t \\over d \\tau} = {\\partial \\mathcal{H} \\over \\partial t}.", "2a038c9d224f26aaf9086161745a6e79": "V^{\\pm} = \\{v\\otimes 1 \\mp Jv\\otimes i: v \\in V\\}.", "2a03dc6b20b74d4eaa1fafbd8543aab1": "{\\mathbf{X_{01}}} = \\sqrt {\\mathbf{Z_{01}}^2 - \\mathbf{R_{01}}^2}", "2a03ed6ac60237c50405ad8e82beeb84": "S = \\{(\\alpha_0,\\dots,\\alpha_n) : \\alpha_0+\\dots+\\alpha_n=1\\ \\land\\ \\alpha_0\\ge0\\ \\land\\ \\dots\\ \\land\\ \\alpha_n\\ge0\\}", "2a03f34949c21b232237b4194c68e071": "\\wp(z; \\tau) = \\pi^2 \\vartheta^2(0;\\tau) \\vartheta_{10}^2(0;\\tau){\\vartheta_{01}^2(z;\\tau) \\over \\vartheta_{11}^2(z;\\tau)} + e_2(\\tau).", "2a0425a0b40c0cdbda1a43eb25990ff4": "\\frac{d{\\mu}}{dx} = e^{\\int p(x)\\, dx} \\cdot p(x) = \\mu p(x)", "2a048f11a305d08f8ac6bc015f938230": "V_r(x,t) = \\rho A \\sin (\\omega t + kx).\\,", "2a04a2171cf1f2d3a0cf2f69d6fe8f86": "\\varepsilon \\left[ M \\right]=o\\left( {{M}^{1-\\frac{2}{p}}} \\right)", "2a04c019eb5a51d86079db2163d0d44d": "c_2=h \\cdot c/k", "2a05479cb65e17d82d562c560ef107db": "\\rho_{\\tau}(y)=|y(\\tau-\\mathbb{I}_{(y<0)})|", "2a059703f1c405e866e6fd354437ab56": "\\partial_\\mu S^{\\alpha\\beta\\mu}=T^{\\beta\\alpha}-T^{\\alpha\\beta} \\neq 0", "2a0597799ac58a1213b461f3b229867f": "r \\ge 1", "2a05ce002c5814fee05d7b4579713294": "\\mbox{gl dim } A < \\infty\\,", "2a0602ca899f8a71ccdd53405c776cf6": " \\{ H_n(\\theta): n \\geq 1 \\} ", "2a06266cd2d3b95317eb18afd1a241cd": "e_I^\\alpha e_J^\\beta \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "2a0633b49324bf06c396598d05ce25ab": "\\rho(A) \\le \\|A\\|_{op}.", "2a067d747105c32309235ae55bbeab33": "\\mathbf{H}_{C}", "2a06ff52783e14f7386a3642823c4533": "\n S_{11} > 0 ~,~~ S_{22} > 0 ~,~~ S_{33} > 0 ~,~~ S_{44} > 0 ~,~~ S_{55} > 0 ~,~~ S_{66} > 0\n ", "2a072ad48af6ef5cb0569c1f85874e13": "= \\frac{3}{4}", "2a0733331943dd7ff2ba7bb94edf29cf": "\\varphi_{\\alpha\\beta} = \\varphi_\\beta\\circ\\varphi_\\alpha^{-1}|_{\\varphi_\\alpha(U_\\alpha\\cap U_\\beta)} \\colon \\varphi_\\alpha(U_\\alpha\\cap U_\\beta) \\to \\varphi_\\beta(U_\\alpha\\cap U_\\beta).", "2a07379d17e777b111088aa41e8bdc13": "\\tbinom nk q^{qn}(1-q)^{n-nq} \\geq \\tfrac{1}{n+1}", "2a074e02b6cb15461dac3af316621704": "\\,\\omega_s=\\frac{2\\pi}{T}", "2a07a8c7c8526c0aaa416c6f9f4cffcf": "f'''(x)=24x", "2a07ab92f737138ae6837c9b865a1a9f": "\\scriptstyle p_1=(x_1,y_1)", "2a07c2aa5ce1da54d8a89e39ebb8a44f": "\\mathbf{F_P}", "2a08254d4c3aa6c7a3b959bf1f0f5e82": "m=m'", "2a0855fe073b9a390090b8dc9a83cfdb": "P = (P_x, P_y) = k_B R", "2a086c27e311c8d3ebdb417db7684d1b": "0 = dU = \\delta W + \\delta Q~", "2a0887bc20dc4d1221038ec9feba6118": "{\\mathcal L}_B", "2a092101337c037f3bdd47716209ff9f": "r_{ii}=1", "2a099f007e8985672723b0b74c07a95f": "\\nu={\\mu}/{\\rho}", "2a09b672ad2518f126efa81bfcefc113": "\\scriptstyle 4\\pi\\varepsilon_0\\;=\\;1", "2a09ce38529affdea1002ae2b788cffe": "A (1-\\omega^2)=\\cos\\phi \\qquad 2 \\zeta \\omega A = - \\sin\\phi.", "2a09d5bc1c2a9251de55e44905fe3349": "\\mu(A) = \\int_A \\theta(x) \\, d\\mathcal{H}^m(x) ", "2a0a71eae84c500d9845f40b5535bb59": "\\int_a^b \\frac{R}{Q}f_mf_n \\, dx=0", "2a0aa1ad9d2a47361305cd11f9444d1b": "Q_F", "2a0ab691ed47e148a5b0fa5eb9081ab1": "p,", "2a0ae57c33fe762651a0b217d425d37e": "-x = (-x_1, -x_2, \\ldots, -x_n) \\,", "2a0af802332425002e0ae81fbcb38eaf": "(x,y) \\mapsto (y,x)", "2a0b14bb03dc2b740a850eb89f60a3c9": "\\boldsymbol{\\tilde{\\Sigma_i}}", "2a0bb188d66715a2c1a8b763acb3de39": "p \\to q", "2a0be5e9afa629daa4a8418822096983": "\\text{Points Earned} = (3 \\times \\text{Number of games won in normal time}) + (2 \\times \\text{Number of games won in extra time or on penalties})", "2a0c17e89a1e27bf1ac931cb38653fe7": "g(x + y) + g(x - y) = 2[g(x) g(y)]\\,\\!", "2a0c4a06db135530c25b1d8e12d7d30a": "[n+1, n+k]", "2a0d03f92be90967b883e2edda8d3d25": "{3 + 2 + 2 + 3 + 1 + 3 + 2 + 2 + 1 + 1\\over 10} = 2", "2a0d2e1abd58ee50bfe24758cd24362d": "\\mathfrak{m}_{\\mathbb{C}}", "2a0d39d4ff7bb579728cc8af7c1d1172": "\\scriptstyle 2 f_c", "2a0d731765a7aa2f37e65dbf4dcbe581": " I \\approx Q_N \\equiv V \\frac{1}{N} \\sum_{i=1}^N f(\\overline{\\mathbf{x}}_i) = V \\langle f\\rangle", "2a0da101a1b20a739aa871c2a9d89941": "[-L,L]", "2a0da321f5d5f1dd3924dda608a812dc": "-R_2 / R_1", "2a0dc5b5410fbd41ed13867438bda5de": "x^n=0\\in \\mathfrak{p}", "2a0e13fc5c779fc5e914a430e120efcd": "(Animal(f(x)) \\lor Loves(g(x), x)) \\land (\\lnot Loves(x, f(x)) \\lor Loves(g(x), x))", "2a0e5cb95dfb2b211b6564934f11f63f": "\\left |\\xi-\\frac{m}{n}\\right |<\\frac{c} {n^2}", "2a0e95afa169414ef3e9f81f54539123": "\\begin{bmatrix} 1 & -1 & -1 \\\\ 1 & -2 & 1 \\\\ 0 & 1 & -3 \\end{bmatrix}.", "2a0f218f60cc5fb910aa8cee09a423aa": "0.73\\angle125^\\circ\\,", "2a0f2b25c46244f339b3d97fe6d7b3b9": "c_4 = -0.100254, \\,\\!", "2a0f5df6369b239f21948cdf4b2c1e83": "\\frac{\\psi_{2n}}{y}, \\psi_{2n + 1}", "2a0f7d6a91657354e867a4a3b343e2d5": "LI = L = \\sum_{i=1}^{n} L | e_i \\rangle \\langle f_i| = \\sum_{i=1}^{n} \\lambda _i | e_i \\rangle \\langle f_i | . ", "2a0ff71e5bb68443924eb8346b14fdb9": "\\gamma \\tau^{-1}", "2a101e3f7f09584ad11208df40442090": "r^2\\omega ", "2a10330413b400af48747413a6d6c5ac": "F_{load}=600 N", "2a108e2c355bdb82665d8c89e582f78c": "\\gamma=-\\frac{b}{a-d}.", "2a109bb87315b272bb4c00716022bf9a": "\\alpha = 1-\\frac{D_o}{D_e} = 1 - \\frac{\\textstyle \\sum_{u=1}^N \\frac{m_u}{n}D_u}{D_e}", "2a10cab56704fab8fc92b6ba4588f855": "e^{-i\\theta_k}", "2a110e307086cf0675378f848bf0d906": "\\int_{-\\infty}^{\\infty} c_{\\omega}\\,y_{\\omega}(t) \\, \\operatorname{d}\\omega\\,", "2a112021339ec31b69c609e6d0d1912e": "\\Delta Q = -\\frac{\\Sigma r Q_0^n}{\\Sigma n r Q_0^{n-1}}", "2a118a110cdf18e8884229cdfb59cf74": "2^{O(r\\cdot \\sqrt{n})}", "2a119d34b3589590359717eaebcd36d7": "c < \\exp{ \\left(K_3 \\operatorname{rad}(abc)^{\\frac{1}{3} + \\varepsilon}\\right) } ", "2a11a26dfffb0530ef613d076de5d170": "\\nabla\\cdot(\\varphi \\mathbf{F}) \n= (\\nabla\\varphi) \\cdot \\mathbf{F} \n+ \\varphi \\;(\\nabla\\cdot\\mathbf{F}). ", "2a11e3b702a0279023cd2aa7e0f21d9d": "\\displaystyle{\\beta(L(a,b),L(c,d))=(L(a,b)c,d)=(L(c,d)a,b).}", "2a1206c741908d2eeb108bca8f31cac0": "{a\\pi\\over 3}\\ {b\\pi\\over 3}\\ {c\\pi\\over 2}", "2a122e33791df989da31a78f1df4f589": "\\mathbf{x} = (x,y,z)^T", "2a124bb756f61906771d0a7fe188bb83": " F_{perf} = sinc(v_{blood}c/\\pi )\\approx (1- v_{blood}c/6) \\,", "2a126b850b1dc6dbf3bdf729cd334f0c": "\\langle \\alpha,\\varphi \\rangle = \\sum_k^ri_k\\varphi_k=a,", "2a128e48ec749192f02fe455fc6b17b6": "\\sec A : \\sec B : \\sec C = \\cos B \\cos C : \\cos C \\cos A : \\cos A \\cos B)", "2a12aa0923ef7220acb3058680d82c20": "\\langle [\\delta \\mathbf{u}(r)]^2 \\rangle \\propto r^{p-1} ", "2a1300619f4d4b404d5ef9386aa0e618": "\\mathbf{\\nabla}\\cdot\\mathbf{E} = \\frac{\\rho}{\\varepsilon_0}", "2a1311e6a2e702cc68aacd02781d1aba": " a_{21} ", "2a1348a54844db9b63f8e0171ed6f21f": " \\rho_{2} = 1.69202 ", "2a137e96c3cf44d2be1eb26c5f9d8758": "\\Psi : \\mathcal{B} \\rightarrow \\mathcal{A}.", "2a13a2f39b7666537aceb343c43b3b9f": "T = \\frac{\\pi}{2k}", "2a13b18399ca345ed7cfff8833cce297": "T([f]\\times[g])(p)=T[f](p)\\times T[g](p), \\qquad \\forall p\\in X;", "2a13fbaa67b1c06ac34cecab5aa032d3": "R[t_1, \\ldots, t_n]", "2a1424ac1b195892b6e31f2b96addc5d": "\\phi^{-1}(0) = \\{\\tilde{u}\\}", "2a14360d2c9a61e149be088761b0fe08": "Q: {\\mathbb R}^d \\to {\\mathbb R}^d", "2a14af3725949a52d7ea111be4972acb": "A^{\\alpha\\beta}=g^{\\alpha\\gamma}g^{\\beta\\delta}A_{\\gamma \\delta}", "2a14cce0da8b8a4008aba2978a2e4c07": "p(x_0)", "2a156644224affb3d1ff7b1683421600": "F(x, y, z)=0", "2a15854eb2591cc6db96f96ae38b2893": " \\mathbb E\\big(\\| X\\|_\\alpha^2 \\big) < \\infty ", "2a159bfb8252f125f2538d4c0d0af3bb": "\\lambda^{-\\nu} K_\\nu (\\lambda z) = \\sum_{n=0}^\\infty \\frac{(-1)^n}{n!} \\left(\\frac{(\\lambda^2-1)z}{2}\\right)^n K_{\\nu+n}(z). ", "2a15ca0a73dc55e41ebeb135b2585dc8": "X_{1},X_{2},...X_{i}", "2a161447a5842e347ac781fd174568c7": "\\delta : n \\mapsto \\delta[n] = \\begin{cases} 1, & n = 0 \\\\ 0, & n \\ne 0 \\end{cases}", "2a1686dc373f447bef0e394fde25f5a4": "\\langle\\cdot, \\cdot\\rangle_2", "2a16a4aef7763390d6267b29778503e3": "\\delta\\mathcal{S} = \\delta\\int_{t_1}^{t_2} L(\\mathbf{q},\\mathbf{\\dot{q}},t) dt = 0\\,,", "2a16a5d416c0cd9846409e6b416fd2b5": "A(j\\omega)=\\frac{V_o}{V_i}", "2a176e15710dc46af94140d340ff3244": "\\left| \\tilde{f}\\left( z_i \\right) \\right| \\geqslant 2", "2a177949895f9be63ceef2adf0b060c0": " \\sum_x p_x |x\\rangle \\langle x|", "2a1786ba018249c4b6440ec7e6c02ff7": "\\frac{I}{I_0}=\\frac{(\\alpha_n-\\alpha_p)e^{(\\alpha_n-\\alpha_p)d}}{\\alpha_n-\\alpha_p e^{(\\alpha_n-\\alpha_p)d}} \n\\qquad\\Longrightarrow\\qquad \\frac{I}{I_0}\\cong\\frac{e^{\\alpha_n d}}{1 - ({\\alpha_p/\\alpha_n}) e^{\\alpha_n d}}", "2a179c287400fa993565ef6adf2af945": "(A\\otimes(B\\,\\wp\\,C))\\multimap((A\\otimes B)\\,\\wp\\,C)", "2a17b3817ad1f85615219e42734e3af0": " v_S : 2^S \\to \\mathbb{R} ", "2a17d627748617a75b96e1dc5ee05761": "v(i,j)", "2a1834ee05a892bc747f70cd6430883a": "N_{Fe}", "2a1838b2926f061a3f98e4d975fe00b6": "{x^* \\in X}", "2a184b589e2066618886c9bf88900301": "0.1\\lambda/n", "2a184e832776184b8b414fbd2d97f054": "h : G \\rightarrow X", "2a187dda552510e4f35a73fca3795538": "\\frac{X-a}{b-a} \\sim {\\rm Beta}(1-\\alpha,\\alpha) \\ ", "2a18a789ecde925eb504154d1d7f0871": " \\frac{{\\delta f}}{{f_{res} }} = \\frac{{Z_0 }}{{2R_{rad} }}\\frac{d}{W} ", "2a18b35c8fac54eec46a8dd78e60d0e6": "\\phi_{\\lambda} (x) \\rightarrow \\phi (x)", "2a198b88047a7cb65c08eeb51458a5ef": "\nX'_{ROMM} = \\begin{cases} 0; & X_{ROMM} < 0.0 \\\\ 16X_{ROMM}I_{MAX}; & 0.0 \\le X_{ROMM} < E_t \\\\ (X_{ROMM})^{1/1.8}I_{MAX}; & E_t \\le X_{ROMM}<1.0 \\\\ I_{MAX}; & X_{ROMM} \\ge 1.0\n\\end{cases}\n", "2a1999005c2ec5db3202c50dcd984cf6": "\\operatorname{cl}(A)", "2a19cb144741fdfa5755ba47123e2adf": "\nD =\n\\begin{bmatrix}\n ~4 & -1 & ~0 & ~0 & ~0 & \\ldots & ~0 \\\\\n -1 & ~4 & -1 & ~0 & ~0 & \\ldots & ~0 \\\\\n ~0 & -1 & ~4 & -1 & ~0 & \\ldots & ~0 \\\\\n \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n ~0 & \\ldots & ~0 & -1 & ~4 & -1 & ~0 \\\\\n ~0 & \\ldots & \\ldots & ~0 & -1 & ~4 & -1 \\\\\n ~0 & \\ldots & \\ldots & \\ldots & ~0 & -1 & ~4\n\\end{bmatrix},\n", "2a1a083734b79351c959ba63d046ae07": "a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3 + \\cfrac{1}{\\ddots}}} ", "2a1a380e4ca82780e8af933df26a69bf": "EVI= G \\times \\frac{(NIR-RED)}{(NIR+C1 \\times RED-C2 \\times Blue+L)}", "2a1a3ac63d3878feb2926da45c60e852": " \\mathrm{li}(x)\\;=\\;\\mathrm{Ei}(\\ln{x}) \n\\color{white}........\\color{black} ", "2a1a527209cfe5365bc92bbdb42d22f7": "\\frac{\\partial w}{\\partial t} = \\tfrac{1}{2} \\frac{\\partial^2 w}{\\partial x^2} - u V(x) w.", "2a1ab2b7d7b224cd7517ff13528444de": "\\langle A\\rangle=\\int_{PS} A_{\\vec{r}} \\frac{e^{-\\beta E_{\\vec{r}}}}{Z} d\\vec{r}", "2a1ad090b56b1b1fe8cf6a1be9f6fc72": "x=2a\\sin^2{\\psi \\over 2},\\,y=a\\frac{4\\sin^4{\\psi \\over 2}}{2\\sin{\\psi \\over 2}\\cos{\\psi \\over 2}} = 2a\\frac{\\sin^3{\\psi \\over 2}}{\\cos{\\psi \\over 2}}.", "2a1adf42d7e04ffc5f6055e755f50a29": "x^3+12x=6x^2+35", "2a1ae89bf007f3ee3266a26263786b8d": "\\begin{smallmatrix}D=\\frac{1329}{\\sqrt{p}}10^{-0.2H}\\end{smallmatrix}", "2a1b077fbc2883c2e7f98d2a03172456": "b^{1/m},", "2a1b71b785c0cb14bb665bfef5a37b68": "a=1.", "2a1b8feb8bbe0e107437fb9f56427db9": "\n\\vec{\\nabla}\\cdot\\left[\\epsilon(\\vec{r})\\vec{\\nabla}\\Psi(\\vec{r})\\right] = -4\\pi\\rho^{f}(\\vec{r}) - 4\\pi\\sum_{i}c_{i}^{\\infty}z_{i}q\\lambda(\\vec{r})e^{\\frac{-z_{i}q\\Psi(\\vec{r})}{kT}}\n", "2a1c0185866bec3f9bc2b533bbf1f31c": "\\scriptstyle\\ell", "2a1c67367c24eefa5c95a18f78bb7ea8": "R:=|Z|", "2a1d20e9fc66eaedf131229280ed9b34": "1-\\frac{T_2}{T_1}", "2a1d475049738d9337a1e856643db61b": " (r - 3)(r^{2} + 2r + 2)^{2} = 0 \\, ", "2a1ded6238bb9b4778914101e2311b2c": "Z(N) \\, = \\, \\frac {\\zeta^{N}_{L}}{N!} \\, \\frac { N_{S}!} { (N_{S}-N)!} ", "2a1df486a87e8737941cc2b2e2a1f0a0": " X,Y,Z,ZZ ", "2a1e295ebebb18b4e00c488c6a71e8ec": " \\frac{dE_1}{dt} = \\beta(t) I_1 - (\\nu +a ) E_1 ", "2a1e31153dee09700368cbbf60b1ce2d": "u(t,x)\\,", "2a1e5fbca0bab65beb68f6f9396d8fee": "\\left\\{\\mathcal{M} f\\right\\}(s) = \\left\\{\\mathcal{B} f(e^{-x})\\right\\}(s)", "2a1e86ae570225942f043e0e7561ccb2": "T(m) = \\mathbf{A}m + \\mathbf{t}", "2a1ea1961cf5f449ab972b7ea2463efe": "| \\psi(t) \\rangle = e^{-iHt / \\hbar} | \\psi(0) \\rangle.", "2a1eccdb3aee49b77b42ad170c7691d2": " x_0\\in U ", "2a1f64638e9b7a8b668fa58e2ed6a75a": " \\begin{align} \n\\hat{T}_{xx} & = \\frac{\\hat{J}_x^2}{2I_{xx}} \\\\\n\\hat{T}_{yy} & = \\frac{\\hat{J}_y^2}{2I_{yy}} \\\\\n\\hat{T}_{zz} & = \\frac{\\hat{J}_z^2}{2I_{zz}} \\\\\n\\end{align}\\,\\!", "2a1f88876c43ac03f02ba150f54d3e65": "u, v \\in \\Sigma^*", "2a1fa2da6a2457eac5e2ffedd4cc09ce": "10\\uparrow\\uparrow 5", "2a1faf84f280f6650fba68e7af1a311c": "\\sigma_0=1", "2a1fc25e8f5512d17e1b177425e9253a": "X \\in \\Gamma^{\\infty}(TM)", "2a1fc8f936915481be3f1b92ab81c56c": "[K(u+z,v+w)-K(u+z,v)-K(u+z,w)-K(u,v+w)-K(z,v+w)-K(v+z,u)+K(u,w)+K(v,z)]-^{}_{}", "2a1fe29cf99afa19c36d6b5506080bec": "s_1 \\approx t_1 \\land \\ldots \\land s_n \\approx t_n \\rightarrow s \\approx t", "2a200635c87b488187f354388fee9588": " \\overrightarrow{a b} \\ \\overleftarrow{c d} \\ \\widehat{d e f}", "2a20df7e8905936961a0b65a72dfd8ad": "(\\gamma_1)^2 =\\frac{(\\operatorname{E}[(X - \\mu)^3])^2}{(\\operatorname{var}(X))^3} = \\frac{4}{(2+\\nu)^2}\\bigg(\\frac{1}{\\text{var}}-4(1+\\nu)\\bigg)", "2a2101b79eb22e27e57abd6749794772": "(y^2)", "2a2132050c4a42d99836ee4101746879": "\\tilde{Y_k}= \\mu + x_k^T\\beta +\\xi_k ,", "2a214600959b181ba2930489f8a5b038": "\\beth_1=\\aleph_1.", "2a215a9bc9019a8fbbb763cb52a49a1d": "{}_kp_x \\, q_{x+k} = {}_k p_x - {}_{k+1}p_x", "2a2191e419f7ef956051cbe5799b0e42": "F(y) = -2 \\int_y^\\infty f'(r) \\sqrt{r^2-y^2} \\, dr.", "2a2199e97cce0d372e8a775def72b41f": "\\alpha\\beta = m_{1} \\,\\!", "2a21fbc692dc434483692daa2f48e75a": "\\Phi( x_1 + x_2 ) = \\Phi( x_1 ) + \\Phi( x_2 )", "2a221bd9da57ef127f66b82db3569391": "PPP: {\\Delta}E_t(S_{t + k}) = {\\Delta}E_t({p^$}_{t + k}) - {\\Delta}E_t({p^c}_{t + k})", "2a2228ff151f9860936d3a0a3b93bdf2": "\\sigma(t) > t", "2a2280388e1b1f0171ce24e2189d8ee6": "\\langle a \\rangle \\phi", "2a228a4fecc0a269a081cf31860d1dcf": "\\rho = \\sum_i p_i v_i v_i ^* \\; , \\; \\sigma = \\sum_i q_i w_i w_i ^*.", "2a22b57e4a169f4bc04652d1adc75c28": "\\frac{\\partial s_c}{\\partial t}= \\mathbf{J}_u \\cdot \\nabla (1/T)+\\mathbf{J}_\\rho \\cdot \\nabla (-\\mu/T)=\\sum_\\alpha \\mathbf{J}_\\alpha \\cdot \\nabla f_\\alpha ", "2a22cf5d815cb316ab923e1749d5da5c": "X^{\\{q\\}}", "2a22e4cf766ad85a305047b4ef10cfee": "A \\subseteq A\\,\\!", "2a22f4692252da78b9f6bc072647c903": "\\mathrm{PbO + 2 \\ NaOH \\longrightarrow Na_2PbO_2 + H_2O}", "2a22fbb73ffd19809361256cb6014985": "\nP_{\\nu-\\frac12}^\\mu(z)=\\frac{\\sqrt{2}\\Gamma(\\mu-\\nu+\\frac12)}{\\pi^{3/2}(z^2-1)^{1/4}}\\biggl[\n\\pi\\sin\\mu\\pi P_{\\mu-\\frac12}^\\nu\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)+\\cos\\pi(\\nu+\\mu)e^{-i\\nu\\pi}Q_{\\mu-\\frac12}^\\nu\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)\\biggr]\n", "2a23149ae45f29affd62741f0e16717a": "p_B(x,y)", "2a233cdc29af0cedbdea918ab539552d": " \\hat{C} = (C, D) = c_0 + c_1 i + c_2 j + c_3 k + d_0 \\epsilon + d_1 \\epsilon i + d_2 \\epsilon j + d_3 \\epsilon k, ", "2a2347918b9338393240fd477f4940c4": "x \\leqslant \\mu -\\sigma/\\xi\\,\\;(\\xi < 0)", "2a236cfddec3c24d408cc331ba9c4b1e": "|\\Psi\\rangle_\\nu= |\\Psi_0\\rangle_\\nu \\oplus |\\Psi_1\\rangle_\\nu \\oplus |\\Psi_2\\rangle_\\nu \\oplus \\ldots = a_0 |0\\rangle \\oplus |\\psi_1\\rangle \\oplus \\sum_{ij} a_{ij}|\\psi_{2i}, \\psi_{2j} \\rangle_\\nu \\oplus \\ldots ", "2a2370ba5c243a1354df30333feec6ac": "P(G,t)", "2a237e54504442e3d483a39f75df7bfa": "s_k", "2a23d286d2354cfea2824c27c85b66b5": "\\ 0 \\leq y < 8", "2a241647568728b294d9b8b755b6f9ec": "\\textstyle{\\frac {\\log(12)} {\\log(1+\\varphi)}}", "2a243a2434fc20d587a2c25614d75da0": "((p \\to q) \\land (r \\to s) \\land (p \\lor r)) \\vdash (q \\lor s)", "2a24435e307c6dd596d0467c056f3ac7": "\\Omega(n \\log_2 n)", "2a249ba0a743a2975e5ad0efddbdfd52": "F_1(x,y)=x^2 y", "2a24c0cbcb1df0aeee33fd03af66e9e2": "Q_B(l_A a_B + l_B) l_B", "2a2521975c37318cf1835c1311f429f9": "\\vec Q", "2a2532a8da2a32e04a43fc1536ab8fbf": "\\sqrt{\\frac{4}{5}}\\!\\,", "2a2593f473c1be71656904b8225124ba": "\\beta(a,i) = a_i", "2a2605f59efd53f99ecb1c14b8db16cd": "\n L^{-1}(x) \\approx x \\frac{3-x^2}{1-x^2},\n ", "2a260c36fb9d72d172b6e96f239f16cf": "C^1(\\mathbb{T})\\subset A(\\mathbb{T}).\\,", "2a2615c4659cbd4b6a48765a54a7c36f": " X'=(x_1',...,x_d')\\in\\mathbb{F}_{q^n} ", "2a262bf55f06f61f1bacd8e5f5d1083a": " t\\in T ", "2a2636b59bef361875e3131c89a6dae2": "\\frac{\\lambda }{-i t+\\lambda }, \\quad \\frac{\\lambda }{i t+\\lambda }", "2a266d274c85c650d1d139542c834a93": "{\\varphi}^2 - \\varphi - 1 = 0.", "2a269500dc2af75c1e5cc8201606130e": " C_1 \\, ", "2a2695306051c7f6c67a2f1a6d13ae9d": "\nH_{r-U}=\\mathbb{I}\\otimes H_U+S_{x_r}\\otimes S_{x_U}+S_{y_r}\\otimes S_{y_U}+S_{z_r}\\otimes S_{z_U}\n", "2a26cd71d68082b8edb871857aa10ff9": "e^{ix} = r (\\cos(\\theta) + i \\sin(\\theta))\\,.", "2a2737b7ebbf5a9952100be22d45dace": "\\frac{\\partial p}{\\partial T}=\\frac{\\beta _p(T,p)}{\\kappa _T(T,p)}", "2a273f8d688f7d23c5f6a690da604a45": "\\Gamma \\vdash id(n):int", "2a27a4874f5739de5d2947d12ac81d4b": "\\textstyle k=0", "2a27c741425676342d9efd1e7d149aa1": "00, \\sum_{i=1}^n a_i=1, f_i(\\cdot ; \\theta_i)\\in J\\ \\forall i,n\\right\\}", "2a2e0493a64d66081c0e53aa193b533a": "\\frac{A}{H} = \\frac{A^{2}}{G^{2}} = \\frac{G^{2}}{H^{2}} = \\phi \\,", "2a2e2c4b9013a65c5cb3adbd9ec69e9a": "8b + 6 = 2(4b+3)", "2a2e73d268028059550780e1f8a4f43b": "Q(A \\times B)", "2a2eb3387bb1414cad18eaa6ce3eb087": " u(t) \\ ", "2a2eb6c3280ccb17bf08670edd5faff3": "\n \\boldsymbol{\\nabla}f = \\cfrac{\\partial f}{\\partial r}~\\mathbf{e}_r + \\cfrac{1}{r}~\\cfrac{\\partial f}{\\partial \\theta}~\\mathbf{e}_\\theta + \\cfrac{\\partial f}{\\partial z}~\\mathbf{e}_z\n ", "2a2f6f3190c37c035cbc9488c0fbf2c3": "E[2n]\\setminus E[2]", "2a2fc404ec52a7ca954d49edb8dbbd7c": "E_u", "2a2fc85de6a7b367222deb1550fee5c9": " I = I_{ion}^{sat}(-1+e^{(V_{pr}-V_{fl})/(k_BT_e/e)} )", "2a303c6890f1cc421acacfa3cb8db5f8": "a,\\, b,\\, c\\!", "2a3043b425c974a5969651011b021a68": "\\displaystyle \\partial_t u + \\partial_x^3 u \\pm 6u^2\\partial_x u = 0", "2a304a1348456ccd2234cd71a81bd338": "link", "2a30b349d0a4a081dd38636783cd0afd": "\\dim~S'<\\dim~S", "2a30c37c4c2a8c7668198e20af85032d": "\\text{side }a : \\text{side }b = {a^2 - 1 \\over 2} : \\text{side }c = {a^2 + 1 \\over 2}.", "2a30cdd7db377caaaa18b1cb7567b95a": "x^2 + 10x + 28.\\,\\!", "2a314a60425dcbe24fdde0537b8a4d62": "g^{M_1\\times M_2}_{(p,q)}\\colon T_{(p,q)}(M_1\\times M_2)\\times T_{(p,q)}(M_1\\times M_2) \\longrightarrow \\mathbf R,", "2a31a2bfb978c1998651fd245c5ea09c": "f = p \\circ h", "2a31dbb265a3a7f8fb10b06493f47efb": "x_{n}=\\sum_{k=1}^d b_k x_{n-k}", "2a31ee186aa992782442a7800e620727": "m^2 = E^2 - p^2 \\,\\!", "2a320fe5796207c988ccd190b1e21019": "B \\equiv -{\\rho^\\prime\\over\\bar\\rho}g = g{T^\\prime_v - \\bar T_v \\over \\bar T_v}", "2a326a3bca92d53a862c650f6e9a9bfb": " \\begin{align}\n&Z_{\\mathbf C} \\cdot \\mathrm{SL}(m, \\mathbf C)\\cdot \\mathrm{SL}(n, \\mathbf C) &&\\subset \\mathrm{Aut}(\\mathbf C^m\\otimes\\mathbf C^n)\\\\\n&Z_{\\mathbf C} \\cdot \\mathrm{SL}(n, \\mathbf C) &&\\subset \\mathrm{Aut}(\\Lambda^2\\mathbf C^n)\\\\\n&Z_{\\mathbf C} \\cdot \\mathrm{SL}(n, \\mathbf C)&&\\subset \\mathrm{Aut}(S^2\\mathbf C^n)\\\\\n&Z_{\\mathbf C} \\cdot \\mathrm{SO}(n, \\mathbf C)&&\\subset \\mathrm{Aut}(\\mathbf C^n)\\\\\n&Z_{\\mathbf C} \\cdot \\mathrm{Spin}(10,\\mathbf C )&&\\subset \\mathrm{Aut}(\\Delta_{10}^+)\\cong \\mathrm{Aut}(\\mathbf C^{16})\\\\\n&Z_{\\mathbf C} \\cdot E_6(\\mathbf C)&&\\subset \\mathrm{Aut}(\\mathbf C^{27})\n\\end{align}", "2a3285cded628c0dd59f324e00303014": "CandS_{k+1} = CandS_{k+1} \\setminus \\{cand\\}", "2a32afbc281a76e84b416bd326c4460a": "\n \\Gamma^{(\\lambda)}(\\mathbf{x})= \\Gamma^{(\\lambda)}\\Big(R(\\mathbf{x})\\Big)\n", "2a32b516ee81b14cd9c135de6a4f74ce": "\\ \\Psi(r)=A e^{i (k r-\\omega t +\\phi)}/r", "2a32bf086267d6a8f6f685d591c5b40f": "E=\\int_{t_1}^{t_2} C(t)\\, dt", "2a32c428db3941d8c057740f4ed706e4": "\\forall x\\,(x \\neq \\emptyset \\rightarrow \\exist y \\in x\\,(y \\cap x = \\emptyset))", "2a3350d47bc44d98bb52964794c00e62": " \\log_3 (243) = \\log_3(9 \\cdot 27) = \\log_3 (9) + \\log_3 (27) = 2 + 3 = 5 \\,", "2a33563d68456c3d6e3cd1b5dfd24171": "s =\\frac{2}{T} \\frac{(z-1)}{(z+1)}", "2a33590a99ada7fbabb125ba1340ed85": " f(u) = a u ", "2a3395f4cb18929ebd6c171c969c9c28": "U(r) = r^2\\log r^2\\!", "2a33be8803306fbe7cccb5bd56610b76": "(W_n)_{n\\ge1}", "2a33c3a2ebf635211c858740d228f94d": "\\sigma : \\Delta\\ ^p \\rightarrow\\ X", "2a341744bed2921189a23f5d3f6f3604": "y^{(p-1)(q-1)/r} \\not \\equiv 1 \\mod n", "2a344855a5932d56d2ed1796e2f66907": "\\scriptstyle\\left(\\omega,k\\right)", "2a34592c8a3a9fa7839362bd86e29cc9": "\\displaystyle{ |f^\\prime(0)| \\le |g^\\prime(0)|}", "2a346327e7c2023d9fe0b4ea1f0c3679": "E(\\ln(x^2\\!+\\!\\nu))=\\log \\left(\\nu\\right)\\!-\\!\\psi \\left(\\frac{\\nu}{2}\\right)\\!+\\!\\psi\\left(\\frac{\\nu\\!+\\!1}{2} \\right)\\,", "2a34697290f9c5e7bb542e2da79e1f8c": "\\delta_j=r_{j+1}+r_{j+2}+\\cdots +r_k", "2a354343b2cf4634669c5fb8748a88f4": " s=(d, \\sigma) \\in S ", "2a356a8532983440e83689f05d71b674": "\\sum_{i=0}^n{i^2\\binom{n}{i}^2}=n^2 \\binom{2n-2}{n-1}.", "2a362c5a5a602ac3a32b5b0d4eeabbf7": "\n\\begin{align}\n&x = r \\cos \\theta \\cos \\varphi \\\\\n&y = r \\cos \\theta \\sin \\varphi \\\\\n&z = r \\sin \\theta\\,,\n\\end{align}\n", "2a363ab6c105e662753977ea73bd04db": "\\bigcup_iF_i(\\lambda)=I(\\lambda).", "2a36882390fea32260733176794f1bac": "w\\Vdash B[e]", "2a369c8fbe22ab1ab35f4d91cb3e4211": "X \\widehat{\\otimes}_\\pi X", "2a36a04036a0d4bacb30391ac66e09a3": "\\frac{2n\\overline{x^2}}{\\chi_2^2} \\leq \\widehat{\\sigma}^2 \\leq \\frac{2n\\overline{x^2}}{\\chi_1^2}", "2a36a0f82e31112cc28423568c49170e": "\\dfrac{-\\sum_{x\\in R}F(x+h)G(x)}{\\sqrt{\\sum_{x\\in R}F(x+h)^{2}}\\sqrt{\\sum_{x\\in R}G(x)^{2}}}", "2a36a53f0b64ec6479f174c368d6da27": "G//K", "2a3715f2d8fc7fbac2c16b90b5fad89c": "k=a^2+b^2", "2a374d881c356c7775efa0b97598f24e": "\\hat{\\mathbf{x}}_2", "2a37995ff7462fd1d62e5d5d0e6170ec": " m \\frac{d^2u}{dt^2} + m \\frac{\\Omega^2}{4}[a_u-2q_u\\cos (\\Omega t) ]u=0 \\qquad\\qquad (3) \\!", "2a379ec74d628cd01b518cd8fab49f1e": "\\begin{align}\n & X_n \\ \\xrightarrow{d}\\ X,\\ \\ \n X_n \\ \\xrightarrow{\\mathcal{D}}\\ X,\\ \\ \n X_n \\ \\xrightarrow{\\mathcal{L}}\\ X,\\ \\ \n X_n \\ \\xrightarrow{d}\\ \\mathcal{L}_X, \\\\\n & X_n \\rightsquigarrow X,\\ \\ \n X_n \\Rightarrow X,\\ \\ \n \\mathcal{L}(X_n)\\to\\mathcal{L}(X),\\\\ \n \\end{align}", "2a37a48106b92dbb748d6187734ceeab": "y_{v}=-\\left( a + \\frac{b}{f} \\right)v+b", "2a37b293d3ac6e7b42f6e3591beebc09": "_k", "2a37ba234a38c875d7139571a9153e21": " u(l) ", "2a3819243835a96b2459410b9c637205": "\\ddot u_i=\\left(\\frac{f}{m\\ \\Delta x} \\right) \\left(u_{i+1} + u_{i-1}\\ -\\ 2u_i\\right)", "2a384c8b38f71bfc1598605c096c4257": "\\alpha^{-1}", "2a3874e4d874d895a318ab6211199d81": "[X, Y]_x := \\left.\\frac12\\frac{\\mathrm{d}^2}{\\mathrm{dt}^2}\\right|_{t=0} (\\Phi^Y_{-t} \\circ \\Phi^X_{-t} \\circ \\Phi^Y_{t} \\circ \\Phi^X_{t})(x) = \\left.\\frac{\\mathrm{d}}{\\mathrm{d} t}\\right|_{t=0} (\\Phi^Y_{-\\sqrt{t}} \\circ \\Phi^X_{-\\sqrt{t}} \\circ \\Phi^Y_{\\sqrt{t}} \\circ \\Phi^X_{\\sqrt{t}})(x)", "2a38b64927a3b10a68c4e761db59cce6": "\\delta(\\vec{x}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\rho(\\vec{x})}{\\bar{\\rho}} - 1 =\n \\int \\text{d}k \\; \\delta_k \\, e^{i\\vec{k} \\cdot \\vec{x}},", "2a38b8f545f1188b3bab6bdf4eda0600": " \\displaystyle C^\\bullet(X) \\times C^\\bullet(X) \\to C^\\bullet(X \\times X) \\overset{\\Delta^*}{\\to} C^\\bullet(X) ", "2a38ef290abb87764f021370cb3dfd84": "\\textstyle v\\in P_{m-1}", "2a393aac500bfd2040114fe98bb74054": "(+,---)", "2a39488897631d241787a9e3884af9ce": "=-\\frac{1}{4\\pi}\\left[\\boldsymbol{\\nabla}\\left(\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\cdot\\boldsymbol{\\nabla}\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\\right)+\\boldsymbol{\\nabla}\\times\\left(\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\times\\boldsymbol{\\nabla}\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\\right)\\right].", "2a395ec4064136d07e6d0ffaa27a4965": "\\mathrm{SNR} = \\frac{ | {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{-1/2}s) |^2 }\n { {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h) },", "2a3968e83c005fbfc1c323612b2a2a6d": "- \\int \\underline {A}(a, \\lambda)\\underline {B}(b^\\prime, \\lambda)[1 \\pm \\underline {A}(a^\\prime, \\lambda)\\underline {B}(b, \\lambda)]\\rho(\\lambda)d\\lambda \\qquad (6)", "2a39802a7874922d89108370900605ae": "U^* W_m U = V^* W_n V = I.", "2a3a1c09703280cb029e7b11ef54f78f": "B_3=(3-x)^2/2 \\qquad 2 \\le x \\le 3", "2a3a1f084488c8e8fb9bbe6d75386635": "\n\\begin{cases}\n(P_1=1) \\to (P_{4}=1) \\\\\n(P_2=2) \\to (P_{4}=1) \\\\\n(P_1=2) \\and (P_2=0) \\to (P_{4}=1) \\\\\n(P_3=0) \\and (P_2=0) \\to (P_{4}=1) \n\\end{cases}\n", "2a3a35b54519f3565335141886993b1c": "\\int \\cosh ax \\cosh bx\\,dx = \\frac{1}{a^2-b^2} (a\\sinh ax \\cosh bx - b\\sinh bx \\cosh ax)+C \\qquad\\mbox{(for }a^2\\neq b^2\\mbox{)}\\,", "2a3a45e29c643e8437e88e1e005f60c3": "\\forall i \\in \\{1,\\ldots, n\\} : X_i \\sim \\Gamma(k_i,\\theta) \\qquad \\Rightarrow \\qquad \\sum_{i=1}^n X_i \\sim \\Gamma\\left(\\sum_{i=1}^nk_i,\\theta\\right).", "2a3ad4248c66e755cc198ee651f37b68": "-\\frac{\\frac{\\partial V}{\\partial p_{1}}}{\\frac{\\partial V}{\\partial Y}}=-\\frac{-\\lambda x_{1}^{m}}{\\lambda}=x_{1}^{m} ", "2a3b6415d38cf1d2012afd7d40bc4d69": "y(t) = \\sum_{k=1}^{K} r_k(t) \\cos\\left(2 \\pi \\int_0^t f_k(u)\\ du + \\phi_k \\right).", "2a3b875f824a3f2dec71c2b23c7022bb": "1-1\\sqrt{2}=-0.41421\\ldots", "2a3bc78c3510586a33349a8f539c588f": "cm^{-1}", "2a3be5560b30f344f5f07a87a0af61f1": "x < z_1 < y", "2a3bf6b3b5db9cf06de87cfaadc45cd0": "\\text{IDF}(q_i) = \\log \\frac{N - n(q_i) + 0.5}{n(q_i) + 0.5},", "2a3bfecf3c9362c0d2d0b4038e1a2349": "\\mathrm{hub}(p)=\\displaystyle\\sum_{i=1}^n \\mathrm{auth}(i)", "2a3c08f92ea6d6f639bdc5cf6c1f44f4": "m_g-m_{od} \\over m_{od}", "2a3c58261a2494c9a818927ebdd9c3c2": "K_{\\rm a} = {k_{\\rm on} \\over k_{\\rm off}} = {[{\\rm RL}] \\over {[{\\rm R}]\\,[{\\rm L}]}}", "2a3ce4e2236ba2897f23a36d04174090": "\\psi>0", "2a3df9da42b701568e349fab75f8ea3c": " x = \\int \\sqrt{1-4u^2} \\, du ", "2a3e2b82284e2e4ab2188bedd66ad7d8": "x^TAx < \\hat{x}^TAx", "2a3e8c2471bd4f461e8359fe941c113e": "(\\eta,\\eta)_{K}", "2a3efc0bfd33de261779d6b1a6fb1327": "\\phi(x,y,z)", "2a3f1306e4b08a4dc43e8f9b48558070": "(x,y)\\in \\Delta^+", "2a3f3c8ece6c12826f13743ad7160bdc": "\nE_{\\alpha ,\\beta }(z)=\\sum\\limits_{m=0}^\\infty \\frac{z^m}{\\Gamma (\\alpha\nm+\\beta )},\\qquad E_{\\alpha ,1}(z)=E_\\alpha (z). \n", "2a3f5329af5a0607bd119abc335c6c76": "N=\\{a\\}^*\\times\\{b,c\\}^*", "2a3f9c3d8d53d76b0041781d02ebf3cd": "|z| \\ge R", "2a3fa0b1e1b34e0345f79f250a51a4cc": "\\dim W + \\dim W^\\perp = \\dim V.", "2a3fb318540c1499daca21180db95122": "\\mathbf{q}(s)=t\\cdot\\mathbf{s}_u + n\\cdot\\mathbf{n}_u", "2a3fd8936124f579c04718b81898d7db": "e^{-t\\tau}=e^{-tr^2}", "2a3ffaecf926bc2eeb31a2392da23927": "v_k \\operatorname{adj} v_i", "2a3ffc1fe6deda81ca6de6db7ad40f74": "a=\\sqrt{2Rd-d^2}", "2a4029806a1ecfa1c6baccdbc1bf7095": "W^0 = -\\vec{J}\\cdot\\vec{P}.", "2a40a507b92764b40bccdc7203cf06ba": "\\langle \\cdot,\\cdot\\rangle ", "2a40d165750d0c78dbe4fcd08b7ecd90": "\\tfrac{1}{2}Q_3", "2a411213fdbba14372884857b0e003be": "\\Delta_{so} ", "2a4161d939f1fafa65a5d72cfe5b8788": "f_{Y_{[r_1:n]}, \\cdots, Y_{[r_k:n]} }(y_1, \\cdots, y_k) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^{x_k} \\cdots \\int_{-\\infty}^{x_2} \\prod^k_{ i=1 } f_{Y\\mid X} (y_i|x_i) f_{X_{r_1:n}, \\cdots, X_{r_k:n}}(x_1,\\cdots,x_k)\\mathrm{d}x_1\\cdots \\mathrm{d}x_k ", "2a41a2a2dd2679b71d3b3babef699eb0": "x^i(t),\\; i \\in \\{1,2,3\\}", "2a41cd9346229d40f454fc43a0e2ccbb": "p(k) = \\frac{\\lambda^k}{k!} e^{-\\lambda},", "2a41cf9d12a073c59ddb6de8a3885491": "R_{max-load} = \\frac{5\\,V}{3\\,A} = 1.67\\,\\Omega", "2a41de6eee1ff5a07b78400aa841db24": "W(x)=\\ln x-\\ln\\ln x+o(1).", "2a4211efe0861b6366f554fca984a92e": "U = \\int_0^{\\sqrt[3]{N}}\\int_0^{\\sqrt[3]{N}}\\int_0^{\\sqrt[3]{N}} E(n)\\,{3\\over e^{E(n)/kT}-1}\\,dn_x\\, dn_y\\, dn_z\\,.", "2a42a0832f156c34d25b49e3f2f9b820": "C_{ij} : M \\times J \\to [0, + \\infty]", "2a42a2aa4ae9ab76c00e6eff9d8393d3": " {\\rm div}_{\\rho} X := \\frac{(-1)^{\\left|x^{i}\\right|(|X|+1)}}{\\rho} \\partial_{i}(\\rho X^{i}) ", "2a42a7dddb1283af40fccb7e2493731f": "H^1(X, \\mathcal O_X) \\longrightarrow H^1(X, \\mathcal O_X^*) \\longrightarrow H^2(X, \\mathbb Z) \\longrightarrow H^2(X, \\mathcal O_X) ", "2a42b9f45f87aa44a54e02721cb288ea": " -\\infty < t_0 < +\\infty \\ ", "2a42f56a25307f4c31009a0ca00fd0e7": "G_{ik}=\\frac{1}{4\\pi \\mu r}\\begin{bmatrix}\n1-\\frac{1}{2b}\\frac{z^2}{r^2}&0&\\frac{1}{2b}\\frac{\\rho z}{r^2}\\\\\n0&1-\\frac{1}{2b}&0\\\\\n\\frac{1}{2b}\\frac{z \\rho}{r^2}&0&1-\\frac{1}{2b}\\frac{\\rho^2}{r^2}\n\\end{bmatrix}\n\\,\\!", "2a4302f02a95266b1edb3c8f196d21ac": "N_\\mathrm{P} = \\frac{\\text{Restoring force}}{\\text{Adhesive force}}", "2a431161fe196cee8d870ded9c1bab24": " S'' = R_{space-time}^{-T} \\, S \\, R_{space -time}^{-1} = \\begin{bmatrix} \\lambda_1 & & \\\\ & \\lambda_2 & \\\\ & & \\lambda_3 \\end{bmatrix} ", "2a4332eb0de0ef99e96384f718140a91": "t\\leq T", "2a435f36835a6edfa0b631fc50d60821": "g : \\mathbb N^* \\to \\mathbb N", "2a43afcd1378e6d04de06f4a1ecf85d0": " g(x)=\\mathbf{1}_{x\\geq 1-\\alpha}", "2a4449ec8cea431598c825568415e5b3": " \\rho^L_{i + \\frac{1}{2}} = \\rho_{i} + 0.5 \\phi \\left( r_{i} \\right) \\left( \\rho_{i+1} - \\rho_{i} \\right),\n \\rho^R_{i + \\frac{1}{2}} = \\rho_{i+1} - 0.5 \\phi \\left( r_{i+1} \\right) \\left( \\rho_{i+2} - \\rho_{i+1} \\right),", "2a44675cdf1710be632ad44cc0682632": " |0 \\rangle \\text{ and } |1 \\rangle ", "2a4468fb520206b5a5a5cc17f08a43a6": "\\sum c_n", "2a447b5b915e69f3ffa745f88f7bd310": "H^2(\\mathfrak{g}; M)", "2a44934587ee17ff223642c01e1331fe": "|\\mathbb{Z}_k|^{|\\mathbb{Z}_k^n|} = k^{k^n}", "2a44a0bac794bacf440be5efcc8890e2": "\n[S]=\\frac{-j}{\\sqrt{2}}\\begin{bmatrix}\n0 & 1 & 1 \\\\\n1 & 0 & 0 \\\\\n1 & 0 & 0 \\\\\n\\end{bmatrix}\n", "2a44afe63ca679434e408b2606cb671a": "\\log(1+x) = x - \\frac{x^2}2 + \\frac{x^3}3 + {O}(x^4)\\!", "2a4500a399b4ea862c00ba50fc2d5dd5": "\\textbf{false}\\mapsto 0; \\textbf{true}\\mapsto1", "2a452dba97c3bfc2abeecd88e02bdee2": "\\mathbf{u}(\\mathbf{x}) = \\iiint_{\\mathbb{R}^3} \\widehat{\\mathbf{u}}(\\mathbf{k})e^{i \\mathbf{k \\cdot x}} \\mathrm{d}^3\\mathbf{k}", "2a453924de2f097fc356dd0a2ac57885": "G_{1} =\n \\begin{bmatrix} c & -s & 0 \\\\\n s & c & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}", "2a4543fb5c57cd83b2ca569cb6ecd2a6": "H_p = T (C + \\log(t)) \\,", "2a4581735f8db6e7c1aa08eea51c380c": "[U]_e=[V]_e ", "2a46119ba5f308c53638841a3b418ded": " {V_k} ", "2a465165b1c2a8e2034ddc514a7e67d9": "\n\\begin{array}{c|c|c}\n\\times & a & bi \\\\\n\\hline\nc & ac & bci \\\\\n\\hline\ndi & adi & -bd\n\\end{array}\n", "2a46534e6cdc3f649210abc0fc44ff83": "\\mathrm{vec}(\\mathbf{X}) \\sim \\mathcal{N}_{np}(\\mathrm{vec}(\\mathbf{M}), \\mathbf{V} \\otimes \\mathbf{U})", "2a46b6a1dfe075c926884c495b948f6d": "\\displaystyle E", "2a47048f5bb09dae91cd8173b3d2a127": "\\xi_{i,1}^2(t) + \\cdots + \\xi_{i d}^2(t)", "2a473d8505c91c6d8dd44c21f0ba5e5d": "1+z^2", "2a4790de836da858b34777c7d4198ec6": "\\zeta(2)=\\pi^2/6", "2a47a8122adc7296a507df690ecb562a": "R_\\mathfrak{p}", "2a47c7bb3d64482d81f55e0a6db6cf33": "W(t_0,t_1) = \\int_{t_0}^{t_1} \\phi(t_0,t)B(t)B(t)^{T}\\phi(t_0,t)^{T} dt", "2a47ef83398cedddf2e09eae964550af": "\\frac{\\mathrm{d}}{\\mathrm{d}\\alpha}\\,\\varphi(\\alpha)\\,=0\\,", "2a4824f296090685b534e8ecf63349da": " \\triangle AXX' \\sim \\triangle CYY'',\\, ", "2a48696bc40bb8d792c9cb2ed7aa2e65": "\n |h(t)| = \\frac{A}{2} |Q(\\omega)|\n", "2a486f564376bc8458a88df60db9193f": " \\text{(1)} \\qquad d U + \\delta W = \\delta Q = 0, ", "2a4898b44c27db92276711efa6fb4694": "\\scriptstyle{t_{r_S}}", "2a489ca0d1bab37890226fc79d12258e": " ^{a^{b^{b^{c}}}}", "2a48c169963677cec121e97f5bd8777c": "\\Phi=(\\phi_0/\\sqrt{2})e^{i\\omega t} ", "2a48c179fc89edd634acc61b1f189c71": "d ( \\lambda ) = m+n", "2a494032a175d54e0161cefaac7bc587": "\\partial{(\\bullet)}/\\partial x_j", "2a496865b327b4822c6acffa80b05e45": "(\\Psi_t)(s)=C{\\rm e}^{As}x", "2a49e8041a74b6041606c6ecb22b720e": "118_{11} \\ ", "2a4a3db9a5bb6d9c2af07fd472ddc2eb": "\\{ \\rho_2^k \\}", "2a4a85f2834625ee045316599e5771aa": "\\ln K= \\frac{nFE^0}{RT}", "2a4a9b70841f05ac509a904297b0dbf7": "L(s,\\chi)=\\frac {1}{k^s} \\sum_{n=1}^k \\chi(n)\\; \\zeta \\left(s,\\frac{n}{k}\\right).", "2a4aa7ba14d7e2bbf0870b61308bddc4": "\\Omega_x=\\Omega_y=0", "2a4ac447deccccc4073cfba3562dd003": "\n u = c(x) + d(x) y + L_{y}^{-1} \\rho(x, y) - L_{y}^{-1} L_{x} u - L_{y}^{-1} N u \n", "2a4ad1e773e0c5ab17189b0db306d397": "\\omega^1,\\dots,\\omega^r", "2a4b92dd4bfed320ae9431354237116e": "E(K) = \\{(x, y) | y^2 = x^3 + ax + b\\} \\bigcup \\{O\\}", "2a4be3cbea7aebd97bce5b82beed8fc8": "\\sum\\limits_{k=1}^n \\mathbf{v}_k c_k", "2a4bf0b2ba2a7b2e34c49ea7d41d7830": "{Z}_{{{\\mathbf{k}}}}(S)", "2a4bfb2622f6767547d7c6424d22442a": " Z(u,v) = r \\, v \\, \\cos u, ", "2a4c42e9440116184cd9d06ef16fc765": "m=(m_v, m_e)", "2a4cea84aa395d1d3dabbe3b0ccf9122": "\\sinh \\frac{x}{2} = \\sqrt{ \\frac{1}{2}(\\cosh x - 1)} \\,", "2a4cf356ab7c9c7a82c24abff667eac9": "t = \\frac{\\bar {X}_1 - \\bar{X}_2}{s_{X_1X_2} \\cdot \\sqrt{\\frac{1}{n_1}+\\frac{1}{n_2}}}", "2a4cfdfa1e1d7e559cb51939d64d61a5": "D_1\\cap D_2=\\varnothing", "2a4d16d2ddfec11769e10ec14cc307bc": "\\begin{array} {l}\nf'(x_0)=\n\\frac{f\\left(x_0 + h\\right) - f\\left(x_0 - h\\right)}{2h} + O\\left(h^2\\right)\n\\end{array}", "2a4d32fcd9c3574d17517aa3440f10d1": "{\\operatorname{d}Q\\over\\operatorname{d}t}", "2a4da84c0f3b35c8135389d847e04a4c": "[0,s(t)]", "2a4dd0d47640166fb29c653355a667e6": "\\mathcal{O}_{f^{-1}(f(x)),x}", "2a4df816076c6631641858c965fb51e3": "\\textstyle{\\frac{\\log(7)}{\\log(3)}}", "2a4dfbe648099168d5ad6ab27b9040bc": " V^{2}+D^{2}=1", "2a4e31d3ff21d1ff7e6fa23d203188f5": "cos\\Theta=(g_{11}+g_{22}+g_{33}-1)/2", "2a4e45a8d8e238bbfe776d2eee4c3178": "O(nh)", "2a4eb43f620faad7884591ceb39c3939": "\\mu(U ) = \\sup_{C\\subset U}\\lambda (C)", "2a4ec146b01a1e24f410e6574e06e9fe": "(1-x^2)^{3/2}=1-\\frac{3x^2}2+\\frac{3x^4}8+\\frac{x^6}{16}\\cdots", "2a4ee704fa05f339818f2013b42c7f56": " g(x_0) \\gg x_1 g'(x_0) ", "2a4f29cba2201f3c9758a9ca47068f64": "\\scriptstyle{\\bar k}", "2a4f2f8b6adc6686456f4d9152e97758": "f(v, q) = 0 \\,", "2a4f3b398f4355d8e982bc88050a6bb4": "|G| = |Z(G)| + \\sum_{i}d_i\\;", "2a4f955c219734ed64a8b01817e2a308": "\\lfloor (n+1)r\\rfloor-\\lfloor nr\\rfloor", "2a4ff5d81ccee839cf099a3651458d76": "\\color{red}z", "2a501498b8fe27704fc352a5e0d04fb0": "\\scriptstyle\\binom{M}{N}", "2a50ca89f24722237d9a60fafda138b1": "\\psi(x) = \\frac{\\lambda \\mu}{c}e^{-\\left( \\frac{1}{\\mu}-\\frac{\\lambda}{\\beta}\\right)x}.", "2a50d1f92a0dba81d64e2ff9233c04aa": "F_2\\left [\\frac{Sin(\\beta )}{Sin(\\alpha )}Cos(\\alpha )+Cos(\\beta )\\right ]=F_{load} \\,", "2a5117fd3a65d38ed99d04f101783ab4": "{AE}_{n}", "2a511ebaa78f3818e0a1b62482af7d09": "\\zeta(s) = \\frac{2^{s-1}}{s-1}-2^s\\!\\int_0^{\\infty}\\!\\!\\!\\frac{\\sin(s\\arctan t)}{(1+t^2)^\\frac{s}{2}(\\mathrm{e}^{\\pi\\,t}+1)}\\,\\mathrm{d}t,", "2a5143e0c7150f127d219582d8a68991": "n_S", "2a514b3878f2778b326f79236faae69e": "\\sigma = \\eta \\frac{d\\varepsilon}{dt}", "2a5154a5922c0fe043f53499092e9f73": "\\sqrt{n}\\theta/\\sigma\\,\\!", "2a517b9b580700b1b899b729e954ae20": "\\lambda\\ = B\\left(\\frac{n^2}{n^2 - m^2}\\right) = B\\left(\\frac{n^2}{n^2 - 2^2}\\right)", "2a5182d0a8ad6e18ef9ef481bc333fc1": "\n \\lim_{kh \\downarrow 0} \\mathcal{S} \n = \\frac{3}{32\\,\\pi^2}\\, \\frac{H\\, \\lambda^2}{h^3}\n = \\frac{3}{32\\,\\pi^2}\\, \\mathcal{U},\n", "2a51a6f7099f74d15319f41731c665a1": "\\partial_v", "2a51fdbbaa75421b4ff6ba35e8f6f20d": "\n A(t)=\\Pr[X(t)=1]=E[X(t)].", "2a520d199495532b2fc1b1e6e69475ca": "\\log \\log(n + 1) - \\log\\frac53 < \\sum_{p \\le n}{\\frac{1}{p}}", "2a5250a09e961452a75991f1165901c5": "\\Delta G_S^\\circ ", "2a525127c6bf6dfb96d92eb63343e140": "\\chi_A(x) = \n\\begin{cases} \n 0, & \\mbox{if }x\\notin A \\\\\n 1, & \\mbox{if }x\\in A\n\\end{cases}", "2a5271c118492b7bb2274dd278a033ba": "FG", "2a52859d462bdb208eeb4ef9c4830363": "|\\mathbf{u}|", "2a52afbae2e0c2143de30b0f058abe82": "\\operatorname{Pr}(H_i) = \\operatorname{Tr}(\\hat{C}_{H_i} \\rho \\hat{C}^\\dagger_{H_i})", "2a52b572526d7d9ab9781ca03131e480": "\nb_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{a_i}{b_i} = \nb_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{1}{c_i b_i}\\,\n", "2a53327a6fe2218a7212af817494f319": "A_iM_j \\subseteq M_{i+j}.", "2a536065a63877c5d1310224d9125cb1": "b(f_1,f_2) = \\frac{\\left| \\sum\\limits_{n} F_n(f_1)F_n(f_2)F_n^*(f_1+f_2) \\right|}{ \\sum\\limits_{n} |F_n(f_1)F_n(f_2)F_n^*(f_1+f_2)|} ", "2a5401275b5c30cea5e646e5f794c155": "m=[12.3, 7.6]", "2a5407e856e9fbfe3ee11f98cbab4125": "D_1=\\{(-1,1),(0,1),(1,1)\\}", "2a54671552370ef6f31c5d3eb3c067f4": "P = P_{(n_1,\\ldots,n_r)} = M \\times N", "2a54748fef870a547df66c96f4e4559c": "W(\\mathbf{q})", "2a54d14a352f5b4264366f290c400822": "\\scriptstyle 1/\\sqrt{a}=\\sqrt{1/a}", "2a54d7dc0a8e6fb656cab89646143533": "\\scriptstyle hf = \\Phi + E_k", "2a54efb08202d823a988aca1c4292f9f": "\\sigma_f = \\frac{F L}{\\pi R^3}", "2a553ac57487378d501037382a40e3f2": " \\rho_G = 1/N ", "2a555c494f1ae90e33bb77bc0de8a01d": "S_l(t+", "2a55b005defe9929b6ec8664a620ca2f": "{wL}/{Y}", "2a55ceb4d4c2f40f7eb129ab7bf2f6bb": " \\lambda x . y ", "2a55f0ed6542ae36fc08bfd43e6d5937": " \\iint g(x)g(y)\\,dx dy = \\int\\! g(x) \\,dx \\int\\! g(y) \\,dy = \\left(\\int\\! g(x) \\,dx\\right)^2", "2a56058435a3f2862e21665eb5d02737": "C = \\{ AN|N\\in \\mathbb{Z}, 0 \\leq N ", "2a56ff6fede086c326c3c75b83b3eea2": "(x - [[z]]) \\cup\n[[w]]", "2a575510cf03e310a80845b800f1a161": "V(q) = q^{-1} - q^{-2} + 2q^{-3} - q^{-4} + q^{-5} - q^{-6}. \\, ", "2a57993e0eb693047c78e4e63444baf2": "Y \\subseteq \\bigcup_{\\alpha \\in A}U_{\\alpha}", "2a57a92f7f0fa0e04746b8748d96b499": "(H, \\mu, \\eta)", "2a582983e1974256482aee85b73dea01": "\\text{Principal} \\times i = C", "2a585b485e638370d85a6f12cafa77f0": "\\alpha, \\beta \\ge 0 ", "2a58755d4380151d227e251268630443": "|R|<|S|", "2a588e18905e3e52d5541223afc4e9ee": "X_{1}=x", "2a58c442915ccbc2287755d7562f8f0f": "\\mathbb{Z}[q]", "2a5906cfab8d4b8fa48fd78dd97aab55": "\\scriptstyle dy \\;=\\; dz \\;=\\; 0", "2a5937b6d937ef3d69ebb65078f4e848": " \\mathrm{L} = \\sum_{i=1}^n n^{n-i} C_i { \\prod_{k=1}^{i-1} f_k } ", "2a595fbab6351f74976bdecacf8654d0": "P_\\lambda(A)=\\sum_{n=0}^\\infty P(A\\mid N=n)P(N=n)=\\sum_{n=0}^\\infty P(A\\mid N=n){\\lambda^n e^{-\\lambda} \\over n!},", "2a5980e31495b4a336ee6b135cefc60a": "\n \\hat{p}_j^{\\mathrm{after}} - \\hat{p}_j^{\\mathrm{before}} = \\hat{p}_j^{\\mathrm{b.a.}}\n = \\frac{2}{c}\\hat{\\mathcal{W}}_j \\,,\n", "2a598cdb9feedb4ee7a4bfa2d0d82fce": "b_k |0\\rangle=0", "2a59a977a493506456782c88d19622c5": "NL \\left [ u \\right ](i)= \\sum_{j \\in I}w(i,j)v(j),", "2a59f39beb154237f05379a4535b7de7": "W=[w_1,...,w_n]^T \\in \\mathcal{R}^{m \\times k}", "2a5a105f5edb4407ca9b65b1d08e7798": "\\lambda\\in \\mathbb{R} ^m", "2a5a1eba8d63ae2123117ad9292759bb": "| \\alpha\\rangle", "2a5a221dbf22f6c25de2ce481301360a": "m \\cdot 3:m\\ ", "2a5a352d72d8d7d46d2dfe1df6e0f5c9": "\\boldsymbol\\tau=\\boldsymbol\\Omega_{\\mathrm{P}} \\times \\mathbf{L}.", "2a5a465c08a86ab0914024d4ed65f196": "x_1M_1+x_2M_2", "2a5a465e7269129b2b5b7a9dfc96ad10": "Y:\\mathcal{A}\\rightarrow\\mathcal{V}^{\\mathcal{A}^{op}}", "2a5a6f8753c0138686e0eae19dc4cce6": "\\forall x\\,(Mx \\rightarrow Lx)", "2a5aad397bffbdcd0a3169d00c2f58d1": " i = { 6 {\\rm \\% \\ per \\ year} \\over 12 {\\rm \\ months \\ per \\ year} } = 0.5 {\\rm \\% \\ per \\ month} ", "2a5abcfb804496504ce67cbf9ad79236": " \\varphi_1( z ) = \\frac{ 1 - e^z }{ z },", "2a5ad5cacdf5de2a79d16e0895d6786a": "S^{-1}: \\mathbb{R}^2 \\hookrightarrow S^2", "2a5aec6002ed7bb864d5bbf5ce15bc9f": " u_i \\in [\\underline u_i, \\overline u_i] ", "2a5b995623e4ab087dd51fa989f23859": "Z_{2}", "2a5bfa1f1a77607d35d5121929663478": "\\nu=3/2", "2a5c215e87fc06b2a46e2b6ecfd05f43": "{NRI}", "2a5c40ce1f84451d40d4f38523217824": "T {f}(x) = \\frac{1}{2 \\pi} \\int_{\\mathbf{T}} {f}(y) g( x - y) \\, dy.", "2a5ce2fa3ecc2667f97523dc1ee57bc9": "M,w'", "2a5d60b373ee37c632874b597eb02dfe": " s=(\\ldots, (s_{i},t_{si}, t_{ei}),\\ldots)", "2a5d8d55d309750a51c2dfb64974bda2": "\\Delta I_{L_\\mathit{on}}=\\int_0^{t_\\mathit{on}}\\frac{V_L}{L}\\, dt=\\frac{\\left(V_i-V_o\\right)}{L}t_\\mathit{on},\\; t_\\mathit{on} = DT", "2a5d9ef7ca93ab6dc08eab7aa0e3e756": "h:=x^{q^{n_i}}-x \\bmod f", "2a5da739866e2ab531e8dd1db72868c1": "\\log N, 2\\log\nN, \\ldots, \\log^2 N, \\ldots", "2a5db89b743177f29b3893f1ac48f7ad": "f_L = 0", "2a5de88e93a141b4119d13b3cffa842e": " \\| Tx \\|^2 = \\| Ax \\|^2 + \\| Bx \\|^2 ", "2a5e04a0fbf393d19dfb46fbe36eb4c1": " ZZ=Z^2 ", "2a5e091b1a7890a43167826332e2915e": "N=\\frac{16\\,\\pi V}{c^3h^3\\beta^3}\\,\\mathrm{Li}_3\\left(e^{\\mu/kT}\\right).", "2a5e142687cc63c7872379caff4e771d": "I(\\lambda_{ex})", "2a5e96afa716d8ef986634f5fa81aad6": "\\Omega_n(k) = c_n k^n", "2a5e96c954f67a5759259484aad80930": "\\Phi_c", "2a5ec3fe91f3a4f8da2ff7da9ea288c0": "\\textstyle 2^{l-1}", "2a5ece3316ca4224e7c2a33704c41627": "a_{mn}=\\begin{cases}1 &\\text{if } n = 0, \\\\ \\frac{1}{n!} &\\text{if } m=1, \\\\ \\frac{1}{n}\\sum_{j=1}^{n}ja_{m,n-j}a_{m-1,j-1} &\\text{otherwise} \\end{cases}", "2a5fde7e460cfd1aa43e8efee042297c": "n_e = n_p", "2a6039655313bf5dab1e43523b62c374": "MA", "2a60ba74d36d391eb082a070a09abcb4": "\\frac{P(x)}{Q(x)} = E(x) + \\frac{R(x)}{Q(x)},", "2a60dbafd4973df2b942c470834121b5": "1 \\le p/q < 2", "2a60e11874910e0a94b78c18e1f4be26": "|\\mathrm{GHZ}\\rangle = \\frac{|000\\rangle + |111\\rangle}{\\sqrt{2}}.", "2a60f882193bc5a1440a8e89c4031433": " (P2) \\min_{W\\in R^{n\\times (n-q)},\\gamma \\in R^{1\\times (n-q)}} \\|AW - e\\gamma \\|_F^2, ", "2a61758957df6bbb7f16c32159a3c710": "\\left( \\frac{\\pi}{R+\\delta} \\right)^2", "2a6178e77b29db7570e5aa245fcf672c": "q_a \\, u^a = 0, \\; \\; \\pi_{ab} \\, u^b = 0 ", "2a617950a8cd53f6ee37b31d4f3b228a": " A \\in M(m,n; \\mathbb{K})\\,\\!", "2a61cc7dcb043ce927cf8be4ef509237": "A\\cup B;", "2a625644f1d1da502865823c6bdf9d29": "\\lambda_{\\mathrm{vac}}\\!", "2a62898a538627ad52b69b43d06d0261": "\\mathcal{I'}", "2a629642453bf27ac5a66d096ce5b072": "\\mathcal{L}_{X_H} \\omega= 0", "2a629ca7967aa55ca0758ac1979689cd": "b_{2}^{*}= b_{2}- \\mu_{2,1}b_{1}^{*}= \\begin{bmatrix}4\\\\5\\\\4\\end{bmatrix}- \\frac{13}{3}\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}=\\begin{bmatrix}\\frac{-1}{3}\\\\\\frac{2}{3}\\\\\\frac{-1}{3}\\end{bmatrix}", "2a62c673a06412a31f0dba00a08a9a8e": " \\psi_1(N(x), x) = N(x) - \\psi(x) < 0, ", "2a62f5c11dd22d7d7de84933e241e337": "\\hat{\\mathbf{x_i}}' = \\hat{H}\\mathbf{\\hat{x}_i}", "2a632e9253f9e0720c71175da7ee02bc": "p_1/q_1", "2a6345f2932480d5c17d19121a419a2e": "\\left(\\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ -\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "2a634ad6d1d7b8d2b5a1fe2ac6d98006": "r^2\\dot \\theta = nab ", "2a637ef3983c1f8f22e557c6f85c4e0e": " (i, j)", "2a639a363111bfcbb98de18c6293b608": " 2~r^{-3}~\\cos\\theta \\,", "2a639fd50012aebeb9ec6cf0c222c1d1": "\\overline{p}", "2a63da71c4c438e663d78c4115650c88": " A_m\\, ", "2a63f984fe0e7baded3685ec54f6d9f7": "\\mathfrak{H}' = \\mathfrak{H}^n", "2a640eda63db34cb21c14bf556749d4e": " \\mathbf{T} = \\mathbf{e^i e^j} ", "2a64278e630b5024b6e5d82c2dc6103a": "\\left\\vert A \\cup B \\right\\vert = \\left\\vert A \\right\\vert + \\left\\vert B \\right\\vert \\,.", "2a6433cd7a2488121c981b4e9a592984": "(\\sin(\\alpha))^{n}\\,", "2a643fa17cd88672f73e0795014f1a5e": "i \\in \\{0,\\ldots,n\\}", "2a6475f2933cecaef518814bffd76d0f": "\\operatorname{Cl}_2(\\varphi)=\\sum_{k=1}^{\\infty}\\frac{\\sin k\\varphi}{k^2} = \\sin\\varphi +\\frac{\\sin 2\\varphi}{2^2}+\\frac{\\sin 3\\varphi}{3^2}+\\frac{\\sin 4\\varphi}{4^2}+ \\, \\cdots ", "2a64893329955f91bae96068b7d60e00": "F(M\\oplus M') = F(M) F(M')", "2a654456d3b16db914a804b3047386db": "\\frac{\\partial \\sigma}{\\partial x^{1}}\\frac{\\partial \\sigma}{\\partial x^{2}} - 2x^{2}\\sigma = 0. \\,", "2a65448218cbf1b769e721bd6cdc8561": "\\widehat\\delta", "2a655284b049e8a633bfa0e20291463f": "{B}", "2a65b7131a6a273409d55b881e8b7dca": "\\hat{g}(f) \\cdot \\hat{u}(f) = (g * u)(t) ", "2a65bb244ac7f1bf38353feab7088255": "O(A_1:A_2|B) = O(A_1:A_2) \\cdot \\Lambda(A_1:A_2|B) ", "2a65e37180cafa69c6a3e3e5d00a9399": "\\lambda_1\\le\\lambda_2\\le\\cdots\\le\\lambda_n", "2a664103a5654b45cfb8ef3d92a1dfbe": "h \\le 1", "2a6674bb3191ad9a4de3eecad1095431": "f _c\\,", "2a66e4b2228514a9ce24a723e99410d9": "\\tbinom{3}{2}=3", "2a67067b0e60b1df13b5cd48ad7e3a3a": "n>0\\,", "2a672604595b76a5fc2f70b065456e79": "\\scriptstyle{a=6.105\\ \\mathrm{millibar};\\quad\\;b= 17.27;\\quad\\;c= 237.7^\\circ \\mathrm{C}:\\quad 0^\\circ \\mathrm{C}\\le T\\le +60^\\circ \\mathrm{C}\\quad (\\pm0.4^\\circ \\mathrm{C})}", "2a6752a26013753cafae812288993f55": "\n\\mathbf{G} \\mathbf{F} \\mathbf{L} = \\mathbf{L} \\boldsymbol{\\Phi},\n", "2a67626253be046f40c7650f783b6089": "O(h^2)", "2a678bc5857c58bf7dc68e379b929975": "n = p_1^{r_1} \\cdots p_k^{r_k}", "2a679fcf0217b5c99ab467df92d8a8e4": "I_{k_\\nu , k_{\\mu}}\\,,", "2a67b601e3d5eb9ef1980c3ceecb36e8": "f(x_1, x_2, \\dots, x_n),", "2a687589335f9bc189e7e4bd97c0d162": "10_{35}", "2a68fedf26ce546090357c95dfaadfd4": "\\mathbf{G} := \\begin{pmatrix}\n1 & 1 & 1 & 0 & 0 & 0 & 0 & 1\\\\\n1 & 0 & 0 & 1 & 1 & 0 & 0 & 1\\\\\n0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n1 & 1 & 0 & 1 & 0 & 0 & 1 & 0\n\\end{pmatrix}_{4,8}", "2a690d52cc726cf6a79e01a363d902f8": " r^2(a^2+b^2)-c^2\\ge0 \\ .", "2a691d5256eb58d953363628d4420b45": "c(t) = A\\cdot \\sin(\\omega_c t + \\phi_c),\\,", "2a69c45edbc076145ba75ac4c2b8c639": "\\Gamma_{ij}{}^k = \\Gamma_{ji}{}^k,\\quad \\cfrac{\\partial \\mathbf{b}_i}{\\partial q^j} = \\Gamma_{ij}{}^k\\mathbf{b}_k ", "2a69e0ebad3a871e842e493029cf1826": "L>d", "2a6a264a95932b110f901b86158a2abb": " \\gamma_r = {{\\gamma} \\over {\\gamma_c}} ", "2a6a26cb6bf2ae0de1bf0322dde45502": "c \\pm r\\omega ", "2a6abf0ac97de43b03a2293980ced801": "{\\Bbb Z}\\rtimes {\\Bbb Z}_2", "2a6ad4a1830d5d612a404bc838b7d11a": "W_i = W''_i", "2a6b30bf1129252e8a1fb7c970ae2960": "A v_k = \\lambda_k v_k.", "2a6b71472a45883a4b270ec4caf2e9f8": "v_{i}^{*} = w_i, c_{a}^{*} = c_a ", "2a6b9c37c4770580b2ab389848c314d5": "\\frac{dN}{dt} = \\frac{I}{eV} - \\frac{N}{\\tau_n} - \\sum_{\\mu=1}^{\\mu=M}G_\\mu P_\\mu", "2a6bcabdde7126769beb7908a70fb911": "V = \\frac{\\hbar}{2 e} \\frac{d \\delta}{d t} ", "2a6bd46c68493ddc7c20271c9adb2091": "g \\in (\\mathbb{Z}/n\\mathbb{Z})^*", "2a6be38f61cb366b8b2bb7da82e00910": "\nE[B(t) | Q(t)] \\leq B\n", "2a6cc0016e25e5bf0aab51a081bde525": "0 < |z| \\ll \\sqrt{\\alpha + 1}", "2a6ccaea07db0dd364bbe96c2ca411ec": "\\alpha,", "2a6cef7585a38a0c0ab47592712549b7": "e(g^x,g^y)=e(g,g)^{xy}=e(g,g)^{z}=e(g,g^z)", "2a6cfca0305322f04f3183a0cb3a6f99": "(L',B')", "2a6d07eef8b10b84129b42424ed99327": "dist", "2a6d0b5857cc8e2bad7e752c1bf91fea": "1 - \\frac{1}{n}", "2a6d3232a1ef35c71cab0f317bd7331b": "W=Q_{hot}-Q_{cold}", "2a6d3b067f9033031ad929ac17ad76c0": "c(n,k),", "2a6da40bebb98e0b96eeb9c760fef06c": "y'(t)", "2a6dde7173cf813b60db51e9c412316d": "\\hat{x}=W_{L_\\xi}\\hat{X}W_{L_\\nu}\\,", "2a6de1cb4be32e7c582576f385a184bf": "\\gamma_\\xi=\\frac{1}{4}\\xi\\left(2\\chi_\\xi^2+\\chi^2\\right)=A^2,\\ \\gamma=A^2\\left(\\xi-\\xi_0\\right)+\\mathrm{const}.", "2a6de213a5e0e4fe624c61e62e85a3bb": "\\|m_k - x^*\\| \\;\\approx\\; \\|m_0 - x^*\\| \\times e^{-ck}\n ", "2a6df51d72b9b83d6e024a2f8eb063ee": "\\forall{x}(P{x} \\to Q{x})", "2a6e44594769636af4d6426373dbcbb7": " (-1)^k\\psi^{-1,(k)}(t;\\theta) \\geq 0 \\,", "2a6e643e02068a17f961da5c641f687c": "\\textstyle \\lambda_{PP} \\leq \\lambda_{BP} < \\lambda_{NP}", "2a6f05c3208e3f9d91a31b12912545d1": "\\bar{6} \\bar{6}_H 15", "2a6f503ed0a1a14de0c925f33644438c": "\\textstyle \\omega_s ", "2a6f5c01789a88aa9b51cf16cb7ebf80": "\\leq_{IITA}", "2a6f87377bab699d295ee4b4e91ea333": "\\frac {2(a-b)} {a+b}", "2a6f891c4b1e2ea99d54c7455501b1f8": "x_{1}^{0} = f^{0}(1) = 6 + 2 \\times 1 = 8", "2a7035a190839c7e07ded343e68ca0de": " I(X;Y|Z) = D_{\\mathrm{KL}}[ p(X,Y,Z) \\| p(X|Z)p(Y|Z)p(Z) ]. ", "2a7076bdb083d29bdbe7d5f5a4a7e6a8": "\\angle ABE \\cong \\angle CDE", "2a709ef11afd6ede51fed391ccd81477": " \\and T_7 = [\\_, S_7, A_7]::[\\_, S_6, A_6]::K_1 ", "2a70ac26a70e94370af5f3a1bab33f6b": " ^s ", "2a7106e20ecc9bfa8a0a727d86e985a1": " B=A+P_RV \\qquad \\mbox{(6)} ", "2a71b4fa93d8b0834bd4590a480b1ead": " (\\phi \\psi)(g) = \\sum_{k\\ell=g} \\phi(k) \\psi(\\ell)", "2a71b7070beb473a388326ea0f5f9981": "\\left\\{ F \\colon \\mathbb{R} \\to \\mathbb{R} \\,\\left|\\, F \\text{ is continuous, } \\lim_{x \\to - \\infty} F(x) = 0, \\lim_{x \\to + \\infty} F(x) \\in \\mathbb{R} \\right. \\right\\}.", "2a71ca5cf673c67701a3ae2e2a607acb": "(a+b)'=a'+b',", "2a71dbfb3c5e4bb2c8f0fcf568f326d0": "\\vartheta(w, q)=f(qw^2,qw^{-2})", "2a720418c14cdfb6a94d32e966736c20": "\\mu=m", "2a721f4873ef1ca9910cc29371e51b62": "T_p", "2a723f7184dad90db43ba0fd8b5532e0": "P_{LCU} = P_0 \\cdot P_4 \\cdot P_8 \\cdot P_{12}", "2a72b26cdbe08265bc584591341d900b": "\n \\cfrac{1}{\\sqrt{3}}~\\sigma_c = A - B~\\sigma_c ~.\n ", "2a72d6d079e44dc57ba49bb8b276706a": "T_L", "2a72f9a58e14858159ce6cb126dc31c5": " \\left( \\forall^p L \\right)^{\\rm c} = \\exists^p L^{\\rm c} ", "2a73f74a6bfeed863434251333b4906c": "f\\in \\Eta ", "2a741a315a5dd97975d879b485785a10": " E(k) = \\frac{\\hbar^2 k^2}{2m^*} ", "2a7435ccb639171f6b4a69ff7a3f76a5": "n_s=\\sum_{i=1}^n I_{H_s}(y_i)", "2a744b9fff8b536490048e87693fa8b0": "{T^{i\\ldots j}}_{k\\ldots l}", "2a74595f5a1be505836933eee6dd6cbe": "\\mathcal{F}_{\\tau}\\subseteq\\mathcal{F}", "2a7468a2e373debb4f428175e332266f": "SU(n)", "2a74d6fd6ac3cb4be7ab870199d97a8c": "\\delta t=3.1\\pm5.3\\ (\\mathrm{stat.}) \\pm8\\ (\\mathrm{sys.})", "2a752bd879dad8bc3eb9b22caf9cd063": "\n\\langle \\Phi _E (f) h , h \\rangle = \\int _X f d \\langle E(B) h, h \\rangle \\leq \\| f \\| \\cdot |E| .\n", "2a7536b2639177116bdd19b371642fbd": "\\Delta x_{\\rm meas}(t_{j+1}) = \\Delta x_{\\rm meas}(t_{j}) \\equiv \\Delta x_{\\rm meas} = \\Delta\\phi/(2k_p)", "2a75c54cef6ea7c742dc7f6308ffcdc3": " \\mathbf{F}_j = m_j \\mathbf{a}_j, \\quad \\mathbf{T}_j =[I_R]_j\\alpha_j + \\omega_j\\times[I_R]_j\\omega_j,\\quad j=1,\\ldots,M.", "2a75f2c7d6635e570d78938c5cd7d26a": " O ", "2a76148d3b52e08765637383ab05ec29": "\\mathbf{R}\\,\\!", "2a76167264b964a77b496c8fc08d2bb3": "\\frac{-t}{1}", "2a762cb3ebc794b1ee5db61741c38781": "\\Sigma _{YY} ^{-1/2} \\Sigma _{YX} \\Sigma _{XX} ^{-1/2} c", "2a76563da73a2dcd8651fb0cef8b4b93": "\\mu = \\mu'_1", "2a766519746c71512e964889a159348e": "\\operatorname{Pr}(T\\le t)=F(t)=1-R(t),\\quad t\\ge 0. \\!", "2a767723b58d4842f042a615f850b1b7": "\\exp(t)=1\\in G,\\,\\,\\lambda(t)\\in 2\\pi i \\mathbf{Z}", "2a77b6482ac5f92bed0fd61e74616942": "E(h_n) := E(\\ h_n,t_n,y(t_n)\\ )", "2a77bd779b3d40a80b4c805eb2fe4b2f": "c_g=\\frac{1}{2}\\sqrt{\\frac{g}{k}}=\\frac{1}{2}c.", "2a77c44b9954b15ae0310095b3ef129f": "(g\\cdot f)(h)=f(hg)", "2a77cc1c4d9d0f8bc9499557e0762a32": "\\rho = 28 ", "2a77e3dedf9fe84e8e46b4ba266301c2": "\\leq 9", "2a789025c1d171f9aef8b9334cee67af": "C_v=\\frac{\\partial U}{\\partial T}=k(\\alpha+1)\\,U\\beta", "2a78df6d411d398f44fd73ae9b325f34": "\\nu(x)=\\gamma x^{-1}\\exp(-\\lambda x)", "2a78f4843366558d69f5a1051bf347ea": "\\sigma_1, \\dots, \\sigma_l", "2a791f62e7bce20da550f5c9cfac3b42": "\\{1,2\\cdots,k\\}", "2a7926ba2be1e577147e29cf964fe738": "y^2=x^3+ax+b", "2a79617dee9555139a40d6722de6e7f5": " \\Omega\\subset\\R^2,", "2a798a116e80c5eeb2f842de028dcf60": "\\sigma_{\\text{realised}}^2", "2a7a0cd67d239534f630df073d744f9c": "10 \\cdot 10 \\cdot 2,268 = 226,800\\,", "2a7a62ba0cbb4c7641c5fcc3633249d2": " b_n= \\sigma \\log(\\tfrac{1}{n}) \\, ", "2a7a81d7b3c0449d3dcb82de3315c42a": "\\overline{g}", "2a7ab5f467b6e0f387df26e544132a89": "0\\leq x\\leq2^{2^k}", "2a7abd7956bdd2c3d5c3b96c7abe0a59": " \\exp(-\\hbar\\omega/kT) ", "2a7acaad6af2a47a5431177f05c5a529": "p_{G, \\alpha}(f) = \\sup_{x \\in G} p_{\\alpha}(f(x))", "2a7aef3d17acd959a02a6b6d410af839": "A_{y} = \\int_a^b 2 \\pi x \\, \\sqrt{ \\left( \\frac{dx}{dt} \\right)^2 + \\left( \\frac{dy}{dt} \\right)^2} \\, dt", "2a7af878abca78099c16c23df1e15fc6": "\\beta(X', X)", "2a7afd03b2f58ba6ce9948aec18421ea": "f\\circ\\gamma(t)-\\lambda t^2", "2a7b97977d2ed4ae158784d2a7aa60f0": "\\overrightarrow{ab} = \\overrightarrow{cd} , ", "2a7bb9fd795d56305ecaea9e0d34e806": "\\cos(b) = \\mathbf{u} \\cdot \\mathbf{w}", "2a7becdcbdb99d5f62d53d281024a2ab": "\\sigma_{ji,j}+ F_i = \\rho \\partial_{tt} u_i\\,\\!", "2a7bf99b9d79fa0ca5c5847c56cb4d68": "s(M) = \\,\\!", "2a7c42a6b2bfb18ae86f15595278ebf0": "\\sim 3.2 U_p", "2a7c653acfdfe41d1f8ced0b9e3e165f": "n^s", "2a7d7cada4d4e6f7b545fb19c733b588": "d_h= \\lambda_d d_h^{\\mathrm{(el)}} = \\frac{\\lambda_d e F}{g \\sqrt{h}}. \\qquad\\qquad (13)", "2a7e0d1a70ad2e15c1f2f52e87658596": " L_n^{(\\alpha)}(x) := {n+ \\alpha \\choose n} M(-n,\\alpha+1,x).", "2a7e27c44f9921e6e8d72f04a832f4fb": "\\vec{v}_{\\nabla B} = \\frac{\\epsilon_\\perp}{qB} \\frac{\\vec{B}\\times\\nabla B}{B^2}", "2a7e8d603ff40fd94475c4fe48dc7ad4": "\\text{period of } \\tfrac{1}{p}= \\text{period of }\\tfrac{1}{p^{2}}= \\cdots = \\text{period }\\tfrac{1}{p^m}", "2a7ec6e1645e87250524f99795cfed6a": "(-\\infty,0)\\cup(0,\\infty).", "2a7f2c8b625d2d4f8122befbd00a20b6": " \\scriptstyle ROI = \\frac {Profit}{Investment}\\ ", "2a7f838425f1493cec66571a2ce02016": "u = U f \\left( \\eta \\equiv \\dfrac{y}{(\\nu t)^{1/2}} \\right)", "2a7fb302a91e8782a8514d3205f295e7": " K\\times N ", "2a8001563d8ba65bd13d20e1b51883ad": "X(\\Omega) = \\frac{\\Delta t}{\\Theta} \\sum_{n=1}^{N}x(n\\Delta t)e^{-j\\Omega \\theta(n\\Delta t)} \\omega(n\\Delta t)", "2a803e57ce6388e5f9d2fc5024ed72c1": "\\phi(x,t)=\\frac12\\, c\\, \\mathrm{sech}^2\\left[{\\sqrt{c}\\over 2}(x-c\\,t-a)\\right]", "2a80b7605946153fe04540e9430c610c": "E_{2} ", "2a80c61ab6182bd1bc2cf5dda87378ff": "\\theta_a=\\arctan\\ (c_a,s_a).", "2a80c8a07c9183d64f837b756749f8cf": "{1\\over T} = {\\partial S\\over\\partial U}.", "2a80edc1c49cd88d79b20a1247b9c39a": "\\epsilon = -3.0", "2a81522850c423d83f5f3fa95ed66a21": "-10^{b_0}[OH^-]_{0^{ }}", "2a819fe8d5be464a36821588c0703f35": "\\bowtie", "2a81b406f775337fc1a7c7e458a07eb9": "\\mathrm{Pr}_\\mathrm{t} = \\frac{\\epsilon_M}{\\epsilon_H}.", "2a81e1df0323809710a94c365b9f82d6": "\\hat{y}(k) = \\left(C - D K\\right) \\hat{x}(k)", "2a81e5c2568e7fe9559cc5c0f37bb5a8": "\n p = \\langle p\\rangle + \\tilde{p} ~;~~\n \\rho = \\langle\\rho\\rangle + \\tilde{\\rho} ~;~~\n \\mathbf{v} = \\langle\\mathbf{v}\\rangle + \\tilde{\\mathbf{v}}\n ", "2a81ecb199d1592b35e13185ea263ebf": " P_n = v ", "2a81f512690ea1a983ab91e1525697a5": " \\langle x, \\alpha y + \\beta z \\rangle = \\alpha \\langle x, y \\rangle + \\beta \\langle x, z \\rangle", "2a8212ff191b526a10f2aeec031569f5": "1-i", "2a823acbd6dddaaa1e291d499041e657": "\n\\Lambda =\\sqrt{\\frac{h^2 \\beta }{2\\pi m}}\n", "2a829b0db439260037f9d4f518b2a709": " v_r = H D \\ . ", "2a831091fc603aacd03b10f3b777d376": "(\\pm 1,0,\\pm 1,0)", "2a832f596284bd7d02afc851471ac48f": "S_{\\mathrm{sos}} = \\frac{10}{H_{\\mathrm{sos}}},", "2a835102c6b349dbe1fcdc6304846c34": " t_n = t_0 + nh ", "2a8358c429e73ebb716857392e1b215c": "g(x) = 1+x", "2a8385381f69f0a4a0b24d9e5732c448": " n_1 m_1 \\equiv 1\\mod m_2 ", "2a839627b54004a4a6f9e9a04c79a22c": " i | d\\Psi(t)/dt \\rangle = \\hat{L} | \\Psi(t) \\rangle ", "2a83e581661bc494d23d4600a5584430": "m = \\theta_2/\\theta_1 \\,\\!", "2a83eeaa6741f907d8557f4c16ae4d07": "\\gamma_A(\\lambda_i)", "2a840ff13c3ecfa60137e94f4c155023": "\\psi(r) = \\frac{e^{ikr}}{4 \\pi r}", "2a846e9054ccd3b4e9fcbe80e44581af": "\\psi~", "2a84728a916aa71d81f59ac1b5246538": "P_{ij}(l)", "2a849fecdbbd5cecd98042d472bd454e": "U^{(k)}\\mathbf{x}_w^{(k)} = \\mathbf{x^{(k)}}", "2a84e89aaf7e97e715bec8a332420a6e": "=\\frac{1}{2a}", "2a8532ea1ca6ea2c8db148ee0258c5b4": "\\mu = G*M\\,\\!", "2a8593ed4d53b45b9ee18d5fdb19d5cd": "p_{\\theta} = mr^2 \\dot {\\theta} = l", "2a85a36a7718c1ca7d15a4d1cdcfcdac": " \\phi(z) = \\frac{\\det(I-zA+zeb^T)}{\\det(I-zA)}, ", "2a860dbeb572afab7716d08fd363ef17": " \\mathcal{A}_2, ", "2a8630d8d1ee63caa538c203033e6b40": " P(Z \\ge k) \\le \\frac { 1 } { 2 k^2 } .", "2a863ed2cc3fc745c5b85dc76bcac0ef": "(K-S_{T})\\vee 0", "2a8698c0000e7e42dc47d5672ea4507c": "|c_{k'}(t)|^2= |\\lang\\ k'|H_1|k\\rang |^2\\frac {sin ^2(\\frac {E_{k'}-E_k} {2 \\hbar}t)} { ( \\frac {E_{k'}\n-E_k} {2 \\hbar} ) ^2 }\\frac {1}{\\hbar^2} ", "2a869b5583016ed060e1a94765ce0024": " \\lambda \\ = \\ \\sqrt{\\frac {r_{m}} {(r_{i}+ r_{o})}} ", "2a86bbf417096e9f310c8cc49e9f993c": "A:m\\times n \\mid m>n", "2a86d128a9ea5da21d45a154aa75f80d": "(x_1,y_1)", "2a86f85f8070afcd0f203c6e95ce5ccd": "\\scriptstyle f(u)=v(\\boldsymbol{x})u(\\boldsymbol{x})", "2a8732630a98a37bac2634bd9003373e": "|\\psi_0\\rangle=|n_0\\rangle+\\sum_{k\\neq0}V_{k0}/(E_0-E_k)|n_k\\rangle", "2a87488669c138ffb3d2cfe89201684f": "E_{AB}", "2a87659d3e126ccaf9e7f1ffcdc63ed7": "\nT = \n\\begin{pmatrix}\n0 & 1/2 & 1/2 \\\\\n1 & 0 & 0 \\\\\n1 & 0 & 0 \\\\\n\\end{pmatrix}\n", "2a87bce2da09bb89e55371d244c833e2": "\\eta_N=4\\beta-\\gamma-3", "2a87dc295eccddaee148d839f8b02311": "\\hat{y}= y_1\\hat{I} + y_2\\hat{J} + y_3\\hat{K}", "2a87f14ca94ad3e0b32e434dbed8128d": "\\frac{P_t}{P} = \\left(1+ \\frac{\\gamma -1}{2} M^2 \\right)^\\tfrac{\\gamma}{\\gamma - 1}", "2a8821c3a73ad1966392812e5b1f2f6d": " \\mathbb{Z}_{m} ", "2a882fc0667b778a11108f0f3dee5bbb": "\\textstyle z(0) = 0", "2a884df2fa831c156ee400eb18576fb0": "\\sum_{i=1}^d a_i = \\sum_{i=1}^d b_i", "2a8860286f0d772f4aec1be8199124b9": "\\pi^* =\\begin{pmatrix}\n 1 & 2 & ... & k & k+1 & ... & n & x & y\\\\\n 2 & 3 & ... & 1 & k+1 & ... & n & x & y\n\\end{pmatrix}.", "2a887e69f793de8dc3c7b04802741e1f": "T=\\frac{x_\\mathrm{m}}{U^{\\frac{1}{\\alpha}}}", "2a888245f25203e96ed65876711a6e76": "T_{\\text{i}}", "2a88ca3d1bfa56809368122147d02b83": " \\frac{d}{d\\varepsilon} V[u + \\varepsilon v]|_{\\varepsilon=0} = \\iint_D \\nabla u \\cdot \\nabla v \\, dx\\,dy = 0.\\,", "2a88ce2acbb0fc1bd77a9c553e87e5c9": "v_o \\approx \\sqrt{\\mu \\over a}", "2a88d17372526670cc2269762cae9bde": "\\scriptstyle (\\frac {2}{3} \\pi R^2 D,", "2a88e21c4730d2e1de35ecd2412c8a4c": "m^*(n_{2D}(T_c)) =\\frac{2\\hbar}{\\alpha c}\\sqrt{\\pi n_{2D}(T_c)} \\approx 2.9767\\times 10^{-33}\\;\\mathrm{kg} \\ ", "2a88f64f1f47ca42eb470fd75775e9c9": "x^iy^j", "2a8943268d5841d4a5625a57bfb6ee29": "\\frac{dN_1(t)}{dt}=-\\lambda_1 N_1(t)", "2a8961aa884c1a9b1d3cc4c212ccf631": "\\mathrm{d}B", "2a896b3f23736f8493ff113592331fbb": "H_{norm}", "2a897cdf7b8a5c783c1118ad51f3c8fb": " \\begin{align}\n|\\operatorname{Cov}(X,Y)|^2\n&= |\\operatorname{E}( (X - \\mu)(Y - \\nu) )|^2 = | \\langle X - \\mu, Y - \\nu \\rangle |^2\\\\\n&\\leq \\langle X - \\mu, X - \\mu \\rangle \\langle Y - \\nu, Y - \\nu \\rangle \\\\\n& = \\operatorname{E}( (X-\\mu)^2 ) \\operatorname{E}( (Y-\\nu)^2 ) \\\\\n& = \\operatorname{Var}(X) \\operatorname{Var}(Y),\n\\end{align}", "2a89bd4332c39fbab9cc7296508434a4": "(1-a) b", "2a89d078ab9af0b59598e58043594eb3": "G:\\mathcal{L}\\to R\\operatorname{-Mod}.", "2a8a85a98252f4dc64df81d49883de9f": "C_8", "2a8abb4240e07da3a09e8d37df28f62e": "GO^2=R^2-\\tfrac{1}{9}(a^2+b^2+c^2).", "2a8adab37672e01e0ad9cfdd5deb4ec6": "\\mathbf{M}(\\mathbf{r}, t) = \\frac{1}{\\mu_0} \\int {\\rm d}^3 \\mathbf{r}'{\\rm d}t' \\;\n\\hat{\\chi}_m (\\mathbf{r}, \\mathbf{r}', t, t'; \\mathbf{B})\\, \\mathbf{B}(\\mathbf{r}', t'),", "2a8b21dcd25004b46a7510f6754b5655": "\\scriptstyle x\\in V,\\;y\\in W", "2a8b3520f7e8179941881de50adb74b0": "\\,k_{1,m}=cos\\psi_{1,m}/cos\\phi_{1,m}", "2a8c5cf93e8753064cc42e5cb7058ef2": "\\mathbb{S}^3", "2a8c632505bb47bd69ed4a7e5631471c": "(A\\leftrightarrow B)\\to(A\\to B)", "2a8c839ead01ef2cadfd1d88350c31af": "{I}_v(\\kappa)", "2a8cc0e13fa5eb22dd36334a7c56de65": "4 \\pi\\,r^2", "2a8d53f4ca9627f4bc8f32777ab5764a": "j=1,2,\\cdots,k-1", "2a8d6d794376c6a037b96301d446b002": "\\sum_{n=1}^\\infty (-1)^n/\\sqrt{n}", "2a8e086ffb106ca9bffda1caefa40d26": "{}+ (a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2)i", "2a8e397e2af5b0c85a92e84f13943115": "\\sin(\\theta+\\pi/2)=\\cos(\\theta)", "2a8e5b678786c2c43db2a374b4a80d01": "(\\mathbb{Z}/p^2\\mathbb{Z})^*", "2a8e95f35080ecc3ede934d1ea11845f": "P=[p_{ij}] \\text{ } \\forall \\text{ } i, j \\in \\mathbf{S}", "2a8ec651e54093e89a8859e59a5518f4": " X_t = \\varepsilon_t + \\sum_{i=1}^p \\varphi_i X_{t-i} + \\sum_{i=1}^q \\theta_i \\varepsilon_{t-i} + \\sum_{i=0}^b \\eta_i d_{t-i}.\\,", "2a8edd14fdfcd0e25accde5d50bab615": "\\sum_{i=1}^{\\log_d n} \\left(\\frac{n}{d^i}+1\\right) O(d) = O(n).", "2a8f0b8ac4d7cadfd583e71063751efb": "x=const.", "2a8f107b83256e4b2336b4963e226bbb": "\\operatorname{S}_\\pi(\\xi, \\eta)", "2a8f15b3aaa518232eef417092ce5f59": "\\mathbf{x}\\mapsto A\\mathbf{x}+\\mathbf{b}", "2a8f454144000048f105503fb3bf6308": "(12)\\;\\;\\quad ds^2=-e^{2\\psi(\\rho,z)}dt^2+e^{2\\gamma(\\rho,z)-2\\psi(\\rho,z)}(d\\rho^2+dz^2)+e^{-2\\psi(\\rho,z)}\\rho^2 d\\phi^2\\,.\n", "2a8f4d033f38af120efc779fc32dfae7": "V.", "2a8f8bf11bde3c11e3f656712f0c6b7c": "\na \\left( 1 \\pm e^{2} \\right) = \\ell = \\frac{L^{2}}{mk} ~,\n", "2a8faeabeeaf1beaeeb0794e925f4763": "\\varphi_{F(p)}(y) = Q(p,y)", "2a90047b3ea783d426d35f0d688c8632": "\\Delta p_m", "2a90450f46ea513e091b74ffa516fbee": "\n\\mathbf{B}=\\begin{bmatrix}\nb_1 & b_2 & \\ldots & b_n \n\\end{bmatrix}.\n", "2a906f163eccd34a19b7f2ce41da3e0d": " \\frac{4}{3} \\pi R^3 \\rho_s g = \\frac{4}{3} \\pi R^3 \\rho g + \\frac{1}{2} \\C_d \\rho \\pi R^2 w_s^2\\,", "2a910292e84ab87e5c9657c92eaadd72": "\\{\\mathcal{F}_t\\} = \\{\\mathcal{F}^W_t\\}", "2a9122fc4dbd7e47b96f50ec758c2426": " \n\\Gamma=exp\\left[i\\omega \\delta t/2\\right]\\hat f \\hat f+ exp\\left[-i\\omega \\delta t/2\\right]\\hat s \\hat s\n", "2a922826fc423534c618ac64644f41e8": "f : a \\rightarrow (b \\times b)", "2a926f076c49debc9e93ebae83ab8287": "\\textbf{V}_P = [\\dot{T}(t)][T(t)]^{-1}\\textbf{P}(t) = \\begin{Bmatrix} \\textbf{V}_P \\\\ 0\\end{Bmatrix} = \\begin{bmatrix} \\dot{A}A^T & -\\dot{A}A^T\\textbf{d} + \\dot{\\textbf{d}} \\\\ 0 & 0 \\end{bmatrix}\n\\begin{Bmatrix} \\textbf{P}(t) \\\\ 1\\end{Bmatrix}=[S]\\textbf{P}.", "2a92880290acdc3991227d20c677a6a7": "\\scriptstyle{k\\, \\in\\, \\left\\{\\max{(0,\\, n+K-N)},\\, \\dots,\\, \\min{(K,\\, n )}\\right\\}}\\,", "2a92a7a0ca0f9f0f0999a8969c6ce64b": "\\begin{smallmatrix}10^{-0.5}\\ =\\ 0.316\\end{smallmatrix}", "2a92d36a6e2b2afdfca8965db08968d2": "\\tfrac{bu}{acre}", "2a930a6f7374df500add2ec913ba5295": "C_\\mathrm{linear}=\n\\begin{cases}\\frac{C_\\mathrm{srgb}}{12.92}, & C_\\mathrm{srgb}\\le0.04045\\\\\n\\left(\\frac{C_\\mathrm{srgb}+a}{1+a}\\right)^{2.4}, & C_\\mathrm{srgb}>0.04045\n\\end{cases}\n", "2a934aa5267676c5e62137fbc06aa7e3": "{{\\sin \\angle BDA}} = {\\sin \\angle ADC} ", "2a937cebbd00578960516654885f78b8": "\\chi_+^{-k} = \\delta^{(k-1)}.", "2a937df51ae98308cc330f2c1d188499": "L(s)\\equiv\\sum_{n=0}^{\\infty}\\frac{(-1)^n}{(2n+1)^s} \\qquad\\qquad\nL(1-s)=L(s)\\Gamma(s) 2^s \\pi^{-s}\\sin\\frac{\\pi s}{2}, \n", "2a938a962bc222ca2f1f4d53689d6656": "\\iint_S {\\mathbf v}\\cdot \\,d{\\mathbf {S}} = \\int_S {\\mathbf v}\\cdot {\\mathbf n}\\,dS=\\iint_T {\\mathbf v}(\\mathbf{x}(s, t))\\cdot \\left({\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}\\right) ds\\, dt.", "2a93ae39c92e8397dcaaa77ea4bb9953": "\\begin{align}\n -R_l^l & =\\frac{\\left(\\dot a b c\\right)\\dot{ }}{abc}+\\frac{1}{2}\\left (a^2b^2c^2\\right )\\left [\\lambda^2 a^4-\\left (\\mu b^2-\\nu c^2\\right )^2\\right ]=0,\\\\\n -R_m^m & =\\frac{(a \\dot{b} c)\\dot{ }}{abc}+\\frac{1}{2}\\left(a^2b^2c^2\\right )\\left [\\mu^2 b^4-\\left(\\lambda a^2-\\nu c^2\\right)^2\\right]=0,\\\\\n -R_n^n & =\\frac{\\left(a b \\dot c\\right)\\dot{ }}{abc}+\\frac{1}{2}\\left(a^2b^2c^2\\right)\\left[\\nu^2 c^4-\\left(\\lambda a^2-\\mu b^2\\right)^2\\right]=0,\\\\\n\\end{align}", "2a93dc555a03a48433d63ff1bf1aab59": "\\ x_i, y_i,", "2a940ff4dd998527714630f392f49979": "\\sqrt{1+y} = \\sum_{n=0}^\\infty {\\frac12 \\choose n} y^n = 1 - 2\\sum_{n=1}^\\infty {2n-2 \\choose n-1} \\left(\\frac{-1}{4}\\right)^n \\frac{y^n}{n}.", "2a94220087ffb5d03e08c851c8b06c6c": "\\mathcal{C}_x^0\\,", "2a943aef1f3a7fcc346a30c808aa136a": "a_f^\\dagger (k)", "2a946c616c4d872bd8c74f94f711cf43": "\\bigstar ||\\bigstar|\\bigstar", "2a94764624cee4b4cc4ad948f9e1b2d4": " \\tfrac49 ", "2a949b4fdaff8478c3890b42174e121b": "\\begin{array}{rl} E_{n\\,j} & = \\mu c^2\\left(1+\\left[\\dfrac{Z\\alpha}{n-|k|+\\sqrt{k^2-Z^2\\alpha^2}}\\right]^2\\right)^{-1/2}\\\\ &\\\\& \\approx \\mu c^2\\left\\{1-\\dfrac{Z^2\\alpha^2}{2n^2} \\left[1 + \\dfrac{Z^2\\alpha^2}n\\left(\\dfrac 1{|k|} - \\dfrac 3{4n} \\right) \\right]\\right\\} \\end{array}", "2a94d9b7dc9ecd04839a4796acd66f5a": "2^{\\aleph_\\omega}=\\aleph_{\\omega+2}", "2a94dfe0adcc351bfdfe39297011d764": " y(t) = v t \\sin \\theta - \\frac{1} {2} g t^2 ", "2a95653f9864c27726fc38021f45274f": "R_{\\theta JC} = 1.5 \\ ^{\\circ}\\mathrm{C}/\\mathrm{W} \\,", "2a95791557f176dab883f49e8220a29a": " e^s =\\sqrt{\\frac{c+v}{c-v}} ", "2a95aaaf954c2187999c6357b04a58dd": "\\hat{x}", "2a961ce94679e29aeb2d11207942aa5a": "(f\\oplus b)(x)=\\sup_{z\\in B^{s}}f(x+z)", "2a969fca6ec5dfb552bd7de5decbbeaf": "n_\\eta(-\\xi)=n_\\eta(\\xi)+2\\xi c_\\eta(0,\\xi)", "2a96c003f9320b9493b7229b2bddc82c": "(n_1, n_2, \\ldots , n_r)", "2a96cd4e9c2a2d71b114892dd5d607a1": "\n(Eq. 7) \\text{ } \\text{Subject to: } Y_i(x) \\leq 0 \\text{ } \\forall i \\in \\{1, ..., K\\} \\text{ }, \\text{ } x = (x_1, ..., x_N) \\in A\n", "2a96e8cc801ffa98afe6c1c32a1f1f84": "\\mathbf{E} = \\frac{1}{\\sigma} \\mathbf{J}", "2a97134c7fd69805eeb10c242e5ac99b": "\\lambda^{[l]}_{\\alpha_l}, l = 0,2k-2", "2a975030ee51042a5bc97b8e9c4ca650": "M(127)", "2a989e25f7a7b36ab2dc5b3f1ce5ced3": "\\frac{\\chi(V(nH))}{\\hbox{rank}(V)} < \\frac{\\chi(W(nH))}{\\hbox{rank}(W)}\\text{ for }n\\text{ large}", "2a98f30f094aa30e09074c57aae6998c": "\\Psi\\left(x,z,t\\right)=Ae^{-\\alpha|z|}\\exp\\left[i\\alpha\\left(x-ct\\right)\\right]=A\\exp\\left(\\alpha\\sqrt{\\frac{g\\tilde{\\mathcal{A}}}{\\alpha}}t\\right)\\exp\\left(i\\alpha\nx-\\alpha|z|\\right)\\,", "2a99213a095c559b80a1935109285223": "\\int_{-1}^1 T_n(x)T_m(x)\\,\\frac{dx}{\\sqrt{1-x^2}}=\n\\begin{cases}\n0 &: n\\ne m \\\\\n\\pi &: n=m=0\\\\\n\\pi/2 &: n=m\\ne 0\n\\end{cases}\n", "2a9943083cba71268a49ed2075484cd4": "\\kappa = 1~", "2a99624a64431f000f6ead90371e9145": "S \\subseteq \\Sigma", "2a99cef721cb5cf56c9f2f078dbe0cde": "[x_1, x_2] / [y_1, y_2] =\n[x_1, x_2] \\cdot (1/[y_1, y_2])", "2a9a0c86f46faf6e18c49b602e407676": "\\scriptstyle l_\\phi", "2a9a0f7545cdbca8151bb401ca2a9c00": "U=\\alpha_0 I + \\sum_{i=1}^3\\alpha_i J_i", "2a9a3c15ece026cdeaa0ba7cd7124ac6": "- [F , G]^{IJ} = *[F,*G]^{IJ}", "2a9a49cb594b29f12fdac7e4cf6c2e22": "u^TX", "2a9a964b4332a7cd9489397f3d748317": "\\ \\displaystyle \\psi(q,\\alpha,u)\\ ", "2a9ad848470cd716761139b9309ab157": "A = False, B = False , C = True ", "2a9ae35efddfa2e701344c07e390dcb7": "f_0(n)", "2a9b050ece4e1a4cca260afd125d5895": "11 \\div x", "2a9b38c3efbfd9bbb90cb3c4c4d0f41c": "(n-p-1)\\mathbf{V}\\text{ for }n \\geq p+1", "2a9b65e67356089b94063cec5c1f2b1d": " (-\\pi,\\ \\pi]", "2a9bf297c62d7bd1ee025151da42a271": "\\Delta^{0,Y}_0", "2a9c01b36b5eb584a60b36f121edea6e": "L_{QA} = \\frac{4\\pi R_H }{\\omega_B} \\ ", "2a9d4a7521fd9cbaa72d92c13982fda9": "\\frac{4r}{3\\pi}", "2a9d734a0d949b587738549b76b43d1e": "z = z_1 z_2 \\ldots z_i \\ldots z_j \\ldots z_n ", "2a9dbcd52e4de4ccfd47dd46cd17c198": "\\frac{2\\pi^2}{3}", "2a9e045cc57eee43501ee1f72c2e9781": "\n\\sum\\limits_{m=1}^{M} {\\beta _m } \\varphi _{j + m}^{n + 1} = \\sum\\limits_{m=1}^{M} {\\alpha _m \\varphi _{j + m}^n }. \n\\quad \\quad ( 1) ", "2a9e82d41d6e8704ffcc771278dfbc93": "\\det(\\mathbf{A}+\\mathbf{uv}^\\mathrm{T}) = (1 + \\mathbf{v}^\\mathrm{T}\\mathbf{A}^{-1}\\mathbf{u})\\,\\det(\\mathbf{A}).", "2a9e97363b5ac5c18461cb70dc1c5111": "x'' \\rightarrow 0", "2a9ec1e9e0a2c39a5cbe27f63ccf125f": "\n\\frac{F L}{k_\\theta} = \\frac{3}{2} \\mp \\frac{\\sqrt{5}}{2} \\approx \\left\\{\\begin{matrix} 0.382\\\\2.618 \\end{matrix}\\right.\n", "2a9ed4e68c4c27f2d97fe4a077796b4b": "N(d_\\pm)", "2a9f6203fffed4bb3212b3041bcbd805": "\\gamma_\\mathrm{SL}", "2a9f7382f41e627d5979b1b1f220363c": "\\frac{1}{\\beta}\\sum_{i\\omega}=\\int_{-i\\infty}^{i\\infty}\\frac{\\mathrm{d}(i\\omega)}{2\\pi}", "2aa08255fdcd351562106e9a489744f3": "\\{J_k\\}", "2aa0b826e3048e572e5a7948262b622a": "1/\\tau_0", "2aa1093ce6d464a444dd0809ba6bd934": "\\Theta=\\mathbb{R}\\times\\mathbb{R}^+", "2aa1641e9f7637f0b710246b1d1c003d": "tI_n - A", "2aa184ab7cbf4caf95b94f18de4bea6a": "s-a=n(mn-k^{2}) \\, ", "2aa1a740076747afb6d6d11a66d1908f": "N = \\sum \\nu_i n_i n_i^T", "2aa22b7503b78ecbaf8cbbbd542c398c": "P_{\\mathbf{v}}(x) = \\langle \\delta(x - \\mathbf{s} \\cdot \\mathbf{v})\\rangle_{\\mathbf{s}}", "2aa285f09f973620f6cc72cf32760c83": "i \\mapsto \\begin{pmatrix}\n i & 0 \\\\\n 0 & -i\n\\end{pmatrix}", "2aa2d87103abac4b2f57595f89374813": "\\hat{A}_{n,\\alpha}", "2aa30944cc78997afc4fd145b842b018": " n \\to \\infty ", "2aa33393c676cbd358614f9de3682b20": "QE_i(\\lambda) = 1 - e^{-\\alpha(\\lambda) d_i}", "2aa3b05792eadac83a2b827863592175": "\\lambda<\\lambda_c", "2aa3da8926064b6326155b057cdcf6b6": "\\scriptstyle \\frac{T\\, V_i}{L}", "2aa3db2646aeb9ce3d1bf22c94bef858": "\\rho_{12}(z)= \\sin z \\,;", "2aa447f3144cccc2865e6268c583f0f3": "\\alpha=0", "2aa4710c8f13378ad5b0405df19afff3": "T=T_{\\rm Kep}+T_{\\rm Gvm}=T_{\\rm Kep}\\pm{S}/{Mc^2},", "2aa479eb551f493fa021e8fccc8776ec": "G = \\sqrt{A H}", "2aa49d7ea921edada76f4f880f39a6dd": "(b_i) := (c_i) * (a_i)", "2aa4a5e19ac8d3a302933e089d300a1c": "\\mathbb{}H_*", "2aa4be5c9b9e23117bf2914f6e61c03e": "c=\\frac{\\sigma_{\\varepsilon}}{V}=\\frac{\\varepsilon}{d}", "2aa530a2648aea0801754dfb329bee07": "b_1,b_2\\in \\mathcal{B}", "2aa57663aae1af808a2360b13e576610": "MacD = 0.5 \\cdot \\frac{9.804} { 130.462} + 1.0 \\cdot \\frac{9.612} { 130.462} + 1.5 \\cdot \\frac{9.423} { 130.462} + 2.0 \\cdot \\frac{9.238} { 130.462} + 2.0 \\cdot \\frac{92.385} { 130.462}= 1.777 years ", "2aa5d276b6c99c65ae04e6a163369447": "m_2=\\frac{a^2+ab+b^2}{3}, \\,\\!", "2aa60a54e2059a74703db397498466e5": "n!/(2\\cdot i \\cdot (n-i)!)\\le n^i/2", "2aa63691fcf488315e2bae19dd15ee00": " \\tilde {u} ", "2aa681088fce8ffe79dcba2c345953b1": "\\textstyle{\\left|E_\\perp\\right|=2\\left|E_{\\theta_1}\\right|\\left|\\cos\\left({kd\\over2}\\sin\\theta\\right) \\right|}", "2aa6b39616936641d69c88eb76db8f3a": "\\eta(s) = \\frac{1}{2} \\int_{-\\infty}^\\infty \\frac{(c + i t)^{-s}}{\\sin{(\\pi(c+i t))}} \\, dt.\n", "2aa6c76b423c0f0aff2fef7cfbfe9269": "\\pi_3(S^3)=\\mathbb{Z}", "2aa736f44362841c7dba0c9a179eda83": "\\int_0^1\\frac{1}{\\sqrt{(1-t^2)(1-k^2t^2)}}\\,dt \\,\\equiv K(k)=\\frac{1}{2\\xi_d}", "2aa7b358c31cf800fbc18661321a1993": "n\\leq \\frac{f(b)-f(a)}{\\alpha}\\ ", "2aa7ebf9421b49c408a1e1849ace3b1b": "b \\notin\\{-2,-1/2,0,\\pm1\\}, a^2=-(b^2+2b) ", "2aa83f808858a2eebd9f45c6f4e62b2d": "\\frac{d \\sin \\theta }{ds} = \\frac{d}{ds} \\frac{y'(s)}{\\sqrt{x'(s)^2+y'(s)^2}}", "2aa8966a374e33fc98b0324f1d369878": "v=\\frac {\\ell} {n}", "2aa9172279ead0518f40513fb0ea8d0b": " \\iota_H d\\alpha = Y^i \\frac{\\partial^2 L}{\\partial\\xi^i\\partial x^j} dx^j - X^i \\frac{\\partial^2 L}{\\partial\\xi^i\\partial x^j} d\\xi^j ", "2aa953630345393aa961cdcc13afc0ad": "10^{6.5}", "2aa957c16701b48eb76009841387ad03": "\\Omega \\times \\mathbb{R}^m \\times \\mathbb{R}^{mn}", "2aa96a859d3d260793eb173dd2c6e185": " (\\mathbf{b} \\in \\ker(A^T)^{\\bot}) ", "2aaa448effea8549a87a92e58fb9b356": " c = \\exp\\left(\\frac{E_1 - E_\\mathrm{L}}{RT}\\right), \\qquad (2)", "2aaa4b9d98b506e0397b70b8aa2b2981": "\n\\begin{align}\n{d^2\\theta\\over dt^2} & = {1\\over 2}{-(2g/\\ell) \\sin\\theta\\over\\sqrt{(2g/\\ell) \\left(\\cos\\theta-\\cos\\theta_0\\right)}}{d\\theta\\over dt} \\\\\n& = {1\\over 2}{-(2g/\\ell) \\sin\\theta\\over\\sqrt{(2g/\\ell) \\left(\\cos\\theta-\\cos\\theta_0\\right)}}\\sqrt{{2g\\over \\ell} \\left(\\cos\\theta-\\cos\\theta_0\\right)} = -{g\\over \\ell}\\sin\\theta\n\\end{align}\n", "2aaa93a4f1279ab07083f0efa8ed3f4e": "\\left (\\frac{1}{n}\\right)\\sum_{i=1}^{n} \\left [ \\mathbf{X_i} \\right ]= \\frac{1}{n}\\begin{bmatrix} \\sum_{i=1}^{n} \\left [ X_{i(1)} \\right ] \\\\ \\vdots \\\\ \\sum_{i=1}^{n} \\left [ X_{i(k)} \\right ] \\end{bmatrix} = \\begin{bmatrix} \\bar X_{i(1)} \\\\ \\vdots \\\\ \\bar X_{i(k)} \\end{bmatrix}=\\mathbf{\\bar X_n}", "2aaa9ef0ecb3415c0c37eef4a7462c48": " f \\preceq g\\iff f \\in O(g) ", "2aaada173bef2fc8b0e6514e7148a209": "NNZ < (m(n-1) - 1)/2", "2aaafa79fdf730a0d19f42e7d5f336cb": "T_{o+}^{TM}=E sin(\\frac{m\\pi }{a}y)e^{-jk_{xo}(x-w)} \\ \\ \\ \\ (22) ", "2aab1a14dc3ef4bfaba444cd6f31b293": "m_i(x) = m_{2i}(x)", "2aab1ce963c0bac2ae83e3e757025c5e": "\\mathcal{G}(\\infty,\\infty)", "2aaba4310a457c469e7004d173134c38": "\\mathbf e_B := \\sum_{i \\in B} \\mathbf e_i,", "2aabaa5dc3da78805ead00bbb3f36b4e": "F(x,cy)=|c|F(x,y)\\qquad\\forall c\\in\\mathbb{C},\\quad x\\in M,\\quad y\\in T_xM", "2aac185ddcf00df8e8bca676690c6a1e": "e^2/h", "2aac18839ce5306a6b3570b648ce7f0d": "\\sum_j \\beta_j B_j \\to \\sum_i \\alpha_i A_i", "2aac215da85d3da8fa8a643a41385c96": " L[t] = - m c^2 \\sqrt {1 - \\frac{v^2 [t]}{c^2}} - q \\phi [\\vec{x}[t],t] + q \\dot{\\vec{x}}[t] \\cdot \\vec{A} [\\vec{x}[t],t].", "2aac320c268e0792915b6908991e3933": " f_\\text{0 dB} = \\beta A_0 f_1. \\, ", "2aac56b75d568e334531493a18595ffa": "x=[0,1,2]\\,", "2aac5783c5bc6b19398614f43519fade": "V=2.217 Y^{0.352}-1.324", "2aad2861e72205f637e8530553a7be90": " \n{2 \\over n!}\n\\int_0^{\\infty} { dr }\\;r^{2n+1}\\exp\\left( -r^2\\right) J_{0} \\left( kr \\right)\n=\nM\\left( n+1, 1, -{k^2 \\over 4}\\right)\n . ", "2aad79e0856ad26be51452e254a3a8ef": "M(H) \\approx n \\mu L\\left(\\frac{\\mu_0 H \\mu}{k_B T}\\right)", "2aad8f47855489267a7f8a68e793ed89": "S \\otimes T\\ ", "2aadb1f4ab9ddc02bedddaceeabd7cc4": "\n\\Pi_\\alpha = -i\\hbar \\sum_{s,t=1}^{3N-6} \\zeta^{\\alpha}_{st} \\; q_s \\frac{\\partial}{\\partial q_t}\n", "2aadc8aa8094ce17a491abfdf111281f": "D_v", "2aae4f58d5f6b1c2433c3230532ef7ed": "S \\otimes T", "2aae8fc6ff175890f322ad88749a694f": "\\sum_{n=2}^\\infty z^n/(n\\ln n)^n.\\ (\\sigma=0)", "2aaeffe00e305903073e4796ead73441": "R_\\infty = \\frac{m_{\\rm e} e^4}{8 \\epsilon_0^2 h^3 c_0} = \\frac{m_{\\rm e} c_0 \\alpha^2}{2 h}", "2aaf16e8678ae83fb20f1abe1ea6352f": "\\prod_{d|12} f(d) = f(1)f(2) f(3) f(4) f(6) f(12).\\ ", "2aaf322e15569ff81e9237a5955dd8b0": "\\mathbf{N}_3 = \\{x ~|~ x \\in \\mathbf{Z} \\land x \\ge 3 \\} = \\{ 3, 4, 5, \\ldots\\}.", "2aafbda9ee07461b5c25eba984a2d15a": "\\delta_{jl}\\,", "2aafea8a5fa25450786bc0fed036a605": "H=\\overline{\\bigoplus\\nolimits_{\\lambda\\in\\sigma(T_0)} \\ker(T_0-\\sigma)},", "2ab00bf2729e74acfe58ee2a0754ca19": "\\frac{11}{3}", "2ab015976676c4d36750d727572b246a": "(n-1):", "2ab035511379b8923d41162ff9ffb2d7": "{}_{\\frac{1}{3}}", "2ab04e1ee9397ff30ac58db81715f630": "J=\\int_{x_1}^{x_2}F(x,y,y')\\mathrm{d}x", "2ab06faa0bb128f3d1f52c416f13b212": "\\left \\langle N, e\\right \\rangle", "2ab0c8fa8b330b5b9500e250a7076977": "M_\\text{true} = \\frac{M_\\min}{\\sin i} \\, ", "2ab0de0384be56d8b300f213a1e8a562": "\n \\oint_{\\mathbf{X}_A}^{\\mathbf{X}_B} (\\boldsymbol{\\epsilon} + \\boldsymbol{\\omega})\\cdot d\\mathbf{X} = \\int_{\\Omega_{AB}} \\mathbf{n}\\cdot(\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon}+\\boldsymbol{\\nabla} \\times \\boldsymbol{\\omega})~da = \\boldsymbol{0}\n", "2ab1562fc4d4afc40a701908fe73ce85": "\\angle A+\\angle B+\\angle C+\\angle D=360^{\\circ}.", "2ab1940193a2f77520e9b9f73ef44086": "\\varphi _j^{n + 1} \\le \\varphi _{j + 1}^{n + 1} \\le \\cdots \\le \\varphi _{j + m}^{n + 1} \\ ", "2ab19a97d646cbf88b7eac45f266005c": "\\zeta(-1)=-\\frac{1}{12}", "2ab1a623c2058f334a1812dfb41c7706": "n=2^m", "2ab1d8efc486f74009b31e6f64106011": "\\mathbb{R}^m", "2ab1e4dfc14b007cd9ecace9bffa95b7": "(u,v,\\phi)\\in(-\\pi,\\pi]\\times[0,\\infty)\\times[0,2\\pi)", "2ab23c6010b4c6aba995e059fed7d97f": "1/(\\log k)^c", "2ab24fca87da57a422f251984aaba813": "{{S_a}^b}_{;b} = 6 \\, \\pi \\, T_{;a} ", "2ab2dfc94b31d493042def5e3e1ab0b2": "\nY = L(X)\n", "2ab2f0bb9ae934fd9be85647f6dd81b5": " Q = \\Delta U + W\\, ", "2ab312a58ff3a690175b014f1a0644d9": "c = \\frac{\\sin^2\\theta}{2\\sigma_x^2} + \\frac{\\cos^2\\theta}{2\\sigma_y^2}", "2ab322ebcafae922dba39abecfcfece7": "\\lim_{\\eta\\to 0^+}\\int_{-\\infty}^\\infty \\left|f(\\xi+i\\eta)-f(\\xi) \\right|^2\\,d\\xi = 0.", "2ab323f3506cf9a6f2d30173ea40ae94": "S_0(q) = \\sum_{n\\ge 0} {q^{n^2} (-q;q^2)_n \\over (-q^2;q^2)_n}", "2ab32bfa6e67197709cc33b52e2b1b48": "X=X_1^2+\\ldots+X_k^2", "2ab3521d8b2682234ebdf071bd58da66": "Z = i \\omega L + \\frac{1}{i \\omega C}", "2ab3987e00fa60b8843fc6680661eb19": "\\left\\{ \\left[\\!\\! \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{array} \\!\\!\\right] \\in K^n : \\begin{alignat}{6}\na_{11} x_1 &&\\; + \\;&& a_{12} x_2 &&\\; + \\cdots + \\;&& a_{1n} x_n &&\\; = 0& \\\\\na_{21} x_1 &&\\; + \\;&& a_{22} x_2 &&\\; + \\cdots + \\;&& a_{2n} x_n &&\\; = 0& \\\\\n\\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && \\vdots\\,& \\\\\na_{m1} x_1 &&\\; + \\;&& a_{m2} x_2 &&\\; + \\cdots + \\;&& a_{mn} x_n &&\\; = 0&\n\\end{alignat} \\right\\}. ", "2ab39f158915df4bb29adbb86a510c17": "N_u", "2ab3c7e819ef8ec6af1263e7f7f8c75f": " \\kappa = \\begin{bmatrix} \\kappa_x & \\kappa_y & \\kappa_{xy} \\end{bmatrix} ^T ", "2ab3e2ae2586c2c9dc761f1a50a709a7": "T = \\{ 0 \\le \\rho \\le 3, \\ 0 \\le \\phi \\le 2 \\pi, \\ -5 \\le z \\le 5 \\}.", "2ab401e997c6ad8055eeb81ff724d91f": " x(\\lambda) = x_0 \\cos(\\lambda) - y_0 \\sin(\\lambda), \\; y(\\lambda) = x_0 \\sin(\\lambda) + y_0 \\cos(\\lambda) ", "2ab433fb17b6cc5efdfb5f99e3c80323": "x=\\log_g y", "2ab474b073a9deb53a4308dae1b1b502": "n = \\left[ n^{\\star} + \\frac{1}{2} \\right] ", "2ab4bfcb41e8a48ba96fa4f7e40cee0d": "\\pi(x) = \\frac{1}{\\Pi(x)} = \\frac{1}{\\Gamma(x+1)}", "2ab4db43994333f7aa9f384570913644": "y_f", "2ab500824107426c7d636ee060ebd49c": "\nS=2\\pi c^2+ \\frac{2\\pi ab}{\\sin\\phi}\n\\left(E(\\phi,k)\\, \\sin^2\\phi + F(\\phi,k)\\, \\cos^2\\phi \\right),\n", "2ab522d5b29182d9819e72c7a6514ec5": " M=6n=6(N-1), \\!", "2ab52ffcb30e699a50bd11deefdcf982": "\\,\\tfrac{-1}{2} + i\\tfrac{\\sqrt{3}}{2}\\,", "2ab54f430ef91e78f90549de0e0ad00e": "q\\times\\alpha\\div\\alpha=q", "2ab5dfe95aa225316ca6404c206964d7": "\\rho_{SB}(t) = \\hat{U}(t)\\rho_{SB}(0)\\hat{U^{\\dagger}}(t)", "2ab64773c1e8af2465b32469022b47eb": "143 >(167^{1/4}+1)^2", "2ab6740a68525342e334015ff60beb77": "ATR_t = {{ATR_{t-1} \\times (n-1) + TR_t} \\over n}", "2ab681b401b7f44fc18d0e9097b0cfa0": "\nI_{-} = -I_{e} e^{-e V_{-}/(k T_{e} )} + I_{ion}^{sat}\n", "2ab6d74c766ca4966684e8d291221e9e": "\\theta^{-1}", "2ab70f5d2765ec9f7028829b885a7f54": "\\frac{\\pi}{2} - 2 \\sum_{k=1}^{N/2} \\frac{(-1)^{k-1}}{2k-1} \\sim \\sum_{m=0}^{\\infty} \\frac{E_{2m}}{N^{2m+1}}\\!", "2ab716e4147125b8796aad8ef8493260": "\\gamma \\approx 0.5772156649", "2ab7df6a2fda5928d325878234a6fb23": "(x_n-\\alpha)(\\alpha-x_{n+1})>0\\,.\\qquad(*)", "2ab82159594870a16c8771c30c8c071f": "\\hat{F} \\Phi_0 \\equiv \\sum_{k = 1}^{N} \\hat{f}(k) \\Phi_0 = 2 \\sum_{i = 1}^{N / 2} \\varepsilon_i \\Phi_0.", "2ab8240d27b20f3fda8746cca8c7fda8": "(1, -1, 0, \\dots, 0),\\ ", "2ab831351c94a6e40e7bfb49c31d7a08": "(X, \\mathcal{B}, \\mu, T)", "2ab847e05de2d0b074776e66e6c7e358": "X,Z", "2ab85e62bc5d5f71c181d85f48ed6872": "RIRP: i_$ - {\\Delta}E_t({p^$}_{t + k}) = i_c - {\\Delta}E_t({p^c}_{t + k})", "2ab8b6b621622a842a986f1c2ef99aa8": "\nS_y = A \\bar x = \\sum_{i=1}^n {x_i \\,dA_i} = \\int_A x dA\n", "2ab8d2adc941c04e9c6984206e267782": "\\begin{Bmatrix} p , q \\end{Bmatrix}", "2ab8e1cd560fe0b0416ee57a8427b349": "\n\\int_S \\liminf_{n\\to\\infty} f_n\\,d\\mu\n \\le \\liminf_{n\\to\\infty} \\int_S f_n\\,d\\mu.\\ \n", "2ab9813d92f9a2e7cf0d7835a5b1aba1": "y_{i-1} \\oplus x_i = y_i", "2ab99048531b34968b29f416c320ce9d": "m+1 \\frac{1}{n(\\omega)}", "2ac506dd6fc4b65c751e48c088f939b8": "1/H(z)", "2ac53831f888e35f6752351fa0cffedf": " \\boldsymbol{\\Omega} \\mathbf{\\times r_B} ", "2ac58cd2147f539b6d6e5729d574cae1": "{{\\rm d}S} \\ge {\\frac{{\\rm \\delta}q}{T}}.", "2ac608b647444902499bf54b8ff1bffb": "0, 1,... ,p - 1", "2ac67612d1419dc7efc68e10fdc7d188": "\\mathrm{sech}\\,\\theta = \\frac{1 - t^2}{1 + t^2},", "2ac68f500c0a57b6c23d79c25ff6cf96": " \\{ g \\in G: \\mu^{(g)} = \\mu \\}", "2ac6a61c3fa99e90f11ecf1dafe25227": "z=\\zeta(x,y,t)", "2ac7e714b53cf5fc35cd5b84c7d4e1db": "\\begin{matrix} {2 \\choose 1}{3 \\choose 3} \\end{matrix}", "2ac8c11785a2f261abe3d7c52c02c781": "f(x)^5+f(x)^4+x=0", "2ac8f384512b9be6d2b115e20c61dc47": "\\mathcal{A},", "2ac9428e3809fdeec51294122a3ceb40": "\\mathbf{P}_{B} = \\mathbf{P}_{A} + (\\mathbf{P}_{B} - \\mathbf{P}_{A}) = \\mathbf{P}_{A} + \\mathbf{R}_{B/A}. ", "2ac94b1018d0b8d62983b6e9f4b64b93": "x^4+x^2y^2+y^4=x(x^2+y^2) \\,", "2ac95f96f41969a1ac25f322d832dd11": "i_{pa}", "2ac9677144a532ed4c13fa7b86389289": " g^* ", "2ac9c01380677f70d4c4caf61d43270c": "P_H \\; f(V) | _H = f(T)", "2ac9dca45c3b669698000fe7d97a4c16": " {} = \\begin{vmatrix} 1 & 2 \\\\ 5 & 6 \\end{vmatrix} \\cdot \\begin{vmatrix} 11 & 12 \\\\ 15 & 16 \\end{vmatrix}\n - \\begin{vmatrix} 1 & 3 \\\\ 5 & 7 \\end{vmatrix} \\cdot \\begin{vmatrix} 10 & 12 \\\\ 14 & 16 \\end{vmatrix}\n + \\begin{vmatrix} 1 & 4 \\\\ 5 & 8 \\end{vmatrix} \\cdot \\begin{vmatrix} 10 & 11 \\\\ 14 & 15 \\end{vmatrix}\n + \\begin{vmatrix} 2 & 3 \\\\ 6 & 7 \\end{vmatrix} \\cdot \\begin{vmatrix} 9 & 12 \\\\ 13 & 16 \\end{vmatrix}\n - \\begin{vmatrix} 2 & 4 \\\\ 6 & 8 \\end{vmatrix} \\cdot \\begin{vmatrix} 9 & 11 \\\\ 13 & 15 \\end{vmatrix}\n + \\begin{vmatrix} 3 & 4 \\\\ 7 & 8 \\end{vmatrix} \\cdot \\begin{vmatrix} 9 & 10 \\\\ 13 & 14 \\end{vmatrix} ", "2ac9fd87738008efb0011ad34180eebc": "\\left\\langle {\\sigma _\\alpha (r)\\sigma _\\alpha (0)} \\right\\rangle = \\frac{1}\n{{\\beta J}}\\int\\limits_{}^{1/a} {\\frac{{d^d k}}\n{{(2\\pi )^d }}\\frac{{e^{i{\\mathbf{k}} \\cdot {\\mathbf{r}}} }}\n{{k^2 }}}", "2aca60404237e0989bcd7b347ad79a39": "\\scriptstyle x \\;\\equiv\\; y \\pmod{N}", "2aca760a79905a1d244bee91c18a8656": "U_i\\cap U_j = V_i\\cap V_j = \\varnothing", "2aca762b8ccb56f4dc5353afad363609": "r_i = y_i - \\frac{\\beta_1x_i}{\\beta_2+x_i}", "2aca7c16a5e7a8dc77514314d8c7b58e": "g(\\mathbf{a}, \\mathbf{b}) = g(\\mathbf{b}, \\mathbf{a}).", "2aca8c098d7b4ca74091da66cb4ad0a5": " x^{\\prime} = F(x) + \\alpha(y-x)", "2acac1c2bcbb69e13007667712c1d6f3": "\\phi =2\\pi\\delta/\\lambda", "2acaca4f6eb29a242292daf584064b79": " I \\ddot \\theta= m g \\ell \\sin \\theta\\,\\!", "2acb653dd2d982527cef6cb823a90b0f": " W \\propto\\ \\Omega^{-1} ", "2acbee256591a0fb912f21130327527c": "f_*([M])=[X]\\in H_n(X)", "2acc3b16683ad0bbd4016c1c8b5c68bb": "{x^2 \\over a^2} + {y^2 \\over a^2} - z = 0 \\,", "2acc44a0835805438a094a08efcc8799": " S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\\eta(\\varepsilon (r)) ", "2acc54644d221d63992f956970a2e3ad": "\nC1=111.40\\cdot (B^{\\prime\\prime}-Y^{\\prime})+156\n", "2acd22c290731f352ce37554a235852c": "V=s^3", "2acd7ae799d4ea5ed2fb606d5051cac3": "\\text{Ranking points} = 100 \\times \\left( \\text{Result points} \\times \\text{Match status} \\times \\text{Opposition strength} \\times \\text{Regional strength} \\right)", "2acdb7e575a476eaf59dc700f641986e": "\\textbf m={\\textbf M}/{\\mathrm M_S}\\,.", "2acdba006f8c8eaf07d8b2fa3330790e": " v\\Big({\\textstyle \\sum_{k=1}^n } \\mathbf{1}\\Big) \\le 1\\ ", "2ace1a74091e3d118c3c43b878bbbb12": "\\frac{\\text{d}C_i}{\\text{d}t} = \\text{k}_{1(i)} E \\overline{S}_i - [\\text{k}_{2(i)} + \\text{k}_{3(i)}] C_i \\qquad \\qquad (8b) ", "2ace34d9cdfb2bf654da3ffdfb9bc245": "\\begin{align}\n\\|f\\|_k^2 & = \\langle f,f \\rangle_k, \\\\\n& = \\left\\langle \\sum_{i=1}^N c_i k(\\mathbf{x}_i,\\cdot), \\sum_{j=1}^N c_j k(\\mathbf{x}_j,\\cdot) \\right\\rangle_k, \\\\\n& = \\sum_{i=1}^N \\sum_{j=1}^N c_i c_j \\langle k(\\mathbf{x}_i,\\cdot), k(\\mathbf{x}_j,\\cdot) \\rangle_k, \\\\\n& = \\sum_{i=1}^N \\sum_{j=1}^N c_i c_j k(\\mathbf{x}_i,\\mathbf{x}_j), \\\\\n& = \\mathbf{c}^\\top \\mathbf{K} \\mathbf{c}.\n\\end{align}", "2ace785b7c4b39b211d953c387a5ac6b": "t\\theta", "2ace93c51667a69850b7e6c724198ae1": "\\log A_{t+1} = \\log A_t + \\bar{g} + \\tilde{u}_{t+1}+\\tilde{v}_{t+1}", "2aceeac022c67107b8c12defda23b332": "\\mathcal L_n(w)", "2acf233eaa490789cf6795ac7fd8f094": "V_D \\approx 0.53 V", "2acf3c85b4a5bc88e2459a00cab7597c": "P(S, t)", "2acf634fe6f16673d650144a51f06839": "\nX_{2}=\\ [2,4],\n", "2acfbea678d1ba37d269fb1710c5ffca": " \\oint_{\\partial \\Sigma(t)} \\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{F}/q(\\mathbf{r}, t) = \n- \\iint_{\\Sigma(t)} \\mathrm{d} \\mathbf{A} \\cdot \\frac{\\partial}{\\partial t} \\mathbf{B}(\\mathbf{r}, t) +\n\\oint_{\\partial \\Sigma(t)} \\!\\!\\!\\!\\mathbf{v} \\times \\mathbf{B} \\,\\mathrm{d} \\boldsymbol{\\ell}\n", "2ad0058c821240ac8ede4b2e83b1f6a9": "\\|\\mathrm e^{tA}\\|\\leq 1", "2ad00df4f5ae9b468daee4141a0429f2": "T^a = \\eta^{ab} T_b \\,", "2ad02db5be56b7ccbe6e11ecaca929ac": "\\pi\\left(x+x^\\theta\\right)-\\pi(x)\\sim\\frac{x^\\theta}{\\log x},", "2ad04b938a0ef1e4a13db74cc2d2beda": "w_{ij} ", "2ad05b5d6ac7a494c5bfa74262cb9225": "M \\rightarrow \\{0,1\\}^l", "2ad0cd78e4909be6617cad2d531f89ca": "k_i \\to \\infty", "2ad113504ddcdbd6bc1fffba24c36b91": "d\\tau = \\frac {1}{c}\\sqrt{g_{00}}\\,dx^0,", "2ad11d73ed372af2d4cb11ab89428e50": "u^{\\alpha} = \\frac{d x^{\\alpha}}{d t} \\frac{d t}{d \\tau} = v^{\\alpha} u^{t} \\,", "2ad19cefbda1f854a110de88e6acbf74": " F(\\varphi \\setminus \\alpha) = F(\\varphi, \\sin \\alpha) = \\int_0^\\varphi \\frac{d \\theta}{\\sqrt{1-(\\sin \\theta \\sin \\alpha)^2}}.", "2ad1ce0ffdd37aeccf5b79763c09b639": "A\\triangle B", "2ad1e2b89e62ad8a57922daf32ab8414": "\\ C_3^2 (3)=\\frac{16}{50}", "2ad243da8feb1f0734f178d3986bc441": "\\ M_y = \\frac{u_y}{a_y} = \\frac{W + u_1 - u_2}{a_2}.", "2ad252f1722204cd1975685dc92eb547": "\n\\overline{\\psi}(i\\partial\\!\\!\\!/-m)\\psi\n", "2ad2de47569ccef1ae97674f804973d8": "S_{mn}(c,\\eta)=(1-\\eta^2)^{m/2} Y_{mn}(c,\\eta)", "2ad2f2bd24c6aa48280bc376a878169f": "\\phi = \\frac{V_V}{V_T} = \\frac{V_V}{V_S + V_V} = \\frac{e}{1 + e}", "2ad301423a8169cd216ad7614ba0e6e2": "c^2 = a^2 + b^2 - 2ab\\cos\\gamma\\,", "2ad32247f715403a07c80199088f88b3": "I_{low}", "2ad374d911c12be7beb0652f93047eef": "\\tau_{\\pi^+}", "2ad376239d10f6bd8124bffd3a1dfcc3": "f(z) = \\begin{cases}\n\\exp(-z^{-4})&\\mathrm{if\\ }z\\not=0\\\\\n0&\\mathrm{if\\ }z=0\n\\end{cases}", "2ad3832ccccae402701f85dcd58d9ccf": "{}_2F_1 \\left(a,-a;\\tfrac{1}{2};\\tfrac{1-x}{2} \\right ) = \\frac{(1-x)^a+(1-x)^{-a}}{2},", "2ad3bc2e778989639a71ac77f8ac7fcb": "[n,\\infty)", "2ad3cd1cb45679c5a774e4111383f7dd": "(|B_s|)_{s \\geq 0}", "2ad4137a5ba0967fcd03ebe73622b044": "\\|\\mu\\| = |\\mu| (X)\\, ", "2ad432de74fd1b2d03e2fe8dce5d018e": "\\scriptstyle \\beta l = 3 \\pi /2", "2ad45c528ebf4bd0e36eb33ee8ca74be": "(\\bold{r}_j, \\bold{r}_k)", "2ad470a3b1efaa069031a87a963dba89": " T_1[i,j],T_2[i,j]", "2ad490d2c03e3e9aa90d61c0842a95dd": "X, Y \\,", "2ad4f5b5fb717ed6348f6b9a51c0e274": "\\sigma_y(\\tau) = \\frac{1}{\\sqrt{2\\tau}}\\sqrt{h_0}", "2ad5f125c7d9762ba184326650122925": " \\frac{1}{\\tau} = \\sum_{k'} S_{k'k}^{Op}=\\frac{2\\pi}{\\hbar} Z_{DP}^{2}\\frac{\\hbar \\omega}{2V\\rho c^{2}} (N_{q}+\\frac{1}{2} \\pm \\frac{1}{2}) \\sum_{k'} \\delta _{k', k \\pm q}\\delta [E(k')-E(k) \\pm \\hbar \\omega] ", "2ad65b086783defe580085878b7e4a4f": "g_\\alpha = -\\frac{g_{0 \\alpha}}{g_{00}}.", "2ad6eee4908dec4131abf20c65545fa0": "U=\\frac{\\partial \\Omega}{\\partial \\beta}=\\alpha PV", "2ad6f5855b9383cf4cc0659ddce778c4": "\\omega\\sqrt[3]2", "2ad73e192a6094ff6333208374b8be6a": "\\vec{v}_{A \\mathrm{\\ rel\\ } B} = 110 - 90 = 20 \\text{ km/h.}", "2ad7433e0a27aa76cbac129798549c4b": "1-\\delta\\,\\!", "2ad7be897952707252014fca342489ee": "\\Pr(A>\\mathbf x) \\le \\Pr(B>\\mathbf x)\\text{ for all } \\mathbf x \\in \\mathbb R^d ", "2ad7fd086cf42dcb837743384aa3d743": "G=F^{-1}", "2ad845b25bb3c937b6ddd77e0d66ee88": "\\scriptstyle{\\hat{H}(t)}", "2ad8736f4a92c3c68fc198f016ad9298": "U_q(\\widehat{\\mathfrak{sl}}(n))", "2ad8980c140fc3a29f7f0995598de212": "\n\\ln_k(x)=\n\\begin{cases}\n\\ln(x)&\\text{for }k=1,\\\\\n\\ln(\\ln_{k-1}(x))&\\text{for }k\\ge2.\n\\end{cases}\n", "2ad8a248bca763409ae5bd1acc121f38": "\\langle\\ell,\\ell'\\rangle=0", "2ad8a72731a0de9a8bd2dff49a738488": " \\Phi_G=-\\frac{GmM}{r} ", "2ad8aa57412fc7dc1cc5fcc80a718640": "\\Delta((y, (P", "2ad8ccdb761a116df350dcd3f1655bd5": "H(V)", "2ad8e851974f25e9d46ea5e41293f281": " g_{\\mu 0,\\nu}+g_{\\nu 0,\\mu} - ( g_{0\\mu,\\nu}+g_{0\\nu,\\mu}-g_{\\mu\\nu,0} ) = 0 \\,", "2ad90c51add16244700a4891170ccc74": "(f * g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau)\\, d\\tau", "2ad924112482661ef2b3601678e3a535": "\n F(x|\\alpha, \\beta, \\theta)=\n\\theta I_z(\\alpha,\\beta) + \\frac{(1-\\theta)(x - a)}{b-a} \\quad \\quad \\mathrm{ for}\\ a \\le x \\le b,\n", "2ad93339e9f11ffd37dfeee05835cbb2": "\\textstyle f : \\Omega \\to \\mathbb{R}^n, ", "2ad95a9884a6928b280947a65668a841": "13^2+7^3=2^9\\;", "2ad999cae78863b547aa5c0382561779": "Y=\\left(Y_1,\\ldots,Y_k\\right)", "2ad9cd11f6bd5e421f114a7c35b91f81": " \\chi = \\int_{t_e}^t c \\; {\\mbox{d} t' \\over a(t')} ", "2ada480fa5e1e5c93ecf5ce917d6c24d": "g=f|_T\\,", "2ada7d5ade6350015f22c8e836dab541": "\n \\boldsymbol{\\nabla}\\cdot\\mathbf{v} = \\cfrac{\\partial v^i}{\\partial q^i} + \\Gamma^i_{\\ell i}~v^\\ell\n = \\left[\\cfrac{\\partial v_i}{\\partial q^j} - \\Gamma^\\ell_{ji}~v_\\ell\\right]~g^{ij}\n ", "2adb3804879aab009b1004b012e7db54": "f_{out}(x) = p f_{in}(x) + q f_{in}(x-1)", "2adc60b1d0826d62b767a44f80cc13f0": "K_B = 0.114", "2adcf5940bf2d38d19bc2356445efdf4": "P\\setminus P'", "2adcf891ae4c70f8857bf7cd26384a23": "Z(\\beta)", "2add78e2315ec08c86ad01c56c02329b": "\\textstyle \\beta_i", "2add7b0edc34a915e324e806d61dadfb": "\\int_0^{2\\pi} \\left|f(x)\\right|^2\\,dx<\\infty.", "2ade1d01234dc5c09819436e03103c37": "a_1,\\ldots,a_m \\in R", "2ade2f02787cf29d6807dd068263c5b5": " \n{1 \\over c} {{\\partial \\varphi } \\over {\\partial t }} + \\nabla \\cdot \\mathbf{A} = 0\n\n", "2ade3fa6c8588a050b0702ede777ddca": "u_1^2+u_2^2+u_3^2+u_4^2+u_5^2+u_6^2+u_7^2+u_8^2 = x_{9}^2+x_{10}^2+x_{11}^2+x_{12}^2+x_{13}^2+x_{14}^2+x_{15}^2+x_{16}^2\\,", "2ade5364600aed042f316de3abd169e8": "\nB_{ij} = A_{ij} - \\frac{k_i k_j}{2m}.\n", "2adeef26fae971ea0d72c6429da95e73": "\n\\sum_{a \\in A_i} g_i(\\sigma^*, a) = 1 + \\sum_{a \\in A_i} \\text{Gain}_i(\\sigma^*,a) > 1.\n", "2adf23a7ed2e46a3413c2106216b784d": " L[(\\lambda V.E)\\ F:=E[V:=G]] = L[(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)) := f\\ ((x\\ f)\\ (x\\ f))] = \\lambda f.f\\ ((x\\ f)\\ (x\\ f)) ", "2adf320af7ca2c256219d969ae0d7857": "P \\neq Q", "2adf3a1259ceee9a74a0324202e12a43": "A_r=1", "2adfa03b38d46a4a0df8b5c1f0521a49": "T(4k)", "2adfb51ba286fa0ee7aa247278c4eac8": "\\mbox{mex}(\\left \\{0, 1, 2, 3, \\ldots\\right \\}) = \\omega", "2adfc291a78aafc44fac7aefbd2d1f8b": " = 9 N k T (T/T_D)^3 \\int_0^{T_D/T} \\,{x^3 \\over e^x-1}\\, dx\\,,", "2ae01fb1e5b6ad4d1dee0c3464184053": " m = \\frac{2\\varepsilon_0\\varepsilon_m R^2T^2}{3\\eta F^2 D}", "2ae036ae30b555588900077711e00a51": "v = M \\operatorname{sgn}( \\hat{x}_1 - x )", "2ae04b38af41b03b60a2a453f585438d": "{n\\choose r} = {{n!} \\over {r!(n - r)!}} = {52 \\choose 5} = {{52!} \\over {5!(52 - 5)!}} = 2,598,960", "2ae069332e09d1135cf57fc045e82f98": "\\zeta(1) = 1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+ \\ldots ", "2ae076b682fbc63849470fd84e022d30": "p_0, p_1, \\dots, p_m", "2ae115740fc351f48a3c66c708482b47": "\\Sigma_x \\equiv \\sigma_x \\ \\rho_A", "2ae1a1a07250711d91239dbb291dd3ee": "S \\supset R", "2ae1a9251e71771b5c3d26ba0def72f8": "\\textstyle W(y|x, s) = W'(y|s)", "2ae20e920d8c00d24454f1a44b0e8df0": "\\mathfrak{b}^c", "2ae217131be12cbf64a6f768a8f73669": "\\mathbf{p}_0", "2ae22911dd718e7b2deebb7c8ac563e0": "\\omega(x, 1)= \\limsup_{H\\to\\infty} \\omega(x,1,H).", "2ae22e25fe6fcd42a1b921b4ba2b251d": "\n G \\ge G_c\n ", "2ae232350ab24ebe2590273530411bd0": "C_{AB}=r(v^0)\\Delta{t}", "2ae2ee5d8e8925bb7c34ccd9516782af": "\\! w", "2ae2f9e8c865467bfb01a81043b7aa13": "\\frac{V_1}{V_2}=\\sqrt{\\frac{Z_{I1}}{Z_{I2}}}e^{\\gamma_1}", "2ae36411e865ff89b33cddabe04d49f5": "c_1 \\mid \\uparrow \\rangle + c_2 \\mid \\downarrow \\rangle \\to c_1 \\left( \\mid \\downarrow \\rangle\\right) + c_2 \\left(\\frac{3i}{5} \\mid \\uparrow \\rangle + \\frac{4}{5} \\mid \\downarrow \\rangle \\right) ", "2ae37984c28cf667b676dd3b3bbdacdc": "w\\equiv v", "2ae38202843405064032e3a2d496f2e7": "(1-z)^{-b}={}_1F_0(b;;z)=\\,_2F_1(1,b;1;z)", "2ae3879ccc3c46274b1eb20b4b776fb3": " \\mathbf{E}_{\\rm est} ", "2ae39ad53f32c9eaa92f1b4b9df933fd": "l_\\mathrm{sum~of~pieces}", "2ae39b9352d885f3b629adc1e35ad87c": "A = \\frac{14}{4}a^2\\cot\\frac{\\pi}{14}\\simeq 15.3345a^2", "2ae443d33eeebaebf1b81c251ab8ee63": " n(a) < n(b) \\Rightarrow (a, b) \\notin R ", "2ae45703e5f6c646502f622420084688": "\n\\mathrm{SNR_{dB}} = 10 \\log_{10} \\left ( \\frac{A_\\mathrm{signal}}{A_\\mathrm{noise}} \\right )^2 = 20 \\log_{10} \\left ( \\frac{A_\\mathrm{signal}}{A_\\mathrm{noise}} \\right ).\n", "2ae46afb8539497b6cff4d1f0afbc8c7": "\\mu + \\frac{\\sigma( 2^{\\xi} -1)}{\\xi} ", "2ae4ae37f0d8a213fb347987b5a5fd67": "D=17.87\\sqrt[3]{\\frac{d}{f^2}}", "2ae4d9ef7d030aee02147e5d8b41ff91": "\\begin{matrix}\nF_{\\mathbf{K}} & = & f \\left[ e^{-i\\mathbf{K}\\cdot\\vec{0}} + e^{-i\\mathbf{K}\\cdot(a/2)(\\hat{x} + \\hat{y})} + e^{-i\\mathbf{K}\\cdot(a/2)(\\hat{y} + \\hat{z})} + e^{-i\\mathbf{K}\\cdot(a/2)(\\hat{x} + \\hat{z})} \\right] \\\\\n& = & f \\left[ 1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} \\right] \\\\\n\\end{matrix}", "2ae59c473dcaa7b6e5b67abd046c7b8c": "FV = DCF \\cdot (1+i)^n", "2ae5e2b0d6017c3da3d7609efe513995": " Z_{\\mathrm{F}} =z_{\\mathrm{S}} {d_{\\mathrm{F}}}^2= {\\lambda_d}^2 (z_{\\mathrm{S}} e^2 g^{-2}) \\phi^{-1} F^2 = {\\lambda_d}^2 a \\phi^{-1} F^2, ..........(24) ", "2ae672043fd8e67ebdc8b909312590b5": "P_i\\langle x - a_i\\rangle", "2ae6e8fd83d8e85b9317d2a3eca3ec37": "\\mu_2 = 1", "2ae71be33718b71cef8afd1b8be9a0a0": "Q=\\sum_{i=0}^n a_ix^i", "2ae72a91be16772672b81c0d3370ec16": "\\nu = 0.33", "2ae73e290023aa9cb8b6bb0b9730fc88": "c(u,v) - f(u,v)>0", "2ae73ee96467c0f3a9d34756439c0965": "\\bold j = \\frac{1}{2m}\\left[\\left(\\Psi^* \\bold{\\hat{p}} \\Psi - \\Psi \\bold{\\hat{p}} \\Psi^*\\right) - \\frac{2q}{c} \\bold{A} |\\Psi|^2 \\right] + \\frac{\\mu_S c}{s}\\nabla\\times(\\Psi^* \\bold{S}\\Psi) \\,\\!", "2ae7440fd40945ae1dc72f8da42fefcf": "=1/ \\epsilon", "2ae75598e4dd3fc3394f567059cfd6d6": "xwy\\equiv xvy", "2ae7b7da34a5a76d7d644ca6f0b44fe4": "\\bigvee \\left( A \\cup \\emptyset \\right)\n= \\left( \\bigvee A \\right) \\vee \\left( \\bigvee \\emptyset \\right)\n= \\left( \\bigvee A \\right) \\vee 0\n= \\bigvee A", "2ae7c8738202abec47b7e4a13a38165e": "\n\\hat{f}(\\mathbf{x}') = \\mathbf{k}^\\top(\\mathbf{K} + \\lambda n \\mathbf{I})^{-1} \\mathbf{Y}.\n", "2ae8426360010a5b01b2781b70dcfed6": "\\left| \\tilde{f}\\left( z_i \\right) \\right| \\geqslant 1", "2ae894a70584fce67861aadc517c8b3f": "\\frac{\\|f + g\\|_p}{\\|f + g\\|_p^p}.", "2ae8d266a2fdd5a1be72c35a8ec1a121": "\\overset{A}{\\mathop{\\left[ \\begin{matrix}\n -1 & 2 & 2 \\\\\n -2 & 1 & 2 \\\\\n -2 & 2 & 3 \\\\\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c \\\\\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_1 \\\\\n b_1 \\\\\n c_1 \\\\\n\\end{matrix} \\right],\\quad \\text{ }\\overset{B}{\\mathop{\\left[ \\begin{matrix}\n 1 & 2 & 2 \\\\\n 2 & 1 & 2 \\\\\n 2 & 2 & 3 \\\\\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c \\\\\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_2 \\\\\n b_2 \\\\\n c_2\n\\end{matrix} \\right],\\quad \\text{ }\\overset{C}{\\mathop{\\left[ \\begin{matrix}\n 1 & -2 & 2 \\\\\n 2 & -1 & 2 \\\\\n 2 & -2 & 3\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_3 \\\\\n b_3 \\\\\n c_3\n\\end{matrix} \\right]", "2ae93fa7ac8653d667d2504ce44a73aa": "(g, g)", "2ae983f2037d50d57d252a9de4e0bab6": "T_\\mathbf{\\delta} f(\\mathbf{v}) = f(\\mathbf{v}+\\mathbf{\\delta}).", "2ae9da9873d6bc07140dd523a8d0505f": "\\Omega = \\rho g Q \\cancelto{1}{L} S", "2ae9e53b95578ba5f5401556977294a8": "\\begin{align}\n 146 & ~/~ -3 = & -48, & ~\\mbox{remainder}~ 2 \\\\\n -48 & ~/~ -3 = & 16, & ~\\mbox{remainder}~ 0 \\\\\n 16 & ~/~ -3 = & -5, & ~\\mbox{remainder}~ 1 \\\\\n -5 & ~/~ -3 = & 2, & ~\\mbox{remainder}~ 1 \\\\\n 2 & ~/~ -3 = & 0, & ~\\mbox{remainder}~ 2 \\\\\n\\end{align}", "2aea1da6bef67f35ed417b5b1d4cc2d6": "J_{\\alpha,l}", "2aeaa6e03c2b03150f4764f2aea82e27": "\\mathrm{Tr}_{15} : V\\otimes V^*\\otimes V^* \\otimes V\\otimes V^* \\to V^* \\otimes V^*\\otimes V", "2aeb2048f6295e530a2c9506fc08340f": "\\bar{\\mathbf{Q}}", "2aeb5982521cf856b91316cde147693f": " \\ v_{ \\bar{x} }' ", "2aebb4235bd1aa4c205a4cdf265cfdd8": "\n s_{ih} r^k - v_i \\geq b_{ih} - c_{ih}^k\n", "2aebb753dc0ccc947dd1fbc9ecc6f2a3": "\\textstyle \\Z", "2aebfa18783d63011fb6826ca3ee6a3c": " E(k) = E(0) + \\left. \\frac{\\partial E(k)}{\\partial k} \\right|_{\\mathrm{k=0}}\n\\cdot k + \\frac{1}{2} \\frac{\\partial^2 E(k)}{\\partial k^2} \n\\cdot k^2 ", "2aec268444233bf766ea62e51926d4bb": "60^o", "2aec44998c2b10ea1a438c10fdabd64a": " A_1 < A_2 < \\cdots < A_N. ", "2aec765cf2d34c4eeebccec0715ac0ee": "\\| \\Gamma \\varphi_1 - \\Gamma \\varphi_2 \\|_\\infty = \\|(\\Gamma\\varphi_1 - \\Gamma\\varphi_2)(t)\\|", "2aec8429edf9c43f5b287d761841c793": "\\omega_{\\text{i}}", "2aec8d0fed15f50a6046e69e41750e12": "\\theta^{\\pm \\alpha}= u^{\\pm}_i \\theta^{i\\alpha}", "2aecb1dc57e87620a373d19b0a889efb": "a_i", "2aecb6568b9f2139dea1e80c5285d0e8": " r_{it} - r_f = \\alpha_i + \\beta_i(r_{mt} - r_f) + \\epsilon_{it} \\,", "2aed5cd31170da951506a5b4f015f9a1": "\\int_0^1 f(x)\\,dx. ", "2aed75181c71ea18fcd96d620e4b34cd": "|a|<|b|", "2aed774a0227aa44670dd494730e0c7e": "\\mu(G)=U\\textrm{-}\\lim1=1", "2aedb7de77e07919731b1f349b44c29e": "\\mathfrak{H}(\\beta; \\infty) =\n\\begin{pmatrix}\n1 & \\beta \\\\\n0 & 1\n\\end{pmatrix}", "2aedbd33f22fbe3bf8bb85f655ee7447": "(\\mu_4 - \\sigma^4)/n", "2aee4b99d9aaeeea01fb29be39525025": "\ng(r) = 2 r \\int^{r_\\infty}_r\n \\left(1-\\frac{r^\\prime}{r}\\right)W(r^\\prime)\n", "2aee549a324df1d2cd03f3a8115e4f47": "R(\\lambda; K)= (K-\\lambda \\operatorname{Id})^{-1}.", "2aee683d90dd8e51c7536621717fa793": " b_{i} ", "2aef157a400779b60d921eb7fa4ef467": "dA = -S dT - p dV + \\sum_{j}\\mu_{j}dN_{j}\\,", "2aef433f85840dc47821bb50e85d24a1": "\\frac{12\\sqrt{6}\\,\\zeta(3)}{\\pi^3} \\approx 1.14\\!", "2aef4cb82bac4a4eb5e7506ed4e7805b": " O(Tm) = O(n^2 m \\log n)", "2aefbaae465b22c9018c62bb1736d01e": " \\rho = \\frac{ x_2 }{ x_1 } = \\left( \\tfrac{x_2}{x_0} \\right) \\diagup \\left( \\tfrac{x_1}{x_0}\\right) ", "2aefdc802ac98bd72ff71a976aea0e53": "\n\\widehat{\\kappa} =\n\\left( \n\\frac{P_b - P_a}\n{ \\left(1/a^{\\widehat{\\theta}}\\right) - \\left(1/b^{\\widehat{\\theta}}\\right)}\n\\right)^{ 1/\\widehat{\\theta}}\n", "2af006ab4b98315ab5d642e4ac607dca": " D[p] = \\_ ", "2af01e05e8e89ab7b4332b94a83f6e35": "{q}", "2af02b1a964afa714af687301fb48f9e": "e^{-\\gamma|t|-\\sigma^2 t^2/2}", "2af03c8e9aeed7cf49333c8c77e0b3d5": "\\exists \\varepsilon_0 > 0 \\ \\forall \\delta > 0 \\ \\exists x, y \\in M : \\left (d_M (x,y) < \\delta \\wedge d_N (f(x) , f(y) ) \\ge \\varepsilon_0\\right) .", "2af05cc9bf0632b6dbe4779ecd43eef3": "(r,z,\\phi)", "2af084b912c9d75dc719fb5f952fe373": "r_p\\,\\!", "2af08b5e0d99b2daa9f553ba8d2be592": "{BC}^{2}+{AC}^{2}={AB}\\times {BH}+{AB}\\times {AH}={AB}\\times({AH}+{BH})={AB}^{2} ,\\,\\!", "2af0991f34d17d9b1e04b599b9da4329": "S_{pure}", "2af0be69609875b7b2d6f21c4fc19631": "(x+c)^2 + y^2 = 4a^2 - 4a\\sqrt{(x-c)^2+y^2} + (x-c)^2 +y^2", "2af0c2a394c6df9de80e1aef85579d77": "\\rho_e(z) ", "2af0c91b90ad9da13f320c2f7043ce01": "\\langle Vf, \\phi_n \\rangle \\approx \\sum_i w_i V(x_i) f(x_i) \\overline{\\phi_n(x_i)}.", "2af0e95d92162794f1ff16887877c518": "U = \\left(\\frac{r}{2GM} - 1\\right)^{1/2}e^{r/4GM}\\cosh\\left(\\frac{t}{4GM}\\right)", "2af14af7ee1c8239b5882b4db11359bc": "\\rho = \\rho_{X,U}", "2af16ef66e7ee8ed728a8e7e21458995": "\\text{Crosswind} = \\sin[30^\\circ] \\cdot 15 \\mathsf{knots} \\approx 7.5 \\mathsf{knots} ", "2af1aa553a3734c6e3d3bb5142193f44": " \\mathcal O_X^\\mathrm{an}(U) ", "2af1ab8897e5d40783e69cf214d85fdd": "g(z)^n/|G|.", "2af1b50946930b53b8224a64440f7ea6": "~k(\\gamma)=\\frac{2\\pi}{T(\\gamma)}~", "2af1be905adfe8b04352f8722f908bb2": "M \\hookrightarrow W", "2af2006d468c326b8ed85454f3478179": " \\operatorname{ var }( \\bar{ y } ) = m_x^2 \\operatorname{ var }( r ) = \\frac{ N - n }{ N } \\frac{ 1 }{ m_x^2 } \\frac{ \\sum_{ i = 1 }^n( y_i - rx_i ) }{ n - 1 } \n\n= \\frac{ N - n }{ N } \\frac{ ( s_y^2 +r^2 s_x^2 - 2r \\rho s_x s_y ) }{ n } ", "2af2197b99a61eb499a5883d36bffcaf": "c = d/1730", "2af282f7787dd52bcab7d7975a359093": " \\{a^n b^m c^n d^m | m,n \\geq 0 \\}", "2af2dbf196abe94e8e6db6957d9e96d7": "\\vec{p} =\\frac{m_{2}\\vec{p}_{1}-m_{1}\\vec{p}_{2}}{M}", "2af31c46eca5bb20bf8152b7875857f4": "\\boldsymbol{m}=\\boldsymbol{N}", "2af3904f3753e3ebbd64b104284897f5": "1/c", "2af400fbff73cf50d58ba9e6b5a904ff": "\\begin{matrix}\\operatorname{Ta}(2)&=&1729&=&1^3 &+& 12^3 \\\\&&&=&9^3 &+& 10^3\\end{matrix}", "2af47ec624dd07092f5c1cd1d52313b6": "P_{n+1}^{[r+1]}(x) = P_{n-1}^{[r+1]}(x) + (2n+1)\\,P_n^{[r]}(x).\\,", "2af484fdab05cb8c609da6a5cd82eb49": "\\frac{1}{89}=\\sum_{n=1}^\\infty{F(n)\\times 10^{-(n+1)}}=0.011235955\\dots\\ .", "2af491ec2f20cfd8f4a98643b93b765f": " \\boldsymbol{T}", "2af4b02257e83f70e68ceb74e811b70d": "f(x;\\mu, k,\\theta)=\\int_0^\\infty\\int_0^\\infty\\ \\mathrm{N}(x| \\mu, \\sigma^2)\\mathrm{Exp}(\\sigma^2|\\psi)\\mathrm{Gamma}(\\psi|k, 1/\\theta^2) \\, d\\sigma^2 \\, d\\psi,", "2af4c659196d1706b9eb39f4362ae625": "(a_n)_{n \\geq 0}", "2af50fdad72007832297920ef356b4d7": "n^a", "2af56756a5b0489bd9c682c7935ad5a3": "\\varphi_X(t) = \\operatorname{E}\\left[\\exp({i\\,\\operatorname{Re}(\\overline{t}X)})\\right], ", "2af5789da1195386202608c7dba93805": "V_t=\\sqrt{\\frac{\\mu}{p}}\\ (1+e\\ \\cos\\theta)\\,", "2af57b518af7778474ce5735fea25695": " E\\ =\\ \\hbar\\ \\omega \\left ( k \\right )\\ \\nu ", "2af58b0ce385e9ed2b5531d5c637af07": " \\underset{x}{\\operatorname{min}}\\ 1/2x^TAx - b^Tx.", "2af59a5e9b7d6456abe5e51bdfc9806e": "TC = {\\frac{DK}{Q}} + {\\frac{hQ}{2}} + cD ={\\frac{h}{2Q}}(Q - \\sqrt{2DK/h})^2 + \\sqrt{2hDK} +cD, ", "2af59fd3a8c25f832b9f40cf4c88afa5": "\\ \\mathrm{S=\\Omega^{-1}=kg^{-1}A^2m^{-2}s^3}", "2af5ae2cf311d58a9d58bd3f2cbbbf83": " \\Psi(\\mathbf{r},t) = \\frac{1}{(\\sqrt{2\\pi})^3}\\int\\Phi(\\mathbf{k})e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)}d^3\\mathbf{k} \\,\\!", "2af5f676d4d498993e8846e32b022eac": "f(x,y) = \\sum_{j,k \\in \\mathbf{Z}\\text{ (integers)}} c_{j,k}e^{ijx}e^{iky},", "2af5ff6b6343aa8fbc165e3e0f12a72b": "(\\mathbb{F}_2)^d = \\{ (y_d, \\ldots , y_1) | y_i \\in \\mathbb{F}_2 \\} ", "2af60f6b92684e0cf3161346fdbc2bd3": "\\scriptstyle\\sqrt{11.2^2\\ +\\ 42.1^2}\\ \\mathrm{km}/\\mathrm{s}=\\ 43.56\\ \\mathrm{km}/\\mathrm{s}", "2af6672f66b3fb804f162040aad96c06": " \\begin{align}\nx_i &= \\beta_i x_{i-1} + (1-\\alpha_i - \\beta_i)x_i + \\alpha_i x_{i+1} \\\\\n\\beta_i (x_i - x_{i-1} ) &= \\alpha_i (x_{i+1} - x_i ) \\\\\n\\gamma_i \\cdot y_i &= y_{i+1}\n\\end{align}", "2af6f4e955dd20aca5329ebd77141296": "\\frac{\\partial \\tau_y}{\\partial z} = \\rho A_z \\frac{\\partial^2 v}{\\partial z^2},\\,\\!", "2af70e9c42d28a1c934f7f473ab7268e": "\\displaystyle s_0=\\Delta x_0", "2af727b2cf2e476255d40a594595cea1": "e_1 = V_s", "2af72bc29eface3ad4b5e4251681b10e": "~\\hbar \\rightarrow 0~", "2af75bf44574096548405d1b196a56b1": "\\delta = 503 \\,\\sqrt{\\frac{2.44 \\cdot 10^{-8}}{1 \\cdot 50}}= 11.1\\,\\mathrm{mm} ", "2af790c0fb66059592c711647b1f4d28": " \\mathbf{P}'(t) = Q\\mathbf{P}(t) ", "2af7bf4cb09844b4d0e9416caff27573": "\\mathbf{A} =\n\\begin{pmatrix}\nA_{1 1} & A_{1 2} & \\cdots & A_{1 m} \\\\\nA_{2 1} & A_{2 2} & \\cdots & A_{2 m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n 1} & A_{n 2} & \\cdots & A_{n m}\n\\end{pmatrix} = \\begin{pmatrix}\n\\mathbf{a}_1 \\\\ \\mathbf{a}_2 \\\\ \\vdots \\\\ \\mathbf{a}_n\n\\end{pmatrix},\\quad \\mathbf{B} = \\begin{pmatrix}\nB_{1 1} & B_{1 2} & \\cdots & B_{1 p} \\\\\nB_{2 1} & B_{2 2} & \\cdots & B_{2 p} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nB_{m 1} & B_{m 2} & \\cdots & B_{m p}\n\\end{pmatrix}\n= \n\\begin{pmatrix}\n\\mathbf{b}_1 & \\mathbf{b}_2 & \\cdots & \\mathbf{b}_p\n\\end{pmatrix}\n", "2af7ce5b187a004f74a5efddf42b74a8": " \\lambda^{2q} \\chi_T (\\lambda^p t, \\lambda^q H) = \\lambda^d \\chi_T (t, H) \\,", "2af7fef739acf6aa3d1ff3f821692d86": "L,M,N", "2af89369aaf34140a7f1a90bb09c972b": "(11) \\ \\beta = \\arcsin \\left( \\frac{\\lambda f_0}{n_0 \\nu} \\sin \\alpha + \\varphi \\right)", "2af8a74750d392319f4facc39c37287a": "\n\\left(\\frac{2}{p}\\right) \n= (-1)^\\tfrac{p^2-1}{8}\n=\\begin{cases}\n\\;\\;\\,1\\mbox{ if }p \\equiv 1\\mbox{ or }7 \\pmod{8} \\\\\n-1\\mbox{ if }p \\equiv 3\\mbox{ or }5 \\pmod{8}. \\end{cases}", "2af8bed9cd2995b5eac69d26f2084587": "M=N_{\\text{A}}\\bar{m}=\\frac{N_{\\text{A }}}{3k_{B}T}\\left[ g_{J}^{2}J(J+1)\\mu _{B}^{2}\\right]H", "2af8c508d93fdcf63dc3edea6e56451f": "\\frac{\\partial \\mathbf{u}}{\\partial t} \\sim \\frac{V}{T}.", "2af938c5c8153d2f3543ca4e9a8cedf8": "\n\\frac{1}{2} \\cdot \\frac{1}{2} \\cdot \\frac{1}{2} \\cdot \\frac{1}{2} = \\frac{1}{16}\n", "2af97d674ba1035f04c4b9e66b7affb2": "\\frac{\\text{(Equity Market Value + Liabilities Market Value)}}{\\text{(Equity Book Value + Liabilities Book Value)}}", "2af9f12e9be1764cba4521ff07ec785d": "N\\sim e^{/l_0}", "2af9f8496be1624878a27d5f09f23a89": "N = \\sum_{k=0}^\\infty Q^k = (I_t - Q)^{-1},", "2afa544ea1e72c0fe2732b45f9856dc7": "|i\\rang |\\epsilon \\rang ", "2afa66adb54119574b2f5a00ce89432c": "\\vec{m}", "2afa79b87bf7c407498a3ce818ff89e4": "\\displaystyle{g^{-1}=\\begin{pmatrix}d^t & -c^t \\\\ -b^t & a^t\\end{pmatrix}.}", "2afa7b60f5ed3fcef2120f6c3a7f59b8": "A^1", "2afb4117158b73d650797fb4e61c8859": "\\displaystyle{v_\\xi= U(e^\\xi).}", "2afb712ceff871058be99920040e766a": "B_n \\,", "2afb7472799f5a832607ca632b3fddfc": " \\and (S_7 \\implies (\\operatorname{equate}[A_7, p] \\and V[F_7] = p)) \\and D[F_7] = D[p] ", "2afb9b5be92a04bde3c5a1b39b952c8c": "\\{a^n b^m c^m d^n | n, m > 0\\}", "2afbea83a23f3b37fe653b9a4486fde5": "\\mathrm{Hom} (M, \\mathcal O(\\mathbf C))", "2afc0c0c29d58a92e173b6b6b7f02f89": " \\int_{\\Sigma} \\left\\{ \\mathbf{F} \\left(\\nabla \\cdot \\mathbf{G} \\right) - \\mathbf{G}\\left(\\nabla \\cdot \\mathbf{F} \\right) + \\left( \\mathbf{G} \\cdot \\nabla \\right) \\mathbf{F} - \\left(\\mathbf{F} \\cdot \\nabla \\right) \\mathbf{G} \\right \\} \\cdot \\mathrm{d}\\mathbf{\\Sigma} ", "2afc77685861e7b7b78dd0bdadb84292": "\\scriptstyle C_\\mathrm d\\,", "2afcbb3d57298e65c3e717138872add6": " T_1 = {{m_1 g (2 m_2 + {{I} \\over {r^2}})} \\over {m_1 + m_2 + {{I} \\over {r^2}}}}", "2afcbfc10a85acda77703c6d55161c4a": "\\text{Prim}=V\\,\\!", "2afcdd123eed8c2dc46d4987865f448f": "z^3=1", "2afe085aff483939fd50ae29a336f44b": "ln(M_2/M_1)", "2afe2725ca06c5a86b44e3083e162547": "p\\in S", "2afe29f08465e79cdff46754bd4d0b4e": " k \\sim o(\\sqrt{n}) ", "2afe4c3a244f9131f05721a4ecb21f9a": "PR:RQ = n:m", "2afe7d6ae4a9f591856ce2377ea8097c": "O \\in \\mathcal{A}", "2afe833f3e4f781e8e80e0ca26fa9247": "P=\\begin{pmatrix}\\vec{\\alpha}_{1} & \\vec{\\alpha}_{2} & \\cdots & \\vec{\\alpha}_{n}\\end{pmatrix},", "2afeaa654a36bbfebb11fe5c8019b1f2": "10^{-17}", "2afedcbe6b6635c85a6f0788b8991002": "(\\operatorname{Tr}_{12} \\circ \\Phi ) (\\rho \\otimes \\omega) = \\rho\\,,", "2afefb66d44ecc79d494dc8f2bc1f71e": "\\int_0^{\\infty} f_a dx", "2aff3873a09fbeb31b2018224730f2ca": "((w,b),(j',i'))", "2aff79f129decf75c688f509a8fda0c5": "O(dr)\\,", "2affa8a7b3c752fa243812cc5ae01680": "\\Delta u=g(x+\\Delta x)-g(x)", "2affe5dcf9824af04d669f55d6628eb2": " L^p(X,\\mu)", "2afffe4b624abdd27735b7626f7a810d": "y=3", "2b00477ad8cc529c2aaf53b5212ef84c": "\\textstyle n2^{l-1}", "2b008536d99e2ef4505a292444b3f507": " \\lceil b/2 \\rceil ", "2b00ac589211f1163d9485b10f30e3f3": "R(w) = \\|w\\|_1", "2b00b1b463999b3c099a8146f02796e1": "N(\\cdot)", "2b00bd19ed9849501da59248e19005bf": "\\pi_{0k}", "2b00c1190f0b2a2e3d4f003fbee93cf9": "\\Omega_x \\subset \\mathbb{C}^n", "2b00ea6b388ebb7850d547397fb9b3d7": " G^+_r=\\sum (a_{-m} + i b_{-m}) \\cdot e_{r+m},\\,\\,\\,\\, G_r^-=\\sum (a_{r+m} - ib_{r+m}) \\cdot e^*_{m}", "2b0128039ee52e277768813ff2fe0d63": "O(\\frac{n}{m})", "2b0157484d85d55c4f50a7d5c7f784e7": "R_\\text{v} = \\min_{i} y_i", "2b0189c3bd053c5de6bb25ada39ea01b": "\\varphi = [1; 1, 1, 1, \\dots] = 1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\ddots}}}", "2b01963d71779575ad6a7231675c3972": "p_1(1-p_2)\\,", "2b01c7440cf72e9b0b108264176a4541": " p \\approx p_t/4 = 1000(SG-1)/4.", "2b021d0739bd2b9035c5ef6fd86fc4ff": "2^3 + 2^2 + 2^0", "2b0239075850078563ec7e976b50e1f8": "(0)", "2b0273238e9fe6b2d8c7a74de68d3a02": " {\\textbf{(3)}} \\quad \\hat{\\textbf{r}}_{k} \\leftarrow \\textbf{x}_{k} - \\hat{\\textbf{x}}_{k} ", "2b02ba8805659e89ae71e8f42136007c": "F=(\\mu I_1 I_2 l)/(2\\pi r)", "2b02d95d60239da14295991e72c011ae": "\\mathbf{w}\\cdot\\mathbf{x}_i - b \\ge 1\\qquad\\text{ for }\\mathbf{x}_i ", "2b02da5b6be96942e2de9da14763822f": "\\|\\beta\\|^2", "2b02e4b4481443b866b12996ca745443": "e_2=CB", "2b02fd4e28568f8fd8b6d756daa83c46": "\\tbinom n m ", "2b03af58c26bdf0b9e96fefd4a9fad07": "P_{sc} = 2\\left(1 - \\frac{1}{\\sqrt M}\\right)Q\\left(\\sqrt{\\frac{3}{M-1}\\frac{E_s}{N_0}}\\right)", "2b03fecf425316f3ded0004c865209e4": " \\frac{f_o}{N} = f_r \\Rightarrow f = Nf_r", "2b04184719db0d9ef0f3ae3408905cd5": "\\boldsymbol{\\alpha} = (\\alpha_1, \\dots, \\alpha_T)", "2b04203cfb163920260b870d4735068d": " \\csc \\theta =\\!", "2b0433fa412f8faf0c016e3c1af3f183": "\\sqrt{3/2} \\approx 1.225", "2b044a7a1c4e47fa901018bc74d48b99": "\\boldsymbol{\\beta}(t) = \\frac{\\mathbf{v}_s(t)}{c}", "2b04bfb3e08d7dd078915da0f5c535d2": "\\tfrac{M \\nu}{1-\\nu}", "2b04da3faaaf98aef1310a608ad1c1ab": "\n -Px - M_{xx} = 0 \\implies M_{xx} = -Px \n ", "2b04ec7cce9f0689b574fa2060b40b34": " \\begin{align}\n\\lim \\limits_{N\\rightarrow \\infty} k_i \\approx -N^2 \\left[ (1-p) \\ln(1-p) + p \\ln(p) \\right]\n\\end{align}", "2b04f67f8a33c71859b72b74f38853cc": "\\nu=1", "2b04f740c3153caf190a6885e9af69a6": "(\\frac{1}{2},0)\\otimes(0,\\frac{1}{2})", "2b050ab75ec831c5c71787c9ff5a9c67": "(f_0,\\ldots,f_{[\\frac{d}{2}]-1})", "2b05202cabde0e8890f74d93617d0cb6": " e^{-r \\tau} K\\Phi(-d_2) - Se^{-q \\tau}\\Phi(-d_1) \\, ", "2b057193e8623317f74fa19518ea2d67": "\\mathbb{E} \\left [(H\\cdot M_t)^2\\right ]= \\mathbb{E} \\left [\\int_0^tH^2_s\\,d\\langle M\\rangle_s\\right],", "2b0574ec72f33c0290f8d5d11976217d": "\\ln(a)", "2b058adedae0734e7b16b512c176ba5b": " = \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\bigg(-\\left(\\frac{\\partial}{{\\partial x_i'}}\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\right)F_j(\\vec{r}') + \\left(\\frac{\\partial}{{\\partial x_j'}}\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\right)F_i(\\vec{r}')\\bigg)d\\tau'}", "2b05b94186d9e5bade285cfd9a4d05ab": " \\dot{V}_A = \\ (V_T - V_{DSphys})* f", "2b05d2d02a24017f361d16912ae7354e": "\\mathbf{P}^{NL}= \\varepsilon_0 \\chi^{(2)} \\mathbf{E}^2(t).", "2b05d74bd3fa5b291e53c9a44ec89c5e": "\\Psi \\subseteq \\Phi", "2b05eb114bd5a987ee0f11f29c5a03d5": "\\phi \\wedge \\neg \\phi\\,", "2b06069c21d224eeea3b6d9188defb1e": "\\sigma_v(p_v) = \\alpha_v (p_v) \\prod_{u \\operatorname{adj} v} \\mu _{v,w} (p _{v \\cap w}) ", "2b060ea995743d96589e02e019141d0b": "\\gamma_A(\\lambda_i) = \\mu_A(\\lambda_i)", "2b0682870eb7e6c00b42abf01021ab1c": "\\{X_1, X_2,...,X_n\\}", "2b06932f9eaf8d861b6821d861b371d4": "\\bar\\psi = \\psi .", "2b06af33cb9fcf29bec886458945d2e8": " \\begin{align} \n\\sum_{l}x_l x_{l+m}&={1\\over N}\\sum_{kk'}Q_k Q_{k'}\\sum_{l} e^{ial\\left(k+k'\\right)}e^{iamk'}= \\sum_{k}Q_k Q_{-k}e^{iamk} \\\\ \n\\sum_{l}{p_l}^2 &= \\sum_{k}\\Pi_k \\Pi_{-k} .\n\\end{align}", "2b06c4dc99eb4c77b6fd6db142da8bcb": "F_{\\alpha\\beta} = (\\eta_{\\alpha i} \\eta_{\\beta 0} - \\eta_{\\alpha 0} \\eta_{\\beta i} ) F^{i 0} + \\eta_{\\alpha i} \\eta_{\\beta j} F^{i j}\\,", "2b07fef7d3640b619c165a6589ffe5fe": "|X_{t+1} - X_t| \\le k", "2b080f172e2f2a9c8431964764bb7d12": " c = 1/\\sqrt{\\epsilon \\mu} \\,", "2b084e793f8c6ae18c426bc182b1ebc8": "\nf (v) = 4 \\pi \n\\left( \\frac{m}{2 \\pi k_{\\rm B} T}\\right)^{3/2}\\!\\!v^2\n\\exp \\Bigl(\n\\frac{-mv^2}{2k_{\\rm B} T}\n\\Bigr)\n", "2b08576408bad3b5b5d4d1aac279f0aa": "\\bar{g}(w,c) = \\sum_{k=1}^\\infty p_k \\ln \\left(\\frac{w-c+D_k}{w}\\right)", "2b0880819e428fc4f244dbde904f7ca5": "\\begin{bmatrix}\nc_3 c_2 c_1 - s_3 s_1 & - c_2 s_1 c_3 - c_1 s_3 & c_3 s_2 \\\\\nc_3 s_1 + c_1 c_2 s_3 & c_3 c_1 - c_2 s_3 s_1 & s_2 s_3 \\\\\n -c_1 s_2 & s_1 s_2 & c_2\n\\end{bmatrix}", "2b08ac316a7b57bfa16653458ce15d8a": "\\langle F, G \\rangle := \\int F(\\rho,\\varphi)G(\\rho,\\varphi)\\rho d\\rho d\\varphi.", "2b08b6ec1e0c42db79bb0218fa27e081": "\\nu = \\omega/{2\\pi}", "2b08fa50cae4cfbea8fedbcb65c5a062": "T(n) \\leq 10 \\cdot c \\cdot n \\in O(n).", "2b0930fed64c7acf76399b05e19424ad": " \\{\\mbox{scalar fields on }U\\} \\;", "2b0987402d3c8e5e6724f0b056147860": "\\text{If }\\tau_A = \\inf\\{n\\geq 0: X_n \\in A\\},\\text{ then } P_z(\\tau_A<\\infty)>0\\text{ for all }z.", "2b0a0e03c1eb2d6d0a81438505feed64": "V_\\mathrm{out}=\\frac{1}{2}AB\\cos(\\boldsymbol\\varphi)+ \\frac{1}{2}AB\\cos(2\\boldsymbol\\omega t +\\boldsymbol\\varphi)", "2b0a296bd460508d4ff3c11fb538d7cb": "O(\\min(V^{2/3}, E^{1/2})E)", "2b0aaa01e9f095f36cfe89a90a42834f": "p(x) = x^3 - 7x +7", "2b0aab82d63af2cd2c15ea078c75a1ed": "\\Pi^1", "2b0b0942e4648261e45b3e2fe50381f7": "a + c < b + d", "2b0b0f7ee478f4477589483de6dd09fa": "p^6", "2b0baff354b9ee69ac5ca127dc2c6c4d": "\\zeta(0,x)= \\frac{1}{2} -x.", "2b0bea0f4af259bcbaa9a601af6b3cca": "\\sum_{i=1}^N k_i c_i", "2b0c15c4b0a9a98a92979bd48120d460": "(x_1 + y_1 \\omega)(x_2 + y_2 \\omega) = \\left( x_1 x_2 + y_1 y_2 \\left(n^2-a\\right)\\right) + \\left(x_1 y_2 + y_1 x_2 \\right) \\omega = 1", "2b0c700ad713e22f55912654fb81376e": "m a_m = M a_M", "2b0cbf565f994ede6882aeba5345c409": "p=\\xi n+1", "2b0cc0ca81b21da318b3c9bd95ca1c72": "\n\\log \\left( \\frac{P_i }\n{1 - P_i } \\right) = v(x_i )\n", "2b0ce316084219149dc3d25e99696f2b": "\\begin{pmatrix}\\alpha & \\beta \\\\ \\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix}", "2b0cfc05ef03ef7b06d850f41d62d530": "\n\\begin{array}{lcl}\n\\theta_{1,\\dots,\\infty} &\\sim& H() \\\\\nz_{1,\\dots,N} &\\sim& \\operatorname{CRP}(\\alpha) \\\\\nx_{i=1,\\dots,N} &\\sim& F(\\theta_{z_i})\n\\end{array}\n", "2b0d0ff9aa159a0a0e172e85162e02c4": "K = \\frac{RT - S^2}{\\left( 1 + P^2 + Q^2 \\right)^2}", "2b0d7c52aab36b8d818dee5aba4d43e3": " \n\\text{(Eq. 6)} \\qquad Q_n^{(c)}(t+1) \\leq \\max\\left[Q_n^{(c)}(t) - \\sum_{b=1}^N\\mu_{nb}^{(c)}(t), 0\\right] + \\sum_{a=1}^N\\mu_{an}^{(c)}(t) + A_n^{(c)}(t) \n", "2b0ddbf1e72fcb683bf6c6ca2882ae40": "\\pi_\\mathbb{G}", "2b0e1534527c3756c0d8c124b8d1e17d": "\\ln (n\\#) \\sim n.", "2b0e2d4467f39458abea49a562a21ad9": "[x]_Q", "2b0e85314a5616e4e4324c6c1793c4ab": "f(k) \\cdot |x|^c", "2b0eed6eba350d6792122515e54bc5a0": "\n\\lambda_{\\rm min} = \\frac{hc}{eV}\n", "2b0f1c55b13317563a7a3be1b219e069": " \\bar{F} = - e \\cdot \\bar{E} = m \\cdot \\frac{\\;d\\bar{v}}{\\;dt}", "2b0f2b7af4e9dc9b84352ea81f1cbf36": " Q^*_{16} = (Q_{11} - Q_{12} - 2 Q_{66})\\cos^3\\theta \\sin \\theta - (Q_{22}-Q_{12}-2Q_{66})\\cos \\theta \\sin^3 \\theta ", "2b0f7da26c0b9cc138c6260420c53ffc": "\\left(\\begin{array}{cc}\n \\omega_n & 0 \\\\\n 0 & \\overline{\\omega}_n\n \\end{array}\n \\right)\n \\mbox{ and }\n \\left(\\begin{array}{cc}\n 0 & -1 \\\\\n 1 & 0\n \\end{array}\n \\right)\n", "2b0f9929f56e64d1524ad7a1804f205e": "\\sqrt[p]{E[|\\xi+\\eta|^p]}\\leq \\sqrt[p]{E[|\\xi|^p]}+\\sqrt[p]{E[\\eta|^p]}", "2b0fb2c8399f1092cdd64ad28617b1a3": "\\scriptstyle C : F \\rightarrow \\mathbb{R}", "2b0fe2e0bba3fbb76cdfc01eb894b103": " \\hat{\\bold n}", "2b0fe87877d3e282226acf7c679d0358": "\\{s_m(A)\\}_{m=1}^M", "2b0ff8896f3b2e879a3c4c2d332b2a74": "y = x\\cdot e^{x} \\,\\!", "2b104a9921adcf167bf11c35cf83ddb3": "\\log_x (x+1)\\,", "2b10b8f73bd19d64f9061127ad8f21ac": "\\rho^{(n)}(\\mathbf{r}_{1}\\, \\ldots, \\, \\mathbf{r}_{n}) = \\rho^{n}g^{(n)}(\\mathbf{r}_{1}\\, \\ldots, \\, \\mathbf{r}_{n}) \\, ", "2b11038dceba2273b8a33f9af77543b1": "{{V_1^2} \\over {g}} = {{y_2({y_2 + y_1})} \\over {2y_1}}", "2b111d02dca450bc354ffb81e7b1de03": "\\ Cxx.", "2b1169e53f4a7d7a7a5e3b9a2d7a2fe7": "V_{2k+1} = \\frac{2(2\\pi)^k}{(2k+1)!!} = \\frac{2 k! (4\\pi)^k}{(2k+1)!}", "2b11748fd89ba6babe5f4e642c2f235b": " | y | = e^Be^{0.85t} ", "2b11c9a5df365f904698027aa96d84c9": "\\frac{dW}{d\\omega}=\\oint \\frac{d^3 W}{d\\omega d\\Omega }d\\Omega\n=\\frac{\\sqrt{3}e^2}{4\\pi\\varepsilon_0 c}\\gamma\\frac{\\omega}{\\omega_\\text{c}}\\int_{\\omega/\\omega_\\text{c}}^{\\infty}K_{5/3}(x)dx", "2b120c3dd812d231d2d8c851ac76dd70": "\\scriptstyle \\vec{F}_{R}", "2b1261f0c10f8f9f50a4f65298d9da1c": " \\lim_{\\rho,z\\to\\infty}\\Phi=0", "2b12636a798e0c014916ffabd210020e": "\\left|y\\right\\rangle", "2b1273c3c70114308a2cdc45cf73f47b": "Z_1 Y_2 X_3 = \\begin{bmatrix}\n c_1 c_2 & c_1 s_2 s_3 - c_3 s_1 & s_1 s_3 + c_1 c_3 s_2 \\\\\n c_2 s_1 & c_1 c_3 + s_1 s_2 s_3 & c_3 s_1 s_2 - c_1 s_3 \\\\\n - s_2 & c_2 s_3 & c_2 c_3 \n\\end{bmatrix}", "2b12794e1251c3c0bc5c2ad7a747fcb9": "{\\mathbf e}_1 = (1,0,0),\\ {\\mathbf e}_2 = (0,1,0),\\ {\\mathbf e}_3 = (0,0,1)", "2b129c03a20f924151aca25f040015c5": "(1+CD)/(RR+CD).", "2b129ea32ce085137cd7b6ca726bd414": "p(y|\\xi)\\,", "2b12a12aa32077a34ada435e949e5728": "S = \\cfrac{\\pi\\left(r_2^4-r_1^4\\right)}{4 r_2} = \\cfrac{\\pi (d_2^4 - d_1^4)}{32d_2} ", "2b12dc997300ecbf4be5dc306cd72ed9": "\\mathcal C", "2b13a18bbc5eded2e368adea7e409d09": "i = 1,2", "2b13ae3fb8702a192579b1507428e814": "\\frac{g(z)^n}{|G|}.", "2b13f237edd01df4b1eabe0cee8adf6b": "\\sum_{j=0}^n {n\\choose j} j^{n-k} a_j", "2b142b24dd5c00b41d44e55b10a4e3fa": "J_2 = K,1 ", "2b1469913fb04640715e807dbf1552fc": "\n \\begin{array}{lcl}\n 0 = x^4 + ax^3 + bx^2 + cx + d & = & (x^2 + px + q)(x^2 + rx + s) \\\\\n & = & x^4 + (p + r)x^3 + (q + s + pr)x^2 + (ps + qr)x + qs\n \\end{array}\n ", "2b14993e2c33d8f948b0b0ec165391a4": "VA^{-1} = (C^{-1}+VA^{-1}U)Y", "2b14d518e41f311327c2ecaf2f6d0ea8": "k_{L}v", "2b1579d779d7438528cca09bde02b5bf": "\\frac{\\partial \\textbf m}{\\partial t}\\, =\\, -\\gamma \\,\\textbf m\\times \\textbf{H}_{\\mathrm{eff}}\\, +\\, \\alpha\\,\\textbf m\\times\\frac{\\partial \\textbf m}{\\partial t}\\,.\\qquad ", "2b1604847176edb7305873e51d9c5035": "\\ \\mathbf U(\\mathbf x,t) = \\mathbf b(\\mathbf x,t)+\\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad \\text{or}\\qquad U_J = b_J + \\alpha_{Ji}x_i - X_J \\,", "2b1635be4452142b05b35ae4ff90c298": "\\beta = \\frac{1}{4\\alpha} - \\ ", "2b166972c6821a3044092494f40f8ac8": "P_{4}^{-3}(x)=-\\begin{matrix}\\frac{1}{5040}\\end{matrix}P_{4}^{3}(x)", "2b16bd8348260ab9ea7d73b2be3ab908": "F_{\\chi} ", "2b16d7ce78843aeba24b9580806947e0": "(R, \\Theta)", "2b17004cc94de5517901e458056f1fc8": "\\displaystyle \\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=\\frac{\\sqrt{25Rr-2r^2}}{4Rr}.", "2b17364a8345f84d13051e241249dda6": "Y_U = \\operatorname{Proj}\\, S(U),", "2b1775926a59cfc85884887ca5e10b93": "A=\\mathbb{Z}[[\\mathfrak{h}^*]]", "2b179d356b5b2b1b56f6636e00fed0f9": " |SC| ", "2b17a5ce7732555ffe3b5684e0b53571": "\n(1)~~~~~FG~=~\nF_{\\rm smooth}~G_{\\rm smooth}~+~\nF_{\\rm smooth}~G_{\\rm singular}~+\nF_{\\rm singular}~G_{\\rm smooth}.", "2b17e1c903b61bcc761a826d2e10deec": "\\tau^{a b}\\,", "2b180dd2cf6c1d27156c0495b62cddfd": "\\Omega(\\alpha(n))", "2b186aa03ccaac5fddfc541707f83b34": "\\chi(\\tau,f) = \\exp[j2\\pi \\tau f]\\chi^{*}(-\\tau,-f) \\, ", "2b18c1e1117cc457b11c50d819145f25": "(x - a/4)", "2b193c21976478162e25107ff9d6d603": "\\sin(x+y)", "2b1985e4d724599756f6bcf2fb9dc343": "\\sum_{n=1}^\\infty\\mu(A_n)<\\infty,", "2b199911c3ce499aca1fec013ab3d248": "\\tau*=\\frac{\\tau}{(\\rho_s-\\rho)(g)(D)}", "2b19c0728c21633cedfa63d74bbc9f2c": "d", "2b1f720a6b4821971c409f175802c457": "\\Pr \\left( X_1,\\cdots,X_N \\isin D \\right) \n = \\int_D f_{X_1,\\cdots,X_N}(x_1,\\cdots,x_N)\\,dx_1 \\cdots dx_N.", "2b1fb341c2f458786517da96e1fe64a6": "c : E \\to S.", "2b1fc11a546d0315daf4dc3dca3d99d8": "\\left.\\right. |z|<1", "2b1fe5db2484a1796c84be07ff4c5fc2": "m^{e^d} \\equiv m^{e d} \\equiv m^{(e d - 1)} \\cdot m^1 \\equiv m^{k(p-1)}m \\equiv 1^km \\equiv m \\pmod{p}", "2b202963ab1baac8fe975d63e4b63f7a": "c_\\mathrm p\\,", "2b20603c6a90cb17a0a94e0d07a801e8": "t^*r+tr^*=0", "2b2070ab5631f19aaea7dd654dbc0586": "\\Rightarrow_G", "2b20aef62e90baef5737b237e62d2e5d": " \\sqrt{\\mathit{N}} \\lambda_\\mathit{C} ", "2b213b699c4f036513fa9136070ede9d": "\n \\ \\ <^d (C)\n", "2b21404339bf7ef8e7c4339cf1636efd": "u : X \\rightarrow \\textbf{R}", "2b21568ad9ec4fd276d0ac18a67a811d": "1/\\xi", "2b21625d33e4e8d4bd99f8ede682a0e9": "\\Phi = - \\frac{1}{\\sqrt{r^2 + \\epsilon^2}},", "2b21c91fb0e8c31d3a972f0396b6da7f": "\\dim(R^\\theta_{T})= {\\pm|G^F| \\over |T^F||U^F|}", "2b21fc2409ec505ab7c5bf07e8aac7f3": "\n a^3 = \\cfrac{3 F R}{4 E^*}\n ", "2b22108e97b0d06a1024f652a1db7256": "\\Psi = c_1(\\phi_1 \\mp \\phi_2) \\,", "2b2237a057b260ea85c03b7532d5d214": "y_{t_2}(t) = x(t-t_2)*h_{t_2}(t).", "2b22510578a45e6e2186c68963d0561b": "\n\\tan\\chi = \\sqrt{\\frac{1+e'^2}{1+k^2\\sin^2\\sigma}}\\tan\\omega,\n", "2b227a04c61618fc8fd068ebefe8a238": "\\operatorname{curl}(\\mathbf{F})=\\nabla\\times\\mathbf{F}", "2b228f5bb2611eba6e9d0bc6bc066d55": "\\ AB=BA=I_n ", "2b22916264c403da4559a71f586749af": "V_n(P,Q)", "2b2295498acc3afe3adb5b878746420f": " \\{ v\\in V\\;:\\;\\forall w\\in C, \\langle w,v \\rangle \\ge 0 \\}. ", "2b22a98f0ed20c61bafc836ed7a3480a": "B^*.", "2b22af6264f841ceeb262293399a9f5b": "M_A^i", "2b22eb5f386b3f640caa3abe730d0faf": "\\mathbb E[v_n]", "2b2325a5b3d18acfdc0d1738df56b0b8": "\\Sigma r (Q_0 + \\Delta Q)^{n}", "2b232f2b2be3e1f5558cf398420cd5dd": "F(t)= n \\int_{-\\infty}^{\\infty} g(x)[G(x+t)-G(x)]^{n-1}\\text{d}x.", "2b23446862998654ada2045b70c745d0": "\\mathrm{Factor} = \\frac{\\mathrm{Days}(\\mathrm{Date1}, \\mathrm{Date2})}{360}", "2b23462c0e6d64dee04cc56dd8ddc969": "X(A_n)", "2b239a4452197d795b2cab91ec544609": " var( s^2 - m ) = \\frac{ 2t^2 } { n - 1 } ", "2b23d04af47255d503dc3a122cda0f2b": "\\beta = \\frac{-\\Delta H_r D_{TA}^e C_{AS}}{\\lambda^e T_s}", "2b247dad291481f8c910230aa8763a98": "\nc_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!} ,\n", "2b2490e4ebab91dfd0111edff4bd2469": "\\displaystyle (\\partial_t+\\partial_z^3+\\partial_{\\bar z}^3)v+\\partial_z(uv)+\\partial_{\\bar z}(uw) =0", "2b2507e7ff31f7305092692d65463203": "\\textstyle u(1/V(z))", "2b2511494ed89c95b959e7d21fc4ce08": "C_s = \\frac {W^2 L}{24 P^2}", "2b2550ddb6ced4a482087deccb84025b": "x_1 * x_2 \\rightarrow x'_1 + x'_2", "2b25ef6a80486e851ee5e4b720dd90fd": " d(uv)=u\\,dv+v\\,du", "2b2627eead963ca3ea7d12bb1ed2bd15": "\\mbox{reachability-distance}_k(A,B)=\\max\\{\\mbox{k-distance}(B), d(A,B)\\}", "2b2628be24fec845fb4629f2ab6368bb": "N_i=i^2", "2b262ba6a188441d85be24f7585334d6": " \\bar X \\sim \\mathrm{Gamma}\\left(\\alpha=n\\cdot k /2, \\theta= 2/n \\right)", "2b263aeb7147167028a24c08e346f0f4": " V(x) = \\begin{cases} K - x & x\\in(0,c] \\\\(K-c)(x/c)^\\tilde{\\gamma} & x\\in(c,\\infty) \\end{cases} ", "2b266e7a6da8942981adece915b4f97d": " \\mathfrak{g} ", "2b2696f768181cc3b9f1b250bcdb6a71": "L(H)", "2b26a4e84aecb454be02473360204d19": "p_\\mathrm o\\,", "2b26def518410990e836975a1e48242a": "\\begin{cases} du/dx = dv/dy \\\\ dv/dx = -du/dy. \\end{cases}", "2b270ee9125979c4f56bccb5935fadba": "p_3 = x_1^3+x_2^3+x_3^3\\,,", "2b274daa9f7fb1791a228f4717e131a8": "\\Phi _\\alpha \\left( {A_\\alpha ^1 , \\ldots ,A_\\alpha ^n } \\right)", "2b2756bcff2ca058161af8ec6a67d4a0": "t=u-\\frac{p}{3u}\\,.", "2b27640772a3ffb02c8d1e7980227908": " \\mathcal{C}_{Y\\mid X} = \\mathcal{C}_{YX} \\mathcal{C}_{XX}^{-1} ", "2b27e7abc223df964f64cae8077d60ab": " q(a \\otimes_B a') = (f_1(a)a',\\cdots,f_n(a)a')", "2b2839eb1829499e9fcc23257d080413": "A \\rtimes_{\\sigma} G,", "2b284faf3522ab16cd2fe9e52aca107c": " a_2 \\, = a_3 = a_4 = 0 ", "2b287f25bf880d279f420adcd47a94dd": "f(t) = {{d^2 t^4 + d^2(1 + d + d^2) t^3 + d^2(1 + d + d^2)t^2\n+ d^2 t} \\over {t^4 +\nd^2 t^2 + 1}} + t^{1/2},", "2b28991e9ef6bfc0bd5ecd5d693335de": "(\\vec{B_{2}}-\\vec{B_{1}})\\cdot \\hat{n}=0", "2b2923229fdabdf6566b2a8d6b9e3153": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T \n - \\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T\\cdot\\boldsymbol{\\sigma} \n", "2b29273c33a03080c91adf000bcca1e3": "\n(f_1\\ ,\\ f_2\\ ,\\ f_3)\\ = \\ - \\frac {\\mu} {r^3}\\ (x_1\\ ,\\ x_2\\ ,\\ x_3)\n", "2b295768432a63bcc69a5b098767bcf6": "2821 = 7 \\cdot 13 \\cdot 31 \\qquad (6 \\mid 2820;\\quad 12 \\mid 2820;\\quad 30 \\mid 2820)", "2b2961465bc0eb89a7671232f27640c4": "N_{RO}", "2b29dc9ddc45b3532e8856e8e009644d": " F(x) = \\sum_{k=0}^{\\infty} \\frac{(-1)^k \\, 2^k}{(2k+1)!!} \\, x^{2k+1}\n = x - \\frac{2}{3} x^3 + \\frac{4}{15} x^5 - \\cdots", "2b29f51966b7082c9048a7287855ae17": "\\Delta y=f(x+\\Delta x)-f(x).\\,", "2b29ffa263893a4a984daeb059e1b07b": "G_{nm}(\\mathbf{x}',\\mathbf{x})", "2b2a0a4b3fbc78a61cec549a40912932": "o_t", "2b2a0d7473e6b35ca6625e94b3b4a2f6": "g = f^{*} g', \\, ", "2b2a5cc83f6ea6da731e34f14708e33a": " \\frac 1p= \\sum_{k=1}^n \\frac {\\theta_k}{p_k}.", "2b2a85b21446c5513dfe9520a196a282": "\\hat R_{\\mu}=\\frac{2G}{c^3}\\hat P_{\\mu}=\\frac{2G}{c^3}(-i\\hbar )\\frac{d}{dx^{\\mu}}=-2i\\,\\ell^2_{P}\\frac{d}{dx^{\\mu}}", "2b2a86bc7d8afb2dcbc5c81229cc5422": "\\tan \\delta' = \\frac{-1}{\\gamma \\cdot \\beta}\\;\\approx -\\frac {1}{\\beta} \\quad \\rightarrow \\quad \\cot \\delta' = -\\beta \\quad \\rightarrow \\quad \\delta' = \\arccot \\beta \\approx -\\frac{\\pi}{2} + \\beta \\quad", "2b2aa8c4d40a99fad24003e934d489b8": "\\omega_{Y|\\overline{X}}", "2b2ab5d721bf76536fa5fbd981a975a4": " T_2 ", "2b2acbb5e965e7b589b388fca00b7707": "\\mathcal{A}_i", "2b2b28526d02f790272375d54c55fe7d": "V=\\frac{\\mathrm{Re}_L \\nu}{L_c}", "2b2b610efc4ccde7e479c5112db538f4": "\\scriptstyle \\mathsf{Boolean}", "2b2bda28283be64a94f65b2795d0d00c": "C = R - \\frac{1}{n-2}\\left(\\mathrm{Ric} - \\frac{s}{n}g\\right) \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc g - \\frac{s}{2n(n-1)}g \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc g", "2b2bf15cb520fbb73f10ed8bed6f627e": "\\ d[\\mathbf{x}(i)\\,\\mathbf{x}(i+3)]=0B) = P(B>C) = P(C>D) = P(D>A) = {2 \\over 3}", "2b44667aa6418a1a3e89ccff47d83378": "V_{oi}", "2b44ad3d36aeeec2adf77918b3aa9143": "p(t)\\approx ", "2b44c56022bcfacc973aa08f2080b25f": "\\left \\| A^r \\right \\|^{1/r} \\ge \\rho(A), ", "2b44d5c01cd2ba515597a30acb82e7b0": "H_2(z)=1.\\,", "2b453a6f26f04bbed33a8f408aed7d49": "\\text{Semiperimeter}=m(m+n) \\, ", "2b455bbee6d72fa5b13f1c2b8a46ccbc": "i, j \\in S", "2b4574e2827d7d9d9bfddb7d18748b40": "g_1 = \\frac{\\sqrt{n} M_3}{M_2^{3/2}},", "2b457960b52ffc44fc1f5305d812125e": "Z'=\\frac{1}{Z}=\\frac{I}{V}", "2b45d45c3752b07f2dae3e33934e27aa": "1\\times 3^{-1\\,\\,} + 2\\times 3^{-2\\,\\,\\,} + {}", "2b45ded4e24b8614d1314fd62dcdddae": "height=D \\times \\sqrt{\\frac{H^2}{W^2+H^2}}", "2b45eac5e242c6a427f7ff2c8833828e": "C^{(k)}(GIP)\\leq c {n\\over2^k},", "2b4605daf2a6d83f3096e4e3428269ac": "f(\\mathbb{I} + \\eta A^{-1}B) = \\mathbb{I} - \\eta A^{-1}B + (-\\eta A^{-1}B)^{2} + \\ldots = \\sum_{n=0}^{\\infty} (-\\eta A^{-1}B)^{n}", "2b4616ff89ebbac86c56f7d37f9c0173": "\\sigma'(n) = n", "2b4688afda51439cf768289e16b7be64": " X^\\alpha \\rightarrow X^\\alpha + b^\\alpha ", "2b469e19de8c2d749b42187e799a50f1": "\\mathbf{x} \\in \\mathbb{R}^6", "2b46b5d95c66bb32a05308ae3b9adee4": "2^\\mathfrak c = \\beth_2.", "2b46f117277f2126b0b06b3f447ffe0d": "\\mu^{-1}(0)", "2b4788112f64436c3dd46bad56b37c1d": "M_{NHL}", "2b478c52d9fb222bbad5e4e336e56d02": "\\log_{10}120=\\log_{10}(10^2\\times 1.2)=2+\\log_{10}1.2\\approx2+0.079181.", "2b47a2af300f3700208ea2d17de9ae1f": "R(T')", "2b47af5fa2911d75bcf164bf1868ec7b": "c = \\frac{1}{d} [0.4024+0.00081t]", "2b47ba4d377403255f24b4d891fac3d4": "S(a,b) = \\frac{P(a,b)}{Q(a,b)}.", "2b486db789b595ce1e505a28eba54588": "m(t) = {{k_d C_m \\rho} \\over {\\lambda}} \\left( t \\lambda_c + {{\\lambda_r} \\over {\\lambda}} \\left( 1 - e^{-\\lambda t} \\right) \\right)", "2b487790aee387267bd157330818d03d": " V_f = \\{(x,y)\\in\\mathbb{C}^2\\mid f(x,y) = 0\\} = \\Bigl\\{ (x,y)\\in\\mathbb{C}^2\\mid(x,y) = \\Bigl(t,\\sum_{n=3}^{\\infty}t^n\\Bigr)\\Bigr\\}", "2b48d685669544a3277d4614052c23ac": "(f*g)(a, b)=\\sum_{a\\leq x\\leq b}f(a, x)g(x, b).", "2b48ea30332522ed910a7b58b2075a56": "P_S", "2b49203ff489ae15921308ceadaf398b": "\\lambda \\in R", "2b4955f277432abdbf4e601fbde1d913": "\\tfrac33", "2b495da090fa27cbac5d732777f25f0a": "\\varphi_m( \\boldsymbol{r- R_n})", "2b49a3970fe31d616aa86bb28f3e858e": "f(n) = L(n/148)^{1/3}", "2b49c32664b4a30d7a43b2ee4e10a4b7": "\\displaystyle d\\alpha(x) = W(x)dx", "2b4a005ceb07be7cba325cbde58665f7": "f(x)>o(g(x))", "2b4a1b122fbdf4ce43753b882d7b2c90": "\\sup\\{f(x) \\mid x \\geq t_0\\}", "2b4a4ca3445e802bdddc171245ac2f69": "G, K\\,\\!", "2b4a75eaaedae773ba89fb01a0a20ba9": "T_{1} = R_{1A}(x)", "2b4a96f7f465f71e9c04b73ae1674095": "\n {\\rm Ind} (-S_{xx}''(x^0))= \\frac 12 \\sum_{j=1}^n \\arg (-\\mu_j), \\quad |\\arg(-\\mu_j)| < \\pi/2.\n", "2b4ad2202458f5b1c4017178368d13d0": " det((J\\varphi)_{({u}_{1},{u}_{2})})\\neq 0", "2b4b200843070e44b9064037f813175b": "\\, {} + p_1x'\\varepsilon + 2p_2xx'\\varepsilon + \\cdots + np_n x^{n-1} x'\\varepsilon", "2b4b4896971128b5b850e33e57a6ece4": "\\begin{matrix}\n x_1 & = & x_0 - \\dfrac{f(x_0)}{f'(x_0)} & = & 0.5 - \\dfrac{\\cos(0.5) - (0.5)^3}{-\\sin(0.5) - 3(0.5)^2} & = & 1.112141637097 \\\\\n x_2 & = & x_1 - \\dfrac{f(x_1)}{f'(x_1)} & = & \\vdots & = & \\underline{0.}909672693736 \\\\\n x_3 & = & \\vdots & = & \\vdots & = & \\underline{0.86}7263818209 \\\\\n x_4 & = & \\vdots & = & \\vdots & = & \\underline{0.86547}7135298 \\\\\n x_5 & = & \\vdots & = & \\vdots & = & \\underline{0.8654740331}11 \\\\\n x_6 & = & \\vdots &= & \\vdots & = & \\underline{0.865474033102}\n\\end{matrix}\n", "2b4b9d14f9b1b2bcebde31f0212d68a0": " 2E_{n+1} = \\sum_{k=0}^n \\binom n k E_k E_{n-k}\\text{ if }n\\ge 1.\\qquad\\qquad(1) ", "2b4c286855d442392aab59b91384732c": "C_G(G^\\prime) = \\frac{F_G(G^\\prime)}{\\sum_i F_G(G_i)}", "2b4c5b4c6e429c13983c0d87327af018": "\\delta \\mathbf{x}_i = \\sum_{j=1}^N \\epsilon_{ij} \\mathbf{x}_{0j} \\qquad(4)", "2b4caa5690c4b7b200e778a9ade78592": " : \\hat{f} \\, \\hat{f}^\\dagger : \\,= -\\hat{f}^\\dagger \\, \\hat{f} ", "2b4d5e0fa198e5cd30ceb943adb0f53b": "\\delta(P,Q)=\\sup_{ A\\in \\mathcal{F}}\\left|P(A)-Q(A)\\right|. ", "2b4d7505f918bd09c5c1320d5887b473": " \\mathcal{S}\\subset \\mathbb{R}^k ", "2b4d75420b7940aa59ae0c263d0ebdbb": "KL_{i,2}", "2b4da9ebb6b4f77d01ab6fb7489b24c2": "\\operatorname{Arg}\\left(z_1 z_2\\right) \\equiv \\operatorname{Arg}(z_1) + \\operatorname{Arg}(z_2) \\pmod {(-\\pi,\\pi]}", "2b4e284ddf19322cf5191a5fee7017dd": "\\mathbf{P}(\\operatorname{Hom}(G,\\mathbf{Z}/p)).", "2b4e867458bcf32464b4698e329e50c5": "\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}}.", "2b4eb64deed721adea1dee6ad50be70b": "y(x) = \\phi(\\ln(x)) = \\phi(t). \\,", "2b4f039dffb8e4334abad6f21f18a820": "B(\\boldsymbol{u},\\boldsymbol{v}) = -\\int_A \\sigma_{ik}(\\boldsymbol{u})\\varepsilon_{ik}(\\boldsymbol{v})\\mathrm{d}x \\qquad \\boldsymbol{u},\\boldsymbol{v} \\in \\mathcal{U}_\\Sigma", "2b4f09e6023f2bc602e37fea76b0380f": "\n {\\rm li} (x) =\n \\gamma\n + \\ln \\ln x\n + \\sqrt{x} \\sum_{n=1}^\\infty\n \\frac{ (-1)^{n-1} (\\ln x)^n} {n! \\, 2^{n-1}}\n \\sum_{k=0}^{\\lfloor (n-1)/2 \\rfloor} \\frac{1}{2k+1} .\n", "2b4f5d799d352299c8dead42f4a306b3": "P_{gap} = \\frac{3R_r^{'}I_r^{'2}}{s}", "2b4f9783881fcb33f9811c0c7884cec6": "\\left[ f_\\text{a}, f_\\text{b} \\right]", "2b4fd6fb47484c94ff6ce1aff1c5466d": "A | B", "2b504b2d024570c7574335acc579fac7": "j^k(f_p)(x)=x'", "2b506b92da93799c0277fe227293abe7": "\\|E\\| = \\sup_{\\sigma\\in\\Sigma}\\|E_\\sigma\\|", "2b50878ccbc9204fe4097de39764e93c": "U^{2^{n-1}}", "2b508d2eaafca228c88132a2a998c68c": "\\begin{bmatrix}K\\end{bmatrix}=\\begin{bmatrix}2000 & -1000\\\\ -1000 & 2000\\end{bmatrix}.", "2b50d2200d00b97ac7f7f3e4252019a7": "k \\equiv \\frac{2\\pi}{\\lambda} = \\frac{p}{\\hbar}= \\frac{\\sqrt{2 m E }}{\\hbar} ", "2b51168616ba88332f746de79e0bee88": "\\langle e_\\mu, e_\\nu \\rangle = \\eta_{\\mu \\nu} ", "2b511cc0ccbaa62d162256e79e354b31": "Yth.g=H_t= c_2u.u_2 - c_1u.u_1 ", "2b51c7551a4173a11ea4a9381bdd3df1": "\\,\\!\\phi_i", "2b522492c73f9c1b6facf172f78ef363": "\\,\\Phi_{PSII}", "2b5256cdf46c27f9ab0b8b8c009a9c52": " x_{ij} \\in \\{0,1\\} \\qquad i=1, \\ldots, m, \\quad j=1, \\ldots, n", "2b52f81f99e448266353f77d4b55c3e4": " C \\!", "2b53581dc0e322428627ba087935b08e": "\n\\omega C \\gg \\frac{1}{R}\n", "2b5373d3b024efd6f71a01c73fb530b2": "\\{\\langle\\tilde v_i, \\cdot\\rangle = 0: i \\in I \\}", "2b53b593e5641ad92b1932a6273d9e03": " \\mathrm{actual\\ speed} = \\frac{\\mathrm{camera\\ frame\\ rate}}{\\mathrm{projection\\ frame\\ rate}}\\times\\mathrm{perceived\\ speed}", "2b53d0fe05ac9337a5960b60387a81f2": "{f'}^2 = 1 - 2\\delta f^2 + \\epsilon f^4", "2b5406a1a4067dce26eeb3db949a7591": " gL \\cdot P = 0 ", "2b5438c6e4fc4cd86a850f0bca49accc": " \\bold{F} \\cdot \\delta \\bold{r} = m\\ddot{\\bold{r}} \\cdot \\delta \\bold{r}. ", "2b544059a0b20f04bc2e05070744f7f7": "\\tbinom{L}{2} = \\tfrac{L(L-1)}{2}", "2b545140317f796f121ef6725859dbdc": "\\Alpha", "2b545d8aae52ff07c5835d8626546a12": "\\widehat{QP2A}=\\frac{1}{2}\\widehat{P2O2Q}", "2b557e3d330a1540c65c620b53268dd4": "\\frac{dE}{dt}=\\vec{v}\\cdot\\frac{d\\vec{p}}{dt}", "2b558191781dfa5932153edb95f79225": "\\frac{ \\Delta \\phi}{\\omega_{\\rm orb}} \\approx -\\frac{\\pi m}{r}", "2b55a0e5840c7d4917c80a9406018ecc": "t_M", "2b55af4b1ff0becfa4b7d03cbd72debe": "h : \\beta \\rightarrow \\beta'", "2b5602bc4b1ec9b6f96bab8641a3888c": "1_C(n)", "2b5680af414476a1fd43d83d0b7096de": "SA \\subseteq S \\times R", "2b56aa5f4d741747caca06af31660a58": "\\langle S \\mid R \\rangle\\,\\!", "2b56c6776eba2bec0ce6252740ca7962": "i=1,...,n, \\ x_i=M", "2b56d810101eb658a177ccc7c38631f8": "\\begin{align}\n\\int \\frac{1}{\\sqrt{a^2+x^2}}\\,dx &= \\int \\frac{a\\cosh{u}}{\\sqrt{a^2+a^2\\sinh^2{u}}}\\,du\\\\\n&=\\int \\frac{a\\cosh{u}}{a\\sqrt{1+\\sinh^2{u}}}\\,du\\\\\n&=\\int \\frac{a\\cosh{u}}{a\\cosh{u}}\\,du\\\\\n&=u+C\\\\\n&=\\sinh^{-1}{\\frac{x}{a}}+C\\\\\n&=\\ln\\left(\\sqrt{\\frac{x^2}{a^2} + 1} + \\frac{x}{a}\\right) + C\\\\\n&=\\ln\\left(\\frac{\\sqrt{x^2+a^2} + x}{a}\\right) + C\n\\end{align}", "2b56e82768e2a48853d832c56a953544": "KR^?_*,LR^?_*", "2b570298fc730bb700d7e09fa9cf4b2d": "\\displaystyle \\Delta_K= h^{-1}\\circ \\Delta_T \\circ h + \\|\\rho\\|^2,", "2b575d8f5cac49645c4911afbb74c7fa": "\\;\\varphi\\left(n^m\\right) = n^{m-1}\\varphi(n).\n", "2b577d61e90eb1b45da17ad0270824c2": "\\Phi(x) = \\int_{-\\infty}^x \\phi(t)dt = \\frac12\\left(1 + \\operatorname{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)\\right)", "2b57beacdd4be7cae8c614ca8600773d": " I \\ = \\ p v \\ \\propto \\ \\frac{1}{r^2}. \\, ", "2b58120f52f652a193b7e11df4a17ad4": "V={d}{i_x}", "2b5893a22ca256d719098624f913f409": "\\rho=\\frac{m}{V}", "2b58bfe04ad5ce9b9845813d37156398": "p_Z(z) = \\begin{cases}\n\\left[ \\phi(0) - \\phi(z) \\right] / z^2 \\quad & z \\ne 0 \\\\\n\\phi(0) / 2 \\quad & z = 0, \\\\\n\\end{cases}", "2b58fc1e592a5b63e849ab8e95dbce31": "y[n] = x[n] * h[n] \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{m=-\\infty}^{\\infty} h[m] \\cdot x[n-m] = \\sum_{m=1}^{M} h[m] \\cdot x[n-m],\\,", "2b58fda982c80f93ebed75bee71568c9": "\\ |\\mathcal Z|=(n+1)n(n-1)", "2b59032fa186808069a9f137d5a057c9": " P_{\\rm max} = \\begin{matrix} \\frac{16}{27} \\cdot \\frac{1}{2} \\end{matrix} \\cdot \\rho \\cdot S \\cdot v_1^3 ", "2b59372591153c0d791c890bcf9ec66c": "E(v)=kT\\left[1+\\frac{1}{12}\\frac{hv^2}{kT}+O\\left(\\frac{hv}{kT}\\right)^4+\\cdots\\right]", "2b5992d40627476e19a7c83841bbef07": "\\nabla\\cdot\\mathbf{B} = 4\\pi\\rho_m", "2b59a4f3ab0f855b871b7cbd715e5c49": "(\\mathbb C\\otimes \\mathbb H)P^2", "2b59cb283b30748a316b0a960a8ccb2d": "\n\\operatorname{Li}_2(z) = \\frac{\\pi^2}{3} - \\frac{1}{2}(\\ln z)^2 - \\sum_{k=1}^\\infty {1 \\over k^2 z^k} - i\\pi \\ln z \\qquad (z \\ge 1) \\,.\n", "2b5a40a8ad707e3cf94dd06488e69289": "HA \\rightleftharpoons H^+ + A^- ", "2b5aa08f2a8e76c72c63ae2a40a87b99": "y = - x - \\frac{1}{2}x^2 + \\frac{1}{8}x^3 + \\cdots", "2b5af0f0cb502d64bd5a4e56fbca463b": " f(x) = \\int \\limits_a^b K(x,t)\\,\\varphi(t)\\,dt. ", "2b5b084224e451c9ad2153fa0e9cfa8e": "\\operatorname{succ}(C,[\\ ]) = C", "2b5b30cb20934e61f0a626bdb35f04a6": "\\mathcal{O}(t^{2}(n))", "2b5b7c36a6c91db8906ccb19eff213f4": "C = \\{g \\in L^{\\infty}(P): g \\leq f \\; \\forall f \\in K\\}", "2b5ba967fcbf63a2a2cbbdd6bea687a5": "T_b=\\frac{\\pi I \\lambda^2 L_c}{4kc \\ln{2} }", "2b5bca535a0fca73c56ac7ce74d25a02": " W_{1\\to 2} = \\int_{V_1}^{V_2} P \\, dV, \\, \\, \\text{negative, work done on system} ", "2b5bcb4ed9c834040ac8fd10de1c8b3c": "\\epsilon^\\mu(q,z) \\,", "2b5c5c798d1539b9837e89a745afc42a": "\\widehat{\\theta}", "2b5cbbfa689c307efe429c007c317959": "\n\\bar{\\Phi}(s,L)=1+\\sum_{c=1}^{[L/2]}(-1)^c\\bar{h}(s,c;L)\n", "2b5cfaa11c783e78c3d585bc9a417121": "\\mathbf{B}' = \\frac{\\mathbf{E} \\times \\mathbf{v}}{c^2\\sqrt{1- \\left(v/c\\right)^2}} \\,.", "2b5d0eed8e52431940ab02079dca8f91": "(x_i, x_i^-)", "2b5d225ec41685cbe9b2d714d0385632": "\n \\Pr\\!\\big[\\; \\ln f(x\\,|\\,\\theta) \\;\\in\\; \\mathbb{C}^0(\\Theta) \\;\\big] = 1.\n ", "2b5d40831078dbdeeeb61ea672806867": "N=<0,1>", "2b5d5e5f08b82ce61c44168af840a4d3": "\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{n}})\\psi(r_i,t) = \\psi(R_{ij} r_j , t) = \\psi(r_i - \\varepsilon_{ijk} n_k \\Delta\\theta r_j , t)", "2b5d99294ac88e606bfc15db12d00a92": "\\gamma_{-k} = \\gamma_k", "2b5e345a17d1134411e4db9fbe083aaa": "\\Phi(r)={I \\over 4 \\pi r \\sigma}", "2b5e706b12ba01fa2b0fb7bcc3a3b465": " \\mathcal{M}^i_j ", "2b5ebe5a352e6b6f679c4e56e797d8dc": "\\frac{1} {2} FD(1-x)*T", "2b5ed277dfc905d39b4fe05f9efbc9e2": "y = r", "2b5f5e92782a0807e8696c02abb03975": " \\mathbf{R} \\, [\\mathbf{t}]_{\\times} = \\mathbf{U} \\, \\mathbf{W}^{-1} \\, \\mathbf{V}^{T} \\, \\mathbf{V} \\, \\mathbf{W} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} = \\mathbf{U} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} = \\mathbf{E} ", "2b5fc25e6691c4c89cf2735b19ab6f3d": "G_\\theta \\subset L_\\theta", "2b608ccb4a886dd659e650a446d19a26": "f(z) = \\frac{az + b}{cz + d}", "2b60bb3e37a2903b2729ec96fd901bde": "( , )", "2b60e9cee176e142dda364798ccaf05e": "\\begin{align}\nV_k(\\mathbb R^n) &\\cong \\mbox{O}(n)/\\mbox{O}(n-k)\\\\\nV_k(\\mathbb C^n) &\\cong \\mbox{U}(n)/\\mbox{U}(n-k)\\\\\nV_k(\\mathbb H^n) &\\cong \\mbox{Sp}(n)/\\mbox{Sp}(n-k).\n\\end{align}", "2b613f0205067091b0a3d3cec5097a72": " \\widehat{\\mu} = (2 \\pi)^{-n/2} \\times \\mbox{Lebesgue measure}.", "2b61ca8801efbfa5890a886bf3b37491": "{dy \\over dx}=\\frac{1}{\\sqrt{1-x^2}}", "2b61ce02c0f624c4d362b54646035994": "s_{n+1} = 0", "2b6247580075b9862a27e019710d19df": " r = \\sqrt{\\frac{1}{2}(\\alpha^{2}-a^k_{k+1,k}\\alpha)} ", "2b62758fc61e722085b9b81fb77c6ee8": "\\mu_i, \\sigma^2_i, \\pi_1, i=1,2)", "2b628bf32ac7e6314223546a773e6783": "P(z) = a_0 + a_1z + a_2z^2 + \\cdots + a_nz^n", "2b635581eeda83bc67477b4f70c595df": " \\simeq 5/9", "2b6386f0a4ccb45749c2dae759a0e242": "\\displaystyle{(S,D)=\\iint_{\\Omega \\cup \\Omega^c}\\nabla S\\cdot \\nabla D= -\\int_{\\partial\\Omega} S\\partial_n D + \\int_{\\partial\\Omega} S\\partial_n D=0.}", "2b6390fa05f05c42c0f0e11c876e8246": "N_\\text{max}", "2b63a6c8da99997f1c0edafddb34fe0b": "2\\le i \\le 10", "2b63f12c0bb2d43af369d576bd6091b7": "\\begin{align}\n\\Delta =256a^3e^3-&192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e\\\\\n&-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2.\n\\end{align}", "2b63f18cd887f62b9ad190ad666c4dca": "N_2(t+1)=f(N_2(t))+\\epsilon_2(t)", "2b64111f71a7899bf746267546c146b3": " H^q(M, K_M\\otimes L) = 0 ", "2b646265ceeceeb1d0bd6c8d9785b60a": " \\nabla \\cdot \\mathbf{B}=0", "2b64ad9f250e484f992bb579b707050d": "\\scriptstyle \\sum_{i \\in I}\\alpha_i f_i \\;=\\; 0", "2b64b6a48fe0c2b96a294b1fcfbb804b": "\n\\begin{align}\n\\operatorname{tri}(t) = \\and (t) \\quad \n&\\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\max(1 - |t|, 0) \\\\\n&= \n\\begin{cases}\n1 - |t|, & |t| < 1 \\\\\n0, & \\mbox{otherwise} \n\\end{cases}\n\\end{align}\n", "2b64caca7b8182c657fa9cf959cd1df4": "B_1B_2", "2b652d903dbff4935efe3c7ce45d6a26": "S^1 \\times D^3", "2b65a753a782a74decce5390d6083e76": " \\theta = \\int_0^\\infty {{u(y)\\over u_o} {\\left(1 - {u(y)\\over u_o}\\right)}} \\,\\mathrm{d}y ", "2b65bfc18f697104e1ed6f7891de31a5": "\n\\begin{bmatrix}\n1 & 1 &\\cdots & 1\\\\\n\\end{bmatrix}\\mathbf{K}=\\begin{bmatrix}\n0 & 0 &\\cdots & 0\\\\\n\\end{bmatrix} ", "2b65cc7e1a4ef951cce484486d2166ef": "V+\\Delta V=(L+\\Delta L)^3 = L^3 + 3L^2\\Delta L + 3L\\Delta L^2 + \\Delta L^3 \\approx L^3 + 3L^2\\Delta L = V + 3 V {\\Delta L \\over L}", "2b65e219a9d545ea780bfba31b846290": "S_n=(-1)^n\\left[1+\\sum_{k=1}^n \\zeta(k+1) \\right] ", "2b6608208e9756debbca6a9751d49ed5": "1_Z(-c)[-2c] \\to i^!1_S,", "2b66163f5f0b30fcf1e1d1598c23ec0a": "\\Pr[\\mathcal{A}(D_{1})\\in S] \\leq \\exp(\\epsilon)\\times ( \\exp(\\epsilon)\\times\\Pr[\\mathcal{A}(D_{3})\\in S]) = \\exp(2 \\epsilon)\\times\\Pr[\\mathcal{A}(D_{3})\\in S] \\,\\!", "2b66710d2b176ad768f175c20d9f8a03": "\\gamma_r", "2b6674b093bb8cac65129556873de118": "(\\tau \\omega_1,\\tau \\omega_2)", "2b66b96a6f8cfa01890188021e5b16af": "\\alpha\\delta - \\beta\\gamma = \\pm 1", "2b67112bcc160144baa2a1773985ae59": "\\delta=\\frac{\\gamma_0\\mu_0}{|\\pi_0|}\\leq \\tfrac{1}{2}.", "2b6724db242b846967ab79b35ade148b": "D + g", "2b6753f407f5f039916e2c871d6dc247": "\\phi(\\tau, \\tau) = I", "2b67beeefbbbe81907e2fc881c4974c1": "\\vec{k}", "2b67f6654510757df4d79690c799ff83": "k^2 + k'^2 = 1\\,", "2b6813aa218cc74d7811874d8ccbbce2": "\\rho(\\vec{x})", "2b681cbcbb334c74e29bab720ff39a8e": "\n \\omega_\\varphi = \\frac{\\partial v}{\\partial z} - \\frac{\\partial w}{\\partial r}\n = - \\frac{\\partial}{\\partial r} \\left( \\frac{1}{r}\\frac{\\partial\\psi}{\\partial r} \\right) - \\frac{1}{r}\\, \\frac{\\partial^2\\psi}{\\partial z^2}.\n", "2b6831ba12a7fbc5011030826d59b132": "\\pi\\colon T^*M\\rightarrow M", "2b684822857d8f0cd2c39179e867dacc": " \\Delta_n(s) = \\mathbb{P} \\left\\{ x_n \\leq s \\right\\} - \\log_2(1+s) ", "2b686069fdc08d77ee65af9c54a8300e": "\n2\\cfrac{\\partial W}{\\partial I_1}(3) = \\mu \n ", "2b68d401b585a234163db5f789046058": "\\mathbf{f}\\mapsto \\mathbf{f}' = \\left(\\sum_i a^i_1X_i,\\dots,\\sum_i a^i_nX_i\\right) = \\mathbf{f}A", "2b68f4b1b3afacdf6efd9e8b1077c42d": " j = \\frac{\\hbar}{2mi}\\left(\\Psi^* \\frac{\\partial \\Psi }{\\partial x}- \\left(\\frac{\\partial \\Psi^* }{\\partial x}\\right)\\Psi \\right) ,", "2b68f5099cb941cd4b8ca7723341cd82": "\\rho = \\psi^* \\psi = |\\psi|^2. \\,", "2b697b764dd4c84051a85e43b08b0045": " f(p) = \\frac{1}{4 \\pi m^3 c^3 \\theta K_2(1/\\theta)} e^{-\\gamma(p)/\\theta}", "2b697dc45f69afccb26f2538015e5770": "x_0, x_1, ... x_n", "2b69ac4af322b5f3af40f43833a4b5f3": "\\varepsilon \\sim \\frac{1}{a^2 b^2},\\ u_{\\alpha} \\sim \\sqrt{ab}", "2b69cc2ebaa1551c8bb8115f095e043a": "\\frac{\\partial^2\\psi}{\\partial\\theta^2} + \\cot \\theta \\frac{\\partial \\psi}{\\partial \\theta} + \\csc^2 \\theta\\frac{\\partial^2\\psi}{\\partial\\phi^2} + \\lambda \\psi = 0", "2b69d5d0c5b88bbc901d940a8c3e6093": "[F : k]", "2b69d80e85a31bc1d3486598c56c34af": "x(t_0)=x_0 ,u(t_0)=u_0", "2b69dfb1cf947cab246ef0456283315f": "s\\Big)", "2b69e00804fc3dd99c7b3e1343702b0e": "\\sin(n\\theta)", "2b69f59325606441cfd1d8d78a7bbe7d": "h(r) = r^{2} \\cdot \\log \\log \\frac1r.", "2b6a2301b29a29efd332d27beea635d0": "aX+bY+cW = 0 \\Rightarrow L(a,b,c) \\cdot P(X,Y,W) = 0", "2b6a3933f711c9971cbc975a12f5ca6d": "Dv=1", "2b6a57c198ed70ac163c3b978364c53c": " \\left| \\nu_{\\alpha} \\right\\rangle = \\sum_{i} U_{\\alpha i} \\left| \\nu_{i} \\right\\rangle\\,", "2b6a87a30a7a2a6e1c924301cf0d2f92": "\\scriptstyle \\Pr(\\mathrm{Head})+\\Pr(\\mathrm{Tail})=1", "2b6a8f3ad66efa7d3bad9ec746fd7a64": "\nr_{q} = \\sqrt{\\frac{q^{2}G}{4\\pi\\epsilon_{0} c^{4}}}\n", "2b6ad055434c0eaf3462bdb372c25857": "\\frac{\\partial S}{\\partial t} = D\\ \\nabla^2 S\n", "2b6ae5783fa5b27018d291ca959e7abf": "x\\geq0", "2b6b72ba3fa691418712246d40f07e74": "\\displaystyle{\\varphi=\\partial_{n+}u - \\partial_{n-}u.}", "2b6b73af1f8a47b25c5f201bf1bebe2b": "r_{k+1} \\leftarrow r_k - \\alpha_k \\cdot A p_k\\,", "2b6b74c70daf419bf265fc09ff39ac8c": "\n\\begin{array}{llr}\n\\min\\limits_{x\\in \\mathbb{R}^n} &g(x)= c^T x + E[Q(x,\\xi)] & \\\\\n\\text{subject to} & Ax = b &\\\\\n\t\t & x \\geq 0 &\n\\end{array}\n", "2b6c34f7832fa912c37abe7bc55d993e": " q \\ne 0", "2b6c7a332918f2cae15deda6886aeef6": " f(x_i,\\boldsymbol \\beta+\\boldsymbol \\delta)", "2b6ce0787965df2899cbbceec06fbd8d": "\\begin{align}\n\\sigma^{i0} &= \\frac{i}{2}\\biggl(\\begin{matrix}\n\\sigma_i & 0 \\\\ \n0 & -\\sigma_i \\\\\n\\end{matrix}\\biggr),\\\\\n\\sigma^{ij} &= \\frac{i}{2}\\epsilon_{ijk}\n\\biggl(\\begin{matrix}\n\\sigma_i & 0 \\\\\n0 & \\sigma_i \\\\\n\\end{matrix}\\biggr)\\\\\n\\end{align}.", "2b6d012903d59ef2fef0856fd75379e5": " x\\in [x_{0}^{(e)}, x_{1}^{(e)}]. ", "2b6d440c0d39606b989ea1762932d221": "\\mathbf{Q}_{\\mathbf{X}}", "2b6d448c3e967cec5ac5d4806d51c272": "u_{m}", "2b6d6306c7a1a94c760ea49fd76fdfde": " \\frac{dN(t)}{dt}=rN(t) \\left(1-\\frac{N}{K}\\right) ", "2b6d7987f65dee6e3fb7115ed8cfb56b": "\nL(\\vec \\omega, \\vec v) = {1 \\over 2} (A \\vec \\omega,\\vec \\omega) + (B \\vec \\omega,\\vec v) + {1 \\over 2} (C \\vec v,\\vec v) + (\\vec k,\\vec \\omega) + (\\vec l,\\vec v). \n", "2b6dffe7eac20e7bc6cf9b6f30524eb1": "\\operatorname{P}[E_2]", "2b6ec977c5a42feb0fe5cbcdf120c788": " = \\frac {dx}{dt} ", "2b6f035b3fc6721eaa26dcacd3ebc714": "(E_{(\\infty)}, \\mathcal{C}|_{E_{(\\infty)}})", "2b6f28068a000385ac3c86106947d4b5": "\\sum_{n=0}^\\infty c_n = AB.", "2b6f7e029ed41225b8f4f1a9f0a45721": "\\langle c\\rangle", "2b6fdc4f43e096b4ce75e5798b0a641e": " Q_s(t) = C_s \\cdot V_s(t)\\, ", "2b701ed5a3e7c97ad314a779720af3c9": "t_2=1.05 \\colon", "2b7021abbe24836afb36ded6db4d826d": "C_{th}", "2b70651f8e9bd5fef7e4e585c3e3e1e4": " \n\\begin{array}{clrcl}\nT_{p,p} & = & & H_{p,q} & - \\ \\ \\ \\mathrm{Re}\\{H_{p,q} e^{-2i\\theta}\\}, \\\\[8pt]\nT_{p,q} & = & & \\frac{H_{p,p} - H_{q,q}}{2} & + \\ i \\ \\mathrm{Im}\\{H_{p,q} e^{-2i\\theta}\\}, \\\\[8pt]\nT_{q,p} & = & & \\frac{H_{p,p} + H_{q,q}}{2} & - \\ i \\ \\mathrm{Im}\\{H_{p,q} e^{-2i\\theta}\\}, \\\\[8pt]\nT_{q,q} & = & & H_{p,q} & + \\ \\ \\ \\mathrm{Re}\\{H_{p,q} e^{-2i\\theta}\\}. \n\\end{array}\n", "2b70b759c8b786e3ec848f4ffbad88cd": "20%($10,000)+80%($0) = $2000 > 100%($1000) \\, ", "2b70ff78713ce5ee4bcaeff8e6a2fd3c": "|E| = |F|", "2b717ed75c8baa5e72e1d78bff7d2ca0": "\\zeta(s) = \\frac{s}{s-1} - \\sum_{n=1}^\\infty \\left(\\zeta(s+n)-1\\right)\\frac{s(s+1)\\cdots(s+n-1)}{(n+1)!}.\\!", "2b718313d243af2d59eb14838e315c38": "LN[n]", "2b71ce68a3b7fdb85271bbe05d19a9be": "\\frac{1}{(s+\\alpha)(s+\\beta)} = { P \\over s+\\alpha } + { R \\over s+\\beta }.", "2b71e72cebac2c3545448ce73724156a": "\\textstyle \\zeta_G ( \\alpha )", "2b71f2cf071e9963fce0e8b341bc103e": "\n\\psi^\\dagger(k) \\psi^\\dagger(k') - \\psi^\\dagger(k')\\psi^\\dagger(k) =0\n\\,", "2b71f3aec5ae940c9b8b49998d673e44": "1/H", "2b71f746bffad82d2389282dab08fbd3": "\\left [\\begin{smallmatrix}2&-2\\cos(\\pi/p)\\\\-2\\cos(\\pi/p)&2\\end{smallmatrix}\\right ]", "2b72009e207aac17362549c0865f89c6": "T = \\frac{P/B}{k_B}", "2b720ecf1d63f94dc4bcfafd9d123f68": "\\ell = {1 \\over 2} {1-\\sqrt{k'} \\over 1+\\sqrt{k'}} =\n{1 \\over 2} {\\vartheta - \\vartheta_{01} \\over \\vartheta + \\vartheta_{01}}.", "2b72b63fdca66197cfa83d0081cec585": "\\Phi (a) = \\sum _{i = 1} ^{nm} V_i a V_i ^*.", "2b72c06eadcd412d0b0dac5642977c3b": " \\nu\\ : \\Gamma\\ \\to \\mathbb{R}^\\mathrm{N} ", "2b72c932e18771bf16d697fbff99815a": "\\scriptstyle T_{01} \\,", "2b73337614b82ad74c50e3a62c4cb834": " \\mathbf{m}=\\frac{1}{2}\\, q\\, \\mathbf{r}\\times\\mathbf{v}", "2b734e4ec4b3f76ea43a62df68dbf2de": "c=100%", "2b735992c8452503187e2eabc132f69f": "\\int f_{n_k} g\\,d\\mu \\to \\int f g\\,d\\mu", "2b73b12ed73d503395e4802a77b3bb0c": "\nPoss(pickup(o),s)\\wedge location(s)=(x,y)\\rightarrow location(do(pickup(o),s))=(x,y)\n", "2b740a101769b073eac66b74d8d43831": "x_t = x_{t-1}=x*", "2b74699235067195d3861c1f17d418d2": "\\scriptstyle \\sqrt {ab}", "2b74917f7669afec03941fa9a7e5ceba": "2 \\sqrt{ac}/b = 2 \\times 10^{(0.6192290 + 1.0576927)/2 - 0.9618637} = 1.505314 ", "2b74b421dc915de3637944a875bc9b4c": " v_{\\rm turb} \\approx c_{\\rm s} ", "2b74ec58479df54770fe46e8bf3b8adc": "K_i, K_j", "2b74fdf55a16118c2204288972899fea": "v_A \\in [x_a,y_a]", "2b7511394bb0406c1b7809f61f7a11a1": "M*Kp + NB*Kp + NB*NB", "2b7564727a202c21f2989e08a7d390f4": "T(Y)=\\dfrac{1}{2N}\\sum_{i,j}|Y_{i}-Y_{j}|^{2}", "2b75efb6dc1b4417ad2991a276e13e79": "Z_{12} = {2 S_{12} \\over \\Delta_S} Z_0 \\,", "2b760da4590af9851630eb4a1b55cfe4": "\\Omega(x)", "2b766c0636c6257ffa9361fcfe84d7d0": "\\pi=\\frac{1}{Z}\\!", "2b76a8bd1a45349c3d2bb3e0732f5523": "\\scriptstyle a/R,", "2b76b46b9e3c6c7dd1572acddd6c6a22": " y(x)= C x^{-2} . \\,\\!", "2b76f867786b28ee3bee14834d46f65e": " \\alpha > 1 ", "2b778c83f2a4fef6ffe930f83be0e5c9": "k_0 n", "2b77c80e49239213b02e66e96e653e1a": "d\\mathbf x=\\mathbf R \\,d\\mathbf x'\\,\\!", "2b77f2695c503837d3738b7593043d6c": "x_{k+1}=f(x_k).", "2b78b6003c417219cce6f390e26a4fa1": "a^{N-1} \\equiv 1 \\pmod{N}", "2b790723bea5d55cd0aff51577633a0f": "\\mathbf{P}^0,", "2b7944b83780309eedd12c5e9f514467": "u_1 = u_1(x_1), u_2 = 0, w = w(x_1)", "2b7963ff1807875e87486a60973de567": "[L_{ij},C_k]=i[\\delta_{ik}C_j-\\delta_{jk}C_i] \\,\\!", "2b79c8b7b72bf6c2b95af2159302fe56": "\\sup_{y_1,y_2\\in Y, \\|y_i\\|=1} \\frac{|y_1|}{|y_2|} \\ge \\lambda", "2b79e90a8ff03b9b07ba86565237a999": " X_m (i)={ \\{ x_i , x_{i+1} , x_{i+2} , . . . , x_{i+m-1} \\} } ", "2b7aa49d633b16cc0b4ecbeed827a981": "A_{\\gamma-norm} = \\left( L_{pp, \\gamma-norm} - L_{qq, \\gamma-norm} \\right)^2 = t^{2 \\gamma} \\left( (L_{xx}-L_{yy})^2 + 4 L_{xy}^2 \\right). ", "2b7abff7d82316136b5b9845732d270b": "\\,(p,A\\alpha) \\rightarrow_M^1 (q,\\alpha\\gamma)", "2b7b2bc9ec03532c3c72baba335e195a": " \\sqrt{1 - \\omega^2 \\, r^2} \\; \\frac{dt_+ - dt_-}{2} = \\sqrt{dz^2 + dr^2 + \\frac{r^2 \\, d\\phi^2}{1-\\omega^2 \\, r^2} }", "2b7b3316470523106f66699ddb3df6f8": "H_i(X; \\mathbf{Z}) =\n\\begin{cases}\n\\mathbf{Z} & i = 0 \\mbox{ or } i = n \\mbox{ odd,}\\\\\n\\mathbf{Z}/2\\mathbf{Z} & 00\\}.", "2b89b92839ea7f69face762659a9c5ba": " \\frac{-p}{1-p} ", "2b89e4146cd3e651b0f3e4eabbfff172": "{\\alpha \\choose \\beta}", "2b89eb9f42ecf805a2e194b3982c7dc0": "SS'", "2b8a3187247ef723262c41833151cb5d": " I = \\int_S \\mathbf{J} \\cdot \\mathrm{d} \\mathbf{A} \\,\\!", "2b8a3fc0728ec10781b6b088fcfc8e32": "(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2.\\,", "2b8a508c342d5d14201646d59b550428": " \\rho_{f,p} \\simeq \\rho_{g,p}. ", "2b8ad27070465ce0677134b546289e11": "f=v(1+t^2)(1+u^2)", "2b8b168850e3e94c14b12a422fabdaae": "E={bp\\over ip}", "2b8b3509098fb1c4b89d7e5819b8314b": "\\hat{D_B}", "2b8b4abf662e0f8e331ef0eb771da6ab": " \\forall y,z \\left\\{ A(x_1, \\ldots ,x_{n-1}, y) \\wedge A(x_1, \\ldots ,x_{n-1}, z) \\implies y = z \\right\\}.", "2b8ba37090275d42c052517313b53699": "Q(2 n,q)(n\\geq 2)", "2b8bf7a21c8567fa2168c3d136830596": " f(2) = (2)^3 - (2) - 2 = +4 \\,.", "2b8c20299a38eda2d32e121e43fdab6e": "S_-", "2b8c2b029ceef8e921296e52333a926b": "a \\wedge x \\wedge a | x \\in S ", "2b8c32b624fb9d6a2882fc3d7da4d7aa": "[\\omega]^{\\omega}\\to Q", "2b8cce392d06b58a6810cdcb73e578c9": " \\chi^2 = \\sum {\\frac{(O - E)^2}{\\sigma^2}}", "2b8d090d86e2b75f528810d16a0371e6": "T T^{-1} = T^{-1} T = I. ", "2b8d943c9059ba310b25154ce5b0d053": "\\sum_i A \\sigma^x_i + B \\sigma^z_i \\sigma^z_{i+1}", "2b8d984b646e25a4e338adb3fa8acdc7": " u(t,x,y,z) = t M_{ct}[\\phi]. \\,", "2b8d9d453108727b0ab75eeb6afada13": "k_1, k_2, \\mbox{and } k_3", "2b8da6268e10bcd57cc79a192dbce45f": "B(\\lambda v,w)=\\lambda B(v,w)\\ \\quad \\forall \\lambda \\in K,\\forall v,w \\in V", "2b8e09844e4f19374464deac3008245b": " \\frac{ ( \\mbox{10.5} \\times \\mbox{117} )}{113} = 11 ", "2b8e7adbf714c0c5fed8e05f1eff51fc": "\\Rightarrow, \\nRightarrow, \\Longrightarrow \\implies\\!", "2b8e8300da6fb20b0e5c5c3431a0508a": "\\mathrm{CD}=k\\frac{\\lambda}{\\mathrm{NA}}", "2b8eea5fc49bfc35d030c87f75c985e8": "F = Q \\cup \\{ \\text{init} \\}", "2b8efffb8e96744d45d9701f92954f7f": "s_2=\\alpha^{-7},", "2b905170b1ec297968ac628f03af91f5": "\\min_{x \\in S_k, \\|x\\| = 1}(Ax,x) \\le \\lambda_k.", "2b90889ce479afc0ad6a6f625429c71b": "Q^{(ab\\ldots n)}", "2b90b0163a6cf6c7fc34e7103716ee2e": "\\begin{align}\n \\mu_M & \\approx (n+1)(\\ln(n+1)+\\gamma)-\\frac{1}{2}-\\frac{1}{12(n+1)},\\\\\n \\sigma^2_M & \\approx (n+1)\\left ( \\frac{\\pi^2}{6} -1 \\right ) -\\frac{1}{2}-\\frac{1}{6(n+1)},\n \\end{align}", "2b90d4544bce05885a2300a5e74d330e": "\\! v_{rec}", "2b90f936011ebd769ffaff31fec69356": " \\frac{d}{dx}\\left( \\ln x\\right) = {1 \\over x} ,\\qquad x \\ne 0", "2b9163e6de8c1abda6c0eafb914ea0c6": "f = \\Delta N/\\Delta t \\,\\!", "2b919f35fa797091fc1fb0d499278f4b": " \\frac{1}{\\pi} = 12 \\sum^\\infty_{k=0} \\frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}.\\!", "2b92149e5c97659dd184e3ff935df1fb": "\n\\left\\lfloor \\frac{\\left(x+y\\right)^2}{4} \\right\\rfloor - \\left\\lfloor \\frac{\\left(x-y\\right)^2}{4} \\right\\rfloor =\n\\frac{1}{4}\\left(\\left(x^2+2xy+y^2\\right) - \\left(x^2-2xy+y^2\\right)\\right) =\n\\frac{1}{4}\\left(4xy\\right) = xy.\n", "2b922ba644ce5cfa7d5365e975944335": " \\sin \\beta = \\sgn(Lxy) \\sqrt{\\frac{1}{2} \\left( 1 - \\frac{L_{xx}-L_{yy}}{\\sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2}} \\right)} ", "2b924cc680ffce9d2ba278e7f885acb2": " \\Psi(\\bold{r},t) = \\left({a \\over a + i\\hbar t/m}\\right)^{3/2} e^{- {\\bold{r}\\cdot\\bold{r}\\over 2(a + i\\hbar t/m)} } ~.", "2b929ea98d4b01a69ac1ab745a0e7202": "f(z)=\\sum_{k=0}^\\infty \\varepsilon_k\\alpha_k (z-z_0)^k", "2b92c925e3aa9e60f6e27c882acf6307": "A = \\alpha_{0} \\alpha_{1} \\dots \\alpha_k \\dots \\alpha_{n}", "2b9315000a57c20964c2aa68a86de7b7": "a_n > 0", "2b933819fc16d2477cba28c9640b31ae": "\\frac{P}{K}= \\frac{g_n}{(1-t_p)s_c}", "2b933f65987df2866131a8a7b34ba76f": "{\\scriptstyle\\frac{1}{720}} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \\,", "2b93a4956d1c6d25a3829d7d428b4ac9": "D_{B_\\ast}(V)=H^i_{\\mathrm{dR}}(X/K).", "2b93c3dd82f1093f916d54cc4bb6e6c3": "\\sin \\theta_t = 1", "2b93f2d79af88e47304e24945f6dcf53": "T_{i,l}", "2b94056f1f1a6d7e452d95a3b31ea135": "\\textstyle P_{A}^{\\perp}=I-E_{s}E_{s}^{*}=E_{n}E_{n}^{*}", "2b94404b757162d98a0d0590480292d0": "\\scriptstyle\\frac{2}{3}\\pi", "2b9499f94205733889814920fe816630": " \\vec q ", "2b94c50e28f292c5629cf5038fb71dec": " \\frac{ \\xi(s)}{\\xi(0)}= \\frac{\\det(H-s(1-s)+\\frac{1}{4})}{\\det(H+\\frac{1}{4})} ", "2b94d5fb84e5c97ad6c4097df711cdfe": " O =\n \\begin{bmatrix}\n {1\\over \\eta} & {1\\over \\eta} \\left({ {1\\over 2} + {\\sqrt{ 5} \\over 2} }\\right) \\\\ {1\\over \\eta} \\left({ -{1\\over 2} - {\\sqrt{ 5} \\over 2} }\\right) & {1\\over \\eta}\n\\end{bmatrix}. \n ", "2b94f8459c94ef7090fcb70f9422d3ef": "\\begin{align}\n&\\left(\\frac{1}{4},\\frac{1}{4},\\frac{1}{4},\\frac{1}{4}\\right)\\\\\n&\\left(\\frac{3}{4},\\frac{1}{4},-\\frac{1}{4},\\frac{1}{4}\\right)\\qquad\\text{2-term truncation}\\\\\n&\\left(1,0,0,0\\right)\n\\end{align}", "2b9501c50d6d349734031714fceb1bd8": "9 \\cdot 2^{2n - 1} - 1", "2b950c683c606f7e2b4252f1f00ce21a": "\\mathbf{C_q}", "2b951b573b57ae2b9b015144d77f5c42": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 4.215616 \\log_e(T+273.15) - \\frac {7522.806} {T+273.15} + 46.78064 - 2.450859 \\times 10^{-7} (T+273.15)^2", "2b95531f1ae4208e8e5f0b27495271cf": "z^{*}", "2b965a433c0217c4a5607eee0f9944f3": "\\sqrt{(x-c)^2+y^2} = -{1 \\over 4a} ((x+c)^2+y^2-4a^2-(x-c)^2-y^2)", "2b965daca051e393c90c235723451c37": "\\langle a\\rangle \\phi", "2b975bcb301025dc64baf365bcdbb82e": "\n\\delta = \\arccos\\left[\\cos(\\alpha) + \\cos(\\beta) - \\cos(\\alpha + \\beta) - \\frac{1}{2} \\right]\n", "2b9785775dc56e399027ab94cd920d02": "\\frac1{\\mu \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\mu(y) \\xrightarrow[r \\to 0]{} f(x),", "2b978deeaa9cc22afd6ecdc73105a42b": "S^2\\mathbb C^n", "2b97b98b99144494d71cbfd7a0fbb90c": "\\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{4} }} = \n \\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ ^{^{^{4}4}4}4 }}", "2b9861f578d13ad99bc2ac2a8819d078": "\\prod_{j=1}^m\\ j^2\\ \\equiv(-1)^{m+1} \\pmod{p}.", "2b9871275241778b7f392524d1c75f62": "\\Delta m_e v_\\perp", "2b98adfd927d76d38ca68b5fa3ff8ded": "H=0.866025 * P", "2b98dc8752e99597a4b38b8b9002f198": "dx/dt=-x^3", "2b990ff6a0361a0dc90e990e547dde03": "i=1,2,\\ldots,n.", "2b99139fc784b42c4c1ec88ebc94cb87": "y_1, \\dots, y_m", "2b9948740748eae305d8bbd20eb5d9fb": "VC(b) < VC(c)", "2b994d020c67f5198428bdd7dbecc3ff": "v_\\infty", "2b9977ee41c7e316d989620c3f5b4fbd": "\nf_k(x)=\\left.\\frac{e^{-ik\\cdot x}}{(2\\pi)^{3/2}(2\\omega_k)^{1/2}}\\right|_{k^0=\\omega_k}\n", "2b99e084972b9d650e94f42c73f1ee8d": "\\omega_{\\alpha}", "2b99ecf4d83d40634e6e640ba70688b0": "X_M", "2b9a129b6681fd5ae1436c7be77eae76": "F=ne \\nabla \\phi", "2b9a507eeb1c17576978d38a50a6c968": "\\mathrm{RM} = \\frac{e^3}{2\\pi m^2c^4}\\int_0^d n_e(s) B_{||}(s) \\;\\mathrm{d}s", "2b9ac11697718125bfa564e28a868c77": " A\\cos(\\frac {k l_w} {2}) = B \\exp(- \\frac {\\kappa l_w} {2}) \\quad \\quad (5)", "2b9b0c0c362482cf6a15d75436e91478": "M({}^{12}\\mathrm{C}) = N_{\\mathrm{A}} m({}^{12}\\mathrm{C}) \\,", "2b9b33cefb0eb88782e190664e290b1e": "\nP_\\mathrm{sat}(T_\\mathrm{drop}) = P_\\mathrm{vapor}\n", "2b9b41849eb1ba3eea96b2caea9f72a7": "L_w = \\frac{\\rho_0}{\\omega_w}. \\ ", "2b9b684c9dcdae1be11a9de17732a1b6": "Y = sin\\alpha sin(\\omega t)", "2b9b72899f8334420474fc5e06749699": "\n s^2\\ \\sim\\ \\frac{\\sigma^2}{n-p} \\cdot \\chi^2_{n-p}\n ", "2b9baa0124c039de50d4f17f858faa0b": "K\\in\\mathbb{R}", "2b9c27f0d538fde84b41d11723a4a0b8": "65539 (2^{16} + 3)", "2b9cd399b02522d3afd05c9f3505ded2": "\\sum_{a,b\\in P}w_{a,b}d_{f(a),f(b)}", "2b9d067544a9e6531cca2a5be1d565eb": "\\chi_{\\textrm{1}}", "2b9d58fad47ec797fbdcb7b73e8201c9": "C_n={1 \\over n+1}{2n \\choose n}.", "2b9d5a13f5ccd2dcf21252f18a52264c": "\\chi_{\\text{e}}\\ = 0", "2b9d785576281543b8912068d3c6c7fb": "\\mathbb{Q}^{*}/\\mathbb{Q}^{*2}", "2b9dcd854a5cd63c7c5d255d97862594": "e^{-E/kT}", "2b9dd5d16e395a1357946d1c3495f2a1": "h_{\\mathrm{o}}", "2b9dfe3d0d12b77f06bd40f151c231c7": " \\prod_{i = 1}^{N}\\mathbb{E}^{-S_{ij}} \\simeq 1 - \\Omega^{-1/2} \\sum_i S_{ij} \\frac{\\partial}{\\partial \\xi_i} + \\frac{\\Omega^{-1}}{2} \\sum_i \\sum_k S_{ij} S_{kj} \\frac{\\partial^2}{\\partial \\xi_i \\, \\partial \\xi_k} + O(\\Omega^{-3/2}). ", "2b9e37c4c18507be001c09da9b02156c": "P=P(X),", "2b9e4ecafb578423c3220b195feda58a": "\\log (X_i)", "2b9eaf860911b001a161383e7c0a4cac": "V_i=\\bar V_{\\text{L}} + \\bar V_S", "2b9ec1bc1d81a9818fac2724d7dd01fa": "e(t)={\\rm exp}(2\\pi\\cdot it)", "2b9f34d5240253a2c4e96fcd868ee10f": " \\mathrm{Var}(f)\\equiv\\sigma_N^2 = \\frac{1}{N-1} \\sum_{i=1}^N \\left (f(\\overline{\\mathbf{x}}_i) - \\langle f \\rangle \\right )^2. ", "2b9f69555791a2b14d611d91cfbba0cb": "w\\,R\\,v \\Rightarrow v\\,R\\,w", "2ba0143d5929b0c3a0062432c4a2b28f": " (A f)(y) = \\inf\\{ f(x) : x \\in X , A x = y \\}. ", "2ba0161483d595bb0c0849b90a32a1f2": "T^r_s (V) \\otimes_K T^{r'}_{s'} (V) \\to T^{r+r'}_{s+s'}(V).", "2ba06ba645cc0b1e8d938747312ceea8": "S(q)=\\sum_{n=1}^\\infty a_n \\sum_{k=1}^\\infty q^{nk} = \\sum_{m=1}^\\infty b_m q^m ", "2ba093d9425950e969bb1d9d1bfd5ead": "\\tfrac{6240321451}{470184984576}", "2ba0d6a27ce2f21ae7990f30a6072394": "| x\\rangle", "2ba0fc4d63209c3fb3bdbf0fdc0d2702": "K_\\xi\\colon \\mathfrak{g\\otimes g}\\to \\mathbb{K}:(X,Y)\\mapsto \\xi([X,Y])", "2ba1045921831dd8a03493b71ac908a2": "c^2=\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_S", "2ba17038a877a5dd7a54b8792ac4e664": " \\int^\\oplus_X H_x \\ d \\mu(x) ", "2ba179a1e122630f9a6dde257da732d9": " Y \\sim \\textrm{EV}(\\mu - \\sigma \\log(\\tfrac{1}{n}),\\sigma) \\,", "2ba18e9ca56ee8302ba5611e58bf89f8": "RL_\\mathrm{in} = 10\\log_{10}\\left| \\frac{1}{S_{11}^2} \\right| = - 20\\log_{10} \\left| S_{11}\\right|\\,", "2ba19f0c85ccbc6dba3ed3f6ead48bfc": "P((N > xn)\\mid n) = \\int_{xn}^{\\Omega} \\frac{1}{N \\ln(\\frac{\\Omega}{ n} ) } \\,dN = \\frac{ \\ln(\\Omega)-\\ln(xn) }{ \\ln(\\Omega) - \\ln(n) } ", "2ba1b3acb05d4cc60dedf59890f964f6": "h=0", "2ba1bf0f3287a05f5da336a088407f95": " f(A) = \\sum_{i=1}^k f(\\lambda_i) A_i ~,", "2ba1cfe9fa4a12cd50aa2fd1bdf53fba": "\\mathit{M}_{t}", "2ba1d90498ff55dc0799ad2de5defdc9": "f(V)", "2ba23687ee375623d78513a36a210c13": "C_2^* \\subseteq C_1^*", "2ba2a61c494139ec7586f9359c57dfba": "R = \\frac{k L S}{A^3}", "2ba2b36a5344acbe64fc666eabc14fcd": "\\ q < p < 2q ", "2ba2fee1721a3ccd1c957ba34fb834d3": "\nW_{0} (x) = L_1 - L_2 + \\frac{L_2}{L_1} + \\frac{L_2 (-2 + L_2)}{2 L_1^2} + \\frac{ L_2 (6 - 9 L_2 + 2 L_2^2) }{6 L_1^3} + \\frac{L_2 (-12+36L_2 - 22 L_2^2 + 3 L_2^3)}{12 L_1^4} + \\cdots\n", "2ba3df156ef48de6d98f8d3949421025": "\\overline{a}:=(a_1,a_2, ... ,a_n)", "2ba40800adc7500f42afbb88ab2e004d": "(K_{s_1}\\cup K_{s_2}\\cup K_{p_1,q_1}\\cup \\cdots \\cup K_{p_k,q_k} ) \\vee K_r", "2ba524d8fe51ec51307b6256e9ce1a7a": "\\psi(\\cdot,t)", "2ba53f84096a86edbc2451a77d03a281": "1 + a = 0", "2ba56f6c8f883d24ac776d15be31c1f9": " \\Box (B(x,p) \\rightarrow \\exists q \\ne p (B(x,q)) ", "2ba5b0c3a5616bca52fb9a98a8f20df0": "{\\rm R} + {\\rm L}\\rightleftharpoons {\\rm RL}", "2ba5d72ee96440fc56c716e5bef85bb4": "S=S_g+S_s+S_v+S_m\\;", "2ba5edfb747ec0e843a3e7015bafda11": "\n\\frac{d}{d E_1} \\ln \\Omega_1 = \\frac{d}{d E_2} \\ln \\Omega_2 \\quad \\mbox{at equilibrium.} \n", "2ba5fa9229a149a614ffebe6eaa46676": " 2ax + b = \\pm \\sqrt{b^2-4ac} ", "2ba613d464590cf567c5ead0cd1ade3f": "\\lambda x\\, \\lambda y.\\, y", "2ba62738a345eaf9b06845b9036d4024": "f(z^{-1})^{-1}", "2ba6430b40dddb8c9ae6fbefe251a42f": "|f(x_1)-f(x_2)| < \\epsilon", "2ba645e392273619f512105b0a20a7a3": "\\ G=\\frac{I_{tot}}{\\delta I_b}=\\exp\\left[Apd \\exp \\left(-B \\frac{pd}{V} \\right) \\right]", "2ba667bbb13d48f1af981688d17ee179": "\\mathfrak{P}^{72}", "2ba6c5c5db71b9bdf5e9e2f98baee0aa": "\\ 2 \\pi n_{2} R_{2} = m_{2} \\lambda_{2}", "2ba6eba62641f6697ffe4d80e7b1a632": "H^*(B^G, \\mathbb{K}) \\cong \\mathbb K(\\mathfrak g^*)^{Ad(G)}.", "2ba74b50e2b71656ddc8233bc7acb7c1": "d=n", "2ba74b9146c5911458ecc35f8abc71f4": "G = (N, T, S, P, R, F)", "2ba78f936071c155dbd0e40bb1e7b7fb": "\\psi(\\hat{\\alpha}) \\approx \\ln(\\hat{\\alpha}-\\tfrac{1}{2})", "2ba7ca0d8f32e07cef816681ad19e8e2": "\n\\begin{pmatrix}\nU^0 \\\\ U^1 \\\\ U^2 \\\\ U^3 \n\\end{pmatrix} = \n\\begin{pmatrix}\n\\gamma c \\\\ \\gamma v_x \\\\ \\gamma v_y \\\\ \\gamma v_z \n\\end{pmatrix}\n", "2ba803f461279798a920bf01590a2404": "u^{}_{\\alpha}", "2ba8645250919f695e08546d04c3d220": "\n\\rho = \\frac{p M}{R T} \\,\n", "2ba88b584536b568e5d7ae18d93fd68b": " D: \\phi \\mapsto \\frac{1}{i} \\phi' ", "2ba8b508203035803dcebcc55e2de4b6": "q(x) = (x-a_1)^{j_1}\\cdots(x-a_m)^{j_m}(x^2+b_1x+c_1)^{k_1}\\cdots(x^2+b_nx+c_n)^{k_n}", "2ba8cfc4a088820f9b0dde65b30a79c0": "f_\\text{wav} = \\frac{P_\\text{cvx}}{P}", "2ba8d0f4bb48333c4ca22d39e285035c": "\\phi (g) = \\langle g, 1 \\rangle.", "2ba8d21859edcec19a287ea99a89e2e6": " \\psi (0)=\\psi (L). \\,\\! ", "2ba9220619f3f6e180816590c2e59786": "3\\left(\\frac{\\mu}{\\lambda}\\right)^{1/2} ", "2ba9488d2cc10d718e109f35331b5eff": "A=(4+\\sqrt{3})a^2\\approx5.73205...a^2", "2ba95e22e7aa79f5834e663df20053d5": "I_1 \\, ", "2ba96b98e279fa5c690820cf7951108c": "\\text{cont} (pq)=\\text{cont} (p)\\,\\text{cont}(q).", "2ba9cdc4df08d0104fe3f99920a95204": "\\psi (\\dots, \\,\\mathbf r_i,\\sigma_i, \\, \\dots, \\,\\mathbf r_j,\\sigma_j, \\,\\dots) = (-1)^{2S}\\cdot \\psi ( \\dots, \\,\\mathbf r_j,\\sigma_j, \\, \\dots, \\mathbf r_i,\\sigma_i,\\, \\dots)", "2ba9f9dd0076b63a0d987e638fdd9db7": "U_i=1", "2baa1a8aeead4b291647449fef7d50ed": " \\ v_{ \\bar{x} } = \\ v_{ \\bar{x} }' ", "2baa38b3862b8518f135e97a290eaf6b": "\\text{subject to} \\Pi \\ge \\Pi_0 +T \\,", "2baa571eed918c1489e7dd070df24b6a": " \\hat V ", "2baab323af1be15969115119a3022456": "E = \\gamma m c^2", "2baad45ed0d59158b3c9f603ae2e3628": "\\alpha=\\beta=\\pm 1/2", "2bab15dc6bee87630d7f1a23f772002a": "\\oint_{\\partial V'} G(\\vec r, \\vec r')\\; d\\vec S' \\; f(\\vec r') = \\int_V [\\nabla' G(\\vec r, \\vec r')] f(\\vec r') = -\\int_V \\delta(\\vec r - \\vec r') f(\\vec r') \\; d\\vec V =- i_n f(\\vec r)", "2bab4029a5ff267b047dd386ffc6a60a": " \\lambda_n = {n \\choose 2} \\frac{1}{N} ", "2bab67ab94d1801fd3db6d195cd27943": "\\textstyle \\Delta", "2bab6d0e985126b5268381fc91b8560e": "t^{(k)} \\gets t^{(k)} / t_k", "2babf91354550c88c973297159012842": "(x,y,z)\\;", "2bac1fa3e4accb2d64515e1c3de86097": "\\boldsymbol{\\tau} = \\mathbf{r} \\times \\mathbf{F},", "2bac26fa33cf7f6eaef62907f4e500aa": " S_2 = \\frac {\\sum_{i} ( \\sum_{j} w_{ij} + \\sum_{j} w_{ji})^2} {1} ", "2bac617357435cafb4eb7622ff3e6af3": "\\hat{\\ell} = \\frac{1}{N} \\sum_{i=1}^N H(\\mathbf{X}_i) \\frac{f(\\mathbf{X}_i; \\mathbf{u})}{g(\\mathbf{X}_i)}", "2bac6ca7801d4fe5bd270fb2c7df1467": "\\sigma^{-1}(X_{n-k}-X_{n-k-1})", "2bac7e733326ddbb7d2ccaaef7d16c2a": "\\rho ^A", "2bac8d1190e1e8b9815dbc5a36a5e05f": " K=\\{(t,tx): t\\in\\mathbb{R}, x\\in\\Omega \\},", "2bacd55350c55ffc640b59950182a0d0": " \\mu \\neq 0 ", "2bad33702c5188e47980e997f28a32e6": "\\overline{GM}", "2bad7297759e39125f76389101483b28": "\n \\sup_{\\theta\\in\\Theta} \\big| \\hat{Q}_n(\\theta) - Q_0(\\theta) \\big| \\ \\xrightarrow{p}\\ 0.\n ", "2bae0253e5062ff082d33fb930b837aa": " a_B = x_B \\gamma_B", "2bae09ab6f1e0dc8213da8a4c07dcaf3": " y'_{1} = \\frac{x'_{1}}{x'_{3}} = \\frac{\\mathbf{r}_{1} \\cdot (\\tilde{\\mathbf{x}} - \\mathbf{t})}{\\mathbf{r}_{3} \\cdot (\\tilde{\\mathbf{x}} - \\mathbf{t})} = \\frac{\\mathbf{r}_{1} \\cdot (\\mathbf{y} - \\mathbf{t}/x_{3})}{\\mathbf{r}_{3} \\cdot (\\mathbf{y} - \\mathbf{t}/x_{3})} ", "2bae0c301af5b035fe2b19d4e68c54f9": "\\delta_{S}(t)=\\left( \\frac{R_{S}(t)}{R_{std}}-1 \\right) 1000, \\qquad \\qquad (4a) ", "2bae840e042251ebc8081f19b56a6ca0": "{100 \\over \\sqrt{3}}", "2bae89011208ef2b29e2bdba12f1c0f0": " W_i \\sim \\Gamma(\\beta_i, 1), \\quad i=1,\\dots,k, \\qquad Z \\sim \\Gamma(\\alpha, 1), \n", "2baec53b2b0fb7a38673a858aaf56b8d": "\\eta\\;", "2baf83438522224e34b624c10c04ea3c": "I_\\text{ion}", "2bafac8cfa1c9c5adf658e628e876414": "g=\\alpha^2\\begin{pmatrix}-(c^2-v^2)&-\\vec{v}\\\\-\\vec{v}&\\mathbf{1}\\end{pmatrix}", "2bafaf3a094cf29050ebd0932543e4cb": "f = {nv \\over 2L}", "2bafc2893131cd85d4ab987d6c9c12f7": " A\\subset\\mathbb{R}", "2bb02256a80c83d0af6e0868c5386fe8": "t_a,t_b", "2bb02c561812670baa4c92edaec44cd1": "\\Sigma_k^{\\rm P} = \\Pi_{k}^{\\rm P}", "2bb04d4f537e35464e7afa2872e59c6a": "\\operatorname{E}[s] = c_4(n)\\sigma \\,", "2bb0aea5e7d7c4ff9623900a0369e79e": "{}_{V \\subseteq L(N,M_0)}", "2bb0b64c18b40886966c7909ffc49fd7": "\\, \\epsilon_{abcd}", "2bb0c8ead3ee72b78b91e7c87325b517": "\\min(\\textrm{rank}(M'_f): M'_f\\in \\mathbb{R}^{2^n\\times 2^n}, (M_f - M'_f)_\\infty\\leq 1/3).", "2bb0d3cb48843eec863f64e1dac77c8f": "\n\\begin{align}\n R &= \\sum_g \\int (Y_\\textrm{gth}-Y_\\textrm{gexpt})^2dE/\\sum_g \\int (Y^2_\\textrm{gth}+Y^2_\\textrm{gexpt})dE,\n\\end{align}\n", "2bb0e3cd5bded7a86588a7e8456437fd": "v = \\nu n \\lambda/\\sin \\theta", "2bb0fdcd9915e258666cf2ce9c9c8270": "10_{22}", "2bb18b9c77976b073c691e22dcf268a3": "\\frac{d^p T_n}{d x^p} \\Bigg|_{x = \\pm 1} \\!\\! = (\\pm 1)^{n+p}\\prod_{k=0}^{p-1}\\frac{n^2-k^2}{2k+1}.", "2bb1b14b1eb6072f64189e8abbccaeb2": "\\Pr\\Big(n \\text{ coin tosses yield heads between } (p-\\epsilon)n \\text{ and } (p+\\epsilon)n \\text{ times}\\Big)\\geq 1-2\\exp\\big(-2\\epsilon^2 n\\big)\\,.", "2bb1b565148eca2b76becd345ead1e66": "\\mathrm{SNR} = \\frac{|h^\\mathrm{H}s|^2}{E\\{|h^\\mathrm{H}v|^2\\}}.", "2bb228f2d537539518148c2b538b67a2": "T'(E)", "2bb2469f6f046ae50f8481ed33cf54ab": "\\begin{pmatrix}h & 0\\\\0 & -h\\end{pmatrix}\\begin{pmatrix}0 & 1\\\\-1 & 0\\end{pmatrix}", "2bb2c8319e3d7a4884f0fd118ece49c8": "\\sigma_0 = (2D/3L)f'_t", "2bb2db233568cb882336b94521b0c235": "e_B = i_B r_B + { {d \\varphi_B} \\over {dt}}", "2bb2fe26ea5644599b0350063a9cfc7d": "C=4L_c \\,", "2bb2ff50d3b00d5f265f6e62b5c12964": "V(r) = {1\\over 2} \\mu \\omega^2 r^2,", "2bb30ad33cdfd4629839a555918d45ef": " k = \\frac {K}{\\gcd(K,G)} ", "2bb32277b93c1a223d51f292267b91ce": "ds^2 =-(1+2\\Psi)dt^2+a^2(t)(1-2\\Phi)\\delta_{ab}dx^adx^b,", "2bb327e9a30ede3a45120c6d55e09e42": "E=E_1(x,t)+E_2(x,t)", "2bb337673e09e0c51061787e3988de6c": "\\kappa(\\alpha^*) = \\alpha", "2bb33810d69c15c24e61544827c8b45f": " \\boldsymbol{\\psi} ", "2bb39a7307eccef6f8e2d86399aa4cfb": "ax^2+by^2+cz^2=0.", "2bb39c3e9f96cbbfa6e7af6e4f048092": "P\\left(a < \\frac{X}{\\omega}\\right) = 1-a^n = \\alpha.", "2bb414ce77bfb7757849bf8626310083": "e^{\\pi}\\,", "2bb45631dcb987b4dca1be9f3b233eb0": "\\log \\gamma = k_m I\\,", "2bb46fa9fd945092515c7042242cd8c6": "\\tbinom{n+k-1}{n-1}", "2bb4b6577189be8920dd75add81e9371": "\\begin{align}\nL_{x}&\\approx -I_{1}|\\dot{\\psi}|\\approx\\mathrm{constant}\\,,\\\\\nI_{2}\\ddot{\\alpha}&\\approx -I_{1}|\\dot{\\psi}|\\Omega\\sin\\delta\\,\\alpha\\,.\\end{align}", "2bb5440837fa50deec3293be39753af9": " \\mathbb{E}(A) = \\int_\\mathbb{R} \\lambda \\, d \\, \\operatorname{D}_A(\\lambda).", "2bb5484daf162aad0bc4ed899af4b042": " Z_1 ", "2bb56c0566c2db071c936ae8d233b2cd": "a,b,c \\in A", "2bb5e45ffdc9242f20a0fb191547144b": "D = \\pi \\bar{\\pi}", "2bb600ac4c0768739ebaa7808d98f6de": "P=F \\cdot v", "2bb60a181cca2ef1b3190b93e14469b5": "J^2", "2bb681eaf107fd54b5881930c8ca5d99": "\\vec \\psi_P", "2bb68b8939160701da95cb887e1e140c": "1=\\sum_{K}\\frac{\\frac{2m}{\\hbar^2}\\frac{A}{a}}{\\frac{2mE_k}{\\hbar^2}-(k+K)^2}", "2bb68f085af9c1dd6aa934dac855bbc8": "R_0 = R", "2bb6adc38c29c6afbc86156bca76a625": ",_{\\alpha \\beta} = \\frac{\\partial^2}{\\partial x^{\\alpha} \\partial x^{\\beta}}\\,", "2bb7049648661fb532941a7403c9472e": " g(u,v) = \\omega(u,Jv) ", "2bb7584403db32161659455ddbdcba78": "X= {F_0 \\over k} {1 \\over \\sqrt{(1-r^2)^2 + (2 \\zeta r)^2}}.", "2bb78c29b81699ac3d861af5570dd886": "e:M\\to G", "2bb7c6e95fc8863ff17027c61391f2b1": " h_R(t) = {R \\over L} e^{-t \\frac{R}{L}} u(t) = {1 \\over \\tau} e^{-\\frac{1}{\\tau} t} u(t) ", "2bb806d33034c4356620c941454cb060": "J_i = J_{0i} \\left(e^{\\frac{qV_i}{kT}}-1\\right)-J_{SCi} \\Rightarrow V_{OCi} \\approx \\frac{kT}{q} \\ln(\\frac{J_{SCi}}{J_{0i}})", "2bb81ca0f002764034e304de335b9117": " x=X+XY+\\cdots \\text{ and } y=Y+2Y^2+X^2+\\cdots", "2bb8294ab63ffd349d590de43f6cdfe8": "\\left( quot = \\frac{V}{s+1}\\right)", "2bb83b8757819112dd08e8f66e5f2f40": "\\scriptstyle(8.0\\pm9.5)\\times10^{-32}", "2bb867665e20a8377375565c0b2dd5e8": "\\mathrm{^{244}_{\\ 96}Cm\\ \\xrightarrow[]{(\\alpha,p)} \\ ^{247}_{\\ 97}Bk}", "2bb8ec30d9e3e7ef17beacf092408a44": "\\scriptstyle G \\;\\to\\; \\mathrm{Aut}(G)", "2bb905e898b44e125bdd45f37912a45b": "{\\tilde{C}}_{6}", "2bb971a6f29d6fdd8ccd37636b7e29ff": "\\psi^{(-2)}\\left(\\frac14\\right)=\\frac18\\ln(2\\pi)+\\frac98\\ln A+\\frac{G}{4\\pi},", "2bb9a063b2e826697311d56d1e925131": " P_0 = S_0 - K e^{-rT}", "2bb9be8404107663bfee19c3d2261043": "d(B,C)", "2bba15800e2ddebc63b19d4bdb4b66a7": "X_1^3+ X_2^3-7=-2h_1(X_1,X_2)^3+3h_1(X_1,X_2)h_2(X_1,X_2)-7.", "2bba354bb9aaa1d73f20d3b0aa20c84d": "|\\phi^+\\rangle_{AB} = \\frac{1}{\\sqrt{d}} \\sum_{x_A,x_B} |x_A\\rangle |x_B\\rangle", "2bba76d8038d0470eda6c21dd6dad5c3": "P = \\frac{x}{47} \\times \\frac{y}{46} \\times 2 = \\frac{xy}{1081}.", "2bbb22dc10c4b9b97b417cd355ca7594": "\\Phi_a=-\\frac{2}{3}H_{ab}{}^b,", "2bbb3dd931cb4ef342402a06e7067df9": "[p\\cdot g,x] = [p,\\mathrm{Ad}_{g^{-1}}(x)]", "2bbb61fc32cefc0d0f2b3d5c74d21968": "\\mbox{THD} = {\\sqrt{V_2^2 + V_3^2 + V_4^2 + \\cdots + V_n^2} \\over V_1}\n", "2bbb9e05dd7b6ff19aabf47d17a871df": "D(p||q)+D(q||r)-D(p||r)=\\langle\\mathcal{E}^{-1}(p),\\mathcal{E}^{-1}(r)\\rangle + o(max\\{||\\xi(p)-\\xi(q)||,||\\xi(p)-\\xi(r)||\\}^3)", "2bbbe22e15175c330cdb6e418c3de548": " \\rho = \\sum_{p} \\frac{1}{2^p} = \\sum_{n=1}^\\infty \\frac{\\chi_{\\mathbb{P}}(n)}{2^n}", "2bbc034e6d0cb4945fc2db4b92fb2e16": "\\Kappa \\, \\kappa \\, \\varkappa \\,", "2bbc60df3bc8756050b53b6f4892994b": "v^*=\\frac{kT_c}{P_c}\\Phi", "2bbc923750bd37017842fc564eacc4db": "-\\infty < \\mu < \\infty ", "2bbca97d8ad4d61ab4f3cb072ea6c3f4": "\\tbinom {12} 6", "2bbcb6738a79e65db1fdc09d5f021d29": "\\{\\sigma_k\\}", "2bbcb9b453420b8ba593fbaf03c8eb94": "\\scriptstyle 2B", "2bbe6d1824f4e1a4710cbbf7058d4c84": "\\tilde{x}_i^{(t)}", "2bbe76d4f1290e8afe5109085c345ad9": "(\\hat{m}_{ij})", "2bbeb1e116dda9e2f6c5cc1db1b33a37": " B_1\\left(\\frac{R}{2}\\right) = \\frac{\\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}", "2bbeea2b43ee57ffe7ba0ec27de96d92": "x \\in D\\,", "2bbf2768dcd3bce8685533f36d4edcad": "SU(2)_W \\times U(1)_Y", "2bbf4034e1064ed70d608e95144787d7": " \n\\text{Choose } x_i(t) = x_{max,i} \\text{ if } Vc_n + \\sum_{i=1}^KQ_i(t)a_{in} < 0\n", "2bbf6e3fbc9897ee02058a79d5315a81": "0 < \\left| \\sqrt{2} q_{n + 1} - p_{n + 1} \\right| < \\frac{1}{2^{2^n}}.", "2bbfd56a5b3897eca966ce39d98c707e": "\\mathbf{access}", "2bbfde8e959503376a97daacb89477a5": "DC={\\Delta}_{uv}=\\sqrt{(u_r-u_t)^2+(v_r-v_t)^2}", "2bc058ae62168f2a252e7147e29fe769": "\\mathrm {P \\left( 1 + T \\right) ^ n = ERV}", "2bc084094ccd453b316b77de8691adea": "\\scriptstyle \\ x_i, y_i", "2bc0b051df61f0af1f2a5dc07cf6a1ff": "\\scriptstyle\\langle N\\rangle = -\\tfrac{\\partial \\Omega}{\\partial \\mu}", "2bc11d2f750362cbc32d75d1c92e71de": "j_1 = j_2 = J/2", "2bc12f7854c7b4d07aa9e07b0b0b2268": "\\omega_0 = \\sqrt{g/l}", "2bc144c878d42b4ba8c742caeeafa2e5": "\\mathbf{c} = c_i~\\mathbf{e}_i ", "2bc18f7f9e752337a1556b284465edb5": "O(L(kt+dnP_2^k))", "2bc1a64fec925c3f589441400989cc0a": " \\mathbf{c} = \\lambda\\mathbf{a} + \\mu\\mathbf{b},", "2bc1e8e4efa1443098f13f525bed2a19": "f(z)=\\frac{z^2}{z-z_1},", "2bc20d713193d48a21a401f737f71489": "h(\\alpha)=f(\\mathbf{x}_k+\\alpha\\mathbf{p}_k)", "2bc20f70d67e54cf50823c1be20332c5": "f_t(x)=x^3-tx,\\,", "2bc232146219f5758232521c75c959fe": "~ n_2=\\frac{W_{\\rm u}}{W_{\\rm u}+W_{\\rm d}} ~", "2bc24fa4537798d0f3c0b9968c95ec18": " \\tan \\psi \\quad = \\quad \\left ( \\frac {r - 1}{r} \\right ) \\tan \\phi \\quad = \\quad \\left ( \\frac {1 - n}{1 + n} \\right ) \\tan \\phi", "2bc2ce9224c9725c02387cbea714613b": "K=\\,", "2bc2d1e04eb3092ee1c121b64ce835dd": "S = \\int_{t_i}^{t_f} L \\, dt.", "2bc2d418ebd8a8ecafaec3a845291c68": "\\frac{ 1 }{ \\sqrt{V} } \\frac{\\operatorname{d}V}{\\operatorname{d}t} \\leq -\\mu,", "2bc2e374e9c00ce4d4b5f16fd6832c45": "\\mathfrak{a}^{ec} \\supseteq \\mathfrak{a}", "2bc2ec3d2b4f7ad11fb7e1095e805c21": "H_{ba} P_i = R P_i - t", "2bc32fe1f0ec51a633eb3a41311ea942": "b(f_1,f_2) = \\frac{\\left| \\sum\\limits_{n} F_n(f_1)F_n(f_2)F_n^*(f_1+f_2) \\right|}{ \\sqrt{\\sum\\limits_{n} |F_n(f_1)|^2|F_n(f_2)|^2|F_n^*(f_1+f_2)|^2}} ", "2bc346b31e6f771b76879cd790156205": " P = 500 + \\left( {22 \\cdot LBM} \\right)", "2bc35591a8d6922f167be78e20ccb262": "u_o", "2bc3681cbb77a95eec309ef13be59204": "w_m = (1C1)^{-1} C1", "2bc39066d6727a9dfbfad989ff578815": " \\lambda_d(E) \\leq 5^{d} \\sum_{j\\in J'}\\lambda_d(B_{j}). ", "2bc3d3a7db7acf7adebd83e87544b073": "\\scriptstyle V_0 = 1", "2bc3e630eddf0a32136485a096368c8e": "\\phi(a, b, n)\\,\\!", "2bc3e85e03dce7c9d601fc322cc2645d": "J_G(q)", "2bc463851e338d59be8dabd2ea3432ff": "T_d = 1 + gh/c^2", "2bc4774ed5330c8149e9fe4f09fdceb6": "A \\smallsetminus B", "2bc4791ddbb6d9f2069095e3ab4e9351": " V = w \\ l \\ h ", "2bc4c66376ebd1e5e9b239d5d952a8af": " f_\\theta(x)=h(x) \\, g_\\theta(T(x)), \\,\\!", "2bc5086fe2061920efc896312f3d120a": "\\begin{bmatrix}\nX^{-1} & A \\\\ A^H & X-Q\n\\end{bmatrix}=0", "2bc51323dae5a26045b93e490dac661f": "t=N*C/f", "2bc51b181918dc67bc8602d1010f7211": " t_{crit}= \\frac{1}{k_2-k_1} \\ln {\\left[ \\frac{k_2}{k_1} \\left({1- \\frac{D_a(k_2-k_1)}{L_a k_1} }\\right) \\right]}", "2bc536b7d5893eb5158c4776170adb7d": " e^{i S/\\hbar}", "2bc55f5a9556a3113d7e0cd1e03e751d": "M^\\text{T}", "2bc57de1f231219253f2261dc0fc2de5": " -\\frac{\\hbar^2}{2I} \\frac{d^2\\psi}{d\\phi^2} = E\\psi ", "2bc599ef07d29ac1375e1c84f25b63f9": " Y(s) = H(s) X(s) \\, ", "2bc59c534cd4249364654c31eeb65e09": " B(\\mu_f) ", "2bc627d6e919d7b7e33e5b57f3ed7128": " cost(l) = \\beta^{usage(l)}", "2bc68069addba5edd3ab2c28cfc523d5": "\\cot 2\\theta = \\frac{\\cot^2 \\theta - 1}{2 \\cot \\theta}\\!", "2bc697f43049f8a950a145b0a5f2a503": "\\mathbb{S}_3\\;", "2bc6e29aa9692d361d09350bd801ec38": " \\mathcal L", "2bc73afe26c2ed7efccf91ea50d0c1bc": "\\Omega\\wedge\\theta=0", "2bc75a3ab84a742598cc7745c2c7997a": "g: D \\to k", "2bc75b0442795c2ea2d329cbc860d02a": "\\oint_C \\mathbf{E} \\cdot d \\boldsymbol{\\ell} = -\\int_{\\partial C} {\\partial \\mathbf{B}\\over \\partial t} \\cdot {\\rm d}\\mathbf{s} = - \\frac{{\\rm d} \\Phi_D}{ {\\rm d} t}", "2bc76e87d2e9a277ba554d0b8838d10b": "\\dot{x} = A x + B u \\, ", "2bc775ec49cbbf74269249c3e7ccd606": "m_v: V\\to V'", "2bc7da4df33f21f52381ca4d60f2843c": "d(X,Y)=\\mathbb E\\left[\\min(|X-Y|, 1)\\right]", "2bc7f1b36e03fa1cae2d3413890ab691": "L_\\text{V} = \\frac{3.9}{C \\sigma_\\text{scat}}.\\ ", "2bc7fe8a43279f839adbd65cf189e565": "z(b)\\quad", "2bc803bec2aaac4f5523e23e25efea69": "(u_1,\\dots,u_n)^t", "2bc820253db52a311c45ffd61e87c1ab": "\\xi=\\xi_0\\exp \\left[\\frac{2a_0^2}{\\xi_0}\\left(\\tau-\\tau_0 \\right)\\right].", "2bc8d62df50d15d08420e57e97b21af2": "|f(re^{i\\theta})|\\le Me^{\\tau r}", "2bc8ffbbf5efbf42f65e45755ea99026": "\n\\begin{bmatrix}\nc\\,t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\gamma&-\\gamma\\,\\beta_x&-\\gamma\\,\\beta_y&-\\gamma\\,\\beta_z\\\\\n-\\gamma\\,\\beta_x&1+(\\gamma-1)\\dfrac{\\beta_x^2}{\\beta^2}&(\\gamma-1)\\dfrac{\\beta_x \\beta_y}{\\beta^2}&(\\gamma-1)\\dfrac{\\beta_x \\beta_z}{\\beta^2}\\\\\n-\\gamma\\,\\beta_y&(\\gamma-1)\\dfrac{\\beta_y \\beta_x}{\\beta^2}&1+(\\gamma-1)\\dfrac{\\beta_y^2}{\\beta^2}&(\\gamma-1)\\dfrac{\\beta_y \\beta_z}{\\beta^2}\\\\\n-\\gamma\\,\\beta_z&(\\gamma-1)\\dfrac{\\beta_z \\beta_x}{\\beta^2}&(\\gamma-1)\\dfrac{\\beta_z \\beta_y}{\\beta^2}&1+(\\gamma-1)\\dfrac{\\beta_z^2}{\\beta^2}\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nc\\,t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix}\\,.\n", "2bc9102e77be265054ce9f7bd321c853": "\\mathbf{r}_{\\mathrm{mean}}", "2bc91c3a751ac20b9190a5793f13e16d": " n \\mid \\varphi(a^n-1)", "2bc92f07147774e517bbd4b252fb09b8": "N = \\sum_{i=1}^t n_i", "2bc93f29a61194efd82beabc2b528867": "q_0", "2bc9633672017cf9eeb19b25fa9de1ef": " \\mathbf{B} =\\frac{\\mu_0 q \\mathbf{v}}{4\\pi} \\times \\frac{\\mathbf{\\hat r}}{r^2} ", "2bc9a5b0f5faa6d767e9850003ca6c73": "\nC^{S_1}_{E_3} = \\varepsilon^{2}_2 / D\n", "2bc9c523bff923ec443eccc372edcc41": " \\scriptstyle C ", "2bca2701bbfb91e38ed5e2acce330ef5": "\\!t_1", "2bca6a12864301b4d8eb5c2b73f0aaaa": "+(22639.55+4586.45)''\\sin(l)", "2bcaad84d01209225255b02cb1e25f96": "M(G_1)=G_2\\,", "2bcab1722c917e6a41cd63898d9cab5e": "C_n, \\mathfrak{sp}_n(\\mathbf{C}): \\mathfrak{sp}_n(\\mathbf{R})", "2bcad2384c02426e5beedc88db70cfe0": "\\mathbb{E} \\big[ \\langle \\mathrm{D} F, v \\rangle_{H} \\big] = \\mathbb{E} \\big[ F \\delta v \\big]", "2bcad6d5fcb2b0ad49a21f49ce23fe9e": "\\frac{150}{26.33} \\approx 5.70", "2bcae4b742f40c7d10514fe2b83c88cf": "+S_z \\otimes S_z", "2bcb09dfd6aeeb90037406555f30d408": "g=f_1g_1+\\cdots +f_kg_k.", "2bcb09fdfc6deb4023c0bc0f7bc8902e": " \\langle \\mathbf{e}_j \\bar{\\mathbf{e}}_k \\rangle_V = -\\mathbf{e}_{jk}", "2bcb28de35a527a004a3d67b6689003d": " x^3 -x -1 = 0.\\,", "2bcb46cf6ff65cc2d05d026f10fffa2e": "\\dot{x} = f(x(t),u(t),t),x(t_0)", "2bcb5a4cdd07dc25b0cbb683d688d61c": "\\Delta(x) > Kx^{1/4}", "2bcb5c69107f8eb308fe6d8af34e7ffb": "sid_1 < sid_2", "2bcbabf8a4acaef3d1a3a21a3ffe672f": "\\psi^{(m)}(z) = (-1)^{m+1}\\; m!\\; \\zeta (m+1,z).", "2bcbb807a59f15a8756a90f343275cbd": "\\mu(\\pi) = \\frac{(\\mathrm{dim}\\,\\pi)^2}{|G|},", "2bcbbb7a8bcb5a428701fd1d5bccbe23": "G[\\mathbf{f}A] = A^\\mathrm{T} G[\\mathbf{f}]A", "2bcbbfccfb3d94fc1133e82ca7760ddc": "\nz = x + iy = |z|\\left(\\cos\\theta + i\\sin\\theta\\right) = |z|e^{i\\theta}\\,\n", "2bcbe887a616134fc6732c7e59de47fa": "\\scriptstyle | i\\rangle", "2bcc025ba7d47f19032c3db92b14cb50": "\\frac{\\operatorname{d}^2\\! F_0}{\\operatorname{d}\\!k^2}+\\frac{1}{k}\\frac{\\operatorname{d}\\!F_0}{\\operatorname{d}\\!k}", "2bcc16fc38f8d88fea5476921c7acd85": "\\vec{h}_0 = \\frac{1}{\\sqrt{1-3m/r}} \\, \\partial_t + \\frac{\\sqrt{m/r^3}}{\\sqrt{1-3m/r} \\, \\sin(\\theta)} \\, \\partial_\\phi ", "2bcc29df5a1bdc240e5004d31461c66f": " \\mathbf{x} = \\mathbf{0}\\,", "2bcca903adea22445fc4b8b733dfb301": "\\nexists t \\, (t\\ E\\ t)", "2bccac3ec414a5664c9c984e18e41458": "a_1 y_1(t) + a_2y_2(t)\\,", "2bccb81961c422a35171af5d85078cee": "2.5 \\log_2 r", "2bccf45f3508132f51b9273b10ad464a": "\\hat{y}=X\\hat\\beta=Py=X\\beta+P\\varepsilon", "2bccfed0c70a038cf4f923bb18075491": " \\mbox{If } x,y \\in F \\mbox{ then } x \\and y \\in F,", "2bcd64e3eae547dd389bcf4c34bea964": "\\nabla \\cdot \\mathbf{E} = \\hat{\\mathbf{k}} \\cdot \\mathbf{E}_0 f'\\left( \\hat{\\mathbf{k}} \\cdot \\mathbf{x} - c_0 t \\right) = 0", "2bcdba7649095936cf4a3c44c283f776": "\\pi, \\overline{\\pi}, \\rho, \\overline{\\rho}", "2bce2bc2c1a6f15f74ed7f6f7ea23ccf": "a^* = g^* = \\frac{1} {L^*}", "2bce6b40cdfdf2232a2063cae528e9b3": "\\partial_1(c_1) = (b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3]", "2bce6d0c9de953dfbea88bc2b699d4a5": " \\sum_{i} \\boldsymbol{M_{O_{1}}^{u_{i}}} = (\\boldsymbol{O}-\\boldsymbol{O_{1}}) \\times \\left ( \\sum_{i} \\boldsymbol{u_{i}} \\right ) = (\\boldsymbol{O}-\\boldsymbol{O_{1}}) \\times \\boldsymbol{R} = \\boldsymbol{M_{O_{i}}^{R}}", "2bceebbc8acbf7e3358d24bdff26cf97": "d(\\mathbf{u},\\mathbf{v}) = \\cosh^{-1}(\\mathbf{u} \\cdot \\mathbf{v}).\\,", "2bcf166a00c370a69b39c590c9b4059c": "v_i \\in S\\,", "2bcf3406601ccb73e48737436a9d012f": "\\langle K_P \\rangle = \\frac{3NP}{2\\beta}- \\langle U_{\\mathrm {spring}} \\rangle ", "2bcf36f5d069a0a3d52db83f5f07d906": "\\Delta s_i = |\\mathbf{r}(t_i+\\Delta t)-\\mathbf{r}(t_i)|\\approx|\\mathbf{r}'(t_i)|\\Delta t.", "2bcf538f0e9bfbbca880466ecfdd9df2": "2p(1-p)", "2bcf735e255e421c65b6216ba4bad2ab": "CT_{min} = \\begin{matrix}max\\\\j=1,M \\end{matrix}\\lbrace \\tau_j/N_j \\rbrace", "2bcf7b3d083288f955a476935e17f168": " \\dot{\\mathbf{P}} = -{\\partial K \\over \\partial \\bold{Q}}, \n\\quad \\dot{\\mathbf{Q}} = +{\\partial K \\over \\partial \\bold{P}}. ", "2bcf89f9a314f17862d235403336959a": " D[p] = [q, S_4, A_4]::[x, S_3, A_3]::K_2 ", "2bcfa516555726d009aec7676ee2c672": " y_{2ss} = y_c ", "2bd0052793d9f019feac23d5724eafca": "z_i \\ne 1", "2bd0287317f4ed6fc8b99fee886e0fd0": " R_{\\theta} = \\frac{x}{A \\times k}", "2bd0335453845ccb1b572a9ca27b07fc": "\\textstyle {N}(B), ", "2bd075f2452fe1b98a2c1798d2b0e2d2": "B = Z_1\\cdot Z_2 ", "2bd0a4075f437a70c4e3c7a311203bdc": "G_\\mathrm{voltage} = {V_\\mathrm{out} \\over (V_\\mathrm{in+} - V_\\mathrm{in-})} = R_\\mathrm{load} \\cdot g_\\mathrm{m}", "2bd0fd643b1d86e79f34fe908abe1b28": "k_B T", "2bd19389c9a647a93d53b6e13e1242e6": "\\overrightarrow{f(P)~f(Q)} = \\varphi(\\overrightarrow{PQ})", "2bd197e157bb4f73baa394527e273cf4": "\\log _{10} 616", "2bd26e8cbdee5ee3d5fdf6ee4c06b794": "R(r) = r^{\\lambda}", "2bd295bb87058e47c7d746fcf097761d": "B>1", "2bd2c9bc5c511c258d429d0b260e29c1": " x= \\left(\\sum_i I-\\hat v_i \\hat v_i^\\top\\right)^{-1} \\left(\\sum_i (I-\\hat v_i \\hat v_i^\\top) p_i\\right).", "2bd2d63b85cbc022c6e0a76d1a510432": "A=\\begin{bmatrix}\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\\dotsb & +1 & \\dotsb & +1 & \\dotsb\\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\\dotsb & +1 & \\dotsb & -1 & \\dotsb\\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n\\end{bmatrix}.", "2bd30f1d44448a2db68fbbb9d7b5ca58": "A(x) = \\frac{1}{\\hbar} \\sum_{n=0}^\\infty \\hbar^n A_n(x),", "2bd3b434b0780fe8e5b9c1e148c5ee12": "y'' = -k^2 y.\\,", "2bd3ed97638e7ebf58e1d076a748ecef": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\; - {4\\over 3}\n{1 \\over \\vec k^2 + m^2}\n", "2bd433fc1aafd3782bf50334df7839da": "e_0=1", "2bd44f855cfac39c88a31df4c4cbfdd0": "\ng(s) = \\sqrt {2 / \\pi} \\int_0^{\\infty} (st)^{1/2} \\, K_{\\nu}(st) \\, f(t) \\; dt,\n", "2bd4724d8c0c8b1983a98cf27eee1407": "l_2= a_{00} - \\mathcal{L}(p_6)+p_3p_6, \nl_3= a_{00} - \\mathcal{L}(p_9)+p_3p_9, \nl_{31}, ....", "2bd473f1569dc1d90d9a5862a7d87ae3": "2^{K-1}", "2bd4b55c9539155fb4b0da36be8424a1": "[x=i]", "2bd4eb8be0a69f90e8b257e9028f0900": "q = \\frac{Q}{w}", "2bd501d811d0f0b10afcdc20132a12f6": "M(n)=\\left\\{\\begin{matrix} n - 10, & \\mbox{if }n > 100\\mbox{ } \\\\ M(M(n+11)), & \\mbox{if }n \\le 100\\mbox{ } \\end{matrix}\\right.", "2bd5172d41c47e0c63b5330bc90ed4e8": "\\frac{k_BT}{x_\\beta}", "2bd5553896cc65569d754c30e05ea565": " \\frac{dy}{dx} = \\frac{du}{dx} + \\frac{dv}{dx}. ", "2bd590d99de833d70ca46cd15ffd8fbf": "\n\\left|\\frac{1}{N} \\sum_{i=1}^N f(x_i)\n - \\int_{\\bar I^s} f(u)\\,du\\right|\\le\n\\|f\\|_{d}\\,{\\rm disc}_{d}(\\{t_i\\}),\n", "2bd5a21b24352352621b082c99694c46": "\\displaystyle z = \\frac {cd}{{a^2+b^2+c^2}}", "2bd5c0e339983136534c05cc04e18d39": "y=a_{0}\\sum_{r=0}^{\\infty }{\\frac{(c)_{r}(c+1-\\gamma )_{r}}{\\left( (c+1-\\alpha )_{r} \\right)^{2}}s^{r+c}}", "2bd60873a9adeb779098c774abc6a055": "a(n)\\approx \\frac{n!}{2(\\log 2)^{n+1}}.", "2bd62f7c6ff5988a4602fba6c8873a46": "(x_0,y_0,u_0)", "2bd64602a4022d602816328af8bf5b0c": "P(x)=2x^3+3x^2-4\\,\\!", "2bd657d27e14d83868a7aa77cff27af7": " E", "2bd6b1cae83a5cfbafc45fa0f2692c1c": "{\\mathbf Y} = \\left ( \\begin{matrix}\n0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 \\end{matrix} \\right )", "2bd7100657b58fe4a663d26b2328a5c0": "(n,k,2t+1)", "2bd713e5885d2f8dfc7a69257b1bed40": "|\\psi(t)\\rangle = \\int d\\varepsilon \\, | \\varepsilon \\rangle \\langle \\varepsilon | \\psi(t)\\rangle = \\int d\\varepsilon \\, \\psi(\\varepsilon,t) | \\varepsilon \\rangle ", "2bd791cd5d96fd53d7bdeeacecc6cbae": " \\Xi = S - \\frac {U + P V} {T} = \\Phi - \\frac {P V}{T} = - \\frac {G}{T}", "2bd7b15b9e48ca753dba53bc72be2ee9": "\\left[J_\\pm, V_\\pm\\right] = 0 ", "2bd7d25bc0c8d830e0204af57417130e": "\\mathrm percentage \\ protonated = {1.9 \\times 10^{-5} \\over .20} \\times 100\\% = .0095\\% ", "2bd7d268d63ace5c6845342797692af9": " y=f(x) ", "2bd7e799360a9ce798b455bf4b36e75d": "u(S_t,t)", "2bd81930b1627e87961371d44c64603c": "\\beta_0 \\,,", "2bd8303c957f3f0259fed3d531b299a5": " W^T = W^+ - W^-", "2bd84efefa7482950c0ac08e4c713822": "A(z) = \\sum_{k = 0}^{\\infty} \\frac{B(z)^k}{k!} = \\exp(B(z))", "2bd87932a2634c14ef180ac7bcc843df": "\\hat\\beta=-1.817.", "2bd880153e9619b6f91a09cb56d81e1b": "X_i \\mapsto A_i", "2bd89c34a04eb3fe965e2567f94fc877": " E,F \\subseteq X ", "2bd8b3daa067725895d0a287e8c0ae26": "+h(p_i)", "2bd8c4904481c60b44894ced365700a9": "B=\\frac{2\\pi}{\\lambda}\\int \\! n_2I(z)\\,dz \\,", "2bd8e70ffb9ba57a82bdeb859216f9a4": "\\mathrm {in\\, thermodynamic\\,equilibrium,\\,when}\\,\\,T=T_X=T_Y\\,\\mathrm {,\\,it\\,is\\,true\\,that}\\,\\,\\alpha _{\\nu , X,Y}(T,T) = \\epsilon _{\\nu , X}(T).", "2bd948809b0b98d2685d4386d9e95240": "P_n(x)= \\frac 1 {2^n} \\sum_{k=0}^n {n\\choose k}^2 (x-1)^{n-k}(x+1)^k=\\sum_{k=0}^n {n\\choose k} {-n-1\\choose k} \\left( \\frac{1-x}{2} \\right)^k= 2^n\\cdot \\sum_{k=0}^n x^k {n \\choose k}{\\frac{n+k-1}2\\choose n},", "2bd953e829739a498f458c8c3ed516e8": " W(x,p)\\stackrel{\\mathrm{def}}{=}\\frac{1}{\\pi\\hbar}\\int_{-\\infty}^\\infty \\psi^*(x+y)\\psi(x-y)e^{2ipy/\\hbar}\\,dy ~.", "2bda87067c701b76c65568d0311417d8": "\\mathrm{C}=[L_A]_\\beta^\\gamma", "2bdaa4e3288a85d081bf24e27cd5007b": "EOA(\\mathrm{cm}^2) = \\frac{Q_{rms}}{51.6\\sqrt{\\Delta p}}\\ ", "2bdacee015d845bbdcc6639a1a5d5e57": " x_1, ... , x_n ", "2bdbb1a6b1c6c7a53a848966191b45e6": "0 \\leq r < 2^b-M", "2bdbded1cbca657bdc5a3b81f55b91b4": "\\Delta P = \\frac{8 \\mu l Q}{\\pi r^4}", "2bdbfdd4e57d0f759ecbcb2876f5b2df": " C = \\max_{f} I(X;Y).\\! ", "2bdc28579ec714a837a44e8bc628170e": "\\psi\\left(x,y\\right)", "2bdc581b31815a01ff8911de35b305c8": "\\text{lim}_{n \\rightarrow \\infty} |\\mathbf v_n - \\mathbf v| = 0.", "2bdc61521c6f9f0ff7ea96d276bd14c4": "\\sum_{x \\ge 1}^{\\Re}f(x)=F(0)\\,", "2bdc711e9802194e6d1c3e6d3f212d32": "(1,1,1).", "2bdc79556f8269cc9594c5b0604e8890": " \\zeta(s) = \\prod_{p\\ prime} \\frac{1}{1 - {p^{-s}}} = \\sum_{n = 1}^\\infty \\frac{1}{n^s} ", "2bdc83b1dd1599cfb208f3d658feb157": "\\gamma = \\sum_{k=1}^\\infty \\left[ \\frac{1}{k} - \\ln \\left( 1 + \\frac{1}{k} \\right) \\right].", "2bdc9d790432f7d4ef06e6107f8c44fa": "\n= - {{\\frac{1}{1}} {\\frac{10^4}{10^4}} } - 10 + 2 \\times 10^{-4} \\times 10 + 10^{-5} \\left({\\frac{10^{-2}}{10^{12}}} + {\\frac{10^{-2}}{10^{12}}} + {\\frac{10^{-2}}{10^{8}}} \\right).\n\\qquad (2)", "2bdcdcba9be5b0db6de8177ddf18e6f8": "\n \\mathit{GM}\\left(\\frac{X_i}{Y_i}\\right) = \\frac{\\mathit{GM}(X_i)}{\\mathit{GM}(Y_i)}\n", "2bdce42832d32e3a33288af84bfa5f49": "\nM^m_\\ell \\equiv \\langle \\Psi | Q^m_\\ell | \\Psi \\rangle.", "2bdd0d8ee7169c734091665d8ad3dae4": "{g}'", "2bdd1576cff0ed6f8e3f5bb1e5b663dd": "\\hat{H}_0 \\equiv \\hat{F}, \\qquad \\hat{V} \\equiv \\hat{H} - \\hat{F}.\n", "2bdd637721a6d942c2d1ea2569941279": " T_R= \\left | \\frac {1}{({\\frac {\\omega}{\\omega_0}})^2-1} \\right |", "2bddab4d4e2a7737f3e37e069ee5a700": "\\alpha = \\arccos(-Z_2 / \\sqrt{1 - Z_3^2}),", "2bdde03f6b9c93b8f56a9f77a1a5136c": "2h", "2bde29441eb5c9c7ba897429e58fc438": "BJ(J+1)", "2bde47c0427c11fefadb51df6e31ec43": " u(x,t) =\\frac{e^{-\\frac{k_0^2}{4}}}{\\sqrt{1+2it}}e^{-\\frac{(x - \\frac{ik_0}{2})^2}{1+2it}}.", "2bde5ac56e955880b0b2043631b94cdb": "\\{1, \\vartheta, \\vartheta^2\\}\\; {}_p F_q (a_1,\\dots,a_p;b_1,\\dots,b_q;z),", "2bde663b39d9fd00e1ad48dfd651a2b8": "\\delta S= \\delta\\int_{x_{3A}}^{x_{3B}} n\\left(x_1,x_2,x_3\\right) \\sqrt{1+\\dot{x}_1^2+\\dot{x}_2^2}\\, dx_3", "2bde77d570453a1530fcf86c559d5aef": "\n \\begin{align} 16\\,\\operatorname{cos}{2\\pi\\over17} = & -1+\\sqrt{17}+\\sqrt{34-2\\sqrt{17}}+ \\\\\n & 2\\sqrt{17+3\\sqrt{17}-\n \\sqrt{34-2\\sqrt{17}}-\n 2\\sqrt{34+2\\sqrt{17}}}.\n \\end{align}", "2bdea78915d6e6b039d779aa9f18bf70": "\\Delta P \\equiv P_\\text{inside} - P_\\text{outside} = \\gamma\\left(\\frac{1}{R_1}+\\frac{1}{R_2}\\right),", "2bdeb98f1ed8a9b8c41779dd1aa25f34": "p_{t}=\\sum_{i}{p_iq_i}\\,", "2bdedbf053f3f91045d80cac6aefc7ac": "\n =e^{-(\\mu_1+\\mu_2)}\\sum_{n=-\\infty}^\\infty\n {{\\mu_1^{k+n}\\mu_2^n}\\over{n!(k+n)!}}\n ", "2bdef31c0c0aebb18bce2ec8a71288f6": " \\forall k \\in \\mathbb{N} : \\phi(x,k+1) \\leq \\phi(x,k) ", "2bdf258080aec973de9b9daa7df45430": "\nR = {i \\over 2\\pi} \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} \\,.\n", "2bdf7027746f7baa16627126feb410e8": "{5 \\choose 3}{4 \\choose 2} = 60", "2bdf8b0635ff669e9c0fe5bcaa5dc5ae": "\nq_{xx} = \\frac{\\sum (x-\\bar{x})^2 w(x-\\bar{x},y-\\bar{y}) I(x,y)}{\\sum w(x-\\bar{x},y-\\bar{y}) I(x,y)}\n", "2be0c6d44ec53662f8e1adaed8e902e4": "\\Omega_c^0(\\mathbb{R}^n)\\equiv C^\\infty_c(\\mathbb{R}^n)\\,", "2be10881641f7cb065dddda28d20cb98": "D_4^+", "2be10edf5cffbd76c35d32dd92b17b15": "f(x;\\mu,\\sigma,\\xi) = \\frac{1}{\\sigma}\\exp\\left[-\\left(\\frac{x-\\mu}{\\sigma}\\right)\\right] ", "2be1282ef94e4d75670d42eda332fe0c": " 0 < y < \\pi \\, ", "2be182bf8836efa71beb4573659d25a0": " ds^2 = 2 \\, du \\, dv + dx^2 + dy^2, \\; \\; \\; -\\infty < u, \\, v, \\, x, \\, y < \\infty ", "2be1be3010a570679a6f93ee410c4c54": "\\langle a,b,c\\rangle", "2be1e88a11310bf78de3d98940b2c98b": "G(a_n; x) = \\frac{A(x) + B(x) (1- x/r)^{-\\beta}}{x^{\\alpha}} \\,", "2be25636d7cc3d6dc3c7bb709a780cd2": "\\mathbf{\\hat{\\mu}}", "2be28a862cdaa3313ce6f90f43102f6c": "\nH_\\alpha \\phi (\\mathbf{r}) = E\\phi (\\mathbf{r}), \n", "2be29a9b786cbebfcb441c61e43a2469": "U\\cap I'_x = \\boldsymbol{\\varphi}(I_w)", "2be2d3f3ce83f1f89e16b5a4614f115f": "N\\bar\\psi(x)ie\\gamma^\\mu\\psi(x)\\bar\\psi(x')ie\\gamma^\\nu\\psi(x')\\underline{A_\\mu(x)A_\\nu(x')}", "2be2dc442c6c9fd8447c763836b31009": "B^* \\widehat{\\,\\otimes\\,}_\\pi B ", "2be3373f705f0771d81830cd225c6dd9": " \\hat{x}=W x \\, ", "2be33bc932c138b144bd34e88897458c": "G:D\\to C", "2be3a20aaa7a57fb4e435f4d6220bb1c": " \\rho_1 = 1-z^{-1}-5z^{-2}-34z^{-3}-267z^{-4}-2037z^{-5}-\\cdots", "2be3bb09f4bd5da3bcd24262a1c323d5": "e^{\\frac{hc}{\\lambda kT}} \\approx 1 + \\frac{hc}{\\lambda kT}.", "2be3f5cdd14a18a54c7a0661e2d0e9b7": "o(f)o(g) \\subseteq o(fg)", "2be44599d3e9e754290b944b7a5c29ee": "f(t) = \\int_0^\\infty e^{2\\pi i st} \\hat f(s)\\, ds", "2be461d51fa6a86cc30a9d5418786918": "\\delta C < \\delta S", "2be4b36b3a5871a7c2dc5792dd048483": "\n \\boldsymbol{\\mathcal{E}} = \\varepsilon_{ijk}~\\mathbf{e}^i\\otimes\\mathbf{e}^j\\otimes\\mathbf{e}^k\n", "2be4f024b1704e4202f67660b01b3654": "a = r\\alpha,\\!", "2be4f2ab08607b35a3e30b3fa772fc59": "\\delta S = S - S_{0} = k_{B} \\ln{\\Omega} ", "2be4f6eb42bf891e886637313e27c621": "m~", "2be511e1a2f2fef8103dda9d8a07d5b3": "\\tau=(\\lambda +1) t", "2be51c5f7a67cf2d96e11b364f427b97": "R^\\theta(w) = \\sum_i(-1)^iH_c^i(X(w),F_\\theta)", "2be52aa5c07cd18601b8ada0ac127192": "\n\\hat{\\varepsilon}(\\omega) = \\varepsilon_{\\infty} + \\frac{\\Delta\\varepsilon}{(1+(i\\omega\\tau)^{\\alpha})^{\\beta}},\n", "2be55362e6a52b207dcadff82933093f": "z \\equiv x^r \\mod n", "2be554adabb307dbc3c99a4379754f44": "\\begin{matrix}\nV_{\\mathrm{eff}} = - mgl \\cos \\varphi_0 + m (\\frac{a\\nu}{2}\\sin \\varphi_0)^2 \\;.\n\\end{matrix}", "2be56ee27eac29eceb113b970f88a203": "\\ddot u_i=\\left(\\frac{f}{\\rho\\ {\\Delta x}^2} \\right) \\left(u_{i+1} + u_{i-1}\\ -\\ 2u_i\\right)", "2be574e26a9cb2933be68cb9027635f1": "\\frac{d^3 W}{d\\Omega d\\omega}=\\frac{e^2}{16\\pi^3\\varepsilon_0 c}\n\\left ( \\frac{2\\omega\\rho}{3c\\gamma^2} \\right )^2\n\\left ( 1+\\gamma^2 \\theta^2 \\right )^2\n\\left [ K_{2/3}^2(\\xi ) + \\frac{\\gamma^2 \\theta^2}{1+\\gamma^2 \\theta^2}K_{1/3}^2(\\xi)\\right ]\\qquad (10)\n", "2be5a767b5dddc085abf2427346a6095": "\\lim_{p\\to 0} S_p(x,y)", "2be5b256d829e5e1748ecdeb08545921": "f(\\vec{x},\\vec{z})\\rightarrow g(\\vec{x},y,\\vec{z}) -\n{\\rm for }~y\\in\\vec{x}\\cup\\vec{z}", "2be5c999dd8c12d43b8ac28160442601": "usage(l)", "2be5cd347ab50a2c415263cca16877eb": "r M \\subset I M", "2be5d3f579dca7f0259cecb061bae794": "\\frac{\\text{d}[{^0_2}S]}{\\text{d}t} \\simeq - \\frac{\\text{k}_{3(1)} E_0 {^0_2}S }{ ^0_2S + K_1 \\left( 1+ \\dfrac{{^1_2}S^\\beta }{ K_2} + \\dfrac{ {^1_2}S^\\gamma }{ K_2} \\right)} ", "2be6893302d2dee82a7e0e43ef61a824": "O(k^4n)", "2be6cda43fb50b8af01f2000b9f6c939": " \\psi = \\begin{pmatrix} {a+bi}\\\\{c+di}\\end{pmatrix}.", "2be6ceb30f983f406e4f6003b8c5ccea": " \\Delta E = \\Delta M c^2\\,\\!", "2be7012f612ea5ff480c46dcce355f12": "F_{MG} \\; = \\; G_M", "2be73ac358c64d306baec0369d514a4f": "u^* = \\arg \\min_{u \\in U} J(u).", "2be741d9ff5773e24cf8ca73993109fb": "\\scriptstyle \\frac{1}{\\sqrt{1-l}}", "2be750ed4925be71538fd1e3a8ddbabc": "\\begin{matrix}I_{\\frac{1}{2}}^{[-1]}(\\alpha,\\beta)\\text{ (in general) }\\\\[0.5em]\n\\approx \\frac{ \\alpha - \\tfrac{1}{3} }{ \\alpha + \\beta - \\tfrac{2}{3} }\\text{ for }\\alpha, \\beta >1\\end{matrix}", "2be78f7ff62dedafd5c8c422e43843f6": "T_L=\\frac{T_N}{cos(\\theta)}", "2be7dd1a80b5d300b201b672f6095734": "\\hbox{d}A_2", "2be8029c31c814bee6330993d9577008": "\\Sigma n r Q^{n - 1}", "2be841fa5da071b1a359f40ab720f008": "\\frac{\\partial u}{\\partial t} = F(u,t)", "2be84572d36b00053603b828ff809e51": "\\textstyle a_i = 0", "2be869272642dc466202dab32f80104e": " p^2+2pq+q^2=1 ", "2be98563e0e0e8f54969bc2893388e15": "RT = a + b\\log_2(n + 1)", "2bea0d70b36b0e2821653e29deeed48e": "\\gamma : \\mathbb{R} \\rightarrow \\mathbb{R}^n", "2bea291989c5827c71b1f8cd994fa647": "T_c = \\frac{2J}{k_B\\log(1+\\sqrt{2})}.", "2bea3e0dabcf074015bacab4e9cf648a": "s+t=2p+2", "2bea40709956af630f4944f36ed1de04": "h(w) = w", "2bead19b3d0b616d86904342dee36e3c": "(1-x^2)^{1/3}=1-\\frac{x^2}3-\\frac{x^4}9-\\frac{5x^6}{81}\\cdots", "2beaeabcb9ac83b3c1731e378dee8d05": "\\overline{x}_i = \\frac{1}{\\sum_{j=1}^N \\alpha_{ij}} = \\frac{1}{\\alpha_{-1} + 1 + \\alpha_1}.", "2beaf1fa7ae4501f1003675d968ce5d1": "\\rho_i = x_i \\rho \\cdot \\frac{M_i}{M}", "2beb28c6781e92453cb5f66af25debf3": "\\frac {1}{Z_0} = \\sqrt {\\frac {1}{{R_1}^2} + \\frac {2}{R_1 R_2}}", "2beb5cbf95d916747cf5f2cfdae4815b": "Q_{ii} = - {\\sum_{\\lbrace j \\mid j\\ne i\\rbrace} Q_{ij}} \\,,", "2bebaf10a54d0ac5d8adb0603f42af21": "10\\log_{10}\\left(\\tfrac{1}{2}\\right) \\approx -3.0103\\, \\mathrm{dB}", "2bebfb5fa47b2e9e9fc2b39dc64c7343": "Z_1^{p,q}", "2bec19f500df74a2f8b05677e162d56c": "\\frac{1}{+ \\infty} = 0", "2becd9ab1c0b284e915bf01f69b91893": "x\\mapsto 2x-1", "2bece9482b2ce39965793d41375a259e": "\\dot{\\tilde{\\mu}} = D\\tilde{\\mu}\\Leftrightarrow \\partial_{\\tilde{\\mu}} F(\\tilde{s},\\tilde{\\mu}) = 0\\Leftrightarrow \\delta_{\\tilde{\\mu}} S=0", "2bed40c8b3b728f1a71aea3e3ccbc008": "\\beta (t) ", "2bed5864f7150bc296be14315dfe6ec2": "\\text{PI}=\\frac{height}{\\sqrt[3]{mass}}", "2bedd1a073144979a3c643ec228edd03": " \\langle x | \\hat{p} | x' \\rangle = -i \\hbar {d \\over dx} \\delta (x - x') ", "2beddd0b48d9a43a26c422172cff56df": "\\scriptstyle x_2 = \\frac{B^2 - k}{A} < A", "2bee4e720ec41536f104295753dbd16e": "q_1 \\vee (q_2 \\wedge q_3)", "2bee75a7d51df79351610adbc0b76461": "u_0\\in H", "2bef3ac8ec18d728447f8828d3b82cc3": "g(x+y) = g(x)g(y)-f(y)f(x)\\,\\!", "2befac151f6311713146a5f1d429cee2": "\\omega\\gg \\omega_\\text{c}", "2beff7e18032cb272a113afd977ac45e": "v_\\mathrm{max}=\\sqrt{2\\left({P_\\mathrm{atm} \\over \\rho}-gh_B\\right)}", "2bf03b334535d4019bc89f2db8f02a08": "H^0(M,\\mathcal{N}) \\to H^0(M,\\mathcal{N}/\\mathcal{I})", "2bf05ae457a41489624759db015746e4": "x^2 \\frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0.", "2bf0794e4aa228c23e49e7efc3e92d25": "v \\in R^{2^n -1}", "2bf07c42a4d90e2efa8edecbf2fbf298": "\n \\lim_{x_i \\to x_{i-1}}\\frac{\\int_{x_{i-1}}^{x_i}f(t;\\theta)dt}{(x_i-x_{i-1})} = f(x_{i-1},\\theta) = f(x_{i},\\theta), \n ", "2bf0812361f619f1c48392edaab7409d": "a_{n-1}+2\\,a_{n-2}\\,u+\\cdots+(n-1)\\,a_1\\,u^{n-2}+n\\,a_0\\,u^{n-1}", "2bf08d58651a1e3501948d3313f3bb04": "w(n,3) = w(n,2) + w(n-1,2) + \\cdots + w(1,2) + w(0,2)\n", "2bf11835bb7576b96adab4ba07e1360f": "x\\ \\dot{=}\\ 0 \\rightarrow [x++;]x\\ \\dot{=}\\ 1", "2bf1710ff14a71f883277287a4c3bd7c": "\\lim_{t\\to 0}\\frac{\\|x+ty\\|-\\|x\\|}{t}", "2bf194991177aa461176c3447a5f46d7": "X \\cup \\{\\alpha\\}", "2bf255f9e13aa8e60c4c39bf86b84972": "X_{jk}", "2bf2610882186bae3993423f0cc2d739": " \\operatorname{Var}[X] = np(1 - p).", "2bf2752c0d33f5bda67a1036e4e8c605": "vD=v'", "2bf36c6b57bd1e0e9c642fbd5c6797e6": "S=\\bigcup_i A(x)\\oplus t_i", "2bf41ecef0bb04ca6b9f4fac02f5cec5": " \\Delta p_x \\Delta x \\geq \\frac{\\hbar}{2}, \\quad \\Delta E \\Delta t \\geq \\frac{\\hbar}{2} ", "2bf420771a697d9c37f4c035b631fb4c": "I_2(3) \\cong A_2, I_2(4) \\cong BC_2,", "2bf42146dae269a8b7e505ceeaac1d9a": "\\scriptstyle \\deg(P) \\;<\\; n \\;:=\\; \\sum_j\\nu_j", "2bf4297c50597f8b1383824ad74cd81c": " -2~r^{-3}~\\cos\\theta \\,", "2bf49b7d5123a28113753775ec1eb4ef": "x=(...,2,3,1,2,3,1,...),", "2bf4c298ef0242c9d1388e84402ff621": "V'_{a}:=\\bigcup_{b\\in B_0(a)} V_{a,b}\\supset B", "2bf4d3dfabb6cf0dafe0cc5799791029": " {\\Vert \\mathbf u \\Vert}^2 = \\mathbf u \\cdot \\mathbf u", "2bf5170d06f2e0b88da03a254960ca4f": "\\textstyle k, q \\geq 0", "2bf5de398a5e23676a13343eb9f2acc8": "f(x) = (1+x)^{\\deg(p)}p \\left (\\frac{a+bx}{1+x} \\right )", "2bf5f178bb2d5b5036d400df9e05f9ef": "X \\sim {\\chi}^2(\\nu)\\,", "2bf604bd3ffc65c4d9b6e05538b33e13": " U^{\\{ a,b\\} } =\\left[\\begin{array}{cccc} {1} & {0} & {0} & {0} \\\\ {0} & {1} & {0} & {0} \\\\ {0} & {0} & {1} & {0} \\\\ {0} & {0} & {0} & {-1} \\end{array}\\right]", "2bf62329ab763e07e3bbe327a5d1823c": "\\lfloor x \\rfloor \\le \\lceil x \\rceil,", "2bf628776c4a56f890f2c9890d5ec83b": "r^4+18a^2r^2-27a^4=8ar^3\\cos 3\\theta\\,.", "2bf6381f5058873761579f20200faf9f": " x_\\mathrm{m} e^Y", "2bf63999a2fdec284f12b1824dc22c43": "Y^{S}(L^{D})", "2bf6b55807e0d7af64caf487a4d75648": "\\Delta V^{\\ddagger}", "2bf72a2a691adf1e4cf3d2c6111be74b": " \\mathbf{E} = - \\nabla \\varphi - {\\partial \\mathbf{A} \\over \\partial t}\n ", "2bf7398499fd6791f07852822c09d4d2": "\\|f\\| = \\left(\\int_0^\\infty |x^\\nu f(x)|^p\\, \\frac{dx}{x}\\right)^{1/p} < \\infty", "2bf755f732360500d9ec0401645688c3": "G = 16 \\left(\\frac{L} {\\pi d^2}\\right)mf_t^2", "2bf79013ba38cc1591d94f31562f7a54": "u=\\frac{dx}{dt}=\\frac{\\gamma \\ (dx'+v dt')}{\\gamma \\ (dt'+v dx'/c^2)}=\\frac{(dx'/dt')+v}{1+(v/c^2)(dx'/dt')}=\\frac{u'+v}{1+u'v/c^2} \\ .", "2bf7a113f99e737bc67f3abb681a355f": "\\rho_{2} = 1.69202 ", "2bf7be75bdf9d7425f833044d65a9f8e": "A\\Rightarrow\\bot", "2bf7ff1eb23abae18f8fec9d7e82f779": "E \\{ \\hat{x} - x \\} = 0.", "2bf800e0fc5a704a732d355a81b0dc52": "\\,g(X,Y) = g(Y,X)", "2bf8012257927b54d6fb0f1206a5fe75": " \\mu_{3,2}= \\mu_{3,2} - 1 = \\frac{-1}{14}(< \\frac{1}{2}) ", "2bf8185ea517c9b4c58e03c5e26e014a": "N(\\bar{X},s^2)", "2bf87c19918203a1153232e72ebce9ae": "u-u_{xx}-u_{yy}=f(x,y)=xy", "2bf892c4d85f9de80213baf925f191c3": "\\rm{bit}(a,b)=(\\rm{DTC}_{a,b,a',b'}\\psi)(a,b,1,0)", "2bf930105109c98fc46b6fe0a10b9faf": "\\frac{\\lambda^{2}}{4\\pi}", "2bf931457015fdf95c8d60522c677a41": "(\\rho^{\\prime}/\\rho)<1", "2bf95ec6edaa424dfbbf7c41d1e20f15": "a\\approx\\ell+\\frac{s^2}{r}", "2bf96b99ca6a467c2dc8df8d85df9f74": "s_M=\\sum_{i=1}^m x_i; s_{\\Sigma}=\\sqrt{\\sum_{i=1}^m(x_i-\\bar x)^2}", "2bf9c55c0d42761c3ecf2309893a300f": "\\lim_{x\\to 0^+} x \\ln x =\\lim_{x\\to 0^+}{\\frac{\\ln x}{1/x}}\n=\\lim_{x\\to 0^+}{\\frac{1/x}{-1/x^2}}\n=\\lim_{x\\to 0^+} -x= 0.", "2bf9fcf9df073f9cfadc502b9900632f": "d t/d \\bar{t}", "2bfa2b93037eab9cea0ad30ed0bd7963": " f'(c) = \\frac{f(b) - f(a)}{b-a} \\, .", "2bfa68e474f438db01ced9e22bb0415f": "[\\omega\\wedge\\eta]", "2bfac60e6407de588492717cdd341220": "\\phi(v_i)=U(v_i)\\tanh(v_i)", "2bfbd732ae73cf470ace012aca9c78ca": "N(d,d)", "2bfbdf5fb68c1dadc0838b279fe80e3a": " \\lor ", "2bfc525087db7ea9be25a6902144be0a": " (\\partial U)_T=-(\\partial T)_U=T\\left(\\frac{\\partial V}{\\partial T}\\right)_P+P\\left(\\frac{\\partial V}{\\partial P}\\right)_T", "2bfc54f42c06025494560dcc4b6ee04a": "\\dot q = -\\mathcal K \\mathbf J \\cdot \\boldsymbol \\nabla T", "2bfc8c413f736ccdf2f49714ee614153": "a_i=0", "2bfcbb45c67df4bed29cc68170f0dd7b": " \\frac{dX}{dt} = \\Delta_T \\exp \\left( \\frac{X - X_T} {\\Delta_T} \\right) ", "2bfcbd03e05c7feb87f44437afb11c8c": "s(t) = \\int_a^t\\sqrt[3]{\\det\\begin{bmatrix}\\frac{d\\beta}{dt} & \\frac{d^2\\beta}{dt^2}\\end{bmatrix}}\\,\\,dt. ", "2bfce324e9d3a3f1d2f0916c339773ce": "\n\\mathbf{D}^{-1}+\\mathbf{D}^{-1}\\mathbf{C}(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1}\\mathbf{BD}^{-1} = (\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\,\n", "2bfd5d7d12afa6e00447050caba22823": "\\Psi = \\frac{1}{\\sqrt{2}}(\\phi_1 - \\phi_2)\\,", "2bfd720cc91e19cc705ee2faf0d46154": "ATP + H_2 O {\\rightleftharpoons} ADP + P_i + heat", "2bfde2a56f30f65444ff4dbb38622892": "\\ y_2 ", "2bfe14c74dcb33be13b9ff27678bfc5f": "k(u^2+v^2)", "2bfe358894372fb714028fff18e25ebd": "\\operatorname{corr}(dZ_{1}, dZ_{2}) =\\rho", "2bfe82416f4c13e8e75ec839071d7657": " \\vdash P(x) ", "2bfed8fd7d2a743614815f532f3e682d": "H_p = \\frac {(273.15 + T)}{1000} \\cdot (C + \\log(t)) ", "2bfefbbe691dcf5f2dc51292191c1567": "\n \\left(\\hat{C}^{(1)}\\right)^\\dagger = (\\hat{H}\\hat{A})^\\dagger-(\\hat{A}\\hat{H})^\\dagger\n = \\hat{A}\\hat{H} - \\hat{H}\\hat{A} = -\\hat{C}^{(1)}.\n", "2bffdbf157436f175f5ead26fabc2fbe": " f_x = \\frac{ \\mid \\mathbf{E} \\mid^2 \\cos^2\\theta }{ \\mid \\mathbf{E} \\mid^2 } = \\psi_x^*\\psi_x = \\cos^2 \\theta ", "2c000305517c9244527a43667d67a98c": "s\\!:\\!\\sigma", "2c0027346876a3309d9cf7d77ecf65c5": "\\begin{align}\np\\\\\nq\\\\\n\\therefore \\overline{p \\wedge q} \\\\\n\\end{align}", "2c002df67f3d0d255056be9376dec760": "\\{\\Omega_a(A),\\Omega_b(A)\\}", "2c00c9e82047dfcfcbbf4b2750101f00": "T_p(x) = (-1)^{p/2}\\ F(x)\\,", "2c0119bb0bcc39bcdb5f235dfd5310e7": " p(x,t) ", "2c01ef8263089042b4e92b3de3e55255": " \\beta_c^{XY} \\ge \\ln(1 + \\sqrt{2}) \\approx 0.88~.", "2c027dcaa7c40262034dbf88ddb9c96d": " \\lim \\int f_n \\geq \\alpha \\int \\phi ", "2c02f21e9b6c08653463fe9ac84ac684": "N(H)= C \\cdot \\frac{H(\\log H)^6} {4\\times 6!} + O(H(\\log H)^5).", "2c03500540cb421119152569b98f8f02": "C_3 = G_2 + P_2 \\cdot C_2", "2c0350f64e7bc0893b7654bab54cd626": "d\\Gamma = \\frac{ \\left| \\mathcal{M} \\right|^2}{32 \\pi^2} \\frac{|\\vec{p}_1|}{M^2}\\, d\\phi_1\\, d\\left( \\cos \\theta_1 \\right). \\,", "2c03998b19d0d35c66d7a211c81340ae": "\n\\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\\\\nC_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\\\\nC_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\\\\nC_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\\\\nC_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\\\\nC_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \\end{bmatrix} = \n\\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & -C_{14} & -C_{15} & C_{16} \\\\\nC_{12} & C_{22} & C_{23} & -C_{24} & -C_{25} & C_{26} \\\\\nC_{13} & C_{23} & C_{33} & -C_{34} & -C_{35} & C_{36} \\\\\n-C_{14} & -C_{24} & -C_{34} & C_{44} & C_{45} & -C_{46} \\\\\n-C_{15} & -C_{25} & -C_{35} & C_{45} & C_{55} & -C_{56} \\\\\nC_{16} & C_{26} & C_{36} & -C_{46} & -C_{56} & C_{66} \\end{bmatrix}\n ", "2c03a6b07f4c6b56fb385ba6d9c2d196": " \\operatorname{E}(X f(X - 1)) = \\lambda \\operatorname{E}(f(X)). \\,", "2c03ade89925939a19e813c50f27e401": "\\sum{\\sum{ (L(r,c) - R(r,c-d))^2 }}", "2c03d11e5684684dfbe65a4402d50a4e": "\\frac{x}{a} + \\frac{y}{b} = 1,\\,", "2c03ea2346c55356e480bc01d48de135": " L(h)", "2c043520e007fe24a1a1871f8759b9d0": "\\scriptstyle (X',\\tau_1',\\tau_2')", "2c044a0cdf068d1cfaaa3e2e09cbbe24": " \\frac{d}{dx}\\ \\operatorname{sech}\\,x = - \\tanh x \\ \\operatorname{sech}\\,x \\,", "2c045f888901740a66203789dfb683af": "Z\\left(t\\right)", "2c0557fba5188bde502ec50a974169ca": "\\mathrm{rad}_t", "2c05d449132f0039a886cb069a81be2f": "\\displaystyle{\\mu(x,y)={E-G +2iF\\over E+G +2\\sqrt{EG-F^2}}}", "2c062681340a343756287012f65a0989": " \\Omega^1_X(\\log D) ", "2c063369eec89ccd6e0de4a4ef07d965": "(4,2,1)_H", "2c06428026e33fd742f07b33b97ce2c2": "x + y = (x_1 + y_1, x_2 + y_2, \\ldots, x_n + y_n) \\,", "2c06962451cf79e79f6c58148986377c": "\\textstyle q:={1+\\alpha}", "2c076167365c419d18df41d26a982bc6": "\nu = u_{1} + \\left( u_{2} - u_{1} \\right) \\, \\mathrm{sn}^{2}\\left( \\frac{1}{2} \\varphi \\sqrt{r_{s} \\left( u_{3} - u_{1} \\right)} + \\delta \\right)\n", "2c0774a5436d9dee6431cfa765662b98": "\\scriptstyle \\frac{1}{2}", "2c07bc3653d2a56186e8ecd03ca35c2f": "E_u(x,y,z)=\\int\\!\\!\\!\\int E_u(k_x,k_y) ~ e^{j(k_x x + k_y y)} ~ e^{\\pm j z \\sqrt{k^2-k_x^2-k_y^2} } ~ dk_x dk_y ~~~~~~~~~~~~~~~~~~(2.1) ", "2c07f3bfc2cd5078b3d7263ff71701d1": " \\ v_1+u_1=u_2+v_2", "2c082054eb1b702c057306449cc34619": "\nK(x-y;T) \\propto e^{i(x-y)^2 / (2T)}\n\\,", "2c085d4c2b058b6d07b90e2d8c1be954": "\\displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a).}", "2c087bc74cc7732cd1b737243e6b7e73": "\\mathrm{Pr} = \\frac{\\nu}{\\alpha} = \\frac{\\mbox{viscous diffusion rate}}{\\mbox{thermal diffusion rate}} = \\frac{c_p \\mu}{k}", "2c08aa7867e126c4d08be8b07180a3ad": "\\text{SG at}~60^\\circ\\text{F} = \\frac{141.5}{\\text{API gravity} + 131.5}", "2c08e4a9e71ab6a358753a585256c6e9": "y_0\\in\\bigg[\\ x_0'\\hat\\beta \\pm q^{\\mathcal{N}(0,1)}_{1-\\alpha/2}\\!\\sqrt{\\tfrac{1}{n}\\hat\\sigma^2x'_{0}Q_{xx}^{-1}x_{0}}\\ \\bigg]", "2c08f6201280b147922c390596239426": "\\frac{\\mu}{\\sigma} ", "2c09218dde3b37f0e479ef2e63c9559f": "f^*(x):=\\sup_{n\\geq1}\\frac{1}{n}\\sum_i^{n-1}|f(T^i(x))|.", "2c094e6095e36c33f080a9fd348b0cd6": "\\alpha = a/(a+b)", "2c09a79e42676a0caf57f0130372b772": "a \\leq x \\leq b", "2c09b56d166e2f375c8dc6f38a27b7e0": "a=i", "2c09f8da9ed95956ea57b8456763146d": "A_1 \\cong B_1 \\cong C_1", "2c0a08333126736f61043882adc64986": "E[XZ] = E[U] = 0", "2c0aa3d0054608b0fe74c379e8ed9627": "\\frac{G}{C} \\ll \\frac{R}{L}.", "2c0b0b9c912b6294628a338dff587901": "\\pi_1(M)", "2c0b64b8b374096ecaee3d4733aef7c1": "\\{\\tilde\nJ_k\\}=\\beta(\\{ J_k \\})", "2c0b80b190db817b09432b3718800cf2": "\\begin{align}\nc^2 & {} = (b-a\\cos\\gamma)^2 + (a\\sin\\gamma)^2 \\\\\n& {} = b^2 - 2ab\\cos\\gamma + a^2\\cos^2\\gamma+a^2\\sin^2\\gamma \\\\\n& {} = b^2 + a^2 - 2ab\\cos\\gamma,\n\\end{align}", "2c0bcf2d5fb15c5ff2ef2fea05c930d9": "(-i\\gamma^\\mu\\partial_\\mu + m) \\psi = 0\\,", "2c0c035daeb4f2bbb1b1d19f4144759f": "p_{\\alpha,y}:V\\to\\mathbb{R}:x\\mapsto p_\\alpha(x-y)", "2c0c1202d98a9047b61bb67d38fccf95": "S*(T*\\varphi) = (S*T)*\\varphi", "2c0c4d1195d69a42fd865ef1122bcfb0": "|\\varepsilon|_\\nu", "2c0c8d88d7fd03070c14e0913635c45d": "[0,c)", "2c0cb09fa6b315609f7bbd2373a9aaf7": "m_n=E(z^n)= C_n +i S_n = R_n e^{i \\theta_n}\\,", "2c0cc1895f01391642f951a47a0ef391": "\\boldsymbol{\\mathsf{E}}=\\tfrac12 \\left( \\nabla\\mathbf{v} \\right) + \\tfrac12 \\left( \\nabla\\mathbf{v} \\right)^\\text{T}", "2c0ccc2f971c90e418e74fad20e879fb": "\\frac{1}{2}\\frac{\\partial f}{\\partial a_{01}}=", "2c0d2e20f14bb0e745747303d69cd3a6": "r^{\\frac{3}{4}} \\exp(\\tfrac{0 \\pi i}{4}) (3-r)^{\\frac{1}{4}} \\exp(\\tfrac{2 \\pi i}{4}) = i \\, r^{\\frac{3}{4}} (3-r)^{\\frac{1}{4}}", "2c0dad19d8f269e56274001029452473": "A_1, B_1, A_2, B_2, \\cdots A_n, B_n", "2c0e0a1c9b5c05c38368ea86c09a3852": " [ b^{-1} , a^{-1} ] ", "2c0e21746edf2873e85ac93c14f34edb": "W_j", "2c0e395e0af98cfe9fff7602e22547cc": "(a,b,c,d)\\times (w,x,y,z)=(aw-bx,ax+bw,cy+dz,cz+dy).", "2c0ec45cc34f9a29aef64ab855737776": "\\bar{W_{U}}\\subseteq U\\,", "2c0ef0cb9e173767760641004f41caf1": "q(q-1)/2 = O((\\log Q)^2)", "2c0f4b62fb7390dc382dcdec900558e6": "x^{**} \\in B_{X^{**}}", "2c0f6b8291773ba2ed97abe4ec8d7945": "u_0\\equiv\\,u.", "2c0f89e6d48dd8fceb24176f7e971cb7": "\\ \\tau_F", "2c0fccd60188b4a85730baac41696c9d": " \\delta_{ }^{ }[c] = [a].\\, ", "2c0ff73e8c6a6a4c8893e93aa3427982": "\\ \\varepsilon_t", "2c101b6540af71f066539a848e87c071": " \\lambda=\\max_{i>0}\\{\\mid \\lambda_i \\mid \\}", "2c106632412bb48b513bb3d57f09f556": "\\overline{P}(Cl_t^{\\leq})", "2c10837c9b824863c6e15ce9f1b286b5": "\\sum_j P_{i,j}=1.\\,", "2c10b5ef113ecabfb09923a13320437c": "\\Lambda_g=\\pi_*\\omega_g.\\,", "2c1101abee13be81daa3d2a7e9a415fa": "\\mathrm{Adj}(A)A = \\mathrm{det}(A)I\\,", "2c118a2f2db31ed08394ca8b53e04930": "v\\otimes v - Q(v)1", "2c119c64bce7cb809cac7d02084c25a6": "\n \\hat\\beta\\ \\xrightarrow{p}\\ \n \\frac{\\operatorname{Cov}[\\,x_t,y_t\\,]}{\\operatorname{Var}[\\,x_t\\,]}\n = \\frac{\\beta \\sigma^2_{x^*}} {\\sigma_{x^*}^2 + \\sigma_\\eta^2}\n = \\frac{\\beta} {1 + \\sigma_\\eta^2/\\sigma_{x^*}^2}\\,.\n ", "2c11ab20bc79d4c467655df1977ca28e": "\\Delta F(A \\rightarrow B)\n = \\int_0^1 d\\lambda \\frac{\\partial F(\\lambda)}{\\partial\\lambda}\n\n = -\\int_0^1 d\\lambda \\frac{k_{B}T}{Q} \\frac{\\partial Q}{\\partial\\lambda}\n\n = \\int_0^1 d\\lambda \\frac{k_{B}T}{Q} \\sum_{s} \\frac{1}{k_{B}T} \\exp[- U_s(\\lambda)/k_{B}T ] \\frac{\\partial U(\\lambda)}{\\partial \\lambda}\n\n = \\int_0^1 d\\lambda \\left\\langle\\frac{\\partial U(\\lambda)}{\\partial\\lambda}\\right\\rangle_{\\lambda}", "2c11b8539de53c8671a9b24a868eb590": "T \\propto H", "2c11fb54cfe0e56c7318829492ecf9fd": "\\, G(1+z)=\\Gamma(z)\\, G(z) \\,", "2c128f10ed5541a6b9a7330f6af2e180": "\\bar{\\lambda} <1", "2c12c966683017bc6c0d3026f5158e6c": "\\nabla_{\\bold{v}} f(\\bold{p}) = \\left.\\frac{d}{d\\tau} f\\circ\\gamma(\\tau)\\right|_{\\tau=0}", "2c12ce0448c8893c8d85aa2736662a11": " \\dot{x_1} = \\frac{dx_1}{dt} = f_1(x_1, \\ldots, x_n) ", "2c1308e5475f3719687b2b4a0ae2d404": "\\scriptstyle{v_{LZ}}", "2c13603c78800c4bdf805cfd443a3bc2": "AB^2 + AC^2 = 2(AD^2+BD^2).\\,", "2c13856050ba584064590c04546ddf19": " M^1(B)=\\Lambda(B), ", "2c13909cb89f194e20af11d9148ec5b8": " \\hat{p} \\pm z_{\\frac{\\alpha}{2}} \\sqrt{ \\frac{ \\hat{p} ( 1 -\\hat{p} )}{ n } } .", "2c139afba036d00232f79aadb425c395": "\\ Z_L = j\\omega L", "2c1428ac121aa3bec21fa6c9ce23070e": "[\\mathbf{X}-\\operatorname{E}[\\mathbf{X}]][\\mathbf{X}-\\operatorname{E}[\\mathbf{X}]]^T", "2c145f55cb97c1a52b05c5a99a22ac85": "\\frac{\\partial}{\\partial t}\\left(\\nabla^2 \\psi\\right) + \\frac{\\partial\\left(\\psi, \\nabla^2\\psi \\right)}{\\partial\\left(y,x\\right)} = \\nu \\nabla^4 \\psi.", "2c1460ad0a6aeee97dfac15be62f5bc9": "dx^i\\wedge dx^j\\wedge dx^k.", "2c149919e989da08fa3c9986049f258a": "\\frac{1}{2}mv^2,", "2c14e2c5530e88d93d1f572f5169d361": " \\frac{dR_{T}}{dt} = -s_{Oy}-s_{Ty} \\,\\!", "2c14fc57334c9afc555bc854fe67696c": "\\rho = {R^2 + L^2\\over 2R}", "2c14ffcfcdf5a2c4a2ea25cb8279a2b9": "N(x)=[y_k]+[{y}_{k}, {y}_{k-1}](x-{x}_{k})+\\cdots+[{y}_{k},\\ldots,{y}_{0}](x-{x}_{k})(x-{x}_{k-1})\\cdots(x-{x}_{1})", "2c16007b85d2291d99ef76964c3e8930": " z = \\langle x, y \\rangle = \\frac{(x + y)(x + y + 1)}{2} + y ", "2c16366b8df3bd877a2877e7b678f073": "D_{ik}=\\frac{1}{T}\\sum_{j\\geq 1} L_{ij} \\left.\\frac{\\partial \\mu_j(n,T)}{ \\partial n_k}\\right|_{n=n^*}", "2c167c7385b6101f84c082d165142219": " |\\phi_1 \\cdots \\phi_N \\rang = \\sqrt{\\frac{\\prod_j N_j!}{N!}} \\sum_{p\\in S_N} |\\phi_{p(1)}\\rang \\otimes \\cdots \\otimes |\\phi_{p(N)} \\rang,", "2c168833791aaa77c4f7cbee455cc73a": " E_0 = mc^2 = \\hbar \\omega_0 ", "2c16ced613a829b017f0fb30c6929f55": " \\Delta w_i = \\alpha (t - y) \\varphi' x_i ", "2c170aead522a2f53612d5fa4c8e4a2e": "\nw(n,2)=\\frac{(n+1)!}{n!1!}.\n", "2c17344694f966eac49055de7008ac6c": "a \\times n = \\underbrace{a + a + \\cdots + a}_n", "2c1743a391305fbf367df8e4f069f9f9": "alpha", "2c177cd09a388153f245143138986bc6": "C^k(\\mathbb{R})", "2c17a7a7b07f2ee66d60210e94c9c656": "C = 2\\pi a \\left[{1 - \\left({1\\over 2}\\right)^2e^2 - \\left({1\\cdot 3\\over 2\\cdot 4}\\right)^2{e^4\\over 3} - \\left({1\\cdot 3\\cdot 5\\over 2\\cdot 4\\cdot 6}\\right)^2{e^6\\over5} - \\cdots}\\right]", "2c17e8d531d9ef49fbe86d3ec6bb6143": " \\mathbf{p}_{k} = \\mathbf{r}_{k} - \\sum_{i < k}\\frac{\\mathbf{p}_i^\\mathrm{T} \\mathbf{A} \\mathbf{r}_{k}}{\\mathbf{p}_i^\\mathrm{T}\\mathbf{A} \\mathbf{p}_i} \\mathbf{p}_i ", "2c18715627f0fe6e203e12a59386ac31": " R_i= \\frac{D_i}{\\sqrt{8\\pi W}}, ", "2c18fe9ab9491723b2c9ca89215ba2c8": "\n \\cfrac{d}{dt}(\\boldsymbol{\\sigma}_r) = \\dot{\\boldsymbol{\\sigma}}_r = \\dot{\\boldsymbol{Q}}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{Q}^T +\n \\boldsymbol{Q}\\cdot\\dot{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{Q}^T + \\boldsymbol{Q}\\cdot\\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{Q}}^T \\ne \\boldsymbol{Q}\\cdot\\dot{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{Q}^T \\,.\n", "2c194672d0edcaf2bc7651cb3975d4e4": "\\sqrt{(x_0-x_1)^2+(y_0-y_1)^2}", "2c199db44d42fd296744de4895351024": "\\pi(a)", "2c1a190670191ff9f9f9c04c83cd1d0a": "\nM \\left ( n + 1, 1, -{k^2 \\over 2 } \\right)\n.", "2c1a3767a15777e0a7539aee4aebcac3": "0 < m < n", "2c1a631c77538279bf7bd00cd731edaa": " C_\\varepsilon( v ) = \\left\\{ x \\in \\mathbb{R}^N: \\sum_{ i \\in N } x_i = v(N); \\quad \\sum_{ i \\in S } x_i \\geq v(S) - \\varepsilon, \\forall~ S \\subseteq N \\right\\}. ", "2c1aa81ca67482c62facbc107aba9788": "\\int f_i''(x) f_j''(x)dx", "2c1ab6fb9e412f83c40be302216da1c0": " y(t) = y(t_0) + \\int_{t_0}^t f(\\tau, y(\\tau)) \\,\\textrm{d}\\tau, \\, ", "2c1b0381d632f2d7aef58a12303ccbde": "\\frac{\\Gamma \\vdash a : X/Y \\qquad \\Gamma' \\vdash b : Y}{\\Gamma; \\Gamma' \\vdash ab : X}[/E]", "2c1b41020be180415d652e84899fe477": "(v_e-v_e^{\\prime})[H^+]_{0^{ }} = 2v_0[CO_2]_0", "2c1b4add8a56402bb1f5543cb841fc6a": "P_v(N,t)=\\frac{N\\cdot x(N)(1 - (1 + \\frac{r}{N})^{-Nt})}{r}", "2c1b8d1328fb80cb1367bb171aff59de": "Y=G+i\\omega C\\,\\!", "2c1b9064d942a7318b127786d40b69cd": " r_2 ", "2c1c15617e9e78ac69413dd02eab8b89": " C_{n+1} =\\sum_{k \\,\\le\\, n/2} 2^{n-2k} {n \\choose 2k} C_k. \\,", "2c1c26efdc44f14609299b98b273cc55": "[0,3] \\times [0,3]", "2c1c7b0ff0b6a6dadb74ab93f8f5c896": " r = \\sqrt{(x^2+y^2)}, \\quad \\theta = \\arctan(y/x)", "2c1cd22f96e3d744a59d699779da9d19": "x^y = x[x,y].\\,", "2c1cf3f2ebd04b4baf464f933bb0859c": "\\frac{d^2\\beta}{dt^2}-\\left(\\frac{Y_\\beta}{mU}+\\frac{N_r}{C}\\right)\\frac{d\\beta}{dt}+\\left(\\frac{N_\\beta}{C}+\\frac{Y_\\beta}{mU}\\frac{N_r}{C}\\right)\\beta=0", "2c1d47f5c1f1e9153841481fb599b259": "\\sqrt{c}\\, W\\left(\\frac{t}{c}\\right)", "2c1d4e87fc975c6402b257b0f87afe23": " \\hat{T} \\Psi = \\frac{-\\hbar^2}{2m}\\nabla\\cdot\\nabla \\Psi \\, \\propto \\, \\nabla^2 \\Psi \\,.", "2c1d84fa405ee001d4e3860da274cfa2": "F = \\sigma T^4 (\\tfrac{R}{D})^2", "2c1e5ac65e8a877aad1c6b6a81d4d18b": "diag: \\mathbb{R}^k \\longrightarrow \\mathbb{R}^{k\\times k}", "2c1e6cb923ab7b5d63ce41ed77fe8c36": "A\\wedge B", "2c1e8ab42b1241a34f099d4ff15a6f9c": " \\mathbf{B} = \\frac{\\mu_0I}{4\\pi}\\int_{\\mathrm{wire}}\\frac{d\\boldsymbol{\\ell} \\times \\mathbf{\\hat r}}{r^2},", "2c1eaab59bd8f77c28a26fe27f23dfcc": "\\, \\varphi_{tt}- \\varphi_{xx} + \\sin\\varphi = 0.", "2c1f1dcaecbc63ffd231734441f33e64": " Z = P\\left( \\bigcap_{ i = 1}^2 ( - k_1 < X_i < k_2 )\\right) ", "2c1f4522424e6bfdb1180aee2f7d1a42": "\\vec{s} \\in \\mathbb{R}^{n}", "2c1f5c6e11aa600cf167539faf06f016": "\\gamma \\in \\mathcal{O}_k, \\;\\;n\\gamma\\not\\in\\mathfrak{p},", "2c208267c2a9b51905a10ac27b04f93e": " C_x = \\frac{\\sum C_{i_x} A_i}{\\sum A_i} , C_y = \\frac{\\sum C_{i_y} A_i}{\\sum A_i}", "2c2188cc5cb5ecd3d8d8993ab533bcbc": "e = (u, v)", "2c21b60fb684bf3d5b795bdb4ab5fc58": "\\bigl( \\begin{smallmatrix}\\\\ 2&-4\\\\ 1&-3\\end{smallmatrix} \\bigr)", "2c21ed250a1fc217f1eb7fd981fb9fa1": "Z = X \\times Y", "2c21f2db1b68c54b2cac81726e89d016": "\n\\begin{align} \n q_1 &= \\frac{1}{2}\\sqrt{1 + A_{11} - A_{22} - A_{33}}\\\\\n q_2 &= \\frac{1}{4q_1}(A_{12} + A_{21})\\\\\n q_3 &= \\frac{1}{4q_1}(A_{13} + A_{31})\\\\\n q_4 &= \\frac{1}{4q_1}(A_{32} - A_{23})\n\\end{align}\n", "2c221dc931261ec6ad750ede5d6b72dd": "\\nabla(z) = 1-2z^2, \\, ", "2c222e9133fdcc64d5a8fc2b510c582f": "\\exists n P(r \\ge k ; k n_2, \\frac{1}{n_2})", "2c2952a4a0116edc22dd4d2f7385e99d": " \\psi_{\\pm}(-{\\mathbf{r}}) = {} \\pm \\psi_{\\pm}({\\mathbf r}) \\; . ", "2c29603fe026110bba0a67b4d05a3837": "\\nu(dx) = \\left( \\frac{C_1e^{-\\lambda_+x}}{x^{1+\\alpha}}1_{x>0} +\n\\frac{C_2e^{-\\lambda_-|x|}}{|x|^{1+\\alpha}}1_{x<0}\\right) \\, dx,\n", "2c2979956ceb701da79d1ab9e3f8e13e": " \\left(\\frac{m (Rt / k)^{0.5}}{p D^2}\\right)", "2c2981f7ba2101c2e62d0f648cb13c15": " U_2(x) = 4x^2 - 1 \\,", "2c2a0e8278f776f73c6a08f5b5f7ec07": "k=n_1-n_2", "2c2a230391a1e25b4728e0a7a6fd0a01": "\\sigma_\\xi(D)", "2c2a26e91717953a25b688debe70c973": "A~\\Delta~B", "2c2a5ee005766d3896affbf4c5c0ad00": "\\gamma_1+\\gamma_2+\\gamma_3=0", "2c2a63be35accc18fc43d284aa4b41a7": "k_4", "2c2a64438d27eb3bc5a0d54e3280c956": " \\leq\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\left(\nI-\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\right) \\rho_{X^{n}\\left(\nm\\right) }\\right\\} +\\left\\Vert \\rho_{X^{n}\\left( m\\right) }-\\Pi\n_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}\n^{n}\\right\\Vert _{1}\\right\\} ", "2c2aa48a2264392c2434c3d3a2fc9c87": "T_{tot}", "2c2aefc45b3cdf3b0b5eaf0ff8933ace": "L=1\\hbar", "2c2b0ef656d724f6a2bea7ab3d8c5f63": "I_{H_B}", "2c2b25295f59b1205db9ff692b39606b": "\\frac{x^2}{2}\\,", "2c2b65e257cf2ac7d8b76373b3bb9d7f": "a_R", "2c2bdb18737a8870f39854144cb8f708": "x=As+n", "2c2be74850deb0b85ec7810135f26f30": "C \\in \\mathbb{R}^n", "2c2c79ddd484f109b808d5152c8200f2": "\\sum_{n=1}^\\infty\\frac{1}{n^r}", "2c2c882338222096379822d3b80c13c7": "\\textbf{z}_{k} = h(\\textbf{x}_{k}) + \\textbf{v}_{k}", "2c2d0b89217cb16a77832a37ead046de": "-\\log \\frac{n(q)}{N} = \\log \\frac{N}{n(q)}.", "2c2d0c95dd377bb0191746ba66d111db": "\\{1...\\lambda\\}", "2c2d85ca7495fab9e2b0c6ff35a48ddc": "t = \\gamma \\tau \\, ", "2c2dbf8afc36377240a32601964634a1": "\\ \\sigma = \\pi^2/60 ", "2c2de5d0579db2f0224e8013f62be0c3": "n = \\frac{\\sqrt{(8s-16)x+(s-4)^2}+s-4}{2s-4}.", "2c2e1824dfb623c0be8690626efbbb24": "v_20 \\end{cases}", "2c3bc7095913d2d9da7d7b1acfb10647": "f({\\bold x}) = x^2 + 3 x y + 2 y^2.", "2c3c0d639fdc08cb71a8ee09ba539b92": "S_C(n)", "2c3c153e6020ae5c8ac25bede47e8946": "\\tilde f_0 = \\tilde f|_{X\\times\\{0\\}}\\,", "2c3c548547bc4f1a24e7850f4957fc19": " \\textbf{H} = \\begin{bmatrix}\n D_{xx} & D_{xy} \\\\\n D_{xy} & D_{yy}\n\\end{bmatrix} ", "2c3c63808876c807ec271b01000ffdaf": "\\lbrace e^{ar} :\\ 0 \\le a < \\pi \\rbrace", "2c3cb67aa65f21635b09d5752e57aa60": "\\varepsilon_s", "2c3ccb7a59d5ad82b0ba07597da95cdd": "dS = -k_{\\mathrm B}\\sum_{j}\\ln\\left(P_{j}\\right)dP_{j}", "2c3ce480c8efed761529d9870b816172": "\\begin{align}\nT_w(x) &= \\sum_{n=0}^\\infty w^n s(2^{n}x)\\\\\n &= s(x) + \\sum_{n=1}^\\infty w^n s(2^{n}x)\\\\\n &= s(x) + w\\sum_{n=0}^\\infty w^n s(2^{n+1}x)\\\\\n &= s(x) + wT_w(2x)\n\\end{align}", "2c3d0db7dac148c613c39d0ca852fd36": "\\big| \\mathbf{E} [ \\langle \\mathrm{D} F, u \\rangle_{H} ] \\big| \\leq C(u) \\| F \\|_{L^{2} (\\Omega)}.", "2c3d212c58c98411813b038e7edbd1a0": "(d_1, d_2, p_1, p_2)", "2c3d331bc98b44e71cb2aae9edadca7e": "[a,b]", "2c3d511ba0036255289e04a743ba270b": "\\{[0]\\} \\times \\{\\operatorname{id}\\} \\; \\triangleleft \\; \\mathbb{Z}/(2) \\times \\{\\operatorname{id}\\} \\; \\triangleleft \\; \\mathbb{Z}/(2) \\times A_3 \\; \\triangleleft \\; \\mathbb{Z}/(2) \\times S_3", "2c3d59af663d18c932d8ea03c6f03803": " \\sum_{k=1}^{n}q_{k}=-p ", "2c3d6b861a0faf9135a5dd865f81845b": " [\\ddot U] ", "2c3d744f2b5bafc838499b518d97fb5f": "f(s) - \\frac{1}{s-1}", "2c3d840554492a44d0497c18a557b2da": "\n \\begin{align}\n u_\\alpha(\\mathbf{x}) & = - x_3~\\frac{\\partial w^0}{\\partial x_\\alpha}\n = - x_3~w^0_{,\\alpha} ~;~~\\alpha=1,2 \\\\\n u_3(\\mathbf{x}) & = w^0(x_1, x_2)\n \\end{align}\n", "2c3d95224a55448f2e2343f683e89823": "\\dot{\\mathbf{x}} = \\mathbf{Ax} + \\mathbf{Bw},", "2c3db681686c1b080e21688bf57b256a": "r=0", "2c3dc944039e4965539d6dce278025fb": " (\\partial S)_P=-(\\partial P)_S=\\frac{C_p}{T}", "2c3e2b1f6da0ef25c9a98f6099769a74": "V_n(P,Q)=P\\cdot V_{n-1}(P,Q)-Q\\cdot V_{n-2}(P,Q) \\mbox{ for }n>1, \\,", "2c3e94774f3e9dd033525806e4049aa9": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 26\\cdot 5.25)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 29.4\\cdot R_{\\bigodot}\n\\end{align}", "2c3ecc2981633e35268ae00490bb2347": " \\frac {1}{c^2} \\frac{\\partial^2}{\\partial t^2} \\psi - \\nabla^2 \\psi + \\frac {m^2 c^2}{\\hbar^2} \\psi = 0 ", "2c3f18747892d5d6528ea70f9934e968": "-\\frac{e^2}{4 m \\omega^2}", "2c3f3fdacffa88e5ac1184bbb1336557": "\\text{PointsPlus} =\\max \\left\\{ \\mathrm{round} \\left( \\frac{\\text{protein}}{10.9375} + \\frac{\\text{carbohydrates}}{9.2105} + \\frac{\\text{fat}}{3.8889} - \\frac{\\text{fiber}}{12.5} \\right), 0 \\right\\}", "2c4002cf8c6282573e8065627c274dbb": "\\ \\sum_{j=0}^t \\gamma_{j,i} = 0,", "2c4015ae6af065dc34a89e3e689c6f65": "\\mathbf x[k+1] \\approx (\\mathbf I + \\mathbf AT) \\mathbf x[k] + T\\mathbf B \\mathbf u[k] ", "2c40291c4172cfdce20dc1e24b68abd1": "\\mathbf{J}=\\frac{\\partial\\mathbf{Y}}{\\partial \\mathbf{y}}", "2c40510ceb5dc73f8a8955879ab207ed": "\\overline{X}_n", "2c4079dc872d3c0ef6f7d1617e981990": "\n \\begin{bmatrix}\n 1 & 2 & 3 & 4 & 0 \\\\ \n 0 & 3 & 2 & 1 & 10 \\\\\n 0 & 2 & 5 & 3 & 15\n \\end{bmatrix}\n", "2c409e336b069367778d4ae491d9373b": "p(c_{irrational}|f_{p\\mbox{-}int}) = 0\\ ", "2c40a410316365b8f84245b446ce959a": " E = h f = \\frac{h c}{\\lambda} = m c^2 \\ ", "2c4107ead0b52f043b7fee4aa31c3d98": "A= 250 ", "2c41743fb7dbb9984e76bc4819ba5378": "\n\\left(\\frac{11}{9907}\\right) \n=-\\left(\\frac{9907}{11}\\right) \n=-\\left(\\frac{7}{11}\\right) \n=\\left(\\frac{11}{7}\\right) \n=\\left(\\frac{4}{7}\\right)\n=1 \n", "2c4181c51874d7a6964edd79d1ce96d3": "= \\mathbf{ \\Omega \\ \\times } \\left( \\mathbf{ \\Omega \\times X}_{AB}\\right) ", "2c41ccfaeb3c4101df8cb11fd76b8bdf": "\\dot{\\varepsilon}_{\\mathrm{eq}}", "2c41d1c31c43be9c5bb35d418b05f1e9": "-\\frac{1}{\\sqrt d_{v}}", "2c420b1877e104218dfa9d83a86b7b4c": "~\\delta_{kk'}", "2c42415da96af5aa01c487e280788e65": "T/P", "2c42bf94bf055e62271e0a797104beb6": "q_\\max = \\frac{1}{0.67 (C_s \\lambda^3)^{1/4}}. ", "2c42f23cdaa2029a913d9b75c2a1a044": "\\Pr(|X| \\geq a) \\leq \\frac{\\textrm{E}(|X|)}{a}.", "2c430dd4a0324d0372210cea70688253": "\\displaystyle R_{ab} - {\\textstyle 1 \\over 2}R\\,g_{ab} = \\kappa T_{ab} ", "2c434e393154b4ccbdb63ce8926b7c9d": "G=\\left[\\begin{array}{cc}\n\\mu S_X & \\left(1-\\mu\\right)W\\\\\n\\left(1-\\mu\\right)W^T & \\mu S_Y\n\\end{array}\\right]\n ", "2c434ff2fc1b83bf3c300858c4c9b399": "\\mathfrak{p} ", "2c43583029bc04110d0b27c8749c2269": "{\\mathbb F}={\\mathbb R}", "2c4396b94d24b4b9c27b3140a9a04e28": " \\prod_{r=1}^5 \\Gamma(\\tfrac{r}{6}) = 4\\sqrt{\\frac{\\pi^5}{3}} \\approx 40.3993191220037900785", "2c43a1eaa42f0a8c05243e11f663f6e6": "B = x^2 + y^2.\\ ", "2c449f424400e1568a33bf97759669e1": "\\frac{dy}{dx}=\\frac{1}{x}\\frac{d\\phi}{dt}", "2c453c28d87523f83d36a71485b61343": "\\sum_{k=1}^\\infty (\\zeta(2k) -1) = \\frac{3}{4}", "2c4640bd7eb394de9098376528f50582": "\n W(r) = C_1 J_0(\\lambda r) + C_2 I_0(\\lambda r)\n", "2c4650d1f6c24e22394530b3a2910260": "\\frac{\\text{Debt}}{\\text{Equity}}", "2c46b6b009651908e409b72047c7b46b": "\\color{Black}\\tfrac{\\bar 6}{m}", "2c473edc4e4726db29e0f8ac8ea857a0": "\nx = \\underset{1}{\\overset{\\infty}{\\mathrm K}} \\frac{1}{z} = \\cfrac{1}{z + \\cfrac{1}{z + \\cfrac{1}{z + \\ddots}}}\\,\n", "2c4777b0afbe12426fe9ccb4ae8fb5b8": "\nm\\mathbf{\\ddot{r}}\\cdot\\delta \\mathbf{r}-\\mathbf{F}\\cdot\\delta \\mathbf{r}=\\sum_i \\left[{\\mathrm{d} \\over \\mathrm{d}t}{\\partial{T}\\over \\partial{\\dot{q_i}}}-{\\partial{(T-V)}\\over \\partial q_i}\\right]\\delta q_i = 0.\n", "2c47c7deb5a699e3e5fa0006600cca8c": "\\Gamma(i) = (-1+i)! \\approx -0.1549 - 0.4980i.", "2c47cef7d743c3107a3bc117d7671dd5": " M_X(z) ", "2c47d537a70a0bc5afad857677726be1": "(\\ \\cos \\theta\\ ,\\ \\sin \\theta\\ )", "2c48043b246f4c6bd429629aee35067e": " \\omega_0 ", "2c4818a4680eabc4bd4dd13a9328fdbf": "{\\color{OliveGreen}\\frac{2}{1}}, \\frac{7}{3} , {\\color{OliveGreen}\\frac{9}{4}} , \\frac{20}{9} , \\frac{29}{13} , {\\color{OliveGreen}\\frac{38}{17}} , \\frac{123}{55} , {\\color{OliveGreen}\\frac{161}{72}} , \\frac{360}{161} , \\frac{521}{233} , {\\color{OliveGreen}\\frac{682}{305}} , \\frac{2207}{987} , {\\color{OliveGreen}\\frac{2889}{1292}}, \\dots", "2c48470e3658431db0e4f0ca0a4a0f00": "= a C \\frac{\\sin\\left(\\frac{ka\\sin\\theta}{2}\\right)}{\\frac{ka\\sin\\theta}{2}} \\frac{\\sin\\left(\\frac{Nkd\\sin\\theta}{2}\\right)} {\\sin\\left(\\frac{kd\\sin\\theta}{2}\\right)}e^{i\\left(N-1\\right)kd\\frac{\\sin\\theta}{2}} ", "2c489bc3b299a23ffc9f2b783ba15940": "\\displaystyle J_n (x)", "2c48c970078f685e684f4ac0266ec68b": "k_\\text{e}", "2c49280e9a88e312ad20eeec8eba8370": "\\text{Cl}_{2m}\\left( \\frac{q\\pi}{p}\\right)=", "2c492ac1072f77f5ef0bff57c08864f9": "\n \\delta (q) ~ \\star ~ \\delta(p) = {2\\over h} \n\\exp \\left (2i{qp\\over\\hbar}\\right ) ,\n", "2c499a574dfc2fe6609c9ee082b341e5": " sBD_J(S)=\\sum_{j=2}^JP\\left[S\\in sB(S_1,\\ldots,S_j\\right]. ", "2c4a1a1dc7572a5e284588cae4645b5f": "x^4 - 4x^3 - 6x^2 + 4x + 1", "2c4a1da77006ed3a89bb336c8eb3f2b0": " A\\,\\! ", "2c4a36681c7da4af740e606388d87a29": "\\displaystyle V_{LJ}", "2c4a5161355da01735321231df81a5a2": "x=\\frac{2\\,t}{1+t^2}", "2c4a924d318ebc8115cf2428d490ae42": " \\Delta G^{0}_{T} ", "2c4ab4cd3689b0586e889eaa61012ae5": "f(x)=\\frac{\\sup \\{f(y): y\\sim x\\} + \\inf \\{f(y): y\\sim x\\}}{2}", "2c4ae0247a4a3297b7d09ccc8ade7b62": "\\dot{\\mathbf{q}}", "2c4b066a220cf70d4930fef5d93c5426": "\n\\frac{\\Delta^2\\vec x_n}{\\Delta t^2}\n=\\frac{\\frac{\\vec x_{n+1}-\\vec x_n}{\\Delta t}-\\frac{\\vec x_n-\\vec x_{n-1}}{\\Delta t}}{\\Delta t}\n=\\frac{\\vec x_{n+1}-2\\vec x_n+\\vec x_{n-1}}{\\Delta t^2}=\\vec a_n=A(\\vec x_n)\n", "2c4b78e0663d3497be9b8916ee66d52d": "E=h\\nu=\\frac{hc}{\\lambda}=\\frac{(4.135 667 33\\times 10^{-15}\\,\\mbox{eV}\\,\\mbox{s})(299\\,792\\,458\\,\\mbox{m/s})}{\\lambda}", "2c4be0f4fdd359bea99ec4f47b9b7ad9": "H_x(A, B) = \\frac{A^x B^{1-x} + A^{1-x} B^x}{2}.", "2c4c3f059a5b1da50b26db1d5fcc9515": " g(\\mu) ", "2c4c5ea8752e15fad5d35ee06c018505": "d_0:\\ M \\to M", "2c4c8a670f38447be2fd3eef8eda1078": "x\\in \\mathbb{R}^n", "2c4caf18cec37fca57aafaa8400b11a3": "C_n=\\overline{\\{x_k:k\\geq n\\}}", "2c4cdbe53aa3d9364d44a98c6ca8e790": "\\Rightarrow \\ q_2 = \\frac{a - q_1 - \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2}", "2c4cf6962f885c2c75ab11ce45063565": "[u]", "2c4d12cb6888b5f05948c12bf968179e": " u = e^{\\sigma\\sqrt {2\\Delta t}}", "2c4d4eecdc693a802ad052b9d37c916a": " u_{j}^{n+1} = (1-2r)u_{j}^{n} + ru_{j-1}^{n} + ru_{j+1}^{n} ", "2c4d771ae3cf87e790e37c5465060c9c": "M_1 \\cup M_3 \\cup \\cdots ", "2c4d7b7ea229fb9443ccd1c42152608c": "\\mathrm P(G,S,R)=\\mathrm P(G|S,R)\\mathrm P(S|R)\\mathrm P(R)", "2c4dc1bf0a38bd0cf4bb1795843d807a": "u_n = \\sum_{j=1}^n u_je_j", "2c4e1d507f43ae80551afa76ce2818ab": "P_{\\rm SE} = P_{\\rm S\\ emt} \\left( \\frac{\\pi R_{\\rm E}^2}{4 \\pi D^2} \\right) \\qquad \\qquad (2)", "2c4e33c881f5ea6572bcd9ac030e2e07": " \\Vert T f\\Vert_{L^2}^2 \n\\le \\alpha \\int_Y \\left(\\int_X p(x)K(x,y)\\,dx\\right) \\frac{f(y)^2}{q(y)} \\, dy\n\\le\\alpha\\beta \\int_Y f(y)^2 dy =\\alpha\\beta\\Vert f\\Vert_{L^2}^2. ", "2c4e50a14f77890f659b31e07142aa78": "Q (q) = \\frac{\\hbar^2}{4m} \\{ S ; q \\}", "2c4ebf0b7b44de2dddf4a6b6d6bd78ba": "p^\\infty", "2c4ecb62e6adacc8e130443c66abe0b4": "m_k\\frac{d q^k}{dt} (t) = \\hbar \\nabla_k \\operatorname{Im} \\ln \\psi(q,t) = \\hbar \\operatorname{Im}\\left(\\frac{\\nabla_k \\psi}{\\psi} \\right) (q, t) = \\frac{m_k \\bold{j}_k}{\\psi^*\\psi} = \\mathrm{Re}\\left ( \\frac{\\bold{\\hat{P}}_k\\Psi}{\\Psi} \\right ) ", "2c4f36ae4e561a03df6101056c9cf1f1": "\\exp\\left(\\sqrt{\\log N \\log \\log N}\\right)", "2c4f94b5007d42829fcb8370745f6c3c": "f(t)=\\sin(t^2)/t,\\; t>0", "2c4f9717006d8d31bf7e0195114a5afe": "-m\\bar{\\psi}\\psi", "2c4fabe435e125af8894dff31b5ed4b6": "\\overline{S}_{nm}", "2c4fba1ff16e5ce67bd93247b8265924": "PA\\to\\lnot O\\lnot A", "2c500fa8d0873cc45030e31b8b1f0895": "\\mathbb Z/n\\mathbb Z [ \\sqrt t]", "2c50163b6fb19accf5decaa5cb2be393": "D+2aA+Bc=0,\\, E+2Cc+Ba=0,\\,", "2c5068fa98fc25715ba28ae3090b8178": "\\{\\mathrm{in}(f) | f \\in N\\}", "2c508f2204affb26db69b42a0c323e5a": "K_\\mathit{rw}^o = 0.6", "2c5109bdbbbd7b93d3f47fb312eeae5a": "x^2-ay-1=0", "2c512f96c5e670bcb01d444b8dd0c2e0": "{Z_\\mathrm{iT}}^2=Z^2 + k^2", "2c515f4ac538143f3a024be372aac2cb": "\\begin{pmatrix}x&y\\\\0&z\\end{pmatrix}", "2c516f5d279de5e5a5402264bdc3e822": "f : \\mathbb{R}^n \\rightarrow \\mathbb{R}^m", "2c51864be4727f2db9fdcb6b2423ef40": "x^4 + 324 . \\,\\!", "2c51ca2e864bb8390bf34d0e3491da4b": "\\pm q_{1} \\equiv 0.42e", "2c51e21104734214d103575ddb878006": "m_i \\in M, i \\in I", "2c51fabebb22f98f7960e09ee0724462": "n_1>n_2", "2c52192741e08420282483e207818eed": "(\\bar{\\mathbf{3}},\\mathbf{1},\\textstyle\\frac{2}{3})", "2c521b71872dfca0ddf2a1aa48b8e06f": "\\frac{\\partial \\mathbf{b}^{\\rm T}\\mathbf{A}\\mathbf{x}}{\\partial \\mathbf{x}} =", "2c5221b5a8e8be398397082e38719071": " C_{\\infty} = \\frac {\\dot{m}}{K} \\qquad(3a)", "2c523d446306bb128071f971552f7b75": " \\theta_b ,\\, \\theta_c ", "2c525b96e9fcf24fae8e4da8747db9df": " \\vec{L} = \\vec{r} \\times m \\vec{v}, ", "2c527d34c4ac7ba90b8b7cc1178b3363": "2^n - 1", "2c5288c4018030ef22658ec69b078c9f": " \\log(1+x) ", "2c52ba8a2a41b406e8b87c1c82d0116f": "P(y) \\ge 0", "2c52fe74cd2a291bc7daff92fd3ec066": "\\vec{S}_t=\\vec{S}\\wedge \\vec{S}_{xx}. \\qquad ", "2c53d6b37237cd1d6019044c812617fc": "y^2 = 2 x^6 + 4 x^5 + 36 x^4 + 16 x^3 - 45 x^2 + 190 x + 1241", "2c542e63a6bd3b9099d48df091123778": " G^{\\alpha\\beta} = 8 \\pi \\, T^{\\alpha\\beta}.", "2c548775178ce3e55e65425a013604a1": "\nL = \\frac{1}{2} M v_1^2 + \\frac{1}{2} m v_2^2 - m g \\ell\\cos\\theta \n", "2c548f5be1f824ac513c29224af31bec": "BG(n)", "2c54984bbb7de01ab5ecd3b641c7e44f": "\\boldsymbol{F} + \\boldsymbol{F_{cf}} + \\boldsymbol{F_{Cor}} = m \\ddot{\\boldsymbol{r}} \\ , ", "2c54fd98b6ab71f1cc9a3e282983304d": "\n {1\\over n!}\\det\n \\begin{pmatrix}\n v_1-v_0 & v_2-v_0& \\dots & v_{n-1}-v_0 & v_n-v_0\n \\end{pmatrix}\n", "2c55120835fd160d53097e4200221e78": "\\{\\mathbf{B}_k\\}", "2c5573f5a0c03384ffc77c6114379b2c": "\\frac{1}{T_{1/2}} = \\frac{1}{t_1} + \\frac{1}{t_2} + \\frac{1}{t_3} + \\cdots", "2c5577578e48da124473a80af92c0333": " \\operatorname{de-lambda}[F\\ P = E] ", "2c5580e2e86ede22417523460b3edf78": "\nF_z =\\ -J_3\\ \\frac{1}{r^5}\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right)\\ \\cos i\n", "2c55e1828aa3ad84a5548cc6bae274e2": "k_{\\mathrm{e}} = 1 / 4\\pi\\varepsilon_0 \\,", "2c565faf856ae4bbdf6f08e86bdc292b": "r_a(\\mathfrak{u},\\mathfrak{s})", "2c566bd8f9eace0cff6b0a5b54007c2a": "a_1v_1+\\cdots+a_nv_n=0", "2c56a71e880b053aa5efc99572dbaba2": "\\mathrm{QSym} = \\bigoplus_{n \\ge 0} \\mathrm{QSym}_n, \\, ", "2c56de0a6b20e88bc64a124a0c38a5a4": " \\sum_{\\langle i,j\\rangle} = \\frac{1}{2} \\sum_i \\sum_{j\\in nn(i)}", "2c57347922cbc7724d278bd7252b1b11": "t_{ik}(x) = t_{ij}(x)t_{jk}(x) \\qquad \\forall x \\in U_i \\cap U_j \\cap U_k", "2c573607612114b7149b03d335694575": "a_i,b_i\\in{\\mathbb Q}(x,y)", "2c577df6ef00831d3dc30ec41d7e328a": "e^{-\\epsilon\\alpha n/2}\\,\\!", "2c57890dd35a5281dc1f43fecf94b94b": "D_P\\,\\!", "2c579bb8f3bafba85dab34b08bc155ac": "\\begin{align}\n\\frac{dS}{dt} & = - k \\sum_i \\left(\\frac{dp_i}{dt} \\ln p_i + \\frac{dp_i}{dt}\\right) \\\\\n& = - k \\sum_i \\frac{dp_i}{dt} \\ln p_i \\\\\n\\end{align}", "2c57d8be899ae5a76b7bdb18482700e7": "n_1 , n_2 , n_3, \\ldots", "2c57f39fe0e4238387fbcadd5a438ea8": "n*a+\\frac{1}{2*1}*n*(n-1)*b+\\frac{1}{3*2*1}*n*(n-1)*(n-2)*c", "2c5869cbd3dcde89e26fd3e958533bbc": " x_\\perp^\\mu =(\\eta^{\\mu \\nu }-P^\\mu P^\\nu /P^2)x_\\nu, \\, ", "2c58715c84521c5659c24801f7a1e7fc": "\\int_V \\mathbf{\\nabla}\\cdot(\\phi\\epsilon \\mathbf{\\nabla}\\phi) d^3 \\mathbf{r}= \\int_V \\epsilon (\\mathbf{\\nabla}\\phi)^2 \\, d^3 \\mathbf{r}", "2c58876f9dcd8f57629830988d78d1a4": "|\\psi^\\prime\\rangle = U_\\alpha |\\psi\\rangle", "2c58b16d1c76fbce2c420298e916072f": "\\alpha^m - \\alpha^n = \\alpha^m + (-\\alpha^n) = \\alpha^{m + Z(e+n-m)} ", "2c58c988d806ce7d23f5bd7e169f82bd": "T_{s}^{-1} = T_{-s}", "2c58d72f14d9b039ad8c53e65aa64a32": "\n \\begin{align}\n & bD \\frac{\\mathrm{d}^4w_x}{\\mathrm{d}x^4}\n = q_1(x) - n_1(x)\\cfrac{d^2 w_x}{d x^2} - \\cfrac{d n_1}{d x}\\,\\cfrac{d w_x}{d x}\n - \\frac{1}{2}\\cfrac{d n_2}{d x}\\,\\cfrac{d \\theta_x}{d x} - \\frac{n_2(x)}{2}\\cfrac{d^2 \\theta_x}{d x^2} \\\\\n &\\frac{b^3D}{12}\\,\\frac{\\mathrm{d}^4\\theta_x}{\\mathrm{d}x^4} - 2bD(1-\\nu)\\cfrac{d^2 \\theta_x}{d x^2}\n = q_2(x) - n_3(x)\\cfrac{d^2 \\theta_x}{d x^2} - \\cfrac{d n_3}{d x}\\,\\cfrac{d \\theta_x}{d x}\n - \\frac{n_2(x)}{2}\\,\\cfrac{d^2 w_x}{d x^2} - \\frac{1}{2}\\cfrac{d n_2}{d x}\\,\\cfrac{d w_x}{d x}\n \\end{align}\n", "2c58f56b23320fc640c7008b17ef9824": "I(X;Y) \\le H(X),\\,", "2c5901db67135a8b7fb3e980fa3cbcf5": " q_{jk}=\\frac{1}{N}\\sum_{i=1}^N \\left( x_{ij}-\\operatorname{E}(X_j)\\right) \\left( x_{ik}-\\operatorname{E}(X_k)\\right), ", "2c59027ea041aa18982ddb6d14ebfe25": "\\begin{align}\n\\operatorname{Pr}\\left(\\left|(X_n,Y_n)-(X,Y)\\right|\\geq\\varepsilon\\right) &\\leq \\operatorname{Pr}\\left(|X_n-X| + |Y_n-Y|\\geq\\varepsilon\\right) \\\\\n&\\leq\\operatorname{Pr}\\left(|X_n-X|\\geq\\tfrac{\\varepsilon}{2}\\right) + \\operatorname{Pr}\\left(|Y_n-Y|\\geq\\tfrac{\\varepsilon}{2}\\right)\n\\end{align}", "2c59df2ad4c70541ce7a46213cc5cf6b": "\\psi(\\Omega^{\\Omega^3})", "2c59f131e35c0e0be8509d052fda35b9": "S_1, S_2, \\dots, S_n", "2c5a2263c6d54e252a9ffa9bfff5b595": " \\mathrm{sys}(\\Sigma_g) \\geq \\frac{4}{3} \\log g,", "2c5a45bb1c14b3a16761a836c7974c35": "\\mathrm{Gal}(\\bar{\\mathbf{Q}}/\\mathbf{Q}) \\rightarrow \\mathrm{GL}_2(\\mathbf{Z}/l^n \\mathbf{Z})", "2c5a75c9f78815bece69985ee63173b2": " \\psi = \\psi(\\mathbf{r}_1,\\mathbf{r}_2). ", "2c5aa2bc6142d90de0c5ec3f83630fee": "\\tbinom{n}{3}", "2c5ac7acc5a1e935c27f2c8dfa5e8871": "H(m) \\equiv H(M) \\pmod{p-1}", "2c5af9243cb4b7d232a7d3c2e2e11f8b": "\\lim_{n\\to\\infty}{ \\left|\\{\\,s_1,\\dots,s_n \\,\\} \\cap [c,d] \\right| \\over n}={d-c \\over b-a} . \\,", "2c5b1c8116f5e4d4d7edd0d6d0399a33": "n = N-1,...,0", "2c5b32270b32814f7aa76b5416c5ca2e": "(Q,*)", "2c5b560e3cbae0812811025c90be3926": " \\frac{1}{\\sqrt{L}}. ", "2c5b637e4785f165943c32e3c0ac6815": "\\mathbf{\\gamma}:I \\to {\\mathbb R}^n", "2c5b912c1a4f37eeaa8c40117bf4ce85": "\ng^{(2)}(\\tau)={{\\langle a^{\\dagger}(0)a^{\\dagger}(\\tau)a(\\tau)a(0)\\rangle}\\over{\\langle a^{\\dagger}a\\rangle^2}}.\n", "2c5b9a592e4b433b4c40845c767d37c3": "\n\\begin{align}\n\\bar{X}_1 & = (X_{1,1}+\\cdots+X_{1,n_1})/n_1 \\\\[6pt]\n\\bar{X}_2 & = (X_{2,1}+\\cdots+X_{2,n_2})/n_2\n\\end{align}\n", "2c5be1687ad74e97c3f496384fafeb68": "\n(a^{\\tfrac{p-1}{2}}-1)(a^{\\tfrac{p-1}{2}}+1)\\equiv 0 \\pmod p.\n", "2c5bf389a39b91217437572948d9714a": "a_0 = M, \\; \\; a_1 = 0, \\; \\; a_2 = M \\, \\left( \\frac{M^2}{3} - \\alpha^2 \\right) ", "2c5bfad289ffcf449cb33e444369aac9": " b_n= \\frac{2}{\\pi} \\int_0^\\pi \\cos(x)\\sin(nx)\\,\\mathrm{d}x\n = \\frac{2n((-1)^n+1)}{\\pi(n^2-1)}\\quad \\forall n\\ge 2 ", "2c5c0579684e122427ac7a5066a2a347": " p^2q^2=a^2c^2+b^2d^2-2abcd\\cos{(A+C)}.", "2c5c09a487e3d853de1b45a571f04daf": "n-1,", "2c5c4a657fa24066ef23955cfffd15bd": "\\begin{matrix}{9 \\choose 3} = 84\\end{matrix}", "2c5c575ca31d0649ec929f7328940cf8": "\\scriptstyle Q \\;:=\\; \\prod_{i=1}^{r}(x \\,-\\, \\lambda_i)^{\\nu_i}", "2c5c5b368825d717ddce76daf646c06e": "\\int_0^{2\\pi} \\Re g(e^{i\\theta}) \\,d\\theta =1", "2c5ca1463241ed97d94a6dc30e6218a7": "R_\\Gamma \\cup \\bigcup_{k=0}^{p-1} ST^k(R_\\Gamma)", "2c5caf36ec7710009bd8fda949e4e8fa": "R\\mathsf{G}", "2c5cb1d7f01ba309d6bf946ed3f41555": "\\scriptstyle \\beta() = \\text{Beta function} , \\quad \\scriptstyle \\Gamma() = \\text{Gamma function}", "2c5cbe3217a93a8d27f3913dd7afe341": "\\approx, \\thickapprox, \\approxeq, \\asymp, \\propto, \\varpropto \\!", "2c5ce5f5d242c3dc27755e85f51c1e8f": "\\int\\operatorname{arsinh}(a\\,x)^n\\,dx=\n -\\frac{x\\,\\operatorname{arsinh}(a\\,x)^{n+2}}{(n+1)\\,(n+2)}\\,+\\,\n \\frac{\\sqrt{a^2\\,x^2+1}\\,\\operatorname{arsinh}(a\\,x)^{n+1}}{a(n+1)}\\,+\\,\n \\frac{1}{(n+1)\\,(n+2)}\\int\\operatorname{arsinh}(a\\,x)^{n+2}\\,dx\\quad(n\\ne-1,-2)", "2c5ce8ce1594f26ec50b1c3392629673": " \\frac{110000-100000}{100000} = 0.1 = 10%.", "2c5d2e8f878598d840068d095d744cf2": "T_y f(x)=\\int_{-\\infty}^\\infty P_y(x-t)f(t)\\, dt,", "2c5d4ca401407c74122035a1d56b53a5": "\\Lambda > 0", "2c5d67be1276f4449af15c20b27a41f0": "\\sqrt[5]{-2} \\,= -1.148698354\\ldots", "2c5d7834bfdcc921d155839327ec87c3": "a_0 + a_1\\zeta + a_2\\zeta^2 + \\cdots + a_n\\zeta^n = 0\\,", "2c5d91bf3866653bd98d024778403778": "p_2 = B\\rightarrow bB", "2c5dab827d89743791348f1317bad89a": "H_\\alpha^{(1)} (x) = \\frac{J_{-\\alpha} (x) - e^{-\\alpha \\pi i} J_\\alpha (x)}{i \\sin (\\alpha \\pi)}", "2c5dbd75736ca3952a9f68bec5b211fa": "A,A\\equiv B\\vdash B", "2c5de33f9d60ae5fe0b7ccd85e6f18c0": " \\Theta_- ", "2c5dfea3aa032413a6e5295f61922eb6": "F = \\sqrt{\\frac{C^{*^4}_1}{C^{*^4}_1+1900}} \\quad T=\\begin{cases} 0.56 + |0.2 \\cos (h_1+168^\\circ)| & 164^\\circ \\leq h_1 \\leq 345^\\circ \\\\ 0.36 + |0.4 \\cos (h_1+35^\\circ) | & \\mbox{otherwise} \\end{cases}", "2c5e26ed6fabd684c802f903c252a324": " \\left \\langle N_i,e_i \\right \\rangle ", "2c5e8a5c39e7f40d5cf021c95e5b0cb9": "11 \\mbox{ meters/second} \\times 9 \\mbox{ seconds} = 99 \\mbox{ meters}", "2c5e9bbcc3b3fd8f22ec4cfefaf56c65": "L^{B/A} \\otimes_B C \\to L^{C/A} \\to L^{C/B}.", "2c5ea2c5421b9cca5e03bcb4dff5fbdc": "r=5, \\ \\theta={\\pi \\over 9}, \\ h=3", "2c5ee02d90144c36c01101b12a9e1978": " P_3 = P_2 P_1 = P_1 P_2\n = \\left[ \\begin{matrix} \\exp \\left((\\beta+i\\theta)/2 \\right) & 0 \\\\ \n 0 & \\exp \\left(-(\\beta+i\\theta)/2 \\right)\n \\end{matrix} \\right] ", "2c5eed95fcf3b935dcf310fd42ec5656": "2^{2 |G|}", "2c5ef63487aa38948d36af46024ee9dd": "A = \\left(6+12\\sqrt{3}\\right) a^2 \\approx 26.7846097a^2", "2c5f0c1561650d9d0a337d5ac4a01b07": "G_{S_N}(z) = \\sum_{i \\ge 1} f_i \\prod_{k=1}^i G_{X_i}(z).", "2c5f2a349fd4a2a6556fc26792510e2d": "\\chi_{f(x)>r}", "2c5f493c295edef5db9451e70b6ea242": "\\forall W_1 ... W_n \\exist Y \\forall x [x \\in Y \\leftrightarrow (\\phi(x, W_1, ... W_n) \\and Mx)].", "2c5f54755a369c2718cd77b2e89dca90": "SG_{\\rm H_2O} = 1.000000 ", "2c5f73e6b70756e24a1f42d9656635ac": "\\xi_{i+1}", "2c5fe25f4d1827b2c03ddbf798ca6982": "\n\\epsilon_\\mu^1(n) = {1 \\over \\sqrt{2}}(1,i,0,0), ", "2c5ff3e545a8aa91f547d03d450bba02": " \\operatorname{de-lambda}[E = F] \\equiv \\operatorname{de-lambda}[E] = \\operatorname{de-lambda}[F] ", "2c600e8af2cbdc0c6afde1255dfa057a": "\n\\epsilon_\\mu^1(n) \\!= \\!{1 \\over \\sqrt{2}} \\left(\n{{-i n_1 n_2 \\!+\\!1 \\!+\\!n_3 \\!-\\!n_1^2} \\over {1 + n_3}},\n{{- n_1 n_2 \\!+ \\!in_1^2 \\!+ \\!in_3^2 \\!+ \\!in_3}\n\\over {1 + n_3}},\n\\!-n_1 \\!- \\!i n_2, 0 \\right), ", "2c6013b5bdd332f82774bd7c910ec9e2": "\\forall~ x\\ge a", "2c60214a7c38b9a65bb8171f3071fb84": "f \\in L^{2} (\\Omega; \\mathbb{R})", "2c6082c39e23b977fbdbf8155d1878ba": "[\\emptyset]", "2c60df58ca598a39ff7c275881c9ea74": " \\tau", "2c6104de0f197f26512a8680598da387": "1<\\beta \\leq 2", "2c6132a387cca147baba360b138c58c2": " \\prod_{k=1}^{n-1} \\sin\\left(\\frac{k\\pi}{n}\\right) = \\frac{n}{2^{n-1}}", "2c61c383f6c6b026275a15ae476c7cf2": "F(\\mathbf u,\\lambda_0)=0", "2c620e47e313d2372020f10fbaae9bd4": "=E|\\psi\\rangle", "2c62415b2576f777c538416f544364e1": "\\mathbf{M} = \\begin{bmatrix} \\mathbf{m_1} & \\mathbf{m_2} & \\mathbf{m_3} & \\cdots & \\mathbf{m_n} \\end{bmatrix} \n= \\begin{bmatrix} m_{1}(1) & m_{2}(1) & m_{3}(1) & \\cdots & m_{n}(1) \\\\ m_{1}(2) & m_{2}(2) & m_{3}(2) & \\cdots & m_{n}(2) \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\m_{1}(L) & m_{2}(L) & m_{3}(L) & \\cdots & m_{n}(L)\\end{bmatrix} ", "2c625749e95c47a8944536f022ad81ac": "U_\\epsilon(t,t)=1", "2c628de6bd849a21112832303eecce1f": "\\lim_{n\\to\\infty} |x_n-y_n|=0\\,", "2c629235daf48b8ccce8241024fd3195": "H\\,\\!", "2c6298dcd932f71624581b5f9a65c214": "O(\\log \\log n)", "2c62b76fb894025295135edbc8a735a8": "\\ V^2 = \\frac{4 G}{3 D_3}", "2c62bd08a28e56862af4872fca4cc539": "W_i=2^{i-1}\\ \\bmod \\ {11}", "2c62e5c9247407c684b92eeb7391902a": " (V,\\delta\\;) ", "2c6309892734889ad0fd065b9d935ad1": "\\,z = e^{Ts}", "2c63289a5e5a9b00c062446cc55c9c6c": "A=\\{a_1G_1,\\ \\ldots,\\ a_kG_k\\}", "2c635c7fc8fce57ea535666f2b9ccdca": "\\Psi(t)=e^t", "2c6378c8486bf43cee321d40297b73d8": " U_{AB} = q_A q_B \\left[\\frac{1}{r_{AB}} + \\frac{(\\varepsilon_{RF} - 1) r_{AB}^2}{(2\\varepsilon_{RF} + 1) r_c^3} \\right]", "2c639a34c5d3df13d2fa693c6c957787": "\\tan \\frac{\\pi}{5} = \\tan 36^\\circ = \\sqrt{5 - 2\\sqrt 5} ", "2c639dc04dcd1a7ce12a467540d20085": " f(x) = 2x^3-4x^2+3x ", "2c64198fc4a9adeeba9e95f2082a4522": "\\mathbf{w} = [w_1, w_2, \\cdots, w_N]^T", "2c64b25a9b05e9d3ae18b28678b262c5": "A(i\\omega)=e^{-\\gamma}\\,\\!", "2c64c5cf613d8b9f4f7f3980d29aca10": "AO", "2c64c6023cfd9ee89c01ba3b4d0846a2": "(x_i,\\;y_i)", "2c64cc23ccbfdd3ada18a512dd938a1c": "\\left [\n\\begin{smallmatrix}\n 1 & \\infty \\\\\n \\infty & 1 \\\\ \n\\end{smallmatrix}\\right ]", "2c64df36098da569d7539e502489f630": "X^{\\rm T} \\hat{\\mathbf r}=X^{\\rm T} \\mathbf y- X^{\\rm T} X \\hat{\\boldsymbol{\\beta}} = X^{\\rm T} \\mathbf y- (X^{\\rm T} X)(X^{\\rm T} X)^{-1}X^{\\rm T} \\mathbf y= \\mathbf 0", "2c65a7b3c480f5d75173c5e88648ad50": "\\Phi (\\mathbf{x},t) = -G m \\int_{}^{} \\frac{ | \\Psi(\\mathbf{y},t) |^2}{|\\mathbf{x} - \\mathbf{y}|} \\, d^3 \\mathbf{y}", "2c65bac1283f83966a0b0304eac25c5c": "2^k{|X|\\choose k},", "2c65c8cffd86d56eea8480b28258a0ea": "P_c < 10^{-4}", "2c66173bab5c8dd1b790e79bab546ea9": "\\frac{1+z}{(1-z)(z^2 - 34z + 1)} = 1 + 36z + 1225 z^2 + \\cdots.", "2c6617c40ba89f7cc91b599e4c8a6800": "I = I_0 \\sin \\left(\\frac{4\\pi (x_0 + v \\Delta t)}{\\lambda}\\right) = I_0 \\sin \\left(\\Theta_0 + \\Delta\\Theta\\right) \\quad \\begin{cases} x = \\text{distance from radar to target} \\\\ \\lambda = \\text{radar wavelength} \\\\ \\Delta t = \\text{time between two pulses} \\end{cases} ", "2c6641501ca3bd8ae3caa4244ac618b5": "\\angle NMP + \\angle PML = \\angle ACP + (180^\\circ - \\angle ACP) = 180^\\circ", "2c669dd78ed2e8a845379f5240b83553": "h = \\frac{8\\alpha}{\\mu_0 c_0 K_{\\rm J}^2}.", "2c67501ce2abe9dd98fa309825d1afb5": " \\frac{dL}{dx} = L_0 \\frac{\\frac{dE}{dx}}{1+k_B\\frac{dE}{dx}}.", "2c67a7bd173c2005c97dfb9a2a3fae5b": "\\{ ww : w \\in \\{a,b\\}^{*}\\}", "2c67cd8b4fe9d5abfe9c8ce1f1e39217": "\n \\displaystyle \n PV \n =\n \\frac\n {2}\n {3}\n K.\n", "2c67cdabbfb426e7607b4e0fdb943a78": "{\\mathbf{x}}_{i+1} = A_i\\mathbf{x}_i + B_i \\mathbf{u}_i + \\mathbf{v}_i", "2c68493a744a968437afb3d9e7478adf": "\\mathbf{u}^{\\rm T}\\mathbf{A}\\frac{\\partial \\mathbf{v}}{\\partial \\mathbf{x}} + \\mathbf{v}^{\\rm T}\\mathbf{A}^{\\rm T}\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}} ", "2c6858e473617049942cd300042489f4": "Z=Z(n_1,n_2,\\cdots)", "2c686a8da305a457f6328054ad2c6f5b": "\\sum_{i \\in I} a_i = \\sum_{k=0}^{+\\infty} a_{i_k},", "2c68770860bd10cf266ebf1ac1602975": " \\bar{\\lambda} ", "2c68ce222fa73e471c4e8cb35f633a2a": "f(t)q(t)", "2c68e1d50809e4ae357bcffe1fc99d2a": "ka", "2c68fea90f6b5dff500345f40eaffcb0": "(1-c)\\mathrm{OPT}", "2c69144e02c5a191df4f86743c7b2bdb": "\\oplus_{i \\in I} M_i", "2c691a5b02ff21427f7b254ce511d740": "\n\\begin{align}\nw_k &= (G_i)_k \\\\\ny_k&=w^T_k\\times r_i \\\\\n\\hat{s}_k&=sign(y_k) \\\\\nr_{i+1}&=r_i-\\hat{s}_k(H)_{ki} \\\\\nG_{i+1}&=((H^H_iH_i)+\\sigma^2I_{Nr})^{-1}H^H_i \\\\\nk_{i+1}&=\\arg \\min \\left\\| (G_{i+1})_j \\right\\|^2 \\\\\ni &\\leftarrow i+1\n\\end{align}\n", "2c6941dbfd6b5566a4c5d7ba7fe4dc56": "E_r^{p,q} = \\frac{Z_r^{p,q}}{B_r^{p,q} + Z_{r-1}^{p+1,q-1}}", "2c69519fe7efa74c8c1341417a35341e": " \\frac{\\partial c_n}{\\partial t} = \\frac{-i}{\\hbar} \\sum_k \\lang n|V(t)|k\\rang \\,c_k(t)\\, e^{-i(E_k - E_n)t/\\hbar} ", "2c6982d7a3f12aa12e6c38bb6c55d24c": "A() \\to \\epsilon", "2c699ffbc6ac11869d5df9e306f7a1a9": "\\scriptstyle f^*", "2c69c9b2db5eb9b738a2a0f6e9950c13": "\n\\beta_{\\kappa\\mu}=\\frac{\n \\prod_{(i,j)\\in \\kappa} B_{\\kappa\\mu}^\\kappa(i,j)\n}{\n\\prod_{(i,j)\\in \\mu} B_{\\kappa\\mu}^\\mu(i,j)\n},\n", "2c69ce3fffe921ae1c3a0ff4cbf7ed01": "(a_dx^d + a_{d-1}x^{d-1} + \\cdots+a_1x+a_0)' = da_dx^{d-1}+(d-1)a_{d-1}x^{d-2} + \\cdots+a_1.", "2c69dd2b0571ca3a8e738ac8c260dd6b": "x^1", "2c6ae10b754a34d0c8a90bda14b2f958": "\\Delta f_\\text{Scherzer}=-1.2\\sqrt{C_s\\lambda}\\,", "2c6afe0da1ef8bc10e7dec81ae6ee8e1": " U_{nit} = (\\bar{\\beta} + \\sigma \\eta_{n}) X_{nit} + \\varepsilon_{nit} ", "2c6b4630d5e0a8a5e741136eb1522963": "[\\mathrm{ad}^k_{\\mathbf{f}}\\mathbf{\\mathbf{g}}]", "2c6bcf0e4fe871e3382f80c72fee6e77": "\\exists k, x \\in A_k", "2c6bfcc8528a9a7feab76c4c3868a5bd": "{\\bar{P}}_3", "2c6c38fc5ffc979ed11741ae363397a0": "\n\\left(\n\\frac\n{\n10630\n+ 674\\gamma \n+ 695.2419\\gamma^2 \n+ 191.4489\\gamma^3 \n+ 16.86221\\gamma^4 \n+ 4.082607\\gamma^5 + \\gamma^6\n}\n{10630 + 674\\gamma + 2467\\gamma^2 + 303.2428 \\gamma^3+164.6842\\gamma^4 + 36.6434\\gamma^5 + 3.9596\\gamma^6 +\n0.8983\\gamma^7 +\\frac{16}{\\pi^4} \\gamma^8}\n\\right)^{1/4}\n", "2c6d5aba2909a751645580ccd1b54dad": "d(v) < H[i+1]", "2c6da4bcb8d0e0c52441ca8b86507ba4": "\\delta_{pitman} \\ne \\delta_{ML}", "2c6dbb4d3cf80d109bee6a216cbbb9ba": "\\begin{align}\\boldsymbol{\\tau}^{(t+1)}\n&= \\underset{\\boldsymbol{\\tau}} {\\operatorname{arg\\,max}}\\ Q(\\theta | \\theta^{(t)} ) \\\\\n&= \\underset{\\boldsymbol{\\tau}} {\\operatorname{arg\\,max}} \\ \\left\\{ \\left[ \\sum_{i=1}^n T_{1,i}^{(t)} \\right] \\log \\tau_1 + \\left[ \\sum_{i=1}^n T_{2,i}^{(t)} \\right] \\log \\tau_2 \\right\\}\n\\end{align}", "2c6e334f4dc526c39c92b7fda745e76d": "H_{\\omega^\\omega + \\omega}(1) - 1", "2c6e6c2e42b12fa46b6682b5853b2466": "x^{2} - 3x + 2 = 0", "2c6e7fe24cb2068987008ca95ac3fe45": " - \\eta^{\\mu \\nu} \\partial_{\\mu} \\partial_{\\nu} \\psi + \\frac {m^2 c^2}{\\hbar^2} \\psi = 0", "2c6ebc1a07cb22686f71ba84ab71a5a9": "\\frac{d M_z(t)} {d t} = 0 ", "2c6ebcc4e4decfc72174a9016c1b5058": "J(u_n-c_n\\Delta_n)_i", "2c6efd11fea7fe153da7e83824fd000a": "\nS(\\omega_k) = H(\\omega_k)H(\\omega_k)^T\n", "2c6f0853d926d95f8872fff43f753217": "F=M\\frac{dV}{dt},", "2c6f123a32f7f4f7a72bb0493c617e84": "s^3 = y", "2c6f2f79296edabb2f9b6d2b959d62d7": "f(1,3) = 0", "2c6f3b6c16df97a1b00e04ff17e4906e": "L_1", "2c6f4b7e036def130725f5668385fc34": "f U g \\or G f", "2c6f79f3f267c6651268ded524878662": "\\textstyle{\\tfrac{\\pi}{2}}", "2c6fa5f7fd94ed05176ef2fa711787cb": "{\\mathbf A}_\\mu = A_\\mu^a\\lambda_a.", "2c6fabd15d10baf6a6bdbb2064c24d6b": "x_1 = m,\\ x_2 = m + 1", "2c700c7e03fb93847cc33e8712cbcdf6": "\\left(\\frac{a}{b}\\right)' = \\frac{a'b-b'a}{b^2} \\ .", "2c7063a472a44e7c43f74cd6968644d0": "n\\in{\\mathcal{S}}", "2c70a347e14258df376fd1928ea9e965": "|m\\rangle", "2c70bc2db15dfb96002e5156a0a970a2": "\n\\begin{align}\n| 1, 1\\rangle_{ab} & = \\hat{a}^{\\dagger} \\hat{b}^{\\dagger} |0, 0\\rangle_{ab} \\rightarrow \\frac{1}{2} \\left( \\hat{c}^{\\dagger} + \\hat{d}^{\\dagger} \\right) \\left( \\hat{c}^{\\dagger} - \\hat{d}^{\\dagger} \\right) |0,0\\rangle_{cd} \\\\\n& = \\frac{1}{2} \\left( \\hat{c}^{\\dagger 2} - \\hat{d}^{\\dagger 2} \\right) |0,0\\rangle_{cd} = \\frac{|2,0\\rangle_{cd} - |0,2\\rangle_{cd}}{\\sqrt{2}}.\n\\end{align}\n", "2c7165d678ebf7b8bea22571c21801e4": "m = 0", "2c71b242cf76191cae4aa6e88b9dc0eb": "\nU = \\exp\\left\\{\\frac{i}{F} \\begin{pmatrix} \\pi^0 & \\sqrt{2}\\pi^+ \\\\ \\sqrt{2}\\pi^- & - \\pi^0 \\end{pmatrix}\\right\\}\n", "2c71d42cc5dbbd985d8ded9e7997c72d": "d_p(\\mathbf{x}_1,\\dots,\\mathbf{x}_n) = \\|(d_1(\\mathbf{x}_1), \\dots, d_n(\\mathbf{x}_n))\\|_p", "2c722dbfeeb458e291948422605eb75d": "\\ f(x) = {1\\over 2} x^2. ", "2c728aeb434af1509027749513c3f0d1": " ax=b ", "2c72e84161498f305fcf7c764576584a": "[abababbca]_D = \\{abababbca\\,,\\; abababcba\\,,\\; ababacbba \\}", "2c732a0e9e1712afe74cc54130a931be": " E_{i} = E_{s} + \\Delta E ", "2c73555b7b7ec8a6156f2761aa0345ce": "v_n\\circ\\sigma\\neq 0", "2c751688a4c900d887e55b3775e3c9b7": " ( c_0, \\ldots, c_{n-1} ) ", "2c75449c2efe8d51e62ebc8df1375d3a": "\\mathcal{E}(X)_t=\\exp \\left ( X_t - \\frac{1}{2} [X]_t \\right ),", "2c7565b20e181da84b24e8353d596141": "O(\\Delta t)", "2c756b4a8bea0f5dcdc63cae68f6ae42": "\\displaystyle\\mathbf v_1,\\,\\ldots,\\,\\mathbf v_N", "2c7576f23b54b83be6fb075f04b3fe4a": "q_{\\mathrm{dual}}(X)=Q_A(X)", "2c758a43fe2b9e43b07f3d0376abef28": "r \\cdot s^2 ,\\qquad r \\cdot s^2 \\cdot s^2 , \\qquad r \\cdot s^2 \\cdot s^2 \\cdot s^2 , \\cdot ", "2c759da1ed76f995a1d4674ef1656dd6": "2^{\\mathbb R}", "2c75b1beb91813cbbc8c9611b92d53a6": "\\bigg. J = -D \\frac{\\partial \\phi}{\\partial x} \\bigg. ", "2c75b85822d343279399ca7269ec9a6d": "T = h^2 + h \\cdot k + k^2", "2c7656fa64c4d3511d9138e56c191ea1": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 126\\cdot 0.43)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 12\\cdot R_{\\bigodot}\n\\end{align}", "2c767ad2c7318464797a0b95fa5a27f0": "2 g(\\nabla_XY, Z) = \\partial_X (g(Y,Z)) + \\partial_Y (g(X,Z)) - \\partial_Z (g(X,Y))+ g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X).", "2c76c3bea5fb99394518943463608e5c": "t\\mapsto P-{ c'(t) \\cdot (P-c(t))\\over|c'(t)|^2} c'(t)", "2c7752bb029fa4a021953444db33237f": "\\big(\\mathcal{B(\\mathcal{H})}\\big)", "2c77b594762c169ceb3bbc5e0e22de81": "v = 0.5 - \\frac{\\arcsin(d_y)}{\\pi}", "2c77fe5fd03548a105c31f80f3e9d536": "y_2=Ay_1=AAy_0 = A^{2}y_0,", "2c7882c08b3aa81e1f1bf918e5d4901a": " \\hat{T} = \\frac{\\bold{\\hat{J}}\\cdot\\bold{\\hat{J}}}{2I} \\,\\!", "2c78958a15f3c49037ed468f1beefea7": "\\mathbf{M}_{4} := \\mathbf{A}_{2,2} (\\mathbf{B}_{2,1} - \\mathbf{B}_{1,1})", "2c78e1c241362bf4f22c58ed4dc68c73": "x \\chi_+^\\alpha = \\alpha\\chi_+^{\\alpha+1}.", "2c7922288f22804234279a2e0759e6ed": "~A \\uparrow B", "2c792c982c189a0881f79b2583979956": "X(x_0)=\\{g(x_0):g\\in G\\}", "2c79860ead70d89c25f6810dfb28f58d": " W(f,\\Phi_0)(0)=0.", "2c7a2ea345d781a42e2a5e8dfe6182f7": " f(x)=x^k\\;.", "2c7a7841f945d3734da1d7bf7b0ee90b": "\\mathbf{P'}_i = \\frac{i}{n + 1}\\mathbf{P}_{i - 1} + \\frac{n + 1 - i}{n + 1}\\mathbf{P}_i,\\quad i=0, \\ldots, n + 1", "2c7a849ccbf1429af83cb80180b1b815": "X_i \\sim \\operatorname{N}(\\mu,\\sigma^2)", "2c7afa576ea23113afaf8e137a11bd1a": "V_-=Be^{\\frac{-1}{RC}t}", "2c7b3ea2834e5b4fcb8ade77289ee83d": "S_m=0", "2c7c2576cfe24f4c4592a7ab45d9522f": "d_r^{p,q}", "2c7c2ade1e68ccaedd16440db0483a3d": " \\alpha A + \\beta B \\dots \\rightleftharpoons \\sigma S + \\tau T \\dots", "2c7c492cbeecf547b976e3eb07631884": " \\int_a^z \\ dx\\ \\int_a^x \\ h(y) \\ dy \\ = \\int_a^z \\ h(y)\\ dy \\ \\ \\int_y^z \\ dx = \\int_a^z \\ \\left(z-y\\right) h(y)\\, dy \\ .", "2c7c64aeb8a1e9bd5f632dd5a5165f98": " a\\, = \\frac {v^2} {r} \\, = {\\omega^2} {r} ", "2c7c7f628c8ca9e7858d65d31a0c7147": "\\vert \\pi^+\\rangle = \\vert u\\overline {d}\\rangle", "2c7cc24fdb69dbcb96d445711c3d563e": "u(0,t) = u(L,t) = 0", "2c7ccfb9fd6d9b15dbfcff70c72e21b6": "H_\\alpha\\,\\!", "2c7ce716a22174a84ed6711cb3b522ed": "r=C_S^2 \\frac{K_1K_2C_AC_B}{(1+K_1C_A)^2}", "2c7d031233100f698c0be7983e9b3794": "X_\\bullet(T)", "2c7d7196773300992c20a6b9087acdb6": "\\, g_{\\mu\\nu}=e^{2\\omega}G_{\\mu\\nu}\\!,", "2c7dc41ae805087ae5dd1902aacc18f5": "u_s : \\tilde D \\to \\tilde C", "2c7e46df9fe9383cc82665629add0e98": "\\beta_2=K_M.", "2c7e8434d99906e22bf51097431fe44f": "\\mathbf{r}_{dx}(n)", "2c7e89587d405b837f5077c841909e40": "D_c\\left (\\frac{d^2C}{dx^2} \\right ) = \\frac{2k_1C}{r}", "2c7ed2109acc28cacda0d9306f4f65cf": "S_N =\n\\begin{bmatrix}\n0 & I_N \\\\\n-I_N & 0 \\\\\n\\end{bmatrix}", "2c7ee9ae9afa7388fa120492e2936bff": "\\{1, i_1, \\dots, i_{2^n-1}\\}", "2c7eeefd67278497457671bc4a40e3eb": "\\left[M\\frac{\\partial }{\\partial M}+\\beta(e)\\frac{\\partial }{\\partial e}+n\\gamma_2 +m\\gamma_3\\right]G^{(n,m)}(x_1,x_2,\\ldots,x_n;M,e)=0", "2c7efcc885adb111ac1d0c77ba6bbc6f": "\\zeta(3) = \\frac{5}{2} \\sum_{k=1}^\\infty (-1)^{k-1} \\frac{k!^2}{k^3 (2k)!}", "2c7f0194bf19bb02bd85ad74c78f6abd": "\n \\boldsymbol{\\nabla} \\left(\\cfrac{1}{T}\\right) = \n \\frac{\\partial }{\\partial x_j}\\left(T^{-1}\\right)~\\mathbf{e}_j = \n -\\left(T^{-2}\\right)~\\frac{\\partial T}{\\partial x_j}~\\mathbf{e}_j\n = -\\cfrac{1}{T^2}~\\boldsymbol{\\nabla} T.\n ", "2c7f052d5a72daa03dcd8c0df846cf2a": "ds^2 = dr^2 + r^2 d\\theta^2 \\ . ", "2c7f322263147458872dfd35227a9025": "\\lambda_i-\\alpha\\mu_i=0", "2c7f9259d82848d57c44ec4a172c4aa4": " y_{1'} = {C_1 e^{i k x} - C_1 e^{-i k x} \\over 2 i} = C_1 \\sin (k x). ", "2c7fa0e5dfeb74c1298ad57861da6111": "\\mathbf{\\boldsymbol\\mu}_0,\\, \\boldsymbol\\Lambda_0", "2c7fb3fd9da3a393a06b34c5934260ee": "\n\\begin{array}{lcl}\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& \\text{Categorical}(\\boldsymbol\\theta_{z_i})\n\\end{array}\n", "2c8071a32223688b8b6d524989820f6b": "\\varrho_A^\\lambda", "2c808601637fff6201537baf2e0d98d5": "\n \\cosh(\\beta_n L)\\,\\cos(\\beta_n L) + 1 = 0 \\,.\n ", "2c809429be28381334dfe67de895a412": "R_{F_0}, R_{F_1}", "2c80a0dff9725865e9de3cca24b6e149": "a^{({n-1})/q}\\ \\not\\equiv\\ 1 \\pmod n \\, ", "2c80c5fb781c2dc8849a127ebf63e84e": "v_\\mathrm{rms} = \\sqrt {{3 k_{B} T}\\over{m}}", "2c80c7d7d08c2190d8840883ce3d3463": "x \\in \\mathbb{F}_q^k", "2c8185bcefdc5338675027e3edd8ae67": "28.24 \\pm 0.08", "2c81a0f117184ab3c10cab4abec843d0": " R^T ", "2c81a9a2e4ce6af427a194f149b1c72e": "E = \\gamma E_0 \\,\\!", "2c81a9ff2d3952f869cc0e2e0e784c0f": "\\bar v_N", "2c81c979074a048990b9c925d32078cc": "\\tilde{\\cdot}", "2c82474c21eeea5ce61f9bd7f16680ea": " Q = \\Delta U + \\Delta (p\\,V) = \\Delta (U + p\\,V) ", "2c824bef4b5f849bfd15f7d5003f4db8": " P(R) = P_B - \\frac{4\\mu_L}{R}\\frac{dR}{dt} - \\frac{2S}{R} ", "2c82512ba2590bbe2a1691d4d2b04c2d": " - \\frac{\\hbar^2}{2m} \\nabla^2 \\ \\psi(\\mathbf{r}, t) = i\\hbar\\frac{\\partial}{\\partial t} \\psi (\\mathbf{r}, t) ", "2c82693acbae86fbf47f0a348bc7d5d2": "\\tau : S \\longrightarrow B", "2c826cb6be5992d3c0cd292e9ce6b009": "\\scriptstyle X_0", "2c82914cb93af37cc966da30cbc5877c": "x_0 \\in \\partial\\Omega", "2c82f26d68e22f3f37905fe80bfa4bf9": "[R], [G]", "2c831e619d68fe62abd001a1da989f03": "\nm= u-u_{xx}+ \\kappa \\,\n", "2c8324714931c841d44493bde6c2539a": "\\phi_D", "2c833b3c74c08b9e588cf82acd869d85": "P_0(x) = 1,\\quad P_1(x) = x", "2c835718bcc81faf4151d8e6a1605340": "(N,S)", "2c837d450b584071d42899f801fe8dac": " 1,1,1 ", "2c837ee516202270b69a00647b4cc16a": "U_i^2 = \\delta U_i", "2c837f7a32b19354c7f8642f6581b454": "\\forall parameters : \\varphi", "2c83bea3b8825ea3655580a90207ec21": "h: [0, \\infty)\\rightarrow R", "2c83ed4c49ebab1d60d8f3ef92f245b1": "{\\rm Var}\\left[ {\\bar x} \\right]\\,\\,\\, = \\,\\,\\,{{\\sigma ^2 } \\over n}\\,\\,\\gamma _2 \\,", "2c840aac32b7e6b65b9835e5a6332f0c": "t = \\frac{k}{s} ", "2c841dfe704157f5723eb63f11e1d307": " M_{n_i}(D_i) ", "2c84445a3bbba8225bdd5ed8ca83fbdf": "\\| f \\|_{\\mathcal{C}^1} = \\| f \\|_{\\infty} + \\| f' \\|_{\\infty}", "2c844f2d0f45bcaa421348cd5a5dafda": "C_{\\alpha I}^{\\;\\;\\;\\; J}", "2c851073ffca8e996195b8c88841e8f5": "\\left\\{ \\left[\\!\\! \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{array} \\!\\!\\right] \\in K^n : \\begin{alignat}{7}\nx_1 &&\\; = \\;&& a_{11} t_1 &&\\; + \\;&& a_{12} t_2 &&\\; + \\cdots + \\;&& a_{1m} t_m & \\\\\nx_2 &&\\; = \\;&& a_{21} t_1 &&\\; + \\;&& a_{22} t_2 &&\\; + \\cdots + \\;&& a_{2m} t_m & \\\\\n\\vdots \\,&& && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; & \\\\\nx_n &&\\; = \\;&& a_{n1} t_1 &&\\; + \\;&& a_{n2} t_2 &&\\; + \\cdots + \\;&& a_{nm} t_m & \\\\\n\\end{alignat} \\text{ for some } t_1,\\ldots,t_m\\in K \\right\\}. ", "2c8514bb13dc6c24693842e49fceec25": "\\varphi=\\frac{-1 + \\sqrt{5}}{2}", "2c851847c9bf42d477e42776a446c041": "\\varepsilon _{ijk}", "2c853eac79be77f8e83a4ddd853931e7": "\\gamma(Z(x_1),Z(x_2)) = \\gamma(Z(x_i),Z(x_i+\\mathbf{h})) = \\gamma(\\mathbf{h})", "2c856ad4332cec0a69554937eeb536c4": "s_{0i}", "2c85abfa59bc47bd648e5377a7a0e3ec": "Z = \\sum _j g_j e^{- \\varepsilon _j / kT},", "2c86348a6675bdc2ac7ab7f1a18eb3dc": "V=\\frac{5\\sqrt{2}}{3}a^3\\approx2.35702...a^3", "2c8664ee2cc187d31c1bd346f1516a88": "m = \\frac{Money Stock}{Monetary Base} = \\frac{Deposits + Publicly Held Currency}{Monetary Base} = \\frac{1 + \\gamma}{\\alpha + \\beta + \\gamma}", "2c868088a53d38a7aa7a1999c4a9fd55": "(a, b) \\in U", "2c86870e6422265f537443586ee15406": "[\\![ \\sigma \\to \\tau \\, ]\\!]", "2c869a88d3a9f21f7a18cba4f67f5a3f": "\\ f_0", "2c86d512e72fa8f128fbbbc129636bd3": "\\displaystyle f(z) = \\int_{\\partial D}f(\\zeta)\\omega(\\zeta, z) - \\int_D\\overline\\partial f(\\zeta)\\and\\omega(\\zeta,z).", "2c870a191423d39d6155173de7ce020f": "\\eta_{th} = 1 - \\frac{1}{r^{\\gamma-1}}\\,", "2c8797884926b2cb107c7d8b27f81bd2": "\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx.", "2c87aae5a1d76c47808d0e2aeadfab89": "p_{K}\\left( f \\right) = \\sup \\{ \\| D^{j}f\\left( x \\right)\\|, x\\in K, 0\\leq j \\leq m \\}", "2c87c1dc9fb53164abac1b012a671550": "M_4=\\left.2\\mu+12\\mu^2\\right..\\,", "2c87d7e179ea674cc9bda54e41be5b84": "P_\\text{inst}(t) = V(t) I(t)", "2c8806b07bf3c258cba87a5bf8d290e9": "\\|Df^{-n}v\\| \\le c\\lambda^n \\|v\\|", "2c882316568cf3c4adcccf249ed39b67": "\\displaystyle \\left|\\sum_{i,j}(STv_i,v_j)\\right| \\le \\|\\sum_i v_i\\|^2,", "2c883e429623b2dd011652432cd67d42": " E_{k} = \\rho_{kij} J_{i} H_j \\,", "2c8849ba7419106a7a7fa0d80044d824": "|s\\rangle", "2c8857e37adb8705d04e40c01e5c4d70": "\\lang i,\\epsilon_i|j,\\epsilon_j\\rang = \\lang i|j \\rang \\lang \\epsilon_i|\\epsilon_j\\rang= \\delta_{ij} \\lang \\epsilon_i|\\epsilon_j\\rang = \\delta_{ij}", "2c8859ae4e5ec0488556ba19bd864e66": "1\\,\\text{slug} = 1\\,\\text{lb}_F \\cdot 1\\,\\frac{\\text{s}^2}{\\text{ft}}", "2c88d02deb0d27de9b20842e06923166": " \\vec{e}_3 = \\partial_y ", "2c88d97a88c2b47d232fa6dbabb17c8c": "\\begin{matrix}I\\end{matrix}", "2c88f93df9b3aa945f93136cfe39360d": "\\sum_{i\\in V} f_{iv} \\le c(v)", "2c890ffe3f25171d0cf545a207f30dbe": " K =\\frac{[SP]}{[S^*][P]}", "2c89b4ae65d4a438a1f594e44e1dc825": "\\sum_{k=0}^n{s_k \\over {n \\choose \\lfloor{n/2}\\rfloor}} \\le \\sum_{k=0}^n{s_k \\over {n \\choose k}} \\le 1,", "2c89cbfe84561b6ffc7a62d522eaf03e": "p : M_p \\to N", "2c8a4b04feba126f91db75445c549e8f": "\\,\\!x", "2c8aa2a4366ea54105c9a6d0c8925250": "A' + B' \\subset A + B ", "2c8ab3acb42345a7ad5291faf8f11918": "{\\scriptstyle L(C)\\, =\\, \\operatorname{Len}(C)/ \\tau(C)}", "2c8afb690d728afb30bae80aa6321625": "x \\mapsto 1/x", "2c8b4d473ed86dce1f1c8d9e0cdc4e3d": "E' = E", "2c8b7e57eec0f047e3c3e26b3d9244e0": "\\epsilon({{{{D}}}}) = 1 - \\prod\\limits_{n=1}^{N-1}(1-\\epsilon_n) \\prod\\limits_{n=1}^{N-1}\\frac{1-2\\epsilon_n}{1-\\epsilon_n} = 1 - \\prod\\limits_{n=1}^{N-1}(1-2\\epsilon_n)", "2c8bc149f04a35b1c7f1160c009225c5": " \\Pr( O_n ) = \\frac{ |\\lang n | \\psi \\rang|^2}{\\lang \\psi | \\psi\\rang} = \\frac{ | c_n |^2 }{\\sum_k | c_k |^2} ", "2c8bcc4a2988aa1cc38ddb3f005a5152": "\\scriptstyle D_F(5\\rightarrow 3)= 4(0)-2+2-0=0", "2c8bd02ab53e08afc575e4df13150297": "(Q_Ix)(\\phi x,\\psi x)\\equiv (Q_Lx)(\\phi x,\\psi x) \\land (Q_Lx)(\\psi x,\\phi x)", "2c8bfb21add382d87d4af56ad8b7a494": "confidence_{i}> 0", "2c8c911e41368caf77db4e0930230b5e": "p=\\bigg(r+\\frac{1}{\\lambda_n}\\bigg)B_0", "2c8c9b530c7f0897e88e71ac457c0817": "{\\mathbf X} \\sim N({\\boldsymbol \\theta}, I). \\, ", "2c8cdbb73bcaf0d47e35c827e7fcca4e": "q 2 ^ q", "2c8cee7b205b91f69a524f0d4777ff06": "\\mathbf{74}_{10} = \\mathbf{7} \\times 10^1 + \\mathbf{4} \\times 10^0", "2c8d582b1e15b6c62524457f62bbcd2d": "V_{ik}=\\langle m_i|\\hat{V}|m_k\\rangle", "2c8d6b871894cb2f3d86e12fbbc14099": "C_1 := \\emptyset", "2c8da0658028f94a2d289ed07c9a430a": "a*b=\\overline{ab}", "2c8df2e929706878f2e9d969e85e9ca7": " \\dim(\\ker( f ))+ \\dim(\\operatorname{im}( f ))= \\dim( V ).", "2c8e6313f5d5ef2835ac286c2065ad92": "X_1+X_2=k", "2c8eff29e075234a2dac523ff63c9b0c": "k_1(s) = l_0 + l_1 s^1 + u_2 s^2 + u_3 s^3 + l_4 s^4 + l_5 s^5 + \\cdots \\,", "2c8f44fb261a8a2ac5b681fec5b02ace": "\\max(\\dim U,\\dim W) \\leq \\dim(U + W) \\leq \\dim(U) + \\dim(W).", "2c8f51d2bf8977c45de7729f79b6a288": "x + y = \\left(\\frac{x}{y} + 1\\right) y.", "2c8f62d2a2adea1a7c8ea531b4c10988": "(1, F_{2n - 1}, F_{2n + 1}),\\,", "2c8f80525866e03777e87a9b2f254ea6": "\\frac{\\neg \\Box A}{\\neg A}", "2c8fbaf24f6ece2ae22769874816a7dc": "L(\\mathcal{G}, t)", "2c8ff837410b8ec4aab0183e501fbc24": "\\scriptstyle f:\\; M\\, \\rightarrow \\,\\mathbb{C}", "2c9077ab825f14213f7fd4cae39b1f0a": " q_{\\nu} ", "2c90dc94eefa75b90f9fcd8b21b7d7ef": "g.", "2c910edb742ea035e013a80f51696a62": "[4,\\infty)", "2c912cb5f0324e0dfc2ed7c98acfd73d": "\\|f\\|'_\\Phi = \\inf\\left\\{k\\in (0,\\infty)\\mid\\int_X \\Phi(|f|/k)\\,d\\mu\\le 1\\right\\},", "2c91edc7dc2deb57d8b5298fcbeea43d": "\\tfrac{a}{b} = \\tfrac{c}{d}", "2c925b9136658e4e637fd5acc46892e3": "n_a", "2c925e11d4c58fe7e441b4270160c636": "\\mathrm{d}B^\\dagger", "2c935f22af41eae02e23b15dbd7e655c": "\\displaystyle{(V \\pm iW)u =(a\\mp i b)(\\partial_x \\pm i \\partial_y)u,}", "2c9379ba57a8a14c2ef9145c1f82b00b": "\\phi_i(x)=\\phi^+_i(x)+\\phi^-_i(x)", "2c93aa3c897801134eab0bced37c7273": "\\mathbf{v} = v_r~\\mathbf{e}_r+v_\\theta~\\mathbf{e}_\\theta+v_z~\\mathbf{e}_z", "2c93c5d202f07fc84b313aec1cc2096d": "\\text{frk} \\geq \\text{acc} \\cdot \\left ( \\frac\\text{acc}{q} - \\frac{1}{h} \\right).", "2c93c7418fb326747c1d2fd9610a16d9": "M_i = 0", "2c93f3c65cf116e52d1617c6ab149afa": "pg(s,t,\\alpha)", "2c93f8116dd884278dd9b07edd5a950e": "v \\mapsto c(v)", "2c940e5c88db0268d577ed9a7750bba2": "\\vec{\\pi}_A", "2c9414e934a4fd06ac808433721a1ad3": "x \\mapsto x^\\frac1\\lambda", "2c94a394c54a98f894f4efbe48354965": "\\rho (f, g) = 0.", "2c94d6de06e87435cde9e63d8f066f03": "\\sigma=\\sigma_\\lambda^\\mu(x^\\alpha)dx^\\lambda\\otimes\\partial_\\mu \\qquad\\qquad (6) ", "2c9501a797a63b5dae82d01230e901a1": "f \\to S(f^{-1}).", "2c95259477c6a7af3a96e436a71025b2": "(t_k,x_k)", "2c95ac9ea3f813edfc691f3260542f0d": "\n \\frac{\\partial\\Phi}{\\partial t} \n +\\, \\frac12\\, \\left| \\boldsymbol{\\nabla}\\Phi \\right|^2\\, \n +\\, \\frac12\\, \\left( \\frac{\\partial\\Phi}{\\partial z} \\right)^2 \n +\\, g\\, \\eta\\, \n =\\, 0\n \\qquad \\text{ at } z\\, =\\, \\eta(\\boldsymbol{x},t).\n", "2c95cd461c77dcc163b0e6f920580922": " b^{\\log_a d} = d^{\\log_a b} ", "2c95e8343061309fba260210869b2180": "\\nu \\to +\\infty \\!", "2c95f582d5b41d9a0b69c2e86692fc6c": " f(x)g(x) = \\left(\\sum_{n=0}^\\infty a_n (x-c)^n\\right)\\left(\\sum_{n=0}^\\infty b_n (x-c)^n\\right)", "2c96018dddaf7a8627be147876394bc6": "2x(p-q)=p^2-q^2.", "2c963678c0e210fd038256e0b0e095ef": "\\text{Var}\\left(\\hat{\\alpha} + \\hat{\\beta}x_d\\right) =\\sigma^2\\left(\\frac{1}{m} + \\frac{\\left(x_d - \\bar{x}\\right)^2}{\\sum (x_i - \\bar{x})^2}\\right).", "2c96c2e7d4b5b59c5f019686596bc6d4": "P = a \\cdot (M - b)^c", "2c96d171582e1ec3b30af07e90b124a8": "\\ni", "2c9708a5df09cc23b92ab5c7d82d774a": "e^{wh}", "2c972210bcfe3772abb8f42c5df3998f": "D_h(x^n) = x^{n - 1} + h x^{n - 2} + \\cdots + h^{n - 1}", "2c972c9f4392ff9828d03acaafc9b85c": "\\begin{align}\nd\\mathbf x&=d\\mathbf X+d\\mathbf u \\\\\n&=d\\mathbf X+\\nabla_{\\mathbf X}\\mathbf u\\cdot d\\mathbf X\\\\\n&=\\left(\\mathbf I + \\nabla_{\\mathbf X}\\mathbf u\\right)d\\mathbf X\\\\\n&=\\mathbf F d\\mathbf X\n\\end{align}\n\\,\\!", "2c972d232f48ae3d29130b2f6d918468": "n(r) \\propto r^{-3.5}", "2c97c92d9fddec3204121da6a578adc1": "\\displaystyle \\log \\left| x \\right|", "2c97ed36e1c39816a53945ce66f6b6ac": "W (t, \\vartheta_{s} (\\omega)) = W (t + s, \\omega) - W(s, \\omega)", "2c981f153f4b1dfa4461627bab0f440b": "\\frac{45}{23}", "2c984fee7828d309d9327337a2924a6a": "\\scriptstyle s=\\sigma+i\\omega", "2c98662db598ab6eb123f3af185fa5fe": "~\\frac{1}{\\tau}~", "2c98b8d94cb8fbea98d26200d622f3d2": "D_2 = h/2", "2c98dfeb901d979b5f090b18e0cd00ff": "cos\\theta", "2c990475e6e990fcc4a7890bb42cea79": " f(x; a_1, \\ldots , a_n, b_1, \\ldots , b_n) = \\sum_{i=1}^n \\, w_i \\, p(x;a_i,b_i) ", "2c996c7883c63a4062621079046f168f": " \\mathbf{J}_{\\rm d} = \\epsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t} \\,\\!", "2c9982c8feb70567c318213d92855920": "A_i(x) = \\frac{A(x)}{A^\\prime(x_i)(x-x_i)} \\quad\\text{and}\\quad B_i(x) = \\frac{B(x)}{B^\\prime(y_i)(x-y_i)}, ", "2c99b365ce3901451ec415b08fc13c0e": "d(a, b) \\le d(a, c) + d(c, b) ", "2c99ba2ec83e7fabe107beb28bc5e80a": "D_{N}^{*}(x_1,\\ldots,x_N)\\geq c_s\\frac{(\\ln N)^{s-1}}{N}", "2c9a046f377a05f89374e17ce5437bcc": "\\mathcal{L}=\\bar\\psi(i\\hbar c \\, \\gamma^\\alpha D_\\alpha - mc^2)\\psi -\\frac{1}{4 \\mu_0}F_{\\alpha\\beta}F^{\\alpha\\beta},", "2c9a4460e04aac05c17227292710df46": "Q(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)}) = \\operatorname{E}_{\\mathbf{Z}|\\mathbf{X},\\boldsymbol\\theta^{(t)}}\\left[ \\log L (\\boldsymbol\\theta;\\mathbf{X},\\mathbf{Z}) \\right] \\,", "2c9a65e49c13284653d15b8ad51c283c": "I_n = I(x_n>z)", "2c9ab61b76108dcbd18f7a241ed29a32": "A=X^TCX=\\int_{-1}^{1}\\int_{-1}^{1}[x_1^j x_2^i e^{-c(x_1-x_2)^2}]_{i,j=0}^{i,j=m-1}dx_1dx_2", "2c9ac50a0573ad6ff9df33a145edf5b0": "\n\\{\\phi_j, H\\}_{PB} + \\sum_k u_k\\{\\phi_j,\\phi_k\\}_{PB} \\approx 0.\n", "2c9b11b7e2874e314a55bbf1be2d322f": "t_0 = 0 < t_1 < t_2 < \\dots < t_m = T", "2c9b682412689d6723e3b31653b5774c": "EF", "2c9b95ac271103b6b77cbe6503c4f74e": "\\mathfrak{der}(A)", "2c9bc1d77931836928ebef02738f7d80": "\\int_0^\\infty \\left[\\vartheta (n,it) -1 \\right] t^{s/2} \\frac{dt}{t}=\n2\\ \\pi^{-(1-s)/2} \\ \\Gamma \\left( \\frac {1-s}{2} \\right) \\zeta(1-s)\n=2\\ \\pi^{-s/2} \\ \\Gamma \\left( \\frac {s}{2} \\right) \\zeta(s).", "2c9c47e0686da29fdedd0684371917da": "A=\\{a_{ij}\\}=P(X_t=j|X_{t-1}=i)", "2c9c8703cd3feb234a6827475159fdf5": "i_1, \\cdots, i_n", "2c9c87ad1d385b66b208d0fc1cccfaa9": "\n\\lim_{\\mathrm{Re}(s) \\rightarrow -\\infty} \\operatorname{Li}_s(-e^\\mu) = \\Gamma(1 \\!-\\! s) \\left[ (-\\mu - i\\pi)^{s-1} + (-\\mu + i\\pi)^{s-1} \\right] \\qquad (\\mathrm{Im}(\\mu) = 0)\n", "2c9cd14355a907b498f06e52e27d23f8": " I_n = \\frac{1}{a} \\left ( x^ne^{ax} - n I_{n-1} \\right ) , \\,\\!", "2c9d3704a52a3a3eeaf878c2e3c36487": " G = \\frac{\\omega \\mu_0 \\int{|\\overrightarrow{H}|^2 dV}}{\\int{|\\overrightarrow{H}|^2 dS}} ", "2c9d5a0f4dfca75393815073b7e34172": "\\begin{matrix} {1 \\choose 1}{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "2c9d8c7f6d173f611f5c49f025babead": "\\frac{H_{N,s-1}}{H_{N,s}}", "2c9d998428d3448135d57bbf29cee3cb": " -r_{A} ", "2c9dad314ee53cbda8a0dd73060442eb": "\\ M_{heel_{max}} > D_{heel} \\times lift \\times (cos(\\beta) +{(L/D)_{\\alpha}} ^{-1} \\times sin(\\beta))", "2c9dbd152e0aff55fcaf5ec10b210881": "g, N \\in \\mathbb{N}", "2c9e3e573f2d8491cd44b4f433311654": "{13 \\choose 7} - 8 - \\left[2 \\cdot {6 \\choose 1} + 7 \\cdot {5 \\choose 1}\\right] - \\left[2 \\cdot {7 \\choose 2} + 8 \\cdot {6 \\choose 2}\\right] = 1,499", "2c9e6077c6c9569389dae545977e91c7": "\\mathbf x(t) = e^{\\mathbf At}\\mathbf x(0) + \\int_0^t e^{\\mathbf A(t-\\tau)} \\mathbf B\\mathbf u(\\tau) d \\tau", "2c9e7558ac15b0fdc230a93ed93aafb0": "f'=q'(u+v)(u'+v') - q'(u-v)(u'-v')", "2c9ebead81fab1de4e8ebb1317c1e0ef": "\\ddot{\\theta}_1 m_2 L_1 l_2 \\cos(\\theta_2)\n+\\ddot{\\theta}_2 (m_2 l_2^2 + J_{2zz})\n+ 1/2 \\dot{\\theta}_1^2 \\sin(2 \\theta_2) ( -m_2 l_2^2 - J_{2yy} + J_{2xx} )\n+b_2 \\dot{\\theta}_2\n+g m_2 l_2 \\sin(\\theta_2)\n= \\tau_2", "2c9ed2cc9d5438b07a8409540a627d6f": "P=\\rho RT + \\left(B_0 RT-A_0 - \\frac{C_0}{T^2} + \\frac{D_0}{T^3} - \\frac{E_0}{T^4}\\right) \\rho^2 + \\left(bRT-a-\\frac{d}{T}\\right) \\rho^3 + \\alpha\\left(a+\\frac{d}{T}\\right) \\rho^6 + \\frac{c\\rho^3}{T^2}\\left(1 + \\gamma\\rho^2\\right)\\exp\\left(-\\gamma\\rho^2\\right)", "2c9ee195d8d30ae3943f7531b01b2216": "\\frac{1}{B(E,0)} = 1", "2c9ee883f86168396c624a2a761bcd0e": "e = \\frac{1}{0!} + \\frac{1}{1!} + \\frac{1}{2!} + \\frac{1}{3!} + \\cdots.", "2c9ef1f8f7155f640e695ecf874e786d": "(X_1, \\Sigma_1)", "2c9f0275b7b750809fd66a9d60ab2695": " \\pi r \\left(r + \\sqrt{r^2+h^2}\\right) = \\pi r(r + s) \\, ", "2c9f068ef6b29fb7e16a9fac4b2a9045": "N=\\frac{{\\rho}^*_{xx'}(1-{\\rho}_{xx'})}\n{{\\rho}_{xx'}(1-{\\rho}^*_{xx'})}\n", "2c9f696b04f0bd99d432da30ab36e4a0": "P_{j}(\\theta_0) = \\frac{\\partial P}{\\partial \\theta^j}(\\theta_0)", "2c9f7dbc041c44735f3c26a8a9b2e69a": "p=r\\sin \\psi", "2c9fa76724018ff01fd2b049708d7d4a": "\\alpha_T = \\frac{1}{R(T)} \\frac{dR}{dT}.", "2c9fbc2f0af12e6f33fa30c2644cd335": "\\ \\mathbb{E}", "2ca01ee78cad7c6a1936411131fc8c51": "S(t,u)", "2ca05a29a4fdcf787567c8aad71a256a": "k_0=1", "2ca07b19582fdd79838106386e70acdf": "\\sigma(r) = \\sqrt{\\frac{1}{N-1} \\sum_{i=1}^N (x_i - r)^2}.", "2ca0d85d00450f09aeffdd17734346a7": "k-", "2ca137edf9a6f6be2b0b21e815b36f0d": "25812.807 \\,", "2ca199347e652a98cc020b23b328d3be": "\\frac{\\phi_s(n)\\zeta(s+1)}{n^s}=\\frac{\\mu(1)c_1(n)}{\\phi_{s+1}(1)}+\\frac{\\mu(2)c_2(n)}{\\phi_{s+1}(2)}+\\frac{\\mu(3)c_3(n)}{\\phi_{s+1}(3)}+\\dots.\n", "2ca1a82539b223670995703e5eaf531e": "a_{21} w x_1 p_2", "2ca1ad241a2751273b4476cc0839c857": "K_j=\\{s \\ | \\ W_j \\mbox{ is a host of } s\\}", "2ca219690787ebde7a8fbeac52226265": "\\kappa \\approx \\left|\\frac{d^2y}{dx^2}\\right|", "2ca22343348d45c97c71bfb81d8c9f14": "f(x) = \\exp(\\lambda x)", "2ca2244ff91dd55e68af8a27cffb7f05": "t_i - i\\beta", "2ca239d0a0fb82998f9a86c3ebfa5dfa": " \\mathfrak{a} \\supseteq \\mathop{\\mathrm{ker}} f", "2ca248b8bfab146bbcdca00216b86a5b": "K+P", "2ca28caadb2f9618fbd04232594b8e1d": "\nQ_{z^2} \\equiv \\sum_{i=1}^N q_i\\; \\frac{1}{2}(3z_i^2 - r_i^2),\n", "2ca29a8b5eadf8a94f38080a6d57b6da": "\\ p_t", "2ca2c5e45e624951b2b022e70a7e09cc": "D_{total}=D_{p-r}+D_{braking}=v t_{p-r}+ \\frac{v^2}{2 \\mu g}", "2ca39eb71f556c356acc32663a2747e5": "\\nabla c", "2ca3c75c07010a798a7b2729634dd504": "\\Delta E_{system}", "2ca435bef607b35046d4e07341832038": "\n\\langle O_1(\\lambda x_1) O_2(\\lambda x_2)\\ldots\\rangle=\n\\lambda^{-\\Delta_1-\\Delta_2-\\ldots}\\langle O_1(x_1) O_2(\\lambda x_2)\\ldots\\rangle\n", "2ca435f084796d1356271772bc8f77ad": "\\scriptstyle e_i \\;=\\; s_iN/n_i", "2ca535f7a818017a46097702658829cd": "\\operatorname{Div}(X) \\to \\operatorname{Pic}(X).", "2ca5adcbe32c0541446a39d768a3932e": "kME_i=cor(x_i,ME) ", "2ca5af00b3435695bd9e9e2e0bc9b348": " \\varphi (L) = 1 - \\sum_{i=1}^p \\varphi_i L^i\\,", "2ca5dfa2151fc7857b507f78e49bc10e": "y = V t + W.\\,", "2ca66b81f75f33d087b4bd4503fbcee6": "M \\otimes_R N = \\operatorname{coker} (N^J \\rightarrow N^I)", "2ca682ac2a4a2ab19ac053501fd0499e": "f(\\cdot)\\,", "2ca713799524a76da9a97543126ee60e": " q(a, b, c) = \\frac{ \\log(c) }{ \\log( \\operatorname{rad}( abc ) ) }", "2ca72d72a9a767587101db1a9c5f5856": " \\ k ", "2ca7728fa4ea21d13e1d47f106abb674": "0 < V^2 - U^2 < 1", "2ca7ab55a4f1a35f9d05e50482541270": "\\mathbf{rank}_q(x)", "2ca7f7962bbe2436c5e590037176b9b8": " \\{ x \\} ", "2ca81036d6296a36e306cc9ac000c4b2": "g_a(u)=u a", "2ca8d4a03c55b71ae142054a6797f845": " v_{\\rm turb} ", "2ca981d3695b6fb130921ce2ca8bb8f1": "kXv=kXk^{-1}kv=\\left(kXk^{-1}\\right)kv.", "2ca9ad83915a83f4df0cb283f718ef0a": "G_{RX}", "2ca9f4dabc56a88ed93454f8aacd5767": "{C_8}^*", "2caa516bb23ca855cb8fd568ca95b5a1": " F(x) = \\mathrm{constant} \\cdot x^m", "2caabe0a081d3a2707ace9b31b06e32f": "Flies(Penguin)", "2caac7defd6d4de9ebb0f90e89a4b73a": "\\int f\\, d\\mu_n \\to \\int f\\, d\\mu", "2caacb6d3a37cc9e2f1d54ef3a53d220": "\\mbox{rank}\\,(A) \\leq \\mbox{rank}_+(A) \\leq \\min(m,n),", "2caae0145f7808182c3fcc066df41a09": "\\tau_i", "2cab2067b603ee49f1e68b892eaec455": "\\, V", "2cab2ceac0d9bb25747ec86bf7cd27b7": "\\textstyle\\beta=\\textstyle\\beta_0", "2cab34c76a12ee9496ae89ebb72ec4b7": "\\frac{\\partial \\mathbf{r}}{\\partial t}=0\\,\\!.", "2cab827b5f4f2748148a80886dc7b7e3": "\\textstyle R_{\\mu \\nu} - {1 \\over 2}g_{\\mu \\nu}\\,R + g_{\\mu \\nu} \\Lambda = 8 \\pi G \\, T_{\\mu \\nu}", "2cab9f11ba3ffe2c9799fdcdc0a96e8a": "|n|_{\\ast}=n^\\lambda=|n|_\\infty^\\lambda", "2caba086659bc1970a8461b287096533": "1\\!+\\!0\\!+\\!9\\,=\\,10", "2caba5bd7272c61c10ebe601ea6a77cd": " n\\mathbf{p} = [n\\underline p,n\\overline p] ", "2cac1511f42d60f84603dc2ab1f45828": "a^2=1+b^2 +x^2 +y^2 ", "2cac2b106bf41de328df49036c8d6be1": "\\mathrm{T}_0^\\infty=\\bigcap_{k=1}^\\infty\\mathrm{T}_0^k", "2cac49a45f09450fe162122b4999e8ad": "\\langle T_f x, y \\rangle = f(S_{x,y}),", "2cac4a0d581c123888a32ef4590342c8": "s \\in \\Sigma", "2cac8955628d7530e90014ae49aa70f3": " X_{ki} = \\lambda_0 + \\lambda_1 X_{1i} + \\lambda_2 X_{2i} + \\dots + \\lambda_{k-1} X_{(k-1),i} ", "2caca4c004f6de52553f029bd9cdf52f": "Q_n=P_n, Q_{n-1}=P_n+P_{n-1}", "2cacaeb3905570a4449ab8164266fbbd": "t_{zerocross}=\\frac{(\\alpha -\\beta )}{(\\alpha +\\beta -2)}\\sqrt{\\frac{\\alpha +\\beta +1}{\\alpha \\beta }}.", "2cacb2c03fba66719caa56aa9e453509": "\\hat e_z = \\hat e_x \\times \\hat e_y", "2caccaa32147ac4b3c55add526c71c34": "S1\\ \\delta^f\\ S2", "2cace222b8a51d762393a176cb63e472": " \\theta_{Ei} ", "2cacefbc53fd23ff26c5ef4e93497b04": " a_n>a_{n-1}>\\cdots>a_0>0,", "2cacf78a8d173af245a13bf85116bb73": "\\hat{Y} = \\sum_{i=1}^n W_i(X)Y_i", "2cad121eef5a7d9b80ca8b256b5ad001": "\\vec{X} = \\vec{e}_0 ", "2cad49f3fa51e2c7c9d2737a602a9548": " {{E(b)} \\over {E(e)}} \\approx {{b \\ln (e)} \\over {e \\ln (b)}} = {{b} \\over {e \\ln(b)}} \\, ", "2cada09652934a72c171e4cdf62faddb": "\\mathrm{Ma} = \\frac {U_\\infty}{c_s}", "2cadc0fb0b1f60c22a339f4316a91da8": "Q_c", "2cadc8beb4ceb26bb232b33093907cc2": "\\,\\Phi(\\alpha) = {\\Phi}' (1 - \\alpha)", "2cae26f59d5e2158fa9c8c77f946cca0": " \\operatorname{lambda-free}[M\\ N] = \\operatorname{lambda-free}[M] \\and \\operatorname{lambda-free}[N] ", "2cae3a624a12d3b4b5fa6507773e4caa": "\\mathbf{A}^{-1}\\frac{\\mathrm{d}\\mathbf{A}}{\\mathrm{d}t}", "2cae6d5ab19d1814d32d33add6a73683": "R^m_n(\\rho) = \\! \\sum_{k=0}^{(n-m)/2} \\!\\!\\! \\frac{(-1)^k\\,(n-k)!}{k!\\,((n+m)/2-k)!\\,((n-m)/2-k)!} \\;\\rho^{n-2\\,k}", "2cae9e70a6bcefc24f236c8bb1cce6b8": "\\sqrt{\\sigma_S^P}\\sqrt{\\sigma_L^P}", "2caecd93436581c64b444feea952444f": " x \\in T ", "2caf6623a46275608d82afb17ce70ea9": "\\frac{a+b}{2}", "2caf6719cca62f904882ba498000b6dd": "_{dual(q'p)\\,}\\!", "2cafefe9ef24691a514fbb59d7342337": "\\theta_{ik}", "2cb046ac408438ddb96b4e50050d2c5c": "\\mathcal{C}\\in \\mathbb{N}[\\mathfrak{A}]", "2cb066752cdddd500024a6f89edac41f": "f(x) = x + \\sin(x)", "2cb06aa56c680ef5bbdddd6de7f21249": "\\nabla\\, \\times\\, \\boldsymbol{\\kappa}\\, =\\, 0.", "2cb0caf518732be6650609e41b4331a2": "v_{0.5}", "2cb0dd99d31264371d7af56d063a44d2": " \\operatorname{drop-param}[(g\\ m\\ p\\ n), D, \\{p, q, m\\}, \\_] ", "2cb15d0c9d8c306532372eadfd9d13c5": "\nRD=\\frac{\\rho_\\mathrm{object}}{\\rho_\\mathrm{ref}}\n= \\frac{\\frac{\\text{Deflection}_\\mathrm{Obj.}}{\\text{Displacement}_\\mathrm{Obj.}}}{\\frac{\\text{Deflection}_\\mathrm{Ref.}}{\\text{Displacement}_\\mathrm{Ref.}}}\n = \\frac{\\frac{3\\ \\mathrm{in}}{20\\ \\mathrm{mm}}}{\\frac{5\\ \\mathrm{in}}{34\\ \\mathrm{mm}}}=\\frac{3\\ \\mathrm{in} \\times 34\\ \\mathrm{mm}}{5\\ \\mathrm{in} \\times 20\\ \\mathrm{mm}} = 1.02\\,\n", "2cb15eba49897f03b84ad9e77d825c4b": "\\mathrm{12 Mg_3Si_2O_5(OH)_4 + 4 Fe_3O_4 + CH_4}", "2cb17e69baebaeed67190ad729bb2601": "1_5", "2cb18672238e91479d291478cb302c68": " \\langle \\xi | F(A) | \\xi \\rangle \\, . ", "2cb1919fda4e8392db1d424cdc94fa0a": "[a,a^{\\dagger}]=1,\\qquad[N,a^{\\dagger}]=a^{\\dagger},\\qquad[N,a]=-a, ", "2cb238ba84c1aa0a15ccb9f2ce09cba5": "\\frac{1}{96} + \\frac{1}{192} = \\frac{1}{64}", "2cb2568d3499efb768f2c421682316e8": "\\lambda\\,\\!", "2cb286802bb4aabb89e7c62b0648eff2": " f^{*} = \\frac{\\text{expected net winnings}}{\\text{net winnings if you win}} \\! ", "2cb28ec4013dadbc8063aadd8e187c27": "Initiates(a,f,t_1)", "2cb28ed4be529aa2166eee8d7955c037": "E=\\frac{p^2}{2m}", "2cb2b2183f61a154ab87e0698b0d6f8a": "\\chi =\\frac{M \\mu_0}{B}", "2cb2b9b0418bc5b18ad1982a75e241bc": "\\omega = \\frac{-1}2 + i\\frac{\\sqrt3}2.", "2cb2bb097139064aa7383fd6cf7e70ea": "\\|x\\|_{\\theta,q;J} := \\inf_v \\bigl\\{ \\Phi(v) \\;:\\; x = \\int_0^\\infty v(t) \\, dt/t \\bigr\\}.", "2cb2bf69c58a4619180baeffd5b6b4fa": "\\hat x,", "2cb34d5a09a2ed8a380bfda2900f3494": "H = -c_0-2\\sum_{m=1}^\\infty \\phi_m c_m = -\\ln\\left(\\frac{1-e^{-2\\gamma}}{2\\pi}\\right)-2\\sum_{m=1}^\\infty \\frac{e^{-2n\\gamma}}{n}", "2cb3a33b3b68da1111ea1e658bdd0e93": "|q_1\\bar{q_2}\\rangle = -|\\bar{q_1}q_2\\rangle", "2cb4007a0e6dfb0895e5f1a9b56e0b35": "ce(ab)", "2cb413c7403f08c3cc2dc2a113abfab6": "h'_1 = 1", "2cb460c7d9efd82b6b42aa2dd220825a": "r=|\\sqrt{(|6^2+16^2|)}|\\equiv|\\sqrt{|13|}|\\equiv", "2cb46753c4627a45a8d8365eba91b1d9": "t_i=\\alpha^{k_1}s_i-s_{i+1}=\\alpha^{k_1}\\sum_{j=0}^{n-1}e_j\\alpha^{ij}-\\sum_{j=0}^{n-1}e_j\\alpha^j\\alpha^{ij}=\\sum_{j=0}^{n-1}e_j(\\alpha^{k_1}-\\alpha^j)\\alpha^{ij}.", "2cb54e572fc2cceea8bdd74fb7d7c695": "s\\rightarrow-\\infty", "2cb55e3f31d5539c992377194a06888e": "\n\\mathbf{\\hat{b}_{t-1:T}} = c_t^{-1} \\mathbf{T}\\mathbf{O_t}\\mathbf{\\hat{b}_{t:T}}\n", "2cb5c136b4a833eb1b066f747c49781a": " \\operatorname{lambda-lift-op}[S, L, P] =P[L := \\operatorname{lambda-lift}[S, L]] ", "2cb6255fc9a886da9dd8429698c524f5": "\\delta_{ab\\dots}^{cd\\dots}", "2cb643ca3b0cc1f8aa61c06ade47ead3": "\\clubsuit_{S}", "2cb6b33cd55830e6a9da9a0e2262b448": "\\lim_{x \\to 0} \\frac{1 - \\cos x}{x} = 0", "2cb6e0f4f55f2748be3d7c1648f78d21": "\\operatorname{\\phi}", "2cb6f2385ee1c7f379ef15901f726a5e": "\nK(r,r^\\prime) + g(r,r^\\prime) + \\int_r^{\\infty} K(r,r^{\\prime\\prime}) g(r^{\\prime\\prime},r^\\prime) \\mathrm{d}r^{\\prime\\prime} = 0\n", "2cb6fac2a4f26cf5e9928511372cb889": "\n\\| \\mathrm{D}^{j} u \\|_{L^{p}} \\leq C_{1} \\| \\mathrm{D}^{m} u \\|_{L^{r}}^{\\alpha} \\| u \\|_{L^{q}}^{1 - \\alpha} + C_{2} \\| u \\|_{L^{s}}\n", "2cb73edb2a9c27a81a1eed84176c017c": " \\cos(\\omega t) \\cdot u(t) \\ ", "2cb765d47678896ee672f509cc4515c5": "\nH_\\alpha (\\mathbf{p},\\mathbf{r})=D_\\alpha |\\mathbf{p}|^\\alpha +V(\\mathbf{r},t),\n", "2cb76d336a8f35306d8d1efd0363c23c": "\\begin{align}\n {}^{\\frac{1}{2}\\pi}e &\\approx 5.868...,\\\\\n {}^{-4.3}0.5 &\\approx 4.03335...\n\\end{align}", "2cb7a2ed07d97fd3790a48310b4390d8": "\\left| {A_\\varepsilon}^{(n)} \\right| \\leqslant 2^{n(H(X)+\\varepsilon)}", "2cb7b3d830eb38f6d821cb5c755cc936": "R+P_1+P_2=P_\\infty", "2cb8861bdd6c1b68220bf0f2619dc40b": "\\,P\\left( 1;\\;0,\\;z_1, \\;z_2, \\;z_3,\\ldots,\\;z_{n-1}\\right)=0\\,", "2cb8a78db1abb5fb2c74c106662a0922": "O(\\operatorname{pcr}(G)^{\\frac{7}{4}}\\log^{\\frac{3}{2}}\\operatorname{pcr}(G))", "2cb8cf6761a3e79ea6ee05cc7fe7fb8a": "\n\\ln W=\\ln\\left[\\prod\\limits_{i=1}^{n}\\frac{g_i^{N_i}}{N_i!}\\right] \\approx \\sum\\limits_{i=1}^n\\left(N_i\\ln g_i-N_i\\ln N_i + N_i\\right)\n", "2cb8efc0092cf5acc457ad5eafe60ed4": "\\mathbf{A} \\cdot \\mathbf{x} = \\mathbf{b}", "2cb943bf7d005eba0b98d183eb805437": "u^\\dagger h u = \n\\begin{bmatrix}\n\\gamma_1 & 0 & \\cdots & 0 \\\\\n0 & \\gamma_2 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\gamma_{N^2-1}\n\\end{bmatrix}\n", "2cb95290c18d6c9a5351e651b2a7b765": "\\scriptstyle 1:\\sqrt2", "2cb9571382b4ac95d87889b2265c7950": "\\#(n)\\le C(n^k+1)", "2cb962466d94da15002772a945563003": "P_\\text{max}=\\frac{1}{4}\\cdot\\frac{V^2}{R_i} ", "2cb9927638ba93a5c8d353f984ff23ff": " \\log(AB) = \\log(A) + \\log(B).\\,", "2cba5a927f19db825adce6fa1448b91c": "\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}=-\\gamma \\mu_0 \\mathbf{m} \\times \\mathbf{H_{eff}} + \\frac{\\mathbf{m}}{m} \\times \\left( \\hat{\\alpha}\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}\\right)", "2cba5ed265afa65b81795f4b13b2d7bb": "2^\\kappa\\not\\rightarrow(3)^2_\\kappa", "2cba79170314b8c489cd649e08e88b16": "\\frac{d^2y}{dt^2}+\\left(\\theta_0+2\\sum_{n=1}^\\infty \\theta_n \\cos(2nt) \\right ) y=0. ", "2cbad4b9554d42c82002863c4bf2c4bd": "|+\\rangle=(|0\\rangle+|1\\rangle)/\\sqrt{2}.", "2cbae925b2b0ff0ea1ba2e70dd39db88": "S_N=\\sum_{n=1}^NX_n", "2cbb3ef450b0ba33115341b7d34650f7": " X=X^{i}\\partial_{i} ", "2cbb5cd5c60e425a29e25295f0fe329c": " w_i = V_\\infty \\sum_{n=1}^{\\infty} \\frac{n A_n \\sin(n \\theta)}{\\sin(\\theta)} \\qquad (8)", "2cbb62e7a3af6051ce982fbce733df65": "x^* = \\frac{x_{m}} {x_{p}}", "2cbb75d93e3fcb6fb019ec8d4ed5adde": "g(x)=c_1 \\varphi_\\lambda + c_2 \\theta_\\lambda + (\\lambda^\\prime-\\lambda) \\int_c^x (\\varphi_\\lambda(x) \\theta_\\lambda(y) - \\theta_\\lambda(x)\\varphi_\\lambda(y))g(y)\\, dy.", "2cbbd82131d1095100807d22cb228d18": "\\scriptstyle\\frac{c}{a}=", "2cbbe80566a48e82761a3aac046796ac": "\\mathcal{F}_{t} \\subseteq \\mathcal{F}", "2cbc0011405f6f81e746427301463188": " \\tan( h_{rgb}) = \\frac{\\sqrt{3}\\cdot (G - B)}{2\\cdot R - G - B} ", "2cbc11f5b7b98e5268a4a689227c7b51": "A_i \\subseteq \\N", "2cbc3a27f5ad35c9c86afb8effbc7e33": "O(d^5n\\log^3 B)\\,", "2cbc78592a8333162d6c233522d52780": " pI = {{pKa_1} + {pKa_2} \\over 2} ", "2cbc7e4d2581eea5c058827b09fb712a": "Eq \\; 3", "2cbca36861b4fb128dd6f22771a4659b": " 1-F(x) = o\\left(\\frac{1}{x}\\right)", "2cbce0a7eeea7c3a911f3830e2c0ae45": "X^3 - X^2 - 2X - 8.", "2cbd1773379fd277f7e0a222a1991b6c": "\\langle L\\rangle=Lclm\\big({\\mathbb L}_{x^2}(L),{\\mathbb L}_{y^2}(L)\\big)", "2cbd281bfb858d31f2e2ba76581fa5ea": "W=-\\int_0^{x_0} P (A dx)", "2cbd2b819b9390836aa2087eff3c3a32": "C_2 = G_1 + G_0 \\cdot P_1 + C_0 \\cdot P_0 \\cdot P_1", "2cbd9e36306966dbc0729e4e062d2210": "\\ \\frac{d}{dt}[P_{ij\\ldots}(\\mathbf X,t)]=\\frac{\\partial}{\\partial t}[P_{ij\\ldots}(\\mathbf X,t)]", "2cbdb4717d3d6516594dc52900873d1b": "\\left(A^{*}\\right)_{\\sigma|\\sigma'}=\\delta\\left(\\sigma_{1},\\sigma_{1}'\\right)A_{\\sigma_{2},\\ldots,\\sigma_{m}|\\sigma_{2}',\\ldots,\\sigma_{m}'},", "2cbeaf79634f10e8b0c9d8bbda4c5dda": " T_p' ", "2cbee6b4ab470a7a6813690b12b8d04c": " c_{23},c_{34},c_{24}", "2cbef96910428e860f9221c181a75613": " \\sigma_\\mathrm{tot} = \\int \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\mathrm{d} \\Omega ", "2cbf49356801deeaedf81dae14ce996c": "X^*_{\\tau(X^*, X)}", "2cbfedafd41fd8ce5e1a3a28f92a0fda": " y\\in \\bar{\\mathcal{S}} ", "2cc0219f84ed005292ee8a83706a06d9": "\\bar{X}_n=\\frac{1}{1-p}", "2cc02c5b5eb68413feadb0ab1b79d12e": "V_{BE}", "2cc03256b67450a68c31a86312b9014a": "\n \\mathbf{p} = q\\mathbf{d}\n", "2cc05c66618e235204b695e3bde9fef0": "\\varepsilon^{\\alpha\\beta\\gamma\\delta}\\,g_{\\alpha\\kappa}\\,g_{\\beta\\lambda}\\,g_{\\gamma\\mu}g_{\\delta\\nu} \\,=\\, \\varepsilon_{\\kappa\\lambda\\mu\\nu}\\,g \\,,", "2cc06b2fe411c92273d1de9d5f52e429": "\\langle \\sigma_{j}\\sigma_{j+N}\\rangle=\\left[\\frac{e^{\\beta J}- e^{-\\beta J}}{e^{\\beta J}+ e^{-\\beta J}}\\right]^N", "2cc07a47c67242717d3338cbda2a6b56": " \\Gamma_s = \\frac{q_s^2}{4\\pi\\epsilon_0 kT_s}\\sqrt[3]{\\frac{4\\pi n_s}{3}} ", "2cc07eb6965f2cd77a5f10f0d96a9b9e": "Mon: V_2 \\to V_2", "2cc098138ba50339593c537615aa5013": "U=U_{+}U_{-}=U_{-}U_{+}", "2cc0d0192f0e5172aea649fb67869e34": " f(p, q, r) = 0 \\,", "2cc127c427e6eaa0b2bc0869ee887da1": "(\\theta_r,\\phi_r)", "2cc12eb681e60182cfb2f52635b9a478": "r_{\\text{in}} \\triangleq \\frac{v_{\\text{in}}}{i_{\\text{in}}}\\,", "2cc169058cfbc7e858cfc1d4428b30e2": "\\frac{H}{R} = tan(\\mu) ", "2cc17958a59ceff58acd65300cd29f93": "H:\\mathcal{E}(G) \\to \\mathcal{V}(G)", "2cc1a4e95596704e4d6a68128a894bec": "\\lim_{n \\to \\infty} (\\mathbf I - \\mathbf A)^n = 0", "2cc1cc00bfc54f51398ef508e2daa72b": "{\\bar{HV}}_3", "2cc1d90da171c79c061ce41897930d30": "GL^+ < GL", "2cc1fce95f96d8ab9d8c27bda66c43dd": "J_2/J_1 \\,", "2cc202484a55e15bf695c7bf9d4223ba": "\\Lambda^1", "2cc2539b67edd709a77fb7169b0e4229": "x=\\hat{x}+e\\,", "2cc27db294a0fea1c078bd33d1c61c21": "\\frac{}{} E_{01} \\approx \\hbar \\omega_p", "2cc2cb7138d7e86e50fd8f61c1cf6cc1": "\na^{\\tfrac{p-1}{2}} \\equiv\n\\begin{cases}\n\\;\\;\\,1\\pmod{p}& \\text{ if there is an integer }x \\text{ such that }a\\equiv x^2 \\pmod{p}\\\\\n -1\\pmod{p}& \\text{ if there is no such integer.}\n\\end{cases}\n", "2cc306ccf5d35c4b805c52dc84ae8cdd": "\\sin i \\ \\cos u \\ \\hat{t}\\ +\\ \\cos i \\ \\hat{z}\\,", "2cc332931ef2e06b23ba77fc32689626": "f(x_1, ..., x_n)", "2cc35c0fb01e2ac3113bbfd22e547b32": "I = \\omega", "2cc36ddc959aaedaa7073c975fe15d9a": "M=(Q,\\Gamma,\\delta,q_0,g)", "2cc3b9aeafe53cdcb505086e27cebeab": "\\frac{y^2}{x^2}", "2cc3c9b5a5d4ea159e4395b941789c35": "\\omega_1 = -\\omega_2,\\ |r_1| = |r_2|", "2cc3ee1f8296933513198b1db917024c": "\\ell^1 = Y \\oplus \\ker Q", "2cc40e35ce3d11913ed4a3b26ba5446e": "\\mathbb{R}^k (k>1)", "2cc41a8c520f6d465991586c261f359f": "f(x;\\mu,c)=\\sqrt{\\frac{c}{2\\pi}}~~\\frac{e^{ -\\frac{c}{2(x-\\mu)}}} {(x-\\mu)^{3/2}}", "2cc4262b81118186a9b823bdc487192f": "n^{\\mu} = \\frac{1}{\\sqrt{2}} \\left( \\hat{t} - \\hat{r} \\right)\\ ,", "2cc4bfa54b0fd58be1d758b8aeae2a5d": "\n\\hat A D\\left ( x - y \\right ) = \\delta^4 \\left ( x - y \\right )\n", "2cc4c6e05691f91faf68b1e999913ad5": "\\sum_{d\\mid{n},\\; d 0", "2ceec8da1f443f213ccf3d84f5ab2392": "\n \\delta U = \\int_{\\Omega} [\\boldsymbol{\\sigma}:\n\\tfrac{1}{2}\\{\\boldsymbol{\\nabla}\\delta\\mathbf{u}+(\\boldsymbol{\\nabla}\\delta\\mathbf{u})^T\\} + \\{\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}+\\mathbf{b}\\}\\cdot\\delta\\mathbf{u}]~{\\rm dV} ~.\n ", "2cef299b2b4bfc8fe15ea4e822367d14": "I_z = I_x + Ar^2", "2cef33650b27ef223c174180a7f695d0": " (\\neg A \\or \\neg C)", "2cefcfbcd80b0e4d715135f87eace8a6": " X \\sim N(0,1)\\,", "2cefcfd498dc00b1044027b4abb057a6": "x/(x-1)", "2cf0db4d5c5370730eb5f58b5970ae3d": "\\frac{x^2}{2|\\varepsilon|}+\\frac{|\\varepsilon|}{2}.", "2cf0f146c475b92aecb7054ef612186d": " \\ddot{q} = M_A^{-1}Q_b + M_A^{-1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b). ", "2cf10d4dca47b97241aa425e8972081a": "V_T^n(V_S^n)", "2cf119822d369d767ac0eb5c0ff9be3b": "\\left|\\;\\frac{3i}{5}\\;\\right|^2=\\frac{9}{25}", "2cf150befe0140c9364e89cfc9bc8b5b": "f(x^1,\\ldots,x^m) = (x^1,\\ldots, x^k,0,\\ldots,0)\\,", "2cf1578fb7499d9b71de95886c493ee5": "(x+y)^p < x^p+y^p.\\,", "2cf174ffbaf29ca543408830f48c072c": " \\displaystyle{gW=(AW+B)(\\overline{B}W +\\overline{A})^{-1}}", "2cf17869c17f2d1bc9248be69c07324b": "\\cos x dx = d ( \\sin x) , \\,\\!", "2cf19559ee2796a95516caafe57be392": " \\ k(x,\\cdot) ", "2cf1a5cb48af17b16f03f007a5845856": "\\begin{array}{rcl}weight_{ingredient} &=& \\frac{weight_{flour}\\ \\times \\ baker's\\ percentage_{ingredient}}{100%}\\\\ &=& {weight_{flour} \\times baker's\\ percentage_{ingredient}}\\\\\\end{array}", "2cf1b49acdfe4c343d43829cbccebace": "\\mathbf{P}^2\\#\\mathbf{P}^2", "2cf1ddf495a51a905ed1a8debc22ab96": " \\sqrt[5]{2}-\\sqrt[5]{2}^2+\\sqrt[5]{2}^3-\\sqrt[5]{2}^4", "2cf1f636f13e5d0c024e5bb0865e31df": "Pr(V<64)=\\int_{27}^{64}{dV \\over 3 V log({5 \\over 3})}={log({64 \\over 27}) \\over 3 log({5 \\over 3})}={3 log({4 \\over 3}) \\over 3 log({5 \\over 3})}={log({4 \\over 3}) \\over log({5 \\over 3})} \\approx 0.56", "2cf22309facd370f1a74ce180bf5264f": " {f_i}=a_1 \\cdot sin(\\omega t)", "2cf2712a053b43f67a67aa80f195b59c": "{[x_1, x_2]}^n = [x_1^n, x_2^n]", "2cf27d04fa49b901c2bb191e347274a9": "a_n \\leq c_n \\leq b_n", "2cf2c137d80bcfd463709c3fbcf1b200": " z = |z| e^{i \\phi} = e^{\\ln |z|} e^{i \\phi} = e^{\\ln |z| + i \\phi} \\ ", "2cf2d0bb0242970c7ea041d49634ba6d": "20 * 50% = 10", "2cf30ae4e0d01219d3f7e29810d91365": "P(x)=p(x|r)=\\frac{p(r|x)p(x)}{p(r)}.", "2cf31bb4391a9cc4ef4961858ffb1f93": "n \\leq n^*", "2cf3403a76d38aaa6155faf8602134bc": "\\ \\begin{align}\n\\sum F_{y'} &= \\tau_\\mathrm{n} dA + \\sigma_x dA \\cos \\theta \\sin \\theta - \\sigma_y dA \\sin \\theta \\cos \\theta - \\tau_{xy} dA \\cos ^2 \\theta + \\tau_{xy} dA \\sin ^2 \\theta = 0 \\\\\n\\tau_\\mathrm{n} &= -(\\sigma_x-\\sigma_y) \\sin\\theta\\cos\\theta + \\tau_{xy} \\left( \\cos^2 \\theta -\\sin^2 \\theta \\right) \\\\\n\\end{align}", "2cf37cf9798fd88e291a932d543a92cf": "\\mathcal{A}(U)", "2cf3ead023ebc44c92396ccb9cfaa87f": "\n\\frac{1}{N} \\sum_{i=1}^N f(x_i)\n - \\int_{\\bar I^s} f(u)\\,du=\n\\sum_{\\emptyset\\neq u\\subseteq D}(-1)^{|u|}\n\\int_{[0,1]^{|u|}}{\\rm disc}(x_u,1)\\frac{\\partial^{|u|}}{\\partial x_u}f(x_u,1) dx_u.\n", "2cf3fe456a4e63e36fdd339faa275cf6": " A = \\sum _j \\lambda_j u_j u_j ^\\dagger ", "2cf412fc59bde512bd62a6bf1a0797ee": " W = \\int_C \\vec{F} \\cdot d\\vec{r}", "2cf416d50cf53e512bf41aafcc4e7509": "x_i = \\frac{\\alpha_i - 1}{\\sum_{i=1}^K\\alpha_i - K}, \\quad \\alpha_i > 1. ", "2cf457674df1a6835b284a8ab63927df": "7, 11, 19, 43, 67, 163", "2cf529b2dba2c2084c7476e42961a346": " N^{2+z}", "2cf53a45fbef5ef35580e234fb12347c": "R^q \\widehat{f}_* \\widehat{\\mathcal{F}} \\to \\varprojlim R^p f_* \\mathcal{F}_n", "2cf54007ca85d1c7f09f118c4c4c643b": "\n B (a\\,e (b\\,e-2 c\\,d m+b\\,e\\,m)+b\\,d (-3 c\\,d+b\\,e-2 c\\,d\\,p+b\\,e\\,p))+c\\,e(B (b\\,d-2 a\\,e)-A (2 c\\,d-b\\,e)) (m+2 p+4) x)\\left(a+b\\,x+c\\,x^2\\right)^{p+1}dx\n", "2cf58e51f8e94dadf39cf0b738c2544c": "\\textstyle 4+\\log_2(n)+\\log_2(2+\\log_2(n))", "2cf595abda56b944601b1455b7987329": "B_n=B_n(1)", "2cf5aeaa65d43a79c97947a44efd31b3": "\\begin{align}\nx & {} = \\frac{5}{4}\\left(1-\\frac{v}{2\\pi}\\right)\\cos(2v)(1+\\cos u)+\\cos 2v \\\\ \\\\\ny & {} = \\frac{5}{4}\\left(1-\\frac{v}{2\\pi}\\right)\\sin(2v)(1+\\cos u)+\\sin 2v \\\\ \\\\\nz & {} = \\frac{10v}{2\\pi}+\\frac{5}{4}\\left(1-\\frac{v}{2\\pi}\\right)\\sin(u)+15\n\\end{align}", "2cf61c489f5b6046419c361bd77f0013": " D_4(x,\\alpha) = x^4 - 4x^2\\alpha + 2\\alpha^2. \\,", "2cf6f95525476a3af9a36c1aca488a9b": "\\displaystyle\\omega = 2\\pi f", "2cf6ffef74971c367152f9d1771f3a7e": " {(J/m^2)} ", "2cf7060d7e548fc977bb934ff309e593": "\\delta(\\varepsilon) \\ge c \\, \\varepsilon^q, \\quad \\varepsilon \\in [0, 2].", "2cf722dc16073457a600659322b8065b": "\\Lambda^{*}", "2cf75c65107ad312e64d48fcc24dad9f": "\\mathcal{H}_{2} =\\pi _{2}^{2}+M_{2}^{2}.", "2cf77b516e7e862a3db86b65eec08a08": "\\overline{X}_{\\mu \\leq \\epsilon} \\cup_V \\overline{X}_{\\mu \\geq \\epsilon}.", "2cf7aac6fc6529e1d80553cf735f58f7": "\n\\begin{array}{lcl}\np(\\mathbb{W}^{k}\\mid z_{dn}) &=& p(w_{dn}\\mid\\mathbb{W}^{k,(-dn)},z_{dn})\\,p(\\mathbb{W}^{k,(-dn)}\\mid z_{dn}) \\\\\n&=& p(w_{dn}\\mid\\mathbb{W}^{k,(-dn)},z_{dn})\\,p(\\mathbb{W}^{k,(-dn)}) \\\\\n&\\sim& p(w_{dn}\\mid\\mathbb{W}^{k,(-dn)},z_{dn})\n\\end{array}\n", "2cf80b65abe71b156a3849a4b8481eb8": "{\\rm CAGR}(0,3) = \\left( \\frac{13000}{9000} \\right)^\\frac{1}{3} - 1 = 0.13040381433805558731822357533153 = 13.040381%", "2cf8a9681effaf98c7d806a7ae57298f": "(x^2 + 1) (x + 1) (x - 1)^2 = 0\\,", "2cf8c12f117afd1ea0950bfd05a267b3": "\nQ = \\frac{1}{R} \\sqrt{\\frac{L}{C}} \\,\n", "2cf8ce6b0f2b888ab7f083a7b031731b": " f_n(t) = n f_0(t) + \\frac{1}{2 \\pi} \\varphi_n^\\prime(t). \\, ", "2cf8e836b38e4d3bf9852ed169cca397": " h_0(n) = n,\\, ", "2cf907cfd1e593cc5c8695ba4b61fa80": "\\xi(t)\\equiv i\\sum_n\\kappa_n\\sqrt{\\frac{\\hbar\\omega_n}{2}}\\left(-a_n(t_0)e^{-i\\omega_n(t-t_0)}+a^\\dagger_n(t_0)e^{i\\omega_n(t-t_0)}\\right)\\,,", "2cf96113c3c1593214e0c17ab3f9b07e": "\n \\begin{bmatrix}\n 1 & 0 & \\tfrac{7}{3} & \\tfrac{1}{3} & 0 & 0 & -\\tfrac{4}{3} & 5 \\\\ \n 0 & 1 & -\\tfrac{2}{3} & -\\tfrac{11}{3} & 0 & 0 & -\\tfrac{4}{3} & -20 \\\\ \n 0 & 0 & \\tfrac{7}{3} & \\tfrac{1}{3} & 0 & 1 & -\\tfrac{1}{3} & 5 \\\\\n 0 & 0 & \\tfrac{2}{3} & \\tfrac{5}{3} & 1 & 0 & \\tfrac{1}{3} & 5\n \\end{bmatrix}\n", "2cf9637554fe1240b48b2a753291fab6": "\\Sigma^{0,Y}_{n+1}", "2cf98f35c0b3ede6135cd6bf50da8f8e": " \\sigma^0_h = (N + 1/2) e^2 / h ", "2cf9acdc0601cbe31258ca8805e948b1": "\\begin{align}\nt' &= \\gamma \\left( t - \\frac{vx}{c^2} \\right) \\\\ \nx' &= \\gamma \\left( x - v t \\right)\\\\\ny' &= y \\\\ \nz' &= z\n\\end{align}", "2cf9c6b3b17aa97fe9f6c6e1fd400a27": "\\mathfrak{P}^{115}", "2cfa63bec7822850743dad43d1ba1282": " \\int \\frac{e^2 \\rho(\\vec{r'})} {|\\vec{r} - \\vec{r'}|} d^3r' = \\frac{e^2}{4\\pi\\epsilon_0} \\frac{1}{|\\vec{r}_1 - \\vec{r}_2|},\\, \\text{ and }\\, V(\\vec{r_1}, \\, \\vec{r_2}) = \\frac{e^2}{4\\pi\\epsilon_0} \\Bigg[\\frac{2}{r_1} + \\frac{2}{r_2} \\Bigg] ", "2cfa79930da193e93258a17e1f5c3ced": "\\hat{\\mathcal{P}}_{S_l^k}", "2cfad96e35caf76ab6593ced65253183": " y(x) = c_{1}e^{(a + bi)x} + c_{2}e^{(a - bi)x} \\, ", "2cfb01e4a9c4752ea06099e719b8e047": "G \\over c^2~", "2cfb4ce53011eb09aa9bceb9da461381": "\\operatorname{DSPACE}\\big(f(n)\\big) \\subsetneq \\operatorname{DSPACE} \\big(f(n) \\sdot \\log(f(n)) \\big)", "2cfb5df57348f49e330e2993e5569c3e": "D(\\omega)", "2cfbcf5c324a383edb541d15d3c7582f": "\n \\{\\psi^1(\\mathbf{x}), \\psi^2(\\mathbf{x}), \\psi^3(\\mathbf{x})\\} = (q^1, q^2, q^3) \\equiv (r, \\theta, z) \n = \\{ \\sqrt{x_1^2+x_2^2}, \\tan^{-1}(x_2/x_1), x_3\\}\n ", "2cfbd93d0ec70698a529c084250b0e3d": "C(\\alpha,\\rho) \\cap \\Omega = \\rho", "2cfc361539fc998def2cff4d062a71df": "1.\\overline{3}", "2cfc725d8eaea32e1a9065708f9c3c18": "\\operatorname{ad} = d(\\operatorname{Ad})_e:T_eG\\rightarrow \\operatorname{End} (T_eG).", "2cfce1464233f2894cdad2122baea728": "T_{new}=T_{old} \\times \\frac{1.4388}{1.435} = 2848\\ \\text{K} \\times 1.002648 = 2855.54\\ \\text{K}", "2cfcfc1b6d19d385994f9ef7adb8a107": "\\text{0/1 reactance} = \\frac{\\text{kva base used in reactance in studied calculation}}{\\text{system short-circuit kva}}", "2cfd2f9a2417acdce73e41abf6f3f59b": "H^0(X, \\mathcal{M}^*/\\mathcal{O}^*)", "2cfd4650b6937b11ebe69e4f1771b198": "\n \\begin{array}{lcl}\n p^2(b + p^2)^2 - c^2 & = & p^2(s + q)^2 - p^2(s - q)^2 \\\\\n & = & 4p^2sq \\\\\n & = & 4p^2d\n \\end{array}\n ", "2cfd91cb922bc7a63485cb9ecaeb5f55": "F \\subseteq Q", "2cfd942da52a0304e27e196b99e9b11c": "f(t)=K\\left[{\\psi \\, \\Delta\\theta\\over F(t)}+1\\right].", "2cfe1b885dd32b252bf2e752ac053abd": "k = \\frac{\\ln 2}{t_\\frac{1}{2}} \\,", "2cfe80dd523284817290e9cb1383d494": "=p_n-\\frac{(p_{n+1}-p_n)^2}{p_{n+2}-2p_{n+1}+p_n},", "2cfe82399acd8c18a37c498e11a8adad": "t_1 \\le t_2 \\le t_3 \\le \\cdots \\le t_N. ", "2cfef040efacdff66a980914de1e0204": "\\operatorname{drop-params-tran}[L] \\equiv (\\operatorname{drop-params}[L, D, FV[L], []]) ", "2cff114027d519f32081bfaa1e3917c5": " 55n^3+2n+10", "2cff41febda34fba20c210f8f57f9881": "S = \\frac{tE}{Ot}", "2cff514d97ee42ec763fa226cfa5e942": " x= \\rho \\cosh\\sigma", "2cff553346aabb472e454541473a7171": "K_{e\\ sys} = {1 \\over 2} mv^2 + {1 \\over 2}(-m)v^2 = {1 \\over 2}[m+(-m)]v^2 = {1 \\over 2}(0)v^2 = 0", "2cff90207d1223f9bf18b0399f39cee0": "\\mathrm{d}A = - S\\mathrm{d}T - p\\mathrm{d}V\\,", "2cffbc5f072b56c2768e1ac4889efa1f": "\\scriptstyle\\|\\;\\|_{BV}:BV(\\Omega)\\rightarrow\\mathbb{R}^+", "2cffc76c5e67e9396032f44cce5135e4": "\n\\mathcal A = \\frac{d^2}{dx^2} - \\frac{d}{dx}\n", "2d000daf8678aec6bff5b41ff1e73341": " MRP_L = 3600 - 80L ", "2d005606558e87688f9a540eeb7e09e0": "i, j, k \\in I", "2d005b6c4b2b788d2b29183c6c647d12": "(1 - p_1) N(1-R) \\delta_1\\,", "2d007d32601db65a9f6f456dc0dc2441": "a + \\cfrac{1}{b + \\cfrac{1}{c + \\cfrac{1}{\\ddots}}} ", "2d00857e6f704ad6ca4bee9434b85980": "\\left(1+e^{-i 2 \\pi k/N} \\right)^{N-1}\\,", "2d00a44d4cba9302d14dc9eb8fceac97": "\\psi_{2n+1}", "2d00da8b06dfb0fb80a8cda1b63d0024": "A(z) = z\\frac{d}{dz}B(z)", "2d0109072f0086f3967e911f28b17584": " Y_{0,1} ", "2d01277f3982fdd7087d16555ee214fa": "\\sigma_{i + 1}^{2} \\leq \\frac{\\sigma_{i}^{2}}{i^{\\alpha}}", "2d018f97ec621a6c88fd74d30441b3ad": "S = k\\ln{W }\\,", "2d01c3b713ffbabd76a047ef37ab45c0": "C_e", "2d01cb9bc1958ef96c9940f3f1323216": "\\frac{\\partial \\boldsymbol{\\hat{\\varphi}}} {\\partial \\varphi} = -\\cos \\varphi\\mathbf{\\hat{x}} - \\sin \\varphi\\mathbf{\\hat{y}} = -\\sin \\theta\\mathbf{\\hat{r}} -\\cos \\theta\\boldsymbol{\\hat{\\theta}}", "2d01dcca30255056e6007b587533666f": "\\frac{1}{3}\\cdot B_1(t)", "2d02595cd359f3c112e4a82a5dadba92": "\\vec{e}_i = (\\vec{e}_{ix},\\vec{e}_{iy})", "2d02689dd5eedd6245e7bfa10207a89c": "\\langle l_1, r_1 \\rangle_w \\cdot \\langle l_2, r_2 \\rangle_w = \\langle l_1, r_2 \\rangle_w", "2d028f38b90e313aa0d784390ddf9ee7": " \\frac{y''}{\\sqrt{1+{y'}^2}}=\\frac{V_t}{V_d(A_x-x)} ", "2d029573e0d81a5d5f8d9a2d944ddec6": "fl()", "2d03155ec3cadf9523c51f2ec26b8227": " I_k ", "2d0319e36a3855fb72479a0d42521f39": " \\psi^{(\\pm)}", "2d0381725852f8ca3cb44f56ca874747": "\\vec {dR}", "2d03979981e0dfdedf53a6d46953edef": "\\int_2^Y\\left(\\sum_{2>1", "2d120fb1330e19e94c51ac36c40c79ff": "\\int_0^T X_{t} \\circ \\mathrm{d} W_t", "2d1289277a5361a836c4f7e8cd1530cf": " (2n-k)!", "2d12d1ec4110e92db3844d551c5f426c": "\\mathcal{F}^0 = \\mathrm{Id}, \\qquad \\mathcal{F}^1 = \\mathcal{F}, \\qquad \\mathcal{F}^2 = \\mathcal{P}, \\qquad \\mathcal{F}^4 = \\mathrm{Id}", "2d12da350b0bc23e0b78b4dfcacf68a6": "= \\frac{T(0,t)-T_\\infty}{T_i-T_\\infty}", "2d12ed149449f04aac77f188162df7f4": "\\{l^a\\,,n^a\\}", "2d1338ac19b95d96d88994baac778878": "T^k(v) = 0.\\!\\,", "2d133bbd468c3a0538444efc2747e6ad": "\\rho_2=\\begin{cases}\n (-2)^{\\tfrac{\\nu}2} & \\text{if } \\nu \\text{ even,}\\\\\n (-2)^{\\tfrac{\\nu-1}2}\\mathrm i & \\text{if } \\nu \\text{ odd.}\n\\end{cases}", "2d1372d1c01dc3786529e8db4d2b52d1": " 16 \\rightarrow 10_1 \\oplus \\bar{5}_{-3} \\oplus 1_5.", "2d138175bc656825216043350942dabd": "e=\\{x,y\\}", "2d13c549f3a33609b9db38b8352e5a62": "10^{2}", "2d13eeeb4077553dae90d3db63fd2a67": "\\operatorname{mod}\\sigma_y^2(n\\tau_0) = \\frac{1}{2}\\left\\langle \\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\bar{y}_{i+n}-\\bar{y}_i\\right]^2 \\right\\rangle", "2d13fa83fba559d69251a071832e99d6": "[X\\; Y] = [U_X\\; U_Y] \\begin{bmatrix}\\Sigma_X &0 \\\\ 0 & \\Sigma_Y\\end{bmatrix}\\begin{bmatrix}V_{XX} & V_{XY} \\\\ V_{YX} & V_{YY}\\end{bmatrix}^* = [U_X\\; U_Y] \\begin{bmatrix}\\Sigma_X &0 \\\\ 0 & \\Sigma_Y\\end{bmatrix} \\begin{bmatrix} V_{XX}^* & V_{YX}^* \\\\ V_{XY}^* & V_{YY}^*\\end{bmatrix}", "2d1421a5e6f73d736bbef3594ab49503": "O(nLkt)", "2d142a61cda3f216629643e7abba8adc": "t_{i,1} < t_{i,2} < \\ldots < t_{i,n_i}", "2d145b12f6ef1b314adf6135d4b2bdf5": "\\mathbf v = \\frac{1}{\\mathbf u}(\\mathbf u \\cdot \\mathbf v) + \\frac{1}{\\mathbf u}(\\mathbf u \\wedge \\mathbf v)", "2d145e3684093dda8dbfe869afa543f9": "P_{2}", "2d1473a7790431c7ac1f151c2ef74958": " S=F^{-1}P ", "2d14a1e261d260af3cce1d50a94f0043": "L = T - V = \\frac{1}{2} m \\dot{x}^2 + m g x.", "2d14f84a09dd9ebda163938000e5a73a": "\\textstyle Y ", "2d14f85ab1aa0c8c9c082c713ef25423": " S_2-S_1 = nC_p \\ln\\left(\\frac{T_2}{T_1}\\right) - nR\\ln\\left(\\frac{p_2}{p_1}\\right)", "2d1511957caf6fb24907aa51cf526ad2": " \\mathcal E_T ", "2d15a320a0674ba887895589bf6f4865": " k_\\mathrm{DM} = \\frac{k_e e^2}{2 \\pi m_\\mathrm{e}c} \\simeq 4.149 \\mathrm{GHz}^2\\mathrm{pc}^{-1}\\mathrm{cm}^3\\mathrm{ms}", "2d15af3a3fd45412adb07ebe71cadcdd": " H =Ri", "2d15bfcb6c9dbc4f5bd0e8d4acb5522a": "H_{ij}(0) \\neq 0", "2d15f9e346bdb40fc3241b4bc3335a87": "X_\\mathrm{L} = -X_\\mathrm{S}\\,\\!", "2d16254778efcc84da6dcaf339160f7d": "\\lg n", "2d162f16d55bd7bc041c45ca9857b62a": "\n \\cfrac{\\Gamma, A, A \\vdash \\Delta}{\\Gamma, A \\vdash \\Delta} \\quad (\\mathit{CL})\n ", "2d16350487dffa72dd4d61b95d93054a": "\nD^{2}U = 0\n", "2d165c2f6d09a08010737e7efdd0455b": "\\scriptstyle e^{bj} \\mapsto e^{(b-a)j/2}.", "2d168d22250dd578d12811dc75b34292": "v\\in E^u", "2d1699b520aea2577bff3c74595c3ac4": "j=1...m", "2d169b74c4f8c96a701411a08e57f782": "\\beta= 0", "2d16a47461bbc1019888aa19499802d9": "\\frac12+\\frac{x\\sqrt{R^2-x^2}}{\\pi R^2} + \\frac{\\arcsin\\!\\left(\\frac{x}{R}\\right)}{\\pi}\\!", "2d16b7626519813abc437f7fb5193dc5": "f(x) = \\arg\\min_{i\\in 1\\ldots m} A[x,i]", "2d16e8ce9c902269f4264751cf4f7aa8": "X_1, X_2, X_3\\ldots", "2d1747352900ef3aada5a9568805184e": " \\rho \\frac{\\partial u}{\\partial t} = -\\frac{\\partial p}{\\partial x} + \\mu \\left(\\frac{\\partial^2 u}{\\partial r^2} + \\frac{1}{r} \\frac{\\partial u}{\\partial r}\\right) \\, ", "2d17788a687fd488c3affec6a7eff584": "F_{t}=(f^{\\top}_{t},\\dots,f^{\\top}_{t-q})", "2d177daf06921a4e21ca94aead0ea36a": "g'(x) = \\lim_{h \\to 0} \\frac{g(x+h)-g(x)}{h}", "2d179a9573928c5bfb7e6d200bf92722": "\\sigma_y^2(n\\tau_0, M) = \\text{AVAR}(n\\tau_0, M) = \\frac{1}{2n^2(M-2n+1)} \\sum_{j=0}^{M-2n} \\left( \\sum_{i=j}^{j+n-1}\\bar{y}_{i+n}-\\bar{y}_i \\right)^2 ", "2d179ada557afde817d21eb98fa9e635": "a_n=n^{-\\tfrac{1}{\\alpha}}\\,", "2d17b3ab12cb4d68843b1d2521ebe85c": "\\frac{2 e^{-k}}{(1+e^{-k})^2}\\!", "2d17b77f49a4aeb76318e4d1f54fcafa": "\n \\boldsymbol{R} :=\\boldsymbol{\\nabla}\\times\\boldsymbol{F} = \\boldsymbol{0}\n ", "2d17ce0a9457e53a51633a87ca82cdfa": "\\mathrm{[H^+] = 7.74\\times 10^{-4}}", "2d17d6795eb2573f2fcf81695ee2ca22": "i^2>0", "2d1824aa42ab738c8fc56b5c93308b3e": "0\\le c\\le 1", "2d183fb64a403992e7fec0ec63b04ec6": "\\Theta_p:T_p\\mathbf{R}^3\\to\\mathbf{R}", "2d18906342cac6b849033226e423bbff": "Q \\colon A \\to A/B", "2d18f351df12266a370f27bbb3e87c49": "q \\ ", "2d1905a26c78e7bec9ebe7bd77fd24f9": "= {p_1(1-p_1) + p_2(1-p_2)}", "2d19115cf2cb000da08c947bf8469ca1": " f(x)=a^{f(x-1)} \\; \\; \\text{for all} \\; \\; x>-1, \\; f(0)=1,", "2d192263f4efcd97c6c7d2463fdd4ed6": "p = 7, q = 11, N = p^2q = 539, d = N^{-1} \\mod \\text{lcm}(p-1,q-1) = 29", "2d19a526d9f50f8285db72a27a51afa3": "\\sigma_x \\sigma_z", "2d19d205912dbbff1da14ecabc244e2d": "\\tau(n)\\equiv 705 \\sigma_{11}(n)\\ \\bmod\\ 2^{14}\\text{ for }n\\equiv 7\\ \\bmod\\ 8", "2d19d27ef591bb3986e2c5958d4353ec": "f(x,y) = 0.26 \\left( x^{2} + y^{2}\\right) - 0.48 xy.\\quad", "2d19e1977f68df47b49d03c262a6a726": "n/(pi*r^2)", "2d19e36d28fc21ace57cd66192f4012c": "Z=R+jX, \\quad Y=G+jB", "2d1a076849c1d997f1d49de051e8a0b0": "G(z\\mid \\cdot )", "2d1a0a5126233ccdf491f771b387e485": "\n f : A\\times (B\\times (C\\times 1)) \\to D\n", "2d1a8b4165cb5a38d9d05aa6821c0a0e": "u_2 = \\begin{bmatrix}{\\ }1\\\\-1\\end{bmatrix}.", "2d1a93d177a199f3261d845123bcf1ee": " M_{\\rm limit} = \\frac{\\omega_3^0 \\sqrt{3\\pi}}{2}\\left ( \\frac{\\hbar c}{G}\\right )^{3/2}\\frac{1}{(\\mu_e m_H)^2},", "2d1abd34379b1160fd44236763bb3299": "1/R^2", "2d1ad34c0cf34cbc8ecf7c5427a4320e": "\\psi_{\\alpha}(t)", "2d1b001293bae3da8bbbac98141b7b94": "S_x(t) = e^{-(At + B(c^{x+t}-c^{x})/ln[c])} = e^{-At} g^{c^{x} (c^{t}-1)} ", "2d1b0277fb58ff6532118b554ccfba9f": "x_2^\\prime = 0", "2d1b12fc4edbb51feeeae17aea601294": "\\circ_l", "2d1b16cd638c18188e937f892a37c1cb": "2k = 2", "2d1b2a11ff4a816536a8937f2ece2e9c": "\\le", "2d1b47eefb2038553eb556fff108bacd": "\\forall \\alpha. \\alpha \\to (\\alpha \\to \\alpha) \\to \\alpha", "2d1b60e3933612ca6f95883dc307ca49": "L : \\left[B\\ C\\right] \\to \\left[\\left[A\\ B\\right] \\left[A\\ C\\right]\\right]", "2d1b9ab066c4cf1e7090904dcce58e27": " \\theta: \\Omega \\rightarrow \\mathbb{R}", "2d1bc94a8026970aa9d96695abcd53e6": "\\| u \\|_{L^{\\varphi} (X)} = \\| u \\|_{L^{p} (X)}", "2d1bf66f26095f2039b4c0374ceb1c3d": "f_1(z)= z+d/c \\quad", "2d1ce0346208904ea9a16ff3f610c646": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} C_n(x; \\mu)C_m(x; \\mu)=\\mu^{-n} e^\\mu n! \\delta_{nm}, \\quad \\mu>0.", "2d1dc88200d501549f9d6edae3d6c195": "a\\times b", "2d1e1af1bd466185d6dbb512de144ef0": " x^{(5)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.8127 \\\\\n -0.6646\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8121 \\\\\n -0.6650\n \\end{bmatrix}. ", "2d1e305093ed7d1f8e732f52e33405e1": "f(j)M_n(i,j)", "2d1e369c482c72d644fb228f9c8bbba6": " f'(x) =\\frac {f(x+\\Delta x) - f(x)}{\\Delta x} + O(\\Delta x) ", "2d1e778d657e53ae82cab578dfc070a9": " B = [B_1, \\cdots, B_K] ", "2d1e92926ba5c2c728cb16b33181f96d": "\n{} + 4000ps^2 + 320r^3- 1600rsq\n", "2d1eb707af9686c3b5b0e08c56d8518d": "ax^2 + bx + \\frac{b^2 - 2b}{4a}", "2d1eced7612903b8a46f16583e79bf7e": "\\sin b = \\sin\\delta \\sin 27^\\circ.4 + \\cos\\delta \\cos 27^\\circ.4 \\cos (192^\\circ.25 - \\alpha)", "2d1f0a95c036edbea7b4a4339f071117": "W:=f^{-1}([c-\\epsilon,c+\\epsilon])", "2d1f179bf212f7d85ecd01bbc1b1f12e": " X_t = \\mu + \\varepsilon_t + \\sum_{i=1}^q \\theta_i \\varepsilon_{t-i}\\,", "2d1f1f94982ad0bb733101efb6837577": "0/1", "2d1f83c0d9fcee4d785056865984573b": "\\mathbf{t} = \\mathbf{n}\\cdot\\boldsymbol{\\sigma}", "2d1fb1548f01996c8ba1730eff41c09e": " K=\\frac{eB_0\\lambda_\\text{u}}{2\\pi mc} ", "2d1fded43b92205091869469c5f8ca93": "\\mathbf{P} \\mathbf{P}^H", "2d1ffd3286d21193cbc374c59132f7e6": "u(t, x) = \\mathbf{E}^{x} [ f(X_t)],", "2d20225af1f138bbce52b0366828a412": " W^u ", "2d20313b390b862c5cd04f73c36f087c": "\\theta_A", "2d2063253bbed943ebfdbb067c67961a": "s^\\delta(\\lambda) = \n\\sum_{n\\le \\lambda} \\left(1-\\frac{n}{\\lambda}\\right)^\\delta s_n ", "2d2122385144218728a5908ff23489bb": "I_d = \\frac{1}{12} m\\left(h^2+w^2\\right)", "2d212330308a5ade40ef7b58a0616530": "Q = \\begin{pmatrix}-\\alpha & \\alpha \\\\ \\beta & -\\beta \\end{pmatrix}", "2d2151eb416092522c4225a24b85094e": "\\mu_B = \\frac{e \\hbar}{2 m_e} = 1/2", "2d21a6e87c247c5a98bb48f969ded5c2": "dE_\\theta(t)={-d\\ell j\\omega \\over 4\\pi\\varepsilon_\\circ c^2} {\\sin\\theta \\over r} e^{j\\left(\\omega t-{\\omega\\over c}r\\right)}\\,", "2d21cdc32b24c094a873a5baebe859a3": "f_{\\text{r}}(\\omega_{\\text{i}},\\, \\omega_{\\text{r}}) = f_{\\text{r}}(\\omega_{\\text{r}},\\, \\omega_{\\text{i}})", "2d21dc273099a22902cc6955164c0a19": " x=\\sqrt n\\, \\sin\\,t ", "2d22058c70f4cb488f126ca7c1e12f02": "\\Delta w'=0", "2d22074222eab40144b9606c40cd19e1": "\\log \\frac{\\bar{Y}}{1-\\bar{Y}}", "2d223b88052c9181392df2e19bd6a1e2": " f\\cdot d = \\frac {d\\cdot c}{\\lambda},", "2d22887a2c558dc6fa160e797932d93e": " R^{\\rm T} \\mathbf z = X^{\\rm T} \\mathbf y,", "2d229c805629226c7a0759894e72ff23": "g_{ij}\\mapsto \\Omega^2 g_{ij} \\Rightarrow P_{ij}\\mapsto P_{ij}-\\nabla_i \\Upsilon_j + \\Upsilon_i \\Upsilon_j -\\frac12 \\Upsilon_k \\Upsilon^k g_{ij}\\, , ", "2d22ca40065a990af40f11f203debe3c": "\\phi : G \\to H,", "2d22e87414875f6b365510a2663e03f6": "\\frac{y_0 - n}{x_0 - m}=\\frac{b}{a}.", "2d22ebf928cc39f6be31923afcccde59": "\\ PF^T = J\\sigma, ", "2d2350a78c072408a29b1b8dc0b3503c": "\\delta_{i},\\ i=1,...,m", "2d235ac0e7c8871ee0a9c96064e35997": "\\begin{align}\nD_{a}=-\\ln\\frac{\\sum \\limits_l \\sum \\limits_{u} X_{u} Y_{u}}{\\sqrt{(\\sum \\limits_{l} \\sum \\limits_{u} X_{u}^2)(\\sum \\limits_{l} \\sum \\limits_{u} Y_{u}^2)}}\n\\end{align}\n", "2d23acd550741e43f199f3acbf97c829": " F_i |\\psi\\rangle. \\, ", "2d23d71122f0da29993aa779050dd7ec": "n=\\langle n|N|n\\rangle=\\langle n|a^{\\dagger}a|n\\rangle=\\left(a|n\\rangle\\right)^{\\dagger}a|n\\rangle\\geqslant 0,", "2d23ec027c648bbdbff083cd0ba55547": " \\alpha^\\dagger_{A,B,C}=\\alpha^{-1}_{A,B,C}:(A\\otimes B)\\otimes C\\rightarrow A\\otimes (B\\otimes C)", "2d240867019834b4005492e3a7d38662": "L=\\begin{cases}\n\\dfrac{V}{4.5} & V < 0.081\\\\\n\\left ( \\dfrac{ V+0.099 }{ 1.099} \\right ) ^{\\frac{1}{0.45} } & V > 0.081\n\\end{cases}\n", "2d24b375c9897230e123dae2f769f085": "\\hat{z}=\\hat{f}", "2d24c9eb86e23bd911e8e2c214447e6e": "X,Y \\in \\Gamma(E)", "2d25a62e37c21b0dd164bac2b7de642e": "\\epsilon_{s\\infty}", "2d261e21d9763d2c5793877d91b7f4a4": "\\kappa 0", "2d2a9955218458a175b555df2a159957": "d=h(d). \\, ", "2d2abe911ee9473c1a9180a87d3a8eac": " \\frac{\\partial L}{\\partial x}=0", "2d2aef689e28e29d1a71fdbf1072b22c": "V_{D}^{\\mathrm{low}}", "2d2af31f46c09e36dc36c3826d910650": " y_i = \\alpha+\\beta x^{*}_{i} +u_i ", "2d2b1ca2113743e485e4e294eeb095c7": " b_3 = b_1 \\times b_2 ", "2d2b64445cb639566a526e774eb3e2c6": "\nS = \\sqrt{2 \\Omega_{ij} \\Omega_{ij}}\n", "2d2b701bbd2888966f9ae78c71b60095": "d=1", "2d2b87013e56e07e512eed432f314166": "I_{hkl} \\propto |F(hkl)|^2", "2d2b9250957e7d0cb8ab1d521e1432eb": "\\omega,\\omega", "2d2b93d129eca1e260553bf3ede9c01d": "(a_n)_{n=1}^\\infty", "2d2bdf68e991adc215eaca1adf1ff649": "V_\\mathrm{A}=\\left\\{ \\begin{matrix}V-V_\\mathrm{E} & \\mbox{if}\\quad h\\geq D+d\\\\V-V'_\\mathrm{E} & \\mbox{if}\\quad h< D+d \\end{matrix} \\right. ", "2d2c2ec5f274424a8af2d93a70f8fb5e": "Q'_x(b,a)", "2d2c32a56e73b87c4627f2ee949744b1": "\n \\int x^m \\left((m+n (p+1)+1) A\\,b^2-a\\,b\\,B(m+1)-2(m+2n (p+1)+1)a\\,A\\,c+(m+n (2p+3)+1)(A\\,b-2 a\\,B) c\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}dx\n", "2d2ce37d432f21510decfd5d0b090841": "\\frac{dQ}{dP}", "2d2cfe9ec6b10171498084f20a44241e": "\\vec B", "2d2d627e4cb0d6a324952d5eadefec46": "\\epsilon_\\mathrm{Total}=\\epsilon_D+\\epsilon_S", "2d2d6be566c44f4bf03a4e9a88dd9c61": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi}-t_h\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ 3\\ \\sin^2 i\\ \\sin^2 u\\ du\\ =\\ \n-\\frac{9}{2}\\ \\sin^2 i\\ \\int\\limits_{0}^{2\\pi}\\ \\left(1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u\\right)^2\\ \\ \\sin^2 u\\ \\cos u\\ du\\ = \\\\ \n&-9\\ \\sin^2 i\\ e_g\\ \\int\\limits_{0}^{2\\pi}\\sin^2 u\\ \\cos^2 u\\ du\\ =\\ -2\\pi \\frac{9}{8}\\ \\sin^2 i\\ e_g \n\\end{align}\n", "2d2db0d720c38e2d85e39aad292c26cb": "f = {{f^\\circ }} \\exp \\left(\\frac{\\mu - \\mu ^\\circ}{RT}\\right)", "2d2e1441b60cb23080c76585d3234036": " \\det \\begin{bmatrix}g&k&p\\\\\ne&i&m\\\\f&j&n\\end{bmatrix} : \\det \\begin{bmatrix}g&k&p\\\\\nf&j&n\\\\d&h&l\\end{bmatrix} : \\det \\begin{bmatrix}g&k&p\\\\\nd&h&l\\\\e&i&m\\end{bmatrix}.", "2d2e2dba46b9ff906ee45da46e7e51c2": "d(\\pi_x(R))=x\\,", "2d2ec6d35daffadfb7e7614d5a598125": "\\beta_2 = s = ( F + \\Omega )", "2d2eca8e3c91543f842d75169a89dd0d": "\\theta_t", "2d2ee43b7c8b6c78dec4c5be87f1b965": "M_{12}=M_1+M_2", "2d2f16f9b87165823951dd1be17b1a01": "(\\mathbb{X}, \\mathcal{A})", "2d2f17967c7e1c235178042b1204c6fd": "\\mathcal{L}_X\\omega = i_Xd\\omega + d(i_X \\omega)", "2d2f2fd4eb09a2bd3442b225e1b7a311": " \\frac { \\mbox{density of sphere}} { \\mbox {density of gas} } = \\frac { \\mbox{weight of sphere}} { \\mbox{weight of sphere} - \\mbox{weight of immersed sphere}}\\,", "2d2f9a1a72c4c76ca300bd7ae5364d7e": "A =\n\\pi r^2 \\cdot \\frac{L}{2\\pi r} = \\frac{r \\cdot L}{2}\n", "2d2fcde765a2ba8503e435e5c64211f0": "\\ddot{x}_j=c^2\\left(\\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\\right)[1+\\alpha(x_{j+1}-x_{j-1})]", "2d2fdb1f3076b3bd4b4cd132bc2531e3": "2^3P", "2d300433d40d716331c8189cfdff9491": "(3\\times 3^4:2\\cdot A_6): 2", "2d304ba9bc1a1f8bff982fa62e17c0f8": "\n\\arctan \\frac12 = \\frac{1}{1\\cdot 2} - \\frac{1}{3\\cdot 2^3}+ \\cdots +\n\\frac{(-1)^{r-1}}{(2r-1)2^{2r-1}} + R_1 ,\n", "2d30738deac595c68ff6d5c49147dfa9": "\\pm\\left(\\pm\\sqrt{10},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "2d308fada3c167f8a006392de39831b6": "\n\\mathcal{G}(\\mathbf{k},\\omega_n) = \\int_{-\\infty}^{\\infty} \\frac{\\mathrm{d}\\omega'}{2\\pi}\n\\frac{\\rho(\\mathbf{k},\\omega')}{-\\mathrm{i}\\omega_n+\\omega'}.\n", "2d30917a59f259d26543c73da3e53f7d": "T=10^{0.2 \\times (ITG-1)} \\cdot (0.45 \\times \\sqrt[3]{D}+0.001\\times D)", "2d309a00396d369a741f59ad0e317084": "(\\sigma)", "2d30bfb430fa4394f508a6d4612dad2e": "d\\sigma\\,d\\varepsilon = d\\sigma\\,(d\\varepsilon_e + d\\varepsilon_p) \\ge 0", "2d30c8e47ae88fde7385a8c874cbd8df": "\\Delta H \\equiv \\Delta H_\\text{L} + \\Delta H_\\text{T} = {\\mu_\\text{B}\\over \\hbar m_\\text{e} e c^2}{1\\over r}{\\partial U(r) \\over \\partial r} \\boldsymbol{L}\\cdot\\boldsymbol{S}. ", "2d30d8a264fbcf33cc425b4e289bb113": "u(x(t)) = u(x(0)) + \\int_0^t \\xi(x(s))\\cdot \\dot x(s)\\, ds.", "2d310e3e8df49219beb4fc5e9fbbfaef": " H = H_{\\rm entropic}+H_{\\rm enthalpic}+H_{\\rm external} = \\frac {1}{2}k_B T \\int_{0}^{L_0} P \\cdot \\left (\\frac {\\partial \\vec r(s) }{\\partial s}\\right )^{2} ds + \\frac {1}{2}\\frac {K_0}{L_0} x^{2} - xF ", "2d317777bbf66371d631b3de577f21b5": "\\gamma = \\tfrac{1}{\\sqrt{2}} (1 - \\sigma_1 \\sigma_2) \\,", "2d31a51359d127f5664c0d0eeeaec316": "RT=\\left(P+\\frac{a}{V_m^2}\\right)(V_m-b)", "2d31ace0a977c59fc799d4caf90b93fb": "K(\\lambda)", "2d31ddb3af34e15a2e904e467f34c892": "{\\mathbb R}^n", "2d323b1c8e8404cb7609d2caa3d6c5ea": "x^2z^2 > xy^2z > z^2 > x^3", "2d32d2ece92c482f85e1247fc1a1962e": "[7,3,4]_2", "2d330094fb0b8ee0a53eab07a73d052c": "h \\in \\oplus _{n = 1 }^k \\mathcal{H}_k ,", "2d3322be4fdf132ddc31b4c826faee65": "HF(L_0, L_1) \\otimes HF(L_1,L_2) \\rightarrow HF(L_0,L_2), ", "2d334632384dcb6cdca897c7792dc339": " \\min_x \\|y-Ax\\|^2_2", "2d336d8f353725b22d0d64d13dc8d78c": "X^*\\in TS^*", "2d33d3bc4cc4604c0931e7eccae0b661": "V_{LAB}-\\frac{c}{n} = \\frac{\\frac{c}{n}+v}{1+\\frac{v}{cn}}-\\frac{c}{n}=\\frac{\\frac{c}{n}+v-\\frac{c}{n}(1+\\frac{v}{cn})}{1+\\frac{v}{cn}} ", "2d33f128928bcbb6ae624447bcd87376": "PostCaP~ICO_{PH} = PreCaP~ICP_{PH} * LH+ = 0.00875", "2d34497e2447752d12883317fce7def8": "\\mathrm{NapLog}(xy) = \\mathrm{NapLog}(x)+\\mathrm{NapLog}(y)-161180950", "2d34658b8076346e29bce5436e3d6bc6": " F_n = {{\\varphi^n-(-1/\\varphi)^{n}} \\over {\\sqrt 5}}. ", "2d34c8de4de58045a7236b680f0f8965": "I(\\lambda)", "2d353a0c8b02273ec2f6c6068981f7a5": "\\mu_f(\\theta) = \\theta", "2d35b8a2efc51963df2e255e749f3915": "(8)~~ ~~ V(p)=W(p-p_0)", "2d35fdbabc01d0483489c068b1a58bab": "0=\\gamma_\\mathrm{SG} - \\gamma_\\mathrm{SL} - \\gamma_\\mathrm{LG} \\cos \\theta_\\mathrm{C} \\,", "2d360bd81e349f70d9a3c1141d6ed0ee": "\n S(x) = 1 - \\frac{x}{\\omega}, \\qquad 0 \\leq x < \\omega.\n ", "2d364680595876f6acaee7fb621f3dba": "\\delta x_{wP}", "2d37325e703e8be7c3d34512f56cae3c": " \\vec{v}", "2d3735226b3f946b953893fc640b85ff": "\\scriptstyle 2 \\sqrt{2}", "2d37b6688264fbc64d80e11ec1caa507": " \\left\\lfloor \\lg (n) \\right\\rfloor", "2d37cc0a81d638e4d2ce257be80fa392": " \nR(D) = \\lim_{n \\rightarrow \\infty} R_n(D)\n", "2d37d83d5462ed556e45c72aadaf9b3d": "v(b)=v(a)+\\frac{1}{C}\\int_{a}^{b}i(t)dt.", "2d383d6fc883ff9865b30e8a7d64c2a6": "a(t) \\equiv (1+z(t))^{-1}\\!", "2d3860573cefc7922a1b8fd37cc73c93": "Y = b_0 + b_1A + b_2B + b_3A*B + \\varepsilon", "2d388bc60a1c3761d95cb2c3c873b625": "\\mathbf{F}^e", "2d3891389ad0e8e28fa0998af3abd087": "E^\\mathrm{damping}(\\mathbf{x}_j,t)=\\frac{e}{6\\pi c^3}\\frac{\\mathrm{d}^3}{\\mathrm{d}t^3}x", "2d38ab3ed41c709130ec2e96c7e1fba9": "\\varphi_{n+1}(x) = o(\\varphi_n(x)) \\ (x \\rightarrow L)", "2d38be949f6227285305008f7273fd1b": "dG = - SdT + Vdp = \\frac{\\partial G}{\\partial T}dT + \\frac{\\partial G}{\\partial p}dp \\,\\!", "2d38d55661959f6e8f8b8f5c059e5e79": "p = \\frac{1}{f} = \\frac{1}{k \\cdot v}", "2d39843d7c2f6fa6bbfe0ef6a9ab3e95": "\\textrm{Cov}\\left(m,t\\right)", "2d39fa3f8c8b934388f9bacd5deccf11": "\\textstyle (x, x', y,s)", "2d3a16c1c35d1cee2ed21031f176ea48": "\\Omega^G_*", "2d3a1b318c0185d938000a85b9866332": "[x_1 : x_2]^{-1} = [x_2 : x_1].", "2d3a263e4223d9c578667932730a2356": "\n\\overline{\\mathbf{PT}}^2 =\n\\overline{\\mathbf{PM}}\\times\\overline{\\mathbf{PN}} =\n\\overline{\\mathbf{PA}}\\times\\overline{\\mathbf{PB}} =\n\\left(s - r \\right)\\times\\left(s + r \\right) =\ns^2 - r^2 = h. \\, \n", "2d3a30e408f26b5e0799099610b00ef2": "\\rho \\theta + \\rho u u =\\sum_i f_i \\vec{e}_i \\vec{e}_i.", "2d3a44510b3dc981f21d363b5af2956b": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} = \\Delta_rG + RT \\ln Q_r~", "2d3a832625034bf023e49358c6222a00": " \\big. R = \\frac{1}{U} = \\frac{\\Delta x}{k} = \\frac{A\\, (-\\Delta T)}{\\frac{\\Delta Q}{\\Delta t}}, \\quad", "2d3b07f4be78707110251174655f1734": "10^{5}", "2d3b350c2146e50727d103b247366344": " |B_H (t)-B_H (s)| \\le c |t-s|^{H-\\varepsilon}", "2d3b3650e38eb746ed32b485bb6dc587": "E_r=(-1/2mc^2)[E_n^2+2E_ne^2\\langle 1/r \\rangle + e^4\\langle 1/r^2 \\rangle]", "2d3b6d393e8c44652294f6aee0b92250": "\\scriptstyle\\varphi|_{\\partial D}", "2d3b85fc69f443605fba5102f44d8420": "\\lesssim, \\lnsim, \\lessapprox, \\lnapprox \\!", "2d3ba77999aaa4615bfc20174cc19002": "\\scriptstyle \\lim_{t \\to 0^+} f(t) \\;=\\; \\lim_{t \\to 0^+} g(t) \\;=\\; 0", "2d3c2ea4e5f4830b8a743b7c5865ea02": "V_n(R) = V_{n-1}(R) \\int_{-R}^R (1 - (x/R)^2)^{(n-1)/2} \\,dx.", "2d3c3d0d3e437b7ffa52cb0cb3a3b111": "\\Gamma(1-z) \\; \\Gamma(z) = {\\pi \\over \\sin \\pi z}.", "2d3c7230d0e29417491f500dae927128": "m^p \\equiv m \\bmod p", "2d3cbf7e9967b74d1d9f93b69bae4d6c": "x_{dqo} = Kx_{abc} = \\sqrt{\\frac{2}{3}}\\begin{bmatrix} \\cos(\\theta)&\\cos(\\theta - \\frac{2\\pi}{3})&\\cos(\\theta + \\frac{2\\pi}{3}) \\\\\n - \\sin(\\theta)& - \\sin(\\theta - \\frac{2\\pi}{3})& - \\sin(\\theta + \\frac{2\\pi}{3}) \\\\\n\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2} \\end{bmatrix}\\begin{bmatrix}x_a\\\\x_b\\\\x_c\\end{bmatrix}", "2d3cdfffcabc3a4f8506db979065b231": "\\partial_\\mu\\left[\\frac{\\partial \\mathcal{L}}{\\partial[\\partial_\\mu \\varphi]}\\right] = \\frac{\\partial \\mathcal{L}}{\\partial \\varphi}\\,,", "2d3ce365958f27e32adbafa92ee3ff93": "\\sum_{n=1}^\\infty (b_n-b_{n+1})", "2d3d1aacb6e2febf071b07cc5839f093": "\\kappa < -1", "2d3d1f9e9749ba3c8e33b34b9c505d40": "q_{n+1} = c_0q_0 + ... + c_nq_n =\\{q_0, q_1, q_2, \\dots, q_n \\}c", "2d3d31315270b7ae9de1920ed3508ba2": "\\mathbf{m_1} = \\begin{bmatrix} m_{1}(1) \\\\ m_{1}(2) \\\\ m_{1}(3) \\\\ \\vdots \\\\m_{1}(L) \\end{bmatrix} ", "2d3d49f1f56dc4d85fc018fdf1608909": "x' = x-1, \\ y'= y", "2d3d54e6e6576d8fd63a79a80e898b6e": "e_{ij}(s)", "2d3d573f0dc5fba4b4f13f2a0e89d136": "(\\theta - \\beta)=\\frac{\\theta H_k}{B+H_k}", "2d3d5d8666e2ab6292804cb34d931be6": " x_1(t)= d + v_1 t = 200\\ + \\ 22t\\ ; \\quad x_2(t)= v_2 t = 30t ", "2d3def01faba620d601cae0136062121": "\t\\begin{array}{rrr|r} \n 1x^2 & -9x & -27 & -123 \n\\end{array}", "2d3df6de1622c3423727ccba9b889252": "\\ln[x_0,\\dots,x_n]", "2d3dfc2eb4424bbc0afe45cace61ee6b": "\\operatorname{cvs}(\\theta)", "2d3e1a4d3de6df91a330849b8a1c057a": "i\\le n,", "2d3eb3aceb3cb48a9397a0c0d1895ba9": "\\sigma_{A}^{2}\\sigma_{B}^{2}\\geq \\left(\\frac{1}{2}\\mathrm{tr}(\\rho\\{A,B\\})-\\operatorname{tr}(\\rho A)\\mathrm{tr}(\\rho B)\\right)^{2}+\\left(\\frac{1}{2i}\\mathrm{tr}(\\rho[A,B])\\right)^{2}", "2d3f1b24c9e782a93c540037098824e1": "{{I}_{SC}}", "2d3f706a2fc7d9a94dcd1e44a4501ce6": "a_s = G_{11}a + G_{12} b", "2d3fa6c5dfc8905510c645518ad6d7e8": "\\tfrac{n}{2\\pi}\\sin{\\tfrac{2\\pi}{n}}", "2d3fdc651d296cf7a5bde9d58fa58c47": "v\\,", "2d403341f643ab5e550bde6f6fbebf1b": "\\mathcal{L}_\\text{approx}(\\theta \\mid x \\text{ in interval } j)= f(x_{*}\\mid\\theta), \\!", "2d4039fa89275bf5381bf6750c5327ca": "f^{\\star} \\left( x^{*} \\right) := \\sup \\left \\{ \\left. \\left\\langle x^{*} , x \\right\\rangle - f \\left( x \\right) \\right| x \\in \\mathbb{R}^n \\right\\}", "2d4085a22ed7bd4d9cf0389f8e1dc86a": "[b_{ij}]", "2d40a687553bf45544ff1db6bd70179c": " {v^2 \\over r} = {GM \\over r^2} ", "2d40ccc8c6d93d54c947af3f9262fe48": "\\pi(x + x^\\theta) - \\pi(x) \\sim \\frac{x^\\theta}{\\log(x)}", "2d40f295331433ae2dce5b9e9098c2c7": "\n\\text{minimize} \\quad \\text{over } \\widehat D, P \\text{ and } L \\quad \n\\operatorname{vec}^{\\top}(D - \\widehat D) W \\operatorname{vec}(D - \\widehat D)\n\\quad\\text{subject to}\\quad \\widehat D = PL\n", "2d40f77e577fb3cc7f721b16ba116733": "\n\\tan(\\phi) \\ \\stackrel{\\mathrm{def}}{=}\\ v / u\n", "2d4106d0b5ea3fb84dcb8246ce79b56c": " E_0 = m_0 c^2 \\,,", "2d4160c375b76a312580a705c7a8a59c": "(\\pi-\\psi)/2", "2d41647e10dd7cac1c4878fe43dd9aa5": "X \\subset P^4", "2d417401580207d19f4160c2da6e63ad": "F(x)\\le\\liminf_{n\\to\\infty} F_n(x_n).", "2d418c08a8c11f04333c8ed81a89dadf": "F(x, y, z) = f(x,y) - z", "2d41946f41003c6692cbfbb3a9497c43": "\\frac{\\mathrm{D}}{\\mathrm{D} t} = u_x \\frac{\\partial}{\\partial x} + u_y \\frac{\\partial}{\\partial y}", "2d41db0ecbe06541f61ea6f8fe9a34bc": "\\mathbf E_{1s}^{rel}", "2d4203997f112df53831ee97c6dc7f17": " \nF = \\frac{\\mu_0^2 \\sigma v m^2}{128 \\pi L^3}\\hat{e}_z\n", "2d4264e5d9b4de72b08afa30a2c851ea": " \\mu_{p}", "2d42820c95f16c3398e84b88d4e2ebee": "\\omega_2'=\\lambda\\omega_2", "2d42eea58f3b1ba2fd01b587353e18c8": " \\Delta_{p \\times p} = \\text{diag}\\left[\\delta_1,...,\\delta_p\\right] ", "2d438c97082899ad151bf3ea669a31e8": "\n\\frac{d}{dt}\n\\left(mr^2\\dot{\\theta}\n\\right)\n-mr^2\\sin\\theta\\cos\\theta\\dot{\\phi}^2+\nmgr\\sin\\theta =0\n", "2d43b26a4d315715f55e6e5b084c3450": "v_d = j/ne", "2d43b9f87fb847d68802f3b6e2c1d2b9": "P(n>0) = 1 - P(0)", "2d43f9b9f80ff85809360c40a7ccb1ef": "U_{0}", "2d440b9e21c830e6d4f4acdcf054890d": "h(t)=\\begin{cases} \\frac{1}{{m\\omega _d }}e^{ - \\varsigma \\omega _n t} \\sin \\omega _d t, & t > 0 \\\\ 0, & t < 0 \\end{cases}", "2d441478cadb4a49f04ffbaa8be47581": "F = \\frac{Id}{k} ", "2d4456b8bb81e65c2a5480ae73297108": "-\\frac{d[A]}{dt}=(k_1+k_2)[A]", "2d447ab127581097bce7da6e66280435": "\\mu = c\\sqrt{\\pi/8}\\;\\frac{\\chi e^{-\\frac{\\chi^2}{4}} I_1(\\tfrac{\\chi^2}{4})}{ \\Psi(\\chi) }", "2d44b820a513b1765e42427714c0ccdc": "\n\\begin{pmatrix}\nI & S_1 & \\cdots & S_1 \\\\\nS_2 & I & \\cdots & S_2 \\\\\n\\vdots & & \\ddots & \\vdots \\\\\nS_p & \\cdots & S_p & I \n\\end{pmatrix}\n\n\\begin{pmatrix}\nf_1\\\\\nf_2\\\\\n\\vdots \\\\\nf_p\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nS_1 Y\\\\\nS_2 Y\\\\\n\\vdots \\\\\nS_p Y\n\\end{pmatrix}\n\n", "2d44c72375b0f0efea5430a3d2ba73d8": "\\mu=\\frac {4\\pi^2a^3} {\\tau^2}", "2d44dc72213ed0e6a45c45edb53fc144": "\\sum_{n=1}^\\infty \\frac{\\lambda(n)q^n}{1-q^n} = \n\\sum_{n=1}^\\infty q^{n^2} = \n\\frac{1}{2}\\left(\\vartheta_3(q)-1\\right),", "2d44e85f40c6b1bdadffa9765bb123b4": "\\operatorname{OE}[a](t) = 1 + \\int_0^t a(t_1) \\, dt_1+ \\int_0^t \\int_0^{t_1} a(t_1) a(t_2) \\, dt_2 \\, dt_1 + \\cdots.", "2d44eb7630a94deeeb8d9b1699d36012": " \\|\\mathbf{A \\times B}\\|^2 +(\\mathbf{A \\cdot B})^2 = \\|\\mathbf A \\|^2 \\|\\mathbf B \\|^2 ", "2d45b61b1a15b5331b8dd5c002c880cf": "{\\mathbf{}}\\tau_i\\hat{P}_i=\\hat{P}_i\\tau'_i=\\hat{P}_i, \\tau'_i\\hat{S}_i=\\hat{S}_i\\tau_i=\\hat{S}_i", "2d45c7ddd24e0c92f7a17de17378a10b": " x = A \\sin\\left ( \\omega t + \\phi \\right ) \\,\\!", "2d45ffa96c1c604b2793f7b62e320803": "\\lbrack\\mathbf y\\rbrack = \\lbrack\\mathbf y\\rbrack_1 + \\lbrack\\mathbf y\\rbrack_2", "2d4606b9be1bb1dd4e83019ed47ce1b1": "P(a)=P(b)=false", "2d461ee51ac7f06021d2d2c0d4918582": "y'(\\phi)=\\cos\\phi", "2d4664f54c5d238fda150e9a0b93764e": " (u_1-x_1)^2+(v_1-y_1)^2+\\cdots ", "2d4674dea1cfd5cc3b21f06a852bfc5d": "\\vec{L}", "2d469b9a772e4874308289d354a21f98": "\\{\\to,\\bot\\}", "2d469beb8b725a3d5e0d7a7b5b146c7a": "c_{n,k} = \\frac{1}{\\omega_{n-1}}\\frac{2k+n-2}{(n-2)}.", "2d46b3ccec6eafc91694a69af222c4b8": "(x',y') = ((x \\cos \\theta - y \\sin \\theta\\,) , (x \\sin \\theta + y \\cos \\theta\\,)).", "2d46b7232649565ce757db2037a50d33": "X \\cap \\lambda", "2d46be36ed05737f3b303a4e27bc66d5": "x^2-h=b^2", "2d475a5db4fe51ecfad3d2adad0d3abd": "\\beta_{k-1}", "2d4788f6bc2683035aa414a58ab4bd2a": "\\textstyle \\int \\!f\\,dx = \\sum\\limits_{i=0}^n \\alpha_i \\ell(A_i),\\,", "2d47c3b48318fcef886f7705dfd2b566": "a=0,", "2d47efd7391442c127889f3e8fbd572e": "w(x,y)", "2d4822b9cbfc7984c47223239821f296": "x(0) = 0", "2d482c556f628df9501325e97510bd8a": "\\frac{\\epsilon^2}{\\delta^2}S_0'^2 + \\frac{2\\epsilon^2}{\\delta}S_0'S_1' + \\frac{\\epsilon^2}{\\delta}S_0'' = Q(x).", "2d4830b062d8db1fa19ad7d6f81aa869": "\\frac 1 2 \\mathcal I_X(\\theta)\n \\ge \\frac 1 {2\\mathrm{Var}(X_\\theta)}\\left(\\frac {d\\mu_\\theta}{d\\theta}\\right)^2,", "2d48628a4d0da03c7658aec367f134db": "\\mathcal A,\\mathcal B\\vdash\\mathcal C", "2d48e27674c00187c063ee55161e363f": " \\frac {{\\dot{m}_O} - {\\dot{m}_S}} {\\dot{m}_O}\\ ", "2d48e90b6d0bb3eca1fbfcb131a8e53a": "\\overline{X}_n \\, \\xrightarrow{\\mathcal D} \\, \\mu \\qquad\\text{for}\\qquad n \\to \\infty.", "2d4900c9afc1a0590cda257855479f95": "\\phi(t;\\lambda) = \\int_0^1 \\exp (\\,i t\\,Q(p;\\lambda))\\,dp", "2d4957509147949423fc62650ce11e8d": "xp_{r-1}", "2d499300a09da719e5e7f70b3bee817c": "\\begin{align}\n \\Rightarrow \\eta'(0) &= -\\zeta'(0) - \\ln 2 = -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\ln n-\\ln (n+1)\\right] \\\\\n &= -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\ln \\frac{n}{n+1} \\\\\n &= -\\frac{1}{2} \\left(\\ln \\frac{1}{2} - \\ln \\frac{2}{3} + \\ln \\frac{3}{4} - \\ln \\frac{4}{5} + \\ln \\frac{5}{6} - \\cdots\\right) \\\\\n &= \\frac{1}{2} \\left(\\ln \\frac{2}{1} + \\ln \\frac{2}{3} + \\ln \\frac{4}{3} + \\ln \\frac{4}{5} + \\ln \\frac{6}{5} + \\cdots\\right) \\\\\n &= \\frac{1}{2} \\ln\\left(\\frac{2}{1}\\cdot\\frac{2}{3}\\cdot\\frac{4}{3}\\cdot\\frac{4}{5}\\cdot\\cdots\\right) = \\frac{1}{2} \\ln\\frac{\\pi}{2} \\\\\n \\Rightarrow \\zeta'(0) &= -\\frac{1}{2} \\ln\\left(2 \\pi\\right)\n\\end{align}", "2d49f2f1a448ea1b7e48943391adda20": "\\mathbf{H} = \\mathbf{R}_R^{1/2} \\mathbf{H}_w (\\mathbf{R}_T^{1/2})^T", "2d4a0d7d4301d2d54665fe85386594aa": "|\\vec{r}_1-\\vec{r}_2|", "2d4a3ae66b80eb6f2cb99c8cd32b350f": "\\mathit{L}_{D}", "2d4a58c72c0982aee0ac328d9eb5cc7d": "\\mathbf{Q}_{\\mathbf{X}} = \\frac{1}{n} \\mathbf{M}_{\\mathbf{X}}^T \\mathbf{M}_{\\mathbf{X}}, \\qquad \\mathbf{Q}_{\\mathbf{XY}} = \\frac{1}{n} \\mathbf{M}_{\\mathbf{X}}^T \\mathbf{M}_{\\mathbf{Y}}", "2d4a64f2756a60e4b2ca053679035522": "\\frac{\\rho^2}{(\\rho-1)^2\\;(\\rho-2)}\\,", "2d4aa00f619db6af83d8371f2f8b0654": "df(x, \\Delta x) \\stackrel{\\rm{def}}{=} f'(x)\\,\\Delta x.", "2d4aac63a7541e4163bf1a6959f6be24": "\\scriptstyle p_{CO_2}=3.5\\times 10^{-4}", "2d4b50be148315bbfff6576e910e7a08": "\\frac{2 (1 + \\frac{2}{\\sigma^2 \\lambda^2} + \\frac{3}{\\lambda^4 \\sigma^4})}{\\left( 1 + \\frac{1}{\\lambda^2 \\sigma^2} \\right)^2 } - 3", "2d4bd68fc6d2e0b14f05019eba861af8": "\\nu ,\\, \\boldsymbol\\Psi", "2d4bef5260c8b13ce02df9857af5f33d": "s \\in Q", "2d4bf0a68946ca2a7bd69d9ef45f0589": "X^2 - 1 = 0", "2d4bf94f35e9e3f6767f0b10e1bb3c06": " \\Delta\\bold{v} = \\bold{v}_1 - \\bold{v}_2 = \\Delta\\bold{u}.", "2d4c1f32979a5ef851ad059cc19a8f88": "\\sum_{p}\\frac{1}{p - 1} = \\sum_{m=2}^\\infty \\sum_{n=2}^\\infty \\frac{1}{m^n} = 1.", "2d4c6ad61b81b43d4c93640b23eee38b": "s_a(t)\\cdot e^{j2\\pi f_0 t} = s_{ssb}(t) +j\\cdot \\widehat s_{ssb}(t).\\,", "2d4cb2cc4ee93713ce4f6b6a84f68204": "d_{j,k} = \\langle S(n),\\phi(n)\\rangle ", "2d4cf901c3a76c066da597fb26c92729": "\\pi_*(BG)=\\pi_{*-1}^S. ", "2d4cff13071d954c01a1f421805c3293": "\\nu(p)", "2d4d28c1a2b239fe663f057b1ddfda6d": " \\Lambda(t) = \\int_0^{t} \\lambda(u)\\,du", "2d4d3aad0a000fa0b356d0b5b148e1ef": "\\mathrm{N}_{\\mathrm{train}} \\le 30", "2d4d66f65c87775d67fdb970f117b3eb": " E[u] = \\frac{1}{2}\\int_\\Omega|\\nabla u|^2 \\, \\mathrm{d}x - \\int_\\Omega fu \\, \\mathrm{d}x", "2d4d94cc9293b6facde18376641f1299": " W = 0 ", "2d4d9ac74d20a8feb0ed95530f5dbe9f": " \\left|\\frac{u}{w}\\right|^t + \\left|z\\right|^t \\leq 1", "2d4da21e925bdd381065b7b78265dff9": " \\{ 1, \\cos (x), \\sin (x), \\cos(2x), \\sin(2x), \\cos(3x), \\sin(3x), \\ldots \\}", "2d4dcf10084570378af72846cd24eee5": "k = 2", "2d4e03f98166e5f4e9aef0a6c7aa6236": "\\ \\phi _j (x) ", "2d4e1ea23b03f85f828381dacdb7b9ce": "c^\\dagger", "2d4e3ef7ec94676615d34403ccf98d05": "2^{-\\Omega(n)}", "2d4e518e8cc1d029a5dc0cab8bd91550": "c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) . ", "2d4e8dd593af2ada5d342b0a530de953": "A=(a_{ij});\\quad a_{ij}\\geq 0, \\quad i\\neq j.", "2d4f2678d6ee0ba958c1b75109a3f160": " f(t) = \\frac{1}{2 \\pi} \\phi^\\prime(t).", "2d4f276baf710f4df47ff7f693d9a17e": "\\mathfrak{e}_{7(-25)}", "2d4f8c39ae268412a2f39bc226ed3505": "m_n = E \\left( X^n \\right) = M_X^{(n)}(0) = \\frac{d^n M_X}{dt^n}(0).", "2d4fe78ec16bb66067541957a78c7ea9": "A (x) = M(u) = \\sum_{n \\le x} \\mu (x) \\,", "2d5003d9fd5aae2d986f987bab342f89": "\n |\\langle m | \\hat{A} | n \\rangle|^2 = \\langle m | \\hat{A} | n \\rangle \\langle m | \\hat{A} | n \\rangle^\\ast\n = \\langle m | \\hat{A} | n \\rangle \\langle n | \\hat{A} | m \\rangle.\n", "2d500b3ce90d2c1d9f46f50c2b69e5f9": "e<1", "2d5011cd2b50fbd38ce8a22892073dfe": "\\scriptstyle \\bar V_{\\text{L}}", "2d503a4878280766a1452d21cb30029f": "R^\\rho{}_{\\sigma\\mu\\nu}", "2d5075519b7bacc12e2bfbbd18070823": "\\omega_{ce} = eB/m_ec = 1.76 \\times 10^7 B \\mbox{rad/s} \\,", "2d5093c3eb39a1a5cb7b44996ebcf28a": "\\liminf_ {n\\to\\infty} x_n\\le\\phi(x) \\le \\limsup_{n\\to\\infty}x_n", "2d50b00bea673d5ed02a4918f1e3d7de": "\n \\boldsymbol{A}\\cdot\\mathbf{f} = \\boldsymbol{K}\\cdot(\\boldsymbol{A}\\cdot\\boldsymbol{d}) \\implies \\mathbf{f} = (\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{K}\\cdot\\boldsymbol{A})\\cdot\\boldsymbol{d} \n ", "2d50b2fd76a3bda07839c4c2dbff8135": "\\rho:\\pi(M)\\rightarrow\\mathop{GL}(E)", "2d50ed1be96dfd3ad9e040e3c1f703aa": "h(\\cdot ; \\omega): \\mathcal{X} \\to \\{-1,1\\}", "2d51826f07e4e78feec1d8f551a2bd81": "MI = \\int_R \\operatorname{min}[\\pi_1f_1(x), (1 - \\pi_1)f_2(x)]\\,dx ,", "2d51f0e5d6272a8bf68107799d6151ac": "T = F r \\,", "2d52297356b4ef6fc15ce91cccc637cc": "x^{-3} \\cdot 2^2 = \\frac{4x}{5}", "2d524386e4420cba4a5e5b7bf7954658": "\\phi : \\mathcal{O}_X^n \\to \\mathcal{M}", "2d52521b637f10e0e5c87fbee00d212d": "x > 4", "2d5274450386a9215597026b8f293aba": "\\|f\\|_{H^p}.", "2d5283452e95f41c03d0aa9a22721867": "\\sum_{j \\in J} a_{ji} m_i = 0", "2d52f3166a465eec82789be1ef196994": "\\left( \\frac{a_1}{a_0} \\frac{a_2}{a_1} \\cdots \\frac{a_n}{a_{n-1}} \\right)^{\\frac1n} = \\left(\\frac{a_n}{a_0}\\right)^{\\frac1n}", "2d533d3bd44c02a67dbcd80d7c3e7172": "\\sup \\sum_i \\|g(t_i)-g(t_{i+1})\\|_X < \\infty ", "2d53c4418aee7867b92d275c803ae071": "a(u), b(u)", "2d53c6d4ee8272ecfaab88263d56ee8c": "s(t) = \\int_0^t |x|_{y(\\tau)} d\\tau = \\int_0^t \\frac{|x|}{1 + \\tau x} d\\tau = |x| \\int_0^t \\frac{d\\tau}{1 + \\tau x} = \\frac{|x|}{x} \\ln|1 + tx|", "2d53cae08ee6d6869ad7978d19365bd7": "y=2/3 + \\kappa e^{-3x}. \\,", "2d53d451e24d71323dad3e0ced17fc5c": "(\\vec f \\cdot \\vec g)' = \\vec f\\;'\\cdot \\vec g + \\vec f \\cdot \\vec g\\;' \\,", "2d546fa8d48c16cbcaf86346ad575e19": "\\alpha = 0.35", "2d5483811d8b0cefd7fbcc828fb64e8d": "\\tan \\frac{\\delta'-\\alpha}{2}=\\sqrt{\\frac{1-\\beta}{1+\\beta}}\\cdot\\tan\\frac{\\delta-\\alpha}{2}", "2d549df9a2578421315a74e3739d7034": "m= \\frac{\\text{rise}}{\\text{run}}= \\frac{\\text{change in } y}{\\text{change in } x} = \\frac{\\Delta y}{\\Delta x}.", "2d54cbecd8a5052623c3a7a0f2ba1e76": "\\forall x \\forall y Lyx", "2d554b48b6da0bb876b0265e2db81891": "B_n \\sim \\frac{1}{\\sqrt{n}} \\left( \\frac{n}{W(n)} \\right)^{n + \\frac{1}{2}} \\exp\\left(\\frac{n}{W(n)} - n - 1\\right). ", "2d5579efa0a170b7e98991bca1b768a0": " f(a) \\equiv 0 \\,\\bmod{f'(a)^2\\mathfrak m}", "2d55f3bc1df1432c46fcf10206a1aaaf": "e^{ikx}", "2d55f68c36c4843d7c87fdd6307fbb59": "\\pi_n(S^k)=0 \\,", "2d5632ba5b4c06bc0d7b92293c9e3897": "\\left( x - a \\right)^4 + \\left( y - b \\right)^4 = r^4", "2d5660afc060aacad224aa94b5286028": "\n \\ \\ a_m <_m b_m\n", "2d576253b957f13ba1410513b3d8caf9": "A = P\\cdot J\\cdot P^{-1}", "2d5766d1781648e82d99cb199f06de6e": "\\frac{\\partial f}{\\partial\\bar{z}} = \\frac{1}{2}\\left(\\frac{\\partial f}{\\partial x} + i\\frac{\\partial f}{\\partial y}\\right)=0.", "2d576accdf2a5528dbc0f872cce68c82": "\\nabla^2 \\varphi - \\frac 1 {c^2} \\frac{\\partial^2 \\varphi}{\\partial t^2} = - \\frac{\\rho}{\\varepsilon_0}", "2d578a8c5129cf38127a3f34e7942ebd": "\\beta_{S}", "2d57d2889387be8f40fbd1860bc01f0a": "g_{rs}", "2d580337ec6425ab84877ab676ef5790": "s'\\in\\mathcal S", "2d58b313a10fc394548ccdbdc3d52cc6": "\\,0 < q/p < 1\\, ", "2d58b5000ce7031be4480849afda36e4": "|6| > |1| + |2|", "2d58b83810125be6c2ee066d6461aa30": " H(x_0,\\phi(x_0),D\\phi(x_0),D^2 \\phi(x_0)) \\geq 0 ", "2d58cee94395d1382cf692ab85d309d3": "\\left| l m n \\right.\\rangle", "2d58d278ae65e12238cccca944152150": "\\vec{t_1}\\bot_{\\vec{t_3}} \\vec{t_2}", "2d59086efa64db31d373fbfaf55e91f1": "Z^4_4", "2d59087e133f4039711d6e35030740cb": "e^{\\pi\\sqrt{163}}=12^3(231^2-1)^3+744-7.499\\cdots\\times 10^{-13}\\,", "2d590d0ffb03bb146d0a1b50c77539cb": "\\delta_B = \\frac {q L^4} {8 E I}", "2d59155fda9ef655fa88ec525b49a05b": "E_{2} = \\Delta x \\Delta y + \\Delta y \\Delta z + 2 \\Delta z \\Delta p + \\Delta p^{2} + 2 \\Delta p \\Delta x + \\Delta x \\Delta z + 2 \\Delta y \\Delta p", "2d594ed2940cd254842adaa54b4a1da1": "y e^{\\int_{s_0}^{x} P(s) ds} = \\int_{t_0}^{x} Q(t) e^{\\int_{s_0}^{t} P(s) ds} dt + C", "2d59666d7b703211ac78c5723bcabfa1": "Y_{2}^{-2}(\\theta,\\varphi)\n={1\\over 4}\\sqrt{15\\over 2\\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\quad\n={1\\over 4}\\sqrt{15\\over 2\\pi}\\cdot{(x - iy)^2 \\over r^{2}}", "2d59c11420f25516ad06e560cae944d1": " n_j(\\mathbf{r}) = n_j^0 \\, \\exp\\left( - \\frac{q_j \\, \\Phi(\\mathbf{r})}{k_B T} \\right)", "2d59cca0f2a8aaf19f947e09a1b2bebb": "I_i(s) = e^s \\sum_{j=0}^{np-1} f_i^{(j)}(0) - \\sum_{j=0}^{np-1} f_i^{(j)}(s),", "2d59d7935812d4ee82fb9035ef553d0c": "\\varphi(t) = \\sum_{j=1}^m b_j \\varphi_j(t).", "2d59ebc2a0612ee1b7656ad8eebef3b4": " \\mathbf{x} \\in \\Omega ", "2d5a149567f03a3794eaf7d9986e6614": "Y=\\frac{5}{s(2s+1)(s+1)}.", "2d5acf70a63d8b692cc0d4f6763a5d0a": "U[\\mu_j]=U-\\mu_jN_j\\,", "2d5ae07b66a2a54d8f4b1e2ba9888298": "g = det(g_{ij})", "2d5b419d0e665636f29d58f9d197353c": "M_p \\times_{G} EG", "2d5b50e35c7354f527016d9e695e2741": "Y_{9}^{1}(\\theta,\\varphi)={-3\\over 256}\\sqrt{95\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(2431\\cos^{8}\\theta-4004\\cos^{6}\\theta+2002\\cos^{4}\\theta-308\\cos^{2}\\theta+7)", "2d5b7214b66490e55344adf6d8b46caa": "S_6 \\in H(3,6)", "2d5b7825eb293f7508afa2fba8483b1a": "\\text{H}=\\,\\theta_{a}-\\,\\theta_{r}", "2d5b82076da2c7a2fbfc5c5cc004d957": "\\mathbf{k} = (k_1, k_2, \\dots, k_d)", "2d5b981a1a617d5cd8a9f87289e6d68b": "\n\\widehat{Var[X u]} = \\frac{1}{n} \\sum_i X_i X_i' \\hat u_i^2 = \\frac{1}{n} \\mathbb{X}' \\operatorname{diag}(\\hat u_1^2, \\ldots, \\hat u_n^2) \\mathbb{X}\n", "2d5bd635e3ea3f8c7341190866c565f6": "x \\in [-R;+R]\\!", "2d5c1c0659de98877137bd86760d01ec": "\n(\\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}})^{[2j]} = (i |\\hat{\\mathbf{p}}|)^{2j}e^{-i\\pi\\mathbf{J}^{(j)}\\cdot\\mathbf{n}}\n", "2d5c595a747a8339234cc30af84a66c0": "u=-2i \\frac{\\partial \\psi} {\\partial \\bar{z}}", "2d5c5df1bf25086fab057d863b07e659": "W_0 = P S_3, \\qquad W_1 = m S_1, \\qquad W_2 = m S_2, \\qquad W_3 = \\frac{E}{c^2} S_3,", "2d5ca393671aab75c69614f703855270": "d \\sin \\theta = n \\lambda", "2d5d003085bfdbd9d720aa93ebc907e9": "\\,^{z_{10} = x_{10} y_1 + x_9 y_2 + x_{12} y_3 - x_{11} y_4 + x_{14} y_5 - x_{13} y_6 - x_{16} y_7 + x_{15} y_8 + x_2 y_9 + x_1 y_{10} + x_4 y_{11} - x_3 y_{12} + x_6 y_{13} - x_5 y_{14} - x_8 y_{15} + x_7 y_{16}}", "2d5d27b11bd311d71f5fde87d2106d07": "I_{ion}", "2d5d3683b76c23886eb429f889d91178": "U \\left( t_2 - t_1 \\right)", "2d5e14e8e658457b1f79204736b23248": "T(R) = T_{eff}", "2d5e5ae684fc78b08aa1cd71d9559516": "\\gamma = \\sum_{k=1}^n \\frac{1}{k} - \\ln n -\n\\sum_{m=2}^\\infty \\frac{\\zeta (m,n+1)}{m}", "2d5e6722858ba5c16891703874a797c2": "Q(s,a) = \\sum \\limits_{i=1}^d \\theta_i \\phi_i(s,a)", "2d5e7c8d184eddae03117d20c8b59329": "g_J = 2", "2d5ea6579c0b479449f75e626ac65e48": " 9 ", "2d5eb2cb1b2a4dc2c9441ddc15d460b0": "(u,v) \\in p", "2d5f154aa04442d30a20a028d7838843": "\\mathsf{\\Sigma \\Tau \\Upsilon \\Phi \\Chi \\Psi \\Omega}\\!", "2d5f1a66c41d32add33a242832d216ab": "2^{\\aleph_0}=\\mathfrak c", "2d5f6ad3dec11f67b905b84cd22599c7": "\\mathrm{d}\\mu_i = RT \\frac{\\mathrm{d}a_i}{a_i} = RT \\mathrm{d}\\ln fC_i\\,,", "2d5fcf5f071c272cadb5b0d5bc248942": "x_{me},x_{ht}", "2d6049f021d4c73a60d9c06937290e25": " \\sum_{n=-j}^j \\sum_{k=-j}^j U_{np}^* U_{kq} = \\delta_{pq}.", "2d6073019376f8e3b8f16a0bec669360": "\\, x = a \\left( \\cos t + t\\sin t \\right) ", "2d60967d280fb1e2e63c93f8039cb180": "\\psi(\\Omega^{\\Omega^2 \\omega^\\omega})", "2d60efa231d6f6ea43f988c3aabfbd21": "f_r = f_t \\left( \\frac{1+v/c}{1-v/c} \\right)", "2d6103fa27312997dbfa184305550ac1": " \\mathcal{L} \\supset m^2_0 \\phi^\\dagger \\phi", "2d61244ed803773b76e814391ee3fd3d": "d_\\mathbf{A} e = de + A \\wedge e = 0", "2d61418dc78ed5a8d4b84f718005b01d": "ROC1 = (1-Price/Price(X1))*100;", "2d619e0e562d297fdde091b2189e56a5": " \\sum_{h=1}^H \\left\\vert{ \\sum_{n=a}^{b-h} e(f_h(n)) }\\right\\vert \\le b-a \\ . ", "2d61feb7c64113202a3dda18d9d53325": "f'(z_0) = \\lim_{z \\to z_0} {f(z) - f(z_0) \\over z - z_0 }. ", "2d6237aaf4441574d1c6583d1a8e6020": "\\ddot{\\theta}_1 \\left(J_{1zz} + m_1 l_1^2 + m_2 L_1^2 + (J_{2yy} + m_2 l_2^2)\\sin^2(\\theta_2) +\nJ_{2xx}\\cos^2(\\theta_2) \\right) \n+ \\ddot{\\theta}_2 m_2 L_1 l_2 \\cos(\\theta_2) \n- m_2 L_1 l_2 \\sin(\\theta_2) \\dot{\\theta}_2^2 \n+\\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(2 \\theta_2) (m_2 l_2^2 + J_{2yy} - J_{2xx}) \n+b_1 \\dot{\\theta}_1\n= \\tau_1", "2d62c3e247db23ce35f9be4f9f357f5b": "2\\operatorname{Cl}_2\\left(\\frac{\\pi}{3}\\right)= 3\\operatorname{Cl}_2\\left(\\frac{2\\pi}{3}\\right)", "2d62e21f0a949b939346c93e8f3d0886": "\\alpha < \\omega_1", "2d634c33b823da67f0ea0a4eeb80fcee": "\\sin(U+V)", "2d63641ea7d5340a03d94a69718eb933": "\\sin c \\,\\sin b \\, \\cos A + \\cos c \\,\\cos b", "2d638638175e38e7841709a50ca26818": "v_k:\\bigoplus^kT^*M\\to\\bigwedge^kT^*M", "2d63cb7df29983c98aa751d7010a84b1": "\\Omega = \\mathrm{E}(e_t e_t') = \\mathrm{E} (B_0^{-1} \\epsilon_t \\epsilon_t' (B_0^{-1})') = B_0^{-1}\\Sigma(B_0^{-1})'\\,", "2d63e034e7c374f515ba4bdcc6a687a8": "S \\left ( \\mathbf {R} \\right ) = k_B \\ln \\Omega {\\left ( \\mathbf R \\right) } ", "2d641aaeeaa3ac3ec7352455a665cc6f": "\\frac{[\\Gamma(\\tfrac14)]^4}{128\\pi^3} = \\frac{1}{\\sqrt{u}} \\sum_{k = 0}^{\\infty} \\frac{(6k)!}{(k!)^{3}(3k)!} \\frac{1}{(u\\sqrt{2}(1+\\sqrt{2})^2)^{3k}}.", "2d642af994f334b31442a1d0a0d70c4b": "L_2 + \\tfrac{3}{2}L_1 \\rightarrow L_2", "2d643bbf47480b4083e4e17608576754": "{b^{\\dagger}}_{\\nu_j}", "2d64c3337501c83c56c4e55be36d0c52": "F_u(\\theta) = \\begin{bmatrix}\n {cos[(\\theta(t)+\\phi)\\frac{d}{D_u}]}\\;m_u\\;r_u\\;(\\frac{dN \\pi}{30D_u})^2\\;\\eta_x \\\\\n {sin[(\\theta(t)+\\phi)\\frac{d}{D_u}]}\\;m_u\\;r_u\\;(\\frac{dN \\pi}{30D_u})^2\\;\\eta_y \\\\\n -m_u\\;g\\;\\eta_z \n \\end{bmatrix} ", "2d6511d21642f5a9bd85941aac22b8db": "\\color{Black}\\tfrac{n}{m}", "2d655b3b515c41b326e2a43c8237c8a1": "R^I", "2d65dd09add42f40abcacdc5289a90c1": "f(f(u, v), v) = f(u, v)", "2d66299761b4f9de573cd33637c26c59": "a_0 \\oplus \\!", "2d66d9749a7040f01ebf66e2667832ad": "\\delta(r_1 r_2)=(\\sigma r_1)\\delta r_2+(\\delta r_1)r_2. ", "2d6701cc39afce8b946e6a1e2e78d0a8": "n(j)M(j,j)", "2d6712861a24c5c689d7bdcb5b368d3a": "\\Delta \\, \\delta \\,", "2d6714e82c164f845eada1520f1610f1": " z' = x y = r^2 \\, \\cos^2 \\theta \\, \\cos \\phi \\, \\sin \\phi, ", "2d675161e57dad9d8d54ca35218d1ab9": "S_1 := \\sum_{i=1}^n {\\mathbb P}(A_i),", "2d67572377b44d0cccba503abd46cb79": " n_+ \\mapsto M \\otimes A^{\\otimes n}. \\, ", "2d676a3b36d696eea47bf439175146c8": "\\displaystyle{a_n(0)=0,\\,\\, a_n(\\infty)=a_n.}", "2d679fcc628a16a07a403dad2d2c4048": "\\partial=\\pi^{p+1,q}\\circ d:\\Omega^{p,q}\\rightarrow\\Omega^{p+1,q},\\quad \\bar{\\partial}=\\pi^{p,q+1}\\circ d:\\Omega^{p,q}\\rightarrow\\Omega^{p,q+1}", "2d682762e2aedcd263476707281a584f": "\\mathcal{I}(\\theta)= \\mathrm{E}[\\mathcal{J}(\\theta)]", "2d68596d96054aa260c2625a598b7556": " M = \\begin{bmatrix} 1 & 1 \\\\ x_1 & y_1 \\\\ x_2& y_2 \\\\ x_3 & y_3 \\end{bmatrix} , ", "2d688271a5e2fda276eb826b16e68a71": "\\bigcup T = \\{w \\vert [w]_D\\in T \\}", "2d68cb4403cec0195c19762926626141": "p^2\\langle\\sigma v\\rangle/T^2", "2d68f2b633bbfd9f494436ed3c80b2d9": "\\langle Ux, Uy \\rangle = \\overline{\\langle x, y \\rangle}", "2d696439ce5b60ea41e45f54a00e707e": "\\frac{\\partial F}{\\partial x} \\frac{\\partial x}{\\partial u} +\\frac{\\partial F}{\\partial y} \\frac{\\partial y}{\\partial u} = -\\frac{\\partial F}{\\partial u}", "2d69d72bffc44970697b67cef1b3e50b": "64=8^2", "2d6a00666fb0bc199f15856953fe4c5b": "\\tau_b = \\tau_d\\;", "2d6a121b6e8db50cddbea018b82c7187": "y_{n+k} - c_{n-1}y_{n-1+k} - c_{n-2}y_{n-2+k} + \\cdots - c_{0}y_{k} = 0", "2d6a33f9f23b74f7a6ab9c5f5c71b728": "\\cot\\frac{7\\pi}{60}=\\cot 21^\\circ=\\tfrac{1}{4}\\left[2-(2-\\sqrt3)(3-\\sqrt5)\\right]\\left[2+\\sqrt{2(5+\\sqrt5)}\\right]\\,", "2d6a46eb8618e5d2018fe5095c3f4bff": "\\sqrt{3\\over 2}", "2d6a5ce3764457e9c904050270a0c85d": "4\\sin{18^\\circ}=\\sqrt{6-2\\sqrt5}=\\sqrt5-1. \\,", "2d6a95105472e24221c1af0fe98ff765": "r,\\theta,\\varphi", "2d6ab1b887c8a7064cbbd7926a2b2b4d": "\\Sigma^0_2", "2d6ac54725c881a6b53a23c20b9c5fcc": " z_{n+1} = A + B z_n e^{i (|z_n|^2 + C)} ", "2d6ae0208d4bbbbacb0e9a05299af9e1": "\\frac{dP}{dt}=rP\\left(1 - \\frac{P}{K(t)}\\right)", "2d6af7dc28f6c003c441f5ed7ccd204e": "I_{k}\\ ", "2d6b00bb40a848d41023b636f5e806c9": "u_{n}(t)=e^{-\\alpha\\frac{n^{2}\\pi^{2}}{L^{2}} t} \\left (b_{n}+\\int_{0}^{t}h_{n}(s)e^{\\alpha\\frac{n^{2}\\pi^{2}}{L^{2}} s} \\, ds \\right).", "2d6b0f6c1cffb164c0c3c501aa4f0fd0": " A = -\\Delta_rG. \\,", "2d6b7e6adb293a0f4564c800c7e46c78": "f_\\mathrm{v}=v\\frac{\\partial^2 u}{\\partial x^2}.", "2d6bb06f2c9ee058f2481234d098e996": "\\Sigma = L \\mathrm{Cov}(F) L^T + \\mathrm{Cov}(\\varepsilon),\\,", "2d6c37fae1ada408563477ae5e78a12f": "\\left\\lfloor\\frac{3\\Delta}2\\right\\rfloor", "2d6c5851f2ca073f4d22166e9e9adeca": "\\psi(x,t) = [A \\sin(kx) + B \\cos(kx)]\\mathrm{e}^{-\\mathrm{i}\\omega t},", "2d6c5c63d2ff53f1ebc135788e871051": "\\text{MPa-}\\sqrt{\\text{m}}", "2d6c6630a75196f2d238b55129a8ddcc": "\\Phi(h_n) := \\Phi(\\ h_n,t_n,y(t_n)\\ )", "2d6c7d46da757c7cab902ba84da84fd5": " f_{\\sharp} (\\alpha) - g_{\\sharp}(\\alpha) = \\partial P(\\alpha),", "2d6c9b7a3ea3a0308e23a0fdc924aab8": "FV_T(X)", "2d6cc4975e723c090b4e8726c8b868d4": "f(x_1,\\dots,x_n)=x_i", "2d6cc4a6ccc631ff9a03b65253b73804": "\ng(x) p_{xy} = \\frac{1}{d}\\frac{ g(x) g(y)}{\\sum_{z \\in \\Theta: z \\sim_j x} g(z) }\n= \\frac{1}{d}\\frac{ g(y) g(x)}{\\sum_{z \\in \\Theta: z \\sim_j y} g(z) }\n= g(y) p_{yx}\n", "2d6d2c9d9293ff8ce461ae16db38dda6": " t_i = \\frac{- \\text{tr} \\Lambda^{-1-2i}}{1\\times3\\times5\\times\\cdots\\times (2i-1)}", "2d6d40c5ebd77374b13fd5b1f4980bf6": " P \\ge P^{*}= 1-\\frac{1}{R_0} ", "2d6d64b6c45117a2383bda3c7626a8aa": "\\mathcal{L}\\left\\{f(t) + g(t) \\right\\} = \\mathcal{L}\\left\\{f(t)\\right\\} + \\mathcal{L}\\left\\{ g(t) \\right\\} ", "2d6d8ca9c44c164d278417fa93431d8f": "\\overline{T}", "2d6dde8cbb79ecab005e4555cff5f7da": "\\beta_S = \\beta_T - \\frac{\\alpha^2 T}{\\rho c_p} ", "2d6e0373564b8b14e9d47090dce8566b": "\\alpha (x) = x + (\\cdots, 0, 0, 1)", "2d6e50a8c98c612f60d089315e6a9103": " |L\\rangle .\\, ", "2d6e69afd102750f81368f6abab3b98e": "\n\\det(\\mathbf{B}) = \n\\sum_{\\pi \\in S_N} (-1)^\\pi B_{1,\\pi(1)}\\cdot B_{2,\\pi(2)}\\cdot B_{3,\\pi(3)}\\cdot\\,\\cdots\\,\\cdot B_{N,\\pi(N)},\n", "2d6ea7613ba70fabd0201310cf0f708e": "x = \\{1,2,3,4,5\\}; \\bar{x} = 3", "2d6f3980341670a0c18727b25d3f726d": " \\scriptstyle \\gamma \\, \\text{= Euler–Mascheroni Constant} = 0.5772156649\\ldots ", "2d6febfec6934f12a161aaa9211fa5a1": "m < 0", "2d705a641d006566a2ba3624bd88862a": "\\nabla_{A'}^{(A}\\Omega^{B)}=0 ", "2d7083329769d83cc53ff01c20c36077": "S=\\begin{pmatrix}I_p & 0\\\\ 0 & -I_2\\end{pmatrix}", "2d70947859a21385d74ca709657e4982": "T::=\\text{s}\\,\\!", "2d70da379b3ffb56bd104b348ba21c55": "P_2", "2d710c1f178b76c2723909374b7d1db1": " [M] ", "2d71423177c016ec5c93f8ec319994d6": "\n\\begin{align}\nA&=\\frac{Z^2\\alpha_{fine}^3c^2}{(2\\pi)^2\\hbar}\\frac{|\\mathbf{p}_+||\\mathbf{p}_-|}{\\omega^3},\\\\\n\\Delta^{(p)}_1&:=-|\\mathbf{p}_+|^2-|\\mathbf{p}_-|^2-\\left(\\frac{\\hbar}{c}\\omega\\right)\n+ 2\\frac{\\hbar}{c}\\omega|\\mathbf{p}_+|\\cos\\Theta_+,\\\\\n\\Delta^{(p)}_2&:=2\\frac{\\hbar}{c}\\omega|\\mathbf{p}_i|-2|\\mathbf{p}_+||\\mathbf{p}_-|\n\\cos\\Theta_+ + 2.\n\\end{align}\n", "2d71a63e19a34bbadf5153276ac88ac2": "\\mathcal{F}^{n}[f] = \\mathcal{F}[\\mathcal{F}^{n-1}[f]]", "2d71bd6bda8cda22bbb61a49da1a0780": "\\scriptstyle{\\phi^l_i}", "2d71cdaaf7500a50829a8dbec80552d4": "\\hat{\\mathbf{z}}", "2d72149d03b10fbb7a5613e945f6e954": "r^\\mu \\rightarrow (x^0, x^1, x^2, x^3)=(ct, x, y, z) \\ ,", "2d7244f02ae41e8782bd8a0a98b3f44c": " (2^r+1) \\mid (2^n+1). ", "2d7284fbd1a894bcc5333f9f4c94bb66": "I_\\mathrm{sat}", "2d72a7686ab1d88aad8bff560fc2126f": "L / G", "2d72ecad43d176bfbf8712e9324ef2ec": "O_\\infty(G)", "2d72f72db433eb27e97cb4a942677cc5": "z=x+y", "2d731b772ac6bc3c81a0f4dad0539a92": "\\pi_1, \\pi_2, \\ldots \\in \\mathbb{P}", "2d731c720573044f0f5ae7dd93052c1e": "C_e = C_X - C_{\\hat{X}} = C_X - C_{XY} C^{-1}_Y C_{YX},", "2d732a4ff46b35e555e670c8ba0e9ea7": "\\mathbf{s} \\in \\mathbf{C}", "2d73c05f44287672cca10bc36fdfa785": "\nJ^\\pi(b_0) = \\sum_{t=0}^\\infty \\gamma^t r(b_t, a_t) = \\sum_{t=0}^\\infty \\gamma^t E\\Bigl[ R(s_t,a_t) \\mid b_0, \\pi \\Bigr]\n", "2d7405ae3e3b6128583f6fd6a1e42d82": " \\theta_{a} = \\lambda_{p}(\\theta_{a,0}+w)+(1-\\lambda_{p})\\theta_{a,0} ", "2d7405bf6722cd64b6113ac6c3c583b8": "\\Rightarrow v=\\sqrt{2gz}", "2d742f43c13d93b204919d0eb5589eed": "(i\\gamma^\\mu\\partial_\\mu^\\prime - m)\\psi^\\prime(x^\\prime,t^\\prime) = 0.", "2d745d84c0c8016fbfc12fd476e6b984": "p=\\rho \\left(\\gamma-1 \\right)e,", "2d748fa42abdf5d8d84eb3beac40535c": "k_t", "2d74e1750aeb7034aa7647212d92889b": "\\text{Visibility}(\\text{real})=\\frac{\\text{amplitude}}{\\text{average}}.", "2d74f049373dc568dad70205e8033889": "\\mathrm{not}~s \\equiv \\mathrm{true}", "2d7525d57035982ce6454e9054ff7279": "f_{QH}(0)=1, \\, ", "2d75312cb87fd7d5bb19a1b74f5abcb6": "\\mathbf e_i\\,\\!", "2d75775a2fc33eec3da8ab127225c043": " (\\log m)/\\log\\log m ", "2d758a825635aabe3aedbdc1daca1ec8": "\\log \\Gamma (x)", "2d7616976eeff7d7dd56512675ccb59b": "x_2,", "2d763cbaec5b289ae38fb7de9de2ee75": "Q^2, [-10..10,-10..10]", "2d768dc1e7d9cc704af4189c8b3d11a3": "\\nu+n,\\, \\frac{\\nu\\sigma_0^2 + \\sum_{i=1}^n (x_i-\\mu)^2}{\\nu+n}\\!", "2d76b138bdb9ddc02e73dcc88ce548c7": "\\arcsec (1/x) = \\arccos x \\,", "2d76f7e939b68cfdda4ace8b955c5a12": "(2k-1)!! = \\prod_{i=1}^k (2i-1) = \\frac{(2k)!}{2^k k!} = \\frac {_{2k}P_k} {2^k} = \\frac {{(2k)}^{\\underline k}} {2^k}.", "2d773566c100e49883c59c205d2ac908": "\nD =\n\\begin{bmatrix}\n ~4 & -1 & ~0 \\\\\n -1 & ~4 & -1 \\\\\n ~0 & -1 & ~4 \\\\\n\\end{bmatrix}\n", "2d77984fa57ed26acc9aace5be1e537e": "\\displaystyle{Se_i=\\lambda_i e_i}", "2d77a3b8fa5754639abc81be975832b5": " y'(t) = f(t,y(t)), \\quad y(t_0) = y_0, \\, ", "2d77c502aa960b36db0e69c27a725fd9": "\\frac{W_{Below}}{W_{Total}}", "2d77cdc9584d57c13524742171249ae5": "\n\\frac{\\partial P\\left(A,t\\right)}{\\partial t}=\\sum_{i,j}\\frac{\\partial}{\\partial A_{i}}\\left(-k_{B}T\\left[A_{i},A_{j}\\right]\\frac{\\partial\\mathcal{H}}{\\partial A_{j}}+\\lambda_{i,j}\\frac{\\partial\\mathcal{H}}{\\partial A_{j}}+\\lambda_{i,j}\\frac{\\partial}{\\partial A_{j}}\\right)P\\left(A,t\\right).", "2d78054ec8f896d599b807e8c12fc682": "\\text{Fib}(1)=1\\text{ as base case 2,}", "2d7820667d79d00950a16cfae0a39a90": "R_n(t) :=\\int_{[a,t)}u(s)\\mu^{\\otimes n}(A_n(s,t))\\,\\mu(\\mathrm{d}s),\\qquad t\\in I,", "2d786a2d990be58de342803679449d18": "\\mathbf{y} \\leftarrow \\alpha A \\mathbf{x} + \\beta \\mathbf{y} \\!", "2d789d2f9eca0f3a969058911b5ed273": " = 1\\,\\mathrm{slug}\\,\\mathrm{\\tfrac{ft}{s^2}} ", "2d78a5d6f771c0a1864b195a0c5ff49f": " p=-3", "2d78d03140c236573478717341e20e53": "\\frac{\\partial s_c}{\\partial t}", "2d794098310441581676ae099228c643": "\\mathbb{N} = \\{1,2,3,4,5,\\ldots\\}", "2d799503ecd36c7216d45b7618195051": "s(X)=\\sup\\{|Y|:Y\\subseteq X ", "2d7a95832b9ee4f1fdf76a8264c0122c": "2bu\\pm(1-b)\\frac{\\partial u}{\\partial x}=0 \\quad\\text{on }x=\\pm1.", "2d7ab2157588a5c34a8e2c724c43cb37": " (A \\or B \\or \\neg C )", "2d7b0611aa5e6b8adda9052cf3110184": "F(n+1,k)-F(n,k) = G(n,k+1)-G(n,k)\\, ", "2d7b120bfc87f94360acc1698389a2e7": "Y_{7}^{7}(\\theta,\\varphi)={-3\\over 64}\\sqrt{715\\over 2\\pi}\\cdot e^{7i\\varphi}\\cdot\\sin^{7}\\theta", "2d7b389dcf84b24b34f78c7f28951d0e": "t'=t", "2d7ba743f33757d406c92ed290016d51": "n \\gtrapprox N", "2d7ba92a205ceeb3921a8fe6d8730ebb": "a_{eff}", "2d7bb33c273f506b0d510ff1c2063748": "G_{ab}\\, =0", "2d7be952b93411fc9df9c535a973b236": "\\|x\\|_{bs} = \\sup\\nolimits_n\\left|\\sum_{i=1}^nx_i\\right|", "2d7bf1aabcbd5b812126069b1cb6628c": "\\mu_0 - \\ ", "2d7c55cefc8a78c7d98a7fce4e9e9abc": "\\left[{13 \\choose 5} - 10\\right]{5 \\choose 2} \\cdot 2,268 = 28,962,360", "2d7c5cac50c80bcd05c7f89807a5ca4a": "r^e", "2d7c64f715ed2751ac953079c69f9462": "\\mathbb{Z}_6 = \\mathbb{Z}_3\\times\\mathbb{Z}_2", "2d7cad30feb6b9bb4128fb95e5abfbe0": "A_1 \\angle \\theta_1 + A_2 \\angle \\theta_2 = A_3 \\angle \\theta_3. \\, ", "2d7cb5b17473fa930460bcd1e5cd48a6": "\\eta(k_h)", "2d7cc8232e26d65126f8515eae37457d": " v_B = v_A \\tan\\alpha, \\!", "2d7d6d4a327ba125185f84f4de67e9f6": " V = \\sigma \\, v \\, dt ", "2d7d9833fc01f4b6863ac60f07c192d6": "e^x = \\sum_{n=0}^\\infty {x^n \\over n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots", "2d7e0782e31fd859083af1351156bd4b": "(\\mathfrak{R}\\mathfrak{R})", "2d7e2cd73a53c303108e6af5e0c5b596": "1\\le p \\le q", "2d7e53c6717e2736d4078f0ca6d8dde3": "\\rightarrow_W", "2d7e71209eeb79cd985435fec51a06ac": "\\Delta p\\, =\\, p_1\\, -\\, p_2\\, =\\, -\\, \\rho\\, \\frac{A_1}{A_2} \\left( 1\\, -\\, \\frac{A_1}{A_2}\\right)\\, v_1^2,", "2d7e786cef05cde83a4cf802ac976758": "\\mathbf{r}\\rightarrow \\mathbf{r} + \\mathbf{v}t", "2d7ec50a54a974ac0e8fce9e2ce16f4c": "\\scriptstyle \\Pi_A", "2d7ed5d44a54ab6ed2691fc7d6bc1971": " X\\left( t\\right) =\\left\\{ x\\in X:g\\left( x,t\\right) \\geq\n0\\right\\}", "2d7f0d0904170ac41dcfc0207fdc3db1": "\\left\\| \\nabla\\Phi \\right\\| \\rightarrow 0", "2d7f40adda28335364f7fa00bd892d1b": "\nE[U|SI] = \\int_Z E[U|z] p(z) dz = \\int_Z \\max_{d\\in D} ~ \\int_X U(d,x) p(z|x) p(x) ~ dx ~ dz\n", "2d7f4a26cc074fb0895c9783b5c42c39": "f(x + \\mathbb{Z}) = e^{2 \\pi xi}", "2d7fddfc809e114b88fdf8e285c1b89b": "3x \\equiv \\pm 7\\pmod {17}", "2d802c2b15c88354fa1fba50e419601e": " u_{i - 1/2} = 0.5 \\left( u_{i-1} + u_{i} \\right). ", "2d80c0b335333ce1b97f6b9f86acb06d": "{_uM_d}=\\frac{10^2}{(32.2)6}+ \\frac{6^2}{2}=18.518ft^2", "2d811dade4fb53e14d8b40bfa6fa9e09": " \\scriptstyle NPV = \\sum_{n=0}^{N} \\frac{C_n}{(1+r)^{n}} = 0", "2d812bbdcdccaf7c6eaa58eaaadfa6ba": "\\int\\operatorname{arsinh}(a\\,x)^n\\,dx=\n x\\,\\operatorname{arsinh}(a\\,x)^n\\,-\\,\n \\frac{n\\,\\sqrt{a^2\\,x^2+1}\\,\\operatorname{arsinh}(a\\,x)^{n-1}}{a}\\,+\\,\n n\\,(n-1)\\int\\operatorname{arsinh}(a\\,x)^{n-2}\\,dx", "2d81e06bab0b203e7773f8615548ab29": "\\nu^2/\\lambda^2", "2d81f0afcd9f23da6fb8ebe3fe150cda": " x = \\lim_{N\\to \\infty} (1+{\\ln x\\over N})^N", "2d8277bb07e78f2f04ad013b6cef1b83": "H(x_1,x_2,\\dots)", "2d834e67ab4fb959dcc28cd6be88fbea": "\\text{extend}: ((M \\rarr A) \\rarr B) \\rarr (M \\rarr A) \\rarr M \\rarr B = f \\mapsto g \\mapsto m \\mapsto f \\, (m' \\mapsto g \\, (m * m'))", "2d838fc2872825c85aa6700590019ee2": "C_s", "2d83ac78cf54d0be7c288db895989b3c": "\\scriptstyle \\|a_i-a_j\\|=2\\leq\\|x_i-x_j\\|", "2d83bc7b538c3c6273b7c65a25804fa2": "{p}", "2d83c34211e9abe1245c6a1456017aec": "x\\vee y= x", "2d84126ea3bd89c1d7ca64dacd53a2a8": " R=\\frac{1}{2} d_\\Gamma\\Gamma=\\frac{1}{2} [\\Gamma,\\Gamma]_{FN} = \\frac12 R_{\\lambda\\mu}^i \\, dx^\\lambda\\wedge dx^\\mu\\otimes\\partial_i, ", "2d841ee8fa46a3095eacb4272ea2bfcf": "\\beta~", "2d8451a3a7a8db2a0050c7602f63ff46": "P_n(x) := \\sum_{m=0}^n \\left\\langle \\!\\! \\left\\langle {n \\atop m} \\right\\rangle \\!\\! \\right\\rangle x^m ", "2d84641f470678842adad62ce32826ff": "\\phi=\\varphi_0", "2d849ec9544a27fa7c910c4c87073731": "A,B \\in \\mathcal{F}", "2d84c5a2330b5c2fd0639f4b541c3afa": "e\\phi_{DL}/k_B T_e \\approx 0.1", "2d84d4a2460cb54d5e21c2e2a02f4d54": "\\rho \\left(\\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{u} \\cdot \\nabla \\mathbf{u}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{u} + \\mathbf{f}.", "2d84e13f37b232f10ca88f93eb319ee9": "\\mu_{3,2}= \\frac{\\langle b_{3}, b_{2}^{*} \\rangle}{B_{2}}=\n\\frac{\\begin{bmatrix}3\\\\5\\\\6\\end{bmatrix} \\begin{bmatrix}\\frac{-4}{3}\\\\\\frac{-1}{3}\\\\\\frac{5}{3}\\end{bmatrix}}{\\frac{14}{3}}=\\frac{13}{14}(> \\frac{1}{2})", "2d850b3cd60e31d95f5daae7edf0dae0": "p^3>3 q^3 \\Rightarrow D(p/q)=\\mathrm{false}.\\;", "2d850f534a89d09a6790252266f17119": "\\exists! n \\in \\mathbb{N}\\,(n - 2 = 4)", "2d85622e6a93368ed2e5dfa898333863": "\\beta=\\mathbf{V}q", "2d857e290fe8f5903d488801a7e4a1aa": "1.1530", "2d860eb0c6d9fda8dacddce915f29274": "\\mathcal{Q}_{Hur}", "2d86249d0b1a90648dde8afdbe47eba2": "x_C = \\tfrac{1}{\\omega C_M}", "2d865133f76c019895ace255212b6112": "\\frac{d^2w}{dz^2}+\\left(-\\frac{1}{4}+\\frac{\\kappa}{z}+\\frac{1/4-\\mu^2}{z^2}\\right)w=0.", "2d867e70764962b4d917b06ebf926075": "\\boldsymbol{ \\nabla \\times}\\left( \\boldsymbol {\\nabla \\times B} \\right ) = \\mu_0 \\epsilon_0 \\frac {\\partial}{\\partial t} \\boldsymbol {\\nabla \\times E} \\ . ", "2d868b5cea0b7154b5df0bd6b7635604": "x_{k+1}=1-\\sum_{i=1}^kx_i", "2d87a941404069994c792f22d91f351d": "\\{A_i : i\\in I\\}", "2d87d0ca1626718126276ad301c57ff2": "\nx \\in R^{n}\n", "2d8834960d1f099ad111ccacfd55f6b6": "\\frac{4\\pi}{3\\sqrt{3}} \\approx 2.418", "2d885e896bc214dde33e740847d17428": "Q_{\\varphi}=\\frac{|\\{x\\in D:\\varphi(x)\\}|}{|D|}\\,\\!", "2d891afbf7add7244a6b01f819280fa6": "F\\colon C\\to D", "2d8938768e21d58a7ab9799671c72d2b": " \\textbf{x}_{c} = \\textbf{x}_o+\\rho(\\textbf{x}_{o}-\\textbf{x}_{n+1})", "2d8a26745f8778255ff9ef7b27c4d8eb": "a_{23}+a_{24}+a_{33}+a_{34} = 34 ", "2d8a80a30a3b87973de29ade4105cd6d": "X:=\\liminf_{n\\to\\infty}X_n", "2d8a87a414d36a2d76a8649322feb45b": "\n\\begin{pmatrix}\n {[a_{11}]} & \\cdots & {[a_{1n}]} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n {[a_{n1}]} & \\cdots & {[a_{nn}]}\n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\n{x_1} \\\\\n\\vdots \\\\\n{x_n}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n{[b_1]} \\\\\n\\vdots \\\\\n{[b_n]}\n\\end{pmatrix}\n", "2d8afa2579a702791ee93d437418a856": " \\ X ", "2d8b7d716f84d6208a911768825efcd9": "N_{Y}\\left(E\\right) - N_{Y}\\left(E+\\delta E\\right)\\,", "2d8bc69e229ec4149a383bc5de497b05": "S_C = 1+K_1 C^*_1", "2d8bd7a100098909240cf1b9562378c0": "i_1, i_2, \\dots, i_m", "2d8c062ac427efc34c3995f84a9380d7": " I_{32} ~,~ \\Gamma_\\text{chir} \\Gamma_{a_1 a_2} ~,~ \\Gamma_{a_1 a_2 a_3}\n~,~ \\Gamma_{a_1 \\dots a_4} ~,~ \\Gamma_\\text{chir} \\Gamma_{a_1 a_2 a_3} ", "2d8c0b6007d7ba7579c11eac5213dcd4": " a=0, \\dots, d-2 ", "2d8c3024529fc375e00e43760bab364c": "Lk = Tw + Wr", "2d8c6d7e1618a9035ac5c332d5af97bb": " X \\sim \\textrm{Beta}(a,1) \\, ", "2d8cc5b6fec7c7e5017feaf5a6bd04f3": "\\{*\\}\\hookrightarrow[0,1]", "2d8d4c7364d71d0a8ab7b5f24cb1fd3d": "\n\\bar{T}_\\text{lost} = \\frac{-4N_ep}{1-p} \\ln p.\n", "2d8d7ae2939406c1f2355cc93f5b042c": "\\widetilde{\\gamma}", "2d8dcc1f4f2251dc6a592fb79de74aba": " \\mathbf{x} ", "2d8de4bff11a281b6563c971c9eaa367": "\\left( \\pi _{2}^{2}+M_{2}^{2}\\right) \\psi =0,", "2d8de946f578d194630e144baedd8269": "\n|\\psi (t) \\rang = \\sum_n |n\\rang \\left\\langle n | \\psi ( t=0) \\right\\rangle e^{-iE_nt/\\hbar}\n", "2d8e1675751ade3677085ffa20bef452": "u_1 = 1.175, a_1=\\left \\lceil \\frac{1}{1.175} \\right\\rceil = 1; \\, ", "2d8e17200939795e1cae220ee76287f9": "E_1(z)", "2d8e410ca26b199c53507d58ddb6d669": "\n[x](k)\\rightarrow x \\Rightarrow C([x](k))\\rightarrow \\{x\\}\\cap X.\n", "2d8e50a3159b21f8e5c1fedbb057b4df": "\\phi \\leq \\psi\\,", "2d8e7de0240f070909dc73ed2069c341": "V_1(\\mathbf{x}, e_1) \\triangleq V_x(\\mathbf{x}) + \\frac{1}{2} e_1^2", "2d8e7e6137175c519dbf7ee713f09f39": " \\eta_{therm}=\\frac{\\dot{W}_{turbine}-\\dot{W}_{pump}}{\\dot{Q}_{in}} \\approx \\frac{\\dot{W}_{turbine}}{\\dot{Q}_{in}}.", "2d8ed4506e8c2297dc8c1ea19906f9ee": " {y_2 \\over y_1} = -{1 \\over 2} + {1 \\over 2} \\sqrt {\\left(1+8{F_{r_1}^2}\\right)} \\quad \\Rightarrow \\quad y_2 = -{y_1 \\over 2} + {y_1 \\over 2} \\sqrt {\\left(1+8{F_{r_1}^2}\\right)}", "2d8f05688ec0c50e9458371750aa0475": "\\mathbb{Z}/n", "2d8f0764068b0d5b7b841069dca7bd22": "\\tan(\\delta)", "2d8f1dcfbbf1f4172b25a5174c169e46": "\\nu=\\frac{\\partial V}{\\partial \\sigma}", "2d8f2a8f544f4eb5341e520fe9b11278": "\\left\\lceil\\frac{|E(H)|}{|V(H)|-1}\\right\\rceil", "2d8f2d73eb79015309de771c8adcac7f": " \\lim_{x \\to p^-}f(x) = L ", "2d8f3355ff528e7ebcb4ef8b41b30b77": " x_{n+1} = \n\\begin{cases}\n\\frac{x_n}{2}, & x_n \\ne 0\\\\\n1, & x_n=0\n\\end{cases}", "2d8f98288f76c34d07fe3373745e83d3": "\\operatorname{Var}(W_t) =E\\left[W^2_t \\right ] - E^2[W_t] = E \\left [W^2_t \\right] - 0 = E \\left [W^2_t \\right ] = t.", "2d8fa9a59e46ac653a331eb27aa41559": "+p_1 p_2 ( 1 - p_3 ) [ N(1-R) \\delta_3 - \\frac{Nc}{4} (\\delta_1 + \\delta_2) ]", "2d8fbadf510a97bed51c63888740ee66": "\\bigwedge", "2d8fd33d20988f1b3e10530eab986b3e": "T_B = \\csc^2{\\left( A/2 \\right)} : 0 : \\csc^2{\\left( C/2 \\right)}", "2d902c0a750a7a22b5c341ce09466a59": "\\Theta=(\\theta^1,\\dots,\\theta^n)", "2d9068cb8a4e92ac980b4543b6a1e5ab": " \\sigma_t^2 ", "2d906d6f20574a7e94118ced8cfbd455": "\\mathrm{1 \\, sb = 1 \\pi \\, L = 10^3 \\pi \\, mL = 10^4 \\, \\pi \\, asb = 10^4 \\pi \\, blondel = 10^7 \\pi \\, sk = 10^{11} \\pi \\, bril}", "2d909f962fe2020b4cf5a275be0c9e4f": " A \\and B", "2d90a28c320a7984e2554316bfa6cb8d": "\n \\frac{\\partial{L_0}}{\\partial\\xi}\n - \\frac{\\partial}{\\partial{t}}\\left( \\frac{\\partial{L_0}}{\\partial(\\partial\\xi/\\partial{t})} \\right)\n - \\frac{\\partial}{\\partial{x}}\\left( \\frac{\\partial{L_0}}{\\partial(\\partial\\xi/\\partial{x})} \\right)\n - \\frac{\\partial}{\\partial{y}}\\left( \\frac{\\partial{L_0}}{\\partial(\\partial\\xi/\\partial{y})} \\right)\n = 0.\n", "2d90c4908d5b9f1e8aa9410757c2a01a": "r := Ax - \\rho(x) Bx,", "2d90d33260a670f799385bf27acc66a4": "\\ell(s_1s_2)=\\ell(s_1)+\\ell(s_2)", "2d910dff7d863a33c6730953c5b3faa7": "\\|\\mathord{\\cdot}\\|_2", "2d915eeb9fb3883f00e5840bd45a0f20": "\\tan \\psi = \\frac {u' + v'} {S} \\,,", "2d916ea2379e1fc0c8b0bd3db94d73c8": "B_{n+1}", "2d91aa4760cbba56000a943ba5d36de3": "\\vec{E} = {\\vec{F} \\over{q}}", "2d91b9f37f9ba90082e6504b32b98f23": "M=(Q, \\hat{\\Sigma}, \\Gamma, \\delta, q_0, F)", "2d9214aa12c6793e06a78863743b8837": "c_0 = {1 \\over \\sqrt{\\varepsilon_0 \\mu_0}}\\ ,", "2d921c8a230bfb50caf5c5a772088fb0": "\\bar{i}", "2d9279bc6ae29cb616087b4459fe0a9f": "\\phi_i = \\frac {V_i}{V}.", "2d92a8030263c55b929a2d7f8f532c8e": "U\\subset {\\Bbb R}^{n+d}, V\\subset {\\Bbb R}^{n}", "2d92b383141d63fb537cdf7e9eb564a2": "\\begin{align}\n\\Delta\\sigma &=\\arccos( \\mathbf n_1\\cdot \\mathbf n_2 ) \\\\\n\\Delta\\sigma &=\\arcsin\\left( \\left| \\mathbf n_1\\times \\mathbf n_2 \\right| \\right) \\\\\n\\Delta\\sigma &=\\arctan\\left( \\frac{\\left| \\mathbf n_1\\times \\mathbf n_2 \\right|}{\\mathbf n_1\\cdot \\mathbf n_2} \\right) \\\\\n\\end{align}\\,\\!", "2d92d873570e9ea98a9ae5a9f4b8b3d1": "\\omega^a{}_{bc}", "2d92f694046a49cb7f5c49a90dce7962": "Q(t,s) = h(t,s) + h(s, t) - \\int^T_0 h(\\lambda, t)h(s, \\lambda)d\\lambda", "2d92fa2d2a3d941e8066d26516d2fa5f": "y^\\prime = y/\\gamma", "2d934fa36157a9db93449f25a431d263": "0 \\to \\Omega^0(M)\\ \\stackrel{d}{\\to}\\ \\Omega^1(M)\\ \\stackrel{d}{\\to}\\ \\Omega^2(M)\\ \\stackrel{d}{\\to}\\ \\Omega^3(M) \\to \\cdots", "2d9362c6490cded6ecd93e0bf42a25bb": "\\geq", "2d936447e9b5f3a799558adb6fd27e05": "\\frac{I(X;Y)}{\\min\\left[ H(X),H(Y)\\right]}", "2d938859f50d1c4cc402fc62a7b44e0f": "\\|x\\|_2 = \\left(\\sum_{i=1}^n |x_i|^2\\right)^{\\frac{1}{2}}", "2d93d6c38cf71241343192d956decba7": "E = (e_{i} - e_{a})g/P", "2d940c28ca11fb037068dcf687a18b11": "S_1(t) = .2^2 = .04", "2d944a29cfe8160ac93ab215526b05c8": "\\mathbf{j}(\\mathbf{r},t)\\cdot d\\mathbf{S} + \\Sigma(t)=\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\frac{\\partial \\rho(\\mathbf{r},t)}{\\partial t} dV,", "2d9456c5f700021fa951e4f49c3ddeae": "\\mathbb{Z}[\\omega]", "2d947c7ccf80a41b6f634013f773ca0c": "\n\\text{Let }a, b, \\text{ and }c \\text{ be integers. Then for every prime } p \\text{ that divides }abc,\n", "2d94dba48c44e9995fc6b58f88d8c7c9": "\\tau = \\|\\mathbf{r}\\|\\,\\|\\mathbf{F}\\|\\sin \\theta\\,\\!", "2d9535a545e474ae67182b74c30753bb": "\\cup[a_i, b_i]", "2d9595f7bb99c79e83fa1b25810975d0": "\\,J^+(x) = \\{ y \\in M | x \\prec y\\}", "2d95c90eaa211c6d64b42848b52d0730": "L = RMV", "2d9683f521919c43e08c7baa07e99deb": "\\Omega(R_0)=\\Omega_0\\ ,", "2d96b0420298e2b4eecc3d6ed5038d4c": "\\langle\\mu_s\\mid s\\in{}^\\omega Y\\rangle", "2d96b0d579ac1a03a64f6f8c6fba0801": "E_\\mathrm{LO}E_\\mathrm{sig}", "2d96b7f44c0e1097ad3f0c8918dc66c7": "\\mathbf{(u^1-u^2)G=0}", "2d96c3c99d7b1477fc474f5c0d078de5": "V_k(\\mathbb F^n) = \\left\\{A \\in \\mathbb F^{n\\times k} : A^{\\ast}A = 1\\right\\}.", "2d96e69d39718d2e4c03d865000bfe5a": "175 = 1^1 + 7^2 + 5^3", "2d97027fe6cc45dff4b19fba0885d987": "B_{n+1} - C_{n+1} = \\sum_{i=0}^n {2i+1 \\choose i} C_{n-i} = \\sum_{i=0}^n \\frac{2i+1}{i+1} B_i C_{n-i}.", "2d971626129b17ce71282335b68e1e1b": "\\mathbf{q}^{om} ", "2d971806210897e2ee12a6154d504afc": " \\bold A = \\bold D - \\bold U - \\bold L, \\quad (6) ", "2d9724675da07dd23751ba900cbb5803": "f^{\\dagger} \\rightarrow f^{\\dagger}b", "2d9735251f1f84097b769ebd6322a5f2": "\\frac{1}{n} * u[n] = \\sum_{m=-\\infty}^\\infty \\frac{u(m)}{n-m}\\qquad m \\neq n", "2d974a5ef0ed321faed8423e6ccd7edb": "1 < p < 3", "2d975f7cd7feff653b24984d8820ea17": "\\scriptstyle \\left({{\\partial P}/{\\partial V}}\\right)_{T,N}>0 ", "2d97c42b7bc519ab7b1c430611ca2c5a": "g(X*Y,Z)=g(X,Y*Z). \\, ", "2d981eb4f29965ebde8b7e10ab3c04af": "\\approx\\frac{2L}{c}\\left(1+\\frac{v^{2}}{c^{2}}\\right)", "2d984d7fc3a9de5765999f0b53c7721e": "\nc=\\frac{1-P_n}{\\sum_{k=1}^nP_k}\n", "2d9879e2c5835aaeaaca1e04b8511ede": "\\alpha = \\frac{1}{\\left |(d(f(f(\\cdots f(z))))/dz)_{z=z^*} \\right |} \\qquad ( > 1)", "2d98c66cdeb76a8ac5be1e342de500d1": "\\overline{S}_{rsi}^{-1}", "2d9924f7494604f05d87230be2f5108e": "(\\lambda, \\mu)", "2d99250f009cf08ffe4af46c72cd4562": "\\begin{align}\n & z\\left( {x_1 \\,\\,x_2 } \\right)\\,\\,\\,\\, \\approx \\,\\,\\,z\\left( {\\bar x_1 \\,\\,\\bar x_2 } \\right)\\,\\,\\, + \\,\\,\\,\\,{{\\partial z} \\over {\\partial x_1 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_2 }}\\left( {x_2 - \\,\\,\\bar x_2 } \\right)\\,\\,\\, + \\,\\,\\,{{\\partial ^2 z} \\over {\\partial x_1 \\partial x_2 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)\\left( {x_2 - \\,\\,\\bar x_2 } \\right) \\\\\n & \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, + \\,\\,\\,{1 \\over 2}\\,\\,{{\\partial ^2 z} \\over {\\partial x_1^2 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)^2 \\,\\,\\, + \\,\\,\\,\\,{1 \\over 2}\\,\\,{{\\partial ^2 z} \\over {\\partial x_2^2 }}\\left( {x_2 - \\,\\,\\bar x_2 } \\right)^2\\end{align}", "2d997ec13230fa38fa0eb3249b52713a": " \\frac{\\dot{a}^2 + kc^2}{a^2} = \\frac{8 \\pi G \\rho + \\Lambda c^2}{3} ", "2d99a6be35a1472a0717b7a537f634aa": "n=1,\\dots,M_{s}", "2d99e0a4f0d4dd71144501e4f544778e": "(\\mathbb{Z}/n\\mathbb{Z})^*,", "2d9a35ee9c9e20ab4c1c3aad3205abcc": "\\boldsymbol{y}_i=\\boldsymbol{H}_i^{-1}(\\lVert\\boldsymbol{r}_0\\rVert_2\\boldsymbol{e}_1)", "2d9a4ac390ba82d3af7c6071cba41b63": "y_{\\xi\\tau}-\\left(\\frac{\\alpha h}{2}\\right)y_{\\tau\\tau}=-y_\\xi y_{\\xi\\xi}-\\left(\\frac{h}{24\\alpha}\\right)y_{\\xi\\xi\\xi\\xi}.", "2d9a53e34449a5a212044e4b4cdb0b19": "\nf = \\frac{676170.4}{47.06538 - e^{0.08950404 \\cdot \\mathrm{ERBS}(f)}} - 14678.49\n", "2d9a67fe6d0f1def73642faf4c976c63": " {\\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} {d \\over d\\tau} (g_{\\lambda \\nu} \\dot x^\\nu + g_{\\mu \\lambda} \\dot x^\\mu) - (g_{\\lambda \\nu} \\dot x^\\nu + g_{\\mu \\lambda} \\dot x^\\mu) {d \\over d\\tau} \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} \\over -g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} = {g_{\\mu \\nu , \\lambda} \\dot x^\\mu \\dot x^\\nu \\over \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\qquad \\qquad (4) ", "2d9a90c0446fa4b20e5b5c77405151a9": "\\displaystyle{|a_{1}|^2 - |a_{-1}|^2 >0.}", "2d9b670d37e8a3f67f3ba96ea8234aab": "c(\\overline{23})=7", "2d9b6a9f8c71253528de3aba2758c7d7": "g_{11}=A\\left(r\\right)", "2d9b6c19c6f4e4bdba49fec0e8f8859e": "1^3+2^3+3^3+6^3=(1^3+2^3)(1^3+3^3)=252.", "2d9bcce3fc04e75dbf0c72b73ad3612f": " \\neg \\neg R ", "2d9bd8c7e4715146f2f607202306c03b": "EL(\\Gamma)=\\infty", "2d9bfd3d0378bef087bd5ce048207009": "U^{(k-1)}", "2d9c14719efc496fa88e1db8a73e3cc5": "a_{I}(t)", "2d9c1884a7e510154738f45f2718b2ce": "f(\\mathbf{X}) = [f(\\mathbf{x}_1),\\ldots,f(\\mathbf{x}_n)]^\\top", "2d9c1cb7c7badb2c45fdca8788fda56b": "1/\\|\\mathbf{x}-\\mathbf{x'}\\|_2", "2d9c9652aaf534f2e815f912fca79383": "u'_{n}(t)+\\alpha\\frac{n^{2}\\pi^{2}}{L^{2}}u_{n}(t)=h_{n}(t),", "2d9cb0e996211d0b0cc58f543012e359": "\\scriptstyle \\left(w_0\\left(n-\\frac{N}{2}\\right)\\right)", "2d9cb725ffbfa04a1422744fa8e4c074": "\\tilde v_i", "2d9d295d8dae2434d5b6ad4b1b6ee2c4": " (y_i, x_{i1}, \\ldots, x_{ip}), \\, i = 1, \\ldots, n ", "2d9d4442b791fff397ba3cbce9b2aa94": "\n \\nabla^2 \\varphi = \\cfrac{1}{\\sqrt{g}}~\\frac{\\partial }{\\partial q^i}\\left(g^{li}~\\frac{\\partial \\varphi}{\\partial q^l}\n~\\sqrt{g}\\right)\n", "2d9d5b0075b0905cbd32e4d84bfac856": "a \\to \\infty", "2d9dbf9947cdfb86bca90a6d0c7f4f4a": "U ( \\rho )", "2d9dedeeb8455cb54359f5e698b1bf2a": "\\Psi = \\Psi(\\mathbf{r},t) = \\Psi(x,y,z,t)\\,,\\quad \\Phi = \\Phi(\\mathbf{p},t) = \\Phi(p_x,p_y,p_z,t) ", "2d9e378a37d2cc1a5374889995e0462f": "Q=I(V){\\tau}_F ", "2d9e53a493196e713595cc03ccc28810": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "2d9f109d719d685cd7ea1f978265b1bc": "D_$", "2d9f5b2cc9ffeb1470f60c73207d5035": "\\operatorname{Ext}^n_R \\left (\\bigoplus_\\alpha A_\\alpha,B \\right )\\cong\\prod_\\alpha\\operatorname{Ext}^n_R(A_\\alpha,B)", "2d9f7e0d6eb438e239bb145d02ae18df": "\\frac{1}{\\sqrt{n}}\\|A\\|_\\infty\\le\\|A\\|_2\\le\\sqrt{m}\\|A\\|_\\infty", "2d9fa264b33bcc416f8966e593afe8e6": "y^2=\\frac{x^3}{2a-x}", "2d9fa741ed7494037dab1a5e77747643": "\\left| x_r - x_n \\right| > \\left| x_s - x_n \\right|", "2d9fe9d1300a4e52536554d3ce333d3e": "y_i^{(k)}, i=1,...,m", "2da0288a8a19c259910b3d3d0c655df5": "\n\\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix},\n\\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix}", "2da02a41f67dda20c03c9eab92adf340": "\\Vert c_{jk} \\Vert^2_{l^2} = \\sum_{jk=-\\infty}^\\infty \\vert c_{jk}\\vert^2", "2da080ed8686c5a27e7ea18dd44e32d1": "\\rho_\\Psi \\left ( t \\right )", "2da0831a622577b1351f8ed78c64b56e": " c(x,y,z)\\max\\{|x-y|,|y-z|,|z-y|\\}", "2da0b84082dd24ef3b7a9addc881d6eb": "\\int f \\, dA= \\int_{-\\infty}^{\\infty} f(t) \\, |\\sinh t| \\, dt.", "2da0f3dd340c81e68553fe79dbe9e9a3": " h_{g; k_1, \\dots, k_n} ", "2da11332ee25699ed2398312c9b1e102": "E \\rightarrow {\\mbox{Div}}^0(E)", "2da1257afa6a65b47f1756fc0d927910": "\\xi: \\Omega \\to \\mathbb{R}", "2da14540509a1600a84a50fd613e66ec": " -\\frac{\\hbar^2}{2 m} { \\nabla^2 \\psi} + U \\psi = i\\hbar {\\partial \\over \\partial t} \\psi ", "2da156584ab3fb2faa2f2a7f6b0624d6": "\\Gamma = \\Gamma_L \\exp\\left(\\frac{-4 j \\pi}{\\lambda}l\\right)\\,", "2da176a659be7cf91e33b59eccf6a7f5": "S({\\mathfrak g}) \\to C^{\\infty}(M) ", "2da1787d6b00540d845ec4089cb8f02b": " \\textbf{c} = -1+X^3-X^4-X^8+X^9+X^{10} ", "2da1b787fecd597ce63607a07f6521cd": "\n f_a(F_D,\\, u,\\, A,\\, \\rho,\\, \\nu)\\, =\\, 0. \\,\n", "2da1fde4d64ea4109cf09913fd5dc88b": " q \\left(q + \\left(x - \\frac{1}{4}\\right)\\right) < \\frac{1}{4}y^2. ", "2da2367b737b5cbd13025e15b519bf29": "f(x)=\\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} \\cos (2\\pi (x-y)\\cdot\\xi) \\, f(y)\\,dy\\,d\\xi.", "2da23bedfdbee0830438c492b83e9360": "\\theta = (\\tan^{-1}1.505314) / 2 = 28.20169^{\\circ} \\text{ or } -61.79831^{\\circ} ", "2da23fa8308ae2596e3498bb1eaecfd7": "\\frac{dx}{dt}", "2da24a3c45e66253317b87f8cefddaea": " Z_3 = T_1Z_1T_2Z_2 - kX_1Y_1X_2Y_2 = -8\\sqrt{3}", "2da26034a8f21066eebb8675ee440ade": "\\int\\mathbf{\\Psi}_{lm}\\cdot \\mathbf{\\Phi}^*_{l'm'}\\,\\mathrm{d}\\Omega = 0", "2da2c1186e28b5a45661105791baaad0": "\ne^2 = \\frac{4n}{(1+n)^2}= 4n(1-2n+3n^2-4n^3+\\cdots).\n", "2da2e3dd8b87309eed8d41abfd75975b": "\\frac{\\partial \\tau_{xy}}{\\partial x} + \\frac{\\partial \\sigma_y}{\\partial y} + \\frac{\\partial \\tau_{zy}}{\\partial z} + F_y = 0\\,\\!", "2da2e5a0e596de801b06ffa9d7aa4cdc": "\\Pi^{-1}", "2da326923cb46335c8217fcf48ea9d05": "\n\\mathbb{P}(x \\mbox{ received} \\mid y \\mbox{ sent}) \n", "2da35dd23a36fd68bfcc9095a7c5f736": "~ \\hat A =\\hat U^\\dagger \\hat a \\hat U~ ", "2da3a4120e61751393a788cb437a7fc9": "\nE_{sr} = \\sum_{i,j} \\varphi_{sr}(\\mathbf{r}_j - \\mathbf{r}_i)\n", "2da3d6c09f9386b4bc03b51d3661b6e6": " \\begin{matrix}\nI &=&|E_l|^2+|E_r|^2, \\\\\nQ&=&2\\mbox{Re}(E_l^*E_r), \\\\\nU & = &-2\\mbox{Im}(E_l^*E_r), \\\\\nV & =&|E_l|^2-|E_r|^2. \\\\\n\\end{matrix} ", "2da483ed15b9b1d1c92a976261c2f00d": "(0,1,2)", "2da49dd09a3624b0b40dc6bb64006aad": "\\mathit{l}", "2da51078891f99391553787d878d6d75": "( \\forall x\\,Px \\and \\exists x\\,(x = a)) \\Rightarrow \\exists x\\,Px", "2da535549526a832e750f7b879691c3a": " f(\\lambda x) = \\lambda^n f(x)\\,. ", "2da540fc05c72e55334f13afe09661b2": "\\mathbf B=\\mathbf C \\mathbf A\\mathbf C^T", "2da57e502e5be0ebf11548bf1d9911d3": "\\boldsymbol{r}", "2da5a2c1045c8a6463e1b32b01010954": "{y}", "2da5cc88d8ac3f1e8110603921707bcf": "\\sec\\left(\\frac{\\pi}{2} - A\\right) = \\csc(A)", "2da5d9b2f73cae95564fb462de45d1f9": "\\left( e^{-2\\phi} \\right)_{,uv} = - \\lambda^2 e^{-2\\phi}e^{2\\rho}", "2da60921be7994a7081baeff93d28361": "D_{KA}", "2da60940ef8d47f684e9f047afab9a8e": " DK_i ", "2da60b2de22c8d46d381ba358286b3ed": "W(u,v)", "2da60db3bd4e0616db93273213aa5e5e": "x*y = \\left(x_1 y_1,\\dots,x_d y_d\\right)", "2da66b90dcc52b54d347c1364d3d14ce": "\\Delta z = \\operatorname{cov}\\left(v_i, z_i\\right)", "2da6924da566e3e965565e39be33c1a0": "\nDF = \\frac{Z_\\mathrm{load}}{Z_\\mathrm{source}} \\,\n", "2da6db8304836dbf0c1d0de5ac4121de": "\\tfrac{1}{2}\\pi", "2da75fc7890beca9479e5615be92bef0": "\\{0,1\\}^\\omega", "2da78d3a8dd028a702fded171af112ad": "\\{\\mathbf{u}_1, \\ldots, \\mathbf{u}_m, \\mathbf{w}_1, \\ldots, \\mathbf{w}_n\\}", "2da7a67a772fd27ce897bb21a51635c8": "T_n(x)= \\frac 1{{n-\\frac 1 2 \\choose n}} P_n^{-\\frac 1 2, -\\frac 1 2}(x)= \\frac n 2 C_n^0(x),", "2da7bdaf5c7e3383bbf3353dbb3ce8a7": "|x-y| = 2^{-\\nu(x-y)}", "2da814ca560218474a377491fdb0a9f0": "R=n\\ln{\\left(1+r/n\\right)}", "2da83e33a481dd7bdeb7003b3558004f": "\\neg\\exists p: \\mathcal{B}p \\wedge \\mathcal{B}\\neg p", "2da83e6ecc3e7a75aff40fe551d58d74": "\\mathbf{y} = \\Phi(\\mathbf{x},\\mathbf{u},\\dot{\\mathbf{u}},...,\\mathbf{u}^{(\\alpha)})", "2da842e2aa6a7e79f7703f285c148e74": "{\\nabla}^2 \\varphi = \\kappa^2 \\varphi", "2da8e5712501b235315436fc05fba867": " \\theta_{co} ", "2da92b3455bd050ceabbf35dfc7fdd79": "r = \\lfloor 0.5 + \\mu_{3,l} \\rfloor =5", "2da99c964f06aa1c872c17b9a42683d3": "{}_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 \\left (a, c-b;c ; \\tfrac{z}{z-1} \\right )", "2da9b3ec8ea2aaf8b288fdea58628240": " \\tilde{a} \\in \\mathbb{Z}^{n}_{q} ", "2da9f7ca3cb6971912ebb5c3d2797553": "B = \\rho_f V_\\text{disp}\\, g, \\,", "2daa2a6250c577d9c7e16b7e95ef6ba0": " = \\sum_k q^{k(k-1)/2} \\binom{m + n}{k}_{\\!\\!q} x^k.", "2daa2cf0ba60d8abdb7f2717fdd0772a": " C\\,", "2daa7b9f39d6cd0ee39ba78af90a4cf9": "\n\\boldsymbol{\\alpha}=(\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4,...,\\alpha_n)\n", "2daa8338296fecb772d3c4a05db622ac": "=119", "2daa847dd4ee37e8538b7854f518f44f": "\n{{\\Delta \\hat g} \\over {\\hat g}}\\,\\,\\, \\approx \\,\\,\\,{{\\Delta L} \\over L}\\,\\,\\, - \\,\\,\\,2\\,\\,{{\\Delta T} \\over T}\\,\\,\\, + \\,\\,\\,{{\\sin (\\theta )} \\over {4\\,\\sqrt {\\alpha (\\theta )} }}\\Delta \\theta", "2daa9e47bcc9ba0c226d674db84e4288": "U_1\\cap U_2", "2dab76067d51ae814a5e03f0480bdf66": " \\min\\{m,n\\}", "2dabce6ea407e4350012a8472c52fcc8": "\\scriptstyle \\mu = r", "2dabd4a2c9254eebf8d1534862db8651": "\\sqrt{\\pi}", "2dac0b7ad07a6357ebaad1e28bc4ef2b": "\\{0,1\\}^n \\rightarrow \\{0,1\\}^k", "2dac0d14755d9873bc5be1ededb5fdc2": "D_{odd} := \\oplus_{n \\, odd} \\, D_n", "2dace6f161d7f3e1a4b6bb1d51c7dbe0": "\\begin{bmatrix} \\dfrac{1}{h_{12}} & \\dfrac{-h_{11}}{h_{12}} \\\\ \\dfrac{-h_{22}}{h_{12}} & \\dfrac{\\Delta \\mathbf{[h]}}{h_{12}} \\end{bmatrix}", "2dacef5de504b1e46b4c5a620940fc62": "D(a,b)=0", "2dad0b29cd0b4d5bc45015d56ceb269e": "(N-x-1, 0)", "2dad5f89d9e72ade6227981c2bcc9e02": "\\frac{1}{d_\\mathrm{o}}+ \\frac{1}{d_\\mathrm{i}} = \\frac{1}{f}", "2dad78dff4d50c6622ff9f162637a130": "\\scriptstyle p( ", "2dad8b7a6c4cba14e048534a88ad94b2": " S^{\\mathrm{core}} \\,", "2daddc12621117703856ee88bf531906": "\\text{Fib}(0)=0\\text{ as base case 1,}", "2dadde5e9afc0e82d96f0a8e6498d95f": "v=2+9+24+50+90+147+224+", "2dadf17e239f5188ddc4b99a318d7a16": "r = -v/i \\qquad\\,", "2dae5ccf8f2621a5aa3760eba0294f73": "\\sup(f(\\mathbb{M})) = \\sup(\\mathbb{M})", "2dae5ed8053cf9ce7d55d77ce02d4a0f": "M^{z+} + Red_{solution} \\stackrel{\\text{catalytic surface}} \\Longrightarrow M_{solid} + Oxy_{solution}", "2dae69d189ab5b1c513a8524e3c795c8": "M_\\infty \\simeq 1", "2daeae02d37a6c5fb04eef6f47b7a9f7": "3, \\tfrac{5}{2}, \\tfrac{8}{3}, \\tfrac{11}{4}, \\tfrac{19}{7}, \\tfrac{30}{11},\n\\tfrac{49}{18}, \\tfrac{68}{25}, \\tfrac{87}{32}, \\tfrac{106}{39}, \\ldots\\, .", "2daeb37db05c37b2b0c4f50e1617a58c": "10^d", "2daf6a0b920929b7524635db4f3c5a1b": "H':", "2dafb2b50e137128b8c16311873fc191": "\\mu(X_1\\times X_2)=\\mu(X_1)\\mu(X_2)", "2dafe2a2e7ad3d0013c4bcbcbc5cc76f": "\\forall x\\in I_1\\quad\\lim_{h\\to 0}\\frac{(x+h)f(x+h)-xf(x)-\\left(F(x+h)-F(x)\\right)}{f(x+h)-f(x)}=x.", "2db0495e59746de76f4dd18c442d9c51": "RD_\\mathrm{new/ref} \\approx 1 + \\frac{A \\Delta x}{m} \\rho_\\mathrm{ref} ", "2db058e63118bb94b5f6756316ac5d2b": " e^x = 1+x+\\frac{x^2}{2!} + \\ldots + \\frac{x^9}{9!} + R_9(x), \\qquad |R_9(x)| < 10^{-5}, \\qquad -1\\leq x \\leq 1. ", "2db06c3f8379de2996eb67feb69d986c": "T_i (T_j f(x)) = T_j (T_i f(x))", "2db0bd69cb67c5537699acafeaf0c383": "D(p||q)=D^{(-1)}(p||q)=D^{(1)}(q||p)=\\sum{p\\log\\frac{p}{q}},", "2db1b95fa03f819d59916ddb618ba361": "\\hbar / \\Delta E ", "2db21a6f6189ec9291a4606962433a23": "\\lambda=\\pm 1/2", "2db273c4c505e961307e3b3550c5bfab": "C_z", "2db28658d9a413243b2e370be113b6c4": "\nProgram\n\\begin{cases}\nDescription\n \\begin{cases}\n Specification (\\pi)\n \\begin{cases}\n Variables\\\\\n Decomposition\\\\\n Forms\\\\\n \\end{cases}\\\\\n Identification\\ (based\\ on\\ \\delta)\n \\end{cases}\\\\\nQuestion\n\\end{cases}\n", "2db2a2aebda2b8111c3991defd0224c2": "P\\approx \\frac{Li}{1-e^{-ni}}= \\frac{L}{n}\\frac{ni}{1-e^{-ni}}", "2db2b6ce7e86bad05141addc8b8e7719": "\nf_i = {1\\over L_i}\\left[kKT\\left({L_i\\over L_i^0}\\right)^2-pV\\right].\n", "2db2f4d5ccabcdac41222dca0f27049b": "x_f(t)", "2db304ed01e1754c48ed4a750cb9bbbb": "\n\\frac{1}{\\sqrt{\\lambda}} = -1.8 \\log \\left[\\left(\\frac{\\varepsilon}{3.7D}\\right)^{1.11} + \\frac{6.9}{Re}\\right]\n", "2db33aceb1235e01003f98fc2fc97634": "c_{12}=x+b_{7}=2*x+(b_{7}-a_{7})+(b_{8}-a_{8})=2*x+161", "2db3add6404b8856b25b49da8f5e691c": "\\Theta(m^{2})", "2db3fa6e33d7d7300bdaf7b939df121b": "(\\ |\\ ) \\!\\,", "2db40c934ed38c92a81557fcf25b85f5": " \\displaystyle{[H_m,E_n]=2E_{m+n},}", "2db46f629bc6062c6907d04ae7f64c7d": "\\scriptstyle \\sum_{n=m}^\\infty(m\\mid n,\\, k) \\;<\\; \\infty", "2db475aac45055a9e4778e68254cb2f9": "\\beta(s) = \\frac{1}{\\Gamma(s)}\\int_0^{\\infty}\\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\\,dx.", "2db48b14203690a6edb9012a15772782": "\\mathbf{p}=m\\gamma \\mathbf{v}", "2db57be549cb7dad5d9b431e862e9396": "L_{1} [\\perp] L_{2}", "2db5de4af795c45107b9650365ed0caf": "x^3 \\equiv \\alpha \\pmod{\\pi}", "2db5e63c450d37cc4d9a2570bd9e102c": "d(c_i,c_j)", "2db5fd0331bea4d4fffb55b0267a0fd1": " MC = \\frac{\\Delta C}{\\Delta Q}", "2db615699378c62e510dc1af0bc1b821": "~r_0~", "2db67514f2dfc8e7138480e166f1c973": "k=1, 2, ..., p", "2db6d16aafe62055890d004414d361d9": "K\\to 0", "2db6fdc1f0b42c4f6c9c7b08e8b53d7c": "\\mathbf{J}=\\epsilon_0\\int \\left(\\mathbf{E}\\times\\mathbf{A}\\right)d^{3}\\mathbf{r} +\\epsilon_0\\sum_{i=x,y,z}\\int \\left({E^i}\\left(\\mathbf{r}\\times\\mathbf{\\nabla}\\right)A^i\\right)d^{3}\\mathbf{r} ,", "2db71c632a19229ef394d7a758bb50ae": " X_{g} ", "2db72c29a275a14ef54913216441b2be": "W = {\\Pr}_{\\text{random }G} [wt(mG) < d]", "2db786687f3d9e271bfceae0741e80ff": " U_E(r) = k_e q \\sum_{i=1}^n \\frac{Q_i}{r_i}", "2db7901bf19851caf815e6d58f49cbb8": "2d 8 ", "2de457575bdb54980644b3c63d6a416c": "= 3 \\rightarrow 3 \\rightarrow 65 \\rightarrow 2\\, ", "2de49512423708052c68139ce022bf46": " k_2", "2de4eac68c7332020bf638c6d57a9621": "f(1/m)^{m}=f(1)", "2de4f17aeddca608248b3f3fa6139098": "\\left(b \\rightarrow Q\\right)", "2de4f7f91cde00f4789c0a12ea394b9d": "\\tau =\\xi^{\\,z}", "2de510e4036c40a6b8648594ad43688d": "PR=\\frac{P_2}{P_1}", "2de561725beaad13706d654577022ec6": "\\Theta(n^{3/2})", "2de5623efca914934a750fc416ad44b7": "\\log_{10} \\mbox{ year} + 7.50", "2de5c176f60de730499030e01f2f591c": "\\mathbf C^{-1}", "2de5e041fe771b7f1c6bbb7a0dac78cf": "\\frac{1}{\\mu} = \\frac{1}{\\mu_{\\rm impurities}} + \\frac{1}{\\mu_{\\rm lattice}}", "2de5e1962896bb2e7133cbdb5085efac": " \\mathbf{r}_0 \\cdot \\mathbf{\\hat{n}} = \\left | \\mathbf{r}_0 \\right | \\cos \\alpha_0 \\,\\!", "2de6bfe657f54f8e7cb672ca05ddb0fb": "\\displaystyle{ \\|U(f)\\|,\\, \\|V(g)\\| \\le 1}", "2de714093fb95d77b6175184310f0acf": "T _{1/2}", "2de794c24be07a33e58ad1b667ce6736": "\\mbox{backward reaction rate} = k_{-} [S]^\\sigma[T]^\\tau \\dots \\,\\!", "2de7b7c8e80fd7a8948a9ff818063da3": "T_\\mathrm{sol-air} = T_o + \\frac{ (a \\cdot I - \\Delta Q_{ir})}{h_o}", "2de7cb9330fb2a34d55b70a3c81234d4": "S \\rightarrow A: \\{N_A, K_{AB}, B, \\{K_{AB}, A\\}_{K_{BS}}\\}_{K_{AS}}", "2de7fefaece84b872d02d273f58c3794": "R_i = \\sum_{j\\neq{i}} \\left|a_{ij}\\right|", "2de81104dae2912d1efdc790ffcb769f": "\\Delta g_{21}", "2de8805266a82e973b6931291823fc89": "\\,S", "2de8f43b7d82e53fb10d6ba925cfc26d": "Y_{i-1}", "2de932fb6f2dbaab8f9b951c23f6dbca": "\\frac{\\partial}{\\partial t}(\\rho V_i)\n+ \\frac{\\partial}{\\partial x_j}(\\rho V_i V_j+P_{ij})\n- nF_i=0", "2de960c6ab3cef39a9a2298fb58e503d": "TG", "2de97d70e3f9a8fb2f5c63872f2c830f": "e^{\\mu t + \\sigma^2 t^2 / 2} \\left[ \\frac{ \\Phi(\\frac{b - \\mu}{\\sigma} - \\sigma t) - \\Phi(\\frac{a - \\mu }{\\sigma} - \\sigma t) }{\\Phi(\\frac{b - \\mu}{\\sigma}) - \\Phi(\\frac{a - \\mu }{\\sigma}) } \\right] ", "2de9a2c40fa797e7d175c0bf1d027709": "f \\in L^1(\\mathbb{R}^n)", "2de9a7563091ed8aa3a8926550555790": "tq", "2dea0a73d1bedadf2e94e8308308c82e": "c^a(x)\\,", "2dea3a06d0765436ddc57be253e22a48": "\n\\frac{\\partial \\bold u}{\\partial t} + \\bold u \\cdot \\nabla \\bold u + \\nabla\\frac{P}{\\rho}=F_D (\\bold u) -F_C (\\bold u)\n", "2dea479bcb4af08b3118f5ad1192baf6": "(\\forall \\varepsilon>0) (\\exists p_0) (\\forall p>p_0) (\\forall n) \\left|\\frac{x_{n}+\\ldots+x_{n+p-1}}p-L\\right|<\\varepsilon.", "2dea58189aa3e52b910b1b6d70bb4523": "W_j(tK) = t^{n-j} W_j(K)~, \\quad t \\geq 0~. ", "2dea6ee5f684928e73b3edf0e550a3b2": "\\kappa\\mapsto 2^\\kappa", "2dea7d719b1ba45850de66b7eee1c3be": "\\textrm{hacovercosin}(\\theta) := \\frac {\\textrm{covercosin}(\\theta)} {2} = \\frac{1 + \\sin (\\theta)}{2} \\,", "2dea918edd4b88c060f36fd40b9fcfd4": "z^n=r^n e^{in\\theta} ,\\bar{z}^n=r^n e^{-in \\theta}", "2dea93a2d6fe5db40e86e62ab3b63eb5": "\n\\mathbf{H}_{\\alpha -1}(x) - \\mathbf{H}_{\\alpha+1}(x) = \n 2\\frac{\\mathrm{d}\\mathbf{H}_\\alpha}{\\mathrm{d}x} - \n \\frac{{(x/2)}^\\alpha}{\\sqrt{\\pi}\\Gamma(\\alpha + \\frac{3}{2})}.\n", "2dea9da82a429e3b642ff749f9a74027": "~n~", "2deab4cafc143e547359cb27b863e7d9": "\\{\\varphi_\\alpha\\colon U_\\alpha \\to V_\\alpha\\}", "2deaee6e3b1c0627a6031f2d6cee37dc": " a_t", "2deb04a8bdc6b9951efff5f047fb936d": "\\prod_{1\\le i c(\\varepsilon) x^{1/2-\\varepsilon}.", "2e064bc9047e4c7b085d6ce1ef0a4839": " f \\in C(G) ", "2e0701fe026c01f075c281b5a085da51": "\\mathbf{X}=[x_1,\\dots,x_n]", "2e0784e1e30639e82c330f85b761bf53": "F(y) = 0.5\\,[1 + \\sgn(\\log(y)-\\mu)\\,(1-\\exp(-|\\log(y)-\\mu|/b))].", "2e0791465f26519495e827b5d00dc13f": "F_\\mathrm{tot} = ma = m \\frac{d^2x}{dt^2} = m \\ddot{x}.", "2e07c88394f8e905fba80e2a3d465b3b": "E_x^{\\rm PBE,LR}(\\omega)", "2e07d54647a29ee8601ff5005f56ec49": "E=\\bigcup_k (a_k,b_k)", "2e0835364a15b2272a4f612ff91a75d8": "2\\pi/c_{ij}", "2e087fe7a54cb70b15f5a359dbb2bebb": "A \\rightarrow B: M,A,B,\\{N_A,M,A,B\\}_{K_{AS}}", "2e0881896b03eb822c9b4d0dc5343753": "\\zeta(3).", "2e089c7618acf7b2647ad6865a1936e6": "360^\\circ", "2e08fbd02a4f18f9b6cc4dc729ca38c2": "\\left(i_1,i_2,\\ldots, i_n\\right)", "2e090022d19aeaf8dfddf008bcddc796": "\\frac {1}{R} = \\frac {d\\theta}{ds} \\propto s ", "2e0989895bb47fe0a88d52b9c2429668": "A^{\\circ\\circ}", "2e09a33ee5c1f885d7b55654fc2b4725": "d_e(t_0) \\rightarrow \\infin", "2e09fc5d19884968971c7a0191142d18": "\\gamma^2 = \\frac{1}{1-v^2/c^2}", "2e0a1a325a1a0a033d384bd9bae009b5": "C_n = [-\\frac{1}{n}, 1+\\frac{1}{n}]", "2e0a378e260c5e81e4cd190260dce2a5": "R_1 = 50 \\ \\Omega\\,", "2e0a82654dc383f4a0139d97eb34ceaf": " \nx(t) = \\begin{cases}\n\\cos(2\\pi t) & \\text{for } t \\le 0, \\\\\n\\cos(4\\pi t) & \\text{for } t> 0.\n\\end{cases}", "2e0a8f7435e9e48c3f86c8d72266a034": "6n", "2e0aa3ae9ad7dddafde1a70e644042e3": "I_x = I_y = \\frac{1}{12} \\pi\\rho h\\left(3({r_2}^4 - {r_1}^4)+h^2({r_2}^2 - {r_1}^2)\\right)", "2e0b0c992de03ce289051f8807499e18": "\n|X-O|^\\alpha+|O-Y|^\\alpha = c\n", "2e0b18410eb8f2bcb04bc8772e447546": "f(\\theta)=0", "2e0b18dd5050796002d82add1fc33666": "A\\in\\mathbb{R}^{m\\times n} \\ ", "2e0b1feecc9742bd4623c36c0dc6829e": "m-2n0 \\,", "2e1f153819256ddf8430d122596b63c4": " \\mathcal{H}(X,h) = \\{ F_{\\bullet} \\mid X F_{i} \\subset F_{(h_i)} \\text{ for } 1 \\leq i \\leq n \\} ", "2e1f1960d5a7a7ab4f986dbed4105657": "(f * g)[n]\\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{m=-\\infty}^\\infty f[m]\\, g[n - m]", "2e1f2799c11b1ed1f122b569efb1d575": "\\langle \n\\mathrm{RPA}|\\mathrm{RPA}\\rangle=\n\\mathcal{N}^2 \\langle \\mathrm{MFT}|\n\\mathbf{e}^{z_{i}(\\tilde{\\mathbf{q}}_{i})^2/2}\n\\mathbf{e}^{z_{j}(\\tilde{\\mathbf{q}}^{\\dagger}_{j})^2/2}\n| \\mathrm{MFT}\\rangle=1\n", "2e1f5ba68c93afe57766b671cfb74e28": "{[f(x) g(\\theta)]}^{h(x)j(\\theta)}, \\qquad {[f(x)]}^{h(x)j(\\theta)} [g(\\theta)]^{h(x)j(\\theta)},", "2e1f9fff3d25d411646aa2062603fa35": " |U(\\omega_0)|=\\gamma ", "2e1fa0e9f6e9521c8de073a05e1a98f4": "f=\\frac{n}{N},", "2e207d845f7effb325d7c9754b9e4638": " \\mathrm{li}(x) = \\int_0^x \\frac{dt}{\\ln t}, ", "2e209036670d6d5baa66198bb203e71d": "p^j", "2e209658ba6193296a434278d0aa6340": "\\operatorname{lcm}(8,9,21) = 2^3 \\cdot 3^2 \\cdot 7^1 = 8 \\cdot 9 \\cdot 7 = 504. \\,\\!", "2e20daa72395379239910fb99321174e": "\\rho(r) = \\frac{3 [v(r)]^2}{4 \\pi G r^2}", "2e21dc3a6a7fb1cfc13bd293875c0e33": "K = \\lambda + \\mu", "2e21dcd02be3b9fd92c827415aefc9b6": "c_p\\rho u_t - k u_{xx} = 0. \\,\\!", "2e21ef359e26093570733b0d48851198": " \\frac{1}{2} R_{\\frac{\\lambda}{2}} I_0^2 = \\frac{1}{2} R_{fd}\\left( I_0/2 \\right)^2. ", "2e220003cb9b15bb63fd7e8ff254bac8": "Ab\\,", "2e22f3682485648a9125dfb49f96f70b": "\n\\xi \\mapsto (x^1,\\ldots,x^n,\\xi^1,\\ldots,\\xi^n)\n", "2e232dcb4cdc6e5b1a548623eadc0b08": "1^7 + 2^7 + 3^7 + \\cdots + n^7 = {12a^4 -8a^3 + 2a^2 \\over 6};", "2e23702ca06edbe1bd8f5dab5efcbd8f": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 19.54\\cdot 5.99)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 25.2\\cdot R_{\\bigodot}\n\\end{align}", "2e2388f5a8060a5823bdc7c38c9dad00": "\\sum_{s\\in \\Omega }h_{[s]}(\\mathbf{\\pi })=q\\left( \\sum\\limits_{t=1}^{S}\\pi\n_{[t]}\\right) =q(1)=1 ", "2e239829c956284f99b283670e734b42": "\\tau_M", "2e244a6374e75d6d6b2be2973417b374": "\\mathrm{E}", "2e246a9cbc57942473a22afabbf3fa06": "J(0,0) = \\begin{bmatrix}\n\\alpha & 0 \\\\\n0 & -\\gamma \\\\\n\\end{bmatrix}.", "2e24b61c2de3c7469ba5390c5fdf758c": "\\scriptstyle i \\;=\\; 1,\\, \\ldots,\\, k-1", "2e24bd3466c5db16659fb5777157f593": "\\max\\{w_C(c):c\\in C\\}", "2e251029fb7ef8dbd6a602a351240973": "\\frac{V(t)}{V_0}=\\left(1-e^{-\\frac{t}{\\tau}}\\right)", "2e252e30f93c61550ceaa2d512177cc2": "\\mathbf{T} = -m \\omega_I^2 R \\mathbf{u}_R \\ . ", "2e2549fab28307d2ed6747b5993a7a9c": "\\{B \\mid A \\sim B\\}", "2e257f44ecc7383955eacd5abe65dd2e": "\\widehat{a}_j\\rho \\rightarrow \\left(\\alpha_j + \\kappa\\frac{\\partial}{\\partial\\alpha_j^*}\\right)\\{W|P|Q\\}(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)", "2e25bae147ded8d13c50fe6b0a375a1f": "\nL(x,v)=\\tfrac{1}{2}|v|^2.\n", "2e26318393d9208639aa32df8992a7d5": "3x \\equiv 2 \\pmod {7}", "2e268f91be81edf2879b918a404ffc48": "\\operatorname{conv}(A\\backslash \\{x\\})\\cap \\operatorname{conv}(B\\backslash \\{x\\})\\not=\\varnothing,\\forall x\\in X. \\, ", "2e26f363b00e46112d9e289aa3d4ae29": " R\\simeq K[[x_1,\\ldots,x_n]]/I ", "2e26f3fdd7ae42bbf191756b3efe7114": "b(z) \\equiv \\sum_{n \\in \\mathbf{Z}} b_n z^{-n-1} \\equiv \\sum_{n>0} \\left( x_n z^{n-1} + n \\frac{\\partial}{\\partial x_n } z^{-n-1} \\right),", "2e27044a09d2fe9ca1bd7abdbe46bcd7": " - \\left(\\eta_1+\\frac12\\right)\\ln\\left(-\\eta_2 + \\dfrac{\\eta_3^2}{4\\eta_4}\\right)", "2e27c8fcd358666fcc0e27fbc1724ac4": "d-\\beta c", "2e27f8ebc2502d48debea014d392b832": "\\|f\\|_k = \\sum_{i,j} k(\\mathbf{x}_i,\\mathbf{x}_j) c_i c_j\n", "2e28187d49603aa401ea57c0dbf06a53": " \\mathbf{x}'(t)=\\mathbf{A}[\\mathbf{x}(t)-\\mathbf{x}^*]", "2e283be13199e97aa5aec133c40c0767": " M-1 ", "2e283bf0602cf2e7bb10ecf0b791e5e1": " X \\sim NEF [\\mu, V(\\mu)] .", "2e2876850796ffe2b73190f30fd6245f": "N(0_k,I_k)", "2e28dd4f94e72560c518ebba12725d8f": "_{p \\nleftarrow p=0}\\!", "2e2924f964ba29a1c7f6f283b11cedea": "\\frac{\\partial G}{\\partial x} \\frac{\\partial x}{\\partial v} +\\frac{\\partial G}{\\partial y} \\frac{\\partial y}{\\partial v} = -\\frac{\\partial G}{\\partial v}.", "2e2925926513b61815f7132355942978": "\n\\begin{align}\n\\sigma_0(p) & = 2 \\\\ \n\\sigma_0(p^n) & = n+1 \\\\\n\\sigma_1(p) & = p+1\n\\end{align}\n", "2e2938ef578167f1bb063dabe0f4640b": "\n\\sum_j P_{ij}x_{ji} - X_{ij}p_{ji} = i \\sum_j 2m(E_i - E_j) |X_{ij}|^2 = i\n\\,", "2e298a5b4335f6797306c5d43d345aca": "W = 330 + 0.452\\,T + 0.00415\\,T^2 ", "2e29caacdfe2c08fd1adecdd0299fb0e": "{\\hat{a}}_{i}", "2e29f266dc084b1e636cd8677e4d40a1": "b_k(x)", "2e2a25d8d4a013b8e804f1857402d8ca": "P(E_n|M_m)", "2e2a5c5bc595a473ee1ab1e73ff956bc": " A=(X+ t^2 P) ", "2e2b0ad913499deae4669f03343782c0": "\\alpha \\neq \\beta", "2e2b359239f9151646777fe01e841183": "\\begin{pmatrix}\n 1 & 1 & 2 \\\\\n 1 & 0 & 2 \\\\\n 0 & 2 & 1\n\\end{pmatrix}", "2e2b5e37283ec196cb7aee760feb4dbf": "{\\Bbb R}^3\\ltimes{\\Bbb R}", "2e2b92ca2638aa14c9bd6452a9b16f59": "\\mathbb{P}(Z_\\pi(T) \\geq Z_\\rho (T)) \\geq 1", "2e2b95068d9ddb49a3053a8d2608b33b": "\\mathrm{COMPOSITE} = \\left \\{x\\in\\mathbb{N} | x=pq \\;\\text{for integers}\\; p, q > 1 \\right \\}", "2e2ba7f56f560ebd0acff02fa11bfa60": "\\sigma_k(n) = \\prod_{i=1}^{\\omega(n)} \\frac{p_i^{(a_i+1)k}-1}{p_i^k-1}\n= \\prod_{i=1}^{\\omega(n)} \\left(1 + p_i^k + p_i^{2k} + \\cdots + p_i^{a_i k}\\right).\n", "2e2bdc70f981081f884913d230823856": "\\operatorname{f}_1(x) \\le \\operatorname{wnchypg}(x;n,m_1,m_2,\\omega) \\le \\operatorname{f}_2(x)\\,,\\,\\,\\text{for}\\,\\, \\omega < 1\\,,", "2e2bf0399b223af8be2270542f15bbbe": "f''(x) = {\\mathrm{d}^2 \\over \\mathrm{d} x^2} f(x)", "2e2c0cd55f1d70471553e62a56e61704": "Y \\to Z", "2e2c0dbff1cb1abcf66d0c981d6daf09": " AH = {m_w \\over p_{net}}. ", "2e2c187ff7aadba9ccd2f6b06ee8b3d8": "d(u, v) = \\arccos \\left(\\frac{|u \\cdot v|}{\\|u\\|\\ \\|v\\|}\\right);", "2e2c2bcfcad89a5c1f4ddd8346c6fce8": " y_l ", "2e2cc4b01bf9dac629110c090050d8d6": "\\eta_c(s) = \\frac{c \\mu}{k \\mu + s + \\lambda-\\lambda \\eta_{c}(s)}.", "2e2cce52e69e6a12923fbd8201489104": "\\omega_{jk,h} ,", "2e2d0bd08298bd5ded6ddb8f60e512cd": "f(z)=1/\\Gamma(z)", "2e2d33e91c4b7a8d222f72a7f5bf7094": "Q^+(3,q)", "2e2d46c8fbde14df8921995d3d0d10dc": "\\left\\{\\beta_k\\right\\}_{k=1}^\\infty ", "2e2d8f6ab2cfbc2488bf7ef27155dcbd": " \\frac{1}{\\pi} = 12 \\sum^\\infty_{k=0} \\frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}. ", "2e2de9ab89408fe2fd9de341074a9ca4": " \\mathbb{C}[\\operatorname{Hom}(\\pi,G)]^G.", "2e2e8e68f68f30af0a16e810d9e8a362": "\\begin{array}{rl}\nx_{n+1} =& x_n + 1\\,\\frac {\\left(1/f\\right)(x_n)} {\\left(1/f\\right)^{(1)}(x_n)}\\\\[.7em]\n=& x_n + \\frac{1}{f(x_n)}\\cdot\\left(\\frac{-f'(x_n)}{f(x_n)^2}\\right)^{-1}\\\\[.7em]\n=& x_n - \\frac{f(x_n)}{f'(x_n)}.\n\\end{array}\n", "2e2edde6df694960751bd4fb39dad5e6": "\\frac{c}{a} - 1 = -(31 \\pm 1) \\times 10^{-4}", "2e2ee7c55dd66a7a0b10aa846170d12c": "\\displaystyle{\\Delta_1=\\Delta -[\\Delta,\\psi]=\\Delta + (p\\partial_x + q\\partial_y - \\Delta \\psi)=\\Delta +X.}", "2e2f1dd80533a5a93cd4a6735e837d9c": "x + y \\sqrt{n}", "2e2f94ae0c38b71549b5b0d04a7ee66b": "\\pi \\rho_C^2 = \\pi x^2. \\,", "2e2fbbe2df05310b9fa54bd275feea80": "\\mathrm{0.\\overline{6}}", "2e2fc61ccc418156c05951ac53d73aa3": "g(p) = \\ln \\left( { p \\over 1-p } \\right).", "2e2fd171b1831f977203f287d644237f": "P(a_{T+1}|\\hat{a}_{1:T}, o_{1:T}),", "2e2fdc520d90c353040394817ad1f9a7": "^1_2", "2e3005d966f1de0fd828ab932d678b6d": "M \\to M''", "2e300be4e736fbef4fa8052ab59f91fd": " v(t,x,y,z) = \\frac{\\part}{\\part t} \\left( t M_{ct}[\\psi] \\right), \\,", "2e304ed36710d74cdd97193da9b7ac11": "\\,\\overline{A}_x\\!", "2e307793e0e63655107f60902ca04123": " \\it f ", "2e308e759a683e0a099d52d841404483": " X_3 = X_1\\cdot (E-F) ", "2e30a63fc38de87fa1e8b3ee53d2c21b": "\\mathbf{T}^{(\\mathbf{e}_1)}= T_1^{(\\mathbf{e}_1)}\\mathbf{e}_1 + T_2^{(\\mathbf{e}_1)} \\mathbf{e}_2 + T_3^{(\\mathbf{e}_1)} \\mathbf{e}_3 = \\sigma_{11} \\mathbf{e}_1 + \\sigma_{12} \\mathbf{e}_2 + \\sigma_{13} \\mathbf{e}_3,", "2e30ad414c4d5be3f65e6c2ebb1c8324": " T^2 \\propto a^3 ", "2e30f893ba4582c6cc7c4950840783a0": " (p\\times p)", "2e3116eaeb0cb02a2acea0f98ed7af70": "I_S = {h \\nu \\over \\sigma(\\nu) \\cdot \\tau_S }", "2e313a977f64d0fd8ca289c817c82ea1": "L_{ij}=L_{ji}", "2e31469fe0df83f0c0cbb98891ab0213": "E_u=100{\\left ( \\frac{U}{Ff_x} \\right )}=100{\\left [ \\frac{f_x-o_x}{f_x(1-o_x)} \\right ]}= \\frac{E_o-(1-f_x)}{E_o f_x}", "2e315a432bde91f8c0ef033f0f4e2b25": "\\mbox{T}(a)|x\\rangle = |x + a\\rangle", "2e32010971e687821005c8de4e62f233": "|x_{t}|", "2e3228f4b7c31705d4c8b6ce400019fc": "k-l", "2e322f159d47537e571ab62ea916fcc4": " L_{a} ", "2e325708b052936bf0ddbd0f08a6d34f": "{m \\choose 1}_q ={m \\choose m-1}_q=\\frac{1-q^m}{1-q}=1+q+ \\cdots + q^{m-1} \\quad m \\ge 1 \\, .", "2e329fc2098a42a17530b00bae750fa1": "f(\\xi )\\star g(\\xi )=f(\\xi )\\exp (\\frac{i\\hbar }{2}\\mathcal{P})g(\\xi ).", "2e3321e33bd09ee62d5288df70d74081": "W_{ij}(n)", "2e33470fa73a9b5f275ca98556ce5036": "e^{x/y} = 1+\\cfrac{2x} {2y-x+\\cfrac{x^2} {6y+\\cfrac{x^2} {10y+\\cfrac{x^2} {14y+\\cfrac{x^2} {18y+\\ddots}}}}}", "2e338cbd788b811a26600f7ad0bd8b84": "\\operatorname{Cl}_2\\left(\\frac{\\pi}{6}\\right)=\n2\\pi\\log \\left( \\frac{G\\left(\\frac{11}{12}\\right)}{G\\left(\\frac{1}{12}\\right)} \\right) -2\\pi \n\\log \\Gamma\\left(\\frac{1}{12}\\right)+\\frac{\\pi}{6}\\log \\left( \\frac{2\\pi \\sqrt{2} \n}{\\sqrt{3}-1} \\right)", "2e33eb1956fc06e3e9581c393b0598c5": "f_D", "2e3413d5aaf6389ddc9a4cca8a080cb3": "r_c = \\frac{r_m}{( 1 - t )} ", "2e341e81c8ad27612ea5b9725d7d4443": "T_{JMAX}-(T_{AMB}+\\Delta T_{HS})", "2e3425b85da1fe18ab698c2158f1345c": "(1-\\delta_s)\\|y\\|_{\\ell_2}^2 \\le \\|A_s y\\|_{\\ell_2}^2 \\le (1+\\delta_s)\\|y\\|_{\\ell_2}^2. \\, ", "2e3474bf078937a353fd5d9a5617c30c": "X = \\sec\\, z \\,.", "2e35773540b553826c1ff92653c22938": "O_{{k} + \\frac{N}{2}} = O_k", "2e359db53235f4746228274e3440293c": "\\varprojlim R^p f_* \\mathcal{F} \\otimes_{\\mathcal{O}_S} \\mathcal{O}_S/{\\mathcal{I}^{k+1}}", "2e35b69c0e1615bdc404bf21b9d4606b": " i \\bar{\\partial} \\Psi\\mathbf{e}_3 + e \\bar{A} \\Psi = m \\bar{\\Psi}^\\dagger ", "2e35d326f55cade5d2a0314815c17ef6": "\\varphi^*:\\mathcal{X}\\to\\mathbb{R}", "2e360704306e665fbfc9ec6d8a57899c": "\\Delta Q = c_p\\rho \\, \\Delta u \\,", "2e3622fadc70e3e5b107efb4864f6b25": " P = 15 \\ kN ", "2e366b582afbb959b5cb7401c27915bf": "u \\wedge b \\wedge u : u \\in A", "2e3684c89b9ce0dbcff9054b91f475a9": "A=\\left(\\frac{1}{2}\\left(5\\sqrt{3}+\\sqrt{10\\left(65+29\\sqrt{5}\\right)}\\right)\\right)a^2=\\left(\\frac{1}{2}\\sqrt{5\\left(145+58\\sqrt{5}+2\\sqrt{30\\left(65+29\\sqrt{5}\\right)}\\right)}\\right)a^2\\approx22.3472...a^2", "2e369612de21984ef02901dfa6d96a70": "\nb = \\frac{L \\, c}{E}\n\\,,", "2e3738c9e33e8159ca8c7803ee5682e7": "\\bar{E}_{4}", "2e375cc2767e50f1c87aba9cb71d38ba": " \\frac{\\partial u}{\\partial t} + t\\frac{\\partial u}{\\partial x} = 0. ", "2e3769cadefb1ed492a6c915498aaeb9": "I=\\frac{2E}{2Z_{oc}}", "2e380034bc144de9335f8bd1b099c7b4": "\\operatorname{Ind}_\\mathfrak{h}^\\mathfrak{g}", "2e385c67bc15868a171e09c6df907c75": "{\\Gamma^j}_{ii}=\\begin{Bmatrix}\n\\,j\\,\\\\\n i\\,\\,i\n\\end{Bmatrix} = -\\frac{h_i}{{h_j}^2}\\frac{\\partial h_i}{\\partial q_j} \\ ,\n", "2e3880a828b0c4c39ff01837bf1e9404": "p_4(x)=-64x +48x^2-12x^3+x^4;", "2e38a64a07715c751c2c9df3ee610be6": " \\int_{f(U)} \\phi(\\mathbf{v})\\, d \\mathbf{v} = \\int_U \\phi(f(\\mathbf{u})) \\left|\\det(\\operatorname{D}f)(\\mathbf{u})\\right| \\,d \\mathbf{u}.", "2e390378489da8e10fa9f055324e3a21": "\\mathbf{B}=\\oint \\sigma \\, d\\mathbf{A}.", "2e3937bc15d1f90da346de04c2ce9d0a": "e= \\lim_{n \\to \\infty}\\left (1+ \\frac{1}{n} \\right )^n.", "2e394385915874db6e16d6d8c66dff79": "BO \\to BG", "2e3952ad5411d8bddc652fab21b65838": "R \\oplus R", "2e397077ac148862e357b89db2f5d7a7": "a\\neq 0\\,.", "2e3971d8eb2feb8592f1dfd2f8785aa7": "\n\\begin{align}\n \\varphi^{n}(x=0) &= f_{1}(y) \\\\\n \\varphi^{n}(x=x_{l}) &= f_{2}(y),\n\\end{align}\n", "2e3971ed4b36045739ea1f29bee75251": "l\\cos\\theta", "2e399ec9513db68b34b49c4cc3d28870": "\\mathbf{T} = T \\mathbf{u},\\,", "2e39ebabd88e8968ccbbf7a743b30a2d": "p_{1T}", "2e39ecb7952636c8e7f44bb4bed5c5f3": "\\gamma:[a, b] \\to X,\\,", "2e3a36fc4442c4ade2278c8b3e55ec4c": "45 = 9 + 9 + 9 + 9 + 9", "2e3aa69144f69535fc5f8818e81df0ab": "\nf^n\\left(x\\right) = f^n(a) + (x-a) f'(a)f'(f(a))f'(f^2(a))\\cdots f'(f^{n-1}(a)) + \\cdots\n", "2e3afa9e08f0c73ee373200f983a0340": "Z^{I}_{i,0} = 0", "2e3b3f94e98291c4b2904442def1ebec": "A_k(x)", "2e3b78ab7b8ee7fed6fa4deb8dab5243": " q_{ult} = 1.3 c' N_c + \\sigma '_{zD} N_q + 0.4 \\gamma ' B N_\\gamma \\ ", "2e3bb12d5371774944bc10e9c9241502": "\\mathrm{NapLog} (x) \\approx 9999999.5 (16.11809565 - \\ln x)", "2e3bbb5b87d0e8b3aa2d8cd20c8301f7": "k_i\\in J(R)", "2e3c12199380423b829f05d5bef183f5": "\\log_{10}(n+1)-\\log_{10}(n)", "2e3c256ee064e8ae1cf6aa67ac6cab4e": "\n\\nabla^2 \\Phi =\n\\frac{1}{a^2} \\left( \\cosh \\tau - \\cos\\sigma \\right)^2\n\\left( \n\\frac{\\partial^2 \\Phi}{\\partial \\sigma^2} + \n\\frac{\\partial^2 \\Phi}{\\partial \\tau^2}\n\\right)\n", "2e3c2cf47fb2f097c85b7855aae8924d": "x=a \\cosh v \\cos\\theta ", "2e3c4acb6f07e9a5a1cf9c541114978e": "k = 0,1 \\dots", "2e3c61df8733b6a917cc4bc39f8a26c8": "z \\in y.", "2e3d13efe37f3bd9472665ae8598334d": " g(x) \\propto x", "2e3d6b2e19c17a148615211c0f521c45": "p^*\\xi\\colon p^*E\\rightarrow Y", "2e3de05ed537907a2ac400781bd6ff4c": "\n|x\\rangle = \\frac{1}{\\sqrt{\\mathcal{N}(x)}}\\sum_n \\overline{\\phi_n(x)}\\, |e_n\\rangle\\, ,\n", "2e3df4e7e3e74234f9b78f437391cf8f": "\n \\begin{align}\n \\mathbf{J}^2 |(j_1j_2)JM\\rangle &= \\hbar^2 J(J+1) |(j_1j_2)JM\\rangle, \\\\\n \\mathrm{J}_z |(j_1j_2)JM\\rangle &= \\hbar M |(j_1j_2)JM\\rangle,\\quad \\mathrm{for}\\quad M=-J,\\ldots,J.\n \\end{align}\n", "2e3e32531151b4fadda4b053ae57a0f3": " \\operatorname{inc}\\ (\\operatorname{inc}\\ \\operatorname{const}) = \\operatorname{value}\\ (f\\ x) ", "2e3e567710413e9934543648b94d55fb": "\\int_C {\\sqrt{z} \\over z^2+6z+8}\\,dz=I.", "2e3e6c8fc31d963bc42216f4f2a58a52": "\\mathcal{G}", "2e3e6e37af4e4cc2ef15e88b6c9b592d": " \\!\\ {1 \\over {S_m^6 - \\lfloor S_m^6 \\rfloor }} + \\lfloor S_m^6 - 1 \\rfloor = S_{(m^6 + 6m^4 + 9m^2 +1)}. ", "2e3e75bfdadb9b380e7184f3bf4d550b": "v_{(G; c)}(\\{1\\})=11", "2e3eda4ee12aa2b2602671a19e9e517f": "e^{ita}p_a+e^{itb}p_b+\\frac{1-p_a-p_b}{b-a-1}\\frac{e^{(a+1)it}-e^{bit}}{e^{it}-1}", "2e3efa1792f101bf7a910e5474a0f03c": "f_\\alpha", "2e3f485799dc65228c72e72b177be949": "r^2=x^2 + y^2", "2e3fa50e3ec9590d6ec8b0ed2acded87": "a_{k+1}", "2e3fd7759c1d481d8c1e8210a333b34c": "A, B, C,...", "2e402c8cacf64700e296635950c81e48": "\\mathrm P(X_1=x_1, \\ldots, X_n=x_n) = \\prod_{v=1}^n \\mathrm P \\left(X_v=x_v \\mid X_{v+1}=x_{v+1}, \\ldots, X_n=x_n \\right)", "2e4075d194151038f03f4682e32a52ee": "\\sigma(a + b) = \\sigma(a) + \\sigma(b), \\quad \\sigma(1) = 1, \\quad \\sigma(a^{-1}) = \\sigma(a)^{-1},", "2e40f9fe5023f9a9d7acd2f8b4099d55": "f_x(y) = d(x,y)", "2e413af5d0071d1577254b2970702ccf": "H_9 =0\\,", "2e41525e9484df8910a3062eba9dda27": "\\tilde{\\omega}^1,\\tilde{\\omega}^2,\\dots,\\tilde{\\omega}^n", "2e41875b5357afbaa6c87434885a3b83": "U\\left(z\\right)=z", "2e41b7f2e7f4da62a933eb540f4fe238": "\\Pi (t,f) = q(t)\\, W(f) ", "2e41e2733e3be2d227f3a36a05eebfa3": "\\chi(\\omega) = \\frac{\\chi_{sp} + i \\omega \\tau \\chi_b}{1 + i \\omega \\tau}", "2e41f993e8adeeb4c0415c401eece2dc": "m^{\\prime}", "2e423030d33624ca94b130bd3052bf9a": "S_0=\\{0,1\\}", "2e4248cbafe4c914c8e0e1284ab87ba1": "t = \\frac{X_o^2}{B} + \\frac{X_o}{B/A}", "2e426000b92cfbc7286b0e2cc2a37482": "b_0", "2e428acbfd71871549ec8efff8c959fb": "\\sup_{R>0} \\int_{R<|x|<2R} |K(x)| \\, dx \\leq C,", "2e42e10274bda8a7455aed8c2e364004": "S_t(z)= e^{\\alpha t} \\frac{z-\\beta}{\\alpha-\\beta} + e^{\\beta t} \\frac{z-\\alpha}{\\beta-\\alpha} ~, ", "2e4332b418ec0be4d5bea12b1acc8fff": "t \\in \\{0, 1, 2, \\ldots\\}", "2e4344887fe97fea2404fa0ea7c5dc70": " t_{1} ", "2e43627bd382166318307005815a5ce2": "{\\mbox{Div}}^0.", "2e43eae76e21dbcc69978a98c39037ef": " K \\gg \\delta ", "2e43fd65a59304de26a68eee41549094": "PV(P) \\ = \\ { A \\over i } ", "2e444649f11585e783f27018c058e13d": " f(x_1,\\dots,x_k) = a(a+1)\\cdots(a+k-1) \\left(\\prod_{i=1}^k \\theta_i \\right)^{-1} \n \\left(\\sum_{i=1}^k \\frac{x_i}{\\theta_i} - k + 1 \\right)^{-(a+k)}, \n \\qquad x_i > \\theta_i > 0, a > 0, \\qquad (1)\n", "2e448d816bdae5b2ba1a6e64d5eb22e0": "\\mu(\\lambda) = \\frac{(f^\\lambda)^2}{n!}.", "2e44b7c5fdfb7e34cd0168298334c532": "p, q \\in A", "2e44f3a4383351cd109758d7b8082c1f": "a = \\sqrt{2/\\phi}", "2e44fd540bb9d2b52cb540e58d014e02": "0=v_{bullet}\\sin(\\delta\\theta)t-\\frac{1}{2}g t^2 ", "2e4542eaee05f9da0a5f60218df017cc": "\n\\begin{align} \n& A=\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}-0|_{R_{0}}\\right)=\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}\\right) \\\\\n& B=-\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}+0|_{R_{0}}\\right)=-\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}\\right) \\\\\n\\end{align}\n", "2e455682768dfaaf09a05d56ead9d2f1": "\\displaystyle{\\mathbf{t} =\\dot{\\mathbf{v}},\\,\\,\\,\\,\\,\\,\\mathbf{n}=(-\\dot{y},\\dot{x}).}", "2e45f17b1f155da2244eb0b20812ed78": "\\left( J_{an} \\, X^n \\right)_{;b} = J_{an;b} \\, X^n + J_{an} \\, {X^n}_{;b} = J_{an;b} \\, X^n + J_{am} \\, {J^m}_b ", "2e460276220868565be261e9063feff8": "if (C^{\\{a\\}} \\neq \\emptyset)", "2e461013f6f98370565d633941061490": "K_{B,C}=\\frac{K_{AC}}{K_{AB}}", "2e46251953890a79af4f88599b7b05c7": " \\frac{\\epsilon_{e,\\omega}}{\\epsilon_{e,\\infty}} = 1 + \\sum_j\\frac{\\omega_{\\mathrm{LO},j}^2 - \\omega_{\\mathrm{TO},j}^2}{\\omega_{\\mathrm{TO},j}^2 - \\omega^2 - i\\gamma\\omega} ,", "2e4693eb2c15237c53e05ed26161c01b": "\\lim_{n\\rightarrow \\infty, n\\in \\mathbb(Z)} \\frac{1}{(2t)^{2n}+1} = \\lim_{n\\rightarrow \\infty, n\\in \\mathbb(Z)} \\frac{1}{1^{2n}+1} = \\frac{1}{1+1} = \\frac{1}{2}", "2e46beabbf2d0ea3b218780c29b7733d": "\n{{d^2 i(t)} \\over {dt^2}} +{R \\over L} {{di(t)} \\over {dt}} + {1 \\over {LC}} i(t) = 0\n", "2e471d5315d797ae3a49bf585671e544": " \\mathbf{D} = \\varepsilon_0 \\mathbf{E} + \\mathbf{P} = \\varepsilon_0 \\varepsilon_{\\text{r}} \\mathbf{E} \\ ,", "2e47324cd256f73484af59f8c7952f97": "\\Omega(x) = S(x)\\,\\Lambda(x) \\pmod{x^{d-1}}", "2e474f33b2f9d1f4c4a496286c6508f6": "H_\\gamma = \\int D_\\gamma (L)Q(L)dL", "2e4761f3200dc9bc2a8f740572316fd9": "\\text{join} \\colon {A^{*}}^{*} \\to A^{*} = l \\mapsto \\begin{cases} \\text{nil} & \\text{if} \\ l = \\text{nil}\\\\ \\text{append} \\, a \\, (\\text{join} \\, l') & \\text{if} \\ l = \\text{cons} \\, a \\, l' \\end{cases}", "2e4784770985deaffc22f84594754564": "\n \\vec w^* = \\sum_i{\\alpha_i y_i x_i}\n", "2e479183b1666f987064a507fdbbf8b0": "\\textstyle \\frac{1}{n-1}", "2e47b1a3627f73de25044f54b282dc81": "f^\\rightarrow", "2e47f72df945687aa81ef4284e37bbb0": " \\mathbf{\\hat{p}} ", "2e4854b20a5da5a334a1af51f4466561": "\n\\mathrm{DQE}(u) = \\frac{\\mathrm{NEQ}(u)}{q} = \\frac{q G^2 \\mathrm{T^2}(u)}{\\mathrm{W}(u)}\n", "2e48865712cd9880f2e9d6870fa7d877": "r\\rightarrow\n\\infty", "2e48c5c5bd89b9b96f6a6d22e9e45dcf": "U(s) = C(s) E(s)\\,\\!", "2e48edc08231e975e593419fed08b035": "s = w + z j \\ ", "2e49038ea6bf3ff97790d35e0e82b2fb": " \\| A^* \\| _{op} = \\| A \\| _{op}. ", "2e495c8d33073d56279e6e6fb3bfeb92": "d_\\infty", "2e4972d2e74fb457b8a67accc544f714": "\nI = \\frac{p^2}{Z} = Z v^2 = \\xi^2 \\omega^2 Z = \\frac{a^2 Z}{\\omega^2} = E c = \\frac{P_{ac}}{A}\n", "2e498e5955fa38bd41d0b017a5b71266": "\\forall s\\in {{R}^{+}},\\theta \\left( su,\\frac{\\xi }{s} \\right)=\\theta (u,\\xi ),", "2e49ca46e2a3b1dd2203c258f308bc22": "\\sum_ {} \\vec{F} \\ne 0", "2e4a08f3a907df0c163b623c6f227cc1": "P(R=1|x,q) + P(R=0|x,q) = 1", "2e4a29aa8742658b76c705139cb31296": "\\lim_{n\\rightarrow \\infty, n\\in \\mathbb(Z)} \\frac{1}{(2t)^{2n}+1} = \\frac{1}{+\\infty+1} = 0, |t|>\\frac{1}{2}", "2e4a384d8fe94fdd0ab88e05111a817f": " H = ", "2e4a71d01464412d52e3014328dad944": "b=-0.5", "2e4aa13c901558fe000cc35d91b16295": "f(\\xi ) \\rightarrow \\acute{f}(\\xi ) \\equiv Tr[\\hat{B}(\\xi )\\hat{U^{+}}f(\\hat{\\xi})\\hat{U}] =\\sum_{s=0}^{\\infty }\\frac{1}{s!}\\frac{\\partial ^{s}f(0)}{\\partial \\xi\n^{i_{1}}...\\partial \\xi ^{i_{s}}}q^{i_{1}}(\\xi,\\tau )\\star ...\\star q^{i_{s}}(\\xi,\\tau) \\equiv f(\\star q(\\xi ,\\tau)).", "2e4b0c8ff703bf1ddfa71a9ad922a913": " r_2 \\ ", "2e4b6f5a90619f4e5cdf37c0ade3ff05": "T_q^*M=\\{T_q\\rightarrow\\mathbb{R}\\}", "2e4c1cb4a578f6a39148638b929f85d0": " [n=q-1,k]_q", "2e4c306a2ea2b91005ad6e58075493d1": "\\frac{g}{2\\pi} T", "2e4c6d223ec8332583bc255533e89398": "x=s+\\epsilon", "2e4cb63afede83da50e7f945608dce08": "\\cos{\\nu} = {{\\cos{E} - e} \\over {1 - e \\cdot \\cos{E}}}", "2e4cdf757228dbd6405929eb2fc70d12": "\\chi\\colon \\mathrm{Ob}(\\mathcal{A})\\to X", "2e4cede94d13b50e379f249b83a7f921": "\\mathcal{Y}=\\operatorname{mod}-B", "2e4cfef43b52c6a1a01a61ce697de698": "(\\mathcal{H},F_0,\\Gamma)", "2e4d28f6982d8fa0d9c8ae3f2272522c": " D = ( a n^{ 1 - b } T^{ b - 2 } )^ { 1 / 2 } ", "2e4d45abb3ef5543d904a77063f59ec1": "M \\ne R_+M", "2e4d5fa46a37847c063d92bb199f5c99": "F^{\\mu \\nu} = \\eta^{\\mu \\alpha} F_{\\alpha \\beta} \\eta^{\\beta \\nu} \\,", "2e4d72fe8e7f7742d9eff78ac4d70d5c": " \\sum_p \\frac{1}{p - 1}>\\sum_p \\log\\left(1+\\frac{1}{p-1}\\right)=\\quad \\log \\left( \\sum_{n=1}^\\infty \\frac{1}{n}\\right)", "2e4dafb079ee01e1be494bd078642d61": "\\mathsf{PP} = \\Sigma_2 \\not \\subseteq \\mathsf{SIZE}(n^k)", "2e4dd5133148568557eb1025ce6d8a86": "Z_0 \\approx 120\\pi \\approx 377\\ \\Omega", "2e4e0a7b37b0f30e8e3ed11ed7896f48": "J := J \\setminus S_I", "2e4e12c05b5e15388bcd04c87fbad0e1": "\\{ a^{2^n} : n \\geq 0 \\}", "2e4e670988721eef62a65e4b181a0465": "\n-\\mathrm{E}\n\\left[\n \\frac{\\partial^2}{\\partial A^2} \\ln p(\\mathbf{x}; A)\n\\right]\n=\n\\frac{N}{\\sigma^2}\n", "2e4ea1dc45bf009e8c93f7f591721253": "dA = |X_u \\times X_v| \\ du\\, dv= \\sqrt{ \\langle X_u,X_u \\rangle \\langle X_v,X_v \\rangle - \\langle X_u,X_v \\rangle^2 } \\ du\\, dv = \\sqrt{EG-F^2} \\, du\\, dv.", "2e4ea306a555908c92d05291f4a0ba87": "\\textit{opendoor}", "2e4ec008b327e826672f66dff571058f": " f' = f \\frac{1-u/c}{\\sqrt{1-u^2/c^2}}=f \\frac{\\sqrt{1-u/c}}{\\sqrt{1+u/c}} \\,", "2e4ed2cf5111bdb6ad3348bad623a822": "\\frac{k-\\frac{1}{k}}{2i} = x", "2e4f09bad1187f85d2f8722051954f4a": "X^\\#", "2e4f134344550686b0a3f0016eb2830a": "\\lambda_{\\epsilon,g}", "2e4f1565b7137d0425c37ddd8ba39342": "\\frac {R}{G} = \\frac {j \\omega L}{j \\omega C} ", "2e4f3f1e2a9f88e8bb7bd3d1713a05a3": " [1_K,\\chi\\downarrow^G_K] \\leq 1", "2e4f5a045412046aef854fca7e1f565b": "\n\\epsilon_r = \\frac{\\pi \\sigma_t }{2}~I\\,n\\,\\cos^2\\chi\n", "2e4f947b3da456c32b9ee10c8d009fe6": "\\ln \\left ( \\frac {P_2}{P_1} \\right ) = \\frac {L}{R} \\left ( \\frac {1}{T_1} - \\frac {1}{T_2} \\right ).", "2e4f9bdbcaf8db9efd7dc5c4545f28e4": "\n\\pi_{n}=\\mathrm{Tr}\\left[\\left(\\hat{\\rho}_\\mathrm{retr}^{[n]}\\right)^{2}\\right].\n", "2e4feab97afd662b7c0da0d658cc6d2f": "_G^D", "2e4ffb2837dd4ef7761604bff772b5cf": "\\displaystyle\\gamma^\\mu\\gamma_\\mu=4 I_4", "2e502531b324bc391f23aba97bf89051": "B > 0, B \\to +\\infty,", "2e5048c34b04af3bc04b90e1804b5b63": "\nW_{12}=\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & 1\\\\\n0 & 0\\\\\n0 & 0\\\\\n\\end{bmatrix}, \\quad\nW_{23}=\n\\begin{bmatrix}\n1 & 0 \\\\\n0& 1\\\\\n\\end{bmatrix},\\quad\nW_{34} =\n\\begin{bmatrix}\n1 \\\\\n0 \\\\\n\\end{bmatrix}.\n", "2e506da678d49d33eb8b1329f379fe4f": "\\left( x_2,y_2 \\right)\\ = \\left( \\frac{-b_2m}{m^2+1},\\frac{b_2}{m^2+1} \\right).\\,", "2e507162813964f28f43dc3cbafe5925": "2^3 + 4^3 + 4^3 = 136", "2e50e09772f8d2ee260a6e673745fb4c": "\\hat{\\rho(t)}=\\sum_n |n(t) \\rangle\\langle n(t) |e^{-\\beta E_n}", "2e50e9f20135b8be1da4a3250aa2632f": "\\log_{2} (c)", "2e511777431cd1608afe15d311285b50": "\\displaystyle h(e^X) = A_\\rho(e^X)", "2e511ca6a39f11bb0c76d15b733aea23": "(2j_2+1)", "2e512a81d467a1c1463701b487dbffa5": "G_L", "2e514f5606757bb3da1076567849c72c": "F(x)\\;=\\;\\Phi\\left(\\frac{x-\\mu}{\\sigma}\\right)\\;=\\; \\frac12\\left[1 + \\operatorname{erf}\\left(\\frac{x-\\mu}{\\sigma\\sqrt{2}}\\right)\\right] ", "2e5188e839940930c6f8357c834436c6": " \\exp\\{-sT\\} =\\frac{\\exp\\{-sT/2\\}}{\\exp\\{sT/2\\} } \\approx \\frac{1-sT/2}{1+sT/2} ,", "2e5193d1515cca185cdfca9187e52121": "\\Phi_n(g(z))=z^n + \\sum_{m\\ge 1} c_{nm}z^{-m}.", "2e51c5f4a1a937e0a7b92032631c8410": "\\partial_x^2 \\mathbf{1}_{x>0}", "2e51d823733c06c9b6351138b404fe94": "n_{vi}", "2e51e90a23597e01a1fd42e06744afdc": "\\frac{\\mathit{MSbetween}}{\\mathit{MSwithin}}", "2e5225480dd75aaa90fb826fd15a6de3": "i\\geq 3", "2e529f0772f635e6dedaca9af64d9e2a": "\\frac{v_x-v_{\\text{out}}}{Z_2}=\\frac{v_{\\text{out}}}{Z_4},", "2e52e51f6351db0db6c4620049dd3d1e": " \\frac{ (\\beta/\\alpha)(x/\\alpha)^{\\beta-1} }\n { \\left (1+(x/\\alpha)^{\\beta} \\right)^2 }", "2e5310d520c35844bef3bad0407f7707": "d(x,c_x)\\leq d-1", "2e5351ff45fcf536e1034c33a3d3241b": " E_2 + E_3 \\mbox{ and } C_1 + C_2 + C_3 + \\cdots.", "2e5372e1435d72c069c427c8090b4be4": "v_1\\cdots v_k \\mapsto \\sum_{\\sigma \\in S_k} v_{\\sigma(1)}\\otimes \\cdots \\otimes v_{\\sigma(k)}.", "2e5375a8b07b544cd990abb33524b337": "\\mathrm{1\\,Bi\\cdot s = 1\\,abcoulomb = 1\\,emu\\, charge= 1\\,s\\cdot\\sqrt{dyne}=1\\,g^{1/2} \\cdot cm^{1/2}}", "2e538ff2bd75d7456cda384844a17ffe": "\n \\boldsymbol{\\tau} = J~\\boldsymbol{\\sigma} = -p\\boldsymbol{\\mathit{1}} + 2C_1~\\mathrm{dev}(\\bar{\\boldsymbol{B}})\n ", "2e53b0e5b8c15dac7fddee199dc01ef5": "d \\omega^k", "2e53c008b2ab606140d51389a551557c": "\\max_{Y\\subseteq X} \\ \\{|Y|: g(x)\\le 0,\\forall x\\in Y\\}.", "2e53df2de3d80bf81c0e8a0e20ed04d4": "x=\\pm j\\infty", "2e53e26fb10c9ab8a109c9ee27e31a99": "H_*(W,M;\\mathbb{Z})=0", "2e53fa0fc3f9d4cb08d308b1e12649c6": " s = (\\ldots,(s_i, t_{si}, t_{ei}),\\ldots) ", "2e54159d97c626570402fc2bc1d39b50": "f_e(v)", "2e541d5da91b97eaf9d8169960781174": "y \\ge 1", "2e5461b7ce3512b26b15907b8a3a5f4a": "\nr=r(0)e^{-\\frac{\\kappa}{2} t}\n", "2e54e65373e4577e15e390b408a3dc29": "x,z\\in\\mathbb{Z}", "2e551361477b228c77b4729fb8bd170d": "m=0 \\Rightarrow e^{-m r}= e^0 = 1.", "2e5599801da5d36e39e4d701ba5efcf3": "f(x)=ax^4+cx^2+e.", "2e55cb797ed4fd97aa29fae939cd622a": "c_s^2 = \\gamma p/ \\rho ", "2e561e56d67205b73574a96d73048975": "43^2", "2e563a63f6907a31e4f89810d7f5c336": "\\bar{A_i}", "2e566ff83dbaa811992649fc2666a958": "\\left(2\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "2e5685bd750b5dbd04f8196c00e36215": "_{metric}\\alpha = 1 - \\frac{D_o}{D_e} = 1 - \\frac{\\sum_{c=1,k=1}^{v} o_{ck} {_{metric}} \\delta_{ck}^2}{\\frac{1}{n-1} \\sum_{c=1,k=1}^{v} n_c n_k~{_{metric}} \\delta_{ck}^2}", "2e56a37dc0a7c5667587566ff3852ba4": " x = \\tfrac{1}{2}(1 - y^2). ", "2e56ad571749195304a178a0d3c9b4f9": "{\\it{N}}", "2e56c5bfafd2948b3e18cdeba9e8e043": " \\widehat{\\boldsymbol{\\beta}}_{L} ", "2e56d979a5d0a4a31e5c1bb6af9ff4b3": "\nf(t) = \\begin{cases} x & t=0 \\\\\np & t\\in(0,1) \\\\\ny & t=1\n\\end{cases}", "2e570e49cbd6bfa94f516297a8625770": " b = - (\\delta C_{mgH} / \\delta \\alpha_{H}) \\times ( \\rho S \\mathit{l}/2) \\times \\mathit{l}_{H}\\,", "2e5735728c615c26d5174ab0fbf1339b": "A \\rightarrow S: A,\\{T_A, K_{ab}, B\\}_{K_{as}}", "2e5751b7cfd7f053cd29e946fb2649a4": "0 ", "2e576cc927d44374f24da20f3524778f": " 2 k' \\cdot p' \\,", "2e57c4bd162183232574946fd722e42b": "k_{eq} = \\frac{k_1 k_2 }{k_2 + k_1} .\\,", "2e57fea4cb3a8d497c06fc84db3db79b": "\\int_{[a,t)}|u(s)|\\,\\mu(\\mathrm{d}s)<\\infty,\\qquad t\\in I,", "2e58074a6babad4424dc1c2ebef22975": "\\min\\{wx\\ :\\ x\\in{\\mathbb{Z}}_+^{l\\times m\\times n}\\,,\\ \\sum_i x_{i,j,k}=a_{j,k} \\,,\\ \\sum_j x_{i,j,k}=b_{i,k}\\,,\\ \\sum_k x_{i,j,k}=c_{i,j}\\}\\ ,", "2e5839ba2ab85b2f62c36d97cc0970fa": "c\\,t_1<0", "2e5860742f2e1f1594a4e13d393d321f": "T[i]=j", "2e58719105bca666de8d4a016cdb0d0a": "\\phi(t) = 2 \\pi \\int_{-\\infty}^{t} f(\\tau)\\, d \\tau \\ ", "2e58d733fa8bc3128551bd1bb648f873": "|z-c|^2 = z\\overline{z}-\\overline{c}z-c\\overline{z}+c\\overline{c}", "2e58e85524e9878101bcbfc7f59c0378": " x_{i} = y_{r(i)} . \\,\\!", "2e595a242d2c6f04d0b54dd42d4a8cd3": "{\\frac{\\partial{S}}{\\partial{E_{0}}}} = const", "2e59c3db4934ef5e297d2bd473a48b60": "\\boldsymbol{p}(t) = h_{00}(t)\\boldsymbol{p}_0 + h_{10}(t)\\boldsymbol{m}_0 + h_{01}(t)\\boldsymbol{p}_1 + h_{11}(t)\\boldsymbol{m}_1", "2e59f7010fb60e96933d180485aa8523": "x=\\sqrt{-2\\ln(u_1)}\\cos(2\\pi u_2),\\quad y=\\sqrt{-2\\ln(u_1)}\\sin(2\\pi u_2)", "2e5a2980ec48e869f27b7079fa2ebcae": " Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 ", "2e5a55907b5834d46388615bd29942ec": "\\phi_0\\,", "2e5ab87e584b4ff2aa36963c10277310": "Y = y\\frac{d}{dx}.", "2e5adc58c80ad33182fb0594066db48f": "7^3 = P(7,4)K(3,1) + P(6,4)K(3,2) + P(5,4)K(3,3) = 84*1 + 56 * 4 + 35 * 1 = 343", "2e5af6d5b631d437156f5fb65d120244": "\\nabla \\times \\vec{V}=0", "2e5b15fb7479c879d26b13556ff75a0f": "\\left\\{ \\begin{matrix} \\mathrm{d} X = f(X) \\, \\mathrm{d} t + \\varepsilon \\, \\mathrm{d} W (t); \\\\ X (0) = x_{0}; \\end{matrix} \\right.", "2e5b411a5d396652ed81e24fbad29d47": "\\frac{Gm}{r^2}", "2e5bc05e64ded6fc77784f95f0ff6bf7": "\n g := \\det[g_{ij}]\\,\n ", "2e5c0ef2238b67c165c8291d9d600401": " \\hbar = c =1", "2e5c2692219d5974d6dd4e604bc36041": "RP_k", "2e5c78517022550c2753da0b989b4fd7": "f(a+t) = \\lim_{h\\to 0^+} \\int\\limits_{-\\infty}^\\infty f(a+x)dP_{t/h,h}(x).", "2e5c7d50a541ce4fa43752e2b2e7e716": " E = p, G = (\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f)), H = p, Y = (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "2e5cbef371fac59f5cc43ff689ed830c": "\\mathcal{D}^{\\mu\\nu}u_\\nu= c^2\\epsilon F^{\\mu\\nu} u_\\nu", "2e5d1a1eff7441aef4d1945cc3b7f257": "r=2M\\,\\!", "2e5d21b3239292d451c678f27f07d2f2": "\\left \\lfloor 1000/(2^{\\frac{n-1}{2}})+0.2 \\right \\rfloor", "2e5d692dbaff0a262ef4c1557bb8b65b": "\\mbox{Hess}(f)(X,Y)= \\langle \\nabla_X \\mbox{grad}f,Y \\rangle ", "2e5d71b73fdeae288e4aef2beb8e5aaf": "m_o", "2e5dccacfe1f77b313ae1c32d821ea40": "\\dot{\\psi} = -\\xi^{-1} + \\frac{1}{4}\\xi \\left ( \\varkappa_a^b\\varkappa_b^a + \\lambda_a^b\\lambda_b^a \\right ).", "2e5e516b06e8b3f82fdeb254f667219b": "\\mathrm{d}E = T\\mathrm{d}S - P\\mathrm{d}V\\,", "2e5e59ddf5f724ba4e7c82f9467ea3cf": "M_{23}", "2e5ef95d27a827fcb853b3206145b1f0": "\\Psi :\\mathcal{B}_1 \\rightarrow \\mathcal{A}_1", "2e5effe1f5a52d0d955632b265acddda": "a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+ \\cdots", "2e5f0b88160822a65279ddf5d213895b": "\\scriptstyle u+P_U+max_V\\{D_F(U\\rightarrow V)\\}", "2e5f0c5925489bfc4f6ebf911d42a33f": " 0 < \\tau < 2/( \\|A\\|^2 ) ", "2e5f113f0a075c709ed73cd9ee6defc5": "\\sim_S \\;\\,=\\, \\{(s,t)\\in M\\times M \\vert S/s = S/t \\}", "2e5f7fb2477861e5bdf2cf6921a25acb": "\\cot(y) = x \\ \\Leftrightarrow\\ y = \\arccot(x) + k\\pi", "2e5f8f1b2e9223632accf82d44244ffc": "R=\\frac{1}{4} \\sqrt{\\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}},", "2e5fadfa891d3cdd24abc130531d95e0": "\\left\\langle M_C\\right\\rangle", "2e5fae1d18836aebc5fd8967e9f0e5af": "(3)\\quad ds^2=-\\Big( 1-\\frac{2M}{r} \\Big) du^2-2dudr+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;;", "2e60000527032f6f87378e84947cdd0c": "\\varepsilon \\rightarrow \\widehat{\\varepsilon}(\\omega)", "2e600c1d39322e86c07e96515de8ad39": "1.\\overline{09}", "2e601cb9d3bb45eee2c79eeec5aa318c": "\\epsilon(p,t)", "2e604e407e6288a98e179e719f1bed23": "\\widetilde{\\mathbf{x}}", "2e605114f9916a40f6da0c4ebd5b0cd4": " \\mathbf{T}^{(\\mathbf{n})} = \\mathbf{T}^{(\\mathbf{e}_1)} n_1 + \\mathbf{T}^{(\\mathbf{e}_2)} n_2 + \\mathbf{T}^{(\\mathbf{e}_3)} n_3.\\,\\!", "2e6071bfb72258abc545e1f203653899": "\\mathit{{c}_{v}}\\mathit{ln}\\left(\\frac{{T}_{2}}{{T}_{1}}\\right)-\\mathit{R}\\mathit{ln}\\left(\\frac{{V}_{2}}{{V}_{1}}\\right)=0", "2e608ccab17a2c4e30d583a6f9d23b0e": "\\mu' = \\kappa\\mu\\kappa.", "2e612bb2c85245c37a76d1734a16c7fe": "\\left( \\mathrm{D} g (u_{0}) \\right)^{*} (\\lambda) = \\lambda \\circ \\mathrm{D} g (u_{0}).", "2e613dc9c8a36b5630c84fad18046b13": " | 1 1 \\rangle \\mapsto | 1 \\rangle U |1 \\rangle = | 1 \\rangle \\left(x_{01} |0 \\rangle + x_{11} |1 \\rangle\\right) ", "2e61aacac7378de4255b1f7e812168fb": "(-1)^m \\frac{g^{(m)}(1)}{m!} = \\sum_{n=0}^\\infty G_{mn} (-1)^n \\frac{f^{(n)}(1)}{n!},", "2e62a19d1590f1a6bd9617eecf81040d": "\\aleph_{\\omega+1}", "2e6303564a5d8ff9864e2bcb8c3a20ec": "c = \\frac{\\sqrt{5}-1}{2}.", "2e631d21bf2a6cb85a6e49d593f59d13": "\\nabla \\times \\nabla \\times \\mathbf{A}", "2e632bccfcc68a2aea35cf66ffd7c8be": "k=\\frac{\\lambda }{\\lambda - 1} ", "2e634ac9d17220b27085069579e657ce": "\\tan \\frac{\\delta'-90^\\circ}{2}=\\sqrt{\\frac{1-\\beta}{1+\\beta}}\\cdot\\tan\\frac{\\delta-90^\\circ}{2}", "2e63b8591bbc1098ebbe97041c73b436": "A=U\\Sigma_1 [ X, 0] Q^*", "2e63cb8be5f902b4be7382b5b948408e": "\\hat\\sigma (\\xi)", "2e63ebb2b8606d595132452671718e4e": "\\chi_A = \\phi^B_e", "2e641e19a4058d97689397f6ddeb4841": "\\mathbf{A} = (\\mathbf{A}_x, \\mathbf{A}_y, \\mathbf{A}_z)", "2e64dab31831caddf6596db73c2d2aec": "r^2 + s^2 = t^2.", "2e6615c17b2ca7edeb57313def30bd13": " = y - \\delta \\beta + b (\\alpha + \\delta) - b a \\ ", "2e6656c5bf1811a2c1440e816fa8579e": "A = 1.", "2e665e1b8c868754b34354519fd0caee": "10+13+7+4=34", "2e6676ce59f5925198529ef21bea8774": "\\|f\\|_k = \\sup\\{|f^{(k)}(x)|: x \\in [0,1]\\}", "2e66a001f9537949375c83eeda12ae32": "0 < \\alpha \\leq 1", "2e671e1fa507b4a45df48f6ec7a20452": "(X_1^i,X_2^i,\\dots,X_d^i), \\, i=1,\\dots,n", "2e67398ae2e73a85e1ebd97db8fa61f2": "D_X(f^*N) \\cong f^!(D_Y(N)),", "2e676bad89f23d38ce64c14471ce77e1": "\nV = E * L = E * L \\propto L\n", "2e676ce2d4a11759d3cc55d37555b057": "\\omega_{B} (\\delta) = \\sqrt{2 \\delta \\log (1 / \\delta)}", "2e67770c43f4f89c7d993d1f02511a19": "({\\mathcal C},\\otimes,I)", "2e680559709996400d998dd14b7b4d48": "\\scriptstyle R_{x\\rightarrow y}", "2e682d5a53e5bd9a4937f1fdb5694d75": "ay'' + by' +cy = 0", "2e6852d8616ec9d2a2f99de72b80bcd9": "r_\\mathrm{E} = \\frac {1} {g_m} ", "2e68c3610bba093fa61463616e798cca": "\\arg\\min_{\\mathbf{w},b } \\max_{\\boldsymbol{\\alpha}\\geq 0 } \\left\\{ \\frac{1}{2}\\|\\mathbf{w}\\|^2 - \\sum_{i=1}^{n}{\\alpha_i[y_i(\\mathbf{w}\\cdot \\mathbf{x_i} - b)-1]} \\right\\}", "2e696a5b49656353732fe6112d9b3ba8": " M^n(B_1\\times,\\dots,\\times B_n)=E [{N}^n(B_1\\times,\\dots,\\times B_n)], ", "2e6a9f2b24475f7fc243584cab1b21d2": " \\mathbf{X} = U \\Delta V^{T} ", "2e6afdeaab16dfe26ee403c1f7505573": "\\tau =\\tau_y(H) + \\eta\\frac{dv}{dz}, \\tau>\\tau_y", "2e6b54229ce23608f22eb71ef593a14f": "r\\ c_i < r\\ c_j", "2e6b581815395417c09f8fdf470e862f": "\n f_j([\\vec{X}])= k_j \\prod_{z=1}^N [X_z]^{s_{zj}}\n", "2e6b84e8a2f9d1160e21f6f94c05402c": "u(\\mathbf{x}) = \\int_S \\rho(\\mathbf{y})\\frac{\\partial}{\\partial\\nu} P(\\mathbf{x}-\\mathbf{y})\\,d\\sigma(\\mathbf{y})", "2e6ba679134279ef5fb692d30f5b0527": "\\frac{dX}{dt}", "2e6bb73e666a86a6e406d976acad6798": "\\ E_{total} = E_{bonded} + E_{nonbonded} ", "2e6bc9c71480a8bb449c7fb9002694f2": " I = I_0 \\left [ \\frac{ \\sin \\left( \\phi/2 \\right ) }{\\left( \\phi/2 \\right )} \\right ]^2 \\,\\!", "2e6c097364c25cec869e0f3c4efa4405": "\\psi_{1}", "2e6c0d924be14f713bb3baff61c8c483": "\\Omega=p/q", "2e6c25b07d2099e72b460aeb774a3b5b": "m^{-\\alpha}", "2e6c6a47a9099b725187a7881d925220": "\\ell_\\min", "2e6c7140e9a0d333ffebc4b5be32102d": "\\pi\\colon E\\to B\\,", "2e6c825b321f5f39bc4eb1cdf58af9a7": " \n\\lim_{n\\rightarrow \\infty} \\mu(A_n)=\\infty.\n", "2e6cee4774419a80bd5f70fe3f6a2f86": "A=\\pi^2\\times C \\times \\rho_1 \\times \\rho_2", "2e6d103f39a4bbf1f622d071c8d94417": " \\boldsymbol\\mu|\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Lambda \\sim \\mathcal{N}(\\boldsymbol\\mu|\\boldsymbol\\mu_0,(\\lambda\\boldsymbol\\Lambda)^{-1}) ", "2e6d3af25af4a6f0a127ed9c2a5c050f": "Q(AB)^2 = (a^2 + b^2)^2\\,", "2e6d85f3c0768de2aa135c8237cc43cf": "\\ u=U(x)", "2e6d9074478523747f11244919c7d3c0": "2^2\\cdot 3^2\\cdot 5\\cdot 7", "2e6dd42b79e1f7a2a81176e4450dd093": " \\beta=6 ", "2e6de5054dd5a2d7e2e4d65347df8e4b": " f: \\C^2 \\to \\C", "2e6e2e168425f3ad1461a5ef410d235e": "\\int x^m \\exp(ix^n)\\mathrm{d}x =\\frac{x^{m+1}}{m+1}\\,_1F_1\\left(\\begin{array}{c}\\frac{m+1}{n}\\\\1+\\frac{m+1}{n}\\end{array}\\mid ix^n\\right)\n=\\frac{1}{n}i^{(m+1)/n}\\gamma(\\frac{m+1}{n},-ix^n),", "2e6e884ef7348ef1b80f6097175b1309": "(C,u)", "2e6f372a56c81f04c5890e44e6653457": "\\mbox{isqrt}( n )", "2e6f3cb4c74a33d5483f5297555e550e": " \\lim_{(x,y)\\to(0,0)} \\frac{xy}{x^2+y^2} ", "2e6f4a864d3327192eeb5bd46fdb193f": "C=\\lbrace 20,4,7,13,19,11\\rbrace", "2e6f98ddf92fe71e30dc4c38dbf492e7": "p \\in \\mathcal{P}, c \\in \\mathcal{C} ", "2e6fe122878ebc5fff702466064d4548": "\n\\delta = \\ln \\left (\\frac{I_1}{I_2}\\right ) = \n\\sum_i \\beta_i \\left (\\sigma_{i2} - \\sigma_{i1} \\right )\n=\\sigma_i \\beta_i \\, \\Delta \\sigma_i\n", "2e701f63423a97ef60eb3797982c959b": " \\vec T ", "2e7044fba347b6a38c11faace6a1ab20": "{a \\over 2} + {x \\over 3}", "2e70535c9b74ed6407e12a91536040b8": " 1 + \\frac{\\gamma}{\\alpha} + \\gamma +\\ln \\left( \\frac{s}{\\alpha} \\right) ", "2e705b78db0ddc039a43705ff425b9c6": "\\{ , \\}", "2e70bdb0c38c3fd8c2f7a6f2bc1a2f33": "K_p = K_c \\cdot \\alpha", "2e70c8f3924a24765a954bb7ef1de751": " S=S_1(q_1)+S_2(q_2)+\\cdots+S_N(q_N)-Et. ", "2e70cf1795b8f2f1538a2b183bb395ed": "X \\to BO", "2e70fbc537cb9c236e25db7315ab57b8": "y = \\frac{1}{2} \\left( \\tau^{2} - \\sigma^{2} \\right)", "2e713029becf7c824e2be5faeeb2bb1a": "\\text{CR}=\\frac{V_1}{V_2}", "2e7131f3c821eaed055be588cf21302b": "\\scriptstyle\\operatorname{Var}[\\ln X] = \\psi_1(\\alpha)", "2e714328adde52a2a6f7895f2757d881": " [M]^B = [M]_{E}^B. ", "2e718d8a3cf43afe57c2d4524a3737e9": "X \\cup \\bigcup X", "2e7263476caffbc05599558fcd8d2ec2": "f(0,0)=0, {\\partial f\\over \\partial t}(0,0)=0, \\dots, {\\partial^{k-1} f\\over \\partial t^{k-1}}(0,0)=0, {\\partial^{k} f\\over \\partial t^{k}}(0,0)\\ne0.", "2e727d630f648be853e1921d6764a4f4": "\\begin{matrix} \\frac{25}{9} \\end{matrix}", "2e7282b21eab69352035c30b1ff59234": "i_\\alpha:\\Lambda^kV\\rightarrow \\Lambda^{k-1}V.", "2e73e41f52d050ce22790edabeb22195": "\\ D_j", "2e73f9a6ca2916211b59c83ccc35fea9": "r(\\epsilon) = q_0", "2e73fcfac6838b81070effdde5b70b72": " \nH = \\hbar \\omega \\big( a^\\dagger a + \\tfrac{1}{2} \\big)\n", "2e741dd0f49de072888f88ce549b3810": " S_j ", "2e7443977ca2fa1b7ab17a574052d002": "q_p(p-1)\\equiv 1 \\pmod{p} ", "2e746e24d7ec4827c87c4f69cbd4db36": "\\aleph_{\\lambda} = \\bigcup_{\\beta < \\lambda} \\aleph_\\beta.", "2e748133e4e0013ded27397f910d81bc": "\\left\\vert A \\right\\vert = \\kappa", "2e749f3118ffaa7e450fedc3169a6139": " \\min \\left \\{X_1,\\dots,X_n \\right \\} ", "2e75053db89e14ec54832bc1a73209b5": "L(\\sigma) \\ge 2l+53", "2e7554dc5082a25b618df9695e6d2b54": "r + \\mathrm{d}r", "2e7557ae9db337d5471671dfbdeb9dac": " E_j-E_i-C e^2 (\\epsilon g(E_f)/V)^{1/d} >0 ", "2e756f94996dedc53eed15de1f6e9219": " \\omega(G) \\leq \\vartheta(\\bar{G}) \\leq \\chi(G), ", "2e75d027d9a822524dc50d07cd51ab9f": "d(x, y) = \\operatorname{arccosh}\\, B(x,y).", "2e75ffb025c83af5961fe4255f36b65a": "S_{IL}=J~F^{-1}_{Ik}~F^{-1}_{Lm}~\\sigma_{km} = J~\\cfrac{\\partial X_I}{\\partial x_k}~\\cfrac{\\partial X_L}{\\partial x_m}~\\sigma_{km} \\!\\,\\!", "2e76073082cfae2555010f99e0e3dd04": "w\\leq w'", "2e764713ebb2e43fab971a8c5deeece8": "\n\\text{Def}(X) := \\Bigl\\{ \\{ y \\mid y \\in X \\text{ and } (X,\\in) \\models \\Phi(y,z_1,\\ldots,z_n) \\} ~ \\Big| ~ \\Phi \\text{ is a first-order formula and } z_{1},\\ldots,z_{n} \\in X \\Bigr\\}.\n", "2e76d4f7c0ea9c5fe54dfd50ed1fa0db": "\\tau_G^{-1}=B\\frac{v}{d_G}", "2e76ed75f75434523c93c297f9396ed1": "\\begin{align}\n\\int_{-\\infty}^{+\\infty} \\frac{\\partial^2\\mathbf{1}_{a 0 \\;\\wedge\\; \\mathrm{Im}(y) > 0 \\;\\vee\\; \\mathrm{Im}(x) < 0 \\;\\wedge\\; \\mathrm{Im}(y) < 0 \\;\\vee\\; \\ldots) \\,.\n", "2e8442e68ff636aaa51ce3dfbc324914": "\\sqrt\\varphi", "2e8450077ae1ffc1f0013aa0fe32e14b": " \\bar{F}=-i\\oint_C p \\, d\\bar{z}.", "2e8464c2a7948165d4e8de27fe011191": "S = \\lbrace 0, 1\\rbrace^{\\mathbb{Z}}", "2e847be707c6c7a342115e973a98c848": " \\mathbf{v} = (V_1/V_0, 0, 0) ", "2e850037e213544e424ff16f74bb7372": " R_{in} = \\frac {R_1} {1 + { \\beta }_{FB} A_{OL} } \\ , ", "2e850289550206e6c6d48e6f43a8f562": "\n\\begin{matrix}\nS&S\\\\\nS 10_H \\overline{10}_H&S 10_H^{\\alpha\\beta} \\overline{10}_{H\\alpha\\beta}\\\\\n10_H 10_H H_d&\\epsilon_{\\alpha\\beta\\gamma\\delta\\epsilon}10_H^{\\alpha\\beta}10_H^{\\gamma\\delta} H_d^{\\epsilon}\\\\\n\\overline{10}_H\\overline{10}_H H_u&\\epsilon^{\\alpha\\beta\\gamma\\delta\\epsilon}\\overline{10}_{H\\alpha\\beta}\\overline{10}_{H\\gamma\\delta}H_{u\\epsilon}\\\\\nH_d 10 10&\\epsilon_{\\alpha\\beta\\gamma\\delta\\epsilon}H_d^{\\alpha}10_i^{\\beta\\gamma}10_j^{\\delta\\epsilon}\\\\\nH_d \\bar{5} 1 &H_d^\\alpha \\bar{5}_{i\\alpha} 1_j\\\\\nH_u 10 \\bar{5}&H_{u\\alpha} 10_i^{\\alpha\\beta} \\bar{5}_{j\\beta}\\\\\n\\overline{10}_H 10 \\phi&\\overline{10}_{H\\alpha\\beta} 10_i^{\\alpha\\beta} \\phi_j\\\\\n\\end{matrix}\n", "2e851e621247761b859309c42c803b4c": " \nv_{k+1} = 2 \\alpha v_{k} - v_{k-1}\n\\,\\!", "2e85379770cc2e711751c0901fb07117": "J(u) = \\frac{mk}{L^{2}} u^{1-\\beta^{2}}", "2e8636139d51aa25f55c22d554ebd17c": "E_{xy,xy} = 3 l^2 m^2 V_{dd\\sigma} + (l^2 + m^2 - 4 l^2 m^2) V_{dd\\pi} +\n(n^2 + l^2 m^2) V_{dd\\delta}", "2e864152fe341e0547eed339242cb24c": "(\\Omega, \\mathcal{F}, \\mathbb{P}, \\vartheta)", "2e86bd1601ffa7db87d3017072ba9bb8": "\\mathbf{p}_k", "2e8704dd2ef7d0f3dbf477e4937f8410": " \\alpha \\ll \\omega_0. \\, ", "2e871e63fb9aec4a3a92058063fd30d8": "\\sigma_f^2 = a^2\\sigma_A^2 + b^2\\sigma_B^2\\pm2ab\\,\\text{cov}_{AB}", "2e87695f6877ba24c3dfef2a479557ff": "({\\rm curl}\\,\\mathbf F)\\,_3=\\frac{1}{a_1a_2}\\left (\\frac{\\partial (a_2F_2)}{\\partial u_1}-\\frac{\\partial (a_1F_1)}{\\partial u_2}\\right )\\,", "2e8808af07f035d340b4024b8a564ce2": "\\ddot{r}-\\omega_S^2 r = 2\\omega_S \\omega_R r + \\omega_R^2 r ", "2e8820ab80392eb9515a54e5012e231b": "\\mathfrak{c}^2 = \\mathfrak{c},", "2e886df462da79391cb4f1765e68cd99": "A_\\mathfrak{p}", "2e892e13c176a5a6afd926b48d0a2070": "p(n)=\\sum_k (-1)^{k-1}p\\left(n- k(3k -1)/2\\right)", "2e8954d4e6cd2f2b087eef29a71f47f6": "\\textstyle\n\\left(\\sum_{i\\in\\N} a_i X^i\\right) \\times \\left(\\sum_{i\\in\\N} b_i X^i\\right) = \\sum_{i,j\\in\\N} a_i b_j X^{i+j},\n", "2e898467e9a3660765c801e5cf586d4a": "[x]_1 = x - \\lfloor x \\rfloor", "2e899571104dfc35f2ff155c7954f588": "P=t^n+a_{n-1}t^{n-1}+\\cdots+a_2t^2+a_1t+a_0", "2e89d0ec6c1f8057599002e36d39c689": " \\operatorname{drop-params}[g, D, V, [F_3, S_3, A_3]::[F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_]\\ \\operatorname{drop-params}[m, D, V, \\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "2e89d1ef3e519b01e24f1ce683c6e766": "\\begin{bmatrix}\n r * (2 * i + (j\\ \\mathrm{mod}\\ 2) + (k\\ \\mathrm{mod}\\ 2))\\\\\n (j + (k\\ \\mathrm{mod}\\ 2) / 3) * \\sqrt{3} * r\\\\\n k * \\sqrt{6} / 3 * 2r\\\\\n\\end{bmatrix}\n", "2e8a112c69b983aac00f93eee6a989a1": "\\textstyle u", "2e8a17c3d97e2dc229fc8af547cdcce1": " K(t) = \\frac{1}{2} mv^2(t) = \\frac{1}{2}m\\omega^2A^2\\sin^2(\\omega t - \\varphi) = \\frac{1}{2}kA^2 \\sin^2(\\omega t - \\varphi),", "2e8a4623ad5f1b29cc0627ae394e5c33": "M_y\\,\\!", "2e8adeea1036fef2752f593b32da1d24": "\\lambda_1 , \\lambda_2 , ..., \\lambda_n", "2e8aeb5f895e7741f4349d9c69ac3b72": "-\\frac{\\xi}{2}(1+2\\eta n_\\eta(\\xi))", "2e8b2c15e74956cee49e222466fd205a": "\\int_{\\mathbf{R}^n} f \\,dV = \\prod_{i=1}^n \\Big(\\int_{-\\infty}^\\infty \\exp\\left(-x_i^2/2\\right)\\,dx_i\\Big) = (2\\pi)^{n/2},", "2e8b528709040d911b6acaed2687b94a": "\\left[ std \\frac{K \\cdot t}{V} \\right]^{-1} \\propto \\frac{C_o}{\\dot{m}/V} \\qquad(9)", "2e8b6d670e08ab962ad147108a4b695a": "n!! = n \\times (n-2)!!", "2e8b81bc3ee597a18781dda5749dac69": "{\\frac{m}{e}}<2.35", "2e8bae982f169ec4c5bea5fa8af93e2b": "0 < \\text{median} < 1-\\tfrac{1}{\\sqrt{2}}", "2e8bafd38e5b538ce96a4867a5716f40": "\\mathcal{B}(\\mathbf{R}) = H^1_{\\mathbf{R}}(\\mathbf{C}, \\mathcal{O}),", "2e8bbca858ca1d0544bd95d5011d9029": "H^{II}_j(H^I_i(C_{\\bull,\\bull}))", "2e8bc4e40058ed488e20218ba90f9629": "\\scriptstyle \\frac {RQ} {P} + P \\ln \\left ( \\frac {R+Q} {P} \\right ),", "2e8c03febde730d5d10a9abac616348f": "\\frac{err(prune(T,t),S)-err(T,S)}{|leaves(T)|-|leaves(prune(T,t))|}", "2e8c15c8cb3911ea0e6eda01c693c13d": "\\mathfrak{m}^{n+1} \\subset x^{-1} A", "2e8c3962a75c96f6cdf035979a2bfe43": "n_2^2", "2e8c6496dbfa1eec32c55c519a10c722": "[H_\\lambda,Y_\\mu] = -(\\lambda,\\mu)Y_\\mu,", "2e8c8437d91d98c3f8104f4064d15550": "\\theta_M", "2e8c9ec8447b3ed9bc5bb8f3c5981cb7": "\\scriptstyle\\int \\frac{\\partial}{\\partial t} ...\\to \\frac{\\mathrm d}{\\mathrm dt} \\int ...", "2e8ce251f544d9692cb6174c20cf6a03": "n = 10^m \\cdot n_m + 10^{m-1} \\cdot n_{m-1} + \\cdots + 10^0 \\cdot n_0.\\,", "2e8d719ea6745e8fa8ad7fd23d273e3c": "[0,1]_{\\ast}.", "2e8da19bb5d7fafc52c98236f6ccb5e9": "\n \\begin{array}{rcl}\n \\vec{\\omega} \n &=& \\nabla \\times \\vec{v} \\;=\\; \n \\left(\\frac{\\partial}{\\partial x},\\frac{\\partial}{\\partial y},\\frac{\\partial}{\\partial z}\\right)\\times(v_x,v_y,v_z)\\\\\n &=& \n \\left(\n \\frac{\\partial v_z}{\\partial y} - \\frac{\\partial v_y}{\\partial z},\\; \n \\frac{\\partial v_x}{\\partial z} - \\frac{\\partial v_z}{\\partial x},\\; \n \\frac{\\partial v_y}{\\partial x} - \\frac{\\partial v_x}{\\partial y}\n \\right) \n \\end{array}\n", "2e8da96a01e92b2124ad3acbd3dbcaab": "\\mathbf{u}_1=\\mathbf{v}_1=\\begin{pmatrix}3\\\\1\\end{pmatrix}", "2e8e07bf61408b851b57388ca47280c9": "\\mathbb{\\hat C}=\\mathbb{C}\\cup\\{\\infty\\}", "2e8e40011439ec178c02312fb1906369": " r < \\frac{x}{c} <= 1.0 ", "2e8e6518468a09d064e8f265a346552b": "\\int \\log_a x\\,dx = x\\log_a x - \\frac{x}{\\ln a} + C", "2e8e6d56ad89927bfbb296a5f944227c": "(x^n)'=nx^{n-1}", "2e8e81b95bbff03ea9cf01849fe9c0e4": " f( k ) = \\frac{ 1 }{ 2 } \\left( \\delta \\left( k - 1 \\right) + \\delta \\left( k + 1 \\right) \\right). ", "2e8ea17d95651054c2cb3c5a4b25b5d2": "w_r", "2e8eb92a88f9d803f149240b11d21d44": "\\langle y, S(h)x\\rangle = \\langle hy, x \\rangle.", "2e8ee6fecbb6251c018d41c1d559d99a": "\\frac{\\mathrm{d}\\varphi}{\\mathrm{d}\\alpha}=\\int_a^b\\frac{\\partial}{\\partial\\alpha}\\,f(x,\\alpha)\\,\\mathrm{d}x.\\,", "2e8effd028aecfcaed9b6ada52667e41": "\n\\begin{align}\n0 & = \\frac{d I'}{d \\epsilon} [0] = L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_2] T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_1] T \\\\[6pt]\n& {} + \\int_{t_1}^{t_2} \\frac{\\partial L}{\\partial \\mathbf{q}} \\frac{\\partial \\phi}{\\partial \\epsilon} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial^2 \\phi}{\\partial \\epsilon \\partial \\mathbf{q}} \\dot{\\mathbf{q}} \\, dt.\n\\end{align}\n", "2e8f48d4b466faadeff6fd78d627faf1": " \\theta^{*} = \\theta^{ML}", "2e8fc63c18189ac83e457cc03ac98716": "\\lambda_1\\!", "2e8ff5a4e297e1d84cfffd31b6a0d936": "\nF=\\sigma T^4\n", "2e9025010009dbc541bf077668fdcecd": "r_t = \\ln (P_t / P_{t-1})", "2e906dab373600cbd65931c4721c43bf": "[a_0; a_1, a_2, \\ldots, a_m]", "2e908bbe13834adc89aa803b0e7d3f17": " E_2'", "2e9100b35219c82bac707bcbd117ccd4": "w\\,R\\,v\\Rightarrow w=v", "2e910bf3d7b24f5512cf977bec7c45d5": "(x,y)\\in \\mathrm{IND}(P)", "2e9122923258bf28a2325726930d68ab": "\\lambda_K = \\sqrt{D_s c_o/R}", "2e912784ce8999b0cd7c5b38474190f1": " R_1 ", "2e91459ea1d1ca57d184d51d08db2131": "f^e_\\mathbf{k}", "2e915193a749b4c75c28ba2634d86322": "\\widehat{\\alpha \\vee \\beta}:= \\hat{\\alpha} + \\hat{\\beta} - \\hat{\\alpha}\\hat{\\beta}", "2e918a222972a4d251222d2a010029af": "6+90+336+900+", "2e9213bcf8c03fb380bb56e08a97be0d": "\\mathcal{E} = \\mathbf{Ab}", "2e92188e7c11a459b95df0c38859104a": "J/K\\}", "2e921a6cd17bcc4f0fbd592a54cc8034": "\\Big( \\pi \\models \\Phi \\Big) \\Leftrightarrow \\Big((\\mathcal{M}, s_0) \\models \\Phi\\Big)", "2e9226f9a9507799c8d1f22e49e778aa": "m^1,...,m^d \\in R_+^{\\bar{k}}", "2e9231f743fbdecee7e13c706eddeded": " y_j = \\mu_j + z_\\alpha \\cdot \\sigma_j", "2e92366541d48d68118505c81c584a7c": "\\begin{align}\n s &= \\frac{3{x_P}^2 - p}{2y_P}\\\\\n x_R &= s^2 - 2x_P\\\\\n y_R &= y_P + s(x_R - x_P)\n\\end{align}", "2e92877f8cd6c0435bd611b5e4850dc9": "{\\tau_2}", "2e92ae03d4339b2c840cd665f6b91f11": "e^{\\delta f}\\in A_{p}", "2e9346ef063b6c2a13bec4d63287bc46": "\\phi(r)=0\\text{ and }\\sigma(r)=\\sigma_0\\Big( 1-\\exp(-(r-R)/l)\\Big)\\text{ for }r > R, \\, ", "2e9350bb520fa7966c9a8f73a3d3e86c": "[V_i,V_j]", "2e9360d79c99c2a446a74806ccd64c70": "\\begin{align} \\nu(M) \n& = \\int \\frac{\\sqrt{M^2-1}}{1+\\frac{\\gamma -1}{2}M^2}\\frac{\\,dM}{M} \\\\\n& = \\sqrt{\\frac{\\gamma + 1}{\\gamma -1}} \\cdot \\arctan \\sqrt{\\frac{\\gamma -1}{\\gamma +1} (M^2 -1)} - \\arctan \\sqrt{M^2 -1} \\\\\n\\end{align} ", "2e93dd493755c1ad3b4fea803d0c23e0": "\\hat C", "2e93f86858f5f758b2b1519489ca1447": "t= \\frac{5}{256}\\, \\frac{c^5}{G^3}\\, \\frac{r^4}{(m_1m_2)(m_1+m_2)}\\ ", "2e9481a2e14cf099357c6a34e82fa7c9": " T= \\frac{1}{2} m\\mathbf{v}\\cdot\\mathbf{v} = \\frac{1}{2} m (\\dot{x}^2+\\dot{y}^2) = \\frac{1}{2} m L^2\\dot{\\theta}^2.", "2e9494ab95685023632a425294596f88": "z_\\mathrm{res} = E_\\mathrm{res}-i\\Gamma_\\mathrm{res} \\, ", "2e94a81586eda7f9317bc6d994098fee": " C_x(t_1,t_2) = \\left\\langle \\left(x(t_1)-\\mu(t_1)\\right) \\left(x(t_2)-\\mu(t_2)\\right)^* \\right\\rangle \\, , ", "2e94bcdf5956e8c8aae3ab57e701c02a": " \\frac{\\bold{r}\\cdot\\bold{\\sigma}}{r} ", "2e94be7f73248c217b328230680e7945": "\n{1 \\over \\pi t}\n", "2e94ca350c31f00f82247011e6fc0506": "\\det(A \\circ B) \\ge \\det(A) \\det(B). \\, ", "2e94d7330c97fbf05dc0a403bdf65d52": "H_{\\omega^{\\omega + 1} + \\omega^\\omega + 1}(1) - 1", "2e94de376916bf098627b1f25d3a25f4": "j^k:\\Gamma^\\infty E\\rightarrow J^kE", "2e94fb85847ba4a0566db434003edb6e": "\\int_{|x|\\le\\rho} |\\hat{u}(x)|^2\\,dx \\le \\rho^n\\omega_n \\|u\\|_{L^1}^2", "2e952430967b78572be6d22cd3293f84": "n! = \\sqrt{2\\pi n} \\;n^n e^{-n}\n \\left(1 + \\mathcal{O}\\left(\\frac{1}{n}\\right)\\right)", "2e95909d45fdf5e4197bf661237ffeaf": "A = A = A;", "2e95c4c6eeedcbfea530eb97f3bfe466": "(-1)^k(\\Delta^k m)_n = \\int_0^1 x^n (1-x)^k d\\mu(x),", "2e95cd2b14736d0dc11238e25985beb9": "(A,a)", "2e962d638f3af7fbefee9a39b223504f": " v_1\\wedge\\cdots\\wedge v_k.", "2e967f6e5c476f95141f6e9d2de2d9ac": "\\mathbf{F}\\cdot(\\sum_{i=1}^n \\mathbf{R}_i \\times \\mathbf{F}_i )=0.", "2e968384421b7e54f84f727f87894dac": "d^2\\alpha (x\\wedge y\\wedge z) = \\frac{1}{3} \\left(\\alpha([[x, y], z]) + \\alpha([[y, z], x])+\\alpha([[z, x], y])\\right).", "2e96f72ddded5410b10850e7cc364e0b": "\\tan (2 \\theta) = \\frac{2 \\tan \\theta}{1 - \\tan^2 \\theta} = \\frac{2}{\\cot \\theta - \\tan \\theta}\\,", "2e96ff186eb7ad78760f585c9cfd8074": " \\bar{h}(t) ", "2e973044decd6c4e32ed6fe6fea60d0a": "dU = dQ", "2e974df15f245e66274d7a3e36900c44": "\\mathfrak{so}_9", "2e975a172f1fc56a597976802a35520c": "\\Omega_{k}", "2e980002b1d84133465a1a66985260f1": " \\{,\\} ", "2e981f9689ab60de4e0c7d17b0d8ff9d": "\\alpha_{,\\zeta\\eta}=0", "2e985237c8ceeb687effae7a3301d0e7": "(\\Phi(n) \\land \\forall i (\\Phi(n+i) \\to \\Phi(n+i+1))) \\to \\forall i \\Phi(n+i)\\,\\!", "2e988411e6dda41c0a7c81e21918bdf7": "\\Pi_{\\gamma\\gamma}", "2e98ad28b2b8663c2d415add94d3ab3d": "\\mbox{Ann}(\\mbox{Ann}(\\mbox{Ann}\\,(S))) = \\mbox{Ann}\\,(S)", "2e991e5ff32eb5bbb250194b271a1d33": "\\, p_{i_{1}}", "2e9953dda5add19918ced3e59520d5da": "wp(S_1;S_2,R)\\ =\\ wp(S_1,wp(S_2,R))", "2e99b7c2da51e27658ea655a120a5535": "\\dot{V} ( \\mathbf x)", "2e99c8f6c33835be7f9bea60b599b063": "\\psi\\ = f_4(\\phi,\\beta),\\,", "2e99db714f09c4bac0fa77f833eafcf5": "\\begin{bmatrix} z\\\\ v \\end{bmatrix} = \\mathbf{P}(s)\\, \\begin{bmatrix} w\\\\ u\\end{bmatrix} = \\begin{bmatrix}P_{11}(s) & P_{12}(s)\\\\P_{21}(s) & P_{22}(s)\\end{bmatrix} \\, \\begin{bmatrix} w\\\\ u\\end{bmatrix}", "2e9a8c8c42f49eda3f2c546a12a8b1ec": "\\mathcal{O}(D^{1/2+\\varepsilon})", "2e9aad608d4055052f3935a104b86e7c": "\n\\int^1_0 S(tx)\\,(1-x^2)^{n-2}\\,x\\,dx\\leq t^\\lambda\\text{ for all }t\\in[0,+\\infty),", "2e9ad80425376edae4d61a50aa7c6174": " g(k) = \\int_k^\\infty \\!xf(x)\\, dx ", "2e9b301636db7a91edf6dccb41215243": " U_\\omega |\\omega\\rang = - |\\omega\\rang ", "2e9b51c517fce2fac5afbf76cbc140e3": "C_y = \\frac{1}{6 A} \\sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i).\\,", "2e9b956c5df7b6f64727094daf733d7e": "d\\triangleq\\max_{i}\\deg(f_{i}).\\,", "2e9c12a7cd031689093714e77c410d9c": "\\frac{ \\int^\\infty_0 y_\\lambda J_\\lambda d\\lambda } { \\int^\\infty_0 J_\\lambda d\\lambda },", "2e9c23c4d754e38e4b358a1744420252": "x_0.", "2e9c2628164c08b8f9a6f8885b0df39f": "\\{q\\in\\mathbb{Q}:a_q\\neq 0\\}", "2e9c2fd61990c8d266f411bc43e0f938": "FBf = BFf", "2e9c772a47ec257aa3fedef85c61f0a7": " \\frac{1}{\\sigma^{2} + \\tau^{2}}\n\\left( \\frac{\\partial^{2} f}{\\partial \\sigma^{2}} +\n\\frac{\\partial^{2} f}{\\partial \\tau^{2}} \\right) +\n\\frac{\\partial^{2} f}{\\partial z^{2}}\n", "2e9c82689a8823de629316a2f6119a22": "J_{ij}=\\frac{\\partial f(x_i,\\boldsymbol\\beta)}{\\partial \\beta_j}", "2e9d4b5bc797dc6558b29e7af91f2910": "v_{Ti} = (kT_i/m_i)^{1/2} = 9.79\\times10^5\\,\\mu^{-1/2}T_i^{1/2}\\,\\mbox{cm/s}", "2e9de85d6f9dcf6aeee9d67ba80c49db": " |K_{g_\\epsilon}| < \\varepsilon", "2e9e2a627cdd47c0782f2fee37e08ed5": "1/4 + 1/16 + 1/64 + (1 + 2/3) ro", "2e9e68bf5c07717de4e755fba72152a5": "\n\\begin{align}\nx& = [a_0;a_1,a_2,\\dots,a_k,\\dot a_{k+1},a_{k+2},\\dots,\\dot a_{k+m}]\n\\end{align}\n", "2e9ebf4a419fd45075ca3f905c44ba3c": "N=\\frac{B}{2*\\mu_{0}*m*c^2}", "2e9ef3d6ef62a48d70720728d3e90e31": "\\Omega", "2e9f96210c1711639750549787720014": "Pwo = 1 - \\dfrac{2}{5} = \\tfrac{3}{5}", "2e9fadeeec8263501f38ce4e64696447": "{}^\\mathsf{T}", "2ea07336044034928b5effa9b212ea22": "\nP(k) \\ \\sim \\ k^\\boldsymbol{-\\gamma}\n", "2ea0866e5307485cc29aa82f2f8eb394": "\\phi^{17}=\\frac{3571+1597\\sqrt5}{2}\\approx 3571.00028\\,", "2ea0fedba934e21cd895f0c1aa3994bb": "\\lambda _{p} = \\frac{1.24}{E_{g}}", "2ea1623817601f2124776d94211d34c6": "u_1(\\vec x,\\vec x/\\epsilon)", "2ea16cd8b70693ea28ec65050d77def1": "Q(x) = \\frac{x}{\\zeta(2)} + O\\left(\\sqrt{x}\\right) = \\frac{6x}{\\pi^2} + O\\left(\\sqrt{x}\\right)", "2ea196a0cdbb3fc3fe9ac84b25fcffe4": "\\rho = \\sum_i p_i v_i v_i^*", "2ea1e6567339609248feeac7130ba172": "{}^t \\! X_i", "2ea213c46d7ce21d3aaab109e0a07241": "f(x)=\\Omega(g(x))", "2ea287966683cfaa4e9ca4e2a4d2a012": "\\omega = 5\\cdot \\omega_c", "2ea28a68be977be8471f2e28e7dfc196": "\\operatorname{Tr}(Q\\sigma) ~\\geq~ \\operatorname{Tr}(Q\\rho) - \\delta ~\\geq~ \\epsilon - \\delta~.", "2ea2b74065f189ea80c08e14080d5486": " \\tau_p ", "2ea2caefba3c8d4a3ffafe2493861a4c": " \\sum_{k=0}^n \\frac{1}{k!(n\\!-\\!k)!}\\sum_{\\pi\\in S_{n}}(-1)^{\\left|a_{\\pi}\\right|}\\Phi^{n-k+1}\\left(\\Phi^{k}(a_{\\pi(1)}, \\ldots, a_{\\pi(k)}), a_{\\pi(k+1)}, \\ldots, a_{\\pi(n)}\\right) = 0. ", "2ea32ce0299b19e50cd16c2ec660500c": "\n\\vec{u^{\\star}} = (1+r) ( \\widehat{\\Sigma} )^{-1} ( \\widehat{\\vec{r}} - r )\n", "2ea377c9483023febdbe9d41f8770cb8": " \\alpha_s = \\int {\\left| M_s \\right|^2 N_s (E)N_s (E + \\hbar \\omega ) \\times [f(E) - f(E + \\hbar \\omega )]{\\rm{ }}} dE ", "2ea4d24c80e51a8255bd3211a5d67d2a": "\\gamma=\\frac{C_p}{C_V}", "2ea4d63bdd6ded1b9c9d57339fb2e0ef": "L(x) = 1/\\tanh(x) - 1/x", "2ea5293a5229de02c20b99e26e8fa3c1": " x \\in I ", "2ea56b110f902917027bc699cb575b54": "T_n(T_m(x)) = T_{nm}(x)\\, ;", "2ea56d4051a1f7f2658f48e9bf873568": "\\eta=(\\nu^{3}/\\varepsilon)^{1/4}", "2ea5858c65490a7756bb437a54e9bb88": "\\frac{ df(T)}{ dT} < 0.", "2ea5decb39b61337c65ca4590faea001": "R_{sw}", "2ea5e33d96947f1a479a25a5337beae2": "F_n = \\dot{m}\\;v_\\text{e} = \\dot{m}\\;v_\\text{e-act} + A_\\text{e}(p_\\text{e} - p_\\text{amb})", "2ea60ba6a365cef3e61591193aeedc4b": "\\underset{\\Theta}{\\operatorname{argmax}}\\left( \\log p(\\{x_i,y_i\\}_{i=1}^l | \\theta) + \\lambda \\log p(\\{x_i\\}_{i=l+1}^{l+u}|\\theta)\\right) ", "2ea687fef834cdcc41f39de3198f7133": " X \\leftarrow \\Gamma ", "2ea69f2606abecd37697892ed6c8a330": "\\langle \\mathbf{u}, \\mathbf{v}\\rangle", "2ea7034bf07bccb1473a84bc54a76015": " \\frac{d^2y}{dt^2} + (\\alpha^2 + \\omega^2 \\sgn \\cos(t)) = 0 ", "2ea732c0ba8aea3defec144b080e736e": " \\mathbf{b} = b_1\\vec{r}_u + b_2\\vec{r}_v", "2ea734c4089a3f63a1cc0957ef71bd28": "\\left(\\frac{\\Delta\\;H}{T}\\right)", "2ea79eea8cbcedd9af4fb6b2735d3dfb": " T_\\varphi : f \\in H^2(\\mathrm{T}) \\rightarrow P(f \\varphi) \\in H^2(\\mathrm{T}). \\, ", "2ea7c77abda00f6fabb5dcb737e9520b": "b - \\sqrt{b^2-4ac} \\approx b - b + \\varepsilon. ", "2ea815cd550303e2eac6cccce8be10dc": "A = \\frac{1}{4} \\pi d^2", "2ea8866757eb09a42255500343ad0e96": "max(0, due\\;date - receipt\\;date)", "2ea88cfe6cb6ce8441baf27d609a74a6": "\\sum_{i = 1}^{m} \\frac{\\log_2 {n_i} }{T_i}", "2ea8b282b4cd3a7e2dcd4c5fff82a444": " \\begin{align} \\langle\\psi|\\mathbf{\\hat H}|\\psi\\rangle & = \\langle\\psi_{\\varepsilon}|\\mathbf{\\hat H}|\\psi_\\varepsilon\\rangle \\\\\n & = \\langle\\psi|\\mathbf{\\hat T^ \\dagger} (\\varepsilon) \\mathbf{\\hat H} \\mathbf{\\hat T} (\\varepsilon)|\\psi\\rangle \\\\\n & = \\langle\\psi|\\left( I + \\frac{i\\varepsilon}{\\hbar}\\mathbf{\\hat P}\\right)\\mathbf{\\hat H}\\left( I - \\frac{i\\varepsilon}{\\hbar}\\mathbf{\\hat P}\\right)|\\psi\\rangle \\\\\n & = \\langle\\psi|\\mathbf{\\hat H}|\\psi\\rangle + \\frac{i\\varepsilon}{\\hbar}\\langle\\psi|\\mathbf{[\\hat P, \\hat H]}|\\psi\\rangle + O(\\varepsilon^2) \\end{align} ", "2ea8bbe3f567e0e4e5f3045555388c9a": "\\phi(b)", "2ea8bc30febc3c0127442d328b81b86d": "k \\in \\{ 1,2 \\}", "2ea8cf923be5182b007631f2f01ea0f1": " p(\\theta | D) = \\frac{p(D|\\theta)p(\\theta)}{p(D)}\\ ", "2ea8ec3e2793f717aac070155fb42608": "\\mathbf{g}=-{G M \\over r^2}\\mathbf{\\hat{r}}", "2ea8fa0541220afe440592ebdb696fb8": "\\sum_{n=2}^{\\infty} \\zeta(n,\\bar{a},\\bar{b}) = \\zeta(\\overline{a+1},\\bar{b})", "2ea9080cf348564cdfb211b8b603803f": "{\\tilde{D}}_{n+2}", "2ea956151a60ada5ea6573f91dde3782": " \\Pi(x) = \\sum_{n \\leq x} \\frac{\\Lambda(n)}{\\log n}. ", "2ea956ae94b1f5c5a554989642c90536": "\\mathcal{I}|_{U_i}", "2ea9d09d07466794a98ce26eb57a909a": " \\sin \\theta\\!", "2ea9d5d60ac66bc49b6a098786249b9b": "P_{\\text{Electric quadrupole}}=\\frac{c^2 Z_0}{1440 \\pi}k^6\\sum_{\\alpha, \\beta} Q_{\\alpha \\beta}^2", "2eaa45d1db458263f011640d34ea2f08": "L:\\Gamma\\to[0,+\\infty]", "2eaa55429c807c21d05df4bf4bdc01ec": "k=\\frac {M} {\\theta} ", "2eaa59c4b0360527d4fcefb27c7a8ac8": " c(\\theta) = c_{root} \\cos(\\theta) ", "2eaa674ca2cc914b77ab76f56ebe5bb6": "\nS = \\frac {1} {R_0}.\n", "2eaa78aa54a152d67e437b439945f02f": "e^i \\left (p,u^i \\right ) - f^i (p)", "2eaa7f0aafd2e33b32a1d91204a4a96e": "\\Lambda = \\tfrac{1}{\\sqrt 2}\\left\\{x \\in \\mathbb Z^n : x\\,\\bmod\\,2 \\in C\\right\\}.", "2eaa8bc2dc9921b215ce5a3609985b02": "A/\\mathcal{J}", "2eaaa7f3cda8df832848028afd6dc4ac": "{1 \\over T}\\int_0^T \\left| \\zeta\\left({1 \\over 2} + it\\right) \\right|^6\\,dt \\sim {42 \\over 9!}\\prod_p \\left\\{1-{1\\over p}\\right\\}^4 \\left( 1 + {4 \\over p} + {1 \\over p^2} \\right) \\log^9 T, ", "2eaab4c0e2ac63095520cfd679ece8f0": "(f^{-1}F)(Y) \\cong \\text{Hom}_{\\mathbf{Top}/X}(f, \\pi)", "2eab0283837d3c33bd90b913a0c1f8d3": " U_{ni} = \\beta_n x_{ni} + \\varepsilon_{ni} ", "2eab0deb7fa851fcaef7f3cd0c655fd8": " D", "2eab4b05e8b40c1692374088c71447b9": " \\int_0^\\pi f(x)\\sin(x)\\,dx=F(0)+F(\\pi)", "2eab639a093a9c51130691f9af6c4f14": "\\begin{align}\nJ_X=\\sum \\limits_{l} \\sum \\limits_{u} \\frac{{X_u}^2}{r}\n\\end{align}\n", "2eab7b1699fe7847772f5a5e44524340": "r_{B,h}(n) > 0 ", "2eab96c47d9f2678d7e6b19f4c816bba": " \\operatorname{build-param-lists}[N, D, V, D[N]] ", "2eabcaa35749d20f3f7e6b5d26a05654": "-b \\pm \\sqrt{b^2 - 4ac} \\over 2a", "2eac1cf98e9bfb2890b1ce300b692ba6": "\n R_n(x_0 + h) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!} (h)^{n+1}\n", "2eac9dd9288b17f5bcdf816720e8b514": "c_j = \\cos^2 \\left [\\frac{\\pi j}{2n} \\right ]\\qquad\\mathrm{j = 1,2,3, \\ldots, n}.", "2eacae3f90157adde5d47afe00badcfd": "T_2 ", "2eacafc1d7dbf10684a114d8f2503fb3": "\\frac{1}{s^{\\lambda}}", "2eacbdd70f924546ab57667c1f0372c2": "r_i=y_i-f(x_i,\\boldsymbol \\beta)", "2eacd71ba8716473b4ab0298e4730b34": "I(X;Y) = \\mathbb E_{p(y)} \\{D_{\\mathrm{KL}}( p(X|Y=y) \\| p(X) )\\}.", "2ead1261faef91edf5992f869237ae83": " u_{1}, \\ldots , u_{n} ", "2ead1c1254e877cc316d732be98ad275": "Z^2 = Z(Z - 1) \\ .", "2ead303ace8c930946484aff05265c5f": "\\beta'(x;\\alpha,\\beta,1,q) = \\int_0^\\infty G(x;\\alpha,p)G(p;\\beta,q) \\; dp", "2ead7b0e02bea8d681e9ed80b86aedc4": " \\mathcal{G}(3,0) ", "2ead93898d8d966d6c0dc5d6c94e94ad": "T_pM\\cong \\mathbb R^n", "2eae0fbf607564d9c92ab629ce6d1117": " SampEn(m,r,N) ", "2eae1fa87f6ff5b2770d0f152f073152": "x(x+1)(x+2) = {\\color{Red}6}x + {\\color{Red}6}x(x-1) + {\\color{Red}1}x(x-1)(x-2).", "2eae27c37062e6131c68c0cff0aa59e5": "T_R=p_R=1", "2eae370109bbb63d935167a7d3a65f39": "(\\mathit{n},\\mathbb{Z}_p)", "2eae5eab30eaa3c2e46be8851e6029a5": "\\alpha_3=\\frac{A_{21}-A_{12}}{2}", "2eaeb6c11ae982ce133f616cc744e15a": "\\mathbf{\\hat r}", "2eaed040fdff9ea0e0f4cbf7ac877773": "A \\mapsto \\mathrm{Lie}(A)", "2eaf477756a2673df22ad5f861097f51": "P(x) + P^{(1)}(x)x'\\varepsilon", "2eaf84fc002c5e754b6d7e9551ea5685": " \\mathbf{A} \\cdot \\nabla \\mathbf{B} = \\begin{bmatrix} A_x & A_y & A_z \\end{bmatrix} \\nabla \\mathbf{B} = \\begin{bmatrix} \\mathbf{A} \\cdot \\nabla B_x & \\mathbf{A} \\cdot \\nabla B_y & \\mathbf{A} \\cdot \\nabla B_z \\end{bmatrix}", "2eafab11d5e1e9e3f29613946ad15217": "F:B_n\\to S^n", "2eafe0196c67de3aedc713547be9d71e": "g=:\\sum b_{j,k}e^{i(jx+ky)}", "2eb0003343ca90a4452c6e58ef4de5fb": "{Q^k}_k", "2eb01475a665a2a0e39af8641ab0d1fe": "\\log(\\exp(z))=z", "2eb0b1a294d7a5694d696dfae6be68fc": "S(X,Y)", "2eb0d210c417353ffcbbbc5620ad48a0": "E_{j,i}=\\hbar\\omega_j(i+\\frac{1}{2})", "2eb0de7d04f3099637524923cb3481c3": "f_i(r_1,\\dots,r)", "2eb12d200eb430e403fb54c251e57ec3": "\\frac{2\\,\\text{bits}\\cdot(15+7+6) + 3\\,\\text{bits} \\cdot (6+5)}{39\\, \\text{symbols}} \\approx 2.28\\,\\text{bits per symbol.}", "2eb1b4fa9ce56429a59d61e46d7d2432": "\\mathcal{I} \\models \\Phi", "2eb21fcc8b5eb542568338405a20cc43": "Cl_A(S) = A\\cap Cl_X(S)", "2eb255cf1e94328d9405d186fa18c8cb": "\nH = \\frac{1}{2}\\epsilon_0\\iiint_V \\left( E(\\mathbf{r},t)^2 + c^2 B(\\mathbf{r},t)^2 \\right) \\mathrm{d}^3 \\mathbf{r} =\nV\\epsilon_0 \\sum_\\mathbf{k}\\sum_{\\mu=1,-1} \\omega^2 \n\\big(\\bar{a}^{(\\mu)}_\\mathbf{k}(t)a^{(\\mu)}_\\mathbf{k}(t)+ a^{(\\mu)}_\\mathbf{k}(t)\\bar{a}^{(\\mu)}_\\mathbf{k}(t)\\big).\n", "2eb26afacd6f64bfb557519d6be5dd8b": "\\{a, b, c, \\dots\\}", "2eb2d979358c187a9630809700219bf8": "\\langle x *_a y, z \\rangle := \\sum_n \\sum_A \\frac{1}{n!} GW_{0, n + 3}^{X, A}(x, y, z, a, \\ldots, a).", "2eb2e6a8f685be7045fd09f00edb05a2": "E=\\frac{1}{8}\\rho g H^2=\\frac{1}{2}\\rho g a^2.", "2eb30d54c72a7e567dac6637ca7dd789": "f : \\mathbb R^2 \\to \\mathbb R^2", "2eb330cc3bf7a46ec256d12dac0b6ad8": "P(Fa)", "2eb3a49099e6055d3c331c49346ff910": " \\displaystyle{G^+_r=(\\ell/2+ 1)^{-1/2} \\sum E_{-m}\\cdot e_{m+r},\\,\\,\\, G^-_r=(\\ell/2 +1 )^{-1/2} \\sum F_{r+m}\\cdot e_m^*.}", "2eb491e2260c21cc8baceeaed53a1a7d": "\\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2} = c(s-t) \\quad", "2eb4dee24ef1b5d9b36a141b2e806104": "\\operatorname{pd}_R M = 0", "2eb5111ae2da3b59c2e9049907abe226": " \\begin{align}\n\\phi(a+a',b)=\\phi(a,b)+\\phi(a',b) \\\\\n\\phi(a,b+b')=\\phi(a,b)+\\phi(a,b') \\\\\n\\phi(ar,b)=\\phi(a,rb)\n\\end{align} ", "2eb5135d4ff37e514b71bacafdf84daa": "|s(k_\\text{surf})|^2=\\frac{1}{4 \\pi} \\sigma^2 \\delta^2 \\exp \\left( - \\frac{\\sigma^2 k_\\text{surf}^2}{4}\\right)", "2eb51f40f46733857b2b800176e40865": "f_{*} (\\mu)", "2eb5249db7f86320d2d0a108d9e4ad2b": " o \\, ", "2eb53a28a2a2bf1899ab0f3d4f251138": "10^6 ~ S/m", "2eb5a42dad575a43734827743c895e33": "F(6)", "2eb624df2ae935bac213611f47b9a268": "w_i^TBw_j=\\delta_{ij}", "2eb62946b85986e16689cf47fdf6ab91": "\nP = \n\\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 0 \\\\\n\\end{bmatrix}\n", "2eb69307ba87ee6a404b8529d823249a": " (D^2 + k^2) y = 0. ", "2eb704ab9b10c935af45edef9ae68456": "\\text{If }(\\neg Q) \\not\\in K*P\\text{ then }(K*P)+Q \\subseteq K*(P \\wedge Q)", "2eb7509abff27101cd096814f866b905": "\\lim_{x\\to a}f(x)=L", "2eb7539f95ad67faea606c3310a8ae49": "(\\Gamma,L,M\\{\\mbox{·}|B\\})", "2eb771e5be94a43cd31183a7cce759f4": "G(z)= z f^{-1}(z)-F\\circ f^{-1}(z)+C.", "2eb7a767982d5762b5b3291647ad8c0b": "A_{\\mathbf{s'},\\chi}", "2eb7c4c7fb151e83a9a1f936e23675cc": "(R, \\Theta) = \\left(2 \\cos(\\phi / 2), \\theta\\right),", "2eb7c88c8d54da89038ba4eb96a1750a": " \\sigma^3 = g(r) \\, r \\, \\sin(\\theta) \\, d\\phi", "2eb7d2ebace37fe00d469d9ca598a7e0": "{\\mathit l \\over n} = {1\\over 2} \\pm {1\\over 2n}", "2eb814896346111a59e20be00fb34665": "x_p=v_pt=\\beta\\,ct", "2eb81f2c9aa439eedf0e28ede29efba2": "i, \\omega_i = \\min(\\Delta(\\mathbf{y_i'}, \\mathbf{y_i}), {d \\over 2})", "2eb83a64b9e8a94fb899de183461a033": "\\theta_4\\,", "2eb883232ab5c302b563d0c2d400fbe7": "\n\\varphi_2(x,y)=x-y,\\psi_2(x,y)=x-\\bar{y},\n {\\mathcal E}_2(x,y)=-\\frac{1}{x+y}.", "2eb894ee6e99ae91f3f7cd61203959a4": "S^* M (S^{-1})^* = N. \\,", "2eb8a8fcb9b959086087dd6d2aa39636": " 2^2 = 4,\\ 2^3 = 8,\\ 3^2 = 9,\\ 2^4 = 16,\\ 4^2 = 16,\\ 5^2 = 25,\\ 3^3 = 27,\\ ", "2eb8ba42886caeb52cfbdf5aee160582": "\\text{Real option value} = \\text{average} \\left[\\max\\left(\\text{operating profit}\\right)-\\left(\\text{launch costs}\\right),0)\\right]", "2eb91a8668deb31f4e17f50f655e3b1a": "\n\\frac{Y(z)}{X(z)} = \\frac{1 + 2z^{-1} +z^{-2}} {1 +\\frac{1}{4} z^{-1} - \\frac{3}{8} z^{-2}}\n", "2eb92e15452837abe36ce3c85b7047f0": "r^2 = \\frac{9 \\eta v_1}{2 g (\\rho - \\rho _{air})}. \\,", "2eb93d6091fdb1e3b82a341f5b864f77": "\\exp:A \\to G", "2eb975a8660bf7a0b099992d7fa00a89": " w = \\frac{F_m + vx}{x} \\qquad\\qquad (3) ", "2eb978110fb17b8584b52980baf5a422": "N_1(k) = 2", "2eb97e5b18d03429337666797b0b36b8": "\\lambda_k = \\sup\\inf \\frac{\\int_\\Omega |\\nabla u|^2}{\\int_\\Omega |u|^2}", "2eb9b7fbf3ee88a6ad2f5f2be90efe62": "w := w - \\alpha \\nabla Q(w) = w - \\alpha \\sum_{i=1}^n \\nabla Q_i(w),", "2eba055fad44cdbae5ec890c4f3db1ae": "\\nabla \\times \\left( \\bold{D} - \\bold{P} \\right) = \\boldsymbol{0} \\ , ", "2eba8580de369a5415b4b1f3ac1438a9": "\\scriptstyle \\| z \\| \\;\\le\\; 1,", "2eba8953acf964531d912833844cb2a5": "\n\\tilde{R}_{ij}(s)=\\frac{1}{N_T}\\sum_{r=1}^{N_T}R_{ij}^{(r)}(s)=\n\\frac{1}{N_T}\\sum_{r=1}^{N_T}\\frac{1}{N_S}\\sum_{t=1}^{N_S}X_{i}^{(r)}(t)X_{j}^{(r)}(t+s)\n", "2ebb2d54f8b76b8bf2293f5a6823c211": "\\alpha = \\alpha^\\prime = 0\\,", "2ebb36b4ee93be472dcef091e9e2bfd1": "\\mathbf S(p)", "2ebba22928d73430f1b446f9796668c4": "D(P)(f)(h) = h'", "2ebcca5eed569041d8bc9954c04f5716": "S = \\{\\mathbf{X},\\mathbf{Y}\\}", "2ebccba3a8bdb9d1c8ff17d37947c8a5": "\\ \\Psi_n(x,t)=\\psi_n(x) e^{-iE_nt/\\hbar}", "2ebcd1623d7c2e6e326c18f816c30156": "\\left(n+1\\right)\\times\\left(n+1\\right)", "2ebd5e7f47f012fb683132da9aa474b8": "\\mathbf A = \\mathbf 0.", "2ebe052820e1b7f73533d45649a39a08": "\\alpha=\\frac{g\\mu_BE_0}{2mc^2}", "2ebf0b15c687b66e4a79fa7f27993e41": "\\mathfrak{D}_0", "2ebf242e0f48557d21dd539b8a721035": "\\exp", "2ebf328e0a87921a25271cf7173235ca": "x*z \\le y*z", "2ebf633373d823c80dcca90c617fd494": "U(t) := u[X(t),t]", "2ebf6d7d7fc41bfba9544e6057a0f56e": " \\nu_1, ... , \\nu_m ", "2ebf7dfd433dc48cffd39580bcf4156d": " \\Pr(X_{n}=j) = \\sum_{r \\in S} p_{rj} \\Pr(X_{n-1}=r) = \\sum_{r \\in S} p_{rj}^{(n)} \\Pr(X_0=r).", "2ebf8695ea7b54448469dbca631797c0": "\\ddot{\\delta \\mathbf{r}} = \\mathbf{a}_{\\text{per}} + \\mu \\left( {\\boldsymbol{\\rho} \\over \\rho^3} - {\\mathbf{r} \\over r^3} \\right),", "2ebf8bf804569d675766eeefd636ee39": "\\frac{d^2x}{dt^2} = f(x, x')", "2ebfca183296fde7fe56f1af766029af": "d=n\\sqrt[3]{FD}", "2ec030dc98820f99243c0c7dc30deb2c": "t_n", "2ec0346b57d2381aeecbf4637a8371af": "\\frac{\\mathrm{d}^2 x^a}{\\mathrm{d}\\tau^2} + \\Gamma^a_{bc} \\, \\frac{\\mathrm{d} x^b}{\\mathrm{d}\\tau} \\,\\frac{\\mathrm{d} x^c}{\\mathrm{d}\\tau} = 0", "2ec0629a927ef7d2385df7b8664ab23e": "y\\in x'", "2ec0d7f083aadfee3f989b9e67d22a36": "x\\ne y", "2ec0fc3bc0230ded8191d16162579dd9": " Re \\leq 0.1", "2ec14c658460eb93f3c831cf06791aaf": "\\#(n)\\sim n^{k_0}", "2ec1738014b5f48e6f07c71bc66be8cf": "m=1/\\tfrac15=5", "2ec19ada120bdb4bda9a4a515144f830": "1\\le j\\le m(i)", "2ec1de1ff03e25e4dfc660c3b4bc3420": " R_\\infty = \\frac{E_h}{2 h c} \\,", "2ec1ecc9aa0291d745f0fbc81246a7b1": "x_1 = x_2 \\,", "2ec20ef0e0f01b44a6023396030c858a": "G(x)=\\int_{x_0}^x \\frac{dy}{w(y)},\\qquad x \\geq 0,\\,x_0>0, ", "2ec2449537da3e342490a480fcc8dcca": "L^* = 116 (Y/Y_n)^{1/3}-16", "2ec26518e89a8c21c7797444c17626ae": " y \\in \\arg \\min \\limits_{z \\in Y} \\{ f(x,z) : g_{j}(x,z) \\leq 0, j \\in \\{ 1,2,\\ldots,J \\} \\}", "2ec2896e30e5270c373accbf106b49c1": "\\mathbf{Q_{\\parallel}}", "2ec28979d417cf42c6ac87d66e61ecb2": "\n\\begin{array}{lrclr}\n\\max\\limits_{x_{t}} & E[Q_{t+1}(W_{t+1})] & \\\\\n\\text{subject to} & W_{t+1} &=& \\sum_{i=1}^{n}\\xi_{i,t+1}x_{i,t} \\\\\n &\\sum_{i=1}^{n}x_{i,t}&=&W_{t}\\\\\n\t\t & x_{t} &\\geq& 0\n\\end{array}\n", "2ec2aabfb0bbf5e4d08f3e182b2efdef": "k^*_s - k", "2ec2c5a908cca02f993ead625fb84ff9": " {\\partial L(t,y(t),\\dot y(t)) \\over \\partial y} = {d \\over dt} {\\partial L(t,y(t),\\dot y(t)) \\over \\partial \\dot y} .", "2ec2eaeae71986f5c79bef8d411d81cb": "dim: N \\rightarrow \\mathbb{N}\\setminus\\{0\\}", "2ec3ac9e73e5f5933fb3fee5ead0c22d": "\\tilde H_n(X/A)", "2ec4c1c33ab3837a6d21d4266c9509c1": "N = \\sum_i \\lambda_i P_i \\,", "2ec4dcb3f1a4c226257417a57d476da8": "R = \\frac{\\mathbb{F}_q[x]}{\\langle f(x) \\rangle}", "2ec4fa53486776b705707792ba268d68": "A(1+\\rho)", "2ec4fb88a7f3fa2a0706cfde094daa19": "s \\cdot s = s", "2ec5018c336e2605782e69939dfe4a04": "f(x, y) = (1-x)^2 + 100(y-x^2)^2 .\\quad ", "2ec520bba83203b0f483035ab0373342": "48 \\times 27=36 \\times 36=1296", "2ec54d38015144798e86d8df49f9945b": "u\\in U, v\\in V", "2ec57556586256ccb0fede8578033060": "\\text{Per-unit ohms}=\\frac{\\text{ohms}}{\\text{base ohms}}", "2ec58857fe6ec5af3311b8ddee3fff1d": "n=\\pi(y)", "2ec5a220e37905b274c28a2e1d9e6f82": "(ab)(cd)", "2ec5c006e1f47afd858ba0c2420d1eca": "\\boldsymbol{\\mathsf{E}}", "2ec5f2bf24dc0b2d3e6bb89b9a8a42b6": "\n\\psi_{\\alpha,a,b}(t)=\\frac{1}{\\sqrt{a}}\\psi\\left(\\frac{t-b}{a}\\right)e^{-j\\frac{t^2-b^2}{2}\\cot\\alpha}\n", "2ec6262da13b2113423e7bba63592f2c": "F(z) = \\int_{-\\infty}^\\infty f(x) \\, \\mathrm{e}^{z x - x^2} \\, \\mathrm{d}x = \\sum_{n=0}^\\infty \\frac{z^n}{n!}\\int f(x) x^n \\mathrm{e}^{- x^2} \\, \\mathrm{d}x = 0", "2ec6342f19a1e5ca300c6c6d67637543": "p_K(r)=2\\pi \\sin_K r", "2ec674f2e3fd61d7d3883b7325175cb9": "\\int x\\arcsec(a\\,x)\\,dx=\n \\frac{x^2\\arcsec(a\\,x)}{2}-\n \\frac{x}{2\\,a}\\sqrt{1-\\frac{1}{a^2\\,x^2}}+C", "2ec67ef0f41dc7965e85c6b29ca3d718": "\\sum_{j=0}^n {n\\choose j} j^k a_j.", "2ec72713b40d6d622e06b9e0700739fe": "Y/Y_n=(6/29)^3 \\approx 0.008856", "2ec72d87951c5a6da31617f3cff00eca": "f(k,0)\\approx k!/e", "2ec7597642b706974427a3ece46b8a8a": "h_1, h_2 \\in H", "2ec7b0a5090f94e8c50b6d19c0568964": "H^i(X, \\mathcal{F}) \\to H^i(X^\\text{an}, \\mathcal{F})", "2ec812f20a24aa286507f38437351bbd": "\\sum_i S(b_i) \\alpha c_i = \\varepsilon(a) \\alpha", "2ec8338b8442a3b167cc072f137ac1fa": "K_w = [\\mbox{H}^+] [\\mbox{OH}^-]", "2ec87703acc48f6456665a80f84b4c43": "A^{T}M + MA + N = 0", "2ec8c76406269ac4fa45142eadb52294": " \\frac{d}{dx}\\cos(x) = -\\sin(x).", "2ec93ad04a6b2c7cde1f94ff126bd435": " F = G \\frac{m_1 m_2}{r^2},", "2ec970507af90d1eaa44cea4361334c6": "\\gamma(0)\\gamma(1) \\neq 0. \\, ", "2ec9e24b18a9971b191a6a27dd2b37f9": "\\lambda_3=1-\\lambda_1-\\lambda_2\\, .", "2eca583a8c364e2895448b87187097b9": "\\lim_{x\\to c}\\frac{f'(x)}{g'(x)}", "2eca6d915cd0b089d78ae63f5be8f63f": "\n\\frac{\\partial}{\\partial p} L_p(x) =\n\\frac\n {\\sum_{j=1}^{n}\\sum_{k=j+1}^{n}\n (x_j-x_k)\\cdot(\\ln x_j - \\ln x_k)\\cdot(x_j\\cdot x_k)^{p-1}}\n {\\left(\\sum_{k=1}^{n} x_k^{p-1}\\right)^2},\n", "2eca7c5231dfb2cd53fc6aa1a12493bd": "\\mathrm{Tr}_{12} : V\\otimes V^*\\otimes V^* \\otimes V\\otimes V^* \\to V^* \\otimes V\\otimes V^*", "2ecb941784f16607362551b35ea29b82": " \\mathrm{d} \\mathcal{L} = \\sum_i \\left( \\frac{\\partial \\mathcal{L}}{\\partial q_i} \\mathrm{d} q_i + p_i \\mathrm{d} {\\dot q_i} \\right) + \\frac{\\partial \\mathcal{L}}{\\partial t}\\mathrm{d}t\n\\,.", "2ecb9478541a533c1cfc64b38566caf4": "p_j,\\,j\\neq k", "2ecbb5d9ddd35495e4562dccbd68a7dd": "E_f = \\frac{\\hbar^2}{2m}(3\\pi^2 n)^{\\frac{2}{3}} ", "2ecc4f573e6fe6b147b5bb1507eacd6f": "x'=x-vt,\\quad y'=y',\\quad z'=z,\\quad t'=t-\\gamma^{2}vx^{*}/c^{2}", "2ecc5ca16aefe2be33cb0460e0c7bd73": "x\\in f_{s}^{-1}\\left( \\left\\{y\\right\\} \\right)", "2ecc9a972c5c6127f18427ce75d3028d": "A[[X]]", "2eccc5330e830e62a4636f95a7dd3afe": "\\displaystyle \\operatorname{sinc}^2 (a x)", "2eccde1bd68d71eac37aad5979ee3393": "[\\mathtt{App}]", "2eccf08ea3db9388c5d5402fb2020986": "\\real\\,(z)\\leq 0", "2ecd3ca624fc3ede9f35c5bd1d80fa06": "\\displaystyle d^2=R(R-2r)", "2ecd44675c1a43bac6f3b474e5968242": " Ax=b ", "2ecd58e3486da1425757d6ef755901af": "\nu(z,\\bar z) = \\frac{1}{2} \n\\ln \\left(\n4 \\frac{ |{\\mathrm{d} f(z)}/{\\mathrm{d} z}|^2 }{ ( 1+K |f(z)|^2)^2 }\n\\right)\n", "2ecd627c4cd686f5e2e07076db5a90df": "\\mathcal{L}\\left[\\frac{df(t)}{dt}\\right] = s\\mathcal{L}[f(t)].", "2ecd80736f853941830c4e9807720fa6": " F_v/r ", "2ecd846d205c9218aa8a3e3354fa99b6": "\\Gamma'(x) = \\int_0^\\infty t^{x-1} e^{-t} \\ln t\\,dt", "2ecd8b1b850236b2139516e5a5b74bcf": " \\gamma_eT_e \\sum_i Z_i^2f_i[m_iv_s^2-\\gamma_iT_i]^{-1} = \\bar Z(1+\\gamma_ek^2\\lambda_{De}^2)", "2ecda7a0252b442ac6ecf47462119f51": "DM", "2ecdba6ee26c97a14552e556704f7344": "\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots=\\frac{1/2}{1-(+1/2)} = 1.", "2ecdde3959051d913f61b14579ea136d": "ABCDE", "2ece01538aea1bc9b8c570e123bd7b0c": "i_t = r_{t+1} + \\pi_{t+1} + r_{t+1} \\pi_{t+1}", "2ece13f663b3faa4106f240a319d2476": "\\mathbf{F}", "2ece3dd42aa1f0f7ff511ed9fe85f530": " \\langle \\psi(x) \\rangle := \\frac{\\text{Tr}\\ \\psi(x) \\exp^{-\\beta H}}{Z} ", "2ece450cf96088aad55a5ae474827293": "\\|x_i - x_j\\| \\approx \\delta_{i,j}", "2ece89940b9b715ca692f61ebb7a8f36": "kR_i \\rightarrow R_i,\\ \\mbox{where } k \\neq 0", "2eceefa14cf0e6dd9196eb7f9058da05": "|x_k|\\ge\\rho\\,|x_{k+1}|", "2ecf1173f9fec762ea48fab7cf42c37d": "H_0(j_{xy}) : H_0(M_x) \\rightarrow H_0(M_y)", "2ecf12b00ac24527c6c485c4d92cdfb8": "F\\equiv 1", "2ecf1db4d69e695ec08620bcdc39f024": "I = \\sum_{i} \\left | i \\right \\rangle \\left \\langle i \\right |", "2ecf2d99ce1ffd3603f5d1be02a8e9fc": "\\rho_\\text{bulk}", "2ecf573a20f2f8a4922205d9bd69a3b1": "\\theta = s - \\lfloor s \\rfloor \\in (0,1)", "2ecf6bfe981f130f092e7c4f4f60c8d6": "s=\\sqrt 2 \\,l", "2ecf723d20fc127143558d70637982e0": "\n\\begin{align}\n\\ln q_\\mu^*(\\mu) &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\{ \\sum_{n=1}^N (x_n-\\mu)^2 + \\lambda_0(\\mu-\\mu_0)^2 \\} + C_3 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\{ \\sum_{n=1}^N (x_n^2-2x_n\\mu + \\mu^2) + \\lambda_0(\\mu^2-2\\mu_0\\mu + \\mu_0^2) \\} + C_3 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\{ (\\sum_{n=1}^N x_n^2)-2(\\sum_{n=1}^N x_n)\\mu + (\\sum_{n=1}^N \\mu^2) + \\lambda_0\\mu^2-2\\lambda_0\\mu_0\\mu + \\lambda_0\\mu_0^2 \\} + C_3 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\{ (\\lambda_0+N)\\mu^2 -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\mu + (\\textstyle\\sum_{n=1}^N x_n^2) + \\lambda_0\\mu_0^2 \\} + C_3 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\{ (\\lambda_0+N)\\mu^2 -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\mu \\} + C_4 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\left\\{ (\\lambda_0+N)\\mu^2 -2\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}(\\lambda_0+N) \\mu \\right\\} + C_4 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\left\\{ (\\lambda_0+N)\\left(\\mu^2 -2\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N} \\mu\\right) \\right\\} + C_4 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\left\\{ (\\lambda_0+N)\\left(\\mu^2 -2\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N} \\mu + \\left(\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}\\right)^2 - \\left(\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}\\right)^2\\right) \\right\\} + C_4 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\left\\{ (\\lambda_0+N)\\left(\\mu^2 -2\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N} \\mu + \\left(\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}\\right)^2 \\right) \\right\\} + C_5 \\\\\n &= - \\frac{\\operatorname{E}_{\\tau}[\\tau]}{2} \\left\\{ (\\lambda_0+N)\\left(\\mu-\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}\\right)^2 \\right\\} + C_5 \\\\\n &= - \\frac{1}{2} \\left\\{ (\\lambda_0+N)\\operatorname{E}_{\\tau}[\\tau] \\left(\\mu-\\frac{\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n}{\\lambda_0+N}\\right)^2 \\right\\} + C_5 \\\\\n\\end{align}\n", "2ecfe5c0a5c1e0abeab782d5bc17f20e": "P=\\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix}", "2ecff1a0cbcfe8f0d736d974e1f403b5": "\\frac{1}{2} \\frac{\\partial^2 a}{\\partial \\tau^2} + i\\frac{\\partial a}{\\partial \\zeta} + N^2 |a|^2 a = 0 ", "2ecfff9e005a21b6a0da4eb3fd76b622": "\\hat{h}(\\xi) = \\hat{f}(\\xi-\\xi_{0}).", "2ed02e9385b333f65da18d5325dbec19": " \\mathbf{L} = \\begin{pmatrix} 1 & 0 \\\\ \\frac{-1}{f} & 1 \\end{pmatrix} ", "2ed03e1443770ccfdcb9aea3a8ad10b7": "T={B\\over { {\\ln{(R / r_\\infty)}}}}", "2ed058fb4666ce97af3cfab8c8793314": "(x^2-2az+a^2z^2)^2 = x^2+a^2z^2.\\,", "2ed06dffb2d589dd4302971f8abbb697": "H^I_i(H^{II}_j(C_{\\bull,\\bull}))", "2ed0714c5c8fb4dd9bd4bf11de9d865f": "{}314\\frac{64}{625}<100 \\times \\pi <314 \\frac{64}{625} +\\frac{105}{625}", "2ed0cf369a76e8f7e1e3a68dcd0566d2": "H_\\gamma (L) = Q(L)D_\\gamma (L)", "2ed0e7f3c3288f84bd7c72b6ac07ae03": "\\pi_k = \\begin{cases} \n \\frac{(\\lambda/\\mu)^k}{k!}\\pi_0 & \\text{for } k=1,2,\\ldots,c \\\\\n \\frac{(\\lambda/\\mu)^k}{c^{k-c} c!}\\pi_0 & \\text{for } k=c+1,\\ldots,K.\n\\end{cases}\n", "2ed1127e79d77effa603e00f561a507d": "t: \\ z^t", "2ed129d041b187ba81f20c2f1ac6a1dd": " f(x) \\le f(x_0) - \\eta ", "2ed1d83c95ccbda981a45d027c0d2e07": "(D^n_q f)(0)=\n\\frac{f^{(n)}(0)}{n!} \\frac{(q;q)_n}{(1-q)^n}= \n\\frac{f^{(n)}(0)}{n!} [n]_q!\n", "2ed1eb26d88f9828d6168dd9b35d9f17": " E_{ss,us} = y_{g} + \\frac{q_{ss}^2}{2gy_{g}^2}=0.5 + \\frac{10.0^2}{2(32.2)0.5^2}= 6.71 \\text{ ft}", "2ed1f4e2e0467e01823aaa7a659b04ba": "\\varepsilon > \\tau", "2ed2446c513337ceb3640a687f68f77e": "q_{max}\\ = C_{min} (T_{h,i}-T_{c,i})", "2ed25f2100838840e30910fbfe76aab1": "\\Delta_{i+1}^{\\rm P} := \\mbox{P}^{\\Sigma_i^{\\rm P}}", "2ed27c1fbb8d7d8b6008cffd635bdb5e": "|Q^{(j)}_b \\rangle", "2ed297e055867a508298cb2cd7b4234f": "\\cong_{\\mathcal{B},\\varepsilon}", "2ed2b5587d7508693e4d37cb6668cc62": "\\vec{N}", "2ed2f20b3b0461e2317a31477c07e4c2": " (\\;9) \\quad\\quad u_s\\left(w_1 - w_2\\right) = f \\left( w_1 \\right) - f \\left( w_2 \\right),", "2ed31542cf78b85b19398f2251355c82": " A^*\\cap B^* = A^* + B^* - A^*\\cup B^*", "2ed3207262d70fa38c763f50d805d12e": "ds^2=-dt^2+\\delta_{ij}dx^i dx^j", "2ed369f29972ca71dcdd798cc31b43f6": "I=\\mathfrak{m}", "2ed37640e8101bc21723dd39f48b062a": " \\delta >0 ", "2ed442760fccaad9ac18382348e1cb84": " \\int d^3 r \\ \\psi^* (\\boldsymbol{r}) \\psi (\\boldsymbol{r}) = 1 ", "2ed4477d8fa3f3d67239f2b3ed5d080a": "\\delta = (1 - R)(H_q^{-1}(1 - r) - \\varepsilon)", "2ed451096b64c09c632533f4e765f558": "\\mathbf{P}:=\\mathbf{I}-\\mathbf {X_1} \\mathbf {X_1}^+", "2ed46b5bd5e5ce4bb758acdef9459b4a": "\\mathbf{F}=\\int_0^\\infty \\! \\rho \\mathbf{f}\\,dz \\,=-\\int_P^0 \\! \\frac{\\mathbf{f}}{g}\\,dp \\,", "2ed4d61ea8ad21db718cbcb8911d9b76": "\\mathbb{Q}(\\sqrt{7})", "2ed4d85f77792edfc0747560dff502f2": "\\int_E f \\mu", "2ed51fdcfe451bdbad47ad7c97c5e1db": "H(z,u)= (1+z)^u = \\sum_{n=0}^\\infty {u \\choose n} z^n = \\sum_{n=0}^\\infty \\frac{z^n}{n!} \\sum_{k=0}^n s(n,k) u^k\n= \\sum_{k=0}^\\infty u^k \\sum_{n=k}^\\infty \\frac {z^n}{n!} s(n,k).", "2ed54438e45aa52d36d7beab0b9b5d1c": "c_{2,1}(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta) = \\alpha x \\beta \\gamma \\widehat{y} \\delta", "2ed5c95e59a882e4133a29e15fc80dda": " \\begin{align}\n\\frac{\\text{mean abs. dev. from mean}}{\\text{standard deviation}} &=\\frac{\\operatorname{E}[|X - E[X]|]}{\\sqrt{\\operatorname{var}(X)}}\\\\\n&\\approx \\sqrt{\\frac{2}{\\pi}} \\left(1+\\frac{7}{12 (\\alpha+\\beta)}{}-\\frac{1}{12 \\alpha}-\\frac{1}{12 \\beta} \\right), \\text{ if } \\alpha, \\beta > 1.\n\\end{align}", "2ed5e76fe394f621dcea2910cf09ffd4": "\\delta_{L/K} = \\{ x \\in O : x \\mathrm{d} y = 0 \\text{ for all } y \\in O \\} . ", "2ed627f47c310b840e0ce6557b738920": "\\int_a^b \\sqrt{\n\\frac{\\partial\\theta^j}{\\partial t}\ng_{jk}\\frac{\\partial\\theta^k}{\\partial t}} dt =\n\\sqrt{8}\\int_a^b \\sqrt{dJSD}", "2ed6d403859258d9cb1a9a2699b477ab": "1.4608", "2ed6d8953c244f0a44cf90c351794067": " P = { { R \\; T } \\over { v - b } } - { { a \\; \\alpha(T) } \\over { v ( v + b ) } }", "2ed6ee0904a3e8d3e7c4ae33fbdd7f18": "{I_{AB}}", "2ed70f19ed250ff944b86fdee9526208": "\\omega_{\\mu\\nu\\rho}", "2ed7dbc29c3e3d7b45be7d9df26619c0": "{ 1 \\over \\lambda (3-2q) } \\text{ for }q < {3 \\over 2} ", "2ed8031858da2babf786e10b1f4d7b65": "\\scriptstyle [x_0,\\ x_0+P],", "2ed8178fecf32061901d30dd923fbdb9": "m_{ij} = a_i b_j = (\\gamma a_i)(\\frac{1}{\\gamma}b_j)", "2ed894a6afbe7a02d5a862ed917a5f5d": " H = \\frac{1}{2m}\\nabla^*\\nabla ", "2ed8ede8a25086c7b151cbd339c79b59": "N(x) := \\sum_{j=0}^{k} a_{j} n_{j}(x)", "2ed906ac1d584554c5ae2a6a8ab22a39": "\n \\lambda := \\sigma_0\\left(\\cfrac{9R}{2\\pi\\Delta\\gamma{E^*}^2}\\right)^{1/3} = 1.16\\mu\n ", "2ed9867852ec4d6394b61f0a75a01448": "\\operatorname{arsinh} \\;u \\pm \\operatorname{arsinh} \\;v = \\operatorname{arsinh} \\left(u \\sqrt{1 + v^2} \\pm v \\sqrt{1 + u^2}\\right)", "2ed9b6ba085bcb5e658353dff4593f06": "\n\\mathbf{A} =\n\\begin{bmatrix}\n\\mathbf{A}_{11} & \\mathbf{A}_{12} & \\mathbf{A}_{13} & \\; \\\\\n\\mathbf{A}_{12}^* & \\mathbf{A}_{22} & \\mathbf{A}_{23} & \\; \\\\\n\\mathbf{A} _{13}^* & \\mathbf{A}_{23}^* & \\mathbf{A}_{33} & \\; \\\\\n\\; & \\; & \\; & \\ddots\n\\end{bmatrix}\n", "2ed9cc36151d1ee09b2dff41418574f7": " G = \\frac{d}{d+1} \\left( \\mathcal{I} + I \\right) ", "2eda2628825f5fb9b24a360b106b3f70": "\\begin{align}\n x_2 + 2x_3 &\\le 3\\\\\n -x_4 + 3x_5 &\\ge 2\n\\end{align}", "2eda3a55111349e4845fa3b4be1f759c": " \\frac{d}{dx}\\left(e^x\\right) = e^x", "2eda44af6cbd9eaf0398f80dfcc42173": "\\tfrac{2G(1+\\nu)}{3(1-2\\nu)}", "2eda571947c51dcb15d6a4dc9557f3a1": "\\frac{M-M_e}{M_0-M_e}=e^{-t/\\tau}", "2eda825edcfaace89115b6b490e7e1a7": "M_1f(x,y,r)=r^{1/2}\\cdot f(x,r)", "2eda97472b42ce2669d4e51f0daec106": "\\int\\limits_X |\\psi(x)|^2 d\\mu(x) < \\infty ;", "2eda9d106baf0241f9178e45c70c112a": "\nh(x,n) = \\left\\{\\begin{matrix}\nn+1 \\ (\\bmod \\ p) & x\\in S_0\\\\\n2n \\ (\\bmod \\ p) & x\\in S_1\\\\\nn & x\\in S_2\n\\end{matrix}\\right.\n", "2edab38918075cb4ac65f5ac3e42282f": "\n\\mathbf{X}=\\begin{bmatrix}\n X(z_0)\\\\\n X(z_1)\\\\\n \\vdots\\\\\n X(z_{N-1})\n\\end{bmatrix},\\quad\n\\mathbf{x}=\\begin{bmatrix}\n x[0]\\\\\n x[1]\\\\\n \\vdots\\\\\n x[N-1]\n\\end{bmatrix},\\text{ and}\\quad\n\\mathbf{D}=\\begin{bmatrix}\n 1 & z_0^{-1} & z_0^{-2} & \\cdots & z_0^{-(N-1)}\\\\\n 1 & z_1^{-1} & z_1^{-2} & \\cdots & z_1^{-(N-1)}\\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n 1 & z_{N-1}^{-1} & z_{N-1}^{-2} & \\cdots & \\cdots & z_{N-1}^{-(N-1)}\n\\end{bmatrix}.\n", "2edb5f999cf76f7860210f4d689cd8fd": "A_{proj} = \\pi r^2 \\cos{\\beta} ", "2edbd3324682a55c70e500cfd0400f82": "(i,t)", "2edbdfe07065082a6784b219cc5d298d": "x^2+y^2-1=0. \\, ", "2edbe3f824dec419df5440cb4b1656ef": "V_x=I_x ( r_O +R_2) +I_b (R_2-\\beta r_O)\\ , ", "2edc2e98d403ae9043ec1b4a733044e6": "\\Delta t = \\frac{2 L}{c}.", "2edcf2c4612bfb5e438c201cce6226e8": "\\mathrm{d}U=\\mathrm{d}U_{in}+\\delta Q-\\mathrm{d}U_{out}-\\delta W\\,", "2edcfc4318e413938de5706470d8f671": "dS=\\delta Q_{rev}/T = 0\\,\\!", "2edda0c3ecc162d1c5b91afab56938b9": "\\pi^* \\sigma=\\begin{pmatrix}\n 1 & 2 & ... & n & x & y \\\\\n 1 & 2 & ... & n & y & x\n\\end{pmatrix}.", "2eddb6bb7afbb7754bfdca873c7f20a0": "i: X \\to \\mathbf{P}^n_S", "2eddf7bfffeb4976a8663014c558a01b": "\\tfrac{8}{3}", "2eded1e35b0c17f37348bdd11e9dfeac": "\\cos(\\alpha + \\beta) = \\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta", "2eded61aea6c0bb658aca4dc65aef943": "{\\rm cof}(I)=\\min\\{|{\\mathcal B}|:{\\mathcal B}\\subseteq I \\wedge (\\forall A\\in I)(\\exists B\\in {\\mathcal B})(A\\subseteq B)\\big\\}.", "2edef963655ee7fe005294cd44599dc4": "(1-a) (1-b) c", "2edf0dab6191e8fe71ca51123a7b4493": "\\mathcal{P}(\\mathbb{R}^k)", "2edf3aaa190d248721811d6e139bc840": "{I_0}=4\\pi\\varepsilon_{0}\\cdot\\frac{m c^3}{e}\\approx 17 \\mathrm{kA}", "2edf615fcb6f083facf0c86806bd2640": "(1-d^n)y", "2edfa1b14f3f0ca7c758b9c4d2fb07a2": "s_{127} = 0", "2edfdf8464715517a9d7e4ddfcbaad4c": "H^0_n=a_1+a_2+\\cdots+a_n \\, ", "2edfe92216c6d681c18474f508e2cd21": " \\delta \\mathcal{S} = 0 ", "2ee056bcf2a1f3530dee3b21e76c8104": "Nx^2 \\pm\\ 1 = y^2", "2ee06e89e5ca519c5f93c7f9663774c6": "\\pi_i^\\text{pr} F", "2ee0c5f85dbfa6f21b565d770b3a751a": "\\vec{\\alpha}_{i}", "2ee105a0c4d2048443954de6b492183f": "\\int \\sqrt{a^2+x^2}\\,dx,", "2ee10963c085c4c5dd9d2c32594c0eb1": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 4.817221\\log_e(T+273.15) - \\frac {4563.180} {T+273.15} + 46.19124 + 4.829056 \\times 10^{-06} (T+273.15)^2", "2ee110ae07aef2d6695d7cc926a71a04": "\\frac{2.34\\times10^2}{5.67\\times10^{-5}} \\approx 0.413\\times10^{7} = 4.13\\times10^6 ", "2ee113c95e623cbe1fe11195b5572868": ", \\mathcal{P}\\ ,", "2ee1173490a9b553e3a40b2b02c5f125": "\\Gamma^\\alpha{}_{\\beta\\mu}\\,", "2ee129822848384f675ca544e898203b": "\\,\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\,", "2ee1a13e89d01fa3a9badefc5ce283aa": "q(x_1)=y_1", "2ee1fddffa5d7c14ebeda7565bb56517": "\n\\begin{array}{c|c}\n0 & 0 \\\\\n\\hline\n & 1 \\\\\n\\end{array}\n", "2ee20aecf992b557a752a595fbd1ba02": "I_{X_i\\leq x}", "2ee20c26f018f7dc8333058eed944a0c": "\nh(\\mathbf{Y})=-\\frac{1}{N}\\sum_{t=1}^N \\ln p_\\mathbf{s}(\\mathbf{y}^t)+\\ln|\\mathbf{W}|\n", "2ee20d4a60926b145e38db2795c622d3": " \\{ x\\in V: \\|x\\|\\le 1\\}.", "2ee20e444e6f615e9699f2298818e3b2": "\n\\mathrm{j}_\\pm = \\mathrm{j}_x \\pm i \\mathrm{j}_y. \\,\n", "2ee2128a0b47ab53bcfc1d37fec1c7e8": "\\displaystyle f'", "2ee220641fd11cb94fbb7511be281b28": "\\tau_*", "2ee231c85c20db8298f330dc972ccbdd": "\\vert \\Omega \\rangle", "2ee28ea5399bfa7563820d1ba14f8434": "\\frac{1}{1-p}=1+[COOH]kt=X_n", "2ee296ef516c124a624cc9ee9ff156fe": " i_S = i_R \\left( 1 + \\frac {R_f}{R_1} \\right) +\\frac {v_E} {R_1} \\ . ", "2ee2a45d525997f7c283a420327db1bc": "+V_{\\rm peak}", "2ee2c2e0907d29dcf0b7cb526252ecb0": "\\left\\{H\\cdot X_t:H{\\rm\\ is\\ simple\\ predictable\\ and\\ }|H|\\le 1\\right\\}", "2ee325f28bc3ee7c98b16aa105cdd6b1": "\\overline{\\mathbf{P}_{1}\\mathbf{P}_{3}}", "2ee33977851189f85f1afd91b4d9b009": "\\mathrm{^{238}_{\\ 92}U\\ \\xrightarrow [-2\\ \\beta^-]{+\\ 6\\ (n,\\gamma)} \\ ^{244}_{\\ 94}Pu}", "2ee33dddb58dae884c1450b635de7104": "\\frac{H^2}{P^2}=\\frac{2P^2-1}{P^2}\\, ", "2ee355d3c6f657edd9d805529ab3a3d1": "2^{15}", "2ee392ccc02795d7639dce9471180d5c": "x^{*} M x \\geq 0", "2ee3a4f64567a93f342cedcf0cb2abfe": "\\scriptstyle \\hat y", "2ee3b6f1fc6f954223949d26a0eb9816": " \\sum_{k=0}^{s-1} a_k = -1 \\quad\\text{and}\\quad \\sum_{k=0}^s b_k = s + \\sum_{k=0}^{s-1} ka_k. ", "2ee3bce7bab60f3e4a37da95cf16adba": "EMA_3(%D)", "2ee3e22403e8af3300510e5b2b59426a": "\n\\begin{pmatrix}\n a & -b \\\\\n b & \\;\\; a\n\\end{pmatrix}.\n", "2ee44a076695c45cffea9ae93869585f": "e_0={L^2}/{T_{M}c_{p}}", "2ee454a724a532505f61100e9c8730a7": "(s + \\vec{v})^{-1} = \\frac{(s + \\vec{v})^*}{\\lVert s + \\vec{v} \\rVert^2} = \\frac{s - \\vec{v}}{s^2 + \\lVert \\vec{v} \\rVert^2}", "2ee49f690c5c01c6babd2c2129e2d7be": " \\mathbf{A} = \\lim_{\\Delta t \\rightarrow 0} \\frac{\\Delta \\mathbf{V}}{\\Delta t} = \\frac {d \\mathbf{V}}{d t} = \\dot{\\mathbf{V}} = \\ddot{\\mathbf{P}} = \\ddot{x}_p\\vec{i}+\\ddot{y}_P\\vec{j}+\\ddot{z}_P\\vec{k}.", "2ee4e3630bf306a5f04018c3957bd6b1": "\\Pi (i \\omega_n )=-\\frac{1}{\\beta }\\sum _{i \\omega_m } \\frac{1}{i \\omega_m +i \\omega_n -\\epsilon }\\frac{1}{-i \\omega_m -\\epsilon '}=\\frac{1-n_F(\\epsilon )-n_F\\left(\\epsilon '\\right)}{i \\omega_n -\\epsilon -\\epsilon '}.", "2ee5ea8baaab29570ebbf0e89d40cbd4": "\\sqrt{\\frac{\\prod_j N_j!}{N!}}", "2ee61a24b719e24ee4f5932de77bc8ac": "\\chi_{0,0} = \\frac{1}{\\sqrt{2}} (\\chi_-(1)\\chi_+(2)-\\chi_+(1)\\chi_-(2))", "2ee65c04977b1768efe79923c05f3c1e": "U \\cap U'", "2ee675b91248990e73861297708e7f95": "[A][x] = [x]\\lambda", "2ee67ca083270461d81083c912e3e314": "{A^2}_1 = {R^2}_1 + {S^2}_1 = (C)-(D)-(F)-(H)", "2ee764dce6e046ae60622630203e9c01": "pn + mq = b. \\,", "2ee77d381f4028409120648f11d2f4b1": "\\Gamma_8' = \\left\\{(x_i) \\in \\mathbb Z^8 \\cup (\\mathbb Z + \\tfrac{1}{2})^8 : {{\\textstyle\\sum_i} x_i} \\equiv 2x_1 \\equiv 2x_2 \\equiv 2x_3 \\equiv 2x_4 \\equiv 2x_5 \\equiv 2x_6 \\equiv 2x_7 \\equiv 2x_8\\;(\\mbox{mod }2)\\right\\}.", "2ee793160fafe2165a81b8e7f718087c": "\\lambda(-\\ln(1-X))^{1/k}\\,", "2ee79d70f30f50dd0d19211d5dab8dce": "I( X ; Y = y)", "2ee79e3264f55d01cc3a75aaae2480f1": "\\rho(z)=|f\\,'(z)|\\,h(|f(z)|)", "2ee7d8105daa0b66c80bfe3d95045d4b": " S = \\int_1^2 C g^{\\alpha \\beta} \\acute{R}_{\\alpha \\beta} \\sqrt{-g}\\, d\\Omega + \\int_1^2 \\mathcal{L}_e \\,d\\Omega ", "2ee8048158e35abd45e0592f83a906b0": "\\langle 111 \\rangle", "2ee853a6c5914a261ae8d2145f0db7e0": "Z_\\phi (s,\\chi)", "2ee8be499c7d295b16064ace2e348754": " b = 2mn, \\, ", "2ee8c23751c7837342d4e5b1ce2e07be": "I_{\\mathcal Q}(1)\\in Q", "2ee94eab3aab29d5c2f7248eebe33d1d": "\\mathbf{p} \\approx m\\mathbf{v} \\, .", "2ee9510aeffa0f57dd7ee41112d1e57f": "\n\\begin{align}\n& (\\sin(x)^3 = \\cos(\\log(y)\\cdot x) \\vee b \\vee -x^2 \\geq 2.3y) \\\\\n& \\wedge \\left(\\neg b \\vee y < -34.4 \\vee \\exp(x) > {y \\over x}\\right)\n\\end{align}\n", "2ee9c56975e3128d869e532b823ba60f": "T \\equiv \\sum_{i=1}^k t_i (N_{1i} R_2 - N_{2i} R_1),", "2eea6d4188d589f22583e40c3146eece": "58.6 \\%", "2eeaa25c1d08888e7a55db040f1fcfcf": "\\mathbf{q} = \\begin{bmatrix} q_0 & q_1 & q_2 & q_3 \\end{bmatrix}^T", "2eeaf17c345937de75ea8dbe447ed9a4": "\\eta(-1/\\tau) = \\eta(\\tau)\\sqrt{\\tau/{\\rm{i}}}\\,", "2eeafa92a387aabfa4d1e3449bab5363": "\\Phi (a) = P_H \\; \\pi(a) | _H.", "2eeb0ff753793f2b3a2905f8379441e5": "\\langle x \\rangle = \\left\\langle \\sum_{i=1}^N S_i \\right\\rangle", "2eeb5818d5a553a49afd8e2743356abd": "|\\nabla K(x)|\\le\\frac{C}{|x|^{n+1}}", "2eeb912053c0285042d2cc04ca151ed7": "x \\vee (y \\wedge z) = (x \\vee y) \\wedge (x \\vee z)", "2eec0f1f2086272a9832272f298c8b55": " V(|r|) = \\begin{cases} 1, & |r| = 1 \\\\ 0, &\\text{otherwise.} \\end{cases} ", "2eec1e927b492aa9d8dbfcc078fa128c": "M_\\text{w}", "2eec3aebb7b773a20fabb9aecb51f438": "|\\operatorname{tr}|/2 = 1,", "2eed1c47c0645e47e29a093c1c50b6f5": "\\begin{alignat}{1}\nx^2 + 6x + 11 \\,&=\\, (x+3)^2 + 2 \\\\[3pt]\nx^2 + 14x + 30 \\,&=\\, (x+7)^2 - 19 \\\\[3pt]\nx^2 - 2x + 7 \\,&=\\, (x-1)^2 + 6.\n\\end{alignat}\n", "2eed40c44c1696160d04074665120a20": "O(\\log \\log n)\\,", "2eedfb87b3da0bac752e40441cd8d244": "{\\vec a} \\oplus m^{\\prime} \\oplus {\\vec c}", "2eee2a5461611e69ed2f1c4d0972b3b7": "\\mathbf{X}\\boldsymbol{\\beta}=\\Phi^{-1}{\\left(\\mu\\right)}\\,\\!", "2eee2deaf00c0ef2e9fc6095d43e4ba1": "\\frac{1-m^2}{m}Y.", "2eee46c036979e6568efe0026881312b": "\\Delta(fh) = \n\\delta\\,\\mathrm{d}fh = \n\\delta(f\\,\\mathrm{d}h + h\\,\\mathrm{d}f) = \n*\\mathrm{d}(f{*\\mathrm{d}h}) + *\\mathrm{d}(h{*\\mathrm{d}f})\\;", "2eeebd4f87a03678a18e651ea543103b": "\\frac{T}{2}a = \\int_{t_1}^{t_1+T}f(t)\\cos(kt)dt", "2eeecd72c567401e6988624b179d0b14": "sup", "2eef6f27603ff5e4fd64de43918abd34": " f(\\gamma)", "2eefafbad54a0eac83e6255147f52e4a": " I_p(\\rho, K)=\\frac{1}{2}{\\rm Tr}[\\rho^p, K^*][\\rho^{1-p}, K],", "2ef00d9e306608031cbb78ba45698987": "\\frac{P_r}{P_t} = G_t G_r \\left( \\frac{\\lambda}{4 \\pi R} \\right)^2", "2ef024b5e27c50b2fe584a4ce1c03145": "C_0 = 2^{-1.5} (4 \\sqrt{2} E ( \\sin\\pi/8) - (1 + 2\\sqrt{2}K(\\sin\\pi/8))^2 \\approx 1.86518.", "2ef03f4d7b9a6238dd1883b389a29f0a": " \\Phi(B) =\\#( B \\cap {N}), ", "2ef07ec3b75b9809a42c676eec179127": "\\langle \\varphi,f \\rangle", "2ef08ac31cf8b47ce5596bc9d8333233": "A \\to a, B \\to bB, C \\to cC", "2ef0deb28c4bbc1ce6c11e4cce7e75b1": "\\equiv ", "2ef1100a7f5455a00e534c192a625c06": "v_n\\in V_n", "2ef1226e9ac9f119917940f66349ddca": "E^u", "2ef16f8731bcffa244f6fe7af7ca2ed1": "T^3+4T^2+T-I_3", "2ef2139b76304d12658e1cbc711e7dfa": "E_\\mu \\{ \\mathrm{SURE}(h) \\} = \\mathrm{MSE}(h),\\,\\! ", "2ef22e1d04ed3cc4f03bd277c51b6db5": "\\begin{align}\nT_\\varepsilon(\\Omega)f(w) &= -\\frac{1}{\\pi} \\iint_{D\\backslash V_\\varepsilon} \\left [{\\varphi^\\prime(w)\\varphi^\\prime(z) \\over (\\varphi(z)-\\varphi(w))^2}f(z)\\right ]dxdy,\\\\\n T_\\varepsilon(D)f(w) &=-{1\\over \\pi} \\iint_{D\\backslash U_\\varepsilon} {f(z) \\over (z-w)^2}dxdy,\n\\end{align}", "2ef23f55a718e42e89515e134ea0ad7d": "(J_1)^2, (J_2)^2, J^2", "2ef24484fbfe6da2f983a5f4ef1c077e": "k(a_1, ..., a_m)", "2ef2633cc900debcd04d22ce5ea2095d": "r_2^2 = p^2 - 4q\\!", "2ef2cad963520e6734b5e598a2398768": " \\mathcal Z:=\\{\\{(x,y)\\in K^2 \\ | \\ y=ax^2+bx+c\\}\\cup\\{(\\infty,a)\\} \\ | \\ a,b,c \\in K\\}", "2ef2e37d476715a3de83e4f4f0ef516e": "\\beta \\le -3", "2ef2f8d1aa2de0bebf0cbd96a3854862": "\\frac{R_1}{R_1 + R_2} = \\frac{r_1}{r_1 + r_2},", "2ef2fbe9e002b175d4f71b2a93763fbc": "\\textstyle \\sin^2 x + \\cos^2 x = 1 ", "2ef31fc69c7005714b9547f2a96a6e69": "U_k", "2ef32a05666137413fba2bd55955ba30": " \\equiv C_L = \\dfrac{L}{qS} = - \\dfrac{1}{S} \\int_\\Sigma [ (-C_p) \\mathbf{n} \\bullet \\mathbf{k_w} + C_f \\mathbf{t} \\bullet \\mathbf{k_w}] \\,d\\sigma ", "2ef378e1ce7a42e9647773d9c44195b6": " \\mathbf{W}_{N \\times 1} ", "2ef38136c0d57cffe11761c1b40929fe": " \\lambda {1\\over k_1^2} {1\\over k_2^2} {1\\over k_3^2} {1\\over k_4^2}. ", "2ef38ffc2f2572af44bbb49534498e10": "\\lambda_{q+1} = \\lambda_{q+2} = \\cdots = \\lambda_p = a,", "2ef3b9c9ae290bb1868e0a3e1ab6dfdd": "F\\subseteq Q", "2ef3ba44cafdfb4a1510393b952efe24": "f \\in \\pi_n(X)", "2ef3d4ceeec117a4b7c42e6172569125": "f_i=\\alpha+\\beta q_i \\,", "2ef3f535dc07358ddaf47d3c58052d69": "n_2 < 0", "2ef4317e6ba671f8799e7394dcb42d7c": "\\Bigg(\\frac{q^*}{p}\\Bigg)_4= \\Bigg(\\frac{\\sigma(b+\\sigma)}{q}\\Bigg).", "2ef44e7d67c5ec8d07da54d4dcc036b5": "x_{\\text{min}} \\le x_0 + u \\Delta x \\le x_{\\text{max}}\\,\\!", "2ef493cd473660c75023be18c8b74dba": "{\\text{engine}}\\;\\overset{\\textstyle\\tau}{\\underset{\\textstyle\\omega}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;\\text{wheel}", "2ef4cf36178cc4e35fe4e9e9bbf3a956": "\\mathcal{C}(X,\\mathbb{R})", "2ef4e7b2689a9fa0211f56b1ed5061e5": " \\mathbf{B}\\cdot\\mathrm{d}\\mathbf{S} \\,,", "2ef5340982b6840b0b89a0749ebdc5c5": "SD_\\text{within}:=\\sqrt{\\frac{SS_\\text{within}}{df_\\text{within}}}=\\sqrt{\\frac{(n_1-1)SD_1^2+(n_2-1)SD_2^2}{n_1+n_2-2}}.", "2ef57369a0a4d90017043f0958f622e8": "\\sqrt{2} - 1", "2ef57395074d8649f1112b723549f990": "\\alpha_c\\,", "2ef5f021c7bc135c267a5f8a91a6e553": "y^2=1-2r^2+x^2", "2ef662a28d24d926771929b6b384dc01": "\\Delta^0_n \\subsetneq \\Pi^0_n", "2ef6b1cb9daa23632b95e184762622c3": "t=\\frac{-(2\\mathbf{v}\\cdot\\mathbf{d})\\pm\\sqrt{(2\\mathbf{v}\\cdot\\mathbf{d})^2-4(\\mathbf{v}^2-r^2)}}{2}=-(\\mathbf{v}\\cdot\\mathbf{d})\\pm\\sqrt{(\\mathbf{v}\\cdot\\mathbf{d})^2-(\\mathbf{v}^2-r^2)}.", "2ef6bcead12d47a27fa38e53923ee71d": "I_n= \\frac{x}{2a^2(n-1)(a^2-x^2)^{n-1}}+\\frac{2n-3}{2a^2(n-1)}I_{n-1}\\,\\!", "2ef76f67dca19b9e867e95a0219cc661": "\\left({\\frac{\\alpha}{\\beta}}\\right)_n \\left({\\frac{\\beta}{\\alpha}}\\right)_n = \\prod_{\\mathfrak{p} | n\\infty} (\\alpha,\\beta)_{\\mathfrak{p}} \\ . ", "2ef78cf9724ca9bdf706ec164d6b6323": "\\zeta(1) = 1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots = \\infty;\\!", "2ef7b605e032266fbb00099b1d102ffd": "\\mathbb{E} = \\sum_x\\Pr[X = x]", "2ef7dd926040acb2ba83d2aec8fa783f": "\\mathbf{f} = \\epsilon_0 \\left(\\boldsymbol{\\nabla}\\cdot \\mathbf{E} \\right)\\mathbf{E} + \\frac{1}{\\mu_0} \\left(\\boldsymbol{\\nabla}\\times \\mathbf{B} \\right) \\times \\mathbf{B} - \\epsilon_0 \\frac{\\partial}{\\partial t}\\left( \\mathbf{E}\\times \\mathbf{B}\\right) - \\epsilon_0 \\mathbf{E} \\times (\\boldsymbol{\\nabla}\\times \\mathbf{E})\\,", "2ef831ed77dd8fc500bfaf0d0f5fb63f": "h\\left(n_1,n_2\\right)=a(l_1,l_2)-\\sum_{k_1=0}^{K_1-1}\\sum_{k_2=0}^{K_2-1}b(k_1,k_2)h(n_1-k_1,n_2-k_2)", "2ef851d71a120211aec5fda06537b4e1": "\n \\mathbf{v}(x, y, z) = \\frac{A}{x^2 + y^2 + z^2}\\begin{pmatrix} x \\\\ y\\\\ z \\end{pmatrix}, \\qquad\n p(x, y, z) = -\\frac{A^2}{2(x^2 + y^2 + z^2)}.\n", "2ef89d27af74a98995d341c77088df1c": "z=v", "2ef8acedb21ce40a66e29c09dbfb321b": "H_1 (X) = - \\sum_{i=1}^n p_i \\log p_i. ", "2ef967ec5b83906bf630d43b33aa1e68": "X_F(t^{}_n,j)=X_P(t^{}_n,k^{}_j)", "2efa106881029b2a0f21eef463482911": " F(x) = \\begin{pmatrix}\n F_1(x), & F_2(x), & \\ldots, & F_n(x)\n \\end{pmatrix}^T ", "2efa5dd4167221acc849bfd7433e454a": "\\mathbf{w}_{(k)}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{arg\\,max}} \\left\\{\n\\Vert \\mathbf{\\hat{X}}_{k - 1} \\mathbf{w} \\Vert^2 \\right\\}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\left\\{ \\tfrac{\\mathbf{w}^T\\mathbf{\\hat{X}}_{k - 1}^T \\mathbf{\\hat{X}}_{k - 1} \\mathbf{w}}{\\mathbf{w}^T \\mathbf{w}} \\right\\}", "2efaa3f2c4d77b939575a7aa5e79647c": "S_k = K_{k,1}", "2efad51919428efa5a473ec6e1c9aa8b": "i\\in \\{1,\\ldots,n\\}", "2efb7861e1fbccfcffacd16000245c75": "H(x(t))", "2efc454b87bb11d1d130c6936dd87834": "\\overline{dz}", "2efc5b4de3c60a1b9367cbc8ff7fb891": "\nc_{p^k}(n) = \n\\begin{cases}\n0 &\\mbox{ if }p^{k-1}\\nmid n\\\\\n-p^{k-1} &\\mbox{ if }p^{k-1}\\mid n \\mbox{ and }p^k\\nmid n\\\\\n\\phi(p^k) &\\mbox{ if }p^k\\mid n\\\\\n\\end{cases}\n.", "2efcb72e2f1725138189d77db2512147": "S= (EG-F^2)^{-1}\\begin{pmatrix}\neG-fF& fG-gF \\\\\nfE-eF & gE- fF\\end{pmatrix}.", "2efd8b7dd62f37f7d1d3a8a485b7470b": "v(c)=f(\\mu(c))", "2efd8f54195894da6e54f9dddd12edac": "\\Delta > 0", "2efd923ff714e805035f50f75aeed596": "X=x", "2efdb23e773774076bf2d20bbfa22929": "x_{18}", "2efdd4a79835dcc31a45c73e41a872a6": "\\mathbb{A}_{n\\geq 5}", "2efe3bbe88d2709b2e6d601a4ba2f95c": "x=e(x_1,x_2)", "2efe9531de025277dab67e789ad5b092": "\\ v_i", "2efee8ec2f5d284440a330615ed3cfe7": "z = z_1", "2eff51a5fc933b24e34c5617176696c9": " \\sigma Q_i v_i", "2eff5f1ede1c72d90e0b81584d067010": "t=(p_1-p_3)^2=(p_2-p_4)^2 \\,", "2effa881c50a8d1f71111b1a8f579459": "\n \\sum_{m_1m_2} \\langle J M|j_1 m_1 j_2 m_2\\rangle\n \\langle j_1 m_1 j_2 m_2|J' M'\\rangle\n = \\langle J M | J' M'\\rangle \n = \\delta_{J,J'}\\delta_{M,M'}.\n", "2f00329bbd468cd2b760883a4f0b96a0": "z_{t+1}", "2f0039f2d3c0dbfc10ecb2184e4e49ad": "T\\equiv T_1+i T_2,", "2f0070be52c3ebc87a0422be498576c9": "\\Delta M_i^{-1} = - \\alpha \\sum_{n=1}^N D_i \\left[ n \\right] \\left[ \\sum_{j \\in C \\left[ i \\right]}^{} F_{ji} \\left[ n-1 \\right] + Fext_i \\left[ n^{-1} \\right] \\right]", "2f00bc1fcbee85773a65ed059048d18b": "\nf(x,y; \\alpha,\\beta)=\\left(\\beta-1\\right)\\left(\\pi\\alpha^2\\right)^{-1}\\left[1+\\left(\\frac{x^2+y^2}{\\alpha^2}\\right)\\right]^{-\\beta} , \\,\n", "2f010820fb5b594a7f9d90920b7b9fdd": "\\bar{u_i}", "2f010926e1c636d66426f78f45deaab0": "s\\leq \\max S", "2f015aa93430777bf8f689ca974a1658": "(\\eta_z^{(i)},\\eta_z^{(j)}) = {\\rm Im}\\, M_{ij}(z)/ {\\rm Im}\\, z, ", "2f015fa72632d445c7158d20bec5d898": "K^{(N)}_{ij}", "2f01942cdbaa563d2d5b83f0f178cf1c": "\\iint_D\\left(\\nabla\\cdot\\mathbf{F}\\right)dA=\\oint_C \\mathbf{F} \\cdot \\mathbf{\\hat n} \\, ds,", "2f01ab97b69082f4191eb984ed342ff1": " i(t) = D_1 t e^{-\\alpha t} + D_2 e^{-\\alpha t} \\,", "2f024379b1541e5c5444516896df5bed": "= (7 * 64) + (5 * 8) + (6 * 1)", "2f0266fe7db543942f057380daec11c8": "{\\hbar(1+1.141)}/2", "2f02abb8832195d9948e8f21bf5dbcb4": "e^{-\\sqrt t r}", "2f02b6a9ad218b5ed3694705328cdf80": "q'\\ =\\frac {y_2-y_1}{x_2-x_1} +(1-2t)\\ \\frac {a\\ (1-t) + b\\ t}{x_2-x_1}\\ +\\ \\ t\\ (1-t)\\ \\frac {b-a}{x_2-x_1},", "2f02e9a245652195f34fe31e959c00b8": "Y_t(u) = T+Y_c(u)", "2f038e682252b60eda8c05107afb795d": "0 = h (\\sigma_x^2+\\sigma_w^2) + cm - \\sigma_x^2", "2f03abce3afa321f6461ad6f618efc1b": " \\lim_{y \\to b} E(u(X)|X>y) = u(b) ", "2f03e7e900b792466136c80ac8cb2e37": "\\beta = bi\\,", "2f03e9f5fdee5cc973a5e31d135d2dc7": " s(E) = \\prod_{i=1}^{k} \\frac {1} { 1 - x_i } = s_0 + s_1 + \\cdots ", "2f04054630786a2354fea2085f46c3a1": "C_3 = 0", "2f041a5a52cd42b266ce43bcb3c4521b": "\n \\int_0^{2\\pi} d\\alpha \\int_0^\\pi \\sin\\beta d\\beta \\int_0^{2\\pi} d\\gamma\n D^J_{MK}(\\alpha,\\beta,\\gamma)^\\ast D^{j_1}_{m_1k_1}(\\alpha,\\beta,\\gamma) D^{j_2}_{m_2k_2}(\\alpha,\\beta,\\gamma)\n = \\frac{8\\pi^2}{2J+1} \\langle j_1 m_1 j_2 m_2 | J M \\rangle \\langle j_1 k_1 j_2 k_2 | J K \\rangle.\n", "2f0421c5c1336ab8b65f5187d53b17e3": " H_0 |k\\rang = E(k)|k\\rang ", "2f045f57e3d498b71d617ae668fdddf1": "N = p^2q", "2f047dc92216c346791c9298c09c8c6d": "[x,y]=-(-1)^{ij}\\,[y,x]", "2f0488c0232021c86f3e9505c68a1258": "\\cos(\\beta x)", "2f04c3384b44e5e25de482ea73b39484": "\\lim_{\\Delta \\rightarrow v (\\Delta)} h_{s_k} = I - A_c s_k^c", "2f04e4d019e8facbb7d21cfb033864aa": " \n \\cfrac{\\Gamma \\vdash \\Delta, A \\qquad A, \\Sigma \\vdash \\Pi} {\\Gamma, \\Sigma \\vdash \\Delta, \\Pi} \\quad (\\mathit{Cut})\n ", "2f051c2fa40be7903872cf5b777edebe": "K_1 e^{-s_1t}", "2f057445b853031aa0d93c1eebe1008d": " H(s) = \\frac { s C R } {C^2 R^2 s^2 + 3 C R s + 1 } ", "2f057e26c504d5495289d9a064cfdf1a": "n(n-1)(n-2)\\dots(n-k+1)=n^k+O\\left(n^{k-1}\\right)", "2f059c493dcff9f1e30f6fc347a33576": "\\{(n_1,\\dots,n_k)\\in \\mathbb{N}^{k}| n_1+\\cdots+n_k=n\\}.\\,", "2f05d1a0d684f9d829293470b820a1a7": "\\left(-4\\sqrt{\\frac{2}{5}},\\ -2\\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ 1\\right)", "2f05d1d2d14c256537ed59496cacfa51": "y^3 = x.\\ ", "2f05f4451a5b8d0815fad1bc767707ea": "5 \\Delta x/\\Delta t = 135 \\text{km/h}", "2f06403d01ceb5677b97d9dc8bf56768": " R(K_X) :=\\bigoplus_{d\\geq 0} H^0(X,K_X^d). ", "2f06c56d1fe8171eaa853f4d24dff6f8": "\\clubsuit", "2f06e32c126aa764b16757966d6e3f81": "h\\sim{2 \\times 10^{-13}/\\sqrt{\\mathrm{Hz}}} ", "2f0713c13087a777ee1bc0fd2e697b5f": "\\{\\partial_1,\\ldots,\\partial_n\\}", "2f0765c21d34c467f4109beb50868898": "\n\\delta\\mathbf{Q} = 2\\pi r\\delta r \\times \\rho U_{\\infty}(1 - a) \\times 2a'r\\omega\n", "2f07a2b7456726273321fc410c1288b1": " \\cos(iy) = {e^{-y} + e^{y} \\over 2} = \\cosh(y) ", "2f07a99121fd21f57408aa0187b4bb0e": "\\mathbf{r}(0)", "2f07bdeb071c9881a9acc1b2e751e3b5": "r \\ge 2^b-M", "2f0822e2d6976a5ec59a8ef99549e978": " D = \\frac {var_{ obs } } { var_{ bin } } = \\frac{ s^2 } { n p ( 1 - p ) }", "2f08ed5fe85daf647a4f0e9fbd01a9eb": "G(e^t) = E[e^{tX}] = M_X(t).\\,", "2f08f3d7d8585e8e05daa35140fd116e": "x^{\\prime}=\\gamma(x-vt),\\quad y^{\\prime}=y,\\quad z^{\\prime}=z,\\quad t^{\\prime}=\\gamma\\left(t-vx\\right)", "2f09314669e7363150274bc88f59b2d8": "\\begin{alignat}{5}\n2x &&\\; + \\;&& 3y &&\\; = \\;&& 6 & \\\\\n4x &&\\; + \\;&& 9y &&\\; = \\;&& 15&.\n\\end{alignat}", "2f09635197f07c5ddd7da62006c6d643": "v/s", "2f098b0d3fa827d2d238289591b85518": " K^{\\cdot,\\cdot} = C^\\cdot \\left (\\mathfrak g,\\Lambda^\\cdot {\\mathfrak g} \\otimes C^{\\infty}(M) \\right ) = \\Lambda^\\cdot {\\mathfrak g}^* \\otimes \\Lambda^\\cdot {\\mathfrak g} \\otimes C^{\\infty}(M). ", "2f09aaf768ccfe16c119fb5a287b6ef9": "e^{\\Lambda t} = \\begin{pmatrix}\ne^{\\lambda_1 t} & \\ldots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\ldots & e^{\\lambda_4 t}\n\\end{pmatrix}\\,.\n", "2f09ad9ad5dfd17e4eb2620875c77ed3": "n^2 = 1 - \\frac{X}{1 - iZ - \\frac{\\frac{1}{2}Y^2\\sin^2\\theta}{1 - X - iZ} \\pm \\frac{1}{1 - X - iZ}\\left(\\frac{1}{4}Y^4\\sin^4\\theta + Y^2\\cos^2\\theta\\left(1 - X - iZ\\right)^2\\right)^{1/2}}", "2f09bcf2e08210fc52aec614250d1e86": "\\mathit{EQ_1}", "2f09db097b32b256d7d84f73089634a1": "\\mathbf{g}(n)=\\mathbf{P}(n)\\mathbf{x}(n)", "2f0a0e05f00409ed7a46235cdb8516eb": "D(D/R)^2/24", "2f0a1b4a5621734e7cc46a559cde69f7": "A = B \\triangle N = \\left( B \\setminus N \\right) \\cup \\left( N \\setminus B \\right).", "2f0a4c76155cd82c4709ca3fea9d0037": "A(x) \\, dx", "2f0a6a428c6d7e45bb354ffef13cdf41": "cs_5 = \\sqrt{\\gamma \\frac{P_R}{\\rho_R}}", "2f0a85a055a85d5e03b6f73d21e94ba3": "\\mathcal{H}^q(E)", "2f0a9ec6a288dae6f2367d245b3cb84d": "\\mathrm{d}U = T\\, \\mathrm{d}S - p\\, \\mathrm{d}V + \\mu\\, \\mathrm{d}n \\!", "2f0ab0ed23060389d74294c9b6b6a6d7": "\\mathrm{hub}(p)", "2f0b03b240e8913a8ae254bd58052826": "b_t\\;", "2f0b37ba8b2d3d501682fcc3e7da528e": " \\mathbf{J}_\\mathrm{f} ", "2f0b41e681de598490b2c44b70e3013e": " {n \\choose k_1, k_2, \\ldots, k_m} = \\frac{n!}{k_1!\\, k_2! \\cdots k_m!}", "2f0b5fa4291e82be36ad77dd4983ed1c": "B = -\\frac{1}{3}", "2f0b693e9bd129a00b9a9d9515b7c1bb": "I \\subset J", "2f0b9cae9f2166c4d583a98a70cea6ee": "\\operatorname{Var}(X-Y)=\\operatorname{Var}(X)+\\operatorname{Var}(Y)-2\\, \\operatorname{Cov}(X,Y),", "2f0be39f4156f4a7d751a5fc8bf2a864": "Z(G^\\prime) = \\frac{F_G(G^\\prime) - \\mu_R(G^\\prime)}{\\sigma_R(G^\\prime)}", "2f0bedab8b4287d2e518a5766f16ce83": "I = \\int_\\Omega |\\phi_y\\rangle\\, \\langle\\phi_y|\\,dy", "2f0c3563c84bfdf4d60967301430d68d": "\\rho\\sum_{i=1}^n\\left(\n\\frac{\\partial\\dot{q}_i}{\\partial q_i}\n+\\frac{\\partial\\dot{p}_i}{\\partial p_i}\\right)\n=\\rho\\sum_{i=1}^n\\left(\n\\frac{\\partial^2 H}{\\partial q_i\\,\\partial p_i}\n-\\frac{\\partial^2 H}{\\partial p_i \\partial q_i}\\right)=0,", "2f0ccf590b3e916a03ea2288ed377508": "S \\, \\subset \\{1,\\ldots, n\\}", "2f0cdcd5b49a4859bdba3970594c14d0": "[x_1, x_2] + [y_1, y_2] = [x_1+y_1, x_2+y_2]", "2f0ce6ac51452ced9d0007c99e5615f2": "x^{-2} \\cdot 2^2 = \\frac{4 \\sqrt{5}}{5}", "2f0d10de3d02e595536ef7f3523afebf": " \\tau = \\tfrac{1}{2} \\sigma^2 (T - t) ", "2f0d18d16c9ba3064a36dae0f3553bf9": "\\operatorname{Vec}(A, n)", "2f0d2a8ed3fe34fbb004716e631b2e4c": "a(v, v)", "2f0d7e9425dae7170314e146546f454e": "[0,1),", "2f0d90842a5e7016704ac23524f09c7e": "l_a=(-\\frac{G}{2},1,0,0)\\,,\\quad n_a=(-1,0,0,0)\\,,\\quad m_a=\\frac{r}{\\sqrt{2}}(0,0,1,\\sin\\theta)\\,.", "2f0e0965e5c54f9a615e577bce1c3909": "H = -t \\sum_{i} \\sum_{\\sigma} \\left ( \\left [ 1+ \\left(-1 \\right)^i \\delta_d \\right ]c^{\\dagger}_{i,\\sigma}c_{i+1,\\sigma}+ h.c \\right)+ U \\sum_i n_{i,\\uparrow}n_{i,\\downarrow} + V \\sum_{i}n_i, n_{i+1} ", "2f0e239780b886c14d65b4f6d48817ef": "u,v:j\\to i", "2f0e539f1e3af5e2f4ecfb6aa12dd4d5": "\\mathbf{\\hat{H}}_e = - \\frac{\\nabla^2}{2} - \\frac{\\mathbf Z}{r}", "2f0e805d5975706769672cc8ab14518d": "\\phi(n) = \\begin{cases}a&\\mbox{if }n=1;\\\\ b&\\mbox{if }n=2;\\\\ c&\\mbox{if }n=3.\\end{cases}", "2f0e8cf956b7e2b7e8db5af3b15386f5": "R_{max}", "2f0eccd16db10e247c58e45f24136fca": "e^\\pi \\approx 23.14069263277926900572908636794854738\\dots\\,.", "2f0f0d332cce3eee13ec201dfd7d3418": "[\\phi(x),\\phi(y)] = 0, \\ \\ [\\pi(x), \\pi(y)] = 0, \\ \\ [\\phi(x),\\pi(y)] = i\\hbar \\delta(x-y).", "2f0f0dcf2a0cb52426a857426fff7b02": "|\\Omega | ={p^km \\choose p^k} = m \\prod_{j=1}^{p^k - 1} \\frac{p^k m - j}{p^k - j} = m \\prod_{j=1}^{p^{k} - 1} \\frac{p^{k - \\nu_p(j)} m - j/p^{\\nu_p(j)}}{p^{k - \\nu_p(j)} - j/p^{\\nu_p(j)}} ", "2f0f47604e39c7f85fcc594add8066a9": "\\scriptstyle 2 \\,\\times\\, 2", "2f0f5aac4a9e7231295024559fa30be6": "\\int_0^{T_r}\\,dt = -\\int_{2M}^0 \\left ( \\sqrt{\\frac{2M}{r}} \\right )^{-1}\\,dr. \\, ", "2f0f95a7e071a7bd98131a903340956e": "\\forall r,s \\in R, m,n \\in M", "2f0fbb1c781344500e56e475ef1cb105": " \\frac{d\\sigma}{d\\Omega} = \\frac{b}{\\sin{\\Theta}} \\left|\\frac{db}{d\\Theta}\\right| ", "2f1048f2b8edbdf8641402e5618408c2": "R_{ab}l^a l^b=8\\pi \\cdot T_{ab}l^a l^b=8\\pi \\cdot T^a_{b} l^b\\cdot l_a\\,\\hat{=}\\,0", "2f105ea570cbe3c0917895d4fee463dc": " U_\\rho <\\rho> ", "2f10d2f247d0850ccd1aa3e6f0f37582": "\\sum_{n=0}^\\infty\\,\\frac{F_n}{k^{n}}\\,=\\,\\frac{k}{k^{2}-k-1}.", "2f10ea6aea5bc925aec714682983ccfa": " P(X \\ge \\varepsilon) \\le e^{ -t \\varepsilon } \\operatorname{ E } (e^{ t X }) ", "2f110987562cc2bd69e1cc023e6e621a": " a_1 = |\\mathbf{a}| \\cos \\theta ", "2f11314baa0a92d77c77d25530b99e03": "\\frac{V_1}{V_2}=\\frac{N_1}{N_2} = a ", "2f113201ac91d30faf4d135f8dfebf09": " \\|A\\|^2 = \\sup_{x\\neq 0}{\\frac { \\langle Ax, Ax\\rangle }{ \\langle x,x\\rangle }}\\,; \\qquad \\mu(A) = \\sup_{x\\neq 0} {\\frac {\\real\\langle x, Ax\\rangle }{ \\langle x,x \\rangle }} ", "2f1136d4d82cfaf9a7979b0cee0764c1": "T=+\\left(n_\\text{t}-n_\\bar{\\text{t}}\\right)", "2f113724f466d9a3e4072e31607fb879": "\\Delta/2", "2f118d046b0db6da8bf37d3e55371f34": " \\rho(t) = \\cos(6\\pi t)[\\cos (2\\pi t)+ i\\sin(2\\pi t)], 0\\leq t \\leq 1 ", "2f11c3f4b4fceb89653e4509a20b10f2": "k \\sqrt{2} ", "2f11c56a6040784a45c3cdc7eb66ff09": "\\mathbf B_d = \\left( \\int_{\\tau=0}^{T}e^{\\mathbf A \\tau}d\\tau \\right) \\mathbf B = \\mathbf A^{-1}(\\mathbf A_d - I)\\mathbf B ", "2f11cc19f234981318d09524adfc875d": "t \\equiv x^0", "2f11f730f53fe6d8d92e473f4075f649": " \\mathbf{J}(\\mathbf{r}, t) ", "2f11fed8279bc324ed5eb4be270b86d6": " \\sum_{k=1}^{\\infty} \\frac{x^{-2k}}{-2k} = \\frac{1}{2} \\log ( 1 - x^{-2} ). ", "2f12795f442051ab47f823c320abff9f": "\\mathbf{A}\\left(\\mathbf{r}\\right)=\\frac{1}{4\\pi}\\int_{\\text{all space}}\\frac{\\boldsymbol{\\nabla}'\\times\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'", "2f130a43f7767a4629ca1e4d2a430c37": " [EI] = \\frac{K_m[I][ES]}{K_i[S]}", "2f1318b6e808dfea3ff04d77483de0d0": "\\sqrt{x^2 + y^2 + z^2} = vt \\,", "2f132768c18981e5a5c50426ec1cd128": "O(l)", "2f133bdfe58cb3bcf78d05e41da08f79": "L = \\lim_{\\boldsymbol{x} \\rightarrow \\boldsymbol{a}} f(\\boldsymbol{x}), ", "2f13d0a85e60e8eb5b1029cc63548f8d": "\n\\kappa = \\frac{G(x)^3 Q(x^5)^2} {H(x)^2}\n\\prod_{n\\ge 1} \\frac{(1-x^{3n-2})(1-x^{3n-1})} \n{(1-x^{3n})^2}\n", "2f13ee8ada859b15d32322c4be3aee82": "\\cos(\\theta)", "2f1432a210cd602667d980061a2a0dcd": "A = \\left( \\frac{\\sqrt{25 + 10 \\sqrt{5}}}{4} + 5\\frac{\\sqrt{3}}{4} \\right) a^2 \\approx 3.8855\\,a^2.", "2f144486818b893c7c74da995adc560a": "\\Box (p \\to q) \\to (\\Box p \\to \\Box q)", "2f146a0d0d33f3d6673382174e206e4e": "\n \\begin{align} \n \\hat{\\alpha} & =\\frac{nm_1-m_2}{n(\\frac{m_2}{m_1}-m_1-1)+m_1} \\\\\n \\hat{\\beta} & =\\frac{(n-m_1)(n-\\frac{m_2}{m_1})}{n(\\frac{m_2}{m_1}-m_1 - 1)+m_1}.\n \\end{align}\n", "2f147dd200398b0f9bff6f2901b030da": "(16)\\quad Z^c\\nabla_c \\omega_{ab}=-\\frac{2}{3}\\theta\\omega_{ab}-2\\sigma^c_{\\;[b}\\omega_{a]c}\\;.", "2f15045b499216c6abeeef74e4582402": "\\Delta\\gamma", "2f15176e5978f7240649952660c4b7fb": " \\frac{1+zf(z)}{1-zf(z)}=F(z)=\\int\\frac{e^{i\\theta}+z}{e^{i\\theta}-z}d\\mu.", "2f153a7322a27e38f4b5e2527466e8c8": "\\theta(\\psi) = \\theta_r + \\frac{\\theta_s - \\theta_r}{\\left[ 1+(\\alpha |\\psi|)^n \\right]^{1-1/n}}", "2f154fb2062bd1ed26e21ecd9d1d2ee3": "{D}_4", "2f15517fb82b7aaf2d245437f8837341": " U = \\left(\\begin{array}{c} u^1 \\\\ u^2 \\end{array}\\right)", "2f155ec638e3538af4927d414fa8306e": "d=\\lambda/(4n_{\\rm coating})", "2f1576df4876b396778b73d78dced590": " K\\cdot K ", "2f15b9470f80a609ad69abda82d23543": "\nx_3=x_1+x_2 \\Rightarrow x_3 \\in [6,\\infty ] \\cap ([-\\infty,5]+[-\\infty ,4]) =[6,\\infty ] \\cap [-\\infty ,9]=[6,9]. \n", "2f15ea6c7074be52a1fb8f71aa8c9f58": "t = n - 1 \\ ", "2f162f3197d7429564a18adab0cb7e0e": " y_i = \\alpha+\\beta x^{*}_{i} +(\\varepsilon_i - \\beta\\nu_i) ", "2f1649f36959dc83be9f132d10a9d0db": "f^{\\circ n}\\,", "2f16937f31078d090324bcec91387fbc": "\\mathbf{y}^1,\\mathbf{y}^2\\in Y", "2f16e1a75d80eba64bbfb06098fe4997": "Y = a + bX", "2f171fed3814607a3956630ae3569a3c": "\\left(\\boldsymbol\\Sigma_0^{-1} + n\\boldsymbol\\Sigma^{-1}\\right)^{-1}\\left( \\boldsymbol\\Sigma_0^{-1}\\boldsymbol\\mu_0 + n \\boldsymbol\\Sigma^{-1} \\mathbf{\\bar{x}} \\right),", "2f172676c742fe2840980204b61c51f0": "0(P)", "2f1745467e687e290c97eec60776a4f8": " \\operatorname{var} \\{ X \\}_s = \\frac {(a-b)^2\\mu\\lambda}{(\\mu+\\lambda)^2}.", "2f182ae0e5496cd0e561e144c3629fe7": " A \\rightarrow A \\otimes I_B,", "2f18628462a2ba94082311821d43f09b": "\\sum_{i=1}^n a_i X_i \\sim N\\left(\\sum_{i=1}^n a_i \\mu_i, \\sum_{i=1}^n (a_i \\sigma_i)^2 \\right).", "2f1863556d9403df957db1729d2a6ad7": "C \\equiv D", "2f18b45687b45fd4222d531642b77a2f": "b = s^2\\cdot k", "2f190645d2ee2d543339aa64d35d8c2f": "r_{i,n} = p_{\\frac{it}{n}}-p_{\\frac{(i-1)t}{n}},\\qquad i=1,\\ldots,n.", "2f1906887c3f8f0bda8b631bf528aa84": "\\overline{T_w}:=T_{w^{-1}}^{-1}", "2f190ea75d30f80eddaf092761123122": "\\phi_\\lambda(i(a)x,y)=\\phi_\\lambda(x,ay)", "2f1915aa772336f3bdc96244f8c87133": "dz_1, ... dz_n", "2f19831558dc8247c677b31ffde35bcb": "\\tau = \\frac{\\gamma E}{2(1+\\nu)}", "2f19cc606da0d0e38f00c4a0fb3aafa2": "\\mathrm{d}H = C_{\\mathrm{p}}\\mathrm{d}T + V\\left(1-T\\alpha\\right) \\mathrm{d}P.", "2f19e17b5cd4c5b4aebda5554a5153b9": "\\phi_1 > \\phi_0", "2f19f860988b83b78b9f1b769023cdc8": "\\begin{align}\n h_3(X_1,X_2,X_3)&=m_{(3)}(X_1,X_2,X_3)+m_{(2,1)}(X_1,X_2,X_3)+m_{(1,1,1)}(X_1,X_2,X_3)\\\\\n &=(X_1^3+X_2^3+X_3^3)+(X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_3^2+X_2^2X_3+X_2X_3^2)+(X_1X_2X_3).\\\\\n\\end{align}", "2f1a8ce867d32ca02039d4dc84de8741": "x = (x_1, \\dots, x_n).", "2f1aacf967e52afbc36cf0642b740c5f": "= 3 \\rightarrow 3 \\rightarrow 64 \\rightarrow 2;\\, ", "2f1aaef131afefd87480a5f76cb6fb0a": " \\Rightarrow(y_1 + \\frac{10^2}{2(32.2)(y_1^2)} = 6.20)", "2f1ab00e151d9d629887064afb259a61": " x, y ", "2f1ab8374b60e97e6046bec24119ecee": "2^{12}", "2f1bdacce3182b749b0dfbc7d9fea5b5": "\\frac{{}_{(1)0}\\partial x^2}{\\partial x}=x\\,\\!", "2f1be6c9a27356d38bf666f7fe807d48": "\n\\left[ \\begin{matrix}\n \\mathbf{X} & \\mathbf{Y} \\\\\n -\\mathbf{\\tilde{N}} & {\\mathbf{\\tilde{D}}} \\\\\n\\end{matrix} \\right]\\left[ \\begin{matrix}\n \\mathbf{D} & -\\mathbf{\\tilde{Y}} \\\\\n \\mathbf{N} & {\\mathbf{\\tilde{X}}} \\\\\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n \\mathbf{I} & 0 \\\\\n 0 & \\mathbf{I} \\\\\n\\end{matrix} \\right]\n", "2f1befefaef90f298fc73eba26c138bc": "\\frac{d\\psi}{dt}=\\frac{4k}{MV}(\\theta-\\psi)+2k\\frac{(b-a)}{MV^2}\\omega", "2f1c3e6903acb39d8a95cccb6e8d6242": "(\\eta,\\sigma) \\in \\mathcal{E}", "2f1c4c877c81eeaf3e300b07160918b4": "\\frac{16}{64} = \\frac{16\\!\\!\\!/}{6\\!\\!\\!/4}=\\frac{1}{4}.", "2f1c5789bf49d31f74a8c4f57ddc942b": "\\lim_{b\\to\\infty} \\int_a^bf(x)\\, \\mathrm{d}x, \\qquad \\lim_{a\\to -\\infty} \\int_a^bf(x)\\, \\mathrm{d}x,", "2f1c65bcc9f7240a380e7da3a8b7a04b": "J_n \\cdot (\\vec{x}_n-\\vec{x}_{n-1})\\simeq \\vec{F}(\\vec{x}_n)-\\vec{F}(\\vec{x}_{n-1}),", "2f1c94ccdaecbf6cde17783d98ce1cca": "\\varphi_{ij}^1:\\mathbb{R}^{n-p}\\to\\mathbb{R}^{n-p}", "2f1ca6406931a1bbb9097f9d467831c5": "\n \\sigma = \\int_{r_\\mathrm{obs}}^{r_\\mathrm{atm}} \\frac {\\rho\\, \\mathrm d r}\n {\\sqrt { 1 - \\left [ 1 + 2 ( n_\\mathrm{obs} - 1 )(1 - \\frac \\rho {\\rho_\\mathrm{obs}} ) \\right ]\n \\left ( \\frac {r_\\mathrm{obs}} r \\right )^2 \\sin^2 z}} \\,.\n", "2f1cb736246edbffab184631d0e1962d": "\\Sigma^0_1, \\Pi^0_1", "2f1cc0244657dc0cbc271eb848e92e36": "\\psi(z+1)= -\\gamma -\\sum_{k=1}^\\infty \\zeta (k+1)\\;(-z)^k", "2f1ccb1391a856a19183d12669813c16": " V_0 a=\\frac{\\lambda^2}{m^2}", "2f1cf5b1a8dcab6352da7533008baca1": "a=\\|u_t\\|, \\,\\, b=u_t\\cdot v, \\,\\, \\alpha=-b/a^2, \\,\\, \\beta=(a^2-b^2)/a^2,", "2f1d1387f8272ad7d2cc2b7f4dd38de6": "\\left [\\begin{smallmatrix}\n0 & 1 & 0 \\\\\n-1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{smallmatrix}\\right ]\n", "2f1d1d97ec01e11dee56e51786f291e5": "j/r = k/s", "2f1df22222d71e0922ea0033268adfa6": "L= \\frac{\\Phi}{i}", "2f1dfd9d435bfc7c5d42fabe04d7a259": " \\sigma_0=", "2f1e317f9a82f8c47ea345544c85a10d": "C_p = -(2u +v^2 +w^2) ", "2f1e3ca965e5325df64675f647ffa547": " \\sum_{s=-1,1} \\mid a_s \\mid^2=1. ", "2f1e8d5af98f2c9643b0334c73ccc25b": "W\\,", "2f1eb6618c3d643e2d33cda26f1a91a2": "{{K}_{cpu}}", "2f1f3e8ebd105030f20d75eeafaf5bcb": "\\beta(n_i, \\tilde{n}_i)=\\frac{\\tilde{n}_i}{n_i+\\tilde{n}_i+4b^2 n_i\\tilde{n}_i}", "2f1f7c6dbe6fe2fa3c84f31bae92ca54": "\\frac{\\lambda q m d^2}{2 \\pi \\epsilon_0 \\hbar} ", "2f1f8d944f531ada31ebbdaf38e1b024": "|x|_p \\le 1", "2f1fafc64ea82ec21dfb7d039c484bce": "\n\\hat{\\alpha}_i = \\bar{Y}_{i\\cdot} - \\bar{Y}_{\\cdot\\cdot}\n", "2f1ff70def5bd32c13b278c3b660b337": "\\mathcal{L}=\\bar\\psi(i\\gamma^\\mu D_\\mu-m)\\psi -\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}\\;,\\,\\!", "2f200acf3704ccd02b8bd795def7f593": "Z_{G}\\,", "2f202d0c3d91cf010f3c4ab386fc4eb1": "p_n = 197418203 \\times 2^{25000} - 1,\\quad p_{n-1} = p_n-6090,\\quad p_{n+1} = p_n+6090.", "2f20a17d52d79e0b1070dd0d6a802814": "e\\wedge", "2f20b5b64aacea06dd09dd96616adbb0": "\\nabla\\cdot{ \\vec{A}} + \\frac{1}{c^2}\\frac{\\partial\\varphi}{\\partial t}=0.", "2f20c39d1c409de26096133fcfb3636b": " = (-1)^\\text{sign}(1.b_{22}b_{21}...b_{0})_2 \\times 2^{e-127} ", "2f20cb1171334c6c6abbfc55979d7047": "\\displaystyle{Q(a,b)=L(a)L(b)+L(b)L(a) -L(ab).}", "2f20f040fac553e512c28f33267b572a": "\\delta(n)=n-\\left(\\tfrac{D}{n}\\right).", "2f212f59befa079209eff36c570c5e64": " \\mathbf{T} = \\int_V (\\mathbf{r}-\\mathbf{R})\\times \\mathbf{f}(\\mathbf{r}) = \\int_V (\\mathbf{r}-\\mathbf{R})\\times (-g\\rho(\\mathbf{r})dV\\vec{k} )= \\left(\\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R})dV \\right)\\times (-g\\vec{k}) .", "2f213d5ac45ee4805cc87c6dd82fd3c6": "(C)\\int\\, f d\\nu + (C)\\int g\\, d\\nu = (C)\\int (f + g)\\, d\\nu.", "2f2164ec5caa6c3869def259cf012afb": "\\mu=5\\log_{10}(d)-5", "2f21ec4b6c52f0afd7045de2c81c849d": "n-1\\choose k-1", "2f222ff6321dda0151611975ea575bfe": "\\boldsymbol{ \\nabla \\cdot } \\left( \\boldsymbol{\\nabla \\times B}\\right ) = 0 = \\mu_0 \\left( \\nabla \\cdot \\boldsymbol J_f +\\frac {\\partial }{\\partial t} \\boldsymbol {\\nabla \\cdot D } \\right ) \\ , ", "2f225d3c377973f063874b7d8dfe1db2": "\\delta=k_j(-w)\\tau_2=k_jL_D", "2f22ac11583dfea7c5f77622cd09363c": "\\nu \\,\\!", "2f22b46c3f034b8d68b7386dc479cb82": "S_k := \\sum_{1\\le i_1<\\cdots t f(x) + (1-t)f(y)\\,", "2f23a23e003b3cc2266908ca4eccd6d0": " p(x|M) = \\int p(x|\\theta, M) \\, p(\\theta|M) \\, \\operatorname{d}\\!\\theta ", "2f241df7ac39a0c081689342faaf55ce": "x^{(s)} \\exp \\left | \\alpha^{(s)} \\right | < 1,", "2f24357e5379ee435aa95463618db8ce": "\\Gamma^*_n \\subseteq \\overline{\\Gamma^*_n} \\subseteq \\Gamma_n.", "2f243c0d69502236e9091b5d93ab56bd": "E \\in \\operatorname{FV}[G] \\and E \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] ", "2f2461c4eb8aca03a240e7c68255501d": "h(I_1,I_2)", "2f2492f08d0943e48def5329dff7907c": "F\\xrightarrow{\\;F\\eta\\;}FGF\\xrightarrow{\\;\\varepsilon F\\,}F", "2f2567a1682e00ae46bb8cd9f3d9f84e": "\n\\beta(u) = \\int_{s \\in \\mathbb{S}}|u^Ts|^\\alpha \\mathbf{sign}(u^Ts)\\Lambda(ds)\n", "2f2584975edc8097952b22d497659846": "\\Omega=\\sqrt{\\Omega_c ^2+\\Omega_p ^2}", "2f25b7bce0db5507290f0b3cc630f150": " t\\ ", "2f25c0a5bef1faa86e3dade442e49b5f": "\n\\Delta V \\approx V_{\\rm bin} = \\sqrt{GM_{12}/a}\n", "2f25c4e49bab7f83838e1e2e4d7fafc0": "c : 2\\ 3", "2f25d90ae508cf499930ab9677d722fc": "\\chi(G)(\\chi(G)-1) \\le 2m.\\,", "2f262a2067062ff83330e6e1b53d811e": "\\mu^* = \\max_k \\{ \\mu_k \\}", "2f2630e8b009760d24c2d14bab9f0219": "J_k", "2f26364b7906d2d1d4de8169fee82ad6": "\\{{e_{(a)}}^{\\mu}\\}", "2f2681ce6918f12272750cef21779c62": " R = \\frac{\\left|\\mathbf{v}\\right|^3}{\\left| \\mathbf{v} \\times \\mathbf{ \\dot v} \\right|},\n\\qquad\\mbox{where}\\quad\n\\left| \\mathbf{v} \\right| = \\left| (\\dot x, \\dot y) \\right| = R \\frac{d\\varphi}{dt}.", "2f269c82d942114ecf6dc308d515a590": "\\frac{d}{ds}\\Big|_{s=0}\\mathcal S(\\gamma_s)\n= \\Big|_a^b \\frac{\\partial L}{\\partial\\xi^i}X^i - \\int_a^b \\Big(\\frac{\\partial^2 L}{\\partial \\xi^j\\partial \\xi^i} \\ddot\\gamma^j\n+ \\frac{\\partial^2 L}{\\partial x^j\\partial\\xi^i} \\dot\\gamma^j - \\frac{\\partial L}{\\partial x^i} \\Big) X^i dt,\n", "2f26edb641f39ff345c3f35e6dff5fa7": "T^{-1}y=Sx \\text{ for each } x \\in \\ell^2(\\mathbb{N}),\\,", "2f26f83cdc7949c23c29e48386c2a0ed": "\\scriptstyle m/\\sqrt{1 - v^2/c^2}", "2f2707e04200f5bd2fc87923134bbf11": "\\begin{align}\n\\mathbf{D} = \\mathbf{B^{-1}C} = \\begin{pmatrix}\n0 & \\frac{1}{3} & \\frac{1}{2} \\\\[4pt]\n\\frac{1}{4} & 0 & \\frac{1}{2} \\\\[4pt]\n\\frac{3}{5} & \\frac{1}{5} & 0\n\\end{pmatrix}, \\quad \\mathbf{B^{-1}k} = \\begin{pmatrix}\n\\frac{5}{6} \\\\[4pt]\n-3 \\\\[4pt]\n2\n\\end{pmatrix}.\n\\end{align}", "2f27cefe216987b76af74bc9d1859258": " \\frac{\\partial h_i(\\mathbf{p},u)}{\\partial p_j} = \\frac{\\partial x_i(\\mathbf{p},e(\\mathbf{p},u))}{\\partial p_j} + \\frac{\\partial x_i (\\mathbf{p},e(\\mathbf{p},u))}{\\partial e(\\mathbf{p},u)} \\cdot \\frac{\\partial e(\\mathbf{p},u)}{\\partial p_j}", "2f27e3601dbcda2e28e43fbc7cf322c2": " X_{ni} = x \\in \\{0,1\\} ", "2f2802823bdf2c9d142fc8700dd20f51": "c_i \\ne C_\\text{in}(y_i')", "2f2842dfddaccb50ab557a1cd5f4c0a2": "(M, \\delta)", "2f2863dd01a5671877a175620cc93ca6": "\\frac{\\text{d}C_3}{\\text{d}t}= \\text{k}_{1(2)} {^1_2}S^\\gamma E - (\\text{k}_{2(2)}+ \\text{k}_{3(2)}) C_3", "2f28711ef403007b89dbacd32117cf30": "(2n-2)(2n-1)", "2f2890be70388a4cc5c58998e34b5020": "\\boldsymbol{\\phi}\\,[\\,\\textbf{x}(t_0),t_0,\\textbf{x}(t_f),t_f\\,] = 0", "2f28beaccbe619e66d6a2fb1de676d2d": " X_o ", "2f28c7c93932981717e3713f19fa6f3a": "\n f(k;\\mu,\\mu)\\sim\n {e^{-k^2/4\\mu}\\over\\sqrt{4\\pi\\mu}}.\n ", "2f290e1f123fbe54c472f6ed692e1306": "G= e^{-(log_e 2) x^2}.", "2f29258266dc6504277060f73ada98ee": "\\textbf{z}", "2f29729dace3c70ace11100bceb991c8": " w_2(x) = \\prod_{i=1}^{20} (x - 2^{-i}) = (x-2^{-1})(x-2^{-2}) \\ldots (x-2^{-20}). ", "2f29765d0a7e46e760209e33cd93716d": "2p x_1", "2f297b890d36e8ba1e608e8f7712669f": "n(n+2\\alpha)\\,", "2f299032f40822381ced110b4170389c": "\n \\overset{\\diamond}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{l}\\cdot\\boldsymbol{\\sigma} + \\boldsymbol{\\sigma}\\cdot\\boldsymbol{l}^T\n", "2f2a1352e6508cab3f4e5d04272990ef": " ( a \\circ b ) \\circ c = a \\circ (b \\circ c) + a \\circ (c \\circ b) . ", "2f2acb31e4646baae7e5233e8e85b81c": " z = r(\\cos \\varphi + i\\sin \\varphi ).\\,", "2f2af7f3ff6317363cda535146d5dfb0": " \\langle k|H'|k' \\rangle = \\frac{1}{Vol} \\int_\\mathrm{Vol}\n\\psi_k (r) H' \\psi^*_{k'} (r) \\, dr ", "2f2afb6438345f51cb44fe7d6c3d8f6b": "D(X,Y)= \\max_{x\\in X, y\\in Y} d(x,y)", "2f2b589a5cf7d2d1a4369f3ee412ae0c": "j, a,b \\in \\Z", "2f2b6a246db95d3e284983e01837de5a": "\\left(x_1 + y_1 \\omega \\right)\\left(x_2 + y_2 \\omega \\right) = x_1 x_2 + x_1 y_2 \\omega + y_1 x_2 \\omega + y_1 y_2 \\omega^2 = \\left( x_1 x_2 + y_1 y_2 \\left(a^2-n\\right)\\right) + \\left(x_1 y_2 + y_1 x_2 \\right) \\omega", "2f2b80ada6560ac79b3e469111af44e9": "\\overline R", "2f2bb5e5df34890957113426140db500": " \\and (S_3 \\implies (\\operatorname{equate}[A_3, n] \\and V[y] = n)) \\and D[y] = D[n]) ", "2f2be35152ed584a84aecccfa7de1149": "\\mathbf{C}^n = \\bigoplus _{i = 1}^k Y_i.", "2f2bea87e8082280d027ab8fb190aa5b": "b=a_{0} + a_{1} p^1 + a_{2} p^2 + ...", "2f2c8df2ff0112dccd7a56322042d34e": "D^2\\,.", "2f2c9d879756a524767a9b671d93ff5f": "U = \\sum_{i=1}^{n} \\frac{C_i}{T_i} \\leq 1,", "2f2ca6b8c9368ac12bf3b063c821df1d": " -(N_A)_{ij}^k=A_i^m\\partial_m A^k_j -A_j^m\\partial_mA^k_i-A^k_m(\\partial_iA^m_j-\\partial_jA^m_i).", "2f2d458ca3165cb518f71fab78e493da": "\\theta_{\\text{min.}} = 6 \\times 24 = 144", "2f2d59994a7267bef28c1b326d1606a8": "k = 1, q = 3, C_{out} = \\{(0,0,0), (1,1,1), (2,2,2)\\}", "2f2d64339ae9f0e3287030d0b740a071": "\\displaystyle{\\|(T+iI)x\\|^2= \\|(T-iI)x\\|^2 + 4(\\mathrm{Im}(T)x,x).}", "2f2d6ea2d21c7f7bdfc4e80e40716949": "S_m \\,", "2f2de31c48c0487d83b8a9a2ad536421": "d(R) = d", "2f2df30cbcc232575bbac2ea3407a78c": "\\sqrt{\\frac{1}{126}}\\!\\,", "2f2e5ba603cb7d232422b25df0f28679": "a_{1}-a_{10}", "2f2e70caf53966d1d3c10f22b5eb96c2": "R=[x_0-a,x_0+a]\\times [y_0-b,y_0+b]", "2f2e9c340fc1b5fa5252362bc994cb9d": "V_\\text{out} = - \\frac {I_\\text{p} R_\\text{f}} {1 + {1 \\over A_\\text{OL} \\beta}}", "2f2ea61a22f6f65a1e927a96b6acec1b": "(m, R_1)", "2f2eaf8c14dea25a948a4b30b4d39e12": "(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)) ", "2f2ede3a5ae97e2aeb4556cc460b2254": "e_2 = \\frac{c_2}{a_1^2+a_2}", "2f2eed8f5763e849ad71fcc4dddaa33d": "Dt", "2f2efe1f4d6f56cc868813f727fc0422": " {\\mathbf{a}}^2 = Q(\\mathbf{a}) = \\epsilon_{\\mathbf{a}}{\\left |\\mathbf{a}\\right|}^2", "2f2f3e52bbd74f04d873f4f6ee531988": "0 \\le w \\le 256^{21}", "2f2f851a2b3f631ae8f95c80ccfd89b7": " X_{ji} ", "2f2fa70c629c7d17d311cbe6a89b7aa4": "Z>z_\\alpha", "2f2ffa5f9bd35f1d97c54868a4b7aace": "L^{4k+1}", "2f2ffcffe62b966f00802345433ae28f": "\\mathrm{ot}\\,(A)", "2f30008dab325dd617829e3a7558987c": "EPC = {g_{ACh}} ({V_m}-{E_{rev}})\\,", "2f300bfa3388192954713cd1d297ee56": " \\vec{F} = \\vec{m}\\cdot\\nabla\\vec{B} ", "2f30125f9d6e40de2e9f2eead80e088e": "g(\\vec{x},\\vec{e}_i,t)\\,\\!", "2f3094ea59a08f53bf9d2fb185d7c4f4": "\\mathcal{K}\\ne T\\mathcal{K}.", "2f30b79b241ae5609985acf671818a5b": "dX_t = dW_t.", "2f30ee3d699bdbd3d742bc9ff06e35c1": "x^4-x^2+y^2 = 0.", "2f3139d012085fa5c00b3c4d11e15654": " \\begin{pmatrix} y_1 \\\\ y_2 \\end{pmatrix} = \\frac{f}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} ", "2f316026e11879965801f7e089b7cd62": "\\textbf{u}", "2f317624f1749c394984fec010c91f3b": "A = B \\setminus \\{x\\}", "2f319f76df7dc672a0f2ca4d77f6b9aa": "\\frac{T''(t)}{c^2T(t)} = \\frac{R''(r)}{R(r)}+\\frac{R'(r)}{rR(r)} + \\frac{\\Theta''(\\theta)}{r^2\\Theta(\\theta)}=K", "2f31e7974287bae87d01ae04953114ef": "\\text{yield}(k)", "2f3284cdde9faeaa5cd8590527aaf28f": " H_{B_1}", "2f32a05698f27557bf93d014571078f8": "\\ln (1 - z^{n})", "2f32da2c65236683e459bb827da055bf": "\\left(\\sum_i a_i x^i\\right) + \\left(\\sum_i b_i x^i\\right) = \n\\sum_i (a_i+b_i)x^i", "2f32fca732bbd494671252fe7fd56f47": "\n\\psi(x) \\rightarrow e^{-if(x)} \\psi(x)\n\\,", "2f32fe2274c0c02d314c54b2a218d6be": "BV(\\Omega)", "2f336da9e8a13ffd0503d7d502cbf361": "|| \\Phi(t) (I - P) \\Phi^{-1}(s) || \\le Le^{-\\beta(s - t)}\\mbox{ for }s \\ge t > -\\infty.", "2f33b100f8d6e009095724ac8b4b1941": "\\varphi\\circ\\vartriangle_X=\\vartriangle_Y\\circ\\varphi^\\vartriangle", "2f340182cfbdf69f6ea8e319e3926222": "v^* = \\frac{L^*} {T{*}} = 1", "2f340df8a61bfd5264d5188134adcd12": "\\frac{1+X}{2}", "2f348a4d94114e007840ee63c02ccaf9": "H_k(X) \\cong H_k(X_n) ", "2f348f3305bf4dba5afdc3852251ea88": "d \\le 5", "2f34ad0220940feded7db4959a73c9a8": " e_0 ", "2f34da4d3b75be341291aff8d49b926a": "f_c(x) = \\begin{cases} 1, & x \\in \\Omega_0^{-} \\\\ 0, & x \\in X \\setminus \\Omega^{-} \\\\ \\mbox{smooth interpolation}, & x \\in \\Omega^{-} \\setminus \\Omega_0^{-} \\end{cases}", "2f353267cef666de023aaef6cb97718b": "X_{T'} = (X_{t_1}, X_{t_2},\\ldots, X_{t_k})", "2f357faf053fe6f6d464a12b806e1056": "\\lim_{\\bold{h} \\to 0} \\Phi(\\bold{h}) = 0 \\qquad \\text{and} \\qquad f(\\bold{a}+\\bold{h}) = f(\\bold{a}) + M(\\bold{h}) + \\Phi(\\bold{h}) \\cdot \\Vert\\bold{h}\\Vert", "2f35933efe272a6e592b27cd52df58a3": "\n\\max \\{ \\min \\{ R_A(x) \\mid x \\in U \\text{ and } x \\neq 0 \\} \\mid \\dim(U)=k \\} \\leq \\lambda_k\n", "2f35fe7b55f3a6ec1a45fdf0777c15b7": "\\chi - \\lambda \\exp{\\frac{\\sigma^{-2}}{2}} \\sinh(\\frac{\\gamma}{\\sigma})", "2f36473f6d38e3b7366f83a2e7a211a8": "\\scriptstyle \\sum_{n=-\\infty}^{\\infty}f(nT)\\cdot \\delta(t-nT),\\,", "2f36857f582b85f36947634f96612b33": "\n\\begin{align}\n\\mu'_x & = \\sin\\theta\\cos\\varphi \\\\\n\\mu'_y & = \\sin\\theta\\sin\\varphi \\\\\n\\mu'_z & = \\cos\\theta \\\\\n\\end{align}\n", "2f369d5323d45bd28d257de7e9328238": " P(x_i,x_j) ", "2f36b2e001448b54a5df5a5cfff7c366": "a^b \\pm 1", "2f36bac7c0c44f9073c23e983bddb047": "\\varphi_{\\beta}(0) [n] = \\varphi_{\\beta [n]}(0) \\,.", "2f377f91e1852556531b97722ef92528": "\\textit{int} \\subseteq \\textit{float}", "2f378dc2c03aaa068503b8df01761ade": "4p^2 + 1", "2f38889d152c28b8beebea090f6a9ac1": "\\frac{1}{\\sqrt{1-x^2}}", "2f38c9ab693cbca9b7052a73073abc13": "G_{\\mu\\nu}=8\\pi\\frac{G}{c^4} T_{\\mu\\nu}", "2f38e76bfcff9c2ff2c50ddc0635c041": "u_t = Lu,\\ ", "2f38ff8cd3a1dc5126cc25f701392f92": "110 = 5^2 + 6^2 + 7^2", "2f39ac3a8f3f3f248920f7cea3566682": " N \\geq 2 \\, \\,", "2f39d1df5631a4e05ad0b428212f0ded": "\\scriptstyle\\sqrt{2\\big(1-\\frac{1}{k} \\big)}", "2f3a39a5254512cf6c4de9a367028936": "\n \\begin{align}\n \\mu \\left( z^n,l^k,r^k \\right)[z] = \n \\frac{\\Big|\n \\left\\{ k+1\\leq i \\leq n-k \\,\\,|\\,\\, ( z_{i-k},\\ldots,z_{i+k})=l^k z r^k \\right\\}\n \\Big|}\n {\\Big|\n \\left\\{ k+1\\leq i \\leq n-k \\,\\,|\\,\\, l^k(z^n,i)=l^k \\text{ and } r^k(z^n,i)=r^k\\right\\} \n \\Big|} \\,.\n \\end{align}\n ", "2f3a9e898ce91c8f4eee3ac3623e12a4": "\\frac{\\partial y}{\\partial \\mathbf{x}}, \\frac{\\partial \\mathbf{y}}{\\partial x}, \\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}}, \\frac{\\partial y}{\\partial \\mathbf{X}}", "2f3ac425cc625b61941d87de5816c6e9": "E_{AB}(r)=m_1(r-r_1)", "2f3b575b35cf32b4c6464ea3d9a1b4a7": " [I_S] = [I_R] - M[d]^2,", "2f3b671dae3de0f657c38ec48ee58e92": "W \\in Gr_n (V)", "2f3b72ddb7d382b7055a9905c669a2c5": "x+x^{-n}=2\\,", "2f3bb6e1cb9db43d02e059d005d5a974": "\\delta_T,\\delta_c", "2f3c16002cfbc823207a459ecf7514aa": "\\left\\{ y = \\frac{\\alpha}{\\beta}, x = \\frac{\\gamma}{\\delta} \\right\\},\\,", "2f3c2d49c4ec89799485c93842db70ec": "l_adx^a=\\frac{\\omega_0+\\omega_1}{\\sqrt{2}}\\,,\\quad n_adx^a=\\frac{\\omega_0-\\omega_1}{\\sqrt{2}}\\,,", "2f3c4e6bf56b93e33e1fbdb6e39a0dd7": "g\\;", "2f3c571145a314eacaecf234960d820c": "T_2^{*}(K)=(P,B,I)", "2f3cabf581dbc7b6946819bc8e60cf7e": "x^5+x+a=0\\,", "2f3d194653f8b4c5ed4cbcc596264bab": "\\mathbf{X}' = \\boldsymbol{\\Lambda}(\\mathbf{v})\\mathbf{X}.", "2f3d3e06d7156f4a34d05c7b86f589ec": "\\partial:R \\to R\\,", "2f3d664c14c79aee3c4119377346dea3": "\\rho_c(\\mu^2)=0", "2f3dc5b806042e426d38fb8b9cf90e8e": "\\frac{\\partial L}{\\partial \\dot{q}}", "2f3dce05f27bd4b341460cfb373ee507": "A_{n} \\in \\mathcal{R}", "2f3e35d839f5933caa3e303ae75b64bf": "\\gamma _{1}", "2f3f2dffe88d1e6b78cf6ee511ef5e83": "f(z)=e^{1/z}.\\,", "2f3f2eb5aea17d38fdad8c669f9a0009": " f(x) = \\frac{x}{1+x^2}", "2f3f4abd018c06802b021a5397b932b2": "d p/d x", "2f3f909e886ca066316655ad771bbd53": "ds^2 = - \\left( k^2r^2 + 1 - \\frac{C}{r^{d-2}} \\right)dt^2 + \\frac{1}{k^2r^2 + 1 - \\frac{C}{r^{d-2}}}dr^2 + r^2 d\\Omega^2", "2f3fa80ab1790d972d6114268950bf59": "r=D_K[F(K,L)]\\,", "2f3faf4374b4629da530eb58e9542bf9": " \\lambda_k = \\max_{{\\rm dim} \\, G = k-1} \\, \\min_{f\\perp G} {(Df,f)\\over (f,f)}. ", "2f40557e9a82c74ea2110f87eab5b44d": "v'(w)=1/w'.", "2f406fb8f71290bc6a7d409f3e1a597e": "z=\\pm c \\cosh v", "2f4080535aa09e72beaecabbae2803bd": "h_{p}(\\tau_{1},.\\,.\\,,\\tau_{p})", "2f408f3aa4a17d26088cb13984f698f8": " F(X_t) = F(X_0) + \\int_{0}^t F'_{-}(X_s) dX_s + \\frac12 \\int_{-\\infty}^\\infty L^x(t) dF'(x), ", "2f409312f4da642dbc554c2ce604fcc0": " \n{m} = {\\rho}{V} ", "2f409f6ff941375bd33e62f1dc9c1bfd": "\\scriptstyle\\Sigma\\subseteq\\hat\\Sigma", "2f412e80685ca75233d14bcdb73fa5e2": "1/\\sqrt{|a|},", "2f414f61f46e0c19372923777d695ad9": "\\scriptstyle p_g", "2f41663c6dec30994825078552bedde4": "\n\\frac{\\partial S}{\\partial \\alpha_{r}} = Q_{r}\n", "2f4168e5c4dc15946f85cb8466747b17": "\n\\langle \\Delta i_1\\Delta i_2\\rangle=\\frac{E^4}{8}\\cos(2\\phi),\n", "2f41839abfb8e1504ddf131721f8faf2": "X_2 \\sim \\mathrm{Pois}(\\lambda_2)\\,", "2f419107815ee04c06779eb2c073913b": "\\mathbf{v}[\\mathbf{f}A] = A^{-1}\\mathbf{v}[\\mathbf{f}].", "2f41c640f8df7961c4335341779822e6": "Z_\\infty^{p,q} = \\bigcap_{r=0}^\\infty Z_r^{p,q},", "2f420ef5343b6c746674058ae08296c8": " M_{\\mathbf{\\Xi} \\cup \\xi}(\\mathbf{x}) = \\int_0^1{M_{\\mathbf{\\Xi}}(\\mathbf{x}- t \\xi) \\, {\\rm d}t}", "2f4246febd6c445aab4811d7ce84f120": "c \\in \\Bbb{R}.", "2f42ca7eb715ad82f90b89a69e59ba69": "\\boldsymbol{r} = \\boldsymbol{x}_1 - \\boldsymbol{x}_2 ", "2f431d07722f0c9ec93b5fdcb090b1d9": "v_{bullet}", "2f437b5780a95ea0e1b5c618abe2ef67": "U_L=1-\\frac{{\\langle s^4\\rangle}_L}{3{\\langle s^2\\rangle}^2_L}", "2f4396bab5869c1e0c9f8a7620bf2518": " D ", "2f43a56fe6e1d28ff9c565362447bc1f": "2.\\mu_{8,4}(p_{4}) = \\Sigma_{p_{2}} \\alpha_{9}(p_{2},p_{4}) ", "2f43b42fd833d1e77420a8dae7419000": "...", "2f43bcce0431981685894042ffa0e661": "\\tau(0)=z_0", "2f43d6fcaa0c6e7283874db47d92049b": " \\mathbf{R}_B = R_B\\mathbf{e}_y", "2f43d71e12f22467c50749632adf3d5a": " f(P) \\approx \\frac{y_2-y}{y_2-y_1} f(R_1) + \\frac{y-y_1}{y_2-y_1} f(R_2). ", "2f4400b73b2cad5954c673eb3e2cd003": "\\begin{align}\n i &= 1 \\cdot e^{\\frac{1}{2} i\\pi + i 2 \\pi k} \\big| k \\isin \\mathbb{Z} \\\\\n i^i &= e^{i \\left(\\frac{1}{2} i\\pi + i 2 \\pi k\\right)} \\\\\n &= e^{-\\left(\\frac{1}{2} \\pi + 2 \\pi k\\right)}\n\\end{align}", "2f4410fed6b3e7374deacebc02474592": "8^2+8^2+1^2", "2f4451302ddca7050860d69d50f78d61": "\n\\frac{I_{theory}(d_{1})}{I_{theory}(d_{0})}=\\frac{I_{exp}(d_{1}+d_{\\textit{f}})}{I_{exp}(d_{0}+d_{\\textit{f}})}\n", "2f448d442cb3a140593d00c56f79067a": "\\sqrt[8]{5}\\sqrt[3]{35} = 4.00000559\\ldots \\approx 4", "2f44af2ae2b062cce48a24593b70872f": "\\mathcal{E}(x^x)=\\mathcal{E}(x^{\\frac{1}{x}})=\\mathbf{Q}\\setminus\\{0\\}.", "2f44d29c662385c935e349578f7a8897": "\\boldsymbol{A}", "2f44d5051514cb9eadc870c8e9eb7025": "\\Phi_n", "2f4560462b09c37baab9fd7e12703973": "\\Pr(M=m\\mid N=n,K=2) = (m|n) = [m \\le n]\\frac{m - 1}{\\binom{n}{2}}", "2f458373db8cb4014158420466c05d81": " F\\colon T_y N\\times\\cdots \\times T_y N\\to \\R.", "2f45a55cc2df1282962e349d81d4bcef": "\\frac{E_{average}}{h_{average} + s_{average}}", "2f45b1f1637df6417c59efdad2d9c4d9": "\\scriptstyle Z_\\mathrm L", "2f45e502a0b9eb06cb7565c7ef9afed6": "\nL = - \\frac{u^3_*\\bar\\theta_v}{kg(\\overline {w^'\\theta^'_v})_s}\\ \n", "2f45e790920b506bd8cecc073973f5de": "\n \\sum_{1\\le k_i \\le n} = \\sum_{1\\le k_1 \\le n} \\ldots \\sum_{1\\le k_i \\le n} \\ldots \\sum_{1\\le k_n \\le n} \\, \n", "2f45ef2bf5c8ccc3214dfe7641fcb38f": "(S)\\rho \\wedge (S')\\rho'", "2f45f3f1f254ec150dfaa13b90e328e7": " \\alpha + \\gamma = \\pi /2 \\ . ", "2f4646b7e5857e2bf32375cf9aad9275": "\\ln z = \\ln r + i\\theta.\\,", "2f464e46959ad36f6e87b853a23aef61": "f=f(y+1)", "2f47060cb4bd33c1036a5d262ec8241f": "I(Y_1) \\subseteq I(Y_2) \\subseteq I(Y_3) \\subseteq \\cdots", "2f470a1b31a40164f7f60b21d9b75212": "\\Delta t < \\frac{1}{2^q \\pi f_0}", "2f471b238b816e22d62470cfc8916f3c": "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 2ik \\frac{\\partial u}{\\partial z}.", "2f4775b46fa6599c499a74fc3c076c4a": "\\frac{\\varphi^2}{\\sqrt 3}", "2f4786b8df7d5aaef1d0defbb16720a8": "\n\\begin{array}{rl}\nEVSI & = E[U|SI] - E[U] \\\\\n& = \\left(\\int_Z \\max_{d\\in D} ~ \\int_X U(d,x) p(z|x) p(x) ~ dx ~ dz\\right)\n - \\left(\\max_{d\\in D} ~ \\int_X U(d,x) p(x) ~ dx\\right)\n\\end{array}\n", "2f478c96e02fbdfc7f783684ade1461b": "\\int_\\text{magnet 1} \\mathbf{M}_1\\cdot\\mathbf{H}_\\text{d}^{(2)} dV = \\int_\\text{magnet 2} \\mathbf{M}_2\\cdot\\mathbf{H}_\\text{d}^{(1)} dV.", "2f47f205b31ae9c5c8ceae9630e5324b": "\\Sigma_{n \\mathbin{:} {\\mathbb N}} \\operatorname{Vec}({\\mathbb R}, n)", "2f482173368d8eb6d045fc97b45e56ad": "y_n=\\frac{1}{2}\\,x_{n-1}+\\frac{1}{2}\\,x_{n+1}", "2f48690cd733179013597dd35fe96887": "d=\\frac{(u^2+1)^3(u^2-4u+1)}{(u-1)^6(u+1)^2}, u\\notin\\{0,\\pm1\\}.", "2f488ae91128307b326343cb9944dac2": "G^{\\mathrm{R}}", "2f48956b46c2be0926200f6833f1c459": "{d{I_{1z}I_{2z}} \\over dt}=-R_z^{12}2I_{1z}I_{2z}", "2f48cf2227b3a3e0e61ff36291bdfc0c": " I_\\mathrm{C} = I_\\mathrm{S} \\left( e^{\\frac{V_\\mathrm{BE}}{V_\\mathrm{T}}}-1 \\right) \\left(1 + \\begin{matrix} \\frac{V_\\mathrm{CB}}{V_\\mathrm{A}} \\end{matrix} \\right) ", "2f48e5328874193a205f814ca24ec699": "\\Bbb{R}((x))", "2f4904b3db603ff696b9e44441102b93": "\\scriptstyle \\mathcal S", "2f493b7434802b5a3574d1d766c889e7": "\\begin{cases}\nQ_1 = 15 \\\\\nQ_2 = 37.5 \\\\\nQ_3 = 40\n\\end{cases} ", "2f49b6747f6a8dccad461365f7f1ea5b": " K(s,t) = \\sum_{j=1}^\\infty \\sigma_j \\, \\phi_j(s) \\, \\phi_j(t) ", "2f49d7f7234e89d6ab7270ae2af7158f": "\\{(x_i)_{i\\in \\Bbb{N}} \\mid x_i=x_j\\text{ for some }i\\ne j\\}", "2f49f6410a0430625eac6b121db26602": "\\left|\\psi_1\\psi_2\\right\\rangle = e^{i\\,\\theta}\\left|\\psi_2\\psi_1\\right\\rangle\\,,", "2f4a0054079c04a8a75b35d16d212582": "dev(D_2)", "2f4a122898d94cbf1670ed7ea4c24e4a": " \\alpha = \\frac{\\pi_i(\\theta^{(j)})\\pi_j(\\theta^{(i)})}{\\pi_i(\\theta^{(i)})\\pi_j(\\theta^{(j)})}\\ ", "2f4a260326e5da29f699d4b671e66379": " \\begin{pmatrix} \nv_\\text{x} \\\\ \nv_\\text{y} \\\\\nv_\\text{z}\n\\end{pmatrix} = \\begin{pmatrix} \nT_\\text{xx} & T_\\text{xy} & T_\\text{xz} \\\\ \nT_\\text{yx} & T_\\text{yy} & T_\\text{yz} \\\\\nT_\\text{zx} & T_\\text{zy} & T_\\text{zz}\n\\end{pmatrix}\\begin{pmatrix} \nu_\\text{x} \\\\ \nu_\\text{y} \\\\\nu_\\text{z}\n\\end{pmatrix}\\,,\\quad v_i = T_{ij}u_j ", "2f4a3123528e5a5e72cf1bc78ba28f8b": "E(Z/nZ)=\\{(x:y:z) \\in P^2\\ |\\ y^2z=x^3+axz^2+bz^3\\}", "2f4a50214865bf4b4a8d532f1d5eb05d": "\\beta : [0, a) \\times [0, \\infty) \\rightarrow [0, \\infty)", "2f4aa28e017383429287b6fe7befd1f7": "a(t) = {h*H}(t) = {H*h}(t) = \\int_{-\\infty }^{+\\infty} h(\\tau) H(t - \\tau ) \\, d\\tau = \\int_{-\\infty}^t h(\\tau) \\, d\\tau.", "2f4ac0eb5075bd469b2ed74ba3e24122": "V_{\\alpha_1}", "2f4b78d8f732bdaa4192e48a79f91730": "\\frac{\\partial s_c}{\\partial t}= \\sum_\\alpha\\sum_\\beta L_{\\alpha \\beta}(\\nabla f_\\alpha)(\\nabla f_\\beta)", "2f4b91fd1984a658d88cf304cea36b91": "\\hbar = 1", "2f4bb3aafb34cdcdfb7ac24e4dff432c": "\\omega(k)", "2f4bb4935b65f8f87f1054194610f78e": "\\dots-1", "2f4bf53c3f63429f6003ed0dae6fc2cf": "\\delta: L\\rightarrow L", "2f4bf83e36c830f3ba86e7fb521cbbe8": "f, g : \\mathbb{N} \\to\\mathbb{N}\\,\\!", "2f4c15d0d663acd340b1af4690088366": "|S_2|=11", "2f4c2bb9a852671673c65a502b83eb45": "\\scriptstyle{R_F}", "2f4c618b5d7a25db3beeeaebb878c0c7": " \\frac{\\vec{F}_{12}}{L_1} = \\frac {\\mu_0 I_1 I_2} {2 \\pi D}(0,-1,0) ", "2f4c9cc2ef8ae6a346a12e3e3a8db552": "F_n = \\dot{m}\\;v_{e}", "2f4cd15d2e0ed2f39d4b7d8ce83b28be": "\\scriptstyle\\vec L\\cdot\\hat p\\,=\\, 0", "2f4cdb9618340625e1fbc0f4380e548a": "B=\\left( \\frac{2e^2 N}{\\epsilon_0\\epsilon_rkT} \\right)^{1/2}", "2f4cf63d49db0f3b79ab3ee50e183862": " 2r^2\\theta \\, ", "2f4d5602e3c1042a37c3d13ed0789c44": " \\Pr(H=h | r, N=h+t) = {N \\choose h} \\, r^h \\, (1-r)^t. \\!", "2f4d785214a2b760894481335711b176": "\\rho_q(\\bold{r})=\\sum_{i=1}^N\\ q_i\\delta(\\mathbf{r} - \\mathbf{r}_i)\\,\\!", "2f4d852d2cfc663aad67601ae2efaae2": " B(x)", "2f4db7f9d9829052d77224f60f323a0d": "r = a \\frac {\\sin \\theta_1}{\\sin \\psi} = a \\frac {\\sin [q \\theta + \\theta_0]}{\\sin [(q-1) \\theta + \\theta_0]}\\!", "2f4dc7629a8bf976ba233212d7f17230": "r^{2} = a \\sec 2\\theta", "2f4de4c3e6f7134db7524447e05d719c": "([X], \\alpha) \\otimes ([Y], \\beta) := (X \\times Y, \\pi_X^{*}\\alpha \\cdot \\pi_Y^{*}\\beta), \\qquad \\pi_X : (X \\times Y) \\times (X \\times Y) \\to X \\times X, \\mbox{ and } \\pi_Y : (X \\times Y) \\times (X \\times Y) \\to Y \\times Y", "2f4dfa2194275405697742c5f335fccf": "\\frac{-e^2}{R_x}+\\Delta E_0 = 0", "2f4e61c69bc1018c5ab74a7a16a87344": "\\mathrm{d}\\, {*\\mathbf{F}} = \\mathbf{J} .", "2f4e65d26f4154b6e8bbb03e4b534056": "j:\\left\\langle\n{\\mathcal{A}}_{i_{n}=i},{\\mathcal{A}}_{i_{n}=j}\\right\\rangle\n=0", "2f4ed68ecce3bb85959121ee51bb8fde": " \\langle Au,v \\rangle = \\langle f,v \\rangle, \\, ", "2f4fbc3d2a361b53418496af3250e01c": "\\tfrac{1}{6}(n^3 + 3n^2 + 8n)", "2f4fbf7a8cd12a0541e1e9ecb2eefbeb": "\\left[S\\rightarrow aAbBcC\\right]", "2f4fc21cce71ab51549af7847d6466b3": "\\mathbb{R}^{p+q}", "2f4fc5bcf0ba8ce9bbb933605fbf0a1f": "\\left\\{1,...,P-1\\right\\}", "2f504912c28754b8fecd16b8862c6179": "\\left|\n\\begin{array}{ccccc}\n 1 & 1 & 1 & \\cdots & 1 \\\\\n x_1 & x_2 & x_3 & \\cdots & x_n \\\\\n x_1^2 & x_2^2 & x_3^2 & \\cdots & x_n^2 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \\cdots & x_n^{n-1}\n\\end{array}\n\\right|=\\prod _{1\\leq i 4\\sqrt{q}", "2f5d5e32ef49cf1dcd57c1ea2c7489ea": "a \\uparrow^{n-2} b\\,\\!", "2f5d81f1e7b42fb2cb8895615a91107f": "-668''\\sin(l')", "2f5d9c0912265190ceab5f534c1c3e5b": "M(q(t))=\\cfrac{\\mathrm d\\Phi_m/\\mathrm dt}{\\mathrm dq/\\mathrm dt}=\\frac{V(t)}{I(t)}", "2f5e284bb419535268d6664d51884725": "(\\tilde{\\mathcal{P}})", "2f5e4868c2f203f54a98f080278ac4c7": "P_{y,w}", "2f5f30cd22fc721ad301befab63ef0ac": "\\displaystyle A(x\\rightarrow x')", "2f5f4b1ae912904d0c0d9dab69de1538": "y = y_{0}\\left(1-\\frac{x^2}{a^2}\\right)^m", "2f5fbce179a4d8f6591591932fcbe05a": "\\displaystyle s=2R+r.", "2f6026c3b208f222e2701ec3b6bb5a0a": "f:M \\to \\mathbb R", "2f60383d7020cc7833cf810557669740": "R(x_{i_1}, \\ldots , x_{i_n})", "2f60936ddee1edf826d27d792d1c78b7": "\\frac{x_j}{x_k}=\\frac{y_j}{y_k}", "2f60bbc4347ade9c30c13b2d9af56eb2": "\\mathrm{d}p_c/\\mathrm{d}S = 0", "2f6133140d772610228014d96d45c74b": "Risk = \\sum_{i=1}^N E_i R_0 (age_i, gender)", "2f6189a90a22cd271ac24634b720b405": "\\frac{d(fg)}{dx} = \\frac{df}{dx} g + f \\frac{dg}{dx}.", "2f6195326f4dc34968171fafd2271129": "\\beta'_{h,v}", "2f61a4696eb25a4fec2f9194f0266e50": "\\vec E_{||}\\frac{\\partial f}{\\partial\\vec v_{||}}+\n(\\Delta\\vec v_\\perp\\times\\vec B)\\cdot\\frac{\\partial f}{\\partial\\vec v_\\perp}\\approx0", "2f61d7588d9968c90a67a0d586f16fc0": "\\mathbf{F}=\\frac{\\rho_\\mathrm{c}V_\\mathrm{p}}{2}\\left(\\frac{\\mathrm{D}\\mathbf{u}}{\\mathrm{D}t}-\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t}\\right),", "2f61e2fe93e860b8e28862e34703cd9a": "(Margin % / (100% - Margin %) ) * Annual Inventory Turns", "2f61eced18221977b270158072c4132e": " F_u = \\dot{m}(V_{w1}-V_{w2}) ", "2f6215dbd585fc4fa932d58d11a1af4a": "c\\cdot f", "2f624a0717ec5e8860ed97e0a4b43331": "(\\hat p -2\\sqrt{0.25/n}, \\hat p +2\\sqrt{0.25/n})", "2f624f67356f67a8eb2e23cab1fd0d5b": "T \\mapsto -T^{\\mathrm T}", "2f62c1885d8e5b44bc6d1cee16ca29be": "red(f) = \\pi_1(\\Psi_{fRep}^{-1}(f))", "2f62c2a0417b9c1eb766695d24e94ff7": "v_x \\,", "2f62c97d571be6a7233a730270a7484e": "(C_\\beta)_{\\beta \\in Sing}", "2f62d49278419c70a2397d0157bdbedc": "\\int | f_n | \\, d\\mu \\to \\int | f | \\, d\\mu", "2f62e38051a64ddcfb228c71b56738d3": "l \\gets k", "2f634365e065e0316b70ab4443e12d3e": "\\int_\\text{crystal} \\phi_{\\mathbf{R}}(\\mathbf{r})^* \\phi_{\\mathbf{R'}}(\\mathbf{r}) d^3\\mathbf{r} = \\frac{1}{N} \\sum_{\\mathbf{k,k'}}\\int_\\text{crystal} e^{i\\mathbf{k}\\cdot\\mathbf{R}} \\psi_{\\mathbf{k}}(\\mathbf{r})^* e^{-i\\mathbf{k'}\\cdot\\mathbf{R'}} \\psi_{\\mathbf{k'}}(\\mathbf{r}) d^3\\mathbf{r} = \\frac{1}{N} \\sum_{\\mathbf{k,k'}} e^{i\\mathbf{k}\\cdot\\mathbf{R}} e^{-i\\mathbf{k'}\\cdot\\mathbf{R'}} \\delta_{\\mathbf{k,k'}} = \\frac{1}{N} \\sum_{\\mathbf{k}} e^{i\\mathbf{k}\\cdot\\mathbf{(R'-R)}}=\\delta_{\\mathbf{R,R'}} ", "2f63691982912f343d68fb878a61a260": "x_i = \\sum_{j\\rightarrow i}{1\\over N_j}x_j^{(k)}", "2f63c6f45dee7ebebce588b3ebf2e1af": "\\lambda_{CWL}(M)=\\left\\vert\\mathrm{torsion}(H_1(M;\\mathbb{Z}))\\right\\vert\\left((a\\cup b\\cup c)([M])\\right)^2", "2f64080351562c6731934e3be1f60d9b": "f_l", "2f6415bafd31e04c56e2f04a4e186854": "\\boldsymbol{R}=\\sum_{i} \\boldsymbol{u_{i}}", "2f6475dd2dea6a7eca83765f330769df": "\\{P_5\\}", "2f647748903d0d6ddfcb689dccfa6082": "C \\propto \\tau^{-\\alpha}", "2f648772909e091c312c4da97800bf8e": "\\sigma = \\frac{3FL}{4bd^2}", "2f64dc1dfd7adc08ca799a868ef8f207": "(10\\uparrow)^n", "2f65650f2762bc89f38953fed7c53217": "\\Psi (\\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N) = \\prod_{i=1}^N\\psi(\\mathbf{r}_i) = \\psi(\\mathbf{r}_1)\\psi(\\mathbf{r}_2)\\cdots\\psi(\\mathbf{r}_N).", "2f65925079212d9d9b701a497a53c54d": "\\Theta_h = {i | i\\in S, j_i = h},", "2f6659eb5d19657e2b44c92f5bb64f22": "b \\approx 15 AU", "2f6662f7e2e5bd80ac45f9e0019ab14b": " n\\# = 2 < 16 = 4^2.", "2f66813cb2fd46bbf8ac583f4da2e0ff": "\\chi_\\mathbb{Q}", "2f66a47dff483e36c3efdd73566cb219": "S(0)=x~,", "2f66abee9eae3ab5665459b14b788a22": "(b_{1}/a_{1})=8(1/3)", "2f66ac5dbda017472e84c22cfe6c911c": "\\{ w \\in \\Sigma^* \\mid S {\\Rightarrow_G}^* w \\}", "2f66b3bbdd988335ee60e2428a4046c2": " F_x(t) = X cos(\\omega t + \\phi) ", "2f66cc0b50530d881dbc4c21f32066b5": "\\frac{\\partial f_i}{\\partial x_j}", "2f672195dae63584151b33d2c7a4dde5": "o_{ik}", "2f674787c29e112fdf132bba1026cc42": " f(cx) \\ne cf(x) \\ ", "2f678e6b07f5f4756b323345d6a59100": "{C_x = C_Lsin(\\alpha)-C_dcos(\\alpha)}", "2f67940000b9e2ca9bf6cd9ed97cf577": "r_s/(1+r_s)", "2f67d4f72f8a8fd0fca4b5226e3855d7": "\\mathfrak{g} = \\mathfrak{s} \\oplus \\mathfrak{a};", "2f67e954f2e3858e5bfb672b1d75ed18": "AUC_{k,l} \\ne AUC_{l,k}", "2f684802287a2df49e8b3c5fba97e44f": "\\displaystyle{\\mathrm{Ad}\\, \\sigma(x)=(\\mathrm{Ad}\\,(x)^{-1})^t.}", "2f68aeaaa9c49dbda7d1b3ae8337f037": "\\int_0^z \\frac{1}{w \\pm (k + \\frac{1}{2})\\pi} dw = \\log\\left(1 \\pm \\frac{z}{(k + \\frac{1}{2})\\pi}\\right)", "2f68c9e0356b40e8c7149436dfc1eef9": "x^3-x-1 = 0", "2f69728dde9b845d58f0bb76053705ab": "T_{\\mathrm {2n}}", "2f6987a9402371a2b3ced22c80811eb7": "3 \\ .\\ 5", "2f698a12af8802c87e5e9d4fbb93ceee": "B = \\sum_{i = 0}^{n - 1} t^i B_i", "2f699e8519006fc161c39888ac24e598": "\\mathrm{Ker}(\\lambda - A)^{\\nu(\\lambda)} = \\operatorname{Ker} (\\lambda - A)^m, \\; \\forall m \\geq \\nu(\\lambda) .", "2f69c05c53597e75f23f29aa7384aef5": "f(x_0) + f'(x_0)\\cdot (x-x_0).", "2f69c828b42356ba81b85a55cfb4b67c": "c(Y^*, Y)", "2f69e38558dede2980e3fbb36d57033c": "N_{\\ell,m}^\\text{N3D} = N_{\\ell,m}^\\text{SN3D}\\sqrt{2\\ell+1}", "2f6a1ddfe115bcd6763a52ba2c37ee7b": "\\operatorname{E}(X_1^2)=\\beta^2\\alpha(\\alpha+1).\\,", "2f6a35de7cc13bca05e2355b05f0008f": "\nE = E^0 - \\frac{0.05916\\mbox{ V}}{z} \\log_{10}\\frac{a_\\text{Red}}{a_\\text{Ox}}.\n", "2f6a78bd1170d3339282e6cd21123967": "A\\mathbf{x} = \\mathbf{b}", "2f6abb1e2d671619e11ff9eac933bca8": "P(X\\leq x) = P(X p(\\neg S\\vert D)", "2fa6b8d0614f28c88ebd638f2cd83621": "h_C\\left(x^*\\right) = x^*\\left(x_0\\right)", "2fa6e1a5937d65ac9e67f692cebcddef": " \\beta ", "2fa6fedfaba2271914fd7f34f90f015c": "T^3 - V", "2fa71b79156ecf79c394bb6152aee624": "\n\\begin{align}\nN_{xx} & = \n A_{xx}\\left[\\cfrac{\\mathrm{d}u_0}{dx} + \\frac{1}{2}\\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x}\\right)^2 \\right] -\n B_{xx}\\cfrac{\\mathrm{d}^2w_0}{\\mathrm{d}x^2} \\\\\nM_{xx} & = \n B_{xx}\\left[\\cfrac{du_0}{\\mathrm{d}x} + \\frac{1}{2}\\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x}\\right)^2 \\right] -\n D_{xx}\\cfrac{\\mathrm{d}^2w_0}{\\mathrm{d}x^2} \n\\end{align}\n", "2fa7651e71283110c8955961cbd01f33": " (\\operatorname{ask}[S] \\and FV[A] \\subset V) \\to \\operatorname{drop-formal}[[F, S, A]::Z, \\lambda F.Y, V] \\equiv \\operatorname{drop-formal}[[F, S, A]::Z, Y[F:=A], L] ", "2fa77579e133a523fd39ef62b4affd69": "\\sigma _{i}^{2}=\\int_{-\\infty }^{+\\infty }(\\tau -m_{i})^{2}\\cdot p_{i}(\\tau )d\\tau ", "2fa7c288b233719e22533aebdd471eb1": "\\nabla \\cdot \\boldsymbol M = \\sum_{i=1}^n \\frac{\\partial}{\\partial x_i} M_i, ", "2fa7e424c5656f17067e6631acbeb8f1": "\\exp\\left(\\sum_{n=1}^\\infty {a_n \\over n!} x^n \\right) = \\sum_{n=0}^\\infty {B_n(a_1,\\dots,a_n) \\over n!} x^n,", "2fa80e4cf4c92abade4f84b43e4f82cc": "L \\circ P", "2fa8580e5e65a705bc42cbe47dbe6ba5": " D\\nrightarrow C", "2fa85aa31df54ab5c0f5266769bc1682": " \\widehat{\\Omega} ", "2fa864c10566f8e8eb5d7d45d38d01a1": "x^{10} = a", "2fa898a027ae42c35e701e900562f2fb": "\\theta_{up}(X) < \\theta_0", "2fa8a17e0d0a85409a9c3ee3b65239c3": "\\mathcal{F}^{-1}g(x):=\\lim_{R\\to\\infty}\\int_{\\mathbb{R}^n} \\varphi(\\xi/R)\\,e^{2\\pi ix\\cdot\\xi}\\,g(\\xi)\\,d\\xi,\\qquad\\varphi(\\xi):=e^{-\\vert\\xi\\vert^2}.", "2fa97a489a13f6e8f9bb856a8966911a": "\\Delta \\phi = \\phi.\\ ", "2fa9f8cec6c2cacd7f27d66cecb987d0": "k_{m+1} = \\frac{(C_e)_m a_{m+1}}{\\sigma^2_{m+1} + a^T_{m+1}(C_e)_m a_{m+1}}.", "2fa9fd5251274820e6347d205a006869": "\\scriptstyle(X,\\,\\mathfrak{F})", "2faa53fdae3e082677437f0ec8878e10": "F_{\\sigma,x}(z)\\equiv \\sum_{n\\ge 0} \\sigma^n e_n(z) H_n(x)= \\pi^{-{1\\over 4}} e^{-\\frac{x^2}{2}}\\sum_{n\\ge 0} {(-z)^n \\sigma^n \\over 2^n n!}{d^n e^{x^2}\\over dx^n} = \\pi^{-\\frac{1}{4}} \\exp (-{x^2\\over 2} +\\sqrt{2} xz\\sigma -{z^2\\sigma^2\\over 2}).", "2faa6031c118d7f392880d5050ebffc3": "X_\\mathbf{k}", "2faaa492f90dd7cfd4f0d1aae45a8788": "b_i^k=\\left\\lfloor\\frac{k}{2^i}\\right\\rfloor - 2\\left\\lfloor\\frac{k}{2^{i+1}}\\right\\rfloor.", "2faab86691acffc58833437ebac258a2": "t(U) = \\frac{L}{\\sqrt{2U}} \\sqrt{\\frac{m}{q}}\\ + \\frac{2 L_{m}\\sqrt{2U}}{U_{m}} \\sqrt{\\frac{m}{q}}\\ ", "2faab8b2b660704720e4913f15954071": "\\eta_r = \\frac{\\eta}{\\eta_0} = \\frac{t \\rho}{t_0 \\rho_0},", "2faae9d24775c41b860055375539e6c5": "C_\\mathrm{linear}=\n\\begin{cases}\\frac{C_\\mathrm{srgb}}{12.92}, & C_\\mathrm{srgb}\\le0.04045\\\\\n\\left(\\frac{C_\\mathrm{srgb}+0.055}{1.055}\\right)^{2.4}, & C_\\mathrm{srgb}>0.04045\n\\end{cases}\n", "2fab0c2c1260d4eab832ce495d9a945f": "1 ms", "2fab2e25cbc18f06679db797cdb15c86": "Q (r)", "2fabb65c41fae8021ac7db39d4caf51a": "\\arcsin x = -i \\ln \\left(ix + \\sqrt{1 - x^2}\\right) \\,", "2fac295cd153fde9c5872016a3ce1a88": "\\|f*g\\|_r\\le C_{p,q}\\|f\\|_p\\|g\\|_{q,w}", "2fac34c009ccbc4867856b9c9e26c875": "n \\ge1", "2fac397bc6ef67a6de110d958af3af2e": " \\mathcal{L} = \\mathcal{L}_p + \\mathcal{L}_e ", "2facd14784c3303c466eaf725b427fae": "\n\\begin{matrix}\n\\mathrm{I} & a_1 & \\quad & a_3 & \\quad & a_5 & \\quad & \\cdots\\\\\n\\mathrm{II} & \\quad & a_2 = g(\\langle a_1\\rangle)& \\quad & a_4 = g(\\langle a_1,a_2,a_3\\rangle) & \\quad & a_6 = g(\\langle a_1,a_2,a_3,a_4,a_5\\rangle) & \\cdots .\n\\end{matrix}\n", "2fad210191c65753fe181753d9ecd5bd": "Z(z)=i\\sqrt{\\pi}w(z)", "2fad27a28fe70b9b0734c895e09b27dc": "\\hat{\\epsilon}_j", "2fad7f6cf61719bbc2b77d316ed920ec": "RT(a) < RT(b)", "2fadbe0b0391933e41852c27d45b77df": "g:M_2\\mapsto M_1", "2fade8f72244a531ff2d29cfaed0754f": "\\displaystyle D_\\mu F^{\\mu\\nu}=0, \\quad F_{\\mu \\nu} = A_{\\mu, \\nu} - A_{\\nu, \\mu }+ [A_\\mu, \\, A_\\nu]\n", "2fadf4e95c39b54c34098c3d37cb531e": "6\\pi", "2fae011e2533a7316e3c9fe38e786a91": " t_s \\in \\mathbb{Q}_{[0,\\infty]}", "2fae2a00da0ae17d956294ff4d5b7c03": "P_{i+1}(x)", "2fae3f794e61875fe46d0aac78042bd2": "V_{\\rm w} = {4\\over 3}\\pi r_{\\rm w}^3", "2fae4c7b59824ff50cde1a0bb265a126": " \\neg \\neg P ", "2fae6bca5961f03e82ce7882a9bd24e9": "I_{x,A}", "2fae8abd1f6ad5222399bd426727042f": "\\ddot{P} = H_{i,j} P ", "2faeca5f22c405ecdf84b4feb0d0b055": "0.1045\\ldots = e^{-2.25\\ldots}", "2faefa0b7830ab437065311cf1d6c131": " H(x,p) = x^2 + p^2 ", "2faf274b10f0ff83b38748a840899db7": " op_1 \\circ T(op_2,op_1) \\equiv op_2 \\circ T(op_1,op_2) ", "2faf4519707bea4ff1aee37df938ed83": "\\text{N}_2\\text{O}", "2faf530867087c1a4e6fddf738d29cfc": " 100_2 \\rightarrow 1. ", "2faf9914439e1fb86dda70c51dc23dba": "f(\\boldsymbol{\\sigma}, \\boldsymbol{q})", "2fafb37a78939ab5e7b1fa2cb6b433aa": "Re(s)=1", "2fafe6f350ca972205b120b6f32c29c3": "\\mathrm{O_2 + uv \\ energy \\longrightarrow 2O}", "2fb00656f0a78c4e12459876908ce055": "g(k) = \\int e^{-2 \\pi i k x} d \\mu(x)", "2fb0091c313feb2f2c6461e90e05ded0": "\\bigl\\||f_1\\cdots f_{n-1}|^r\\,|f_n|^r\\bigr\\|_1\n\\le\\bigl\\||f_1\\cdots f_{n-1}|^r\\bigr\\|_p\\,\\bigl\\||f_n|^r\\bigr\\|_q.", "2fb070160c75d55f7cf2d7028d4905d0": "L \\subseteq\\Sigma^*", "2fb0fce3d5a8b87b2cf4a21849728a98": "Q = \\iint \\mathbf{q} \\cdot \\mathrm{d}\\mathbf{S}\\mathrm{d} t \\,\\!", "2fb135405d09801f288f2a70bfc11efc": "\\begin{bmatrix}\nV_x\\\\[10pt]\nV_y\n\\end{bmatrix} \n=\n\\begin{bmatrix}\n\\sum_i w_i I_x(q_i)^2 & \\sum_i w_i I_x(q_i)I_y(q_i) \\\\[10pt]\n\\sum_i w_i I_x(q_i)I_y(q_i) & \\sum_i w_i I_y(q_i)^2 \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\n-\\sum_i w_i I_x(q_i)I_t(q_i) \\\\[10pt]\n-\\sum_i w_i I_y(q_i)I_t(q_i)\n\\end{bmatrix}\n", "2fb136641d3b2ff1b696e79fea1dd921": "\\hat{Z_b} = \\sum_{c \\in \\mathbb{F}_d} \\chi (bc)|c \\rangle \\langle c| ", "2fb1785fb03ffc1b21c5fbc150f96fff": "G(a_n;x)=\\sum_{n=0}^{\\infty}a_nx^n.", "2fb1a3b8ae3c5bf2e040af88e46c0aa1": "L_{pl}(d,m)", "2fb22093247fb5d6c896381da0695987": " S \\cdot X = T. ", "2fb2bcb6e6bcd15767ba6835747f647e": " \\mathbf{e}_j = \\frac{1}{\\sqrt{D_{j-1} D_j}} \\begin{vmatrix}\n\\langle \\mathbf{v}_1, \\mathbf{v}_1 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_1 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_1 \\rangle \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_2 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_2 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_2 \\rangle \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_{j-1} \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_{j-1} \\rangle & \\dots &\n\\langle \\mathbf{v}_j, \\mathbf{v}_{j-1} \\rangle \\\\\n\\mathbf{v}_1 & \\mathbf{v}_2 & \\dots & \\mathbf{v}_j \\end{vmatrix} ", "2fb2c5ea48e27131f4f698c44242c88d": "-e/2+127", "2fb2c80f6016498bbfb88fb480dd50d4": "(X,Y) \\prec (X',Y')", "2fb2cc31ea7d84895a1c39e8dd87853e": "\\langle (\\vec{v}+\\vec{0})*a,\\vec{w}*b \\rangle", "2fb2cf3fb763dcffb2285b8cdaaf1124": " \\kappa = 3 + \\frac{ 6 \\alpha^2 ( 93 \\alpha^2 + 41 ) }{ ( 5 \\alpha^2 + 4 )^2 }", "2fb2fa6e7289b6199be3311866cb3f0b": "| g' (x^*) | < 1,", "2fb3051262c2bf17898f56d347671f6d": "\\ker\\left(P^{-1}~T_0(P-\\alpha)\\right), \\quad \\ker\\left(P^{-1}~T_0(P-\\alpha) \\right)^{\\bot}", "2fb3f456438a6d23f4f9fbdf439fd9d1": "\\lim_{x \\to \\infty^{-}}{f(x)} = L", "2fb486f6b57d32684bde30b0d98e7d74": "\\scriptstyle d(f_n(x),f(x)) \\le 2^{-m}", "2fb4fc141f21a00c448a3c71c581587c": "\\operatorname{pred} = \\lambda n.\\lambda f.\\lambda x.n\\ (\\lambda g.\\lambda h.h\\ (g\\ f))\\ (\\lambda u.x)\\ (\\lambda u.u) ", "2fb55582b31926ea5f4421c04f890c92": "x \\to \\left [ (c-a)^{-1} - (b-a)^{-1} \\right ]^{-1} x\\,", "2fb576dae97789422930b7de3be01ce7": "T_{q*q^*}", "2fb5a6170796904344701549c13a4c0d": "f(x) = \\frac{\\alpha x_m^\\alpha}{x^{\\alpha+1}}", "2fb5b3486901b0ef2e6891d453f6ffc1": "P(X_{(k)}= 0.02 ", "2fc8bafb597a231eb8426b1e1033c8ce": "z = \\frac{\\lambda_{\\mathrm{obsv}} - \\lambda_{\\mathrm{emit}}}{\\lambda_{\\mathrm{emit}}}", "2fc8ee89482b2d86833f2f764914a1f4": "\n I = \\int |x,p\\rangle \\, \\langle x,p| ~ \\frac{\\mathrm{d}x\\,\\mathrm{d}p}{2\\pi\\hbar}\n", "2fc9114ba56f7cdc5ea4dc8f0e19d7b0": "\\Box\\varphi", "2fc922c177b2d09d15844fd0f3e4a8e3": " -(n+1)(n-2)~r^n~\\sin(n\\theta) \\,", "2fc96414caaeda383d1fe28c80b29ce7": "\\Lambda(\\bigcup_{i=1}^\\infty A_i) = \\sum_{i=1}^\\infty \\Lambda(A_i)", "2fc9a9e67e09785d7030856b6ae537ca": "\\lim_{x \\to 0^-} \\frac{1}{x^r} = \\begin{cases} -\\infty, & \\text{if } r \\text{ is odd} \\\\ +\\infty, & \\text{if } r \\text{ is even}\\end{cases} ", "2fca0b317883374c5ed1ce6f6dd8f69a": "C_p - C_v = nR", "2fca0f297a30fed2a2d77402864cdabd": "\\mathrm{(SNR)_{C,FM}} = \\frac{A_c^2} {2 W N_0}\n", "2fca40e0061c08cf4af4c1b2ef4f3686": " { 1 \\over {\\beta (5-3q)}} \\text{ for } q < {5 \\over 3}", "2fca4fb53162a009bda951cbd4886417": "_{ ^{c}}\\!", "2fcaa1d0a6a1f998c64107000e39fe80": " \\mathbf u = \\frac{\\int \\vec v f d^3v}{\\int f d^3v}= \\frac{1}{n}\\int \\vec v f d^3v\n", "2fcabad8935fc2ec56f66f89125c516e": "\\Delta \\vec{F}\\!", "2fcb201a41b94e69c7db654e03aec1a3": "\\frac{\\partial}{\\partial t}\\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{\\boldsymbol{u}} \\right] + \\nabla \\cdot \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{\\boldsymbol{u}} \\otimes \\overline{\\boldsymbol{u}} + \\mathbf{S} + \\frac12 \\rho g (h+\\overline{\\eta})^2\\, \\mathbf{I} \\right] = \\rho g \\left( h + \\overline{\\eta} \\right) \\nabla h + \\boldsymbol{\\tau}_w - \\boldsymbol{\\tau}_b,", "2fcb2300eff93e5d20ad6d51b5d9ff41": "e_s = e \\left[ T_{wet}\\right]", "2fcb31bde9e8a1f17aab32066c090416": "\n\\psi(k)\\psi^\\dagger(k') - \\psi(k')\\psi^\\dagger(k) = \\delta(k-k')\n\\,", "2fcbcb6fd24b207860df1733348d847d": "H_{\\mathrm b}(p) = 1 - \\frac{1}{2\\ln 2} \\sum^{\\infin}_{n=1} \\frac{(1-2p)^{2n}}{n(2n-1)} ", "2fcbccd54c5fb9297e5f5a0e992ed098": " |n^{(k)}\\rang = \\frac{1}{k!}\\frac{d^k |n\\rang }{d \\lambda^k}. ", "2fcc2b74ba35fcc5943304a6c9b35eb7": "\\frac{900000}{1000000}={0.90}", "2fcc82319892dcf19e716487858ce98d": " p(\\textbf{x}_k \\mid \\textbf{x}_{k-1}) = \\mathcal{N}(\\textbf{F}_k\\textbf{x}_{k-1}, \\textbf{Q}_k)", "2fcc929f4087979f3966e0feb70d5bc2": "(x_{n+1},y_{n+1})", "2fcccda953787297f9bb9902e44da760": "(I - \\alpha A^T)", "2fccd85c34ae63045c73fa17faeae711": "\\Delta c\\rightarrow 0 ", "2fcd04c84e53b33b8f1b759d08b83f80": "\nP(k) = {\\mathrm{B}(k+a,\\gamma)\\over\\mathrm{B}(k_0+a,\\gamma-1)},\n", "2fcd0d70d162ddbb6f75e77dfa8aea21": "A \\rightarrow \\ldots | \\epsilon | \\ldots ", "2fcdfcc68cc40f5d4d1b486a79798cbf": "\\sigma_1^2- \\sigma_1\\sigma_2+ \\sigma_2^2 = 3k^2 = \\sigma_y^2\\,\\!", "2fce1252ef526dfe6a96031228b5a5b3": "\\beta_{E}(Q)=\\frac{1}{\\ell(Q)}\\inf\\{\\delta:\\text{ there is a line }L\\text{ so that for every }x\\in E\\cap Q, \\; \\text{dist}(x,L)<\\delta\\},", "2fce12d2b05d64c3b9e96764a4e68d1f": "= a C \\frac{\\sin\\frac{ka\\sin\\theta}{2}}{\\frac{ka\\sin\\theta}{2}} \\sum_{j=1}^{N-1} e^{ijkd\\sin\\theta}", "2fce3568693cf96b8572e0d3b5b35963": "P_{\\rm{80}}", "2fce8ffc3bd36491230a3468eb6d0a83": "p(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_1x + a_0 = 0 \\, ", "2fceb4c553dd62765c3380f3e53bdad2": "(x+y)^{n+1} = \\sum_{k=0}^{n+1} \\tbinom{n+1}{k} x^{n+1-k} y^k,", "2fcebcb59d50af30668866eaf314678b": "\\mathbf e_j\\,\\!", "2fcec97225550c63d5633354a8c7f249": "\\sin \\pi z = \\pi z \\prod_{n\\neq 0} \\left(1-\\frac{z}{n}\\right)e^{z/n} = \\pi z\\prod_{n=1}^\\infty \\left(1-\\frac{z^2}{n^2}\\right)", "2fcf083318ec3669e7a40b42dcc24d73": "\\Delta E = E_{-} - E_{+} ", "2fcf0a44a475f531bc47a8345df7263d": "(x_1 x_2 + y_1 y_2,\\ y_ 1 x_2 + y_2 x_1 )", "2fcf0e4e516537d98d794892a10db3cd": "\\left(\\frac{\\partial P}{\\partial T}\\right)_S \\left(\\frac{\\partial V}{\\partial S}\\right)_T -\\left(\\frac{\\partial P}{\\partial S}\\right)_T \\left(\\frac{\\partial V}{\\partial T}\\right)_S", "2fcf6997a4952dee39713dd0830805bd": "{i, j}\\neq g", "2fcf938401405211bcfa0024634a6967": " u_L ", "2fcfbb9ac9cb010d557b8a829293a189": "\n\\begin{align}\nF^{(0)}(s) & = 1, \\\\\nF^{(1)}(s) & = \\int^s_{-1} F^{(0)}(u) \\, du=\\int^s_{-1} 1 du=s+1, \\\\\nF^{(2)}(s) & = \\int^s_{-1} F^{(1)}(u)du=\\int^s_{-1} (u+1) \\, du={s^2 \\over 2!}+{s \\over 1!}+{1 \\over 2!}={(s+1)^2 \\over 2!}, \\\\\n& {} \\ \\vdots \\\\\nF^{(n)}(s) & = {s^n \\over n!}+{s^{n-1}\\over {(n-1)!1!}}+{s^{n-2} \\over (n-2)!2!}+ \\dots +{1 \\over n!} ={(s+1)^n \\over n!}, \\\\\n& {} \\ \\vdots\n\\end{align}\n", "2fcfc6b99df809d80bed555d24c49ec3": "\\frac{\\partial}{\\partial t}\\left( \\frac{E}{\\omega_i} \\right) + \\boldsymbol{\\nabla} \\cdot \\left[ \\left( \\boldsymbol{U} + \\boldsymbol{c}_g \\right)\\, \\frac{E}{\\omega_i} \\right] = 0,", "2fd01712009c29e0fa5ca948fe413a4a": " S_{i,j}^{t+1}=1 ", "2fd0727c8ce90f1ef7758ec81be58953": "dV_g", "2fd07babdef1c42c521307f1bb6577db": "\n\\varphi(s)=\\frac{1-(1-ps)^{-r}}{1-(1-p)^{-r}}\n\\qquad\\text{for }|s|\\le\\frac1p.", "2fd095bcb69ab2d84de1e1267025bc21": "p_2(x)=x^2-4\\,={(x-2)(x+2)}", "2fd0a1c592087f9e4b894ca2fbb9abe0": "\\mathbb{D}^q(\\sin(t))=\\sin \\left( t+\\frac{q\\pi}{2} \\right) ", "2fd140124fbabd0ea9b3e70ea9f98902": "a_{-i}", "2fd15c147369019913f7bc1e0446d16b": "k(x,y) = k(y,x), \\, ", "2fd17715277941473a88e6ad1ed6702f": "\\begin{smallmatrix}E(\\alpha,m)\\,\\!\\end{smallmatrix}", "2fd1e5643b7cba2bd59a2659f4178920": "T(x_1)=T(x_2)", "2fd221a2139b01ff1e6b9f734e70331a": "x_2 \\le x \\le x_3", "2fd2480311be30d2c3ee064dee326536": "E=h\\nu", "2fd270b12904d4c84fcaafbeb941dd50": " (\\mathbf{y}')^\\top \\, \\mathbf{E} \\, \\mathbf{y} = 0 ", "2fd2b23c7b17b6eaeed2d21f81bd53be": "\\frac {a_1 +16a_2}{(a_1+4a_2)^2}", "2fd34b983ce595b2d4e6caced8b6516a": "\\textrm{ZPP} = \\textrm{RP} \\cap \\,\\text{co}\\,\\textrm{-RP}, \\,\\!", "2fd38cda3736f53f66268402fea816c0": "\\exp\\langle u,u'\\rangle = \\langle \\exp u , u' \\exp u \\rangle ", "2fd3a67ad8388b0d03b9221491127c45": "{-\\hbar^2 \\over 2m} \\psi_{xx} = E \\psi \\,\\! ", "2fd3f0ccd3c58ff4888a3bc86f72fcaa": "|z| < R", "2fd43bf634acc450868238d20bce8075": "\\tfrac{30}{7}", "2fd45bea761631a582ef8a2daeb52a7f": " SubCipher_n=DEC_{b_n}(k_{b_n},s_n) ", "2fd4835b28466a6a7784581a52356c7a": " f(x) = f(a) + \\int_a^x f'(t) \\, dt ", "2fd48bbd1d4aacd3302c319c973a3cf1": "\nF = \\frac {G m M_M} {d^2} ,\\quad W = \\frac {G m M_E} {r_E^2}\n", "2fd52120752d96c279a39e2d4cf394ea": "y_r(n)", "2fd52c26776e9edbc721897c0eb84e60": "~v=\\sqrt{2gh}~", "2fd5321c083c3e63e67f72621cca2085": "\n\\ell^\\prime(\\beta) = \\sum_j \\left(\\sum_{i\\in H_j} X_i -\\sum_{\\ell=0}^{m-1}\\frac{\\sum_{i:Y_i\\ge t_j}\\theta_iX_i - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_iX_i}{\\sum_{i:Y_i\\ge t_j}\\theta_i - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_i}\\right),\n", "2fd54e399812df4e0ccb07cc3516dab1": "P(z)", "2fd593050728e81866cad82201b02342": "\\scriptstyle \\ell\\left( \\left[ J_F \\right] \\right) \\, > \\, 0", "2fd59415d6e1e1e102e502d544034c9f": "\\langle\\xi_k(t_1)\\,\\xi_l(t_2)\\rangle=\\delta_{kl}\\delta(t_1-t_2)", "2fd5b619a2f55031a9fee7c072739276": "\\scriptstyle C_c^1(\\Omega,\\mathbb{R}^n)\n\\subset C^0(\\Omega,\\mathbb{R}^n)", "2fd5baa5eb8e505bca19748e6d85bdd9": " = \\lim_{h \\to 0} \\frac{f(x+2h)-2f(x+h)+f(x)}{h^2},", "2fd6033bf8321320740469dfdeb00005": "\\textstyle N\\rightarrow\\infty", "2fd643629d4f482a445cf8e7dbf30c00": "\\overline {\\Delta q}\\,", "2fd64c5379d30ab1974fde2ddc5d788f": "\\left\\vert H_A(f)\\right\\vert^2 = \\frac{2\\sin^4\\pi \\tau f}{(\\pi \\tau f)^2}. ", "2fd66a7acf4fbc2cc07a2ca47f2bc547": "[x,y,z] = (xy)z - x(yz).\\,", "2fd68321ee8969ca69ebbce452866be3": " x_T(t) \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{k=-\\infty}^\\infty x(t - kT) = \\sum_{k=-\\infty}^\\infty x(t + kT).", "2fd6ee14e2f03941ad550b05e1f09f53": "DABDDB", "2fd71c9a97b31ec3f657e56b70cd564d": "x' = x1^{2^{|x|^c}}", "2fd8231498e26c1614f5f116a96a871f": "x_i^*=x_{i-1}", "2fd8623ebd443dea6e7f3df6eb8f97f2": " \\lambda = \\lambda_1 + \\cdots + \\lambda_r ", "2fd8b8022654f47831ca5228c5744a5a": " \\psi (\\xi, \\tau) = \\left[ 1-\\frac{4 (1 + 2 i \\tau)}{1+4 \\xi^2 + 4 \\tau^2} \\right] e^{i \\tau} ", "2fd90ef7806248eaa88c3d4e0a0104d7": "\\Rightarrow r=\\frac{1}{T}\\ln{\\frac{M_aT}{P_0}}", "2fd93c862a857e2951bf7cc3c411b1fa": "S\\subseteq \\mathbb{N}", "2fd9b27cd6c2be3c45a52f249376ac30": "A{\\to}(B{\\to}(C{\\to}D))", "2fda02bb210d97ba61a3faacfc29edb0": " \\boldsymbol{\\theta} ", "2fda16735dfa3d2db5d55d14d841fe5b": " \\begin{align} \\mathcal{G}: \\mathcal{B}(\\mathcal{H}) &\\rightarrow \\mathcal{B}(\\mathcal{H})\\\\\n A &\\mapsto \\displaystyle \\sum_\\alpha |\\psi_\\alpha \\rangle \\langle \\psi_\\alpha | A |\\psi_\\alpha \\rangle \\langle \\psi_\\alpha | \\end{align}", "2fda4a5384d9f819d6b1ad5e61a0122b": "H(w_1+w_2)= Hw_1 + Hw_2 =2Hw_1=0", "2fda624a3f396b4d0cac177f170f512d": "\\lim_{z \\to a}f(z)", "2fda65908b837086f8dc97f9b93b0113": " \\mathbf \\zeta=\\mathbf Z.", "2fda8d4d3a6e86b5eb5556ab29681158": "a(b(cd))", "2fdaa31fb7ad316383070a80a8cc00cd": "X = \\begin{cases}\n2 & \\text{with probability } a, \\\\\n1 & \\text{with probability } b, \\\\\n0 & \\text{with probability } c.\n\\end{cases}\n", "2fdafad1217b31e9779862786ba8b933": "F(r) = \\frac{GMm}{4r^2 R} \\int_{r-R}^{r+R} \\left( \\frac{1}{s^{p-2}} + \\frac{r^2 - R^2}{s^p} \\right) \\, ds", "2fdb4993cd8b5beda68f4dac578cf71b": "j_\\mu=-i(\\Phi^*\\partial_\\mu\\Phi-\\partial_\\mu\\Phi^*\\Phi), \\, ", "2fdb7153102f2287a4271efb63e97d5c": "\\sum |c_\\nu|^2\\log(\\nu)^2<\\infty", "2fdb81ed57f07ce0d215b01bbd8e8c4d": "P_b = \\frac{1}{2} \\operatorname{erfc} \\left( \\sqrt{\\frac{E_b}{N_0}}\\right)", "2fdb8a83997a98ad0cd6f05921f8e3d1": "(Pxy \\and Pyx) \\rightarrow x = y.", "2fdc49a712ff088d30e9c4eb013f54dd": " (g,g^R,h^0) ", "2fdcc037a0c17930f347d20bb9ecab8d": "x_i\\in\\{0,1\\}", "2fdcc834091dd767a53398b76f069fc5": "(A_1,i)", "2fdd54fbfde985ccc74bc39d1ec18423": "f: E \\mapsto \\Re", "2fdd71953992171a32539a8d323b9975": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{z}{w} \\right) =\nw^{1-a_1} \\sum_{h=0}^{\\infty} \\frac{(1 - w)^h}{h!} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1-h, a_2, \\dots, a_p \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n \\geq 1,\n", "2fdd72be85c047b4e269fcc1a3f74c8b": "\n \\sigma_{11} = 2~\\left(\\lambda^2 - \\cfrac{1}{\\lambda^2}\\right)~\\cfrac{\\partial W}{\\partial I_1} ~;~~ \\sigma_{22} = 0 ~;~~ \\sigma_{33} = 2~\\left(1 - \\cfrac{1}{\\lambda^2}\\right)~\\cfrac{\\partial W}{\\partial I_1}~.\n ", "2fddb3b9f0173b79b558555201da05b1": "\\mathbf{C}_{2,2} = \\mathbf{M}_{1} - \\mathbf{M}_{2} + \\mathbf{M}_{3} + \\mathbf{M}_{6}", "2fdddb6cf14a009dafdcc8355dbd23c7": " \\lim_{x \\to c} (f(x) - g(x)) = \\lim_{x \\to c} \\frac{1/g(x) - 1/f(x)}{1/(f(x)g(x))} \\! ", "2fde287f8c4ad963a82b7d373309f8ab": " U = U_{rotating}+U_{applied} \\,;", "2fde34c1180e8e2e7c13b8271a28f714": "\\frac{d\\theta}{d\\zeta}\\xi_d^2\\left(\\frac{d^2\\theta}{d\\zeta^2}\\right)+\\frac{d\\theta}{d\\zeta}\\sin{\\theta}\\cos{\\theta}=\\frac{1}{2}\\xi_d^2\\frac{d}{d\\zeta}\\left(\\left(\\frac{d\\theta}{d\\zeta}\\right)^2\\right)+\\frac{1}{2}\\frac{d}{d\\zeta}\\left ( \\sin^2{\\theta}\\right)=0", "2fde457286969c1ddc8f60a9e00decd4": "H=H_n", "2fde83597d1a4901378ecd021770a9a8": "J(N,P) = 1 - \\frac{\\ln N}{N \\ln(\\frac{1}{1-P})} \\left[ 1 + O(\\frac{1}{N}) \\right]\\rightarrow 1. ", "2fde8e10aea782ec74b4c10995fc78be": "X_{i:j}", "2fdebb11ebf97006a738420af05450ac": "\\frac {\\partial V_x} {\\partial y} = \\dot \\gamma ", "2fdebf238b45e7838484d09ec6dbe7b9": "x(t) = r(t) \\cos(t) = ae^{bt} \\cos(t)\\,", "2fdfb3131492421611bccba6f4779f9d": "I\\mathcal{Q}_{Hur})\\}", "2fdffbe31b2c974d4b20b43194399f0e": " \\left\\vert u\\left( x\\right) -v\\left( x\\right) \\right\\vert \\leq C\\left( b-a\\right) ^{2}\\sup_{\\left( a,b\\right) }\\left\\vert u^{\\prime\\prime }\\right\\vert. ", "2fe0a497c264b2a26b2bdd603c95f279": "\nv_{\\mathrm{combined}} = \\sqrt{v_{1}^{2} + v_{2}^{2} + \\cdots + v_{n}^{2}}\n", "2fe0a97ccd8aaeb3c5a659b3b54ab9f0": "v = H_0 D\\ ", "2fe10f0805f24cabd641caab70aa7acb": "Z(x)\\leq \\sum_{T_{-I} <\\cdots < T_{-1},\\; T_0>\\cdots > T_j} 2 \\left(\\prod_{k=-I}^j B_{T_k}^x\\right) = 2\\left(\\prod_{T>0}(1+B_T^x)\\right)^2<\\infty.", "2fe10fe1483ade5174243852e21cd472": "I = \\frac{bh^3}{12}+bhr^2", "2fe1165589655deda071af67a793f105": "(t_0,x) \\in \\Omega", "2fe1358d4e9bbcc7d6eaddec5706fe8d": "\\partial\\Omega\\setminus\\Gamma", "2fe14aa42dc210141dca523aae138e29": "i\\!i", "2fe1f10658dcb413e749e84fefeb0a87": "\\mathbf{v} = \\boldsymbol{\\omega} \\times \\mathbf{x}", "2fe229220da5d24731f228b99de8360f": "\\textstyle b_{0} = p(t_x, a_{(-1,0)}, a_{(0,0)}, a_{(1,0)}, a_{(2,0)})", "2fe25c9ce0f84c7ba125eda87e38fb96": "fg-(-1)^{|f||g|}gf=i(f,g) \\,", "2fe27c3e9972e34e293654e7dedf152c": "\\hat{h}", "2fe2c6b425a9180ec91ed2ea75971f73": "\\begin{align}\n\\sigma_\\mathrm{avg} &= \\tfrac{1}{2}(\\sigma_x + \\sigma_y) \\\\\n&= \\tfrac{1}{2}(-10 + 50) \\\\\n&= 20 \\textrm{ MPa} \\\\\n\\end{align}\n", "2fe2e793dcefaa1e8810e32c4a44c41e": "\\frac{\\delta T}{\\delta q_j}=\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac{\\partial (L+V)}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial (L+V)}{\\partial q_j}\n=-\\frac{\\delta L}{\\delta \\dot{q}_j} - \\frac {\\partial V}{\\partial q_j}\n", "2fe2f4ef82723de838441abce516a3d2": "\\begin{align}\n C(F, \\tau) &= D \\left( N(d_+) F - N(d_-) K \\right) \\\\\n d_\\pm &=\n \\frac{1}{\\sigma\\sqrt{\\tau}}\\left[\\ln\\left(\\frac{F}{K}\\right) \\pm \\frac{1}{2}\\sigma^2\\tau\\right] \\\\\n d_\\pm &= d_\\mp \\pm \\sigma\\sqrt{\\tau}\n\\end{align}", "2fe4138058ff1e0332bed7571b8275b0": " Coefficient~of~variation = \\frac{s.d.}{m},", "2fe419d0fe825a57c263fe389f220021": "\\frac{c}{n(\\omega)}", "2fe41c71800372d709624f005741c476": "\\frac{x}{e^x-1} = \\sum_{n=0}^\\infty B_n \\frac{t^n}{n!}", "2fe433984d5588d8aba5a1bafa879b4f": "d(p,q)=\\frac{1}{2} \\log \\frac{|qa||bp|}{|pa||bq|}", "2fe444b6024f3028e7608cad7913f58a": "\\aleph_{\\beta}", "2fe46c9a976cd6d388a51a27c5d1c4a3": "\\forall A: A \\subseteq \\varnothing \\Rightarrow A = \\varnothing\\, .", "2fe4826578dd0d6be924e38a8ed93ea5": "\\|f\\|_{[k+1/2]}^2 = \\|f\\|_{(k)}^2 + \\int_0^{2\\pi}\\int_0^{2\\pi} {|f^{(k)}(s)-f^{(k)}(t)|^2\\over|e^{is} -e^{it}|^{2}}\\,ds\\, dt.", "2fe4bd09ff07eedd743fc8885f9a0f56": "\\tfrac{1}{2}(\\ln(1)+\\ln(n)) = \\tfrac{1}{2}\\ln(n),", "2fe4c7924507068b38aec880c5e18d6c": "{p(S\\vert D)\\over p(\\neg S\\vert D)}={p(S)\\over p(\\neg S)}\\,\\prod_i {p(w_i \\vert S)\\over p(w_i \\vert\\neg S)}", "2fe4e4534ec47fcd4728de11dd66f23d": "V = v_{i,j}", "2fe62a3cbf20ca46cea4140360579186": "\\begin{matrix}{4 \\choose 1}^2{2 \\choose 1}\\end{matrix}", "2fe66cc393c55cf67cc136e701f8c990": "f = f \\circ \\pi", "2fe6a4a2fc85f54b8b94c95e725b1ff6": "\\Phi(E)=e^{\\beta(E-\\mu)}+1.\\,", "2fe6b11ab657e9e21f7169411b5aeafb": "M_D=P\\cdot L(r,Y)", "2fe70fe4e2b6e16b24cb1d20dfbdfa06": "u_i (g)= \\sum_{j\\ne i:j \\in N^{n-1}(g)} b(l_{ij} (g))-d_i(g)c", "2fe745aa4b761086ea09323386c9def5": "\\sum_{n=1}^\\infty\\frac{1}{n(n+1)}", "2fe7b02e529aa6bd86181d2d29565e5a": "V_1 = \\frac{\\left( \\frac{V_S}{R1} + I_S \\right)}{\\left( \\frac{1}{R_1} + \\frac{1}{R_2} \\right)}", "2fe7f12e4ef7bf7a0cba00696e5441f7": "\\left( \\frac{3}{2} \\right) ^2 \\times \\frac{1}{2}", "2fe8907fd3cbd36afb7d93bd0a1b180a": " V \\,", "2fe90c3650bab7a89364b133eb68f78b": " x_n=(ax_{n-r}+c_{n-1})\\,\\bmod\\,b,\\ c_n=\\left\\lfloor\\frac{ax_{n-r}+c_{n-1}}{b}\\right\\rfloor,\\ n\\ge r,", "2fe9db78488b62c3a073ba8512dfbbe0": "B \\smallsetminus A = \\{ x\\in B \\, | \\, x \\notin A \\}. ", "2fe9f3cde94ca5b533d9d4080fbefa57": "\\rho:\\mathcal{X}\\times\\Theta\\rightarrow\\mathbb{R}", "2fea0be78a9c1df4cae67b511abdfa41": "\\tilde{H} = \\epsilon_{ijk} \\tilde{E}_i^a \\tilde{E}_j^b F^i_{ab} = 0", "2fea1210a35115c600c1bdb2cf0c8986": "\\boldsymbol{\\hat{v}} = \\boldsymbol{v} / \\sqrt{x}", "2fea1caf39f22284817c2c31272c966a": "\nE_{\\ell r} = \\int d\\mathbf{r} \\ \\rho_\\text{TOT}(\\mathbf{r}) \\ v(\\mathbf{r}) \n", "2fea59b7e470acac5ec1589d504da14e": "Completion", "2feb2507d95e666f726c5036aa8b79fd": "N^{a}", "2feb64e1de49496557181bd861eacca6": " > 3", "2feb73d08fd5ae2882f4d322515676ed": "A(\\rho)<\\infty", "2feb80df08430dfc2766f262cbdb54f5": "su (2)", "2fec231b56869374b6b1e58146f54cc3": "1.9593", "2fec2ff59c011ffbbaad9b35b299b08d": " {\\| z \\| \\over \\| z \\|^2} = {1 \\over \\| z \\|},", "2fec34c1201499909dd8a699a9f46e8a": "\\mathcal{L}_X \\phi^A", "2fec59f13952c3c0e029b2bb6c01ceab": "\\{x \\mid \\phi(x)\\}", "2fec7f708b941d65fb6d4c84c5e6c0ed": "100\\frac {\\bar{X}_m - \\bar{X}_f}{\\bar{X}_f + \\bar{X}_f} ,", "2fec827636cc2b51ee3dde49c0e55a10": "\nJ_{\\mu\\nu}^{ } = 4 g_{\\mu \\nu}\n", "2fecd795e5da2dde788c5b3cd4815c0b": "e^\\Lambda", "2fed157aa818b478d84e2304e40f634f": " \\langle \\psi | \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} \\psi_x^* & \\psi_y^* \\end{pmatrix} = \\begin{pmatrix} \\quad \\cos\\theta \\exp \\left ( -i \\alpha_x \\right ) & \\sin\\theta \\exp \\left ( -i \\alpha_y \\right ) \\quad \\end{pmatrix} ", "2fed258999c744eac992f41220906e15": "P = F/A", "2fed5dca0849fbfa95ae7503018d7657": "\\displaystyle{\\mathcal{U}H_0 = E_0.}", "2fed69cec4da0fbdfb587af5e757e477": "\\int x^{2} R\\,dx= \\frac{6ax-5b}{24a^{2}}R^3+\\frac{5b^{2}-4ac}{16a^{2}} \\int R\\,dx", "2fed8ce93a24e357a456ff755c13781c": " x_{k-1}-x_l ", "2fedd7c6fb5e588cea7f1c6c5558e1a5": "\\iint_D v\\nabla \\cdot \\nabla u \\,dx\\,dy =0 \\, ", "2fee16e828ba2f48cfdb48821185f5a5": " uLv - vLu = - \\frac{d}{dx} \\left[ p(x) \\left(v\\frac{du}{dx} -u \\frac{dv}{dx} \\right ) \\right]. ", "2fee47592325ffec70a6704c4dfa7dce": "\\psi_n^{(m)}(x) = \\sum_{k=0}^m{m \\choose k} (-1)^k 2^{(m-k)/2}\\sqrt{\\frac{n!}{(n-m+k)!}} \\cdot \\psi_{n-m+k}(x) \\cdot {\\mathit{He}}_{k}(x).\\,\\!", "2fee754e6692515003cceb224cf748ff": "\n\\frac{1}{t}\\sum_{\\tau=0}^{t-1} \\sum_{i=1}^NE[Q_i(\\tau)] \\leq \\frac{B}{\\epsilon} + \\frac{E[L(0)]}{\\epsilon t} \n", "2fee7d8ecaa16317766372c0e6d5af1f": "p_j={A_{ji} \\over k_i}", "2fef1bfe5a2a14ec370270cbbab899f8": "\\mathbf{t}_0", "2fef63545923b2fd1fbc527dfd27a71f": "\\nu:\\Sigma\\times\\mathit{L}\\to[0,1],~(p,a)\\mapsto[0,1]", "2fef9ed06442f58d4a5feaa5c48cfe22": "\\mathbf{p}_k \\in P", "2fefc74f1f837079888b8d65bb9db631": "E_\\mathrm{gap}", "2fefe486d5ae5f3aa59f452abbe2d535": " \\psi(x)= \\int_{x_0}^x p(t)^{-1/2}\\, dt", "2feff5a17fbe9b0aeeeb4f185b489ca4": " \\left\\{\\frac{p}{q}\\right\\}=\\left\\{\\frac{q}{p}\\right\\} ", "2ff0142c2d605ba638d917925f11989f": " c_5 = -0.000850208, \\,\\!", "2ff02a48bc6249b1cae22aa11f1eeace": "B^*\\subseteq B", "2ff03f7225ff111c65816dfff90415c6": "P_{A,B, \\Lambda}", "2ff058b166cecd7ef715b38ab1772e36": "p_2(x,t) = K \\cdot P_c^2 \\cdot \\frac{\\partial^2}{\\partial t^2} E^2(x,t)", "2ff0619d0ef47dce3888a415d0cea28b": " \\widehat{\\mu}_{Y \\mid x}^\\pi = \\widehat{\\mathcal{C}}_{YX}^\\pi \\left( (\\widehat{\\mathcal{C}}_{XX})^2 + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\widehat{\\mathcal{C}}_{XX}^\\pi \\phi(x) = \\widetilde{\\boldsymbol{\\Phi}} \\boldsymbol{\\Lambda}^T \\left( (\\mathbf{D} \\mathbf{K})^2 + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\mathbf{K} \\mathbf{D} \\mathbf{K}_x ", "2ff086c8883e2812528ff63b96ef2ede": "j = 3g-3-n - \\sum d_i ", "2ff0ae4fd297e2968ee6b0ec5d73592d": "N[(1 + K)^M - 1]", "2ff0d703978891af4758cdcf23e37535": "x^2 - 10x + 18 = 0.\\,\\!", "2ff16c88148c268bdddd3fa6e8f2bd51": "D= D_x-deg(D_x ) \\infty_2+v_1 (D)(\\infty_1-\\infty_2)", "2ff193f585cef575e63a1fe862a977bb": " E' = y' + \\frac{1}{2y'^2} = 3.1 + \\frac{1}{2(3.1)^2} = 3.1", "2ff1cd2c428c1e8682095263be5c7cdd": "S=\\{1,2\\}\\cup\\Sigma", "2ff23cf669e66ce3e4aa6437e9d39d11": "= C\\sum_{j=0}^{N-1} \\int_{-\\frac{a}{2}}^{\\frac{a}{2}} e^\\frac{ikx\\left(x^\\prime - jd\\right)}{z} \\,dx^\\prime", "2ff246f41423071a7c8024653707109e": "\\frac{v_{in}}{R_1}", "2ff2d3baac6ea933d7662b6a38871e52": "f = \\frac{1.875^2}{2\\pi l^2} \\sqrt\\frac{EI}{\\rho A}", "2ff2d45e983c58e96c9ba33d69721ce8": "3N", "2ff2eda97da4d0bc661b3ae8957869a1": "E_{1} c_{2} = c_{1}", "2ff2ef334ed433b9b2af99960031f146": "f : X \\to X' ", "2ff30a1b2ba101d3ff69256d569d4b8b": "|g_1\\rangle", "2ff36961e712800c5d43080821794db8": "y'' + p(x)y'+q(x)y=g(x)\\;", "2ff37c23ea7d5779bdb3f2566d339626": "A\\cdot e_i + A^{-1}\\cdot 1", "2ff382949f992e73195b6de034d08468": "L\\subseteq \\bigcup\\limits_{q=2}^\\infty V_{n,q}.", "2ff38dcc1b2070b5ed014f211b30290a": "HA \\rightleftharpoons A^- + H^+", "2ff3a5a07ba5994f68c37224f799bc8a": " \\int_X |f|^2 d \\mu < \\infty, ", "2ff3fa23d4845ac52c5b0d8691c4f2e9": "A_2 = L_2 U_2 P_2", "2ff4044fe1eabef5e7cf8b3451148a2e": "\\frac{v^2}{c^2}", "2ff4228dc552241c046a1869b3a1f666": "n(\\vec r )\\ \\stackrel{\\mathrm{def}}{=}\\ n_s(\\vec r)= \\sum_i^N \\left|\\phi_i(\\vec r)\\right|^2. ", "2ff43879d48c748f8da51d6a77b95832": "\\mathbf{k} \\cdot (\\mathbf{\\epsilon} \\, \\mathbf{E_0}) =0", "2ff4ab3ecc0b93c0ecf0aeb204f15486": "\\left(1-\\frac{\\mu_1}{m_1}\\right)^{1/\\omega_1} = \\left(1-\\frac{\\mu_2}{m_2}\\right)^{1/\\omega_2} = \\ldots = \\left(1-\\frac{\\mu_c}{m_c}\\right)^{1/\\omega_c}", "2ff4b4048096e3531a9fd1c2027b168c": "\\sigma = \\sqrt{\\frac{{n}_1^2+\\cdots+{n}_m^2-m}{12}}", "2ff4df9abfdd78947cff09b2d7574de8": "W_n = \\frac{Wo}{P V f} \\frac{(T_H + T_K)}{(T_H - T_K)} = B_n \\frac{(T_H + T_K)}{(T_H - T_K)}", "2ff4f4279d56b3609cad380c90381950": "(p,0,Z,p,AZ)", "2ff51d7ef8d9b81e7ce28af268166bb2": " z_t\\sim iid~ N(0,1) ", "2ff51e576f8c97e84260b58e2aad3664": "D^{(0)}", "2ff53b92b76afc9eac3e4f692fa66262": "(\\mathbf{p}-\\mathbf{p_0})\\cdot\\mathbf{n} = 0", "2ff573d174cbfcc9047955f3f085582d": "c_{14} = -2.18429 \\times 10^{-8},\\,\\!", "2ff589545ebb9e4484bc7bca5555fae8": "\nn_i=\\sqrt{\\epsilon_i}\n", "2ff5b435550ad6b9603e15464b05eb2a": "\\delta_0", "2ff5cdd459dac42e5a9606bec8ea6eb4": "Y_{3}^{3}(\\theta,\\varphi)\n={-1\\over 8}\\sqrt{35\\over \\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\quad\n={-1\\over 8}\\sqrt{35\\over \\pi}\\cdot{(x + iy)^3\\over r^{3}}", "2ff5da9c694bd024cb40054834798d51": "\\triangle f = -\\nabla^i\\partial_i f", "2ff5ea6379c68e4993f42958a242a06d": "\\bar{r_i}", "2ff5ecc26132fe12630e68ff8e6ddfa9": "g(s) = (1-s)^{a-1}", "2ff6381403bbb311f21908b1857f2726": "\\left| y(t) \\right| \\le \\sqrt{1 + \\tfrac12 \\varepsilon} \\quad \\text{ and } \\quad \\left| \\frac{dy}{dt} \\right| \\le \\sqrt{1 + \\tfrac12 \\varepsilon} \\qquad \\text{ for all } t.", "2ff649216c4a0b6def510057fe33199d": "\\mathrm{TSS} = \\mathrm{ESS} + \\mathrm{RSS},", "2ff6603d504faa8c20cccc3ca1e71238": "x(t_0 + 2\\Delta t) = 2x(t_0 + \\Delta t) - x(t_0) + \\Delta t^2 x''(t_0 + \\Delta t) + O(\\Delta t^4) \\, ", "2ff765c77da59e50599944aca398ac34": "P(t) = \\prod_{i = 1}^{g}{(1-a_it)(1-\\bar{a_i}t)}", "2ff76786da0fd198456c6f569ab08988": " \\eta_{V_k}=(\\eta_j)_{j\\in V_k} ", "2ff7ad2d04bc1875e7fd478ec0441da2": "\\frac{45}{32}", "2ff819ef5e09d220533c6071f282631b": "\\frac{\\partial f(M, T)}{\\partial T} < 0.", "2ff82eef42693c8e64054c3741fd94f4": " h>0 ", "2ff848321efec32e56935a32b17d2325": "\\gamma_1^2 = \\gamma_2^2 = \\gamma_3^2 = {-1}", "2ff8a8221b3c9164dced0bc1f751997a": "\\forall (s_i, t_i) \\in \\Gamma", "2ff8d2c43e7e50fb40352ace5d9a60c8": "\n\\begin{matrix}\nv_0 & = & (1,1,1,1,1,1,1,1) \\\\[2pt]\nv_1 & = & (1,0,1,0,1,0,1,0) \\\\[2pt]\nv_2 & = & (1,1,0,0,1,1,0,0) \\\\[2pt]\nv_3 & = & (1,1,1,1,0,0,0,0). \\\\\n\\end{matrix}\n", "2ff8d3c16b1cc0367a849bf7b5f139c8": "\\frac{d}{dx}{\\mathcal C}_n(x) = {\\mathcal C}_{n+1}(x).", "2ff8d823dbd595f7aa853c8ecedf441f": "\\mathbf G=\\langle G,S,W\\rangle", "2ff9358720c48ea257c046e8afc01227": "\\mathcal{L}_1 = \\mathcal{L}(\\Alpha,\\Omega,\\Zeta,\\Iota)", "2ff94130e401690c185a3dbed9ae151e": "U^{\\{ a,b\\} }", "2ff95d73f19734d2e8a53d03933e9231": "k[x_1, \\ldots x_n]", "2ff9b08d263db5e0a90632e313bb9709": "\\boldsymbol{F} = {q'\\over 4\\pi\\varepsilon_0}\\int dq {\\boldsymbol{r} - \\boldsymbol{r'} \\over |\\boldsymbol{r} - \\boldsymbol{r'}|^3}.", "2ff9cb6b35eb8593450f01ecab16328f": "\\|I_\\alpha f\\|_q\\le C\\|f\\|_p.", "2ff9d2bc34ec2599fa782682d49ad2a5": "I_L = I_O \\left ( 1 - \\frac{V_S}{V_O} \\right )^{k_O}", "2ff9e1d35e858ebcbbbaf4b66ae72c91": " [ 2 w_m w_a \\rho_{am} \\sigma_a \\sigma_m] \\quad ", "2ffa134d25782856aa86424869a45a9d": "h_{\\overrightarrow{X}}", "2ffa2c3bb52726c3a7077e0052d81260": "\\mathbf{N}^{-1}\\mathbf{M}", "2ffa81875a46150ca10e13e53937def5": "C_{p min}= \\frac{SFD k_a}{V_d(k_a-k)}\\times\\{\\frac{e^{-k\\tau}}{1-e^{-k\\tau}}-\\frac{e^{-k_a\\tau}}{1-e^{-k_a\\tau}}\\}", "2ffaea3ef0cb7f7a98ce59402167b506": "\\delta P = Z_h \\, Q", "2ffb02144416c401af18b6b40724ed2e": "X=(X_{i,j})\\in\\mathbb{R}^{(n)}", "2ffb32fa8376afe2f69bf237dca621ce": "\\epsilon_{LJ}", "2ffb579f0eca962da8a9175dcc079735": "x \\equiv y \\pmod p. \\,\\!", "2ffbb9d67bb1caf88915f2a98881997f": "\\begin{align} \\nu(M) \n& = \\int \\frac{\\sqrt{M^2-1}}{1+\\frac{\\gamma -1}{2}M^2}\\frac{\\,dM}{M} \\\\\n& = \\sqrt{\\frac{\\gamma + 1}{\\gamma -1}} \\cdot \\arctan \\sqrt{\\frac{\\gamma -1}{\\gamma +1} (M^2 -1)} - \\arctan \\sqrt{M^2 -1}. \\\\\n\\end{align} ", "2ffbc6b4d34fd286f210da4a6e4c0f75": "\\ RMSE = \\sqrt{\\frac{\\sum_{t=1}^N {E_t^2}}{N}} ", "2ffbc770af8c888fb388bc386ac61ba4": "\n\\text{Current yield} = \\frac{\\text{Annual interest payment} }{\\text{Clean price} }.\n", "2ffbf869b6d2a1110a674fb9c947f38c": "1^3+3^3+\\dots+9^3 = (7\\cdot 5)^2 \\,", "2ffc3b987f0e4dec359ad8e612663805": "\\frac{65}{64}", "2ffc3f669facf3ed29a5b410a4a60521": "\\left(\\frac{dr}{ds},\\ r\\frac{d\\theta}{ds}\\right)", "2ffc8370e55f5ce61206a2a2b36ceebb": "\\mathfrak{P}^{109}", "2ffcb0d7ef633be0124c366b71242f7b": "x=x_1", "2ffcd9846b346abb319e395f4c0d43f6": " \\ p ", "2ffcdaeb3b54269f4fe4e3fc6138c040": "S^{n}=G/P", "2ffd2c406f7501afe058c502c7865df2": "r_1 = (S \\to X X, \\emptyset, \\{Y, A\\})", "2ffd2d291a711d09242d543c8b01a4de": "\\overline{X}_{i}", "2ffd313cc3312b27d72e02e92ae2cdc1": "n = \\frac{A}{{4\\over 3} \\pi R^3}", "2ffd6a8b5985ae143c39f2f6190a1e5a": "c_1=c_2=1000", "2ffd7c842745d22ed59221897703b821": "f(m, x_1, \\ldots, x_n) = \\sum\\limits_{i=0}^mg(i, x_1, \\ldots, x_n)", "2ffe22542970f244f35ebe1b7ff961f0": "(a;q)_\\infty^{-1} = \\prod_{k=0}^{\\infty} (1-aq^k)^{-1}", "2ffe4e77325d9a7152f7086ea7aa5114": "max", "2ffe85c942cd9ee66e321e7f80f32878": "f(y) - f(x) = \\nabla f ((1- c)x + cy) \\cdot (y - x)", "2ffe90a7b96bb1ea85ca00d07202cf94": "h(x,\\lambda)=\\sqrt{(2x+2x\\lambda)^2+(x^2-1)^2}\\approx\\sqrt{\\left(\\frac{\\Lambda(x+\\epsilon,\\lambda)-\\Lambda(x,\\lambda)}{\\epsilon}\\right)^2+\\left(\\frac{\\Lambda(x,\\lambda+\\epsilon)-\\Lambda(x,\\lambda)}{\\epsilon}\\right)^2}.", "2ffec2cdfb1a33b413a207446d8e0d94": "Z=\\sum_{n=0}^{\\infty } \\frac{(280n+19)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^3{99}^{2n+1}}\\!", "2ffee9504446816353da3c9bddfb65de": "I_{s}", "2fff148be97b3cecd8bf3a0aef0c0fa9": " \\hat H = {\\hat{p}^2 \\over 2m} + V(\\hat q ) ", "2fff8777de7ed0cfe19066961fbc52d6": " M_{00}(z)= {a(z)b(z)\\over a(z)-b(z)},\\,\\, M_{01}(z)=M_{10}(z)={a(z)+b(z)\\over 2(a(z) -b(z))}, \\,\\, M_{11}(z)= {1\\over a(z)-b(z)}.", "2fffb56c61f0fec868c7a53110cdc71d": " F(a_1, \\ldots, a_n) \\ge F(g_1, \\ldots, g_n) ", "3000221cba1c6d37e99cb5269be7ebc4": "\\langle X \\rangle", "30003c635497e6db01fd876cdb26f782": " \\textbf{a} = c\\textbf{g} +c\\textbf{f} \\pmod q ", "300074053859a37dbe79745bcc05a4f0": "f:\\mathbb R^n\\to\\mathbb R", "3000c2a6f7247b62971fbeb341e4df8f": "c_{2n-1} = c_{n-1} \\cdot c_n - c^p \\cdot c_n^p + c_{n+1}^p", "30015dea778039899b185c0a72e24807": " \\big. H_0=\\sum_i^N \\frac{p_i^2}{2m}", "30018972f9840b2cb119e6890a0f770e": "\\Omega_{1,-1/2}\\propto\\binom{(x-iy)/r}{z/r}", "3001a25207c588a16b4060e335e85668": "2 \\, ", "3001ee72b5ba15218662aafced267336": "=\\frac{1-q}{1-q} \\cdot \\frac{1-q^2}{1-q} \\cdots \\frac{1-q^{n-1}}{1-q} \\cdot \\frac{1-q^n}{1-q}", "30026c9b76f1b91d5781e70efa239dc0": "(\\lambda f.(\\lambda x.f\\ (x\\ x)) (\\lambda x.f\\ (x\\ x)))[\\lambda x.f\\ (x\\ x) := p\\ f] ", "30028a8f529f2d27c179a7153397490e": "\\text{x quus y}= \\begin{cases} \\text{x + y} & \\text{for }x,y <57 \\\\[12pt] 5 & \\text{otherwise} \\end{cases} ", "3002b5f0d4420da35d3bc2a8d6d55688": "g \\vec{J} = \\left\\langle\\sum_i (g_l \\vec{l_i} + g_s \\vec{s_i})\\right\\rangle = \\left\\langle (g_l\\vec{L} + g_s \\vec{S})\\right\\rangle,", "3002db16ba226d0981aab42a6abbbf91": "\n\\tan \\theta = \\frac{\\cos \\lambda}{\\sin \\eta \\sin \\lambda + \\cos \\eta \\cot(15^{\\circ} \\times t)}\n", "3002e3ace33d68b105925d11e65fac66": "k[y_1, ..., y_d]", "30035302408b26dccdd0337f4ec3cda6": "\\left(\\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "3003bb50a70e73e40ab1e7c0a2ce5f80": "2\\pi/17", "3003c39d34fc481a1edd41c45563b384": " x_1 + x_2 + x_3 = - \\frac{b}{a}, \\quad x_1 x_2 + x_1 x_3 + x_2 x_3 = \\frac{c}{a}, \\quad x_1 x_2 x_3 = - \\frac{d}{a}.", "3003eba93600f9d86772f53e34bc7edf": "O(1,3)", "3003ef787d5e828c8ca9526116fa88df": "\\tfrac{1}{2} \\sqrt{N}.", "3004643daada22c17cc1f92b689e12bf": "(\\nu,\\,M)", "30048b4d894b3daf2ad85f038457b23d": "a_n = c_1a_{n-1} + c_2a_{n-2}+\\cdots+c_da_{n-d},", "30048c4a1572d25b4ac0ef4db9feac76": "dp_e = \\frac{dm_{fuel} \\ (1 - \\eta) \\ v_e}{\\sqrt{1 - \\frac{v_e^2}{c^2}}}", "30050d2c7f3d97bdda84ead65cd7b737": "2^{10} = 1024 \\approx 1000 = 10^3", "30054ebd0a5a2c6e77082907124b666e": "a_{ijs}", "30058eab31d947d8f5f580a78dd6ffe3": "^T", "300598bf321db79fc510b096fa3dd5ea": "h_L \\;", "3005c74c9ec8c4caf662618096d59cbd": "\\mathbf{N}:\\left(\n\\mathbb{Z}_{2}\\right) ^{2n}\\rightarrow\\Pi^{n}", "3005ece02a4905586f61337e6adb35f9": "F(4) + F(6)", "30061277aea7b970dfb9ef9fae9d12d2": "\\pi_1(\\mbox{Spin}(p,q)) = \\begin{cases}\n\\{0\\} & (p,q)=(1,1) \\mbox{ or } (1,0) \\\\\n\\{0\\} & p > 2, q = 0,1 \\\\\n\\mathbf{Z} & (p,q)=(2,0) \\mbox{ or } (2,1) \\\\\n\\mathbf{Z} \\times \\mathbf{Z} & (p,q) = (2,2) \\\\\n\\mathbf{Z} & p > 2, q=2 \\\\\n\\mathbf{Z}_2 & p, q >2\\\\\n\\end{cases}", "30065108651ecee569ee75050adde8fa": "\n\\begin{align}\n\\mu_z=-1\n\\end{align}\n", "30069e8370775dca3e04bcafe86fe8ea": "\n\\begin{align}\nN_{h-1} & = [a_1\\dots a_k]_b \\\\\nN_{h-2} & = [a_{k+1}\\dots a_{2k}]_b \\\\\n& {}\\ \\ \\vdots \\\\\nN_0 & = [a_{l-k+1}\\dots a_l]_b\n\\end{align}\n", "3006a4b7a6ef9d629a7c218855542887": "\\partial\\mathcal{M}", "3006e1b36dfefc07c813556ecc6dd925": "\\begin{bmatrix} \n1 & 0 & -1 \\\\\n2 & 0 & -2 \\\\\n1 & 0 & -1 \n\\end{bmatrix} = \\begin{bmatrix} \n1 \\\\\n2 \\\\\n1 \n\\end{bmatrix} \\begin{bmatrix} \n1 & 0 & -1\n\\end{bmatrix} = \\begin{bmatrix} \n1 \\\\\n1 \n\\end{bmatrix} * \\begin{bmatrix} \n1 \\\\\n1 \n\\end{bmatrix} \\begin{bmatrix} \n1 & -1\n\\end{bmatrix} * \\begin{bmatrix} \n1 & 1\n\\end{bmatrix}", "30070731588e28f008d5f2e480e43606": "\\tau(\\ldots,\\sigma_{k},\\sigma_{k+1},\\sigma_{k+2},\\ldots) = \n(\\ldots,\\sigma_{k-1},\\sigma_{k},\\sigma_{k+1},\\ldots)", "3007384978e1ca3dd77ce45238c479d3": " \\tan ( \\theta_1 ) = \\tan \\left( \\theta_2 + \\tfrac{\\pi}{2} \\right) ", "300776c04f95f8b63ef74a62efbd6b26": " Output(\\omega) ~ = ~ H(\\omega)~ F(\\omega) ", "30077a52f1f3e32dd89d83233efabf48": "T_{clock}", "3007987254cada94e12cf134e14dc35b": "\\hat{x}_{j-1} ", "3007d71982cf30a9fc24383ce87aa196": "p(\\boldsymbol\\theta|\\mathbf{D}) = \\left[\\prod_{i=1}^n p(y_i|\\boldsymbol{x}_i,\\boldsymbol\\theta) \\right] p(\\boldsymbol\\theta).", "3008984aa94f35d8b2b0c97daacc4b37": "p_{t=0}(x,p)=p", "3008e33aab5692a255ca43a8f5f8d566": "\\tfrac{G(3M-4G)}{M-G}", "3009076753a8a18e607e1f13989be2e5": "v=\\frac{q}{n}", "30090acbe7da1bd05bb58e5453823e20": "I_2=\\int f(\\theta) d^3 x", "300938f40503d4af57c5f93c804f7b82": "f(3)=6", "3009bfc30b3b7a1b18d4851731e58fb8": "\\operatorname{E}(X_{(k)}) = {k \\over n+1}.", "3009c2caf5045eb338ae8767e0a282fd": "90 \\cdot a_n", "3009f064277ec1ed3a85dbc8d03bb77b": "d_1=x_2^{(t)}-x_1^{(t)}\\,", "300a0cf313760dc4a9d49717ffe8b3d5": "\\hbar\\,", "300a6c64d12e7838d14855112b06ecab": "\\begin{align}\nS[k]\\ &\\stackrel{\\text{def}}{=}\\ \\frac{1}{P}\\int_0^{P} s_P(t)\\cdot e^{-i 2\\pi \\frac{k}{P} t}\\, dt\\\\\n&=\\ \\frac{1}{P}\\int_0^{P} \n \\left(\\sum_{n=-\\infty}^{\\infty} s(t + nP)\\right)\n \\cdot e^{-i 2\\pi\\frac{k}{P} t}\\, dt\\\\\n&=\\ \\frac{1}{P} \n \\sum_{n=-\\infty}^{\\infty} \n \\int_0^{P} s(t + nP)\\cdot e^{-i 2\\pi\\frac{k}{P} t}\\, dt,\n\\end{align}", "300a7a4b9534da0650a265ff003679b2": "Ax^2+ Bxy + Cy^2 + Dx + Ey + F = 0 ,\\,", "300a96f73b9546b09a4c02cf9b74e685": "\\Delta\\left[ \\ln (S)\\right] = \\rho/\\sigma \\cdot g_0 \\cdot r_0 \\cdot ( 1 + x/2 - 3/2 \\cdot x^{1/3} )", "300af5cfbd7f35fee4ff4c6c683f2641": "\\rho(\\omega_{S,i}, \\omega_E) = |\\omega_{S,i} - \\omega_{E}|", "300aff41947bb68290a4a57f0181d62f": "\\overrightarrow{V} = \\overrightarrow{Va} + \\overrightarrow{Vr}", "300b40d9a25b9585a87e6926a95f59a5": " d\\langle E \\rangle = T dS - \\langle p\\rangle dV .", "300b616517bb0223ca2ca57594a6999b": " \\qquad =\\begin{cases}\n\\displaystyle \\frac{\\,\\pi\\,}{2n}\\tan\\frac{\\,\\pi\\,}{2n}\\ln2\\pi + \\frac{\\pi}{n}\\sum_{l=1}^{n-1} (-1)^{l-1} \n\\sin\\frac{\\,\\pi l\\,}{n}\\cdot \n\\ln\\left\\{\\!\\frac{\\Gamma\\!\\left(\\!\\displaystyle\\frac{1}{\\,2\\,}+\\displaystyle\\frac{l}{\\,2n}\\!\\right) }{\\Gamma\\!\\left(\\!\\displaystyle\\frac{l}{\\,2n}\\!\\right)}\\right\\}\n,\\quad n=2,4,6,\\ldots \\\\[10mm]\n\\displaystyle \\frac{\\,\\pi\\,}{2n}\\tan\\frac{\\,\\pi\\,}{2n}\\ln\\pi + \\frac{\\pi}{n}\\!\\!\\!\\!\\!\n\\sum_{l=1}^{\\;\\;\\;\\frac{1}{2}(n-1)} \\!\\!\\!\\! (-1)^{l-1} \\sin\\frac{\\,\\pi l\\,}{n}\\cdot \n\\ln\\left\\{\\!\\frac{\\Gamma\\!\\left(1-\\displaystyle\\frac{\\,l}{n}\\!\\right) }{\\Gamma\\!\\left(\\!\\displaystyle\\frac{\\,l}{n}\\!\\right)}\\right\\} ,\\qquad n=3,5,7,\\ldots \n\\end{cases}\n", "300b65625dcf5e29bfc3b6fe68d05a0a": "x \\wedge y\\,", "300bad34711aac76a3df6681a453308d": "\\tau_1, \\dots, \\tau_n", "300c2099bcd658b291fcbe156b0fca72": " \\vdash (r,\\epsilon,Z)", "300c628c41e89381bf2580cbe7f80b2c": " S_0 = \\emptyset ", "300c968f69abd467df49c77c1f318821": "\\langle E\\rangle = u = -{1\\over Z}\\partial_{\\beta}Z", "300cd23f44369c8b7c41d898b09b3206": "s_1 = r_1 - cx_1 (\\mathrm{mod}\\,q) ", "300d1382aba94013adb09cf94bfbf830": "\\varphi = \\left(1 + \\sqrt 5\\right)/2", "300d2be5b5272ac8a93b8035432c2380": "\\mathbf{N}\\left( \\mathbf{v}\\right) ", "300d5f7635da62a64de30572520315b4": "r_E = r_0 + \\frac{D}{E}(r_0 - r_D)", "300d859282168b6e293229dfb7fc2459": "K_{-0}", "300da5c324315cbf0d6714558c1501f9": "\\ln(I_l) - \\ln(I_0) = (- \\sigma \\ell N + C) - ( - \\sigma 0 N + C) = - \\sigma \\ell N \\,", "300dd8ecbf409a3a1a345444e8108505": "D[g(\\xi_j)] \\equiv D(\\xi_j) = e^{ i \\xi_j D(X_j)}", "300ddebcd7cbafd039c7953d36424062": "I=\\{i_1,\\ldots,i_p\\}", "300e1f8827f0fa20ee899bb9e5c99c85": "\\langle v(t),i(t) \\rangle", "300e6f76e5b5dd1ead149cf958fe6039": "D_{2n} \\cong C_n \\rtimes C_2", "300e7687b5ef3b5e7b57bbd1b7ca701e": "\\mathbf{v} = \\left[ \\begin{matrix} v_1 & v_2 & \\cdots & v_{n - 1} & v_n \\end{matrix} \\right] = \\left( \\begin{matrix} v_1 & v_2 & \\cdots & v_{n - 1} & v_n \\end{matrix} \\right)", "300e8f5ec4863b76b9c7e4e76b349545": "p_0", "300ee3ff236042f53b377dab7d92d577": " d^2 - {d_0}^2 = kt \\,\\! ", "300f2dafc9a9a125d2c07b9875acd105": "H^2 = \\left(\\frac{\\dot{a}}{a}\\right)^2 = \\frac{8 \\pi G}{3} \\rho - \\frac{kc^2}{a^2} + \\frac{\\Lambda c^2}{3}", "300f45a0e6e4f0d0e8c162731ee67351": "\\ \\sum_{m=0}^{M-1} a_{m}y_{n-m} = \\sum_{k=0}^{n-1} b_{k} x_{n-k}", "300f81b4bfd15869d59b03e706553d57": "\n\\begin{align}\n& \\left(\\csc\\left(A - \\frac{\\pi}{6}\\right), \\csc\\left(B - \\frac{\\pi}{6}\\right), \\csc\\left(C - \\frac{\\pi}{6}\\right)\\right) \\\\\n& = \\left(\\sec\\left(A + \\frac{\\pi}{3}\\right), \\sec\\left(B +\\frac{\\pi}{3}\\right), \\sec\\left(C + \\frac{\\pi}{3}\\right)\\right)\n\\end{align}\n", "300f94f4be28c3b0b846bd5df82d5baf": "\n\\Phi_B = \\iint_S \\mathbf{B} \\cdot d\\mathbf S.\n", "30102e8c3a5b4d7533795927851d52fd": "\\displaystyle{{1\\over 2\\pi i} \\int_{-\\infty}^\\infty {f(s)\\over s-z}\\, ds ={1\\over \\sqrt{2\\pi}} \\int_{-\\infty}^\\infty f(s) \\widehat{g_z}(s) \\, ds = {1\\over \\sqrt{2\\pi}} \\int_{-\\infty}^\\infty \\widehat{f}(s) g_z(s) \\, ds= V_yPf(x).}", "3010813eaba3bcbb3df370efcfc672e6": "(\\lambda, \\mu, \\nu )", "3010899268c88de01caea0a53e7dfa8c": "u_e = u - \\rho_w g z", "3010a5ff579e03431358aa31c0dd551d": "\\tilde{h}\\in\\mathfrak{a}^{\\ast}\\not\\ni h\\,", "3010f096ba2bf0e1c6cbf7237fcd7228": "v = \\sum_i x_i \\alpha_i", "301110e731b0bf70cf9385538d0cb268": "Eq.5", "301116cb9997258d8c36457b1032c9b4": "\\Pi_f=1-e^{-BR}", "301120c8202d96f6dc51ff76c5525d96": "W \\setminus \\operatorname{int}\\, K \\cong \\left(M \\setminus \\operatorname{int}\\, A \\right) \\times \\left[0,1\\right],", "301122d1b550712cd9d82979d8e4ab86": " \\langle \\mathbf{u} \\rangle = \\frac{1}{n}\\sum_i n_i \\mathbf{u}_i = \\frac{1}{n}\\sum_i \\mathbf{j}_{{\\rm n}, \\, i} ", "3011494cbe81f0e05d5cdf271db3f6ed": "s > \\sqrt{3 / 2}\\,l", "301159a0a71fcd5111917ab41edc94a1": "h_{-1}=1\\,", "30116ade2f332a03c337e5510169c76a": "f_1\\ ,\\ f_2\\ ,\\ f_3", "301267dc4eafc60623d61388fa8c7033": "\\!\\mathrm{I}(x,y)= \\langle x,y \\rangle.", "3012cd6e2f3255888c36e704b368015e": "\\bold n = ( \\bold p_2 - \\bold p_1 ) \\times ( \\bold p_3 - \\bold p_1 ), ", "3012d8f8d1b7ca6a7eb6081643201785": "f(x) = \\sum_{\\nu=1}^nc_\\nu\\rho \\left(\\frac{\\theta_\\nu}{x} \\right)", "30137d49b2f86d106b9b2f167bef0fee": "\\epsilon(S)", "3013926625245a744f3ac7af79ed4d8a": "\\langle \\Psi_1 , \\Psi_2 \\rangle = \\int\\limits_{\\mathrm{ all \\, space}} d^3\\mathbf{r} \\, \\Psi_1^*(\\mathbf{r},t)\\Psi_2(\\mathbf{r},t) \\,,", "3013f3794e2ef8364d89f80e1ac63d7f": " \\mathbf{G} = e^\\mathbf{F} =\n\\begin{bmatrix} \\dots & \\mathbf{A}_d^{-1}\\mathbf{Q}_d \\\\\n \\mathbf{0} & \\mathbf{A}_d^T \\end{bmatrix}.", "30141a8d92975f5471b6f240c3efd611": "x = 1,2,3,4", "301422e016dea8b6e7a3c2cc2c4945f9": " Z(s) = \\operatorname{Tr}(\\Delta^{-s}) = \\sum_{n=1}^{\\infty} \\vert \\lambda_{n} \\vert^{-s}.", "3014323601491a678511ab5a192381d0": "\\mathbf{K}=k_1\\mathbf{b_1}+k_2\\mathbf{b_2}+k_3\\mathbf{b_3}", "30144febe3f8ea4e1b711bf4a38ebe46": "\n{\\rm PDF}_z \\,\\,\\, \\sim \\,\\,\\,{1 \\over {2\\sqrt z }}\\,\\,\\,{1 \\over {\\sqrt {2\\pi } \\,\\,\\sigma }}\\left[ {\\exp \\left( { - \\,\\,{{\\left( {\\sqrt z - \\mu } \\right)^2 } \\over {2\\,\\sigma ^2 }}} \\right)\\,\\,\\, + \\,\\,\\,\\exp \\left( { - \\,\\,{{\\left( { - \\sqrt z - \\mu } \\right)^2 } \\over {2\\,\\sigma ^2 }}} \\right)} \\right]", "301462ca1b5c4e3170538e4097986f43": "K_\\mathit{rw}(S_{w})=K{_\\mathit{rw}^o}S{_\\mathit{wn}(S_w)}^{N_\\mathit{w}}", "301478f06ed61c5989e5947c1e5d0565": "\\phi= \\operatorname{atan2}( \\rm{BC} \\sin \\alpha, \\rm{AC} \\sin\\beta )", "30148788b5fe535a7f367d28d58b2413": "h_s = \\frac{\\left(1-t^2+t^4\\right)^{1/2}}{1+t^2}, \\,", "3015c4dbb84c38c777bdcce61aacdbda": " \\kappa\\equiv\\lambda/e^2", "3015f2a47b9c01e97ff3d57622ac7ae5": "\\begin{bmatrix} 1 & 0 \\\\ -\\frac{1}{R} & 1 \\end{bmatrix} ", "30160e84e65559d8404511cbceed6d4a": "E_k=\\frac{1}{2} I \\omega^2", "30164d983bead66f93423b995bbd0794": " q = \\frac{ \\Theta_i - \\Theta_a }{ R_T }", "3016552723a6dff53fb2521adf11227b": "(q(T_w))(x)", "301665c93bc034441f5eb9276bf0edc0": "\\Delta t_i = \\frac{(i-1)d \\cos \\theta}{c}, i = 1, 2, ..., M......(1) ", "301678c45cc810adc925eaf7e14e2425": "P_p~dp = \\frac{Vf}{N}~\\frac{4\\pi}{h^3\\Phi_p}~p^2dp", "301695a13c9df84a8ab370bcb381ad97": "h=g + i\\omega", "3016a841ff0cc6e4e09606aef63db74e": "\\begin{align}\n-k & = i j k k = i j (k^2) = i j (-1), \\\\\n k & = i j. \n\\end{align}", "30173e1130bb35935f6871f3f46f2149": "u_{k+1}", "30175172277c76201c91c42da8032954": "k=R/N_A", "3017b2315225c6c85be4a0d82b973dd2": "O(n \\, {\\log |\\Sigma|})", "3017cf55a214b0966264bea1e75381a9": "H_{\\tau} \\in -K_0 - \\sum_{k=1}^{\\tau} L_d^p(K_k)", "3017d5e3a7c513e44ee26a278a1aa45a": " Y ", "3017d84961ccd2397da6ac0da36d9e43": " Fx(K_1,K_2,\\ldots,K_n)= \\sum_{n_1=0}^{N_1-1} \\cdots \\sum_{n_m}^{N_m-1} fx(n_1,n_2,\\ldots,n_N) e^{-j \\frac{2 \\pi}{N_1} n_1 K_1 -j \\frac{2 \\pi}{N_2} n_2 K_2 \\cdots -j \\frac{2 \\pi}{N_m} n_m K_m} ", "3017d911efceb27d1de6a92b70979795": "dt", "301864ef03fa7322558cb03fedd82fa2": " M_4 = \\left\\{\\frac{1}{2}(1,-i,-1,-i),\\frac{1}{2}(1,-i,1,i),\\frac{1}{2}(1,i,-1,i),\\frac{1}{2}(1,i,1,-i)\\right\\} ", "301888645b39c6e21ad354617a28d161": "x=(1,0)", "30189fd62384cb0e3db7b67e4a4e6117": "v_M", "3018c07bcce5731866d679e864d11e92": "\\mathbf{J}= \\frac{\\epsilon_0}{2i\\omega}\\int \\left(\\mathbf{E}^\\ast\\times\\mathbf{E}\\right)d^{3}\\mathbf{r} +\\frac{\\epsilon_0}{2i\\omega}\\sum_{i=x,y,z}\\int \\left({E^i}^{\\ast}\\left(\\mathbf{r}\\times\\mathbf{\\nabla}\\right)E^{i}\\right)d^{3}\\mathbf{r} .", "301922334ffddcaf0f90cf47dac73a24": "\\mu_0 \\left ( \\bigcup_{n = 1}^\\infty A_n \\right ) = \\sum_{n = 1}^\\infty \\mu_0(A_n)", "301952e30356f7cacd913a854e259e38": "m_{\\nu} << m_D << M_{NHL}", "30196618d3415436daa5966e5003f9e4": "R^2 = -2\\cdot\\ln U_1\\,", "30196ad51c1f631829f795f80488d831": "\n\\varepsilon _n=-\\left(\\lambda -n-\\frac{1}{2}\\right)^2\n", "3019bbab300fccf6e269ba16af42dda8": " A = \\frac{4 \\pi}{3} N_A \\alpha, ", "3019e80aedd05692ff5400fe46c27ec3": "x^2(\\sin(1/x))", "301a1ba01ce71cf88155635dc27a597e": "|a-c|=|b-d|.", "301a1f6f096e5e5d396b6290fb45ea29": "f_{\\mathbf{k}}^h", "301a4c55453cb66ea259cd93067a72c3": "C_0 = 1 \\quad \\text{and} \\quad C_{n+1}=\\sum_{i=0}^n C_i\\,C_{n-i}\\quad\\text{for }n\\ge 0.", "301a5579e0d26cb2f5ee49b30b1eff78": " [\\mathbf{A}]\\cdot \\mathbf{x} = [\\mathbf{b}]\\mbox{,}", "301a8fc17de4980c57990ab01eae7423": "x(t) = \\int_0^t {p(\\tau )h(t - \\tau )d\\tau } ", "301a97a1f26efe7a623700ba7194c2a9": "C, \\lambda_+, \\lambda_->0, \\alpha<2", "301af0943d5b8a6a26ec29eda31f44f4": " 100 \\ \\mbox{g}\\,NH_3 \\cdot \\frac{1 \\ \\mbox{mol}\\,NH_3}{17.034 \\ \\mbox{g}\\,NH_3} = 5.871 \\ \\mbox{mol}\\,NH_3\\ ", "301b3cc96b449726583b1c2e86033576": "\\kappa = \\frac{6 E_1 E_2 (h_1 + h_2)h_1 h_2 \\epsilon }{E_1^2 h_1^4 + 4 E_1 E_2 h_1^3 h_2 + 6 E_1 E_2 h_1^2 h_2^2 + 4 E_1 E_2 h_2^3 h_1 + E_2^2 h_2^4}", "301b9b0e3cd4fa69cd84e5a3de17fab1": "\\rho(u)\\approx u^{-u}", "301c349a0d80e44af6cec86fabede15b": "G=\\mathbb{Z}_n", "301c97ca4f0ed4e9903061ddc14b45e4": "\\mu^2 \\to \\mu^2 \\frac{ e^{\\gamma_E} }{4 \\pi}", "301cdead9973c7580e9b5b588d72379b": "\\int_0^1 \\frac{dx}{1+x} = \\ln 2,\\text{ or, equivalently, }\\int_1^2 \\frac{dx}{x} = \\ln 2.", "301d01b8e1a84ddc93fb5c16f3f9de28": " k_x^2+k_y^2 > \\frac{\\omega^2}{c^2} ", "301d251459701f60c27be3229f1c4122": " b ", "301d98e99daa3f96443c2dc92042b9e8": "y=x^{[x^{[x(\\cdots)]}]}", "301dee8289e8b2f122e09b0f0022f4b0": "\\frac K{2-K} = e^{-4\\phi_0} - 1", "301e77c1563bec8e58e72a448e6e8728": "(1+x)^3=1+3x+3x^2+x^3 \\,", "301ef4d46b0fcde73075f6d8bc876509": "(x_1,x_2,x_3)\\cdot (y_1,y_2,y_3) = x_1y_1+x_2y_2+x_3y_3.", "301f0bcf3fff607c98113f83735dcd57": "\\mathfrak{e}_6\\oplus\\mathfrak{u}_1", "301f1943545ff6d496183db27b14d790": "a^2-b^2 = (a+b)(a-b)", "301f363fc6ed0c468478751c76c7362e": " g 0)", "302625dd512c6ad605a718ea0fd1f5d6": "\\Pi_e", "30263fa5d4aa32dc07845778a8a195b8": " \\displaystyle{ 0.41\\, W } ", "3026d8b1691aaf1bcb1cbdab78b2ca12": " f(x) = x^2\\,\\!", "302710281b976889205e8f77325fd9d9": "\\left|p\\right|", "3027281a758ee3f28ac30998db7173c2": "\\left\\{ w \\right\\}", "30274688dfaac461d2d0c6f04b55969d": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "302858688b147aafbb36b014754af304": "\\frac{x}{x+1}\\,", "302858fe47285ea7590d7c9836b4121f": "x = x_{1}Ax_{2}", "302865b5240b7c5ff424841a17746a34": "X_u = \\begin{pmatrix} -\\sin u \\sin v \\\\ \\cos u \\sin v \\\\ 0 \\end{pmatrix},\\ X_v = \\begin{pmatrix} \\cos u \\cos v \\\\ \\sin u \\cos v \\\\ -\\sin v \\end{pmatrix}.", "30291b0149bfe1721d9dbdbd3429a656": "\\operatorname{perm}(J - I) = \\operatorname{perm}\\left (\\begin{matrix} 0 & 1 & 1 & \\dots & 1 \\\\ 1 & 0 & 1 & \\dots & 1 \\\\ 1 & 1 & 0 & \\dots & 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & 1 & 1 & \\dots & 0 \\end{matrix} \\right) = n! \\sum_{i=0}^n \\frac{(-1)^i}{i!},", "30292373fa031eca5f4d99809b55cdc7": "d={{R}_{\\text{E}}}\\left( \\psi +\\delta \\right) \\,,", "3029242024b5fc716eb29ddeae7fada5": "\\mu(\\{x\\in X\\,:\\,\\,f(x)\\geq t\\}) \\leq {1\\over g(t)} \\int_X g\\circ f\\, d\\mu.", "302968cd5bc367a28d4659a8b8b8fdcd": "\\partial^\\mu \\partial_\\mu = \\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\nabla^2 ", "302979cb700fc53fd4b963f0715d2dae": "\\begin{smallmatrix}L\\ \\propto\\ M^{3.5}\\end{smallmatrix}", "3029ceaff88dcc08da331b2c1d29cc94": "\\sum U_i", "3029e4849de9447a70e4528a7663b6d3": "\\scriptstyle I_c", "3029e7eaeeb9984e0d74316703b1aa39": "\\ell/t^2=\\ell^1t^{-2}", "302a0fec41071ac62ca003101be83560": "p_i^2 | (n - p_i)", "302a1f56261e699d0c3bfaff0ad4cd55": "H = 1 + \\ln\\left(\\frac{\\sigma}{\\sqrt{2}}\\right) + \\frac{\\gamma}{2}", "302a828e3e0ca2e258c4181091bf1b46": "\\forall\\,\\underline{b} \\in\\R^p:\\,E[b^\\top X|\\beta_1^\\top X=\\beta_1^\\top x,\\ldots,\\beta_k^\\top X=\\beta_k^\\top x)\n==c_0+\\sum_{i==1}^{k} c_i\\beta_i^\\top x", "302ab1eeeeaa148aa41591e765008aab": "e^\\xi-1=u\\xi.\\,", "302ad4d8b770142ab35ccdeff057cbef": "_kV^r_1 = 0.5\\left(_kV^i_1 + _kV^i_2 + _kV^i_3 - _kV^i_4\\right)", "302ae1991d4fddeba282c4be7cd78ae3": "\n\\begin{pmatrix}\n 1 & 1 & 1 & 0 \\\\\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 1 \\\\\n 0 & 0 & 1 & 1 \\\\\n\\end{pmatrix}\n", "302af0b6df73e4bf73297e6d21b95101": "\n\\begin{align}\n a & = \\sqrt {xYZ} \\\\ b & = \\sqrt {yZX} \\\\ c & = \\sqrt {zXY} \\\\ d & = \\sqrt {xyz} \\\\ X & = (w - U + v)\\,(U + v + w) \\\\ x & = (U - v + w)\\,(v - w + U) \\\\ Y & = (u - V + w)\\,(V + w + u) \\\\ y & = (V - w + u)\\,(w - u + V) \\\\ Z & = (v - W + u)\\,(W + u + v) \\\\ z & = (W - u + v)\\,(u - v + W).\n\\end{align}\n", "302b401d6ff8fde78811a1f2336acacc": "\\sum _{i=1} ^n \\alpha _i ^2 \\lambda _i", "302b73fdcdf9404920ec438ee212a45a": "-js=\\cos(\\theta)", "302baaa84fd62987cca0542e7db7a9b8": "k_2 = \\frac{\\sqrt{2m(E-V_0)}}{\\hbar},", "302bc3e237fb71d2c5cf16a0893665a3": "\\displaystyle{G_{\\mathbf{C}} = G\\cdot P =G \\cdot \\exp i\\mathfrak{g},}", "302c00387deb45f41aafc3d3ae0c8df5": "T = 2 \\pi \\sqrt{ \\frac {I}{\\kappa} } \\,", "302c2108588c511c8315badd0e2b442e": "t_m^2", "302c7295bdab28faac277a476b5a5ac3": "\\mathbf D (\\mathbf r , \\ t) = \\int_{-\\infty}^t dt' \\int d^3\\mathbf r' \\ \\varepsilon (\\mathbf r, \\ t ; \\mathbf r' ,\\ t') \\mathbf E(\\mathbf r', \\ t') \\ . ", "302c84145ed993859837a0b2ac318984": "T=A + \\partial B", "302c84d83546c808f498b7b6d857a096": "\\mathbf{b_{T:T}}", "302dd1940a97b0335d0e027585234919": "d\\mathbf{x}\\cdot d\\mathbf{x} = \\cfrac{\\partial x_i}{\\partial q^j}\\cfrac{\\partial x_i}{\\partial q^k}dq^jdq^k\n ", "302e69459c879569d5d0bfb52fce2bf0": "f\\colon L \\rightarrow L", "302e7279d116d7f6b12080f6f7edd3c7": "R=I/T", "302eac2a23461c34115618528753da36": " \\mathbb{Z}/p ", "302ebb84ddb662387434dead0edcaebf": "\\iint_S K\\, dA = 2\\pi\\chi(S) - \\int_{\\partial S}k_g\\,ds", "302f3b99c0b4368de2591f1e1c3161a8": "S_xV = \\{\\beta \\in T^*_xV - \\{ 0 \\} \\mid \\ker \\beta = \\xi_x\\} \\subset T^*_xV", "302f3e9c1979605d3796592a47bdc61c": "f(w)={1\\over 2\\pi i} \\int_\\Gamma {f(z) \\, dz\\over z-w}", "302f3f6673003062d13ac890555676af": "r_1 = r_2 > 1 \\,", "302f7e97b0f3aae2b3506f4d6c630761": "I_B << I_2 = V_B / R_2 \\ ", "302fce2c6b7954d1bf1338db7e7b545f": "\\langle v, \\varphi(x_i) \\rangle = 0", "302febbd1c6b060141a9616e6a478a1d": "{d\\over dx} \\left ( \\int_{y_0}^{y_1} f(x, y) \\,dy \\right )= \\int_{y_0}^{y_1} f_x(x,y)\\,dy", "3030867e5792681a851ad33bc79022c8": "\\mu \\pm 3 \\sigma = np \\pm 3 \\sqrt{np(1-p)} \\in [0,n].", "303093f3055d2ee403dc45b127b84be5": "Dis(X,U_{d})", "3030b498bdef4ac80f344cbcf0b867b7": "u()", "3030e3adae145cb168b24db22ab64280": "\\varphi + \\mathrm{d}\\varphi", "3030e68bce07fb6b296e45dc679578ad": "y_{1,t} = c_{0;1} - B_{0;1,2}y_{2,t} + B_{1;1,1}y_{1,t-1} + B_{1;1,2}y_{2,t-1} + \\epsilon_{1,t}\\,", "3030ea8961e17aa4cdb77a4632b8f4a7": "\\Omega(M_x,N_y)", "30310e799ee1611493ef2a876e218c34": "\\frac{1}{(-x,-qx^{-1};q)_\\infty}", "30312f7042e4c027a03cc2967f04ab83": "1000_2", "30318ee7595ea6edb92908c349efead6": "exp(-iz)", "3031b5123ff4d00ff94642205136aab5": "\\left(\\rho u A \\right)_{r} - \\left(\\rho u A \\right)_{l}", "3031bb9574a736e821efb28fb581268f": "D=\\{S_1,S_2,\\dots,S_K\\}", "3031d3a326112345d350f33100c241fb": "\n\\int_{S=S_{\\infty}+S_D+S_A}\\left[\\rho\\,u\\,\\vec{q}+\\left(p-p_{\\infty}\\right)\n\\vec{i} - \\vec{\\tau}_{x}\\right].\\vec{n}dS\\,=\\,0\n", "30322420470dd712f5df0be79cd1a299": "v_i(k)\\,\\!", "3032314835149f560de3785992605747": "\ne^{-n\\lambda} \\lambda^{(x_1+x_2+\\cdots+x_n)} \\cdot\n{1 \\over x_1 ! x_2 !\\cdots x_n ! } \\,\n", "303260cfb2b33a93539afc6cbc293b91": "Pr[M^w(1^k)=1]=1", "3032a68a2900c0ca4ffabc9019557d1e": "C=\\frac{N}{2}\\log_2 N", "3032e2e551d99390789dfcc1556ca8f6": "\\pi(x, a, q) = (\\text {number of primes } \\leq x \\text{ such that } p \\text{ is in the arithmetic progression } a + nq, n \\in \\mathbf Z), ", "3032f39aa66c1bfde95937f7a05af5fa": "x_2,x_4,x_3", "3033c74292142e1d12436553586ac545": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 2}{4 \\choose 2}^2 \\end{matrix}", "303450f0f570d8e237388c3436a7ae1f": "nR=N k_B\\ ", "30348795989a681f80a06a2c73098545": "J \\approx \\frac{\\pi a^3 b^3}{a^2 + b^2}", "3034b9f076881994c8f6222e8b347864": "\n\\sum\\limits _{m}p_{m}\\nabla q_{m} = \\left(\\mathbf{P}\\cdot\\nabla\\right)\\mathbf{Q}+\\mathbf{P}\\times\\nabla\\times\\mathbf{Q},", "3034bf021705c3776eb0f4f08ce319cd": "F\\left (x,y,y',\\cdots y^{(n-1)} \\right )=y^{(n)}", "3034de8ce3aff37c317808a292414c7f": " pq = \\tfrac14 \\left(L^2+ 27M^2\\right).", "303517a2dc2b8cb23e32160b8f52fdff": " \\Psi_i ", "303524c9a333de5baad295c06c66ff17": "((x,y),R)", "30353560b89031ef7ce1be8d4a087d50": "\\scriptstyle F(x_1, x_2, \\ldots, x_n) = 0", "303543f04a98454cc47238e717c1bb25": "{\\bar{DH}}_4", "303548e1bb91aa9d81329d3807ee1be1": "E_{\\rm d}({\\rm AB}) =[E_{\\rm d}({\\rm AA}) + E_{\\rm d}({\\rm BB})]/2+(\\chi_{\\rm A} - \\chi_{\\rm B})^2 eV", "30358bc6a06347469626958a0c599232": "\\Delta f = \\nabla^2 f = \\nabla \\cdot \\nabla f ", "3035a2d1e5855d3085a073873c6e9e6c": " (Rl) ", "30361dac5243c2e4b3652d8ae8b4cba7": "{dC_n\\over d\\beta} >0", "3036494fc398d8c0de5527bb202d350f": "\\begin{align} 2\\cdot R_*\n & = \\frac{(18.03\\cdot 5.46\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 21.2\\cdot R_{\\bigodot}\n\\end{align}", "3036793df3bd715a22f209b20066cb1a": " \\ln \\,\\operatorname{cov_{G{X,(1-X)}}} =\\operatorname{cov}[\\ln X, \\ln(1-X)] = -\\psi_1(\\alpha+\\beta)", "30367d56ddef4ed0e4381714c6577a57": "a \\sim \\xi^{\\frac{1+k}{2}},\\ b \\sim \\xi^{\\frac{1-k}{2}},\\ c \\sim \\xi^{-\\frac{1-k^2}{4}},\\ t \\sim \\xi^{\\frac{3+k^2}{4}}.", "3036b772bc0c6f367da7437b52dc4b80": "X^*_E", "3036cd5840039a4833569e99466f17aa": "-\\sqrt{s}", "3036d28ecb8b801c8718647f9a247d1e": " r(k-1) ", "3036e6b40b7c696738353394d637a2d3": " f_E = 5^1 \\cdot 2^{-2} \\cdot f_C = {5 \\over 4} f_C ", "3036f8ddc38fd0ff7b75f60741f374ae": "\\frac{W^2}{L\\lambda} \\ll 1", "303708cc915a18a41a9bac5fec73b0db": ": \\!\\,", "30371a8662a4ff63990778f968da5da9": "\\mathrm{Ad}_{T}(iH) = TiHT^{-1} = -iH", "303771e6481197431756cc2ccfd6820a": "\\psi = \\frac{1}{\\sqrt{2}}\\left(\\begin{matrix}1 \\\\ e^{i A \\sin 2 f t} \\end{matrix} \\right) ", "3037ef9114dfd8c5927282ae588db61b": "\\Delta R \\leq \\log_2 g", "30380f49509226a6b721e8368516e86e": "\\mbox{Yield Gap} = \\frac {\\mbox{Yield Ratio of Equity}} {\\mbox{Yield Ratio of Bond}}", "303831bfb5258a89ec0c1a71a3606a5d": "\\rm NADH + Q + 5\\; H^{+}_{matrix} \\rightarrow NAD^+ + QH_2 + 4\\; H^+_{intermembrane} \\!", "30384ea00ffa5f589b8bf7b8de3d92b0": " t=2 \\int_0^\\infty dz, ", "303875cd6d48ab4b4acd2ad7e72ea667": " f'(x) = \\alpha (1 + x)^{\\alpha - 1}.", "30388477e915931013b9734160836ac1": "\\sum_{n=-\\infty}^\\infty a_n e^{2\\pi in\\theta}", "3038b746500a60565f0ee7755855b9b9": "\\mu u_{i,jj}+(\\mu+\\lambda)u_{j,ij}+F_i=0\\,\\!", "3038bc0582cbfd3ce5aac101aa2e6177": "\n\\int^T_0\\dot{\\phi}(t) \\, dX_t = \\dot{\\phi}(T)X_T - \\int^T_0\\ddot{\\phi}(t)X_t \\, dt,\n", "3038d5f47724a87e857eb365072596bb": "\\tfrac{dM}{dT} = B - \\delta MS - \\mu M ", "30390634b4a288d78f0add5b32a2dfc0": " a=r_0", "303907d574694487bd998f33ca8381d3": " \\langle x | s \\rangle = \\sum_{j=1}^\\N\\, \\Psi(r_j) e^{-i \\Omega _j}", "303952ffa8691e1daf7ed427a905877a": "\\pi_if_i", "303958642dfc3754d86634a09751df25": " \\lambda \\neq 1 ", "3039704dcde5e8ec0b92acc4e4a82797": "RC = \\frac{A\\;\\times\\;B}{C}", "303982f35ebc9ffe82ef51f4680d614d": "Physics \\leq medium", "303993f5f690cd7752eca33760a7115b": "\\frac{36}{35}", "303a308280817207bd14d7d862c9d526": "\\vec{\\sigma} ", "303a5e9a40b0b59eac98a053d6a6f5ce": "I_\\mathrm{s} = \\frac{V_\\mathrm{s}}{R}", "303a6fd0a5b06ce760c82ecb48e51360": "\\dot{g}(s)", "303ac8e17da87b1f55883ee16a8866db": " \\rho\\, _{vapor} ", "303ad43608a461b153e1ef6012623583": "\\scriptstyle \\int_{\\mathbf{R}^n}\\chi_r\\, dx=1", "303ae0dcc9bbb6ef7220931af5949a27": "1-R=\\frac{(Z_1 + Z_0)^2- (Z_1 - Z_0)^2}{(Z_1 + Z_0)^2}=\\frac{4 Z_0 Z_1}{(Z_1 + Z_0)^2} = T", "303b35f8b96d115f38427dc4b97d1910": "|\\mathbf{B}-\\mathbf{u}|^2 = r^2", "303b5836ab53f2e6c494d864a2a90766": " VV^T=I, UU^T=I ", "303ba0ca616f8b478cd17d20be0de400": "\\pi_i^* = \\gamma_i(1)", "303bb8fc39339c1d98dfee6242348831": "\\displaystyle \\frac{2}{i\\nu }", "303bc7f59f38ec85b9ed89389018a22e": "\\nabla\\times\\mathbf{B} = \\frac{\\mu_0}{4\\pi}\\nabla\\iiint_V d^3r' \\mathbf{J}(\\mathbf{r}')\\cdot\\nabla\\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right) - \\frac{\\mu_0}{4\\pi}\\iiint_V d^3r' \\mathbf{J}(\\mathbf{r}')\\nabla^2\\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right)", "303be97ae0ab22136878144d7fa5772f": "E_d = - \\frac{dQ}{dP} \\cdot \\frac{P}{Q}", "303d468a6c5949ab12bcd6bd575811c5": "P = i ^2 r ", "303d7b59a8816b8a31c72b41a360aae6": " X_1^n(i) \\, ", "303da119f4911ead84ef92f6ea9a3113": " \\frac { \\partial \\mathbf{A} } { \\partial t }", "303e82f01a62db0ce3d2da66509c0a1b": "\\{x^i\\}", "303ecc95c84d81f29cd610cb4885cded": "[A]_\\text{seq} = \\overline{A} \\, ", "303f46d06220bdb8b92137ff7e3ee768": "X=QR \\ ", "303f675d93213da5f92178e85d6d4856": " R = e^{\\frac{B}{2}},", "303f6cad937309e1842bfaf4e7ac1a25": "\\, \\omega", "303f86057df8ac8bb0755fe6b38da356": "\n\\begin{align}\n{\\mathbf F}^{+} \\left({\\mathbf r} , t \\right)\n& =\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\sqrt{\\epsilon ({\\mathbf r} , t)} {\\mathbf E} \\left({\\mathbf r} , t \\right) \n+ {\\rm i} \\frac{1}{\\sqrt{\\mu ({\\mathbf r} , t)}} {\\mathbf B} \\left({\\mathbf r} , t \\right) \\right) \\\\\n{\\mathbf F}^{-} \\left({\\mathbf r} , t \\right)\n& =\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\sqrt{\\epsilon ({\\mathbf r} , t)} {\\mathbf E} \\left({\\mathbf r} , t \\right) \n- {\\rm i} \\frac{1}{\\sqrt{\\mu ({\\mathbf r} , t)}} {\\mathbf B} \\left({\\mathbf r} , t \\right) \\right)\\,.\n\\end{align}\n", "3040baa80bd114df8d6b4e38cd5be698": "V(r,\\theta,\\phi) = \\sum_{l=0}^\\infty\\, \\sum_{m=-l}^{l}\\, C^m_l(r)\\, Y^m_l(\\theta,\\phi)= \\sum_{j=1}^\\infty\\, \\sum_{l=0}^\\infty\\, \\sum_{m=-l}^{l}\\, \\frac{D^m_{l,j}}{r^j}\\, Y^m_l(\\theta,\\phi) .", "3040bad664fb0c9a2068494b87dcff42": "\\; H_A", "3040bdbd9d5196d442878919f3faddf9": "C_i = N_i/V\\,\\!", "3040cbc30e7c98195c0660eb48d7922c": "y'(t)\\,\\!", "3040e425d8dc7889054619f292269dfc": "T \\propto \\bar{E_{\\rm k}}.\\,", "3040e4eba80a2decba5b853effbf9c6b": "2\\,", "304145013c7c70cb30e735ee7e17459e": " \\mathbf{u}", "304161f9c4b73158752d2d5b1cdc23d0": " 2\\sqrt{\\int_0^{t'}D(t')dt'} ", "304290a0b9a93a2152cbfaca771518f5": " P(i)={\\rm tr}(\\rho \\hat{\\pi}_i)={\\rm tr}(\\rho \\hat{\\pi}_A \\hat{\\pi}_i \\hat{\\pi}_A),\\; ", "3042a691ae211890cbb5a9826a199bad": "K_{Ic}^2/E", "3042ac0c065e6532e265077e553d3da4": "\\Delta u = f \\text{ in }\\Omega,", "3042b20a20fa733365a67bbb264d21bd": "\n=\\frac{\n \\mathrm (P(G=T|S=T,R=T)\\mathrm P(S=T|R=T)\\mathrm P(R=T))_{TTT} + \\mathrm (P(G=T|S=F,R=T)\\mathrm P(S=F|R=T)\\mathrm P(R=T))_{TFT}\n}\n{\n \\mathrm (P(G=T|S=T,R=T)\\mathrm P(S=T|R=T)\\mathrm P(R=T))_{TTT} + \\mathrm (P(G=T|S=T,R=F)\\mathrm P(S=T|R=F)\\mathrm P(R=F))_{TTF} + \\mathrm (P(G=T|S=F,R=T)\\mathrm P(S=F|R=T)\\mathrm P(R=T))_{TFT} + \\mathrm (P(G=T|S=F,R=F)\\mathrm P(S=F|R=F)\\mathrm P(R=F))_{TFF}\n}\n", "3042cffba56a37f8990d35ddbb757a6a": "\n D = F \\left( 1 - \\textstyle{\\frac{1}{3.6}} e^{-(L_A + 42) / 92} \\right)", "3043078ba537fe47695ec3d0c758693b": "c_{k'}^{(1)}=\\frac{\\lang\\ k'|H_1|k\\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\\hbar})", "304325ecfaacf6a5b16064efb611de30": "-\\frac{117}{25}X^2-9X+\\frac{441}{25},", "30435fea513541991a6eb093398d854e": "\\vec{x} = (x, y, z)", "30438b8ab51e4170e238f5a5f1ea39df": "\\tilde{E}(M)=\\{x\\in E(M) \\mid \\forall f\\in S, f(M)=0 \\implies f(x)=0\\}\\,", "304390a8488b857bd4ab61ea190b6aca": "\\frac{A}{\\sin (\\pi - \\alpha)}=\\frac{B}{\\sin (\\pi - \\beta)}=\\frac{C}{\\sin (\\pi - \\gamma)}", "3043f08647e205e166280469df067c91": " F_T = \\frac{GMu}{(d-r)^2}-\\frac{GMu}{d^2}", "3043fe2d07bfe5b7b028799faabf1b40": "(M,f)\\ ", "304404e1e2d27ac329587258890da97d": "\\boldsymbol{a} = \\frac{d\\boldsymbol{v}}{dt} =\\frac{d^2\\mathbf{r}}{dt^2} = (\\ddot r - r\\dot\\theta^2)\\hat{\\boldsymbol{r}} + (r\\ddot\\theta + 2\\dot r \\dot\\theta)\\hat{\\boldsymbol\\theta} \\ , ", "30441ac8d703dd19d1857b95f77a6469": " 5 \\pm 3i. \\, ", "30441b551470328d7d7313cd4c9c10e2": " \\epsilon_1=\\epsilon_2=\\epsilon_3=\\epsilon_4>0, \\epsilon_5=\\epsilon_6=0.", "30444c6b9452c595bddf7d270313c63e": " e^{\\lambda(x-1)} \\,", "3044563f6daf1ef84e8eb17fff1e3ed9": "J_{\\text{mag}}(\\mathbf{p}_z)", "3044576469dd37b9a1c279cacd53f5ea": " \\frac{1}{2}\\cdot \\left(-32\\frac{\\text{foot}}{\\text{second}^2}\\right)\\cdot t^2 + \\left(500\\frac{\\text{foot}}{\\text{second}}\\right)\\cdot t. ", "3044c895eb3c63f33ca6a8ac4a97fcbc": "f(a,b)", "3044cbedbb87048500ce7a33bf2c281f": "p(k+1)", "3044f0746d26eeb2fc605aa85a064c5f": "1-10^{-154}", "3045149b73dc42d45b736c866c9d5cc0": "B(x)=\\sum_{n=-\\infty}^\\infty b_ne^{inx}", "304563d369e19fd039ac01fee4470100": "\\begin{align}\nT\\left[\\phi(x_1)\\cdots \\phi(x_n)\\right]=&:\\phi(x_1)\\cdots \\phi(x_n):\n+\\sum_\\textrm{perm}\\langle 0 |T\\left[\\phi(x_1)\\phi(x_2)\\right]|0\\rangle :\\phi(x_3)\\cdots \\phi(x_n):\\\\\n&+\\sum_\\textrm{perm}\\langle 0 |T\\left[\\phi(x_1)\\phi(x_2)\\right]|0\\rangle \\langle 0 |T\\left[\\phi(x_3)\\phi(x_4)\\right]|0\\rangle:\\phi(x_5)\\cdots \\phi(x_n):\\\\ \n\\vdots \\\\\n&+\\sum_\\textrm{perm}\\langle 0 |T\\left[\\phi(x_1)\\phi(x_2)\\right]|0\\rangle\\cdots \\langle 0 |T\\left[\\phi(x_{n-1})\\phi(x_n)\\right]|0\\rangle\n\\end{align}", "30456e9166c8741666c7140fb3044226": "{e^{-1}}(1-w)=0.39(1-w)", "30459d283e88a8caf572bf3cf2102612": "\\overline G_k(X)", "3045f47c9c8cb22561f22525ba196cba": "a_2 = \\frac{R^2}{r}", "304606696e06d8912ec54dc25e3d0b02": "A\\subseteq(\\mathbb{R}^n\\setminus E)^\\circ=\\mathbb{R}^n\\setminus E^-", "30468da8d32a373d9e37facbf24e5805": "\\text{ Put } = \\text{ Call } + \\text{ Present Value of Strike Price } - \\text{ Underlying security } \\,", "3046b95f04ad97e36f8368e37f22de94": " = (a_1\\mathbf{u}_1^T + a_2\\mathbf{u}_2^T + ... + a_n\\mathbf{u}_n^T)\\mathbf{U\\Sigma}^k\\mathbf{U}^{-1} ,", "304744c98855db68f82a8ca6851c550d": " \\vec{E}(\\vec r)\n=\\frac{1}{4\\pi \\varepsilon _0}\\sum_{i=1}^N \\frac{\\hat\\mathfrak r_i Q_i}{\\left \\|\\mathfrak\\vec r_i \\right \\|^2} ,", "30474ae543605fb5422e3c9f78dd1d9b": "{\\color{Blue}~6.11}", "304751750933113b7951f6d6d246d11a": "\\mathcal{E}(b) = x^b r^2 \\;\\bmod\\; m", "30479511517dee6b9c93a87d2e776341": "-\\ell(\\theta|X,Y)", "3047be66378857aa726ae324a9703c21": " \\mu_\\mathrm{N} ", "3047d9d8fc0816c19b0aa8d84f31c5bc": "f(n) \\in o(g(n))", "30481ce6c5c91fcae4165dc793326d72": "S_{FI} = \\xi \\int d^4\\theta \\, V", "30482ef07e9b2aeb6161e2b9d64d6460": "\\displaystyle u_t - u_{xxt} + 2\\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx}", "30483aa0e854484e4c6965c881bc46ea": "~ \\hat b~ ", "304864ac62fbc37949dc9e2b3fa5d442": "\\textstyle \\{|e_i\\rangle\\}", "304885a785d31edade3e47c6e3272b2e": "\\mathrm{d}\\mathbf{\\Sigma}^2 = \\mathrm{d}x^2 + \\mathrm{d}y^2 + \\mathrm{d}z^2.", "30489299cb5b2121871466d8c398bd55": "y_1,\\dots,y_n", "3048c6447c744152a0468b186a43e2b5": "\\|.\\|_1=\\|.\\|_\\infty", "30490ed30a97a77c6649595d39595be8": " (A_0, E'', F'', G'', A_1) = (A_0, H''', I''', J''', A_1). ", "30495021e1626fad3e2e408c0c1a2afc": "\\mu_{k,i}=\\frac{1}{2}(\\mu_{k,i+1}+\\overline{\\mu_{k,i+1}})", "304977231bf76677a1b88e4d25f40fad": "\\gamma[1] \\gamma[2] \\ldots \\gamma[L]", "30498425adfcde8f004ef1a96daf531f": " t_s = \\infty ", "3049f16673dbd8eaed4cff29d1cac2b6": " D_{CFA} = D_{LA} \\left(1 + \\frac{z/2}{(g+1)^2+(z/3)(2g-1)}\\right) ", "3049fa0940a2e3af42fef6bd692fb477": " |\\hat{n}|=1 ", "304a6a3272eb4aff6dd23905fc5b6a29": "{1 \\over 1}+{1 \\over 2}+{1 \\over 4}+{1 \\over 8}+{1 \\over 16}+{1 \\over 32}+\\cdots = 2.", "304a8bf0b4a312eb04961827e92042c6": " \\frac {p_{02} - p_{01}} {{\\rho} {u^2}}\\ ", "304b14470b7f08452353b8a20c469cdd": "Y_{9}^{-8}(\\theta,\\varphi)={3\\over 256}\\sqrt{230945\\over 2\\pi}\\cdot e^{-8i\\varphi}\\cdot\\sin^{8}\\theta\\cdot\\cos\\theta", "304b7b561eee553687c6f4ab5acf715a": "\\frac{d f}{d x}(x)", "304b94a1d4c958c20fa94cea7b7751f3": "1-\\sum_{j=1}^{m}P_{ij}", "304be4013aaa4bd414d4c1d891dd24ff": "\\mathbf{N}_m = \\{ x ~|~ x \\in \\mathbf{Z} \\land x \\ge m \\} = \\{ m, m + 1, m + 2, \\ldots\\}.", "304c37c50d192acad69d3f09b24265e9": " \\sigma_y\\sqrt{a} = C ", "304c47f00162f336d414ab8080fa5e55": "x \\cdot y = mx+ny ", "304c679e0126951d73d1d9a5bb782a4f": "\\theta \\leftarrow \\theta + \\eta \\mathbf{F}^{-1}\\nabla_{\\theta} J(\\theta) ", "304c6c155724d608b00cd3f1b0ce72ca": "\\int_\\Omega\\vert D u\\vert = V(u,\\Omega)", "304c7dbf2177ec46d2ea3015e08720df": "\\mathcal{L}_{\\mathrm{Yukawa}}=-\\lambda_f\\bar{\\psi}H\\psi", "304ca4f6ae12f1845130642dbfdf545d": "\\forall P\\,\\forall x (x \\in P \\lor x \\notin P)", "304e07ec563b7bc46ceab666455b60db": "g(w, x) = f_{\\langle X | R\\cup \\{w\\} \\rangle}(x).", "304e2be18b17f00b1645d4e380ba7a4b": "{\\tilde K}_1(\\mathbf{Z}[G])", "304e79d2bcced283f5a3704f54dad762": "E(\\bar{x})= c. \\, ", "304e8c6e45a1e97ec10869b697e59af6": "U\\subset Y", "304eb765f268cda3ddf28657f04bb317": "\n\\theta\\circ(\\theta_1*s_1,\\ldots,\\theta_n*s_n) = (\\theta\\circ(\\theta_1,\\ldots,\\theta_n))*(s_1,...,s_n)\n", "304f91633295e7b7eb3f8a28619ae5a9": "X|Y=r \\sim P_r", "304fa9b663a54e1865508830d117965b": "s_\\mu = -\\sum_{i=1}^n \\pi_i \\sum_{j=1}^n p_{ij} \\log p_{ij}", "304fbf093064cebf77aaa923ab11788a": "r^{-1} \\mathrm{OPT}", "304fdeec57eab6be7f71ac8ab3f8da23": " \\psi_j(t+dt)^{ } - \\psi_j(t) = -i \\sum_k^{ } H_{jk} \\, dt \\, \\psi_k(t) ", "30501601dd8acbce722e5140c79a69c3": "\\langle Pu,u \\rangle > 0, \\forall u \\neq 0", "305024e7a02d45175464cff4a69c1d61": "\\psi_{\\nu_j}", "305084737a47e5762698ee603872f019": " y^2+a_1xy+a_3=x^3+a_2x^2_2+a_4x+a_6", "3050ac29c3d8f29c8a65373635174f62": "(20)\\quad L+M=r\\,,\\quad l_+ + l_- =2\\sqrt{M^2-Q^2}\\,\\cos\\theta\\,,\\quad z=(r-M)\\cos\\theta\\,,", "3050afcef8810132dc5cceedab63c00d": "\\mathbf{b}_{i,j} = \\mathbf{b}_{j,i}\\quad\\Rightarrow\\quad\\Gamma_{ijk} = \\Gamma_{jik}", "3050c685f6ee6f8c0bfd16c4cdea48a2": "q, 2q, 3q, \\dots\\ ", "3050c6a5197d77a1f3998f2c0441494d": " = (1 + 28 +99 + 24 + 4 - 38)\\ \\bmod\\ 7 = 6 = Saturday", "3050e015e3bebf0a702e52127cfc9746": "\\frac{1}{\\tau_{ph-e}}=\\frac{n_e \\epsilon^2 \\omega}{\\rho V^2 k_B T}\\sqrt{\\frac{\\pi m^* V^2}{2k_B T}} \\exp \\left(-\\frac{m^*V^2}{2k_B T}\\right)", "30510598c14f1395e34c06366348cb5e": "\\begin{pmatrix}\n\\mathbf{T'}\\\\\n\\mathbf{t'}\\\\\n\\mathbf{u'}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n0&\\kappa_g&\\kappa_n\\\\\n-\\kappa_g&0&\\tau_r\\\\\n-\\kappa_n&-\\tau_r&0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{pmatrix}\n", "30514f3fa065815388008d5e0ee4a97d": " \\sum_{p}^{}{(T_i-T^0)(-\\Delta S^0_i)} ", "3051a04b58e4cd4ce5430705c351c449": "P^{a}\\beta P^{b} = \\sum_i (-1)^{a+i}{(p-1)(b-i) \\choose a-pi} \\beta P^{a+b-i}P^i+\n\\sum_i (-1)^{a+i+1}{(p-1)(b-i)-1 \\choose a-pi-1} P^{a+b-i}\\beta P^i", "3051daa1d754ec2a3a5c005d9542a72a": "R_{hs}=R_{b} + R_{f}", "305233189afcb3b93ae1fe26320daddc": "\\vec F =- \\vec \\nabla (\\mu + M_4gz).", "30524a32779919a30cedbdcd07e81cf6": "\\psi(x, t+dt)=F^{-1}[e^{-idtk^2}F[e^{idt\\hat N}\\psi(x, t)]]", "3052589dbb35e7332b609a1daa171bc8": "\\displaystyle{g(x)= \\mathbf{Av}_{J_n}(f)\\,\\,\\, (x\\in J_n), \\,\\,\\,\\,\\,g(x) = f(x)\\,\\,\\, (x\\in \\Omega^c).}", "3052b65a277fa1e48ef1b5b9e8f1a8f5": " f_{kj}(A_j)=f_{ki}(A_i)", "305304d37e9ccbdcd20a91d5279701c3": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi \\epsilon_0} \\iint \\frac{\\delta(t' - t_r')}{|\\mathbf{r} - \\mathbf{r}'|} q\\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t')) \\, d^3\\mathbf{r}' dt'\n", "3053410730b4de2acbd279a1b1de0260": "\\tfrac{5-\\sqrt{5}}{8}", "305395d9a9c8a064ed420b92212622a6": "C_\\text{R} = \\sqrt{\\frac{\\text{bounce height}}{\\text{drop height}}}\\,.", "3053a82326a073234f279b12b9193fa8": "\\Pi(n) = n!,", "30548117cc5d6bc8e5c5c81890e3e743": "\\Sigma^E_k", "3054aa0bf4cdb4e80f02e4189cf62104": "\\Phi(1/2 + it) = \\varphi(t)", "3054b02026aa93c8f0f8e066e5a05770": "\\Omega_A", "3054bcc085f7a650bd89e43dd1071ca1": "\n\\begin{align}\n\n (\\nu x) ( \\; & 0 \\\\\n | \\; & x(y). \\; 0 \\\\\n | \\; & \\overline{x}\\langle x \\rangle . \\; 0 \\; )\n\n\\end{align}\n", "30551e638ad4ec03dd5bfd20629ddb0b": "\n n \\to \\infty\\,\n ", "3055d1b41789ad66a43150ccba2f93d6": "U_i(x)", "30565a8911a6bb487e3745c0ea3c8224": "0.0", "30565c6c8437eed67a77f9867c510184": "\n \\left( A \\rightarrow \\left( B \\or C \\right) \\right) \\vdash \\left( \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\rightarrow \\lnot A \\right)\n ", "305669a48a8222b2681cdfe68b8fcab4": "P_w \\circ T", "3056ad61fdb24a85e30fc1a542d2b662": "\\varphi(p)=p", "3056c1369734bda834ba47a86a745552": "\\sigma=\\frac{1}{\\rho}. \\,\\!", "3056d69fade7e9ab39c5db5e313ec59b": "\n\\begin{align}\n\\pi = \\frac1{2^6} \\sum_{n=0}^\\infty \\frac{(-1)^n}{2^{10n}} \\, \\left(-\\frac{2^5}{4n+1} \\right. & {} - \\frac1{4n+3} + \\frac{2^8}{10n+1} - \\frac{2^6}{10n+3} \\left. {} - \\frac{2^2}{10n+5} - \\frac{2^2}{10n+7} + \\frac1{10n+9} \\right)\n\\end{align}\n", "3056eceb1aa7bfabc389631b20496951": "\\bar X_1 - \\bar X_2", "3056f2e962771b4f3ac71cc4d8d6344e": "2\\tfrac{3}{4}", "305775f54c9f6f8b8b0700700521c141": "= {1 \\over \\sqrt{2}} \\left|1,V\\right\\rang \\left|2,V\\right\\rang + {1 \\over \\sqrt{2}} \\left|1,H\\right\\rang \\left|2,H\\right\\rang ", "30581a2baf6db4ad7e4eb871ed2575a5": " \\rho_{n}- \\rho ", "305837540b14c05ec56167e74aea3132": "1/r", "30583a562cfc0ae019ceb255b8df3cda": "a^l B \\subset a^{l+1}B + A.", "30584846908e0183bda833b4ff53884e": " x \\le y \\or y \\le x.", "30584f098be067805d1d7ea76b2aba88": " {\\rm det}\\, (I+ zA) = \\sum_{k=0}^\\infty z^k{\\rm Tr} \\Lambda^k(A)", "305852b400410541be32e8b9cff65958": "l_{ij} = \\log(\\mathrm{tf}_{ij} + 1)", "3058ab72c2fb5a027884e5b236f31fa2": "a = 2, b = 2, c = 0", "3058c3b1c82262d745e8051aa4330a46": "q = \\exp (\\pi i \\tau)", "3058db699c83c93f991a1fca23e274e7": "u,v,w\\ge 0", "3058db792ee6a66dbbf4b997a626f7b0": "Y=Y^{s}(Z_2)", "305913056d775349317e07813a6799cb": " P = \\sum_{s_{zN}}\\cdots\\sum_{s_{z2}}\\sum_{s_{z1}}\\int_{V_N}\\cdots\\int_{V_2}\\int_{V_1} \\left | \\Psi \\right |^2\\mathrm{d}^3\\mathbf{r}_1\\mathrm{d}^3\\mathbf{r}_2\\cdots\\mathrm{d}^3\\mathbf{r}_N\\,\\!", "30596e929d1c459da5570f77ee30b6e2": "\\varphi=0\\text{ on }\\partial \\Omega", "3059a4b5e5255269dde62e4f679d3055": " \\ \\text{AR} ", "3059ae9a4a04fc2398bc6d48e373874e": "y_1,\\ldots, y_n\\in\\mathbb C", "3059c9a5b2ee3de307a3f48da4c5d01f": " (2\\pi\\hbar)^{-1} ", "3059cbfe1393fa920467352c34c001e6": "\\left\\vert Z \\right\\vert", "3059f82c477f749c30eaec51481406f9": "c_0, c_1, c_2, c_3", "305a00819caf09fa3952dd4eeecbeba9": "\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n=\n\\begin{bmatrix}\n 1 & 0.9563 & 0.6210 \\\\\n 1 & -0.2721 & -0.6474 \\\\\n 1 & -1.1070 & 1.7046\n\\end{bmatrix}\n\\begin{bmatrix} Y \\\\ I \\\\ Q \\end{bmatrix}\n", "305a3a1c54f6aab3553b99d81980ac4c": "\\otimes: \\Phi \\otimes \\Phi \\rightarrow \\Phi,~ (\\phi,\\psi) \\mapsto \\phi \\otimes \\psi\\,", "305a466c7b64d0684e010385f3dd7d6c": "a<-1:", "305b217916aecfb843ab01dd13b05943": "\\mathfrak{N}_*=\\mathbf Z_2[x_i \\vert i \\geq 1, i \\neq 2^j -1 ]", "305b2ef581e762c6da9f4f35cd29a5c6": "L_{p\\times k}", "305b3edf3d7e172af574be0057bc60a1": "E_\\text{total}", "305b4a7113fc51854e43f395a1392ac8": " \\langle a, b\\rangle", "305b735eef301f7b258c2136d1dab160": "Z(YW-Z^2)-W(XW-YZ)", "305b859dfe755e42927ce807497a1fcc": "{\\tilde u}", "305b9ba37aa6e4060f8b2730359cd72c": " \\mathbf{F}_\\bot = - m \\omega^2 R \\bold{\\hat{e}}_r= - \\omega^2 \\mathbf{m} \\,\\!", "305ba2175b933c4f159967769c9fa746": "\\mu_i =0", "305be03cee4c7ee4ed6a2feba97816b0": "(A \\leftrightarrow B)", "305cf16cbc4cf3f9ea3e38b18d8fb5d7": "\\langle \\overline{R}^2\\rangle=\\frac{1}{N}+\\frac{N-1}{N}e^{-2\\gamma}.", "305d0b226634c4f79dd5e05bfd3195a4": "F_{X}(x_{t_1 + \\tau}, \\ldots, x_{t_k + \\tau})", "305d7fc8621cece57aa45cdd43828519": "m_\\mathrm{Al} = \\left(\\frac{85.0 \\mbox{ g }\\mathrm{Fe_2O_3}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{Fe_2 O_3}}{159.7 \\mbox{ g }\\mathrm{Fe_2 O_3}}\\right)\\left(\\frac{2 \\mbox{ mol Al}}{1 \\mbox{ mol }\\mathrm{Fe_2 O_3}}\\right)\\left(\\frac{26.98 \\mbox{ g Al}}{1 \\mbox{ mol Al}}\\right) = 28.7 \\mbox{ g}", "305d9a2ffbf9368c7462367cf02e8f55": "s < \\lambda", "305e160524deb0e58772c7d5bb1c3de2": "t'=\\gamma \\left(t - \\frac{v}{c^2} x\\right), ", "305e43ed02b261de610602fdc7b67021": "\\int_{X\\times Y} |f(x,y)|\\,\\text{d}(x,y)", "305e551fb5101baa1e850df251482f55": "\n \\boldsymbol{\\mathcal{E}} = \\mathcal{E}_{ijk}~\\mathbf{b}^i\\otimes\\mathbf{b}^j\\otimes\\mathbf{b}^k\n = \\mathcal{E}^{ijk}~\\mathbf{b}_i\\otimes\\mathbf{b}_j\\otimes\\mathbf{b}_k\n", "305e7f7a89eea769f9d6016e653612f2": "\\frac{dz(t)}{dt}=x(t)*y(t)-b*z(t)", "305ed1ac5a03be6d054b0ab79937092d": "\n \\displaystyle \n w(n,g) \n =\n \\sum_{k=0}^{n}\n w(n-k, g-1)\n =\n w(n, g-1)\n +\n w(n-1, g-1)\n +\n \\cdots\n +\n w(1, g-1)\n +\n w(0, g-1)\n", "305f05f5e8f98f051749c9f85bd60910": "N_2~", "305f37f9c4a4e7db19a7281b390a077d": " \\lim_{n \\to \\infty} \\mathbb{P} \\left\\{ k_n = k \\right\\} = - \\log_2\\left(1 - \\frac{1}{(k+1)^2}\\right)~.", "305f4e9b65a2f73c8ba7419afbf1532e": "1_L = \\bigvee L", "305f66c93881bb9a47b1a536fad4b679": "p = \\frac{e}{k_1} + \\frac{f}{k_2}", "305f683eee64503125f3c86d1cd8df48": "\\Delta_N(x)", "305f9b0335f038d2cefa866e3d1c82ec": "\\check{a}, \\breve{a}, \\tilde{a}, \\bar{a} \\!", "305fd942201657a64661ae288fe96536": "x_{\\pi(0)} \\leq x_{\\pi (1)} \\leq \\dots \\leq x_{\\pi (n)}.", "305feb0f7c49c1399bed082b780f4747": "\\|x\\|_{p+a} \\leq \\|x\\|_{p}", "30609db9ced255784c518e6ffd9ae0bb": "\\begin{matrix} {9 \\choose 1}{4 \\choose 2}{8 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "3060ea3d0b4ae3da7dc61404b129ed22": " \\begin{align} \n\\sum_{l}x_l x_{l+m}&={1\\over N}\\sum_{kk'}Q_k Q_k'\\sum_{l} e^{ial\\left(k+k'\\right)}e^{iamk'}= \\sum_{k}Q_k Q_{-k}e^{iamk} \\\\ \n\\sum_{l}{p_l}^2 &= \\sum_{k}\\Pi_k \\Pi_{-k}\n\\end{align}", "306155acd9f4e474b927d68ee00083d5": " \\hat q_{\\theta}", "3061acdcdf2d99238db769632d90e038": "{{1/2}\\choose{k}}={{2k}\\choose{k}}\\frac{(-1)^{k+1}}{2^{2k}(2k-1)}.", "3061d9d4c6a31e5d85fa22e6e826bcab": "f: \\mathbb{R}^d\\rightarrow \\mathbb{R}", "306256ac7217f5634cd2be72b5be486a": "\\ v_i = \\sqrt {2GM\\Big( \\frac{1}{r}-\\frac{1}{r+d} \\Big)}\\ ", "306259a0cb34e517d8b08bc8b524fcb4": "|\\nabla p|\\sim \\frac{p}{L_p}", "30626dcaa18ca321be83d80feddefebd": "d ( \\mu ) = m , d ( \\nu ) = n", "306283320badaa6a023b614fcb99c980": "\\bar{5}\\rightarrow (\\bar{3},1)_{\\frac{1}{3}}\\oplus (1,2)_{-\\frac{1}{2}}", "3062aabe9a46b0840dcef0e1850cbba3": "\\scriptstyle\\gamma^\\mu", "3062cab52760e50c147a79919c2051b3": "\\frac{p}{1-q},", "3062d8e5997a35d2cae89ded27e0320e": "\\frac{x^2}{\\left(\\frac{x}{2}\\right)},", "306345e617d64933a9fce675be33ee05": " \n \\int_0^{2 \\pi} {d\\varphi \\over 2 \\pi} \\exp\\left( i p \\cos\\left( \\varphi \\right) \\right)\n=\n\\mathcal J_0 \\left( p \\right)\n ", "3063a56aa381e008933b972732cd3a28": "z = z \\times z' = z'", "3063c2eb495da131a6a5815a08356324": "S(S+1)", "306483507fedf5961d17e1543f73dce3": " \\gcd(1,x) = y,", "30648f9b0621f4d2bf4d030ab073aa74": "f(x;\\theta)", "30649319951cebf20ca33f6df53aebf1": "\\scriptstyle{w = 1/2}", "3064b8f20167dfe3aae90a435ee934b4": "u\\Vdash B[e]", "3064d4403cf1e2efcbb57a89eb59d9a9": "\\textbf{G}_c = K", "3064e3a3a8596e3d263822a26201d9d0": "\\Phi = \\frac{\\hbar}{e_s}\\oint_{\\partial S}\\mathbf{\\nabla}\\phi\\cdot d\\mathbf{l} = \\frac{\\hbar}{e_s}\\Delta\\phi = \\frac{2\\pi\\hbar}{e_s}n. ", "3064fb1538ff102311c2cd31c4b99ee7": "0.02(V_{s0})^2", "30652042ff4b0a3e94009f2b72da93d3": "Z_\\mathrm {nom}=\\sqrt \\frac{L}{C}", "30655f6bf6eb98cefc7ec485fc354b7d": "p(n;d) = \\begin{cases} 1-\\prod_{k=1}^{n-1}\\left(1-{k \\over d}\\right) & n\\le d \\\\ 1 & n > d \\end{cases}", "306581d1c3cd6fc9d7c31cc759157755": "X_\\theta", "3065a385742e59407f7c8aa68957f4fd": "dz=dx+i\\,dy.", "3065bd63302259788f1bcf2e158e070f": "\\not\\in", "306640dbfa15f15f322dd5bc82a48973": "\\dot{\\gamma}_{ph,a}", "30666cbea154a1959f14200a68c803f2": "\\mathbf\\sigma", "306699bab168a61b692d04756b5ac9e0": "{n \\choose r} < 2^{{r \\choose 2} - 1},", "3066a256c2f7f4feb3a578b9d964e863": " \\mu\\left(\\bigcup_{n=1}^\\infty A_n\\right) = \\sum_{n=1}^\\infty \\mu(A_n)", "3066bc4ee6867b3d1f4560a920b7ab07": "{\\mathcal L}_{xx}^3", "3066fdc7ae15e230caa51ec0dcb66a7b": " L = - V \\frac{{\\mathrm{d} t}}{\\mathrm{d} I} \\,\\!", "3067f939d420241acb6bf46461f88c24": "i, j \\in \\mathbb{N}", "30680b6e82730af15d5b1ba4505b80f7": "\\frac{F_n}{\\dot m} = (V_{jfe} - V_a)", "306840ec02d896a9382b51085d10f398": "F(a) + c = G(a) = \\int_a^a f(t)\\, dt = 0,", "30686fb800049303f2364b5be52cbd91": " T_d = T_m \\times \\frac {MTTR + MLDT + MAMDT}{MTBF} \\begin{cases} T_d = Down \\ Time \\\\ T_m = Mission \\ Duration \\\\ MTTR = Mean \\ Time \\ To \\ Recover \\\\ MLDT = Mean \\ Logistics \\ Delay \\ Time \\\\ MAMDT = Mean \\ Active \\ Maintenance \\ Down \\ Time\\\\ MTBF = Mean \\ Time \\ Between \\ Failure \\end{cases}", "30688af1db42d7792fb6d7b08bdfb706": "\\alpha \\beta \\neq 0", "30688f6770d14f9943ec74f553463809": "L_1\\!", "3068911b5e3a2556407a8d11ec48c6dc": "e = q_d - q", "306928fec89340eb953d813bc64bcd9a": "\\lambda\\,", "306a7969ef33b3dd15bd08c22610df40": "\\mathrm{3Fe_2SiO_4 + 2H_2O \\rarr 2Fe_3O_4 + 3SiO_2 + 2H_2 }", "306a923d2a6f9f26c41e199a5deedb84": "{S}_{1,2}", "306ad0dc3852532ad2b6703dff1db9a5": " l^\\infty(\\mathbf{N}) \\to C(\\beta \\mathbf{N}) ", "306b07841be642d8265b841ee767e14f": "(1-x^2)\\,y'' - 3x\\,y' + n(n+2)\\,y = 0 \\,\\!", "306b2103388f45623d050964ce5323b1": "| x - y | > \\varepsilon;", "306b8b2e2a790022d26e2b9abf6bdf4c": "X=\\{X_t:t \\geq 0\\}", "306ba5302e77b6ddd435d68a4d81d4ec": "\\le 5", "306ba61845f916670564a01ff366bc76": "8_{V} \\times 8_{V} \\,", "306baabaa273c92348ae32dce939e4cd": "P_n(0,\\rho)=\\rho^n\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,\\rho^{-n}\\,", "306bec2e23c90b45b25ddc6fe99b7e5f": "\\mathrm P(R|do(G=T)) = P(R)", "306c35220780187e7b3276b3a33bbbb8": "r_{12} \\!", "306c9bb327cedea1bd7320de8e467682": "\\lambda G_1' \\subseteq G'", "306cb228473d02c0b9b7d4fff280a072": "{M+N \\choose M}_x", "306d168848056ca4e7cf2f171e63081a": "\\det(M \\circ N) \\geq \\det (N) \\prod\\nolimits_{i} m_{ii}.", "306d2d62a0da1ac069036909deee2aec": "\\exist s[s0 ", "30968ebb68853a3ba5dd0d4eaf62a666": "V_{r1}\\,", "3096dc90f5b79e6d3f23b40a58db98ea": "\\alpha^0 = 1 \\,,", "3096e3ad7c44fcd362294dc9f02b0adb": " a = \\frac{v^2}{r} ", "30973acfe9fc43baa59338d53669627a": "\\big(x_{ie}\\big)", "30974dda4a254daa18d00f39b2671a3d": "\\kappa= \\frac{y''}{(1+y'^2)^{3/2}}", "309786b488f5be40a54f2336fc04f2e3": "v \\in \\mathbb{R}/R\\mathbb{Z}", "309790ae4b89f046e60645e438efe982": " r=(\\frac{3\\pi \\gamma R^{2}}{K})^{\\frac{1}{3}}", "309793eb6321257fb4d612fd80c6e5d4": "a_{r,s}=0", "3097a15a0dbc5d0aa00ceac1a65b2035": "\n\\mu=\\ln\\left(\\frac{m^2}{\\sqrt{v+m^2}}\\right), \\sigma=\\sqrt{\\ln\\left(1+\\frac{v}{m^2}\\right)}\n", "3097e46bfb5a1d235230d92d5f2d09e8": "p=\\frac{R\\,T}{V_m-b} - \\frac{a\\,\\alpha}{V_m^2+2bV_m-b^2}", "309806881a059002fe1ed362817f8701": "R\\geq 0", "309891d3ad9a69852489d876e59097c2": "\\int_{\\Omega} d\\omega = \\int_{\\partial\\Omega} \\omega \\,\\!", "3099018cb5590b7b69f0d3005264013f": " e^{i \\omega t}", "3099613170e0b4da215e4c174498d3a3": "H=(H_r)", "30996c870d83350a077837bf4e37d067": "Q(u)=\\frac{1}{2}\\int_{\\R}|u(x,t)|^2\\,dx", "309979c4ca5e62f6f1d1ff8c416d4b8c": "\\{p_{i}(t)\\}", "30999c127716cadeafa1024ae521272c": "\\hat{w}_i = 1 + \\epsilon 0; i = 0,3", "30999e3910f793c56f53875edde581e7": "E^k_p", "309a2a4414ce8f990a698d121adfb0cf": " \\forall X \\in M(n,p; \\mathbb{K})\\,\\!", "309a340be1c5c4bf00e0e85b7577607f": "\\mathcal{Z}(_R R)", "309a48ed7ce33dc0c70f3336a5e13f11": "{1}", "309a79679f76fdbbb60f042ad103b4f0": " \\lambda +1 = \\prod_{i=1}^n (1+\\lambda g_i) ", "309a85546fa2a9d224804762e9e89c68": "O((k+\\lambda)^{-2})", "309a94a72b38c9534312053a7761d819": "h_1 \\ge \\cdots \\ge h_k \\ge \\cdots", "309b2a15dd8d422bcdcc5d7dc80fc3c7": "g(f(x)) = \\sum_{n=1}^\\infty\n{\\sum_{k=1}^{n} b_k B_{n,k}(a_1,\\dots,a_{n-k+1}) \\over n!} x^n,", "309b4257e133a6a610871fb649d3851d": "\n[x_3] =[x_3]\\ \\cap \\ \\frac{[b]}{[a]} \n", "309b702a5b8b47e3e36dca9ae9d97a6a": "(C_\\alpha, C_\\beta)", "309be6648fa0cac2e086e57925861d2a": "\\left\\{(x_i)\\in\\mathbb Z^4 : {\\textstyle\\sum_i} x_i \\equiv 0\\;(\\mbox{mod }2)\\right\\}.", "309c1ce524ab33a1db75c5cb5e25a3ab": "\\mu(\\xi) \\sim 1", "309c4620395be6fe87fe2da64ab17d90": "A =\\begin{bmatrix} 0 & b \\\\ 1 & 1 \\end{bmatrix}", "309c7644e38882901abfa8c8b5b1cf8b": "\\left\\{1+\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{R}_{x}^{-1}(n-1)\\mathbf{x}(n)\\right\\}^{-1} \\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{R}_{x}^{-1}(n-1)", "309c841120526801bf9984eb99d7226c": " g(\\omega) = \\frac{\\gamma}{\\pi[\\gamma^2 + (\\omega - \\omega_0)^2)}", "309cba158a0a1ef46d52500d73770747": "f(x) = 1-x+x^2-x^3+\\cdots = \\frac{1}{1+x}.", "309cc44174b5ffb00e2581cf208d1695": "\n\\begin{pmatrix}\nu'\\\\x'\\\\y'\\\\z'\n\\end{pmatrix}\n=\n\\begin{pmatrix}\np&-q&-r&-s\\\\\nq&\\;\\,\\, p&\\;\\,\\, s&-r\\\\\nr&-s&\\;\\,\\, p&\\;\\,\\, q\\\\\ns&\\;\\,\\, r&-q&\\;\\,\\, p\n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\nu\\\\x\\\\y\\\\z\n\\end{pmatrix}.\n", "309ccc03cb1274dc7d974ecbb9d7a5ab": "A^\\text{WSM-score}_i = \\sum_{j=1}^n w_j a_{ij},\\text{ for }i = 1, 2, 3, \\dots , m. ", "309cf0417d64455390d28c51b17634e9": "k[y_1, \\dots, y_n] \\to k(X), \\quad y_i \\mapsto x_i/x_0", "309d0467b199f4da7280ba3505aab3c0": "\\Phi'_{i,k}=\\sum_{j=1}^n a_{i,j}\\Phi_{j,k}\\,,\\qquad i,k\\in\\{1,\\ldots,n\\},", "309d1cec9d9605ec209ab811c6bdf228": "\\smile", "309d6fb694c906fca29a1ae29d1d168a": "\\iint\\limits_S \\mathbf{E} \\cdot d\\mathbf{A} = \\frac{1}{\\epsilon_0}(Q + Q_{induced}) = 0 \\,", "309de7e6fa1e93dda1809095e3a88366": "\\beta_m =k n_1 \\cos \\theta_m", "309e08dfc784296b311b300fb3b238c6": "\\phi=\\rm{gd}(\\mathit{m} (\\lambda-\\lambda_0)).\\,", "309e59186a5379dec1d9849c61e0739b": " X \\sim \\text{Pois}(\\lambda)", "309e7a359d8c623a5725d9de2e3ea40b": "{k} = \\frac{V_{r2}}{V_{r1}}", "309e9ab69f146dca333e8d0fe7c3e4dd": "D_{\\mathrm{KL}}(P\\|Q) = \\sum_j p_j \\log \\frac{p_j}{q_j} = \\sum_j (- \\log \\frac{q_j}{p_j})(p_j).", "309eac347f671b5dcbff1a96936c15c3": "\\{p_n\\}", "309ee06904692d3484a1b48bce79e864": "S\\ =\\ \\mathbf{if}\\ \\mathbf{true} \\rightarrow x:=0\\ [\\!]\\ \\mathbf{true} \\rightarrow x:=1\\ \\mathbf{fi}", "309f13ea702ae53ce2b42771529ddb70": "x^x y^y z^z \\ge (xyz)^{(x+y+z)/3}.\\,", "309f31b831fd38d38173d545dd657991": "\\scriptstyle{\\gamma_n(t)}", "309f32572f84ee55b5bbf5ba6b594614": "g(x) = c x^k", "309f3c5daf8615b7bbe82a7cac3a8eef": "\\varphi(x^a)=a", "309f43e5d6c0568412f938bf55546823": "\\text{CAI}=\\exp \\left( 1/L \\sum_{l=1}^L{\\log \\left(w_i(l)\\right)}\\right)", "309f6a86fb2e2cdf0ed863fecb04e55c": "\\alpha^{\\beta+1} = \\alpha^\\beta \\cdot \\alpha \\,,", "30a01dee23b5e68da0e26d751f33ed15": "k = |V_H|", "30a0e13deee2ad135badfc985c8c7126": " {d \\over d\\tau} \\left( {g_{\\lambda \\nu} \\dot x^\\nu + g_{\\mu \\lambda} \\dot x^\\mu \\over \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\right) = {g_{\\mu \\nu , \\lambda} \\dot x^\\mu \\dot x^\\nu \\over \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\qquad \\qquad (3) ", "30a0e36f8a85597b8923285f447a974c": "\n\\left.\\frac{d}{dv} \\left(\\frac{1}{1+z}\\right)^{-v}\\right|_{v=1} =\n- \\left.\\log \\frac{1}{1+z} \\left(\\frac{1}{1+z}\\right)^{-v}\\right|_{v=1}\n= -(1+z)\\log \\frac{1}{1+z}.", "30a0e46b59b8df74959881712fa8dc6c": "\\rho_c\\!", "30a104bb00c59c8ff7610b37084a9db5": "\\sum_{e \\in E} w(e_j) \\cdot y_j ", "30a10863cf0a1c1ab9edb6321a56506a": " B_{1}\\cup B_{2}\\cup\\ldots \\cup B_{n}\\subseteq 3B_{j_{1}}\\cup 3B_{j_{2}}\\cup\\ldots \\cup 3B_{j_{m}}", "30a15ff29d52b80ac6835cfebfe467d1": "S_{\\rho}(z)=\\sum_{n=0}^{n=\\infty}\\frac{m_n}{z^{n+1}}.", "30a1662abeaa3495f2a0f32c6e48c964": " {\\mathbf u}={\\mathbf \\nabla} p \\frac{z^2-H^2 }{2\\mu} \\, ", "30a19fc8d8ea6eb3146333dc3436951c": "\\frac{d}{dx} \\sin x = \\cos x", "30a1acd8c7cf72ca7fee83d551f64766": "sR_{AW} = {R_{AW}}{V}", "30a1c7fc78ce56ed2a0409c2079f8df8": "CVP", "30a23625c220eae6413737aad1f0d8f8": "f(x_1,\\ldots,x_n)", "30a25380d8b35d64e56a0441a8359fc8": "f\\ x = y \\equiv f = \\lambda x.y ", "30a26f343bd126010bb3573bae00271b": "\\chi = \\frac{\\mu_0 N \\mu}{B} = -\\frac{\\mu_0 N Z e^2}{6 m}\\langle r^2\\rangle.", "30a2775578c5fe9b957a22b9cdf3759f": "d((M,\\varphi),(N,\\psi))", "30a2b37e7c2b7ae806fc2c3bb9d9f089": " \\mathbf{m} = \\mathbf{r} \\times \\mathbf{q} \\,\\!", "30a2b7857f3f780abef557f392585870": "2I \\to S_5", "30a35f3e61b51bb5b030e1b8a2291d4e": "\\tbinom n5", "30a380cd234cfdce09a518326a8d4bf1": "\\frac{4}{3}\\pi r^3 = \\int_0^r A(r) \\, dr.", "30a3d803b268b4c528041fe4f935f924": "f: D \\rightarrow D^{'}", "30a45c001f8a163811400301dcecf46b": "i_0+i_1+\\cdots+i_n=2n-2", "30a488fc373ecf551f36b15f90eedcd3": "\\mathbf{M} = \\frac{C}{T}\\mathbf{B}", "30a4ad407fa580fc00a4fe12952d7933": "v=-u-1.", "30a4fd3fe20d986fe605badc3dee6ff9": "\\phi=\\frac{\\phi_0}{r_0^2} \\big( \\lambda x^2 + \\sigma y^2 + \\gamma z^2 \\big) \\qquad\\qquad (5) \\!", "30a585f3d6c42a17bc1c23f0c6dc03a3": " g\\ m\\ n ", "30a588f1fba399b0f37d44756cd96ffa": "\\frac{99}{36}=2 \\frac{3}{4}", "30a5935834cf38068fe1d59572c39ed0": "G^*=G^*_m\\frac{1+3\\phi H^*}{1 - 2 \\phi H^*}", "30a5b94533c2d69dd1d8028b40eba839": "\\sigma A > 0", "30a60ec1e11ad5b866a246ac573637b9": "x \\mapsto l_x", "30a6251ad6263e1dda6ea345510e4e95": "S(\\tau)= \\int w(t+\\tau)f(t) \\, dt ", "30a6bb05d8c0da53d70e6a601554fe35": " \\text{Cov}(X) = \\Sigma_{p \\times p}", "30a6e5d17caa46727c1da2f44322db95": "\\mu=\\mathbf{X}\\boldsymbol{\\beta}\\,\\!", "30a715b904da8599dd7e0f1f12dded8f": "\\scriptstyle n/2", "30a7646e5c4aeaafc20ef44ebc5d0ebf": "f\\colon \\{0, 1\\}^{\\omega} \\to \\{0, 1\\}", "30a76b0a948aef5a33a6ca565a4e3d84": "m\\in S", "30a7b1b094899243a6860f2bb44de0e8": "\\mathcal{L}_1", "30a7b5ba50ad70421e386d8af318f30a": "P(\\mathbf{y} | \\mathbf{x}) = \\sum_\\mathbf{h} P(\\mathbf{y}|\\mathbf{h}, \\mathbf{x}) P(\\mathbf{h} | \\mathbf{x})", "30a7bc086793c2a820c75d5f3a315eb2": "G^{\\mathrm{R}}(\\mathbf{k},\\omega)", "30a7caec25d4d20101d2c692c8385c59": "\\,\\varphi(\\alpha)=\\int_0^\\pi\\,\\ln(1-2\\alpha\\cos(x)+\\alpha^2)\\;\\mathrm{d}x \\qquad |\\alpha| > 1.", "30a7d35f50e800659c69362295934c49": "A = 2b \\cdot r .", "30a7d7c6e8ec72b57dc1adf7d66056f0": "\\begin{align}\n T_\\beta &= \\int_{\\mathbb{R}^k} \\left| {1 \\over n} \\sum_{j=1}^n e^{i\\mathbf{t}^T\\widehat{\\boldsymbol\\Sigma}^{-1/2}(\\mathbf{x}_j - \\bar{\\mathbf{x})}} - e^{-|\\mathbf{t}|^2/2} \\right|^2 \\; \\boldsymbol\\mu_\\beta(\\mathbf{t}) d\\mathbf{t} \\\\\n &= {1 \\over n^2} \\sum_{i,j=1}^n e^{-{\\beta \\over 2}(\\mathbf{x}_i-\\mathbf{x}_j)^T\\widehat{\\boldsymbol\\Sigma}^{-1}(\\mathbf{x}_i-\\mathbf{x}_j)} - \\frac{2}{n(1 + \\beta^2)^{k/2}}\\sum_{i=1}^n e^{ -\\frac{\\beta^2}{2(1+\\beta^2)} (\\mathbf{x}_i-\\bar{\\mathbf{x}})^T\\widehat{\\boldsymbol\\Sigma}^{-1}(\\mathbf{x}_i-\\bar{\\mathbf{x}})} + \\frac{1}{(1 + 2\\beta^2)^{k/2}}\n \\end{align}", "30a7dd89936a32d62ff89e68bcb985a9": " \\phi= \\pi ", "30a7e474a563f5151f01e6795a77ccbb": "\n G^{\\star}_n = G_n \\, \\varrho_o^{1/2} \\, N^{-1}\n", "30a7eac9688d54fa672757ea6fd4f1a5": "V=\\{\\}", "30a826d7ba8793ff42aea15b328ca06a": "m=\\sum_{i=1}^{N}m_{i}", "30a82b724c38d01c7081a0e03f96f466": " \\forall x' \\in K, \\ \\ j(x)(x') = x'(x).", "30a843c79460e73359719a656ad5a61c": "\\mathbf A = \\begin{pmatrix}\n\\sqrt{c_1} & 0 & 0 & \\cdots & 0\\\\\nn_{21} & \\sqrt{c_2} &0 & \\cdots& 0 \\\\\nn_{31} & n_{32} & \\sqrt{c_3} & \\cdots & 0\\\\\n\\vdots & \\vdots & \\vdots &\\ddots & \\vdots \\\\\nn_{p1} & n_{p2} & n_{p3} &\\cdots & \\sqrt{c_p}\n\\end{pmatrix}", "30a8a9318e1b34b668c420ad3895b7cf": "\\|A\\|\\leq 1", "30a8feed888292fe685e30ce2c5c86a7": " \\operatorname{build-list}[o\\ x\\ y, D, V, L_3] \\and D[g] = [x, \\_, \\_]::[o, \\_, \\_]::[y, \\_, \\_]::L_3 ", "30a91b1476720eef5e14bb514fe1312c": "p_1 D_L[F_1(K,L)]=w=p_2 D_L[F_2(K,L)]\\,", "30a91f34b9e39dc9c4eb5265ecc6cb8d": "\\mathcal{O} = \\{a\\in F: |a|\\leq 1\\}", "30a91fb853ce333c03f1e096584d4ba1": "(7)\\quad \\hat{B}_{ab}:= \\nabla_b k_a\\;,", "30a9326d92dd92d7f23514f1c3658210": "T^{i_1\\dots i_n}_{i_{n+1}\\dots i_m}[\\mathbf{f}]", "30a9e2e46072bdf788ef0d9771b5736c": "n \\rightarrow\\infty", "30aa2e74d3a9776a88fc586aa8ad5ef1": "F_\\ell(\\mathbf{P},\\mathbf{K})", "30aa5d201327a1a6a1619f2ef2d5eadf": "\n\n\\begin{align}\n\\begin{bmatrix}\n\\dot{e}_1\\\\\n\\dot{e}_2\\\\\n\\vdots\\\\\n\\dot{e}_i\\\\\n\\vdots\\\\\n\\dot{e}_{n-1}\\\\\n\\dot{e}_n\n\\end{bmatrix}\n&=\n\\mathord{\\overbrace{\n\\begin{bmatrix}\n\\dot{h}_1(x)\\\\\n\\dot{h}_2(x)\\\\\n\\vdots\\\\\n\\dot{h}_i(x)\\\\\n\\vdots\\\\\n\\dot{h}_{n-1}(x)\\\\\n\\dot{h}_n(x)\n\\end{bmatrix}\n}^{\\tfrac{\\operatorname{d}}{\\operatorname{d}t} H(x)}}\n-\n\\mathord{\\overbrace{\nM(\\hat{x}) \\, \\operatorname{sgn}( V(t) - H(\\hat{x}(t)) )\n}^{\\tfrac{\\operatorname{d}}{\\operatorname{d}t} H(\\hat{x})}}\n=\n\\begin{bmatrix}\nh_2(x)\\\\\nh_3(x)\\\\\n\\vdots\\\\\nh_{i+1}(x)\\\\\n\\vdots\\\\\nh_n(x)\\\\\nL_f^n h(x)\n\\end{bmatrix}\n-\n\\begin{bmatrix}\nm_1 \\operatorname{sgn}( v_1(t) - h_1(\\hat{x}(t)) )\\\\\nm_2 \\operatorname{sgn}( v_2(t) - h_2(\\hat{x}(t)) )\\\\\n\\vdots\\\\\nm_i \\operatorname{sgn}( v_i(t) - h_i(\\hat{x}(t)) )\\\\\n\\vdots\\\\\nm_{n-1} \\operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\\hat{x}(t)) )\\\\\nm_n \\operatorname{sgn}( v_n(t) - h_n(\\hat{x}(t)) )\n\\end{bmatrix}\\\\\n&=\n\\begin{bmatrix}\nh_2(x) - m_1(\\hat{x}) \\operatorname{sgn}( \\mathord{\\overbrace{ \\mathord{\\overbrace{v_1(t)}^{v_1(t) = y(t) = h_1(x)}} - h_1(\\hat{x}(t)) }^{e_1}} )\\\\\nh_3(x) - m_2(\\hat{x}) \\operatorname{sgn}( v_2(t) - h_2(\\hat{x}(t)) )\\\\\n\\vdots\\\\\nh_{i+1}(x) - m_i(\\hat{x}) \\operatorname{sgn}( v_i(t) - h_i(\\hat{x}(t)) )\\\\\n\\vdots\\\\\nh_n(x) - m_{n-1}(\\hat{x}) \\operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\\hat{x}(t)) )\\\\\nL_f^n h(x) - m_n(\\hat{x}) \\operatorname{sgn}( v_n(t) - h_n(\\hat{x}(t)) )\n\\end{bmatrix}.\n\\end{align}\n", "30ab4323e9b649a85212607c3dcf5748": "\\Delta _{i,j}=\\frac{\\left( a_{i}-d_{j} \\right)+\\left( d_{i}-a_{j} \\right)}{2}", "30ab47fac1888aa2965002280e4c20f7": "f\\colon\\mathbf F\\to\\mathbf G", "30ab833efca65f40fb803cafdfe28f38": "\\sigma=T_{11}-T_{22}=T_{11}-T_{33}", "30abc9955e0969bc8412d7642428ac0f": "H_4 = \\frac{1}{\\sqrt{4}}\n\\begin{bmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & -1 & -1 \\\\ \\sqrt{2} & -\\sqrt{2} & 0 & 0 \\\\ 0 & 0 & \\sqrt{2} & -\\sqrt{2}\\end{bmatrix}\n", "30ac3c6e0f54b550e1a4fbaa2c493b21": "W\\!", "30ac4211262f8ea51cbf7666fcc01a07": "\\cdots\\rightarrow P_2 \\rightarrow P_1 \\rightarrow P_0 \\rightarrow A\\rightarrow 0", "30ac6db34d10326e889e80692b0f7205": "X \\sim \\beta^{'}(\\alpha,\\beta,p,q)\\,", "30ad699794926716d9f092b3f7c1f3ba": "\\Delta \\nu = \\frac{c}{2\\sum_i n_i L_i} = \\frac{c}{2}\\left[ \\frac{1}{n_1 L_1 + n_2 L_2 + n_3 L_3 + \\ldots} \\right]", "30ade8772c9d65bfa58ab7e7bcd430e1": "O_X(1)", "30adee242d4eaf0996dd518899e4a0ba": "G (r) \\approx \\frac{1}{r^{d-2+\\eta}}\\exp{\\left(\\frac{-r}{\\xi}\\right)}\\,,", "30ae0b80adec9059dfdaf1bd276667b0": "q=r", "30ae2a79dbbad608a0cc9590c5f33fb1": "Q=(1-D)/4", "30ae314327b6b95281e131cba5a8168f": "K=\\rho \\frac{\\mathrm d P}{\\mathrm d \\rho}", "30aed477a1bcf5656ff3bbdca2b4501f": "\\mathbf{A}, \\mathbf{B}, \\mathbf{C}", "30aee72f22279496acb8418aef654217": "\\int_{-\\infty}^\\infty \\delta (\\xi-x) \\delta(x-\\eta) \\, dx = \\delta(\\xi-\\eta).", "30aef2036114adf0cd85fcc7f7abe4be": "\\begin{align}\n \\Rightarrow\\frac{2}{\\pi} &= \\prod_{n=1}^{\\infty} \\left(1 - \\frac{1}{4n^2}\\right) \\\\\n \\Rightarrow\\frac{\\pi}{2} &= \\prod_{n=1}^{\\infty} \\left(\\frac{4n^2}{4n^2 - 1}\\right) \\\\\n &= \\prod_{n=1}^{\\infty} \\left(\\frac{2n}{2n-1}\\cdot\\frac{2n}{2n+1}\\right) = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdots\n\\end{align}\n", "30af58bf98d1cedaa2232f0eb94cb4cc": "A = \\{a_j\\}", "30afaf146f33f32867b8260d018a89ad": "\\epsilon, \\alpha", "30b0e43878851121dedb7f04ffd5c214": "f \\approx 0.381966. \\,", "30b1106cfbd1f38a3fffebd84e5ce891": "\\begin{pmatrix}T_3\\end{pmatrix}\\,", "30b15e4e8b64eaed41ff2e08f5710fb6": "V_n = \\prod_{1\\le iu|C=c)\\le \\Pr(B>u|C=c)", "30b352be5b5f15415d5ffb6df1c548ea": "\\{p\\}\\cup A", "30b3a070af9471e6d2f13a8f7e58d38c": "V_n = U_{n+1} - Q U_{n-1}=2U_{n+1}-PU_n \\,", "30b3b4d0511808c9468a0a0eceba68d9": "P\\in {\\mathcal Q}", "30b3f2ee326b296fd51ed808e4b3f6be": "\\sigma = {m_e^*}/{m_h^*}", "30b407905900b9ec5085b5ddef32bebd": "\\lang k^{(0)} | V | n^{(0)} \\rang ", "30b442dbf320b26c10f727ffac23103c": "\\mathbf{B} = \\frac{1}{c_0} \\hat{\\mathbf{k}} \\times \\mathbf{E}", "30b5058d41eb5dc79e923606e15c3fe7": " \\cos({{2\\pi}/5})={{\\sqrt{5}-1}\\over 4}>0.\n", "30b54e56d098343c1ea749559d2b3cc5": "e\\in\\mathbf{N}^{+}", "30b577f9d4f7974defa90b4a4f8f0dff": "\\forall A\\in \\mathcal{B}\\;\\; \\mu(T^{-1}(A))=\\mu(A) ", "30b5b9a41beac0a0a25ab9cff6bbc3ed": "\\delta \\sqrt{-g}= -\\frac{1}{2} \\sqrt{-g} g_{\\mu\\nu} \\delta g^{\\mu\\nu}", "30b61c86880c6643dfd12d8340179b0c": "\\dot{x}^a = \\frac{\\partial H}{\\partial p_a} = g^{ab}(x) p_b", "30b6baa880581278e7a68b0739500e36": " P = C V^2 f ", "30b6d5d9958c2006d8049c20dbb83301": "P[X_1,\\ldots,X_n]=\\prod_{i=1}^nP[X_i|pa_i]", "30b7672aeb46afdcd80860d94554b561": "\n\\mathcal{A}^t=\\sum (-1)^{|\\alpha+\\beta|}\\binom{\\alpha+\\beta}\\alpha (\\partial^\\beta a_{\\alpha+\\beta})\\partial^\\alpha.\n", "30b7ad68b66bcf3be3f86d897109d305": "\\left\\langle\\mathbf{P},\\mathbf{P}\\right\\rangle = |\\mathbf{P}|^2 = P^\\alpha g_{\\alpha\\beta}P^\\beta \\,,", "30b7b0838e957973948a52e687516bb7": "\n\\begin{align}\n\\lambda &= 2 \\arctan \\left[\\frac{zx}{2(2z^2 - 1)}\\right] \\\\\n\\phi &= \\arcsin(zy)\n\\end{align}\n", "30b87e39242db9ee088d98bd1a547200": "H_{rc}(f) = H_{rrc}(f)\\cdot H_{rrc}(f)", "30b887e5fd5f2a470b085af1fcba817e": " 0 < c < \\infty ", "30b9092c24dcbbfd7deb7355ad28d7cf": " \\psi_1(1) = \\frac{\\pi^2}{6}", "30b94bbbad526eeb6dd345afdaeaccf8": "x'", "30b96a563018da33a0a1e0683f36d80f": "kA_c\\frac{d^2T}{dx^2} + k\\frac{dA_c}{dx}\\frac{dT}{dx} + Ph\\left (T-T_\\infty\\right) = 0.", "30b9b685833cc089c77820a5c4606c2a": "{5 \\choose 1}{4 \\choose 3} = 20", "30b9bad0939e2a357c93df0686030596": "2^{H(p + \\epsilon)n}", "30b9e0f6087e4db1225f0700b2da07fa": "\\displaystyle{(a+c)^b= a^b + B(a,b)^{-1}c^{(b^a)}.}", "30b9f00ad9ca04c5a9a2f859862ad48e": "Z_{S_3}(t_1,t_2,t_3) = \\frac{t_1^3 + 3 t_1 t_2 + 2 t_3}{6}.", "30b9f1c79e1f47916041e42ecb12146c": "\\mbox{Aut}(\\widehat{\\mathbf C}) \\cong \\mbox{PSL}(2,\\mathbf C).", "30ba4967e49a1fa6b0ff725a7e246fab": "\\nu / \\text{GHz}", "30baa94172ecfb17ec68cbc06b1f3353": "\\max_\\alpha\\{\\mathsf{H}((1-\\alpha)(1-p)) - (1-\\alpha)\\mathsf{H}(p) \\},", "30baf51e8c1973814afbf4b3e37adea8": "\\phi_* : \\mathfrak g \\to \\mathfrak h", "30bb02bcee09bf21849c2ec8bf878288": "X \\to Y \\to Z \\to X \\to Y", "30bb174e68c09b2a883d522c8a108fd1": "(L+H)/2", "30bb9e6c82f3955605075c42ca659a5f": "\\sqrt{3}, \\sqrt{5}, ...,", "30bbd121659e35781c38cae828f9f236": "(x,y)\\in U\\times V", "30bc0994d6f74007cd5a5264339b555a": "f(z) = \\frac{1}{g(z)} + b", "30bc1bd0aa0913255739ce0c398f986d": " 0 \\le n < N_\\text{ZC}-1, \\,", "30bc3843b0f94cd2fa655273f196cfb5": "\\hbar k_3 \\approx \\hbar k_1 + \\hbar k_2 ", "30bc7a044bd0dd3457f8333ccdadbab0": "\\begin{cases}\n a_n =a_{n-1}-b_{n-1}\\\\\n b_n =2a_{n-1}+b_{n-1}.\n\\end{cases}\n", "30bc8207ecf7325ce91b55fc3b8ed661": "l_c=M\\times l_m", "30bcb186064efcd404f1e4544e3fef11": "2x^2-5x+4x-10 = 2x^2-x-10.", "30bcb22b9b3d9051517ecec9c5947fd9": "\n\\frac{1}{2}e^{-x}\\,\\ln\\!\\left( 1+\\frac{2}{x} \\right)\n< \\mathrm{E_1}(x) <\ne^{-x}\\,\\ln\\!\\left( 1+\\frac{1}{x} \\right)\n\\qquad x>0\n", "30bcdf66559e1fc77330ecf2105a09a0": "\\theta_N(x) = \\begin{cases}\n0 & \\text{if } x \\leq N/2, \\\\\n1 & \\text{if } x > N/2. \\end{cases}", "30bd0d012940fc02fb33ecf916ba4d2d": "\nu_\\epsilon(\\vec x) = u(\\vec x,\\vec y) = u_0(\\vec x,\\vec y)+\n\\epsilon u_1(\\vec x,\\vec y)+\\epsilon^2 u_2(\\vec x,\\vec y)+O(\\epsilon^3)\\,\n", "30bd20a69a73a6da036e54ec2778883e": "\\hat x = \\sqrt{\\frac{\\hbar}{2m\\omega}}(a+a^{\\dagger})", "30bd2d0c9721acba5b3fdc91b4801b4d": "b^e = b^{\\left( \\sum_{i=0}^{n-1} a_i 2^i \\right)} = \\prod_{i=0}^{n-1} \\left( b^{2^i} \\right) ^ {a_i}", "30bd5674617b21a6b5ed424b0173f7a2": "\\wr \\!\\,", "30bd8a388b4617b4d2d94c7e7740e7cd": "\\mu_{\\mathrm{sf}}", "30bdca55147a90704782a99db3de26ec": "\\{X(t) : t=1,\\ldots,N\\}", "30bdd34f34ebf6304108c62b9e98d2b0": "\\scriptstyle{v/c}", "30be1d4e6765d7c03184b7aa4a37e4c5": "(1-c)^n \\to 0", "30be712385617332bcb9538e06642297": "\n\\frac{\\pi\\,(n-1)}{2\\lambda}\\prod_{k=1}^{n-1} \\Bigl(1+\\frac{\\lambda}{2k}\\Bigr)=\\frac{\\pi\\,(n-1)}{\\lambda^2}\\cdot\\frac{1}{\\Beta(\\lambda/2,n)}\n", "30bef351a43a5f2b5832c27b21d10a04": "C.C'", "30bef4eb54f79d4d5f0270b13d9c3a36": " \\sum_{n,k=0}^\\infty \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\} \\frac{x^n}{n!} y^k = \\textrm{exp}(y(e^x-1))", "30bf1e5d5e189febb1f26f939aaba76d": "motor = (left, neutral, right)", "30bfa9c6abb6a3b1ad3c68da6509ed96": "\\mathrm{1\\ Square\\ of\\ Land} =(\\frac{\\mathrm{77\\ acres}}{\\mathrm{3\\ Squares\\ of\\ Land}}) = 25.41\\ acres ", "30bfdcfbd9a4e20bba0148762f16c555": "u(x,t)", "30bfe14e090ef8e144bb2f7538361a3c": " U = I - i \\epsilon G + O(\\epsilon^2) \\,", "30bfe28cdea44a4e0b17820fcc93b80b": "l_{in}", "30bfe38acd4dc753654f5035c2baed6f": "6\\times(c\\bmod 7)\\bmod 7 + \\rm{Sunday = anchor}.", "30bff8ac7fc6fc268325cad0d5f52878": "z^{4M} + az^{2M} + 1 = (z^{2M} + \\sqrt{2-a}z^M+1) (z^{2M} - \\sqrt{2-a}z^M + 1)", "30c022a948e84419a0e283718a0af23e": "t=\\infin ", "30c0323d87380aa6bae0b3b0377e740e": "\n\\qquad C ::= in \\ n \\; \\mid \\; \\overline{in} \\ n \\; \\mid \\; out \\ n \\;\n\\mid \\; \\overline{out} \\ n \\; \\mid \\; open \\ n \\; \\mid \\;\n\\overline{open} \\ n", "30c09623549cbba8ed16b5c313cc0200": "[f(U) \\in U] \\wedge [f(V) \\in V].\\,", "30c1d2beca17ba8c34118203db278d92": "\\tfrac{r^2}{2} = 18", "30c1dfbec6848b15b4c4a920c02aee44": " G_{\\sigma} \\nabla ^2 I ", "30c23552ee38ee6d7ce05b094886f58e": "\n\\frac{\\mathrm{cyt~c_{red}}}{\\mathrm{cyt~c_{ox}}} = \\left(\\frac{[\\mathrm{NADH}]}{[\\mathrm{NAD}]^{+}}\\right)^{\\frac{1}{2}}\\left(\\frac{[\\mathrm{ADP}] [P_i]}{[\\mathrm{ATP}]}\\right)K_{eq}\n", "30c24016df2b868da4e3a8ec58e45ce7": "{\\scriptstyle \\partial \\Omega }", "30c2bf61a2e085c8884e51ff9807b339": "\\mathrm{Re} z = 1", "30c2c56aff861322bd4d6d5b9885a712": "PC = \\left(\\frac {\\eta_H \\cdot \\eta_O \\cdot \\eta_R}{\\mbox{appendage coefficient}}\\right) \\cdot \\mbox{transmission efficiency}", "30c2fc859935295aab217aa2894c7845": "d r^* = 0 + \\frac{\\Delta x}{2}i + \\frac{\\Delta y}{2}j + \\frac{\\Delta z}{2}k", "30c304680c618bcb0fdc318bb66a95c7": "g: U \\rightarrow \\mathbb C", "30c304fe704846b8b570ffeb3421d8f2": "\\Delta(t) = t^3 - t^2 + t - 1 + t^{-1} - t^{-2} + t^{-3}, \\, ", "30c32e3f749b51d312f8619c87a784cf": "M_{\\mathrm{right}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\frac{q dx \\, x^2 (L-x)}{L^2} = \\frac{q L^2}{12} ", "30c3586fa1bb7f21b9333dcda2618a9b": "\\dot{p}_k=dp_k/d\\sigma", "30c361d3c262ebcc599bf0a4923569a0": "\\Delta f=0", "30c3ade6aea06ba15bafc6c9838683eb": "16_1\\oplus 10_{-2} \\oplus 1_4", "30c3b54ee7acacdd1643da67a2b6c2f1": "w = 1 + 0 + 5 + 0 \\ \\bmod\\ 7 = 6 = Saturday", "30c44d94140b8a6575d51842818f5936": "= \\nabla \\varphi", "30c4df53d08813d9ec9bb9d05117bae0": " \\and S_6 \\implies A_6 = n ", "30c4e1b8febe27eca69f6dd5f0ecaa8b": "1-\\exp\\left(-\\eta\\left(e^{bx}-1 \\right)\\right)", "30c50147f5cbab5bb5a9bef8a4bb267d": "\\sum_{k=-n}^n e^{ikx}=\\frac{e^{-(n+1/2)ix}-e^{(n+1/2)ix}}{e^{-ix/2}-e^{ix/2}} =\\frac{-2i\\sin((n+1/2)x)}{-2i\\sin(x/2)} = \\frac{\\sin((n+1/2)x)}{\\sin(x/2)}", "30c523f40ed2f83b45190ea2cb68e630": " - 2c ", "30c543a66b6b2274b99008aec22ffd70": "\\mathbf{k} = \\sigma_2 \\sigma_1, \\mathbf{i} = \\sigma_3 \\sigma_2, \\mathbf{j} = \\sigma_1 \\sigma_3", "30c5d832ad31f16cba0ff01dc8f1ac98": " M_Q = M \\otimes_R Q, ", "30c64c1ed3ae2e916d5e0cabc31820d4": "\\boldsymbol{\\mathit{1}}", "30c697053aa8ae154a0031564fcf204a": "\\mathrm{Var}(X)= \\begin{cases} \n\\infty & \\alpha\\in(1,2], \\\\ \n\\left(\\frac{x_\\mathrm{m}}{\\alpha-1}\\right)^2 \\frac{\\alpha}{\\alpha-2} & \\alpha>2. \n\\end{cases}", "30c6a90d2e73c6840b3a29a3827844c0": "(\\and) \\frac{A \\wedge B}{\\begin{array}{c} A \\\\ B\\end{array}}", "30c709eb596de0e476b56132ffb1623e": "\n \\Delta = \n W_4W_2^{15} - W_3^3W_2^{12} + 3W_4^2W_2^{10} - 20W_4W_3^3W_2^7 +\n 3W_4^3W_2^5 + 16W_3^6W_2^4 + 8W_4^2W_3^3W_2^2 + W_4^4.\n", "30c74c25cf3a4f12a177406694611c36": "P_0=200000", "30c78fa095bc45f0ae33c1999df20300": "\\mathbf W", "30c7b5c2666e566fadf8928f559406fd": "\\mathbf{R}^n \\to \\mathbf{R}^{n+1}", "30c872662b356aa720d1971361b45724": "a_2", "30c87382e282d104cfa19578cc03c3eb": " \\mho^i(G) = \\langle \\{ g^{p^i} : g \\in G \\} \\rangle. ", "30c8a52b726e9d2e20d9b7949b89b72d": "\\psi - \\widehat{W_r}", "30c90c622beace6b93029594c161113d": "\\mathrm{T}/\\mathrm{m}", "30c99e0905d25f0367bea55d8733b821": "h(t) = \\mathcal{L}^{-1}\\{H(s)\\}.", "30c9cff064c33d912cd79b3b3e1959ea": "\\hat S_n", "30c9fe33284be39a5b086ba808ec5044": "U(f)(z)=zf(z)", "30ca267f4622df0b997e82aa9680685a": "\\bullet \\rightrightarrows \\bullet", "30ca2bad700053da1bc7e745bb6b7062": "(x_i, y_i, z_i, \\dots)", "30ca34ca84255725a89014af3f664fb0": "2 x U_n\\left(1-2x^2\\right)= (-1)^n U_{2n+1}(x).", "30cacab82939d882f763ac25fd2f6eb2": "C_{0} ([0, T]; \\mathbb{R})", "30caf60b93e8faf4b2885a4bdf9b2345": " \\mathrm{\\frac{9\\,\\rm{L}}{100\\,\\rm{km}}} =\n \\mathrm{\\frac{9\\,\\rm{L}}{100\\,\\rm{km}}}\n \\mathrm{\\frac{1000000\\,\\rm{\\mu L}}{1\\,\\rm{L}}}\n \\mathrm{\\frac{1\\,\\rm{km}}{1000\\,\\rm{m}}} =\n \\frac {9 \\times 1000000}{100 \\times 1000}\\,\\mathrm{\\mu L/m} =\n 90\\,\\mathrm{\\mu L/m}", "30cbb1065f387292b688fa0050eb117b": "r_1 = \\sqrt{ x^2 + y^2 + z^2 } \\,,", "30cc2425382385ff47fbf316ea9fe13a": "\\mathrm{{}^{89}_{38}Sr}\\rightarrow\\mathrm{{}^{89}_{39}Y} + e^- + \\bar{\\nu}_e", "30cc3542760cd566653c57fdaae5c70f": "\\vdash P \\to A(x)", "30cc36cda3d61c124e35bb4a1f17b7f8": "V=\\frac{1}{3}l^2h.", "30cc94029d6c51fdce5f0dbfe3e35f9c": "\\omega_Y = i^*\\omega_X \\otimes \\operatorname{det}(\\mathcal{I}/\\mathcal{I}^2)^\\vee,", "30cca842f733487cd7077a39040dba05": "N_R=n_e^2 \\beta_2(T_e)", "30ccdeeff5cf0c2877b0833789a6df90": " v:\\mathcal{T} \\rightarrow \\mathbb{R}^+\\cup\\{+\\infty\\} ", "30cd547249cf986fa54bbb1e3a74362e": "a \\to b \\to 2 \\to 3 = g_3(2) = g_2^2(1) = g_2(g_2(1)) = f^{f(1)}(1) = f^{a^b}(1)", "30cd5f322ada11977a9e828c01b5df2f": "\\, e^{-i\\omega t}", "30cd64cab6e0d3b632a1581b6457a7ca": "\\int \\sin{x}\\, dx = -\\cos{x} + C", "30cdc99ee93597b9d1660b5c650bc4c3": "\\mathbb{C} \\to \\mathrm{End}(V)", "30ce66bc0df9ab0293759328424e8e09": "\\alpha \\simeq 2", "30cf12a55eb4bd52792cb7fec9fca893": " \\frac{256}{81} \\approx 3.16049", "30cf6e33e33c08c826091ffebf6e33ea": "{\\mathbf{}}F", "30d0214cd391f2d445555bffef4db917": "\\frac{\\sqrt{15}}{2}\\cos(\\theta)\\sin(2\\phi)", "30d06d46d6b93199755caa2b3fa957e6": " \\min_{x\\in\\mathcal{H}} F(x)+R(x)", "30d094f3b5d112010e5c51f76be9347c": "\nv(\\mathbf{R}- \\mathbf{r}) = v(\\mathbf{R}) - \\sum_{\\alpha=x,y,z} r_\\alpha v_\\alpha(\\mathbf{R}) +\\frac{1}{2} \\sum_{\\alpha=x,y,z}\\sum_{\\beta=x,y,z} r_\\alpha r_\\beta v_{\\alpha\\beta}(\\mathbf{R})\n-\\cdots+\\cdots\n", "30d0ac91e371859f1742798b0806893c": "(2n+1)(n^2+n+1)", "30d0b909311c53c1df690262daa7a976": "q+1", "30d0c6c37919a0a947fde7c000a2d2af": "\\boldsymbol\\beta\\cdot\\dot{\\boldsymbol\\beta}", "30d0ca57dac4a9b67ed463d4ea0c7b38": " H\\left(\\bold{q},\\frac{\\partial S}{\\partial \\bold{q}} \\right) = E. ", "30d15731506533f20bd1225da1c58aac": "\\vec{r}", "30d19d9d93d3234024f66ff910120657": "\\rho(\\mathbf{x},t)=\\rho(\\mathbf{x})e^{-i \\omega t}", "30d1de939aeedeb21f0544f00376e080": "\\Theta(k)", "30d1e3402c0c5bf2f5081c7762700187": "q_x=q_x(x,y)", "30d2868522d9ecd2f73993a8dd5a3189": "\\left[\\frac{\\Delta \\lambda_B}{\\lambda_B}\\right] = C_S\\epsilon + C_T\\Delta T", "30d29ff22a2b8e397afb6e71a59b6673": "\\forall U \\in T \\mbox{ s.t. } U \\neq \\emptyset, \\mu (U) > 0.", "30d2e8349c908ee667cbfa3866278a98": "\\theta \\sim", "30d2f37875c7120973128c0776e9ae5a": "D - \\widehat D", "30d37d685c0cd34f1ba5aff0ff6bc7b4": "a_{n-1}x^{n-1} + \\cdots + a_1x + a_0.\\,", "30d3f66588dac29f72fb4f529d41a9f6": "\\scriptstyle \\sum_N", "30d42d8b561d1f1ecfe73007a92a6861": "\\nu_2=B", "30d439695ce5517da7db6748140487c9": "b=24 ", "30d44c2bdc6ac118b09d06ea8f81e115": " A_p = \\frac {T_m}{T_m + T_d} \\begin{cases} A_p = Predicted \\ Availability \\\\ T_m = Mission \\ Duration \\\\ T_d = Model \\ Down \\ Time \\end{cases}", "30d4502fe5254e16560bc0ff3e0bd693": "\\cos\\theta_2", "30d452a4edde7ab41d9fbedb08802dc0": " \\frac{{\\rm d}a}{{\\rm d}N} = C \\Delta K^m ", "30d499c9f4b14c6139eea1db2a959673": " h\\in H_1\\setminus \\{w_{1,\\beta_m},1\\} ", "30d4a4668b976b2686c976ae4087b7d4": "\n (x_1, x_2, x_3) = \\mathbf{x} = \\boldsymbol{\\varphi}(q^1, q^2, q^3) = \\boldsymbol{\\varphi}(r, \\theta, z)\n = \\{r\\cos\\theta, r\\sin\\theta, z\\}\n ", "30d500e292b7fba8bc36c7feb57885a6": "n = 1,2,3", "30d5a7fd27584d1b42f0632ed39d2c4a": "x_i=\\sqrt{\\rho}M+\\sqrt{1-\\rho}Z_i", "30d5cae1d21bae1b6af6b843d3f590c3": "i = p < \\lfloor n/2 \\rfloor", "30d5e3b531334410fab5280f81cdb877": "s = \\frac{g \\sin \\theta}{\\cos \\theta} = g \\tan \\theta", "30d5edae27b10e87eb446524e2a638e5": "\\begin{matrix} {10 \\choose 2}{4 \\choose 2}^2 \\end{matrix}", "30d607ed3f74dc2d3b51f9769a71ae97": " W_{ij}(t;L)", "30d669733fb6fed9e820ec497dc651a8": "5 \\times 0 = 0", "30d683c4cc6f702695b899c923054c40": "\\bigstar \\bigstar \\bigstar |||", "30d6b215e9b2750129e4cc1f9c23291f": "a^2 - c^2 = d^2 - b^2", "30d6f0336fda1a078c867b27c3da10cd": "p\\mid q", "30d71d17889d0cc589a3cd7de704e5e6": "L^{(d_k)}_k", "30d7359003a00ef45ae5c940409e8a4c": "r^2 - 2 r r_0 \\cos(\\theta - \\phi) + r_0^2 = a^2\\,", "30d744bc1ac345e5d92cd26bfc8642fc": "B_{p,1}(z) = 1+zB_{p,p}(z)", "30d7b6e54296af46da744cb2995dbc5c": "p_1,c_1,sid_1", "30d7d4b813466e903e442f0ce5676f5f": "\\sqrt{\\frac{4}{105}}\\!\\,", "30d847ec9b2a7870fe86333a0b3a49c3": "\\ y,", "30d8cacc886d2cd01df9a05a485404dd": "\\frac{-1}{\\pi k} \\sum_{n=1}^k \n\\sin \\left( \\frac{2\\pi nm}{k}\\right) \\psi \\left(\\frac{n}{k}\\right) =\n\\zeta\\left(0,\\frac{m}{k}\\right) = -B_1 \\left(\\frac{m}{k}\\right) = \n\\frac{1}{2} - \\frac{m}{k}", "30d9270cbd043bcedaf8c62b8d8d5233": "\\operatorname{E}(1_B \\,\\operatorname{E} (X\\mid N))= \\operatorname{E}(1_B \\, X)", "30d92a75889f3605781c52c9ea26310e": "c = (a^2 + b^2 - p^2)/(2p)", "30d949f818dc6b2c3a1db5d4848d25bc": "\n \\begin{cases}\n \\bar{x}=a_1+2a_2 \\\\\n f_0 =\\exp(-(a_1+a_2))\n \\end{cases}\n ", "30d994274cce009f1d9a4bc8ccb63702": "\np_{k,i}^C={\\mathcal P} \\left[ C_k(s_1\\ldots s_i)=s_{i+1} \\right | s\\in S_k\\text{ with probability }\\mu_k(s)]\n", "30da9beb908a84599b19328b1ab879f1": "L = mr^2\\dot{\\phi} \\, ", "30db0ca02e5630940200478cb7350642": "E=F[\\alpha]", "30dbfa4c7499b31f1ebf241eb1e9de3a": "{\\tilde{A}}_{3}", "30dc36fd0d3ca5356f27b818b1dadca7": "\\varphi_j(a), j=s,i,r", "30dc5eda329ba47bf90283bb517077e8": "\n\\frac{\\partial x}{\\partial u} = \\frac{\\begin{vmatrix} -\\frac{\\partial F}{\\partial u} & \\frac{\\partial F}{\\partial y} \\\\ -\\frac{\\partial G}{\\partial u} & \\frac{\\partial G}{\\partial y}\\end{vmatrix}}{\\begin{vmatrix}\\frac{\\partial F}{\\partial x} & \\frac{\\partial F}{\\partial y} \\\\ \\frac{\\partial G}{\\partial x} & \\frac{\\partial G}{\\partial y}\\end{vmatrix}}.\n", "30dc758bcc59fcf6d5b65ff4737ff81f": "V={2 \\pi r^3 \\over 3}", "30ddda3aa1296e114a3be2d99e69ce66": " \\scriptstyle 440\\, Hz. \\textstyle 2^\\frac{1}{12} \\, 2^\\frac{2}{12} \\, 2^\\frac{3}{12} \\, 2^\\frac{4}{12} \\, 2^\\frac{5}{12} \\, 2^\\frac{6}{12} \\, 2^\\frac{7}{12} \\, 2^\\frac{8}{12} \\, 2^\\frac{9}{12} \\, 2^\\frac{10}{12} \\, 2^\\frac{11}{12} \\, 2 ", "30de0c4272cbd477a98afd49ce5d2e94": "T \\approx {1 \\over 5} \\theta_\\mathrm{D}", "30de205332e97dcaf1fd744ee45e94da": "F[y]= -\\frac{y}{y'}", "30de2818efbf6f0a64a3b224126adfb0": "\\kappa_p(\\theta) = \\dfrac{\\alpha-1}{\\alpha} \\left(\\dfrac{\\theta}{\\alpha-1}\\right)^\\alpha", "30de6e8e3be04e94ba9651acd8cb64ae": "\\mathbf y", "30de70ea76476c4deb8b436a710d56da": "x(u,v)= \\left(1+\\frac{v}{2} \\cos \\frac{u}{2}\\right)\\cos u", "30de72b0f36388ec7972e23040f60c03": "T_n = {n(n + 1) \\over 2} = {n^2 + n \\over 2} = {n+1 \\choose 2}", "30dea157bea2678b39a47402abba5df7": "e_v = H(M || r_v) = H(M || r) = e", "30dea6c67dcaf6b2c247cef764fcaff6": " \\left|\\int T\\bigl(\\tau^x(\\varphi_r)\\bigr)(y) \\tau^x(\\psi_r)(y) \\, dy\\right| \\leq Cr^{-n}", "30deae1c16c7583df4d65915db2285a4": "\\kappa=|r''(s_o)|", "30deaf48b9fce3167a3d251244e9a58a": "E=-m L^2 V_0^2/2\\hbar^2", "30dedb313b931e71e0cac2a1eab2780b": "h_i = \\left. s_i^2 \\right/ HHI.", "30dee156d82b3b4c99cb093c99ae6810": "a_n=n2^n+1\\, .", "30df4d5892a7a68a5db6390e3d313b1e": " \\mathbf{g} = \\frac{\\mathbf{S}}{c^2}\\,.", "30df50062477d862a3d5778d698f5699": "\\Sigma^0_\\alpha", "30df7e1804b10bd721959b9cba222987": "\\tfrac{5}{3} \\scriptstyle{\\sqrt{11}}", "30df95604f1d39718ec7aea5a8886404": "E_{11}\\,\\!", "30dfc05aafeaca80c9eadee8ec823f84": "\nL\\frac{dI}{dt} = - V\n", "30e00d3fd2390567cfc04abe05a1ab2a": "(x-1)x(x^{16}-26x^{15}+325x^{14}-2600x^{13}+14950x^{12}-65762x^{11}+", "30e01efb536d82297befe5ade16d0ade": "\\alpha \\not\\in \\mathbb{Z}", "30e023b407a0f52837d8a991e821716c": "\\Pr(A)=\\sum_n \\Pr(A\\mid B_n)\\Pr(B_n),\\,", "30e028e205aeda492c5fe95d4d07d006": "y = D/C", "30e02a33c100b8643e5f38b25ec9b8d2": "J^\\mu\\Big.", "30e030438f30a7e6fc5608c76f613301": " H =G\\cdot \\exp(C) =\\exp(C)\\cdot G.", "30e0a0ba48863fd2902b22c481cd7b7a": "R< 2 \\land b > 0 \\}.", "30e73356b62f92de11d5b6c3a220e52e": "\\vec F(\\vec x) = \\frac{1}{3} \\vec x = (\\frac{x_1}{3}, \\frac{x_2}{3}, \\frac{x_3}{3})", "30e80b0cc7cf1927635b7f66af7321cb": "\np(V)=3K_0\\left(\\frac{1-\\eta}{\\eta^2}\\right)\\exp\\left[\\tfrac{3}{2}\\left(K_0^\\prime-1\\right)(1-\\eta)\\right]~,~~ \\eta:=(V/V_0)^{1/3}~,~~ K_0^\\prime := \\frac{dK_0}{dp}\n", "30e8646325124e0ae64d45f21684ce1b": "\\triangledown_X: X\\to X^\\triangledown", "30e87a12af010023c094d753ec289726": "|\\det M| = \\sqrt{\\det P} \\le 1.", "30e8835ada4384e53bd7510d3c1df188": " \\|\\mathcal{H}f\\|_p \\leq A_p \\|f\\|_p", "30e8b26d867f67ab37696e4a94ff83a2": "\n(u_t + u u_x)_{xx} = u_x u_{xx},\n", "30e8df6310b53fae51f5c6fc23f56681": "\\mathbf{a},\\mathbf{b}\\in\\R^2", "30e90ec9019810043703ef46781af4e2": "SO_2=\\frac {C_{HbO2}} {C_{HbO2}+C_{Hb}}", "30e91b52e72f1bcb7f9c308a8106fbeb": "\\displaystyle \\tau_\\phi", "30e93bc05217eb0680b606aec6826365": "z=r\\,e^{i\\theta}", "30e98579aa3804e1396194586a8b7708": "1-\\frac{1}{\\sqrt 2}\\approx 29.3\\%", "30e9976a71701ed65a001c546fa71bdd": " \\lambda \\geq 1 ", "30e99bb5ad9855c7ef775dfec895a8bf": " V/\\!\\!/G=\\operatorname{Spec} A=\\operatorname{Spec} R(V)^G.", "30e9b10f7efbb855e5577040819a89c6": "\\vec E=\\rho\\vec J", "30e9c388436c18afabe87aa7c5f2170d": "n = p_1^{\\alpha_1}p_2^{\\alpha_2} \\cdots p_r^{\\alpha_r}", "30e9e80b775a7d2c69d5d9bd64b4f407": "\\Pr\\left[X \\le \\min(U_A,U_B) \\right] \\!", "30ea7a2fd25764ffe16b0e6ce9b02b82": "z=f(\\zeta)=\\int^\\zeta \\frac{dw}{(w-1)^{1-a} (w+1)^{1-b}}.", "30ea8baaed2092bc526f008cec7d6118": "V_G = I_1 R_G\\,", "30ea9462eff4e0f7209428280eb2e1f9": "\\langle.,.\\rangle", "30eaa39b08a11bf0f9745efda6f9e55c": "\\Delta A", "30eb15651099123b7315e37415163213": "f(x)\\sim\\frac{c^\\alpha (1+\\mbox{sgn}(x)\\beta) \\sin(\\pi \\alpha / 2)\\Gamma(\\alpha+1)/\\pi}{|x|^{1+\\alpha}} ", "30ebe778df5395d44823da59e442581e": "W(3,q)", "30ec1b0f4b9fe32eb357c1210d2e6a65": "0=\\boldsymbol{\\eta}(0)", "30ec3b9d0b34c0276eae80227d626022": "\\frac{(n+\\alpha)(n+\\beta)(2n+2+\\alpha+\\beta)}{(n+1)(n+1+\\alpha+\\beta)(2n+\\alpha+\\beta)}", "30ec61a9660e3c13689e396e80025249": " ds^2 = d\\mathbf{x} \\cdot d\\mathbf{x} = dx_1^2 + dx_2^2 + dx_3^2, ", "30ec882d10eb911dc311a175a6c6becc": "\\alpha S = 4 s_w^2 c_w^2 \\left[ \\Pi_{ZZ}^{\\prime}(0) - \\frac{c_w^2 - s_w^2}{s_w c_w} \\Pi_{Z \\gamma}^{\\prime}(0) - \\Pi_{\\gamma \\gamma}^{\\prime}(0) \\right]", "30ec8e86c62c10ff7de9da05522cbc63": "R_{\\theta JC}", "30ed01c88122c19784be7251d03ede51": "F(y)", "30ed0a31bfe620753611025ad27cd41f": "p\\times 1", "30edbda429a56e764b668062fcf3ee63": "(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\\,", "30edd4d7dce5f7f2be6fd9aa8ead5172": "(\\mathbf{Z}/2)^k", "30ee960be398aa04b9e4b770c0af4d9c": "T_{ig}\\, ", "30eeb203d66f0c29522e851b605d8a9e": "[4,5]", "30eedc04bea2ad89cede72ee759a3d55": " \\lim_{x \\to 0}\\frac{\\sin x}{x} = 1 ", "30ef813557556c45a3712da079503fc3": "C_Y=\\left[\\begin{array}{ccc}\nE[x_{1},x_{1}] & E[x_{2},x_{1}] & E[x_{3},x_{1}]\\\\\nE[x_{1},x_{2}] & E[x_{2},x_{2}] & E[x_{3},x_{2}]\\\\\nE[x_{1},x_{3}] & E[x_{2},x_{3}] & E[x_{3},x_{3}]\\end{array}\\right]=\\left[\\begin{array}{ccc}\n1 & 2 & 3\\\\\n2 & 5 & 8\\\\\n3 & 8 & 6\\end{array}\\right].", "30effa7cde6860073de58031d6a71e16": "\\begin{array}{cc} \\begin{array}{rrrr} \\\\ j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} & nj & nk & nl & nm & & & \\\\ a & b & c & d & e & f & g & h \\\\ \\hline a & & & & & & & \\\\ n & & & & & & & \\\\ \\end{array} \\end{array}", "30f0b037fd976005cc2f7f55f3af71c7": "p_h", "30f0e44836fda72687afcf57a8673b24": "\\mu + \\frac{\\delta \\beta K_{2}(\\delta \\gamma)}{\\gamma K_1(\\delta\\gamma)}", "30f10c6b4a126b1ffc5b861f9152575f": "V(y) = \\sum\\limits_n\\left|y_{n+1}-y_n \\right|.", "30f1100cee2561134b06fffd06a7f98e": " \\hat{H}_\\mathrm{nuc} = -\\frac{\\hbar^2}{2}\\sum_{i=1}^N\n\\sum_{\\alpha=1}^3 \\frac{1}{M_i} \\frac{\\partial^2}{\\partial R_{i\\alpha}^2} +V(\\mathbf{R}_1,\\ldots,\\mathbf{R}_N) ", "30f12ab79cd5cea36783a5f029209645": "p_{y}", "30f1321d3116b1ec9838e95df57a94ac": "\\scriptstyle{\\vec{d}_{i,j} = \\vec{d}_{j,i}^*}", "30f1553eb0cffb9ef997a111cb1df61b": "\\mathrm{d}\\rho_J(t)=-i[H_\\mathrm{sys},\\rho_J(t)]\\mathrm{d}t+\\mathrm{d}t\\mathcal{D}[c]\\rho_J(t)+\\mathrm{d}W(t)\\mathcal{H}[c]\\rho_J(t)\\,,", "30f15e2a21c934357c480efbda6af54c": "a_{n,r}= \\frac{\\Gamma(n+1)}{\\Gamma(n-2r+2)}.", "30f1d01a9d7fb0cb7cd00884f5305227": " \\begin{align} d &= \\mathrm{Tr}(I^2) \\\\\n&= \\displaystyle \\frac{1}{d^2} \\sum_{\\alpha,\\beta} \\mathrm{Tr}(\\Pi_\\alpha \\Pi_\\beta) \\\\\n&= \\displaystyle \\frac{1}{d^2} \\left( d^2 + \\mu^2 d^2 (d^2-1) \\right) \\end{align} ", "30f278eec3725b87d46c2fe6f73b5dd7": " \\frac{K(n+1,(a_0,\\ldots,a_n))}{K(n,(a_1,\\ldots,a_n))} . ", "30f27c44db3efa6860d9ca7fcddb7550": "1, -1, 2, -2, \\frac{1}{3}, -\\frac{1}{3}, \\frac{2}{3}, -\\frac{2}{3}\\,.", "30f2806234f71f0aefb2fc07cb6afce0": " T_a: L^2(R)\\rightarrow L^2(R), (T_af)(x)=f(x-a)", "30f28b9a1d65e7f2bfaf6cce9266507b": "\\textbf{u}(t)=-\\textbf{K}(t)\\textbf{x}(t)", "30f2a29981d82f1ff16d330115f401b3": "Q(p;\\lambda) = \\frac{-\\ln(1-p)}{\\lambda}, \\!", "30f2a5ad6611a557f840ce66b3504b0c": "\\iint_S \\nabla \\times \\mathbf{F} \\cdot \\mathbf{\\hat n} \\, dS = \\iint_D \\left(\\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y}\\right) \\, dA. ", "30f2dcd83b4c30152cc3ed2bc3c0c9fa": "\\ \\vec{V} = \\mu_{obs} \\vec{E} \\qquad(6)", "30f304f1a6c09b99db39af5b46d6d368": "\\hat{X_a} = \\sum_{c \\in \\mathbb{F}_d} |c + a \\rangle \\langle c| ", "30f305ad7705fa21a8ac968f847c14a1": "\n\\widehat{\\mathbf{v}}=\\frac{i}{\\hbar }(H_{\\alpha }\\widehat{\\mathbf{r}}\\mathbf{\n-}\\widehat{\\mathbf{r}}H_{\\alpha }),\n", "30f3259ec9d2b80c76e0cd99349e4886": "a;b\\,\\!", "30f3299d576c2633d90891e2675b1cda": "\\left[u(x)v(x)\\right]_a^b = \\int_a^b u'(x)v(x)\\,dx + \\int_a^b u(x)v'(x)\\,dx. ", "30f34b4c2112ee7ad8c2f71e6f393b5c": "\n \\sum_{i,j} B_{ij} C_{ij} = \\sum_{i,j} B_{ji} C_{ji} = - \\sum_{i,j} B_{ij} C_{ij} = 0.\n", "30f36833b3ee0f1209497204b5928d2e": "(f, p)", "30f3cf2ef0ee99a467f037c7612c8cfb": "\\rho_T", "30f44decfe3a48e02d465396b10d91d0": "A.wwxy = \\begin{bmatrix} 0&0&0&1 \\\\ 0&0&0&1 \\\\ 1&0&0&0 \\\\ 0&1&0&0 \\end{bmatrix}\\begin{bmatrix} 1\\\\ 2\\\\ 3\\\\ 4\\end{bmatrix} = \\begin{bmatrix} 4\\\\ 4\\\\ 1\\\\ 2\\end{bmatrix}", "30f4691683f23f7822add85bd5ea3fcb": "\\dot H^{s/2}", "30f472e83f0dfaef84a81d8808751ab9": "N\\sigma\\ << 1", "30f4d030907c1b7bf5aafe3b12e4de3c": "X \\subseteq On", "30f558be5fcc6980ccef601f5ac5a956": "E_i=\\{2^{2^i}\\}", "30f57c0d80ddfaa3b1c2a1b5d5857b28": "\\scriptstyle \\mathbf{S}^1", "30f5ba95dc8089538d8e3644768b896b": "\\ \\partial :C_i((X))\\to C_{i-1}((X)) ", "30f5ce3d7363dd5a5ec82c34813220a5": "\\rho_P \\propto \\frac{1}{r^2}.", "30f5d346a59dcbd6f8bb63768ba8b770": "2\\mu", "30f62e6f74a3a0567de9bfe9ca91e5a8": "f:(X,x) \\to (Y,y).", "30f6607e3af225f09116b5e816c58594": "\\|y\\|\\le 1,", "30f6b2b35c338ee2711fb9b34ad4bc2d": "\\frac{\\Gamma \\vdash a : X \\circ Y \\qquad \\Delta, b : X, c : Y, \\Delta' \\vdash d : Z}{\\Delta, \\Gamma, \\Delta' \\vdash d[b := a, c := a] : Z}[\\circ E]", "30f6d35cc1f79e1cb95b9a781bdea8c1": "C_k = 2^j + C_{k - 1} + 1\\,", "30f6dfa7e9727a142b3c6684595d0f4c": "X\\subseteq \\mathbb N^n", "30f767aa191cd5d261e767fd78393607": "kA", "30f7714d3469087b4238f1145670d34c": " h\\equiv k\\pmod {\\varphi(p^{a+1})}", "30f79f767e706c566e8ae4d53419396b": "\\overline{v\\otimes z} = v\\otimes\\bar z", "30f79ff25952f8139539223dfe30013b": "{B_g}^2", "30f8b4451a5f8b313ef4e541ebae9009": "N\\subset M", "30f908cd992a9f2a581535081768ddba": "\\vec{v}_\\mathrm{A}", "30f9401b4ed670e94ea0616d1eb40554": "^{+6}_{}", "30f9ccc690bb6e15f08fc4260659cafa": "(\\forall R.C)^{\\mathcal{I}} = \\{x \\in \\Delta^{\\mathcal{I}} | \\texttt{for} \\; \\texttt{every} \\; y, (x,y) \\in R^{\\mathcal{I}} \\; \\texttt{implies} \\; y \\in C^{\\mathcal{I}} \\} ", "30f9ce8f35f48de3f7eb1943d01c8db0": "q_0 = -15, q_1 = 8, q_2 = -1, e_0 = -3", "30fa0fc20712e321a28e4f01026aac14": "\\left.\\right. A^2_\\alpha ", "30fa37fe8e841dd5add7b998e0eb83c5": " {d\\over dx} W(f,g) =0, ", "30fa997f7b3d43ccdf1fd08776dec047": "\\lim_{\\delta \\to 0} \\limsup_{n \\to \\infty} \\mu_{n} \\{ f \\in C | \\omega_{f} (\\delta) \\geq \\varepsilon \\} = 0", "30fae9b7853616fc64981da45dfeb9a8": "\\scriptstyle{\\tau_{int} = 2\\pi\\hbar/E_0}", "30fb1a967ac9c0e9292822195bd1d13c": "p = p(\\rho)", "30fb317fcf2095f3619030cf9063f8f2": "R \\!", "30fb53308b70db0b16554a5a52a05275": "g \\mapsto (u_{ij}(g))_{i,j}", "30fb5e9c6028dc0ba68943917c43cef8": "\\left(\\int_{\\mathbb R} |g(y)|^q \\,dy\\right)^{1/q} \\le \\left(p^{1/p}/q^{1/q}\\right)^{1/2} \\left(\\int_{\\mathbb R} |f(x)|^p \\,dx\\right)^{1/p}", "30fc017998d04f18cb22ff535836bb92": " 1/ \\lambda = {1 \\over\\sum_{i=1}^R p_i^2} = {}^2D", "30fca94faf1b16c1ecb2f17c937c59d3": "S(K,m) = \\left\\{ \\mathbf{k} \\in \\mathbb{Z}^m \\text{ such that } \\sum_{i=1}^m k_i = K \\text{ and } k_i \\geq 0\\quad \\forall i\\right\\}.", "30fd24313349d9b13dca6bf497f53d39": "(V\\otimes W)_1 = (V_0\\otimes W_1)\\oplus(V_1\\otimes W_0).", "30fd7e7173154c80f81ae16f02fd735a": "3{2p \\choose p} \\equiv 6 \\pmod{p^3}.", "30fd8e3ce7c998323e9e656a00aea4ff": "\\phi(k_i)", "30fda2d21f19d27df866ebce3b5b2b85": "\\frac{k_x}{k} = \\sin \\theta \\cong \\theta", "30fde1c07ff44ce042ef873f6a5ff7d8": "\\zeta(8) = 1 + \\frac{1}{2^8} + \\frac{1}{3^8} + \\cdots = \\frac{\\pi^8}{9450} = 1.00407... \\dots\\!", "30fde338496f0f6bc82d837ff80a4ca0": "\\int_{y=0}^1\\left(\\int_{x=0}^1\\frac{x^2-y^2}{(x^2+y^2)^2}\\,\\text{d}x\\right)\\,\\text{d}y=-\\frac{\\pi}{4}", "30fdef27a0f8fde2a1df2511dfa6371f": "kT", "30fe16536656287735c1431564ddfcd6": "\\frac{1}{S_1} + \\frac{1}{S_2} = \\frac{1}{f} ", "30fe247a7a21b007ed7001c1211d193f": " X, V ", "30fe79c4b9fb7edcf77193fb47dc736f": "A + B \\rightarrow A' + B'", "30fe851af53efd9a12d4a21c7110d3e9": "M+nH_{2}O\\rightarrow MO_{n}+2nH^{+}+2ne^{-}", "30fe91ae44a7c0af1c159e7db254efd2": "W = \\mathbf{F} \\cdot \\Delta \\mathbf{r} \\, .", "30ff156bda6810cf7a339914988c1623": "Q = c^i \\left(L_i-\\frac 12 {{f_{i}}^j}_k b_j c^k\\right)", "30ffbbd8b5a23231ea19444e31806296": " \\Box A^{a} - {A^{b;a}}_{ b} = - \\mu_0 J^{ a } ", "30ffcd7f54341025ca2cea3f9058ac70": " \\chi(0) = \\frac{1}{12}\\left(c_1(X)^2 + c_2(X)\\right)", "3100054b304de93a4438e8152467acd5": "\\Phi(H)", "310024a0af75cf9366ea0e1311725379": "X_t^{\\tau_{k}} := X_{\\min \\{ t, \\tau_k \\}}", "31003f447b3ed828faf385d3e7ef8b6c": " P [(N(b) - N(a)) = k] = \\frac{e^{-\\lambda_{a,b}} (\\lambda_{a,b})^k}{k!} \\qquad k= 0,1,\\ldots.", "3100444f41c50d93532ad6bb249ac72f": "\\frac{\\partial U}{\\partial p}\\ ", "3100b00e6204036e1907d22c832f1c52": " S := \\left\\{ \\sum_{i=0}^k \\lambda_i\\bold{y}_i : \\lambda_i \\ge 0 \\ \\text{and} \\ \\sum_{j=0}^k \\lambda_j = 1 \\right\\} .", "3100d1429487cd38ab8b7ed25d08e4e0": " \\frac{d\\omega}{dt}\\propto \\frac{2\\pi}{\\hbar}|\\langle \\psi_{f}|\\hat{H}'|\\psi_{i} \\rangle |^2 \\delta (E_{f}-E_{i}-h\\nu) ", "3101cf7315c185465307a8b4f82380cb": "Q_{out}", "3101d3a4c9bb616624b0ee3a12094632": "\\frac1{32}\\int_0^1 x^{12}(1-x)^{12}\\,dx=\\frac1{2\\,163\\,324\\,800},", "3101d55bbff433433335f23ed0ff9ee1": "h(x_1^n)", "310229c36abfdc7158c6539e9483a3e7": "s_P^2, s_N^2", "31022c3b75c63a238e5243a64061d05c": "\\kappa_i = \\kappa_{\\mathit{ri}}\\kappa", "31023e0bea43e76e141df621337ef533": "y^{(k)}(t_0)=y_k", "310249cbaabf1169d59639deb6819b58": "\\{A_i \\rightarrow B_i : i\\in\\{1,\\dots,n\\}\\} \\cup \\{B_i \\rightarrow A_i : i\\in\\{1,\\dots,n\\}\\}", "31029bab60df500404b920f4c1c1470a": "2^{226}", "3103a257a1b38e5dad3d34ed7f8884ae": "(M_2,d_2)", "3103b4a339eca1268c20ccb613173660": "n=1018", "3103e9625413351a51e40fd21ac98af9": "S_N= \\sum_{n = 1}^N X_n,", "3103f2d6de00249b24c165088b7f7caf": "\\mathit \\Gamma\\,\\!", "3103fffddf0c9ee708c260320ffdab3f": "F(\\mathbf{q}) = \\left|F(\\mathbf{q}) \\right|\\mathrm{e}^{\\mathrm{i}\\phi(\\mathbf{q})}", "31042c24d2847ad6ec4375c8f208d61a": "\\check{H}^*(\\mathcal U,\\mathcal F) \\to \\check{H}^*(\\mathcal V,\\mathcal F).", "310487358a2803a0a38dd7c770ec0164": "\\mu_2^{'}= 2\\sigma^2+\\nu^2\\,", "3104bcc503b3b8c16ffa2940b56aaf1c": "e_1", "3104c9df72dc9f06fed0ebbe9d0ad5cc": "\\sqrt{\\frac{0.25}{BAF}}", "310523fe5e745ba9037a7bb3fe5f1f18": "-\\frac{d[A]}{dt} = k", "31052b0eeee0cdaecc035210be7a1af4": "\\ell_{(N,\\psi)}(\\bar x,\\bar y)>\\ell_{(M,\\varphi)}(\\tilde x,\\tilde y)", "31054130f4c7873bd63967e82aa95584": "j = 1,...,5", "3105bf74e1ef21a8be7f9879c773203d": "(-\\infty,\\infty)\\,", "3105cf960ac5bfd66424f1a0a5c890c7": "\\scriptstyle L^s_{\\mathbf\\xi}(\\mathbf{x})", "31060a38bb2ed1cf9174dedc50aaef3c": "g_i \\ (i = 1, \\ldots,m)", "310621fc837aca3e8989c029eac691ac": "41+29\\sqrt{2}=82.01219\\ldots", "3106507f2aab72a23d1a733715d5d32a": "\\mathbf{x} = \\sum_{i=1}^K \\mathbf{w}_i s_i,", "3106743dc07b62e58bd5c1f71866ee6b": "J_2=\\begin{pmatrix}\n1 & 1 \\\\\n1 & 1 \n\\end{pmatrix};\\quad\nJ_3=\\begin{pmatrix}\n1 & 1 & 1 \\\\\n1 & 1 & 1 \\\\\n1 & 1 & 1\n\\end{pmatrix};\\quad\nJ_{2,5}=\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 1 & 1 & 1 \n\\end{pmatrix}.\\quad", "3106853a82bd9a00d7a08c8ac6a61f29": "\\pm 6.289835988\\ldots", "310699529871a381df21573adbc5d1eb": "\\phi\\colon [-\\tau,0]\\rightarrow \\mathbb{R}^n", "31069cc885fc8d3a68d81256eb86e338": "\\theta=\\delta-90^\\circ", "310723828816eb7c084f241295e1a9ec": "\n\\left(\\frac{\\lambda}{2\\pi}\\right)^3 , \n", "310745bad9828080890c0186cb989f5b": "\n L_\\epsilon u_\\epsilon = - \\phi^\\epsilon(x), x = (x_1, x_2, \\dots , x_n) \\in \\Omega\n", "310752b67d72cdd42a6a5065104b54a8": "{ad} = {cb}", "31076b4bebd8eea3e3d9cef5749bf1b8": "\\textstyle D' = (11001,6)", "31078709de6c82e4149e5b780bd8c04c": "l_2z=0,\\ldots", "31079163095f689e8c6cd259b00d9ac2": "y_\\text{crossover}", "3107a66b0ce0b20ba0330df998604150": "\n\\overset{\\{0\\} }{\\bigcup}X_{i} =\\bigcup X_i\n", "3108044f3f275001ed92a40c20baee1e": "(3) \\qquad \\frac{\\partial\\eta}{\\partial t}\\, =\\, \\frac{\\partial\\Phi}{\\partial z} \\quad \\text{ at } z\\, =\\, \\eta(x,t).", "310824f58104e017737ddc5e478e9ff5": " | p | ", "31084c1c18ab07f4e7dd7ec2c8b830d5": "\\lambda = \\exp \\left ( \\left |-\\alpha^{(0)} \\right | \\right )\\sum_{s=0}^\\infty \\left [ \\left ( \\pi^2 / 6 \\right ) \\exp \\left ( \\left |-\\alpha^{(0)} \\right | \\right ) \\right ]^s \\approx \\exp \\left ( \\left |-\\alpha^{(0)} \\right | \\right ).", "310860960ca3a71243d151f6f594060d": "L_R \\equiv \\frac{(gD)^{1/2}}{f_0}", "3108ae37c8320e406edc080e263e08af": "(x,z)\\in S\\circ R", "3108de0fd4e1c113f0190553adea3dd0": "G(k) = \\int d\\tau e^{- (k^2 - t)\\tau} = {1\\over k^2 - t}", "3108f24f84e524aa98eababdd6b9243f": "\n\\begin{align}\n &\\frac{1}{T}\\int_0^{T}\\left(e^{j2\\pi k_1t/T}\\right)^*\n\\left(e^{j2\\pi k_2t/T}\\right)dt \\\\\n = &\\frac{1}{T}\\int_0^{T}e^{j2\\pi (k_2-k_1)t/T}dt = \\delta_{k_1k_2}\n\\end{align}\n", "3108f442a33f7d4d414bfd3bbbe4a7af": "s_i\\in S_i", "31092a425f30dff20e7ba6c6b16712d5": "w = \\frac{p}{\\rho} = \\frac{\\rho_mC^2}{\\rho_mc^2} = \\frac{C^2}{c^2}\\approx 0", "3109324b20178e6234e60c30d0a973a7": "|\\underbrace{x+\\cdots+x}_{n\\text{ terms}}| > 1. \\, ", "310958bb830b1ab19364356db85c8c5b": " 1 = k(Cq_1 + D) \\, ", "310a08953d998eb0c1cc38628101cea0": "w^\\dagger=w_k^\\dagger w_{k-1}^\\dagger \\cdots w_{2}^\\dagger w_{1}^\\dagger.", "310a16a6c29796545f9b5d4a5c97e952": "\\lim_{k \\to \\infty}A^k=\\lim_{k \\to \\infty}VJ^kV^{-1}=V(\\lim_{k \\to \\infty}J^k)V^{-1}=0", "310a9404ed6a3fd9c1fd80740067ac2f": "\\dot{\\phi}", "310a9c5ef81b93cbb3e853550b91a669": "\\frac{|A(x)|}{|R|} \\le \\frac{1}{2^{|k|}} \\implies \\neg \\exists t_1,t_2,\\dots,t_{r}", "310ab7986264220ad225b578a2b577eb": "\\left (\\frac{x}{1+x^2} \\right ) \\varphi(x) < Q(x) < \\frac{\\varphi(x)}{x}, \\qquad x>0,", "310ad130edc20fe3470bcdc1703513e3": "\n-m < a_0 \\leq m \\mbox{ and } -m \\leq a_1 \\leq m \\mbox{ s.t. } a = a_0 + 2ma_1.\n", "310ad65355b55fa0037905bba079f9e1": "\\alpha = \\phi - \\beta", "310ae8dc1a3defe310d45e5b50dfdf91": "T_2 =L/(c+v)", "310af7b007cfebe5f709ac40220cd34f": "\\{H[k]\\}", "310b0108727569abe327d98c65f1b2e7": "(r\\times r)", "310bb467153106b8d5ae554a0a959d7d": "\\ D", "310bde351e68a887b88e0a119afb96b9": "\\sum_{n=0}^{\\infty}(ax)^n={1\\over1-ax}\\,.", "310bded5931f5c78cfb7b708b54b8fee": "xF(u)y \\text{ iff } xF(u')y", "310c5cb9a5569ce0037379d7fa98ec9b": "\\phi \\leq_{x} \\psi\\,", "310d17738769cf89815e88c8726eb6fe": " \\phi_{ij}\\circ\\phi_j=\\phi_i,", "310d309a6fb4daa06c5cab3053effb68": "c_p", "310d57d068d00f818d4ee34173b8b537": " \\mathbf{F} = -\\nabla P ", "310d9f39f3eeb242f601dc52ceb1beca": "\n \\begin{bmatrix}\\varepsilon_{\\rm xx} \\\\ \\varepsilon_{\\rm yy} \\\\ 2\\varepsilon_{\\rm xy} \\end{bmatrix} = \n \\begin{bmatrix} \\frac{1}{E_{\\rm x}} & -\\frac{\\nu_{\\rm yx}}{E_{\\rm y}} & 0 \\\\\n -\\frac{\\nu_{\\rm xy}}{E_{\\rm x}} & \\frac{1}{E_{\\rm y}} & 0 \\\\\n 0 & 0 & \\frac{1}{G_{\\rm xy}} \\end{bmatrix}\n \\begin{bmatrix}\\sigma_{\\rm xx} \\\\ \\sigma_{\\rm yy} \\\\ \\sigma_{\\rm xy} \\end{bmatrix} \\,.\n ", "310dd49f6bb7ecff4c4402da53f7aa4e": "H^x", "310e1da29e5c35af6585ceeb38c794c4": "\n\\psi (\\mathbf{r},t)=e^{-iEt/\\hbar }\\phi (\\mathbf{r}),\n", "310ee8f3deb2987f0225a2a1d561b127": "\\lambda _1", "310ef2cfb1f83bf22bd3c1fb61623f14": "C(x,y)=C_s(y-x).\\,", "310f580d34263a4e580dbef9b49fecf3": " h_1 , h_2 ", "310f626ef7b78f87cf3c9ef0b43cfe62": "\\scriptstyle \\Bbb{Z}_p\\,", "310f9e00e314a560c70da27d991c593c": "\\sum_{\\ell=0}^\\infty (1+\\ell^2)^s S_{ff}(\\ell) < \\infty,", "31109a826bb81690ac8108edc23148d3": "d=\\pm\\sqrt{\\frac{a\\pm R}{2}}\\text{ and }e=\\pm\\sqrt{\\frac{a\\pm R}{2c}}\\,", "3110c0a01f1ec692fd7b33637c089d26": "V_{in} +V_{GS}", "3110c409be85a7b83fb922e150e4518c": "L \\preceq M\\preceq N", "3110edd32646a95d736e9265ecd0f449": " | V_\\parallel(\\beta) | ", "311136fbc818e16d0ba814f01cb5e30e": "\\operatorname{E}([x=i]) = p_i", "31118e890ac439d21ced05a1cf808222": "SU(M+N)\\times SU(M)", "311241c1d472d2a55e53dda5082bf679": "F^1,\\,\\ldots,\\,F^n", "311252810d89819716cf4e6668e93df6": "A_{arbelos}=\\frac{\\pi-\\pi r^2-\\pi+2\\pi r-\\pi r^2}{8}", "31126cf7c91f3cfa7e43af4612aadc98": "p = p_0 \\times \\text{CR}^\\gamma", "31128222b41752c942cfb6aa731ca0ba": "\\overline{\\delta}_D(k,i) = \\overline{\\delta}_D(k-1,i)\\sqrt{(1 - \\overline{e}_b^2(k,i))(1 - \\overline{e}^2(k,i))} + \\overline{e}(k,i)\\overline{e}_b(k,i)", "3112e38c179a86ed68db14da673ef114": "N = Integer \\ Between \\pm \\left(\\frac {0.5 \\times Bandwidth}{PRF} \\right) ", "3112e9bbdc531f793d5231881c302702": "\\lim_{n\\to\\infty} T_n = T", "3113103b03b94f1bee4b66e996bb246b": " \\Delta\\sigma_{y} = {Gb\\sqrt{\\rho_\\perp}} ", "3113f3e18e3ad51db3a79a7977fc51ad": "x_i= A_i \\cdot B_i + \\overline{A}_i \\cdot \\overline{B}_i", "31141bf0eac62e31729b6b7e1ea2f1b0": " A_\\mu^a(x) = \\bar\\eta_{\\mu\\nu}^a \\Pi(x) \\partial_\\nu \\Pi^{-1}(x) \\quad\\text{with} \\quad \\Pi(x) = 1+\\frac{\\pi\\rho^2T}r \\frac{\\sinh(2\\pi rT)}{\\cosh(2\\pi rT)-\\cos(2\\pi rT)} \\ ,", "311426dab159f9f8a407ebdb6362177d": " \\operatorname{ var }( r ) = \\frac{ 1 }{ n ( n - 1 ) } \\sum_{ i \\ne j }^n ( r_i - r_J ) ", "31143feb2b4c300c50c8393c0f2f5ab3": "\\omega=\\{0,1,2,...\\}", "31144742932fbd5792dbe69b817daca9": "A\\Gamma", "31144b707dd4549314ee88be6eb4ce02": "\\begin{array}{lclll}\nA^+ &=& A^+ & A^{+*} & A^*\\\\\nA^+ &=& A^* & A^{+*} & A^+\\\\\nA &=& A^{+*}& A^* & A \\\\\nA &=& A & A^* & A^{+*}\\\\\nA^* &=& A^* & A & A^+\\\\\nA^* &=& A^+ & A & A^*\\\\\n\\end{array}", "311462360b34c91958f85affb3584ead": "\\hat{O}", "31149fd68e7c4891b32611ffd820ba1d": "\\ F_H ", "3114e113b5092e48d5d7b98bbfd95c8f": "T \\approx 16 \\frac{E}{U_0} \\left(1-\\frac{E}{U_0}\\right) \\exp\\left(-2 L \\sqrt{\\frac{2m}{\\hbar^2} (U_0-E)}\\right)", "3114f1112bde2839ac017ccf67c0a7ae": "A_n(s,t)=\\{(s_1,\\ldots,s_n)\\in I_{s,t}^n\\mid s_1s", "31388310e43721debb918dbaf6a4bb79": "\\sum_{u\\in V} \\frac{1}{d(u)+1} ~\\ge~\\frac{|V|}{D+1}.", "3138b8e6061fdf7cfaaa79279175cbcd": "LG", "3138fda4dcb20dd45f186ffcbd804178": "(\\mathbf{M}\\cdot[\\mathbf{A}])\\cdot \\mathbf{x} = \\mathbf{M}\\cdot[\\mathbf{b}]", "313938b045b8f23a0569cb34086c32fa": "\\displaystyle P_s=\\delta_{s^{-1}w_0} d.", "31398d0b3f17562b072aa89acebe2003": "\\vert\\upsilon(\\gamma)\\vert = 1", "3139f13e1ebcb125fa0a2154669b5aec": "\\scriptstyle\\pi(W_n)\\, =\\, n", "313a62fdc0389687c8d43501527cd1a6": "K_\\text{fw}(s) = \\frac12 \\nu \\text{e}^{-\\nu s}", "313aad4c9d4833bac04953e040492daa": "x_1\\wedge x_2\\wedge\\dots\\wedge x_k,\\quad x_i\\in V, i=1,2,\\dots, k.", "313b1cfc7133423cbad21bf00a31fb8c": "\n\t\\begin{matrix}\n\t\ta^2 + b^2 &=& c^2\\\\\n\t\t\\frac{ab}{2} &=& n.\n\t\\end{matrix}\n", "313b82b982efb86b97c860ac885d0de2": "x^{x^x} \\ge x.\\,", "313b9811b0862fec934dd5ed8476974a": "dT = 0.4\\frac{T}{P}dP", "313bb3d288bc3f00b0fe9271e9d1970b": "\n\\frac{1}{2m} \\left( \\frac{\\mathrm{d}S_{r}}{\\mathrm{d}r} \\right)^{2} + U_{r}(r) + \n\\frac{1}{2m r^{2}} \\left[ \\left( \\frac{\\mathrm{d}S_{\\theta}}{\\mathrm{d}\\theta} \\right)^{2} + 2m U_{\\theta}(\\theta) \\right] + \n\\frac{1}{2m r^{2}\\sin^{2}\\theta} \\left[ \\left( \\frac{\\mathrm{d}S_{\\phi}}{\\mathrm{d}\\phi} \\right)^{2} + 2m U_{\\phi}(\\phi) \\right] = E.\n", "313bbf486185a7f8fa62b9656c838e4d": "\n \\begin{bmatrix}\n \\mathbf{U}_1 & \\mathbf{U}_2\n \\end{bmatrix}\n \\begin{bmatrix}\n \\begin{bmatrix}\n \\mathbf{}D^\\frac{1}{2} & 0 \\\\\n 0 & 0\n \\end{bmatrix} \\\\\n 0\n \\end{bmatrix}\n \\begin{bmatrix}\n \\mathbf{V}_1 & \\mathbf{V}_2\n \\end{bmatrix}^* =\n \\begin{bmatrix}\n \\mathbf{U}_1 & \\mathbf{U}_2\n \\end{bmatrix}\n \\begin{bmatrix}\n \\mathbf{D}^\\frac{1}{2} \\mathbf{V}_1^* \\\\\n 0\n \\end{bmatrix} =\n \\mathbf{U}_1 \\mathbf{D}^\\frac{1}{2} V_1^* =\n \\mathbf{M}\n", "313cf3be6824b65496a0da74ed7a24a2": " i^{*}(\\mathcal{O}(1)) \\cong L.", "313cf838701341ae2b48e3175baf8327": "V_\\text{max}", "313cfc6d103a607e83aac2007d354026": "C = D N(d_+) F - D N(d_-) K", "313d86ccb4c8b3f5ee1392cda1251aee": "C \\leq 2^j \\leq n/C", "313d8d6e5b86a50f985edfdbb6749ad5": "\\textstyle p = {\\rm tr} \\left((A - qI)^2 / 6\\right)^{1/2}", "313da25d4aa45856627982e9152881fa": "D_{i+1} = \n\\left\\{\\begin{matrix} \nD_i\\ &\\mbox{if}\\ S_i\\ \\mbox{is not consistent with}\\ D_i \\\\\n\\langle D_i,h_1,h_2,\\dots,h_n \\rangle &\\mbox{if}\\ S_i\\ \\mbox{has a solution in}\\ H\\supseteq D_i\\ \\mbox{with}\\ x_j\\mapsto h_j\\ 1\\le j\\le n\n\\end{matrix}\\right.\n", "313e010a9d88a67cf1ae357dcfb4958b": "\\omega_{SP}=\\omega_P/\\sqrt{1+\\varepsilon_2}.", "313e8f6d09edda2b6e63872149909a9a": "\\chi(x) = \\chi_{\\infty} (x_{\\infty}) \\prod_p \\chi_p(x_p) \\rightarrow e^{-2 \\pi i x_{\\infty}} \\prod_p e^{2 \\pi i \\{x_p\\}_p}", "313eddfb5939a1fa71fd5b79065cc14e": "\\frac{6}{3}", "313f03bee79a30f1633d34b29c696304": "b_{i+128}=s_i+b_{i}+b_{i+26}+b_{i+56}+b_{i+91}+b_{i+96}+b_{i+3}b_{i+67}+b_{i+11}b_{i+13}+b_{i+17}b_{i+18}+b_{i+27}b_{i+59}+b_{i+40}b_{i+48}+b_{i+61}b_{i+65}+b_{i+68}b_{i+84}+b_{i+88}b_{i+92}b_{i+93}b_{i+95}+b_{i+22}b_{i+24}b_{i+25}+b_{i+70}b_{i+78}b_{i+82}", "313f5e5249d5991750a894ce1801e414": " \\mathbf{\\gamma_2}(\\phi(t)) = \\mathbf{\\gamma_1}(t) \\qquad (t \\in I_1)", "313f621f29ff04fe92c75c690af6172c": "\\frac{[x]_{1}}{[x]_{2}} = \\text{constant} = K_{N(x,12)}", "313fbc2c72f16067134afe154298dd61": "\n \\mathbf{V} = \\mathbf{R}~\\mathbf{U}~\\mathbf{R}^T =\n \\sum_{i=1}^3 \\lambda_i~\\mathbf{R}~(\\mathbf{N}_i\\otimes\\mathbf{N}_i)~\\mathbf{R}^T =\n \\sum_{i=1}^3 \\lambda_i~(\\mathbf{R}~\\mathbf{N}_i)\\otimes(\\mathbf{R}~\\mathbf{N}_i)\n\\,\\!", "313fc3eedf6f8ab61175760d070436cc": "\\frac{\\partial L}{\\partial q} = 0", "313fed2cf93d2b1bc1a101202bdf9fd6": "\\sum_{i=1}^n (c_i \\cdot \\boldsymbol{H_i}) = \\boldsymbol{0}", "313ffcf0c9a1b31571f7d920ca9f0176": "\\tilde{d} =1", "31400a219aa3ac523aa7cf5aa347aff7": "j \\geq 0", "314048acb55f815916376bed2e97a124": "\\exp(\\sum_{d|m} x^d/d)=\\sum_{n\\ge 0} \\frac{a_{m,n}}{n!}x^n,", "31404db1b4e273436cf3f2b023df11ff": "\nQ_i (x,y) = \\begin{cases}\n\\left[ x,x+a \\right] \\times \\left[ y,y+a \\right] & \\mbox{ if }i = 1\\\\\n\\left[ x-a,x \\right] \\times \\left[ y,y+a \\right] & \\mbox{ if }i = 2\\\\\n\\left[ x-a,x \\right] \\times \\left[ y-a,y \\right] & \\mbox{ if }i = 3\\\\\n\\left[ x,x+a \\right] \\times \\left[ y-a,y \\right] & \\mbox{ if }i = 4\\\\\n\\end{cases}\n", "31404e198a63d4b75ce3b0a05493ec3b": " f \\mapsto f^*u ", "3140c318ceaf9a1e5ebd21cf29a6d5af": "\\sum_{i=1}^n \\frac{1}{i^k} = H^k_n", "3140ef997e23c49c5e93ca4ef2a39973": " \\vec{R} = \\vec{r} - L \\hat{e} _z ", "31412382991770f9336d869227e6631b": "2^{2^j}", "31416ffacd9756ec38c0388b617f814e": " p\\lor q=(p^{\\prime} \\land q^{\\prime})^{\\prime}. ", "3141f04c68c463d821a8123e5efcb887": "s_p\\left(t\\right)", "314223170efd831aceae75da50f66569": "\\psi_{n_x,n_y,n_z} = \\sqrt{\\frac{8}{L_x L_y L_z}} \\sin \\left( \\frac{n_x \\pi x}{L_x} \\right) \\sin \\left( \\frac{n_y \\pi y}{L_y} \\right) \\sin \\left( \\frac{n_z \\pi z}{L_z} \\right) \\quad (22)", "314226223666300bc4e340b57f1eb88d": "Y_{10}^{8}(\\theta,\\varphi)={1\\over 512}\\sqrt{255255\\over 2\\pi}\\cdot e^{8i\\varphi}\\cdot\\sin^{8}\\theta\\cdot(19\\cos^{2}\\theta-1)", "314232db1c628cd1e450eb6294a20e8f": "f(x, \\boldsymbol \\beta) = \\sum_{j=1}^{n} \\beta_j \\phi_j(x).", "3142f5f38a832323bc3acb1d5b488138": "m\\{x:\\, |Tf(x)| \\ge 2\\lambda\\} \\le m\\{x:\\, |Tg(x)| \\ge \\lambda\\} + m\\{x:\\, |Tb(x)| \\ge \\lambda\\}.", "314375d589789b95c04bedaa0f30c5e2": "\\Phi_{01}=\\overline{\\Phi_{10}}:= \\frac{1}{2}R_{ab}l^a m^b\\,\\hat{=}\\,\\frac{c}{2}\\,l_b m^b\\,\\hat{=}\\,0", "3143773d006494cb61c84d349ac0b66b": "\\operatorname{E}[g(X)] = \\int_{-\\infty}^\\infty g(x) \\, dF_X(x) ", "3143a7a99be1ce2565d949b10cb1805b": "c_i = l_A a_i", "3143f1506d05adc5668b5bba972ea33f": " + \\left( 2\\dot{r}\\dot{\\theta} + r\\ddot{\\theta} + r\\dot{\\zeta}^2\\cos\\theta\\sin\\theta + R_0 \\dot{\\zeta}^2 \\sin\\theta \\right) \\mathbf{e}_\\theta ", "3144bc6735538be2629ac4055b852c0f": " M^0_0 = \\sum_{i=1}^N e Z_i, ", "3144c2e12ad3d0dd44ce9117f9f73379": "\\hat{N} = m\\left(1 + k^{-1}\\right) - 1", "3144f546cd98f0ccc147d3fe08174a1f": "\\nabla _{\\vec Y} \\vec X = X^a{}_{;b}Y^b \\frac {\\partial} {\\partial x^a} = (X^a{}_{,b} + \\Gamma ^a _{bc}X^c)Y^b \\frac {\\partial} {\\partial x^a}", "31450c38fa85b0cf1bd3486d0429cfe7": "a \\cdot b^{-1}", "314537cbb08f89b9c21346ee9fcb3b3f": " \\Delta(x_{ij}) = \\textstyle\\sum_l x_{il} \\otimes x_{lj}, \\quad \\varepsilon(x_{ij}) = \\delta_{ij}\\quad\\ ", "314556932fefe8b9140850a5c475b338": "a_j\\in \\{0,1\\}", "3145873b212b207ae42316a6d3cdfa82": "T_\\text{hold}", "31459efc1532736f9a87097423158d14": "\\frac{dr}{dt}=(1+r)-(1+r)^2=-r-r^2", "3145bbfb67faef595b8c48837dc61023": "\\sigma_v = \\sqrt{3}|\\sigma_{12}|\\!", "3145caea8844c052ab9252b5084720f2": " \\mathbf{k}^e = \\int_{V^e} \\mathbf{B}^T \\mathbf{E} \\mathbf{B} \\, dV^e \\qquad \\mathrm{(11)}", "31463def79f37c6495ebd762831a0fb6": "g:=h^{\\frac{q-1}{2}}- 1 \\pmod f.", "314669d10218b2085283f3e46ebbcb9f": "\\omega(x)=1", "31466c2f727c9d33129f808a67b05429": "\\,^{238}_{92}\\mathrm{U} + \\,^{nat}_{28}\\mathrm{Ni} \\to \\,^{296,298,299,300,302}\\mathrm{Ubn} ^{*} \\to \\ \\mathit{fission}.", "314691117a1f548782c1fcaeb17b13c3": " \\|x\\|_p = \\bigg(\\sum_{i\\in\\mathbb N}|x_i|^p\\bigg)^{1/p} \\text{ resp. } \\|f\\|_{p,X} = \\bigg(\\int_X|f(x)|^p\\,\\mathrm dx\\bigg)^{1/p} ", "3146adbe7c160cc5119db716b25db1c7": "\n\\begin{align}\n\\hat g &= {{4\\pi ^2 L} \\over {T^2 }}\\alpha (\\theta ) \\\\ \\\\\n{{\\partial \\hat g} \\over {\\partial L}}\\,\\, &= \\,\\,\\,{{4\\,\\pi ^2 } \\over {T^2 }}\\alpha (\\theta )\\\\ \\\\\n{{\\partial \\hat g} \\over {\\partial T}}\\,\\, &= \\,\\,{{- 8\\,L\\,\\pi ^2 } \\over {T^3 }}\\alpha (\\theta )\\\\ \\\\\n{{\\partial \\hat g} \\over {\\partial \\theta }}\\,\\, &= \\,\\,{{L\\,\\pi ^2 } \\over {T^2 }}\\,\\,\\sqrt {\\alpha (\\theta )} \\,\\,\\sin (\\theta ) \\\\ \\\\\n{\\mathbf{\\,\\,\\,\\,Eq(10)}}\n\\end{align}\n", "3146d35c742147f8f5086e5f2f420708": "\\frac{d}{dz}\\langle B\\alpha^\\mu_z(A)\\rangle = i\\langle B\\alpha^\\mu_z\\left(\\left[H-\\mu N,A\\right]\\right)\\rangle", "3146e34fe9673c4303b68c188d786269": "i=1, 2,\\dots, m.", "31472dbdbe72498463f89227ff6e74b9": "\\sup_{j \\in \\mathbb{Z}} \\left( \\int_{-2^{j+1}}^{-2^j} |m'(\\xi)| \\, d\\xi + \\int_{2^j}^{2^{j+1}} |m'(\\xi)| \\, d\\xi \\right) < \\infty", "314745b9638eaba8ae73aff567727778": "\\phi(x+y) = \\phi(x)+\\phi(y)\\,", "3147899da0c50d43843b3e3a98e3ad1c": "R = \\frac{a}{\\cos \\theta}", "314789c0b289a9a04c51e90e0eac360b": "t=2,\\dots,T", "314791a68dde6c0780af2173cf413976": "\\alpha>2", "3147ae39444953115994289d5e29dfd4": "v_2 '", "3147fcd7e4f4e5e11e33dcbaf8230955": "V_T =\\frac{nT}{M}", "31483e9ea4623b96da33114ea5b1a4fe": "5 = 2^2 + 1.", "31485a93bd7ea8158c229a548862f04a": "\\delta = \\delta(\\epsilon) > 0", "31486f58b9990988a4e874da2badceae": "\\varphi_i", "3148f9df144c7dea5e651dd1a12eb184": " ds^2 = (1+m/r)^{-2} \\, (-dt^2 + dr^2) + r^2 \\, (d\\theta^2 + \\sin(\\theta)^2 \\, d\\phi^2 )", "31490737ba5a11065d06909354390f08": "z(t) = n_r (t) + \\sum_{n = -\\infty}^{\\infty} v[n] \\cdot g (t - n T_s)", "31490b6ccb6ce0771bc93f78ad4f1e14": "\\phi = V - \\frac{W}{e}", "3149296042af1848fa053179eb0e1ac7": "\\Delta^+\\ ", "314941387a65aa7a40ff33eb75f1d4c8": "x \\in (A \\cap B)^c", "3149776ec90b8975965a31126ce174de": "\n\\sum_{\\stackrel{1\\le k_1, k_2, \\dots, k_s\\le n}{ \\gcd(k_1,n)=1}} \\gcd(k_1-1,k_2,\\dots,k_s,n)\n=\\varphi(n)\\sigma_{s-1}(n).\n", "31498a7209df445c9ac12c7bc62997db": " 1^n", "3149a960cdc5752e7c4d4b800dc0b737": "\\operatorname{mfnchypg}(\\mathbf{x};n,\\mathbf{m}, \\boldsymbol{\\omega}) = \\operatorname{mfnchypg}(\\mathbf{x};n,\\mathbf{m}, r\\boldsymbol{\\omega})\\,\\,", "3149eb3900c599b6e89d4ccef659db8f": " f(1) = (1)^3 - (1) - 2 = -2 ", "314a0376e7fb15c0a7df246e2c871aa2": " \\sigma", "314a48a1a6797ecf4791a4a4a81f1fad": "I_{M}^2", "314a5a4df843d9af4eb3cf6c0bd883b5": "t \\in \\{0,1,...,T-1\\}: \\rho_t(X) = \\rho_t(-\\rho_{t+1}(X))", "314ab9e53b7933e50b8dfa2dc270f070": " \\vec{p}", "314adc1345446207904212bd8cd1b1ef": " E' = y' + \\frac{1}{2y'^2} ", "314b067ae0ef1c68d17959ebe560a8c9": "Z = i \\omega L\\!\\,", "314b0bcdf5d2876b928f2a2ceaa688de": "\nx\\rightarrow\\rho_{x}.\n", "314b464469056bb4af63d448af0c80de": " f(k; r, p) = \\frac{-p^k}{k\\ln(1-p)},\\qquad k\\in{\\mathbb N}.", "314b7547116ba73dd956a83c62fb2ea3": "v_i\\in\\{1,-1\\}", "314b8b21d70ec6b0f867040d5ff946b6": "J_c\\,", "314bd37d48aa31ee670ae3c5c94e4663": "\\trianglelefteq \\ntrianglelefteq \\trianglerighteq \\ntrianglerighteq \\!", "314c51d5b508d05991ee2ed1f2822d42": "V_n = V", "314c5c15229eab36894ea9034f1f332a": "\\operatorname{Pr}(X=x_k) = Cr^{x_k} \\quad\\mbox{ for } k=1,\\ldots, n", "314c6f8e819856f252ce34d9a485d35e": "\n \\int_{\\Omega} \\boldsymbol{F}\\cdot(\\boldsymbol{\\nabla}\\cdot\\boldsymbol{G})\\,{\\rm d}\\Omega = \\int_{\\Gamma} \\mathbf{n}\\cdot(\\boldsymbol{G}\\cdot\\boldsymbol{F}^T)\\,{\\rm d}\\Gamma - \\int_{\\Omega} (\\boldsymbol{\\nabla}\\boldsymbol{F}):\\boldsymbol{G}^T\\,{\\rm d}\\Omega \\,.\n ", "314c8a59a9bd9dea48a632b75b216120": "(M[f]_{x_0})^n = M_x[x - x_0]M[f]^nM_x[x + x_0]", "314ca0968f6053a516aa09a653d8df90": "A_D(t; y)", "314cd9946528e43f66e209edbcda8c7f": "\\cos x = \\sum^{\\infty}_{n=0} \\frac{(-1)^n}{(2n)!} x^{2n} = 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\cdots\\quad\\text{ for all } x\\!", "314ce419a0f51ac745c2008cb5bd28fc": "\\begin{align}\nx&=\\cos t\\\\\ny&=\\sin t,\n\\end{align}", "314d2ec8c7183d9cc59fef5139b84fe1": "A(c)=-\\frac{u''(c)}{u'(c)}", "314d82e8f1596e9e42dbfc31b7163cff": "\\omega(t) = \\frac{d\\theta}{dt}.", "314dd4bf8526aa7615162641e79b8148": "\\mathbb{R}\\times S^2", "314e06813a4145a34dcb9715e2c05c97": "n = 2\\;", "314e13d08981f777641805beb4868e32": "\\omega_{\\lambda} \\,,", "314e22e71cedf656d1c4df907304ac8b": "I^{n} \\cap N = I^{n - k} (I^{k} \\cap N).", "314f60d8bf0952ad6a0c56dc21474153": " \\operatorname{lift-choice}[(\\lambda x.f\\ (x\\ x))\\ (\\lambda y.f\\ (y\\ y))] ", "314f64cf38272a488079f31925f3580a": "\\ C=\\frac{P_tG^2\\lambda^2}{1024(\\log2)\\pi^2R^2}c\\tau\\eta", "314f9482f927df3ef2ad929ac70380e7": "\\left|\\frac{\\mathbf{L}}{m}\\right| = M = u r \\cos (\\phi) + \\Omega r^2 \\cos^2(\\phi) ", "314fa7f338754ff9270492a1bf521759": "\\epsilon_c", "314facdb2d47ccf1b200cd3d1e8637ec": "u(1)", "314fbb913bcfe0ce5caab3275f09d194": "x^{\\left [n \\right]} = x^{n}(\\log(x) - H_n)", "314fd2b727e1de5ebbfd897ae4f354ab": "\n A = \\pm \\cfrac{\\sigma_y}{\\sqrt{3}} ~;~~ B = \\mp \\cfrac{1}{\\sqrt{3}}~\\left(\\cfrac{\\rho}{5~\\rho_s}\\right)\n ", "31500f8b39d052f2236fa3cb53329dd6": "([K_0]+[\\delta K])(\\mathbf{x}_{0i} + \\delta \\mathbf{x}_{i}) = (\\lambda_{0i}+\\delta\\lambda_{i})([M_0]+[\\delta M])(\\mathbf{x}_{0i}+\\delta\\mathbf{x}_{i}),", "315018422d58b7f48048aea3bf07ce78": "I_D = I \\ ,", "3150864cd01e80e497359bc74a71e556": " -1 \\le s(i) \\le 1\n", "31508751d31ab041fe63f3e7895ab84b": "\\vec{I}", "3150bc3c6d06b1b58a55773fd1033f50": " \n\\pi (t,p_{-i})-\\pi (0,p_{-i})=\\int_{0}^{p_{i}}x_{i}^{\\ast }(s,p_{-i})ds, ", "3150eafe3deed3e440b397884ac63ede": "P(E)=\\sum_{x\\in E} f(x)\\,.", "315165e1af111d7a0b308741f7fc4e2f": "\n\\begin{align}\nP & = \\frac{(14 - r) \\times 4}{50} \\times \\frac{3}{49}\\\\\n& = \\frac{84 - 6r}{1225}.\\\\\n\\end{align}\n", "315172a5d1add95ecb8ffb289d0882b8": "\nh_\\text{eff}=\\begin{pmatrix} |\\vec p|&0\\\\\\\\0&|\\vec p|\\end{pmatrix} \n+\\frac{1}{2|\\vec p|}\\begin{pmatrix} (\\tilde m^2)&0\\\\\\\\0&(\\tilde m^2)^*\\end{pmatrix}\n+\\frac{1}{|\\vec p|}\\begin{pmatrix}\n\\widehat{a}_\\text{eff}-\\widehat{c}_\\text{eff} & \n-\\widehat{g}_\\text{eff}+\\widehat{H}_\\text{eff} \\\\\\\\ \n-\\widehat{g}_\\text{eff}^\\dagger+\\widehat{H}_\\text{eff}^\\dagger &\n-\\widehat{a}_\\text{eff}^T-\\widehat{c}_\\text{eff}^T\n\\end{pmatrix} ,\n", "31517ba475d50a10147b5c6a06551e32": "S \\subseteq \\{1, 2, \\ldots, n\\}", "3151de77f8a21edca21aa0bc47100f54": "\\|F_n*f \\|_{L^p([-\\pi, \\pi])} \\le \\|f\\|_{L^p([-\\pi, \\pi])}", "3151f84a0b25bdd84aae08941b095a0f": "w_2 = (w_2 \\sqrt{T_3}/P_3) * (P_3/P_2) * (\\sqrt{T_2/T_3}) * (P_2/\\sqrt{T_2}) \\,", "31527c773714a50fa2598305259d9dc1": "\\scriptstyle C=E[(x - f(x))^2]", "3152879fd068c33240b80458de6ace4e": "f(a) = \\int \\frac{-da}{a^2+1} = A - \\arctan a,", "3152a2ce3182bb918891d7d932994936": "A\\left(\\omega\\right)", "3152a3733e95f6f7b52cdba416c1974d": "\\forall x Pxy \\to Pty", "3152b9a9fcc7656fd7d09ac358061b23": "m_{Planet}", "3152d12a0d4d211cd050883a9b4eeb2c": "\n E_x=-\\frac{1}{j\\omega\\epsilon} k_x k_z \\cos k_x x \\sin k_y y \\sin k_z z\n ", "3153081a00d5e2adc23b526ae11ebad2": "\\,\\{x_p\\}_p", "31530f18f4c42e0039259855b47f5fdc": "\\ker \\chi_{\\rho} := \\left \\lbrace g \\in G \\mid \\chi_{\\rho}(g) = \\chi_{\\rho}(1) \\right \\rbrace, ", "3153335822e3b148d8c655f4564c3278": "\\psi(\\mathbf{r},t)", "3153a005d9a83b176da52bab0c247ef4": "\\scriptstyle S/x\\,", "31544c3ecce74bf142e2a2cef9577233": "f_s = \\frac{8}{7} B", "31549d7fe9a705e8e982cef2e7184ea2": "a_{n} = \\sum^{A}_{n} (U^{A}_{mn} - E\\delta_{mn})a_{n} = 0, m \\in A ", "3154b2d00425f6e2651a8b23a3bb7cb6": "A=\\frac{1}{4}(60+\\sqrt{10(190+49\\sqrt{5}+21\\sqrt{75+30\\sqrt{5}})})a^2\\approx33.5385...a^2", "3154b33bbb967bb5ec7b9906ee7a0a84": "\\textstyle\\hat{Z}_{3}", "31551bddec65cd42f01e520cb0667f2c": "\\Bigg(\\sum_{i=1}^k a_iC_i\\Bigg)A=\\sum_{i=1}^k a_iC_iA", "315524a5a16223f305d56b887b4ae600": "[B, -]:\\mathcal{C}\\to\\mathcal{C}", "31553052a41a5f96119545b76b2bd3d4": "P_A O_2 = \\frac{P_E O_2 - P_i O_2 \\frac{V_D}{V_t}}{1- \\frac{V_D}{V_t}}", "31556ff342d6a574bccbdcbff48c908a": "u(x,y,z,t)", "31558650a4b03be9854f2960f4b53cc5": "\\Gamma(\\alpha \\, , \\, \\beta) \\,\\,", "3155ac82cf6130c637a11ae72377307e": "\\text{A} \\mapsto 3, \\text{B} \\mapsto 2, \\text{C} \\mapsto 1,", "3155bb010c944139b177cce15d910906": "\\scriptstyle \\operatorname{aut}(M)", "3155d00798010b0df1b0564ff9e75140": "\\sigma_f^2 = A^2\\sigma_B^2 + B^2\\sigma_A^2 + \\sigma_A^2\\sigma_B^2 ", "315649c07fc9cf63d351163e3f527d2e": "\\forall i \\in \\omega", "315651a5e0d2aac2c9cdd46afbbf3cfa": "h[n]=Th_c(nT)\\,", "3156551826c43b8a713233f83394a8cb": "{\\boldsymbol S}(c)", "31574e46378a232f6c25d4b174183160": "\\left \\langle v\\right \\rangle", "31576f59419d16f1527beb380ce5f708": "\n \\begin{bmatrix}\n a_{11} & 0 & 0 \\\\\n a_{21} & a_{22} & 0 \\\\\n a_{31} & a_{32} & a_{33}\n \\end{bmatrix}\n ", "3157762bacddbf349b5b1a7bd0d4aaf9": "E(A,B)", "3157f9d96bc708977652f5c6c66ad87f": " Re^{-3/4} ", "31580802dec4347b27ae70e6afd4778e": "3 \\cdot b_n\\ \\mathrm{dB}", "31583d963264f617d978d96925d3aaf5": "X_{C}= K \\exp\\left(-\\frac{a}{\\alpha}\\right).", "31588284c5c88cff30fa2fc65806471c": "\\left(\\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "3158c559e6a1af9ce4f7b6632a79ae94": "\\hat{C}_{H_i} := T \\prod_{j=1}^{n_i} \\hat{P}_{i,j}(t_{i,j}) = \\hat{P}_{i,1}\\hat{P}_{i,2}\\cdots \\hat{P}_{i,n_i}", "3158e76bde97f851204519ad83026cd9": "S_2 \\wr S_6", "3158fcabbda1174894ae955e3c6f576f": "p(28995244292486005245947069k + 28995221336976431135321047) \\equiv 0 \\pmod{29}.", "31592ec4f0b781e68ce6169f66e6ffba": "v_T = t", "31594acce96a82c6fcb5cc2ead8fa06b": "\\alpha=\\,", "315988446647336058c55a7bfbf63c78": " \\mathcal{A}_2 ", "315a7ff435fc1ef2c1659576080fbf61": " y' = y, \\quad y(0) = 1. ", "315a886e888a6df0c687f0b9d6f5dbdd": "\\limsup_{x\\to x_0}f(x) =f(x_0)", "315abdc7a2c3a027e9fcf906e3bec055": "\\left(x, y\\right) \\in R", "315ac72770f7b5c73b8e4901c441a16b": "t\\begin{Bmatrix} q , p \\end{Bmatrix}", "315ada329611a87553e58d5b0ee6224f": " \\{ (x_j^{te}, y_j^{te}) \\}_{j=1}^m ", "315ae15b81f0ea49975a83b61cd3deee": "- RT \\frac{1}{n}\\nabla n=- RT \\nabla (\\ln(n/n^{\\rm eq})).", "315af77a18ec3c6c456e3a53c4a04f45": "\\operatorname{dom}f = \\{x \\in X: \\exists y \\in \\mathbb{R}: (x,y) \\in \\operatorname{epi}f\\}. \\,", "315b2c09d73293b5fa2cffb7133a8452": "\\color{Lavender}\\text{Lavender}", "315bb178d04858d286ccc961b62ada9b": "a_n = \\left (b_1\\lambda_1^n + b_2n\\lambda_1^n + b_3n^2\\lambda_1^n+\\cdots+b_r n^{r-1}\\lambda_1^n \\right )+\\cdots+ \\left (b_{d-q+1}\\lambda_{*}^n + \\cdots + b_{d}n^{q-1}\\lambda_{*}^n \\right )", "315bb94ac80a31c6265d86897022e18a": "n_{yy} = (1 + \\chi_{yy})^{1/2} = (\\varepsilon_{yy})^{1/2} .", "315c82a57164f0e70880d5170b94a4d4": "\\alpha>0", "315cce0bc0a011e86b47bda396e9ef4c": "\\Rightarrow C_V = T\\left(\\frac{\\partial S}{\\partial T}\\right)_V", "315cf51544bcf663e1b19cb16e75c13c": "\\sigma : \\Sigma^* \\to D", "315d19b22a3b6c348c844d3e4f1f3e37": "r(t) \\triangleq \\frac{1}{S_0(t)}\\frac{d}{dt}S^a_0(t), ", "315d80adafd10545c83d66b989c2c900": "\\omega_{I}", "315dcf547e3e38fe15a0a763f6318d15": "R = \\left\\{ z \\in \\mathbf{H}: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\frac{1}{2} \\right\\}", "315dd04b7a64ad63eb321272bfa4c0a9": " \\begin{align} W(p,q) W^\\dagger (p,q) &= S^p T^q T^{-q} S^{-p} \\\\\n &= Id \\end{align} ", "315e2f0bbd4ab319edbdd0e9eabf2eee": "C_V = {\\partial U\\over\\partial T} = {N^{\\prime}\\varepsilon^2\\over k T^2}{e^{\\varepsilon/kT}\\over \\left(e^{\\varepsilon/kT}-1\\right)^2}", "315e42775a013a4d6a1a1b47be713d06": "\\omega_2'", "315ec1dfd99556846c3c2a6ddf3843bc": "\\scriptstyle x_i \\,\\in_R\\, \\mathbb{Z}_q", "315ef8f6ccb15912fdf023e104d37d5d": "\\scriptstyle \\leq5\\times10^{-24}", "315f31198f7df9047b172e1627af40d6": "O(n) = \\bigotimes_{i = 1}^n O(1)", "315f3a9afe09b9883cbce0867347228b": "M_s\\,", "315fd60a25e3858814db98cec3f56c72": " {\\rm ZodiacalLight}/{\\rm S}_{10} = 140 - 90\\sin(|\\beta|)", "315fd7c4470e4771cfe9e1115ee6bcc0": "f_{n} \\to f", "315fea4b67319474a50f4a16ab959acc": "\\ O(n)", "315ff37558d1be249e8de15480afe60f": " \n w(x,y) = \\frac{a^2}{2\\pi^2 D}\\sum_{m=1}^\\infty \\frac{E_m}{m^2\\cosh\\alpha_m}\\,\n \\sin\\frac{m\\pi x}{a}\\, \\left(\\alpha_m \\tanh\\alpha_m \\cosh\\frac{m\\pi y}{a}\n - \\frac{m\\pi y}{a}\\sinh\\frac{m\\pi y}{a}\\right) \n", "31600c749d57f25ef544d962b3ff7278": "\nV(g) = \n\\begin{cases}\n l & \\text{if } g \\text{ is an input} \\\\\n l(V(g_1), \\dotsc, V(g_i)) & \\text{otherwise,}\n\\end{cases}\n", "31600fe5e0a5f994aa2489053105638d": " T_{00} = T_{11} = \\frac12 \\left( \\dot{X}^2 + X'^2 \\right) = 0 ", "31604ae6ed6d2b83c5336e8f4db3c279": "\\mathbf{q}^m = \\mathbf{f}^m \\mathbf{Q}^m + \\mathbf{q}^{om} \\qquad \\qquad \\qquad \\mathrm{(1)}", "3161976361b830118b3b73fd24f09ba1": "C\\ ", "316262a58948f8875e348b06caf0cdd1": "\\tau_s \\equiv \\ln \\left ( \\frac{\\Omega^{(s)}}{\\Omega^{(0)}} \\right ) = \\sum_{p=0}^{s-1} \\xi_p ", "31627603c1e93810b91a30329762a051": "\\ln R", "316293fa92d12a475d942688bce18979": "l \\le t", "316302e9569f8214fc0ae40a87b60931": "L(n) = \\sum_{k=1}^n \\lambda(k) \\leq 0 ", "3163374f2cd8435e752d741c258140ae": "\n \\operatorname{E}\\left[|X|^p\\right] =\n \\sigma^p\\,(p-1)!! \\cdot \\left.\\begin{cases}\n \\sqrt{\\frac{2}{\\pi}} & \\text{if }p\\text{ is odd} \\\\\n 1 & \\text{if }p\\text{ is even}\n \\end{cases}\\right\\}\n = \\sigma^p \\cdot \\frac{2^{\\frac{p}{2}}\\Gamma\\left(\\frac{p+1}{2}\\right)}{\\sqrt{\\pi}}\n ", "316337ae7b2b495afc977bd7d3d143b6": "\n1\\ {\\rm dB} = \\frac{1}{20 \\log_{10} e}\\ {\\rm Np} \\approx 0{.}115129254 \\ {\\rm Np}. \\,\n", "316367744d0424c40ec0585471e86ae4": "({\\mathbb P},\\sqsubseteq)", "31638839a35a449800d5f42923f31631": "A=\\pm e^{-C}", "3163c4123dfac4e3118643026efbbbbe": " \\mathbf{\\hat Q} ", "31645929703e0ca8727296992f4dc763": "ZW", "31647972e82311108bc942895e21f2cc": "\n\\begin{align}\nds\n&= \\frac\n{\\sqrt{b^2\\sin^2\\beta + c^2\\cos^2\\beta}\n \\sqrt{(b^2-c^2)\\cos^2\\beta - \\gamma}\\,d\\beta}\n{\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}}\\\\\n&\\quad {}+ \\frac\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega}\n \\sqrt{(a^2-b^2)\\sin^2\\omega + \\gamma}\\,d\\omega}\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}}.\n\\end{align}\n", "31648546ee736f5324bd89fa79433f51": "F_A^+", "3164d285ceb15f3d64bfb58269970575": " Q_y=\\int_0^b q_y dz = -k b\\frac{\\partial h}{\\partial y}", "316520c46def317f704c2f025b96c6f8": " L_z \\mid l, m \\rang = \\hbar m | l, m \\rang ", "31657ae25278a6f07f70a789332f654d": "\\varepsilon = \\frac {\\Delta L} {L_0}", "3165c9e3ad2f986ac67095d2fed0b8af": "\\mathrm{shoe~size ~({Paris~points}) = {\\frac{3}{2}}\\times{last~length}\\left({cm}\\right)}", "3165d143ee2cd1433e38a1ffc1653f31": "G m_\\text{P}^2=\\hbar c", "3166646256c50a312de319f862ab1d9d": "\\langle O(x) O(0)\\rangle", "316689b1edd88c439d8d9aab19228eb0": "{\\mathbf t}_n= y_n/q \\in [0,1)^k", "316691b71685e9009db3303693f7cf77": "m = \\frac f {s - f}\\,;", "3166ae66c3af6f528144fcf1c347ba82": "g(k; p) = p \\, (1-p)^{k-1}\\,", "3166c382054fb02f9a6abbed46af8d4d": "V_\\mathrm{P}^{(2)}=\\sqrt \\frac{K_\\mathrm{sat}^{(2)}+\\frac{4}{3} \\mu_\\mathrm{sat}^{(2)}}{\\rho^{(2)}} ", "3167513b86817b9e9052b902b4e1291e": "\\Gamma \\vdash A", "316788f6fad0f0bebddc97830e6401d4": "w\\,=\\,g_1g_2...g_n", "316799a17e084a5141faa77538e39aa2": "x\\cos(y)", "3167a0965dd245daa74ffc40bd13bbfc": "(\\alpha_1, \\alpha_2,\\ldots,\\alpha_k)", "3167d2f6a7b2bd3e3a134cf919c0be60": "Ax_i", "3167ef6c6e00affd47dedd153d8444b4": "R= \\int_0^\\infty I(\\lambda)\\,\\overline{r}(\\lambda)\\,d\\lambda", "3168335dc13a1830271ac33be6b6f278": "\n d\\boldsymbol{\\varepsilon}_p = d\\lambda\\,\\frac{\\partial f}{\\partial \\boldsymbol{\\sigma}}\n ", "3168489fa9f26de2f3ec9c5c005fba00": "M_{max}", "3168896dc328278a0854790c86bd8d01": "H = -\\ln G + \\text{constant}\\,", "316899044e7f90734cb8836450620ebb": "\\mathrm{primary}", "3168a2f0bd25e0593fd8b61ec901c5af": "p \\times p", "316982f225c05801e716ef4eee3d65b8": "a_r", "3169f9549d62a94a6528d5f8050fe1a7": "X,Y \\in L", "316a0d5fe0288af7383276b01764f582": "1+2+3+\\cdots = -\\frac{1}{12}\\ (\\Re)", "316a45a39d708ec6eecfca94e9419392": "\\eta^R", "316a896274df89405f656a639d410b3d": "R_n(x)=\\sum_{m=0}^{n} \\frac{(m!)^2}{(2m)!}{n+m-1 \\choose m}{n \\choose m}\\frac{(-4)^m}{(x+1)^m} ", "316aacab643d4e1b6d1dcae54d74b17c": " f(x) = f(x_1) + \\cfrac{x-x_1}{\\rho(x_1,x_2) + \\cfrac{x-x_2}{\\rho_2(x_1,x_2,x_3) - f(x_1) + \\cfrac{x-x_3}{\\rho_3(x_1,x_2,x_3,x_4) - \\rho(x_1,x_2) + \\cdots}}} ", "316b087d1a07825cb227a9bcc350d25b": "4\\times\\begin{pmatrix}6\\\\2\\end{pmatrix}", "316b6e3bfedb067fff7eac356f0a817e": "m^{\\text{th}}", "316bd6751e9457a96efbde8ca3a85db5": " u_1 =0.06236 ", "316bde49aa9bc7f60c189d55155480f4": " \\forall i, a_{ii} \\ne 0 \\,", "316be4b2fd51d1cb9a8edfa0216fb74c": "KP(x,y) \\leq KP(x) + KP(y|x^{\\ast}) + 657", "316c495d68f35e944a3b1ddfda3be9a8": "\\int x\\cosh ax\\,dx = \\frac{1}{a} x\\sinh ax - \\frac{1}{a^2}\\cosh ax+C\\,", "316ccdf8491a75f84f9b4d8e95ddeba1": "|g\\cap \\mathfrak o|=1", "316cd6e09399cbfc127654c38d7fa46a": "\\mathbf R^n", "316d3edd31559663347a8a527b203010": " \\Rightarrow m_1\\left(v_1-u_1\\right)\\left(v_1+u_1\\right)=m_2\\left(u_2-v_2\\right)\\left(u_2+v_2\\right)", "316d54adc51e49e14bf5d6601d3aaff6": "{\\it t} + d{\\it t}", "316d57faf29824eabbd503267b153bea": "ds^2=e^{\\nu(r)} c^2 dt^2 - (1-2GM(r)/rc^2)^{-1} dr^2 - r^2(d\\theta^2 + \\sin^2 \\theta d\\phi^2) \\;", "316e0788036616d4f67e4ac81cc66a13": "\\textstyle \\sum_e x_ed_e(x_e) \\leq \\frac{\\lambda}{1 - \\mu} \\sum_e x_e^* d_e(x_e^*)", "316e8570223f1dcc13544c7c1f5e6a68": "T\\subseteq X", "316f173a8cede6e2e495dd181f18f166": " \\widehat{\\boldsymbol{\\beta}}_k = V_k \\widehat{\\gamma}_k \\in \\mathbb{R}^p ", "316f65d17bd115193a39e54b8aa6203e": "X=Y", "316f99a39b73c02981f3fda29a8f0ba8": "\\psi_n(x,t) =\n\\begin{cases}\nA \\sin(k_n x)\\mathrm{e}^{-\\mathrm{i}\\omega_n t}, & 0 < x < L,\\\\\n0, & \\text{otherwise,}\n\\end{cases}\n", "316fcda137854b31fb571a8a53fa9d0e": "\\mathrm{res}(P',Q)=\\mathrm{res}(P,Q)", "31701e846e6ed7bf4453099c64dd78db": " \\mid \\phi \\rangle ", "3170312fb7ad6352e17c20192a785d84": "(N-1)", "31703e22881ddbb6863a21be9dd49c76": " a_0 v + a_1 T(v) + \\cdots + a_{d-1} T^{d-1} (v) + T^d (v) = 0", "317050ebed3ec854e284052485b94678": "(n-k)", "3170912bf9317ea19dae7cccd6baacc3": "m = 1, n = 4", "31709158b2c95bc8c6982a3a764824a2": "X \\sim \\operatorname{Logistic}(0,1)", "3170ed1a2d495f9ba66aade9b11f2c18": "non outs = {\\mathrm{unseen}\\,\\,\\mathrm{cards}} -\\mathrm{outs}", "3170fe13439bef850d6ac8c155b84998": "\\kappa\\geq\\mu", "31714644e4de3667b4659c73b36aa767": "O(nD^{-1/2}\\ln^{3/2}D)", "317151f8c49b10a688f6a02e159deb54": "r_{n-1}-c_n=r_n.\\,", "3171bbe6cf6aeb95c466f44a2ac3afbc": " \\{0, i, i-1, -i, i-1, -i...\\} \\,", "3171cc8e059e9f9689197ceb33aa9253": " f = \\left(A - \\frac{(B - A)^2}{C - 2B + A}\\right)^{-2}", "31721f1c603f480dcf810059fde9f0d7": "R_{ab}l^b\\,\\hat{=}\\,cl_a", "31723546f08e8a52dc31b951e45ce1f8": "\\underbrace{\\sum_{n=-\\infty}^{\\infty} s(t + nP)}_{S_P(t)} = \\sum_{k=-\\infty}^{\\infty} \\underbrace{\\frac{1}{P}\\cdot \\hat s\\left(\\frac{k}{P}\\right)}_{S[k]}\\ e^{i 2\\pi \\frac{k}{P} t }", "31724a83bd96657c3cc40084e65179e7": "\\mathbf{Y} =\\mathbf{X}\\mathbf{B} + \\mathbf{E},", "317281751af53193f7c0c7864340daed": "E = 0- {0.05916 V \\over 2} \\log {0.05\\over 2.0}= 0.0474{ } V\\,", "3172fd89e1f82dcd02ed5506596aa141": "{\\tilde{A}}_5", "317304efa54d1c7f1b7e4cd09cd185bb": "\\eta = \\frac{P_{out}}{P_{in}}. \\, ", "317365928e0fa1ebbd1a9e4e933ff441": "{C}_{4}^{(1)}", "317387f873f345092f458b812f25e823": " a^{\\varphi(n)} \\equiv 1\\mod n.\\,", "3173a05b61f8e62d8be34025fb169896": "\\beth_{\\alpha+1} = 2^{\\beth_{\\alpha}},", "3173c3fe1db6e0528187405ced252015": "\\begin{align}\nW_R &= 0.299 \\\\\nW_G &= 1 - W_R - W_B = 0.587 \\\\\nW_B &= 0.114 \\\\\nU_{Max} &= 0.436 \\\\\nV_{Max} &= 0.615\n\\end{align}", "3173e6d54bea898285f9f89cebb5a721": " C(x) = C_{L} \\ x^{L} + C_{L-1} \\ x^{L-1} + \\cdots + C_2 \\ x^2 + C_1 \\ x + 1 ", "31744e5b40989dbb07df6ff4cb1d60be": " W_\\emptyset = \\{1\\}", "31745cfabe1d4305c4d3b36d36862587": "F(d,a,n) = a^3+(a+d)^3+(a+2d)^3+\\cdots+(a+dn-d)^3", "31749c3d795f5a6d66f7096025477615": "\\scriptstyle \\nabla \\vec X", "3174b21373495b15be7ef15fa08b8ac7": " {R} = \\left \\{r_1,...r_N \\right \\} ", "3174c84361511a6cff8b242167a7029a": "z(r,\\theta) = (4/3)r^{3/2} \\cos(3\\theta/2).", "3174e62b274613f9fe60ae0e4286619d": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x\\right) & = x^{2} \\\\\n f_{2}\\left(x\\right) & = \\left(x-2\\right)^{2} \\\\\n\\end{cases}\n", "31751c2aff482c00fe6882ed1d5fb952": " u = 0 \\mbox{ on } \\Gamma \\times (0,T)", "31752bf7720f79d24bf89bef1e376850": "\nB' (\\cdots, a_{-2}, a_{-1}, {\\hat a_0}, a_1, a_2, \\cdots) = (\\cdots, - a_{-2}, {\\hat a_{-1}}, a_0, a_1, a_2, \\cdots).\n", "31756f13a0f57fbf4b639d69f63e9295": "40^2", "31758d439f21616a6b682ceb7b3f2c84": "H_z=0", "3175dd5673e8ae68aae78caeca2038c7": "dT", "3175f729107dec2f3c0f80654468c7ce": "\\mu : \\Sigma \\to M", "3176083241fd664a5da1450cf3b740e9": "P(P_3, k)", "317624937dc3e0efb60f2c8a85d9c440": "6) B \\rightarrow A : E_{KU_A}[S_{KR_{KDC}}[N_A || K || ID_A || ID_B ] || N_B]]", "31764d74512644e4b2ca16d35f3d7f72": " \\boldsymbol{x} = (x,y,z) ", "31766c4996aad6f12af71de8aa428f34": "f(t) = {1 \\over \\sqrt{2 \\pi}} \\int_{- \\infty}^{+ \\infty}{g( \\omega )e^{ i \\omega t } \\,d\\omega }. ", "3176ec5d847259a23e8227421c2999a6": "\\|f-f_0\\|\\le 2^{2k-N}\\,nk\\,100/98", "31774e36b55254a3109bb658cde99a4c": "\\alpha_3=4\\beta_1-2\\gamma-2-\\zeta", "317793b20f3dc496b7b51bbf003903cd": "\\ 1", "3177a2f0a521ef742277307703272c4f": "K(j \\omega)=\\frac{1}{1+j \\omega RC}=\\frac{1}{1+\\frac{j \\omega}{\\omega_0}},", "3177e3b1e868855bdba2fe1c0e682c9e": "Nc", "31782393322cb7cc104fb9759705d82a": "\\mathbf{P}(\\tau,\\mu | \\mathbf{X}) \\propto \\mathbf{L}(\\mathbf{X} | \\tau,\\mu) \\pi(\\tau,\\mu)", "317891a7cf94d4525b2b4797441d8116": "(A + B) \\cdot \\overline{A \\cdot B}", "3178ab4bbfc5c3aac094da1b92d5708f": "\n\\begin{align}\n&\\Delta \\bar{e}\\ =\\frac {J_3}{\\mu\\ p^3}\\ \\sin i\\ \\cdot \\\\ \n&\\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ 2 \\ \\sin u\\,\\ \\left(5\\sin^2 i \\ \\sin^2 u\\ -\\ 3\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right) \\ \\cos u\\right)du\n\\end{align}\n", "3178b56ae7dcccb49d939093920ec71c": "y^2 = x^3 + 3kc^2x + 2kc^3", "3178ea3f6f614a6dbb69ecaa0fc2f9fe": "x=(x_i\\in H_i|i\\in I) \\in \\prod_{i\\in I}H_i", "31791d10c5effa4a82f50314d889709e": "\\dot u_i=0", "317924785a841bd16ffbd63ca81535a7": "2^m -1", "31797c5b06528c02114872cc3e713edb": " \\mathbf{E}_\\mathrm{LH} \\ = \\ {\\mathbf{E}_\\mathrm{G} \\over \\sqrt{4\\pi}} ", "3179870ee978f1197d7468e4f5c8e858": "\\lambda_{\\mathrm{min}}(M)", "3179c1086458d06bd94e961cc50f7c1a": "n_x\\times n_y \\leq \\mathrm{N}^{0.6} ", "3179ca7c653b5ff532b7da92f29ae502": "a_{ij}(x)=a_{ji}(x)", "3179d9b0c15acb3a1882bab037ba7748": "\\mathcal{M}(z)", "317a0a87c3f527858fd3f2a192c82f45": " t = \\frac{5}{256} \\frac{c^5}{G^3} \\frac{r_0^4}{(m_1m_2)(m_1+m_2)}\\ ", "317a0ba61425d9ac59e3f8c67f67ed15": "*[F,*G]^{IJ} = - [F , G]^{IJ} = *[*F,G]^{IJ} ", "317a3b0259a3bd765036040ec7271f40": "G \\setminus K \\subset G", "317a89b58e5a21d84ea14af53f7e28e9": "c_{l,design}", "317ab54c6bfd266e8973d4ceb883fefc": "\\frac{|a|}{a^2+b^2}", "317af09855262828d573eb7409218c86": "\\left [\\begin{matrix} \\ell+m \\\\ \\ell + 1 \\end{matrix}\\right ]", "317b0173960732d42b154710aa37d790": " \\phi \\ge M ", "317b2f8a8ad67f1c2195c445434119e0": "q(\\lambda) = \\operatorname{perm}(\\lambda I_n - A)", "317b56abffeab17825f8aeb4cd6fc2fb": "\\Delta \\tau \\approx D L \\, \\Delta \\lambda", "317c288b056d42f497d2d72d2a0ea5df": " \\frac{d}{dx}\\left(c^{ax}\\right) = {c^{ax} \\ln c \\cdot a } ,\\qquad c > 0", "317ccf38153e6501adff079af594bf23": "\\mathbb R^6", "317cd5f95eeb7952200de41171543ca7": "2^{-j/2}", "317d3a5d68b9cb91947504087e79f837": "|g(\\xi)|\\le \\frac{M}{(y_0+\\lambda)^N}", "317d7a5ba07054a0a01b375f4daa6224": "\n| \\psi _{\\rho} \\rangle = \\sum_{i=1}^n (\\rho^{\\frac{1}{2}} | e_i \\rangle) \\otimes | e_i \\rangle \\in \\mathbb{C}^n \\otimes \\mathbb{C}^n \n", "317d8aa1fae33b822b4c7969ad16c6d6": "\\frac{5}{12}=\\frac{1}{4}+\\frac{1}{10}+\\frac{1}{15}=\\frac{1}{5}+\\frac{1}{6}+\\frac{1}{20}.", "317d8e302b5b0a18f69c181ebd07722c": "R_\\Delta = \\frac{R_P}{R_\\mathrm{opposite}}", "317de8921acd6a198d551e04f122fdfc": "P_0 = F(P_1)", "317dfd44679e9bffa1e449967fefa406": "\\hat{\\textbf{t}}_i^T = \\textbf{t}_i^T V_k^{-T} \\Sigma_k^{-1} = \\textbf{t}_i^T V_k \\Sigma_k^{-1}", "317e32fd7aedbc645fa4b7ab0ca2192d": "\\surd x", "317e51f5b781a5b2e776fcb14acd0f30": "\\Delta t = t_1 - t_2 = \\frac {4 \\pi R^2 \\omega }{c^2-R^2 \\omega^2} . ", "317eaac40225bc01a15ce7a705ec3183": "|\\dot{x}| + k + 1 > |\\dot{x}| + |a(t,x,\\dot{x})|", "317eb7b7118834800f9f8c810c711bae": "b\\in U", "317ef18dce6a1953b410a7875228bc6f": " \\Pi_4 = ", "317f36ef888cd47d9f8feecceda0b13a": "\\kappa^+ = 2^{\\kappa} \\,", "317fe076b9d7fa77aa614531e185c89c": "I_{AB}", "3180345eb4fce04bf4066a4eb51fc972": "R_1\\,", "31807e3aa3e0198af8c5c34138c94989": "G = K_P", "3180c3986aec302a50203fe81ac9d523": "\\ {A_q} = \\frac {L} {(L/A)(1-q)} = \\frac {AL} {L(1-q)} = \\frac {A} {1-q}.", "3181190a6b003bcde90eefc93fcdb7ec": "p=\\left\\lfloor\\frac{7 + \\sqrt{1 + 48g }}{2}\\right\\rfloor", "31811cc72a7a6ae35ae5df628e8e3d91": "N_\\text{min} = \\frac{k_1V}{Q_0k_f \\nu^2 P^{1/2}} (Eq. 2)", "318133f984a8b3d74b94dff73b25a8bf": "\\{ \\, 1, (111) \\}", "3181421e9ed8b5a2184af52ef46952b5": "T_d(X,Y) = -{\\log} _2 ( T_s(X,Y) ) ", "31816472553e45e9f1075e925e32c9ac": " P_\\mathrm {a,b} \\ ", "3181b33a32c500d01973777881f525f0": " h^i = \\bar{\\mathcal{M}}^i_j \\bar h^j ", "3181ba1545be4e44ebe30a44a2e2b42a": "\\psi : S^1 \\rightarrow S^1", "3181f427104bc90abc5dcd4c10713035": "G_{\\mu\\nu}^a = \\partial_\\mu A_\\nu^a - \\partial_\\nu A_\\mu^a + g f^{a}_{bc} A_\\mu^b A_\\nu^c. ", "318222bbe0488e8b63b5e8b63872d681": "R_{\\text{f}} \\| R_{\\text{in}} \\triangleq R_{\\text{f}} R_{\\text{in}} / (R_{\\text{f}} + R_{\\text{in}})", "318257bab41b0d6ffc493fca67cac337": "x_{0}=A^{-1}\\mathbf{b}", "31826b257c7b7995c45adcf851ada290": "e^\\alpha_I e^\\beta_J \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "3182824ede1f50b7c862f8daafb877bd": "\\zeta : X \\rightarrow \\mathbb{R} ", "318282dd740beaa72c460574b371df67": " \\mathbf{p} = m\\mathbf{v} = \\mathbf{P} - e \\mathbf{A} ", "3183335b2467d35d7df5377817646e98": "\\binom{p+q+3}{2}", "31833399df717720267b13532d454327": "D_\\mathrm{min} = D_\\mathrm{maj} - 2\\cdot\\frac58\\cdot H = D_\\mathrm{maj} - \\frac{ 5 {\\sqrt 3}}{8}\\cdot P \\approx D_\\mathrm{maj} - 1.082532 \\cdot P", "31835662ff7c0a84e748775b2105bf8c": "\nr = \\frac {h_c} {k_y c_s}\\,\n", "3183833af6ba0d9c1a74ec974f8a7759": " E = \\frac{d}{A} \\quad (9)", "31839f5f22738adca0c5b8d4919fd0e1": "\\begin{bmatrix} 1-\\lambda^2/2 & \\lambda & A\\lambda^3(\\rho-i\\eta) \\\\\n -\\lambda & 1-\\lambda^2/2 & A\\lambda^2 \\\\\n A\\lambda^3(1-\\rho-i\\eta) & -A\\lambda^2 & 1 \\end{bmatrix}. ", "3183d3bd1591b9255b8774ba06af5d65": "g = \\frac{1}{2}(d-1)(d-2) - \\sum_P \\delta_P,", "3183e7c23fd4f7463ec8a7d748729627": "\\lim_{x\\to c}{\\frac{f'(x)}{g'(x)}} = L.", "31840f3cc1edd726ca7d11c0f61450e8": "\\sum_{j=0}^\\infty c_j", "3184289192f5af64477da117e7700cdb": "\\frac{1}{1 + X} = 1 - X + X^2 - X^3 + X^4 - X^5 + \\cdots.", "3184cf682d320381b17ea556123edd1b": "\\sgn(z) = \\frac{z}{|z|} ", "3184d253c60a5b5b30808956d648c40c": "G_i G_{i+1} G_i=G_{i+1} G_i G_{i+1},", "3185042583ac2513c8033e131f6dcc5b": "\\Delta L = \\varepsilon L = \\frac{\\sigma}{E} L = \\frac{F}{A E} L = \\frac{F L}{A E}", "318522954fc860fb30f295ce3674ccdd": " d ", "3185516d4b570f4ce403d793eb858142": "1/\\sqrt{5}", "318588de4f41bdfaff8495d9c9ae4dd0": "\n+ \\Gamma^{\\lambda}_{\\alpha \\beta}[g^{hi}g_{hj},\\lambda - g^{hi}g_{mj}\\Gamma^{m}_{h\\lambda} -\\Gamma^{i}_{j\\lambda}]\n", "3185aae32d0370183e8a4fde63f751db": "u = -K x \\,", "3185d7ea9334d35c0fc57532f1a0d204": " \\mathbf {u \\times v} = -(\\mathbf u \\wedge \\mathbf v ) i \\,,\\quad \\mathbf u \\wedge \\mathbf v = (\\mathbf {u \\times v} ) i \\ . ", "3185f5712e0576184be1ca6b89762227": "\\hat\\varepsilon", "31860ca6fb862b6b7e9a8d819f9d829e": "\n\\begin{align}\nY_{\\ell m} &=\n\\begin{cases}\n\\displaystyle {i \\over \\sqrt{2}} \\left(Y_\\ell^{m} - (-1)^m\\, Y_\\ell^{-m}\\right) & \\text{if}\\ m<0\\\\\n\\displaystyle Y_\\ell^0 & \\text{if}\\ m=0\\\\\n\\displaystyle {1 \\over \\sqrt{2}} \\left(Y_\\ell^{-m} + (-1)^m\\, Y_\\ell^{m}\\right) & \\text{if}\\ m>0.\n\\end{cases}\\\\\n&=\n\\begin{cases}\n\\displaystyle {i \\over \\sqrt{2}} \\left(Y_\\ell^{-|m|} - (-1)^{m}\\, Y_\\ell^{|m|}\\right) & \\text{if}\\ m<0\\\\\n\\displaystyle Y_\\ell^0 & \\text{if}\\ m=0\\\\\n\\displaystyle {1 \\over \\sqrt{2}} \\left(Y_\\ell^{-|m|} + (-1)^{m}\\, Y_\\ell^{|m|}\\right) & \\text{if}\\ m>0.\n\\end{cases}\n\\end{align}\n", "318610995749a389e6111263e67f3547": " \\theta= \\alpha -(I-P')\\mu ", "3186c42bebf77f4a416b874dbf538838": "\\mathbf{Z_{01}}", "318723dbf7f20b3cecfcb481c655cf8c": "\n\\Bigl\\langle \\sum_{k} q_{k} \\frac{\\partial H}{\\partial q_{k}} \\Bigr\\rangle = \n\\Bigl\\langle \\sum_{k} p_{k} \\frac{\\partial H}{\\partial p_{k}} \\Bigr\\rangle = \n\\Bigl\\langle \\sum_{k} p_{k} \\frac{dq_{k}}{dt} \\Bigr\\rangle = -\\Bigl\\langle \\sum_{k} q_{k} \\frac{dp_{k}}{dt} \\Bigr\\rangle,\n", "318773441261df183ee20b6299c27894": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{sinc}\\left(\\frac{\\xi}{a}\\right)", "318783d1d021fee8e0163c01992387dc": "\\ P_n", "318799ac5fc0ed5f86902e202961c3e3": "{\\Bbb P}^{2n-1}{\\Bbb R}\\hookrightarrow {\\Bbb C}^{n,*}/{\\Bbb Z}_2", "3187ccead87cf277bf0f11a74d8ea6e4": "R_A^\\prime = R_A + K(S_A - E_A).", "3187ee81d17bf9a1e910f292374243c0": "(A, C)", "31884a3ed8d58014d998a35a64840a7f": "|G(\\chi)|=\\sqrt{N},", "31886bc20577846531373b6dece67a02": "n_{pas}", "31887e55477de748d453cf7f196e4489": "u, u' \\in U", "3188e527a58d60d72e1e638d18ff334a": "[{0.5}, {2}]", "3188ed8e9885db19a32efd3008de1f96": "A^2=I \\quad\\quad\\quad\\quad (1)", "31892bf973d005be7233c837432d001b": "\\scriptstyle f\\in\\mathcal{F}(U)", "318930f516dbeb658e279273151555c5": "\\scriptstyle -I_c", "318955d4580f56a695b9fefe6491b484": "-\\lambda^3+\\lambda^2(a+x-c) + \\lambda(ac-ax-1-z)+x-c+az =0\\,", "3189b637ad0da2ae3d2db6bf0b35e162": "A:\\; V \\mapsto V", "3189c479b7bc8852a94b09553cfdab87": "C_D^{(\\beta)}", "3189fb4a0e1eb3cf8ace71ab5ecd4866": "p(m)", "318a0b1fcf0fd8bdf7b88085f9acda62": "F= F_4(p, P, t) + qp - QP \\,\\!", "318a4126bfd5669d0e9fc88658ab1a98": "(t,t^2)", "318a6bb87840dbea85331df602dd3bad": "S = \\sum_{i=2}^{18} a_i \\cdot W_i", "318a9c46b7d50d1fa35b00df84de0c28": "b_2=1", "318adeac0d35e70ba8460e67f7bac515": "\\mathbb{E}\\left[\\mbox{ Charles }|\\mbox{ folding }\\right] = \\mathbb{E}\\left[\\mbox{ Charles }|\\mbox{ calling }\\right]", "318b330df13f06370c5f5d9c433a5670": "d_{\\varrho_i}\\,", "318b37ad07d5b216fc785d99bdaa9030": "2.405 \\sqrt{\\frac{g+2}{g}}", "318b534d21bc2203188cad9d6a8f9b55": "\\pm d_0.d_1d_2d_3\\dots\\times 10^n", "318b7ea06679b5703e20b20843e8f08b": "{\\nabla}^2 \\varphi", "318beea988ccf38c764e0ce510ce3dd6": "\\chi(H) < 1 + \\frac{m}{\\alpha(H)}\\ln m.", "318c02cac7900492f8fd8ba7dfb57d02": "31.5% + 48.2925% = 79.7925%", "318c2ff853710ac53ffc1da7f461045d": "\\begin{matrix} {13 \\choose 4}{4 \\choose 3} \\end{matrix}", "318c775307f910ddd3a802e14b00f557": " \\{ \\cdot,\\cdot\\}: {C^{\\infty}}(M) \\times {C^{\\infty}}(M) \\to {C^{\\infty}}(M) ", "318c9648c67a618e17d3252167898a4a": " \\dot{m} =\\,", "318d1f57e47f9af987bcc62d607ef753": "\\operatorname{Spec}R", "318e11851d451154cada9f4282b373bf": "X \\times_k \\overline{k}", "318e2b806d7287cda271f64748c4f7fc": "N\\subset\\mathbb R^k", "318e7089a9dd83efcb53dd99cb8a2d27": "F(X) = E(F(X))+\\int_0^1 E (D_t F | \\mathcal{F}_t ) \\, d X_t .", "318e829af860e4b6d95bff3d0c1f819f": "m_{\\text{h}}^*", "318ea3bb322e227122be88ca81babe2e": "sT^{n-1},", "318ee9040d75fa2c3207820b080d0333": " y^+ > 11.63\\,", "318f00789852993c0b91768d91cbe922": "\\gcd (a,n) >1", "318f27fdcbbc0e1f90227d5c16f47c21": "\\{0,1\\}^{n}", "318fa26b5f9a090c6b304be088bb68e0": "C_w' = \\sum_{y\\leq w} P_{y,w}(v^2) T_w", "319025222eb9c29a591d224ee1be8ecf": "\\approx failed \\ 50 \\ minutes/year", "3190903705e16dd50a6c307388fec31d": "V_0a^2", "3190aa1abda529c0c98c95f900fc1090": " E >> m", "3191dedb6443466319a51714176487c2": "\\langle a,b \\rangle", "3192307d1801b077b094875ed1424799": " (6)\\,", "31923927021375e504c760ed7f797cde": "\\xi > 0", "3192a040dae903310dae2e949bdd0109": "\\sum_i\\max\\{p_i+\\Delta\\, ,0\\}=1.", "3192b2a82d07668e0538fdf052f6b7f8": "(I : J) = \\{r \\in R | rJ \\subset I\\}", "3192b31daf3d2348b49727895316720f": "\\scriptstyle \\delta t_{\\text{clock},i} (t_i,\\, E_i) \\;=\\; \\delta t_{\\text{clock,sv},i} (t_i) \\,+\\, \\delta t_{\\text{orbit-relativ},i} (E_i)", "3192b3212a021d7f0a5e65f97a961746": "\\langle x, u_\\varphi\\rangle = \\varphi(x)", "3192de16de3c9c090ff75d45abffaef5": "\ny(\\mathbf{x}) \\, = \\, \\sum_{i=1}^N w_i \\, \\phi(||\\mathbf{x} - \\mathbf{c}_i||) + \n \\mathbf{v}^T \\, \\begin{bmatrix} 1 \\\\ \\mathbf{x} \\end{bmatrix}\n", "31932b774acf3983b373bc160d014392": "\n \\mathbf{F} = e^{\\boldsymbol{l}\\, t}\n ", "319351953123ae45b8e1e86b8d554b3a": "\\mathbb P(A \\cup B) = \\mathbb P(A) + \\mathbb P(B) - \\mathbb P(A \\cap B),", "31939c9910b4f15ecd33c11602c470ad": "\\begin{align}\n g_{k\\, k} &{}= 1 \\qquad \\text{for} \\ k \\ne i,\\,j\\\\\n g_{i\\, i} &{}= c \\\\\n g_{j\\, j} &{}= c \\\\\n g_{j\\, i} &{}= -s \\\\\n g_{i\\, j} &{}= s \\qquad \\text{for} \\ i > j \n\\end{align}", "3193d0f0a67308ae179f343ba53d8be4": "\\psi_W(t)", "3193ddf4788bc3741168858a71c51079": "P = \\frac{q^2a^2}{4\\pi c^3}\\frac{\\sin^2 \\theta}{(1-\\beta \\cos\\theta)^5},", "3193e8ba520b108345656538b7415f20": "\\textstyle w_j", "319495391f718395b69f092f49330016": " \\mathbf{u} = \\begin{pmatrix} \\frac{d y_{1}}{d t} \\\\[2mm] \\frac{d y_{2}}{d t} \\end{pmatrix} ", "31950a757d3dec220cc86632f712dd92": "\\scriptstyle\\mathbb R^n", "319529eaffbf696a9bdc3ae9a686b57e": "(x - 1)(x - 1) = 0", "31955d45985a982117f76c7d5cde5ceb": " \\textstyle \\underset{ p,\\, p+2: \\text{ prime}}{\\sum(\\frac1{p}+\\frac1{p+2})} = (\\frac1{3} \\! + \\! \\frac1{5}) + (\\tfrac1{5} \\! + \\! \\tfrac1{7}) + (\\tfrac1{11} \\! + \\! \\tfrac1{13}) + \\cdots ", "31959e2988dd063ec876cf985dbc716b": "{{{h^a}_{bc}}^d}_e \\in {{{V^a}_{bc}}^d}_e = V\\otimes V^*\\otimes V^* \\otimes V\\otimes V^*.", "3195b212f1b39456946afcc8c9b7c558": "x/y \\cdot z", "3195b5a18be8630999e16e801117c127": "(x,y) = (1151,120).", "31960937d368ed587326b458601e21d6": "\\vec{R}_2 ", "31961148ba669ff2b425f45ba8957759": "(\\mu_{ab}(t)) \\in \\Gamma_{S(t)}", "3196169c4260f30c3d8a3c666bbbaa8a": "\\langle\\ | \\!\\,", "3196ae8a227b61ace9953559a5f0ec37": "P(e\\mid Ctx=[q_1,q_2,q_3]) = P(-e\\mid Ctx=[-q_1,-q_2,-q_3])", "3196c06aef1010d724c72a16e58cd727": " \\mathbf{D} = \\varepsilon_0 \\mathbf{E} + \\mathbf{P} = \\varepsilon_0 \\mathbf{E} + \\varepsilon_0 \\boldsymbol{\\chi} \\mathbf{E} = \\varepsilon_0 (I + \\boldsymbol{\\chi}) \\mathbf{E} = \\varepsilon_0 \\boldsymbol{\\varepsilon} \\mathbf{E} .", "3196cd48582e4687efb80ff1607b4f00": "\\left |\\Delta X\\right |=\\left |\\frac{\\partial f}{\\partial A}\\right |\\cdot \\left |\\Delta A\\right |+\\left |\\frac{\\partial f}{\\partial B}\\right |\\cdot \\left |\\Delta B\\right |+\\left |\\frac{\\partial f}{\\partial C}\\right |\\cdot \\left |\\Delta C\\right |+\\cdots", "3197008fd17978a510425f49785fd160": "\nx_0 = \\frac{A_0}{B_0} = b_0, \\qquad\nx_1 = \\frac{A_1}{B_1} = \\frac{b_1b_0+a_1}{b_1},\\qquad\nx_2 = \\frac{A_2}{B_2} = \\frac{b_2(b_1b_0+a_1) + a_2b_0}{b_2b_1 + a_2},\\qquad\\cdots\\,\n", "3197070ba34a331caea2b72062980747": "q \\geq 2 ", "3197349f5acc1a0108fba95667bcecfd": "\\scriptstyle {n \\choose i}", "3197829e0f2935f9d56a1b7cc04ff0fb": "\\xrightarrow{C2H}", "3197b66713ea288f3cd9c537f8fccefe": " Fujii(x,y) = \\sum_{k=1}^{N} \\frac{ I_k (x,y) - I_{k+1} (x,y) }{ I_k (x,y) + I_{k+1} (x,y) }\\,\\!", "3197d8f233b548f21ee290bc9bfd6352": "\\nabla_{\\dot{c}} \\dot{c} = 0.", "3197dfa14afac95b0cac8f062f1df9c4": " \\mathbf{x}(3), \\mathbf{x}(6), \\mathbf{x}(9),\\ldots,\\mathbf{x}(48)", "319855b20e2ab83125b5a18bf8086b49": " a_{SIW} ", "319856ce304dc3161e558ec0a1efce90": "x \\wedge \\bigvee_{s \\in S} s = \\bigvee_{s \\in S} (x \\wedge s).", "31986ba8b8f8ef3906665f5a5a31f514": "V_{sw}", "3198d4de037f6ad12e691a497606b659": " p(\\lambda)= \\det ~(\\lambda I_n -A) = \\lambda^n \\exp (\\operatorname{tr} (\\log (I_n - A/\\lambda))). ", "31990fa8be1641b2050590f7accf40c4": "[L_z,L_+] = L_+,\\quad [L_z,L_-] = -L_-, \\quad [L_+,L_-] = 2L_z.", "319960620191ad6044e734352ef9eb3b": "p = (\\sigma(n) - \\varphi(n))/4 - \\sqrt{[(\\sigma(n) - \\varphi(n))/4]^2 - [(\\sigma(n) + \\varphi(n))/2 - 1]}, \\, ", "3199b20a7157e2471459141259f2b1bd": "g(w)=w", "3199d24343edc9be1ca8d0bacd27aa4f": "\\alpha=\\tfrac{2}{3}", "3199de26665a9c6baea313c0cdb02666": "\\|x+y\\|+\\|x-y\\| \\le 2 + \\epsilon\\|y\\|.", "319a4f5f969f222589133ed2d77920e0": "u^{\\pm i} \\to e^{\\pm i \\phi} u^{\\pm i}", "319a56a7997dc81ce1bdf56519e3501c": "[\\mathbf{A}^\\mathrm{T}]_{ij} = [\\mathbf{A}]_{ji}", "319a5b3eb785e0d87600fcb923515eff": "V_{\\text{Coulomb}}(r)= -g^2 \\;\\frac{1}{r}.", "319a61709935b563d731764514d546cd": "O | \\psi \\rangle = | h \\rangle ", "319ae89dffb44cf55d3e14b087de1d43": " = Prob", "319af4b2b42ab097c95f7520ef9b7a1b": "\\displaystyle \\chi_{[0,1]}(|\\mathbf x|)(1-|\\mathbf x|^2)^\\delta", "319b20a64d61fb45ded44177c0276e9c": "\\mathbf{Ab}", "319b38fb11307d76f5b89f3f8fa6ebe0": "f(t)=\\alpha", "319b8a578bac0220640e4f97a1e07f4a": "w=\\sigma v", "319bc45c64685ca3e52b7fdcf4b58b4a": "\\!E_\\mathrm{h} / (ea_0) ", "319c03c74c099ff7fc61b5de4ee9c135": "V = \\frac{1}{|\\mathbf{x}|}", "319ca960cf975a947139f5cd6087637c": "H = ", "319d0e6cad92c2a784b6582f5ba9bd6c": "N = \\frac{7758\\ A\\ h\\ \\phi\\ \\left(S_o \\right)}{B_{oi}}\\ \\ ", "319d33ccb0eef57a6a4f687cc1874b38": "\\sum_{i=0}^n \\left( a_i\\times b^i \\right)", "319d584a4a5166ee6c51f4b8348856ea": "\\odot", "319d6fcc8c7840a1bf557e81dca7d175": " V(t) =\\sup_{x\\in X}f(x,t)", "319dc380f6d945b874ce4289452b4d66": "x=y=z=0", "319dff86f4db95858b6d469fb31f06c4": "0.1(-100)+0.3(-20)+0.4\\cdot 0+0.2\\cdot 50 = -6. \\, ", "319e045f536a6cab308f01ebd20900da": "F_2(a, b) = a\\cdot b", "319e4a595497513f176e0b2cee45cda9": "\\partial_t L = \\frac{1}{2} \\nabla^2 L", "319ea487df6978733fe15299cade866b": "\\mathcal{L}_X(f\\otimes Y)=\n(\\mathcal{L}_Xf) \\otimes Y + f\\otimes \\mathcal{L}_X Y", "319ea7564ba5ea4fe854f358bbc2e927": "\\mathbf{U} = \\gamma \\left( c, \\vec{u} \\right) ", "319f067655c2d62c4b2652f305916956": "(\\Sigma,\\Lambda)", "319fcdce8e2934fee4372bc5cad8a49f": "{\\Gamma^{\\alpha}}_{\\mu\\beta} \\!", "319fd980a52346406b9e15d79292ebc9": "\\alpha(t) = vt\\ ", "319fe9d0288b2275ece5834aca551c2d": "\\sigma^{2}_{f}= \\sum_i^n a_{i}^{2}\\sigma^{2}_{i}.", "319ffd6b54a07bfcd53830b94463c480": "m = \\sum_{i=0}^n \\frac{2^i}{p_i-1} \\times \\left( \\gcd(p_i,c^s \\mod p) -1 \\right)", "31a012e955ae736aef7d59a579ea2cb0": "R(1+\\langle L_i\\rangle)^{-1}", "31a070c878e288f2df653a604c2b8699": "\\operatorname{E}(X_1)=\\alpha\\beta\\,", "31a0adb80632258032bc2dcb5beecc5d": "{{20 \\times 19 \\times 3} \\over 2} + 64 - 1 = 633", "31a0be21cffe55f57d46e4cc44bea46f": "EVEN\\cap \\overline{\\{2\\}}\\cap \\overline{PRIMES+PRIMES}", "31a0e66aaa0b7945b875eedfb4fa10ca": "s^2 = \\frac{1}{n-1} \\sum_{i=1}^n \\left(y_i - \\overline{y} \\right)^2 ", "31a13eeef2751c78ee33b6cef1b70cd6": "Q_{t_m}", "31a157dcf38aa80e2b4de365aeea5e71": "\\int_{-\\infty}^\\infty \\varphi^*(t - k) \\cdot \\varphi(t - k') \\, dt = \\delta_{k,k'}", "31a1867feea7d54ab45570e37db555dc": "\\pi - \\tan^{-1}\\left(\\frac{2}{11}\\right)", "31a1af50f98da32fc9dc843def2e5f7d": "ax+by+cz=d", "31a1bd3f0bd44297867a374c42724777": "4!/(4!0!) = 1", "31a2c626e7abbb63a1e45232009023af": "\\mathrm{rd}(z)(t) = \\frac{\\dot{z}(t)}{1 + \\| \\dot{z}(t) \\|}", "31a2c7ec01a36c9ef9844b90a7c86abc": "F(\\Pi^{\\mathbb{Z}^{+}})", "31a31240f00969830793bafb3e6e26c1": "f:G \\rightarrow G'", "31a317ad5484710b254952bd0ab0cd6d": "\\displaystyle{e^{-Dt}H_n=e^{-(2n+1)t}H_n.}", "31a34b8c7c0041a83fb65a3803e81ec1": " f(k)", "31a3e254c106dadfea8dadd52a0cf905": "-\\frac{18}{y}=-\\frac{18}{8}=-\\frac{9}{4}", "31a40160c376be7247493fa39c48e71d": "d \\psi = -\\varphi_y\\, dx + \\varphi_x\\, dy.", "31a46ee96513915e0bcc3947c0b8d932": "|F(z)| \\leq e^{C |z|^\\rho}", "31a4f2ba6125650d7fc2051c4bd38ab2": "-\\pi\\le x,y \\le \\pi", "31a5c71d413c41c2b60f72ba7702b6b9": "(r^{\\prime}/r)<1", "31a5fea5806a89e1e8c1a7eb72ee08f4": "\\operatorname{gr}_F R = \\oplus_{n=0}^\\infty I^n/ I^{n+1}", "31a641d0e46d844053931ed1ec2ebbdb": "({L^*_1},{a^*_1},{b^*_1})", "31a6518d1b04a090784938645b4a6633": "b-a", "31a689944b71f072e85e85e536c2952d": "x_{n+1} = x_n - \\frac {f(x_n)} {f'(x_n)} \\left[1 - \\frac {f(x_n)}{f'(x_n)} \\cdot \\frac {f''(x_n)} {2 f'(x_n)} \\right]^{-1}", "31a69bcab9102ab6a84a575c595a542c": "\\phi_i(x)\\downarrow", "31a6b6db7bf4c57ca662ca5278dd66ba": "U(\\alpha,\\tilde{u})", "31a72049d5de5def0d57660e0e8c3dad": "\\coprod_i U_i = N", "31a726dea6c07068cfac5da2f1d9fed5": "\\Delta p_{ref}", "31a80177a676fe0f3691e2283acb4cc4": "r = \\frac{1}{t_2-t_1} \\log_e \\left [ \\frac{x_2 (1-x_1)}{x_1 (1-x_2)} \\right ]", "31a83c58565db6fc6999b7134dd9d9a7": "f_1(x),f_2(x), \\dots, f_m(x)\\,", "31a8a2091bd1e4026e52f27b979c33de": "M(x) \\cdot x^n = Q(x) \\cdot G(x) + R (x) ", "31a8a80936dd80842aa6832ab8d14957": "\n\\bold A_y= \\left[ \n\\begin{array}{c c c c c}\n0 & 0 & 1 & 0 & 0 \\\\\n-vu & v & u & 0 & 0 \\\\\n\\hat{\\gamma}H-v^2-a^2 & -\\hat{\\gamma}u & (3-\\gamma)v & -\\hat{\\gamma}w & \\hat{\\gamma} \\\\\n-vw & 0 & w & v & 0 \\\\\nv[(\\gamma-2)H-a^2] & -\\hat{\\gamma}uv & H-\\hat{\\gamma}v^2 & -\\hat{\\gamma}vw & \\gamma v\n\\end{array}\n\\right].\n", "31a8e27dcea5882443f096773d24ebed": "N = mg", "31a8ec14730cf10e2dace66316c95f03": " f(x) = \\begin{cases}\n-0.1522 x^3 + 0.9937 x, & \\text{if } x \\in [0,1], \\\\\n-0.01258 x^3 - 0.4189 x^2 + 1.4126 x - 0.1396, & \\text{if } x \\in [1,2], \\\\\n0.1403 x^3 - 1.3359 x^2 + 3.2467 x - 1.3623, & \\text{if } x \\in [2,3], \\\\\n0.1579 x^3 - 1.4945 x^2 + 3.7225 x - 1.8381, & \\text{if } x \\in [3,4], \\\\\n0.05375 x^3 -0.2450 x^2 - 1.2756 x + 4.8259, & \\text{if } x \\in [4,5], \\\\\n-0.1871 x^3 + 3.3673 x^2 - 19.3370 x + 34.9282, & \\text{if } x \\in [5,6].\n\\end{cases} ", "31a9286393f8bfff6c532704849ee83b": "\\varphi(\\tau)\\,", "31a949d6f01d1efcc45a6056a5f32e8b": "0\\leq\\mu(A)-\\mu(A_1)\\leq\\varepsilon", "31a95ac473201bf4c0e21a59c4037669": "f(N) \\geq 1 + 2^{N-2},", "31a99b3c2409917eb687db4910b04fab": "h_j =\\frac{1}{i\\hbar} \\int_{j\\varepsilon}^{(j+1)\\varepsilon} dt \\int d^3 x \\, H(\\vec x,t). ", "31a9a6ab9db8e02f196abf2088a3d3d8": "\\nu_t = \\Delta x \\Delta y \\sqrt{\\left(\\frac{\\partial u}{\\partial x}\\right)^2 + \\left(\\frac{\\partial v}{\\partial y}\\right)^2 + \\frac{1}{2}\\left(\\frac{\\partial u}{\\partial y} + \\frac{\\partial v}{\\partial x}\\right)^2}", "31a9b68d9b837c971431e82221a124e6": "G(t) = N_1t +N_2t^2/2 + N_3t^3/3 +\\cdots \\,", "31a9f69f3afc6093a4325c331c1b29fc": "I_i=\\frac{2e}{h}\\int \\frac{1}{1+e^{\\frac{E-E_f}{k_BT}}}dE ", "31a9ff0062e0692d9bf381cd2ba32075": "d=6", "31aa33606f9bbaa4bf36407bade07a32": "f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \\frac{1}{2} \\,", "31aa55ca4d655fb603cfa01fe3f99b62": "\\mathrm{[H^+] = \\mathit{K_a} \\frac{[HA]}{[A^-]}}", "31aa69bbf365fcbd133ddfa0a973e5ef": "K_1 = -1", "31aaa04ff3bad3812c0900d1d556be89": "e\\downarrow m", "31aaa80dece46e6eb66bb86d02c8692b": "\\begin{align}p(\\lambda) &\\propto \\sqrt{I(\\lambda)}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{d}{d\\lambda} \\log f(x|\\lambda) \\right)^2\\right]}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{n-\\lambda}{\\lambda} \\right)^2\\right]} \\\\\n& = \\sqrt{\\sum_{n=0}^{+\\infty} f(n|\\lambda) \\left( \\frac{n-\\lambda}{\\lambda} \\right)^2}\n= \\sqrt{\\frac{1}{\\lambda}}.\\end{align}", "31aaf4ac7af1fc5ca7005dfcf8c0c8a7": "(m,n,p,q)", "31ab466b1920eed6aead52539ac18985": "\\varphi(G_1)=G_2", "31ac62e236ff656ead51125610307247": "(d + 1)(r - 1) + 1\\ ", "31ac6445a8cde76c17e5878da0402a1c": "\\displaystyle u_t+Hu_{xx}+uu_x=0", "31ac762c6b62922799130720362238ee": " a_i (t+1) = a_i(t) + \\nu \\big [ y(t) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ] u \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big ) ", "31ac904e12542a88339b1233202b8d7d": "L(x) = \\Sigma_{s=0}^{r-1} \\alpha_s x^{q^s}", "31aca882831fdc8ab3e29258a3fcea1f": "H = -J{\\sum}_{}\\mathbf{s}_i \\cdot \\mathbf{s}_j", "31acb019c50bb4761c2098b5a8268c4f": "{{z}_{out}}=\\frac{{{v}_{test}}}{{{i}_{test}}}=\\left( 1+\\frac{\\beta }{2} \\right){{r}_{O3}}\\approx \\frac{\\beta }{2}{{r}_{O3}}", "31acf44c1559dd11f26aa30abf80c1fa": "\\ (4,3) = 4 e_1 + 3 e_2 ", "31ad11ef14d8d2fbd3410446f0415b45": "\\displaystyle{\\theta(S_a)=T_{a^*},\\,\\,\\, \\theta(j)=j,\\,\\,\\, \\theta(T_b)=S_{b^*},\\,\\,\\, \\theta(W)=(W^*)^{-1}.}", "31ad2c53a186fc1c764da03a453854f6": "\\mathrm{\\Beta}(x,y) = \\int_0^1 t^{x-1}(1-t)^{y-1}\\,dt = \\frac{\\Gamma(x)\\,\\Gamma(y)}{\\Gamma(x+y)}.", "31ad68e532bf959d796dd63edfd75360": "\nc \\int_{t_\\mathrm{then}}^{t_\\mathrm{now}} \\frac{dt}{a}\\; =\n \\int_{R}^{0} \\frac{dr}{\\sqrt{1-kr^2}}\\,.\n", "31ad7f101a3727392fa5b402153eb822": "f(x) = \\frac{\\tau}{\\sqrt{2\\pi}}\\, e^{\\frac{-\\tau^2(x-\\mu)^2}{2}}.", "31adb4e380fabf1f8b836dec41846b91": "g^-=1_{\\Omega'\\backslash U}g_0^-", "31ade271460c5b04981fc44ac5d16a1f": "\\psi(\\Omega 3)", "31ae15695fb539b5eb0a156b362df31c": "dy = \\frac{dy}{dx}\\, dx", "31afc243fb527724e8d6c91c2f745002": "\n\\zeta(2n)=\\frac{(2\\pi)^{2n}(-1)^{n+1}B_{2n}}{2\\cdot(2n)!}\n", "31aff82cd8741d4a254cf765bf4aec3c": "\\mbox{CAR} = \\cfrac{T_1 + T_2}{a}", "31b058d54005634b89e775b872205913": "\n\\begin{align}\ns_1& = x_1\\\\\nb_1& = x_1 - x_0\\\\\n\\end{align}\n\n", "31b07fd7f53c5bfb555cf9250b586c37": " W'_{out} = Q'_{in} - Q'_{out} \\,", "31b0bfccb76ede4ddfc4dc2b84b06fc2": "T_{final}", "31b0c435bb9ae22b8eb0ed9fdcecf303": "C_{ijklmn}", "31b0c6660460c40921c26ce2e11362c2": "z \\in \\mathbb{C}\\backslash \\Pi_A", "31b121cb9801f16a3fa59792a4e3d457": "\\frac{\\dot{H}}{H^2}=-(1+q).", "31b12c9755fee1f3526a9de74f22a611": "u_{x,r} = \\frac{1}{|B_r|} \\int_{B_r(x)} u(y) dy", "31b1655db72b052e59b55208a4ca99a4": " \\, +i,-1,-i,+1,+i,\\ldots \\, ", "31b2360b779f4b6c1ace4bf28331aee7": "\\partial_i\\psi=\\eta_i", "31b2816c2492f243280b149c4b46ee09": "\\|x\\| = 1", "31b2e74dd2aa9945f3c55400a09b3b82": " \\phi_g(t)=\\int dE\\, e^{-iEt} g(E)\\phi", "31b307a8eaa1cd0ff4f886c10bb07f33": "D^t", "31b312aa5b3a63abef742a145f4d1d69": "s_i = -1", "31b32316c7018218af8a72fd978df505": "PS_k(X)", "31b34440298c587112445da972cc6618": "\\ln(Y(t))= \\alpha \\ln(K(t)) + (1-\\alpha)[\\ln(L(t))] + [\\ln(A(t))] + \\varepsilon. \\, ", "31b349afd65f07de9a0ab86d9cbb7b82": "Q_D\\left(n\\right) = O(Q_S\\left(n\\right)\\log\\left(n\\right))\\,\\!", "31b43cc7945d36d6b650fc100bc1021a": "\\begin{matrix} {12 \\choose 1}{4 \\choose 3}{44 \\choose 2} \\end{matrix}", "31b46333443f2d6b5128de5a598fcf1c": "A\\in\\mathbf{H}_n", "31b4654c91986fdd6114f7d03fa1ae61": "x''=x-x'\\,", "31b47fbaf3aa04f587a193f3c580e7b3": "\\color{Salmon}\\text{Salmon}", "31b501ff1b4a41afe6db74fa03cd03d0": "V(x(t)) - V(x(0)) \\le \\int_{0}^{t} u(\\tau) \\cdot y(\\tau) d \\tau", "31b51042cf37f55e329f59a07980c538": "\n \\int_{\\mathbb R} \\frac {\\vert\\widehat\\psi (k)\\vert^2}{\\vert k \\vert}\\; dk < \\infty\\; ,\n", "31b51ac65a31db14cde1c1446b1de809": "\\,\\ \\sec x", "31b557a3adcb144549ea6c36c519e839": "\\max(a,b) = \\lim_{h\\to 0}h\\log(e^{a/h}+e^{b/h}).", "31b5c0ddd70f8cbaa076b0a682b926c7": "\\frac{\\Delta f}{f} = -\\frac{2 \\mathrm{DIBL}}{V_{DD}-V_{Th}},", "31b61df1aa484156f742cd1c11ddee83": "\\sigma^2=1/\\nu ,", "31b62921b3a6ba1cc7f80f090dd6620e": "\\bigg( (\\mathcal{M}, s) \\models E[\\phi_1 U \\phi_2] \\bigg) \\Leftrightarrow \\bigg( \\exists \\langle s_1 \\rightarrow s_2 \\rightarrow \\ldots \\rangle (s=s_1) \\exists i \\Big( \\big( (\\mathcal{M}, s_i) \\models \\phi_2 \\big) \\land \\big( \\forall (j < i) (\\mathcal{M}, s_j) \\models \\phi_1 \\big) \\Big) \\bigg)", "31b67f9ee134f074df33e082be581d34": "BW \\ = \\ F_0 / Q ", "31b6c18aeefeca7c8a327a28114f32bb": "T = \\frac{\\displaystyle \\exp\\left(-2\\int_{x_1}^{x_2} dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}\\,\\right)}{\\displaystyle \\left( 1 + \\frac{1}{4} \\exp\\left(-2\\int_{x_1}^{x_2} dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}\\,\\right) \\right)^2}\\ ,", "31b6c74b29d45c3523418138e47ac1e0": "\\ g^1(q;\\tau)= \\exp(-\\Gamma\\tau) \\, ", "31b6cb95c57c4071d2ec9a7970193cf9": "\\det(AB) = \\sum_{S\\in\\tbinom{[n]}m} \\det(A_{[m],S})\\det(B_{S,[m]}).", "31b6e3c888e67d9e5cfa51de9084c9cd": "|\\hat{n}(t) \\rangle", "31b6e628cbe5b78c5af9fe170db2bcb8": "y = n - 1", "31b6eb924de49df799c3016cfea5a36b": "O_1", "31b6ecfe797a444b849d0c06af6ce4b7": "C_6 = \\frac{3}{4} \\alpha_1 \\alpha_1 \\hbar \\omega", "31b6f73d006440c6620acf47105a4f9c": "\\omega_L", "31b7454e36a47ac08305fbf28d44024e": "ax=b.", "31b7e7ae265befcc6fbae1ec880a6061": "\\theta = a \\, t \\, \\exp(a^2 r^2/2)", "31b88a15bb360f4a03f3a5e0af6414a0": "X_{\\mathrm{f}}", "31b8b18bfe61930ff2f358e29590e2f4": "\\frac{1}{W(x)} \\ \\frac{d^n}{dx^n}\\left(W(x)[Q(x)]^n\\right).", "31b8ffa5be432da7a2a681f5a026bbf1": "\\Lambda^{-1}", "31b9071d5f19a1fd4434111b335ca788": "f:\\partial V\\longrightarrow \\R", "31b9089cd6e0d897c3637489c054be7c": "f\\colon 2^S \\to R", "31b948ca5a689cfb4db519b041e54d47": "y_t\\,", "31b975c924984c16b33df4f81ef3b1bf": "B(L(T)),B(R(T))", "31b97701762039d6f6c48450b7e8cbf7": "S=B-A", "31b99e0f0f07a77335af122790f63309": "\\Psi_\\pi", "31b9c1db92f976e43f2ac3f27d9262cc": "| \\Psi \\rangle \\in H^{{\\otimes} N }", "31b9d0c0293c81d674c653f13e2e6e30": "\\operatorname{Pr}(K\\leq K_\\alpha)=1-\\alpha.\\,", "31ba1a207e46d39b88a0c88728121cd2": "Y_i=a+bx_i+\\varepsilon_i\\text{ for } i=1,\\dots,n", "31ba38b44626443b938b7bf4fccce4b6": "\\{\\mathrm{milk, bread}\\} \\Rightarrow \\{\\mathrm{butter}\\}", "31ba4cac5f9eb22c6f87cfa6235c3f39": "U = \\int_0^{\\nu_m} \\,{h\\nu^3 V F\\over e^{h\\nu/kT}-1}\\, d\\nu\\,,", "31baf9aced4849bb9ff4dd1c8eb513a5": " R_{BCS} \\simeq 2 \\times 10^{-4} \\left( \\frac{f}{1.5 \\times 10^{9}} \\right)^2 \\frac {e^{-17.67 / T}} {T} ", "31bb09e0474a3ddd1483459edda1778d": " E_{z}=\\frac{k_{t}^{2}}{j\\omega \\varepsilon }\nL \\ T^{TM}=\\frac{k^{2}-k_{z}^{2}}{j\\omega \\varepsilon } L \\ T^{TM} \\ \\ \\ \\ \\ \\ \\ (29) ", "31bb12bd77fabb78041b4b5a6526081a": "dT/dx", "31bb2239389f72ecb35cff00bb10012c": "\\displaystyle{\\int \\varphi =\\int f,}", "31bb4231e94034b6b7e96f4162bfd37f": " F_X(b)=F_Y(b) ", "31bb47713a684eef852865473139e3d3": "\\frac{1}{(i\\omega-\\xi_1)(i\\omega-\\xi_2)}", "31bbaf08a4625a14a7b7b3fb475efcf8": " y\\prec x.", "31bbbe9460c71a3361eb8d270ea02e0d": " (k+2)(1-\\alpha_0^2-\\alpha_1^2-\\cdots-\\alpha_{13}^2) > 0 ", "31bbc6fa42d18ad2d7d459a87cbdffed": "\\textstyle \\left\\vert \\alpha\\right\\vert =", "31bc16c37fa7164f3f170c734d189160": "\\displaystyle \\frac{1}{\\sqrt{2 \\pi} (a + i \\omega)}", "31bc2bdce202e14a0219e0797463ed37": "f(x) = \\int_a^x g(t)\\,dt", "31bc6cb2f97d3dde9698558fffa4c458": "S(n) = \\sum_{k} c_{j0,k}\\psi_{j0,k}(n) + \\sum_{j>j0}\\sum_{k} d_{j,k}\\phi_{j,k}(n)", "31bc983824afe9b0954d3451eab88fb5": "R[X,Y,Z,W]/(XY-ZW)", "31bca469977f343f11f94defbe8f020e": "P \\ne Q", "31bcb41d889c77f31fccb3a2eb1b91e6": "\\frac{1}{n}\\sum_{i=1}^n C_i", "31bcc6f5d9b9b33541491fc4d6426b08": "\\mathbf{F}^\\alpha\\ \\mathbf{C}^\\alpha\\ = \\mathbf{S} \\mathbf{C}^\\alpha\\ \\mathbf{\\epsilon}^\\alpha\\ ", "31bcd4d17083fbecc48cce28ebf383b0": " T_f ^* = T_{{\\bar f}} .", "31bcdd4b5adc4de17b20f2fc90472fff": "\\frac{2t}{\\sqrt{t^2-u^2}}", "31bce7a670a1a5cd9c96d4fad38978a9": "\\pi/c_{ij}", "31bd1e55ad69c74dc6a42d9a02cb1975": "\\scriptstyle (1,0,0,\\dots,0),\\dots,(0,\\dots,0,1) ", "31bd57ca89b177785706f1a92faba588": "X^k\\leftarrow \\min_X \\tau(X,Z)", "31bd919372a206bd95621048a765aeb7": " d_n(x_n,p_n)\\le C", "31bdcf184cb515402b8fe25a9b4b0760": "\nT_2 = \\frac{T_1/\\left(t - 1\\right)}{\\left(bk-b - T_1\\right)/\\left(bk - b - t + 1\\right)}\n", "31bdf7b7bf8bc39795dc60b162b87e38": "u > u_{mf}", "31be3ed05cfea78fce1c55b1ab07cca3": "(\\phi \\land \\psi) \\,", "31be4d43b431dfb15e284c0742c47fa6": "1 \\over 12", "31bed2733f858205331c22258e9fb387": " \\tfrac{d H(x)}{dx} = \\delta(x)", "31bf05ffa44e7239b0f67e933e404757": "E_{0}=3.28\\times10^{-20}", "31bf5573d0f26d8e8a21c347fac3c80e": "1-p", "31c01dc1eddf304072c1dc2ad2ce172a": "T(x,y) = t(x,y) - \\overline{t}", "31c051aa89c6235c4e269e74a446c946": "N_{R_1}", "31c06d49de8444876dbe9146a6a0aef5": "C^{abc}", "31c085d3f134831362d5704241f8328a": " \\ = \\left \\{ \\begin{matrix}\n-\\frac{1}{4 \\pi} \\delta(s) + \\frac{m}{8 \\pi \\sqrt{s}} H_1^{(1)}(m \\sqrt{s}) & \\textrm{ if }\\, s \\geq 0 \\\\\n -\\frac{i m}{ 4 \\pi^2 \\sqrt{-s}} K_1(m \\sqrt{-s}) & \\textrm{if }\\, s < 0.\n\\end{matrix} \\right. ", "31c0acdc007a38f23ccac91f7e67bfc8": "\\mathbf{E} = c \\mathbf{B} \\times \\hat{\\mathbf{r}}", "31c0de2e6b0d2eccad6bd2f7328b5abf": "\\text{append}\\colon A^{*} \\to A^{*} \\to A^{*} = l_1 \\mapsto l_2 \\mapsto \\begin{cases} l_2 & \\text{if} \\ l_1 = \\text{nil} \\\\ \\text{cons} \\, a \\, (\\text{append} \\, l_1' \\, l_2) & \\text{if} \\ l_1 = \\text{cons} \\, a \\, l_1' \\end{cases}", "31c0e7c1abb87f8749e43c1930fda8e4": "R = K[\\theta_1, \\cdots, \\theta_N]", "31c18a3e3e33dbf9fd770cfe30efc790": "a|n\\rangle = \\sqrt{n}\\,|n-1\\rangle", "31c1c1378a799628a7f997d120ea5618": "\\ \\mathcal{L} = \\frac{1}{2} (\\partial_\\mu \\Phi)^T \\partial^\\mu \\Phi - \\frac{1}{2}m^2 \\Phi^T \\Phi ", "31c1fffc568e4f82c1fa159b35f8825d": "(\\tfrac{2}{p})", "31c20e1d918b83e03c0a6f6da346af24": " A_1", "31c2382d23b8c62c57239ab829d986ed": "RP_s", "31c253a12f9fabd7a097efaf931fdabd": " t_1, t_2, \\dots, t_n ", "31c26868ab2b650f33cc0f6ff643c06f": "\nN_P=\\min\\{N_U\\ ,\\ N_D\\}=\\min\\{N_{P_1}\\ ,\\ N_{P_2}+(x_D-x_M)k_j\\} \\qquad (10)\n", "31c298c879a6d498b11f6f2fc022c36d": " = 8*10000 + 2 (10000/500) + 0.16 (500/2) = $80080 ", "31c2a03163360aa364a70610199ddb98": " W_n = \\int_0^{\\frac{\\pi}{2}} \\sin^n(x)\\,dx, ", "31c341fbd3a3f15ca0db67df36fedc42": "\\left(\\frac{\\mathit{Q}_{4-1}}{{m}}\\right)=\\mathit{u}_4-\\mathit{u}_1", "31c35e51d8f3b01eb8b9552ddc244c3c": "0\\leq\\mu(A)-\\mu(A_0)\\leq\\varepsilon", "31c366d4e0f3337eeb0fc4e144343d95": "V \\sim \\chi_k^2", "31c36ff95823365ea31fe291b3f42297": "\\binom nk = \\frac{n^{\\underline{k}}}{k!} = \\frac{n(n-1)(n-2)\\cdots(n-(k-1))}{k(k-1)(k-2)\\cdots 1}=\\prod_{i=1}^k \\frac{n-(k-i)}{i},", "31c38cae46719a9aa134d46eeccbf01d": "A_6 \\cong \\operatorname{PSL}(2,9).", "31c393bb80556808c2ac1477fb8a9936": "\\mathfrak{sl}_{2+1} = \\mathfrak{sl}_3", "31c3a8ddfa1987e7247c2bd755d1b965": "S \\in R^N", "31c3c1a8af1230b5a20ecf58c182bd96": "\n\\frac{\\partial p}{\\partial r} = \\rho \\frac{v^2}{r}~(>0),\n", "31c40307ab021c9bddefb527b9bffd89": "p \\ne q", "31c416b06d95b04e658d8b001178b9c3": "\\begin{align}\nr_1 : \\mathbb{R} \\rightarrow \\mathbb{R} & \\quad r_2 : \\mathbb{R} \\rightarrow \\mathbb{R} & \\cdots & \\quad r_n : \\mathbb{R} \\rightarrow \\mathbb{R} \\\\\nr_1 = r_1(t) & \\quad r_2 = r_2(t) & \\cdots & \\quad r_n = r_n(t) \\\\\n\\end{align}", "31c44edc47faa2398e80ca72a122b3fa": " \\mathfrak{so}(4,\\mathbb C) \\cong \\mathfrak{sp}(2,\\mathbb C)\\oplus\\mathfrak{sp}(2,\\mathbb C)", "31c4912924373176e558decc45591f2d": "\n\\phi(f^*f) \\phi(g^*g) - | \\phi(g^*f) |^2 \\geq 0 \\quad \\text{i.e.} \\quad \\phi(f^*f) \\phi(g^*g) \\geq | \\phi(g^*f) |^2. \\,\n", "31c4ad4026ccebbbc889d3dcae8e97ad": "\\frac{\\partial u}{\\partial t} = \\alpha\\frac{\\partial^2 u}{\\partial x^2}.", "31c4e95f07fe3929a44ae56620913993": "i(2 \\sin \\varphi \\cos \\varphi) + \\cos^2 \\varphi - \\sin^2 \\varphi\\ = \\cos 2\\varphi +i \\sin 2\\varphi", "31c53ae34518ab1d1b79e0a444af3ff2": "\\operatorname{Tor}_{i+1}^R(k,M) = 0.", "31c556037eec78d84312f790bebf3b9e": "\\left({12 \\choose 9}+{12 \\choose 10}+{12 \\choose 11}+{12 \\choose 12}\\right)\\left({1 \\over 2}\\right)^{12}", "31c5c3737bc7894bf96858aa9cbb00b3": "\\displaystyle \\pi\\left(\\delta(\\nu-a)+\\delta(\\nu+a)\\right)", "31c5cae850427988e18b0077cbf23dfc": "f(T)= \\sum_{i=0}^{\\infty} a_i T^i.", "31c5db12de6fe52e3389b6e16188f7d4": "\\psi_{a,b} (t) = \\frac1{\\sqrt a }\\psi \\left( \\frac{t - b}{a} \\right),", "31c5f1a854199b13112ec07e941411ea": "\\Delta(t) + Vp(t),\n", "31c65ac18a6630ea2edabdd3b60f76bf": "(H,k)\\in \\Pi ", "31c673c8c68b05f86c1e7d6a65a2e61c": "{\\rm E}(V) = 0", "31c6b3fdfaaa80dba2dbf92a4600524c": "Sign", "31c721a7efd30689e2d0824dc1367533": "3^{3^{3^{3}}} = 3 \\uparrow \\uparrow 4 \\approx 1.26 \\times 10^{3,638,334,640,024} \\approx (10 \\uparrow)^3 1.10", "31c72296def4262236dc4ec3f2748851": " L \\or \\neg K_1 \\or \\cdots \\or \\neg K_m ", "31c745b04a17e235cb02d09bd30726d0": "i=0,\\ldots,n\\ ", "31c746a8a4715b0e0c0514cfb280355d": "\\scriptstyle(\\Omega,\\mathcal F,\\mathbb P)", "31c75b9efeb18d6e773339a2aa5f8ba3": "\n\\frac{\\partial}{\\partial t}\\left(\\nabla^2\\psi\\right)-\\left[\\left(\\nabla\\psi\\times \\mathbf{\\hat z}\\right)\\cdot\\nabla\\right]\\nabla^2\\psi =0\n", "31c77ee281804697804c2eefd1b18418": "h'_{0}", "31c7d91e5aeaff39d1983a6f2f67ed5e": "\\rho(A,x) = \\max_{\\alpha, \\|\\alpha\\|_2 \\le 1} \\ \\min_{y \\in A} \\ \\sum_{i~:~x_i \\neq y_i} \\alpha_i", "31c7e9a5afcad22d81abbbb7d15c1cfe": "r>r'", "31c80ef69b7967e73cd3b15d05a9b440": "\\nabla \\cdot \\mathbf{F} = \\nabla \\cdot (-\\nabla U) = -\\nabla^2 U = 0.", "31c8238472bcb473feb28c960cd94226": "\nx = 1+\\frac{1}{1+x}.\\,\n", "31c8239b8755afe07707ab8d16583508": "\\nabla \\times \\mathbf{E} = -\\dfrac{\\partial \\mathbf{B}} {\\partial t} = 0", "31c82d470a8b24b8f714157bbd0b81fc": "\\mathbf{r}_\\parallel = r_\\parallel \\dfrac{\\mathbf{v}}{v} = \\left(\\dfrac{\\mathbf{r}\\cdot\\mathbf{v}}{v}\\right) \\frac{\\mathbf{v}}{v}", "31c872b28087b7e82a7c4954baa5bea0": "f_{in}/2", "31c928d0e7454de82e02c7ef77d7e2e7": "S \\times \\{1\\}", "31c94c106677e2ed3fd751afbe79dea6": "\\int_{0}^{t_1} E(t)\\, dt", "31c97374e7d0e464cdef7d254758ddbc": "{m \\choose r}_q=\\frac{[m]_q[m-1]_q\\cdots[m-r+1]_q}{[1]_q[2]_q\\cdots[r]_q}\\quad(r\\leq m),", "31c97f55e27c6cd70cc572c176492156": "C\\!", "31c9a71f3b0dc15afe1fb1c56ac1924f": "\\alpha < \\beta \\,,", "31c9aa7e7455bf8d7d07c5501b8d65d0": "[a_i,a_i+\\lambda_i)", "31c9c4abd29a76e218d66551d3f3a0b4": "H*", "31c9cee0c6834bad86f756ed3280b38d": "\\overline{CD}=\\frac{\\overline{EC}\\cdot\\overline{BD}-\\overline{ED}\\cdot\\overline{BC}}{\\overline{BE}}.", "31c9e510be6ebba6a8497ad4366a7dba": "\\mu/(sh)", "31ca2c4e1a40d2d9026dbfc294acc76c": "u_x^2", "31cac3152b03d0401d23864f3e5b2b47": "\\mathcal{S}_{\\alpha}", "31cb398fccc2524e4e1b66c4a7cc7806": "\\vec e_\\text{1,2,e}", "31cba769dc426d04e3a3aac55864f305": "\\frac{f^{(n)}(0)}{n!}", "31cbdfb081e3c0a4650b47b934085305": "\\operatorname{cr}_H", "31cbefda72a70964df0b731e5cdab2be": "\nt_{{\\rm max}} = {1\\over b}\\ln \\left({a} \\right).\n", "31cc2302a79d43cc1f91e8a8385edad1": "2^\\ell", "31cc40b06c525c29e9a006dd5fb6a1c8": "\\theta_m ", "31cc50be532b4ff430e6dc282df7e705": "\\mathbf J \\cdot\\mathrm{d}\\mathbf S = \\int J dS \\cos\\theta = I.", "31cccc8340e15e692a4bce429e723143": "A_1, A_2, A_3, \\ldots \\vdash B_1, B_2, B_3, \\ldots", "31cd39ad9cfcae4b69912444ef93c1a4": "\\zeta s_2", "31cd3a1782b6e9ae8c662ca30cc7befd": "\n\\begin{align} &\nn^2 H_n^2 - n + \\sum_{k=1}^{n-1} \\frac{k^2}{(n-k)^2} =\nn^2 H_n^2 - n + \\sum_{k=1}^{n-1} \\frac{(n-k)^2}{k^2} \\\\ = & \\;\nn^2 H_n^2 - n + n^2 H_{n-1}^{(2)} - 2 n H_{n-1} + (n-1).\n\\end{align}\n", "31cd5f982f5b431afc8dc9bfe77c4598": " -\\frac{\\mathrm{d}}{\\mathrm{d}x}\\left[p(x)\\frac{\\mathrm{d}y}{\\mathrm{d}x}\\right]+q(x)y=\\lambda w(x)y, ", "31cd755aca779a3b7dd8a5664c85e5ba": "|\\langle x|\\psi(t_1)\\rangle|^2 = |\\langle x|\\psi(t_0)\\rangle|^2\\quad", "31cdf9bc83df9d2f60b8b5a6448f2e4a": "= m\\tilde{\\boldsymbol{a} } ", "31cebd3702a8460b788eea72536235cc": " Y(x_1,\\dots,x_n)=a_0+\\sum\\limits_{i = 1}^m a_i f_i", "31ceeb448945f07c7dbb35beeb18b7c5": "\\{0, 1, 2, 3, 4, 5\\}", "31cf0d45fd80d722cf84d9d882e2ed58": "\\operatorname{recc}(A) = \\{y \\in X: \\forall x \\in A, \\forall \\lambda \\geq 0: x + \\lambda y \\in A\\}.", "31cf165e1be8255a24f07c23ecbba9e5": "\\scriptstyle{\\simeq 1}", "31cf1c073e06976935c2dcfa3007d78f": " \\mathbf{s}(x)", "31cf31d1df82a3a303fad634bdddc2f7": "\\bigwedge\\Gamma\\to\\bigvee\\Delta", "31cf5d0de7a1ff1bf057e8c31117cdc2": " V_{\\alpha} \\,\\!", "31cff1f0a3d31f235ecc605a12c4c804": "W_{21} > W_{32}", "31d00452577682ee7bdde695b1d6b752": "I = I_1\\times I_2 \\times \\cdots \\times I_n", "31d03e96c143957ba8d870961c1bb2cd": "l_{i,n}", "31d0a1b709a4b6fe6c56d0b9c1d7e75f": "\\textit{SENTENCE}", "31d0c211480143a2a3b2e6370db30762": "w_e", "31d101e2bcb900aa0d17946ed85adf02": "\\frac{n(n+1))5n-2)}{6}.", "31d1498ac42063bd1603d7aaa89585cf": "p_{2}\\Psi =\\left( \\frac{\\varepsilon _{2}}{w}P-p\\right) \\Psi ", "31d16cdcaf60309fd3fe15617ba1d3b1": "\\psi(x,t) \\mapsto \\psi_{[v]}(x,t)=\\psi(x+vt,t)\\; e^{-iv(x+vt/2)}.", "31d1839bc6b46f49327d9a17271a6afb": "d=3 ", "31d1d479a347e250699c84d7e31176d4": "(A \\oplus C)", "31d1dcbcabd20c912348e4d5094bdf8c": "\\mathrm{res}^{|\\partial_j \\sigma|}_{|\\sigma|}", "31d205bdc68d129a7f8400c5725e8bd9": "\\tan\\frac{\\pi}{3}=\\tan 60^\\circ=\\sqrt3\\,", "31d21142c91658907272c0af1ded09cb": "\\mu + 1/\\lambda", "31d21c6a624138677d40cc74071e1de8": "\\displaystyle M_{1}", "31d251b9aa35da821bda371964bdc4ce": "\nS = \\frac{1}{2} \\sum_{k=1}^{N} m_{k} \\left( \\mathbf{a}_{k} \\cdot \\mathbf{a}_{k} \\right)\n= \\frac{1}{2} \\sum_{k=1}^{N} m_{k} \\left\\{ \\left(\\boldsymbol\\alpha \\times \\mathbf{r}_{k} \\right)^{2} \n+ \\left( \\boldsymbol\\omega \\times \\mathbf{v}_{k} \\right)^{2} \n+ 2 \\left( \\boldsymbol\\alpha \\times \\mathbf{r}_{k} \\right) \\cdot \\left(\\boldsymbol\\omega \\times \\mathbf{v}_{k}\\right) \\right\\}\n", "31d26629338e85ac246190a8d6d9013b": "V_D = 600\\,\\mathrm{mV}", "31d28100e40e039cd36ade6df908fe92": "\\mathfrak{g}=T_eG\\cong \\{\\hbox{left-invariant vector fields on G}\\}.", "31d2aa4a0a124b74093d012254353767": "V_0\\times V_1\\times \\dots\\times V_k,", "31d2bc5ebde6c65b39a544bb84f1f02c": "\nV=\\frac{1}{L}\\frac{\\partial L}{\\partial\\theta} = \\frac{A}{\\theta}-\\frac{B}{1-\\theta}.\n", "31d2d38862f6ab575396c0bb439b617c": "TP = 2 \\pi \\sqrt{a^3/\\mu}\\,", "31d2eded7004728637911506b98234de": "B_{n}^{(-k)}=\\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},", "31d317c9de94f63c10c4f645b02d872d": "\\tfrac{2^{4031399}+1}{3}", "31d3621cb0877fd772780f47ec88ee86": "\\pi_2(M^3)\\ne 0", "31d385102919a84ca59f292f9092e036": " \\begin{align}\n\\|A\\| &= \\max\\{\\|Ax\\| : x\\in K^n \\mbox{ with }\\|x\\|= 1\\} \\\\\n&= \\max\\left\\{\\frac{\\|Ax\\|}{\\|x\\|} : x\\in K^n \\mbox{ with }x\\ne 0\\right\\}.\n\\end{align} ", "31d3eef820ff8620612d966650ff1e05": "|p_g\\rangle=g|p_0\\rangle", "31d460b42f98db32adcd4d3bca91d376": " Q = \\frac{1}{(1/Q_L + 1/Q_C)} ", "31d4852a9932c936561b880635bc5084": "r_p=r_{p+ik}", "31d4c9591335569acce5a604d01c77e6": " I(X;Y) = \\sum_{y \\in Y} \\sum_{x \\in X} p(x,y) \\log \\frac{p(x,y)}{p(x)\\,p(y)}, ", "31d4f796d58dc1fb2ae80657aa1594a0": "\\,\\mbox{T}(a + da) = \\mbox{T}(a) \\mbox{T}(da) = \\mbox{T}(a)\\left(1 - \\frac{i}{h} p_x da\\right) \\Rightarrow", "31d515fddf1b4b26b4551dbfb7eaa4d4": "\nV(t)=V_0 e^{-\\frac{t}{RC}} \\ ,\n", "31d558c279964ee427911af484aa2cc1": "S[k] =\\frac{1}{P}\\cdot S\\left(\\frac{k}{P}\\right).\\,", "31d59c4912a6f5803bc1c91aed4cdfe0": "\\mathbf{x_1}+\\mathbf{x_2} \\in \\mathbf{C_1}", "31d5ab24699b204d5f86784de41202b1": " H_n = \\int_0^1 \\frac{1 - x^n}{1 - x}\\,dx. ", "31d5b353a842799f158bfd4ce8a66529": "u_1 < u < u_2", "31d6274517076f2254efa138f3a3bc80": "\\mathbf{U}, \\mathbf{V}", "31d63a3bd9bd62356cdc14df15d2495c": "A+\\delta A", "31d6865867f28b9e97ea1f6f2dc97fd1": "V^a:=-T^a_b l^b", "31d68fecbd44fc7336cc45611dfff885": "\\frac{132,450 \\mbox{ MWh}}{(8760 \\mbox{ h/yr}) \\times (63 \\mbox{ MW})} \\approx{24%}", "31d6cce0f4d343f11a386d6640d385a6": " \\mu + \\sigma \\sqrt{{2 \\ln (k+1)} \\over {k+1}}", "31d70db222a258bdbd93e4cd2f203088": "\\boldsymbol{\\eta}", "31d712e2b2c0b7bf9a4e7fcb887c1c3a": "a_1, \\ldots, a_n", "31d715fe5b2120e296a03d74661882e3": "\\sigma_{fail}=K_{IC}/\\sqrt{\\pi a}", "31d769d645c3067a305bc6076276fa5b": "\\mathbf{j}(\\mathbf{r},t) \\cdot d\\mathbf{S}", "31d7b98931703da7eae997c9b152e6b2": "\\arctan\\left(\\frac{1+\\alpha}{1-\\alpha}\\cdot\\tan\\left(\\frac{x}{2}\\right)\\right)\\,\\bigg|_0^\\pi=\\frac{\\pi}{2}\\,", "31d808aa395f0f68aee53e2de2bc6272": "\\Delta(c^1,c^2)", "31d82f6bcee3ecc465c3476bfa6469a5": "F(s) =\\int_0^{\\infty} e^{-st} f(t) \\,dt", "31d87cca9556e101f919619a8813db68": " \\operatorname{E}\\,\\hat\\sigma^2\n = \\tfrac{1}{n}\\operatorname{E}\\big[\\operatorname{tr}(\\varepsilon'M\\varepsilon)\\big] \n = \\tfrac{1}{n}\\operatorname{tr}\\big(\\operatorname{E}[M\\varepsilon\\varepsilon']\\big)", "31d8c6b2d27b30af5abe3866d5c7496b": "(J^3_0f)\\circ (J^3_0g)=-\\left(x-\\frac{x^3}{6}\\right)-\\frac{1}{2}\\left(x-\\frac{x^3}{6}\\right)^2-\\frac{1}{3}\\left(x-\\frac{x^3}{6}\\right)^3\\ \\ (\\hbox{mod}\\ x^4)", "31d8e4792d862a82d4f0a374b0207826": "A_{s,\\chi}", "31d8f4f7bdf9a2cd1cc5fc48bbc26610": "\\psi_L = \\langle L|\\psi\\rangle = \\frac{1}{\\sqrt{2}}(\\cos\\theta\\exp(i\\alpha_x) + i\\sin\\theta\\exp(i\\alpha_y)).", "31d912dec1d3d6298fd00ff64ecbe3a9": "\\oint_C \\mathbf{B}\\ \\boldsymbol{ \\cdot}\\ \\mathrm{d}\\boldsymbol{\\ell} = \\mu_0 I_D \\ .", "31d939c83e055cdb32d64f40c92807c0": "m(z)= i \\cdot \\frac{1+z}{1-z}", "31d93fa0bd536ba6d3020c56f296da46": "\\frac{dE_{r}}{dx}", "31d94cdc52ad7a2a97136f0c938f8bb4": "f(c x) = a(c x)^k = c^k f(x) \\propto f(x).\\!", "31d94fd0ad62434adadfa4657e7b7060": "P(v\\in K)=p^n. \\, ", "31d9671c33809c804d097dd40c216643": "\\sum_{i=1}^n w_{ni} = 1", "31d97523cc3eaceb2a51856176fd4a3c": " \\rho = \\rho^{123}, \\sigma = \\rho^1\\otimes \\rho^{23}, T= 1_{\\mathcal{H}^{12}}\\otimes Tr_{\\mathcal{H}^3}", "31d9e008ed225531f9a73de91d82d044": "(dG)_{T,P} = -\\sum_k\\mathbb{A}_k\\, d\\xi_k \\,.", "31d9fcbd00c77f25ece1fbe75ecb9c7f": "W_{A \\rightarrow B} = \\Delta F ", "31da23e3c8fc04b9cafc31b2b5b4d9d5": "(\\gamma A)^\\circ = \\frac{1}{\\mid\\gamma\\mid}A^\\circ", "31da69555962a4c419ce7bb6b094d6df": "0.\\overline{307692}", "31da6f87f19d9cd2264061a0afc2cbb1": "(A,B)", "31da9ab1eb0627cdf7f6fe4670d1a463": "\\int x^2u\\;dx = -\\frac{x}{4} u^3+\\frac{a^2}{8}(xu+a^2\\arcsin\\frac{x}{a}) \\qquad\\mbox{(}|x|\\leq|a|\\mbox{)}", "31daa41998be6227e923d5aea91f1218": "FC = V - (N + {B \\over 2}) ", "31db3965c96f2106e0648673e2574631": "\\nu_{hi}", "31db5c73c911cf22834f23b046096cc4": "C^\\infty(X) ", "31db6c6a7f22dcbe9e7d03ea68320c98": "\\langle Q, \\Sigma, \\Gamma, \\delta, q_0, F \\rangle ", "31db6d13c2a23d7adb0634eba50677c0": " C(a,b) = { {(a+b)^2 - 2ab } \\over {a+b}} = {{a^2 + b^2} \\over {a+b}} ", "31db718ec7641ef4526998cf11b45dbe": "\n\\begin{align}\n& \\frac{a}{p} = [0.\\overline{a_1a_2\\dots a_\\ell}]_b \\\\[6pt]\n& \\Rightarrow\\frac{a}{p}b^\\ell = [a_1a_2\\dots a_\\ell.\\overline{a_1a_2\\dots a_\\ell}]_b \\\\[6pt]\n& \\Rightarrow\\frac{a}{p}b^\\ell = N+[0.\\overline{a_1a_2\\dots a_\\ell}]_b=N+\\frac{a}{p} \\\\[6pt]\n& \\Rightarrow\\frac{a}{p} = \\frac{N}{b^\\ell-1}\n\\end{align}\n", "31dbf570e685055246f35e6b6833a4b9": "C\\subset\\mathbb{P}^2", "31dbfb7517ebbabe2906c898d5f06a06": "(0,0),(-b_1,a_1)", "31dc18b755c83946f4d371451a590abe": " v=|\\dot{\\mathbf{X}}|=\\sqrt{\\dot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}}.", "31dc20062ad221ea46689bfaf81df3c8": "E(X)=k", "31dc959999b025bb9d08d108ac679cf0": "V_\\mathrm r = \\mathit \\Gamma V_\\mathrm {iL}e^{-\\gamma x}\\,\\!", "31dce583003943cd8898a4233bcb3342": "\\star \\Box A = \\mu_0 J ", "31dd0e8c6a9d05024490cb4ec117c223": "\\gamma\\geq 0 ", "31dd1411c20b27b708048cba65b3e91e": "\\ D_{pitch} ", "31dd3d31419387f2f4576097abf48deb": "S, T:\\mathcal A \\to \\mathcal C", "31dd55b8a633f63565e491e3ca81b00d": "x(t) = \\frac{1}{2}a_0(t) \\ + \\ \\sum_{n=1}^\\infty \\left[r_n(t)\\cos \\left( 2 \\pi \\int_0^t f_n(\\tau)\\, d\\tau + \\varphi_n(0) \\right) \\right]", "31dd9dcc0adaf2e36dd2735f87a74322": " P = \\frac{1}{V}\\frac{m}{\\mu m_\\mathrm{u}} kT = \\frac{k}{\\mu m_\\mathrm{u}} \\rho T .", "31dda5de91617af9fcdc0d325388a2a4": "{\\Delta p}_D = 0.0050 \\,", "31de5b0b28710a285ef2f33449108f71": "E(\\nu )\\propto k^2", "31de65407c803fb3532516053cdebe2f": " u(X^1, X^2) = X_1^a X_2^{(1-a)}\\,", "31def1497dcc3a6379c98f570f848f8f": "d_r^{(k)}", "31df5ed3b8f99cdc0db0d1823243d7ad": "\\, \\! T=2\\ln(3)RC", "31df77552f09d5e6194dbbd6a710bd6c": "\\mathrm{NA}=\\eta\\sin\\theta", "31dfc48d80b5a46f984aaa9fafa2c1f3": "\\langle\\Phi\\vert(H-E_{0})e^{S}\\vert\\Phi\\rangle = 0", "31dff7cb3efc3648a580fc05c4284e87": "sim(q_{or},d)=\\sqrt{\\frac{w_1^2+w_2^2}{2}}", "31e02c24a80c250578d0a321d0a87228": " N/2", "31e0492fd1481f1bf829d80d1f704d55": "{\\tilde{B}}_2", "31e04ffb7afaf8ad707155f5e5fbb46b": "n: \\mathbf Z \\to \\mathbf Z", "31e0762f24b20c267755538901bd1be1": " \\delta:R\\to R ", "31e0b254cdfe97fbdbd8a963699b3b48": "X \\subset \\mathbf P^n", "31e0c45fb614cf383f08a0ef65073c8d": "T^{-1}(|\\{0,1\\}^A|)", "31e0ed50a71b8499a3ea0ecc13c2a19a": "\\{O_{3},O_{7},O_{10}\\}", "31e18a4993b5ad2b0d5a567c815a2f2e": "Q(x,y)", "31e23ea51f87563bf4a59d0940f23023": "\\sqrt[3]{x}_s", "31e2dee685f4275d04f4f15ee3ba5d94": "V_{EDL}= \\pi \\varepsilon_0\\varepsilon_ra_p\\bigg\\{2\\psi_p\\psi_cln\\bigg[\\frac{1+exp(-\\kappa h)}{1-exp(-\\kappa h)}\\bigg]+(\\psi_p^2+\\psi_c^2)ln \\big[1-exp(-2 \\kappa h)\\big] \\bigg\\}", "31e2f0a9a89044793712b17fa6be7f5e": " \\Pi_{xy} = \\sum_{i}\\vec{e}_{ix}\\vec{e}_{iy}\\left[ f_i^{eq} + \\left( 1 - \\frac{1}{2 \\tau} \\right) f_i^{(1)} \\right] \\,\\!", "31e3147800f15210be48a251e4d71298": "\n{\\rm supp}\\,\\phi\\ast \\psi\n\\subset\n\\mathop{\\rm supp}\\,\\phi\n+\\mathop{\\rm supp}\\,\\psi\n", "31e342b8d09355c1fd7f478fe970f735": "T'\\,", "31e375fa6d887b7f6c613a183347c571": "\\scriptstyle \\mathrm{distance\\ in\\ parsecs}=\\frac{1000}{\\mathrm{parallax\\ in\\ milliarcseconds}}", "31e387a4e6015e8bd24a576813b38b6a": "\\sin\\theta\\approx\\theta", "31e3a6d8140af8b51118f1a468120663": "\\begin{matrix} {13 \\choose 2} \\end{matrix}", "31e3e0645832fe2d572c0b41f00dd796": "M_x=\\int^z_0 \\rho u dz,\\,", "31e3e22e87623ab1f99a8141d5d69ba0": "\\int\\left(\\int (x+y) \\, dx\\right) \\, dy", "31e44d38e12c5621475d65cb097974a0": "(f(x)\\cdot b)'=f'(x)\\cdot b", "31e45ddc3cd0f22f53b790525f9cd761": "f^\\sharp(x) = \\sup_{x \\in B} \\frac{1}{|B|} \\int_B |f(y) - f_B| \\, dy", "31e50445d285488055f45ba20b024516": " \\gamma=dx^\\lambda\\otimes (\\partial_\\lambda +\\Gamma_\\lambda^m\\partial_m + (A_\\lambda^i +\nA_m^i\\Gamma_\\lambda^m)\\partial_i) ", "31e551d6cd65406b37747ff400e0ffd7": "E=\\frac{\\hbar^2 |\\vec{k}|^2}{2m}", "31e56c49a2d81757c478c59fafdc3f31": "f(x_0 + h)", "31e58836f95754f3d8546ae29e4cfa4a": "{\\nabla}^2 \\phi = - {\\rho\\over\\varepsilon_0}.", "31e5a0690bbe89e3f26e87369d11d4f1": "\\tau_R=\\frac{\\mu_0 a^2}{\\eta}", "31e5acc78e6fff18b17884ab508c8238": "X'^2 - aY'^2= P^* (T') Z'^2 ", "31e5bd8aeab553eda6f8656072f278c0": " \\lim_{\\tau \\to 0}\\frac{f(x_0 + \\tau v) - f(x_0)}{\\tau} = f'(x_0) v", "31e5eb9b5796bfab036471aa5df46567": "\\left[\\delta_{\\epsilon_1},\\delta_{\\epsilon_2}\\right]\\mathcal{F}=\\delta_{\\left[\\epsilon_1,\\epsilon_2\\right]}\\mathcal{F}", "31e6179ef8e6aee7bd1f26391fc7e692": "[K_{i+1} : K_i]", "31e6396a5835607da74fbf508e667a41": "in(e)", "31e677ff5b00cd0b1ec7a0882f8a7045": "\\pi_i=q_1^{a_1}\\,q_2^{a_2}\\cdots q_n^{a_n} \\, ", "31e68c68f966044ab7e6b9922656c572": "I_{n,m}= I_{m-2,n-1}+a^2I_{m-2,n}\\,\\!", "31e69a3fe1e95d744e8ec2cded5a66ad": "(b^2-4ac),", "31e6aa03268403b16581e76161cf9fa3": "F(\\alpha,k-1,N-k)", "31e710e39e8a35e17361a35abf938111": "\\sum_{n=1}^\\infty \\left\\vert a_n \\right\\vert", "31e74d9bac73985725119227002829dc": " = \\sum_{k_1+k_2+\\cdots+k_{m-1}+K=n}{n\\choose k_1,k_2,\\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\\cdots x_{m-1}^{k_{m-1}}\\sum_{k_m+k_{m+1}=K}{K\\choose k_m,k_{m+1}}x_m^{k_m}x_{m+1}^{k_{m+1}}", "31e768271a6c7b9abc014c998b691f31": "tI", "31e7a4b28ad3df86b847b450c793f6b7": "\\scriptstyle f_c", "31e7c1a1be22afbdad604da0bd3a167c": "\\psi^*(t) = \\overline{\\psi(-t)}", "31e7f7ec0f010799c88a7324f6073531": "C_m=\\frac{1}{(1.52) \\omega Z_0} = \\frac{1}{(1.52)(2 \\pi f) Z_0}", "31e80fd40662db30040f73a9259c762a": "d\\mathbf{y} = \\mathbf{a}\\,dx", "31e8abd3e893772e301cfca823b41930": "\\frac{UC^3}{OC^3}=\\frac{BC}{CA},", "31e917ea7dfa9193e279a9412c27cbf2": "\\left(\\frac{p}{q}\\right)", "31e96ed97290d5cb748097b6a33fc061": " a=[1,8] ", "31e99a7ea4b71ebf536ac6164b234076": " Sp(2m,\\mathbb C)\\times SO(3,\\mathbb C) \\quad\\mathrm{on}\\quad \\mathbb C^{2m}\\otimes\\mathbb C^3", "31e9b37ce56117ceae303ef5b57c05c0": "\\begin{align} g_{ij}(\\beta) \n& = \\langle \\left(H_i-\\langle H_i\\rangle\\right)\\left( H_j-\\langle H_j\\rangle\\right)\\rangle \\\\\n& = \\sum_{x} P(x) \\left(H_i-\\langle H_i\\rangle\\right)\\left( H_j-\\langle H_j\\rangle\\right) \\\\\n& = \\sum_{x} P(x)\n\\left(H_i + \\frac{\\partial\\log Z}{\\partial \\beta_i}\\right)\n\\left(H_j + \\frac{\\partial\\log Z}{\\partial \\beta_j}\\right)\n\\\\\n& = \\sum_{x} P(x)\n\\frac{\\partial \\log P(x)}{\\partial \\beta^i}\n\\frac{\\partial \\log P(x)}{\\partial \\beta^j} \\\\\n\\end{align}\n", "31ea1ea3c5474d5ac60cfa0d6a997260": "\n S(\\boldsymbol{\\eta}(x)) = {x}_1^2 + \\cdots + {x}_r^2 + \\sum_{i,j = r+1}^n {x}_i {x}_j W_{ij} (x).\n", "31ea319361d1b03c9f77c6fbdb6833bd": "S\\!", "31ea5778129c6b5a68177d33a6e62374": " \\lambda_\\mathit{C} \\varpropto {(\\mathit{x}_\\mathit{e} \\mathit{n}_\\mathit{b})}^{-1} ", "31ea810c3f8fb26f7f8faeeb53413b7b": "\\sum_{y\\in A} |\\langle x,y\\rangle|^2 \\le \\|x\\|^2.", "31eb275ba29b9ee4c03f7a90415ae6b4": "\\tau_\\mathrm{oct}\\,\\!", "31eb3eed62cd1e23e61bf0544e30766e": " \n\\begin{align}\n\\bar{X_i} &= \\frac{1}{n_i} \\sum_{j=1}^{n_i} X_{ij}, \\\\\nA_i &= \\sum_{j=1}^{n_i} (X_{ij} - \\bar{X_i})(X_{ij} - \\bar{X_i})', \\\\\n\nS_i &= \\frac{1}{n_i - 1} A_i, \\\\\n\n\\tilde{S_i} &= \\frac{1}{n_i}S_i, \\\\\n\n\\tilde{S} &= \\tilde{S_1} + \\tilde{S_2}, \\quad \\text{and} \\\\\n\nT^2 & = (\\bar{X_1} - \\bar{X_2})'\\tilde{S}^{-1}(\\bar{X_1} - \\bar{X_2}).\n\n\\end{align}\n", "31eb45c6ab0fac8755e76e4af6203280": "m_{1,2}(\\varnothing) = 0 \\, ", "31ebd9d9679d559153864af3fec7ff3e": "{R_2 \\over R_1} = {\\mbox{cell voltage} \\over V_\\mathrm{S}}", "31ec0b5c4dae8c6c1a4df5ac50689f21": "\\mathbb{Z}_k^n \\to \\mathbb{Z}_k", "31ec3dba8191559c5efee4cb7ac1bc99": "x \\wedge y = y = y \\wedge x", "31ed3b2e01e4ae97e6df1f58e6ec57df": "\\operatorname{Id}(n)=n", "31ed654c8ed4eb265fd458cd7805a3af": "\n\\begin{array}{lllll}\n(z-1) & & \\\\\n(z-1) & - & \\frac{(z-1)^2}{2} & \\\\\n(z-1) & - & \\frac{(z-1)^2}{2} & + & \\frac{(z-1)^3}{3} \\\\\n\\vdots &\n\\end{array}\n", "31edb5b8c442344ad09bba2758bbd17d": "\n \\psi = - \\frac{1}{2}\\, V\\, r^2\\, \\left[ \n 1 \n - \\frac{3}{2} \\frac{R}{\\sqrt{r^2+z^2}} \n + \\frac{1}{2} \\left( \\frac{R}{\\sqrt{r^2+z^2}} \\right)^3\\;\n \\right].\n", "31ee2b5083c47b455adcfae32038bd9c": " \\mathcal{L}_p = C g^{\\alpha \\beta} \\acute{R}_{\\alpha \\beta} \\sqrt{-g} ", "31ee4c3600f35b1cf861f76ef2c8d814": "\\frac{\\beta }{z}\\,\\,\\, \\approx \\,\\,\\,\\frac{{b^2 \\sigma ^2 }}{{2\\,n}}", "31ee73f1a9fa7e3c3ec76bae7d1ad0f1": " R < C \\,", "31eee6f6d774d064239355581539e0b9": "S^n = \\mathbb{R}^n \\cup \\{ \\infty \\}", "31ef2affe045f9dd74b0f0bb08678b7c": "\\arg\\tilde{\\chi}(\\omega)", "31ef453d390c339cfe3c4bbace6d613f": "F(x,y)=\\begin{bmatrix} x^2 y \\\\\n 5x + \\sin(y)\n\\end{bmatrix}. ", "31ef7d1f7ed1f063b9f78f1f13e92d52": "S \\in \\mathcal{F}", "31ef9109ebcbe3e1560fde74a4138b74": "\\widehat{P1QP2}=\\pi", "31efa1706284cad57e0de433f85733a2": "w(a)", "31efa61a3c13e4cb3a789bac03d9a3d9": " \\displaystyle{U(s)V(t)=e^{-is\\cdot t}V(t)U(s).}", "31efacc2f9099e435540c861ea51dd2f": "\\Omega_{X/Y}=0", "31efcd1a455b4ea069b940acc1e11985": "\\mbox{normalized margin of victory} = \\begin{cases}0; & w \\le \\frac{c}{2} \\\\ \\frac{w - \\max\\{r, \\frac{c}{2}\\}}{\\frac{c}{2}}; & w > \\frac{c}{2} \\end{cases}", "31efd1c0a26f6e326cf016d512f07df8": "d \\Phi = d S - \\frac {1} {T} d U + \\frac {U} {T^2} d T", "31f051ad6b0d6bbf07d89ad6e4aa2f6b": "\n H_x=-\\frac{1}{j\\omega\\mu} k_x k_z \\sin k_x x \\cos k_y y \\cos k_z z\n ", "31f06a27ca5d896d11fca1e125d950aa": "AB^2 = CA^2 + CB^2 + 2 (CA)(CH)\\,.", "31f0b363b5d386537f6f190fdfd2070f": "2^N = k", "31f108b0fa43680227bd069360a63ff6": "h = h_{ab}.\\,", "31f138358287453c4d6aea0543e739a5": "\\mathbf{\\hat{t}}", "31f15b9cce20577b009a3634a4abf335": "\\phi_1,\\phi_2, \\cdots \\phi_m ", "31f15cfe1de9c1b47ea7ff973c784da4": "V = bT", "31f1612076965ca53f35c7731adef0b2": "\\scriptstyle N_0", "31f17ad22eb35ee1daee996c0a745ca4": "y_d", "31f19e9eaabc8b09613559ee29bcec8d": "(a,b] = \\{x \\in \\mathbb{R} : a < x \\le b \\}", "31f1c8453ec6d27b54205107e5b8aac8": "U(a,2a,x)= \\frac{e^\\frac x 2}{\\sqrt \\pi} x^{\\frac 1 2 -a} K_{a-\\frac 1 2} \\left(\\frac x 2 \\right),", "31f27d070a793f6b0a3220bc30acf761": " \\frac{d}{dx}\\left(u \\pm v\\right) = \\frac{du}{dx} \\pm \\frac{dv}{dx}. ", "31f2ae496e296e97027e920d682c4cef": "\\Pi = \\mathbf{E}^2\\backslash\\{\\mathbf{0}\\}", "31f2e5acb1f38ac3d154a24da322e257": " yxyxy = x^2 ", "31f2ea4c6d499a45cf1a8a9a283ff3bd": "\\textrm{pK}_{a}", "31f327f1a34ecd8ec98088f768f08710": "V_J", "31f3345723193f18e86b9d9b09e51115": "\\left\\{\\mathcal{M} f\\right\\}(s) = \\left\\{\\mathcal{B} \nf(e^{-x})\\right\\}(s) = \\left\\{\\mathcal{F} f(e^{-x})\\right\\}(-is).", "31f38786d4b3393b2ea63e30a29bb39a": "\\varphi_{ij} =\\phi_j \\phi_i^{-1}", "31f3b7e4ff8fc028bcfbf0ccd075c8bf": "\\displaystyle \n0=\\frac{\\partial}{\\partial t}\\left(\\nabla^2\\phi-\\phi\\right)", "31f3e49adc5bdf97512efb171c5eed6b": "\\beta < \\Gamma_{\\beta} \\,", "31f4a0f5b07bf2ce0ac50b3fd81b5612": "{}_{\\ 83}^{212}\\mathrm{Bi} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 81}^{208}\\mathrm{Tl}\\ \\mathrm{(61\\ m,\\ 0.78\\ MeV)}", "31f4af069f58d75df28b5703f86ce827": "\n u = \\frac{a_{n-1}}{a_n} = \\frac{11}{6} ; \n \\quad \n v = \\frac{a_{n-2}}{a_n} = - \\frac{33}{6}.\\,", "31f4cde4ac065a12e719b6b1f96b5e8f": "p_{2}=x_{21}+x_{22}", "31f4e6d8a572ce20bbf8022a08c8112d": "\\frac{d}{dx} \\int_0^x t^3\\, dt = f(x) \\frac{dx}{dx} - f(0) \\frac{d0}{dx} = x^3.", "31f4f3df08567d6879d7a447f0a09664": " \\phi_j(x_1) ", "31f52928747e9f263f1137248ad605f9": "\\Pi_k\\rm{P}", "31f58c685184304391022911dd686d6a": "\\operatorname{refl} \\mathbin{:} \\Pi_{a \\mathbin{:} A} a = a", "31f590634e64a5c67cbe07064710bb9a": "\\oint_{\\partial \\Sigma} \\mathbf{B} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = \\mu_0 \\iint_{\\Sigma} \\left(\\mathbf{J} + \\varepsilon_0 \\frac{\\partial \\mathbf E}{\\partial t} \\right)\\cdot \\mathrm{d}\\mathbf{S}", "31f5c82422a9900da1a2926bc6f5f622": "|x_2-x_1|=|f(x_1)-f(x_0)|\\leq L|x_1-x_0|", "31f616893fd4ec2b72767808d2006c85": "a=\\dot v =i\\dot \\omega z +i \\omega \\dot z =(i\\dot \\omega -\\omega^2)z", "31f68bb6fb5cf5ab01d63a8d5f532a39": "\n\\int \\chi^*_{nlm}({\\mathbf{r}})\\frac{1}{|{\\mathbf{r}}-{\\mathbf{r}}'|}\\chi_{n'l'm'}({\\mathbf{r}}')d^3r\n=\n4\\pi\n\\int\n\\frac{d^3k}{(2\\pi)^3}\n\\chi^*_{nlm}({\\mathbf{k}})\\frac{1}{k^2}\\chi_{n'l'm'}({\\mathbf{k}})\n", "31f6dfb65cfe40cbcb51570ea16096a6": "\\qquad \\sum^{\\infty}_{n=0}a_n = \\infty \\quad \\mbox{ and } \\quad \\sum^{\\infty}_{n=0}a^2_n < \\infty \\quad ", "31f7115b8f8c19ecd8efdb98e7b18b02": "I^p = 0", "31f76f4b66bfed4061d4465f2198adf3": " \\Delta Z = -n_0F_0 \\ , ", "31f882021f4c6944cbf17737566d3deb": " \\sum_{j=1}^n \\left(\\frac{x_j - \\mu}{\\sigma_v}\\right)^2 + \\left(\\frac{\\mu-\\mu_0}{\\sigma_m}\\right)^2.", "31f89150e01c7ced0816a76a81a5a9ba": "= [\\, a(x, \\sigma(x), \\sigma'(x)) + b(x, \\sigma(x), \\sigma'(x))\\sigma'(x) + c(x, \\sigma(x),\\sigma'(x))\\sigma''(x)\\, ]dx \\, ", "31f8ab7133455a04eb2cfffc476cd39c": "y=b\\sin (t)\\cos(t)", "31f8ef7af1c129ea068e60528b0206fb": " \\left [ \\hat{A}, \\hat{B} \\right ] \\psi \\neq 0, ", "31f95274a784a123d0654984d087adf4": "z = a + bh,\\quad a,b \\in R,\\quad h^2 = -1.", "31f97da1c75a8efb645811153aa5ffcc": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\int_{\\Omega(t)}\\omega=\\int_{\\Omega(t)} i_{\\vec{\\textbf v}}(\\mathrm{d}_x\\omega)+\\int_{\\partial \\Omega(t)} i_{\\vec{\\textbf v}} \\omega+\\int_{\\Omega(t)}\\dot{\\omega},\\,", "31f9a9c887743d4e6432fa652b082812": "\\scriptstyle IJ", "31f9f9e4366d2a2d3ec43a95aec10676": "2(x_2-x_1)x+2(y_2-y_1)y=r_1^2-x_1^2-y_1^2-r_2^2+x_2^2+y_2^2. ", "31fb1302455f1dc9ea14478547905e7a": "\\sum_{k=1}^n H_k = (n+1) H_n - n.", "31fb3f8e97bdd4f08e5794158180b42c": "\\pi^i", "31fbd7c1e84110e7bca6b6a4dfe3a65b": "\\begin{smallmatrix}A=4\\pi r^2 \\end{smallmatrix}", "31fc103ab54d69d6f01b7431ea746d95": "{\\theta}=\\operatorname{arccos}(z/r)", "31fc245a59b8707dc7cb2f646ef785be": "\\Delta_1 = 0,", "31fc25ce663d2b4d77e41e1fd904e11c": "z_0, z_1, \\ldots, z_{N}", "31fc82c113f560e09fba5f9a45bf6e3c": "\\langle 2,2,1,0,0\\rangle", "31fcf746eeda0d8f8e17097276ff6f70": "\n\\mathcal{L}={i\\over2}\\left(\n\\bar{\\psi}\\gamma^\\mu\\partial_\\mu\\psi-\\partial_\\mu\\bar{\\psi}\\gamma^\\mu\\psi\\right)-m\\bar{\\psi}\\psi\\;.\n", "31fcfd153743359bbe367fb2ed2d9c51": "\\nu = \\frac{\\tau-1}{\\sigma d}\\,\\!", "31fd0a532df882518ede84350f77116b": "B_i\\subset\\{1,2,\\dots,n\\}", "31fd21072c7773694625c3785fc93b9a": "\\omega_{max}=2\\pi c/\\lambda_{min}", "31fd2e24dd94f4f5aebe11431413f141": "\\scriptstyle r(\\boldsymbol{r}_i,\\, \\boldsymbol{r}_{\\text{rec}}) \\;=\\; | \\Omega_{\\text{E}} (t_i \\,-\\, t_{\\text{rec}}) \\boldsymbol{r}_{i,\\text{ECEF}} \\,-\\, \\boldsymbol{r}_{\\text{rec,ECEF}} |", "31fdd4aacad47ad76dbebc2cac5395eb": "a = \\frac{T_0}{\\lambda_0 g}\\,", "31fdda4ef2168e0de3f5ec98675a1a56": "\\tfrac{|AC|}{|AB|}=\\tfrac{|AB|}{|BC|}=\\Phi ", "31fde3cdef241c625d6e71d9225e4556": "\\langle E\\rangle", "31fe1767f031d48a477cb9d5c963c7c5": "m = 3.4", "31fe3a0f66f6a073c3731327a84efe4e": "G_G", "31fe540b5a0afadd0d29f562c0784a0f": " SL(2,\\mathbb C)", "31ff082b80d03389026020eaada1e4c2": "\nR\\ln x_i = - \\frac{H_i ^\\circ }{T} + \\frac{H_i^\\circ }{T_i^\\circ }\n", "31ff1ffa63f316524d7b32d12f024936": "\\scriptstyle \\Gamma(1/2) = \\sqrt{\\pi} ", "31ff411aff5d8c4e82dee0a0de14a953": "\\sigma \\approx J_p^+(\\hat{x_0})\\delta p", "31ff4335fdac207eb1f6bf09b1a62303": "y=-x+(x_1+y_1)", "31ffbae11e8d4edaf5a06b1fa42b3d7c": "n^{}", "31ffed96e86c68ce2e4a10d546178763": "\\Omega_w", "32001540d2e0f100c534470fa201465b": "d_k:=(0,...,0,1,0,...,0)", "320054869872b88a5c9c90bccc5120d9": "n\\rightarrow \\infty", "320054975f9bab07554add65fdd16453": " D = 2h^3E/[3(1-\\nu^2)] = H^3E/[12(1-\\nu^2)]", "320061d6f5cbefd3b19ce83a5decbdcc": "m_j^i\\simeq P^iw_j", "3200f78f59aa4cec131601bacf781350": "\\left| \\psi \\right\\rangle", "3201037aa01a0e50c7d469f08739c59b": " \\sum_{m=0}^{3}{(-1)^m{3\\choose m}m^{2n}}=3 \\cdot 2^{2n}-3^{2n}-3\\equiv0 \\pmod 4 \\!", "32010a6b14d74054b8e6bdf2b9bab247": "\nV_{ij}(r_{ij}) = V_{repulsive}(r_{ij}) + b_{ijk} V_{attractive}(r_{ij}) \n", "320116c35104a6269f4b60d6cb247a88": "\\begin{align}\n\\boldsymbol{x}_i&=\\boldsymbol{x}_{i-1}+\\alpha_{i-1}\\boldsymbol{p}_{i-1}\\text{,}\\\\\n\\boldsymbol{r}_i&=\\boldsymbol{r}_{i-1}-\\alpha_{i-1}\\boldsymbol{Ap}_{i-1}\\text{,}\\\\\n\\boldsymbol{p}_i&=\\boldsymbol{r}_i+\\beta_{i-1}\\boldsymbol{p}_{i-1}\\text{.}\n\\end{align}", "3201b145548b8a270c8a56e4ac3e53dd": "X_T-X_0=\\int_0^T Y_{t} \\circ \\mathrm{d} W_t + \\int_0^T Z_{t} \\,\\mathrm{d}t", "3201bbc64bf06ac46c42356b24d163db": "\\sum_{a,b} P \\left ( { {a, b}{|}{A, B} } \\right ) = 1 \\quad \\forall {A,B}", "320207496c7ff5b21ede8c6a983e910f": " \\mathbf{H}_\\alpha(x) = \n \\sum_{m=0}^\\infty \\frac{(-1)^m}{\\Gamma(m+\\frac{3}{2}) \\Gamma(m+\\alpha+\\frac{3}{2})}\n {\\left({\\frac{x}{2}}\\right)}^{2m+\\alpha+1} ", "32020cd8255607e7b98467882171f378": "\n w^+ = \\operatorname{min}\\left\\{1, \\frac { 2n\\hat p + z^2 + [z \\sqrt{z^2 - \\frac{1}{n} + 4n\\hat p(1 -\\hat p)-(4\\hat p - 2)}+1] }\n { 2(n+z^2) }\\right\\}\n", "320237300ae2cd9d5f7d2dc0f4c411af": "{D}^i_{\\it nk}", "32027dd546f8fd6c5f2699e773509006": "\\tbinom n4", "320281f055f45d5af0398da0137fd9ee": "\n\\langle p \\rangle_V = \\frac{1}{2}(p - \\overline{p})\n", "3202d439a0a0f2a9732ed8da21bf4096": "\\kappa_6=\\mu'_6-6\\mu'_5\\mu'_1-15\\mu'_4\\mu'_2+30\\mu'_4{\\mu'_1}^2-10{\\mu'_3}^2+120\\mu'_3\\mu'_2\\mu'_1-120\\mu'_3{\\mu'_1}^3+30{\\mu'_2}^3-270{\\mu'_2}^2{\\mu'_1}^2+360\\mu'_2{\\mu'_1}^4-120{\\mu'_1}^6\\,.", "320355ce404f4f5b2f34cfe9c2af0cea": "s(\\xi) = \\sum_{k=0}^\\infty s_k \\xi^k ", "3203aec7b5ce82e5b13636e39a3d315e": "R = UTS/FS", "3203cb31f3d54f7b0b38e0afc58007a0": "\n \\begin{pmatrix}1&-h\\\\0&1\\end{pmatrix}\\frac{\\partial}{\\partial (q_0,p_0)}\\Phi_{{\\mathrm{sE}},h}(z_0)\n = \\begin{pmatrix}1&0\\\\-h\\cos q_0&1\\end{pmatrix},", "3203d3185206b53a61936e3c49a8c87a": "\\hat{\\theta}\\in\\Theta", "3203daff3ecd1703712dcbb11725d273": "\\Rightarrow_{X \\to Y}\\ YXXX \\ \\Rightarrow_{X \\to Y}\\ YYXX \\ \\Rightarrow_{X \\to Y}\\ YYYX \\ \\Rightarrow_{X \\to Y}\\ YYYY", "3203e214015828d66b57298cbb64035d": "h \\leq k", "320408f5b0a8fdd26ae4649cddd80500": "A(i\\omega)=\\frac{V_o}{V_i}=\\sqrt{\\frac{Z_{I2}}{Z_{I1}}}e^{-\\gamma}\\left[ \\frac{\\tau_{I1}\\tau_{I2}}{1-e^{-2\\gamma}r_{I1}r_{I2}} \\right]", "3204406dc4e0966ab055c22fefcd5360": " H_\\alpha = H_{\\alpha-1}+\\frac{1}{\\alpha}\\, ,", "320444536238202ef1ed7540d03fd818": "w=W(z)", "320454beb77259b7832f16754fcac3bf": "C(X,S)\\cong \\mathcal{T}(X)", "320470daba798e1d1d53761a5ec89a2e": " \\alpha = \\begin{cases}\n - \\frac{4-d}{d-2} & \\ \\mathrm{if} \\ 2 4 \\end{cases} ", "32047bc5bba768430b34dcd4ce4648b0": " 0^0 = 1 ", "320596e4adf19cbd066d9b3a2f677a90": "v_{incident}", "3205cf890178fc6881999839f637531e": "A_m(2,4) = 1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440,\\ldots", "320608a4411549442025190f0024ef04": "P_\\pi = \\begin{bmatrix} \\mathbf e_{\\pi(1)} \\\\ \\mathbf e_{\\pi(2)} \\\\ \\vdots \\\\ \\mathbf e_{\\pi(m)} \\end{bmatrix},", "32060c70b4b33c106794bd61ae85f2d3": " ed = K \\times \\operatorname{lcm}(p-1, q-1)+1 ", "32060e2ad882f5861f8e1029dbda441e": "\n\\begin{align}\n\\left\\langle \\frac{\\delta F[\\varphi(x)]}{\\delta\\varphi(x)}, f(x) \\right\\rangle \n&= \\int \\frac{\\delta F[\\varphi(x)]}{\\delta\\varphi(x')} f(x')dx' \\\\\n&= \\lim_{\\varepsilon\\to 0}\\frac{F[\\varphi(x)+\\varepsilon f(x)]-F[\\varphi(x)]}{\\varepsilon} \\\\\n&= \\left.\\frac{d}{d\\epsilon}F[\\varphi+\\epsilon f]\\right|_{\\epsilon=0}.\n\\end{align}\n", "320666167e419ff28bf99469397cedb0": "D\\{x[n]\\} \\ \\stackrel{\\text{def}}{=}\\ x[n-1]", "320674205bf4ef7a33f8202399aef988": "\\int_{\\mathbb{R}^3} \\vert \\mathbf{v}(x,t)\\vert^2 dx V_0", "3219ded4ef5ca7f8833d58577d797573": "dF(u;\\cdot):X\\rightarrow Y.", "321a0a8afb3f53040d1b24024997eb2c": "f(k;N,q,s)=\\frac{1/(k+q)^s}{H_{N,q,s}}", "321a13a1ebdfeea80f6ed66226a2be80": " B(\\mathit{G}) ", "321a2629f9f7ccdbdcecd9977d95d402": "x^3+y^3=3axy", "321a283cd527e0fab38c12704c692d74": " f_1, f_2, \\ldots", "321a35c70051345529fe904b6abf8691": "g_{ij} = {^{(4)}}g_{ij}\\,\\!", "321a83cb4f8143583db1451694365bab": "\\begin{align}\n\\mathbf u(\\mathbf X,t) &= \\mathbf x(\\mathbf X,t) - \\mathbf X \\\\\n\\nabla_{\\mathbf X}\\mathbf u &= \\mathbf F - \\mathbf I \\\\\n\\mathbf F &= \\nabla_{\\mathbf X}\\mathbf u + \\mathbf I \\\\\n\\end{align}\n\\qquad \\text{or} \\qquad\n\\begin{align}\nu_i& = x_i-\\delta_{iJ}X_J \\\\\n\\delta_{iJ}U_J &= x_i-\\delta_{iJ}X_J \\\\\nx_i&=\\delta_{iJ}\\left(U_J+X_J\\right) \\\\\n\\frac{\\partial x_i}{\\partial X_K}&=\\delta_{iJ}\\left(\\frac{\\partial U_J}{\\partial X_K}+\\delta_{JK}\\right) \\\\\n\\end{align}\n\\,\\!", "321b04f9b08ea05109ee153f82c0a78a": "d(Oj, Q)", "321b95e469a67cf838c9626b206d5ffb": "r_1 + r_2 - 1.", "321bb2cb4aa223b8d1323b65c374aaa8": "\\Delta E_i = \\sum_j w_{ij} \\, s_j + \\theta_i", "321bce255aec780828c51ea5ed2153f3": "10^{24}", "321d4c52a6f41c491d3c16be50f5fc7a": "|E(a, b) - E(a, b^\\prime)| \\leq 2 \\pm [E(a^\\prime, b^\\prime) + E(a^\\prime, b)],", "321d6bd204d42f42383fe2f44fbe7671": "\\{X_i\\}_{i \\in I}", "321df856d5716418058780e9cc0bb658": "\\int\\frac{\\sinh^m ax}{\\cosh^n ax} dx = \\frac{\\sinh^{m-1} ax}{a(m-n)\\cosh^{n-1} ax} + \\frac{m-1}{n-m}\\int\\frac{\\sinh^{m-2} ax}{\\cosh^n ax} dx \\qquad\\mbox{(for }m\\neq n\\mbox{)}\\,", "321e053cec894a96b7c3b66ee5dd3685": " \\| A \\| _{op} := \\sup \\{ \\|Ax \\| : \\| x \\| \\le 1 \\} ", "321e11bb36c10e96f343d08a60a1d2a1": "j \\in N_i", "321e147b175ebdbe5c8045d396e81a1c": "\\mathbf{x}_L", "321e2bdf585c0ec44c75ff357a920d0e": "f_{t'}", "321e2d0c12262ba9c2de328665d5852a": "\\delta \\sum_{j}\\gamma_j O_j = \\sum_{j}\\gamma_j \\delta O_j = 0\\,\\!", "321e41408e428ec6d922a2f3b7ef5125": "x^{n} \\in y^{n+1}", "321e5be4b81fd861e2aa5bbf4c619d62": "\\chi_{A} (x) = (+ \\infty) \\left( 1 - \\mathbf{1}_{A} (x) \\right).", "321e6f12baad2ef9d5cdea8995ff43d1": "C_2 + C_4 + C_6 + \\cdots\\,", "321e7721b16c15b248ac82fa7a217778": "f_B = \\frac{1}{|B|}\\int_B f", "321e893eb6f5231a88d9b4780672cdb5": "\\frac{[S]}{K_M} = W \\left[ F(t) \\right]- \\frac{V_\\max}{k_{cat} K_M}\\ \\frac{W \\left[ F(t) \\right]}{1+W \\left[ F(t) \\right]}\\, ", "321e8fff249b227f4d3a1d444913dee7": " \\alpha < \\beta ", "321ed30317fda7212eacc18993d6aabc": "BP=\\frac{R}{K_i}", "321ee795f434e81932936cbcd471f469": " \\tan \\theta\\!", "321f026389863e1303ab03f9d4f27a08": "{\\hat h}(i)", "321f5f4434262c0dd74e627df324e341": "{}_0^1 \\{ x \\}", "321f80322ef63e22634ea9531726f8c8": "\\Omega(M,E) = \\bigoplus_{p=0}^{\\dim M}\\Omega^p(M,E)", "321f93d9c67abf903b38f1f9175ef248": "\\displaystyle \\hat{f}(\\nu) \\hat{g}(\\nu)\\,", "321fb7d98204790629a633469a00755b": "\\theta_2 < \\theta_0", "321fef7a2aafaf175d69464a14fce1b2": "Y(k)", "321ff638d576f8661ad97d9dea0f063b": "\\alpha\\leq\\phi_3(0)", "3220521deaccab4cd1f1d216b27e71a9": "\n\\frac{A \\wedge B\\hbox{ prop} \\qquad A\\hbox{ true} \\qquad B\\hbox{ true}}{(A \\wedge B)\\hbox{ true}}\\ \\wedge_I\n", "322091167feadb36d50c27529ded002b": "\\Bbb{Z}_3", "3220ad9b76ee15a57a944fb9b5ba2272": "f_H = f_0 \\cosh \\left(\\frac{1}{n} \\cosh^{-1}\\frac{1}{\\varepsilon}\\right)", "3220c9c91791c29ab426395e5051707d": " \\rho = \\rho_\\text{f} + \\rho_\\text{b} ", "32212affcf7946254d6d2695583f7e22": "\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}.", "32214d971d91028ed81109ffc5c6139c": "l_1 + l_2 = m ", "3221a55ce696ae7e0769ee1a2b5db59d": "C^2_k", "3221a820cb6822556f32784cfe2fbd88": "\\operatorname{pos}", "3221d0aaea7c28e6e78171580a2b8733": "A e^{i \\varphi_n}", "32222c9b8708e16000de704abbf05b66": "\n\\mathbf{x} = \n\\begin{bmatrix} \\frac{1}{11} \\\\\\\\ \\frac{7}{11} \\end{bmatrix}\n", "322237f01f545887ac671033abee29cb": "EMV = BMV \\times (1+R)+ \\sum_{i=1}^n F_i \\times (1+R \\times \\frac{T - t_i}{T})", "32223ab9a60d9b1999748856a0d96c04": "\\tau^{a}{}_{b;c}\\equiv \\partial_c \\tau^{a}{}_{b}+\\Gamma^a{}_{c d}\\tau^d{}_b-\\Gamma^d{}_{c b}\\tau^{a}{}_{d}", "322241cfa8b14cf7c74febf04f51ed7b": "K(p_1,p_2,p_3,\\cdots) = Q(z_1)Q(z_2)Q(z_3)\\cdots", "322264fb3b287c837adde22eb69c4e61": "q = (\\sigma(n) - \\varphi(n))/4 + \\sqrt{[(\\sigma(n) - \\varphi(n))/4]^2 - [(\\sigma(n) + \\varphi(n))/2 - 1]}. \\, ", "32228ecbc477c8e67d7d890b8d006c6e": " A = CS_3 ", "3222c37e9f9804ddfd6b9aa78cd405f1": " D_{n, m}\n= \\frac{n!}{m!} [z^{n-m}] \\frac{e^{-z}}{1-z}\n= \\frac{n!}{m!} \\sum_{k=0}^{n-m} \\frac{(-1)^k}{k!}.", "3222f6bd8b90425412ecc9f96f658c65": "v(x,y) = ax+by+cxy+d", "32235a66041e86e3cc9d406927434dcb": "= (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_I + \\nabla_\\alpha (C_{\\beta I}^{\\;\\;\\; J} V_J) - \\nabla_\\beta (C_{\\alpha I}^{\\;\\;\\; J} V_J)", "3223850ac5bed269f62f4d59055165d0": "\\mathfrak{c} = \\aleph_1", "322425087fcc9d629592bad9ef52b011": "\n \\begin{array}{ll}\n \\dot{x} = Ax + b\\varphi(\\Delta\\theta), &\n \\Delta\\dot\\theta = \\omega^2_{free} - \\omega^1 + Lc^*x,\n \\\\\n \\Delta\\theta = \\theta^2 - \\theta^1. & \n \\end{array}\n", "32243df76d72bb735d1277bc8d210887": " P( | Y | < k ) \\ge 1 - \\frac{ \\sigma_Y^2 + \\mu_Y^2 }{ k^2 } \\quad\\text{ if }\\quad \\mu_Y ( k - \\mu_Y ) \\le \\sigma_Y^2 \\le k^2 - \\mu_Y^2 ", "32246d021ae59644424d9030d7bd2e76": " = 0.2 + 0.02 \\times 4\\,", "322500c7d2330c8e00bf1913209892e0": "\n\\left( \\frac{dr}{d\\tau} \\right)^{2} = \\frac{E^{2}}{m^{2}c^{2}} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( c^{2} + \\frac{h^{2}}{r^{2}} \\right).\n", "32258108b350f2c63fc2e7ba348b37d8": "\n(b_1 - b_q) \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1+1, b_2, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right) +\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1, \\dots, b_{q-1}, b_q+1 \\end{matrix} \\; \\right| \\, z \\right), \\quad 1 \\leq m < q,\n", "3225a038ac075a791dba1c1c11420ddf": "\\mathcal{L}=\\overline{\\psi} \\left(i\\partial\\!\\!\\!/-m \\right) \\psi + F\\left(\\overline{\\psi} \\psi\\right)", "3225a3aa44d4dff1e6cc0d70b199f54b": "(2n+1)\\times{(n^2+n+3) \\over 3}", "3225d5286a19aad8f41431625497f2c3": "\\nu^*=1", "32261407b8dded7dc2d35387dc3e9257": "0 \\le K' \\le 1", "32264acff3e1ab6fd089fb6f1605e934": "f(x;p)=p^x (1-p)^{1-x} I_{\\{0,1\\}}(x)", "32269bfc6018092928bbbbab54f25aa3": " P_{a \\leq \\varepsilon \\leq b} = \\frac{1}{\\|\\psi\\|^2}\\int_a^b d\\varepsilon' | \\psi(\\varepsilon') |^2 = \\frac{1}{\\|\\psi\\|^2}\\int_a^b d\\varepsilon' | \\langle \\varepsilon' | \\psi \\rangle |^2 \\,, ", "3226a2b84a1d7e821a8465c7f5e6e223": "\\left.\\frac{\\partial T_L}{\\partial x}\\right|_{x=0} > \\frac{\\partial T}{\\partial x}", "32270d045a4f997a6907fc2e6895816c": "r_1 = (-a) \\frac {\\sin [(1/q) \\theta_1 - \\theta_0 /q]}{\\sin [(1/q-1) \\theta_1 - \\theta_0 /q]}\\!", "32275dda14f7b86910e21a0b354f0197": "|0\\rangle_A\\otimes|0\\rangle_B", "3227801bcb8201f8faac2502933afe50": "\\frac{\\partial}{\\partial x_i}(f * g) = \\frac{\\partial f}{\\partial x_i} * g = f * \\frac{\\partial g}{\\partial x_i}.", "3227a0643b9d88d21b9b609ff0230979": "\\alpha\\|\\beta\\|^2", "3227cb10ce9cb1c3b0b6fa187702b9b3": " C_T=\\frac{k^T_{tr}}{k_p} ", "3227ef2f07da318b33b55baadf98694a": "\\pi_1: \\mathrm{TopCov}(X) \\to \\mathrm{GpdCov}(\\pi_1 X)", "32282f17334238e61d844730874a5265": "W_{2}^{A}(x,x)\\geq W_{2}^{B}(x,x);", "322869917bb87fc967a8720f95b1983f": "x_0 \\le x_1 \\le x_2 \\le x_3 \\le ...", "3228f25fbf5c86aa25fad261e3f7e066": "P_1= \\frac{1-(\\frac{q}{p})^{n_1}}{1-(\\frac{q}{p})^{n_1+n_2}}", "3228f7a2336878854aef134af8470844": "n=k^{1+\\epsilon}", "32291d864bda72ba3e5d71ec0dcffda5": "w \\models P", "322925d352b9350cc50e3473cf26eb39": "r = \\sqrt{\\frac{1+c}{1-c}}", "322944c66476ec02726fc056762d3afa": "P(Spam)", "32294c0764369f6909d17337e87c29f2": "\\eta = \\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix}", "322962df4bc866629723d9a9ff4a2392": "\\theta_1, \\dots, \\theta_n", "322984ceca6e8a9ecaaa689e8b79b032": "\n \\int_G | g \\rangle\\langle g| \\; d\\mu (g) = I_{\\mathfrak H}\n", "3229932f3c71a1e4edadd45ea3fd14d6": "S^{n-1} = \\{\\mathbf{x}\\in\\mathbf{R}^n\\,\\mid\\, |x|=1\\}", "322a0d24a578352020bf6ed5153836fb": "\n C_{ijkl}=C_{jikl}=C_{ijlk},\n ", "322a568bf60dd53e044d14beabd494a7": "\\mathbb P(\\theta_{k+1}=\\vec s|\\theta_k=\\vec r)=\\Lambda^1(s_1|r_1)\\prod_{\\ell=2}^N\\Lambda^\\ell(s_\\ell|r_\\ell,s_{\\ell-1})", "322aa853ad815e373cdd4353ec7150aa": "\\Gamma \\backslash \\mathrm{GL}_2^+(\\mathbb{Q}) / \\Gamma", "322ae6bc03543c8faf3a53a3b71d10a6": " y\\in\\mathbb R", "322aef055ad94097e8b8acfdf3ef0015": "\\sqrt{\\sum_{i=1}^k \\left(\\frac{X_i}{\\sigma_i}\\right)^2}", "322af76e7dc4025d43336ceefb317dc4": "G_M", "322b227441558f0ed48aba836d0caf6f": "w_{ij}", "322b4b40d0213403d6422c33a8fb9c42": "f \\sim g_1 + \\cdots + g_k", "322bc512ff51348444b19c7ce8113492": " \\sum_{j=1}^N \\sigma_j^2 = N ", "322be2beed7f4ad21b4f1b367813aa9d": "{\\hat{\\alpha}}(q, {r_{\\rm c}}) > {\\hat{\\alpha}}(q^\\prime, {r_{\\rm c}}).", "322c1e2d0b74b0a7e73c52e7bc2837be": "\\delta\\;", "322c22e1ffe9dfedf159853fa2a79b3a": "r_2(m_2)", "322c39c761dcbcbb554f942191e12aa7": "{\\to_G}^*", "322c97e8acab45dd45506cffa259ce24": "p(t) =\n\\tfrac{1}{2}\n\\begin{bmatrix}\n\n1 & t & t^2 & t^3 \\\\\n\n\\end{bmatrix}\n\\begin{bmatrix}\n\n0 & 2 & 0 & 0 \\\\\n-1 & 0 & 1 & 0 \\\\\n2 & -5 & 4 & -1 \\\\\n-1 & 3 & -3 & 1 \\\\\n\n\\end{bmatrix}\n\\begin{bmatrix}\n\na_{-1} \\\\\na_0 \\\\\na_1 \\\\\na_2 \\\\\n\n\\end{bmatrix}\n", "322caa5df538b00d14200b197279b486": "{-\\log} P(x)", "322cb1e0731a111e7540ce0dc50c508d": "-u_x", "322cb7ff2d1e6cef47a1f13b13b39ff7": "- W \\, \\Gamma^{\\delta}_{\\delta \\alpha} \\, \\mathfrak{T}^{\\mu \\dots}_{\\nu \\dots} \\,", "322db1a640c2746c2df143aa18a68af1": "\n\\begin{align}\nx(t) * h(t) &= \\int_{-\\infty}^{\\infty} x(\\tau)\\cdot h(t-\\tau) \\,\\operatorname{d}\\tau\\\\\n&= \\int_{-\\infty}^{\\infty} x(\\tau)\\cdot O_t\\{\\delta(u-\\tau);\\ u\\} \\, \\operatorname{d}\\tau,\\,\n\\end{align}\n", "322de4c79073f0afe09f597c1f7ab075": "1+i =(1+\\rho )\\times (1+\\pi^e)", "322e1e9ae4d5d1133a5b4768bc633295": "\\sum_{{\\sigma \\in \\sum_{k}}}S(i_{\\sigma(1)}, \\dots, i_{\\sigma(k)}) = \\sum_{\\text{partitions } \\Pi \\text{ of } \\{1,\\dots,k\\}}c(\\Pi)\\zeta(i,\\Pi)", "322e2d026c8f0b7e50fe8e189544ba11": "P_0\\equiv \\frac{L}{n} ", "322e337b5cd253f0e5c600501e9ae4ce": "e_1: LG \\to G", "322ec962935d405a5eaee90a7e6a1abc": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 41.0\\cdot 2.96)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 26\\cdot R_{\\bigodot}\n\\end{align}", "322ef63377ad2b52f53e838ac1302c8f": "K = C + (L - a E)^{2}", "322f304590b247f3b5bbb42cc3b8f59e": "\\gamma\\equiv\\sqrt{k^2-Z^2\\alpha^2}", "322f5460eb9b33392720dc13fc8095cb": "\\hat\\mu = 27.49, \\hat\\sigma = 4.51.", "322fb3b7ab4cdb8ce18f7214aef4f082": "y=(x-C)^2/4", "322fe870ca791a4838b3a121ae2569cb": "\\displaystyle{f_{x,y}(g)=(T_g^{-1}x,T_g^{-1}y),}", "323041a845c977f167a79d958d5cc427": " x^2 - p^* = cq \\,\\!", "32304e12846acbe8ffb13cb57ca5c656": "J_{XY}", "3230cc69d2eedd1ba071b9e225af8705": "E_{u,v}(\\tau,1) = -2\\pi\\log|f(u-v\\tau;\\tau)q^{v^2/2}|,\n", "3230d321d81cfb77d4e81e4fba3ecc70": "f(y) \\leq f(x) + f'(x)[y-x]", "32315894246bf8c8cb20fd1406f46569": "\\{x_1,x_2,\\ldots,x_n\\}", "3231699dce4ae9712725d37b4f2db7a5": "f(i)= \\boldsymbol\\beta \\cdot \\mathbf{X}_i,", "3232103b34c371deb19c7cb3c30b7edd": " U(s) \\subsetneq U(t) ", "32322f20b490490cdfedd0806e892111": "\\ldots d_3d_2d_1d_0.d_{-1}d_{-2}d_{-3}\\ldots", "3232675e5978bfc97081bb6e1bbeb9ca": "y^*=x^{**}=x", "32326ae55c80812a6ede5c642355e4be": "E(\\bar{\\mathbb{F}_{q}})", "3232afb83079bd74f5771818f644f859": "\\Delta H > 0", "3232b923f32eb15fb228ac2a75e8f20a": "\\Pi(t)\\ \\stackrel{\\text{def}}{=}\\ \\begin{cases} 1 &\\text{if } |t| < \\frac{1}{2},\\\\ 0 &\\text{if } |t| > \\frac{1}{2}.\\end{cases}", "32330d7f0c5c70d1f7adc8998145b3bb": "\n B \\or C , \\lnot C \\vdash B\n ", "32337393066877ee291ebdf89dfb1047": "\\gamma_{\\mu, \\sigma^{2}}^{n} (A) := \\frac{1}{\\sqrt{2 \\pi \\sigma^{2}}^{n}} \\int_{A} \\exp \\left( - \\frac{1}{2 \\sigma^{2}} \\| x - \\mu \\|_{\\mathbb{R}^{n}}^{2} \\right) \\, \\mathrm{d} \\lambda^{n} (x).", "32338ca2e593fb6910029dbff097d064": "x_0 \\leq \\cdots \\leq x_k \\geq \\cdots \\geq x_{n-1}", "3233dd115e2ad853e3bce5b98820df89": " \\nu_t = c \\cdot k^{1/2} l_m. ", "32340b95fcc0594d47be1460504970f0": " \\operatorname{build-param-lists}[x, D, V, T_5] \\and \\operatorname{build-param-lists}[q\\ q\\ x, D, V, K_5] ", "32348f6d15c93f1a7993bd81857e6128": "F(x) = \\sum_{k=0}^{n} \\sum_{m=0}^{k} \\alpha_{mk} R^ky^{k-m}(1-y)^m", "3234a23e906df3618dd313fb1101491d": " m \\lambda = d ( \\sin{\\theta_i} - \\sin{\\theta_m})", "323518c5ff59c19b723bd8dd57b19af9": "B \\rightarrow S: B,\\{A, N_A, N_B\\}_{K_{BS}}", "3235a43bd451e3b477d95ff5e9b58a7c": "\\nu_{11}, \\nu_{12}, \\nu_{22} ", "3235c5a19312b76169c1866a53c2aa3e": "\\textrm{Value\\;of\\;an\\;MP\\;vote} = \\cfrac {\\mbox{Total value of all MLAs' votes}} {\\mbox{Total number of MPs}} ", "3236701562f367b0b27d268ec623aaee": "\n\\sum_{r=1}^{s} \\gamma_{r} = 0.\n", "32368cb57ae0da8fb934801ba31e8288": "\\mathcal{H}_i=p_i^2+m_i^2+\\Phi_i (x_1,x_2,p_1,p_2)\\approx 0, \\, ", "32369d27a0d2f388f4fcd3662e8e03b6": "S(p_x)", "3236a91af645e6f354ac8bc51b3ca031": "\\tau_1 \\neq \\tau_2 \\neq \\ldots \\neq \\tau_P", "3236b5d3dc3cff533e7c5e5fb5752f64": "Value=benefit.value/money", "323761d9a1f39a861bb17cf0429c11ed": "r=r_0, \\theta=\\theta_0, \\phi=\\phi_0", "32378c8750d0283973f4e0a279cf2e6f": "A^{3^n}", "323791a4bd28cb39dfabb2b259f233aa": "P_y", "3237b19d04b12f350ad7e3e9d125ee67": "\\pi_{\\mathbf{f}}|_{SO(2n)}= \\bigoplus_{f_1\\ge g_1 \\ge f_2\\ge g_2\\ge \\cdots \\ge f_{n-1}\\ge g_{n-1}\\ge f_n \\ge |g_n|} \\pi_{\\mathbf{g}}", "3237b2e9a664b881e4e2d1c3fd75b063": "e = \\lim_{n\\to\\infty} \\left( 1 + \\frac{1}{n} \\right)^n", "3237e4f18120b6fa3dd8518a2ba2b244": " \n x + y = INT\\_MAX + 1\n", "32383a8fa26f9b02f37a69eb803f80de": "E=\\sum_{x}\\left [F(Ax+h)-G(x)\\right ]^2.", "323868e41767e5bc4a03ef9e25b8c5fd": "x' = x / z", "32386a91cd41ee1bad290e9febc68106": "V = Mgl\\cos\\theta.", "3238a175cec2a25314e9e91f7c8c516e": "t_i \\ge 0", "3238bb4de0c7b2d2378036fa6100a7ee": " {} =\\frac{15!}{2}\\cdot {\\left( \\frac{4!}{2} \\right)}^{14} \\cdot 4", "3238bdb2edbfdd6d25f524e356cd0221": " 2/3 ", "3238f7ccbf6e2d07f18b77a989517586": "h=\\sqrt{pq} ", "3239335fd87f14735e4375875246d7e4": " \\binom p n = \\frac{p \\cdot (p-1) \\cdots (p-n+1)}{n \\cdot (n-1) \\cdots 1} ", "323937ac6988b940f395030bb6c4aeb4": "B_{r'} (x; d_{1}) \\subseteq B_{r} (x; d_{2})", "3239440b9c35cdadd0c29c2725b93197": "((d_1,d_2),(e_1,e_2))\\mapsto d_1 e_2 + d_2 e_1", "32395ea6126a3ff4b6b6a923dfad272b": "P(x)=p(x|r),", "32398457e626dabf231926a1618b0cf3": "\\beta = d_2 / d_1", "3239e2ececfadea9f92737a49198ae79": "I(x) = I_0 e^{-\\alpha\\,c\\,x} \\,", "3239f03881aeb340b6c1c0bccd00b5f9": "1 \\rightarrow K \\rightarrow \\pi_1(M^{-}) \\rightarrow \\pi_1(M) \\rightarrow 1", "323a42ac91b7a9bd824ab44e63e97859": "\\operatorname{qri}(A) := \\left\\{x \\in A: \\overline{\\operatorname{cone}}(A - x) \\text{ is a linear subspace}\\right\\} \\,", "323ac53f3e2462f3dfa7ca14a440aec8": "\\ c^T x \\ ", "323afa834eef06b203db3b5ea02f221a": "\\left(D_{ds}\\right)", "323b16ed73868ad41804a416d4a8a340": " x \\mapsto \\{\\langle e_k, x\\rangle\\}_{k \\in \\mathbb{N}} ", "323b380b337c88cb5ed1e595e61a9889": " \\boldsymbol { \\mathcal{L} } = \\mathbf{r} \\times \\boldsymbol { \\mathcal{P} } = {1 \\over 4\\pi c } \\mathbf{r} \\times \\left [ \\mathbf{E}( \\mathbf{r}, t ) \\times \\mathbf{B}( \\mathbf{r}, t ) \\right ] ", "323b515dec6e9a6563cad1790f7590bc": "D_1", "323b635867df2acfc6e97575c063050d": "\n\\zeta_{k+1} = \\frac{1}{\\zeta_k - a_k}.\\,\n", "323b8a512398d2c3264342c3e482cc87": "(18)\\quad ds^2=-\\Big( 1-\\frac{2M(u)}{r}+\\frac{Q(u)}{r^2} \\Big) du^2-2dudr+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;,", "323bede1c981f5f3592aad4da442bba3": "0 \\oplus L \\subset \\bar{G}", "323c2ed1bc13bdff794e8b06e901017f": "\\operatorname{E}(\\mathbf{1}_A (\\omega)) = \\operatorname{P}(A) ", "323c3914237236d0d51186ce9e8a4c2d": "\\bar u_i = \\frac {\\sum_{j=1}^n \\mbox{no. of defects for } x_{ij}}{n}", "323c5f97105643bc61e288fe596194ca": "a>0", "323c83efe22b3463a3d51f0bf7b4ff1b": " \\hat{Z} = \\sum_{k=0}^{d-1} \\omega^k |k \\rangle \\langle k| ", "323c848cce3537c1228c1332b8bf0294": "a \\uparrow \\uparrow \\uparrow b", "323c87d4b8120c005513cffdf2db9a3c": "\\left\\langle \\overline{R}^2\\right\\rangle = \\frac{1}{N}+\\frac{N-1}{N}\\,e^{-\\sigma^2}\\,", "323ce121ebbf941d59a5ece5fd96a15c": "E^\\beta", "323da1fae4ed5d3ce2708de4700eed08": "\\mathbf{e}\\mathbf{m}\\mathbf{p}", "323dc112059635f02a24b7aa0e352014": "C=\\begin{pmatrix}1 & i \\\\ i & 1\\end{pmatrix}", "323dda5eb514a414ad4ff7074abead76": "8x = 16 \\,", "323e4591a16f78f5be15699971581f48": "(\\cup_{n<\\omega}E_n \\in \\mathfrak{E}) \\Rightarrow (\\exists \\; m < \\omega \\; | \\; E_m \\in \\mathfrak{E})", "323e45b0ae13911b01584dae2082acc1": "(u,v) \\in E", "323eb0030aa635de257dad4459167a40": "G = T S - p V + \\sum_i \\mu_i N_i + p V - T S\\,", "323ee51b72a7bf19f8480ff9c5dc5216": "S={S_1,S_2,\\dots,S_n}", "323ef90d3a6f37ce35cdec4a762d5ebf": "\\mathbf {a} \\ \\wedge \\ \\mathbf{b} = \\mathit i \\ \\mathbf {a} \\ \\times \\ \\mathbf{b} \\ ,", "323f3188ad1cd3bce2cdde3e5e525cd2": " h(f(z))= f^\\prime(0) h(z).", "323f3a3274a0266414ed763463390919": "a(b+c) = ab + ac ", "323f5d5dc0814cdbf0e4cd276e6022a0": "\\sigma_{1t},\\sigma_{2t},\\sigma_{3t}", "323f8ee059110dd96459308c87215684": "R(\\theta,\\hat\\theta)=E_\\theta(\\theta-\\hat\\theta)^2.", "323fa0980761de1663a2017f9e593df4": "\\frac{1}{[A]^{n-1}} \\ \\mbox{vs.} \\ t", "323faab4a73717b48c30724621309606": "2.8985", "323fc42c0009d603aef94b67fd2e831c": "\\phi(r) = \\frac{1}{1+(\\varepsilon r)^2} ", "32402eda3ff6a208be87b5362c83ba6a": " \\mathbf{a} = \\mathbf{c} \\times \\mathbf{d} = \\begin{pmatrix} \nc_2 d_3 - c_3 d_2 \\\\\nc_3 d_1 - c_1 d_3 \\\\\nc_1 d_2 - c_2 d_1 \\end{pmatrix}\n", "32403b0575df29990bb82027b209b92c": " \\frac{P_2}{P_1}=\\left(\\frac{V_2}{V_1}\\right)^{-\\gamma} ", "324047e74b3535469340ebfba0e6c7de": "\\forall w\\,[B(0,w)\\land\\forall x\\,(B(x,w)\\to B(x+1,w))\\to B(n,w)]", "3240510e5e05de236971d5f812e94b39": "\\beta-1", "324075542ff06ebebf3b8d23e6adf902": "X=L_\\beta", "32408ef7568e61d5a363577d29d42f08": "\\rm \\ F_2 + HClO_4 \\rightarrow FClO_4 + HF", "3240a515cefc1f3aa42c1495882c1bbd": "\\hbar \\sqrt{6}", "3241dffa48d876c65a16b0a415f8eb82": "\\sqrt{ \\frac{ n\\Sigma x^2 - \\left( \\Sigma x \\right) ^2 }{n^2} } \\ . ", "3241ef4561b3ed52dbb624bc567f2ac6": "\\sum_{i=1}^N e_i < \\frac{Dd}{2} \\qquad\\qquad (1)", "3241ff16ccc559d2a035f97e5000c937": "pino(\\rho)", "3242306d451ea3e4be4c4fa936e5eaa1": "PV\\,=\\,{A \\times n \\over 1+i} ", "3243181ca76803322f723012f5c900bc": "\\alpha_t(Q) = \\operatorname*{ess sup}_{X \\in A_t} \\mathbb{E}^{Q}[-X \\mid \\mathcal{F}_t]", "3243292b104d2bce5bcf01192c85b5a1": "\\{\\,e_\\lambda a: \\lambda \\in \\Lambda\\,\\}", "324342a39bb452b67d228efc45d41a9a": "n^2 = (n^2 - a^2) + a^2", "3243730fbfbf5881cb9c3fc9ab231e04": "x = \\frac{n-p}{n} \\times 100", "3243de5856ccbf6c181c427ba28de992": "f_\\mathrm{G}=2\\sigma\\sqrt{2\\ln(2)}.\\,", "3243f33962ea80c29926e6774f1b0ace": "X \\times_Y Z \\to Z", "3243f9645314ac825f89f1989cdc6b52": " \\lVert z \\rVert_\\infty \\leq 1 ", "324475c961e2e11d00834691a77dbf5d": "E_k = \\frac{1}{2}m_0 v^2 + \\cdots ", "3244c6ef6cbd3ed5306eaf64d01cb834": "\\frac{\\hbar^2}{2m}\\frac{a}{A}=\\sum_{n=-\\infty}^{\\infty}\\frac{1}{\\alpha^2-(k+\\frac{2\\pi n}{a})^2}=-\\frac{1}{2\\alpha}\\sum_{n=-\\infty}^{\\infty}\\left[\\frac{1}{(k+\\frac{2\\pi n}{a})-\\alpha}-\\frac{1}{(k+\\frac{2\\pi n}{a})+\\alpha}\\right]=-\\frac{a}{4\\alpha}\\sum_{n=-\\infty}^{\\infty}\\left[\\frac{1}{\\pi n + \\frac{k a}{2}-\\frac{\\alpha a}{2}}-\\frac{1}{\\pi n +\\frac{k a}{2}+\\frac{\\alpha a}{2}}\\right]=", "32451117259e0bca3855ca5eab99e82c": "\\pi_{20}", "32453bdfa1e163b17d3ee1710c418611": "e_{min}", "3245430d4ad186d50e5f7c0005b4ae19": "x \\not\\in R", "32455e9da4c3072d5e3d2a043ae988a9": " a_k = \\frac{1}{\\pi} \\int_{-\\pi}^\\pi f(x) \\cos(kx) \\, dx.", "32456bc118145c0de0f32c12acd78908": "\\mbox{Transformity} = \\frac{\\mbox{emergy input}}{\\mbox{energy output}}", "32457e44f646818cf414ba16cb525808": "|\\psi (t)\\rangle =a(t)|A\\rangle +b(t)|B\\rangle ", "3245ebc546f836ed0e2d80dcb0968ec8": "\\varepsilon _{t-1}", "3245ef4df4d41ae74886458677dddef7": "\\theta \\in [0,2\\pi]", "32461c3b439535bd4d2c110fa39286ea": "\\langle \\alpha,\\beta \\rangle = \\det(\\langle \\alpha_i,\\beta_j \\rangle)", "3246206abb5551ae6571aa1f8c666511": "(p \\to r)", "32463182e0c79e57753430544803c7f3": "\\frac{dy}{dx}={g(x) \\over h(y)}", "32467c39fb705e5dcf5b9d47a8befcad": "\\sqrt{x^2 + y^2}", "32468a6f601f84fa78af39a91087a0e6": "\\angle xyz", "32469c69cee5e763210ca5649828028c": " C \\sim M^{2-\\gamma} R^{3\\gamma - 4}", "32476a9c65881ba0db0837f575b0f434": "\\mathbf{k} = (0,0,1).", "324793c876bf2c240c0f041b9438a5ae": "M \\neq N", "3247c2c70622f8350697bea5e465d4e8": "S(t) = {\\tilde \\rho}_{\\mathrm{eff}}( {\\vec k}(t) ) \\equiv \\int \\mathrm{d}\\vec x \\ \\rho( {\\vec x} ) \\cdot e^{2 \\pi \\imath \\ {\\vec k}(t) \\cdot {\\vec x} } ", "3247cbe014fdd7ebd7f03ed847d06e2f": " { 1 \\over q_f } = \\frac{C+D/q_i}{A+B/q_i}", "3247cfc71b9218de19fadb8a9c66ce7e": " \n \\int_0^{2 \\pi} {d\\varphi \\over 2 \\pi} \\exp\\left( i p \\cos\\left( \\varphi \\right) \\right)\n=\n J_0 \\left( p \\right)\n ", "3247e7ca70ce2e07ef752045690a92d2": "BV_{loc}", "3247f9d19d3a2fb98aae7d3eae4b39af": "\\,|b+\\rangle", "3248e329662a204ff22ee458a0763c9c": "h(\\mathbf{K}) = \\frac{1}{\\mathbf{a_1}\\cdot(\\mathbf{a_2} \\times \\mathbf{a_3})}\\int_{C} d\\mathbf{r} f(\\mathbf{r})\\cdot e^{-i \\mathbf{K} \\cdot \\mathbf{r}} ", "3248fbc8e2e926a309cd68f4b1e2e30f": "\\|f\\|_p = \\max \\Bigl\\{ \\Bigl| \\int_S f g \\, \\mathrm{d}\\mu \\Bigr| : g\\in L^q(\\mu),\\,\\|g\\|_q \\le 1 \\Bigr\\},", "3249158726cb92d12e728f47402b4e87": "f(0) = 1,", "3249dcc313f952505fb8abbd9cdaadbf": "\\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w", "324a075ccd9c4c87b8d2e3acc8162cf9": "Reject \\,", "324ad25bf7198f67f78fcaa4f37f993c": "R_{\\text{in}} = -R_3 \\frac{R_1}{R_2}", "324ade543f3f71d4683b0081bbe7f2d7": "2\\tfrac{3}{4}=\\tfrac{8}{4}+\\tfrac{3}{4}=\\tfrac{11}{4}", "324b20a12b0117e5925bc0c437486a85": "R.K", "324b9eec102b655d51d4e014d74195c4": " u(c_t^t,c_t^{t+1}) = U(c_t^t) + \\beta U(c_t^{t+1}),", "324bad7ab0beb3b7a9873811faa697ac": "\\frac{1}{c(q)}\\int_{-\\nu}^\\nu E_{q^2}^{-q^2 x^2/[2]} \\, x^{2n} \\, d_qx =[2n-1]!! ,", "324beccff5396de4a5fb8ab1836211b4": "\\Sigma_{A}^{+} = \\left\\{ (x_0,x_1,\\ldots):\nx_j \\in V, A_{x_{j}x_{j+1}}=1, j\\in\\mathbb{N} \\right\\}.", "324bf8b12421efc1aee30cff8e570496": "1\\to\\{\\pm 1\\}\\to 2O\\to O \\to 1.\\,", "324c92ee94507daab9e4a22e227b5520": "L^2 \\sqrt{3}", "324d1a316aa36d248dd3dc3bfdb9c9a6": "A_\\text{e}", "324d4b66108fbe9d44ce9ccdce5706ce": "\n\\delta =\\delta (k,d)<1 ", "324d60d9017385925d4bf11572c9105b": "\\mathbf{y}(t)", "324d710483ce72e2345948b7be8ed40a": "RN_i[j]", "324dd49b4d04402de8d3f1c55457baf4": "\\langle \\cdot,\\cdot \\rangle", "324e944b64f7ac9ab257b455d16ce139": "\\langle Tx \\mid y \\rangle_2 = \\langle x \\mid Sy \\rangle_1", "324e94f7f21043b6d1e9105844afd456": " \\hat f(n) := \\frac{1}{L} \\int_0^L f(x) e^{-2i\\pi n x/L}\\, dx", "324ecbe16634bb06ad17558b61dda8f3": "\n\\begin{align}\nW_x(t,f) & {} = \\int_{-\\infty}^{\\infty}\\delta(t+\\tau/2)\\delta(t-\\tau/2) e^{-i2\\pi\\tau\\,f}\\,d\\tau \\\\\n& {}= 4\\int_{-\\infty}^{\\infty}\\delta(2t+\\tau)\\delta(2t-\\tau)e^{-i2\\pi\\tau f}\\,d\\tau \\\\\n& {} = 4\\delta(4t)e^{i4\\pi tf}\\\\\n& {} = \\delta(t)e^{i4\\pi tf} \\\\\n& {} = \\delta(t).\n\\end{align}\n", "324ee4bdae788a84d10371d9bf2ce3f3": "(\\Omega, \\mathcal{D})", "324f7540a5d06e210e602fec3a558374": "k(x,y) \\leq \\Lambda |x-y|^{-n-s}", "324fdfbc5c58346688fdca1d375230d2": "\\mathrm{id}", "3250217d1859f58be5d7d1b237abfb60": "x = \\frac {90\\ \\mathrm {miles} \\times 7\\ \\mathrm {hours} } {3\\ \\mathrm {hours}} = 210\\ \\mathrm {miles}", "325050ce7c99ef5fe0f99647ef7ca620": "\n \\boldsymbol{\\sigma} = \\eta~\\dot{\\boldsymbol{\\varepsilon}}_{\\mathrm{vp}} \\implies\n \\dot{\\boldsymbol{\\varepsilon}}_{\\mathrm{vp}} = \\cfrac{\\boldsymbol{\\sigma}}{\\eta}\n ", "32505e3c9dde84903ca32f8893236008": "V_{loop}*dF_{O_2loop}=((Q_{dump}+V_{O_2})*F_{O_2feed}-V_{O_2}-Q_{dump}*F_{O_2loop})dt", "32506be2ce3825af6187ca21dd2b23fa": " G = x\\ f ", "325096f42c6cefe55b1db4956471801f": "\\left( \\frac{91}{66}, \\frac{11}{13}, \\frac{1}{33}, \\frac{85}{11}, \\frac{57}{119}, \\frac{17}{19}, \\frac{11}{17}, \\frac{1}{3} \\right)", "325218926522f3492b36d2c74142f532": "\\tfrac{\\sqrt{5}-1}{2}", "32523a94dbb39a4e93f521af70247dd4": "7 \\cdot 9^{(7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 + 7 \\cdot 9^3 + 7 \\cdot 9^2 + 7 \\cdot 9 + 7)}", "32524f31228faf2428b237d5b9cf687c": "Itd(1-d)^{n-1}", "3252d4accaf38a6e1f4aa20a22e8e2fa": "\\Omega(\\sqrt{n})", "32535f6330b30d9de661087e340736a1": "T_P = T_S\\sqrt{\\frac{R_S\\sqrt{\\frac{1-\\alpha}{\\overline{\\varepsilon}}}}{2D}}", "325367aaccfed4dd8e7edf977738eac1": "\\frac{m_A}{m_P}=2.512^{(5.05-3.62)}=3.73", "3253aaf4145b25d78d8a62cc3afdb594": "C = p_t + {p^2 \\over 2m} + {1 \\over 2} m \\omega^2 x^2 = 0.", "32545f2052840fd375847812f244017e": "x > c\\ ", "325481606dd191eafd7042f67a60dcf9": "\\frac{60}{5}=12", "32548937abbe0fcdba82404ac40d4714": " -5x^2y\\,", "3254d0b75aab912d9ab19403b3b15ead": "\\bar{\\mathbf{e}}_i\\cdot\\bar{\\mathbf{e}}_j=\\mathbf{e}_i\\cdot\\mathbf{e}_j=\\delta_{ij}\\,,\\quad\\left|\\mathbf{e}_i\\right|=\\left|\\bar{\\mathbf{e}}_i\\right|=1\\,,", "32550e6adf19510e78d0b83d8437ed8b": "\\frac{n 2.4 GeV}{c^{2}}", "325530e29e4797974f8e063db0b952fb": "y_{5}=2-0", "3255842bfc857c915f0fa6dfcfb58e39": "\\rho = 28", "3255ef35c51a333edf7e08423b4cf388": "u_i=\\alpha_{iJ}U_J \\qquad \\text{or} \\qquad U_J=\\alpha_{Ji}u_i\\,\\!", "325607668d10926ec0bd81d2705e850a": "f(x, y, z)=z^k g(x/z, y/z).\\,", "32563fa8109a3e1f90f53c0157c89829": "S\\subseteq \\alpha<\\kappa", "3256cb4d5ae1b376f2169c50163ca2de": "A \\prec B", "32570d8c65efeec524626e60adb2d237": "\\sigma_P (\\xi) = \\sum_{|\\alpha|= k} a_\\alpha\\xi^\\alpha", "325789a2070a67f38ff188dcbfbe53ef": "y(a)=\\alpha \\ ,", "3257be96e6f71b887c29490aff41d3f6": "\nV(j,t) = \\frac{1}{2\\pi i} \\int^{a_0+i\\infty}_{a_0-i\\infty} e^{tp} \\frac{V^\\ast(j,p)}{p} \\, dp\n", "3257c9d1680a6cc4cf4d79bfa04303ea": "U = \\{0, \\dots, u-1\\}", "3258901736e6374d2175b9ef12e586e2": "x^3 + y^3 = 3 a x y \\,", "325954669a29d2cd057abf35938669cd": " E = -{Zk_\\mathrm{e} e^2 \\over 2r_n } = - { Z^2(k_\\mathrm{e} e^2)^2 m_\\mathrm{e} \\over 2\\hbar^2 n^2} \\approx {-13.6Z^2 \\over n^2}\\mathrm{eV} ", "3259986fe0415384605d5444bc8c731f": " (1,0, . . . ,0) \\in \\mathbb{Z}^n ", "3259edc0e29d801a65df219feff2f596": "f_n(z_n+\\rho_nz)\\to f, \\, ", "3259f91a39c02f4d654de0a4f02dcc7b": "\\gamma\\approx-{\\pi\\omega_p^3 \\over 2k^2N} f'_0(v_{ph}), \\quad\n N = \\int f_0 {\\rm d}v", "325a84d29fbb6df52cfcc1ea168673f6": "\n\\vec x_{n+1}=2 \\vec x_n- \\vec x_{n-1}+ A(\\vec x_n)\\,\\Delta t^2.\n", "325aa079399b3df1ef52442d3df73859": "x_0,x_1,\\ldots,x_n\\,", "325ac622023e1057c7eb109b45317ad3": "G(\\phi,\\theta)", "325af3ed986b487f913a3fa5459859c6": "\n\\begin{align}\n\\textbf{a}^* &=\\frac{2\\pi\\textbf{b}\\times\\textbf{c}}{\\textbf{a}\\cdot(\\textbf{b}\\times\\textbf{c})}, \\\\\n\\textbf{b}^* &= \\frac{2\\pi\\textbf{c}\\times\\textbf{a}}{\\textbf{b}\\cdot(\\textbf{c}\\times\\textbf{a})}, \\\\\n \\textbf{c}^* &= \\frac{2\\pi\\textbf{a}\\times\\textbf{b}}{\\textbf{c}\\cdot(\\textbf{a}\\times\\textbf{b})}\n\\end{align}\n", "325b140bc3691c64eb373f1c228f6c3c": "\\{W_{U}:U\\in\\mathcal{O}\\}\\,", "325bf1e28c561dce8a95fff98eb985d8": "N^{(c-1)/2}\\pmod c", "325bf9900b03a50dd3cadb0223bca3a8": "0,0 \\Leftrightarrow 0", "325d6ff99e5b13d23978a3591890fecd": "\\cos \\theta_s \\,", "325d740888ce183dcfd492a7db777d46": "\\begin{pmatrix}a & b/2\\\\ b/2&c\\end{pmatrix}.", "325d7f408e8fba82979d2df305fe2d67": "\\lambda u_{k,ki}+\\mu\\left(u_{i,jj}+u_{j,ij}\\right) +F_i=0\\,\\!", "325d9df628f5662f51dfa5cc2bd7008f": "\\mathrm{Area} = \\pi \\left({f \\over 2N}\\right)^2", "325dcd6eb0e0297ecc93ea4a802ae613": "\nY_{t} = Y_{0} \\left \\lbrace 1 + \\alpha_{0} \\frac{\\lambda_{k}\\beta_{k} + \\lambda_{c}\\beta_{c}}{\\lambda_{k}\\beta_{k}} \\left \\lbrack (1 + \\lambda_{k}\\beta_{k})^t - 1 \\right \\rbrack \\right \\rbrace\n", "325ddc25e99dff3dc71900ed02db0c6e": "\\beta^r", "325e4b2d2cb5414a1125b650a1343a0c": "\\frac{1}{T}\\frac{\\partial f}{\\partial t^\\prime}+\\frac{V}{L}\\vec v^\\prime\\cdot\\frac{\\partial f}{\\partial \\vec r^\\prime}+\\frac{q}{m V}(\\vec E+V\\vec v^\\prime\\times\\vec B)\\cdot\\frac{\\partial f}{\\partial\\vec v^\\prime}=0.", "325e4b428b21d03f4dfa82c84d0e43e2": " I_t", "325e964393b63dc9e758dcb25d95dc1f": " \\mbox{vec}(ABC)=(C^{T}\\otimes A)\\mbox{vec}(B) ", "325eba243517e142d176495514ca7d4d": " K_{i2j} ", "325edee69c7c2aa05b07751b80de8389": "L^2(\\mathbb{R}).", "325f16a5b136084464c151e585612c37": "\\left(\\binom{n+2}{2} = \\frac{n^2+3n+2}{2} \\right)", "325f29b0a96fe53759c25d15f715661e": "\\lim_{n \\to \\infty}\\vec{S}(n) = b_1 K_1^n\\vec{\\xi}_1 = b_1 3^n\\vec{\\xi}_1.", "325f3126ab5e1ce8e19f9856475e9c7c": "y \\in (\\mathbb{Z}/n\\mathbb{Z})^*", "325f32d5cde223899e1e92990c4c9f7d": "\\; t \\geq 1\\,", "325f6059229cb95fff3d3edf8b44f0dc": "x^5 - 10Cx^3 + 45C^2x - C^2 = 0.\\,", "325f7fb3d85b04d415c806348ad2f74a": "s = n^2", "325f85f3623ef834f8946add630e7cf0": " t_c = \\frac{a}{b} ", "32605483744d3c3c6ef3b51c60a15b24": "\\scriptstyle\\bar{z}", "326087b6011d0c0b28701c169f82f1fe": "\\mathbf V=\\mathbf V^T\\,\\!", "3260a445fdecedea0852c4f29ba706bc": "u_S = \\text{id}_S \\otimes u", "3260b8fe9dcbb0173860b972f20684f9": "\nF^{-1}(y) = \\sgn(y) (1 / \\mu ) ((1 + \\mu)^{|y|}- 1)~~~~-1 \\leq y \\leq 1\n", "3260ba77c3f538a763b07d429a740cd1": " \\mathbf T^n = \\begin{bmatrix}\n t_0 & t_{-1} & t_{-2} & \\dots & t_{-n+1} \\\\\n t_1 & t_0 & t_{-1} & \\dots & t_{-n+2} \\\\\n t_2 & t_1 & t_0 & \\dots & t_{-n+3} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n t_{n-1}& t_{n-2} & t_{n-3} & \\dots & t_0 \\\\\n \\end{bmatrix}. ", "32610bc996ee8f7e81f4a2f13fe36b0b": "AI=\\left.\\sum_{i=1}^n \\left(s_i- {1 \\over n}\\right)^2 \\right/n.", "3261b14f1f93f3d70c95453be3269d78": "C_{\\text{CB}}\\,", "3261b2154b90fc981dcc5dbceba3bf5c": "s=x+y, t=xy", "3261ca3e642402ed6554198d549fe080": "\\overline{X}_n=\\tfrac1n(X_1+\\cdots+X_n). ", "3262c3c939e317dddddd24f07bb85803": "(4x - 2)/(x^2 + 1)", "32630152aaf0f3be9606ced3abf506fe": "\nL = \\psi^\\dagger \\left(i {\\partial \\over \\partial t} + {\\nabla^2 \\over 2m} \\right)\\psi\n\\,", "3263a2b2d76593006346a910c41ec3ef": " x = 2\\pi t \\cos(2\\pi t)+ 2\\pi [(\\cos^3 2\\pi nt+\\sin^3 2\\pi nt) \\cos(2\\pi nt)- \\cos(2\\pi t)], ", "3263de8367216fa4f35b5603c5bfbbff": "0 = \\frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)", "3263f5906bf4d6636379cd06e00c5172": "P\\subset\\mathbb{R}^{n+1}", "3264328dab799159ae84ef72534b66d0": "f:\\mathcal{D}\\rightarrow\\mathbb{R}^{d}\\,\\!", "32645ab889d2cde8fe6a344788e7a280": " g\\otimes x\\mapsto -x", "3264a7a3d2a788420f2593395676e900": "\\left| A_n^\\epsilon \\right| \\leq 2^{n(H(X)+\\epsilon)}, n.(H(X)+\\epsilon)\\; ", "3264af99646af0b0c76ad8aeffdf9b94": "I_w = \\frac{1}{12} m\\left(h^2+d^2\\right)", "3264b0c41e0e85e8f6026f2e2fefc666": " = \\sum_{i=1}^n \\left| x_i - y_i \\right|", "3264ef6f5a9b54e46e584b57f5788e1a": "\\ell{\\ge}m", "3264f8e8e30fbf21dc13ecc82a2e6072": "\\Delta I_{L_{On}} + \\Delta I_{L_{Off}}=\\frac{V_i D T}{L}+\\frac{\\left(V_i-V_o\\right)\\left(1-D\\right)T}{L}=0", "326515d77b1e7e9a24552a57baca7e70": "\\left[\\begin{matrix}0 & 1 \\\\ 2 & 3\\end{matrix}\\right] +\n\\left[\\begin{matrix}1 & 2 \\\\ 3 & 4\\end{matrix}\\right] =\n\\left[\\begin{matrix}1 & 3 \\\\ 5 & 7\\end{matrix}\\right] =\n\\left[\\begin{matrix}1 & 3 \\\\ 0 & 2\\end{matrix}\\right]", "32651cbceaad70cd1802a8e61a93ec00": "\nT_s=\\left[ \\frac{S_0(1-\\alpha_p)}{4\\sigma} \\frac{1}{1-{\\epsilon \\over 2}} \\right]^{1/4}\n", "326540e3b1669908cc1ef1d57bbd2535": "Q=0.", "3265463f4b79fa96f9f5bca863690f9c": "C_{abcd}\\, k'^bk'^d=\\gamma k'_ak'_c", "32656f5236344e2a1d51eb9e7e74b725": "(p_1 - p_2)/2", "32658fc3574738ff063b748cb7aa514f": "P = P_0\\exp(-\\frac{z}{H})", "3265d0999c92153c48a6902054b9f531": "\\left| a \\right\\rang", "3265f8879f17367c7956ae3dca827210": "P_{\\mathrm{error}\\ 1\\to \\mathrm{any}} \\le M^\\rho \\sum_{y_1^n} \\left(\\sum_{x_1^n(1)} Q(x_1^n(1))[p(y_1^n|x_1^n(1))]^{1-s\\rho} \\right) \\left(\\sum_{x_1^n} Q(x_1^n)[p(y_1^n|x_1^n)]^s\\right)^{\\rho}.", "32660957358768dc5d7b8786b219cf36": "\\left | f^{\\prime\\prime} (p) \\right | \\neq 0", "32664edf66569c5f6b100768bc6554de": "\\Im(\\tau) > 0", "32666b9b9d8e4e8b31f5b83716dd7908": "\\sum_{n=-N}^N (1 - \\frac{|n|}{N}) \\hat f(n) e^{int}", "3266e24077c238c01d0f8ff02d9538c2": " \\left(\\frac{1} {R}\\right) ^{j-n} E_{t+n} {y_{t+j}}", "326710221e782863ac86e9abd3a1a135": "\\begin{align}\n1/(1+x) &\\approx 1-x,\\\\\n(1+x)(1+y) &\\approx 1+x+y.\n\\end{align}\n", "3267abbb5de330483738f7c41fd18ddb": "\\mu = \\mu_r \\mu_0", "3267f4a67c2e58fe5b53101ec5506cb5": "I(p_{t_m},p_{t_m},\\alpha \\cdot q_{t_m},\\beta\\cdot q_{t_n})=1~~\\forall (\\alpha ,\\beta )\\in (0,\\infty )^2", "3268244c7e40a95a117a7acf43fe7fd7": "c_t = D \\Delta c, \\quad ", "32692bf81c7c19cb4c86dbae1c2d7933": " yM(xy)\\,dx + xN(xy)\\,dy = 0 \\,\\!", "3269307d79f00824bc49d7c8d0fd2ab9": "\\gamma=\\frac{F}{2L}=\\frac{F \\Delta x}{2 L \\Delta x}=\\frac{W}{\\Delta A} ", "326956a6fae7dcae1ab2865ff4e6b2ef": "p_{1}\\Psi =\\left( \\frac{\\varepsilon _{1}}{w}P+p\\right) \\Psi ", "32697be5483ebb3a61cafde84669bb70": "l(v,w) \\leq f(v,w) \\leq u(v,w)", "326a09f14133186997950f295c7bcbf6": "x_1 y_2 + x_2 y_1)", "326a23d0c990111320e7ebf1d0420aef": "X \\sim \\mbox{Inv-Gamma}(\\tfrac{\\alpha}{2}, \\tfrac{1}{2})", "326a46ae64ab797c72581444faf8c42d": "K \\approx n_P + n_N - 3.48 ", "326aaf6517b5dcf6a7fed3d202e653cc": "\\,k=2", "326b24aa71fb73a5701739d2cb3a3b64": "\\displaystyle{M\\rightarrow\\mathfrak{k}^*\\rightarrow \\mathfrak{t}^*.}", "326b86b3d03dc9f1642e3ca5d4f6e235": "\\mathcal{H}^{12}=\\mathcal{H}^1\\otimes \\mathcal{H}^2", "326b8753b1e02cdaac0dbcc0db7d898b": " \\Psi = \n\\begin{pmatrix}\n\\psi_{1,2} \\\\\n\\psi_{3,4} \\\\\n\\end{pmatrix}\\,\\quad \\psi_{1,2} = \n\\begin{pmatrix}\n\\psi_1 \\\\\n\\psi_2 \\\\\n\\end{pmatrix}\\,\\quad \\psi_{3,4} = \n\\begin{pmatrix}\n\\psi_3 \\\\\n\\psi_4 \\\\\n\\end{pmatrix}\\,.", "326bbe5c43ba40289ebf60437146a8ab": " (\\mathcal{C}_{XX} + \\lambda \\mathbf{I})^{-1} ", "326bcfc6d264acd4bbd7ce940a5558e3": "(\\mathbf{S})\\int_{t_0}^tg(t^\\prime)\\mathrm{d}B(t^\\prime)-(\\mathbf{S})\\int_{t_0}^t\\mathrm{d}B(t^\\prime)g(t^\\prime)=\\frac{\\sqrt{\\gamma}}{2}\\int_{t_0}^t\\mathrm{d}t^\\prime\\,[g(t^\\prime),c(t^\\prime)]\\,.", "326bd34b6500754ae0412cbedab58f95": "T h = \\sum _{i = 1} ^n \\alpha_i \\langle h, v_i\\rangle u_i \\quad \\mbox{for all} \\quad h \\in U ,", "326bdaaf2409dc1d4ad2aa29e8c3e17f": "k x", "326c74adb28b74232f983850dc109d87": "s,t : Mor \\to Ob ", "326c7be4a13855518b400c36a8a3d3d3": " x=h=-\\frac{b}{2a} ", "326ca9f337acbb36bc99059e19c980c7": "t_\\frac{1}{2}", "326cdaab41c7b705fa87aad17bd025a5": "g(n)\\,\\, = O(m^n)\\,.", "326d0dbd5c7b9c358e042e3abed6a1ca": "\\ d = G(m)", "326df1e5cc97070f4c9c56a28db31dd0": "Ef(b,x,y)=f(b,x). \\, ", "326df453296abc0a46ced68a8d262bc3": " S(i_1,i_2,i_3)=\\zeta(i_1,i_2,i_3)+\\zeta(i_1+i_2,i_3)+\\zeta(i_1,i_2+i_3)+\\zeta(i_1+i_2+i_3)", "326e3c72a3a9d1e608de9be84b23476e": " A\\Gamma =dx^\\lambda\\otimes[\\partial_\\lambda + (\\Gamma_\\lambda{}^\\mu{}_\\nu(x^\\alpha)\\widetilde x^\\nu + s^\\mu(x^\\alpha))\\widetilde\\partial_\\mu], \\qquad s^\\mu = - \\Gamma_\\lambda{}^\\mu{}_\\nu a^\\nu +\\partial_\\lambda a^\\mu, ", "326e5eca023c926b0e2c1cb52bffd02f": "A_B=\\|\\mathbf A\\|\\cos\\theta", "326e994a9259ec04209a15edb09492e2": "f = 0.079 \\mathrm{Re}^{-{1 \\over 4}}", "326e9d04b8a10da6dfa7192b9d3c614a": "\\frac{\\zeta(s-k)}{\\zeta(s)} = \\sum_{n=1}^{\\infty} \\frac{J_k(n)}{n^s}", "326ec0fbb92a5511319afbcc32e3663a": "U=e^{\\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\theta} = \\cos \\theta + \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta =e^{\\mathbf{\\gamma} \\cdot \\hat{p} \\theta} = \\cos \\theta + \\mathbf{\\gamma} \\cdot \\hat{p} \\sin \\theta ", "326f17d90ffd4fb490113b77c2dde159": "\\tau_\\mathrm{n}", "326f1807042ef3ce82d3820e02e9bedb": "\\begin{align}\n\\bar{F}_{\\alpha \\beta} & = \\frac{\\partial \\bar{A}_{\\beta}}{\\partial \\bar{x}^{\\alpha}} \\, - \\, \\frac{\\partial \\bar{A}_{\\alpha}}{\\partial \\bar{x}^{\\beta}} \\\\\n& = \\, \\frac{\\partial}{\\partial \\bar{x}^{\\alpha}} \\left( \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} A_{\\gamma} \\right) \\, - \\, \\frac{\\partial}{\\partial \\bar{x}^{\\beta}} \\left( \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} A_{\\delta} \\right) \\\\\n& = \\, \\frac{\\partial^2 x^{\\gamma}}{\\partial \\bar{x}^{\\alpha} \\, \\partial \\bar{x}^{\\beta}} A_{\\gamma} \\, + \\, \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} \\frac{\\partial A_{\\gamma}}{\\partial \\bar{x}^{\\alpha}} \\, - \\, \\frac{\\partial^2 x^{\\delta}}{\\partial \\bar{x}^{\\beta} \\, \\partial \\bar{x}^{\\alpha}} A_{\\delta} \\, - \\, \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\frac{\\partial A_{\\delta}}{\\partial \\bar{x}^{\\beta}} \\\\\n& = \\, \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\frac{\\partial A_{\\gamma}}{\\partial x^{\\delta}} \\, - \\, \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} \\frac{\\partial A_{\\delta}}{\\partial x^{\\gamma}} \\\\\n& = \\, \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} \\, \\left( \\frac{\\partial A_{\\gamma}}{\\partial x^{\\delta}} \\, - \\, \\frac{\\partial A_{\\delta}}{\\partial x^{\\gamma}} \\right) \\\\\n& = \\, \\frac{\\partial x^{\\delta}}{\\partial \\bar{x}^{\\alpha}} \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} \\, F_{\\delta \\gamma} \\ .\n\\end{align}", "326f3476e959eb675ef9ece52c24b789": "I_{E}(f) = - \\sum^{m}_{i=1} f_i \\log^{}_2 f_i", "326f7b20a9d0a3afbdc63db2265bbcd1": "-T^a_b l^b\\,\\hat{=}\\,cl^a", "326f8cebde1a1c636f900588980dded8": "\\tau_{eq}", "326ff472d8acd695fc216836cff1279f": "dr_t = (\\theta_t-\\alpha r_t)\\,dt + \\sigma_t \\, dW_t", "32701e187ca778cb26d21ed3f4835ac7": "D = {d \\over dx};", "32704f528b439347ec842ac69d1c218b": " {\\eta_{i,j}}(\\mathbf{x}) = \\sum_{k} c_{ij}^{k} \\langle \\mathbf{x},\\mathbf{e}_{k} \\rangle ", "32708f124a986869b50cd9492dec4c0a": "\\sqrt[6]{3}\\approx 1.2009", "3270b8e1921091149924230e80d46450": " \\xi(t)=A\\Bigl(1 -e^{-\\frac{t}{\\tau}}\\Bigr)", "3270d1415dae492d581d35b6af47cce5": "z(m,n;s,t)=O(nm^{1-1/t}+m)", "32714816377b1802327db618dafabf18": "\n \\begin{align}\n \\theta_s & = \\frac{L_s} {2R_c} \\\\\n & = \\frac{100} {2 \\times 300} \\\\\n & = 0.1667 \\ \\mbox{radian} \\\\\n \\end{align}\n", "32719a93c81da41ceba01c3f2e0ff300": "t^2\\,d\\theta + 2t\\, dt \\cdot \\theta\\,", "32719b895937ec8549cfe9f819e6ccae": "F_\\text{load} \\,", "3271c5742e8c29c399020485fc6f9109": "\\mathbf{r}_i = 0 ", "3272134e8ae179231a5ac205ef27e92b": "\n2 \\uparrow 2 \\uparrow 2 \\uparrow 2\n", "327223a7a6ccf2924481beaf1469f6a9": "\\mathfrak c = 2^{\\aleph_0} > \\aleph_0 \\,. ", "32728b830b61f60f5112852f3791118b": "F(t) = \\int_{0}^{t} f(s)ds ", "327296c434f899a19db212502f1f59b9": "\\sigma_k ^{\\downarrow} = \\max_{S_k} \\min_{x \\in S_k, \\|x\\| = 1} \\| Mx \\|.", "3272cd000f4450ff6737776a52c2fd00": "c_{m,n} : 0 \\leq n < m", "3272e727bcb340c057f388ff69290c13": "\\textstyle \\delta < b < 2l -1", "3272e80ebc305acf7812090b699d155c": " 1 \\leq s_1 < \\cdots < s_k ", "3273b9330f52da54fac30b5cf71f2330": "\\Sigma_\\hat{\\alpha} = \n\\begin{pmatrix}\n\\psi_{\\alpha}\\\\\n\\bar{\\chi}^{\\dot{\\alpha}}\\\\\n\\end{pmatrix}\n", "3273d8cbd7af2b463d75b976a6662b38": " \\|AX - B\\|_F \\ge \\|AZ -B\\|_F", "3273f60e828d7b7d25e0fc39576fcffe": "\\ln m! = \\sum_k \\ln k \\approx \\int_{1}^{m}dk \\ln k = m\\ln m - m ", "32744aba77fd24c2010e67cdd6f15a7d": "c^2=(dx/dt)^2+(dy/dt)^2+(dz/dt)^2+(cd\\tau /dt)^2", "3274ce9c1b09d1273c316cd5400a4a55": "R_{i,n}(u) = {N_{i,n}(u)w_i \\over \\sum_{j=1}^k N_{j,n}(u)w_j}", "3275033aaa85866cf43db5a74e163e0e": "\\scriptstyle \\mathcal{E}", "3275256b09b3199024656be90d4f6de3": "j^2 = -1", "327547fd918ce9ac5bbfc0e27cebefb1": "\\{x_\\alpha\\}_{\\alpha\\in I}", "3275943e6dda314f2064ba7f8c949713": "\\zeta(X,n-s)=\\pm q^{\\frac{nE}{2}-Es}\\zeta(X,s)", "3275c642f99263b1fd7b0b27566db53b": "m(t)=\\sum_{n=0}m_n\\frac{t^n}{n!},", "3275eac180c0a97313f2ce5f51dd9078": "\\mathbf{l}_a + (\\mathbf{l}_b - \\mathbf{l}_a)t = \\mathbf{p}_0 + (\\mathbf{p}_1-\\mathbf{p}_0)u + (\\mathbf{p}_2-\\mathbf{p}_0)v", "3275f8851c07a970299dddd3d250b68d": " S(\\Psi) = \\tfrac{1}{2} \\langle \\Psi |Q_B |\\Psi \\rangle + \\tfrac{1}{3} \\langle \\Psi,\\Psi,\\Psi \\rangle ", "3276b57213f8ad4b651e2525fd5134dc": " m = 5 ", "3276ca19417dce72b0ba3676df8661c6": "\\varphi\\hat{\\mathbf{a}} = \\varphi(a_1, a_2, a_3)", "3276ed82199e03d8caaaace30ffd03ec": " B_{j_{1}}, B_{j_{2}}, \\dots, B_{j_{m}} ", "32773f2cf7e04172892228400c7aa47c": "\\mu(S^{-1})=k\\nu(S)\\,", "327741610f06cd1922f6795a4f458689": "(\\neg A\\to B)\\to((A\\to B)\\to B)", "3277589ec3431a3694492f4f2e807814": "M(t)=\\frac{\\Gamma(\\beta-t) \\Gamma(\\alpha+t) }{ \\Gamma(\\alpha) \\Gamma(\\beta) }, \\quad -\\alpha0.\\!", "32863f8ade0ba708acf33229ec6f0770": " \\langle A x | y \\rangle = \\langle x | A y\\rangle ", "32864f80e5e774ce9ae4a7c1cd890a5c": "R_F/R", "3286b7a01ac70c2d2cde6a61fa9ecbbd": "\\boldsymbol{F}\\cdot\\mathbf{e}_1 = \\mathbf{e}_1", "328721da9f3f90112b11e4aa3376087d": "S^{\\ell^*}", "328752ba6dc146333f53e73d8d5e6e73": "\\rho_{12}\\mapsto S(\\rho_1)-S(\\rho_{12})", "328753052299cda5739b4cfbd09fd111": "R^{\\mu}(\\tau)", "3287667d2b507ab6b00ca1523fea5da8": "S(x)\\Gamma(x)\\Lambda(x){\\textstyle{\\{k+v,\\ldots,\\,d-2\\}\\atop =}}0.", "32879bbf43555388c35342ba347ee242": "C_\\rho(u,v) = \\Phi_\\rho \\left(\\Phi^{-1}(u), \\Phi^{-1}(v) \\right) ", "3287fa44794f79b0e0310cf4919d5da3": " \\operatorname{st}(x + y) = \\operatorname{st}(x) + \\operatorname{st}(y) ", "3288bb0ac39a210cc35c1f29506c3188": "\n y_i = X_i\\beta_i + \\varepsilon_i, \\quad i=1,\\ldots,m,\n ", "3288fa0d91d50376e6acfccbb9d79991": " O(N^2 \\log T)\\, ", "32892c62ca55a035ee7ae126f4992bc6": "\\widehat{\\Omega} = {U(R)}^\\dagger \\widehat{\\Omega} U(R) = \\exp\\left(\\frac{i\\theta}{\\hbar}\\hat{\\mathbf{n}}\\cdot\\widehat{\\mathbf{J}}\\right) \\widehat{\\Omega} \\exp\\left(-\\frac{i\\theta}{\\hbar}\\hat{\\mathbf{n}}\\cdot\\widehat{\\mathbf{J}}\\right) ", "32898ef8a2fa7726f0f2b040e86c5271": "U{}^{r-1}_{n-1}", "328a13a295642b8c283f7a6b862c56a3": "0.479", "328a160cf0349c760137fe335888f667": " \\zeta_G ( \\alpha ) = \\sum_r S(r)/r^{\\alpha}. ", "328a1d028c1229eee33eee777a071230": "N (x)", "328aac5ae6cfbd4f1163e361a3429b33": "x \\in L_{1},\\,\\, y \\in L_{2}", "328ac50fa84c721d22e0a54dad346034": "\\mathbf{Max}", "328adc54f09b10d692bff6651a191c65": " O_{bg} ", "328ae7b7501e6ce5f9a643b14aeff31d": "\\mathbf{d}\\cdot(\\mathbf{k}-\\mathbf{k^\\prime})=2\\pi m", "328b1f9fc7ce8a4013827d29571eaa41": "\\nabla p = \\mu \\nabla^2 {\\mathbf v} ", "328b3bdb363e550ec8588b8abb437b83": "\\sqrt{-1}\\int_{X_0} A\\bar{B}\\gamma^* \\wedge \\bar{\\delta}^* + \\bar{A}B\\bar{\\gamma}^* \\wedge \\delta^* = \\int_{X_0} \\operatorname{Im}\\,(2\\bar{A}B \\bar{\\gamma}^* \\wedge \\delta^*) > 0", "328b4a37f28f5b445b490f4161569418": "s=c", "328b6bf4ddc2557b38af5db5c373d105": " \\mathbf{L}_{\\nu}(z) = {\\left({\\frac{z}{2}}\\right)}^{\\nu+1}\n \\sum_{k=0}^\\infty \\frac{1}{\\Gamma(\\frac{3}{2}+k) \\Gamma(\\frac{3}{2}+k+\\nu)}\n {\\left({\\frac{z}{2}}\\right)}^{2k} ", "328b9a98c9505933e289fb938d63f91f": "I \\rightarrow T", "328bcff261956941908d58d6c21eb789": "D:=[{a}_{1},{b}_{1}]\\times[{a}_{2},{b}_{2}]", "328bf32542851a5863e70be77f94dab6": " \\left\\{{n \\atop n-k}\\right\\} \\sim \\frac{(n-k)^{2k}}{2^k k!} \\left( 1 + \\frac{1}{3} \\frac{2 k^2 + k}{n-k} + \\frac{1}{18} \\frac{4k^4-k^2-3k}{(n-k)^2} + \\cdots \\right).", "328cab36997a7288687cc3dd14af35ea": "\\Delta R_F", "328cdcb338fe745569f6c0383db82e9a": "\\biggl( \\sum_{i=1}^k p_i e^{t_i} \\biggr)^n", "328d0ec1283bcf0f04168fd0dc458d92": "-\\Gamma^d{}_{d c} T^{a_1 \\ldots a_r}{}_{b_1 \\ldots b_s}.", "328d422232417cedbbf7b6f24745bcc1": " e^X \\geq 1+X \\, ", "328d54d966b184ef9ece56d96c018a02": "-P=(X: -Y: Z: ZZ) ", "328dc64d0af58e2e26f8d57e26645ddd": "\\frac{\\sqrt{114}+4}{14}=1+\\frac{\\sqrt{114}-10}{14}=1+\\frac{14}{14(\\sqrt{114}+10)} = 1+\\frac{1}{\\frac{\\sqrt{114}+10}{1}}.", "328e07cd87e634f71e8482f031733b16": "I(X;Y|Z) = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z)\n = I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(Z)", "328e4c0ec26f7645be9e000fd993affe": " a_2 | N_1, N_2, N_3, \\dots \\rang = \\sqrt{N_2} \\mid N_1, (N_2 - 1), N_3, \\dots \\rang,", "328ec6fae52d419476ddb74fa186ff8d": "L'(t) = r_B \\left( L_\\infty - L(t) \\right)", "328f4d3f093d196f0a7009f73a1ceec2": "\\text{ iii) Inv}(s_\\Lambda,\\boldsymbol u_i) =\\sum_{j=1}^m(-\\log u_{ij})/s_\\Lambda", "328f60cdedc62366ffcf2744cfd8f882": "s = \\int_{a}^{b} \\sqrt { [X'(t)]^2 + [Y'(t)]^2 }\\, dt. ", "328fe11bb3775ff989d6361089a55b49": "\\begin{align}\n \\mathrm{d}{\\omega} &= \\mathrm{d} (f_I \\mathrm{d}x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d}x^{i_k} ) \\\\\n &= \\mathrm{d}f_I \\wedge (\\mathrm{d}x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d}x^{i_k}) +\n f_I \\mathrm{d}(\\mathrm{d}x^{i_1}\\wedge \\cdots \\wedge \\mathrm{d}x^{i_k}) \\\\\n &= \\mathrm{d}f_I \\wedge \\mathrm{d}x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d}x^{i_k} + \n \\sum_{p=1}^k (-1)^{(p-1)}f_I \\mathrm{d}x^{i_1} \\wedge \\cdots \n \\wedge \\mathrm{d}x^{i_{p-1}}\\wedge \\mathrm{d}^2x^{i_p} \\wedge \\mathrm{d}x^{i_{p+1}}\\wedge \\cdots \n \\wedge \\mathrm{d}x^{i_k} \\\\\n &= \\mathrm{d}f_I \\wedge \\mathrm{d}x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d}x^{i_k} \\\\\n &= \\sum_{i=1}^n \\frac{\\partial f_I}{\\partial x^i} \\mathrm{d}x^i \\wedge \\mathrm{d}x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d}x^{i_k} \\\\\n\\end{align}", "32902e3f2795423f64ec261afa98079d": " \n\\Omega = - JF_e V/{kT}\\ \n ", "32902edfdccb5f853b2d0d8b776ec74a": "\\varepsilon:=\\frac{|f(b)-f(a)|}{2(b-a)}>0.", "329043ac4415488e9b0746575fcaeb10": "x \\mod 1", "32905c98de732060111b5fad5274c1af": "B^{-1}", "329075819864fecb38e83fde2b9f70dc": "\\frac{\\partial k_{i}}{\\partial t}=\\frac{k_{i}}{2t}", "3290b14f14ea641125044c698552a56b": " \\vec T = u_1\\vec a_1 + u_2\\vec a_2 + u_3\\vec a_3 ", "3290fa92e878f7e51ef960d48105023c": "M_3=\\left.\\Delta\\right.,\\,", "329139b51ef5ecf9ade5d4204c63590e": "\\|u\\|_{H^p}=\\|f\\|_{L^p}", "32915bb63a2e0bbb2598549916530b83": " |b(t)|^2", "3291698d59638bae6a2f90137d1401f4": "{\\rm SO}(2) = \\left\\{ \\left(\\begin{matrix}\\cos\\theta&\\sin\\theta\\\\ -\\sin\\theta&\\cos\\theta\\\\ \\end{matrix}\\right)\\,:\\,\\theta\\in{\\mathbf R}\\right\\}.", "32917f74311bc9da197c9f4d805ed576": " b= \\infty ", "32918c58b60ce2b8eda931464a0549b5": " a \\uparrow \\uparrow \\uparrow b = \\underbrace{^{^{^{^{^{a}.}.}.}a}a}_{b}", "32920e2a99a26f529dc64aff6abf8ae3": " b_{n,\\mathbf{R}}", "329242e93ceb744698654e6cc08f8ad3": "\\rho = \\sqrt[3]{1 + \\sqrt[3]{1 + \\sqrt[3]{1 + \\cdots}}} \\, .", "3292770590b89c0e6292e6a1b4132802": "(S,A,P_\\cdot(\\cdot,\\cdot),R_\\cdot(\\cdot,\\cdot))", "329284a56515e2085156cf1b121ed130": "\\textstyle U(s|x)", "329288b161d38b85b05263b85467b9ca": " \\{ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \\ldots + a_1 x + a_0 | a_n , a_{n-1}, a_{n-2}, \\ldots , a_0 \\in R \\}", "3292ce38762d15d02a3c65d64183e55d": "L=L_1\\cup L_2\\cup L_3", "32933707c0132898c2affece4dac114f": " \\sigma = \\mathrm{constant} \\cdot \\frac{Z^n}{E^3} ", "32936811efe1ed8ed38483fec2e46000": "C(x,y,z)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\ G( x-x',y-y',z)S(x',y')\\, dx'\\,dy'. \\qquad(1) ", "3293749b3529ddbcb5058f752419381e": "\\xi=\\frac{\\dot Q_L}{P}.", "3293765967d00230b7f8f3bda30b2cd6": "(i, j)", "32940400c6723a3a973682ac22464560": "(\\Delta x)^4", "3294083ee3fedba8f5d54799166bada7": "\n \\mathrm{Re}\\, =\\, \\frac{u\\,\\sqrt{A}}{\\nu}\n", "32943f5032952b50cc8d08dc01b93a0e": "U^2", "32944629138a6e8df671d088cc0e5e53": "\n\\begin{array}{lcl}\n\\hat{\\mathcal{J}}_1 &=& i \\left( \\cos \\alpha \\cot \\beta \\,\n{\\partial \\over \\partial \\alpha} \\, + \\sin \\alpha \\,\n{\\partial \\over \\partial \\beta} \\, - {\\cos \\alpha \\over \\sin \\beta} \\,\n{\\partial \\over \\partial \\gamma} \\, \\right) \\\\\n\\hat{\\mathcal{J}}_2 &=& i \\left( \\sin \\alpha \\cot \\beta \\,\n{\\partial \\over \\partial \\alpha} \\, - \\cos \\alpha \\;\n{\\partial \\over \\partial \\beta } \\, - {\\sin \\alpha \\over \\sin \\beta} \\,\n{\\partial \\over \\partial \\gamma } \\, \\right) \\\\\n\\hat{\\mathcal{J}}_3 &=& - i \\; {\\partial \\over \\partial \\alpha} ,\n\\end{array}\n", "329460f2879a7fc6659be1ca107f0fb8": "\n0 = \\frac{d^{2}t}{dq^{2}} + \\frac{1}{w} \\frac{dw}{dr} \\frac{dt}{dq} \\frac{dr}{dq}\n", "32949b44e15fa048377fccaaae2f4a6d": "\\mathcal{M}", "3294dff896d06fea4749ce99aa43eb28": "\\{a,b,c\\}", "3294eb0109ec927619e11c758cfb56f5": "-4\\pi", "32951c85133bfb7a557d6a38b7ea9dbb": " \\delta g_B = 2\\pi\\rho G H, ", "329595a445ad0baf969c590b394fb716": "{\\mathbf{}}E", "3295d850606bf204986b4a8934850029": "+2^5 -2^3 -2^0", "3295eb834562e87004c98d2af99229a4": "\\mathbf{e_z} \\times -\\mathbf{e_x} = -\\mathbf{e_y} \\ . ", "329623ddda785cc2a341c997a0a4ea1a": "~n_2=N_2/N~", "3296672671ed55ae1401c846c89580c1": "(\\lambda x.t)", "3296677df0f9ce12ccee5d22bc65ca24": "\\vec{R}_0", "329678139aa125c4c9ed3677686573c1": "[O_2]~=~Dissolved~oxygen~concentration", "3296be8983c05193dbc755a0fcd01332": " |\\tau_n| < Ch^{p+1} ", "329738770fe303f523ff4c57ce8b0ecb": "j \\mapsto \\begin{pmatrix}\n -1 & 1 \\\\\n 1 & 1\n\\end{pmatrix}", "32974722d07f212a88ad4ce19467db2c": "\\mathbf{\\Omega} = \\begin{bmatrix}\n\\frac{1}{1-\\rho^2} & \\frac{\\rho}{1-\\rho^2} & \\frac{\\rho^2}{1-\\rho^2} & \\cdots & \\frac{\\rho^{T-1}}{1-\\rho^2} \\\\[8pt]\n\\frac{\\rho}{1-\\rho^2} & \\frac{1}{1-\\rho^2} & \\frac{\\rho}{1-\\rho^2} & \\cdots & \\frac{\\rho^{T-2}}{1-\\rho^2} \\\\[8pt]\n\\frac{\\rho^2}{1-\\rho^2} & \\frac{\\rho}{1-\\rho^2} & \\frac{1}{1-\\rho^2} & \\cdots & \\frac{\\rho^{T-2}}{1-\\rho^2} \\\\[8pt]\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\[8pt]\n\\frac{\\rho^{T-1}}{1-\\rho^2} & \\frac{\\rho^{T-2}}{1-\\rho^2} & \\frac{\\rho^{T-3}}{1-\\rho^2} & \\cdots & \\frac{1}{1-\\rho^2}\n\\end{bmatrix}.", "3297540ddf15586af735e57dca329020": "\\mathcal F=\\{f_i: X \\to K, i \\in I\\}", "32979a55fd8fcbce8a85224c4366fc19": "A=U^* B U", "3297a9c4412a631503f6ac6059159925": "HP_B=HP_A+HP_C.", "3297ccc1ddcb4d70d56592e2b84e66a9": "\\{x \\in V\\mid x\\not\\in x\\}", "329810e1508a8b5dd16c8bdba3aa283a": " [\\mathcal{H}_2,\\mathcal{H}_{1}]=0, \\, ", "329816c252757d1cf6b67d12faa7989f": "\\Gamma_* = \\Gamma_{k\\rightarrow\\infty}", "32981a13284db7a021131df49e6cd203": "js", "329843a2dfe51a1fc9016986cb9fc0e4": "A \\oplus B \\ne \\N", "329857cd0d3c9d4d03277c0d734cf731": "(X;T)", "329858159a8e128639895320506a7df9": "\\frac{d}{dx} e^x = e^x", "32985a2ca08b3a278d7062e1496558d5": "\\epsilon_k = \\frac{\\hbar^2 k^2}{2m}", "329893e73df41936b38fb9df965c5f02": "b_1, b_2, \\cdots, b_m", "32991ec5a24eaa328efad850d8069861": " \\left\\langle {\\overline \\Sigma _t } \\right\\rangle \\ge 0,\\quad \\forall t. ", "32993772b26021a52c521eecd7c59040": " R \\geq 1 - (1 - r)m \\, ", "32993a265f6aac965d854b1383e1bae0": "[\\mathbf{x}]\\cap \\left(\\mathbf{y} - [J_f](\\mathbf{[x]})^{-1}\\cdot f(\\mathbf{y})\\right)", "329948c421f020ba70d18723cd872f60": "|\\nu_a\\rangle", "3299be22160db6a27402bc9825c11a71": "\\vec s =\\frac {d \\vec j} {dt}=\\frac {d^2 \\vec a} {dt^2}=\\frac {d^3 \\vec v} {dt^3}=\\frac {d^4 \\vec r} {dt^4}", "3299c05ec0d8c879603729e373ddf836": " f_0(z)=z", "3299de5fc8cb2575966976a2353dddc9": "a^k_{\\;1} \\approx -y_1", "3299f7b7c9238f3ffbe0ba0a2f759de9": "\\theta \\to 0", "329a162ea33dceb754bf722113ffaf9e": " \\hat{f}(k,y)=\\int_{-\\infty}^{\\infty} f(x,y)e^{-ikx}\\textrm{d}x. ", "329a3e0931952c16ca19d05974bc7fcb": "c_i(t)", "329a419a68733f2ff400a12318a96c5b": "{\\mu}\\,\\,", "329a4863186ef6ed7914726896ede726": "H = \\frac{1}{2} \\sum_n x_n D x_n", "329a8848845afa8814a5949087e2954f": "V \\otimes k \\cong V \\cong k \\otimes V", "329a9ed96a990f3dad064def9e6b8057": " \\times (\\sigma^{2})^{-k/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\boldsymbol\\beta -\\boldsymbol\\mu_0)^{\\rm T} \\boldsymbol\\Lambda_0 (\\boldsymbol\\beta - \\boldsymbol\\mu_0)\\right)", "329abbfd5cf0a4f2c22496daba4d8b24": "\\boldsymbol\\theta^*", "329af53681e882b193eaaf7f3b215880": "y(\\theta, \\varphi) = (R + r \\cos \\varphi) \\sin{\\theta}", "329c51e5d9183d1ac304ed7c08200130": " \\bar{x} < \\sigma^2 < 2 \\bar{x}", "329ca49c874e7b9c08cae1ca3f6c2f80": "( \\varnothing, (0, 1))", "329cee8e9e5260488ef8bb15cf362eb3": "\\{v,r,\\theta,\\phi\\}", "329d00e30e024c641720fdb85812bebc": "z_2 = x_2 y_1 + x_1 y_2 + x_4 y_3 - x_3 y_4 + u_2 y_5 + u_1 y_6 + u_4 y_7 - u_3 y_8", "329d023c48fa641a9d2ff3c056d0a8e6": "RC^{real}", "329d3ffe2de18c3b6e6134b9150254d9": "M \\colon \\mathbf{B} \\to \\mathbf{C}", "329d4d4739da7902ccd87a338ceac81e": "\\text{Earnings per share} = \\frac{\\text{Net income} - \\text{Preferred stock dividends}}{\\text{Weighted average of common stock shares outstanding}}", "329d7712648d6b3d31c507ddd97fab0d": "p \\in \\mathbb{R}^L_+ \\ ,", "329d9fbe92a70e01fe5ebf21a9942fbd": "(e_i,f_i)=1", "329e06a626ffc823b7577e48f389d4eb": "h(x) \\le d(x,y) + h(y)", "329ed74402850551fa1f81e814d28e87": "\\begin{pmatrix}\n0 & 0 \\\\ 0 & 1\n\\end{pmatrix}", "329ef1d0941d6c4f7b95093d4bee8e06": " f(t - a) u(t - a) \\ ", "329f6ed98f7567714f111072a7bbf885": "\\vec{p}\\!", "329fb2a46c1e3ddb439a4d7c8d5ceaf2": "\\operatorname{E}(\\mathbf{1}_A)= \\int_{X} \\mathbf{1}_A(x)\\,d\\mathbb{P} = \\int_{A} d\\mathbb{P} = \\operatorname{P}(A)", "32a015278afb9e1a1c17fd3b38d10077": "T/(\\mathfrak{p}_1 \\cap T)", "32a031728626f21d3e27c421c1b7044c": "c_\\mathrm{max}", "32a04bb4db7c60ceeb9b55138c028c51": "S_n-K", "32a07d0eb5aeca32b6e0a165a677227e": "|J_1\\dots J_n|\\ge(p-1)!^n", "32a174c4f28a31428b4422b4a3af1ab7": "F(x) = 1 - \\frac{p^2}{2!}x^2 + \\frac{(p-2)p^2(p+2)}{4!}x^4 - \\frac{(p-4)(p-2)p^2(p+2)(p+4)}{6!}x^6 + \\cdots ", "32a20ebbed3d5d75d9f4b52eac2ab0ce": "_{s.7 \\,}\\!", "32a243f63d69a7733f7eb3a8e3859bff": "\\scriptstyle t ", "32a29129bd11742180b580c54eae2d84": "R_x\\left(\\tau \\right)", "32a2bee8d99b73d3bd633dc46cd944af": " y(s,t) = {3 s + 3 t + {1\\over 3} (s^2 + s t + t^2)^2 \\over t (s^2 + s t + t^2) - 3} ", "32a32c0d03f3255f6245043e7d62d9a7": "X^{(1)} = X'", "32a39466fa2e2c0eb14b6b351e32bdf8": "\ng'(\\alpha)\n= \\mathbf{x} \\cdot \\nabla f(\\alpha \\mathbf{x})\n= \\frac{k}{\\alpha} f(\\alpha \\mathbf{x})\n= \\frac{k}{\\alpha} g(\\alpha).\n", "32a3e5cb042bc3a893638a18c7b3aa4f": "3\\uparrow\\uparrow\\uparrow\\uparrow n", "32a3f938055d1ec828e9c3efbd87ecf2": "(A\\otimes B)^T = A^T \\otimes B^T.", "32a44830f2f54405edeacfb2295e88bb": "d\\mu(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2/2}\\, dx", "32a44938a7f08cf19d638ddbbbe812d8": "t\\mapsto t+\\theta\\mod 1", "32a44a7bb9b87aa20e414da2c9caa858": "(0.4 \\cdot 1 \\cdot 1.78 + 0.1 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.0961) = 1 \\cdot 1", "32a45b38ee9f99c997814fc2d30cf2f0": "h=L/m", "32a465c972b0e482ea0bfa40c93aaf79": "\\scriptstyle\\phi\\in C_c^1(\\Omega,\\mathbb{R}^n)", "32a56d64a653f7991d54ed0b9d7c0902": "\\mathbf{T}_{K,i}", "32a5739588257da5da58748a35f8bfac": "\\mathrm{curl}\\ \\mathbf{x}=\\mathbf{0}. \\, ", "32a57eca1911724547c1614ef7a26971": "N^j", "32a58da6fb09e7362aa1cc1e52b0ba75": "\\boldsymbol\\beta-\\boldsymbol\\mu_n", "32a5bb121f133e58c27732c7aacad03e": "\\alpha_i\\ge 0 ", "32a5e0a6024fd4e73b8d21858c650160": "x_k[n]\\,", "32a622eadae57a402c590ebb6c694c2c": "g_F", "32a6dc66b308a2c2b89295c21574b1c8": " \\Delta G^{0}_{T} < \\Delta H^{0}_{T} ", "32a7235d98b434d8fdb94e825796c60c": " f=f_1", "32a77e7ad36add10501177510e2aaecd": " x_{i} ", "32a7cc0f56361b338d113ad2b0634269": "n > 1.", "32a867a3e27b3beb10120dbcc243e727": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{versin}(x) = \\sin{x}", "32a8a066d4507cd46852aaa757d9c230": "x_1-x_3", "32a8efa0da15638a0472b859a9998e59": " p = \\frac{RT}{V_\\mathrm{m}}.", "32a8fd18eb4026234f1446d0aee92ee8": "\\cos C\\cosh a=\\sinh a\\coth b-\\sin C\\cot B.", "32a930c603f01a49aaf90457bb1dc3d4": "\\mu(E_{n_k,k}) < \\frac\\varepsilon{2^k}.", "32a936e90eb2399324368c35e9190255": "\\frac{dE}{dy} = y - t ", "32a9a7d125696f6ac401e36a61e854c2": "u[j] = {1 \\over N} \\sum_{i=1}^n X[i,j] ", "32a9e36d922c8608391f0b90ea84dfeb": "D_2 + D_4 = P_1 + P_2 \\bmod 2\\,", "32a9f2a414f5316f992abe102cc7c5b3": "\\Phi\\left(\\mathbf{r}\\right)=\\frac{1}{4\\pi}\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\frac{1}{4\\pi}\\oint_{S}\\mathbf{\\hat{n}}'\\cdot\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'", "32aa02e47f1ae60f2840275196e1942d": "(HA + ECT_0 \\vdash \\phi) \\leftrightarrow (HA \\vdash \\exist n \\; (n \\Vdash \\phi))", "32aa5af90c679ef555f9995fe6eba985": "\\alpha \\in (0,2\\pi/3)", "32aa686a2cfbe4a46f6c5635527e0648": "\\psi \\varphi ", "32aa88c95e374ad86ae74082feaaeb25": "\\left(a \\rightarrow a \\rightarrow STOP\\right) \\Box \\left(a \\rightarrow b \\rightarrow STOP\\right)", "32aac338bd80c540249e7db67a1a0dda": "\n F =\n \\underbrace{\\rho\\, V \\dot{u}}_{a}\n + \\underbrace{\\rho\\, C_a V \\left( \\dot{u} - \\dot{v} \\right)}_{b}\n + \\underbrace{\\frac12 \\rho\\, C_d A \\left( u - v \\right) \\left| u - v \\right|}_{c}.\n", "32aae338a7571f548c9c262fab6df3ca": "{\\int_K|f|\\,\\mathrm{d}x}={\\int_\\Omega|f\\chi_K|\\,\\mathrm{d}x}\\leq\\left|{\\int_\\Omega|f|^p\\,\\mathrm{d}x}\\right|^{1/p}\\left|{\\int_K \\mathrm{d}x}\\right|^{1/q}=\\|f\\|_p|\\mu(K)|^{1/q}<+\\infty,", "32abb683cb18d5f5d3b6718d7a676f09": "\\lbrace \\vec{b_i} \\rbrace", "32abbba1988cfeedbcc41dfa8fa0e3b9": "\\mu(\\emptyset)=0", "32abdc2838a32e83fed633dac1c0b143": "\\|A_0-A_0 A A_0\\|<\\|A_0\\| ", "32abe7f59bf491c0bc7b6d6f5c76e38a": "\\begin{align} &[D,K_\\mu]=-iK_\\mu \\,, \\\\\n&[D,P_\\mu]=iP_\\mu \\,, \\\\\n&[K_\\mu,P_\\nu]=2i\\eta_{\\mu\\nu}D-2iM_{\\mu\\nu} \\,, \\\\\n&[K_\\mu, M_{\\nu\\rho}] = i ( \\eta_{\\mu\\nu} K_{\\rho} - \\eta_{\\mu \\rho} K_\\nu ) \\,, \\\\\n&[P_\\rho,M_{\\mu\\nu}] = i(\\eta_{\\rho\\mu}P_\\nu - \\eta_{\\rho\\nu}P_\\mu) \\,, \\\\\n&[M_{\\mu\\nu},M_{\\rho\\sigma}] = i (\\eta_{\\nu\\rho}M_{\\mu\\sigma} + \\eta_{\\mu\\sigma}M_{\\nu\\rho} - \\eta_{\\mu\\rho}M_{\\nu\\sigma} - \\eta_{\\nu\\sigma}M_{\\mu\\rho})\\,, \\end{align}", "32ac908af753ca5c008dc40b7e870e29": "for\\, each\\, cluster\\, cl \\in C^{bestSubspace}", "32accbcf83c3378ef0a2a922ae42a1a4": "\\psi_{\\alpha,a,b}(t)", "32acd9d2d9208dea681f8b749f191c2a": "x^6+x^3+1", "32ad8282a984717ae84e97127f362a03": "\\alpha(i)=\\sup_{\\tau>0} \\frac{R^\\tau(i)}{Q^\\tau(i)}", "32ada9fb05d53f8ee0cf428aa6700105": "|X_1|\\le k", "32adae3da87af1fd901480b147505b36": "\\frac{\\sin\\theta_1}{\\sin\\theta_2} = \\frac{v_1}{v_2} = \\frac{n_2}{n_1} .", "32ade69d37f06df0a6ea4f6289c96d8a": "\\gamma^m(t):=\\gamma(mt)", "32ae06af571f5593eb08bb3ba8b58721": "|\\nabla _{t}f(x,t)|\\leq B", "32ae3a3668be2fbbc123c48267bbe305": "\\cosh^2 x - \\sinh^2 x = 1", "32ae863df70260be583c1d4223b85550": "F_2(x_0) < M", "32aef280785330eee045657f2d26bdf8": "^2 R,\\ ^2 C,\\ ^2 H \\!", "32af113e8b3e1d515f08334443d3a14e": "q_j\\,", "32af19debe9ff8d97f8787e35b068719": "(\\mathbf{a\\times b})_1 = a^2 b^3-a^3 b^2\\,,", "32af1fb52d9c1c3e294b707e9eddf995": "\\!k", "32af7619b19e9fb7d3823e5baa5ee988": "agh+cef=beh+dfg", "32af9dbf641d6541a5b4572878f150de": "\\scriptstyle z' \\,=\\, z", "32afaa66657e87b5d3579fc8941f91ae": "v_{e}= - \\sum_{i=1 ,i \\neq e }^{k+1}v_i.", "32b008ea7f1339d9f44cfc7f89a1a40c": "\\mathfrak{P}^{59}", "32b02b63ec9577f59ff50b8b22eeb7b9": "\\frac{2\\pi d}{\\lambda} \\ge \\pi", "32b06f12f9e316c00b60ae1332299baf": "(R,cR,P_1,P_2)", "32b08edb43969810694e7e39d2af5e9f": "\\begin{cases}\n\\Phi_V:\\mathbf{R} \\times (E\\setminus 0) \\to (E\\setminus 0) \\\\ \n(t,v)\\mapsto \\Phi_V^t(v) := e^tv.\n\\end{cases}", "32b0b334d298072c76ef7228772704dd": "\\sum_{n=0}^{N}a_n(\\alpha cos(\\omega t + \\phi))^n ", "32b0be0fb864a66b9573b7f45577099f": "I = \\sum x^2 ", "32b0c7a2a11d218d081b46ac00d1208c": " d=100", "32b0d9aa48f26fa30275096e680db226": "f_{j} = \\mathbf{1}_{\\pi_{j} (E)}", "32b0e26269e0e3936a389d47e309bed4": "f : \\Gamma^* \\times I \\rightarrow \\Gamma^* \\cup \\{\\empty\\}", "32b0f1a905ff94ad876c42244a749bd2": "\\! G", "32b14838cee67b577c7efa7a95a4e048": "S(a,b)\\,", "32b1b00f054bb5e3542823e232b5c8c3": "r_i = (\\mu_{i,j}: j\\in J)", "32b1b3cab536897af71bc481d31f8286": "g(q)=reject", "32b1d36094c7fa09e4c28aca1d62ad79": "^\\ddagger", "32b1d8529e997f6c7ae7cc0dd4bd748e": "V=\\sum_{i=1}^K Y_i\\sim\\operatorname{Gamma} \\left(\\sum_{i=1}^K\\alpha_i, \\theta \\right ),", "32b1dd4037d91ec6062ff6c109c57bc8": "C^\\infty(Z)", "32b1e021a02393d68d064d7dc844a4d1": "x_C \\notin U_a ", "32b25a26d6e2bca8cec629ee4129e8fe": "S_{d(k-2)}=1 \\oplus \\bigoplus_{j=1}^{d-1}S_{j(k-2)}\\oplus \\bigoplus_{j=1}^{d}K_{j1(k-2)}", "32b26d7e73af6ebdeacc1029d1aaa5eb": "\\sin\\frac{\\pi}{8}=\\sin 22.5^\\circ=\\tfrac{1}{2}(\\sqrt{2-\\sqrt{2}}),", "32b29632b6d50b869853c79ca32e31fa": "\\mathbf{a}\\cdot(\\mathbf{b} + \\mathbf{c} ) = a_i ( b_i + c_i ) = a_i b_i + a_i c_i = \\mathbf{a}\\cdot\\mathbf{b} + \\mathbf{a}\\cdot\\mathbf{c} ", "32b2bd745e27ae803dca208fa4eafdb2": "\\Delta = \\nabla^2", "32b2f05af37f3bf21476494a43fe3e9c": " C_{ijkl} = -C_{jikl} \\qquad C_{ijkl} = C_{klij} ,", "32b329eb483aa2953bcf5c1481bd73ea": "\\sqrt{\\frac{9}{20}}\\!\\,", "32b37ee266befec85bd4cbf265c6ac75": "K \\otimes C(X)", "32b3884295e4d053b15c77a24d736fe6": "a_\\max", "32b390148a40318c6a192a564482dffb": "J^{\\alpha} = ( c\\rho, \\bold{J} ) ", "32b39f59175d6b4fcf0510bdaad5f4b6": "\\lambda = \\frac{\\varepsilon_1 - \\varepsilon_2}{\\varepsilon_1}.", "32b3c09b0f168aeb3aa2e7fdb513d8f3": "p_{0 \\tfrac{1}{2}} \\leftarrow 64x^3+384x^2-1024x+512", "32b409be0b0508cb2611eb1ad91ad0dc": "\\phi(z) = b_1z+b_3z^3+b_5z^5+\\cdots", "32b426114460790711d6074bd8444b8d": " \\mathbf{H}(x) = -\\mathbf{A}(x) \\cdot \\nabla u (x) ", "32b44081119de2eb646066f510ffdfcf": "r(a)=s(a)", "32b4c451ccdccaa6fa1c786383b2f33a": "\\frac{\\partial (\\mathbf{U}+\\mathbf{V})}{\\partial x} =", "32b54363b2ccf86a2276588e57389b00": " f(x) = \\frac{1}{1-x} + 1 \\, .", "32b5516a8767b24c80488132fc0f139f": "\\phi_X \\, ", "32b56f6f7bec29ad6ec60616f742f825": "\\tbinom{p}{n}", "32b58132d28dfce68875c251db0cca0b": "{_4^2}\\text{S}^{\\beta} \\rightarrow {_2^0}\\text{P} + {_2^2}\\text{P}", "32b58186bab052cfdb19e7c7b91b21a9": "X_0,X_1,X_2,\\dots", "32b5f02a68dbaeba5365ebd54b5192a7": "x = r\\cos\\theta = 2a\\sin^2\\theta = \\frac{2a\\tan^2\\theta}{\\sec^2\\theta} = \\frac{2at^2}{1+t^2}", "32b611ace5971a9ed5f27c37c9f81769": "1 \\over {w + 1}", "32b61ad5f73d12596d5f8ae4efb5d28c": " \\tau ", "32b6e30af02e2bae277a1002a5279e3c": "1, 2, 3, 4, 5, 6.\\,\\!", "32b70d62522f5a86f5651e0b538200a3": "\\mathrm{sinc}(0) = 1", "32b71954576c907f47297c3a78b0109b": "\n\\begin{align}\n\\Pr(Y = 1 \\mid X) &= \\Pr(Y^\\ast > 0) = \\Pr(X'\\beta + \\varepsilon > 0) \\\\\n&= \\Pr(\\varepsilon > -X'\\beta) \\\\\n&= \\Pr(\\varepsilon < X'\\beta) \\quad \\text{(by symmetry of the normal dist)}\\\\\n&= \\Phi(X'\\beta)\n\\end{align}\n", "32b7389c49430928131da1265fc87201": "\\tfrac{26}{11} = 2 \\mbox{ remainder } 4.", "32b7f8623531f63aafc9bc1c59a10fbd": "\n\\Omega_{\\alpha \\beta I}^{\\;\\;\\;\\;\\;\\; J} V_J = (\\mathcal{D}_\\alpha \\mathcal{D}_\\beta - \\mathcal{D}_\\beta \\mathcal{D}_\\alpha) V_I ", "32b822d72a0310566d18be8d2edecd31": "\\oint_S \\mu_0 \\mathbf{H} \\cdot \\mathrm{d}\\mathbf{A} = \\oint_S (\\mathbf{B}- \\mu_0 \\mathbf{M})\\cdot \\mathrm{d}\\mathbf{A}= (0 - (-q_M)) = q_M,", "32b83959807b49e3b7db040e93c72d84": "U_\\epsilon C", "32b8443e6b33223eaf858b0d99d990c0": "[x_1,x_2,...,x_k]", "32b84f248915d98f08a872734a5db0d9": "B_0 = I", "32b88f648b7a7b9acd8fb2b059d2dea2": "\\log p(f(x)|\\theta,x) = -\\frac{1}{2}f(x)^T K(\\theta,x,x')^{-1} f(x) -\\frac{1}{2} \\log \\det(K(\\theta,x,x')) - \\frac{|x|}{2} \\log 2\\pi ", "32b8a8c50a006ad18c2c232aeb22d2a3": "\\sum_i N_i=N\\,", "32b8cb320da17c2f1c37d60cced2f000": "(u,v) \\in R", "32b8d95ba75b25a33cb8d2d82a47ecd3": "(\\neg P\\or P)\\and (\\neg P\\or Q) ", "32b91568e9f836668d90fa7b1c13a981": " C = \\sum_{i=1}^8 \\lambda_i \\lambda_i = 16/3 ", "32b9a58edce9c2ea11793090294e1670": "b \\in (\\dots,-2,-1,0,1,2,\\dots), b \\ge a", "32ba18abae12383b9ceee37d0ffc6846": "x_1=x_2=\\dots=x_{N-1}=x_c", "32ba2938ef05478c4c67d6d5e7a61983": " 7^3 + 13^2 = 2^9", "32ba4e8d3ff1431ae027ebc28f1962e9": "[\\hat{a}(t),\\hat{a}^\\dagger(t)] = 1 \\ ", "32ba6f7323fb02f7748f26d37e380246": "\\dim_{\\mathfrak p}:K_0\\left(A\\right)\\to \\mathbf{Z}", "32ba82bdec4aff0796f4c058678707cd": " U_\\mathrm{E}^{\\text{single}}", "32ba8415b0687f3e7bd4575ee6e97ee2": "\\int_a^\\infty f(x)\\, dx=\\lim_{b \\to \\infty } \\int_a^b f(x)\\, dx ", "32bac3a0a42ed120b8dc0cbf969f209e": "H_X (\\Beta(\\alpha, \\beta) )=H_{(1-X)}(\\Beta(\\beta, \\alpha) ) \\text{ if } \\alpha, \\beta > 1 ", "32bb02d3efcc33bcefd4cf759b06c2bd": "P^{m}\\{|Q_{P}(h)-\\widehat{Q_{x}}(h)|\\geq\\epsilon\\,\\!", "32bb4df06d1ee7d374fa13d1c0c70e4a": " X_{i,i'}\\, ", "32bb56677810492704c3e326a8fe4dfc": " R = E\\{x(n)x(n)\\}", "32bb64137deef555998247cca2bee915": "A_1=\\cdots=A_n=A", "32bbd4a286ac4d26a80c9c10182eead0": "t \\in [0,T]", "32bc3532733185c29398a8dfb3ceed01": "\\scriptstyle \\geq7.2\\times10^{17}", "32bcc493b0e630065dae078696515a9a": "T^2(\\Omega)<\\Omega", "32bcfe0802231a253afdfd0705b63e92": "f_i(x+h) - f_i(x) = \\nabla f_i (x + t_ih) \\cdot h.\\,", "32bd66c1a66539cd64468266665bda0b": "\\scriptstyle-\\infty", "32bdb035773fe3f338b36937883f9aab": " P(Y_3=0)=\\left(1+\\sum_{n=1}^\\infty \\left(\\frac{1.25}{10}\\right)^n\\right)^{-1}=\\frac{7}{8} ", "32bdbdd3ebada86ee3d46fd4f47444e2": "z^{1-a}\\;{}_0F_1(;2-a;z),", "32bdbf33ed529d17ea71859a8d2c5e9f": "g={1\\over 2}(d^*-1)(d^*-2)-\\delta^*-\\kappa^*", "32bdc37d6f3121c4ed8a31b6ec93fa8f": "\\sum_{i=1}^n w_i\\, \\|x_i-y\\|,", "32be90544179de03a9ea07b3303d2083": "\\alpha_{\\mathbf{v}} = \\frac{1}{\\gamma_\\mathbf{v}} = \\sqrt{1-\\frac{|\\mathbf{v}|^2}{c^2}}", "32be96e2c59d87ea0b2f1e5de81105a3": "\\alpha=0.9", "32be9d724d521b6acffd01bd4b1b5d1b": "P_{seq}=C_{total}*V_0^{2}*f", "32befb2cad3a6978975c81c24342f53c": "\\phi_{Pk}=\\phi_{wk} - \\left(\\frac{\\delta x_i}{2}\\right) \\left(\\frac{\\partial \\phi}{\\partial x}\\right)_{wk} + \\frac{1}{2!}\\left(\\frac{\\delta x_i}{2} \\right)^2 \\left(\\frac{\\partial^2\\phi}{\\partial x^2}\\right)_{wk}+\\cdots.", "32bf07adbe1303f98ac28efda6009a0c": "\\mu_{nb}^{(c)}(t)", "32bf51c6c05bbc7ca2f751d51ff5fed8": "Q_{AdiabaticWalls}=0", "32bf79476e5914e8ba78d1518fb9990b": "f(t) = \\sum_{j=-\\infty}^\\infty \\sum_{k=-\\infty}^\\infty \\left[ a_{j,k} w(2^j t - k) + \\tilde{a}_{j,k} w^*(2^j t - k)\\right],", "32bfaa55fd07f4a7dc539ba31c06a21f": "\\left. \\cdots \\right|_{\\mathrm{scatter}}", "32bfc8b188f4ffa1068c1fcfcc62f509": "e_u=\\frac{g}{h}", "32bfec255d44328daa57cc8fb2890fe2": "A=\\frac {1}{2}\\left|\\int{\\left(xy'-yx'\\right)dt}\\right|", "32c01e8eef1221d19ec1e635a3099cb5": "Y_i=\\rho_i/\\rho", "32c0788383d5e4a2b60c8d74c9424924": "\\lim_{a \\to \\infty}\\pi \\left( 1 - {1 \\over a} \\right) = \\pi.", "32c0eee889a723de0d4bd57903791d68": " \\displaystyle \\mathfrak{f}(\\tau) = q^{-1/48}\\prod_{n>0}(1+q^{n-1/2})", "32c1145e6baaf5179224fea38a89553b": "\\Psi_g:G\\to G", "32c13b97268f1917b5b6e43fd258ba96": "R_{\\mathrm{g}} = \\frac{1}{ \\sqrt 6\\ } \\ \\sqrt N\\ a.", "32c1b332ec93459d8e503144414283cf": "F_1=1-F_2=\\frac{r_1 f_1^2+f_1 f_2}{r_1 f_1^2+2f_1 f_2+r_2f_2^2}\\,", "32c202f318e8d31fc9b3b237a599a138": "\\eta = \\frac{y}{y_{max}}", "32c21a9c19f155e32a06efae5da63694": "\\frac{1}{2}\\sum_{\\mathrm{v}} | P[A = v] = P[B = v] |", "32c21f740ee95de3e06b35af4d4b67c1": " D_0 = \\frac{\\alpha^2 h}{3n}I_s", "32c2510fa5f0fb99301f16c388c5260f": "q^{1/24}", "32c26825714e5e6f541a28fdac9f953a": "d(x,c_x)", "32c2bf797519d0190f3c98dc6c504d65": "\\eta = y + Y", "32c30d3fff5b16326ed48c3fd072ba94": "0 \\leq i \\leq n/m-1", "32c3535413e7766a9f628bc399686451": "\\textstyle \\theta = 0.5", "32c379abc6ccf674a5d3abb48df323cf": "\\Delta^2/12", "32c39b16e0f32ba002bf74436aff344b": " = (\\lambda p.p\\ (\\lambda x.\\lambda y.x))\\ (\\lambda z.z\\ a\\ b) ", "32c39ec602a980d626ca0fcfe9c8f62c": "\\begin{align}\n(v_1, w) + (v_2, w) - (v_1 + v_2, w),\\\\\n(v, w_1) + (v, w_2) - (v, w_1 + w_2),\\\\\nc \\cdot (v, w) - (cv, w),\\\\\nc \\cdot (v, w) - (v, cw),\n\\end{align}", "32c40cfd37b23082f3bb04dfab58335a": "Im\\left( df_x \\right) + T_{f\\left(x\\right)} Z = T_{f\\left(x\\right)} Y", "32c4890dd3c35b07382ce0a75c467bb5": " \\ell (f) = \\int_{a}^{b} \\sqrt{1+(f'(x))^2}\\,\\mathrm{d}x,", "32c4afb0400a96fe369391e9757ffc07": "\n\\gamma_{\\pi}^*(t) := \\frac{1}{2} \\sum_{i=1}^n \\pi_i(t) \\sigma_{ii}(t) \n-\\frac{1}{2} \\sum_{i,j=1}^n \\pi_i(t) \\pi_j(t) \\sigma_{ij}(t)\n", "32c4bb04dd3a056780e0b522117bdcc3": "\\begin{align}\n(1 - B)^d &= \\sum_{k=0}^{\\infty} \\; {d \\choose k} \\; (-B)^k \\\\\n& = \\sum_{k=0}^{\\infty} \\; \\frac{\\prod_{a=0}^{k-1} (d - a)\\ (-B)^k}{k!}\\\\\n&=1-dB+\\frac{d(d-1)}{2!}B^2 -\\cdots \\, .\n\\end{align}", "32c4e84c7dfc002f62230de1c75e52d4": "\\mathbf{\\nabla}\\times \\mathbf{B} = \\mu_0\\mathbf{J} + \\underbrace{\\mu_0 \\epsilon_0 \\frac{\\partial }{\\partial t}\\mathbf{E}}_\\mathrm{Maxwell's \\ term}", "32c4f752ff622b85e33401959f2d6ae3": "|d(\\varphi_1-\\varphi_2)|^2", "32c5ca3d6496a7091cf725631ee7a776": " rN\\left(1-\\frac{N}{K}\\right) = H ", "32c5f2f717fc7202b44dbd0e650e18f0": "c_{p+1} \\in GF(p)", "32c5f3e9a408da54e2077f32157a4b2e": "\\le -d(v)/T", "32c60e468f5bab27e5cfc501a748f08d": "\\|A^*A\\|_{op} = \\|A\\|_{op}^2", "32c64e570a3b92171cb6bd0c09b98125": "x_k\\,\\!", "32c68845a2d1430e6bd3ecda98363bea": " \\psi(z)=U \\,:\\,\\exp \\, \\int \\phi(z) \\,:", "32c6a28a298dd990b9e2f74455d43033": "e^{i\\varphi} \\rightarrow P_r(\\theta - \\varphi).", "32c71da83a223d87f462e376ab7edacb": "p \\approx 2^d", "32c7658f4a24ad2203512cf0543e83cb": "\n\\hat{{\\Z}}^\\times \\to G_\\Q^{\\rm ab} = {\\rm Gal}(\\Q(\\mu_\\infty)/\\Q), \\quad x \\mapsto (\\zeta \\mapsto \\zeta^x), \n", "32c815efb87c51d75db817528febc86f": " J_{n} (u, v) ", "32c820e8988eca5ac9f7a3a1b2322e21": "e^{-z}/z", "32c83355a4da15892e1eba6f7b20748a": " g = \\bigoplus_{j\\in\\mathbb{Z}} g_j,\\quad [h, a]= ja {\\ \\ }\\textrm{ for } {\\ \\ } a\\in g_j.", "32c83554016c921b9277c03a12775063": " = \\operatorname{Tr}\\left( p\\!\\!\\!/' \\gamma_\\mu p\\!\\!\\!/ \\gamma_\\nu \\right) + m^2 \\operatorname{Tr}\\left(\\gamma_\\mu \\gamma_\\nu \\right) \\,", "32c84d79eab0a09108d366cdf4e8de07": " y = scos{\\theta}", "32c866674ad9dfeeac744fddc779c522": "q \\geq 1", "32c87239c63d19afa3b346c0318c89fd": "V_{\\text{R}_{\\text{b}}}", "32c88715f6e14fb5d54adab0632bd420": "Lu_G(x)=0", "32c88b304219490696a9571e06cd9217": "I[p]", "32c89ba98aaf75a06138bdc0ac482dcf": "\\sum_{i=0}^{n-1} i", "32c9075e5421d596e006bc02d84f89b1": "f\\colon A\\rightarrow B", "32c9ca8dfa410eebe9ce78161f1c45ff": "f(z)={1 \\over (z^2+1)^2}={1 \\over (z+i)^2(z-i)^2}.", "32ca87d9b1a9f1b35230dadc23373c41": "I(x; t) = h(x; t)*I(x)", "32cabdbf0a9e2188becf28c75a03007d": "{\\bar{O}}_4", "32cba23503c750f48829c1b624155ac2": "w(x+iy)=V(x,y)+iL(x,y)", "32cbc44c7f7df6cfd14d00cfee2e7afa": "\\lceil X \\rceil_\\infty", "32cbccdd6d28301eb6d682b808f0bbe2": " \\sum_{F \\subseteq E|i \\in F} M(F) \\geq 0", "32cc1ba613157e7e9e05786fc090dd1c": " a(\\omega b)=(a\\omega)b,\\ \\forall a,b\\in A,\\ \\omega\\in\\Omega^1", "32cda2b34c7414b104cecb0194ec6e6a": "\\exp : \\mathfrak g\\rightarrow G ", "32cdd13b449700e24e4e36ee6aaafc65": "b(b(H)) = H", "32ce1d56d3962830cbdf854582446641": "V_f = V_e \\ln(M_0/M_f)", "32ce2120a0cc07b7fdd1363014d32f81": "= [1]_q [2]_q \\cdots [n-1]_q [n]_q", "32ce3bf5a52e0ca264e0ae115cd73c88": "\\epsilon_{SR}^{-1}\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial w_t}{\\partial L_t}\\frac{L_t}{w_t},\\qquad\\epsilon_{SRL}^{-1}\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial w_{t+1}}{\\partial L_t}\\frac{L_t}{w_{t+1}}\\,\\!", "32ce444c315f012914b5705b55570e37": " \\lambda(x) ", "32ce54fe2e5704177c19ecbb539a7b8d": "a_2b_0", "32ce5d8c8dd07bb78b15a698cbb8d6f4": "\\ e=\\frac{\\ell-L}{L}=\\lambda-1", "32cea533fd26d97152c2bc219e853afc": " M_{X_1} ", "32ceaf854b40c96da8d5e1ffd1c5dcff": "\\nabla\\times \\mathbf{B} = \\mu_0 \\mathbf{J}", "32ceb515a29523e46272ecd21044d6ec": "i\\frac{\\partial}{\\partial t}\\rho_{1} =PL\\rho_{1}+PL\\rho_{2},", "32cee5f6e7aa0903b07a57e7e0d57da6": "\n\\frac{N^{-1}\\sum_{n=1}^N(\\bar{x}_n-\\bar{x})^2}{s^2},\n", "32cf047087f33d8c2f3c4eadb949a5eb": "T_m(z)=T^\\sharp_m(z), m=n,n-1, \\ldots , 0 ", "32cf0832308d42a21029de42436e8825": "\\int \\frac {-\\sigma_my}{c} dA = 0 ", "32cf14a3658161a9d3d4748b94e85f9b": "\\mathbf{v}[\\mathbf{f}] = \\begin{bmatrix}v^1[\\mathbf{f}]\\\\v^2[\\mathbf{f}]\\\\\\vdots\\\\v^n[\\mathbf{f}]\\end{bmatrix}", "32cf2b96306858043fd6f979820f92dd": "\\sin\\frac{2\\pi}{5}=\\sin 72^\\circ=\\tfrac{1}{4}\\sqrt{2(5+\\sqrt5)}\\, .", "32cf56a1080355325170638c9d84e58e": "T=\\frac{1}{4}\\sqrt{2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}", "32d06f3555fa6ec560c760fe34102077": " Z/p^2Z", "32d0c24118e54db23e2c5df17fb585ae": "\\int_r^\\infty \\int_r^s \\frac{-2 y}{\\pi \\sqrt{(y^2-r^2) (s^2-y^2)}} \\, dy f'(s) \\,ds = \\int_r^\\infty (-1) f'(s) \\, ds = f(r).", "32d0f00d529e80c59a88137ee9ff0ec8": "Q_L=\\frac{\\omega'_0L} {R}", "32d1347b4c050e03cdfb8f0e2aa9dd5a": "\\displaystyle s= O\\left(\\frac{n}{\\epsilon^2 \\log (1/\\epsilon)}\\right)^{5/4}", "32d173c94af5b632d258bcc8141fd10d": "a \\in \\mathbb Z", "32d191d6f6608081fcd234a3ae0ffd37": "\\beta_1<\\alpha", "32d1f1316f2b3af70ba8ed64f319802e": "\\sin(nx) = \\sum_{k\\text{ odd}} (-1)^{(k-1)/2} {n \\choose k}\\cos^{n-k} x \\sin^k x.", "32d20e69200d8b80a0cfb9c6ba299656": "Q_A = 10^{R_A/400}", "32d22df8ff86b451ba5f0f4b1ebe8f97": "\\Theta=\\mathbb{R}_{++}^l \\times \\mathbb{R}_{++}", "32d2da3092d0829c51adcf9f41616c5a": " v_i\\mapsto g_i\\otimes v_i", "32d2e2a5f6eff6054788f6705d42bdf5": "\n\\ v(t) = \\sgn(\\cos[t])\n", "32d2e99b6034495bfa1c4a3b62dc3af5": "\\tfrac16\\eta'' + \\left( 1 - c \\right)\\, \\eta + \\tfrac34\\, \\eta^2 = \\tfrac14 r, \\, ", "32d2f115e25de9131be3cc8769577ae2": " 3.1 = y' + \\frac{1}{2y'^2}", "32d2ffa60be268acf28656e4d895cb02": " \\lim_{T \\to 0} \\Delta S = 0 ", "32d33a82f8a552416ca088374269f257": "\\hat f(\\mathrm{rect}(ax)) = \\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{sinc}\\left(\\frac{\\xi}{a}\\right)", "32d34408e9071ce83e3d3a19061e4be9": "{}^\\mathrm{N}\\boldsymbol{\\alpha}^\\mathrm{B}.", "32d3475ca5926b9825142dc68f07ab86": "- (\\beta-1)[\\psi(\\beta) - \\psi(\\alpha + \\beta)] \\, ", "32d37afa9a486558e4480c53bfee927f": "2 \\mathit{9} 46\\, ", "32d38b5fc068d9df72db5c5bc010f913": "\\mathbf{g} = \\mathbf{F}/m \\,\\!", "32d3cd035b86876d2c4a72bf755510e6": "\n{\\partial E\\over\\partial t}+\n\\nabla\\cdot(\\bold u(E+p))=0", "32d3ea3744282f7c34b0b02832e5e882": "\\widehat{x}(n)", "32d45335eae0ba7d6d49013fb805d0cf": "\n3Nk_{B} T = - \\biggl\\langle \\sum_{k=1}^{N} \\mathbf{q}_{k} \\cdot \\mathbf{F}_{k} \\biggr\\rangle.\n", "32d48543136ae6ceba4689af67a62c9a": "x_{cp}", "32d4b02d328027a3397743444a182e6b": "\\textstyle{\\frac {1} {2+\\varphi}}", "32d4ba4180f9e8e140618c878699f5e6": "\\vec{\\sigma}", "32d4d2301848ab92b4cd0cbb80769bf7": "(\\mathbf u_0,\\lambda_0)", "32d4eca3d327c5c9c333a1c56c13ff5f": "\\Delta_\\alpha:=\\bigcup_ {s^\\alpha+t^\\alpha=1} T_{s,t}.", "32d510f6cafd87814d71250d56394784": "\\lambda_c = \\lambda_0 \\sqrt{1-\\frac{\\sin^2\\theta}{{n^*}^2}}", "32d52dbf690c65a3cec25c3b0fb29077": "CRF \\to i", "32d55e054cd1f08af577a0ad7fbbb531": "y_{16}, y_{15}, ... , y_{2}, y_{1}", "32d56064797c37a0b6f608322d5db9be": "\\gamma = \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}", "32d5b4db0098662f997b18da9bc2699c": " M_0=\\Phi^{-1}(0) ", "32d5e93414fe979be185ca16f007edde": "\\langle x_j,x_k\\rangle=0", "32d61a756991470dbf503851a046b5d3": "\\omega = \\sum_{i=1}^{n}dq_i\\wedge dp_i.", "32d6b502e62a8ddcf963c28f482c8d4a": "\\begin{align}[]\n \\left\\lbrack\\mathbf{b}\\right\\rbrack_1 &= \\begin{bmatrix} 1 & -R \\\\ 0 & 1 \\end{bmatrix}\\\\\n \\left\\lbrack\\mathbf{b}\\right\\rbrack_2 &= \\begin{bmatrix} 1 & 0 \\\\ -sC & 1 \\end{bmatrix}\n\\end{align}", "32d6e8f530647deef915856db9d1a344": "\n\\begin{align}\n& \\frac{m_a u_a + m_b u_b - m_b v_b}{m_a} = v_a \\\\\n& v_b = C_R(u_a - u_b) + v_a \\\\\n\\end{align}\n", "32d6ee718cfa00736af7d1d707af831c": "\\scriptstyle +V_{CC}", "32d70be63e01606bd8a6f02484823c8b": "\\langle L_1 L_2 \\mid x^n \\rangle = \\sum_{k=0}^n {n \\choose k}\\langle L_1 \\mid x^k\\rangle \\langle L_2 \\mid x^{n-k} \\rangle.", "32d7aa136f7800242174c895f0612ab4": "V = \\frac { |\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c})| } {6},", "32d7abff10127788e21a9bd4b2db3eba": "\\varepsilon_t", "32d7bcd148dad4f943af93e38373c6bf": "h_{Preucil\\ circle} = 60^{\\circ} \\cdot \\left( 4 + \\frac{R - G}{B - G}\\right)", "32d7c030459c75b14f4b849b1dfdbfd1": "\\psi\\ \\leftarrow CEncode(L_B(P),s,u_1,u_2,v,e)", "32d7d6d977783ab14231d4c195640879": "=-\\left ( \\frac{1}{3}\\right ) \\left ( \\frac{1}{5}\\right ) \\left ( \\frac{2}{7}\\right ) \\left ( \\frac{9}{23}\\right )", "32d825850a009e35612b3f424389c207": "\\scriptstyle G(s)", "32d8893266bc6de53cb71cb0418bc57c": "P_K(\\ell,m)=P_{Mirror Image(K)}(\\ell^{-1},m),\\,", "32d8c01d2f4af980d8fc53d70ae0cc98": "\\begin{vmatrix}\\cfrac{K-P}{P}\\end{vmatrix}=e^{-kt-C}", "32d8cc7151587b1a7dc7d53e1e6f8f75": "\\overline f (\\overline v) = \\overline{\\,f(v)\\,}.", "32d90adc7d3834127c2cd2585ed666c7": "\\dot Q_L=\\frac{T_L}{T_a-T_L}(P-T_a\\dot S_i) .", "32d91c0b5b0b01a28433978ab37d0621": " var [ \\log ( t_a ) ] = \\frac{ \\left( \\frac{ s_a^2 } { t_a^2 } + \\left( \\frac{ x_a - \\bar{ x } } { t_a } \\right) ^2 - 1 \\right) } { n }", "32d96889872664da9134a346a35add48": "g_n > \\frac{c\\log n\\log\\log n\\log\\log\\log\\log n}{(\\log\\log\\log n)^2}", "32d96e6010d5b1288521be509d4097a2": "\\Delta y=f(x+\\Delta x)-f(x)", "32d9f966990b000180bab0501289edca": "u = \\,", "32da1412c2e9d82b18f554b32e4add65": "(q, \\delta, \\epsilon)", "32da5524d8b9c4940fe65daba3121815": "|f(\\sigma(t))-f(s)- f^{\\Delta}(t)(\\sigma(t)-s)|\\le \\varepsilon|\\sigma(t)-s|", "32da7c0b0d9dbe989228260bb31daecb": "t=1\\,", "32dad88b7a834813f5f925c608873f7a": "= \\, \\frac{1}{3} \\left( \\partial_\\lambda F_{\\mu \\nu} + \\partial _\\mu F_{\\nu \\lambda} + \\partial_\\nu F_{\\lambda \\mu} \\right) = 0 \\,", "32dafa9a16d5ef04e7e0726a1de4218c": "\\frac{\\rho\\,\\!S-1}{1 - e^s}", "32dafd8c571aad493301f5f997bf85ca": "(P^{-1})_{IJ}^{\\;\\;\\;\\; MN} = {\\gamma^2 \\over \\gamma^2 + 1} \\Big( \\delta_I^{[M} \\delta_J^{N]} + {1 \\over 2 \\gamma} \\epsilon_{IJ}^{\\;\\;\\; MN} \\Big).", "32db33ae74baef297ae053099deb2410": "\\textstyle \\ell = \\sum_{i=1}^k l_i 2^{i-1}", "32db6d11e585810cd3e82b13ecd71f5e": "t(C_n^{1,2}) = nF_n^2 ", "32db711d97f8d297355b804da2c3ecb5": " H_0\\psi^{(0)}(\\vec{r}_1, \\vec{r}_2) = E^{(0)} \\psi^{(0)}(\\vec{r}_1, \\vec{r}_2) ", "32db81bf3df172ddfec79ee59c288b49": "\\langle Y, \\psi_i\\rangle", "32db9598b78c1da686752e5b8f37301a": "\\tau_B = \\frac{n_c-n_d}{\\sqrt{(n_0-n_1)(n_0-n_2)}}", "32dba180942bb805e51c1bedcfbe38e7": " (x^*)^* = x,\\quad ", "32dc9608bb3ad5fb35d7723c57d0e629": "(x+y)", "32dc9fdb336a877a9929a1b161432909": "{{200 \\choose 115}q^{115}(1-q)^{85}}.", "32dca7355826b20bdbcf52cfc24beee8": "Y_{-(m+\\frac{1}{2})}(x) = (-1)^m J_{m+\\frac{1}{2}}(x). ", "32dce4b03787fdc4e2710bfcdfdecac7": "\\psi(\\Omega^2) = \\phi_3(0)", "32dce7805c8646cc40b3a749c1076dca": "F_Y(y) = F_X(\\sqrt{y}) - F_X(-\\sqrt{y})\\qquad\\hbox{if}\\quad y \\ge 0.", "32dcf995844fa3b40ff84f07ccabef46": " \\theta = \\nu(M_2) - \\nu(M_1) \\,", "32dd49aa08e1f402f5a212e3352ac5f4": "{\\eta_c} = {{useful\\,heat\\,received\\,by\\,the\\,coolant}\\over{incident\\,radiation\\,on\\,the\\,collector}} = {Qu\\over Qc}", "32ddd2860fcea0c37cc9dcd83b1fd36a": "n \\ge \\left ( \\frac{3}{\\delta} \\right )^2 \\bar p(1-\\bar p)", "32ddddc9bb96a4f774eb62fd48e90071": "\\nabla f= \n\\frac{\\partial f}{\\partial x} \\mathbf{i} +\n\\frac{\\partial f}{\\partial y} \\mathbf{j} +\n\\frac{\\partial f}{\\partial z} \\mathbf{k}\n = 2\\mathbf{i}+ 6y\\mathbf{j} -\\cos(z)\\mathbf{k}.\n", "32de43379f7350fb9036ef2a8c8da54f": "a * b + a * c = a * (b + c)", "32de61f19a154c6856ef0c99984540dc": "v/w \\cdot (x/y \\cdot z) = (v/w \\cdot x)/y \\cdot z", "32de72d818ed712c58c10962085a9011": "1/y_i", "32dea576c7ddab283997cfd429692b6b": "0<\\omega<2 ", "32df19869eaecb9e239af70b3d3ad37b": "f(x)=\\frac 12\\left(\\frac ax + x\\right)", "32df6466d16c12fb3b869ee77d531ac1": " \\beta = \\omega \\sqrt {LC} ", "32df6dbaacb0dbb3d9996baf9b34f2d1": "\\mathbf{\\hat n}", "32dfc3bd76caa17748ccd8868b9c1623": "\\frac{\\textrm{d}[\\textrm{H}^+]}{\\textrm{d}t}= k_1[\\textrm{CO}_2] - k_{-1}[\\textrm{H}^+][\\textrm{HCO}_3^-] + k_2[\\textrm{HCO}_3^-] - k_{-2}[\\textrm{H}^+][\\textrm{CO}_3^{2-}], ", "32dfc6123638db878782168b8da0772b": "O([7+o(1)]^n) = O(N^{\\log_{2}7+o(1)}) \\approx O(N^{2.8074})", "32dfc6da066c7ac80b98924ce45d5a53": "\\mu_i N_i", "32e00dc048ed06366c9bcabd9c678f7f": "F^n \\rightarrow +\\exist F^n", "32e01d7aaa7a0839594240a8a2321dd2": "\\sum_{i,j=1}^n \\phi(x_i-x_j)\\ge0\n", "32e04092faac477a8363c88f67cdcfaf": "v_{2}= \\begin{pmatrix}0.0002422 - 0.1872055i \\\\ 0.0344403 + 0.0013136i \\\\ 0.9817159 \\\\\\end{pmatrix}", "32e04b4d671e8a75252d094b054b932b": "P^3", "32e08ae72e15444f06bcf1fac95e2e59": "t_1=t_1',\\ldots,t_n=t_n'", "32e0938d4f1df2b833ad61b7a452c9cd": "cum f(k_{i}) = f(k_{i}) + \\sum_{j} cum f(k_{j})", "32e0a4efabef13993d1133dc497aac77": "A \\rightarrow bB", "32e0dc67a5940b9dd99d7b79b9e06425": "y = j \\sin(d t) - \\sin(e t)", "32e0f758366dd48d01cbcc0df2049d61": "\\exp\\left(\\frac{v}{u} \\log \\frac{1}{1-uz}\\right)\n= \\left(\\frac{1}{1-uz}\\right)^{\\frac{v}{u}}.", "32e122f20bafce122a1356380de75906": "\\mathbf{v}_{\\mathrm{refract}} = \\left( \\frac{n_1}{n_2} \\right) \\mathbf{l} + \\left( \\frac{n_1}{n_2} \\cos\\theta_1 - \\cos\\theta_2 \\right) \\mathbf{n}", "32e1647804b3e14dc5516816d082389f": "A^\\top K A \\mathbf{x} = \\begin{bmatrix}1 & -1 & 0 \\\\ -1 & 2 & -1 \\\\ 0 & -1 & 1\\end{bmatrix}\\begin{bmatrix}2 \\\\ x_2 \\\\ x_3\\end{bmatrix} = -A^\\top K L = \\begin{bmatrix}-1\\\\-1\\\\2\\end{bmatrix}", "32e1babb8e4d4d333f5eb4226d53df9e": "dU = nC_vdT = -pdV\\,\\!", "32e1c44d1917be7ab7fe695b3a9a1832": "\n\\begin{align}\nU(I,~J) & = \\frac{H(I)U(I|J)+H(J)U(J|I)}{H(I)+H(J)} \\\\[8pt]\n& = 2 \\left [\\frac{H(I) + H(J) - H(I,~J)}{H(I)+H(J)} \\right ] .\n\\end{align}\n", "32e1eda46297f7b32676a31a037284a5": "[\\![\\nu Z. \\phi]\\!]_i = \\bigcup \\{T \\subseteq S \\mid T \\subseteq [\\![\\phi]\\!]_{i[Z := T]}\\}", "32e1ef0914cda296247deeec8cb7f6e8": "\n\\begin{align}\n1 & = (1, 0, 0, 0), \\\\\ni & = (0, 1, 0, 0), \\\\\nj & = (0, 0, 1, 0), \\\\\nk & = (0, 0, 0, 1),\n\\end{align}\n", "32e21ab25f45c25c27a33c789f0307b2": " \\int_S (\\nabla \\times \\vec{F}) \\cdot \\mathrm{d}\\vec{a} = \\oint_C \\vec{F} \\cdot \\mathrm{d}\\vec{r} ", "32e260da9baea75811ae4b8289412289": "\\frac{f(t)}{1-F(t)} = (p + {q}F(t)) x(t)", "32e26b3dfefb78b0bc208932b25fab50": "\\mathbf{F}_{\\text{p}}=", "32e2922f185b8f05d4c5fb94b097e4fd": "[\\widehat{R}(\\theta, \\hat{\\mathbf{n}})]_{ij} = (\\delta_{ij} - n_i n_j) \\cos\\theta - \\varepsilon_{ijk} n_k \\sin\\theta + n_i n_j ", "32e2a03bd786c203874e8da86452785c": "\\mathbf{B}=\\mathbf{C}=\\begin{bmatrix} 1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\ \n0 & 0 & 1 \\end{bmatrix} = \\mathbf{1}\\,\\!", "32e2b86e09287397962385d0dec8c166": "\\succcurlyeq, \\curlyeqsucc \\,", "32e2f32e15d114f998c6afe03bff9711": "\\Rightarrow e^{i\\mathbf{K}\\cdot\\mathbf{R}}=1", "32e30cbd18f32990b4a60c8af33796ed": "v_\\text{r} \\,", "32e34b60b715d8c34dbcf5878b5c5823": "P_{F} (X) = \\mathbf{E} \\big[ X \\big | F \\big].", "32e363b1656f9ee792789c3f109b9877": "s_1 \\smallfrown s_2 \\in traces\\left(P\\right) \\implies s_1 \\in traces\\left(P\\right)", "32e3ec531cd2580bf7b832f2eeb738e0": " m_X=E[X]=\\frac{\\overline p + \\underline p }{2}, \\sigma_X=\\sqrt{Var[X]}=\\frac{\\Delta p}{6} ", "32e3f2c395b6fd0b67374866e1c3abd4": "x=\\sqrt{ 2n+1}\\cos(\\phi)", "32e40a3fd067a5dad39497aba30b28cc": "\\sum_{n=2}^{\\infty} \\zeta(n,\\bar{k}) = -\\phi(k+1)", "32e43a00e991e5176f4593d81b235226": "\\Theta(m^{2}) = \\Omega(m^{1})", "32e495f8ae784a03d0e61481e16685cb": "\\Psi_{k,l,\\mathbf{r}} = (-1)^{r_k-1} \\sum_{\\mathbf{i} \\in\n\\Omega_{k,l}} \\prod_{j \\neq k} \\Big( \\!\\!\\!\n\\begin{array}{c}\ni_j + r_j-1\\\\\ni_j\n\\end{array} \\!\\!\\! \\Big) \\Big(\\frac{1}{\\sigma^2_j}\\!-\\!\\frac{1}{\\sigma^2_k} \\Big)^{-(r_j + i_j)},\n", "32e4ccded52650b6e2d6f83653d6b76a": "f(x):\\mathbb{R}^n \\rightarrow \\mathbb{R}", "32e53b1f5df8209ea37a0351aa3c7bb0": "{dQ_g \\over dt} = F_g (C_{art} - {{Q_g} \\over {P_g V_g}})", "32e53f3d077d750979c8eae5589b0444": "\\arg(X_k) = \\operatorname{atan2}\\big( \\operatorname{Im}(X_k), \\operatorname{Re}(X_k) \\big),", "32e555b1b79156c639f3dfdf389eeb02": " p_{X=1, Y=1}-p_{X=1}p_{Y=1}", "32e59cb0a8d2cb76d77590a953d508fa": "L(\\partial_t, \\nabla_x) u := \\left( \\frac{\\partial^2}{\\partial t^2} - c(x)^2 \\Delta \\right)u(t,x) = 0, \\;\\; u(0,x) = u_0(x),\\;\\; u_t(0,x) = 0", "32e5aea35faf1255dd2d15e88b73b504": "0 \\le ~_{metric} \\delta_{ck}^2", "32e5b417764147b5f0b97cd58dca511d": "{D}'(\\omega )=\\frac{\\sinh \\left( \\frac{T}{2K\\tau } \\right)}{\\sinh \\left( \\frac{1-j\\omega \\tau }{\\frac{2K\\tau }{T}} \\right)}", "32e6bf7fb746cda2f97b136342546c04": " \\nu\\ : 2^{\\mathbb{P}(N)} \\to \\mathbb{R}", "32e704750f0c59898c739ba7f58aae4a": "P = AA^+\\,\\!", "32e7ad81bd1e918ee592046f1dddfa22": "D(c^*) = - \\frac{1}{2 t} \\frac{\\int^{c^*}_{c_R} x \\mathrm{d}c}{(\\mathrm{d}c/\\mathrm{d}x)_{c=c^*}}", "32e7c4f462ea4f7c252078091a8c802b": "\\delta = (t_3 - t_0 ) - ( t_2- t_1 )", "32e7df502af86f2529aa867a60a85b0f": "\\bar u_k", "32e7ffa8274af2d7c3b23694ca8ed5c8": "O(M + N\\log N)", "32e8096d028b5e5d630996c4dad32968": "\\theta_1 \\; d_{\\rm S} = \\theta_{\\rm S}\\; d_{\\rm S} + \\alpha_1 \\; d_{\\rm LS}", "32e86be254ae5b040254471a77aeed1b": "\\lambda_n(L)", "32e8c5f9f57d3837373e032568376036": "\nr_{\\mathrm{outer}}^{3} = \\frac{G(M+m)}{\\omega_{\\varphi}^{2}}\n", "32e8deed0400b2c45cf9fe329acc2c76": "2X\\rightarrow X_2", "32e8f6c76bce1433524df36a258d0223": " \\operatorname{E}(z_t) = 1", "32e900f991e8dbf16b2c3ed1bb59ead2": "-D_1\\frac{\\partial C_1}{\\partial y}", "32e9486d196f3329a7371958749422df": "\\;\\;n=p_1^{k_1}p_2^{k_2}p_3^{k_3}\\dots, \\;", "32e9547dc28697783eb044c8b433daf2": "\\displaystyle 4r^2 \\le K \\le 2R^2.", "32e9dc7550ab662d62bfb1094f768878": "o_1,o_2,n_1,n_2,n_3,n_4\\,", "32e9f4c3d91c786695c3f2dbdc1418d5": " ds^2 = (1-m/\\rho) \\, \\left( -dt^2 + d\\rho^2 + \\rho^2 \\, ( d\\theta^2 + \\sin(\\theta)^2 \\, d\\phi^2 ) \\right) ", "32ea088ce5e6e20581871e1d87383442": "\\Delta H_0", "32ea2d17134fdb71d9a4ab77ee8c3e93": "v(x) = -2x+10", "32ea5f7ec3a37980d8f8de0ade7d11c3": "\\sigma_1 c \\sigma_1^{-1} = \\sigma_2 c \\sigma_2^{-1}=c", "32ead57e055834e4cd77ac901c2401ef": "p_i = \\frac{k_i}{\\displaystyle\\sum_j k_j},", "32eb18cc8e6323c2f7792c504ecee1c0": "|\\lambda|+|\\mu|=|\\nu|", "32ebb042e89e1a29408329bae505522e": "K_{\\rm{eq}} = \\frac{[{\\rm{H_3O^+}}] [{\\rm{OH^-}}]}{[{\\rm{H_2O}}]^2}", "32ebc7ea9bf93d26efe2b1fffca61c52": "z\\rightarrow -\\overline{z}", "32ebe5ce73396017f5c85980984dddd9": "\n\\begin{align}\na_0 & & b_0 \\\\\na_1 & = \\frac{a_0+b_0}{2}, & b_1 & = \\sqrt{a_0 b_0} \\\\\na_2 & = \\frac{a_1+b_1}{2}, & b_2 & = \\sqrt{a_1 b_1} \\\\\n & {}\\ \\ \\vdots & & {}\\ \\ \\vdots \\\\\na_{N+1} & = \\frac{a_N + b_N}{2}, & b_{N+1} & = \\sqrt{a_N b_N}\n\\end{align}\n", "32ec0e5c3ba3c30d47da14cd565e6e15": "10^{16}", "32ec2712f4205527fe5338c8e321749f": "\nV:TM\\to TTM; \\qquad V_\\xi := \\operatorname{vl}_\\xi\\xi,\n", "32ec5325cbff9fc230721336eb349148": "qr-ps=1", "32ec6e52b6f2b1a7866078e64d0676fc": "(x, y, p)", "32ec7db33df41d1b273fc5bfd1b406a2": " x^5-25s^3x^2-300s^5", "32ecaf1b7a6c6331065653f6fef0bc41": "a^x\\ln a\\,", "32ed1a5786e7650445cc39ac0d4f2207": "(x \\cdot y) \\cdot z", "32ed3b49e8a0844e7a2d397f1953898f": "\\Theta(n_1", "32ed4e1538c330231507ecefe77b6f06": "[1]^F x = x.", "32ed51537f2052f5bfd269d33071b496": "\\log2", "32ed69cb7d821e506a1d7240f4b01980": "\\mathop{\\mathrm{Models}}(\\bar{q}(T)) = \\{ q(R)\\,| R \\in \\mathop{\\mathrm{Models}}(T) \\}", "32ed9f8e41d9e04c9a464e68b988c456": "\\vartriangle_X:X^\\vartriangle\\to X", "32eda6ff0cfbc6c4fd570aacd2f93402": "E(X^n) = \\sum_\\pi E((X)_{|\\pi|}).\\,", "32edb962de0799c62775fcef87b5cc2c": "v=\\frac{6 \\times 0.4031}{-2 \\times 0.4402 + 12 \\times 0.4031 + 3}=0.3477", "32edbc31d761a85e4c581cf078aeab06": "T_n\\,", "32edefc151c5136930a76f1359cc14d6": "\\omega^2\\, x''(\\tau) + x(\\tau) + \\varepsilon\\, x^3(\\tau) = 0\\,", "32edf1b0f6a8a14a6534a992191d66d5": "\\frac{e}{m} = \\frac{2 \\, U}{r^2\\cdot B^2}", "32ee1fdecde4bc1c085e1ceaaa21abd7": "\n\\cos\\left(2\\pi\\cdot\\frac{1}{42}\\right)+\n\\cos\\left(2\\pi\\cdot\\frac{5}{42}\\right)+\n\\cdots+\n\\cos\\left(2\\pi\\cdot\\frac{37}{42}\\right)+\n\\cos\\left(2\\pi\\cdot\\frac{41}{42}\\right)\n", "32ee375c58ba83786b688bd251d9cb29": "\\left( \\mathbf{A}\\right) ", "32ee44ae51ef7e45da7dbedebf04807e": " \\pi_{2}(X,x) = \\pi_{2}(\\Pi_{2}(X,x)) .\\!", "32ee6a5dec4daca25cd20bfdb79b2809": "\\forall x (x \\in a \\leftrightarrow x\\in b)\\rightarrow a=b", "32eee8356902b032f70ad5db5b12d33f": " [S] = \\begin{bmatrix} \\Omega & -\\Omega\\textbf{d} + \\dot{\\textbf{d}} \\\\ 0 & 0 \\end{bmatrix}", "32ef07742ae99d34a1500a586f444bef": "d=\\frac{78.5 - 54.7}{360}\\times 2 \\pi\\ \\times 3400\\mbox{ km} = 1412\\mbox{ km} ", "32ef336ac1502a3e5eb5ecdedd265d18": "\\mathrm{Stk} = \\frac{\\tau U_o}{d_c}", "32efe540736aeb87596d48be3fd43859": " \\sum_{k=0}^n\\sigma(3k+1)\\sigma(3n-3k+1)=\\frac19\\sigma_3(3n+2).", "32efeba172cce609c4bb413e95bd33af": "f=f(\\epsilon)=f\\frac{\\phi_U-\\phi_{UU}}{\\phi_D-\\phi_{UU}}", "32f00b3c6187a75403f58ce5c8613623": " {dE \\over dx} = {v_x^2 \\over K} ", "32f03677a37353666d23a6ba57b350dc": "-(E - e\\phi) \\psi_- + c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) \\psi_+ = mc^2 \\psi_-", "32f06d7919063506f4dde117b02e9f36": "c_d(k)", "32f16a102ad4c1ca99ea88509e5cf328": "M\\cup_\\Sigma M^*", "32f17f38b692929d270e240d5bce1f02": "P(x_1,x_2,\\ldots,x_n)", "32f18a0a38f37be393cfd0bd6c4de685": "[HG]_{eq\n}", "32f1b70218ae7f2ef7e3ed7c114043f1": " \\sigma_2 \\otimes \\sigma_0 ", "32f1bf614e6babb62fbb6b9b9c0dd390": "\\sum_i \\int_{S_i} (\\phi\\epsilon \\, \\mathbf{\\nabla}\\phi) \\cdot \\mathbf{dS} =\n0", "32f1f541a8851b48e228cb94e46c2d31": " f(t,(x,v))=(x+tv,v).\\ ", "32f1fdd17906da65a4b167c5b5445444": "f(x)=\\frac{1}{2}a_0+\\sum_{n\\ge 1}\\left(a_n\\frac{\\sin nx}{\\sqrt{\\pi}}+b_n\\frac{\\cos nx}{\\sqrt{\\pi}}\\right),", "32f2627172ad7831e01c98a4452201c8": "P_\\mathit{GATEDRIVE} = Q_G V_\\mathit{GS} f_\\mathit{SW}", "32f26d49107694fd9bc226e8fc916c97": "\\sigma(x) \\sigma(p) \\ge \\frac{\\hbar}{2} \\,\\!", "32f276734c14dbae99333a582511aefa": "N = \\frac fD \\quad \\xrightarrow {\\times D} \\quad f = ND", "32f2c61f7959f44bb009587d130ca8cf": "mk = O(m)", "32f2f18bb693ab2dd632a020db143df8": "PM \\,", "32f303ff95f76de6449aa7272886209f": "N = N_0\\,e^{-{\\lambda}t} = N_0\\,e^{-t/ \\tau}, \\,\\!", "32f31504ec4ac3af9dccbbe856fb4684": "T(P \\vee Q)\\ \\Leftrightarrow\\ (T(P) \\vee T(Q))", "32f342f502afd36f7337a147a6702d95": "\nP_t(a) = \\frac{\\exp(q_t(a)/\\tau)}{\\sum_{i=1}^n\\exp(q_t(i)/\\tau)} \\text{,}\n", "32f343569da2e71243d95f9ce2e91d01": "{\\ni_X}\\subseteq PX\\times X", "32f3c953531d4d692516a09a37b8e50d": "\\vec{j}_{\\text{diffusion}} = -D \\, \\nabla c", "32f3d92a1b3c299150a7a93d068ec66a": "\nZ(\\lambda,\\nu) = \\sum_{j=0}^\\infty \\frac{\\lambda^j}{(j!)^\\nu}.\n", "32f3eb826389928f7587497e2f91d87d": "1.2 * 10^5 M^{-1}", "32f435a1388d24ccb6c1082a7ca32a04": " \\lambda ", "32f4565d2cd5046241de95890b8044f1": "v,", "32f461032ffd9f053916d28c4be4c6c4": "a_i\\le b_i", "32f46ed96db8f030d5b75647467b300d": "\\scriptstyle Q_j \\;:=\\; \\frac{Q}{(x \\,-\\, \\lambda_j)^{\\nu_j}}", "32f484757085ae27f2b4122e451e5194": " M = \\left[ 1 - \\sinh^{-4}(2 \\beta J) \\right]^{1/8} ", "32f4c1c88e09e5d6ec0fafdb3089c4bc": "\\nabla\\times\\bold{B} = \\frac{1}{c^2}\\frac{\\partial\\bold{E}}{\\partial t}. ", "32f4fe09d1e75eb8fc9bd6d520cfa115": "\\psi^{(-2)}(1)=\\frac12\\ln(2\\pi)", "32f5240d0dbf2ccbe75ef7f8ef2015e0": "x^2", "32f54eb960f1989a59e7fb430591bf45": "y=b \\sinh v \\sin\\theta ", "32f5d8c6b9d5a4a37ad73a7d66d8f8e8": "\\log_a x\\,", "32f653868da87655fdf0dd43d1f6a4c4": "\\lambda_{i,j}", "32f670f7ad4c348b3c5cbb1bc425f9cf": " \\nu = \\frac {[\\mbox{M}]_o} {[\\mbox{I}]_o} \\rho ", "32f69c19572d443d5df0557de2d20160": "\\operatorname{End}_R(R^n) \\simeq \\operatorname{M}_n(R)", "32f69c509e0dbc9737f51d1b370db9b1": " \\mathbf{P} - e\\mathbf{A} = \\frac{m\\dot{\\mathbf{r}}}{\\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2}} \\,\\!", "32f6d3748b783f8fc7b798917a90b401": "\\sqrt{T}", "32f6e8bac19a7263ad688c6374853a78": " \\phi_X(t,p) = \\gamma_p(t) ", "32f701ec0b9e79083fd04ccaa700b6fa": " \\left\\{ k \\right\\}_{k=0}^{d-1} ", "32f784fe62b2f61e39971ea80989d510": "m_{b}/f", "32f7934a75b0d6bf1f5c7a35656095cb": "\\epsilon = n_2 - n_1 = n_2 - 1", "32f7ad8a651ea3673f3cd595f243fc10": "h(n)", "32f7d14e59c51a89ec22a54e2861a3a1": "Es = Ei^{*} \\cdot \\frac{Es^{*}}{Ei}", "32f7e55105410deb187d27f457c58dbd": "\\ \\begin{align} l^* & = \\dot{F}^*(F^*)^{-1} \\\\\n & = (\\dot{Q}F+Q\\dot{F})(QF)^{-1} \\\\\n & = (\\dot{Q}F+Q\\dot{F})F^{-1}Q^T \\\\\n & = \\dot{Q}FF^{-1}Q^T + Q\\dot{F}F^{-1}Q^{-1} \\\\\n & = \\dot{Q}Q^T + QlQ^{-1} \\\\\n & = \\Omega + QlQ^{-1}. \\end{align}", "32f81a51b6fcaf178d9abbaa5de4a92f": "c \\in \\mathbb{Z}^n", "32f8641420e007dc0b0c6823fe3e2ea4": "A = i + \\frac{b}{2} - 1.", "32f89c2bc5c18565ea84bb670282b324": "\\{ 1, 2, 3 \\} \\,", "32f8af845c887e3564e59c99d779e2ee": "\\scriptstyle\\vec{f}_0", "32f8f5b0d98ac66403ab2f655ec6897a": "\\displaystyle i\\partial_tu + \\Delta u= V(u)u", "32f8fa35722f9efe17a55da72a82a09b": "0 \\leq x_1 \\leq 690 - 1.5 \\cdot x_2 \\;\\land\\; 0 \\leq x_2 \\leq 530 - x_1 \\;\\land\\; x_1 \\leq 640 - 0.75 \\cdot x_2", "32f929d2a90d7f4880b9f9ea487bc17f": " = z + \\frac{\\rho^2}{2 z} - \\frac{\\rho^4}{8z^3} + \\cdots", "32f9596b925ac12588aa5622d40bbeb6": "i\\partial_t|n(t) \\rangle=\\hat{H}(t)|n(t) \\rangle", "32f98f9735dc572b7bf69e05344983c1": "\\vec\\omega", "32f9c40a8383afe90b9e916f4d2204d3": "\\exists n_1 \\exists n_2\\cdots \\exists n_k \\psi", "32fa18785e8031f9ac6069f17421a508": "P_r =P_t ( {\\frac{\\lambda G}{4\\pi d}} ) ^2 \\times (-j \\Delta \\phi)^2", "32fa677742dd1f466239fda1f77b0c0a": "(\\Omega^{-1} - 1)", "32fa8888837db1f21edce54b559f57b2": "2y + h(x) = 0", "32fa98bae8ef8b3af5e2781c84abb95f": " [0,\\,1]", "32fb01c19fe8cb8f473c327bcd2e17d5": "\\sum_{r\\neq s}\\dfrac{u_r\\overline u_s}{\\lambda_r-\\lambda_s},", "32fb1312e3c9ee8759b698cdde3a6fe5": "R_{h}", "32fb32de88e22a330c85c3fffba82ad9": "A = 3\\sqrt{25+10\\sqrt{5}} a^2 \\approx 20.645728807a^2", "32fb859b306ce86390c58c4571f4ea08": " \\Gamma \\subset G", "32fba04c8b42962a44627cbe28321ff5": "\\gamma(s, z) = \\int_0^z t^{s-1}\\,e^{-t}\\,{\\rm d}t, \\, \\Re(s) > 0. ", "32fc223fa3038785d719a1d226f08669": "\\mathbb{M} = \\{ \\bot, f(\\bot), f(f(\\bot)), \\ldots\\}", "32fc284301a73d73d54768c9ca1426d4": "\\phi_{CTS}", "32fc7cf3c8d4168e4fc9eb0e0936176a": "x_0^2+x_1^2+x_2^2+\\cdots < \\infty, \\,", "32fc8dbe4d77cff5044aafb06c7a3e19": "(3, 1), (3, 4)", "32fc9f91638999456bf655a1d04d9120": "\\frac{a}{\\sqrt 2}", "32fd0962056aa8de2920046fac4a9a26": "\\displaystyle{C(x)={x-i\\over x+i}}", "32fd31dcd5d3080a73004923258235f3": "e^N = I + N + \\frac{1}{2}N^2 + \\frac{1}{6}N^3 + \\cdots + \\frac{1}{(q-1)!}N^{q-1} ~.", "32fd75250822e17b071c2225a7146850": "\\bar{y}(t, \\tau) = \\frac{x(t+\\tau)-x(t)}{\\tau}", "32fda19c3c124c8b47669335cd3dc98e": "x^{\\mu}", "32fdcb0ee9890f36105a3771e66d5028": "\\operatorname{E}_{Q}[\\log P(\\mathbf{Z},\\mathbf{X})]", "32fde068da20f8fa9da37c56ea1aac09": " \\varepsilon^{v_1}_S = 0 ", "32fe1018949b396d09160d0b639872e9": "i =", "32fe21c7481d93249c0e87e176b270a1": "{S^2}_2", "32fe4b7334b2b431cf7ea2ade5bd7e57": "V(q) = q^2-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}. \\, ", "32fe963dfaff3f05de3b6993b91599f8": "\n M_{\\alpha\\beta,\\alpha\\beta} - q(x,t) = J_1~\\ddot{w}^0 - J_3~\\ddot{w}^0_{,\\alpha\\alpha}\n", "32fed3c20872d870c2f03cf18f0fb15d": "j(E')", "32ff223f4b9214ce44a3f7cac5abe8bf": "\\exists", "32ff4ba249f15df8298566102a15baf6": "log_{2-\\alpha} n ", "32ff846892af3ff98a18de22edffea61": "I_d - V_g", "32ffa50432a0a8cb71f0f73dc909d3d1": "\\frac{\\partial y}{\\partial \\mathbf{X}} ", "32ffa9b6cf188a624091854982f0b6d5": "f_1 = 2x^2 + 3x^3", "32ffabcccdeb7afc502f0a00d5f9d859": "f:A\\longrightarrow B", "3300003ac2e0938fe7d470c1867975b6": "\\operatorname{Alt}(\\omega)(x_1,\\ldots,x_k)=\\frac{1}{k!}\\sum_{\\sigma\\in S_k}\\operatorname{sgn}(\\sigma)\\,\\omega(x_{\\sigma(1)},\\ldots,x_{\\sigma(k)}).", "330028512e1a6bbb4d014b64a0becb96": "q=\\infty", "33008d367d78c30912395475b6ff07bf": "n = 2^k", "3300aac672b6aeef9c0fbb9d146ce02b": "\\frac{\\partial(y_1, \\ldots, y_k)}{\\partial x_i} = \\sum_{\\ell = 1}^m \\frac{\\partial(y_1, \\ldots, y_k)}{\\partial u_\\ell} \\frac{\\partial u_\\ell}{\\partial x_i}.", "330108d390400be6a2e16229b6b66d90": "R = \\sqrt{x} u", "33013b0ac0504f5e7ae5c9e55a4e18d2": "\\frac{1}{2}\\, c_p\\, \\left( 1\\, +\\, k\\, h\\, \\frac{1\\, -\\, \\tanh^2\\, (k\\, h)}{\\tanh\\, (k\\, h)} \\right)", "33016d488aa1f918d94e99ec0ae1d9a0": " \\mathbf{S}_{2}\\psi \\equiv (\\mathcal{S}_{20}\\cosh (\\Delta )+\\mathcal{S}\n_{10}\\sinh (\\Delta )~)\\psi =0, \n", "3301e56042e0f2bb2bfa782d66c70bf8": "\\left|\\frac{\\Delta \\beta_j}{\\beta_j}\\right|<0.001, \\qquad j=1,\\dots,n.", "3301e95da95f7373b36758d54da6c568": " |A|^{d-1}\\leq \\prod_{i=1}^{d} |P_{i}(A)|", "33023ba8c87eca17162ed6e6b7142f5e": "f(x)=\\tfrac{1}{x}", "33028989a8b412687d3218ccc4fcf243": "\\scriptstyle n_3 \\,-\\, n_2", "3302a6779c408deee0f33ed1e4a3d674": "(1,1), (1,2), \\ldots, (2,1), (2,2), \\ldots,", "3302abb4a67e9827d2a536ca1c663659": " W(t) = (1+t) B\\left(\\frac{t}{1+t}\\right).", "3302e6a1c237e417e64981ce12e556e5": "\\, \\kappa_0", "33030cfb0c12f52e14c6686d300697eb": "f(\\theta)=\\sum_{\\ell=0}^\\infty (2\\ell+1) f_\\ell(k) P_\\ell(\\cos(\\theta)) \\;,", "33034a4b7ae125b51c20d4510b7fe7f8": "\\langle S(x_1) S(x_2) ... S(x_{2n})\\rangle = C^n \\sum G(x_{i1},x_{j1}) G(x_{i2},X_{j2}) \\ldots G(x_{in},x_{jn})", "33036d790495c8f856c6b442a043bfd3": "\\tfrac{1}{256}", "33037baa92c83aa6c37e834f694cceb8": "k_2= 1.09 + 2.84\\times 10^{-2}\\,T - 9.04\\times 10^{-5}\\,T^2 ", "3303886915a241c40fc85896d513e282": "\\Omega_x \\equiv \\frac{\\rho_x}{\\rho_c} = \\frac{8 \\pi G\\rho_x}{3 H^2}", "33038a2a8022423509621d303eb3ba34": "R+T", "3303adea1446306738bb01b1598b9669": "S \\rightarrow aA", "3303b47b629bd85d28c26ac7ca94332d": "f(x) = (2x + 8)^3 .", "3303f96fde3c59e55553ef09ff4c712a": "V_{mat}", "3304359f0b5c607219d69eccd61cca3c": " u_2 =0.21495 ", "33046307fd87b7daa357aad43f815454": "\\xi=\\frac{zFQ_f (C_{inlet}^d - C_{outlet}^d)}{N I}", "3304f0ce276d58148c3fbcaff0713391": "\\mathcal E_m(M)", "3304f8ea2003c971ed66d836220f7e31": "\\displaystyle f(x) g(x)\\,", "33051ecdc62ae45cf31cbffd944442e8": "\\mbox{Grade of Service}=\\frac{\\left(\\frac{A^N}{N!}\\right)}{\\left(\\sum_{k=0}^N\\frac{A^k}{k!}\\right)}\\qquad(2)", "330597deb3acbf59c094f5dd6e0d8387": "\\sup_{y \\isin F}", "3305e50ebb0c8daa75033fcb523fb2b1": "\\nabla \\times \\mathbf{A} = \\mathbf{B}", "33060d83bbec381975b6f4bace214924": "d(i, j) \\ge 0, d(i, j) = d(j, i)", "33061b51c661c064be82b410e5cf7509": "~\\varepsilon(x)~", "3306a0e29b9a6d6e3454c717601aaf05": "E = 120/100= 1.2", "3306b8b7225c9c88002c394f9ce77bbb": "R_x=R_2 \\cdot \\frac{R_3}{R_4}+R \\cdot \\frac{R_3 \\cdot R'_4 - R'_3 \\cdot R_4}{R_4 \\cdot \\left( R+R'_3 + R'_4\\right)}", "3306cd4a4300e0da0d00faadb62a5a0a": "W (n,N) = {N \\choose n} = {{N!} \\over {n!(N - n)!}}", "3306edaa27b8f9d315a63e075c12d3bd": "\\Gamma_{ij,k}^{(\\alpha)}", "3307059110ed6699ea6716b37f97a54c": "\\ldots, -2\\hbar, -\\hbar, 0, \\hbar, 2\\hbar, \\ldots", "33078b162c9c75258c68fed51a971d6d": "1 \\times 2 = 2", "3307a11581a36d70eddacd32aacc0ffa": " \\vdash \\ \\ \\lnot \\lnot A \\rightarrow A ", "3307d924aa714cec04365a6b408eb768": " \\omega =2 \\pi f ", "3307ee316553c1b76189f3fce50e7bbe": "\\begin{align}D(\\omega, \\Omega) &= \\mathfrak{F}^{-1}[\\widetilde{E}^{ac}(\\widetilde{t}-\\tau)]\\\\\n&= \\sqrt{I(\\omega)I(\\omega-\\Omega)}e^{i[\\phi(\\omega)-\\phi(\\omega-\\Omega)]}e^{-i\\omega\\tau}\\end{align}", "33080007745dc1169e6194b817ec1469": "\\phi_F = (kT/q) \\ln{(N_A/N_i)}", "3308770b832b803a21fc1b35d8043810": "f''+{a_1(z)\\over a_2(z)}f'+{a_0(z)\\over a_2(z)}f=0.", "33088463e4eee340ddf5e8dfd66bb695": "R=\\sqrt{X^2+Y^2}=V_{sig}", "33095c38e05758ecbe88e516bf660a13": "\\sum_{n=N}^\\infty a_n", "330975f2aeb13afa1b0b0e8d00ae36b6": "\\Phi(t_1)-\\Phi(t_0)", "330990bfecf66470570139a468b7c586": " \\hat{a}\\hat{c}=(a, b)(c, d) = (ac, ad+bc). \\!", "330a4b047e8975277911e8121e9075a4": "\n\tZ_{CO}^j = \\sum_{i=1}^n e^{-\\hat{y_i}(g_j^{t-1}(\\boldsymbol{x_i})+\\alpha_t^jg_t^j(\\boldsymbol{x_{j,i}}))}\n", "330a97115b92c2c918372159377785a3": "R = \\frac{Z/3-Z}{Z/3+Z} = -0.5", "330aac726995cb6eaa0acd58157ea7f0": "S_{Th}(P,V,T,...)_{(eqm)} = k_B \\, S_I(P,V,T,...)", "330af575fb137c50f2220d778be5dae8": "\\nu=\\sinh r", "330b0983a9871107188bd5b8c7a56604": "\\frac{dU}{dV}", "330b38537ee16ef9e9c6be3818df9ed4": "\\rho>0\\,", "330b872ae330b91b2ce536ab5f225ff3": "\\sigma^2_b(t)", "330bbccaad8fb8614910cb362c729903": "P V = nRT", "330bbf1a8213c0556e58fe527df3a928": "\\log g = \\log jI - {B_B \\over f_h}", "330c10558ba92ab928600ca91a9b205c": " \\bar u^i \\ \\stackrel{\\mathrm{def}}{=}\\ {D \\bar h^i \\over Ds} ", "330c39e67f3ba21a13bb2a58d7c165c3": "p_{k,n}", "330c588f96f98eef90b1dbfe4957b0cc": "Y=C+I+G+(X-M)", "330c70ad9c6fec484fbc5a60f2182abd": " A = \\{ x | P(x) \\} ", "330c9709097048bdf785b88f7e25b125": "V=Z(v;T)", "330cadab35eb826424fdc2818372c9b6": "a\\not\\succ_Wb \\wedge b\\not\\succ_Wa", "330cb5a53c3929f84a204369c3884661": "\\phi_c = 1 - e^{-\\eta_c} ", "330cd78049f93290f51fbbf0d5253fdb": "\\Psi_4", "330ce35bc68f4719b06fc50b953fd6ad": "I_a = \\frac{m (b^2+c^2)}{5}\\,\\!", "330d4789780aecb28de10e7d03824a90": "w(x^{q}, y^{q})", "330d69987765d08a72fea7843d55f7cc": " H^q(X,L^{\\otimes-1}\\otimes\\Omega^p_{X/k}) = 0", "330d7ab9866777dc011f8ff91632e276": "\\nabla\\bigl(\\mathbf e_g \\otimes \\mathbf e_h\\bigr) = \\mathbf e_{gh} \\,,", "330e9467ff9e765d440c86fe7e035e7f": "P=P_b \\cdot \\exp \\left[\\frac{-g_0 \\cdot M \\cdot (h-h_b)}{R^* \\cdot T_b}\\right]", "330f363bee8dd6ef12caa812704c641e": " \\operatorname{build-param-lists}[n, D, V, T_2] \\and \\operatorname{build-param-lists}[g\\ m\\ p\\ n, D, V, K_2] \\and \\operatorname{build-param-lists}[g\\ q\\ p\\ n, D, V, K_1] ", "330f3b182db9995c1d7c6ca279ab776f": "\\frac{1 - 1/(1.05)^3}{0.05}", "330f722206b9b06e7f6dd507f239929d": "h_i(x) = h_i(y)", "33101ce78224053ebaef6cd9374c6ae3": "dp_1 \\ ", "331024221a8c52d9530348b0fdf5f0ab": "x_{n}\\in\\mathbb{R}^{k}", "3310270307f11cdf20640741e7379c8a": "25^2 + 26^2 + 27^2", "33103bea0056c2ba419e26428ba12940": "E_i \\,", "331074d12536b83864417202e504def5": "\\lceil \\sqrt n \\rceil", "331087595c84694cf19b21b8e1b65699": "{1\\over (4,q^n-1)}q^{n(n-1)}(q^n-1)\\prod_{i=1}^{n-1}(q^{2i}-1)", "3310b391d40aba80e2e005168e6a6421": "S^+_xV = \\{\\beta \\in T^*_xV - \\{0\\} \\,|\\, \\beta = \\lambda\\alpha,\\,\\lambda > 0\\} \\subset T^*_xV,", "3310e4adbc42c9d4bd421ffc6ce2be89": "(Rv_1R^{\\dagger}) \\cdot (Rv_2R^{\\dagger}) = v_1 \\cdot v_2", "331139484b93db6aec843079e312f84e": "Z({\\Bbb P}^n,t)=\\prod_{i=0}^n\\frac{1}{1-{\\Bbb L}^i t}", "33119571aac1f62cecd4cc0adb27211f": "F(x,0) = f(x)", "3311ef81019302f9b68076d9c8c67740": "F I([0,\\infty))", "331233c9a690139e31773a6793226992": " Q^*_j = -\\left(\\frac{d}{dt} \\frac{\\partial T}{\\partial \\dot{q}_j} -\\frac{\\partial T}{\\partial q_j}\\right),\\quad j=1, \\ldots, m.", "33125b2ded05ac533dfa867282103cab": "\n L = \\frac{1}{2}m \\left( \\dot{r}^2 + r^2\\dot\\varphi^2 \\right).\n", "3312eeb398433357a317a4609bcf3d12": "\\textbf{A}(\\textbf{r}) = \\frac{1}{4 \\pi} \\iiint \\textbf{J}(\\textbf{r}^{\\prime}) \\ G(\\textbf{r}, \\textbf{r}^{\\prime}) \\ d\\textbf{r}^{\\prime} \\,", "3312f78d6851cd4de8446ed5703e4307": "D_\\alpha = \\lim_{\\epsilon \\rightarrow 0} \\frac{\\frac{1}{1-\\alpha}\\log(\\sum_{i} p_i^\\alpha)}{\\log\\frac{1}{\\epsilon}}", "3313323bac35ad2ef531ad361e1a5170": "\\alpha^{-1} = 137.035\\,999\\,173(35).", "331395b4b7d91f829cd983940ea132b3": "[\\mathfrak{a}_i, \\mathfrak{a}_i] \\subset \\mathfrak{a}_{i+1}", "3313c4cc97b949721277c584174441ac": "\\langle y|L\\rangle\\langle L|x\\rangle\\dots", "33149e694dd3f9bb480434b154f232dd": "\\underbrace{s_P(t) = \\sum_{k=-\\infty}^{\\infty} S[k] \\cdot e^{i 2\\pi \\frac{k}{P} t}}_{\\text{Poisson summation formula (Fourier series)}}\\,", "3314b371c36bb2d63538518695bb1a46": "GL_n(\\mathbb{Z})", "3314ba8dd446819bebb3cfff880a5519": "O(\\min(\\sqrt{m},n^ {2/3})mn\\log(n^2/m+2))", "3314cc501416d8260380de35b945f50e": "\n\\sigma(X) = \\sqrt{\\frac{N}{N-1}} \\sqrt{E[(X-E(X))^2]}.\n", "3314ea02c502dd4af9e4b7845d859480": " T: (a_k)_{k=-\\infty}^\\infty \\mapsto (a_{k+1})_{k=-\\infty}^\\infty.", "3314ed94e07228874f52efca733ed811": "\\tau=L/v", "3315371a986a68d28fc1e6fa9a772a79": "C_n^{(\\alpha)}(z)=\\frac{(2\\alpha)_n}{n!}\n\\,_2F_1\\left(-n,2\\alpha+n;\\alpha+\\frac{1}{2};\\frac{1-z}{2}\\right).", "331550cdf7454ecf3a5a300694898d72": "m_{k} = m_{k-1} \\exp \\left( e_{k-1} \\right)", "33155d2fe2cad847c44c69d25486bacd": "L(E(\\mathbf{Q}), s) = \\sum_{n>0}a(n)n^{-s}", "33155e05782504abd40435ef84051e07": " i=1, \\ldots, r-1 ", "33159745f6e7cd5c4d158b310b5c2b10": "\n\\begin{align}\nG_X(s) & = \\frac{s\\,p}{1-s\\,(1-p)}, \\\\[10pt]\nG_Y(s) & = \\frac{p}{1-s\\,(1-p)}, \\quad |s| < (1-p)^{-1}.\n\\end{align}\n", "3315f166493bfbbae2d2006b02174951": "\\mathcal{E}(x) = x^e \\;\\bmod\\; m", "33160b81f955f1200dc700122985a874": "|s\\rang", "33161afc5f21a72eddc57339ddc89b3c": "\\begin{align}\n(3) \\quad \\int_0^\\delta e^{-xt} \\phi(t)\\,dt &= \\int_0^\\delta e^{-xt} t^\\lambda g(t)\\,dt \\\\\n&= \\sum_{n=0}^{N} \\frac{g^{(n)}(0)}{n!} \\int_0^\\delta t^{\\lambda + n} e^{-xt}\\,dt + \\frac{1}{(N+1)!} \\int_0^\\delta g^{(N+1)}(t^*)\\, t^{\\lambda+N+1} e^{-xt}\\,dt.\n\\end{align}", "33162e10a6463451dfbc1fc70d78ce42": "\n\\begin{align}\nC_M & = C_C \\left( 1 - \\frac {v_{\\ell}} {v_i} \\right) \\\\\n & = C_C \\left( 1 - A_v \\frac {R_L} {R_L+R_o} \\frac {1+j \\omega C_C R_o/A_v } {1+j \\omega (C_L + C_C ) (R_o//R_L) } \\right ) \\ . \\\\\n\\end{align}\n", "331647049787e3478e7b6f58b1b8e668": "(a_n)_{n\\in\\N}", "33164949ffe12e0ff6aeda141d5d44ec": "\\displaystyle{\\|y\\|^2= \\|P(T)y\\|^2 + 4(\\mathrm{Im}(T)x,x).}", "33167636e88dda807d807894c3db18f8": "\\tbinom{n+1}{m+1}", "3316875ea4451ff3212f2662278722d5": "\\beta_1 \\ge \\beta_2 \\ge \\cdots \\ge \\beta_n", "3316b7a59be3cf01e464b00065393266": "S[0..n-1]", "3316dceca558f9eca43c9bb314f7013f": " \\qquad \\qquad \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\epsilon_{e,r,\\omega} = 1 + \\frac{4}{\\pi}\\mathbb{P}\\int_{0}^\\infty \\mathrm{d}\\omega'\\frac{\\omega'\\epsilon_{e,c,\\omega'}}{\\omega'^2-\\omega^2}.", "331724be2336abc08713b93feb03f028": "\\left |\\beta\\right |=1", "33174f76fa278de6abfa82643b1ee668": "\\frac{\\part \\rho}{\\part t} +\\nabla \\cdot \\rho \\vec{u} = 0 ", "331764ef195a213bf2243c9905638395": "\\|I^\\alpha f\\|_p \\le \\frac{|b-a|^{\\operatorname{re}(\\alpha)/p}}{\\operatorname{re}(\\alpha)|\\Gamma(\\alpha)|}\\|f\\|_p", "33176a8555dffc941697d7c893be7c87": "\\text{Current Yield} = \\frac{F \\times r}{P} = \\frac{$100 \\times 5.00%}{$95.00} = \\frac{$5.00}{$95.00} = 5.2631%", "3317be201a19bdfab342ece73cc17492": "\\ \\gamma\\, = 1.4000 ", "3317d29db4b05b3d1136a5f75203de3e": "2n-2", "3317d2ce846d6b0a17ce7e5491a0e6b1": "v = \\tfrac{|\\mu|}{\\sqrt{\\lambda}}", "3317d45d253f508a1e36273c020817c7": "\\epsilon_\\mathrm{back} = -\\frac {\\sigma(t_1)} E = \\epsilon_0 \\exp \\left(-\\frac{E}{\\eta} t_1\\right). ", "3317dc11e492a7233057f743e7e4ff8e": "\\| \\cdot \\|_{ab}", "3317f9f6560150533ad6b714d4e9f475": "\\frac{265}{153}<\\sqrt{3}<\\frac{1351}{780}", "33185f211c0db2e4765013820fefdf3c": "{}_sY_{\\ell m} = \\sqrt{\\frac{(\\ell+s)!}{(\\ell-s)!}}\\ (-1)^s \\bar\\eth^{-s} Y_{\\ell m},\\ \\ -\\ell\\leq s \\leq 0;", "331981c1d6497ef785342812e866c83b": "\\forall E\\in\\mathfrak{F}\\forall 0<\\varepsilon\\in\\mathbb{R}\\exists N\\in\\mathbb{N}_1\\forall N\\leq n\\in\\mathbb{N}_1, \\left|\\frac{\\#\\left\\{i\\in\\left\\{1,2,\\dots, n\\right\\} : f(i)\\in E\\right\\}}{n}-P(E)\\right|< \\varepsilon", "3319d0e2a11589a31066ed9ea96f2e8e": "(\\Omega,\\mathcal{A},P)", "3319dff092b511661e7cb4a8cb0203e8": " u(\\mathbf{r} + \\mathbf{a}_i) = \\mathrm{e}^{-\\mathrm{i}\\mathbf{k} \\cdot (\\mathbf{r} + \\mathbf{a}_i)} \\psi(\\mathbf{r}+\\mathbf{a}_i) = \\big( \\mathrm{e}^{-\\mathrm{i}\\mathbf{k} \\cdot \\mathbf{r}} \\mathrm{e}^{-\\mathrm{i}\\mathbf{k}\\cdot \\mathbf{a}_i} \\big) \\big( \\mathrm{e}^{2\\pi \\mathrm{i} \\theta_i} \\psi(\\mathbf{r}) \\big) = \\mathrm{e}^{-\\mathrm{i}\\mathbf{k} \\cdot \\mathbf{r}} \\mathrm{e}^{-2\\pi \\mathrm{i} \\theta_i} \\mathrm{e}^{2\\pi \\mathrm{i} \\theta_i} \\psi(\\mathbf{r}) = u(\\mathbf{r})", "331a0256e7dda7035d98f6b308e241fc": "\\sigma_{n,\\cdot}", "331a0b6b3a6433add741c8cb752f8a6d": "p(t,f(A))", "331a21c61f45484e20fb7126198ebe1e": "\\theta = \\frac{price\\,of\\,primary\\,products}{price\\,of\\,manufactures} ", "331a250719477aca91b59932d9ac62f4": "\\left\\{a_m\\right\\}_{m\\in\\mathbb{N}}", "331ad8e1673b015a2b9e142e28f3bd80": "\\begin{alignat}{7}\n x &&\\; + \\;&& 3y &&\\; - \\;&& 2z &&\\; = \\;&& 5 & \\\\\n3x &&\\; + \\;&& 5y &&\\; + \\;&& 6z &&\\; = \\;&& 7 & \\\\\n2x &&\\; + \\;&& 4y &&\\; + \\;&& 3z &&\\; = \\;&& 8 &\n\\end{alignat}", "331b27eaabf29ba63361730373ac7648": "\\int_0^t f \\, d^+g = \\text{ucp-}\\lim_{\\varepsilon\\rightarrow\\infty}I^+(\\varepsilon,t,f,dg).", "331bd3dcaf3b1a017ef15b4c8809890b": "a\\,\\mathcal{L}\\,b", "331bfdd1ed082abbb4d5a811f365961d": "\\{a^n b^n a^n | n \\ge 1\\}.", "331c01a3c142014a084a54837911ad75": "\\beta'>\\beta", "331c55d1eb9a4293b2f1be8217b687ae": "P \\mapsto P(a)", "331cc01b889e448a6b13678006a2c9d8": "\\scriptstyle{[-2\\pi,2\\pi]}", "331cd888032e64105a5e42ecbb076160": "h\\nu \\ll kT", "331cdc8791e63d593e400a61f95d3585": "{\\Delta p = f(\\dot{m})}\\ ", "331ceb5e3809744f0071aa2d289de6e9": " U = U_{21} + U_{22} + P_{11} \\cdot (U_{11} - U_{21}) - P_{12} \\cdot (U_{22} - U_{12}) ", "331d15f6e1a19c18c55c85f8d685b6c8": "D_0 = \\lim_{\\epsilon \\rightarrow 0} \\frac{\\log N(\\epsilon)}{\\log\\frac{1}{\\epsilon}}.", "331d3cc96b6091ac6953b6113b4cdae8": " a^n + b^n = (a+b)(a^{n-1} - ba^{n-2} + b^2 a^{n-3} - \\ldots - b^{n-2} a + b^{n-1} ).\\!", "331d492d6e2f86b83c5d66b0c6154f57": "{2p-1 \\choose p-1} \\equiv 1 \\pmod{p^4},", "331d7c25e916505dbc25eddbc23c1b70": "D_{\\mathrm{KL}}(P\\|Q) = \\sum_i \\ln\\left(\\frac{P(i)}{Q(i)}\\right) P(i).\\!", "331da804140c0b838c0348e8396ccd76": "\\mathcal{SROIQ}^\\mathcal{(D)}", "331dc341a76ee9485c7fa237a1d05fd8": "(1523/2, 195/2, 1)\\,", "331dc38495d78be5093f4a76f8308908": "V\\to V", "331dcfa5fba83996d157ff022bf61bbb": "n_B(z)=(e^{\\beta z}-1)^{-1}", "331de3da70105c51e08741a120d11693": "\\mbox {Apply}(f,y)=f(y)", "331dfb75eb5e87a716e6afb321dbf6c7": "\\sigma(\\mathbf{x})=0", "331e309f321b1491d4d64bb652922f3c": "\\mathrm{A}_{x}\\mathrm{B}_{y}", "331e49c48167cc8b957a9acb50e910b3": "\\Psi(\\mathbf{r},t) = \\frac{1}{\\sqrt{\\left(2\\pi\\hbar\\right)^3}}\\int\\limits_{\\mathrm{ all \\, space}} d^3\\mathbf{p} \\, e^{i \\mathbf{r}\\cdot \\mathbf{p} /\\hbar} \\Phi(\\mathbf{p},t). ", "331e89bd8b6777110c7bb884229391b9": "\\chi(M) \\geq \\frac{3}{2}|\\tau(M)|,", "331ea28743b1eb447f45f0469417c0fc": "f(x).", "331ec581b5bd5880c7ccc1d80494394a": " d = \\frac {v \\cos \\theta} {g} \\left [ v \\sin \\theta + \\sqrt{v^2 \\sin^2 \\theta + 2 g y_0} \\right]", "331ec897e736949fac78e18b7b7cbd89": " \\tilde{\\nabla} \\widehat{E}_\\theta(f)", "331eeae403a4932c0ee8d4c26f9644a8": "l(\\varphi)", "331f2f92c06ede6779fb673832adb08b": "r = \\frac{v}{\\omega} = \\frac{\\beta c}{\\omega} = \\frac{\\gamma \\beta m_0 c}{q B}", "331fadf1a5b227e6095b0449d764dfec": "\\mathbf{x}=\\kappa_t(\\mathbf X).", "331fb54fc4caebc6b15b849ccec4d014": "\\Delta {U_f^\\circ}_{\\mathrm {reactants}}", "331fce1451b64c1b310075afb17a9ba1": "m^{-H}L^{-\\frac{1}{2}}(m)\\sum_{i=(j-1)m+1}^mk (X_i-E(|X_i|)),~j=1,2,\\ldots,[\\tfrac{n}{m}]", "331fcf3c53785236da67957a1a1f1a23": "= 18.", "331fec19c96c11f996e23218522c3ab7": "d \\approx \\sqrt{12.74h} \\approx 3.57\\sqrt{h} \\,,", "331ff468e4c0f28c8ed746ba524fe47a": "B(t) = \\sum_{i=0}^n \\beta_i^{(0)} b_{i,n}(t) \\mbox{ , } \\qquad t \\in [0,1]", "332016f63549216c697314d0ee85e3b5": "\ndW = \\sum_{r=1}^{D} Q_{r} dq_{r} \n", "3320699403515250e73239ec9c86a202": "k=\\lfloor -n/2 \\rfloor,...,\\lfloor (n-1)/2 \\rfloor", "3320776410330651c9333257c3e46c9c": "S\\circ R", "33208f3768d18b313396668551e8e275": "\\{|u_i\\rangle\\}", "3320911c44868edd9dab4a94308070fe": "\ny_t=a+bx_{1t} + cx_{2t} + \\varepsilon.\\,\n", "3321100528d3c94923eb58289ba86176": "\\scriptstyle a(\\cdot)^2", "332185dd64bd3fd26dc06514f73c2ad8": "(f^k(x))_{k \\in \\mathbb{N}}", "332192141919935e75d1af5b2a9facb2": "b_2=0", "3321d2caf5af37872f443c6197b67702": "a_k = \\pi_1^k + \\pi_2^k \\ . ", "3321e115def41b3c82e39948eaca65ac": "E^2 = P^2 c^2 + m^2 c^4", "332230c8c7caf871741da91edd3e640b": "\\sigma_j\\,\\!", "3322c202b88c1a69a70208f19e02edeb": "F = \\mu \\frac{W}{2}", "3322d6f3137dafc6bdd5a4a28b78da50": "Rate=k*[A]", "3322f104aee5e8ceacfab491afb7ef8b": "\\Gamma^{[D]}.", "332317e26b7c62296146b4639aedfadf": "\nZ_\\mathrm{in} (l)=j Z_0 \\tan(\\beta l). \\,\n", "33238a7aa18efa79bdbf8906a7e7a4be": "\\sigma(x).", "3323ef8c4dbee3476f2db993edeaa4b3": " \\left\\vert{ \\sum_{n=a}^b e(f(n)) }\\right\\vert \\ll \\frac{b-a}{\\sqrt H} \\ . ", "33253489ccdedca24d6b99af3e03ac43": "p_n = \\tfrac{3n^2-n}{2}", "332557d24bd3a869b8215a50b0234f9a": "\\lim \\frac{d(f(x),f(y))}{d(x,y)} = r. ", "3325785a880562176004ee588d181194": "v_{\\text{in}}(t) - v_{\\text{out}}(t) = RC \\frac{\\operatorname{d}v_{\\text{out}}}{\\operatorname{d}t}", "332587b715dbafc25cea05c0c72a93c0": "\n \\frac{\\partial f}{\\partial \\boldsymbol{A}} = \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T ~.\n", "3325fd5bf989c354be02e9456dc0bb11": "\\scriptstyle A_i", "332608082e8a2b41596fae6d21145d34": "\n\\mu_0=\\int_a^b \\omega(x) dx.\n", "3326199624180185e5aa678077d06ac5": "T^{\\mu\\nu} = \\frac{\\hbar^2}{m} \\left (\\eta^{\\mu \\alpha} \\eta^{\\nu \\beta} + \\eta^{\\mu \\beta} \\eta^{\\nu \\alpha} - \\eta^{\\mu\\nu} \\eta^{\\alpha \\beta} \\right ) \\partial_{\\alpha}\\bar\\psi \\partial_{\\beta}\\psi - \\eta^{\\mu\\nu} m c^2 \\bar\\psi \\psi .", "3326744e21d32e52982353b2cc5750de": "V_i > C", "3327064236781901a13f63e40f147a3d": "\\rm\\frac{1}{s}", "33277c186d3a1f22c62a7f8de0b36036": "x(t) = \\sin(2 \\pi ft + \\theta) \\ ", "3327d4ef92b29f22763f2d97efc8bb87": "\\mathrm{dist} (B, A) := \\sup_{b \\in B} \\inf_{a \\in A} d(b, a).", "3327f3a691462711db979d98694b9bde": "\\alpha=0.05", "3328072701e07dda11ecf088f85c19ca": " \\cos x = 1 - \\frac{x^2}{2} + O(x^4) \\text{ as } x \\to 0\\ ,", "33281692bcb0171ab7736c1220532be5": "s_i\\colon\\Omega\\rightarrow A_i", "33283160fc2c8963d5515a8b03baf62d": "(x-3)^2+4", "33288a688cd0b5516e03e78adcc23787": " \n(B.5)\\quad \\zeta_{,\\,\\Phi\\Phi}-2=0.\n", "3328919f7fd601b18c20abba6b54477f": "\\{0,1,\\dots,p-1\\}", "3328e97a37f931f81f68e0312f42ea35": "\\int_a^b f(x)\\,dx = \\lim_{\\Delta x\\to 0}\\sum f(x_i)\\,\\Delta x,", "3329049d4a4527130bf280a5649e2233": " H(x, i) = (H(x) + i) \\pmod n.\\, ", "332909dcdb7382eb995611985667d2cd": "\\Pr(s|w)", "332927dc660c1efc8d861f26d47ef710": "_{1}\\!", "332948dee0dfdc2d684e3e206e37dead": " \\mathbb{E}(X) = \\int_\\mathbb{R} \\lambda \\, d \\, \\operatorname{D}_X(\\lambda) ", "3329e673f2ae887f22ffefb3d484b182": "\n\\dot{x}_k = u(x_k),\n\\qquad\n\\dot{m}_k = -(b-1) m_k u_x(x_k)\n\\qquad\n(k = 1,\\dots,n).\n", "332a38e364976a612445be2e3676af21": "\\sum_i\\,\\delta_i\\,\\frac{\\sigma_i - \\sigma_e}{\\sigma_i + (n-1) \\sigma_e}\\,=\\,0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)", "332a7f4068bfa47959c5ee06c45b64a9": "\\Delta T_{HS}", "332a8340faaca17795abd58c56129240": "(u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\\sigma) \\,", "332a914e05b8187f17395c5dfdaafaed": "\\overline{AB} \\cong \\overline{DE}\\, ", "332b1aa8bc083accff33fd5b34f8a8dc": "A_n=a_{n,0}=pub", "332b5347135c19297caed3b83da39db8": "{1 \\over \\varphi} = \\varphi - 1,", "332bbcdcd84662d5d8b8a3a0e2ee013b": "D = {Y_1}^2 = 3", "332bbfe5d644bcb7829b0bdffc4fc091": "e_i^{(0)} = f_i^{(0)} = 1", "332bd55c3c4d2abc062914611ea2e01e": " A \\Lambda = \\begin{bmatrix}1&-3\\\\1&2\\end{bmatrix} \\begin{bmatrix}\\lambda_1 \\\\ \\lambda_2 \\end{bmatrix} . \\,\\!", "332bf7d71ae5f5f9a0a58fea80790bb4": "{}^{II}E^2_{p,q} = H^{II}_q(H^{I}_p(C_{\\bull,\\bull})).", "332bf9b79338accb1bda8ac4bb91bdcd": " D_5", "332c216e5ca18c81eadbbf102ac7b741": "\\langle f(0), f(1), \\ldots, f(n-1)\\rangle", "332c29c22fde9e20ecada05c8652dbbf": "(x-a)^2+y^2=a^2. \\, ", "332c6457109f8737e21a446bbbf6705f": "T^{ab}{}_{,b} \\, = 0", "332c64bd4da6d65d79ce265054c2d921": "\\varphi(\\boldsymbol{x},t)=\\Phi(\\boldsymbol{x},\\eta(\\boldsymbol{x},t),t)", "332c799c80fd55f746272870fd04f01b": "a^{(b-1)/2}\\equiv +1 \\pmod b\\;", "332c837bc94bd00ddde7249c70a8c9ac": "\\|s\\|= \\min_{m\\in\\mathbb{Z}}|s-m|", "332ca88b786b7cce12cdcf52ed59ff87": "f(z) = \\sum_{n=0}^\\infty c_n (z-p)^n", "332d4dee044659b44c73bde3ea77acd2": "1, 2^{4n}, 3^{4n},\\dots, (4n)^{4n}", "332d528960a5abbd63b0bbb435243264": "\\Pr[\\mathcal{A}(D_{1})\\in S]\\leq\ne^{\\epsilon}\\times\\Pr[\\mathcal{A}(D_{2})\\in S]\\,\\!", "332da297d9d2ec60e6abd2fd286aef09": "a_{\\alpha_1, \\alpha_2, \\dots, \\alpha_n}", "332db92aa68664bdddb37bee90f07ff8": "\nL_\\mathrm{int} = \\int L \\ dt.\n", "332dea629a60f88053ee2b77e6c1fd50": "A_\\text{OL}", "332df4cf2a8fe8891977364be2a5724e": "(1 + i_\\$) = (1 + \\rho_$) \\times E(1 + \\pi_$)", "332e35f08bef56683b6c5e32b7ae7d16": "\\mbox{inversion in the unit semicircle.}\\,", "332e694fa7b1a68541d6e5d1467f6745": "\n\\Sigma(Z)=\\omega^{A}\\bar\\pi_{A}+\\bar\\omega^{A'}\\pi_{A'}\n", "332e6dcecbdeafef73e117025be27c0a": " f(z) = z^2 + c ", "332e7882bd72da1622cdee63568227c9": "\\frac{UF}{W}", "332eac2f11479d587fadc5e5f50af0c7": "0.123412341234\\ldots \\;=\\; \\frac{a}{1-r} \\;=\\; \\frac{1234/10000}{1-1/10000} \\;=\\; \\frac{1234}{9999}.", "332eb90a9705926d8365701b9304ea28": "\\xi>0", "332ef52e02f20390e4b04658119bbe3b": "c \\cdot \\sum_i \\Big(\\sum_\\alpha a_{i\\alpha} X^\\alpha\\Big) \\otimes b_i = \\sum_i \\Big(\\sum_\\alpha a_{i\\alpha} X^\\alpha\\Big) \\otimes b_i c.", "332ef838ab920b016016b288748db1d8": "1/4,", "332f5c6f620c02cb8bf498eaf42b9a28": "P(X = x)", "332fa7999b01688a90356f9bb6c2b75c": "P(X_k\\ |\\ o_{1:t}) = P(X_k\\ |\\ o_{1:k}, o_{k+1:t}) \\propto P(o_{k+1:t}\\ |\\ X_k) P(X_k\\ |\\ o_{1:k})", "332fb858c265370265150d5713f6bde3": "\n\\operatorname{div}(\\nabla\\varphi) = \\Delta\\varphi.\n", "332fcea1af664e86f39fc0bd9395a9ee": "y(\\theta) = \\{ x(\\theta), t(\\theta) \\}, \\ x \\in X, t \\in T ", "332ff87a86ddc02193e08a779e13c341": " \\prod_{i=1}^n f(x_i; \\theta) = g_1 \\left[u_1 (x_1, x_2, \\dots, x_n); \\theta \\right] H(x_1, x_2, \\dots, x_n). \\,", "332ffc7d02f82e337e8091bc2634f342": "x=\\sqrt{2n+1} \\cosh(\\phi)", "33311c5adb7d15919f68772086929900": "q^{O(kn)}=q^{O(k^{2})}=N^{O(log N)}", "33312691cebfe65abf306fcdd5085a91": "\\displaystyle{N(x)={1\\over 2 \\pi} \\log |z|.}", "3331272c6105181da1d51cc71c2188ce": "|\\bigcup_{i \\in I} A_i|", "33327140e872dc97a272f0eced21e908": "n_w", "33332f4b95242d69c3d396a5cbf4bf20": " f([g_1,h_1] \\cdots [g_n,h_n]) = [f(g_1),f(h_1)] \\cdots [f(g_n),f(h_n)] ", "3333c3d9b039be512b329f24a87c3882": "F_\\text{P}", "3333f89f93b500c7d1eb98db7a188a0a": "S = \\frac{1}{2\\pi} \\int d^2x\\, \\sqrt{-g}\\left\\{ e^{-2\\phi} \\left[ R + 4\\left( \\nabla\\phi \\right)^2 + 4\\lambda^2 \\right] - \\sum^N_{i=1} \\frac{1}{2}\\left( \\nabla f_i \\right)^2 \\right\\}", "333432a32219fb556e1dd047a937e2ef": "\\begin{array}{rl} E_{j\\,n} & = -m_\\text{e}c^2\\left[1-\\left(1+\\left[\\dfrac{\\alpha}{n-j-\\frac{1}{2}+\\sqrt{\\left(j+\\frac{1}{2}\\right)^2-\\alpha^2}}\\right]^2\\right)^{-1/2}\\right] \\\\ & \\approx -\\dfrac{m_\\text{e}c^2\\alpha^2}{2n^2} \\left[1 + \\dfrac{\\alpha^2}{n^2}\\left(\\dfrac{n}{j+\\frac{1}{2}} - \\dfrac{3}{4} \\right) \\right] , \\end{array}", "3334351ff44a2ccead241d0973b177e6": "Y(t)= \\exp \\left( \\int_{t_0}^{t}A(s)\\,ds \\right) Y_{0}.", "333460f83df05a4c8ce33e0ec3f9b8ee": "T_\\text{r}", "3334799ce1d9e0f1660f22220f221fa1": "\nd(x,y) = \\begin{cases} 0 & \\text{if }x\\text{ and }y\\text{ are in the same partition} \\\\\n1 & \\text{otherwise},\n\\end{cases}", "3334a63bfc14aa64e7f8d5a099a696b3": " x_1, x_2, x_3 \\,", "3334cb656863447024b2930e28b619e8": "\\langle\\mu_s\\mid s\\in{}^{<\\omega}Y\\rangle", "3334d9f06d292db1632d1d4469dba8dc": "\\lambda x(1-x)", "3334effc6a7cee96c24181c0ad36c1b2": "Contrast \\propto \\frac{log I_a}{log I_b}", "333586c06e4b91594830a8f9ec213c95": "b^{p^e} \\equiv a^{N-1}_p \\equiv 1 \\pmod{v}", "3335e6732679bbb8b7100a3b7fb06b96": "R_i=100-4.6 \\Delta E_i", "33360c5b721110e72f0543cf4488439a": "C^{bnn}_{n,n'}", "333622d5f69eb8a39b44101d9bbc24f8": " f:A\\rightarrow S^n ", "3336338e063d68dfb567143b9c15fb4d": "x\\leq a", "3336e4fdc666105d9bed2c304700c6f0": "=\n\\zeta(s+1)\n\\left(\nn+\n\\frac{T_2(n)}{2^{s+1}}+\n\\frac{T_3(n)}{3^{s+1}}+\n\\frac{T_4(n)}{4^{s+1}}\n+\\dots\n\\right)\n", "33374b70ae5360e50375c05134d1cf5c": "\\ log ", "3338080f7607e5edcca548decc5fec79": "\\scriptstyle<2\\times10^{-27}", "33380d8df9002a16e35b9e047bc76a2e": " x_t = x_0 e^{-\\theta t} + \\mu(1-e^{-\\theta t}) + \\int_0^t \\sigma e^{\\theta (s-t)}\\, dW_s. \\, ", "3338150a797da7f029a498c29012bc91": "\\left\\{\\begin{array}{ll}\\infty & r = 1\\\\ 1 & r = n\\\\ 2 & \\text{otherwise}\\end{array}\\right.", "33381c5a918902267dac37a7beed7653": "\n \\Sigma_1 := \\cfrac{\\sigma_{1c}+\\sigma_{1t}}{2\\sigma_{1c}\\sigma_{1t}} ~;~~\n \\Sigma_2 := \\cfrac{\\sigma_{2c}+\\sigma_{2t}}{2\\sigma_{2c}\\sigma_{2t}} ~;~~\n \\Sigma_3 := \\cfrac{\\sigma_{3c}+\\sigma_{3t}}{2\\sigma_{3c}\\sigma_{3t}}\n ", "3338d3c10521175c1823908402163026": "s \\ = \\ \\sigma + j \\omega ", "33396abbdf04a15c88a46a16dcf564d6": "\\Sigma = \\left\\{ \\left. \\left(x, y, \\log \\left( \\frac{\\cos (x)}{\\cos (y)} \\right) \\right) \\in \\mathbb{R}^{3} \\right| - \\frac{\\pi}{2} < x, y < + \\frac{\\pi}{2} \\right\\}.", "3339bd716df0f05d893e1453b17192e9": " \\{\\cdot,\\cdot,\\cdot\\}_-\\colon V_+\\times S^2V_- \\to V_-", "3339e88d6ed941102bc56ad58af456a9": "w_i = 1/m", "3339ebace629ad9f8287b0a5fb9b6bff": "A(t)", "333acf5f864bdb2d54c0a7d8b7dce275": "\n f := (J_2^0)^3 - \\alpha~(J_3^0)^2 - k^2 \\le 0\n ", "333b2cc5aa97d437a99a2d49d9e39c16": "\\binom{n-1}{n-x}", "333b7b2847bc908e11021abda943424a": "\\sigma_p(0) = \\frac12\\zeta(-p).\\;", "333bbf5aaf6b439388eaa342cb4539e1": "\\tau_1 \\leq \\tau_2", "333be19453568e049bdedd294192b889": "\\mathbf{x} = \\mathbf{v} t ", "333c2aa754fc2c331dfb2509e67cb4e5": "\\Delta H_1 = - R * slope_1,", "333c4b94e6e214a852ff34f11ee7013b": "\\hat{F}(x)", "333ce31287200d302d5dbeaa6ca35e99": "(\\pm 1,0,0,\\pm 1)", "333d40b44797415e2253d7ca2e2aeb75": " G(t)=-\\ln p(\\tilde{s},\\tilde{x},\\tilde{u}\\vert m) ", "333d49b851c0d4b7c22800ebf1dfa9ec": "s = W x\\,", "333d4f3d33bd2f3e853f32fca5b9f704": "C(x,y,z)\\,", "333d6ae68fcd0d3674ea58dbcff0999b": "dx = 0", "333d8e5b5e85be37b741033f99292963": "c \\downarrow 0", "333e2b2a706364fbe8d6788542e7004e": "\\begin{align}\n\\mathbf{M} & = \\begin{pmatrix}\nM^{00} & M^{01} & M^{02} & M^{03} \\\\\nM^{10} & M^{11} & M^{12} & M^{13} \\\\\nM^{20} & M^{21} & M^{22} & M^{23} \\\\\nM^{30} & M^{31} & M^{32} & M^{33} \n\\end{pmatrix} \\\\\n & = \\left(\\begin{array}{c|ccc} 0 & - N^1 c & - N^2 c & - N^3 c \\\\\n\\hline\n N^1 c & 0 & L^{12} & -L^{31} \\\\\n N^2 c & -L^{12} & 0 & L^{23} \\\\\n N^3 c & L^{31} & -L^{23} & 0 \n\\end{array}\\right) \\\\\n & = \\left(\\begin{array}{c|c} 0 & - \\mathbf{N} c \\\\\n\\hline\n\\mathbf{N}^\\mathrm{T} c & \\mathbf{x}\\wedge\\mathbf{p} \\\\\n\\end{array}\\right)\n\\end{align}", "333e52febd30349f75045be16b27398b": "\\iff \\neg (\\neg B) \\or \\neg A ", "333e863732f4808d0d083a0a5c34ddfc": "\\theta =\\frac {k_1 C_A}{k_1 C_A + k_{-1}+k_2}", "333ea43a55c15ae3333f8c90ab8c5c0a": "x \\Rightarrow_{p} y", "333ec1eb46ca28b3672fc5de959f857b": "\\mu(q) = \\sum_{n\\ge 0} \\frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q^2;q^2)^2_{n}} ", "333efed01706df60469afcb03293799a": "L(d) = \\frac{L(10)}{(\\frac{d}{10})^2} ", "333fb0f12c9a0147652a0939d8398ba4": "H_{k+1}=B_{k+1}^{-1}=", "334034904ad72a29e2b7f95042dbbe0a": "a_{\\mathrm cA}", "33405054d27036a9a629fa6e2d23a300": "\\mathbf U=\\mathbf U^T\\,\\!", "33408b50a09a374c08d03f5acd0e6e0f": "e^{-nx}\\sum_{k=0}^\\infty{\\frac{(nx)^k}{k!}f\\left(\\frac{k}{n}\\right)}", "3340be2c928efc3591922ecb1d2a5955": "\\partial_\\mu E_n=\\langle n|\\partial_\\mu H | n\\rangle", "3340c05761c3bfe6f6b30fae3ea3ffb2": "P \\in \\Bbb Q^3 \\iff P' \\in \\Bbb Q^2", "3340ebd077effa2faccb6e40a842150f": "c_{FR_n} = 1 - (1 - \\beta)n + n0 = 1 - \\epsilon(1-\\beta)n ", "33413902509d983a7874287cc94b366c": "q_0=\\Omega_0/2", "3341e6a8ea58095ce01fd0022d275991": "E=-\\nabla V", "3341ff6dc50411fad5c3c85a8832b0e2": "|\\mu|(A)=\\sup \\sum_{i=1}^n \\|\\mu(A_i)\\|", "33426c40a028bde531f11f15e160c0a4": "\\mathbf \\nabla \\cdot \\mathbf A + \\frac{1}{c^2} \\frac{\\partial \\varphi}{\\partial t} = 0\\,.", "33427c68b1f785d1caa76374bf7e1da3": "R = a\\cos\\beta", "3342a23faa89b251a4f07911782e6a83": "\\hat{f_j}", "3342b5462b201b9e74c9e5a731660d65": "b(x) = 0,\\ x\\in F", "3342cec1fe8100d843dd192a31d4c3ba": "\\text{Diagnostic odds ratio, DOR} = \\frac{TP/FN}{FP/TN}", "33433587643488f3878e892f3257569c": "(23) \\in S_4", "3343451f059789535090f23bb85d315a": "\\frac{2\\cdot\\pi}{3\\sqrt{3}}", "3343779cf1ae5ae24080685bd68782f1": "\\! \\lim_{n \\to \\infty} \\! \\left( \\! 1 \\! + \\! \\frac {1}{n}\\right)^n \\! = \\! \\sum_{n = 0}^\\infty \\frac{1}{n!} = \\frac{1}{0!} + \\frac{1}{1} + \\frac{1}{2!} + \\frac{1}{3!} + \\textstyle \\cdots ", "334390e960aa73941d28dfb74586cdbb": "({\\mathcal O}(n + m))", "3343e099d787d511f9226ef5ee36beca": "\\sum_{k=0}^\\infty a_k = \\infty\\!", "334406afaa00980e7e3a00f34a3714c7": "S_r'(r)", "334498cabe7d8964d9adfdd22e152d38": "\\operatorname{rank}(CA) = \\operatorname{rank}(A).", "33449e01c30ed9a6c59fe439f2a2b9ad": " \\frac{s}{s_0}=[1+\\frac{P_d'\\theta}{(P_a+P_d)(1-\\theta)}]^{-1} ", "33456ce3c89a82342451e67818b6165f": "f(t).\\,", "334578de9ceca742390c164b8fc1ab30": "e=\\mathrm{10.T0111TT0T0T111T0111T000T11T}...", "3345e90ac50fa0bccbc2025d8eb8d5a0": "\\Pr(X_{n+1}=x|X_n=y) = \\Pr(X_n=x|X_{n-1}=y)\\,", "33460ddd465fa14641d4c76e3ae6350b": " \\mathbf{v}\\,\\!", "334638ed7159a1bd0ea2f0ffdcd5146a": "V^* h = 1 \\otimes h", "3346a9223752ee71073bb0888a4c0f3d": "g_p \\colon T_pM\\times T_pM\\longrightarrow \\mathbf R,\\qquad p\\in M", "3346b08a98adac7b6522f1168c4ea9ee": "\\lambda f.(f\\, \\textrm{true}, f\\, \\textrm{0})", "3346b303af4e5faca1b86f7836bc195a": "\\Delta V/V", "334719af51ab028d4a76fd7d5d6c3a03": " \\lambda_t \\in \\mathbb{C}, t = 1, 2, \\ldots , n ", "334726250fc591e3813b65d7ba6fb0a3": "\\mathbf{J}=\\rho \\mathbf{v},", "334730a7306f25d9d5fa7ddedf89402d": "\\cos z=\\lim_{n \\to \\infty}((1+iz/n)^n+(1-iz/n)^n)/2", "33476f0b23cb938af79a091ad6d87bbb": "(C_\\beta) < \\beta ", "33479d5458c96c5de91349588edf1a27": "\\displaystyle{\\widetilde{\\mu}(w)= (w^\\prime/ \\overline{w^\\prime})\\cdot\\mu(z).}", "3347db00ca54fd7d1967c1ff568ed727": "T = \\rho^\\nu_\\mu (x-x')_\\mu(x-x')_\\tau\\rho^\\nu_\\tau.", "334804985b95eababda35b8de8a8b0d9": "= \\lim_{N \\to \\infty} N b \\log_2 \\left( \\frac{A}{N W} + 1 \\right)", "334838e8f60baa6edad5b4664b38f749": " \\gamma_w \\,", "33483de42230321208b81b20497244dc": "A^\\lambda = \\Gamma^\\lambda {}_{\\mu \\nu}U^\\mu U^\\nu = \\{0,GM/r^2,0,0\\}", "3348a97aabf9d7a47858036a6f30c047": "\\frac{\\pi}{4} \\approx 1 - \\frac{1}{3}+ \\frac{1}{5} - \\cdots (-1)^{(n-1)/2}\\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)", "3348d64bf8670bb42ef788098ca64fdd": "y_0=\\frac{\\nu}{9 u_\\tau}", "3349522a6e5bbc3e51242248b9a349b4": "d_i = e_i", "3349702875554af3e4114e126071be28": "t:\\textit{Tom}", "3349c1564c04064b3d60f7d388ffbd3a": "\\psi(w,z) = \\overline{\\varphi(z,w)}.", "3349e422bb02a34e6a04f969f579d600": " R = \\sqrt{Z_1^2 + Z_2^2}, \\qquad \\Theta = \\tan^{-1}(Z_2/Z_1)", "3349faae9b584b0b0d50a2d71675ea78": "\\rho = \\frac{\\sqrt[3]{108 + 12\\sqrt{69}} + \\sqrt[3]{108 - 12\\sqrt{69}}}{6}\\, .", "334a03c20fea03d9d12627e46cbeddb8": "\\begin{bmatrix}\n0 & 1 & 2 & \\dots & n-2 & n-1 \\\\\n1 & 2 & 3 & \\dots & n-1 & k\\end{bmatrix}", "334a60b2d5c6f09c8da93c6063604195": " a_r = \\frac {v^2}{r}", "334b8b51ab1892b7ad007461b4a23eab": " 2 \\uparrow^{n - 1} 2 ", "334bc293063f4c1a4b8c30347d345813": " Q_\\text{surf}=Q_\\text{sec,man}+Q_\\text{rad}+Q_\\text{cmb} ", "334bd430def30dc98e1df2a10a2f6ca9": "\\sqrt{\\frac{1}{63}}\\!\\,", "334c142d1630fdd424c6c5791c08bb30": " \\mathit T = \\frac {\\mathit CF}{\\mathit P} + \\boldsymbol {\\Phi} g + \\frac{\\Delta PB}{PB} \\mathit(1 + g)\n", "334c6e244e93cb1e84e8a48ccfc0aab7": "s=\\sqrt{M/Q}", "334c7d5961db31fc276e163c989d3284": "\\lambda_2\\,", "334ce9eb79df1178b0380461c9eaa09e": "\\ell ", "334d07311a29500d566e41e32ccd9606": "\\sigma ^{\\infty} = \\frac{e^{2}}{k_{B}T}\\sum z_{i}^{2}D_{i}n_{i\\infty }", "334d0f359a888ad603daec033b210155": " v(S) + v(T) = v(S \\cap T) + v(S \\cup T)~.", "334d26a764d833b66be36fc75645adaa": "E^o = 1.229\\,V", "334d3bb5ad8d78e01ae8bc60cb62d846": "\n\\frac{\n\\Gamma \\; \\vdash \\; t \\; : \\textbf{nat} \n, \\quad \\quad\n\\Gamma \\; \\vdash \\; s_0 \\; : \\sigma \n, \\quad \\quad\n\\Gamma \\; \\vdash \\; s_1 \\; : \\sigma \n}\n{\n\\Gamma \\; \\vdash \\; \\textbf{if}(t,s_0,s_1) \\; : \\sigma \n}\n", "334d5a8137509d84bd169203ff86cc0e": "Y/Y_n > 0.01", "334d7696bf053233a10d4cdc4eeef281": " W(M) = \\int_M H^2 \\, dA. ", "334da94629adc05454c0b86ebc607101": "H^\\uparrow=\\{e\\quad|\\quad\\forall h\\in H, e\\downarrow h\\}", "334dc0e14ff95ded15522c527a031859": "f(x,y) = A \\exp\\left(- \\left(\\frac{(x-x_o)^2}{2\\sigma_x^2} + \\frac{(y-y_o)^2}{2\\sigma_y^2} \\right)\\right).", "334dce61464a4ffec70fba9c6db45c58": "{\\partial u}/{\\partial n}", "334de1ea38b615839e4ee6b65ee1b103": "\\gamma ", "334e190d393b9ba3ff12e254121a6846": "f(\\cdot|\\theta)", "334e1f764d77e666040adaa9dc81df1c": "\\scriptstyle \\vec U", "334e4656f0c490a4489cd71f348380d9": "\\lnot C\\land I", "334eb84f8f82eee58abfc12d0b69fd09": "\\cong \\mathop{\\textbf{y}}\\downarrow P", "334ec9aef1d2463d059a7e92cc25b944": "E_A", "334f60e0f83184f75d3950fc7c79f5a3": " \\Delta\\ V = 2 \\cdot\\ G \\cdot\\ \\theta\\ ", "334fc46b0aaa16ccf554e6b6a4022bf2": "\\chi_{A} (x) := \\begin{cases} 0, & x \\in A; \\\\ + \\infty, & x \\not \\in A. \\end{cases}", "335004c7b3a6ee0b131547528dbeb57f": "\\,{}^{x}a", "335024f5cc92ab390b36c188aad93c14": "x_{p,ni}", "335049bdb1aa2ec9ac068d09dcd36d34": "\\scriptstyle k(s)", "33505068cd4b55d13199b1a78fca1851": "[e,f]=h", "335077973ddbf636c3270b029fd38a72": "b_n=b_0+\\frac{1}{2}(\\mathbf{y}^{\\rm T}\\mathbf{y}+\\boldsymbol\\mu_0^{\\rm T}\\boldsymbol\\Lambda_0\\boldsymbol\\mu_0-\\boldsymbol\\mu_n^{\\rm T}\\boldsymbol\\Lambda_n\\boldsymbol\\mu_n) .", "33507cd51419c3c00d2dfeb0c8262eb0": "\\mathcal O(N^2)", "335120a9d53e0900283d4c118c167f63": "CL_o", "335128b17c65f9aff6ac9d58ea7187c3": "\\mathcal{}MU_*", "33514655762953d99cbab12f22d222f1": "v \\in N(S)\\,", "3351720313e739d5b0780f131b42de51": "s(n) > c_1 4^n n^{-{5 \\over 2}},", "3351bb26a1bcd09c5d2ae10c9fa51622": "x \\underline{x}^{-k} = \\underline{x}^{-k+1},\\quad\\text{if }k>1.", "3351be9174b2b690609516a0df4dcfd4": "B\\in \\mathbb{M}_n", "3351d58faeda5f653f16e3e49b0eac25": "\\frac{p_{n}(x)}{x-x_{i}}", "3351d6ec9442ed3779d2914c79ee28fa": "\\lambda\\geq 2", "3351efc01139705a739ca4bc2acc9a2a": "\\begin{align}\\hat{a}|\\alpha\\rangle&=\\alpha|\\alpha\\rangle \\\\\n\\langle\\alpha|\\hat{a}^{\\dagger}&=\\langle\\alpha|\\alpha^*. \\end{align}", "3352547126e21c8c7988b4f73873aa4d": " \\textstyle {3 \\over 2} a^{1/2} ", "335275e964fdf43a686f2a23733da67f": "2-\\sqrt{2}", "33529715aa8907611d0b4cd850bf2e81": " \\mathbf{B} ( \\mathbf{r} , t ) = \\hat { \\mathbf{z} } \\times \\mathbf{E} ( \\mathbf{r} , t ) ", "3352f0273293cbb04dedd802017ab0a4": "w= \\sum_{i=1}^n w_i", "33530f218a2036e53aea9776ed2f0bb8": "b_i\\;", "3353289f4800a71f505ce266fcf551b0": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} =\n \\cfrac{2Eh}{1-\\nu^2} \\begin{bmatrix} 1 & \\nu & 0 \\\\ \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix} \n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix} \\,,\n", "33532f13b849a095415b290fe5a96ee1": "P_x = x Y_x - (x-1) Y_{x-1} ", "33533d8021b390532d0c960323ad4a8e": "f = \\mathrm{eval}_{A,B}\\circ(h \\otimes \\mathrm{id}_A).", "3353401b9589b4c080c0d3c1c99fd7fc": "\\lambda_S(t) = \\int_S \\lambda(x,t)\\,d\\mu(x).", "3353906f01daddc63d0b4a3521962c97": " a_0, \\ldots, a_{s-1} ", "3353a12a200868733bb2214d075530ef": " \\theta = \\int_0^\\infty {{\\rho(y) u(y)\\over \\rho_0 u_o} {\\left(1 - {u(y)\\over u_o}\\right)}} \\,\\mathrm{d}y ", "3353a5bf7588eb5542eda03871d09fb1": "e^{-i\\omega t}\\phi(x)\\,", "33545006b3a15fe9b9c3046a6a6320bd": "\\phi(\\cdot,z)", "33545f2e1612ea82228d7c59b2751c6e": "dm_{fuel} \\ c^2 = \\frac{dm_e \\ c^2}{\\sqrt{1 - \\frac{v_e^2}{c^2}}}", "3354c5c738a80a959894cdd94d7e2bde": "f(m) = m", "3354ea5a0942b4a637058b13036c615a": " \\operatorname{lambda-anon}[V] = \\operatorname{false} ", "3355aaede9d11be17de2d86d58d89e4a": " \\int f_k \\, \\mathrm{d}\\mu = \\int_{X \\backslash N} f_k \\, \\mathrm{d}\\mu, \\ \\text{and} \\ \\int f \\, \\mathrm{d}\\mu = \\int_{X \\backslash N} f \\, \\mathrm{d}\\mu, ", "3355abbb384809ef5e0c3516f8726db0": "\\delta (x) \\mapsto \\frac{\\sin \\bigl[ \\frac{\\pi}{2}(1+x/h) \\bigr]}{ \\pi (x+h) }~,", "3355c87ed495a642e5f4d7013cc08827": " \\xi=\\sum_{i=1}^N \\delta_{X_i}, ", "3355e6e968934034b57d8d3feb1d39c0": "\\scriptstyle<3.7\\times10^{-31}", "3355f94aff566e7afd58293f2549d246": "\\left| k - {m \\over n}\\right| < {1 \\over n^2 \\sqrt 5}.", "335620ed27066a04701d2a7e4ee97b72": "t ", "33562d783a5ad8a4db440e3fbaa4cc15": " e (t) ", "335654f70a338e4e456704e0059ad317": "\\pi^{ab}/tor ", "335694f99a23400be8fc8b6ab398f968": "(\\dfrac{\\alpha \\delta_o^2}{4})(1-\\in) N ", "33569848446dd095dec59c68a84c67a1": "F(t,x)", "3356a1e042c258d12552c49286b7825f": "\\sum_{i=1}^n w_i x_i > \\Theta ", "3356c901b58e1b1ff4b797986f74a598": "u_i = a^{2^i} + a^{-2^i}", "3357457cf475900b028308e097d7af53": "AF \\times BD \\times CE= - FB \\times DC \\times EA .", "335748264ddb4922c5a901c9fb059477": "Q_H", "3357530e5ab1f3f99fa0aadf13ef600f": "\\,\\Delta w_i = \\eta x_i y, ", "3357876df4a771e76581d70b0b3f5572": " {\\mathbb F_p}^*", "33580c70f9b843f9e485f4fea1197394": "V_{ds}", "33584f166497934bbce813dea6796394": "\n\\begin{align}\n& {} \\quad t(t+1)+ t( x_1 \\partial_1 + x_2 \\partial_2) \n+x_2 x_1 \\partial_1 \\partial_2+x_2 \\partial_2 -\n x_2 x_1 \\partial_1 \\partial_2 - x_2 \\partial_2 \\\\[8pt]\n& = t(t+1)+ t( x_1 \\partial_1 + x_2 \\partial_2)=t^{[2]}+ t\\,\\mathrm{Tr}(E).\n\\end{align}\n", "33586776f59ba6d68244d1435d8d4fe4": "f_v(k,r)", "335895eae86cdc65bd7289fee8bbfbf0": "S_{xx}", "3358a48823c9e4b1366791d092853a7d": "({v_0+v_i})10^{-b_1E_i}", "3358aac47c72d987c0a1fc66b2f9816b": "\\mathbf{E} = -\\mathbf{\\nabla} \\phi - \\frac{1}{c} \\frac{\\partial \\mathbf{A}}{\\partial t} ", "3358dd4a355f03e7e0f00d8102f653a1": "S_{\\lambda} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{j_{\\lambda}}{\\kappa_{\\lambda}}", "3358e1d96b2fb13244414db75c166b01": "\\displaystyle{2Q(ab,a)=L(b)Q(a) +Q(a)L(b).}", "3359379344b632757444328c5ce17b4a": "\\,{}^{x}e", "33594aa6832621e55a1fa708f505f99d": "[\\textbf{x}^*(t^*),\\textbf{u}^*(t^*),t^*]", "33596476aa076e8e38183bb48ec6173f": "j d^2 = (j_e + j_i) d^2 = \\left( 1 + \\sqrt{\\frac{m_e}{m_i}}\\right) \\frac{4\\varepsilon_0C_0}{9} \\sqrt{\\frac{2e}{m_e}} \\phi_{DL}^{1.5},", "3359a837f5edb773e122443cc897d56b": "(2vy)^2+(2xz)^2 =\\, (v^2-x^2+y^2-z^2)^2 ", "3359cf3aefc33d3bf5fdc630d878c053": " p = k \\ln{S} + C, \\,\\!", "335a02f922e4fa39f79a19187663b48f": "\\subseteq B", "335a0cb203ae7bc7d670052e01a85b62": "-\\mu(-A)", "335a9ee97b02ce4322a5bbc635bb31a6": "{\\tilde{C}}_{2+}", "335aa43cc281fd1b7a869b1e92f5b707": "M(a,b,z) = \\frac{1}{2\\pi i}\\frac{\\Gamma(b)}{\\Gamma(a)}\\int_{-i\\infty}^{i\\infty} \\frac{\\Gamma(-s)\\Gamma(a+s)}{\\Gamma(b+s)}(-z)^sds", "335af3892a491de828bdfcb673bba25a": "\\tan\\frac{\\pi}{10}=\\tan 18^\\circ=\\tfrac{1}{5}\\sqrt{5(5-2\\sqrt5)}\\,", "335af92f7c371c4a93e74fed1179fa66": "\\zeta(\\bar{a},\\bar{a})=\\tfrac{1}{2}\\Big[\\phi^2(a)-\\zeta(2a)\\Big]", "335b1b47d70a67acc9ecea0b17656e5f": "x_1=-1\\,\\!", "335b260d8dac5d0a821835153817b8bf": "\\Psi\\colon V \\times V \\to K", "335b5612763431dc825cd81bf916a40d": "\n(\\forall X\\subseteq U_p)(CUM(X)\\iff \\exists x,y(X(x) \\wedge X(y) \\wedge x\\neq y) \\wedge \\forall x,y(X(x) \\wedge X(y) \\Rightarrow X(x \\oplus_p y)))\n", "335b931ef1e9c14595858f1be914a2dc": "\\text{length} (\\gamma)=\\sup \\left\\{ \\sum_{i=1}^n d(\\gamma(t_i),\\gamma(t_{i-1})) : n \\in \\mathbb{N} \\text{ and } a = t_0 < t_1 < \\cdots < t_n = b \\right\\}. ", "335c2b8edc9f3f1368c622252d97eb2c": "a_{k}", "335c2d7c2ccfc1766a245197da8da7c8": "[-1,+1]", "335c37f22b49d3eb330119d395c73732": " \\{ \\tilde{\\mathbf{e}}_{k} \\}", "335c41cb986e230301d5b713d782b417": "a^{(n-1)/2} \\equiv \\pm 1\\pmod{n}", "335c63dbac556876e19e17abc88e727c": "\\alpha,\\alpha'", "335c76bfd3b54a918105c9b2a092a66b": "\\left(a, p, v\\right)\\succsim \\left(b, q, w\\right)", "335caf66766edbf9dbe5dfd4866019e5": "S(x)\\Gamma(x)", "335d135de25fe2dbf0ccd0f20e0150e1": "E_+", "335d1d91f0b4ca4cde3037a0518503af": "\\isin", "335d571a800e87abab3a451c4b336b2b": "EER=(662-(9.53*Age))+PA*((15.91*wt)+(539.6*ht))", "335d5bf2780db870acd5349d619472c2": "\\textstyle {n \\choose n}", "335d706d435b81162df7574175609595": "\\det\\begin{pmatrix}A& B\\\\ C& D\\end{pmatrix} = \\det(A) \\det(D - C A^{-1} B) .", "335dcdaf50f8976befa31e2b74ab268c": "\\langle{y_1,y_2}\\rangle = \\int_a^b w(x)y_1(x)y_2(x) \\, dx", "335e11db9f728db3854c52d7b61bb381": "\\oint_\\gamma f(z)\\,dz = 0. ", "335e56093ff9d750b6a2d341dcbf07ed": "f^{*}\\colon T_{f(x)}^{*} N \\to T_{x}^{*} M", "335ea4282af1f421e9fe9dbdc29d4cef": "\n Q(A) = \\int_A f\\,\\mathrm{d}P, \\,\n ", "335ea4846fe18e60d534d4e10abb098e": "\\frac {1}{2\\sqrt {\\pi}}=0.28209\\dots\\,.", "335ed076f24b12ca03bc11a40fdce1b2": "\nL\\bar{L} = \\bar{L} L = 1\n", "33609b5bf3d36b4b5697cbb089ca77d5": "S=\\frac{1}{2\\kappa_0^2}\\int d^Dx\\sqrt{-G}e^{-2\\Phi}\\left[-\\frac{2(D-26)}{3\\alpha'}+R+4\\partial_\\mu\\Phi\\partial^\\mu\\Phi+O(\\alpha')\\right],", "3360e6a8d9590945f72749f6e5482244": "P _{1/2} (\\cos\\theta _0)\\,", "3361616c1f1670025342c74da73ea564": "\\widehat{a}^{\\dagger}_j\\rho \\rightarrow \\left(\\alpha_j^* - (1-\\kappa)\\frac{\\partial}{\\partial\\alpha_j}\\right)\\{W|P|Q\\}(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)", "3362285e44ed00916456a98f8039e856": "n=q^m-1", "3362c1cf63008ef712ba7942abfbed2e": "f'(z_k)", "33633fbbcc0b982b788cbb56c3420321": "\\frac{1}{5} + \\frac{1}{20} = \\frac{1}{4}", "33643d038fd57cc8ce81c2c0be49fe14": "V |k^{(0)}\\rangle = \\epsilon_k |k^{(0)}\\rangle \\qquad \\forall \\; |k^{(0)}\\rangle \\in D. ", "3364534c444cb41a2eef9b3af1de3d07": "J_1=2\\left[\\left| A \\right|^2-\\left| A' \\right|^2\\right], \\quad J_2=2\\left| B \\right|^2 \\,", "3365154e49051067dc53fb4772f93c80": "\\, k_B", "33652f1992eaad8e2714be4a98fbbb10": "x\\nleq b.", "336557087f5c2478a46cd6a3bd06ccb9": "\\,(z-(\\gamma+1))\\,", "336589123b6682d1e4625748db32ecee": "\\gamma = \\sin^2(\\theta)", "3365ef36bbf0aa1faaa96dcff395cac2": "\\omega_1, \\dots, \\omega_g", "3365fc0b805aaa396066b58c2912d4a6": " \\operatorname{let-combine}[\\operatorname{let} V : \\operatorname{de-lambda}[V\\ F = E] \\operatorname{in} V] ", "33660c3dc68b0b6c1d15e85deb903c68": "(p \\land q) \\vdash q", "33662cba1f032622548bc60a22a13e40": " -\\infty < t, z < \\infty, \\; \\; 0 < r < \\frac{1}{\\omega}, \\; \\; -\\pi < \\phi < \\pi", "33662d44fdbdb0de47eaaed7be1e0538": "~\\begin{align}\n{{\\rm d}X}/{{\\rm d}t} & = KXY-UX \\\\ \n{{\\rm d}Y}/{{\\rm d}t} & = - KXY-VY+W\n\\end{align}\n", "33663abf2b191cda151b5f98e0687917": " \\partial y^2 / \\partial x = \\cos( x / r). ", "3366661c02842cedff47fb1ea75a1c62": "\\mathcal{G}_n^d", "3366a9cd4aeb6b4bf1e5babac5818992": " \\int_{-\\infty}^{\\infty} \\exp\\left( -{1 \\over 2} a x^2 + iJx\\right ) dx = \\left ( {2\\pi \\over a } \\right ) ^{1\\over 2} \\exp\\left( -{ J^2 \\over 2a }\\right ) ", "33670f622012286dee14dc9df88bb483": "a_r = \\ddot r - r \\dot \\theta^2= \\ddot r - r \\left(\\frac{nab}{r^2}\n\\right)^2= \\ddot r -\\frac{n^2a^2b^2}{r^3}. ", "33673d3fc4512d67aca54d1bb5286ba7": "\\sigma_3 = 0 ", "336778ade288c18e56813785e67841d6": " F = M m f(r) ", "336781e7e0648ec031eaad9bb58dcb16": "\\{\\;<, \\le, =, \\ge, \\;>\\}", "336794e85fb8a0e29d7d04b4274dc7cb": " \\frac {1} {m^*}", "336816136cc2a5e6cc187558c433792c": "\\gamma_\\mu a\\!\\!\\!/ b\\!\\!\\!/ \\gamma^\\mu = 4 a \\cdot b \\,", "336839a757c3bc8f40fe309daddde7a4": "\\delta'=\\delta+kv-\\frac{\\mu'B}{\\hbar}", "336863d2959c13734c9d32b2760c8195": "g(\\chi)", "33688a93fc3e85473b5ba449af880462": "V_D/(kT/q)", "3368c23db622906d475df948c01c5576": "\\Pi=-(RT/V)\\ln(\\gamma_s x_s) ", "3368d4e330142d20babf9bd65aaec824": "u_{\\min}^{(s)}=x^{(s)},\\ u_{\\max}^{(s)}=k^{(s)}+x^{(s)},", "3368fd6fb10694249edf4052718f7e34": "\\displaystyle{\\delta(X)=[T + A,X],}", "33690e97dbeaa68d615e68dba167db2a": "P(A|B)", "3369228b91298529286f7fea8359fdf4": "\\delta Q = T\\,dS\\, .", "33697ce7dfa48ba80980d298c8089378": "O(N)", "33697f4fcf476e1a12b18fff178a4756": "\n\\Delta r \\approx \\sqrt{r^2 + \\lambda \\frac{g b}{g+b}} - r.\n", "3369ed71dab07d945e58e0b1cbaf0c19": "v_\\text{e}=0.6275 \\Delta v", "336a145bd7054e8a64767885c377630e": "\nG_{p+1,\\,q+1}^{\\,m,\\,n+1} \\!\\left( \\left. \\begin{matrix} \\alpha, \\mathbf{a_p} \\\\ \\mathbf{b_q}, \\beta \\end{matrix} \\; \\right| \\, z \\right) =\n(-1)^{\\beta-\\alpha} \\; G_{p+1,\\,q+1}^{\\,m+1,\\,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p}, \\alpha \\\\ \\beta, \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad m \\leq q, \\; \\beta-\\alpha = 0,1,2,\\dots,\n", "336a3472a60299556434a984e2edcd4c": "(x,t)+E_\\lambda\\subset\\mathrm{dom}(u)", "336a35e8cf39f50818f772518354ab70": " F_i = \\sum_{j=1}^m \\mathbf{F}_j\\cdot \\frac{\\partial \\mathbf{v}_j}{\\partial \\dot{q}_i},\\quad i=1,\\ldots, n,", "336a463d750f0f005449496e6655e3ef": "-1 = i \\cdot i = \\sqrt{-1} \\cdot \\sqrt{-1} = \\sqrt{(-1) \\cdot (-1)} = \\sqrt{1} = 1", "336a534e581caeabc6ca1d872a512363": "W = \\frac {{p_1} {V_1^n}} {1-n}\\ ( {V_2^{1-n}} - {V_1^{1-n}} ) ", "336a5cf2c98dc961c8bea7e22a3819e8": "\\textstyle P(A\\mid[x]) \\leq \\gamma", "336a6d6c8ba7a0fae62fb9f9aad52e43": "\\int_{-\\infty}^\\infty g(x)\\exp\\left(-(x-2\\theta)^2/4\\right)\\,dx.", "336a97d18d66c095def5ab62b9889a8b": "\\lambda >0", "336abdf8e26e5d1155dc90d9c8c1c6ac": "\\bold p_Q=-(\\hbar/2m)^2 \\rho \\nabla\\otimes\\nabla \\ln \\rho", "336b1b57931a60280ae6e54eb39c193b": "C_n(K) \\cong \\mathbb Z_n \\oplus C_n^0(K)\\,", "336b96c5b216dd2c75b2dc51d174214f": "\\scriptstyle p_\\phi", "336bb9c8758032e25ed1ca250b722198": "\\,1,g_2,g_3, g_2g_3", "336bd744d4e960445cce709476b7a0e7": " \\varepsilon_0 = \\frac {1}{\\mu_0 c^2} = \\frac {e^2}{2\\alpha h c}\\ ,", "336beddaf4008f9fd3f9959eec8fab6f": "\n\\begin{align}\nY_{2,-2} & = d_{xy} = i \\sqrt{\\frac{1}{2}} \\left( Y_2^{- 2} - Y_2^2\\right) = \\frac{1}{2} \\sqrt{\\frac{15}{\\pi}} \\cdot \\frac{x y}{r^2} \\\\\nY_{2,-1} & = d_{yz} = i \\sqrt{\\frac{1}{2}} \\left( Y_2^{- 1} + Y_2^1 \\right) = \\frac{1}{2} \\sqrt{\\frac{15}{\\pi}} \\cdot \\frac{y z}{r^2} \\\\\nY_{20} & = d_{z^2} = Y_2^0 = \\frac{1}{4} \\sqrt{\\frac{5}{\\pi}} \\cdot \\frac{- x^2 - y^2 + 2 z^2}{r^2} \\\\\nY_{21} & = d_{xz} = \\sqrt{\\frac{1}{2}} \\left( Y_2^{- 1} - Y_2^1 \\right) = \\frac{1}{2} \\sqrt{\\frac{15}{\\pi}} \\cdot \\frac{z x}{r^2} \\\\\nY_{22} & = d_{x^2-y^2} = \\sqrt{\\frac{1}{2}} \\left( Y_2^{- 2} + Y_2^2 \\right) = \\frac{1}{4} \\sqrt{\\frac{15}{\\pi}} \\cdot \\frac{x^2 - y^2 }{r^2}\n\\end{align}\n", "336bff37d4e3f200bf38168a0af3be1f": "''e_{ex}''", "336c2fb50db82c847d381f49f522d52e": "c_2\\cap{\\boldsymbol S}(c_1)\\neq\\emptyset", "336c3989d856c9bafa3441e0a114c6d8": " \\lfloor b/2 \\rfloor ", "336d4c8fac60d2380a35dabd664d9a2e": "\\frac{4ab}{\\sigma^2}", "336d54b8c53666c19eb10e3cd88e2ded": "R_\\mathrm{static} = \\frac {V}{I} \\,", "336d5ebc5436534e61d16e63ddfca327": "-", "336d62419816ea0881ddb21e5efe8506": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ -1\\end{pmatrix}", "336d719a5d4b06704b6aede1f814904f": "\ns^2 = \\begin{bmatrix}c \\Delta t & \\Delta x & \\Delta y & \\Delta z \\end{bmatrix}\n\\begin{bmatrix} -1&0&0&0\\\\ 0&1&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{bmatrix}\n\\begin{bmatrix} c \\Delta t \\\\ \\Delta x \\\\ \\Delta y \\\\ \\Delta z \\end{bmatrix}\n", "336e0670ed5955cc0dc455f7bb3a1eef": "R \\equiv 0 \\mod 4", "336e5beada03a46b25ceedc28f7ca110": "\\mathcal{S} = \\int{L \\, \\mathrm{d}t}", "336e7f2f91666bc7655176922facb287": "f_{\\lambda}(\\theta)", "336e850d6b6ff95f20a5ab95fafb814d": "\\mathbb{Z}/n\\mathbb{Z} \\simeq \\mathbb{Z}/p\\mathbb{Z} \\times \\mathbb{Z}/q\\mathbb{Z}", "336ec8bf580d82a0b2dc1c20fd42b718": "\\lim_{\\mu_{xy}\\to 0} D_{NN}(X, Y) = D_{NN}(X, X) = \\frac{2\\sigma}{\\sqrt\\pi}.", "336f0a212a95d85553224d19b1225870": "w_i = \\frac{\\alpha_i/\\sigma_i^2}{\\sum_{j=1}^N \\alpha_j/\\sigma_j^2}", "336f317a067517466b5f43545fb198f1": " \\frac{du}{dx} = cu+x^2. ", "336f8961e0170b30fd99734cce20e13e": "C^{1}_{k+1}", "336fc83ba6b7a87832ec91af47c9aa2d": "d\\Pi=\\Delta dS + \\Gamma \\frac{dS^2}{2} + \\kappa d\\sigma + \\theta dt = \\Gamma \\frac{dS^2}{2} + \\kappa d\\sigma + \\theta dt\\,", "33702e5818a3ad9d31e3183a8a075d1a": "n \\geq n_0 \\Rightarrow 0 \\leq f(n) < c \\cdot n", "33706289dfcf731db3308f653de977d5": "\nV=I\\cdot R\n", "3370868eb2d63e51c8a23feebfc72022": "n = N/V", "337086991059860ccdd83d75fd72b501": "z=x+e", "3370ee21127f46122a4e0b6bdbde9aa3": " K\\subset \\mathcal{P}(S)", "33718fe6ddc32e79e70e5309d6ac45bd": " I_1(\\boldsymbol{A}) = \\text{tr}{\\boldsymbol{A}}.", "3371d0807d522f5d469abc7cbfb93642": "(20)\\quad\\quad 2 \\left( h_2 - h_1 \\right) = \\left( p_2 -p_1 \\right) \\cdot \\left(\\frac{1}{\\rho_1} + \\frac{1}{\\rho_2}\\right),", "3371e2ab2e9aaf92fd95b174af7b5158": "Z=\\sum_{n=0}^{\\infty } \\frac{(-1)^n(28n+3) \\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} { (n!)^3{3^n}{4}^{n+1}}\\!", "3371ea230f042a74f1709748ecc5f9e7": "t = \\frac{2p}{(1-p)^2}.", "3372265185b64334c98df5ec9298763d": "\\omega_\\text{c}=\\frac{3}{2}\\frac{c}{\\rho}\\gamma^3", "3372f21599e172e1b9b18b7ffc2b4756": "p_0 = -d e_0/d V", "3372f75be20a5bd9a1912557772c6775": "\\lnot\\ \\exists{x}{\\in}\\mathbf{X}\\, P(x) \\equiv\\ \\forall{x}{\\in}\\mathbf{X}\\, \\lnot P(x)", "33737b6877b2f079251004bc6b1c8f5c": "\\frac{\\partial}{\\partial z},\\quad \\frac{\\partial}{\\partial w}.", "33739cc4f3ef7aa3ba4d8c5a86a446fc": "y_3 \\in Y", "3373d79699661333da0aea0c879ac1dd": "|0| < |1| + |-2|", "33740fd63c5e76a70a3ee82f4137e4d1": " \\psi = \\psi(x)", "33742910d1bc68a72c89a6ed298e8737": "\\begin{align}\n\\text{root} & \\simeq 4 - \\frac{16 - 15}{2 \\times 4} \\\\\n& \\simeq 4 - 0.125 \\\\\n& \\simeq 3.875 \\\\\n\\end{align}\\,\\!\n", "3374296a6f6825cff72318873fbfd582": "\\operatorname{Var}(Y|X)", "337437a837d53f6d497c90acdb635176": "\\partial W", "337453336b6e88b01c090f5ec5683b47": " P(I)dI = \\frac{1}{\\mu} \\exp( -I / \\mu) ", "3374e8304eeed370dbf15241860b18b0": "\\mu\\frac{dy}{dx}+\\mu p(x)y = \\mu q(x)", "3374ec4d090caef97078dd45ea689a03": " d_J(A,B) = 1 - J(A,B) = { { |A \\cup B| - |A \\cap B| } \\over |A \\cup B| }.", "3374f9cb67c99abf9a5fc61e60d14ae2": "I = \\frac{bh^3}{3}", "3375451ba5986e32112fa52776d1928a": "\\mathbf{A}\\mathbf{x} \\le \\mathbf{b}", "33754a90ce3498034c4e86ccd0258cdd": "\n \\varepsilon_{ijk}~\\mathbf{b}^k = \\cfrac{1}{J}(\\mathbf{b}_i\\times\\mathbf{b}_j) = \\cfrac{1}{\\sqrt{g}}(\\mathbf{b}_i\\times\\mathbf{b}_j)\n", "33758268503a0d2f1cab6d73c89b3261": "(a_1b_8 - a_2b_7 + a_3b_6 + a_4b_5 - a_5b_4 - a_6b_3 + a_7b_2 + a_8b_1)^2\\,", "3375af1d095e866dfeaf5f8077f1b0d2": "\\pi_1 \\circ (t, u) = t, \\pi_2 \\circ (t,u) = u, (\\pi_1 \\circ t, \\pi_2 \\circ t) = t", "3375d0df162d82d2176f990f7b380e88": "\\nabla\\phi = \\sum_{l=0}^\\infty \\sum_{m=-l}^l\\left(\\frac{\\mathrm{d}\\phi_{lm}}{\\mathrm{d}r} \\mathbf{Y}_{lm}+\n\\frac{\\phi_{lm}}{r}\\mathbf{\\Psi}_{lm}\\right)", "3375d705bc42b90cbd38976690fc50c4": "N_{t+1} = N_t e^{r(1-\\frac{N_t}{k})}", "337606625630fc5e3124a9fe1a3701d7": "x_{nj}", "33767217f542acb3dc08a524d24a1ae4": " y'(t) = H(t), \\quad y(0) = 0, ", "33769c83e44bfc64584e61f4ec0d9764": "\\textstyle\\Phi = \\sum_{e \\in E} \\sum_{k=1}^{x_e} d_e(k)", "3377053aae580228bedd3f6fb3421321": "\\Pr[Z \\ge \\rho] > 0", "33774525da315f53102d05abd59d2bf2": "r_2 = \\frac{T_2}{r_2}", "337778f4c628d9c80045adbc3a9a81b0": "\\pi(i)", "337823ba7a04daeab35b5fb36257b564": "\\frac{\\nu_c ({}^{12}{\\rm C}^{6+})}{\\nu_c ({\\rm e})} = \\frac{6A_{\\rm r}({\\rm e})}{A_{\\rm r}({}^{12}{\\rm C}^{6+})} = 0.000\\,274\\,365\\,185\\,89(58)", "33787ca5d9731a2e0d08567cfb1519d4": " p_j = \\frac{\\exp(x_j)}{\\sum_k \\exp(x_k)} ", "3378cac0c6db030ff0365097cd7dd863": "\\forall a,b \\; \\exists u \\; \\forall z ((\\forall x \\in a \\; \\exists y \\in b \\; \\phi(x,y,z)) \\to \\exists v \\in u \\; (\\forall x \\in a \\; \\exists y \\in v \\; \\phi(x,y,z)) \\wedge (\\forall y \\in v \\; \\exists x \\in a \\; \\phi(x,y,z)))", "33792fbf9c0eeb81a172de01779f79e0": "\\displaystyle{E(f) = \\sum_{m=0}^k a_m f(-x/(m+1)).}", "3379445dfc692afa10933961884bc182": "f(x) = g(x)h(x)", "337958450405622380ed4db10d30ea58": "\\mathbf{\\nabla}\\times\\mathbf{E}(\\mathbf{x})=ikZ_0\\left(\\mathbf{H}(\\mathbf{x})+\\mathbf{M}(\\mathbf{x})\\right)", "33796c2f5ceefa1e5615dfdc769c9762": "\\mathcal P = \\empty", "33798ad4a7e7dfda6c3f83f9451cb38f": "coaction: ~~ End(A) \\otimes A \\leftarrow A, ", "3379b0e3f567aafed0655f76768f8e31": " x_i = \\alpha A^T_{i,j}x_j + e_i \\, ", "3379ec0cdf1dfd06e8945abbc3d409e8": "\\mathfrak{H}_1 = \\begin{pmatrix}\nz_2 - z_3 & -z_1 (z_2 - z_3)\\\\\nz_2-z_1 & -z_3 (z_2-z_1)\n\\end{pmatrix}", "337a316249899b8c134945ca45b1409e": "\\mathbf{t}", "337a335c86a2888500ba4e4251052c23": "\\textstyle \\mathbf{r}_1", "337a7299c26d5a726ef10e653ca6304b": "T_{\\mbox{graph}}\\in O((v,e)\\mapsto v\\cdot\\log e)", "337aa19c96401566a17bb1148ef51def": "N=\\int_V n(\\mathbf r) \\,\\mathrm{d}V", "337aa66f19b3e29637382bf2d45782c6": "-1\\le u\\le 1\\,,", "337aa95cf02620ec5fee5f689ec78376": "I = \\int L \\left(\\boldsymbol\\phi, \\partial_\\mu{\\boldsymbol\\phi}, x^\\mu \\right) \\, d^4 x", "337afd813b14137226c62a73aade6478": "q = e^{\\pi i \\tau}", "337b6ba2bd420e56fb3c526f86cc0cdf": "\\mathrm{Ka} = k t_c", "337b6cbccadd2e5cae7ca1b6e54f6b77": "\\cos\\frac{\\pi}{10}=\\cos 18^\\circ=\\tfrac{1}{4}\\sqrt{2(5+\\sqrt5)}\\,", "337bf2267bf8ab95d16c59b76b1bac7b": " L_{i,j} = \\frac{1}{L_{j,j}} \\left( A_{i,j} - \\sum_{k=1}^{j-1} L_{i,k} L_{j,k}^* \\right), \\qquad\\text{for } i>j. ", "337bfdddd753a7dc54c7872efc0b462d": "\\mathbf{C}(s) = \\cos(s)\\mathbf{i} + \\sin(s)\\mathbf{j}.\\ ", "337c041c962a3f9d9f091d01584a4dbf": "x^2 \\frac{d^2 y}{dx^2} + x \\frac{dy}{dx} - (x^2 + \\alpha^2)y = 0.", "337c2ee6395fc79534f13cefc3f3aa60": "\\rho_\\Psi(0)=|\\Psi\\rangle\\langle\\Psi|", "337c59482a3dbfb8aa492229c61e6bad": "n \\leq 0", "337cb5cd6d52d7ace2264ff1b6d8117f": "\\check H^q(\\mathcal U,\\mathcal F)", "337d9b361ea3e4ca74e85e056ee3f979": "F(t)\\,", "337d9d5a5a7210302e393507b96fbd0f": "F_{m+n}(x)=F_{m+1}(x)F_n(x)+F_m(x)F_{n-1}(x)\\,", "337de0a77683aecc9875950f5f3e20e9": "\nL^* = \\prod_{n = 1}^N {P_i ^{Y_i } } \\left( 1 - P_i \\right)^{1 - Y_i } \n", "337de1b3a03af8bd8b9ceb4a5f8df613": "\\textstyle r\\rightarrow \\infty", "337de3b8addca41802bdc4bd521828f1": "\\text{data-ink ratio}=\\frac{\\text{`ink' used to display the data}}{\\text{total `ink' used to display the graphic}}", "337e06f824be2d17ad5b04ffb647dcf6": "s_6=\\alpha^{-7}.", "337e0baf3e541cbf3f5a0147fdac6606": "T\\subseteq (X\\cup X^{-1})^*\\times (X\\cup X^{-1})^*", "337e32ad88507dd65a214740059ce0fa": "h_2(x)", "337eec7332282ec1c1844233ca8487ef": "{\\widehat{BB}}_3", "337f36d3174ddd7b953eefcb5b1661da": "{\\tilde{D}}_{9}", "337f37ae9719f38fea5db9c001ed1dbc": "l \\sigma", "337f5967270a10261e7dfd498bcf79e1": "T = \\frac {b^{2}(\\sin \\alpha)(\\sin (\\alpha + \\beta))}{2\\sin \\beta},", "337f679fb17b4625aebb7e425bd449eb": "\\mathbf {J^TJ}", "337f6a74447cdc82a097590314731225": "(x,y)\\in\\R_+^2", "3380195a8703c35a0552323381e606ef": "x\\in A", "33802639f5a3763e565b90722be3cd14": " \\begin{align}\n\\frac{d}{dt} p(n,t) &= e^{-(q(n,t) - q(n-1,t))} - e^{-(q(n+1,t) - q(n,t))}, \\\\\n\\frac{d}{dt} q(n,t) &= p(n,t),\n\\end{align} ", "338056aa8ee981914790b0d3739d298c": " \\lambda_+ \\lambda_- = 1", "33807c22b3c5d6c2cd3f32fcd82b363e": "\\mathbf u = 0", "3380c74bfe5be689f4abfc7076cd3f33": "\n \\delta n =\n{e \\varphi \\over \\hbar \\omega_c A_M L_B}\n", "3381370a02fa7a44c8c11abbfcd9d8cf": "\\mathcal D_A (\\rho) = I (\\rho) - \\max_{\\{\\Pi_j^A\\}} J_{\\{\\Pi_j^A\\}} (\\rho) = S (\\rho_A) - S(\\rho) + \\min_{\\{\\Pi_j^A\\}} S (\\rho_{B | \\{\\Pi_j^A\\}} ) ", "33818b8e5f2d5acbb55d1de115540352": " p(a,d) < d^2. \\; ", "33818fca48d744de8cd323390b3e50e3": "\\lambda(L)>\\beta", "33831059ae8f9f213bdf04c383b8f4c7": "\\sum\\nolimits_c\\partial M_c=\\varnothing", "3383175a3e0d96a9c91a33bccee3089a": "\\displaystyle{ W(x,y)=e^{ixy/2}U(x)V(y).}", "33834ce4ec12c0fed152eea6c06e014c": "S=\\alpha\\ln{\\left(1+{N\\over{\\alpha\\,\\!}}\\right)}", "338350e7fb182a3af9ac53ae0019ca5c": "-\\Phi = \\Phi \\ ", "338388fa195d85afbba26ee001a41fe6": "x_a(t) = \\frac{e}{m} E_0 \\left(\\frac{e^{i\\omega t}}{\\omega_a^2 - \\omega^2 + 2i\\omega/T} + \\mathrm{c.c.}\\right)", "3383a99f8b5ad85e9d508f097a6e28e6": " \\partial_x - z \\, \\partial_z ", "338429eb25240b7c3f4583fea1e76d12": "k_n=0", "33847e47b0cc66b845fe0b0ac085a828": "\\scriptstyle{\\mu}", "3384dcd031e317235cb97eee3082ab9e": "f \\left(\\text{Hz}\\right)=RPM\\frac{\\text{P}}{120}", "3384e91e83f96b4e19331d8f721958cc": "p_t = p_s + \\left(\\frac{\\rho V^2}{2}\\right)", "3384f661b169bf7de0aa3a387edcd0ee": "\\pi_T = T \\left ( \\frac{\\partial p}{\\partial T} \\right )_V - p", "338515c0b9b84621d237f122b01bb358": "U = -\\frac{Gm_1m_2}{r}", "33851cb148c344c282329cc66c12de17": "\\hat u_R (t)", "33854224b3e9ace124dfb2069b2a365d": "{\\Bbb Z}^2", "33858fdb9e7d718a835e5a8edf766b3c": "\\partial_+f(a):=\\lim_{{\\scriptstyle x\\to a+\\atop\\scriptstyle x\\in I}}\\frac{f(x)-f(a)}{x-a}", "3385d6f79ce2091931b47be82bf92bce": "(\\tfrac{1}{2}, 1)", "3385de244f8010d5ff14e753b201346d": "S \\rightarrow B: M,\\{N_A,K_{AB}\\}_{K_{AS}},\\{N_B,K_{AB}\\}_{K_{BS}}", "338631919da2076dce967d9568ac6fe3": "B \\rightarrow A: N'_B, \\{N'_A\\}_{K_{AB}}", "3386706e3f649fdcbc84fae5e5e49c63": "\\forall s_{-i}\\in S_{-i}\\left[u_i(s^*,s_{-i})> u_i(s^\\prime,s_{-i})\\right]", "33867c2904370b0b611f4273e936cd22": " - \\alpha He^{- \\alpha L/2} = - k B \\sin(k L/2)", "3386918168e2cd4590e74e2798e024f0": "\\forall p \\text{ prime dividing}\\ k,\\quad x^{k/p} \\not\\equiv 1 \\pmod{n}.", "3386941fd11b38ed7eb7eefc8180d3f5": "\n\\Pi(x) = i \\psi^\\dagger.\n\\,", "33874045a8bd824819a40ea911b5eb8b": "(\\mathcal{F}_s,\\ s \\in I)", "3387564970d17b23be6d615adc6a7402": "\\det (A^{-1}) = \\frac 1 {\\det (A)}.", "33876f5eabe796067ea2800ee02e9078": "\\sum_{0\\le j\\le s(M)} s_j \\le \\frac{32M^2}{s(M)} + 4M.", "338774114becbd4f51587ca6f32a82f6": "~\\cos^{3}(x)~", "3387b6ee5836ce12ec49ed92452be3cf": "\\langle A,\\wedge,\\vee,-\\rangle", "3387b7142146f625ef87e32b853995ef": " \\mathcal{L}(B)\\subseteq \\mathbb{Z}^n", "3388ae4e2931c5b52ef3ee1a2b81a53b": "(P ^ a f)(x) = f'(x) \\,", "3388b52205dd8eaae690c433b748eaaf": "\\left[ [ \\mbox{mi} ] [ \\mbox{ned} ] \\right]", "3389b7de38f150491b520c0b02d5ebb8": "\\psi(\\Omega+1) = \\varepsilon_{\\zeta_0+1}", "3389dae361af79b04c9c8e7057f60cc6": "*", "3389e536a57fe51a0f39ade5765c4695": "z_n = a + nW", "338a7e42eaceb086a658edd1b29e6caf": "\\{L=0\\,,\\dot\\theta=0\\,,\\dot\\phi=0\\}", "338a920f4713bc5b84a928072561b4f7": "q_{n_1 + n_2} = 0", "338b40659ce3656eb2a0baf9919dc5f3": "{\\mathcal L}_{xy}^5: L=Exquo\\big(L,\\mathbb{L}_{x^m}(L)\\big)\\mathbb{L}_{x^m}(L)=\n \\left(\\begin{array}{cc} 1 & 0\\\\ 0 & \\partial_y+A_1\\end{array}\\right)\n \\left(\\begin{array}{c} L\\\\ {\\mathfrak l}_m\\end{array}\\right);", "338b5106ec99c65ea5ad8d406d708e63": "\\mathbf{D} = \\varepsilon_0\\mathbf{E} + \\mathbf{P}", "338b728ff7d4b6a37745b90a31f62cc7": "\\{ A_1,\\dots,A_m \\}", "338b93658232c10749f6d5930d025385": "\\hat x_{4}", "338bb6042d071ab79e32f86dc5b749e8": "\\tfrac{59}{4}", "338bccb0506c69f6aa9e0ccf02fae9b8": "\nT_n= \\frac{1}{(n!)^{2}} \\sum_{i_1,i_2,\\ldots,i_n} \\sum_{a_1,a_2,\\ldots,a_n} t_{a_1,a_2,\\ldots,a_n}^{i_1,i_2,\\ldots,i_n} \\hat{a}^{a_1} \\hat{a}^{a_2} \\ldots \\hat{a}^{a_n} \\hat{a}_{i_n} \\ldots \\hat{a}_{i_2} \\hat{a}_{i_1}.\n", "338be88b94f8ffb74bef23112be7c399": "\\sum_a W_{ia}h_a", "338bf6c0a280b6693b985e0f9e5db666": "n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\\dots", "338c0a706df2bcbe541babcbcc9b23a3": "u_k 0 ", "33921c465210e0a3b41d07b704412164": "G(\\mathbf{x},\\mathbf{\\eta})={-1 \\over 4 \\pi\\|\\mathbf{x} - \\mathbf{\\eta} \\|}.", "339220b174b89ef9c0481121a34355ed": "\\mathbf{u}' = \\mathbf{u} - \\mathbf{v} \\, .", "3392c22d8cb5c359f0592f2765d982b0": "\\Delta \\geq \\frac{\\zeta(n)}{2^{n-1}},", "3392fe5a4103bf041f0ef7f66b36a64d": " |n^{(1)}\\rangle = \\sum_{k \\not\\in D} \\frac{\\langle k^{(0)}|V|n^{(0)} \\rangle}{E_n^{(0)} - E_k^{(0)}} |k^{(0)}\\rang, ", "33932d330e92bba255d588ea47c160d6": "r = a\\sec^3(\\theta/3).", "3393b34f9de1ebdaa102bb8e4fd55a29": "D_{med} = \\sigma \\sqrt(2/\\pi) ", "3393eb70b25bae387b10eaff0e1af98a": "(\\infty,\\infty)", "33940fa4ed9841d44102bc2a8d953eb9": "\\left(\\frac{c}{R}\\right)^2 = \\left(\\frac{a}{R}\\right)^2 + \\left(\\frac{b}{R}\\right)^2 + O\\left(\\frac{1}{R^4}\\right)\\text{ as }R\\to\\infty\\ .", "3394c411848ce55f5b9479f800d4258d": "\\displaystyle{Q(e^a)=e^{2L(a)}.}", "3394d7ac6a2316be62000822c448ab64": "D(y) = -2 \\Big( \\log \\big( p(y|\\hat \\theta_0)\\big)-\\log \\big(p(y|\\hat \\theta_s)\\big)\\Big).\\,", "3394df9ee6a83f100842ed3fecc200c7": "(y-x^2)(y+x^2) = 0.", "3394e9094dcaba034fa1ce795272d7c2": "R_j = ", "3395068d792bd179ce24cd3416455d6e": "\\begin{align}\n&\\mathbf{T}\\mathbf{a} = \\mathbf{v}\\\\\n\n\\Rightarrow\n&\\begin{bmatrix}\nR_w[0] & R_w[1] & \\cdots & R_w[N] \\\\\nR_w[1] & R_w[0] & \\cdots & R_w[N-1] \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nR_w[N] & R_w[N-1] & \\cdots & R_w[0]\n\\end{bmatrix}\n\n\\begin{bmatrix}\na_0 \\\\ a_1 \\\\ \\vdots \\\\ a_N\n\\end{bmatrix}\n\n=\n\n\\begin{bmatrix}\nR_{sw}[0] \\\\R_{sw}[1] \\\\ \\vdots \\\\ R_{sw}[N]\n\\end{bmatrix}\n\n\\end{align}", "33959c708996fd5725a5617a19bb354b": "N/3", "33961948df4cc2471479c4d7c25fc4a3": "C_p - C_V = R", "339625dab09ba2573e611c52f7b7ef17": " [\\mathbf{\\hat T} (\\varepsilon),\\mathbf{\\hat U} (t)] = 0 ", "3396298a3a5647de449449530db14c30": "m_B^{-n}", "339633f076c9650ba16afb64067cdbe0": "r=|z|", "3396821f7e3c90f312e6dee56b6177ff": "J_k:=\\bigoplus_{j=1}^k U_j", "3396abf171cdab5ceeee9028a3ee924b": "A\\rightarrow B", "3396b789716f8ad2f618ff26bee3a467": "\n S(\\boldsymbol{\\varphi}(w)) = S(z^0) + \\frac{1}{2} \\sum_{j=1}^n \\mu_j w_j^2, \\qquad\n \\det\\boldsymbol{\\varphi}_w' (0) = 1,\n", "339719796c1d3794dba7825693df2dac": "O(N \\log^2 N)", "33972dfb37ac04699f143d5be8c4a26a": "\\arctan x ", "339738fa69b599b24066c821a7c1e172": "C \\in G", "33977246030f49816d64199ec39ada9f": " q_* ", "339774dba7f70386f630bb95b93807c0": "\\pi / \\sqrt \\delta", "339782e52a054edbe8ba8317c873e1b9": "\\mathcal{N}=(\\mathbb{N},+, \\cdot, <)", "33978b66e233eb60124fe84db57b16fa": "\\lambda _1, \\lambda_2, \\ldots, \\lambda _M", "3397c44a21ad197f85a29c25d7cf28de": "(x+1)^n=\\sum_{i=0}^n a_i x^i.", "3397c463582e367ab0a94ec92d994fa7": "n=\\,-2", "3397d316e486116e3055496badfb4ca4": "A = BC ", "3397f4a8365281a04d0830259ae6eade": "n_{t+1} \\; = \\mu u_t^a v_t^b + (1-\\delta)n_t", "33980bbdf0f8ab15a77d9d0a314bfa00": "C^\\infty_c(K_0\\backslash G_0 /K_0)", "33987098433056749c755262bc7a3f68": "\\rho_0 \\ ", "339870c42d32b96e05b346e927a44907": "\n\\begin{align}\n\\text{jiva } \n& = R\\cdot \\frac{s^2}{R^2(2^2-2)} - R\\cdot \\frac{s^4}{R^4(2^2-2)(4^2-4)}- \\cdots \\\\\n& = \\left(\\frac{s}{C}\\right)^2 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^2}{2!} \n- \\left(\\frac{s}{C}\\right)^2 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^4}{4!}\n - \\left(\\frac{s}{C}\\right)^2 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^6}{6!} - \\cdots \\Big]\\Big]\\Big]\n\\end{align}\n", "339883a7a30ce237f7b94f0e6882c76e": "Y_f=m\\frac{dv_f}{dt}", "33989f111ed40a179b8303a604f63a6e": "C(t) = -g(e^*, c(t)) \\, ", "3398d4d70ffcaeba6b67d24c8fee94cb": "E_A+E_B=E", "339975aac50e804d2a57473f5e060bec": "\\scriptstyle U_j\\subseteq U_k", "3399902eecaf22f8c87522f9b573b1f3": "I_\\text{i}", "3399c5399e25ee7292113ac4adb233c9": "=P_t ( {\\frac{\\lambda}{4\\pi}} ) ^2 \\times ( \\frac{G_{los}} {l} + \\Gamma(\\theta) G_{gr} \\frac{e^{-j \\Delta \\phi}}{x+x'})^2 ", "3399c79266bb8138a18e48cd2b8483ac": "m_{H}\\,", "3399d4a86ccb02a3b7f4d53b81cd8738": "y_i^*\\,\\!", "339a7395526bed6811b3b2410ae516c4": "\\mu_2 \\neq 0", "339a95eb7b489248b7c8e3e1bb4d4d93": "\\hat\\mu(x)", "339aaf57447490c9ff162c2cae882eac": "\\alpha^{+\\infty} = \\infty", "339ab7dbc61608f4a5bb9ab74fd5ceb7": " |e;0\\rangle ", "339b1cbee90ba6ac2a8db1f11c736b74": "\\delta f_1 = \\delta f_2 = 0", "339b226ada99fe33b56479d27e4b9638": "\\nabla f(\\boldsymbol{x}) = \\left(\\frac{\\partial}{\\partial x_1}, \\frac{\\partial}{\\partial x_2}, \\ldots, \\frac{\\partial}{\\partial x_n} \\right) f(\\boldsymbol{x}) ", "339b52e95483d28b281ab96765b3070b": "x = M x_{\\mathrm {base}}", "339b94da3a2f15984617424f03a01dba": "E = \\frac{0.1288\\cdot A\\cdot P\\cdot M^{0.667}\\cdot u^{0.78}}{T}", "339ba9324e4c1872374a241a651e8bf8": "\\implies P(A_i|B) = \\frac{P(B|A_i)\\,P(A_i)}{\\sum\\limits_j P(B|A_j)\\,P(A_j)}\\cdot", "339c585614ce27d39856ff479c502da2": "\\mathbf{r}=z", "339c7e5d36ebd8a80f030ec3fe5825ec": "T dS = C dT\\,", "339ca42ad4fcf2671da564a91ba63b80": " \\mathfrak{g} = \\mathfrak{h}\\oplus\\bigoplus_{\\lambda\\in\\Delta} \\mathfrak{g}_\\lambda", "339cf08c7bac22b4d8aeb920487bc710": "\\hat \\beta", "339cf695ce04efca44db4239cadad58f": " f_k(x) = \\mathbf{1}_{[0, 1]}(x - k) - \\mathbf{1}_{[0, 1]}(x + k), \\ \\ \\ k > 0.", "339d0486508a73d7c6b6765513fb358a": "\\mathrm{d}\\rho_I(t)=\\left(\\mathrm{d}N(t)\\mathcal{G}[c]-\\mathrm{d}t\\mathcal{H}[iH_\\mathrm{sys}+\\frac{1}{2}c^\\dagger c]\\right)\\rho_I(t)\\,,", "339d34c6080d41cc985b5ba81c6b1c4f": "{a}^{\\dagger}=\\begin{pmatrix} \n0 & 0 & 0 & \\dots & 0 &\\dots \\\\\n\\sqrt{1} & 0 & 0 & \\dots & 0 & \\dots\\\\\n0 & \\sqrt{2} & 0 & \\dots & 0 & \\dots\\\\\n0 & 0 & \\sqrt{3} & \\dots & 0 & \\dots\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\dots\\\\\n0 & 0 & 0 & 0 & \\sqrt{n} &\\dots & \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots &\\ddots \\end{pmatrix}\n", "339d3e18a6501fec5720ed5466087555": " \\forall w\\,\\exists!u\\, w\\,R\\,u", "339d5af1aa88eb3c9a202b2f382aa3f1": "U(X,Y) = 2R = 2\\frac{I(X;Y)}{H(X)+H(Y)}", "339d71a8ccb5cb1599b3aecfd4317268": "p_{11}", "339d83204f70d2c32513a57fe6b5901d": "M_{frac}", "339de5167a9699f5cf0335cdca356e07": "1 - (d+1)\\delta", "339df84c31e983f29a5cd3553cdbe8de": "A \\in B", "339e8f657d69b2390a8dafa620c8a87d": "H^n(C_\\phi) = \\langle\\alpha\\rangle", "339eb71e1c3c654ff297e33f5e24bbb2": "\\nabla\\cdot\\mathbf{v}=0.", "339f4232af1862029428ffb950a3985a": "f\\in L^p(\\mathbb{R})", "339fa06fc97eb9049e84d71f3b8ea88f": "\n X = U\\sqrt{\\frac{-2\\ln S}{S}}, \\qquad Y = V\\sqrt{\\frac{-2\\ln S}{S}}\n ", "339fa7bddaeec237e65f009868c7a576": "\\begin{alignat}{7}\n x &\\; + &\\; 3y &\\; - &\\; 2z &\\; = &\\; 5 \\\\\n3x &\\; + &\\; 5y &\\; + &\\; 6z &\\; = &\\; 7 \\\\\n2x &\\; + &\\; 4y &\\; + &\\; 3z &\\; = &\\; 8 \n\\end{alignat}", "33a013abbd29642948a2ca854856c687": "[g(\\theta_i),\\theta_i]", "33a14ed36a7f07917080a558afe03447": "H+K\\subseteq_s M", "33a176459a2584212beaa5a7e59ca6d6": "m_s=-e \\vec{S}/m", "33a1ed8e6be4fd0803905d45ac930f94": " \\tau_{crit}", "33a205b3c109ada9a4d868a8df9443c1": "\\int \\frac{d y}{\\sqrt{\\frac{6 A}{5} y^{5/3} + C_0}} = t + C_1;", "33a2363bd072f265e79effe5e02b38d2": "\\mathbf{X}_i=\\sqrt{\\lambda_i}U_i V_i^\\mathrm{T}", "33a254e5f1847c50043958a230dd571f": " f = \\frac{b}{a+b} = \\frac{1}{1+\\varphi}.", "33a263bfc1379fb2a00241aeb7bb507f": " R_\\pi = (m_e/m_\\mu)^2 \\left(\\frac{m_\\pi^2-m_e^2}{m_\\pi^2-m_\\mu^2}\\right)^2 = 1.233 \\times 10^{-4}", "33a2740bb3cb857145f192fd40a7a335": "\\psi_{n}^{2}", "33a278be9a603e0a0cfecb2602b0f848": "\\Sigma_{imp}(i\\omega_n) = (G_{imp}^0)^{-1}(i\\omega_n) - (\\mathcal{G}^0)^{-1}(i\\omega_n)", "33a2cd6fb275a6bbacfd4e72af812d9c": "T(n) = 64T\\left (\\frac{n}{8}\\right )-n^2\\log n", "33a2fecceaf0dc6f65c81b66008407fb": "V_I(f)", "33a344e1b5d58cbfe0a529d141964c85": "D^n_+", "33a3543dfb80102959ce7edeb310ba52": "n = F_{2i+2} F_{2i+3} - 1,\\,", "33a36466db7f7b208a902aea947ea547": "G = (N, \\Sigma, P, S)", "33a3a5748ee6954816f7af533ed17c56": " \\varepsilon_{ij} = \\frac {1}{E} \\left [ \\sigma_{ij}(1+\\nu) - \\nu \\delta_{ij}\\sigma_{kk} \\right ] ", "33a3ed24c8453a8e5c3c6bae5de8012e": "\n W = \\sum_{i=1}^n C_i~(I_1-3)^i ~.\n ", "33a4f44034985125b2f7906c23bc55dd": " \\begin{bmatrix} x'_{i} & y'_{i}\\\\x'_{j}& y'_{j} \\end{bmatrix} = \\begin{bmatrix} x_{i} & y_{i}\\\\x_{j}& y_{j} \\end{bmatrix} \\begin{bmatrix} \\lambda_{00} & \\lambda_{01} \\\\ \\lambda_{10} & \\lambda_{11} \\end{bmatrix} . ", "33a54f4faf315de12e54cee7323ea97e": "\\{v_4\\}\\}", "33a569bb1558cc64f55264d421205912": "\\phi = S_{(2,1,1)} - S_{(3,1,0)} + S_{(4,0,0)}.\\,\\!", "33a56e20c2520914b887b125f5b10585": " \\log\\left( {\\theta\\over 1-\\theta} \\right) = n\\log{[L]} - \\log{K_d}.", "33a582171ced85c9db1f55afd83176d5": "Q_j = \\frac{\\delta W}{\\delta q_j}= \\sum_{i=1}^n \\mathbf {F}_i \\cdot \\frac {\\partial \\mathbf{r}_i} {\\partial q_j}.", "33a5979f073f4123ae6288959384500f": "\n\\left [ Z-Z_{t} \\right ]\n", "33a5dd09d795d60e5ebe49e58dd579a7": "{p}={\\rho}R_dT_v \\, ,", "33a682c9050ec8ad65794c4e88242682": "E_\\mathrm{kin}=\\frac{\\pi\\rho l^3 U^2}{24}", "33a6bf0d5a6891d45b5450b2b7dab513": "\\tau(\\omega) = \\frac{1}{1 + Y(\\omega)}", "33a6c0ce9ce90019f9be0ed6aa3e89dc": "q_k = -k A \\frac{dT}{dx}", "33a6c19dacef9405712f1ea30a49e871": "Z_0=\\exp{W[0]}=1", "33a6d8b80975d9c918839cb0afb26bcd": "y\\in S", "33a6e50a833c1f1a2962c59d3ac79837": " \\mathbf{P}={\\begin{bmatrix} \\mathrm{R}_1 \\mathrm{R}_2 \\mathrm{R}_3 \\mathrm{R}_4 \\mathrm{R}_5 \\mathrm{R}_6 \\mathrm{R}_7 \\cdots \\mathrm{R}_L\\end{bmatrix}} \\qquad \\text{(1)} ", "33a72fd9be496f4ab3e6523c8dd2d92c": "\\Phi_i=J A^i J',", "33a73c2723eb937bec7ca484fefb277a": "\\alpha\\to 0,\\ \\beta\\to 0", "33a748abe27707a7d2def6d0d9852ca2": "\\begin{Bmatrix} 3 \\\\ \\infin \\end{Bmatrix}", "33a77a4f16067eab0461a1449b2c3d21": "\\lambda\\rightarrow", "33a78880b62a6d3802edeae5775096b3": " A=DP. ", "33a799a559894d39418a7ec1955d65a0": "B^{-1}C\\text{d} a", "33a7d8744edb4b3a0d450e2df492f59b": "x=\\sqrt{{{{\\left({b\\over{3c}}\\right)}^3}+{{y^2}\\over 4}}}", "33a8159e75110d58fd12f5778cd7d3c1": "x_2 .", "33a82d84738d5f15407d36c78ef1a85c": "-\\eta ", "33a85082a893f555d838d76f5251fcaf": "{m+n \\choose r}.", "33a8e50bd045b80a5ada7c655bf9b0ab": "\\mathbf{c} = (c_1,\\ldots, c_N) \\in C_\\text{out}\\circ{C_\\text{in}} \\subseteq [q^n]^N", "33a90101a3aeca20a5fe3ca7493cb085": "\\nu >2\\,", "33a92adf5b0108b792a9d51e481fcae7": "\\operatorname{P}(n) = 2^{-n}\\,", "33a957fd9975701998a2cd402d1841ee": "{\\mathfrak l}_m\\equiv\\partial_{x^m}+a_{m-1}\\partial_{x^{m-1}}+\\ldots+a_1\\partial_x+a_0", "33a95aefb1c48844a52ade70f7d2d8ec": "\\Lambda^2(T)", "33a9a2ed312ab9aecba4760a825811fc": "b_s = G_{21}a + G_{22}b", "33a9b07dbb3c9e27b6aaf8a7eb7a41c0": " \n\\lim_{n\\to\\infty}\\mu_n(A_n)=\\mu(A_n)=\\infty.\n", "33a9c85bf97587d6be86b102c30b3b7b": "\\, \\phi", "33a9cf1725d80e11c00e276f971962f3": "\\begin{align}\nX &= T P^{\\top} + E\\\\\nY &= U Q^{\\top} + F,\n\\end{align}", "33a9d18d79362eca5b0c0b81bb35f52d": "g_8=-x+119x^2-490x^3+105x^4;", "33a9dc1632037180972df0e00d97e42d": "=\\left(\\frac{1}{4} + \\frac{1}{5}\\right)", "33aa58d9d2b0bdcebff56e66cad68df8": "(0, \\pm a \\sqrt{e^4-1})\\quad(e>0).", "33aa75aa4239d1e6c8a89bb437507664": " \\gamma = \\cosh \\varphi = \\frac{1}{\\sqrt{1 - \\tanh^2 \\varphi}} = \\frac{1}{\\sqrt{1 - \\beta^2}} \\,\\!", "33aa9d9a4806237b24335569f39b0a64": "\\begin{align}\na_{n-1}&=-x_1-x_2-\\cdots-x_n\\\\\na_{n-2}&=x_1x_2+x_1x_3+\\cdots+x_2x_3+\\cdots+x_{n-1}x_n = \\textstyle\\sum_{1\\leq i} {1\\over 2} (\\phi(x) - \\phi(y) )^2\\,,", "33ac38248f756c206658df33d70efc31": "\\cdots ", "33ac698790eb1ba5cb17e5e41e5d3b25": "\n \\vec F_{FK} = - \\iint_{S_w} p ~ \\vec n ~ ds,\n", "33acb72c4ce51f64c4ad198fd85c3dae": "C + D \\cdot K = P + S. \\,", "33acca00845da91e2f97855602375059": "R_\\mathrm{internal}", "33ad4de802b60b96c68f035d75632aaf": "a = \\frac {bc} {d}.", "33ad989fb45de98c72dbf945779b8c9c": "A_e(j_f,i_e)", "33adbe4fbf8c0df07fe53777823d00f4": "\n(\\mathbf{w}^{\\text{T}}\\mathbf{S}_B\\mathbf{w})\\mathbf{S}_W\\mathbf{w} = (\\mathbf{w}^{\\text{T}}\\mathbf{S}_W\\mathbf{w})\\mathbf{S}_B\\mathbf{w}. \n", "33adfd3136e240bdcc9acea27ffa37e3": " \\bold A(x,y,z) ~ = ~ \\sum_{mnp} ~ \\bold A(\\alpha_m,\\beta_n, \\gamma_p) ~ e^{j(\\alpha_m x + \\beta_n y + \\gamma_p z)} ~~~(2.1c) ", "33ae2bbe9699f67dbd46b48bc6c6ff85": "\n\\Sigma=\\mathrm{E}\n\\left[\n \\left(\n \\textbf{X} - \\mathrm{E}[\\textbf{X}]\n \\right)\n \\left(\n \\textbf{X} - \\mathrm{E}[\\textbf{X}]\n \\right)^{\\rm T}\n\\right]\n", "33ae809f8a7965906ccf44cc374e4dd2": "\nII = LL - RA\n", "33aeae3a9508aa30cc435c82b85f8f07": " \\begin{array}{lll}\n p + p & \\rightarrow & p + n + \\pi^+\\\\\n p + n & \\rightarrow & n + n + \\pi^+\\\\\n \\end{array}\n", "33af4ac4137c100330bd305027659ee2": "H = U\\Lambda{U^T}", "33af66011c9154b492fe79ecfba6143e": "\\ p (h)=p (0) e^{-Mgh/kT}", "33af7ffebd5b00b87f711758ba2b4f58": " M=m ", "33af85400477f58820fb0f11b502253f": "A \\# kG", "33afce9320f735d6189bf02e435ed938": "y[n] = (h*s)[n].\\,", "33b0015e84099b5d4cfe5456ce095aa5": "N(1-R) \\delta_2\\,", "33b03cbc42f22ddf1f028ef90eab4298": "= - \\left ( t_j-y_j \\right ) g'(h_j) \\frac{ \\partial h_j }{ \\partial w_{ji} } \\,", "33b0d66601284bae8ed06b61676e5d7e": "\\varphi(u,v) = u\\otimes v = cu\\otimes c^{-1}v = \\varphi(cu, c^{-1}v).\\ ", "33b0f737842f60e0008b618b83b02535": " v_1^2+v_2^2=2E_A \\qquad v_3^2 = 2E_B", "33b0f7e7a7b9f4ee1a9ac223e6e55e21": "\\mathbf{C}_{2,2} = \\mathbf{A}_{2,1} \\mathbf{B}_{1,2} + \\mathbf{A}_{2,2} \\mathbf{B}_{2,2} ", "33b11c3098043b5da9629e59ec19b697": "\\tfrac{1}{n^{\\epsilon}}", "33b160fe7c3c701dbee6f9f617949bcb": "\\omega \\in X^{-1}((u, v])", "33b1ec7fd1197483c99da85ffae6bccf": "\\sum_k m_k\\mu_k^\\ominus-\\sum_j n_j\\mu_j^\\ominus =-RT \\left(\\sum_k \\ln {a_k}^{m_k}-\\sum_j \\ln {a_j}^{n_j}\\right)", "33b2123d826f89c5e9d793a199d11f82": "\\mathcal{F}(\\mathbf{x})", "33b23a68e62ea273bc7986883476e838": "\\Sigma (\\Omega )", "33b24d7893e5f8a8860ade784f5778c2": "Q(X,Y_1,Y_2,\\ldots,Y_s)=A_0(X) + A_1(X)Y_1 + A_2(X)Y_2 + \\cdots + A_s(X)Y_s", "33b2955cc56e924cefb6bf272a6ce8d8": "T_1 \\le t \\le T_2", "33b2a07f2015e7db818223f8afc86f72": "\\mathbf{c}_{x,y,z}", "33b2c52a58ecdf565ea4adfd95b73d27": "2q,", "33b3486a1eaadf5475d819c13914642c": "\\delta \\approx (n - 1) \\alpha\\ ,", "33b36db00c3b97101f67f2cb723950ab": "p=PMT(rate,num,PV,FV,) = PMT(r,n,-B_0,B_n,)\\;", "33b396491608481abe43f1f4c2f0a3bc": " S_E = \\sqrt[12]{2} = 100 \\ \\hbox{cents}. ", "33b3a767950d5a08b3d5fdd20b5e36b6": "\\textstyle\\frac{3}{4}+\\textstyle\\frac{\\lambda}{4}", "33b3bd0a06621059d4fd6bcc263cecea": "f_x(w;\\theta)=\\sigma_\\epsilon^2 f_x(w;(1,\\eta))", "33b44396eafde65d9765c5f2625b92ef": "\\oplus_{n \\ge 0} \\overline{I^n} t^n", "33b46d5309e83fbcea81a44f8fc3571a": "x\\in[0,b)\\subset\\mathbb{R}", "33b479b5ab46eea7d68021c38486dcd7": "w\\in\\{0,1\\}^n", "33b48094b11d24f95ef2827fcc5d8665": "A = \\frac{ns^{2}}{4}\\cot \\frac{\\pi}{n} = \\frac{ns^{2}}{4}\\cot\\frac{\\theta}{n-2}=n \\cdot \\sin \\frac{\\pi}{n} \\cdot \\cos \\frac{\\pi}{n} = n \\cdot \\sin \\frac{\\theta}{n-2} \\cdot \\cos \\frac{\\theta}{n-2}.", "33b492ab9f61a4ebd8967959c0a81507": "g_{00}=-c_0+2\\alpha U\\,", "33b4c3bc47e7c8d6c2cee747fe031883": " x_{com} = x_{max} \\frac{ \\overline{\\theta}}{2 \\pi} ", "33b522916eea63a1d7a9027439ff4936": "q q^* = t^2 - x^2 - y^2 - z^2 \\!", "33b581561561a337b0d5f12c311e3567": "\\nu=5/2", "33b591b8ec8977b2151f23a2171ee4cd": "L^1(\\mathbb{R})", "33b5dad0d1fb97ca6a42d3e0775c3c9e": "F(a_1, a_2,\\ldots, a_n) \\le G(a_1, a_2,\\ldots, a_n)", "33b5f3512fb8c08fa0ecfbf0ac6ba339": "\\|T\\|\\leq\\lambda", "33b62caf94bae9aa158d3c1448f8ad41": "\\chi_+ = \\begin{bmatrix}\n 1\\\\\n 0\\\\\n \\end{bmatrix}\n", "33b62f513980a0edd29c493225418a83": " 1/c^{10} ", "33b68651a9b3d65c92471d0ecc4a4af5": "\nP=EI\n", "33b71449198072730bfdeab49b41fb98": "L \\; p", "33b74395e52e1ad2cc1ddd2ab5ca6176": "\\frac{1}{D^\\alpha}\\,.", "33b76edcb0b0b5ae879f61f7912b40db": "r<0", "33b7f4bb5a62c7f88cc156e2d117f27c": "\\Delta r_i\\mathbf{e}_i = \\mathbf{r}_i-\\mathbf{C},\\quad \\sum_{i=1}^n m_i\\Delta r_i\\mathbf{e}_i=0,", "33b83bd834aa9496214b02429d75b687": "\\lambda_{n-1}\\cdots \\lambda_2", "33b8922a6e1090febb6b813541d6575a": "\ny(\\mathbf{x}) = (\\mathbf{w}\\cdot\\phi(\\mathbf{x})) = \\sum_{i=1}^l\\alpha_ik(\\mathbf{x}_i,\\mathbf{x}).\n", "33b8c0e80bed585e98b49d5f600b55fe": "t=x+\\tfrac{b}{3}", "33ba3bf6bec0365dfcb2c950408b31d8": "r = x-qM-1", "33baf4563f57485c6e859b241b4af115": "T = 2 \\pi \\sqrt{\\frac{I} {mgL}}", "33bb0057f31ced00956aa7a991d53979": " | x_1, x_2, \\cdots,x_n \\rangle \\quad ", "33bb0fe000b612b087ff2e668da86585": " \\frac {1} {N_f^{oxidation}} = \\left(\\frac {h_{cr}\\delta_0} {B\\Phi^{oxidation}K_p^{eff}}\\right)^{\\frac {-1} {\\beta}} \\frac {2(\\Delta \\dot{\\epsilon_m})^{\\frac {2} {\\beta} +1}} {\\epsilon^{1-\\alpha / \\beta}}", "33bb7741d674c527afa5f8745a829161": " \\gamma(\\tau) ", "33bb874cc6707d57b722b5718c9e1b64": "S_A = D - C A^{-1}B", "33bb94cdbc150d53ba974e99cda20500": "\n s = \\sqrt {\\left ( R_\\mathrm {E} + y_\\mathrm{atm} \\right )^2\n - R_\\mathrm {E}^2 \\sin^2 z}\n - R_\\mathrm {E} \\cos\\, z \\,,\n", "33bbc74c08a91c5962a22a4cdaa0059e": "\\Gamma\\vdash\\Delta", "33bbd5bd979cb129586a1594c12f581f": "\\gcd(a,b) = \\prod_p p^{\\min(a_p, b_p)}\\;", "33bbf9df920e0fc48edc0facf5c231fa": " F= \\left(m_{p}g\\right) +2(t_{p} + w_{p})\\gamma_{LV}\\cos(\\theta) - \\rho_lV_pg ", "33bc899a67d79fd2ee7f2fb8754b4dc7": "MD(\\top) = 0", "33bc94648173f4587cdbad6a6048d005": "V_{TOT}= \\pi \\varepsilon_0\\varepsilon_ra_p\\bigg\\{2\\psi_p\\psi_cln\\bigg[\\frac{1+exp(-\\kappa h)}{1-exp(-\\kappa h)}\\bigg]+(\\psi_p^2+\\psi_c^2)ln \\big[1-exp(-2 \\kappa h)\\big] \\bigg\\}-\\frac {Aa_p} {6h} \\bigg[ 1+ \\frac {14h} {\\lambda} \\bigg]^{-1}", "33bcad6000f497c7ebfeea9d4d5fbf1c": "X \\mathbf{\\operatorname{=}} Y", "33bcfb67aeb6ca74886aee93dc05cc9f": "(1-Q)", "33bd2129c821d11bae0240566e7cfd85": "\\tilde c_n = \\langle Vf, \\phi_n \\rangle", "33bdbe9bd1c9f57a2bd08c9ff4d2e26a": "(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge 0},\\mathbb{P})", "33bddc3d43160cab108458bb6e268545": "\\rho^c(\\lambda)", "33be2ef33f35d3e66fb40d42973632a6": "\\operatorname{tr}(AB)\\neq\\operatorname{tr}(A)\\operatorname{tr}(B)", "33be87e20a929a42ef2affb96af0a6c2": "\\mathbf{B}(\\omega) = \\begin{vmatrix}\n \\mu_{1} & -i \\mu_{2} & 0 \\\\\n i \\mu_{2} & \\mu_{1} & 0 \\\\\n 0 & 0 & \\mu_{z} \\\\\n\\end{vmatrix} \\mathbf{H}(\\omega)", "33beb3e82839cd962b2213633d3c5da6": "\\mathcal{P}_{C \\cap D}", "33bec8dd8396fbac78fb1f0290367866": "\\mu^'_4=(k+\\lambda)^4+12(k+\\lambda)^2(k+2\\lambda)+4(11k^2+44k\\lambda+36\\lambda^2)+48(k+4\\lambda)", "33bf1568a80b36aac7ba7f36c2636c21": "\\int_{0}^{1} f(x) \\, \\mathrm{d} x \\neq 0.", "33bf207f0dcfbf5a32b395b1475e80b7": "\\dot{v}_2 = {1 \\over C} i_2 ", "33bf3aa8d2068041a3e53cc5c835e6dc": "\\varepsilon^r_A:A\\otimes A^r\\to I", "33bf49da8800d0e78a3d68da32c0f801": "Z_{i 1}^2 = Z^2 + \\frac{Z}{Y}", "33bfbed0ffcac9aab48b96299e2d24c5": " C_M=\\frac{k^M_{tr}}{k_p} ", "33bff4945a7294054e1151c4cc83adbb": "\\gamma=-0.55268+0.959456i", "33c025c04d22a7f43bd4cc8ffd361d0a": "\\begin{align}\n\\int \\sin^2 x \\cos 4x \\, dx \\,\n&=\\, \\int \\left(\\frac{e^{ix}-e^{-ix}}{2i}\\right)^2\\left(\\frac{e^{4ix}+e^{-4ix}}{2}\\right) dx \\\\[6pt]\n&=\\, -\\frac{1}{8}\\int \\left(e^{2ix} - 2 + e^{-2ix}\\right)\\left(e^{4ix}+e^{-4ix}\\right) dx \\\\[6pt]\n&=\\, -\\frac{1}{8}\\int \\left(e^{6ix} - 2e^{4ix} + e^{2ix} + e^{-2ix} - 2e^{-4ix} + e^{-6ix}\\right) dx.\n\\end{align}", "33c03d202de43c6fcd968d32cd1f6d88": " \\eta ^2 = \\frac{SS_\\text{Treatment}}{SS_\\text{Total}} .", "33c0436d2d57b16d99d6b2fac834a34d": "\\begin{smallmatrix}\\frac{M_{Jupiter}}{M_{Neptune}} \\ =\\ \\frac{1.90 \\times 10^{27}}{1.02 \\times 10^{26}}\\ =\\ 18.63\\end{smallmatrix}", "33c0843d52530e18ae8a002728868df2": "D_{0},\\ldots,D_{n}", "33c0849460ed9ca8df0913109dd1de21": "H(a,u) = \\frac{U(u/a,1/4 a^2)}{a\\sqrt \\pi}", "33c0aee043ec2b96f8657b5d9c89b91e": "n-\\frac{3}{2}", "33c0c51d55bbe3f9c8e0a201f6b9f6ce": "x \\in A^c", "33c0eca7ffe300a178750e78545c00f0": "\\mu \\in R", "33c13eeaa0eb9a0169ce5b766795e751": " \\mathbb{U}(\\mathcal{H})", "33c1511dfb36fadacc4fd71b40aa8412": "\\biggl\\|\\prod_{k=1}^n f_k\\biggr\\|_r\\le \\prod_{k=1}^n\\|f_k\\|_{p_k}.", "33c1642479e94fc3144b2a2c21856a77": "A =P\\cdot\\frac{r(1 + r)^n}{(1 + r)^n - 1}", "33c177eb1cfbc4259eb1a5ec6a29b542": "\\delta \\rightarrow 0", "33c180793bb63cd1dd8f331580e1e445": "|f|_{0;\\Omega} = \\sup_{x\\in\\Omega} |f(x)|", "33c19f0afba993a99a1343a7d3ac0375": "\\frac{\\partial\\eta}{\\partial t}+u'\\frac{\\partial\\eta}{\\partial x}=w'\\left(\\eta\\right).\\,", "33c1a7553db63854d41b14b6ad75b632": "-e, \\sqrt{2}, 3, \\pi\\,\\!", "33c1f083b2657b2c95cf2b780b3741ac": "d=d(p,q)", "33c1f5a204e6b0a8d5a0f0b0cf8bcd02": "p_{\\rm total}", "33c2d68c3530f82c8dca42d9febd9f6f": "\\tfrac{1}{e^2} \\,", "33c2e583578c3686ab3a9e58518594ea": " |z| =1 ", "33c30d46dde2f1574db15a048771bfaf": "k \\leftarrow k+1", "33c32f11feccdcac5d71e8e0ac1ad09a": "\\partial_\\mu \\partial^\\mu A^\\nu = \\mu_0 j^\\nu", "33c33a93452bb8259598ce98b2cb360e": "f(z)=z^m e^{g(z)} \\prod_{n=1}^\\infty E_{p_n}\\!\\!\\left(\\frac{z}{a_n}\\right).", "33c3796e7ce0ec0904d303cb135a9846": "\\Delta E_{pot} + \\Delta E_{kin} = 0", "33c39ff5a4710bdd934746ba4a9e9166": "a \\uparrow\\uparrow b", "33c3ddcd4b85103d32e11c0f9988e088": "I(\\omega ,\\tau )=|\\int e^{ -i\\omega t }E_{ 1 }(t)|E_{ 2 }(t-\\tau )|^{ 2 }dt|^{ 2 }", "33c3e9b110a4418c6085b5ebb55aa327": " |1 \\rangle ", "33c445569f9120120961a5ec9ff73dd7": " \\frac{ | m - x_r | }{ s } \\le \\text{max}\\left[ \\sqrt{ \\frac{ ( n - 1 )( r - 1 ) } { n ( n - r + 1 ) } } , \\sqrt{ \\frac{ ( n - 1 )( n - r ) }{ nr } } \\right] ", "33c49bae20a03c40622430afbe8fcd26": "\\omega=\\pi c/\\lambda_{SHG}", "33c4ac5e6882f371ea24669f6f1717e3": "\\alpha,\\beta,\\dots", "33c4b4c14b9380e16e8f975de2e0e5cf": "\\displaystyle \\hat{f}(\\boldsymbol \\omega)=", "33c4c1f3ab7b4185259701bdb0b53a75": "X=\\frac{1}{2}IT =\\frac{1}{2} \\times .045 \\times 30 = .675", "33c4fee491e9b10411ecb67787db7902": "\\frac{\\mu-\\mu_\\text{baseline}}{\\sigma}.", "33c502501e30113b1a332b3feddb1864": "\\int_{I_O}^{I_L} \\frac{dI}{I_L} = -k_O \\int_{0}^{V_S} \\frac{dV}{V_O - V_S}", "33c55ab1ceeeb23c3344a24be01da926": "Z_{TE}\\approx{K_{TE}^2R_s+jX_s}", "33c5a460951b89f0d54ce4265df15cd5": "\\Pr\\Big(n \\text{ coin tosses yield heads at least } (p+\\epsilon) n \\text{ times}\\Big)\\leq\\exp\\big(-2\\epsilon^2 n\\big)\\,.", "33c604a8d29954f88a0520abbf46debc": "-724\\pm 2.7%", "33c6193fb437d4f4fbdc9450cd7a0e15": "\\frac{1}{\\tau} = \\frac{1}{\\tau_{\\rm impurities}} + \\frac{1}{\\tau_{\\rm lattice}} + \\frac{1}{\\tau_{\\rm defects}} + \\cdots", "33c6334abe665991ed45dcf0aad7b1d8": "\\operatorname{E}[X^k] = (-i)^k \\varphi_X^{(k)}(0).", "33c64686d9230910fe8e4f07bd722da8": "\\sigma_z\n= \\frac{\\partial^2\\Phi_{yy}}{\\partial x \\partial x}\n +\\frac{\\partial^2\\Phi_{xx}}{\\partial y \\partial y}\n-2\\frac{\\partial^2\\Phi_{xy}}{\\partial x \\partial y}", "33c6562bec8e942737ca8cb5b6c18f79": " H_{\\frac{1}{3}} = 3-\\tfrac{\\pi}{2\\sqrt{3}} -\\tfrac{3}{2}\\ln{3}", "33c67fbe72d6685edbd27e6397f518a9": "\n\\Psi(x) = {1 \\over \\sqrt{V}} \\sum_\\mathbf{k} \\left\\{\n\\left[ a_1(\\mathbf{k}) u^{+1}_{+1}(\\mathbf{k}) + a_2(\\mathbf{k}) u^{+1}_{-1}(\\mathbf{k})\n\\right] e^{i k x} \\right. ", "33c683de23a4c513c51fcb59420d7a66": "g(x) = x - a", "33c68c919171e55104c6ebbe83726d0b": "g \\geq 1", "33c7d9841361f488eb9f43f5b43e964b": "dY=(G+i \\omega C)dx=Ydx\\, .", "33c7dee71a650d7b96457714f74ab9c7": " \\alpha( g\\cdot v ) = g \\cdot \\alpha(v)", "33c81c66ab10ae4e714ac9960c40f2d2": "\\scriptstyle{K(k)}", "33c865d4a9406756df9d3d79ac8cfc80": "e^{-rT}", "33c8807c7a724d182935b8be2912a5e2": " \\beta_3 ", "33c88f3762d8bc5c3956a242c5aa7f6a": "\\vec{\\Omega }^{'}", "33c8c41cdbd1da02af1b71eb8eb7d267": "\n\\operatorname{Li}_s(z) = {\\pm i\\pi \\over \\Gamma(s)} \\,[\\ln(-z) \\pm i\\pi]^{s-1} - \\sum_{k = 0}^\\infty (-1)^k \\,(2\\pi)^{2k} \\,{B_{2k} \\over (2 k)!} \\,{[\\ln(-z) \\pm i\\pi]^{s-2 k} \\over \\Gamma(s+1-2k)} ~,\n", "33c8d2d036b937b4727324b7fff5308d": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left(\\frac{p}{\\rho^\\gamma}\\right) = 0,", "33c8e3dfc560cc022ff8fb0b7671b712": "E (\\mathcal{A}f)(W)=0", "33c96208744bff68452a410ce71a846b": "z_0 \\in \\mathbb{C}^n", "33c97fc3987afb5f1ae81e570c3f8afd": "\\partial X.", "33c9909f6c54c0f582549c895c0697c4": "\\mathfrak{m}_K", "33c9d99f7d9ca5b990f8221fe7a654ab": "J_{\\mathbf{a}}(f \\circ g) = J_{g(\\mathbf{a})}(f) J_{\\mathbf{a}}(g).", "33c9f36ed53d8e34086c168a667fc9c3": "L_{z}", "33ca1390b6448b407568705b065c7b66": " \\operatorname{E}(X \\mid a0,\\ \\mathrm{Re}(y)>0 \\! \n", "33f16f5b7611b79ae6daf3c8cc51d93c": "^*[\\,\\cdot\\,]", "33f1797f13a7083d84875c01f215c4cf": "\\scriptstyle\\downarrow", "33f1887efc60d29f73091ae37a3327bd": "\\mathrm{d} Y_{t} = b(t, Y_{t}) \\, \\mathrm{d} t + \\sigma (t, Y_{t}) \\, \\mathrm{d} B_{t},", "33f1893a26884faf29d7e360c0778332": "\\bar{g}_n", "33f192394b1470c1a04cac256a91c887": " \\sigma_\\theta \\! ", "33f1de5a9b0563fe0bda51fa6782c235": "J_0(x)", "33f1e3cbb51b56b833e52c4bfef068b0": "\\mu(X)<\\infty", "33f28b64dda2f218e98efc1e64a286f9": "A + A + A... \\rightarrow AAA ...", "33f2cac0dda008591d92a86fd7a02a24": "F(x_1) = \\int_{a}^{x_1} f(t) \\,dt", "33f3143d994add8c7a1958828975b940": "G \\times X \\to X", "33f376bcefa6ae45cc504d31184d6098": "\\tau_b=\\tau_c", "33f3b752ea3fcc6a61dd33b6bf04edb5": "= f(a) + \\mathbf{E}^{a} \\left[ \\int_{0}^{\\sigma_{k}} \\frac1{2} \\Delta f (B_{s}) \\, \\mathrm{d} s \\right]", "33f3b9afdd8b8a10cbe7f0a63814ef7b": " \\oint_{\\partial \\Sigma} \\mathbf{E} \\cdot d\\boldsymbol{\\ell} = - \\frac{d}{dt} \\int_{\\Sigma} \\mathbf{B} \\cdot d\\mathbf{A}. ", "33f3c4355875c7d1fea218b01b768b73": "\\{1,2,...,d\\}", "33f3c71fb533dcbe10c540d89aa2cf4e": " D_{KL}(Q||P) \\geq 2 H^2(P,Q). ", "33f3e9fab5ade9d4bd4a4eef34a40b9c": "y_x \\approx \\frac {u'} u_\\mathrm{h} J \\,,", "33f4155011b0540619e8e2c49f12b4f7": "\\sum_{i = 0}^n {2n \\choose 2i} - \\sum_{i = 0}^{n - 1} {2n \\choose 2i + 1}\n= \\sum_{j = 0}^{2n} (-1)^j {2n \\choose j}\n= (1 - 1)^{2n}\n= 0 ", "33f42fb612482854236b13798e601e8b": " f \\in {C_{c}}(\\mathbb{R}) ", "33f43bbe474efd8e3355fdd2b837892c": "\\displaystyle{f(x) = {1\\over \\omega_{n-1}} \\int_{|y-x_0|=R} {R^2 -r^2\\over R|x-y|^n}\\cdot f(y)\\, dy,}", "33f44d5c20730c65c824a2c749617072": " E_D \\approx T + V_C(R) + \\Delta(\\vec{P_0}) ", "33f49920bd0d49bfde802d7419e66d4c": "\\psi_{2m+1}\\in\\mathbb{Z}[x,A,B]", "33f4bd5910ac8d513f9cd9fa4309f6e9": "n\\geq 8", "33f4fd126ab7d5855e82e7cfc8c41409": "a=1,2,3", "33f55ae702c53213017e9673cbf8ae66": "i_{\\text{E}}", "33f579c6e06fe716cf623cbb52d25235": "V(z)", "33f5fb595d488719e43394fa151420d6": "\n a_{mn} = \\frac{4q_0}{mn\\pi^2}(1 - \\cos m\\pi)(1 - \\cos n\\pi) \\,.\n", "33f60fefd3914a8a699946eea0006d8a": "G_{2}(2)", "33f61986b17a85928a8be44de18cddb3": "s(x)\\,", "33f6eea06cd97cf6ac1d1460280da326": "w_{ij} = \\frac{n'_{ij}}{n_{ij}} = k - a z_{ij} + b z_i", "33f6f37d364b61ec08365aa4acef4ab4": "\n \\Delta\\mathcal{W}\\Delta\\phi \\ge \\frac{\\hbar\\omega_p}{2} \\,.\n", "33f7445dacf4a611771760ad5d8aef0b": "\\zeta(n)>2^{n/2}", "33f749ce6ae32250d8e91af779ac19f4": "\\Delta g=\\frac{\\partial g}{\\partial x_{1}}v_{x_{1}}+\\frac{\\partial g}{\\partial x_{2}}v_{x_{2}}+\\,\\,\\,\\cdots \\,\\,\\,+\\frac{\\partial g}{\\partial x_{N}}v_{x_{N}}=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\Rightarrow \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\nabla g^T\\,\\,\\centerdot \\,\\,\\,v\\,\\,=\\,\\,\\,0.", "33f771c8947ac9376a613ee6e7ea570d": "q+2", "33f79bda86c81f90c39a88a41e256b31": "\\log 2 = 1", "33f79db6028a0c4235d9c817bfd3dc5b": "\\left(X_{i_1}X_{i_2} \\cdots X_{i_m}\\right) \\cdot \\left(X_{j_1}X_{j_2} \\cdots X_{j_n}\\right) = X_{i_1}X_{i_2} \\cdots X_{i_m}X_{j_1}X_{j_2} \\cdots X_{j_n},", "33f7b577fb84598c03e51ba4fc55c76f": "\\psi(\\alpha) = \\varepsilon_\\alpha", "33f7ca0cc33e6db0d6b44c3184931ddc": "T^*M=\\bigcup_{E \\ge 0} M_E", "33f845cefec00bdf328b704344a17900": "I=(a,b)\\ -\\infty\\leq a(1-\\epsilon)\\phi(x)~\\forall k\\geq n\\}.\n", "33fd5d247fbdfa77dc33fd3233f21aff": "f = f^\\prime", "33fdb9d9d351cb24e38f1c9403964e05": "Y=\\sum\\limits_{m=0}^{N-1}{\\left\\langle Y,{{g}_{m}} \\right\\rangle {{g}_{m}},}", "33fdd84f807f4b1564dc95bc14be7aff": "-(x-1)^5 (x^2-x-10) (x^2+3 x+1)^2.\\ ", "33fe2d75d1921ed1cd5cfa76ba11389d": "Y \\mathbf{\\operatorname{>}} X", "33fe51c4da51cfcaeef19e87e95b3921": "\\ c(u,v)", "33ffab90add13cf9737b0afce20162cc": "V_{in} = \\dfrac{R_{i} \\left (-\\dfrac{N_{p}V_{ref}t_{\\Delta}}{R_{p}} + \\dfrac{N_{n}V_{ref}t_{\\Delta}}{R_{n}} - C ( V_{out2} - V_{out1} ) \\right )}{N t_{\\Delta}}", "33ffb8de286c1ef1a63f663f0d6e760e": "\\lambda_{CWL}(M)=\\tfrac{1}{2}\\left\\vert H_1(M)\\right\\vert\\lambda_{CW}(M)", "33ffc6ed8eda45fd63f10432f1a4c986": "\\sin \\left( \\frac {\\pi} {10} \\right) = \\sin 18^\\circ = {\\sqrt{5}-1 \\over 4} = {\\varphi - 1 \\over 2} = {1 \\over 2\\varphi}", "33ffc878810b5ccb9c78a0cff1d8c8b8": "S^0 \\to S^n \\to \\mathbf{RP}^n", "34002f7eceb3e03df184634de706d1f4": "ana$", "3400551844db2cf6f8203a1789a881f7": " p > 0 ", "34006d4375831164f0eb7a7f4ba95f29": "R =\\begin{pmatrix}\nr_w^2+r_x^2-r_y^2-r_z^2 &2r_xr_y-2r_wr_z &2r_xr_z+2r_wr_y \\\\\n2r_xr_y+2r_wr_z &r_w^2-r_x^2+r_y^2-r_z^2 &2r_yr_z-2r_wr_x \\\\\n2r_xr_z-2r_wr_y &2r_yr_z+2r_wr_x &r_w^2-r_x^2-r_y^2+r_z^2\\\\\n\\end{pmatrix}.", "340075cb2be0ff549b8e70babe93580c": "Z=Y_1+Y_2\\!,", "34007b45573a5ffc8e93906559734145": "V(S, T) = H(S)", "34008452b579cc2980771d59fbcbfe25": "r^e \\bmod N", "3400a1bbbd0c696534db53206c768079": "\\text{VEV }v\\approx 246 \\text{ GeV}, \\qquad\\qquad |\\langle H \\rangle|= v/\\sqrt{2}\n ", "34013b55aa2789c81283c871390babda": "\\mathbf \\zeta_{non rel}", "34014041eb011e2583786ec25da0f6c9": "1 \\to \\mathcal O^*_X \\to M^*_X \\to M^*_X / \\mathcal O^*_X \\to 1", "3401b1a24e6781b7dea0ef61984977bc": "a, b \\mathbin{:} A", "3401b6a40141a415dddfe4afe2d76903": "= 2.69 \\ast 10^{25} molecules \\cdot m^{-3} \\cdot 10^{-5}m = 2.69 \\ast 10^{20} molecules \\cdot m^{-2}", "3401ee34b25e1ceb70fed5fb87d7e603": "L_{z}=-i\\hbar\\frac{\\partial}{\\partial\\phi,}", "34021565743a45b1c1056e93d43ed073": "f = 1.64\\,\\mathrm{MHz} \\cdot \\left(\\frac{E}{E_0}\\right)^2 \\cdot \\exp\\left( -8.5 \\frac{E_0}{E} \\right), \\quad ", "34023105d19a008597cbea5a98ac4cfa": "p=e^{-\\Delta F/kT}", "340273c16adbe86fe43bb5ace901e503": "U(\\mathbf{r}) = -\\frac{1}{4\\pi}\\int_\\text{volume} \\frac{\\nabla'\\cdot\\mathbf{M\\left(r'\\right)}}{|\\mathbf{r}-\\mathbf{r}'|}dV' + \\frac{1}{4\\pi}\\int_\\text{surface} \\frac{\\mathbf{n}\\cdot\\mathbf{M\\left(r'\\right)}}{|\\mathbf{r}-\\mathbf{r}'|}dS',", "34029003e8365c12499176019eaad177": "P = - \\frac{dE}{dt} = \\frac{K_{\\operatorname{ev}}}{M^2} \\;", "3402bff6c5204a9b33da91e7f2d38a2b": "\\frac{1}{(1-2xt+t^2)^\\alpha}=\\sum_{n=0}^\\infty C_n^{(\\alpha)}(x) t^n.", "3402c967e8770155a39518982868ea49": "x_T = \\frac{X_T}{X_T+Y_T+Z_T}", "34030f43b89797ace7df37cc550f14c4": "x+\\sqrt{b^2-y^2}= a \\ln \\frac{b+\\sqrt{b^2-y^2}}{y}.", "3403301765a000b8f560eafc76ae1c71": "\\angle 1 = \\angle 2 \\,", "3403dba4709b91b5c0e292ac87f3c1cc": "\\begin{align}\n&\\left\\lfloor \\frac{x}{n} \\right \\rfloor +\n\\left\\lfloor \\frac{m+x}{n} \\right \\rfloor +\n\\left\\lfloor \\frac{2m+x}{n} \\right \\rfloor +\n\\dots +\n\\left\\lfloor \\frac{(n-1)m+x}{n} \\right \\rfloor\\\\=\n&\\left\\lfloor \\frac{x}{m} \\right \\rfloor +\n\\left\\lfloor \\frac{n+x}{m} \\right \\rfloor +\n\\left\\lfloor \\frac{2n+x}{m} \\right \\rfloor +\n\\dots +\n\\left\\lfloor \\frac{(m-1)n+x}{m} \\right \\rfloor.\n\\end{align}\n", "3404218ef79c17d4a2608fd9ac704ac2": "\\langle \\eta | \\psi \\rangle = \\int \\eta(\\bold{r}) \\psi(\\bold{r})d^3\\bold{r} ", "340433dcbc94418177907d07edbf2832": "\\mathbb{RP}^{\\infty} = K(\\mathbb{Z}_2, 1)", "34046b8f08e9d877ba2be26ed0812f6e": " \n\\langle\\,\\chi_{k'}(\\mathbf{r};\\mathbf{R})\\,|\\, \\chi_k(\\mathbf{r};\\mathbf{R})\\rangle_{(\\mathbf{r})} = \\delta_{k' k} \n", "34049e460f01dc9dea5e4450e5271091": "\\max_{k=1,\\ldots,n}\\frac{\\sigma^2_k}{s_n^2} \\to 0", "34054d3cc0080d549caf8e387d3de61e": " \\left | x_0 \\right |< \\left(\\prod N_i \\right )^{\\frac{1}{q}}", "3405a223d22b06d1b1f02c2276eeb6fa": "\\epsilon \\leq 1/2^{k-1}", "34064658f48279e3b1c6d594821d8a42": " y_1 = A_1 e^{-i k x}. ", "3406472e01d8cec031d54b0f90ef8999": "(n,\\delta n)", "3406a5a68cf0256ea7760c9dd71ce03b": "\ns=\\sin\\phi,\\qquad\nc=\\cos\\phi,\\qquad\nt=\\tan\\phi.\n", "3406c75a0b70064090b2808e3362e29d": "E(X)= \\begin{cases} \\infty & \\alpha\\le 1, \\\\ \n\\frac{\\alpha x_\\mathrm{m}}{\\alpha-1} & \\alpha>1. \n\\end{cases}", "340702b3ea7a97ec50d57ec9511b604a": "s \\,-\\, \\frac{2}{3}s \\;=\\; 1,\\;\\;\\;\\mbox{so }s=3.", "3407091028d5baa1513b07bbb4713de9": "a_\\max \\leq w_\\min \\leq t ", "340755ce165bc062e0c01b4a3a1f5b1d": " \\gamma = E - TS ", "34077fa3548beaf863497b72121d59be": " = 2^{(z-1)/2} \\frac{\\Gamma\\left(\\frac{z}{2}+1\\right)}{\\Gamma\\left(\\frac{1}{2}+1\\right)}\n= \\sqrt{\\frac{2^{z+1}}{\\pi}} \\Gamma\\left(\\frac{z}{2}+1\\right) = \\left(\\frac{z}{2}\\right)!\\sqrt{\\frac{2^{z+1}}{\\pi}} \\,.", "3407c942faf77ea25262436b75867e02": "\\!\\mu_2(v_2)", "34085f114e99e63ba15023c502f3c306": "0\\leq\\operatorname{dCor}_n(X,Y)\\leq1", "3408ec1ea583b1a8c8ccc6c41efdb487": "\\mathbb{R}^n \\times \\mathbb{R}^n", "340911c4b697294304b326ce8814863f": "p = \\frac{{(r \\cdot V_t)}^2}{\\mu }", "340920ad2005772be010df559301da3f": " (A \\times B \\times I) / R, \\, ", "34094a17c4bcf46649808ad1cad170d6": "\nA_q(n,d) \\geq \\frac{q^n}{\\sum_{j=0}^{d-1} \\binom{n}{j}(q-1)^j}\n", "3409872c907f805ceafc1120324eb32a": "B_0 \\gets q_0 - {P^{(0)}}^T B", "3409ced3394056414a6ca0879bf489e2": "|{\\psi^{\\bot}_{D}}\\rangle = \\frac{1}{\\sqrt{\\epsilon_n}}\\sum\\limits_{{{\\alpha}_n}={\\chi}_c}^{{\\chi}}\\lambda^{[n]}_{{\\alpha}_n}|{\\Phi^{[1..n]}_{\\alpha_n}}\\rangle|{ \\Phi^{[n+1..N]}_{\\alpha_n}}\\rangle", "3409dae4df48195d2d70ce8dd4b7914f": "y_k = h(x_k) + v_k \\,", "3409dff74c33c2b619a3e2a05178c728": "\\beta(g) = \\frac{\\partial g}{\\partial \\log(\\mu)} ~,", "3409e7cead6b458818955ca563288a3e": "Q/A", "3409ecb394048b226fdbed6ca7c9affe": "\\bigstar\\bigstar", "340a28fe95379a0b4109e88865dbda4c": "F(G) \\subseteq S_G", "340a52545223e258b02ccf4105ae4f4f": "\\sum{\\Delta{\\vec{p}}}=\\Delta{\\vec{p}_{1,2}} + \\Delta{\\vec{p}_{2,1}} = 0", "340a70228d46809ef1728e7ea9b9b743": "\\prod_{k=1}^n (x_k + y_k)^{1/n} \\ge \\prod_{k=1}^n x_k^{1/n} + \\prod_{k=1}^n y_k^{1/n}", "340aa0a997def5a0da71d867606355be": "v\\!", "340ad4d6f6f3201b046392802b688751": "e_n = \\bar{\\eta}(E)=\\int \\limits_{-\\infty}^{E_n}dE^\\prime \\bar{\\rho}(E^\\prime) ", "340afc610db16e4e8fd1aaaf8c8fecc2": "[C_i,P_j]=i\\hbar M\\delta_{ij}", "340b40ce10c289f67eea64b5b54d3b75": " \\pi = \\{t_i\\}_{i\\ge 0} ", "340b62ab75f7b1b64a440f201370f410": "\\prod_{i=a}^b", "340b7748adb01ba5380c5e8d7cd9a21f": "\\begin{align}\n&5x^6-96x^5-10x^3+1=0\\\\\n&5x^6-960x^5-10x^3+1=0\\\\\n&5x^6-5280x^5-10x^3+1=0\\\\\n&5x^6-640320x^5-10x^3+1=0\n\\end{align}\n", "340bd2501e3fbe6eb35bec6a287ff2e9": "\\frac{(c^T x)^2}{d^Tx} ", "340be632a771ae9365f2398c0341594a": "\\left ( \\frac{t'}{2} \\right )^2 = \\frac{h^2}{c^2 - v^2}", "340beb4cdbc2d1ce5a87374aed6eb4e4": "G = K\\rtimes H", "340c6eefd9ff9a882f06c2da757c4acf": "B^{T}=\\left( \\begin{matrix}\n 1 & 1 & \\dots & 1 \\\\\n X_{1} & X_{2} & \\dots & X_{N} \\\\\n\\end{matrix} \\right)", "340cab2f8b8639377e2fc1b423c068f8": "p_{-1}(x)\\equiv 0", "340cdad28343dbdd52c7762274e98484": "d=-c", "340d3ae057d394b8831dcbc0b3164b92": "\\,I^-(x)", "340db402cfebcd48f4c4d044ad39e85a": "\\mathcal{L}^{\\rm local}(\\mathbb{R},\\mathbb{R}^q)", "340dcbf7b2e92c9b2897cfc0619dc79d": "p \\cdot x_i \\geq w_i", "340dd844027e99c15012b8d6793e3e60": "\\beta=\\sqrt[n]{\\alpha}", "340e1e895dd4ef967434ccebbd0e8165": "e^{[a}_M e^{b]}_N \\delta^M_{[I} \\delta^K_{J]} C_{bK}^{\\;\\;\\; N} = 0", "340e409ffd764c2c6170270e16889b47": " \\begin{align} t_{n+1} & = 1 + 2r_n \\\\\n u_{n+1} & = (9r_n (1 + r_n + r_n^2))^{1/3} \\\\\n v_{n+1} & = t_{n+1}^2 + t_{n+1}u_{n+1} + u_{n+1}^2 \\\\\n w_{n+1} & = \\frac{27 (1 + s_n + s_n^2)}{v_{n+1}} \\\\\n a_{n+1} & = w_{n+1}a_n + 3^{2n-1}(1-w_{n+1}) \\\\\n s_{n+1} & = \\frac{(1 - r_n)^3}{(t_{n+1} + 2u_{n+1})v_{n+1}} \\\\\n r_{n+1} & = (1 - s_{n+1}^3)^{1/3}\n \\end{align}\n", "340e40a11e14c550665b45dad84246c7": " B_1=-G^{32}", "340ea62420f0bc2df87e26c3d8f2b42c": " \\Omega^0(M)\\ \\stackrel{d_0}{\\to}\\ \\Omega^1(M) \\to \\Omega^2(M) \\to \\Omega^3(M) \\to \\cdots.", "340ef0e1c2f39d52da1cbf30fc16bec8": " \\frac{W}{m^{2}K} ", "3410e6a3bbfaeeaf7ca27bc551e43ff0": "\nq_\\star= \\frac{2\\pi B_0 a^2}{\\mu_0 R_0 I} \\left( \\frac{1+\\kappa^2}{2} \\right).\n", "3410ec251bfee0424507aa970cbd41e6": "E_{Fe}=E_{Fe}^0 + \\frac{RT}{nF} \\ln \\frac{[Fe^{3+}]}{[Fe^{2+}]}", "3410fcdc4b7c2e72b8c62030ca14d507": "\n\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} G( \\boldsymbol{\\xi}, t^{\\prime} ) d^3 \\boldsymbol{\\xi} dt^{\\prime} = 1.\n", "341115a649c4b637c6f9f99c5a47f785": "\\left\\Vert \\mathbf x - \\mathbf c \\right\\Vert^2=r^2.", "341167e9ebc25035133fada5b2bfc896": "t_{orbit}", "34119e15d9bb0878cc65376db7c70957": "\\frac {d \\bold {M}(t)} {d t} = \\gamma \\bold {M} (t) \\times \\bold {B} (t) ", "3411c989772903dadefc6c39aaf1c4c5": "h_1^2", "3411e117a41f0e9cf0607286208c6fe7": "\\theta(\\tau)= \\sum_{n=-\\infty}^\\infty \\exp(\\pi i n^2\\tau).", "341278e7f02770b893f769dc0a0726b9": "P_B(\\lambda_B) \\approx \\tanh^2 \\left[\\frac{N \\eta (V) \\delta n_0}{n}\\right]", "3412903f8db9023300710fea91a76c46": "\\gamma^*(T;F) := \\sup_{x\\in\\mathcal{X}}|IF(x; T ; F)|", "341295a96382d9cb9c5f24c6923f3058": "\\mathbf{A} = \\mathbf{u} \\otimes \\mathbf{v}", "3412d37ba9528123a0d61edb1cb917f4": "l(t,s)=c(t,s)=1", "3412e4858db890008818cb754d0b46c8": "K(X,Y) = K(X) + K(Y|X) + O(\\log(K(X,Y)))", "341302929d546ebc527247768f52ec27": "\\sum_{r=0}^{n_0} y \\cdot (p_r (M^r x)) = 0", "34136dc77fbd614445f2e4027daac134": "\\zeta(a)", "3413a9616baf906e569447f3c401d820": "-5 \\sqrt {2} /16", "3413aac66a355f6f0912b004f4dd03cc": " \\mathbf{a} = -\\frac{\\kappa v^2r}{p} \\mathbf{e}_r + \\frac{(h^2)'}{2p^2} \\mathbf{e}_t = S_r \\mathbf{e}_r + S_t \\mathbf{e}_t . ", "3413ce51f3f5fe95bcf9936959d83549": "s=x \\circ i", "341401c79b7db875cb6eed8260574cf1": "\\textstyle 2.\\ Use\\ the\\ langrange\\ method:", "3414080c4e9636b54f817d5968fd37c6": "n_1, n_2, \\dots, n_j", "34145d9419d8ca51df5612f8525042ef": "\\textstyle g(\\mathbf{r})\\equiv g^{(2)}(\\mathbf{r}_{12})", "341478b11257acdd5982f6f0bbcab212": "1 \\over 10", "3414abca2f6d833744495c38aa536287": "B(\\rho,\\hat{u})", "34151f77cdab704d2c5aced3ed9abeef": "y = A \\left( \\cosh \\frac {Cx}{L} - 1 \\right) \\quad\\Leftrightarrow\\quad x = \\frac {L}{C} \\cosh^{-1} \\left( 1 + \\frac {y}{A} \\right)", "341526650188ef9c0706bd45ec48b166": "j(j+1)\\hbar^2", "34152dde3896d6d41a86411f2b236baf": "Q=\\lambda 1 ((\\lambda 1 1) (\\lambda \\lambda \\lambda \\lambda \\lambda 1 4 (3 (5 5) 2))) 1", "34161ac268095aa6ac1da67fa0a65286": "f(x_{1},x_{2}, \\dots ,x_{n})", "3416370c16f44b79e0881ba50c114c66": "[x]_R=\\{y \\in A \\mid x R y\\}", "341637fbab02cf417220165fc7225f39": "\\mathrm{Tr}[\\mathcal{E}(\\rho)] \\leq 1", "3416482536161e9bf50bd9dbe8a432b1": "f = 2", "34164fd6bff0d89fac9c87e8dbd3a0bb": "U_{\\alpha}", "3416535181f4dd5f8eb11a1731978a49": "(r_E + R_E)", "341689304fbf3ca21362a76abe3448c2": "\\epsilon_p", "3416e47947c43342837cd7bcb4fd22fa": "\\sigma_\\mathrm{max}\\,\\!", "34173cb38f07f89ddbebc2ac9128303f": "30", "3417416dc2ae864f5a94519bd5c4504a": "(X, \\frac{d}{n}, p_n)", "3417744e6d409bc66b40c950bac03153": "H = -\\sum J_{jk}S_jS_k-\\sum z_jS_j", "3417763d806d5d62708fb82991be08e5": "\\sum_k V_{ik}V^*_{jk} = 0.", "341796fbad28edad1f5cb82198ecb9ca": " f'(r) \\equiv 0 \\,\\bmod{p},", "3417cf9c14249d4f131674fd733118b0": "P(A \\mid m, s)", "3417ddc8785dbf6054092a39906ceab3": "\\frac{\\partial}{\\partial q}(\\mathbf a \\times \\mathbf b) = \\frac{\\partial \\mathbf a }{\\partial q} \\times \\mathbf b + \\mathbf a \\times \\frac{\\partial \\mathbf b}{\\partial q}.", "3417f2b5af773bf23079239798eca296": "\\bigg( (\\mathcal{M}, s) \\models \\Phi_1 \\Leftrightarrow \\Phi_2 \\bigg) \\Leftrightarrow \\bigg( \\Big( \\big((\\mathcal{M}, s) \\models \\Phi_1 \\big) \\land \\big((\\mathcal{M}, s) \\models \\Phi_2 \\big) \\Big) \\lor \\Big( \\neg \\big((\\mathcal{M}, s) \\models \\Phi_1 \\big) \\land \\neg \\big((\\mathcal{M}, s) \\models \\Phi_2 \\big) \\Big) \\bigg)", "3419a19aa8920f41d80663d65cf08ce5": " O(|w| (l + w |A|^2)) ", "3419e9ffb6a4af3fc758509a226f9e99": "\n\\varphi_f(t;\\sigma,\\gamma,\\mu_\\mathrm{G},\\mu_\\mathrm{L})= e^{i(\\mu_\\mathrm{G}+\\mu_\\mathrm{L})t-\\sigma^2t^2/2 - \\gamma |t|}.\n", "3419ef804db382ed07cdd81624e19fff": "I = \\liminf_{n\\to\\infty} x_n", "341a3a84103679426ed211a6afae4461": "\nW\\left( \\lambda_1,\\lambda_2 \\right) = \\sum_{p=1}^N \\frac{\\mu_p}{\\alpha_p}\\left( \\lambda_1^{\\alpha_p} + \\lambda_2^{\\alpha_p} + \\lambda_1^{-\\alpha_p}\\lambda_2^{-\\alpha_p} -3 \\right)", "341a6bf5bb31ae8ec734c46281a63aaf": "\\nabla \\times \\mathbf{F} \\cdot \\mathbf{\\hat n} = \\left[ \\left(\\frac{\\partial 0}{\\partial y} - \\frac{\\partial M}{\\partial z}\\right) \\mathbf{i} + \\left(\\frac{\\partial L}{\\partial z} - \\frac{\\partial 0}{\\partial x}\\right) \\mathbf{j} + \\left(\\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y}\\right) \\mathbf{k} \\right] \\cdot \\mathbf{k} = \\left(\\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y}\\right). ", "341a98dd7e6f323951aa4194e062f785": "R_s\\,\\!", "341b0b505a3383e41687602138afe92c": "\\begin{smallmatrix} R_H = a\\left ( \\frac{m}{3M} \\right )^{\\frac{1}{3}} \\end{smallmatrix}", "341b1051d2a1c2943cd9945f05e1fe74": "\\boldsymbol{\\sigma}^{+},\\boldsymbol{\\sigma}^{-}", "341b14cbd40bd3e493290bc234dfbe41": "(ayz)^n+(bxz)^n=(cxy)^n", "341ba0c111c2222fc302ec677c599b95": "\\scriptstyle (f - Nf_s),\\,", "341ba0fd2bef414898c6f15cce4d91a7": "(H+W)/2", "341c666e2c38f867befe5804d1117efd": " E = h \\nu = h f", "341c7908aff2c71c4782fd23e55ae465": "r_{p}", "341c853067a8055c6f6f3d51e2371a47": "f: B \\cup \\beta \\to (B \\cup \\beta) \\times (B \\cup \\beta)", "341cec5a3a404a5d0af5313bb50c6a9e": "\\pi(x_i) = \\frac{\\exp(x_i)}{\\sum_i \\exp(x_i)}", "341cf3a9685dfdb7b73ff923c55ebcac": "x''", "341d43c84f8f6b21d1f136372617ef81": "\\begin{array}{lcl}\na \\equiv b \\pmod n \\text{ and,}\\\\\n0 \\le b < n.\n\\end{array}", "341dd72582340660e598a95213a201c7": "\\sqrt[55]{2}", "341de38403902f0e9213da4a660a1764": "\\Delta x = 0", "341df894fbe5c7f4423ea97d67a9ecd1": " \\textstyle x = \\sum_{n=1}^\\infty \\frac{1}{n}\\ ", "341e11edc9ca1a2eef6c0c0ffa50acb0": "\\left( \\boldsymbol{U}+\\boldsymbol{c}_g\\right)\\, E", "341e3b5b93a4edc311edae2320d1d236": "\\textstyle M_{\\mathrm e} R M_{\\mathrm e}^{-1} = T", "341e509b55d2b6a4c59b79102238f588": " \\frac{d}{dt} (e^{-At}\\mathbf{y}) = e^{-At}\\mathbf{b}~.", "341e6be4fffb1bc5d2edd2920ec8e992": "\n\\begin{align}\n \\frac{\\partial}{\\partial t}\\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_x \\right] \n &+ \\frac{\\partial}{\\partial x} \n \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_x \\overline{u}_x + S_{xx} + \\frac12 \\rho g (h+\\overline{\\eta})^2 \\right] \n + \\frac{\\partial}{\\partial y} \n \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_x \\overline{u}_y + S_{xy} \\right] \n \\\\\n &= \\rho g \\left( h + \\overline{\\eta} \\right) \\frac{\\partial}{\\partial x} h \n + \\tau_{w,x} - \\tau_{b,x},\n \\\\\n \\frac{\\partial}{\\partial t}\\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_y \\right] \n &+ \\frac{\\partial}{\\partial x} \n \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_y \\overline{u}_x + S_{yx} \\right] \n + \\frac{\\partial}{\\partial y} \n \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{u}_y \\overline{u}_y + S_{yy} + \\frac12 \\rho g (h+\\overline{\\eta})^2 \\right] \n \\\\\n &= \\rho g \\left( h + \\overline{\\eta} \\right) \\frac{\\partial}{\\partial y} h \n + \\tau_{w,y} - \\tau_{b,y}.\n\\end{align}\n", "341e77235c703e5a1976888025b04143": " x_i = \\sum_{j \\ne i} a^j p_{ij} , \\qquad i = 0 \\ldots 3 . \\,\\! ", "341ea3c22a44ffeccaaeb58fec23bf5a": "P = \\frac{T_H-T_a}{T_H}\\dot Q_H - T_a \\dot S_i.", "341ed523fe8e00978135855868103989": " \\dim_{\\mathbb R}V^+ = \\dim_{\\mathbb R}V^- = \\dim_{\\mathbb C}V\\,", "341f160048888f17eb4c3836ef872042": " \\mathbf{C_1 =}\\begin{bmatrix}\n1 & 0 & ...& 0 \\\\\n0 & 1 & ... & 0 \\\\\n... \\\\\n0 & 0 & ...& 1 \\end{bmatrix}.", "341f1de8d2031b804c5a1745f2e630c0": "u'(x) = \\int_{y_0}^{y_1} f_x(x, y)\\,dy ", "341f44f79743a2415a4a372def05a473": "\\sum_{k=0}^\\infty (-1)^k {n \\choose k} {2k \\choose k} 4^{n-k} = {2n \\choose n} ", "342025a56acb2d88515e8d10c3bb0a9c": " A \\lor (B \\land C) \\to (A \\lor B) \\land (A \\lor C)", "342052f37c08b5846002c2e43f897b7d": "\\Theta_i", "34205bd6cbbc1b7358092e9ca9523926": "x_1,...,x_d", "34213011163b02938930154749efe435": "\\zeta\\in", "3421655559729edc56e8a03af97db2bb": "\\Delta m\\, \\frac{p_{1}}{\\rho} - \\Delta m\\, \\frac{p_{2}}{\\rho} + \\Delta m\\, g z_{1} - \\Delta m\\, g z_{2} = \\frac{1}{2} \\Delta m\\, v_{2}^{2} - \\frac{1}{2} \\Delta m\\, v_{1}^{2}", "34221f9f807d8f3b10cf3c0c6cf97910": "\\alpha \\approx 1", "34222e8e267af7aa275f1e6b4db80412": "{C} = 2D\\tan (\\theta/2) ", "342259579ad43e3e78d8ae74eb92fe71": "\n\\begin{align}\nP(a+b\\varepsilon)=&p_0 + p_1(a + b\\varepsilon) + \\ldots + p_n(a + b\\varepsilon)^n\\\\\n=&p_0 + p_1 a + p_2 a^2 + \\ldots + p_n a^n\\\\\n& + p_1 b\\varepsilon + 2 p_2 a b\\varepsilon + \\ldots + n p_n a^{n-1} b\\varepsilon\\\\\n=&P(a)+bP^\\prime(a)\\varepsilon,\n\\end{align}\n", "34226ab63c1a214975a62f14ffe0d86f": "\\mathbb{R}\\times S_1", "3422753da8356018aac77ed9816284ce": "F(x) = \\begin{cases}\n0 &:\\ x < 0\\\\\nx &:\\ 0 \\le x < 1\\\\\n1 &:\\ 1 \\le x.\n\\end{cases}", "3422914c84db7840d19dbb6d102f4df4": " \nR_m = \\left | \\vec P_{m} - \\vec E \\right | = \\sqrt{(x_m-x)^2+(y_m-y)^2+(z_m-z)^2}\n", "342296823becd44b3c2ceaf23066face": "\\mbox{div}\\,(\\mbox{curl}\\,\\vec v ) = \\nabla \\cdot (\\nabla \\times \\vec v)", "3422ada8e93ef8dce2efc8d257923214": "x' = \\sum_{k=1}^{\\infty}\\left\\langle x,e_k\\right\\rangle e_k, ", "3422c9ce796fc3cf5f52fe8d2b30095b": "\n\\begin{align}\n\\vec{s}^A_{i} &\\equiv \\vec{f}_i \\\\\n\\vec{s}^A_{i+3} &\\equiv \\vec{f}_i \\times\\vec{R}_A^0 .\\\\\n\\end{align}\n", "34239bdae2c922419f64638250075a15": "\\mathbf{a} \\wedge \\mathbf{b} \\wedge \\mathbf{c}. ", "3423a6cddc4fa2a34756d57f82f7e645": "\\scriptstyle\\varepsilon^{-2}\\log\\varepsilon^{-1}", "342400ffc49952adf98d480a9de8f069": "m_{BC} = \\frac{y_B - y_C}{x_B - x_C} = \\frac{\\sin \\theta}{\\cos \\theta - 1}", "3424cf380281dfc30c8acfdd5ec62fc9": "\\mathbf{B}'(t) = 2 (1 - t) (\\mathbf{P}_1 - \\mathbf{P}_0) + 2 t (\\mathbf{P}_2 - \\mathbf{P}_1) \\,.", "3424f2a157f75c5c0bddc3258db765d4": "\\frac{\\text{Operating Income}}{\\text{Sales}}=\\frac{\\text{Unit Price} - \\text{Unit Variable Cost}}{\\text{Unit Price}}", "3425849a0bb8c78790ed1e26f9e5316b": "\\subsetneq", "34261195456d751dabec19b7d2da0ec9": "C: y^m = f(x)", "342649d01588c8001f807cd9ef5650fa": "|\\bar{z}| = |z|, |(\\bar{z})^n| = |z|^n, \\arg(z^n) = n \\arg(z)", "3426587428bdcc2f60b443c1cb3ef522": "X_{lc}(\\bold{r})", "34266e51bc44754324b55211661b5518": "B(x,\\mu) = f(x) - \\mu~ \\sum_{i=1}^m\\ln(c_i(x))~~~~~(2)", "34268237d1094b3593f0fe09e3ca265f": "n_2 = 4 \\pi \\int_{r_1}^{r_2} r^2 g(r) \\rho \\, dr. ", "3426b56dda77603ec148f2f1b4884f8c": " \\mathrm{d}V =\\frac{\\delta W}{P},", "3426e2003d46ab11fd54f6c33989df3e": " -\\frac{1}{4}\\boldsymbol\\eta_1^{\\rm T}\\boldsymbol\\eta_2^{-1}\\boldsymbol\\eta_1 - \\frac12\\ln\\left|-2\\boldsymbol\\eta_2\\right|", "3427814ecb1c732f47b9f5648bc0f3ce": "Z_0 \\approx 120 \\pi", "342792a6bb74dd6f756293914089031e": "(p \\land (q \\lor r)) \\vdash ((p \\land q) \\lor (p \\land r))", "3427ec6fef45fd52e085cbfde182e4cb": "\\left(\\frac{\\partial x}{\\partial y}\\right)_z = - \\frac{\\left(\\frac{\\partial z}{\\partial y}\\right)_x}{\\left(\\frac{\\partial z}{\\partial x}\\right)_y}", "3428245251607e0367b9898aec0b77a1": "\\varepsilon_{i_1 \\dots i_n}\\varepsilon^{i_1 \\dots i_n} = n! ", "342837c8da31d692b065bcbc1b2d8dc8": "\\delta=\\frac{4 \\pi}{\\lambda} n t \\cos \\theta_t", "34283c219af6e4efd445a73ad1017e8c": "\\ A^* \\wedge B^*", "34285db2fb7a4a8fc8d0948b7bfc58f0": "A = ( X \\cdot \\overline{S})", "342a013e73e97fee0072ad64f73411e0": "|{\\psi_{T}}\\rangle", "342a3ae9d6b1c7d99413068af367c619": "\\begin{matrix} 4 & 3 \\\\ 2 & 1 \\\\ 1\\end{matrix}", "342a3dfcd13c3c3c5e0fcf20e5d3bf61": "O(k(\\#t+\\#s))", "342a44f9b0b2d84a07f62e75e0d906ae": "\\varphi(h(y),t) = h\\psi(y,t)", "342a5238a17c9ad23a1ecc8e0e70af64": "g : E\\times_M E\\to \\mathbf{R}", "342a5f29d12a916e1f4e499d55e753fa": "e^k f^k v=k!h(h-1)...(h-k+1) v", "342a92bb3c000d3e78e4b5ad895c307a": "{{\\theta }_{T}}\\left( x \\right)=\\left\\{ \\begin{align}\n & x,if\\left| x \\right|\\ge T \\\\ \n & 0,if\\left| x \\right|0, ad=m}\\frac{1}{d^k}\\sum_{b \\pmod d} f\\left(\\frac{az+b}{d}\\right), ", "3449a1209e401d90f110efb01e7bd6cf": "\\int_{0\\searrow 1}\\frac{e^{-i\\pi u^2+2\\pi i pu}du}{e^{\\pi i u}-e^{-\\pi i u}} = \n\\frac{e^{i\\pi p^2}-e^{i\\pi p}}{e^{i\\pi p}- e^{-i\\pi p}}", "3449a556c7665dedef2f7c3deec474a0": " \\tilde{p}= \\frac{ n_1 + \\frac{1}{2} z_{\\frac{\\alpha}{2}}^2}{ n + z_{\\frac{\\alpha}{2}}^2 } ", "3449e242f3e35597f529d00cb5663355": " \\acute{u}^{\\mu} = { {d \\acute{x}^{\\mu}} \\over {d\\tau}} ", "344a0f27d6371e53e6703b4068e0f410": "\n \\delta K = \\int_0^T \\int_{\\Omega^0} \\int_{-h}^h \\rho \\left(\n \\dot{u}_\\alpha~\\delta\\dot{u}_\\alpha + \\dot{u}_3~\\delta\\dot{u}_3\\right)\n ~\\mathrm{d}x_3~\\mathrm{d}A~\\mathrm{d}t\n", "344a423a2ab300d1b26b483da2cf3745": "\\widehat{\\mathbf{C}} = \\mathbf{C}\\cup\\{\\infty\\}", "344a7c25a1a8b57763cd56039855c641": "\\begin{align}\n f(j\\omega) & = a_0(j\\omega)^n+b_0(j\\omega)^{n-1}+a_1(j\\omega)^{n-2}+b_1(j\\omega)^{n-3}+\\cdots & {} \\quad (20)\\\\\n & = a_0(j\\omega)^n+a_1(j\\omega)^{n-2}+a_2(j\\omega)^{n-4}+\\cdots & {} \\quad (21)\\\\\n & + b_0(j\\omega)^{n-1}+b_1(j\\omega)^{n-3}+b_2(j\\omega)^{n-5}+\\cdots \\\\\n\\end{align}", "344a84e4a55338343ab86bbfed06fe89": "\\overline{I^{n+l}} \\subset I^{n+1}", "344b097d1247f5e8c7cf58c875763633": "\\mathit{alg}", "344b1e117bb2d290339ea5b5a63e69ad": "I(\\omega) = \\frac{1}{2} \\int_{0}^{T} | \\dot{\\omega}(t) |^{2} \\, \\mathrm{d} t", "344b5e68f3c1e87185cbc894db32b0fe": "\n\\begin{align}\nf(x,y,z)\n&= 6 \\cdot \\frac{ \\frac{x}{y} + \\frac{1}{2} \\sqrt{\\frac{y}{z}} + \\frac{1}{2} \\sqrt{\\frac{y}{z}} + \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} + \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} + \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} }{6}\\\\\n&=6\\cdot\\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6}\n\\end{align}", "344b8bb60a3d26394895aa3510f424f7": "u \\in R^d", "344bcc7d26d1bca3935aea2c5959c7ef": "\\Delta^n a_0 = \\sum_{k=0}^n (-1)^k {n \\choose k} a_{n-k}.", "344c08c5f19dcc7ad675de52e464bea3": "R_1 - R_2", "344c0d58ba0799ef97aa8e6cddde355d": "n-\\phi(n)", "344c4104e021bccaf69cd2eb7cf96fe8": "(p_k,q_k,0)\\,", "344c53dededcba6afe79104348a635cd": "w = w_1w_2\\ldots w_n", "344cf2f710cc2c8703b825497477c73a": "\\int_0^1\\int_0^1 \\left|\\frac{x^2-y^2}{(x^2+y^2)^2}\\right|\\,\\text{d}y\\,\\text{d}x=\\infty.", "344d1e98647dad8cad07877f27744865": "G_{1} =\\frac{ 2 A_{1} }{ \\gamma }", "344d24b20b9a93f9496b7a72248f2431": "p_\\mathbf{k}", "344d92c8a4b165cab21a79226681b38a": "\\Pr[\\mathcal{A}(D_{2})\\in S] \\leq \\exp(\\epsilon)\\times\\Pr[\\mathcal{A}(D_{3})\\in S]\\,\\!", "344d982a57ded4e906dd47f8c772b078": "P_i =\n\\varepsilon_0 \\sum_{j=1}^{3} \\chi^{(1)}_{i j} E_j +\n\\varepsilon_0 \\sum_{j=1}^{3} \\sum_{k=1}^{3} \\chi^{(2)}_{i j k} E_j E_k +\n\\varepsilon_0 \\sum_{j=1}^{3} \\sum_{k=1}^{3} \\sum_{l=1}^{3} \\chi^{(3)}_{i j k l} E_j E_k E_l + \\cdots\n", "344dd2ed64e4523b75ee092d0a3dc3e6": " p = \\sum_{i=1}^n x_i \\otimes y_i", "344ddfe53eae4c52df320282e33b9d94": " \\varepsilon_m = \\int d^3 r \\ \\psi^* (\\boldsymbol{r})H(\\boldsymbol{r}) \\psi (\\boldsymbol{r}) ", "344df017cd973e2b68f81471620a88c9": "y_{k}", "344e1722f5ba9ae91a7ee03b7c97e482": "\\hat{c}= (\\hat{c}- \\hat{a}) + \\hat{a} ", "344e2283d4284246d7c377da79d87754": "\n\\frac{\\partial T (m,s,x) }{\\partial s} = \\ln x ~ T(m,s,x) + (m-1) T(m+1,s,x)\n", "344e32846c5161233df71b7216bbbfb1": "\\lim_{x \\to x_\\pm} \\|y(x)\\| \\rightarrow \\infty\\,,", "344ec9eee6d8a52e883e8b84b4bf350e": "\\alpha^\\mathrm{T}[\\mathbf{f}A] = A^\\mathrm{T}\\alpha^\\mathrm{T}[\\mathbf{f}].", "344ef81fb23063d3c45cddfc67d56b85": "\\Delta y =f(x+\\Delta x)-f(x)", "344f3b928e36ed62742c770fff906d31": "M \\times \\{0,1\\} / \\sim", "344f455c1d79fba0fdebb77a1268303c": "\\hat{\\textbf{x}}_{k\\mid k-1} = \\textbf{F}_{k-1}\\hat{\\textbf{x}}_{k-1\\mid k-1} + \\textbf{B}_{k-1} \\textbf{u}_{k-1} ", "344f8fcdf73a76bf23f7a937eb37d328": "T^{\\mu'} = \\Lambda^{\\mu'}{}_{\\nu} T^\\nu", "344fc3459b8cb15499101a4322e5a1b7": "A_Y", "34500019343db984329e582647235673": "\nC^+ = \\max_{0 \\leq \\beta \\leq 1} \\min \\left\\{ \\frac{1}{2} \\log(1 + (1 - \\beta) (c^2_{21} + c^2_{31}) P_1), \\frac{1}{2} \\log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \\sqrt{ \\beta c^2_{31} c^2_{32} P_1 P_2}) \\right\\}\n", "34503692bbbd8a4c72b948a26ad590e1": "H(Re^{i\\theta},t)=p((1-t)Re^{i\\theta})", "3450587edfef25cedd0c48a80c96aae8": "\\int_\\Omega u_j u_k = 0", "3450d9ab8d0b586bf680007b1902729a": "\\mathcal{Q}(u)", "3451158d81940fad5405216068015554": "u_i \\in \\mathbb{R}\\cup\\{\\infty\\}", "3451318245d5bd6de0f96c1232041304": "\\tilde{O}(\\log^4 q)", "345163f8112c5fae997a00d27c9108e6": " A=\\begin{pmatrix}1&-2\\\\1&-1\\end{pmatrix} ", "3451fd3d2684169606c18431dcf87ae7": "\\sqrt{\\frac{1}{7}}\\!\\,", "34524a6075024bfa2f8c208ec521d623": "\\tau<0", "3452863532e1772752ab8a016b50b69d": "R ", "34528afc9ef6032ffafae53e1d95f6aa": "x_1^iP_i", "3452991fb6030866c664e3d6befbce56": "\\{C(a,b),c,d,e\\}", "3452c536b840ebf3102a9190f150208f": "b = |n-a^2/4n|", "34530f3e82f5242b0454ccd7954a32fd": "N/\\mathcal{Z}(_R R)=\\mathcal{Z}(_{R/\\mathcal{Z}(_R R)} R/\\mathcal{Z}(_R R))\\,", "3453268c1326dc70ba6d90947c8a2665": "\n X\\ \\sim\\ \\mathcal{N}(\\log (RR),\\,\\sigma^2). \\,\n ", "345328052668fd0ca485a24771c5c4e8": " \\beta = \\frac{ - ( b + a ) k }{ b - a }. ", "34539e1f695aea1431260acbc61177d5": " \\frac{1}{2}v_a^2 = \\left( \\frac{GM}{r_a} - \\frac{GM}{r_p}\\right) \\left( \\frac{r_p^2}{r_p^2-r_a^2} \\right) ", "3453ec396cc234dcf2105b27e536a314": "\\mu_{M} \\,\\to\\, \\mu", "34547502cc34a468f1fd7bb4e0964115": "\\delta_D(k)", "34549139498844de805feea49d49eea0": "\\frac{-0}{-\\infty} = +0\\,\\!", "34549913a3cf2cdba5d04d18937432d1": "(D-C A^{-1}B),(A-B D^{-1}C)", "3454aeb5d42b4391c7ecc3cb316c177b": "T_c = t_2 - t_1.", "34552230f53e04c6481775eaef9311e2": "\\overline{a_i}", "34555086e76f0f0bfd2d9e56d7c9c649": " T_{m} = {RI_{m} \\over (r-g)} ", "34557ddac12bee213e000350956a83a6": "\\rho'_{AB}", "3455a8b2cda63e350819dc447d4e02f7": "\\int \\log_a x \\, dx = x(\\log_a x - \\log_a e) + C", "3455d86c8dd5c3cee39fa2a0db68b471": "\\mathbf{F}=-\\boldsymbol{\\nabla}\\Phi+\\boldsymbol{\\nabla}\\times\\mathbf{A}", "345676c3e3d14a6774b68f56d1733e96": "\\begin{cases} 3x + 5y + z \\\\ 7x - 2y + 4z \\\\ -6x + 3y + 2z \\end{cases}", "3456a50aa2ff226042123bbe35469ea3": "1 = \\int_x g(\\boldsymbol\\eta) h(x) e^{\\boldsymbol\\eta \\cdot \\mathbf{T}(x)} dx = g(\\boldsymbol\\eta) \\int_x h(x) e^{\\boldsymbol\\eta \\cdot \\mathbf{T}(x)} dx = g(\\boldsymbol\\eta) Z.", "3456bc4f3615ed07e832aec0fa5ecdaa": "T \\equiv \\frac{Z}{\\sqrt{V/v}} = \\left(\\overline{X}_n-\\mu\\right)\\frac{\\sqrt{n}}{S_n},", "3457b0ced2d5a06997b48057d1239662": "x \\in \\{1, 2, \\ldots, N\\}", "3457ef2c966888a3427c969e499ce0cf": "\\textstyle K = \\frac{N\\pi kT}{3\\times 2^d \\alpha^d}", "3457f56742f1e6e1b26088ff8cb81a78": " \\nabla\\times(\\mathbf{A}+\\mathbf{B})=\\nabla\\times\\mathbf{A}+\\nabla\\times\\mathbf{B} ", "3457fedb2a657b62bd2f2b67f8ddc52a": "E_8", "345807c16e69ebeb5dce70e3a4503466": "\\frac{W_x}{W_o}=\\frac{xN_xM_o}{N_oM_o}=\\frac{xN_x}{N_o}=x\\frac{N_x}{N}\\frac{N}{N_o}", "34581717f5262330a59cf4ed53dac613": "\nP = T N {2 \\pi}\n", "34581874ef1f4237559efc3c8da046b7": "\\langle x, y\\rangle={\\|x+y\\|^2-\\|x-y\\|^2\\over 4}+i{\\|ix-y\\|^2-\\|ix+y\\|^2\\over 4}.", "3458343a810730e2ff0283cc9b733b06": "F(m) = \\sum_{k\\in\\mathbb{Z}} \\exp\\left(-\\frac{\\pi\\cdot(m+N\\cdot k)^2}{N}\\right)", "34583ed75680bbc166f3f7921309886a": "\\displaystyle{ye=1.}", "345867382942d15300e5224efba95d29": " A_{n+1} = \\sqrt{\\frac{n}{n+2}} \\ A_{n} ", "34587362d3393cc45196e1950af777f8": " \\pi(z; h\\lambda) = (1 - h\\lambda\\beta_s) z^s + \\sum_{k=0}^{s-1} (\\alpha_k - h\\lambda\\beta_k) z^k = \\rho(z) - h\\lambda\\sigma(z). ", "345884510db9f41bd1a7f523eee658f8": "\\mathfrak{Q}", "3458cfed0b9ecdd33067c19431ee9b4d": "z^k \\ne 1 \\qquad (k = 1, 2, 3, \\dots, n-1 ). ", "345922d8b401fdedc580ba6e7fde9cb3": "x = 11 + 12s = 11 + 12(5u) = 11 + 60u", "3459317ae96b1936371df9e80ad03f05": " \\displaystyle\n u = \\log_b (n).\n", "3459746a9859af8a85f1b765f94eb22a": "S(x)=\\frac{1}{1+\\exp[-a(x-\\theta)]}-\\frac{1}{1+\\exp(a\\theta)}", "345982ec75cf86546b276aff87363d8e": "F_2(x)=\\sum c_ip_i(x)", "3459f623fcbda43995c78cfd7bd24def": " y^m(\\mathbf{x},\\boldsymbol{\\theta}) ", "345a13575aeec6b816e6727166f1b2ed": "A'A''", "345a55a08889a778accce74eff9844f6": "{\\delta_x \\choose \\delta_y}", "345a9ace1ecad967f81455cb12f3f1d7": " \\mathbf{\\hat T} (\\varepsilon) = I - \\frac{i\\varepsilon}{\\hbar}\\mathbf{\\hat P} ", "345aaed5e78c8fe72e84452a74fb82ca": "4(|R_1|+|R_2|) \\ < \\ 2^{-n}.", "345af5d84017315ed57574fecac56fa3": " S = { 8 : 5^{5/4} }, \\ ", "345afe175d6730ef80f9300c30743a6c": "\n \\mathbf{R} \\mathbf{U}\n\\begin{pmatrix}\n\\frac{1}{\\sqrt{2}} & \\frac{i}{\\sqrt{2}} & 0 \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{-i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}\n= \\mathbf{U}\n\\underbrace{\n\\begin{pmatrix}\n\\frac{1}{\\sqrt{2}} & \\frac{i}{\\sqrt{2}} & 0 \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{-i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} & 0 \\\\\n\\frac{-i}{\\sqrt{2}} & \\frac{i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}\n}_{=\\;\\mathbf{I}}\n\\begin{pmatrix}\ne^{i\\phi} & 0 & 0 \\\\\n0 & e^{-i\\phi} & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n\\frac{1}{\\sqrt{2}} & \\frac{i}{\\sqrt{2}} & 0 \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{-i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}\n", "345b379f3b55d1c45fe0e6967dd5402e": "\\mathrm pH \\approx 14.00 - 4.7 = 9.3", "345b83529633036f06ab024cab79d087": "\\bold {n}_i", "345b8cc37845f1179d5f5184d4a979ab": " : \\hat{b}_1^\\dagger \\,\\hat{b}_2 : \\,= \\hat{b}_1^\\dagger \\,\\hat{b}_2 ", "345bb2c6212d262aaae0cac46b61ec2e": "\\left(\\sqrt{1/55},\\ -3\\sqrt{1/5},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "345c1c6275f80f1c89c52c71a1f383a7": "Df(x)", "345c1cf71e3649d31f7463721e2da5d6": "\\beta = \\frac{v}{c} , \\gamma = \\frac{1}{\\sqrt{1-\\beta^2}}", "345c720431a2db8ae6bb440ec91117bb": " \\left\\langle e^X \\right\\rangle\n= e^{\\langle X \\rangle} \\left\\langle e^{X - \\langle X \\rangle} \\right\\rangle ", "345c75fc500e06775283d268679b0531": "\\gamma_\\mathbf{u}", "345c9667996f7da407e8a2af5289c5e6": " G = \\frac{3}{2} = \\mathrm{1.76\\ dBi},", "345ce702b08aeb4224fc728bec486815": "\\frac{e}{N} = \\frac{17993}{90581} = \\cfrac{1}{5 + \\cfrac{1}{29 +\\dots + \\cfrac{1}{3}}} = \\left [0,5,29,4,1,3,2,4,3 \\right ]", "345cff10f4de7e77996423e6309e27b1": "\\zeta(z,q)=\\frac{\\Gamma (1-z) \\left(2^{-z} \\left(\\psi \\left(z-1,\\frac{q}{2}+\\frac{1}{2}\\right)+\\psi \\left(z-1,\\frac{q}{2}\\right)\\right)-\\psi(z-1,q)\\right)}{\\ln(2)}", "345d19b7779a844fc5b41d42600f54ae": "L^2(\\mathbb{R}^n),\\ell^2(\\mathbb{Z})", "345d7b89ce889a3c82ea2055d5f7b117": "\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}", "345d824e9af58eccda106bdb561b8195": "\\pi:E\\rightarrow M", "345dc3752dde684d80362a1fa1593017": " L_{uv} = 0, L_{uu}^2 - L_{vv}^2 \\geq 0 ", "345e12661f51f0e4a3a1422b55725c9c": "\\sum_{i=1}^n L(a_i) ", "345e215390cc7389820093e9c0be03e8": "\\frac{d^2y^k}{dt^2}+|\\dot\\gamma|^2\\sum_j y^j(t)\\langle R(e_j(t),e_1(t))e_1(t),e_k(t)\\rangle=0", "345e3dc525125b6e3c835f9f37327f00": "\\beta_2 = \\alpha_1 \\alpha_3 + \\alpha_2 \\alpha_4", "345e8355babdc937574179f46b2cd447": "\\epsilon_f = \\frac{6Dd}{L^2}", "345efc282d0750c460881152cc1057de": "a=0.1", "345f014365ca786ba5b7aad8aad016e9": "\\mathrm{MTF}(\\nu) = \\left\\vert \\mathrm{OTF}(\\nu) \\right\\vert ", "345f0e91fc971b32251f01f670098e83": "\\frac{1}{2^n} \\sum_y \\sum_x e^{-2\\pi i x\\cdot y/2^n} e^{ix\\theta} \\left| y \\right\\rangle \\otimes \\left| \\psi \\right\\rangle = \\frac{1}{2^n} \\sum_y \\frac{ e^{i 2^n \\theta} - 1 }{ e^{i\\left( \\theta - 2\\pi y/2^n \\right)} - 1 } \\left| y \\right\\rangle \\otimes \\left| \\psi \\right\\rangle", "345f120ec9f6b74cd69b1aa5811ad282": "\\textstyle B_{p}", "345f17b7687e2be3a67882f6ebefdbd9": "O(c^n),\\;c>1", "345f43c27e0b8e3f08f506efd298a081": "v_f^2 = v_i^2 + 2av_it + 2a(\\Delta d - v_it)", "345f612874834e92a6a06936beb1ad79": "{A}_{4}^{(2)}", "345f8e74c469df04d36bd9378619ec11": "\\,y_i\\,'s", "345f8ec07cb6dae76435da271f034038": "\\limsup m(n) > 1.06", "345f9bafa1c80e5bf7b6cc8800b98b55": "(K) \\not=2", "345fcba73798a7fe5ca49b7dfadc01ae": " \\lbrace \\cosh a + j \\ \\sinh a : a \\in R \\rbrace ", "345fd06e7e93a0e1a71a1f21dadefb84": " \\pi = \\lim_{n \\rightarrow \\infty} \\frac{4}{n^2} \\sum_{k=1}^n \\sqrt{n^2 - k^2} ", "346026f03a55c2093586460b9a13cfbf": "\\begin{bmatrix} \\dfrac{b_{22}}{\\Delta \\mathbf{[b]}} & \\dfrac{-b_{12}}{\\Delta \\mathbf{[b]}} \\\\ \\dfrac{-b_{21}}{\\Delta \\mathbf{[b]}} & \\dfrac{b_{11}}{\\Delta \\mathbf{[b]}} \\end{bmatrix}", "3460a4e1467cc7bafd59e0d65ca2f1d8": "(6/5)^3*(7/6)*(2/1)^{-1}", "3460b75f716a5b6cdbddc1c06497bf65": " \\displaystyle{U(s)f(x)= f(x-s),\\qquad V(t)f(x)=e^{ixt} f(x).}", "3461006c342661d9bee54b19533eeaaa": " \n\\text{Subject to: } \\lim_{t\\rightarrow\\infty} \\frac{1}{t}\\sum_{\\tau=0}^{t-1}E[y_i(\\tau)] \\leq 0 \\text{ } \\forall i \\in \\{1, ..., K\\}\n", "346103618f440c38bf2c9feea403cbf6": "\\begin{align}\n\\dot{v} &= g(n - \\cos{\\beta}) \\;,\\\\\nv \\dot{\\beta} &= g \\sin{\\beta}\\;. \\\\\n\\end{align}", "3461181427f5736eccc949903458f147": "1/\\omega_g< \\mu(A_2) > \\mu(A_3) > \\cdots > 0. ", "346261d2e20303df03a84a3626eb384a": "((r + c)(r + c - 1) + \\gamma(r+c)) a_r + (-(r + c - 1)(r + c - 2) - (1 + \\alpha + \\beta)(r + c - 1) - \\alpha\\beta) a_{r - 1}= 0", "34626e9e8cd2bdce9b63d706d1bb9922": "\\operatorname{GL}_n(F)", "346276159cfd3aca2a8198514f180e61": "f(x) = \\frac{1}{\\left(2+x^2\\right)^{\\frac{3}{2}}}.", "3462c892cca6f86c687d28fa58ef5f3f": "G(z) - G(z)^2 = z", "3462deba7be3450b361045f91c3efc1e": "\\epsilon = H/L", "34634fa76ea1af84c5dd50d6c0ac1593": "\\delta<\\alpha", "34636373db2729f7533380932b205c83": " g(y_n|X_n,t_n,\\theta)", "34638b0604db43728af19d1bfc86c7ce": "V_{\\text{CE}}", "3463b1d3345c5c85e6f74b77cf001df2": "\\ B", "3463df0532a111faeb5f3c4749329c33": "\\mathbb{CFM}_\\mathbb{N}(R)", "3463e9abeab360ac3940a5c02e864e87": "b(i)", "3463f992a93b8770dbadda02649c0144": "\\mathbf{x}_{0i} ^\\top", "3463fdadfdd2308351f42986ed370680": "13.75\\pm0.11 \\times10^9", "34645f77d9e273ac7b4e78c62f47aa20": "\\{f_\\alpha : X_\\alpha \\to Y\\}", "3464bf3b4bf4f72de2a9e1f1223ee466": "\\{ N \\subseteq M | P \\not\\in \\operatorname{Ass}(N) \\}", "34654a7362e3e53d4c69412a9a01fbfc": "h_n(G)", "346578334a58f0255a4edf34c360e0c0": "P'= P\\times \\frac{T}{T'} ", "3465b52572581418c2fafa310614f17e": "0 \\leq \\theta < 2 \\pi", "3465c40ec7ca5f3c39fed5b3047f5183": " \\lambda_{x}(\\alpha) =\n\\min_{S} \\{\\Lambda(S): S \\ni x,\\; K(S) \\leq \\alpha\\},\n", "3465e0e74947cb9e1c2e525a0f49974a": "\\scriptstyle{d=\\sqrt{2}}", "3465ea9c5b51d2603c25d6be83498449": " f\\in M^{p,q}_m(\\mathbb{R}^d) ", "34662c3de49f4545de9a642452e186a1": "h(x)=\\frac{-\\beta(1-p)e^{-\\beta x}}{(1-(1-p)e^{-\\beta x})\\ln(1-(1-p)e^{-\\beta x})}.", "34671787937aaa7e007bc26151866b23": "M(x,y)\\,dx + N(x,y)\\,dy = 0 ", "3467edc895f7c7b3983c2eb5ff654d88": "\\sin A=\\frac{\\textrm{sinh(opposite)}}{\\textrm{sinh(hypotenuse)}}=\\frac{\\sinh a}{\\,\\sinh c\\,}.\\,", "34680fd4a0342bb5937be0264e74f080": "\\{F_i\\} ", "34681a4d0a43076e354c75d42440a8ec": " \\mathbf{\\Pi} = \\mathbf{p} + \\frac{\\hbar}{4m_{0}^{2}c^{2}}\\bar{\\sigma} \\times \\nabla V ", "34681dd25bee11319be8aeeef19f8750": "\\neg Q(f(g(x), y, 2), x)", "34682b5bbefa44bf706750bd8e8f01fc": "\\mathop{\\mathrm{ind}} Df_s\\left(x\\right)> 1", "346af3369de794f4118401ee0532f962": " TU = \\sqrt{\\frac{DU^3}{G*M}}", "346b37efe035066851a088ee63bc5d9d": "f(x_1, x_2, \\dots, x_n) = (f_1, f_2, \\dots, f_m),", "346bae93bc64c636cc4d8d5dfd29f429": "\\eta_R", "346bd7f15e284e882901be95764b7e44": " y'= \\frac{1}{2}\\left(\\frac{y'(0)+\\sqrt{{y'(0)}^2+1}}{(1-\\frac{x}{A_x})^{\\frac{V_t}{V_d}}} - \\frac{(1-\\frac{x}{A_x})^{\\frac{V_t}{V_d}}}{y'(0)+\\sqrt{{y'(0)}^2+1}}\\right) ", "346bdbba3c8118b45e90c511401e47a8": " \\frac{1}{\\sqrt{2}} (|0 \\rangle + |1 \\rangle ) ", "346cefe88c97f31488f9444db65dbe7a": "x_{i+1}= c", "346d1c7045b5daf07020d06072aaed07": " \\epsilon_A \\eta_B + \\epsilon_A \\sigma_B + \\sigma_A \\eta_B \\ge \\left| \\frac{1}{2i} \\langle [\\hat{A},\\hat{B}] \\rangle \\right|", "346d2b4efea7dc68a1c34f6d2296fc81": "g_J = 4/3", "346d6b126348a2b94291fef27191946f": "\\Phi : N \\to C^{\\infty}(M,\\mathbb{R}).", "346daaab6e94a6a30948399afc17da94": "\\cdots \\rightarrow X_1\\rightarrow Y_n\\rightarrow \\cdots ", "346dcba56a54decf7ef85fab1ea5762c": "\\left\\{ \\bar{X}_{i},\\bar{Z}_{i}\\right\\} = 0\\ \\ \\ \\ \\ \\forall i.\n", "346dd3c73c2385e7ce90dd7f0413940c": "\\log^2(p)", "346e011c79a9fe1eddcfbd3df70fe3ab": "p(X,A|\\theta) = \\sum_{\\omega=1}^{\\Omega} \\sum_{\\textbf{h} \\in H} p(X,A,\\textbf{h}, \\omega | \\theta). ", "346e3e715f1c4355ccdfa0fc7d7e06b8": "E_n = \\hbar \\omega_c (n+1/2)", "346e49f29e956eb17b4b78df21139b77": " \\sum_{k=0}^\\infty B_k(a_k - a_{k+1})", "346e6db9711239d8948a32886b97da63": "\\text{kva base} \\approx\\text{ horsepower rating}", "346ee638f6b3b8e3adf8a5fde7ff55e4": "\\bot \\equiv (\\top | \\top)", "346f08fc761cb452f526866a5adc0394": "\\scriptstyle \\boldsymbol R", "346f1988073471dedf640088c34e2ab8": "F(a,b;1+a+b-c;1-z)", "346f2ccd3c19fdad378b18b3cc190e66": "\\varphi(Q)=\\psi(x^1(Q),\\dots,x^n(Q))", "346f62f4f4a99fcc48193d712fd39854": "b=3m^2+n^2\\, ", "346fa1ec380a24a72df7673bae5a22b1": "\\partial/\\partial y_{j}", "346fe8b0cd2602fc9a93a90bb4bdc95d": "e^{-\\mathbf At} \\mathbf{\\dot{x}}(t) = e^{-\\mathbf At} \\mathbf A\\mathbf x(t) + e^{-\\mathbf At} \\mathbf B\\mathbf u(t)", "346fe9b43fb2f0496ba99b737914d395": "\n{N\\choose k} (1/2)^N\\!\n", "3470210d24bacd6bfac46e3716b959d7": "{\\textit{VAR}_\\text{tot} = SS_\\text{tot}/n}", "3470650792fd13932b8109a28fe54aa4": "(a, \\tfrac{1}{2}(a+b))", "34708a43504dbfeef8174675a3a23e28": "c_\\alpha|\\alpha\\rang+c_\\beta|\\beta\\rang", "3470968f6b7375d3207fa922c77e7a2f": "(\\hat{x}+\\hat{z})/\\sqrt{2}", "3470ab2af4eb1854295c8f49570f7502": "|\\mathbf{AB}\\times\\mathbf{AC}|,", "3470cbc65364cf9e376629b470de8c75": "c_L\\!", "3470f85e53168a678ee9c627b11dd237": "\\mathcal{F} = \\mathcal{N}^{\\otimes N}", "3471c52702ce7942071161bcf7e7869e": "\\scriptstyle {D : \\mathbf{Set} \\rightarrow \\mathbf{Set}}", "3471ec464d3daf4285e2bb244eabf71f": "\\mathbf{T}_L = \\mathbf{U}_L\\mathbf{\\Sigma}_L = \\mathbf{X} \\mathbf{W}_L ", "34720372ea6730e7152392d478176fe5": " V_R(n) = Kn^\\beta ", "347225f3d9acb05409342c2d930513d9": "u_3 = u_5 + \\frac{(P_3 - P_5)}{\\sqrt{\\frac{\\rho_5}{2}((\\gamma+1)P_3 +(\\gamma-1)P_5)}}", "3472b04ab9334c7f962b5999aa7e64c9": "8 * 2 = 16", "3472fa2707af28560e7c15e441556b91": "\\forall m,n > N, x_n x_m^{-1} \\in H_r", "347312315c2db7b8c2aed0bbfaddacd5": " n = N/V ", "347321e2bddcaa1d82bfe9cbac83d571": "a^2 \\cos^2 \\alpha + b^2 \\sin^2 \\alpha = p^2,\\,", "347322995cb3b1d878c3b33c829078d8": "\\scriptstyle{|\\alpha |}", "347324be2d30634c17e2e8962717ea88": "\n-\\biggl\\langle\\sum_{k=1}^{N} \\mathbf{q}_{k} \\cdot \\mathbf{F}_{k}\\biggr\\rangle = P \\oint_{\\mathrm{surface}} \\mathbf{q} \\cdot \\mathbf{dS},\n", "34732b3f234b485c1f1ea46f8f6a617e": "h(P)=(h_0,h_1,\\ldots,h_d)", "34732f8cd5842f3d3e8d5f89e58d06b7": "- \\nabla^2 [\\Delta\\phi(r)] = \\frac{1}{\\epsilon_0} [Q\\delta(r) - e\\Delta\\rho(r)]", "3473fc4664556f5a704d50203e61afa2": "y_i^*", "34740505e6c140ebd09c9c2d80b5e680": " \\operatorname{def}[F_2] = \\operatorname{false} ", "3474792e86f2c79b0fb5ad74dae7d709": "\\textbf{E} = -j \\omega \\mu \\textbf{A} + \\frac{1}{j \\omega \\epsilon} \\nabla (\\nabla \\cdot \\textbf{A})\\,", "3474b06ad9adfb3d8387ce2f04e39767": "8_9", "3474e773c66080d9fa3fc7c6983a020a": "{}^nx", "3474f04cbcfbf55c620a766f2cf99b94": "M\\mathbf{x} = \\mathbf{r}", "34753304296f070a45754ed6e7ff5ff2": "\\phi_{1}=r", "34754e28cd0348bd217a93f6532331e5": "\\Phi = \\iiint L_\\nu \\mathrm{d} \\nu \\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) \\mathrm{d} \\Omega \\,\\!", "347555308536b091f7f01ab13ad5f2e9": "s(d_1) \\odot... \\odot s(d_n) \\leq s(\\bold{h}(d_1,...,d_n))", "3475bc3c7d9c0c83d798f936b5ada16d": "p(X,Y) \\,", "3475d42c7b6a4fbb2ce2995bf0e206df": "E[\\textbf{w}_k\\textbf{w}_k^T] = \\textbf{Q}_{k}", "3475df6e25947a67051f172acb259606": "\\displaystyle{\\lim_{h\\to 0} \\frac{\\int^{x+h}_{x-h}|f(t)-f(x)|\\, dt}{2h} \\to 0.}", "3475f90289f441ca93cfaf1212ecf51c": "v = \\frac {2 v_{\\mathrm N} v_{\\mathrm F} }\n{v_{\\mathrm N} + v_{\\mathrm F} } \\,,\n", "347606ce1c02921cb4d69ae998bc70ec": "[2]P=P+P", "347608584dcd23d707f6b2a9a42a1ee7": "\\frac{1}{q_\\theta} = \\frac{1-\\theta}{q_0} + \\frac{\\theta}{q_1}", "347610ee725933277c8f90e0ada3a530": "x^5+d_1x+d_0 = 0\\,", "3476a60d2df489890c997c01e9a4355c": "\\scriptstyle D_F(1\\rightarrow 2)= 4(1)-1+1-3=1", "3476c8f5522318d32044702418afd897": " \\psi_1(z) = \\zeta(2,z). \\frac{}{}", "34771262fc69eff49fd92b33def8646e": "\n\\det(M') = a_{1,1} a_{k,k} - a_{k,1} a_{1,k} = \\det(M_1^1)\\det(M_k^k) - \\det(M_1^k) \\det(M_k^1),\n", "34775505d0379bf7fd4a6eeeca7afeee": "\\delta(\\tau) \\delta(f)", "347770a29463dff05feb21a7152a0e4c": "\n \\alpha = \\alpha_0 - \\alpha_1 \\ln(\\dot{\\varepsilon_{\\rm{p}}}); \\quad\n \\beta = \\beta_0 - \\beta_1 \\ln(\\dot{\\varepsilon_{\\rm{p}}}); \n", "34792be22b07e22826981a15398c71ad": "a \\cup b\\,\\!", "34794b5c8b97f41e2e1331aaa1cf8477": " {1 \\over 4 \\pi r } ", "34795e02be2afb46672a3de6146ff97e": "\\{x_1,x_2,x_3,\\ldots,x_n\\}", "3479c13302faa23e7e9aae38cfb14fe1": "\\frac{2\\pi}{\\sqrt{ 4 A C - B^2 }}", "347a1c04a574eefec5021b64cf29a8ab": "\\lim_{n\\to\\infty}g_n\\to\\infty", "347a200dd14c3d28dcba659f56a19563": "0\\to X\\to I^0\\to I^1\\to I^2\\to\\cdots", "347a45996c65d726c6d530ad51ef71cf": "\\gamma_1 = \\varphi_1 \\gamma_0", "347a69382ce125b32095f3bfd5e57826": "w = P_B = 1", "347a81a76b3d4574511732e2391fda38": "|x_1, x_2, x_3 \\rangle \\mapsto \\frac{1}{\\sqrt{2^3}} \\ \\left(|0\\rangle + e^{2 \\pi i \\, [0.x_3] }|1\\rangle\\right) \\otimes \\left(|0\\rangle + e^{2 \\pi i \\, [0.x_2 x_3] }|1\\rangle\\right) \\otimes \\left(|0\\rangle + e^{2 \\pi i \\, [0.x_1 x_2 x_3] }|1\\rangle\\right).", "347a9e41c70ab69f9f22e5457262bb32": "x(t) = Asin(wt)", "347abb6ae2196e9a58b01c4e8162bc51": "H_\\alpha(X) = \\frac{1}{1-\\alpha}\\log\\Bigg(\\sum_{i=1}^n p_i^\\alpha\\Bigg)", "347ad2e82c4398c581ea7dc26be365f2": " \\tbinom {12}4 ", "347b9941cd46eeb9ed8735fc15adfd09": "(Q, \\sigma)", "347b99be8c291ade0c6b4d680e18916a": "x^a", "347bdf9fb628c503e103dfc6009c0e98": "s,t\\in M(\\lambda)", "347be7cd9351094422a187a37327f2a7": "10\\uparrow\\uparrow 65,533", "347bf6d398841cf2a0ce8acdb63a64d8": "\\det(\\mathbf{I}+\\mathbf{uv}^\\mathrm{T}) = (1 + \\mathbf{v}^\\mathrm{T}\\mathbf{u}).", "347c6e9bca0df04d1bbac562d6cf20ac": "\\psi_{n^{*}s}(r) = r^{n^{*}-1}\\exp\\left(-\\frac{(Z-s)r}{n^{*}}\\right)", "347cbca54fc09db908ac3af9e27dc002": "\\hat{\\mathbf{x}}", "347d2c504cb1cf0d7b0443d755655f32": "\\displaystyle E_H=-HM_s\\cos(\\theta-\\gamma)", "347d85b2a9f16e87b223e67021b13eaa": "\\,{|x_p|}_p", "347da48edd35629590c118c97623bbfd": "\\det\\left[s\\textbf{I}-\\left(\\textbf{A}-\\textbf{B}\\textbf{K}\\right)\\right]", "347dea0c4d6132082a6d25a34d0608c1": "\\text{3. }\\sum_{\\omega \\in \\Omega} {P(\\omega|B)} = 1.", "347dffb54b0d6ddd15fabc8ee272eaca": "t_1,t_2", "347e6ee599d9eb0b4eab91ce2aa2b1c1": "\\sigma^2_{i=1 \\dots N}", "347e86b61e6d3cd7de89c29d237e7f06": "\\psi\\to\\psi^\\prime=U\\psi", "347ec1947be6d801802f62ae2216ba01": " U = \\int_V \\mathrm{d} \\mathbf{m} \\cdot \\mathbf{B} ", "347ed35afef036e343d7669bbbac9091": " a_m =\\frac{p_m}{N}\\sum_{n=0}^{N-1} u(x_n) T_m (x_n) ", "347f1121658d2aad97c0c604a41287f8": "p'_\\infty(x,y)=P'_z(x,y,1).", "347f1e27f17e2b3dd37070399f6492be": "h,k,l", "347fbc7978160eba13de9e7c187abd7d": " \\mathit{q} = \\mathit{q}_{1}\\mathit{q}_{2} .... \\mathit{q}_n ", "347fd6e63b160c898e1ae0265663765d": "\\frac{1}{[A]} \\ \\mbox{vs.} \\ t", "34801b3ede1df9f9009bd75d69c96d34": "\\pi:\\; M \\mapsto X", "3480222dddbdae5626c5c9b812596e92": "\\lim_n Ax_n = \\lim_n Ay_n", "34802771809f4c163ffdb4865c950401": " \\operatorname{dim}_{\\mathrm{Haus}}(X) \\geq \\operatorname{dim}_{\\mathrm{ind}}(X). ", "3480d34017ccb63461cf6c53f67db5e3": "a_x = \\frac{-kv_x}{m} = \\frac{dv_x}{dt}", "3480d63a43aab214e28df7bdd9f2224f": "\\scriptstyle\\sum\\limits_{n=0}^\\infty a_n", "3480dfcd068e4c18232dbfe6e0baf348": "J^k_p(M,E)", "3480e988f45c82e9b16d4ad61c5b6bba": "s_{\\gamma_i}", "3480ef2d9df820490813b7d5b35c3357": "R_c = R ", "34813c708d2f8720ca70288c64e15763": "P_\\mathrm{net}=1-\\prod^N_{k=1}(1-P^{(B,k)}) ", "348144617c253369841ce986e53f0b70": "An^2", "3481527737370a576ca4fdedc62d0a9f": "\\begin{align}\n\\int_0^1x^k(1-x)^l\\,dx\n&=\\frac l{k+1}\\int_0^1x^{k+1}(1-x)^{l-1}\\,dx\\\\\n&=\\cdots\\\\\n&=\\frac l{k+1} \\frac{l-1}{k+2}\\cdots\\frac1{k+l}\\int_0^1x^{k+l}\\,dx\\\\\n&=\\frac{1}{(k+l+1)\\binom{k+l}{k}}.\\qquad(**)\n\\end{align}", "348186d68ec1a3c6f1343cab347e4674": "(1/2!)\\pi^2 = (1/2)\\pi^2 ", "34820b300b015443cc45433cdbb09fe1": "r_{im} = -\\left[\\frac{\\partial L(y_i, F(x_i))}{\\partial F(x_i)}\\right]_{F(x)=F_{m-1}(x)} \\quad \\mbox{for } i=1,\\ldots,n.", "34823dfede3bbaf2cfae359524e9ba30": "n=p_1^{k_1}p_2^{k_2}...p_l^{k_l}", "3482a2824d7fc9a839a03eea94115340": " f(x) = \\sqrt[x]{x}", "3482ac2bcf3c826629a785d4859fb6bb": "(a_0, a_1, a_2, \\ldots)", "348335fcc8bc8c770714670bbec89a10": "\\sum_ {} \\vec{F} = 0", "34834d01de1fdbf274c2a815b17334dc": " v_{ij}\\big ( \\mathbf{x} - \\mathbf{c}_i \\big ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{cases} \\delta_{ij} \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) , & \\mbox{if } i \\in [1,N] \\\\ \\left ( x_{ij} - c_{ij} \\right ) \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) , & \\mbox{if }i \\in [N+1,2N] \\end{cases} ", "3483ad8a9ddd995fc0da09c887e7c4c7": " \\sum ", "3483b43d1ed7ecb44bb8834b18729275": "\\operatorname{span}(S) = \\left \\{ {\\sum_{i=1}^k \\lambda_i v_i \\mid k \\in \\mathbb{N}, v_i \\in S, \\lambda _i \\in \\mathbf{K}} \\right \\}.", "3483b762409c965248a8bda75755440e": " \\frac {\\theta_L} {\\theta_C} = \\frac {g_0 \\Delta_0} {\\Delta} \\, ", "3483bbfe1eb473e594e76a26afb94dd0": "\\phi(x)=E_q(x)^{-1}", "3483c75130ea4c8349581099ff099933": " E(X_j)= \\mu=(\\mu_1, \\mu_2)^T, \\qquad\n \\Sigma = \n\\begin{bmatrix}\n \\sigma_{11} & \\sigma_{12} \\\\\n \\sigma_{21} & \\sigma_{22}\n\\end{bmatrix}\n= \\begin{bmatrix}\n \\sigma_1^2 & \\rho \\sigma_1 \\sigma_2 \\\\\n \\rho \\sigma_1 \\sigma_2 & \\sigma_2^2\n\\end{bmatrix},\n", "348406538c2f8ae3d19fd7bc19b09774": "EC_{50}", "34841f02cc1fc2d01f5acead64a3c48d": "n^*=(-1)^{(n'-1)/2}n", "3484223db15532348f636ab3c3ce0ba6": "C = 2 \\sum_{i}^{\\rm core} h_{ii} + \\sum_{ij}^{\\rm core} \\left( 2 \\left\\langle ij \\left.\\right| ij\\right\\rangle - \\left \\langle ij \\left.\\right| ji\\right\\rangle \\right) - 2 \\sum_{i}^{\\rm core} \\epsilon_i", "34843c5421e2190b542d2f06612ba674": "a_{ij} = \\langle\\psi_i | {a} | \\psi_j\\rangle", "3484a708587ade84c4fdbfd67b2f4bbe": "\n0_{K_{m,n}} = \\begin{bmatrix}\n0_K & 0_K & \\cdots & 0_K \\\\\n0_K & 0_K & \\cdots & 0_K \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n0_K & 0_K & \\cdots & 0_K \\end{bmatrix}\n", "3484f36957206475308586469207ecd2": "\\tfrac{ad}{bd}", "34855742bf08b7d0ea536d74253d2502": "r = 2M + u^2/2M", "34856085e07fefc2e33effb479db4441": " \\sigma_1 = \\sigma_2 ", "34857097d7ef3ccebb0108cb52991ce1": " \\frac{| \\theta - \\nu |}{ \\sigma } \\le \\sqrt{ 3 } ,", "3485d38d84e450e40fad06c189734630": " (E - e\\phi) \\psi_+ - c\\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) \\psi_- = mc^2 \\psi_+", "3486016c26776863081fc5596fe2aff1": "H_v[i]", "34861c5477d59e6f68c1f62be0974a88": "\\hat{\\textbf{y}}_{k\\mid k-1} = \\textbf{P}_{k\\mid k-1}^{-1}\\hat{\\textbf{x}}_{k\\mid k-1} ", "34861d975079cedb63fd25e2322a3478": "Q = \\left(1 + \\frac{R_4}{R_q}\\right)\\left(\\frac{1}{2+\\frac{R_1}{R_g}}\\right)", "348676dcb7e2b1701ffd2fae61845a45": "\\displaystyle T(X,Y) = J[hX,hY] - v[JX,hY) - v[hX,JY]. ", "34869e3936ac8386ebe82c91b558eefb": " \\qquad \\qquad \\mathrm{nonlinear,\\ polyatomic\\ ideal\\ gas } \\ \\ \\ \\ \\ \\ \\ \\ \\ c_{v,f} = \\frac{R_g}{M}\\{3+ \\textstyle\\sum_{j=1}^{3N_o-6}\\displaystyle (\\frac{T_{f,v}}{T})^2\\frac{\\mathrm{exp}(T_{f,v,i}/T)}{[\\mathrm{exp}(T_{f,v,i}/T)-1]^2}\\} .", "3486ec359c8ddd01ad8426eb10ef7e44": "T''_n = n \\frac{n T_n - x U_{n - 1}}{x^2 - 1}", "3487073b527e0cf2ffed8069149f10d5": "\\text{Level 4:} \\ \\ 266 = 2 \\uparrow\\uparrow (2 + 1) \\uparrow 2 + 2 \\uparrow\\uparrow 2 \\times 2 + 2", "34870fa657e4b70f43c3b2593df0c136": "R \\models S", "3487140ea79fad139a62f0bae3110662": "\\mathrm{_6^{12}C} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{8}^{16}O} + \\gamma + Q", "348719b113d5b60c87ee28c5446ac4ff": "z,\\,", "34873f86421784407be90f51a7711ad1": "A_\\mathrm{fb} = \\frac{V_\\mathrm{out}}{V_\\mathrm{in}} = \\frac{A_{OL}}{1 + \\beta \\cdot A_{OL}}", "34874c8fae55e6571117f384035d1144": "X \\wedge Y = (X \\times Y) / (X \\vee Y). \\, ", "3487517eb4011d90a7182421be1949cd": "C=\\frac{Q}{V}", "3487c4b4a7493b84b9dfda1de99b913e": " \\hat{A}_0\\left[ f^2 \\right] ", "3488526777371b632d29a358f769a364": " \\tau = \\frac{4 \\eta/3 + \\eta^\\mathrm{v}}{\\rho V^2}", "34885acf801891666673a8dbd76cf709": " \\frac{\\partial \\sigma}{\\partial P} = E\\frac{\\partial \\varepsilon}{\\partial P} =E\\frac{1}{L} \\frac{\\partial u}{\\partial P} = \\frac{1}{A} >0 ", "3488b83e930ae6cb6f803de130ee04d1": "f:\\Omega \\rightarrow \\mathbb{C}^n", "3488c589add2af2c7b28970f948e3f96": "\\mathbb{Z}[x]", "3488c694bad7ffe610d5eb1850cf4e44": "h \\ge k", "3488e2d7462470588be9d08377d1aeda": " D=\\left|\\;\\frac{|C_A|^2-|C_B|^2}{|C_A|^2+|C_B|^2}\\,\\right|", "3488e44aa7cccb503992f3d10de01d10": "v_u", "348951ea496b3e29390857eddd4e78f9": "\\textstyle N_\\text{RPM} ", "34895d8b206f88bb308ad81bd794bf4d": " | \\langle f_k|f_k' \\rangle | < \\delta \\qquad \\text{ for } k \\neq k' \\and 0 \\le \\delta \\le 1 ", "34896a70fe17171aebce38a09a8c19b5": "\\sigma(f)\\in L_n(\\mathbf{Z}[\\pi_1 X])", "348980352b61abfb7eafbac04bc43cd0": "\\Re[S(z)]", "3489d2c5344b148a500ac3d3213336ca": "r^*q", "348a02440b09bf361e6cb03d02244b92": "1\\le i < r,s\\le n", "348a063ae4a6107ad91ac6588bc453f8": " {\\partial^2 \\mathbf{H} \\over \\partial t^2} \\ - \\ c^2 \\cdot \\nabla^2 \\mathbf{H} \\ \\ = \\ \\ 0", "348a255423ccf0c109425aa69b11e687": "\\boldsymbol\\Psi \\in\\mathbb{R}^{D\\times D}", "348ad3ca9b8e64f693ac84777ca7c160": " m^2 = \\left(\\sum E\\right)^2 - \\left\\|\\sum \\vec{p} \\ \\right\\|^2", "348ae582823edf73299e22a75ae4fe75": " U_{Max} =\\frac{q_1 q_2}{4 \\pi \\epsilon_0 r}(1-\\frac{\\mathbf{v_1}\\cdot\\mathbf{v_2}+(\\mathbf{v_1}\\cdot\\mathbf{\\hat{r}})(\\mathbf{v_2}\\cdot\\mathbf{\\hat{r}})}{2 c^2}) ", "348b5f54328bef161673616850cf846b": "SFM = \\text{stock diameter (in)} \\times \\pi \\times \\frac{1}{12} \\times \\text{RPM} \\approx \\text{stock diameter (in)} \\times 0.2618 \\times \\text{RPM}", "348b63988c046ca4136b923d2d4654aa": " \\cfrac{\\Gamma \\vdash s = t \\qquad \\Delta \\vdash t = u}\n{\\Gamma \\cup \\Delta \\vdash s = u}\n", "348bb8d83eb1e82c072cf4c9ea2c1d4e": "r \\equiv s \\pmod{p^k}", "348c41254b5e4cf844d7af2a71e996f0": "U_{bias}^{LE}(\\mathbf{Q};n\\Delta t) = \\sum_{i=1}^{n} k_{LE} F(\\mathbf{Q}-\\mathbf{Q}_i)", "348c65128c930fe2773b3112c37f1b53": "v_A", "348c65b1f43b87b3a1404480f422b87f": "\\int_1^\\infty \\frac{1}{x^2}\\,\\mathrm{d}x=\\lim_{b\\to\\infty} \\int_1^b\\frac{1}{x^2}\\,\\mathrm{d}x = \\lim_{b\\to\\infty} \\left(-\\frac{1}{b} + \\frac{1}{1}\\right) = 1. ", "348c6a38b43cfec39120360d36f61a16": "\\mathbf{a}i = \\left(a_1 \\mathbf{e_1} + a_2 \\mathbf{e_2} +a_3 \\mathbf {e_3}\\right) \\mathbf {e_1 e_2 e_3} \\ ", "348c8df53e6eceb923f90ab39367a9da": "G = 1", "348cd5110a66a41c458021763189e402": "\\mu^{h} (E) = \\lim_{\\delta \\to 0} \\mu_{\\delta}^{h} (E),", "348d550ad595457315fd3300fca87416": "a \\times b \\times h", "348d9920beacbdc1e12397e5319ef1d0": " ( (X_0, X_1)_{\\theta, q} )' = (X'_0, X'_1)_{\\theta, q'}, \\ \\ \\text{where} \\ \\ 1/q' = 1 - 1/q.", "348dcdc90d238a0eebc4ed2a3149a089": "P = \\frac{N}{d}", "348dd3d31709d7655609104eca8cb31b": "\\scriptstyle [-1,1]\\times [-1,1]\\subset \\mathbb{R}^2", "348e4abb2499159d9d4d95d23f56d4e3": "\\sum_{\\text{x}}P(x|\\theta)=1", "348f509058a93c3416d19b77cbb90c88": "\\begin{align}\n\\ln\\Omega_{E,\\ell} &= \n\\ln\\left(\\ell^n E^\\frac{n}{2}\\right) -\\frac n 2 \\ln n\\\\\n &= n\\ln\\ell + \\frac n 2 \\ln\\left(\\frac E n\\right)\\\\\n\\end{align}", "348f5b07c785638510e5ba09fc0bbe3f": "\\Omega = \\bigcup_{n=1}^{\\infty}\\Omega_{n}", "348f7be9975694ecc87ebb164180e0c3": "y_1,y_2,\\cdots,y_M", "348fa3c07571c69a554b186b2b03597a": "i\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r},\\,t)=-\\frac{1}{2m}\\nabla^2\\Psi(\\mathbf{r},\\,t) + (\\mathbf{E}(t)\\mathbf{.r}+V(\\mathbf{r}))\\Psi(\\mathbf{r},\\,t)", "34900a821d23d4b5195044c7279ff0e0": "\\mathbf{r}_{X} = \\mathbf{F}_{XX} \\mathbf{X} + \\mathbf{r}^o_X = 0 \\qquad \\qquad \\qquad \\mathrm{(7b)} ", "34900ce5bcfcaf325b5f2bcade625c34": "(\\cdot)^+", "349014eb971b01339c8397eb4ec8dfde": "Q(x, y )", "34901e5ca1263970adc64f9ed0fc4afa": "\\widehat{\\boldsymbol \\theta} = \\widehat{\\boldsymbol \\theta}({\\mathbf y})", "34903401fef0e49bbde668ae3e36d715": "\n \\sigma_1^2 + \\cfrac{R_0~(1+R_{90})}{R_{90}~(1+R_0)}~\\sigma_2^2 - \\cfrac{2~R_0}{1+R_0}~\\sigma_1\\sigma_2 = (\\sigma_1^y)^2 \n ", "34907cdc671a5f20f81595fd6bacdaf4": "\\Box f = 0", "349133b7dc93e440b80fec958eeff0ec": "T_{ijk}=\\frac{1}{2}E[\\partial_il\\partial_jl\\partial_kl]", "3491546fcfdf44d3b7847741e2635e6b": " \\square X^\\mu = \\eta^{ab} \\partial_a \\partial_b X^\\mu = 0 ", "349182dcf78daab119bdaf5da9e1fcac": " M = 6(N - 1 - j) + \\sum_{i=1}^j f_i = 6(4-1-4) + 4 = -2, ", "349188302ddeb135d9dd16648c1577d7": "\\mathbf{} b ", "3491c07cbeb154ffb4c49d5def52cedc": "\\rho(\\theta) \\,=\\, \\frac{1}{1 - \\cos \\theta}.\\,", "3491c16af67f7a95670b33203324b3ed": "\\cos(x) = \\frac{e^{ix} + e^{-ix}}{2} ", "3491efb1666661a0d85f5d5f1df075b6": "B(y,z)", "34922134d9c00a38fbdece296e83e3d1": " x=\\sqrt{R^2+r^2+r\\sqrt{4R^2+r^2}}. ", "349225bef6d7a19ebafd3676bc84a659": " (C)_{i,j} = \\sum_{r=1}^n A_{i,r}B_{r,j}.", "349227e7e78e4b1a7c11cbacedb2f1d3": "p= mv_\\text{cm}.", "349228a6c68a41dbc40c5bb3bf729824": "K_k", "3492566f2611f94747df245d126ef3f1": "2y''+3y'+y=5.\\ ", "3492877c45a066c887d3e97ea20ada50": "V_\\mathrm T\\,", "349289d68b24f17c0b1f2fe3c382539e": "109_{11} \\ ", "3492df25d4a173722f0deff245ac2d28": "(X^*_{\\mathcal{G}})^*", "3492f47dc460a6923e5963b4832f941e": "F_\\text{net} = 0 = m g - \\rho_f V_\\text{disp} g \\,", "34931042dea451d8e9dccc313613618d": "\\sum_{i=0}^{\\infty} f_i", "34934b89bc410a43d2b78e9563cd3db2": "\\sqrt{\\log n}", "349359e6dfdddab6c396d85d7d0ca862": "a\\times b\\neq b\\times a", "349380da5ac6134256069e69776b1679": "J \\circ df = df \\circ j,", "3493b17930f1cbb6cff62b924bb6e30c": "F \\subseteq 2^{N \\times (N \\cup T)^\\ast}", "349411fb9ab0071ae0963d9e3e113a1e": "C_3", "349414ee509ddd32f723991c714a8d1a": "\\tau_{ij} = \\lambda\\delta_{ij}\\partial_ku_k + \\mu \\left( \\partial_i u_j + \\partial_j u_i \\right)", "34943f19309d7c417805baa6efe6c79c": "\\mathbf{N}_{L/K}(l) = k", "34946b20102794bef2340663b442b541": " \\Delta\\omega = \\omega - \\omega_{0} ", "34946e6937dfda9f404c9cd71561317c": "\nv_2^2= \\dot x^2 -2 \\ell \\dot x \\dot \\theta\\cos \\theta + \\ell^2\\dot \\theta^2 \n\n", "349475bf0fe76464ad837adbead7ca36": "(1+x)^{-0.5} = \\textstyle 1 -\\frac{1}{2}x + \\frac{3}{8}x^2 - \\frac{5}{16}x^3 + \\frac{35}{128}x^4 - \\frac{63}{256}x^5 + \\cdots", "3494799cd4278917aa2da74a6c1fbf4a": "-90^\\circ+\\delta < \\phi < 90^\\circ - \\delta", "3494c0daad8a1b48ff74f9b1aa0fc193": "\\mbox{true range} = {\\max[(\\mbox{high} - \\mbox{low}), \\mbox{abs}(\\mbox{high} - \\mbox{close}_\\mbox{prev}), \\mbox{abs}(\\mbox{low}-\\mbox{close}_\\mbox{prev})]}\\,", "3494c45e4b9e2f6d77faea92517cc509": "\\mathop{\\darr}x = \\{y\\in X: y\\leq x\\} = \\textrm{cl}\\{x\\}", "3494defa7a15e5f24299f20275621808": "(Q, \\ell)", "34950c3f8dc1a1977f822ef5e0f5e363": "x^2 - Dy^2 = \\pm1", "3495146876bfa6a2d37332c88819cbfa": "L_0(x) = 1\\,", "349539e1dc9a7564fea7ae210ffe2145": " P_{extractable}^{max}=\\frac{16}{27}.P_{arriving on sail}", "3495c480f7a6e8155df3bbb5af001136": "\\int_{-\\infty}^\\infty \\varphi(x)\\,dx=\\int_{-\\infty}^\\infty \\frac{1}{\\sqrt{2\\pi\\,}} e^{-x^2/2}\\,dx=1 ", "349602f4653af63f4ddf370bd6fa5900": "f_{xx}(a,b)=e^x\\log(1+y)\\bigg|_{(x,y)=(0,0)}=0\\,,", "3496708542a40a2fb11019b22170cf95": "ab > 0", "349705437e8414e21dd715a84024ae9c": "c_1(t)=\\frac{\\Omega_c }{\\Omega},\\qquad c_2(t)=-\\frac{\\Omega_p }{\\Omega},\\qquad c_3(t)=0,", "349833159c453697e5b4cf832fed5ed2": "\\chi_P\\,", "349835402a79335027cddf795a15abaf": " Q_d =Q_1-Q_2", "34983e9e94620e13a64af1bbcbba2ced": " \\Omega^2(k)\\, =\\, \\left( g\\, +\\, \\frac{\\gamma}{\\rho}\\, k^2 \\right)\\, k\\; \\tanh\\, ( k\\, h ),", "3498513e0f529878ecd072c200f7b649": "\\sigma/\\sqrt{N}", "349866fa36a74db44077912783f87317": "V^{-1}X", "349882226c3accf77589d5ccf6dd5d97": " \\psi(x) \\sim \\exp(-x^2) ", "3498a1a59ec3931fc73a353f2f5cda81": "g.e", "3498e3ae94cd45e2cbd9a6199210716d": "\\nabla^2 u=0", "3499097209bd3e8a8db7224e5c6d2d25": "\n\\overline{A} = \\sum_{\\alpha=1}^{D}|c_{\\alpha}|^{2}A_{\\alpha \\alpha} + i \\hbar \\lim_{\\tau \\to \\infty} \\left [ \\sum_{\\alpha \\neq \\beta}^{D} \\frac{c_{\\alpha}^{*}c_{\\beta}A_{\\alpha \\beta}}{E_{\\beta}-E_{\\alpha}}\\left ( \\frac{e^{-i \\left (E_{\\beta} - E_{\\alpha} \\right )\\tau/\\hbar}-1}{\\tau} \\right ) \\right ].\n", "3499126e390da5dd016fb2bae4a8a691": "S_1(c)\\!\\ =(3, c, c^2-2c^p) ", "349918071f500ebd74d9a041c003c670": "\\! V", "34996f955909e5277bc8db5c16a03bcf": "2^b-M \\leq r < M", "34999b3844747f7ae0be7ab5675e633a": "S(x)", "3499a292dcf738391e84e80137c715ae": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ +0.01\\end{smallmatrix}", "3499e0a9e508ccd7d19fb993c1980aac": "f_Y(y; \\theta) = \\frac{2\\theta}{\\pi}\\exp \\left( -\\frac{y^2\\theta^2}{\\pi} \\right) \\quad y>0", "349a0c15ff6827cf32fedd1c144ea4b3": "xy = yx,", "349a25432dfc09c35455d713f3702d3a": "|R_1| \\leq \\frac45\\frac{1}{2r+1}\\frac{1}{2^{2r+1}}; ", "349a5708e331986d04a7c6c7f7ae28a1": "\\lambda = \\frac{\\det(A)}{\\det(A+B)} = \\frac{1}{\\det(I+A^{-1}B)} \\sim \\Lambda(p,m,n)", "349a5b8a3443d55f08b3f5dda8cafe82": "\\hat{x}_{m+1}", "349b08efa36b33be5573212961ec75fe": "C_0=1", "349b2e517d008bea9b7741e070f05912": " {a\\!\\!\\!/}^\\mu \\rightarrow {\\Lambda^\\mu}_\\nu {a\\!\\!\\!/}^\\nu. ", "349b324b622753911d200bdb830577d8": "\\begin{smallmatrix}M_\\odot\\end{smallmatrix}", "349c076aabfe32f8e21f752d9ea4d0dd": " a*b + a*c ", "349cc9214579102330fd4e9b8915a09f": "\\sigma_{21} = \\frac{d \\ln (\\frac{x_2}{x_1}) }{d \\ln MRTS_{12}} = \n \\frac{d \\ln (\\frac{x_2}{x_1}) }{d \\ln (\\frac{a}{1-a} \\frac{x_2}{x_1})} =\n \\frac{d \\ln (\\frac{1-a}{a}\\theta) }{d \\ln (\\theta)} = \n \\frac{d \\frac{1-a}{a}\\theta}{d \\theta} \\frac{\\theta}{\\frac{1-a}{a}\\theta}=1\n\n", "349d3556ce5e1452fd9bff0077ccb1ae": "G_n(\\omega) = \\left | H_n(j \\omega) \\right | = \\frac{1}{\\sqrt{1+\\varepsilon^2 T_n^2\\left(\\frac{\\omega}{\\omega_0}\\right)}}", "349d5fa743e4acfde8c43f691ce5143f": "D_i(g_t)= \\frac{d}{dt} g_t ", "349d6e88bd9bc3475f869af83eb1b2d8": "\\lambda\\overline{\\psi(y)} -\\int_a^b \\overline{\\psi(x)} K(x,y) \\,dx=0.", "349d999dfdf7428ce9a362f6f6b6083a": "\\displaystyle 144(x^4+y^4)-225(x^2+y^2)+350x^2y^2+81=0.", "349dd097cae1bf7de361b89b77f76b6d": "p_n > n \\cdot \\ln n. ", "349dda33330424d479265bc9097d255c": "v_j = 0", "349e03de465b11881df66caa042b1f6d": "(Z,A)", "349e3d218bb68c21f3017f8834f2f4b1": " E(\\bar{\\mathbf{Q}}) ", "349e9052b656adda571f2e09da9c701d": "\\forall \\alpha", "349ea6bb4079e3c2f876e322361b8dca": "E_0(x,y)=x+y", "349ed502ed09accb4e29a594b205ad11": "\\scriptstyle f(n)=\\lfloor rn\\rfloor-n", "349f0205f285c0b096d7587b8b14f58e": "\\underline{f}(\\left\\langle A \\right\\rangle_r) = \\left\\langle\\underline{f}(A)\\right\\rangle_r ", "349f0d0958b8fbb4c6520e16c1e69a9f": "\\{x\\in X \\,|\\, f(x)\\in I\\} = \\bigcap_{k\\in \\mathbb{N}} \\{x\\in X \\,|\\, f_k(x)\\in I\\}.", "349f11fa093eeba260a5e650688d7a7e": " \\mathcal P:=K^2 \\cup ", "349f6f9b447d6c11d4ec1b5aba2c6e78": "3\\eta=12\\beta-3\\gamma-9+10\\xi-3\\alpha_1+2\\alpha_2-2\\zeta_1-\\zeta_2", "349f9884ef47bf446773eb39396def7e": "e(x) = 0", "34a02d510cceb1e17d1e7d94dc811721": " Y_t = \\mathbb{E} ( \\delta(W_1) | F_t ) ", "34a085d49a08720a8f25b3891008dd93": " \\ c_{01} = V[x_0,y_0, z_1] (1 - x_d) + V[x_1, y_0, z_1] x_d ", "34a10459dd6cb85a4245b9f02833e318": "\\Theta=\\tfrac{1}{2}\\pi", "34a1072d1ec654d1dca08f65148e5229": "\\;^-T^{IJ} := {1 \\over 2} \\Big( T^{IJ} + {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} T^{KL} \\Big)", "34a1174217d5525c1da1ca77df03fbf1": "A_i\\subset\\{1,2,\\dots,n\\}", "34a12f98fc464a0e40d8f2e3890e0035": " \\mathbf{F} = \\nabla W ", "34a130aa4c9bc6cb437621b6e99c2dc9": "y(\\phi)", "34a1325ef0873a7c93ec5d54b4344c6e": "f_r = f_0 \\frac{V \\pm v_r}{v \\mp v_0}\\,\\!", "34a1daa525357407b6d649a9abf72941": "\\mathcal{F}.", "34a226c688852123e77cff99b12225d0": "\n\\begin{align}\nh\\left(f\\right) & = -\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\cdots\\int_{-\\infty}^\\infty f(\\mathbf{x}) \\ln f(\\mathbf{x})\\,d\\mathbf{x},\\\\\n& = \\frac12 \\ln\\left|(2\\pi e) \\boldsymbol\\Sigma \\right|,\\\\\n\\end{align}\n", "34a22863872328ac1ba691a27767216e": "\nF(t) = Z_G(1+t,1+t^2,1+t^3,1+t^4) = \\frac{(t+1)^6 + 9 (t+1)^2 (t^2+1)^2 + 8 (t^3+1)^2 + 6 (t^2+1) (t^4+1)}{24}\\,\n", "34a228eb6b3c385a4ea4a1f98ba2c3ed": "\\frac{\\exp\\left(\\frac{ \\lambda t}{1-2t }\\right)}{(1-2 t)^{k/2}}", "34a24d822996023ac88f7c6b02d90547": "\\alpha_i > 0 ", "34a24ebda2f0e31a548b7623905d30f5": "\\mathbf{q}~=~-k~\\nabla\\theta ,", "34a2b0194990d23a233298c101b40d5c": "n < 2^w", "34a2c30d9054827e5d4a3f7f1fd896fa": "\\mathbb C\\otimes_\\mathbb{R}\\mathbb C", "34a357778c881f2d84ba125200d47d83": "\\binom{x}{n} = \\frac{x(x-1)\\cdots(x-n+1)}{n!}", "34a35a536cde3dc5779c160c3b3bb92c": "\\scriptstyle{}\\qquad{}\\times(1-r_{\\rm IT})+r_{\\rm M}\\Big)\\times P/12.", "34a36049de88a6d27fbd1064354c288e": "T=P^{-1}", "34a49451d39cf1d843b3e6491a08d5d8": "\\langle x,y \\rangle = 0 \\Rightarrow q(x) + q(y) = q(x+y).", "34a4b5496da6483ccc231089ff4f1652": "\\nabla\\phi = \\left[\\frac{\\partial \\phi}{\\partial x},\\,\\frac{\\partial \\phi}{\\partial y}\\right]^\\mathrm{T}, \\quad\n\\nabla\\times\\phi = \\left[\\frac{\\partial \\phi}{\\partial y},\\,-\\frac{\\partial \\phi}{\\partial x}\\right]^\\mathrm{T}.", "34a4fc7f932bcfb7f33cbeb60dd430ef": "w\\,R\\,u\\land w\\,R\\,v\\Rightarrow u\\,R\\,v\\lor v\\,R\\,u", "34a55ed875b8827dacbb2689a090c63e": "R(t) = \\left| \\frac{(232\\cos(t)^4-97\\cos(t)^2+13-144\\cos(t)^6)^{3/2}}{6\\cos(t)(8\\cos(t)^4-10\\cos(t)^2+5)} \\right|\\,.", "34a5e45779ec15d8da27571fb967b676": "d_{3/2,-3/2}^{3/2} = - \\frac{1-\\cos \\theta}{2} \\sin \\frac{\\theta}{2}", "34a6366f7f8694edee8fa2cf5c37da57": "I_n = (p,v_1,\\dots, v_n)", "34a656e51b1b7cec57fc92fa5eb6f1b0": "\\mathbf{v} = \\frac{1}{\\|u\\|^2}\\mathbf{u}", "34a6a2a9c9b8ff1f5d16fdaed3875d0e": "\\widehat{\\mathbf{L}}\\cdot\\widehat{\\mathbf{S}} = \\widehat{L}_x \\widehat{S}_x + \\widehat{L}_y \\widehat{S}_y + \\widehat{L}_z \\widehat{S}_z \\,. ", "34a6c8bbcecb7d1fd43c621a3c97f4df": "b=2j-2k-a\\,", "34a6e19625040089f3a91548a06321d1": "|\\underline {A}| \\leq 1, |\\underline {B}| \\leq 1 \\qquad (5)", "34a748e5ce7c869cbdee613a33863d0c": "N_j", "34a78552d15bd27667589a0bfa879e98": "y_{i+1}", "34a793f93975934ce6a945f916a619d9": "X(\\omega) = \\operatorname{rect} \\left( { \\omega \\over 2\\pi W } \\right)", "34a7dfd7c0641c8db387b73c1605fc37": "N = 2^3 - 1 = 8 - 1", "34a817fa5dc1d91fdb6b39d2f258a690": "D(f) \\leq Q_1(f)^2Q_2(f)^2", "34a818566f3eb46431b3c40ff60060d6": "V_s(\\vec r) = V(\\vec r) + \\int \\frac{e^2n_s(\\vec r\\,')}{|\\vec r-\\vec r\\,'|} {\\rm d}^3r' + V_{\\rm XC}[n_s(\\vec r)]", "34a823f35e447fa6eb1d7610cc4e0d65": "E^0(\\mathbb{Q}_p)", "34a83bcbf460b6459fcbd1fad789f29c": "\\lambda_1 > 0, \\lambda_2 = 0", "34a88e1120baf1873aee8c4535bd6040": " E_{eq,K^+} = \\frac{RT}{zF} \\ln \\frac{[K^+]_{o}}{[K^+]_{i}} , ", "34a8cf13ca3174e687595972dbc4d4e8": "R(\\lambda) = (\\lambda I - T)^{-1} \\,", "34a8d3abd58270cda650cd67632457da": "u_i = \\overline{u_i} + u'_i", "34a924e165496588cf9b504e3a64b97e": "a_i\\;", "34a92f7fbd06a29d3316790ddeb95478": "\\begin{align}\n U_2 &= U_1(1 - a)\\\\\n U_4 &= U_1(1 - 2a)\n\\end{align}", "34a9ac4aabd091b451100f4987f63b18": "R\\textbf{u} = \\textbf{u}", "34a9d4e202d18a4688af711f0f92559c": "\\mathbf{r} = \\mathbf{r}(\\mathbf{q}(t),t)", "34aa1dabb7f04d192f7404a81a85592c": "I \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_V l^2(m)\\,dm = \\iiint_V l^2(v)\\,\\rho(v)\\,dv = \\iiint_V l^2(x,y,z)\\,\\rho(x,y,z)\\,dx\\,dy\\,dz \\!", "34aa3d3711994a8cd50b73a281ac94c4": "\n\\,\\!V_C(t) = V\\left(1 - e^{-t/RC}\\right)\n", "34aa46c565e77de80edc4ccf49845021": "\\mu_0\\boldsymbol {J=\\nabla\\times B}\\ ,", "34aa9171f5a0835a22e349668c34df74": "f_{ref}=1.023", "34aada0951f6af5f9444bb0a3b958259": "x\\in[-\\infty,+\\infty)", "34ab125d6124dcce1085979e61a19377": "V = \\frac{1}{3} A_0\\,h \\,", "34ab3894db40306d383cdbf5754e5cfb": "d{\\Theta}=\\cfrac{1}{K}dp", "34ab45460f7b8382193ac14fc8557d47": "s_\\pi = O(M(n)\\log^2 n). \\,", "34aba90f307cb2722e58c6aba180ac65": "E\\subseteq\\Gamma\\subseteq\\mathbb{R}^{2}", "34abc86a1de9dcfeff8cac4362df848f": "V(x) \\ge 0", "34abe092aa7ca301595d9689c2a8beb6": " \\bar { \\mathcal{N} }(\\epsilon) = \\frac{g(\\epsilon)}{e^{(\\epsilon-\\mu) / k T} + 1} ", "34abf3115c85f7e3ddc7a95dfe9539b5": "\nE_\\text{TOT} = \\sum_{i,j} \\varphi(\\mathbf{r}_{j} - \\mathbf{r}_i) = E_{sr} + E_{\\ell r}\n", "34ac11dc83cfa9f0df32d6d461e00c46": "\\mathfrak{L}_{NT}\\,", "34ac519adf549e07e7530a3cfee7d047": " \\left| \\frac{1}{N} \\sum_{i=1}^N f(x_i)\n - \\int_{\\bar I^s} f(u)\\,du \\right|\n \\le V(f)\\, D_N^* (x_1,\\ldots,x_N).\n", "34ad032f9c2fb947998d3de8496fe0d0": " e^{i p q(t+\\epsilon)} \\,", "34ad23f7ec456c5ed792ab47337cfe04": "\\mbox{1 Gaussian year}= \\frac {2\\pi} {k} \\,", "34ad456387dac60a79e04c7ba06cb798": "[Y,Z,X]\\subseteq N", "34ad470de094c8294318405f738f532c": "\\scriptstyle q_e\\,", "34ad635e9915fcdf8b6282fb6a765f67": "\\vec{v}\\ \\stackrel{\\mathrm{def}}{=}\\ \\nabla \\varphi", "34ad8f4108c8712231d220305318e16a": " Dv=\\sum_k e_k\\otimes(dv^k) + \\sum_{j,k}e_k\\otimes\\omega^k_j(\\mathbf e)v^j.", "34adab386c6c4b1ded56d531b3a536a6": "\\|y\\|\\leq\\delta", "34adf7ae2b3595e1baab994434eac0df": "supp(A \\Rightarrow 0) = P(A \\and 0) = P(A)P(0|A) = P(0)P(A|0)", "34ae07e405d0b28a9b1a7f6a224bb4f8": "R_{out} = ( \\beta +1) r_O \\ ,", "34ae1ef723f842ef90c7a1844a757b07": " G = \\frac{E}{2(1+\\nu)}. ", "34ae44a5f2c3279d15de80fe7b8fe26d": "a_1 \\ge 1", "34ae652f8247274abea236d70124bf18": "e_z < 0\\,", "34ae67b8e045538e068a59bc777a05ad": "d(v) \\leq H[i+1]", "34aea25264dd2531bc419456b6109061": "\\int \\frac{dx}{1+ax}=\\frac{1}{a}\\ln(1+ax)+C \\ ,", "34aeaba751f4b633306fa3539220447b": "\\operatorname{im}\\, \\kappa \\times \\operatorname{im}\\, \\sigma", "34af545a62eafc00d314f8547c4ae3ec": "\\ f(0)=0 ", "34af78524b1834fa9cc3dcf02b28acf8": "\\Delta x \\neq \\Delta p", "34afa165180a49550d57923dedd62be9": "\\scriptstyle\\pi(12)", "34afd4595794273f6ab1d424f43e88f8": "K_{m,1}", "34afdf267039eb0060df1a60b9fb57e8": " L_i = \\epsilon_{ijk} r_j p_k .\\,\\!", "34b04797be9e92346630f9451e8aea5c": "\\mathrm{d} X_{t} = - \\kappa ( X_{t} - m) \\, \\mathrm{d} t + \\sqrt{2 \\beta^{-1}} \\, \\mathrm{d} B_{t},", "34b0820aa51a71afa28351b25f3a765b": "m(t) = M\\cdot \\cos(\\omega_m t + \\phi).\\,", "34b0f7e5aee54122a013518b0ec5be43": " 2<\\alpha < 4 ", "34b10c5d71c60b08cbd0752a7140666b": " (k-2)T(r,f) \\leq \\sum_{j=1}^k \\overline{N}(r,a_j,f) + S(r,f),\\,", "34b114cdf159977cdf1665074a107a61": "\\mathcal{K}_{\\infty}", "34b12e12e9c9f199a06bfc4a915c34ec": "{X = X_L - X_C}", "34b17d305fd868ac0bf3eca5a717ea60": "K_m =\\frac{ET + GR - 2FS}{\\left(1 + P^2 + Q^2\\right)^2}", "34b1a90e13078fa351121e708fe8f770": "\\Delta W_{n} = W_{\\tau_{n + 1}} - W_{\\tau_n}.", "34b1b4cba11d09d2c0bedbd35d358782": "G\\times G \\ni (g,h) \\mapsto gh \\in G.", "34b1e557168a6ef635ceb3486982de26": " \\kappa = \\frac{1}{R}.", "34b1fe757202769680bd51219702c22d": " g_jH ", "34b2456c8c7901ea4b2f7dc6f4093baa": "\\vec v \\ast (\\vec e_\\text{1} - \\vec e_\\text{2}) = 2 v \\sin \\varphi", "34b25ac4f8de271395bc1fcc75174627": " a_n = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) e^{-inx}\\, dx. ", "34b28214c3b641e6057d7064be9dde9d": "v \\in A", "34b28ca9b3c996991567fb3c2a797236": "\\scriptstyle 5\\tau", "34b30c44612b8d15bf81498010e290c2": "Lk_o=bp/10.4", "34b35838110c3553557d1e168f33dcb0": "\\frac{1}{2}LI^2 = \\int_V \\frac{B^2}{2\\mu} dV", "34b3a745038e5fc80ea036484ac7413e": "n^2 + n + 41", "34b3ebbc72159090ebd3ea7d953df1b4": "\\pi_{2m+k}(S^{2m})(p) = \\pi_{2m+k-1}(S^{2m-1})(p)\\oplus \\pi_{2m+k}(S^{4m-1})(p)", "34b4396d48e01cb230517c8125019069": " \\displaystyle f=u+iv ", "34b43b3ab3d5b4602008cc9510f55fb4": "U\\rightarrow V", "34b4488fda519191bb56c8cf2639ec62": "d^{(2)}(N_f)", "34b46bebe969b32914dec41aeffb3361": "\\langle v_j,v_k \\rangle", "34b5036a961181df331b7e804467e080": "\\omega = c_0 k ", "34b506d4a8cb0a7bc03701bec2c7691c": "y>0", "34b5277f9a3abca548633dfc40aaf1eb": "\\nabla F(\\mathbf{x}^{(0)})= J_G(\\mathbf{x}^{(0)})^\\mathrm{T}G(\\mathbf{x}^{(0)})", "34b544ed14ed7e2c55ddf7b78cb4f75e": "\\Box(A\\to B)\\to(\\Box A\\to\\Box B)", "34b5635b5ffce9d3214bfe26d4e43378": "2^5", "34b59e8ff6e35059bf4e93ea614bcbab": "y = x/a", "34b606b91595abc898ae75d0ef29d58f": " L_T(\\lambda) = L_r(\\lambda)+L_a(\\lambda)+L_{ra}(\\lambda)+TL_g(\\lambda)+t(L_f(\\lambda)+L_W(\\lambda)) ", "34b72e7753e8a1461011720ecac94d84": "\\sqrt{I}", "34b74f67c345abf2959bfd5c4262a938": "G \\cap M,", "34b755e1aedfdc3e184834e4d7e19391": " = \\max \\left(|x_1 - y_1|, |x_2 - y_2|, \\ldots, |x_n - y_n| \\right).", "34b779c1473f6110a19de9d453ce89ec": "\\Delta\\,v = u\\,\\ln\\left(\\frac{m + M}{M}\\right)", "34b78bf4f9f56c61f7a5265b8a9e81d2": "Hw'=0", "34b7ba4f21aba12dd77d1d78b8f087e2": " \\ell + h \n\\approx \\frac{\\varphi}{\\theta} L \\sin \\theta + L \\sin \\theta \n= \\left(\\frac{\\varphi}{\\theta}+1\\right) L \\sin \\theta\n", "34b828a51f7032ad230913d538b91a72": "\\mathrm d/\\mathrm dt^+", "34b83a6dbd812c8a39f94a18e8c87de3": "a\\, \\cos\\, \\theta(\\boldsymbol{x},t)\\,", "34b8495984f9b8f638fa87ef95affd7d": " \\left(\\frac{\\mathrm{d} y}{\\mathrm{d}x}\\right)^2 = (1-y^2) (1-k^2 + k^2 y^2)", "34b854429000ff6927279cd760b13cc3": "\\mathcal{B}\\subseteq \\mathbb{W}^\\mathbb{T}", "34b886a033dcdd892f8084920ab93f7d": "\\left\\{\\begin{align}\n\\frac{\\partial}{\\partial z_1} &= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x_1}- i \\frac{\\partial}{\\partial y_1} \\right) \\\\\n&\\qquad\\qquad\\vdots \\\\\n\\frac{\\partial}{\\partial z_n} &= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x_n}- i \\frac{\\partial}{\\partial y_n} \\right) \\\\\n\\end{align}\\right.\\quad,\\quad", "34b88aa3217225c13d0958cf2f256bc6": "V_{\\text{out}}", "34b8c3ca588dd2d3d8a228a011367540": "y^2 = x^3 - 3kc^{2r}x + 2kc^{3r},\\text{ where } k = \\frac{j}{j-1728},", "34b8ca2b7ad19a2f5746ef4bb8c07a49": " v_{i} = R_{i}v_{i-1}\\ ", "34b94b9a4ab42a804aa6c238d4164cd3": "Ext_{A}^p (R,A)", "34b94fd36364f30ad34c4e1ffd260a2f": "\\Delta f(p) ", "34b979a5a012e6db97dbd03340a9b501": "\\mathfrak{f}\\left(\\mathbf{Q}(\\sqrt{d})/\\mathbf{Q}\\right) = \\begin{cases}\n\\left|\\Delta_{\\mathbf{Q}(\\sqrt{d})}\\right| & \\text{for }d>0 \\\\\n\\infty\\left|\\Delta_{\\mathbf{Q}(\\sqrt{d})}\\right| & \\text{for }d<0\n\\end{cases}", "34b990a88b7e9adbed8a73c9dd92a12c": "(x - h)^2 = 4p(y - k) \\,", "34b9a93124aee3915fd882014e840461": "{ 1 \\over 1+(x/\\alpha)^{-\\beta} }", "34b9e4d8ea689a696bd247dece5acde3": "\\mathrm{Diff}^k(X)", "34babf5e55f43ff9143ad25362b3f7b4": "ds^2 = r^2 \\left[ d\\psi^2 + \\sin^2\\psi\\left(d\\theta^2 + \\sin^2\\theta\\, d\\phi^2\\right) \\right]", "34bad392b02f670bd68cf9e2df2daf5a": "M_1=\\frac{-1.3515-1.7703x+5.9114y}{0.0241+0.2562x-0.7341y}", "34bb0832483731d880211de222e4169a": "-W = \\Delta \\mathrm{KE}_{system}+\\Delta \\mathrm{PE}_{system}+\\Delta U_{system}", "34bb3164c088946a3be9ccd1263a42c0": "y_1(t),\\,", "34bb423fec7d6b3611567ccf23fcc103": "P(x)=ax^3 + bx^2 + cx + d", "34bb61e804f27408b486700049a413e4": "|x_i|\\leq |y_i|", "34bb6b889c39a306abb1559aa1303465": "\\mathbf{v}_k=(v_k,\\underbrace{0,\\cdots,0}_{n-1})\\,\\!", "34bbbcfd69d83f56ced6a553d7db785d": "\\sqrt 2=1.4142", "34bbe9a3757da829f2bf1a8a14802919": "\\mathrm{NH_2Cl + Cl_2 \\longrightarrow NHCl_2 + HCl}", "34bc13d2928512ee1afa8453feccdf0c": "\\textstyle\\exp(X)=\\sum_{n\\in\\N}\\frac{X^n}{n!}", "34bc2c764af2b042c3c54570430c223f": "Q=n^\\nwarrow.n^\\searrow(~)", "34bc56017bdbf8b7a06d22b88d5b8e78": "\\deg(D_1+D_2)=deg(D_1)+deg(D_2)=16.", "34bcd315be1cf3d831e34eae3f30a335": "\\boldsymbol{\\mu} = \\boldsymbol{\\mu}_L + \\boldsymbol{\\mu}_S", "34bcd8604a4894d0776939aa97fec155": "\\scriptstyle n\\geq 2", "34bd15b4c342798e2f0d5565621b9c69": "\\pi_k(\\operatorname{Sp})=\\pi_{k+8}(\\operatorname{Sp}) ,\\ \\ k=0,1,\\dots . \\,\\!", "34bd2cde498b1b08bd7e9781f7731e9b": "10^{-b_0}[OH^-]_0/K_{w^{ }}", "34bd38a22fedd4b7ee18d43bcf9c6cf3": "q_{2}=x_{12}+x_{22}", "34bd3b57e66968d2dfdd421854aef1c7": "x_\\mathrm{rms} = \\sqrt {\\langle x^2 \\rangle} = b \\sqrt N. ", "34bd467615bea640f5614113c62b861f": " Z = \\sum_{i=1}^{100}{1 \\over (i + a)^2}", "34bd4eb4d481e847eb7410ec149a4150": " E_j-E_i <2\\epsilon ", "34bd505f270d455a7fb60b80b74f9168": "(\\mathbf{y}-\\mathbf{x})'\\mathbf{A}(\\mathbf{y}-\\mathbf{x}) + (\\mathbf{x}-\\mathbf{z})'\\mathbf{B}(\\mathbf{x}-\\mathbf{z}) = (\\mathbf{x} - \\mathbf{c})'(\\mathbf{A}+\\mathbf{B})(\\mathbf{x} - \\mathbf{c}) + (\\mathbf{y} - \\mathbf{z})'(\\mathbf{A}^{-1} + \\mathbf{B}^{-1})^{-1}(\\mathbf{y} - \\mathbf{z})", "34bd610213066af518c399eda1251e27": "\\dot{\\mathbf{x}}(t) = \\mathbf A \\mathbf{x}(t) + \\mathbf B \\mathbf{u}(t) + \\mathbf{w}(t)", "34bd7ec2549984a1ecce6eb41e66eeb9": "M(t) = \\exp\\left(\\int_0^t r(s) ds\\right)", "34bd8f47ae1eec88e4a3135e4ff7f571": "\\scriptstyle \\sqrt 2", "34bd9da3488e99d59cb78074ac5b76bb": "T_n = ", "34bda91b89b5293fef0bc04b41d98411": "\\int\\!R(x,\\sqrt{ax^2+bx+c})dx ", "34bde88cef2b543b2da22f8b342729e3": " \\left(H_0 + \\lambda V \\right) |n\\rang = E_n |n\\rang . ", "34be0023827e06f8e5a1c7e775fa7a11": "x = 0.99R\\lambda \\qquad y = 0.99R\\ln \\tan\\! \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right)\\qquad k\\; = 0.99\\sec\\phi.", "34be4359ecb4da3c1135c599103e4d71": " { n! \\over s^{n+1} } ", "34be43dd2c6255a6952bc0fb8c13c613": "\\mu = {{m_e m_p} \\over {m_e + m_p}} = \\frac{m_e^2}{2m_e} = \\frac{m_e}{2},", "34be6db3665d3ba336b6a8de3abe5d48": "\n\\mathit{LCSuff}(S_{1..p}, T_{1..q}) =\n\\begin{cases}\n \\mathit{LCSuff}(S_{1..p-1}, T_{1..q-1}) + 1 & \\mathrm{if } \\; S[p] = T[q] \\\\\n 0 & \\mathrm{otherwise}.\n\\end{cases}\n", "34bef7898e41af1de2ec3cee9a10d7fc": "\\qquad\\mathcal{O}_{[(1234)]}\\;", "34bf0522e62b696431d4229a7ac6e1ec": "S(\\mathbf{q}) = \\underbrace{\\frac{1}{N_p} \\left \\langle \\sum_{jk = 1}^{N_p} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_{j} - \\mathbf{R}_{k})} \\right \\rangle}_{S_1(q)} + \\frac{N_c - 1}{N_p} \\left \\langle \\sum_{jk = 1}^{N_p} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_{1 j} - \\mathbf{R}_{2 k})} \\right \\rangle", "34bf16770508ada6db26cfc5d9361896": "r_l", "34bf5a2e5d86e5762a5b39e61aa13a68": " A = 4 \\pi r_0^2", "34bf97f325fd8a759855ca33f4ca8936": "(\\boldsymbol\\beta - \\hat{\\boldsymbol\\beta})", "34bfa19069fb26d553a9bdbdfc972877": "S_i = -\\frac{Q_H}{T_H}+\\frac{Q_a}{T_a}.", "34bfbfa4f883179ef5aceffaf55b1245": "\\mathbf{x}_\\mathrm{A} =\\mathbf{X}_\\mathrm{AB} + \\sum_{j=1}^3 x_j \\mathbf{u}_j \\ . ", "34c058a8e99291451a64259bfc77dea5": "A_{reduced}", "34c0c3ba4843bbef17f5269b39855d1e": "\\frac{1}{-2 - 2} = \\frac{3}{-2 + 2} - \\frac{6(-2)}{(-2 - 2)(-2 + 2)}\\,.", "34c0ef16ec24c7e6d30165ace7da5e46": "S^{4m+2+k} \\to S^{2m+1+k}", "34c111424ac89412b4b4d3cc563c78e2": "f(x,y) = 0", "34c11382f483a66695d2e7e7396ff14e": "\\langle x_a,t_a|x_b, t_b\\rangle", "34c13048034846cb8e5d199caa253b63": "L | e_i \\rangle = \\lambda_i | e_i \\rangle \\, ; ", "34c140a4ff5188e027ac5fa3dc5646e4": "\\mu\\in\\sigma(A)", "34c14119d1fb38e150b02987816e0513": "v_{(G; c)}(\\{2\\})=5", "34c15e33568dc563c4bdc95f7b953183": "(a,b)\\longmapsto a\\cdot b", "34c17382c598968b6923e9e1acd2fc87": "1/\\mathrm{poly}(|x|)", "34c17740aed209bbf0c74c3aba0a65ff": " \\begin{matrix}\n\\gamma_a^2 &=& +1 &\\mbox{if} &1 \\le a \\le p \\\\\n\\gamma_a^2 &=& -1 &\\mbox{if} &p+1 \\le a \\le p+q\\\\\n\\gamma_a \\gamma_b &=& -\\gamma_b \\gamma_a &\\mbox{if} &a \\ne b. \\ \\\\\n\\end{matrix}", "34c19c098c25e16557d74b9237ea880e": "\\pi_1(A \\cap B)=\\pi_1(S^1)", "34c1a1dabb26a10d46ae89b32ec5fad1": "x^0 \\cdot 2^0 = 1", "34c1d173d638ceb8fb5bec184c055549": "k_i", "34c1e85d7325c95bf7a549cb59a507e9": "F_r = {d_f \\over L}mg + {h_{cm} \\over L}ma", "34c2832f71dd85986b9dfdca241710a9": "\\Omega^2(TM)= \\Omega^{2,0}(TM)\\oplus \\mathrm{im}(\\tau)", "34c2a5f99941921ac08c4c8aca5077de": " z = r \\ \\operatorname{cis} \\ \\varphi. \\,", "34c2ba9207ccfaf3a70b5b41e36099cd": "(\\overline{A} \\vee \\overline{B} \\vee C) \\wedge (A \\vee \\overline{C}) \\wedge (B \\vee \\overline{C})", "34c2d3dd390eb9117c17bc860cfeecce": "j(q) -744 = {1 \\over q} + 196884 q + 21493760 q^2 + \\cdots.", "34c303098dea59e9bac734a8ea7e3fc2": "Z = \\frac{Z_0}{\\pi \\sqrt{\\epsilon_r}} \\, \\operatorname{arccosh}\\left(\\frac{D}{d}\\right)", "34c3a19117bbd3505169959fa659cf4c": "10^k + 1", "34c3c5c884508e35b64a9ce7ea727831": "X_H=-\\sum a_{ij} q_i\\partial/\\partial p^j. ", "34c431042417e9e86177acfc6ce9ee68": " P=DEC_{k_1}(DEC_{k_2}(C)) ", "34c4b5b10acfece7e0b87142464493cb": "R_g \\sim N^\\nu", "34c4b912904c74bc8818b7ed0add54da": " \\cosh(\\alpha t) \\cdot u(t) \\ ", "34c4c6a054d77c000eacbc222a488de1": "\\boldsymbol\\phi", "34c4df1370420cb417634638ebd886a4": "q \\succ _{\\rm w} q^\\prime", "34c4e5f010708439074be4b8a050d7c3": " g : \\mathbb{R}P^2 \\rightarrow \\mathbb{R}P^2 ", "34c515c58b0999b9a1d8ef96810d8f6b": "a(t) = (4\\cdot 2^t-(-1)^t)/3", "34c51c160df2e1f4459e76f3e648492d": "A\\in\\Gamma", "34c52e1b9c5220e776e839f3e42c8036": "\\rho_{QR}' = \\phi[\\rho_{QR}] \\quad", "34c552ae5e83d9eb44b628b7812a706e": "p\\Vdash\\phi\\iff p\\leq||\\phi||", "34c57d926366b010dce0d376e1ae4e28": "S_\\pm = S_x \\pm i S_y ", "34c5a3a001c0eedf2a2337760c70a5a9": "\\displaystyle{\\mu=|A|^{-1/2} \\sum_{x\\in A} Q(x).}", "34c5e365d4b81b797ad8e2ffda97ac9a": "O(n^2 M(n^2)/\\log{n}) = O(n^{4+o(1)})", "34c616121f1a80d7349402343d3b2c42": "\\langle X_ix_k\\rangle=\\delta_{ik}", "34c640901c7b881cce2459ce6926ca0e": "\\gamma=\\frac{P_s-P_d}{P_s+P_d}\\ ,", "34c6416b511dde27bc3e33ab75856109": "\\frac{d}{dt}\\, dA = H \\,dA,", "34c6ad06da4d24c8d3ba96a95ff74511": " (-1,0),", "34c6e2a0161c8e0fc9d38dbb9cac7a1e": "L=\\sum_{i+j\\leq n}r_{i,j}(x,y)\\partial_x^i\\partial_y^j", "34c6f0c13e5f280703cc730b532a7f2f": " (1, 0,\\dots , 0) \\in \\mathbb{Z}^n ", "34c832707f5aedb4f728a43fb1e0fa7e": "pH_{i^{ }}", "34c83e765b28da1b77c988ccc43b0860": "{I}_{0}=D\\left(b\\right)f\\left(b\\right)", "34c86bbef54cb567db3fb47af942ad7e": "\\bar\\lambda_m = \\frac{M}{L}", "34c96d0c0998ce24e8cb86d819d0181d": "\\sqrt{n}log n", "34c9e4a33b1b9a1eeb65aca5426fa50d": "A= \\pi r^2.", "34ca05b5eb722e09a4f17bd21b7bfbfe": "\\lim_{n \\to \\infty \\atop m\\to \\infty} \\sum_{k=n}^{n+m} a_k = 0.", "34ca240edac6a63d2f7c1a3e67114267": "\\delta(q,a,x)\\,", "34ca6e91743267c3a0df0fd904665148": "x+y=-a", "34ca70ae08fd2631c5b4d564b5d7fda9": "\\begin{pmatrix}\n1 & -1\\\\\n1 & 0\n\\end{pmatrix}", "34cadf112ba1bb1f473d1a0340c3cff1": "\\Gamma_a^i = \\epsilon^{ijk} \\Gamma_a^{jk}", "34cb00f31a78a5c9c63460e5ba711930": "f_Y(y; \\sigma) = \\frac{\\sqrt{2}}{\\sigma\\sqrt{\\pi}}\\exp \\left( -\\frac{y^2}{2\\sigma^2} \\right) \\quad y>0", "34cb12641b5164288f594f7eedf08ef7": "\\scriptstyle{ \\{ i\\} }", "34cb6977b0c678de5f3d660980ee7e73": "\\tan(\\beta l)", "34cb9142f9eeec16caad583b4cb07972": "d_A(z) \\propto \\int_{0}^{z}\\frac{dz'}{H(z')}\\!", "34cbbfbf2100f04cffadd5ddfcac005f": "M[f]_{x_0} = M[g]", "34cc1a50a7058e610c8d8db3ca61eed6": " g_s(K) \\ge ({\\rm TB}(K)+1)/2. \\, ", "34cc208e1cc7270498d62076f690c748": "\\int_{0}^{T} g(t)\\, dt", "34cc40d40336db23c57f5b6257d73b4e": "0 < T \\leq \\infty", "34cc81047696064f935ea73560bf35a5": "\\begin{align}\nx_1 &= \\frac{-b - \\sgn (b) \\,\\sqrt {b^2-4ac}}{2a}, \\\\\nx_2 &= \\frac{2c}{-b - \\sgn (b) \\,\\sqrt {b^2-4ac}} = \\frac{c}{ax_1}.\n\\end{align}", "34cca8e8f9660b406743786520e5ca13": "(r,\\phi),", "34ccfa997f6f946333a1d428d779b1c7": "\\{x_1, ..., x_n\\}", "34cd2e5700b4d4657378b8b78104538d": "\\operatorname{Out}(S_n)", "34cd4d92d4bda37875363a3b65d04dbf": "X_L = \\sqrt{(50)(50-1000)} = \\sqrt{(-47500)}= j\\, 217.94\\ Ohms", "34cd4fd6708e507c6b3caa3cd51be416": " \\tilde{n} = 0 + .01 j ", "34cd7ed2c60df96ca5e466a63b470982": "\\textstyle{\\frac{1}{2}}\\left(-p \\pm \\sqrt{p^2 - 4q}\\right)", "34cdc21534f773e5bff0717179ee2d8b": "\\omega = i\\partial\\overline{\\partial}\\log |\\mathbf{Z}|^2", "34cdf49fb4e244c7bfe8497485e8e709": "E= - \\tfrac{1}{2} \\sum_{ij} J_{ij} S_i S_j - \\sum_i h_i S_i", "34cdfaad90c05dccc70542844f5b150d": " r = \\frac{h}{m c} = \\sqrt{\\frac{2G h}{c^3}} ", "34ce36fd20510577b276b69366a85fbd": "\\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}\\left(x-x_{j}\\right) = \\frac{\\prod_{1\\leq j\\leq n} \\left(x - x_{j}\\right)}{x-x_{i}} = \\frac{p_{n}(x)}{a_{n}\\left(x-x_{i}\\right)}", "34ce429b91b70f9572e2e16a78bb12df": " = \\{ n (\\ln(r) + (\\varphi + k2\\pi) i) | k \\in \\mathbb{Z} \\}", "34ce7224526c6bc4eed7ccd32f22eaad": "= 2u_{1}(\\theta_{1} + u_{2}dx) - \\phi(x,u,u_{1},u_{2})dx + u_{2}(\\theta + u_{1}dx) \\,", "34ce9a47bfa59a9c809e4b5eda5adc6e": "\\frac{1}{2\\pi}\\,\\frac{\\sinh(\\gamma)}{\\cosh(\\gamma)-\\cos(\\theta-\\mu)}", "34cef76ef97069057045ad378db5abff": "S''_{xx}(x^0)", "34cefde33c245bef44f991511cf52150": "\nf(x,y) = \\begin{bmatrix} \\ 0.85 & \\ 0.04 \\ \\\\ -0.04 & \\ 0.85 \\end{bmatrix} \\begin{bmatrix} \\ x \\\\ y \\end{bmatrix} + \\begin{bmatrix} \\ 0.00 \\\\ 1.60 \\end{bmatrix}\n", "34cf1539789524a94e58d1e2ef5e364a": "q_1,\\ldots,q_c \\neq 0 \\mod p", "34cf1b5997a578b1dc714807dc945435": "\\Omega(n)\\,\\!", "34cf51797a60c391560f53f55d3e69b1": "\ndet^{column}\\begin{pmatrix} \nA & B \\\\\nC & d \\\\\n\\end{pmatrix} = \ndet^{column}( A)det^{column}(D-C A^{-1}B) = \ndet^{column}( D)det^{column}(A-B D^{-1}C). \n", "34cfaefa752b883f15647b163ed92b6e": " L(\\theta,\\widehat{\\theta}) = \\begin{cases}\n a|\\theta-\\widehat{\\theta}|, & \\mbox{for }\\theta-\\widehat{\\theta} \\ge 0 \\\\\n b|\\theta-\\widehat{\\theta}|, & \\mbox{for }\\theta-\\widehat{\\theta} < 0\n \\end{cases}\n", "34cfe56d86a59e30b75458b41133a975": " I_{xx},\\,I_{xy},\\ldots", "34cffd88964d6a99e7f5aef95d8cfe05": "h=\\|y-x\\|", "34cffde2351756df8c4fc158b280f751": "\\bold{w}=\\bold{0}", "34d0757dcf97d9e1599e0bac00aebfd7": "H/9 = 9 \\cdot \\frac{H}{IP}", "34d07aa3d6bbbbb15282175e9718b63b": "(xy)^m", "34d0ba55a98dede12a2d87f45a65005a": "{R_{\\mathbf{B}}}^2 = e^{\\frac{\\mathbf{B}}{2}}e^{\\frac{\\mathbf{B}}{2}} = e^{\\mathbf{B}},", "34d103de4b000af0b13830561fd0c350": " |\\tau_n| \\le \\tfrac1{12} h^3 \\max_t |y'''(t)|. ", "34d13d172597f98a0a9fa841c364ed20": "x^{-a}", "34d147aeeafeda41b984b401e9908b19": "E_n = -h c R_\\infty / n^2 ", "34d174a0bed140da9478723823b0cf71": "z_1=(y+w)^2", "34d21e1cddee73c829a86c274f9ef287": "\\displaystyle(\\sigma(e^{X})v,v)= \\sum_{n=0}^\\infty (\\sigma(X)^n v,v)/n!", "34d232cc95101c2c3d171624d2bc07ae": "\nr = \\sqrt{\\frac{r_1^2+r_2^2-2a^2}{2}}\n", "34d24a76722fbf13301507439c71b094": "f(x)=|x|^{k+1}", "34d27a574325c99290a769794173ee5f": "P_1 \\sqsubseteq P_2 \\quad\\text{if and only if}\\quad \\left[ P_2 \\Rightarrow P_1 \\right]", "34d31963a10c461a069431ea44976d6a": " w(p) = \\sqrt \\frac{\\pi}{2} \\sum_{k=0}^{\\infty} \\frac{d_k}{(2k+1)}(2p-1)^{(2k+1)} ", "34d33ec4e755dca31f7a400a6e4cf04a": " \\ u_{1}", "34d3c03c70d81bbeea97f656b7f0cb3c": "a_0 + a_1 \\zeta + \\cdots + a_{p-1} \\zeta^{p-1}\\, , ", "34d3c3ac35c49a68c602166e22d2beae": "!P \\equiv P|!P", "34d434e48d04b59e12fdc07216d79d96": "e_{\\langle R \\rangle} = \\frac{\\operatorname d Q/Q}{\\operatorname d P/P}", "34d4950912740178f7d66a29697a95bc": " (\\mathbf{a_{1}}, \\mathbf{a_{2}}, \\mathbf{a_{3}})", "34d49e49af1427790dc1c0f27cef42b4": "\\vec{t}", "34d51528a0bd526c81e951163f63c59d": " Coverage = \\frac{Faults \\ Detected \\ By \\ CBM + Faults \\ Detected \\ By \\ PMS}{Total \\ Possible \\ Faults}", "34d56dfb571a8385930f174d2d29bff4": " dU = dW + dQ ", "34d57774449ae0a49bd2f9224377cb3e": "\nf(x)=\\boldsymbol{\\alpha}\\exp({S}x)\\mathbf{S^{0}},\n", "34d5c5a3b77ca4b0f5dd3044010f77a5": "f(z) = \\frac{6z + 1}{2z - 10}", "34d5e49df0216203f753912ada94b64e": "\\left(-2\\sqrt{\\frac{2}{5}},\\ 0,\\ -2\\sqrt{3},\\ 0\\right)", "34d75657d49b53426abb5ffbf0be8b55": "S(W)", "34d75e3b34c8b9e5849805c02c8ba6a7": "\\ A^{-1} \\left (A^{-1} \\right )^T \\left (A^T A \\right ) = I ", "34d795487894db9ed949f06d6bfa6775": " = \\sqrt{ m^2 c^4 + c^2 \\mathbf{p}^2 } ", "34d7bcfe8b70384c22d107853797c7f4": "x=\\frac{m_1-m_2}{l_1m_2-l_2m_1},\\,y=-\\frac{l_1-l_2}{l_1m_2-l_2m_1}.", "34d7bd41d8af4d6830dab5ece9098395": " \\mathbf{d}_{\\ell-n},\\mathbf{d}_{\\ell-n+1},\\dots,\\mathbf{d}_{\\ell} ", "34d7ef11d21c11802f02afefd5dac286": "X \\in \\mathbb{F}^{r \\times r} ", "34d84e711392d1848ffc3f32aa32af41": "\\nabla E_{snake}^*\\approx \\displaystyle \\sum_1^n \\nabla E_{snake} (\\bar v_i) ", "34d874e2048e7f485f596a81d1ff88c7": " c^{2s}(H^{s}|_{E})<\\infty", "34d90068aea677f93700caa8dad5c7cb": "\\color{Violet}\\text{Violet}", "34d91c4c57ea28939ba16eafeeb49896": "b^n=\\Sigma_{i{<}m}c_i a^i", "34d9664391159f4d93242d7dee444dc6": "\\phi(7)=6", "34d97e2c35d4e8d9451a5a3c94ca9567": "4 N \\log_2 N - 6N + 8", "34d9edc27e73993ea76b050523a868b4": "m= \\frac{112}{3}", "34da118c192f100cc5f5e63425626ab8": "\n\\begin{align}\nx & = \\frac{W}{2\\pi}\\left( \\lambda - \\lambda_0\\right), \\qquad\\quad\ny = \\frac{W}{2\\pi}\\ln \\left[\\tan \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right) \\right]. \n\\end{align}\n", "34da34dcbed121958b21913f169109e0": "0_M", "34da777f58659683c59140d045748aec": " - {3e^2 \\over 2 \\pi^2 \\hbar} \\left ( ln \\left [ {(4/3)B_{SO} + B_\\phi \\over B}\\right ) - \\psi \\left ({1 \\over 2} + {(4/3)B_{SO}+B_\\phi \\over B} \\right ) \\right]", "34db5c5c1a49af75d15500e5260af739": "N(h)=\\{h\\}+N(0)", "34db6278501f3db93d9ca9474ed26935": "\\{v_i\\}_{i\\in\\mathbb{N}}", "34db6483aa7b900bef7c7fb876e824f2": "\\left(\\textrm{ad}(e^i) \\circ \\textrm{ad}(e^j)\\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n ", "34db6a93b17a0d89eaef7ecbd4324ac0": "\\frac{\\mathrm{d}E(k)}{\\mathrm{d}k}=\\frac{E(k)-K(k)}{k}", "34db703b935674108ff87492ae1c2c45": "I_{C1} = \\frac{\\beta_1}{\\beta_1+1} \\left( I_{R1}-I_{C2}/\\beta_2 \\right)\\ . ", "34db92c289ec862852bf80f68ef16924": "\\ \\xi' ", "34dbc4e99ac6c3f3c783655365ddf073": "-5.9", "34dc2389afcd9ee31825d5390d2ab546": "\\theta = \\pi/2", "34dc764b6dd45bb02ba501df21cf96b3": "|e\\rangle|n-1\\rangle\\leftrightarrow|g\\rangle|n\\rangle", "34dcc4ca1cf2a795347517afb4713bbd": " H_n(x) = (-1)^{n} \\mathrm{e}^{x^2} \\frac {\\mathrm{d}^n}{\\mathrm{d}x^n} \\Bigl( \\frac {1}{2\\sqrt{\\pi}} \\int \\mathrm{e}^{isx - s^2/4}\\, \\mathrm{d}s \\Bigr) = (-1)^n \\mathrm{e}^{x^2}\\frac {1}{2\\sqrt{\\pi}}\\int (is)^n \\, \\mathrm{e}^{isx- s^2/4}\\, \\mathrm{d}s. ", "34dccfe133a0fe16e3d10578e84fd9cd": "(\\nabla_a\\,\\nabla_b -\\nabla_b\\,\\nabla_a)\\,\\mathbf{\\xi}^d", "34dceaac227294bc76fc03dd8e1f174b": "\\displaystyle{T_x(a,T,b)=(a+Tx -Q(x)b, T-R(x,b),b),}", "34dcf08f843bb25b361649185c49dff0": "R = \\mathfrak{a}_1 \\oplus \\cdots \\oplus \\mathfrak{a}_n, \\quad \\mathfrak{a}_i \\mathfrak{a}_j = 0, i \\ne j, \\quad \\mathfrak{a}_i^2 \\subseteq \\mathfrak{a}_i", "34dcf9defa519b7e7a251374f677571a": "\\varphi(K_n)=n-1", "34dd1ef47563bad867ae1c4626708cc0": "C_{2v}", "34dd32456ea40000a3474860769e2b7b": "R = \\alpha \\xi_1 ", "34dd49ccb415f2d1fdfb16ae3e4428b5": "a_w=1-\\frac{P^{vap}}{P^{vap}_{max}}", "34dd5a4bd509bc01a3673419179712dc": "I(x, y)\\frac{\\mathrm{d}y}{\\mathrm{d}x} + J(x,y) = 0,", "34dd66203ecceb8df574375c412b67e8": "H(q,N)", "34dd6c954af8e8ce1527eb916091bf8f": "z=z_0", "34dd871ca6ee4bb1f6cc809c6f8bd141": "\\frac{1}{2\\pi i}\\int_{c(r)}\\frac{p'(z)}{p(z)}\\,dz,", "34dd94fa6c67865b64b22da5f03ea32f": "\n\\begin{align}\nU^\\dagger S_x U &{}= S_x \\left[ 1 - \\frac{\\theta^2}{2!} + ... \\right] - S_y \\left[ \\theta - \\frac{\\theta^3}{3!} \\cdots \\right] \\\\\n&{} = S_x \\cos\\theta - S_y \\sin\\theta\\\\\n\\end{align}\n", "34dd9766c9ebd04a3af7e186ec6d86f4": "\n\\begin{array}{lcl}\n (n+1)^2 - n^2 & = & ((n+1)+n)((n+1)-n) \\\\\n & = & 2n+1\n\\end{array}\n", "34ddb29c8633620b2861a085286be50d": "\\Delta \\rho=\\frac{\\pi}{\\lambda n}g_{lk}l_{l}l_{k}=\\frac{\\pi}{\\lambda n}\\Delta G=\\frac{\\pi}{\\lambda n}(\\gamma _{lkm}E_{m}+\\beta _{lkmn}E_{m}E_{n})l_{l}l_{k}", "34ddb7f6f8e0396057e478ceee6f9221": "n\\log n + {O}(n)", "34ddc791e02ae31a44cce8c56adfca82": "\\mathcal{L}=\\frac12 (D_\\mu \\phi)^* D^\\mu \\phi - U(\\phi^*\\phi) -\\frac14 F_{\\mu\\nu}F^{\\mu\\nu}\\ ,", "34ddf9d47b637ca2a208eda8a53e132b": " \\sgn(x) = 2 H(x) - 1 \\,", "34de14e322fc35afab7202d4f3751c03": "{\\partial \\det(A) \\over \\partial A_{ij}} = {\\partial \\sum_k A_{ik} \\mathrm{adj}^{\\rm T}(A)_{ik} \\over \\partial A_{ij}} = \\sum_k {\\partial (A_{ik} \\mathrm{adj}^{\\rm T}(A)_{ik}) \\over \\partial A_{ij}}", "34def1ccc897f2a33e158c357bb29bb0": "\\sigma_{ij}(\\lambda) = (\\xi^{(i)}_\\lambda, \\xi^{(j)}_\\lambda),", "34df00b3ad2435c67f9d598ee6d1cf2b": "\\lim_{n\\rightarrow\\infty}\\frac{1}{10^n} = 0.", "34df1a533e78b7e37bbc10230cda415d": "\\omega = -\\beta/k, \\, ", "34df271eb69f5877f5a3ec9357d9b31a": "\\psi_1\\frac{\\partial \\psi_2}{\\partial x}-\\psi_2\\frac{\\partial \\psi_1}{\\partial x}=constant", "34df6bd3187440caabda865b37cb703e": " X/\\mathord{\\sim} := \\{[x] | x \\in X\\}", "34df865ee5aed69e037b66a2b4d3efa0": "\\int\\frac{dx}{\\cosh ax} = \\frac{1}{a} \\arctan (\\sinh ax)+C\\,", "34e0add88d127eabc7150e8b668c1cf8": "m^2+3n^2.", "34e0cca5def7bfc13db3d7fd0d07ad9f": "n_{rel} = \\sqrt{1 - \\frac{\\omega_p^2}{\\omega^2}}", "34e0d292dfd1f4adf8b043ae058c0539": "c_B(a,0)=\\frac{\\beta}{4}\\mathrm{csch}^2\\frac{\\beta a}{2}", "34e0dd96a8dd5eb8005796e961b163ef": "(a_{1}, a_{2} \\ldots, a_{n})", "34e146d97083f919cf98c3e7bdc5959b": "q\\ p\\ f = (p\\ f)\\ (p\\ f) ", "34e1487729a59f63debe5a3d35c140a0": "\\textstyle\\left(\\frac{1}{2}, \\frac{1}{4}, \\frac{1}{8}, \\frac{1}{8}\\right)", "34e18e7cadb4248f12c0b0b6734113a0": "\n\\begin{align} |0_{E}\\rangle &= |0\\rangle_{1}\\otimes|1\\rangle_{2} \\\\\n |1_{E}\\rangle &= |1\\rangle_{1}\\otimes|0\\rangle_{2}\n\\end{align}", "34e194fe979786fb61336ee2803ca45e": "(\\text{skewness})^2-2< \\text{excess kurtosis}< \\tfrac{3}{2} (\\text{skewness})^2", "34e1ea3224240a1d3f69640b11f24adb": "h\\nu = \\epsilon_{2} - \\epsilon_{1}\\,", "34e1fbfd60515ed9e7cf278929280ed6": "v(x)=\\text{True}", "34e23575a0e84e820d632e47d407a257": " \\sigma_{ij}(\\lambda) = (\\xi_i(\\lambda),\\xi_j(\\lambda))", "34e25f8623d16293b70f3b35a1ec8173": " G_\\text{I} = biL_\\text{p} ", "34e2e980a1d752ebabccc6b4d0235d54": "\\text{last}(l) = (2^{l + 1} - 1) - 1 = 2^{l + 1} - 2", "34e34c238668cf7ce0c608f7c40837ce": "\\mathrm{CR}", "34e38cdd10ac47f6aaf6e1fdd20e6b44": "W_S \\leq F_i-F_f", "34e399b60434cdda5881637a4895cb87": "\\mathcal{\\dot{H}}_{1}\\approx \\mathcal{\\lambda }_{2}\\{\\mathcal{H}_{2}, \\mathcal{H}_{1}\\}\\approx 0. \\,", "34e3b11c6eb6e3efc0b85546f24d7a6b": "(X_1, Y_1)", "34e3b46110049dc54931d404860cd067": "i_d = env_d \\times 2r_z l_z", "34e3efac246ea5b9dcc880be251ca4ba": "\n\\Omega = \\begin{vmatrix} \\frac{\\partial}{\\partial x_{11}} & \\cdots &\\frac{\\partial}{\\partial x_{1n}} \\\\ \\vdots& \\ddots & \\vdots\\\\ \\frac{\\partial}{\\partial x_{n1}} & \\cdots &\\frac{\\partial}{\\partial x_{nn}} \\end{vmatrix}.\n", "34e47922677b04ad12a57fe0051e1c28": "f: \\Bbb{R}^n \\to \\Bbb{R}", "34e4a17d8fb926bbebf6b048c10388fd": "\\mu^-(A)=\\mu(A^{-1})", "34e52303cd56eebb32e4245f8636afdb": "v_1,v_2,\\ldots,v_n", "34e55971f7f3ddc1bd0e9cabcf387cdb": "\\frac{\\part u}{\\part t} + a\\frac{\\part u}{\\part x} = 0, ", "34e56aa9ba835d5e06962ab6195391a8": "c= (1-s) y\\,", "34e5d3c24409a4efa590ed3f99168051": " \\text{true price } (\\$) = \\text{cost } (\\$) \\cdot \\dfrac{\\text{consumption } (\\$) + \\text{depreciation } (\\$)}{\\text{credit } (\\$) + \\text{production } (\\$)}", "34e5fed09f3444791784d0814c161a8b": " F(\\widehat{\\theta }(x)|X) = \\frac{a}{a+b}. ", "34e6556bafca8647ea50e6dad33ea9f4": "\\tfrac{M-\\lambda}{2}", "34e687de3754eb7441fa125e41ee66c1": "\\sigma_\\varepsilon", "34e688e9e594b4c5d7789210662a0d95": "E^2 = p^2 + m_0^2 \\,.", "34e6e161026450c9f52038cc3ee3ee0e": "\\vec{a}_{rot} = \n\\vec{a}_{o} - \\vec\\omega \\times (\\vec\\omega \\times \\vec{r} ) - 2 \\vec\\omega \\times \\vec{v}_{rot} - \\frac{d \\vec\\omega}{dt} \\times \\vec{r}\n", "34e6f2bac9a14892f0e1c77a73d09776": "v^\\prime\\lesssim 1", "34e6f763da54128a2f2b6b96076ae24e": "X\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{2}\\eta^{\\mu\\nu}\\partial_\\mu \\phi \\partial_\\nu \\phi", "34e71ff4b1289d982d0746ac286eab6a": "0<\\alpha<1.\\,", "34e7469bd18148816f3aa3a53d0bbc13": "B_k=\\underbrace{B\\oplus\\ldots\\oplus B}_{k\\mbox{ times}}", "34e7b60755b5e8541572996ad0d72014": " \\phi = b - a ", "34e7cd48fc435d619d6f18e078264ddd": "\\mathrm{Res_0}\\big(u(1/V)\\big)=\\mathrm{Res_0}\\big(v(1/U)\\big)", "34e7cd9972cfb1f936f30cff225faa91": "I(t) = 0", "34e7e6a104a1f236765a8e44f524306a": "P_2^0", "34e8331e9acbddf9774732f8e4f7e0b1": "\\cos\\frac{2\\pi}{15}=\\cos 24^\\circ=\\tfrac{1}{8}\\left(\\sqrt6\\sqrt{5-\\sqrt5}+\\sqrt5+1\\right)\\,", "34e834920ad5fc13166c5278b198729f": "\\int_0^1 f(x)v(x)dx", "34e8718924b741f431cdbe10a58a4ebc": "((b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15}))^2 \\over 4", "34e89b5564c2f1691ce3159c1ccbe637": "\\textstyle\\frac12\\rho_0v^2", "34e8b70f5fdedbd7571e7e024c2527e1": "\\ y[n] = h[n] * x[n]", "34e8c88d9e61c0a4cdc56c03bbb350ef": "v(x) = -4x + 12", "34e8d887c8d71c54b37afc1a7bb4dc02": " X \\to Y ", "34e92c4d1a918e6fe73f53ae96cbda4d": "M_t\\sup_{0\\leq s\\leq t} X(s)", "34e93296714f789c95ae50f1ce0d6828": "U_{2n} = \\frac{2 U_{n} u_{n}}{ U_{n} + u_{n}}", "34e95a4b221efe149d61e5f9309142bc": "-\\sin(\\pi \\nu)\\mathbf{E}_\\nu(z) = \\cos(\\pi\\nu)\\mathbf{J}_\\nu(z)-\\mathbf{J}_{-\\nu}(z)", "34e98c66bb27b323ab3ba1f9e28a5f17": "\\sigma\\circ\\Delta = \\Delta", "34e995ba1d93519b2ffd1111673315ba": " \\mathbf{\\hat{e}}_{\\bot} \\,\\!", "34e9c4c4ef793826aeabbafccf50738b": "4 \\uparrow \\uparrow 3 = 4 \\uparrow 4 \\uparrow 4 = 4^{256}", "34e9e57d76ca07b1458f3ea1f851f546": "\\mathbf{A}^\\dagger \\,\\!", "34ea20fed7b79140b5af42c6601bb566": "\\varphi:\\mathbb{R}\\times X\\to X", "34ea6619cf97a8b4c09645e110cba87b": "x > 0 \\Rightarrow x \\geq 1", "34ea68cb7f5967515a973581c3f258fb": "J = \\int_a^b p_\\parallel d s", "34eaba71856df9f5ab8dc7527c5f165a": "\\langle f,g \\rangle = \\int_a^b f(x)g(x)\\,dx", "34eb26fea0096329f7878b9336001f7b": "\\tau_\\mathrm{p,n}", "34eb2c5fcbaf62d1bb5fcb2ee0cdc7fb": " S = 125348 = 12.5348 \\times 10^4", "34eb45d8070d136a4732d863511b7f1f": "\\frac{}{} g(z) dz = g_0 dh ", "34eb80b197a1ce7719a499636884e84f": "g(E)=2\\int\\frac{d^d\\vec{k}}{(2\\pi)^d/V}\\delta\\left(E-E_0-\\frac{\\hbar^2\\vec{k}^2}{2m}\\right)=V\\frac{d\\,m^{d/2}(E-E_0)^{d/2-1}}{(2\\pi)^{d/2}\\ \\Gamma(d/2+1)\\hbar^d}", "34eba7d4d3e3a37b734943edba0e350e": " x \\ = \\ b^y ", "34ec6ea7079aa41b5b6a514716b11e13": "X = \\mathbb{R}^n", "34eceb9b0edd6cbd7c8ddaae2fa5b3a3": "\\dim_{\\mathbb C}\\Lambda^{r}\\,V^{\\mathbb C} = {2n\\choose r}\\qquad \\dim_{\\mathbb C}\\Lambda^{p,q}\\,V_J = {n \\choose p}{n \\choose q}.", "34ed867bc8123d4304bfd90805db3d9a": "\\begin{align}\n\\delta\\mathcal{L} & = \\frac{\\partial \\mathcal{L}}{\\partial \\phi}\\delta\\phi + \\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)} \\delta (\\partial_\\mu \\phi)\\\\\n& = \\frac{\\partial \\mathcal{L}}{\\partial \\phi}\\delta\\phi + \\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)} (\\partial_\\mu \\delta \\phi)\\\\\n\\end{align}", "34ed885e7a9840fab1f8276f39fe2be8": "L^{C/B} \\to (L^{B/A} \\otimes_B C)[1]", "34eda40e2e9227299983d7eda368130d": "f:S^n \\to S^n", "34edca209d2fc5c20b05c4011ec42dac": "L_c(s,t)+L_d(s,t)-B(s,t). \\, ", "34edfa864704665178806ec5e4298240": "C\\ell_{p+k,q}(\\mathbf{R}) = C\\ell_{p,q+k}(\\mathbf{R}) .", "34ee7cb1ad877f0e38e05126e4771a2b": "\\gamma_n:=(E(\\gamma_n)\\to\\mathbf{P}^n(\\mathbf{R}) ),", "34ee7dc4f93c86a83ae2f3c56e836f03": "y_{1}=3-1", "34ee82230ed2fd0de084091e38a5736a": "\\Lambda_q", "34ee9380f3aef87393a23cc84023da2c": "\\sqrt[3]{x}\\!\\,", "34ee9f2c9d133866c661cd76b7c5cff1": "f = a \\log(A)\\,", "34eee02f3d536d2ff5a0d9949fcbf723": "U^\\left(-1\\right)=U^\\left(-2\\right)=0", "34ef77acc1208a501938f5792b3ddc78": "N \\cong \\operatorname{Hom}(M_S,S_S)", "34ef8a35dc3f4d0a7e2d577f86847d27": "X\\,\\sim\\,\\textrm{Scale-inv-}\\chi^2(1,c)", "34ef9b460034f2c4f8f1b95cd85eb291": "\\sum_{n=1}^\\infty z^n H_n = \\frac {-\\ln(1-z)}{1-z},", "34efce864152a9e9a92c25ee37c4a700": "P_h", "34effbc4aa6110672e1cdf1a3c04d3f8": "8+2+7+8=25", "34f0185b9703da670ac78e353ed9320a": " \\vdash ", "34f044b9b9dc92f0ab34e07ca1bfbb27": "r = r_{min}", "34f063a43c1388e904f373bb2fe74c98": "L_M(f(u))=\\int L_M(u,u')f(u')du'", "34f0b9be1d0818f66bf389318127978a": "E=\\tfrac32gy-\\tfrac12L^2\\lambda=\\frac12(u^2+v^2)+gy", "34f0cf3a26ad8e4fbf696ced74b43476": " {\\rm{cn}} u\\,", "34f1076f8ea4d4ca0348645d984a1ad8": " \\omega \\propto \\Delta v ", "34f14d72ee675fdc94bff107a1087417": "0 = \\sum_{n=1}^\\infty 0 = \\sum_{n=1}^\\infty (1-1) = 1 + \\sum_{n=1}^\\infty (-1 + 1) = 1\\,", "34f17b02f18aea41f896a4a1002dba0d": "n = 61 \\times 53 = 3233", "34f1e1c1e359511d516222c78caa0c09": "{D}", "34f23ee5465844cc3cb6988e6f2ea3db": "\\forall x,y\\in \\mathbb{R}^d ", "34f263c71a0bccfecd46926bec405627": "\\mathcal{P}(s) = \\left \\lbrace \\dfrac{\\prod_{i} (s + z_i)}{\\prod_{j} (s + p_j)}, \\forall z_i \\in [z_{i,min}, z_{i,max}], p_j \\in [p_{j,min}, p_{j,max}] \\right \\rbrace ", "34f265085aa30dbb52d64352a75cfc03": "2^{O(T(n))}", "34f2826a69d660f1698c712812a5afbd": "Y/2N", "34f2999681f40cead4ddee9bb67e7777": "T_{max} \\le S \\le M T_{max}. \\ ", "34f2b052ab20515cc884f608548e770b": "\\sum_{i=m}^n i^3 = \\left(\\sum_{i=m}^n i\\right)^2 + m(m-1)\\sum_{i=m}^n i", "34f2b3fea6a141d282ad0d2aa5f6cbf9": "\\tfrac ac \\ =\\ \\tfrac 1k.", "34f2bbc08d87dfd3596fbedb401cdc36": " 8^{8^8} - 1 = 7 \\cdot 8^{(7 \\cdot 8^7 + 7 \\cdot 8^6 + 7 \\cdot 8^5 + 7 \\cdot 8^4 + 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 7)}", "34f2c3c6f2a2bce189ea4132e7caabb9": "\\lim_{A_0\\rightarrow \\infin} \\frac {R_f} {R_b} = 2 \\, ", "34f2c74cd8114f6ccb867b84542ae6ca": "x>N_{\\epsilon}", "34f32233a6645418742a4e6ef6a5147b": "\\displaystyle{\\varphi=f_-|_{\\partial \\Omega} - f_+|_{\\partial\\Omega},\\,\\,\\,\\, \\psi=\\partial_{n}f_-|_{\\partial\\Omega} -\\partial_{n} f_+|_{\\partial\\Omega}.}", "34f339ea5710509307a6bd2d0e27d901": "\\partial (u + v) = \\partial u + \\partial v ", "34f3b2d2ef058a80ed574b9aa883d0ce": " g\\in U", "34f3c93e4f25b45e80b108d8b97a472a": "\\alpha Q'_0 = Q'_{in}", "34f3ccc5f83de7dd9c3b046d5f2f3789": "ACGTCC", "34f3f5e0aba3f6286dedb2b8b81873e4": "\\text{Rank }(A) \\leq \\text{Rank }(F_1)", "34f46e271340f720d96580e4a3d7aeae": "A_\\varepsilon^{(n)} = \\{(x^n, y^n) \\in \\mathcal X^n \\times \\mathcal Y^n ", "34f49cbe205ee2a1945aa7814d47d7bf": "F_2(s) = \\det(I - A_s)\\,", "34f4e3fdeb9a99cb8a5365b98fa8310d": " \\psi=\\psi(\\mathbf{r}) ", "34f5221f7d384dfbae90b6055d3f296d": "\\gamma_{yz}=\\gamma_{zy} = \\frac{\\partial u_y}{\\partial z} + \\frac{\\partial u_z}{\\partial y} \\quad , \\qquad \\gamma_{zx}=\\gamma_{xz}= \\frac{\\partial u_z}{\\partial x} + \\frac{\\partial u_x}{\\partial z}\\,\\!", "34f5c69069a7051b4d5f4b535f58bcf2": "\\Phi_{risa} = E_{si} E_{ra} \\Psi_m^{(0)}", "34f5cc6e3a404acc1541349f24eab15d": "Lf(\\alpha,s) \\equiv \\frac{\\partial^2}{\\partial s^2} f(\\alpha,s)", "34f5ce41899c9ad9a13b6219b55787be": "\\left(R - \\sqrt{x^2 + y^2}\\right)^2 + z^2 = r^2,", "34f5f3035bf4c4a9e4aa9094280b447a": "1^l = 1 = 1^r", "34f6846a2fde8384c811486c3077f5ac": "\\langle au,v\\rangle= a \\langle u,v\\rangle.", "34f6914bf4f79e609fe6ce8326a99dd2": "\\sin(t) \\over t", "34f69b5c49f2973baab7e09481611d98": "-1.1101", "34f6a08cf11b71ec1098a04b9dfcb5c2": "V_\\mathrm {rms}=\\frac {V_\\mathrm {peak}}{\\sqrt 2} ", "34f6aa249bbdae8caea3fa0e23b0856d": "\\partial_t \\rho+\\nabla\\cdot(\\rho\\bold u)=0,", "34f70b23ec881e96a24eeb1f359f57cf": "b^2 + 5 c^2 = 2", "34f712727a194c26680c3b2ca480c15c": "y_{unk}-\\bar{y}", "34f751cc18f076610a7caebfbe1483f3": "a^m = a^{b(p-1)}\\cdot a^n \\equiv \\left({a^{p-1}}\\right)^b \\cdot a^n\\equiv 1^b\\cdot a^n \\equiv a^n \\pmod p .", "34f787d90daeb3407bb661480ced8e34": " p = \\frac{\\part L}{\\part \\dot x}. \\,", "34f7b0810a3ec47155cb467771b15005": "\\forall i < n \\left( a_i < m_i \\right)", "34f7ce770e2b6fb11db0f43940417cdf": "c \\cdot \\mathbf{v}", "34f7dc7f1b495e26677db1ea885d397c": "R_\\mathrm{S} = R_\\mathrm{L}. \\,\\!", "34f8330f375a86e3c8a7c0af231606fb": " m \\longleftarrow \\lceil \\log n \\rceil ", "34f850109a0f6ee651f2d9c9fedf9b16": "E = X_1^2 + X_1 X_2 + 2X_2^2", "34f8a1fd1c174608f3ff03fbc25734ff": "K=\\tfrac{1}{2}(ac-bd)\\tan{\\theta},", "34f9af0b57f872351b92769e0dc83c6c": " k_{f_n} ", "34f9bc6e1a2abfa97366f8cba975636e": "\\sigma_{A}(R \\times P) = \\sigma_{B \\wedge C \\wedge D}(R \\times P) = \\sigma_{D}(\\sigma_{B}(R) \\times \\sigma_{C}(P))", "34f9c165b9a41c86762900824287cb5c": " H=K\\oplus iK,", "34f9db8058aedec60cc050a42cfd7625": "\\scriptstyle p_1^{r_1} p_2^{r_2} \\cdots p_k^{r_k}", "34fa0d24da5824b5ade36353e364de25": "\\mathrm{SO}(m+2,\\mathbb C)", "34fa0d4fba29afb9022d5d6df6ce22fa": "Q_\\theta(A) = \\frac{\\int_A e^{\\theta x}Q(dx)}{\\int_{-\\infty}^\\infty e^{\\theta x}Q(dx)}\n = \\frac{1}{M_Q(\\theta)} \\int_A e^{\\theta x}Q(dx)", "34fa3d8216369c8ccdab0d03471a594a": "\\ \\Delta^r, \\Delta^s, \\Delta^t ", "34fa434e961284dfc596e29ed0b5c65b": "v = \\frac{x - \\bar x}{h(x)}; w = \\frac{y - \\bar y}{h(y)}", "34fa5316414c384df816317cda14dd31": "\\mu_\\mathrm{B} = {{e \\hbar} \\over {2 m_\\mathrm{e}}}", "34fa662f4d53156c0d8fe54c7e3e0a20": "\\mathfrak{sp}_4", "34fa907d2ad56abe3ce6123ab3bfe2a3": "\\neg \\textit{open}(0)", "34fa91b006f6e4f276c26b28f422d062": "A = \\left(A^1, \\dots, A^n\\right).", "34fb3b4bf6bfcca1c68270b53c1c3a4f": "\n\\overset{\\{q\\}}{\\bigcup}X_{i}=\\bigcap^{\\{m-1-q\\}}X_i\n", "34fb4d1ec6e012d1578794a0d8d4f95b": "((a \\rightarrow b)", "34fb7491c8970dc92fc7293ed96a52c7": " \\bold{D} = \\kappa \\epsilon_0 \\bold{E} \\ ,", "34fbb177705bf7df7f4c32ce6fe59763": "\n|\\psi\\rangle = \\sum_{x,y} A(x,y) |x,y\\rangle\n", "34fbde9c6597394804ec4de143672cc5": "x(t) = A \\sin (tf + p) e^{-dt}, \\,\\!", "34fbe178ec9ab8566624624b92ba361b": "D_q(x^n) = \\frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\\ x^{n - 1}", "34fc4f8ca64b6f87f44ea875f02a0276": "s_n = \\left\\lfloor E^{2^{n+1}}+\\frac12 \\right\\rfloor,", "34fc83b94147f340c09668606bfde9bc": "k_0 \\ell_0", "34fc881620a59b1f9622d8df134cc55d": "O(n\\log m)", "34fca34ca67c333c0e6c5f0c845180ba": " \\mathbb{R}^{3} ", "34fca7b904d8dfba808e97e01d626446": "\\begin{array} {ccc}\nf^{\\ast}E & \\stackrel {\\tilde f} {\\longrightarrow} & E\\\\\n{\\pi}' \\downarrow & & \\downarrow \\pi\\\\\nB' & \\stackrel f {\\longrightarrow} & B\n\\end{array}", "34fcc7d1924b1f00c2281def03d9499f": "B \\text{d}x + C \\text{d}a = 0 \\,", "34fcf9d51eeedc16467c6854b3e3845e": " \\left(\\frac{\\partial}{\\partial \\rho} + i\\frac{\\partial}{\\partial \\theta}\\right)f(e^{\\rho + i\\theta}) = 0 ", "34fd0fedbea98eb5b141651b37dcb23b": "\\bot_{\\mathrm{sum}}(a, b) = a + b - a \\cdot b", "34fd148aeb596d451719450113af993b": "L_{sd} = \\frac{h \\eta_m}{2 \\eta_f}", "34fd3a0e8f2711ef8eb351a5d4035de7": "Q_A x_1 x_2 y_1 z_1 z_2 z_3(\\phi(x_1 x_2),\\psi(y_1),\\theta(z_1 z_2 z_3))", "34fdafc1bbdaed45e3ca75e8ebd76c31": "T\\mathrm FM\\,", "34fe4b73521a4e77286606fb71e6f211": "\\sigma_L\\ (\\cos\\theta + 1) \\over 2", "34feb4dde4042657d13843930645bd1f": "I_x = I_{x'} + A d_y^2", "34ff05780ba1af91c68c1985fb914264": "(z_1,z_2)", "34ff0d4b44ea8d9e7b42d836629e95e7": "\ns(w) = \\frac{A_{k-1}w + A_k}{B_{k-1}w + B_k}\\,\n", "34ff6b95cd87ca5bb00db97429fe960b": "(CPT)H(CPT)^{-1}=H^{\\dagger}", "34ff93d8c0b5dc9b22bb4293626199a0": " c = m^2 + n^2, \\, ", "34ffa88eba315eb75cdfae563f4d38b2": "L(A)=\\emptyset", "34ffad1bb635359183de09dccf0804a2": " h_x'(x,y,z,t) = h'(x)h(y)h(z)h(t)", "34ffe49aa50632cd6b04e2f436c56d79": " \\mathbf{X}\\sim \\mathcal{W}^{-1}({\\mathbf\\Psi},\\nu)", "34ffff4b42a62e96d46717d764556ada": "\\sqrt{1-BC}", "350005e31d5b45a1b25e0a2ad40f62a0": "\\begin{align}\n\\frac{d\\mathbf L}{dt} &= \\mathbf M \\\\\n\\frac{d}{dt}\\int_V \\mathbf r\\times\\rho\\mathbf v\\,dV&=\\int_S \\mathbf r \\times \\mathbf t dS + \\int_V \\mathbf r \\times \\mathbf b\\rho\\,dV. \\\\\\end{align}", "3500c2e77d95073f1a5f9c672b427456": " 1/R ", "3500e2e269bd39142056412dff139c95": "\n\\begin{align}\nY_{4,-4} & = g_{xy(x^2-y^2)} = i \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 4} - Y_4^4 \\right) = \\frac{3}{4} \\sqrt{\\frac{35}{\\pi}} \\cdot \\frac{xy \\left( x^2 - y^2 \\right)}{r^4} \\\\\nY_{4,-3} & = g_{zy^3} = i \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 3} + Y_4^3 \\right) = \\frac{3}{4} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{(3 x^2 - y^2) yz}{r^4} \\\\\nY_{4,-2} & = g_{z^2xy} = i \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 2} - Y_4^2 \\right) = \\frac{3}{4} \\sqrt{\\frac{5}{\\pi}} \\cdot \\frac{xy \\cdot (7 z^2 - r^2)}{r^4} \\\\\nY_{4,-1} & = g_{z^3y} = i \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 1} + Y_4^1\\right) = \\frac{3}{4} \\sqrt{\\frac{5}{2 \\pi}} \\cdot \\frac{yz \\cdot (7 z^2 - 3 r^2)}{r^4} \\\\\nY_{40} & = g_{z^4} = Y_4^0 = \\frac{3}{16} \\sqrt{\\frac{1}{\\pi}} \\cdot \\frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4} \\\\\nY_{41} & = g_{z^3x} = \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 1} - Y_4^1 \\right) = \\frac{3}{4} \\sqrt{\\frac{5}{2 \\pi}} \\cdot \\frac{xz \\cdot (7 z^2 - 3 r^2)}{r^4} \\\\\nY_{42} & = g_{z^2xy} = \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 2} + Y_4^2 \\right) = \\frac{3}{8} \\sqrt{\\frac{5}{\\pi}} \\cdot \\frac{(x^2 - y^2) \\cdot (7 z^2 - r^2)}{r^4} \\\\\nY_{43} & = g_{zx^3} = \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 3} - Y_4^3 \\right) = \\frac{3}{4} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{(x^2 - 3 y^2) xz}{r^4} \\\\\nY_{44} & = g_{x^4+y^4} = \\sqrt{\\frac{1}{2}} \\left( Y_4^{- 4} + Y_4^4 \\right) = \\frac{3}{16} \\sqrt{\\frac{35}{\\pi}} \\cdot \\frac{x^2 \\left( x^2 - 3 y^2 \\right) - y^2 \\left( 3 x^2 - y^2 \\right)}{r^4}\n\\end{align}\n", "3500ee648d81c57404d3b658315b8217": "N_{t+1} = \\lambda N_t e^{-aP_t} \\,", "350105ba546bdbb28f6449932e485958": "v_{AB}", "3501371619260065f831f2eb8c5a3728": " \\circ ", "35013eff1347180116e1bf5845085d28": "0 \\le \\nu < \\delta", "35018d3cffa14e946bfcfbd4979b1b9f": "f = \\left( \\frac{c + v_\\text{r}}{c + v_\\text{s}} \\right) f_0 \\,", "3501b2d14bf51ead96afadd45afe3ea3": "\\Delta S=Q_2/T_2", "3501c1493af999f5ed65927e7debf285": " L_\\text{f} = \\frac{1}{2} m_1 v_1^2 + \\frac{1}{8c^2} m_1 v_1^4 + \\frac{1}{2} m_2 v_2^2 + \\frac{1}{8c^2} m_2 v_2^4, ", "3501f92c2ef7e1237b0b4ffd2a2c3a07": "l < n\\,", "3501fee1b724f3629a0fda569cac7c91": " \\rho={3\\over4}, \\quad \\eta=0, \\quad \\xi=1, \\quad \\xi\\delta={3\\over4}.", "35020bb65b8596970b763de8da6c611b": "srg \\left (q, \\tfrac{1}{2}(q-1),\\tfrac{1}{4}(q-5),\\tfrac{1}{4}(q-1) \\right ).", "3502189012a263868b9046e774cbbae8": "\\det(L_fR_g)=\\det((L_f)_{[m],S})\\det(R_g)_{S,[m]}).\\,", "35022dbaf78dcc44d6259ccd737555e2": "\\gcd(a,0) = a", "350251f00bef6f5bc3e1a3b4fb01159f": " \\mathbb{T}", "350252ed650cf08950966633496fd27f": "\\displaystyle P(w,x|c,z)", "35029f1a20f5141a07dcdd66874f9e08": " \\nabla^2_{\\mathbf{r}_i} \\equiv \\boldsymbol{\\nabla}_{\\mathbf{r}_i}\\cdot \\boldsymbol{\\nabla}_{\\mathbf{r}_i}\n= \\frac{\\partial^2}{\\partial x_i^2} + \\frac{\\partial^2}{\\partial y_i^2} + \\frac{\\partial^2}{\\partial z_i^2} ", "3502b2c3adc3cef0a39ae9ef2ae0dcd2": "M(t_0,t_1) = \\int_{t_0}^{t_1} \\phi(t,t_0)^{T}C(t)^{T}C(t)\\phi(t,t_0) dt", "350351d1297cd7c95101f04464ce5535": "\\boldsymbol{Av}_{i-1}", "3503a8b3dffc085a49347b02c3ac0d46": "T(x',z,t) = T_1 \\cdot \\operatorname{erf} (\\frac{z}{2\\sqrt{\\kappa t}})", "3503abe25ef67b5b061a2ed5f03ad15b": "\\gamma_{SL}\\ ", "3504272f896c27dd9d5db7abdeea76c4": "\\sum_{i=0}^{n-1} \\mathbf{X}^i\\mathbf{A}\\mathbf{X}^{n-i-1}", "35045b9ac9878df31e12ac7b016abc89": "\\sqrt 3a", "350463ff34fc86d0ce903d9c4abb3159": "u^2+u ", "3504813709871785456005a439cdf6a3": "\n \\lambda_k = \\lim_{m\\to\\infty}\\frac{1}{t_{n+m}-t_n} \\sum_{l=1}^m \\log r^{(n+l)}_{kk}\n", "350489bd53c98e86278345a7b18f34b7": "\\frac{\\partial n}{\\partial \\mu} = \\frac{3}{2}\\frac{n}{E_f}", "3504bbe227356affc26b0e292437199a": "x' = \\frac{x}{||x||}", "350533f8ecb1395a55e84a52558c9452": "\\Delta(f)(x,y) = f(xy)", "35054e262220636c94ebaa64adf10bb8": "L(s,\\pi,r) = \\epsilon(s,\\pi,r) L(1-s,\\tilde{\\pi},r)", "350569bf0200884dfbd8a2a5254fb11e": "\\sqrt{n}(\\bar{D}-\\tau)/\\hat{\\sigma}_D", "3505ab1f6551dd9cf73ebf5923b180d4": "(y,w)\\in R_\\sigma", "3505d13a9962f489b16bba4f9436a8a3": "\\sin(t) = y. \\,\\!", "3505fc0c92d63e49ed80f30076cac6fe": " \\nabla_X Y - \\nabla_Y X = [X,Y] ", "35065d4bfea5a7b4151c368e0fde9d67": "\n G^{ij} = \\frac{\\partial x^i}{\\partial X^\\alpha}~\\frac{\\partial x^j}{\\partial X^\\beta}~g^{\\alpha\\beta}\n", "3506629dabbac4f8fe406452c85d3507": " \\nu = \\nu_0", "35067d647c9dfb475fe864e521bc9887": " J_z = J_3 = i\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & -1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{pmatrix}\\,.", "3506d7f1ccfe9b47a772c3ef9716f06c": " u_c=\\frac{ \\Delta P R^2}{ 4\\mu_p L }[1-(\\frac{ R-\\delta}{R})^2-\\frac{\\mu_p}{\\mu_c}(\\frac{r}{R})^2+\\frac{\\mu_p}{\\mu_c}(\\frac{ R-\\delta}{R})^2]", "3507020e45c32f1819df9e515d11e34d": "h=b-a", "350736998462b2e2050e985ddb863de1": "\n\\Omega_{ab}^{\\;\\;\\;\\; IJ} - R_{ab}^{\\;\\;\\;\\; IJ} = 2 \\nabla_{[a} C_{b]}^{\\;\\;\\; IJ} + 2 C_{[a}^{\\;\\;\\; IK} C_{b] K}^{\\;\\;\\;\\;\\; J}\n", "35075eeec6f9295fae61309d4bbb4dd6": "TK_4", "350778958462b36b1e34b7fc0932946f": " E_{kinetic} = h\\nu - E_{binding} = \\hbar^{2}k^{2}/(2m) = (2\\pi)^{2}\\hbar^{2}/(2m\\lambda^{2})", "350787502cedffa6c872211774311e57": "\n _{(x)}\\Gamma_{ijk} = \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~\\frac{\\partial X^\\gamma}{\\partial x^k} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} + \\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k}~g_{\\alpha\\beta}\n", "3507d6595caa42eb784e8dfde1897bf4": "P_H \\; V | _H = T", "350863e9dd58629b0e3c60315d8d7ae4": "C_X\\, ", "3508bf1465161b6fdd00fda39310ecc2": "xv = 0 \\forall x \\in \\mathfrak{g}, v \\in V", "35090618812842a39d4b9ea67a3db60e": "\\scriptstyle(-1.8(2.8))\\times10^{-25}", "3509416fd2b07c00eeac324549e3f435": "\\mathbf{X} = \\{ n_1, n_2 \\}", "350950fd7e9fa21284b0d2272ba84bcd": "\\displaystyle \\vec{F}(\\vec{x})=\\vec{0}", "3509a7b603f6de3170200f26c6fe091d": "\\theta = \\kappa t + \\alpha", "3509c504d239414a770f7ee61b26dde6": "\\Phi(t_0,t_1)=\n\\int_{t_0}^{t_1}P\\frac{dV}{dt}\\,dt\n+\\int_{t_0}^{t_1}V\\frac{dP}{dt}\\,dt\n=\\int_{t_0}^{t_1}\\frac{d(PV)}{dt}\\,dt=P(t_1)V(t_1)-P(t_0)V(t_0).\n", "350a3a40aafc2fa9a0f469fd2c18e6f7": "\\scriptstyle{\\{\\emptyset,i\\}}", "350aaf941f00a36e7a5386b199e55a3f": "\n \\left(\\frac{\\partial \\rho}{\\partial t} + \\boldsymbol{\\nabla} \\rho\\cdot\\mathbf{v} + \\rho~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}\\right)\n ~\\eta +\n \\rho~\\left(\\frac{\\partial \\eta}{\\partial t} + \\boldsymbol{\\nabla} \\eta\\cdot\\mathbf{v}\\right)\n \\ge -\\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) + \n \\cfrac{\\rho~s}{T}.\n ", "350ae536a92a31e5ad5148f661c43bc4": "a_1 \\mathbf{v}_1 + \\cdots + a_n \\mathbf{v}_n=0. \\,", "350b89f1a4314ececa61bf1120072c81": "\\mu_S = 2.83 \\mu_B", "350c0dc6116a22fb0a4357dbd268fba2": "\\sum_{n=1}^{\\infty} \\frac{1}{n}", "350c2effef1d52f932484b7ffbfd2acb": " C_I=\\frac{k^I_{tr}}{k_p} ", "350c3f4c5df25d01d170ae3b56a491b7": "\n\\mathbf{m}_{\\rm spin}=\\frac{-g_s\\mu_{\\rm B}}{\\hbar} \\, \\langle\\Psi \\vert\\mathbf{S} \\vert\\Psi\\rangle\n", "350c508b3d2a2b80438e95b156fa4aed": "T(A\\to B)=\\Box(T(A)\\to T(B)).", "350c78a7be6998d611029ab4db1a4982": "\\varepsilon_{\\color{BrickRed}{1}\\color{RedViolet}{4}\\color{Violet}{3}\\color{Orange}{\\color{Orange}{2}}} = -\\varepsilon_{\\color{BrickRed}{1}\\color{Orange}{\\color{Orange}{2}}\\color{Violet}{3}\\color{RedViolet}{4}} = - 1", "350d312c3edd97b8315b8c15b0ea0521": "v_1 \\otimes v_2", "350d859bcb43b202f9f2b967ec8a7ad8": "{DE}_{n}", "350da9bb2c95be13aaa33af85ddfec35": " L_\\text{int} = L_\\text{C} + L_\\text{D}, ", "350dc2364d46461608205bc20cc5155e": "2i + 1", "350e047c2414520d61556e16d8518173": "q= - \\kappa \\nabla T ", "350e1622be265583f8005f3a0a433731": "N\\gg 1", "350e29e7c95be5fb96de5422ef9abc65": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2\n\\end{smallmatrix}\\right ]", "350e722b6d3497f437c923cf1691eef0": "(Q_k)^{k-1}", "350eb1ce045806243aa68da4a3ef81a1": "\\mathbb{E}[X_\\tau]\\le\\mathbb{E}[X_0],", "350eb3ad03921c1a445b40a232456a49": "\\vec{a}_{shell} = \\vec{a}_o - g \\hat{r}", "350eb87ebac771aad64c211aa710dcb1": "((A\\to B)\\to C)\\to(\\neg A\\to C)", "350ef3d1696d84ccf09e5a7f19e9a9f9": "\\mathcal{O}_X \\simeq f_* \\mathcal{O}_Y", "350efeacf45f54ee3bef9c278dfc0747": "r(\\sigma)", "350f109d6f38c8708ce323e58d18627d": "{\\rm cof}({\\mathcal K})", "350f15a0a14accb1dafa2ec96368818a": "\\bold{J} = \\bold{0}. ", "350f33b0579171099432b1d489e56b25": "r_{12}=\\frac{n_{1}-n_{2}}{n_{1}+n_{2}}", "350f39115a60f08bea34161ee9623338": " R = {GM \\over { r^3} } ", "350f399ecdab7db1ef63bce48bc9ea1b": "d^2 = 0", "350f4c116c786d972442d674c91febf6": "\nr_{ij}^{(t+2)} = \\frac{r_{ij}^{(t+1)}}{r'}.\n", "350f7e7652bc8b5ad56c3b9cce6e9065": " \\frac{ 1-z^{-1} \\cos(\\omega_0)}{ 1-2z^{-1}\\cos(\\omega_0)+ z^{-2}}", "350f8fcc3cd566cd76e9906ea8e5936b": "A_B", "350fe027f4f799b10ba2e35bcf571cc6": "\\hat{\\alpha}<\\hat{\\beta}", "35105f91c192f77efc2a460999381b59": "\\langle A,B \\rangle", "3510816ab83cb89f2d2b9a4488a023a8": "- \\int \\frac{dQ'}{T_{surr}} + S'_{gen} = 0", "3510afa0910d9dfe8f0083fcd5d8d780": " (M_2/M_1)^2", "3510ed348c75dad31e0c08491d38a75a": "(\\kappa-1)~r", "3511ce03c9e4cb235bcdfc21a09cf162": " \\psi^\\dagger \\gamma^0 ", "3511f2d9e413f5853a4ec19bd7896bb4": "\\left(d, p, u\\right)\\succsim \\left(b, t, x\\right)", "35123c89a5742cc7c60e331f226addd1": "\\left[ \\begin{matrix} \\cos \\left( \\frac{\\theta}{2} \\right) & -\\sin \\left( \\frac{\\theta}{2} \\right) \\\\ \n \\sin \\left( \\frac{\\theta}{2} \\right) & \\cos \\left( \\frac{\\theta}{2} \\right) \\end{matrix} \\right]", "35126713e22441cead47954f767ab78e": "(P \\to (Q \\to R))", "351295f1bad57b1d5a4130dd7d753643": "a(t)", "3512b6bd0ebe877ca8acb16cbed7f0e7": "{\\Bbb P}(V\\oplus K)", "3512b769f3e8c5b77f6d457133ee31b7": "f_{12}=E_z/c\\,,", "351349213f32d71fe23624288915b308": "\\aleph_n, \\aleph_\\omega, \\aleph_a", "351371d00a08d899952aab029195f4e1": " K_{ij} = \\begin{cases}\ne^{-||x_i -x_j||/\\sigma ^2} & \\text{if } x_i \\sim x_j \\\\\n0 & \\text{otherwise}\n\\end{cases}\n", "35138278adb80e94f1efd28297fafe3a": " W(m, P) \\approx \\frac{m}{\\sqrt{P}},", "3513f622611e88ffba76bd631a408fbb": "a + b x + c y + d z = 0,\\ ", "3514aacf1878c00d8a318af296450717": "\\mathrm{eval}\\colon (Z^Y \\times Y) \\to Z", "3514d68cb9abf36811412b7bc83ddb12": "\\;\\approx 1.16624", "351529084e6f687f5404246eb9211099": " f_\\alpha(n) = f_{\\alpha[n]}(n) \\,\\!", "35153c2966c114410449f06003592c31": " X^\\#_p = d_{(e,p)}A\\left(X,0_{T_p M}\\right) ", "35158d4c4e14362352aeaabbc0f04651": "\n \\eta = \\lambda\\left[\\cfrac{\\lambda}{||\\boldsymbol{\\sigma}||}\\right]^{N-1}\n ", "3515b983da6feba92b417659ed6b8a02": " \\begin{bmatrix} \\dfrac{d x_1^*}{d p_1} \\\\[2.2ex] \\dfrac{d x_2^*}{d p_1} \\\\[2.2ex] \\dfrac{d y_2^*}{d p_1} \\end{bmatrix}\n=\n\\begin{bmatrix}\n\\dfrac{\\partial^2 U_x}{\\partial x_1 \\partial x_1 }\n&\n\\dfrac{\\partial^2 U_x}{\\partial x_1 \\partial x_2 }\n&\n\\dfrac{\\partial^2 U_x}{\\partial x_1 \\partial y_2 }\n\\\\[2.2ex]\n\\dfrac{\\partial^2 U_x}{\\partial x_1 \\partial x_2 }\n&\n\\dfrac{\\partial^2 U_x}{\\partial x_2 \\partial x_2 }\n&\n\\dfrac{\\partial^2 U_x}{\\partial y_2 \\partial x_2 }\n\\\\[2.2ex]\n\\dfrac{\\partial^2 U_y}{\\partial x_1 \\partial y_2 }\n&\n\\dfrac{\\partial^2 U_y}{\\partial x_2 \\partial y_2 }\n&\n\\dfrac{\\partial^2 U_y}{\\partial y_2 \\partial y_2 }\n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\n-\\dfrac{\\partial^2 U_x}{\\partial p_1 \\partial x_1 }\n\\\\[2.2ex]\n-\\dfrac{\\partial^2 U_x}{\\partial p_1 \\partial x_2 }\n\\\\[2.2ex]\n-\\dfrac{\\partial^2 U_y}{\\partial p_1 \\partial y_2 }\n\\end{bmatrix}\n", "351675aef4d0a4197a79ff6d874c792f": "n^2-2n+2", "35168372436ce9fec8a9ef153532f106": "f(k,1)=\\sum_{i=1}^k(-1)^{i+1}{k \\choose i}i(k-i)!", "351712efa34fe7743331eed7028e1d2b": "G|_{S} \\circ H=G\\circ H|_{S\\times D}", "351731515075a03d45cbde2af40b5463": "\\mathrm{Pr}", "35174a3f760e6d81bd5dba04df263a4a": " \\mathrm{Sc} = \\frac{\\sum {\\left | {Q} \\right |}}{\\bar p V_{sw}}", "35174a76fc342cb1f0cc079e6c55e49d": "I_{FLIC}", "351768dfe98790b4c17c6ebc393fda26": " \\underline{\\hat{\\mathbf{y}}}(\\ell) = \\mathbf{G}_1 \\underline{\\mathbf{X}}(\\ell) \\underline{\\hat{\\mathbf{h}}}(\\ell-1) ", "3517967d4a75127c7b10e8134df74a7f": "\\int f \\, dg=\\int fg' \\, ds", "3517d84ee0cc0267048bc5e3dfd53771": "1+x+\\frac{x^2}{2!}\\left(1-\\frac{1}{n}\\right)+\\cdots+\\frac{x^m}{m!}\\left(1-\\frac{1}{n}\\right)\\left(1-\\frac{2}{n}\\right)\\cdots\\left(1-\\frac{m-1}{n}\\right)\\le t_n.", "351808c0486899b0eb9ef6530f233ac7": "\n\\hbar \\frac{\\partial}{\\partial t} f^{e}_{\\mathbf{k}}\n=\n2 \\operatorname{Im} \\left[ \\Omega^\\star_{\\mathbf{k}} P_{\\mathbf{k}} \\right] + \\hbar \\left. \\frac{\\partial}{\\partial t} f^{e}_{\\mathbf{k}} \\right|_{\\mathrm{scatter}}\\,,\n", "3518157d6f9ae7cdc7f1ae14e277e7e8": " \\frac {I}{I_0} = \\frac {1}{2}\\quad", "351817db51eb8cc94c78bc6c3dce86fb": "f_a=f_b=f", "35185164c589e3582f26e8862fef60af": "\n\\operatorname{prob}(\\psi \\Rightarrow \\varphi) = \\sum_i \\operatorname{prob}(\\psi \\Rightarrow i \\Rightarrow \\varphi)\n", "3518b96e285735091965b852a08c9d77": "E M_+ = \\sqrt{\\pi/2} \\approx 1.25331 \\ldots, \\, ", "3518c7cfa03ffd4bcde61cd6813915cf": "T={2 g m_1 m_2 \\over m_1 + m_2}", "35192b2fd21c8d594d452b9ffd02f0f4": " a_2 | N_1, N_2, N_3, \\cdots \\rangle = \\sqrt{N_2} \\mid N_1, (N_2 - 1), N_3, \\cdots \\rangle,", "35198a7e7f1bfeab4adff8b715ae1f9b": "\\rho = \\mu /\\sigma ", "3519a5efef81423650d45cab719b115a": "\\langle f,g \\rangle = \\sum_{e\\in E} \\int_{0}^{L_e} f_e^{*}(x_e)g_e(x_e) \\, dx_e,", "3519f3a0804bc340108d3cc4dc1e2b7f": "\\mathbf{P}=(E/c,\\mathbf{p})\\,,", "351aa08de7cbdd9670e9bd312d5c3499": " rate= -\\frac{d[R 1]}{dt} = k[SH^+] [R 1] [R 2]", "351aa52f2d84647230b42835fae8bac4": "e_i \\ne e_j", "351ba78696aa406443e310e28e47db24": "\\lambda_1 = k_1", "351bb0689462de26eb226d3322e63654": " X_1, X_2, \\ldots ", "351c0da34b680f7d907cd7f857856b65": "\\frac {v- v_\\mathrm F} {v_\\mathrm F} = \\frac c d\\,.", "351c3c455adc0a7870a9a6226441aa2f": "C_\\alpha^{\\;\\; IJ} = 0", "351c4a18bcd3e855dd1b133a6ae8579e": "\\langle X+\\xi,Y+\\eta\\rangle=\\frac{1}{2}(\\xi(Y)+\\eta(X)).", "351d14cb280b86ce3015e044cb17048b": " \\sum_{x=0}^{10} \\mathbb{P} ( Y=y | X=x ) \\mathbb{P} (X=x) = \\mathbb{P} (Y=y) = \\frac1{2^3} \\binom 3 y ", "351d19a8cf268e7231921250bf24e9f8": " hD = \\log(1+\\Delta_h) = \\Delta_h - \\tfrac{1}{2} \\Delta_h^2 + \\tfrac{1}{3} \\Delta_h^3 + \\cdots. \\, ", "351da5f1a192d6a5040fb6ba6131fbe8": "\\frac{\\displaystyle\\Box\\Bigl(\\Box q\\to\\bigvee_{i=1}^n\\Box p_i\\Bigr)\\lor\\Box r}{\\displaystyle\\bigvee_{i=1}^n\\Box(q\\land\\Box q\\to p_i)\\lor r},\\qquad n\\ge0", "351ddd6f65b88e8f1c7ea0d13a113592": "f(kX_1, kX_2) = k^n f(X_1, X_2)", "351e2768a7b425533751a22c62870bbf": "\\gamma'", "351f048f8ca2e27289333a67b6f92dda": "\\frac{\\partial v}{\\partial t} = - g \\frac{\\partial \\eta}{\\partial y} - f u.", "351f1f5c6c2c30f5071fc0a90012d975": "\np_i=\\frac{1}{Z}B(E_i).\n", "351f5a2fa3ccdc5cd58120a187a30494": "\\scriptstyle =(2.5\\pm3.0)\\times10^{-43}", "351f97931e3d497b1b666f517bf12bd6": "\\int_{X} f(x) \\, \\mathrm{d} \\mu (x) = \\int_{\\mathrm{supp} (\\mu)} f(x) \\, \\mathrm{d} \\mu (x).", "351f9945b9e04734ffa4d0825ceaf42c": "f\\,{*}'\\,g = D((D^{-1}f)*(D^{-1}g)).", "351fa183703da70bcba2ef526f4f8cf3": "b(H)", "351fa514f90a28010d0c8189b09554ab": "\n\\begin{align}\nx_N * y & = \\scriptstyle{DTFT}^{-1} \\displaystyle \\big[ \\scriptstyle{DTFT}\\displaystyle \\{x_N\\} \\cdot \\scriptstyle{DTFT}\\displaystyle \\{y\\} \\big] \\\\\n& = \\scriptstyle{DFT}^{-1} \\displaystyle \\big[ \\scriptstyle{DFT}\\displaystyle \\{x_N\\}\\cdot \\scriptstyle{DFT}\\displaystyle \\{y_N\\} \\big],\n\\end{align}\n", "352021fd472e0c6af1f73bcedc394e2d": "\\sin(\\theta - \\phi) = v/c", "35203aadf8629076d3489c704b2e27aa": "f(x) = \\frac{e^{-x}}{(1 + e^{-x})^2}", "352059b6cc26180c6055264f22d6e224": "79.9\\pm 2.3%", "352108f8e2cade9f2d012f107246574e": "x'_{r,s}=1", "35213d00b1d3a7129f4dee1a83b81f78": "R_{s}", "35218d0578525095c7fc34c66b393aee": "n^2 + n + 41\\,", "3521b5b712257bf13d4c4d3a14489e66": "a=\\sum_{i=0}^m {2^{wi}a_i}\\text{ and }b=\\sum_{j=0}^m {2^{wj}b_j}.", "3521eaa701c0a10c69afe7ff3f9ebb1f": "g_{S3}=\\begin{bmatrix}\n-0.376 & 0.756 & 0.536 \\\\\n-0.770 & -0.577 & 0.273 \\\\\n0.516 & -0.310 & 0.799 \\\\\n\\end{bmatrix}", "3522601caecf6b14dfcb6c83cfaf982c": "\\ln x = \\int_1^x \\frac{1}{u}\\,du.", "352271f775e3aea76453665b69d7893a": " \\zeta(s) = \\prod_{p\\in\\mathbb{P}} \\frac{1}{1-p^{-s}} ", "3522cb43028bb8cd551eb77cc247d5ec": "-\\frac{N_c^2}{N_f}d^{(2)}(N_f)", "3522db7e1ba61167aad1979b559088fd": "Q(s)", "3522dfd69af9b318d866878947375e06": "F(x):=\\int_{-\\infty}^x\\,dp", "35235634b11fc4e873305d41138fd9fb": "\\mathcal{O}_{\\text{Spf} A}(D_f) = \\widehat{A_f}", "3523742cdda9fd3ebd213d2ad39d80ae": "\\begin{align}\n\\boldsymbol{\\nabla}\\phi &=\\cfrac{\\partial \\phi}{\\partial r}~\\mathbf{e}_r + \\cfrac{1}{r}~\\cfrac{\\partial \\phi}{\\partial \\theta}~\\mathbf{e}_\\theta +\\cfrac{\\partial \\phi}{\\partial z}~\\mathbf{e}_z \\\\\n\\boldsymbol{\\nabla}\\mathbf{v} &= \\cfrac{\\partial v_r}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_r + \\cfrac{1}{r}\\left(\\cfrac{\\partial v_r}{\\partial \\theta} -v_\\theta \\right)~\\mathbf{e}_r \\otimes \\mathbf{e}_\\theta + \\cfrac{\\partial v_r}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_z +\\cfrac{\\partial v_\\theta}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r +\\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r \\right)~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta \\\\\n&\\quad + \\cfrac{\\partial v_\\theta}{\\partial z}~\\mathbf{e}_\\theta \\otimes\\mathbf{e}_z + \\cfrac{\\partial v_z}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_r + \\cfrac{1}{r}\\cfrac{\\partial v_z}{\\partial \\theta}~\\mathbf{e}_z \\otimes\\mathbf{e}_\\theta + \\cfrac{\\partial v_z}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_z \\\\\n\\boldsymbol{\\nabla}\\boldsymbol{S} & = \\frac{\\partial S_{rr}}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_r + \\cfrac{1}{r}\\left[\\frac{\\partial S_{rr}}{\\partial \\theta} - (S_{\\theta r}+S_{r\\theta})\\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{rr}}{\\partial z}~\\mathbf{e}_r \\otimes \\mathbf{e}_r\\otimes\\mathbf{e}_z + \\frac{\\partial S_{r\\theta}}{\\partial r}~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r \\\\\n&\\quad+ \\cfrac{1}{r}\\left[\\frac{\\partial S_{r\\theta}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta}) \\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{r\\theta}}{\\partial z}~\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z + \\frac{\\partial S_{rz}}{\\partial r}~\\mathbf{e}_r \\otimes \\mathbf{e}_z \\otimes \\mathbf{e}_r + \\cfrac{1}{r}\\left[\\frac{\\partial S_{rz}}{\\partial \\theta} -S_{\\theta z} \\right]~\\mathbf{e}_r\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta \\\\\n&\\quad+ \\frac{\\partial S_{rz}}{\\partial z}~\\mathbf{e}_r \\otimes \\mathbf{e}_z\\otimes\\mathbf{e}_z + \\frac{\\partial S_{\\theta r}}{\\partial r}~\\mathbf{e}_\\theta \\otimes \\mathbf{e}_r \\otimes \\mathbf{e}_r + \\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta r}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta}) \\right]~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{\\theta r}}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_z \\\\\n&\\quad+ \\frac{\\partial S_{\\theta\\theta}}{\\partial r}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_r +\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta\\theta}}{\\partial \\theta} + (S_{r\\theta}+S_{\\theta r})\\right]~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{\\theta\\theta}}{\\partial z}~\\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta \\otimes \\mathbf{e}_z + \\frac{\\partial S_{\\theta z}}{\\partial r}~\\mathbf{e}_\\theta \\otimes \\mathbf{e}_z\\otimes\\mathbf{e}_r \\\\\n&\\quad+ \\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta z}}{\\partial \\theta} + S_{rz} \\right]~\\mathbf{e}_\\theta \\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{\\theta z}}{\\partial z}~\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_z + \\frac{\\partial S_{zr}}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_r + \\cfrac{1}{r}\\left[\\frac{\\partial S_{zr}}{\\partial \\theta} - S_{z\\theta} \\right]~\\mathbf{e}_z \\otimes \\mathbf{e}_r \\otimes\\mathbf{e}_\\theta \\\\\n&\\quad+ \\frac{\\partial S_{zr}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_r\\otimes\\mathbf{e}_z + \\frac{\\partial S_{z\\theta}}{\\partial r}~\\mathbf{e}_z \\otimes \\mathbf{e}_\\theta \\otimes \\mathbf{e}_r + \\cfrac{1}{r}\\left[\\frac{\\partial S_{z\\theta}}{\\partial \\theta} + S_{zr} \\right]~\\mathbf{e}_z \\otimes\\mathbf{e}_\\theta \\otimes \\mathbf{e}_\\theta + \\frac{\\partial S_{z\\theta}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_\\theta\\otimes\\mathbf{e}_z \\\\\n&\\quad+ \\frac{\\partial S_{zz}}{\\partial r}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_r + \\cfrac{1}{r}~\\frac{\\partial S_{zz}}{\\partial \\theta }~\\mathbf{e}_z \\otimes \\mathbf{e}_z \\otimes\\mathbf{e}_\\theta + \\frac{\\partial S_{zz}}{\\partial z}~\\mathbf{e}_z\\otimes\\mathbf{e}_z\\otimes\\mathbf{e}_z\n\\end{align} ", "352383e4e798148b05ec354cff23c231": "h(0)=0", "3523cdb512ced866edb22c031c620e2e": "p = f(c t - x) + g(c t + x)", "3523dd6f4bbb995f61b69b7cb8334716": "\\tan \\theta_B = n_2/n_1\\,\\!", "3523f11027314eed85da8a41a8914c96": "\\scriptstyle {\\mathbb R^3}", "3523fd342c9996e2fef20319fc5cb3e9": "H_{ij}\\,", "3524291cb7b6b409d49f6e91e3e6c3e2": " ds^2 = g \\, dx^2 = dx^2. \\, ", "35246f08875ae55d66c050a505f6d689": "AE=\\sqrt{d^2+g^2}", "3525124f3aad9cebdb2ab6a6b48ce5a0": "\\dot Q_H\\ge 0 ", "352540839ce9919d10bb41237f887fd2": " = \\mathbf{xU\\Sigma}^k \\mathbf{U}^{-1} ", "352552e80f073cfd9b7af490a8289770": "x \\in D", "3525b78f4332e856e8cf8408c915096e": " o(H)", "35260f9a93758ea2c9a9a586fb653a0e": "F r_2^2=\\frac{y_2^2}{2 y_1^2}+\\frac{y_2}{2 y_1}.", "35263dcf658764b852b405531f59ef55": "w^5 = ~1 - w.", "35264f2c71f5f2fb473690cc3122d70a": " \\frac{x+z}{y}+\\frac{y+z}{x}+\\frac{x+y}{z}\\geq\\frac{6}{1}", "35266882fce017c48e9fb169e58f5e37": "\\operatorname{Spec}(R/I) \\to \\operatorname{Spec}(R)", "352686c35f935f06a133a861bf5f6dd9": "x^{i-k} + 1", "3526b1801bf591bc776bac5a56b00cd1": "x_0^2 = \\left(a+\\omega \\right)^{p+1} = (a+\\omega)(a+\\omega)^{p}=(a+\\omega)(a-\\omega)=a^2 - \\omega^2 = a^2 - \\left(a^2 - n \\right) = n", "35274c8c97d10b1f797646bfc7f604bf": "\n\\begin{bmatrix} \n0&0&1&0\\\\\n1&0&0&0\\\\\n0&1&-1&1\\\\\n0&0&1&0\n\\end{bmatrix}.\n", "35275efcc4b7e6e41b92f2d838a93129": "{\\rm \\mbox{d} v^{-1}(t)/\\mbox{d} t}", "3527a4de4bca1be4a7906a5b0f65a975": "\nr^\\ell\\,Y_\\ell^{0} \\equiv \\sqrt{\\frac{2\\ell+1}{4\\pi}}\n\\bar{\\Pi}^0_\\ell .\n", "3527bf5c4ba178c2f0a23161ca901d96": "O(h^p)", "3527f906da4cc225568cdcdc82542b90": "(I) \\quad s_V(\\mathcal{R}) = s_{V_1}(\\mathcal{R}) + s_{V_2}(\\mathcal{R})", "352810d923e5b2f9b69a71559b39f043": " \\{x_i \\}", "352811caf082efd09c53758e531f2946": "\\gamma_\\mathrm{S} ", "35281b81be1cbb116582b2326e875899": " Z_q(V_o,T)=\\bigg(\\frac{1}{\\beta - \\beta _o }\\bigg)^{\\alpha}-1 ", "35289819faa7615184b948200312ae09": "\\boldsymbol B", "3528b06160b33f88fd7bd2d955abbdb7": " (x_1,...,x_n)\\in \\mathbb{F}_q^n ", "3528e2ea5d657403d97a95b2edaf93da": " = [F_6, S_6, A_6]::[F_5, S_5, A_5]::[F_4, S_4, A_4]::\\_] ", "3528fe167d6dd743b46460ace71f47cb": "h(x)-h(y)=0", "3529095e3e700f4c71ad4f205082297b": " f: M\\to N ", "35293d46a4828c5e5dd69de5d96ae77d": "T_{em}\\approx\\frac{2T_{max}}{\\frac{s}{s_{max}}+\\frac{s_{max}}{s}}", "3529b3271c012af71f86fa9f97ec3f4e": "y(t) = e^{- t} \\cdot \\cos(2 \\pi t)", "3529b62f32ea78b0e1395213df8f5fb0": "\\left( x,y \\right)", "3529c5ed8625e2ab530a497567fec9be": "SL\\left(2,\\mathbb{Z}\\left[\\frac{1}{p}\\right]\\right) \\subset SL(2,\\mathbb{R})\\times SL(2,\\mathbb{Q}_p), \nS=\\{p, \\infty\\}.", "3529d7b8bff21ac0c75470524cf7bbea": "\\alpha_2\\le\\frac{131}{416}\\ ,", "3529ede45daeec7141e93eef3ffc0638": "FW", "352a50486d57e260bde876d7a7a02d02": "\\cup_{n=1}^\\infty A_n\\in D", "352ab6636ac62619008da5323aae753a": "u(c_t)=\\ln(c_t)", "352abf4ad30956905c0dabc7299f7f21": "\\begin{pmatrix} -A_0 & 0 \\\\ 0 & I \\end{pmatrix} \\begin{pmatrix} \\mathbf{x} \\\\ \\mathbf{y} \\end{pmatrix} = \\lambda\n\\begin{pmatrix} A_1 & A_2 \\\\ I & 0 \\end{pmatrix} \\begin{pmatrix} \\mathbf{x} \\\\ \\mathbf{y} \\end{pmatrix}\n, ", "352af17bb3a84efcc54bfaefd45129b8": "\\frac{\\mathrm{d}\\,\\Delta T}{\\mathrm{d}\\,z}=\\frac{\\mathrm{d}\\,(T_2-T_1)}{\\mathrm{d}\\,z}=\\frac{\\mathrm{d}\\,T_2}{\\mathrm{d}\\,z}-\\frac{\\mathrm{d}\\,T_1}{\\mathrm{d}\\,z}=K\\Delta T(z)", "352b8a259cf89a7d0f8af04dfd8cc049": " \\frac{H_t}{ H_0 }=\\frac{1}{2-(1-(\\frac {\\delta}{r_0}))^2}", "352b91e1501611f1c2c0045a481f881e": "\\Gamma_{12}(u, v, 0) = \\iint_{\\textrm{source}} I(l, m) e^{- \\frac{i \\omega}{c} \\frac{2 \\pi c}{ \\omega} (ul + vm)} \\, dl \\, dm", "352b9b32a87a1daa405ba3dcc6336e62": "k^s\\,\\zeta(s,kz)= \\sum_{n=0}^{k-1}\\zeta\\left(s,z+\\frac{n}{k}\\right)", "352bb8db4db18795d952e3a9e4988469": "u_c= u_p+u_o", "352bd6b55af3802c3748941a57cf356a": "-\\omega^2", "352c7225736ead83ad352ca540db4d32": "L(R)=\\{x\\mid \\exists y R(x,y)\\}. \\, ", "352c940e1e39a2f8b67bf688464d72eb": "q \\equiv s \\or t", "352caed5866a2004666689b2898a7524": "\\psi(t),\\psi(2t),\\psi(4t),\\dots,\\psi(2^n t),\\dots", "352d21f11edd87083bfd3ba3d5077bd0": "\\eta_{\\mu \\nu} \\,", "352d807effd6e38899c88d40fd60d283": "\\frac{1}{g(r)}", "352d9b413eaf3e6c9d6155540ae9ccc7": "\\mathcal{G} \\subseteq \\mathcal{F}", "352d9f99ff7e996a51de564edd9da9b5": "e^T x^2+\\frac{1}{T}xy+\\sin(T)z -2 =0", "352dd724d6b7e2f6db4517d0bc75f7d5": "\\mathbf{a} = {\\mathrm{d}\\mathbf{v} \\over \\mathrm{d}t}.", "352de3e508111d9aa412b6e14a6b0ffd": "\\eta \\lim_{\\Omega\\rightarrow\\infty}\\left[\n\\int_{-i\\Omega}^{i\\Omega}\\frac{\\mathrm{d}(i\\omega)}{2\\pi} \\left(\\ln(-i\\omega+\\xi)-\\frac{\\pi\\xi}{2\\Omega}\\right)-\\frac{\\Omega}{\\pi}(\\ln\\Omega-1)\\right]\n=\\left\\{ \n\\begin{array}{cc}\n 0 & \\xi\\geq0 \\\\\n -\\eta\\xi & \\xi<0\n\\end{array}\n\\right.,\n", "352e0849e9e856d0c91c98ced543becc": "\\tbinom 83", "352e3674d78fb98eb2f7c10deff41d76": "Z + O(1)", "352e4fb7e17e05205f307b5c7b587710": " E_{l}^{k} = \\frac{1}{N_l^k} \\left\\langle \\Psi_{l}^{k} \\left| \\hat{\\mathcal{H}}^D \\right| \\Psi_{l}^{k} \\right\\rangle", "352ec56f2300264f4ee77833b5e5165e": "\\scriptstyle \\sqrt{5}/2", "352f2e1fb34185d576d5cce04a539276": "x \\equiv a_i \\pmod{n_i} \\quad\\mathrm{for}\\; i = 1, \\ldots, k", "352f3abed92bd813f3e2fe37814414e0": "\\Delta = (1 + Z_0 Y_{11}) (1 + Z_0 Y_{22}) - Z^2_0 Y_{12} Y_{21} \\,", "352f41dcd72d2c68f6365e9000b9802d": "0<\\lambda \\leq 1", "352f5dd5f9007aab2fea0cb1d6c3a532": "\\sum_{j=1}^{N} t_{j \\, (\\mathrm{processor})} < t_{i \\, (\\mathrm{execution}\\!)}", "352f685529f9a8a3cbe083a19bb8cc4f": "F=h^2/2", "352f81bdaa3374796597a78a051533c3": "\\displaystyle{W^\\prime(z)=\\lambda(z) UW(z)U^*,}", "352f9a63d13a0b1874dde3fc0e83f6a1": "x^n + a_{1} x^{n-1} + \\cdots + a_{n-1} x^1 + a_n", "352fc1fe56854299f412ed32aa86c67b": "r_{avg,n}=-\\frac{\\Delta A}{\\Delta t_{p}}=\\frac{A_{n-1}-A_{n}}{\\Delta t_{p}}", "352fe3806f70beacd49e188d291bb3f7": "\nm_j^i\\simeq P^iw_j\n", "35301a2fb70b38d6a7106e61f7ba3835": "k(X)", "353082cd14a9bba759768fef65cf8570": "\n\\left(\\frac{-b/a}{p}\\right)=-1.\n", "3530a17fd3d38c0aec9104d060e3a81b": " 1 = \\frac{a_1}{1\\cdot i_1} + \\frac{a_2}{i_1 \\cdot i_2} + \\frac{a_3}{i_2 \\cdot i_3} + \\dots + \\frac{a_n}{i_{n-1} \\cdot i_n} + \\frac{1}{i_n} ", "3530d472664efd11e185975813659d80": "t = t_1=\\Delta t", "35319b20c8f5bafb68f47ba9b9e4fb92": "v\\in V \\setminus \\{s,t\\}.", "35319e6954e9987cf704e924e3b930f2": "\\log(x)", "3531b9bc34301876aa5058bf726b1c30": " \\delta W = \\sum_{i=1}^n \\mathbf{F}_i\\cdot (\\vec{\\omega}\\times(\\mathbf{X}_i-\\mathbf{d}) + \\dot{\\mathbf{d}})\\delta t. ", "3531ba8f366b561c91a7367864d5b26b": " \\gamma_n = \\frac{e}{2m_p}g_n = g_n \\mu_\\mathrm{N}/\\hbar,", "3531d3bfb348585f608adf1b4f48dd8e": "\\xi\\in H", "3531d727b5235c202337135cc39b8daa": "\\operatorname{Li}_n(z)+\\operatorname{Li}_n(-z)= 2^{1-n}\\operatorname{Li}_n(z^2).", "353264257e56cbd997a3bd02bfa6e1b8": "\\frac{h}{e^2},", "35326fe47aebff52e4af26bb717d6b78": "a_{ij}^*=\\frac{\\sum^{T-1}_{t=1}\\xi_{ij}(t)}{\\sum^{T-1}_{t=1}\\gamma_i(t)}", "353276fec810072340bd0bff2392acf2": "C\\subseteq X", "35336c72bcfed17c1697679d6cd5c8c8": "i = \\max\\{{\\lfloor}{\\lfloor}\\log_b m{\\rfloor}/k{\\rfloor}, {\\lfloor}{\\lfloor}\\log_b n{\\rfloor}/k{\\rfloor}\\} + 1.", "353395884d52082381280dff1487e4d4": "F_1 = 1, F_3 = 2, F_5 = 5, F_7 = 13, F_9 = 34 = 2 \\cdot 17, F_{11} = 89, F_{13} = 233, F_{15} = 610 = 2 \\cdot 5 \\cdot 61.", "3534041834da3d173986f88e981a826b": "\np = 69 + 12\\log_2 {(f/440)}\n", "35347f81df8a024ef9af6a4a741f9b9d": "{2a_{11} \\times b_{11} \\over b_{11} - a_{11}+c_{11}}=d", "3534b605c2dd114b6573691df2adc56f": "\\mathrm{^{238}_{\\ 92}U\\ \\xrightarrow {(n,\\gamma)} \\ ^{239}_{\\ 92}U\\ \\xrightarrow [23.5 \\ min]{\\beta^-} \\ ^{239}_{\\ 93}Np\\ \\xrightarrow [2.3565 \\ d]{\\beta^-} \\ ^{239}_{\\ 94}Pu}", "3534b7ff51156f7a8d30fa4541568863": "f_1, f_2, \\dotsc \\colon I \\rightarrow \\mathbf R", "3534d87c87fcad094f7ccbde2a76c097": "q = e^{i \\pi \\tau}", "3534e31efdc557c7a57141adfb49927c": "p,\\ q,\\ r,\\ s", "3535c00e79b6e6ab828dd4620c90f350": "\\textstyle k_i>0", "3535d0492beeb6b4c8f50e8b267db51e": "\\sum_{k=0}^n a_k \\sim n.", "3535fa237476bba278af6fbd28942f11": "\\hbox{RawScope}=\\left(\\tfrac{\\hbox{Victims}}{\\hbox{Population}} + \\tfrac{\\hbox{Monetary Losses}}{\\hbox{GNP}}\\right)^W", "35361eb0e9e740b8b6e4c33cecd7168c": " J(A,B) = {{|A \\cap B|}\\over{|A \\cup B|}}.", "3536200fa14b9e1741b93ff63ccd18d5": "\\beta_e=", "353627e1424904e51d6f35323ad70e95": "\\scriptstyle TL_{t_f}", "353648cf5494c6774ec5b8261fcc6298": "L_*.", "35367091c09e64bd7813d8171862da4f": "v_n(t+\\tau)", "3536ba655433c687015156ca75e9db1c": "\\Theta(\\log n)", "3536d82fbd208de97662d366c114efcf": "\\mathbb{T}^3=\\{(\\theta_1,\\theta_2,\\theta_3): 0\\le \\theta_i<2\\pi\\,,\\quad i=1,2,3\\}.", "3536fb1af2f3b415af7827dd04aa8e5a": "\\frac{dx}{dt} = \\alpha x - \\beta x y.", "3537253503fd613c88ff54a38a778bf0": " r_{12}", "35372b910ecfe7f8f6d28367a99bd96a": "g'^*", "35374fc805f4751f53cb9ff53ffbbb3f": "\\Delta\\,", "353750094108e9af20e39e2217c14284": "\\varepsilon = 0.001", "3537bc77881f93b55250d96ab5c03241": "\\omega_{abc} = - \\omega_{bac}\\, ,", "35380882415ac7c7f7d0ab133442c206": " R_{a, \\theta}^T = R_{a, \\theta}^{-1} ", "353811885716684a4d3ac249c8ef2f1e": "\\scriptstyle\\vec a", "353836a83a18044a1d4f00a69c1865e4": "M(t) = M_0 \\sum_{i=1}^n{a_i}\\mathrm{e}^{-t/T_2}", "35384f394f09133e6b2c5c4cd8cf0ed4": "\\zeta(9) = 1 + \\frac{1}{2^9} + \\frac{1}{3^9} + \\cdots = 1.002008\\dots\\!", "3538774b62dc45eef1d784965f2d6692": "g(x)=\\frac{f(x)}{\\sqrt{f'(x)}}=0.", "3538b97f4dc7c20b611ad81ca426c745": "X_l^{(3)},", "3538eb9c84efdcbd130c4c953781cfdb": "\\varphi ", "3538fef3e7b4617c8e84ef2b73fc788d": "\\delta=\\frac{\\rho_ap}{k}=\\frac{2\\pi\\rho_aZr_0}{k^2}", "35392730e0dfc97980af7b4998ed551c": "a_{13}+b_{13}+c_{13}=a_{1}+b_{1}-c_{1}", "353936c3395a2361d003ddfe08fea8c3": "z_\\infty", "3539476280ded0b8ae128f624f212b68": "e^{-x}", "3539536258440100bdbb9675f430d6d0": "\\omega^{\\beta_1} + \\omega^{\\beta_2} + \\cdots + \\omega^{\\beta_k}", "3539847f5b92e0b94e62ffbbb9d293a4": "\\tilde{Q^c}\\,(\\overline{N_f},1)_{1/(N_f-N_c),N_c/N_f}", "3539fc6e9600054365ba085f10717061": "y = \\frac\\lambda{\\sin(\\lambda)}(\\sin(\\phi)\\cos(\\lambda) - \\tan(\\phi_0) \\cos(\\phi))\\,", "353a0f7e49075a02c55c360d04fa366f": "\\textstyle \\mathrm{d} \\mathbf{r}_1", "353a54c74b52e0cf33e91bc3b10833ce": "r = v/(-i) = -v/i \\,", "353a55edd80e3c13779c09b622012bee": "m = 1337", "353a9fbeeb5d084a73e67e9628d9ecb7": "(k \\ll m)", "353abe7e328f014d53ecd08ed335ac68": "z^q-z \\neq \\pm 1", "353accafb38d1f2a25dca91ddaafe414": "\\mathrm{Bin}(r|n,p)", "353ad29f0c084ef40697cf8e64d9994f": "\\rho = \\frac{1}{q(n\\mu_n + p\\mu_p)}", "353aeba39197ba782db1436f53fcf83b": "\n\\begin{align}\n\\frac{1}{N_e^{(F)}} = \\frac{1}{4T}\\left\\{\\frac{1}{N_0^f}+\\frac{1}{N_0^m} + \\sum_i\\left(\\ell_{i+1}^f\\right)^2\\left(v_{i+1}^f\\right)^2\\left(\\frac{1}{\\ell_{i+1}^f}-\\frac{1}{\\ell_i^f}\\right)\\right. \\,\\,\\,\\,\\,\\,\\,\\, & \\\\\n \\left. {} + \\sum_i\\left(\\ell_{i+1}^m\\right)^2\\left(v_{i+1}^m\\right)^2\\left(\\frac{1}{\\ell_{i+1}^m}-\\frac{1}{\\ell_i^m}\\right) \\right\\}. &\n\\end{align}\n", "353b0189853de96dadf010553bfb8c10": "\\alpha=2", "353b93096ed0d467b0595cebd384032c": "K(v_i,x)=\\phi(v_i)", "353bc11682f8977a37188a6da87cb885": "\\nabla^2 \\mathbf{F} = -\\lambda^2 \\mathbf{F}.", "353bc8f28e231837dec3374e7d01bbad": "C_Y W^T=C_{YX}", "353bd7e2a0cc7480c5725b826a0046e6": "c_n=\\frac{2}{L}\\int_0^L f(x) \\sin \\frac{n\\pi x}{L} \\, dx, n\\in \\mathbb{N}", "353c6e52c4c429199d8dde9db6cba50f": "\\forall t,s", "353cd7203f0c30a677ae92e8d3e1f279": "\nr_{pb} = \\frac{M_1 - M_0}{s_{n-1}} \\sqrt{ \\frac{n_1 n_0}{n(n-1)}},\n", "353d07ef906fb2e5aebd55f5571d80a6": "\\mathbf{k^\\prime}=\\frac{2\\pi}{\\lambda}\\hat n^\\prime", "353d2f06ca7b92fab27046d2f9d45ff1": "P=\\operatorname{Tr}(\\rho^2)", "353d791262da5b4e78af2fc18bf1ee7b": "\\mathbf x = \\mathbf y + u \\mathbf r. \\, ", "353d944e6c8ab3afb66df0e801845d21": "\\int_0^1 \\operatorname{sinc}(x)\\ dx", "353d976a1ea31a999abdd456f925ffbd": "(6)\\qquad D\\sigma-\\delta\\kappa=(\\rho+\\bar{\\rho})\\sigma+(3\\varepsilon-\\bar{\\varepsilon})\\sigma-(\\tau-\\bar{\\pi}+\\bar{\\alpha}+3\\beta)\\kappa+\\Psi_0\\,\\hat{=}\\,0\\,,", "353dee9ea646a3446bc70b9cc512185d": "V(-\\infty)\\,", "353e714cb6437ea19d6aa62693c4ea77": "0 \\not\\in S", "353e96798f7e9ab89ecae73ae171b167": "\\frac{x'}{\\gamma} = x - v t", "353edcf1dff6ad8cb268f397fe614a99": "\\mathbf{J}_\\nu(z)=\\frac{1}{\\pi} \\int_0^\\pi \\cos (\\nu\\theta-z\\sin\\theta) \\,d\\theta", "353ee46f57f1b0722142fa0cb9cb3e0f": "\\Delta: A \\rightarrow A \\otimes A", "353ef03bd2e380b6b651193777776187": "\\det(\\Delta_n) > 0\\ \\mathrm{and}\\ \\det\\left(\\Delta_n^{(1)}\\right) > 0.", "353efe5ee95c89242c3840388dbc9a7b": "\\tan \\gamma = \\gamma", "353f153ed9cdf0db2735ddc16e69a38d": "ln(8)", "353f2947146159ac1084e84811cb1dc9": "\\ddot{\\bold{r}}\\times\\bold{H}=\\mu\\dot{\\bold{u}}", "353f4afef0c626234f521760351023af": "X \\sim \\mathcal{N}(\\mu, \\sigma^2)", "353f52268daeb6c4e9db5681bdccaa0b": " x_{k+1} = y_{k+1} ", "354008542aef0ffc6201d773038b8520": " n\\ge 2", "35406acfb99ecb753274fd4e18828686": "\\tau_1>\\dotsb>\\tau_m\\geq 0", "354090157759a225c95e67986d148981": "\\begin{align}\nc_i&=b_i/d_{i-1}\\text{,}\\\\\nd_i&=\\begin{cases}\na_1 & \\text{if }i=1\\text{,}\\\\\na_i-c_ib_i & \\text{if }i>1\\text{.}\n\\end{cases}\n\\end{align}", "3540bdf6e7514f63921e1cc9c3d7f581": "v_0 = \\sqrt\\frac{2GM}{r_0}\\,", "3540cba84f647cf4e2a686bcb1c98b3f": "! \\!\\,", "354172df5ba5e293699b680c706b7610": "\\mathbb{F}_{q}.", "35417bbf7f6aa1cd421904b37cb7cef2": "\n\\begin{align}\n S & = \\mbox{current total cost} - \\mbox{past total cost} \\\\\n & = 22 - 20 \\\\\n & = 2 > 0.\n\\end{align}\n", "354180e61fb166e1f2241710674d6645": " (fV)(p) := f(p)V(p)\\,", "3541a997d3f9baf4e20237cbbc5f4fe7": " d \\ln(r_t) = \\theta_t\\, dt + \\sigma\\, dW_t", "3541d1d8f242608359ff7b5ac58e6098": "f'_c=", "3541e873dc6352e46aa89ed21d7370a3": " m_1 ", "3541eb4b1ef7f78c472465ad0e5b25e5": "c_g: G \\to G", "35421bc60c136215d07d7adb24f54a41": "\\lim_{x \\to \\infty}", "3542531659630f6d8b31de61042c063e": "\\lambda > 0 \\,", "3542b227e86fb5c906b9ec8e5466a0e9": "D\\ll R", "3542bf355e9f4d4faa4eb81b7bc76430": "\\begin{align}\nv & = u + at \\quad [1] \\\\\ns & = ut + \\frac{1}{2} at^2 \\quad [2] \\\\\ns & = \\frac{1}{2}(u + v)t \\quad [3] \\\\\nv^2 & = u^2 + 2as \\quad [4] \\\\\ns & = vt - \\frac{1}{2}at^2 \\quad [5] \\\\\n\\end{align}", "354303f4b7041104de9eb455519b9ea6": "e_n = \\mp e_1 e_2", "3543073454609d120be07459eb004278": " \\frac{1}{\\sqrt{2\\pi}}\\int_{z_{.025}}^\\infty e^{-x^2/2} \\, \\mathrm{d}x = 0.025.", "3543675722b2f495be3467b080e36013": "\\Delta=(2\\pi)^{12}\\eta^{24}", "3543815a5298ea7a38d601bdf27f48d1": "L_{x}=i\\hbar\\left(\\sin\\phi\\frac{\\partial}{\\partial\\theta}+\\cot\\theta\\cos\\phi\\frac{\\partial}{\\partial\\phi}\\right), ", "3543915049ac46cf9be9ef4c86fa9bde": "E=hf ", "3543949c29a3b3076d57b8ffa3a3ab33": "(\\scriptstyle\\ = \\pm 1)\\,", "354394a571a65a23468f3f3de78116df": "\\Delta\\ge\\frac{n}{3}", "35440bdec4326a134d205a396e764c03": " \\hbar k ", "35444059a528d8b39950a7c2e269c47c": "MC = TC - FC", "354495cba588d740ace491d01d8e8615": " f_1(x) = \\frac{P_1(x)}{Q_1(x)}, ", "35457d09280385ab23f6ba3f1b02f9fe": "2^\\kappa=\\gimel(2^{<\\kappa})", "354594aa63e9e5e1356cc3ead1d74b53": "p \\circ q", "3545c9b1227f837c9b5687e05fa7edea": "f(n) \\ge \\log~n", "3545edb294ca5676a13744dfee4e3243": "I_{im+} (x',y')=2I_0[1+cos f (x, y) ]", "3546167c129846088ff0011f03a98c60": "(x + i0)^\\alpha[\\varphi] = \\lim_{\\epsilon\\downarrow 0} \\int_{\\mathbb{R}} (x+i\\epsilon)^\\alpha\\varphi(x)\\,dx.", "35464d72bd984fd60980595904424245": "~\\ddot x=\\frac{{\\rm d}^2x}{{\\rm d}z^2}~", "35468189f32c620041fb87a4ad1fbd24": "R = \\sqrt{\\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}};", "354698f1f99288d3479d98bc359e2190": " \\int_0^\\infty \\int_{\\mathrm{simplex}} u^n e^{-u(v_0 D_0 + v_1 D_1 + v_2 D_2 ... + v_n D_n)} dv_1 ...dv_n du\\,, ", "3546a646d4b334c321e43052b58bd558": "a=a_2", "3546dcfaae74c6579a972f56ff2f5418": " L^2-4\\pi F\\geq \\pi^2 (R-r)^2. \\, ", "3546ddbe469e475ce765594557cba680": "mn\\frac{\\mathrm d}{\\mathrm d t}\\mathbf{u}=-\\nabla\\cdot\\mathbf{p}+qn\\vec{E}+qn\\mathbf{u}\\times \\vec{B}", "3546fe87e538a0e57601e9f37d6c8324": " Q^{j+1}_i = \\left(1-0.9\\right) 9.6 + 0.9\\left(10.3\\right) = 10.2 \\text{ m}^3/\\text{s}", "35473ae64c186dc564d24bb3363d00c5": "[q-1,k] ", "3547983d5ac593a48b6d1b9568511117": "f_X(x;\\nu,L)= \\frac{2}{x} \\left( \\frac{L \\nu x}{\\mu} \\right)^\\frac{L+\\nu}{2}\n \\frac{1}{\\Gamma(L)\\Gamma(\\nu)} \n K_{\\nu-L} \\left( 2 \\sqrt{\\frac{L \\nu x}{\\mu} } \\right), ", "35482868bfde9bcc6ca8414bd1d71bb4": "\\frac\\sqrt{3}{4}s^2\\,\\!", "3548641dac215f1ce38382b97b9d711e": "d\\phi/dx", "354883eb73ac2dddddd040e0df499032": " c_{t+1} - c_t = (1-R^{-1}) \\left[A_{t+1} -A_t + \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} E_{t+1} y_{t+j+1} - \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} E_{t} y_{t+j} \\right] = (1-R^{-1}) \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} (E_{t+1} - E_{t}) y_{t+j+1}", "3548b5d5706b507d480921652700b4af": "D_o", "3548c56395d9701bee20a7eb6d58ae35": " g^{bn} g^{am} (R_{abmn;l} + R_{ablm;n} + R_{abnl;m}) = 0,\\,\\!", "354940dd367802382434ea13090b0858": "\\mathbb{E}f=\\int_\\mathcal{S} fdP = P (f) ", "3549cc87160c1b63e85ecafeb0c5aba8": "S_{a/b} S_{b/\\$}", "3549d6007e511efc51de295b0c8a83f4": " \\frac{dy}{dx} = \\frac{1}{2y}", "354a044043dd098cacacc5b0c041e759": " \\nabla \\times (\\mathbf{\\xi} \\times \\mathbf{B}) = \\mathbf{\\xi} (\\nabla \\cdot \\mathbf{B}) - \\mathbf{B} (\\nabla \\cdot \\mathbf{\\xi}) + (\\mathbf{B} \\cdot \\nabla) \\mathbf{\\xi} - (\\mathbf{\\xi} \\cdot \\nabla) \\mathbf{B} \\ . ", "354a363cfe76483944c2b2835986750f": " r(t) = \\bigg| \\frac{1}{N} \\sum^N_{j = 1} e^{i\\theta_j(t)}\\bigg| ", "354a571f82fc05e9e0e292f08b90e3e5": "{\\psi}", "354a5ff593f96bb6b3542f8f80f4a8b8": "\n\\Delta z\\,\\, \\approx \\,\\,\\left( {a\\,\\alpha \\,\\mu _1^{\\alpha - 1} \\,\\mu _2^\\beta } \\right)\\Delta x_1 \\,\\,\\, + \\,\\,\\,\\,\\left( {a\\,\\beta \\,\\mu _1^\\alpha \\mu _2^{\\beta - 1} } \\right)\\Delta x_2", "354a7d619fff8f600dde1c59612d620a": "3 * 2 = 6", "354a9f79a63e5229af0b3646c2b2bb38": " M_X(H)=\\sum_{m=1}^{n-1}(H_{m-1}-P_{m}H_{n-1})X^m-P_0H_{n-1}\\,,", "354ad03592a73527f8ce9af6d2ba9af2": "\\mathsf{Pad}(M_{1})", "354b620c97e8929c3e3add63f585075f": " P_i = \\Pr( \\varepsilon_i - \\varepsilon_a > (C_a - C_i) + \\alpha P_a - \\alpha P_i + \\beta D_a - \\beta D_i ) ", "354b75e4a39ce5ad28a1608b318d083b": " \\ PV \\ = \\ {A \\over e^r - 1} ", "354bb2c18341d5e7fb8feef2d9d55677": "\n\\begin{bmatrix}\n1 & 0\\\\\n0 & 0\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\n0 & 0\\\\\n0 & 1\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\n0 & 1\\\\\n0 & 0\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\n0 & 0\\\\\n1 & 0\\\\\n\\end{bmatrix}\n", "354bbbe4b53505b99aee9d697d41953a": "1/\\cos(\\theta)", "354c5b04fb56617e3b82fdbcec148a95": "i,i'\\colon X \\hookrightarrow \\mathbf{R}^m", "354c63ee0e9a7990aab4c368353f019f": "f(u) \\sqsubseteq f(v) \\iff u \\le v . ", "354c808b3ac7ff5bd75faa26bcc36220": "c=", "354ca610b1f4549680960e8f2eae3c42": "K \\subseteq A \\leftrightarrow \\exists H: \\exists J:[\\langle H,J,K \\rangle \\in R_0 \\wedge H \\subseteq B \\wedge J \\subseteq \\alpha / B ],", "354cb8a4542c3a05cc51ba59a396aacb": "\n\\frac{1}{n} \\sum_{i=1}^{n}(f(\\mathbf{x}_i)-y_i)^2 + \\lambda \\|f\\|_k^2,\n", "354ce78b8380fa73c81d77d786303564": "D^nf \\;", "354d341d957fca39a96dd1f1cf5c65e4": " ModD \\approx - \\frac{1}{V} \\frac {\\Delta V} {\\Delta y} \\rArr \\Delta V \\approx - V \\cdot ModD \\cdot \\Delta y ", "354d3a6f3df8bbd6a3e622e13c0ccb96": "\\min \\left\\{ d_i \\right\\}", "354d92f00f48fe0e3e98420688f2ad88": "\\mathbf{u}'", "354db1fa54b292268a4838f9fd213516": "v_{t} = \\sqrt{ gd \\frac{ \\rho_{obj} }{\\rho} }. \\,", "354db3c09cae082f37a7fff74a2d59fe": "\\operatorname{Aff}(G) := V \\rtimes G", "354dc46af91a5c73153bb74db0e42765": "f:[1,\\infty)\\to\\R_+", "354e26d929008cedbcdf85a6a7692980": " \\mathbf{x} ", "354e6a4865094e2c27a8f8fa7f551c7c": "(\\mathcal{P}(S),d_0)", "354e9c3575f49779a85719a7d531b824": "k(s)", "354ec17c4cb396c6b423c9c2d358a76f": " f(x) = cx \\ ", "354f2392dc10a0d907447a7980e4ec29": "\\Delta x = x-x_n = - \\frac{f'(x_n)}{f''(x_n)}", "354f35a3fe99a0294670f96d5fdaec79": "\\nabla^4 \\psi = 0", "354f8563e2dadcdd212ab8dcb40659f9": "d-b_{15}", "354fb8cbb82aa847ac6ac1139ef990cd": "\\sigma_\\bar{C} = g(\\bar{C}, \\cdot) \\,", "354fe8f2781e36bdbfcb6ff03dcf8b77": "p_A\\left(z\\right) = {\\rm det}\\left( zI - A \\right) = \\prod_{i=1}^k (z - \\lambda_i)^{\\alpha_i},", "35501c87660d51d4e84d0d54bb9dbce1": "F(t) = \\frac{t^4}{4} ", "35504c7ffd1511b99861004de9e41799": "t_l \\equiv 0", "3550993cea08163f3527a968a481ef75": "\nf=f'=0\n", "3550b51f7726b2f50187d581cf58c2b1": "\\neg loaded(0)", "35510615fa17fbfa6d2a3aef21f7945d": "R= \\frac{\\omega_s}{\\omega_a}=-\\frac{N_a}{N_s}.", "35516d636936b4e46571640135c4537d": "\\begin{align} 2\\cdot R_*\n & = \\frac{(41.3\\cdot 0.465\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 4.13\\cdot R_{\\bigodot}\n\\end{align}", "35517a0f60b33d1e7a941391b8cda88c": "H_n(a, b) = a\\uparrow^{n-2}b \\text{ for } n \\ge 3\\,\\!,", "355242b393b156cca981697e1e734413": "H|E_n, l, m\\rangle=E_n|E_n, l, m\\rangle", "35528703c4fae5f8f9c27de7a8c69441": "\\textstyle \\mathbb{R}, ", "35529aa10eea1ee1ae8f3a097c9b0abb": "EL(\\Gamma)\\ge EL(\\Gamma^*)", "35535da969257f534879a80e00421a86": "y,z\\in E", "355367431411b9ee1ac07f4dcd993ba1": "R(W)\\;", "35536ea0735866086ff647ece7e2ff1a": " \\begin{bmatrix} V_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} 0 & n \\\\ -n & 0 \\end{bmatrix}\\begin{bmatrix} I_1 \\\\ V_2 \\end{bmatrix}", "3553731ed5b915e89ff7a6c5e955984a": " a\\!\\!\\!/ := a_\\mu\\gamma^\\mu ", "35541495614a59ac596fbe6bcc198591": "O(n^2 \\log n \\cdot\\log \\frac1\\delta)", "3554197d281b72e764109789aee2cf5a": "X_{t}^{\\tau} (\\omega) := X_{\\min \\{ t, \\tau (\\omega) \\}} (\\omega).", "355424a3cf7c7301940d13805aaa1de7": "\\frac{0.0703}{0.0620}= 1.13 = {} ", "355446ef7dccb257f1a8dcae46b3cee6": "\\begin{align}\n A &= \\int_{t=0}^{t=2 \\pi} y \\, dx = \\int_{t=0}^{t=2 \\pi} r^2(1 - \\cos t)^2 dt \\\\\n &= \\left. r^2 \\left(\\frac{3}{2}t - 2\\sin t + \\frac{1}{2} \\cos t \\sin t\\right) \\right|_{t=0}^{t=2\\pi} \\\\\n &= 3 \\pi r^2.\n\\end{align}", "3554a9348ad41b2bf17eb8dd24f226e1": "f:M \\rightarrow \\mathbb{R}", "3554c4b1825e5a7a00286f07b24b3f1c": "1/10=0.1", "3555bd7e34672ef9d74668e1e019cbce": "IV_0=1", "3555cd9fca4cc729b8f07f8ed152174b": "N_t N_r", "3555eb2a7d0595d9588c3e2e7a1af629": "\n\\zeta = \\frac{1}{2 Q} = { \\alpha \\over \\omega_0 } = { 1 \\over \\tau \\omega_0 }.\n", "355612cb828fb570f1e88ffc0abd798c": "{\\Delta}^{*}:W\\mapsto\\,P", "3556224bb15ea33f500369142015b521": "\\omega \\in \\Omega_{Z,[t_l,t_u]}", "35563f298f3c43cff8a4fe86634df5e3": "J_n = \\int \\frac{dx}{\\cos^n{ax}}\\,\\!", "3556f6bf4ced3d231bb7b9603a4ea471": "|n^\\wedge A|\\ge\\min\\{p(F),\\ n|A|-n^2+1\\},", "35574612bdfea44de1c820da8fbacb33": "K_a = \\frac{x^2}{C_a - x}", "355762c7f6789c0f8b95679f5b5e43ab": "\\sum_i \\sgn(\\pi_i) \\pi_i", "3557dccc6f1e247ba99cf2d89d1c9d2f": "1 - \\Chi^2_r(G)", "3557f1c68dc5ba900bbcde5728ced362": "(Y,Z)", "35586d4dec8d6c7a8b4cc77a6ef09ad1": " E_t = {{e a \\sin(\\theta)} \\over {4 \\pi \\varepsilon_0 c^2 R}}.\n", "3558903d0386d797b4f7859d89d9170e": "\n\\int\\left[\\left(\\nabla\\phi\\right)^2+\\left(\\nabla^2\\phi\\right)^2\\right]\\,dV.\n", "3558a20498812c49171c9da37f238999": "P_AdA", "3558b198a2ffeb18995afd101849e17c": "(A+2)/4", "3558c7c7169da78644a3af9641c4ea39": "A \\rightarrow A\\alpha_1\\,|\\,\\ldots\\,|\\,A\\alpha_n\\,|\\,\\beta_1\\,|\\,\\ldots\\,|\\,\\beta_m ", "3558ccf3711a825ca88aba3d6d8d8e89": "b_n(t) + \\mathbf{\\delta}_n(t) - r(t) = \\sum_{d=1}^D \\sigma_{n,d}(t) \\theta_d(t)", "3558f29389ce20b305a52cc55e7d2bc2": "\\left|{ x - \\frac{p}{q} }\\right| > \\frac{c}{q^2} \\ . ", "3559400533459bb2cc52c1dea1e58ed2": "\\int x^m\\,\\operatorname{arsech}(a\\,x)dx=\n \\frac{x^{m+1}\\,\\operatorname{arsech}(a\\,x)}{m+1}\\,+\\,\n \\frac{1}{m+1}\\int\\frac{x^m}{(1+a\\,x)\\sqrt{\\frac{1-a\\,x}{1+a\\,x}}}\\,dx\\quad(m\\ne-1)", "355981b142e09d7e0886aa28d34b1a6d": "T_H>0", "355a505a98a555ad4d375d31f699d23c": "r = R", "355a81e39b7866e4d89b5cdd7c19616d": "\\sigma=\\Delta{Lk/Lk_o}", "355a93a4c8564d06ce9f620f45b7fb06": "e(\\mathbf Z-1)", "355aaff0379498aa2dfa53f8e7fd11f1": "F_{p^2}", "355accc488765d1750c1160898484c56": "\\Gamma_L\\,", "355ad8af63c7f28d924d95c4367dfec9": "\\Gamma_{kij}=\\frac12 \\left(\n \\frac{\\partial}{\\partial x^j} g_{ki}\n +\\frac{\\partial}{\\partial x^i} g_{kj}\n -\\frac{\\partial}{\\partial x^k} g_{ij}\n \\right)\n =\\frac12 \\left( g_{ki,j} + g_{kj,i} - g_{ij,k} \\right) \\,,\n", "355b05da04392443da3ead59ff82c584": "\\left(\\frac{\\partial S}{\\partial V}\\right)_{T} = \\left(\\frac{\\partial p}{\\partial T}\\right)_{V}.\\,", "355b0729e5004cbd2395379c9d653a4b": " \\hat x = R_1^{-1} (Q_1^T b) ", "355b078226649f1cf244ee0f59eeaf3a": "H(x)", "355c20b11ba76beabed7abf4b5263211": "M^{N/2}", "355c9d511f156b1b79876fc2adbdda91": "\\begin{align}\n\\frac {\\mathrm{d}}{\\mathrm{d}t} f(p,q,t) &= \\frac {\\partial f}{\\partial q} \\frac {\\partial H}{\\partial p} - \\frac {\\partial f}{\\partial p} \\frac {\\partial H}{\\partial q} + \\frac{\\partial f}{\\partial t} \\\\\n&= \\{f,H\\}+ \\frac{\\partial f}{\\partial t}.\n\\end{align}", "355cf11b18bd423f72df29c0716a82c9": "\\begin{align}\n\\log \\left (pe^{(1-q)t} + (1-p)e^{-qt} \\right ) &= \\log \\left ( e^{-qt}(1-p+pe^t) \\right ) \\\\\n&= \\log\\left[e^{-q \\log\\left(\\frac{(1-p)q}{(1-q)p}\\right)}\\right] + \\log\\left[1-p+pe^{\\log\\left(\\frac{1-p}{1-q}\\right)}e^{\\log\\frac{q}{p}}\\right] \\\\\n&= -q\\log\\frac{1-p}{1-q} -q \\log\\frac{q}{p} + \\log\\left[1-p+ p\\left(\\frac{1-p}{1-q}\\right)\\frac{q}{p}\\right] \\\\\n&= -q\\log\\frac{1-p}{1-q} -q \\log\\frac{q}{p} + \\log\\left[\\frac{(1-p)(1-q)}{1-q}+\\frac{(1-p)q}{1-q}\\right] \\\\\n&= -q\\log\\frac{q}{p} + (1-q)\\log\\frac{1-p}{1-q} = -D(q \\| p).\n\\end{align}", "355d59892fd34b70e4a43597c3f247c2": "\\bar{f}(n) = \\langle f(0), f(1), \\ldots, f(n-1)\\rangle.", "355d79aafeab37016a894a5b1fd16d98": "h_{01}(t)", "355d85319354ff3646b70d07a8a98c54": " P(t)", "355daf29156581f9b0608fce8789af00": "F(r) = k \\frac {m_1 m_2}{r^2} (1+ {\\alpha \\over {r^3}})", "355e0c69f5fdad5c6bce2d10f74ac53a": "t \\in IS(s,T)", "355eda2485f300286172362e78a6b6e6": "\\mathcal{L} (\\alpha, \\beta \\,|\\, x) = \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha-1} e^{-\\beta x}", "355f207521a73f548fd7137a3fe33734": "\\mathbb{Z}_2 = \\mathbb{Z}/2\\mathbb{Z}", "355f821e22cc9c0b6c4b3378f2b35d35": "\n\\overline{\\delta}=\\frac{1-\\overline{R}_2}{2\\overline{R}^2}\n", "355f9037b74a69dde02f1caae5eeaf8e": "y_1, y_2, \\ldots, y_m", "355f940f85e43c219b3294e694e67e36": " \\mathbf{L} = I_C \\omega \\vec{k}.", "355fc057dc79f9ff44a0c2c7e4874922": "\\bar{\\bar{C}}", "35602de1a2a00b12ec8aba068c462b5e": "{\\rm RL} \\to {\\rm R} + {\\rm L}", "3560856db396f6864ff7340f9d0045b0": "\\int_{-\\infty}^\\infty (1-(1-\\Phi(x))^n-\\Phi(x)^n ) \\text{d}x.", "3560957cfdf10ad727857b78470042fa": "= [\\partial_\\sigma[\\partial_\\mu V_\\nu] - \\Gamma^\\rho{}_{\\mu\\nu} \\partial_\\sigma V_\\rho - \\Gamma^\\rho{}_{\\sigma\\nu}\\partial_\\mu V_\\rho - \\Gamma^\\rho{}_{\\sigma\\mu}\\partial_\\rho V_\\nu] - [\\partial_\\sigma [\\Gamma^\\rho{}_{\\mu\\nu} V_\\rho] - \\Gamma^\\alpha{}_{\\sigma\\nu}\\Gamma^\\rho{}_{\\alpha\\mu}V_\\rho - \\Gamma^\\alpha{}_{\\sigma\\mu}\\Gamma^\\rho{}_{\\alpha\\nu}V_\\rho] ", "3560d3a8b91dd9bb3687772fcaab6043": "N_2O_2 + H_2 \\rarr N_2O + H_2O ", "356110daf994b384592be048df5a9080": "\\langle\\sigma v\\rangle/T^2", "356123a3dab7b319b9e1a70304bb78ac": "0 \\rightarrow X \\rightarrow Y \\rightarrow Z \\rightarrow 0", "35613de03f45c238a055b39850308626": "c\\nu\\in \\mathrm{Tan}(\\mu,a)", "3561beb054a95d4ead43a8451708286c": "\\gamma \\,\\!", "356265b4e979b85a70b1cd8f1f35278f": "\\delta=\\frac{\\mu - x_m}{x_{m+1} - x_m}", "356282043fe477aa88095493bff6827d": "\\begin{matrix} {12 \\choose 1}{4 \\choose 2}{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "3562901aed36416f2a0edd63301d0b73": " \\ln {Q_{10} \\over 10} ", "3562d40c3a2acd87f1f1788bf1b9fd92": "f(x;\\alpha)=\\frac{\\alpha \\exp(-x)}{\\left(1+\\exp(-x)\\right)^{\\alpha+1}}, \\quad \\alpha > 0 .", "3563636f98deac38e35487d9b94c4689": "L_{n+m} = L_n L_m - (-1)^m L_{n-m} \\,", "356418b73e0fe5dcf7474fe40ba3b98a": "K_B = 0.0722", "35647e8da5aa448b6759b2b459280359": " \\textstyle\\ \\mbox{rate(prop)} = k_p[\\mbox{M}^-][\\mbox{M}] ", "356489a5e54761be314a6206f8cf8bdf": "\\int x^m (\\ln x)^n\\; dx = \\frac{x^{m+1}(\\ln x)^n}{m+1} - \\frac{n}{m+1}\\int x^m (\\ln x)^{n-1} dx \\qquad\\mbox{(for }m\\neq -1\\mbox{)}", "3564e5d5b4e2e00c1a8afd174a80fe74": "T,P,K", "356503b94107888d1692bf414e750e30": "{13-r \\choose 1}{6 \\choose 1}{4 \\choose 2}{4 \\choose 1}^5 = {13-r \\choose 1} \\cdot 36,864", "35654ecae23ec313b59cd53d1cb37ea5": "u_4 = u_3", "35658f46897a25955b54960fac1a31be": " \\operatorname P [a \\leq X \\leq b] = \\int_a^b |\\psi(x)|^2 \\, \\mathrm{d}x ", "3565bd8a990a4ef43dafafa3b9d5a403": "\\sum_{i} \\frac{i}{1+ \\lambda_i^2} = 1", "3565e76644f4aba1313adf95ecd27025": "\\Psi (f \\otimes I)", "35664d74b6648bc8a04c607d7b5008c9": "\\operatorname{Cone}(f)", "356682eea8f008a86523007e84b4d580": " \\nabla = d + i A\\,", "3566d9472d0243025274c6851aa083e9": "y = x * e^{rt}", "3566f689b9de0acf1ed7d99a952d43fd": "\\frac{1 - \\cos \\theta}{\\theta} = \\frac{1 - \\cos^2 \\theta}{\\theta (1 + \\cos \\theta)}\\,", "35670db6772eb1105a25fa52154410ad": "\\mathrm{Wb=V\\ s=kg\\ A^{-1}m^2s^{-2}}", "35672b76f61e5b776c58df9619af4a97": "R(\\theta,d') = \\mathbb{E}_\\theta\\left[ \\left|\\mathbf{\\theta - X} + \\frac{\\alpha}{|\\mathbf{X}|^2}\\mathbf{X}\\right|^2\\right]", "3567414e3eeb4b9cb8b0dde8f9d62de1": "\n\\chi = \\frac{q_{xx}-q_{yy} + 2 i q_{xy}}{q_{xx}+q_{yy}}\n", "356789882b2ca768b451a1c69f6a38bc": "B \\to {\\rm Fr}(M)", "3567cfa4db3cd8246df0ec60126cb928": " \\frac{1}{\\gamma } = \\sqrt{1-\\frac{v^2}{c^2}} ", "3568193ea6eb78730fe114f2a12bd602": "\\textrm{E}[\\tilde{\\textbf{y}}_k] = 0", "35684df7328fb192dffc9d90feb0dfe7": " 0 = v_0 t_d \\sin(\\theta) - \\frac{1}{2}gt_d^2 ", "3568bea4a06db07c80df4a85a94a456e": "(8)\\quad \nds^2=-e^{2\\psi(\\rho,z)}\\frac{L-M}{L+M}dt^2+e^{2\\gamma(\\rho,z)-2\\psi(\\rho,z)}\\frac{(L+M)^2}{l_+ l_-}(d\\rho^2+dz^2)+e^{-2\\psi(\\rho,z)}\\frac{L+M}{L-M}\\,\\rho^2 d\\phi^2\\,.\n", "3569200cb28ffdcfd4df7c24159ff74b": " \\boldsymbol{\\epsilon} ", "35693778d9eb9302dd887454d2ed00e8": "\\displaystyle{K(z)=G(hz)H(z).}", "35693f5b7ef5d83751a2caa740d07b7d": "\\sin x = x - x \\frac{x^2}{(2^2+2)r^2} + x \\frac{x^2}{(2^2+2)r^2}\\cdot\\frac{x^2}{(4^2+4)r^2} - \\cdots ", "3569556ab90a750d69655464613592c5": "\\begin{matrix}\\operatorname{Hom}(Y,-) &\\rightarrow& \\operatorname{Hom}(X,-)\\end{matrix}", "3569bf1a9c94e7f73d64563b83fce5d4": "\\frac{d}{dt}\\frac{P}{K}=r\\frac{P}{K}\\left(1 - \\frac{P}{K}\\right)", "356a62e061722ede4ebd2651528b8877": "\\tilde{f}_i(s_i,X)=s_i P(X)-C_i(s_i) ", "356b0d243784ea2cc50407afc50bb10b": "\\mathrm{(2n+1)H_2 + nCO \\rarr C_nH_{2n+2} + nH_2O}", "356b378cbded37a3fb968870fd3a0bd7": "\\tilde O((\\log n)^3)", "356b49e22e0fde73d8052a7c220bce68": "1-h^0(X,K)=1-g.", "356be4db01823d9b26c6dacccddd11b1": "\\alpha|0\\rangle + \\beta|1\\rangle", "356c1ea618fb48dad828a941f84a73ee": "v_{i}(\\mathbf{X})=u(X_{i},\\mathbf{X}_{-i}),", "356c24dff22c4a812a5fe07d3d56b042": "\\lambda\\in\\Lambda", "356c97e2c4cc7758e5130c2c706baca4": "\\beta_{max}", "356cc96dcf1f1c0d74e9b43fe0406a47": "\\mbox{kg/mm}^2", "356d02f97acefaf9474bcab72fa74a37": "\\mu(k/d)", "356d3f7388b9f795bb474aa930616bff": "\\beta_c + \\beta_a = 1", "356d6af5a67fbc276c88e691c754ceea": "\\nabla f = \\frac{\\partial f}{\\partial x_1} \\hat{x_1} + \\frac{\\partial f}{\\partial x_2} \\hat{x_2} + \\frac{\\partial f}{\\partial x_3} \\hat{x_3}", "356d813066a6db254ad6e849ec20b9ec": "u=0\\text{ on }\\partial \\Omega", "356d9cd8c6d077f8eec7a67bbaf6cbd9": " m^{-1/2 - \\epsilon}", "356db6cf02c972c3a8ab563678e8812d": "y \\in y \\iff y \\notin y", "356db87e6f8bd3cbf0725cbc3a96cf2f": "\\scriptstyle\\frac{3\\ \\,7\\,\\,2}{4\\,\\,12\\,\\,3}\\,5", "356dc4c04f59a4a897338a945e7d8c24": "K_{sp} = x^x y^y \\frac{{(-1)}^x {(-1)}^y {(N_{AxBy(\\Delta)})}^x {(N_{AxBy(\\Delta)})}^y}{V^{(x+y)}}\\,", "356e3ae845875e152d439239c77d82e0": "\\, D_0", "356e5e0fa122463b59fb3c6398f2ef50": "C_r = C_m\\cdot\\frac {r}{m}", "356ebb6267cba93a286adeb4c538e6b4": " a(t) ", "356f1ab89c2bd5732cd704671491b8ee": "2^{75}", "356f25bd3b6fc3d092b7eea25c01ad0f": "\\textstyle\\prod_j (1 - \\alpha_{ij}T)", "356f430db02e96ac8afa139d371b13c1": "H_R=\\frac{J\\bar{g}}{2}\\sum_i\\mathbf{L}_i^2-J\\sum_{\\langle ij\\rangle}\\mathbf{n}_i\\cdot\\mathbf{n}_j", "356f5f5ec154b74cce29f783acd602c7": "\\frac{\\partial (\\mathbf{u} \\cdot \\mathbf{A}\\mathbf{v})}{\\partial \\mathbf{x}} = \\frac{\\partial \\mathbf{u}^{\\rm T}\\mathbf{A}\\mathbf{v}}{\\partial \\mathbf{x}} =", "35702e6db962f64ea201735f95fbe68e": "E^1 = \\cup_{i=1}^k \\Lambda^{e_i}", "35706ea2aaa059a659f14ce02f1635c7": " \n\\pi(\\mu,\\tau) \\propto \\tau^{\\alpha_0-\\frac{1}{2}}\\,\\exp[{-\\beta_0\\tau}]\\,\\exp[{ -\\frac{\\lambda_0\\tau(\\mu-\\mu_0)^2}{2}}].\n", "3570793ec72acadb9b5b2e78983e9234": "PO(2k+1) = PSO(2k+1) \\cong SO(2k+1)", "35707e01cfacd1c5d441f2df30f17559": "E_{t-1}(U_{t+\\tau}) = U_{t-1}", "35709ffcc7bbb407c40c3c602331389e": "{\\tilde{D}}_7", "3570c83793775778b5acefb780e7bac7": "\\mathfrak{b}^{ce}=\\mathfrak{b}", "35710ca87b0fe8ae3352e43c51b7a836": " \\ell +1", "35712d28e4f10fac641e03351baf554e": "\\langle \\delta \\vec{r} \\rangle _{vac} =0", "3571e490f36867ff77854baea4d4ab25": "f : R \\to S", "3573170b762d9abcebc0a63117bd3496": "|B_{i+1}-B_i| \\le c_i", "357332e3846aa3ffd2aabb873688f3ac": "\\sin \\phi", "3573334ad0e1717a937d7a60988c3cad": "\\frac{1}{4\\pi}\\int_{S^2} n(v) \\, dA", "35733d0d87d61e606c0ad0b4ebec2ed3": "\\scriptstyle\\hat{s}(t)", "35737ce722f12ea0e42b78504b941e10": "\\left(\\begin{smallmatrix}\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n1 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\\\n1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\n\\end{smallmatrix}\\right)", "3573915668f8af4bdd33e556157d8ce0": "f(y') = x \\odot y'", "3573c3a7dad5f24d26979feeccc670a4": "\\mathbb{E}[M]=\\mathbb{E}[|X_0|]+\\sum_{s=0}^\\infty \\mathbb{E}\\bigl[|X_{s+1}-X_s|\\cdot\\mathbf{1}_{\\{\\tau>s\\}}\\bigr]", "357424692f692fccefe372ff70e93bcd": "\\lambda<\\liminf_{n\\to\\infty}x_n", "357476b46b07a9565b1376994d85223c": "1 \\over 23", "35748a295d7fe82fb240fcbcb14535fb": "a_1x_1+a_2x_2+\\cdots+a_nx_n>b", "3574a57761ac26929d60b06e1484b86c": "\\rho=\\frac{-r+2M(v)}{2r^2}\\,,\\quad \\mu=-\\frac{1}{r}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\varepsilon=\\frac{M(v)}{2r^2}\\,.", "3574adabcf3f3a2322921e855672054c": "\\bold(I)=", "3574e790d9f9a595c17e7dc76fbdb6f4": "x_1 x_2^3 x_4 x_5^2", "35752e8e764279fe596bb3136d6ee8be": "\\textstyle \\frac{0.99}{0.01} = 99:1", "3575f3631fc8dd320a32d63f4e2e01a1": "\\textstyle C_n=\\sum_{i=0}^{n-1}C_iC_{n-1-i}", "357646108e7028b45a15cb75d7ceb78b": "\\mbox{F}\n= \\,\\mathrm{\\frac{A \\cdot s}{V}\n= \\dfrac{\\mbox{J}}{\\mbox{V}^2}\n= \\dfrac{\\mbox{W} \\cdot \\mbox{s}}{\\mbox{V}^2}\n= \\dfrac{\\mbox{C}}{\\mbox{V}}\n= \\dfrac{\\mbox{C}^2}{\\mbox{J}}\n= \\dfrac{\\mbox{C}^2}{\\mbox{N} \\cdot \\mbox{m}}\n= \\dfrac{\\mbox{s}^2 \\cdot \\mbox{C}^2}{\\mbox{m}^{2} \\cdot \\mbox{kg}}\n= \\dfrac{\\mbox{s}^4 \\cdot \\mbox{A}^2}{\\mbox{m}^{2} \\cdot \\mbox{kg}}\n= \\dfrac{\\mbox{s}}{\\Omega}}\n", "3576649a422a5be09fc56e10896853c8": " q(n;d) = 1 - \\left( \\frac{d-1}{d} \\right)^n. ", "35768c22c6b8ca6b3e739753c6ba1135": "\\sigma \\propto \\exp \\left(-T^{-\\frac{1}{d+1}}\\right)", "3576b848a368ab6d4567136bc36cdc94": " \\displaystyle 2 \\times 10^{-7} ", "3576db31c762060db03c2c0002f0f2c8": "x_{k} = A x_{k-1} + B u_k \\,", "3576f774abb01d9f6f54c600b3968e0b": "V/n", "35770baff0878f93870b2af43a84c2ee": " \\frac{m}{n} = 2eV_1 \\left(\\frac{t}{f}\\right)^2 ", "357718e7b61abe85f0e16fe37c960802": "\\psi_{-}(q) = \\sum_{n\\ge 1} {q^{n}(-q;q)_{2n-2}\\over (q;q^2)_{n}}", "35771fcd469b26ff0b5e3f2fcf47fbdd": " \\delta W = \\sum_{j=1}^m \\mathbf{M}_j\\cdot \\left(\\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_1} \\delta q_1 + \\ldots +\\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_n} \\delta q_n\\right) .", "35774bcd065c9bccd34537f8f62f64eb": "Y=2k(\\phi(rear))=2k(\\theta-\\psi)+2k\\frac{b}{V}\\frac{d\\theta}{dt}", "357751c653a998ead1eb05c46b84bb1f": "p = 2 \\quad \\mbox{or} \\quad p \\equiv 1 \\pmod 4.", "3577b9e8d5f7d07d98bc94f5e857ed21": " F_i\\,\\!", "3577ce8c5b6fdb06e1b33c0cc596ad26": "\\vec{v}_{\\bot}", "3577d940e6a92c96451350ecf9c007cb": " \\cos A = -\\cos B \\, \\cos C + \\sin B \\, \\sin C \\, \\cos a ,", "3578083cc73105782c62482ca44535bd": "(p\\sin\\theta\\cos\\phi,p\\sin\\theta\\sin\\phi,p\\cos\\theta)", "3578781d07cdf4bdf2baa98cc520558f": "\\frac {0.55 \\cdot UF}{V}", "35790d991e8a52bef8204c0d7e808a23": "\\langle a,\\;b \\mid aba^{-1}=b^{-1}\\;\\rangle", "357979a922bc184cfad645b49540b498": " C_p\\rightarrow C_p+d\\Lambda_{p-1}+H\\wedge\\Lambda_{p-3} ", "357a0a12d9aa5504d21475b220be55b9": "w-c+D_k = 0,", "357a4b5f5752d7aec6cd34baba9eb434": "\\mathrm{height}(v) < k", "357a7e85a69a1fafa45949c9dad7b428": "t_m=z_1^m+\\cdots+z_n^m", "357aa4f9628667b8d0985224e42e5aed": "\\vec x|n", "357afa8b684d223270670f2b1f76d429": "y'(t) = f(t,y(t)), \\qquad y(t_0)=y_0, \\qquad\\qquad (1)", "357b4cbc9ed61005529a5ced543f9886": "A < B", "357b927bc48cd6db8e6d3cc54b4d6ab7": "\\frac{1}{1+z} \\; g(z) = h(z) \\; \\frac{1}{\\sqrt{1-z^2}}", "357b9b413643098d163a190ec7ae5a02": "\\ T(n)= O(n^2) \\, ", "357bae902e8d1abc41b4f6ae78566a24": "B = a_2ZZ_1(X_1+ZZ_1) = 2", "357c133a1e701336cf7772be4baf6675": "\\scriptstyle \\mathbb R", "357c1d58fb75295febeb238dc3ff43db": "\\neg [a] \\neg \\phi", "357cb45050f7fbdf72dd7d3a822646fe": " 0\\leq s\\,", "357d5bf876f8c907aeca237bdb1f673f": "V = \\int_0^{2\\pi} \\! \\int_0^\\phi \\! \\int_0^r\\! r^2sin{\\phi} \\, \\mathrm{d}r \\, \\mathrm{d}\\phi \\,\\mathrm{d}\\theta = \\frac{2 \\pi r^3}{3}\\int_0^\\phi \\! sin{\\phi} \\, \\mathrm{d}\\phi = \\frac{2 \\pi r^3}{3}(1-\\cos\\phi) ", "357d7bc6b1c595045dda7e6cecd0f4c2": " L(x,v,t)", "357d90e9c46e76377d8ba49664443ade": " \\tau_n = y(t_n) - y(t_{n-1}) - h A(t_{n-1}, y(t_{n-1}), h, f). ", "357d9e94db05375b4a8b053cd71bcb2b": "\\beta<\\kappa", "357dfe2c3cd418fb704ea5cf6e34b2da": "r = |x - y |. \\, ", "357e564112c8be448ad371ce602b0658": "A=0", "357eb8575037fc2d9edb86222e22843d": "c_p dT - \\alpha dP = 0", "357f04a3c2c66ac3ed741fee3079b005": " \\vec{B} = -q \\, \\sin(\\omega u) \\, \\vec{e}_3 ", "357f30c14d25e4a8bb979a1632eb2b9e": "\\partial_i=\\frac{\\partial}{\\partial x_i}", "357fbcfa16a135954040bd594fc3ba78": "\\pi_2(x) \\sim 2 \\prod_{p\\ge 3} \\frac{p(p-2)}{(p-1)^2}\\frac{x}{(\\log x)^2 } \\approx 1.32 \\frac {x}{(\\log x)^2}.", "357fcb8f78a86ebbd90c855ecbdc8893": " P = \\boldsymbol{\\tau} \\cdot \\boldsymbol{\\omega},", "358032b2c8ce73c72b80abb875e6d6f6": "Q_{dump}", "3580410f50f68c2e5c05f1de0f6e9dbd": "D+Y = \\frac{t^2+t}{2} ", "35809f77151e52ed2d2ccfd05eff9e93": " K >1 ", "3580dcf791bc48d05aaeea10008677b2": "\\frac{\\Delta S}{R}", "358114d845fad905741a4bbd0da935de": "\\vec{F}_{1,2} = \\frac{\\mathrm{d}\\vec{p}_{1,2}}{\\mathrm{d}t} = -\\vec{F}_{2,1} = -\\frac{\\mathrm{d}\\vec{p}_{2,1}}{\\mathrm{d}t}", "35814ce4bb90218c38eab10a26a33d23": "\\scriptstyle \\mathcal{F}", "3581511ef344bf7189ad9ccff71c8a47": "X \\to 1 \\to (q + 1) = X", "3581631cacc60aefc7d7934e475d0dc9": "k = \\frac{d^4G}{8ND^3} \\,", "35818e35385dfbfee3b2158830a18a58": "J \\equiv T \\cdot \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\hat\\theta)\\bigg)' \\hat{W}_T \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\hat\\theta)\\bigg)\\ \\xrightarrow{d}\\ \\chi^2_{k-\\ell}", "3581a0bd0f3d154aad723ac61264e02b": "\\displaystyle{E\\times E \\rightarrow E,\\,\\,\\, a,b\\mapsto ab =ba,}", "3581f3d62f6629abd9b479d77dc749a5": "\\scriptstyle x^\\mu", "3581f46d6ad261fa6047c1963f945f32": "\nQ_i(t+1) = \\max[Q_i(t) + a_i(t) - b_i(t), 0] \n", "358256853d44f2225e9fe3dd42152cbb": "\nY_\\ell^m Y_{\\ell'}^{m'} = \\sum_{\\ell''} \\hat{A}^{\\ell''}(\\ell,m,\\ell',m',) Y_{\\ell''}^{m+m'},\n", "3582721ad1c11c3384969d94d1fbefeb": " \\nabla^2 \\mathbf{A} = (\\nabla^2 A_x, \\nabla^2 A_y, \\nabla^2 A_z) ", "35828405152ba156758f6d2dc9dd80a3": "K_{\\mathrm{eff}} =\\frac{1}{2}\\frac{m}{L}\\int_0^L\\left(\\frac{vy}{L}\\right)^2\\,dy", "358309636ec04fdee23e29c054e5baa9": "x^i=\\mathbf{x}\\cdot\\mathbf{e}^i \\qquad (i=1,2,3)", "35830ac000d4a77743366a4927a09105": "\\begin{array}{cc} \\begin{array}{rrrr} j &k & l & m \\\\ \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} a & b & c & d & e & f & g & h \\\\ \\hline a & & & & & & & \\\\ \\end{array} \\end{array}", "3583240b29a3bc846084d2eb0ceb6592": "m(j) = \\frac{\\sum_{i=1, i\\neq j}^{20}A(i,j)}{n(j)}", "35832496b109048a50da3f8f692279db": " \\mu_0(\\bigcup_{n=1}^\\infty A_n) = \\sum_{n=1}^\\infty \\mu_0(A_n)", "358362ad4b84f0fee10ff04d6abd2c04": "\\kappa^2 = \\sum_i \\frac{z_i^2 q^2 n^{0}_i}{\\varepsilon_r \\varepsilon_0 k_B T} = \\frac{2 I q^2}{\\varepsilon_r \\varepsilon_0 k_B T}", "35837b46a96e56a17719c9a6de889d64": "A \\subset [A]_{\\text{seq}} \\subset \\overline{A}", "3583b9907439d6e30eeaa5356871a1fe": "\\sigma \\approx 1.4826\\ \\operatorname{MAD}. \\, ", "3583cfe92e81469255c631912aab832c": " e^{-r \\tau} \\frac{\\phi(d_2)}{K\\sigma\\sqrt{\\tau}} \\, ", "3583e0cf543a398e4030562f0a8fe216": " \\rho= \\sum_i \\lambda_i |\\varphi_i\\rangle\\langle\\varphi_i| ", "358415c0d13910b070f1bfb554719b97": "L^{a,b,c}\\ ", "3584ebfb9bd59ea33a5f170c616e9bed": "k \\neq l", "35859204fdaddd123cd800d5105d4db4": " (I+A)", "3585ac5d81da761544db271c908c1ced": "\n\\begin{align}\n \\operatorname{cn}\\, (z|m) \n &= \\Bigl( 1 - \\tfrac{1}{16}\\, m - \\tfrac{9}{16}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, \\alpha\\, z\\;\n \\\\\n &+\\; \\Bigl( \\tfrac{1}{16}\\, m + \\tfrac{1}{32}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, 2\\, \\alpha\\, z\\;\n \\\\\n &+\\; \\Bigl( \\tfrac{1}{256}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, 3\\, \\alpha\\, z\\;\n +\\; \\cdots,\n\\end{align}\n", "358643ee2cd9c02b8782183bc4a014d5": "e_\\lambda = \\frac{n!}{\\prod_{(i,j)\\in Y(\\lambda)}h_\\lambda(i,j)}.", "35866a6d2c652dbd52d473eae390c9bb": " A_i \\cap A_{j} ", "358680e8ca72ae23015955e7c27e7f66": "G(x) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}} e^{-\\frac{x^2}{2 \\sigma^2}}", "3586be6d9ac9a6d340bfa64c18fb7aa4": "1 = 0.02x", "3586c021dff29da0a8585957ce0dd749": "p-(D/p)", "35870b14e47851a8a24d5ff4f5127380": "\nx=\\frac{-b \\pm \\sqrt {b^2-4ac\\ }}{2a}.\n", "3587308b96da299e5550fd34a763e870": "\\lambda_{i}", "35875e34e4e7663e7696750f0c28dd21": " T \\equiv {\\bar x_1 - \\bar x_2 \\over \\sqrt{s_1^2/n_1 + s_2^2/n_2}} ", "3587e651838f64d01620aa760490cd03": "\\|x\\|_2 = \\left(\\sum_{i=1}^n |x_i|^2\\right)^{1/2}", "3587e6a89649a49de6623a342b03d203": "R_a(s,s')", "3588cae06fd72ca92d47d791b9a36287": "H^2 + r_p^2 = (H + l_p)^2 = H^2+l_p^2+2Hl_p", "35892667e58774ac40723c582f92de3b": "(A \\to B)\\and \\neg B", "358962d68da1562fa71078a35a431e38": "MA = \\frac{F_B}{F_A} = \\frac{V_A}{V_B} = n.\\!", "358a210fe66082c5280da06472be74db": "\n p(s)= a_0 + a_1 s^1 + a_2 s^2 + ... + a_n s^n\n", "358a2f40342c30f498450ff599596ad5": "O(m\\log n \\log \\log n)", "358a6bb68273a3b23dc1a0970672401f": "x_n = F_n / F_{n-1}\\,", "358a6ef3bb7c41ea1d04af6803352051": "s_{average}", "358a9f4f27380235f96bae49050a3716": "(e^{\\frac{k}{m}t}v_y)^\\prime = e^{\\frac{k}{m}t}(-g) ", "358aa202e04351b35e2c464b34a0f839": "\\frac{1}{P(D)}(e^{ax}y)=e^{ax}\\frac{1}{P(D+a)}y.\\,", "358ab925088b581ec685350d4446a4eb": "(k=0.2,\\;\\sigma_0=5)", "358ae3fe0ad919912b8b123e213bbcc6": " \\rho_q ", "358b4a85d005d17cece65388251c3771": "\\alpha + \\frac{n}{2},\\, \\beta + \\frac{\\sum_{i=1}^n (x_i-\\mu)^2}{2}\\!", "358b68c82a50406e08753737132628a5": "\n\\langle m | [\\hat{A}, \\hat{C}^{(k)} ] | m \\rangle = \n\\begin{cases} \n 0, & \\mbox{if }k\\mbox{ is even} \\\\\n 2 \\sum_n (E_n-E_m)^k |\\langle m | \\hat{A} | n \\rangle|^2, & \\mbox{if }k\\mbox{ is odd}.\n\\end{cases}\n", "358b6f45371fac64cd8707f0364e88e1": "H^+/H^-", "358bafe296bacad5703ace8a613a141d": "\\scriptstyle{\\mathrm{R}^- \\in \\mathrm{R}^-}", "358bc806b93f0a31b5e41d288a1e1bf1": " a + -\\infty = -\\infty. \\, ", "358bf37edaa144f7538c3163450b99d3": "5 = 25 - \\phi(25)", "358c42bf286a8f0627c579a5517961e0": "\\! ds^2 = -c^2 d\\tau^2 = - c^2 dt^2 + a(t)^2 \\left( \\frac{dr^2}{1 - kr^2} + r^2 \\left(d\\theta^2 + \\sin^2 \\theta d\\phi^2 \\right)\\right)", "358cbf68e6ca7b8848c1f9508270591d": "\\sum_x \\operatorname{artanh}\\, a x =\\frac{1}{2} \\ln \\left(\\frac{(-1)^x \\Gamma \\left(-\\frac{1}{a}\\right) \\Gamma \\left(x+\\frac{1}{a}\\right)}{\\Gamma \\left(\\frac{1}{a}\\right) \\Gamma \\left(x-\\frac{1}{a}\\right)}\\right) + C", "358daa2dad0bea3d31a223be9636056f": "h_{n+1} = \\frac{2}{\\frac{1}{a_n} + \\frac{1}{h_n}}, \\quad h_0=y", "358e058a1878f86666e03738f28dad0b": "\\pi=\\sum_{n=0}^\\infty \\left(\\frac{4}{8n+1}-\\frac{2}{8n+4}-\\frac{1}{8n+5}-\\frac{1}{8n+6}\\right)\\left(\\frac{1}{16}\\right)^n\\!", "358e18f3fffdff70ed6e5244a5e4c6f7": "N_1 A = A N_2", "358e31d1eb01d89055acba2620fa6303": " \\mathbf{x} \\leftarrow \\mathbf{E}\\mathbf{D}^{-1/2}\\mathbf{E}^T\\mathbf{x} ", "358f057ec0031142cd9e162c12be83a9": "\\varepsilon_i\\sim\\mathcal{N}(0,1)", "358f0a1acafbf7395a13e0429ad5825e": "Y = a_{N_p} (X_1 + X_2 + \\cdots + X_{N_p}) + b_{N_p}", "358fdb36fb35429ea307967749ca1923": " z_{\\mathrm{S}} f_{\\mathrm{FD}} \\mathrm{d}{\\it{\\epsilon}} \\mathrm{d} K_{\\mathrm{p}} ", "358fef8bbc0cacb87139a79767f3252b": "\np_6(x) = (x-3)(x+3)(x+5)(x+8)(x-2)(x-7)\n", "35905378aeab43d87c1add9fa0bfe631": "\\varphi_\\pm = \\frac{1\\pm \\sqrt{5}}{2},", "3590cb2fb11b30b235dd4d27f8cf914f": "\\int f'(x)e^{f(x)}\\;\\mathrm{d}x = e^{f(x)}", "3590cb8af0bbb9e78c343b52b93773c9": "'", "3591136299003fed028e64ac3a5fb111": "i+r+16=L(M)\\,", "3591182e3c98d5ce6cec2acd183d9d0d": "F = \\{ \\mbox{HALT} \\}", "35914c0f72fdd9d66f220f6e8cbe4396": "\\displaystyle iv_t+u+v|u|^2=0", "35917aa1198c40a7e09113d40bb55c4b": "\\mathbb E \\big( \\big| \\mathbb E(X|Y') - \\mathbb E(X|Y) \\big| \\big)\n \\le \\sqrt { 2 \\log 2 \\, I(X;Y|Y') },", "35917f822b72fb3cc8eb0e18838f4971": "(m_0, \\dots, m_{L-1})", "35918883f4789e89111a38f447924179": "\\,\\!V_{\\rm XC}", "3591ce9b1acdd8c1e2503c8e6d002ddf": "c(a, x) \\geq 0", "3591f27afd878f657dde228c391aaa82": "V_v", "35924363d7a9611ed2ba73e1f3e432c6": "B\\mathbf{x}_0=\\mathbf{b}", "359258bb7f69c36ec2db31de29e357ed": "\\alpha_1\\neq 1;\\alpha_m=1", "359276970cdade9f539ade75f504c919": "|x_n-x|<\\varepsilon u", "35927d4654e39eb602195fca0d1f7dd8": "\\mathrm{VSWR}= \\frac{1+|\\mathit \\Gamma|}{1-|\\mathit \\Gamma|}", "35927d80e873ed3f203b55095e38ac43": "\\ \\displaystyle \\mathcal{U}(\\alpha,\\tilde{u})\\ ", "359293d9873d4a8f6991d9500d12094b": "\\dot{\\mathbf{x}} = \\overbrace{f(\\mathbf{x},t) - B(\\mathbf{x},t) \\left( \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} B(\\mathbf{x},t) \\right)^{-1} \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} f(\\mathbf{x},t)}^{f(\\mathbf{x},t) + B(\\mathbf{x},t) u} = f(\\mathbf{x},t)\\left( \\mathbf{I} - B(\\mathbf{x},t) \\left( \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} B(\\mathbf{x},t) \\right)^{-1} \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} \\right)", "3592b4a8516b55711652bff1287d9088": "k = ?", "3592d1edefcf847d82f1a481229938cd": "\\, e^{\\lambda(e^{it}-1)}", "3592f552bfd6c8d1ccb0b5150377a87b": "\\oint_{\\Gamma_k} \\left|\\frac{\\tan(z)}{z}\\right| dz \\le \\operatorname{length}(\\Gamma_k) \\max_{z\\in \\Gamma_k} \\left|\\frac{\\tan(z)}{z}\\right| < 8k \\pi \\frac{\\coth(\\pi)}{k\\pi} = 8\\coth(\\pi) < \\infty.", "359315a774157e310c5b0cbf2acb34bc": "\\Psi_4(t,r,\\theta,\\phi) = - \\frac{1}{r\\sqrt{2}} \\sum_{l=2}^{\\infty} \\sum_{m=-l}^l \\left[ {}^{(l+2)}I^{lm}(t-r) -i\\ {}^{(l+2)}S^{lm}(t-r) \\right] {}_{-2}Y_{lm}(\\theta,\\phi)\\ . ", "359327c86f07c078ce78c457eeb3ce27": "M\\left(a,b,\\frac{x y}{x-1}\\right) = (1-x)^a \\cdot \\sum_n\\frac{a^{(n)}}{b^{(n)}}L_n^{(b-1)}(y)x^n", "359339a9bdedf9f534914d23a6290b85": "\\mathcal{T}_\\mbox{seq}=\\mathcal{T}", "3593484a55b98360932984cc2ab55cf2": "\\langle (\\delta \\vec{r} )^2\\rangle _{vac}=2\\frac{\\Omega}{(2\\pi )^3}4\\pi \\int dkk^2\\left(\\frac{e}{mc^2k^2} \\right)^2\\left(\\frac{\\hbar ck}{2\\epsilon_0 \\Omega}\\right)=\\frac{1}{2\\epsilon_0\\pi^2}\\left(\\frac{e^2}{\\hbar c}\\right)\\left(\\frac{\\hbar}{mc}\\right)^2\\int \\frac{dk}{k}", "359377698eff2752d2b4f7b78a6da141": "L = \\{z : \\forall x. \\exists y. \\phi(x,y,z)\\} \\, ", "35937982097be483dcf10fbbb4b6488b": "g_{n,-n}(r)=A(n+\\gamma)\\rho^\\gamma e^{-\\rho/2}", "35938fe01f4e315457759033e1cd53ef": "\\Re{s} > 0", "3593a27a69182c9ed76014ee89bee1e5": "A = \\left( \\frac{1}{\\sinh{\\gamma}} - \\frac{1}{\\gamma} \\right) \\ \\epsilon_{ijk} \\frac{x_j}{\\gamma} \\sigma_k \\, dx_i, ", "3593e4ed0c2a8bb2747be99c0c01a85c": "ln(1+i)\\approx i", "3593fa55bbbdb1467a3af14340beb5b6": "PSL(2,\\mathbb{Z})", "3594984abcf5c27271a899a7d0b01163": "H_n'(x)=2nH_{n-1}(x),\\,\\!", "359531e8a1566eb8bddf79f15a28e93f": "2^{2^{|Y|}}", "359540f6e85a1af73bacf562415ff544": "p(x_1^n)", "35955eec82ff87b118d9f3d124b295cb": "\\gamma = \\frac{1}{\\sqrt{1 - v^2/c^2}} = \\frac{1}{\\sqrt{1 - \\beta^2}} = \\frac{\\mathrm{d}t}{\\mathrm{d}\\tau} ", "3595bd6ceb17da4fda97b9bf12551ff4": "Q_M(a,b)=\\langle a\\smile b,[M]\\rangle.", "3595fb58037030b76bc1655de1775e76": "t^+ ={1\\over {2}}(t+\\sigma t)", "35964324711c9c58baafff1a9449a88a": "\\text{left}", "3596d6219f16a348ecb4cfff815a6649": "J_{-n}(x) = (-1)^n J_{n}(x).\\,", "359708896fdbba1d0dac50c395db347d": "\\tau(n)\\equiv\\sigma_{11}(n)\\ \\bmod\\ 691.", "359714a7b7aa1689a5af5d89ed7976db": "\na \\in [-\\infty ,\\infty ] ,\n", "35971fd4e10e134849767b0bb092e63a": " \\nu \\times \\lambda = c", "359748a6bf0d8334aa3799bb199647ac": "\nR=|\\langle z \\rangle| = \\frac{\\lambda^2}{1+\\lambda^2} .\n", "35974de9f8e00db179e482056921d1b1": "\\frac{1}{3} s^2 h\\;", "3597512f62c14e61ee31938aa8078fa3": "\\delta(g(x)) = \\frac{\\delta(x-x_0)}{|g'(x_0)|}.", "35976e1ce4f81b169311e9971ca4b25e": "| \\eta | \\ll 1", "35977cce8785e6a8aa2015aadc67777b": "F'(0) > 0", "3597a22dda9ff6868bf1fa39afcb0fe7": " C_0 = 2 \\sqrt{ 3 } - 3 \\quad ( \\approxeq 0.464 ) ", "3597a88033922fc8bae846669eb8a24f": " \\frac{Z^4}{n^3(j+1/2)} 10^{-5}\\text{ eV}", "3597d73249e96d54ec1a3e55b13b64ea": "a_1\\geq a_2\\geq\\dotsb\\geq a_i.", "3598084200830a04c2a8fe91fba62c43": "P[N=k] = p_k= (a + \\frac{b}{k}) \\cdot p_{k-1},~~k \\ge 1.\\, ", "35982c872d87920734e082ade0460868": "(x,y)\\in R_{i}", "3598337728aa0ae0584ca32a2d63b548": "1-e^{-\\alpha\\log{\\frac{x}{\\sigma}}-\\beta\\log[{\\frac{x}{\\sigma}}]^2}", "35984c8fa1bf0aadd4ef26b6432a4ef2": "r \\geq \\phi(L)", "359858c2abe3eec62cd565937e9e42c6": "f=(\\text{red},\\text{green},\\text{blue})", "3598a443597f79780be002981bc7a22f": "\\int_1^a x^n\\,dx = \\tfrac{1}{n+1} (a^{n+1} - 1) \\qquad n \\neq -1.", "3598e66291dd06fb2451fdaedb0aee61": "\\pi_1(N)", "35990b77bf9527f97a6d251d541f125e": "dr =\nd\\theta = d\\phi = 0 ", "35992876dc12a198e52f7ed6bffe93ba": "\\tan^2(\\alpha)/m^2=\\frac{1-\\cos^2(\\alpha)}{\\cos^2(\\alpha) m^2}", "35992ef6574500b450892bf7371a914a": "a_N > c - \\varepsilon ", "3599c7d313c41cd6c5ec6e15af850535": "v_\\mathrm{IN}(t)=V_\\mathrm{IN}+v_\\mathrm{in}(t)", "3599f36890211ac946c042907539553b": "\n W = C_{1}~(\\bar{I}_1 - 3) + D_1~(J-1)^2 ~;~~ J = \\det(\\boldsymbol{F}) = \\lambda_1\\lambda_2\\lambda_3\n ", "359a2f88f3f5b03a0e9807b8b50a9269": "BU", "359a6d539512a397eb739bde6c21325b": " =\\frac{\\tfrac12\\times\\tfrac13}{\\tfrac12\\times\\tfrac13+0\\times\\tfrac13+1\\times\\tfrac13} = \\tfrac13.", "359ac317a372f52d501e42a1ca919144": "g^{(k)} = \n\\begin{cases} \n \\partial f_0 (x) & \\text{ if } f_i(x) \\leq 0 \\; \\forall i = 1 \\dots m \\\\\n \\partial f_j (x) & \\text{ for some } j \\text{ such that } f_j(x) > 0 \n\\end{cases}", "359aca466bb88780aac5e374aa1aac47": "\\omega_r=\\frac{d\\theta_r}{dt}", "359ae1bd4001301d4c85c136733244c0": " D\\phi_2 -\\bar{\\delta}\\phi_1=-\\lambda\\phi_0+2\\pi\\phi_1+(\\rho-2\\varepsilon)\\phi_2\\,, ", "359b612a3a4829ab174371364e826ad5": "d (x^2(x^2 + y^2)-a^2y^2) = 0 ", "359ba363d5679236f45e0ce220caaf42": "R(\\omega; x,y) = \\sum_n \\frac{\\psi_n^*(y)\\psi_n(x)}{\\omega_n - \\omega}", "359bdaabe31a75934e0c84e1ca09dc35": "\\textstyle\\frac{2}{4}", "359c10400b6f5e1e850e89bb0c1c8241": "\\textstyle {x=a\\cos(t)-\\frac {a\\sin^2 (t)}{\\sqrt 2}, \\qquad y=a\\cos(t)\\sin(t)}", "359c191e7bdbec65238451a60a935986": "a= d\\,(\\mu^2\\nu)^{\\frac{1}{3}}\\,\\!", "359caf8c1e59db0c87af97987c2a366b": " P(a) \\land P(b) \\land P(c) \\equiv \\neg (\\neg P(a) \\lor \\neg P(b) \\lor \\neg P(c)) ", "359cd6454e3a5666d94161fa84204d81": "\nd^j_{m'm}(\\beta) = (-1)^{\\lambda} \\binom{2j-k}{k+a}^{1/2} \\binom{k+b}{b}^{-1/2} \\left(\\sin\\frac{\\beta}{2}\\right)^a \\left(\\cos\\frac{\\beta}{2}\\right)^b P^{(a,b)}_k(\\cos\\beta),\n", "359d325ac0829219cdfa6767c26be2b4": "X_t= \\frac{1}{\\phi (B)}\\varepsilon_t \\, .", "359d34521e181e1fccd8b14af37ec985": "\n\\widehat{B}_x \\equiv \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_x) = \\begin{pmatrix}\n\\cosh\\varphi & \\sinh\\varphi & 0 & 0 \\\\\n\\sinh\\varphi & \\cosh\\varphi & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix} \\,,\n", "359d50b71049313d1915956b44bd95f6": "1 \\leftarrow 2", "359d87bf79ae16778f6dde8e85d42e53": "Gv", "359d8e866352317aebee8bbebac1a051": "C^y -C^{y -1}+-\\cdots \\pm C^0 \\ge b_y(M)-b_{y-1}(M)+- \\cdots \\pm b_0(M).", "359db0b2bc09498f82abb845eb7e69f8": " \\hat{\\boldsymbol{\\jmath}}", "359e6573ea13693035aca9551903414f": "x^2-680x+96000=2x^2", "359edf0ec3f1b98ce7254c50fac3139b": "= \\delta\\int_{\\sigma_A}^{\\sigma_A} L\\left(x_1,x_2,x_3,\\dot{x}_1,\\dot{x}_2,\\dot{x}_3,\\sigma\\right)\\, d\\sigma=0", "359f08db3d42ba455b020e5ee1474212": "\\epsilon = {\\mu \\over{2a}}\\,\\!.", "359f15107a4947e57956fabb9ad82d62": "T^{\\alpha\\beta} = \\begin{pmatrix} \\epsilon_{0}E^2/2 + B^2/2\\mu_{0} & S_x/c & S_y/c & S_z/c \\\\ S_x/c & -\\sigma_{xx} & -\\sigma_{xy} & -\\sigma_{xz} \\\\\nS_y/c & -\\sigma_{yx} & -\\sigma_{yy} & -\\sigma_{yz} \\\\\nS_z/c & -\\sigma_{zx} & -\\sigma_{zy} & -\\sigma_{zz} \\end{pmatrix}\\,", "359f3852856a696641fa5aef07f2bede": "W(x,p;t)=W(m\\omega x \\cos \\omega t - p \\sin \\omega t , ~ p \\cos \\omega t + \\omega m x \\sin \\omega t ;0) ~.", "359f390941c0d654a85d55bb26d63e1e": "Ind(C)", "35a0164cb251f1b238e292491905750a": "\\langle\\langle\\sigma_A(\\sigma_B-\\sigma'_B)\\rangle\\rangle", "35a0324ac6032d3851f2a4a572bdb0c6": "\\le \\sqrt[3]{n}", "35a05c400b7a2fd61fc4eb75e29d6b79": "c_1,c_2\\in\\Sigma^n", "35a06a302874bae72dbf35525dbf010d": "\\textstyle Q_{ID} = H_1\\left(ID\\right)", "35a0b7deb6b93e8eb78971b977a950b6": "\\{1\\} = H_0 \\triangleleft H_1 \\triangleleft \\cdots \\triangleleft H_{s-1} \\triangleleft H_s = G", "35a0c4ed0513a4080f45198aba7dcd3d": "f_{X_1,\\cdots,X_n}(x_1,\\cdots,x_n) = f_1(x_1)\\cdots f_n(x_n),", "35a0c6f36b7f4e003289bd2fd0de7613": "x_i\\in \\mathcal{O}_{\\mathbf{C}_p}/(p)", "35a0f5fab8dc61de04af2143f4347f91": "a = a", "35a12043c9d524c88c078fcea9e29e59": "E_{in}", "35a1238de4bbf319179d80473eaad021": "\\forall t\\in (-\\delta,\\delta): f(\\mathbf \\gamma(t)) = c .", "35a186f28368f66db3c3ecd010ad08ea": "B_n(x) = -n \\zeta(1-n,x)", "35a18ab2c2fcf8bd5fbd2bde6d16cae9": "\\|P\\|=\\int_{-\\pi}^{\\pi}\\!|P(x)|\\,dx.", "35a18af4916b9c0ee3bb2e5abbb063ed": "y=e^{i\\pi z}.", "35a198e0a1ce2d1c0a5576d575ed2df9": "\\vec{F} = \\frac{\\mathrm{d}\\vec{p}}{\\mathrm{d}t} = \\frac{\\mathrm{d}\\left(m\\vec{v}\\right)}{\\mathrm{d}t},", "35a1efd6a68f175550a25a19edbd374f": "d\\epsilon_x d\\epsilon_y", "35a28b765fdbe858789e9ea17eae2830": "\\Omega(g) = \\frac{1}{\\tau_g} (g_i^{eq}-g_i)", "35a2b9e90232fc04af31fff20b4a2cc9": "S_{n,m}(u) = \\sum_{k=0}^n s_k\\,u^k", "35a2ce94acb3f700a62cab451c9dbaf6": "\\hbar |\\mathbf{k}|", "35a2e8897e899c2230bfd6c83a82ca3e": " \\sum_{i=1}^n m_i\\,x_i\\mod p", "35a327fbfe70603c2cee36bdae5b16bc": "f(x+2h) - f(x-2h) = 4hf'(x) + \\frac{8h^3}{3}f^{(3)}(x) + O_2(h^4). \\qquad (E_2).", "35a374b204ceba175ccbb88264d52bfa": "S_t=e^{at}\\ \\frac{z-b}{a-b}\\ \\Bigg(1+\\left(t+\\frac{1}{b-a}\\right)(z-a)\\Bigg)+e^{bt}\\ \\frac{(z-a)^2}{(b-a)^2}\\quad.", "35a3996b6e8e0b234a1e85f1e7313f26": "\\textstyle{\\int x^{n-1}\\,dx = \\tfrac{1}{n} x^n}", "35a39c754c75d530020aeca13c469bde": "\\{5\\}", "35a3cc4777a03b3cc3a8d2b102228ef9": "J_n(x)=x^{2n+1}I_n(x)=n!\\bigl(P_n(x)\\sin(x)+Q_n(x)\\cos(x)\\bigr),\\,", "35a3e49f0d55eb333b6533eedd3a1fa4": "x>0,", "35a4318a9958a31a69015f4fe2e0419d": "\\displaystyle{(D(X),Y\\circ Z)+(D(Y),Z\\circ X)+ (D(Z),X\\circ Y)=0.}", "35a45b2712caac41fc6dfcfdad110750": "\\begin{bmatrix} \\dfrac{\\Delta \\mathbf{[h]}}{h_{22}} & \\dfrac{h_{12}}{h_{22}} \\\\ \\dfrac{-h_{21}}{h_{22}} & \\dfrac{1}{h_{22}} \\end{bmatrix}", "35a4cc20c6d7d5a1bf9117427c2ac679": "NE(X) = \\left\\{\\sum a_i[C_i], \\ 0 \\leq a_i \\in \\mathbb{R} \\right\\} ", "35a501c665b9e0c1454cd84344cdefdf": "= (1 + i 2\\pi fT) \\left( \\frac{1 - e^{-i 2\\pi fT}}{i 2\\pi fT} \\right)^2 \\ ", "35a5415c00d6604913e2aa1791b7d123": "\\mathbb{D}^{1,2}", "35a549dd33a9f688088c779ccdfe0a02": "\\left( \\frac{u^2}{4R^2} - 1 \\right) \\frac{d^2S(u)}{du^2} + \\left(\\frac{3u}{4R^2} - \\frac{1}{u} \\right) \\frac{dS(u)}{du} + \\frac{1}{u} S(u)= E S(u)", "35a5cb48f5abf9a375a0fbaa764c9761": "\\begin{align}\n \\dot{\\hat{\\mathbf{r}}} &= \\dot\\theta \\hat{\\boldsymbol{\\theta}} \\\\\n \\dot{\\hat{\\boldsymbol{\\theta}}} &= - \\dot\\theta \\hat{\\mathbf{r}} \\\\\n \\dot{\\hat{\\mathbf{z}}} &= 0 \\end{align}", "35a5fa87e884efbd3963ad54edf6d790": "M=(Q,\\ \\Sigma,\\ \\Gamma,\\ \\delta, \\ q_{0},\\ Z, \\ F)", "35a613caf6b2a4a175d53bc5c2c0fa9b": " P(z)= \\alpha \\prod_{i=1}^n (z-a_i) ", "35a6273dfef0567597e993abcf318090": "u(x)v'(x)-u'(x)v(x)\\neq 0.", "35a67c6a164fdcec2b4b5bfe7b196886": "\n\\gamma_n=\\int_\\mathcal{S} d\\mathbf S\\cdot\\mathbf\\Omega_n (\\mathbf R).\n", "35a6a4087fbdf8597d17108e5470d1dd": "\\vec{e}_1,\\dots, \\vec{e}_n", "35a6aaeb732cdb124b3739b798a09a78": "f_b(x) = \\left\\{\\begin{matrix}1&p(1|x)>p(-1|x)\\\\-1&p(1|x) \\frac{1}{2}(\\pi_0+\\pi_1)", "35b4e9218499d01f8a9f4665a540ddf0": "\\frac{| \\triangle SCA|}{|\\triangle CDA|}=\\frac{|\\triangle SCA|}{|\\triangle CBA|}", "35b54bc9ad7785c7a0e59b16f984c4f4": "a_n = r\\,a_{n-1}", "35b5a2b6ead996b28fd125687b62d7d8": "\\scriptstyle(-6.9(2.2))\\times10^{-16}", "35b5a2f2cf013651824ef4e680515fd1": "F[p]", "35b5b6dd65208db47c721cee4974dcc1": "{I_\\mathrm {v}=\\frac{I_\\mathrm {d}}{3}+ \\frac{I_\\mathrm {ac}}{2} }", "35b5f4987a0dab341e642fe7e0a09f40": " T(A \\and B) = T(A) \\and T(B) ", "35b60fdc7ba4b4c7b10a5a1330b60591": "A = C_1 \\cup C_2 \\cup \\cdots \\cup C_n", "35b670ed7ac850e14d7c2df35608d3c6": " x_i\\,", "35b73ce0a6793a4e2ed116bb0318e9c0": " z^2+3z+2 ", "35b7547b1382a81bc524d3308cc5b7a8": "\\mathfrak{o}(2l+1, F) = \\{ x \\in \\mathfrak{gl}(2l+1,F) | s x = - x^t s , s = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & I_l \\\\ 0 & I_l & 0 \\end{pmatrix}\\}", "35b7799597d4c4c05308a28b2e4c5cc4": "P(z) \\neq 0", "35b82d68ca6433b6eb55c932c438eded": "M(x)\\cdot\\frac{\\delta E}{\\delta x}(x)=0", "35b83b2d961943a840d07dc62c8c5e46": "f(x) = a_0x^n + b_0x^{n-1} + a_1x^{n-2} + b_1x^{n-3} + \\cdots \\quad (19)\\,", "35b86ac7ed6a62cb1f322bbc897ab8bb": "\\textstyle b_n = y^n/n!\\,", "35b8b0b12b9f4f200da7bbc4d360c137": "\n\\begin{align}\n f&=\\frac{a-b}{a}, \\qquad e^2=2f-f^2,\\qquad b=a(1-f)=a\\sqrt{1-e^2}.\n\\end{align}\n", "35b8bb84e3921b3fcbcb08a34035aebc": "\nG(\\mathbf{k},z) = \\int_{-\\infty}^\\infty \\frac{\\mathrm{d} x}{2\\pi} \\frac{\\rho(\\mathbf{k},x)}{-z+x},\n", "35b91ca198e8c587aa8ea8494b65f69d": "D(s)=s^5+4s^4+2s^3+5s^2+3s+6.\\,", "35b93b30541b73e817fafd6827c04afe": "\\mathfrak{Y}", "35b961751ec533be17ea34132807d680": "\\theta=(\\theta_1, \\theta_2, \\cdots, \\theta_n)", "35b99698c57488ee527b62d52ecd0784": "P_i=\\sum \\alpha_{ij}H_j + \\sum \\beta_{ijk}H_jH_k+\\ldots", "35b99b7e37a7a92eba5d6d2e0495ccd4": "K_{h_\\lambda}(X_0 ,X) = D\\left( \\frac{\\left\\| X-X_0 \\right\\|}{h_\\lambda (X_0)} \\right)", "35b9bc9d85ed421961cf58a42512dba1": "\\frac{\\partial u}{\\partial y}", "35b9e1c9cee6233645730b00e522d6c5": "x(t) = 0", "35ba1302be36f03a9418d49e4402d673": "f(a)= b' g(a) {b'^{-1}}", "35ba133570e4c20a7db751709ee16a92": "(k'+p')^2 \\approx \\,", "35ba571ac65bbaed38c08c3ae2f13299": "E(Y) = \\sigma \\sqrt{2/\\pi},", "35ba731133b682e258113310dfd6cee3": "p = p_s + \\vec{p}_v,", "35baed8ff9f7b939498be3d560772682": "\\textstyle ((0,1),\\operatorname{mes}) ", "35bbff8a35b00fe88392a36f046cde0a": "\\exists z[x\\le z\\and y\\le z].", "35bc00cfde5772144a4184703ce7026e": "\\frac{p}{p+q}\\frac{p-1+1-q}{p-1+1}=\\frac{p-q}{p+q}", "35bc10c4a5eef91cecf438f16a11ea32": "\\quad \\frac{\\partial T}{\\partial \\dot{q}_j} = \\sum_{i=1}^n m_i \\mathbf{\\dot{r}}_i \\cdot \\frac{\\partial \\mathbf{\\dot{r}}_i}{\\partial \\dot{q}_j}.", "35bcb340ee24c06d8351e0d969fb81af": "\\left( \\cos\\frac{\\theta}{2}\\ ,\\ \\sin\\frac{\\theta}{2} \\right)", "35bcebf658e1219d1e783b244a868ad9": " z-\\bar{z}=2iy ", "35bd1f1d508457210ae64c51a41daf91": "\\alpha + \\beta + \\gamma = -\\Omega,\\,", "35bd2f37fc7ffeb187778582c4a6fa8d": "g_i = \\frac{1}{1 + k_i z g_{i+1}}", "35bd4a240958ebc35fd2c9cb6c1d4979": "G_{LD}", "35bd9d14efddde838aa2b0a8c9426964": "2f", "35bdddb1a863195c1da1d5782a4e016d": "q^4", "35be182140bc7bf1d1b9fb0ac2d97274": "\\nabla \\times \\bold{E} = \\boldsymbol{0} \\ , ", "35be2898b0a3243e55be1af18d41d587": "\\|\\cdot\\|_{\\infty}", "35be3b1a35301c7ffe8b76285d4aa994": "\\mathbf{w}\\cdot\\mathbf{x}_j + b > \\gamma ", "35be7b9f9f2bc0385607e05b4c7ad12d": "U = S_r \\cdot r^2", "35be80c435f519fdd02bd3a6f72d4113": "\\int_{-5}^5 dz \\int_0^{2 \\pi} d\\phi \\int_0^3 ( \\rho^3 + \\rho z )\\, d\\rho = 2 \\pi \\int_{-5}^5 \\left[ \\frac{\\rho^4}{4} + \\frac{\\rho^2 z}{2} \\right]_0^3 \\, dz = 2 \\pi \\int_{-5}^5 \\left( \\frac{81}{4} + \\frac{9}{2} z\\right)\\, dz = \\cdots = 405 \\pi.", "35be9afa9f58dc7d9b35c6f9fbffdd67": "|A| \\leq |B|", "35bf1685d9802c39a6f4364a00f156d9": "H\\,\\mathrm{char}\\,G.", "35bf19398c494cfcc45b3c4767024096": "z \\in \\mathbb{D}", "35bf1b13465d57b38ef3c37b8ab253cf": "\\mathrm{Fr} = u_f/\\sqrt{ g' h }", "35bf40c521e0d5675a114059e8b5caac": "E_{ij} = \\sum_{a=1}^m x_{ia}\\frac{\\partial}{\\partial x_{ja}}.", "35bf62fdd9963665da900e9efbb8ed38": "\n\\begin{array}{c|ccc}\n0 & 1/6 & -1/3& 1/6 \\\\\n1/2 & 1/6 & 5/12& -1/12\\\\\n1 & 1/6 & 2/3 & 1/6 \\\\\n\\hline\n & 1/6 & 2/3 & 1/6 \\\\\n\\end{array}\n", "35bfa94b3f6721194c1183642212b9fd": "\\sigma_{il}", "35bfbbc3f265f5615f4e0db98c7c7a91": "\n\\begin{align}\n E &=& \\frac{\\hbar^2}{2m}\\left(\\frac{\\pi}{a}+\\kappa\\right)^2\\pm |V|\\left[-\\frac{\\hbar^2 \\pi \\kappa}{m a |V|}\\pm \\sqrt{\\left(\\frac{\\hbar^2 \\pi \\kappa}{ma |V|}\\right)^2+1}\\right]\n\\end{align}\n", "35c0730760c67abadbe19fc16b53abf3": "G\\in{\\mathbb R}^{I\\times J}", "35c0804c862a635e2fe8371dc43e25d0": " 1 ", "35c0c8de16896165ddcfbcf7b4d1d416": "\\Delta \\rho \\ ", "35c1112bac5ed5d938ba8feb031494df": "\\Gamma[\\phi]=S[\\phi]+\\frac{1}{2}Tr\\left[\\ln {S^{(2)}[\\phi]}\\right]+\\cdots.", "35c1ccc273c07465975527994feac74a": "_{p\\not\\subset q}\\!", "35c1d1edc69ec53132451ac951170e89": "\\sum_{i\\in I} r_i^d<\\delta ", "35c22770d2ad34405bdeee5534345aa6": "R_{ab} = k\\,g_{ab}.", "35c22ae721884bd02415e0d0b04a595a": "(-2^{j+1}, -2^j) \\cup (2^j, 2^{j+1})", "35c24ef4412706f2db824bc58583ad8f": "z(u, v) = r\\sin{v}+{P\\cdot u \\over \\pi}, ", "35c274cb1de79ae3a7170266fe85fb35": " \\frac{\\partial c}{\\partial t} = - \\left( \\frac{1}{N_v} \\right) \\left( \\frac{\\partial J}{\\partial x} \\right) = \\left( \\frac{m}{N_v} \\right) f'' \\frac{\\partial^2 f}{\\partial x^2}", "35c2a4658d5aa630da476ac74d63159a": "\\left( \\frac{1}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0 \\right)", "35c2e20a64de6a0c02884825cd4851c0": "\\sharp=\\sharp^g\\colon T^*M\\to TM", "35c3206ffd8e51a8d9cfd1c92502c9fe": "A2:= 27.399", "35c323307002994d5f3b4017fa33f56f": "\\operatorname{Cl}_{4}(2\\theta) = 8\\operatorname{Cl}_{4}(\\theta) - 8\\operatorname{Cl}_{4}(\\pi-\\theta) ", "35c364f0cfd447ab0d6d14950aad21e1": "C^T\\begin{bmatrix}a_k\\\\\na_{k+1}\\\\\n\\vdots \\\\\na_{k+n-1}\n\\end{bmatrix}\n= \\begin{bmatrix}a_{k+1}\\\\\na_{k+2}\\\\\n\\vdots \\\\\na_{k+n}\n\\end{bmatrix}.", "35c371ce23ce5759209b9993a5a02423": "\\partial_\\alpha F^{\\beta\\alpha} = \\mu_0 J^\\beta ", "35c3839de4076ccac50e465b8d801a53": "p_2 = \\{ s, v_2, v_3, v_4, t \\}", "35c389306c71a50ba2cd0b4211d4c375": "q(y)\\in \\mathbb{Q}[y]", "35c3c1337808e74459f2c3d04ef6f925": "f \\in L^p \\cap L^q", "35c3dda4bb05adb46e820a53dd06348b": "\\frac{c\\mu}{n}", "35c3ef103d026dc8a2ac2f6f3883dfdc": "h_j: \\mathcal{X} \\to \\{-1,1\\}", "35c40069652f311986d35c56b1e14c40": "\\log \\Gamma", "35c444f8235a8ce49143704189bfe741": "y_c = \\biggl(\\frac{q_\\mathrm{trans}^2}{g}\\biggr)^\\frac{1}{3} = \\biggl(\\frac{(16.4)^2}{(32.2)}\\biggr)^\\frac{1}{3} = 2.02\\text{ ft}", "35c47e7cf9d839effcd4f53c968d3d59": "b_1b_2\\dots b_n", "35c49d3053bf90cc640df66ac9698058": " {G^a}_b \\, {G^b}_c \\, {G^c}_d \\, {G^d}_a = R^4/4 ", "35c4ce89f68497c3762829060da2f6d9": "\\mathfrak{p}_0 \\subsetneq \\mathfrak{p}_1 \\subsetneq \\cdots \\subsetneq \\mathfrak{p}_n", "35c5479645a3e2c3f00497d10a8fd03d": "A_n':=\\frac1n\\sum_{i=1}^n (1-x_i),\\qquad G_n'=\\biggl(\\prod_{i=1}^n (1-x_i)\\biggr)^{1/n}", "35c55912eb58e196d5a1467e1f655dca": "i_r\\,\\!", "35c589913aba709f88cef25928f3ebbe": "D_{\\mathrm{KL}}(X_1,X_2) = D_{\\mathrm{KL}}(X_2,X_1), \\text{ if }h(X_1) = h(X_2)\\text{, for (skewed) }\\alpha \\neq \\beta", "35c59afed5d590046546453876dabf4d": "\n\\mathcal{F}_{n}\\left(\\psi_{tar}\\right)=\\langle\\psi_{tar}\\vert\\hat{\\rho}_{retr}^{[n]}\\vert\\psi_{tar}\\rangle,\n", "35c5af513d921f05a70bf9c9a7ccd3a3": "\\epsilon = \\Pi", "35c5fb5cb8a4ba85ffead83683036df5": "A^{\\prime}", "35c6bb91e5473dedc47995fdfa3be85e": "T_{\\mathrm {1n}}", "35c6cbad8d343135146e98a8070b319f": "39.5 \\pm 0.4", "35c6cde55c291bbc26bd4f76b624d6f6": " M^*\\Phi_{2\\lambda}", "35c6d881e7c2c988d14744b26e14953c": " z_{k+1} = z_k + hf(z_{k+1}) \\, ", "35c716637fdaff17b903cb3ab98ca75c": "\\delta_{ijl} = \\bar{J}_{jl} - \\bar{J}_{il},", "35c75ae302a4dae267e4665117952455": "a=\\frac{P}{\\omega g}", "35c7b20f7792b8f186d76b15005d1859": "p_2^2 = m_2^2.", "35c7c94e52a3ef21a16e6b6c559bbc1d": "C_g f = f \\circ g.", "35c7d6aa89da0a8db8ed96dc4b343234": "NaN + a = NaN", "35c812625cf7ef280c9386f61f01d8d4": "\\langle \\lambda_n \\mid n \\in \\mathbb{N}\\rangle", "35c83acab8432a44ffa8e273a5765c7e": "\\tfrac{1}{12n}(b-a)^2", "35c8625cf5c6af4f7420f86d33323dc1": "\n\\begin{align}\n \\psi &=\\ln(\\sec\\theta+\\tan\\theta),\\\\\n {\\rm e}^\\psi &=\\sec\\theta+\\tan\\theta,\\\\\n\\sinh\\psi &=\\frac12({\\rm e}^\\psi-{\\rm e}^{-\\psi})=\\tan\\theta,\\\\\n\\cosh\\psi &=\\sqrt{1+\\sinh^2\\psi}=\\sec\\theta,\\\\\n\\tanh\\psi &=\\sin\\theta.\n \\end{align}\n", "35c87d5f0c424abd103def3f7fe1e1f7": " y[k] = x[k] \\cdot h [0] + \\sum_{n \\neq k} x[n] \\cdot h[k - n] ", "35c88dfc0edb4e1bf36223eb65a23933": "C = C^{oo}", "35c8dff47071020ea9f4e6ccc054429e": " M^{-1}AM = D \\, ", "35c8eb657f586fb1854fed7720a2bfa0": "A^{-} = -\\sqrt{\\frac{2}{3}}\\ \\sigma^{-}", "35c8f303558746f6c1dafca8d727531c": " \\hat{\\mathbf{r}}_{21} = \\frac{1}{\\sqrt{(x_1-x_2)^2+D^2}}(x_1-x_2,D,0)", "35c8fe281ce04ccc0fbcc3c88a9fcfec": "p_{\\mathbf{y}}=p_s", "35c904a2e1094859530cb14119591087": "((P \\to Q) \\and \\neg Q) \\to \\neg P", "35c916481097298679690a8dec9b9dcb": " (w_{j_1}, v_{i1}) ", "35c9208c3e8768cfc0ca872eecc5520c": "\\begin{pmatrix}C&0&\\cdots&0\\\\U&C&\\cdots&0\\\\\\vdots&\\ddots&\\ddots&\\vdots\\\\0&\\cdots&U&C\\end{pmatrix}", "35c9274d518f4f8602c65bdd7cd7a8ad": "\\mathrm{NPV}(i) = \\sum_{t=0}^{N} \\frac{R_t}{(1+i)^{t}}", "35c93e12454f766989aab11ab85c22a0": "\\begin{align} \nSS_{\\rm err} & = \\sum_{i=1}^N\\;(y_i - \\widehat{y_i})^2\\\\\nSS_{\\rm tot} & = \\sum_{i=1}^N\\;(y_i-\\bar{y})^2 \\\\\nSS_{\\rm reg} & = \\sum_{i=1}^N\\;(\\widehat{y_i}-\\bar{y})^2 \\text{ and} \\\\\n\\bar{y} & = \\frac{1}{N}\\sum{}_{i=1}^N\\;y_i.\n\\end{align}", "35c9681f09cd73a40d574787e4e8392c": "S = \\left(\\frac{D}{0.8}\\right) \\times 0.2", "35c9c752390acc923ec16b30edca62c6": "\\Delta(u, v)", "35c9cbf9c6d6922380765f18c4ea5c92": "(p+q)", "35c9f9bba93a340efd124f565997b318": " \\textbf{P}_{k|k} ", "35ca6b20b3da964bb80078c6b8d5420d": "C_k\\supseteq C^1_k\\supseteq C^2_k\\supseteq \\dots", "35ca823f44995eb713f1402b3c633675": "Fr = \\sqrt{(\\frac{y_c}{y})^3}", "35cad2b27888e8aef98728ab271828ff": "\n[X_i ,P_j ] = i\\delta_{ij}\n\\, .", "35cae7b126e47c3b841873d5ea24b03e": "A^+A A^+ = A^+\\,\\!", "35cb1cc3b47a87defae1481971562082": "\\lim_{\\theta \\to 0^-} \\frac{\\sin\\theta}{\\theta} = \\lim_{\\theta\\to 0^+}\\frac{\\sin(-\\theta)}{-\\theta} = \\lim_{\\theta \\to 0^+}\\frac{-\\sin\\theta}{-\\theta} = \\lim_{\\theta\\to 0^+}\\frac{\\sin\\theta}{\\theta} = 1 \\, . ", "35cb2c10140222109d601d91b2b261b2": "F(a,b,\\ldots) = Z_G(f(a,b,\\ldots),f(a^2,b^2,\\ldots),f(a^3,b^3,\\ldots),\\ldots,f(a^n,b^n,\\ldots)).", "35cb4a78a2ac5c38468dac6025f9dc20": " H_k + \\frac {\\Delta x_k \\Delta x_k^T}{y_k^{T} \\, \\Delta x_k} - \\frac {H_k y_k y_k^T H_k^T} {y_k^T H_k y_k}", "35cbc8c03d2ab3e19b7eb3096ad62a77": "\\displaystyle{\\sum_{m\\ge 0} \\partial^\\alpha F_m}", "35cc32438acfeda0cd86ee44794eb90e": "\\pi_1(M_X,P)\\ ", "35cc9ae5d5e10f6caa3db66d0c91b1c1": "\\sqrt{\\frac{3}{28}}\\!\\,", "35cca852d4c7b5568eabea39c9144d59": " K = -V \\left(\\frac{\\partial P}{\\partial V}\\right)_T", "35cd3f9364d1919be6aaa334aebf6ccf": " \\sum_{1 \\leq i < j \\leq N} \\|x_i - x_j \\|^{-s} ", "35cd9ddc52a77a4bd8acc1c0a6638105": " erfc \\left ( \\frac {FDt}{\\beta\\,^2f^2} \\right )^{\\frac{1}{2}} ", "35cdaad648b1e2507b5c5ac8f66302d0": " T(s+t)= T(s) \\circ T(t), \\quad \\forall t,s \\geq 0. ", "35ce1c0b78f3922bcbae1d3900bba28a": "u\\Vdash\\Box A", "35ce6a7dad26f49226c58d3daf889a56": " g\\phi_n(g_1,g_2,\\ldots, g_n)= \\phi_n(gg_1,gg_2,\\ldots, gg_n).", "35ce912343841e11d28d99781ca736a7": "F_{O_2loop}=\\frac{(Q_{dump}+V_{O_2})F_{O_2feed}-V_{O_2}}{Q_{dump}}", "35cec7f100f73753f9b1fe7e90fe1e62": "\n E=\n \n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_{B}^2 }\n\\; M \\left ( \\mathit l + 1, 1, -{k^2 \\over 4} \\right) \\;M \\left ( \\mathit l^{\\prime} + 1, 1, -{k^2 \\over 4} \\right) \\;\\mathcal J_0 \\left ( k{r_{12}\\over r_{B}} \\right)\n", "35cef435c8a1bf5299255c0747900f9e": "\\mathit{dr}(n)=n \\Leftrightarrow n \\in \\{0,1,2,3,4,5,6,7,8,9\\}.", "35cf0445cf4b73823473c545d5a2d72d": "\\mathcal{A} = \\mathfrak{P}\\{\\mathcal{B}\\}", "35cf15d6cb7ed54ea50e052b8353db49": "\nx^2 = 2\\,\n", "35cf1b9ca8cc6af8258dc5c5e1e5f26b": "\\left|x_1, x_2\\right\\rangle", "35cf4854f0456a6fc98ddfeed664b854": "\\Delta t \\leq 0.1 \\omega_{pe}^{-1},", "35cf56e8543daa5703e82cce4812594c": "n^c_{\\mathbf k'}", "35cf63b242b184b32f026f8937ab11f8": "\\mathbf{u}_j", "35cf772de5a2c56f0508a5bcc245cce8": "X=U^2-T^2,\\quad Y=T\\,(T+2\\,U),\\quad Z=T^2+TU+U^2.", "35d0480617f173e857fd32bac389a901": "\\mathbf{A}\\frac{\\partial \\mathbf{u}}{\\partial x}", "35d05b4b6e8f2ad328f3ca2a8e4261be": "T(UTM) = H\\oplus V", "35d08a57e1340e92425e0663eecd98c0": " n.(H(X)+\\varepsilon) ", "35d0ebd316d9ff86c40b53420629becc": "\\vert \\bar{\\mathbf{\\gamma}}'(s(t)) \\vert = 1 \\qquad (t \\in I).", "35d1248dab65509b5e838c9fd3be6300": "f(Aa)", "35d15323df58fe4dad855c18417bc940": "x,y,r \\in \\mathbb{Q}, r > 0, x \\neq y", "35d1748582a31397678ea0b77873969b": "\\mathbf{ x}(3) = [u(3)\\, u(4)\\, u(5)]=[89\\, 85\\, 80]", "35d1a63fb7e2e19ba5355d8776329ee8": "f(t) = 6t^5 - 15t^4 + 10t^3", "35d1b04892902d86e1916d106e0c07e8": "\\scriptstyle\\operatorname{cov}", "35d1b6e9c8a75a0207e87207b269461b": " Q = 90L - L^2 ", "35d1f1f069a8ac22e377dd92290e11a6": "\\mathfrak{a}^*", "35d1f8e82fd1ed0aa9178e4bb36c9453": "X_1 = \\,\\!", "35d2028ec761bebd3708948ff8e416d8": "x = (x^0, x^1, x^2, \\ldots, x^d).", "35d20fffba2fd532a848c37e6585e7b5": "\n\\sqrt[n]{z^m} = \\sqrt[n]{(x^n+y)^m} = x^m+\\cfrac{my} {nx^{n-m}+\\cfrac{(n-m)y} {2x^m+\\cfrac{(n+m)y} {3nx^{n-m}+\\cfrac{(2n-m)y} {2x^m+\\cfrac{(2n+m)y} {5nx^{n-m}+\\cfrac{(3n-m)y} {2x^m+\\ddots}}}}}}\n", "35d23fa01af557f521e1ef77279323da": "N^*N = QV^*VQ = Q^2 = \\begin{bmatrix} P^2 & 0 \\\\ 0 & P^2 \\end{bmatrix}.", "35d240eefc1914c8e9036183156a875e": " \\frac{\\beta N}{\\gamma} \\le 1 \\Rightarrow \\lim_{t \\rightarrow +\\infty}I(t)=0 ", "35d2708b2854b898aec3b2b71faa6b59": "\\mathcal H ", "35d2cc619033891d8eaaab6924cbdf8e": "E = E^\\circ - \\frac{R T}{n F} \\ln Q_r \\, \\,", "35d2d7b3868065c411863aba4fd3e93c": "L_0 =\\begin{pmatrix} {1\\over 2} & 0 \\\\ 0 & -{1\\over 2}\\end{pmatrix},\\quad L_{-1}=\\begin{pmatrix}0 & 1 \\\\ 0 & 0 \\end{pmatrix},\\quad L_{1}=\\begin{pmatrix} 0 & 0 \\\\ -1 & 0\\end{pmatrix}. ", "35d31ef89468f6eeda5bccb11ef38d0f": "Y \\mathbf{\\operatorname{oi}} X", "35d31f79e8f4383072c68ce0bf8f6acf": "S^1 \\times D^4", "35d37aa98c2e870dea1a54d03fbf73e6": "\\{x, y\\}", "35d3c26972b3a60499c31f0987b9b325": "\\displaystyle (\\hat{f} * \\hat{g})(\\omega) \\over \\sqrt{2\\pi}\\,", "35d49047280940e7b507a96c77c03d01": " \\psi(\\bold{r} | \\bold{r}') = \\frac{e^{ik r}}{4 \\pi r} e^{-ik ( \\bold{r}' \\cdot \\bold{\\hat{r}})}", "35d4a26e8839ee3849200deafbab7836": "f=\\sum_i^n a_i x_i: f=\\mathbf {a x}\\,", "35d4e1749f2e7fb87be45f90740df33d": "|{\\Phi^{[1]}_{\\alpha_1}}\\rangle", "35d54d752e26d2010d52dc679e0c1023": "h = \\sqrt{e^2-\\frac{1}{2}a^2}=\\frac{\\sqrt{2}}{2}a", "35d56f8645dc1fb5b3869d6ec7ccd22b": "\\theta(Z)", "35d5717af9a6cbd56f914d7eb7b9f579": "\\lambda_4 = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix}", "35d58260d0eed091a1f59ddc6c63dbf6": "k=1,..,i", "35d5bf3e5ea6a6ef6b33b3d860b3af94": " \\sum_{k=0}^5 e^\\frac{2i\\pi 6^k }{31}", "35d5d0ea66431ad62d1e39f8422023af": "M_t= O( v/\\sqrt{2})\\approx 174 \\text{ GeV}~,", "35d5ec1c55ee01bc496a5f12882cf54d": " f:X\\rightarrow X ", "35d5f270f0188a359121911b7e9b345b": "x = 5 +\\frac{1}{x'}", "35d611bf01493ad8e7d5475081eb1453": "\\digamma^{[1]}(\\nu)", "35d67b9902e584446b1d8a05b905945f": "Q_d = \\frac{1}{\\tan \\delta}\\,", "35d68624c521e205143cbe6c4a7feaef": "\\scriptstyle\\delta t_{\\text{clock,sv},i} (t)", "35d69bf0e3460a31a646ad06fbb890b9": "\\bigstar\\bigstar\\bigstar\\bigstar", "35d6d520f4b033763d34b7970b8836e7": "\\frac{7128}{(1+0.10)^4}", "35d71568e9dfcc7690c0481f698a6bd2": "(a,b]=\\{x\\,|\\,aH^{-C}", "35e6d21d865ae46a3f6c18aa113ffdd0": " C^{\\infty}(M_0) ", "35e6f8e633ee7607af8d10a8af2772ab": "P_{\\beta}, P_{\\leq \\beta}", "35e70fa77ab76c2b852b61f554006b98": "\n\\begin{align}\n\\hat\\Omega_n &= (\\frac{1}{n} \\sum_i X_i X_i' )^{-1} (\\frac{1}{n} \\sum_i X_i X_i' \\hat u_i^2 ) (\\frac{1}{n} \\sum_i X_i X_i' )^{-1} \\\\\n&= n ( \\mathbb{X}' \\mathbb{X} )^{-1} ( \\mathbb{X}' \\operatorname{diag}(\\hat u_1^2, \\ldots, \\hat u_n^2) \\mathbb{X} ) ( \\mathbb{X}' \\mathbb{X})^{-1}.\n\\end{align}\n", "35e7102e11a2be5219fe1e0a296c04f2": "x_0 = \\sqrt{2}\\frac{|\\mathbf{y}|^2}{1+|\\mathbf{y}|^2}, x_i=\\frac{y_i}{|\\mathbf{y}|^2+1}, x_{n+1}=\\sqrt{2}\\frac{1}{|\\mathbf{y}|^2+1}.", "35e73ed5c6600739ba310c65664317b6": "B_1=B", "35e75ae4aa1b91968a1e45143e3a5836": "\\pi=\\frac{4\\sqrt{5}}{5Z} \\!", "35e80180ff30c97a721a29c8464d9111": "\\overline{op_1}", "35e81f3c2409b1310f79164bc38211dd": "\n\\operatorname{Li}_s(b,z) = \\frac{1}{\\Gamma(s)}\\int_b^\\infty \\frac{x^{s-1}}{e^x/z-1}~dx.\n", "35e8e2b326eccc1f80520ab3af04ca5e": "n_b \\leq n^*", "35e91f529083cf0e3197ad97d68c9bf7": "0<\\alpha<1\\,\\!", "35e925de94608bdf17a9438aa10f70f2": "P_n(X) = \\langle X,\\dots,X \\rangle = X'^n\\,", "35e93cbbb7f68dbbd5cb02bb5060cea5": "\n\\Delta n = 2\\pi \\frac{\\mathrm{stitch\\ gauge}}{\\mathrm{row\\ gauge}}\n", "35e96eea1c9abd640a6b437aa153a08f": "R_\\mathrm{L} \\,\\!", "35e985d710477a356d19570d41e41111": "\\beta_j", "35ea61d607452a2dbf00d22f2af45bb7": " y(x) = \\int_a^x \\! {[y(a) \\delta(t-a)+g(t)] e^{-\\int_t^x \\!f(u)du}\\, dt}\\,.", "35ea65dc6539eb2e40830c59e59caa66": " B_{i,j} ", "35eaadcf2010ef9e47e296e17f4c4d19": " \\Delta K=\\Delta\\sigma Y \\sqrt{\\pi a} ", "35ead047c025adf2b850a8817eaf5575": "\\Delta_t", "35eb109f37227061aaed6a16699fd557": "x + (\\pm 0) = x\\,\\!", "35eb90b5e569a35a028a33f5fb510cf8": "G^a_{\\mu \\nu} = \\partial_\\mu \\mathcal{A}^a_\\nu - \\partial_\\nu \\mathcal{A}^a_\\mu + g f^{abc} \\mathcal{A}^b_\\mu \\mathcal{A}^c_\\nu \\,,", "35ebc55587fa47e7d06b71995a8da02f": "z = b (e^{it} + e^{2it}) = b e^{3it\\over 2} (e^{it\\over 2} + e^{-it\\over 2}) = 2b \\cos {t\\over 2} e^{3it\\over 2} ", "35ebdc8f22f8830f45d55cc0cc576e03": "{\\mathbf e}_i", "35ec0a75c562819bf202e7f78ed978b7": "\\frac{N^2}{4ch}+O(N)", "35ec8fadc07ba56dc29e0a99fccf1885": " x^2 + y^2 = 1.\\,", "35ec9d9ef8185888a1861824cd192a56": "\\exists c\\forall (x,z)\\;V(c,z)\\wedge c(s(x)) \\, ", "35ecc90126c94b7b4e02557d7716d035": "a_\\text{Koch}=\\frac{1}{2} + i\\frac{\\sqrt{3}}{6},", "35ed1540cc140838f25fdd370adb3908": " \\exists x \\in A, \\Phi(x, \\alpha_1, \\ldots, \\alpha_n) ", "35ed76752eea8933a457bc9805a22f13": "T \\setminus T_d", "35eda21cb28b6a77354720bda7018fa9": "\\scriptstyle e^z", "35ededb33effa7fa45bbd6e76472f160": "\\eta(-\\tau^{-1}) = \\sqrt{-{\\rm{i}}\\tau} \\eta(\\tau).\\,", "35ee3e33a7381a2c86cebb4a8710d74d": "\\tilde{\\kappa}_{o+}", "35ee820d5ac4ad8a295afa580af03bee": "^{\\;}\\{c^{i}(\\xi,\\tau), c^{j}(\\xi,\\tau)\\}=\\{\\xi ^{i}, \\xi ^{j}\\} ", "35eef394e40c53b77eddffb9a4e457b2": "\\eta_0 = \\left(\\frac{\\rho \\cdot Bl^2 \\cdot S_{d}^2}{2 \\cdot \\pi \\cdot c \\cdot M_{ms}^2 \\cdot R_{e}}\\right)\\times100\\ %", "35ef0050270ff892b6f96f1657b95a2e": "A_{2N} + D_{2N}.", "35ef8319425e40b3d29ddd303b6c771e": " \\delta=\\frac{\\varepsilon_0^2 m \\zeta_2}{N^2q^3} ", "35ef8996da47fba9f05f341a31902b6e": "2^{p-1} \\equiv 1 \\pmod{p}\\,\\!", "35f0075f148f6b7af028c997c77a80c3": "\\scriptstyle{\\ell_\\circ}", "35f0aa43cddacabdb849f6e227f91d91": " S( aX + b ) = \\operatorname{sign} (a)\\, S(X) ", "35f0fc44ca88332d3518b43c16e31213": "N(a) = O(\\log a).\\,", "35f103c38397fe0073588cac5f331e27": "\\mathfrak{sp}_1", "35f10748856fc7459f17a495459e8768": "M^*\\Phi_{2\\lambda} =b(\\lambda) \\varphi_\\lambda", "35f114ff393ceb8f11f3196c072ffdfe": "\\begin{align} x &= r \\cos(\\theta) \\\\ y &= r \\sin(\\theta) .\\end{align} ", "35f132bea2777d6b98e0252c620f2b63": "y(u,v)= \\left(1+\\frac{v}{2} \\cos\\frac{u}{2}\\right)\\sin u", "35f15a93840e2c033bc2b9b3efd8802a": "K_M^B", "35f163dc7bc14fa2bb180199fd5cda87": "O(V^{2.376})", "35f196f677897a6b21276614541c17c4": "x_i = 1", "35f19787a577886bcb304ebceff2c526": "\\ker{(A-4I)}^2 = \\operatorname{span} \\, \\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 1 \\\\ 0 \\\\ -1 \\\\ 1 \\end{bmatrix} \\right\\}.", "35f1a418644c77a9b5f45a514f56ce39": " (\\sin x)' = \\cos x \\,", "35f1b0ab9e7bc2d3d783a573c18c730d": "\\mathbf{P}(n)", "35f1de2b33b7f8b510411b05cb566c2d": "u^{\\alpha} \\, = \\, \\frac{d x^{\\alpha}}{d \\tau} ", "35f1ed468e545a1aec1c3702ac14b6eb": "\n\\left| A_p \\right| = (n-1)!\\; , \\; \\;\n\\left| A_p \\cap A_q \\right| = (n-2)!\\; , \\; \\;\n\\left| A_p \\cap A_q \\cap A_r \\right| = (n-3)!\\; , \\; \\ldots\n", "35f254b6ebe5e4674f72a2baa32ea6bb": "\\mathrm{Bo} = vL/\\mathcal{D} = \\mathrm{Re}\\, \\mathrm{Sc} ", "35f288f4b0d3ada1b6be0964797d1074": "\\limsup_{n\\to\\infty} \\left(\\frac{|F^{(n)}(x)|}{n!}\\right)^{1/n}=+\\infty\\, ,", "35f2f11c29e44d63e4abf61ff8405888": "\\hat{w}=w+\\varepsilon w^0", "35f3044d4ee8330e208deccca429c40a": "\\epsilon_{kk}", "35f30b0c62832aaed2e63c7d8115a671": "\\mathrm{sign}(\\mu)", "35f39d2f3b3198bb8d1d5e3b2876b323": "Insulin = \\frac{glucose-TR}{CF}+\\frac{carbohydrates}{KF}=\\frac{KF(glucose-TR)+carbohydratesCF}{{CF}\\times{KF}}", "35f3c2a6cf4a33facd9af64c4b49060e": "\n [u_\\epsilon(0)] = 0,\\ \\left[\\frac{du_\\epsilon}{dx}\\right] = 0\n", "35f3c435a5782f5ab9a9ac47a2f159bc": "t=t_\\mathrm{then}", "35f40c75e1944e0c18d216d777e3a07b": "F_0(q) = \\sum_{n\\ge 0} {q^{2n^2}\\over (q;q^2)_{n}}", "35f441ac7c1dda40bf59c53fad3001ca": "-\\arccos {\\left(\\frac{1}{e}\\right)} < t < \\arccos {\\left(\\frac{1}{e}\\right)} ", "35f44b554996fc2e94cf37eed05faae7": "\\{0,1,2,...\\}", "35f4a8d465e6e1edc05f3d8ab658c551": "78", "35f4b27b98e349d226e84ef08e358d2c": "SO(4) ", "35f4be4131646848746c5215ca37a06f": "\\mathbf{F} = \\gamma(\\mathbf{v}) m \\left( \\mathbf{g} + \\mathbf{v} \\times \\mathbf{H} \\right) ", "35f526a28c1927dfaa7090a4fd6dc0e8": "\\mathbf{P}^n_A", "35f530da057fa53e36342253aae15733": "\\operatorname{Tr}\\; (\\Phi_x \\otimes I)(\\omega) \\cdot F_y = \\delta_{xy}", "35f5af46d35e7ed8fc79732ab2fb9414": " \\sqrt{ k_B T / C } ", "35f5c26c8bb98f17d57341191f1d0e60": "_{t,d} ", "35f5e3a4f6b8e50c6b763cd3f5e264ab": "\n \\varepsilon_{\\mathrm{eq}} = \\sqrt{\\tfrac{2}{3} \\boldsymbol{\\varepsilon}^{\\mathrm{dev}}:\\boldsymbol{\\varepsilon}^{\\mathrm{dev}}} = \\sqrt{\\tfrac{2}{3}\\varepsilon_{ij}^{\\mathrm{dev}}\\varepsilon_{ij}^{\\mathrm{dev}}} \n ~;~~ \\boldsymbol{\\varepsilon}^{\\mathrm{dev}} = \\boldsymbol{\\varepsilon} - \\tfrac{1}{3}\\mathrm{tr}(\\boldsymbol{\\varepsilon})~\\boldsymbol{1} \n ", "35f6022a9710d1a7b152f0d2a102cdb7": "\nV_\\mathrm{S} = 10\\ \\mathrm{V}", "35f6128bfd0ef25ac409597f279b4928": "\\lambda_{1}>\\lambda_{2}>\\lambda_{3}>...>\\lambda_{N}", "35f62f37522f7f599b7fd6a969dbb9db": "\\alpha = \\theta_L - h ", "35f64eddaf4a35b4d55a37a111c8fffa": " \\frac{1}{\\cot \\theta}\\! ", "35f6ab38818181b8de968a58908e2ba4": "m_i/V", "35f6bf402b47d87ac35c07f70c038f85": "N <= X", "35f6c1596b48ed9ad23c2289370fa6a8": "g(x) = p(x)(1 + x)", "35f6ea3b44c15ff06c3312a9551e825e": "\\nabla\\left(\\kappa\\left( T\\right) \\nabla T\\right)+g=\\rho C\\frac{\\partial T}{\\partial t} ", "35f6f2430feba101e3e7a07c1026028e": "p_G(\\boldsymbol{\\eta}|\\boldsymbol{\\chi},\\nu) = f(\\boldsymbol{\\chi},\\nu)g(\\boldsymbol{\\eta})^\\nu e^{\\boldsymbol{\\eta}^{\\rm T}\\boldsymbol{\\chi}}", "35f70c510bafad0cef0e4cb824aca237": "(1 \\Rightarrow y) = y.", "35f7ceeb08a53c28168f83b8b7c89c0f": "f(x)\\,=\\,x^6 + 1", "35f801461095a13a3e1b9a4bc76549df": " F(t) = f(g_1(t),g_2(t)),", "35f8144d8b5717b67a9f3629976086c5": "2 z({\\frac{\\pi}{2}} )", "35f83ab9f00b4d2bcd21963302547173": "H_i(E_f)", "35f843a3278d8ef1ff9dc6fa80719386": "{\\bar \\alpha(\\theta_i)}", "35f84a053f78d68720be38b9958e0a8d": "|S|-1", "35f8523e1e67028c910b4a5a560df743": "V(a, r) = \\max_{ 0 \\leq c \\leq a } \\{ u(c) + \\beta \\int V((1+r) (a - c), r') Q(r, d\\mu_r) \\} .", "35f852ff35a6d1e56418ef43de7712de": " g(\\theta_m|\\theta_m)= f(\\theta_m)", "35f8614275fd12fa9a48a43ea04a50e4": "\n F_{11} F_{22} - F_{12} F_{21} = 1\n ", "35f8ac534dc0d352947de4649c7df0fd": "E_i = \\int E(t) dt", "35f8c89c86bbebf02f0672e202eca760": "\ny_t - \\beta x_t = u_t \\,\n", "35f8f12925bf245de631f18cecfcc2e4": "\\mathcal{D}/\\mathcal{D}_0(S):=Num(S)", "35f90c7a3909a566f3bf0fe540bb1942": "\\mathrm{Ra}_{L} = \\frac{g \\beta} {\\nu \\alpha} (T_b - T_u) L^3", "35f92e4c8f49783929cb8b80a1b405a3": "\n\\begin{align}\n\\mu &= 0 \\\\\ns &= \\sqrt{\\frac{7}{9} \\frac{\\pi^2}{3}} \\\\\n\\nu &= 9\n\\end{align}\n", "35f9eb0cddca810dc0dfb3ba0367917f": "\\psi_{n_x,n_y}(x,y)=2/\\sqrt{L_xL_y}\\sin(n_x\\pi/L_x)\\sin(n_y\\pi/L_y)", "35fa02fecac289964a01f2888de93978": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2} + V_0 ", "35fa0e58083ffc123515dce29efba5cc": "f_W = a(n-1) = 3(6-1) = 15", "35fa33fa9db2096279f8330cd1f6b06d": "i = n+1,\\ldots,N", "35fa51dfc8af53db2ce5e32a56a1b560": "\\Delta^0_\\beta", "35fa8abbcbeebd8c021c7c6d05d6fac3": "\\alpha = 2 \\cos^{-1} w = 2 \\sin^{-1} \\sqrt{x^2+y^2+z^2}.", "35fb2d0a63d3515fa4f1adf387f68011": "I = \\int_0^3 \\frac{x^{\\frac{3}{4}} (3-x)^{\\frac{1}{4}}}{5-x}\\,dx.", "35fb6a07df9b8029fe4f10e14529b38a": "\\ell \\gg a", "35fb6bf3ee08dbf349831a449a6afc98": "t_1\\,\\!", "35fbd38646a71f4ab0b9cd9864728675": "2+\\sqrt{2}", "35fbda892f5ebd6aae3de239bbd20cc6": "{\\rm Riesz}(x) = \\sum_{k=1}^\\infty \\frac{(-1)^{k+1}x^k}{(k-1)! \\zeta(2k)}\n= \\sum_{k=1}^\\infty \\frac{(-1)^{k+1}x^k}{(k-1)!} \\left(\\sum_{n=1}^\\infty \\mu(n)n^{-2k}\\right)", "35fc1ad23402dc404fffd248eb3889d3": "\\,_1\\psi_1", "35fc3d8d93bab0249e49ad011bbfbb9b": "s_\\theta = -1, s_\\zeta = +1", "35fc427401caf6b076f818cdb4e5530b": "\\frac{1}{(n-2)\\;\\omega_{n}\\;\\|x-y\\|^{n-2}}", "35fc9412b82b0501dc52eaff900ed622": "\\alpha_i(t)=P(Y_1=y_1,...,Y_t=y_t,X_t=i|\\theta)", "35fd15c7d77b3c109c6ed41ec5bb8607": "\\rho = p_\\text{sym} \\frac{2}{d^2 + d} P_\\text{sym} + (1-p_\\text{sym}) \\frac{2}{d^2 - d} P_\\text{as},", "35fd5a7228957fb41e19be18f96dd6aa": " R = 24 \\pi \\, T, \\; \\; \\; C_{abcd} = 0 ", "35fd8fe10084ff88d75e76f17f799a3f": "\\mathrm{T}_1", "35fda5cb6133f0f674073559fbb4e6ae": "\\mbox{O}(1) \\to S(\\mathbb{R}^{n+1}) \\to \\mathbb{RP}^n", "35fdc967984ed11a8c0a94e2b5f9721c": "\\textbf{P}_{x_{k}z_{k}} = \\sum_{i=0}^{2L} W_{c}^{i}\\ [\\chi_{k\\mid k-1}^{i} - \\hat{\\textbf{x}}_{k\\mid k-1}] [\\gamma_{k}^{i} - \\hat{\\textbf{z}}_{k}]^{T} ", "35fe0a8fd18915f80bc0c07b9f0f3e44": "R_\\mathrm{S}", "35fe6d81a516366aff88082a020a070a": "\\boldsymbol{\\theta} = \\left[ \\theta_1, \\theta_2, \\dots, \\theta_d \\right]^T \\in \\mathbb{R}^d", "35fe814f77e3f284aa60935184f236a6": "\\delta: S \\times \\Sigma \\rightarrow S", "35fe9f96d5cf5ef3c5fc0a9eae1b315a": "G|_{\\mathbf{R}^{n}}=\\sum_{j=1}^{n-1}e_{j}R_{j}", "35fead5bea94fd90ea75ecd0025f3b15": " \\mathrm{det}(A) \\ge 0", "35fec6076d46f5521d7df3c59dda5e6a": "\\{c^a\\}", "35feeadcb3a5dbfae1a7262ef812c690": "a_t^{n=0, \\ldots, N-1}", "35ff1d98a00941ee0d2dc265596f1e39": " [E]_0 = [E] + [ES] + [EI]\\,\\!", "35ff1ef4b44708150182c4acf5af0971": "\\Delta(C^*(m_1),C^*(m_2)) \\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k \\cdot T", "35ffa7e788f46a03a48956915c2541d5": "\\bold S = [s(t_{1}), ......, s(t_{M})]", "35fffc72fc89f7555a9f2cb4e8efb2d2": "Tf = \\phi(f)", "3600078a9b537c4db59075a17186b53d": "{d \\over dt}\\left\\{ X_2 \\right\\} = \\left\\{A \\right\\} + \\left\\{ X _2\\right\\}^2 \\left\\{Y_2 \\right\\} - \\left\\{B \\right\\} \\left\\{X_2 \\right\\} - \\left\\{X_2 \\right\\} + D_x\\left( X_1 - X_2 \\right)\\,", "36008f416808148263561a6150f26bb4": "p_{01}/(p_{01}+p_{00})", "3600e62561e7bc9f79633202544b76f9": "r,s\\in {1\\over 2}+{\\Bbb Z}", "3600faa6c6e1bdd89925ed754450c38b": " i_p \\leq i_{p+1} ", "3601146c4e948c32b6424d2c0a7f0118": "Price", "36012fb47def5388342af00da1636a5b": " \\vec{\\xi}_6 = y \\, \\partial_v + \\int \\frac {du} { C \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right) } \\, \\partial_y ", "36018c6a4ba643b9665f5d02a38efc80": " |\\partial_\\xi^\\alpha \\partial_x^\\beta P(x,\\xi)| \\leq C_{\\alpha,\\beta} \\, (1 + |\\xi|)^{m - |\\alpha|} ", "36018c8fa329eb8a4668a945179c6279": "h \\approx \\dfrac{\\sum_{x}\\dfrac{G(x)-F(x)}{F'(x)}}{\\sum_{x}1}.", "3601a3700069809b201f46d237cca4ce": "\\ell_{a} \\sim 1/\\alpha m_e", "3601bce2f720544b0b8aaecaddcf32e0": "2^{m_1-1}", "362050d49b7311c9d74d62ad13ac56ed": "\\textstyle \\gamma=d/\\rho_\\max", "362097ac7f58d6853eeb3d94fa8f52a6": "\\alpha_1(\\theta_1) = \\frac{d_{\\rm S}}{d_{\\rm LS}} (\\theta_1 - \\theta_{\\rm S})", "3620e93959b1ead8446a7c355538e48a": "f(x,y) = 0.5 + \\frac{\\sin^{2}\\left(x^{2} - y^{2}\\right) - 0.5}{\\left(1 + 0.001\\left(x^{2} + y^{2}\\right) \\right)^{2}}.\\quad", "36211d2acd533abfcd83e4800233497a": " \\Im (\\alpha) ", "36212b3a7e58916970efabae5475ca09": "b_{0^{ }}", "36216b4d462c2207ca631bbd629801b3": "{\\rm depth}({\\mathbb B})=\\sup\\big\\{ |A|:A\\subseteq {\\mathbb B}", "3621d29730f5754d5e2e584509f1d620": "[-1,s)", "3622092fb2295e4bde48634a45f33176": " D(x)=\\left|\\begin{array}{ccc}f(x) & g(x)& h(x)\\\\ f(a) & g(a) & h(a)\\\\ f(b) & g(b)& h(b)\\end{array}\\right|", "36227cfd6efaba297b58cf6738286736": "u(x, y)=1-x^2-y^2.", "362287e21488358317a0880155fce4e7": "\nR=\\begin{bmatrix}\nYusuf & Height, & 178cm \\\\\n & Weight & 98kg \n\\end{bmatrix}", "3622a067332db06abc355613a8480594": "F_2(a, b) = a^b", "3622dca43d90437fc609e7255ecf82ff": "\n\\begin{align}\n& \\left(\\csc\\left(A + \\frac{\\pi}{6}\\right), \\csc\\left(B + \\frac{\\pi}{6}\\right), \\csc\\left(C + \\frac{\\pi}{6}\\right)\\right) \\\\\n& = \\left( \\sec\\left(A -\\frac{\\pi}{3}\\right), \\sec\\left(B -\\frac{\\pi}{3}\\right), \\sec\\left(C - \\frac{\\pi}{3}\\right)\\right)\n\\end{align}\n", "362320abd29b325e7bafe8b83dcb4d26": "\\sum_c\\Pr(C=c)=1", "3623bb70868b6323bafeb354df269a92": "\\mathbf{J} = \\Delta\\mathbf{p} = m\\Delta\\mathbf{v}.", "3623cd336bc349b52b7ae73915719db8": "p'_{c,t}", "3623d0b1cb53baced0af20349b832394": "y_0, \\ldots, y_m", "3623e735c945df520dbde8b845d9b57b": "\\epsilon^{ijk} B_k = -F^{ij} \\,", "362403ff978a57e3ba2fb08b7a51834a": "c = \\arccos \\left(\\cos a\\cos b + \\sin a\\sin b\\cos\\gamma \\right)", "362449cb0ab90176aec4e293b2699609": "S_2^P \\subseteq \\mathsf{ZPP}^{\\mathsf{NP}}", "36250cbc0d12cdb1e7e764d65f51c343": "ds^2 = h_1^2(q^1)^2 + h_2^2(q^2)^2 + h_3^2(q^3)^2 ", "3625149b89ef78abf5eb7a0099cdb692": "L_\\rho(\\Gamma):=\\inf_{\\gamma\\in\\Gamma}L_\\rho(\\gamma).", "36252f810b85f1b657d180b8d50cc45a": "\\mathcal{L}^* = \\{ \\mathbf{v} \\in V \\quad | \\quad \\langle \\mathbf{v},\\mathbf{x} \\rangle \\in R, \\forall \\mathbf{x} \\in \\mathcal{L} \\}", "36258f71124c08d7506fddeb8d55d974": "\\textstyle N(C) > 0", "3625bd290d1a47b3645de510883e2e3f": "r||s\\,\\!", "3626033b99f6df6d8a54d26c512701fc": "\\tilde g^{\\alpha\\beta}\\;", "362610d82d6effcd5ee955f79648d689": "f''(x_0)<0", "3626169c2ae7eebbdf332a34441d427b": " \\Delta F \\leq W ", "36262036ef50aab9296af763a0afbc04": "\\psi()", "36262a9ad3ba877ba11aa3d62f201f4b": "V_{J_{35}} = 4.9550988153084743549606507192748", "36262def14abfaf773982945edb5003c": "(-\\infty, a)", "36267719468c63fa34d9a390f875ef14": "E = \\left(\\frac{\\rho_{o}}{t^2}\\right)\\left(\\frac{r}{C}\\right)^5", "3626a6f77456bf8427537bcc83405a04": "r_{1l}r_{2l}<|z|0", "36364864a9f9550fb11abe57ecac6eb1": "d([L]\\mathbf{v}, [L]\\mathbf{w})^2=([L]\\mathbf{v}-[L]\\mathbf{w})\\cdot([L]\\mathbf{v}-[L]\\mathbf{w})=([L](\\mathbf{v}-\\mathbf{w}))\\cdot([L](\\mathbf{v}-\\mathbf{w})).", "36368d14b51dd2d1294679308fab0959": " \\psi(x) = \\sum_{p^k\\le x} \\log p.", "3636b2024b6af6e4924058e4ce4d5058": "\\lambda x.y", "3636bb81514ee0ab8cc120633e1ebc97": "F_{2n} = F_{n+1}^2 - F_{n-1}^2 = F_n \\left (F_{n+1}+F_{n-1} \\right ) = F_nL_n", "3636bfe2325455221aa7b0659fcf372c": "b_j", "363717ae6c1e28d08542c8c4e9c8f741": "\\, \\hat{B} = \\begin{bmatrix}B_{r\\overline{o}} \\\\ B_{ro} \\\\ 0 \\\\ 0\\end{bmatrix}", "3637353791280c14199229f1b46da521": "GBWP = \\frac{{{H_0}}}{{\\sqrt {\\frac{{\\omega_c^2 + 25{\\omega _c}^2}}{{\\omega_c^2}}} }}\\cdot5{\\omega_c} = \\frac{5}{{\\sqrt {26} }}{H_0}\\cdot{\\omega_c} = 0.98\\cdot{H_0}\\cdot{\\omega_c}", "3637ba821e82a613138f5598cf0019eb": "\n\\delta L = \\epsilon mk\\frac{d}{dt} \\left( \\frac{x_{s}}{r} \\right)\n", "3637e3b053b29e8af0c774153ca65bff": "\\tau = i \\frac{\\,_2F_1\\big(\\tfrac{1}{2},\\tfrac{1}{2},1,1-\\lambda\\big)}{\\,_2F_1\\big(\\tfrac{1}{2},\\tfrac{1}{2},1,\\lambda\\big)}", "363843bae674e0073c4f569fa03d0dfb": " \\psi(\\vec{r}) = e^{i(\\vec{k}\\cdot\\vec{r})} ", "363894d9d589e63df69936f857ee7ee7": " 2 NO(g)\\ \\overrightarrow\\longleftarrow \\ N_2O_2(g)", "3638a7008ddcb9e633dfe07798bb5463": "9 + 4 = 13", "36395d46d86ffc4654e8203c7296c48e": " F = \\frac{4R}{{(1-R)^2}}", "3639d5bf44ae3baeb35e90088cd3ca9d": "t_0 \\sim t_1", "3639d6d4df738a1b5cebaa87bdc1def0": " 0 < y < \\pi", "3639ee45aa3983d6dcf2f12e9f2be491": "\\frac{V}{n} = k", "363a7a90ac26ed962f9d1384be8a9c1f": "c_f(u,v) = 0", "363a9c646ba6dac7bcc4f64222e2bf7d": "\\rho(i)", "363ad5f6483b928b467cf9f043ce9c79": " P = \\begin{cases} \\frac{-1+\\sqrt{p}}{2}, & \\text{if }p=4m+1, \\\\[6pt] \n\\frac{-1+i\\sqrt{p}}{2}, & \\text{if }p=4m+3. \\end{cases}", "363b122c528f54df4a0446b6bab05515": "j", "363b3985eae6762e140fa77f685a4fcc": "\\mathfrak{f}_4^{\\mathbb C}", "363b51fe8037f04a452448f4e1050fd5": "\\phi_i:U_i \\to \\R^n", "363b7114a9d7e96b5b75ef926766911a": "\\frac{dL}{dt}v+LMv=\\frac{d\\lambda}{dt}v+\\lambda Mv.", "363b712b47056b2deee9300a9b5d48ff": "\\mathcal{EL^{++}}", "363b7daa0e25c4c5916eadd2c60843c0": "\\lim_{c\\rightarrow \\infty} \\lim_{x,H} f_{x,H}(c)=0", "363b8ccacad9b587b50f430ac629885c": "\\mathit{K} = [0, \\mathit{\\bar{K}}]", "363bad9ffe6a6163e08d33fd614740bb": "\\frac{1 + \\cos (\\theta)}{2}", "363bb4d77a2dfaf2b1b15e667489aa87": "d_\\mathbf{A}B=0", "363be1a1694c88fafb86bd7b8f3bfffc": "x=-2", "363be7c9e65904d4355c9e94d5000bfb": "\\mathrm{Cov}(\\varepsilon)=\\mathrm{Diag}(\\psi_1,\\dots,\\psi_p)=\\Psi\\text{ and }\\mathrm{E}(\\varepsilon)=0. \\, ", "363c35bc91746586c46d74a19a74cb63": "\\mathbf{g} = \\varepsilon_0 \\chi^{(m)} \\mathbf{H}", "363c876af82bb2828e6f64759d58e390": "H(p) = -\\int_{-\\infty}^\\infty p(x)\\log(p(x))dx", "363cc0e17e4f9baa17099d1c300701e7": "p \\colon P \\to M", "363defda565d0d3a0ca29135a5e1076c": "S_0=0", "363e86128933f2a2a534981bf23e1725": "\\binom{f}{k} = \\frac{(f)_{k}}{k!} =\\frac{(f-k+1)\\cdots(f-2)\\cdot(f-1)\\cdot f}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots k}", "363e89d32dcc021ec47e78f3f8998317": " b_2 = (\\frac{\\rho_c-\\rho_w}{\\rho_m-\\rho_c}){h_w} ", "363eb66407b6db7adb5bc60bbfccd888": " \\nabla^2 G(\\mathbf{x},\\mathbf{x'}) = \\delta(\\mathbf{x}-\\mathbf{x'})", "363f193488ed43f4ed4eea3c93d48247": " H_2 = \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}.", "363f4a0879a1077fa41b82eee80d9d31": "K(P)", "363f5310e4e6467d28510c3d44a0d37b": " \\left \\langle \\mathbf{x} |A| \\mathbf{y} \\right \\rangle + \\left \\langle \\mathbf{x} |B| \\mathbf{y}\\right \\rangle = \\left \\langle \\mathbf{x} |A+B|\\mathbf{y}\\right \\rangle", "363f5ce16327cd2c04bd497ed04f706f": "[\\alpha]\\psi", "363fdc5697d28c069adc5cd031ccf155": " ds^2 = -g_{tt} \\, dt^2 + \\mathrm{quadratic\\ form}(dx \\, dy \\, dz) \\,", "363fdca884e07f98361da5fd476d2c57": "5x+8", "363fea886c5b6d5316299495bd057d38": "v\\le d-1,", "363fed82aca62b6a4c12d57845dea6aa": "\\sum_{i=1}^n \\partial_i^2 f(x)^{s+1} = 4(s+1)\\left(s+\\frac{n}{2}\\right)f(x)^s", "36402a0aa57ca1ba48eb9e3d121f79a5": "(\\tfrac{-4}{n})\\ ", "36403012746a8afa6903bb4eea1da9b9": "z \\in Z", "364032cb427a0ca23c9768be223d940e": "0= (-1)^kke_k(x_1,\\ldots,x_k)+\\sum_{i=1}^k(-1)^{k-i} e_{k-i}(x_1,\\ldots,x_k)p_i(x_1,\\ldots,x_k),", "36405dbed10628f3e5fc6574a8cb4d20": "c = \\varphi ^ \\frac{1}{90} \\doteq 1.0053611", "36408296faa216881d098e81a9ce993a": "(X_1^k,\\dots,X_d^k)=(F_1^{-1}(U_1^k),\\dots,F_d^{-1}(U_d^k))\\sim H\\;\\;(k=1,\\dots,n)", "3640dc5d21288b81a27d98875f8a704d": "\\partial\\omega_2+R_\\nabla=0,", "3640e8ab82e93465a91f143b72161cf4": "\\ln_q( e_q(x) ) = x .", "3640edb505b0cfae28cb80a5dfe22209": "\\rho(k) \\sim k^{-\\alpha}", "3641072df10af4c1df60158176caaa63": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{3 \\choose 1}{3 \\choose 1}{36 \\choose 2} \\end{matrix}", "36417610ce220e5b2ebfcb1990317322": " u(0) = u^0 ", "36417a0f88d72d317035880231781c5a": "3/4\\ ", "3642162715367a65ed6b52548f900fd5": "\\textstyle (C,\\; r+s+1)", "36424a435f21a4412b6e833c0a2e9c00": "M(x) =-\\tfrac{q(L^2-2Lx+x^2)}{2}", "364251d837e2655e9b1ecc94d7651509": "ax+by+c=0,\\,dx+ey+f=0,\\,gx+hy+i=0.", "364256afa166297e62b3c27665b315e8": "\\frac{\\pi}{4} = \\arctan\\frac{1}{2} + \\arctan\\frac{1}{3} ", "364268cb5a51cedcc809fb59f5f814b7": "\\frac{P}{T}=k", "364335e170316dbaa083ba3211f74d1b": "\\|x+y\\| \\leq \\|x\\| + \\|y\\|.", "364348cfeac118db242ea9b55e260581": "\\operatorname{Pr}(\\sigma) = \\Vert P U_{\\sigma_k} \\cdots U_{\\sigma_1} U_{\\sigma_0}|\\psi\\rangle\\Vert^2 ", "36435a7a3d0a6b8ebc15b3676da9aa44": "\\theta_p", "36438311496a15ea643e7d16060bd0ce": "\\frac{\\partial L(\\vec{r},\\hat{s},t)/c}{\\partial t} = -\\hat{s}\\cdot \\nabla L(\\vec{r},\\hat{s},t)-\\mu_tL(\\vec{r},\\hat{s},t)+\\mu_s\\int_{4\\pi}L(\\vec{r},\\hat{s}',t)P(\\hat{s}'\\cdot\\hat{s})d\\Omega' + S(\\vec{r},\\hat{s},t)", "364389944c65bc8c14e67543e6cebc29": "I_{L2}=I_P\\sin\\left(\\theta-\\frac{2}{3}\\pi-\\varphi\\right)", "3643bcdf3f91e5b9bb44117847f23ff8": "\n\\mathcal{L}_1 = u_x u_t + u u_x^2,\n", "3643d6055b31486321d3d8ee2f42753b": "u\\in S", "364401c5f4ec6f8c76540e47548dac57": "R_n = \\frac {\\pi W_n}{k T_0} ", "364453a1e1f7bd7774c0a2485c94245f": "r_1, ..., r_n", "3644a036fc8501f8edf38dec70e3bff2": " \\ \\textbf{f}_p ", "3644a437ae256b5072a5124e13eacae4": "(h\\leftarrow k)^* = h^* \\rightarrow k", "3645224702906807e4cc9ff44ffa058f": "w\\in[0,1]", "364533ee119f1333d0c620fcc03a0a89": "\\Pi(k) = p ", "364589b3d1b7624042a4bf618f72c728": "\n\n\\,A_{\\stackrel 1 x :{\\overline 3|}} = 0.1(1.06)^{-1} + 0.09(1.06)^{-2} + 0.081(1.06)^{-3} = 0.24244846,\n\n", "3645db79c666c9d37c9030740f0f000e": "R(\\lambda) = \\sum_{i=1}^{16} w_i R_i(\\lambda)", "3646048be1ff7ad971a5a7ab6c190169": "\nS(\\omega_k) = S(\\omega = \\omega_k)\\Delta\\omega/2\n", "36466953bffc5ee99d7df9cb70d761c5": "\\mathbf{U} \\mathbf{x} = \\mathbf{b}", "3646a3552ca7efbd88c935a80cdceb98": "C=2 \\pi r", "3647147392ca3049178fe8f3b902acfb": "\nU_{\\epsilon I} (t_1,t_2) = e^{i H_0 t_1/\\hbar} U_{\\epsilon}(t_1,t_2) e^{-i H_0 t_2 /\\hbar}.\n", "364725f390351ba5604af0faaac01a1e": " {\\rm trig}(M)=(1,2g-1,g_3) \\,", "3647c29ba2c6e4b31846b49c16105ff1": "N=pq", "364802af78c962968c655e30904270f3": "d\\tau\\,", "3648040e7fb7d3db625f3ebf66a546dd": "\\text{return} \\colon T \\rarr E \\rarr T = t \\mapsto e \\mapsto t", "364847807e2fe0ddd3bf80e0e79b53d3": "g(x)=1+x^{32}+x^{37}+x^{72}+x^{102}+x^{128}+x^{44}x^{60}+x^{61}x^{125}+x^{63}x^{67}x^{69}x^{101}+x^{80}x^{88}+x^{110}x^{111}+x^{115}x^{117}+x^{46}x^{50}x^{58}+x^{103}x^{104}x^{106}+x^{33}x^{35}x^{36}x^{40}", "364879001e02a3b075d08ebe4e9f7a38": " {(s)} ", "3648965980e17a26917980d48abfc8e8": "\\scriptstyle{K(m^{2})}", "3648ffe65eaeafaee251be503e3c2721": "= p(C) \\ p(F_1\\vert C) \\ p(F_2,\\dots,F_n\\vert C, F_1)", "36491593d1ea6419a110a436913af491": "\\{ x\\in E\\mid x\\le y\\}", "36493a726db872fd64e58ea66f68c18e": "1.63661632\\ldots", "36494eff953794305de3473a345d83a4": " \n(a)\\quad \\ \\ (b)\\quad \\ \\ \\ (c) \\qquad (d) \\ \\ \\ (e)", "36499f3e3b87ec33f7b4a1f9dd8112d4": " ln(\\gamma_2^\\infty) ", "3649e89906ab2a41b4d74a3ec23d7efa": "R_2 \\ ", "3649f90314c7e2e0a46f7307aa7369c7": "C_{sv}", "3649f97c609d694cc9109529c3c22403": "\\eta^{\\mu\\nu}\\partial_\\mu\\partial_\\nu\\phi+V'(\\phi)=\\partial^2_t\\phi-\\nabla^2\\phi\n+V'(\\phi)=0", "364a58241910a547a95c24b9e867f2e1": "\\pi = \\lim_{r \\to \\infty} \\frac{1}{r^2} \\sum_{x=-r}^{r} \\; \\sum_{y=-r}^{r} \\begin{cases}\n1 & \\text{if } \\sqrt{x^2+y^2} \\le r \\\\\n0 & \\text{if } \\sqrt{x^2+y^2} > r. \\end{cases}\n", "364a592d02d76ebf910990733e299214": "\\alpha_\\text{scat} = \\frac{8 \\pi^3}{3 \\lambda^4} n^8 p^2 k T_\\text{f} \\beta", "364a9fa6421709962043b1cf0ef06506": "1 \\rightarrow A \\rightarrow B \\rightarrow C \\rightarrow 1\\ ", "364ad777f81bce11f665446c5b51d757": "|z|^2=|z^2| = |z^2+1-1| \\le |z^2+1|+1,", "364ae190dc5c09e1486c00cf009b76cf": "H_{trt}=\\frac{GM}{r^2}", "364ae2df75f9c599054f338f4e57bff7": " k-1 \\, ", "364b053664290bc982e5412e665084d3": "(\\tau_1\\to\\tau_2)\\to\\tau_3", "364b51c4a4ba33e4ecad9e08732475cc": "\nd\\nu_t = \\kappa_t(\\theta_t - \\nu_t)\\,dt + \\xi_t \\sqrt{\\nu_t}\\,dW^{\\nu}_t \\,\n", "364bc140bd9daa87f8cbcce31afba52e": "\\tfrac{1}{2}, \\tfrac{1}{4}, \\tfrac{3}{4}, \\tfrac{1}{8}, \\tfrac{5}{8}, \\tfrac{3}{8}, \\tfrac{7}{8}, \\tfrac{1}{16}, \\tfrac{9}{16}, \\tfrac{5}{16}, \\tfrac{13}{16}, \\tfrac{3}{16}, \\tfrac{11}{16}, \\tfrac{7}{16}, \\tfrac{15}{16}, \\ldots", "364bf9538bd61718e55d3125eb483e79": "h[n] = {\\phi}_{sy}.\\,", "364c43ce885ecfc3586e8f7a3fc7377f": "E[P]=P^*", "364cd397cbb7090e63ef18a7756608be": " \n(Eq. 3) \\text{ } E[P(\\alpha^*(t), \\omega(t))] = p^* = \\text{ optimal time average penalty for the problem} \n", "364cfe3e9324887e707fa52cc1aee546": "\na^{d} \\not\\equiv 1\\pmod{n}\n", "364dbe9605dfc712232fcbb7ceb4147c": "\\beta^+=\\frac{cov(r_i,r_m |r_m>u_m)}{var(r_m |r_m>u_m)}", "364e0a98fc167ee2428b1da6b2b0485f": "t_4", "364e2fdd0619573489156b2959d2b26f": "w = (d + \\lfloor 2.6m - 0.2 \\rfloor + y + \\left\\lfloor\\frac{y}{4}\\right\\rfloor + \\left\\lfloor\\frac{c}{4}\\right\\rfloor - 2c)\\ \\bmod\\ 7,", "364eca6e2cafba23623dc6006e87650e": "f(ka)=kf(a)", "364f033b35a31c4d87bf1cd273379340": " t = T(x) ", "364f3ec244b713e820775b668005bc07": "P_1=|100\\rangle\\langle100|+|011\\rangle\\langle011|", "364fa1a4ebb050320e9e5dd8e62413cf": "s=t", "364fcb0228a90c83c86d4354146c25ff": "\\left[{4\\atop 2}\\right] = 11", "364fea4fe1fd681a04ef8fc6482c773a": "\nu(x,t) = F(x-vt) + G(x+vt). \\,\n", "365006af6e51207556b23a99359e8b95": "\\int_{-\\infty}^{\\infty} c_{\\omega}\\,x_{\\omega}(t) \\, \\operatorname{d}\\omega", "3651a23336e00bfda6a5efba53d81fd9": "\\cos[\\arcsin(x)]=\\sqrt{1-x^2} \\,", "3651aa8b1b207a4773205bf73c90ffc1": "\\operatorname{artanh} t = \\frac{1}{2}\\ln\\frac{1+t}{1-t}.", "3651f9b707821bb29ced33668c19de2a": "\\zeta_X(s) = \\zeta_U(s) \\zeta_V(s).", "36520709a7d1697c8c468ae77f451dea": "V_{x} = \\int_a^b \\, \\pi \\, y^2 \\, \\frac{dx}{dt} \\, dt", "36521094a3baefd5486184c54c7bf0b0": "\\frac{dp}{dr}=-\\frac{G\\rho(r) M_{enc}(r)}{r^2}", "3652a05ea7008e8b0efc57b88cb78c63": " \\mathrm{d}f_p = \\sum_{j=1}^n D_j f(p) \\,(\\mathrm{d}x^{j})_p .\\,", "3652b8cb030dd16f687663d9749b9cac": "V_{\\mathrm{out}}(x,t) \\approx V_{\\mathrm{in}}(t - \\sqrt{LC}x) e^{- \\frac{\\sqrt{LC}}{2} \\left( \\frac{R}{L} + \\frac{G}{C} \\right) x }. \\,", "3652ca8202e9c0aee450ae1de5bf8bcf": "H_c(s) = \\sum_{k=1}^N{\\frac{A_k}{s-s_k}}\\,", "36537566edf075d2cf13c650fd2b57b1": "\\nabla^2 y + y = 0", "3653ead102baa7042e29dc68dc595699": "\\scriptstyle y\\,\\not\\le\\, x", "36543462e38617d3dc65d458a8632a24": "\\nabla \\times \\left( \\nabla \\times \\mathbf{V} \\right) = \\nabla \\left( \\nabla \\cdot \\mathbf{V} \\right) - \\nabla^2 \\mathbf{V}", "365473bb6b282dbdd39dc4fffcf38e67": "k\\sqrt{\\frac{1}{3}}", "3654cf6587eb495d65452e33583dfc57": "\\{s, 2, 4, t\\}", "3654d8599290a8c33c6b0f5ebbea59ef": "\\vec a.", "365509da15afa15767b7a0bb23dbe57b": "m_p=aE(e).", "3655300c95bd9a5bb5be8bc4024755bb": "A_i \\in M", "3655f2038e1fb5e44ceb740e9f282371": " \\textrm{Gamma}(\\alpha_i, 1) = \\frac{y_i^{\\alpha_i-1} \\; e^{-y_i}}{\\Gamma (\\alpha_i)}, \\!", "36568de72ee2a9b55ec10bfaec99712a": "(n + g + \\delta)k(t)", "3656d355b6db2fb860329ce163ac8376": "Z_i^{(m)}=(Y_{im-m+1}+...+Y_{im})", "3656f2d631acf140d365b6cebd5e29df": "Z=\\sqrt{{ESR}^2 + (X_\\mathrm{C} + (-X_\\mathrm{L}))^2}", "3657ae2eeda99023259ee7dfbf28d8b7": "\\mathbf{p}_k\\in\\mathbb R^n", "36584a32c2c203747a6ad9efc89add20": "o(T)=0", "36585fd6a974f7ebd39bb71bb2614369": "c_\\text{l}", "36589c0963bd9597f22f7a7e659cae3b": "\\pm i", "3658a0e72653ded8c1959fdee38f52a3": "\\{X_\\beta^n\\}_n", "3658a80c418cb6f7d8e1dbf684e26905": " \\Delta t < \\rho c \\frac { \\Delta x^2} {K} ", "36592486afab3d3752e97f7217bbf8cc": "\\textstyle \\le (n+1)/2", "365951fc31a863044bcbbad3adb058ac": "\nE = \\frac{p^{2}}{2m} - \\frac{k}{r} = \\frac{1}{2} mv^{2} - \\frac{k}{r} ~.\n", "3659639ca4763bf4a2bb7c97cf592556": "A_{n-1}(1) = \\frac{2\\pi^{n/2}}{\\Gamma(\\frac{n}{2})}.", "36599bf31d9880a409874ae60beee82a": "A=\\oplus_0^\\infty A_i", "3659cae307f8efa4ec9b1f323b5d7f96": "2^{N/2}", "365a40eddf1a25eee63c01d245bb75b8": " \\Sigma ", "365a49189606a793ba57fac2938d7696": "\\, = \\det(x) ", "365af4ba1b1c063b787b0e767524fb3d": "[W_T] \\gg [E_1]", "365af74e19cd56c8910014710fc7ed3d": "\\mathfrak{nil}(\\mathfrak g)", "365b208cb5c4067d0ae69f0ad9067d5e": "X[x, y] = x - \\frac{x'}{\\sqrt{x'^2 + y'^2}} \\int_a^t \\sqrt{x'^2 + y'^2} \\operatorname{d}t", "365b94e0c49f7e6ceb4c88c0675bb52e": " F_k = \\rho \\int_A \\sum_i (u_k u_i n_i - v_i u_i n_k) \\, \\mathrm{d} S. ", "365ba21b9905ba8a4edee3cb969ae265": "\n\\frac{d}{d t}\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\beta}}=\\frac{\\partial \\mathcal{L}}{\\partial \\beta}\\Rightarrow 0=-\\mu B \\sin{ (\\theta-\\beta)}+2K_uV\\underbrace{\\sin\\beta \\cos\\beta}_{\\approx \\beta} \\Rightarrow\n", "365bda8acae765e942dd3fb06cd2e9ea": "{Li_{k}(1-e^{-x}) \\over 1-e^{-x}}=\\sum_{n=0}^{\\infty}B_{n}^{(k)}{x^{n}\\over n!}", "365bebffa7d97ac1659fa8c2ff888823": "\n\\begin{matrix}\n\\circ: P(n)\\times P(k_1)\\times\\cdots\\times P(k_n)&\\to&P(k_1+\\cdots+k_n)\\\\\n(\\theta,\\theta_1,\\ldots,\\theta_n)&\\mapsto&\\theta\\circ(\\theta_1,\\ldots,\\theta_n),\n\\end{matrix}\n", "365c0b3ff8a6fa58b7ae709949b55608": " k ", "365c1a2d7a8f9068dafc9fb000287926": "\\mu \\colon X \\wedge X \\to X", "365c271993bd6d29a2b34682914c38ab": "\\frac{\\partial k}{\\partial t} + \\frac{\\partial \\omega}{\\partial x} = 0", "365c40fe649ab2597fc588f2ef17eaa2": "\\displaystyle\\frac{1}{2}", "365c5b6e42e4f11145c628e62aed237a": " \\delta(C_i,C_j) ", "365c72c1275fdc3fe9a5d10da42eacd6": "{ e.g. } 729 = (7+2)^{\\sqrt{9}}, \\text{ }4096=\\sqrt{\\sqrt{\\sqrt{\\sqrt{4}+0}}}^{96}", "365c7c48471be2019a9454412729393b": "\\pi_1 = \\frac{q \\varphi(r)}{k_b T} = \\Phi(R(r))", "365cb3cf736fdc446ff67ef86dab0ac6": " \\operatorname{let-combine}[\\operatorname{let} V : E \\operatorname{in} F] \\equiv \\operatorname{let} V : E \\operatorname{in} F ", "365cfdac89f06a7239611f43353529b1": "{\\rm d}", "365d81ec96d4572a4bcba65263793464": " \\Box A^{a} = - \\mu_0 J^{ a } + {R^{ a }}_{ b } A^{ b } ", "365d89d9a799eab5a96790722783ab57": "r=\\frac{j-i}{2}", "365dec92f55a09774ef001084deef306": "\\lim_{x \\to 0^+} \\frac{1}{x^r} = +\\infty", "365e22e38485488297561ae550b73419": "F(t,x,z)\\,", "365e811f85e9fcf113e51e5bf28b6f30": "\\mathbf{x}' = \\mathbf{A}\\mathbf{x}^0", "365ec9f28ec9177ec76679b0cf192bf5": "Z = \\{ n | \\varphi(n) \\}", "365ee058a4c9f0542cfce97079f6ee88": "FDR_{+1}", "365f0066164ee9f3fea7a89c07218c68": "S^2 := \\{ (x,y,z) \\in \\mathbb{R}^3 \\mid x^2 + y^2 + z^2 = 1\\}", "365f81b1febb9b8e67ae39de768bfff8": "[\\pi\\!/2\\,,\\,\\beta\\,,\\,0", "365f88fdd3d9a6ccc8ee9ea1290cb913": "\\sum_{i=1}^n \\left(x_i-\\mu\\right)^2", "365fc0aa888a2922441eb5f72ba9d285": "U:H\\rightarrow H", "365fca1639b768ffa6c66cf7e64953f8": "p(n)\\approx 1-e^{-\\tfrac{n^2}{{2x}}}", "365fd40c2ff4f942e0e87e975c1287d7": "D_{L_{O_2}} =\\frac {\\dot{V}_{O_{2}}} {P_{A_{O_2}} - P_{a_{O_2}}} \\simeq \\frac {\\dot{V}_{O_{2}}} {P_{A_{O_2}} - P_{v_{O_2}}}", "365ffeda3b70605330e2166c19539801": "PQ=\\frac{|AD+BC-AB-CD|}{2}.", "366001c2114bf6d2bcfa3562e67143f7": "j_9=0", "36600a3773feead2e07b7c978e7f6029": "\\tilde{n}_i", "366079351917e73ec73d590a324bb33e": "b_{15}+c_{13}", "3660883361a3ba169973983d1e34ddc4": "\n\\frac{a \\vee b, \\quad \\neg a \\vee c}\n{b \\vee c}\n", "3660cf1fbfbedf07969ce57d7ed210ad": " \\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda y.f\\ (y\\ y)) ", "3660e0e03733c5bf2a2416ad888740e7": "\\operatorname{dist}(q_1, p) \\cdot\\operatorname{dist}(q_2, p)=b^2.\\,", "36610b7294002ec5fc17253134d5c811": " (2+2r)u_j^{n+1} - ru_{j-1}^{n+1} - ru_{j+1}^{n+1}= (2-2r)u_j^n + ru_{j-1}^n + ru_{j+1}^n ", "366126b4c1e4b59858528c3fc7cf7936": "P_i = wl_i", "3661aaa7f2e2ce490e35a79818caf922": " \\begin{align}\n\\Pr \\left (\\min\\{X_1,\\dots,X_n \\} > x \\right ) & = \\Pr\\left(X_1 > x \\wedge \\cdots \\wedge X_n > x\\right) \\\\\n&= \\prod_{i=1}^n \\Pr(X_i > x) \\\\\n&= \\prod_{i=1}^n \\exp(-x\\lambda_i) = \\exp\\left(-x\\sum_{i=1}^n \\lambda_i\\right).\n\\end{align} ", "3662542de6c5d8b898aabe611af2aacb": " v_1^2 = \\mathbf{v}_1 \\cdot \\mathbf{v}_1 = { { \\mathbf{p}_1 \\cdot \\mathbf{p}_1 c^6 } \\over { E_1^2 } } ", "3662acd029554ced52f8189ce023e926": " p_{2,4}(x) \\, ", "3662b919af6cb270eb8717f93120b49e": " (1+\\tfrac{0.129949}{12})^{12} - 1 ", "3662d1b8ffef0fbd71fd78db205242d3": " Q(I,\\vec{P}) ", "3662e2e9613226d35f440b22664a4ef2": "\\frac{\\partial \\tau_x}{\\partial z} = \\rho A_z \\frac{\\partial^2 u}{\\partial z^2},\\,\\!", "36634c655f47d3d4c675529d6b1191d0": " \\sum_{n=1}^\\infty \\frac{1}{n^2} z^n ", "36637e5b2efe4e451d146218d4ff9f5d": " \\omega_{L} = \\omega_{0} ", "3663867b6861b55da0e3e012196ef788": "\\xi<0", "366386fc22cb9ea6afe7aa806dd4ac15": "\\frac{x^{2}}{c^{2}\\cos^{2} \\theta} + \\frac{y^{2}}{c^{2}\\sin^{2} \\theta} = 1", "36640d8992132ef08325c99f587b7c8e": "T \\, \\propto \\, R \\,.", "36642ee692431b71f3ee49d0ba2812ae": "f,g: M \\to G", "36644e97756f76b26077c7c828edc492": "A \\cap B", "36653014f0942b5a929595be092f340d": "R = \\mathcal{O}_K", "366593a49a7f8e114474100b2632b284": "\\scriptstyle\\Box^2", "3665b2615e59cca70575b1be9dbf0030": " E(x) = x^2, \\, ", "3665b71a0cd0452dc44e337fc35ff23c": "R(0)", "3665b9e333e6eaf5752d3cb37ef9f16c": "\\{(1,\\tfrac{3}{2}),(\\tfrac{3}{2},2),(2,4)\\}", "3665d26c76005f7275d6b5a68660a586": "\\alpha=0\\quad\\quad\\quad\\quad\\sum_{i=1}^n\\frac{x_i^{n-1}}{\\Pi_i(x_1,\\ldots,x_n)}=1.", "3665f40d496b616baad4230c62d5e2b2": "H = -\\frac{\\hbar^2}{2m} (\\nabla_1^2 + \\nabla_2^2) - \\frac{e^2}{4\\pi\\epsilon_0} \\left(\\frac{2}{r_1} + \\frac{2}{r_2} - \\frac{1}{|\\mathbf{r}_1 - \\mathbf{r}_2|}\\right)", "366626aa52fbde2fa7d2ec76d06c06c6": "G'=(V, E+e_{i,j})", "36665e01a33706ac2173a5899bfbb662": "\\gamma=1/\\sqrt{1-(v/c)^{2}}", "3666cc1d9cea9de5c78b95dbca2db1a0": "\\sigma_\\bar{C} = g(S, \\bar{C}, \\cdot) \\,", "366744e9ccc475249fc9f7fe43d4df80": "E(R/\\mathfrak{p})\\,", "36674840810e689aa55237f782fd8de9": "\\mu_{i,j}:= \\frac{\\langle b_{i}, b_{j}^{*} \\rangle}{B_{j}}", "36681ab37468a20c471ab1a5ae7511ce": "e_{g}", "3668272a5890ad9cbb7c4a019f1f5413": " \\chi_i ", "36682ff07280a707c64bed581b05ec84": "\\Box A^\\mu = \\mu_0 J^\\mu", "36688c58ea5220416a20abebb836e31f": "S^n/{\\sim} = S^n \\vee S^n ", "3668bbb494d9939850c256967e546deb": "\\scriptstyle 1.1m", "3669c4cca9af5b4f7b799a78e79fdb1b": "P_{f}", "3669d6f247847da38e9604fc051f97ff": "(3,5,7)", "366a11828906f64b8516b1613a7964eb": "f(gT^2/L) = 0.\\ ", "366a13e13016802c75c4ec662ea600ac": "\\omega=e^{2\\pi i/p}", "366a714e26ea6edc77be8d58b3cb387b": "\\sum_i \\alpha_i p_i", "366ac618bd2cffd79169861f890684bc": "(s_1, s_2)\\in R^j", "366b042e3eeed773ea729e37e1685664": "7 \\times 6 = 42", "366b1773318329866e7a5d7836c55e42": "\n \\mathbf{A}=\\begin{bmatrix} \n \\color{red}{1} & 2 & \\color{red}{3} & {\\color{red} 4} \\\\ \n \\color{red}{5} & 6 & {\\color{red}7} & {\\color{red}8} \\\\\n 9 & 10 & 11 & 12\n \\end{bmatrix} \\rightarrow \\begin{bmatrix}\n 1 & 3 & 4 \\\\ \n 5 & 7 & 8 \n \\end{bmatrix}.\n", "366b56078dcb5ddc925107a67abcbf55": "\np_{X,A,B}(x,a,b) = p_X(x) p_{A,B}(a,b)\n", "366b5a34e091513052414b60e8b2b36d": "\\scriptstyle \\left(\\frac {(x_i-x)} {R_i}, \\frac {(y_i-y)} {R_i}, \\frac {(z_i-z)} {R_i}\\right)", "366b6d99fb8389a0aefbbd2ddc5a03f6": "\n\\frac{d}{dx}\\left((1-x^2) \\frac{d\\Theta}{dx}\\right)+\\left(\\lambda -\\frac{m^2}{1-x^2} \\right)\\Theta=0\n", "366b7bfbfe86a23e872eb88663024675": "\\mu.\\!\\,", "366b8a46e0e544b12c0e915a83d27bbf": "\n\\langle L(t) \\rangle\\equiv \\int^{\\infty}_{-\\infty} L(x,t) p(x,t) dx,\n", "366bd765cd2c4f4582633a0066dc6327": "k \\leftarrow 0", "366c001abd870c5b367025fe49dd07c7": "Q \\ge 0,", "366c1f64d2266475f402459c01c45e1b": "T^n \\times P^n / \\sim, \\, ", "366c2f3c808208839cd6a0ea7a7b52ae": "\\bigcup_{n \\in F} (S-n) = \\mathbb{N}", "366c4230dade9728fa07a6c9b66f1b85": " \\operatorname{build-param-lists}[o, D, V, T_{10}] \\and \\operatorname{build-param-lists}[x, D, V, K_{10}] \\and \\operatorname{build-param-lists}[y, D, V, K_{10}] ", "366c44b4351134aa2c2f109f756c1bf2": "({V}_{4}/{V}_{3})", "366c8f1acaafe2f7e9ee930ab130924c": "S_U", "366c94f4de6ef4c380c948567c3c9f3b": "\\wedge^* A", "366cbe069e4544037f009ecddb90e3ae": "x_1, x_2, \\ldots, x_n,", "366cd587c5494b057ab5ac85b51427d9": "\\bar{\\mathbf{A}} = \\mathbf{A}-\\mathbf{B}\\mathbf{K}", "366cd5f3e5fd3df073aa16940e3c1aa7": "\\displaystyle e^{-2\\pi i a \\xi} \\hat{f}(\\xi)\\,", "366ce1a6a4e7b617df26ea023e9a2183": "\\infty,", "366d310ca9a631d37403d80b446f21cd": "t_0\\,", "366d3a4641adabb0c5fbc3a7a2ab7d63": "1 < p \\leq 2 ", "366db8eb88feab4a6fe7dca68cb161ad": " \\frac{\\partial}{\\partial x} F(x) = H(x) + C.", "366dd436d3bdaecda222cb0899364952": "\\kappa_0(\\mathcal B)", "366e317236e5a67f9e23661d541c8f4a": " p=3", "366e3bea68f46ae29772bf00fcde7217": "D=P^2 - 4Q", "366e5f4376f909a7eeb836b73fe8ed9c": " \\tilde{W} ", "366eb54dd2b677357130a8df45b9f061": "12n", "366ecb6da0f2fd4fea1a6d0e1a4e38f3": "T_T=T_L=T_s = 2 H \\, \\operatorname{arctanh} \\left( H \\right).\\,", "366f1a07a58104483db85f74c1623a55": "T_z {\\mathbb{C}}^n", "366f4a79709a54a95ad8cdf9627a50a4": " \\pi", "366f6878acc4fb87759479b107ecb619": "\\mathbb{Z}_n \\rtimes_\\varphi \\mathbb{Z}_2", "366fb0b515e05a65f34e3c64428368c9": "V = \\pi R^2 h", "366fe7cce292b844ba45398c270eb32c": "D^{1/2} 1", "3670241ce7408a4daee4739c75a96406": "\\sum_{n=0}^N(-1)^n\\,{x^{-2n}\\over n!}", "3670e0418ee0711d481bdc5f5d5a152b": "f_i = \\frac{A_i} {A_{st}} f_{st} ", "36721cc9b99b0aafd59b8b32c34e035e": "xy * z", "3672db497f943b1526d06f401e5d469d": "HP = \\{(u, v) : u \\in \\mathbb{R}, v > 0 \\}", "36735f535bc37a8597d95a5f74f24e20": " \\begin{align} \\int \\cos^n x dx & = \\cos^{n-1} x \\sin x - \\int \\sin x d(\\cos^{n-1} x) \\\\\n& = \\cos^{n-1} x \\sin x + (n-1) \\int \\sin x \\cos^{n-2} x\\sin x dx\\\\\n& = \\cos^{n-1} x \\sin x + (n-1) \\int \\cos^{n-2} x \\sin^2 x dx\\\\\n& = \\cos^{n-1} x \\sin x + (n-1) \\int \\cos^{n-2} x (1-\\cos^2 x )dx\\\\\n& = \\cos^{n-1} x \\sin x + (n-1) \\int \\cos^{n-2} x dx - (n-1)\\int \\cos^n x dx\\\\\n& = \\cos^{n-1} x \\sin x + (n-1) I_{n-2} - (n-1) I_n ,\n\\end{align} \\,", "36736c0b5c0fd6da5e5f2df96c2c389b": " \n\\pi = \\sqrt{12}\\left( 1 - \\frac{1}{3\\cdot3}+\\frac{1}{3^2\\cdot 5} -\\frac{1}{3^3\\cdot 7} +\\quad \\cdots\\right) \n", "3673884f333877720a16f2aba1df669d": " \\forall a,b\\in Y, a 0", "367a49e2b90eb0c38205e30e19b7722b": "\\zeta(z) \\; \\Gamma(z) = \\int_{0}^{\\infty} \\frac{u^{z-1}}{e^u - 1} \\; \\mathrm{d}u,", "367a788d924c710df016d8dce2d1a36d": "r=(4.5\\mu t^2)^{1/3}\\!\\,", "367aa96838ac5d9620d51fba98e1f505": "\\scriptstyle I_1 \\,\\cap\\, \\cdots \\,\\cap\\, I_k", "367aebd319e68b5de7f73d61908fb863": "H_2=\\{B^n|n\\in \\mathbb Z\\}=\\left\\{\\begin{pmatrix}1 & 0\\\\ 2n & 1 \\end{pmatrix} : n\\in\\mathbb Z\\right\\}.", "367af0b267195acb7f8a80e753681a2b": "U^n", "367b14174fc4814df71695f2ee51538d": " \\mathit P + \\Delta \\mathit P = (\\mathit A + \\Delta \\mathit A ) ( \\mathit B + \\Delta \\mathit B ) \\,\n = \\mathit AB + \\mathit B \\Delta \\mathit A + \\mathit A \\Delta \\mathit B + \\Delta \\mathit A \\Delta \\mathit B \\,\n ", "367b34fabcea47cfe6438038e37ae9bb": "f^{(k)}_j.", "367b4d7a349c263ac79b2dc0b3a63356": "n=2t + 1", "367be354b41647a9f5fd6d5a4d9ef574": "\\mathrm{d} X_t = \\sgn (X_t) \\, \\mathrm{d} B_t,", "367bec59c257e50dd369206e57471f22": " z \\rightarrow rz,\\ \\ \\infty \\rightarrow \\infty \\quad , ", "367c2b888098b7da6f461dabf02d1535": "\n\\hat{G}(\\boldsymbol{k}) = \\frac{ \\sin{( \\frac{1}{2} k \\Delta )} }{ \\frac{1}{2} k \\Delta }.\n", "367c39921058afc050b0ceca063de712": "\\overrightarrow{AD}", "367cf4d2857d71947ed518430dbbf119": "\n \\begin{align}\n \\lambda^3&\\left(I_0~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_0}{\\partial \\boldsymbol{A}}\\right) + \n \\lambda^2\\left(I_1~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_1}{\\partial \\boldsymbol{A}}\\right) + \\\\\n &\\qquad \\qquad\\lambda\\left(I_2~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_3}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_2}{\\partial \\boldsymbol{A}}\\right) + \n \\left(I_3~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_4}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_3}{\\partial \\boldsymbol{A}}\\right) = 0 ~.\n \\end{align}\n", "367d653791b1b4252ca2e46a722ae947": "\\|u\\|", "367d9efae6750a944e87c4e31cfce4ad": "a_{x_1, x_2, \\dots , x_n}", "367e02b6ed2dea7be1885d4c36001e32": " f(1)= i ", "367e2ca67d8daeb88b90142c02a58076": " \\textstyle \\Delta E = 0 ", "367e5c6fb0bf4d9da903c8a09548522d": "\\lambda_1,\\lambda_2,\\ldots", "367e6ef56c46ac9591c99aaf09b3605f": "\\vdash_M^*", "367e77a610da17a4405c94da7547a1ed": " Q[\\varphi] / R[\\varphi], ", "367ec8911122e39e2a70ef7a62075677": "\\partial g/\\partial x", "367eda521dbb6d69f008750b9a2a8e0a": "\\operatorname{cons} \\equiv \\lambda h.\\lambda t.\\lambda c.\\lambda n.c\\ h\\ (t\\ c\\ n)", "367ee3803cafd6adcb7a68a3fe46a4cd": "N = R^{*} ~ \\times ~ f_{p} ~ \\times ~ n_{e} ~ \\times ~ f_{l} ~ \\times ~ f_{i} ~ \\times ~ f_{c} ~ \\times ~ L", "367f1325931b77838ffb478d97356e75": "F(t) = e^{-2\\nu t}", "367f2e885b70e173081966860037a13e": "g^{(1)}( \\tau)= \\frac{\\left \\langle E^*(t)E(t+\\tau) \\right \\rangle}{\\left \\langle\\left | E(t)\\right |^2 \\right \\rangle }", "367f8ef5617e549a1bb78e26a56fff29": " T = \\frac{1}{2}\\partial_\\mu \\Phi \\partial^\\mu \\Phi + \\frac{1}{4g^2}F_{\\mu\\nu}F^{\\mu\\nu} + i \\bar \\psi \\gamma^\\mu \\partial_\\mu \\psi", "367fa00310ed23a7b0d9491b0e728827": "(\\rho_0,\\varphi_0,z_0)", "367fd333611032d9be9d3823a8edabda": "w(e) = w(u,v) - w(\\pi(v),v)", "3680a1540f04451d3c50ae03656923c6": " J_2 ", "3680d0d2e6036717d116957d1b4c43a9": "T \\circ T \\rightarrow T", "3680ecd1e51f420b6d2910bb6ce1621f": "(\\tfrac{p}{q})=1. ", "36815f2f67c6cdc9da02b77a0da5a439": "\\phi^{2}(P)", "36817eec34c99910b4ea62dd30c3f041": "y^2 = x(x-1)(x+2)", "3681b0551b3ab85da5f509fb056b98fe": " p_{ij}", "36821474e7024612e7f46ef4a40b1383": "\\mathcal{P}(\\kappa)", "368255f59256a8fc38f3092cb39fb22d": "f_0 = {1 \\over 2 \\pi R_0}\\sqrt{{3 p_0 \\over \\rho}+{4 \\sigma \\over \\rho R_0}}", "3682a126c0c2c609bf74b583ada506f5": "B^{ij}= \\mathrm{tr} (\\mathrm{ad}(e^i)\\circ \\mathrm{ad}(e^j)) / I_{ad}", "3682aaf47602cb4942ff4bac1fb3ff24": "d(f_{n}(x), f_{m}(x)) < \\varepsilon", "3682c87c8dd5dc1124642efd404e7460": "f^n(x)\\notin E", "3682da780703ce557ff8a6d2b11f7bbd": "\\text{Phase2}: AP > MP \\,", "3682ebce0e2e75512abeafd7a5078a6a": " d\\tau = \\sqrt{dx_\\mu \\; dx^\\mu} = \\sqrt{g_{\\mu\\nu} \\; dx^\\mu \\; dx^\\nu}.", "3682ff1f1ddafa95f8d811018045464d": "P(W_n|Spam)", "368329ec0dafd4434ac6342750f1b340": "u_a(t)", "3683527d2d4549c2fafcc46ab0ef6090": "\\mathrm{He}_n(x) = (-1)^n e^{\\frac{x^2}{2}}\\left(\\frac{d}{dx}\\right)^n e^{-\\frac{x^2}{2}}", "368356c3a6f9748ed72fc7ca2c886e1e": "\n\\operatorname{Li}_2(z) + \\operatorname{Li}_2(1/z) = \\tfrac{1}{3} \\pi^2 - \\tfrac{1}{2} (\\ln z)^2 - i\\pi \\ln z \\,.\n", "3683617bac73c7724f153ff53cf4b1cc": "z^2", "36838c59d7f683ce4a650efc81b587e0": "\\mu\\in\\mathbb R", "3683d6e2967dae7bf17fe2e3845f3e09": "\\textbf{x}_{k} - \\hat{\\textbf{x}}_{k\\mid k}", "3683d705417e5fdf17ad6d80fa7a2465": " ({C^{\\infty}}(M),\\{ \\cdot,\\cdot \\}_{M}) ", "3684063038c67f346f2bf373eb20be94": " [\\sigma_{am}] / [ \\sigma_{mm}] \\quad ", "36846beeac36108b775a0f9520268569": " 2 \\times 3", "36848381d1ec882b37699acf59045a3c": "\\sigma_{ij} = \\begin{pmatrix}\n\\sigma_{xx} & \\tau_{xy} & \\tau_{xz} \\\\\n\\tau_{yx} & \\sigma_{yy} & \\tau_{yz} \\\\\n\\tau_{zx} & \\tau_{zy} & \\sigma_{zz}\n\\end{pmatrix}\n=\n-\\begin{pmatrix}\n\\pi &0&0\\\\\n0&\\pi &0\\\\\n0&0&\\pi \n\\end{pmatrix}\n+ \n\\begin{pmatrix}\n\\sigma_{xx}+\\pi & \\tau_{xy} & \\tau_{xz} \\\\\n\\tau_{yx} & \\sigma_{yy}+\\pi & \\tau_{yz} \\\\\n\\tau_{zx} & \\tau_{zy} & \\sigma_{zz}+\\pi \n\\end{pmatrix}\n= -\\pi I + \\mathbb{T}\n", "3685077a551572c1d3101af75b3056e8": "\\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \\phi_x", "36850ee57354ba30cd6123325aa0147b": "5x^2-4", "36852bcd02906a8d2d777d9de3e17412": "\\ \\varepsilon={|cs|\\over a},", "36857ef5261600017893bce8aaa1800a": "\n \\varepsilon_{ij}=\\tfrac{1}{E}(\\sigma_{ij}-\\nu[\\sigma_{kk}\\delta_{ij}-\\sigma_{ij}]) ~;~~\n \\boldsymbol{\\varepsilon} = \\tfrac{1}{E}(\\boldsymbol{\\sigma} - \\nu[\\mathrm{tr}(\\boldsymbol{\\sigma})~\\mathbf{I} - \\boldsymbol{\\sigma}])\n ", "3685c183c2c8d209fcde49c7363d6edb": "{\\mathfrak M}", "3685dee3b9e0c38dbbb07af1fe997b68": "\\frac{7}{15}=\\frac{1}{3}+\\frac{2}{15}=\\frac{1}{3}+\\frac{1}{8}+\\frac{1}{120}.", "3685e7f027c19a18778cc0fec39c5d05": "\\delta_{ext}:Q \\times X \\rightarrow S \\times \\{0,1\\} ", "368618fb550075274993d5f63f7bc147": "f(i)f(j)", "36862d48acaa576aafe2812dcc3933c6": "L(s\\otimes n)=n.", "368642918cf9bd33228643bdc52b21f8": "X \\times Y", "3686434670d870a7603e7bb16c6fba69": "\\lambda x . x \\mathrel{:} A \\to A ", "36866b6f1f1c9314018c81ecd650ee03": " \\partial_{\\overline{z}} f (z) =\\mu(z)\\partial_zf(z)", "36870090e84598e83cc977ad089752bb": "y_2,\\ldots,y_q", "36873bdee7c57710ab168a3c0d466c85": " \\frac{\\partial}{\\partial q_k}\\sigma(\\textbf{q}, i) = \\dots = \\sigma(\\textbf{q}, i)(\\delta_{ik} - \\sigma(\\textbf{q}, k))", "3687428c8c0b57c17f36111d3a39a2d0": "C_J - \\ ", "3687c8074b592918e86ef15de1442a15": "\\kappa\\vdash k", "3688e8ee69786024a9a941b46fa9e058": "\\scriptstyle H_\\infty(X) \\;=\\; -\\log \\max_x \\Pr[X=x]", "3688f8de18956871df0073fd890ece70": "\\nabla E_{internal}(s)=\\nabla \\bigg[(\\alpha\\,\\!(s)\\left \\| \\mathbf{v}_s(s) \\right \\Vert^2 + \\beta\\,\\!(s)\\left \\| \\mathbf{v}_{ss}(s) \\right \\Vert ^2)/2 \\bigg] ", "368921719b793a3cd4d79026bfcd20e9": "\\bold{Set}^{\\mathbb{T}}", "36895c025a84ff9ce964f1b75880254c": "u \\oplus 1_{\\ell^2} \\sim u \\oplus 1_{\\ell^2} \\oplus 1_{\\ell^2} \\oplus \\cdots \\sim u \\oplus u^{-1} \\oplus u \\oplus u^{-1} \\oplus \\cdots \\sim 1_{\\ell^2} \\oplus 1_{\\ell^2} \\oplus \\cdots (u \\in {\\rm U}(H))", "3689b1917c68993b2a21e79a9b47bc16": "\\begin{pmatrix} i \\\\ j \\end{pmatrix} = \\begin{pmatrix} 3 & 2 \\\\ 1 & 0 \\end{pmatrix}^t \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}", "3689b54576708079edf58dce301e41e8": "\\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c}) = \\det \\begin{bmatrix}\na_1 & a_2 & a_3 \\\\\nb_1 & b_2 & b_3 \\\\\nc_1 & c_2 & c_3 \\\\\n\\end{bmatrix}.", "3689bec8a44fbd9d5ebfb8c4eb73d570": "G_n(z) = (z-z_0)(z-z_1)\\cdots(z-z_n)", "368a0b31f5d75b5346f5d37a60f62a08": "m(E,a_x,a) = \\frac {m_0 (E,a_x,a)}{DDREF} \\frac {x_D x_s x_T x_B}{x_{Dr}}", "368a20f28909f2d0b57194a3f390cea6": "-512\\le x,y \\le 512", "368a2ceb76b6dbdfc1adfda72fe98269": "\\mathbf{M_{B}(100% normalized)} = \\begin{bmatrix}\n39.216 \\\\\n16.012 \\\\\n10.242 \\\\\n16.022 \\\\\n4.699 \\\\\n7.276 \\\\\n6.533 \\end{bmatrix}", "368a5bea782878aacabe60e1281d7cdb": "\\operatorname{E}( X ^ 2 ) = \\sigma^2 + \\mu^2.", "368a6b3e283b3b3ad38739f776e37ef5": "\\frac{\\partial}{\\partial\\theta} \\ln Q = \\frac{1}{Q}\\frac{\\partial Q}{\\partial\\theta}", "368a8dde476fdb8b83c82f19a99d38ab": "0 \\leq i < \\omega", "368ad68fdcae431f90fcbfded7a62f93": "\\begin{bmatrix}\n c_3 c_1+s_3 s_2 s_1 &\t-c_3 s_1+s_3 s_2 c_1 &\ts_3 c_2 \\\\\n c_2 s_1 &\tc_2 c_1 \t & -s_2 \\\\\n-s_3 c_1+c_3 s_2 s_1 &\ts_3 s_1+c_3 s_2 c_1 & \tc_3 c_2\n\\end{bmatrix}", "368b4ae0bf4d106cc3cf845e4d60b69d": "n \\ge n_0", "368b5d3de95add5b1078bcde515b0e4e": "P_{0}^{0}(x)=1", "368b74a2b46d30cd068b670a55f5ab33": "I(X_{i};X_{j(i)})", "368c18c83c5f6c1f8ae9a6ce3ff0306c": " (\\Delta f)(x) = f(x + 1) - f(x)\\, ", "368c2a4013d75437f5d7f13e41ead2de": "\n \\rho_0~\\cfrac{\\partial v_r}{\\partial t} + \\cfrac{\\partial p}{\\partial r} = 0 ~;~~\n \\rho_0~\\cfrac{\\partial v_\\theta}{\\partial t} + \\cfrac{1}{r}~\\cfrac{\\partial p}{\\partial \\theta} = 0 ~;~~\n \\rho_0~\\cfrac{\\partial v_z}{\\partial t} + \\cfrac{\\partial p}{\\partial z} = 0 ~.\n ", "368c2c25dacbc3052414ff6632d7b735": "a,c_n\\in\\mathbb{C}.", "368c4176a510d82106436c9963c5d323": " B = -2\\log_{10} \\left({\\varepsilon\\over 3.7 D} + {2.51 A \\over \\mbox{Re}}\\right) ", "368c9d467392b9136c60f6926d6c93a0": "\n \\tilde{\\epsilon}_{\\mathbf{k}} \\phi_{\\lambda}^{\\mathrm{R}}(\\mathbf{k}) - \\sum_{\\mathbf{k'}} V_{\\mathbf{k}-\\mathbf{k'}}^{\\mathrm{eff}} \\phi_{\\lambda}^{\\mathrm{R}}(\\mathbf{k'}) = \\epsilon_{\\lambda} \\phi_{\\lambda}^{\\mathrm{R}}(\\mathbf{k}) \\,,\n", "368ce7d0ec19d6b8ac348830c7d4aa3f": "x^- \\to x^-", "368cf28514704aa5796ef6ded08653ae": "\n P\\left[sgn\\left(m\\right)\\Delta S > Z_\\lambda \\sigma \\left(\\frac{V}{V_\\sigma}\\right)^\\frac{1}{2}\\right] = 1 - \\lambda \\;.\n ", "368dbb4905d22c57078d66bde29adccb": "H_1 C_2 P_n \\simeq \\mathbb Z^{n+1}", "368de71c30f2913ab75e3394d83c9ab1": "M_W = \\frac{v|g|}2,", "368e60dc7f39166213730ccff8c7c8b5": "\\chi_m", "368e68a76a9975aaf702557f897464cd": "{\\rho}=\\frac{p-e}{R_dT}+\\frac{e}{R_vT}\\, .", "368e7baa36f717a6d334503182a19c5b": "\\int dk {1\\over k^2} - t \\int dk { 1\\over k^2( k^2 + t)} = A\\Lambda^2 b + B b t", "368f88b4d393787d15bb833c0830a181": "\\forall \\alpha.(K_1^1[\\alpha/S]\\rightarrow\\dots\\rightarrow \\alpha)\\dots\\rightarrow(K_1^m[\\alpha/S]\\rightarrow\\dots\\rightarrow \\alpha)\\rightarrow \\alpha", "368fb57cec959022678a3fc22286f5e4": "e^{2f}\\left(R+\\left(\\text{Hess}(f)-df\\otimes df+\\frac{1}{2}\\|\\text{grad}(f)\\|^2 g\\right) {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} g\\right)", "368fd02b1ed157128d35b2e6c605349a": "|\\mathcal U|", "368fd68d211952bba93f557afa6030ca": "S(t)", "368ffca28a113aa7a0f2ce19c8824970": " \\{f_n\\}_{n=1}^{\\infty} ", "369036202f4d3e6efa221af25bed2964": "\n\\nabla \\phi =\n\\frac{\\hat{ \\mathbf e}_1}{h_1} \\frac{\\partial \\phi}{\\partial q^1} +\n\\frac{\\hat{ \\mathbf e}_2}{h_2} \\frac{\\partial \\phi}{\\partial q^2} +\n\\frac{\\hat{ \\mathbf e}_3}{h_3} \\frac{\\partial \\phi}{\\partial q^3}\n", "36907218fa5f88c4c5ded9192398e4c5": "\n\\lambda(\\rho, \\theta) = \\int dz \\ f(\\rho, \\theta, z)\n", "36907514062a0df94e391b0b0a7c4079": "\\displaystyle \\frac{2\\, \\operatorname{rect}(\\pi\\xi)}{\\sqrt{1 - 4 \\pi^2 \\xi^2}} ", "3690e953365f79ee7e3e7078db522320": " \\and T_3 = [F_3, S_3, A_3]::K_2 ", "369168a0707541e7468399d27cfc56c3": "su(4)", "3691d7e67c8ad452a18d1e2a8c9b16aa": "d\\mathbf{m} = \\mathbf{M} \\, dV", "3691ebee608b6bfcc3f22728e5716e28": "i_1+\\cdots+i_k=n", "36927a3198293f00a6d3829def1a8536": "\\frac {D_{T1}} {D_{T2}} = \\frac {T_1} {T_2} \\frac {\\mu_{T2}} {\\mu_{T1}}", "369289bebca96507aca216b9ba04e29b": "\\theta_{33}", "369289da63d5353010fc92dd7fc8e972": "\\theta_{\\mathrm{eq}}", "3692a23078fc1b542ca887877fae4c0b": "\\langle A\\otimes B,R_b\\rangle = \\langle B,\\Omega_b(A)\\rangle", "3692c9bd348e4067f74f6e910420a06b": "g_i=\\frac{4a_{i-1}a_i}{b_{i-1}g_{i-1}}, (i=2,3,...n),", "36930e0c7648bea2d7f406a62830721f": "M=\\mathbb Z", "36932b8112ab9ad0f0903e5ac8483066": " \\text{Risk} = (\\text{probability of the accident occurring}) \\times (\\text{expected loss in case of the accident})", "36934ea70d1c4360544dad4810f78036": "\\|aA\\|_{op} = |a| \\|A\\|_{op} \\quad\\mbox{ for every scalar } a ,", "36934ea7ac6a7be8633175454f4f729e": "\\underline{x}^{-k}", "36936516e7f9dd4f5fdf896c0178ac42": "\\mathcal{L}=-\\frac{1}{16\\pi}(\\partial^\\mu A^\\nu-\\partial^\\nu A^\\mu)(\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu)+\\frac{m^2 c^2}{8\\pi \\hbar^2}A^\\nu A_\\nu.", "369388d1fc7480de1e630a71aff28d85": "g(r) = \\exp \\left [ -\\frac{u(r)}{kT} \\right ] ", "36938a7d2f2e096aee26065ce4941c5b": "\\mathrm{d}V=r^2 \\sin \\theta \\,\\mathrm{d}r\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\varphi.", "3693cdd318a4dcd1b7af530e1a53288b": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2}\\Psi + V\\Psi ", "369401c5334f6b1f20fad8995ff8749c": " w_a^2 \\approx 0 ", "369440d0e5bafbf8f9b34d978fc114c2": "1-c\\,X", "36947cd0e3b899d3d53b9d93437dbfc1": "{{{10}}}", "3694affb824b9354260809f6a7199fc6": " [M_i,N_j]=-i\\epsilon_{ijk} N_k", "3694cbac39ad0a248abbdd56d08b9b5e": "\\|Av\\| \\le \\|A\\|_{op} \\|v\\| \\quad\\mbox{ for every } v\\in V .", "369514ada29bef1dc5f2e17454ba63ff": "\\ \\sigma = h(b). ", "36954c5bfefda82c13f78e5531a1da8d": "\\nabla_{\\vec{X}} \\vec{X} = 0", "36955dcd4880ae90257ffd6289f68200": "f(x) =\\pi(x)+\\frac{1}{2}\\pi(x^{1/2})+\\frac{1}{3}\\pi(x^{1/3})+\\cdots", "3695720a845778a68181307ca1f1d628": "F \\subset A \\subset U \\subset V \\ \\ \\text{and} \\ \\ \\mu(U) - \\mu(F) = \\mu(U \\setminus F) < \\varepsilon", "36957fe3536e05d000a2e05874a69d04": "\\frac{\\mathrm{d}X(t)}{\\mathrm{d}t} = \\begin{cases} r_i & \\text{ if } X(t)>0 \\\\ \\max(r_i,0) & \\text{ if } X(t)=0.\\end{cases}", "36958295118fbd8924804c411c4ee8a5": "r = (L \\leftarrow K \\rightarrow R)", "369593105c68955d0735bb457087d0e3": " v( S \\cup T ) + v( S \\cap T ) \\geq v(S) + v(T), \\forall~ S, T \\subseteq N.\\, ", "3695be1fd3cea16fca320573092dc516": "(\\ell+1-m)\\,P_{\\ell+1}^{(m)}(x) = (2\\ell+1)x\\,P_\\ell^{(m)}(x) - (\\ell+m)\\,P_{\\ell-1}^{(m)}(x).\\,", "3695e6fbfe72cb6a6203b13be8001aac": "(A \\wedge B) \\rightarrow (A \\otimes (A \\rightarrow B))", "36960ed139ec514bd98a317e7a88450b": "\\operatorname{prox}_{\\varphi}:\\mathcal{H}\\to\\mathcal{H} ", "369634a210792a491358393007f36711": "L\\prec M = N", "36965c7c942c93d793b887459a3799cb": "R(x)=\\sum_{i=0}^{n-4} f_i x^i", "36967af788101efeab37f42c9d0df75a": "f(x) = \\frac{1}{2}(x + \\frac{2}{x})", "3696e4bc8511726c49557117c98f1d9c": "\\partial f/\\partial\\boldsymbol{\\sigma}", "36970197fda3fa578593a4a25fe428a5": "\\lim_{x \\to p^{+}}{f(x)} = L", "369735af74ffae994df661ed7d76e58e": "H(s) = \\left( \\frac{1}{\\beta-\\alpha} \\right) \\cdot \\left( { 1 \\over s+\\alpha } - { 1 \\over s+\\beta } \\right). ", "369767b06d20e71faa16ff18ba46b5f7": "a_0 + a_1 b_1 + a_2 b_1 b_2", "369788750fc35d388f050814acc59773": "\\mathcal{I}_{c, c}", "3697d1e2eae9e680bc14928d5ae897bc": " X \\gtrdot a \\iff \\begin{cases} A \\to \\alpha B Y \\beta \\in P \\\\ B \\Rightarrow^+ \\gamma X \\\\ Y \\Rightarrow^* a \\delta \\\\ A, B \\in V_n \\\\ \\alpha , \\beta, \\gamma, \\delta \\in (V_n \\cup V_t)^* \\\\ X, Y \\in (V_n \\cup V_t) \\\\ a \\in V_t \\end{cases} ", "3697d28002e9d9644378eccc1b08d546": "0 \\leq N_i \\leq b^k-1. \\, ", "3697ee11b905ff20da2f727c1eb48f85": " \\langle\\alpha|\\widehat{p}|\\alpha\\rangle = i2^{-1/2}(\\alpha^{*} - \\alpha) = p_{\\alpha} ", "3697fcf668cddffcda7c397e5acfa907": "V = \\sqrt{600 \\, \\Omega \\cdot 0.001\\,\\mathrm W}", "3698b58f46c47815cac4671618ca27ab": "a\\!\\!\\!/a\\!\\!\\!/ =a^{\\mu}a^{\\nu}\\gamma_{\\mu}\\gamma_{\\nu}=\\frac{1}{2}a^{\\mu}a^{\\nu}(\\gamma_{\\mu}\\gamma_{\\nu}+\\gamma_{\\nu}\\gamma_{\\mu})=\\eta_{\\mu\\nu}a^{\\mu}a^{\\nu}= a^2", "3698c4ec6b7adca50389f62bbe635cbc": "\\sum_{i=1}^m|F_n(i) - \\tfrac{i}{m}| = O(n^{\\frac{1}{2}+\\epsilon})", "3698ef53d769babc3959444bdad82776": "10^-8", "369941e2811cd8a723e56db75bec1606": "\\sec^2(x)", "36996c558db6d8c8518f4cb488e90c82": "IS(p,P)", "3699c63f25dcc992b21c89f62802690a": "(M,g) ", "369a02d054855a949e5d76cce7f2633b": " \\rho \\int_0^\\infty {u(y) \\left(u_o - u(y)\\right)} \\,\\mathrm{d}y ", "369a154d5347e70114104fec91cf72fe": "(1,2)", "369a2c79230347522ea1a41a2bf60e62": " x^{6972593} + x^{3037958} + 1. ", "369a635758b09cf2633053eb9c6d5194": "SD(X) = \\left(\\mathbb{E}[(X - \\mathbb{E}[X])^2 1_{\\{X \\leq \\mathbb{E}[X]\\}}]\\right)^{\\frac{1}{2}}", "369a74ac8e5a1a73a9ccea416ec4e45d": "\\alpha = \\lambda - \\sin^{-1}[y\\sin(\\alpha + \\lambda)]", "369aa7a0a07fd443ddb6c37dc89ba7e3": "\\ \\Delta G", "369aada125ca0266d9cbc465345311f0": "\\scriptstyle N", "369ac045ed08533a2cdca233187dcb23": "\\alpha = (\\alpha_1, \\alpha_2, \\ldots , \\alpha_k)", "369ac7ed88848a49d1ffeebd48e7cd8f": "W={m_\\mathrm{0} c^2} \\left( \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} - 1 \\right) - k \\frac{Z e^2}{r}", "369b1ca57980a7586aeeb9403232162d": "\\underline{\\varphi \\quad \\quad \\quad}\\,\\!", "369b3c7e81945d53649bfbf7329472f7": " |f_{k_b} \\rangle ", "369b455d269a21363f69edb87629c056": "\n\\mbox{Peak-to-peak} = 2 \\sqrt{2} \\times \\mbox{RMS} \\approx 2.8 \\times \\mbox{RMS} \\,\n", "369b4b090b0a803dbacee1cb36f1a5f7": "h=N_{v}\\frac{e^{-(E_{Fp}-E_{v})}}{k_{B}T}", "369bf27b34ec7a025199f929ff923ecd": "R_1 = \\frac13(2a+b) \\approx 6371\\,\\mathrm{km}", "369c92b6c67bef6f7cd48ced0c34b573": "\\Bigg(\\frac{a}{p}\\Bigg)_4= \\pm 1 \\equiv a^{\\frac{p-1}{4}} \\pmod{p}.", "369cb57cc9c44a0cf5a741b8f7d54119": "[H_{\\alpha_i},E_{\\alpha_j}]=\\alpha_j(H_{\\alpha_i}) E_{\\alpha_j}", "369ce6b458c20b596b90fa154a7dcd06": "\\Box\nA\\to\\Box\\Box A", "369d565a0a9db1277c2f3d573b155bf3": "na + \\frac{n(n-1)}{1\\cdot 2} \\Delta a + \\frac{n(n-1)(n-2)}{1\\cdot 2\\cdot 3} \\Delta^2 a +\\cdots,", "369db02bb9f399a957a6c7a0236181c7": " k \\propto \\frac{1}{d} \\ln{\\frac{1}{\\delta}}", "369dcaa83bc27467a5737ef235eb4f44": "q=e^{-\\sigma^2}", "369e71dc3a67c2eaf110bc9b557c4ad4": "\\Gamma \\vdash \\varphi(y)", "369e80633567e18824ce714c20ab2425": "= \\frac{e^4}{(k-k')^4}\\operatorname{Tr}\\left( \\Big(\\sum_{r'} v_{k'} \\bar{v}_{k'} \\Big) \\gamma^\\mu \\Big(\\sum_{r}v_{k} \\bar{v}_{k} \\Big) \\gamma^\\nu \\right) \\operatorname{Tr} \\left( \\Big(\\sum_{s} u_p \\bar{u}_{p} \\Big) \\gamma_\\mu \\Big( \\sum_{s'}{u_{p'} \\bar{u}_{p'}} \\Big) \\gamma_\\nu \\right) \\,", "369e98a090b6c21a9c245730972702c3": " \\beta_n ~ = ~ k_0 ~ \\sin \\theta_0 ~ \\sin \\phi_0 ~ + ~ \\frac{2n\\pi}{l_y} ~~~~~~~~~~~~~(2.2b) ", "369ece26dec3aaba057c84241eea7a56": "E_{1,2}(p)", "369f096129117edab73174c013ad7644": "PR(p_i; 0) = \\frac{1}{N}", "369f163a785752215a8a4ba7871d2d2c": "L^0(\\mathcal{F}_T)", "369f207f4a1252ede6c059af9cdaafcb": " \\langle q|\\mathbf{\\hat T}(-\\varepsilon)|\\psi\\rangle = \\psi(q) + i\\frac{\\varepsilon}{\\hbar}\\langle q|\\mathbf{\\hat P}|\\psi\\rangle + O(\\varepsilon^2) ", "369f6f9a52a4ea87bd6122211d2bc2c2": "e^{bj} \\!", "369f9ddae201721c25c5ffb18081448b": " \\Box \\Box p ", "369fcce846c499dae2c9b6ed7e1e7d5b": "A^\\circ", "369fe591f35a691199f4cec80793344c": "\\operatorname{Tr}\\, K^2 = \\iint K(x,y) K(y,x) \\,dx\\,dy", "36a034059c79c04673c00f87df09ed25": " R(1) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} S(\\omega) e^{i\\,\\omega\\,1} d\\omega. ", "36a0396f882f0f9260ed9c6b3a3a07a9": "C\\,\\!", "36a065b2893c60ee6b7996f7a93b729d": "\\mathbf{F}_{i_1,i_2,\\ldots,i_N} ", "36a071c40544a6852863d6ec0c92e825": "v=\\frac{BV \\cdot GF \\cdot \\epsilon}4", "36a0abf844ab3ebb053f12b97dca4d70": "X\\in\\mathcal{H}", "36a0c25eef123d54a64f09fc52b01950": "\\triangledown ^2 \\zeta\\,= 0.", "36a105560395b7d90267d820959449aa": "\\ln(4/3)/\\lambda\\,", "36a131f6c44a0c50a4fd055029b64fbd": " s = -\\alpha \\pm \\sqrt{ \\alpha^2 - {\\omega_0}^2 } ", "36a136468a04b99b26a8158401fa4c19": "\\inf\\varnothing=\\max(\\{-\\infty, +\\infty \\} \\cup \\mathbb{R})=+\\infty.", "36a13d87675c53fd56bd1bb08c39824a": " E_i\\frac{x^i}{i!} ", "36a154417995cfa2c1818d7381b6f546": "\\alpha_1,\\alpha_2,\\ldots,\\alpha_m", "36a16670546694f6235174819f3de6fb": "\\mathrm{tr}\\, \\sigma^i\\sigma^j = 2\\delta_{ij}", "36a1704b90e7cefb0d72e8af7ca4ee22": "p_j=\\frac{1}{j!}\\exp\\left\\{-\\frac{\\mu^2}{2}\\right\\}\\left(\\frac{\\mu^2}{2}\\right)^j,", "36a1819d02a742f4ece9ddae516843e1": " f_X(x|\\boldsymbol \\theta) = h(x) \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(x) - A({\\boldsymbol \\theta})\\Big)", "36a1fe0206b183faf88de78b7650c677": " {(M_1^* F)}^\\sim(\\lambda) = \\tilde{F}(\\lambda).", "36a22d1ecd947717945da51bd4d0e553": "|r^\\prime(s)|=1", "36a24662b386b8212aa9eee86d1b6d0c": "\\delta_\\text{c}\\colon Q \\times \\Sigma_\\text{c} \\to Q \\times \\Gamma", "36a28436a39c45abf81d14f8363fe46d": "\n \\Beta(x,y) = \\frac{x+y}{x y} \\prod_{n=1}^\\infty \\left( 1+ \\dfrac{x y}{n (x+y+n)}\\right)^{-1},\n\\!", "36a297660994f781aa60fc2456e34b9f": "\\rho(\\mathbf{E}|\\boldsymbol\\Sigma_{\\epsilon}) \\propto (\\boldsymbol\\Sigma_{\\epsilon}^{2})^{-n/2} \\exp(-\\frac{1}{2} {\\rm tr}(\\mathbf{E}^{\\rm T} \\boldsymbol\\Sigma_{\\epsilon}^{-1}\\mathbf{E}) ) ,", "36a3285b3e35c7e9b44d9e21e7b7172c": "G_1 \\leftarrow ", "36a336d023ddcef0879b255bcf416f73": "x/y = x - xyx", "36a33798a3201a8da602d8e196d27f7d": " \\langle \\phi , \\psi \\rangle = \\frac{1}{|G|} \\sum_{g \\in G} \\phi(g) \\psi(g^{-1}) ", "36a36099606fb8ba095ecfddc9e7d5d2": "B \\rightarrow bB", "36a3f34f1579f5f52803f170e98d27b4": "t_{1-\\alpha/2}\\,\\!", "36a482bf717809fba0a5cd30f52da6b9": "g^{f}_{V}", "36a4b47b16933ed3dbf9195c151b9175": "\\sigma_{a \\theta b}( R )", "36a504c4cba5d2ca4622f136a1682150": "L\\;", "36a54bda84a86cb22fabf1683f07351e": "Q_j = \\sum_{i=1}^n \\mathbf {F}_{i} \\cdot \\frac {\\partial \\mathbf {V}_i} {\\partial \\dot{q}_j}, \\quad j=1,\\ldots, m.", "36a579ed17b1aedc5169886c7f3107c7": " s = s(x) \\in R_q ", "36a5b1443842a910493df04238cbea7b": " [0,1]^\\ast = \\{\\,x\\in\\mathbb{R}:0\\leq x\\leq 1\\,\\}^\\ast", "36a5cc33d8f3a1064b11e8dd94d983e4": "S_\\eta=-\\frac{1}{\\beta}\\sum_{z_0\\in g(z)\\text{ poles}}\\text{Res}\\,g(z_0)h_\\eta(z_0)", "36a6be14e625cfb4d7ea0023592b3922": " - \\,\\! ", "36a77daa0ef50922c4f8014d6f0a6473": "L \\rightarrow { L \\over 2 }", "36a7b52e429ff2c025a2343a3eb3dcd9": "F_i = i/n\\,", "36a8416cdd61f711ad336379d05b76b4": "x_{\\rm i}", "36a8bfc36f6f45263b1a3dafa64db3a4": "D \\frac{\\mathrm{D} u_z}{\\mathrm{D} t} = 1 + \\nabla^2 u_z ", "36a8d570962edd86eb838839914f8bd1": "d_x = d(x, \\partial \\Omega)", "36a8e664d36796aa80e41fb8e8daed82": "\\textstyle x^k-1", "36a8f4931a8867a3c1529551f8e1b721": "0 \\div 0 = NaN", "36a90c34befbde43341c2567b5ea273d": "\\forall x[x\\in y' \\leftrightarrow x\\in z'] \\rightarrow [y'=z']", "36a931964f1192a6aeb888dcb7ba90bb": "f(x')", "36a95010c7700deed92fc2bb698a96a8": "\ny = f(x) = \\int\\limits_{0}^{x} p_x (u) du\n", "36a95e778aafc17ea70108aa25dbbfa1": "P + P \\subseteq P", "36a98fe4884030a2fb369c68c66930cc": "\\hat{w}_2(F)", "36a9eb1d7df76cd5b709d91043cd5bea": "a+d < b+c.\\,", "36aa233bf0f64061e5377bee7bde48b6": "B = \\mu_0 \\mu_{\\mathrm{eff}} \\frac{N I}{l} = \\mu \\frac{N I}{l},", "36aa3b802e9ce1caf62f24a19426f9d6": "\\frac{\\part \\overline{u}}{\\part z} = \\frac{u_*}{\\overline{\\xi'}}", "36aa5a19631d196a0c181cdbf8b8d083": "B = \\frac {\\mu_0 I}{2 \\pi r}\\,", "36aa630d5bd4e437ea5e94d1f4896d67": "C \\ ", "36ab083a52d653b714edf12772f5effd": "(Q_h/R \\approx \\mbox{2700 K})", "36ab4489fbcd618d6013a5a3b035182e": "G \\cap M.", "36ab5a8bfb765a782f5eb4bc3f327545": "a=R/2\\,\\!", "36ab68a364c45d5923b7ef005ccb13a1": "S = \\prod_{i=1}^s \\frac{\\mathbb{F}_q[x]}{\\langle p_i(x) \\rangle}", "36ab7f2cef52ef04bc4f38ad4e0959c5": " \\mathbf{r}_1 = \\frac{m_2 \\mathbf{r}}{m_1 + m_2} , \\mathbf{r}_2 = \\frac{-m_1 \\mathbf{r}}{m_1 + m_2}.", "36abc46ccc95f16617f2fca7447a27d1": "b(\\theta)=0", "36abe4e7e09e1a65bcd041b97b37c449": "VCA(64x^3-448x+448,(0,1)) \\cup VCA(64x^3+192x^2-256x+64,(1,2))", "36ac0d28f159c320d83c39aa8309c057": "\\, h", "36ac26b694d663c705602d73c3875ecd": "Y_{ij} \\sim N(\\gamma_{00},(\\tau_0^2+\\sigma^2))", "36ac291c2c70427f23a7a23d848f6775": "\\textstyle \\mathrm{length}(P_1) + \\mathrm{length}(P_2) \\le n + 1", "36acbdda0c3b399621ca7d2d9f1541bc": "I_t \\, ", "36accc318bf89a22a48709123eca9ed8": "\n\\Lambda = \\prod_{n} \\Pr\\{(x_{ni})\\mid r_n\\} =\\frac{\\exp(\\sum_i -s_i\\delta_i)}{\\prod_{n} \\gamma_r}\n", "36acde23cf5b7609f39e0ff51e872e1c": "\\Gamma_{ij,k}^{(\\alpha)}=E[(\\partial_i\\partial_j\\ell+\\frac{1-\\alpha}{2}\\partial_i\\ell\\partial_j\\ell)\\partial_k\\ell]", "36ad62f5fc392d0178dc6840dc68807d": "\\sum_{k = 1}^n \\left[a + (k - 1) d\\right] r^{k - 1} = a + [a + d] r + [a + 2 d] r^2 + \\cdots + [a + (n - 1) d] r^{n - 1}", "36ada4352b0e10934630d813beee46d1": "f_{WN}(\\theta;\\mu,\\sigma) =\\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty q^{n^2/2}\\,z^n", "36adb81bec09f02a0266d2aa4d55f036": " a,b ", "36ade947ef1c19cf9a5eacb1ef7a109b": "g(-\\tau).", "36ae1c009f31983696aefbc37785b500": " \\int_{a}^{b} f(x) \\, dx \\approx \\frac{b-a}{6}\\left[f(a) + 4f\\left(\\frac{a+b}{2}\\right)+f(b)\\right].", "36ae1e433776770cc1b8b14f99c29cd0": "\\, t", "36aec2b0ebd6b61f744f1ff3653faa3a": "\\pi_1 \\cdot (U_{11} - U_{21}) \\cdot p(y|H1) - \\pi_2 \\cdot (U_{22} - U_{12}) \\cdot p(y|H2) > 0 ", "36aec88940969ca924436afa202c97e4": "\\Lambda(B)", "36af08b431ce5cbd80e6bd3e4c015047": "(Y,f)", "36af5277352ff61236723406e7a9afce": "\\int\\limits_C f\\, ds = \\int_a^b f(\\mathbf{r}(t)) |\\mathbf{r}'(t)|\\, dt.", "36af78615d775dc8ef41ed6b4d241d60": " x \\oplus 0 = x,", "36af817554cf9a5fd9db068e7000f0d4": "F\\dashv G.", "36afde2972efdb93b5e03329f845b012": "\\Delta(A) = \\kappa(A,A') + \\sum_{A \\stackrel{\\rho}{\\to} A'}m(\\rho) A'", "36b04f736acc64652a84982c092d8598": "\\phi(p)", "36b06aab48227a5dc48bbeb52e685db3": "\\lim_{x\\to{+\\infty}}f(x)=0,", "36b06fa963ca1d43db8c68242c480cad": "L(z) = f'(x)z", "36b073b331fbf276b4b8c4059bd9a7d1": "\\gcd(a,q)=1", "36b0a56a9b19ad28ff547bba74c787fa": "{\\delta}", "36b0ba265b0db4b6f73fd6ca280f6185": "\\varphi=\\frac{13}{8}+\\sum_{n=0}^{\\infty}\\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}.", "36b109c367d82549c6dcb23386f1cb86": " \\hat T(V) \\hat\\otimes \\hat T(V) = \\prod_{j,k\\in\\mathbb{N}} T^jV \\otimes T^kV,", "36b17fffb9facc0d21d3cf2216ce4f1d": "F_0,F_1", "36b18b24ab4e3cc4f4ae33386e55627c": "\\begin{align}\nL_{x} &=I_{1}\\dot{\\psi}=\\mathrm{constant}\\\\\nI_{2}\\ddot{\\alpha}&=0\\,.\\end{align}", "36b19725a28ec00c7369114b9659221c": "RMS_{DC}", "36b19b0423419b0bf8ed7a500f5c3d40": "\\langle\\Phi| \\ \\ \\ |\\Psi\\rangle", "36b19ef4cdba32920ed9c0c6f69e87b1": "x^0=(x_1^0,\\ldots,x_n^0)", "36b1d68d2868f28a467042355c736c39": "1i} g^{x_j}", "36b499cfaa7d94d83428cc5ee296bcab": "=, \\ne, \\neq, \\equiv, \\not\\equiv \\!", "36b4b0d9b55d2c9c9059d082b4d7a0a8": "X_i = 0", "36b4f6b49303b1b96294960acce165f6": "\\operatorname{GL}_2", "36b52ec5df1b72bb1b7eb6b60cbb3893": "W = \\frac{1}{T} \\int_0^T Ri^2\\,dt=\\frac{R}{T} \\int_0^T i^2\\,dt=\\frac{RI^2}{2}", "36b5731162bcadea42ce941f64780b0e": "s_{n+1} > \\sum_{j=1}^n s_j", "36b5ef3757485064ab7dca4248db29b4": " -0.96x^2 +56.4 x -810", "36b6360bd9979879cd7a9ad5e3913c35": "B(Y,Z)\\circ K(X,Y)\\circ B(W,X)\\subseteq K(W,Z).", "36b639df9d73f5d030bb1efbfccf6f8a": "\\begin{align}S(\\omega) &= |E(\\omega) + E(\\omega-\\Omega)e^{i\\omega\\tau}|^2\\\\\n&= I(\\omega) + I(\\omega-\\Omega) + 2\\sqrt{I(\\omega)I(\\omega-\\Omega)}\\cos[\\phi(\\omega)-\\phi(\\omega-\\Omega)-\\omega\\tau]\\end{align}", "36b6433a33c009ee81e6b8c270c08d9d": "{d^n \\over dx^n} f(g(x)) = \\sum_{k=1}^n f^{(k)}(g(x))\\cdot B_{n,k}\\left(g'(x),g''(x),\\dots,g^{(n-k+1)}(x)\\right).", "36b6d4ac71b482f77aabd497f65d01e3": "\\cos 2\\theta = m/ \\sqrt{m^2+|p|^2} ", "36b70211034c8c5e22f4daed98d1f9d0": "\\scriptstyle(-1.8\\pm1.5)\\times10^{-11}", "36b737813932638220635ba601484acb": "\n\\pi = 6 \\sin^{-1} \\left( \\frac{1}{2} \\right) \n= \\sum_{n=0}^\\infty \\frac {3 \\cdot \\binom {2n} n} {16^n (2n+1)}\n= \\frac {3} {16^0 \\cdot 1} + \\frac {6} {16^1 \\cdot 3} + \\frac {18} {16^2 \\cdot 5} + \\frac {60} {16^3 \\cdot 7} + \\cdots\\!\n", "36b7614fbf7e4e43b99ac4388c73eab2": "\\chi \\times \\mathbb{R}_{+}, g, q", "36b76ca57e3aa99f2dd4d8c87aeabbe7": "(y_j)", "36b7a5a0150dd4e04b4078a7f3aeeac7": "A_k", "36b7b4f59f4937081bcaa6d927511eee": " \\left(\\frac{a}{c}\\right)", "36b7f86541ddb9019d5e10203694e127": "y = f'(x_n) \\, (x-x_n) + f(x_n),", "36b87e3f5c3e285ccb3370e401a36574": "d(x, y) = |x-y|", "36b8a131346bf66bc07cd1c6bbe649bd": "\\begin{smallmatrix}\\binom{n}{r}\\end{smallmatrix}", "36b8c465a78de9b419caaa90f7598ba0": " \\mathbf{p}\\cdot\\mathbf{p} \\propto \\mathbf{k}\\cdot\\mathbf{k} \\propto T \\propto \\dfrac{1}{\\lambda^2}", "36b8ef4436ec49cce9eb2e959dad54e7": "\\beta^{(q^m-1)/(q-1)} = 1", "36b9279de8ea7c46ddc3acc3e70d8dec": "\\frac{1}{\\sqrt{1 - x^2}}", "36b92d4bc1bb06f85c2d8a6652a3104f": "\\begin{cases}\nF_1(x,y), \\quad p_1 \\\\\nF_2(x,y), \\quad p_2 \\\\\n\\dots \\\\\nF_n(x,y), \\quad p_n\n\\end{cases}", "36b943baa74685b2881e413adbb77838": "\\begin{align}\n \\frac{\\mathrm{d}}{\\mathrm{d}x} \\int_{\\sin x}^{\\cos x} \\cosh t^2\\;\\mathrm{d}t &= \\cosh\\left(\\cos^2 x\\right) \\frac{\\mathrm{d}}{\\mathrm{d}x}\\left(\\cos x\\right) - \\cosh\\left(\\sin^2 x\\right) \\frac{\\mathrm{d}}{\\mathrm{d}x} \\left(\\sin x\\right) + \\int_{\\sin x}^{\\cos x} \\frac{\\partial}{\\partial x}\\cosh t^2\\;\\mathrm{d}t \\\\\n &= - \\cosh\\left(\\cos^2 x\\right) \\sin x - \\cosh\\left(\\sin^2 x\\right) \\cos x\n\\end{align}", "36b97c0067d36b267493aec3382c64fb": " S_r = -f(r) = -\\frac{\\kappa r h^2}{p^3} , \\qquad S_t = 0 . ", "36b9a44573c568e089ea388dac713099": "E = \\begin{bmatrix}\nT1 & T2 & T3 \\\\\nR(X) & & \\\\\n & R(Y) & \\\\\n && R(Z) \\\\\n\nW(X) & & \\\\\n & W(Y) & \\\\\n && W(Z) \\\\\nCom. & Com. & Com. \\end{bmatrix}", "36b9b9b27627e7f62421c48b9b1efb41": " F(k;n,p) \\leq \\exp\\left(-\\frac{1}{2\\,p} \\frac{(np-k)^2}{n}\\right). \\!", "36b9e4877a40d04398ab93554021bf29": "\\Delta U", "36b9f0f37993adc1e29e8e29c6cbd268": "{\\sigma(\\omega) = \\sigma_{1} - i\\sigma_{2}}", "36ba154fc7b34cb808389fa5f0fb801e": "\\mu_B = \\frac{e \\hbar}{2 m_e c}=\\alpha/2\\approx 3.6\\times 10^{-3}", "36ba1fa88efe1ff7854c808a324dd2dc": " Z^{\\alpha'} = \\Lambda^{\\alpha'}{}_\\alpha Z^\\alpha \\,.", "36baad583e43bce9e380d71d5a90186d": "x^5-5s^3x^2-3s^5 ", "36bbc8356a94af0a1576642cd1fdb09f": "T1 = \\frac{a_1}{\\sum_{h=1}^{H}{a_h}}", "36bc2255153352a31010715d1d32782c": "\\int_L^*", "36bca35fe7c901ed5cba1aa7711f1949": "dy^2=\\sum_i dy_i^2", "36bcb9cb022c17febf9757cb2d0952fe": "\\ \\mathcal{L} = \\mathcal{L}_\\mathrm{loc} + \\mathcal{L}_\\mathrm{gf} = \\mathcal{L}_\\mathrm{global} + \\mathcal{L}_\\mathrm{int} + \\mathcal{L}_\\mathrm{gf} ", "36bcc72a2db8488e084ecd3a4c6f83a1": " d = a + c ", "36bd02b8afcf89eb9068358563e60945": "\n\\langle 2;x_1 x_2 | 2;y_1 y_2\\rangle = \\delta(x_1-y_1)\\delta(x_2-y_2) \\pm \\delta(x_1 -y_2)\\delta(x_2-y_1)\n\\,", "36bd1e5803f7271b5d1419cf8fe36bb3": "\\begin{align}\n\\varphi &= \\arcsin\\left[\\cos c \\sin\\varphi_0 + \\frac{y\\sin c \\cos\\varphi_0}{\\rho}\\right] \\\\\n\\lambda &= \\lambda_0 + \\arctan\\left[\\frac{x\\sin c}{\\rho \\cos \\varphi_0\\cos c - y \\sin\\varphi_0 \\sin c}\\right]\n\\end{align}", "36bd29ea98e231521005878b51efa99d": "(a,a)", "36bd4b7cb2892c59cec557e88dc234ca": "A^2 = -n-S \\,,", "36bdc31d0b972859d921f1fbd328fb22": "F_{res}=F_e-F_p=q\\cdot Z^*\\cdot E=q\\cdot Z^*\\cdot j\\cdot \\rho\n", "36bdcbdcd21c117efc52c1a62e78a799": "\\left(\\mathbb{Z},+\\right)", "36bde6d3e056c50474b0d27d733bdec2": "G/PL", "36be5f66f2c93cd555197affde9c2309": "\\mathfrak p\\otimes_{\\mathbb F}\\overline{\\mathbb F}", "36be9fbe06aaabd4a24216d855fe0656": "C_1 = pq - qp + Tr[PQ] - Tr[QP]", "36bea98a8375fb6013986ea9721b1ffd": "e' > e", "36becb23ce35084d630eddf08c2fa9a5": "\\theta = 4N_e\\mu", "36bf019823bc6e613f4978c9bece9721": "P_\\ell ^{-m} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!} P_\\ell ^{m}", "36bf96765014a8fc85240614f3f4c9f4": "x_{-1}(z) = z", "36bfbf17b5de143c10e2c39a85d2a1f2": "f\\bullet b=(f\\oplus b)\\ominus b", "36bfce780d9b8f2312c4c0893d32c95b": "\\alpha : I \\subset \\Bbb{R} \\longrightarrow M", "36bffc46ae75de4244a869839ba48dfd": " \\ (((A)B)(A(B))), ((A)(B))(AB) \\ ", "36c03ce5ab3dcf66cf36495eb0586335": "\n\\begin{align}\nm(s_1, s_1) &= 0, \\quad\n\\left.\\frac{dm(s_1,s_2)}{ds_2}\\right|_{s_2 = s_1} = 1,\\\\\nM(s_1, s_1) &= 1, \\quad\n\\left.\\frac{dM(s_1,s_2)}{ds_2}\\right|_{s_2 = s_1} = 0.\n\\end{align}\n", "36c09b8761daa4ba7d3594fe9703d552": "I \\subseteq P", "36c14f2040fa0df6448a95e93bb6e72a": "(K,\\,M)", "36c15555f17bad31373e3eb26f4bd1f2": " J^1Y\\to_Y (Y\\times_X T^*X )\\otimes_Y TY, \\qquad (y^i_\\mu)\\to dx^\\mu\\otimes (\\partial_\\mu + y^i_\\mu\\partial_i), ", "36c1b255cd8bc68b539ef22da8a646a0": "E = \\frac{hc}{\\lambda}.\\,", "36c1c0fd89017f14e6ff7dc422223ee9": "i = 1,...,k", "36c1ec401a344150c416570207f58679": "q=(y,\\rho(y))", "36c215933ddcb6458250279077bf0122": "(0, 1), (0, 4)", "36c2608050aa56b3b7c9dcb186c83afa": "u_L", "36c2deea9829635d18c86640ba4fb49d": "M_{w'\\cdot\\lambda}\\to M_{w\\cdot\\lambda}", "36c2f18502b0d6092a01ac1bbd4972ce": "\\int_a^b dx", "36c304327f5525da3d6ad0aa45f670ce": "|\\mathbf{a_1} \\times \\ \\cdots \\ \\times \\mathbf{a_k} |^2 = \\det (\\mathbf{a_i \\cdot a_j}) = \n\\begin{vmatrix}\n\\mathbf {a_1 \\cdot a_1} & \\mathbf {a_1 \\cdot a_2} & \\dots & \\mathbf {a_1 \\cdot a_k}\\\\\n\\mathbf {a_2 \\cdot a_1} & \\mathbf {a_2 \\cdot a_2} & \\dots & \\mathbf {a_2 \\cdot a_k}\\\\\n\\dots & \\dots & \\dots & \\dots\\\\\n\\mathbf {a_k \\cdot a_1} & \\mathbf {a_k \\cdot a_2} & \\dots & \\mathbf {a_k \\cdot a_k}\\\\\n\\end{vmatrix}\n ", "36c350216560d688dc185012665ce1e6": "y^{-1}N=\\{r\\in R \\mid yr\\in N \\} \\,", "36c35c5a1beaf2bfc638c693317631f3": "\\scriptstyle{2^{\\aleph_0}}", "36c385ae8dd6a8232b007e3ff3b7022b": "\\begin{matrix}\n & & & & 1\\\\\n & & & 1& 1&1\\\\\n & & 1& 2& 3&2&1\\\\\n &1& 3& 6& 7&6&3&1\\\\\n1&4&10&16&19&16&10&4&1\\end{matrix}", "36c3898fe6d60f8811f71b6c1057f4de": "R_{}^{} =r_1+r_2+1", "36c3a9d1a8c73b825d45c41f8703e342": "1_{c}, f \\colon c \\to c", "36c416f405a8cc1a7a0e5aa684449ce7": "{\\boldsymbol \\theta} = \\left(\\theta_1, \\theta_2, \\cdots, \\theta_s \\right )^T.", "36c4261752df818ff48b8459fe7f1b17": "R=K[x_1,\\dots,x_n]", "36c42e4f934e3ac171ac13983ccf8042": " Z = \\ln(Y) ", "36c43fbdddd5e7a40e96f2317cb67d2c": "\\scriptstyle G^\\alpha {}_{\\mu\\nu}\\!", "36c448db69439941d88538fb047b2f01": " A^*_{ij}=\\int_{(0,1)^n} A(\\vec y)\\left(\n\\nabla w_j(\\vec y)+\\vec e_j\\right)\n\\cdot\\vec e_i\\, dy_1\\dots dy_n , \\qquad i,j=1,\\dots,n\n", "36c44ac39c3d0458e6e1feae39195e71": "\\begin{array}{lll} |\\gamma'^2(t)| &=& \\rho^2 h'^2(t) \\\\ \\gamma'(t) \\cdot \\gamma''(t) &=& \\rho^2 h'(t) h''(t) \\\\ |\\gamma''^2(t)| &=& \\rho^2 (h'^4(t) + h''^2(t)) \\end{array}", "36c484ed3fae13ddae68f4c29aad7207": " \n S = \\int_0^L \\left[ \\frac{1}{2} \\mu \\left( \\frac{\\partial w}{\\partial t} \\right)^2 - \\frac{1}{2} EI \\left( \\frac{ \\partial^2 w}{\\partial x^2} \\right)^2 + q(x) w(x,t)\\right] dx.\n", "36c4b6429045312d45e9e952afc7107a": " \\alpha \\in \\Lambda^k(V^*) \\mapsto i_\\alpha\\sigma \\in \\Lambda^{n-k}(V). \\, ", "36c4c35aa048fea10a3cafad47d723a4": "\\theta=\\theta_0", "36c50f1c397b68d81e62b0f93d9c871c": "p(\\tau \\mid \\lambda, \\beta)~d\\tau = \\frac{\\lambda}{\\Gamma(1 + \\beta^{-1})}~e^{-(\\tau \\lambda)^\\beta}~d\\tau", "36c51065400d48e4e2ad62231aaa8eee": " \\text{E}(X) ", "36c5191c121c8205afda6248936aa487": "H = \\max\\left\\{ \\eta(x\\,-\\,c_p\\,t) \\right\\} - \\min\\left\\{ \\eta(x - c_p\\,t) \\right\\}, \\, ", "36c52296fb4f5dcf5cd0d58420c31fd1": "Q \\quad = m_s C_{ps}(T_i - T_o)", "36c527a66c350d34900106740e66c88c": "L\\in\\Gamma", "36c5615dc2147f506898bf5dbf46eede": "\\displaystyle \\sin^2{A}+\\sin^2{B}+\\sin^2{C}=2.", "36c5a48bddb2ec7a61d99375a750c777": "\\; \\mathcal{H}_{A_1 \\ldots A_m}=\\mathcal{H}_{A_1}\\otimes\\ldots\\otimes \\mathcal{H}_{A_m}", "36c5ddb3478562727931a4cd0c7cbb02": "F(s) = (\\mathcal{L}f)(s) =\\int_0^\\infty e^{-st} f(t)\\,dt.", "36c6001c960033c366ccac819dff2292": "\\textstyle {x=a\\cos(t), \\qquad y=\\frac {a\\sin(t)}{\\sqrt {2}}}", "36c606b8ad788f3292302ad885461518": "(-1,+\\infty)", "36c60f3022eaf3891c952d190761707b": "-\\vec{\\omega }\\times\\vec{R}", "36c6513d5108673dba211e986b11e243": "\\frac{1}{3+1} = 0.25", "36c6ab0ffe8fdd8ce7d885ed033709be": "S \\subset T\\,", "36c6b058e9e505e5195629081f88ef37": "o(\\lambda^{(d-1)/2})", "36c6e4915fb56031a34a3564d99828bb": "\nE(t) \\ge \\sup_{\\tau \\ge 0} \\{A(t+\\tau) - A(\\tau) \\} = (A \\oslash A)(t).\n", "36c6ee19141098b643b235b763d6a10d": "{C} = {Q \\over m \\Delta T}", "36c754f91129574349fa23471b526d87": "L=\\frac{\\Theta}{360^{\\circ}} \\cdot 2 \\pi R \\, \\Rightarrow \\, \\Theta = \\left( {\\frac{180L}{\\pi R}} \\right) ^{\\circ}", "36c768014f09c7980cbf450b2a09a261": "U^0 = \\frac{dx^0}{d\\tau} = c \\gamma ", "36c76cc7f88e76ac945fcc57026b29ae": "\\vert\\nu(\\gamma,z)\\vert=\\vert cz+d\\vert^k", "36c77152d40675d488ce7039c13cb20b": "\\frac{-d[M_1]}{dt} = k_{11}[M_1]\\sum[M_1^*] + k_{21}[M_1]\\sum[M_2^*] \\,", "36c774a4f6bb1c556242e27528f742ce": "\n\\Phi(z,s,a)=\\frac{1}{2a^s}+\n\\frac{\\log^{s-1}(1/z)}{z^a}\\Gamma(1-s,a\\log(1/z))+\n\\frac{2}{a^{s-1}}\n\\int_0^\\infty\n\\frac{\\sin(s\\arctan(t)-ta\\log(z))}{(1+t^2)^{s/2}(e^{2\\pi at}-1)}\\,dt\n", "36c77dcd7c6f99fc7f6300dc3dcb33fa": "G(K) = \\sum_{\\mathbf{k} \\in S(K,m)} \\prod_{i=1}^{m} \\left( \\frac{e_i}{\\mu_i} \\right)^{k_i} ,", "36c7946bfa1d5cdf2f1510f201868250": "C_0 = -1\\;,\\quad \\sum_{k=0}^n\\frac{C_k}{n+1-k} = 0,\\quad n=1,2,3,\\dots", "36c79b51044a2bb4eefd7ae7a4f52bd2": "\\lor_{\\gamma < \\delta}{A_{\\gamma}}", "36c8701d680d388a320c037abb188498": "F(x;\\mu,\\beta) = e^{-e^{-(x-\\mu)/\\beta}}.\\,", "36c8f4c9bb5c6472785244a302fdd125": "t(\\tau)", "36c971674a761c6a91990f7343614843": "c = ", "36c975111919019de25bfe246cf200e2": "\n\\Phi(r)= 1 -2m/r +ar +br^2\n", "36c9d79bcadf9acfab45e16df89743b9": " \\langle \\omega \\rangle = \\frac{\\omega_1 + \\omega_2}{2} \\,\\!", "36ca1ae51842039fb8e4c8f1d70a57b4": "k_{x\\text{Region}_1} = k_{x\\text{Region}_2}", "36ca3c026cd62440689788d13c47913e": "\\operatorname{F}(z,k)", "36ca9cd718cbe95d74c66830ac828e79": "23 = 3", "36cb6ed5283e452800c275dd6934aecc": "n.", "36cb6ef012c8a8a27955946a67c294c7": "Y_1\\sim GIG(r_j,\\lambda_j;p)\\!", "36cb72f1edba9ac8b5c321fd9a6f6e02": "V^2 - U^2 < 1.\\,", "36cbc5d2a2cbf579bf0bac4f900bdb24": "{ T }_{ eq }={ T }_{ \\bigodot }{ \\left( 1-a \\right) }^{ 1/4 }\\sqrt { \\frac { { R }_{ \\bigodot } }{ 2D } } ", "36cbd928012ae151ff8879f979e4c80c": "(1-r^2) \\sum_{k=0}^{n} ar^{2k} = a-ar^{2n+2}.", "36cbf46351d579d04a0fd76ef14d9bca": "3, 6, 2, 5, 1, 4,\\,\\!", "36cc6b5af57a521b70c8476a39800a20": " r^{\\frac{3}{4}} \\exp(-\\tfrac{3 \\pi i}{4}) (3+r)^{\\frac{1}{4}} \\exp(\\tfrac{0 \\pi i}{4}) = r^{\\frac{3}{4}} (3+r)^{\\frac{1}{4}} \\exp(-\\tfrac{3 \\pi i}{4}).", "36cca18e855a35d4efcfafda218e9a8d": "H^1(X, \\mathcal{O}_X)", "36ccb2b74cddf0fd0d81c7476c6db392": "\\displaystyle \\nabla^2\\log u_n = u_{n+1}-2u_n+u_{n-1}", "36ccbf8e3912e3ccbf202bde749897b4": "b * a = (-1)^{\\deg (a) \\deg (b)} a * b.", "36ccc5a3bb07e02f16eca6667c10f2ab": " \\mathbf{v} = W\\mathbf{r} ", "36ccf0601094628b0a1decd745fcefe3": " f: \\textbf{R}^d\\rightarrow \\textbf{R}", "36cdaa79d6c5dceb3cf08642084d19b2": "\\mathfrak c = \\beth_1.", "36cdb542e015cb6f807b48a1403b0b25": " i = 1,2 ", "36cdbc6305ed5b0a8e37005061343b62": "\\partial_p = \\sin \\beta \\partial_x - \\cos \\beta \\partial_y, \\partial_q = \\cos \\beta \\partial_x + \\sin \\beta \\partial_y ", "36cdeef9089fb089d284d7b1cd59698c": " y_1 = x_1 \\, ", "36ce24bc85a16465e22d610223875ebd": " N_3(k) = 4 \\pi k^2 ", "36ce45743c6044b77098c2977de0ab69": "(x_1,y_1)\\ne(x_2,y_2)", "36ce935afe9233b627a92acdd3c1e03a": "\\theta = \\arcsin \\left( \\frac{\\text{opposite side}}{\\text{hypotenuse}} \\right)", "36ce93d20e093e4f15ea139c24cb6855": "g_1,~g_2,~g_3", "36ceb4b06b9bd0ca78caaa930681ca5a": "C: \\operatorname{ker}(ABC) / \\operatorname{ker}(BC) \\to \\operatorname{ker}(AB) / \\operatorname{ker}(B)", "36cf40d2cbbab385c933851a7d8695db": "\\sqrt{2} \\approx 1.414", "36cf85fc6cff1b45b0060ec3f146f26e": " \\lambda a, b, c.c\\ (\\lambda x.\\lambda a, b, c.b\\ (\\lambda a, b, c.a\\ f)\\ \\operatorname{mse}[x\\ x])", "36cff2a1df9f1ef2ee6a06d267ddae1f": " g(\\theta) = \\frac{1}{\\theta^{2}} ", "36d016857406fd0cefe0b8a4e7d35107": "\\dot{q}_i = \\{ q_i , H \\}", "36d032c22e9fe80bacf877afb8515a49": "E_n = n + \\tfrac{1}{2},", "36d03fdb8e5b84dc896679c61dbf0e68": "\\frac{g(x)}{g(y)}", "36d09a796b68a104e77ccb486438b751": "B \\Omega", "36d1417ad7f466d6878729ff5e81e2d9": "10\\uparrow\\uparrow\\uparrow\\uparrow 3=10\\uparrow\\uparrow\\uparrow(10\\uparrow\\uparrow)^{8}(10\\uparrow)^{10}1", "36d18991b9aa36755bd500577d86fdfb": "\\Phi_{S}=0", "36d1c9320e37316e13ad7601bc656128": "\\mathbb Z/p", "36d1d506176ce995943bdd0321e07435": "T = \n\\begin{bmatrix}\n0 & \\; & \\cdots & T_z \\\\\n\\frac{1}{2}I & \\ddots & \\ddots & \\; \\\\\n\\; & \\ddots & \\ddots & \\vdots \\\\\n\\; & \\; & \\frac{1}{2}I & 0 \n\\end{bmatrix},\n", "36d23b8ac087b17d2ce754a4b957acb6": "\\frac{E}{I}=Z_{oc}", "36d246108e385b254809b61ebb5ae110": "\n \\begin{align}\n u_\\alpha(\\mathbf{x}) & = - x_3~\\varphi_\\alpha ~;~~\\alpha=1,2 \\\\\n u_3(\\mathbf{x}) & = w^0(x_1, x_2)\n \\end{align}\n", "36d27c101fe56abcce893635c576ae9c": "{D \\; \\over Dt}", "36d288db1fd4e91af1d7e91abe336800": "(I_x^2 + \\alpha^2)u + I_xI_yv = \\alpha^2\\overline{u}-I_xI_t", "36d2f3094dee6aeae8ae4512d66962de": "Q_2(f)", "36d323c18bc1a8d6ac87607524f1306b": "(2a_1a_2+2b_1b_2)^2,", "36d3592fe96f5e2607f6539ce5fc0358": "\\textstyle \\gamma_2=a_2/N_2", "36d3abeb2cfb8f9e46dee52e256238ac": "a_{-1}", "36d3b8a5ff3ce96276a1de8189afec0e": "\\forall{n\\in\\mathbf{N}},|n|_{\\ast}\\leq 1", "36d42dd913ff1c97deb9763cd34fe584": "A \\cup B = \\{ x: x \\in A \\,\\,\\,\\textrm{ or }\\,\\,\\, x \\in B\\}", "36d4347020680654fa419ccfdb84fd1f": "\\displaystyle\\int_{-1}^1 4(1-y^2) \\, dy. ", "36d446d7ea529978be4f827a85223b75": "P=\\mu", "36d461d7637c139e2f731cf7be024ca1": "\\nabla \\times \\mathbf{H} = \\frac{1}{c}\\mathbf{J}_{\\text{f}} + \\frac{1}{c}\\frac{\\partial \\mathbf{D}} {\\partial t}", "36d488e2645b2927457a844934ae5ce0": "\n \\forall x\\in X\\qquad \\exists A\\in {\\mathcal A}\\qquad x\\in A,\n ", "36d4b107c58922b32438964940c31854": "\n\\theta_3 = \\arctan\\left(\\frac{A_1 \\sin\\theta_1 + A_2 \\sin\\theta_2}{A_1 \\cos\\theta_1 + A_2 \\cos\\theta_2}\\right) \n", "36d508a76a846567f6d340446fbc7b6e": " d = p_1^{n_1} p_2^{n_2}...p_k^{n_k} ", "36d51e58b6e2f17ef60f3e176c887407": "1 + u_{1}u_{1} - \\rho(x,u,u_{1}) = 0 \\,", "36d52c6ab71add72e70ac71f3db55187": " T_2\\cos\\theta - T_1\\sin\\theta, \\, ", "36d55aaf351ae90d1e50987b52aafe3d": "\\mathcal{S} \\subset \\mathbb{Z}_q^n", "36d595c48faed65f1b100e598ca4b000": "a_{\\overline{10|}}", "36d5b0ddd0b4007ef964c64e8bdef925": "\n\\begin{bmatrix}\n \\alpha - E & \\beta \\\\\n \\beta & \\alpha - E \\\\\n \\end{bmatrix} \\times\n \n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n \\end{bmatrix}= 0\n", "36d5fa15df3ef29751860f9f94ad0ebe": "\\overline{\\mathfrak{p}}", "36d6427f73e861b749d2f82a615511f3": "\\left\\langle\\tilde{H}\\right\\rangle = \\left\\langle H\\right\\rangle \\,", "36d64dc53fe6ce1f6ff4472b5ec47fd2": "f(z)=\\sum_{k=0}^\\infty \\alpha_k (z-z_0)^k", "36d6645efa97ff92133c15292ca73311": "J'={ H^\\prime \\over H_\\max^\\prime }", "36d67061ae80f7fed16105508fe2471c": "k_m \\propto 1/l_c", "36d71bd588fc02c35a321c75a3f4c682": "R = \\cap_\\mathfrak{p} R_\\mathfrak{p} = \\cap_\\mathfrak{m} R_\\mathfrak{m}", "36d75e7bf40e156ae464a96d9c2086df": "{\\color{Blue}~2.18}", "36d7be1f2a85e2c0e80dbd581fd75db9": "\\textrm{Kendrick~mass~defect}= \\textrm{nominal~mass} - \\textrm{Kendrick~exact~mass}", "36d80b2711a56fdee01f7c6401af53a9": "f_n = 1", "36d81f59851ebfe0e493f2e6b3738316": " \\forall A \\exists X X = A ", "36d83cb4cdfc0776836ffd58771cb3fa": "|\\phi_i\\rangle", "36d890286ba67d6985b9e6e00eb0b2bf": "(N+1)\\,\\operatorname{sinc} \\left(\\frac{(N+1)t}T\\right) - N\\,\\operatorname{sinc}\\left( \\frac{Nt}T \\right).", "36d8c0b5f1d55c81ede5a345648f185a": "\\hat{Z}(x_i)=Z(x_i)", "36d8cf4e7b3f89b45d3c3ba29270bac3": "\\Delta=\\omega_a-\\omega_r", "36d9830abfb6c02d2e85cf42952e2dd4": "O(KN\\log N)", "36d9f09f6eda0675d81e5b5a3200726e": "D_F\\{\\cdot\\}", "36da1dc8800de3d4dc442343afdf4763": "\\ x/L", "36dabf01f56a4a23a039ef4ad9af008a": "d_L^C", "36dadfc7841228ba73100252a9e3e416": "\\Sigma^0_0", "36db79d3fdbe6dd5ec18547560526992": "i x_n^*", "36db93b4dede851087799254b33967a9": "(p q) r = p (q r).\\ ", "36dbb45a7741ddb0d3cc8ec367c05afe": "\\mathbf{f} = \\frac{1}{\\epsilon} \\mathbf{D}", "36dc812e628f085662a6a482d86c2b2e": " V(x_0) \\; = \\; \\max_{ \\left \\{ a_{t} \\right \\}_{t=0}^{\\infty} } \\sum_{t=0}^{\\infty} \\beta^t F(x_t,a_{t}), ", "36dc85f6b40f8b342a66e665cb039418": " \\frac{\\partial \\bar{u_i}\\bar{u_j}}{\\partial x_j}", "36dca565d3516c9b99a8ee226928ec0e": "u(r) = \\frac{n}{n+1}\\left(\\frac{dp}{dz}\\frac{1}{2K}\\right)^\\frac{1}{n}\\left[R^\\frac{n+1}{n}-r^\\frac{n+1}{n}\\right] ", "36dcd7652da658611ed3f9dc12d7b083": "\\frac{\\partial^2 r_i}{\\partial \\beta_j \\partial \\beta_k}", "36dd02378266a6d431bedeaa900bd59b": "\\overline{\\mathsf{f}}^{-1}(X) = \\frac{I^{-1}\\underline{\\mathsf{f}}(IX)}{\\det\\mathsf{f}} = [\\underline{\\mathsf f}(I)]^{-1} \\underline{\\mathsf f}(IX) .", "36dd20a5095aff8b4a7c264950784993": "P_{4}", "36dd36783349067d6c236c621bd06b55": "0\\le\\biggl|\\int_S \\liminf_{n\\to\\infty} f_n\\,d\\mu\\biggr|\n\\le\\int_S \\Bigl|\\liminf_{n\\to\\infty} f_n\\Bigr|\\,d\\mu\n\\le\\int_S \\limsup_{n\\to\\infty} |f_n|\\,d\\mu\n\\le\\int_S g\\,d\\mu", "36dd50aec48ef2c03e47af8b9c815c34": "\\langle\\Delta_z(\\mathbf{r})\\Delta_z(\\mathbf{r'})\\rangle = \\Delta^2\\exp\\left(-\\frac{|\\mathbf{r}-\\mathbf{r'}|^2}{\\Lambda^2}\\right)", "36dd59c612746e07f89fb12439633409": " {\\Gamma_{kj}}^i =\\bar\\Gamma_{kj}{}^i + {K_{kj}}^i,", "36dd95e66aa6b1ab9b838163b708e9fe": "T_{1} = M_{1}a=M_{2}b =T_{2},\\!", "36ddbcc8a7f344c73deb0616dfc146bf": "(2B)^2 - 4AC = 4(B^2-AC),", "36dde132f4241a67c47064a40560b072": "\\delta\\mathcal{L}=0", "36de93c88793d148750c72124827c559": "n(d)-\\sqrt{2d\\ln2}", "36deae75e7e508f42a4b9e16bbddb2e7": "D_h = \\sqrt{2 \\times H \\times \\left( \\frac {4R_e}{3} \\right)}", "36df8bbdb154a7d15495e8f2f72d5739": "t^5 + t - a=0\\,", "36e00a94c82368dfa896c02854497d78": "\\tilde{H}_n\\left(X\\right)\\cong\\delta_{1n}\\,(\\mathbb{Z}\\oplus\\mathbb{Z}_2)=\\left\\{\\begin{matrix} \n\\mathbb{Z}\\oplus\\mathbb{Z}_2 & \\mbox{if } n=1\\\\\n0 & \\mbox{if } n\\ne1 \\end{matrix}\\right.\n", "36e0139b61088d4c0df10906331c481f": "13 = 2^3 + 2^2 + 2^0", "36e024aeb5072d67446baedae0ff03d8": "\\mathbf{\\Lambda}_k", "36e05bb03ab657580e809fec3f43c514": "E_{y}=\\frac{1}{j\\omega \\varepsilon }\\frac{\\mathrm{dL} }{\\mathrm{d} z}\\frac{\\partial T}{\\partial y}^{TM}-L\\frac{\\partial T}{\\partial x}^{TE}=\\frac{-k_{z}}{\\omega \\varepsilon }L\\frac{\\partial T }{\\partial y}^{TM}-L\\frac{\\partial T}{\\partial x}^{TE} \\ \\ \\ \\ \\ \\ (28) ", "36e0822b2cf04cb268ad5407abd5b95c": "\\sigma A=0 \\Rightarrow \\forall \\epsilon>0\\ \\exists n\\ A(n) < \\epsilon n.", "36e09571353af50c9ba4157d7bb04fe5": " \\Delta = \\exp(-2 \\, p(x,y)) \\left( D_x^2 + D_y^2 \\right). ", "36e0b7b074c4f89ad11d05afc2b84a3e": "\\textstyle {\\rm gap}\\left ( A \\right ) = \\sqrt{{\\rm tr}^2 (A) - 4 {\\rm det}(A)}", "36e1072955ac10cffdb35db918efdb57": "f:S_0\\to S_1", "36e11f4aa67c6edbb1fcca180c1a8f89": "/m", "36e1d5d3f2650702fddf52681643161f": "\\ln 2 = \\sum_{k\\ge 1}\\left(\\frac{1}{3^k}+\\frac{1}{4^k}\\right)\\frac{1}{k}.", "36e22762b09a40e285711d0df34dd7cf": "{\\tilde{D}}_{2n+1}", "36e249bd85bbed5842eea0b925219ea4": "\\mathbf{A}=\\begin{pmatrix}-1 & 0\\\\ 0 & 1\\end{pmatrix}", "36e26ef355c6613933b09da2794e41e4": "x \\in [\\mu, \\infty)", "36e2fde4e8836f6eeef49733257c7fd5": "\\mathrm{im}(\\partial_{n+1})\\subseteq\\ker(\\partial_n)", "36e30935ad02493e18dea29bd5ecb28a": "A_R = \\frac{1290}{2670} = 0.483", "36e360c6b0a3ec6dda30e61f3e0269ea": "\\mid n,\\pm \\rangle=\\frac 1{\\sqrt 2}\\left(\\mid g\\rangle \\mid n \\rangle\\pm \\mid e\\rangle \\mid n-1\\rangle\\right)", "36e392bf4d5f52a91e9f9f869319b0ee": "1/G", "36e3b3572ee5dc3da94e5ae2926c2f98": " w \\,", "36e3b6900cb6dc5a082819cf4fcf06ef": "OPT-2t\\,\\!", "36e3eddc3b63fa3aa5dcd63327f9a45d": "a u(0) - bu'(0) =g(0)\\,", "36e42065882092f55e5393c9b2951dad": "F_X^{-1}(Y)", "36e47563357cbb7e1313434d2b7ba22b": "\\mathcal{E}(x_1) \\cdot \\mathcal{E}(x_2) = x_1^e x_2^e \\;\\bmod\\; m = (x_1x_2)^e \\;\\bmod\\; m = \\mathcal{E}(x_1 \\cdot x_2)", "36e47747fc43d38993f9de0f84a99e38": "\n J = \\det\\boldsymbol{F} = \\left[\\mathbf{b}_1,\\mathbf{b}_2,\\mathbf{b}_3\\right] = \\mathbf{b}_1\\cdot(\\mathbf{b}_2\\times\\mathbf{b}_3)\n", "36e4aff3b6f8c6ceb6f9ae2bf0caa589": "\\int R\\,dx= \\frac{2ax+b}{4a} R+ \\frac{4ac-b^{2}}{8a} \\int \\frac{dx}{ R}", "36e4bcc814b5b7a983da0ece0f74cb12": "\\mathbb{R}^{2n}", "36e4d9e37ff9a5d4bd60b3203fce52e1": "g^{x_4}", "36e4e0b1639cf0b1e8e62d956f36c041": "=\n \\sum_{n=1}^{\\infty}\n \\frac{1}{n}\n \\left(\n \\frac{1}{(n-1)!}\n \\lim_{w \\to 0} \\left(\n \\frac{\\mathrm{d}^{n-1}}{\\mathrm{d}w^{n-1}}\n \\phi(w)^n\n \\right)\n \\right)\n z^n,\n", "36e5402da52fe6df1b10d50a399c855c": " -\\frac{\\pi}{T} < \\omega < +\\frac{\\pi}{T}. \\ ", "36e5506352274e378fd187f28b3bb8a7": "\\zeta(z)", "36e5b77de21275ba24668f3447aac8ab": "M = m", "36e5ba526445decb3266108d095cfba5": "\\tau=-1/\\ln(\\varphi)", "36e5c512999dbf4650729d9996c3278d": "\\mathbf{D}^2_{xy}=\\begin{bmatrix}0 & 1 & 0\\\\1 & -4 & 1\\\\0 & 1 & 0\\end{bmatrix}", "36e5dd7637b20f57d08b23d3c3cd8de8": "\\widehat{U}(\\Delta t)\\psi(\\mathbf{r},t) = \\psi(\\mathbf{r}, t + \\Delta t)", "36e603683a4d36ffaed277cd36fc94d1": "H (K \\otimes L, X \\times Y)", "36e60ea26635c3b178ac4966d7ceaabc": "p = ~~\\frac{\\partial F_1}{\\partial q} \\,\\!", "36e617e22b9cd62469d6ef0129fe850c": "f(0,y) \\simeq g(y),\\,", "36e69b04e8b0d0efb9d283583cc3e139": "\\frac{1}{2} \\psi(0)", "36e6a40194797b8899d9a9fccd709015": "\n \\sigma_{\\mathrm{eq}} = \\sqrt{\\tfrac{3}{2} \\boldsymbol{\\sigma}^{\\mathrm{dev}}:\\boldsymbol{\\sigma}^{\\mathrm{dev}}}\n ", "36e72404457441c35c6c17fd2708a380": "P_\\text{as} = \\frac{1}{2}(1-P),", "36e755e6e5e69e0ade90413e487c5462": "u:\\mathbb{C} \\rightarrow A", "36e75ed0fb4b04061d46f411bcfd3185": " \\lim_{\\theta \\to 0}\\frac{1 - \\cos \\theta}{\\theta^2} = \\frac{1}{2} ", "36e7939366daac411c44fe7428b6fd15": "x^\\mu \\to x^{\\bar \\mu}", "36e79f63a863237ed7edebfede497e91": " \\ k = a/b \\,", "36e7b31b8b763876cbcd70512738eb38": " 0.05 H_s ", "36e7d3298663374bb7d265fdebddb0d0": "\\Psi =\\Delta \\exp(i\\theta)", "36e83d970f0b73b52ec613c1b04af60f": "Bird(X)", "36e8a83680de90d68e836c2cb141a003": "Y = m X + C", "36e8c4f88918b0c86c91c71310b96115": "D^\\ell(\\alpha \\beta \\gamma)", "36e8f5c33a319a2fb5f5aebdd9c38c87": " \\bar r_1 \\times \\bar r_2\\ ", "36e90c37b339710b170c2011ee765d69": " e^{i \\int_{\\partial \\sigma} A} = e^{i \\int_{\\sigma} dA} = e^{i\\int_\\sigma F}", "36e9bb983a28559738f41e227a5358dd": " \\operatorname{Var}(X) = \\alpha^2 \\left( 2b / \\sin 2b -b^2 / \\sin^2 b \\right), \\quad \\beta>2.", "36e9c5651e88bcbcaaa08632ec1c72d5": "G = CF_4 = 4", "36e9ff605b28504f6a78f2195625cccb": "\\forall x [\\exists y Animal(y) \\land \\lnot Loves(x, y)] \\lor [\\exists y Loves(y, x)]", "36ea010efd3629c1149fb30ca55f36e1": "d\\mathbf x=\\mathbf V \\,d\\mathbf x'\\,\\!", "36ea31c7c63cefd8933688f5d7b40056": "8a(n+1)I_{n+\\frac{1}{2}} = 2(2ax+b)(ax^2+bx+c)^{n+\\frac{1}{2}} + (2n+1)(4ac-b^2)I_{n-\\frac{1}{2}}\\,\\!", "36ea64f79ef4425cafb253c7fa865fa4": " \\frac{1}{r_n} ", "36eabceb6d7f4b66b6f208e95f66845c": "|f_n(x_k) - f_m(x_k)| < \\varepsilon/3,\\quad n,m \\ge N.\\,", "36eaf02f0ef35397728c554f77c6cf2c": " x > y ", "36eb47224e3dd3a356b984fe92696928": "\\begin{pmatrix} 2 & 1 \\\\ 1 & 1 \\end{pmatrix}^n = \\begin{pmatrix} F_{2n+1} & F_{2n} \\\\ F_{2n} & F_{2n-1} \\end{pmatrix}\\, ,", "36eb5438630897242efe268ee4e80685": "f^{\\mathbb C} : V^{\\mathbb C} \\to W^{\\mathbb C}", "36eb7f87cc8918fb3caf4888546fea21": "Lv=\\lambda v", "36eb99904389e96c6e83771decb8a664": "\nZ_G(t_1,t_2,t_3,t_4) = \\frac{t_1^6 + 9 t_1^2 t_2^2 + 8 t_3^2 + 6 t_2 t_4}{24}.\n", "36eb99fe7d7885ddfad71a2a1afe3058": "i = (a+bi)^2\\!", "36ebaafa4ddf88ac7b3b29e132a2133c": "\\|x\\| = \\sqrt{x^*x}.", "36ec1615213f42dc15a236b62a1e0545": "X \\sim \\textrm{Weibull}(\\sigma,\\,\\mu)", "36ec536574d361152859c40006383521": "Attr_i(U)", "36ec9cf4ece7abb4332ed095764a3518": "{\\phi}=\\arctan \\left( {\\frac{y}{x}} \\right)= \\arccos \\left( \\frac{x}{\\sqrt{x^2+y^2}}\\right) = \\arcsin \\left( \\frac{y}{\\sqrt{x^2+y^2}}\\right)", "36ecaedbdde05a6a888166a8f60c345a": "\n \\mathbf{A}_{x,0} = \\mathbf{P} \\mathbf{\\Lambda} \\mathbf{P}^{-1},\n", "36ecb1a755c7ccfc708cdd5a4dcb1308": "\\int\\frac{dx}{s^3}=-\\frac{1}{a^2}\\frac{x}{s}", "36ece3c0f5676a9834b52907c9b84a92": "V_\\mathrm T = V_\\mathrm i + V_\\mathrm r = V_\\mathrm {iL}(e^{\\gamma x} + \\mathit \\Gamma e^{-\\gamma x})\\,\\!", "36ed3c3f69497de2b7f0134243c01f84": "-\\frac{\\Delta\\chi}{2}[\\mathbf{H}\\cdot\\mathbf{\\hat{n}}]^2", "36ed5b557f9412fee8d9d26acf8db621": "\\mathbf{x} = \\boldsymbol{\\chi}(\\mathbf{X},t)", "36edb8d805e725c63fde2211d47577a1": "\\ d,f", "36edfd67662f63dc0b43725aa84f5de0": "V\\alpha^{3}", "36eed66ba8b67d04dbf98ab36309c5a7": "\\pm 1/L_n(\\xi)", "36eef78a1e883db642a28e9dac69b236": " P=B^2\\left(\\frac{r}{r_c}\\right)^4 r_c c=\\frac{B^2 r^4 \\omega^2}{c}", "36eef8fb33ada8aa000954f3d670b5af": "{\\mathit{He}}_n^{[\\alpha]}(x)=\\sum_{k=0}^n h^{[\\alpha]}_{n,k}x^k\\,\\!", "36ef511c9c82388e120e41b5aca92830": "\\tilde A_n(R)=\\frac{n\\pi^{n/2}}{(n/2)!} R^{n-1}\\,.", "36ef85a6927088bad7f930fceae5d869": "\n \\Pr(Y=1 \\mid X) = \\Phi(X'\\beta),\n ", "36efd8cd6fd37632749615c9d5bb94ab": "\\int\\frac{\\sin ax\\;\\mathrm{d}x}{\\cos ax(1-\\sin ax)} = \\frac{1}{4a}\\tan^2\\left(\\frac{ax}{2}+\\frac{\\pi}{4}\\right)-\\frac{1}{2a}\\ln\\left|\\tan\\left(\\frac{ax}{2}+\\frac{\\pi}{4}\\right)\\right|+C", "36f08972d6421277828b98634e18e975": "\\lceil \\log_2(17) \\rceil = 5", "36f0c633622d831b937e7009c6ecd257": "F_m(x) = F_{m-1}(x) + \\nu \\cdot \\gamma_m h_m(x), \\quad 0 < \\nu \\leq 1,", "36f0d4c5c03c26ffc4684441369142b6": "h_{\\beta}^k", "36f0d56ca1e7c1df3a8ad7d6efee3de3": " \\ -\\textbf{f}_2 \\cdot \\textbf{h} \\pmod q ", "36f0d952ab871d3c7ff0606246073853": "\\frac{\\partial N_1 \\ (net)}{\\partial t} = \n- \\frac{\\partial N_2 \\ (net)}{\\partial t} = \n B_{21} \\ \\rho (\\nu) (N_2-N_1) =\n B_{21} \\ \\rho (\\nu) \\ \\Delta N ", "36f0dcd495ca145f9ec15a9adc45f694": " \\int_{u,v} u e^{-u ( k^2+m^2 + v 2p\\cdot k + v p^2)} = \\int {1\\over (k^2 + m^2 + v 2p\\cdot k - v p^2)^2} dv ", "36f0e315c0590fadde67e8ca84b22ca7": "\\xi^d", "36f0e459b32ebf4b4e348335abd4af0f": "y_{3}=1-4", "36f16740667d44d0ae64ec25ad3660ac": "d:\\Gamma(V)\\rightarrow\\Gamma(W)", "36f169eb3a212b1065e22f1d5b449fcc": " \\frac{ \\langle fg\\rangle_\\mu }{\\langle g\\rangle_\\mu} = \\langle f\\rangle_1 \\ge \\langle f\\rangle_2 =\\langle f\\rangle_\\mu, ", "36f1d02f31e43a725a9160fa7451891e": "x_k(\\zeta,\\theta) = \\Re \\left\\{ e^{i \\theta} \\int_0^\\zeta \\varphi_{k}(z) \\, dz \\right\\} + c_k , \\qquad \\theta \\in [0,2\\pi] ", "36f1f845a4101a9c30080f258edea65f": " f = \\frac{[M_1]}{[M_2]} \\,", "36f25e816686ce0f38508bbb346444a1": "-\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}\\psi", "36f26c3740b0139b995649a90cd81395": "a_n z^n + \\dotsb + a_1 z + a_0 = 0", "36f285a3d82ff934307d7442a1febeaf": "\\mathrm{Hydrogen:} \\frac {\\mathrm{mass \\ of \\ air}}{\\mathrm{mass \\ of \\ H}} = \\frac {\\frac {1}{4} (4.773) \\times 28.96} {1.008} = 34.28", "36f29ea99fc068aaf311616071250849": "\\omega_\\mathrm{LO}", "36f29ef9da51d5e340e7406a8fa6db8f": "\\,\\epsilon(x_0-y_0) = 2 \\Theta(x_0-y_0) - 1.", "36f30f872275572dcd10c686d7fd5ee9": "\\{W_t\\}", "36f31df6a8f081fd497108faf1b773ac": " (x_1,y_1) ", "36f3378e5f9d5a32b55d8ece890677ba": "\\{ XZ - Y^2 , YW - Z^2 , XW - YZ \\}.", "36f3502affeb8b7e0bddb50163799470": "\\begin{align}\n\\text{WAL} &= \\sum_{i=1}^n \\frac {P_i}{P} t_i\\\\\n\\text{WAL} \\times P &= \\sum_{i=1}^n P_i t_i\n &&= \\sum_{i=1}^n P_i \\frac{i}{12}\\\\\n\\text{WAL} \\times P \\times r &= \\sum_{i=1}^n iP_i \\frac{r}{12}\n &&= \\frac{r}{12} \\sum_{i=1}^n iP_i\n\\end{align}\n", "36f383d7a1bb8562e151973989ae36ef": "t_\\mathrm{ev} = 8.407 \\, 16 \\times 10^{-17} \\left[\\frac{M_0}{\\mathrm{kg}}\\right]^3 \\mathrm{s}\n\n\\ \\ \\approx\\ 2.66 \\times 10^{-24} \\left[\\frac{M_0}{\\mathrm{kg}}\\right]^3 \\mathrm{yr} \\;", "36f385f70473621fa622143d7209cc26": "\\tan (\\Psi)", "36f3a81119789c95c8994998a7968735": "{\\partial A_x \\over \\partial x} + {\\partial A_y \\over \\partial y} + {\\partial A_z \\over \\partial z}", "36f3a812cd44644e7cc35f0740d49125": "h_{inv} (n) = 0 \\,\\, \\forall \\, n < 0", "36f3b62f690b051b137066ab20a1f459": "P(G)", "36f3c529d3cc3935ebaa0f70b009026f": "V* = EPS \\times (8.5 + 2g) ", "36f4a7abc30b275df28534bce790e1a4": "(\\delta', P')", "36f4ab53d79f76013d32873de0e96e02": "[T] = [Z_{1,j}][X_{1,j}][Z_{2,j}][X_{2,j}]\\ldots[X_{n-1,j}][Z_{n,j}],\\quad j=1,\\ldots, m.\\!", "36f4d36304ca8be1794d92bd044b95cf": "C^T(p)=\\begin{bmatrix}\n0 & 1 & 0 & \\cdots & 0\\\\\n0 & 0 & 1 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 1\\\\\n-c_0 & -c_1 & -c_2 & \\cdots & -c_{n-1}\n\\end{bmatrix}", "36f50195cfc4ce44ef44a051c4a24dc0": "pp_.", "36f579d1cee2d26bf838b4623097af6b": "e_2 = 0", "36f57eeddb90b835dba08fada167164d": "\n \\left| \\int_{C} f(x) e^{\\lambda S(x)} dx \\right| \\leqslant \\text{const}\\cdot e^{\\lambda M}, \n \\qquad \\forall \\lambda \\in \\mathbb{R}, \\quad \\lambda \\geqslant \\lambda_0.\n", "36f5b8bf64a7e16aa30e02a292884c73": "\\frac{V_2^2}{2}", "36f5cc19e624bbfb9598d46707a0d3ed": "\\sigma_0 \\otimes \\sigma_3 \\otimes \\sigma_1 ", "36f635120254c19b1b9101f73cdadce5": "\\ell^*(t)", "36f6716c34549a279a59d47f9c58e3be": "F(x)=\\mu(-\\infty,x]", "36f68c8286b2d6e700a41fcef2f1b3ac": "\\begin{bmatrix} \\dfrac{-y_{22}}{y_{21}} & \\dfrac{-1}{y_{21}} \\\\ \\dfrac{-\\Delta \\mathbf{[y]}}{y_{21}} & \\dfrac{-y_{11}}{y_{21}} \\end{bmatrix}", "36f6afa68d06b4a3f8f41907fc576415": "\\upsilon_{a}", "36f6b5c431cb0b749e8bcd84ada847f8": "[\\mathbf{k}]_\\times \\mathbf{v} = \\mathbf{k}\\times\\mathbf{v} ", "36f6f11f3dab2cd8b1efae9981b07366": "\\frac{d^4y}{dx^4}", "36f738c22e5be0aceff66d9f692197cd": "A^4=A^3A=(27A+10I_2)A=27A^2+10A=27(5A+2I_2)+10A=145A+54I_2\\, .", "36f773e833abbe0e5a529ba0988407a2": "\\psi^\\dagger", "36f78ae9573c2f81a35c933913e8c9e2": "\n\\cfrac{\\partial \\mathbf{b}^i}{\\partial q^j} = -\\Gamma^i{}_{jk}\\mathbf{b}^k,\\quad\n\\boldsymbol{\\nabla}\\mathbf{b}_i = \\Gamma_{ij}{}^k\\mathbf{b}_k\\otimes\\mathbf{b}^j,\\quad\n\\boldsymbol{\\nabla}\\mathbf{b}^i = -\\Gamma_{jk}{}^i\\mathbf{b}^k\\otimes\\mathbf{b}^j\n", "36f7a4d66088eddf68ad443830a8abde": "A = \\bigoplus_d A_d.", "36f7d0b68898b9ccfdfc43d86019d101": "f\\left( p \\right)", "36f86fec5ad234defc1efc06be194e7b": " \\langle \\hat{L} \\rangle ", "36f8a149a7daacad0af5c30195183c97": " 1.3 \\cdot 10^{5}", "36f8ab13b58305976bf8ffecd28190f2": "\\mathcal{H}=\\int \\mathrm{d}^d x \\left[ \\frac{1}{2}\\rho(\\vec{\\nabla} \\varphi)^2 +e(\\rho) \\right],", "36f8ae4c86b69d52d037a6802d91cc4a": "\\cdot ", "36f8d3506fa2476f013c584699fd54c3": "\\begin{cases} u_{t}=ku_{xx} & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ u(x,0)=g(x) & IC \\\\ u_{x}(0,t)=0 & BC \\end{cases} ", "36f9030e459625905f274698e9fff2b6": "u(x, 0) = u_0(x) \\equiv K(e^{\\max\\{x, 0\\}} - 1)", "36f909a3680e4a66a9060a31b8da8511": " \\arg(M(s))", "36f97cc312e85c4330eaeb0cd4e4f54c": " VWS = \\frac{\\Delta V}{\\Delta Z} ", "36f99f6027e5684462a9b893d5f322df": "\\int_{0}^{t}x_m(\\tau)d \\tau = \\frac{A_m \\cos (2 \\pi f_m t)}{2 \\pi f_m}\\,", "36f9d6153a579a3575668630361054e6": "\\mathbb{M}_n (K)", "36fa2b9e139531af526eaef58d928963": "\\Omega^*_G(X)", "36fa3e76528ec170e8c004aef7e4bcac": "\n [u_\\epsilon(0)] = 0,\\ \\left[k^\\epsilon(x)\\frac{du_\\epsilon}{dx}\\right] = 0\n", "36fabf6c452883719fadf3241c105b2d": "\\mathbf d", "36faf2223bf483e6592f45d0d66c5e3c": "t_{cr}\\,", "36fb090fca710f3359b4138f824def16": "d|S| \\geq |N(S)| \\geq |U(S)| \\geq d(1-2\\varepsilon)|S|\\,", "36fb22d333eadcc4c54fc4edc6f6d045": "D\\cdot F = S", "36fb66b0ae4379c587c281754a1b8666": "\\|f\\|_{L^{1,\\infty}}", "36fb8faf2632b6ad6d8f467ebf8cd866": " \\Delta \\omega = \\frac {R_\\mathrm L}{L} ", "36fc081e489bdab4f91d13e6dd0664dd": "\\mathfrak{M}", "36fc179000b3bac7b0ad1663fa2599db": " TREE_d ", "36fc19717b891bb1a5e3490a7aaa4d0f": "\\vec{v}_D = -\\frac{\\nabla p\\times\\vec{B}}{qn B^2}", "36fca5f3c449b2e95a7d019fb3f4670a": " -(p u')' + q u -\\lambda r u =0 \\quad \\hbox{for} \\quad x_1 < x < x_2.\\,", "36fcca82be8e90e22917a3e854f7b14b": "F_{t,T} = (S_t - I_t) e^{r(T-t)} \\, ", "36fcea4efbfd5900df0b7f950ca8a00f": "\\displaystyle{F(r,\\theta)=r \\exp [i\\psi(r)g(\\theta) + i(1-\\psi(r))\\theta] ,}", "36fced6a311119d663179553f9f508cb": "\n\\int p dq = n h\n\\,", "36fd0abf4df2346b6f79abac13c88ddb": "\\lambda_1 \\langle x,y \\rangle = \\langle Ax, y \\rangle = \\langle x, Ay \\rangle = \\lambda_2 \\langle x, y \\rangle", "36fd119db654c143c13f8204f937fe9e": " \\equiv q = \\tfrac12\\, \\rho\\, V^{2}", "36fdad025e83703fa4cd2ceb703e6480": "\\mathfrak{g}_x", "36fe06a44b2b5eb3d95266785569f0ac": "g^* = \\frac{1} {L^*}", "36fe25c348db25d90d654f4aa0ee5748": "\n\\begin{align}\nm(t) & {} = \\mathbb{E}[X_t] \\\\[12pt]\n& {} = \\mathbb{E}[\\mathbb{E}(X_t \\mid S_1)] \\\\[12pt]\n& {} = \\int_0^\\infty \\mathbb{E}(X_t \\mid S_1=s) f_S(s)\\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\mathbb{I}_{\\{t \\geq s\\}} \\left( 1 + \\mathbb{E}[X_{t-s}] \\right) f_S(s)\\, ds \\\\[12pt]\n& {} = \\int_0^t \\left( 1 + m(t-s) \\right) f_S(s)\\, ds \\\\[12pt]\n& {} = F_S(t) + \\int_0^t m(t-s) f_S(s)\\, ds,\n\\end{align}", "36fe46376d569d3d92ea27639ba4b969": "\\Bigg[\\frac{\\pi}{\\theta}\\Bigg] =-\\left[\\frac{\\theta}{\\pi}\\right]. ", "36fe535b600ea891fcc717c21ace4f4c": "\\bar{u}=\\left(\\begin{matrix} E_x \\\\ E_y \\\\ H_z \\end{matrix}\\right),", "36fe5bed2b3caa08c5b8c81c734827ef": "\\!\\, \\phi=tan(2 {\\pi} f {\\tau})", "36fe749312d270b1265ba9a4446866e3": "\\left[a, b\\right]", "36feecbe774966a9d30b22a7bfc12f6c": "\\tan \\beta = \\tan \\theta - \\frac{D_{ds}}{D_s} \\left [\\tan \\theta + \\tan \\left (\\hat{\\alpha}-\\theta\\right ) \\right ]", "36feeecd79e31ae152b286e913b75ed8": "\\mathrm{^{A}_{96}Cm\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{A+1}_{\\ \\ 96}Cm\\ +\\ \\gamma}", "36fefa79cc0a5c0cdf970bab36a219d9": "\nS= \\int_k ( (1-\\cos(k_1)) +(1-\\cos(k_2)) + ... + (1-\\cos(k_d)) )\\phi^*_k \\phi^k\\,.\n", "36ff4f3128a6bc6b86171e3054c875d0": "\\mathrm{DOF} \\approx \\frac {2Nc} {m^2} \\,.", "36ffe4d4943b9bc775258885c5ec680a": " (\\lambda E.G\\ H)\\ Y ", "3700060e0192f73fa45492cf40f97846": "(\\alpha_{i},\\beta_{i})", "3700159b1b43544b243b9c57f286217e": "c\\geq 0", "370189e552c0779ca43fe3e4bf3bff9a": "\\forall \\psi(\\boldsymbol{x}) \\in p(\\boldsymbol{x}), \\varphi(\\boldsymbol{x}) \\rightarrow \\psi(\\boldsymbol{x})", "37019133ef72d170ced7d5a3c72f75ea": "\\| y_n - y_m \\|^2 \\; \\le \\; 2\\left(\\delta^2 + \\frac 1n\\right) + 2\\left(\\delta^2 + \\frac 1m\\right) - 4\\delta^2=2\\left( \\frac 1n + \\frac 1m\\right)", "37019e344b62952a33d921a4ad11ec1c": "\n W = C_1(\\lambda_1^2 + \\lambda_2 ^2 + \\lambda_3 ^2 -3) ~;~~ \\lambda_1\\lambda_2\\lambda_3 = 1\n ", "3701b90015c57885f27045edc67527ca": "C_1, C_2, \\dots ", "3701c1fe9121bce4b7b9dc73128cf1fc": "I/I_o", "3701db61637710d6ca648f8c18b4d147": "= m_1^2 + m_2^2 + 2\\left(E_1 E_2 - \\textbf{p}_1 \\cdot \\textbf{p}_2 \\right). \\,", "3701dced97eb88fea97f45030471083f": " \\, h=\\phi - \\epsilon + K_{\\mathrm{p}}...........(19a)", "370283db608d16e7b57e2172a739955c": "\\Gamma(z)=\\lim_{n\\to\\infty}\\frac{n^zn!}{\\displaystyle\\prod_{k=0}^n (z+k)}. \\!", "37028c7698f272648d8c9d3cc07f1b3d": "\\;_{1}\\phi_0 (a;q,z) = \n\\frac {1-az}{1-z} \\;_{1}\\phi_0 (a;q,qz).", "370346eef6336cad37eae1e046b9b8ae": " \\begin{bmatrix} x' \\\\ y' \\end{bmatrix} =\n \\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix}", "37038b5a2fdbed9fdd0a4908aa5fd9c4": "\\Bigg(\\frac{a}{p}\\Bigg)_4= 1.", "37039c391216899d383d85511de7427f": "t=\\sinh\\left(\\tanh^{-1}\\sin\\varphi-\\frac{2\\sqrt n}{1+n}\\tanh^{-1}\\left(\\frac{2\\sqrt n}{1+n}\\sin\\varphi\\right)\\right),", "3703a8c92e299f49e4108b4a43c775d4": "\\begin{align}\n\\ln\\Bigl(\\frac{w_1x_1+\\cdots+w_nx_n}w\\Bigr) & >\\frac{w_1}w\\ln x_1+\\cdots+\\frac{w_n}w\\ln x_n \\\\\n& =\\ln \\sqrt[w]{x_1^{w_1} x_2^{w_2} \\cdots x_n^{w_n}}.\n\\end{align}", "3703deb2963125be2d5783ef656a3579": "D(p)", "3703feb9138ef86b99d111f8e3c605fc": " = \\frac{1}{2} (\\eta_{\\mu \\nu}\\gamma^\\mu \\gamma^\\nu + \\eta_{\\nu \\mu}\\gamma^\\mu \\gamma^\\nu)", "3704045f16043cddbafbda41f0e7b524": "\\begin{align}\n& \\bold{A}_{[p]} = (A_{i_kj_\\ell})\\\\\n& {1 \\leq i_k, j_\\ell \\leq n \\text{ for }1 \\leq k, \\ell \\leq p}\n\\end{align}", "370414a3d191b8dc88c9a93c3e112c7a": "\\mathrm{Ad}_T(J_i) = TJ_iT^{-1} = -J_i, \\qquad \\mathrm{Ad}_P(K_i) = TK_iT^{-1} = K_i.", "3704330783b83277f468dcabe737b35d": "\\pi^- + C\\to \\bar\\Sigma^- + K^0 + \\bar K^0 + K^- + p^+ + \\pi^+ + \\pi^- + nucleus~~~recoil", "370449617eb974b40a3f03b8f1a9e774": " 1\\leq p \\leq P ", "37046f649bcc35eb1f30195e8aff307e": "S_F=\\pi D^2_F/4", "37048599fef1997d666d579afe2d20f8": "\\notin", "370510bc600a8871d079853ff7625d24": "y_k|x_k \\sim p_{y|x}(y|x_k)", "370530aab70852005e4544c7c54d9a7a": "e^{\\left(\\frac{-\\Delta E}{kT}\\right)}", "3705fad623b8acf6704b325b972155ae": "\\int_{U(K)\\backslash U(\\mathbf{A})}f(ug)\\,du=0", "3706005edbb2cff7544b272477f83bce": "{{|z_1-z_3|\\cdot |z_2-z_4|}\\over{|z_1-z_4|\\cdot |z_2-z_3|}}=\\epsilon {{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}=\\epsilon\\, \\operatorname{cr}( z_1,z_2,z_3,z_4) ", "370609c7aa7c7ca290f27ebe6c931144": "(b>a)", "370617037f94175f140245fe8b3451ee": "f(x)=x^p", "3706178b6c91b9eeb131673aab657dc2": "\\mathcal{H}_{Phys}", "37064af2fd54031c9ce5f1e22d8028ed": "T_{sample} \\ne T_{clock}", "3706b2b21f121ab5fc79dacaddfa98e7": "N = \\mathit{p}^k - 1", "370748fc1b09d58fba1f89cfe465c896": " y \\in F, \\ \\ x - y \\perp F.", "37075501c02b19203c9d10dd307de98d": "|{\\Phi^A_i \\Phi^B_i}\\rangle=| {\\Phi^A_i}\\rangle\\otimes | {\\Phi^B_i}\\rangle", "3707a8082e715300ddd439f4332de465": "F = F_1 + \\frac{F_2 - 1}{G_1} + \\frac{F_3 - 1}{G_1 G_2} + \\frac{F_4 - 1}{G_1 G_2 G_3} + \\cdots + \\frac{F_n - 1}{G_1 G_2 G_3 \\cdots G_{n-1}},", "3707f5bf166ff99d41d057f820ecdab0": "\\lambda = \\mu, \\!", "3707fba9071dc9bf971530f2c9affad0": " r_\\mathrm{out} = \\frac{v_\\mathrm{out}}{i_\\mathrm{out}}", "3707ffdc759cbd52180de467fb825cab": "\\sin \\theta \\pm \\sin \\varphi = 2 \\sin\\left( \\frac{\\theta \\pm \\varphi}{2} \\right) \\cos\\left( \\frac{\\theta \\mp \\varphi}{2} \\right)", "3708241afd07c7b7b24a9104518fdf74": "\\ |u(2)-u(3) |>|u(1)-u(2) |", "37082c2336926759b5804e5d8f60bfed": "\\frac{1}{2a}", "3708785470ff55644fa48004f312a969": "\\sum_{i=1,2,3}\\varepsilon_i = \\frac{1}{E}((1+\\nu)\\sum_{i=1,2,3}\\sigma_i - 3\\nu(\\sum_{i=1,2,3}\\sigma_i)) = \\frac{1-2\\nu}{E}\\sum_{i=1,2,3}\\sigma_i", "3708841a17495fddcb7628621d9fca7f": "\n a={\\sqrt{\\gamma R_{sp} T_s}}\n", "37089c391e5cc3a77513a0c74456d16f": "\\int_{-\\infty}^{\\infty}k\\,e^{-f x^2 + g x + h}\\,dx=\\int_{-\\infty}^{\\infty}k\\,e^{-f (x-g/(2f))^2 +g^2/(4f) + h}\\,dx=k\\,\\sqrt{\\frac{\\pi}{f}}\\,\\exp\\left(\\frac{g^2}{4f} + h\\right),", "3708b32db40f4eaeef201909131ea8aa": "\\theta(\\lambda r)", "3708b9c821dc971aec44badb321fb9e7": "k_2/Q=0.0011", "3708cf0f68ad38add50c442fdc08e5ba": "x^i[\\mathbf{f}](v) = v^i[\\mathbf{f}].", "3708fd54c1df5ca5f2a2bbb64e771f6b": "(v_1+v_2)^2 = v_1^2 + 2 v_1 v_2 + v_2^2", "3709013d89817db1f7ad40b27e90ea55": "x = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3 + \\cfrac{1}{\\ddots\\,}}}} ", "37093a41ede3b3dd9cffe1f93ce16ef5": " \\begin{Bmatrix}u, v, \\phi \\end{Bmatrix} = \\begin{Bmatrix}\\hat \\phi \\end{Bmatrix} e^{i(k x + l y - \\omega t)} ", "3709bbcc1356e9062b7f20507e98d6dc": "\\Omega(n\\log n)", "370a1e6675f7f4d3b0e94e5fd7e171c3": " W^{k,p}(\\Omega) = \\left \\{ u \\in L^p(\\Omega) : D^{\\alpha}u \\in L^p(\\Omega) \\,\\, \\forall |\\alpha| \\leq k \\right \\}. ", "370a953b5ee194ed7525ca940040ada9": " \\dots \\to C_{i+1} ((X)) \\to C_i ((X)) \\to C_{i-1} ((X)) \\to \\dots ", "370b63ac0274d3ce2677e42c7b113fe1": " c_- ", "370b836da0efe44fd5bdf1b1ef2270a0": "f(z)={1 \\over (z^2+1)^2}", "370b9c84249b2f4fdc3199fc31275f09": "\\scriptstyle C=E\\left[(f(x) - y)^2\\right]", "370bb734cf8a45e61114f2094998d72f": "( a, b )_{\\text{01}} := \\{\\{0, a \\}, \\{1, b \\}\\}.", "370be54c0b9e2bc69b9281e743835040": "x \\backslash y", "370bf1e03b911b63a31be76d055814dd": " c_v ", "370c234fcf863671c681442ea1236a92": "x=x_1\\ldots x_r", "370c247f51c8b5febff0f8acd9b18c31": "X_w", "370c2e6878fad465cbe1eb1a247b41d9": "c_n = \\sqrt{n}", "370c362572e6881b00e6eb2b08f05c59": "\n\\begin{alignat}{2}\n\\operatorname{Ker}(A^+) &= \\operatorname{Ker}(A^*). \\\\\n\\operatorname{Im}(A^+) &= \\operatorname{Im}(A^*). \\\\\n\\end{alignat}\n", "370c505b245f364e9b88184cf93e8a3c": "\\mathbf{P}\\cdot\\left[\\nabla\\left(\\nabla\\cdot\\mathbf{Q}\\right)\\right]-\\mathbf{Q}\\cdot\\left[\\nabla\\left(\\nabla\\cdot\\mathbf{P}\\right)\\right]=\\nabla\\cdot\\left[\\mathbf{P}\\left(\\nabla\\cdot\\mathbf{Q}\\right)-\\mathbf{Q}\\left(\\nabla\\cdot\\mathbf{P}\\right)\\right].", "370c63870d5e6ed3481821e2e203ccfa": " 0 < {\\lambda_1}^{+} < {\\lambda_2}^{+} < {\\lambda_3}^{+} < \\cdots < {\\lambda_n}^{+} < \\cdots \\to \\infty; \\, ", "370caff28087e5d3d804baaa47e7501b": "l_{i,n} := -\\frac{a_{i,n}^{(n-1)}}{a_{n,n}^{(n-1)}}", "370d9c6ef59a7d1e0694c457da04a2ec": "M = \\int d^3 x \\epsilon^{abc} \\{ A_c^k , \\sqrt{V}_\\epsilon \\} F_{ab}^k (x) \\int d^3 y \\chi_\\epsilon (x,y) \\epsilon^{a'b'c'} \\{ A_{c'}^{k'} , \\sqrt{V}_\\epsilon \\} F_{a'b'}^{k'} (y)", "370db3d2f553e38b4e1fcbcc92bdc29b": "{P}={2}\\pi\\cdot{r}.\\!", "370dc79752bb6a6a0197db9e64965e93": "x_ix_j", "370e0a1ea4b39f6b1e3b79727836639c": "d^{RC}", "370e340edb15ba895e37f2951f727c0a": "T^\\alpha{}_{\\nu\\mu}=D_\\nu\\sigma^\\alpha{}_\\mu -D_\\mu\\sigma^\\alpha{}_\\nu\n", "370e39e90eab83469e8949fae30cd204": " n= \\frac{c}{c_0} = 1+\\frac{2 GM}{r c^2} ", "370e3f2c0b84a9a9ea5db09ce36ec737": "\\begin{bmatrix} 100 \\\\ 50 \\end{bmatrix}", "370e4187b244b4a263e89e0b84a8ae4d": "H^2(\\operatorname{Gal}(F/k), k^*)", "370e6951782904573ae9325116ff34aa": " \\and (S_6 \\implies (\\operatorname{equate}[A_6, x] \\and V[F_6] = A_6)) \\and D[F_6] = D[x] ", "370ead48df93fd8f2a3cdb6029446533": "L_{0}=T\\cdot v", "370ed22e60c50f0c4e6b0be9272aa375": " {N}(B)= \\sum_{x\\in {N}}1_B(x) ", "370ee23154c15be6453fdbdeef36146d": "\\ell\\,=\\,1", "370ee5b2c4fc1cf3a0aa88fcf223c92b": "\\begin{bmatrix}\nr_{11} & 0 & . & 0\\\\\n0 & r_{22} & . & 0 \\\\\n. & . & . & . \\\\\n0 & 0 & . & r_{nn} \\end{bmatrix}.", "370ef06949f32d88ea22eb636af56791": "\\frac{(P \\and Q) \\to R}{P \\to (Q \\to R)}.", "370f3b9bba6221f9725aa36eb9175e2a": "\\frac{\\partial}{\\partial y} = \\sin \\varphi \\frac{\\partial}{\\partial r} + \\frac{1}{r} \\cos \\varphi \\frac{\\partial}{\\partial \\varphi}.", "370f5d3db87bc8314e2c8934f6c95a50": "A \\to aA | bB ", "370f6ebd9dd74d91e94b40977ddd9f97": "\\cos \\theta + i\\sin \\theta = e ^{i\\theta }. \\,", "370fb3eabb73c124b211f9846d6fe1c6": "t' = t-\\frac{x}{c}", "370ff543543ca6a9ed7449a3bcf608e9": "f: Y \\rightarrow X", "371032163727b885cfc2311d44be0efe": "(\\rho, \\rho\\dot{q}_i,\\rho\\dot{p}_i)", "37108b65d48e38b2bd6f56fbf0223277": "d(\\ln S_t) = \\frac{d S_t}{S_t} -\\frac{1}{2} \\, \\sigma^2 \\, dt\\,.", "3710c39392e34e3f6c555981640bf1d3": " D = \\frac{k_b T}{6 \\pi \\eta a} ", "3710cddc97f2f282776ab240f149b2a1": "R_b = \\frac{t_b}{kA_b}", "3710faa5cb7a67ec2d865427ba76cc96": "\n0 = c \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\delta t \n- \\int { \\frac{d}{dq} \\left[ c \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\right] \\delta t dq }\n\\,.", "371166d7a2d68fa1cdda33b4965cdb2f": "S^1 \\times S^1 = \\{ ( \\cos{\\theta}, \\sin{\\theta}, \\cos{\\phi}, \\sin{\\phi} ) \\, | \\, 0 \\leq \\theta < 2\\pi, 0 \\leq \\phi < 2\\pi \\}.", "37117256039b0134f2a1b0362bf34f4a": " i \\colon A \\to X", "3711c3b85d3d43996fa88d977e393492": "\\theta_V = \\operatorname{Ad}(h^{-1}_{UV})\\theta_U + (h_{UV})^* \\omega_H ", "3711ebf023a160c31173bf03e78cba66": "\nf = a_1 a_2^2 a_3^3 \\cdots a_k^k \n", "37120ac1778e9b8561450eff2b7bd26d": "\\cosh\\theta = \\frac{1 + t^2}{1 - t^2},", "37123fc695e00c2ccdecec6feeae0fc9": "\\rho v_1 h_1", "37123ff27f93442f633c8171ceb04fa8": "K_b=\\mathrm{[C_5H_5NH^+] [OH^-]\\over [C_5H_5N]}", "371258011ee0f8d1832b9346153845fc": "\\boldsymbol{u}_f.", "3712c3f9af1b29ad48a309e5e1ecbaf3": "\na = \\frac{p(x)}{p'(x)}\\cdot\n \\left(\n \\frac1n+\\frac{n-1}n\\,\\sqrt{1-\\frac{n}{n-1}\\,\\frac{p(x)p''(x)}{p'(x)^2}}\n \\right)^{-1}\n", "3712ef06e01b3603f0b2d912369d1c10": "\\mathrm{M}_n(R)", "3713562fb7cf319d5936078b00ec4805": "\n\\frac{1}{4}\n\\begin{bmatrix} \n 1 \\\\ 2 \\\\ 1 \n\\end{bmatrix} \n*\n\\frac{1}{4}\n\\begin{bmatrix} \n 1 & 2 & 1\n\\end{bmatrix}\n\n=\n\n\\frac{1}{16}\n\\begin{bmatrix} \n 1 & 2 & 1 \\\\ \n 2 & 4 & 2 \\\\\n 1 & 2 & 1\n\\end{bmatrix}", "371363463d07b1634509f2c3872926dc": "\\mu_c \\ne 0", "3713d4c9b6750ea6783779e94be8d0c4": "\\operatorname{D}(P) = \\sum_{e} \\ell(e)\\otimes (\\theta(e)+\\mathbb{Q}\\pi)", "3714745e7d7bd78d4c9a24d526bbb45e": " \\dot m = \\rho \\cdot A_1 \\cdot v_1 = \\rho \\cdot S \\cdot v = \\rho \\cdot A_2 \\cdot v_2 ", "371484473d367465862a7c65220745a7": "\\Delta x_{\\mathrm{b.a.}}", "3714878e9e07938379ca367c604d2b04": "\\ i", "3714c3b7ae3e476bb4da6eab80de9935": "\\Phi=(\\phi^1,\\dots,\\phi^n)", "3714d5d40d2639d0a002c6a97d5b0890": "\\gamma=\\frac{1}{3}A", "3715516b959941d21c0f78d1a12808e9": "\\color{red}\\exists z", "37156eab56663f7c19fe9a79fe972f6f": "G(R) = \\sigma^+_R - \\sigma_R", "3715cacb191d7c5210627dfac724d4d3": "{z}\\,", "3715d01ac631f46d2ffe8d70827070c7": " \\hat{U}^{\\dagger} \\hat{U} = I ", "37164a4d3a727a0e477cf6f4d673a2a0": "{{v}_{IN}}={{v}_{BE2}}+{{v}_{BE3}}", "37165e1062aa7398655dedc85431efab": " \\mathbf{J_m} = \\nabla\\times\\mathbf{M} ", "37166d8eb0f8b5274e29af029f23b836": "((A\\equiv(B\\equiv C))\\equiv C)\\equiv(B\\equiv A)", "371675c000e180a87e7045787981dca5": "\\tilde{f}^{*}", "371699971396ffc50294d2dc57639a7e": "1024 \\le m \\le 16384", "3716a0380972e0ee1afadd1be32492fe": "\\psi'_e", "3716e8a8a9165c5eba880fd6fc0eba62": " \\psi_1(z) = -\\int_0^1\\frac{x^{z-1}\\ln{x}}{1-x}\\,dx ", "3716f0bbf0a597a297baf3ed20fca748": "-\\beta < \\Im{z} < 0", "371708f2cdd7ea58548ddb37fc0c0800": "\\int_{{-c}}^{{c}}\\cos {x}\\;\\mathrm{d}x = 2\\int_{{0}}^{{c}}\\cos {x}\\;\\mathrm{d}x = 2\\int_{{-c}}^{{0}}\\cos {x}\\;\\mathrm{d}x = 2\\sin {c} \\!", "371740803fed6054fc33068849515e08": "{1\\over2}(\\pi - \\sqrt3)s^2", "37174f704d20f1a23d063db624642b1d": "d = x - \\frac{1}{m} Z_{1,m} X,", "371768de8fba18722c46e066c46a9eb6": "\\ 0 \\leq u < 8", "37177b3555af50c28fbeea3662beb243": "\\Gamma(\\tfrac14) / \\pi^{-1/4}", "3717805dbd8b5d1ffeb3371b4f995feb": "\\bar{\\mu}_i=\\mu_i ", "3717b8fbc8f856b474ce829c0f2d86d9": "e^{\\pi\\sqrt{43}}\\approx 884736743.999777466\\,", "3718244322f541ca3e42586bac3167f5": "\\lfloor \\ldots \\rceil \\!\\,", "371828366dc9296f8a68366e676e31b5": " f.\\check{f} ", "37182d89f38ee0f05dea235dbe5b3d0d": "X_0 * X_1", "37184e05f4c22d4b37e634bb34f1ed84": "A_{\\varepsilon}^n(X,Y)", "3718891debe833a83adc941ea876902b": "v=\\left|X\\right|=\\binom{\\ell}{n}", "37188e83dbfc1b4fe07452bcb163fc06": "S = S_q + (S_t - S_q) \\frac{M-M_q}{\\overline{M}_t-M_q} ", "3718b6393b5677759b023fa1460d5d56": "\\sqrt{a^2+b^2+c^2}", "3718b87c208c41fc7100b88603faf408": "\\mathrm{d}\\gamma = \\Gamma_1\\mathrm{d}\\mu_1 +\\Gamma_2\\mathrm{d}\\mu_2", "3718cf7632fe776fcf690b554c3d7384": "E[\\sup \\rho_{t}] = \\rho + (1-\\rho)(1-e^{-\\lambda \\theta (t-1)}) ", "371918b091e4f081f2c33e5b90b7a121": "\\textstyle d(x)", "371924b24411206eab92d28df666abe7": "\n\\chi = \\frac{m_\\ell}{m_g} \\sqrt{\\frac{\\rho_g}{\\rho_\\ell}},\n", "37192ce753904e53f410ee670f57778e": "dW_r = JN d\\sigma_r = JN \\frac{d\\sigma_r}{d\\Omega} d\\Omega", "37193bb781b1f7b0f12c2852942426c4": "L-f'\\frac{\\part L}{\\part f'}=C \\, ,", "371957113e17ddc923d9da6100285cbd": "\n\\begin{align}e^{\\varphi(p)/p_1} & \\equiv (g^x)^{\\varphi(p)/p_1} \\pmod{p} \\\\\n & \\equiv (g^{\\varphi(p)})^{a_1}g^{b_1\\varphi(p)/p_1} \\pmod{p} \\\\\n & \\equiv (g^{\\varphi(p)/p_1})^{b_1} \\pmod{p}\n\\end{align}\n", "37197e51e2b2c43ed6d19a7fd4c00c48": "A = A( \\zeta, \\omega) = \\text{sign} \\left( \\frac{-\\sin\\phi}{2 \\zeta \\omega} \\right) \\frac{1}{\\sqrt{(1-\\omega^2)^2 + (2 \\zeta \\omega)^2}}.", "3719ae8be3b5b81c2f631d8d0e2bd1a8": "-\\mathbf x = (-x_1, -x_2, \\cdots, -x_n).", "371a0aaee3a09a266bbba626e59cf396": "A^j = \\sum_{k = 1}^n a_k^j E^k", "371a1af3f88f3ac87c383646c2961bcc": "-\\Vert x\\Vert^2 + \\Vert y\\Vert^2", "371a4c3a6d0c5125ed8dffbf188dd68d": "\nP_{n+1,j} = 1P^{\\pi_n}_{n,j-1}, 0P_{n,j}\n", "371add69ec1f300a3312d480493e8d4b": "P(D)=D^2-4D+5", "371af8563eddd96a0c9c4aa84180ea4a": "\\mathrm{Sp}(2n,\\mathbb C)\\,", "371b27a5c5c1b2e0100ab2493f25059a": "\n \\tau^{\\mathrm{core}}_{xz}(x,z) = \\cfrac{Q_xE^c}{D}\\int_z^{h} z~\\mathrm{d}z + C(x)\n = \\cfrac{Q_x E^c}{2D}\\left(h^2-z^2\\right) + C(x)\n", "371b7399444ee9a22c3034136c8e943e": "[g^{ij}] = [g_{ij}]^{-1}", "371ba6a703bd43913fc1bbca23e824f4": "x^{19} + x^{18} + x^{17} + x^{14} + 1", "371bb712213ef52811eca06fb5ed958b": "p_d", "371bd05cbed4fecb9395ee3f58aaf8f6": " i <_{\\mathcal{O}} j \\,,", "371bdbe89e3145f2b744188b51545fc5": " \\bold A \\bold x = \\bold k, \\quad (1) ", "371bf208d289ff03b3db102eb9988ad3": "C(0,u) = C(u,0) = 0 ", "371c8a9e952f80bc88f1d7394c932651": "\\lim_{\\theta \\to + \\infty} \\frac1{\\theta} \\log \\int_{A} e^{- \\theta \\varphi(x)} \\, \\mathrm{d} x = - \\mathop{\\mathrm{ess \\, inf}}_{x \\in A} \\varphi(x),", "371cd86d4b0ede88a7961d269234c08a": "Y_{2}^{1}(\\theta,\\varphi)\n={-1\\over 2}\\sqrt{15\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot\\cos\\theta\\quad\n={-1\\over 2}\\sqrt{15\\over 2\\pi}\\cdot{(x + iy)z \\over r^{2}}", "371cdd49ee92259b5d09b14f877b54d2": "C\\land I", "371cdf61ddec9811a333f70919b0905f": "S = \\frac{1}{\\sqrt{2}}\n\\begin{pmatrix}\n 0 & 0 & 1 & -1 \\\\\n 0 & 0 & 1 & 1 \\\\\n 1 & 1 & 0 & 0 \\\\\n -1 & 1 & 0 & 0\n\\end{pmatrix}\n", "371ce60d4c05fc4cb23a67a606abc741": "G(s) = \\frac{H(s)}{S_x^{+}(s)},", "371d0aa1367f300d10b54bc2b9327123": "\\int_{\\gamma} y dx+x dy=\\int_{\\gamma}\\nabla(xy) \\cdot (dx,dy)=xy|_{(5,0)}^{(-4,3)}=-4 \\cdot 3-5 \\cdot 0=-12", "371d24b8f93d7f2cdad066ba22fe8674": "\\scriptstyle\\partial A\\setminus\\Sigma ", "371d559e618c1205f4f7d183f453af2b": "\\mathsf{L}=(\\omega, \\mathbf{d}\\times\\omega),", "371db5a48382163a6eb345c84f4e8203": "\\sigma_{A}\\sigma_{B} \\geq \\left| \\frac{1}{2i}\\langle[\\hat{A},\\hat{B}]\\rangle \\right| = \\frac{1}{2}\\left|\\langle[\\hat{A},\\hat{B}]\\rangle \\right|.", "371e1bd6a51eae02402978216cc2adaa": "{\\mathbf{w}}_i = \\left( {w_{i1} ,..,w_{id} } \\right)^T ,i = 1,..,M", "371e54dd69a34f4e1282ee595cf847f0": " \\phi ", "371e94ed6be67b2fe9f78bfcd967a2d4": "Q_D", "371eb6f914c4cfd1bfe8b76140ff6edd": "\\lambda a=\\alpha_{0n},", "371ec92ebd938ce77a1156005040d7f6": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "371ec9a061ac05f67637fb494b9e31d8": "d^2E(Q)/dQ^2<0", "371ee05a17f8e3d464c8a1cecc4552b1": "| \\overline{\\ell , m} \\rangle = \\sum_{m'} D_{m'm}^{(\\ell)} U(R^{-1}) | \\ell , m' \\rangle\\,,\\quad | \\overline{\\hat{\\mathbf{n}}} \\rangle = U(R) | \\hat{\\mathbf{n}}\\rangle ", "371f11ee97f28a0df6ebd99d05381ad0": "f(X)\\not\\le_Q f({_*}X)", "371f51413597b0da0fa87a133089a352": "\n\\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{dt}{d\\tau} \\right) = \\frac{a}{b}\n", "371f7f4faa2b553260641a2071a5e9e2": "\\frac{d[S]}{dt}=-k_S[S][B^*]\\implies\\log[S]=-k_S[B^*]t+\\log S_0 ", "371f818a91e2fd4db34d78b878060187": "P = A_s j^{\\star} = A_s \\epsilon \\sigma T^{4} \\;", "371fa2925f4d75aa966d864644ab3d04": "{F}_ij", "371fa67f35f9d9c86a15dd37be272a39": "\nk = \\sqrt{\\frac{a + b}{am + bn}}\n", "371fa97542c58e777e1cf82fa4db6070": "\\frac{\\partial F}{\\partial x}(x, y) = I", "371fc03edf5224d80282e54d487c2f68": "(t,x) \\in \\mathbb{R}\\times\\mathbb{R}^n", "371fe9b647f57493b754f2346220433a": "\\hbox{Der}(A)=\\bigoplus_l \\hbox{Der}_l(A)", "372007ab41b2227caf6e71f24dd97cf8": "0 = \\begin{vmatrix}\n\n\\cos \\theta_{11} & \\cos \\theta_{12} & \\cos \\theta_{13} & \\cos \\theta_{14} \\\\\n\\cos \\theta_{21} & \\cos \\theta_{22} & \\cos \\theta_{23} & \\cos \\theta_{24} \\\\\n\\cos \\theta_{31} & \\cos \\theta_{32} & \\cos \\theta_{33} & \\cos \\theta_{34} \\\\\n\\cos \\theta_{41} & \\cos \\theta_{42} & \\cos \\theta_{43} & \\cos \\theta_{44} \\end{vmatrix}", "372017ca0399603d34d5a6c0af1ba2c5": "\\displaystyle{\\mathrm{Aut}\\,C =\\{g\\in \\mathrm{GL}(V)| gC=C\\}.}", "372051aec5d3809caa502a47ed77f12f": "1 - \\frac{1}{2} - \\frac{1}{4} + \\frac{1}{3} - \\frac{1}{6} - \\frac{1}{8} + \\frac{1}{5} - \\frac{1}{10} + \\cdots", "3720a14f2fe0c6a875924db46a733d0c": "\\wedge^*A", "3720b2acb6208b3b935542cc3fe1cc78": "H_{w}", "372107b86d71268819309cb5a4219f81": " \\mathit T = \\mathit g + \\frac{\\mathit ROE - \\mathit ROE (1 - D/E)} {\\mathit PB} \n", "37213b8c8cc7531d00e4953e00fff8e1": "\\begin{array}{ccc}\n\\textbf{Vect}_K & \\rightarrow & \\textbf{Set} \\\\\n\\uparrow & & \\uparrow \\\\\n\\textbf{TVect}_K & \\rightarrow & \\textbf{Top}\n\\end{array}", "3721816687d32065d23c1513861f0e02": "s_c = e_1 \\alpha^{c\\,i_1} + e_2 \\alpha^{c\\,i_2} + \\ldots", "37219d9a564e56013270ec6551d61b48": "(\\ast\\downarrow P)^{\\rm op}", "37219f139305dddb45e456f3f80f3315": "\\mathbb {R}^3", "3721b8824b5d5feb2ecc11f21a4e0d9a": "\\hat{C}(\\mathbf{k})", "3721edcf104a71d235f56d1fafa3249c": "\n\\tan \\beta = \\sin \\chi \\sin \\eta \\frac{\\tan \\lambda \\cos \\chi + \\sin \\chi \\cos \\eta}{\\cos \\chi - \\tan \\lambda \\cos \\eta \\sin \\chi}.\n", "3721f7d70f78fdaad8cad60f5caad4cb": "x(t)=\\frac{K}{(t_c-t)^2}.", "372228501367dabcc6e77ab6375baf2c": "F=N\\varepsilon_0-Mk_BT\\log k_BT+Mk_BT\\log\\hbar\\bar{\\omega}. \\, ", "372291a45f9a526cbedee8b00543252a": "\\,127 = -1 + 2^7", "3722997540f5a5ba660bea0b5a66c5be": "\\begin{align}\\phi(\\omega_0 + N|\\Omega|) &= \\begin{cases}\n-\\sum^N_{n=1} \\theta(\\omega_0 + n\\Omega) &\\text{if}\\, \\Omega>0\\\\\n\\sum^{N-1}_{n=0} \\theta(\\omega_0 + n|\\Omega|) &\\text{if}\\, \\Omega<0\n\\end{cases}\\end{align}", "37229e4092df265d7c8caf8c358b3e42": " \\langle \\mathbf{s}_i\\rangle^\\theta= M(\\beta) (\\cos \\theta, \\sin \\theta) ", "3722ad9d34b080e8560361ff78c81ab0": "\\frac{-100,000}{(1+0.10)^0}", "3722e8b63393ba09d82f8bc5e02ad078": "\\beth_{\\lambda}=\\sup\\{ \\beth_{\\alpha}:\\alpha<\\lambda \\}.", "37230d06fefd44e36582d33012b9b4b1": "s \\in S, r_1,r_2 \\in R", "3723f1d11982e0b3c41fcd713ea70ec5": " \\lambda(a)x=ax,\\, \\, \\rho(a)x=xa.", "372414684d01f06a41450946f737acc8": "\\langle r,s \\mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}+1}\\rangle\\,\\!", "372447f818e3defc902af6d33127d607": "\n\\mathrm{Hg} = -\\frac{1}{\\rho}\\frac{\\mathrm{d} p}{\\mathrm{d} x}\\frac{L^3}{\\nu^2}\n", "372536303f055858c919784cfc43697d": "\\sqrt[19]{100}^3 = 2.06914...\\approx 2", "37254e14e0789a2524e287ca1d1e1f11": "\\begin{smallmatrix}\\left[\\frac{\\alpha}{Fe}\\right]\\end{smallmatrix}", "37255a60389ca49924f0a41ab1681f1b": " \\tan (\\theta + \\psi) = \\frac { \\tan \\theta + \\tan \\psi } { 1 - \\tan \\theta \\tan \\psi } ", "3725c68e2dde844cc38a0f07a071500e": "entry(O_{j})", "3726009a5eb31000a565b29c6362c72f": "\\pi_0=-10", "3726129b58adf1dcaa29ec9f8d21c599": "\\operatorname{E}(X)", "37263ac7a9b3eac77ebeee7ff25614fc": "e_x\\mapsto x", "372693fc7da86462211124c7b11c84ae": "(ab)c= a(bc)", "37269598017958dd32d706b6eecd63a6": "\\tau_{c}", "3726ed4d21194fca3ff86f3cf5fc2307": "\\sum_{(v,w) \\in E} c(v,w) \\cdot f(v,w).", "3726fc44456ecf49b76ada47188955e4": "(x,x^2,x^3,\\dots,x^n);", "372712f15064fe31e71e4d2329b235db": "Q(x)", "37272ee2c35e544bce90658fde1b33b8": "\n \\begin{align}\n EI\\dfrac{dw}{dx} &= \\left[\\dfrac{Pbx^2}{2L} + C_1\\right] - \\cfrac{P\\langle x-a \\rangle^2}{2} \\\\\n EI w &= \\left[\\dfrac{Pbx^3}{6L} + C_1 x + C_2\\right] - \\cfrac{P\\langle x-a \\rangle^3}{6} \n \\end{align}\n ", "37275e6b21e790c3a7ae322cb68174a8": "W_{TOT}", "37276db83a13eae922b39458abbcdba2": "\\mathbf{E=E_0}\\exp \\left[i(\\mathbf{k \\cdot r}-\\omega t)\\right] \\,", "372786847fd6ad08d130bd81f8cd2a7c": "\\sum_{k=0}^n k^2 \\tbinom n k = (n + n^2)2^{n-2}", "3727c0529ba47668695332ff25d4b289": "\\omega=0", "3727d7a80621ed0aa85a0d57f5576159": "\\xi \\in C_i(\\alpha)", "37282046fad54abb82dd9de397d57e21": "\\mathbf{D} = \\{(\\boldsymbol{x}_1,y_1),\\dots,(\\boldsymbol{x}_n, y_n)\\}", "372987221f3d9bb582bcd1995b6e7db3": "\n\\begin{bmatrix}\n u_{1,j-1} , & u_{2,j-1} , & \\ldots, & u_{i-1,j-1} , & u_{i,j-1} , & u_{i+1,j-1} , & \\ldots , & u_{m,j-1}\n\\end{bmatrix}^{T}\n", "372987742d32e3f3136f7d8f40dce261": "y(x) = \\sqrt{r^2 - x^2}", "372a00253dd1ae3c3cbaa595eefb6cf5": "\\!\\models^-", "372a827360a0734e73270ee5fe3cfe6e": "{\\rm Range}\\{\\,X_1,\\ldots,X_n\\,\\} = X_{(n)}-X_{(1)}.", "372abad71f8253060b38785ad2e291b7": "\\scriptstyle (r(x), 1)\\,", "372adbc7fdcace888ae24b298f376e0b": "U(t+\\bigtriangleup t,w)=U(t,w)\\exp \\bigg( \\frac{|w|\\bigtriangleup t}{2Q(w)}\\bigg) \\exp \\bigg( i w\\bigtriangleup t\\bigg) \\quad (2)", "372b05e7d1bf45e391ad87b9c970a1b1": " \\mathcal O_X^\\mathrm{an} ", "372b3423d55651c10d464162a331183a": "y'\\,", "372b4d707bc29e1aee2ef1a13ce2ee84": " \\text{d} R / \\text{d} Q", "372b871f6b5cc9e77be96421e96581ee": " y=Ce^{x}-x-1\\ ", "372be7f7705418ec44fb128de8965f62": "\\operatorname{MSE}(\\overline{X})=\\operatorname{E}((\\overline{X}-\\mu)^2)=\\left(\\frac{\\sigma}{\\sqrt{n}}\\right)^2", "372c097da256128db62037fe920c4c47": "S_n \\approx a_n+\\Xi b_n. \\, ", "372c1221a2478ff8ebb4a92b62a82443": "t \\geq \\frac{[b-Ac]_i}{[Av]_i} \\;\\;\\;{\\rm if}\\;[Av]_i < 0 ", "372c2588bf5e7741c903162606eb9bbb": "y'(\\phi)=a\\sec\\phi", "372c39c38ada99173dac6698335740e9": " A= A_0 + \\epsilon^1 A_1 + \\epsilon^2 A_2 + \\cdots", "372c3e5210ada3e8e813eac16e039892": "D = Y_1\\cdot Y_2 ", "372c8c0974114a670f97d71577e77b98": "A^2", "372cca60f434aed72958c792bb927784": "x^2-x-1=0", "372ceb7b7fe06dce4d02a076089a4d15": "\\mathbf R(t')=\\mathbf r-\\mathbf r_0(t'),", "372cfd6ef9f2ee84a52fe0b710aebd05": "(\\epsilon,\\delta)", "372d82078e4ed305fbf3db3940d23da5": "\nm \\ddot{x} + k ( 1 + \\imath \\eta ) x = 0\n", "372dee8ef280343030dccaa391fb54fa": "R_{TP}\\,", "372e0e83c08a9accf873952d02e57f51": "p_{me}", "372e262bfe45f439e2a81bfa774c2b30": " S_h = \\sqrt{Var_h} ", "372e666a541a284d8943a632ab20ff67": "x=P/K", "372e748a1ccc154f1ec6efab7a987838": "I_p(x) = C_a + C_d (L(x) \\cdot N(x)) ", "372e93c95e3be7950544d077849ceee9": "E_i(Z+1)", "372eb428b5d9436bbae1afedce988351": " f(y) = \\frac{y^{\\gamma_1-1} (1-y)^{\\gamma_2-1}}{B(\\gamma_1, \\gamma_2)}, \\qquad 00,", "372edc1bc4ea5fc291fa391066c97414": "\n\\begin{align}\np_r &= \\frac{\\partial{\\mathcal{L}}}{\\partial \\dot{r}} = \\frac{\\partial T}{\\partial \\dot{r}} = (M+m)\\dot{r}\\\\\np_\\theta &= \\frac{\\partial {\\mathcal{L}}}{\\partial \\dot{\\theta}} = \\frac{\\partial T}{\\partial \\dot{\\theta}} = mr^2 \\dot{\\theta}\\\\\n\\therefore \\mathcal{H} &= \\frac{p_r^2}{2(M+m)} + \\frac{p_\\theta^2}{2mr^2} + Mgr - mgr \\cos{\\theta}\n\\end{align}\n", "372f29a9838d2a70504ab484d3c58466": "V = \\frac{64\\pi RC k_BT \\Gamma e^{-Kh}}{K^2}, ", "372f9c8cf62bf056e67ecd50a8759ef8": "Z_\\mathrm B=Z_2\\ .", "37301425d9c4ba34154a480de9e57017": "\\lambda_0(x)", "373071dc71f12d086673b1a3045d9019": "\n\\begin{array}{l}\\displaystyle\n\\bar{x} = \\frac{1}{L^2} \\int\\limits_0^x r^2(x) dx,\n\\quad\n\\bar{y} = \\frac{r(x)}{L} y,\n\\\\ \\displaystyle\n\\bar{u} = u,\n\\quad\n\\bar{v} = \\frac{L}{r} \\left( v+\\frac{r'}{r} y u \\right),\n\\\\ \\displaystyle\n\\bar{U} = U,\n\\end{array}\n", "37307e135eb399f30464cf4f4fb9731b": "\\xi + \\omega\\delta\\sqrt{\\frac{2}{\\pi}}", "3730f866aa607de3222354647dfe6a69": "[M + nH]^{n+} + A^- \\to \\bigg[ [M + nH]^{(n-1)+} \\bigg]^* + A \\to fragments", "3731180bd047cb7cb9f9f71c8c14b379": "\\int_{a}^{b}\\omega(x)r(x)dx= \\sum_{i=1}^{n}r(x_{i})\\int_{a}^{b}\\omega(x)\\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}\\frac{x-x_{j}}{x_{i}-x_{j}}dx", "373146c71e72ad764250c26290253874": "59^4 + 158^4 = 133^4 + 134^4", "373297633ea0304b6a938985ecb86fcb": "\\operatorname E[X_{ik}m(\\vartheta)]=\\operatorname E[\\operatorname E[X_{ik}m(\\vartheta)|\\Theta_i]=\\operatorname E[(m(\\vartheta))^2]=v^2+\\mu^2", "3732a78e9ab729111fdf9a5bc6e0f36a": "\n{e^{-\\lambda} \\lambda^{x_1} \\over x_1 !} \\cdot\n{e^{-\\lambda} \\lambda^{x_2} \\over x_2 !} \\cdots\n{e^{-\\lambda} \\lambda^{x_n} \\over x_n !} \\,\n", "3732bf06826255f71b4f33067759a53b": " [(x+y)^{n+1}]_{jk} = [(x+y)^n]_{j-1,k} + [(x+y)^n]_{j,k-1}. \\, ", "3733080c9c7ec52326228aed64a9ead3": "I = \\frac{1}{2}", "37334942b9c1d4b50557b26bc5338d1a": "\\begin{matrix}-\\frac {g_\\mathrm{m} R_\\text{D}}{1+g_\\mathrm{m} R_\\text{S}}\\end{matrix}\\,", "373373928815de837107272cec9683fe": "\\lim_{n\\to 0} {x^n-1\\over n}=\\ln x.", "37339921732a8f31151f0153348c4d56": "x\\le_Q y", "373399440521ca08d098e0f053126858": "a^2I_{n,m}= I_{m,n-1}-I_{m-2,n}\\,\\!", "3733b02ec5502ab185b29f5803d530a7": "n,k", "3733d0410bfe1955b4a59954df0929a4": "g^{(1)}(\\tau)=g^{(1)}(-\\tau)^*", "3734261de365e56ac468e1643de6cbd6": "\n(\\backslash \\leftarrow) \\quad\n{Y\\leftarrow X \\Gamma\n \\over\n X\\backslash Y\\leftarrow\\Gamma}\n", "37345383dc6d4708ceb127cafcd1fd64": "\\lnot \\phi \\lor \\psi", "3734957293630cec91c99a7d212da7ca": "\\frac{\\textrm{d}[\\textrm{HCO}_3^-]}{\\textrm{d}t}= k_1[\\textrm{CO}_2] - k_{-1}[\\textrm{H}^+][\\textrm{HCO}_3^-] - k_2[\\textrm{HCO}_3^-] + k_{-2}[\\textrm{H}^+][\\textrm{CO}_3^{2-}], ", "3734b04d041dd26ec555c90dea2d46a6": "\\frac{3}{\\sqrt{3}+\\sqrt{5}}", "3734b428648ee0ddcfad61ecc9cb8cd9": "\\textstyle\\frac{1}{2}", "3734bc4bb2aa64fab7a87ee71cc8a79d": "\\boldsymbol s_\\boldsymbol\\Theta", "3734fce7a1324e59540b05c59bf13417": "^6_{}", "3735389fc386df6f482927361f830749": "\n\\frac{d}{d\\tau} \\left[ r^{2} \\frac{d\\varphi}{d\\tau} \\right] = 0,\n", "3735656bb9c5c60edad1bbe7ee6d72ed": "\\varphi(\\varphi(x,t),s) = \\varphi(x,s+t).", "37356ae2cd13fe8ab401f628b5e605a8": "\nT_{grav} = n\\sum_{k = 1}\\cos(nk\\Omega_0t)\n", "3735f15f9168463c88cb23955417514e": " \\sin\\theta_{r} = \\sin\\left({90^\\circ} - \\theta_{c} \\right) = \\cos\\theta_{c}\\ ", "3736691105662fc4b13beaa7fa6b9272": "\\frac{4}{3} \\sqrt{2}", "3736801dbb959f46323fc17afc09359b": "\n 1 = \\int_{-\\infty}^{\\infty} f(g(x)) \\times g'(x)\\,dx\n = \\int_{-\\infty}^{\\infty} f_s(x)\\,dx.\n \\!", "37368baafc510600f02e7c8a53e3e2a1": "x0 ", "3747b177ce7cc9c576ac6bd7dd32fdbd": "p(x\\mid I) = \\frac{1}{Z(\\lambda_1,\\dotsc, \\lambda_m)} m(x)\\exp\\left[\\lambda_1 f_1(x) + \\dotsb + \\lambda_m f_m(x)\\right]", "3748975bcc45fda2da22ca8317f5766b": "\\! \\rho", "3748a1f79e4a2d10b59a9f5125ba94ba": "\\delta P=(\\delta x,\\delta y)", "3748eb647f37c8737828ba30c8d57ec0": "\\kappa( \\mathfrak{g}, [\\mathfrak{g},\\mathfrak{g}] ) = 0", "374949a62d093f6f4faf39a20d6b343c": "f(x) \\propto x^{\\alpha + \\beta x}", "3749caed80a32324237e3586bb2a4dec": "Z = \\frac{\\hat u}{\\hat \\imath} = \\frac{U_\\mathrm{AC}}{I_\\mathrm{AC}}.", "3749cdea803c1f3f543dad8eb7f3ea8e": "4:2\\ ", "3749d6623a467bf93e1bddc9eae394b5": "D_{\\mu\\nu}^{ab}(p)=\\delta^{ab}\\left(\\eta_{\\mu\\nu}-\\frac{p_\\mu p_\\nu}{p^2}\\right)\\frac{p^2}{p^4+M^4}.", "374a1bf6b548fec329d0bbfb325386b7": "|\\mathit{after}\\rang = \\sum_i |i,\\epsilon_i\\rang \\lang i|\\psi \\rang", "374a2f68ead23bc698b424cb65f816eb": "\\partial_n f_*=f_*\\partial_n", "374a4cb5be90ca22823521c66a675df5": "\\scriptstyle n_{R}", "374a9cd1c8ea262a3bd6d65593967212": "\\Gamma(\\alpha)=\\int_0^\\infty x^{\\alpha-1} e^{-x}\\,dx.", "374abe8edbb53a48d08befa9aef58bd0": "K_{3,3}", "374b09fb74d5c70e17069b2cd7329351": " H^G_*(E_{FIN}(G),K^{top}_{l^1})\\rightarrow H^G_*(\\{\\cdot\\},K^{top}_{l^1})=K_*(l^1(G)) ", "374b47700265ba97516d1b34c07e1b90": " P^{-1} ", "374b755006919fa87c4111ab98b4f2fb": "\\displaystyle{\\|v\\|_{(2)} \\le C( \\|\\Delta(v)\\|+ \\|v\\|_{(1)}).}", "374b8fcdb815d81ec0f25dc4053253ad": "\\mathrm{vol} \\big( (1 - \\lambda) K + \\lambda L \\big)^{1/n} \\geq (1 - \\lambda) \\mathrm{vol} (K)^{1/n} + \\lambda \\, \\mathrm{vol} (L)^{1/n},", "374bb78075836e781715163f7da6e72c": " F_X(x) = \\mathcal{L}^{-1}_s \\left\\lbrace \\frac{E\\left[e^{-sX}\\right]}{s}\\right\\rbrace (x) = \\mathcal{L}^{-1}_s \\left\\lbrace \\frac{\\left(\\mathcal{L} f\\right)(s)}{s} \\right\\rbrace (x) \\, ", "374bca837c5fa2d0f99795fcf7ca7ed9": "u'' + \\left(\\frac{\\lambda}{Q} - \\frac{S''}{S}\\right)\\,u = 0.\\,", "374c6272faaab45e43898835c3f9b685": "\\begin{align}\na' & = \\bigg|\\bigg| \\Big(\\begin{smallmatrix}-a\\\\K-b\\\\-c\\end{smallmatrix}\\Big) - \\bigg( \\Big(\\begin{smallmatrix}-a\\\\K-b\\\\-c\\end{smallmatrix}\\Big) \\cdot \\Big(\\begin{smallmatrix}0\\\\\\;\\;1/\\sqrt{2}\\\\-1/\\sqrt{2}\\end{smallmatrix}\\Big) \\bigg) \\Big(\\begin{smallmatrix}0\\\\\\;\\;1/\\sqrt{2}\\\\-1/\\sqrt{2}\\end{smallmatrix}\\Big) \\bigg|\\bigg| \\\\\n& = \\bigg|\\bigg| \\Big(\\begin{smallmatrix}-a\\\\K-b\\\\-c\\end{smallmatrix}\\Big) - \\Big( 0 + \\tfrac{K-b}{\\sqrt{2}} + \\tfrac{c}{\\sqrt{2}} \\Big) \\Big(\\begin{smallmatrix}0\\\\\\;\\;1/\\sqrt{2}\\\\-1/\\sqrt{2}\\end{smallmatrix}\\Big) \\bigg|\\bigg| \\\\\n& = \\bigg|\\bigg| \\bigg(\\begin{smallmatrix}-a\\\\K-b-\\tfrac{K-b+c}{2}\\\\-c+\\tfrac{K-b+c}{2}\\end{smallmatrix}\\bigg) \\bigg|\\bigg| = \\bigg|\\bigg| \\bigg(\\begin{smallmatrix}-a\\\\\\tfrac{K-b-c}{2}\\\\\\tfrac{K-b-c}{2}\\end{smallmatrix}\\bigg) \\bigg|\\bigg| \\\\\n& = \\sqrt{(-a)^2 + \\big(\\tfrac{K-b-c}{2}\\big)^2 + \\big(\\tfrac{K-b-c}{2}\\big)^2} = \\sqrt{a^2 + \\tfrac{(K-b-c)^2}{2}} \\\\\n\\end{align}", "374c8a4a844c66da9fc041988f933243": "\\tfrac{1,2}{1,3}\\,,", "374caa6fc5bcf5b0106543cd2e7876ef": "p(x,y)", "374cbd174bad32fbd1943b40c5ecb983": "x\\ \\exp(-x)", "374cf6840f77fc8ae56f5414685254d6": " U(\\sigma, \\tau, z) = \\frac{U_{\\sigma}(\\sigma) + U_{\\tau}(\\tau)}{\\sigma^{2} + \\tau^{2}} + U_{z}(z) ", "374d2ddf96012ef4dbfda596d0c1481c": " f(z) = \\sum_{k=0}^\\infty a_k z^k.", "374d40af346ba00e53223f28cffe5193": "N=1/(\\Delta f \\, T)", "374d53d8bb8ca12b3d267d6edde8b0bd": "k(x_{j+1}+x_{j-1}-2x_j)", "374da37a3568d6bb000e3bf5cf5fa7df": "\n \\lim_{n\\to\\infty} \\operatorname{Pr}\\Big( \\omega \\in \\Omega : \\sup_{m\\ge n} | X_m(\\omega) - X(\\omega) | \\ge \\varepsilon \\Big) = 0 \\quad\\text{for all}\\quad \\varepsilon>0.\n ", "374da7e4cfd200da5529312b85492312": "E\\left\\{ Y\\left[ n \\right] \\right\\}=\\frac{1}{N}\\sum\\limits_{p=0}^{N-1}{f\\left[ \\left( n-p \\right)\\bmod N \\right]}=0", "374dbdd03ff05ced33b352d1ba8bdb9b": "\n\\left(\\mathbf{A}+\\mathbf{uv}^\\mathrm{T}\\right)^{-1} = \\mathbf{A}^{-1}- \\frac{\\mathbf{A}^{-1}\\mathbf{uv}^\\mathrm{T}\\mathbf{A}^{-1}}{1+\\mathbf{v}^\\mathrm{T}\\mathbf{A}^{-1}\\mathbf{u}}.\n", "374e13d23b40d9ed99bb9c0aa52f6009": "O(n*2^n)", "374e14207296d629fdef99234068f8d0": "\\sqrt{V} \\geq 0", "374e6bcaa78f7d14d4c0d78abbaa8f0f": "\\mathcal V_\\xi", "374e9c88093d2b39ee6c873864c674ca": "F \\to T_kF,", "374efba46fd449a7bcc5dcbcb2d1679a": "C_X = C_A + C_B + (\\bar x_A - \\bar x_B)(\\bar y_A - \\bar y_B)\\cdot\\frac{n_A n_B}{n_X}", "374f2106a3f945c2a3b1037939cd7c3d": "L(\\lambda,{\\hat \\lambda})=\\sum_{i=1}^p \\lambda_i^{-1} ({\\hat \\lambda}_i-\\lambda_i)^2", "374f25cfb9510f2309df3aa23c47c4ae": "B_y", "374f273bfdfcf0c91fdde6ca90098498": " b = \\frac {2 n} { \\left( n + 1 \\right) \\left( 2 n + 1 \\right) } ", "374f48e5eabc30e7e85057e3e959b376": "\\mathtt{in}", "374f78c3bd757dc28e48e21529169e6f": "s^4 :=~(a^2 + b^2)^2 + 2(a^2 - b^2) + 1", "374fbdc7de71eaf99f35262b9eccad2b": " \\forall g \\in K \\ : \\ \\left \\|\\pi(g) \\xi - \\xi \\right \\| < \\varepsilon.", "37501a9680887b89e5492ec12f398cc1": "{c \\over {s+v}}", "37505d78843c3098527a9b591500185a": "y_{match}", "3750666f488c6f07312354a6c4b5862b": " 3,\\, 2 + 5i,\\, 2 - 5i,", "3750726a1e3d5d6e5a88464d34409a7c": "\\forall \\bar{y} (\\varphi(0,\\bar{y}) \\land \\forall x ( \\varphi(x,\\bar{y})\\Rightarrow\\varphi(S(x),\\bar{y})) \\Rightarrow \\forall x \\varphi(x,\\bar{y}))", "375084802a7718d8f60f93d1580cc5f8": "\n \\begin{align}\n \\sigma_{xx}^{\\mathrm{f}} & \\approx \\cfrac{zM_x}{2fh(f+h)} ~;~~&\n \\sigma_{xx}^{\\mathrm{c}} & \\approx 0 \\\\\n \\tau_{xz}^{\\mathrm{f}} & \\approx \\cfrac{Q_x}{4fh(f+h)}\\left[(h+f)^2-z^2\\right] ~;~~&\n \\tau_{xz}^{\\mathrm{c}} & \\approx \\cfrac{Q_x(f+2h)}{4h(f+h)} \\approx \\cfrac{Q_x}{2h}\n \\end{align}\n", "3750b02b0858a6256bf1757db5f09c69": " \\mathbf M ", "3750c050a7bbb0627cfd8476ebc9ef83": "{\\mathbf{q}}_{\\mathbf{0}}", "3751680cf456bec4a7bde8c2668221c3": "y' = -x \\sin\\theta + y \\cos\\theta\\,", "3752002cd164ff906870314e9460690c": "\n\\begin{matrix}\nI(X;Y|Z) & = & H(X|Z) + H(Y|Z) - H(X,Y|Z) \\\\\n\\ & = & H(X|Z)-H(X|Y,Z)\n\\end{matrix}\n", "375212a9abb7aebc17737f3344cbfbe5": "\\omega_2\\,", "3752ab943c2069f0f24e4860b0f3b549": "\\ \\displaystyle \\max\\ ", "3752e70183036d75dc7268a7ac78217e": "\\mathbf{A}^{-1}={1 \\over \\begin{vmatrix}\\mathbf{A}\\end{vmatrix}}\\mathbf{C}^{\\mathrm{T}}={1 \\over \\begin{vmatrix}\\mathbf{A}\\end{vmatrix}}\n\\begin{pmatrix}\n\\mathbf{C}_{11} & \\mathbf{C}_{21} & \\cdots & \\mathbf{C}_{n1} \\\\\n\\mathbf{C}_{12} & \\mathbf{C}_{22} & \\cdots & \\mathbf{C}_{n2} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\mathbf{C}_{1n} & \\mathbf{C}_{2n} & \\cdots & \\mathbf{C}_{nn} \\\\\n\\end{pmatrix}", "37530599881372d1de30ac0ad38ca284": " \\mathbf{} v_t ", "37533f1cef93c7bacf863cfa5921a68a": " \\theta = \\mathrm{contact \\ angle} ", "37535f50e3d2a27c38d3d2a69e48aa12": "\\mathit{MPC}=\\frac{dC}{dY}", "37536c9a0d5c41f1bc5b99ce033a1c53": "x_{k+1} = \\frac{1}{n} \\left({(n-1)x_k +\\frac{A}{x_k^{n-1}}}\\right) ", "3753e174e6a5fc5f28105870a79f20ca": "s_N + c_1s_{N-1} + c_2s_{N-2} + \\cdots + c_Ls_{N-L}", "3753e7512f34929216586548d0e3eec1": "O\\left(w_\\text{kernel} w_\\text{image} h_\\text{image}\\right) + O\\left(h_\\text{kernel} w_\\text{image} h_\\text{image}\\right)", "3753efe4b8a9425f487c5f635b6be1d5": "NE(X)", "3754167469454a99c38f22279e2a8f12": "f_{r}(e^{i\\theta})=f(re^{i\\theta})", "375427bea56c1acb205628e3ae74609a": "\\left (\\sum_iM_it^i \\right )\\cdot \\left (\\sum_jN_jt^j \\right )=\\sum_{i,j}(M_i\\cdot N_j)t^{i+j},", "37542ac4ac8192c85f0e20305e3b9124": " m/(r+h)^2 - m/r^2 = -2m/r^3 \\, h + 3m/r^4 \\, h^2 + O(h^3) ", "3754578eb6983f439c321e78cc763f53": "(-2r)^2=4(r^2),", "3754e682939ef51658ce9aaf4b630570": "\nv(\\mathbf{R}) = \\frac{1}{R},\\quad v_\\alpha(\\mathbf{R})= -\\frac{R_\\alpha}{R^3},\\quad \\hbox{and}\\quad v_{\\alpha\\beta}(\\mathbf{R}) = \\frac{3R_\\alpha R_\\beta- \\delta_{\\alpha\\beta}R^2}{R^5} .\n", "37552975c0a463c204436c7d8c4be9f7": "R = \\frac{R_s+R_p}{2}\\,\\!", "3755392453d1caa5daab26f19273fdb9": "S(T,X) = \\int_0^T \\frac {C(T^\\prime,X)}{T^\\prime}\\mathrm{d}T^\\prime.", "37556120647a11634b30932c22b18bb1": "gcd(a_m,a_n) = a_m", "3755a4f2f0b6d5d01e3904cf86437981": "\\mathrm{Factor} = \\frac{\\mathrm{Days}(\\mathrm{Date1}, \\mathrm{Date2})}{\\mathrm{Freq} \\times \\mathrm{Days}(\\mathrm{Date1}, \\mathrm{Date3})}", "3755c97e26673f857c54238989eafd9d": "\\textstyle{k = \\frac{1}{2\\pi}\\oint_{S^1} \\omega}", "37562aceaa0cbaf82836214a8dedede5": "q = n", "3756366c4850aafa7cf74ec4779fddaa": "\\frac 1 {D_\\mathrm N} + \\frac 1 v_\\mathrm N = \\frac 1 f\\,", "375655b51febe3b37399164539005cca": "ax\\,", "37569746ebd614c38965b140e7ae13c5": "w_f=U\\sin(\\theta-\\alpha)", "3756eacc09121f6dc0d951e30f9ed1a2": "K = B \\cdot C \\cdot \\sin \\theta.\\,", "3756fbe002139f7b0666eb3734861add": "\\mathbf{E}_{-n}(z)=\\frac{(-1)^{n+1}}{\\pi}\\sum_{k=0}^{[\\frac{n-1}{2}]} \\frac{\\Gamma(n-k-1/2)(z/2)^{-n+2k+1}}{\\Gamma(k+3/2)}\\mathbf{H}_{-n}. ", "375758e592af6f5c240a5b04e9ddb700": " \\sin \\left( \\theta \\right) \\equiv -\\left( { a - \\lambda_{-} \\over c \\eta }\\right) ", "37577f28ee6dfc2bddd68ce319c7493b": "O(r^3)", "3757a32be4c2cf03e69fa9deee056a41": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = \n -\\cfrac{2h^3E}{3(1-\\nu^2)}~\\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} w^0_{,11} \\\\ 0 \\\\ 0 \\end{bmatrix}\n", "37589920f59b6b64bc39e6b48eb931a6": "\\begin{align}\na_1 &=\\frac{(c)(c+1-\\gamma )}{(c+1-\\alpha )(c+1-\\beta )}a_{0} \\\\ \na_2 &=\\frac{(c+1)(c+2-\\gamma )}{(c+2-\\alpha )(c+2-\\beta )}a_{1}=\\frac{(c+1)(c)(c+2-\\gamma )(c+1-\\gamma )}{(c+2-\\alpha )(c+1-\\alpha )(c+2-\\beta )(c+1-\\beta )}a_{0} = \\frac{(c)_{2}(c+1-\\gamma )_{2}}{(c+1-\\alpha )_{2}(c+1-\\beta )_{2}}a_{0} \n\\end{align}", "3758e81cec52008e6d56dd380dac7030": " N '_c, N '_q and N '_y ", "3758ea417bb366289ec02538813fb14f": "N<2\\sqrt{m}", "37592e842477ad82af8bf323b8108b40": "\\left( \\frac{\\pi}{H+2\\delta} \\right)^2 + \\left( \\frac{2.405}{R+\\delta} \\right)^2", "37595ad7946086553d6a4dc848b23ebb": " D F = \\mu_0 J ", "37598809671c445f830501cf385fb4d9": "q_2(F_S(s))=q_1(F_S(s-\\ell)", "3759917a3f5e2ce5e731fe10c00da587": "-\\frac{\\pi}{2} < y < \\frac{\\pi}{2}", "3759efd7cf0cf97fdcf80077be67c4c9": "p_{s}\\;", "3759f62da4f0e0292e8e4781a9625e9e": " SG_\\text{true} = \\frac {\\rho_\\text{sample}}{\\rho_{\\rm H_2O}}", "375a2a3733fe302bd291b9bf29553137": "F \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathrm{d}A.", "375ab103de5953b54079ec9b33ee8b2d": "{\\rho}{V^2}\\over L\\,\\!", "375b01648546869fec8390ebb56e803b": "G \\to X", "375b169b9cf29436c83242b33340fd6d": "\n\\operatorname{Li}_{-n}(z) + (-1)^n \\,\\operatorname{Li}_{-n}(1/z) = 0 \\qquad (n = 1,2,3,\\ldots) \\,.\n", "375b330b6b19676102a23ae3e04a6e37": "R_{\\rm S} \\,", "375bc8fa3dd8a7913138ce9ada402049": "\\tfrac{e^{-X}}{k} \\sim \\mathrm{PowerLaw}(k, \\lambda)", "375bf21e18d8aa6458c6adc5228e23f2": "/ \\!\\,", "375c12a7758b281be18061566a6035e5": "n_{j,r}^i", "375c1406d41f564c96611394e657e5a4": " \\hat D ", "375c61473d535f268188ae4389b0fae2": " U U^\\text{T}=I. ", "375c96f45ed0fef4ca6b865af92816f4": "\\mathbf{r}(t), a\\leq t\\leq b, \\mathbf{r}(a)=\\mathbf{r_0}, \\mathbf{r}(b)=\\mathbf{r}.", "375cb0b02b3f5a4fd66abc00146691d1": " \\sigma_\\text{Total} = \\sigma_D + \\sigma_S. ", "375cbd05256d2fd2571fd5112389faa4": "\\det(M M')", "375cd5b80695c9ef2c3952931e0fe2bf": "H(x^\\mu)", "375d1689482a529d5f3254adaf123e95": "f=\\sum_{k=0}^\\infty A_kz^k", "375d2ba57227ceae2f5213149bbfc6e8": " \\epsilon_b^n \\ = \\ \\sum_{i=2}^n \\ M_{1i} \\ b_{i-1}^{n-1} \\ = \\ \\sum_{i=1}^{n-1} \\ t_{-i} \\ b_i^{n-1}. \\ ", "375d5f015b4f9e4f74efeed715ee62d9": "\\omega_2 /\\omega_1 = 1-a/c", "375d9c53ddbe402373c037e8058a058c": "\\mathbf{1}_{A\\cup B} = \\max\\{{\\mathbf{1}_A,\\mathbf{1}_B}\\} = \\mathbf{1}_A + \\mathbf{1}_B - \\mathbf{1}_A \\cdot\\mathbf{1}_B,", "375dca02c3fec3e93c52a5479ce4ee40": "g(\\overline{x})", "375e063bc4213998f6cca88ecc149692": "\\mathbf{D_{xx}} ", "375edb18ed7ef44eb0c1370d49a24c34": " n \\ln (1+e^{\\eta})", "375f69359c85bd38bce9fd6ce46a2894": "V_F + V_R = V\\,", "375fa964cb70aaaac72fe843bc40e031": "P(s,n) = \\frac{n^2(s-2)-n(s-4)}{2}", "375fc04226aac8c4eaa8ceb22a8a8dae": "F[y]={y''' \\over y'}-{3\\over 2}\\left({y''\\over y'}\\right)^2", "37603c9e2e2dd1bf949b54b8a2cedb0b": " \\mathcal{H}^2 ", "376059fc2a27151730ef6b647758a01e": "E_p(x)=\\sum_{n\\ge 0} \\frac{t_{p,n}}{n!}x^n.", "37606a0673d2614b8df23ffe7b6fbf5c": "Z(S_3) = \\frac{1}{6} \n\\left( a_1^3 + 3 a_1 a_2 + 2 a_3 \\right).", "37613b9f38edb711b7b71af3652996a3": "\\scriptstyle\\vec{ab}", "376153327c956d1ed11f7c8fa69d9590": "T = 4\\sqrt{\\ell\\over g}\\,K\\left( \\sin{\\theta_0\\over 2} \\right)", "3761f92062cfe6e5ba58dae7d6f36b32": "V = x", "37626c8ad482927e1568a02fd5e455ca": "\\scriptstyle v(-15)", "3763125204036fab0c943bd16b3721a7": " \\mu = GM \\,", "37637e29c1a802dece39006e67fd61eb": "j(\\tau) = \\frac{256(1-x)^3}{x^2} ", "37638be4c4ab7f35f56800232a8aad15": "\\sin^2 \\theta + \\cos^2 \\theta = 1.\\!", "37638ce6ec062729d7a016dec1ea9cbd": "\\mathcal{Q}^1_{Hur}(I)", "37639816ad8b6259c337136a53ba0dc9": " F_b = -V_p\\rho_f \\frac{V_t^2}{r} ", "37639baff96d51fee1d80114d322d912": "\\mathbf J\\,,", "3763feaba7196c41315408d1d925b454": " |F(a)|< C\\|F\\|. ", "3763ff1c386791fcbe16ba9bd3580a24": "\\scriptstyle -r", "376434ab2f8274114bd06630bd445f54": "a + n", "3764376dadaf96d55ca69376ff0d7bb3": "\\begin{smallmatrix}K_s \\end{smallmatrix}", "376457871c53fc82fce64d0bffe4c3ac": "p = 1/2 \\pm \\sqrt{1/12}", "3764611c24a675e0db51877eb9a6c4d1": "\\textstyle \\bar{x}_k = \\bar{M}_{\\mathrm e} R^k (e)", "3764c2901bc0653ad6e53aa817fda36a": "L_\\mathrm {L1} = \\sqrt{Z_\\mathrm {i T} Y_\\mathrm {i \\Pi}} \\ e^{\\gamma_\\mathrm L}", "3764fbfeedaa2abbe0347bdbf7901ef5": "k = 3\\,", "37651188c3f3d642b3715f3f0cf818e3": " \\lrcorner ", "376554a98079c75469794ae9610340c3": "\nf_{WC}(\\theta;\\mu,\\gamma)=\\sum_{n=-\\infty}^\\infty \\frac{\\gamma}{\\pi(\\gamma^2+(\\theta-\\mu+2\\pi n)^2)}\n", "3765a25de9c7910be6a31f37b589c6af": "\\alpha > \\sqrt[n]{x_1 x_2 \\cdots x_n}", "3765e2c9d0034399d12e753ade64f7d8": " \\limsup_{z\\rightarrow\\pm\\infty} |\\hat{f}(z)| \\leq \\limsup_{z\\to\\pm\\infty} \\left|\\int (f(x)-g(x))e^{-ixz}dx\\right| + \\limsup_{z\\rightarrow\\pm\\infty} \\left|\\int g(x)e^{-ixz}dx\\right| \\leq \\varepsilon+0=\\varepsilon,", "37663c74097db9811aa1427d80ab305c": "\\left(\\!\\!\\!\\binom{4}{3}\\!\\!\\!\\right) = \\binom{4+3-1}3 = \\binom{6}{3} = \\frac{6\\times5\\times4}{3\\times2\\times1} = 20.", "376645d46a888572ee68f5b936538405": " { e^{i\\theta}+ z\\over e^{i\\theta} -z} = 1 +2 \\sum_{n\\ge 1} e^{-in\\theta} z^n,", "3766f29da5606c93c5b3b9e80d0d0b40": "\\,\\phi(c,\\bar{c}) = c(c-\\theta_M) ~~~ \\textrm{and} ~~~ \\theta_M = \\langle c^2 \\rangle = \\frac{1}{\\tau}\\int_{-\\infty}^t c^2(t^\\prime)e^{-(t-t^\\prime)/\\tau}d t^\\prime,", "3767576b59165ca48c26425fa499fe88": " \\{ |\\phi_n\\rang\\} ", "3767da4261a848eab8ed014158355f67": "\\sum_{n=1}^\\infty 1", "3767fdac1003fab73a74bbd357d2d291": "\\left(\\frac{\\partial y}{\\partial x}\\right)_z\n=\n1\\left/\\left(\\frac{\\partial x}{\\partial y}\\right)_z\\right.", "376813b19013d8625f6fab5e465c3f28": "(32)\\cdot(200) = 80^2 \\equiv (41^2)\\cdot(43^2) \\equiv 114^2 \\pmod{1649}", "37684dfa77ed4677863e434574397d6c": "\\gamma=0.5", "3768524ee8a578203ecc93b223b32438": "\\mathcal{L}_{D} = \\bar{\\psi}_{a}(i\\gamma^{\\mu}_{ab} \\partial_{\\mu} - m\\mathbb{I}_{ab})\\psi_{b}\\,", "3768fe3adb60ac7f72b75e3a81f33e11": "\\sigma_2\\sigma_3=\\sigma_1\\sqrt{-1}", "37693ce1ecf6e16eaf9e2a4b6b0bad02": "(p/q,1/(2q^2))", "37693cfc748049e45d87b8c7d8b9aacd": "23", "37694d861ba53c0c850a70ebb20d4ea8": "f_c \\,", "3769aad9bac16187c143609d8e5efdd0": "d(\\varphi,\\beta) = 1-\\cos(\\varphi-\\beta).", "3769b8f895d88c378fcf69adaff0d15b": "B(T) = X_1B_1 + X_2B_2 + X_1X_2B_{1,2}\\ ", "3769b9e153a006665bc690eb114cbb7b": "1\\leq i \\leq n", "3769be04eb137dfe6f4e189ebe5b5eb0": "\\,y'(t)=-15y(t),\\quad t \\ge 0, y(0)=1. \\qquad \\qquad \\qquad \\quad \\quad (1) ", "376a2095fdca3d20d837b89a7c74145b": " v_1(x) ", "376a5706f6661a21e978fac180093054": "S(a, b) = \\mathbb{Z} \\setminus \\bigcup_{j = 1}^{a - 1} S(a, b + j).", "376a8c21e80c422bd154797aa466f9a7": "(\\forall x \\,\\phi x \\and E!y \\,\\phi y) \\rightarrow \\phi z", "376a919c481f949f0b442e76e09cf23c": "2t+1", "376a9c9a759054bc00e7825649a2684f": "\n\\int_{\\phi(a)}^{\\phi(b)} f(x)\\,dx = \\int_a^b f(\\phi(t))\\phi'(t)\\, dt.\n", "376abc8dc38b29d1f0f8b785622ee341": "\n F(t) = \\text{Re}[ A e^{i\\omega t} + B e^{-i\\omega t}] \\,.\n", "376ac4daa5f0dd60f5b59195e3013974": "\\lim_{x \\rightarrow \\infty}\\left(1 + \\frac{1}{x}\\right)^x = e", "376b074ca702dce2cba994d558d0a8e9": "\\sum_{i=0}^n {m+i-1 \\choose i} = {m+n \\choose n}", "376b22f0b52085daca29d64b182d6c78": "\\exists B: \\frac{E_0}{N} > -B, \\, ", "376b48e82c4feb1fdbe5a6eea573f07e": "M_{B//B}", "376b67db120b72d67f1e3c601d608708": "|V\\rangle", "376b971e765958925bfdfd202656472f": "\\rho^{\\Gamma_A}", "376bd38253328fec24b627161bb9228a": "f^n(U)\\cap V\\neq\\emptyset", "376c71ad25a224838c8e1e6a996b7d1c": "\\mathrm{\\mathbf{x}}", "376cd0b471dc8f7cfd7f082f6c489789": "\\forall a \\in T : \\{ a\\} \\in Con", "376cee160c742e2091d16ba33f2e944e": "2^k + 1", "376cf18ef5b518ce3a341df76380e2ea": "{\\rm T}_{n,p}(\\alpha,\\beta,\\mathbf{M},\\boldsymbol\\Sigma, \\boldsymbol\\Omega)", "376d277469ac3b6cadfa7946d6547ac5": "\\sum_{n=1}^{N} n^2 = M^2\\;", "376d795ac1cce468a68209c06ff60fe9": "\\mathbf{F}\\cdot\\mathbf{n}\\,{d}S = 2\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W\\, dV = \\frac{8\\pi}{3},", "376d9082db3391046216e21f424a8f7d": " \\left\\vert A_{pq} \\right\\vert = \\frac{1}{f(p)f(q)} X + R_{p,q} ", "376d9eef69bfece854b644734dbbf634": "\\frac{m_0}{m_1} = \\exp\\left[\\frac{at}{I_{sp}}\\right]", "376dc3f91e0c61782b038665ead9e387": "n=\\sqrt[3]{\\frac{1}{0.4\\sigma} } ", "376de435676cee75ea7cab9598ed5a71": "o(\\cdot)", "376df5b0c41a4fd3fd0cb22e9f35b5d8": "\\det(D)", "376e138b54166ed1c742834fc56c7ee6": "\\int_{\\mathbb{R}^4}\\operatorname{Tr}[\\bold{F}\\wedge\\bold{F}].", "376e171dee1a9f766e2eb0712107bb4a": "-cI_n =\\frac{1}{x^{n-1}(n-1)}+ bI_{n-1}+aI_{n-2}\\,\\!", "376e39b97d0fb1bbf8fa68d6feeaa180": " - \\tan a \\tan b \\cos C \\;= \\; 1 - \\sec a \\sec b \\cos c", "376e6af15ef7d9d7ae7a7754a682a951": "V = \\frac{1}{3} h(a^2 + a b +b^2).", "376ea97f86a07e2fa36b64c4148e5efd": "\\Delta P = \\gamma^\\circ (\\frac{1}{R_1}+\\frac {1}{R_2})", "376f7f83c85c015bcad7bc50ba6d4651": "\\vec f^n = {1 \\over { 1 - \\epsilon_b^n \\epsilon_f^n }} \\begin{bmatrix} \\vec f^{n-1} \\\\ 0 \\end{bmatrix} \n - { \\epsilon_f^n \\over { 1 - \\epsilon_b^n \\epsilon_f^n }}\\begin{bmatrix} 0 \\\\ \\vec b^{n-1} \\end{bmatrix}", "376fc9a1204a50272a54358491efd617": " F_w = 2 F_s cos \\theta \\Leftrightarrow \\rho A_s L g = 2 \\gamma \\, L \\, z \\, cos \\theta ", "376ff215ac39a62f0f506be81758b228": "k^*_s", "37705de0752d1027f8fc3b3f390c448d": " 1", "37709bd94086a30f1254effd99951289": "R=\\frac{EMV - BMV - C}{BMV +{C/2}}", "3770a42fc684cc32c387923b591414f1": "z_1 = -0.4121345\\ldots + i 0.5978119\\ldots", "37712419931f26f9fe4df2fd6e8be6f6": "E^o = 0\\,V", "37714cc3de22142df739abd4f373acca": "f = \\frac{q B}{2\\pi \\gamma m_0} = \\frac{f_0}{\\gamma} = {f_0}{\\sqrt{1-\\beta^2}} = {f_0}{\\sqrt{1-\\left(\\frac{v}{c}\\right)^2}}", "377164a5bff8c56373a0d668ad6ec1e7": "P(\\boldsymbol{r})", "37716b12cca6797aefec9cad49b2384b": "Z=79", "3771b416f0692b894abda5ef0ce59397": "\\tbinom{5}{0}", "3771f8588f393d309b44af29df886590": " \\scriptstyle \\overrightarrow{k}", "377235871b7995309b6bad3db3c6a9ed": "F_m F_{n+1} - F_{m+1} F_n = (-1)^n F_{m-n}", "37724153e239e9c998c8612e0eff4a84": "\n\\nabla_{x_i}\\mathcal{S}_{\\alpha}\\left(\\left\\{x_i\\right\\}_{i=1}^{n}\\right) = \\frac{e^{\\alpha x_i}}{\\sum_{j=1}^{n}e^{\\alpha x_j}}\\left[1 + \\alpha\\left(x_i - \\mathcal{S}_{\\alpha}\\left(\\left\\{x_i\\right\\}_{i=1}^{n}\\right)\\right)\\right] \\text{,}\n", "37727d55e4f65cce87642b323580d089": "(\\mathit{argument}, \\mathit{label})", "377290bb2625172c9dcdc9557c7e9727": "\\textstyle\\ f_{f_{(p)}}(I)=\\underset{x\\in I}{max}f_{f_{(p)}}(x) ", "37732b0f8909384d5f1568b1db2f322c": "\\vdash A\\to B\\Rightarrow\\ \\vdash OA\\to OB.", "37735b2b7b7b75384895ea72c1953221": " w ( u \\wedge v) = \n\\sum_{i \\sigma(m)", "37cd3fb325f82668680da48b5d0b5938": "\n\\begin{array}{rcll}\nu_t - \\Delta u & = & 0 & \\mbox{ in } \\Omega \\times (0,T), \\\\\nu & = & 0 & \\mbox{ on } \\Gamma \\times (0,T),\n\\end{array}\n", "37cd728221ef256632d19460e33c623a": "c = \\sqrt{gh}\\, \\left[ 1 + (\\kappa\\,h)^2\\, \\frac{4}{3\\, \\pi^2}\\, K^2(m)\\, \\left( 1 - \\frac12\\, m - \\frac32\\, \\frac{E(m)}{K(m)} \\right) \\right],", "37cdcc157308d74dae5614cb3defce43": "\nr_{Q}^{2} = \\frac{Q^{2}G}{4\\pi\\epsilon_{0} c^{4}}\n", "37cde68b3bfd13887ca3f884d6aef472": "\\Lambda^{-1} x", "37cde8ce5d6a97beecff85de099884b5": "\\alpha_l=\\frac 1l\\frac lT", "37ce119dd10330ffc54c3fdc027e32b3": "A = \\pi - A'", "37ce84ad7633805d9f64e65d9871bf7c": "\\rho + p \\ge 0 .", "37ce9ee43cbe4edb9d4719beedac7392": "\\text{Base ohms }=\\frac{\\text{ohms * base kva}}{kv_{L-L}^2 * 1000}", "37cedfe847c4fe614fd37caff9563282": "\\lim_{x \\to -\\infty}{|f(x)|} = +\\infty", "37cf3794e052254f8ce63edd6b85066f": "\\mathbb{P}(X \\geq a) \\leq \\frac{\\mathbb{E}(X)}{a}.", "37cf3f2ea2b3f4827f5e6c7f48d7cd4f": " E_c=\\frac {1}{2}k (\\frac {F_c}{k})^2=\\frac {F_c^2}{2k}", "37cf4c5df4bdd1035d547a646be37c56": "2 z_{0} z_{1}^{\\ast} \\cdot 2 z_{0}^{\\ast} z_{1} + \n\\left( \\left| z_{0} \\right|^{2} - \\left| z_{1} \\right|^{2} \\right)^{2} = \n4 \\left| z_{0} \\right|^{2} \\left| z_{1} \\right|^{2} + \n\\left| z_{0} \\right|^{4} - 2 \\left| z_{0} \\right|^{2} \\left| z_{1} \\right|^{2} + \\left| z_{1} \\right|^{4} = \n\\left( \\left| z_{0} \\right|^{2} + \\left| z_{1} \\right|^{2} \\right)^{2} = 1", "37cf80e91abceb3d73cc5f6c4ebe7495": "C_{yy}(A.!send)=\\phi, C_{yy}(B.!send)=\\phi", "37d00b3aa858f9daf1626e83217bf047": " \\frac{\\partial F}{\\partial n_1}=0 \\qquad \\frac{\\partial F}{\\partial n_2}=0 \\qquad \\frac{\\partial F}{\\partial n_3}=0 \\,\\!", "37d07146e2cc67740573f3e022e942e3": "\\mathbf{w}^{\\mathrm{MRT}}_k = \\sqrt{p_k} \\frac{\\mathbf{h}_k}{\\|\\mathbf{h}_k\\|}, ", "37d0726965cc25ce78b4e7b1f16a21e5": "\\scriptstyle\\Delta f \\,=\\, \\frac{1}{1\\,\\mathrm{ms}} \\,=\\, 1\\,\\mathrm{kHz}", "37d07a6bd713f72e5922f2ea67f97fdd": "\\mathsf{c}", "37d087ebfeec5f7c499fefc7c4d7d358": "Z_n(c) = e^{-c\\beta} \\sum_{s_0,\\ldots,s_n \\in Q} 1", "37d1595a5f3206bbce18d48cfec6adfa": "\\mbox{NPV} = -1000 + \\frac{-4000}{(1+r)^1} + \\frac{5000}{(1+r)^2} + \\frac{2000}{(1+r)^3} = 0", "37d15f847733719b9b02b7465a3ac9d3": "\\ J_{ab} = \\frac{1}{2} (E_+ - E_-) = \\frac{J_{ex}- CB^2}{1-B^4}", "37d1b00eaea1baf58489fb680a976a9f": "P \\propto \\frac{1}{\\alpha}\\ ,", "37d1ea21c6bfb33d22d3151cd14e2cba": " z^{ideal}_i=\\inf_{x\\in X}{f_i(x)}\\text{ for all } i=1,\\ldots,k.", "37d22e00aff8c39ccb9209d2d914ca58": "\n p {\\left( r \\right)}=\\frac{Ed}{\\pi a\\left(1-\\nu^2\\right)}\\ln\\left(\\frac{a}{r}+\\sqrt{\\left(\\frac{a}{r}\\right)^2-1}\\right)\n= \\frac{Ed}{\\pi a\\left(1-\\nu^2\\right)} \\cosh^{-1}\\left(\\frac{a}{r}\\right)\n", "37d237e1ae7e3d0dea86ab8ae6d831b8": "m = 20", "37d243f32d111b91db683c7b3ee835e2": "\n\\text{Tr}\\left\\{ \\Lambda\\rho\\right\\} \\geq1-\\epsilon.\n", "37d2b2c16c51054427e437d87119e5e2": "\\mathrm{Sp}(3)\\cdot \\mathrm{SU}(2)\\,", "37d3e7356175640fc78712ec618a71a4": " \\ = m \\mapsto k \\mapsto s \\mapsto (k \\ t \\ s') \\quad \\text{where} \\; (t, s') = m \\, s", "37d425fe89676ca629955e3d02b4742e": "S(t) = 1 - F(t) = [1+(t/\\alpha)^{\\beta}]^{-1},\\, ", "37d44016fcdc3b979e3a58ad48e23fca": "A, B\\in 2^{\\mathbf{X}}", "37d465140ecb8499f8c19e6863e6dd94": "R(u,v)", "37d4673926d6939b439502aef425c481": "a \\cdot \\partial F = a \\cdot \\nabla F", "37d486a5dcee9f862d5d9c96953cf46d": "\n \\begin{array}{lcl}\n P(D|M) &=& \\int P(D|\\theta,M) P(\\theta|M) d \\theta \\\\\n &=& \\int P(D|\\theta,M) dP(\\theta|M)\\\\\n \\end{array}\n ", "37d4a18768a9b8040e919f85a7d3e4b8": "\\frac{d\\theta}{dt}=1 +\\frac38\\varepsilon r^2 -\\frac{15}{256}\\varepsilon^2r^4 +\\mathcal O(\\varepsilon^3).", "37d4a2a39f5ce8099bc6e0ac7a796fe2": "n \\geq 3, n \\neq 6", "37d4aa79feea5f9b2151c54cbde457a5": " \\Delta E / E ", "37d4ca2aecea4ea38bb0647493adc33b": "f(x, y) = \\frac{x + y}{2},", "37d4d4550942478edf631bf63e8df4cf": "75 / (.25 + .75)", "37d4f8cfa8034ee9192dd9e437e5382f": "S_{35}\\,", "37d50b2c732ea6592f470cc2bf0675dc": "\\begin{matrix} {4 \\choose 1}{3 \\choose 3}{45 \\choose 2} \\end{matrix}", "37d54336c73664c5529820113e9b4b33": "\\mathrm{Hol}_p^0(\\omega, U)\\subset\\mathrm{Hol}_p^0(\\omega, V).", "37d57cb816d5311ef7c71f7fbd865f7b": "\n \\operatorname{cost}=-\\sum_{j=1}^N \\sum_{i=1}^{M+1} P^{\\text{old}}(i|s_j) \\log(P^{\\text{new}}(i) p^{\\text{new}}(s_j|i))\n", "37d5a7f871a62dd8ed1e8cbf47232ff5": "\\lambda_{3}= -5.68718 \\,", "37d5c49ab01f1575b45c5a7d55a05e94": "(x + x'\\varepsilon) \\cdot (y + y'\\varepsilon) = xy + xy'\\varepsilon + yx'\\varepsilon + x'y'\\varepsilon^2 = xy + (x y' + yx')\\varepsilon", "37d63692744083c88d2c319aae477713": "\\varphi:M \\longrightarrow W", "37d63e9bbf716322fc99704427c9fdc6": "J = \\nabla v", "37d65bad100a4c6bedd5c4ead1cbee6a": "\\delta Q_N\\approx\\sqrt{\\mathrm{Var}(Q_N)}=V\\frac{\\sigma_N}{\\sqrt{N}},", "37d6e7833bb4b074cfa562effe4326f7": "\nt = RC \\times \\ln(2) \n", "37d6ee24ead42c2a52a75dbf0550ce10": "\\left(\\dfrac{(\\dfrac{2nd}{2}) \\left(\\gamma^2 + (\\dfrac{\\lambda}{d})\\gamma \\left(1-\\gamma\\right)\\right)}{\\gamma n}\\right)", "37d6f4f095e8d3ec2b41bd45f1a3b0c9": "\\operatorname{Ric}(\\xi ,\\eta) = \\operatorname{Ric}(\\eta ,\\xi).", "37d7ca75fdf956dcdfe8f9e595abf087": "5 > 4", "37d7f7699d9999bf5a1439767581e870": "Z_i^2 \\rightarrow \\frac{Z}{Y}", "37d7fd1c0234c63a605600983c5bdb8f": "x=-1.", "37d81161d12b7493c443f9bfa8a6c50d": "\\scriptstyle U^a = \\frac{dx^a}{d \\tau}", "37d867ff0004d32de8a79829f97fc33e": "\\Q ", "37d86b0b9a92f3dbb2092f21ec43acc3": "g_i\\gg N_i", "37d887582f8de95c11db2da5c2ddbf5f": "\\mathcal{L}=-b^2\\sqrt{1-\\frac{E^2-B^2}{b^2}-\\frac{(\\vec{E}\\cdot\\vec{B})^2}{b^4}}+b^2", "37d898a5c9400843fc6bd5e9ebb50dd0": "m = \\{ p_1, p_2, ..., p_n \\} \\in M", "37d8aa1a8e8c41d98593e6f061d59a14": "w \\in W", "37d8e72ae1170b9e7cd720973aaed83b": "(1-(1-0.00245)^n)", "37d9125a4851aadb6201f2844a3b180a": " T_t = e^{t \\, {d}/{dx}}. \\quad ", "37d9dc30f34789341c1ed673f226a5e4": "(f,b) \\colon M \\rightarrow X", "37d9e328b24ee5601a25c0923dc30db3": "\\sqrt{2n}", "37da5edf7e3381fc6646588709858946": "\\begin{align} \\\\\nR_Y(\\theta) =\n\\begin{bmatrix}\n\\cos \\theta & 0 & \\sin \\theta \\\\\n0 & 1 & 0 \\\\\n-\\sin \\theta & 0 & \\cos \\theta\n\\end{bmatrix}\n\\end{align}\n", "37db728c8ed330c1413e4cd3bdce1d74": " \\langle \\varepsilon | \\varepsilon' \\rangle = \\delta (\\varepsilon-\\varepsilon')\\,.", "37dbbd5a44a74cb4767ed6011f8e5704": "h(x)-h(y) ~\\bmod~ m", "37dc09a2b334eb9f45b15d240ba67472": "\\mu_1", "37dc9a574640d6454b408990346044d5": " \\cfrac{\\Gamma, A \\vdash \\Delta \\qquad \\Sigma, B \\vdash \\Pi}{\\Gamma, \\Sigma, A \\or B \\vdash \\Delta, \\Pi} \\quad ({\\or}L)\n ", "37dcb2a830ecc1127e5039e6cdab566a": " \\frac{\\mbox{Trace}(UU^\\text{H})}{N}=\\frac{\\sum_{j=2}^N|\\lambda_j|^2}{N}=1, \\mbox{det}(UU^\\text{H})=\\prod_{j=1}^N |\\lambda_j|^2=1. ", "37dd1324e2bdf9fc7a5bff79ce0e1052": "D \\sim \\ell,\\ T \\sim t,\\ V \\sim \\ell/t.", "37dd7b72953b9a08dd23cb9e30be46d3": " \n\\int _X {\\bar f} \\; d \\langle E(B) h_1, h_2 \\rangle,\n", "37dd9fcb868668ceaa0a58a64905f1f9": "m_v \\circ s = s' \\circ m_e", "37ddf695cdbcbac27c2bc1a7ae299b96": "\\mathbf{x}_A", "37ddfb2224664fbef3e4b0f9d7bc8866": "g_{t}", "37de38c59359db02be94b26ce8495e9d": " K_2 = \\frac{d_2 + k_2}{a_2[W_T]} = \\frac{K_{M2}}{[W_T]}. ", "37de794dc642328ecb9448638ebf4cb9": "\\mathcal{A}(i_{U,U\\cup V})", "37dee75f2439230cba9047737eba8771": "\\lnot", "37df076bdeb41a43485ed7c8cb6e1e8f": " \\hat{\\Sigma}", "37df590695e059ea75667f441a500808": "AR = {b^2 \\over S}", "37df75eab372edd6809d82110f5e834d": "(x-a)H(x-a)\\,", "37df7ed4bc471b318be0ff8f664a404c": "0 \\to \\bigoplus_{i + j = k} H_i(X; R) \\otimes_R H_j(Y; R) \\to H_k(X \\times Y; R) \\to \\bigoplus_{i + j = k-1} \\mathrm{Tor}_1^R(H_i(X; R), H_j(Y; R)) \\to 0.", "37df7ed595d326146a238dd2156e7f81": "\\gamma(u) = \\gamma u \\gamma^*\\,", "37dfc9951cdf64d1ec6cbe27e2bdcc76": "t_n=\\begin{cases}2P_n^2&\\mbox{if }n\\mbox{ is even};\\\\H_{n}^2&\\mbox{if }n\\mbox{ is odd.}\\end{cases}", "37dfcfc70910f6ff30896e6ca8d6124a": "g(\\mu)", "37dfdb5bb9e684136bdb60921cc4eb76": " \\frac{1}{3} \\sqrt{2}\\, s^3", "37e01aa071925fb7364763a7df7a4ed2": "b_i\\to 0_x\\in B", "37e03ea899921edee3d7fd09d94e1da7": "\\begin{align}\\operatorname{arcsch}\\, x = \\operatorname{arsinh} \\frac1x & = x^{-1} - \\left( \\frac {1} {2} \\right) \\frac {x^{-3}} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {x^{-5}} {5} - \\left( \\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} \\right) \\frac {x^{-7}} {7} +\\cdots \\\\\n & = \\sum_{n=0}^\\infty \\left( \\frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \\right) \\frac {x^{-(2n+1)}} {(2n+1)} , \\qquad \\left| x \\right| > 1 \\end{align} ", "37e0993ffc765b6133f8ad72ee01afae": "v_{e^{ }}", "37e09b032c722e8066fe83637a39a441": "b_0 = c_{1}v_{1} + c_{2}v_{2} + \\cdots + c_{m}v_{m}.", "37e0c1ee62953f5a248899d63fb05ad9": "\\mathfrak{e}_{8(-24)}", "37e0cb4dca22f5edc8d11abcf79926f3": "e = 4.80 \\times 10^{-10} \\mathrm{esu}, m_e = 9.11 \\times 10^{-28} \\mathrm{g}, c =3.00 \\times 10^{10} \\mathrm{cm/s} \\, ", "37e0d890e644ad10a98f6d21ea9a74db": " F(t)=E_q [G(\\tilde{s},\\tilde{x},\\tilde{u})]-H[q(\\tilde{x},\\tilde{u}\\vert \\tilde{\\mu})] =-\\ln p(\\tilde{s}\\vert m)+D_{KL} [q(\\tilde{x},\\tilde{u}\\vert \\tilde{\\mu})\\vert \\vert p(\\tilde{x},\\tilde{u}\\vert \\tilde{s},m)] ", "37e14740072c5c41b4612f14bfef74e2": "f: X \\to \\mathbb{R}", "37e18c3d4f2dc52ecca059239e632f72": "\\displaystyle{(1-\\delta)^2={|\\alpha +\\alpha^{-1}|\\over |\\alpha| +|\\alpha^{-1}|}},", "37e192648b57c434baf6d9e0b4c385e5": "\n\\begin{align}\n\\lambda_i & = \\mathbf{v}_i^T L \\mathbf{v}_i \\\\\n& = \\mathbf{v}_i^T M^T M \\mathbf{v}_i \\\\\n& = (M \\mathbf{v}_i)^T (M \\mathbf{v}_i). \\\\\n\\end{align}\n", "37e1f9b538c7740371a58048962bde84": " p_B ", "37e234a2cff6849469d24dc5dc0d5396": "\n\\psi = \\sqrt{Z_2} \\psi_r \\;\\;\\;\\;\\;\nA = \\sqrt{Z_3} A_r \\;\\;\\;\\;\\;\nm = m_r + \\delta m \\;\\;\\;\\;\\;\ne = \\frac{Z_1}{Z_2 \\sqrt{Z_3}} e_r \\;\\;\\;\\;\\;\nwith \\;\\;\\;\\;\\; Z_i = 1 + \\delta_i\n", "37e256c3e4f187d0e2ac536446e15e9e": "\\boldsymbol{\\mu} = \\boldsymbol{\\mu}_e + \\boldsymbol{\\mu}_N = -e\\sum\\limits_i \\boldsymbol{r}_i + e\\sum\\limits_j Z_j \\boldsymbol{R}_j.", "37e2a697fe6b98bfc1070f1001294c4f": "\\chi^2 \\left (p - 1 \\right )", "37e2c48d507da874e7bf049896884d84": "\\mathbb{B}", "37e2e11bd6a33d9cb21ff1ae5994a720": "K_{ab}", "37e33a52f648b76ba57f278cb0c18b50": "Initiates(a,f,t) \\leftarrow \\cdots", "37e37c626849aebd87e94c559af02a87": " GeneralFormula.truckloads:Nec.intake(Calories)/Max.trans.capacity(Calories/Truckload)=Truckloads.Nec.Intake ", "37e3c3d275699df2883bdc76d7a1b110": "\n\\begin{align}\n\\tan x & {} = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1} 2^{2n} (2^{2n}-1) B_{2n} }{(2n)!}\\; x^{2n-1},\\,\\, \\left |x \\right | < \\frac {\\pi} {2}\\\\\n\\tanh x & {} = \\sum_{n=1}^\\infty \\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}\\;x^{2n-1},\\,\\, \\left |x \\right | < \\frac {\\pi} {2}.\n\\end{align}\n", "37e3d7ea9301e77e878623f1df3d28d2": "d z = {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y \\, d x + {\\left ( \\frac{\\partial z}{\\partial y} \\right )}_x \\,dy.", "37e40946cfc7f6b3b58f4f1d729999c4": "\n \\cfrac{\\cfrac{\\cfrac{{}}{A \\ true} u \\quad \\cfrac{{}}{B \\ true} w}{A \\wedge B \\ true}\\wedge_I}{\n \\cfrac{B \\supset \\left ( A \\wedge B \\right ) \\ true}{\n A \\supset \\left ( B \\supset \\left ( A \\wedge B \\right ) \\right ) \\ true\n } \\supset_{I^u}\n } \\supset_{I^w}\n ", "37e40a232925388fc5067662f7fba306": "{\\lambda}\\to 1", "37e4504746558b652c3fda0f3fea5154": "\\begin{bmatrix}\nx_0^n & x_0^{n-1} & x_0^{n-2} & \\ldots & x_0 & 1 \\\\\nx_1^n & x_1^{n-1} & x_1^{n-2} & \\ldots & x_1 & 1 \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\\nx_n^n & x_n^{n-1} & x_n^{n-2} & \\ldots & x_n & 1\n\\end{bmatrix}\n\\begin{bmatrix}\na_n \\\\\na_{n-1} \\\\\n\\vdots \\\\\na_0\n\\end{bmatrix}\n=\n\\begin{bmatrix}\ny_0 \\\\\ny_1 \\\\\n\\vdots \\\\\ny_n\n\\end{bmatrix}.\n", "37e48302eda55a840ea6acf0002b33d9": " f(z + \\omega) = f(z)+az+b \\ ", "37e4aeab0759386cca27403844de9e02": "\\displaystyle \n\\frac{\\partial u}{\\partial t} = r u - (1+\\nabla^2)^2u + N(u)\n", "37e4b8a43fcaf49f42bacf159be54680": "\\rho^{T_A} = (\\rho^{T_B})^T", "37e4ca31523b5859a386ab3abc6880af": "\\ f^T x \\ ", "37e508e08b8b1e1d10f4890176f960e6": "\n\\begin{align}\n D_\\text{fitted}- D_\\text{null} &=-2\\ln \\frac{\\text{likelihood of null model}} {\\text{likelihood of the saturated model}} -\\left(-2\\ln \\frac{\\text{likelihood of fitted model}} {\\text{likelihood of the saturated model}} \\right)\\\\\n &=-2 \\left(\\ln \\frac{\\text{likelihood of null model}} {\\text{likelihood of the saturated model}}-\\ln \\frac{\\text{likelihood of fitted model}} {\\text{likelihood of the saturated model}}\\right)\\\\\n =& -2 \\ln \\frac{ \\left( \\frac{\\text{likelihood of null model}}{\\text{likelihood of the saturated model}}\\right)}{ \\left( \\frac{\\text{likelihood of fitted model}}{\\text{likelihood of the saturated model}}\\right)}\\\\\n =& -2 \\ln \\frac{\\text{likelihood of null model}}{\\text{likelihood of the fitted model}}.\n\\end{align}\n", "37e516eb1688fdd9b77844ecfbfb6719": "C_1 + C_2 + C_3 + \\cdots\\,", "37e57f9678b33ad590ab6e8d1186b9b2": "\\textstyle W^n: X^n \\times", "37e59244202153fe9aeb2e3a239e3e63": "R_F(x,y,z) = \\tfrac{1}{2}\\int_0^\\infty \\frac{dt}{\\sqrt{(t+x)(t+y)(t+z)}}", "37e59da5affe06505974223fee6d18b0": "y = A(1 - \\cos \\phi) ", "37e5c93da4be07069236da73bf75032f": "B_1(N, r, \\mu) = \\frac{1 + \\sum_{n=1}^{N-1} \\frac{N-n}{N(N-1)}\\left [ 2\\left (rn\\right )^{\\mu+2} - \\left (rn+1\\right )^{\\mu+2} -\\left |rn-1\\right |^{\\mu+2}\\right ]}{1 + \\frac{1}{2}\\left [ 2r^{\\mu+2} - \\left (r+1\\right )^{\\mu+2}-\\left |r-1\\right |^{\\mu+2}\\right ]}.", "37e5e2322916e6cd354d326d857bdf14": " p_0 (x_1, x_2, \\dots,x_n) = n ,", "37e5e4a10c421aecf3b3ddbf224176fb": "\\{a\\}\\times B\\subset U'_a\\times V'_a\\subset N", "37e65ade1119882e9f830c72a1669b7a": " Ext(w,x) ", "37e65fca526d587f13d6ef11297bb468": "V \\otimes W \\cong W \\otimes V.", "37e6b3f7269b61845fc0220619fcacf5": "u=e/e_0 + h(\\phi)/2", "37e6cd4eb77587de8e20241b3cee093a": "\ne\\left(\\widetilde{X}\\right) = \\left(\\frac{1}{N}\\right) \\left(\\frac{\\pi}{2N}\\right)^{-1} = 2/\\pi \\approx 64%.\n", "37e7041e481d86c03307a2ac97a8afb9": "\\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{\\partial \\beta^2} = -N\\frac{\\part^2\\ln \\Beta(\\alpha,\\beta)}{\\partial \\beta^2}<0", "37e71c45d6321e4befebf4a5c8f45613": " \\Pi(k_i) \\propto k_i(t-t_i)^{-\\nu},", "37e720d18b291d5cd2e453345475297a": "{{z}_{in}}=\\frac{2kT}{q{{I}_{IN}}}", "37e797b7cebf8ba75a9731a031192302": "+(22639.55-4586.45)''\\sin(l)", "37e7de2e1db43adf181b1d698a4c7535": "n^{2/3}", "37e8748fc43cdaa438f4185f78d1bb98": "\\left(D^2-k^2\\right)\\Psi=0,\\,\\,\\,\\ D=\\frac{d}{dz}.", "37e899226c2ac7de7494ea666bba71b1": "f'(t)=1+i\\sinh(t) \\qquad r'(t)=\\cosh(t).", "37e8a25c0efb6aff85a2990073bd64d6": "[v'] = B^{-1}AB[u'] = A'[u'].", "37e8aaef326b7a2e694347790cf6503d": "(x^\\lambda, \\dot x^\\lambda) ", "37e8ed29e77b21d896e9ce4923b69ef5": " -n_i S_F (eZ_i/M_i ) = C_i ", "37e9537858269739f05d7928d23d2617": "\\begin{align}N&\\in \\left\\{0,1,2,\\dots\\right\\} \\\\\n K&\\in \\left\\{0,1,2,\\dots,N\\right\\} \\\\\n n&\\in \\left\\{0,1,2,\\dots,N\\right\\}\\end{align}\\,", "37e955184aa973da6e85e8ce2d8cd491": " K_p = \\frac{\\prod_{j=1}^p p\\left ( {\\rm Y}_j \\right )^{\\eta_j}}{\\prod_{i=1}^r p\\left ( {\\rm X}_i \\right )^{\\nu_i} } \\,\\!", "37e9a11ca058217d0fa56e3aa4eea17f": " \\Psi = \\sqrt{\\rho(\\mathbf{r},t)} e^{iS(\\mathbf{r},t)/\\hbar}\\,\\!", "37e9d329e64ba57402742152e002abee": "\\textstyle\\int_a^b f(x)\\,dx = F(b)-F(a).", "37e9de606e2afbd9cee0fbe769978822": " CGPA \\,\\! = {\\sum_{i=1}^N C_i . {GP}_i \\over \\sum_{i=1}^N C_i}", "37ea3df962c28561e0d5c382c24abde8": "\nG_\\mathrm{dB} =20 \\log_{10} \\left (\\frac{V_1}{V_0} \\right ) \\quad \\mathrm \\quad\n", "37eae6077b8bb89afae0db293bf3a3b2": "C = \\{ c_1, . . . , c_c \\}", "37eae7acc9bbfc11978d8a81f9bb05af": "{0^2 \\over 2}+g(0)+{P_\\mathrm{atm} \\over \\rho}={v_B^2 \\over 2}+gh_B+{P_B \\over \\rho}", "37eafc0ac8da42b2a079b39314654bab": "\\ M_{heel_{max}} ", "37eb0ceb0f99643973ed67fe9084736c": "\\begin{align}\\int |\\alpha\\rangle\\langle\\alpha| \\, d^2\\alpha\n&= \\int \\sum_{n=0}^{\\infty}\\sum_{k=0}^{\\infty} e^{-{|\\alpha|^2}} \\cdot \\frac{\\alpha^n (\\alpha^*)^k}{\\sqrt{n!k!}} |n\\rangle \\langle k| \\, d^2\\alpha \\\\\n&= \\int_0^{\\infty} \\int_0^{2\\pi} \\sum_{n=0}^{\\infty}\\sum_{k=0}^{\\infty} e^{-{r^2}} \\cdot \\frac{r^{n+k+1}e^{i(n-k)\\theta}}{\\sqrt{n!k!}} |n\\rangle \\langle k| \\, d\\theta dr \\\\\n&= \\sum_{n=0}^{\\infty} \\int_0^{\\infty} \\sum_{k=0}^{\\infty} \\int_0^{2\\pi} e^{-{r^2}} \\cdot \\frac{r^{n+k+1}e^{i(n-k)\\theta}}{\\sqrt{n!k!}} |n\\rangle \\langle k| \\, d\\theta dr \\\\\n&= 2\\pi \\sum_{n=0}^{\\infty} \\int_0^{\\infty} \\sum_{k=0}^{\\infty} e^{-{r^2}} \\cdot \\frac{r^{n+k+1}\\delta(n-k)}{\\sqrt{n!k!}} |n\\rangle \\langle k| \\, dr \\\\\n&= 2\\pi \\sum_{n=0}^{\\infty} \\int e^{-{r^2}} \\cdot \\frac{r^{2n+1}}{n!} |n\\rangle \\langle n| \\, dr \\\\\n&= \\pi \\sum_{n=0}^{\\infty} \\int e^{-u} \\cdot \\frac{u^n}{n!} |n\\rangle \\langle n| \\, du \\\\\n&= \\pi \\sum_{n=0}^{\\infty} |n\\rangle \\langle n| \\\\\n&= \\pi \\hat{I}.\\end{align}", "37eb24d7d733fd349c495213ec9993ad": "x+y=55, xy=16", "37eb3f8cf0d4f1018d780e0cd90abd68": "P_{N,N}=1", "37eb620534abdd5a12549b1d3c7f9a2b": "R(n,k)", "37eb70644e91fd061cda742258549a76": " E\\left \\{ \\mathbf{x} \\mathbf{x}^{T} \\right \\} = \\mathbf{E}\\mathbf{D}\\mathbf{E}^T", "37ebf0dcdf5ac38d109c5a06ed44d0b7": "V^{\\otimes d}", "37ec2a86b54381dcc048dd70fa9964db": "F(y,x)", "37ec31c20f8289d8b3d7a9b2de7c6ed9": "\n\\left\\{c_i , c_j \\right\\} = 0 \\quad,\\quad\n\\left\\{c_i^\\dagger , c_j^\\dagger \\right\\} = 0 \\quad,\\quad\n\\left\\{c_i , c_j^\\dagger \\right\\} = \\delta_{ij}.\n", "37ec46bc458fcfd6451f00798c251268": "\n 2\\Big[ (\\overline{Z}-\\overline\\mu)'\\overline{P^{-1}}(Z-\\mu) -\n \\operatorname{Re}\\big((Z-\\mu)'R'\\overline{P^{-1}}(Z-\\mu)\\big)\n \\Big]\\ \\sim\\ \\chi^2(2k)\n ", "37ec5fc47f31fad6708108cf40ad6cfc": "=\\mathbf{J} + \\varepsilon_0 \\frac{\\partial \\mathbf E}{\\partial t} \\ ,", "37ec6f5ad65423ea54171dac5a746b32": "{3 \\choose 3}{4 \\choose 2} = 6", "37ec7a122139e461aea48a62e2233476": "\\ \\cos^3(x) = \\frac{3}{4}\\cos(x) + \\frac{1}{4}\\cos(3x)", "37ec88825fbddb522a1ccf0d8039b955": "-90^{o}", "37ec9468f019dab337f20a4c155e6c7f": "r \\geqslant 0", "37ecbd4f16cb15ead9a63c1c671aec38": "ds^{2} = -\\left(1-\\frac{2GM}{r} \\right) du^2 - 2 du dr + r^2 d\\Omega^2.", "37ece954e37e3ca54fd1b4eb6e894d25": "\\varphi_a", "37ecec6d21f6fb2a4a3d655374628e0a": "\\displaystyle q=\\sqrt{\\frac{f+h}{e+g}\\Big((e+g)(f+h)+4eg\\Big)}.", "37ed0b4f7bf7f783fa461a547a2a58c2": " X \\sim \\mathrm{Weibull}(\\tfrac{1}{\\lambda},1)", "37ed36663e09947bed638ec5e41b2839": "d_{k+1}/d_k \\rightarrow 1", "37ed7a9f9499b6e664b0b24f5fe4ce32": "\\displaystyle{H_{g\\circ f \\circ h} = h^{-1}\\circ H_f \\circ g^{-1}}", "37eda311ad28fbbe3c18544460faf2f1": "P(T\\omega)=P(\\omega)", "37eda52f8f4a66a60b878afe24b4f7ae": "S_{G}(v,r)=\\{x\\in V: d(v,x)=r\\}", "37ede6c5c8e3044eda83301160bb3951": "\\ \\Delta^t(\\Delta^r(\\Delta^s(\\phi_{1,1,1}))) ", "37ede802b35c9b7a7d6a4c47f9acbc2c": " \nL_n=\\Gamma_n= \\Gamma_n^* =\\overline{\\Gamma_n}^*\n", "37ee4aea089269fa2c926ef9ef59b166": " (\\mathbf{a \\times b}) \\mathbf{\\times} (\\mathbf{c}\\times \\mathbf{d}) \\ ,", "37ee65780f4ca6321749c1eb660f1505": " \\frac{b}{c} = \\sin (\\beta)\\,", "37ef0b847ddca313175f73a80bca4065": "\\left\\langle n_i\\right\\rangle\\approx\\begin{cases}1 & (\\epsilon_i<\\mu) \\\\ 0 & (\\epsilon_i>\\mu)\\end{cases}.", "37ef6d4bb19c097d52d61ac09d4949ac": "ax^2+by^2+cz^2=0", "37effd0af7dfe473c83c1563de660285": "\\vec F = -\\frac{c}{12\\pi\\sigma}\\vec \\nabla U,", "37f07d2153aec67fa6eb0b067b4c2790": "w[1] w[2] \\ldots w[L] \\in W^*", "37f0eb181c851621e6420b38e7beba77": "\n1 - p_1 p_2 - p_1 p_3 - p_2 p_3+ p_1 p_2 p_3 = 0\n", "37f0f5d515b8e40b901bdd7904b7780c": "\\delta T\\ ", "37f13de5df3f1c2c57ae2e79fe916ff9": " \\tfrac{1}{t}\\bigl( \\begin{smallmatrix}\\\\ \\pm r&\\mp s\\\\ \\mp s&\\mp r\\end{smallmatrix} \\bigr),", "37f14e7725ae8fec0cacf956a69e3f6e": "x_c", "37f15302b7cf483248ee5c4bba1ce2e2": "\\mathcal{U}(G, N) = \\{f \\in Y^T : f(G) \\subseteq N\\}", "37f1fa74502b2325eefa594d414c2d12": "\n \\boldsymbol{\\varepsilon} = \\sum_{i=1}^3 \\sum_{j=1}^3 \\hat{\\varepsilon}_{ij} \\hat{\\mathbf{e}}_i\\otimes\\hat{\\mathbf{e}}_j \\quad \\implies \\quad \\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\begin{bmatrix} \\hat{\\varepsilon}_{11} & \\hat{\\varepsilon}_{12} & \\hat{\\varepsilon}_{13} \\\\\n \\hat{\\varepsilon}_{12} & \\hat{\\varepsilon}_{22} & \\hat{\\varepsilon}_{23} \\\\\n \\hat{\\varepsilon}_{13} & \\hat{\\varepsilon}_{23} & \\hat{\\varepsilon}_{33} \\end{bmatrix}\n ", "37f24184c5c356493ff44cec5b53b4c8": "{\\mathcal I}_{\\boldsymbol \\eta}({\\boldsymbol \\eta}) = {\\boldsymbol J}^{\\mathrm T} {\\mathcal I}_{\\boldsymbol \\theta} ({\\boldsymbol \\theta}({\\boldsymbol \\eta})) {\\boldsymbol J}\n", "37f27460e16c96f3d4c4d07469337e60": " Eq \\equiv Bqv \\,\\!", "37f28b7e2ec966a7bd622da55dabf287": "\\displaystyle{C(z)=i{1+z\\over 1-z}=-i +{2i\\over 1 - z}}", "37f335c8de6e25ac44ca76a71aed1afc": " x \\preceq z ", "37f345146ce89076913d9554564f12e8": "b_n(t)", "37f38ea9c4210d65ec9dde56a5182aa7": "F_\\bold{X}(x) = \\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{i=1}^n I(X_i \\le x).", "37f3f8155aa78df0d2e8babcdbba5dbb": "Y := \\bigcap_i \\ker \\phi_i = \\ker \\Phi", "37f4a7482f484ff931ba103361f39253": "\n\\begin{align}\nT &= r^2\\left(\\cot(A/2)\\ \\cot(B/2)\\ \\cot(C/2)\\right) = r^2\\ \\frac{s-a}{r}\\ \\frac{s-b}{r}\\ \\frac{s-c}{r} \\\\[8pt]\n&= \\frac{(s-a) (s-b) (s-c)}{r} \\\\[8pt]\n\\end{align}\n", "37f4e5d2a9d3699014f26d4d237cce04": "\\partial_k \\Gamma_k = \\frac{1}{2} \\text{STr} \\,\n \\partial_k R_k \\, (\\Gamma^{(2)}_k + R_k)^{-1},", "37f4e7ef46f1b1f91c4f052787495f4d": "\\bar{x}^j \\bar{\\mathbf{e}}_j = x^i \\mathsf{L}_i{}^j (\\boldsymbol{\\mathsf{L}}^{-1})_j{}^k \\mathbf{e}_k = x^i \\delta_i{}^k \\mathbf{e}_k = x^i \\mathbf{e}_i ", "37f51f6162e5688eb2238ccc7f737543": "\\int^\\infty_0 r^2 |R(r)|^2 \\, dr = 1.", "37f5237ae5f67141cd6869fa20a0c29b": "\\delta(q_s,\\varepsilon) \\to q_t", "37f53320e446898498d7786216c9bacb": " \nE_{act} /{RT_m}^2 = \\sigma_0 v_2/\\beta* e^{-E_{act}/RT}\n", "37f55ad3dae44da1cc2a7a5eca9a22b7": "\\tilde{{\\Pi }}(x,u,\\theta )", "37f5858bf9f6fe8dec563f68fb62a488": "p_{k,i}^{\\mathcal M}={\\mathcal P}[M(s_1\\ldots s_i)=s_{i+1} | s\\in S_k\\text{ with probability }\\mu_k(s)]", "37f593659cc996c4991266398ba13af6": "\\Delta G", "37f5c54a1e37ce72b00bc4e52f3903aa": "R_{x_1,x_2}(z)=X_1^*(\\tfrac{1}{z^*})X_2(z)", "37f5dbbd40ea4b3e37bd2c7232681549": "v = { \\frac{1}{\\sqrt{LC}} } \\ ", "37f5ed18a4361ee0f8f36f32a6bf5976": "\\zeta(6) = 1 + \\frac{1}{2^6} + \\frac{1}{3^6} + \\cdots = \\frac{\\pi^6}{945} = 1.0173...\\dots\\!", "37f63675bf365230b0709162a50d81a8": "P=\\frac{2\\pi}{\\hbar}\\sum_f |M_{fs}|^2 \\delta (E_i + \\hbar \\omega - E_f),", "37f68f8ccc6226bc81de4291a82d8381": "t_1^2", "37f6c40cb53112a58c90bc90bb08d151": "\\langle E(s) \\rangle = \\frac{1}{2} \\sum_n \\hbar |\\omega_n| |\\omega_n|^{-s}", "37f72916b9c791984df1655732b9d724": "E_1^{\\cdot,\\cdot}", "37f790347045bc6a7bb3ef87ed0ff441": "\n\\operatorname{Li}_{s}(z) = \n\n\\int_0^\\infty {t^{-s} \\,\\sin[s \\,\\pi /2 - t \\ln(-z)] \\over \\sinh(\\pi t)} \\,\\mathrm{d}t \\,.\n", "37f7d1e81d1c7ce34ab3366c743794cd": "\\text{if }gV\\subseteq V\\text{ for all }g\\text{ in }G,\\text{ then either }V=0\\text{ or }V=\\mathbb{C}^n.", "37f7df0c0ae521e23edb0776d40dac5e": "NEFD = \\eta \\frac{NEP}{A \\nu}", "37f80b980ebf8e01f027c01498287d8d": "\\sum_{i\\in N}\\phi_i(v) = v(N)", "37f825d3659d687e684351ee9a8b736b": " x_1 = x_2 ", "37f8754ba5df463e386bc69a051f36ca": "\\ge H_q^{ - 1} (\\frac{1}{2} - \\varepsilon )", "37f87cdb7a15ab0e4618c321679c22fd": "H_k(S) = Z_k/B_k\\, .", "37f8c1d068ee40cc4719dbf73db61268": "h_c=\\sqrt{pq} ", "37f8d024276ea91bd5ae327bc56e39e7": "\\mathbf{w}\\cdot\\mathbf{x}_i - b \\le -1\\qquad\\text{ for }\\mathbf{x}_i ", "37f8eaea284c67c93be34cb60e5d9881": "S_c = \\left [ - \\beta , \\beta \\right ]\\,", "37f965da69954e3a744cda526a36c46b": "A \\leftarrow B_{1},\\dots,B_{m},\\hbox{not } C_{1},\\dots,\\hbox{not } C_{n}", "37f96a9602a5c292e5583f07a12c924f": "w'_i = 2^{(n+1)} w_i + 2^i", "37f9aa0747ea2354739a640c44d04cd8": "k_2/k_1=10", "37f9bf656e812e25243be1f93fbc4cde": "f(\\omega)", "37f9ce13dcdec78ae58ab9433f530708": "\n\\mathcal{Z} = \\mathrm{tr} \\left\\{ \\prod_{k=1}^{N} \\mathbf{W}_{k} \\right\\}\n", "37f9dbaee0508a223fdbbcb02d39fddc": "r_x \\,", "37f9e098b382a125717ef0035a967b0c": "h\\to \\int_c h", "37fa0655ff5d3cee289b2e7a15f454bd": " \\mathrm{div} V = V^\\alpha {}_{;\\alpha} = 0 ", "37fa480193158ba218baaab09a78a340": "\\left(3\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ 0\\right) \\pm \\left(0,\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "37fa6672907bef8a177b3dddb43a0eec": "\n\\gamma_n=\\int_\\mathcal{C} d\\mathbf R\\cdot \\mathcal{A}_n(\\mathbf R)\n", "37fa66e08aa415b333bc45df8b29f8b9": "\\Sigma _{XX} = \\operatorname{cov}(X, X)", "37fa93c040c22929dbfb442d47eed891": "f = (f_l : A_l \\rightarrow B_{l+n})", "37fae9170aad8af00ce57259fbbfe5cb": "[C_{Op} | x_{1} = v_{1} \\land \\dots \\land x_{n} = v_{n}]", "37fb5b55b3b90624f01bbab5a98c5071": "\\|f\\|_1 = \\int |f|\\,d\\mu", "37fb9954e667423d31e8cc877e2c5256": " (\\kappa+n+1)~r^{n+1}~\\sin(n\\theta) \\,", "37fbf6771c8985253295082ac78ab46b": "\\sec^2(x) - \\tan^2(x) = 1\\ ", "37fc15caebf4b63f75ac3febad1f6ddb": "\\vec a_0=A(\\vec x_0)", "37fc1bba4ef127b31fd78f1d241d4b13": "\\mathbf{y}(t) = (y_1(t),...,y_m(t))", "37fc5cbde8c0ca6622f294c2c7d7009a": " D_{F}(x-y) = \\int \\frac{d^{4}p}{(2\\pi)^{4}} \\frac{i(p\\!\\!\\!/ + m)}{p^{2}-m^{2}+i \\epsilon}e^{-ip \\cdot (x-y)}", "37fc642518650b4227803609141fc120": "\\ v_a", "37fc6e0e34b74568623625f2131fe1f1": "GNI=GDP+(FL-DL)+NCI=C+I+G+(X-M)+(FL-DL)+NCI", "37fc8a8ca397849496d4ca084f909822": "|\\lambda|\\ne 1", "37fd6ce21ba852a585f7f0c65eceb19d": "\\mathbb Q", "37fdbbb1effb41debec95aaf8a4c8fa5": "C^*=\\bigcup_{S\\in C}S.", "37fdfa08e3af2c8a194b7500e6d5e41b": " (Tf)(u) = \\int \\limits_{t_1}^{t_2} K(t, u)\\, f(t)\\, dt", "37fe483906dad542c334d6226bf9cb45": "1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots", "37fe690f58fe1eef959c712567a37082": " F = q\\mathcal{F}\\cdot v", "37fe924910f8b9551489af5fb1655b23": " Z(S_n) = \\frac{1}{n!} \\sum_{g\\in S_n} \\prod_{k=1}^n a_k^{j_k(g)}\n= \n\\frac{1}{n!} \n\\sum_{l=1}^n {n-1 \\choose l-1} \\; \\frac{l!}{l} \\; a_l \\; (n-l)! \\; Z(S_{n-l})\n", "37fe9e4554867cd6acf99f390a430c08": "[S:T]=[0:1]", "37ff257fc19be1c4c20406802bfd7b87": "B ' \\subseteq B", "37ff40bc7f980968907bdd0976ad4af2": "n \\in \\mathbb{N};", "37ff54fb303c5a5630d54758e52b29d7": "\\Omega^\\mathrm{spont}_{\\mathbf{k},\\omega}", "37ff997bc0eafe1856099b2f7cf1fd14": "\\ Q", "37ffad85509d4b58c0ffe2de07c59fc5": "u = \\left( \\begin{matrix} \\alpha & \\gamma \\\\ - \\gamma^* & \\alpha^* \\end{matrix} \\right),", "37ffc3b3ce70f227cd63ca9bd5b40533": "\\sigma_{\\text{realised}}^2 = \\frac{A}{n} \\sum_{\\text{i=1}}^{\\text{n}} R_{\\text{i}}^2 ", "3800ac3fabb2f6c3bee18dcba92bb23c": "BaZn_2L(ClO_4)_2", "3800ef960fa6adeca275ee31d5dd50f5": "f \\circ\\phi", "38012bdb2dc08193736ab609d6b766aa": "\\mathbb{F}_p^*", "3801e4b7b974f30c222a16bbcdef509b": "G(z) = \\textrm{e}^{\\lambda(z - 1)}.\\;\\,", "38021e087284ee0c172f69e3e44bcc6d": "f(t) = t^3 \\,", "380229c5d779ee87bd65de66bd414823": "P\\left(n\\right) = {p^n} + {q^n} + {r^n}. ", "380232719a3fa592a838a905dac2aa14": "H = x \\cdot \\dot{y} - y \\cdot \\dot{x}=a \\cdot b \\cdot ( 1 - e \\cdot \\cos E) \\cdot \\dot{E}", "380284e659ef1534563228b6d438ca8b": "\\mathrm{d} \\tilde{B}_t = \\sgn (X_t) \\, \\mathrm{d} X_t.", "3802a9b06cee49da9bbf0ace69a81b3b": "\\bar{N}", "3803705fb9d2330eb9a569b318ec21f9": "x, y, z \\in M", "38037269b104b5e9a0a49b747bb4decd": "\\langle [\\delta \\mathbf{u}(r)]^n \\rangle = C_n \\varepsilon^{n/3} r^{n/3}", "38037300c9a9843b81afee9435069b13": "b_3 = 2, b_2 = 0, b_1 = 2, b_0 = 5 ", "38039fd132d48b54dde15cfa25ee9632": "\\begin{bmatrix}\n0&1&-1&0 \\\\\n0&0&1&-1 \\\\\n0&0&0&1 \\\\\n\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}\\\\\n\\end{bmatrix}", "3803d294c9911ddd6cafcf30a42f3b57": "\\left| S \\right|", "3803f2c0be641eee0d4b0e81155b8f90": "h_{\\mathcal{T}}(t) = t\n\\cdot h(t)", "380474d6aee633135fb3d9cc8545bb03": "p\\!\\,", "38047ef0db5db953639926b983eb134d": " \\mathrm{isotopic \\ ratio} \\ (\\mathrm{i:j}) = \\mathrm{isotopic \\ percent} \\ (\\mathrm{i}) : \\mathrm{isotopic \\ percent} \\ (\\mathrm{j}) \\ .", "3804841a62a3de37a566d60d7d614568": "\\beta\\mapsto\\omega^\\beta", "3804bd444fc4fde68f81025421374b0c": " \\frac{\\pi}{4} = 4 \\arctan \\left(\\frac{1}{5} \\right) - \\arctan \\left(\\frac{1}{239} \\right) ", "3804d40da913cc7832d1547bcb80dec6": "\\iint_D (x+y) \\, dx \\, dy = \\int_0^1 dx \\int_{x^2}^1 (x+y) \\, dy = \\int_0^1 dx \\ \\left[xy + \\frac{y^2}{2} \\right]^1_{x^2}", "3804f4ad2fc5e3d2a6dd9615d7a940d0": "\\theta = \\{\\theta_1, \\theta_2, \\dots, \\theta_n\\}", "3805511b33a03bb146b9790a1a6f6a68": " \\kappa:= -m^aDl_a=-m^a l^b \\nabla_b l_a\\,,\\quad \\tau:= -m^a\\Delta l_a=-m^a n^b \\nabla_b l_a\\,,", "380579c10625cc3294e9afe4527e0d6d": "p_1, p_2, p_3, \\ldots, p_{32}", "3805934cfdbaf0778a2b942aa190d8ab": "\\left( 7,24,25 \\right)", "3806ac0f2804e392fc07ff30a3bdd8d4": "\\frac{|\\operatorname{aut}(M)|}{\\sum_i|\\operatorname{fix}(M,Q_i)|}", "3806b02f3c6ba3211ccd23b4f40ddf08": " d_\\lambda (E(\\lambda)\\eta_z^{(i)},\\eta_z^{(j)}) = |\\lambda - z|^{-2} \\cdot d_\\lambda \\sigma_{ij}(\\lambda).", "3806f5463e540487c0c37f26f96e005b": "\\boldsymbol\\tau = I_C\\alpha\\vec{k}.", "3806ff188e1447e5afbb0cefa3f25776": " \\int\\!\\!\\int |f(z)g(z-x)|\\,dx\\,dz=\\int |f(z)| \\int |g(z-x)|\\,dx\\,dz = \\int |f(z)|\\,\\|g\\|_1\\,dz=\\|f\\|_1 \\|g\\|_1.", "380705a45df95d2971dfca86449aea01": "M_{M_p} = 2^{2^p-1}-1", "380708a46049d1559dffd58718507385": "\\ M=E-\\varepsilon\\cdot\\sin E", "38073430ef9405ac2c4af288dcb970fd": "F_{22}", "38077c9acb0d6448eaae44a31d0bd0d6": "G_{liquid} < G_{solid}", "3807b0519c510a8d94764e2ccc1116b5": "\\Phi_b(z)=\\exp\n\\left(\n\\frac{1}{4}\\int_C\n\\frac{e^{-2\\sqrt{-1} zw }}\n{\\sinh (wb) \\sinh (w/b) }\n\\frac{dw}{w}\n\\right)", "3807e2ac22cd3a80af3f4da0eeafe2a5": "\nV(t) = V_0e^{-t/\\tau} +A \\tau \\left( 1 - e^{-t/\\tau}\\right).", "3807e757ee0b1d26e929e7d8bfe11ac6": "\\neg Abnormal(...)", "38081df8178ce41c0482783a56be21f5": "a\\ge 0.8", "3808c3a0a62554f6caf38e4ac8693a18": "V = x^3 + ax\\,", "3809025dfeab6201685149e85d462eee": "E_D = E_D(q_{0,D}) + 3 f_D(\\Delta q_D)^2", "38091abd587ee6a3bf85b78bbdd8f994": "f(x,y)= x^2+y^2(1-x)^3,\\qquad x,y\\in\\mathbb{R},", "38097de95833510e3ada339469ed518d": "\\Phi(h) = \\int_0^h g(\\phi,z)\\,dz\\, ,", "3809838ab030ea446cf6294aaa63ade0": " \\mathcal{L} = p_1 \\partial_x + p_2 \\partial_y ", "380a30baf64656d0026b07be16e7d06d": " \\langle v_{i,j}, v_{k,l}\\rangle = -(1-t)(1+qt)(q-1)^2t^{-2}q^{-3}\n\\left\\{\n\\begin{array}{lr}\n-q^2t^2(q-1) & i=k \\frac{N}{S(0)} ,", "380dbdc957b19f6fcec42cbb0bf1aaf3": "Y^TA_jY", "380e60629701fe7d2ea2e80a68fd52aa": "\\left\\langle 10, Z_{10} \\right\\rangle", "380e74c19188eacdebf657be3813f8bc": "b\\rightarrow a", "380ea7bb80496fa02cf365e6adc50489": "\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\end{pmatrix}\n", "380f2d83fdcb13145e0b5b7f8f3c1de6": "\\scriptstyle{X}\\,", "380f40674dcc13ab964f970e5c5750bb": " V", "380f4499d942e9844a1a4f6f54dcc121": "\\textstyle H_1: \\theta \\neq 0.5", "380fb8847117d77cff05c024effb7f3c": "\\pi \\oplus \\sigma = \\operatorname{rev}(\\operatorname{rev}(\\sigma) \\ominus \\operatorname{rev}(\\pi))", "380fd6794845378461dde9d202acaa5f": "\\mathcal{F} \\cup \\mathcal{G}", "380fec2d6e0151815fb13482f7d2b8c3": "f(2) = v", "38100adf3140fab9a226240ff9fc93df": "\nd^{(k)}(z):=(\\downarrow 2)(b^*(z)\\cdot s^{(k+1)}(z))\n", "38106a5fb447faa69b36b5db061dbf45": "2\\mu e^{\\mu\\pi} \\ge 1", "381071d3d2b058af79c7a460b04fdf39": "\\scriptstyle P^*", "381113bdd1ae21a91f41f3b5bd89c78d": "\n\\begin{cases}\nx^2-1=0\\\\\n(x-1)(y-1)=0\\\\\ny^2-1=0\n\\end{cases}\n", "38114e00ab54d5bbd1939cb9a9f6f0ee": "Ky_1(t)+Ky_2(t).\\,", "381163b87b0e2f7a758de9d20ca52d7f": " q = \\beta\\operatorname{sign}(z(t)\\dot{u}(t))+\\gamma", "3811abd6b41a0d97f7a1d7531ba6b7ee": "\\omega' + \\omega", "3811be553c476c2e4c6de6d99678d741": "f_{\\theta_1}(x_1) f_{\\theta_0}(x_0) \\geq f_{\\theta_1}(x_0) f_{\\theta_0}(x_1). \\, ", "3811c648fff42774a4ee1b9c4e58917c": "C_Z = 0", "381287485c594f0c2fd9b015caf69238": "\\varrho_P(f,g)=(E(G_P(f)-G_P(g))^2)^{1/2}", "3812ef5429c6452454e6955c130fd4af": "\n\\Bigl\\langle p_{k} \\frac{dq_{k}}{dt} \\Bigr\\rangle = -\\Bigl\\langle q_{k} \\frac{dp_{k}}{dt} \\Bigr\\rangle = k_{\\rm B} T.\n", "38131de8a71a9a2437b215982ea40c76": "H_n\\,", "38133d093f9f8e098ddb76bdf87fcb81": "\\underset{i}{\\min}\\Pr(R_{i}<\\underline{R})", "3813726fdb46bb013edc8b7ac4018be8": "(\\bar{r} , \\bar{v} )", "3813a5f5a1d3761b19bde3d7302b334c": "\\phi=0\\ ", "3813a8c1d9b0c33b88388e59283263a3": " i \\in \\{ 1,\\dots,M \\} ", "38140818917a35d5ca255526823b7ec1": " F_n = \\frac{\\varphi^n - \\psi^n}{\\varphi - \\psi}", "38141d5d60ed2a86fa4af6a96f9df414": "\\mbox{PU}(2) = \\mbox{PSU}(2) = \\mbox{SU}(2)/(\\mathbf{Z}/2) \\cong \\mbox{Spin}(3)/(\\mathbf{Z}/2) = \\mbox{SO}(3)", "38143f36753edf99fb733f46d9a447c6": "n\\ge0", "38144545c06aab3d2754df39e5725490": "(Poincare)_{3}\\times SO(8)", "381478f325776cbeb79f5fa9097d3773": "1\\le a< 10", "3814966e13c4484bd5f2435095fb0ad0": "\\begin{align} \\arctan\\frac{1}{b} &\n= \\frac{1}{b} - \\frac{1}{b^3 3} + \\frac{1}{b^5 5} - \\frac{1}{b^7 7} + \\frac{1}{b^9 9} + \\cdots \\\\ &\n= \\sum_{k=1}^{\\infty}\\left[ \\frac{1}{b^{k}} \\frac{ \\sin\\frac{k\\pi}{2} }{k} \\right]\n= \\frac{1}{b} \\sum_{k=0}^{\\infty}\\left[ \\frac{1}{b^{4k}} \\left( \\frac{1}{4k+1} + \\frac{-b^{-2}}{4k+3} \\right) \\right] \\\\ &\n= \\frac{1}{b} P\\left( 1, b^4, 4, (1, 0, -b^{-2}, 0) \\right).\n\\end{align}", "3814df7a1eccfb6accb60518fdd38e1c": "\\lambda > 0 ", "3814eac4dd84ab19163124c0dd618308": "[0]=\\emptyset", "3815448bff559faf10449c7031f01a09": " \\det \\mathfrak{H}=\\det\\mathfrak{H}'=1 ", "38157c3fe68072610b54fc4a70fd87c6": "a_0b_0x^n + a_1b_1x^{n-1} + \\cdots + a_{n-1}b_{n-1}x + a_nb_n = 0. \\, ", "3815b2da03290a10220a801a02286c03": "\\text{monad}(x)=\\{y\\in \\mathbb{R}^* \\mid x-y \\text{ is infinitesimal}\\}.", "3815bb6e931b81ffdb58e334b4aeb0e3": "\\mathrm{ad}_x(y) = [x,y]\\,", "3815df8b18d8af4b2274b41690a8509d": "\n\\sin z\\,\n", "38163d02af3142224c82158c59bff16f": "1/\\sqrt{S}", "381652259731e5a1621688e55bc55e84": " \\mathbf{\\bar F} = ((\\mathbf{T}')^{T})^{-1} \\, \\mathbf{F} \\, \\mathbf{T}^{-1} ", "3816644568a1bdab08f4dc08803084a5": "z_{n+1}=z_n^2+c", "38169713d9667e1056b14c7a0f72b65f": "\\mathbf{r} = \\frac{\\mathbf{s}}{|\\mathbf{s}|}", "3816bc8cb1caf997950b98fa168d7f75": "\n\\begin{align}\n G_{u_g}(s) &= \\frac{ \\sigma_u \\sqrt{\\frac{2L_u}{\\pi V}} \\left(1+0.25\\frac{L_u}{V}s \\right)}{1+1.357\\frac{L_u}{V}s+0.1987\\left(\\frac{L_u}{V}s\\right)^2} \\\\\n G_{v_g}(s) &= \\frac{ \\sigma_v \\sqrt{\\frac{2L_v}{\\pi V}} \\left( 1+2.7478\\frac{2L_v}{V}s + 0.3398\\left(\\frac{2L_v}{V}s\\right)^2 \\right)}{1+ 2.9958\\frac{2L_v}{V}s + 1.9754 \\left(\\frac{2L_v}{V}s\\right)^2 + 0.1539 \\left(\\frac{2L_v}{V}s\\right)^3} \\\\\n G_{w_g}(s) &= \\frac{ \\sigma_w \\sqrt{\\frac{2L_w}{\\pi V}} \\left( 1+2.7478\\frac{2L_w}{V}s + 0.3398\\left(\\frac{2L_w}{V}s\\right)^2 \\right)}{1+ 2.9958\\frac{2L_w}{V}s + 1.9754 \\left(\\frac{2L_w}{V}s\\right)^2 + 0.1539 \\left(\\frac{2L_w}{V}s\\right)^3} \\\\\n G_{p_g}(s) &= \\sigma_w \\sqrt{\\frac{0.8}{V}} \\frac{ \\left( \\frac{\\pi}{4b} \\right)^{\\frac{1}{6}} }{(2L_w)^{\\frac{1}{3}} \\left(1 + \\frac{4b}{\\pi V}s \\right)} \\\\\n G_{q_g}(s) &= \\frac{ \\pm \\frac{s}{V}}{1+\\frac{4b}{\\pi V}s} G_{w_g}(s) \\\\\n G_{r_g}(s) &= \\frac{ \\mp \\frac{s}{V}}{1+\\frac{3b}{\\pi V}s} G_{v_g}(s)\n\\end{align}\n", "3816ec5ec9ddaa983983a01900c1be56": " D_w = max[ \\frac{ c_i }{ K } - \\frac{ i }{ N } ] ", "38174291c5af9470d9468ad65e434e92": "r=\\liminf_{n\\to\\infty} \\left|a_n\\right|^{-\\frac{1}{n}}", "381772e262f83553288d44003a919c27": "J=J^{-1}=J^{*}", "381778189cc513f32127d10a7bce1973": "\\left[00,\\alpha\\right] = -\\frac{\\varepsilon}{2} \\gamma_{00|\\alpha}", "3817d89bdb507358e321966891091e3f": "y_3 = \\frac{y_2}{2}(-1 + \\sqrt{1+8\\frac{q^2}{gy_2^3}}) = 3.46 ft", "381804db063836a4e9254fc4620c30bd": " l^2", "38186df4de1011e4f5b18153368a3d6c": "z \\gg \\big|\\left(x - x^\\prime\\right)\\big|", "3818822583332d510f6a326d141ed4a9": "2 > \\sqrt{2} > 1", "3818ac2d220e2c6bd5cc707e5ee8a492": "l^an_a=n^al_a=-1", "3818bb8bb4beee88ef1ce0cfb874c2d2": " x_k-x ", "3818ca09e747634e1f71aa79e2d3494a": "i\\in[0,infty)", "3818f1d2f7cfd3379a741827be37b832": "\\langle\\phi|A|\\psi\\rangle", "381900602a7be1e48f1eb9d5805e87cc": "t_{worst} = O(k \\cdot N^{1-\\frac{1}{k}})", "381911dbf33c1ff7dc49681ab553db91": "\\,kp+2,\\, ", "3819232ea3b94b43fdf58e4690f44858": "\\ [0,1] ", "38197af171b1e0726e8f0e3923875cdb": "\ny = a_0 + a_1 x + \\varepsilon, \\,\n", "381998227639a0323532d5bc9ca7b37f": "\\mathbf M= \\mathbf M_B + \\mathbf M_C", "3819998087b8f8b58db46533ef27b9e8": "3^{\\log_{1/\\alpha}n + 2}-2 \\le\n9n^{\\frac{1}{\\log_3(1/\\alpha)}}", "381a3c31bb704569c0552a187d7cf823": " EG(a_n;x)=\\sum _{n=0}^{\\infty} a_n \\frac{x^n}{n!}. ", "381a3e79bfaf490b484df75ab5f1172e": "\\langle\\partial_i,\\partial^j\\rangle=\\delta_i^j", "381a94fb209c12be113de0e6bc353464": "W_0 = m_0c^2 - \\ ", "381a9b6f442461734a54ab0248a4f765": "\nf_\\mathbf{v} (v_x, v_y, v_z) =\n\\left(\\frac{m}{2 \\pi kT} \\right)^{3/2}\n\\exp \\left[-\n\\frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}\n\\right],\n", "381a9d2b31f67402be7b59ae3b9d9dbb": "q=\\frac{m^2 g l^3} {\\hbar^2} ", "381abe33724d714091a6176fafef8a26": "49^2", "381b1f2e0fb27667905eb6a36f1e2815": "\\mathcal{O}(1) = g^*(\\mathcal{O}(1))", "381b8b623d738ba72c8a6ff830287a6b": "(g,g') \\mapsto (gN, g'N')", "381bdbf57a8f636e7c7bb27bba979239": "v_0 = s, v_1, v_2, ..., v_k = t", "381bdcd0110fa2ee245753241cd071fa": "D(\\omega)=-\\frac{d\\phi}{d\\omega} =\n\\frac{6 \\omega^4+ 45 \\omega^2+225}{\\omega^6+6\\omega^4+45\\omega^2+225}. \\,\n", "381bdfbb95906552fff72d0f42a71b4a": "A\\mathbf{x}=\\mathbf{0} \\;\\;\\Leftrightarrow\\;\\; \\begin{alignat}{6}\na_{11} x_1 &&\\; + \\;&& a_{12} x_2 &&\\; + \\cdots + \\;&& a_{1n} x_n &&\\; = 0& \\\\\na_{21} x_1 &&\\; + \\;&& a_{22} x_2 &&\\; + \\cdots + \\;&& a_{2n} x_n &&\\; = 0& \\\\\n\\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && \\vdots\\,& \\\\\na_{m1} x_1 &&\\; + \\;&& a_{m2} x_2 &&\\; + \\cdots + \\;&& a_{mn} x_n &&\\; = 0. &\n\\end{alignat} ", "381bef04b289b9ccf6d23e87a021b0d4": " k(a,a_1;t) ", "381c0fb8d115ebbbcfc885605bdb3dd7": "N^{-2}", "381c4c588af87e756aec56eb7f50ea96": "f(x, y) = (u(x, y), v(x, y))", "381c5db2e753ecf49cd0b66abd925ace": "\\frac{\\zeta(2s)\\zeta(3s)}{\\zeta(6s)} \\ ", "381dc21cdb086529f56e2c5ef59b5fde": "X_R y + x Y_R - X_R Y_R", "381e106901ececf4f53ac9bd36ad9870": "\\frac{t}{1+\\lfloor (l+m-2)/m \\rfloor}", "381e3ed8236941989ac4ebe0161070b6": "L_n(x) = \\begin{cases}\n2, & \\mbox{if } n = 0 \\\\\nx, & \\mbox{if } n = 1 \\\\\nx L_{n - 1}(x) + L_{n - 2}(x), & \\mbox{if } n \\geq 2.\n\\end{cases}", "381e89efcfe4a69df971062fcebc260d": "\\mathbf{Gr}(r, \\mathcal E_T)", "381e935dadccf85eeddeb19e4812e2ac": "n = -39\\log_{92} \\left( \\frac{d_{n}}{0.005~\\mathrm{inch}} \\right)+36 = -39\\log_{92} \\left( \\frac{d_{n}}{0.127~\\mathrm{mm}} \\right)+36", "381e93832b2eb01b272e822e8ae5555f": "D \\subset C", "381ea5b6109226f7f570f54c6464598a": "\\frac{d\\Gamma}{d\\cos\\theta} \\sim 1 - \\frac{1}{3}P_{\\mu}\\cos\\theta.", "381ee55b1a00157fe8e76eabf63cde40": "O\\left( n^{-1/2} \\right) ", "381ee7472382f9faa4f60f06ee337a20": " \\beta = u, ", "381efdab637f544ec40d330523aa5fcc": "n_T", "381f0d854ecf3610d4f9e133240a1caf": "e\\!\\!e", "381f3cf30ba195b671f632eeeeeb390d": " \\int_0^t \\frac{d S_t}{S_t} = \\mu \\, t + \\sigma\\, W_t \\,, \\qquad\\text{assuming }W_0=0\\,.", "381f82f898ca11fdd6987215ed343098": "\\text{Im}[Y_\\ell^m] = 0", "381fb9a512ab2667a0bb3c8f828479fa": "w-1,\\ldots,w-M+1", "382003d39f83b395c74d9c4d9a267f60": "\\sum_{k=1}^K r\\left(H_{X_k} \\right) = r(H)", "382019a55780a5ebb631082eb5b04753": "r=\\frac{f}{f_n}.", "3820244a59ffa1981e1d29bfba9bab22": " e = \\left ( \\frac{2}{1} \\right )^{1/1} \\left (\\frac{2^2}{1 \\cdot 3} \\right )^{1/2} \\left (\\frac{2^3 \\cdot 4}{1 \\cdot 3^3} \\right )^{1/3} \n\\left (\\frac{2^4 \\cdot 4^4}{1 \\cdot 3^6 \\cdot 5} \\right )^{1/4} \\cdots ,", "382078b62ab81e1bc907cc9b03a9be4e": "b=(k-1)/2", "382090fc0531d142c055de6bf3fcc8bd": "a/(bc)", "3820f8a85a6c50cf490684e419b7179a": "\\tan\\phi = - \\frac{2 \\zeta \\omega}{ 1 - \\omega^2} = \\frac{2 \\zeta \\omega}{\\omega^2 - 1} \\Rightarrow \\phi \\equiv \\phi(\\zeta, \\omega) = \\arctan \\left( \\frac{2 \\zeta \\omega}{\\omega^2 - 1} \\right ). ", "3821450ac3feec285fefeb2553ba5dc9": "\\R^3", "3821549ba920f328092d1533d2b3f28b": "\\alpha \\alpha^* + \\mu \\gamma^* \\gamma = \\alpha^* \\alpha + \\mu^{-1} \\gamma^* \\gamma = I,", "382177ca835d3f70b4dd96f08fd12287": "\\textstyle c_i \\in \\mathbb{Q}", "3821f95504a0f80f74dcce473fde6154": "M = \\int_{V}\\rho\\ dV", "38220485ba588fda9f8c7adcec3e50b3": "L_i*w_i", "38226e855535a578e9346133e1d0b72e": " \\boldsymbol{.}:H\\otimes R\\to R ", "382272412e41b637181b36376279c082": " \\rho_\\text{P} = \\frac{m_\\text{P}}{l_\\text{P}^3} = \\frac{c^5}{\\hbar G^2}", "3822ce8d9342cab9f537947d1e8a1a78": "\\begin{align}\n M_i = \\sup_{x\\in[x_{i-1},x_{i}]} f(x) , \\\\\n m_i = \\inf_{x\\in[x_{i-1},x_{i}]} f(x) .\n\\end{align}", "3823157e4d64e6cfbeec4a885dcf42b1": "\\bigcup_{x\\in X} \\{p,x\\} ", "38233e7e3f3719dc058a95ed06c87800": "\\mathrm{height}(s) = |V|", "382367d0e688e2df2ce5db2ca0a186c9": "mda(L) = max_{i=1,L-1} (\\Delta \\Phi_{i+1} , \\Delta \\Psi_i)", "3823827af4d59e5c7348bcb6fad28f14": " \\Gamma\\, ", "3823a928ded53e8b0c76a9fcbf4bcc5c": "f_1(\\mathbf{x}, \\Omega) = \\frac{n_1}{\\Omega} = x_1", "3823e53383b1a8159ddce61186cde2ef": " Y_i = \\beta_0 + \\beta_1 X_{i1} + \\beta_2 X_{i2} + \\ldots + \\beta_p X_{ip} + \\epsilon_i.", "38240d4a0afc954c2f4286f9eac48828": " f(A) = \\int_{\\mathbb{R}} f(\\lambda) \\, d \\operatorname{E}(\\lambda).", "382455e8431870f406a5f716533b6793": "1-1/e \\approx 63.2\\,\\%", "38245fd7bdd1d165368715423bef284b": "\\phi = v e^{i\\theta} \\,", "382482c0d0bd3142cc6f80cd8866b7a4": "F(r) + m r \\dot\\theta^2 = m\\ddot r ", "3824b4f5a595633fec3ca073e82a9e8c": "p(a,b)=p(a|b)p(b)=p(b|a)p(a) \\,", "3824c44809460bbafb0573ee6893dab2": "{\\mathcal I}_{\\mathcal P}", "3824d949535f3305ed4b334e7cecd684": "nan^{-1} \\mapsto nAn^{-1},", "382501548aaf3cbef148d1a00f0984c5": "\\Pi=R-C-S \\,", "38250fec8abeb1ba511d1f82b17e27f5": "b = 6 \\pi \\eta r\\,", "38257b5ef1a8d68de5c416b02e7a9705": "\\left( \\begin{smallmatrix} 16 & 0 \\\\ 0 & 16 \\\\ \\end{smallmatrix} \\right)", "38257ff13d867ef072fd5f1b680eaf8d": "e^{\\epsilon\\Delta q}\\,\\!", "38260fa290797577dbb70ecdbe58b157": "\\overline{\\operatorname{Span}}(E)", "382623d04eb37cc9fc7e6df299311ee2": "\\mathbf{1}_A", "38265fe0407d0054fa880a0686395de8": "V_{\\mathrm{pk}}", "382671b5a6868a12bd1cc97065499f77": " k_{x1}, k_{y1}", "38271c433a59d632d91254b647567994": "<^*\\,", "3827a54fb3ab4f71a5da63f0d0b20d12": "t_{1}=t_{2}", "3828181b7f99ce04b335380414e5cdc4": "n=\\left(\\sum_{i=1}^{k}{d_i}\\right)^3\\, ,\\text{ e.g. } 512 = (5+1+2)^3 \\, .", "38281f2a7f97cd7397cfb8be938099b9": "\\mathbf{\\chi}_\\mu^A \\ ", "382832a3aa2cc31ebeab13022b72462e": "(c^2-d^2)^2+(2c d)^2 = (c^2+d^2)^2", "38286620918d94bf96b3ca055380f41b": "\\Gamma (x)", "382894c3f73788aa45046f0b0d0071fb": "e_q(t)=\\sum_{n=0}^\\infty \\frac{t^n}{[n]_q!}=\n\\sum_{n=0}^\\infty \\frac{t^n (1-q)^n}{(q;q)_n}.", "38289ce7c4f0589e8fd16c7d43db30a4": " = \\vec{\\nabla}_{\\vec{r}}\\bigg(\\frac{1}{A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{\\vec{r}-\\vec{r}'}{|\\vec{r}-\\vec{r}'|^n} \\bullet{} \\vec{F}(\\vec{r}')d\\tau'}\\bigg)", "3828f01313d537434c6c73d597763691": "R = r_0 A^{\\frac{1}{3}}.", "38291befa849d46b26b91bbd136de343": "\\mathrm{\\ 2 NbF_4 \\rightarrow NbF_5 + NbF_3}", "3829673c79dfd59697ddb4380bd63430": "\\ D_S = 0 ", "3829722047eba109619219b3cd937278": " \\forall \\varepsilon > 0\\ \\exists \\ \\delta > 0 : \\forall x\\ (0 < |x - a | < \\delta \\ \\Rightarrow \\ - \\varepsilon < f(x) - L < \\varepsilon).", "38297b7ea05ab096bd964abce5f5c2f3": "\nec(K_{n,n}) = (n/2-1)!^{2n} 2^{n^2-n+1/2}\\pi^{-n+1/2} n^{n-1} \\bigl(1+O(n^{-1/2+\\epsilon})\\bigr).\n", "3829bd6f2d122352d4e0f821c318e01f": "Q(\\lambda)\\,", "3829cd3cd54ccd526d7ef7fbe3f7b1d1": "K(e,f) = \\langle I^e,I^f \\rangle ,\\quad e,f \\in E", "3829d0a69d25e4582ed9aaafd00a6bec": "\nE = E_0 + \\frac{1}{2} K_0 \\frac{(V-V_0)^2}{V_0}.\n", "3829d0a7f30a6527c3e0f8511f7b2e6a": "\\scriptstyle \\frac{1}{T}", "382a2abdb5652214a96d94632b55b88a": "[D \\rightarrow [D^{'} \\rightarrow D^{''}] ", "382a40a28afe641fb229feff97720f63": "\\partial f/\\partial x", "382a72eec5c4d8c20d3bb3f153db4fad": "-\\frac{1}{n} \\log p(X_1, X_2, ..., X_n)", "382a83234a2e98a4812fc5f0e752f84a": "w^{j}_{i}=0\\,\\!", "382aaa59c5984c8e3481fda2f6f83702": "{x_N} = {x_N} \\cdot {x_M}^q", "382ab7e6627339a02d65213a10358bc0": "R(\\lambda)", "382b6a468fb1213e1842e79ceb8e7bb9": "x^3-Tr(g^n)\\!\\ x^2 + Tr(g^n)^p x -1", "382b6ed73aed2e3c795866451dadda04": "x_{n-1}^\\ast", "382b837abd0809ba0804d0a2754c1b24": " \\frac{k_2[L'][Int]}{k_-1[L][Int]+k_2[L'][Int]}", "382bae4e949d6bcd52e07436b80b2d2a": " f_V(v; \\sigma) ", "382beea31e7e4e48963a8d012e488f79": "x\\in[0,\\,1]", "382c456b4b8563663e7075259125a143": "\\partial_R", "382c52009bfcca37c45249419741efad": " T \\Delta S_{int} - Q \\ge 0 \\,", "382c9c875695ab67582e306de79c1b2b": "\\Pr(H)", "382d112083bb837e65ec69efa1529805": " \nAC = [A^+] C = \n\\begin{bmatrix}\na_0 & -A_3 & A_2 & A_1 \\\\\nA_3 & a_0 & -A_1 & A_2 \\\\\n-A_2 & A_1 & a_0 & A_3 \\\\\n-A_1 & -A_2 & -A_3 & a_0\n\\end{bmatrix}\n\\begin{Bmatrix} C_1 \\\\ C_2 \\\\ C_3 \\\\ c_0 \\end{Bmatrix}.\n", "382d6a1136276a1933715198396155d3": "\\frac{0.2394}{0.2730}=0.8769", "382da15dfcfa571b3973cb5ae2223f76": "pq", "382da7dec3705a92b99575d8b6d9d005": " r \\rightarrow m-r ", "382daf57778a8036f20c352cac2f97a0": "E_k(\\Lambda) = \\sum_{\\lambda\\in\\Lambda-0}\\lambda^{-k}.", "382de830abeb457dd05a6b4bc237e5be": "\\frac{\\Gamma \\left(\\frac{\\nu +2}{2}\\right)}{\\pi \\ \\nu \\Gamma \\left(\\frac{\\nu }{2}\\right)}= \\frac {1} {2\\pi}", "382e006155ec0dab923c90a0ea73d1d7": "X=S \\cdot \\cos\\theta\\cos\\phi", "382e21de6061d26665c0c4792f7eca4d": "\\left(\\mathbf{A} + \\epsilon\\mathbf{X}\\right)^{-1}\n= \\mathbf{A}^{-1}\n- \\epsilon \\mathbf{A}^{-1} \\mathbf{X} \\mathbf{A}^{-1} + \\mathcal{O}(\\epsilon^2)\\,.", "382e58c8abbb06fb0515612c715048af": "\\displaystyle F_n(x)", "382e8e2eee3b81e76a3ea91f81810792": "\\{0,1,2,\\ldots,9\\}.", "382ed96e588f80f82e9f432d1340f78d": "10\\uparrow\\uparrow\\uparrow 4=(10 \\uparrow \\uparrow)^4 1", "382edd29842e5a846af4871c314aabc6": " AU = b ", "382ee832afb10df4dc6c3c46f2a29eb7": " \\text{Sl}_4(\\theta)= \\frac{\\pi^4}{90}-\\frac{\\pi^2\\theta^2}{12}+\\frac{\\pi\\theta^3}{12}-\\frac{\\theta^4}{48} ", "382f3ecffd0460613c3bd44296b53cd5": "\n W = \\rho_0 \\psi \\;.\n ", "382f7ea53786d6cedebbddbedb4de3ce": "E = \\frac{N_1 \\cdot N_2}{d}", "382f86a30efd79e3863d4bab8a0a9545": "x^2 = 0", "382fc1b91654d70b61b5039227bb03d6": "\\ell^1(S) = \\{\\varphi:S \\rightarrow \\mathbb{C}: \\|\\varphi\\| = \\sum_{x \\in S}|\\varphi(x)| < \\infty\\}.", "382fedaaf4d5c1c61f9f3f7731c44d1b": "{\\mathcal{A}}_{i_n = i}", "382ff7786e12a637b98c4134ad773143": "x^{'}_{i} \\leftarrow x^{'}_{i} + \\delta", "38302cd7d55fdaa052ae2e5172263c7d": "RangeSearch(*ptr(T(O_{r})),Q,r(Q))", "383065e91a46da470276bade7cbc7db6": "\\xi=\\beta\\,", "3830a160e9aa4406bedc576802e8a116": "S(i) \\neq S(j)", "3830ceda6f9a964187778507fb49b307": "(1 - |\\xi|^2/R^2)_+^\\delta", "3830d128df9e5fdaab401afccdaba582": "\\overline{X}_n\\pm A\\frac{S_n}{\\sqrt{n}}", "3830d214307569d5a5ff0b973c41efcd": "11 = 2^0 + 2^1", "3831841c52bb51bc857ee36663901e54": "f_k(t)=t/k^2", "383245a3a088537ff9d7beddb7ca4ddd": "\\frac{dP(r)}{dr} = - \\left( \\frac{T_{00} g^{00} + T_{11} g^{11}}{2} \\right) \\frac{d\\nu(r)}{dr} = - \\left( \\frac{\\rho(r) c^2 + P(r)}{2} \\right) \\frac{d\\nu(r)}{dr} \\;", "383264ada1c4cef958b499fd383080bb": "x^3 + a^2x = b", "3832f3568d5597ffc5fd9db9fb4ea39a": "\\sum_{m=0}^n P(2m)=P(2n+3)-1", "38334b71b16ec3cb8182e144575d3712": "\\bold{J}=\\bold{L}+\\bold{S}.", "38335b779c576f5c0c9f8aca4f4ec30c": "p_1 - p_2 = \\frac{\\rho}{2}\\left(v_2^2 - v_1^2\\right)", "38336832e29e7a5663888da0e8f10324": " d_H = c/H_0 ", "38337fb7cbea2ac86296bcd28ccfa578": " \\lambda_1 \\geq \\lambda_2 \\geq \\cdots \\geq \\lambda_n ", "3833864e4c7ecb82199534fb89099e49": " F = \\beta(t) \\frac{I}{N} , \\beta(t+T)=\\beta(t)", "38339895baa631cf2ef1f70a86d2d6c9": " =\\frac{E_\\mathrm{sig}^2}{2}\\left( 1+\\cos(2\\omega_\\mathrm{sig}t+2\\varphi) \\right)", "3833a7eec5e02d7cc3061a1e2cb08512": "Tp_n(x) = x^n.\\,", "3833d435a08583d9327a25b57a85cf3a": "f^*:X_T\\to Y_T", "38351a05c4e17295f6247226c142bed6": "APC=\\frac{C}{Y-T}", "3835236097ec362cbeb5d9c55abf6305": "f^*([Y']) = [f^{-1}(Y')]\\,\\!", "38356d40bdc9c7d6ced31fe49c32c6b5": "\\delta \\mathcal{S}", "3835a0672692f74b690c2cf7df8658d3": " \\Theta_{*}^{s}(\\mu,a)=\\liminf_{r\\rightarrow 0}\\frac{\\mu(B_{r}(a))}{r^{s}}", "3835aac5437142ed0b7ae7b24b639423": "U(r+\\bigtriangleup r,w) =U(r,w)\\exp (ik\\bigtriangleup r) \\quad (1.2)", "3835bbab7e602d84566858547b552787": "(uv)^{(n)}(x) = \\sum_{k=0}^n {n \\choose k} \\cdot u^{(n-k)}(x)\\cdot v^{(k)}(x).", "383617863693615c1fc96ac51b840147": "x = \\frac{3u}{2u - 8v + 4}", "383665098b89ffe741127f7c75a966c9": "\\mathcal{F}_\\alpha (u)", "38366eae4ea15d8ea41e801c8d7530a7": "h(u_1, \\ldots , u_D) = \\sum_{1 \\leq i\\leq D} (u_iP_i)", "383673a0724ddba37e569ad2bd364f6c": "V\\, ", "3836f7c8180134147b4273d40032a783": "(x_1, \\dots, x_n)", "3837a6aca328172f7fade3825b787a84": "\\begin{bmatrix}\n-2c & 0 \\\\ c & d \\\\ \\end{bmatrix}^{-1} \\begin{bmatrix} 1 & 0 \\\\ 1 & 3 \\\\ \\end{bmatrix} \\begin{bmatrix} -2c & 0 \\\\ c & d \\\\ \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\\\ \\end{bmatrix}, [c, d]\\in \\mathbb{R}", "3838acf1d90707f0c1130abbb76a8844": "Q =\\int_R d^3 \\mathbf{r} \\, \\rho_q =\\int_R d^3 \\mathbf{r} \\, q \\delta(\\mathbf{r} - \\mathbf{r}_0) = q \\int_R d^3 \\mathbf{r} \\, \\delta(\\mathbf{r} - \\mathbf{r}_0) = q ", "3838fa633e4feb75d791a5d46222a9ed": "x_1 = 2 x_2\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;x_3 = 5x_4.", "38394e7c7e0446fdea5305fa6bb454c8": "\\rho_{B}(0) = \\sum_{j}a_{j}|{j}\\rangle\\langle{j}|.", "383975092bd653b66ed28aea4ae936d5": "f(t) = \\beta S I", "3839ae1f4420928d89b33c7e7d012424": "f:K\\to L", "3839c2c8509c03c1130912e72e49b2fb": " \\gtrsim, \\gnsim, \\gtrapprox, \\gnapprox \\,", "383a12b597a7734518ec78c23a737a52": "P_k:= \\mathbf{u}_k \\left(\\mathbf{v}_k^* A \\mathbf{u}_k \\right)^{-1} \\mathbf{v}_k^* A,", "383a610b529b8b80858f9aced1a5fb18": "k e^{a x}\\!", "383a8b1d9eb9b6c056fe96e9e2431b9b": "D_\\alpha(x)=\\rho_\\alpha(x)/\\alpha", "383adb9e05e18c12050d113cfaf34f54": "R\\left[ n,k \\right]={{R}_{Y}}\\left[ n-k \\right]with,", "383ae1982ba1bd3de4d6ee2ce36637be": "\\frac{dI/dV}{I/V}=\\frac{d\\left(logI\\right)}{d\\left(logV\\right)}=\\frac{\\rho_s\\left(r,eV\\right)\\rho_t\\left(r,0\\right)+A\\left(V\\right)}{B\\left(V\\right)}\\ ,\\qquad\\qquad (9)", "383af41bbb4d44cfb23c16d0755e8461": "\\sigma_t^2 = \\sigma_R^2 d_t^2 = TDOP^2 \\sigma_R^2", "383afc1e56b8d24c42b0a0973b49dbcc": "f: F \\to G, g:H \\to G", "383b20a71a82d719aa50d820834a5947": "(g_{kl})", "383b7a2015a2a00b548e1e35f7cf8999": "\\exp(-E it/ \\hbar),\\,", "383bc1184317597ab5ca878ec5237a39": "S \\to c", "383bf8ddc91e5cd69aee64fa6cff3d1f": "\\,b", "383c04eb109868daab740d4a941a4d4e": "\\vec a\\ ,\\, \\vec b\\ ,\\, \\vec c ", "383c3335b929578893ae74409a7d0fce": "\\tau H_i M", "383c6e68865441ad252ee93f7de86a65": "\\int_{-\\infty}^{+\\infty} e^{-(ax^{2}+bx+c)}\\ dx=\\sqrt {\\frac{\\pi}{a}}e^{(b^{2}-4ac)/4a}", "383d211ff44adda7d4b5817e75fce61f": "I_{abs} = I(t)^{-\\beta z}", "383d8d8e98fc2a5b9537585c35a86e64": "m_k = \\lfloor k \\phi^2 \\rfloor = \\lceil n_k \\phi \\rceil = n_k + k \\,", "383daec1c0125a0728ef310b599d1646": "\\Psi \\in \\mathcal{H}_{Kin}", "383dcd57793cb626804952562e100c87": "\\boldsymbol{\\hat{r}_{21}}={\\boldsymbol{r_{21}}/|\\boldsymbol{r_{21}}|}", "383e0ec3539a17a68294125ba0e9597e": "a\\cdot c\\cdot 100 + (a\\cdot d+b\\cdot c)\\cdot 10 + b\\cdot d", "383e4933a314b5d082806f515beea53a": " h:T(TM\\setminus 0)\\to T(TM\\setminus 0) \\quad ; \\quad h = \\tfrac{1}{2}\\big( I - \\mathcal L_H J \\big),", "383e6f4d6df20452644d89d25ec2c627": " -(n+1)(n-2)~r^n~\\cos(n\\theta) \\,", "383ea4b61faf3b347bf9c8aeb9c2ed62": "\n - (\\Sigma _{11} )^{ - 1} \n", "383f1e180832a93dc4980f7f0ec8a4e0": " K_b= K_a *t", "383f334fa579f244d92859ced777bcb2": "{M^\\mu}_{\\nu\\lambda}= (x_\\nu {T^\\mu}_\\lambda - x_\\lambda {T^\\mu}_\\nu)+ {S^\\mu}_{\\nu\\lambda}.", "383f3dc553b9ad037d575acc7f297356": " F_{vac} = C_f\\, \\dot{m}\\, c^*", "383f64f07ec50417f386eeb2e9ede436": " \\int e^{{1\\over 2} A_{ij} \\psi^i \\psi^j} D\\psi = \\mathrm{Pfaff}(A)", "383f8b057417aab6f8409fdf1f461795": "\\Phi(\\vec {r},t) =(\\phi(r)/\\sqrt{2})e^{i\\omega t}", "383f9a2f806343e37f8473a49aab80c3": "mx+k=\\frac{x_0-x}{m}+y_0.", "383fa2c77da5288196c02acdf0ac1d2e": "\\operatorname{pf}(A^\\text{T}) = (-1)^n\\operatorname{pf}(A).", "383fd1e71f8d976ce1a433359e2a2080": "C_L ", "383fd6d4845b9ed3a5af883031643f2d": "V_2=0", "383fe2146550430b2171b96e0133cffb": "\\displaystyle D_q f(g(x)) = D_{q^k}(f)(g(x))D_q(g)(x).", "383fe8612803581b329f3197470eaf95": "\n \\sigma_{11} - \\sigma_{33} = \\sigma_{22} - \\sigma_{33} = \\lambda_1~\\cfrac{\\partial W}{\\partial \\lambda_1} - \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3}\n ", "3840011430996a4c8b2642e64ff9b2ff": "(\\phi_n)_{n<\\omega}", "38402a2c47dea5e8582d739d017db4c6": "a^4+b^4 =\\, c^4+d^4 ", "38402b6de78b143619462bc2303cbc4d": " \\frac{\\ell}{t} = \\frac{1+x}{x(1+n)}", "384116acd07af4357a31cbf9bd7ae219": "\nf(\\boldsymbol{x})=P(\\boldsymbol{x},f_{1}(\\boldsymbol{x}),\\ldots,f_{r}(\\boldsymbol{x})),\\,\n", "3841399c377e6409d8bdea97c2f8bf08": "S=S_s b \\,", "38413ef8fa18b04d8544ad6a39becab9": "\\frac43\\, \\left( \\frac{\\text{d}\\psi}{\\text{d}\\xi} \\right)^2 = \\left(\\eta_1 - \\eta_3 \\right) - \\left( \\eta_1 - \\eta_2 \\right)\\; \\sin^2\\, \\psi(\\xi), ", "38417d091086aa35a1c02abd5a8076dd": "z_n\\rightarrow 0", "38417f63037240daab1a2ad48ac94daa": "\\delta W = \\sum_{i} ( \\mathbf {F}_{i} - m_i \\mathbf{a}_i )\\cdot \\delta \\mathbf r_i = 0.", "38419018347d8c11b9747b9fa8f858ed": "\\varrho(X) = \\sup_{Q \\in \\mathcal{M}(P)} \\{E^Q[-X] - \\alpha(Q)\\}", "3841ab3adcf00d8ccea954cad4b7990a": "\\sqrt[3]{-27i} = \\begin{cases} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3i \\\\ \\ \\ \\frac{3\\sqrt{3}}{2}-\\frac{3}{2}i \\\\ -\\frac{3\\sqrt{3}}{2}-\\frac{3}{2}i. \\end{cases} ", "3841afcde206d631db3b74b2b1da9358": "\\delta_n = \\delta_n(x_1,\\ldots,x_n)", "3841fc0c9b38518168acd09dfd5f5793": "{d{I_{1z}} \\over dt}=-R_z^1(I_{1z}-I_{1z}^0)-\\sigma_{12}(I_{2z}-I_{2z}^0)", "38427e9a748868785765a20a6e06f7b8": "r = \\frac{R}{t}", "3842c9064b0375709c225b507877e052": "S(Z,X,JY)=-S(JZ,X,Y)", "3843439214bbb0fd909410688f4f5474": "V(+\\infty)\\,", "384354563897a2a1fa84d9fe5dd68671": "\\langle\\Delta \\tilde{r}^{2}(s)\\rangle", "3843b047b42c3746e1c377969c37c7e0": "\\, \\begin{bmatrix} T_{r\\overline{o}} & T_{\\overline{ro}}\\end{bmatrix}", "384425428a857b74bf6d663ff5f4a13e": "\\alpha^2 = \\beta^2 + \\gamma^2 + \\delta^2 - 2\\left(\\beta\\gamma\\cos\\left(\\widehat{\\beta\\gamma}\\right) + \\gamma\\delta\\cos\\left(\\widehat{\\gamma\\delta}\\right) + \\delta\\beta\\cos\\left(\\widehat{\\delta\\beta}\\right)\\right).\\,", "384451f6c8edff99e55df1465505d19e": " \\alpha_\\mathrm{\\{per\\ comparison\\}}=\\bar{\\alpha}/n", "384487812cb62eea0e0e6eccdabe8835": "\\Phi_{i+1}=[\\Phi_{i}:\\varepsilon]", "38448aba6d8a69017d5ce68c7b7ad706": "\n\\lambda\n=\\left(\\frac{\\textrm{mes} E}{2C \\mathrm{mes} J}\\right)^{n-1}e^{-\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J}\\max_{x \\in J} |p(x)|\n", "3844d65a5d7b5fa49dc96b6782b6eaf8": "p = a_0^2 + a_1^2 + a_2^2 + a_3^2\\ ", "3844f42df538bb2095245ec1b700797c": " c (\\mathbf{A}+\\mathbf{B})=c\\mathbf{A}+c\\mathbf{B} ", "38451634aeba48683d07185295326f12": "a, a + \\Delta x, a + 2 \\Delta x, \\ldots, a + (n-2) \\Delta x, a + (n-1) \\Delta x, b.", "38455af16fb78e6da9b78f481c59a7ed": "\\log(z)", "3845cb00abe3eeab586e0d932a89a624": "\\omega(x \\oplus \\eta, y \\oplus \\xi) = \\xi(x) - \\eta(y).", "3845ebcf4ae0887953c83568bd257dad": " s_n ", "3845f595f7ae96f9ffde254e85be52f9": "\\mathit{n}\\mathit{I}", "384610f0cfd9247a80ed8f9a934ed06f": "= \\left( \\frac{1 - e^{-sT}}{sT} \\right)^2 \\ ", "38461dbcbaa5e1b425c24bde57205543": " E_s = (n + s)\\hbar \\omega - W,", "38467b629acb35f8fa4f0989b3e384fb": "\\Delta G^{\\ddagger} = \\frac{(\\lambda_{o} + \\Delta G^0)^2}{4 \\lambda_{o}}", "38467e910b545eea4fe33ea42a6bcea9": "A= \\exp i \\mathfrak{t}, \\,\\, P= \\exp i \\mathfrak{u},", "38471c1541bc8e5e55060f6f88ced515": "\nZ_i \\sim N(0, n)\n\\,\\!", "3847278728fac89386bff38cf5577596": "|F(z)|0, \\exists N\\in\\mathbb{N}: \\forall k\\geq N \\Rightarrow \\rho(A)-\\epsilon < \\|A^k\\|^{1/k} < \\rho(A)+\\epsilon", "384c8a3ede8bdda1af217100758b633d": "Z=\\frac{1}{2}\\rho U^2 c_L S_w=W", "384d29d278c35ab6e44133b8839019ea": "\\ln ([A]) \\ \\mbox{vs.} \\ t ", "384d38d56e079495094671133fc243a4": "\\alpha_i^\\vee", "384e709be1b74df77bee711c70210789": "\n\\text{(Eq. 8)} \\qquad E\\left[\\lambda_n^{(c)} + \\sum_{a=1}^N\\mu_{an}^{*(c)}(t) - \\sum_{b=1}^N\\mu_{nb}^{*(c)}(t)\\right] \\leq 0 \n", "384ec47de24a5d9f8051935cee02b649": "\\textstyle N_2=\\frac{C_2}{C}", "384ec61c02dd103fa1b957c682654932": "g_{\\rm centrifugal} = -\\left(\\frac{2\\pi}{T}\\right)^2 r_{\\rm eq}", "384edd1871e80d06bedf200d4509e7a0": "Y_{t-i}", "384efea71b814c44eeec7e18b9096563": "(i\\cdot n) \\hat{f}(n)", "384f15f8393a50ee9960225d3a603222": "Dynamic \\ Range = Smaller \\ Of \\begin{cases} \\tfrac{Carrier \\ Power}{Noise \\ Power} & \\text{Transmit Noise, where bandwidth is} \\tfrac{PRF}{Filter \\ Size}\\\\ \\\\ 2^\\left(Sample \\ Bits + Filter \\ Size \\right) & \\text{Receiver Dynamic Range} \\end{cases} ", "384f324969e988ba956ca3e12fd55532": "1/CharLength^\\alpha , \\alpha < 1 ", "384f43137b3594a940708ed9f9c1698a": " r_1 = -e x - a\\,\\!", "384f4d6ffb816e9ce44c17965484a42c": "\\begin{vmatrix} A & B/2 & D/2\\\\B/2 & C & E/2\\\\D/2 & E/2 & F \\end{vmatrix}", "384f5fdf3053b4c7a7a3969eb7a795ea": "O(n ^ {\\log_b a})", "384f6030041d7eea9d2a3b0b30f9e2e7": " \nE[\\Delta(t)] + Vp_{min} \\leq B + C + Vp_{max} + \\sum_{i=1}^KE[Q_i(t)](-\\epsilon) \n", "384f92ae73d0af481b8a952b5ad65ea7": "ee''=\\frac{[R']-[S']}{[R']+[S']}=\\frac{ee(1-c)}{c}\\implies ee=ee''\\frac{c}{1-c}", "384fae692eb8f75a4648a67ccd2dc11c": "\\alpha_m=90^\\circ +\\frac{\\alpha_r+\\alpha_b}{2}", "385041c67c1082b017f748c68e3998e8": "2^\\kappa", "385050f14524a5c46e56de108af90865": "x = 1+t = 2, 0, \\frac{4}{3}, \\frac{2}{3}", "38508a0cc50810223190efb350aec9bf": " \\mathfrak{H}", "3850a1770a11bd36bda02e5b02736796": "\\dot{\\hat{z}} = A \\hat{z}+ \\phi(y) - L \\left(C \\hat{z}-y \\right) ", "3850bddd94cbcdf18109896e0b83584f": "dn_s / dlnk", "3851159900b50b54d95aab54a7e09468": "\\pi=\\pi_a\\pi_b", "3851179cc562d7281eb0ec2b0356d50f": "q=\\exp(2 \\pi i \\tau)", "3851194559a2c218ca89eeb201ce5890": "A_{N}", "385119c12bd872e7359e3a5a518223d0": " f_X(x) = \\frac{d}{dx} F_X(x) .", "3851522e6ab8314cc0511ae43e2c6098": " c_1 = \\frac{ \\beta^2 ( 1 - p_0 ) }{ n p_0 \\log_e( p_0 )^2 } ", "3851e44a582563287a9674ca243616ad": " \\varepsilon_0 = 1/\\mu_0 c^2 \\,", "3851fb20d15439f8851a225255c075c8": "f(\\cdot)=\\langle \\cdot,y \\rangle", "385238d95a979c334f7722085c31929f": "\\begin{align}\n I_{\\text{E}n} &= \\left. -q A D_{\\text{E}} \\frac{d \\Delta_{\\text{E}} (x'')}{dx} \\right|_{x''=0''} \\\\\n I_{\\text{C}n} &= -q A \\frac{D_{\\text{C}}}{L_{\\text{C}}} n_{\\text{C}0} \\left(e^{\\frac{q V_{\\text{CB}}}{kT}} - 1\\right)\n\\end{align}", "3852543e416dd8de113be61788fe943c": "g_{ab}\\,g^{bc} = \\delta_a^c = g^{cb}\\,g_{ba}", "385283616774ae20d4452432d48a5f93": "\\frac{f(x)}{g(y)}", "38529f6ab344b40a57d13dea8bcc03ce": "t = RC\\ln(3) \\approx 1.1 RC", "3852c24fc5f632a5b0e1901eb334d777": " M= \\frac{\\hbar ^2}{2 m} \\int_{z=z_0} ( \\chi*\\frac {\\partial \\psi}{\\partial z}-\\psi \\frac{\\partial \\chi*}{\\partial z}) dS ", "3852f29111da472f358dab01f1fce3f0": "-A", "385303fcef97c7fa2dcf4a50c2d3e6ce": "ref1", "38531a276799a675d43412a468341a32": "h_p", "38535a643ea0ae97acf2ec36eef4b1d4": "A \\leq B_1, A \\leq B_2", "3853b49dd9208d346253a4955bf038f7": "m_1,...,m_k", "3853e7f0a1409aac8e8efe70507abc3e": "-(-2)=2", "3854145920df804da9a73f425cb39c73": "Wo = \\frac{}{}B_n P V F ", "38541c981e1fa3cd164f376ec49abd40": "\\sum_{l=0}^{N-1}\\tilde{V}^2_N(\\omega-\\frac{2\\pi l}{N})=1", "3854f7542f619e1f2fb0a35abb16c442": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) \\ge 1,\\,", "3854faf1c4dbb0085221ff17d14fb865": "\\mathrm{C^{\\alpha}_{i}}", "3854fc720743a13946e2a53b3f117f98": " I_\\mathrm{p} = \\mathrm{d} \\left | \\mathbf{p} \\right |/\\mathrm{d} t \\,\\!", "38551a2d373efb813b85ba25b65b4217": " \\textstyle P(\\theta \\mid k, n) = \\mathrm{\\Beta}(k + 1, n - k + 1).", "38552dcf8262a9664630682994cd9f7f": "{\\mathbf y}-{\\mathbf x}=(y_1-x_1){\\mathbf e}_1 + (y_2-x_2){\\mathbf e}_2 + (y_3-x_3){\\mathbf e}_3.", "385608af7801f4549eb099fa7553a2fb": "df=n-1 \\ ", "38560d3d403d928e8b6a1e4e8f9e93e1": "(3)", "38562b7849d60a5eec1425bc1f75b83d": "v(x, y) = \\frac{1}{\\pi}\\int_{-\\infty}^\\infty f(s)\\frac{x - s}{(x - s)^2 + y^2}\\,ds", "38566bde510fe12cece471fc9119522b": "\\frac{1-\\sqrt{1-4x}}{2x}\\,.", "38567b08fb70a968bb9219636447fb90": "\\langle\\varphi_n,\\psi\\rangle\\to \\langle S,\\psi\\rangle", "3856a5100d91c318a8e245a2c533f611": " Upper~limit = e^{Log_e (upper~limit)} = e^{1.85} = 6.4", "3856e11333f843f1ef8ffd943cb196d7": "w(z) = w_0 \\, \\sqrt{ 1+ {\\left( \\frac{z}{z_\\mathrm{R}} \\right)}^2 } \\ . ", "3856e374759891cee2b252b9205956e3": "I_{k} = \\frac{\\lambda}{k} \n\\frac{\\cos k\\theta^{\\prime}}{\\left( \\rho^{\\prime} \\right)^{k}},", "38570ec6519bdfa5f2715757610c82a3": "L_{\\mathrm{core}}+L_{\\mathrm{gap}}\\,", "38571f00fc6d6cef046de899ac4fd45a": "1-fa", "38576b4ebfc21f33aa082956cf3f5cb6": "\\begin{align}\n\\frac{\\partial P_1}{\\partial v} - \\frac{\\partial P_2}{\\partial u} &= \\left\\langle \\frac{\\partial (\\mathbf{F}\\circ \\psi)}{\\partial v} \\bigg| \\frac{\\partial \\psi}{\\partial u} \\right\\rangle - \\left\\langle \\frac{\\partial (\\mathbf{F}\\circ \\psi)}{\\partial u} \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle \\\\\n&= \\left\\langle (J\\mathbf{F})_{\\psi(u,v)}\\cdot \\frac{\\partial \\psi}{\\partial v} \\bigg |\\frac{\\partial \\psi}{\\partial u} \\right\\rangle - \\left\\langle (J\\mathbf{F})_{\\psi(u,v)}\\cdot \\frac{\\partial \\psi}{\\partial u} \\bigg|\\frac{\\partial \\psi}{\\partial v} \\right\\rangle && \\text{ Chain Rule}\\\\\n&= \\left\\langle \\frac{\\partial \\psi}{\\partial u} \\bigg|(J\\mathbf{F})_{\\psi(u,v)} \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle - \\left\\langle \\frac{\\partial \\psi}{\\partial u} \\bigg |{}^{t}(J\\mathbf{F})_{\\psi(u,v)} \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle \\\\\n&= \\left\\langle \\frac{\\partial \\psi}{\\partial u}\\bigg |(J\\mathbf{F})_{\\psi(u,v)} - {}^{t}{(J\\mathbf{F})}_{\\psi(u,v)} \\bigg| \\frac{\\partial \\psi}{\\partial v} \\right\\rangle \\\\\n&= \\left\\langle \\frac{\\partial \\psi}{\\partial u}\\bigg |\\left ((J\\mathbf{F})_{\\psi(u,v)} - {}^{t} (J\\mathbf{F})_{\\psi(u,v)} \\right )\\cdot \\frac{\\partial \\psi}{\\partial v} \\right\\rangle \\\\\n&= \\left\\langle \\frac{\\partial \\psi}{\\partial u}\\bigg |(\\nabla\\times\\mathbf{F})\\times\\frac{\\partial \\psi}{\\partial v} \\right\\rangle && \\left ( (J\\mathbf{F})_{\\psi(u,v)} - {}^{t} (J\\mathbf{F})_{\\psi(u,v)} \\right ) \\cdot \\mathbf{x} = (\\nabla\\times\\mathbf{F})\\times \\mathbf{x} \\\\\n&=\\det \\left [ (\\nabla\\times\\mathbf{F})(\\psi(u,v)) \\quad \\frac{\\partial\\psi}{\\partial u}(u,v) \\quad \\frac{\\partial\\psi}{\\partial v}(u,v) \\right ] && \\text{ Scalar Triple Product}\n\\end{align}", "3857cc8717e691ed9d1ec3eab2814fc2": " T_{00} = - {T^{0}}_0 = \\sum_{i=1}^N \\left ( {\\gamma_i m_i c^2 \\over V }\\right ) ", "38580b412e0455229bc5d43498b3de74": "1 \\over 1 - \\alpha", "385847b1b83f8b7f4e8102a639eb2967": "\\frac{ln[A]_{f}}{ln[A]_{i}}=-kt", "38584e8667385b38dc8122c72a8254db": "2\\leqslant i\\leqslant k", "38586d50128e37bb94e79220ec33cf83": "\n\\int_{-\\infty}^{q_0} P(q | \\vec{y}) \\, dq = \n\\int^\\infty_{q_0} P(q | \\vec{y}) \\, dq\n", "38594d73ee432b280ae2c0131e29c60c": " q_0 \\in Q", "3859830072bc2a21d07d6982edebeeaa": "\\chi\\alpha", "3859b3a7030a5ad0c4852aced228b502": "f_X(x)", "385a1942e07744ce247863b197452f34": "f(x) = a(x - r_1)(x - r_2) \\,\\!", "385a30f7d7bffe12aa197e66923d84b8": " Pr = \\nu/\\alpha = 1", "385a3b1f5e4192df941cde71cc1427a1": "\n\\left[ {\\begin{array}{*{20}c}\n q & {q'} \\\\\n \\bullet & \\bullet\n\\end{array}} \\right]\n", "385a4584abcaba4455c3d9a8f17caf3f": "\\begin{bmatrix}1 & 2 \\\\1 & 1 \\end{bmatrix}. ", "385a4fb1140dba383515b1505e78b1fe": "\\frac{25}{18}", "385a5466c829b27f3e2f542902e09fae": "w\\Vdash\\bot[e]", "385a57b96cbf28a730eca82fb6504086": "[[a,b,c],d,e] = [a,b,[c,d,e]] .", "385a7163b74bc051409752b9e42a072a": "(\\mathbf{\\xi} \\cdot \\nabla) \\mathbf{B} = 0 ", "385a93dd9550fe22644b408b05f8bd98": "G(z) = \\frac{z}{1 - G(z)}", "385ab15acc80d4d3f5fe3925ec188a1e": "\\omega_{\\xi}, \\omega_{\\eta}", "385adf76ea3d067f98b9dd9b5d196fa6": "T =\n\\begin{cases}\n\\frac{1}{2\\pi} \\arctan{\\frac{r'}{g'}} + \\frac{1}{4}, & \\mbox{if}~g'>0 \\\\\n\\frac{1}{2\\pi} \\arctan{\\frac{r'}{g'}} + \\frac{3}{4}, & \\mbox{if}~g'<0 \\\\\n0, & \\mbox{if}~g'=0 \\\\\n\\end{cases}\n", "385af59cb03f5e953f37d56df507e0dd": " f_i (\\Delta ,x) = \\operatorname{Tr}(Q_i \\Delta ) + 2g_i^T x + d_i. ", "385afc0df30ab25975fca0a197edb6fb": "x = 10 - 6 = 4", "385b453d51dd8328584c601748de21d8": "p_s = \\rho_m\\, R\\, T,\\,", "385b9b794a04da5227f5a01aa6ebfc2f": "\\overline{r}(\\lambda)", "385c0fffc7f8ab273c8ae16139aed1a9": " \\dot{\\chi}=-i[H,\\chi] ", "385cd905b6675a4c824888d96c7a2c90": "R_{ab}={R^c}_{acb}", "385cef4e97758d43527cef3312e702aa": "Q_3 = L_{14} + L_{23}.", "385d56d7a62e73571674580065f0b79f": "T_{n+1}(x)=x\\sum_{k=0}^n{n \\choose k}T_k(x).", "385da14bfd978737d2d99f2a45b2afa4": "H(x_1,\\dots,x_d) = C\\left(F_1(x_1),\\dots,F_d(x_d) \\right), ", "385da8fae621976717e2e95901beedcf": "j_0(x) = \\sin(x)/x", "385de06d8f0f20448a5caa7b934ff2c3": "x^2-y^2=a^2,", "385de5d9d2b06e3c9d9cfc9bd18ecb81": "\\epsilon_m = \\frac{c}{\\rho}", "385e0bc095ecc941feaba9085d6ee1f0": " dH = \\iota_{X_H}\\omega ", "385e0c52f3257781f9736850f867b8ef": "0.2 K_u", "385e4ca92db44c9d6af60806573f6e32": " H_d ", "385e584fe3b3b16bc3e3bf0799a3afd7": "O\\left(h^3\\right)", "385ea949af784d4f7e2a617e58390059": "\\mathrm{d}\\mathbf{F} = \\mathrm{d}q\\left(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}\\right)\\,\\!", "385ec26380d0f2688060ee7622c0dfd1": "r(x,t)", "385f68edc6baf1d0f8a36e096e97228f": "u(c)=1-e^{-a c}", "385f7c4b3fbfe92286109d95087cc946": "2 \\pi r ( r + h ) = 6 \\pi\\,r^2", "385fe1db2487671a77450f1410c9c8fe": "d\\int \\langle \\rho(g)\\cdot v, w \\rangle \\overline{\\langle \\rho(g)\\cdot x, y \\rangle}dg =\\langle v, x \\rangle\\overline{\\langle w, y \\rangle}", "385fe479d14e8d3f9d7e5b58a65d1ef9": "t \\rightarrow Q(t)", "385ff5d72bd0434b8dfc9a799dbdbfee": "(a \\rightarrow c)", "385ffa4757e20a3f137f4e7af6185af9": "(P_1)", "3860124fd647104561eae5ae0ef4ec05": "\\scriptstyle\\hat{Y}_{\\!-i}", "38601490e3a6810e7626cd0d68ad09bf": "U{}^r_{n-1}", "38604fe6772afac0f0b19b7a5d5ae73c": " \\lambda_{\\pm} = \\sigma^2(1 \\pm \\sqrt{\\lambda})^2. \\, ", "38606b7488b494641e3896a87505f8f8": "a_{44}+a_{41}+a_{14}+a_{11} = 34 ", "386083bab670bfcd2356b7bf838d05da": " \\textbf{G}_\\textrm{hkl} = h\\textbf{a}^*+k\\textbf{b}^*+l\\textbf{c}^*", "3860d8022b6b002501188fb56e6ce081": "\n\\bar{G}_{ij}(s)= \\bar{\\Gamma}_{ij}(s)\\frac{\\bar{\\Phi}(s,\\tilde{L})}{\\bar{\\Phi}(s,L)}\\bar{\\Psi}_i(s).\n", "386162afe7aabe57c5397fb28cba17d9": "\n\\Delta f = {2v\\over c} f \n", "3861698a9b858de916a20df4a7f2ba0a": "B \\land h", "386170c87e7f0d113e4fa68678c16b01": "x=\\frac{-\\tfrac{1}{2}\\pm\\sqrt{(1/2)^2+4(1/2)(F r_1^2)}}{2(1/2)}=-\\frac{1}{2}\\sqrt{(1/4)+2(F r_1^2)}", "3861b74cf2716b71c2a02eae2edf8076": "M(S, K, \\tau, r, \\sigma),", "3861b88a922062bdbd2d6c5ac666bd65": "|Q_{P}(h)-\\widehat{Q_{r}}(h)|\\geq\\epsilon\\,\\!", "3861d21058b637948fd524842a634730": "p_{i}(t)", "386258088e7bf8695f9a0cd3f61d1d9a": "({\\lambda}_{1},{j}_{1},{c}_{1})=({\\lambda}_{2},{j}_{2},{c}_{2})", "3862b6f831e2a704b2dae104b0c7fd7d": "\\textstyle l_2-1 < l", "3862bdca0a465a52651c85b23c6f4fba": " AP^{-1}Px = b", "3862e7218719a6312afcba83fd594a3e": "\\; T w = \\alpha v_1", "3862fe39559e77e6674121665851097c": " \\left\\{ z \\in \\mathbb{C} \\left| \\ \\left| \\tfrac12 \\Big(1 + \\tfrac32 z \\pm\\sqrt{1 + z + \\tfrac94 z^2}\\Big) \\right| < 1 \\right.\\right\\}.", "3863067e6d7d7cf442cf51d04f8bbd3d": "y \\in \\mathbb{R}^{n}", "3863d020e2fe4c2055dc3a1d67b3c17d": "\\|f-f_k\\|_{\\infty,X}=O \\left (k^{-\\frac{\\alpha}{1-\\alpha}} \\right ).", "3863e3abd5ef566c266c3b88531f288b": "E_z=-\\mu_B g_j B m_j", "386431af344e79a4cff5e05f96d436f6": "\\beta(\\xi\\cdot\\varphi,\\psi) + \\beta(\\varphi,\\xi\\cdot\\psi) = 0", "38644a80fbc2296dce6b24eee7071142": "d\\Phi_E = \\mathbf{E} \\cdot d\\mathbf{S}", "386468d86e0433214ca10326155be475": "P \\pi = \\pi \\quad \\text{ and } \\quad \\mathbf e^\\text{T}\\pi = 1", "386503f6ffc577b2743f162b4f780fbd": "C \\approx \\frac{a}{2} \\sqrt{93 + \\frac{1}{2} \\sqrt{3}}", "3865137da9e00dfa5cb51f1a96b8c38d": "p_1\\in K_1", "386558efedd72de8253d993467bba0bc": "\\nu=0.35", "3865d1a4a48e2b1b7d2ece55704f295e": "B \\leq 1", "38664097aee46e7819326b306ed39896": "x^2 + y^2 + z^2 = 1 \\,", "3866aaa043367ab05017cad8c16d97f8": "Nx^2 \\pm\\ C = y^2", "3867029c5a09be48a64b12cafae19aa5": "L_{q}\\left[1/4+\\epsilon,c\\right] ", "38671ccbd2f9c6fa8824cf253b0c50de": "u(\\mathbf{r},t) = \\mathrm{Re} \\left\\{ \\psi(\\mathbf{r}) e^{j\\omega t} \\right\\} ", "386778ec8358692789d50db6186566d4": "\\frac{c\\Delta t}{2}", "38677ba48626d0292892a7522b9bde4b": " m_e ", "38678598a687fc44824e6f7f4e89744c": "m>30%", "38679ee18640029bc60ea53933d104fa": "\\textstyle Q_jA_j", "3867b90ef4d9850cafd116953e690c94": "\n\\begin{array}{ll}\n \\dot{x} = Ax + bf^1(\\theta^1(t))f^2(\\theta^2(t)),& \\dot\\theta^2 = \\omega^2_{free} + Lc^*x.\n\\end{array}\n", "3867c4747187048ae499898c008fab15": "{\\hat \\theta_w}", "3868d400bf2c6ca4f3847bfe9ee5e9d2": "c+d\\omega", "386903e73dd3197030a944eb34711ade": " \\Phi^4", "386916d7836d709a5a8b7a5ba9fa463d": "h_{00}, h_{10}, h_{01}, h_{11}", "386926c119ceed413a0c2b49294e0edc": "\\lim_{t\\rightarrow 1}\\gamma(t)=e^{i\\theta}", "38694eb4a26963cfa7a1385172c54ec9": "(a_x, a_y, a_z).", "3869de19c5b3bae48838106220fa1399": "(\\dot{r},\\ \\dot{\\theta})", "386a43542bd6753030e6b71d9aa1803c": "\\sum_\\text{sym} x_1^{\\alpha_1} \\cdots x_n^{\\alpha_n}", "386a80605bbbaaef5e9776f22424b46f": "0\\leq g(t) 1_{A_t}\\leq g(f(x))\\,1_{A_t},", "386ad91225dc94f2cf32ceac3268940c": " S_1 ", "386b201cd228e5ac7f298251affd73ff": "(w\\div a)\\equiv (v\\div a)", "386b41023cc10cd7c2217b04373f14b6": "8 |E| > |V|^2 / 8", "386ba1b12ce219a2245c030c0d2f3f7a": "\\Gamma\\left(z\\right)", "386ba3ac7486f1dfb14c0455768739ac": "C-\\frac {Yi} {2{N^2}}=0", "386bb34735b6ffe7749170d6bfa8e4e0": "\\left(\\frac{\\sigma_f}{f}\\right)^2 \\approx \\left(\\frac{\\sigma_a}{a}\\right)^2+\\left(\\frac{\\sigma_b}{b}\\right)^2+2\\left(\\frac{\\sigma_a}{a}\\right)\\left(\\frac{\\sigma_b}{b}\\right)\\rho_{ab}.", "386bc4001c95e516922702c1a7aac455": "\\varepsilon = \\mu", "386beafa9df7b09cae57b20a1b61b3db": " 20(0.6)(0.5)= 6 ", "386c00765aecee74c37330e2a66932aa": "\\vec{a}\\cdot\\mbox{grad}\\,f = a_x {\\partial f \\over \\partial x} + a_y {\\partial f \\over \\partial y} + a_z {\\partial f \\over \\partial z} = (\\vec a \\cdot \\nabla) f ", "386c17f9c122f90e2e92c4c093b5d663": "\\,= \\frac{ \\overbrace{ e^{-i[\\omega - \\phi] z}+e^{-i[\\omega + \\phi] z}}^{negative.frequencies}+\\overbrace{e^{i[\\omega - \\phi] z}+e^{i[\\omega + \\phi] z}}^{positive.frequencies}}{4}", "386c33f6475e6edf7e46cffc7fd44f37": "s^2_n = \\frac{M_{2,n}}{n-1}", "386c66e518994f22c36f3e2f90dafd55": " |S_{ij} | \\le |p| ", "386cbc801dced23c5e3229291c2d4ccc": "\\beta\\gg 1\\!", "386cf9b00ded52f4fd40bb381e64ba25": "\\int_0^{|E(2\\omega)|l}{\\frac{d|E(2\\omega)|}{E_0^2-|E(2\\omega)|^2}}=-\\int_0^l{\\frac{i\\omega d_{\\text{eff}}}{n_\\omega c}dz}", "386d0bda1eb7aa12fca538b013ed6495": "\\alpha_7", "386d154d4864eb36fb5e338897a59ea5": "\\begin{align} \nx_0 &= \\text{distance radar to target} \\\\ \n\\lambda &= \\text{radar wavelength} \\\\ \n\\Delta t &= \\text{time between two pulses} \n\\end{align} ", "386d7da3ef4d138d69b6504345559df0": "\\frac{1}{2\\pi i} \\int_{-i\\infty}^{i\\infty} \\Gamma(a+s)\\Gamma(b+s)\\Gamma(c-s)\\Gamma(d-s)ds\n=\\frac{\\Gamma(a+c)\\Gamma(a+d)\\Gamma(b+c)\\Gamma(b+d)}{\\Gamma(a+b+c+d)}.\n", "386d9a31a4cf6bc974cf3e430facfb36": "\\exists_f S \\subset Y", "386d9b96b67d29388ae134d1747b68cd": "\\textstyle x^i b(x)", "386e159ed91368d3a9207e37d78daa15": "ax^2 + bx + \\frac{b^2 - 2b - 8}{4a}", "386e1b364daff05be13192f96508e458": "q (1 - \\cos \\theta) \\hat{z}", "386e3286e8f4d7933dc8c5b5925c88fd": "s\\in\\Sigma^*", "386e61dd6d1bba7ff4d6c8d42f1c17b8": "\n\\begin{align}\n \\frac{1}{9} & = 0.111\\dots \\\\\n 9 \\times \\frac{1}{9} & = 9 \\times 0.111\\dots \\\\\n 1 & = 0.999\\dots\n\\end{align}\n", "386ea2a8bf9bfd7120d4b734c954671b": "0\\to 2\\pi i \\mathbb{Z}\\to \\mathbf{O} \\xrightarrow{\\exp} \\mathbf{O}^* \\to 0", "386fa62da0f4775d463768320afd6960": "q^{3}", "38702b826bb904c3e7f169f5f293d594": "\\mathcal{M}=(S,\\rightarrow,L)", "387068dcb2f4f6873878cb8dc38936d5": "\\mathbf{r}_i\\,\\!", "3870d8e5e4240020e0cbd3988e5f5d5c": "p_1 0", "38a1d619722d707d6818066d41a2af01": "\\{i\\} \\,=\\, \\gamma_{i} \\, [i] ", "38a239f9018e4bd069c66f982c58ff54": "f^*", "38a2941519702c5b5c31ca57a815fdbc": "\\frac{a-b}{a+b} = \\frac{\\mathrm{tan}[\\frac{1}{2}(\\alpha-\\beta)]}{\\mathrm{tan}[\\frac{1}{2}(\\alpha+\\beta)]}.", "38a2c0c5166004cdc693a6331808ef03": "\\cup_{v\\in B} E_v", "38a2d6fd932e6c994d69be3d8baafbd4": " \\lbrace . , . \\rbrace ", "38a2ff378349e07a2b2e3826987080fe": " - \\textstyle \\frac {1}{2} \\Delta E (T^3) + \\Delta F + \\Delta H^\\circ_{form 298} ", "38a378c1d415844628d9d2c95b3cb196": "\n\\mathbf{j} = (\\mathrm{j}_x,\\mathrm{j}_y,\\mathrm{j}_z)\n", "38a38a2711350b8c1828bc29822d3d22": "L = \\frac{P}{1+i}+\\frac{P}{(1+i)^2}", "38a3a0736d73c85ae1aedc5375a8057f": "f(x)=72 \\Rightarrow \\int 72 \\, dx = 72x+c", "38a3ba8abeaf509e465af81a5c14d80f": "E=\\sum_\\gamma w_\\gamma H_\\gamma", "38a3be6d5ea9af5ea394614bd3034408": "B_1 = \\{ |a_{i}\\rangle_{i=1}^d \\} ", "38a3ce1c52e483bc04b6c0af08e4ea9c": "F_y = \\gamma m a_y \\,", "38a3f47f99878442cc83b2cca72c6849": "u_{tt} = c^2 \\left[u_{rr} + \\frac{2}{r} u_r \\right].", "38a3f48b79fd5aa74c5cd8696596e0e2": " \\sum_{j=1}^k m(r,a_j,f) \\leq 2 T(r,f) - N_1(r,f) + S(r,f). \\,", "38a3fd2e7fd149052b3de877e76d05bf": "\\tilde{P} ", "38a4bf700a471b63e61078111853fa55": "\\scriptstyle \\frac{12\\sqrt{5} - 7}{61} ", "38a4c7b8e561372de443559376ebbf8d": "\n\\boldsymbol\\mu_{X,Y}\n=\n\\begin{bmatrix}\n \\boldsymbol\\mu_X \\\\\n \\boldsymbol\\mu_Y\n\\end{bmatrix}, \\qquad\n\\boldsymbol\\Sigma_{X,Y}\n=\n\\begin{bmatrix}\n \\boldsymbol\\Sigma_{\\mathit{XX}} & \\boldsymbol\\Sigma_{\\mathit{XY}} \\\\\n \\boldsymbol\\Sigma_{\\mathit{YX}} & \\boldsymbol\\Sigma_{\\mathit{YY}}\n\\end{bmatrix}\n", "38a50ba862b2f62cc9367250cc1c2738": "n_2 > 0", "38a535eb8bc5d1d3e5128e30d098fad7": "R_\\mathrm{int}", "38a547b9b720022539717951a51ab8e7": "r=2s", "38a5a187202ab5f1161b927270569533": "C_*", "38a5e5620676feb3b54b2c112dac8e2a": "\\displaystyle{U(x)V(y)=(x,y) V(y) U(x).}", "38a5e779a1c7285e6e1fededde57376c": "H= G + ST \\,\\!", "38a5f53dba4a9c078459892cd421dc98": " t_a = t + \\frac{|\\mathbf r - \\mathbf r'|}{ c}", "38a60bd36ee6bd2db79c33d79189d67e": "\\mathrm{GL}(n,\\mathbb{C}) \\subset \\mathrm{GL}(2n,\\mathbb{R})", "38a60cc41b6fd00d9ef297fdf4a1d267": "x^5+x^4-4x^3-3x^2+3x+1", "38a6bd777c56272a6572ea5caed5f00a": " \\scriptstyle \\mathbf{\\hat{e}}_\\theta \\,\\!", "38a6d761d234b8e864b78f3ca55da7e6": "s_\\lambda=\\sum_{\\rho=(1^{r_1},2^{r_2},3^{r_3},\\dots)}\\chi^\\lambda_\\rho \\prod_k \\frac{p^{r_k}_k}{r_k!},", "38a70c2609c8a85d144db90be4dff9dd": "a_0, a_1, \\ldots, a_{1,2,\\ldots,n} \\in \\{0,1\\}^*", "38a77f8ec08631da0791f235dc7fb144": "m(\\cup J_n^*)\\le \\sum m(J_n^*)=2\\sum m(J_n) \\le 2\\lambda^{-1} \\mu^{-1} \\|f\\|_1.", "38a7f9b6700351d11f0aea0ec32c0c54": "x,y\\in \\bigcup_{i\\in I} A_i", "38a7fcc6170eeef069f2941ef1c00414": "\\min \\left\\{(3 - \\cos \\theta)^{-1} (2 + (2/\\pi)\\theta) \\,:\\, \\pi/2 \\leq \\theta \\leq \\pi \\right\\} = 0.943...", "38a8003e7b038fe687e0dfb4c1a6da69": "(\\beta_1,\\beta_2,\\ldots,\\beta_n)", "38a88da6b738067deae4d6c3a5bd8771": " \\wedge \\varphi'.\\ ", "38a88f2c517939cb9e2b065184a9a14d": "t_a", "38a8bdc11ec771582b67553c71ae7cc1": "\n \\tfrac12 K_{ijk} + J_{j,ik} = \\operatorname{E} \\bigg[\\; \n \\frac12 \\frac{\\partial^3 \\ln f_{\\theta_0}(x_t)}{\\partial\\theta_i\\,\\partial\\theta_j\\,\\partial\\theta_k} +\n \\frac{\\partial\\ln f_{\\theta_0}(x_t)}{\\partial\\theta_j} \\frac{\\partial^2\\ln f_{\\theta_0}(x_t)}{\\partial\\theta_i\\,\\partial\\theta_k}\n \\;\\bigg].\n ", "38a8f16555c7333aa9f6e7240daf41e8": "\\chi(\\tau,f)=\\int_{-\\infty}^\\infty s(t)s^*(t-\\tau) e^{i 2 \\pi f t} \\, dt", "38a907a4cf285fbed210620833a78294": "\\ ASA_{unfolded}=a_{polar}*ASA_{polar} + a_{aromatic}*ASA_{aromatic}+ a_{non-polar}*ASA_{non-polar}", "38a95cae2041f74c426221d76b1381f8": "c\\boldsymbol{\\beta}", "38a981aa7bcc360d35a2db66904f3efa": "(\\sigma,", "38a9c8f18346eea34d292cc246b467dc": "P(i)=\\frac{|U_i|}{N}", "38a9d4ca302f0aecbe966785c0f70b87": " q' q = \\gamma:\\alpha = OC:OA . ", "38a9eaf2c46a436e17cf4f419e123bf6": " [n,k_2]", "38aa6bc95a2b62f79d61b6691d54080f": "d=5", "38aa843ab94960e5a7e8a9c1d4d44dd0": "\\dot{f}_0(x)=0", "38aae1aa56ba5bdcfc0ba0f97c3c5aeb": "m_H c^2 = \\frac {Gm_e^2}{r_e}", "38ab2ffafebbbeba1993e94b3c349a3c": "F(2) + F(4)", "38ab48f00a2a07136f6e12e3a4b692ce": " c = {1 \\over \\sqrt {\\mu\\epsilon} } ", "38ab54605c4e9cac922c64d45fe7955d": "1 \\to SK_1(A) \\to K_1(A) \\to A^* \\to 1,", "38ab78d9081ae776f4720faa3d1c4b03": "W_x(t,f) = W_y(t-af,f) \\, ", "38abaaa88fcb3ee8f25f18416ec9e9ca": "n\\to0", "38ac200f4e030a992813068c94730c57": "\\int\\limits_{-\\infty }^{\\infty }{{{P}_{\\theta }}f(u,\\xi )}.du={{\\left| \\hat{f}(\\xi ) \\right|}^{2}}", "38ac2f07134d7fc8e2462df1dbd28304": " F(L,K) = K^2 + L ", "38ac917cf4bf51a0460480aeae962c49": "H^3(M, \\mathbb{Z})", "38ac99273e80fa66c35402bb58ea0c8f": "\\textstyle \\mathcal{C} ", "38acd8cb822369148faa6d4d70b881fd": "\\epsilon(p)", "38acf8641060e25b2f98403accf0ec74": " H_A \\otimes H_B ", "38ad1ba86318f6b93670cad322de4468": "\\begin{align}\nf(s) & = 1 - \\left(\\frac \\mu {r_0}\\right) s^2 c_2(\\alpha s^2), \\\\\ng(s) & = t - t_0 - \\mu s^3c_3(\\alpha s^2), \\\\\n\\frac{df}{dt} & = \\dot{f}(s) = -\\left(\\frac{\\mu}{r r_0}\\right)s c_1(\\alpha s^2), \\\\\n\\frac{dg}{dt} & = \\dot{g}(s) = 1 - \\left(\\frac{\\mu}{r}\\right)s^2c_2(\\alpha s^2)\n\\end{align}", "38ad88e1dc14aac34bf8de5e3c3d76cd": "B=X^TNX=\\int_{-1}^{1}[x_1^{j+i}]_{i,j=0}^{i,j=m-1}dx_1=X^TX", "38add72e7534a393cfa2a7dbb90272c1": "H(\\Sigma_1,\\Sigma_2,\\ldots,\\Sigma_n)", "38ae20530c6210cabe4dd6276a63ab09": "[S_i, S_j ] = i \\hbar \\epsilon_{ijk} S_k", "38aea78f795e25f3f3b47518f4a4c439": "\\int_{\\Lambda^n}\\theta_{n}\\cdots\\theta_{1}\\,\\mathrm{d}\\theta=1,", "38aeb82b9b1376ee53d4f94255c84f1b": " \\operatorname{tr} \\mathbf{A} = \\operatorname{tr} \\mathbf{A}_1 +\\cdots +\\operatorname{tr} \\mathbf{A}_n.", "38aebb31b49fccbbdc04cbd019c30355": "\n\\text{ if } a_i = b_j\n", "38aec0e2ecf719a266720d178d241358": "\\mbox{EF }\\big(r \\mbox{ U } q\\big) ", "38aecddf4168dca86fe5523dbd09007d": "a^\\dagger|n\\rangle = \\sqrt{n+1}\\,|n+1\\rangle", "38af52d33f83fe6be085ab779bee2807": "v=v^i(x)\\partial/\\partial x^i", "38afad761739c68e1811b1e9c4116c1a": "c_g = \\lim_{k_1\\, \\to\\, k_2} \\frac{\\omega_1 - \\omega_2}{k_1 - k_2} \n = \\lim_{k_1\\, \\to\\, k_2} \\frac{\\Omega(k_1) - \\Omega(k_2)}{k_1 - k_2}\n = \\frac{\\text{d}\\Omega(k)}{\\text{d}k}.", "38afce0a93f31ef8e86845baab95ff15": "Lu = \\frac{1}{w(x)} \\left(-\\frac{\\mathrm{d}}{\\mathrm{d}x}\\left[p(x)\\frac{\\mathrm{d}u}{\\mathrm{d}x}\\right]+q(x)u \\right)", "38aff61ad7fb4da6eaa55094b38b28a9": "2\\eta = \\left(\\frac{\\partial \\mu}{\\partial N}\\right)_Z \\approx -\\left(\\frac{\\partial \\chi}{\\partial N}\\right)_Z,", "38b0049431a2eff7bf9b8e5e4aef98a8": "\\log M(r)", "38b1648243beb7fe3cb3fbe5b9f66a31": " {\\rm NP} = \\exists^{\\rm P} {\\rm P} ", "38b173c9d44e8435f4589562ae72d015": "x^2+y^2+z^2=4a^2 \\, ", "38b20bd399bd3dbb86985d685f6cd626": "\\textstyle R = \\left( \\begin{array}{cc} 6 & 1 \\\\ -1 & 0 \\end{array} \\right) = \\left( \\begin{array}{cc} 1 & 6 \\\\ 0 & -1 \\end{array} \\right) \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right)", "38b26fdadb72b647f5fe1ba58fb9eb4e": "b^{s \\dagger}_{\\textbf{p}}", "38b294598bed8e2f58bf515a505f87e9": "\\displaystyle \\phi_{;a}\\equiv \\partial_a \\phi", "38b2b318f6a34e87c6bd0fb9661ba078": "\\int_S H^2 \\, dA ", "38b35dbe17a0a0b55ad91cf76ee15fbd": "(P \\leftrightarrow Q) \\vdash (P \\to Q)", "38b3bf53bce8fb206c5102fa9f10cc2d": "10^{\\,\\!10^{10^{963}}}", "38b3cce9e6689c72f6606c2ce0f3c879": "n/\\log n", "38b3e0d81cce3cd9f57fe74f9f9f2a9a": "\\left| \\mathrm{trace}(AB) \\right| \\le \\sum_{i=1}^n \\alpha_i \\beta_i.", "38b3eeeab7d617978e9faccbf4b5f44d": " \\overline{Y}_{i\\bullet} = \\frac{1}{n_i} \\sum_{j=1}^{n_i} Y_{ij} ", "38b3eff8baf56627478ec76a704e9b52": "101", "38b48ea03f33749fe6835d211a4d83f7": "\\psi \\colon A \\to A \\otimes A.", "38b4a01cd6eb3c9479dda9412f52c2f0": " out_b = \\bigcup_{s \\in succ_b} in_s ", "38b4ee09fc371ed8a6da0023cd833299": "FI(KI,x)", "38b535cefca1e87138b188d1b98c0040": "B = {F \\over{I \\ell}}", "38b53f375f150c3590cda852090525ad": "|W(S,T)|", "38b589a4ca108f6c42cb4449362445b8": "a \\to b", "38b5a27db82a0cd8e08cdd5cb2b6d2a3": "y_{max}", "38b5ae0a172d1901671935a555555c5a": "B\\stackrel{\\mathrm{def}}{=}b^8", "38b5af9b10ddbb59e2aecdcde5f0ab3a": "\\frac{\\operatorname{d}^2y}{\\operatorname{d}x^2}", "38b5d2a017198d6e59f122a28585159e": "\\Phi [\\gamma]", "38b6094aca399d8498f833c673c084e1": "a_2b_1", "38b63ffbac14af037ed87c2159736939": "C\\rightarrow\\infty", "38b64921479e053bfe27291531a42235": "f(t_1, \\ldots, t_n)", "38b6625152c9976405d5890750287355": "\\frac{p/a^d}{\\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}", "38b6ab97393426deb6b5beaaa6dcfd32": "B_1(t) = \\sum_{i=0}^n \\beta_0^{(i)} b_{i,n}\\left(\\frac{t}{t_0}\\right) \\mbox{ , } \\qquad t \\in [0,t_0]", "38b6adf2d626df6c9699b01a6d7eaa18": "[b(\\omega),b^\\dagger(\\omega^\\prime)]=\\delta(\\omega-\\omega^\\prime)", "38b700c0373da644f1c7e85d8e5534a9": " = A_{\\alpha , \\beta , \\gamma} g^{\\beta \\gamma} - A_{\\rho , \\gamma} \\Gamma^{\\rho}_{\\alpha \\beta} g^{\\beta \\gamma} - A_{\\rho} \\Gamma^{\\rho}_{\\alpha \\beta , \\gamma} g^{\\beta \\gamma}\n- A_{\\sigma , \\beta} \\Gamma^{\\sigma}_{\\alpha \\gamma} g^{\\beta \\gamma} \n- A_{\\rho} \\Gamma^{\\rho}_{\\sigma \\beta} \\Gamma^{\\sigma}_{\\alpha \\gamma} g^{\\beta \\gamma} \\,.", "38b7052dedaf0dbefe9883c55aa5f38a": " \\sum_{j\\ne i} |a_{ij}| = R_i", "38b70d6e4dbb5b1470d919f0a0286bec": "T \\cos \\varphi = T_0,\\,", "38b77c4def8136735a57b20ad928432f": "\\det\\begin{pmatrix}A& B\\\\ C& D\\end{pmatrix} = (D-1)\\det(A) + \\det(A-BC) = (D+1)\\det{A} - \\det(A+BC)\\,.", "38b7eae8c6192152be2029e39bc848ef": "V_1, \\ldots, V_e", "38b820e87a488f80207249f8f52fdc8b": "\\mathbf{B}\\cdot\\mathbf{ds} = 0", "38b8274e08010fdb12fa9253ffabaf5c": "A_{\\dot{\\alpha}}=0", "38b84fac519175e4d23157c79a4bef88": "A\\times B\\times \\{0\\}", "38b86d09589600d50b2915c4f05a5808": "\\ q=Mp'", "38b893342e6d9dcdc1073775c026a754": "u=x^2, \\quad dv=xe^{x^2}\\, dx,", "38b8c86597e474b7c7c9167384a8ac68": " v\\in V .", "38b8d3af50c434c65d0f7cb214be1ff5": "\n\\text{the congruence }x^2 \\equiv p \\pmod q \\text{ is solvable if and only if }x^2 \\equiv -q \\pmod p\n\\text{ is.} \n", "38b98564346a48be1261477076bdfdec": "L(\\tilde{k})=64+20L_B(\\left\\lceil (L(M)+8)/64 \\right\\rceil)", "38ba05b73e6d8986280effa8359eab35": "\n \\begin{align}\n G_1 & = 1 - \\text{Re}\\left[\\frac{a+z}{\\sqrt{z^2-a^2}}\\right] \\,,\\,\\,\n G_2 = - \\text{Im}\\left[\\frac{a+z}{\\sqrt{z^2-a^2}}\\right] \\\\\n H_1 & = \\text{Re}\\left[\\frac{a(\\bar{z}-z)}{(\\bar{z}-a)\\sqrt{{\\bar{z}}^2-a^2}}\\right] \\,,\\,\\,\n H_2 = -\\text{Im}\\left[\\frac{a(\\bar{z}-z)}{(\\bar{z}-a)\\sqrt{{\\bar{z}}^2-a^2}}\\right] \n \\end{align}\n", "38ba05e61a2b9941e7eea8044b788c4e": "p_1=\\frac{q_1+q_2}{2q_5}\\ ,", "38ba10ae890917d3b21b64d0f50c864b": " \\mathbf{a} = \\mathbf{b} \\times \\mathbf{c}, \\quad a_i = \\epsilon_{ijk} b_j c_k ,\\,\\!", "38ba2f64470c74021ed2cb6eef6448be": "(f\\otimes f)\\circ\\Delta_1 = \\Delta_2\\circ f", "38ba4209bca7b7d5df27825a51aa8e5f": "J_F = \\left | \\begin{matrix} \\frac{\\partial f_1}{\\partial X_1} & \\cdots & \\frac{\\partial f_1}{\\partial X_N} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial f_N}{\\partial X_1} & \\cdots & \\frac{\\partial f_N}{\\partial X_N} \\end{matrix} \\right |,", "38ba49e1e311bfdde20453e144dc441d": " f_n(x):=x^n ", "38ba7b280b07f087acec590fbd812b4f": "u(x,0)=f(x) \\,", "38babe7c251f304432da487f10eeba85": "\\frac{d \\mathbf{M}}{d t}=-\\gamma \\left(\\mathbf{M} \\times \\mathbf{H}_{\\mathrm{eff}} - \\eta \\mathbf{M}\\times\\frac{d \\mathbf{M}}{d t}\\right)", "38badb7acbfab3acc3c08dd8b6c9c938": "score(x_t,y_t) - score(x_t,y')\\geq L(y_t,y')\\ \\forall y'", "38bb3a8f70d0654ebdb39d9fdc707f79": "DCG_p", "38bb485e02f73bc06f4dab296fa4ac63": "R_{23,41}", "38bb64533c6ab1b7a2e78196f3d1d808": "\\cos\\theta = \\cos\\left(\\theta + 2\\pi k \\right),\\,", "38bbc6ea7a05da48727259039641912b": "u_i=0", "38bc02fd8881af1a6da2da135caf102e": "P_1=2", "38bc37af05d86023bf673db965b5ffe0": "\\lim_{y \\to \\infty} v(x + iy) = 0", "38bc4e073d33a2b1c412d19f4742e8d6": "P_{Tx_{dBm}}\\;=10\\log_{10} \\frac{P_{Tx}}{1mW}", "38bc5c85d39be6ee371d2e8430af5a15": "\\widehat{\\lambda 1} = 0010", "38bd260d29fe1dcaa8ab0e18550ce93f": " | \\psi \\rangle ", "38bd2bb3d1b39d5864c51b84f3d5aa84": "\\phi(x,u,u_{1},u_{2}) = 3u_{1}u_{2} \\,", "38bd560cb617210532d5d82cc635c58b": "\nU(P_1) = \\frac{A e^{\\mathbf{i} k g}}{g} \\frac{b}{\\sqrt{b^2+a^2}} e^{\\mathbf{i} k \\sqrt{b^2+a^2}}.\n", "38bd6c19740e43396bd14b1575d58f60": "[S]", "38bd8bdbe61db248755d31dd1f68f406": "\\scriptstyle \\{(x_i, y_i, z_i)\\}", "38be16dbc908687436916cbcacdf8ed7": " \\frac{\\delta \\langle \\psi(x) \\rangle}{\\delta h(0)} = \\frac{\\phi(x)}{h_0} = \\beta \\left ( \\langle \\psi(x) \\psi(0) \\rangle - \\langle \\psi(x) \\rangle \\langle \\psi(0) \\rangle \\right ) ", "38be22cde8ea9cf9db4e68308b2bf8f2": "\\mathbf{T}^2", "38be3508607309ec0cd157d2f35becdd": "\n|\\Phi \\rangle = a^\\dagger(\\mathbf{k_1})\na^\\dagger(\\mathbf{k_2})...a^\\dagger(\\mathbf{k_n})\nc^\\dagger(\\mathbf{q_1})c^\\dagger(\\mathbf{q_2})...c^\\dagger(\\mathbf{q_m})|0 \\rangle\n", "38be37b4746a50b636a1a0125a07f4d1": "c_1 = 16.923, \\,\\!", "38be3ff1e12646ad3a0c92f5e7ea3d4c": "\\mathrm{DOL} = \\frac{\\mathrm{EBIT\\;+\\;Fixed\\;Costs}}{\\mathrm{EBIT}}", "38be63a48471fd65534e70bd547284d7": "[u_n]", "38be663a3bb72f3d734ec904618c7c3e": "{d\\lambda \\over \\lambda} = {-3 B \\lambda \\over 2} b ", "38be9e82f731b55b53479adf0decec12": "f(x) = x - \\sin x ,", "38beb697680e0b7945aac76b12ad4ab2": "+2", "38bee54c7025d85224061c5c976141f1": "\n \\eta(x,t) = \\eta(x-ct)\n \\quad \\text{and} \\quad \n \\mathbf{u}(x,z,t) = \\mathbf{u}(x-ct,z).\n", "38bf4cb4d8c8412f483466aa6c8941b8": " V_ {\\omega M} \\, ", "38bff45d9cd240cfd7280df745a94d79": " Q = ~~\\frac{\\partial F_4}{\\partial P} \\,\\!", "38c0112769d5a85c60ff005c8f895cc6": " Unseeded: BOD_5 = \\frac{(D_0 - D_5)}{P}", "38c07339e9d3159d60147b0ff56e172b": "\\boldsymbol{\\tau}_{\\mathrm{net}}=I \\ddot \\theta", "38c07ebff5d416b3a5bfe26f69ed2fab": "\\mathbf{b}(\\boldsymbol\\theta)", "38c08f272962b1b5602f2169e440f5fb": "D = P_1 + P_2 + \\cdots + P_n", "38c0f8391562ff33b86a8cb7d0335a1d": "\\sigma_{xx}\\sigma_{zz} - \\sigma^2_{xz}", "38c11205a630f54c504153fb59674261": "\\beta = 1/2 = \\nu ", "38c1dd4b4a17eba504d6c38d8ad25e60": "\\lim_{n\\to\\infty} a_n = \\lim_{n\\to\\infty} b_n = L", "38c23090e73a0909463c883ca9924ddf": " P = \\rho \\cdot S \\cdot v^2 \\cdot (v_1-v_2) ", "38c302ba9f3a36deb0f042e26084d72d": "g: x\\mapsto x^2", "38c3577da86bf5af1e6120995d440402": "p \\cdot x''_i < w_i", "38c35fd4559c9a4666d7fd1a59857f9e": "G\\colon D\\to C", "38c37c6da5f46676a8cbad359e4d726a": "\\operatorname{Tr} (S U^* \\operatorname{E}_{F(x)} U) \\geq 1 - \\epsilon - \\delta.", "38c3b2c554aaa65af1eab65def6691b5": "\\mathrm{up}(c)", "38c3b5291c604e95795adeaee9075c1c": "\\, \\! V_-", "38c3b5facc28223c9552a6bba8affd36": "\\vert \\psi \\rangle", "38c3bde20b7c8903fc74636d2902f8db": "f(1-e)", "38c3c4cf0771ff1ad138cede24ff3eb9": "\\lim_{n\\to\\infty}\\frac{P}{P}", "38c3fffee8d5b8ffc30ef2be39600010": "p_0(x) = x^4+x^3-x-1", "38c464372b32716b96228696e101b101": " \\mathbf{SL} =\n\\begin{pmatrix} 1 & d \\\\ 0 & 1 \\end{pmatrix}\n\\begin{pmatrix} 1 & 0 \\\\ \\frac{-1}{f} & 1\\end{pmatrix}\n= \\begin{pmatrix} 1-\\frac{d}{f} & d \\\\ \\frac{-1}{f} & 1 \\end{pmatrix} ", "38c4658d5308897a92cef9e113aefc3a": "Op", "38c489fc41b0ab49132243e5c5631f12": "x(z)", "38c4a0e552230765cb3b899dbd49394a": "w_{i_j}", "38c4cc3ef1c30fad0bb6f60b54daa2b9": " \\frac{x^2 + ab}{x} = x + \\frac{ab}{x}. ", "38c4d155270b5d9efa4204243320a053": "-m + \\frac{n}{2} < s < \\frac{n}{2}", "38c4da00b2f729a113e407225cd23efe": "Q=\\begin{pmatrix}\n-0.025 & 0.02 & 0.005 \\\\\n0.3 & -0.5 & 0.2 \\\\\n0.02 & 0.4 & -0.42\n\\end{pmatrix}.", "38c5109c1fe3e33d1af2ff9203e1ad54": "WXYZUV", "38c5596387a418cd349b592bd80da6dc": "\\textstyle g(x) = (x^9+1)(1 + x^2 + x^5) = 1 + x^2 + x^5 + x^9 + x^11 + x^{14}", "38c57a6f50f935e4011748a5c1cb1200": "\\mathrm{d}\\Omega^2 = \\mathrm{d}\\theta^2 + \\sin^2\\theta \\, \\mathrm{d}\\phi^2", "38c5e3c4c5a96069c4f7f4c0c8f06636": "\\Phi^+(\\gamma)", "38c60bc4a982638865b0e989e24eeaa7": "\\mathbf{X} = \\left\\lbrace x_1,\\dots,x_n \\right\\rbrace ", "38c66a11a5df236de618cd0348fcf7b3": "f \\colon A \\to C", "38c77c64591f05934aed7cb2bfd42d2d": "\n\\displaystyle f(u-u', x-x', x') = -f(u'-u, x'-x, x)\n", "38c797d31cc729aeb6046765af57d829": " \\chi = 3 \\frac{\\mu_r-1}{\\mu_r+2} ", "38c87957a396416f4dd2c7c056e415bb": "\\int\\frac{\\mathrm{d}x}{1\\pm\\sin ax} = \\frac{1}{a}\\tan\\left(\\frac{ax}{2}\\mp\\frac{\\pi}{4}\\right)+C", "38c89e5f628d9b33ad6a0d3c46de6b5d": "p(C \\vert F_1,\\dots,F_n) = \\frac{p(C) \\ p(F_1,\\dots,F_n\\vert C)}{p(F_1,\\dots,F_n)}. \\,", "38c8cd325ec69d6645a8410b0a087e1a": "\\mathbf{H_2}", "38c8fcdc8dca56e39c036eac8ca0efb2": "[a_\\lambda b]=-[b_{-\\lambda-\\partial}a], \\, ", "38c908fd4c831c3c6550a5f2fa2c27ac": "b_0\\infty^{n-1}\\,", "38c9abaca4c0e762d42281e967e45959": "k=2", "38c9d7e967baaec8d64381e6735e038a": "P(a_1,\\ldots,a_n)\\not=0", "38c9f53977da51e438f05051ea894118": " a_n\\,\\leftarrow\\,\\langle R_n, g_{\\gamma_n}\\rangle ", "38ca988b9e443abd9e0506b8b7de90f6": "\n\\mathrm{THD_R} = \\frac{ \\sqrt{V_2^2 + V_3^2 + V_4^2 + \\cdots + V_n^2} }{\\sqrt{V_1^2 + V_2^2 + V_3^2 + \\cdots + V_n^2}}\n", "38caaaba7a434283428dbd4e546a5a82": "\\,p_x", "38cb10bf4ae14f590d64989356992fc7": "L = \\frac {Q}{m}.", "38cb3de95dbf0a46f30a80eedd815670": "U_2 (\\mathbf r,t) = A_2(\\mathbf r) e^{i [\\varphi_2 (\\mathbf r) - \\omega t]}", "38cb64cbb311e5531933932235c66bfd": "\\begin{align} c^{2} d\\tau^{2} \n= & \\left( 1 - \\frac{r_{s} r}{\\rho^{2}} \\right) c^{2} dt^{2} - \\frac{\\rho^{2}}{\\Delta} dr^{2} - \\rho^{2} d\\theta^{2} \\\\ \n& - \\left( r^{2} + \\alpha^{2} + \\frac{r_{s} r \\alpha^{2}}{\\rho^{2}} \\sin^{2} \\theta \\right) \\sin^{2} \\theta \\ d\\phi^{2} \n+ \\frac{2r_{s} r\\alpha \\sin^{2} \\theta }{\\rho^{2}} \\, c \\, dt \\, d\\phi \\end{align}", "38cb9dee8aaa602fed2d0552bf17f59f": "\\forall i < n (", "38cba550261edfa38441535ef0f70b4c": "1_{A_j^{\\mathrm c}}I +1_{A_j}E\n=1_\\Omega I-1_{A_j}I +1_{A_j}E\n=1_\\Omega I +1_{A_j}\\Delta,\\qquad j\\in\\{0,\\ldots,m\\}.", "38cbbc36d683c98a604b4179b9fa1e06": "\\scriptstyle{p_{ij}^{-}}", "38cbc39e64636df54e39fdbe8d4ac314": "\\textstyle \\cot \\frac {\\pi}{8} = \\tan \\frac {3\\pi}{8} = \\sqrt{2}+1=\\delta_s ", "38cbe00e1d73604b71200c467fd41055": "\\gamma = \\lim_{n \\rightarrow \\infty } \\left( 1+ \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots + \\frac{1}{n} - \\ln(n) \\right).", "38cbf0e8dbd62339b89aa2976c03e83c": "V_\\mathrm{pp} = \\frac{I}{fC}", "38cc3eefd011ff501ceeb30f6195e865": "L^2[0, 2\\pi]", "38cc64c1f8c09604d842a1f58865ef7b": "~D~", "38cc7a1040dab995518aacec7e0d5860": "\\scriptstyle(4.0\\pm3.3)\\times10^{-31}", "38cc96034088b4e4c346cfbf782d7455": "y < z", "38cd07a260fc415e62b59a63a9f88c43": "\\frac{8.095 \\mbox{ mol }}{2.7 \\mbox{ mol }} = 3", "38cd1da34e3b912dafb8ac4dddaa350a": "\\sigma_{\\varphi \\or \\psi}(R) = \\sigma_\\varphi(R) \\cup \\sigma_\\psi(R)", "38cd5bb3c9baaf695313040a3af021a0": "e = (u, v_C)", "38cd8c792d52bea400a4dbec4dff16f0": "1.28 > \\lambda \\ge 1", "38cde4512040b4512ea3dd3a3277e70e": "\\tfrac{1}{p} + \\tfrac{1}{q} = 1.", "38ce2243ab70a74fe57575c14ca37975": "M \\simeq {4 \\over 3} \\pi \\rho a^3", "38ce88386228242591720666b1ba8d44": "2.9\\cdot 10^{-28}%", "38ceb2bf63b55f60683f104debeeb8f2": "\\lim_{n\\to\\infty}p^n=0", "38cec14c92ee06d0add0e9cfebd2610d": " R_n \\hat{\\boldsymbol{\\beta}} =\\left(Q^{\\rm T} \\mathbf y \\right)_n.", "38ced08fad13d55bd01a737c8197be4f": "h_{[s]}(\n\\mathbf{\\pi })", "38cedd38a239520c35c50a016356249a": "\\Phi^t(x) := \\Phi(t,x)\\,", "38cf220b5b741d8b2b51cb2d3e5c5feb": "123.456\\,", "38cf32a2ae06e8855c272b55d4d07cdb": "d(v,u)", "38cf55146f29445f491679de099f72e5": " g_{\\lambda \\nu ,\\mu} \\dot x^\\mu \\dot x^\\nu + g_{\\lambda \\mu ,\\nu} \\dot x^\\mu \\dot x^\\nu - g_{\\mu \\nu ,\\lambda} \\dot x^\\mu \\dot x^\\nu + 2 g_{\\lambda \\mu} \\ddot x^\\mu = {\\dot x_\\lambda {d \\over d\\tau} (g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu) \\over g_{\\alpha \\beta} \\dot x^\\alpha \\dot x^\\beta} \\qquad \\qquad (7) ", "38cf5644e516db381a1f1acbc1864cb8": "\\frac{a}{b} \\cdot \\left(\\frac{c}{d} + \\frac{e}{f}\\right)", "38cf5b61f8710f73a5f77c5be2fb20c2": "((C, j), f, (x_1, \\ldots, x_n))\\,", "38cf66535231eb4e06378771bf00d33a": "e^{-2 \\pi i \\omega}", "38cfcfdd13bbbe318001397fc8612648": "v_\\mathrm{rel}\\ln{\\frac{m_0}{m_1}} = v_1 - v_0", "38cfe7e9d09bdb6a284cc2fa314cf8e7": " u_{k}(j^{k}\\sigma)= \\left.\\frac{\\partial^{k} \\sigma}{\\partial x^{k}}\\right|_{p}\\,", "38cffe8b705e4d5e17260c2907018ee0": "\\psi(\\mathbf{r}+\\mathbf{a}_i) = \\mathrm{e}^{2 \\pi \\mathrm{i} \\theta_i} \\psi(\\mathbf{r})", "38d0b396e0e40827c1b3606390087dde": "M(t)=M\\left(\\frac{k}{2},\\frac{1}{2},\\frac{t^2}{2}\\right)+", "38d0c5bf7b0c209bb27dbb2462b1fba0": "\ndA = a^{2} \\frac{\\sigma^{2} - \\tau^{2}}{\\sqrt{\\left( \\sigma^{2} - 1 \\right) \\left( 1 - \\tau^{2} \\right)}} d\\sigma d\\tau\n", "38d1a023442706e93d586086a710145f": " f_*( \\sigma ) \\frown \\psi = f_*(\\sigma \\frown f^* (\\psi)). ", "38d1ffe687f2179268a7800b700e7558": "E^* = E' + iE'' \\,", "38d200a28caa679661139594d415d387": "(x)_{n} = \\sum_{k=0}^n s(n,k) x^k.", "38d20e551739b1b4d03a08dd9e49c050": "\\{|y_j\\rang \\lang y_j|\\}", "38d2189ce7ace4baa2f7ed33c5021a61": "P_n=P'_n\\oplus P''_n", "38d242f46f6b3b77dfa62d8e9b2e17d9": "r(a) = p(a) \\cdot g(a) + e(a)\\,.", "38d24e43f71046633579acc39ece72ae": "\\|A\\| < r", "38d267abba4894dee0ec7852ad5618b4": "\\delta\\circ g", "38d2ecfde9acb7ffeb0de9e93826899b": "w_2 = T_Z(w_1)\\,", "38d3513cc26bad17744b4f630054aef8": " \\oplus", "38d369e3a2d7e67df3a7d8bf8ba9fb78": "\\prod_a^b f(x)^{dx} = \\lim_{\\Delta x\\to 0}\\prod{f(x_i)^{\\Delta x}}\n=\\exp\\left(\\int_a^b \\ln f(x) \\, dx\\right),", "38d43c5555007d6c3438be81177e74be": "M(u)", "38d4affc25d25a1726a043b1cff51766": "A_z", "38d4c4b52a00ff51743c72181eb1dcba": " \\sum_{t} \\Pr(q = t) (1 - t)^k \\approx (1 - E[q])^k = \\left(1-\\left[1-\\frac{1}{m}\\right]^{kn}\\right)^k \\approx \\left( 1-e^{-kn/m} \\right)^k", "38d4d82002369c1900998b29cfc39291": "|a_{n+i}| < r^i|a_{n}|", "38d4f49655f1e84fe2a30945e1ccce28": " D_e=84.5(0.79+1.602 N_s) \\frac{\\sqrt{H_n}}{60 * \\Omega} ", "38d5417ea5d57b57e349a7011f5103fc": "W_-", "38d57be222f17d85c4016789d34cf3fa": "C^0(X,Y) \\subset \\mbox{Hom}(X,Y)\\,", "38d5d62ebd04bbdfe3b70d52defa4fe4": "\\displaystyle E/V = K_1 \\sin^2\\theta + K_2 \\sin^4\\theta + K_3\\sin^4\\theta \\sin 2\\phi ", "38d5ecc9088490e2eb8e9a6645e4f02c": "N_{ss} = ", "38d62d0f70adb8a1c687ae2a56c7af12": "D_3 \\times \\pm 1", "38d6814e6e807b88b388fb3e3e7894ca": "f(x,y) = y \\oplus x y \\oplus x = x \\oplus y \\oplus x y", "38d69bbcfe14597f37dccf1b8578566c": "u_i^*=h_i^*-pv_i^*", "38d76b597cca6256b79446cf2f62d92d": "e^{-\\frac{A}{k T}} = \\int \\ldots \\int \\frac{1}{h^n C} e^{\\frac{- E}{k T}} \\, dp_1 \\ldots dq_n ", "38d77b3d7b3a1b5060370dc6fd6daeb8": "\n\\begin{align}\n\\exp \\left( \\begin{bmatrix} 16 & 1 \\\\ 0 & 16 \\end{bmatrix} \\right)\n& = e^{16} \\exp \\left( \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} \\right) \\\\[6pt]\n& = e^{16} \\left(\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} + \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} + {1 \\over 2!}\\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix} + \\cdots \\right)\n= \\begin{bmatrix} e^{16} & e^{16} \\\\ 0 & e^{16} \\end{bmatrix}.\n\\end{align}\n", "38d79e2b6fa9910c67aa2c42bda6a60e": "a \\times b", "38d8017aa65bb3f7b8b7b7e9c605433b": "z(t) = I(t) \\cdot \\cos(\\omega_c t) - Q(t) \\cdot \\sin(\\omega_c t)", "38d8ef1377fe2da31016249c1debbc83": "\\chi_{a} \\left(\\mathbf{r}\\right)", "38d9b307e9fc2283e2e55ffee9de2c95": "F^{-} \\cap \\overline{F^{-}}=\\sum_i \\mathbb{Z}\\widetilde{c}_i \\to F^{-}/v^{-1} F^{-}", "38d9b31eaa5c24666baa1ebfdef4eb5b": " \\angle BAC \\cong \\angle EDF\\, ", "38d9b9359cfeaeb24f617d02c2a4ec30": "10_{129}", "38d9d17aa0b6a63d588043c42c2879c9": "\\operatorname{logit}(p)=\\log\\left( \\frac{p}{1-p} \\right).", "38d9f6fb875307496869a7d51bde75a1": "\\frac{d^2\\delta}{{dL}^2} = -\\frac{L}{4 a^2 \\left(\\frac{L^2}{4 a^2}+1\\right)^{3/2}} ", "38da5c67a7ff6518f6720584613e8bab": "\n\\Gamma = -G + \\sum_{i} g_{i} \\left( \\frac{\\partial L}{\\partial \\dot{q}_{i}}\\right)\n", "38da684bb9b477d2aeb6bf4bea720913": "\\textbf{K}(t)", "38daf7a3deaaca9033d9e30f890d0fc3": "P_{mag} = \\frac {B^2} {2 \\mu_0}", "38daf9dc7323a732101629d82f42b50a": "s_0, s_1, s_2 \\cdots s_{n-1}", "38db16ec595afd79e6a57ea6f4c1cd3f": "\n\\begin{bmatrix}\n2 & -1 & 2 & 1 & -3 \\\\\n1 & 2 & 1 & -1 & 2 \\\\\n1 & -1 & -2 & -1 & -1 \\\\\n2 & 1 & -1 & -2 & -1 \\\\\n1 & -2 & -1 & -1 & 2\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n5 & -5 & -3 & -1 \\\\\n-3 & -3 & -3 & 3 \\\\\n3 & 3 & 3 & -1 \\\\\n-5 & -3 & -1 & -5\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n-30 & 6 & -12 \\\\\n0 & 0 & 6 \\\\\n6 & -6 & 8\n\\end{bmatrix}.\n", "38db275b8686558934c32b5b77b16fa5": "k_{e} = \\frac{\\ln 2}{t_{1/2}}\\,", "38db3f81ef6f2a86d813aa29dece5ef3": "q_0=0.5", "38db64a4c0b3fe1121a9f7425b418fb1": "V = \\frac{4}{3}\\pi r^3", "38dc1779c12ef048587355f35037aab3": "\\langle\\mathbf{S}\\rangle = \\frac{1}{T}\\int_0^T \\mathbf{S}(t)dt = \\frac{1}{T}\\int_0^T \\left[\\frac{1}{2}\\mathrm{Re}\\left(\\mathbf{E_c} \\times \\mathbf{H_c}^*\\right) + \\frac{1}{2}\\mathrm{Re}\\left(\\mathbf{E_c} \\times \\mathbf{H_c} e^{2j\\omega t}\\right) \\right]dt.", "38dc8e8b341f3f1a45d786e35ded53af": " X_1^n(i) ", "38dc9817a4b9d8248d907b1e97304df4": "1/2 + |\\mathcal U|/(2\\mathcal U)=1", "38dca1f84b53862c2ea42dbe207cb92a": " V \\leq m_0 ", "38dcbb9bec84ef4065b8d177d8d906ca": "U(W_T)", "38dcc747fa043643dfd88bf6660176cb": "\\frac{\\mbox{Advertising expenditure}}{\\mbox{Sales revenue}} = -\\frac{AED}{PED}\\mbox{ or, symbolically, }\\frac{A}{P.Q} = -\\frac{E_A}{E_P}", "38dd4618d076d9b776e3795fd5e13d08": "\\lim_{z\\to a}(z - a) f(z) = 0", "38dd50c4880d38434e2caca095f816df": "A = \\{ a, b, c, d\\}", "38dd6b341ff9857cef7ecf5bf79817ad": " c_g = \\left | \\{ (i,j) : a_i b_j \\in H g \\} \\right |", "38dd9ab7d40bdaa045770dd6af961e68": "\\textstyle |S_{11}| = |S_{22}|", "38ddbca5a8fb4feb1b9fd5f95c8a6ab8": "\\begin{alignat}{13}\nf(0) &&\\; = \\;&& 0 \\;\\;\\;\\;\\;&& \\Rightarrow &&\\;\\;\\;\\;\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; a_0 &&\\; = \\;&& 0 & \\\\\nf(1) &&\\; = \\;&& 1 \\;\\;\\;\\;\\;&& \\Rightarrow &&\\;\\;\\;\\;\\; a_3 \\;&& + &&\\; a_2 \\;&& + &&\\; a_1 \\;&& + &&\\; a_0 &&\\; = \\;&& 1 &\n\\end{alignat}", "38ddceda06e289a5882b19b739a5223e": "\\scriptstyle{H^2(M,\\mathbb{Z})=0}", "38ddea14f20f51306e9a510919a0fa37": "g(x_2,\\ldots,x_n) = f(0,x_2,\\ldots,x_n)", "38de001706dc07978e7f790ecba6dfa2": "h_{T,P}-h_{T,P}^{\\mathrm{ideal}}=RT_C\\left[T_r(Z-1)-2.078(1+\\kappa)\\sqrt{\\alpha}\\ln\\left(\\frac{Z+2.414B}{Z-0.414B}\\right)\\right]", "38de13ee948a28f7237e0f8d8aaae87a": " \\ddot{\\textbf{r}} = \\textbf{a} = - r \\omega^2 \\cos(\\omega t) \\hat{x} - r \\omega^2 \\sin(\\omega t) \\hat{y} ", "38de176714f4db1a7ba4d78cf26884ea": "\n|\\dot{r}| = \\Big| \\frac{dr}{dt}\\Big| = \\sqrt{\\frac{2}{m}} \\sqrt{E_{\\mathrm{tot}} - U(r)}\n", "38de63671939b399fb9c28594fd4f93b": "\\frac {du/dt}{u} = g_u=g_w-\\alpha", "38de87a5d6fa8c69e4783d3ad7f6048f": "\\scriptstyle+\\sqrt{3}", "38df03d47ae63becb282d2fe35fa5d29": "f'(z_0) > 0", "38df62cd3c8cb66d5f240c4aa0c6e144": "\\vec x^n", "38df693a1baba962d02a642bbc83dfef": "w_i = \\frac{n_i'}{n_i}", "38e06af8805ad2107d8d3a75a21f20f0": "\\log n\\,", "38e0f331727aceb4550cb464687ed80e": "y=mx+k", "38e105cbeed93a8204b9e8ef7434099e": "\\sin^2(x) + \\cos^2(x) = 1", "38e157d0c3bc08369887e9fccb8f0db0": "\\sum_j h_i(x_j) y_j e^{-\\frac{(r_i(x_j)+\\alpha h_i(x_j) y_j + s - t)^2}{c}} = 0", "38e15e2de6f642ddf7e6ff6982e8be73": " [h,e] = 2e, \\quad [h,f] = -2f, \\quad [e,f] = h. ", "38e1feb397f2f1ca5f5197d185bb1807": "\\mathcal J_0", "38e20bd592f2f1b9354e63b284b4f262": "0 < \\left| \\alpha q_n - p_n \\right| < \\frac{1}{b}", "38e20d10e183f1afee1ae6c5d8835d44": "G(x) - G(a) = \\int_a^x f(t) \\, dt", "38e256cb7df6bfe8dc0529c33a320930": "f(x; \\alpha, \\beta) = \\frac{ (\\beta/\\alpha)(x/\\alpha)^{\\beta-1} }\n { \\left( 1+(x/\\alpha)^{\\beta} \\right)^2 }", "38e2bb49cb53492539816a0c0653ee42": "p(x_1,\\ldots,x_n)=0\\,", "38e2c21de6a82df470c34e31a1ce6c19": "\\mathbf{e_z}", "38e2e0cdcb85542b96f5c932a8783864": " \\delta_p = \\frac{1}{\\alpha} ", "38e2ea1ca2bc661c09ef54096d2e48ba": "\n \\varepsilon_{\\alpha 3} = \\cfrac{1}{2}~\\kappa~\\left(w^0_{,\\alpha}- \\varphi_\\alpha\\right)\n", "38e30d58de96009449632e4a97c97d51": "F[y]=\\frac{1}{t-a}\\int_a^t y\\,dt ", "38e30eee47ae5c77815a3975434505e0": "\\mathbf{Z}(p^\\infty) = \\mathbf{Z}[1/p]/\\mathbf{Z}", "38e33aaaecf60bacccbe3ebb31fdea0f": "\n\\langle r^{2} \\rangle \\approx \\frac{6k_{B} T\\tau}{m} t = \\frac{6 k_{B} T t}{\\gamma}.\n", "38e3489045be3472d372ebe477bc8ded": "K_G \\left(i,j\\right) = \\sum_k V_i^G(k) V_j^G (k)", "38e37f6423e97ea4a999d7af5bf5c11d": "\nx_i = 1 \\Rightarrow T = T_i^\\circ \\Rightarrow K = \\frac{H_i^\\circ\n}{T_i^\\circ }\n", "38e3d84162d5f66792c9061a51e78d31": "\\mathrm{id}_x", "38e41200dc59345e60ea2e77474c7c57": "\\psi( \\mathbf r )\\approx \\left( \\sum_{\\mathbf k } F( \\mathbf k ) e^{i\\mathbf{k\\cdot r}}\\right)u_{\\mathbf {k=k_0}}(\\mathbf r ) = F( \\mathbf r )u_{\\mathbf {k=k_0}}(\\mathbf r ) \\ . ", "38e41c4fe04c50b2a2727adcaa532322": "F \\subseteq 2^Q", "38e4320d98d13a1b8778122f42d4d2d0": "\\begin{align}\n\\oint_C {1 \\over 1 + 3 ({1 \\over 2} (z+{1 \\over z}))^2} \\,{dz\\over iz} &= \\oint_C {1 \\over 1 + {3 \\over 4} (z+{1 \\over z})^2}{1 \\over iz} \\,dz \\\\\n&= \\oint_C {-i \\over z+{3\\over 4}z(z+{1\\over z})^2}\\,dz \\\\\n& = -i \\oint_C { 1 \\over z+{3\\over 4}z(z^2+2+{1\\over z^2})} \\,dz \\\\\n& = -i \\oint_C {1\\over z+{3\\over 4}(z^3+2z+{1 \\over z})} \\,dz \\\\\n&= -i \\oint_C {1 \\over {3\\over 4 }z^3+{5 \\over 2}z+{3 \\over 4z}} \\,dz \\\\\n& = -i \\oint_C {4 \\over 3z^3+10z+{3\\over z}}\\,dz \\\\\n&= -4i \\oint_C {1 \\over 3z^3+10z+{3\\over z}}\\,dz \\\\\n& = -4i \\oint_C { z \\over 3z^4+10z^2+3 } \\,dz \\\\\n& = -4i \\oint_C {z \\over 3(z+\\sqrt{3}i)\\left(z-\\sqrt{3}i\\right)\\left(z+\\frac{i}{\\sqrt{3}}\\right)\\left(z-\\frac{i}{\\sqrt{3}}\\right)}\\,dz \\\\\n& = -{4\\over 3}i \\oint_C {z \\over (z+\\sqrt{3}i)(z-\\sqrt{3}i)\\left(z+\\frac{i}{\\sqrt{3}}\\right)\\left(z-\\frac{i}{\\sqrt{3}}\\right)}\\,dz.\n\\end{align}", "38e43f610cfc1d0c66f8e15a1b6bbd88": "\n\\rho_n K_{n\\rightarrow m} = \\rho_m K_{m\\rightarrow n}\n\\,", "38e44e15a5710d4590311237436cad6c": "R(T') = X'", "38e4c93fa37fbf173bd7d707cda0565c": "[(q^m-1)/(q-1), (q^m-1)/(q-1)-m]", "38e4de78fa6cc5c21c84e6574a3b69e8": "Reliability = e^\\left(- \\lambda \\times Time\\right)", "38e5330ce2a347cca6b82d855592f4bd": "\\mathbf{a},\\mathbf{b}", "38e5a745fc5adde148e9a6bab0cf855a": " k\\,", "38e5b3ee566cca072eaa8804796d2df9": "O(\\sqrt{kn})", "38e5e8c778091411414153fece0926c7": "\\scriptstyle H_t", "38e6a363024fe632ea2a61bfaf198454": "\\operatorname{Re}\\{v\\}=\\frac{1}{2}\\left(v + c(v)\\right),", "38e6c155af3b8aa1aa3774c863733573": "\\langle\\cdot,\\cdot\\rangle", "38e6dd061116aad0ded1761da16b7386": "\\boldsymbol{J}=\\boldsymbol{L}+\\boldsymbol{S}", "38e719a3de9d0c2eb9f8476356e26bd4": "\\mathbf{G}_{ij} = k(y_i, y_j), \\widetilde{\\mathbf{G}}_{ij} = k(y_i, \\widetilde{y}_j) ", "38e8ab3f49e9e18db53912c03d06102a": " E_{z} = \\langle \\psi| \\left( H_{0} + \\frac{B_{z}\\mu_B}{\\hbar}(L_{z}+g_{s}S_z) \\right) |\\psi\\rangle = E_{0} + B_z\\mu_B (m_l + g_{s}m_s). ", "38e8ad24b9863e99b3e3671fc817d449": "\\begin{align}\nJ_{XY}=\\sum \\limits_{l} \\sum \\limits_{u} \\frac{X_uY_u}{r}\n\\end{align}\n", "38e8be639a2cc5d50fe11d4bc9c7f345": "(y + \\lfloor \\frac{y}{4} \\rfloor) \\bmod 7", "38e95027e6d9c920e3d2cb29e878f36c": " -p_\\mu p^\\mu = E^2 - P^2 = \\omega^2 - k^2 = - k_\\mu k^\\mu = m^2", "38e95f01d95724b9d6599e41c3b21e70": "I \\approx \\frac{1}{n} \\sum_i f(u_i)+c\\left(\\frac{1}{n}\\sum_i g(u_i) -3/2\\right). ", "38e961c78fb5c49e25a0a89a0e264ded": "\\hat f_s(\\xi)\\ \\stackrel{\\mathrm{def}}{=} \\sum_{y \\in \\Gamma} \\hat f\\left(\\xi - y\\right) = \\sum_{x \\in \\Lambda} |\\Lambda|f(x) \\ e^{-i 2\\pi \\langle x, \\xi \\rangle},", "38e9c1704c47f3814dc13e047aa97e56": "\\rho(A)\\leq \\|A^k\\|^{1/k}\\,\\,\\square", "38e9d358afa039ca0b8f69cfd6f73885": "\\frac{2(2\\alpha+\\beta-1)}{\\beta-3}\\sqrt{\\frac{\\beta-2}{\\alpha(\\alpha+\\beta-1)}} \\text{ if } \\beta>3", "38e9e1d0f2defac5b5b173b5472b2323": "ds^2 = -2 \\, du \\, dv + C\\left(\\frac{q^2}{\\omega^2}, \\frac{2q^2}{\\omega^2}, \\omega u \\right)^2 \\, \\left( dx^2 + dy^2 \\right) ", "38e9f06b2afb8a1668aa471bcd7fde03": "X(z) = \\mathcal{Z}\\{x[n]\\} = \\sum_{n=0}^{\\infty} x[n] z^{-n}.", "38e9f2d51a66e83091b951dac567e099": "U=\\mbox{Im}(L)", "38ea10a96577ea674b7ad8fc09a6c66f": "H_2 (X) = - \\log \\sum_{i=1}^n p_i^2 = - \\log P(X = Y)", "38ea10aa125d8351e0e2495d084730b8": "\\mbox{If } x \\to y = 1 \\mbox{ and } y \\to x = 1 \\mbox{ then } x = y ,", "38ea89401d30e1aadf228266d5fb050a": "1 = \\frac12 + \\frac13 + \\frac17 + \\frac1{43} + \\frac1{1807} + \\cdots.", "38eabe3e19539b06b400ccee3c2025b7": "\\mid F_{in}\\cdot e_{ex}\\mid^{2}", "38eaf7187ecd253fae7bb38be54912bb": "\\Pi\\,\\!", "38eb009b194ea73407972bac2b4a7425": "T=\\left(\\frac{\\partial U}{\\partial S}\\right)_{V,\\{N_j\\}}", "38eb38858b68026b3809f75c078a7e17": "\\epsilon=\\frac{T-T_c}{T_c}", "38eb62ab667f8dd2901c49c7135bd298": "\\alpha^n_{f}", "38eb69401d4ee57b7f9a0ca058fa64a2": " \\bar x_w = \\sum_{h=1}^H W_h \\bar x_h, ", "38eb7099f835eededf9016ff1776293d": " p(x)=x^T Q x", "38ebb2b61023a8bc472a95497237a8f2": " a_u ", "38ebb794820639a9acadb39fa9e5aad1": "aF(K,L)", "38ec00c6a1b669952e198e0581e3d838": " (V+W)(p) := V(p) + W(p)\\,", "38ec37adcf646ea5b276742d723fc02a": "\\left[\\frac{\\alpha}{\\beta}\\right]\\left[\\frac{\\beta}{\\alpha}\\right]^{-1}=\n(-1)^{bd+\\frac{a-1}{2}d+\\frac{c-1}{2}b},\\;\\;\\;\\;\n\\left[\\frac{1+i}{\\alpha}\\right]=i^{\\frac{b(a-3b)}{2}-\\frac{a^2-1}{8}}\n", "38eca44f23da94e467e38d2ccf3b3c20": "\\frac{2 x}{1+x}", "38ecc91f26145c796d43e442aad791de": "\\langle\\phi\\rangle", "38ecd0bb9d52a382eeab14952eddc3b3": "W_\\alpha ^i= \\{w| \\mu_{W_i }(w) \\geq \\alpha \\}, A_\\alpha ^i=\\{ x| \\mu _{A_i }(x)\\geq \\alpha \\}", "38ecd1c73aea0ce6844166f12de4d74b": "\\ M_{pitch_{max}} > D_{pitch} \\times lift ", "38ecd5878b7ea682f38916586491b044": " \\tau_{xy} = k \\left (\\frac{du}{dy}\\right ) ^n ", "38ed1e8d3207d877ce366a430891ef6a": "\n \\begin{bmatrix}\n q_1 & q_2\n \\end{bmatrix}\n \\begin{bmatrix}\n -0.1 & 0.1 \\\\\n 0.5 & -0.5\n \\end{bmatrix}\n = \\begin{bmatrix}\n 0 & 0\n \\end{bmatrix}\n", "38ed8247f0f1e3d6e1530c526ee8b3f8": "Y(g) = \\inf_{f} \\mathcal{E}(e^{2f} g),", "38ed99bac1be0aeba6f5c05a11d0c6d1": " Y = \\left[ \\begin{array}{c} y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right],\\quad Z = \\left[ \\begin{array}{c} z_1 \\\\ \\vdots \\\\ z_n \\end{array} \\right],\\quad U = \\left[ \\begin{array}{c} u_1 \\\\ \\vdots \\\\ u_n \\end{array} \\right] \\in \\mathbb{R}^{n\\times 1}.", "38ee0b1508fa5e924d498aa5f8b5c456": " n V_{2,n} {\\stackrel d \\longrightarrow} \\sum_{k=1}^\\infty \\lambda_k Z^2_k,", "38ee68b3eb2f2d9680557eb148985be7": " T = - \n \\begin{bmatrix}\n 0.0625 & 0.0000 \\\\\n 0.0398 & -0.0909\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0 & 3 \\\\\n 0 & 0\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}, ", "38ee901d36cf7cbd2fd8ae40aced1495": "a_1x_1 + a_2x_2 + \\cdots + a_nx_n = b.\\ ", "38ef19267d5a9ffe4f9062ec23274965": " a = m^2 - n^2 ,\\ \\, b = 2mn ,\\ \\, c = m^2 + n^2", "38ef2b95ce70701b4de9291752d6b73a": "g[B]", "38ef387a977226de10bdfcd678f35ffb": "S = 1 - \\sum_{i}\\frac{a_{i}}{z_{i}}G_{0}^{(i)}(u_{1},...,u_{n})", "38efe86c7daa3b889f6226f613a324b1": "(C \\rightarrow \\overline{A}) \\wedge (\\overline{C} \\rightarrow A)", "38efef5ffb496f8d9d7deff4ff5f6a7c": "B(JX,Y) = B(X, JY) = - J B(X,Y).\\,", "38f0311aae3a7ee5603fc09746eff1bb": "h = \\int_0^{\\infty} h_\\nu d\\nu = 0", "38f0acd8d87fefea1ebb5490f9120bb7": "R_1(x) = f(x)-P_1(x) = h_1(x)(x-a). \\ ", "38f0b78bf8dc7cfe5f306e94ec2a072e": "E_k=h\\nu-E_B", "38f110d9b7843c98d93c0fc03bc76799": " 1 \\mathrm{SLPM} = 1 \\frac{\\mathrm{standard\\,litre}}{\\mathrm{minute}} = 1.68875 \\frac{\\mathrm{Pa \\cdot m^{3}}}{\\mathrm s}", "38f13964f92e9a0163df776b82c46432": "\\boldsymbol{\\mathsf{a}}'", "38f1abb9bd2f8083c0a2ffabb2823254": " M^{(2)}(A\\times B)=M^1(A\\times B)-M^1(A\\cap B). ", "38f2100ec9e0f8aee443e47b817e0529": " \\Delta_\\mathrm{R}H ", "38f2c88001132c58cbb4b406382455b0": "H(f(x))=\\frac{1}{2} \\log \\left(\\frac{b}{a}\\right)+\\log \\left(2 K_p\\left(\\sqrt{a b}\\right)\\right)-\n(p-1) \\frac{\\left[\\frac{d}{d\\nu}K_\\nu\\left(\\sqrt{ab}\\right)\\right]_{\\nu=p}}{K_p\\left(\\sqrt{a b}\\right)}+\\frac{\\sqrt{a b}}{2 K_p\\left(\\sqrt{a b}\\right)}\\left( K_{p+1}\\left(\\sqrt{a b}\\right) + K_{p-1}\\left(\\sqrt{a b}\\right)\\right)\n", "38f2c8afc9b81b1402f99cdfa94e0f8c": "x_n^2=x_{n}\\left(\\pi-\\alpha\\right) ", "38f30b16218364be130f62a23d0f3923": "\\dot{f}\\to 0", "38f36b733901307c52a0cf617a2f8dc5": "0\\rightarrow H_n(D^n,S^{n-1}) \\rightarrow H_{n-1}(S^{n-1}) \\rightarrow 0. ", "38f38f82689ffb3ea1d8e15f7699ba0e": "I_{y} = \\frac{1}{12} \\sum_{i = 1}^{n} ( y_{i+1} - y_i ) ( x_{i+1} + x_i ) ( x_{i+1}^2 + x_i^2 )\\,", "38f39c72722b9c1b98e6482762e427ba": "\\bar{R} \\ \\gg \\ \\bar{\\lambda} \\ \\approx \\ \\frac{h}{\\sqrt{3mkT}} ", "38f3e72da79a40e2093a2744d5fa58c7": " ~\\theta_t~ ", "38f4519b2b0bd44a766ed178bd368d73": "\\left|\\frac{4}{3}\\right| > 1", "38f487f98824f2d5b98b5e2d0eafcf1c": "q\\geq (a/b)z+(c/b)x", "38f495f895098c6392d4e4154b02a3e0": "\\textrm{Bl} \\ ([D]) := \\bigcap_{D_\\text{eff} \\in [D]} \\textrm{ Supp } \\ D_\\text{eff} \\ ", "38f4a4763db247c21516f12053001de8": "\\tau = c_m r_m", "38f4e0aeedc073f9af20a57b52611293": " u^i = {dx^i \\over dt } .", "38f4e984b761f071e29efefb5f942e13": "\\displaystyle{p_k}", "38f4f8604f386a7f0b56021f1be81971": "p-1 = 12 = 3 \\cdot 2^2 ", "38f5726b7a4857369592cddc080e139b": "x_0=r", "38f5728c6b14b00b656ca3db4b0a35b7": "x=\\frac{p^0-p}{w/2},", "38f57de3fb64063c972623b215961803": "\\lceil\\log{n}\\rceil", "38f5d7f31f3b8127927d6ecc99235229": "q = \\left |\\mathbf{q} \\right |", "38f5f39059c37868299e08afb39aefa3": " \\frac{y(x)}{2}-\\sum_{r=1}^N \\frac{B_{2r}}{(2r)!}D_t^{2r-1} y \\left(\\frac{x}{t+1}\\right) + x\\int_0^x \\frac{y(u)}{u^{2}} \\, du = x^{-1}H(\\log x) ", "38f5f8f21c919cf18f83b6a0b8a61572": "K_M\\otimes L", "38f62a976cbee045f65de6c200ef74bf": "Z = V \\times W", "38f65855646be83a0c23a1cc810be2a9": "\\Phi_n(x) = 1-x+x^2-\\cdots+x^{p-1}=\\sum_{i=0}^{p-1}(-x)^i.", "38f6691c3275038763e1d83c483466c6": "{A}_{14}^{(2)}", "38f68a027861c8d4174d3e3c3b7eecd2": "(\\mathbf{a} \\times (\\mathbf{b}\\times \\mathbf{c}))_i = \\varepsilon_{ijk} a^j \\varepsilon_{k\\ell m} b^\\ell c^m = \\varepsilon_{ijk}\\varepsilon_{k\\ell m} a^j b^\\ell c^m", "38f68a82b018f459f96c061a0e68258d": "q(x_0,x_1,\\ldots,x_n)= x_0[1- p(x_0,x_1,\\ldots,x_n)^2].\\,", "38f6a052482d7510956398816f779b77": "\\begin{alignat}{2}\nij & = k, & \\qquad ji & = -k, \\\\\njk & = i, & kj & = -i, \\\\\nki & = j, & ik & = -j, \n\\end{alignat}", "38f6c15ce92726994a4f23e76dad64b6": "GDP = NDP + CCA + Indirect Taxes ", "38f6c1c2cee257118be82ce79655ca52": "I_{ds} \\propto (V_{gs}-V_T)^2", "38f6e7db1339c98331fa5ff332691bc3": " \\mathop{\\mathrm{ker}}\\,h = \\{(a,a') \\in A \\times A | h(a) = h(a')\\}", "38f6f47f0d9edf45f84cd1ca43d47c4e": "\\frac{1}{2a_1} + \\sqrt{a_1(1-e_1^2)} \\cos i_1 = \\frac{1}{2a_2} + \\sqrt{a_2(1-e_2^2)} \\cos i_2", "38f6f517fa3026aca8383d2ee26c4f01": "\\frac{1}{k_{\\mathrm B} T}\\equiv\\beta\\equiv\\frac{d\\ln\\left[\\Omega\\left(E\\right)\\right]}{dE}", "38f713c90cc2183cfdbf26990dee3a02": " u^3+v^3=-q", "38f7189d479a5cee0146ea6317bccfe8": " \\partial_\\phi ", "38f75299fb8312cdfaf531e77831517a": "\ns_{\\bar{Y}}=s/\\sqrt{N}\n", "38f7b4a4185cbce985e863213ee4089e": "\\operatorname{Li}_{0}(z)=\\sum_{k=1}^\\infty z^k=\\frac{z}{1-z}\\!", "38f7ea9e97568ef6182c07ba4f4dd702": "\\partial(r_1 r_2)=(\\partial r_1) r_2 + r_1 (\\partial r_2),\\,", "38f7ec5a824ecc4943341e1e58c2515f": " \n\\int_\\Omega\\chi_E(x)\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x =\n\\int_E\\mathrm{div}\\boldsymbol{\\phi}(x) \\, \\mathrm{d}x =\n - \\int_\\Omega \\langle\\boldsymbol{\\phi}, D\\chi_E(x)\\rangle \n\\qquad \\forall\\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n)\n", "38f8399963c22191c5705823e9c50782": "s = \\frac{a+b+c}{2}.", "38f8451097ad5be5f205fe0780a7e926": "\n\\begin{align}\n\\alpha_1 & = \\frac{a \\rho_L - \\rho_0 u_L}{2a\\rho_0} \\\\[8pt]\n\\alpha_2 & = \\frac{a \\rho_L + \\rho_0 u_L}{2a\\rho_0}\n\\end{align}\n", "38f84605189f1e7db755eee613301d51": "j(E)", "38f88440716fdd98619f11f6b6c7ebac": "x^{(n)}=x(x+1)(x+2)\\cdot\\cdots\\cdot(x+n-1)", "38f8ad427b9e6446ea5a5f98a3c8535c": "D_{E}/D_{NE},", "38f8b92f8f7b132648744b6b4b638034": "\\,u=\\Sigma v", "38f8c80029174a8b9836efcb016e8c70": " A(z) = \\sum_{k=0}^\\infty a_k z^k = \\sum_{k=0}^\\infty a^k \\left( \\int_{0}^\\infty e^{-t}t^k dt \\right) \\frac{z^k}{k!} = \\int_{0}^\\infty e^{-t} \\sum_{k=0}^\\infty a_k \\frac{(tz)^k}{k!}dt, ", "38f910dc401abbf3e3947b3bc578d530": "\\mu T", "38f935f045839f8b221748e9491b7d65": " p_{i,j}(x) = \\frac{(x_j-x)p_{i,j-1}(x) + (x-x_i)p_{i+1,j}(x)}{x_j-x_i}, \\, ", "38f955a766b9ffbdccfa800da5b4a61a": "E(\\lambda) (\\lambda - C)^{\\nu} = (\\lambda - C)^{\\nu}E(\\lambda) = 0.", "38f95833266ec394e7c9d08b8ea082f3": "\\left|\\frac{x}{(1+x^2)\\arctan(x)}\\right|", "38f977706e9360e576f3eefc0451e1c2": "\\frac{dN}{dt} = -\\lambda N.", "38f98ac1e147add9b1f370d6d73883a9": "\\overline{NE(X)} = \\overline{NE(X)}_{K_X\\geq 0} + \\sum_i \\mathbf{R}_{\\geq0} [C_i].", "38f99f2057da5f77013c1f9297c17455": " \\scriptstyle\\sqrt{\\pi} ", "38fa07b74ec0895e688606eb3c445af9": " 0 \\leq Q(\\alpha) \\leq \\frac{1}{\\pi}. ", "38fa0a647052a934ecdd0ea0497bafd7": "\\tfrac{1}{2}+i\\mathbb{R}", "38fac222104a3e39164e0fbe3507afcd": "f(z)\\in f^{-(n-1)}(t)", "38fae0d11286dd89fe794c77bf6fce7f": "U=\\frac{1}{2} L I^2 =\\frac{\\phi^2}{2L}", "38faeb902e1e4ef40f02fc32964d730c": " B-\\text{vertex} = \\csc^2\\left(\\frac{A}{2}\\right) : 0 : \\csc^2\\left(\\frac{C}{2}\\right)", "38fb39e5cc285c42ec403077cc8240e3": "H(\\mbox{Oe})= \\frac{1000}{4 \\pi} \\frac{I(\\mbox{A})}{l(\\mbox{m})}", "38fb8c8565d61b10cc1923f74ebae04b": "\\mathrm{ad}_{X_\\lambda}^{-(\\mu,\\lambda)+1}X_\\mu = 0\\text{ for }\\lambda\\ne\\mu,", "38fb983cbdc84a239ccb754fc79c7e02": " H_2O ", "38fb987bd70c1d2d3d2cff0e71365e2d": " V_\\mathrm{schr}(z) = -k_BT \\cdot \\log(n_q(z)) - V_p(z) + V_0 ", "38fc528e43ec25686b6dd38713787ffa": " P^{\\mathrm{*form}} \\phi = \\sum_\\alpha D^\\alpha (\\overline{a_\\alpha} \\phi) \\quad ", "38fcbfdad5e3805e0213d3d007c4297c": "\\langle z-x, a \\rangle=0", "38fcc4841c162696b8805c2d9796b901": " \\big[...\\big]\\,\\big[\\partial_x-\\partial_y+\\tfrac12(y+x)\\big].", "38fd005725439788ad7ccf3612055fe1": "\nV(x, y) = \\frac{-\\mu_{1}}{\\sqrt{\\left( x - a \\right)^{2} + y^{2}}} - \\frac{\\mu_{2}}{\\sqrt{\\left( x + a \\right)^{2} + y^{2}}} .\n", "38fd38e9126095d053888557cd0cfb9a": "1-Q_c", "38fd3cbc15fdea9170fae1cddcacac35": " \\int_0^x \\frac{dt}{\\ln t} = \\int_0^x \\frac{dt}{\\ln t} - \\int_0^{\\mu} \\frac{dt}{\\ln t} ", "38fd417e37015fbde6b69cf0d4b8b871": " \\begin{align} \n {\\nabla}_{\\!\\theta} E(f(x) | \\theta) \n &= \\nabla_{\\!\\theta} \\int_{\\mathbb R^n}f(x) p(x) \\mathrm{d}x\n \\\\ &= \\int_{\\mathbb R^n}f(x) \\nabla_{\\!\\theta} p(x) \\mathrm{d}x\n \\\\ &= \\int_{\\mathbb R^n}f(x) p(x) \\nabla_{\\!\\theta} \\ln p(x) \\mathrm{d}x\n \\\\ &= E(f(x) \\nabla_{\\!\\theta} \\ln p(x|\\theta))\n\\end{align}", "38fd41a2ccaf938f0d7acf7f67f6eaf5": "P = k_B R = k_B r G = r k_B G = r K_B", "38fdb68ef3faaeac4d016d4006bdd1d8": "(n-1) \\in n", "38fdc3307b610cc480b283e969cd7590": " \\frac{R_{input,coupled}}{R_{input,uncoupled}} = 1- \\frac{r_M}{r+2r_M} ", "38fe0d20f7fce364c09de7c473f09762": "\\hat p", "38fe2e20d2ebfc989208c86aea15c142": "\\text{did} : (S\\backslash W)/S", "38fe355e654eeb55643da3bf0d4c1208": "\\mathbf{1}_{A^\\complement} = 1-\\mathbf{1}_A", "38fe37c2b717250acd7b6da0da8baaea": " {\\hat O} ", "38fe3d02ab24d43519f76510cffd9620": "\\pm 2", "38fe405a966b7e8464ad2e9792652db5": "QH^*(X, \\mathbf{Z}[q]) \\cong \\mathbf{Z}[\\ell, q] / (\\ell^3 = q).", "38fe8d3086ce6666edf218d09781aa2c": "\\epsilon = 0.1", "38feccaee4b71b9e0e59dcbfe7d1f6c5": "\\rho(x)=\\mu(\\chi_{[a,x]}),", "38fed08af63a9b7e44571303ad638529": " \\ |1 \\rangle ", "38fefb20e22cc34ba99a1c5bbf50b880": "\n \\sigma=\\frac{\\ln{(1+x_0+\\sqrt{x_1^2+x_2^2+x_3^2})}}{\\sqrt{1/t}}\n ", "38fefd149eb73ddc8e2434653d843d1e": "z_{match} = - j 1.52 = \\frac{-j}{\\omega C_m Z_0} = \\frac{-j}{2 \\pi f C_m Z_0}\\,", "38ff20e4b7dae5c39a57f038850bcfba": "B_{1,2}=B_{1,2}^s", "38ff50b6fd5200e1e724c17716acf08f": "w{(a)}", "38ffaf3a78f7b0113f3ebecc630db4e6": "y_1, \\ldots, y_k", "39001a5d4d75d999bacfe9674152db50": "\\boldsymbol{T}\\sim \\mathrm{G}\\text{-}\\mathrm{MVGB}(\\delta,\\nu,\\boldsymbol{\\lambda},\\boldsymbol{\\mu})", "3900376f66aadb84d43e345a6942e481": " \\hat{H} = -\\frac{\\hbar^2}{2I_{xx}}\\hat{J}_x^2 -\\frac{\\hbar^2}{2I_{yy}}\\hat{J}_y^2 -\\frac{\\hbar^2}{2I_{zz}}\\hat{J}_z^2 ", "39008416756483df1bcd5ed3153a5a46": "\\int x^2\\,\\operatorname{arcsch}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{arcsch}(a\\,x)}{3}\\,-\\,\n \\frac{1}{6\\,a^3}\\,\\operatorname{arcoth}\\sqrt{\\frac{1}{a^2\\,x^2}+1}\\,+\\,\n \\frac{x^2}{6\\,a}\\sqrt{\\frac{1}{a^2\\,x^2}+1}\\,+\\,C", "3900bec82365b0fdb2e78221b3aeda19": "e^{-iv(x+vt/2)}\\,", "3900c36ea9f889f0ed7f5f789f83b412": "g(f)(x) = (\\int_0^\\infty|\\nabla u(x,y)|^2ydy)^{\\frac{1}{2}}", "3900c50688b39287aea89dd5c03424e1": "M_3(n) = \\frac{n(n^3+1)}{2}.", "3900fbc8059a52910d78ef5e91dd8d2a": "i\\frac{\\partial}{\\partial t} \\rho (x, p) = \\hat{L} \\rho(x, p)", "3900fe9ba66197729f3063edecd5756e": "q \\leftarrow s", "3901257ab27e907ca37576f92983bd35": "\\vec{p}_\\mathrm{in} = q \\hat{z}", "3901387a0c27ebbdbca4f99503780823": "\\mathbf{\\hat{r}}, \\boldsymbol{\\hat{\\theta}}, \\boldsymbol{\\hat{\\phi}}", "39015c300f5377f694bfab9427ed7d63": "\n\nX^o =\n\\begin{pmatrix}\nx_{11},x_{12},{\\cdots} ,x_{1N_1} \\\\\nx_{21},x_{22},{\\cdots} ,x_{2N_2} \\\\\n\\vdots \\\\\nx_{T1},x_{T2},{\\cdots} ,x_{TN_T}\n\\end{pmatrix}\n\n", "3901704c73dd84019da85e12a058fddb": "\\begin{bmatrix}\n1 & \\lambda_2 & \\lambda_2^2 & \\cdots & \\lambda_2^{n-1} \\\\\n0 & 1 & 2\\lambda_2 & \\cdots & (n-1)\\lambda_2^{n-2} \\\\\n1 & \\lambda_3 & \\lambda_3^2 & \\cdots & \\lambda_3^{n-1} \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n1 & \\lambda_n & \\lambda_n^2 & \\cdots & \\lambda_n^{n-1}\n\\end{bmatrix}=\n\\begin{bmatrix}\n\\lambda_2^{n+m} \\\\\n(n+m)\\lambda_2^{n+m-1} \\\\\n\\lambda_3^{n+m} \\\\\n\\vdots \\\\\n\\lambda_n^{n+m}\n\\end{bmatrix}", "390176b36eb62ed1461225433c507970": "-b/a", "39018c800e96ddf60e2bf79fbe546a3b": "X_n = Z_1 + ... + Z_n", "3901d689b60e93261ed6dea7f3b1b651": " \\langle\\phi_k \\phi_k \\rangle = {V \\over k^2} ", "3901f4e6f1709c7317db28581906b8b9": "\n\\epsilon = \\frac{R_{p} - R_{e}}{R_{e}}\n", "39024c4b8e819e2339ef23224055012a": " A \\geq B \\iff A-B \\text{ is positive}", "390256ff70facbc9ff2a132cf1ff3a33": "c=1+6(b+1/b)^{2}", "3903507e2381df1ec885b4aec9bcc238": "\\scriptstyle 0 \\,\\pm\\, \\sqrt{\\mathrm{r.d.f.}}", "39038264174b0e0fd7aaf61767c8eabe": "v^i", "3903c6657dda9391ff04f37cf8cba051": " \\ln((r(\\cos \\varphi + i\\sin \\varphi ))^{n}) ", "390405d3aba003ce069f18299b919784": "\\lambda : T^m \\rightarrow T^n", "39043a64a2b1d4c277a873d1854e3b40": "X: \\,\\,", "39045243ca4f1653639187e4945e8ee8": "f(x) \\approx f_n (x) = a_1 e_1 (x) + a_2 e_2(x) + \\cdots + a_n e_n (x)", "39045541f154a5fa8852643f26becdc2": "k\\in\\{1,\\dots,m\\}", "3904c4af1338b2b20031f1abebae205a": "\\text{Efficiency} = \\frac{\\text{Measured Performance}}{\\text{Ideal Performance}}", "3904ca4217735f0b1c3057dcaa24f57a": "1+r= \\frac{M_2}{M_1 + C_1}", "39052cbfe5ffd1e0bc806a4c8e3a4e6c": "\nY_{ij} = \\mu_j + \\alpha_i + \\epsilon_{ij}\n", "3905e6dd9b90606302602347db3a4f01": "\\mu=\\frac{d\\theta}{2\\pi}", "39061055125339718d198b034134feb7": "N_{p} = 1, N_{n} = N - 1", "3906872a2b312e7e745f9b05db27a133": "G^0=G \\cup \\{0\\}.", "3906c80746caf7e1912b647544a48cad": "\\scriptstyle \\lim_{t \\to 0^+} f(t)^{g(t)} \\;=\\; 1", "3907218c055f899bd057144c9569ff34": "2^{2n}-1", "39077f197b65c3b883bf01263f41550c": "\\omega_{\\overline{x\\land y}} = \\omega_{\\overline{x} \\lor \\overline{y}}\\,\\! ", "390791f0365bc5b7b970d31c82dd56a2": " I \\propto V \\rho_s (0, E_f) e^{-2 \\kappa W} ", "3907c05eab507b9f90ead1144d524c7c": "10^{-7}", "3907f3175228e3e30419e981b1b0e5ab": "\\phi_j", "39084eedab3c6d1c53357b1739fd2488": "\\varepsilon_\\alpha + 1", "390872ebb0f853a5aecb778bc4acc947": "10^{-14}", "39097a5a430398ccbeaa6ae8217a3db8": "\n\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)-1}{8^{2n}} = \\frac{61}{126}-\\frac{\\pi}{16}(\\sqrt2+1).\n", "3909a8b40d4b8dcc2d906b5530f66de3": "B^\\circ \\subseteq A^\\circ", "3909b1df87c265a5e6cda81759c9fba4": "\n\\mathbf{p} = X\\mathbf{i}+Y\\mathbf{j}+0\\cdot\\mathbf{k}\n", "3909be0321e440790774f06f3ce054bd": "\\lambda^{\\Delta}\\varphi(\\lambda x)", "390a0f601f8fe47fa698c8c763bb3dfc": "\\scriptstyle f \\colon (X \\times Y \\times Z) \\to N ", "390a24b16e2a146570b883b7c11f5e5a": " (i \\gamma^\\mu \\partial_\\mu - m) \\psi = 0", "390ae2b71e56f78b435d613dcf1d7c88": "k \\varphi (N) D-1\\,", "390d9c65f3fd4cd3e23760a5171a77c9": " \\text{shaves}(x,x)\\leftrightarrow \\neg \\text{shaves}(x,x)", "390dafacf92bcdd2485bed22982fe389": "\\frac{|2 - 2| + |2 - 2| + |3 - 2| + |4 - 2| + |14 - 2|}{5} = 3.0", "390db5b0553a2f66b5b27375a2b588c0": "p_{t+1}(\\hat{x}) = \\sum_{x} p(x)p_t(\\hat{x}|x)", "390de7cd0b0c39ec27790412705f9b30": "E_n \\,", "390df82bb74d5be880dda84cdb615c74": " q_m = \\iint \\sigma_m \\mathrm{d} S ", "390df8ad48b6ac483eabf02617264b8e": "\\theta_{min}\\,", "390dfa725e7a6bbaabbad720b42124e9": "\\cfrac{x}{1 + \\cfrac{\\cancel{y}}{\\cancel{y}}} = \\cfrac{x}{2}", "390e1c3f882c2dc325c65fd8aebfd3e3": "D_{\\dot\\gamma}D_{\\dot\\gamma}X(t) + R_{\\dot\\gamma}(\\dot\\gamma(t),X(t)) = 0", "390e2265a5cbcd72a2afa5ae8d2aaf81": "\\tau_{\\leq 0}", "390e407ad45af52603e62d8dfc53b1cf": "\\left \\lfloor \\frac{i-1}{2} \\right \\rfloor", "390e9338b9b6635c33c852c9693cb56c": "\\scriptstyle 0 \\;<\\; f \\;<\\; 1", "390e9e23ec469cafc31205fbef932e6f": "\\forall x\\,Fx \\rightarrow (\\exists x\\,Fx)", "390ea1d7d72b01f064d858117d0852e9": "\\scriptstyle{\\sqrt{2}/4}", "390ee719aaf02685ae0f6488c5e988c5": "\\alpha \\neq \\beta,", "390f58c1b120068c8cb7b9c1698a806c": "\n1/\\sqrt{3}=0.577...\n", "390f58c8c1ae31b1b9dcf9f4f2b3e112": "i{\\partial \\over \\partial t} \\psi = {(\\nabla - iqA)^2 \\over 2m} \\psi.", "390f8a4571f043b607a0d27409b97ee0": " \\and T_4 = [\\_, S_4, A_4]::[\\_, S_3, A_3]::K_2 ", "390fe2f234c96decc123b7516dddaeec": "\\|\\mathbf{x}\\|", "390ff8473254a70fd649646494594202": "m_{i j}= \\infty", "39103deb17c867eb97bd72396347d01a": "\\dim R[x] = \\dim R + 1", "3910624d569e5d0fdfaab2549c393fd8": "\n\\left\\langle\n\\left\\{\n\\frac{Republican(X):\\neg Pacifist(X)}{\\neg Pacifist(X)},\n\\frac{Quaker(X):Pacifist(X)}{Pacifist(X)}\n\\right\\},\n\\left\\{Republican(Nixon), Quaker(Nixon)\\right\\}\n\\right\\rangle\n", "39106347832987eabf69d5ab87df4844": " \\tilde{O}(n) ", "391068c8e1dc948a564311b305f1ad36": " \\begin{bmatrix} \\Omega_n & 0 \\\\ 0 & 1 \\end{bmatrix} \\tilde{H}_{n+1} = \\begin{bmatrix} R_n & r_{n+1} \\\\ 0 & \\rho \\\\ 0 & \\sigma \\end{bmatrix} ", "3910a128b78a539e364aa67d195027e1": "\\frac{dy}{dx}=\\tan \\varphi = \\frac{\\lambda_0 gp}{T_0},\\ T=\\sqrt{T_0^2+\\lambda_0^2 g^2p^2},\\,", "3910d48b91976f78a5b7add49c95d30e": "\\{(x,y) \\in \\mathbb{R}^2 \\mid xy = 1\\},", "3910d90507029b4213064a7fd2cae0f9": "c_0, c_1, c_2, \\ldots , c_T", "3910fba866a23a3f305e93404d557ae0": "r_1 = \\frac{r}{2(s-a)}(s+d-r-e-f),", "3910fbf408808578fa12a1c259dc26cf": "\\begin{align}\n \\phi({\\mathbf{r}}) = 1 -i\\frac{\\pi}{E\\lambda} \\int \\int \\limits_{z'=-\\infty}^{z'=z}\n V({\\mathbf{X'}},z') \\phi({\\mathbf{X'}},z') \\frac{1} {i\\lambda (z-z')} \\exp\\left(ik\\frac{|{\\mathbf{X-X'}}|^2}{2(z-z')}\\right)d{\\mathbf{X'}}dz'\n \\end{align}", "391102aec7d7120d15bd79210ce8d69b": "m \\rightarrow 0", "39111ac49f65e6429fd5f656178d5386": "\\mathbf A\\cdot\\mathbf B=0.", "39116627547f8361c4e51ec9a8f7a9a5": "E_{ab} I_{ab}", "39116f7219222e258d7979f1010a29d5": " \\theta_t ", "391179f52144e86d0c4901ed601d55ec": "\\mu_{\\mathrm{kf}}", "39119455a1e928c435ab24f47099815e": "A_2\\,\\neq 0", "3911d4ff739f58239bdee91ef3541b16": "~~~\na_{s,j}(\\omega)=\\sigma_{s,j}(\\omega)\n\\frac{\\omega^2 v(\\omega)}{\\pi^2c^3}~~.\n~~~~~~~~~~~~~~~~{\\rm comparison1}\n~~{\\rm partial}\n", "391212991470ba7875cd8a5a83d4dedb": "a(n) >0, \\quad b(n) \\in \\mathbb{R}.", "39123fd1d771f196d88a07cf74c3307b": "\n\\begin{bmatrix} x' \\\\ y' \\end{bmatrix}\n =\n\\begin{bmatrix} a & b & x_0 \\\\ c & d & y_0 \\end{bmatrix}\n\\begin{bmatrix} x - x_0 \\\\ y - y_0 \\\\ 1 \\end{bmatrix}\n", "391297fb43eba8e61dbca6a2c3a5d109": "C(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iiint s^*(u-\\dfrac{1}{2}\\tau)s(u+\\dfrac{1}{2}\\tau)\\phi(\\theta,\\tau)e^{-j\\theta t-j\\tau\\omega+j\\theta u}\\, du\\,d\\tau\\,d\\theta ,", "3912cc491468cf063d1e8ca26d44f2f7": "m=\\lim_{x\\rightarrow+\\infty}f(x)/x=\\lim_{x\\rightarrow+\\infty}\\frac{\\ln x}{x}=0", "39131d90d7c74c842dc91035c4b6199f": "{\\color{Blue}~2.33}", "391330b019f89085f70fb9b1367a6976": "K=\\frac{[{CH_3CO_2}^-][{H_3O}^+]} {[{CH_3CO_2H}]}", "3913697aa0fc9c430c90229942c9a1cf": "C^{0,\\alpha}(\\overline{\\Omega})", "3913f694eafc6a2afc7473d1a75fb689": "\\gamma(t) = \\exp(tX)", "3914312f233e97528cbd130381632855": "f(y^*)\\,\\!", "3914774ede3c5ab841b901b716e6cdfd": "R(\\varphi,\\partial_\\mu \\varphi) = \\int_\\mathcal{V} \\mathcal{R}(\\varphi,\\partial_\\mu \\varphi)dV\\,,", "3914784069aaebbe492540d7baf24cdc": "f(u) \\odot f(v) = f(u * v)", "3914c5566d52e5e627d8a49d3b6ef77b": "\\Delta\\,P+\\tfrac{\\rho f}{2\\,D}\\,W^2\\Delta\\,X+\\rho\\left(\\frac {2-\\beta}{2}\\right)\\Delta\\,W^2 = 0", "3914d1c222bb7fecb320b2b70a269f7e": "x_{up}^{(jam)}(t)", "3914f35bce4f03d69a51827a1a9e5fa8": "\\int \\phi\\, \\text{d}x,", "39154c23e164b3f19ec990566714f93e": "S_{r_1, r_2}(m)", "3915972221fb842c9f2ab60e91b952d9": "p(D|w,b,\\log \\mu ,\\log \\zeta ,\\mathbb{M})", "3915c3692e08f5940931b0cb256bef7a": "s_{\\lambda},\\quad\\lambda=L,L+1,\\dots", "3915cec6c63e53082c2c84d50041dea4": " \\max\\left\\{ \\mathcal{A}\\phi + L, M-\\phi \\right\\} = 0 ", "3915ed415ca8242fc38414eb4c50739c": "=dv(\\bar{X}) +d v(X_\\xi)+ \\eta(\\bar{X})\\cdot v+ \\xi\\cdot v", "3916033c3d7c14881a9bd634c8d18324": "g(x) = \\begin{cases}f(x) & x\\ne x_0 \\\\ L & x = x_0\\end{cases}", "39164c6b5fb5d38243a88ca1d03d99db": "\\displaystyle M_2(\\tau) = -q^{-25/168}F_2(q) + R_{7,2}(\\tau)", "391662a4923718906f4bedad63cfd9f9": "Q \\propto \\frac{1}{r^3}", "3916fa9b85cb8ae6012c89b92622133a": "(\\lambda_1, \\lambda_2, \\ldots, \\lambda_n)", "39170e0654fd3841129dc36f3050fec5": "\\langle W,\\le\\rangle", "391744832814ebfa5d0050972ccf3a10": "{d^2x \\over dt^2}+x= 0.", "3917a21d46473f43fedefd7859bf98de": "\\rho(r) \\sim e^{-2Z_{\\alpha}r}\\,.", "3917f96f4f959012370b19dc0141dcce": "\\alpha<\\kappa", "39181664ad6905ead69ecdab2910b6b5": "20, 240, 1470, 10640, 83160, 584640, 4496030, 42658440, 371762820, 3594871280,\\ldots", "39185117b6eb32287eb1ee9fdce661e1": " \\scriptstyle x_{n+1}=\\,ax_n(1-x_n)\\quad \\text{or} \\quad x_{n+1}=\\,a\\sin(x_n)", "39186d0f7669a685b6aa6e53767c2583": "{\\bar{DH}}_3", "39187923ef8d5934c13039188399edfc": "\\neg \\Box a", "3918fe68086642f0c653268c8dbde257": "\\iota:X \\to \\omega", "39194959de0f68269b0b2bf460acce48": "\\theta = \\arctan \\left( \\frac{\\sqrt{8a^2(r_1^2+r_2^2 - 2a^2)-(r_1^2 - r_2^2)^2}}{r_1^2 - r_2^2}\\right)\\,\\!", "391952f647ce9ff90012f47ad2425302": "C=f^{-1}(0)", "39196196973eaf8bbd4ae2132679ee98": " B = - \\frac{1}{4 \\eta} R^2 \\frac{\\Delta P}{\\Delta x}. ", "391a6995f036ed809d87fe22d77b2023": "\\cos \\alpha = \\frac{L_x}{\\sqrt{L_x^2 + L_y^2}}, \\sin \\alpha = \\frac{L_y}{\\sqrt{L_x^2 + L_y^2}} ", "391b082e99108fc9211c863a69bd2cc3": "Z_{21} = {2 S_{21} \\over \\Delta_S} Z_0 \\,", "391b39659667cd1a4c7577c2df69c1a8": "\\mathrm{ind}\\,f := \\dim \\ker f - \\dim \\mathrm{coker}\\,f,", "391b9ab937a0d40a416f5c9f9b804b75": "R\\cdot + :\\!SO_2 \\longrightarrow R- \\dot{S} O_2", "391bbca91f91df75da63fd305a8b2368": "u_1 = 1\\ ,\\qquad u_2 = {-1 + i\\sqrt{3} \\over 2}\\ ,\\qquad u_3 = {-1 - i\\sqrt{3} \\over 2}", "391bfdbc7250790e3b1266d5182a6091": " {v_{o} \\cdot w^{n} (s - a_{v})\\over 100} = {c_{o} \\cdot r^{n} \\cdot a_c\\over 100}", "391c609388567c60c231e80368921c08": "\\phi_n:X\\to X'", "391c6454febaa81459a12413e9c994e7": "L=\\tfrac{\\pi t}{6} (2^n + 4)(2^n - 1)", "391c83868c565b9e3994749f1519bcb4": "HF_S\\;:\\;n\\mapsto \\dim_K\\,S_n", "391c8bb42c4c5b98c84e734b17b8d38e": "\n1-F(\\rho,\\sigma) \\le D(\\rho,\\sigma) \\le\\sqrt{1-F(\\rho,\\sigma)^2} \\, .\n", "391cb3174d3f280937e2de1f7f771a14": "e=\\frac{r_\\mathrm{ap}-r_\\mathrm{per}}{r_\\mathrm{ap}+r_\\mathrm{per}}=1-\\frac{2}{\\frac{r_\\mathrm{ap}}{r_\\mathrm{per}}+1}", "391d4e2548480f70fa047f669ad735d0": "\\Gamma(t):= H(\\gamma(t)) \n=(\\Gamma_{1} \\oplus \\Gamma_{2} \\oplus \\Gamma_{3} \\oplus \\Gamma_{4})(t)\n", "391d6e4563e03f2d81fd095505052106": "G.x \\subset X", "391d9dd8558bdfde456b6dbe232a53fe": " \\ln \\gamma_i = \\ln \\gamma^C_i + \\ln \\gamma^R_i", "391da59afa60d25cf0b069afda2dabd5": "{d \\over dt}\\left\\{ Y_1 \\right\\} = \\left\\{B \\right\\} \\left\\{X_1 \\right\\} - \\left\\{ X_1 \\right\\}^2 \\left\\{Y_1 \\right\\} + D_y\\left( Y_2 - Y_1\\right) \\,", "391dd4ac1393adea33313c9a8b1ca7fe": "\\mathcal{O}_k = \\mathbb{Z} \\omega_1\\oplus \\mathbb{Z} \\omega_2. ", "391e189697600a12c3ba0f1f239b891a": "\\scriptstyle g(t)", "391e67cef2156f9c6a2f7f6e513733bb": "1,4,7,2,5,8,3,6,9", "391e7efe4dc8773cb18a870f156c3089": "m \\frac{\\partial^2 x(t)}{\\partial t^2} = -6 \\pi R \\eta \\frac{\\partial x(t)}{\\partial t}-\\frac{F}{l}x(t)+f(t) ", "391e7fd5988144ef4b961193e3deb072": "T_\\mathrm{v,env}", "391e906f1823eee8edad71388b577faa": "\\geq 2", "391e9aa10395e6acdad55e5cecef49b8": "\\Delta l = \\pm 1", "391ea6de10dd1052edcad8ddc84b1195": " F_p= \\frac{G \\cdot m \\cdot M_p} {d_p^2}", "391eb4bc9a6d6f3fda6c4694319d4ad1": "A \\# G", "391ed7953ed9ba2b37c8d7c320c8eab5": " P^{tr}(X \\mid Y) \\neq P^{te}(X \\mid Y)", "391ed9549cf85db617300308074e83df": "\\rho(f) = \\limsup_{r \\rightarrow \\infty} \\dfrac{\\log^+ T(r,f)}{\\log r}.", "391f005102b97ca77f40149ef2eb7d8a": "w_{\\mathrm{max}} = \\tfrac{\\sqrt{3}Pb(L^2-b^2)^{\\frac{3}{2}}}{27LEI}", "391f5e5cab8cded3eea173e699f77370": "\\int_0^{\\pi/2} \\arctan (r \\sin \\theta) d\\theta \n= -\\frac{1}{2}\\int_0^{\\pi} \\frac{ r \\theta \\cos \\theta}{1+ r^2 \\sin^2 \\theta} d\\theta \n= 2 \\chi_2\\left(\\frac{\\sqrt{1+r^2}- 1}{r}\\right)", "391ff69cd181211f0a315e74486d4fd5": "X \\to Y \\to Z \\to X[1],", "391ffc892d273162231e412f49f57e0b": "\\Gamma^{i}_{jk}", "3920386e074ada6dbf7ec88a773471c2": "\\lambda = 1, -5 \\,\\!", "39204c8d4066730cf457ca906ece96f4": "{\\gamma_i}", "392076019b9e5a9a9e993056f1112589": "x = 4.001", "39207f15e7687a95b43aa01988bbd01f": "f(a) > 0", "39208eb0fb0ed2305cce44f5eb4ddafc": "d = d_0 + 2^wd_1 + 2^{2w}d_2 + \\cdots + 2^{mw}d_m", "3920c2e7e1dfc674c4bcc51a9c9e1b65": "\n r_{\\mathit l} = \\sqrt{\\mathit l}\\;r_B\\; \\; \\; \\mathit l=0,1,2, \\ldots\n", "3920e5368cdfbe7cae8da22d0e918586": "\\int \\cosh ax \\sinh bx\\,dx = \\frac{1}{a^2-b^2} (a\\sinh ax \\sinh bx - b\\cosh ax \\cosh bx)+C \\qquad\\mbox{(for }a^2\\neq b^2\\mbox{)}\\,", "3920e546c471accb246edc0a8a946276": "u[2] := 2*atan(\\sqrt((a0+b1)/(a0-b1))*cot((1/2)*\\sqrt(a0^2-b1^2)*\\eta))+(1/2)*\\pi", "39212a161b03121e8a5eb861bea74e14": "B_{j_1},\\dots,B_{j_k}", "39214d06e1a45ff4792d5da9f13002a0": "\nm_2 \\ddot{x_2} - { c_2 } \\dot{x_1}+ { (c_2+c_3) } \\dot{x_2} - { k_2 } x_1+ { (k_2+k_3) } x_2 = f_2. \\!\n", "392164d110c7b1ce39e5a877c3080111": "\\rho_{\\bold{k}}^c(\\bold{r}) = \\frac{1}{2\\Omega} \\left[1 + \\cos(2 \\bold{k}\\cdot\\bold{r})\\right]", "392189924aab26674d10cc9837be6192": "g(\\xi) = \\frac{1-\\xi^m}{(1-\\xi)^n} ", "3921cd577a612cbc761b3a1ae154eb98": "C^\\text{Bayes}(x) = \\underset{r \\in \\{1,2,\\dots, K\\}}{\\operatorname{argmax}} \\operatorname{P}(Y=r \\mid X=x).", "3922210f1bfa265139290efbb057ad4b": "|r| = m = |x|^{O(1)}", "3922d4bc03905bfcd3fee568a44689d7": "[c^{n_1} G^{n_2} \\hbar^{n_3}] = M^{-n_2+n_3} L^{n_1+3n_2+2n_3} T^{-n_1-2n_2-n_3}", "3922eb84e2000c79c06d42d4879320aa": "B \\}", "3922fe2d9241eaff930357e503bad24c": " \\left(\\frac{N D^{0.5}}{(g H)^{0.75}}\\right)", "39231fa3f7bfbff25184f970c15f6cbc": "\\sin(x)+1", "392329114183248e642e96c3307d7dc8": "\\mathbf{n}-\\mathbf{e}_i = (n_1,n_2,\\ldots,n_i - 1,\\ldots,n_m).\\,", "3923564b783cc665aea30b01d7f9cd26": "f(x_1,\\ldots, x_{n})", "39235be9d595ac532a27af6bc2371bfa": "s = \\lg n / 2", "39239d27c323aa43586509b1e57fc618": "A_i \\cap A_j = \\varnothing,\\quad i\\neq j", "39242c63d97a57d37f87b634b7adf888": "\\eta_{th}=1-\\frac{1}{r^{\\gamma-1}}\\left ( \\frac{\\alpha^{\\gamma}-1}{\\gamma(\\alpha-1)} \\right )", "3924952a6dd5d5bdcc9856216fdb911b": "\\max(0, k - m) \\le j \\le \\min(n, k)", "3924bf9f391e65d826f5bd1e580c34ad": "1 + S_1 + S_2 + S_3 + \\cdots\\,", "39252f8e6ca3322bb7c97568bcf405f8": "c=\\frac{\\alpha^2}{\\sigma^2}", "392548da1aeaa0ca2a6b76082c50e283": "(a+b)^3", "39255bca801de77e028991248cf33475": "f_{STUN}", "39259e73816a953048a079ba4ea1454f": "pV = nRT", "3925d695d0a3df0463499053b4f2ec56": " I = \\sigma \\epsilon \\left ( T_\\mathrm{external}^4 - T_\\mathrm{system}^4 \\right ) \\,\\!", "3926265cb758c5e849e6174a838905dd": "E \\not \\in \\operatorname{FV}[G] \\and E \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] ", "3926466be16a3c6a43061cf69c80a7ec": "\\ S", "39266fa658d9e4f6a71fa3b18cf84b23": "z_{n+1}:=z_n-\\frac{p(z_n)}{p'(z_n)}", "3926a93b614dbbd20ddd59640b70c285": "\\pi \\otimes |\\det|^{s/2}", "3927008cf27b5fa6abecb48d1e4c855e": "\\ y' = 0.93", "3927122d7a582379b171dba8b160c3a1": " r\\in \\C ", "39271dcf2300ae325500771700cdac16": "B^{-n} = \\begin{bmatrix}\nx & y \\\\\nty & x \\end{bmatrix}^{-n} = \\begin{bmatrix}\nx_{-n} & y_{-n} \\\\\nty_{-n} & x_{-n} \\end{bmatrix} \\equiv B_{-n}.", "392723a78b3dbee3b7d5d4ba0dd8ec19": "x = \\int_1^y {dt \\over t}", "392735ec26694988cdace4d610e41b6f": "\\frac{A}{B}\\times \\frac{C}{D}", "392751aa19cf6a53b09f34d21ac56db6": "|\\Phi^+\\rangle_{AB}.", "392759609081305b01c2ab642243ffdd": "Qu'(c_t)", "3927b3479503c51397433f5f8fe7bb13": " \\Lambda_n(T) \\sim \\frac{2^{n+1}}{e \\, n \\log n} \\quad\\mbox{as}\\quad n \\to \\infty. ", "39284d124e85e85f828e751873375285": "f_{X\\mid Y=y}(x)", "3928da20f3a79b6c91c3dc54eea9501e": "F(t) = \\sum(A_1e^{-t/\\tau_1})", "3928de08c3f461786c2150c21c048f67": "\\frac{dC_{\\mathrm{A}}}{dt}=\\frac{Q_0}{(V_{t=0}+Q_0t)}\\left( C_{\\mathrm{A},0}-C_{\\mathrm{A}} \\right) ", "392928ede4d2869cbd69ebf89dc5851d": "\\int e^x \\cos x \\,dx. ", "39295c8aa3557cc5bdc0557eddd5f7f2": "\n S \\to aS, S \\to bS, S \\to \\epsilon\n \n ", "3929847537e5f4b8ebcf41847a03139b": "c(q)=2(1-q)^{1/2}\\sum_{m=0}^\\infty \\frac{(-1)^m q^{m(m+1)}}{(1-q^{2m+1})(1-q^2)_{q^2}^m} .", "39299b3259b5c7dd0ab1e73381780839": "H_n(p_1,\\ldots,p_n)=H(X)", "3929ccf4473eb3482f3cd34d3140d1a7": "{\\hat{\\beta}}(q, {r_{\\rm w}})", "3929e27e159288a50e7cb8a5f704c537": "\\phi: R \\to S = \\operatorname{End}_R(V)", "3929eb18966766ab3d98b5ca83ce9eb3": " \\gamma^0 = \\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\ \n0 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & -1 \\end{pmatrix},\\quad\n\\gamma^1 = \\begin{pmatrix}\n0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & -1 & 0 & 0 \\\\\n-1 & 0 & 0 & 0 \\end{pmatrix} ", "392a4c7f97d01776a4c0f42e5e8ab716": "{\\mathfrak G}", "392a562250f6028ecab8a18bd90511e8": " M_{i,j} = 1", "392a595796d57a350f57fa9ab9cd7668": "\\alpha(x) = x", "392a6b9382de2153323fc26f83984771": "I = \\sum |\\phi_n\\rangle\\langle\\phi_n|.", "392a7af322eab0b8c030e32c614216ce": "(x \\odot e^i) = x_i", "392ae1a93e7317f15586970511b6b4a8": "A_2\\in \\mathbb{R}^{m_2,n}", "392b0c07a1077f5264291f7e06976b20": "\\log (V_{out} / V_{in}) = a \\cdot (\\mathrm{CodeValue} + b )", "392b1b61eab49c6c8dfdd511b70d8611": "\\displaystyle{([b^2,a,b],c) =0,}", "392b2fb247c15845885395a2c2d6df0c": "\\|\\mu\\|_{ba} = \\sup\\nolimits_{A\\in\\Sigma} |\\mu|(A)", "392b6ce00dbf613bf6ca65273c7f827b": "\\ O = \\left|k_a\\right| + \\left|k_b\\right| + \\cdots + \\left|k_N\\right|,", "392b7670829021b142cacae095d0e3a7": "\\scriptstyle f_s/2\\ >\\ f", "392b8330047444e99175b9312865af3d": "C_{n}={\\scriptstyle \\pi^{n/2}/\\Gamma[1+\\frac{n}{2}]}", "392b868232b5e6ab3a4e18907ac6c304": "\\xi = 1+q\\lambda", "392bac8a1c43c73094d85f7ca9846248": "\\vec{n}=\\{n_x,n_y,n_z\\}", "392bc39d205f97bd0a2252432ef874ea": "\\bar{m}=\\frac{g_{J}^{2}\\mu _{B}^{2}H}{3k_{B}T}J(J+1)", "392c521009b3e5edadee862661ff9841": "-Q =W", "392c94123c237f6a9d3aad16e3666f29": "~0~", "392caf96ae7ed9b3f4f1bc6b9c663ab8": "K \\,\\!", "392cdf23e4f44c4eb3290e8e689adbb5": " \\sum_{k=2}^\\infty \\varphi^{k} = \\frac{1}{1-\\varphi} - \\varphi - 1= 1", "392db03da33401776a07fa8d9287199a": "312 = (132)", "392db0ceb5366464dc5e66e83978800c": "\n\\frac{d\\mathbf r_i}{dt}=\\mathbf v_i,\\qquad\\frac{d\\mathbf v_i}{dt}=\\frac{\\mathbf F_i(\\mathbf r_1,\\ldots,\\mathbf r_N,\\mathbf v_1,\\ldots,\\mathbf v_N,t)}{m_i},\\quad i=1,\\ldots,N. \n", "392db21665b1f373955c4dc763cedb27": "F(g \\circ f) = F(f) \\circ F(g)", "392e101b5faa015f339ca56522478c18": "\\mathrm{r_{prop} = k_{prop} \\cdot [RH] \\cdot [ROO^{\\cdot}] = k_{prop}\\cdot [RH] \\cdot \\sqrt[\\,]{\\frac{k_{init}}{k_{term}}}\\cdot \\sqrt[\\,]{[ROOH]}}", "392e41c3875d8a6cf51a1c05feb4920c": "\\Gamma(z) = \\lim_{m\\to\\infty}\\frac{m^{z}m!}{z(z+1)(z+2)\\cdots(z+m)}", "392e62b136ffb8fdd8a1f2cf42b4f779": "\nX = \\frac{n_\\text{p}}{n_\\text{p} + n_\\text{H}},\n", "392ef3d35b3af277241fc924edba7662": "\\nabla_XY-\\nabla_YX=[X,Y],", "392ef94cba46761c4011c389e0dbc1e6": " h'(-1) = 1, h'(0) = 0, h'(1) = -1 ", "392f39673c36ae03129004d9f2785336": "F_{C}", "392f40249b16c3aca38f09cb03f894dd": " \\frac{P_r}{P_t} \\propto G_t G_r \\left( \\frac{\\lambda}{ R} \\right)^n", "392f4d1acf225fcada7e712664087653": "c=E_ m={C \\over m} = {C \\over {\\rho V}},", "392fc6dffafe82592490f36812ecd80c": "MAT^\\lambda", "392fcc678028ea75a2af49c857fc6b27": "b_{p,q}", "392ff8d46aa8bbd61810ad878b2e2222": "\\tilde{\\alpha}_{\\rho}(\\lambda)=\\lambda \\alpha_{\\rho}+(1-\\lambda)\\beta_{\\rho}", "39300282214603edc7b46a69b156bdcb": " T ", "393011e14d2531d0e701b31ae716eee7": "\\rho (\\mathbf{r})=\\rho (z)=\\rho_0+\\rho_1\\cos\\left (q_sz-\\phi\\right )+\\cdots \\, ", "39301aec3473618a21e4a33ca7b8cecb": "G \\subset \\mathcal{B}_B ", "39304601836535fd78940883c55bf022": "Q^+\\!", "393053fdb4e0cb6c8be5e0a7600f70a4": "s, s' \\in S", "3930858f1906f3aacbe0f069a069d556": "\\Bigg\\{ \\Pr(h_1|h_2) \\Bigg\\}", "39309f9ce011351f33e6765c3318c65f": "\\eta(12) = {{1414477\\pi^{12}} \\over {1307674368000}} \\approx 0.99975769", "3930a5f40dbb3477eff2dd9b444bc8fa": "\\begin{align}\nK &= K_1\\times K_2\\times\\cdots \\times K_n\\\\\nK' &= K_1'\\times K_2\\times\\cdots \\times K_n\\\\\nK'' &= K_1''\\times K_2\\times\\cdots \\times K_n\n\\end{align}", "3930a9a49d8e3caa916182edbb838f00": "J^ 2", "3930bc9c3b4898d45f3ed13600202c8e": "L_{-}", "39310091f2a358a274fc1c8f418f0132": "[x] \\cap [y] \\ne \\emptyset", "39313c0f85818f4d9e11b9f02891e02f": "P_i^k", "39315270ebe5463d0406e8a1cf439f12": "\\check{H}(X,\\mathcal{F}_A)", "3931ee45dbdf8fd8a7e11f0866a512f0": "\n\\epsilon_{\\uparrow} (k) = \\epsilon_0 (k) + I \\frac{n_{\\uparrow}-n_{\\downarrow}}{n}\n", "39322bba33b04504cf8fb48491b2fc46": "n^{O(1)} = O(e^n)\\, ", "393277e5ad11eb91fd1158acbb906576": "\\log_2(1+\\textrm{SINR}_k)", "39329984d41f89898b2849cd62e05e46": "\\gamma = \\sqrt{\\alpha^2 - \\beta^2}", "39332de423bbfc0f2fdff11958524632": "d\\theta:", "393339f3d10c6dffa678fde4581fc7bb": "\\lim_{\\Delta\\alpha\\to 0}\\int_a^b \\frac{f(x,\\alpha + \\Delta\\alpha) - f(x,\\alpha)}{\\Delta\\alpha} dx = \\int_a^b f_{\\alpha} (x,\\alpha)\\,dx", "39334d29e35ebaf5573067a3908f6ec4": "[n,k,d]_{\\mathcal{F}}", "39335d159c51e45f9cbf88d97bbf3b97": "\\mathbf U(\\mathbf x,t)=U_i\\mathbf E_i\\,\\!", "3933ecbb090765d1718873a52a544412": "z_P-z_0", "393468cbefb77fafdcaf25776697dd5c": "\\mathcal{N} \\models \\varphi(\\underline{\\#(\\theta)}).", "3934791c617f6ab42063ec0e748d6477": "\n\\cdots \\to \nA^{-2} \\xrightarrow{d^{-2}}\nA^{-1} \\xrightarrow{d^{-1}}\nA^0 \\xrightarrow{d^0}\nA^1 \\xrightarrow{d^1}\nA^2 \\to \\cdots \\to\nA^{n-1} \\xrightarrow{d^{n-1}}\nA^n \\xrightarrow{d^n}\nA^{n+1} \\to \\cdots.", "39350a0e4faa59430b3e3099dcd70300": "\\| y-x \\| \\leq \\varepsilon", "393563328a473140ffac9a0170d8ddfd": "\\Phi (\\rho)= \\Phi_2 \\circ \\Phi_1 (\\rho) = \\sum _i \\rho (F_i) R_i.", "3935be146c6aabdde4d4e51d66726575": "f(q)=\\sum_{n=1}^\\infty a_n c_q(n)", "3935cb47a2517e39937b469c76d66c07": " 0 \\le p < q ", "393627e8c49f42c52cde69ff570c472c": "c_1(t') = \\exp\\left({i\\pi \\left(f-\\dfrac{E_{0}-E_{1}}{h}\\right) t'}\\right) \\mathrm{sinc}\\left(t'\\left(f-\\dfrac{E_{0}-E_{1}}{h}\\right)\\right)", "39367b421d2315abe7a1c6a359ffec9c": "A=[a_{i,j}]", "393690d238e810e41b17dd0186c21343": "\\scriptstyle \\lbrack\\mathbf b\\rbrack", "3936cb4c85e8022aa1c4e878bb087590": "ATX\\to X", "3936d19d52f28e9e86cc78839032479f": "5\\%-2\\% = 3\\%", "3936e368d8fa8c76564508c7edfd9fd7": "\n\\mathrm{M} = \\sqrt{5\\left[\\left(\\frac{q_c}{p}+1\\right)^\\frac{2}{7}-1\\right]}\\,\n", "3937159db2bd220a8cb8d813ae714eee": "T_i = \\mathrm F(L_i' - R_i', K_i)", "39372b4a391534a819bf8d6cecf892e5": "\\mathrm{LEC}", "393758cf80caafbbe4d8e81b3037eb10": "{\\operatorname{d}P(t)\\over\\operatorname{d}t}=rP(t)-M_a ", "3937762f9f26ef5fab75c8d0d4e01f97": "\\Pi_{m, n \\mathbin{:} {\\mathbb N}} m + n = n + m", "3937b264bada01389475cc28bac047e4": "\n\\frac{d^{2}u}{d\\varphi^{2}} + u = -\\frac{\\alpha}{mh^{2}}.\n", "393819264c018841dfcdd9b078d95624": "\\sum_{n=1}^{N} \\mathrm {rect} \\left[ \\frac {x'-nS} {W} \\right] ", "39384d6106ac8a16ca942fe5b3b50766": "f = a A \\pm bB\\,", "393866a0daef97e277bf63d0bf8ce2f9": "B_{ijkl} N_j N_l m_i m_k > 0 ", "39386ed70ebcac505df7eb44df1cb791": "n_{j,r}^{i,-(m,n)}", "39386f8dd28d4035fd441981269cfb82": "e=m=1", "3938868e064551c60ece26e8dc5f55ec": "\\Pr \\left [\\tilde f(r_1, \\dots, r_i) = f_i(r_1, \\dots, r_i) \\right ] \\leq \\tfrac{1}{n^2}.", "3938b71c34afe24872773cf7c82fe957": "\\langle T_f,\\varphi \\rangle = \\int_U f\\varphi\\,dx.", "3939103aa1240a4266231b4d30444839": "\\bar t^{(k+1)} = C^T \\bar t^{(k)} ;", "39392e2c5908828d2d61ff747acab75d": " H(S) = - \\sum_{x \\in X} p(x) \\log_{2} p(x) ", "39396ae61d92adb37fb21e753480df93": "\\begin{matrix}{4 \\choose 1}^3\\end{matrix}", "3939bd07c4ca610481f59b8cd74af6a2": "A_1(p,r)=r", "3939d9531969ce0b0e26e8f4b34ad14a": "dx_i/dt", "3939e877d0f96bafed4fba76c2e5a7f7": "\\left(\\frac{t^k}{k!}\\right)", "393a0ec4215ae05583ba6a9a811072f0": " c(r)=g_{\\rm total}(r) - g_{\\rm indirect}(r) \\, ", "393a35c4eee82e028f709d383774e553": "m_1 = [0, 2.40] + [12.3, 7.6] = [12.3, 10.0]", "393a449ba3fabfedba808061777efc6c": "\\operatorname{Perf}_s(f,r') - \\operatorname{Perf}_s(f,r) \\geq t", "393a7c80ad346f1b601149a9c9b6f6c8": "\\int_0^\\infty \\left (\\frac{1}{x}\\int_0^x f(t)\\, dt\\right)^p\\, dx\\le\\left (\\frac{p}{p-1}\\right )^p\\int_0^\\infty f(x)^p\\, dx.", "393a809ad39807b51673e7fb80c238d6": "h={{2 \\gamma \\cos{\\theta}}\\over{\\rho g r}},", "393a9eec45cb762fcc7e216f7877aa58": "\\varphi(1)=1.", "393ac72768bf1e93158c1cc6cb1727e0": "n\\# = (n-1)\\# < 4^{n-1} < 4^n.", "393b27230ece833f30d672f3ccffea44": "\n\\sum_n \\psi_n |n\\rangle\n\\,", "393b5ad2fbc014c77e78c7c7b225157f": "f(k,n,p)=f(n-k,n,1-p). ", "393c09c070e61fea822c4cead9ecf7b9": "L_n(R) \\to L^n(R)", "393c265265b1c1a1b284450bf2ede27c": "\n{R(r+dr) - R(r) \\over R} = {dr\\over r} = g dr\n\\,", "393c8d36362000a23296dd67b0823a88": "u+v = \\int \\left(\\frac{du}{dx} + \\frac{dv}{dx}\\right) \\,dx", "393c8f6da52b58a208e7eb3a82cfb019": "(h^{ij}\\nabla_i \\nabla_j - 2R^{(3)})\\Phi_A = 0,\\,", "393cb29364d46207d27905f0fea6b44e": "f(x_1,\\dots,x_n)", "393cbaeeb57e0b8c3ec38aadbd04a3fc": "\\ M_{pitch_{max}} ", "393d0f067334e16e36dd412bc18ccc0f": "\\textstyle P \\in E", "393d472824ce3945fe2b6af9d7993e4f": " \\displaystyle n \\leq \\sum_{m=1}^n r_{B,h}(m) \\leq |B \\cap [1,n]|^h ", "393d76a65d95e43674eeed831617fe60": "\\sum_{i=1}^t c_i = 0", "393d83043a2601d5164bbc3bc269c82b": "|\\int e^{ -i\\omega t }E_{ 2 }(t)|E_{ 1 }(t-\\tau )|^{ 2 }dt|^{ 2 }", "393dce073c69b0231463012f5baba006": "\\mathit{H}\\mathit{I}\\mathit{C} = \\bigg\\{ \\Big[ \\frac{1}{t_{2}-t_{1}} \\int_{t_{1}}^{t_{2}} a(t) dt\\Big]^{2.5}\\left ( t_{2}-t_{1}\\right ) \\bigg\\}_{max} ", "393de4b2a0f83d63675f9d1d62e80611": "\\lambda = \\frac{\\partial V}{\\partial S}\\times\\frac{S}{V}", "393de6e414be9e20767365423c7a8602": " K_{\\lambda 1^{(n)}} ", "393dfa8906d734ee7d552501f259e5be": "M \\otimes_R N \\to L", "393e0a3caabb86266e196ef14bfec768": "\\bar{x} = \\sigma^2_\\bar{x} (\\mathbf{W}^T \\mathbf{C}^{-1} \\mathbf{X}).", "393e3e44f26091579a546e13966c7aac": "f(x)=AP_n(x)+BQ_n(x)", "393e8cd19fea5bb0ae4e029359735381": "\\hat T = \\sqrt{ \\frac{\\sum_{i=1}^r\\sum_{j=1}^c\\frac{(p_{ij}-p_{i+}p_{+j})^2}{p_{i+}p_{+j}}}{\\sqrt{(r-1)(c-1)}} } ,", "393e92bb7d03ade9d308e85e383b4360": "y(t)=0", "393ed01381fa3a0b84f3b80ea742b372": "\\mathbf{Y}(s)", "393ee034cf0ba963dbdb23fecd6d75fc": " \\sigma_y^2 \\le \\frac{y_\\text{max} (A - H)(y_\\text{max} - A)}{y_\\text{max} - H}, ", "393f4cccdafad6c17845d5c720f68240": "\\delta_t = p + {s \\over {1+rse^{st}}}", "393fd94e9144b658615f7f1e0e879a0c": "0\\leq a_i\\leq b-1", "393fdd9812c3b370975efd675f693e87": "\\cot (A \\pm B) = \\frac{ \\cot A \\ \\cot B \\mp 1}{ \\cot B \\pm \\cot A } ", "39406a7373140f9e1181e529168c8db0": "{^b} \\bar a", "39409e200bd99a2386efe5e8410212e9": " f: \\mathbb{N} \\rightarrow \\mathbb{N} ", "3940e2decf29705d0c158ac76a3a9836": "7x^2-3xy+1.5+y", "39410fc88742015fa52eb2a4ff0f3cd0": " \\approx \\$200,000 \\times 0.01110205 \\ \\approx \\ \\$2,220.41 {\\rm \\ per \\ month} ", "394143311310b5f43f7576aa1781a124": "c_1 \\ne c_2 \\in C", "3941ce22af4cb1c03bf7dfd3f7a88d89": "\\forall x_1, ..., \\forall x_n (S(x_1) \\and ..... \\and S(x_n) \\rightarrow S(h(x_1, ..., x_n))", "3941d927559284894e5d5c8e286dfc12": "\\Sigma\\ :=\\ \\Sigma\\ +\\ m_n", "3941dc6b62a363cbc771655c3a304e21": "(\\sqrt{2} m )^2 =~0", "3941e2013e7f9bc731b2fcc8c54505a1": "uS(u)", "3941e5583c161d33e16206b7bf844df1": "\nH(0,j) = 0,\\; 0\\le j\\le n\n", "3941f061added11f23967d37e298f165": " r= \\sqrt{\\rho^2+z^2} = z \\sqrt{ 1 + \\frac{\\rho^2}{z^2} } ", "394223897cf8e480e3f4e65756960ecf": "\n\\det(\\mathbf{R} - \\lambda \\mathbf{I}) = 0\\quad \\hbox{for}\\quad \\lambda=1.\n", "39433b3e8c047b7727449c88c3cc17ec": "R = X(\\alpha) Y(\\beta) Z(\\gamma)", "394369a5c4b755913959989b028cac3a": "p \\ge \\frac{1 + \\epsilon}{n}", "3943a639f9c0eb86a864d36785a9a8ef": "\\psi_L", "3943feae38b51af92b3aa5c971a2d628": "=\\left(1/b\\right)\\ln \\left(1/\\eta\\right)\\ ", "394400de86488abb0c91e7db891735fc": "\\mathbf{r(x)} \\neq \\mathbf{r(y)}", "39440a2feaac3d48794951ea99794dd5": "\\frac{1}{2} k_1 Y_1^2", "394421f97bf191c78281ff451af80248": "\\begin{Vmatrix}m^*\\end{Vmatrix}", "3944890c65ca83cb5e7f245aad2f9527": " - a^3 {\\dot \\rho} = 3 a^2 {\\dot a} \\rho + \\frac{3 a^2 p {\\dot a}}{c^2} \\,", "3945106e00e528930d273f2ebba93ff7": "S^{\\prime} = S^{\\prime \\prime \\prime}", "394563569a03c782691b5fc76174e69b": "t > t_1", "3945c3e498eefa521abaa7cd4dbfc2f7": " \\nabla E_I = - \\mathrm{div}(g'\\left( \\| \\nabla I(x)\\|^2 \\right) \\nabla I) ", "3945d86d4613a08e958a091597bc01bf": "u, \\ \\mathrm{and} \\ \\bar{u}\\,", "3945eb9c936bf24003305e70cc649fb5": " R^3=\\frac{3r\\gamma}{2g(\\rho_2-\\rho_1)}\\!", "39465e7968eaf709c5d27211a020810b": "{\\partial V}/{\\partial t}", "3946b4da8c5d592cf11032ff1fbde620": "y_{i,1} = \\mathbf{x}_i^{\\rm T}\\boldsymbol\\beta_{1} + \\epsilon_{i,1}", "3946d77747881f8905125b2133603de3": "\\pi_0\\big(PO(2k)\\big) \\cong \\mathbf{Z}/2, \\pi_0\\big(PO(2k+1)\\big) \\cong 1.", "394755da9b56d797f759b98a410e1c4c": "\\phi \\to (\\chi \\to (\\phi \\land \\chi ))", "394770a99ba20293b43cced3690ea3be": "\\mathbf A_2", "39477b5dc2e09356b76fef29cd3bd488": "\\tilde{f}(\\lambda)=\\int f \\Phi_{-\\lambda} \\, dV", "394841f8df03ac0bb908c84dbfe23e93": " G_{u,d}", "39486fbaed190fb7ff6223ed93de609e": "\\{d_0,\\ d_1\\}", "394878432568e8db3f7334163c0db49b": "1 / \\omega_0", "3948b25030d8fd4428c08f59315776b4": "{1 \\over 3}\\times \\left( {2 \\over 3} + {1 \\over 3} + {4 \\over 9} \\right) = {13 \\over 27}", "3948e150856588fa2e0d9f3d93e5925c": " \\frac{d p_1}{d t} = - \\frac{d p_2}{d t}, ", "39490dd1c50a73fdc6bd589c3be48661": "_P", "3949245b055e11343d61779f247aee41": " \\textbf{e}_r = \\cos\\theta(t)\\vec{i} + \\sin\\theta(t)\\vec{j}, \\quad \\textbf{e}_t = -\\sin\\theta(t)\\vec{i} + \\cos\\theta(t)\\vec{j}.", "3949341d7fd5e42f5bd39c2206566821": "I^-[S] = I^-[I^-[S]] \\subset J^-[S] = J^-[J^-[S]]", "394948adb8a37baf044a978088d3009c": "\\omega_d \\,", "394958ed803c8ef21911bc0e1c1e0d15": "v^i[\\mathbf{f}A] = \\sum_j \\tilde{a}^i_jv^j[\\mathbf{f}]", "3949bf315fb021a03a151ab913bb0796": "y=x^p", "394a0eca2bf8108140f9292865f98d1c": " r_{m}= [(r_{t}^2 + r_{h}^2)/2]^{0.5} ", "394b724438ad0024d6ea057f1575e963": "\n\\sin \\gamma = \\cos \\eta \\cos \\lambda. \\,\n ", "394b73160286c7f0257f05acfad2f078": "\\mathrm{u}(1)+3 \\times 6 \\,", "394bc74d738aa3c41e576a7833049bcc": "L = \\frac{d^2 \\Phi}{dS \\cos \\theta d\\Omega } = n^2 \\frac{d^2 \\Phi}{d^2 G}", "394be305090e45a43f4b87c479fb9194": "[\\cdot,\\cdot]^*\\colon \\mathfrak{g}^* \\to (\\mathfrak{g} \\wedge \\mathfrak{g})^* \\cong \\mathfrak{g}^* \\wedge \\mathfrak{g}^*", "394c2c68154b640ba500122b407f5aa6": "\\int_3^6 \\int_2^4 \\ 1 \\ dx\\, dy=\\mbox{area}(D).", "394d0ef0db1bb390e7bbbeafca5218d1": "\\partial S/\\partial E", "394d3d7e61dc1a0de650ddce3f177e6a": " V = \\frac{L^2/r^2 - \\epsilon}{ \\left( 1 + m/r \\right)^2}", "394d6bd0c81cc6fff7a5b95f15c97a17": "C^{(T)}_V(V,T)\\ ", "394dc659e02d3c127a65ffb325d08a59": " m_i \\equiv \\langle s_i\\rangle ", "394de19fd0d501a388c660155bbc2a50": " s \\in [0,t] ", "394de4995e9f077a44b982ecf8319235": "\n C^0_1 = \\frac{C^g_1 \\,C^g_2}{C^g_2 + (T_0-T_g)} \\qquad {\\rm and} \\qquad C^0_2 = C^g_2 + (T_0-T_g) \\,.\n", "394df13c5a180a62b43ac0a16e0ff0e4": "\\left(f^{-1}\\right)^\\prime (y) = \\frac{1}{f'\\left(f^{-1}(y)\\right)} . ", "394e126bd0cc715a2d86107e0a2223b0": "\\Gamma(5 + 3i) \\approx 0.0160418827 - 9.4332932898 i", "394e12fa73cb22115c1187dc4177cf9d": "[(X+E)\\; (Y+F)] = [U_X\\; U_Y] \\begin{bmatrix}\\Sigma_X &0 \\\\ 0 & 0_{k\\times k}\\end{bmatrix}\\begin{bmatrix}V_{XX} & V_{XY} \\\\ V_{YX} & V_{YY}\\end{bmatrix}^*", "394f50b4f4dc97b14ff3d38c1867e65d": "f'(x) = (1/4)x^{-3/4},", "394f51dec0bd9e4256c9397adcd75c6b": "\\mbox{tournament net run rate }=\\frac{\\mbox{total runs scored in match 1 + total runs scored in match 2 + ...}}{\\mbox{total overs faced in match 1 + total overs faced in match 2 + ...}}", "394fa5febceb4589cd18b3321235a1a3": "\\sum_i p_i z^i = \\sum p'_i z^i \\cdot \\sum_i p''_i z^i ", "394fd74537690805c48292ae2f5b0000": "y_0, y_1, \\ldots, y_m", "39503effaaa620168fde6ccb930f60fe": "\\ (\\lambda - k)^2", "395082adeba0a80263e40a919885aadb": "\\Pi_t^j \\equiv P(M_t = m^j | r_1,...,r_t).", "39508bdd9ec8f29c48fc52e24c676463": " y' = f(t,y). ", "3950a2beed8f89819daa872126e7ff5d": "\\mu=0", "3950fa8e26450b5ea437b553a2430821": "g_0=f_k(x)=ay\\in\\mathbb{Z}y\\text{ and }g_1=\\frac ckf_{k+1}(x)=\\frac{bc}ky\\in\\mathbb{Z}y.", "39513ce744622fa2ee3da9ee9012dab8": "\\mathbf{x}_3=\\alpha_j(\\boldsymbol{\\alpha}-\\mathbf{r})+\\boldsymbol{\\delta}", "39513fd050e105807be7740511274d9a": "u(x)\\geq u(y)", "395183fff91a064a30a6128a2c7815a6": "E(k) = \\tfrac{\\pi}{2}\\, _2F_1\\left(-\\tfrac{1}{2},\\tfrac{1}{2};1;k^2\\right)", "39518eab9232fa35b88fd0ae5db710e4": "\\beta\\ge 0", "3951a7244deeec63db14433cca5c20ad": "T_i = \\max\\{0, C_i-d_i\\}", "3951a8654de2004abe46dad93d65e231": "(x,x') * (y, y') = (x+y, j^mf_p(y) + y')", "39525ad922bc0c5712b3e09a2c18e041": "D_{KL}(P\\|Q) \\ge\n \\int_{\\mathrm{supp}P}\\left(\\log\\frac{\\mathrm dQ_\\theta}{\\mathrm dQ}\\right)\\mathrm dP\n = \\int_{\\mathrm{supp}P}\\left(\\log\\frac{e^{\\theta x}}{M_Q(\\theta)}\\right) P(dx)", "39528229ab56b13519c1e0ef51606d57": " \\exists \\xi\\in[x,y] : \\ f'(\\xi) = \\frac{f(x)-f(y)}{x-y} ", "39528305e052eb782b3b64640255b106": "g(h)", "3952ef8fd5e15d74f68160a9742f914e": "\\nabla p ", "3953193faa8672b14cd35d7e5d8e9cb8": "\\Gamma (p,m)\\big\\}", "39535bc9996314618e3efef376096274": "k\\times2^{n+2}+1", "3953a562576ff0e6403b027306af51cb": "\\int_0^t u(s)\\,ds\\in D(A)\\text{ and }A \\int_0^t u(s)\\,ds=u(t)-x.", "39541654b809861fce03b6dd137668a3": "\\boldsymbol{F} - m\\frac{\\operatorname{d} \\boldsymbol{\\omega}}{\\operatorname{d}t}\\times\\boldsymbol{r} - 2m \\boldsymbol{\\omega}\\times \\left[ \\frac{\\operatorname{d} \\mathbf{r}}{\\operatorname{d}t} \\right] - m\\boldsymbol{\\omega}\\times (\\boldsymbol{\\omega}\\times \\boldsymbol{r}) ", "3954c032f39a021bcc01545cb4014b30": "\\pm\\left(0,\\ 4\\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "3954d6068c1d3701df503a35721424ea": "\n\\mu(x; t, s) =\n\\sum_{n \\in \\mathbb{Z}^k} \n(\\nabla I)(x-n; t) \\, (\\nabla I)^T(x-n; t) \\, \nw(n; s) \n", "3955461f26c9cc7db00174adf4e23f9b": "D(z_1,r_1) \\times \\dots \\times D(z_n,r_n).", "39559219ca75828e279b81110321a88e": "y-u \\equiv 0 \\bmod p", "39559b4a8df2a7009ad4cb53ddcc35ff": "V_{loop}", "3955e062fe5ff7adf97dd2c075935848": "Q(p) = \\operatorname{sign}(p-1/2)\\,2\\,\\sqrt{q-1}\\!", "39561302557aa405f1a555d200603c1f": " \\left( 2.62\\,rs^4 \\right)^{1/5} ", "3956816dc2a7c5ceb0a75bb6e1093857": "f_i\\in C", "39569d4e57a95dc7539d25ac35343a25": "\\sum_{k} \\langle A e_k, e_k \\rangle. ", "3956a27a5dfe33292cb125c2925bdffd": "T_m=\\frac{T}{2 \\cdot \\sin \\left ( \\frac{\\alpha_r-\\alpha_b}{2} \\right )}", "3956cdd8e67145601e5e34a45ee87547": "\n\\operatorname{dCov}^2(X, Y) := \\operatorname{E}\\big[d_\\mu(X,X')d_\\nu(Y,Y')\\big].\n", "3956ef3f406b914c8f4a6a33d196e815": " \\mathbf{u} = u_1 \\mathbf{e}_1 + u_2 \\mathbf{e}_2 + u_3 \\mathbf{e}_3 ", "3956f2f542d7bcbbb4ae15bf762e1261": "x_2(t)", "395774abbb25b867bb4d390ee6afb0a6": "N(G)=G\\overline{G}", "39579066fe7836dfc8249e2b67fa3d3c": "(e,w,d) \\leftarrow GenCertPrime(s)\\,", "3957cf02800a6a1d83c53c142c6dca98": "P=A", "3957d544dc16ad2ccf43399b76a0f24f": " \\ln\\left(\\frac{\\rho}{\\rho_0}\\right) = -\\int_{r_0}^r \\frac{g(r)}{\\Phi(r)}dr, ", "3957e3de7b240b71be9ef889550bc2c6": "2\\pi\\textstyle\\int_{0}^{\\infty} \\int_{0}^{\\pi}", "39582221fca601739b31db298430e1ad": "((P \\to Q) \\land P) \\to Q", "39582257d490f7099f77bc8292edf746": "\\Delta _G H", "3958340365c74f8afc1f50a1913605f0": "\\lfloor a \\rfloor", "39583c77003cedd901b685b9b2263fda": "c_{T-1}(k) \\, = \\, \\frac{Ak^a}{1+ab}", "39583fefc225a723bf6aed6a23b27e7f": "\\vec{Q}_r", "3958551a89e2e0d93348772e0ee1c3ad": "v=U \\sin\\beta", "3958650f30d774f1617df0b28e153213": "d=\\frac{\\lambda_\\text{u}}{\\bar{\\beta}}-\\lambda_\\text{u}\\cos\\theta=n\\lambda", "39589e7b5076b6161c620dc79dca803a": "\\lambda = (\\hat{R}_m - \\hat{V})\\cdot(\\hat{R}_m - \\hat{V}) / 2", "3958e403075e4d377381087cba7b26c8": "\\{1\\}", "3959116d1adc8ad245a719a6980cb50d": "D^{(0)}(p||q)=2\\sum{(\\sqrt(p)-\\sqrt(q))^2}=4(1-\\sum{\\sqrt{pq}})", "3959353fb06a885d10f68069e10bf520": "\\omega_1^2 = \\Omega^2(k_1)\\,", "395951d4193b413bee1fc09a920b52c1": "\\ \\mu \\,", "3959b4df7374521725219b8c302dbad4": " \\tau = T-t \\, ", "3959cf55c1fc4d9025e4807e1073d866": "h_0\\left( t \\right)", "3959d032a6541ec2670466b5fffea530": "p \\to \\text{Provable}(p)", "3959d943e848846fb8b5834b43867655": "r(s)=(\\sinh^{-1}(s),\\cosh(\\sinh^{-1}(s)))\\,", "395a12859e629ec5b2bcaf5e9c6bab31": "{d_1 ,\\dots , d_r }", "395a426e64030ab374c17cf6b7e0edea": "V_{out}=A_{o} (V_1 - V_2)", "395a59fc98cef5e9897ced30365afa72": "\\mathrm{Rp}(A,a) := \\forall x(x \\in A \\leftrightarrow x \\in a).", "395a9af17f8642e02a32af8637542947": "N\\times N", "395aa0c525a4cf3525eaf785cd6f7f71": "{n \\choose k}p^kq^{n-k}\\simeq \\frac{1}{\\sqrt{2\\pi npq}}e^{-\\frac{(k-np)^2}{2npq}}.", "395ac7dc7e0cdc515c2da80dede3f1e8": "\\int \\!\\,", "395acd7f4dcdee629e0a9e663083a533": "R^T w", "395aef6a421a24dbd4a1a6ba751e447d": "O(n^{1+\\epsilon})", "395b32ebad77c16e6e86ffe01729f1a2": " \\rho = e^\\varphi", "395b3967a99c5267e91874a218ba30f1": "2^n\\sin\\tfrac{x}{2^n}", "395c31e2e80b114e025a3c0d69dd3b46": "\\scriptstyle{n-1}\\;", "395c623d0f0661e9aa86ee7285d27840": "\\scriptstyle m_i=-1/l_i", "395c7d99723808dade11e9b04768133b": "f\\chi_{\\{|f| \\leq 1\\}} \\in L^{p_1}", "395c91de51975179c7e289bd1bc66e08": " x \\mapsto \\operatorname{tr}~x^2 ", "395d12e531d03592e4ea2d0a8bfda8f2": "BP_2=f_2BP", "395d7c6eb06b90f24c1f7b1345a21ef1": "l_{21}\\cdot u_{12} + l_{22} \\cdot u_{22} = 3.", "395da888c6051cb9c3a92b3d0110851b": "s_4=\\alpha^{2},", "395e2f281c9ac917bb079d11fbd3fb5e": "\\ C_{rr}", "395e3e329ed9c524d0d2c8960d1eb520": " c_{1} = c_{2} = \\tfrac{1}{2} ", "395e5d1a23e980e6e478f4a0919282a3": "(S,\\le_S)", "395e74623300e6700c54eb264c2128e2": "\\frac{dS}{dt} = \\frac{\\dot Q}{T}+\\dot S+\\dot S_{i}", "395e75fa3f1821b82e4c23c2db72d136": "\\scriptstyle - \\frac{10+9\\sqrt{5}}{61}", "395eda733cf60e25a725a8bdfe84428f": "\\operatorname{Tr}(A \\, \\rho) < 0 ", "395f0b73d271f174be98ebcd3db3a16b": " \\pi\\ : \\prod_{i \\in \\mathrm{N}} \\Sigma\\ ^i \\to \\Gamma\\ ", "395f2e68bcc4576eda5eb5e353f631b3": " \\alpha \\ge 0 ", "395f2ee96d1e4d5b65ed8581893e18c6": " l_2= a_{00} - \\mathcal{L}(p_6)+p_3p_6, ", "395f8c9b203bbb863af45bda61d7d834": "\\log\\circ f", "395fd8c757e7ee35440a012cb551341e": "\\langle \\vec{R} \\rangle = \\Sigma_{i=1}^N \\langle \\vec r_i\\rangle = \\vec 0~", "39600258e1c8b01965b60e03bce8063d": "\\Delta\\nu = \\frac{c}{2L}", "396051bfe5db67fec03ffc77253fe9a5": "0\\leq l_1 < 2^{m_1}", "39607e71f76bafd9750b86d4e3b04622": "0 \\rightarrow \\mathrm{Pic}(X) \\rightarrow \\mathbb{Z} \\stackrel{\\delta}{\\rightarrow} \\mathrm{Br}(K) \\rightarrow \\mathrm{Br}(K)(X) \\rightarrow 0 \\ . ", "3961369792e28f50610dc9dbcf27a87c": "\\mathrm{mes} \\left\\{ x \\in \\mathbb{R} \\, \\mid \\, \\Re \\frac{1}{\\pi} \\sum \\frac{a_k}{x - b_k} \\geq t \\right\\} = \\frac{\\sum a_k}{\\pi t} ", "39616775220a7867c3591e99168e87e5": "\\; [R_\\pi(\\varrho_{A_1\\ldots A_m})]_{i_1j_1,i_2j_2,\\ldots,i_nj_n}\\equiv\\varrho_{\\pi(i_1j_1,i_2j_2,\\ldots,i_nj_n)}\n", "3961a8e52749fa84a86b5f76b219524b": "\\mathbf{K}=h\\hat{x}^* + k\\hat{y}^* + l\\hat{z}^*=(2\\pi/a)(h\\hat{x} + k\\hat{y} + l\\hat{z})", "3961ddc2b39b28d035c55872a6f4cabd": "f_{P}", "39622cdff557f12cc5596d3174615ae9": "f: X\\to Y", "39626f87b7a5136efa8971c8c1dd53c3": "e^{-\\frac {1} {2} \\sigma^2} \\epsilon_t", "3962cb9df1b63b37689d933dc66cfe96": "\\alpha:~\\alpha\\in (0,\\infty)", "39632fa8ea4de58c0e10f29fe3ab3102": "x^4+px^2+qx+r=0", "39633b094aa74444b599a8ba1cef3e15": "\\omega^2 = \\frac{SS_\\text{treatment}-df_\\text{treatment} * MS_\\text{error}}{SS_\\text{total} + MS_\\text{error}} .", "39634a0932a87cbd3eae5cf45553b19a": "\nV = \\xi^k\\frac{\\partial}{\\partial \\xi^k}, \\qquad J = dx^k\\otimes\\frac{\\partial}{\\partial \\xi^k}.\n", "3963ad3a73fb7d50865af5df8b872b8a": "{1 \\over 1}+{1 \\over 2}+{1 \\over 3}+{1 \\over 4}+{1 \\over 5}+{1 \\over 6}+\\cdots \\rightarrow \\infty. ", "3963d906e30c959a0450f868af230054": "\\vartheta(p_k)\\le k\\left( \\ln k+\\ln\\ln k-1+\\frac{\\ln\\ln k-2}{\\ln k}\\right)", "3963d9695f05d4a43272ae3f4f5db131": " \\left( \\frac{1+|x|^2}{1+|y|^2} \\right)^t \\le 2^{|t|} (1+|x-y|^2)^{|t|}.", "39646e2a62d7193ee0410d4d2c4e66fc": "b(t;u) = H(u-t)\\cdot e^{-(u-t)r} = \\begin{cases} e^{-(u-t) r} & t < u\\\\ 0 & t > u,\\end{cases}", "396494f882bf299d62ea6a8ca9d5c413": "\\sigma_{ab} = \\theta_{ab} - \\frac{1}{3} \\, \\theta \\, h_{ab}", "3964f2d88803d394d109a6835e1aae46": "\\mathbf a", "396538658b592c08d1a5b618aaf87a6f": "r_0 \\ldots r_n", "3965702b1b5bbd8bbc559ae13a7a8799": " m-E(Q)/Q ", "39657773c473c2172ba215da54d4c914": "\nP = \\frac{V^2_{\\text{S}}}{R_{\\text{L}}} = \\frac{{N^2_{\\text{S}}}{V^2_{\\text{p}}}}{{N^2_{\\text{p}}}{R_{\\text{L}}}}\n", "39657b5521b77602dd43ad004dda7e93": "\\tfrac{f(x)}{g(x)}", "39657f8e132f698562d7eafbf9acb225": "P(t_1,\\ldots,t_n)", "3965c8994a5a96e3f09fa45980c448b3": "{\\rm add}({\\mathcal K})", "396687bea563983b598415a277d51fd3": "\\mu=\\langle(l,s),j,m_j=j|\\overrightarrow{\\mu}\\cdot \\overrightarrow{j}|(l,s),j,m_j=j\\rangle \\frac{\\langle (l,s)j,m_j=j|j_z|(l,s)j,m_j=j\\rangle}{\\langle (l,s)j,m_j=j|\\overrightarrow{j}\\cdot \\overrightarrow{j}|(l,s)j,m_j=j\\rangle}", "3966955ab1ba4480ad501f402feebddd": "\\left\\lfloor\\frac{x}{n}\\right\\rfloor = \\left\\lfloor\\frac{\\lfloor x \\rfloor}{n}\\right\\rfloor", "3966d7ed14a93042110eb313225e3844": "u = \\varphi(\\tau)\\,", "396752b89d2220820c58b5e4fd48e546": " \n\\tilde \\sigma ^M=\n\\langle \\sigma_{11}, \n\\sigma_{22},\n\\sigma_{33},\n\\sqrt 2 \\sigma_{12},\n\\sqrt 2 \\sigma_{23},\n\\sqrt 2 \\sigma_{13}\n\\rangle. ", "3967c060abe49be7b1b2d502427f3c7c": "\\mu>1", "3967d462bd7042548ac8603574ce304a": " x(t) = x_0 \\left( - \\frac{1}{999} e^{-1000 t}\n+ \\frac{1000}{999} e^{-t} \\right)\n\\approx x_0 e^{-t}. \\qquad \\qquad \\qquad (15) ", "396875907c6671e361fd33638e51074e": "\n{{documentation}}\n", "39688a102745a9a07376953559d1afd5": "A\\in\\mathcal F,", "39693f8a43aafb19a0232100bcbaa832": "\\begin{align}\n P_0(x) &= -\\alpha \\beta, \\\\\n P_1(x) &= \\gamma - (1+\\alpha +\\beta )x, \\\\\n P_2(x) &= x(1-x)\n\\end{align}", "3969421d6364be4ee3629ca58298c0e3": "B=-\\Omega", "39695103ab9eebee1e504c5ffaabc0d1": "4.57540\\times 10^{165}", "39695d5251a9457f30f8920c97c56315": "\\pi(x) = \\operatorname{li}(x) - \\frac{\\operatorname{li}(\\sqrt{x})}{2} - \\sum_\\rho \\operatorname{li}(x^\\rho) + \\text{smaller terms} ", "396985064c1b06b227d16761b39b6cdb": "\\sum_{i=1}^n (X_i-\\overline{X})(X_i-\\overline{X})^\\mathrm{T} \\sim W_p(\\Sigma,n-1).", "3969c2defec53208b8a7b0bca7071d43": " 8\\varepsilon a ", "3969ec9332f6c7a2e666b66a881b9d31": "\\Phi(x\\mid\\mu,\\kappa)=\\int f(t\\mid\\mu,\\kappa)\\,dt =\\frac{1}{2\\pi}\\left(x + \\frac{2}{I_0(\\kappa)} \\sum_{j=1}^\\infty I_j(\\kappa) \\frac{\\sin[j(x-\\mu)]}{j}\\right). ", "396a03a3d9bfb3bd7d05baa94d0be459": "\\mathrm{4\\ CmO_2\\ \\xrightarrow {\\Delta T} \\ 2\\ Cm_2O_3\\ +\\ O_2}", "396a0f5a1d974f3157bda621483d9df7": "\\epsilon/\\epsilon_0 = 1+\\chi_\\text{e}", "396a14ce118d50e3c039578de40f10d0": "u_2\\,", "396a26fed2a8752bde33449b5f644dd4": " k = - \\tfrac{F_r}{L-L_{o}}", "396a8c0267f4feffdaadf71c2911d8bc": "10^5", "396b67f535e1d6ada5fb70db279bdff8": " \\text{Area} = 2 \\times \\left(\\frac{8}{9}\\right)^2 \\times (\\text{diameter})^2 = 2 \\times \\frac{256}{81} (\\text{radius})^2", "396b7f3c937e0dbf49f0a7d9545af100": "x_{44}=-x_{41}\\,", "396bdc10c3b51971fee35145b4d751ff": " L(n,1) = n!", "396beab3e8c9ee3b5f01f42e59be7569": "\\mathbf{A} = (q_4^2 - \\check{\\mathbf{q}}^\\mathrm{T}\\check{\\mathbf{q}})\\mathbf{I}_3 + 2\\check{\\mathbf{q}}\\check{\\mathbf{q}}^\\mathrm{T} + 2q_4\\mathbf{\\mathcal{Q}}", "396bf1f360115b03d606ab03df10e194": "(\\partial \\pi)^2 + m_{\\pi}^2 \\pi^2", "396bff37f28fe4d8de25182a0b01c18f": "\\frac{\\partial f}{\\partial t} = \\theta \\frac{\\partial}{\\partial x} [(x - \\mu) f] + \\frac{\\sigma^2}{2} \\frac{\\partial^2 f}{\\partial x^2}", "396c0ceea04af81d187ef7948d65eb59": "\\mathbf{a} \\cdot (\\mathbf{b} - \\mathbf{c}) = 0,", "396ccd13d058d227217def28b6c24fe2": " \\lim_{x \\to c} f(x) = \\infty,\\ \\lim_{x \\to c} g(x) = 0 \\! ", "396d0e63f29f76ecdc50ef52ebab4073": "c(0)", "396d3119e52ad32e6cdda72f5c61d72c": "\\xi^b", "396d4565c8db3ba2e3276d6419e9f992": "n^2-1", "396d4ac5ce7c881b5b07a9bd5c780ebf": "(W_0, W_1, \\cdots)", "396dc0982760244ce7d6107b84670bea": "a_{11} x_1 p_1", "396e6779071bb0dab2643ad0f1c0638e": "|\\Phi_n(q)| = \\prod |q - \\zeta|", "396e966ccda5bf1d351758b8eb855c38": "\\frac{F_B}{F_A} = \\frac{v_A}{v_B}. ", "396e9f2b8188e83f208774a752509ad0": "R_{ik\\ell m}=\\frac{1}{2}\\left(\n\\frac{\\partial^2g_{im}}{\\partial x^k \\partial x^\\ell} \n+ \\frac{\\partial^2g_{k\\ell}}{\\partial x^i \\partial x^m}\n- \\frac{\\partial^2g_{i\\ell}}{\\partial x^k \\partial x^m}\n- \\frac{\\partial^2g_{km}}{\\partial x^i \\partial x^\\ell} \\right)\n", "396ee1263b6ec52791de6eadae4f6b63": "s_\\lambda=\\sum_{i=1}^m x_i", "396ee1b96fa1d5712a05f68ca18a0a58": " \\mathbb{P}(f): \\mathbb{P}(V)\\to \\mathbb{P}(W).", "396ef22611eeb0735236fcb3f5ec416c": "\\sigma_t^2\\sigma_\\omega^2 \\ge 1/4", "396f3393ee122b09d12b2d1fd0046846": " +\\frac{1}{q}\\oint_{C}\\mathrm {\\mathbf { effective \\ thermal \\ forces\\ \\cdot}}\\ d \\boldsymbol{ \\ell } \\ ,", "396f65c9db92e98a80ee0a03ccd53423": "f(x;\\alpha,\\beta,c,\\mu)=\\frac{1}{\\pi}\\Re\\left[ \\int_0^\\infty e^{it(x-\\mu)}\\sum_{n=0}^\\infty\\frac{(-qt^\\alpha)^n}{n!}\\,dt\\right]", "396f97fb6de223e725fed88ee5451e94": "(L=0, \\dot\\theta=0, \\dot\\phi=0)", "396f9b2c272c438a55e0a0f06b610eab": "= 0 + 21 + 42 + 40 + 20 + 27 + 10", "396faf5d717eb15e4f458c63e4e68f0c": "(x \\,\\bmod\\, y) \\,\\bmod\\, y = x \\,\\bmod\\, y.\\;", "396fb8dee795819625ddabc89a6478bc": "z = uA + (1-u)B", "396fdd45f0c0dbaf4d34248bed746cce": "\n \\begin{bmatrix}\n q_1 & q_2\n \\end{bmatrix}\n = \\begin{bmatrix}\n 0.833 & 0.167\n \\end{bmatrix}\n", "396fe69762dd1e728e8f7ec7325f83c7": "g\\colon \\mathbb{Z} \\to \\mathbb{Z}", "396fee0fd7e2ed0d016830a13d4bd835": " Z_{01} ", "396ffc5c603da04a88c3ff51e8ff2d72": "g_N \\ge 0\\,, \\quad p_N \\le 0\\,, \\quad p_N\\,g_N = 0\\,.", "39703948cd6a5159393c0098a53327ce": " b \\,", "397086f257e0867dd5b94a5a8dfe486d": "\\widehat{QO2P2}", "3970dd40dfb8452a10ce6641167b2d4f": "\\operatorname{LMI}(y):=A_0+y_1A_1+y_2A_2+\\cdots+y_m A_m\\geq0\\,", "397100597b579279db93b1c671663efb": "s_{int} = -\\frac{\\mathbf{v}^{\\perp} \\cdot (p_1 - q_1)} {\\mathbf{v}^{\\perp} \\cdot \\mathbf{u}}.", "397148155f21411fe79314b697be462e": "\\Delta T = (T_s - T_o)", "3971851d62863a261dff40f4e90ea498": "N^i", "3971908630b78d7c8971bf040af38a05": "\\{x_1,\\ldots,x_N\\}", "3971aff237c8206e4c5522cb38e2f819": "D(0,u_k)", "3971f90156c84902205579b5fea17aec": "\\pi\\!", "397224ae3afaf47952130d09e687677e": " \\mathbf{r} = q\\mathbf{e}_t - p\\mathbf{e}_n ", "39725ce32e1a39b0c04363e2a6eee14a": "\\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}\\left(x_{i}-x_{j}\\right) = \\frac{p'_{n}(x_{i})}{a_{n}}", "39725f4cdfb4f6089b21881ca9dc097a": "H\\psi = i\\hbar\\,\\partial \\psi/\\partial t", "3972f11488a90964ea72e83ef0a852d9": "\\phi(x,\\lambda,t)", "3973023bd9e6a1d9b0a88d8a3168ada7": "\\begin{align}\nD_{A}=1-\\sum \\limits_{u} \\sqrt{X_uY_u}\n\\end{align}\n", "39731c9a873b674ba5283e4cd0bbd44e": "\\mathrm{tr}\\ (F_i Z) +c_i =0,\\quad i=1,\\dots,n", "39734cddf71060ac3e472ab1368ce6cb": "e_C = i_C r_C + { {d \\varphi_C} \\over {dt}}", "39737b238c7a10765c9ce4b14d6dd793": "W(s) = \\min_{i \\in N} u_i(s),", "39737b29a2a510a24bbf44ccc0df0a8d": " \\chi^2 = \\frac{ \\left[ [n_{PQ} - n_{QQ}]_{PQ \\sim QQ} + 2\\times[n_{PP} - n_{QQ}]_{PQ \\sim PQ} + [n_{PP} - n_{PQ}]_{PP \\sim PQ} \\right]^2}{[n_{PQ} + n_{QQ}]_{PQ \\sim QQ} + 4\\times[n_{PP} + n_{QQ}]_{PQ \\sim PQ} + [n_{PQ} + n_{PP}]_{PP \\sim PQ}}\n", "39738e3955ab9d9c026a99eaddc955ce": "h^0\\rfloor \\mathfrak F=0", "39741cef2ba713dd3fb94d6a3cf1643c": "t/km^2", "397425e8708d880d0b3135a7f6d57602": " F_{e} \\phi_{P}-F_{w} \\phi_{W}\\,= D_{e}(\\phi_{E}-\\phi_{P})-D_{w}(\\phi_{P}-\\phi_{W})", "39742d7a029fa876305956855ad6b5b7": "E \\left(\\left(\\sum_{j=1}^K\\lambda_j(\\widehat\\beta_j-\\beta_j)\\right)^2\\right);", "39742f5b9aeed30f1a04773f10e56068": "e^{ax}", "39744d1918f5377238d0210bfcc64390": "h_B(n)=\\max\\{ \\langle n,x\\rangle |x \\in B \\}", "397479f52f655520d70001a8a1701704": "W_{\\gamma}(a,b)", "39749358278d738e2cf0c2b924b00cd0": "\\ F_{VW}(r) = -\\frac{d}{dr}U(r)", "3974a3be775ca45e914cdf56e291865c": "s,t,u", "3974c6aa28cc49a902f70d4fb92fd3be": "\\aleph_{0} < I(T,\\aleph_{0}) < 2^{\\aleph_{0}}", "397558d7103ddb4a36713636edfa9f9d": "\\varepsilon_{23}\\,\\!", "3975904d87e0e7788c60759de9622b9a": "q^k = n", "3975ba73357834165f5004a1b0ce9166": "(n-1) n!", "3975c6bf2b10fdf6ce73d29b8e3059d3": "P = \\vec{F} \\cdot \\vec{v} ", "3975e863d20970a0f01be27a95ea6cae": "d_k(a_i,a_j)", "3976280a9fbfc207bede4e4e70a168cf": " \\acute{\\mathbf{z}} ", "397710d12e7821fef5695e76f7b8eb42": "E_F = \\frac{\\hbar^2}{2m_e} \\left( \\frac{3 \\pi^2 (10^{36})}{1 \\ \\mathrm{m}^3} \\right)^{2/3} \\approx 3 \\times 10^5 \\ \\mathrm{eV} = 0.3 \\ \\mathrm{MeV}", "397715277168154354c7a42f95890afe": "\\begin{align}\nh_1&=h_2=\\sqrt{u^2+v^2} \\\\\nh_3&=1\n\\end{align}", "3977406a9c38913879d488d5e68c6520": "x^2+y^2-1", "397747c054cf5627a9bcb35318cd799a": " \\exists x \\exists y \\,( \\mathrm{inf}(x) \\land \\mathrm{inf}(y) \\land |\\mathcal{P}(x)| \\neq |\\mathcal{P}(y)| \\land \\forall z ( \\mathrm{inf}(z) \\rightarrow ( |z|=|x| \\lor |z|=|y| ) ) ) ", "3977a81444c9d8799fad00e62a3b3f78": "[G,G]", "3977ba04138b77ee9bbebcd6166ecad3": "e_2 = e_3 = 0", "3977ec5c38af8178d4f5f5656efc4c5c": "|\\psi (0)\\rangle ", "3978a0434eab69edab2a3d3a48de6da5": "A,B\\subset X", "39791804942ea5c0d3d96bc8c0eba43b": "H_{\\mathrm{dR}}^{k}(S^n \\times I^m) \\simeq \\begin{cases} \\mathbf{R} & \\mbox{if } k = 0,n, \\\\ 0 & \\mbox{if } k \\ne 0,n. \\end{cases}", "39795b7123fd06a04bf5e70a6c35e92c": "|\\Lambda|^1_M", "39798c0e50f07dccaa34d65047bf644f": " U_i = C_i + \\alpha P_i + \\beta D_i + \\varepsilon_i\\, ", "39799bfebdc229234cbfd41617efeb4b": "\nV=\\frac{\\partial}{\\partial\\sigma^2}\\log\\left[\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-(X-\\mu)^2/{2\\sigma^2}}\\right]\n=\\frac{(X-\\mu)^2}{2(\\sigma^2)^2}-\\frac{1}{2\\sigma^2}\n", "3979a4eaadd49518a6a3d00b8cbcafb0": " g_{\\mu\\nu} = g_{ab}e^{(a)}_\\mu e^{(b)}_\\nu ", "3979d510dfc57a24595ecb4338cfb863": "= \\int d \\tau \\Big[ {dx \\over d \\tau} p - {dt \\over d \\tau} C' (x,p) \\Big]", "397a496ce7c1a2854d3eeaaf3d599d57": "F^m \\times G^n \\leftrightarrow F^m \\exist^m G^n.", "397ab922ad972a703cdf91adff3e1213": "2 \\frac{dy}{dx}\\frac{d^2y}{dx^2} = \\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)^2 \\,\\!", "397acbd3cefdd83c9448c2b603955792": "n,m=1,\\ldots,l_\\mu", "397b1d4827390d75d263bbe4aa9cf2c2": "g(x) \\geq x", "397b1fbc497aec01ef4e1d766c7cccda": "\\scriptstyle f_{uv} \\leq c_{uv}", "397b26ee6cd1092409fbefb2a6791249": " \\boldsymbol{\\mathsf{a}}\\cdot\\boldsymbol{\\mathsf{b}} = \\eta ( \\boldsymbol{\\mathsf{a}}' , \\boldsymbol{\\mathsf{b}}' ) = \\eta \\left ( \\Lambda \\boldsymbol{\\mathsf{a}} , \\Lambda \\boldsymbol{\\mathsf{b}} \\right ) = \\eta ( \\boldsymbol{\\mathsf{a}} , \\boldsymbol{\\mathsf{b}} )", "397b630876be12192308c7b34a544223": "\\beta (\\omega) \\approx \\beta_0 + (\\omega - \\omega_0) \\beta_1 + \\frac{(\\omega - \\omega_0)^2}{2} \\beta_2 + \\beta_{nl}", "397b6fadee009510a972b52f3f352eac": "E = \\gamma m c^2 \\,", "397b92dd2ee15bd1e7778f725b0fe48f": " c_1 = 0, c_2 = 1 - \\gamma", "397bb657f19cf2ffdfe790fea4101949": "\\left[\\begin{array}{rrr|r}\n1 & 3 & -2 & 5 \\\\\n3 & 5 & 6 & 7 \\\\\n2 & 4 & 3 & 8\n\\end{array}\\right]\\text{.}\n", "397bbc069aba22510b71265f5ed1ac84": "T\\cong 5\\cdot 10^{12}", "397be204941f8cdb2e95e93d63b37d8a": "\\! M = pq \\ldots r", "397beea9e55e8a3493203794a262f22b": "\\delta = k - L = \\left(2a\\right) \\sinh^{-1}\\left(\\frac{L}{2a}\\right) - L ", "397c005c0f5247f1086e330fd5bc66b4": "\nf(n_i)=\\ln(W)+\\alpha(N-\\sum n_i)+\\beta(E-\\sum n_i \\epsilon_i).\n", "397c1985878e0e342917c454be89e960": "E(\\omega) = |E(\\omega)|e^{i\\phi(\\omega)}", "397d1eac4d6a7c20268fb3dc455c29fb": "1+2+3+\\cdots+100\\ = \\sum_{n=1}^{100} n", "397d3a357d84fcb9fc78a4713eeb4c76": "R = D_3 + D_4 \\bmod 2\\,", "397df26292b2408f4fd3d4edb5335bf6": "\\left\\langle\\mathbf{P},\\mathbf{P}\\right\\rangle = |\\mathbf{P}|^2 = (m_0 c)^2\\,,", "397df4ccce4d5c96a8fb4ddcf3b9bf45": "\\hat{H}_{II} = -\\sum_{\\alpha\\neq\\alpha^\\prime}\\boldsymbol{\\mu}_\\alpha\\cdot \\mathbf{B}_{\\alpha^\\prime}", "397e1b60d4eb43a037a5af03dc2c8e0d": " C_{0}^{j+1}", "397e35d4bf9cc608fedc9a5d6cf39556": "\\tilde{F \\times G}(x) = (Z_F \\times Z_G)(x, x^2, x^3, \\dots).", "397e6d94ce03263ca93a6c5b0d27a943": "S \\rightarrow B: \\{T_S, A, K_{AB}\\}_{K_{BS}}", "397e85165da25c74a7b3319d236b1fc3": " - \\Delta u = -\\sum_{i=1}^d \\partial_i^2u\\, ", "397ec068c02f11cc3b5d967bcdf0b130": "\\lim_{(x,y) \\to (0, 0)} -\\left | y \\right \\vert = 0", "397ed61e4fa9d5569ea2c6e1adaf3ea1": "O\\left\\{\\left[log\\left(n\\right)\\right]^2\\right\\}", "397f55f0705bf16ce88ca83244db36b1": "\\frac{d^ny}{dx^n},\\quad\\frac{d^n\\bigl(f(x)\\bigr)}{dx^n},\\text{ or }\\frac{d^n}{dx^n}\\bigl(f(x)\\bigr)", "397f852b1ee27ab02eb4049988f89d4f": "f(x-1, y)", "397fb498e99b253ebe7612ba23e16f05": "\n\\psi(\\phi)=\\tan^{-1}\\left[(1-e^2)\\tan\\phi\\right]\\;\\!.\n", "397fbc81ce5b735b2c476028d5ec0c30": "\\frac{m}{s} = \\frac{n}{t}", "39809b2c484a31ffb76144cbd8c9ad7a": "\\phi^{+}", "3980b94a040af9a9870a9045764eab4f": "y(t)=ae^{be^{ct}}", "3980fc1c7525a140eca70646573c22c2": "\n\\begin{align}\n& {} \\quad \\cfrac{1}{1+\\cfrac{e^{-2\\pi\\sqrt{5}}}{1 + \\cfrac{e^{-4\\pi\\sqrt{5}}}{1+\\dots}}} \\\\ \\\\\n& = \\left( \\frac{\\sqrt{5}}{1+[5^{3/4} (\\varphi-1)^{5/2}-1]^{1/5}} - {\\varphi}\\right) \\, e^{2\\pi/\\sqrt{5}} = 0.99999920\\dots\n\\end{align}\n", "39810b6953c561b51e847d4e7321b4d4": "E = \\{e_1, e_2, ~ \\ldots ~ e_m\\}", "3981367f9a8f4d32dfd4d0190cc30cd6": "(i\\omega-\\xi_1)^{-1}(i\\omega-\\xi_2)^{-1}", "39819c8b741bfa32ad8d88f9a2c505ad": "(c_{i,j}\\mid (i,j)\\in I\\times I)", "3981a77c441bfcefc7b1933b56ef9fb3": "\\{ (\\mathbf{q}(t),\\mathbf{p}(t))\\in\\mathbb{R}^{2N}\\,:\\,t\\ge0, t\\in\\mathbb{R} \\} \\subseteq \\mathcal{P}\\,,", "39820adbf066d28ac6c97db306f9499d": "\\mathbf{r}(t)=f(t)\\mathbf{i}+g(t)\\mathbf{j}", "398215e3505a8def01a15495117d4e1d": "\\Xi(-z) =\\Xi(z).", "39824201e96dd5c5d73b72492d418665": "\\Delta \\tau \\propto {k \\over {d^x}}", "398243008eb9fe7f0401080685949624": "\\rm \\ Cl_2O + N_2O_5 \\rightarrow 2ClONO_2", "39827505837912384063d653633b496c": "\\mathbf{x} \\sim t_{2\\alpha}(\\boldsymbol{\\mu}, \\frac{\\beta}{\\alpha} \\mathbf{V}) \\!", "39833d16c14fb7c0a6d5d91255466867": "s'_1", "398348f569c1186d66a109368039b8ad": "W = \\int_{V_1}^{V_2} p dV = p_1 V_1^n \\int_{V_1}^{V_2} V^{-n} dV", "398358a68a1053efec190106800e1f74": "\\oint_{\\partial V} d\\vec S \\; f(\\vec r) = \\int_V d\\vec V \\; \\nabla f(\\vec r)", "39835a618f8a5faa17222e926eb6e5f1": " P\\left[ (X^n,Y^n) \\in A_{\\varepsilon}^n(X,Y) \\right] \\geqslant 1 - \\epsilon ", "3983666544b3720bb2f7c9464a7f54ed": "\n \\begin{align}\n \\sigma_{11} & = \\cfrac{4C_1}{3J^{5/3}}\\left(\\lambda^2 - \\tfrac{J}{\\lambda}\\right) + 2D_1(J-1) \\\\\n \\sigma_{22} & = \\sigma_{33} = \\cfrac{2C_1}{3J^{5/3}}\\left(\\tfrac{J}{\\lambda} - \\lambda^2\\right) + 2D_1(J-1) \n \\end{align} \n ", "3983744b0393d3aa2d522ba29019d22e": "N=\\sum_{i=1}^{c} K_i", "3983b31102fb3e7f69f7f15f927989ab": "\\nu_{11}\\sigma^2", "3984a5985f7c15cca11cbb81e0ed6261": "\\bold j \\cdot \\mathrm{d}\\bold{S} = 0", "3984b396362315f1bed329979c1a3ee5": "a=\\sqrt[3]{\\mu\\left(\\frac{P}{2\\pi}\\right)^2}", "39850d57e508bdeb89ae5c8f4f655be1": "{\\partial v \\over \\partial x} = -e^x \\cos y", "39852b5f0f73c742f5fddaffef016eeb": "\\delta_1=2n-\\frac{2}{3}n^2-2n^3,\\,\\,\\,\\delta_2=\\frac{7}{3}n^2-\\frac{8}{5}n^3,\\,\\,\\,\\delta_3=\\frac{56}{15}n^3.", "39855adb22f2696d22babc41665cd1ff": "\\Delta G^\\ominus = \\sum_k m_k\\mu_k^\\ominus-\\sum_j n_j\\mu_j^\\ominus", "39858c982fcbd48831980a58d360d6ff": "\\textit{green}", "398598b7505886a6a8cca8f502522798": "\\mathfrak{t}_{\\mathbf{C}}", "39863324bedff976d61768476e8dc5d4": "u_i = v_{i-1}", "39869ed4c561a1c4872b1576b4844fda": "\nV_C \\approx \\frac{1}{RC}\\int_{0}^{t}V_{in}dt\n", "3986aadac6c0541fcfd0d1298fd0796f": "\\scriptstyle f\\colon M \\to \\mathbb{C},", "398700aa8b49344d8233034fcaf40be7": "\\eta: id \\Rightarrow T", "398707fc801d3681b4368b85c0e8ff31": " \\bold d = \\begin{bmatrix} \\bold F \\\\ \\bold C \\\\ \\bold E \n \\end{bmatrix}. ", "39870eee28285047ded6a080bbceefb5": "15 M", "39872420b111e4ea694fddbb38f0b991": "\na = \\frac{L}{m \\, c}\n", "398742813539d55d928bc7cc382007f7": "\\sum_{\\lambda\\in\\mathcal{P}_n} (t_\\lambda)^2= n!", "39878328bfdfbff7b9cb179ee686938e": "\\scriptstyle (\\cdot)^*", "3987deba22f0a388592d0c30bd921bdb": "\n z_{ij}\\ \\sim\\ \\mathcal{N}(\\beta_j,\\, \\sigma_i^2).\n ", "398800e3adb0fbc00c6d45b0eb7d4cda": "x \\succeq z ~\\forall z \\in B'", "398873c1abcb1ce73e8408df3a1a9661": "[(L,u),(M,v)] = ([L,M]+L_{u,v}, L(v) - M(u)).", "3988844060b6d449834538e54a5c34d5": "\\Delta t=1", "39891fbed954b44772dc5603418103a5": "(s + x_n) ", "3989cb09847832cf4d31093a2335968f": "C_P=C_L \\sqrt{1+\\lambda^2} \\left( \\lambda - \\gamma \\lambda^2 \\right)", "3989de848dc36477def3247d1df20ea8": "(S,T,E)", "398a205965b32268e59d7df4777596af": "L^{B/A}", "398a3a4773815191b9447c33323b0c4d": "\\{ f_{ij} \\}_{j \\in \\mathcal{J}_i}", "398a40d4ebd23ca82c5d080cb58d8b38": "f:X\\to Y\\,", "398a553e75f179ba43eb69f29a981980": "[M-nH]^{n-} + A^+ \\to \\bigg[ [M-nH]^{(n+1)-} \\bigg]^* + A \\to fragments", "398a8b32942d3fdfe7dc3e8cfa99d7a2": "z\\log \\Gamma(z)-\\log G(1+z)", "398b8c3e58e961cdea0fe626771f3243": "\\Psi ", "398ba111113d1df47d8fc3d20bb729fa": "x = 0.9996a\\lambda \\qquad\\qquad y = 0.9996a\\ln \\left(\\tan \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right) \\right).", "398bc008b3b26b8912bedc0993534fb2": "\n{dJ \\over dE} = T\n", "398bca62554d9c4bc2094270d655b183": "det \\; q^{(2)}", "398bfae73029566f9b99051dd6ed0dc3": "(A, B, C)", "398c1db9caece18b8da8d3448f8c9e5b": "(\\neg A\\to\\neg B)\\to((\\neg A\\to B)\\to A)", "398c26ceddc9a484efc2c90d4d0eb9e4": " A^n \\rightarrow A \\otimes_B A", "398c439a05ebd0e959437e6979562cdf": "E = \\int_0^T P(t)dt = A^2 T", "398c6d7e3fa8ab1d33995e2f02ac0dac": "n_0 = \\frac{p_0N_{\\rm A}}{RT_0}", "398c9178d1533ca38006f1a5d4601285": "\\omega_{pe}", "398c95d4c7490503110cfc9258018a01": "g.V_h\\subset V_{ghg^{-1}}", "398cd3dc771b7820869f2b9eaeab29fc": "\\nabla^*:\\Gamma(T^*M\\otimes E)\\rightarrow \\Gamma(E).", "398cdd832c4505dc3a3de1c7ad089602": " M_i", "398cddf51428e3fe81d7704d940bae54": "\\bold{A}_{[p]} = (A_{i_kj_\\ell})", "398d223c6efec93bc2abf0cb92b19aba": "f^* E = \\lambda_1 \\oplus \\cdots \\oplus \\lambda_n", "398d2f1533228c9fc0ae7c8f47516d42": "p=\\frac{nE}{c} .", "398d37818a0e8806f309530827a24dfb": " \\Delta G^{\\ddagger}", "398d9aa1cdac24d7c310a1ebb7a4a3c6": "\\Pr[P(r_1,r_2,\\ldots,r_n)=0]\\leq\\frac{d}{|S|}. \\, ", "398ddf387b9012f616a3123a22355a1f": "\n a_{mn} = \\frac{4}{ab}\n \\int_0^b \\int_0^a q(x,y)\\sin\\frac{m\\pi x}{a}\\sin\\frac{n\\pi y}{b}\\,\\text{d}x\\text{d}y \\,.\n", "398e76a39547df1dc3c97a4dd2eb2153": " \\{1\\} \\rightarrow G^o \\rightarrow G \\rightarrow \\Gamma \\rightarrow \\{1\\} ", "398e8512d8498cb97b16cad8e3cc9551": "\\neg(A \\land B) \\to \\neg A \\lor \\neg B", "398efc15bd7a277ea3f1b45d340cebd4": " s^2=\\frac{\\tau \\; r^2}{t \\; d}. ", "398efc3a4d0b12dc505be4de1199ad71": "\\partial \\!\\,", "398f14f88c05e4a4c6555bbf729a5b83": "{}_Z^A\\!X\\to {}_{Z+1}^A\\!Y+ \\bar{\\nu} + \\beta^-", "398f1fdead8ea0b48b79f7c956142162": "\\mu(x) = \\epsilon", "398f517310feeef2f00078e7593a63f8": "\\mathbf{a}_{i}^{\\dagger}", "398f66ff7b8ba7cb84923da2fe3c430f": " = \\sum^b_{r=a} \\int f(r,x)\\, dx", "398f848da020c8aa9d227d5683751862": "\n\\mathbf{a \\times b} = \\mathbf{c}\\Leftrightarrow\\ c^m = \\eta^{mi} \\varepsilon_{ijk} a^j b^k\n", "398fa71208316ea1785bc217b5b6ee3e": "\\beta, n", "398fa777962e0fff536c3f663fc2ea15": "\\frac{c^2}{b}", "398fe251caa777c90fda0bbe8a1e4c8f": "\\Gamma_n,", "39902b3ebd45b890a415f277108b3f6d": "\nQ = \\frac{1}{2 \\zeta} = { \\omega_0 \\over 2 \\alpha } = { \\tau \\omega_0 \\over 2 },\n", "3990a6b2b4a3d44cd0cab3e5c913bad3": "16 | 560", "3990df8c67c8391c1649c7d2c2b21857": "q_n = \\sum_{i = 0}^n 2^{-a_i - 1}.", "3990f49fcfaca61173edea491851e163": "\\pi_1\\otimes\\pi_2(X) = \\pi_1(X) \\otimes \\mathrm{Id}_V + \\mathrm{Id}_U \\otimes \\pi_2(X), \\quad X \\in \\mathbf{g}, \\qquad \\pi_1\\otimes\\pi_2(X, Y) = \\pi_1(X) \\otimes \\mathrm{Id}_V + \\mathrm{Id}_U \\otimes \\pi_2(Y), \\quad X,Y \\in \\mathbf{g},", "39913b7becf0cb139971921bf4c80e99": "s = \\int_a^b \\sqrt{r^2+\\left(\\frac{dr}{d\\theta}\\right)^2} \\, d\\theta.", "399191225002bf95fd287cb121eaf874": "\\begin{align}\n & z_t = \\left( 1\\ z_{t1}'\\ z_{t2}'\\ z_{t3}'\\ z_{t4}'\\ z_{t5}'\\ z_{t6}'\\ z_{t7}' \\right)', \\quad \\text{where} \\\\\n & z_{t1} = x_t \\ast x_t \\\\\n & z_{t2} = x_t y_t \\\\\n & z_{t3} = y_t^2 \\\\\n & z_{t4} = x_t \\ast x_t \\ast x_t - 3\\big(\\operatorname{E}[x_tx_t']\\ast I_k\\big)x_t \\\\\n & z_{t5} = x_t \\ast x_t y_t - 2\\big(\\operatorname{E}[y_tx_t']\\ast I_k\\big)x_t - y_t\\big(\\operatorname{E}[x_tx_t']\\ast I_k\\big)\\iota_k \\\\\n & z_{t6} = x_t y_t^2 - \\operatorname{E}[y_t^2]x_t - 2y_t\\operatorname{E}[x_ty_t] \\\\\n & z_{t7} = y_t^3 - 3y_t\\operatorname{E}[y_t^2]\n \\end{align}", "39919e30a04a9cf0c4665534fb536808": "(I - Q)\\,\\!", "3991a190170e4b38612bbf628ec8a3f4": "D_{k}", "3991adb96f658b3ad89f22cfb227451d": " x_3 = r\\, \\cos\\theta. \\,", "3991ae1cb42347bf2122361bff62ee2b": "m_f v_f + m_p v_p = 0\\,", "3992363c3f34b3fc1449ea4d407399e9": "P_1 - P_2 = \\frac{1}{2}\\cdot\\rho\\cdot \\bigg(\\frac{Q}{A_2}\\bigg)^2 - \\frac{1}{2}\\cdot\\rho\\cdot\\bigg(\\frac{Q}{A_1}\\bigg)^2 ", "399255f8ab4a956cd4c45d986913a431": "E(\\tilde{m}) = 1/R_f", "39925667c31dd5e46118b9ff027bcee6": "x_j(s)", "3992972c0b794d9558f7ff0970ac7236": "Q_c = \\frac{(kad)^3b\\eta}{2\\pi^2R_s} \\cdot \\frac{1}{l^2a^3\\left(2b + d\\right) + \\left(2b + a\\right)d^3}\\,", "3992d5f336f21774af488fda794b5122": "(\\pm 1,\\pm 1,\\pm 1,\\pm 1,\\pm 1,\\pm 1,\\pm 1,\\pm 1)\\,", "39930378728544b537fa3069a38d57ee": "\\scriptstyle\\varnothing", "399354e9e91b6ffa0985c48c4d89baa4": " {{\\bar{v_{n}^2}} \\over {B}} = 4 k_B R T", "39936204f7869860d9cd2a8969ed86ed": "\\vartriangle, \\triangledown\\!", "3993971c5421a3a662ddc6377497e546": " y_i = m_i + m_i / s^2 - 1 ", "3993c145b6286b08cd8a27a2067b8a89": " dt = \\frac{r_0 \\, d\\phi}{1 \\pm \\omega \\, r_0} ", "3994741c6a4049ec5d54f792a6bcedb5": "(M, \\omega),\\,", "399498b46e94a5aab59f215793d15df2": "dw = -w^{-1}(xdx + ydy +zdz) \\,", "399547ec5af636549b624e27470d0d83": "\n\\begin{align}\n\\lim_{k \\rarr \\infty} g'_{k^{-1}\\cdot (k^2\\times \\{+1,-1\\})}(t) & = \\lim_{k \\rarr \\infty} \\frac{d(k^2\\cdot \\log(e^{+t\\cdot k^{-1}}+e^{-t\\cdot k^{-1}}))}{dt} \\\\\n& = \\lim_{k \\rarr \\infty} \\frac{d(k^2\\cdot \\log(2)+2^{-1}\\cdot t^2+\\cdots)}{dt}=t.\n\\end{align}\n", "39957691fa8b7f0518dd36ac0f4d11ee": " U_d \\equiv 0 \\pmod {n} ", "399615d6eafdb9300447bb2c8f88299a": "w\\,R\\,w", "399658972db066ff42bf68507192d805": "H^{2k}(M)\\ ", "39967b529ef8eba1c7ebd9ff32c5e98d": "\\tbinom{[n]}m=\\{[n]\\} ", "3996bc890dbef7e5cf91926cd37dde5b": "T_i \\in L_i", "3996bdb1d887557c16331bf938a66667": "\nP_j = \\left( \\frac{1}{2}\\right) ^{2j}\\frac{\\left( 2j\\right)!}{\\left( j! \\right)^2}.\n", "3998122b9a6840d61cc297beab15649c": "(a,{\\pm {1 \\over \\sqrt{3}} a})", "39984cbf8b161b88b578db582be1257b": "\\mathcal M=\\{A_1,\\dots, A_m\\} \\subset \\mathbb R^{n \\times n},", "39988dabb34768f7df9754486b657eb3": " \\{C_i\\}_{i \\in I} ", "3998c458719b218ffe671786b5f786fa": "\\nabla T \\ll T", "3999310ac1ab6f82e8b4f267b67882b0": "\\mathbf{F}= q(\\mathbf{E} + \\mathbf{\\dot{r}}\\times\\mathbf{B})", "3999339c396f7602410334f6342f8bcb": "\\xi_C = \\frac{W_C}{W_0} = \\frac{2\\pi}{R_H}\\sqrt{\\frac{L}{C}} = 2\\pi \\cdot \\frac{\\rho_q}{R_H}. \\ ", "399942ffc94bb0d2010d9e21973bbd60": "\\forall i \\in \\{1,\\ldots,N\\} \\exists \\lambda \\in P(\\mathfrak{g}) : V_i \\simeq M_\\lambda", "399955ac64a0217485c99cc927102698": "F_H=\\frac {dW}{dx}", "3999a81804c7d2fc635babd4f351cf75": "p_1, \\ldots, p_{|S|}", "3999d2c3ae693d9346f9ac8c50cb7735": "T_{h-1}", "399a3ea5b6aad708a8422721359a7e30": "D_n:", "399a6bf90ef3ae1068484e07a217a0ae": "\\frac{D}{L}\\ge \\frac{35}{Gr_{L}^{\\frac{1}{4}}}", "399a6e22d37b5efa4c2c8ecf5c127a42": "\\ \\mu_{\\delta}=\\delta_1-\\delta \\ .", "399a75c8a4144e01057712d0676ce124": "T^{\\mu}_{\\mu} = 0", "399aea381af7f59e3660a1b926a44cd8": "n=1,2\\,", "399b28ec882b235a76b510e8d7faccea": " \\ell^2(B) =\\big\\{ x : B \\xrightarrow{x} \\mathbb{C} \\mid \\sum_{b \\in B} \\left|x (b)\\right|^2 < \\infty \\big\\}.", "399b378fe4033085f7662c0f5fc46e5d": "E_2 = E_{a2} + E_{b2}", "399b5c976bbf96e256f29a91cd7f8cc9": "\\chi_2(n)", "399b9af70216e223d9f563bc8013889c": "\\frac{dy}{dt} = (x - y ) / tau", "399b9c287a17324af180b3459f7ec379": "\n\\tilde{a}_{00}=a_{00}-\\partial_x a_{10}-\\partial_y a_{01}+\\partial_x^2 a_{20}+\\partial_x \\partial_x\na_{11}+\\partial_y^2 a_{02}.\n", "399bd538e8c8b989b25edcf5a85067d6": "V (\\cdot) V^*", "399be142bf42bb4f253097bfd9c8e802": "\\pi = g^{ij}\\pi_{ij}", "399c74eae62d6a7e1965406c12f28f1a": "\\mathbf{x}\\in\\mathcal{X}^\\infty", "399c90c8914b55ecf7ce374974ba8163": "\\begin{bmatrix}\n2({q_1}^2+{q_4}^2)-1 &2({q_1}{q_2}-{q_3}{q_4}) &2({q_1}{q_3}+{q_2}{q_4}) \\\\\n2({q_1}{q_2}+{q_3}{q_4}) &2({q_2}^2+{q_4}^2)-1 &2({q_2}{q_3}-{q_1}{q_4}) \\\\\n2({q_1}{q_3}-{q_2}{q_4}) &2({q_2}{q_3}+{q_1}{q_4}) &2({q_3}^2+{q_4}^2)-1 \\\\\n\\end{bmatrix}", "399cab48c0cc82eeac68dd000901b475": "\\frac{\\Vert J \\Vert}{ \\Vert f(x) \\Vert / \\Vert x \\Vert},", "399d30284ac7d31be1aa0f5aca69da16": "\\left\\langle\\Delta H\\right\\rangle", "399d82a9178056f1af28d9abe23b99c6": "b/2 < c < b", "399d98b97192ab7f1e5c490bebe3ad5c": "\\sigma_\\varepsilon^2 .", "399d9a4505aafb895f9bb42e8ff3b303": "v = {d\\phi \\over dt} \\,", "399d9a916cfa9b41194b5138c499d095": "|1/\\sqrt{3}|\\,\\!", "399dd3ca54be205724214ea4960e32c0": " (a_k^2 + b_k^2) / 2", "399ddd16cd750e4ca086344986696bbc": "e\\langle \\psi_{1} |x|\\psi_{0}\\rangle ", "399e39c1a0121b1b518242fef7aac4f4": " \\scriptstyle{Q\\, dx} ", "399e996a47feae38c871cf2422d9d050": "(\\forall x P) Q", "399eb3f6a2b3a42b89e7765034460854": "F_{\\mu \\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu ", "399fa9d915bb4012be21c4c584224c4c": "W(x)<0\\mbox{ }\\forall\\mbox{ }x\\in I", "399fd772e6b57af121a0d0742d7bd348": "A(t)\\,", "399ff58312467ad806d5654536837cbe": " B_X = A_{2,0} / (x^3 - 3xy^2, 3x^2y - y^3) \\cong \\R\\langle 1, x, y, x^2, xy, y^2, xy^2, y^3, y^4 \\rangle . ", "39a007f4506564d5a8e27688a537bf90": " y(t+h) \\approx y(t) + hf\\left(t+\\frac{h}{2},y\\left(t+\\frac{h}{2}\\right)\\right). \\qquad\\qquad (4)", "39a04ed91d3bce2e1687a492a36e95b4": " \\gamma_\\mu \\gamma_\\nu + \\gamma_\\nu \\gamma_\\mu = 2 \\eta_{\\mu \\nu} ", "39a0769aa714464568b911fe2672113d": " y_{n+4} - \\tfrac{48}{25} y_{n+3} + \\tfrac{36}{25} y_{n+2} - \\tfrac{16}{25} y_{n+1} + \\tfrac{3}{25} y_n = \\tfrac{12}{25} h f(t_{n+4}, y_{n+4}); ", "39a0e00d8b75e487d0a1ed31dcfe5405": "V'(a) f(x) = f(y) V(a) ", "39a10657b28325f3dda96b7adf1fe0a6": "\\mathcal{N}(0)", "39a11be391304f5c736716e090035b41": "\n\\frac{\\pi}{2}=\\sum_{k=0}^\\infty\\frac{k!}{(2k+1)!!}=\\sum_{k=0}^{\\infty}\\frac{2^k k!^2}{(2k+1)!} =1+\\frac{1}{3}\\left(1+\\frac{2}{5}\\left(1+\\frac{3}{7}\\left(1+\\cdots\\right)\\right)\\right)\\!", "39a1d6626e90ab7ecf2ab41f108b68b2": " \\frac{dI}{dt}(0) <0 ,", "39a2280060efaae037f5318d4d16801f": "\\begin{align}[]\n \\left\\lbrack\\mathbf{a}\\right\\rbrack &= \\begin{bmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{bmatrix} = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\\\\n \\left\\lbrack\\mathbf{b}\\right\\rbrack &= \\begin{bmatrix} b_{11} & b_{12} \\\\ b_{21} & b_{22} \\end{bmatrix} = \\begin{bmatrix} A' & B' \\\\ C' & D' \\end{bmatrix}\n\\end{align}", "39a249a2f625f99ef89abf278bd6b561": " \na_{20}\\ne 0,\n", "39a27818d3f9d12b7dbb517c9c7c3bd8": "\\delta \\int ds = 0 \\,", "39a2b96acc6095b7f63edea9feea389a": "\n\\begin{bmatrix}\n\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}^T&\n\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}^T\\\\\n\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}^T&\n\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}^T\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{bmatrix} ,\n", "39a32a1a698c541b069373f0b640cd75": "SSB = n\\sum_{i=1}^m (\\overline{Y}_{i\\bullet} - \\overline{Y}_{\\bullet\\bullet})^2 \\,", "39a3783a510115aa4e0e382b21beedf4": " \\hbar \\omega_k", "39a3ffc944706409b3b5fa0d5e08ee25": "(R,+)", "39a427e0b250982dd0fab7c404b4e2c2": "F_{1}", "39a4352737af2c28642dc33ac902bcfc": "\\eta:X\\to U(F(X))\\,\\!", "39a44791637ff7a9478f704d7468481d": "\\scriptstyle \\pi \\approx \\frac{(a_n + b_n)^2}{4 t_n}", "39a44d2aec88bf5adefa4883e511782b": "\\boldsymbol \\beta^{(s)}", "39a4f05de67f867cdf14b30f0a2c644f": "X_0 = (\\mathfrak{X}, \\mathcal{O}_\\mathfrak{X}/I), S_0 = (\\mathfrak{X}, \\mathcal{O}_\\mathfrak{X}/K), I = f^*(K) \\mathcal{O}_\\mathfrak{X}", "39a50a758732a2dacb646c07020a1963": "\\left(\\mathbf{J^TJ+\\lambda D}\\right)\\Delta=\\mathbf{J}^T \\mathbf{r}", "39a52441a99eb0a2dd8b0ec579585020": "X_7", "39a56b879a4262de531811a49d88d5e7": "\\scriptstyle \\scriptstyle\\sin^n(x)", "39a58a4dc342eb4bb68b9b3822b37779": "\\textit{on}", "39a5b6d4bd385beff2c36f29be74b2cb": "\n\\begin{alignat}{1}\nR_x(\\theta) &= \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos \\theta & -\\sin \\theta \\\\[3pt]\n0 & \\sin \\theta & \\cos \\theta \\\\[3pt]\n\\end{bmatrix} \\\\[6pt]\nR_y(\\theta) &= \\begin{bmatrix}\n\\cos \\theta & 0 & \\sin \\theta \\\\[3pt]\n0 & 1 & 0 \\\\[3pt]\n-\\sin \\theta & 0 & \\cos \\theta \\\\\n\\end{bmatrix} \\\\[6pt]\nR_z(\\theta) &= \\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta & 0 \\\\[3pt]\n\\sin \\theta & \\cos \\theta & 0\\\\[3pt]\n0 & 0 & 1\\\\\n\\end{bmatrix}\n\\end{alignat}\n", "39a5d9354b47966c15bbb986db2641c8": "\\delta_1 (X) = \\Bbb{E}_{\\theta}\\{\\delta(X') \\,|\\, T(X')= T(X)\\}, \\,", "39a5fc6bfd72ee4739d928a1320eac53": "\\, a_{\\perp m} = a - a_{\\| m} = (a\\wedge m)m^{-1} .", "39a5fdd81677f5e0b6c2bfcb039b3713": "\\chi^2 ", "39a62f827b36ef419af77cfab4eae0b6": "\n\\begin{cases}\nh_0 = 0 \\\\\nh_{k+1}=h_k + \\dfrac{\\sum_x w(x)F'(x+h_k) \\left [G(x)-F(x+h_k)\\right ]}{\\sum_x w(x)F'(x+h_k)^2}\n\\end{cases}\n", "39a657f878347c077e15209da7fde599": "\n( C_1 \\cap C_2)( [x]) =C_1 ( [x]) \\cap\nC_2 ( [x]) \n", "39a67810317d6cbd6d4be50e4817bacb": "\\frac{1}{1+z}", "39a695c016b9189455f18350bb9ead68": "[p+1-2\\sqrt{p},p+1+2\\sqrt{p}]", "39a6a4ab7043470f56854f5d6ef07eb3": "\\mathrm{Rord}\\ ", "39a6c6b5f6944413fe56d2f38c4ebe19": "\\langle p , x \\rangle", "39a6d39c69a61b6be066a0a02e7a81b8": " A_x \\subseteq \\operatorname{L}(H_x) ", "39a72988a755c9be5c7f040e8517d58a": "T = 1 / (\\mu - \\lambda )", "39a75ac6930e1d019eb8a3b44d38093e": "\\omega^2=(e_1e_2\\cdots e_n)(e_1e_2\\cdots e_n)", "39a77a9c8c150b60f8fe282c11ebfeaa": "\\mbox{DSPACE}[s(n)] \\subseteq \\mbox{NSPACE}[s(n)] \\subseteq \\mbox{DSPACE}[(s(n))^2].", "39a7985f92f88e19accecd309a03d427": " e_i := S_{(1)^j} ", "39a7e0eff23f53ef9177222b7461fc28": "f(x+\\delta)-f(x) = f(x-y) [ f(y+\\delta) - f(y)] \\rightarrow 0", "39a7f2d3595c8d41b9ac7e526b7f9928": "(m, 1, m^2 - N)", "39a82215fed5059ccccc72b994eb6ba1": "a(x) = a_0x^3 + a_1 x^2 + a_2 x + a_3", "39a835d3bed1f6a8a1ab906e58232620": "\\langle x + y,x + y\\rangle =\\langle x,x\\rangle + 2\\langle x,y\\rangle + \\langle y,y\\rangle,", "39a8538fa8343f37609e6a00e7acc9b9": "f(x)-f(x_0)\\ge v\\cdot (x-x_0)", "39a865eb0d3bb01298da6be7425832a6": "a_i^H a_i = 1.", "39a88015eb114b07fc2b7ab594366902": "\\displaystyle \\hat{f}_1(\\xi)\\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{\\mathbf{R}^n} f(x) e^{-2 \\pi i x\\cdot\\xi}\\, dx = \\hat{f}_2(2 \\pi \\xi)=(2 \\pi)^{n/2}\\hat{f}_3(2 \\pi \\xi) ", "39a91f293b6d27b69a8c479509450250": "\\{\\omega\\in\\Omega \\mid u < X(\\omega) \\leq v\\}\\,", "39a98bae7831550d6cb3511e02d2f591": "A \\subseteq X \\setminus \\mathrm{supp} (\\mu) \\implies \\mu (A) = 0.", "39a9b3187989efcd01ffa2ae021c1184": " \\text{Dur} = \\frac{1}{P} \\left( C\\frac{(1+ai)(1+i)^m-(1+i) - (m-1+a)i}{i^2(1+i)^{(m-1+a)}} + \\frac{FV(m - 1 + a)}{(1+i)^{(m-1+a)}} \\right ) ", "39a9df5dcec6c391c6ad34b1167ad84f": "48.2 \\times 10^6", "39aa276a5aeed93228d58cd985dca7a0": "\\mathrm{GL}_{n}(\\mathbb{F}_{q})", "39aab78b94b615af85ed29cea0767ab8": "\\frac{\\partial \\eta}{\\partial t} = -\\frac{1}{\\varepsilon_o}\\nabla\\cdot\\mathbf{q_s}", "39ab18f6bb105f5fc8470ac6c5abdc3c": "\\epsilon_{\\rm dry}(\\rho,H)=\\frac{\\rho/H}{Na\\sqrt{1-\\rho^2/H^2}}", "39ab4b729f5b669abe707dd5855d39d7": " y=f_1(x), \\ y=f_2(x)", "39ab50b943525ed179ee367ad4fe3749": "\\alpha=\\arcsin(e)=\\arccos\\left(\\frac{b}{a}\\right).\n \\,\\!", "39ab9fe3b01071b71460f8a3fed799df": "e^{tD}f(x)=f(x+t)\\,", "39abd69c5d8f66b2d5c8e51dd91409af": "(value_{i}, confidence_{i})\n", "39abdd148ab6e6caa2c5315026b99818": " n_{e}=n_{H}", "39ac533fcac2940e3867d8c2edfd65d3": "Q_0=T_0\\oint\\frac{\\delta Q}{T}", "39ad4c51608d1036e1af6ac0fa887371": "\\mathcal{Q}-I", "39ad76dbe69d1d1107fe539398567a07": "\\not\\equiv", "39adf874e14231e8b3ea78585b1d2bf3": "\\scriptstyle 1/\\sqrt{N}", "39ae27784e98df9dbb62a7f1644bdbef": "\\frac{dy}{dx}=2ax+b=1", "39ae3fa1b0c28cbed543c4f64c7cd375": "\\mathbf{P}(0)=\\delta^{-1}I", "39ae4411df81ed2d0550a59e264c6e68": "\\lim_{\\epsilon\\to 0}T_\\epsilon = \\lim_{\\epsilon\\to 0}T\\ast\\varphi_\\epsilon=T\\in D^\\prime(\\mathbb{R}^n)", "39ae4ce7db3f1077587a36f46114c617": "\\operatorname{lcm}(21,6)={21\\over\\operatorname{gcd}(21,6)}\\cdot6={21\\over\\operatorname{gcd}(3,6)}\\cdot6={21\\over3}\\cdot6=7\\cdot6=42.", "39ae585720f6dcf52765be327dcfa831": "\\mbox{ironfly} = \\Delta(\\mbox{butterfly strike price}) \\times (1+rt) - \\mbox{butterfly} ", "39ae73ba9f33ff22caf78f484821c1bb": "\\Delta x = x \\otimes 1 + 1 \\otimes x", "39ae7c7e5c7a005497dc0f8fa76f4587": "H_{10} = -30240\\,", "39aea325274e8fc2965ccddfdb74f27d": " {\\hat x}_{CLS} = \\operatorname*{\\arg\\min}_{x \\in C} \\left\\| y - Ax \\right\\|^2 ", "39aea6d33cddc771c287659e79e72611": " \\frac {1}{c(w)} = \\frac {1}{c_\\infty} -\\frac{2a_2ln(a_3w)}{\\pi (1-a_3^2w^2)} \\quad (1.2)", "39aead914b47f8908626da5efe2146fb": "\\max_N \\left|\\sum_{n=1}^{N}\\left(\\frac{n}{q}\\right)\\right|>\\frac{1}{2\\pi}\\sqrt q", "39aef186a2932eeaca93211132be145d": "H^i(X,L^{-1})=0\\text{ for }i = 0,1.\\ ", "39af258fa1296f97a96bb9d43dfc3939": "\\left( \\frac{\\varepsilon_{eff}-\\varepsilon_m}{\\varepsilon_{eff}+2\\varepsilon_m} \\right) =\\delta_i \\left( \\frac{\\varepsilon_i-\\varepsilon_m}{\\varepsilon_i+2\\varepsilon_m}\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(8)", "39af6774f77d76e3ff486eeb51a6eec6": " E\\subseteq \\mathbb{R}^{n} ", "39af801dc6f9656c720cf184d6de6bb2": "\\mu \\frac{\\partial}{\\partial\\mu} y_\\text{t} \\approx \\frac{y_\\text{t}}{16\\pi^2}\\left(6y_\\text{t}^2 +y_\\text{b}^2- \\frac{16}{3} g_3^2- 3g_2^2 -\\frac{13}{15} g_1^2 \\right),", "39afe24d2d6d18ff0517d4eefa6c1143": "\\pi_*\\circ F", "39b00e7a0bd3692e14a3e69d3dc7a707": "L_{\\mathrm{core}}\\,", "39b03091378283e5edbf840416a45b7a": "C^{\\operatorname{op}} \\to \\mathbf{Rings} \\stackrel{\\textrm{forgetful}}\\longrightarrow \\mathbf{Sets}", "39b04eb72d66634d62b8d749514974f2": "P = \\frac{N\\sigma}{A} = \\frac{x}{\\lambda}\\qquad\\qquad(6)", "39b05243d0caee91fb174ddc07f856f0": "\\langle\\mathcal{O}\\rangle", "39b05a7de356df12ba528d351e59f6e4": " \\operatorname{E}_\\theta\\phi'(X)=1-\\beta'\\leq 1-\\beta=\\operatorname{E}_\\theta\\phi(X) \\quad \\forall \\theta \\in \\Theta_1.", "39b07732247f46ab2c9ff754d06d1250": "\\|(\\lambda I-A)^{-n}\\|\\leq\\frac{M}{(\\lambda-\\omega)^n}.", "39b0f889a35131bce59f35554a5d7e64": "n_{\\bullet0}", "39b118ace4967415e85e2b5e9c78bdea": "\\hat{a}_i^{\\dagger\\bullet}\\, \\hat{a}_j^{\\dagger\\bullet} = \\hat{a}_i^\\dagger\\, \\hat{a}_j^\\dagger \\,-\\,\\mathopen{:}\\hat{a}_i^\\dagger\\,\\hat{a}_j^\\dagger\\,\\mathclose{:}\\, = 0", "39b1a10350a13032032125eeec1ea141": "\\{y_i,y_{i+1}\\} = y_i\\, y_{i+1}", "39b1e12f78aa571bfa2207c4a99bf134": "\n\\begin{align}\n\\nu &= \n\\left[ \\frac{1}{n_1}\n\\left(\n\\frac{\\bar{X}_d'\\tilde{S}^{-1}\\tilde{S}_1 \\tilde{S}^{-1}\\bar{X_d}} \n {\\bar{X}_d'\\tilde{S}^{-1}\\bar{X}_d}\n\\right)^2 + \n\\frac{1}{n_2}\n\\left(\n\\frac{\\bar{X}_{d}'\\tilde{S}^{-1}\\tilde{S}_2 \\tilde{S}^{-1}X_d^{-1}}\n {\\bar{X}_d'\\tilde{S}^{-1} \\bar{X}_d}\n\\right)^{2} \n\\right]^{-1}, \\\\\n\n\\bar{X}_d & = \\bar{X}_{1}-\\bar{X}_2. \n\\end{align}\n", "39b223d238d61a234789ab9939a11c26": "\\ (Ra)^2=4(\\delta_{d2}-\\delta_{d1})^2+(\\delta_{p2}-\\delta_{p1})^2+(\\delta_{h2}-\\delta_{h1})^2", "39b223f8fa139022a8cfed20574a65e1": "f_c = \\sigma_f = \\frac{1}{2\\pi\\sigma}", "39b2350f6d739c5ec5da76a98bf7a60b": "\nP(t) = \\begin{pmatrix} p_{AA}(t) & p_{GA}(t) & p_{CA}(t) & p_{TA}(t) \\\\\n p_{AG}(t) & p_{GG}(t) & p_{CG}(t) & p_{TG}(t) \\\\\n p_{AC}(t) & p_{GC}(t) & p_{CC}(t) & p_{TC}(t) \\\\\n p_{AT}(t) & p_{GT}(t) & p_{CT}(t) & p_{TT}(t)\n \\end{pmatrix} ", "39b24236c46de2116bf5b3a8fc08113f": " Q = 90(44.625) - (44.625)^2 ", "39b2568dfbfa5aea059cf984a5595487": " L(x, y; 0) = f(x, y), ", "39b27e86e67d5680bd4d656393b60259": " L = \\tfrac{1}{2}m\\vec{v}^2 + \\frac{q}{c}\\vec{A}\\cdot\\vec{v} - V(\\vec{r}),", "39b28ea670fe755ca3ec14c8e60db1ae": "\\{X_1,\\,X_2,\\,\\ldots,\\,X_{n-k}\\}", "39b32b89d56d5125f262810683e8d213": "w*\\lambda := w( \\lambda + \\rho ) - \\rho \\,", "39b3483eb2b103f6c660a04e378b83e3": "\\langle \\omega^\\sharp, Y\\rangle = \\omega(Y),", "39b3b0d836f8f4f78f8817ab28020e68": "E \\approx \\frac{h^2}{8 m R^2} + 4\\pi R^2\\alpha + \\frac{4}{3}\\pi R^3P", "39b3e7f2c74bad066ea57c0b0fa2f9a3": "{\\vec c} = {\\vec m} \\oplus {\\vec b}", "39b42af0bfe9f3f61b5ef49963126ce1": "\\alpha_\\ell", "39b469f1c711d538d893ed3506f79855": "\\nu=3", "39b4ac397b0c38d6810bf5aace7a431b": " [u,v,w] + [w,u,v] + [v,w,u] = 0", "39b4c23111c9dfc8fa9a4c9d4a9620f8": "\\Pi = A^{-1}B \\, ", "39b4f89a4653ba69710d624f94105699": "i \\in A", "39b56f6e24a66e0038806ff204c7648f": " \\operatorname{de-lambda}[p\\ f = (\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))] ", "39b57a564b2a471f12f7cd184731490f": "\\mathcal{R}", "39b5867fb91557de8f06028cf2c41ba7": " H = H_{s} + \\epsilon H_{ns},", "39b5af5847b3651d59e8f148734185f3": "I_{m,n}= -\\frac{x^{m-1}}{a(2n-m-1)(ax^2+bx+c)^{n-1}} - \\frac{b(n-m)}{a(2n-m-1)}I_{m-1,n} + \\frac{c(m-1)}{a(2n-m-1)}I_{m-2,n}\\,\\!", "39b5c1035ecdbf8b5fe84046a1591d33": "\\ c_f(u_i, u_{i+1}) > 0", "39b5c70283eee796759f9eb75e6b1615": "\\ \\displaystyle \\min_{d\\in D}\\,\\min_{s\\in S(d)}\\,g(d,s)", "39b5dc0883319ca16c0c66d8a7114ca1": "\\ M_{pitch}= D_{pitch} \\times drag ", "39b61d1faa26f203db3f0b7ac5a0b237": " = (\\nabla \\cdot \\nabla) \\varphi = \\nabla^2 \\varphi = \\Delta \\varphi ", "39b665353a0a1282f5f202e176eaf6f5": "\\Theta_{00}={4\\lambda \\over \\pi D_{00}}", "39b6d251a78a638f541467f090e1f7d1": "\\,R(x)=x^n\\,", "39b6f948311773061ac23fe7b4ae7ac6": " b_n= \\beta \\log(n) \\, ", "39b735659ea035fa265e060139fca14e": "f \\colon {\\textbf{R}}^n \\to {\\textbf{R}}", "39b736d87dd79edae40652206db2cf67": "\n D\\nabla^2\\nabla^2 w = -q(x,t) - 2\\rho h\\ddot{w} \\,.\n", "39b7802cc57e07115914c40e6d365e66": "^{1}\\Delta", "39b7d0aa144d244d192d816e8853a936": "\\tilde G_n^{\\,\\text{cd}}(X)", "39b80501dafe0260c9853e979c3a7ead": "P(q) = \\frac{ {\\mathrm{e}^{-\\frac{E(q)}{k_BT}}}}{Z}", "39b807e962c066b13165d846a0832eda": "P_\\ell^{m}(x) = (-1)^m\\ (1-x^2)^{m/2}\\ \\frac{d^m}{dx^m}\\left(P_\\ell(x)\\right)\\,", "39b812b4d334ab19832d20ec24f8827b": "g(x) = x^2 - x - 1", "39b837eff00a4d2b1d6d0c012e1daba4": "f_{j+1}[n]", "39b856e911e09f191f05b7af373a1cf6": "F(g \\circ f) = F(g) \\circ F(f)", "39b8a85108197cfff25334f69854bd0c": "Y_{1}^{0}(\\theta,\\varphi)={1\\over 2}\\sqrt{3\\over \\pi}\\, \\cos\\theta", "39b8b186afd2faaa4188ef0fc901fca9": "\\oint_{\\partial D}\\,\\underset{\\alpha \\to \\beta}\\lim n_\\beta \\cdot \\nabla_\\alpha f(\\alpha)\\;d\\beta =-\\displaystyle \\int _{\\mathbf{R}^d}\\nabla_x\\mathbf{1}_{x\\in D}\\cdot \\nabla_x f(x)\\;dx.", "39b8c929e9b13b3e382fc179ad5d2fc4": "L^2 Y_{lm} = \\hbar^2 l(l+1) Y_{lm} ", "39b8d664f9d03bf51896856f72931c70": "T = n\\cdot (\\tau_1 - \\tau_2)", "39b8ebcb7ae9b3b0751f61b0aa809770": "\\zeta=1/2", "39ba527984f783beb9c223768ef43aaa": "\\operatorname{Li}_2\\left(-\\frac{\\sqrt5+1}{2}\\right)=-\\frac{{\\pi}^2}{10}-\\ln^2 \\frac{\\sqrt5+1}{2}", "39ba85f758cb409a55c10e105a589f7c": "f_{-j}", "39babc585f46c66bd821958f9d1ddbce": "\n\\begin{bmatrix}\n\\boldsymbol{I} & \\boldsymbol{V}_1\\\\\n\\boldsymbol{W}_2 & \\boldsymbol{I} & \\boldsymbol{V}_2\\\\\n& \\ddots & \\ddots & \\ddots\\\\\n& & \\boldsymbol{W}_{p-1} & \\boldsymbol{I} & \\boldsymbol{V}_{p-1}\\\\\n& & & \\boldsymbol{W}_p & \\boldsymbol{I}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1\\\\\n\\boldsymbol{X}_2\\\\\n\\vdots\\\\\n\\boldsymbol{X}_{p-1}\\\\\n\\boldsymbol{X}_p\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{G}_1\\\\\n\\boldsymbol{G}_2\\\\\n\\vdots\\\\\n\\boldsymbol{G}_{p-1}\\\\\n\\boldsymbol{G}_p\n\\end{bmatrix},\n", "39bb0d0dae737aee2cdec9e261892a14": " \\text{vbl}(A) \\cap \\text{vbl}(B) = \\emptyset ", "39bb270ed9fdbdfbfb736afa7320039f": " E_1(x,\\alpha) = x \\,", "39bb5466a23af9dcc96f78c89374416b": "M_v = \\sum _{k=1} ^m \\alpha_k u_k v_k ^T ,", "39bbd51bf1545e2bffd71872464dbc3c": "c t' = \\gamma c t - \\gamma \\beta x \\,", "39bc1e06816d0a77b575be014e5c7ab5": " \\frac{V_n}{m} = \\frac{1}{\\rho} ", "39bc6dbb41299b4c394f3bd56e1f8d77": "g(x) = h(x) + O(f(x))\\,", "39bcf2d1c2f342bfebaceb4440bad3da": "\nS(\\theta)=\n1+\\theta\\ln\\left(1+\\frac{J-1}{\\theta}\\right).\n", "39bd289fe16b9deea4a72c6720498f47": "\\mathbf{\\Delta}^0_{\\alpha + 1}", "39bd57e3b5c9e36ecf7cafd09a27a20c": "d=50 \\ln\\left({\\frac{1}{1 - 2 \\Pr[\\text{recombination}]}}\\right)\\,.", "39bd8d540ac7640de9370d0034492a86": " v = \\sum_{b \\in B} \\alpha_b b ", "39bdabc5a554229df0cb0d9cbe24c37b": "H^*\\cong G^*", "39bdb106e311d5b009f21c8c81f84321": "\\! V(\\phi)", "39be242b005f0202c68415f93e96d03a": "\nx(t) = A \\cos (2 \\pi f_n t). \\!\n", "39be86a313129d312bd8591784f88233": "\\textstyle \\textbf{R}^{n d}", "39bee9a2052e1aa4040a98c8f16126e6": " x^2 y z^2 < x y^3 z^2 ", "39bf150bb7bd8e3ed58592254828f083": "\n\\frac{\\partial \\mathbf{a}_{k}}{\\partial \\alpha_{r}} = \\frac{\\partial \\mathbf{r}_{k}}{\\partial q_{r}}\n", "39c0016944c74ea0ab51ab9345167fd5": "U*", "39c002c2b6ac00d948d0a4e2168f0202": "\\lim_{x \\to c} x = c", "39c0365709619a2f0b650f3c9ba1384e": "x(t)=\\delta(t)", "39c0498f88f354082a887ca657f08efc": "f(x) - H(x) = \\frac{f^{(K)}(c)}{K!}\\prod_{i}(x - x_i)^{k_i}", "39c080b9486970b9c85a3f0ab0497213": "S=\\frac{1}{\\left\\langle 0|U(\\infty)|0\\right\\rangle}\\mathcal T e^{-i\\int{d\\tau H_{\\rm{int}}(\\tau)}}.", "39c0b1734c51c8349d2efcad70fde005": "(u_x, u_y, u_z)", "39c0b36f38aed3bd87307bb554c74d74": "dae", "39c0c71a643dfdf114a3da74f62419bb": "\\dot{V}_1\n= -W(\\mathbf{x}) + \\frac{\\partial V_x}{\\partial \\mathbf{x}} g_x(\\mathbf{x}) e_1 + e_1\\overbrace{\\left( -\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1 e_1 \\right)}^{v_1}", "39c157ce00ee93abb7fb22e09487aebd": "\\alpha: 1 > \\alpha > 0", "39c15e444dba459655489b154af7d1f0": " p(x)=c_0+c_1x+c_2x^2+\\cdots + c_{n-1}x^{n-1}+x^n", "39c1eacb70fbdeb6d9786652f42c0fa1": " \\theta' = \\theta + \\Omega t \\,\\!", "39c23b783d5b7cf31183d7c658772a6a": "T_n=(-1)^{n+1}\\left[n+1-\\zeta(2)+\\sum_{k=1}^{n-1} (-1)^k (n-k) \\zeta(k+1) \\right] ", "39c2c0fe21139a95377acc0bc8fe78af": "\\operatorname{Ai}' (\\tilde{a}_1 ) \\approx 0.7022", "39c30119bce30e9d22ca7d8f651acae8": "P(x)(f)=\\sum P^i(f)x^i ", "39c35ca65ad0b81822bb1dfcdb7d0af7": "\\begin{matrix}\n 1 & 0 & 1 & 0 \\\\\n 1 & 1 & 0 & 0 \\\\\n 1 & 1 & 1 & 1 \\\\\n 1 & 0 & 0 & 0 \n\\end{matrix}", "39c3ae9d4c3625ec1d8329e70a3226a3": "B = -y\\sin(\\phi)+z\\cos(\\phi)", "39c3f17c4ae4dd59a932cd3fb1562c63": "[a_\\lambda [b_\\mu c]]-[b_\\mu [a_\\lambda c]]=[[a_\\lambda b]_{\\lambda+\\mu}c]. \\, ", "39c3f599cd159184d3526e4de896251f": "E_\\text{Z} = -\\mu_0 \\int_V \\mathbf{M}\\cdot\\mathbf{H}_\\text{a} \\mathrm{d}V", "39c3f70fb3927a519b2f1208543ac745": "0=f^k\\left(x\\right)=f^k\\left(f^k\\left(y\\right)\\right)=f^{2k}\\left(y\\right)", "39c3f805160075cd9d729ee2fa8cdd5f": " \\nu", "39c492ca0f54f5c449349c98579a8676": "\\displaystyle{Jf(t)=\\overline{f(t)},\\,\\,\\, Uf(t)=\\dot{z}(t)\\cdot f(t).}", "39c49c2650fa26e9e69bef58f42503ca": "Y_n(z)", "39c4d708f10a98c37d5932062f0bb17d": "J:\\mathcal{D}\\to \\textbf{Set}", "39c50a0ebb2d3241457f1b6c8c29b2ca": "\\delta E = -\\frac{1}{8} U^2 md^2 \\left(\\frac{2k}{rd}\\right) \\theta^2", "39c52dc799e9dba097162abe9c3ad864": "D_n\\left(E\\right) = \\frac {d}{dE}\\Omega_n(E) = \\frac{n c_n}{p c_k^{n/p}}\\left(E-E_0\\right)^{(n/p - 1)} ", "39c54ca032b21743c91cf95c205fe0c0": "H^* (M) \\to H_{d - *}(M;\\mathbb Z^w)", "39c55a30e73c32eb313dcc9776d11b1d": " j^{\\star} = \\sigma T^{4}.", "39c56a12c1d9ef5553c0ac3ee7e5019f": "\\frac{d\\bigl(f(x)\\bigr)}{dx} = f'(x)\\,.", "39c57a27aee3f50ac89a104d417f5322": "FX/O(1,3)\\to X", "39c5c4d5218866e7d919fb60f294158f": " s_x(x,y) = \\delta_x + u_x^{sphere}(x,y) - u_x^{plane}(x,y)", "39c5e22a7b37356540074af57d31d22b": "\\overline \\Delta_{12}(\\mathbf{p},\\mathbf{p})", "39c5ebe871554aeb52b8fd5fe471e6fe": "r_i = a \\cdot \\frac{\\varphi^2}{\\sqrt{1 + \\varphi^2}} = a \\cdot \\sqrt{1 + \\frac{2}{\\sqrt{5}}} \\approx 1.37638 \\cdot a", "39c617587f732c192c2bc004c4f66fa7": "X_j = \\left. \\frac{\\partial g}{\\partial \\xi_j} \\right|_{\\xi_j = 0} ", "39c6613b285f65131bc42fee3bf0cb09": "\n \\begin{bmatrix} \\sigma_1 \\\\ \\sigma_2 \\\\ \\sigma_3 \\\\ \\sigma_4 \\\\ \\sigma_5 \\\\ \\sigma_6 \\end{bmatrix} =\n \\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\\\\nC_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\\\\nC_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\\\\nC_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\\\\nC_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\\\\nC_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \\end{bmatrix}\n \\begin{bmatrix} \\varepsilon_1 \\\\ \\varepsilon_2 \\\\ \\varepsilon_3 \\\\ \\varepsilon_4 \\\\ \\varepsilon_5 \\\\ \\varepsilon_6 \\end{bmatrix}\n ", "39c6699efa9f8eb8ae8e89d6b20c517a": "f'(a)=\\lim_{h\\to 0}\\frac{f(a+h)-f(a)}{h}", "39c691344640b1a5e859bd7b284949ad": " \\eta = \\frac{\\epsilon}{|v|}\n = \\left| \\frac{v-v_\\text{approx}}{v} \\right|\n = \\left| 1 - \\frac{v_\\text{approx}}{v} \\right|,\n", "39c6a79fe45ebffea0c1a626d317ece5": "(2,2)", "39c6fbd3221281dc4ee0ae576af6dd79": "{\\tilde{D}}_{3}", "39c70174e9570d8a2e049e319d6d8e53": "A = 3", "39c716a1841945477d5fc602a4dc23a2": "K_{B}", "39c72659ccb6f2c23de7908007f9e739": "\\left(E=\\textstyle\\sum N_i \\varepsilon_i\\right)", "39c73cfaf4e368a0cf2813f60806fb0d": "\ny_\\mathrm{d} = x_\\mathrm{u} + y_\\mathrm{u}(1 + K_1r^2 + K_2r^4 + \\cdots) + \n(P_2(r^2 + 2y_\\mathrm{u}^2) + 2P_1 x_\\mathrm{u}y_\\mathrm{u})(1 + P_3r^2 + P_4r^4 \\cdots)\n", "39c76fe80e8d0d77fd137f28d25d9a9d": "([X], \\alpha) \\oplus ([Y], \\beta) := ([X \\coprod Y], \\alpha + \\beta)", "39c80e5e589f40e86a1e3a67199dd5e3": "f\\left(z\\right)", "39c8350300293f6d6377c18708ff13ae": "d(O_{r}^{*}, O_{n})", "39c892367f14eb457cdd929ceda6ca19": " m = M ", "39c8a6e70df0e630fcd204169d1cc85b": "\\mathbb{W}^{k,(-dn)}", "39c92ac75bfaa0adf00ab04a33b01bf3": "f: \\mathbb{R}^n \\rightarrow \\mathbb{R}", "39c949641156b110feb44627bb2bf75d": "\\mathbf{K} = \\left(\\frac{\\omega}{c}, \\mathbf{k} \\right) \\,. ", "39c95527c84827ca4f5a7522c4e89508": "\n\\operatorname{E}[Z] = \\operatorname{E}[ Z \\, \\mathbf{1}_{\\{ Z < \\theta \\operatorname{E}[Z] \\}}] + \\operatorname{E}[ Z \\, \\mathbf{1}_{\\{ Z \\ge \\theta \\operatorname{E}[Z] \\}} ].\n", "39c95b32b1740ada118e3cb8fba8bb80": "\\displaystyle{K(z,w)=\\partial_{n,w} N(z-w).}", "39c99592bcb11a2066aa84110fbcdf40": "a'(t) = \\frac{d}{d t} u'(t) = \\frac{d}{d t} u(t) - 0 = a(t).", "39c9a78cb30ae317519741f4e1d60203": "(\\vec{H_{2}}-\\vec{H_{1}})\\times \\hat{n}=0", "39c9bd7e04bd734ac9d748727931b2cb": "X=\\varnothing", "39c9d656152a7c3b126b30d8d7e0137d": "r = z \\sqrt{ 1 + \\frac{\\rho^2}{z^2} } ", "39c9e3e89b54668de2fb7acd2fa2a2e2": "Q(p) = 2(p-1/2)\\sqrt{\\frac{2}{\\alpha}}\\!", "39c9fe494e5c55aa2277a2e9ab601a99": "\\int_a^\\infty,", "39ca339b1c025e6807f227ad85f9602e": " Y_\\nu (x) = G_{1,3}^{\\,2,0} \\!\\left( \\left. \\begin{matrix} \\frac{- \\nu - 1}{2} \\\\ \\frac{\\nu}{2}, \\frac{-\\nu}{2}, \\frac{- \\nu - 1}{2} \\end{matrix} \\; \\right| \\, \\frac{x^2}{4} \\right), \\qquad \\frac{-\\pi}{2} < \\arg x \\leq \\frac{\\pi}{2} ", "39ca9db303a1eb49efaea71ff631586a": "\\sqrt{S} = S \\cdot (1/\\sqrt{S})", "39caa2b6ef4d8c21801a77d2e0377b8e": "x_2(t)=0", "39caae8cda0b617057fb4f0dd1ce9f5d": "f=m\\Delta_{F}", "39cb276e49ad6bef1d7d31b67c95f1d0": "\n\\begin{align}\nF(A) & = \\sum_{\\sigma\\in S_{n},\\sigma(j_{1})<\\sigma(j_{2})}\\left[\\sgn(\\sigma)\\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^na_{\\sigma(i)}^{i}\\right)a_{\\sigma(j_{1})}^{j_{1}}a_{\\sigma(j_{2})}^{j_{2}}+\\sgn(\\sigma')\\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^na_{\\sigma'(i)}^{i}\\right)a_{\\sigma'(j_{1})}^{j_{1}}a_{\\sigma'(j_{2})}^{j_{2}}\\right]\\\\\n& =\\sum_{\\sigma\\in S_{n},\\sigma(j_{1})<\\sigma(j_{2})}\\left[\\sgn(\\sigma)\\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^na_{\\sigma(i)}^{i}\\right)a_{\\sigma(j_{1})}^{j_{1}}a_{\\sigma(j_{2})}^{j_{2}}-\\sgn(\\sigma)\\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^na_{\\sigma(i)}^{i}\\right)a_{\\sigma(j_{2})}^{j_{1}}a_{\\sigma(j_{1})}^{j_{2}}\\right]\\\\\n& =\\sum_{\\sigma\\in S_{n},\\sigma(j_{1})<\\sigma(j_{2})}\\sgn(\\sigma)\\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^na_{\\sigma(i)}^{i}\\right)\\left(a_{\\sigma(j_{1})}^{j_{1}}a_{\\sigma(j_{2})}^{j_{2}}-a_{\\sigma(j_{1})}^{j_{2}}a_{\\sigma(j_{2})}^{j_{_{1}}}\\right)\\\\\n\\\\\n\\end{align}\n", "39cb89fffe654a7112625fd593c70b22": "\\cos A = \\frac{\\cos a\\,-\\,\\cos b\\,\\cos c}{\\sin b\\, \\sin c}.", "39cb904a3147572bd2c0ed75cf230652": "\\partial_v\\,\\!", "39cbb01268d0dd2dc2587b074905f44b": "d : \\Lambda \\to \\mathbb{N}^k", "39cbb53ecf8597705482869f876cc80e": " \\chi_k^2 \\sim {\\chi'}^2_k(0)", "39cbdc08618fbc62028ee6c7452f474c": "(\\Phi(n) \\land \\forall i (\\Phi(n+i) \\to \\Phi(n+i+1))) \\to \\forall i \\Phi (n+i)\\,\\!", "39cbe564daf7ba0866dc9566aca305a5": "\\rho_1", "39cc0daea08ec51f72936ec6e2c3ff03": "{l_\\text{P}}", "39cc44d33a585f1ac5c02f36f695d658": "\\langle x,y \\rangle = E \\{ x^H y \\}", "39cc4a02b3d6e4ef1f20a9cc969065e1": " G( y ) = \\frac{ b - y^{-1} }{ b - a } .", "39cc5faa60ac13bc6d59f40040edf9b5": "\\iota(g)=g^{-1}", "39cc722163d8db73d4f95218c06b0523": "K[X]/\\langle p\\rangle,", "39cc88dcff054904e246988c12333196": "R = 2GM/c^2", "39cca8d5ff72ca63b0d9356d6f1fca5a": "\\mathbb{T}_1", "39cd065bdf02a2e7692a4611f9c9399b": " V(x;\\sigma,\\gamma) = H(a,u) / ( \\sqrt 2 \\sqrt \\pi \\sigma ) ", "39cd6921d28d1c2f71e43a7fabe524f6": "C_p\n= \\left ( {\\partial Q_{rev} \\over \\partial T} \\right )_p\n= \\left ( {\\partial U \\over \\partial T} \\right )_p + p \\left ( {\\partial V \\over \\partial T} \\right )_p \n= \\left ( {\\partial H \\over \\partial T} \\right )_p\n= T \\left ( {\\partial S \\over \\partial T} \\right )_p ", "39cd8bf00594863309a36840e19c4996": "\\mathbb{Z}/9\\mathbb{Z}", "39cd9595f0481b799a348ef420ae30fc": "F_{k^{1/p}}", "39cdaa9c415841950a4ba217c834fd3b": "\\mathbf{P} = \\begin{bmatrix}\n1 & 1 & 2 & 2\\\\\n1 & 1 & 2 & 2\\\\\n3 & 3 & 4 & 4\\\\\n3 & 3 & 4 & 4\\end{bmatrix}", "39cdbd017ae84271fd53eccba99a8abf": "|A+B|\\ge\\min\\{p,\\ |A|+|B|-1\\}.\\,", "39cdc407f46386bbe121ddc43b4e9ebf": "\\operatorname{E}[\\ln |\\mathbf{\\Lambda}_k|]", "39ce0f3a18ab426452e77ddd68558242": "b_{i-1}=a_i+b_i", "39ce62229ab93fe94bcd37716fc51498": "\\frac{b!}{n!}\n=\\frac1{(b+1)(b+2)\\cdots(b+(n-b))}\n\\le\\frac1{(b+1)^{n-b}}\\,.\\!\n", "39ce701d6adc033f888f11ec856822e2": " \\hat{n}_{\\nu_j}", "39cee986e620f36e94358eb5c7dddc14": "\\scriptstyle |\\phi_n\\rang", "39cf2baa4054f29e29a30e81703613bb": "\\frac{(x'(t),\\ y'(t))}{|x'(t),\\ y'(t)|}", "39cf9e4f2253a2a3c7518e3dc46404c8": "{\\tilde{A}}_{5}", "39cfeaabe5da3e8393aae81adcbcedb8": "\\mathfrak{P}^{7}", "39d002baede3d95d7031c8df0405e4a8": "\\scriptstyle {\\mathbf{Set}}", "39d0cfe5a5726bdcda4426eb4a84a1db": "Z = Y_1 + Y_2 + \\ldots + Y_n", "39d0f4ea0615490696354c25fa7f1e62": "gz+{p_{atm}\\over\\rho}={v^2 \\over 2}+{p_{atm}\\over\\rho}", "39d13b7a2b20cbc2452a6c1fd0f16b1b": "V_\\max", "39d1a95c9fd58b8b4e189f5a60345204": "\\upsilon_M\\,", "39d1dd2733df2faef98c52b682345b83": "\\frac{\\partial u}{\\partial n} = \\frac{\\partial v}{\\partial s},\\quad \\frac{\\partial v}{\\partial n} = -\\frac{\\partial u}{\\partial s}", "39d1f15559176ba37077a30d0111942d": " A_q(n,d) \\leq \\frac{q^n}{\\sum_{i=0}^t {n \\choose i} (q-1)^i + \\frac{{n \\choose t+1} (q-1)^{t+1} - {d \\choose t} A_q(n,d,d)}{A_q(n,d,t+1)} }. ", "39d23fb264e0efc191984795d1636e6b": "\\alpha(2)", "39d2771905906ba5c78b2bb3bc667b13": "a\\in L ^q(\\Omega, \\mathbb{R}^m)", "39d2fd80fcac230fe1250b37b4cb8252": "F(x+1) - F(x) = f(x+1) \\,", "39d358b4eec81f8dc124b01811588fdb": "(CA,BD) \\cdot (CA,DD') = (CA,BD').", "39d3dc7fac8e07b1d83d0f5283eaff71": "\\Pr(A>x) < \\Pr(B>x)", "39d426d27f23f27229932d4666e9d24a": "d(x,y)=|x-y|", "39d42a47905081838b4cad52b3ce9a2b": "\\land,", "39d45d6a17da736ee2bebe2ffaefe0ec": "t \\geq \\log\\{Vol_2(d,n)\\}", "39d4a46e79dee68bcb8947dfe0580f13": "\\! b_{jm}", "39d4aa186ae4b9abd63a95df91880b51": " ax + by + cz + d = 0 ", "39d4c16b4d82a80165519ad56e4678d1": "\\sigma^2=\\left.2\\mu\\right.\\,", "39d55c498edaba1e9cd4d9833223922c": "w''", "39d59efbbe07abd762cd382ca40cbfc1": "(\\wedge)", "39d5c3636d750cd3b719b7a96827851c": "(a_1,\\dots,a_n)", "39d5ea0b9845e7b3ce8b1c1271d6cff7": " (A f)(x) = i \\frac{d}{dx} f(x), \\, ", "39d60f894109d46930b4a08f78248a75": "\\frac{1}{\\tau_C} = \\frac{1}{\\tau_U}+\\frac{1}{\\tau_M}+\\frac{1}{\\tau_B}+\\frac{1}{\\tau_{ph-e}}", "39d60fbc74db0730a6bfac5cdce2faf1": " L_{h} = h^{\\mu}(x)\\frac{\\partial}{\\partial x_\\mu}.", "39d6293847c3eb7a8496ffc24e6e3086": "\\begin{align}\na & = \\cos (\\phi/2); \\\\\nb & = k_x \\sin (\\phi/2); \\\\ \nc & = k_y \\sin (\\phi/2); \\\\ \nd & = k_z \\sin (\\phi/2).\n\\end{align}", "39d6c3f89637295e7ae4bb4db1f5752e": "{\\mu}_s \\in \\mathbb{R}", "39d6ca842f47829a0d4f83f7219f6692": "\\tan (\\alpha + \\beta) = \\frac{\\tan \\alpha + \\tan \\beta}{1 - \\tan \\alpha \\tan \\beta}\\,", "39d74aa1a6c7248b01f697c8a0dd52e5": "C_\\xi \\,,", "39d766338aedf887baf5a7755ea0c23a": "\\chi_n^{\\alpha\\beta}", "39d790ee4bb23bceb79bddc337598446": "\\prod_{i\\in I}M_i / U . ", "39d7e5bd9201a77ed4b8fbfe3c1040fe": "C(v,p)", "39d866f8b7d655fb0ff7f2f876a6ff84": "\\Gamma_i,i\\in I", "39d889318bafb7b064945984ae350f45": "\\hat{u}_{k}(0)", "39d9486fc07e7b1acaeacc3428163521": "u(|\\vec{r}_1- \\vec{r}_2|)", "39d95beecddc47a03f079d6fdaa52ed2": "e^{-\\frac x 2} \\, {}_1F_1(a,2a,x)= {}_0F_1 \\left (;a+\\tfrac 1 2; \\tfrac{x^2}{16} \\right )", "39d994fbb2f69fde8e790c900af4719e": "=-x-\\frac{x^2}{2}-\\frac{x^3}{6}", "39d9c49f3de60f2db5e15bd7439666e5": " \\int_K | f|\\, \\mathrm{d}x <+\\infty,", "39da28a64c0e9529c400f9c25df82876": "c_1 \\mathbf{r}_1 + c_2 \\mathbf{r}_2 + \\cdots + c_m \\mathbf{r}_m,", "39da4951f1397c432e89333161f0edfe": "v*", "39da5652879f515853450008cd007278": "\\mathbf{L}_{i}:=\n\\begin{pmatrix}\n\\mathbf{I}_{i-1} & 0 & 0 \\\\\n0 & \\sqrt{a_{i,i}} & 0 \\\\\n0 & \\frac{1}{\\sqrt{a_{i,i}}} \\mathbf{b}_{i} & \\mathbf{I}_{n-i}\n\\end{pmatrix},\n", "39da6026e2806ea6ad072c6220013481": "\\sum_{k=1}^N \\lambda_k/\\sum_{k=1}^\\infty \\lambda_k \\geq 0.95", "39da79a1446baa1310ccd23221787fc9": "\\delta = (\\dots, 0, 0, \\underset{0-\\mbox{th position}}{1}, 0, 0, \\dots)", "39dad944700a1c8acf7f41c32caa839e": "M(x) \\cdot x^n = Q(x) \\cdot K(x^n) + x \\cdot R(x)", "39db79a3072d31ff845baf325060770c": "\\frac{\\ln\\, \\mathcal{L} (\\alpha, \\beta|X)}{N} = - H = -h - D_{\\mathrm{KL}} = -\\ln\\Beta(\\alpha,\\beta)+(\\alpha-1)\\psi(\\hat{\\alpha})+(\\beta-1)\\psi(\\hat{\\beta})-(\\alpha+\\beta-2)\\psi(\\hat{\\alpha}+\\hat{\\beta})", "39db9e798818a4eeba4af6f6917fc7b3": "\\begin{align}\\mathrm{d}^kX &= \\left(\\mathrm{d}x^{i_1} e_{i_1}\\right) \\wedge \\left(\\mathrm{d}x^{i_2}e_{i_2}\\right) \\wedge\\cdots\\wedge \\left(\\mathrm{d}x^{i_k}e_{i_k}\\right) \\\\\n&= \\left( e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\right) \\mathrm{d}x^{i_1} \\mathrm{d}x^{i_2} \\cdots \\mathrm{d}x^{i_k}\\end{align}.", "39dbf0b4573a724f1d368df8e3890614": "m - M = 5 \\log_{10}(d) - 5 = \\mu ", "39dc0834f85015d474f35863c04b5292": " [x \\psi](x) = x \\psi(x) \\quad ", "39dc09011a7563b45d5165278d85c1ba": "\\mathcal{F}^{-1}(\\{x_n\\}) = \\mathcal{F}(\\{x_{N - n}\\}) / N", "39dc5702c0046be2f058f0f22fd51f00": "(R, \\Theta) = \\left(\\frac{r}{1 - z}, \\theta\\right),", "39dc7ade80c70c99dda54b19a381aaff": "S(\\mu(t)) = -\\sum_{i=1}^{n} \\mu_i(t) \\log(\\mu_i(t)).", "39dcaeadea17634dc77ca88342811ca3": "\\tfrac{\\triangle}{4T}", "39dcc5ecab512bab53547d799b63352b": "\\{-\\frac{1}{p}, -\\frac{1}{p^2}, -\\frac{1}{p^3}, \\ldots\\}", "39dd13722ab903275f6e424250d78554": " \\begin{bmatrix} 1+\\sqrt\\varepsilon & 0 \\\\ 0 & 1-\\sqrt\\varepsilon \\end{bmatrix}. ", "39dd4f272176a6a0af86b4b098231c16": "c_{1}-b_{1}", "39dd64106d02f4f4251576a8d927368a": "I = [a, b] = \\{x \\in \\mathbf R \\,|\\, a \\leq x \\leq b \\}. ", "39dd806042abcf68b9cf900fcdaadfd3": "(x^{10}-23 x^8+188 x^6-644 x^4+803 x^2-101)^2", "39ddb5ef70e65ae2f31867a8de7b5988": "\\begin{align}\n\\operatorname{Cov}_N(X,Y) = \\frac{C_N}{N} &= \\frac{\\operatorname{Cov}_{N-1}(X,Y)\\cdot(N-1) + (x_n - \\bar x_n)(y_n - \\bar y_{n-1})}{N}\\\\\n &= \\frac{\\operatorname{Cov}_{N-1}(X,Y)\\cdot(N-1) + (y_n - \\bar y_n)(x_n - \\bar x_{n-1})}{N}\\\\\n &= \\frac{\\operatorname{Cov}_{N-1}(X,Y)\\cdot(N-1) + \\frac{N-1}{N}(x_n - \\bar x_{n-1})(y_n - \\bar y_{n-1})}{N}.\n\\end{align}", "39ddb89c152a0361f6d0300fc119b5be": "Q_1(z,v) = \\frac{1}{2} Q(z,+1,v) + \\frac{1}{2} Q(z,-1,v) =\n\\frac{1}{2}\\left(\\frac{1}{1-z}\\right)^v\n+\\frac{1}{2}\\left(\\frac{1}{1+z}\\right)^{-v}", "39dddff3ab2dddba982aae46c2abc6c5": "t \\mapsto \\phi_t(x_0)", "39de0e6aea986f843d6f99db270c2fed": "\\mathbb{E}[X_\\tau]=\\mathbb{E}[X_0].", "39de19ce40379e7653c870596978ee76": "\nG = \\sum_{k=1}^N \\mathbf{p}_k \\cdot \\mathbf{r}_k\n", "39de37a70028c62fc8018dd892c0cc5f": "R_2\\,\\!", "39de3cf74fa244887f8920012ccb9587": "\\alpha,\\beta,\\gamma, \\ldots", "39de46d4e7eebf459af2e46db7c3f411": "t_i = a_{L+1}^i", "39de68b579f2a95ad8a12c903fd93c4d": "Z^q(\\mathcal{U}, \\mathcal{F}) := \\ker \\left( \\delta_q : C^q(\\mathcal U, \\mathcal F) \\to C^{q+1}(\\mathcal{U}, \\mathcal{F}) \\right)", "39de99e31c214376d625079cd6b094d5": " \\tau_k = \\min \\{ t : X_t = k \\} ", "39dedc5efa55e69f0438e137c79e194c": "\n\\begin{align}\n z + 2 &= \\zeta + 2 + \\frac{1}{\\zeta}\\, = \\frac{1}{\\zeta} \\left( \\zeta + 1 \\right)^2, \\\\\n z - 2 &= \\zeta - 2 + \\frac{1}{\\zeta}\\, = \\frac{1}{\\zeta} \\left( \\zeta - 1 \\right)^2.\n\\end{align}\n", "39df232f310bbb1bf68d06f7f0c9b0ab": "\\displaystyle \\eta", "39df686dfd9d08eef5f6e098248a4508": "f(S)\\leq \\sum_{i=1}^n \\alpha_i f(X_i)", "39dfd2ab4885f2b783eb4d9978064363": " M(t)=(1-t)(D+I)+tA ", "39e0115aa127747095c047bf2cfb86de": "X_0 = Z_{k-1}^{-1}", "39e01df71dec3491fc9ee837f860fd56": "x^2 \\equiv p \\ \\ (\\text{mod } q),\\,", "39e025d3650431e6767c03d77c1f4158": "U_{XOR} = e^{i\\frac{\\pi}{2}S_L^z}e^{-i\\frac{\\pi}{2}S_R^z}U_{sw}^{1/2}\ne^{i \\pi S_L^z}U_{sw}^{1/2}.", "39e037e4f9586fd367447ecb36c3c637": "dpq", "39e08f01929de832aac443f7b0075a69": "G_{n+1} =\\begin{cases} 1 & \\text{if } n \\text{ odd} \\\\\n \\coth^{2} \\left ( \\frac{ \\beta }{ 4 } \\right ) & \\text{if } n \\text{ even} \n \\end{cases}", "39e0d2428932f31e4608ee40f865b8a9": "\nS_{ijk} \\rightarrow S_{ij}(\\omega_k)", "39e0f09d1ad8b356793874d010b19f6f": "\\sigma = \\sigma' + u", "39e13d9767b36600f1ea774d9f72db7c": "{\\textstyle m \\propto \\sqrt{P}}", "39e14629b39e0b4a095b777c40f2f8ab": " \\delta_p(1) =0 ", "39e15d2cabc255928d468a5d3271c925": "X \\rightarrow \\textrm{Spec} (\\mathbb{Z})", "39e15f25b7013b53b80683ab89c95cd9": "y = a(x)", "39e16161dd24e0c1487a73fb7662803a": "q(x_2)=y_2", "39e19a79fd114430cece73cc0a30d998": "\\alpha_j \\in R", "39e1dba0636e7983364d2f19d8324b0c": "\\; \\rho ' = \\sum_i P_i \\rho P_i.", "39e2a6daacdb62ddc3435555f52f5b9d": "\\ u^2x^2+v^2y^2+w^2z^2-2vwyz-2wuzx-2uvxy=0", "39e321542f61d3da3bf8eee5b5ee5784": " x_1 == s_i ", "39e324715baba38b24018e4adff021c2": "(h * h_{inv}) (n) = \\,\\! \\delta (n)", "39e342e2fb8b220c2fb74816ff9ab4f0": "\n\\sqrt{\\rho_n} R_{n\\rightarrow m} {1\\over \\sqrt{\\rho_m}} = H_{nm}\n\\,", "39e35d987c2d5c8989ffbe155e5bd8bb": "2 + \\frac{12}{\\sqrt{13}}", "39e3d6b204f9b66a5c57ae837917b130": "\\displaystyle{e^\\xi=\\sum_{k\\ge 0} (k!)^{-1} \\xi^{\\otimes k}.}", "39e41061d88ab2474819684ae6827f94": "u, v\\in U", "39e4287675cb149dc2844e3467134234": "C_i \\in C", "39e446eb832ca3701eb5e5fdf5467728": "F(y)=\\mathcal{O}\\left(\\sqrt{\\frac{1}{y}}\\right).", "39e475b2b9fe9809f3fcd2e7b2500bb0": "\n\\int_{N_k}^\\infty\\frac{dx}{x\\ln(x)\\cdots\\ln_{k-1}(x)(\\ln_k(x))^{1+\\varepsilon}}\n=-\\frac1{\\varepsilon(\\ln_k(x))^\\varepsilon}\\biggr|_{N_k}^\\infty<\\infty.\n", "39e4e7015711cf339f4d157167717b16": "\\begin{align}\np=&\\frac{8b-3a^2}{8} &=&\\frac{8a_2a_4-3a_3^2}{8a_4^2}\\\\\nq=&\\frac{a^3-4ab+8c}{8} &=&\\frac{a_3^3-4a_2a_3a_4+8a_1a_4^2}{8a_4^3}\\\\\nr=&\\frac{-3a^4+256d-64ca+16a^2b}{256}&=&\\frac{-3a_3^4+256a_0a_4^3-64a_1a_3a_4^2+16a_2a_3^2a_4}{256a_4^4}\n\\end{align}\n", "39e563bac9b294b8bb33ddaae5c81fa6": " \\vec{F}_{12} = \\frac {\\mu_0} {4 \\pi} \\int_{L_1} \\int_{L_2} \\frac {I_1 d \\vec{\\ell}_1\\ \\mathbf{ \\times} \\ (I_2 d \\vec{\\ell}_2 \\ \\mathbf{ \\times } \\ \\hat{\\mathbf{r}}_{21} )} {|r|^2}", "39e568918468d6551e1354fb02e7005a": " \\rho \\in C^\\infty(U,\\mathbb R) ", "39e62955530352dfb39bd49e56f19640": "\\mbox{E} \\approx 222\\cdot\\frac{\\sqrt{N}}{d} ", "39e7200c07147a97856b038a059d4044": "\\rm CO_2 + H_2O \\xleftarrow{Carbonic\\ anhydrase} H_2CO_3", "39e758789a319101a3a0135fa3e6b3cb": "I:\\mathcal{X}\\to [0,+\\infty]", "39e7c138395e111354d7ec5210c8e834": "\\mathcal{C}_{Y \\mid X}: \\mathcal{H} \\mapsto \\mathcal{H}", "39e825cffb65c7a25f4540d2815e3a84": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "39e8f2d0a6f46cb8df612aca8e608051": "\\rho_{a/b}(R)", "39e9047e28e5ce9cad45b7f1f14bf9b3": "\\{ f_s \\}_{s\\in S}", "39e917dd416613bdb7aae443d0ba51fc": "_{p\\tilde{\\leftarrow}q}\\!", "39e9795162d48c441a8c0b49f36d3adc": "L(\\theta|X,Y) = \\prod_{i=1}^m \\frac{e^{y_i \\theta' x_i} e^{-e^{\\theta' x_i}}}{y_i!}", "39e97cc9c619ccc21f409f8aed9c99f5": "\\Delta E\\ = \\pi r^2\\gamma_{OW}(1-|cos{\\theta_{OW}}|)^2", "39e9a7a20d25a7eb4c2ccedb4a37ea92": " \\qquad \\qquad E_b = \\int_0^\\infty d E_{b,\\lambda} = \\sigma_\\mathrm{SB}T^4\\ \\mathrm{, \\ where} \\ \\sigma_\\mathrm{SB} = \\frac{\\pi^2 k_\\mathrm{B}^4}{60 \\hbar^3 u_{ph}^2} \t\\ \\ \\ \\ ", "39ea089e2ac058db08ee4677d43edc40": "(s_2, s_3)\\in R^k", "39ea40d7e55bc0cf998281bc80f73668": "\\bar{R}=\\frac{I_1(\\kappa)}{I_0(\\kappa)}\\,", "39ea4293fce8bf38aa450b9acd066da0": " \nr_{d,s} = h_{d,s} x_{s} + n_{d,s} \\quad\n", "39ea60acfb310f9588a2342325f62eb5": " P ( \\text{ one hit in B } ) = \\lambda |B| e ^ {-\\lambda|B|} \\rightarrow \\lambda |B| \\text { as } |B| \\rightarrow \\infty", "39eac8ee53d0f9e4f74b44aab300ff89": "\\frac {G/D}{A5/E}", "39eb0477843bdb8e8cddd3e045c68222": "\\det \\left( 1-z\\mathcal{L}_w\\right)", "39eb128cbb659e434ad0f0284ea30ec0": "1-\\epsilon", "39eb5874ca1fe3922c32781c0b8f29c9": "\\eta = \\frac{\\eta_{c}}{\\Iota}", "39ebc8d6c7c2c263c8fe5c8a07c753ec": "\\frac{1024}{729} \\sqrt[4]{2}", "39ec2c18830d1ad375ae3e6d7cb1a81f": "L_M", "39ec357b788a3fcee34690a6c7fcc65c": "\n \\eta = \\sum_{n=1}^{\\infty} A_n\\, \\cos\\, (n\\theta).\n", "39ec4774f124fbe3c381cbcb4403d900": " \\lim_{t \\rightarrow \\infty} e^{-t} (\\mathcal{B} A)(zt) = e^{t(z-1)} = 0. ", "39ec623e9caea727ec73f76bfb74e41a": "S_{22} = \\frac{-T_{21}}{T_{22}}\\,", "39ecfea6491eed68e4940958dfa3ed4a": "g \\circ f \\colon I \\rightarrow \\mathbf R, x \\mapsto g(f(x))", "39ed3bc338f86dea7292abe0d2fa7e9d": " {N \\hbar \\omega \\over V} ", "39ed55c7fa7def35d4d5065b4f01bb27": " q \\equiv ( \\mathcal{B}q \\to p) ", "39ed7eda69dd6486a20401609cdcb199": "P(7,4) = P(7,3) + P(6,4) P(7,2) + 2P(6,3) + P(5,4)", "39ee0366d86b16efc8c87cf7844cb970": "r_e = {e^2 \\over 4\\pi\\varepsilon_0 m_{\\mathrm{e}} c^2} = \\alpha {\\hbar\\over m_{\\mathrm{e}} c} \\approx 2.8 \\times 10^{-15}\\ \\mathrm{m}.", "39ee18367ba7358b90d58b10d7475a22": "{\\mathbf l} \\; \\leftrightarrow \\; {\\mathbf l}^{'} \\; \\leftrightarrow \\;{\\mathbf l}^{''}", "39ef5c416b8a3c86de79e1ec51ab6f38": "s,t\\in T", "39ef7f3183ca61c4f97cb1bbb48a9877": "\n \\langle j_1, m_1; j_2, m_2 | 0 0\\rangle = \\delta_{j_1,j_2}\\delta_{m_1,-m_2}\n\\frac{(-1)^{j_1-m_1}}{\\sqrt{2j_2+1}}.\n", "39efc816ef6eca9944d95eaef503e6d2": "\\Gamma*\\mu(x) = \\int_{\\mathbb{R}^d}\\Gamma(x-y) \\, d\\mu(y)", "39efebdafd28a537d3ac3a4b9cec99f4": "a(n) = 3a(n-1) + 4a(n-2)\\,\\!", "39eff34b619f76c87ec33a73bad1742f": "\\left( \\overline{u_j} + u_j' \\right) \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right)}{\\partial x_j} = \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right) \\left( \\overline{u_j} +\nu_j' \\right)}{\\partial x_j} - \\left( \\overline{u_i} + u_i' \\right) \\frac{\\partial \\left( \\overline{u_j} + u_j' \\right)}{\\partial x_j}, ", "39f016a5797fb79a9dc6f7f032c7b6e9": "\\scriptstyle v_i(U)\\leq v_k(U)\\!", "39f054ed660a92998b67d82a43e2a01a": "(\\neg A)\\lor(\\neg B)", "39f0b6ec7b9e84ffd6083912484e113f": "L^{\\lambda,p}(\\Omega)", "39f0e58238a218d9b464cd3788cff168": "(x',k')", "39f17f2e9005f33655bd0b9636136990": "\\text{hom}(-, -) : C^{op} \\times C \\to C", "39f192054c1b134113d18f6b937c6c4b": "dy = \\frac{\\partial y}{\\partial x_1}\\,d x_1 + \\cdots + \\frac{\\partial y}{\\partial x_n}\\,d x_n.", "39f1a4616319a2eb9793cdd533bf96a0": " \\omega = \\frac{2}{T} \\arctan \\left( \\omega_a \\frac{T}{2} \\right). ", "39f1be227d500fcbf34d21a2b4c84dc1": "((0,0,0), (1,1,0), (0,1,1), (1,0,1))\\,", "39f1d90283e2eaabdd1a03f63f6b394e": "\n G_\\mathrm{bin}(x;p,N) \n \\equiv \\sum_{k=0}^{N} \\left[ \\binom{N}{k} p^k (1-p)^{N-k} \\right] x^k\n = \\Big[ 1 + (x-1)p \\Big]^{N}\n", "39f24f3a16701553d980fd9680f52541": "\\mathbf{IC} = \\frac{\\displaystyle\\sum_{i=1}^{c}n_i(n_i -1)}{N(N-1)/c}", "39f265e1ba03e757f71b5606be3d6cb1": "H^+, H^-", "39f2a481adf8e783bac183ea2be24fb0": "Q(x,y) = \\sum_{j=0}^l \\sum_{k=0}^{m+(l-j)d} q_{kj} x^k y^j", "39f2decec514bbcb030c563c20c888d1": "W_i = V, W_{1-i} = \\{\\}", "39f310aa3bccf217c90a6d7376e61136": "n=\\log(q)", "39f35a7a61cdf35fef8c6d436b6aca40": "D_{1}", "39f38a70fc705ea5b3b1d3d85de983fc": "X\\overset{\\underset{\\mathrm{A}}{}}{\\sim}((1-\\lambda)X,\\lambda X)", "39f3bd89eeb59ddd08c452eb03fe4359": "\\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \nf( (x+ka) \\mod 1 ) = \\int_0^1 f(y)\\,dy ", "39f3cb0020f02966808fef46d41cef22": "= 2u_{1}du_{1} - \\phi(x,u,u_{1},u_{2})dx + u_{2}du \\,", "39f3ccbdd763ce1bc07876b6e5a73b1a": " (n,n) ", "39f44987b74fbfed85cc30aaae7591b0": " ds^2 \\, = (-N + N^i N^j \\gamma_{ij}) dt^2 + 2N^i \\gamma_{ij} dt dx^j + \\gamma_{ij} dx^i dx^j", "39f48154eefd25655d94b497a9a92df5": " \\left( 0 \\le \\epsilon < 1 \\right) ", "39f4846771b7bf2aee2c328bfa10808d": "\nP_\\nu~d\\nu = \\frac{h^3}{N}\\left(\\frac{Vf}{\\Lambda^3}\\right)\n\\frac{1}{2}~\\frac{\\beta^3\\nu^2}{e^{(h\\nu-\\mu)/kT}-1}~d\\nu\n", "39f4caf236a48b676da8c55d254011ed": "\\mathbf{A}\\cdot{\\rm d}\\boldsymbol{\\ell}.", "39f50bbe5615bcdf44d30715b7047ebf": " \\langle u \\bar{u} \\rangle_S, ", "39f54490dc4efa0390e40709c55b4657": "MAP = (CO \\cdot SVR) + CVP", "39f54e8d5d46e17dce6cdc3beb57d120": "\\zeta(1) = 1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots = \\infty\\!", "39f5af420b871cfb401b0965cb8082c6": "t=0,1,2,\\ldots,T", "39f5b334cfff4d6e7698dac9dc343b48": "\\sigma_{ij} = c_{ijk\\ell}~ \\epsilon_{k\\ell}", "39f5c299b71ab132f48d37fd0f00b746": "g(z)=\\frac{z^2}{(z-z_1)(z-z_2)}", "39f5de531dc09c822ecf1c199d247f5e": " (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)", "39f61c9d395e7a6362749c0939ad1495": "\\frac{\\varepsilon_0 }{2}E^2", "39f66aaa7f0a094e028c1f601b4725e2": " \\text{Assets} = \\text{Liabilities} + \\text{Capital} ", "39f6a020334a7999df750982fdf60305": "\\sum_{n \\ge 1} \\frac{1}{n^2}", "39f6d82d30094c72620003c20968d4ca": "x_1^2 = b_1", "39f6d8f59aea314e0592440cbb7b10ff": "\\Delta (f g) = f \\Delta g + 2 \\nabla f \\cdot \\nabla g + g \\Delta f", "39f6dc4a810cbcf079f7544e2e99604c": "\\langle u,\\ v \\rangle = 0 \\,", "39f700021f25f36d94cb2c46090320e2": "G_{\\mu \\nu}", "39f70d3d648e1d25ba24b39ba641d62a": " g - g'", "39f78513b57e831d2a198a06b72935ab": "\\textstyle l_2-1", "39f7b482eb5f1412347f4764d93973ba": "V_S \\,=\\, \\sum_i V_i", "39f7e425f9be59dbb51ad667c325c67b": "G = ({\\rm GF}(q), +) \\oplus ({\\rm GF}(q+2), +)", "39f810d5fddb9f363978d5971e3c64c3": "EL(\\Gamma)=w/h", "39f893feccb5e8ca307d0ad9cc36a43c": "(t,x) \\rightarrow P_tf(x)", "39f90ff3c16f34aaed50c72ccbc58276": "Q \\neq 0,", "39f952375bdf68318ec317d252144ccb": "P_{x^\\nu}", "39f9a9b1b58dffae30a3f4aebb2505a3": "\\widehat{T}^{(1)}_{ij} = \\frac{\\widehat{a}_k \\widehat{b}_k}{3}\\delta_{ij} ", "39f9f506ae298571825be2610f3ad4b9": "\\nu_{peak}", "39fa4730b70f69ceb13660a26e6887aa": "P(X_N - X_0 \\geq t) \\leq \\exp\\left ({-t^2 \\over 2 \\sum_{k=1}^N c_k^2} \\right). ", "39fae02f1d597ffb27a5932582dc824d": "\\lambda x\\!:\\!\\sigma.~t\\,x =_\\eta t", "39fb2451db28585bc2c5c02e5b33491e": "|n^{(4)}\\rangle=\\Bigg[\\frac{V_{k_1k_2}V_{k_2k_3}V_{k_3k_4}V_{k_4 k_2}+V_{k_3k_2}V_{k_1k_2}V_{k_4 k_3}V_{k_2k_4}}{2E_{k_1 n}E_{k_2k_3}^2E_{k_2k_4}}-\\frac{V_{k_2k_3}V_{k_3k_4}V_{k_4 n}V_{k_1k_2}}{E_{k_1 n}E_{k_2 n}E_{n k_3}E_{nk_4}}+\\frac{V_{k_1k_2}}{E_{k_1 n}}\\left(\\frac{|V_{k_2k_3}|^2V_{k_2k_2}}{E_{k_2k_3}^3}-\\frac{|V_{nk_3}|^2V_{k_2 n}}{E_{k_3 n}^2E_{k_2 n}}\\right)", "39fb5d090fc75e4e92ffac369239b3db": "\\Delta S \\rightarrow S_z", "39fb83f4be548857988a72630bf70886": "\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\int_{-\\infty}^{\\infty} |\\phi|^2 dxdydz", "39fb967328f5f9072e76722dc325e8e2": "\\int \\cos^2 x \\, dx.", "39fc3b36603a702ddc3bb9574f4c1a40": "E_{xy,3z^2-r^2} = \\sqrt{3} \\left[ l m (n^2 - (l^2 + m^2) / 2) V_{dd\\sigma} -\n2 l m n^2 V_{dd\\pi} + l m (1 + n^2) / 2 V_{dd\\delta} \\right]", "39fc47f9d158ff93d3e2e75fef9cdc24": " 1 \\leq t^a(d,n) \\leq t(d,n) \\leq n ", "39fc8012c1c2dbd07dae5473d21914b2": "\\int \\frac{x^m\\; dx}{(\\ln x)^n} = -\\frac{x^{m+1}}{(n-1)(\\ln x)^{n-1}} + \\frac{m+1}{n-1}\\int\\frac{x^m dx}{(\\ln x)^{n-1}} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}", "39fcc7718fa01ebd411fd33a8d132219": "\\phi \\wedge \\psi", "39fcd2c59929444e69781dae76b33cee": "B \\rightarrow S: \\left. B, A \\right .", "39fcd3c4fb9654c4f2f08413bd3b34bf": "H^r(X; \\mathbf{Z}),", "39fd00f5f0bb8ed79d077b81c0b5d622": "\\bar{\\mathbf{e}}^j = (\\boldsymbol{\\mathsf{L}})_i{}^j \\mathbf{e}^i = \\mathsf{L}_i{}^j \\mathbf{e}^i", "39fd34949dace1fc9b1078bace92546f": "x (av+bw) = a (xv) + b (xw)", "39fdabbc41e9da8d0d9908a66abae6e2": "UL = I - \\operatorname{diag}(0,\\ldots, 0,1),", "39fe132d9ccfa3762df2da5e509cab42": "\\frac{dx}{dt}=a_1x+b_1y,\\quad\\frac{dy}{dt}=a_2x+b_2y", "39ff25814859a6e67de17a96663d8c09": " F \\to E \\to B", "39ff2d4efdc5eb931b8901d8b02e37a3": "\\left(\\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ \\sqrt{1/3},\\ \\pm1\\right)", "39ff9bfa7c9997e169fd558b34794873": "\\scriptstyle C_2", "39ffaf7c307a99d923c2c19034459c39": "\\Rightarrow^{*}", "39ffd07950070c1ca3508f8066f58702": "t' = E_K(P_n' \\oplus E_K(P_{n-1}' \\oplus E_K( \\dots \\oplus E_K(P_1'))))", "39fff0940c0a7635f7310897d8e743e9": "\\frac{\\pi}{4} = \\arctan \\frac12 + \\arctan \\frac13,", "3a000b1be919c60c409df9c302ce49ef": "\\mbox{Power in dBW} = 10 \\log_{10}\\frac{\\mbox{Power in W}}{ 1 \\mathrm{W}} ", "3a001186940cb5e3bd6e74d80e3beb64": " \\rho_B(t) = (|\\alpha|^2 + (1-\\eta) |\\beta|^2) | \\Downarrow \\rangle_B\\langle \\Downarrow | + \\eta |\\beta|^2 |\\phi_1^{\\prime}\\rangle_B \\langle \\phi_1^{\\prime}|+ \\sqrt{\\eta} \\alpha \\beta^* | \\Downarrow \\rangle_B\\langle \\phi_1^{\\prime} | + \\sqrt{\\eta} \\alpha^* \\beta | \\phi_1^{\\prime} \\rangle_B\\langle \n\\Downarrow | ", "3a0026ea5f1679b315fddb5385e7f310": "n! [u z^n] g(z, u) = n! [z^n]\n\\exp\\left(\\sum_{k=1}^{\\lfloor\\frac{n}{2}\\rfloor} \\frac{z^k}{k}\\right)\n\\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty \\frac{z^k}{k}", "3a00538b4c179923843d516f110e26bf": "\nZ(z) =\\Re \\left \\{ \\frac{1}{k-kz^k} \\right\\}\n", "3a00f7d60f98b3d885aa71a00994e2d2": "\\operatorname{Ric} = \\lambda g", "3a01217af61e2cbdc5b0e40658f6d63c": "C^1(I)", "3a013c266fe893e8abcfdfd2714e2e0e": "\n\\begin{align}\n\\sin(\\tfrac{\\pi}{2} - \\theta) &= +\\cos \\theta \\\\\n\\cos(\\tfrac{\\pi}{2} - \\theta) &= +\\sin \\theta \\\\\n\\tan(\\tfrac{\\pi}{2} - \\theta) &= +\\cot \\theta \\\\\n\\csc(\\tfrac{\\pi}{2} - \\theta) &= +\\sec \\theta \\\\\n\\sec(\\tfrac{\\pi}{2} - \\theta) &= +\\csc \\theta \\\\\n\\cot(\\tfrac{\\pi}{2} - \\theta) &= +\\tan \\theta \\\\\n\\end{align}\n", "3a01495cd1ff8a982aca5ef9b312e155": " X \\cdot Y := \\eta_{\\mu \\nu}X^\\mu Y^\\nu ", "3a01506e5f9da7cf96340718ab0b9829": "\\Gamma, \\Pi \\vdash B", "3a0162068869e4947ae990fe8c6101e8": " NCD_Z(x,y) = \\frac{Z(xy) - \\min \\{Z(x),Z(y)\\}}{\\max \\{Z(x),Z(y)\\}}. ", "3a0190acebf7074e3bc9818c8b5561ce": " v^*(w) := \\langle v, w\\rangle.", "3a0195fdbdec1ad63514af4d082005c7": "\\textstyle |", "3a01a06bd56fb5f9369af0dd36afcd10": "_a^b\\text{S}^\\gamma", "3a01e7258c1540b2549c18c63855fdc4": "i^*:\\Omega_p^1(M)\\rightarrow \\Omega_p^1(N)", "3a02394874fd592deee946eb617f7850": " 32/27 d^2 h = 128/27 r^2 h ", "3a02437e01a7c75cd9e5104637f4faba": "\\mathbf{AXB} \\sim {\\rm T}_{n,p}(\\alpha,\\beta,\\mathbf{AMB},\\mathbf{A}\\boldsymbol\\Sigma\\mathbf{A}^{\\rm T}, \\mathbf{B}^{\\rm T}\\boldsymbol\\Omega\\mathbf{B})\n.", "3a0275c3303ad6baad056e0dd95f7907": "\\sigma(n) = 2 \\times 2^k - 1 = 2n - 1,", "3a029a8af1a5bd3fcb057e1ca9cba00c": "d(i, j) = 0", "3a02afeba140c7ecf20468f0adf8e8f5": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\chi + (\\tilde{u}_{1}- E)\\chi + i\\frac{\\hbar^2}{2m}[2 \\mathbf{\\tau}_{12}\\nabla + \\nabla\\mathbf{\\tau}_{12}]\\chi + i ({u}_{1} - {u}_{2} )\\psi_{2} = 0 ", "3a02ba173637bf4961eeabca04688b1a": "r\\leftarrow q", "3a02cd512065660143f45fa9338a1729": "\\operatorname{E}_{i \\neq j} [\\ln p(\\mathbf{Z}, \\mathbf{X})]", "3a02ddc02cd26080f4d1f087d69b82ef": "\\Gamma\\vDash A", "3a02e4b614dba75a3abb8b18c98e03bb": "{m \\choose r}_q=\\frac{[m]_q!}{[r]_q!\\,[m-r]_q!}\\quad(r\\leq m),", "3a02eec4f4ad584fcdf359c95e0123fa": "\\frac{10\\times9}{2}", "3a036945a4f4bd6af33d6205191dfc2a": "\\underline{\\varphi{(\\beta / \\alpha)}}\\,\\!", "3a036d03c5820bf935e9e9819747b605": "\\mathfrak{g} = \\mathfrak{s} \\oplus \\mathfrak{a}.", "3a038a9fd2b6b4c22c034b762897a965": "\\left(\\frac{7}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "3a03cf459168fc257b299a94ebb46531": "\n\\frac{1}{\\omega_{ci}}\\frac{\\partial}{\\partial t}\\left(\\rho_s^2\\nabla^2\\frac{e\\phi}{T_e}-\\frac{e\\phi}{T_e}\\right)-\\left[\\left(\\rho_s\\nabla \\frac{e\\phi}{T_e}\\times \\mathbf{\\hat z}\\right)\\cdot\\rho_s\\nabla\\right]\\left[\\rho_s^2\\nabla^2\\frac{e\\phi}{T_e}-\\ln\\left(\\frac{n_0}{\\omega_{ci}}\\right)\\right]=0.\n", "3a0464d97991f9bc34eb5e3d647380aa": "s_1(z)=(1-z)^\\mu (1+\\mathcal{O}(1-z))", "3a04a463bbabcb6c269833e898c56eeb": "\n2T_\\mathrm{vib} = \\sum_{A=1}^N M_A \\dot{\\mathbf{R}}_A\\cdot \\dot{\\mathbf{R}}_A\n= \\sum_{A=1}^N M_A \\dot{\\mathbf{d}}_A\\cdot \\dot{\\mathbf{d}}_A.\n", "3a04c8937de52e2a9a05ea392c963a6b": " e^{\\pm i \\frac{r}{\\hbar} \\sqrt{2 m E} \\theta } = e^{\\pm i \\frac{r}{\\hbar} \\sqrt{2 m E} (\\theta +2 \\pi)}", "3a04cf6ab581ddd12153a5ed760146a6": "H^1(G_k, X^\\bullet(T)) \\cong Ext^1(T, \\mathbb{G}_m)", "3a04daefa8f302fd61233c76bc387e42": "\\|f*g\\|_{r,w}\\le C_{p,q}\\|f\\|_{p,w}\\|g\\|_{r,w}.", "3a050d172ec95512420947e341eba080": " \\left|\\int_G f(g) h(g)\\, dg\\right| \\le \\|\\pi(f)\\|", "3a052c7572c1137980c9401efcf98c8e": " f''(x) \\approx \\frac{\\Delta_h^2[f](x)}{h^2} = \\frac{f(x+2h) - 2 f(x+h) + f(x)}{h^{2}} . ", "3a059ba700cea79483e8fc7a94f95e1b": "y = 345", "3a05a5621f2fb4fbbcc87f40b2a76035": "0k\\}", "3a114c967b654a40bbb8750a7ace9571": "p_R(r) = \\frac{2r}{\\Omega}e^{-r^2/\\Omega},\\ r\\geq{}0", "3a114de25250b2b6b1dd9febbd7ef4c9": "p(c)\\ ", "3a116227bba8f531d8cecf3ca0df47a2": "P\\,\\xrightarrow\\tau\\,P'", "3a11983a48759b4a9ba85c1ce897ecad": " 0 = 1 \\,", "3a123cc3b3169ebd79a1f527cec6a01e": " X\\sim U(a,b) ", "3a12581f254b4cef7076594bc0ed326e": "z_1^\\times", "3a125a7627388151ece758e3ec6ddb44": " S = \\int_k {1\\over 2} k^2 \\phi(k) \\phi(-k)", "3a1279318979ddfca16de6026302cce7": "{2a_{3} \\times b_{3} \\over a_{3} + b_{3}+c_{3}}=d", "3a12df3c914e9071375fb4f72b1fdbc5": "f(y) = f(x) + f'(x)\\cdot(y-x) + f''(x)\\cdot\\frac{(y-x)^2}{2!} + f'''(x)\\cdot\\frac{(y-x)^3}{3!} + \\dots ", "3a12f439811e7d1b4ffa2a18cac4caf4": "\nE[U|z] = \\max_{d\\in D} ~ \\int_X U(d,x) p(x|z) ~ dx\n", "3a13b34986cc6110d3eea2d15a42b70c": "( e^t, \\ e^{-t}).", "3a13cd0ab09e0a46dca1b575bb2e32c5": "\\mathcal O_{X, z}", "3a1401648b2712b71613773501773b3c": "SU(3)", "3a141607d33940786a429f62a26faf52": "\\delta-\\epsilon", "3a142ef30936a22add9cab81eaee6078": " \\delta(x-y) = \\int e^{ik(x-y)} dk ", "3a143fe6b5c1e77905ab67d706e7aa13": "\\langle \\cdot,\\,\\cdot \\rangle", "3a1448b65a02b5536406464671f04c10": "M'\\in Mod(\\Sigma')", "3a1488ef0cd93863882c7d7baa3d4440": " \\mathbf{y}' = \\mathbf{H} \\, \\mathbf{C}_{N} \\, \\mathbf{x}' ", "3a14a55a6a396e0102e27e9c8d835a37": " I = I_0 \\left [ \\frac{ \\sin \\left( N \\delta/2 \\right ) }{\\sin \\left( \\delta/2 \\right )} \\right ]^2 \\,\\!", "3a1508a3a6d17a100a41c6c87f75367f": "Z_C", "3a1550413f940f4c04465c826154a376": "\\min(1000-0, 1-0, 1000-0)=1", "3a15b073bdb7b7af6e9638767a139eac": "O(k n + z)", "3a15bc10cf8f7ede033ee6013b2f96ed": "\nX_t=\\sum_{k=1}^\\infty Z_k e_k(t)\n", "3a15e697044df5b1d4e317a5785a8d0b": " \\sigma_1 \\,\\!", "3a160796efe299af3cf0bcbbf5bfee88": "f^*(\\theta)=v(p,w)", "3a162a39a1689e9b448d0b582b31ad7a": " \\int_{x_1}^{x_2} r(x) u_1(x) \\varphi(x) \\, dx=0. \\,", "3a164045cc67603c36d68e5e2dce95f1": " = - \\sum_{i=1}^n \\mathbb{E}_\\theta \\left[ \\frac{1}{|\\mathbf{X}|^2} - \\frac{2 X_i^2}{|\\mathbf{X}|^4} \\right]", "3a1649cf388e354885f4692701d987ba": "\\Delta(t) = t^2 - 3t + 5 - 3t^{-1} + t^{-2}, \\, ", "3a1680f6368892226aac031612277e6b": "\\scriptstyle \\Psi \\partial \\Psi^* / \\partial t \\,\\!", "3a1700f0bd2eaa03baaa0cc2a60fdc33": "\\tilde{\\phi}_a", "3a170f08adcc3de1bdfe47217182b609": "3m'^2=n^2", "3a17246f3599e24b9471d16d7e568644": "c\\cdot t(n)", "3a1755a9e1b6331c1014ea5756abd48f": " d_p = \\frac { CI } { 2p } ", "3a1759051a96c25f773b5116c4c9ddfe": "\\prod_p^{\\infty} \\left(1-\\frac{1}{p^2}\\right) = \\left( \\prod_p^{\\infty} \\frac{1}{1-p^{-2}} \\right)^{-1} = \\frac{1}{1 + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots } = \\frac{1}{\\zeta(2)} = \\frac{6}{\\pi^2} \\approx 61\\% ", "3a17c1f21bfc61699e3bc889d83dd1ce": "\\hat{e}_{k_{1}},\\dots,\\hat{e}_{k_{n}}", "3a17de1c6324700fb0b66533c5d1cf84": "\\{ v, Tv, T^2v, \\ldots, T^kv, \\ldots\\}", "3a17f1537c0f46f468afac657834fa22": "6^3=216", "3a17f57d9af78403b7ac2dd5f82c2d3c": "n>0", "3a18042e3a9708beb216b5f8ffab406b": "Y = \\sum_{i=1}^N k_i (X_i-\\mu)\\,", "3a18281a2852a3c97af8f87a4048a21f": "\\frac{1}{[A]} = \\frac{1}{[A]_0} + kt ", "3a1834872c6e4030224d425f41f813c0": "G_C \\to 0", "3a18480b1b1f935b1ec12b3dcd3be25a": "\\mathbf{S} : V \\rightarrow V", "3a18621136ba0415039ea7fbf8501a07": "\\widehat{di}=Z_{i}\\gamma+\\varphi_{it}", "3a1870dd4e2bc043978cf6184ae5240d": " \\mathbf{F} = \\nabla \\times \\Psi \\mathbf{r} ", "3a18f5349d0fe0e7adc940e109f14f60": "\\leq 0", "3a19b1bfa237afc4c0801268939dc4ca": "E(\\varepsilon_i)=0, ", "3a19cac7554dbfdd00d000e5d8dad519": "\\frac{\\partial g(u)}{\\partial \\mathbf{X}} =", "3a1a9d1c5d4d4920b0b2457e939f39a6": "p = p_a p_b + S \\cdot q_a q_b\\,\\!", "3a1aaf3191e33f32a37b8f12db748008": " \\mu=\\sum_{n=1}^N \\delta_{X_n}, ", "3a1b07cb6c100dfe78e2e394f6010dea": " \\Phi = -4 \\pi G_\\mathrm{eff} \\frac{a^2}{k^2} \\delta\\rho_\\mathrm{m} ", "3a1b79a4fe065ea5d97a5129c1f943f7": "\\lambda_1,\\lambda_2", "3a1bebf8caee9d9d485e6aaaba13df47": "\\boldsymbol{\\epsilon}", "3a1bfb5b300a6846e35edb23ea2534ba": "||(a, b, c)|| = \\sqrt{a^2 + b^2 + c^2 + ab + bc + ca}.", "3a1c76ecaf9a7987f9dd3a4043587bb1": "(\\lambda\\cdot f)[x_0,\\dots,x_n] = \\lambda\\cdot f[x_0,\\dots,x_n]", "3a1cbc720df393bf0f75edbfef15bf9f": "ax^2 + bx \\pm c = 0 ,", "3a1cf606e43a5eb26cfb4df0d454c8f8": "\\Phi_E = ", "3a1d02b033cb804a16f9c2fba3331272": "\\mathbf{A}^{\\mathrm{T}} = \\mathbf{A}^{*} .", "3a1d11e14a71edc9c299f9136f90cdfe": "y^2=x^5-x", "3a1d23d2e678bf763000ad68287d821a": "H=UA", "3a1d2b6cdc2f8b0a681005bf082ef251": "\\ln \\Gamma(z+1)= -\\gamma z +\\sum_{k=2}^\\infty \\frac{\\zeta(k)}{k} \\, (-z)^{k}\\qquad \\forall\\; |z| < 1", "3a1d70cfadcaeb26f9e15a48b86f92c5": "\\phi = A\\,\\!", "3a1d7565a7e56da51290c15162b81cae": "d = t_0 - t_f", "3a1d7e499f7e3b22847bb1193f67a71a": "x\\mapsto L_x", "3a1daa9cfef8cc05be9334f155af98e5": "\\mathbb R^{2^n-1}", "3a1dce31a0bf29a6fcc8378e5aa24add": "\\rho\\mathcal{V}", "3a1dec638811d51c502be52d3cb7c85a": "\\ \\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B}} {\\partial t}\\ ", "3a1e2c23df02021b83600f19c5d4f20c": " \\omega \\ ", "3a1e3aa0202f08e96a8e6089c499d6d8": "f_{av} = \\frac{\\sum N_i \\sdot f_i}{\\sum N_i}", "3a1e7dbbc8f1504169216863eb0684e9": "M= \\begin{pmatrix} a& 0 \\\\ b & a^{-1} \\end{pmatrix}", "3a1ec4b15ec083b10a291537c9ca0637": "B_{m,n}^k = \\bigoplus_{i=1}^n \\R^k[x_1,\\ldots,x_m], \\implies \\dim\\left\\{B_{m,n}^k\\right\\} = n \\dim \\left\\{ A_m^k \\right\\} = n \\left( \\frac{(m+k)!}{m!\\cdot k!} - 1 \\right) . ", "3a1efe59a7b80988fef03b8374a717e2": "z_j\\,", "3a1f53d8565b247d013a649c2ff0eb95": "\\Theta = \\frac{\\theta - \\theta_r}{\\theta_s-\\theta_r}", "3a1ffad437fb95d56b08b1231b052c26": "V(x) =\n\\begin{cases}\n0, & 0 < x < L,\\\\\n\\infty, & \\text{otherwise,}\n\\end{cases},\n", "3a2024147a95e43d6150104cc4022e25": "\\scriptstyle\\hat\\ell(\\theta|x)", "3a205c03fe78c61bf2c6657674a7a2f1": "\\frac{b\\Gamma(1+\\tfrac{1}{a})\\Gamma(b)}{\\Gamma(1+\\tfrac{1}{a}+b)}\\,", "3a207c091558c6fb5dcb92d1b145f8d7": "f: A \\to k", "3a20852bd42744d32749dd0ef9971644": "\\sum_j n_j Reactant_j\\rightleftharpoons \\sum_k m_k Product_k", "3a20b90a53542d36982aa298d6ec17a6": "\\{\\psi_{k,n}(x)=\\sqrt2^k\\psi(2^kx-n):\\;k,n\\in\\Z\\}", "3a20e1f52b12b63b547a8eb394427d99": "\\scriptstyle p \\Rightarrow q", "3a20eac460ee16ec21e81de46df4ef6b": "\\sigma\\, a\\; \\text{e}^{\\displaystyle k\\, z}\\, \\sin\\, \\theta\\,", "3a21cd7317e1445be89999f5d7f62a53": "pub", "3a21e99d56941dcf440b139dd52e97e6": "\n\\begin{align}\n\\tan\\lambda_{01} &= \\frac{\\sin\\alpha_0\\sin\\sigma_1}{\\cos\\sigma_1},\\\\\n\\lambda_0 &= \\lambda_1 - \\lambda_{01}.\n\\end{align}\n", "3a21f22a276a8e64e556cd6e577aaae4": "f: X \\to S", "3a220486c5e7f9e8aed3fdf7e802a3c8": "p(x) \\equiv b_0 + b_1x + \\ldots + b_nx^n", "3a22144773cc9b4c43280315f3ff2022": "X[\\sigma] \\to Y[] Z[\\sigma f]", "3a223a9f61f3b6c110394dc2e6d4217c": "\\frac{dx}{dt} = kx", "3a22a92b4f1012bf095c6c7fb0a0953f": "c\\rightarrow \\infty", "3a22e414b61a36a1804a57efaf4e983f": " f^{*} = p - q . \\! ", "3a22fa43fb1214f98a0b02e04caba980": "Y \\in \\mathfrak{h}", "3a234ad38aa0f413c32fefa0d0ecbb21": "\\rho = \\frac{M_s}{t_V}", "3a236f1c246f60116ccc4310bd7ef27d": "\\Lambda^{p,q}\\,V_J\\;\\stackrel{\\mathrm{def}}{=}\\, (\\Lambda^p\\,V^+)\\otimes(\\Lambda^q\\,V^-).", "3a2389642eea022bbc9dd2c062eb61ad": "\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_1)(\\sigma_\\mathrm{n} - \\sigma_2) \\ge 0", "3a23f2ac136bf023be7ebc6382604435": "\\sigma = \\sqrt{\\frac{ B - N D^2}{N-1}} \\,", "3a23f903a8c6c95599147c8d7597455f": "\\underset{i}{\\overset{2}{x_j}}(t)", "3a245eca23b90c7249cdb251be9d2e20": "((x_{i}, l_{1}, ..., l_{j-1}), l_{j}), l_{j} \\in \\{0,1\\}", "3a246cf84b93583e0951acbfd68cdf08": "\\partial C", "3a24fc0ff0fb396b6e185c9083cf140c": "j(\\tau)", "3a25041a925e09a1e14287e3cf386ec8": "\\lceil \\log_2 1,000,000 \\rceil = 20 ", "3a25125d0703b0a563c1ad26916a7d4e": "\\mathbf{A}^{(i)}=\n\\begin{pmatrix}\n\\mathbf{I}_{i-1} & 0 & 0 \\\\\n0 & a_{i,i} & \\mathbf{b}_{i}^{*} \\\\\n0 & \\mathbf{b}_{i} & \\mathbf{B}^{(i)}\n\\end{pmatrix},\n", "3a2596dd92cece78d24e3c984b7727aa": "z_7=\\chi_{\\psi_{7,7}}(z_7,\\rho_{\\psi_{6,7}}(z_6))=\\chi_1(z_7,\\rho_{6}(z_6))=x_1q_1e^{-x_2}", "3a25a74ff7696becc95a216875d89264": "\\begin{align}\nA = M_R^{-\\tfrac{1}{2}} R M_L^{\\tfrac{1}{2}} \\\\\nx_R^' = M_R^{\\tfrac{1}{2}}x_R \\\\\nx_L^' = M_L^{\\tfrac{1}{2}}x_L \\\\\nx_L^' = R x_R^'\\\\\n\\end{align}\n", "3a26495d8554d0f00cc651f05058facf": "\\mathcal{B}(M) ", "3a26ad0a38d96d157ebfd586299b8a40": "E_{m,n} = 2E_{m,n-1} + E_{m-1,n-1} \\!", "3a276f6f0b9df4014a00f7e56acfa676": "\\left [\\begin{smallmatrix}2&0\\\\0&2\\end{smallmatrix}\\right ]", "3a27a5bf7a77360a436d689f08f93155": "\\scriptstyle f(x)=x(1-x^{2a})", "3a27bbf7ab4971b908c1aa47e1fe0f93": "\\frac{E_1}{E_2} = \\frac{k_1}{k_2}. \\,", "3a27cc5ec88eaf60abf507e4783b9a3d": "\\mathbf{r}_i=\\mathbf{R}+\\mathbf{R}_i\\,", "3a27da81d4cb5ddcb3956d360d16119a": "U_n := U \\times_X U \\times_X \\dots \\times_X U", "3a27f91e85479b3f037e21232bc29e20": "\\begin{align}\nh(x_1^n)= 1, \\quad\ng_{(\\alpha, \\beta)}(x_1^n)= \\left({1 \\over \\beta-\\alpha}\\right)^n \\mathbf{1}_{ \\{ \\alpha \\, \\leq \\, \\min_{1 \\leq i \\leq n}X_i \\} } \\mathbf{1}_{ \\{ \\max_{1 \\leq i \\leq n}X_i \\, \\leq \\, \\beta \\} }.\n\\end{align}", "3a27fabe9f21657d5ae0a761d9e0a16f": "\\int_{-\\infty}^\\infty \\varphi(t - k) \\cdot \\varphi(t - k') \\, dt = 0", "3a283f6901c7d3888ab4dc43d0e4dc65": "\\scriptstyle \\gamma^\\mu", "3a2866050eb07dc1d609bbe3d54f9d49": "q(\\sum x_i e_i) = \\sum a_{ij} x_ix_j", "3a286c94081cc120c0a8350305626bcb": "\nZ = \\sum_{k=1}^{N} \\left| \\frac{d^{2} \\mathbf{r}_{k}}{dt^{2}}\\right|^{2}\n", "3a2889ed96f5736d84e1f2f0905e2ae9": "\nL^{2} = L_{1}^{2} + L_{2}^{2} + L_{3}^{2}\n", "3a28a8ed213a990ad03bdcbd43b2a040": "e^{\\frac{-||x_i-x_j||^2}{\\epsilon}}", "3a28efcbc2f958c616f914a68df3c1b8": "\n{\\mbox{KILL}}[d : y \\leftarrow f(x_1,\\cdots,x_n)] = \\{y\\}\n", "3a293a297a3fd4f001cd99fead9d79f5": "E_k = \\frac12 mv^2", "3a29cd14c7551074472e0d862031c820": "R^2 \\,", "3a29f777d8dbf033ce62c7f852630432": " d + d^*", "3a2a131d4ac03b6d52bbd4399f940bf4": "\n\\operatorname{E}(T) = \\left. \\frac{\\mathrm{d}}{\\mathrm{d}z} G(z) \\right|_{z=1}\n= G(1)\n\\left(\nn\n+ \\frac{1}{n-1}\n+ \\frac{2}{n-2}\n+ \\frac{3}{n-3}\n\\cdots\n+ \\frac{n-1}{n-(n-1)}\n\\right)\n", "3a2a3b131cd0e47215020cc94be20728": "\\mathbf{x}=\n\\begin{bmatrix}x_1 & x_2 & \\dots & x_m\\end{bmatrix}^\\mathrm{T}\n", "3a2a518d597c105a2139045166a77fbe": "\\rho g(r)", "3a2aba2290de677860e52a25a4c3ac38": "E_r^{p,q} \\times E_r^{s,t} \\to E_r^{p+s,q+t},", "3a2b1ff320b8a85087b56e76705c6dd5": "copy\\;m.", "3a2b3420bb999b7ce5d995703d4a967d": "-x = ax", "3a2b89b200e37e4568e207935d52ffe3": "\\sum_{k=1}^p A(i,k)B(k,j)", "3a2b8aa1b314f1d7f54776bb0891a7da": "\\sigma|_{[v_0, \\ldots, v_q]}", "3a2ba3565f62804156a24099c20d3c43": "k \\cdot P", "3a2c297dfcf7430fec3cc8ca18a1ee99": "x^2 + y^2 + z^2 - t^2 = 0", "3a2c4359ac3744bf0d9169e5393153c9": "\\text{Level 3:} \\ \\ 266 = 2 \\uparrow 2 \\uparrow (2 + 1) + 2 \\uparrow (2 + 1) + 2", "3a2c9f5592bbac2cbf060011f01250bc": "\\textrm{havercosin}(\\theta) := \\frac {\\textrm{vercosin}(\\theta)} {2} = \\frac{1 + \\cos (\\theta)}{2} \\,", "3a2ca912c31ddd3a8e29fe5f89a6ac34": "\\sin(666^\\circ) = \\cos(6\\cdot6\\cdot6^\\circ) = - \\varphi/2", "3a2cda536b469d983c858d2fbb7bcdc0": "r(A)=|A|", "3a2ce69f2fcdba980633754feeedd3a7": " L(t) = \\int \\mathcal{L}(\\mathbf{r},t) \\mathrm{d}^3 \\mathbf{r} \\,", "3a2d1ecf836c368b4321e7722495ac8c": " ds^2 = \\exp(2 \\, \\psi) \\, \\eta_{ab} \\, dx^a \\, dx^b \\approx (1 + 2 \\psi) \\, \\eta_{ab} \\, dx^a \\, dx^b", "3a2d3423f9e282037ff33c53a67174e0": "\\frac1r + \\frac1s = 1 \\,", "3a2d4957d8f022a8093450a41fe4996c": "\\mathbb{L_A}=\\{(f^mt^nf^r)^+:1\\leq m\\leq k; 1\\leq n\\leq \\ell;1\\leq r\\leq k\\},", "3a2d714fbbf99322444a7fdf212b632b": "\\text{E} = \\bigcup_{c \\in \\mathbb{N}} \\text{DTIME}\\left(2^{cn}\\right)", "3a2d8d0fdfa62ad35dd4dd7152a87d6b": "B(\\omega)", "3a2dbc76a36164c9a0ec98080d2cf505": "\\tilde\\gamma(1)", "3a2dea694fbb3767e693eda78e9b582f": "Y' \\subseteq Y", "3a2e3b3c183d6debd4a74410f8983176": "\\mathfrak{V}", "3a2e43e59a71e1c497a15668ad2e7749": "\\textstyle (X_{1},Y_{1},Z_{1})", "3a2e788f5d22c914a67aa10da29d85d6": "\nf(t) = \\int_{0}^{\\infty} h(\\tau) s(t - \\tau)\\, d\\tau\n", "3a2e951d31ce198f9dd144cbdf361833": "M^{(2\\eta + 1)} = \\text{diag}(r^{(\\eta+1)})M^{(2\\eta)}", "3a2f35893f374a3414d8a6fb4d4d4fe3": "S:\\mathbf{C}^{\\mathrm{op}}\\times\\mathbf{C}\\to \\mathbf{X}", "3a2f397da2310bc466b7ecb87ae03719": " U=\\sum_{t=0}^\\infty \\beta^t u(c_t) ", "3a2f6e5fd3a7e0af87382a49be2636e2": "\nH \\big|(\\mathbf{k},\\mu)^m; \\, (\\mathbf{k}', \\mu')^n \\, \\big\\rangle = \\left[m(\\hbar\\omega) + n(\\hbar\\omega') \\right] \\big|(\\mathbf{k},\\mu)^m; \\, (\\mathbf{k}', \\mu')^n \\, \\big\\rangle ,\n", "3a3029228c162b0317ac3230c222260c": "L = 10^{(T - T_\\mathrm{Ref})/z}", "3a30477cbea5021cf3db01dd67f5fec3": "x=\\sin \\theta", "3a3071380eefc8cb7aab53c1ae2c850f": "R(j,k,\\ell) = \\sum_{n,q,r} x_{n,q,r}\\,x_{n-j,q-k,r-\\ell}.", "3a3099b50b3ea61f58011ab568ac7199": "x \\in U", "3a30f91edcb085494ae26a4dea5c187a": "\nS [ w ] = \\frac{1}{2 \\beta V^2} \\int d \\mathbf{r} d \\mathbf{r}'\nw (\\mathbf{r}) \\bar{\\Phi}^{-1} (\\mathbf{r}-\\mathbf{r}') w (\\mathbf{r}') -\n\\xi Q [ i w ]\n", "3a3121dca6d4ec45f761f81a3cee351a": "\n \\sigma_{ij}=3K\\left(\\tfrac{1}{3}\\varepsilon_{kk}\\delta_{ij}\\right)\n +2G\\left(\\varepsilon_{ij}-\\tfrac{1}{3}\\varepsilon_{kk}\\delta_{ij}\\right)\\,~;~~\n \\boldsymbol{\\sigma} = 3K~\\mathrm{vol}(\\boldsymbol{\\varepsilon}) + 2G~\\mathrm{dev}(\\boldsymbol{\\varepsilon})\n", "3a313bd7ed5c95c224fbfc6a10a90192": " \\gamma_{r, n}(A)=\\theta_{n}\\{g\\in O(n):gV\\in A\\}.", "3a31bb8df34927a6bfb31fef44151d13": "h(x)=x-x_0\\,\\!", "3a31d3908dec1ded9ffe81b414efadb4": " = \\mathbf{u}_{\\rho} \\left[ \\frac {\\mathrm{d}^2 \\rho }{\\mathrm{d}t^2}-\\rho\\left( \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t}\\right)^2 \\right] + \\mathbf{u}_{\\theta}\\left[ 2\\frac {\\mathrm{d} \\rho}{\\mathrm{d}t} \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t} + \\rho \\frac {\\mathrm{d}^2 \\theta} {\\mathrm{d}t^2}\\right] \\ ", "3a328ccfaf134dd461d0a71fcfe51445": "24x+1", "3a32c78e6424b3fa9dcebedd2add7b52": "x^2 - Ny^2 = k_i", "3a33002d6a897528fcc958ab0df349de": "\\left(\\frac{{p}}{{5}}\\right)", "3a334da52e8d7bc9054c5c97bacd5a03": " S'' < S' ", "3a33758ffd16f497d56d75c24ebdcbdf": "R_3 = \\frac{R_aR_b}{R_T}. ", "3a33901970efc48aa4e5ea37cfb9ddfc": "\\scriptstyle\\frac{1}{h}2e \\cdot U_{DC}", "3a33cebd440754960d272ca7fb8bef65": " \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\frac{\\partial^2 \\mathbf{S}}{\\partial x^{2}} + \\mathbf{S}\\wedge J\\mathbf{S}.\\qquad (3)", "3a33d03fb7ebd5c1c574bccd49afa667": "\\Phi_E = \\,\\!", "3a34273d6895e3298811708c64a3d69b": "n = 0 \\ldots N ", "3a3440d408a891bc2aa6963bc2fe865f": "\nL_\\mathrm{W} = L_\\mathrm{p}-10\\, \\log_{10}\\left(\\frac{2}{4\\pi r^2}\\right)\\,\n", "3a344c89da012e6c6f9cc02d19c3b2cf": " \\frac{1}{n} \\sum_{j=1}^n a_j b_j \\geq \\left( \\frac{1}{n} \\sum_{j=1}^n a_j\\right) \\, \\left(\\frac{1}{n} \\sum_{j=1}^n b_k\\right).", "3a344ce3b08db11a4734d938b0f59d27": "x^6 - 9 x^3 + 8 = 0. \\,", "3a346321dbee7fa71b118dcde9c9c81d": " C = g^2/2\\kappa\\gamma ", "3a346bba481fa2a0627f43d7c54ffe2b": "\\left|\\alpha-\\frac{p}{q}\\right|.", "3a347a9ad004bb93d4ca7b40bda763e2": "A_a^i - \\Gamma_a^i = \\beta K_a^i = \\beta \\{ A_a^i , K \\}", "3a348250d2cda580d099d1c9fb3d6e39": "D_{dB} = 10 \\cdot \\log_{10}\\left[\\frac{D}{D_{reference}}\\right]", "3a3485045f229ff0bd795dd294287dc6": "Fu : _SM \\to N", "3a34e0a13eeb1a495b68bfb4e33d1f1d": "v_2\\, =\\, \\frac{A_1}{A_2}\\, v_1.", "3a34f68457e3a20d2f20551bbfb5611f": "\\tilde{A}^T\\tilde{A}x = \\tilde{A}^T\\tilde{b}.", "3a3560809a595349eac44269f9bf27d2": "{\\mathbf Q}(\\sqrt{13},\\sqrt{17})/{\\mathbf Q}", "3a3560c2f9d0289edbc677d569eede1d": " F_C = \\omega^2 ur = \\frac{GMur}{d^3}", "3a3575f280c24c1fcbc1853b1f568d33": "0<\\lambda\\le 1", "3a357c76da35ab3ccffc4a1f6408e559": "\\sigma^2_2", "3a361247f95d262aaebe6dd63905ccf5": "\\vec{\\xi}_1", "3a3617374146950e0a0840a09447f39e": "E\\left[ x_i^2x_jx_k\\right] = \\Sigma _{ii}\\Sigma _{jk}+2\\Sigma _{ij}\\Sigma _{ik}", "3a362228c972b19460a4c62431f46bf0": "\\mathbf{f}(t)\\in \\mathbb{R}^3", "3a368b7db25c194ee83a3f77d7f862a7": "m_\\text{red} \\leq m_1, \\quad m_\\text{red} \\leq m_2 \\!\\,", "3a369ed15d1e267a3cd230c183da49e5": "\\left| \\left\\langle f,{{g}_{m}} \\right\\rangle \\right|", "3a370c6d927d42341573556ba723f5b8": "\\omega = - \\log_{10} (p^{\\rm{sat}}_r) - 1, {\\rm \\ at \\ } T_r = 0.7", "3a37195efd7f96082db6a27ab69a2e7a": "v_2 \\ge 2", "3a372980110230dc38112ebf2ec96f05": " \\alpha - \\beta \\approx \\sin(\\alpha-\\beta) = \\sin \\alpha \\cos\\beta - \\sin \\beta \\cos \\alpha ", "3a37699c301017d0693f23bddd3adb06": " K^*_i, K_i ", "3a3773617b5bb7df20d9536221d9bf4e": "(g,e') \\in N \\subset H", "3a3775cb23a4bda86eb8d580ca8a3a4e": "\\begin{vmatrix}u_i & u_j\\\\ u_i & u_j\\end{vmatrix} = 0 ", "3a37b8f4e03951d41251345e7b6d866e": "w(\\mathbf{e_1}) \\leq 1", "3a37d46b6b61916ca6b8a182c8b8df83": "5 \\times 3 = 15.\\,", "3a3873f27064129b4e969b253ce9a1d7": "\\hat H = J \\sum_{j =1}^{N} \\vec{S}_j \\cdot \\vec{S}_{j+1} + \\frac{J}{2} \\sum_{j =1}^{N} \\vec{S}_j \\cdot\\vec{S}_{j+2} ", "3a3883dabd64cf8c46504cae36d06da9": "\nA' = U^{\\dagger} A U\n\\, ,", "3a38b4091df9af94d5b3c359fc0c2c4e": "\\bigoplus_{d=0}^\\infty H^0(V, L^d)", "3a39184ade80d72bdef7fcee9dcadc43": "\\displaystyle E_2(\\tau) -3/\\pi y", "3a39633f02b03b27a1b9058b64e02b88": "|\\{1,\\ldots,n\\}| = n", "3a39be4611b8c33fc8e9e5fa784dfaa4": "\\big\\{\\mathbf{A}_{l}\\big\\}", "3a3a31c01221cd0fa25152cb1c38f56c": "k_0", "3a3a7a0b78aeb6a955cafdb7423e8417": "\\sum _x \\log_b ax = \\log_b (a^{x-1}\\Gamma (x)) + C \\,", "3a3aa7836d01f45498efb61574048970": "\\lambda_1=\\frac{(y_2-y_3)(x-x_3)+(x_3-x_2)(y-y_3)}{\\det(T)}=\\frac{(y_2-y_3)(x-x_3)+(x_3-x_2)(y-y_3)}{(y_2-y_3)(x_1-x_3)+(x_3-x_2)(y_1-y_3)}\\, ,", "3a3abe5c575e6db1928aeda244d5c794": "1\\times\\frac{1}{2}+2\\times\\frac{1}{4}+3\\times\\frac{1}{8}+3\\times\\frac{1}{8}=\\frac{7}{4}", "3a3ad1db8ed0abee29ec6e132e95b19d": "q = \\Delta\\varphi.\\,", "3a3af9937c0e327806626cf2ce021153": "\\mathcal{N}_k(x) ", "3a3b086ce9f94b055e109f21564fe23b": "2{{V}_{BE}}\\approx 1.4", "3a3b15a70596b1235d5acfabb1306a61": "\\langle (\\Delta N)^2 \\rangle", "3a3b8ff44028965db560fe4d3e064ad0": "-\\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{r})", "3a3bf5c23b707446c173e1fab65b328f": "J^\\mu\\,\\!", "3a3cb88c01ed7a6b37d60f9172d3a834": "B \\rightarrow B'", "3a3ced2151f5661062da50f359e9b4b4": " P \\,\\ ", "3a3d05731c63e71b2fec3317235be5bb": "C = \\frac{N g^2 J(J+1) \\mu_B^2}{3k_B}", "3a3d21067feba259db12697453ecc86c": "\\frac\\pi4\\!", "3a3d69073bbd7b0cb17baedcb082f750": "\\exp^z ~(x)", "3a3dd3d770701d2afc465a04c9344be6": " \\; \\delta(\\alpha) = \\lambda(\\alpha)[\\lambda(\\alpha)-\\alpha]", "3a3e291da2282f1aadc3875d6888042c": "SO(3)\\times Z_2^T", "3a3e4a5134efb83cde5a935401d05bf2": "Q=F/f = 4000", "3a3e597dceae68e725e6dde54a9f6409": "\\log(1+x) = \\sum^\\infty_{n=1} (-1)^{n+1}\\frac{x^n}n\\quad\\text{ for } |x| < 1", "3a3e6593499aee9706fb5b332c22979a": "f X", "3a3e94092e8d60c093c73ddd9f369940": "f_Q", "3a3e9cf9ef08d9ea9820da83d4c3e7a4": "\\int\\sinh^n ax\\,dx = \\frac{1}{an}\\sinh^{n-1} ax\\cosh ax - \\frac{n-1}{n}\\int\\sinh^{n-2} ax\\,dx \\qquad\\mbox{(for }n>0\\mbox{)}\\,", "3a3ea00cfc35332cedf6e5e9a32e94da": "E", "3a3ece9b0c7ae9c2ce41fe7d8d295592": "AB+AB'+A'B-A'B'= A(B+B')+A'(B-B') \\le 2", "3a3ed01a9f7b8732f1c4cb6db73c0d0a": "t = -1", "3a3ed38c78c2ca39019a7552c9269b33": "3y^2 + 3y + 1", "3a3fa8e7b0ff7ed7bab99de4b8725af0": "\\frac{dQ_T}{dQ_*} = \\frac{1}{B(T) E_{Q_*}[1/B(T)]} = \\frac{D(T)}{E_{Q_*}[D(T)]}.", "3a4062af6b7ecd6c16cf475ff51f2f8f": " \\underset{\\text{For } x = 1/2 \\qquad \\qquad} {\\sum_{n=0}^\\infty \\frac{(\\!-1\\!)^n x^{2n+1}}{2n+1} = \\frac {1}{2} - \\frac{1}{3 \\! \\cdot \\! 2^3} + \\frac{1}{5 \\! \\cdot \\! 2^5} - \\frac{1}{7 \\! \\cdot \\! 2^7} + \\cdots}", "3a407d83d7fde402b250f827e87a045c": "{\\boldsymbol \\theta}", "3a408072dc0a3b3642d6a0497021bf15": "w_i > w\\,\\!", "3a409997853a691a3f400d4abf6df254": "O(2n+1) \\to \\pm 1", "3a4109974975e78e0492325d80e1caec": "A(S) = \\displaystyle\\bigvee_{i\\in S} x_i", "3a4154ba60cb7bd26fb2fb1101e5f2ec": "i_{\\alpha\\beta\\gamma}(t) = Ti_{abc}(t) = \\frac{2}{3}\\begin{bmatrix} 1 & -\\frac{1}{2} & -\\frac{1}{2} \\\\ \n0 & \\frac{\\sqrt{3}}{2} & -\\frac{\\sqrt{3}}{2} \\\\ \n\\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} \\\\ \n\\end{bmatrix}\\begin{bmatrix}i_a(t)\\\\i_b(t)\\\\i_c(t)\\end{bmatrix}", "3a416fb37204609667945165007412b6": "801 = 17^2 + 8^3 = 26^2 + 5^3 = 1^3 + 2^3 + 4^3 + 6^3 + 8^3 = 2^3 + 4^3 + 9^3.", "3a418d9281a9d05f4c348cba5d1cc2c1": "C(X_1,X_2,\\ldots,X_n|Y=y) = \\sum_{i=1}^n H(X_i|Y=y) - H(X_1, X_2, \\ldots, X_n|Y=y)", "3a41cf5669d95c6694aa24b2d0b99f2a": "\n\\frac{1}{\\tau} = \\frac{1}{C}\\oint \\frac{1}{\\tau_l}(s) \\, ds\n", "3a41d37625ac83b0e8c208e07acb6202": " f(z)={1\\over 2\\pi i}\\int_{x\\in I} g(x){dx\\over z-x}.", "3a41e1c19569edbd1e9cd0975e7eaafa": "p(\\mathbf{X}) = h(\\mathbf{X}) \\, g(\\theta, T(\\mathbf{X}))\\,", "3a423744724983d7c2eee5e1c7dcff6c": " \\frac{1}{2B}\\Pi(t/2B) (2B \\,\\mathrm{III}_{2B} * F) = F", "3a427d888687d9dfe0f72040d9fa53a4": "x\\!", "3a42a2cfe8de1c00f8ea6fb0f7825b62": "\n \\boldsymbol{Q}(\\boldsymbol{N})\\boldsymbol{m} = \\rho_0 c^2 \\boldsymbol{m}\n ", "3a42a30b2e689a187abeccbec98f6713": "S=\\left[\\begin{array}{cc}\nI & 0\\\\\n0 & -I\\end{array}\\right],", "3a42e2d3ed20b9a43b65004a9e2f1655": " f^P(m) =a^{-\\rho}\\int_Nf(nm)\\;dn", "3a43521f79638a4480ee153afebe9cdb": "\\begin{align}\n \\Delta f \n&= {1 \\over r} {\\partial \\over \\partial r}\n \\left( r {\\partial f \\over \\partial r} \\right) \n+ {1 \\over r^2} {\\partial^2 f \\over \\partial \\theta^2}\\\\\n&= {1 \\over r} {\\partial f \\over \\partial r} \n+ {\\partial^2 f \\over \\partial r^2}\n+ {1 \\over r^2} {\\partial^2 f \\over \\partial \\theta^2}\n.\n\\end{align} \n", "3a43ae13403148e9f5f01fe5c18f5882": "\\mathbb{D}_{12} = \\mathbb{Z}_6\\rtimes\\mathbb{Z}_2", "3a43d96ff5e70038f629b716f6baf2f4": "\\!\\mu_4(v_2)", "3a4415ebe68cb1396e06b79807c3ce5f": "\\omega_a = \\frac{m e^4}{(4\\pi \\epsilon_0)^2 \\hbar^3}", "3a447ffe4474047badfba35de4c39246": "I(\\theta_1)", "3a44874fb43218357874d27c3674dc02": " = \\vec{\\nabla}_{\\vec{r}}\\bigg(\\frac{-1}{(n-2)A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\bigg(\\vec{\\nabla}_{\\vec{r}} \\frac{1}{|\\vec{r}-\\vec{r}'|^{n-2}}\\bigg) \\bullet{} \\vec{F}(\\vec{r}')d\\tau'}\\bigg)", "3a4496033d47f3b989ca0d9dacef6159": "1+y+y^2+y^3+\\cdots = \\frac{1}{1-y}", "3a44adcdd482aca65adf019ee6ed9141": "\\Epsilon_\\text{mach} = \\tfrac{1}{2} B^{1-P}.", "3a44bada3b331725b7b01c1a59b8c6fd": " \\varphi_X^{(k)}(0) = i^k \\operatorname{E}[X^k] ", "3a44c2bf49d788af102f00a74c0522d4": "H^k(\\mathbb{P}^n\\mathbb{C}) = \\begin{cases}\n \\mathbb{Z} \\quad\\text{for } 0\\le k\\le 2n,\\text{even}\\\\\n 0 \\quad\\text{otherwise}.\n\\end{cases}", "3a44cfdeaea61ac81426d721c79b6edd": "\\tfrac{2}{7}\\scriptstyle{\\sqrt{6(10+\\sqrt{2})}}", "3a44ff551b50c6de6330cbc60eaf1519": "\\varphi \\rightarrow \\varphi -d\\lambda", "3a450fa8829c495be7a0d66dceb5dac5": "L_n^{(\\alpha)}(x)= \\frac {(-1)^n}{n!} U(-n,\\alpha+1,x)", "3a454e0c61b14a26be56f5b98f128b67": "n_p = \\infty", "3a455ba5dd289d2e5fbe5acaddf4585e": "a \\equiv \\frac{U_1 - U_2}{U_1}", "3a45cea07a0c8a46177c87c16626432f": " 1 - (1 - \\alpha)^{N+1} ", "3a468c7d29a3581109aeb9473ec326e6": "p_r = {p \\over p_c}", "3a468eefdd9e8fc34d49983fb95bc70c": "r_e = \\alpha\\left(\\frac{\\lambda_e}{2\\pi}\\right)\\simeq\\frac{\\bar{\\lambda}_e}{137}\\simeq 2.82~\\textrm{fm}", "3a46fa7d4262211116b6d60579549416": " \\langle x| \\Phi [f] |y \\rangle = \\int_{-\\infty}^\\infty {\\text{d}p\\over h} ~e^{ip(x-y)/\\hbar}~ f\\left({x+y\\over2},p\\right) . ", "3a478be0a3487efa896d71540d271d49": "x_m", "3a47ad9dc0aa3add781a869aa44cce42": "s = \\frac{\\alpha}{180}\\pi R = {\\theta} R", "3a47d1a295d85f45c9387ee8d926691b": "E= h\\nu = E_i-E_f=R_\\mathrm{E} (Z-1)^2 \\left( \\frac{1}{1^2} - \\frac{1}{2^2} \\right) \\,", "3a480a49894adabb93949652d9f7f173": "i=1,\\ldots,N", "3a4815ac611b7fc8baca884de37e4c9d": " \\overrightarrow{F} ", "3a48273bf6f90db173c0e655739d73a1": " \\{AAEENRT -> \\{''anatree''\\}\\}", "3a48966e2f260056a3376d0075acceb2": "\\Lambda(S)=\\log|S|+K(S) \\ge K(x)-O(1)", "3a48ba17f3bc21a7ebb40e6f78341038": " \\operatorname{cov}(\\mathbf{X},\\mathbf{Y}) = \\operatorname{cov}(\\mathbf{Y},\\mathbf{X})^{\\rm T}", "3a48ef01b66c9ca2db67ddb99b5e610e": " \\theta = \\frac{[L]^n}{K_d + [L]^n} = \\frac{[L]^n}{(K_A)^n + [L]^n},", "3a4944410f88e8475839578844a8a2c9": "A \\subset \\mathbb{R}^n", "3a4983997b2f47b4d6511da1a4fc49db": "!n = n! - \\sum_{i=1}^n {n \\choose i} \\cdot !(n-i),", "3a4995ff41e44bce1e644cc65f752237": "\\mathcal{O}_{[g]}^\\chi", "3a499e80a15707d42adcefe164052be8": " V(s) = { I(s) \\over sC } + { V_o \\over s }. ", "3a49beedc3c5277a2ef300fe51a4ba0f": "\\ \\phi_i ", "3a4a26d867af44225443afa5ffd1a5f7": "-1.2618", "3a4a48b52480a29e0dc7f309180f3ecc": "f^{-3}", "3a4ae2031f71a5abc4e9dd0bc41f8bd4": "\\mathbf{c} \\cdot \\mathbf{a} \\times \\mathbf{b} = \\begin{vmatrix} c_\\text{x} & a_\\text{x} & b_\\text{x} \\\\ c_\\text{y} & a_\\text{y} & b_\\text{y} \\\\ c_\\text{z} & a_\\text{z} & b_\\text{z} \\end{vmatrix} ", "3a4af71c819f90851e9bf753c0a0fb3f": "\\frac{2\\pi r}{r} \\text{ rad}", "3a4b324ac68dd0bf2802abe457653208": "a=c", "3a4b79a79faf790434105a7c7cb847d9": "S_{F^{-1}}", "3a4b84b472e7a6c30f6f93a8223b75a9": "3\\uparrow\\uparrow 2=3^3=27 ", "3a4bc67c3a42aef5943cc8513da7bfe8": "\\ln \\mathcal{L} (p|H) = H \\ln(p)+ (1-H) \\ln(1-p).", "3a4be82a575eee3638facc804afc2941": "[0, 2 \\pi]", "3a4c3743463ca5a790d53d566420183e": "1 \\Leftrightarrow 2, 3 \\Leftrightarrow 4, 1 \\Rightarrow 4, 3 \\Rightarrow 5", "3a4c780e59b399df7737082b5f2edd08": "\\alpha=-\\pi/2", "3a4d0497cfee9981baf1cd240dc83690": "w(2-i)=0", "3a4d3e4b3a3f5e0e11986bc16a382d72": " B(\\mathcal{H}_B) ", "3a4d503c659f668e864502b6047881f4": "\\rho(x)=\\frac{1}{\\pi\\sqrt{1-x^2}}.", "3a4d5a35ba7d2f8e691f49cefe4348a2": " (n+1)\\alpha=\\ x_1 + \\cdots + x_n + x_{n+1}.\\,", "3a4d5f7668d725d110431d150646f319": "F_k(z) = F_{k,1}(z) F_{k,2}(z)", "3a4d96aea2c8f17ecef6c2bda978777b": "a + b\\sqrt{-1} \\mapsto a + b\\mathbf{U}\\vec{q}_v.", "3a4dbe351d9ab20c2c155e8d92f955ff": "\nz = \\frac{p}{v} = ZS =\\rho D \\,\n", "3a4e257bacb7f98636313963c63ca89b": "m_1 \\times \\cdots \\times m_d \\times n", "3a4ee770718515475bb6641677dc6943": "\\mathbf{\\overline{o}}_{\\mathbf{k}}", "3a4eeedfe038507d035fdec16a8ff7d5": "\nJ_{\\perp} = J_{\\parallel}\\ \\beta\n", "3a4f275290fff3eeb46681d622ca2ba4": "a(Z)\\,a(Z^{-1})+a(-Z)\\,a(-Z^{-1})=4", "3a4f6b5f42663a6a9c76ed2cf7649b97": " Z_2 ", "3a4f6c813bdbe74f86b9a16795e66889": "\\mathrm{R_{A\\alpha}}", "3a4f70ed6e6229a7a03b621da3908a15": "f_{IN}", "3a4f8ccedb6ba90463d83af861cfc042": "V = \\bigoplus_{n=0}^\\infty V_n\\,", "3a4fb25e52756e5c9baf0440b27f0557": "\n\\sqrt{\\frac{1+z}{1-z}}\n\\left(\n\\frac{1}{6} \\left( \\frac{z^2}{2} \\right)^3 +\n\\frac{z^2}{2} \\frac{z^4}{4} +\n\\frac{z^6}{6}\n\\right) =\n\\frac{5}{16} z^6\n\\sqrt{\\frac{1+z}{1-z}}.\n", "3a500760afe5b8ff130d482a7ccccfa8": "{x \\choose k} = \\frac{(x)_k}{k!}", "3a501bcbc7a2fcd5a66382012dd10d8b": "\\textstyle P_2(f(\\Omega_1)) ", "3a501d4228f842517ca488178a4661cb": "\\scriptstyle \\boldsymbol{r}_i \\;=\\; \\boldsymbol{r}_i (t_i)", "3a502e1cc7db566fc7a3b8001e5b523a": "m_{ij}=\\min\\{i,n+1-j\\}", "3a504583a62895949f47dcaec8323e7d": "\n\\mathcal M=-i\\sqrt{\\frac{2\\omega_p}{Z}}\\left\\{\n\\lim_{x^0\\rightarrow-\\infty}\n\\int \\mathrm{d}^3x f_p(x)\\overleftrightarrow\\part_0\n\\langle \\beta\\ \\mathrm{out}|\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n\\varphi(x)\n|\\alpha\\ \\mathrm{in}\\rangle-\n\\right.\n", "3a5073edb7ce9ed44c5710dc24363031": "x = - \\cot ({2\\pi \\cdot T})", "3a5078d571f4b2f69fc7a6a50fd39ec9": "x^8 + x^7 + x^4 + x^3 + x + 1", "3a50aa2a53241de729417d1caf73a14d": " \\frac{\\mathcal{O}(x_1,\\ldots,x_n)}{J_f}", "3a50ad3bf1af96451a9f710e0f5f0220": "(\\mathbb{N}^n,\\le)", "3a510a741b8fd2cb0bdce0cab39b82bb": "\\,\\mathcal{M}\\,", "3a517d0dd5510394fce006cc875ee733": "P_1=(1/2)-\\epsilon", "3a51ac34326f0a61ac4d4272f5980497": "f(x,\\beta)", "3a51d2dae2a4c583a3f70ceed2e17ff9": "Mor", "3a525991674131e140c37ab6815d8c8d": "||\\bigstar \\bigstar \\bigstar |", "3a5281a66068c29c7b7c82f323d6140a": " \\Tau ", "3a52d8a880ce716dfa783233f8b0bc01": " R=\\sqrt{\\frac{p^2+q^2+4x^2}{8}}. ", "3a52f3c22ed6fcde5bf696a6c02c9e73": "DE", "3a53a0860ba7055ac71bcb8b16c3551f": "K/Z", "3a53f37178a6264d7a2224c47bafea26": "[n,k]", "3a53fb6587e63ed7fa611f5f4b2a05dc": "A + B = 1 - X + 5X^2 - 3X^3 + 9X^4 - 5X^5 + \\cdots.", "3a541e0fd62f75882a8e54d4dcd7eb14": "a_{\\mathrm{fast}}", "3a54cf156aa29b51ac5ce681396afc3c": "Y^{D} = \\pi + \\omega \\cdot L^{S}", "3a55270a9c1dd898768529efab205e2c": "\\begin{bmatrix} d+1 \\\\ k\\end{bmatrix};", "3a554f57a9b9ea5c22ecd88176695194": "f(x)=x^2+3x+4+\\frac{1}{(x-1)} + \\frac{1}{(x - 1)^3} + \\frac{x + 1}{x^2+1}+\\frac{1}{(x^2+1)^2}.", "3a55fc90f006e68cd8eef71d430d7072": "\\{U_{ij} = \\text{Spec} \\; A_{ij}\\}", "3a56043bcfa4975abf85a221ff8dbfe9": " u(0)=0, u(1)=1 ", "3a562b6a87d911f46bd7eed1c514b90b": "\\,(\\omega,x_0)", "3a56b50274d14ed600007f16a5b7ff9e": "\\sum_x p(x\\mid y_1,y_2,I) \\log \\frac{p(x\\mid y_1,y_2,I)}{p(x\\mid I)}", "3a56c8c1b9b9c97c23c88cde7d02cf55": "\\cos^{-1}\\left(\\frac{23}{27}\\right)", "3a56ec5a6b14e6aa7c81cbd12e56c0fd": "H^1(\\R^n)", "3a56f126ee15cd4379cb0a59d4e2b061": "\n\\begin{align}\n\\mathbf u(\\mathbf X,t) &= \\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad &\\text{or}& \\qquad u_i = x_i-\\delta_{iJ}X_J =x_i -X_i\\\\\n\\nabla_{\\mathbf X}\\mathbf u &= \\nabla_{\\mathbf X}\\mathbf x - \\mathbf I = \\mathbf F - \\mathbf I \\qquad &\\text{or}& \\qquad \\frac{\\partial u_i}{\\partial X_K} =\\frac{\\partial x_i}{\\partial X_K}-\\delta_{iK} = F_{iK} - \\delta_{iK}\n\\end{align}\n", "3a5704e8c6248db41320aa1df7ed749b": "\\scriptstyle Q(x) \\;=\\; \\frac{1}{\\sqrt{2\\pi}}\\int_{x}^{\\infty}e^{-\\frac{1}{2}t^{2}}dt,\\ x \\geq 0", "3a57c0dfd527c27b57053fea00cf7ee6": "{\\rm PGI} = \\frac{1}{N} \\sum_{j=1}^{N} \\left( \\frac{(z-y_j).1(y_j 0", "3a636b89ca0aec779034de35616d9350": "\\boldsymbol{c'}", "3a63a82a0fee101aaa57ac9f0096a9e3": "m=(4/3)E/c^2", "3a63ab31a28347d467f5a5870f07db65": "m_i(\\hat{x}) > 0", "3a63b67210ceb679fe406e1675610b01": "1/a^2", "3a63d17d587c5c8a8147c67ca16a96a0": " \\operatorname{Pr}(X\\leq a-\\varepsilon) - \\operatorname{Pr} \\left (\\left |X_n-X \\right |>\\varepsilon \\right ) \\leq \\operatorname{Pr} \\left (X_n\\leq a \\right ) \\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr} \\left (\\left |X_n-X \\right |>\\varepsilon \\right ). ", "3a63f79d4f8bec01763d4d192abe66b3": "\\sum a_i f_i \\sum b_j g_j = \\sum a_i b_j f_i g_j", "3a6452911bc4e55e2e71b5c8bfc6710b": "E^\\mathrm{tot}(\\mathbf{x}_j,t)=\\sum_{n\\neq j}\n\\frac{E_n^\\mathrm{ret}(\\mathbf{x}_j,t)+E_n^\\mathrm{adv}(\\mathbf{x}_j,t)}{2}\\ \\text{.}", "3a646a99c08099a2ac062eeb2dbac0d8": "\\nabla F(x, y, z) = \\hat{\\mathbf{k}} - \\nabla f(x, y)", "3a64fc936bcb15c0936ac8d226bb7a02": "\\mathbf{L}_z=m\\hbar .", "3a6520cb7e586ac637b7fd65d6092f63": "O(n^{2.3736}).", "3a656110dd5b5630eeff4aa5dcb2ee3f": "a^2+b^2+c^2+d^2", "3a657bc6e3e46116ee0b8cba8e53e052": " 4z^3+3z^2+2z+1 ", "3a65c687bed4dee390e5d75ddaa5bf7f": "\\scriptstyle I\\cup\\{x\\}", "3a662f13cab29f7a49380cc6d4c85165": "K(x,S)=K(x)+O(1)", "3a6664aae85068e734b50ab33a0496ea": "\\hat{H}\\Psi = E\\Psi", "3a669e27b10edbe7700371af95205487": "\\hat{f} \\,\\hat{f} = 0 ", "3a66bc3702eed43a4162f290a800d3bb": "\\ \\mu_{obs} \\,", "3a66be9adfe65d52187426f6f16f18fc": " \\frac{dR}{dt} = \\nu I ", "3a66c09fc15fa2a1d2d3c40076cc786f": "\\vert\\Phi\\rangle", "3a66d1868beea94c83bb77732d42c022": "n/2+1", "3a6705fde63ddad7186168f6ccaa5100": "(o_1,a_1,o_2,a_2,..,o_T,a_T)_i ", "3a673ff289097ef0c563e966dcf1bbb0": "\\scriptstyle{i \\rightarrow j}", "3a674f6841e93aa837e7473ce2d2ab59": "\\ C \\times S ", "3a67860a1eb80630133ceba131fe6568": "\\lim_{R \\to \\infty} \\int_{C_R} f(z)\\, dz = 0.", "3a67f2441409890cc6ef96c9be284b48": "[\\omega]^{<\\omega}", "3a67fbbc43784a24beff9d4d6994b34a": "\nM(G) M(G)^{T}. \n", "3a67fcf8e778d6813c5cb6f8b45541a6": "y(w+k \\pi) =e^{i \\nu k \\pi}y(w)\\text{ or }y(w+k \\pi) =e^{-i \\nu k \\pi}y(w), \\,", "3a6870616ef5bb86afd08a233013802f": "S_R^\\delta", "3a68995b86610b9d6475947ac1b8e8ef": "y(x)=\\left(\\left\\lceil\\sqrt{n}\\right\\rceil+x\\right)^2-n\\hbox{ (where }x\\hbox{ is a small integer)}", "3a689ba021edbd7199a0663c62dd9ddd": "60 \\times 5 = 300", "3a68c5ad76b071740d228c8aeb927c65": "\\alpha\\leq 0.5", "3a693dfb5b9e96338687bcfaac39101c": " I = \\frac{\\pi a^2}{k_b} \\left( \\frac{P_\\infty}{T_\\infty} - \\frac{P_A}{T_A} \\right) \\cdot C_A \\alpha ", "3a697bb7d1b56305db2a9698f2f227a0": "\\hat T = t_{1+i} \\dots t_{m+i}", "3a69b6d010df152c661ebce6f369eb7d": "\\theta = -\\frac{\\pi}{3}+2m\\pi \\quad[m\\in\\mathbb{Z}]", "3a69d07d2adf42061942779219a83d19": "\\hat \\theta", "3a69f8ec2d35557e44589474bda9a4b0": "{\\Delta x}_{t-1}\\,", "3a69fe4e6aabd924cea71b89098622e2": "G \\to GL(n,\\mathbf{R})", "3a6a7a0c642cadda48d7221d2755bc63": "G(t) \\triangleq \\int_0^t \\left[\\sum_{n=0}^N\\pi_n(t)\\right]\\left(r(s)ds + dA(s)\\right) + \\int_0^t \\left[\\sum_{n=1}^N\\pi_n(t)\\left(b_n(t) + \\mathbf{\\delta}_n(t) - r(t)\\right)\\right]dt + \\int_{0}^t \\sum_{d=1}^D\\sum_{n=1}^N\\mathbf{\\sigma}_{n,d}(t)\\pi_n(t) dW_d(s) \\quad 0 \\leq t \\leq T", "3a6a8ef33d105dfb309c33342f2886d7": "\\gamma_\\text{SL}\\ ", "3a6ab4adf5ccb18133bd69aa5dbe837f": "M''/IM''", "3a6ae2b9adf1335ee568038e658b6963": "\\overline z", "3a6b48411cea6bf2f4e97922dec7d110": "0 \\to \\Theta^{4i} \\to \\Omega^{alm}_{4i} \\to \\mathbb{Z} \\to bP^{4i} \\to 0", "3a6bbffc4b3ae64da668c45deafe1b0a": "H^s", "3a6be2299da7c0ccbde7996f4ca60584": "4x^4-28x^3-7x^2+16x+16", "3a6bf0fb8257df97f05476f0cad2b6b1": "p_1, ..., p_n", "3a6c8cd6c7186ab5791c679dfef74dc8": "\\Delta y=f(u+\\Delta u)-f(u)", "3a6c9d8313b316d19a185ad2e93ecb6b": " H_{ij} = \\frac{1}{i+j-1}. ", "3a6ca3465e7ae5491c25ef8023544c07": "0.63\\angle60^\\circ\\,", "3a6cc49fdd8c33f5f0e00af1d5ce999b": "\\Delta t/2", "3a6d0284e743dc4a9b86f97d6dd1a3bf": "val", "3a6d13b46f880fa93d4c0a4e5fb5e7e6": " e^{i 2\\varphi}=(\\cos \\varphi +i \\sin \\varphi)^{2}", "3a6d1e60984b72d3e45fff86a77fea2a": "\\sum_x \\left| x \\right\\rangle \\left| 0 \\right\\rangle", "3a6d370975c41f8dab2c4e7913ac74c8": "~ M(\\vec x) ~", "3a6d67e35606e824f7f7aa3b37149aed": "f_1(x,y)=0, \\ f_2(x,y)=0 \\ .", "3a6d6e2e7a3cede544ea63e30f78f953": "\\theta,\\beta,z", "3a6d84b90910179b4bda0d74976c8631": " I(\\tilde{\\nu}) ", "3a6df03719bac8af2a1da0ebf867d5e4": "O\\left(h^2\\right)", "3a6df19ae7fc4c5da88b5ae8c5b56bad": "q=p", "3a6e66c1ea17e3fb62418a2aac95d6df": "C^\\infty(\\mathbb{R}^p)\\otimes\\Lambda^\\bullet(\\xi_1,\\dots\\xi_q).", "3a6e7af6d84356b247f6ea07ee7fdde2": "(-\\Delta)^{\\alpha/2} f(x)", "3a6e7d9d970150ec975a816739634506": "T > \\mathbb{S}(\\mu(0))/h,", "3a6e998fb44a19a2c30b6a22dd2957bd": "\\lim_{n\\to\\infty}\\Pr\\left[\\left|-\\frac{1}{n} \\log p(X_1, X_2, ..., X_n) - H(X)\\right|> \\epsilon\\right]=0 \\qquad \\forall \\epsilon>0.", "3a6f26270a5797e3ba3190abb5d4a7fa": "p \\sim \\mathrm{Beta}\\left(\\alpha+n,\\ \\beta+\\sum_{i=1}^n (k_i-1)\\right). \\!", "3a6f3f4baa7be12c116615c67b66f2d5": "\\textstyle\\sum D^2=ND_o", "3a6f459625a429689df9a2e36dd57269": "\\hbar k_y", "3a6f69a0062b6c40b566a41e70043096": "\\begin{matrix} \\frac{cosine \\;of \\;misalignment} {1} \\end{matrix}", "3a704d5a3bb5da226767d0807bdbfd54": "\\int_{P} \\left[\\sum_{k=-\\infty}^{\\infty} s(t-kP)\\right] \\cdot e^{-i 2\\pi \\frac{k}{P} t} dt = \\underbrace{\\int_{-\\infty}^{\\infty} s(t) \\cdot e^{-i 2\\pi \\frac{k}{P} t} dt}_{\\stackrel{\\mathrm{def}}{=}\\ S(k/P)}", "3a7052051ad49c0495233d1af1dbfa99": "\\sum f(T,c)=1", "3a707b9a460e71d4330415b150b6affb": "R_{\\varphi\\varphi}=8\\pi T_{\\varphi\\varphi}", "3a708270791ff1265e46a06419a921f5": "P_B(f)=u", "3a708ef0336cb416d31d9a02d31b60a9": " k \\le 4 ", "3a70adf725f85ba77fc5dec802183547": "a_{ji} n", "3a7129789533cde27ab03e021c2d566a": "T_{mm}", "3a715b38e336282c3e3d33e5f2148f8c": " \\mathbf{F} = \\frac{{\\rm d}\\mathbf{p}}{{\\rm d}t} ", "3a71781b01b9ef718d7d835fb602987d": "F_\\alpha = \\left\\{x | h(x) \\geq \\alpha \\right\\}", "3a71bb3477e13ebe788bf679e92ab08b": "\\frac{\\partial u}{\\partial\\bar{z}}-iz\\frac{\\partial u}{\\partial t} = F(t,z)", "3a71cc8bcf49c9e530f83cddb334ce1d": "Td_3 = (c_1c_2)/24", "3a71e73ba2db70227a86dabb869336bd": "A,B \\subset X ", "3a7229c1e68c5332d2c51e4010cb3680": "0<\\vert z-a\\vert<\\delta", "3a72ffe6c580c54a6f2c8fd3bccf65ce": "\\displaystyle c_g= c_p^2 \\frac{\\partial\\left(\\lambda/c_p\\right)}{\\partial\\lambda}=\\frac{\\partial\\omega}{\\partial k}", "3a73564302b40f8fd7827b92d18d92d1": "t_{1/2} = \\frac{\\ln 2}{\\lambda}.", "3a73831f1d76caae37bd83b812500e11": " [k] := {1,2, \\dots , k} ", "3a73956ab461faffe600a8f9c40e4f3f": "\\left(\\frac{x}{a}\\right)^{2}+\\left(\\frac{y}{b}\\right)^{2}=1;\\,\\!", "3a7395817904f091d857e2690b7cee3f": "\\forall A\\in\\mathcal{A},\\quad N_A", "3a73f8a50e7fce208d317c3b1eced529": "- 35 \\sqrt {2} /256", "3a7431ac085c1a62b79b7c4571198278": "\\rho_\\mathrm{c}", "3a74890c81156baa3d775aaae470ecbd": "r = |z|=\\sqrt{x^2 + y^2}.\\,", "3a748fde7537d18c8a02854028cd1d62": " \\zeta = \\frac{1}{\\sqrt{1 + (\\frac{\\pi}{ln OS})^2}}. ", "3a75cf870f647a0cec58f5b278ac0a46": "{{i}_{IN}}=\\left( \\frac{\\beta +1}{\\beta +2}+\\frac{1}{\\beta } \\right){{i}_{C3}}", "3a763d5bbe41afcd34a8c55fb935e3e4": "\\mathbf{Y}=\\mathcal{A}\\mathbf{X}+b", "3a772d553a907801b4444f55227f9015": "\\mu(1) = 1", "3a77851192b2d2b5b73500bd5f1649cb": "\\frac{p^{2}}{6n^{2}}<1", "3a779a9b679e86d9c1fcfddc29418716": "b_n^{(k)}", "3a77b0e3e514b2d37269eafd9d6ddbb4": "\\dot{q}\\equiv \\mathrm{d}q/\\mathrm{d}t \\,\\!", "3a77b5b33d2418e8d9bdefd01968dcdc": "2 \\cdot \\Delta f", "3a77bdb8058b0348dd2d2752ae5413c5": "\\ e_A", "3a780b2ae6907ea7e704d850837e06d0": "\\beth_{\\beta}(\\kappa) = \\beth_{\\beta}(\\mu)", "3a7822ebe2baaf455e56b6d7fd688374": "a_\\bar{\\alpha} = a_\\gamma L^\\gamma{}_\\bar{\\alpha} ", "3a7863f4142f44111d3db003cb623e45": "\\mathfrak{D}^A_w(s) = \\sum_{a \\in A} \\frac{1}{w(a)^s} = \\sum_{n = 1}^{\\infty} \\frac{a_n}{n^s}", "3a7875ff1df1546c7d85fce00702d5dd": "\\beta = v / c\\,", "3a7882e4105169399068784829b02b09": "\\mu_y", "3a78c7cc5ccf255f40a2a21dc1664daa": "\\sin (2 \\pi f_1 t)\\sin (2 \\pi f_2 t) = \\frac{1}{2}\\cos [2 \\pi (f_1 - f_2) t] - \\frac{1}{2}\\cos [2 \\pi (f_1 + f_2) t] \\,", "3a78c9c9d5d34b54160b73a4fd75f67e": " P_t = P_0 + V_bT_t + \\frac{1}{2}\\acute{A}_0T_t^2 ", "3a797e9a86bc8c4397bf0e17c24ba02c": "f(xy) = \\sum g_l(x) h_l(y)\\,\\!", "3a79804aac44c61c26a4b9b6d9b74be9": "a_1\\wedge a_2\\wedge\\dots\\wedge a_r = \\frac{1}{r!}\\sum_{\\sigma\\in\\mathfrak{S}_r} \\operatorname{sgn}(\\sigma) a_{\\sigma(1)}a_{\\sigma(1)} \\dots a_{\\sigma(r)},", "3a79c2404a723747dae94d92a61f26d9": "F\\in \\bigcap_{n=1}^\\infty G_n", "3a79edd6ce6506eedb518a38793df0a5": "[\\cdot, \\cdot] \\circ ([\\cdot, \\cdot] \\otimes \\mathrm{id}) \\circ (\\mathrm{id} + \\sigma + \\sigma^2) = 0", "3a7a00284e0e8ffbcfcce28b5818ee95": " L(n,k) = \\sum_{j} \\left[{n\\atop j}\\right] \\left\\{{j\\atop k}\\right\\},", "3a7a413b5c98663b0073e86cf350e1e5": " \\mu_L ", "3a7a838589cffa7a0670d7fa2b604056": "\\textstyle{\\frac {\\log(16)} {\\log(3)}}", "3a7aa1138bae0f40128d05efdd8c8768": "E_j(\\mathbf{x},t) = e^{i(\\mathbf{k}_j \\cdot \\mathbf{x} - \\omega_j t)} + c.c.,", "3a7b0b94157ba6cb3667283b2c0c2c25": "\\frac{1}{l}+\\frac{1}{m}+\\frac{1}{n}<1.", "3a7b911ed473fe26e6ff573096013acd": "\\mu_1=\\sqrt{\\lambda}", "3a7b98241839ab8afa9290168bb6f446": "\\frac{1}{\\sqrt{-g}}\\frac{\\partial}{\\partial x^i} \\left (\\sqrt{-g}\\sigma u^i \\right ) = 0,", "3a7bbf53f95515d0c45778ed2922998a": "\\displaystyle{F(z)=\\exp -f(z) -iHf(z),}", "3a7bd88269fda23cd3f0be952f4c5ccf": "C(3, 2) = 3", "3a7c15355f69cf5dd637dcc79a76404d": "(1,1)_{1\\frac{}{}}", "3a7c3f1712b57085f8f83c791e224602": "\\forall N\\in\\mathcal{N}_{f(x)}: f^{-1}(N)\\in\\mathcal{M}_x", "3a7c62e55577090df6cf7b43cfd857f1": "a * (b * c) = a * \\overline{b c} = \\overline{a \\overline{b c}} = \\overline{a} b c ", "3a7cae12d6b8612f799286ee6d569078": "\\left [ X \\right ]_2", "3a7cfc7e53af52c72238baf0ea836026": "\\ln |p(x)| = \\ln |C| + \\ln |x - x_1| + \\ln |x - x_2| + \\cdots + \\ln |x - x_n|. ", "3a7d2077b2a45432b74ae8bdb156557e": "S\\!\\left(x\\right) = \\left\\{x\\right\\}", "3a7dc01d01c63cc8e5301fce15b152eb": "S_{\\mathrm{psi\\ per\\ foot}} = \\frac{P_d}{L} = \\frac{4.52\\ Q^{1.85}}{C^{1.85}\\ d^{4.87}}", "3a7dd58716e0d74c9ffca5d4171248cc": "f=g\\left( \\theta^{\\prime}\\right) \\theta_{1}", "3a7dd810094e28bcca611ae72c7f0352": "\\mathrm{SINR}(x_i) {{=}} \\frac{\\ell(|x_i|)F_i}{\\sum_{j\\neq i} [\\ell(|x_j|)F_j] +N} ", "3a7dddd4e8141e391b7f33db8308f228": "genState \\leftarrow InitGen(k,s) \\in GenState", "3a7dec62c64fe69ae4bdbab70814874b": "\\quad", "3a7e124d6e0fb34efb6264a26b2d79f8": "\\{A \\mid \\exists B,C\\,(B \\in m \\wedge C \\in n \\wedge B \\cap C = \\emptyset \\wedge A = B \\cup C)\\}", "3a7e2c3237516425e320a9b367e3945c": " C^J_{E_1} = 1 + \\varepsilon^{1}_S C^S_{E_1} ", "3a7e619d45ffa4c1281e21bc7ea4ce55": "P = g m V_g (K_1+s) + K_2 V_a^2 V_g", "3a7ec6083fc3d5e3a8b1aac73ef33fe4": " s_1 = -\\alpha +\\sqrt {\\alpha^2 - {\\omega_0}^2} ", "3a7ed6598219ca8f6cba46ecb77b051a": "\\exp \\log y = y", "3a7eee9062c40cddacb1b61458caa670": " \\omega = \\arccos { {\\mathbf{n} \\cdot \\mathbf{e}} \\over { \\mathbf{\\left |n \\right |} \\mathbf{\\left |e \\right |} }}", "3a7fa93a8d8ae209c0cb38ef84cf3fca": "\\operatorname{Bernoulli}(p)", "3a7fda3c543b7f62a3f21d4ccf5a54cd": "20 \\cdot a_n\\ dB", "3a807c5cbc784bab447ef009b61182b5": "{\\tilde{A}}_n", "3a807f134146043055e40e95e3c284e4": " \\omega_d \\approx \\omega_0. \\, ", "3a8099e01f03b066c1eb3e10ee32d113": "= a C \\frac{\\sin\\frac{ka\\sin\\theta}{2}}{\\frac{ka\\sin\\theta}{2}}\\frac{\\frac{e^{-iNkd \\frac{\\sin\\theta}{2}} - e^{iNkd\\frac{\\sin\\theta}{2}}}{2i}}{\\frac{e^{-ikd\\frac{\\sin\\theta}{2}} - e^{ikd\\frac{\\sin\\theta}{2}}}{2i}} \\left(e^{i(N-1)kd\\frac{\\sin\\theta}{2}}\\right)", "3a80ac3701e88bbfa481b11f9acfdcf7": "x=f_{n-1}{\\alpha}^{n-1}+\\cdots+f_1{\\alpha}+f_0", "3a80c400139d10366513b736b66aa605": "F_{X,Y}\\colon\\mathrm{Hom}_{\\mathcal C}(X,Y)\\rightarrow\\mathrm{Hom}_{\\mathcal D}(F(X),F(Y))", "3a8109b6135e1ec2a707dba0d0cbe792": "\\scriptstyle 2 \\arccos\\left(-\\frac{1}{3}\\right)\\approx109.47^\\circ", "3a813bb400993828c908bca42cdfd83d": "\\nabla^2~=~\\frac{\\partial^2}{\\partial x^2}~+~\\frac{\\partial^2}{\\partial y^2}~+~\\frac{\\partial^2}{\\partial z^2} .", "3a831e9ae42375b16ef395656bc360de": "\n\\Lambda^{2} = r^{2} - r_{s} r + \\alpha^{2}\\,\\!\n", "3a835350848f2c3957c758ea5a5d66b8": "\np_x = p_0 \\frac{\\eta_x}{1 - \\eta_w}\n", "3a83bd54a06fe1760cedbe5bded6ec10": "G_n^{(2)}", "3a8405d67254d03fbf14cb70b2c02846": "C=+(n_\\mathrm{c}-n_\\mathrm{\\bar{c}}),", "3a84341711ab951b21f1b4e7295baa0e": "\\mu_0", "3a845edf59b5d0555b863443aa0be2e2": "V_w", "3a845fc95f2753822b2c2bb2586fdeeb": "0 \\rightarrow K \\rightarrow K(\\!(X)\\!) \\xrightarrow{D} K(\\!(X)\\!) \\;\\xrightarrow{ \\mathrm{Res} }\\; K \\rightarrow 0. \\, ", "3a84a96df7e3cca5299daadf546f9865": "J_\\mathrm{diffusion} (x)=-D \\frac{d\\rho}{dx}", "3a8543776ac92eb59e05816f6a2a343c": "\\,g(Y,\\theta)", "3a8545ded563000347a33b0903986626": "p_{i} \\phi_{p,i} = p_i^{\\star} \\gamma_i x_i.\\,", "3a8551e1b2736f2fb74e55ac78f94066": "p^2 \\leq \\Lambda^2", "3a85d286552be938d207625afef320ce": "d= \\sqrt{250,000}", "3a85f6ad83fae0d511dafc241360f942": "U(x,y,z) \\propto \\hat f[A(x',y')]_{f_x f_y} ", "3a86077cda1a46301a635e206721e37d": "\\textstyle |b(o,r)|", "3a864e14af7a0131e847dd439c2f986f": "\\omega = k \\sqrt{\\frac{T}{\\mu}}", "3a86e474b21147201a565edfe3fa046d": "(A+UCV)", "3a86fef7fcd15a9d6d59d8380ffcf6a6": "C_b = K \\frac{A}{b}", "3a8708fe32bfba73caeeb0f47f8a7e0a": "\\Lambda^\\text{op} = (-1 \\rightarrow 0 \\leftarrow +1),", "3a87686d489f13453970a9b3b59b7088": "\\left(s_y\\right)", "3a8799c39f6742ae6fb7176d8f3de6a3": "P,K", "3a87ca3c0b533c907dc9f94b346a9b50": "H^1_0(a, b).", "3a87f93f97d671714bc42106ad9b6a1b": "\\tan \\frac{1}{2}(x-y) = \\tan \\frac{1}{2}(\\alpha+\\beta+C) \\tan \\left(\\frac{\\pi}{4}-\\phi\\right).", "3a881514490f31f8444da1cfe980aa12": "L\\cap K", "3a88156b6e53e7b6c6d3164cc49a1e44": "g_\\text{threshold} = \\alpha_{0} - \\frac{1}{2l} \\ln (R_1 R_2) ", "3a88395a7c1c5344592cc71b601ceef4": "d S = \\frac {\\partial S} {\\partial U} d U + \\frac {\\partial S} {\\partial V} d V + \\sum_{i=1}^s \\frac {\\partial S} {\\partial N_i} d N_i", "3a88699b9a848582f0cf926f832cb802": "x_i^*=\\tfrac{1}{2}(x_i+x_{i-1})", "3a887b53f7ea2eb580eb21c51bbea577": "n\\in\\mathbb{Z}", "3a889f4431b76ac41b97c549af86d88c": "\\begin{align}\ny_{n+1} &= y_n + \\tfrac{1}{6} h\\left(k_1 + 2k_2 + 2k_3 + k_4 \\right)\\\\\nt_{n+1} &= t_n + h \\\\\n\\end{align}", "3a88c48ac0d252d0d8fdd62c6e05bd47": "n = p_1^{c_1} \\times p_2^{c_2} \\times \\cdots \\times p_k^{c_k}\\qquad (1)", "3a88f98aba82389e47e287bef0e048b7": "q = \\rho\\overline{\\rho}", "3a893cc3e478f5accfa5e3ca02ff46f4": "Z^0(G)=Z^1(G)", "3a8956d17cd171548b234fe25e240994": "A_\\rho, A_\\phi,", "3a8965be1ac56626bc1692ae92c93647": "\\scriptstyle |d| < r", "3a896cf58ffe62fc96389e8504e607aa": "\\{0\\},", "3a896d0158ff979e6bfd709d9796996d": "\\ {}^{235}\\mathrm{U} + n \\longrightarrow {}^{95}\\mathrm{Sr} + {}^{139}\\mathrm{Xe} + 2n + 180\\ \\mathrm{MeV}", "3a89936ff05197101b2b57091c7828b8": "P_{em}", "3a8a1514cdebaf43fcd67fce1c8c362e": "S_{z_B}", "3a8a3761eb84831e2e26fd63cc42851c": "C_n:", "3a8a7f5d959a2bac88f83243b21858bf": "M(x) = -\\tfrac{q}{8}(L^2-5Lx+4x^2)", "3a8ae31b05f9e0bc829b7987199ba192": "cz^\\lambda", "3a8b26c327721d577868e63667f980ec": "n \\approx m/2", "3a8b5e24f4a49b57c72df71d6ae4934f": "\\begin{matrix}\\ln\\Beta(\\alpha,\\beta)-(\\alpha-1)\\psi(\\alpha)-(\\beta-1)\\psi(\\beta)\\\\[0.5em]\n+(\\alpha+\\beta-2)\\psi(\\alpha+\\beta)\\end{matrix}", "3a8b62455c6ecded94521bcc130ce281": "\\mathrm{ad}_x", "3a8b656272f0c469b0dd8f229922ae69": "d \\Phi = \\frac {P} {T} d V + \\frac {U} {T^2} d T", "3a8bb050c0ee9f43a9e10cb5ea7f790d": "\\begin{align}Z \\langle \\sigma_A \\rangle \n&= \\int d\\mu(\\sigma) \\sigma_A e^{-H(\\sigma)} \n= \\sum_{\\{k_C\\}_C} \\prod_B \\frac{J_B^{k_B}}{k_B!} \\int d\\mu(\\sigma) \\sigma_A \\sigma_B^{k_B} \\\\\n&= \\sum_{\\{k_C\\}_C} \\prod_B \\frac{J_B^{k_B}}{k_B!} \\int d\\mu(\\sigma) \\prod_{j \\in \\Lambda} \\sigma_j^{n_A(j) + n_B(j)}~,\\end{align}", "3a8bb1ee0740fc2d419b75d518491c13": "\\langle X, \\mathcal{F} \\rangle", "3a8c43c69124bed266cf80c2ff659955": "s = c + 1", "3a8c51fd4977b70fa7b14ffaf57f8dee": "2\\uparrow^m n", "3a8cccd2a57a286d5face12a2714e0ef": "\\pi_1: G \\times H \\to G\\quad \\text{by} \\quad \\pi_1(g, h) = g", "3a8cd1e2641eaf1b838dab42086161eb": "[1,m]", "3a8cfb9d21045027826ab6ee19f6f52c": "\\color{NavyBlue}\\text{NavyBlue}", "3a8d0ee2dd155d3bbf71843c5bc2fd96": "\\min_w F(w) + R(w),", "3a8d3ef25c03257a4dfd0daa3de06a6c": "\\gamma_1, \\dots, \\gamma_{2g}", "3a8d3f95c6abc653bfa78a1b2d236203": "g = (g_0, g_1, \\ldots, g_n).\\,", "3a8d48efd7b509326859dde61f416618": "n'=(3\\cdot n^2 - 2\\cdot n^3) \\ \\bmod{10^{2k}}\\, .", "3a8d8ec90ca2d552f7f63b3627032a7b": "y=ax+b", "3a8d98b5dfeba71a34cfe900c1859271": "\\sum_{i,j \\in V}(LML)_{i,j}\\Omega_{i,j}=-2\\operatorname{tr}(ML)\\,", "3a8de82ec211a84655f127027e8215db": " T_2 = T_1 \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} ", "3a8dfae4bd1bcfb20567eb38a6534730": "\n \\boldsymbol{S} = \\varphi^{*}[\\boldsymbol{\\tau}] = \\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\tau}\\cdot\\boldsymbol{F}^{-T}\n", "3a8e0a621795a7460095f454725099c4": "\\scriptstyle z \\;=\\; i", "3a8e4626d9c3c722003e122c9a2a54df": "\\psi_1(\\psi_1(0))", "3a8e932486badec668d69a480f2bc1fd": "RTS(O_j)", "3a8e9b0430909512398e795b2c4a1e6b": "\\Sigma^y=\\sigma^2 \\mathbf{I}", "3a8ea1f81e88c4a64bc9f4471c7e13af": "\n\\begin{bmatrix}\n-1 & 0 & 0\\\\\n0 & -1 & 0\\\\\n0 & 0 & -1\n\\end{bmatrix}\\text{ and }\n\\begin{bmatrix}\n0 & -1 & 0\\\\\n1 & 0 & 0\\\\\n0 & 0 & -1\n\\end{bmatrix}", "3a8ed117dd98710136027c15ee81d3cf": "IJ(2+J) + IJ(2+I) = I^2J + IJ^2 + 4IJ \\, ", "3a8eecd49c6a694159ef58c51b3d05da": "\n\\mathcal{S}[\\mathbf{q}(t)] = \\int_{t_1}^{t_2} L[\\mathbf{q}(t),\\dot{\\mathbf{q}}(t),t]\\, dt\n", "3a8f1fe00b4a5d5619405d30f09bb903": "M_y=\\{p\\in M:f(p)\\le y\\}\\ ", "3a8f4645dace9bca8a1de93cfbf28b7e": "~~~\nn_2\\sigma_{\\rm e}(\\omega) v(\\omega)D(\\omega)+n_2 a(\\omega)=\nn_1\\sigma_{\\rm a}(\\omega) v(\\omega)D(\\omega)\n~~~~~~~~~~~~~~~{\\rm (balance)}\n ", "3a8f4dd05831985edcc1f6500d2a1cdb": "A\\in\\mathcal S_n(\\mathcal A_{1\\cdots n},\\mathcal C_{1\\cdots n})", "3a8f6191d6b82d32e70bfccb923d1ce1": " \\lim_{N \\to \\infty} Q_N = I", "3a8f9a292ec183accfc58e87cff13574": "q_{\\alpha\\beta} = \n\\begin{pmatrix}I_{k\\times k} & 0 \\\\ 0 & -I_{(n-k)\\times\n(n-k)}\\end{pmatrix}\n", "3a8faa47e71ff2e7683ab141dd220a65": "1-2c_{ij}/c_{ii}\\ ", "3a90188847b802d6fd1905998e265044": "\\det P = r = |\\det A|", "3a9045ac68b481bd0dd422fedfd885a3": " W = \\int_C \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{s}", "3a9105eed28140272ae4c361ad1edb1f": "[t_1,\nt_4]", "3a9116cb5da281e7c68875b4a442ccbb": "R_h=\\frac{\\left|m_i\\right|}{s\\cdot n}", "3a911d9ef04d5a3d96507d7323daa7ff": "G_0 = 2 e^2 / h \\,", "3a9164d4bf12e44717c738c757f252b6": "L(a1,a2,a3,...)=(a2,a3,a4,...)", "3a91a592378e148015a0b43ac50e0048": "f(x) = {\\left \\vert \\Im{\\theta} \\right \\vert \\over \\pi \\left \\vert x-\\theta \\right \\vert^2} \\,,", "3a91afef8cec66278d969ba02e831ad3": "\\dot{z}_2 = f_2(\\mathbf{x},z_1,z_2) + g_2(\\mathbf{x},z_1,z_2) u_2(\\mathbf{x},z_1,z_2)", "3a91c73e850fbb4d0466af260ed8b0bd": "\\Phi_{\\hat{a}\\hat{b}} = \\frac{m}{r^3}\\operatorname{diag}(-2,1,1)", "3a91c91b864360c609f7b80863306e06": "\\mathbf{S} = \\frac{1}{\\mu_0}\\mathbf{E} \\times \\mathbf{B},", "3a926707afe480f51712f1b4caa880d7": "\\phi_{e3}", "3a92726be151afcc41ceeafa5d2cb829": "\\gamma(1)=z", "3a9289b63e6940a3dcd803331d7cea8f": "\n\\bar {v_{n}^2} = k_B T / C\n", "3a929b9aa296da9b223b7eeaee3bf9b7": "f_n\\to f", "3a92b05b240dd32936e180bf3618f6b6": "\\begin{align}\\widetilde{S}(\\widetilde{t}) &= \\mathfrak{F}[S(\\omega)]\\\\\n&= \\widetilde{E}^{dc}(\\widetilde{t}) + \\widetilde{E}^{ac}(\\widetilde{t}-\\tau) + \\widetilde{E}^{-ac}(\\widetilde{t}+\\tau)\\end{align}", "3a92bb1b888857d0eaa890f036fafc73": "\n(H-\\mu N)\\psi_\\alpha|n \\rangle = (E_n - \\xi_\\alpha) \\psi_\\alpha |n \\rangle,\n", "3a9325dfd5edefa5786eabe224e01c11": "\\tilde{\\mathbb{P}}[X(T)=V(T)] = 1", "3a932f8a3ee16a7548ed7ecd9ec4626f": "\\mathbf{E}(\\mathbf{x},t)=\\mathbf{E}(\\mathbf{x})e^{-i \\omega t}", "3a9357810b0b7d9990c10747fac77ee8": "\\gamma(1)=y", "3a93787dc7670fae11098f481fa88021": " \\mathbf{r}_{k} ", "3a937dba36cc8500cb0c2b35b7ef44fe": "q_f", "3a93e247421717ac71c87d6479af951c": "10", "3a9542ef1bad7e6b15bc371db0ead980": "x, x^p, x^{p^2}, x^{p^3}, \\ldots.", "3a9598252f86d48eba2d18ac65a1ce57": "\\left(\\frac{\\alpha}{\\mathfrak{p} }\\right)_n = \\left({\\pi, \\alpha}\\right)_{\\mathfrak{p}} ", "3a96bb0e3410ad5c7fd20335dd3a0a30": "\\forall x \\lnot \\phi", "3a96c26d13326b35a184d2c96c715e2d": "\\varphi\\ ", "3a970230482d365d2b13ed1c14e25c11": " T_B^{\\mu\\nu}", "3a9721e15e5e5d769baefa2506f3968d": "\\mathcal{F}(M)", "3a97586844c71aa5c78faf43668ba12f": "-\\textstyle\\frac{1}{2}(\\textstyle \\sum_{i=1}^8e_i)", "3a97825a59be156cef66fd1fe178ae27": "X_i \\sim N(0,\\sigma_i^2), \\qquad i=1, \\dots, n,", "3a979652830dd7e6ef1be6dcc438726a": "\\textstyle{\\sum_{N}}", "3a97c596017ff8dc12a8932ed026c331": " \\phi_i \\leftarrow (1-\\omega)\\phi_i + \\frac{\\omega}{a_{ii}} (b_i - \\sigma)", "3a97dbf6b33a052a183d7fb02755ebcf": "\\gamma_{\\mathrm{LG}}", "3a97e3ab6434f04b72dcb60ebd0fa625": "B(\\epsilon) \\to \\epsilon", "3a98002bf571b18786c905b8b7a83719": "u_S : _SM \\to _SN", "3a981b49f9f6643b3f4ef3aa6d6b291b": "\\scriptstyle\\mathrm{Distance\\ in\\ parsecs}=\\frac{1}{\\mathrm{parallax\\ in\\ arcseconds}}", "3a98556a93d244ec28e5fa3fcdba8c01": "\\omega_0 ", "3a9862040e205f68fecc082ab29cfa45": " A = U - TS ~,", "3a997370044e175887b2dd6adbea891f": "\\mathbf{A} = * \\mathbf{a} \\,,\\quad \\mathbf{a} = * \\mathbf{A}", "3a997b172551c30d97e4258a7fd0946a": "\\Phi_{J} = \\frac{1}{2}\\lambda^{-1}\\omega.", "3a9996be06f7fbefed019f218c933ea5": "k^{r} \\propto \\frac{1}{y(t)} ", "3a99ebf762b4e80febfc24dab8e530a8": " T_{ab} = 0 ", "3a99ff037ec70a58c2069882f8c3166e": "\\arctan (1/x) = -\\tfrac{1}{2}\\pi - \\arctan x = -\\pi + \\arccot x,\\text{ if }x < 0 \\,", "3a9a5965059dbf07494d342b1bfd6c81": "\n\\widehat H(\\omega) = i H(\\omega)\n", "3a9a6a2e4acdeab5673fa2e2ec46ffd8": "f(x,y) = 2x^{2} - 1.05x^{4} + \\frac{x^{6}}{6} + xy + y^{2}.\\quad", "3a9a942d894e3cbac81256c64014d94e": "\\text{Color Intensity}=\\frac{A*100}{TS}", "3a9ad0adc09b3b8bd1976c84191d3c02": "H_{\\alpha_1}, \\ldots, H_{\\alpha_n}", "3a9aec8351672aa3cc0232f76054a73b": "t^2 = t_0^2 + \\frac{x^2}{v^2}", "3a9b4791b744ac04918f1a885406201e": " S_{m}^{G} ", "3a9b627805f021b6fb35833aa08ffaa3": " Z', i', j'", "3a9b958c8d96c51c046d7c34626eadf4": "d= 0.46", "3a9c0e52292d6e16e516190177a376f2": "g_{ij}", "3a9c312c721a8df7f83640059284241e": "\n\\begin{align}\n R(x)A_{(h_0,\\ldots,\\;h_n)}(x)\\cdot(y_0)^{h_0}\\cdot\\ldots(y_n)^{h_n}&=x^{h_0+\\ldots+h_n}A_{(h_0,\\ldots,\\;h_n)}(x+1)\\cdot(y_0)^{h_0}\\cdot\\ldots(y_n)^{h_n}\\\\\nR(x)A_{(h_0,\\ldots,\\;h_n)}(x)&=x^{h_0+\\ldots+h_n}A_{(h_0,\\ldots,\\;h_n)}(x+1)\n\\end{align}\n\\!", "3a9c678b7cfc2797473b3a5824cc17af": "V^*", "3a9c9c0a69a8f79582f8a687683dcf65": "\\left(\\boldsymbol{U}+\\boldsymbol{c}_g\\right)\\, \\mathcal{A}", "3a9ca1ce27d17c285e350dfe8cddd3bc": "S^3 \\hookrightarrow S^7\\to S^4.", "3a9cbfe10bcf8e3922b752dd9a8934e5": " Y_\\ell^m (\\theta, \\phi) = N_\\ell^{|m|} P_\\ell^{|m|}(\\sin(\\phi))\\cdot\\begin{cases}\n{\\sin(|m|\\theta)} & \\mbox{if } m<0 \\\\\n{1} & \\mbox{if } m=0 \\\\\n{\\cos(|m|\\theta)} & \\mbox{if } m>0\n\\end{cases}", "3a9cda5aa10b47faadefb65e4b684f57": " e^{i \\frac{2\\pi}{N}}", "3a9d47b1239bd2331911b054dbf91374": "C_{QA} = \\frac{4\\pi }{R_H\\omega_B}. \\ ", "3a9d4a0915283215a4551a1fd0b2e732": " \\Phi_{00} = \\frac{1}{4} \\, \\left( H_{xx} + H_{yy} \\right)", "3a9d5162e7f51a4480c857d726e7444a": " \\lambda_0 = \\frac{v}{f_0}", "3a9d548bff4ba5093cf4c4ff76ce3ce3": "(X_n+c)^+=(X_n+c)+(X_n+c)^-\\le X_n+c+X_n^-1_{\\{X_n^->c\\}},", "3a9d776652a831bbc04869d36f4b32ba": " 9^2 -4 \\frac{1}{2} (3) (3) = 63", "3a9d86d6bf3b913d43ae576edaba04c6": "\\pi_{ij}>0", "3a9dae3b8f73a063a2aba317a88ce1f2": "O(n^{\\lfloor d/2\\rfloor}k^{\\lceil d/2\\rceil})", "3a9dcf03e18d2d44d732235ee3dd3304": "\\beta^\\star", "3a9e9bfbf2f5650ca95773971d1fb49f": "\\chi_k = 1", "3a9efbba39b4cbb89488ef743e1ae2d7": "F_{p,n-p, 1-\\alpha}", "3a9f05068280f92dd147a27562752197": "g(x) = x/s", "3a9f128344fff6ea6bb450cc2e35753d": "I(\\mathbf{v})", "3a9f7897636eb992cdf58628e1b94e17": "f(\\mathbf{x})=\\mathcal{S}\\mathop{\\otimes}_{n=1}^N\\mathbf{w}_n(x_n),", "3a9f7b68e5353e4c293ee70f8e210006": "i:A \\to B \\to D", "3aa02ee9e31e23443e1ba37416e6ade9": "\\begin{align}\nQ(\\varphi,\\psi) &= (-1)^n Q(\\psi, \\varphi); \\\\\nQ(\\varphi,\\psi) &=0 && \\text{ for }\\varphi\\in H^{p,q}, \\psi\\in H^{p',q'}, p\\ne q'; \\\\\ni^{p-q}Q \\left(\\varphi,\\bar{\\varphi} \\right) &>0 && \\text{ for }\\varphi\\in H^{p,q},\\ \\varphi\\ne 0.\n\\end{align}", "3aa039df38d7c6c90f666294ce19977f": " \\mathbf{y}_{2} ", "3aa03cbdd2016ad8974e498cffd0c5a1": "\\mathbf{L} = \\mathbf{r} \\times m\\mathbf{v} \\, .", "3aa0792b606fd40433ccfe89df20a9ab": "\\ln \\frac{a \\Gamma(d/p)}{p} + \\frac{d}{p} + \\left(\\frac{1}{a}-\\frac{d}{p}\\right)\\psi\\left(\\frac{d}{p}\\right)", "3aa09933cdbf8241980f0e3669aaeaee": "S=\\mathbf{r^TWr},", "3aa0a620fd7704b9414b5e45dd1cc13d": "\\frac{\\partial \\textbf J}{\\partial a}=-\\int_0^{\\frac{\\pi}{2}} \\frac{\\cos^2 x\\;\\mathrm{d}x}{\\left(a\\cos^2 x+b \\sin^2 x\\right)^2}\\,", "3aa0ca83044b6f0a6b1bb0ca4ae928c7": " Inv Fx_1...x_n \\leftrightarrow Fx_nx_1...x_{n-1}.", "3aa0cb7c81773c3249c2e6494a8ab40e": "h_i(x) = a_i^T x + b_i", "3aa0e8100961d92cd457c598b0b9dfdd": "\\mathbf{N}\\,\\!", "3aa114ccf60c1b8352801a6d33bcbdee": "H_\\alpha", "3aa1618ac126313c8ce86aa2d15a3f76": "\\Delta\\tau = D_\\text{PMD} \\sqrt{L} \\, ", "3aa16ce614f752aaf90af1b8b143c3e5": "g^{0 \\beta}g_{\\beta 0} + g^{00}g_{00} = 1.\\,", "3aa17d21b70c1cd00e8f8f76ec1aeb2a": "m = \\frac{p - 1}{2},\\;\\; n = \\frac{q - 1}{2}.", "3aa2880d43cc36fa052facca599beaa6": "\\stackrel{\\bigwedge}{\\vee}", "3aa28cb968a2ff701bc1f55c28bdb7b0": " E_i", "3aa28fd1661e7192f5cdc3197aa3d668": "f(x_i|\\theta_i)", "3aa2a487d826fcef086a22980a6259f8": "\\{(\\mathbf{a_i},\\mathbf{b_i})\\} \\subset \\mathbb{Z}^n_q \\times \\mathbb{T}", "3aa2c113f40e6bdfb627684b7187b9b8": "\\,R", "3aa2ecb8e4c114e549abbd54c88aae31": "v+w+x+y+z=v^2+w^2+x^2+y^2+z^2=v^3+w^3+x^3+y^3+z^3=0.", "3aa357bbb32de0afe7354de8e6bdfc94": " ~\\epsilon^2 ", "3aa3808c25881f9458e3605cfec53567": "B(L^{1/m};\\theta)", "3aa38f8f4ae9783ed34b29d083b24510": "A_e = A_w", "3aa417c38fc4b45aaa7c2091cf0d9aba": "b_0(-\\infty)^{n-1}\\,", "3aa4b00fd606ad6dd7891b7665cdd49d": "\\phi=\\frac{nx}{\\sqrt{2+x^2}}", "3aa4e39593dc16a8819af52193d89691": "m\\not=n", "3aa4e7cac6382738e7bbabdcaa759034": "x_1, ..., x_t", "3aa51eb929506c09eacb63eca570e66d": "d:\\textbf{Q}\\times\\textbf{Q} \\to \\textbf{R}", "3aa58e93bf2c7e83b3f1e74e220cb1d1": "\\ln n", "3aa599bbe8ad88b940b9321900e6b641": "p_{r+1}(x)=(x-a_r)p_r(x)-b_rp_{r-1}(x)", "3aa5a5dbbcc96cfecb0b1b4c513efbc7": " \\rho = \\frac {1}{2} \\left( \\frac {1} {\\tau_1} + \\frac {1} {\\tau_2} \\right ), ", "3aa5d6354578caa0fcd4532d540bd721": " \\Delta G = 0 \\,", "3aa5fe70e2aec2dbe9ee56956f3440b3": "= \\sum_{n=0}^\\infty a_n x^n ", "3aa6a5712deba5604170c420870b522d": " e^{i( p (q(t+\\epsilon) - q(t)) - \\epsilon H(p,q) )}\\,", "3aa6aa9d552c146738c22c332f17521e": "O_{p'}(G)", "3aa6afa60550bfdfe6eec16cfaaa556f": "\\mathrm{SOS} = \\frac{\\mathrm{2(OR)+(OOR)}}{\\mathrm{3}}", "3aa6d1d4111df237edef8aea1a2ecc9c": "F(x) = 1^n + 2^n + \\cdots + (x - 1)^n.\\,", "3aa6f0dbc630fb2168097dddd6ea699c": "\\mathbf{x}_\\mathrm{B} = R(t)\\mathbf{u}_R, ", "3aa73fc0df280fe0283909c268a0a13f": "( \\bar{r} ,\\bar{v} )", "3aa76af920eee60077d66978e006af5c": "\\displaystyle i", "3aa77a84ced9ec5c54e5d963ec8785f2": "\n \\begin{align}\n \\boldsymbol{\\sigma} & = \n \\cfrac{2}{\\sqrt{I_3}}\\left[\\left(\\cfrac{\\partial\\hat{W}}{\\partial I_1} + I_1~\\cfrac{\\partial\\hat{W}}{\\partial I_2}\\right)\\boldsymbol{B} - \\cfrac{\\partial\\hat{W}}{\\partial I_2}~\\boldsymbol{B} \\cdot\\boldsymbol{B} \\right] + 2\\sqrt{I_3}~\\cfrac{\\partial\\hat{W}}{\\partial I_3}~\\boldsymbol{\\mathit{1}} \\\\\n & = \\cfrac{2}{J}\\left[\\cfrac{1}{J^{2/3}}\\left(\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_1} + \\bar{I}_1~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}\\right)\\boldsymbol{B} - \n\\cfrac{1}{J^{4/3}}~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}~\\boldsymbol{B} \\cdot\\boldsymbol{B} \\right] \\\\\n & \\qquad \\qquad + \\left[\\cfrac{\\partial\\bar{W}}{\\partial J} - \\cfrac{2}{3J}\\left(\\bar{I}_1~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_1} + 2~\\bar{I}_2~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}\\right)\\right] ~\\boldsymbol{\\mathit{1}} \\\\\n & = \\cfrac{2}{J}\\left[\\left(\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_1} + \\bar{I}_1~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}\\right)\\bar{\\boldsymbol{B}} - \n\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}~\\bar{\\boldsymbol{B}} \\cdot\\bar{\\boldsymbol{B}} \\right] + \\left[\\cfrac{\\partial\\bar{W}}{\\partial J} - \\cfrac{2}{3J}\\left(\\bar{I}_1~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_1} + 2~\\bar{I}_2~\\cfrac{\\partial\\bar{W}}{\\partial \\bar{I}_2}\\right)\\right] ~\\boldsymbol{\\mathit{1}} \\\\\n & = \\cfrac{\\lambda_1}{\\lambda_1\\lambda_2\\lambda_3}~\\cfrac{\\partial\\tilde{W}}{\\partial \\lambda_1}~\\mathbf{n}_1\\otimes\\mathbf{n}_1 + \\cfrac{\\lambda_2}{\\lambda_1\\lambda_2\\lambda_3}~\\cfrac{\\partial\\tilde{W}}{\\partial \\lambda_2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2 + \\cfrac{\\lambda_3}{\\lambda_1\\lambda_2\\lambda_3}~\\cfrac{\\partial\\tilde{W}}{\\partial \\lambda_3}~\\mathbf{n}_3\\otimes\\mathbf{n}_3\n \\end{align}\n ", "3aa77f6a01551215b6f23fb3a224ccfc": "u=\\tfrac{1}{2}\\rho\\langle (w_i-V_i) (w_i-V_i) \\rangle", "3aa78859b57f6911cd42f149d87bf798": " z \\mapsto z^d + c . \\, ", "3aa79c0e3ec611c964197281311261d3": "k=k_0\\,", "3aa7bbb459de1a5faf0d8a79387bcce9": "Ext_\\sigma(S)", "3aa80d7d51539d94036ba2e47b2c0c49": "V_i", "3aa82d7156a57d3a350444db496f9d97": "\\mathbf{H} = {1\\over {2m}}\\sum_k \\left(\n{ \\Pi_k\\Pi_{-k} } + m^2 \\omega_k^2 Q_k Q_{-k} .\n\\right)", "3aa877fdbaef3894607b3c5796b77491": "[5, 12)", "3aa8b21f43aaf47e7234b5cab10c75de": "\\rho(x,x_\\epsilon)<\\epsilon", "3aa8c94eeb6229ae05b7781506cda472": "\\frac{a+b+c}{b+c}+\\frac{a+b+c}{a+c}+\\frac{a+b+c}{a+b}\\geq\\frac{3}{2}+3", "3aa92cddbdd78b69b842153f59206c0f": "(\\phi\\to\\psi) \\leftrightarrow ((\\phi\\land\\psi) \\leftrightarrow \\phi)", "3aa949c638cf5b960caa3ab34b15d1f8": "\\oint_S \\mu_0 \\mathbf{M} \\cdot \\mathrm{d}\\mathbf{A} = -q_M", "3aa95c2759ff5b4092ac59686e321d2e": " H= \\int d^3\\vec r \\hat\\psi^\\dagger(\\vec r) \\left ( -\\frac{\\hbar^2}{2m} \\nabla^2 +V_{latt.}(x) \\right) \\hat\\psi(\\vec r) \n + \\frac{g}{2}\\hat \\psi^\\dagger(\\vec r)\\hat\\psi^\\dagger(\\vec r)\\hat\\psi(\\vec r)\\hat\\psi(\\vec r) - \\mu \\psi^\\dagger(\\vec r)\\hat\\psi(\\vec r)\n", "3aaa106af3b38fdb01b16564a3ffefa3": "\n \\overline{u}_S\\, \\approx\\, \\omega\\, k\\, a^2\\, \\text{e}^{2 k z}\\, \n =\\, \\frac{4\\pi^2\\, a^2}{\\lambda\\, T}\\, \\text{e}^{4\\pi\\, z / \\lambda}.\n", "3aaa129b8c07cf841b9a3971fa9a12dd": "f_0 = {1 \\over 2 \\pi \\sqrt {L \\left ({ C_1 C_2 \\over C_1 + C_2 }\\right ) }}", "3aaa734d7bde9b6a8ece80d64088b7bd": "R_q = \\sqrt{ \\frac{1}{n} \\sum_{i=1}^{n} y_i^2 }", "3aaaad391b68ef32975f800fb4472dc9": "H_{1t},H_{2t},\\ldots,H_{c-1,t}", "3aaacbfc55425de4697748d2491a406e": "u: A \\to M", "3aaae7a260eec4c2b5a76241129300a6": "\\mathfrak{gl}_n = \\mathfrak{sl}_n \\oplus \\mathfrak{k},", "3aab350a2b5b876512ab7ee5ae8369b5": "\\Delta(y, E(m^{\\prime})) \\leq \\Delta(y, E(m))", "3aaba9c36d6f074e6866eaa60b37f03e": "h_i = \\frac{(c_i - c_\\text{batch})m_i}{c_\\text{batch} m_\\text{aver}} .", "3aabb34947d33e57086c8c361d0af704": "\\frac{1}{2}(XY-YX)=iZ", "3aabdd99fa487e3c509793eb2acd5ce2": "condition_{j-1}", "3aabeeca0975178c49a8282161d18f2b": " \\overrightarrow{D_j} ", "3aabf03b6a2c7a4ab67c92bd119fa5d4": "\\rho_\\text{fluid}", "3aabf9f2a53ad1c9a68ce60bee5eabff": " S(n,k)=\\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}= S_n^{(k)} \\,", "3aac04e575f45b965a1c6f445d791dc8": "\\Gamma_k ", "3aac8fde102d7a64b248477f3cf37220": "\n\\mathrm{Fr}=\\frac{\\text{centripetal force}}{\\text{gravitational force}}=\\frac{mv^2/l}{mg}=\\frac{v^2}{gl}\n", "3aac944e31e8fe19081ced8e1b40d25f": "\\begin{align}\n\nK_n^{(\\alpha)}(x,y)&{:=}\\frac{1}{\\Gamma(\\alpha+1)} \\sum_{i=0}^n \\frac{L_i^{(\\alpha)}(x) L_i^{(\\alpha)}(y)}{{\\alpha+i \\choose i}}\\\\\n\n&{=}\\frac{1}{\\Gamma(\\alpha+1)} \\frac{L_n^{(\\alpha)}(x) L_{n+1}^{(\\alpha)}(y) - L_{n+1}^{(\\alpha)}(x) L_n^{(\\alpha)}(y)}{\\frac{x-y}{n+1} {n+\\alpha \\choose n}} \\\\\n\n&{=}\\frac{1}{\\Gamma(\\alpha+1)}\\sum_{i=0}^n \\frac{x^i}{i!} \\frac{L_{n-i}^{(\\alpha+i)}(x) L_{n-i}^{(\\alpha+i+1)}(y)}{{\\alpha+n \\choose n}{n \\choose i}};\\end{align}", "3aacb0b01865de89a0221d49394024ff": " X \\pm ( \\frac{ 2 }{ \\alpha } - 1 ) | X - \\theta | ", "3aace47c6e198f82737729e0fcba040f": "GS_{Q}=\\max_{D_1,D_2:d(D_1,D_2)=1}|(Q(D_1)-Q(D_2))|\\,\\!", "3aad1407bbf9495dcbc479519ee9b71d": "\\textstyle F(t)", "3aad20b73c77a2dbd4a3d85f3c772a98": "\\tau_\\mathrm{n} = -\\frac{1}{2}(\\sigma_x - \\sigma_y )\\sin 2\\theta + \\tau_{xy}\\cos 2\\theta\\,\\!", "3aad2b99ca4049e29897e0e19c26d9ec": " W := (M\\times I) \\cup_{\\mathbf{S}^p\\times \\mathbf{D}^q\\times \\{1\\}} (\\mathbf{D}^{p+1}\\times \\mathbf{D}^q)", "3aad639ab0f909d27dc7fefbfc3b1b71": "F_2=aS(a^{-1})\\cup S(a), \\, ", "3aad6aad2201ef193ebec97a7484a1ca": "\\det D", "3aad6d30992c2ae2dbf275a5b7d28197": "\\left(n_1, n_2, n_3 \\right)", "3aad7f222db895c491222cd0bc7247e4": "(1 + 4\\pi^2 |\\xi|^2)^{-\\alpha/2}", "3aad9aec688ad6f000dc2e0b903b8096": "p_1 \\epsilon_1^1(\\mathbf{p}) + p_2 \\epsilon_2^1(\\mathbf{p})\n+ p_3 \\epsilon_3^1(\\mathbf{p})", "3aadb3b9b2a18420300ebcad28caa9ae": "q_2 * (5000-q_1-q_2-c_2)", "3aadbb9f88489c36868bb920a3c35ac4": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 10.38248\\log_e(T+273.15) - \\frac {6904.904} {T+273.15} + 83.96795 + 8.368130 \\times 10^{-6} (T+273.15)^2", "3aadd6e923d44694836234937ad5ba5c": " \\sin(\\theta)/2 = \\sin(\\theta /2)\\cos(\\theta /2) ", "3aade55c0f789a0e9bdf0fccc15f075d": "4^{A-r}", "3aae1aac794d3a7adf594196685c25ae": "\nH = \\sum_{i}h_i Z_i + \\sum_{i0 \\\\\n A e^{-Bx^2 + Cx} & \\text{if}\\ d=0\n \\end{cases}\n ", "3aaff538c1192f2eb24913657d6fb1b5": "M _{CD} ^f = - \\frac{PL}{8} = - \\frac{10 \\times 10}{8} = - 12.500 \\ kN\\cdot m", "3aaffb10a87303e3b5271c99275ba55b": "u'\\,", "3ab0b244a3551a46c4a4af6c56874134": "{ P }_{ in }={ P }_{ out }", "3ab0e089ae6d09686a46c6753faed93c": "\\lambda_n^{(c)}", "3ab106501ed555cb732dc656a9a06528": " \\left|{\\partial \\mathbf{x} \\over \\partial \\lambda} \\right| = \\sqrt{h_{ki}h_{kj}\\cfrac{\\partial q^i}{\\partial \\lambda}\\cfrac{\\partial q^j}{\\partial \\lambda}} = \\sqrt{ g_{ij}\\cfrac{\\partial q^i}{\\partial \\lambda}\\cfrac{\\partial q^j}{\\partial \\lambda}} = \\sqrt{h_{i}^2\\left(\\cfrac{\\partial q^i}{\\partial \\lambda}\\right)^2} ", "3ab10cce9975ec0ca74964a5a8d87d6b": "F_i, i=1,2", "3ab118e9a93fe3a9b998bd41f442b0fe": " \\mathbb{R}^3", "3ab1968c1e1c30ed6013b01797cc08c9": "G = \\frac{I}{V}", "3ab1b43211a55c5aa8e975bfa8039896": " \\frac{D\\rho}{Dt} + \\rho \\nabla \\cdot \\mathbf{u} = 0 ", "3ab2187d7c0eb9f30a344b1403041a73": "\\text{Ch}(F(w\\lambda)) = \\Delta_1\\Delta_2\\cdots\\Delta_ne^\\lambda", "3ab28cb7d71941bc9e8b8f904c15525a": " m ", "3ab2da9c7d9c571da804308ab96ac0ad": "r^{-1}", "3ab2e9d8336c9a9e82bb390cfb103121": "g_{55}", "3ab2f1aa8f7e191038eae36fcbffbe1a": "F^\\times\\cong\\mathbf{Z}\\oplus\\mathbf{Z}/(q-1)\\oplus\\mathbf{Z}/p^a\\oplus\\mathbf{Z}_p^d", "3ab30a74fabc879730d102ca70cf7045": "U_H=TU_R=kU_O|U_R|^2+k|U_R|^2U_R+k|U_O|^2U_R+ kU_O^*U_R^2", "3ab3665f01a0756830f7ac746b16e12e": "0 \\leq \\omega(x, D)(E) \\leq 1;", "3ab3d0ff75d5d08c73624b037fdf8914": "\\begin{matrix} {3 \\choose 1}^2{10 \\choose 1}{4 \\choose 2}{36 \\choose 1} \\end{matrix}", "3ab3f4569cebd730164f196bfd1bd33b": "A = \\pi r^2 \\cdot \\frac{\\theta ^{\\circ}}{360}", "3ab3f58a4d1d0265386666e4a0594a47": " \\phi : (\\mathbb{C}^n,0) \\to (\\mathbb{C}^n,0)", "3ab418988c2192e6e195bc4c5dca7aae": "\\mathbf{select}_q(x)= \\min \\{k \\in [0 \\dots n) : \\mathbf{rank}_q(k) = x\\}", "3ab4c5b6259b9964286e8f722384c404": "\n\\begin{align}\n0 &= E^o - \\frac{RT}{nF} \\ln K\\\\\n\\ln K &= \\frac{nFE^o}{RT}\n\\end{align}\n", "3ab4ff16f8e0e4bcbd6f759380d7743c": "(F_i, y_i),\\ i=1,\\dots,n\\!", "3ab5144beb738f1c2525ba5edf855da9": "\\frac{\\partial A}{\\partial t} = \nA + (1 + ib)\\frac{\\partial^2 A}{\\partial x^2} \n- (1 + ic)|A|^2 A", "3ab55ff781f2928ab795b6c8fdcf3266": "p_{HB}", "3ab581a82f736f497a3b06c25c377d9a": "\\mathrm{\\tfrac{u\\bar{u} + d\\bar{d}}{\\sqrt{2}}}\\,", "3ab58631bad750e815c7d325d14b8d18": "\\frac{\\partial \\log(p(v))}{\\partial w_{ij}}", "3ab59a4ff434f7e6fa659a319d0c5535": "\\mathcal{O}(1)", "3ab5b367be0ce60dac1d9d859b5d43ef": "(a, b) \\in M", "3ab5d003adcd5fa6e7dccfea3a5c404d": "\\frac{d}{d\\theta}\\operatorname{Cl}_2(\\theta) = \\frac{d}{d\\theta} \\left[ -\\int_0^{\\theta} \\log \\Bigg| 2\\sin \\frac{x}{2}\\Bigg| \\,dx \\, \\right] = - \\log \\Bigg| 2\\sin \\frac{\\theta}{2}\\Bigg| = \\operatorname{Cl}_1(\\theta) ", "3ab5fecaf618a07f4cb628edef1dc95e": "\nU_{kl}^{AB} \\ \\stackrel{\\mathrm{def}}{=}\\ E_{k}(r_{k}^{A}) + E_{l}(r_{l}^{B}) + E_{kl}(r_{k}^{A}, r_{l}^{B})\n", "3ab648a7bbea40689e5c539552f3f26e": "E\\, =\\, E_\\text{pot}\\, +\\, E_\\text{kin}\\, =\\, \\frac12\\, \\rho\\, g\\, a^2.", "3ab66bfbb29eaee5e2966b6d7cfd1665": "{}_2F_1 (a,b;c;1)= \\frac{\\Gamma(c)\\Gamma(c-a-b)}{\\Gamma(c-a)\\Gamma(c-b)}, \\qquad \\Re(c)>\\Re(a+b) ", "3ab68f4127f77789ce129046db0d745f": " \\rho = \\tfrac{1}{2} | R \\rangle \\langle R | + \\tfrac{1}{2} | L \\rangle \\langle L |. ", "3ab69d3fb95949646b9914b85116a2bb": "v_1=w_1", "3ab6c2eba22c1f55b1e40a676d6a23ad": "Cr \\lbrace B \\rbrace + Cr \\lbrace B^{c} \\rbrace = 1", "3ab6d95c0a624cfc3ad5d1aef746e111": "\n{\\rm E}[z]\\,\\,\\, = \\,\\,\\,\\mu _z \\approx \\,\\,ae^{b\\,\\mu } \\,\\,\\, + \\,\\,\\,\\frac{1}{2}\\,\\,a\\,b^2 e^{b\\,\\mu } \\,\\,\\frac{{\\sigma ^2 }}{n}", "3ab7199f22debe2451782d696eff1227": "\\vec{V}.", "3ab762429bd8c35fd902895b52322863": "N \\cdot N^r", "3ab7cdda8d543c8c8cf66f493dcdfc5c": "\\ A_q = \\frac {L} {R_q} = \\frac {L} {R_0(1-q)}.", "3ab7cedc455bdeface51e2f2dce7646e": "1+z = \\frac{f_{\\mathrm{emit}}}{f_{\\mathrm{obsv}}}", "3ab7e6ec69d27af8bb74ad58c1e1bc46": "m=\\frac{\\text{change in } y}{\\text{change in } x} = \\frac{\\Delta y}{\\Delta x},", "3ab84b8ae4f4c37a1ece0c6d4d0c9ec6": "\\operatorname{div}(\\mathbf{F})=\\nabla\\cdot\\mathbf{F}", "3ab873c9300cfd78d9f5514e7bb47c24": "(a_R)^\\alpha_{\\bar a\\bar b}", "3ab8a0345c4e6e6a03b1955dbebc7cae": "|{\\tilde{\\psi}_{Tr}}\\rangle = \\sqrt{1-{\\epsilon}^2}|{\\psi_{Tr}}\\rangle + {\\epsilon}|{\\psi^{\\bot}_{Tr}}\\rangle", "3ab8bc56542b323fcf00d7e098cda019": "\\square \\Phi = 4 \\pi G \\rho", "3ab8cbfc10826041e67ab80e8b6b454e": "\nC_\\mu (s,t)=\\sum\\limits_{n=0}^\\infty e^{isn}P_\\mu (n,t)=E_\\mu (\\nu t^\\mu (e^{is}-1)). \n", "3ab950a2a6b7f37d717008369c87fa0b": "E - E_i", "3ab9593682f53ec843155bdacc5edb52": " Ne ", "3ab9acd55b8b497be7379c5d2d0b07f3": "\\Lambda \\, \\lambda \\,", "3ab9bd214b424651d374ee72d7f65e4a": "\\frac{ds}{dt}", "3ab9f1e7105cbb5f05859f7970b4d8a2": "auth_i", "3aba557ab5ab117e3d4b71b9af5f9b7a": "\\mathbf{u}\\;", "3aba8365b5ccd3e6d89813bf64dfd53c": "\\mathbf{F=\\Delta y -\\frac{\\partial f}{\\partial r_x} r_x-\\frac{\\partial f}{\\partial r_y} r_y -X\\Delta\\boldsymbol\\beta=0}", "3ababefe32979f8046a2e13c77355c2d": "0 \\alpha\\} \\bigr) = \\lambda \\bigl( \\{ t > 0 : f^*(t) > \\alpha\\} \\bigr), \\quad \\alpha > 0, ", "3ac7967040f9e21e39f0c2e3d802e128": "X = \\begin{bmatrix}X_{00} & X_{01} \\\\ X_{10} & X_{11}\\end{bmatrix}", "3ac7f2dea11a3404777a69a25223c456": "A_{\\alpha} = \\frac1{\\alpha} ( \\mathrm{id} - J_{\\alpha} ).", "3ac81c7a32c9c6d9a833f2a328654ae0": "\\Theta = z {d \\over dz}.", "3ac855aa5418c019e2dd6edd59bf3b89": "P_e\\,= \\frac{\\rho u L}{\\Gamma}", "3ac89938dc4db2e7f464ec429412490e": " \\operatorname d M = \\frac{\\partial M}{\\partial t} \\operatorname d t + \\sum_{i=1}^n \\frac{\\partial M}{\\partial p_i}\\operatorname{d}p_i.", "3ac907ee323507be16353063342a227f": "=\\frac {1}{8}a^2\\left|\\int{\\left[3\\cos(t)+\\cos(3t)+2\\sqrt {2}\\sin^2(t)\\right]dt}\\right|", "3ac916557ddbde3e7e0f85c02b6e6b7a": "S(q = (2 k +1) \\pi/a) = 1/N", "3ac92505068109e4d653f82df59f5de0": " F_v(J) = B_v J \\left( J+1 \\right) - D J^2 \\left( J+1 \\right)^2 ", "3ac93918b5b3b0fdb5a9ce17f6dc7aed": "\\frac{1}{r}+\\frac{1}{p}=1+\\frac{1}{s}.", "3ac95dc2122cb085edf2793efebbaeec": "q_i=0", "3ac975f7a3359387d330b4608cf1706c": "\n U = \\int_0^a\\frac{bD}{24}\\left[12\\left(\\cfrac{d^2 w_x}{d x^2}\\right)^2 +\n b^2\\left(\\cfrac{d^2 \\theta_x}{d x^2}\\right)^2 + 24(1-\\nu)\\left(\\cfrac{d \\theta_x}{d x}\\right)^2\\right]\\,\\text{d}x\\,,\n", "3aca303062c04a7f1a621f073e15840b": "F(A) \\,", "3aca735bc1c976a037562a10353d134b": "dT=d{\\Pi}=pd{\\Theta}+{\\sigma}d{\\epsilon}=dT_{V}+dT_{D}^*", "3aca9b6e891c72a65576a6ca1b7117e7": " c_n = \\sum_{i=0}^n\\frac{x^i}{i!}\\frac{y^{n-i}}{(n-i)!} = \\frac{1}{n!}\\sum_{i=0}^n\\binom{n}{i}x^i y^{n-i} =\n\\frac{(x+y)^n}{n!}", "3acafa1f0fec543dd7d2e8377e7647d8": "T=\\frac1{2}\\sqrt{a^2c^2-\\left(\\frac{a^2+c^2-b^2}{2}\\right)^2}", "3acb148f862fbeaf8f65b92d7fdea6b2": "\n\\mathcal{A}_3=\\sum_{j+k\\le3}a_{jk}\\partial_x^j\\partial_y^k =a_{30}\\partial_x^3 +\na_{21}\\partial_x^2 \\partial_y + a_{12}\\partial_x \\partial_y^2 +a_{03}\\partial_y^3 +\na_{20}\\partial_x^2+a_{11}\\partial_x\\partial_y+a_{02}\\partial_y^2+a_{10}\\partial_x+a_{01}\\partial_y+a_{00}.\n", "3acb5b5897a9418ff2e32da8ba504abf": "\\mathbf{F} = q\\left[-\\nabla \\phi- \\frac{\\partial \\mathbf{A}}{\\partial t}+\\mathbf{v}\\times(\\nabla\\times\\mathbf{A})\\right]", "3acba0a7a2c3d9454fe2360b5e98be8d": "\\sqrt{\\frac{27}{70}}\\!\\,", "3acc98f04193d72bfb0d49868ca41d21": "U_0\\times V", "3acca9c00aa0229bb23687596e617b2b": "\\Delta p = \\rho \\cdot g \\cdot h_f", "3accae645b72d2ff5310e79fd3809900": "f_2(z)=1-\\frac{(1+i)z}{2}", "3acce971a79b882b69336e7aacaa54d9": "\\left\\langle\\mathbf{P},\\mathbf{P}\\right\\rangle = |\\mathbf{P}|^2 = - (m_0 c)^2\\,,", "3accf0a14b11f87273172486ac2db27f": "-\\frac{\\Delta\\epsilon}{8\\pi}[\\mathbf{E}\\cdot\\mathbf{\\hat{n}}]", "3acd03085a214c7d3cfae4a50818a7be": "\\scriptstyle \\mu_r", "3acd0e54484c78d5d60cf3d5e7e3ee8a": "L \\subseteq [a,b]", "3acd4122359442c49fd5229bf68cabfe": "\\vartheta(0;\\tau)=\\frac{\\eta^2\\left(\\tfrac{1}{2}(\\tau+1)\\right)}{\\eta(\\tau+1)}.", "3acd7667144c5ccb55d5cf5bae7b1ab5": "C_2 = 0.1 10 11 100 101 110 111 1000\\ldots_2", "3acdd91f93c0d155e9fa9a808a07508e": "g_{lk}\\, ", "3acde473a4bc38f73d869ea69f094670": "(2) \\quad \\int_0^T e^{-xt}\\phi(t)\\,dt = \\int_0^\\delta e^{-xt}\\phi(t)\\,dt + O\\left(x^{-1} e^{-\\delta x}\\right)", "3acdf5b3322d3ae5348c9f8f9faea91e": "k_{n}", "3ace166743bc1a941e907c2f82fa8341": "N_{A}\\bar{m} = \\frac{N_{A}\\sum\\limits_{M_{J} = -J}^{J}{\\mu_{M_{J}}e^{{-E_{M_{J}}}/{k_{B}T}\\;}}}{\\sum\\limits_{M_{J} = -J}^{J}{e^{{-E_{M_{J}}}/{k_{B}T}\\;}}} = \\frac{N_{A}\\sum\\limits_{M_{J} = -J}^{J}{M_{J}g_{J}\\mu_{B}e^{{M_{J}g_{J}\\mu_{B}H}/{k_{B}T}\\;}}}{\\sum\\limits_{M_{J} = -J}^{J}{e^{{M_{J}g_{J}\\mu_{B}H}/{k_{B}T}\\;}}}", "3ace5f955386de46f5f7349a9666f5a5": " q \\rightarrow q' ", "3aceb18358f61c77cfa9ad5ac71748db": "D_k=\\nabla_k-\\frac{ie_k}{c\\hbar}\\mathbf{A}(\\mathbf{q}_k)", "3acf23ad40f07f907aa05ec05575fc0a": "\\frac{\\partial n_\\text{mean}}{\\partial t}=D_p \\frac{\\partial^2 n_\\text{mean}}{\\partial x^2}-\\mu_p p \\frac{\\partial E}{\\partial x}-\n\\mu_p E \\frac{\\partial n_\\text{mean}}{\\partial x}-\\frac{n_\\text{mean}}{\\tau_p}", "3acf2f702892e379a527ab2694c36082": "N_{1,1}", "3acfd00f15d1d1b8c4efdb1c2fb1b4b2": "\\beta(T,\\mathcal{A})=\\sup \\{ \\| P^\\perp TP \\|\\ :\\ P\\mbox{ is a projection and } P^\\perp \\mathcal{A} P = (0) \\}", "3ad06cc06001f3c9e591773dff023dee": "H_q(p)=-p\\log_q(p)-(1-p)\\log_q(1-p)", "3ad0c7980565c975781958d77dc0b92d": "\\int \\phi_m(x)\\psi_n(x) d\\mu(x) = 0", "3ad11114122330002ed6f754eea10573": "\\frac{3}{4}(a^{2}+b^{2}+c^{2})=m_a^{2}+m_b^{2}+m_c^{2}", "3ad172ad59c6b11c2fe66256434273b7": "A x e^x + A e^x = A x e^x + e^x", "3ad1768a703e453c0b361a9247501a4b": "\\sigma_0(24)", "3ad257da93e755da8c32d4d45726aa01": "\\displaystyle \\int_{-\\infty}^{\\infty} f(x) e^{-i\\nu x}\\, dx", "3ad28acec2dbd982bf0f8af8871fa41e": "M\\in\\mathcal{M}_0", "3ad2a500b7a4543b6db91d971cbcf4bd": "\\tau_{xz}", "3ad32d96822bfe4e44567f2e7eb50799": "r\\dot\\varphi^2", "3ad33591fe369efb9a04712a3ab18534": "-\\infty < t < \\infty, \\, r_0 < r < r_1, \\, 0 < \\theta < \\pi, \\, -\\pi < \\phi < \\pi", "3ad3544ce676095a21c6f7702f0d9b8f": "\\left(E - m \\right) \\phi = \\left(\\vec{\\sigma}\\vec{p} \\right) \\chi \\,", "3ad398e173a1a40a595c4de4559aba9b": "\\varphi:V^n \\to V^{\\otimes n}", "3ad3a0382f9204c4e951d067faced3ae": "\\lambda_1>\\lambda_d", "3ad3b544137e58ad95a5d779372ffb21": "\\frac{21}{\\alpha (6- \\alpha)} - 3", "3ad40d6e6f20870c6e2e49abfc3a7f2c": "f: R^n \\to R", "3ad4316f31f6df23bebedf720d203daa": "\n\\frac{p(X|S,h,\\Theta)}{p(X|S,h,\\Theta_{bg})} = G(X(h)|\\mu,\\Sigma)\\alpha^f\n", "3ad43a510282438472d5b7952b0b6138": "m - M = 5 \\left( log_{10}(d) - 1 \\right) ", "3ad46db99ebcc4868c9f489feaffdaf0": "I_1 \\cdots I_n", "3ad4984e8b0a64292b3c91f6b2e14e0c": "3F H^{2} = \\rho_{{\\rm m}}+\\rho_{{\\rm rad}}+\\frac{1}{2}(FR-f)-3H{\\dot F}", "3ad4e3265b80dc1db48facfead8d9b73": "\\ g^1(q,\\tau) = \\exp\\left(-\\bar{\\Gamma}\\tau\\right) \\left(1 + \\frac{\\mu_2}{2!}\\tau^2 - \\frac{\\mu_3}{3!}\\tau^3 + \\cdots\\right)", "3ad4e70f9f597fbfc4e9b630bf2b17d3": "1.22 \\lambda / D", "3ad4e71fd5c773db26847b6163c338b7": "R_n(\\xi,-x)=-R_n(\\xi,x)\\,", "3ad4f1e7f5820ef3c8097caed1dc637e": "_{metric} \\alpha = 1- \\frac{D_o}{D_e} = 1 - \\frac{\\sum_{c=1,k=1}^{v} o_{ck} {_{metric}} \\delta_{ck}^2}{ \\sum_{c=1,k=1}^{v} e_{ck} {_{metric}} \\delta_{ck}^2} = 1 - \\frac{\\sum_{c=1,k=1}^{v} o_{ck} {_{metric}} \\delta_{ck}^2}{\\frac{1}{n-1} \\sum_{c=1,k=1}^{v} n_c n_k~{_{metric}} \\delta_{ck}^2}", "3ad58570a8a2a9f43df6e04c68b28b30": "W=-\\frac{bT^4Ax_0}{3} = \\frac{bT^4V_0}{3}", "3ad59e2a174d08540760365eb2cc5d69": " DF(x(t)) ", "3ad5cdc1860c2d6c4f393b3fd7a05ccb": "1=A(x-1)+B(x+3)", "3ad61670e771fa6d644a9ad102120af4": " \\nabla^2\\,\\phi =0\\!", "3ad62ac793b177014ff8847ace252e57": "P(M, t) \\in \\mathbb{Z}[\\![t]\\!]", "3ad64c6e05906725fb51d13e5c573d50": "\n\\frac{\\partial}{\\partial t}(\\nabla^2 \\vec \\psi) + (\\nabla \\times \\vec \\psi) \\cdot \\nabla(\\nabla^2 \\vec \\psi)\n = \\nu \\nabla^4 \\vec \\psi", "3ad6c8cb39b4670b03004fb0ec1e4b61": "e^x = \\sum_{n = 0}^{\\infty} {x^n \\over n!} = 1 + x + {x^2 \\over 2!} + {x^3 \\over 3!} + {x^4 \\over 4!} + \\cdots", "3ad6cf5334932b04af074650d5d34e54": "s+1=12*2^{-(s+1)}-6*3^{-(s+1)}", "3ad6ec54245f76c0479d955e4c0a0af6": "\\gcd(a,n)=1", "3ad787816e09032d484399f224802b08": "h(r) = r^{2} \\cdot \\log \\frac1{r} \\cdot \\log \\log \\log \\frac1{r}.", "3ad7a1c4221fb37117da81d868978382": "\\ \\sgn(x) \\approx \\tanh(kx) \\,.", "3ad83240e5ef6398957a1aabd2261647": "\n\\hat{\\beta}_iX_i \\mathrm{\\ versus\\ } X_i.\n", "3ad8394fca94595dbda60d4422e10390": "F(3) + F(6)", "3ad86bdae949a276bc5e4547e693fe22": " a_1 = 0\\, ", "3ad8b297b67273dee8a87e5ae2086a33": "\\mathrm{Tr_{A,B}^C}(f)=\\rho_B\\circ(id_B\\otimes\\varepsilon_C)\\circ\\alpha_{B,C,C^*}\\circ(f\\otimes C^*)\\circ\\alpha_{A,C,C^*}^{-1}\\circ(id_A\\otimes\\eta_{C^*})\\circ\\rho_A^{-1}:A\\to B", "3ad9089390de524a6d918264233e4fa9": "\\ t ", "3ad911699a98001f2c9e3106ed9a97df": "K_n(RG)", "3ad950d890b359e259f18047eb2c7b5e": "b_{\\nu, n}(x) = 0", "3ad9743e09293286c79a2e7d39943217": "P \\leftrightarrow Q", "3ad9aececfe7a0f6dced9ae1dfae37b4": "\\color{Black}\\tfrac{\\infty}{m}", "3ad9c3d81f70f20fff5935c13a2d5ecc": "\\rho_{gas}=\\frac{m_{fullgas}-m_{evactube}}{V}\\,", "3ad9c71d56b4ff82d2e4802d85023067": "J_3(\\mathbb H)", "3ad9daf7bd31a244e78ba59d954898bb": "\\mathbf{r}_{R} = \\mathbf{B}_R^T \\mathbf{q} = \\mathbf{F}_{RR} \\mathbf{R} + \\mathbf{r}^o_R ", "3ada509751d145ae9df3af055e89cfdc": " \\mbox {Percentage error} = 100\\times(1-\\mbox{new diameter} / \\mbox{standard diameter}) ", "3ada56883bad6004b262212f2df057fd": "\\mathbb{N}_{\\geq 1} = \\left\\{1, 2, 3, \\dots\\right\\}", "3adacdf3847171e374d0c33775514aa7": "f:\\mathbb{R}^n \\rightarrow \\mathbb{R}^m", "3adae8381474c3ef9ead946f4a645a2b": "\\scriptstyle{p(\\mu|\\sigma^2, I) = \\mbox{const}}", "3adaed39c049e2d25099c3f8cf287219": "\\Gamma(n)", "3adaeed4102aeab590cde510a9dc0c27": "2 \\pi R", "3adb96f45a54bb1bf92a8634efc4818b": "\\partial \\sigma := \\sum_{j=0}^q (-1)^{j+1} \\partial_j \\sigma.", "3adb9931d8f8db77ad8a589cd38efa7f": "\\operatorname{Id}\\ S", "3adbbe047dc27be9e3224a4a1dd25a07": "h=\\frac{d}{\\sqrt{(r^2+1)}} \\qquad w=\\frac{d}{\\sqrt{\\frac{1}{r^2}+1}} \\qquad A=\\frac{d^2}{r+\\frac{1}{r}}", "3adbe5bc588b9e532c75f7f1a4fd5202": "t_1 \\ldots t_n", "3adc4f12d69dfd8b1374c03486d1a594": "k=-2", "3adc6e761bc3b8f3525011d884e2382e": "A v = \\begin{bmatrix} 3 & 1\\\\1 & 3 \\end{bmatrix} \\begin{bmatrix} 4 \\\\ -4 \\end{bmatrix} = \\begin{bmatrix} 3 \\cdot 4 + 1 \\cdot (-4) \\\\ 1 \\cdot 4 + 3 \\cdot (-4) \\end{bmatrix}", "3adc7ae57c792b6f2e9d335fa52c911c": "F_4 = F_3 + F_2", "3adcf073d561dbc53f1cac7ce8a87e80": " R^m\\big((\\mathbf{x},\\boldsymbol{\\theta}),(\\mathbf{x}',\\boldsymbol{\\theta}')\\big)=\\exp\\left\\{-\\sum_{k=1}^d \\omega_k^m(x_k-x_k')^2\\right\\}\\exp\\left\\{-\\sum_{k=1}^r \\omega_{d+k}^m(\\theta_k-\\theta_k')^2 \\right\\}. ", "3add06bdf75497efa545174a0cd13be1": "c_i=\\frac{b_1\\times{a_{n-2i-1}}-a_{n-1}\\times{b_{i+1}}}{b_1}.", "3add19ee5bb6328309e1592a5666cf27": "z = \\sqrt{r^2-x^2}", "3add6ed1c69747a930a9ec3cfe17712f": "A_0 \\to A_1\\alpha_1 \\mid \\ldots", "3adda79fbe343211a28feaa964ac4b34": "(x \\backslash y)^{-1} = y \\backslash x,", "3addcdaa59a3aa130fe66dab8f8f4390": "[S] / (K_M + [S]) \\approx [S] / K_M ", "3ade35cd5f8489f3f5f1faf869bb92d5": "\\sum _x f(x)= \\sum_{n=1}^{\\infty} \\frac{f^{(n-1)} (0)}{n!} B_n(x) + C \\, ,", "3ade54d93ecb654129c46242349a26d6": "\\log k", "3adeaf432f47fdae604e14228f69157e": "\\ln : \\mathbf{R}^+ \\to \\mathbf{R} : x \\mapsto \\ln{x}", "3adeef6dacb7a800510728f82c7aaccf": "\\int_0^{\\pi} \\int_0^{2\\pi} \\sqrt{ EG-F^2 } \\ du\\, dv = \\int_0^{\\pi} \\int_0^{2\\pi} \\sin v \\, du\\, dv = 2\\pi \\left[-\\cos v\\right]_0^{\\pi} = 4\\pi", "3adefca883f9745e38c34c00f7389594": "\\Delta_X \\colon X \\to X \\wedge X", "3adf01258ed5aa4fec83eeb8f1497c1e": "\\displaystyle{H_{\\partial\\Omega} g(s) -H g(s) = {1\\over \\pi i} \\int_0^{2\\pi} K(s,t)\\cdot g(t)\\, dt.}", "3adfa2097eb6fb4d265214f7dcd33815": "\n {\n \\rho~(\\dot{e} - T~\\dot{\\eta}) - \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} \\le \n - \\cfrac{\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T}{T} \\qquad \\qquad \\square\n }\n ", "3adfabb56d3015b6a1448ff8b5b5f508": "\\operatorname{pt} \\stackrel{1}\\to R", "3adfea437890887d102eb84758c9b0ca": "( \\lnot", "3adfef1cdbfbe66c94badfa9d0232163": "\n C_p=\\frac{2(\\lambda_2 - \\lambda_3)}{\\lambda_1 + \\lambda_2 + \\lambda_3}\n ", "3ae02f8233e1e8adc95d184c0fe1d924": "\\Phi(\\omega)", "3ae04e65608c99dc4f32b6c72e6b2b82": " DE-2CB=2AD-CE \\ne 0 \\,", "3ae07cf8c32a3263303c6bdfffb03415": " \\langle A \\rangle = \\sum_i p_i \\langle \\psi_i | \\hat{A} | \\psi_i \\rangle = \\sum_n \\langle u_{n} | \\hat\\rho \\hat{A} | u_{n} \\rangle = \\operatorname{tr}(\\hat\\rho \\hat{A}).", "3ae0843ea499abe47bffef51372b0719": "c_i = 0 ", "3ae08d84266106fe780c7e0bd83c324b": "n_A = 2 mol , n_B=1mol , n_C=0 mol", "3ae12ac9bc74f5f35dcd9f20b63f7b1b": " r_n \\ ", "3ae17635b4b85aa5682d37f5777166a2": "\\displaystyle{\\mu_{G\\circ F^{-1}}\\circ F={F_z\\over \\overline{F_z}} {\\mu_G-\\mu_F\\over 1 -\\overline{\\mu_F}\\mu_G}.}", "3ae1aeda2ecb8521f674ba69a9969351": " B_{2n}=I_n-\\sum_{(p-1)|2n}{\\frac{1}{p}} \\!", "3ae1b35618121a45b5fccf595043d95d": "\n m \\mid n \\Longrightarrow W_m \\mid W_n.\n", "3ae20502030e494c797987b667f5ee8b": "\\beta_1(\\alpha) = { \\alpha^2 \\over \\pi} \\left( -{11N \\over 6} + {n_f \\over 3} \\right) ", "3ae21883432f83c6d4e33b1fd8420ea3": "(\\sqrt{3} m )^2 =~-1", "3ae21b5197c389295ae5a522ea22f8da": "\\varphi_n(\\lambda)=\\frac{1}{1-in/\\lambda}", "3ae25dfd850b5d9113ed9b9f9e12374f": "\\Re(z) < 0", "3ae2e447edf572be2656dcbeee1cfe7c": "\\left(\\tfrac{5}{323}\\right)", "3ae38deb8725df9d0e1582d6cb0fd073": "\\iota: A\\rightarrow B, \\qquad \\iota(x)=x.", "3ae39e4cc0109189b91596e70c3a681a": "I_{b+}", "3ae3a363f0d125f90169765876e177b2": "\\scriptstyle\\sigma_y^2", "3ae3a7c3e4df5eac5ee8cf1fe7797cef": "\\frac {1} {2}\\sum_{i} m_i v_i^2", "3ae3c2dc8f13625459386968e2f7fdad": "(\\psi\\phi)(e,c)=\\left(\\coprod_{d\\in D}\\psi(e,d)\\times\\phi(d,c)\\right)\\Bigg/\\sim", "3ae408c2bd192801fad6d85b46a43e34": "d \\mathbf{r}_{2}", "3ae44941af5eec451ce90beb1f32d970": " L(0)=L^{'}(0)=0\\text{ and } L^{''}(h)\\leq \\frac{1}{4}", "3ae45852feab8917559d090999347776": "F_K(k)=1-F_{R_k}(r_K)", "3ae45cfcb52bb104ed069300a6a48840": "\\frac{(n-1)S^2_{n-1}}{\\sigma^2}\\sim \\chi^2_{n-1}", "3ae4cded6e0cd0c784164d755f5c36ee": "\\langle x - y,x - y\\rangle =\\langle x,x\\rangle - 2\\langle x,y\\rangle + \\langle y,y\\rangle.", "3ae4e4e7c7b6340cd153817ffe8284e8": "a^\\dagger_{\\mathrm{in}}", "3ae5105a8bda7b8ae63caaa302a5b8e1": "\n\\begin{align}\nJ_{-1}^{(1)} &= \\dfrac{1}{\\sqrt{2}}(J_x - iJ_y) = \\dfrac{J_-}{\\sqrt{2}}\\\\\nJ_0^{(1)} &= J_z\\\\\nJ_{+1}^{(1)} &= -\\dfrac{1}{\\sqrt{2}}(J_x + iJ_y) = -\\dfrac{J_+}{\\sqrt{2}}\n\\end{align}\n", "3ae55bfb52cdecd012357f5f80c98fd4": " (dW/dt)_i = P_{ci} * dW/dt ", "3ae5811a6f1b52f791691a99c8fd226d": " < r ", "3ae583ff181ee9dedf3079671478be5a": "a \\ll \\ell", "3ae5a91e8534b3339ae0c0edc89f847f": "\\pi_i q_{ij}", "3ae5d1ace0f6ffe17854eb492513b9ea": "U(P) = - \\frac {1}{4 \\pi} \\int_{A_1} \\frac {e^{iks}}{s} \\left[ ik U_0(r) \\cos(n,s) + \\frac {\\partial U_0(r)}{\\partial n} \\right] dS ", "3ae633ca6092d0db03a196f161e144ed": "X=6Y/W+(Z-2)^2", "3ae64e6cc3b0df8ee993c1c850a47131": " Constant ", "3ae6e1ba17995c8eb6f8f8d43c56bb15": "a_{12}+b_{12}", "3ae783fe98dce041a0393f17d52ed667": "Q(x) =\\int_x^\\infty\\varphi(u)\\,du <\\int_x^\\infty\\frac ux\\varphi(u)\\,du =\\int_{\\frac{x^2}{2}}^\\infty\\frac{e^{-v}}{x\\sqrt{2\\pi}}\\,dv=-\\biggl.\\frac{e^{-v}}{x\\sqrt{2\\pi}}\\biggr|_{\\frac{x^2}{2}}^\\infty=\\frac{\\varphi(x)}{x}.", "3ae791f74094cf95bbf0b6f8d4b7875d": "F_g = \\left( \\rho_p - \\rho_f \\right)\\, g\\, \\frac{4}{3}\\pi\\, R^3,", "3ae7948c64c1735e5ea9023908cec6cf": " f(t) = \\int_{-\\infty}^{\\infty} \\delta(t-t') f(t') dt'", "3ae7e929f7aa2bab4255bc2e534ac4d8": "\\frac{A_1,\\dots,A_n}{B_1,\\dots,B_m}\\qquad\\text{or}\\qquad A_1,\\dots,A_n/B_1,\\dots,B_m.", "3ae85ab00e712ea0e693520666b4bf76": " E, E' ", "3ae86747effb6aea8c83c109749376ea": "K_{2/3}^2(\\xi) ", "3ae87397ec1e328f5fe8e08b5b084aec": " \\Sigma^k X \\cong X \\wedge S^k. \\, ", "3ae8b80bf1a6ec5d5ef39cdb78af329d": "(s,t)\\mapsto \\gamma(t) + s\\gamma'(t).", "3ae8e3bbd8dd4821ebce7b8bf3c55360": "Gm = constant", "3ae8eb858c34f84b9a392ba159ed4666": "X^2 = 0", "3ae91fd962db18becc3c524fe215fd74": "\\lim_{\\kappa\\rightarrow\\infty}\nf(x\\mid\\mu,\\kappa)=\\frac 1 {\\sigma\\sqrt{2\\pi}} \\exp\\left[\\dfrac{-(x-\\mu)^2}{2\\sigma^2}\\right]", "3ae95973bf19b1c4797ab1faf04a4815": "\\overline{\\mathbf{X}}", "3ae9624a30d85ca32a450b5600dba656": "b k_B T - 2a", "3ae9760015a3cfad3b89eb30fa331625": "\\sum_{k=0}^n {n \\choose k} = 2^n,", "3ae9cd1d981a59ad517d99bafb3365b1": "(x, y) \\mapsto \\operatorname{tr}(xy)", "3ae9fba8e35648008e7a6a3bec7f799c": "p(k)+q(k)=k-2 \\, ", "3aea0334494f27b280f0276bcb37d7d5": "V(\\mathbf{x}) = -\\int_{\\mathbf{R}^3} \\frac{G}{|\\mathbf{x} - \\mathbf{r}|}\\,dm(\\mathbf{r}),", "3aea6c63a6ed11ce21838bc82322606e": "\\overline{u}(\\boldsymbol{x})", "3aea6f4b7b100cbf798382cc5bb5d7a9": " D\\left ( x - y \\right ) ", "3aea97ba9eec50897eee1330b811ee65": "\\operatorname{Var}(X) =\\sigma^2 =\\int (x-\\mu)^2 \\, f(x) \\, dx\\, =\\int x^2 \\, f(x) \\, dx\\, - \\mu^2", "3aeabcd624dccd85d74aef9d386bfbb6": "1 \\over 10000", "3aeae00c79f832e5febc371ccde5e38d": "x^\\prime=-a/2", "3aeb26145a2eaec9cca65a93b5ef3ace": "\\beta_{mk} \\in D_R", "3aeb66c8fa75eb518e4ec15eb28b80e7": " System Noise Figure = F_1\n+ \\frac{ F_2 - 1 }{ G_1 }\n", "3aeb8a9aa8c3d1317d67056339f5882d": " -\\Delta u = 0 ", "3aeba4a94134694615c619b3056f1446": "\\mathbf e_i\\cdot\\mathbf e_j = 0.", "3aebe29cd9666553813e88c8e5b8d1b4": "\\mathbb{P}^3", "3aec131d1e4ce4778317deb8ab2ca466": "(1-\\langle v_{i}, v_{j}\\rangle)/{2}", "3aec363fb82036b3cdb0afdacec68a1c": " T_s(X,Y) = \\frac{\\sum_i ( X_i \\land Y_i)}{\\sum_i ( X_i \\lor Y_i)}", "3aec528972d59e2aadae0589b703d930": "\\xi \\in \\mathfrak{g}^*", "3aec882e50a2fd2aa401fab46c51cf02": "\nS = \\int{ \\left( -\\frac{1}{2} D^\\mu X_I D_\\mu X_I +\\frac{i}{2} \\overline{\\Psi} \\Gamma^\\mu D_\\mu \\Psi +\\frac{i}{4} \\overline{\\Psi} \\Gamma_{IJ} \\left[ X^I, X^J, \\Psi \\right] - \\frac{1}{12} \\left[ X^I, X^J, X^K \\right] \\left[ X^I, X^J, X^K \\right] + \\frac{1}{2}\\varepsilon^{abc}Tr(A_a\\partial_b A_c + \\frac{2}{3}A_a A_b A_c)\\right) }d\\sigma^3\n", "3aed0aad5db45cf788202560cb82c275": "\n\\begin{align}\np_{\\rm vanna} &= a \\, \\gamma \\\\ p_{\\rm volga} &= b + c \n\\gamma \n\\end{align}\n", "3aed1a2d5bd80f6acde56ffced070f3d": "\\aleph_1", "3aed6265dff176716ad5d40af6a93f12": " \\mathbf{E} = \\mathbf{E}_0 + \\mathbf{E}_\\omega \\cos(\\omega t), ", "3aed6d762172d63ed719991456f98e33": " \\theta = \\tan^{-1}{\\left(\\frac{v^2\\pm\\sqrt{v^4-g(gr^2\\cos^2\\phi+2v^2r\\sin\\phi )}}{gr\\cos\\phi}\\right)} ", "3aedba2550ae1bf0614febf98226c097": "(1+r)P-c", "3aedc839785c0f30a5f0196f95005c92": "\\alpha \\approx_{Y}\\beta\\ ", "3aede926e580d8d35d8d2250f56357f2": "X\\,C_n", "3aee03b0aff51001a4582f2ae2b5f15b": "E_3(x)=x^3-\\frac{3}{2}x^2+\\frac{1}{4}\\,", "3aee210d5d4224328d4837686818b72d": "H: X \\rightarrow \\{l, \\neg l\\}", "3aee70f488438102d35c6c8093478a00": "V^a_k", "3aeec1189e52affe1dea10805385e2a1": "\n(1)\\cfrac{\n \\cfrac{\n (1)\\cfrac{C_1 (1,3)\\qquad {C_8}^*}{C_3 (3)}\n }\n {C_7 {\\color{red}(3)}}\n \\qquad\n (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{C_8 (-1)}\n}\n{\nC_9 (3,5)\n}\n", "3aef128b48cced91213046d5ba421b5f": " \\!\\ S_m^3 = S_{(m^3 + 3m)} ", "3aef16128aecfaaa35320492d957d3f8": "\\mathrm{Si-OH + Si-OH \\rightleftharpoons Si-O-Si + HOH}", "3aef30784658c22c1dedecaa524ebbc3": "X = \\log x", "3aef466304779bec4d5b493a68fbabd4": "\\kappa:\\mathcal{X} \\times \\mathcal{X} \\to \\mathbb{Z}", "3aef6d01ed5af25840c24385908129e4": "\\frac{\\mathrm{d}}{\\mathrm{d}s}\\mathbf{u}_\\mathrm{t}(s) = -\\frac{1}{\\alpha} \\left[\\cos\\frac{s}{\\alpha} \\ , \\ \\sin\\frac{s}{\\alpha} \\right] = -\\frac{1}{\\alpha}\\mathbf{u}_\\mathrm{n}(s) \\ ; ", "3aefb9eb4c6b9f2cea88da3471570a1c": "S_{nk}=S(n,k).", "3aefd974526db2151ef1c310814ebe01": "\\textstyle \\alpha_1", "3aefdc889c13eae3a99a645d06c35c4d": "\\Delta I_{L_{On}}=\\frac{1}{L}\\int_0^{D T}V_i d t=\\frac{D T}{L} V_i", "3aefe2f5074d2e294af538619cefa436": "p=\\frac{E}{nc}.", "3af004fa2b5345277b9f77086c086e7e": "\\mathbf{e}_i\\wedge \\mathbf{e}_i = 0", "3af0ca403f8da5fa79525a15f30a6b4d": "C \\equiv M^3\\, \\bmod\\, N_1 N_2 N_3", "3af0cc959da69a2f373e13cf9e4cff6c": "\\begin{pmatrix}\\cos(\\theta) & -\\sin(\\theta)\\\\ \\sin(\\theta)& \\cos(\\theta)\\end{pmatrix}", "3af0cfcd92402d60a3a54298c50542bb": "\\mathbf{A} =\\left(0, \\mathbf a\\right)", "3af0f69da37fd2629d8bc1650874b4fe": "\\deg(f,\\Omega,p):=\\sum_{y\\in f^{-1}(p)} \\sgn \\det Df(y)", "3af151861a4b13f825c6e383d5129b9c": " \\frac {\\Delta Y}{\\Delta I} = \\frac {1}{(1-c)}", "3af1583e27d1296cca03e318f17398bb": "C_K", "3af15d2e304f4b608987c8ac2cf54a25": "4 +\n3w", "3af1610db41a2e2ece779b4f74d98661": "a < b < 2a", "3af17f863ac49c81c13d9507e226e980": "i,", "3af1e2e0629f1718460c2dbff344b7de": " f^{-1}( (-\\infty , t] ) = \\{ x \\in R | f(x) \\le t \\} ", "3af1fdb6d6950820f3f628a15b5b4e9b": "\n\\mathbf{\\hat{b}_{2:5}} = \\alpha\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.1 & 0.0 \\\\ 0.0 & 0.8 \\end{pmatrix}\\begin{pmatrix}0.6533 \\\\ 0.3467 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.1289 \\\\ 0.2138\\end{pmatrix}=\\begin{pmatrix}0.3763 \\\\ 0.6237 \\end{pmatrix}\n", "3af20ba3a1edc051e2984917f94956e4": " dQ = qdx ", "3af20eac86ed23ba5fa0c9df6aeeedea": "\\begin{align}\n(x + 3)(x + 5) \\,&=\\, x\\cdot x \\,+\\, x\\cdot 5 \\,+\\, 3 \\cdot x \\,+\\, 3 \\cdot 5 \\\\\n&=\\, x^2 + 5x + 3x + 15 \\\\\n&=\\, x^2 + 8x + 15\n\\end{align}", "3af2113616dc264a27fee9053fa0c049": "\\left[{f(x+h)\\over f(x)}\\right]^{1/h}\\text{?}", "3af24042f457b6c6c05b794f6c57d273": " \\dot{\\varepsilon}=k A^{m}S^{n} ", "3af245c88ea87601874cbff8e2e97b08": " \\left\\langle i \\right| p_{z} \\left| f \\right\\rangle ", "3af2b58841794d0d3b7967760e4b8e1b": "A = \\sqrt{2} + \\sqrt{3}", "3af2d1d63ff88b53e144b011324ce80d": " \\sum_j\\left(\\sum_{k,l}\\int_\\Omega a^{kl}\\frac{\\partial\\varphi_i}{\\partial x_k}\\frac{\\partial\\varphi_j}{\\partial x_l}dx+\\int_{\\partial\\Omega}c\\varphi_i\\varphi_j\\, ds\\right)u_j = \\int_\\Omega\\varphi_i f\\, dx+\\int_{\\partial\\Omega}c\\varphi_i g\\, ds,", "3af301412b478a432722f2c2d76cb89f": "n\\lambda=2\\pi r \\; \\; (2)", "3af305f19b502c14cb04a47f7e446cd7": "\\|p_\\sigma\\|", "3af41c38065d2add798c20f64166aa5a": " \\begin{align}\n\\text{areal velocity} &= \\lim_{\\Delta t \\rightarrow 0} \\frac{\\vec{r}(t) \\times \\vec{r}(t + \\Delta t)}{2 \\Delta t} \\\\\n&= \\lim_{\\Delta t \\rightarrow 0} \\frac{\\vec{r}(t) \\times \\bigl( \\vec{r}(t) + \\vec{r}\\,'(t) \\Delta t \\bigr)}{2 \\Delta t} \\\\\n&= \\lim_{\\Delta t \\rightarrow 0} \\frac{\\vec{r}(t) \\times \\vec{r}\\,'(t)}{2} \\left( {\\Delta t \\over \\Delta t} \\right) \\\\\n&= \\frac{\\vec{r}(t) \\times \\vec{r}\\,'(t)}{2}. \n\\end{align} ", "3af423b4dd1399788a9734cdb2b72b20": "\\frac{a}{b} + \\frac{c}{d} = \\frac{ad+bc}{bd}.", "3af42f04ee9a1b3a123e8c0e49a5d0a9": "R^n = \\cup_i P_i", "3af442da4f0a7afc63cb5cfe4afc830c": "\\left \\{ z \\in \\mathbf{C}^n \\ : \\ \\|z\\| < 1 \\right \\}.", "3af481cd6ccf96bc1bdedb7f2715803a": "AH \\cdot HD = BH \\cdot HE = CH \\cdot HF.", "3af4bfc585068686fbcba30ecbf0efdd": "\t\\begin{array}{rr} \n -1x & + 3\n\\end{array}", "3af4cb6e28194b0a7be3ea2f3a2fd205": "n=\\prod_{l=1}^i (p_l)^{a_l}", "3af4d3ef51730bddbc852aba25d283c0": " \n\\text{Choose } x_i(t) = x_{min,i} \\text{ if } Vc_n + \\sum_{i=1}^KQ_i(t)a_{in} \\geq 0\n", "3af4eff04b83d73fc9c14ff5b64b37e4": "2^{w-2}-1", "3af504b9dc986174588743162d4d8c7d": "R^*\\,", "3af513aa69cd995a42f68a12b5fdc5a6": "g = \\begin{bmatrix} 1 & 0 & 0 & 0\\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 \\end{bmatrix}. \\ ", "3af560fe00c6418f7f20134a106a8c89": "N-D", "3af568d8f2d887404255ee26f7e20d15": "-\\pi/4", "3af57c5c9644a153f99b9c6bcebb8afc": "-0.3", "3af656e5c08c17feb1f9ccddb3253efb": "\n\\begin{array}{lrl}\n \\text{Predicate} & = &\\sigma\\sqsubseteq\\sigma'\\\\\n & \\vert\\ &\\alpha\\not\\in free(\\Gamma)\\\\\n & \\vert\\ &x:\\alpha\\in \\Gamma\\\\\n\\\\\n \\text{Judgment} & = &\\text{Typing}\\\\\n \\text{Premise} & = &\\text{Judgment}\\ \\vert\\ \\text{Predicate}\\\\\n \\text{Conclusion} & = &\\text{Judgment}\\\\\n\\\\\n \\text{Rule} & = &\\displaystyle\\frac{\\textrm{Premise}\\ \\dots}{\\textrm{Conclusion}}\\quad [\\mathtt{Name}]\n\\end{array}\n", "3af66fa43723fca59221f98d2562a9c6": "\\begin{align}\n&\\Delta{X}=\\cos(\\phi_2)\\cos(\\lambda_2) - \\cos(\\phi_1)\\cos(\\lambda_1);\\\\\n&\\Delta{Y}=\\cos(\\phi_2)\\sin(\\lambda_2) - \\cos(\\phi_1)\\sin(\\lambda_1);\\\\\n&\\Delta{Z}=\\sin(\\phi_2) - \\sin(\\phi_1);\\\\\n&C_h=\\sqrt{(\\Delta{X})^2 + (\\Delta{Y})^2 + (\\Delta{Z})^2}.\\end{align}\n", "3af672517a91988e56c273f0bce7fee0": "\\textstyle u_y", "3af67e2983ef9c53300bf1119949ad9f": "\\vec x(t_{n-1}),\\vec x(t_n),\\vec x(t_{n+1})", "3af681301a32a2d0a3b456093443bcf4": "\\rho\\, C_a\\, V\\, \\dot{u},", "3af6e3f1c18c452f1966dbfdda70ce6d": "Z_T : V_{w_1} \\otimes V_{w_2} \\otimes \\cdots \\otimes V_{w_k} \\longrightarrow V_{w_0}", "3af74f2df6a32c3ea5bf1f11939e16b4": "(\\mathbf{u}\\cdot\\nabla)\\mathbf{u} \\sim \\frac{V^2}{L},", "3af753e643a1f1472e43f271ab6ff55e": "\\exp\\left( -\\frac{\\Delta E}{kT} \\right) ", "3af8187d0ecca6791ba57ed4a016c0a7": "\\mathcal F = \\int_V \\mathbf a\\,dm = \\int_S \\mathbf T\\,dS + \\int_V \\rho\\mathbf b\\,dV", "3af8555363576789a1a477e64b9d4197": "=q_1(\\mathbf{r}_1-\\mathbf{r})+q_2(\\mathbf{r}_2 - \\mathbf{r}) = q(\\mathbf{r}_+ -\\mathbf{r})-q(\\mathbf{r}_- - \\mathbf{r}) = q (\\mathbf{r}_+ - \\mathbf{r}_-) = q\\mathbf{d},", "3af86a8e1406578727be39757710dffc": " S(a_{i,j}) \\neq a_{i,j} ", "3af8a6f4d6083215dab51b2c41d3b567": " \\mathrm{A} + \\mathrm{B} \\rightleftharpoons [\\mathrm{AB}]^\\ddagger \\to \\mathrm{P} ", "3af8af43bb3ca7c846e50600d0ff4ee8": "\\langle f(\\theta)\\rangle= \\int_0^{2\\pi} p_w(\\theta')f(\\theta'+2\\pi a')d\\theta'", "3af8c6004702d0bc452750c2847f7256": "\\Gamma = \\frac{\\sum (r_j-r_i)(s_j-s_i)}{\\sum(r_j-r_i)^2} ", "3af8c62dbefcd8b65b0aebba64f6767f": "s_{uv}=\\frac{C^*_{uv}}{L^*}=13 \\sqrt{(u'-u'_n)^2+(v'-v'_n)^2}", "3af9039ad3ebe739f1bd45b6457c0fbd": " \\boldsymbol{\\tau}' = \\boldsymbol{\\tau} - \\boldsymbol{\\tau}_\\mathrm{app} ", "3af94ea9027996f7bbbc81bb760e71b7": "\\begin{bmatrix}\n x \\\\\n y\n\\end{bmatrix},", "3af95dc1fb702ef0cef5ea67c3f91543": " D^{\\mathrm{beam}} \\,", "3af982efc49859afa1ff7e33164a0ce5": " \\!\\ K_n = mK_{(n-1)} + K_{(n-2)}. ", "3af9abbe9c173fe1b9f1c6e440ada102": "l_{1,1} x_1 = b_1", "3af9fae23a3bbd8fa5b40d311f06d499": "N_{rr}^*", "3af9fbd987988ce5cd8f29f3b578f389": "f_1(3) - 2", "3afa1a7c9ff0ac4a85039f9b4f78ebf7": "\nS = k N \\ln\n\\left[ V \\left(\\frac UN \\right)^{\\frac 32}\\right]+\n{\\frac 32}kN\\left( 1+ \\ln\\frac{4\\pi m}{3h^2}\\right)\n", "3afa50f87a26648dc0b0b1de0724b1c9": "G = E_{antenna} \\, D = (1)(1.698) = 1.698", "3afa59ec820b105d9ae89cbd55741778": "k_2 \\in B^{2\\left\\lceil l/16 \\right\\rceil+40}", "3afa7a061a4dd43b1ad28a5684330e33": "\\scriptstyle \\gamma_\\mathrm{la}", "3afaf944a5f58bdc0e3d63f43283ea70": " |\\psi\\rangle", "3afb5f3e95e59bb3641441c23febf638": "\n\\cfrac{\n \\begin{matrix}\n \\cfrac{}{A \\ true} u \\\\\n \\vdots \\\\\n p \\ true\n \\end{matrix}\n}{\\lnot A \\ true} \\lnot_{I^{u,p}}\n\\qquad\n\\cfrac{\\lnot A \\ true \\quad A \\ true}{C \\ true} \\lnot _E\n", "3afbc130086faa1f97e27f4a93a5f727": "\\frac{dy}{dx}=\\frac{dy}{dx}+x\\frac{d^2 y}{dx^2}+f'\\left(\\frac{dy}{dx}\\right)\\frac{d^2 y}{dx^2},", "3afbe6839a0267763de01fcb23aedccd": " p>3", "3afbf3bb8778ef6a16fa19a516841e7f": "(\\gamma,p)", "3afc234c04eeed71937c719a2729967d": "A\\mid B\\Rightarrow\\varphi^{-1}(A)\\mid\\varphi^{-1}(B).", "3afc628b542a26a5dd7247df9f87afc6": " f(\\theta, \\varphi) = \\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell f_{\\ell m} \\, Y_{\\ell m}(\\theta, \\varphi). ", "3afc6d08d242f797e740a4f5737f8b44": "\\vec w \\cdot \\vec \\mu_{y=1} ", "3afda815637fcd85a537c42d6f0b8b3b": "T\\,\\mathrm{d}S\\,", "3afe467869b29e04e8d6daf540790c03": "S_{n+1}(s) = 2(1-2s) S_n(s) - S_{n-1}(s) + 2s.\\,", "3afe5ecfbe061e91adb1ee01084b25b7": " C_M", "3afe97ab12af66178cce31175fa5cf42": "\\mathcal{L}=-\\tfrac{1}{2}\\;\\bar{\\psi}_\\mu \\left ( \\epsilon^{\\mu \\nu \\rho \\sigma} \\gamma_5 \\gamma_\\nu \\partial_\\rho - i m \\sigma^{\\mu \\sigma} \\right)\\psi_\\sigma", "3afeb569439c2476c08df1cae48286f2": "G(x,x')=\\dfrac{1}{|x-x'|}.", "3afec4fdf6cbae2264638d4237bc4afd": "\\operatorname{Sym}(\\mathfrak{g})", "3aff1d1140673381bdf2a02d1f3c1227": "\\mathbf{u}(\\mathbf{r}) = \\int \\mathbf{f}(\\mathbf{r'}) \\cdot \\mathbb{J}(\\mathbf{r} - \\mathbf{r'}) \\mathrm{d}\\mathbf{r'}, \\qquad\np(\\mathbf{r}) = \\int \\frac{\\mathbf{f}(\\mathbf{r'})\\cdot(\\mathbf{r}-\\mathbf{r'})}{4 \\pi |\\mathbf{r}-\\mathbf{r'}|^3} \\, \\mathrm{d}\\mathbf{r'} ", "3aff4402290d031bc979f1104756fcca": " \\mu_{A,\\Gamma}\\colon RKK^\\Gamma_*(\\underline{E\\Gamma},A) \\to K_*(A\\rtimes_\\lambda \\Gamma),", "3aff7fc04df1c426e4d0148a06f8a860": " \\mu = \\frac{m_em_p}{m_e+m_p} ", "3affa7be389bdb02ffb0dcc4e3220675": "\\frac{\\partial^{|I|}}{\\partial x^{I}} := \\prod_{i=1}^{m} \\left( \\frac{\\partial}{\\partial x^{i}} \\right)^{I(i)}.", "3affe1adc2dccf32038a3a3730e29882": " \\frac{1 \\ \\mathrm{s}}{1023 \\times 10^3} = 977.5 \\ \\mathrm{ns} \\ \\approx 1000 \\ \\mathrm{ns} \\ ", "3afff2ce7b00d77aa39c106a3dcbd020": "| \\phi (\\mathbf{r})|^2 ", "3b003836cf6562545fe4f8a34daacc32": "\\langle \\alpha'j'||T^{(k)}||\\alpha j\\rangle \\frac{1}{\\sqrt{2j+1}}", "3b003985fc8f3408a7ba7ae90fed498c": "\\epsilon^{\\mu}_r(\\vec h) ", "3b00450dbeb6cc24ed17e0aa2ee0b7ec": "\\det(A) = ad - bc", "3b00944de88f2501991ff00d51227a9e": "\\epsilon\\!\\,", "3b00b8af0b7125d35b5bc942ed3f52aa": "\\chi(-\\omega) = \\chi^*(\\omega)", "3b010db4b74e714aa135589bfc3f3ab2": "\\cos \\theta_1 = -\\frac{(x_0+d) f_x + y_0 f_y}{r_1} \\quad (18)", "3b017dcecc9e2220c5fbe3bade24d4e6": "\nG(\\mathbf{r}_\\perp) = \\frac{1}{4\\pi^2} \\; \\iint \\mathrm{d}^2\\!k \\; \\frac{e^{i \\mathbf{k}_\\perp \\cdot \\mathbf{r}_\\perp}}{k_\\perp^2 + 1 / \\rho^2}.\n", "3b0187799f5e8ca32a75b213485a6a2b": "C^{m-1}(\\Omega_{mr})\\;", "3b01a8e7e490471ceef84130cdd4d696": "Q_1 \\wedge \\dots \\wedge Q_n \\wedge \\neg Q_{n+1} \\wedge \\dots \\wedge \\neg Q_{n+m} \\rightarrow P", "3b01f43f7523dc5b9d0a6290a05066e0": "\\phi(D|m_x, m_y)", "3b02235dddce84d3da23244a370961b3": "\\scriptstyle M_V=m_V-5\\log_{10}\\left(\\frac{100}{\\pi}\\right)", "3b022b6c4f61c3ed1b0fb1c8902f2cdd": "~10^{-4}~", "3b02379e8c5b30100915c623cfda7e8a": "\nf^\\prime (x) = \\sum_{n=1}^\\infty a_n n \\left( x-c \\right)^{n-1}= \\sum_{n=0}^\\infty a_{n+1} \\left(n+1 \\right) \\left( x-c \\right)^{n}\n", "3b026222cccd0c61fdb737a5371bf00d": "\\frac{\\sqrt{105}}{2}\\cos(2\\theta)\\sin(\\phi)\\cos^2(\\phi)", "3b028ab34c600f77dbfdffe8d9e83457": "\\operatorname{Hol}(G)=G\\rtimes \\operatorname{Aut}(G)", "3b029ac45cd5f10df6fd91b5dc7715ea": "\n\\zeta_{G, p}(s) = \\sum_{\\nu=0}^\\infty s_{p^n}(G) p^{-ns}\n", "3b02bb10b32053c0d30905b73580d15a": "\\mathbf{M}=\\frac{N}{V}\\mathbf{m}=n\\mathbf{m}", "3b02ec1f094e5da49ca4e532462b1862": "\\begin{align}\n \\hat\\beta & = \\frac{ \\sum_{i=1}^{n} (x_{i}-\\bar{x})(y_{i}-\\bar{y}) }{ \\sum_{i=1}^{n} (x_{i}-\\bar{x})^2 }\n = \\frac{ \\sum_{i=1}^{n}{x_{i}y_{i}} - \\frac1n \\sum_{i=1}^{n}{x_{i}}\\sum_{j=1}^{n}{y_{j}}}{ \\sum_{i=1}^{n}({x_{i}^2}) - \\frac1n (\\sum_{i=1}^{n}{x_{i}})^2 } \\\\[6pt]\n & = \\frac{ \\overline{xy} - \\bar{x}\\bar{y} }{ \\overline{x^2} - \\bar{x}^2 } \n = \\frac{ \\operatorname{Cov}[x,y] }{ \\operatorname{Var}[x] }\n = r_{xy} \\frac{s_y}{s_x}, \\\\[6pt]\n \\hat\\alpha & = \\bar{y} - \\hat\\beta\\,\\bar{x},\n\\end{align}", "3b02f272bd5ba418e1d9f60989a2b603": "\\sqrt {gy}", "3b0329f9ac67e4e3bcd4e62ef35e3782": "2\\$=2^2=4 \\,", "3b03a364bb0cabac6a527aa3ebf1dda0": " H(0,x)=\\xi(1/2+ix) ", "3b03fa00dfdaddb194103a97acd150eb": "A_{\\hat{1}\\hat{i}}", "3b041a4a1f1dd0ee413442c92cc8facd": " C(r,z)=G_1(0,z)S(r)+2\\pi S_0\\int_{0}^{\\infty} G_2(r'',z)\\,exp\\left [-2\\left (\\frac{r''-r}{R} \\right )^2 \\right ]I_{0e}\\left(\\frac{4rr''}{R^2} \\right )r''\\,dr''.\\qquad(13)", "3b042a4b7091600a01c52675ddb58a6e": "d(x,y)\\le d(x,z)+d(z,y)<2\\cdot\\max\\{a,b\\}", "3b0459dc197d8ca3070bd1f6a33fbf56": "Q=\\frac{\\pi L}{2\\lambda (1-r)}", "3b0464f265662880a1280eaa16d247a9": "\n\\begin{align}\ndf(x_t,t) & = \\theta x_t e^{\\theta t}\\, dt + e^{\\theta t}\\, dx_t \\\\[6pt]\n& = e^{\\theta t}\\theta \\mu \\, dt + \\sigma e^{\\theta t}\\, dW_t.\n\\end{align}\n", "3b04ab7c43c1b79fc2c434066647b471": "g(v,w)=h(df(v),df(w)).\\,", "3b04b60ceec36098cf15cf2b77363123": "\\empty \\!\\,", "3b04db505952dee246e0985865bd3e7f": "M_{k,j}=\\frac{1}{N}\\sum_{i=1}^N I(X_{i,j}=k),", "3b04f8a67d546f537e07bd90a3cd83b6": "\\mathcal{L}(\\mathbf{r},t)", "3b04ff9b17c2ebe412e13c3c2b740b81": "B = \\mathrm{clamp}(( 298 \\times C + 516 \\times D + 128) >> 8)", "3b0520ed6c96a44812ad591769ef48ca": "n_i-\\langle n_i\\rangle", "3b05249eb45e6f288b24906cd09a21a0": "p(x) = \\prod\\limits_{t \\in K} (x-t).\\,", "3b055648135da0614ce7590c403d6aa7": "Z = \\sum_q \\mathrm{e}^{-\\frac{E(q)}{k_BT}}", "3b05968665b07fb7cfe0c9550add8fa7": " \n \\mu(p,T) = \\mu_0 + \\frac{\\partial \\mu}{\\partial p} \\frac{p}{\\eta^{1/3}} +\n \\frac{\\partial \\mu}{\\partial T}(T - 300) ; \\quad\n \\eta := \\rho/\\rho_0\n ", "3b0601a80085c45df9eeae964c2afaab": " P(X \\geq x) \\leq \\frac{e^{-\\lambda} (e \\lambda)^x}{x^x}, \\text{ for } x > \\lambda ,", "3b0619e986d2ebf9acd80c494cb421a4": "\\nu\\,\\!", "3b062c78683c49a313f5944bfcaf2a3b": " \\Bigl\\| \\sum_{k=0}^n \\varepsilon_k \\alpha_k b_{\\pi(k)} \\Bigr\\|_V \\le C \\Bigl\\| \\sum_{k=0}^n \\alpha_k b_k \\Bigr\\|_V. ", "3b0638350539aca3e1db8ccd55b2fd61": " E^{\\star}", "3b0643fa2f6346acad6280685193ca73": "\\textstyle \\alpha_2", "3b06624c360c3854f9a64f437836e209": "f(x+h) - f(x-h) = 2hf'(x) + \\frac{h^3}{3}f^{(3)}(x) + O_1(h^4). \\qquad (E_1).", "3b0689be46524681868639ea9b29e170": "L'\\subset U", "3b069a9ff3bbbb907820074ef747cea4": "\nX_{1}=[1,4],\n", "3b06e0b2ca980677d604d08d5299cfca": "1/R_1", "3b06efd4e270b6de866f84aeecc49cd3": "\\Theta(n^2)\\,\\!", "3b071ff073293043e0e0a869e60a36b7": "\\gamma(c)", "3b07406a3088964cf4ea01b4a4211a03": "\\mathbf{\\left(C^TC\\right)^{-1}C^T}", "3b07b9db830dd2dfe0671de76fc199c7": "p_1 \\ ", "3b07c1adcf319938278f5baabd41040a": "\\Gamma(z) \\Gamma\\left(z + \\tfrac{1}{2}\\right) = 2^{1-2z} \\; \\sqrt{\\pi} \\; \\Gamma(2z).", "3b07f4bc28a5acb3fffb42dbce56c35f": "e_n>0", "3b082dda8bf604e12ded74665a781e83": " a_r = - R\\dot{\\theta}^2, \\quad a_t = R\\ddot{\\theta},", "3b083fd731313fb484a3df32edc6e6e2": "c(W)", "3b08534ffaeac3a530b9eed937256bc7": "\\overline{\\mathbf{x}} = \\frac{1}{n}\\sum_{j=1}^n \\mathbf{x}_j", "3b0873649d7f796d86ac94732db802fd": "\\nu_{\\rm ij}\\,", "3b0892bd2e5779fa0407be18df76f05f": "\\mathbf{P}=\\begin{bmatrix} \\mathbf{p}_1 & \\cdots & \\mathbf{p}_n \\end{bmatrix}", "3b08cd8ca1a1ee8009d2efa8ca6046a8": "g: A \\rightarrow \\mathbb{R}", "3b08d35114a95966e1300b5a2be756d8": "S^{n}", "3b0934c7cf7fe55ab79fc20bfe09abe3": " T_{eq}", "3b0969370f7cf4ec166e3e23a62db266": "\\mathrm{OPT_A}(x)", "3b09c55e900582865fc471bdfc9c9d47": "\\frac{1}{l}+\\frac{1}{m}+\\frac{1}{n}>1.", "3b09f48be5987cae034edc2c8db34f3c": "\\sum_{n=-\\infty}^{\\infty} a_n e^{in \\theta}", "3b0a28e21555d325ef0b018d2253e882": "M\\models\\varphi(\\boldsymbol{a},\\boldsymbol{b}_i)\\quad\\Leftrightarrow\\quad i\\in X.", "3b0aad24ba991ea674e1db1bb9eb3447": "\\Rightarrow v_i' A' A v_i = v_i' \\lambda_i v_i", "3b0bc1af0bfc8ad0b6b6eecf4086b64b": "t_{erosion} = \\frac{ln-ln\\sqrt[3]{M \\over N_a(N-1)*p}}{k}", "3b0bc7c8ba99f02922f95b0ed031d043": " \\frac{p(M_1|x)}{p(M_2|x)} = \\frac{p(M_1)}{p(M_2)} \\, \\frac{p(x|M_1)}{p(x|M_2)} ", "3b0be8bfcacef168f9f319325870a66f": " k\\,n = \\gcd(k,n)\\, \\operatorname{lcm}(k,n),", "3b0c0250b4dae3bd5c213440aa160729": "A, B\\subseteq \\mathbf{X}", "3b0c03a295f7ccb1bf40e93c0855680b": "y_{st}", "3b0cbc7a66385cdb6e660d79d832b88e": "\\pi = 3.14159265358979\\dots.\\,", "3b0cc52f836e4a2060b8b3ea4a9286f8": " p(\\text{State}_x \\rightarrow \\text{State}_y) = \\min \\left(e ^ { - \\beta \\, \\Delta U} , 1 \\right) = M(\\beta \\, \\Delta U) ", "3b0cf1c0bed947bef1ad13b216d92933": "O(n) \\to GL(n)", "3b0d0ee51a08628c8bf4cf5803934f4f": "q q^* = q^* q \\!", "3b0d28ee945540b0319d025d47edd547": " \\widehat{J}(\\theta,\\hat{\\mathbf{n}}) = \\exp\\left( - \\frac{i}{\\hbar}\\theta \\hat{\\mathbf{n}} \\cdot \\widehat{\\mathbf{J}}\\right) ", "3b0db11f7d980f663d41743e015d0b5f": " \\boldsymbol{\\Pi} ", "3b0e288b1adaa5e884af63f766cacf84": "z_n ", "3b0e3f2743431bf4e37f57243cd458e9": "\\Phi_{0}", "3b0e47a9c3b48f4ca037c8ed9aa02e84": "\\tan \\varphi = \\frac{\\mathrm{d}h}{\\mathrm{d}r} \\ , ", "3b0e5e1d97adc256f2b98cd2f744bf36": "H =-\\textbf m \\cdot \\textbf B ", "3b0ed2b8c224323c9013b655cab96c3d": "K_1 \\subset K_2 \\subset \\ldots \\subset K_i \\subset \\ldots \\subset \\mathbb{R}^n", "3b0ef6c5b8211ce98a7b53bd02483b80": "dN/dt = \\alpha N/\\tau + R_{ext}", "3b0f0c1f9691a56064c8ab7060541535": "\\epsilon _r", "3b0f0f8e6e9f9340b93a810c6799d45f": " \\chi", "3b0f12c3103e5b2c8348068d9cda21cb": "f:X' \\to X", "3b0f4ae8948b006034035a219444d8f5": "\\Vdash", "3b0f63e1f204605caf65080079f9768c": "\\alpha_{(X, d, \\mu)}(r) \\leq \\inf_{\\lambda \\geq 0} e^{- \\lambda r / 2} E_{(X, d, \\mu)}(\\lambda).", "3b0fd87374e9f1fb68109f97bd30554c": "\\frac{d \\mathbf{U}_i}{d t} = - \\frac{1}{\\Delta x_i} \\left[ \n\\mathbf{F}^*_{i + \\frac{1}{2} } - \\mathbf{F}^*_{i - \\frac{1}{2}} \\right]. ", "3b10662a8cf9041bde2c6a2c5c04523a": "+ 8.1328 \\times 10^{-3} \\left( 10^{-3.49149 \\left( \\frac{373.16}{T}-1 \\right)} -1 \\right) ", "3b10a44788d879113e9c52d9f2cf9592": "\\tilde{w}_{t+1}", "3b115ea7deabe9c25d0300730054e2b5": "{R} = {d \\over k}", "3b117adf5fbcdcda245a6a1bc3994377": "P_p", "3b11965be31a7037143594edb7b30dfd": "\\begin{align}\n\\dot x&=f(x,y,t)\\\\\n0&=\\partial_x g(x,y,t)\\dot x+\\partial_y g(x,y,t)\\dot y+\\partial_t g(x,y,t),\n\\end{align}", "3b11c09f045420af63f613bb366c1215": "\\Gamma(1-i\\,\\beta\\,t)\\, e^{i\\,\\mu\\,t}\\!", "3b11da5ab8568206a5e1184fbf7f2005": "t'", "3b12100b64fe2bf2b6891c49927b1641": "{\\mbox{Rate}_1 \\over \\mbox{Rate}_2}=\\sqrt{M_2 \\over M_1}", "3b129757f5aff3ee2b9cac3abce03dd6": "\\frac{p}{r}\\ =\\ 1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u", "3b12b4cd3489fdb768f0c0c5612cf901": "RF = \\sum_{n=1}^{100} Abs_i * F_i / (path length * density)", "3b12d83f8b89b21472cf27077e3baae7": " G_{i\\pm1} G_i E_{i\\pm1} = E_i G_{i\\pm1} G_i = E_i E_{i\\pm1},", "3b12e7a9ae284b196241957e00e64f02": "c_L = \\frac{(S_{22} - \\Delta S_{11}^*)^*}{\\left|S_{22}\\right|^2-\\left|\\Delta\\right|^2}\\,", "3b1330e0c2545335e71016fb5f22e413": "\\exp(ar) = \\cosh a + r \\sinh a", "3b13f8bff79a3c1bb02ee3d0e830a2f9": "n^{\\searrow}", "3b14123f58f4887d7d8265174c2f8bff": "\\nabla u = \\xi", "3b14140ad31c06d1b25a3f39d61927e1": "\\mathbf{\\tau}=\\frac{\\mathrm{d}\\mathbf{L}}{\\mathrm{d}t}", "3b141fc310bd0905d591dcaf295dce55": "\n1 \\mapsto I, \\quad\ni \\mapsto i \\sigma_3, \\quad\nj \\mapsto i \\sigma_2, \\quad\nk \\mapsto i \\sigma_1.\n", "3b1420091167ca25fa32ad9dcef064a7": "a_\\bar{\\alpha}e^\\bar{\\alpha} = a_\\gamma L^\\gamma{}_\\bar{\\alpha} L^\\bar{\\alpha}{}_\\beta e^\\beta = a_\\gamma \\delta^\\gamma{}_\\beta e^\\beta = a_\\beta e^\\beta ", "3b1472caccc92a8b1e8d12f50edbebdb": "\n\\mathrm{i}\\hbar\\frac{\\partial}{\\partial t} \\Pi_{\\mathbf{k},\\omega}\n = \\left( \\tilde{\\epsilon}_{\\mathbf{k}} - \\hbar\\omega \\right) \\Pi_{\\mathbf{k},\\omega}\n + \\Omega^\\mathrm{spont}_{\\mathbf{k},\\omega} \n -\\left( 1-f^e_{\\mathbf{k}}-f^h_{\\mathbf{k}}\\right) \n \\left[ \\Omega_{\\omega}^\\mathrm{stim} + \\sum\\limits_{\\mathbf{k'}} V_{\\mathbf{k}-\\mathbf{k}'}\\, \\Pi_{\\mathbf{k'},\\omega} \\right]\n + T[\\Pi]\n", "3b14b00e76a4edd926be4d6e434cea81": "\\Rightarrow \\varphi ad=2az", "3b14efa94daee39473593385cafc0c37": "RT_{60}", "3b14f6f309b25494208f8a0df8dbad95": " \\beta = \\frac{\\partial f} {\\partial y} ", "3b151fae5cce949fc7d80b6bcd36ce03": " Y \\sim a_n X+b_n \\,", "3b158d090867371c5a3184204c485350": "Pr(u 0\\,", "3b282b6179c137cf1d04744dc0f6e389": "{\\rm GEN}[d : y \\leftarrow f(x_1,\\cdots,x_n)] = \\{d\\}", "3b283f9cf5c9ed184db5d0facdfca9e8": " x = -i / 365\\ ", "3b2859a85e2d74b3709dd218dd3fe86a": "\\text{ Theorem (Lyapunov Drift):}", "3b2937ad3a590f457abbd9f057694b91": "\n\\text{mean }\\sim\n\\begin{cases}\n\\log(\\log B /\\log A) & \\text{ if }A \\ge 2p\\\\\n\\log(\\log B/\\log 2p) & \\text{ if } A < 2p\n\\end{cases}\n", "3b29432c37a1b2a1feda408c55b114ad": "\\mathbf{E} \\cdot {\\rm d}\\mathbf{A} = \\frac{Q_A}{\\varepsilon_0}", "3b294daf846d4ec8f5c0fdfb7e2365e7": "M^{-1} = \\Omega^{-1} M^T \\Omega=\\begin{pmatrix}D^T & -B^T \\\\-C^T & A^T\\end{pmatrix}.", "3b2a00e05ecbe4e2be9fa52da2a9b783": "P_i\\not= \\infty_2", "3b2a1c31ae67bcf5b5e1fc95ec1aacf4": " \\mathcal{H} ", "3b2a1ec8379c3c396ea71c84da5654c3": "w= u \\cdot y", "3b2a24ab51ceed02c1f2eed6180cc5d6": "V = \\{v_1, v_2, \\dots , v_n\\}", "3b2a34ca378aef6c17c4a3e243da516a": "h_n= 2n^2-n = n(2n-1) = {{2n}\\times{(2n-1)}\\over 2}.\\,\\!", "3b2a66e71172ac602fcddf2d898ce41a": "\\boldsymbol{a}_k", "3b2ab5c9156cc0235b27d2381e7068d0": "h(x)\\not=0", "3b2add5f3d07adeb536f18957dda3513": "[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0", "3b2b047befafd0c595548b1ea596e665": "\\partial_{\\alpha} F^{\\alpha\\beta} = \\mu_0 J^{\\beta}", "3b2b2f55f494023cc950938ec6cf35b1": "Y ' = \\frac {1} {1 - b + bt - t}", "3b2b4ab47f9f2a3bdd3a12a580b0ef1f": " (\\lambda N.S)\\ L ", "3b2bcfc85199be35933cc4e6863de403": "\\displaystyle{\\theta(a,T,b)=(b^*,-T^*, a^*).}", "3b2c1e66454b51b9cc09092522944edf": "PU(n,q^2)", "3b2c98fba60af916a92914cb7d82a4a7": "\\left| \\int_{a}^{b} f(t) \\, \\mathrm{d} t \\right| \\leq \\int_{a}^{b} | f(t) | \\, \\mathrm{d} t.", "3b2c9d153c64807f201904be7a76c3fc": "Pic^2 C", "3b2cb08fd321d537d08511485ecf9db3": "\\mathbf V^2 = \\frac{(\\nabla \\rho \\and \\mathbf s)^2} {(m \\rho)^2} = \\frac{(\\nabla \\rho)^2 \\mathbf s^2 - (\\nabla \\rho \\cdot \\mathbf s)}{(m \\rho)^2}", "3b2d023b859ded08c8c4c184ecacd022": "\\,^{238}_{92}\\mathrm{U} + \\,^{66}_{30}\\mathrm{Zn} \\to \\,^{304}_{122}\\mathrm{Ubb} ^{*} \\to \\ \\mbox{no atoms}.", "3b2d313c996f7b649bd03ec171e4ca55": "(1-\\Delta)^{-\\alpha/2} f(x)", "3b2d3d0237af5b39491befdb07ea28eb": "\\mathfrak{P}^{42}", "3b2d420c1a4d25644b219ed39d3e5af5": "A=QR", "3b2d4a1471c4ca4f92802ea99fedf192": "\\sum_{i=0}^n {a_i\\over i+1} x^{i+1}+c.", "3b2d4bf09a1522c000aa0ca512680a75": "\\upsilon_i\\,", "3b2d4bffbf92348313f0ba26d1c65aa8": "(\\rho, V) \\mapsto (\\widetilde{\\rho}, \\widetilde{V})", "3b2d83f5259aceba9eda5299818320af": "(\\pi_* \\alpha)_b(w_1, \\dots, w_{k-m}) = \\int_{\\pi^{-1}(b)} \\beta", "3b2d942e1b6b35478eac541a1ba94985": "\\ H(X)", "3b2da26bcb84195111209bbb94892b0f": "xI-A", "3b2ddd65b843e8771a11e3dacfb17b30": "n \\in \\mathbf{F}_p", "3b2de88c6db990a0b1aea350cab43b07": " f(x_i^1, x_i^2) \\sim y_i ", "3b2e0846862c2d68570c2aad976b389a": "(DD^+)^*_{ij} = \\overline{(DD^+)_{ji}} = \\overline{D_{ji}D^+_{ji}} = (D_{ji}D^+_{ji})^* = D_{ji}D^+_{ji} = D_{ij}D^+_{ij} \\Rightarrow (DD^+)^* = DD^+", "3b2e116244d59da1693bca9865a9d109": " SCx \\leftrightarrow ((Owx \\leftrightarrow (Owy \\or Owz)) \\rightarrow Cyz).", "3b2e50c2f191e119bee6d39569506137": "P' \\rightarrow Q'", "3b2e6afe7165b41c42ce46fad306f39f": " \\frac{\\mathrm{d}^2z}{\\mathrm{d}t^2} + 2\\zeta\\omega_0\\frac{\\mathrm{d}z}{\\mathrm{d}t} + \\omega_0^2 z = 0,", "3b2ee0c405069427f5c510ad61e68ea5": "\\lim_{x \\to \\infty^{-}}{f(x)} = \\infty", "3b2f82bbfea9e3a0c6530a3192ee9f94": "\\Phi [\\gamma] = \\int [dA] \\Phi [A] W_\\gamma [A] \\qquad Eq \\; 3.", "3b2fc3c8fe0476a1cec4b4b442ba81bd": "a_{i_1,i_2} = \\sum_{j_1} \\sum_{j_2} s_{j_1,j_2} u_{i_1,j_1} v_{i_2,j_2}.", "3b2ff3a1ee340d0c089db2fc4471873e": "\\gamma^\\mu \\,", "3b30218aa528ffc16aceef143dfffcc5": "\n X_{i+k} = X_{i+k-1}=\\cdots=X_i, \\,\n ", "3b304622880b8a97c2ce6b70b998fa15": "\\lnot\\ \\exists{x}{\\in}\\mathbf{X}\\, P(x)", "3b305b9510c12e5211176d7a9c2f7a9b": "out = (i \\times s+o)^p\\,", "3b30634366e6f1777ab951f24a3bb78a": "\\rho_{XX}= { \\sigma^2_T \\over \\sigma_X^2 }", "3b307654e51df17652081e535ef32516": "\\Gamma_1(N) \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\Gamma_1(N)", "3b30cf8167f3c3b05088083bb74e949a": " \\mathbf{u} \\cdot \\mathbf{v} = u_j v^j ", "3b30f6c2d383613a51049bbab7928c5e": "H^i(X/W_n)= H^i(Cris(X/W_n),O)", "3b318ebe541d1e7815ecb989232ae7c8": "\n \\prod_{i=1}^n f \\left[ w_i(y_1, y_2, \\dots, y_n); \\theta \\right] = \n |J| g_1 (y_1; \\theta) H \\left[ w_1(y_1, y_2, \\dots, y_n), \\dots, w_n(y_1, y_2, \\dots, y_n) \\right].\n", "3b31dc1fd5532e644436a83d6acc0cca": "\\begin{matrix} \\frac{63}{25} \\end{matrix}", "3b31e75d6c3592f5ac8b1ad65daa3e60": "p_{\\text{c}} = \\frac {1}{z - 1}", "3b323244dc03be358ea60e4bac582d36": "P(t') = \\begin{cases}\n 0 & t'<0 \\\\\n \\ (\\frac{t'}{t_p})^\\frac{1}{2} & 0t_p \\\\ \n\\end{cases}\n", "3b324733612987eabfb38a979f6eb19a": "f(x) = g(x) +c \\delta(x)", "3b324db1eaa47e0dda98d48a710bac7c": "\\begin{align}f'(3) &=\\lim_{h \\to 0}{(3+h)^2 - 3^2\\over{h}} \\\\\n&=\\lim_{h \\to 0}{9 + 6h + h^2 - 9\\over{h}} \\\\\n&=\\lim_{h \\to 0}{6h + h^2\\over{h}} \\\\\n&=\\lim_{h \\to 0} (6 + h) \\\\\n&= 6.\n\\end{align}\n", "3b3277d2dd40b31b362ac8e0b404c0a6": "P_n^{[r]} = (ax+b)P_n^{[r+1]} + cP_{n-1}^{[r+1]}", "3b3278c882257d3ac8ab2bb01fa832e1": "\n\\begin{array}{lcl}\n\\kappa_1 = \\frac{ (d_m-d_1) \\alpha_{min} \\alpha_{max} }{\\alpha_{max}-\\alpha_{min}} & \\mbox{and} &\n\\kappa_2 = \\frac{(d_m-d_1) \\alpha_{min} }{\\alpha_{max}-\\alpha_{min}} - d_1 \\,, \\\\\n\\kappa_3 = \\frac{ \\beta_{min} d_3- \\beta_{max} d_2}{d_3-d_2} & \\mbox{and} & \\kappa_4 = \\frac{\\beta_{max}-\\beta_{min}}{d_3-d_2} \\,. \\end{array}\n", "3b32e8f19d038f4eb31277a4748077d1": "\\mathrm{sinc}(t) * \\mathrm{rect}(t)", "3b334f61cb7691400b5c49efba5e1927": "\\sum_{i=1}^{m} f_i(X) \\exp(w_i X) \\ , ", "3b3377d330524dd9a309dc12c3485a63": "P_f = \\frac{C}{{48 \\choose 3}} = \\frac{C}{17,296}.", "3b337a2c86bbc018e9a1090ea01f130c": "a^{-1}", "3b338359f8056df1239509a4b0cc0ea3": "M^{\\nu\\lambda} = \\int (x^\\nu T^{0\\lambda}_B - x^\\lambda T^{0\\nu}_B) \\, d^3x,", "3b3385d534f70fc3076aa6904d06cccf": "\\Sigma_2^{\\rm P}", "3b33a567b762d91b0fe096e466f3eb50": "\\textstyle (\\Omega,\\tau) ", "3b33bcab0b9526d3c61dcf72d1f4b4f9": "f_{cr}\\equiv\\frac{\\pi^3\\textit{E}I_{min}}{{L}^2}\\qquad (9)", "3b33c6b3e6e1cb2ef749405f90b42c4b": "\\omega_0\\tau_c = 3.2\\times 10^{-5} ", "3b34eb7bcc02dc635efb92947a6433bb": "A_i=B_i=\\{i\\}", "3b35a4bf9a4ab88710401950e2b4c373": "s(x)=\\frac{x}{1-x-x^2}", "3b35bdb5290a289774676d2f4d317761": "\n (U(b,a)\\varphi )(x) = \\frac 1{\\sqrt{\\vert a\\vert}}\\;\\varphi \\left(\\frac {x-b}a\\right) \n= \\frac 1{\\sqrt{\\vert a\\vert}}\\;\\varphi \\left((b,a)^{-1}\\cdot x\\right)\\; .\n", "3b360244984f1ac142e381b0a84a8a06": " c_1 = {n \\choose { \\lfloor n/2 \\rfloor}} \\frac{ \\lfloor n/2 \\rfloor +1} { 4^{\\lfloor n/2 \\rfloor}} = \\sqrt{ \\frac{2n+1}{\\pi}} (1 + \\frac{1}{16n^2} + O(n^{-3}) ) ", "3b3614a1e8046e13567bbc548912fc47": "\\left(\\sum_i a_i x^i\\right) \\cdot \\left(\\sum_j b_j x^j\\right) = \n\\sum_k \\left(\\sum_{i,j: i + j = k} a_i b_j\\right)x ^k", "3b36ab597b78b5d8e005480516634617": " Q_{1,2} = \\sum_{k} n \\left ( k,T_1 \\right ) E \\left ( k \\right ) \\alpha \\left ( k,T_1,T_2 \\right ) ", "3b370b69d2d26efed2ebdca9fa922282": "a^{\\dagger}_{q} = \\frac {q} {\\sqrt{2M\\hbar\\omega_{q}}}(M\\omega_{q}Q_{-q}-iP_{q}), \\; a_{q} = \\frac {q} {\\sqrt{2M\\hbar\\omega_{q}}}(M\\omega_{q}Q_{-q}+iP_{q})", "3b3772bbb08acf39967810c3727a989e": "F_s (\\theta) = \\vert F_u (\\theta) + F_1 (\\theta) + \\cdots + F_n (\\theta) \\vert", "3b37a270a57eac29d0de5001b4cae670": "= a_1 \\mathbf{e_2 e_3} (\\mathbf{e_1})^2 + a_2 \\mathbf{e_3 e_1}(\\mathbf{e_2})^2 +a_3 \\mathbf{e_1 e_2}(\\mathbf{e_3})^2 \\ ", "3b37fe86ea2fcf95fd818b7c20878b32": "-\\tfrac12\\,\\lambda \\le (x-ct) \\le \\tfrac12\\, \\lambda,\n", "3b387b16223045fe6b26aa684f0a7abc": "\\Delta \\tau^{k}_{xy}", "3b3898f49f454ae636ee8f845beb46c9": "(\\forall^\\infty n) P(n).", "3b38b310cce2ab26db2ca1961448e42e": "s_{\\hat{C}}^2 = \\hat{\\sigma}_e^2\\sum_{i=1}^r \\frac{c_i^2}{n_i},", "3b38d76686c624c531e43e776eafecf6": "\nQ_p = \\Delta H = \\Delta E + P \\Delta V\n", "3b38e499475ed0414b0b6f6be19f7e80": "D \\,\\ ", "3b393bc6d3f5dc661ddaadf03c2007b9": "n\\ge 3,", "3b3961071623492f0f7954aa8703cf19": "(\\sigma_x,\\sigma_y)", "3b39a2fdc2062bbe0da91a62259eba37": "\\frac{\\operatorname dV}{\\operatorname dr} = \\overbrace{\\frac{2 \\pi r h}{3}}^\\frac{ \\partial V}{\\partial r} + \\overbrace{\\frac{\\pi r^2}{3}}^\\frac{ \\partial V}{\\partial h}\\frac{\\operatorname d h}{\\operatorname d r}", "3b39c294586e81aacc978ef908631425": "\\phi: B \\rightarrow A", "3b39ec83ff7a8148b1d91b96938bd7b7": "1 \\le v < N", "3b39ee79bb70d5adf582c648e7ece84c": "\\scriptstyle c^{\\underline{k}}", "3b39f51b23bf8d422ec3487c7a5856fb": "S_q^{FD}", "3b3a8db2b8c4d79c979644d3ffab0ebb": "F_5(a, b) = (x \\to a \\uparrow\\uparrow (x - 1))^b(0)", "3b3acd877f62a7c88e8ae3ed14aa4c4c": "\np_2(x) = x^2 + 13x + 40 \\,\n", "3b3ad4a373a06cf47c4f5fe221317089": " \\int g_k \\, \\mathrm{d}\\mu \\leq \\lim_j \\int f_j \\, \\mathrm{d}\\mu", "3b3af28dc7c5786eb0bc823abbaa7d92": "\\Pi^P_i", "3b3b0e9de547edc68ceafea06f074f12": "\\frac{\\partial v(\\boldsymbol{x})u(\\boldsymbol{x})}{\\partial x_i} = {\\bar u(\\boldsymbol{x})}\\frac{\\partial v(\\boldsymbol{x})}{\\partial x_i} +\n{\\bar v(\\boldsymbol{x})}\\frac{\\partial u(\\boldsymbol{x})}{\\partial x_i} ", "3b3b3f7ca3d225991ca23ad28820d7c5": "\\eta \\neq 0", "3b3b6176e39feefdcad084245062877c": "\\tbinom {11}4 ", "3b3be736d579f49f089d5b0396f9e696": "y=z-x", "3b3bf39f22a71197852e3887b8c4355c": "(0,1)^T", "3b3c1396f4e117e34019db53e63214fe": "k\\leq 0", "3b3c371e0ea9262f06e4d1a04983270e": " e^{iu \\hat H} ", "3b3c8e8c075018de592b4b7eb3df94ca": "F_j\\,(p_i)\\!", "3b3ccdc6b4cddb317f454ce61084a251": "\\Pi^{i_2j_2}\\partial_{i_2}\\Pi^{i_1j_1}\\partial_{i_1}f\\,\\partial_{j_1}\\partial_{j_2}g", "3b3cdac3c269edd53582e8f4058010e4": "n\\in\\N", "3b3ced53cf8efe05fe595900a23e654f": "z = \\frac{\\lambda_o - \\lambda_s}{\\lambda_s} = \\frac{f_s - f_o}{f_o}", "3b3d146c84932f395735516d07a2abe6": "\\frac{d}{d\\Lambda}S_{\\text{int}\\,\\Lambda}=\\frac{1}{2}\\frac{\\delta S_{\\text{int}\\,\\Lambda}}{\\delta \\phi}\\cdot \\left(\\frac{d}{d\\Lambda}R_\\Lambda^{-1}\\right)\\cdot \\frac{\\delta S_{\\text{int}\\,\\Lambda}}{\\delta \\phi}-\\frac{1}{2}\\operatorname{Tr}\\left[\\frac{\\delta^2 S_{\\text{int}\\,\\Lambda}}{\\delta \\phi\\, \\delta \\phi}\\cdot R_\\Lambda^{-1}\\right].", "3b3d6d85793e4a70ac702c0428a17e21": "\\eta^j", "3b3da96bc86e677130886cf94f7340f3": "\\frac{d\\mathbf{r}}{ds}=\\mathbf{u}\\,", "3b3dccbb803fcab60b3f53f6d025e763": " \\mathcal{R}", "3b3de84a8caf71cda25801b27f149a60": "8_4", "3b3e529c3f47b0a8aa261b88b7a40d58": "\\hat{h}(\\xi)= e^{-i\\,2\\pi \\,x_0\\,\\xi }\\hat{f}(\\xi).", "3b3ecf5b8d7b969717404fa71dad33cb": "e^{\\sqrt{2 \\ln{n} \\ln{\\ln{n}}}}", "3b3f36aa8b47beed747da100f96cdbb7": "\\mathcal{Z}(M)=\\{m\\in M \\mid \\mathrm{ann}(m)\\subseteq_e R\\}\\,", "3b3f4599e6a876470a36d1a51ab7cd4c": " \\frac{1}{|\\mathbf{x} - \\mathbf{x'}|} = \n\\sqrt{\\frac{\\pi}{2RR^\\prime(\\chi^2-1)^{1/2}}}\n\\sum_{m=-\\infty}^\\infty \\frac{(-1)^m}{\\Gamma(m+1/2)} P_{-\\frac{1}{2}}^m\n\\biggl(\\frac{\\chi}{\\sqrt{\\chi^2-1}}\\biggr) e^{im(\\varphi-\\varphi^\\prime)}\n", "3b3f99a5588e1cd0c235b04edc563f25": "\\mathcal{N}(0, 1)", "3b3ff098a241ee9b03e652351d9679a4": "\\frac{\\partial ^2 x}{\\partial s^2}(\\bar v_i) ", "3b4009e99bf3748330c2d88a3f54c08e": "\n \\underbrace{C_k^{-1/2}x_i}_{\\text{represented in the encode space}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!} \n \\sim\\ \\underbrace{C_k^{-1/2} m_k} {} + \\sigma_k \\times\\mathcal{N}(0,I)\n", "3b402a16df1fa257375e59c71643186c": "P_0(unknot)=1", "3b404228079c6fb90a4d3b9e03be9cf2": " \n\\sum_{\\tau=0}^{t-1} E[\\Delta(\\tau) + Vp(\\tau)] \\leq (B+C+Vp^*)t\n", "3b404da201bef933ed8a3f14f27aac7d": "(3)\\; \\tanh(z)=\\frac{e^z-e^{-z}}{e^z+e^{-z}}\\qquad\\text{if necessary, this is an alternate way to calculate }\\tanh(z)", "3b40aaaf53ecdaa26e99ac5e7887feba": "\\Delta H_\\text{T} = - {\\mu_\\text{B}\\over \\hbar m_\\text{e} e c^2}{1\\over r}{\\partial U(r) \\over \\partial r} \\boldsymbol{L}\\cdot\\boldsymbol{S}. ", "3b40b61a22637050e1df8744a5bb4d7d": "Q = 2(\\cos 60^\\circ + \\epsilon \\sin 60^\\circ) \\qquad \\mathrm{where} \\; \\epsilon = \\frac{\\mathbf{\\hat{i}} + \\mathbf{\\hat{j}}+ \\mathbf{\\hat{k}}}{\\sqrt{3}}", "3b411dd2dc1ed5361ea6837422076db8": "I_n = \\begin{bmatrix}1 & 0 & \\cdots & 0 \\\\0 & 1 & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\cdots & 1 \\end{bmatrix}.", "3b412e509ebbe2263833dc6f4e929677": "g(q)=\\vee", "3b41371d89db1e10cbcdb4337ff45eb1": "S(M) = F_{\\mathrm{Spin}}(M) \\times_\\sigma V\\,", "3b41658e8ebb65e67531a1965e7b37ad": "\\tfrac{104GeV}{c^{2}}", "3b417c6d45b752cba04a2e4fcc272ec0": "\\Phi^t(x_1) = x_1 + b t. \\, ", "3b417d4b4b2baf5993f55705765fa9e5": "\\scriptstyle 1 \\;\\leq\\; i \\;\\leq\\; k", "3b4184c4d5244141f02d72c530058a30": "\\gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} \\,\\!", "3b419e6821b41fcf20bd5e9532112eaa": "\\bigcup_{n \\in \\omega} \\left(\\prod_{ie^{0{.}3}\\approx 1.35", "3b538eafeca2a73b8f4a230cc039d74e": "\\mathrm{OPT} + c", "3b53af3e0a3e8fdf0d007855d398d9f7": "\\sum_{i=1}^rk_i=n.", "3b53ca2a50ed178e55a6d222ce820d62": "t = 1,2,3,4,...,T", "3b548677e8e56e1c1ec36ef131402269": "h_\\lambda(i,j)", "3b5499025a83a061ee5bfa7b0b9d9e88": " = \\sum_i \\sum_k \\operatorname{tr}(V_k^* T_i V_k) S_i ", "3b54a99ce841101d2da49d53d21d1aab": "\n\\begin{bmatrix}\n u_{1,j+1} , & u_{2,j+1} , & \\ldots, & u_{i-1,j+1} , & u_{i,j+1} , & u_{i+1,j+1} , & \\ldots , & u_{m,j+1}\n\\end{bmatrix}^{T}\n", "3b54da5cf02a89535bfdb905d78362e9": "x_2 = 0", "3b554d15c82feda50395af470a69d38c": "\\ln(-1) = i \\pi \\,", "3b55bfb694e78619de61e6e81ee4d319": "I_z=I_x=I_y = \\frac{5m s^2}{9}\\,\\!", "3b55d2b028332efa73a15f2eefff1db5": "\\mathcal{I}(y_i^\\ast < 0)", "3b55f3a5f2e5c5028de1bdf58182490a": "\\mu=G(m_1+m_2)", "3b561880168d2eba7b3b535697d1584a": "{\\mathcal L}^2_1", "3b562967bedf212d0241cf964cfbf3b9": "f \\in \\mathbf{B}^{*}_{1}", "3b562ca069c0083147b92a2b76a63a83": "x'_s", "3b568b190e2d7958119297d6765b5a54": "x'= \\gamma\\left(1 - v/c\\right) x , ", "3b576420b0f9041686bf146bb9e05313": "\n \\{x_{i:\\lambda}\\;|\\;i=1\\dots\\lambda\\} = \\{x_i\\;|\\;i=1\\dots\\lambda\\} \\;\\;\\text{and}\\;\\; \n f(x_{1:\\lambda})\\le\\dots\\le f(x_{\\mu:\\lambda})\\le f(x_{\\mu+1:\\lambda}) \\dots, \n ", "3b57cb879a00fe6fa4327709bc343d60": "a_n\\le n\\left(\\exp\\frac{\\log n\\log\\log\\log n}{\\log\\log n}\\right)^{-1+o(1)}.", "3b587fd0fd075523fb1254a40f00bb11": "\n\\frac{\\displaystyle 1}{\\displaystyle 5}\n\\begin{bmatrix}\n1 & 3 \\\\\n4 & 2 \\\\\n\\end{bmatrix}\n", "3b58c0f665aa394bc302541fe7ad910b": "I_t", "3b58c83ae80346eda6d4011c3ecd5b58": "\\log 2 \\approx 0.693", "3b592a1b9ef4db02c2604bcef8d066a9": " (Uf_1,Uf_2)= \\int_0^1 f_1(\\lambda) \\overline{f_2(\\lambda)} \\, d\\rho_\\xi(\\lambda).", "3b596f340b03622801d962ea64fdf78e": "f(x) = x^0", "3b5973da96b05f7c670feafade237ffc": "\\left( f\\circ g\\right)_{*} = f_{*}\\circ g_{*}", "3b59aca92004ccb465fdced3c2637b3b": "c = \\frac{(8-3\\kappa)(\\kappa-6)}{2\\kappa}.", "3b59c29c58b6c2e8549037e1100b5354": "\\Omega(t)", "3b5a01055dab14e30e196edf86ada18e": "q > 2", "3b5afc975bc6a86aaa2fdcb42a32afe2": "p_k > 0", "3b5b13001ff20440df0949d0c98ceccf": " 8 \\pi G \\beta", "3b5b2cbef8271bd4988b52b9ca76e0ce": "C,c", "3b5bf17060cc47ad0b2e0afe3a871766": "\\phi_C \\to -90^{\\circ} = -\\pi/2^{c}", "3b5c0ae668d22d6ddeb35c5d9493397b": " \\bold{H} = \\bold{r} \\times {\\dot{\\bold{r}}} ", "3b5c33ddde965028886a6302773c60a3": "\\left|\\frac{\\partial f}{\\partial z}\\right|^2\\left|\\,dz+\\mu(z)\\,d\\bar{z}\\right|^2", "3b5c67dc509b7eb570e21ed48aad306d": "H_\\mathrm{sat}", "3b5ca8e8b4227088f3c723c17d37a64d": "\\mathcal{O}(n^{-\\frac 4 {d+4}})", "3b5cc95662d4b80545e7f51c1e961110": "\\boldsymbol{r}\\in\\Omega", "3b5cd61f0eeb2afb03f5d976f4c4776b": "(f(u),f(v))", "3b5ce468c9f4d57fab6810e8da9543e8": "\\textstyle l_\\phi", "3b5d433bad8544082ffba17dccdec5b8": " P \\phi(x) = \\sum_\\alpha a_\\alpha (x) [D^\\alpha \\phi](x) \\quad ", "3b5d98f81f5e9b8c05395efeecd7f81f": "\\scriptstyle L \\;=\\; E_K(0)", "3b5ddab97b4b756a33c90f2413708299": "S \\Rightarrow^{ac}_{f} AA \\Rightarrow^{ac}_{f} \\text{failure: f cannot apply, no S to rewrite}", "3b5e15f8b649c3f05e4401b4d963a7ce": "F = m_2g\\,", "3b5e1cdc976e621110fe3cd2ad42e20a": "d\\in K", "3b5e220a3eaa7ee8d067eb53e5646e2b": "{\\boldsymbol{k}}=-i\\nabla-(\\frac{e}{\\hbar c}){\\boldsymbol{A}}", "3b5e2e71c669aeeb7cfc164cd9b0c871": " d=8 k ", "3b5e945e06a92b75da0f1fac1f3c210c": "|s- s_0| = O(a)", "3b5ed22c67975ebb8451c780d15f593f": " W(\\mathbf{E}, \\omega)=lim_{x->\\infty}\\int_0^\\frac{2\\pi}{\\omega} \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\mathbf{J}(\\mathbf{r}, t)\\,dz\\,dy\\,dt ", "3b5ee8024caf15f5d7f78db7a0fc267e": " \n\\mathbb{E}\\left[e^{\\frac12\\int_0^T|X_t|^2\\,dt} \\right]<\\infty\n", "3b5ee9a11d177a1e927a8a1ae3bdcb35": "0.72m", "3b5f05190dadd3aae770bee42654f2ef": " dG = - SdT + VdP -\\sum_k\\mathbb{A}_k\\, d\\xi_k + W'\\,", "3b5f492839c9c578e407002ec5cf24f7": "Y_{4}^{4}(\\theta,\\varphi)={3\\over 16}\\sqrt{35\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\n= \\frac{3}{16} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{(x + i y)^4}{r^4}", "3b5fc447f812a4c1960e7623827de449": "V^-_i (x)", "3b5feba03b6d9da8be8f622c5ce58674": "P = \\lbrace p \\ : \\ w^2 = x^2 + y^2 + z^2 \\rbrace ", "3b6001d8506045a12d7ed9f238d66304": "\\{\\mathcal{F}_t\\}", "3b6017630b6710045c4c052af7e8d838": " 2 |\\psi_{\\rm{first}}| |\\psi_{\\rm{second}}| \\cos (\\varphi_1-\\varphi_2)", "3b6032e21471911fe22eb9a98691a0d8": "\nRi(t) + L { {di} \\over {dt}} + {1 \\over C} \\int_{-\\infty}^{\\tau=t} i(\\tau)\\, d\\tau = v(t)\n", "3b605a361318f03d0bcabe1893b7ce5a": "\\mathbf{N}_\\perp' = \\gamma(\\mathbf{V})\\left(\\mathbf{N}_\\perp + \\frac{1}{c^2}\\mathbf{V}\\times\\mathbf{L}\\right)", "3b6094567cf6cf59f998046d09ce466e": "\\hat{\\mathcal{H}}^D_v = \\sum_{ab}^{\\rm act} h_{ab}^{\\rm eff} E_{ab} +\n\\frac{1}{2} \\sum_{abcd}^{\\rm act} \\left\\langle ab \\left.\\right| cd \\right\\rangle \\left(E_{ac}\nE_{bd} - \\delta_{bc} E_{ad} \\right)", "3b609db315daca08b1ba5798f495ada7": "\\mathbf{a} \\times \\mathbf{b} = \\begin{vmatrix} \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3\\\\a_1 & a_2 & a_3\\\\b_1 & b_2 & b_3 \\end{vmatrix} \\,,\\quad \\mathbf{a} \\wedge \\mathbf{b} = \\begin{vmatrix} \\mathbf{e}_{23} & \\mathbf{e}_{31} & \\mathbf{e}_{12}\\\\a_1 & a_2 & a_3\\\\b_1 & b_2 & b_3 \\end{vmatrix}\\ ,", "3b60c3ad613fadd38d803bf778dd360d": "q_p(a^r)\\equiv rq_p(a) \\pmod{p}", "3b60f7b61a34f4dfa6e4eb1d39f8b643": " Ulcer = \\sqrt { R_1^2 + R_2^2 + \\cdots R_N^2 \\over N }", "3b610ac9cb9c73a0b860de3f19551508": " \\forall n>0, \\quad G_n=F_n/F_{n-1}\\,,", "3b614fd07528b012fb282ec4c13014f3": "Y_0, Y_1, \\ldots, Y_k", "3b61c4dc59c6fe580f8ac917f976e9ca": " \\displaystyle T(f)=Cf(o).", "3b6224c4af77a49bb71a3ad2ee3e32ad": "\\delta^{\\prime}", "3b622665e22696c8abded4af3df4ec78": "l_m~", "3b624c6c26ca71b969fecc43cafed22e": " \\frac{D\\mathbf{u}}{Dt} = -\\nabla p + \\nu \\nabla^2 \\mathbf{u} + \\rho'\\mathbf{g} + 2\\mathbf{\\Omega} \\times \\mathbf{u} + \\mathbf{\\Omega} \\times \\mathbf{\\Omega} \\times \\mathbf{R} + \\mathbf{J} \\times \\mathbf{B} ", "3b62651acb7fe7f344e9b57051699f88": "\\cos b= \\cos c \\cos a + \\sin c \\sin a \\cos B. \\!", "3b62f4d0917da274ef3495cbd3424f41": "\\scriptstyle\\frac{TV_a}{2\\pi NQ}", "3b638484a01e42f1615228d8fb701413": "|a_n|\\le\\frac{\\varepsilon}{3N(\\sup_{i\\in\\{0,\\dots,N-1\\}} |B_i-B|+1)}\\,. ", "3b644c40484114fecd4a61c33b3366cf": "\\scriptstyle g^{c_i y_i}", "3b64750b3d25472bc02615a95a1ad93a": "R_\\mathrm{bc} = \\frac{R_aR_b + R_bR_c + R_cR_a}{R_a}", "3b648282eabb091877b4a804cd1a3f70": "{1 \\over 2} + {1 \\over 4} + {1 \\over 6} + \\cdots + {1 \\over 2 q} = {1 \\over 2} \\, \\gamma + {1 \\over 2} \\ln q + o(1),", "3b648d0253c4c3feb8df7f91a36bab96": "\\varnothing", "3b64f795a2d017501026578e56536178": "\\beta_0 = \\frac{I_\\mathrm{C}}{I_\\mathrm{B}} \\,", "3b6504d8d199c038ed34977e841951eb": "t \\ge 0", "3b652357e59da356dbb6f2105020406e": "M\\,\\!", "3b6587e29600b16050e600d998297c9f": "\\dot{x}(t) = A(t) x(t) + B(t) u(t)", "3b65e49e8cb88bcb30bf25fdea28f31f": "\\begin{align}\n x x^{-1} &= x^{-1} x = 1 \\quad\\text{(two-sided inverse)} \\\\\n (x y) z &= x (y z) \\quad\\text{(associative)} \\\\\n x^{-n} &= \\left(x^{-1}\\right)^n \\\\\n x^{m-n} &= x^m x^{-n}\n\\end{align}", "3b65e61b54b6cee0cd07c0fc1d99d08b": "\\iint\\limits_R \\, \\operatorname{div} \\mathbf{F} \\,dA = \\iint\\limits_R \\left (\\frac{\\partial M}{\\partial x} + \\frac{\\partial N}{\\partial y} \\right) \\, dA = 0. ", "3b6622f13d43749e85e0313d9c4e9c14": "\\scriptstyle{\\left\\langle E\\right\\rangle} - \\frac{\\varepsilon}{2}", "3b662e37216cb244b2ea95d3d0df48bc": "k\\not\\in B_i", "3b663072b176b2e938bb854c110afa39": " \\left (\\frac{\\rho_2}{\\rho_1} \\right )^{\\gamma}", "3b666fdb1b20334ed73a2726a7151cd5": "x,y,z \\in X", "3b6674d8bc09b2363e2d563e5828b6fb": "X_n(\\omega) = \\begin{cases}\n n, & \\omega\\in (0,1/n), \\\\\n 0 , & \\text{otherwise.} \\end{cases}", "3b668fcd7150b31229c08fe26550390e": "{R^0}_{202} = \\frac{-f'}{r \\, f \\, g^2} = {R^0}_{303} ", "3b669556eecc5ef4fc0c943c268fa6c7": " \\pi(n) ", "3b66d842f0650894bd2328e5403250fb": " (E_{1}-E_{2}+W_{1}-W_{2})=F_{1}(\\alpha_{1},\\alpha_{2},\\alpha_{3}.....\\alpha_{k})=0 ", "3b6759743fd82d988ff61e21d4de7191": "TS=U+PV-G\\,", "3b677ed68904b003ee168ea3275f2cf6": "\\mathrm{Var}(x)=\\int_{-\\infty}^\\infty (x-\\langle x\\rangle)^2 P_n(x)\\,dx = \\frac{L^2}{12}\\left(1-\\frac{6}{n^2\\pi^2}\\right)", "3b680635163a9410438cb7b2d5dd2723": "2^\\mathbb{N}=\\{(x_n)\\vert x_n\\in \\{0,1\\} \\mbox{ for } n\\in \\mathbb{N}\\}", "3b68137c98974f598c7bddb342273c38": " \\widehat{m}_h(x)=\\frac{\\sum_{i=1}^n K_h(x-X_i) Y_i}{\\sum_{i=1}^nK_h(x-X_i)} \n", "3b683f8e66ae572ddff8e6632f550404": "\\displaystyle \\ \\mathcal{X}", "3b685aac4197507645024807acc8d407": "N(\\mathbf{p})|0\\rangle = 0", "3b687b0e97a6fe01888142980abd1b28": "\\alpha_i = 0", "3b68b046714fea5e8d9646ff58b1b8b9": "W=Tr[a\\Phi^2+b\\Phi^3]", "3b68b4f85a146dc6488aa71e33ef8af0": "a_0b_1", "3b68d3c13aa96d563c91648bcd6b81c3": "\\mathbf{v}(\\boldsymbol{x},t)", "3b68e50c59aa5f2dd88829a5809df8c6": "d(x, y) = \\lim_n d\\left(x_n, y_n\\right)", "3b69024674eeb7ce815245e669ae26f0": " I = V/R \\,\\!", "3b69126a477842d54f5358584761fd3a": "16-x^2", "3b69861ee4c00ba53f0214944f960fc6": "\\begin{pmatrix}\n0 &1 & 2 & 3 \\\\\n0 &3 & 2 & 1\\end{pmatrix} ", "3b6986c8d2f9dd45e67a04dd8c6d9cf4": "R_{\\text{3}} / R_{\\text{2}}", "3b69d39c74d7d55e7187ae384b119003": "I_a=JS=\\frac{4 \\epsilon_0}{9}\\sqrt{2 e / m_e} \\frac{SV_a^{3/2}}{d^2}", "3b69d97cba57aebc1de7bea72f71c11b": "\n \\frac{\\partial \\rho}{\\partial t}~\\eta + \\rho~\\frac{\\partial \\eta}{\\partial t} \\ge\n -\\boldsymbol{\\nabla} (\\rho_\\eta)\\cdot\\mathbf{v} - \\rho~\\eta~(\\boldsymbol{\\nabla} \\cdot \\mathbf{v}) -\n \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) + \n \\cfrac{\\rho~s}{T} \n ", "3b69fa3cef032d756df32ee5d7e98bb4": "t \\rightarrow \\infty", "3b6a1dafe03dcff604b96c04847103f7": " {K'} = \\frac {\\dot{m}}{C_o} \\qquad(4)", "3b6a29bbf9e7183f3c550fffdfbad8c7": "m^3/s^2", "3b6a4ab25f4561cb4dea3f39f11c8f66": "N(t) = e^C e^{-\\lambda t} = N_0 e^{-\\lambda t} \\,", "3b6ad8ccb16b0ec3c38715f14eed240b": "\\mathbf{S}(\\mathbf{V})", "3b6b49628044e68828254413b29cd6e0": "|X| = 2^{\\mbox{card}(X)}", "3b6b65fb0e6e0d214770e5c370b75b45": "\\frac{v^2}{2g}", "3b6bee88f489226452832aed233c6086": "\\textstyle d_e(n_e) = \\frac{c_e}{n_e}", "3b6c063ff56152b6eca8fe4d25241f22": "\n\\begin{align}\n\\Gamma(s)\\eta(s) & = \\int_0^\\infty \\frac{x^{s-1}}{e^x+1} \\, dx\n= \\int_0^\\infty \\int_0^x \\frac{x^{s-2}}{e^x+1} \\, dy \\, dx \\\\[8pt]\n& =\\int_0^\\infty\\int_0^\\infty \\frac{(t+r)^{s-2}}{e^{t+r}+1}{dr} \\, dt\n=\\int_0^1\\int_0^1 \\frac{(-\\log(x y))^{s-2}}{1 + x y} \\, dx \\, dy.\n\\end{align}\n", "3b6c64b7893f9e40507b6cdf143dfb6d": "\\textrm{Process\\ Productivity} = \\frac {\\textrm{Size}} { \\left[ \\frac {\\textrm{Effort}}{B} \\right]^{1/3} \\cdot \\textrm{Time}^{4/3} } ", "3b6c7b22cd0170a3af8fe7eec695609e": " \\mathbb{A}^{*} = \\{x \\in \\mathbb{A} \\text{ where } x_{\\infty}\\in \\mathbb{R}^* \\text{ and } |x_p|=1 \\text{ for all but finitely many primes.}\\}", "3b6c9e73aff098069988b4306fbfff45": "\n\\delta \\int_{t_{1}}^{t_{2}} \n\\left[ \\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t) \\right] dt = \n\\delta \\int_{t_{1}+\\tau}^{t_{2}+\\tau} \n\\left[ \\mathbf{p} \\cdot \\dot{\\mathbf{q}} - H(\\mathbf{q}, \\mathbf{p}, t+\\tau) \\right] dt = 0\n", "3b6cba45dc4be85b4486509c53055d6f": "\nH= {p^2 \\over 2m} + {m\\omega^2 q^2\\over 2}.\n", "3b6d158685a7ef49d56ebe4150eb871a": "\\lambda_{21}=5.13562", "3b6d189a86849d81b61853c7f77b2d24": "K^*", "3b6d77cf697b7ec22da9da854bad3631": "C_n,", "3b6da8e93643f69fcd300ddc9d68166e": "F(t) = Pr[S_{i} \\leq t]", "3b6df586a309f24673bb4de8865c7664": " U(s) = \\frac{1}{s^2 + 2s + 5} ", "3b6e19f29ef2a925ce3bd20da3049d5e": "x^2+y^2+z^2-1=0.\\,", "3b6e2d94ae7f8dfeb494d2a0c0b628e0": "P(\\mathcal{D}|\\theta)", "3b6e7150f5c9064761394756e64f6d17": " \\sum _{ v \\neq v0} (d(v) - 1) q_v", "3b6e9d28bce32f66e0034fca52ebcd53": "\\lim_{h \\to 0^-}\\frac{f(a+h) - f(a)}{h} = {+\\infty}\\quad\\text{and}\\quad \\lim_{h\\to 0^+}\\frac{f(a+h) - f(a)}{h} = {-\\infty}\\text{,}", "3b6eed3422f8d47f0572b6f33714ad64": "0\\leq z", "3b6f100dd9fad5740728edeb5e411d80": "\n p(x,y)= \\begin{cases}\n 1/2d & \\text{if } |x-y|=1 \\text{ and } \\eta(x)\\neq\\eta(y) \\\\[8pt]\n 0 & \\text{otherwise}\n \\end{cases}\n", "3b6f42bdc4f39e553c506797a501528c": "\\lbrace u e^{ar}:\\ 0 \\le a < \\pi \\rbrace,", "3b6f4576eb2a5a2500d490f8a1f55f58": "Ma_0", "3b6f4d207acf256e7c3e777d4163a505": "U_\\text{eff}", "3b6fecac8d3adb62712c38b7555344e6": " \n\\int_{L} ( \\psi_{+} \\nabla^2 \\psi_{-} - \\psi_{-} \\nabla^2 \\psi_{+} ) ~dV = \\int_{L} ( \\phi_{B}^{} \\nabla^2 \\phi_{A}^{} - \\phi_{A}^{} \\nabla^2 \\phi_{B}^{} ) ~dV \n", "3b6ffa5511447c52673781f801f36537": " P_1 ", "3b7013baa6aacf4bf8441ad23ffcca61": "\nE = {p^2\\over 2m} + {m\\omega^2 x^2\\over 2}\n\\,", "3b7054c486be686d15d610be7f43683c": "(m_0 c)^2 = \\left(\\frac{E}{c}\\right)^2 - p^2\\,,", "3b7095fe5dd23ec44b0d6f085383caeb": "\\lambda_{111}", "3b70afeb717329fd1495e2bd8472d998": "$1M\\cdot 0.5\\cdot \\max(0.03-0.025, 0) = $2500", "3b70c1472aa24cf7c2a93220ba618b4e": "m(n,k,l)", "3b71137b0196a3cef54e8b7fe41a2883": "10^{-10}", "3b7119f1ab1004ea4b5781876eb9942a": " k[\\Delta] = k\\oplus\\bigoplus_{1\\leq i\\leq n} x_i k[x_i]. ", "3b712033f07383f3a4c6adbc1df598ef": "0.98(100 - x)", "3b713d91cc2018fd248da115bff95f7d": "\\alpha_3\\,", "3b7154023ea9a3c86afe4df83460cbde": "\\sigma(T) = \\sigma_p (T) \\cup \\sigma_c (T) \\cup \\sigma_r (T).", "3b7189da1df9416c3986db6082a25f4b": "s \\mathbf{[Z]}= s^2 \\mathbf{[L]} + s \\mathbf{[R]} + \\mathbf{[D]} ", "3b71d4341102f81f06a3478168ee5298": "\\epsilon(v_{ij}) = \\delta_{ij}", "3b7283568ad1fda225bf180f5ff10d87": "\\frac{59!\\times 20!}{2^{11}\\times 6!^{10}} \\approx 4.40\\times 10^{66}", "3b728c7fd357786034661fd198c1d95d": "f(z)=\\sin(\\pi z)", "3b72ae8f8d247bda8c2b8ce76af04959": "\\begin{align}\ndy &= \\frac{dy}{dt}dt \\\\\n&= \\frac{\\partial y}{\\partial x_1} dx_1 + \\cdots + \\frac{\\partial y}{\\partial x_n} dx_n\\\\\n&= \\frac{\\partial y}{\\partial x_1} \\frac{dx_1}{dt}\\,dt + \\cdots + \\frac{\\partial y}{\\partial x_n} \\frac{dx_n}{dt}\\,dt.\n\\end{align}", "3b72b7299628387f771208bf97974cff": "\\beta \\rightarrow 1", "3b736893760b15c3bc7cca718b27698f": "f = (0.79 \\ln Re - 1.64)^{-2}", "3b73d8077efbe78c5970cc3a8df51cbd": "\\frac{L^m\\alpha^m}{m!}<1", "3b73f46e6400eea844177de465282c67": "BC(p,q) = \\int \\sqrt{p(x) q(x)}\\, dx", "3b74e408ca63f40b9cd1087d21697496": "c_{t}=0", "3b74f6308570e55aa4fdc76271b91dd0": "A^H", "3b75715a7ed91bb635dfc7a1050dd497": "y^2=1,", "3b7574916a0854385ae977628eef2c82": "\nT^{2k} = \n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1/2 & 1/2 \\\\\n0 & 1/2 & 1/2 \\\\\n\\end{pmatrix}\n", "3b75b5064c8fa8db8c350d93a8f420ef": "\\Delta t \\to 0", "3b76051f1657f727354db7c73ab987f1": "c \\|u-u_n\\|", "3b761db4deab08cc512451ff4f9d7c81": "\\textstyle r = a^{\\frac{n+5-p-q}{8}} \\mod n", "3b761f186590a09be27d1d84c01201ca": " \\mathsf{S}\\cdot \\mathsf{T}=|\\mathsf{S}||\\mathsf{T}|\\cos\\hat{z}; ", "3b765de55420248ce8a41bf448a2cc2c": "u_{j,iij} = 0.\\,\\!", "3b766ac17d221a7947f874f94c3acea9": "\nc = \\sqrt{\\gamma \\cdot {p \\over \\rho}}\\,\n", "3b76b29c63afe0d67a2cf67bfe6e1ff3": " e^+e^- \\to e^+e^- \\gamma ", "3b76c5decfe529c65e3399aa57fedf2d": "J(t, t')", "3b76d022351cb2d3d7a2ace6c42dd1e6": "\\lambda x.t", "3b76e8d7c12426f0b487b7ea2f250858": " R=\\{(t,y) \\, | \\, |t - t_0 | \\leq a, |y - y_0| \\leq b\\} ", "3b7712dcd76308e342fd3d653f5f5059": "g = -1\\,", "3b774929da5449fa39ac207c039a8610": "h(x) = f(x) + g(x).", "3b775319a266a0c2215b2282b6b9c7da": "|f(n)-g(n)|\\leq\\varepsilon|g(n)|\\qquad\\text{for all }n\\geq N~", "3b776c66df53e33aa870c578575b480a": "p(b) = (1+c)/2,", "3b776e9670e2c1cf2a2c91cb3d41f413": "G\\left(\\epsilon\\right)=G_0\\,\\, e^{-\\left(\\frac{2\\pi\\epsilon}{\\lambda}\\right)^2}", "3b7796d61d2b348e517a64b8671e9ff8": "\\left\\lfloor\\frac{n}{p}\\right\\rfloor + \\left\\lfloor\\frac{n}{p^2}\\right\\rfloor + \\left\\lfloor\\frac{n}{p^3}\\right\\rfloor + \\dots = \\frac{n-\\sum_{k}a_k}{p-1}\n", "3b77c65725f2ea6d76956b3ef3a8d6e0": "\\Delta_1", "3b77cc67becb23b9c58e8102bdc2d4c0": " \\and (S_5 \\implies (\\operatorname{equate}[A_5, q\\ q\\ x] \\and V[F_5] = A_5)) \\and D[F_5] = K_5 ", "3b77d9c7278a5a892c8f28973296ec12": "r < 2^b-M", "3b7800b92e82da8393bea767558407d6": "aN", "3b7842b5827aa101a43fddbd716380c6": "\\sin\\frac{\\pi}{12}=\\sin 15^\\circ=\\tfrac{1}{4}\\sqrt2(\\sqrt3-1)\\,", "3b7879635295b49f3d79a6552974bd71": "\\sqrt{\\sigma_L^D}", "3b78bf233e752b5e02477517842e8270": "\nf_1(\\text{A}) = f_1(\\text{AA}) + \\frac{1}{2} f_1(\\text{Aa}) \n = p^2 + p q = p \\left(p + q\\right) \n = p \n = f_0(\\text{A})", "3b79368bb0b066ce7043e61e8e8c88f2": "t_0 < t_1 < \\cdots < t_n", "3b79505e186d487d7fbbc439be32235e": "\\mathbf{\\nabla \\times E}' = -(\\mathbf{v} \\cdot \\nabla) \\mathbf{B} = -\\nabla\\times(\\mathbf{B} \\times \\mathbf{v}) - \\mathbf{v}(\\nabla\\cdot \\mathbf{B}) = -\\nabla\\times(\\mathbf{B} \\times \\mathbf{v})", "3b796fb9822e4d42a70961f76c1ce66e": "X \\to X/\\sim", "3b79784860802eaf425fe99f7c21560e": " H^{k+1,p}(\\mathbb{R}^n) \\hookrightarrow H^{s',p}(\\mathbb{R}^n) \\hookrightarrow H^{s,p}(\\mathbb{R}^n) \\hookrightarrow H^{k, p}(\\mathbb{R}^n), \\quad k \\leq s \\leq s' \\leq k+1 ", "3b7993d77e4c5fc6813b3723b4eb982e": "\\left(\\hat{\\mathbf{n}}\\times\\mathbf{A}\\right)dS=\\iiint _{V}\\left(\\nabla\\times\\mathbf{A}\\right)dV", "3b79fe19720ea372ebb60a3f79e30773": "1.3\\pi", "3b7a2a08f462bb3de34a3bcab5e0d23a": "P \\langle E_{1,j}^C \\rangle \\times P \\langle E_{2,j}^C \\rangle \\times \\cdots \\times P \\langle E_{m,j}^C \\rangle = \\left(1 - 2 t / G\\right)^m.", "3b7a72acf48cc300034719bed5878c6a": "B=\\frac{m}{P_x}", "3b7a9a32c935e0ed5dd330173183443c": " X\\leftarrow X/Y,\\; Y", "3b7a9c9b5c2e968a73d851184dd7f059": "A_+ = \\{ a^2 \\colon a \\in A \\}", "3b7aaa9299bb33539a27312a7cd507b2": "\\phi(a, b, n)", "3b7aed70437d68aad910eebb5aa5e334": "W_{[a,a+r]} = \\int_a^{a+r} w(t)\\, dt", "3b7af12a1a565b66956664338c99b25d": "\n \\mathbf{x}^{(n)} = \\mathbf{x}^{(n-1)} P \n", "3b7b2387ac3764529052e2e6d75425f5": "w_i = \\sum_{j=2}^{n}\\frac{a_j}{(i+1)^j}, ", "3b7b450c119c48451c189f5b4043d6dd": "G_0 = \\sqrt{\\rho} \\Phi^{-1} (1-\\alpha)", "3b7b4d68f75d2a5c429c99b5295df6c6": "C_*(\\widetilde{X})", "3b7b5dc71dbec58e9fa7913a2f72a720": "\\sin\\frac{\\pi}{6}=\\sin 30^\\circ=\\tfrac{1}{2}\\,", "3b7b5f7f32fa5629bb6da3ba618e30b9": "r,s \\in R", "3b7b6eb8d1b51e8022e0cb39a878902d": "\\scriptstyle{\\{\\emptyset,i\\}\\in R}", "3b7bedcc50588111ea822cf08b7ba870": "P_s(x) = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}\\left(x+\\sqrt{x^2-1}\\cos\\theta\\right)^s d\\theta = \\frac{1}{\\pi}\\int_0^1\\left(x+\\sqrt{x^2-1}(2t-1)\\right)^s\\frac{dt}{\\sqrt{t(1-t)}},\\qquad s\\in\\mathbb{C}", "3b7c3c5fde123e089b1d0ebb58743cc8": "X \\to p \\to 1", "3b7c9d04fca6e853a6b3636d63a976ee": "{\\Delta}TD", "3b7cc0a404505603850ae33f36a4819e": " S = \\int \\partial^\\mu \\phi \\partial_\\mu\\phi + {\\lambda \\over 4!} \\phi^4. ", "3b7ce518c51d6dd2a8a73ce2f25ab3e2": "U_n= \\frac{a^n-b^n}{a-b} = \\frac{a^n-b^n}{ \\sqrt{D}}", "3b7ceeb3140ac37261b6bfc8a93b6092": " \\frac{d}{d-1} (n - s_d (n)) - (d-1 - (n \\mod d)) ( e_d ( \\lfloor \\frac{n}{d} \\rfloor) + 1) ", "3b7d13b85f71353476bd4a5451978b57": " = \\left( 3 : 0 : 0 : {1 \\over \\lambda} \\right) + (0:0:1:1) ", "3b7d47672318591b04e13d1026fdd915": "(\\mathbf{u} \\oplus \\mathbf{v}) \\oplus \\mathbf{w} = \\mathbf{u} \\oplus (\\mathbf{v}\\oplus \\mathrm{gyr}[\\mathbf{v},\\mathbf{u}]\\mathbf{w})", "3b7d604c85a2dfe781d8a56e9aa5f07c": "l\\;=V_g t", "3b7da1ac0c3eaf23b265a1d6ae339de9": "\nf_\\epsilon(\\epsilon)\\,d\\epsilon= \\sqrt{\\frac{1}{\\epsilon \\pi kT}}~\\exp\\left[\\frac{-\\epsilon}{kT}\\right]\\,d\\epsilon\n", "3b7db768127a454939488a9eaa99b98c": "\\gamma K", "3b7dd64879b4e2c12b1ae1051db5f822": " \\frac{WL}{Y} ", "3b7e09f05d1b7d9b7cb2b30c3508feb6": "P\\cdot P \\subseteq P", "3b7f80dc7ee3bbdca42b3864c73fc099": " \\lim_{\\varepsilon \\to 0} (1/\\varepsilon)(F(X+\\varepsilon \\varphi) - F(X) ) = \\int_0^1 F'(X,dt) \\varphi(t)\\ \\mathrm{a.e.}\\ X", "3b7fb094bd7bf2c3cc31e6eac5d3611c": "E_i=N\\pi_i", "3b7fb6d970c0e2b3376bb9f16e96b3bb": "\\mathbf{B} \\cdot {\\rm d}\\mathbf{A}", "3b8003a46e40569c6b40fb29e2f5bb04": "\\text{area} = \\int_a^b{ \\cosh{(x)} } \\ dx = \\int_a^b\\sqrt{1 + \\left(\\frac{d}{dx} \\cosh{(x)}\\right)^2} \\ dx = \\text{arc length}", "3b8007df665bcf2c5a320a5bda88fca1": "\\cong G \\cap H .", "3b80165d944afa2f2ec380d3eb02656a": "F(m) = 2^{2^m}+1", "3b8057a400aeadeb0a09f3108b189009": "n\\ln n", "3b807a7caacff59a37a4e53351104590": " \\begin{align}\nY_{k+1} &= \\tfrac12 (Y_k + Z_k^{-1}), \\\\ \nZ_{k+1} &= \\tfrac12 (Z_k + Y_k^{-1}). \n\\end{align} ", "3b8093e6be2ae36b95a4a8d1b66311c1": "\\begin{align}\n\\hat\\sigma^2 &= \\tfrac{1}{n}(y-X(X'X)^{-1}X'y)'(y-X(X'X)^{-1}X'y) \\\\\n&= \\tfrac{1}{n}(My)'My \\\\\n&=\\tfrac{1}{n}(X\\beta+\\varepsilon)'M(X\\beta+\\varepsilon) \\\\\n&= \\tfrac{1}{n}\\varepsilon'M\\varepsilon,\n\\end{align}", "3b80a0849b4528597eeac052743d1b7f": "\\textstyle \\mathfrak{V}", "3b8122f66388c61c326cc9cb063e53eb": "\\mathrm{Pic}\\ \\mathbf P^n_\\mathbf k = \\mathbb Z", "3b8177a60ea70e8baf0913253cfae4d4": "P = k \\left(h-h_{f}\\right)^m.", "3b81cca4a7052b275a1080205c10bdf2": "a_0 = \\frac{1}{\\alpha}\\left(\\frac{\\lambda_e}{2\\pi}\\right)\\simeq 137\\times\\bar{\\lambda}_e\\simeq 5.29\\times 10^4~\\textrm{fm} ", "3b81e768f760a77fcd2647e83ca8d142": "\\textstyle{s \\choose k}", "3b825608922ca4bd7af4337fbff82497": "h \\equiv 2 (\\bmod 4), h > 2", "3b82cbd3e72ee242cc83afa8ddb254ad": "t(A)", "3b82df0e0dc28b08298fa764686158bf": " Fx(K_1,K_2,\\ldots,K_r ) = \\sum_{n_1=0}^{N_1-1} \\sum_{n_2=0}^{N_2-1} \\cdots \\sum_{n_r=0}^{N_r-1} fx(n_1,n_2,\\ldots,n_r) \\cos { \\frac{ \\pi (2n_1+1) K_1}{2N_1}} \\cos { \\frac{ \\pi (2n_r+1) K_r}{2N_r}}", "3b82faff9b1f00965313eb02d4711375": "\\begin{align}\n\\mathrm{length}(ab) &= \\sqrt{\\left(dx+\\frac{\\partial u_x}{\\partial x}dx \\right)^2 + \\left( \\frac{\\partial u_y}{\\partial x}dx \\right)^2} \\\\\n&= dx~\\sqrt{1+2\\frac{\\partial u_x}{\\partial x}+\\left(\\frac{\\partial u_x}{\\partial x}\\right)^2 + \\left(\\frac{\\partial u_y}{\\partial x}\\right)^2} \\\\\n\\end{align}\\,\\!", "3b8308387ca2866f203d12f098758d96": "\n\\delta x(t+\\delta t) \\approx \\delta x(t) + \\delta t (\\delta x \\cdot \\nabla) \\vec v\n", "3b83512cd3a58ca4266c220691bfd4de": "p^{\\star}_{\\rm B} = 0", "3b83900155de0faf6ed439e7f1115729": "R_2 = \\frac{R_aR_c}{R_T} ", "3b83d2501edc2c1004b5502717827404": "\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1.", "3b842185cbbfe202a835921efe0b41af": "\\scriptstyle \\Pi", "3b8425331e76f78aa58a1f7f30dc609f": "r_i(t)=s(\\vec x_i,t)", "3b84533d90f4cd78b908a31dfef954d8": "T_B^2 = [T_A^2]", "3b846eaa1f4db99e4f1db135b495332e": "\\mu + \\delta \\beta / \\gamma", "3b8476b72589bcbec6aa567e6e39ee13": "[\\;]_{\\text{seq}}: A\\mapsto [A]_{\\text{seq}}", "3b8496cf0fe920a53de41c00fd991090": "R_S=\\frac{v_{Bullet}^2}{g}\\, \\left(2\\sin(\\theta-\\alpha)\\cos(\\theta)-\\alpha)\\right)\\frac{\\cos(\\theta)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\sec(\\alpha)\\,", "3b849b5939fe41b2de2484a00df60d0e": " a,b = 0,1, \\dots, d-1 ", "3b84ae4a99a2dff30fce53e053e1356d": "Y_{9}^{-3}(\\theta,\\varphi)={1\\over 256}\\sqrt{21945\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(221\\cos^{6}\\theta-195\\cos^{4}\\theta+39\\cos^{2}\\theta-1)", "3b84b4936f7278d5b500686e34b1de02": " \\frac {dy}{dz} = y(1-y) ", "3b850588ec2c35a066bd342207d9a169": "h(w) = \n\\left\\{\\begin{matrix}\n0 &\\mbox{if}\\ a=1\\ \\mbox{in}\\ S_w\\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ a\\neq 1\\ \\mbox{in}\\ S_w.\n\\end{matrix}\\right.", "3b85a149146b36ea27855e7e05656385": "T ", "3b85a72f3e6cab8843a90e1702567a51": " \\omega\\approx\\omega_0", "3b85b00b81471f9218acfdc0f4bef056": "G_1 \\; \\stackrel{T_1}{\\rightarrow} \\; P_1 \\; \\stackrel{T_2}{\\rightarrow} \\; P_2 \\; \\stackrel{T_3}{\\rightarrow} \\; G_2 \\;\n\\stackrel{T_4}{\\rightarrow} \\; G_1' \\; \\rightarrow \\cdots", "3b85b80557b6c88b2006b208ac3790f8": "~g(x)~ - ~T(f_{\\rm a})(x)", "3b860b2884047cec466e85b591debe29": "|\\gamma_n| < 0.0001 e^{n \\log \\log n}", "3b866518493ff48a227e4df2ee97e55d": "\\{ f_k \\}_{k=1}^\\infty", "3b86ffeb325eea7506a6dac4120f530d": "m^2=\\frac{hP}{kA_c}", "3b870a1c521ad4d8fd4c9862e87584fe": "H_{1I}", "3b8726444872a7d3f64aac8e44f06903": "H_\\xi : M \\to \\mathbf{R}.", "3b875a6c3134d21c531b0741062552dc": "\\scriptstyle\\rho_n \\,>\\, 0", "3b8772842421c00e09e877f8ac3308f5": "Fred(\\mathcal H).", "3b87cde88677ea0b58103efc321dbb4f": "f_Y(y)", "3b8816f6fe63ff5d010dd05d34428e4e": "p(\\textbf{z}_k|\\textbf{x}_k,\\textbf{x}_{k-1},\\dots,\\textbf{x}_{0}) = p(\\textbf{z}_k|\\textbf{x}_{k} )", "3b888c10a58d89b637613050c8703cfe": "A_R = \\frac{d_\\min}{d_\\max}", "3b88f565084c879e616d79c4a7c3309e": "(\\tfrac{1}{2}, \\tfrac{1}{2})\\in\\mathbb R^2", "3b891168af04adc8d8595dc87f132395": "\\pi-\\theta _0\\,", "3b89554d06bfebe1265aa32bc367a792": "p=\\frac{F}{A}", "3b896625960db2a6b484ba48a9c2945f": " \\text{For all } p\\in|\\Delta|\\text{ and for all }\ni<\\operatorname{dim}\\, |\\Delta| = d-1, \\quad \n\\tilde{H}_i(\\operatorname |\\Delta|; k) = \nH_i(\\operatorname |\\Delta|, \\operatorname |\\Delta| - p; k) = 0. ", "3b897ca0c9c4e194274195e399c67fff": "\n\\sum_j 2m(E_i - E_j) |X_{ij}|^2 = 1\n\\,", "3b899f4a6c109540c7da9eb0de0f5f31": "q(i,j)=\\begin{cases} \\infty & j < 1 \\text{ or }j > n \\\\ c(i, j) & i = 1 \\\\ \\min(q(i-1, j-1), q(i-1, j), q(i-1, j+1)) + c(i,j) & \\text{otherwise.}\\end{cases}", "3b89dfb1854b4dbbba3940ae04cc49b0": "\\mathcal{L}_{yuk}=y \\, \\eta L \\epsilon H^*+...", "3b89ef207177106f43138774c0a5eb0e": " \\delta W_{H,i} = \\rho g\\int \\eta(x)dx\\delta\\epsilon ", "3b8a1fa934a89b0bbb297d9ec0c3b6ac": " \\frac{\\partial^2 \\psi}{\\partial x^2}-u(x,t)\\psi=\\lambda\\psi.", "3b8a9cb05773d251ae3891482c5599da": "R\\,\\!", "3b8b09044660b8aff4962c66489e11f3": "\nQ_{k\\ell} = \n\\begin{bmatrix}\n 1 & & & & & & \\\\\n & \\ddots & & & & 0 & \\\\\n & & c & \\cdots & s & & \\\\\n & & \\vdots & \\ddots & \\vdots & & \\\\\n & & -s & \\cdots & c & & \\\\\n & 0 & & & & \\ddots & \\\\\n & & & & & & 1\n\\end{bmatrix} .\n", "3b8b201bc486dffd7289c290ef68e501": "\\sum_n p_n = \\sum_n v_n i_n = 0 \\,", "3b8b5676c74c7b9962415165143c4e51": "\\| x \\|_{\\infty} \\sum_{k=-\\infty}^{\\infty}{\\left|h[k]\\right|} = \\| x \\|_{\\infty} \\| h \\|_1", "3b8b7a01f99bbdd50dc6d7c3b0d2bbcd": "A = \\int_r^R 2\\pi\\rho\\, d\\rho = \\pi(R^2-r^2).", "3b8b83af6d6eb5328406c616c1d65f54": "\\displaystyle{K}", "3b8ba58286312c3ada4cbc0a490ab96a": "\\textstyle 2.\\ span\\{\\bold A\\}=spane\\{e1,....,e_{d}\\}=span\\{\\bold E_{s}\\}.", "3b8ba5c49bcc6488c50265e005ebf1ab": "\\scriptstyle\\hat\\nu_n", "3b8ba9d3b3daac3266449e77c638a646": " \\Lambda^{-1}) ", "3b8bc61e3c70f3f1c26b832e3491670a": "\\dot{\\underline{x}}=\\begin{bmatrix}0 & 1 \\\\ -2 & -3\\end{bmatrix}\\underline{x}+\\begin{bmatrix} 0 \\\\ 1\\end{bmatrix}\\underline{u}", "3b8c04de47510db9e7299c088a78f4f6": " d = | | Pi - Pj | | ", "3b8c2042ec8500b080ae066c90adf296": "U_1 Q_1 = Q_1 U_3", "3b8c2a4cd474c4bfd6770dce49f6baa4": "S\\equiv\\{e_1,e_2,\\ldots \\}", "3b8c4ed52d9216bfb9041a748425363f": "\\rho_{xx'} ", "3b8ca2b0a5e3ce221ef8043c9d1408a7": "f_b = 3-1 = 2", "3b8cb126a5b4a7d554a98f1393f236b0": "\n D_{i+k}(\\theta) = D_{i+k-1}(\\theta) = \\cdots = D_{i+1}(\\theta) = 0. \\,\n ", "3b8ceee07cd3efa484cddddda5a2319a": "(\\pm\\sqrt{2},-1).", "3b8d16f38cd78d8a4acf00645903928c": "m_1 = m_2", "3b8d29cbbf6be40a0ce368c8b1623c21": "{}^{i - 1}T_i(\\theta_i)", "3b8d2fc1ffe540654e9f4b4b3b6d95d6": "\n\\int_E f\\,d\\mu\n=\\lim_{k\\to\\infty}\\int_E g_k\\,d\\mu\n\\le\\lim_{k\\to\\infty} \\inf_{n\\ge k}\\int_E f_n\\,d\\mu\n=\\liminf_{n\\to\\infty} \\int_E f_n\\,d\\mu\\,.\n", "3b8dbcf440ff8797082704b98e1db4be": "x = x' A x''", "3b8dc5f076d832543d6cb8fcc6f48e21": "P(b)=\\frac{\\left[\\frac{\\left(S-1\\right)!}{N!\\cdot\\left(S-1-N\\right)!}\\right]\\cdot M^N}{\\sum_{X=1}^N\\left[\\frac{\\left(S-1\\right)!}{X!\\cdot\\left(S-1-X\\right)!}\\right]\\cdot M^X}", "3b8dfae028442069c28715e7f5caac3c": " 0 \\leq X_j \\leq 1, j=1, \\dots, n", "3b8e6e382183bbe88f43eb275db5fb08": "U = 4\\sigma T^3 \\frac{1}{N(2/\\epsilon - 1) + 1}.", "3b8ef9d8090aa998a2fb85379c024de4": "A=\\frac{K-P_0}{P_0}", "3b8efb437a2821d1cabea5a4df62dc4b": " \\!\\ E_{n} ", "3b8f255ba3acf18e13dc9c95c60dc67f": "k(\\lambda,\\phi)", "3b8f43c9b356e82294a22bd99c0ec925": " k = q / \\hbar", "3b8f4611209c660c4c3d79617b8aed29": "s(\\ell_1, \\ell_2) = \\frac{Q(B, C)}{Q(A, B)} = \\frac{Q}{R}.", "3b8fc62f5a649da76b0f553e1eb9a0a5": "\\lfloor n/4 \\rfloor", "3b9020bd69ef73365a492474e4ee263f": "P_H = H . r", "3b9026ba75aa62e155a912860faee7ef": "a|0\\rangle = 0 = \\langle0|a^{\\dagger}", "3b90b1a4a7ec05fb122c9f3023c68547": "\n\\Psi \\bar{\\Psi} = -1\n", "3b90bf07fd6aba45285ea189b1cde40c": " H^{MF} = -J \\sum_{\\langle i,j \\rangle} \\left( m^2 + 2m(s_i-m) \\right) - h \\sum_i s_i", "3b9102adc39964186dfe68615b940290": "\\textstyle KSD_{a,b}(X)=\\underset{\\alpha }{max}\\left | cdf(\\alpha |a) -cdf(\\alpha |b)\\right |", "3b916fca8910144357079031cef8bb73": " 2 \\pi r", "3b918429b56fbb229be5d871b0642d25": "L_{m} = \\frac{L}{4}", "3b921426f2c31b2e1406672f4abe985a": "(n+1)\\,L_{n+1}^{(\\alpha)}(x) = (2n+1+\\alpha-x)\\,L_n^{(\\alpha)}(x) - (n+\\alpha)\\,L_{n-1}^{(\\alpha)}(x).\\,", "3b92298f57ac62da7e59d564a8e60412": "\\begin{align}\n 3 x + 2 y + z &= 10\\\\\n 2 x + 5 y + 3 z &= 15\\\\\n x,\\, y,\\, z &\\ge 0\n\\end{align}", "3b927c3b3d05ebecc9ba24dcc498b026": "\\Delta_1 = 2c^3 - 9bcd + 27b^2 e + 27ad^2 - 72ace", "3b929dbd81812f11cfe4da335677d5a9": "\\sqrt{-1}^2=\\sqrt{-1}\\sqrt{-1}=-1", "3b92af8b72ed41896498df1c794bd684": "\\eta=2", "3b92b722a67ff859fa6ad8d27d16dd96": "\nT = k\\left(k-1\\right)\\frac{\\sum\\limits_{j=1}^k \\left(X_{\\bullet j} - \\frac{N}{k}\\right)^2}{\\sum\\limits_{i=1}^b X_{i\\bullet}\\left(k-X_{i\\bullet}\\right)}\n", "3b92c8580c86c21be4e8b03f48275edf": "\\mu=0, \\sigma=1,", "3b92dc8a097194993f807d5e2d987b0a": "\n \\sum_j f_{ij} = N.\n", "3b92ddfbc0bc045363035fa9c2be5842": "[L'_{ij},L'_{kl}]=i [\\delta_{ik}L'_{jl}-\\delta_{il}L'_{jk}-\\delta_{jk}L'_{il}+\\delta_{jl}L'_{ik}] \\,\\!", "3b9320d2df401a118b47038cd0372227": "v(S\\cup\\{i\\}) = v(S\\cup\\{j\\})", "3b9322481c414e414116f6cda9856e49": "2 \\pi k - \\pi", "3b9325b8f640efdeee7a66fbf96948d2": " A_\\lambda = -\\log_{10} \\left( \\frac{I_1}{I_0} \\right)", "3b939526ba8a7ee15cb5d60dd0611197": "f(z) = \\int_D K(z,\\zeta)f(\\zeta)\\,d\\mu(\\zeta).", "3b942154fa7f9d6b30cc3c611389f59a": "\\mathbf{F}=q\\left(\\mathbf{E}+\\mathbf{v}\\times\\mathbf{B}\\right)", "3b94275d178eaa622b70af1a8f0c350c": "\\mathbf{F} = q\\left[-\\nabla (\\phi-\\mathbf{v}\\cdot\\mathbf{A})- \\frac{d\\mathbf{A}}{\\mathrm{d}t}\\right]", "3b94cd377f769240127c2d61dd7b53bc": "\\int_0^{F(t)} {F\\over F+\\psi\\,\\Delta\\theta}\\, dF = \\int_0^t K\\,dt", "3b94f729e24e16872b464aa31bd19993": "{\\mathcal E}_i(x,y)\\equiv\\exp\\Big(-{\\displaystyle\\int} b_i(x,y)\\big|_{y=\\psi_i(x,\\bar{y})}dx\\Big)\\Big|_{\\bar{y}=\\varphi_i(x,y)}", "3b9560596b94ac338cd46ad1f2b3facb": "\\Pi_{i=1}^l p_i^{m_i}", "3b95c34f55f299676b3c2bd4a7f7f852": "=A'(x)u_1(x)+B'(x)u_2(x)+A(x)u_1'(x)+B(x)u_2'(x)\\,", "3b966849b4e915dae0b93a2921fbf8df": " L^2(\\mathbb{R}^n) ", "3b966dee64d7865e0e3b07332cd2f439": "\n\\mathrm{Ri} = \\frac{g \\beta (T_\\text{hot} - T_\\text{ref})L}{V^2}\n", "3b96839bd8d3e3aa5a0c86b9acd10173": " \\boldsymbol{r} =\\sum_{k=1}^{d} q_k \\ \\boldsymbol{e_k} \\, ", "3b96c278c6f5f611b4aafd6e4469849d": "h(y_2,\\dots,y_n|y_1)", "3b9745ae1d425ec08b908eeaa022cf82": "\\scriptstyle V_{(ab)}= V_{(a)}-V_{(b)}; \\;V_{(bc)}= V_{(b)}-V_{(c)}; \\; V_{(ca)}= V_{(c)}-V_{(a)}", "3b974c875ac646f1215f6971d1ee24e7": "\\scriptstyle 1-1/p^2", "3b9753dd9d21f56033761fc33e9b0529": "\\operatorname{var}(x_t)={\\sigma ^2 \\over 2\\theta}. \\, ", "3b975e7148eb902c47962f171f3bf21d": "\\Box_\\kappa", "3b97ecc1bd031c94cf5de620748fdcc8": "\\displaystyle{g=\\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix},}", "3b986b1268656bc26108b60883a9e45d": "\\mathbf{B} = \\sum_{i=1}^{n} a_i \\mathbf{w}_i {\\mathbf{v}_i}^T", "3b9878b56c94891bbdbf7798daea632b": "\\mathfrak{S}_r", "3b98c8238b44fd2d9feb8822d63902c4": " = {i \\over \\hbar} e^{iHt / \\hbar} \\left( H A - A H \\right) e^{-iHt / \\hbar} + e^{iHt / \\hbar} \\left(\\frac{\\partial A}{\\partial t}\\right) e^{-iHt / \\hbar} ", "3b98e2dffc6cb06a89dcb0d5c60a0206": "AA", "3b99383a89de1059fa81c4313994d279": "\\frac{\\partial \\mathbf{u^*}}{\\partial t} = -\\nabla p^* + \\frac{1}{Re} \\nabla^2 \\mathbf{u^*}.", "3b997a4f3d0ac68a12c2e2ed5803bc4d": "S[\\sigma] \\to T[\\sigma]V[]", "3b9992bc32db750956fb4c7cf190d38d": "a^2+b^2 = x p_1p_2\\cdots p_n", "3b9999cc764937d5a883eb5ba176c50e": "\\scriptstyle{\\vec{v}\\text{ }=\\text{ d}\\vec{x}/\\text{d}t}", "3b999cc7c9ddcd2eb9edf36130037154": " -\\frac{\\hbar^2}{2m}\\frac{ \\partial^2\\psi_1}{ \\partial x^2} + V\\psi_1 =E\\psi_1", "3b99ca44f9d9205e3c6691d6ce45bba9": "-\\frac{1}{12} - \\frac{\\beta}{2}.", "3b9a4c0c40788d7fcc9033d376bf52a0": "k_1,\\dots,k_n", "3b9a9ef7735686cce9ac5035bd2f3645": "= \\left( \\frac{\\partial}{\\partial x}, \\frac{\\partial}{\\partial y}, \\frac{\\partial}{\\partial z} \\right) \\cdot \\mathbf{A}", "3b9ab92ee1d5ea64961813bff2b683fd": "|e_i| < q", "3b9ae9803c23a557f85da54c56889b6f": "\n P(B | A) = {P(B \\cap A) \\over P(A)}\n ", "3b9afed00218d18ef3c6257d9d024a08": "\\varphi:\\mathcal{H} \\to \\mathbb{R}", "3b9b3f8cac8e378acdb451431097d083": " \\sqrt[n]{x^n+y} \\approx x + \\frac{y}{n x^{n-1}}. ", "3b9bb7dca8a57ed15f8abb10a4e8beac": "{\\mathcal L}_{xy}^7: L=Lclm\\big(k,\\mathbb{L}_{x^m}(L)\\big);", "3b9c1a8f506ea7f5138628abdb3d9719": "L=\\dfrac{d}{dx}\\left[p(x) \\dfrac{d}{dx}\\right]+q(x)", "3b9c2c3879f417963d860bd7c7435d42": "\\nabla\\times(A\\times B)=A(\\nabla\\cdot B)-(A\\cdot\\nabla)B+(B\\cdot\\nabla)A-B(\\nabla\\cdot A)", "3b9c2eea19a8d15d9c5487a22ae8f13c": "\\Delta\\;v_w =\\, ", "3b9c3daad2e1d41387f3ffba79b1e5cb": "(J_i - J_j)", "3b9c420aa657983befce2ef4cfc00707": "T_\\varepsilon", "3b9c480da3570d7c0c90a7d277069b60": "H_n(x)=2^{n/2}{\\mathit{He}}_n(\\sqrt{2} \\,x), \\qquad {\\mathit{He}}_n(x)=2^{-n/2}H_n\\left(\\frac x{\\sqrt{2}} \\right).", "3b9c562ee667618195d07817921a748b": "\\begin{align}\n\\tau_0' &= \\tau_0 + n\\tau \\\\\n\\mu_0' &= \\frac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0} \\\\\n\\bar{x} &= \\frac{1}{n}\\sum_{i=1}^n x_i\n\\end{align}", "3b9c73ee16bf8bc8ab14d11d7e2ffd5e": "H(\\mathbf{q},\\mathbf{p},t) = \\mathbf{p}\\cdot\\mathbf{\\dot{q}} - L(\\mathbf{q},\\mathbf{\\dot{q}},t)", "3b9c93699ea924189a8e6242a9efb8a5": " Du \\ll 1 ", "3b9cb3ebe1b2d3638b0cd94770ec43d7": "v_i = \\dot u_i", "3b9cc8218b0b72133e0fcab4bfd90a0a": "f_*([X])=\\deg(f)[Y] \\, .", "3b9d05a6865f6f9aa3e1ed43601459bc": "CH_{4}", "3b9d1690b98e3bb71d530c6ada25a1dd": "x^3-t(x+1)", "3b9d5addeffb3b45cdd6d13159d98614": "1\\le n \\le 250", "3b9d64e72ebb52c99342ce198daafcda": " \\frac{y+11(y \\bmod 2)}{2}", "3b9d7606e08518eb2bd06528df690a82": "p=\\frac 1 2 ,", "3b9d8d8727abd593ef1895dadacc5faa": "f(N)=\\Phi N\\,", "3b9d9481d99dbd6f81615ef190867e68": "\n\\Gamma = \\Lambda^T.\n", "3b9e712b9089da8dbe27b53707749a75": "\\scriptstyle N\\,\\times\\,\\ell", "3b9e9c2bb42da2b8802f4b580ab0f979": "\\displaystyle{\\Omega(f)=\\limsup_{r\\rightarrow 1} |T_rf -f|.}", "3b9eb6d8953795c9685a0de869e7ac44": "h_{\\mathbf{a},b}", "3b9eee705efbd4601fde19ddea0082cd": "\n2\\Delta P = v {E\\over c^2}.\n\\,", "3b9f3210da032fb05ecb31f1df4a8116": "j\\phi = j\\left(\\beta - \\alpha\\right)", "3b9f7490b950e291d516820e6f58b05c": "E \\propto {1\\over r} \\propto {1 \\over L^2}", "3b9f7b2eb85955cdacc9a25b30575e38": "\\sqrt{4}=2", "3b9fef728ce9cf7932eed8b794181158": "z_3", "3b9ffe918a564bdb41081bf634dd52b4": "G \\times \\{e'\\}", "3ba03ca613e47da6c5a2e8924da2b581": "i=1,\\ldots,n\\,\\!", "3ba080e3535966558d7d6f93a982ee9e": " \\frac{1}{\\xi^2} \\frac{d}{d\\xi} \\left({\\xi^2 \\frac{d\\theta}{d\\xi}}\\right) + 1 = 0 ", "3ba0ab7ad5d0ad5ca4bcc0c4a35ca4cf": " {\\partial \\mathcal{L}\\over\\partial \\mu_l} = 0 \\qquad l=1,\\dots,m ", "3ba119ad7905b1d07a3df1a84404af35": "\\widehat M", "3ba128a3d4fc84b6ce5954ba9121ada7": " \\beta_i\\beta_j=\\sum_{l=1}^{n}m_{ijl}\\beta_{l},\\ \\ m_{ijl}\\in\\mathbb{F}_q", "3ba128c8b6fdbd1388f8f58c176e1e85": "v+dv", "3ba12d26683d1aca75888ddd313c2cf9": "ad_g(f) = [f,g]", "3ba1349949638171bac2eef43a740c24": "\\frac{P \\and P}{\\therefore P}", "3ba1927266170b0c5b5469f3aa7d7063": "\\scriptstyle\\frac{c}{a}", "3ba1adfafb57333676f87899fe2e85c1": " \\left ( \\frac{-\\Delta\\,G}{RT} \\right ) ", "3ba1d6f4e18cf4a7ad5021855f014573": "\\pi_l(c) := \\begin{cases}\n c, & \\text{if } c \\text{ is an input} \\\\\n \\pi_l(p), & \\text{if } c = p \\oplus_x n \\text{ and } (l = x \\text{ or } x \\notin \\pi_l(p)) \\\\ \n \\pi_l(n), & \\text{if } c = p \\oplus_x n \\text{ and } (l = \\neg x \\mbox{ or } \\neg x \\notin \\pi_l(n)) \\\\\n \\pi_l(p) \\oplus_x \\pi_l(p), & \\text{if } x \\in \\pi_l(p) \\text{ and } \\neg x \\in \\pi_l(n)\n\\end{cases}\n", "3ba1e2b17528b87c07c810b0bcdf5356": "\\begin{matrix} {r \\choose 4}{4 \\choose 1}^4{52 - 4r \\choose 1} \\end{matrix}", "3ba1f80cbac3bf0e5270d7fb1d7c48f1": "\\scriptstyle \\left(a \\,-\\, b\\right) \\,+\\, b \\;=\\; a", "3ba20e60c69e2a817b3158c22304fb96": "\\left( T_{ab} - \\frac{1}{2} \\, T \\, g_{ab} \\right) \\, X^a \\, X^b \\ge 0", "3ba21e041ae88578069b734f4e78fc3e": " E_{+} ", "3ba23238dea5b81ef174e57d448a4616": "-0.62", "3ba24fbd7c73f14bf8c5eae5b6dfaad4": " Q_{C} \\ ", "3ba293e3e5bf44130929e055d8b6c4ab": "\\ \\}", "3ba2ffe43a4d13abecb4b53e6b410081": "{{\\partial u_a \\over \\partial x}+{\\partial v_a \\over \\partial y}+{\\partial \\omega \\over \\partial p}=0}", "3ba339579ea00ba854b92d1bfed52a53": "O(\\epsilon^{-45})", "3ba35f7f8ae063c092822a39cae55657": "\n\\mathrm{var}(T) = \\frac{\\mathrm{var}(X-\\mu)^2}{n}=\\frac{1}{n}\n\\left[\nE\\left\\{(X-\\mu)^4\\right\\}-\\left(E\\left\\{(X-\\mu)^2\\right\\}\\right)^2\n\\right]\n", "3ba3758b2f8a9164b0df90687bb34471": "\\bold{\\hat{p}}=-i\\hbar\\nabla", "3ba38d20e1dcc9a64691f0d819e31447": "\n\\begin{cases}\n\\{O_{1},O_{2}\\} \\\\ \n\\{O_{3},O_{5},O_{7},O_{9},O_{10}\\} \\\\ \n\\{O_{4},O_{6},O_{8}\\} \\end{cases}\n", "3ba3bdbd794899b572168797df4161a2": "{1/c^2}", "3ba3ec0635aa91b09f712e3666b8ec6e": " C = U_n^{*} \\operatorname{diag}(F_n c) U_n = \\frac{1}{n} F_n^{*} \\operatorname{diag}(F_n c) F_n, ", "3ba3fedebad4a18e0d04d17316618bef": " \\left[ {\\begin{array}{c}\n 0 \\\\\n \\omega_x \\\\\n \\omega_y \\\\\n \\omega_z\n\\end{array}} \\right] = 2 \\frac{d\\mathbf{q}}{dt} \\otimes \\tilde{\\mathbf{q}}\n", "3ba410acc25e3b307f511ab2c7b4b2c9": "f(x) \\propto x^{-1} \\quad \\text{ for } 0 E(r_i) + \\sum_{j\\ne i} \\max_{r_j} E(r_i,r_j) ", "3ba820e1afc9d97e41e312c382374794": " G_x(t,f) = \\int_{-\\infty}^\\infty e^{-\\pi(\\tau-t)^2}e^{-j2\\pi f\\tau}x(\\tau) \\, d\\tau ", "3ba8a427510d5d272e7be989de046809": "\\mathbb{CP}^4", "3ba8cdca19e0d724afb8e76e44e5e027": "g_0=1\\, ; \\, g_n = ng_{n-1}^2, \\qquad n > 1, \\, ", "3ba920cdb6a55a634562ea51bb0bc5db": "| v(x) | \\leq C \\left( 1 + \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau_{D}} \\big| g(X_{s}) \\big| \\, \\mathrm{d} s \\right] \\right),", "3ba94bd93ccbc33a57a0dfbe2553d851": "J_1 = 0 ", "3ba997a14e27b775fc33e04f49bfb3ad": " \\frac{d}{dx} \\frac{\\part L}{\\part f'} = 0 \\, . ", "3ba9b325e1b083d73c87577df50994b3": " \\Delta e_n = -\\frac{n^2 \\pi^2}{L^2} e_n. ", "3ba9be0382414353467a01cae25e67e5": "\\scriptstyle \\operatorname{E}\\left[X|\\mathcal {H} \\right]", "3ba9cc1ca6c181387a263d3895923dc9": "b \\times n = \\underbrace{b + \\cdots + b}_n", "3baa0dd71660bdd54eaa5278b5873a94": " \\begin{align}\n\\mathbf U(\\mathbf x,t) &= \\mathbf x - \\mathbf X(\\mathbf x,t) \\\\\n\\nabla_{\\mathbf x}\\mathbf U &= \\mathbf I - \\nabla_{\\mathbf x}\\mathbf X \\\\\n\\nabla_{\\mathbf x}\\mathbf U &= \\mathbf I -\\mathbf F^{-1}\\\\\n\\end{align}\n", "3baa4bf34dddfda9f146ebcb2bf89f55": "Z_{K}\\,", "3baaa288baeadcdddca4604fa169d147": "\\theta=90^\\circ-\\phi.\\,\\!", "3baab2cb056cb8e05176775f7def50d5": "\nR_{ij}=R^\\ell{}_{i\\ell j}=g^{\\ell m}R_{i\\ell jm}=g^{\\ell m}R_{\\ell imj}\n=\\frac{\\partial\\Gamma^\\ell{}_{ij}}{\\partial x^\\ell} - \\frac{\\partial\\Gamma^\\ell{}_{i\\ell}}{\\partial x^j} + \\Gamma^\\ell{}_{ij} \\Gamma^m{}_{\\ell m} - \\Gamma^m{}_{i\\ell}\\Gamma^\\ell_{jm}.\\ \n", "3baafcdfc546db7205eec0e0d9fa3fc3": "\\det(\\Lambda-\\mu I)\\ \\ne\\ 0", "3bab2578b2b7fdd0490186f2fb5d0a3e": "P \\in\\hat C := \\mathbf{Set}^{C^{op}}", "3bab56b53f8eebd8e846f6bacd15c834": "\\frac{1}{|X|^2} \\sum_{U,V \\in X}|tr(U*V)|^{2t} \\geq \\int_{U(d)}|tr(U*V)|^{2t}dU", "3bab9da406135a83ee48f88ae5f069aa": " \\langle\\Psi_1 ,\\Psi_2 * \\Psi_3,\\Psi_4 \\rangle = \\langle\\Psi_1 ,\\Psi_2 ,\\Psi_3* \\Psi_4 \\rangle ", "3babf344ce3bbe896196f0cf8b97ccc7": "m = k +d-2\\log \\left(\\frac{1}{\\varepsilon}\\right) - O(1)", "3bac11d68663a1911f0476f7627d1215": "PWV = P_{i} / \\left( v_{i} \\cdot \\rho \\right)= Z_{c} / \\rho ", "3bac1a3b3b92249bc13edef035f6d47a": " -6 = 7 \\times (-6/7)", "3bac58ae4bc0d4d46cea28049daa9e23": "-\\frac{d\\Psi(x,t)}{dt}=(H-E_0)\\Psi(x,t)", "3bacb4bb1c281eaab114fffb5cd9eed1": "\\frac12n\\log n+\\frac12n\\log\\log n+cn", "3bacda03423194c2f7d9a3a3eed631d2": "k = R/\\mathfrak{m}", "3bacdb1a94347a9fce9da98dc22f642b": "c \\in \\mathbb{N} = \\lbrace 0, 1, \\ldots \\rbrace", "3bacdcf22e43912fe5a6f5d9be96bad5": "\\sum_{n=1}^\\infty \\beta^{n!}", "3bace3add5ea19951673b15d991e16b1": "\\overline{a}_n + \\overline{b}_n = \\overline{(a + b)}_n", "3bacfdb3eacef93568bff23a732a1211": "L_{ind} = C_{cap}", "3bad08aaeb34165f51c6f13c38d3bdfb": "\\gamma={\\dfrac{(nd+1) \\cdot (t-1)+1}{nd \\cdot (n-1)/2}}", "3bad262f14ab52388fe96148c2708e10": " \\neq", "3bad4282f49e182bdf7634660fc85949": "\n \\mathbf{v} = v^k~\\mathbf{b}_k\n ", "3bad4c845584a2ea68cf25a30a4bd1d2": "W_{p} (\\mu, \\nu):=\\left( \\inf_{\\gamma \\in \\Gamma (\\mu, \\nu)} \\int_{M \\times M} d(x, y)^{p} \\, \\mathrm{d} \\gamma (x, y) \\right)^{1/p},", "3badc93629073c87e98bc1186c94d85a": "\\displaystyle f''(x_{0}) \\approx \\displaystyle \\frac{2f(x_{0}) - 5f(x_{+1}) + 4f(x_{+2}) - f(x_{+3}) }{h_{x}^2} + O\\left(h_{x}^2 \\right), ", "3bae019ea80a9ddeea154fac7e0817ba": "\n1 - p_1 - p_2 - p_3 + p_1 p_2 p_3 = 0\n", "3bae062e3aa8359338e2260af5a32aeb": " \\psi(0^{-})=\\psi(0^{+}) \\qquad \\psi'(0^{-})=\\psi'(0^{+}). \\,\\! ", "3bae5d8b3d94a66271b838c4e18af2c1": "X = |\\alpha|", "3bae83abce92a2a00c5227f8b2aeedad": "C_n \\sim \\frac{4^n}{n^{3/2}}", "3bae961d020e0c2f5b0b27fb76f28022": " \\partial_\\gamma \\mathcal{J}^{\\alpha\\beta\\gamma} = 0 ", "3baedf64a66b59aec7270508a09acfe8": "E\\{X_i\\} = np_i", "3baef122d638aa833a641b7c62955c23": "\\bar{y}(t, \\tau) = \\frac{1}{\\tau}\\int\\limits_0^\\tau y(t+t_v) \\, dt_v", "3baf1600ae50930a155f58ae172b51bd": "f(x,y)", "3baf52d5c8951376aff1c7a08d9e23ca": "V_\\text{out} = A_\\text{d}(V_\\text{in}^+ - V_\\text{in}^-)", "3baf55e80447224cb755473b144b1150": "f : (\\mathbf{R}, \\mathcal{L}) \\to (\\mathbf{R}, \\mathcal{B})", "3baf6efb19fddc260aeb696801a07caa": "\\mathbf X=X_I\\mathbf I_I\\,\\!", "3baf70c932893bc11ff8c3fff394cdd1": "M_{32}", "3bafadf030f9dc7d46828d9b8aea3252": "T = K P_{pre} d ", "3bafb0767f6f3a6ea909f0edee9ceff5": " s \\, \\equiv \\, (H(m)-x r)k^{-1} \\pmod{p-1}", "3bafb0a7f8a60581bec259ebc5e3f88f": "\\begin{matrix}{5 \\choose 2}\\end{matrix}", "3baffd623d24688b6229e8808f4dd24a": "\\frac{dy}{dx}", "3bb03308259064b5df9dc4254fa893e8": "t^a", "3bb05a9c98be70883ea385af5541eae7": " x_k = \\cos\\left(\\frac{k}{n+1}\\pi\\right),\\quad k=1,\\ldots,n.", "3bb0643db034bc1dc71a2e2828800e5d": "T_{M^*}(x,y) = T_M(y,x),", "3bb09658812a7d333e9402a555a5c6ca": "k={d \\choose 0} + {d \\choose 1} + \\dots + {d \\choose r}", "3bb0cffcd3aba4edeafcac78046e4cb4": "X_{\\mathcal{D}h}(t,\\omega)", "3bb0d8bdd12b249c3f04666bb9f0f86f": "S = \\int \\mathrm{d}^4x \\sqrt{-\\tilde{g}}\\frac{1}{2\\kappa}\\left[ \\tilde{R} - \\frac{1}{2}\\left(\\tilde{\\nabla}\\tilde{\\Phi}\\right)^2 - \\tilde{V}(\\tilde{\\Phi}) \\right]", "3bb176cec5d9c55f6e83fe9a44f83e8f": "s_b \\,", "3bb1de2477a5d9af23c38f287bf83b4e": "\\langle r \\rangle \\sim 1/n^{1/3},", "3bb1f804e767049bc705a464fd91de0e": "S_y\\frac{\\partial h}{\\partial t} = \\nabla \\cdot (k h \\nabla h) + N. ", "3bb21855a5cbe15790851e7579403bd0": " v_{i,j} = \\begin{cases}\nn^{-\\frac{1}{2}} & \\mbox{if } j = 1.\\\\\nn^{-\\frac{1}{2}} (-1)^i & \\mbox{if } j = n \\mbox{ and n is even.}\\\\\n\\sqrt{\\frac{2}{n}} \\sin (\\frac{\\pi (i-0.5) j}{n} ) & \\mbox{ otherwise if j is even.}\\\\\n\\sqrt{\\frac{2}{n}} \\cos(\\frac{\\pi (i-0.5) (j - 1)}{n}) & \\mbox{ otherwise if j is odd.}\n\\end{cases} ", "3bb22951633431739579ae6ec32782cb": " R\\sqrt{\\frac{a^3-a^2b-ab^2+b^3-a^2c+3abc-b^2c-ac^2+c^3}{abc}},", "3bb29df95c694b26621e73e3d5cdc8c1": "H^q(Y, \\textstyle\\bigwedge^p\\Omega_Y)", "3bb2b95088f040bb200566e75939e9b4": "\\mathrm{C + O_2 \\ \\Rightarrow \\ CO_2}", "3bb32345047727e0b33846ea628135bf": " \\nu_G = \\sum_{j} \\sigma_j \\nu_j ", "3bb397c3343df9f0d0ae4db0651a9e37": "\\nu=c\\left |\n\\mathbf{k} \\right | / \\left ( 2 \\pi \\right )", "3bb3dce4f655cc57d8871dfa27da874b": "P_b(N, A, S) = \\frac{A^N\n{\\left( \\begin{array}{c} S \\\\ N \\end{array} \\right)}}\n{\\sum_{i=0}^NA^i\n{\\left( \\begin{array}{c} S \\\\ i \\end{array} \\right)}} ", "3bb3ed3020874068d7a4562426aee295": "\\ \\int_{p(z_1)}^{p(z_2)} \\frac{\\mathrm{d}p}{p} = \\int_{z_1}^{z_2}\\frac{-g}{R \\cdot T} \\, \\mathrm{d}z.", "3bb438d8fd1d739d05fab3143d5b8615": " \\frac{d^{k}}{dx^{k}}(u) = u^{(k)} = 0 ", "3bb4cac7cea680fd8e548d765636a16f": "P_{R50} = 0.668 G_B G_M \\big[\\frac{h_B h_M}{d^2}\\big] ^2 [{\\frac{40} {f}}]^2 P_T", "3bb501af3afe24816e67aeaa34e33e19": "\\displaystyle i^n \\frac{d^n \\hat{f}(\\omega)}{d\\omega^n}", "3bb521ac4e3d5e68faf90b142e3fed11": "R_1={(1.523-1) \\over -3.0\\ \\mathrm{dpt}} =-0.174\\ \\mathrm{m}", "3bb53165149e2914bed687462b062366": "n_{0\\bullet}", "3bb536f042eca132fc729f64ff824ce1": " \\operatorname{G}_{t+s} = \\operatorname{G}_{t}\\operatorname{G}_{s}. ", "3bb60b98b3f3d5336a586843ca547dba": "\\mathrm{1\\, \\frac{cd}{m^2} = 10^{-4}\\, sb}", "3bb648101e237e353802a2e06a1bb207": "G[\\mathbf{f}A]^{-1} = A^{-1}G[\\mathbf{f}]^{-1}(A^{-1})^\\mathrm{T}.", "3bb64c9cc7f95698205b159cb3275a9c": " -ic ", "3bb65fb60d21e75bb04aaba65615412b": "e_2 = \\frac{1}{r} e_1 \\,", "3bb6c0d0d4585bed5206e86cec41a686": "R_G", "3bb72d292ba41db99a678d2cc8963bd8": "\\Theta(V^3)", "3bb736015bb18f2fb5aac35c9940036c": "\\mathfrak{so}_{2n}(\\mathbf{C}),", "3bb7dcbe6d9191df0d9bdbe0391b35da": "\\partial_t - D\\Delta", "3bb7f7fd3cccebcd4685b2221de307dd": "K(x,y) =\n\\begin{cases}\n\\sqrt{\\theta} \\, \\dfrac{k_+(|x|,|y|)}{|x|-|y|} & \\text{if } xy >0,\\\\[12pt]\n\\sqrt{\\theta} \\, \\dfrac{k_-(|x|,|y|)}{x-y} & \\text{if } xy <0,\n\\end{cases} ", "3bb821f1fdbcbaced3034ee35ac30a80": "-(a_m - a_{m+1})", "3bb83c835479e7583816d5699bc872a0": "\\scriptstyle {a_1,...,a_m} \\in A", "3bb8912f4b0d1065e9a3715e23bbfdf7": "(\\pm 7)^2 \\equiv 49 \\equiv 10\\pmod{13}", "3bb8a7d1081f029da201198366fb16ba": "F_i(G) = -\\log(N_i(G)/T(G))", "3bb8bfeda9ff65547cee02158f2b2819": " G^{\\alpha\\beta}{}_{;\\beta} \\, = 0 ", "3bb901384778ecdf9188c30c79b5c27f": "0 \\le (f*F_n)(x)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(y) F_n(x-y)\\,dy,", "3bb947675e48dc74ced7ee783ac7ffd2": "\\mathfrak{g}_2^{\\mathbb C}, \\mathfrak{sl}(3,\\mathbb C)\\times\\mathfrak{so}(2,\\mathbb C),\n\\mathfrak{sl}(2,\\mathbb C)\\times\\mathfrak{so}(3,\\mathbb C)", "3bb9afc9ec5a6c948c7ad9a95a6cd150": "v = 1 + k + \\left (\\frac{k(k-\\lambda-1)}{\\mu} \\right ),", "3bba0fb3c8642ba99805aebef354e741": " (D - i k) y = 0 ", "3bba4abfbfdabd406bf3f786a4b66701": "O_{/\\sim_{\\mathbb{F}_2}}", "3bbb10b11af37cf0e02ad3633ae6e2b4": "\\mathsf{Pad}(M_{2}).", "3bbb10d68f90c314260e919f7735acd0": "\n\\frac{\\Delta A}{A} = \\alpha_A\\Delta T\n", "3bbbc938e2c58a713096e43e1a736b7e": "\n \\sum_{k=0}^{\\infty} p(7k+5)x^k =\n 7~ \\frac{ (x^7)^3_{\\infty} } {(x)^4_{\\infty}}\n +49 ~ \\frac{ (x^7)^7_{\\infty} } {(x)^8_{\\infty}}\n", "3bbbfd52bfc1d50d34a328e04cb12337": "m = \\frac{1}{2R}(y_2 + y_1)", "3bbc452eb15226c4dcbdfff47a12b057": "\\scriptstyle 0.7\\pm1.4\\times10^{-12}", "3bbc6dc449c529b5218efcedfdd790ba": "\\scriptstyle{X^-}", "3bbc9b2c74f6b052ad33d0ab552360bf": "\\left\\langle \\rho, Z \\right\\rangle", "3bbd112c4b35f2ec1cb4eae9099d8e04": "\\top_{\\mathrm{D}}(a, b) = \\begin{cases}\n b & \\mbox{if }a=1 \\\\\n a & \\mbox{if }b=1 \\\\\n 0 & \\mbox{otherwise.}\n\\end{cases}", "3bbd24510142d19f02d8d45f00202301": "\\mathbf{P(s)}", "3bbd5299222be9f72fab8a4195a25c65": " \\scriptstyle Y = X^2.", "3bbdbf1880a4a001fdc9e8cb9e04e310": "\\dot{t}^2 = \\frac{r}{r-3M}", "3bbdcd8f55c56e43f87d870248d5e8fe": "ac-(b/2)^2=ac-b^2/4.", "3bbdcef68d8ac0576939fe17b85530b7": "V=aM", "3bbe54d91326d8574818a68c7590d369": "q = w + xi + yj + zk \\!", "3bbe826e9df1d349784e16a34294567c": "d = n \\cdot p = \\frac{p^2}{2 \\delta p}", "3bbe96b5026028b0eff1b8610dd70a9c": "\\frac{1}{\\lambda} = R\\left(\\frac{1}{n_j^2} - \\frac{1}{n_i^2}\\right), \\, n_jj}{\\frac{1}{r_{ij}}}+ \\sum_{A > B}{\\frac{Z_A Z_B}{R_{AB}}}\n\\quad\\mathrm{and}\\quad T_\\mathrm{n}=-\\sum_{A}{\\frac{1}{2M_A}\\nabla_A^2}.\n", "3bcd4ae09310741ce101c08cdd14d1f7": " AP = F + \\varepsilon \\, ", "3bcd553dc3173e89157fe36654abdf52": "j \\in \\left\\{ {1,2,\\dots,k } \\right\\}", "3bcd941bce6e411500d714689a331a40": "\\ [A] = [A]_0 e^{-(k_1+k_2)t}", "3bcda3e605515531698dff063f8d3f86": "2^{-(m_1+m_2+\\cdots+m_i)}", "3bcdbbc4d414fbf380fb057e2e32d1f4": "J\\;\n\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup\\!\\!\\!|}}\\;R\n\\qquad \\text{and} \\qquad\nJ\\;\n\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup}}\\;R", "3bcdcbdd9b04a05536e25a8066af0898": "\\rho^{T_B} = \\frac{1}{4}\\begin{pmatrix}\n1-p & 0 & 0 & -2p\\\\\n0 & p+1 & 0 & 0\\\\\n0 & 0 & p+1 & 0 \\\\\n-2p & 0 & 0 & 1-p\\end{pmatrix}", "3bce292372df901868583280548581ef": " p(\\lambda) = \\mathrm{Gamma}(\\lambda; \\alpha + n, \\beta + n \\overline{x}). ", "3bce363e94b707977ff85b759580f144": "G:=D\\cap \\varepsilon \\mathbb{Z}^2,", "3bce427aa2141c9c5e5b42f737ecf3d2": "b^{r \\dagger}_{\\textbf{q}}", "3bceb65afa29ce6e59e20ee0c4f52e1c": "d(\\mathbf{X}\\mathbf{Y}) =", "3bceb7840920b5e48b417dd733e3612f": "{l}/{n}", "3bcebbd41a0b2c5531f9959c2bf70f3d": " \\mathbb{R}^d ", "3bceccb0bc4aeac8a5deb03eaf75de97": "B > R \\ge G", "3bcf1744e54f0ffe0be34343cbf87661": "\\epsilon_f=\\frac{\\dot{Q}_f}{hA_{c,b}\\theta_b},", "3bcf1c15ba3a7fe4171e7823f25f7561": "\\scriptstyle 1\\leq p\\leq\\infty. ", "3bcf49d7e34668e0b3ebe6c4b07986f0": "wx^2", "3bcfde961f0c17f6b8e7b5ecb97b2efd": "\\scriptstyle\\phi^n=\\phi\\circ\\phi\\circ\\ldots\\circ\\phi", "3bd00957eae13ffb637f9a6997b5b22a": "\\mathrm{B}", "3bd0323fd8a33fe923f02cdebec4d3f7": "\n\\begin{align}\n& c= \\frac{dY}{dK}=\\frac{Y(t+1) - Y(t)}{K(t) + sY(t) - \\delta\\ K(t) - K(t)} \\\\[8pt]\n& c= \\frac{Y(t+1) - Y(t)}{sY(t) - \\delta\\ \\frac{dK}{dY} Y(t)} \\\\[8pt]\n& c(sY(t) - \\delta\\ \\frac{dK}{dY} Y(t))=Y(t+1) - Y(t) \\\\[8pt]\n& cY(t)\\left(s - \\delta\\ \\frac{dK}{dY}\\right) = Y(t+1) - Y(t) \\\\[8pt]\n& cs - c \\delta\\ \\frac{dK}{dY}=\\frac{Y(t+1) - Y(t)}{Y(t)} \\\\[8pt]\n& s \\frac{dY}{dK} - \\delta\\ \\frac{dY}{dK} \\frac{dK}{dY}=\\frac{Y(t+1) - Y(t)}{Y(t)} \\\\[8pt]\n& s c - \\delta\\ = \\frac{ \\Delta Y}{Y}\n\\end{align}\n", "3bd0390f74773cfb1ff924afe326e5fe": "P(p;\\alpha,\\beta) = \\frac{p^{\\alpha-1}(1-p)^{\\beta-1}}{\\Beta(\\alpha,\\beta)}.", "3bd0b6cb75086971e7822af0491938e3": "T = \\frac{B}{A-\\log_{10}\\, p} - C", "3bd12457a7a6de4db21670fd351c40d8": "\\mbox{TC}^0 \\subsetneq \\mbox{PP}", "3bd20e653546136c7b111f2fc03eae99": "u \\wedge v = *(u \\times v)\\ \\text{ if } u, v \\in \\mathbb{R}^3", "3bd21a8c0bcee876e4b3aeb537b55147": "\\phi \\mapsto F", "3bd29029fbaf1cd6cded1b81b90789c6": "\n\\begin{array}{rcl}\nx_T &=& \\arg\\max_{x \\in S} (V_{T,x}) \\\\\nx_{t-1} &=& \\mathrm{Ptr}(x_t,t)\n\\end{array}\n", "3bd2a89d6244d9a193562743f22d7f94": "E(B-V) = 0.27", "3bd2d183a058c6feb310a35428be3ceb": " B^2 - A C > 0 \\,", "3bd30d52253c680dd35a1bfb5fc215c0": "\\frac{|\\mu_p+\\mu_\\bar{p}|}{\\mu_p}", "3bd348666142b00216fbdabef76f4f3e": "d \\colon \\mathbb{N} \\to \\mathbb{N}", "3bd36648ca1cebbca95f48ce902375d1": "\\textstyle \\delta + 2l -1", "3bd385c436b67f991dd5a3b919f804fb": "W_s=\\sum^N_{i=1}\\pi_i V_i \\text{ where }\\pi_i = \\text{rank of element }i\\text{ and }V_i\n= \\begin{cases}\n0 & \\text{ for }\\pi_i\\in C \\\\[3pt]\n1 & \\text{ for }\\pi_i\\in T\n\\end{cases}\n", "3bd43b09fa0fe6b5dfc00c06f6a53618": "r\\lesssim\\xi", "3bd4634c9c1933780e0af9591395ad57": " \\sum_{b=0}^{a} \\sum_{\\beta} x_b \\ {_{a}^{b}}\\text{S}^{\\beta} \\rightarrow \\sum_{d=0}^c \\sum_{\\gamma} u_{\\gamma} y_d \\ {_c^d}\\text{P}^{\\gamma}, ", "3bd566627aa479a506985ab6fe865aec": "\n\\mathrm{Ek} = \\sqrt{\\frac{\\nu}{2\\Omega L^2}} = \\sqrt{\\frac{\\mathrm{Ro}}{\\mathrm{Re}}}.\n", "3bd569168ff0ccc6ca86a99339470a79": "\\textstyle z_{11} = z_{22}", "3bd5bf22090fd9dd4b9ab49582c49370": "n \\int_0^1 x(G)[G^{n-1}-(1-G)^{n-1}] \\text{d}G", "3bd5cede9e02cef1f722c784bf7634c0": " J_x = -D_x \\frac{\\Delta C}{\\Delta x} ", "3bd635209d0966c582129952e959c014": "t\\in[0,\\infty)", "3bd7006ed0cc85d6f8afe8f1bde1450d": "\\frac{\\hat{\\mu}^1_{ij}}{\\sum_{j=1}^{N+1} \\hat{\\mu}^1_{ij}}", "3bd73b595d47881820aac884c16f642f": "\\partial_\\mu\\partial_\\nu H =0", "3bd7997ea75a976704b4d4a302184150": "f(x) = \\begin{cases}1, &x = c,\\\\0, &x \\neq c.\\end{cases}", "3bd79b54a06bb236b2eea556112374f7": "\\begin{align}\n \\mathrm{d} \\sigma &= \\mathrm{d}(u) \\wedge \\mathrm{d}x^1 \\wedge \\mathrm{d}x^2 \\\\\n &= \\left(\\sum_{i=1}^n \\frac{\\partial u}{\\partial x^i} \\mathrm{d}x^i\\right) \\wedge \\mathrm{d}x^1 \\wedge \\mathrm{d}x^2 \\\\\n &= \\sum_{i=3}^n \\left(\\frac{\\partial u}{\\partial x^i} \\mathrm{d}x^i \\wedge \\mathrm{d}x^1 \\wedge \\mathrm{d}x^2\\right)\n\\end{align}", "3bd79c1592d6f2980ad8b7309c653346": " \\mu = G(M+m ) \\,", "3bd7fb2f586df6af706fab3735c353b7": "Z_0=\\sqrt{\\frac{R+j\\omega L}{G+j\\omega C}}", "3bd831724c8fc4847337f6d3c53a3ec8": "(A, [e,f])", "3bd863280d09bf9147fadc2087f82cef": "K_{SP}=\\prod_k{{a_k}^{m_k}}", "3bd8676ce52aa89147ccfdd25cfeed87": "\\mathbb{R}\\,.", "3bd896dd0f033ca74977be2ef5978f8c": "hc/\\lambda \\ll kT", "3bd8a259ab2558635ca6a9bb1ff40bf5": " T = 1 \\,", "3bda0afb7b9ddcb71c3342e895bd3c62": "k = k_{\\infty}", "3bda34dd8e94b11c9fea5056723a84cb": "G = \\mathrm{Tor}(G) \\oplus G/\\mathrm{Tor}(G).", "3bda48d38b3d5a29661b43ea67208738": "\\sqrt[1/105]{36}-1=3.4%", "3bda5770881aac4242397e2d8d7c7406": "\\vec E\\!", "3bda5ce4208c330035e3165c8ba37f54": "{ {\\tbinom{10}{0}} {\\tbinom{14}{12}} }/{ {\\tbinom{24}{12}} } \\approx", "3bda7f4303ea1aa374977e814802653d": "\\boldsymbol X=\\{X_1,\\ldots,X_m\\}", "3bda91d711e155f2d1511205752985f8": "\\displaystyle \\Delta=A_0A_2-A_1^2.", "3bdac5598d62fb9882cc8570bdf06bb0": "T_f \\in L(V\\otimes\\cdots\\otimes V\\otimes V^*\\otimes\\cdots\\otimes V^*; W)", "3bdae623152d1db51278a83a9908ce73": " \\theta i ", "3bdae77d40ffd593181ae3a56af22980": "\\pm\\frac{1}{\\sqrt{1 - \\sin^2 \\theta}}\\! ", "3bdb0989dc02a22c0ff2f73b4c16e094": " X_o \\rightarrow S_1 \\rightarrow S_2 \\rightarrow X_1 ", "3bdb10a522303b96d50da448e68b237c": "\n(1-p)^{-r}=\\sum_{k=0}^\\infty{-r \\choose k}(-p)^k\n=\\sum_{k=0}^\\infty{k+r-1\\choose k}p^k,\n", "3bdb2dd24dd49a17f965d8371768bd09": "I(s,\\pi)", "3bdb6b6e5d458bf5ab22517ab34e7d39": "\\mathcal{S}_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int m v \\, ds \\ \\stackrel{\\mathrm{def}}{=}\\ \\int p \\, dq", "3bdb824d3d7e9593b369bdad1153583b": "\\! w=1", "3bdc11e2f0c0b1b5282cbc60e1ceeced": "t^3-2t^2+t", "3bdc6fb38ade4722f75cc657da9d2fc5": "f = (S \\to AA, A^{*}S^{+})", "3bdc946a5194b1af49328ad793845cfa": "\\tfrac{X}{X+Y} \\sim {\\rm Beta}(\\tfrac{\\nu_1}{2}, \\tfrac{\\nu_2}{2})\\,", "3bdcb8847e6aa733e9e4d3c1e1794be9": "\\phi (x) \\,", "3bdd2026a34b5b806faa46ea35662877": "\\qquad\\qquad Y_t = A_t * (N_t, K_t, G_t) ", "3bde0092525f0308985e2349480295f2": "g_n=\\langle\\psi_{0,0},\\,\\phi_{-1,n}\\rangle", "3bde349ae98dc41acb8f9a05d09c735c": "\\beta_{i}(p_{S_{i}}) \\, = \\displaystyle\\sum\\limits_{p_{S_{i}^c} \\in A_{S_{i}^c}} \\beta(p)", "3bde5c71067f2d0732e27d1598d0e3f1": "\\dots", "3bde63214a54e821601294bac9d1ef17": " C = \\partial Q/\\partial T\\,\\!", "3bdf7397fe473fba538342e7bc6642dc": " \\varphi \\equiv c", "3bdfe796be809fd4095d34812973bde4": "GDD = \\frac{T_\\mathrm{max}+T_\\mathrm{min}}{2}-T_\\mathrm{base}.", "3be074294add2e72a254f646268c02db": "\n\\mathrm{d} \\mathcal{H}\n=\\sum_i \\left( \\frac{\\partial \\mathcal{H}}{\\partial q_i} \\mathrm{d} q_i + \n\\frac{\\partial \\mathcal{H}}{\\partial p_i} \\mathrm{d} p_i \\right) + \\frac{\\partial \\mathcal{H}}{\\partial t}\\mathrm{d}t\n.", "3be086fa99c1f7c42065b0b6f4a66f37": "\\cos(\\gamma) = Y_3 / \\sqrt{1 - Z_3^2},", "3be0a20ae79b5abc344d12b44bad01ce": " u_{60}(\\mathbf{r}) = \\bar{u}_{SO}(\\mathbf{r}) = \\left | \\frac{1}{2},-\\frac{1}{2} \\right \\rangle = \\frac{1}{\\sqrt 3} |(X-iY)\\uparrow\\rangle - \\frac{1}{\\sqrt 3} |Z\\downarrow\\rangle ", "3be0a7b133f1ec3e67f4f5438dc8f220": "a={\\operatorname{Cov}(Y,X) \\over \\operatorname{Var}(X)}", "3be0b7127d88a0d20252222d184bacc6": "\\mathrm{Cov}(x-\\mu)=\\Sigma", "3be0e29fdf7bcb1e72cf932516182ddb": "\\ \\mathit{SS} = 1- \\frac{\\mathit{MSE}_\\text{forecast}}{\\mathit{MSE}_\\text{ref}} ", "3be12ec7581d535622c27ebbc775114b": "\\Gamma, \\Lambda", "3be1a6ad0012d22c07380142c10e4419": "\\sigma >0,", "3be1b9e453ae58840e32e1a3c495330b": "1.2 \\times 10^{-8} \\frac{\\text{Sv}}{\\text{Bq}}", "3be200ef8d6b7b3823ff2420ae1f5150": "\\,w = Q_s R_s", "3be27ab6bf88fc02480e312228f3a04f": "\\int e^x\\,dx = e^x + C", "3be2a0fecf11d0df6b9ef19dc6b0e3ce": "=(-1)^{(n-l)/2}\\sqrt{2n+D}{ (D+n+l)/2-1 \\choose (n-l)/2}\\rho^l\n{}_2F_1( -(n-l)/2,(n+l+D)/2;l+D/2;\\rho^2)", "3be2e95f8f41f591176b5d5ebda2f861": "\\zeta(a,b,c) = \\sum_{n > j > i \\geq 1} \\ \\frac{1}{n^{a} j^{b} i^{c}} = \\sum_{n=1}^{\\infty} \\frac{1}{(n+2)^{a}} \\sum_{j=1}^{n} \\frac{1}{(j+1)^b} \\sum_{i=1}^{j} \\frac{1}{(i)^c} = \\sum_{n=1}^{\\infty} \\frac{1}{(n+2)^{a}} \\sum_{j=1}^{n} \\frac{H_{i,c}}{(j+1)^b} ", "3be2f31745f5cf04a9829b7773115d6e": "\\mu_t : x\\mapsto tx", "3be325ea5e9141308dea0dcf61ba405b": "x_2(t) = \\dot{x}_1(t)", "3be33d5a2ad4f327886de5a95deceab8": "\\begin{align}\n \\exp( \\tilde{\\boldsymbol{\\omega}} )\n &{}= \\exp \\left( \\begin{bmatrix} 0 & -z \\theta & y \\theta \\\\ z \\theta & 0&-x \\theta \\\\ -y \\theta & x \\theta & 0 \\end{bmatrix} \\right)= \\boldsymbol{I} + 2cs~\\boldsymbol{\\tilde{u}\\cdot A} + 2s^2 ~(\\boldsymbol{\\tilde{u}\\cdot A} )^2 =\\\\\n &{}= \\begin{bmatrix}\n 2 (x^2 - 1) s^2 + 1 & 2 x y s^2 - 2 z c s & 2 x z s^2 + 2 y c s \\\\\n 2 x y s^2 + 2 z c s & 2 (y^2 - 1) s^2 + 1 & 2 y z s^2 - 2 x c s \\\\\n 2 x z s^2 - 2 y c s & 2 y z s^2 + 2 x c s & 2 (z^2 - 1) s^2 + 1\n \\end{bmatrix} ,\n\\end{align}", "3be342bf1b5acc719c2811c9e7240189": "G(C)=CE", "3be3447555731dfdd6b06a49815b2a90": "\\ell_n", "3be3984c88c66c7ab8508db926e2c50b": "\\,\\!\\phi", "3be41275017a12d0ab47aa3a1ee36ef6": "u={1 \\over r}", "3be4a070107427122db277d78444d372": "\\mathbf{v} = (v_1,\\ldots,v_n).", "3be5369c563e1cde70d056b073e71cc7": "p(\\alpha_i) = \\beta_i", "3be53f6f1810b095a2b82505f45ae617": "c = {{r(1+r)^N} \\over {(1+r)^N-1}} P_0", "3be545246ffe6e6d3e856589da687289": "l=l_1\\pm1", "3be5469f00e5b95352ae200884dd4428": "f:\\R^2\\rightarrow\\R", "3be550a6942f2d4298e05bd28e0e1975": " \\vec p_0=\\frac{d_1(\\vec n_2\\times \\vec n_3) +d_2(\\vec n_3\\times \\vec n_1) + d_3(\\vec n_1\\times \\vec n_2)}{\\vec n_1\\cdot(\\vec n_2\\times \\vec n_3)} \\ .", "3be5b84ab6840e3439568cc832ebfc94": " \\mu(A) < 0 \\, \\Rightarrow \\, \\|A^{-1}\\| \\leq -1/\\mu(A) ", "3be5ba92339cf85701b597bfac197a69": "\\Xi = \\Xi(\\frac {1}{T},\\frac {P}{T},\\{N_i\\})", "3be5f1141e410e5a4a4e922d6dda9975": "\\varprojlim", "3be707dc619bd039323402245085f5df": "\\alpha = \\arccos\\left(\\cos(\\phi)\\cos\\left(\\frac\\lambda 2\\right)\\right)\\,", "3be73f5984f7de05ff249dfa702ac146": "\\exists x \\lnot P(x)", "3be74506e955238b1e187f464e8c253f": "d=\\sqrt{ \\left( {\\frac{x_0 + m y_0-mk}{m^2+1}-x_0 } \\right) ^2 + \\left( {m\\frac{x_0+m y_0-mk}{m^2+1}+k-y_0 }\\right) ^2 }.", "3be78fb8960f0b7f7c5366128104ae9a": " \\tan^2 \\psi \\, \\tan^2 \\theta = \\frac { 2 g y_0 + v^2 } { v^2 } \\frac { v^2 } { 2 g y_0 + v^2 } = 1", "3be790e72cf53564f1671c6dece18dd8": " \\pi - \\arccos{ \\left( \\sqrt{ \\frac{ (5 + 2\\sqrt 5)}{15} } \\right) } ", "3be7a9dd7a942173554b07460865ffa1": "\\omega_r(\\phi,\\lambda) d\\phi d\\lambda", "3be7c6f860e7add653aea1ccf84555f3": "\\frac{dx}{dt} = \\frac{1}{H}\\nu(x),", "3be7d9f83845ca9533c3583541a23807": "v,u\\in\\mathbb{R}", "3be7e97109981f582b4c0d37b48a0af1": "n \\ge 3.", "3be88124f6381b1a1c7de1eefbf674f3": "\n |j_1 m_1\\rangle,\\quad m_1=-j_1,-j_1+1,\\ldots, j_1,\n", "3be88ed050a235cef13944368da47516": "J_\\nu(z)", "3be89c75afb0d52c848ee93ebb6d6154": " f_n \\circ d_i = d_i f_{n+1} ", "3be8b6028180552cec41f2cc4cbff5dd": "\\textstyle X ", "3be8d12e72eb2e473d53f308595ac4be": " \\text{gear inches} = \\text{drive wheel diameter in inches}\\times\\frac\\text{number of teeth in front chainring}\\text{number of teeth in rear sprocket}", "3be93ff90f48894fcf761de2a0fa193b": " GT = D ( 1 - \\frac{ A }{ A + B } )", "3be961ce547e924150e77e15c893a226": "4x - 2y = 2. \\,", "3be9959f000618caaced17645656339b": "n(\\frac{1}{2} - \\gamma)", "3be9967cd3238582313451642bc79b1b": "Re < Re_{crit}", "3be9b1a6725e4e9f57c2351f0be2287a": "Q_{\\ell m}=\\int d^3\\mathbf{x'} r'^\\ell Y_{\\ell m}^*(\\theta', \\phi')\\rho(\\mathbf{x'})", "3be9d9167c71a5d95392c99cf9687ed0": "\\ \\rho_0", "3bea28e79644a4ebaa327b35eedb3b32": "C\\|x\\|_\\alpha\\leq\\|x\\|_\\beta\\leq D\\|x\\|_\\alpha", "3bea7da90949e77fe9e0b1175e95558f": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] W_\\gamma [A] \\hat{O} \\Psi [A]", "3beb3d570ce5de1aef72af9d1321e987": "\\int_a^b f(x) \\, dx = - \\int_b^a f(x) \\, dx. ", "3beb6bca81216e36f1515daa4a7ea544": "t_1=[v_1, v_2]\\,", "3beb7be10f95f1a59e20bc4fab978d98": " \\int_0^r\\frac{dt}{t}\\left(\\frac{1}{\\pi}\\int_{|z|\\leq t}\\frac{|f'|^2}{(1+|f|^2)^2}dm\\right)=T(r,f)+O(1), \\,", "3beb87adf2aae01796cbddcc62f9785b": "\\int g \\frac{\\partial f}{\\partial t}\\,d^3p=\\frac{\\partial }{\\partial t} (n\\langle g \\rangle)", "3bebd23ef092510c82bd03291e21736b": "0 \\leq a < b \\leq 1\\,", "3bebd741f08fe81c9fb015c366bec9ba": "A\\parallel_+ B", "3bebe8d490a2519de0ecdaf4dfdd4a2e": "\\partial_\\mu [f] \\equiv \\frac{\\partial f \\circ \\varphi^{-1} }{\\partial x^\\mu}.", "3bebf3522805c4675da8d6e6ffbce2cf": "\\max_{x \\in [0, 1]} |f_{n+1}(x) - f_n(x)| \\le \\frac 1 2 \\, \\max_{x \\in [0, 1]} |f_{n}(x) - f_{n-1}(x)|, \\quad n \\ge 1.", "3bec3469f48f08a33cda284110df6089": "t\\left\\{p,{q\\atop q}\\right\\}", "3becc5a6d9360858c048d71b60ccc14b": "dF\\left(T,V,N_{i}\\right) = -SdT - pdV + \\sum_{i} \\mu_{i} dN_{i}", "3becc972ac101b09e9d5f720b1f9bc11": "x_{k+1} = \\frac{1}{2}\\left(x_k + \\frac{A}{x_k}\\right)", "3bece6b3118f82279c8f903c0f74add4": "RR^{\\dagger} = abba = ab^2a =a^2b^2 = R^{\\dagger}R", "3bed03dab07e9f2630db8f3a08541049": "\\psi(x)=\\sum_{p\\le x} k \\log p", "3bed4827497e4c9540b3a7fb34cc5b1e": "V_+\\,", "3bed4b9e18e8233e8358c0fd62153a10": "c_{j,k} = {1 \\over 4 \\pi^2} \\int_{-\\pi}^\\pi \\int_{-\\pi}^\\pi f(x,y) e^{-ijx}e^{-iky}\\, dx \\, dy.", "3bed6d6ba303f7b673ae0c1e7021cb49": "T_{b}", "3bed94a4bf7a16d1e9cec3eca3e327c1": " \\bigcap\\mathop{\\mathrm{Models}}(\\bar{q}(T)) = \\bigcap \\{ q(R)\\,| R \\in \\mathop{\\mathrm{Models}}(T) \\}", "3bedd7dbcb1292dbf92e2295119ed5d2": "\\color{Black}\\tfrac{8}{m}", "3bedf66c213ab108534427765e9267d2": "g(\\theta) = (1 + \\cos\\theta)\\;", "3bee39179e34b52b6aabd2abf9894a52": "\\sqrt{a} \\cdot \\sqrt{b} = \\sqrt{a \\cdot b}", "3bee591befadcccbba94c85cc2f67564": " \\mathbb{E} ( \\mathbb{P} (Y=0|X) ) = \\sum_x \\mathbb{P} (Y=0|X=x) \\mathbb{P} (X=x) = \\mathbb{P} (Y=0), ", "3bee6642361479808ef15e09c73a65a6": "1 \\over { 1 - e^{K \\times (1-K) \\over 2^{N+1} } }", "3bee6f4477839f3a51c74f95502f05bf": "f(W)", "3bee9dcafb885b2c5b1e609aebc95502": "Q_N = 4 \\frac{1}{N}\\sum_{i=0}^N H(x_{i},y_{i})", "3beec150d91209e6811fe2bf1a1d7163": "[x,y,z] = x \\cdot y^\\top \\cdot z", "3beeccf251d23054f96ca71766a9f425": "S_N \\,", "3beed98f61cb2ba0e07a14cd52eb4546": "p(x).", "3bef18212ce738ddbc80726456c390af": "\\tilde{\\Omega}", "3bef622ea9d99575b36a9f56d5485b25": "a \\left(\\frac{d-1}{p}\\right)^{\\frac{1}{p}}, \\mathrm{for}\\; d>1", "3bef69b007362d41938da100e1feaf8c": "y^5+y^4-16y^3+5y^2+21y-9 ", "3bef6a8bc55e75361b475c9f410ced16": " D_{B}(p,q)", "3bef8f734bf2bf6910d7af781c6fde1e": "dy = f'(x) dx\\,", "3befe69a634e1994832f85db20eb26c5": "A \\setminus U = \\varnothing\\,\\!", "3bf008493cf5c24f7c5df16e409283fa": "0.3^2+0.4^2+(-0.866)^2=1", "3bf02db8304a736fbe7516c78f3e9de5": " |R\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ {1 \\over \\sqrt{2}} \\begin{pmatrix} 1 \\\\ i \\end{pmatrix} ", "3bf0686d672a0abbd815be931fd6fbbc": "f \\mapsto \\operatorname{T}_\\pi(f)", "3bf0d14f5d98f49eda75cd25297a5e4d": "P(2)=18.39%", "3bf142ee59a614a1ec0e7d4a503994a8": "\\widehat{\\sigma}_m^{2}=\\frac{1}{n}\\sum{(x_i-\\widehat{\\mu}_m)^{2}}.", "3bf15288cef70008596ebf33b3cc6e19": "D_M=d_M+iqA_M", "3bf18d71e2ad374612e45017854a75f4": "U_\\mathrm{E} = \\frac{1}{4\\pi\\varepsilon_0} ( \\frac{Q_1 Q_2}{r_{12}} + \\frac{Q_1 Q_3}{r_{13}} + \\frac{Q_2 Q_3}{r_{23}})", "3bf1c4fca1dd621e6d69bae581a2adfe": "A \\leq_m B \\, \\mathrm{and} \\, B \\leq_m A", "3bf1d69b80e63cad7404100eb1d28683": "E^{n-layers}_{surface} = \\frac{E_{n} - k \\cdot E_{bulk}}{2A}", "3bf210f0c9ca4701dc4e45ea79fb1233": "s_0\\in S^n", "3bf2ecd137c9653aea9ca073531ba336": "{[f(x) g(\\theta)]}^{h(x)j(\\theta)} = {[f(x)]}^{h(x)j(\\theta)} [g(\\theta)]^{h(x)j(\\theta)} = e^{[h(x) \\ln f(x)] j(\\theta) + h(x) [j(\\theta) \\ln g(\\theta)]},", "3bf3000c4c60da703e433801f06d8d91": " S^*(t) = 1- \\frac{p}{1-(1-p)E^{-\\mu T}}E^{-\\mu MOD(t,T)} ", "3bf35f290a96b86db2f7989e972de1c5": "\\begin{bmatrix} \\dfrac{z_{22}}{z_{12}} & \\dfrac{- \\Delta \\mathbf{[z]}}{z_{12}} \\\\ \\dfrac{-1}{z_{12}} & \\dfrac{z_{11}}{z_{12}} \\end{bmatrix}", "3bf41768d0b80b63a57db085debac03c": "{\\color{white}.}\\qquad\n\\displaystyle\\frac{d^2t(s)}{ds^2} = K(s) t(s), ", "3bf43a5ef02d6d034979c8496bcecd07": "W=\\Delta E_k=\\tfrac12mv_2^2-\\tfrac12mv_1^2", "3bf44783d88a5b2394d3d6b232a6f2d1": "\\rho_J = 500 \\Omega. \\ ", "3bf473ebbd9ed37a1918e7fbe77751df": "\\mathfrak c^{\\aleph_0} = {\\aleph_0}^{\\aleph_0} = n^{\\aleph_0} = \\mathfrak c^n = \\aleph_0 \\mathfrak c = n \\mathfrak c = \\mathfrak c,", "3bf4a36efb4ae2b34b98fa71da1445af": " \\frac{E}{y_c} = \\frac{y}{y_c} + \\frac{q^2}{2gy^2y_c}", "3bf4b313355606cbe04ce522d6c20ef8": " x^2 + y^2 = 0 ", "3bf4be6686f5696943fff0cefe20a55d": "\\phi_B = \\frac {q L^3} {6 E I}", "3bf4c9ebb49da1ca5430d8afceb3e658": " w_i ", "3bf4d3942ba12576ccbf133a1d567989": " \\lim_{n\\to\\infty}S_{2m+1}-S_{2m}=\\lim_{n\\to\\infty}a_{2m+1}=0. ", "3bf51757ff48bed797a5f34b296450cd": "\\sum_{i=1}^n y_i \\alpha_i = 0", "3bf59af2da34bf1a631573847a5e4f72": "\\varphi_i\\nVdash p_0", "3bf59d54a6e4203e28481b4c932541fe": "y(t) = 10 \\, x(t)", "3bf5b64d9648667acb8d5e206bd7db34": "\\ 2x^2+3x+1 = 0", "3bf5c46e494a503bd09c2a262e07edb0": "I_c = \\frac{m (a^2+b^2)}{5}\\,\\!", "3bf60182e0902a1ac3c506c8fad5b928": "\\upsilon\\,", "3bf655abc9f5298914260a84831152b4": "\\Vert T_j\\Vert=1", "3bf6744622419e6d623428707b686792": "c_g=\\tfrac{g}{4\\pi}T", "3bf68d3646ba8ed9d695fa0a8abc16e5": " \\mathbf{S} \\in \\mathbb{R}^{C \\times M} ", "3bf6c5057d6af32e89dd6276f4ffd61f": "T:\\Omega\\rightarrow\\Omega'", "3bf6c5445bf8f31d855958c88f5d6b3b": "\\{ | i \\rangle \\}", "3bf722e2852b3798ec1ff1b74146cb81": "\\mathfrak{so}(2,2)\\cong \\mathfrak{sl}(2,\\mathbb R)\\oplus\\mathfrak{sl}(2,\\mathbb R)", "3bf72a6149b3a87e44072b95401d982a": "\\color{Gray}\\text{Gray}", "3bf754e506fd8a569077ae3b8ee211aa": "\n \\frac{\\partial \\boldsymbol{F}}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = D\\boldsymbol{F}(\\boldsymbol{S})[\\boldsymbol{T}] \n = \\left[\\frac{d }{d \\alpha}~\\boldsymbol{F}(\\boldsymbol{S} + \\alpha\\boldsymbol{T})\\right]_{\\alpha = 0}\n", "3bf79935b02e5febf518554cbdec0545": "H_2(S(2k+2,n)) = \\begin{cases} 1 & n = 0, 1\\\\\n\\mathbf{Z}/2 & n = 2\\\\\n(\\mathbf{Z}/2)^2 & n = 3\\\\\n(\\mathbf{Z}/2)^3 & n \\geq 4.\n\\end{cases}", "3bf7b7bf4ccb0cc22bd777453ec13ee0": "0.01K", "3bf8cd0b0d2eda989ed1aae88ac2f3c9": "H_{\\mathcal{S}}", "3bf90cbec810d28c58bdb94293630b15": "\n\\frac{\\mathrm{d}}{\\mathrm{d}t}{\\partial{L}\\over \\partial{\\dot \\theta}} - {\\partial{L}\\over \\partial \\theta} = 0\n", "3bf96112e735a71cf14d759786b57aa9": "|a_n|\\le |b_n|", "3bf979b3da2341d5cac2fda8c89ff507": "e^-e^-", "3bf98357e108d25be3cb04bd267d3a77": "1,\\ldots,E", "3bf9978660c2d8a4508cd6c60db41a7e": "\\mu_\\omega(U)=\\int_U\\omega. \\,\\!", "3bf9e4849205d9ab51a7a2db0ceedaf5": "\\frac{\\$ 100}{(1+I)} \\,+\\, \\frac{\\$ 100}{(1+I)^2} \\,+\\, \\frac{\\$ 100}{(1+I)^3} \\,+\\, \\frac{\\$ 100}{(1+I)^4} \\,+\\, \\cdots.", "3bf9e72d1ea8ff07fa9b060299ff370b": "H^* = (V^*,\\ E^*)", "3bfa4939e544b1883a8f0cd7c7b1ab1c": "\\mathbf{w}_k", "3bfa6d0bfd09d66df860e2b1979410d4": " Z_t ", "3bfb463e74f3e281875191e043099e5a": "(\\mathbf u x )\\times (\\mathbf u y) = x y (\\mathbf u \\times \\mathbf u) = x y \\mathbf 0 = \\mathbf 0", "3bfb599586001767eda7088e423d1371": "\\theta = \\begin{bmatrix}\n \\theta_{1}, \\theta_{2}, \\dots , \\theta_{N} \\end{bmatrix}^{\\mathrm T},", "3bfc5f44892e104a11fc04e4f456bb2d": "g_{\\alpha \\beta , \\gamma} \\eta^{\\beta \\gamma} = k \\, g_{\\mu \\nu , \\alpha} \\eta^{\\mu \\nu} \\,.", "3bfc607a20e1737db455bde3d5b54517": "p_\\text{H} = 0.5", "3bfc6fbfcacfbe10f1322b5e9e8bd109": " T = \\left ( \\frac{1}{e^{SCV/20} \\times S \\times PRF} \\right) ", "3bfc7a78031f1226fb218b450ce5e53a": "1 \\div 2 \\times x = 1 \\times \\tfrac{1}{2} \\times x = \\tfrac{1}{2}x.", "3bfc7c76f9438741796f449eae8ee7f6": "b \\in B", "3bfd2bce930e6f9a87cfdb33b539bff2": "t \\in V \\subset U,", "3bfd5450ea018559bf54a427b3d09539": "\\mathbf \\zeta_{rel}", "3bfd81ef3f619b3b8404581df4803f9e": "L_1 = 0.098 \\lambda\\,", "3bfdc31af02bd879ca1d2404942e6a87": "c \\times b = a\\,", "3bfea20a0ef475f1cb45bd40ba25fdc1": "\\scriptstyle r\\,", "3bfee909adcc3ffd0a56777c815f1698": "s\\tbinom{n+s-2}{s-1}", "3bff48b5cdbcb0f8f5c252641dc1ead9": "T = \\frac{(1 + 2 + 3)^2}{3} + \\frac{(4 + 6)^2}{2} = 12 + 50 = 62", "3bff8891ce36f6b40f7330fdb288efc1": "2\\kappa", "3bff9cb399a294e06a8268b02ac1e929": "[0,L_e]", "3bffb5f8187f1858600a0f63e92b5561": "(\\Box_x + m^2) \\Delta_{\\pm}(x-y) = 0.", "3bffc237cfdbf53cdb83c0d4ef325d1b": " \\langle p(t) \\rangle = p_0", "3bffd1512881072f10210366561cee69": "\\mathit{Conv}1(R) = \\{(b,a)\\mid (a,b)\\in R\\}", "3bfff00112689a49aa8e6368c9ee58e9": "rb > c \\ ", "3c00043a586ab7ee59de6208dcad8880": "g_{UV}:U\\cap V\\to \\operatorname{GL}(k).", "3c000817a33bd696790ca7f48692bbeb": "\\tfrac{P}{k!}", "3c002a629d4997604626089f321deaf2": "\\zeta(x,y,t)=\\Re\\left\\{\\eta(x,y)\\,\\text{e}^{-i\\omega t}\\right\\}", "3c00de993af829a76f4c050b6b5c6903": "\\oint \\frac{\\delta Q}{T}=0.", "3c00e8293a2fd0f3520b363e38589d3a": "\\boldsymbol{H} \\cdot \\boldsymbol{c}^T = \\boldsymbol{0}", "3c010072b5758fc275768e9f8f9f6757": "\\mathbf\\Sigma_i", "3c0111fd75682090d88b42a0a517fb04": "w\\in A_{p},\\;\\; p\\geq 1,", "3c011eabcc4b36c0d5f46e7949a1ec2d": " t' = \\gamma t", "3c014bd49fddc39e0c48a20b3b02008f": "A \\times (B \\setminus C) = (A \\times B) \\setminus (A \\times C),", "3c018a7c680d7b70fda3e1bda55db87f": "\\{1,2,3, \\ldots,n\\}", "3c01e1859483fd96e01a1f266a95d95a": "W = \\min_{m \\in 1, \\dots, r} Y_m\\,", "3c02195e69f7e8e6ec470cccbb489ef8": " C^J_{v_i} = \\left( \\frac{dJ}{dp} \\frac{p}{J} \\right) \\bigg/ \\left( \\frac{\\partial v_i}{\\partial p}\\frac{p}{v_i} \\right) = \\frac{d\\ln J}{d\\ln v_i} ", "3c022be571ff021fe8f2fb4abb2ea673": "\\ \\sigma_{1}^2 + \\sigma_{2}^2 + \\sigma_{3}^2 - 2 \\nu (\\sigma_1 \\sigma_2 + \\sigma_2 \\sigma_3 + \\sigma_1 \\sigma_3) \\le \\sigma_y^2. \\,\\! ", "3c02475463e7345967ab3477116ce20e": "c \\in M \\setminus \\left \\{ \\frac{1}{4} \\right \\}", "3c02546ce7e4001d341b934a9eda9528": "\\mathcal{O} _X", "3c02aef6a8bd83b674f3b78eee7e3275": "V_{T-j}(k)", "3c02bef00cc53ad4778bf1371796bea2": "\\Pr(A_i |B) = \\Pr(A_i)\\frac{\\Pr(B|A_i)}{\\sum_{j}\\Pr(A_j)\\Pr(B|A_j)}.", "3c02d1b56bbbf1580f7834b057fe4cd1": "\n\\begin{align}\nc_0 & = \\sin{1} = {\\int_{-1}^1 \\cos{x}\\,dx \\over \\int_{-1}^1 (1)^2 \\,dx} \\\\\nc_1 & = 0 = {\\int_{-1}^1 x \\cos{x}\\,dx \\over \\int_{-1}^1 x^2 \\, dx} = {0 \\over 2/3 } \\\\\nc_2 & = {5 \\over 2} (6 \\cos{1} - 4\\sin{1}) = {\\int_{-1}^1 {3x^2 - 1 \\over 2} \\cos{x} \\, dx \\over \\int_{-1}^1 {9x^4-6x^2+1 \\over 4} \\, dx} = {6 \\cos{1} - 4\\sin{1} \\over 2/5 }\n\\end{align}\n", "3c03125b42dc16a1f04680bd4771df01": " 136 \\rightarrow 244 \\rightarrow 136 \\rightarrow ... ", "3c03255ed730936e35fc346b0f078beb": "\\ X(s)", "3c032e8f0efdc4ce3d0620e8e8e2c1d2": "S^n=\\{x\\in \\mathbb{R}^n_{+} \\mid\n\\sum_{i=1}^nx_i=1\\}", "3c03ddff0001116011e593c3110265d4": " A = \\{ a_1, \\ldots, a_n \\}", "3c0435ff5802c5bc6290fd44ca550c86": "(x) \\, P \\qquad \\bigwedge_{x} P", "3c04f722dc0ef8f46610cc2254222bc1": "\\scriptstyle{\\underline{\\underline{q}}}", "3c052e866e3f633085f8f6b8718c255b": "P=I*f/N", "3c05301ae0e8616d13dbba8e5e706503": "\\rho_i^B", "3c054d7b33d8a1fde523f8afe381f06b": "\n\\begin{align}\nD=& \\psi_a(1)\\Big( \\psi_b(2) \\psi_c(3) - \\psi_b(3) \\psi_c(2)\\Big)\n- \\psi_a(2)\\Big( \\psi_b(1) \\psi_c(3) - \\psi_b(3) \\psi_c(1)\\Big) \\\\\n& {}+ \\psi_a(3)\\Big( \\psi_b(1) \\psi_c(2) - \\psi_b(2) \\psi_c(1)\\Big) .\n\\end{align}\n", "3c0568644d50c89d2cba53cedac64bcd": "G \\to G_i", "3c056d1fb6579f7ec715588930792840": "g \\bar \\Psi i\\gamma^5 \\phi \\Psi", "3c05a9fda783427e8a7428d053d56a56": " u_n(t) = \\cos(2\\pi nt)+ i \\sin(2\\pi nt) \\, ", "3c05cc18b831f15635b5898ac0f6f59c": "{{i}_{E3}}", "3c05cca7847d9e280c296ca82c852d31": "\\mathbf K", "3c05d2a9fed855630fcbdb138fd0c70a": "n=pq", "3c05ffdae641e9e05896cb093dbdce65": " \\rho(\\mathbf{r},t) = \\Psi^{*}(\\mathbf{r},t) \\Psi(\\mathbf{r},t) = |\\Psi(\\mathbf{r},t)|^2 \\,\\!", "3c062806cef472802b3b29a7d86a33fa": "\\frac {\\rho }{\\rho_0 } = \\frac {p \\, T_0}{p_0 \\, T}", "3c0670fba879ba63aa4baf85d9f85462": "(u+du,1)", "3c06a79c5d0cd4cfeb62c6be02a2aa4c": "I[u]=\\int_a^b L[x,u(x),u'(x)] \\, dx \\, ,", "3c076b8c53577341b6cbdf6627798d0d": "x^0\\rightarrow -\\infty", "3c07fa597ba54b1b7d99556b5d0f1b60": "H=\\mathbb R^n", "3c08188cd360d0945406ba9b1ce852c7": "\\begin{align}\n & g_1 = \\frac{ m_3 }{ m_2^{3/2} } = \\frac{\\frac{1}{n} \\sum_{i=1}^n \\left( x_i - \\bar{x} \\right)^3}{\\left( \\frac{1}{n} \\sum_{i=1}^n \\left( x_i - \\bar{x} \\right)^2 \\right)^{3/2}}\\ , \\\\\n & g_2 = \\frac{ m_4 }{ m_2^{2} }-3 = \\frac{\\frac{1}{n} \\sum_{i=1}^n \\left( x_i - \\bar{x} \\right)^4}{\\left( \\frac{1}{n} \\sum_{i=1}^n \\left( x_i - \\bar{x} \\right)^2 \\right)^2} - 3\\ .\n \\end{align}", "3c0842987c22d1295d958bfbf8ed3bd9": "\\mathrm{2\\ BkO_2\\ +\\ H_2\\ \\longrightarrow \\ Bk_2O_3\\ +\\ H_2O}", "3c084f5d569728398ec0647a5b638780": "x[S]y", "3c086b724946968361007b44e6e3784f": "(\\mathbf{B}_2 - \\mathbf{B}_1) \\cdot \\mathbf{n}_{12} = 0 ", "3c087ce415187c313e53898ee084cea7": "\\widehat H", "3c0885bcea2bc1edee04c50ddae9eea3": " \\tilde \\nu_{J^{\\prime}\\leftrightarrow J^{\\prime\\prime}} = F\\left( J^{\\prime} \\right) - F\\left( J^{\\prime\\prime} \\right) = 2 \\tilde B \\left( J^{\\prime\\prime} + 1 \\right) \\qquad J^{\\prime\\prime} = 0,1,2,...", "3c0910c22846ddb4a6c426c89703823f": "u(L,t')=u(H,t')=t'", "3c093545a21a4d1ff382296391f88029": "y(p)= \\frac {\\partial \\pi (p)}{\\partial p}", "3c098f36119f1c2104b704c3c2b84699": "x=L", "3c0991cc03fb91112375b1dc80fd854d": "\\frac{a^2}{b}\\,\\!", "3c0a0e21ca125735118e5ca789dd1678": "\\vdash \\Psi \\leftrightarrow (\\Box \\Psi \\rightarrow P)", "3c0a30ac2fde1ec5e6d2e0f914cc5304": "L(f)", "3c0a997eee7751b5a3cb1376804cdabf": "R_1 = \\frac{V_{S} - V_{Z}}{I_{Z} + K \\cdot I_{B}}", "3c0aa933b306a9ea1530235ff9eb7f4d": "KIE = \\frac{k_1}{k_2} = \\frac {\\ln(1-F_1)}{\\ln[(1-F_1)R/R_0]}", "3c0ac65b90659fdd19de9e278f9696fd": "(7=+1\\times 2^3+0\\times 2^2+0\\times 2^1-1\\times 2^0 = 8-1)", "3c0aeb6b45fc0c950c83553fef3f4be5": "\\sum_{n=1}^\\infty a_n^{+}", "3c0afd8222ddce2f62e75121025f81e5": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{rect}\\left(\\frac{\\nu}{2 \\pi a}\\right)", "3c0b37491607158da4b9516a32dcf830": "P_\\text{sym} = \\frac{1}{2}(1+P),", "3c0b409b4f308799a50df30f1d4fdc8d": "\\langle(\\hat{\\phi^2}\\rangle-\\langle\\hat{\\phi}\\rangle^2)^{1/2}", "3c0ba77d8b5e41e99e92d89acff59ac0": "\\gamma = L_\\mathrm{A}-L_\\mathrm{B}\\,", "3c0bd458d0cb8653c5b4c3435e3dc77d": " B_{k} < (\\frac{3}{4}- \\mu_{k,k-1}^2)B_{k-1} ", "3c0c7513c0effadf02507ff19c226c83": "\\begin{pmatrix}\n 0 & 0 & 1 \\\\\n 0 & 1 & 0 \\\\\n 1 & 0 & 0\n\\end{pmatrix}", "3c0d9612cf88640dc2f1bb5d00db8ca5": "\\mathbf{B}, \\mathbf{t}\\rbrace, \\sigma^2", "3c0e2173046f48edcb0758bb8b84ecb0": "\\mathcal{O}^\\times = \\{a\\in F: |a|= 1\\}", "3c0e2f89b5262bdb3e73d9c2a736b96f": "\\Delta U = C_v \\Delta T\\;", "3c0e3d04f933ddaf5507d30cf90db20a": "H_1(T^2;\\mathbb{Z}_2)", "3c0e6a94ee4c12b946b55e2fb9ff954f": "\\frac{dy}{dx}=x+y", "3c0e867a7cda48738f93cd2bff03fe0b": " \\mathbb{R}^\\infty \\,", "3c0ec7d22ff6cb1f6d82a18b38458195": "t\\rightarrow\\lambda t.", "3c0efb7850cfc1d57c6aa3803c80c1ba": "x^n-1 = \\prod_{m|n} \\Phi_m(x)", "3c0f0d114d2f27e47e1720139047e56e": "\\nabla \\cdot \\mathbf{E} = 4\\pi\\rho", "3c0f73fd375977467e7c8b2fdfd94114": "j_\\nu=\\kappa_\\nu B_\\nu.\\,", "3c0f8ed2bdc79afa6a83284d9d45de17": "f,g\\in\\pi_n(X,x_0)", "3c0fc4d848447976392789f108184850": "\\sum_{i=1}^{m}\\sum_{s=1}^{n} J_{ij}J_{is}\\ \\Delta \\beta_s=\\sum_{i=1}^{m} J_{ij}\\ \\Delta y_i \\qquad (j=1,\\dots,n).\\,", "3c0fc4e3dd6230e57df721dfcb1d1b85": " \\cos\\theta\\ obs=\\sum_{i=1}^n f_i \\cos\\theta\\!_i ", "3c0ff889aa2040dcbec0805794007c9f": "\\scriptstyle p_1 \\,", "3c10069985ad0513bffdb5081dd4c7b0": "\\ln\\frac{X}{1-X}", "3c101bb9e3d53106f4ac0ab25bda2696": "\\Delta L'", "3c103a72bb230f9d70e4268215acf861": "\\left( \\frac{3}{2} \\right) ^4 \\times \\left( \\frac{1}{2} \\right) ^2", "3c105457e60389d1355290e947b6dc9a": "C =\\sum_{j=0}^{M} \\sum_{ir\\}", "3c2c8f927acb1b0c6fa63f180a7be16b": "z_3 = f_c(z_2) = (c^2 + c)^2 + c\\,", "3c2cd08e0e38b3735126829f75e10ff5": "v_e = \\sqrt{2gr}", "3c2ce4aa1dadb561dedf034952ed2edf": "|\\mbox{Cl}(\\mathbf{Q}(\\tau))|", "3c2d0f9f2f60ff84e5656ba4f4a94674": "\\scriptstyle\\boldsymbol{\\theta}=\\frac{\\boldsymbol{y}-\\boldsymbol{x}}{r}", "3c2e4fa767769efe7878af6ca6a01813": "\\det\\,A = \\det\\,A^\\text{T} = \\det\\left(-A\\right) = (-1)^n \\det\\,A", "3c2ee33f00eef23fd63d6db836b66206": "{V_P}", "3c2ef66cc87a535619e45e004254fc9d": "{\\mathbf e}' = {\\mathbf e}\\, g,\\quad \\text{i.e., }\\,e'_\\alpha = \\sum_\\beta e_\\beta g^\\beta_\\alpha.", "3c2f24e07156a102e1101d1bce965868": "E_4", "3c2f53f5840592b4e02b56747aed5fb1": " \\left|\\Psi\\right\\rang = {1 \\over \\sqrt{2}}[\\left|\\Theta\\right\\rang + \\left|\\Lambda\\right\\rang] ", "3c2fae76763360b556e3a1af4df61707": "\n\\prod_{i=1}^{n}\\left(1-a_{i}\\bar{a}_{i}\\right)\\prod_{i=1}^{n}\\left(1-b_{i}\\bar{b}_{i}\\right)=1-\\sum_{i=1}^{n}\\left(a_{i}\\bar{a}_{i}+b_{i}\\bar{b}_{i}\\right)\n+\\left(\\sum_{i=1}^{n}a_{i}\\bar{a}_{i}\\right)\\left(\\sum_{i=1}^{n}b_{i}\\bar{b}_{i}\\right)+\\sum_{i 0", "3c3a67fdc4637949db0eaedf34795a6b": "v_2 = R i_1", "3c3a83e7127c7ef8e982dd345ff85937": "\\psi(ab)=\\psi(ba)\\,", "3c3b0174d0a69009e3dcae6a8e3bf275": "\\cos A = \\cos a \\cdot \\sin B", "3c3b663efd2df3047fc9cf3845ce2f5a": "\\textstyle{-\\frac{\\log(2)}{\\log(\\frac{1-\\gamma}{2})}}", "3c3b9961eaacea339b70491387ae36db": "\n\\mu=\\frac{2}{\\lambda_{\\mathrm{max}}+\\lambda_{\\mathrm{min}}},\n", "3c3c2314d5588ae445e7340ae35b5760": "\\sigma_x^2(\\tau) = \\frac{\\tau^2}{3}\\operatorname{mod}\\sigma_y^2(n\\tau_0)", "3c3c3389e0b14c225c8dde63e2939e7b": "{n \\choose k}={n(n-1)(n-2)\\cdots(n-k+1) \\over k! }.", "3c3ca3e085b5be18a0747d6f9314c8c7": "T_{A}", "3c3cadd9ca9fc128314cfe0cc8e95c0f": "\\left(\\dfrac{n}{m}\\right) S", "3c3cd7be0dd28ec3a8560562e812866b": "{\\Delta}_{\\rho}^{2}", "3c3cf0c53a61adf0e54479c565ffe7fa": " E \\left[ \\widehat\\mu \\right] = \\mu, \\, ", "3c3cf741d57ef339d240a9141937b930": "L[q(s,t), \\dot{q}(s,t)] = L[q(t), \\dot{q}(t)] ", "3c3d021445d4caa8a08b9e7da715fd99": "z_\\epsilon", "3c3d4b3395dd4e182d95824042b888ed": "x = x_0 + u (x_1 - x_0) = x_0 + u \\Delta x\\,\\!", "3c3e06c0a714a809e2669932659f83be": "\\alpha \\mapsto \\Omega_\\alpha", "3c3e0fad127c53c8c2def5457d0ec766": "\\mathsf{S}(a)", "3c3e522323f86fbb59f150e9d1fea698": "\\beta(\\varphi)=\\tan^{-1}\\left(\\frac{1-n}{1+n}\\tan\\varphi\\right)", "3c3e6a2a4ed31fc7336f07c6f17a7295": "\\alpha = (\\alpha_1,\\alpha_2,\\ldots,\\alpha_n)", "3c3e70f8b5cdd417547a88212df16bbd": "L_{\\xi^m \\eta^n}(x, y; t) = t^{(m+n) \\gamma/2} L_{x^m y^n}(x, y; t)", "3c3e781f7df18399eaa146f189af6e8f": " \\frac {\\log 2} {\\sum \\limits_{n=1}^\\infty \\frac {1}{n}\n \\log\\bigl(1+\\frac{1}{n(n+2)}\\bigr)} = \\lim_{n \\to \\infty} \\frac{n}{\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n}}", "3c3e79ba6a508c1e26f435fa3d1af02a": " (\\mathbf{a_{1}}, \\mathbf{a_{2}}) ", "3c3e9495d1da22e92f3818cc30dbba05": "t\\mapsto \\frac{\\tilde{t}}{\\epsilon}\\,,\\quad r\\mapsto M+\\epsilon\\,\\tilde{r}\\,,\\quad \\epsilon\\to 0\\,,", "3c3e9c24f27618a406fb7626ea227691": "\\ \\mathrm{ApEn} = \\Phi ^m (r) - \\Phi^{m+1} (r). ", "3c3eed36d3c7aecced697a2fee1d99b3": "t_1 \\theta | \\xi |^{2 k} \\mbox{ for all } x \\in \\Omega, \\xi \\in \\mathbb{R}^{n} \\setminus \\{ 0 \\}.", "3c4127701141321081cf785af9e1eded": "Ma^{-1}", "3c4133a5729b1dce5de17592ef4514d6": "\\textstyle n = 2", "3c41425c506daf9c0a7fcdce65a2b60e": "\n W = \\sum_{p,q = 0}^N C_{pq} (\\bar{I}_1 - 3)^p~(\\bar{I}_2 - 3)^q +\n \\sum_{m = 1}^M D_m~(J-1)^{2m}\n ", "3c414996c30afe65599dda7d8daa9400": " \\zeta_A(s) = \\frac{1}{a_1^s}+\\frac{1}{a_2^s} +\\cdots", "3c418695c50f8906a2efb585ff5b9b41": "f(z)=\\sum_{n=0}^\\infty a_n z^n, \\ \\ \\ |z| < 1.", "3c41a4132ba9f4be6873d4847361a8ff": "m \\equiv 0\\pmod{N}", "3c41b7c4a0d4727b402952e5a545f5da": "\\bar R^2 = {1-(1-R^{2}){n-1 \\over n-p-1}} = {R^{2}-(1-R^{2}){p \\over n-p-1}}", "3c41e2ae5eee7a5b8d319b781b701311": " \\Lambda(x,y,\\lambda) = f(x,y) + \\lambda \\cdot \\Big(g(x,y)-c\\Big),", "3c426ec19272bf86995999558a4c085a": "\\phi_i = \\sum_{j = 1}^n p_{ij}Q_j \\mbox{ (i = 1,2,...n)}", "3c4272287fef19fedff91ab81956767b": " e_i = y_i - (w^T \\phi (x_i ) + b).", "3c428fbef68c190d058170f0e05be105": "R \\oplus R .", "3c430fc59ecfcc331a864e17196f7c3b": "x_{A}", "3c43a17af9d26ef02349db4061a6e46b": "L_xL_y \\neq L_yL_x", "3c44319bdc37598282b3ed4d8c759abb": "Z_{i\\Pi m}=\\frac{1-\\left(\\omega/\\omega_\\infin\\right)^2}{\\sqrt{1-\\omega^2}}.", "3c444a7e852bd940da9fb697823fcb24": "L = \\mu_0 \\frac{N^2A}{l}.", "3c4463a53b8017daff0a7d97e21c988a": "gz_1=z_2", "3c446a1a6d88d3cd034439962c89b92d": "\\int \\frac{1}{x \\, \\log x} \\, dx = 1 + \\int \\frac{1}{x \\, \\log x} \\, dx", "3c44700fcbcf41a03a3f13ad85ce58a5": "V(x)=|x|", "3c44c270967e71e684bf31de053a605d": "{1+\\varphi} = \\varphi^2,", "3c451698bc6dc37d3bc43da378320db2": "4.5n", "3c452f518766e72ea9f99eac956603c3": "X=[1,x,x^2,...,x^{m-1}]", "3c454bc47a3950dc33a85b9062022c87": "\\left(r_1+r_2\\right)^2=\\left(r_1+r_3\\right)^2 + r_4^2-2\\cdot\\left(r_1+r_3\\right)\\cdot r_4 \\cos\\theta", "3c45655c36fa94eea12face314c68c1e": "m = \\dim \\mathcal{H}_B", "3c45da85ea36021c77de7a062b58a6c5": "d(R\\bowtie S)=d(R)\\cup d(S)\\,", "3c469e31600d5429268f3a39a9e96133": "\\mathbf{p} = \\mathbf{a}\\times\\mathbf{b}.\\,", "3c46acca1a0dd21a5251ab4793079a51": " x_{start}q} e^{-\\sqrt{k}} k^n+\\sum_{k\\in A\\atop k\\le q} e^{-\\sqrt{k}} k^n\\cos(kx) \\ge e^{-\\sqrt{n}} n^n + O(q^n)\\quad (\\mathrm{as}\\; n\\to \\infty)", "3c4a8d93a1d9f024bda3aa10963c7f10": " \\mathbf{\\hat T}(-\\varepsilon) = \\exp\\left(\\frac{i\\varepsilon\\mathbf{\\hat P}}{\\hbar}\\right) = I + i\\frac{\\varepsilon}{\\hbar}\\mathbf{\\hat P} + O(\\varepsilon^2) ", "3c4a9e3ec90f667697c496cef92f3d5e": " \\mathcal{O}_{X,f(y)} ", "3c4af5d66505d141afff6ab75c173b28": "E_{3z^2-r^2,3z^2-r^2} = [n^2 - (l^2 + m^2) / 2]^2 V_{dd\\sigma} +\n3 n^2 (l^2 + m^2) V_{dd\\pi} + \\frac{3}{4} (l^2 + m^2)^2 V_{dd\\delta}", "3c4b1785f24c3eba6d35553c0c204df8": "K = \\frac{\\gamma \\lambda_u}{2 \\pi \\rho }", "3c4b3914294e226711e1b1146d7228db": "\\alpha+ \\beta x+ \\gamma \\cdot f(\\lambda x+\\delta)", "3c4b41d9d7ec56fb21dbe940e7bde787": "c_k=\\int_0^1 \\frac{\\Gamma(x+1)}{\\Gamma(x-k+1)}dx", "3c4b44fdc86ba2cabf0276e2f76b4ef9": "n=p_1p_2", "3c4b5c573424e020bc49a4b04469e68b": "t_{ij}^{(p)}", "3c4b6454ac48073982dd059ef859df5f": "\\langle a * b, c \\rangle = \\langle a, b * c \\rangle.", "3c4b8da037c9060ede101e88b98455bf": "A_{i_1}A_{i_2}\\ldots A_{i_{k+1}}", "3c4ba83099ff29765005c266d98383b9": "\\begin{cases}\nC_f : \\mathrm{D}(\\mathbf{R}^n)\\to \\mathrm{D}(\\mathbf{R}^n) \\\\\nC_f g \\mapsto f * g\n\\end{cases}", "3c4bb13c778b6da0b7e00dfdf80879bb": "A/I = \\bigoplus_{n\\in \\mathbb N}(A_n + I)/I.", "3c4c34c1f63136a8c80c33c5b2602496": "\\partial_a = \\left[ \\frac{1}{c}\\frac{\\partial}{\\partial t}, \\frac{\\partial}{\\partial x}, \\frac{\\partial}{\\partial y}, \\frac{\\partial}{\\partial z} \\right]", "3c4c3fea013cf8be4899def27972c294": " {{C'_w} }", "3c4c445eb98fe5fc4c890e86fd592eb0": "[L'_{ij},C'_k]=i[\\delta_{ik}C'_j-\\delta_{jk}C'_i] \\,\\!", "3c4c513ea898fdfa18cdf4cadc7ecdc1": "L^q(d\\nu)", "3c4c591b7bc097a7ea8bbe847cb144aa": "\\frac{1}{N}\\sum_{i=1}^N \\mathbf{x}_i = \\mathbf{0}", "3c4c98311c21cb95b8adc943342f7149": "g^{\\star} = \\exp(-0.5ahr) = g^*", "3c4cd28c66e6a756c64ec481c04bfdff": " t^{-1}Ht \\cap K", "3c4cd4e6654b6b2049899e4ec0e72071": "R, \\ ", "3c4d22f5a6de521763085730433f6be8": "|a|:=\\frac{\\mu(aX)}{\\mu(X)}", "3c4d4ce550e45fc172381229526059ad": "A_1\\cup A_1", "3c4d549396cae3b29adb4d10f3540a6e": "\\gamma (t) = A(t)\\cdot e^{j(\\phi(t) - \\omega_0 t)} \\,", "3c4d5a892793f059c6c88cf7df457b42": "\\delta W = \\mathbf{H}\\cdot\\delta\\mathbf{B}.", "3c4d6fe1e794981509b273c892a45491": "d=4k+2=2(2k+1)", "3c4d87b525b1a9757be0ed4bad96b918": "\n\\mathbf{F}_{\\mathrm{centrifugal}} = \n-m\\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{r})\n", "3c4db35b872bc0f9be4422b88521c83b": "\\sup\\{ |a(\\cdot)| \\} \\leq k", "3c4dc1af0e960b74f3607e6b1358890a": " \\mathbf{J}\\mathbf{J}^{\\mathrm{T}} = \\mathbf{J}^{\\mathrm{T}}\\mathbf{J} =\\mathbf{I}", "3c4e0948f04fad46e3f29b523b77c836": "f(1)=y", "3c4e14ff566a1e0dc0fb88c49bc1c992": "\\Pr[x\\text{ is singular}] = \\Pr[w(x) = \\alpha] \\le 1/N", "3c4e39962dc162611eec511dc88481a5": " 2 + 2 ", "3c4e4508b7f0cd19667ce5348412d1a0": "-\\infty < t < \\infty, \\; \\; 0 < r < \\frac{1}{\\omega}, \\; \\; -\\pi < \\phi < \\pi", "3c4e58e48cbb3256b7720302ee74538f": "P_P", "3c4e66b6e242922573e9131fda0151bf": "\\Delta \\circ \\eta = \\eta_2 : K \\to (B \\otimes B)", "3c4e6edcb39693956a78800a2a049a6e": "N_{sd}", "3c4eca27c30ad34adb1efd288003b971": "E_0 = \\hbar \\omega /2", "3c4eef090bb6d8b783973450036d1c70": "e(rm,n)=e(m,rn)=re(m,n)", "3c4f62ac9c136ad97fa26911ee1571a1": "\\mathcal{E}(\\pi)", "3c4f74457633acb2d74e61009ebb7c1d": "j_i\\in J(R)", "3c4f8c9a9dd5b40c53d7801abea77d21": "GWRCL", "3c4fb38e971f32ab9331aa3569457bb8": "R \\ge G \\ge B", "3c4fead8ae4e83487b31075ea45dd204": "\\langle j_1 j_2; m_1,m_2\\pm 1|j_1 j_2; jm \\rangle", "3c50323c06fe4a5704e9506879e17dc0": " \\langle \\varphi, \\psi \\rangle = \\langle u_\\psi, u_\\varphi \\rangle.", "3c507b5f09552ea9094fa83fee9d71de": "(T^{*} T)^{*} = T^{*} T", "3c5154df557f7afae2a5aeabdcb2a7ba": "\\begin{alignat}{2}\n S & = \\sqrt{s(s + 1)} \\cdot \\frac{h}{2\\pi} \\\\\n & = \\frac{\\sqrt{3}}{2} \\hbar \\\\\n\\end{alignat}", "3c51a20d61d93859ed8dffca986f70cf": "A^c~=~U \\setminus A", "3c51ddee8cc5e34eada75c14283210a1": "1-2^{-\\ell}", "3c522a522574afc8cf9caee6c04c3574": " \\ C_i ((X)) ", "3c523571228872986ee59b5502754b9b": "t^{2} = \\frac{d^{2}}{2U} \\frac{m}{q}\\,", "3c52bb6ddefb8b649ce040056e0cae9e": "k_{\\mu} \\mathcal{M}^{\\mu}(k; p_1 \\cdots p_n; q_1 \\cdots q_n) = -e \\sum_i \\left[ \\mathcal{M}_0(p_1 \\cdots p_n; q_1 \\cdots (q_i-k) \\cdots q_n) \\right. ", "3c52cce97a80ef908ba6ccabd0910a58": " \\bar{n}_i ", "3c52d67fdd66f393b5e96a64bf0589c0": "L_g^a + L_m^a", "3c5313e503868b0742881fa84ab9ac04": "Q(z) =\n-{{\\rm e}^{z+1/2\\,{z}^{2}+1/4\\,{z}^{4}}}+{{\\rm e}^{z+1/2\\,{z}^{2}+1/4\\,{z}^{4}+1/8\\,{z\n}^{8}}}", "3c539f07b4130b7df6790b5d6bcec443": "\\partial(\\sigma) = \\sum_{\\tau \\in \\mathcal{X}}\\kappa(\\sigma,\\tau)\\tau", "3c541341de57cf3ddc9d20e982e6cf5c": "\n\\begin{align}\n\\eta &\\approx \\frac{E(N+1)-2E(N)+E(N-1)}{2},\\\\\n &=\\frac{(E(N-1)-E(N)) - (E(N)-E(N+1))}{2},\\\\\n &=\\frac{1}{2}(I-A),\n\\end{align}\n", "3c5445a5d7223ff1ef0fa0d3aa7536b1": " s^2 = (ct)^2 - (ct)^2 .\\,", "3c54d43f83a2517617e62a1885fd2dae": "2p(n)=p(n)+p(n-1),", "3c54f1bf7da427219577cf3d2550f996": "\\hat{\\sigma}^2=\\int_{-\\pi}^{\\pi} \\frac{I(w)}{f(w;(1,\\hat{\\eta}))}\\, dw", "3c54f35c4bf76b3b7173177eb42fc1a9": "\\gamma_1^\\infty \\gamma_2^\\infty > exp(4) \\approx 54.6 ", "3c553b849ce64f31eea7a8de19c3fde5": "\n\\wp(z;\\omega_1,\\omega_2)=z^{-2}+\\frac{1}{20}g_2z^2+\\frac{1}{28}g_3z^4+O(z^6)\n", "3c5583a94266ff4d14e8c65ec2595571": "{\\rm for }\\quad \\operatorname{Re} (x) \\geq \\frac12 \\,.", "3c55912ea1afb92fc17a3b44e7333594": "r_{nk} \\propto {\\tilde{\\pi}}_k {\\tilde{\\Lambda}}_k^{1/2} \\exp \\left\\{ - \\frac{D}{2 \\beta_k} - \\frac{\\nu_k}{2} (\\mathbf{x}_n - \\mathbf{m}_k)^{\\rm T} \\mathbf{W}_k (\\mathbf{x}_n - \\mathbf{m}_k) \\right\\}", "3c55b9872030358b4de86a7fe2668928": "\\mu_r\\,", "3c5628b7dc0b514ff3e06322befd05b0": " N(d,t)0,\\delta>0", "3c6a0159b9e57c45a0d6e79e4468f03b": "x \\in A \\setminus \\{0\\}", "3c6a1e531734e5467e31ee8ed2f357e8": "0<\\int _0^\\infty \\frac {\\sin x}{x}dx < \\int _0^\\pi \\frac {\\sin x}{x}dx < \\pi", "3c6a2bccc6f6924b91207c2d3e6ed3f2": "\\Psi_0=D\\sigma-\\delta\\kappa-(\\rho+\\bar{\\rho})\\sigma-(3\\varepsilon-\\bar{\\varepsilon})\\sigma+(\\tau-\\bar{\\pi}+\\bar{\\alpha}+3\\beta)\\kappa\\,,", "3c6a93fc922fbfa19aea78961760cefb": "\\sigma ^{2}", "3c6a94ccdc4fa80cc126c9c588317265": " M : V \\rightarrow V ", "3c6a9ee3c1bde3947926e68a15928e66": "a/(1+a)", "3c6b70e272c9dad8c01c9e887eb589fc": " A_{rr} + \\frac{1}{r} A_r + \\frac{1}{r^2}A_{\\theta\\theta} + k^2 A = 0. ", "3c6b8eef0e66d4b7b0496637ac62170b": "E(s|r,r_i,RD_i) = \\frac{1}{1+10^{(\\frac{g(RD_i)(r-r_i)}{-400})}}", "3c6b8f1f0e710bc4f19d892897062e2c": " \\sum (\\lambda_n - \\lambda_{n+1}) \\, B_n.", "3c6bb91ad255169ad74417598df6818f": "\\epsilon_{0} \\mu_{0} c^2 = 1\\,", "3c6bd7d0b23ba5b0680f1eaf2aa6dde1": " d_M(z) = \\left\\{ \\begin{array}{ll} \\frac{d_H}{\\sqrt{\\Omega_k}} \\sinh\\left(\\sqrt{\\Omega_k}d_C(z)/d_H\\right) & \\text{for } \\Omega_k>0\\\\\nd_C(z) & \\text{for }\\Omega_k=0\\\\\n\\frac{d_H}{\\sqrt{|\\Omega_k}|} \\sin\\left(\\sqrt{|\\Omega_k|}d_C(z)/d_H\\right) & \\text{for }\\Omega_k<0\\end{array}\\right.", "3c6be365f0e1675ad64c5f5ca81a6407": "\\scriptstyle K_{x\\rightarrow y}(t)", "3c6bec371eda6db7c621815fc724e7ca": "y^*\\,", "3c6c33a7e770860d310d77baa1d55ed3": "\\left\\|x\\right\\|_{bs} = \\sup_n\\left|\\sum_{i=1}^n x_i\\right|.", "3c6c7c8a3b072045c3d5ef1a658e888d": "\\Gamma_{i}(t):= \\varphi(\\gamma_{i}(t)), \\qquad i=1, 2, 3, 4", "3c6cf53da06fca777a1580cdd82c736d": "G(\\vec r, \\vec r') = \\frac{1}{S_n} \\frac{\\vec r - \\vec r'}{|\\vec r - \\vec r'|^n}", "3c6d0a5056da4e9b7f763ae67b9dadeb": "|\\psi\\rang", "3c6d22beed198b24fd2467d5917f7fa1": "\\begin{align}M(x,y) &= \\frac\\pi2\\bigg/\\int_0^{\\pi/2}\\frac{d\\theta}{\\sqrt{x^2\\cos^2\\theta+y^2\\sin^2\\theta}}\\\\\n&=\\frac{\\pi}{4} (x + y) \\bigg/ K\\left( \\frac{x - y}{x + y} \\right)\n\\end{align}", "3c6d5915533ef4571813e474de415fdf": "n\\geq k", "3c6d89c955027e29bc116948dbbd02f8": "B_\\nu(T) = \\frac{ 2 h \\nu^3}{c^2} \\frac{1}{e^\\frac{h\\nu}{k_\\mathrm{B}T} - 1}", "3c6dc870258ef44abd493f8f1ed55a67": "f:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}^{n}", "3c6dd7f44b78a871e0f92f099b3001b5": "\\mathrm{d}U=T\\mathrm{d}S+V\\sigma_{ij}\\mathrm{d}\\varepsilon_{ij}", "3c6debdfebda507899bd23e4346d6d9f": " \\begin{align}\nV(\\mathbf{x}) &= - \\frac{G}{|\\mathbf{x}|} \\int \\sum_{n=0}^\\infty \\left(\\frac{r}{|\\mathbf{x}|} \\right)^n P_n(\\cos \\theta) \\, dm(\\mathbf{r})\\\\\n{}&= - \\frac{G}{|\\mathbf{x}|} \\int \\left(1 + \\left(\\frac{r}{|\\mathbf{x}|}\\right) \\cos \\theta + \\left(\\frac{r}{|\\mathbf{x}|}\\right)^2\\frac {3 \\cos^2 \\theta - 1}{2} + \\cdots\\right)\\,dm(\\mathbf{r})\n\\end{align}", "3c6e0b8a9c15224a8228b9a98ca1531d": "key", "3c6e40053064fd5b7abe1d5644e6c0d1": "S_I", "3c6eb6cc0e64000d0fee51c89f798cb5": "x^4 + x^2 + 1 = 0", "3c6eeaca58e587f52336cf0d5684156a": " D_x(r)=1-P({N}(b(x,r))=1\\mid x). ", "3c6efcc4d7d7053de2b3c068a92b7b4b": "[A]^n", "3c6f262d80be4220d439af7c9eb6f5ee": "w \\in L \\Leftrightarrow w^n \\in L", "3c6f9d03598605efb1686c6a5c655f3d": "i^{\\text{th}}", "3c6fd0fca06e59a7daea5eda4a81c681": " 2^i ", "3c6fe45b68f4a6995d1473391fdf5af7": "\\frac{\\ell x^2 + mx + n}{(x-a)(x-b)(x-c)} = \\frac{A}{(x-a)} + \\frac{B}{(x-b)} + \\frac{C}{(x-c)}", "3c701d7c4e68dde057281b04b0014454": "{dN_1\\over dt}=r_1 N_1 {K_1-N_1-\\alpha_{12} N_2\\over K_1}", "3c704b9b54be087249c6c998a807a80c": "P_L=-\\rho_L g z+p_0,\\qquad P_G=-\\rho_G gz +p_0,\\,", "3c70807775d74ed985a3a26802bdc933": "2l \\le n - k", "3c70dcce69932794aa424832e4a04107": "\nF(\\mathbf{k})=\\iint f(\\mathbf{r})\ne^{i\\mathbf{k}\\cdot\\mathbf{r}}\\operatorname{d}\\!\\mathbf{r}.\n", "3c715b85dd0ba0fae6559dd1dad052c8": "- {1 \\over 2} g_{\\mu \\nu} \\,", "3c71b8aab62b0a7e8cb0f88bd8ad653c": "g^{-1}\\in G", "3c71f9f32d18f31fbaea476fc2444b9c": "F = e^{r\\tau} S = \\frac{S}{D}", "3c7291cb998d58760d6b06259d7b9907": "Con := \\{ X \\in \\mathcal{P}_f(T) \\mid X \\mbox{ is consistent} \\}", "3c729a6f25b6432dca0b5a16270de879": "\\vartriangle^{m-1}_n", "3c72a45dc00151cd97969f1a6b1f2c9b": " \\vec{R_i} = \\langle \\sigma_i \\rangle ", "3c72fa27a899ee564529f9f5afd6364a": "365\\tfrac{1}{4}", "3c73370d00f1292ffea5ca0ad4e6bfd2": "t_s = 1 + r_s", "3c733be97d545b99fae84d09aa0678d9": " \\mathbf{A} + \\nabla m ", "3c7379a32d5bec9272ab8836627c87af": "{A}_{17}^{(2)}", "3c73b5ac1432f53f9e303bf20f0aad55": " W = -kq \\sum_{i=1}^n \\bigg( Q_i \\int_{\\gamma} \\nabla \\left(\\frac{1}{|\\mathbf{r}-\\mathbf{p}_i|} \\right) \\cdot d\\mathbf{r} \\bigg) = kq \\sum_{i=1}^n Q_i \\left( \\frac{1}{|\\mathbf{a}-\\mathbf{p}_i|} - \\frac{1}{|\\mathbf{b}-\\mathbf{p}_i|} \\right) ", "3c7503cc4c8805aa4a1e1db4fafd25d2": "\\sum_{j\\in Z}|\\hat{\\psi}(2^j\\gamma)|^2=A", "3c752b5bb8fad884db07bd6383d40cdb": "\\begin{array}{rclcrcl}\n h_{\\mbox{e}} &=& h \\downarrow 2 &\\qquad& a_{0,\\mbox{e}} &=& a_0 \\downarrow 2 \\\\\n h_{\\mbox{o}} &=& (h \\leftarrow 1) \\downarrow 2 && a_{0,\\mbox{o}} &=& (a_0 \\leftarrow 1) \\downarrow 2\n\\end{array}", "3c75783ac079f23a425fbb2a175f4f36": "A_{il}", "3c75f7b520cd6c6533c0b459da3f0ef0": "t \\rightarrow 0", "3c75fe99ab0f8db750cb48304f0e5516": " \\begin{align} \n\\operatorname{E}[|X - E[X]|] = \\frac{2^{1-\\nu}}{\\nu \\Beta(\\tfrac{\\nu}{2} ,\\tfrac{\\nu}{2})} &= \\frac{2^{1-\\nu}\\Gamma(\\nu)}{\\nu (\\Gamma(\\tfrac{\\nu}{2}))^2 } \\\\\n\\lim_{\\nu \\to 0} \\left (\\lim_{\\mu \\to \\frac{1}{2}} \\operatorname{E}[|X - E[X]|] \\right ) &= \\tfrac{1}{2}\\\\\n\\lim_{\\nu \\to \\infty} \\left (\\lim_{\\mu \\to \\frac{1}{2}} \\operatorname{E}[| X - E[X]|] \\right ) &= 0\n\\end{align}", "3c7636124cd5384e29b794174febff70": "k = \\|DT/ds \\|", "3c76454a4b408d608af6a58c4d67ef69": "d_\\mathrm{opt}", "3c764b576f614ddaefb20c6cd430126b": "T={1 \\over nR}\\lim_{p \\to 0}{pV}.", "3c7665045b6b39633fed58621d4f24ef": "\\mathsf{P}", "3c7672af9dc2bb42790f01dc636dd891": "\\gamma > 0.", "3c769fbf66977c13d10f142a7d5ad4e8": "\\scriptstyle m\\,\\geq\\, n", "3c76d2fb7d23f26cbbdc63a18543ae56": "v(S) = \\sum_{j=0}^n c_j W_j(S)~.", "3c7743c04eb423dc05b132c5db99664b": "\\tau \\succ_C \\sigma", "3c776cf9267605dee6d5de3353b99751": "\\operatorname{tr}(A) = \\sum_{i=1}^n A_{i i} = \\sum_{i=1}^n \\lambda_i = \\lambda_1+ \\lambda_2 +\\cdots+ \\lambda_n", "3c779d92c4304ded0d6d8bae28fd2ab8": "\\kappa(\\mathbf{A})", "3c77f4029be2e609c22bba665f13b101": "fv", "3c78299b49e519531ff7dc2320d6e2a1": "v_{1}:T_{1} \\dots v_{n}:T_{n}", "3c78beadf8afeddf4cc7ae29b15cb73b": "\nW(\\mathbf{C})=\\frac{\\mu_1}{2}\\left(I_1^C -3 \\right) -\\frac{\\mu_2}{2}\\left(I_2^C - 3\\right)\n", "3c79150d42a83c4ec5e44fe4f4cc31d3": "\\alpha= 1/2", "3c79288c64634e8c1c2b9260cdb54875": " \\sigma_{\\epsilon} ", "3c797289a4c4eea8fb13b3721ab75c1b": "\nz\\,\\, = \\,\\,a\\,x_1 \\, + \\,\\,b\\,x_2 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left[ {x_1 \\,\\,x_2 } \\right]\\,\\, \\sim \\,\\,BVN\\left( {\\mu _1 ,\\,\\,\\mu _2 ,\\,\\,\\sigma _1^2 ,\\,\\,\\sigma _2^2 ,\\,\\,\\sigma _{1,2} } \\right)\\,\\,\\,\\,\\,\\,\\,a,b\\,\\,{\\rm constants}", "3c79fb2bcfc529e01752956f79efb7a4": "0.8736", "3c7a0db47deed01e3bf29f58586d5d57": "\n\\begin{align}\nax^2 + bx + c & = \\frac{a^2x^2 + abx\n+ ac}{a} & = \\frac{(ax+p)(ax+q)}{a} \n\\end{align}\n", "3c7a888b38097d37cdb5969ce33561fd": " \\{\\cdot\\}^H ", "3c7aa34470f9e00644d9c209b9c402b4": "X \\subset \\mathcal{P}(X).", "3c7aaf7c044eef64c79c48ce930a2a66": " 0 < z < 1 ", "3c7ac15605dd62803c5061a3ef58a101": "V = \\frac{5}{12} \\left(99+47\\sqrt{5}\\right) a^3 \\approx 85.0396646a^3", "3c7ac4ee42d7aeeb9b140efd9f234f02": "L = 2^{\\mathrm {EV} - 3} \\,.", "3c7b044d7ce75915d5e0e854eb9ff6c8": "X = \\{a, b, c, d \\}, N = \\{1, 2\\} \\text{, then }", "3c7b069f4ee50b25eb571253cd2a73dd": "\\frac{2}{19}=0.\\overline{105263157894736842}.", "3c7b19f15760979f73ee51a333ed38f3": "T(v)=w", "3c7b1c672eddef189f000edc20905daf": " r_\\mathrm{ corr } = r - \\frac{ N - n }{ N } \\frac{ ( r s_x^2 - \\rho s_x s_y ) }{ n m_x^2 } ", "3c7b20f6d57b5f554af4027ab0dc471c": "\\begin{bmatrix} -\\frac12\\boldsymbol\\Psi \\\\[5pt] -\\dfrac{m+p+1}{2} \\end{bmatrix}", "3c7b35c4fbaa028ed9037e3f6c75ca5e": "g(z) = z^p - z", "3c7b3f80267833544ed16170b1185975": "p^*(x) = a_n + a_{n-1}x + \\ldots + a_0x^n = x^n p(x^{-1}).", "3c7b593e3169a01578cbb664eca21517": "\\exp (-ikx)", "3c7bceff589ba8f1c5ad4cf5b2eb6180": "f(\\xi) = \\int_{-\\infty}^{\\infty} f(x)e^{-2 \\pi ix \\xi}\\, dx", "3c7bf7479900eef44bc5b5cda4faa5f5": "\\zeta_i = \\frac{m_i}{m_\\mathrm{tot}-m_i}.", "3c7c06cb076fd5c039ce4c273c093c8e": "\\frac{d\\tilde{\\mathbb{P}}}{d\\mathbb{P}}", "3c7c0fa110296b8d3b906c3f061f0288": "Cm_q", "3c7c10627dd12cd3b796959a1f374618": "\\left\\{ Y \\right\\} = {B \\over A} \\,", "3c7c635c9364ac41581e995e886421bd": " \\nabla \\times \\vec v = \\left( {\\partial v_z \\over \\partial y} - {\\partial v_y \\over \\partial z} \\right) \\mathbf{i}", "3c7cbd8c7eb34b8988f061b0ea5e9a68": " K_\\mathrm{sat}^{(2)} = \\frac{K_\\mathrm{mineral}}{\\frac{1}{S-F_1+F_2} + 1} ", "3c7d02829fd0c07bc1e6035af7fde3d4": "n/k", "3c7d0fc6b358cbc04d3c279f938a6205": "\\epsilon dn(t)/dt = -n(t) + (b^+ - n(t))p^+ - (n(t) + b^-)p^- \\, ", "3c7db49e5e0cbd7a7fb70e2b20d77ed0": "c(E)", "3c7dc99c782b89a1de3208773fb93411": "\\vec \\omega = \\vec {e}_1\\times \\dot{\\vec{e}}_1 = \\vec {e}_2\\times \\dot{\\vec{e}}_2 = \\vec {e}_3\\times \\dot{\\vec{e}}_3.", "3c7de6ed2aa3e6e5fd718788feb7f1c1": "U = \\frac{H\\, \\lambda^2}{h^3} = 62,", "3c7e2c0207be7e9fb8b8ca01b7a6d013": " \\hat{\\xi}_k = \\hat{\\lambda}_k \\hat{\\varphi}_k^T \\hat{\\Sigma}_{Y_i}^{-1}(Y_i - \\hat{\\mu}),\n", "3c7e33611f6fea20e5ca2320392fb016": "y_1(a;z) = \\exp(-z^2/4) \\;_1F_1 \n\\left(\\tfrac12a+\\tfrac14; \\;\n\\tfrac12\\; ; \\; \\frac{z^2}{2}\\right)\\,\\,\\,\\,\\,\\, (\\mathrm{even})", "3c7eb8f4cfa94b3c518df496fb57a57d": "\\sum_{Treatments} (I_j-1)s_j^2", "3c7f33f304c9d0762a92365da96b630a": "\\Sigma_R", "3c7f344087f8ed372dae9f199fb69b5d": "A\\ominus B \\subseteq C\\ominus B", "3c7f48b041b71afbbb322b9a62fbf7ff": " \\alpha = \\frac{\\cos \\left(\\pi/137 \\right)}{137} \\ \\frac{\\tan \\left(\\pi/(137 \\cdot 29) \\right)}{\\pi/(137 \\cdot 29)} \\approx \\frac{1}{137.0359997867}, ", "3c7f55e5c5cfadfd3ea097cbc63f92c0": "U_g(t) = U_{Rp} + U_{Cp}", "3c7fabfe6697e25eaaf8eb51710aab7d": "r=\\sqrt{x^2+y^2}. \\,", "3c7fad004a8fbc50793fb0312cc60d0c": "p(h_j = 1 \\mid \\textbf{V}) = \\sigma(b_j + \\sum_i v_iw_{ij})", "3c7fd137bf5520ef01f9f4f47682ed56": "\\lambda_n(A)", "3c7fdda8be4bddeda6b188f751df3aad": " \\mathbf{A} = \\begin{pmatrix} A_0 & A_1 & A_2 & A_3 \\end{pmatrix} ", "3c7ff99de246a8a8980968d4092c8983": " x = ( x_1 \\ , \\ x_2 \\ , \\ x_3 \\ , \\ x_4 \\ , \\dots ) ", "3c80032ea8f1348ecb4e21f95917018c": "x^3 +\\frac{5}{2}Dx^2+2D^2x = \\frac{P^2}{2D} - \\frac{D^2}{2}", "3c80156f8e7e42554c1b63fe883bdfcd": "\n\\frac{d^2s}{{dt}^2} = - k^2s\n", "3c802819292a9e450a181ff286dba122": "\\scriptstyle \\frac{1}{1-x} = \\sum_{n=0}^{\\infty} x^n", "3c805620c8f37c4859a08dae283033f7": "\\vec{f}_1 = \\cos(\\omega t) \\, \\vec{e}_1 - \\sin(\\omega t) \\, \\vec{e}_3", "3c805a1229cfc7415395c72102460444": "\\zeta_1", "3c80dcdce998cee0ea60363605f5e552": "\\{ \\varphi_i: U_i \\to U \\}", "3c80f3be0f24ea65a2768a7bd304e07f": "\\nabla^2_i", "3c812a8ef51291ae4b7e8b29c67df320": "\\Re(s)=1", "3c81337faf2494dd830dd499e4ecba68": "x_c(t) = A_c \\cos (2 \\pi f_c t)\\,", "3c81d2fbc9681e873cfad2a87507bca9": "W_{ij}", "3c81d3ebab7dad2573f2208c099bd1f1": "\\phi_i: X_i\\rightarrow X", "3c81d756eb6bb6ef79ab9ba1642c8c22": "\\boldsymbol{\\Omega} = 0 \\ . ", "3c81e72d88b961b5be9cf61799122175": "\\lVert g \\rVert_\\infty", "3c81f7eb05fc1abbeb38105e8640d99b": " \\overline{X_t} ", "3c8263f4cfe8f2697af0ed7aabe14a73": "v_{\\text{in}} = 0", "3c8310eff2f7a1828cf49fd8993b100c": "P = \\sum_{t=1}^N \\frac{D_0 \\left( 1+g \\right)^t}{\\left( 1+r\\right)^t} + \\frac{P_N}{\\left( 1 +r\\right)^N}", "3c8321d9f46a814a68a52f6f23304499": "O = (O \\times O)/O", "3c8337b02dd4a935bb1178280444c1f9": "\\{a^\\dagger_i, a^\\dagger_j\\} = \\{a^{\\,}_i, a^{\\,}_j\\} = 0.", "3c838b1ea02a1e000c8446fdc63b0480": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{j}) ", "3c8390130f0f95e9885663be65af8741": "\n\\begin{pmatrix} \n u \n \\\\ \n v \n\\end{pmatrix} = \n\\begin{pmatrix} \n \\displaystyle {\\partial \\varphi \\over \\partial x}\n \\\\[2ex]\n \\displaystyle {\\partial \\varphi \\over \\partial y} \n\\end{pmatrix} = \n\\begin{pmatrix} \n \\displaystyle + {\\partial \\psi \\over \\partial y}\n \\\\[2ex]\n \\displaystyle - {\\partial \\psi \\over \\partial x}\n\\end{pmatrix} =\n\\begin{pmatrix}\n +2Ax\n \\\\[2ex]\n -2Ay\n\\end{pmatrix}.\n", "3c839123ee052fcba5f972dec391be07": "\\leftrightarrows", "3c84538ac04d51943e695237f607876b": "x=y=0", "3c84617edac0700a0be16d60e5633d3e": "n(x,G):= \\min\\{n\\in\\mathbb N:B(x,1/n)\\subseteq G\\}", "3c84791fdf40e6286b16377075715e0d": "-\\Delta\\psi_i^{(k)} = 0, \\qquad \\psi_i^{(k)}|_{\\partial\\Omega} = 0, \\quad \\partial_n\\psi_i^{(k)} = \\partial_nu_1^{(k)} - \\partial_nu_2^{(k)}.", "3c848bfc85e060a00f81eabc99207980": "\\mathbf{\\Sigma}^0_\\alpha", "3c848ed3becec6a0c3b6da183a2b3efd": "\\begin{alignat}{3}\n g_{SP}(t,\\omega) & = \\dfrac{1}{4\\pi^2}\\iint \\dfrac{A_h(-\\theta,\\tau)}{\\phi(\\theta,\\tau)}e^{j\\theta t+j\\tau\\omega}\\, d\\theta\\,d\\tau \\\\ \n & = \\dfrac{1}{4\\pi^2}\\iiint \\dfrac{1}{\\phi(\\theta,\\tau)}h^*(u-\\dfrac{1}{2}\\tau)h(u+\\dfrac{1}{2}\\tau)e^{j\\theta t+j\\tau\\omega-j\\theta u}\\, du\\,d\\tau\\,d\\theta \\\\\n & = \\dfrac{1}{4\\pi^2}\\iiint h^*(u-\\dfrac{1}{2}\\tau)h(u+\\dfrac{1}{2}\\tau)\\dfrac{\\phi(\\theta,\\tau)}{\\phi(\\theta,\\tau)\\phi(-\\theta,\\tau)}e^{-j\\theta t+j\\tau\\omega+j\\theta u}\\, du\\,d\\tau\\,d\\theta \\\\\n \\end{alignat}", "3c84b2ffb073a5d59ce54c519145f1bb": "x=[x_{1},x_{2},x_{3},x_{4}]^{T}", "3c84d672bf64cce93a51abef433d1930": " (\\Delta f )", "3c852865123d7d0c1d0fba2852126f60": "\\vec{v}_g = \\frac{m}{q} \\frac{\\vec{g}\\times\\vec{B}}{B^2}", "3c85833b0f0f379a772e92fad2a06e16": "\\tilde{O}(VCDIM(H)/\\alpha^{2})\\,\\!", "3c85986a8ff68df908c64eae287822e2": " \\lVert M \\rVert_2 \\leq \\gamma ", "3c85a8c3e2b37699821d64785062e6e1": "s=\\frac{R_r^'}{\\sqrt{R_{TE}^2+(X_{TE}+X_r^')^2}}", "3c85ac31a3bd21a93d8ae7d92fd40775": "2.8599", "3c85d4d93b9649f5582d5c7652cb1319": "\\scriptstyle Ma\\, ", "3c8690e8a3d5d57014a373c2c2e2b03a": " \\mathbf{E} = \\mathbf{E}_\\omega \\cos(\\omega t), ", "3c86a4ba28d5bcf0be244d947ae84cc2": "m \\neq j", "3c86aa88610b8c973adf8cccfbac3de2": "[a,b)", "3c86bdc491818a35a6603a86cf324b8e": " 1 \\rightarrow \\mu_2 \\rightarrow \\mathrm{Pin}_V \\rightarrow \\mathrm{O_V} \\rightarrow 1 ", "3c87a9811a48988370addee308893b86": "\\omega_{deg}\\,", "3c87ed20464ceb878dd5bf070ec7e66e": "S=\\sum_{k=1}^6 (j_k-m_k)", "3c8809bbaf7c49647a51d1478a8036fa": "\\sum_{j=0}^\\infty z^j= \\sum_{i=0}^\\infty \\frac{1}{(1+y)^{i+1}} \\sum_{j=0}^i {i \\choose j} y^{j+1} z^j = \\frac{y}{1+y} \\sum_{i=0} \\left( \\frac{1+yz}{1+y} \\right)^i", "3c88106d9fb309804f7ff97a8b9e2805": "\\theta = (k+l-1/2)/2", "3c883797b6eb47d57cc1640b9c6fad6e": " M \\leq 2 \\left\\lfloor \\frac{d}{2d-n} \\right\\rfloor. ", "3c883babeea4980519b0b3eaa34f0f06": "\\mathfrak{k}_a^\\perp", "3c88d7f2f710399a03ae0fae023569a3": "Y_t = \\exp\\left(X_t-X_0-[X]_t/2\\right).", "3c88ef6026bcbc49a95959ec39ec59a2": "V_a[x] = V_b[x] = max(V_a[x], V_b[x])", "3c88fe820fedea41c879f256e30a6462": "sp(4)", "3c89092d30ecaf98e0c2a68cf8cb437a": "\\frac{\\Delta l}{\\Delta r_{N}} = 1.22", "3c890b6de3ff9e971591115432729ccf": "\\sin (\\arcsin x) = x ", "3c890f4678878e68a59fc11776d21c99": "\\frac{19}{4}t^2 \\cot \\frac{\\pi}{19}", "3c891b40100b4c9b7994ad64972e1d34": "R = \\frac{m_1r_1+m_2r_2}{m_1+m_2}", "3c8957a843c56f5b3312fd8d56cd5f1f": "s_2=\\min_{i=1,\\ldots,m} \\{k u_i^{-\\frac{1}{a}}\\},", "3c89924e3eddca48afabf025fd38e34b": " \\operatorname{drop-params}[g\\ m\\ p, D, V, [F_1, S_1, A_1]::\\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "3c8995f9401724478b97fbfe4b201b8f": " \\mathcal{H}^3 ", "3c899a6cb2b2e9d35e9ec7a29c8e8704": "S^{n-3}", "3c89a6aca9ec27193ea90c393e1853af": "j_n = \\frac{\\hbar}{2mi}\\left( \\Psi_n^* \\frac{\\partial \\Psi_n}{\\partial x} - \\Psi_n \\frac{\\partial \\Psi_n^*}{\\partial x} \\right) = 0", "3c89d5fad8bc2520d5aa88ea8aa207d9": "Sum_n", "3c8a07783042d3ca6ee8f422ea1348a3": "\\begin{Bmatrix} q , p \\end{Bmatrix}", "3c8a632b27f87539f81029748f0301e7": "\\zeta(s,t) = \\sum_{n > m \\geq 1} \\ \\frac{1}{n^{s} m^{t}} = \\sum_{n=1}^{\\infty} \\frac{1}{n^{s}} \\sum_{m=1}^{n-1} \\frac{1}{m^t} = \\sum_{n=1}^{\\infty} \\frac{1}{(n+1)^{s}} \\sum_{m=1}^{n} \\frac{1}{m^t}", "3c8ac119db9ce0494fbd6827accaaac8": "S=banana$", "3c8ad59c8c13379328e69ae494767373": " P \\lor Q ", "3c8ad79d15ca3d181a8614a898ac5b4b": "\\begin{pmatrix}0 & b\\\\-b & 0\\end{pmatrix}", "3c8b04b2356fdc861f13ad5fa91a0e4f": "y(t_0)\\,", "3c8b14834a9c8dc89c161d88538b58b3": "\\sum_{k=j}^\\infty\\binom{k+1}{j}a_{p,k}=\\binom{p}{j}.", "3c8b5136c988f16487ad60559220d263": "Z_\\mathrm A=Z_1\\ ,\\!", "3c8b62552f33dad32bc34c373da2f632": "y_i=\\frac{e^{x_i}}{\\sum_{j=1}^c e^{x_j}}", "3c8ba646b62083fa427b937cf626aaeb": "a^{p-1} (p-1)! \\equiv (p-1)! \\pmod p.", "3c8bd59a8e4a170c9204b011cfc618a6": "d \\approx \\sqrt{1.50h} \\approx 1.22 \\sqrt{h} ", "3c8c4737892173ee0211dd70c6b9edbc": "\\operatorname{Var}(aX)=a^2\\operatorname{Var}(X).", "3c8c7d609835ad2667f773237980a8ca": "\\displaystyle\\mathcal{L}f", "3c8cc54d95f422ceb0645eed4a160e46": " \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\left(\\frac{\\partial^2 \\mathbf{S}}{\\partial x^{2}} + \\frac{\\partial^2 \\mathbf{S}}{\\partial y^{2}}+\\frac{\\partial^2 \\mathbf{S}}{\\partial z^{2}}\\right)+ \\mathbf{S}\\wedge J\\mathbf{S}.\\qquad (5)", "3c8cf155f7aebe29e3f568d4d212a897": " e^{y_1z}, e^{y_2z} ", "3c8d1908620d55d5151d17f1b32bd40f": " \\ \\sqrt{gh_0} ", "3c8d2f1f6817f2d4dbeacc60694c4c6f": "(\\textbf{x},Y) = (x_1, x_2, x_3, ..., x_k, Y)", "3c8d33a3f1773aaef6f5d89cef8dde95": "\\log(z^2-5)", "3c8d3635283b0d925c6c9959d4fa24ef": " \\frac{h-h'}{\\lambda}=\\frac{D}{\\lambda}=N ", "3c8e1e06355e495fd70fb4a545811cb6": "(m \\times n) / t", "3c8e642392ce06408c950510e5066f3f": "|\\mathbb{F}_q^n|=q^n", "3c8ebc38f0b99211d4339fe6d7fe32fc": " \\rho = \\frac{\\partial \\Sigma}{\\partial E} ", "3c8ed51958944d4b4fccd08a154c5d00": "w(a) = 0", "3c8efdc5ba81fd310bb2b007e6f89612": "\\Diamond", "3c8f255462469bfb4f02764d29cc32e5": "V(q) = \\frac{1 + q^{-2} + q^{-n} - q^{-n-3}}{q+1},", "3c8f512163adf4860920c222ebc280e6": " {\\partial C \\over \\partial t} + V(y){\\partial C \\over \\partial x} = D{\\partial^2 C \\over \\partial y^2} ", "3c8fbb7ad5345bc02f66b11d26400956": "\\left[ \\begin{array}{ccc|c}\n2 & 1 & -1 & 8 \\\\\n0 & 1/2 & 1/2 & 1 \\\\\n0 & 2 & 1 & 5\n\\end{array} \\right]", "3c903957e7e580d9a5db4ef45d443076": "(u(x_n))", "3c905230f307a1f393faeacc6162f91e": "b(t;u) := H(u-t)\\cdot \\exp\\left(-\\int_t^u r(v)\\,dv\\right)", "3c905d8dc9ecb4ae62e37af54e14f0b0": "\\sum_{n=1}^{\\infty}a_n e^{-\\lambda_n s},", "3c908b0c8f854896dfe8f4de498ed094": "g_\\xi = g_{ij}(x,\\xi)(dx^i\\otimes dx^j)|_x", "3c90986f9dcdb6947b13994a0a218f3c": " \\hbar = c = 1 ", "3c90bdde94ba3f7d41f665e21faa52ff": "R(S^o)=1-\\sum_{i \\in S^o}{f^o_i}", "3c91739faf7a8fdba6abb56e6ab7e7fa": "\\mbox{hyp} f = \\{ (x, \\mu) \\, : \\, x \\in \\mathbb{R}^n,\\, \\mu \\in \\mathbb{R},\\, \\mu \\le f(x) \\} \\subseteq \\mathbb{R}^{n+1}", "3c918a32ea888165e9d8b3b7cfd38316": "\\dot{\\rho} + \\rho~\\boldsymbol{\\nabla} \\cdot \\mathbf{v} = 0", "3c91ebecaf3b279b5ac339356e535288": "\\frac{}{\\Gamma_1, x:\\alpha, \\Gamma_2 \\vdash x:\\alpha}", "3c922edcb10863c6aee3da3aaa75fa01": "\\mu'_{02} = \\mu_{02} / \\mu_{00} = M_{02}/M_{00} - \\bar{y}^2", "3c92d868b89e9a494d65e738dc3b4573": "\nE_{\\mathrm{dip-dip}} = \\frac{1}{R^{3}_{AB}}\\left[ \\boldsymbol{\\mu}^A\\cdot\\boldsymbol{\\mu}^B - 3 (\\boldsymbol{\\mu}^A\\cdot \\hat{\\mathbf{R}}_{AB}) (\\hat{\\mathbf{R}}_{AB}\\cdot \\boldsymbol{\\mu}^B) \\right].\n", "3c933fe4b8737e12a01b9fb305dd160c": "c_{-m}=c_m", "3c935acac0d0558b81de0ca57b0dd142": "2^{O(k)}\\cdot V^\\omega \\in O(V^\\omega)", "3c93d65fea36d1b11ea8ef00f7b1fd41": "C^\\circ = \\{y \\in Y : \\sup\\{\\langle x,y \\rangle : x \\in C \\} \\le 1\\}", "3c93ee29e04ff5b78ad032219204eb6f": "p_i(t)>0\\,", "3c941be493f690f0f49fc3662ff2632d": "\n \\alpha := \\frac{s_0 - \\tau_y}{d}; \\quad\n \\beta := \\frac{\\tau_s - \\tau_y}{\\alpha}; \\quad\n \\varphi := \\exp(\\beta) - 1\n", "3c94427fd6e396cee057356a241286a9": "{}_pF_q(a_1,\\dots,a_j+1,\\dots,a_p;b_1,\\dots,b_q;z),", "3c94b35de9b843b0c4b340302f49651d": "X\\mapsto Xf(p)", "3c94be176a5b7e315c03486babd187aa": " P_g=n m v^2 ", "3c94ca1a7cc6966336d3561652904525": "p(-\\infty < X < x) \\le u", "3c94d884933477acdc14fc70da4b987a": "b=1", "3c94eabdc4f465ebd1a4688078ad6773": "\\pm90^\\circ", "3c953ac81bdd026ec549ca18b27fbe76": "A = \\frac{7}{4}a^2 \\cot \\frac{\\pi}{7} \\simeq 3.633912444 a^2.", "3c9578f510e30ecd9b45e72f32fb2c71": " (1-p)(r-1)\\, ", "3c95b8925c71216707bf81512ebdaadb": "\\textbf{f}_{dyn}", "3c95e4ef75c15fcdff327f2c76336c9e": "\\mathbf{Z}(s)=\\mathcal{L}[\\mathbf{z}](s)", "3c95f09794aec6eaab34edcb9926255a": "\\mathcal{E} = -N{d\\Phi_B \\over dt}\\quad", "3c963d255c8961c79fc52d8a24acdfb6": "\\varepsilon_{\\kappa\\lambda\\mu\\nu}", "3c9651f119e37369fd7952d065854d1d": "H_\\nu", "3c9659c96b2ac256fa26faee3363cdb7": "0 \\to 1 \\to \\infty \\to -1 \\to 0.", "3c96de1ae5f1d8757a959a5ab6d2d71f": "\\mu'_n=\\int_{-\\infty}^\\infty (x - c)^n\\,f(x)\\,dx.\\,\\!", "3c975ee1d55dd95c3da5ffec62d008a5": "y''-(\\alpha+\\beta)\\ y'\n+\\alpha\\,\\beta\\ y=f(t),\\quad\ny(t_0)=y_0,\\quad y'(t_0)=y_1", "3c97e94775bbb36b67e6f11a4ffffdae": "T^{-1}(|\\Pi_{i \\in I}A_i|)", "3c97fb63407ad16036491fb79ae4bf5a": "b=100%", "3c982b4a9aafe14337fd03eb6eff2f14": "\n \\boldsymbol{\\psi}(\\mathbf{x}) = \\cfrac{\\mathbf{q}(\\mathbf{x})}{T} ~;~~ r = \\cfrac{s}{T}\n ", "3c9859584e0205c9b59b16c4b9f1afb0": " \\mathit T = \\mathit g + \\frac{D} {\\mathit P} \n", "3c98869040f1643bef10aa9eb01c1388": "\\left(\\frac{1}{(i\\omega)^2-\\xi_1^2}-\\frac{1}{(i\\omega)^2-\\xi_2^2}\\right)^2", "3c98dfec479fc3dad3c41d4df0095dcc": "X \\sim \\textrm{Gumbel}(\\mu,\\,\\sigma)", "3c98fb0e785d11f96b8f81d50b55ac12": " BS=\\frac{1}{N}\\sum\\limits _{k=1}^{K}{n_{k}(\\mathbf{f_{k}}-\\mathbf{\\bar{o}}_{\\mathbf{k}})}^{2}-\\frac{1}{N}\\sum\\limits _{k=1}^{K}{n_{k}(\\mathbf{\\bar{o}_{k}}-\\bar{\\mathbf{o}})}^{2}+\\mathbf{\\bar{o}}\\left({1-\\mathbf{\\bar{o}}}\\right)", "3c9942247e9ae2a23b76901f0586bef1": "a_n = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(y)e^{-iny}\\,dy.", "3c9983e8f2e79f1ef9b09293b5286c10": "d={2a_{1} \\times b_{1} \\over a_{1} + b_{1}+c_{1}}", "3c999114e14cd95990d4db76e03f2bc1": "\\bar{R}=", "3c99b36f7ce669148df427ad427d6b44": "\\operatorname{ad}_x (y) = [x,y]", "3c99b5c5cd63537e919a8cc311ec7c60": " \\pi_{k\\neq 1}(U(1))=0", "3c99e99067486ed53fddaf3be5eb9562": "\\zeta_4\\;", "3c99fe4d0a467553e33f470ba4db69d4": "\\|x\\|> 0", "3c9a5975942f9130b704855424496d17": "\\sigma_{yy} - \\sigma_{yz} + \\sigma_{xy}", "3c9a65aceb61cc302a60c13b11b2f72d": "\n\\frac{1}{\\sqrt{\\lambda}} = 0.8686 \\ln[\\frac{0.4587Re}{(S-0.31)^{\\frac{S}{(S+0.9633)}}}]\n", "3c9a7e12690d886403fe4cc8702155e1": "g \\approx 360 \\times 0.381966 \\approx 137.508^\\circ,\\,", "3c9a98b1ca60e634f5425ec946632f05": "H_1(x)=2x\\,", "3c9ad19c80b36ecd215b30ca1cc6a93c": "\\operatorname{corr}(x,y;w) = {\\operatorname{cov}(x,y;w) \\over \\sqrt{\\operatorname{cov}(x,x;w) \\operatorname{cov}(y,y;w)}}.", "3c9b873ad8c62bac54f63e0848b169d5": "ex - x \\in \\mathfrak{i}", "3c9b8fe50732b5a24efa44926635abd4": "U_\\mathrm{E}(\\mathbf r) = -W_{r_{\\rm ref} \\rightarrow r } = -\\int_{{r}_{\\rm ref}}^r q\\mathbf{E} \\cdot \\mathrm{d} \\mathbf{s}", "3c9c267a85fd411fe0dc10fd1288f08a": "\\gamma(\\emptyset) = 0", "3c9c42c7eba0625b797d0255f16470e0": "f'(a) = \\lim_{x\\rightarrow a}f'(x)", "3c9c560a1dd421ba6509972c3a9dabe5": "\\beta\\left(t\\right)=e^{\\int_0^t f\\left(s,s\\right)\\,ds}", "3c9c5f123144dc70a7ac19766e38a42e": "z^{-1} = \\frac{\\bar{z}}{|z|^2}", "3c9cb5059e7e6e1e0d2439223aa501e3": "P_{\\text{rg}}", "3c9ccd0b6146d6100ef1289726fb9b4e": " b^c \\cdot b^d = b^{c + d} \\!\\, ", "3c9cde6b5c3a037bbd6478dfbd67571c": "\\langle S \\mid R \\rangle = F_S / N.", "3c9d3ed0a095d035831e02950399697b": " \\sum_{n \\in \\mathbb{Z}^d} |\\psi(t,n)|^2 |n| \\approx D \\sqrt{t}~, ", "3c9d74740f9bbdcaaeac447e0c3d9212": " speed = {distance \\over time} ", "3c9d7634907a823f9ad9d9cb51cdc27a": " \\mathbf{a} \\cdot (\\mathbf{b} + \\mathbf{c}) = \\mathbf{a} \\cdot \\mathbf{b} + \\mathbf{a} \\cdot \\mathbf{c}. ", "3c9dc5a172417c2d36819366e70a18f3": "{a\\pi\\over 5}\\ {b\\pi\\over 5}\\ {c\\pi\\over 2}", "3c9dc940e682a0ccb22c4d905381b984": "p(H2) = \\pi_2", "3c9dd83edb89ada77b33839573c0dcc8": "=\\mathbf{w}_{n-1}+\\mathbf{g}(n)\\left[d(n)-\\mathbf{x}^{T}(n)\\mathbf{w}_{n-1}\\right]", "3c9df6c6a689ffa82eb90a71866f610c": "\\frac{y_n}{z_n}\\frac{Z}{Y}=\\frac{z/z_n}{y/y_n}", "3c9e51fa7b9afb4311bdbc78b8b78fbc": "(r, \\theta)", "3c9e97651e60132f38e89687a843121d": "\\displaystyle P_n(x;a,b,c;q)={}_3\\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q) ", "3c9e993c71d90f5ade41575339d68d4b": "B_{i_1} e^{\\lambda_i t}, ~ B_{i_2} t e^{\\lambda_i t}, ~ B_{i_3} t^2 e^{\\lambda_i t} ", "3c9ebc66fc73c60e8f73a471538961e8": " [\\Theta_0+\\Psi](\\eta_\\ast) = [\\hat{\\Theta}_0+\\Psi](\\eta_\\ast)\\mathcal{D}(\\mathit{k}). ", "3c9f420c6d3829670397162105ef86e2": " \\pi(N) \\approx \\frac{N}{log(N)} ", "3c9f594f81a1d8af0c3053509bac82bc": "f_z=\\frac{1}{2\\pi}\\sqrt{\\frac{L_m}{(L_1L_2-L_m^2)C_m}}", "3c9fafd49ed7ffe7762254187b5291be": "v = v_{Ar}\\cos(\\omega t-kx)+v_{Al}\\cos(\\omega t+kx). ", "3c9fc646b5d7693cf9ea37eb41f1398b": " \\sum_{i \\in I} q_i = 1", "3c9fd08f70c28d3e41cc14aa6473d1fa": "\\mu_n(E)\\to \\mu(E),~\\forall E\\in \\Sigma. ", "3ca07d8e69d2c1864d7cf392da6bd50f": "\\alpha >1", "3ca0c976111e3a8410a287d8575456f6": "(1) \\ \\Delta B_i = p_{ij} a_j, \\,", "3ca0d700e269c9204344880f2537e3e7": "\\chi(\\xi) = -\\frac{e\\varphi(\\xi)}{k_\\mathrm{B}T_\\mathrm{e}}", "3ca0ef10c91d05125fb8a3774241bb4c": "\\bar{\\varphi}", "3ca11bb4332a1b877fc1fc0fc068ce91": " \\{y\\}\\in\\Phi\\;\\;\\mathrm{ and }\\;\\; \\lambda_y\\left((X\\setminus \\pi^{-1}(\\{y\\})\\right)=0 \\qquad \\forall y\\in Y", "3ca1969717604d40b1ab96ef93664033": "E\\left(\\frac{1}{n}E^{-1}(a)\\right)^n=E(E^{-1}(a))=a", "3ca1cec8652aa16eacb4cfcb5c2672f9": "\\boldsymbol x=\\{g_{\\boldsymbol\\theta}(z_1),\\ldots,g_{\\boldsymbol\\theta}(z_m)\\}", "3ca1d16af9f6c0dba0ea5223f7a78173": "W_1(x)\\;R_2(x)\\;W_2(y)\\;C_2\\;R_1(z)\\;C_1", "3ca1f104907a0fba67f0ed4cf307966e": "T_N(x)=c_0 + c_1 x + c_2 x^2 + \\cdots + c_N x^N ", "3ca1f4d9d8c1bc898c1dad9d9d26cf82": " A \\subset 2^{(L^*)} ", "3ca22a7bb301957d8540931f98fb5fb7": "d_j S(t) = E[d_j S(t)] + d J_S(t) = h(S(t^-)) (\\int_z z \\eta(S(t^-),z) \\, dz) dt + d J_S(t).", "3ca2556246764cecae8c2868d91e186b": "n=\\sum_{i=0}^{l-1}n_ib^i \\text{ with } |n_i|k ; \\\\\n\\mbox{Does not exist} ,\n& \\mbox{if }\\nu\\le k .\\\\\n\\end{cases}", "3cbb804bdc60b1dc6e6ce422b0f67470": "R(x_i)=\\mu_i \\pm \\frac{r\\sigma_i}{\\sqrt{n}}", "3cbb83923019bfef4f754cf754748eff": "\\hat{\\bold{\\imath}}", "3cbb860238468a9dd4b86fae9852a47e": "\\frac{\\partial C}{\\partial S}", "3cbbb2011e5a6c750576aacb7476b38f": "CTF(u)=A(u)E(u)\\sin(\\chi(u))", "3cbbcc0685d588d6de7b7fc746e8ddc2": "\\mathbf{C}_1, \\mathbf{C}_2", "3cbc0c8193bc7c56d5b887cdd055503d": "\\Lambda(f)|_{\\Lambda^1(V)} = f : V=\\Lambda^1(V)\\rightarrow W=\\Lambda^1(W).", "3cbc11ac204ea18b8909f7c5dbcf789d": "(4)\\; \\Delta E=\\frac{(y_2-y_1)^3}{4 y_1 y_2}=\\frac{(2.94-0.5)^3}{4*2.94*0.5}=2.47\\;m", "3cbc1283d1959cd18d9964232303362a": " qe^{-q \\tau} \\Phi(d_1) - e^{-q \\tau} \\phi(d_1) \\frac{2(r-q) \\tau - d_2 \\sigma \\sqrt{\\tau}}{2\\tau \\sigma \\sqrt{\\tau}} \\, ", "3cbc1977e8293819e31ed27f4ba68fcc": "V(t_n) = V(t_0) \\times (1 + {\\rm CAGR})^n", "3cbc4b3e08942e985711d2756179b74c": "\\dot{\\rho}=-i[H,\\rho] + 2\\kappa\\left(a\\rho a^\\dagger -\\frac{1}{2}\\left(a^\\dagger a \\rho + \\rho a^\\dagger a\\right)\\right) + 2\\gamma\\left(\\sigma\\rho \n\\sigma^\\dagger -\\frac{1}{2}\\left(\\sigma^\\dagger \\sigma\\rho + \\rho \\sigma^\\dagger \\sigma\\right)\\right)", "3cbc9174140aaaebafaef398033b9355": "(1+x+x^2+x^3)^2=2(1+x^2+x^4+x^6)\\ ,", "3cbcdb040d521cdfa7b10d1e7f4b0172": "a_0 = \\frac{4 \\pi \\epsilon_0 \\hbar^2}{m_e e^2} = \\frac{m_\\text{P}}{m_e \\alpha} l_\\text{P}. ", "3cbce10558d4bb62f47b7855cf143218": "e^L", "3cbd758f14dc94e3b0be208b55e4e989": "Y \\times_X Y,", "3cbdc19079f09c8fc356f9ed643082e4": "V_\\max = k_2 [E]_0", "3cbe12523fdebf3abedb7dd4bee866b9": " \\textrm{mes} \\left\\{ x \\in \\mathbb{R} \\, \\mid \\, |P(x)| \\leq a \\right\\} \\leq 4 \\left(\\frac{a}{2 \\mathrm{LC}(p)}\\right)^{1/n}~, \\quad a > 0~.", "3cbe6fd8f03daf36cbe805380a89430c": "\nH=-h^2 \\Delta + V(x)\n", "3cbe809c6b4e04c4f0ea7a52284bc3f4": "\\frac{\\binom{n+1}{k-1}\\binom{n+1}{k}\\binom{n+1}{k+1}}{\\binom{n+1}{1}\\binom{n+1}{2}}", "3cbf2133b7b5bc8550ea9678f6eabf29": "\n\\Phi (\\rho) = \\begin{bmatrix} \\rho(F_1) \\cdot F_1 \\\\ \\vdots \\\\ \\rho(F_n) \\cdot F_n \\end{bmatrix}.\n", "3cbf87b36d6ff5cda2f5760422888926": "\\frac{\\operatorname df}{\\operatorname dt}=\\frac{\\partial f}{\\partial t} + \\frac{\\partial f}{\\partial x} \\frac{\\operatorname dx}{\\operatorname dt} + \\frac{\\partial f}{\\partial y} \\frac{\\operatorname dy}{\\operatorname dt}.", "3cbf8cd83126ce2d9da5f0e17db6e8cd": "\\text{Z-factor} = 1 - {3 (\\sigma_p + \\sigma_n) \\over | \\mu_p - \\mu_n |}.", "3cbfaa8bb4d286d99b11d4e8fa6ecadf": "\n\\frac{\\partial g_{ab}}{\\partial x^c} = \\left\\langle \\frac{\\partial^2 \\vec\\Psi}{ \\partial x^c \\, \\partial x^a} ; \\frac{\\partial \\vec\\Psi}{\\partial x^b} \\right\\rangle + \\left\\langle \\frac{\\partial \\vec\\Psi}{\\partial x^a} ; \\frac{\\partial^2 \\vec\\Psi}{ \\partial x^c \\, \\partial x^b} \\right\\rangle\n", "3cbfb62729efc1ae00958f6239c0566a": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{J} \\ ", "3cbfcfc699081c55270e56827f09418f": "b=K_b\\left(\\frac{Y/Y_n-Z/Z_n}{\\sqrt{Y/Y_n}}\\right)", "3cbfe2b4c1eb27922cc2d0bf0b13b6ad": "\\omega:", "3cbff7a028d4a8665e54ba6d9548c948": "\\{\\lambda_n(A)\\}_n", "3cbffe91a2eeff394fbd4edcd23b515b": "\\frac {A^2} {T} = LS \\,.", "3cc0321c13726da744f48264d77978bd": "\\varphi \\in C^{\\infty}_c (U)", "3cc0770d1852960d4e2bd42ad45f0c0d": "\\operatorname{ad}", "3cc07d45f8615f2deccd159e5bcc513a": "\\sigma = \\frac{\\sqrt{Ns_2-s_1^2} }{N}", "3cc097798113d1577a840d5b965cdbda": "\\lfloor x_{k+1} \\rfloor=\\lfloor \\sqrt n \\rfloor.", "3cc0faf2944cf1873d7d83fb7b197b64": "A=-TS", "3cc109cf04e21ec6282170ff40ca5fd5": " \\sin(hR) ", "3cc1b026d628e98021dfa19b65300e48": "\\nabla_n = \\mathbf{e}_x \\frac{\\partial}{\\partial x_n} + \\mathbf{e}_y\\frac{\\partial}{\\partial y_n} + \\mathbf{e}_z\\frac{\\partial}{\\partial z_n}\\,,\\quad \\nabla_n^2 = \\nabla_n\\cdot\\nabla_n = \\frac{\\partial^2}{{\\partial x_n}^2} + \\frac{\\partial^2}{{\\partial y_n}^2} + \\frac{\\partial^2}{{\\partial z_n}^2}", "3cc1c03b12822e43c8459217e30ca466": "b^2 = 4^2", "3cc1e75af8e9252331929b3bd20dd915": "\\mathcal S_n(\\mathcal A_{1\\cdots n}, \\mathcal C_{1\\cdots n})", "3cc2214683eb7c03ce4825f93cc3e0a4": "C_S = - \\frac{dI_L}{dV_S}", "3cc255a42109b04c47d5778ac719297b": "\\operatorname{Pr}(\\varnothing)= \\Vert P |\\psi\\rangle\\Vert^2", "3cc27cfa9abd97c2b3f0f141e84abd83": "\n\\begin{align}\\forall a\\in\\mathbb{R}, \\; P^{(L^*)}(X\\in[a;a+da]) &= \\int_{\\omega\\in\\{X\\in[a;a+da]\\}} \\frac{X(\\omega)}{E[X;P]}dP(\\omega) \\\\ &= \\frac{1}{E[X;P]}\\; a\\,P(X\\in[a;a+da]) \n\\end{align}", "3cc2cd4ea54460231578be0a1d311f56": "\\Omega = 4 \\arcsin \\left( \\sin {a \\over 2} \\sin {b \\over 2} \\right) ", "3cc327b38349e1d2ae5cd568375c5c0d": "V_g=V_d", "3cc3c37cf9ccab5d7e5a4f8a5395e633": "K L", "3cc3fbaeb3a60f77c028f7ef2b56df79": "\\scriptstyle \\leq10^{-19}", "3cc4050d5ceb1ed42c2abf7cd4af185c": "A=((1+\\sqrt{2})a)^2-a^2=2(1+\\sqrt{2})a^2", "3cc41ded57fa149c4b1c3117680a1515": "\\sqrt{35\\over{8}}\\cos(3\\theta)\\cos^3(\\phi)", "3cc45bfcc552f403c196ca4c075e3483": "\\frac{\\partial y}{\\partial u}", "3cc49940b2ea1a26347097fa7d3ecbee": "\\color{LimeGreen}\\text{LimeGreen}", "3cc4d25cede82efab5c4c2765a7603e6": " u:\\mathbb{R}\\times[0,T]\\to\\mathbb{R}", "3cc5a10a47cadb15d46a3342a6db1fbf": "\\,\\sum_{w \\in V} f(u,w) = 0 \\text{ for all } u \\neq s, t", "3cc5a131e96b8fd773c6f0a91fef6dd3": "\\phi(a^n) = \\sum_{\\pi\\in\\text{NC}(n)} \\prod_{j} k_j^{N_j(\\pi)}", "3cc5a1b29d6ddde3d1d00cb0876439ed": "\\mbox{SAIFI} = \\frac{\\sum{\\lambda_i N_i}}{\\sum{N_i}}", "3cc5aa4112942559d2704dee52435129": "\n Z_{\\text{dl}}(\\omega) =\\frac{1}{\\text{i}\\,\\omega\\, C_{\\text{dl}}}\n", "3cc5e1f8eb3e39895c385f4e4ce1d42c": "d(g(\\mathbf{X}), g(\\mathbf{Y}))^2 = d(\\mathbf{X}, \\mathbf{Y})^2.", "3cc682b5f7eb105e270e9fb917543327": "L=L_m^{(d_m)}L_{m-1}^{(d_{m-1})}\\ldots L_1^{(d_1)}", "3cc6915887b84c22d5c8818d562a8267": " \\hat y = X \\hat \\beta", "3cc7168064983a3c6d3d0336984d0c44": "F : \\mathrm{Hom}_{\\mathbf{Vect}}(T,U) \\rightarrow \\mathrm{Hom}_{\\mathbf{Vect}}(F(T),F(U)),", "3cc77f630336927c4e76fe17b1455b22": " \\gamma_n = \\lim_{m \\rightarrow \\infty}\n{\\left[\\left(\\sum_{k = 1}^m \\frac{(\\log k)^n}{k}\\right) - \\frac{(\\log m)^{n+1}}{n+1}\\right]}.", "3cc7ce75e38058176ab52a2e6c7eaf0c": "V\\subseteq\\overline{\\underset{=}{A}(\\frac{2kU}{r})}", "3cc7cfdb63ce21351444d3500606166e": "|A_i|", "3cc7d17be0ed14bb7a22affa27e04d77": "\\hat{H} |\\psi\\rangle = i \\hbar \\partial / \\partial t |\\psi\\rangle ", "3cc7f9e1499ab348cae2e584e136ff3f": "E(e) = \\frac {(5% - 7%)} {(1 + 7%)} = -0.018692 = -1.87%", "3cc86f4a90035f71773c4aba1b4cfb4a": "0 \\notin [l_n, u_n]", "3cc87ab81814d43d5dde79c8c5771a55": "c_p=[E(\\mathbb{Q}_p):E^0(\\mathbb{Q}_p)],", "3cc8919b961ded5b0a0df6ef4228ad8d": "\\scriptstyle V_\\mathrm i + V_\\mathrm r = V_\\mathrm o", "3cc8db411dfecd5384ae277987fdf1e9": "(\\lambda \\cup \\left\\{c\\right\\}) / (\\mu \\cup \\left\\{d\\right\\})", "3cc8fdf2a68abf5fe5116749538b1e72": "K(f) = F_\\infty^C.", "3cc94ad1b63e5ca69a0f4a9fd11de0bc": "\\,\\sigma_0 \\in \\Gamma", "3cc9761ba035995ea0d71edfa40e804d": "1/(k_B T)\\,", "3cc99964e714a2a313e8b032c8a8c2eb": "\nS(t)=2(n-1)\\prod_{k=1}^{n-1} \\Bigl(1+\\frac{\\lambda}{2k}\\Bigr)\n\\, t^{\\lambda},\n", "3cc9d00f16e88224aca43258c1bad55f": "f(c^+) = \\lim_{x \\to c}f(x)", "3cc9d712f7eb595e592fe907754a802e": "\\{N\\ge n\\}=\\{X_i=-2^{i} \\text{ for } i=1,\\ldots,n-1\\}", "3cca1b77c9d977678486c18e38497873": "s_v = \\frac{2 | A \\cdot B |}{| A |^2 + | B |^2} ", "3cca4996a9c2f7ad678d0c3cfc3f5186": "Kf_1(t)+Kf_2(t)\\,", "3ccacdf2156cf7bf7ae7470ab7c975f1": "f=\\theta^2/2", "3ccadcbb8f6f1ac3b9a5f106d7e49015": "\\scriptstyle a_2=5", "3ccb0938bf17dc277a65b61f62d0af56": "\\alpha_{15}", "3ccb30881c4fee0fc6a5d0ce261ef1b0": "\\mu_m(\\{\\sigma : \\sigma[0] = s \\land (\\forall i \\geq 0)\\sigma[i] \\models_K f\\}) \\sim \\lambda", "3ccb34d326382c78d24a3bf66046f8ba": "\\delta_x = \\frac {F x^2} {6 E I} (3L - x)", "3ccb428444c9b99516e8647e68dac7ef": "A={\\Bbb C}[t,t^{-1}]", "3ccb4fb73f4b3f135f9549d1140f2289": "\\scriptstyle \\gcd(n_1, n_2) \\,=\\, 1", "3ccb7608b23413d58ed7eeb1248ee285": "\\lim_{x \\to 1} \\frac{x^2-1}{x-1} = 1+1 = 2", "3ccba3267c4a778df5c44de8e6e54004": " \\sum_{R\\in G}^{6} \\; \\Gamma(R)_{11}^*\\;\\Gamma(R)_{11} = 1^2+1^2+\\left(-\\tfrac{1}{2}\\right)^2+\\left(-\\tfrac{1}{2}\\right)^2 +\\left(-\\tfrac{1}{2}\\right)^2 +\\left(-\\tfrac{1}{2}\\right)^2\n= 3 .\n", "3ccbc2929e919061e788d7c64cfdd5b2": "m_1=2,147,483,563", "3ccc0477256ed5b03700f97c0d99c6ad": "E_n^{(1)}=-\\frac{1}{2\\pi}\\int e^{-i n \\phi} \\cos \\phi e^{i n \\phi}=-\\frac{1}{2\\pi} \\int \\cos \\phi = 0", "3ccca3faa0edffae4eb5c2a3b1132817": "\n C_a=C_l+C_p=1-C_s=\\frac{\\lambda_1 + \\lambda_2 - 2\\lambda_3}{\\lambda_1 + \\lambda_2 + \\lambda_3}\n ", "3ccca9d53fa392b65af2fee9e9925999": "\\otimes_i", "3cccccaacf8ae8527f24ad72b4dda3d0": "1 \\rightarrow {\\rm Aut}_0(X) \\rightarrow {\\rm Aut}(X) \\rightarrow {\\rm MCG}(X) \\rightarrow 1.", "3ccd14c3d62d27c3b2c592d0a0689a3f": " \n \\partial_{ z } = \\frac{ 1 }{ 2 } ( \\partial_{ x_1 } - i \\partial_{ x_2 } ), \\quad \n \\partial_{ \\bar z } = \\frac{ 1 }{ 2 } ( \\partial_{ x_1 } + i \\partial_{ x_2 } ). \n", "3ccd2343f355ad19cd5e699f00e6de47": "\\oint_{c_i} \\omega = a_i,", "3ccd634066b03e304450708bb555c514": "-\\frac{d H_\\alpha}{d\\alpha}\n= -\\frac{1}{(1-\\alpha)^2} \\sum_{i=1}^n z_i \\log(z_i / p_i),", "3ccd839325d4afef8cc8cde17bcd4f14": "\\frac{n}{i} x[n] \\!", "3ccd86bd9ef8655e36f877225983256a": "\\frac{\\mbox{Net Credit Sales}}{\\mbox{Average Net Receivables}}", "3ccd8d449a429ee31dc41c1473d69a05": "I(x) / V(x) = Y_0", "3ccdb2b3a72330f2b7f293a03bcb0804": "\n\\begin{align}\n6 \\times 2 & = 12 \\text{ (2 carry 1) } \\\\\n1 \\times 2 + 6 + 1 & = 9 \\\\\n3 \\times 2 + 1 & = 7 \\\\\n0 \\times 2 + 3 & = 3 \\\\\n0 \\times 2 + 0 & = 0 \\\\[10pt]\n316 \\times 12 & = 3,792\n\\end{align}\n", "3ccde38da56fc01d6c67f839210e664e": "\\displaystyle{(Pf)_{\\overline{z}} = f,\\,\\,\\, (Pf)_z = Tf.}", "3cce6af8085235557a79ba8fa0144857": "\\ M_{pitch_{max}} > D_{pitch} \\times F_{forward} ", "3cce7e7dfd5b7b8e28c547c939bdf0ae": "\\textstyle W(y|x, s)", "3cce81a5bebe9aae67f20752f4c1768c": "\n\\frac{\\sqrt{2}}{Y}\\, dt = \\frac{d\\varphi_{1}}{\\sqrt{E \\chi_{1} - \\omega_{1} + \\gamma_{1}}} = \n\\frac{d\\varphi_{2}}{\\sqrt{E \\chi_{2} - \\omega_{2} + \\gamma_{2}}} = \\cdots =\n\\frac{d\\varphi_{s}}{\\sqrt{E \\chi_{s} - \\omega_{s} + \\gamma_{s}}} \n", "3ccec6087c3ae559547624598bff7bb2": "U^\\nu", "3ccee44e8c8ab9017f5503287c2da92c": "\\displaystyle \\frac{\\partial \\theta}{\\partial t}= \\frac{\\partial}{\\partial z} \n\\left[ K(\\psi) \\left (\\frac{\\partial \\psi}{\\partial z} + 1 \\right) \\right]\\ \n", "3ccf1ca711b3173aa911a1d6d1ac3cdf": "P,Q,R,S,T,U", "3ccf2c750382481e1a5604ed228dc645": " p_{\\mu} p^{\\mu} = - m^2 \\,", "3ccf54371a7362f51fd0aced8c1b911c": "f \\in o(k)", "3ccf9ebc33e06923789a0b5265c9163e": "\n\\lim_{\\mathrm{Re}(s) \\rightarrow -\\infty} \\operatorname{Li}_s(e^\\mu) = \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} \\qquad (-\\pi < \\mathrm{Im}(\\mu) < \\pi)\n", "3ccfa70e92d13dc1fb59b7e353594a44": " \\mathbf{T} ", "3cd0252fb0375c2130d30717a487ca90": " \\delta W = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}}\\right)\\delta q = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\cdot \\frac{\\partial (\\vec{\\omega}\\times(\\mathbf{R}_i-\\mathbf{R}) + \\mathbf{V})}{\\partial \\dot{q}}\\right)\\delta q. ", "3cd06abcdc756890b8357ecf88833074": "\n \\begin{align}\nm \\frac{\\partial^{2}w}{\\partial t^{2}} & = \\frac{\\partial}{\\partial x}\\left[ \\kappa AG \\left(\\frac{\\partial w}{\\partial x}-\\varphi\\right)\\right] + q(x,t) \\\\\nJ \\frac{\\partial^{2}\\varphi}{\\partial t^{2}} & = N(x,t)~\\frac{\\partial w}{\\partial x} + \\frac{\\partial}{\\partial x}\\left(EI\\frac{\\partial \\varphi}{\\partial x}\\right)+\\kappa AG\\left(\\frac{\\partial w}{\\partial x}-\\varphi\\right)\n \\end{align}\n", "3cd07124894a9bc46eadf6b9c18f00cd": "\\gcd({S_i,N})>1", "3cd10d1378dfe06fee0aee3416521d74": "\\phi \\colon X \\to M", "3cd17b5b2296f2acc43bbed38b18a55e": "\\sin^2 x = 1 - \\cos^2 x,", "3cd188910b4305c55516cc485ee7899f": " \\mathcal{F}(T)", "3cd1ab059ebf71a2c63986d8c12f14e0": "\\int \\frac{p_j g}{m}\\frac{\\partial f}{\\partial x_j}\\,d^3p=\\frac{1}{m}\\frac{\\partial}{\\partial x_j}(n\\langle g p_j \\rangle)", "3cd1b0c359b7fb6efbc0bcb3e74db598": "S = {4^{1.17}\\, 4^{1.85}\\,Q^{1.85}\\over \\pi^{1.85}\\,k^{1.85}\\, C^{1.85}\\, d^{1.17}\\, d^{3.70}}\n= {4^{3.02}\\,Q^{1.85}\\over \\pi^{1.85}\\,k^{1.85}\\, C^{1.85}\\, d^{4.87}}\n= { 4^{3.02} \\over \\pi^{1.85}\\,k^{1.85}} {Q^{1.85}\\over C^{1.85}\\, d^{4.87}}\n= { 7.916 \\over k^{1.85}} {Q^{1.85}\\over C^{1.85}\\, d^{4.87}}\n", "3cd1b4afba931e1113e83f0f7c5b0bc2": " d p\\, d q =\\frac{\\partial(p,q)}{\\partial(\\theta,\\varphi)} d \\theta \\,d \\varphi = \\left ( \\frac{\\partial p}{\\partial \\theta} \\frac{\\partial q}{\\partial \\varphi} - \\frac{\\partial p}{\\partial \\varphi} \\frac{\\partial q}{\\partial \\theta}\\right) d \\theta\\, d \\varphi \\ ", "3cd1b58deb6140ca966483752ddf2418": " S = n^{ -3 / 2 } \\sqrt{ \\frac{ n - 4 }{ n - 2 } } + O( n^{ -5 / 2 } ) ", "3cd1c632487020c541b31d9bf4cb38b1": " \\mathcal{L}_X(Y) = [X, Y] ", "3cd210c34a7d9ca3598d22e8352ea952": "\\ \\ell", "3cd27887a0e65d935935caa3b0770092": "\\scriptstyle p_A(t)", "3cd2a82e3c192c54be6d46ddf27fa115": "\\mathrm{Fo} = \\frac{\\alpha t}{L^2}", "3cd2d57b8ffc8577776eb4c7b43fa263": "Br_{2(aq)} + 2e^- \\leftrightarrow 2Br^{-}_{(aq)} ", "3cd2fb8ba6cf45793dcd78a3e3a27d51": "m_{UT}=\\frac{1}{4}\\Sigma^4_{i=1}{m'}_i", "3cd32d3341f7caa11c0ac3b68942a294": "C_3 = \\left[ \\begin{array}{rrr}\n1 & 0 & 0 \\\\ \\\\\n0 & 1 & 0 \\\\ \\\\\n0 & 0 & 1 \n\\end{array} \\right] - \\frac{1}{3}\\left[ \\begin{array}{rrr}\n1 & 1 & 1 \\\\ \\\\\n1 & 1 & 1 \\\\ \\\\\n1 & 1 & 1 \n\\end{array} \\right]\n = \\left[ \\begin{array}{rrr}\n\\frac{2}{3} & -\\frac{1}{3} & -\\frac{1}{3} \\\\ \\\\\n-\\frac{1}{3} & \\frac{2}{3} & -\\frac{1}{3} \\\\ \\\\\n-\\frac{1}{3} & -\\frac{1}{3} & \\frac{2}{3} \n\\end{array} \\right] \n", "3cd37d76b427955e5b78f7aa973d7cb3": "x > \\mu-\\sigma/\\xi", "3cd38ab30e1e7002d239dd1a75a6dfa8": "epsilon", "3cd3a6540846d24ae4b90abdf9e86988": "\\gamma\\,_0", "3cd3e79c5b4fb76d9e763cbbd53806be": "ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\\,", "3cd47800ec0a18cad0663c83b442091f": "\\displaystyle h(t)", "3cd4b6943dc83952c0d445b7d17f80cb": "(i+j)", "3cd50f296fb574258c78aac2dc53a9f2": "K = \\tfrac{1}{2} pq \\cdot \\sin \\theta,", "3cd5241bf9937c17d3d4ba286677a2c0": "\\forall x (\\phi \\rightarrow \\psi)", "3cd5738b6fb28020fc0f91babb601976": "h\\nu = g_\\mathrm{e} \\mu_B B_\\mathrm{eff} = g_\\mathrm{e} \\mu_B B_0 (1 - \\sigma) \\,", "3cd6c6e0ec34f309debc630d8094eadb": "\\dot{\\mathbf{x}}(t) = A(t) \\mathbf{x}(t) + B(t) \\mathbf{u}(t) \\, ", "3cd6e0971ef24c60e02723b2bcf6c26b": " a_i (t+1) = a_i(t) + \\nu \\varepsilon \\frac {u \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} {\\sum_{i=1}^N u^2 \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} ", "3cd7422e2278e21a714d0a2e489748ed": "\\scriptstyle T'", "3cd7cf2e5c61d3500adc7415be8c8c34": "=R \\frac {d \\omega}{dt}\\ \\hat u_\\theta - \\omega^2 R \\ \\hat u_R \\ . ", "3cd81f2d7312857fb4a45c00af15fa69": "\\leq\\kappa<2^{\\kappa}\\,", "3cd839624b3d71549911804d4eeb2517": "v_f^2=v_i^2+2ad", "3cd855c161ca1f3eb0d001e9e629bddc": "K\\to\\infty", "3cd8a39984975386f0004ec166439df1": "\\left(a+b\\right)^p = a^p + b^p", "3cd8abe9b1fad834bc5e0c092dce9c60": "\\phi_0:= -F_{ab}l^a m^b \\,,\\quad \\phi_1:= -\\frac{1}{2} F_{ab}\\big(l^an^a-m^a\\bar{m}^b \\big)\\,, \\quad \\phi_2 := F_{ab} n^a \\bar{m}^b\\,.", "3cd8be6c009c176bfc22bc7ebcb98268": " z = \\theta, \\ ", "3cd8ca932263aa8d01017b58135c80dc": "\\mathbf{D}=\\varepsilon_0\\mathbf{E}+\\mathbf{P}\\!", "3cd8d9202e5755182ef10ccf0c05f0a1": "f_{t}^{\\mathbf{z}}", "3cd91a6710f61e4db5006da9438640b3": "g(x)=\\int K(x,y) f(y)\\,dy.", "3cd93d1eeda97e57ccc7b3c5e035ab04": "t=t_0, \\, r=r_0, \\, \\theta = 0", "3cd96db78612a2ed4b07bd5064e8afb4": "Pr(i|j)=P_{i,j}", "3cd9ad7682fd0e30aac5c039d09023ed": "(N,h,x,e^{\\prime},k^{\\prime},s)\\,", "3cd9f239be4fea575b0e29c49253b9e0": "(a \\land b) \\to c = a \\to (b \\to c)", "3cda761ef079df563246fb4adba2d5bf": "A_4, S_4, A_5", "3cdab68ebf92363a2144989ddcc120cd": "\\sigma_i=\\alpha_i/\\beta_i", "3cdad45c175c4057603ddc43b31891cc": "x(t) \\ ", "3cdaf54951dd4b62ef9c6c5ed019b521": "\n\\sum_{i} p'_i = \\sum_{i} m_i v'_i \n= \\sum_{i} m_i (v_i - V_c) \n= \\sum_{i} m_i v_i - \\sum_i m_i \\frac{\\sum_j m_j v_i}{\\sum_j m_j} \n= \\sum_i m_i v_i - \\sum_j m_j v_j\n= 0\n", "3cdb254573fb1a8c805b4d217177b4fd": "E_{kl}", "3cdb5f69b22d32f8de2ecc37dd81b33d": "L_{pp}", "3cdbaa08140d3395ddfaf0da800a4247": "\\Delta_Z = Z_{11} Z_{22} - Z_{12} Z_{21} \\,", "3cdbbd76001396dffa7543b67e39dcd9": "\\mu,\\nu,\\phi", "3cdbdc05d72e68ca621c4aa595d099cb": "\\mathbf{u} \\oplus \\mathbf{v}=\\mathrm{gyr}[\\mathbf{u},\\mathbf{v}](\\mathbf{v} \\oplus \\mathbf{u})\n", "3cdc1f4406a6bbd62779894ffd902ba7": "\\tfrac{17}{29}=\\tfrac{1}{2}+\\tfrac{1}{12}+\\tfrac{1}{348}.", "3cdcd93227d799cf637b8428734e8d5d": "r^{-1}=\\limsup_{n\\to\\infty} \\left|a_n\\right|^{\\frac{1}{n}}", "3cdd017bafd6907ebb595b28cccb1cf3": "|\\arg z|<\\tfrac 3 2 \\pi.", "3cdd280125880508e97c5b6b16332617": "d^I_{p,q} + d^{II}_{p,q} :\nT_n(C_{\\bull,\\bull})^I_p / T_n(C_{\\bull,\\bull})^I_{p+1} =\nC_{p,q} \\rightarrow\nT_{n-1}(C_{\\bull,\\bull})^I_p / T_{n-1}(C_{\\bull,\\bull})^I_{p+1} =\nC_{p,q-1}", "3cdd6d4bb5983959a569ba6f22df9c89": "D:\\Gamma^\\infty (E)\\rightarrow F", "3cddeff48358756be51811c39838413a": "A^j_b", "3cde3984e3f767da92d4eae8e11f3aa9": "\\varepsilon_{ijk} \\varepsilon_{k\\ell m}=\\delta_{i\\ell}\\delta_{jm}-\\delta_{im}\\delta_{\\ell j}", "3cdef3674842db52be4d27b7d99d5ff0": "\\begin{align}\n\\sum_{s,t \\in G}\\langle F(s^{-1}t) h(t), h(s) \\rangle \n& =\\sum_{s,t \\in G}\\langle P \\Phi (s^{-1}t) h(t), h(s) \\rangle \\\\ \n{} & =\\sum_{s,t \\in G}\\langle \\Phi (t) h(t), \\Phi(s)h(s) \\rangle \\\\ \n{} & = \\left\\langle \\sum_{t \\in G} \\Phi (t) h(t), \\sum_{s \\in G} \\Phi(s)h(s) \\right\\rangle \\\\ \n{} & \\geq 0\n\\end{align}\n", "3cdf0c652b3a5b3db182aafbe47690f6": "\\scriptstyle{\\hat{\\epsilon}E_0}", "3cdf10340517077561fc6c33c8e24860": "S(z)=\\sqrt{\\frac{\\pi}{2}} \\frac{1+i}{4} \\left[ \\operatorname{erf}\\left(\\frac{1+i}{\\sqrt{2}}z\\right) -i \\operatorname{erf}\\left(\\frac{1-i}{\\sqrt{2}}z\\right) \\right],", "3cdf3541bc61cfd02fd986187b358fd7": "e\\left(\\frac ne\\right)^n \\leq n! \\leq e\\left(\\frac{n+1}e\\right)^{n+1}.", "3cdf47ada1f6f4e4cd31aa81653ab9a1": "O(R^2/\\gamma^2)", "3cdf502e4e0610e3833094f54b89331b": "\\int_0^\\infty \\sqrt{x}\\,e^{-x}\\,dx = \\frac{1}{2}\\sqrt \\pi", "3cdf699c5f364b87fead38f502c0dde3": "Z_\\textrm{microstrip} = \\frac{Z_{0}}{2 \\pi \\sqrt{2 (1 + \\varepsilon_{r})}} \\mathrm{ln}\\left( 1 + \\frac{4 h}{w_\\textrm{eff}} \\left( \\frac{14 + \\frac{8}{\\varepsilon_{r}}}{11} \\frac{4 h}{w_\\textrm{eff}} + \\sqrt{\\left( \\frac{14 + \\frac{8}{\\varepsilon_{r}}}{11} \\frac{4 h}{w_\\textrm{eff}}\\right)^{2} + \\pi^{2} \\frac{1 + \\frac{1}{\\varepsilon_{r}}}{2}}\\right)\\right),", "3cdff7646c566ef008c26abc107ff79c": "Q(x)<0", "3ce0055f262528e5d26ea0ec65a5a156": "g < \\frac{a + cb}{2cb}", "3ce02d62053556f9fe83cd43a0066831": "f(\\bigwedge A) = \\bigwedge\\{f(a)\\mid a\\in A\\}", "3ce044db75da30951c4a601890fe6bdc": "E_\\text{P}", "3ce0d6de95b31190da7d392396d1643b": "V(S) = \\{(t_1,\\dots,t_n)|\\forall p\\in S, p(t_1,\\dots,t_n) = 0\\}.\\,", "3ce10afcab78e9980da6ae41bc68b13e": "\nk = \\sqrt{\\frac{u_{2} - u_{1}}{u_{3} - u_{1}}}\n", "3ce1304acf9083d42a5a11092292d99b": "V_2 = V_1 \\pm \\ G \\cdot\\ \\theta\\ ", "3ce14b404482fa603d0ce611a38229a9": "P_{runs} - A_{runs}", "3ce1e0b185a7f2c47abeac6b82bc1863": "\n\\begin{array}{l}\n KZ_{m,k=2}[X(t)]=\\sum\\limits_{s=-(m-1)/2}^{(m-1)/2}{KZ_{m,k=1}[X(t+s)]\\times\\frac{1}{m}} \\\\\n=\\sum\\limits_{s=-2(m-1)/2}^{2(m-1)/2} {X(t+s) \\times {a_s^{m,k=2}}}\n\\end{array}\n", "3ce22e3fd3bffd5a63a27c0ffb5c6415": "A^{\\alpha}{}_{\\beta\\gamma} = B^{\\alpha}{}_{\\beta\\gamma} ", "3ce235cb5b486400213d2ded895c3b73": "L = 2\\alpha_{0}l ", "3ce2377cf23a24fe7b39e62fa28e551d": "\\operatorname{MSE}=\\frac{1}{n}\\sum_{i=1}^n(\\hat{Y_i} - Y_i)^2.", "3ce25c2ba6815286010aa04fd0dabeba": "1\\leq i < k", "3ce278f5f6dbfa2bf0fe4f30fb75a6e2": "x\\,R\\,y\\,S\\,z", "3ce29a4030cc387e2bb32def6d93bd9b": "v(x,t)", "3ce2f5fdc05a8af24624966ab62cbabd": "\\begin{align}\n36 x &+ 120 x^2 + 180 x^3 + 170x^4+114x^5 + 56x^6 +21 x^7 + 6x^8 + x^9 \\\\\n&+ 36y +84 y^2 + 75 y^3 +35 y^4 + 9y^5+y^6 \\\\\n&+ 168xy + 240x^2y +170x^3y +70 x^4y + 12x^5 y \\\\\n&+ 171xy^2+105 x^2y^2 + 30x^3y^2 \\\\\n&+ 65xy^3 +15x^2y^3 \\\\\n&+10xy^4,\n\\end{align}", "3ce2fae6a719cdb48d55efbfe921f371": "y'=f_3(x)y^3+f_2(x)y^2+f_1(x)y+f_0(x) \\, ", "3ce3aedf769ff14a7f3d441d2010c368": " X + Y \\longrightarrow XY", "3ce43100c4c051c49d9408965c2bd0d6": " Q_B ", "3ce44732f9477b495834347fd9412c9f": "|z^2 + b^2| \\le 2b^2 < 2a|z|", "3ce4eb62de7c3f908430c93c507bc3cb": " \\left(\\frac{\\Delta S}{\\Delta t} \\right)_{i=2}=I-O=q^{ss}-q^{trans}_{i=2}= 10.0 \\text { ft}^3/\\text {s}-7.16 \\text{ ft}^3/\\text{s} = 2.84\\text{ ft}^3/\\text{s} ", "3ce4fc9d1a5a8779c686520b41c691e4": " \\frac{1}{\\sqrt{2}} (1,1), \\frac{1}{\\sqrt{2}} (1,-1) ", "3ce51fad377b4db5cfc5042d0a766bc3": "\n\\begin{align}\n& R_\\lambda = (\\lambda - A)^{-1}, \\\\\n& A = \\lambda - R_\\lambda^{-1}.\n\\end{align}\n", "3ce5547b5dfeea3123bee3f50b8e1083": "\n\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}/S = \\sqrt{\\frac{\\sigma_1^2+\\sigma_2^2}{\\sigma_1^2+\\sigma_2^2-2\\sigma_1\\sigma_2\\rho}} > 1 ~~\\text{where} ~~ \\rho := {\\rm corr}(Y_{i1},Y_{i2}).\n", "3ce5ad075f9c58c72eaef8626cd501cc": "k=\\sum_{i=1}^n k_i ,\\!", "3ce608fae7869c7bb73add8b215b5910": " \\mathbf{E} = \\mathbf{E_0} * \\left(\\frac{J}{J_c}\\right)^n\\, ", "3ce61db6cc61ca3512872f5c6a4cf114": "\\int_\\mathbb{R}\\left(\\int_\\mathbb{R}|f(x,y)|\\,dy\\right)\\, dx", "3ce677433286e93bcab5efe95335d2a4": "T(\\mathbf{x})=\\sum_{i=1}^nx_i", "3ce6879bcd2d32a4ddda6ec499125dab": "\\left(\\frac{0.27 \\mbox{ mol }\\mathrm{CH_3OH}}{1}\\right)\\left(\\frac{4 \\mbox{ mol }\\mathrm{H_2O}}{2 \\mbox{ mol } \\mathrm{CH_3OH}}\\right) = 0.54\\ \\text{mol}", "3ce6a5dedb52e54f7f7a75f8c523e5be": "3\\sigma_R= \\sqrt{3^2+5^2+2.5^2+2^2+1^2+0.5^2} \\, \\mathrm{m} \\,=\\,6.7 \\, \\mathrm{m}", "3ce6a866dc8e30ae0263bdcf0ea27809": "(e)", "3ce6fee95fbe839a944fec13e303c4d4": "\\mathrm{d} \\mathbf x/\\mathrm{d} t = 0", "3ce7097355a73529c4ff14a45ba4706a": "((p \\to q) \\land (p \\to r)) \\vdash (p \\to (q \\land r))", "3ce70b7acbcc767763b5d570dc8178df": "\\scriptstyle \\left\\langle x^2 \\right\\rangle \\;=\\; \\left\\langle y^2 \\right\\rangle \\;=\\; \\left\\langle z^2 \\right\\rangle \\;=\\; \\frac{1}{3}\\left\\langle r^2 \\right\\rangle", "3ce766990f5aefa3b44c90060e9bf194": "\\sqrt{2}\\lambda> \\xi", "3ce7715ceb828adc36b46fb9f6602b46": "10^{-5} 1 \\\\\nC^q & \\text{if } p = 0 \\\\\nA^{q+1} & \\text{if } p = 1 \\end{cases}", "3cf3ad8e5e07cdddc42b754b089c2e48": "\\left( -\\frac{1}{u^{2}}\\frac{\\partial}{\\partial u}\\left(u^{2}\\frac{\\partial}{\\partial u}\\right) + \\frac{\\hat{L}^{2}}{u^{2}} +\\frac{1}{4}ku^{2} +\\frac{1}{u}\\right)R_{l}(u)Y_{lm}(\\hat{\\mathbf{u}}) = E_{l}R_{l}(u)Y_{lm}(\\hat{\\mathbf{u}}),", "3cf3b70db8c3c5975eb2bdc5bcb7fb53": "n_2=n_3=0\\,\\!", "3cf4038fc447142a944d899dbe48c616": "f = \\text{arg}\\min_{f\\in\\mathcal{H}}\\left\\{\\frac{1}{n}\\sum_{i=1}^n V(y_i,f(x_i))+\\lambda||f||^2_\\mathcal{H}\\right\\} ", "3cf412ba792f633a3768eb028acb18bf": "\\beta(s)=(s,s^2/2)\\,", "3cf4d219b68445faae0a5e19cbe14e9a": "\\displaystyle\\mathbf F_\\parallel", "3cf4d749130bd926993bd3778655faed": "\\frac {\\sigma}{\\sqrt{n}}=\\frac {2.5~\\text{g}}{\\sqrt{25}}=0.5\\ \\text{grams}", "3cf4e615cdfd6d3f4100da63fa54d0cd": "-\\frac{6\\pi^2 - 24\\pi +16}{(4-\\pi)^2}", "3cf51dd1c04a4f01a72934ce2c1bab81": "f(q) = e^{kq}", "3cf53de0cb81f8745c8d545e3571ae93": "q = \\exp(a \\mathbf{r})", "3cf57213dc7d2606f90a4a6bd3486e36": "\\ X_i", "3cf59425e9ca719730bc743f80e882dc": " \n{1 \\over \\pi r_B^2 L_B} \n{1 \\over n!}\n\\left( {r \\over r_B} \\right)^{2 \\mathit l}\n\\exp \\left( -{r^2 \\over r_B^2} \\right)\n.", "3cf5e09cda81d4c656f30eeb64929bd8": "X_0\\ldots X_{n-1}", "3cf5f22da16afe07fee0866df669eb4d": "\\frac{mol O_2}{mol C_6H_6}", "3cf5fe0f51ba655594d7afe6bf118880": "\\partial /\\partial z = 0", "3cf61883a93c3834394b781e648f8247": "f(x+y)=f(x)+f(y),", "3cf63a55b1553737756527527e73c629": "\\pi_0 H(X, Y) \\to [X, Y], \\quad (f, g) \\mapsto g \\circ f^{-1}", "3cf68424e0bf21ddb278ba30f2e14b1c": "\n{} +2250q^2rs^2+108q^5s-27q^4r^2-630pq^3rs+16p^3q^3s-4p^3q^2r^2.\n", "3cf68d8d88c9b1053455cb4352f4d193": "L^{X/Y}_0 = i^*\\Omega_{V/Y},", "3cf69bf17dd5a700659005bd79af6caa": " H_i(X; \\mathbf{Z}) \\cong \\mathbf{Z}^{\\beta_i(X)}\\oplus T_i .", "3cf6d534509e788885cc682ca3479056": "\\tfrac {1}{15} \\pi^2 - \\ln^2 \\phi \\,", "3cf6eca140e03a4d259bbf6548df770b": "A=LU", "3cf745e9df144fe5c580f747a6a0a9a3": "q \\succ_P p", "3cf75fa7a39b1c93e623ba3600aeb496": "\\mathrm{Cl}^-", "3cf7d233a515c83215bd48df374c81fa": "u_{i,j}+u_{j,i}= 2 \\varepsilon_{ij}\\,\\!", "3cf8349e5f59db16793b70f24507c0d8": "y(x)=y(x-x_0)", "3cf83edc09365d3e7a72f7711ddb9d78": " \\mu_B(x,y) = \\mu_B(0,0),", "3cf84af5e69077ddecefa56144968554": "z = -c_2", "3cf86b30e6ac4f0825ccc45ae13537f3": "\\displaystyle B = \\mu_0 Ni/l", "3cf875ed2f1b5b291a8064408a98f4a8": "Q_{3 \\times 3} = \\begin{bmatrix}\\cos \\theta & \\sin \\theta & {\\color{CadetBlue}0} \\\\ -\\sin \\theta & \\cos \\theta & {\\color{CadetBlue}0} \\\\ {\\color{CadetBlue}0} & {\\color{CadetBlue}0} & {\\color{CadetBlue}1}\\end{bmatrix} ", "3cf8ae04c08db375344128287bc32a08": "\\displaystyle{[L(x^2),L(y)]+2[L(xy),L(x)]=0.}", "3cf91f5964e30223d17b7631f9480157": "\n\\begin{bmatrix}\n\\mathbf{P}\\\\\n\\mathbf{e}_1\\\\\n\\mathbf{e}_2\\\\\n\\mathbf{e}_3\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1&0&0&0\\\\\n0&\\cos\\theta&\\sin\\theta&0\\\\\n0&-\\sin\\theta&\\cos\\theta&0\\\\\n0&0&0&1\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{P}\\\\\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}\n", "3cf92a600558c3b988c5c4d3ec3608ae": " \\mathbf{k}_\\perp = (k_r\\cos(\\theta),k_r\\sin(\\theta)) ", "3cf99f0ea8b0d00bdb18103783251177": " F = \\frac{\\left|q_1q_2\\right|}{4 \\pi \\varepsilon_0 r^2}", "3cf9b8a9ff814f9a36f03cb949171f76": "f(x)=z+a_2 z^2 + a_3 z^3 + a_4z^4 + \\cdots", "3cf9d07bf1d5409cabf7bb62c0bfcd26": "\n\\langle F_{thm}(s)F^T_{thm}(t) \\rangle = 2k_B{T}\\Upsilon\\delta(t - s).\n", "3cf9ea3136365304b440b508a6953c62": "{\\rm ci}(x)", "3cf9f5b53a8aab06b9f76359e13d7814": " d_j ", "3cf9fe9eebcd3c99d0ae4ad5a40bfc3a": "d(XY) \\leq d(XZ)+d(ZY)", "3cfa1d162826786014cb0f3109bb05b7": "S \\rightarrow \\varepsilon", "3cfa40609dbbc0e357d5c1201fa25a67": "X^i={e\\in E|x_e^i =1}", "3cfa747424473f2e10d347b4c04a3d1a": "v_{n}", "3cfa7b7016d2fead65bd616d5e289d5d": "\\sqrt{3+2\\sqrt{2}} = 1+\\sqrt{2}\\,", "3cfaab70ea4efcb08021781227ee58e8": "W(s,0)=0,", "3cfaaf74be40fb6d4da609683ebd732b": "\\scriptstyle{\\vec E_\\theta}", "3cfabf85eadad7bc6d2129fee0f6f15e": "f(x) = \\sum_{n=0}^\\infty a_n \\left( x-c \\right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + a_3 (x-c)^3 + \\cdots", "3cfad1f64a7cf6f6bc7b80af2b4d398d": " BV(\\Omega)=\\{ u\\in L^1(\\Omega)\\colon V(u,\\Omega)<+\\infty\\}", "3cfb452cf29121573db07742df143aeb": "\\{n^2 + k : n > 0 \\}", "3cfcb7ddf102082262a79b3d3301c56d": "X \\rightarrow Y \\leftarrow Z", "3cfcc7a8ebf7e138ae996a256ab063a5": "\n\\bar{Y} = \\frac{\\alpha(1+\\alpha)^{n-1}+Lc\\alpha(1+c\\alpha)^{n-1}}{(1+\\alpha)^n+L(1+c\\alpha)^n} \n", "3cfd108bfe4b6de535a3270618d52315": "\\scriptstyle\\varphi = (\\varphi^1, \\ldots, \\varphi^n)", "3cfd7c5593a9171b1082bc3255f2f879": "h(ap,u)=h(p,u)", "3cfda1900f0b3e0393d8428bf06eb190": "L(s,\\pi,r), \\ L(s,\\tilde{\\pi},r) \\ ", "3cfda9ca2853743e5cdd8190ce0356b4": "k\\cdot (X\\cdot v)=(\\operatorname{Ad}(k)X)\\cdot (k\\cdot v)", "3cfde0f65e918dc86167cd78d26ac765": "\\frac{d}{dz}\\langle\\alpha^\\mu_z(A)B\\rangle=i\\langle\\alpha^\\mu_z\\left(\\left[H-\\mu N,A\\right]\\right)B\\rangle", "3cfe89b6a390d09b12f3b4c1543335d8": "\\hat \\beta_j", "3cfeb3ed233367f331d2ab8a9dfac7a0": "\\sigma_{XY}(m)=E[ (X_n-E[X])\\,(Y_{n+m}-E[Y])].", "3cfeb63ca7d1806188d03e5dfba8c883": "(M_n(R))[X]", "3cfee15c4238e7e9350ffc81e106172c": "X_1,\\ldots, X_N", "3cff73c96f7d06fa75d997b32d545710": "\\, 10 (b_L + b_R) + t_L t_R", "3cffdfdbd944fecb808ff46a52175f6a": "n_z = -1", "3cffe2804aa5d238d0be483e745dc882": "\\Delta x_i = x_{i+1}-x_i.", "3cfff6f4c44a91119982b3539dcc9ec3": "\\text{if }M \\models \\varphi\\text{ then }M \\models K_i \\varphi.\\,", "3d00058b0d33658a1a20d34f74e89c02": "[M][U] \\lambda + [K][U] = [0]", "3d003065632219db7d41642b27e073c3": "\\frac{\\partial uv}{\\partial \\mathbf{x}} =", "3d0050e6dbca4602d91430dafae9a127": "W_{in} = \\frac{T_s-T_o}{T_s}\\left(C_k \\rho |\\mathbf{u}|(k^*_s-k) + C_d \\rho |\\mathbf{u}|^3\\right)", "3d00b3db7ac8cf3bae9ae4708c6c1816": "(x_e, y_e) = \\frac{-r_2}{r_1 - r_2}(x_1, y_1) + \\frac{r_1}{r_1 - r_2}(x_2, y_2).", "3d011f69f4a44e47ab50124e86ab19ca": "P_{hs}=\\frac{R\\, T}{V_m - b} = \\frac{R\\, T}{V_m}\\, \\frac{1}{1 - \\frac{b}{V_m}}", "3d0125770d76d50bf322977a89046431": "\\frac{N \\log_2(N) + N}{N-M+1}.\\,", "3d0146ced0908058f94dad0d94acd866": "A \\rightarrow U", "3d016f618c8324cfdc5376b0c550daaa": "\\,C_n", "3d019a13db7d040e74700057f33943da": "\\operatorname{logit}(x) = \\sigma^2(2x-1)+\\mu .", "3d019d1e8e49eed353361c1c9e829750": "(\\phi_0)\\,", "3d01bf7a5a0f861f9b3828e6097fd535": "A = \\overline {\\cup_n A_n}.", "3d01cab4c5b8798916930cc82e1b4925": "\\begin{array}{cc}\n \\begin{array}{rr} \\\\ &3 \\\\ \\text{-}1& \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & \\text{-}12 & 0 & \\text{-}42 \\\\\n & & 3 & \\\\\n & \\text{-}1 & & \\\\\n \\hline \n 1 & \\text{-}13 & & \\\\ \n \\end{array}\n\\end{array}", "3d021647c384c5d57a77b1c8ede5538d": "R_N(t,s)", "3d026c3e9e8ec732b718285dcd1567d7": "\\tilde{f}_{ij}", "3d029b343ceb5f359e6c7a7b05cdf1c5": " \nV\\sum_{\\tau=0}^{t-1}E[p(\\tau)] \\leq (B + C + Vp^*)t\n", "3d02c46a0df4d1d95a046af622664e44": "\\textstyle dy_j\\left(\\frac{\\partial}{\\partial y_k}\\right) = \\delta_{jk}", "3d02e0f3cc12c9f16bf9cbbf961c456d": "P(u)=u_t - k\\Delta u\\,", "3d02f29671032b72f231a7aee663f7ca": "\\beta = \\partial f / \\partial y", "3d02fb982cd25806c25ea3366d3ca4f6": "B_{i_1}(t_{i_1})\\cdot B_{i_2}(t_{i_2})\\cdot\\dots \\cdot B_{i_n}(t_{i_n})", "3d031b4b8a9c3d29952d2c676251181a": "c^2 = \\frac{1}{\\epsilon_0\\mu_0} ", "3d0344a2eccb5cffaed928562e6bad99": "\\cos(\\alpha) = -Z_2 / \\sqrt{1 - Z_3^2}.", "3d03e77325668f4c1806822c15cb2732": "K_N(f)", "3d04025af7d6d76b581ac9367789d9d2": "d=\\det \\begin{pmatrix} z_1w_1 & z_1 & 1 \\\\ z_2w_2 & z_2 & 1 \\\\ z_3w_3 & z_3 & 1 \\end{pmatrix}", "3d042c63f7e9f6c8e2d065324f48d0a5": "E_yL+M_y=\\frac{3}{2}(B-\\sigma)L-\\frac{1}{2}(E_xL+M_x)", "3d043cbc01fbe5a3d1a7b5844f75bc0a": "L_p(s,E)=\\begin{cases}\n (1-a_pp^{-s}+p^{1-2s}), & \\text{if }p\\nmid N \\\\\n (1-a_pp^{-s}), & \\text{if }p\\|N \\\\\n 1, & \\text{if }p^2|N\n \\end{cases}", "3d04855d82fd899ba7cb262c54e318ef": "\\times\\zeta(2)", "3d0496f9b22b0f3e1e06354df54db788": " s_f(n) ", "3d04a9741999c44f616f72a6171b774e": "\\frac{\\partial\\rho(\\mathbf r,t)}{\\partial t}+\\nabla\\cdot\\mathbf j(\\mathbf r,t)=0.", "3d04b1076751f400d1c2b5747d899b18": "i_\\alpha\\circ i_\\alpha = 0.", "3d04c062e0569e19463669d94f53be79": "\\hat{f} = 0\\,", "3d0547459a2f163fe6f1d10546af80f2": "\\sum_{i=1}^n B \\cdot 2^{i-1} = B (2^n - 1)", "3d05a89bb7c7f64a8ee3c7f1d8c321cd": "\\overline{AB} \\perp \\overline{CD}", "3d05bfa5e900b516eaac5b05585c2036": "\\beta = \\arccos\\left(\\frac{\\cos b-\\cos c\\ \\cos a}{\\sin c\\ \\sin a}\\right),", "3d05c00ab214c48720b563b50dfdcc65": " \\! P_{\\text{sys}}", "3d05dfb0a5c03aae3dd585147343b2c6": "R_q=a\\sqrt{q_p/2}.\\,", "3d060dc93acf6eef105264fd2ac37ed2": "\\operatorname{Res}_a(f)", "3d0622b4bd98989be34b272a295b03c5": " - \\phi(-x) R(\\pi)\\phi(x) \\, ", "3d062f7460aa6e8517296a4cc04b0c66": "{R^\\rho}_{\\sigma\\mu\\nu} = \\partial_\\mu\\Gamma^\\rho {}_{\\nu\\sigma}\n - \\partial_\\nu\\Gamma^\\rho {}_{\\mu\\sigma}\n + \\Gamma^\\rho {}_{\\mu\\lambda}\\Gamma^\\lambda {}_{\\nu\\sigma}\n - \\Gamma^\\rho {}_{\\nu\\lambda}\\Gamma^\\lambda {}_{\\mu\\sigma}.", "3d06344fb25a05c9fc08eba95a721a10": "A_{8,5}", "3d0654bdb93f6f43726cf19fcebf7e77": " \\forall i, j", "3d0666bd3690da036b6f5316d7c73bcf": "t \\in I", "3d066e5b8a84d178406ff2d1cb4419fb": "\n\\mathbf{x}^T A_Q\\mathbf{x}=0\n", "3d0696005a58facc0b207d344f1d489d": " C_i = C_j ", "3d06d0d025180d63052fe02142422d10": "A=U D U^* \\;", "3d06fffe0a3cf4a1c85ccd64e0cd8f22": "\\mathfrak P", "3d0717df56421b0532c79c7d41ce748f": "\nf_1^2=a^2-b^2=(a^2+\\lambda)-(b^2+\\lambda), \\,\n", "3d076aceba183982805fdc43d04f1114": "\\sigma_{\\bar x}", "3d076d6ee61301d9e6daed90601246cc": "\\frac{2187}{2048}", "3d0790f65daf99a1ae5c5f72f7de577b": "E(n)", "3d07b3add794342df2c429da64ed93b1": "8,3966667 = \\frac{ 8,11 * 12 + ((14 * 5 / 4) - 7,76)} { 12}", "3d07b59fbf01298bf92d40a2be994a48": "\nx_{ij}= \\frac {S_i-S_j}{ \\sigma} \\,\n", "3d080be00ecb8b1c17b9a304ee4c2e4c": " \\alpha \\cdot u_{n} ", "3d083c458689c052890acf1082b8967a": " \\int_{\\Omega} \\nabla u \\cdot \\mathbf{v}\\, d\\Omega = \\int_{\\Gamma} u (\\mathbf{v}\\cdot \\hat{\\nu})\\, d\\Gamma - \\int_\\Omega u\\, \\nabla\\cdot\\mathbf{v}\\, d\\Omega,", "3d0898af3d5b45a23985840f9ac80ca7": "\\frac{3 + 6 + 8 + 8 + 10}{5} = \\frac{35}{5} = 7", "3d089a5c1faba92ce870d4eb5f71c519": "r(H)", "3d08c397ec8c69cadda2bd8596b52a1b": " \\mathcal{D}(D)", "3d08f01949bb41891d359638fa72ad13": "[-U,U]", "3d09176a8c708001e84d751b483913cf": " 0 \\leqslant \\theta \\leqslant 2\\pi ", "3d0918821d6f98afc2c8455baace3a0f": "\\pi\\circ s={\\mathrm {Id}_U}\\,", "3d09c12bcc39d6879709c038b8dd7131": "\\Phi_B=\\,\\!", "3d09cde44c0050112cbdf4247d376327": "s^{(M)}", "3d0a09533fa600787707c5642a463a4e": " \\sum_{k=1}^n I_k = L(I_1,I_2,\\dots,I_{n-1})", "3d0a0e2f3020e153acd0283f6faa09d7": "MV \\frac {d\\psi} {dt} =Y cos(\\theta-\\psi)", "3d0a48f4f1ec8456e058002e925843a3": "\\tfrac{1}{n} < \\gamma \\leq 1\\,", "3d0adb67d294c11c1b75e434aeb89b51": "\\int_\\Omega | f \\varphi|\\, \\mathrm{d}x = \\int_K |f|\\,|\\varphi|\\, \\mathrm{d}x \\le\\|\\varphi\\|_\\infty\\int_K | f |\\, \\mathrm{d}x<\\infty", "3d0af7efb7a61981e20551bc547e6c22": "\n\\begin{pmatrix}\np_\\alpha \\\\\np_\\beta \\\\\np_\\gamma \\\\\n\\end{pmatrix}\n\\ \\stackrel{\\mathrm{def}}{=}\\ \n\\begin{pmatrix}\n\\partial T/{\\partial \\dot{\\alpha}}\\\\ \n\\partial T/{\\partial \\dot{\\beta}} \\\\\n\\partial T/{\\partial \\dot{\\gamma}} \\\\\n\\end{pmatrix}\n= \\mathbf{g} \n\\begin{pmatrix} \\; \\,\n\\dot{\\alpha} \\\\ \\dot{\\beta} \\\\ \\dot{\\gamma}\\\\\n\\end{pmatrix},\n", "3d0b2738fd5a04d6189b3bb6d2ff1818": "\\upsilon_{c3}", "3d0b29828bc957796348cba02760f386": "\nx_k = -a\\ \\cosh \\mu _k\n", "3d0b5abdc490344119371bacde0f6eb5": "f_n=f(x_n)", "3d0b929f8c6d89066976c5140c694422": "\\mathbf{1}_{A\\cap B}(x) = \\min\\{\\mathbf{1}_A(x),\\mathbf{1}_B(x)\\}. ", "3d0bb3de82ac15f75e90baef7b615e4c": "\np_3(x) = x^3 + 16x^2 + 79x + 120 \\,\n", "3d0bc9a4bab9a29bd456dbdb71e98d19": "\n \\begin{align}\n M_{\\alpha\\beta,\\beta}-Q_\\alpha & = 0 \\\\\n Q_{\\alpha,\\alpha}+q & = 0 \\,.\n \\end{align}\n", "3d0c2f5eea33be4b7f5b70b098f557f3": "L_{\\sigma}(X, Y)", "3d0c52dacadc7446ba94c4db9ccfd7a6": "q_0 = -10", "3d0c712d5347b92a583e8a1d5c87e815": "H_{\\chi^2}(\\theta)=1-F_{\\chi^2_{n-1}}(s^2/\\theta)", "3d0cfc20b92044e8c502bd36cdd6769d": "G_S", "3d0d380ba2178527cf0eede1552dc9cd": " 0 \\leq f_k(x) \\leq f_{k+1}(x). \\, ", "3d0d57b20b6590ee0d0012eaf569002c": "\\langle X,X\\rangle_A\\ge 0", "3d0d60b72052aec3103e5a3da81d9306": "\\beta_A=(S_1-c)+(S_2-c)", "3d0d679dd2f05fd0f0b187365d7e5730": "b=r_{1}.", "3d0d907c459b7e25efda0d1115a9f08d": "\\frac{\\,_0F_1(a+1;z)}{a\\,_0F_1(a;z)} = \\cfrac{1}{a + \\cfrac{z}\n{(a+1) + \\cfrac{z}{(a+2) + \\cfrac{z}{(a+3) + {}\\ddots}}}}", "3d0de84484e2b89ed4dc1e1eb9ce5e3d": "\\frac {d}{dt} \\left (\\iint_{\\Sigma (t)} d \\mathbf{A}_{\\mathbf{r}}\\cdot \\mathbf{F}(\\mathbf{r}, t) \\right) = \\iint_{\\Sigma} d \\mathbf{A}_{\\mathbf{I}} \\cdot \\frac {d}{dt}\\mathbf{F}(\\mathbf{C}(t) + \\mathbf{I}, t)", "3d0e0a3cf34a2557c0bb6e8e4a2038df": "{\\mathcal O(n^3)}", "3d0e43abbd33ff1eb1bea159bd58cb8b": "-1.2557", "3d0e847483543b5c2cfe22ab5a770a1d": "\\mathcal{L} = \\mathcal{L}(\\mathcal{D})", "3d0ec869413cee73fe3e75ba81e4e68f": "=\\int_{V}P^{m}\\{s:\\exists h\\in H,|Q_{P}(h)-\\widehat{Q_{r}}(h)|\\geq\\epsilon\\,\\!", "3d0eebe40f668eb004e52c374b6a1858": "EC_{\\text{rate}}", "3d0eefc02cb43638396ce620933da199": " P_1(x)Q_1(y) + P_2(x)Q_2(y)\\,\\frac{dy}{dx} = 0 \\,\\!", "3d0f060b761fb6c1104f610f9b22fdbe": "E= \\frac{8+7}{2}-3.5+1", "3d0f5a22a856895041056183923d322a": "\n\\hat{\\boldsymbol\\varphi} \\cdot \\mathbf{\\hat{r}} = -\\sin \\varphi \\cos \\varphi + \\cos \\varphi \\sin \\varphi = 0\n", "3d0f8413255b4be7b8dede8e23418e45": "\\mathbf{E} = \\sum_i \\mathbf{E}_i = \\mathbf{E}_1 + \\mathbf{E}_2 + \\mathbf{E}_3 + \\cdots \\,\\!", "3d0f85b09b1a9a125af581d57bfdee00": "\\boldsymbol{f}(s) \\approx \\boldsymbol{f}(s_0)", "3d0fddf563ddaf97438bb5806d134626": "z=(q,p)", "3d107e01002a51b463a8d5e3f38ec7d5": "k^a k_a \\rightarrow -1", "3d108d8e42a7f2e9b9f7143c91805f6f": "B_{21}", "3d10c81d66824eedc29ec676f324a633": "\\Delta_G = E_{PL} - E_P - E_L ", "3d10e9d2ada501bc3f02da1a7a37fd8a": "\\frac{d}{d\\omega}Q(\\omega)<0", "3d10fe6f8a17b8c970ec7a3bc255a76a": "W(t_0,t_1)", "3d11a893606e428d8852bf98b8e160cf": "g_\\varepsilon(x) = \\begin{cases}\n\\frac{x}{\\pi( x^2 +\\varepsilon^2)} & |x|\\le \\varepsilon \\\\\n\\frac{x}{\\pi( x^2 +\\varepsilon^2)} -\\frac{1}{\\pi x} & |x| >\\varepsilon\n\\end{cases}", "3d120400f7e730c30551fda5709098f6": "\\bigl( \\begin{smallmatrix}\\\\ 1&4\\\\ 8&5\\end{smallmatrix} \\bigr)", "3d12187562c8b5b8496588fe19ac5f3e": "c = -0.000015", "3d1221310639d4fa666c85b9fbff873c": "{1 \\over \\det (H)}={{c_{2n}}\\over {c_n^{\\;4}}}=n!\\cdot \\prod_{i=1}^{2n-1} {i \\choose [i/2]}.\n", "3d1221a9bcf2f797b99b660b03a5d4d8": "\\partial_A", "3d1236d4bafdb4f8511437b423bb3638": "x_\\alpha = (ct,-x,-y,-z ) ", "3d124943ea331b822b20d025dfacc7b8": "\\Delta F = - T \\Delta S \\left ( \\mathbf {R} \\right )", "3d129e00920d2077ef5aa9af78a611f3": " C_{n,m}", "3d12c4908bf466ef30e6423bc7b0a123": " Q_{Corrected} = Q_{Measured}*{\\rho_{Out} \\over \\rho_{In}}\\,\\!", "3d12d0d8b95c82a8e8979465c58d0ab9": "A\\vec{\\rho} = \\begin{pmatrix} E_1^\\dagger \\vec\\rho \\\\ E_2^\\dagger \\vec\\rho \\\\ E_3^\\dagger \\vec\\rho \\\\ \\vdots \\end{pmatrix} = \\begin{pmatrix} E_1 \\cdot \\rho \\\\ E_2 \\cdot \\rho \\\\ E_3 \\cdot \\rho \\\\ \\vdots \\end{pmatrix} = \\begin{pmatrix} \\mathrm{P}(E_1 | \\rho) \\\\ \\mathrm{P}(E_2 | \\rho) \\\\ \\mathrm{P}(E_3 | \\rho) \\\\ \\vdots \\end{pmatrix} \\approx \\begin{pmatrix} p_1 \\\\ p_2 \\\\ p_3 \\\\ \\vdots \\end{pmatrix} = \\vec{p}", "3d130767879bc94f2de5cb043f1f2093": "U(t)=\\frac{2mt+l^2-t^2}{t^2+l^2}", "3d13343a4be70e71daec73a55733882f": "\\Delta\\ll\\omega_L+\\omega_0", "3d133f466312f98f3da767c293b0a024": "\\int_0^1 <\\mathbf u(t) - \\mathbf u_0(t),\\mathbf F(\\mathbf u_0(t),\\lambda_0)>\\, dt = 0", "3d13613a9043582791618385ead7d49d": "1/C", "3d1386d34d2fbb4180052b55a768e1ee": "x_1 - \\phi(t_0,t_1)x_0", "3d139a4bba1824ba4937c51ffdcfdb47": "N(n,k)", "3d142d4af432960f32f143a1d4267126": "p-2", "3d145dd35ef1c5ddf5f92a6d888ca542": "\\sum_{i=0}^\\infty (|G_i|-1).", "3d1485b6c996bf7506698b5d150bb80b": "m \\le m_0", "3d1492c90a3fb037f86c3bce31899614": "N(\\varepsilon||F||_{Q,2}, \\mathcal{F}, L_2(Q)) \\leq C \\left(\\dfrac{1}{\\varepsilon}\\right)^V", "3d14d5afdebda78459786eb4ce79d94d": "\\text{N}", "3d150885af1fb153feea861ae1170056": "\\begin{align}\n\\mathbf{A} & = 2\\mathbf{ij} + \\frac{\\sqrt{3}}{2}\\mathbf{ji} - 8\\pi \\mathbf{jk} + \\frac{2\\sqrt{2}}{3} \\mathbf{kk} \\\\\n& = 2 \\begin{pmatrix}\n 0 & 1 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{pmatrix} + \\frac{\\sqrt{3}}{2}\\begin{pmatrix}\n 0 & 0 & 0 \\\\\n 1 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{pmatrix} - 8\\pi \\begin{pmatrix}\n 0 & 0 & 0 \\\\\n 0 & 0 & 1 \\\\\n 0 & 0 & 0\n\\end{pmatrix} + \\frac{2\\sqrt{2}}{3}\\begin{pmatrix}\n 0 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 1\n\\end{pmatrix}\\\\ \n& = \\begin{pmatrix}\n 0 & 2 & 0 \\\\\n \\sqrt{3}/2 & 0 & - 8\\pi \\\\\n 0 & 0 & \\frac{2\\sqrt{2}}{3}\n\\end{pmatrix}\n\\end{align}", "3d1535fced98672af376163b9d7910bf": "\\int { 1 \\over x} dx = \\ln|x| + C", "3d154ee6291197d9f5f55c6843f66ca0": "\\rho = 0", "3d1570def160314bcfc0e921cc64d247": "P_3=P_1+P_2", "3d15b12f9a4ba8b4632a9b0e0d36b43b": "E_{b^{n}}", "3d15e865aafc76f36ba7978f6835f734": "n^{n^n}", "3d15eab400e4efd6610dffbac323d1e6": "D(m,n)", "3d162fb75d15f837f70918e02420080c": "\nf^n(x) = f^n(a) + (x-a)\\frac{d}{dx}f^n(x)|_{x=a} + \\frac{(x-a)^2}{2!}\\frac{d^2}{dx^2}f^n(x)|_{x=a} +\\cdots\n", "3d165100829a9a519e3aa7e85e213a48": " 2S_0'S_n' + S''_{n-1} + \\sum_{j=1}^{n-1}S'_jS'_{n-j} = 0", "3d167369c1499bcecf80e20f39b78b27": "f^{\\mathrm{SW}}_p(x) = \\begin{cases}\n 1 - x & \\text{if } p = 0 \\\\\n 1 - \\log_{1 + p}(1 + px) & \\text{otherwise.}\n\\end{cases}", "3d167e0d9e6578b85df1085b5d0fda26": " A\\,\\!", "3d168302177f3decd941c7dd5055cebd": "\\vartriangle^m_n = \\wedge^{m+1}_n", "3d169edae84af0d0e42b3d36b1ba7abe": " \\sigma_{ij}\n=\\lambda \\delta_{ij} \\varepsilon_{kk}+2\\mu\\varepsilon_{ij}\n\\,\\!", "3d16c6ba1348b1d0a4a221119e53d312": "\\gamma_*", "3d16ea94dff49728d35c51c56d180530": "\n\\frac{\\delta F}{\\delta\\rho(\\boldsymbol{r})} = \\frac{\\partial f}{\\partial\\rho} - \\nabla \\cdot \\frac{\\partial f}{\\partial\\nabla\\rho} \n", "3d170a6abfb923ac28f76e2028ba1a49": "\n\\cos(3 \\vartheta)=2\\cos\\vartheta \\cos(2\\vartheta) - \\cos\\vartheta = 4\\cos^3\\,\\vartheta - 3\\cos\\vartheta \\, ,", "3d175b5756b3f834cbceaa98c96d4707": "\\int_1^\\infty {\\rm Riesz}(z) z^s \\frac{dz}{z}", "3d17607a6fbceaa9680045e2c88e59d8": "B^{*}", "3d17cc37e5ca8a11419aac312f125b7b": "\\begin{align}\n \\int_{11}^{14} (x^2 + 4y) \\ dx & = \\left (\\frac{1}{3}x^3 + 4yx \\right)\\Big |_{x=11}^{x=14} \\\\\n & = \\frac{1}{3}(14)^3 + 4y(14) - \\frac{1}{3}(11)^3 - 4y(11) \\\\\n &= 471 + 12y\n\\end{align}", "3d17e87ce9b3ce788926b40abb6a354c": "\n\\rho_{\\alpha\\beta}(\\omega) = \\frac{1}{\\mathcal{Z}}\\sum_{m,n} 2\\pi \\delta(E_n-E_m-\\omega)\\;\n\\langle m |\\psi_\\alpha|n \\rangle\\langle n |\\psi_\\beta^\\dagger|m \\rangle\n\\left(\\mathrm{e}^{-\\beta E_m} - \\zeta \\mathrm{e}^{-\\beta E_n}\\right) ,\n", "3d18062422860fa61082b947837c9544": "\n L^{(2l+1)}_{k}(x),\\qquad k=0,1,\\ldots , \n", "3d185032598acced189097554edb2282": "v_1, v_2, \\ldots, v_n", "3d186c31f19807004ce4bb5bcf35a1ef": "f_{IN}+f_{LO}", "3d186e3f242c7511741b45d660b6c3cd": "\\epsilon = \\frac{T_s-T_o}{T_s}", "3d18adf0653e02bce72edb43917a9253": "P_M^i \\equiv P^iH = K_i\\left ( R_i | t_i \\right ) ", "3d18dfc853e063b46a6982dc3a954659": "y_0+y'_0=1+A", "3d18ed796e10c204a4d646ba7596b08b": "E = \\nabla G", "3d18fd4c4ab39a914362849499581dc3": "i[j]", "3d18fe3c225b2e5efead33f7cee86326": "\\overline{S}=V", "3d195452aed162e6e43ff4f17246a83c": "S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).\\ ", "3d19746057462d5c6007ead3140fa728": "\\phi(\\omega_N - \\Omega/2) \\approxeq -\\sum_{n=0}^N\\frac{\\omega_{n+1}-\\omega_n}{2\\Omega}[\\theta(\\omega_n)+\\theta(\\omega_{n+1})]", "3d198b27068463e46bb109df41902f17": "D_{IS}", "3d19f7b1ae80cf9985cc548d73298e91": "12 = x", "3d1a289696096a765af5c02641a95812": "\\delta_a[\\phi]=\\phi(a)", "3d1a2f32ac2b9eb3299e96a8698f2f5f": "\\kappa(f(x))", "3d1a5c9a977462ddcf44814a3ab75d3d": "M(V) = w(V/V_d)^{-2/3} + (1-w)(V/V_d)^{1/3}", "3d1a9cee13fe5547bedd4767f7665483": "3^{2n}=2^m", "3d1b2a09c609bcb382acd909fb587624": "f = \\frac{\\Sigma_a^F}{\\Sigma_a}", "3d1b31b722af2bef6c16250305ee68a5": "A_1\\cup A_2=\\mathbb N.", "3d1b810782a7573c13277a7cc33182ff": "f(A)=f(A+t)", "3d1bc7d693319de4b51748856b6d2bdb": " \\int_{T(X)} \\big[ P(A^{-1}B | T=t) - P^A(B) \\big] \\ P_\\theta^T (dt) = 0 \\,", "3d1c35897f3e23e9587e5484f6b8bbcf": " I-H ", "3d1c3e9f8be0361fd5428b06bc2088c1": "\\rho_n=\\rho_n(t)", "3d1d05f38b39faacd6ea7104d0e68207": "\\frac{m}{s} \\frac{n}{t} := \\frac{m n}{s t}", "3d1d2f95b4838d5363807569bfabb85b": " X(z)=\\sum X(n) z^{-n-1}", "3d1d42437bd3b4ffa153c7ce957570db": "\n \\left. \\left[ \\frac{k_{c}}{2}(2H+c_{0})^{2}+\\bar{k}K+\\lambda+\\gamma k_{g}\\right]\\right\\vert _{C}=0\n", "3d1dc394f52796adbfb04d8867060663": " y_L ", "3d1dd701d04947841a0ad62f0ecbd302": "x \\in k\\left[M\\right].", "3d1ded493f814d5516914935ae4d9289": "I_z = \\frac{1}{2} m\\left({r_2}^2 + {r_1}^2\\right)", "3d1dfe70cdc0d574aa6cf3e228a57166": "r_t", "3d1e2d521748b01a9da4715dfa7b0158": "D_{p-r}=D_{braking}", "3d1e44eb8c11c0d422c9447bc5986665": "f(\\bigvee A) = \\bigvee\\{f(a)\\mid a\\in A\\}", "3d1e5ceb78436ac430ca29e7e49ebde2": "W = \\{ Bird(Condor), Bird(Penguin), \\neg Flies(Penguin), Flies(Bee) \\}", "3d1e8961aa7ca83da0944343b2a362b3": "\\int_{\\tau_1}^{\\tau_2} \\mathbf{F}_\\mathrm{rad} \\cdot \\mathbf{v} dt = -\\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d \\mathbf{a}}{dt} \\cdot \\mathbf{a} \\bigg|_{\\tau_1}^{\\tau_2}+\\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^3 \\mathbf{v}}{dt^3} \\cdot \\mathbf{v} \\bigg|_{\\tau_1}^{\\tau_2} - \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^3 \\mathbf{a}}{dt^3} \\cdot \\mathbf{v} dt = -0 + 0-\\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^3 \\mathbf{a}}{dt^3} \\cdot \\mathbf{v} dt", "3d1f037fa85484aef4432841bf0fe410": "\\textit{dau}(m,h) \\lor \\lnot \\textit{par}(h,m) \\lor \\lnot \\textit{par}(h,t) \\lor \\lnot \\textit{par}(g,m) \\lor \\lnot \\textit{par}(t,e) \\lor \\lnot \\textit{par}(n,e) \\lor \\lnot \\textit{fem}(h) \\lor \\lnot \\textit{fem}(m) \\lor \\lnot \\textit{fem}(n) \\lor \\lnot \\textit{fem}(e)", "3d1f0a1b8587e03d384c8d1b86ea51b6": "V_\\alpha\\cap V_\\beta", "3d1f0e9cc6fa99863bc0219e5ee26d38": "\\frac{f_{x}}{f_{y}}=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "3d1f12c272ab8eb82eae1ee19ee4f28c": " {A^\\mu}_{; \\mu} = 0. ", "3d1fca6ec674bda9c79d5e9c8c0f1dbe": "t\\equiv p^i\\ \\pmod{v^*}", "3d1ff53aef61b69542d10165df63d86b": "\nI_C = C\\frac{dV_{in}}{dt}\n", "3d1ffaba461741a7ccfd91c0412c0d17": "|\\omega|=m\\,", "3d206b4a0c34df6583789db84027dfb7": "[0.2, 0.9] \\not\\supseteq [0.16, 0.88]", "3d20984e9bc22629f46ec8d3706ccf32": "\\Delta\\,T_m(x)= \\frac{k_{GT}}{x} = \\frac{k_g \\, k_s \\, k_i}{x}", "3d20a1c5f60c6521ae2b3706dee3ced6": "\\begin{align}\nH_{\\varepsilon,R} f(x) &={1\\over \\pi}\\int_{\\varepsilon \\le |y-x|\\le R} {f(y)\\over x-y} \\, dy ={1\\over \\pi}\\int_{\\varepsilon \\le |y|\\le R} {f(x-y)\\over y}\\, dy \\\\\nH_{\\varepsilon} f(x) &={1\\over \\pi}\\int_{ |y-x|\\ge \\varepsilon} {f(y)\\over x-y} \\, dy ={1\\over \\pi} \\int_{ |y|\\ge \\varepsilon} {f(x-y)\\over y}\\, dy.\n\\end{align}", "3d20d99f8427a16700c4e4db5ed8b723": "x^2z^2", "3d215b669adeaec7e16283c51d86c3f9": " (q _{v \\cap w})(q _{v \\setminus w} -1) = q _{v} - q _{v \\cap w}", "3d2161e8eeb9138d02892c5432ae7d0e": "\\frac {\\partial U_{A_1}} {\\partial n} =\\frac{ae^{ikr}}{r} \\left[ik - \\frac{1}{r} \\right] \\cos {(n,r)} ", "3d219db6930cffcc99acb290819851ea": "\n\\begin{align}\n\\iota \\colon \\mathbf{Gr}(r, K^n) &{}\\rightarrow \\mathbf{P}(\\wedge^r K^n)\\\\\n\\operatorname{span}( v_1, \\ldots, v_r ) &{}\\mapsto K( v_1 \\wedge \\cdots \\wedge v_r )\n\\end{align}\n", "3d21a27f5dbfa0f3e203fac38dddd7dd": "\\left\\langle\\pm 2\\mathrm i,Z_4\\right\\rangle", "3d22370c4d86fbf01e88dc9f12c2564d": "f(x) = \\sqrt x", "3d224966e500b16bbf4f15e59021c1ac": " \\bold{r}\\cdot(\\dot{\\bold{r}}\\times\\bold{H})=\\bold{r}\\cdot(\\mu\\bold{u} + \\bold{c}) = \\mu\\bold{r}\\cdot\\bold{u} + \\bold{r}\\cdot\\bold{c} = \\mu r(\\bold{u}\\cdot\\bold{u})+rc\\cos(\\theta)=r(\\mu + c\\cos(\\theta))", "3d227698e29180c7318cbc70492399ba": "\n\\mathbf{\\Lambda} \n= \\begin{bmatrix}\n\\lambda_1 & 0 & 0 \\\\\n0 & \\lambda_2 & 0 \\\\\n0 & 0 & \\lambda_3 \\\\\n\\end{bmatrix}\n= \\begin{bmatrix}\nu-a & 0 & 0 \\\\\n0 & u & 0 \\\\\n0 & 0 & u+a \\\\\n\\end{bmatrix}.\n", "3d2291f760818d993d274ae1b7e1b31b": "1+\\textbf{G}(s)=0", "3d22a462aa0beefd3e082761cef43a02": "\\omega = \\prod_p p^{n_p},", "3d22dd2e01b4ba26da1dafa8ffed3ce1": "\\mathcal{L}_n", "3d22e1b7028a465050fc11f6c9797251": "\ns = \\pm\\sqrt{\\delta} \\quad \\quad t = \\pm \\sqrt{\\tau + 2 s}\n", "3d22ead21ae294e607325d9ef813c562": "\\mathcal{G}=(V, E, C)", "3d232812ab30bd46383823d023379a37": " F(\\rho,\\sigma)", "3d234c289b7b2cb218442acd316cdf42": "T_{max}", "3d23e8a9f4120c253a7f16361a8e7efe": "Var(X) = V(\\mu).", "3d2425a86329fa423e6ded6511b04eab": " P^{(N)}(\\mathbf{r}_1,\\ldots,\\mathbf{r}_N ) \\, \\mathrm{d} \\mathbf{r}_1 \\cdots \\mathrm{d} \\mathbf{r}_N = \\frac{\\mathrm{e}^{-\\beta U_{N}}}{Z_N} \\, \\mathrm{d} \\mathbf{r}_1 \\cdots \\mathrm{d} \\mathbf{r}_N\\, ", "3d242d5e4f3ab1442edee97ef385def9": "\\mathbf{u}^*", "3d24419fde937c5213ad3ba61bcf144f": " b_{ks}(t)", "3d247bbf2628efd42d310f2c767e9213": "f^{*} \\pi : f^{*} E = M \\times X \\to M", "3d255cb81a1d71b28e364275a46c1073": "I_{\\lambda \\mathrm{sun}}", "3d25a0b47e8e6432215a0a447d1b5eae": "U^*U=UU^*=I", "3d25ab969c4f3512e43f316fc175d3eb": "f^*(x)", "3d25b0d325c664bf0159dc0932a216f3": "x(yx)^{\\rho} = y^{\\rho}", "3d25fd44133daa023e43f61a9bbf93a0": "Sp(m,\\mathbb C)", "3d264c4ee9ab39cf062ca6cfe74f2649": " 0 < M < 0.3 ", "3d267d0f29e4582432b8aaed74885173": "(P,q,g_1,g_2,c,d,h_1,h_2,k_1,k_2)\\,", "3d268568b55ab5b79d802cb9273185bd": " T_{r} = c_2 M ", "3d26a555f4b83572a59c272a1f12d852": "(t - 2)", "3d27213b984e3ba426c12e247122d65b": "\\left\\langle x^2 (t) \\right\\rangle\\propto t^{(3 \\alpha-\\alpha \\beta)/2 \\beta}.", "3d27367ea16a2a7b40b3eb3172a32120": "i\\neq j", "3d275e04f7630bca6630e865f37c78b0": "\\mathrm{D}_{H} F (x) := \\mathrm{D} F (x) \\circ i : H \\to \\R", "3d2773d4cf72f90980eeffabb3ee7a2e": "\\begin{align}\n&\\,\\,\\,\\,\\,\\,\\, x^{71} && &&- x^{69} &&- 2x^{68} &&- x^{67} &&+ 2x^{66} &&+ 2x^{65} &&+ x^{64} &&- x^{63} \\\\\n&- x^{62} &&- x^{61} &&- x^{60} &&- x^{59} &&+ 2x^{58} &&+ 5x^{57} &&+ 3x^{56} &&- 2x^{55} &&- 10x^{54} \\\\\n&- 3x^{53} &&- 2x^{52} &&+ 6x^{51} &&+ 6x^{50} &&+ x^{49} &&+ 9x^{48} &&- 3x^{47} &&- 7x^{46} &&- 8x^{45} \\\\\n&- 8x^{44} &&+ 10x^{43} &&+ 6x^{42} &&+ 8x^{41} &&- 5x^{40} &&- 12x^{39} &&+ 7x^{38} &&- 7x^{37} &&+ 7x^{36} \\\\\n&+ x^{35} &&- 3x^{34} &&+ 10x^{33} &&+ x^{32} &&- 6x^{31} &&- 2x^{30} &&- 10x^{29} &&- 3x^{28} &&+ 2x^{27} \\\\\n&+ 9x^{26} &&- 3x^{25} &&+ 14x^{24} &&- 8x^{23} && &&- 7x^{21} &&+ 9x^{20} &&+ 3x^{19} &&- 4x^{18} \\\\\n&- 10x^{17} &&- 7x^{16} &&+ 12x^{15} &&+ 7x^{14} &&+ 2x^{13} &&- 12x^{12} &&- 4x^{11} &&- 2x^{10} &&+ 5x^9 \\\\\n& &&+ x^7 &&- 7x^6 &&+ 7x^5 &&- 4x^4 &&+ 12x^3 &&- 6x^2 &&+ 3x &&- 6\n\\end{align}", "3d27f0f6f1d403b59552338edd53321a": "\\scriptstyle H_n \\otimes H_m", "3d2862aedf30d0f1c45d4206ad5ef5c7": "p_g(X|B)", "3d28bfd5c33fede375257d71dddc1873": " x_{ij} ", "3d2905d2f31e05dcab53007a0a2deae5": "p(S|W_1)", "3d29080a53fd4fcc345a845ee77bed8a": "r^3\\,", "3d292cc96f41f097ad38a7d72a35d38b": "\n\\nu^* = \\nu_\\mathrm{ei}\\,\\sqrt{\\frac{m_\\mathrm{i}}{k_\\mathrm{B} T_\\mathrm{i}}}\\,\\epsilon^{-3/2}\\,qR,\n", "3d292e38180275bb90c7a8161c523764": "C(G)", "3d2937a79a35dec62e7d7a99d7e65385": "\\scriptstyle{E_{1}}", "3d293d672c91f9c3e18330a75dfb6324": "\\log(d)", "3d29528ad50679e9c85830fb042efe9f": "\\text{apparent immersed weight} = \\text{weight of object} - \\text{weight of displaced fluid}\\,", "3d295498a7aeb8f0fd1fe65229681844": "f'':\\bar{V}\\otimes V\\rightarrow \\mathbf{C}[G]", "3d299a067cc004b36d4d060e68c5b0c4": "a \\in E", "3d29fad9984d024f63e3ebb7a177cf92": "S_m\\,", "3d29fd5ded11cc0d88d75e1b442c345d": "\\lambda_{MFP}", "3d2a0c8e0e51b4a5e770abfe5003c21f": "I(n) = n!\\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\frac{1}{2^kk!(n-2k)!}", "3d2a426f132f2d081bb26e431d914bd8": "v_n(t+\\tau) \\le b_n\\tau+\\sqrt{b_n^2\\tau^2-b_n\\left(2\\left[x_{n-1}(t)-s_{n-1}-x_n(t)\\right]-v_n(t)\\tau-v_{n-1}(t)^2/\\hat{b}\\right)}", "3d2a7c7d467b909e43fe36e62fc9377b": "L_m,", "3d2ac91b2c44afe76f04c7d03343d203": "P(E_i) = g(E_i)/(e^{(E-\\mu)/kT}+1)\\,\\!", "3d2ada4095b1ad3802ae83ef6a28bf47": "O(1/\\sqrt{s})", "3d2ae5172212e1e5bcc2be828978dbb4": "l, m,", "3d2ae8a93326e9d2ec6444a79d1db3e1": "{13 \\choose 1}{4 \\choose 3} + {13 \\choose 1}{4 \\choose 2}{12 \\choose 1}{4 \\choose 1} = 3,796", "3d2af17609ac4fe69cb6f189f3610e9b": "p \\mapsto F(x, y, p)", "3d2b4d870e9071026e01aaad4461e931": "a\\mathrm{R}p", "3d2b93b18cbf0ba0773459f79fcea143": "T = wR", "3d2bdfd51278a70c08db3dbccdcc1e91": "g_{\\tau\\tau}=\\sum\\limits_{\\mu,\\nu}\\frac{\\partial x^\\mu}{\\partial \\tau}\\frac{\\partial x^\\nu}{\\partial \\tau}\\underbrace{g_{\\mu\\nu}}_{\\delta_{\\mu\\nu}} = \\sum\\limits_\\mu\\left(\\frac{\\partial x^\\mu}{\\partial \\tau}\\right)^2=m^2 a^2+2m^2ab\\cos(n\\cdot \\tau)+m^2b^2\\cos^2(n\\cdot \\tau)+b^2n^2\n", "3d2c07864ba32be4f54c38c0b8e15a4b": "s_1,\\ldots s_c", "3d2c15f14853dc884bea9a736ba1943b": "\\langle A | B \\rangle = A_x^*B_x + A_y^*B_y + A_z^*B_z ", "3d2c27c6608a7ce501613aa5c2e76633": "\\text{maximum speedup } \\le \\frac{p}{1 + f \\cdot (p - 1)}", "3d2c9314a4a8874d2ef8113218dbbbdc": "dY=F_A dA+F_K dK+F_L dL", "3d2cc00235db649bab861706dfe6ffc2": " \\operatorname{build-param-lists}[x\\ (q\\ q\\ x)), D, V, \\_] ", "3d2d0559b683296e27a8b70b4b117a3f": "t_p \\neq 1 + r_p", "3d2d0f1d9758932c734f18f65ac96286": " = 2^b\\,", "3d2d24bb4a445d49075ecc2404e71671": "0 = - m \\ddot{\\vec{x}}[t] - m \\nabla \\zeta [\\vec{x} [t],t] ", "3d2d63d422a4250ce53f3123dff0199a": "\\bar{x}, \\bar{y}", "3d2d778319f0fc049bdf15265d2fd119": "\n\\frac{q_C}{q_H} = \\frac{T_C}{T_H}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(6)", "3d2de5dc876b3677516e61b3b8ceaedb": "\n (\\lnot L)\n ", "3d2e676c69f83a402c6085544d9c3f15": "\\langle L, \\wedge, \\vee, \\otimes, \\rightarrow, 0, 1 \\rangle ", "3d2ed8d7755911bb5ddeafc3e7f5abff": "x(t_j)", "3d2f54628bc4de17f9eb96a702dc7191": "M, w \\models \\Box \\Box \\varphi", "3d2f6b19a42d46d7c416c6a9de8620ca": "h:(X, A) \\rightarrow (Y,B)", "3d2f856760d9760f30d8bdba8c16b022": "\\mathbf{S} = \\{\\boldsymbol{x} \\in \\mathbf{S}_0: h_j(\\boldsymbol{x}) \\leq 0, j=1, \\ldots, m\\}", "3d2f89ddb4419bd3267cc3f4d61f8a09": "2^r \\mid \\varphi(n).", "3d2f8c8080d49748403e79ece393b077": " I_i = \\partial{S}/\\partial{E_i}.", "3d2f9e254afe4be361f104d3748e8570": "X.", "3d2fa0929c6ff7e2db14fe2f699226f8": "\n \\lambda_i\\frac{\\partial W}{\\partial \\lambda_i} =\n C_1\\left[-\\frac{2}{3}J^{-5/3}\\lambda_i\\frac{\\partial J}{\\partial \\lambda_i}(\\lambda_1^2+\\lambda_2^2+\\lambda_3^2)\n +2J^{-2/3}\\lambda_i^2\\right] + 2D_1(J-1)\\lambda_i\\frac{\\partial J}{\\partial \\lambda_i}\n", "3d2fa7832a985224cb0e5e425848fbab": " {\\rm Tr}_\\omega(A) = \\omega \\left( \\left\\{ \\frac{1}{\\log(1+n)} \\sum_{k=0}^n \\lambda(k,A) \\right\\}_{n=0}^\\infty \\right), \\quad A \\in L_{1,\\infty} . ", "3d2fac4181ff563830d4233be82bf9d1": "L = \\left \\{ a^{n,n} | n > 0 \\right \\} ", "3d2fc0cfdc7aec5f5c3e9a7490f0d973": "\\mathbf{\\Sigma}^1_{n+1}", "3d302c9c252710dac2d0d006d0389946": "f: \\{0,1\\}^{n} \\rightarrow \\{0,1\\}^{m}", "3d30583f82052825e48e38c97ce0578f": "{dN_2\\over dt}=r_2 N_2 {K_2-N_2-\\alpha_{21} N_1\\over K_2}", "3d31084f130bfce41a6e382beea2158e": "f(t) = \\log(t)", "3d312ede65310f43aca62cc2b432e235": " V_j(K) = \\binom{n}{j} \\frac{W_{n-j}(K)}{\\kappa_{n-j}},", "3d318093b6c57d5e1d2dd3b5cae6728b": "\\theta_i = \\frac{x_i q_i}{\\displaystyle\\sum_{j=1}^{n} x_j q_j} \\mathrm{\\,\\,;\\,\\,} \\phi_i = \\frac{x_i r_i}{\\displaystyle\\sum_{j=1}^{n} x_j r_j} \\mathrm{\\,\\,;\\,\\,} L_i = \\frac{z}{2}(r_i - q_i)-(r_i-1)\\mathrm{\\,\\,;\\,\\,} z=10", "3d31c3f15ac667d672ad2522fcaeda4a": "\\Delta f=\\frac{1}{r} \\frac{\\partial}{\\partial r} \\left( r \\frac{\\partial f}{\\partial r} \\right) + \\frac{1}{r^2} \\frac{\\partial^2 f}{\\partial \\phi^2} + \\frac{\\partial^2 f}{\\partial z^2} =0", "3d322604bb0f49602ef371453430bfa1": "y(t)=v_{bullet}\\sin(\\delta\\theta)t-\\frac{1}{2}gt^2\\,", "3d326cf78f3f78797768c5f5e704ab0a": "T = \\frac {\\bar{X}_1 - \\bar{X}_2 - (\\mu_1 - \\mu_2)} {\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2}}} : t_\\nu ,", "3d32c21637edf9abdd7f2f91818acf34": "(V,E,F,s,t,r)", "3d32e10feb40f7066b5b7158cd7ed342": " \\frac{d\\hat{\\mathbf{r}}}{dt}=\\dot{\\hat{\\mathbf{r}}} = \\dot\\theta \\hat{\\boldsymbol\\theta},\\qquad \\dot{\\hat{\\boldsymbol\\theta}} = -\\dot\\theta \\hat{\\mathbf{r}}", "3d32fe99d72034fac686214d1a626d95": "\\|x\\| \\geq 0", "3d332c5084a90682e8d5ad304081c3ce": "K = \\frac{\\bar P- \\bar P_e}{1-\\bar P_e}", "3d33746670952cbe78d7992319b96dcd": " \\vec{\\nabla} \\cdot \\vec{Q} ", "3d33a2f6fbd77b681ebdad45dd9597e9": " {} = {PX.QX \\over PY.QY}, ", "3d33cea6f3bffaa9f675145f0883294f": "FOV \\propto \\frac{1}{\\Delta k} \\qquad \\mathrm{Resolution} \\propto |k_{\\max}| \\ .", "3d33de251bacdff387b647f0f21926da": "\\operatorname{Majority} \\left ( p_1,\\dots,p_n \\right ) = \\left \\lfloor \\frac{1}{2} + \\frac{\\left(\\sum_{i=1}^n p_i\\right) - 1/2}{n} \\right \\rfloor. ", "3d3491cd9f05df067744409c34b3ca25": "\\theta_{HPBW}", "3d34b677cb98368e640c1ae5788ba0a0": " \\mathfrak g = \\mathfrak g_{+1} \\oplus \\mathfrak g_0\\oplus \\mathfrak g_{-1}", "3d34f2706de646c3a60d99d7d750ec13": " a(x) = -\\omega^2 x.\\!", "3d34f64aef406dcd7cb983347294fd93": "\\begin{align}\nf(u) &= \\log \\cosh (u); \\quad g(u) = \\tanh (u); \\quad {g}'(u) = 1-\\tanh^2(u) \\\\\nf(u) &= -e^{-u^2/2}; \\quad g(u) = u e^{-u^2/2}; \\quad {g}'(u) = (1-u^2) e^{-u^2/2}\n\\end{align}", "3d3519ae418c0240400ab1b0f95f12bf": " \\int_{-\\infty}^\\infty \\Phi(a+bx)^2 \\phi(x) \\,dx = \\Phi\\left( \\frac{a}{\\sqrt{1+b^2}} \\right)-2T\\left( \\frac{a}{\\sqrt{1+b^2}}, \\frac{1}{\\sqrt{1+2b^2}} \\right) ", "3d35324d1940ddf0c5b7f9d33e1c5ceb": "\\{e^{c_1t},e^{c_2t},\\ldots,e^{c_nt}\\}", "3d3533cabe40183813bb2697656b7b7b": "k_\\theta/L", "3d35b47d6a6af99243f4e0b2b8dc1f27": "\\begin{align} p(\\mu) & \\propto \\sqrt{I(\\mu)}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{d}{d\\mu} \\log f(x|\\mu) \\right)^2\\right]}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{x - \\mu}{\\sigma^2} \\right)^2 \\right]} \\\\\n& = \\sqrt{\\int_{-\\infty}^{+\\infty} f(x|\\mu) \\left(\\frac{x-\\mu}{\\sigma^2}\\right)^2 dx}\n= \\sqrt{\\frac{\\sigma^2}{\\sigma^4}}\n\\propto 1.\\end{align}", "3d35ec0dd634b0aa6bef5cd2abcb9334": " \\mathrm{Re}_m = {{\\rho_m {\\mathbf v_m} L_m} \\over {\\mu_m}} = {{\\rho_p {\\mathbf v_p} L_p} \\over {\\mu_p}} = \\mathrm{Re}_p", "3d35fb9307c219666da468472d6b18fc": "Y := \\overline{\\mathcal{M}}_{g, n} \\times X^n,", "3d35ff8959592292fb017b19f03281ad": "\\frac{N_{++}- N_{+-}- N_{-+}+N_{--}}{N_{total}}", "3d360bc26813be5b4e933018d26fd9b9": "q(x) = F \\delta(x-x_0).\\,", "3d3625bfbddd06bc90add0c09abe9f1f": "\\mathbf{\\theta}_{x,y,z} = \\langle 0,0,0\\rangle,", "3d362eda5deafd072e8f061c9890452a": "|\\log P:Q|_\\pi = \\frac{\\sup_{p\\in P^'}|\\log p - \\log \\pi(p)|}{\\log \\min \\left(|set(P)|,|set(Q)|\\right)}", "3d368f27ae91611caadea1b220bcb825": "1.43^{-1} + 2.85^{-1} = 1.051", "3d36e221e558a9c02bb376e006d5c7b5": "\\left(\\sum_{i=1}^n k_i x^i\\right) e^{a x} \\cos(b x) \\;\\;\\mathrm{or}\\;\\; \\left(\\sum_{i=1}^n k_i x^i\\right) e^{a x} \\sin(b x)\\!", "3d370d4473a74ddf072ba9888c5302d1": "\\pm\\sqrt{\\csc^2 \\theta - 1}\\! ", "3d373691f92c5138ff2eb0ffe7e206e6": "S^1 \\wedge X_n \\to X_{n+1}", "3d373edb7813567e055a461a31ff8e9e": "\\Delta\\omega=\\omega_2-\\omega_1\\,", "3d37433f335f66b0aeaf6e68b081fc8a": "u\\Rightarrow v\\,", "3d377e5483773e0afd3cc1962343ec2d": "k\\,\\ ", "3d37805f27d24f342639fc8678618fa1": "X \\sim \\chi_k", "3d37bd88e511f3b9c2742813f4753ef5": "\nD(i,j)=\\frac {1}{N-1}\\sum_{k = 1}^{N-1} DP(i\\rightarrow k|j) \n", "3d384e61398fe5e4991f539ac3b612e5": "(k=3)\\,\\!", "3d38ebc8a036abeed59c19ae3e9c4e19": "V_{j^n(\\kappa)}", "3d38ec0526ad5392422c46707bc4a374": "B \\otimes A", "3d38fa4433da6cea69f521c01155ed49": "\\textstyle(x-1,y-1)", "3d3906035dbe8214815dbeb9e29d2dac": "M: V \\to V ", "3d39b512dbd5652384c2ec0638a60e15": "\\sqrt{4-2\\sqrt{2}}.\\ ", "3d3a06741ec4824f29edf69b1b9b9a79": "b_k = (-1)^{k} a_{N - 1 - k} ", "3d3a3d57751be9f97124fe0f961dd34c": "\\Rightarrow \\frac{\\pi}{2}=\\prod_{k=1}^\\infty \\left(\\frac{2k}{2k-1} \\cdot \\frac{2k}{2k+1}\\right)=\\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\cdots", "3d3a523f564133686d6c79433b8a0013": "\n d\\boldsymbol{\\sigma}:\\frac{\\partial f}{\\partial \\boldsymbol{\\sigma}} \\ge 0 \\,.\n ", "3d3a609fde69e0b922cfa18ecf3d4f6f": "\\xi\\colon X \\to BG", "3d3ab8ab572deef415cfbfd3c4f01b68": "g =\n\\begin{cases}\n- \\sqrt{\\frac{5}{9(x^2+1)}} \\cdot S, & \\mbox{if}~T>\\frac{1}{2} \\\\\n\\sqrt{\\frac{5}{9(x^2+1)}} \\cdot S, & \\mbox{if}~T<\\frac{1}{2} \\\\\n0, & \\mbox{if}~T=0 \\\\\n\\end{cases}\n", "3d3ac89d756750e1c29b920d07690262": "d(\\mathbf{x}, \\mathbf{y}) = \\|\\mathbf{x} - \\mathbf{y}\\| = \\sqrt{\\sum_{i=1}^n (x_i - y_i)^2}.", "3d3b1193f98ffa035e1101bab627817d": " z\\in\\mathbb{R}^d ", "3d3b5260388abc2f3bba927c795da9c5": "f(p_1) = f(p_2)", "3d3c0bb422b55003c65dc2e7d55e4e82": "A - B", "3d3c19db6a19e177fbb1e4a5056c2f72": "\n (CL)\n ", "3d3c1ad223f2b80138a85ae3db89cd26": "\\mu_0 = 4 \\pi \\times 10^{-7}", "3d3c8583dd4b9d0789f88ae143133315": "x_1 \\leq x_2 \\leq \\ldots \\leq x_n", "3d3c8ca50211b309086da3bbc87198cd": "\n\\frac{\\partial^2 G^{\\star}_n}{\\partial x^2} \\, + \\, \\alpha_n^2 \\, G^{\\star}_n = F_n(x)\n", "3d3ccd0caebd5535485d8c8f256024e7": " A-C(W) \\in [0, 1] ", "3d3cde77f2af8d56191f9f03bcdb99dd": "\\varphi(\\lambda_1 x_1+\\lambda_2 x_2+\\cdots+\\lambda_n x_n)\\leq \\lambda_1\\,\\varphi(x_1)+\\lambda_2\\,\\varphi(x_2)+\\cdots+\\lambda_n\\,\\varphi(x_n),", "3d3d1c222c6f03883157a95ca20f1676": "\\sigma_{SS}=E(\\sin^2\\theta)-E(\\sin\\theta)^2\\,", "3d3d38f5bcba4aef72a74acf352c6adc": "r_{o2}", "3d3d6662ffca5e8bc580a39094228103": "\\begin{matrix}\np \\oplus q & = & \\lnot ((p \\land q) \\lor (\\lnot p \\land \\lnot q))\n\\end{matrix}", "3d3da5d1b7abba432d991b6fc6b6f693": "1/L\\,", "3d3db501bdb5255d5d8dedd268585234": "c_1x_1^r+\\cdots+c_nx_n^r=0,\\quad c_i\\in\\mathbb{Z}, i=1,\\ldots,n", "3d3ddbe26b75a04b4c781cf2060b6cc5": " \\text{(1)} \\qquad W = \\oint P \\ dV ", "3d3dddb3715ce48f64df8c3134c711b1": "\\{\\omega=T\\}", "3d3e00e0b84ad6b64a3461fe9092698a": "x>1", "3d3e18d1279fd0f88b1234f65011ed24": " \\Pr(X_1=k_1,X_2=k_2)= \\exp\\left(-\\lambda_1-\\lambda_2-\\lambda_3\\right) \\frac{\\lambda_1^{k_1}}{k_1!} \\frac{\\lambda_2^{k_2}}{k_2!} \\sum_{k=0}^{\\min(k_1,k_2)} \\binom{k_1}{k} \\binom{k_2}{k} k! \\left( \\frac{\\lambda_3}{\\lambda_1\\lambda_2}\\right)^k ", "3d3e27ce0543187883076e790804de73": "\\sigma(X^*, \\hat{X})", "3d3e466db27577fe81631774f83f3890": "S = \\left\\{ \\left. h \\colon C_{0} \\to L_{0}^{2, 1} \\right| h \\mbox{ is bounded and non-anticipating} \\right\\},", "3d3e4c8b3f47e706524b3c425e0804cd": " u_t ", "3d3e8029ff97f4eef21d8b0fd4f10ca5": "u = e^{\\sigma\\sqrt t}", "3d3e80da40f6558d18f9d20e1e79dc3f": "Z_\\infty^{p,q}", "3d3ecd338d020a1a1c7b6cdd92330af1": "\\sum_{i=1}^{\\infty} \\left \\lfloor \\frac{n}{p^i} \\right \\rfloor .", "3d3f5aa8cf079711cfcf4431a58e3338": "\\frac{X_{[nt]} - n/2}{\\sqrt{n}}", "3d3f6aaf2180e465418bd7ecf83d1b5b": "[[A+B]]\\ne[[A]]+[[B]]", "3d3f8ee6b6f8be94900b8104862b1d66": "\\textstyle N_1=f(\\lambda)", "3d3fb4683d5fbc72138fa5ff4e60d1d8": "A:=\\{z\\in\\mathbb C:r_1<|z|0} e^{-2mx},", "3d40cbcc9ff9fa72731815d82c6f655e": "u_1(z) \\le u_2(z) \\le ...", "3d40cca06d99cf97de846df0bd4983e1": "a_k^\\nu=2(\\nu+2k) \\int_0^\\infty f(z) \\frac{J_{\\nu+2k}(z)}z \\,dz\\!", "3d40daa8ca1f02151472c68fe326e25d": " A_{fb} (f) = \\frac { A_{OL} } { 1 + \\beta A_{OL} } ", "3d4101e72fd64feddbfb413d63ec680a": "\\overline{e}_f(k,i+1) = \\frac{\\overline{e}_f(k,i) - \\overline{\\delta}(k,i)\\overline{e}_b(k-1,i)}{\\sqrt{(1 - \\overline{\\delta}^2(k,i))(1 - \\overline{e}_b^2(k-1,i))}}", "3d411de08b6c2e95b701a7f4b505ecd3": "j^+_\\mu = \\overline U_{iL}\\gamma_\\mu D_{iL} +\\overline \\nu_{iL}\\gamma_\\mu l_{iL}.", "3d41570a4699330742c549562acb9326": "t_r \\ll t_f", "3d418e18c7f6fe6e40fdc5b2fa6619fc": "\\int_{-\\infty}^\\infty f(t)\\,dt = \\lim_{x\\to\\infty}\\int_{-x}^x f(t)\\,dt.", "3d41c871e788a099dd7444ee895b946d": "\\mathfrak{n}=\\bigoplus \\mathfrak{g}_\\alpha,", "3d4212cbc0dc55985e9fa3180c96ec23": "f(n) \\geq g(n)\\cdot k", "3d4264d0745e678d568e4f54640dee70": "\\frac{1}{2^\\frac{np}{2}\\left|{\\mathbf V}\\right|^\\frac{n}{2}\\Gamma_p(\\frac{n}{2})} {\\left|\\mathbf{X}\\right|}^{\\frac{n-p-1}{2}} e^{-\\frac{1}{2}{\\rm tr}({\\mathbf V}^{-1}\\mathbf{X})}", "3d4282e1768a3c84a3cd3b663ac3de98": "\\neg \\exists y. \\phi(x,y,z)", "3d42d78cc25f0f0ba2b52ac17801d535": "\\mathbf{(1)}", "3d42dee355a07c5d2e58cd1999a64a21": "(n_0,n_1,...)", "3d42ef7a86af47fa400ce3a3b02188ec": "\\int_{x\\in M}\\varphi(x)N\\!J\\;F(x)\\,dM = \\int_{y\\in N}\\int_{x\\in F^{-1}(y)} \\varphi(x)\\,dF^{-1}(y)\\,dN", "3d430c9db8256f72123c9ff31ae986ad": "p_i \\mid p_j - 1", "3d43446617b5dbf00c534cbc2f0e7d15": "\\neg \\psi = \\forall x_1 ... \\forall x_n \\neg \\phi", "3d435871ff49fbcba0f36fc3b4750b5b": "\\Psi_i(\\alpha)", "3d43a191b0c48a94272125b12bac0610": " n \\geq {t+d-1 \\choose d-1} ", "3d43c25b73a0cb7e7eef2f4270154448": "(h {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} k)(X_1,X_2,X_3,X_4) := h(X_1,X_3)k(X_2,X_4) + h(X_2,X_4)k(X_1,X_3) - h(X_1,X_4)k(X_2,X_3) - h(X_2,X_3)k(X_1,X_4) ", "3d43f7997295c0fe2ffcdcdd41c58441": "\\left(\\frac{D}{N}\\right) = 1", "3d442243ca040e3418c59aacfb70a1a9": "\\frac{Z}{Z_0} = \\frac{Z_0}{Z'}", "3d44597061ad6d1f91f99f3504349537": "\\displaystyle{P_rf(e^{i\\theta})=\\sum_{n\\in \\mathbf{Z}} a_n r^{|n|} e^{in\\theta}={1\\over 2\\pi}\\int_0^{2\\pi} {(1-r^2)f(e^{i\\theta})\\over 1-2r\\cos\\theta + r^2}\\,d\\theta =K_r\\star f(e^{i\\theta}),}", "3d447953f58332ee242d918035bf81cb": "C_{Cr-corrected} = \\frac{{C_{Cr}} \\ \\times \\ {1.73}} {BSA}", "3d44c8e5c90627fca0824ef1038ab81a": "(\\frac {\\partial h}{\\partial t})\\bigtriangledown (h(\\vec {U} + (\\frac {\\vec {t}}{f}) - \\bigtriangledown(E\\bigtriangledown h) = R ", "3d44d347d326cb4a1006c986f555fbaf": " \\vec{m} = \\frac{1}{\\sqrt2} \\, \\left( \\partial_x + i \\, \\partial_y\\right)", "3d44e6dc2bbd62eb39b2b6dec1a2b0a8": " P\\left[ (\\tilde{X}^n,\\tilde{Y}^n) \\in A_{\\varepsilon}^n(X,Y) \\right] \\geqslant (1 - \\epsilon) 2^{-n (I(X;Y) + 3 \\epsilon)}", "3d4522858aa0d076ec4a508364d20abf": "U(t+\\bigtriangleup t,w)=U' exp(iw\\bigtriangleup t)\\exp([-bw+ iH(bw)]\\bigtriangleup t) \\quad (1.7)", "3d45a3451d4dd9bce0822551a6640941": "s = \\frac { m_\\mathrm s + 1 } { m_\\mathrm s } f ,", "3d45b81d1df20166f1957e8e1c3effe6": "\\begin{align}\ne^{\\pi \\sqrt{19}} &\\approx 3^5 \\left(3-\\sqrt{2(-3+1\\sqrt{3\\cdot19})} \\right)^{-2}-12.00006\\dots\\\\\ne^{\\pi \\sqrt{43}} &\\approx 3^5 \\left(9-\\sqrt{2(-39+7\\sqrt{3\\cdot43})} \\right)^{-2}-12.000000061\\dots\\\\\ne^{\\pi \\sqrt{67}} &\\approx 3^5 \\left(21-\\sqrt{2(-219+31\\sqrt{3\\cdot67})} \\right)^{-2}-12.00000000036\\dots\\\\\ne^{\\pi \\sqrt{163}} &\\approx 3^5 \\left(231-\\sqrt{2(-26679+2413\\sqrt{3\\cdot163})} \\right)^{-2}-12.00000000000000021\\dots\n\\end{align}\n", "3d45fd9ab61ebafa54c76f208663b6cf": "\\tfrac1{24}", "3d4632eca643876e2623480dcb7675e2": "f(\\zeta) = \\frac{1}{2\\pi i}\\int_C \\frac{f(z)}{z-\\zeta}\\,dz.", "3d4637081dbe704799a37f5c488d2243": "S_{21} = \\frac{1}{T_{22}}\\,", "3d466f279a72923cf83fde3f677dcde1": "f(x)=\\sum_{k=0}^\\infty\\frac{\\Delta^k [f](0)}{k!}(x)_k", "3d46861a29124660528d979cb7c11711": "\\left|x - a\\right| + \\left|y - b\\right| + \\left|z - c\\right| = r.", "3d46aee09f0ed84515eec52b72ab88f9": "0 < d \\le 1 ", "3d46b37ca497b3ae9e7d437aa0c35154": "m = (\\lambda_1,..., \\lambda_n)", "3d473e40fadfba4226c12802dfd32161": "\\Omega\\,e^{\\Omega}=1.\\,", "3d4748e3d9f5264d389325d2a7469693": "q_{bi}", "3d474950e04b93d75504c4811273fdf4": "7572_{11} \\ ", "3d475731b634152200fa00134df7337a": "\nW(4) = \\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & 1 & -1 & -1\\\\\n1 & -1 & -1 & 1\\\\\n1 & -1 & 1 & -1\\\\\n\n\\end{bmatrix}\n", "3d477f82d13530b442e56a142fa2d921": "\\sum_{j = 0}^{\\infty} e^{-\\lambda/2} \\frac{\\left(\\frac{\\lambda}{2}\\right)^j}{j!}\\frac{x^{\\alpha + j - 1}\\left(1-x\\right)^{\\beta - 1}}{\\mathrm{B}\\left(\\alpha + j,\\beta\\right)}", "3d479b587b46add35404d21729109ba0": "|b\\rangle", "3d481a6a3cf16ddffa1d53f49681add8": "\\displaystyle \\hat{f}(\\nu)", "3d481abc3a7cfd94c7e35bbc4c89e42c": "\\displaystyle{|\\alpha +\\alpha^{-1}|<|\\alpha| +|\\alpha^{-1}|.}", "3d4823cb8879f0fee19328cea101e067": "k_j^j = N", "3d48357f21bc25e26f1c7bfd16cbfd35": "\\begin{smallmatrix} \\sqrt{18^2 + 29^2 + 13^2} = 36.5\\,\\text{km/s.} \\end{smallmatrix}", "3d4852c513067caf506fff12f8d76e1a": "\\scriptstyle r_k\\leq1+\\|a_0-a_j\\|\\leq 1+\\|x_0-x_j\\|\\leq r", "3d4860fba44774e2f922e04bdff0f492": "U_{\\mathrm{breakdown}}", "3d4894f68f7e4a36012bf2881edccf42": "\\Delta x_n=-\\nabla_x f (x_n) ", "3d489d50bf8f5edf3fa763e0c85cafae": " \\;_mF_n()", "3d4937e955005133d7c5763fd6c9595d": "f_1/f_2=-n_1/n_2", "3d494605b0f35571757d9cf1630a528b": "f(t) = \\lim_{n\\rightarrow\\infty}f_n(t)\\text{ for almost every }t, \\, ", "3d49c195c881680675e9e305761ca349": "r < 0 < s ", "3d49fceb11e05aa2aea7fa285afbcd81": "\\frac{C_p}{C_v}", "3d4a0cf2406ffbe76e52827943385e52": "V_i=\\langle w_i\\rangle", "3d4a1c1060e1ff79eac15ddef7e00441": "| \\{ e \\}(k)| = \\gamma_k ", "3d4a5dfbfd4c46ea1ffb59a0478f9e26": "|\\phi_1\\rang, |\\phi_2\\rang, |\\phi_3\\rang,", "3d4a71a77bf3d13b62ca4a2eb5b39957": "y(t) = A \\cdot \\sin(\\omega t + \\phi),", "3d4a82430b7c643f62d07c095dbe4d4c": "x^2 + 2x + 1.", "3d4ab9fb4f7ddc788913b1e8dbfbe991": "\\pi_i q_{ij} = \\pi_j q_{ji}", "3d4ad0da5f3f86054c2ba4d3228e40fd": "\\overline{c} \\langle y \\rangle.P", "3d4aef1473025680b5b9d77453583ff0": "5 \\mid 0", "3d4b36e163788ab140bc08bd50a54953": "\\dot{W}", "3d4b612169b117aa252dc5bddc8c8190": "(r,t,h)=(\\sqrt{13},\\arctan{(3/2)},4)", "3d4b976ae16fa46972b78c4519984010": "\\beta_0,\\dots,\\beta_p", "3d4b99e8768fb5dd646463f86ef1ad9f": "=\\mathcal{FF}(\\mathbf{S}(x,y)) ", "3d4b9fceb4533485de417c94bdb9e6ce": "KL(P||Q) = \\sum_{i \\neq j} p_{ij} \\log \\frac{p_{ij}}{q_{ij}}", "3d4ba6e763e2456e6172a4a0cb8e3de0": " G=U(1) ", "3d4bd8f9fd407ddfb4b64d1a49205f31": " E_{t}u^{\\prime }(c_{t+1})=u^{\\prime }(c_{t}) ", "3d4c05d7fb13058c014faa7ac03fa831": "G_{i-1}", "3d4c7e65ead927df618ef5258270a0a3": " X(t) = K \\exp\\left( \\log\\left( \\frac{X(0)}{K} \\right) \\exp\\left(-\\alpha t \\right) \\right) ", "3d4cd6d69ad92e0e1a6adcd208006187": "(1^*(01^*0)^*)^* \\,\\!", "3d4ce0f6b2ea039ed05d29cce15a107e": "\\begin{align}\n\\frac{d\\vec\\omega}{dt} &= \\frac{\\partial \\vec \\omega}{\\partial t} + (\\vec v \\cdot \\vec \\nabla) \\vec \\omega \\\\\n&= (\\vec \\omega \\cdot \\vec \\nabla) \\vec v - \\vec \\omega (\\vec \\nabla \\cdot \\vec v) + \\frac{1}{\\rho^2}\\vec \\nabla \\rho \\times \\vec \\nabla p + \\vec \\nabla \\times \\left( \\frac{\\vec \\nabla \\cdot \\tau}{\\rho} \\right) + \\vec \\nabla \\times \\vec B\n\\end{align}", "3d4ce37755e237828d507b8832b0c675": "\n\\mathcal C", "3d4d2beeebd5a49405ef67744141a746": " g(x) ", "3d4d5decdd03f34a8aab4a261a32b189": " \\widehat{\\mathcal{C}}_{XY} = \\frac{1}{n} \\sum_{i=1}^n \\phi(x_i) \\otimes \\phi(y_i) ", "3d4da322347772bd1e4ac5b460c29454": "p \\in l", "3d4dd9b228d2b18eece3f25acc073224": "D_2=1P_1+ 5P_2 ", "3d4e402769d7953ff647d680c5858bc5": "H(\\theta,0)=\\int_{\\R^n}\\left(\n\\frac{1}{2}|\\nabla\\theta|^2+\\frac{1}{2}f(\\theta)\n\\right)\\,d^n x", "3d4e49b0a38e600b3c6128d71b89c953": "\\delta(x) = \\begin{cases} +\\infty, & x = 0 \\\\ 0, & x \\ne 0 \\end{cases}", "3d4eb4a5a51e0398441ee5e05a5deb48": "\\bigcup_{\\beta<\\alpha}A_{c_{\\beta}}", "3d4f208a0557bcc09abeea1ae2876638": "\\epsilon_0=1/4\\pi", "3d4f699e01bfdbaedd32697b36350a56": "P_{k}", "3d4f8dfa739d720fea444074732bd2eb": "\\left\\langle\\phi|H|\\phi\\right\\rangle \\ge E_g \\sum_n |c_n|^2 = E_g. \\,", "3d4fa239e05b672851aeb59bea284e51": " \\int_\\Omega \\nabla u \\cdot \\nabla v \\,dx = \\int_\\Omega f v \\,dx.", "3d4fc125dcd91567b7b4e1ce167d07aa": "I_\\text{axis}", "3d5062bc4447c2d12f894f2b212c9fe0": "A P U\\, ", "3d508163bf3af76a4e1244204fc62f24": "f\\in{\\mathbb Z}[x]", "3d50c3e0f546cef0d426d981f8acad06": " y_i = \\beta x_i + \\varepsilon_i, ", "3d50c4f03ccb89abd2e40a076864ab38": "\\alpha = \\sqrt{\\frac{j \\omega \\mu_0 \\eta}{\\rho}}", "3d50eb69310df4edcc07ba9c2944e518": "L_n[1, 1/2] = n^{1/2+o(1)}.\\,", "3d50f5cf3a246944ca23f6b644e99041": "p_n=p\\cos(\\psi)", "3d5111c44e5014c7e0e85e0ffa2d1c17": "\\hat K_j(1)", "3d517eda720f9d81415f4acc8032adf6": "\\mathcal{L} (\\mu,\\sigma) = f(x_1,\\ldots,x_n \\mid \\mu, \\sigma)", "3d51b452ad5d04e65ada739372c8079b": "\\boldsymbol{U} = \\boldsymbol{\\mathit{1}}", "3d5235da298c18f921cce1478a1a1138": " = ([(m + 2 + 1)2.6 - (2.6m - 0.2)])\\ \\bmod\\ 7 ", "3d523603a9bc55b2a8e55c481e4ea037": "\\varepsilon_0>0,", "3d52490731e8fa0758689bcaa6ff8679": "\\frac{\\partial k_{i}}{\\partial t}=\\frac{mM_{}}{m_{0}+t}\\frac{k_{i}}{\\sum_{j\\in Local}k_{i}^{}}", "3d5263e2b52477369880e1ccbee9e785": "\\rho(X+\\alpha)=\\rho_A(X)+\\rho_{A^*}(\\alpha)", "3d5279c2ab513b95b7fc64d130f73e1b": "(11)\\quad ds^2=-\\Big(1-\\frac{2M}{r} \\Big)\\,dt^2+\\Big(1-\\frac{2M}{r} \\Big)^{-1}dr^2+r^2d\\theta^2+r^2\\sin^2\\theta\\, d\\phi^2\\,.", "3d52a361315f4e7ca265c3513c2a6ffa": " \\displaystyle{Df=(3z +z^{-1})f}", "3d53374e69f5ff36b606598d2816dcc2": "c^\\star = - \\frac{\\textrm{Cov}\\left(m,t\\right)}{\\textrm{Var}\\left(t\\right)}; ", "3d53777269d95c34fc977995cddbc3bd": "D_{eff}", "3d53930453ac18d7af4f738e509f3842": "\\Omega=d\\omega+\\tfrac{1}{2}[\\omega,\\omega]", "3d53bea63f17df1d7f1d7cd514ffaed8": "X^{**}", "3d53fc88cd1ab81e067a1f39db7d762e": "u_2 = rw\\,\\bmod\\,n", "3d53fdd2f43f2e1b8970d1015251806c": " C_D\\, =\\, f_c(R_e).", "3d543a27ada580191636e2ef986e5eae": " \\int_0^{\\sqrt n}(1-x^2/n)^n dx \\leqslant \\int_0^{\\sqrt n} e^{-x^2} dx \\leqslant \\int_0^{+\\infty} e^{-x^2} dx \\leqslant \\int_0^{+\\infty} (1+x^2/n)^{-n} dx", "3d545b677f3c760ef8f850a9dbb2c2c9": " S(T) \\!", "3d546056eaca689a8d76a0a143b19223": "\\gamma^2 = \\begin{pmatrix}\n0 & 0 & 0 & -i \\\\\n0 & 0 & i & 0 \\\\\n0 & i & 0 & 0 \\\\\n-i & 0 & 0 & 0 \\end{pmatrix},\\quad\n\\gamma^3 = \\begin{pmatrix}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\end{pmatrix}.", "3d548470c3ac604875fdc0f0af51d049": "\\scriptstyle |c_n|^2. ", "3d54874edf525479ac96d6a5b733d2cc": "\\sigma_{12}=\\sigma_{21}\\,\\!", "3d5498037af995b93519dcf44f313bdc": "F: X^* \\to {\\Bbb R}", "3d54c1b2675650c67b382b68c2d80653": "c(\\zeta, \\tau)", "3d54c7888cd6c27e909a6261ab7a4e4c": "\\alpha_e=", "3d54c9080ff179f4f87432482b5a71c1": "g_6=-x+25x^-15x^3;", "3d54cae6560b365e7fccdc57215e0348": "S_\\gamma", "3d54dca1696bb6b154242be0cf7df28a": "\\frac{9}{5}", "3d550723412a134f1c5039a775df4276": "\\scriptstyle y_q = \\sum(x_i*w_{iq}) ", "3d5512418d50493899e2aecf997146b7": " 10^5 ", "3d557281efe81a5b9684ea3eb7cbfc0b": "{\\Delta z}/{\\Delta t}", "3d557b7d37b8e09396a590ee4037ecda": "\\scriptstyle L=\\mathbf{Q}(\\sqrt{d})", "3d557bc8581fb80415362524907d38be": "\\phi_{A,B}^{}", "3d55bd7892b794d65024477b0ef77042": "Pwo + Pwf = 1", "3d56272188ab318a337df9fd1e38a9cd": " y = z - t \\!", "3d5640ea5749c99d0a574c6ed325cbab": " 2s-t=0", "3d57a78400437a420670d32551963e1c": "\n\\big(G\\gamma _{1}\\cdot \\mathcal{P}_{2}-E_{1}\\beta _{1}+M_{1}-G\\frac{i}{2}\n\\Sigma _{2}\\cdot \\partial (\\mathcal{L}\\beta _{2}\\mathcal{-G}\\beta\n_{1})\\gamma _{52}\\big)\\psi =0, ", "3d57ac14151e3840ec41152e06036fb1": "\\scriptstyle B=\\{1,2,3\\}", "3d57f3df1fdddcfad2c3cd690fe05240": "V(\\mathbf{r}) = \\frac{-k}{r} = -ku", "3d5823b977825e6b22d8b2b15d63c64d": "H = \\frac{kT}{Mg}", "3d58311a009ae7e1f2f8daa4583381f4": "(2^n - 1)^2 - 2", "3d5847a927574c5d02353508e0aa096e": "A_{ce f}", "3d585d7e92925d0d6d38bbadee65c767": " y = \\pm \\sqrt{a x^2 + b x + c} ", "3d588da85f8cf97a67314e4e6b1285f8": " \\operatorname{E}_A (B) = | \\psi_i\\rangle \\langle \\psi_i|, ", "3d589bcf5ebf6a59bb003ae470444ae8": "\n+\\left(\\pi^2/6\\right)+2\\gamma\\ln\\left(\\eta\\right)+[\\ln\\left(\\eta\\right)]^2-e^{\\eta}[\\text{Ei}\\left(-\\eta \\right)]^2\\}", "3d58a16754cb5a3ba81538fe98e5093c": " \\mathbf{C} = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\frac{1}{f} & 0 \\end{pmatrix} ", "3d58d04862eaddc6a8c927e8021bb591": "\\begin{pmatrix} I_p & 0 \\\\ 0 & -I_q \\end{pmatrix}", "3d5901cc504ac52925070e9e4ac99d75": "-\\boldsymbol{\\alpha}(itI+S)^{-1}\\boldsymbol{S}^{0}+\\alpha_{0}", "3d598376cf9c6f38795c754bea1f88c1": "\\mathcal{T}:\\mathcal{A} \\rightarrow \\mathcal{M}", "3d599b9ffc53c3cc21002659ea8fea4d": "(\\frac{C}{m^2})", "3d59a328468fc1d2f4ebf2a32609905e": " \\theta = \\tan^{-1}\\frac{\\Delta h}{\\Delta x} ", "3d59d7aa5b5dc5377b305dfd641f5fb8": "\\lambda \\psi", "3d5af5ddb57a99f8150887a62568387b": " (((P \\rightarrow Q) \\And (R \\rightarrow S)) \\And (\\neg Q \\vee \\neg S))) \\rightarrow \\neg P \\vee \\neg R ", "3d5b7dc34a5ce9e1ec9808a033dfac54": "T(f)(x) = \\int K(x,y)f(y) \\, dy, ", "3d5b8d69b95aa252494a02a66428d2aa": "Q = (x_2:y_2:1)", "3d5bafaa26e4511194c3bbab26fbbba9": "2^{p-1}", "3d5bb5bcf3ea3a27047443dbd4c13374": "\\nu_0", "3d5bd49a5239140346c6d7905b565e24": "RL=\\frac{RL}{L+Ki}", "3d5bd80ebda8616c3da9578ccac22f68": "\nC_6 H_{12} O_6 \\Rightarrow 2 CO_2 + 2C_2 H_5 OH\n", "3d5c095ea58fc3a7732734eaf7c3c511": "\\sum_{n=2}^\\infty 1/ [S(n)]!=1.09317\\ldots", "3d5c209bea662258016b7892174264a6": "f^*EG", "3d5c24f5b48cf8ac8e93fb05439e2fdb": "\\rho_{\\infty i}", "3d5c998feadd76e813ad37e5b3dff9de": "\\; N\\leq s", "3d5cd24f04a70ba9323c87cebb5b688b": " t [ x := s]", "3d5cd2f905ad262ed8025ea9aac52b25": "g^{\\mu\\nu} \\Gamma^{\\sigma}_{\\mu\\nu} = 0 \\,.", "3d5cea2c9678c6d68d59162c68ada924": "(X,=)", "3d5cea3955dcccf3139ee16f7b6699d0": "\\begin{align}\n\\hat{t}(\\tau, \\omega) & = \\tau - \\frac{\\partial \\phi_{\\tau}(\\omega)}{\\partial \\omega} = \n\t\t- \\frac{\\partial \\phi(\\tau, \\omega)}{\\partial \\omega} \\\\\n\\hat{\\omega}(\\tau, \\omega) & = \\frac{\\partial \\phi_{\\tau}(\\omega)}{\\partial \\tau} =\n\t\t\\omega + \\frac{\\partial \\phi(\\tau, \\omega)}{\\partial \\tau} .\n\\end{align}", "3d5cf3d73c4ed91dd7c7b76f61eb18b2": "\n \\mathbf{v}(\\mathbf{X}_B) - \\mathbf{v}(\\mathbf{X}_A) = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} \\boldsymbol{\\nabla} \\mathbf{v}\\cdot~d\\mathbf{X}\n = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} \\boldsymbol{A}(\\mathbf{X})\\cdot d\\mathbf{X}\n", "3d5cff24350f32b9e88c9e3d3e13715b": "\\Gamma(E_0)\\rightarrow \\Gamma(E_1)\\rightarrow\\cdots\\rightarrow\\Gamma(E_N)", "3d5d88282b15332e88fc5603578e0939": " X^{m,s} ", "3d5db75696452266245979a5f92d13b2": "pKa_1= \\log \\beta_{13}-\\log \\beta_{12}\\, ", "3d5dbc5e5a647d219156f2c4faf364d2": "d\\theta = \\frac{1}{r^2} \\left( x\\,dy - y\\,dx \\right)\\quad\\text{where }r^2 = x^2 + y^2.", "3d5e2955ba3f46bb440d11045ab3ca08": "\\displaystyle{L(b)L(a)^2 +L(a(ab))+L(a)^2L(b)=L(a^2)L(b)+2L(ab)L(a).}", "3d5e466ea8bc454cc01775cb7f87a6fb": "\\mathcal{E} = -\\frac{d\\Phi_B}{dt} = B A \\sin{\\theta} \\frac{d\\theta}{dt} ", "3d5ec85b45766cbba97998be5839240b": "P\\left(0\\right)=P_0", "3d5efa1c28aaf6e238f3a407cb31b25b": "\\mathrm{ACA}_0", "3d5f5c556a0fe6b895b6e9690a600934": "\\{\\omega_1, \\omega_2, \\ldots, \\omega_n\\}", "3d5f77baa0d8114b575009259cc44da1": " \\bar r_2", "3d5fb201e4b077a522b73e1330de730e": " L^2({\\mathfrak H}^2)", "3d6032e430e3cd78d145f8189fd09b1c": "(1+x)/(1-x)", "3d60f4fc0d1df841b9b2136e3fd3aa51": "N_G(Q)", "3d6131089b9197d77a07c4343b296eaf": "\\frac{1}{\\Theta}\\ \\cos\\theta\\ \\frac{d}{d\\theta}\\left(\\cos\\theta \\frac{d\\Theta}{d\\theta}\\right)\\ + \\lambda\\ \\cos^2\\theta\\ +\\ \\frac{1}{\\Phi}\\frac{d^2\\Phi}{d\\varphi^2}\\ =\\ 0", "3d614d8ea5b2a617ec9afb1e0fbb3c9e": "u'=v_{A|O'}", "3d623c6832b4c2258dd9506f8fb350a8": "\\Omega\\left(E\\right)", "3d62695063eb4e40ec4850fbfca62804": "d= 0 \\mod \\lambda", "3d62f4f808d5f34e9a771254541cff9c": "\n\\begin{align}\n \\operatorname{cn}^2\\, (z|m) \n &= \\Bigl( \\tfrac12 - \\tfrac{1}{16}\\, m - \\tfrac{1}{32}\\, m^2 + \\cdots && \\Bigr)\n \\\\\n &+\\; \\Bigl( \\tfrac12 - \\tfrac{3}{512}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, 2\\, \\alpha\\, z\\;\n \\\\\n &+\\; \\Bigl( \\tfrac{1}{16}\\, m + \\tfrac{1}{32}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, 4\\, \\alpha\\, z\\;\n \\\\\n &+\\; \\Bigl( \\tfrac{3}{512}\\, m^2 + \\cdots && \\Bigr)\\; \\cos\\, 6\\, \\alpha\\, z\\;\n +\\; \\cdots.\n\\end{align}\n", "3d630418da560838fda5c1e992309c77": "\\frac{5 \\cdot \\pi}{6}", "3d633c75d835511a8223ba36bcb28833": " \\hat{L} X(x) =\\lambda X(x)", "3d644580ec557f3bdd6da0217c58cb9b": " b^n \\left(\\frac{a-b}{b}\\right)^n + b^n n \\left(\\frac{a-b}{b}\\right)^{n-1} =\n(a-b)^n + n b (a-b)^{n-1}", "3d647898faac1664e4c2133d9366ebf9": " \\frac{s}{t} = \\frac{1+x}{1+n}", "3d6494eb9b2dfb1c52c26fe496164e55": "\\frac{d^2y}{dt^2}+\\epsilon(y^2-1)\\frac{dy}{dt}+y=0,\\quad\\epsilon=1", "3d64febb28e60e3bd4519e9356df1443": "f \\circ f^{-1} = \\operatorname{id}_Y, f^{-1} \\circ f = \\operatorname{id}_X.", "3d654df1593743d59bc9f24d729e7675": "T_r", "3d65690a62111f205fed8ec57495f36c": "\\frac{\\partial u}{\\partial t} - v \\beta y = -\\frac{\\partial \\phi}{\\partial x}", "3d6575118139b2a0905089449a9400a5": "m_1 u_1 = \\left( m_1 + m_2 \\right) v\\,,", "3d666a13f119eca06d9ffdfcef79878a": "\\det(M_{11}) = \\sum_S \\det(F_S)\\det(F^T_S) = \\sum_S \\det(F_S)^2", "3d6676b59e170a02ae96852c44ba49ae": "t=\\tan\\theta", "3d6696695cf1c9a0faaa3afac41bed5e": "Y = b_0 + b_1x_1 + b_2x_2 + b_3(x_1\\times x_2) + \\varepsilon \\, ", "3d66b1b1e55fecc41dab5e1bb0f2d458": "\\int \\Psi \\Psi ^* \\mathrm{d}V = N_s.", "3d66b1ba2a230931804ca25caa7127dd": "\\lambda = \\frac{128 - (30 + 98)}{128} = 0", "3d66d380dbb183c61ba5cb83336eda64": "V_{\\text{CB}}", "3d6703a50494e9ee594698831c22d9c5": "\\mathfrak{0123456789} \\!", "3d67391919c52e12a841cb13ff6536f6": "\\frac{\\pi^5}{120} R^{10}", "3d6790e508b36c2a620569091a470d32": "\\textbf{G}(\\textbf{r}, \\textbf{r}^{\\prime}) = \\frac{1}{4 \\pi} \\left[ \\textbf{I}+\\frac{\\nabla \\nabla}{k^2} \\right] G(\\textbf{r}, \\textbf{r}^{\\prime}) \\,", "3d67ae2d7b4929c9e45b87b18b2954e2": " \\text{Cov}[{N}(A),{N}(B)]=M^2(A\\times B)-M^1(A)M^1(B) ", "3d67b13c77c3365927e0669394c70f93": "H_{RISP}", "3d67ca7ee26b8ca27da5e307c091281e": " \\frac{dx}{dt}=rx(1-x)-px,", "3d68288c25a0a4e9ffdcd0f05f5b18f0": " \\vec{s}(C_{0}^{(5)}) = [+1,-1,0,0], ", "3d682f2a28bac21c83f103026f921986": "I_\\nu (z_1+z_2)= \\sum_{k=-\\infty}^\\infty I_{\\nu-k}(z_1)I_k(z_2)", "3d68330182bcc024512a61d46f89b195": "\\mathbf{X^TWX\\boldsymbol\\beta=X^T Wy}", "3d686ccd1c02617d2b8f96bb9246d30e": "j=\\langle C_{j}, R_{j}, I_{j}, A_{j}\\rangle", "3d68828a83778328b5322b16d9d06b94": " c_{t}=\\left[ \\frac{r}{1+r}\\right] \\left[ E_{t}\\sum_{i=0}^{\\infty\n}\\left( \\frac{1}{1+r}\\right) ^{i}y_{t+i}+A_{t}\\right] ", "3d689ffc8202c00f23e9de31383f8cf0": "\\{\\} \\!\\,", "3d68cc5ccb050d1b6567e16e36d7a716": "y = \\frac{h}{b^2} x^2 ", "3d690ab7dadf485ecf1d601ef023997d": "\\{q_j\\}", "3d69498c2421e9b1a5ee7e34c13d08ca": "73^2", "3d699a596e563f7408125eb079dfcb33": "A(x, ayb)", "3d69b91e633095b8bd9f4784517ba556": "2^{-\\Omega(\\gamma N)}", "3d69e3e6b9345260e4884e677be2da71": "\\begin{pmatrix}aa+bc & ab+bd \\\\ac+cd & bc+dd \\end{pmatrix} .", "3d6a0d45dde3e0f55a6f8c19e0711d75": "(\\alpha,\\beta)=\\int_M \\alpha \\wedge *\\beta.", "3d6a7905435d57520bbd19da602b1802": "p_1(x)", "3d6a79e3ddd14b3164cfc6915d991de3": "x \\ll b", "3d6aa8c4995110933fc6d2d322ef1e85": "\n\\frac{P_d}{P_i}=1-\\rho^2\n", "3d6aaef00692265be03f7af579db1420": "\\scriptstyle \\mathcal{C}", "3d6af1eb14bd27e7fb1e39bf085db7ad": "M_{s}+M_{l}=\\frac{M_{s}}{\\phi_{sl}}", "3d6b503412e9fdb650df49d621618af9": "\n\\begin{align}\n\\frac{\\delta F[\\rho]}{\\delta \\rho} &{} = \\frac{\\partial f}{\\partial\\rho} - \\nabla \\cdot \\frac{\\partial f}{\\partial(\\nabla\\rho)} + \\nabla^{(2)} \\cdot \\frac{\\partial f}{\\partial\\left(\\nabla^{(2)}\\rho\\right)} + \\dots + (-1)^N \\nabla^{(N)} \\cdot \\frac{\\partial f}{\\partial\\left(\\nabla^{(N)}\\rho\\right)} \\\\\n&{} = \\frac{\\partial f}{\\partial\\rho} + \\sum_{i=1}^N (-1)^{i}\\nabla^{(i)} \\cdot \\frac{\\partial f}{\\partial\\left(\\nabla^{(i)}\\rho\\right)} \\ .\n\\end{align}\n", "3d6bc1666218514c0589d329692b5aab": "\\!\\mathcal A \\models_{\\{\\emptyset\\}}^+ \\phi", "3d6be4dc1ba53c34040a0d903d4f1bc9": " d_{\\sigma x} = 2 \\sqrt{2} \\left( \\langle x^2 \\rangle + \\langle y^2 \\rangle + \\gamma \\left( \\left( \\langle x^2 \\rangle - \\langle y^2 \\rangle \\right)^2 + 4 \\langle xy \\rangle^2 \\right)^{1/2} \\right)^{1/2} ", "3d6bee862089257f0c5b7b5a0853cf90": " \n\\begin{matrix}\nij & =\\\\\n\\Downarrow & \\\\\n\\alpha & =\n\\end{matrix} \n\n \\begin{matrix}\n11 & 22 & 33 & 23,32 & 13,31 & 12,21 \\\\\n\\Downarrow & \\Downarrow & \\Downarrow & \\Downarrow & \\Downarrow & \\Downarrow & \\\\\n1 &2 & 3 & 4 & 5 & 6\n\\end{matrix}\\,\\!", "3d6c227abb2837fcb821bf887740c23d": "\\frac {PV} {T} = \\mbox{a constant}", "3d6c33a00507a59f8834c203c3c71b2d": "\\Delta S \\left( \\mathbf {R} \\right ) = S \\left( \\mathbf {R} \\right ) - S \\left (0 \\right )", "3d6c4d175616109ceba0b529194b7941": "\\Chi^2(k_0)", "3d6c8522feb5159333614e24164cf1db": "\\rho(\\mathbf{r}) = \\frac{2\\pi^{3/2}}{(8+5\\sqrt{\\pi})}e^{-(1/2)r^{2}}\\left(\\left(\\frac{\\pi}{2}\\right)^{1/2}\\left(\\frac{7}{4}+\\frac{1}{4}r^{2}+\\left(r+\\frac{1}{r}\\right)\\mathrm{erf}\\left(\\frac{r}{\\sqrt{2}}\\right)\\right)+e^{-(1/2)r^{2}}\\right).", "3d6cbd35d42b7154048a22e5c984ff28": "\\frac{\\theta}{2}", "3d6d06eb13a768022d09374d85bfd7c6": "x'_i = \\frac{\\sum_j w_{ij}x_j}{\\sum_{ij} w_{ij}x_j}", "3d6d184038499a038e0ce4b9c56a1795": "z=i\\omega_n", "3d6d5da8a3f12761b367b06357575de0": " -2 \\exp(-x) \\, \\partial_t + z \\, \\partial_x + \\left( \\exp(-2x) -z^2/2 \\right) \\, \\partial_z.", "3d6d685747e40e13332c4ca97a661229": "L_{m,n} = \\frac{2m+1}{m+n+1}\\binom{2n}{m+n} \\qquad\\text{ for }n \\ge m \\ge 0.", "3d6dd38ce8beaa8e6487d83e1fcbfb74": "\\|xx^*\\| = \\|x\\|^2,", "3d6ddfdc7810c83d71290f7ac9b8bb72": "\n \\mathbf{F} = 0\\,\\mathbf{E}_x + 0\\,\\mathbf{E}_y -F\\,\\mathbf{E}_z\n \\quad \\text{and} \\quad \\mathbf{r} = x\\,\\mathbf{E}_x + 0\\,\\mathbf{E}_y + 0\\,\\mathbf{E}_z \\,.\n ", "3d6de401d007ea0cffc99610ad623239": "d\\,", "3d6de4a0bda1b47bcb2b1e85c568765e": "\\lambda \\wedge \\theta = f_{\\lambda}(\\theta) \\, (e_1\\wedge\\ldots\\wedge e_n)", "3d6de6def2dadbd14b325e355e4e227c": " e^{-(n(n-1))/(2\\cdot 365)} < \\frac{1}{2}", "3d6e343d6004163afd685e7869c684d8": "X_1^n=(X_1,\\dots,X_n)", "3d6e862f3830a45de15371c88507e84e": "R_aR_d=R_bR_c", "3d6eb853f8360edad7b26833a1f39d12": "\\nu_k", "3d6ed46e6e42fcd4c082a9df6e1dc7ef": "\n \\omega_1(z,t) \n = \\frac{\\partial u_1}{\\partial z} \n = -\\kappa\\, U_0\\, \\text{e}^{-\\kappa\\, z}\\, \n \\Bigl[\\, \n \\cos\\left( \\Omega\\, t\\, -\\, \\kappa\\, z \\right)\\, \n -\\, \n \\sin\\left( \\Omega\\, t\\, -\\, \\kappa\\, z \\right)\\,\n \\Bigr]\n", "3d6efbcb92feb2b4708c00e60533114d": "v = \\frac{Q}{A},", "3d6f98c258c8e288717677a124a56935": "\\alpha\\beta.", "3d6f9ad0944b502ec36828aada9d6f6e": "PPxy \\rightarrow \\exists z[Pzy \\and \\lnot Ozx].", "3d6fde69f09457e045aa986697293e4f": "Y=jB_C=j \\omega C\\,", "3d70b72d9e0464ead357fbb2ee7d2d04": "(5.d)\\quad \\gamma_{,\\,\\rho\\rho}+\\gamma_{,\\,zz}=-\\big(\\psi^2_{,\\,\\rho}+\\psi^2_{,\\,z} \\big)\\,,", "3d70ba2e00dd77040a5bb11bbf04d563": "C(z) = B(z) - 1", "3d70c519c1189782eef781bb388f60de": "\\forall \\lambda", "3d70dcf93c66c693b841c97435c2d76a": "g(\\mathbf{r}) = \\frac{1}{\\rho} \\langle \\sum_{i \\neq 0} \\delta ( \\mathbf{r} - \\mathbf{r}_i) \\rangle = V \\frac{N-1}{N} \\left \\langle \\delta ( \\mathbf{r} - \\mathbf{r}_1) \\right \\rangle", "3d712108b68a1a149f28a1113573ab29": "\\text{WAL} \\times r", "3d7124cc94d797e04c4b84af7bff52db": "\\mathbb R^n \\rtimes K", "3d719424feee0fd01d78b261a95df149": "f(x)=\\int_{\\mathbb{R}^n} e^{2\\pi ix\\cdot\\xi} \\, (\\mathcal{F}f)(\\xi)\\,d\\xi.", "3d72c2df18fc6e5b07ea95f5ff775442": "S=(z_1, z_2, \\dots, z_m)", "3d7300083d936465e2930ecd0b48908e": "t_l\\in [0,\\infty)", "3d73c4f5882bebf00ae90df422499afa": "\\displaystyle 3(p^{4}+q^{4}+t^{4}+a^{4})=(p^{2}+q^{2}+t^{2}+a^{2})^{2}.", "3d73c6dfc5ea253c84514f9dd6908284": "q(\\mathbf{Z},\\mathbf{\\pi},\\mathbf{\\mu},\\mathbf{\\Lambda}) = q(\\mathbf{Z})q(\\mathbf{\\pi},\\mathbf{\\mu},\\mathbf{\\Lambda})", "3d73fdfdf636106df5ae3ecc53991d79": "\nK(x,y;T) = \\int_{x(0)=x}^{x(T)=y} \\Pi_t \\exp\\left\\{-{1\\over 2} \\left({x(t+\\epsilon) -x(t) \\over \\epsilon}\\right)^2 \\epsilon \\right\\} Dx\n\\,", "3d74187ca2586879c9457d873e0b780a": "[g_1,h_1] \\cdots [g_n,h_n] ", "3d747e4ede3e3f7c257b81012adb2a8c": "P = \\frac{D_0 \\left( 1 + g \\right)}{r-g} \\left[ 1- \\frac{\\left( 1+g \\right)^N}{\\left( 1 + r \\right)^N} \\right]\n+ \\frac{D_0 \\left( 1 + g \\right)^N \\left( 1 + g_\\infty \\right)}{\\left( 1 + r \\right)^N \\left( r - g_\\infty \\right)},", "3d74d9016038196122e319c5b4b4ece9": "s=-5", "3d751c6f7788093e11dc5bf796eeeaf0": "Y\\left(\\omega\\right)", "3d7582dbda6e615461af84773784024f": "M\\times S", "3d75be1910173cc6f40106c3fe3e28dd": "X^A(X-1)R((X-1)^2)", "3d768c39cb284727a15ff29c670e7c4c": "R_{4,3} = 9 r^4-8 r^3", "3d76a6dcbb5e670cbf0991be274a48db": "A(ax, ay, az) \\to A(x, y, z)", "3d77492b872fc3e43bf671f8703df664": "\\widehat{E}", "3d775f4407cdae007605fc8ff63c2421": "\\pi_0(x) = \\sum_n\\mu(n)f(x^{1/n})/n = f(x) -\\frac{1}{2}f(x^{1/2})-\\frac{1}{3}f(x^{1/3}) - \\cdots.", "3d776a6550b9c6e4fa384398ac406134": "\\sum_n |x_n|^p < \\infty.", "3d77996ffa836a5ebc4973e4d576218b": "\\zeta(s)=\\frac{1}{s-1}-s\\int_0^1 h(x) x^{s-1} \\; dx", "3d77e20d5c32b5b00d8180a0a3f944b4": "A = B(B^TB)^{-1}", "3d77eb944bf92c5178b71adf323ae8ae": "\\frac{1}{(1+n_f)^{0.5}}", "3d7828da3c798e4428531af5d61f453e": "y = r \\sin\\theta", "3d7834ccde82996d1e8730b2c93a3efa": "f\\in {{L}^{2}}(\\mathbb{R})", "3d7843695aff5ef5ebea78b4bd68a41e": "a\\mathbb{E}(I_{(X \\geq a)}) = a(1\\cdot\\mathbb{P}(X \\geq a) + 0\\cdot\\mathbb{P}(X < a)) = a\\mathbb{P}(X \\geq a).\\,", "3d784eed95ddf587af4267d62a1c7727": "(1) \\mathit T = \\mathit g + \\frac{\\mathit ROE - \\mathit g} {\\mathit PB} + \\frac{\\Delta PB}{PB} \\mathit(1 + g)\n", "3d78eb71bd6d75961538ed243d5a7606": "z\\log(\\sin \\pi z)-\\int_0^z\\Bigg[\\log(2\\sin \\pi x)-\\log 2\\Bigg]\\,dx=", "3d79343159548a4f15563f9457c438b3": "c(s)", "3d793865fd3aa3c5b3803a08bf2be3ea": "f(\\psi)", "3d796d3a48e8d86890631e9f349aaaa6": " e_1", "3d7991beffd151e986ea7238590f1952": "\\hat{\\mu}_{1,1} = \\frac{x_{1,.}\\times x_{.,1}}{x_{.,.}}=\\frac{34\\times 68}{400}=5.78", "3d79a3ac9573051c921d151e5b6c93c9": "p = 2347", "3d79b1b69171048509ed0f6e20470ff2": "VSP = v \\times (a + g \\times \\sin{\\phi} + \\psi) + \\zeta \\times v^3", "3d79d60ddbc248997c5694d449ef112f": "(E+H\\wedge T) {\\rm d}^2 \\Sigma", "3d7a0808d0e46bdc073344c28caaebac": "\n \\boldsymbol{B} = \\lambda^2~\\mathbf{n}_1\\otimes\\mathbf{n}_1 + \\cfrac{1}{\\lambda^2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2+ \\mathbf{n}_3\\otimes\\mathbf{n}_3 ~.\n ", "3d7a220e06a8484a0556b01496024445": "\\textbf{n}", "3d7af62196ba8e4c6faa7e24131795cf": "\\textbf{t}_i^T = \\begin{bmatrix} x_{i,1} & \\dots & x_{i,n} \\end{bmatrix}", "3d7b044f39e2e325156c73ae0955168a": "\\,q\\,", "3d7bf0d395c25be43129758a49b43b0b": "H = \\sum_{k\\sigma} \\epsilon_{\\mathbf{k}} c^{\\dagger}_{\\mathbf{k}\\sigma}c_{\\mathbf{k}\\sigma} - J \\mathbf{S}(0)\\cdot \\mathbf{S} ", "3d7bff146f6b76aad5dc65c0961051b3": "0,\\,\\,\\lambda > -1/3", "3d7c079cb0da400b43790d850bb2c91a": "\\mathrm{I\\!I}(T,T)", "3d7c0d5e50adbb5d24b71bfbe3d4ea0d": "N_{C}", "3d7c0f953733fbfd4a9f578dd5594c1b": " I_3-I_2-I_1=I_4", "3d7c17f2c7bfc1a9704556642a8c8ed6": "O(\\sqrt{V} E)", "3d7c5842d8cac7f4ce3a2fe1c5b8d994": "\\boldsymbol{\\mathbf{X}} = (ct, \\mathbf{x} )", "3d7cd8aff82d2f6abc5ddfd0dc300ae1": "\\bold{J}_{\\text{free}} = \\sigma \\bold{E} \\,", "3d7d0738248d06d56427d6ee224bd241": " G' = \\frac {\\sigma_0} {\\varepsilon_0} \\cos \\phi ", "3d7d784aa34fcc859c9bd7ca34ab1f23": "\\Delta E_\\text{D} = \\frac{1}{2}\\langle\\hat{A}\\rangle[F(F+1)-I(I+1)-J(J+1)]", "3d7e014ae4d2fb69253452ec8b9d738f": "\\left|\\begin{matrix}a_{11}&a_{12}\\\\a_{21}&a_{22}\\end{matrix}\\right|.", "3d7e37a3b53e6b367606af2e40072c63": "a=\\frac{a_3}{a_4},\\quad b=\\frac{a_2}{a_4},\\quad c=\\frac{a_1}{a_4},\\quad d=\\frac{a_0}{a_4}.", "3d7e6e82fe848c4325994cf84c85b482": "R=R_a R_b^T", "3d7f0549890fad49138adc10b2ce18bd": "d\\Omega=0", "3d7f13d8a7fc36e2f1db05664d74ff57": "\\pi_3", "3d7f370a1a1f06f259b321c5a2f6a07a": "1 \\over 21", "3d7f3c3414d2288eade98e4343fdd731": "H[\\mu_j]=U+pV-\\mu_jN_j\\,", "3d7f4eb1a205b1df3f62800170307926": "\\vec{K}= \\frac{I}{L} \\hat{\\phi} .", "3d7f64c7ac2d53e36481adb1d6b06186": "B_n^{(\\alpha,\\beta)}(x)=\\frac{a_n^{(\\alpha,\\beta)}}{x^{\\alpha} e^{\\frac{(-\\beta)}{x}}} \\left(\\frac{d}{dx}\\right)^n (x^{\\alpha+2n} e^{\\frac{(-\\beta)}{x}})", "3d7fc03a073b5698f894b6300ff30523": "\\begin{align}P(\\alpha)&=|c_0|^2\\delta^2(\\alpha-\\alpha_0)+|c_1|^2\\delta^2(\\alpha-\\alpha_1) \\\\\n&\\, \\, \\, \\, \\, +2c_0^*c_1\ne^{|\\alpha|^2-\\frac{1}{2}|\\alpha_0|^2-\\frac{1}{2}|\\alpha_1|^2}\ne^{(\\alpha_1^*-\\alpha_0^*)\\cdot\\partial/\\partial(2\\alpha^*-\\alpha_0^*-\\alpha_1^*)}\ne^{(\\alpha_0-\\alpha_1)\\cdot\\partial/\\partial(2\\alpha-\\alpha_0-\\alpha_1)}\n\\cdot \\delta^2(2\\alpha-\\alpha_0-\\alpha_1) \\\\\n&\\, \\, \\, \\, \\, +2c_0c_1^*\ne^{|\\alpha|^2-\\frac{1}{2}|\\alpha_0|^2-\\frac{1}{2}|\\alpha_1|^2}\ne^{(\\alpha_0^*-\\alpha_1^*)\\cdot\\partial/\\partial(2\\alpha^*-\\alpha_0^*-\\alpha_1^*)}\ne^{(\\alpha_1-\\alpha_0)\\cdot\\partial/\\partial(2\\alpha-\\alpha_0-\\alpha_1)}\n\\cdot \\delta^2(2\\alpha-\\alpha_0-\\alpha_1).\n\\end{align}", "3d7fc4959508552a3b4de756ab208799": "\\hat{\\tau}(t) = t + \\int_{[0, t]} \\| \\mathrm{d} z \\| = t + \\mathrm{Var}(z, [0, t]).", "3d7fed3e6bea1054b55567e342eac299": "A_g", "3d801aa532c1cec3ee82d87a99fdf63f": "temp", "3d8078482072ab69e478e31d15199d93": "q_s", "3d80e7ce956a8e1748adfe30cb2160ae": "\\sum_{s\\in\\mathcal U} c(S)x^*_s", "3d80fc5438692dc50bc87ee9a57b8258": "V_0 = 0", "3d81d2668e68219373ee24eda6c04bd9": "L \\le ( \\mbox{SAT}, \\epsilon-\\mbox{UNSAT})", "3d81e5e3b48830f054ec101198ec232d": " j > 0", "3d8239f2a362b8a4b4bfa67ea39aa4dc": "\\; R_T(\\lambda) = (\\lambda - T)^{-1}", "3d827b7ed68f3f4a823f09a59ca7ca24": "(f_1(x))^{s_1}(f_2(x))^{s_2}", "3d828531f38192bc0f905545896b3402": "e^\\frac{-ikx^2}{2z}", "3d82dd4398a28d38e650c5cfdaa47c07": "\np_\\mu \\epsilon_\\mu^2(p) = 0. \\quad\\quad\\quad\\quad (3)\n", "3d82e52e9f94f5eda93b0bf173cf9244": "f(q;\\Delta t)", "3d830e73e849316ae7cf3baf328000af": "Y_i\\ast", "3d83458b82f721f69948f911fb42e703": "\\ p = \\rho \\cdot R \\cdot T ", "3d835949def5ff24d6d0ca343a34fc22": " t_g = \\frac {v \\sin \\theta} {g} \\pm \\frac {\\sqrt{v^2 \\sin^2 \\theta + 2 g y_0}} {g} ", "3d83adb63585d7276b6f6ac6bc9e7826": "\\lim_{n \\to \\infty}\\frac {d_n}{d_{n+1}}", "3d83dec2218e9b723b2b32f219bfdeac": " z = 0\\,", "3d844c6339805638f55bad914efb6f67": "Q_{max}", "3d8464f237b744e7099205176fa1b9e4": "|\\mathcal P|=9 ", "3d847e74c4b6b6691954715b5d79ab50": "\nf(x| \\boldsymbol{p} ) = \\prod_{i=1}^k p_i^{[x=i]} ,\n", "3d8491d7c8730849cc0c68f8dcf6abdd": "\\textstyle Q", "3d84966bfeeee78f29647927fa9b3643": "x \\in [x_\\min,x_\\max]", "3d84fa75473eeb4d8d3037b47ff33b2d": "LQ _l = K_l + E1_{l,k} \\times LP^k + E2_{l,k} \\times (LP^k)^2", "3d85692ab5b0dbc9c6b5f61ea75486f7": "\\phi=[1-qf(\\phi)]f(\\phi)^{-r}\\;", "3d85e54769ab40380b5e529cd52b9f47": "a_i = x", "3d862b552555763c4c2c96c4456d8251": "S_n=k", "3d862be38ec0e382ef247dc72304ddf5": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{covercosin}(x) = \\cos{x}", "3d863c4043ca68ef766d1f9450edbbf9": "0 \\in \\operatorname{core}(\\operatorname{dom}g - A \\operatorname{dom}f)", "3d86497acb129e70351e8ae260a249a4": " Re \\leq 1.0", "3d8675b57d60a8ac01c44c8abc3cfcf3": "\\psi(\\omega)=0", "3d86c66fc5e92b6009149a62c54d40cd": "r_p = \\frac{n_2 \\cos \\theta_\\text{i} - n_1 \\cos \\theta_\\text{t}}{n_1 \\cos \\theta_\\text{t} + n_2 \\cos \\theta_\\text{i}}", "3d86d38691f053d7d899eed24b52daca": "\n[\\Pi(\\Lambda_1)\\Pi(\\Lambda_2)\\Pi^{-1}(\\Lambda_1\\Lambda_2)]^2 = 1\\Rightarrow \\Pi(\\Lambda_1\\Lambda_2) = \\pm \\Pi(\\Lambda_1)\\Pi(\\Lambda_2), \\qquad \\Lambda_1,\\Lambda_2 \\in SO(3;1),\n", "3d87209870575024daf59b69a188c445": "\\{7^i \\mod{9}\\ |\\ i \\in \\mathbb{N}\\} = \\{7,4,1\\}.", "3d875a3b4a3535571cc412b261689c66": "\\alpha_1=7\\Delta_1+\\Delta_2-4\\gamma-4", "3d8768de27aff0a301daf60469fc6c66": "\\displaystyle{AT_1=T_2A,\\,\\,\\, BT_2=T_1B.}", "3d879323ba02e150fac13f1073aaa2fc": "\\varphi: A \\rightarrow B", "3d8800d9b50dc5e80ad3c7b8ad5c1e9b": "\\begin{bmatrix} 0 & f(p,q) \\\\ 0 & g(p,q) \\end{bmatrix} z^{-n}", "3d885516fcd6cb0742100ced226c16e4": " (p_x,p_y) ", "3d885bc5b64e188374474a399609ad27": "p \\equiv q \\equiv 3 \\pmod{4}", "3d886fb905f381b2288d4573ca15871f": "\\ M\\,", "3d88f8bf1d5d7024b427135a98e3d90d": "z_{n+1} = (|\\operatorname{Re} \\left(z_n\\right)|+i|\\operatorname{Im} \\left(z_n\\right)|)^2 + c, \\quad z_0=0", "3d891964e537333fd80f67c8a0dcf4a1": "\ns^{2} = L_{1}^{2} - c^{2} \\Delta t_{1}^{2} = L_{2}^{2} - c^{2} \\Delta t_{2}^{2}\n", "3d892d4395352fb5a37092facc277fa0": "\n\\begin{align}\n \\mathbf{v}_{\\mathrm{rot}} &= (I\\cos\\theta) \\mathbf{v} + ([\\mathbf{k}]_\\times \\sin\\theta) \\mathbf{v} \n + (1 - \\cos\\theta) \\mathbf{k} \\mathbf{k}^\\mathsf{T} \\mathbf{v} \\\\ &=\n \\left( I \\cos\\theta + [\\mathbf{k}]_\\times \\sin\\theta \n + (1 - \\cos\\theta) \\mathbf{k} \\mathbf{k}^\\mathsf{T} \\right) \\mathbf{v}\\\\\n &= R\\mathbf{v} \n\\end{align}\n", "3d89f1b146ecfdb5339d0751858512ec": " W_{A_\\text{sample}} ", "3d8a78e23d4628d3bd0592e06dee88c5": "U=\\frac{1}{R}=\\frac{\\dot Q_A}{\\Delta T}=\\frac{k}{L}", "3d8a94bed8b9b59a76fd712341abcbca": "y_{n+1} = y_n + h\\left( {23\\over 12} f(t_n, y_n) - {16 \\over 12} f(t_{n-1}, y_{n-1}) + {5\\over 12}f(t_{n-2}, y_{n-2})\\right).", "3d8b1a4e91da6005d17c34a59ec69479": "\n\\sqrt{x^2} = \\left|x\\right| = \n\\begin{cases} \n x, & \\mbox{if }x \\ge 0 \\\\\n -x, & \\mbox{if }x < 0. \n\\end{cases}\n", "3d8b5fb862e5aaea3f58436bc592e6f3": " {f_o} {v_g} = {\\partial \\Phi \\over \\partial x}", "3d8b6c087be2d4c5e60c0e3dce194eac": "z > z_{critical}", "3d8b73e89d09dffd17cda42a5bce948b": "E_{y,3z^2-r^2} = m [n^2 - (l^2 + m^2) / 2] V_{pd\\sigma} - \\sqrt{3} m n^2 V_{pd\\pi}", "3d8b9505993eeeaaf3e7dfe4180cb84e": "j^\\star = \\frac{2\\pi^5 k^4}{15 h^3 c^2} T^4 ", "3d8bb6abec523d330f821375a84eb417": "\\text{Step 3}", "3d8c1613829f4fe0b24126c713880eb4": "\\int_0^1 e_k^2(t) dt=1\\quad \\implies\\quad A=\\sqrt{2}", "3d8c5c9af6ef06220d03336849728b7f": "-\\nabla'T' = -\\frac{L}{k(T_h-T_c)}q=\\frac{hL}{k}", "3d8cb6b27586cf201c27223035748537": "\\begin{matrix} {1 \\choose 1}{11 \\choose 1}{4 \\choose 2}{40 \\choose 1} \\end{matrix}", "3d8ce58653cd223bebf6c1313dd82a05": "t > t(\\alpha/2,n-k-1^*)", "3d8ce96f8669e6ec48887dbcc4c46829": "[\\mbox{S}]_{i} - [\\mbox{S}]_{o}\\exp(-z_{S}V_{m}F/RT)=0", "3d8cfb5e106ff48dbfb269108b966759": "S(1)=f(x).", "3d8d2b008620a1a82ffc67b53f72f1fb": "\\mu_{1/2},", "3d8d7c6c78fdbdb745e263630da81e29": "(\\langle b \\rangle \\cdot \\langle l \\rangle) \\cdot (\\varepsilon \\cdot \\langle ah \\rangle) = \\langle bl \\rangle \\cdot \\langle ah \\rangle = \\langle blah \\rangle", "3d8d9af50faba0189c0adeb169ebeef3": "\\scriptstyle V_\\mathrm{out}", "3d8da7675c598d2e6278a654863c4091": "N_{j}", "3d8db8eb6a6689baa63323d8e4e92900": " C = True", "3d8e221258ec2079e1e18623bade9b3b": "[SU(3)\\times SU(2)\\times U(1)_Y]/\\mathbb{Z}_3", "3d8e46b3403b8d48cd5380f81094c17b": "{\\rm var}(t(X)) \\geq \\frac{[\\psi^\\prime(\\theta)]^2}{I(\\theta)}.", "3d8e47197465de6a02b92482184c9532": "=\\frac{1-\\varepsilon}{1+\\varepsilon}\\cdot\\frac{1-\\cos \\theta}{1+\\cos \\theta}=\\frac{1-\\varepsilon}{1+\\varepsilon}\\cdot\\tan^2\\frac{\\theta}{2}.", "3d8e503657fab8e2c34f0c4039968091": "F(s) = \\frac{1}{1 + s R C}", "3d8e81e01742ff9a9234a357ef07b231": "p(X,E = e)/ \\sum_{X} p(X,E = e)", "3d8eb5c806f77e71ce90a9c1cc41be77": " x^3 \\rightarrow 1 ", "3d8eec4fa3be844cb5f921227e3dfafd": "{n\\choose k}_2=0", "3d8f4b7f8e50364240ab9c1df60a81de": "(c \\triangleleft b) \\triangleleft a = (c \\triangleleft a)\\triangleleft (b \\triangleleft a)", "3d8f5a94b87dead55a7c7ff04273d5c9": "var(R_{ij}) = \\sigma^2", "3d8f9600fa035c19fc4a389a083a1d51": "c_\\eta", "3d8fc150b8d694f2e96ebade68bdec70": "I_i \\!", "3d90570a1384c46b154a77015c4d0f99": "\\int_0^\\infty \\frac{\\sin(x)}{x}\\frac{\\sin(x/3)}{x/3}\\cdots\\frac{\\sin(x/13)}{x/13} \\, dx = \\pi/2", "3d910851a7ea9052e8a13e2b0f4c0e68": "\\begin{align}\nh_1&=h_2=\\sqrt{u^2+v^2} \\\\\nh_3&=uv\n\\end{align}", "3d9125ea38e32a0fc017e4085cb7f805": " \\frac{dS_{21}}{d\\left(p_2/p_1\\right)} = \\frac{c_2}{c_1} + \\frac{p_2}{p_1}\\cdot\\frac{d\\left(c_2/c_1\\right)}{d\\left(p_2/p_1\\right)}\n = \\frac{c_2}{c_1}\\left[1 + \\frac{d\\left(c_2/c_1\\right)}{d\\left(p_2/p_1\\right)}\\cdot\\frac{p_2/p_1}{c_2/c_1} \\right]\n = \\frac{c_2}{c_1}\\left(1 - E_{21} \\right)\n", "3d91ef727c449e32514afe148d545627": " s = \\ell\\theta\\,", "3d92522a1bfa5345342732d348a18d1e": "\\sum_{k=1}^{n} \\frac {\\varphi (k)} {k} \\sim \\frac{6n}{\\pi^2}\\!", "3d927395a4f01ab9dde5cbbfe06e56f4": "\\langle 0 | \\left\\{ \\Phi(x),\\Phi(y) \\right\\} | 0 \\rangle = \\Delta_1(x-y)", "3d92a29bf38c62d4e589d2583df5097c": "\\begin{pmatrix}\\cosh a & \\sinh a \\\\ \\sinh a & \\cosh a\\end{pmatrix}.", "3d92a7ca9d1727256dd090abfcda2838": "2 \\theta \\,\\!", "3d92b32ba57fa13548df9b483113479f": " t_e \\in [0, \\infty] ", "3d92c23e48d73825408ad29d1f4270da": "e\\mathrel{\\Theta}f", "3d92f141de9420d426ff0064391eeb02": " \\sqrt{\\langle v^2 \\rangle} \\,\\!", "3d92fc477f2b9ed73435f02873702492": "W(y_1,\\ldots,y_n)(x)=W(y_1,\\ldots,y_n)(x_0) \\exp\\biggl(-\\int_{x_0}^x p_{n-1}(\\xi) \\,\\textrm{d}\\xi\\biggr),\\qquad x\\in I,", "3d932a26f72ca02090f5e3dbea15bb98": "x^{q-1}-1=0,", "3d935d79b899c3edac1edf9476adc0c9": "K=GF(q)", "3d9390b81ababc7eed3f0f62c0c7f2a5": "\\ c_k(x)", "3d93ac622a85435719e85bf5d954f7a1": "\\varnothing= \\sqrt{\\frac{4\\times 10^{-6}\\cdot \\mathrm{dtex}}{\\pi\\rho}}", "3d94277ddce363fa0bc6e8206edcec96": "f(x)=\\sum_{k=0}^\\infty (\\Delta^k f)(0){x \\choose k},", "3d94287a30928de3e621fe7c852aabf5": "\\xi= 0", "3d944417b348554908b00b5db974f372": "y = m\\,\\Delta y", "3d9472fdc28b0ec422dd21a4a48e9d0a": "\n\\begin{align}\nf_\\text{D}\n& = f_\\text{e2} - f_\\text{e1} \\\\\n& = f \\left(\\vec v \\ast \\frac{\\vec e_\\text{1} - \\vec e_\\text{2}}{c} \\right) \\left( \\frac{\\vec v \\ast \\vec e_\\text{e}}{c} \\right)\n\\end{align}\n", "3d9538a7b461eef39eabb6eb248843b8": "P_v(t) = \\frac{M_a}{r}(1 - e^{-rt})", "3d956685aa52393f609c168c0b2c860d": "a_{N_p}", "3d95e54414bc7397528a1b6cfc24fb52": "\nQ_{n} = \\sqrt{ k_B T C }.\n", "3d96352fc58ecc19a5bbe260b94b9830": "f(\\rho, \\phi) = \\rho^2 (\\cos^2 \\phi + \\sin^2 \\phi) = \\rho^2", "3d9646a05371f51ce57c4692bcd0ef0c": "Y_{nki}=1", "3d9652206b728352f565e7658728b530": "d'_1 = \\frac{d_1}{b_1}\\,", "3d967cb83e60cabe450674968deb2494": " W\\left ( J \\right ) =\n - T a_1 a_2\\int {d^3k \\over (2 \\pi )^3 } \\; \\; D\\left ( k \\right )\\mid_{k_0=0} \\; \\exp\\left ( i \\vec k \\cdot \\left ( \\vec x_1 - \\vec x_2 \\right ) \\right )\n", "3d971d54f8a99a0e8129376a881fc919": " \\scriptstyle \\beta_2", "3d972699a1f33ad9ff161a985d7cf122": "D_F^q(p, q) \\ge 0", "3d9755f9577d3af429a091af61e846dd": "\\leq k", "3d979348f05673a7a173fe6702bc4dfe": "N_a = \\frac{N_t}{E}", "3d97ceabea5b71808d3c0ac3efeb2b19": "{(\\eta_b)_{max}} = \\frac{2\\rho(\\cos\\alpha_1-\\rho)}{V_1^2-U^2+2UV_1\\cos\\alpha_1}", "3d97dd89a2c974139e019a18483714dd": "(p \\to q)", "3d97eb56e02c2889dd20a89529548180": "f(x)>0", "3d981c58226a42a172a83e5c3517ad85": "\\varepsilon_1' = \\frac{1}{E}\\sigma_1", "3d982a22825db24bfde51caa6f1ee3b9": "B-1", "3d9843f8e490167edd4c8c2cf4628f71": "\n\\begin{align}\n|x|_2 & = 2 \\\\[6pt]\n|x|_3 & = 1/9 \\\\[6pt]\n|x|_5 & = 25 \\\\[6pt]\n|x|_7 & = 1/7 \\\\[6pt]\n|x|_{11} & = 11 \\\\[6pt]\n|x|_{\\text{any other prime}} & = 1.\n\\end{align}\n", "3d985dec5dca58cbfa8294087f8a32b6": " \\lim \\int f_n \\geq \\int f", "3d988fae6a23bf9dedcd469b927c2249": "\\Delta t_{e^2y^2}=\\Delta t_{ey}-(5/4)e^2\\sin 2M+ey\\sin M \\cos(2M+2\\lambda_p)-(1/2)y^2\\sin(4M+4\\lambda_p)", "3d98bd1fa9dfe41c969f4d8df65d55d5": "E I", "3d98cd98e0f3f610be912d0ea57f6cb3": "\\phi_{CTS}(u) = \\exp\\left( iu\\mu\n+C_1\\Gamma(-\\alpha)((\\lambda_+-iu)^\\alpha-\\lambda_+^\\alpha)\n+C_2\\Gamma(-\\alpha)((\\lambda_-+iu)^\\alpha-\\lambda_-^\\alpha)\n\\right),", "3d9913daa09a662359d53ec1191de692": "Q(x + \\alpha,y + \\beta)=\\sum_{i,j} a_{i,j} \\Bigg ( \\sum_u \\begin{pmatrix}i\\\\u\\end{pmatrix} x^u \\alpha^{i-u} \\Bigg ) \\Bigg ( \\sum_v \\begin{pmatrix}i\\\\v\\end{pmatrix} y^v \\beta^{j-v} \\Bigg )", "3d997186b190d522c3e85b72d586dcb5": "\\overline I", "3d99cfc142611723727141e6cc76ffbd": "t \\ge \\tau ", "3d9a3c2aaf6e352e4c3141561010876d": "(f(U) \\neq f(V)) \\to \\neg P.", "3d9ae6d78cecffb391ef7928094b8d97": "\\frac{1}{2^k}", "3d9af2a188d3c6f777821eb03d2cd3ab": "\n C_1 = -\\cfrac{Pb}{6L}(L^2-b^2) ~.\n ", "3d9afa98733a9f4a6912d0de660e78f0": "\\frac{\\text{d} [{^1_2}S^\\gamma]}{\\text{d}t}= \\text{k}_{2(2)} C_3 -\\text{k}_{1(2)} {^1_2}S^\\gamma E", "3d9b8b12a2b51b0cab7bb632930ddd0d": "W(C^\\perp;x,y) = \\frac{1}{\\mid C \\mid} W(C;y-x,y+x). ", "3d9bea2f7456e06355a89c58c1205e97": "\\begin{align}\n D_{\\mathrm{KL}}(X_1,X_2) &= \\int_{0}^1 f(x;\\alpha,\\beta) \\ln \\left (\\frac{f(x;\\alpha,\\beta)}{f(x;\\alpha',\\beta')} \\right ) dx \\\\\n&= \\left (\\int_0^1 f(x;\\alpha,\\beta) \\ln (f(x;\\alpha,\\beta)) dx \\right )- \\left (\\int_0^1 f(x;\\alpha,\\beta) \\ln (f(x;\\alpha',\\beta')) dx \\right )\\\\\n&= -h(X_1) + H(X_1,X_2)\\\\\n&= \\ln\\left(\\frac{\\Beta(\\alpha',\\beta')}{\\Beta(\\alpha,\\beta)}\\right)+(\\alpha-\\alpha')\\psi(\\alpha)+(\\beta-\\beta')\\psi(\\beta)+(\\alpha'-\\alpha+\\beta'-\\beta)\\psi (\\alpha + \\beta).\n\\end{align} ", "3d9bee59fc25ceb289d6653815364ce6": "r_c=\\langle\\hat{e}| \\hat{a}\\rangle-\\langle\\hat{d}| \\hat{b}\\rangle", "3d9c156bf01ecd9747565759e25e9d66": " G(s, t) = \\text{Cov}(X(s), X(t)) = \\sum_{k=1}^\\infty \\lambda_k \\varphi_k(s) \\varphi_k(t), ", "3d9c93033c7b924c068d51d24d8f7d21": " = \\pm\\frac{\\sqrt{1 - \\sin^2 \\theta}}{\\sin \\theta}", "3d9d31cd2c1b6077b72d756caadd69f1": "r = \\|\\mathbf{x} - \\mathbf{x}_i\\|\\;", "3d9d3cfca544dce102c5e7a6a24f5e07": "\\left( \\psi_p(x_i) \\right)_{ip}", "3d9d446b8207f1ad25898e0799dda997": "{R}(\\Delta\\theta,\\hat{\\mathbf{n}})\\psi(\\mathbf{r},t) = \\psi(\\mathbf{r}',t)", "3d9d5131bd69251b4178bf321a1f07bb": "\\int_a^b f(x) \\, dg(x)=f(b)g(b)-f(a)g(a)-\\int_a^b g(x) \\, df(x).", "3d9dd0a64c2f78e8f19e1af2c8b29a8a": "\nU_q(n) = \nT_q(n) + \n\\tfrac12\\phi(q).\n", "3d9e6423c62d5e8dbf8792019dc4cb58": "\\bar{\\sigma}-\\bar{\\epsilon}", "3d9e9372e9d0103329a0fb4fcdb382e3": "\\varpi = \\Omega + \\omega\\,", "3d9ec326ca1f6b34f84bbd4e841869b2": "\\epsilon_{\\mathrm Y}", "3d9ed1c88919c51a557a483b225914f6": "2^k\\sqrt N\\ge N", "3d9ee3e38c4b6c22b8abafdbbbe7d75d": "\\sum_{i\\in\\N}XY^i", "3d9ef18ceb0f8254c9612035e84a4f33": " \\eta(X_1, X_2, \\ldots, X_n) = \\mathrm{E}(\\delta(X_1, X_2, \\ldots, X_n)|T)\\,", "3d9f78d11bb90f3cd58a516318cc48ac": "M_k = \\bar x_k", "3d9f866fb3c75a5fd2f78f09cd1a9002": "1,1 \\Leftrightarrow 2", "3d9fb7bd701b56029ad0d82459f7be7f": "\\operatorname{I}(\\operatorname{V}(J)) = \\operatorname{Rad} (J)\\,", "3d9fd66ac6d982998a6f6e4db8a195d0": "L(\\boldsymbol\\theta; \\mathbf{X}) = p(\\mathbf{X}|\\boldsymbol\\theta) = \\sum_{\\mathbf{Z}} p(\\mathbf{X},\\mathbf{Z}|\\boldsymbol\\theta)", "3d9fe2d7b982eab523d57bcb77c00e02": " F(k;n,p) \\leq \\exp\\left(-nH\\left(\\frac{k}{n},p\\right)\\right) \\quad\\quad\\mbox{if }0<\\frac{k}{n} \\alpha)", "3da84ede03b26ecb9c1a25b1b064662d": "K_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\{ x \\in K : \\langle x,\\,x \\rangle = 0 \\}", "3da86db0ea01bb65a7d9388029887c64": " \\operatorname{Reg}(A/K) = \\det\\bigl( \\langle P_i,\\eta_j\\rangle_{P} \\bigr)_{1\\le i,j\\le r}.", "3da8723962ef2f209a95e76e78550738": " |j \\rangle ", "3da8d04a622a2ac33023ce6442af6f29": "\n\\begin{align}\n4\\Phi_7(z)\n&=4(z^6+z^5+z^4+z^3+z^2+z+1)\\\\ \n&= (2z^3+z^2-z-2)^2+7z^2(z+1)^2\n\\end{align} \n", "3da91ccec27542f482b444ec57cab049": "\\tilde{\\mathbf A}", "3da966b36dcbd68822cf6892afc8780d": "H\\{\\bold{X}\\} = \\lim_{n\\to\\infty} H(X_n | X_0, X_1, \\dots, X_{n-1})", "3da983b04949a441c0ca1c648f0c7d98": " B \\rightarrow X_s \\gamma ", "3da9afce096ef71eca83c12c011ee15d": "t \\times p", "3da9d5e426a4f834e6e6967de69a966a": " 1/2 +1/\\sqrt{\\pi M} ", "3daacd78902eeb0c04bcd34c5166607b": "\\frac{-1}{\\ln(1-p)} \\; \\frac{p}{1-p}\\!", "3dab02233582f69b9c556987babd6c1b": "p(z)=z^5-3iz^3-(5+2i)z^2+3z+1", "3dab03cac24f62541b4d41ac056e969c": " f(0) = f(1) = 0 \\quad ", "3dac11bea98bcc665d46616957f5c331": "\\gamma([a, b])=S,", "3dac3780a7cf30d49ab5aa6ba7fd47ff": "(\\alpha-\\beta+1)=-4.8(3.7)\\times10^{-8}\\,", "3dac6f4e6410d14d94651eaed0f82255": "E_{\\rm XC}^{\\rm LDA}[n]=\\int\\epsilon_{\\rm XC}(n)n (\\vec{r}) {\\rm d}^3r.", "3daca1ba6662f13ffb091166171dad62": "\\mathrm{Ei}(x) = \\gamma+\\ln |x| + \\sum_{k=1}^{\\infty} \\frac{x^k}{k\\; k!} \\qquad x \\neq 0", "3dad1e2a03bcd825b1aee52a284f4e7a": "g^{ab}\\,", "3dad28281778d5ef4b7a78c7bc7a6b09": "x = 0", "3dad2c7e368b3d74ee5110fdeeeea34a": "c_{3,1}(\\widehat{a}, w(c_{3,1}(\\widehat{a}, w(c_{1,0}(\\widehat{\\epsilon}), \\widehat{b}c), \\widehat{d}), \\widehat{b}c), \\widehat{d})", "3dad494c7db66246af706dd096cd7581": "\\tau H_k M \\otimes \\tau H_k M \\to \\Bbb Q / \\Bbb Z.", "3dadc04cebe89d3edd022699b1f1749f": "c_\\beta", "3dadf9e2c54dfbd2ed589726250a1170": "\\sum_{n,\\alpha}E_{0}\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n,\\alpha}", "3dadff7ea5f0b71093ef9d7b2fa28117": "\\frac{46088}{(1+0.10)^7}", "3dae212ec792a1fb503d7332461cdaa3": "E_0 = - \\mu B", "3dae30016a1d53846b27d488a597405b": " \\lfloor . \\rfloor ", "3dae71ecf1779c72bf714f8c0866b87d": "\\boldsymbol{\\varepsilon}_e = 0", "3daeb051c00122207e3b39ce644be094": "v \\in T_x \\mathbb{R} \\mapsto (T_{f(x)} L_{f(x)})^{-1} \\circ (T_x f) v \\in T_0 \\mathbb{R}.", "3daec8dea85232fa2baef18aeb2334d8": "slip", "3daee286e8f72482c383a3525853a538": " \\sigma_+^2 = \\frac { \\sum (x - m)^2 } { n - 1 } ", "3daf3de0c737bcf433ae6d7ddfb1eab1": "|\\mathbf{Z}:n\\mathbf{Z}| = n", "3daf8a521a3b1b21b0ecd92bcc5a98bb": " S_{(2,2,0)} (x_1, x_2, x_3) = \\frac{1}{\\Delta} \\;\n\\det \\left[ \\begin{matrix} x_1^4 & x_2^4 & x_3^4 \\\\ x_1^3 & x_2^3 & x_3^3 \\\\ 1 & 1 & 1 \\end{matrix}\n\\right]= x_1^2 \\, x_2^2 + x_1^2 \\, x_3^2 + x_2^2 \\, x_3^2 \n+ x_1^2 \\, x_2 \\, x_3 + x_1 \\, x_2^2 \\, x_3 + x_1 \\, x_2 \\, x_3^2 ", "3db00d0dd1af7e1f1483bbb061f78874": "a_1=a_2", "3db0538a0ae9e2d239f47cd7703d7dab": "136 / 512 = 25%", "3db09599311c3b7fafceddca23dccd1c": "d^2G_\\Sigma = d^2G_S ", "3db0f3eb183aaffa1ec3f85e5b8d042e": "\\phi \\to \\phi \\lor \\chi ", "3db1700bdbb2f7110aeb5cce4f1b83fa": "\n\\operatorname{E}(X_1 | X_2 > z) = \\rho { \\phi(z) \\over (1- \\Phi(z)) } ,\n", "3db18affaf16b43d4a0e8d3299dff8e2": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}e^{2\\pi i \\ell x_{j}}=0.", "3db1cd895a76b495f7472745dff33eaa": "\\mathbf{p} \\rightarrow \\mathbf{m}/c", "3db26139b78e365a1ba12e7e17fbb461": "r_{k},", "3db2d5fc7a7f28a3eb25daeb0c87c079": " \\{ p \\mid p <_\\mathcal{O} f(e) \\} ", "3db314a0d30959886199c0448767e44a": "\\mu_1=\\sqrt{2}\\,\\,\\frac{\\Gamma((k\\!+\\!1)/2)}{\\Gamma(k/2)}", "3db35b0e3ad0da3479fa5bb201d65bc8": "\\pi_{GC} ", "3db432f32dae9fb4dc4b34f7f3e9e59c": "M\\ddot{X}(t)", "3db4406dafbee3d621c5c2fd2bcb5fc1": "E = E_0 - \\int P \\, dV", "3db49cf2c9e0aed664599a3c177f7088": "C_{4} = T_{4} + \\frac{1}{2}(T_{2})^{2} + T_{1}T_{3} + \\frac{1}{2}(T_{1})^{2}T_{2} + \\frac{1}{24}(T_{1})^{4}", "3db4a0c3f656d4f084d01892a5f6e338": "\\sqrt{x^2 +y^2}", "3db4bb89d26991587fe81731da8bf333": "400 N", "3db4d6a7632e3e58a4ff973d1b735fdd": "\\frac {\\rho}{2}(V)^2 + \\Delta P = \\frac {\\rho}{2}(V^2 + 2 V v + v^2),\\,", "3db4d8a32964b2a6a229559510d5d0c1": "(a_1, b_1)", "3db50a57e42d115cc20ece021c8940bf": "T (3,1)_{-\\frac{1}{3}}", "3db5277d2ef8d61aa82ac96c6bc92643": "V_{t+1} = V_{t} + a_\\text{thrust} \\cdot \\Delta t + a_\\text{gravity} \\cdot \\Delta t", "3db555b3679f01bc58fe40068d7b838b": "\\Gamma,A\\vdash B", "3db58782e901882732bac576a3e1fb1c": "e_2=\\{a,e_1\\}", "3db59fa8423c4518f8f792a6f927894d": "\n\\epsilon_{eff}=\\epsilon_1+\\frac{V_b \\epsilon_1 (\\epsilon_2-\\epsilon_1)/\n(\\epsilon_1+ P(\\epsilon_2-\\epsilon1)}{1-P V_b(\\epsilon_2-\\epsilon_1)/\n\\left [\\epsilon_1+P (\\epsilon_2-\\epsilon_1) \\right ]}\n", "3db5f15b9726a1b5f0bfa94b82c82721": " V^1(K') = V(K_+) - V(K_-)", "3db643053795f4604642f717dc3cfb0d": "k e^{a x} \\cos(b x) \\;\\;\\mathrm{or}\\;\\; ke^{a x} \\sin(b x) \\!", "3db6499cad561fc4871cf307666c108e": "m^{th}\\,", "3db651a4ba590f127712650168ddd131": "(A \\rightarrow B) \\rightarrow ((B \\rightarrow C) \\rightarrow (A \\rightarrow C))", "3db6886157f6bab34fc346c919f08874": "\\liminf_{i \\rightarrow \\infty} A_i = \\bigcup_i \\bigcap_{j \\geq i} A_j", "3db6ad84ca37980484dc8e501e085eb3": "n_d = \\frac {s_d \\cdot t} {t - 2}", "3db6ba9f084c398c9aa7c68decbce412": "L_n(\\beta^r)", "3db6f46c50acd225bb119302642a009d": " \\text{error} = -\\frac{(b-a)^3}{12N^2} f''(\\xi)", "3db6fc6e76fb67568a0a24aac41ac7d2": "\\langle W',R',\\Vdash'\\rangle", "3db73efac284449759c03bfe1b5fd0c8": "X_{2\\pi}(\\omega) = \\frac{1}{1-e^{-i \\omega}} + \\pi \\sum_{k=-\\infty}^{\\infty} \\delta (\\omega - 2\\pi k)\\!", "3db74ae99fc82ba6929ef563491d4b14": " \\!\\ x^2 - mx - 1 = 0.", "3db7908cffe272ce5c2a30962b4ad34e": "\\Delta h = \\frac{v^2}{2g}", "3db854d89a20359d9d65d0453f99055b": "\\scriptstyle s ", "3db87f1fddd24ce71dd2ce08a2eb2087": "(s \\downarrow T)", "3db8c5b7409367c509359f712ddf66e0": "h_{10}(t)", "3db907525d918d019be2144afd5fe9a7": " \\langle n\\pm 1|H|n\\rangle=-\\Delta \\ ", "3db91e4b2d38173f336ed7a2f3790991": "M \\to N", "3db951d6f7c31a983680279a98db2a74": "e_\\mathrm{rms}\\, =\\sqrt{\\, \\frac{1}{\\Delta}\\int_{-\\Delta/2}^{+\\Delta/2} e^2\\, de\\, }=\\, \\frac{\\Delta}{2\\sqrt{3}}", "3db95afcd9cc26f30e3fd98a01b70d62": "O\\sum_k c_k(0) \\exp(-i e_k t/\\hbar)\\,|e_k\\rangle=q_j\\sum_k c_k(0) \\exp(-i e_k t/\\hbar)\\,|e_k\\rangle", "3db98a57c8ea037c7daf98f09fd1fe4a": "dm_e = (1 - \\eta) \\ dm_{fuel}", "3db9b6cc94f0ceb0f49b5a24d301021f": " \\sum_{i,j = 1}^n c_ic_jK(t_i,t_j) = \\sum_{i,j = 1}^n c_ic_j_H = \\|\\sum_{j=1}^nc_jK_{t_j}\\|_H^2 \\geq 0 ", "3dba0d413bd333ae817851b83db90491": "I = \\frac{\\mathcal{E}}{R}\\left ( 1-e^{-Rt/L}\\right )\\,\\!", "3dba25910879b862888d5bd010587858": "[S/G]", "3dba28bb555c801dd115d7cf65f0e778": "\\scriptstyle P_{mmHg}=10^{7.80307 - \\frac {1651.2} {225+T}}", "3dba2b6142e6f471524d270dfca4c4a8": "x = 2.58383... ", "3dba4d13aab23eaf505c6ca67dd403ca": "C = dQ/dT", "3dbaa3676a58949a4f0344d5fee28c22": "d(km) =130(\\sqrt{hr(km)}+\\sqrt{ht(km)})", "3dbb1e907a136237cf92aad4db876ea0": " \\nabla \\times \\mathbf H = j \\omega \\epsilon' \\mathbf E + ( \\omega \\epsilon'' + \\sigma )\\mathbf E ", "3dbb65e61c0f42c00cdc4dd717a33be0": "f(0)=g(0)", "3dbb7a134dd723727300d24590f691ec": "\\mathcal{T}=\\operatorname{mod}-A", "3dbb7e0bc389ccaadbd39a3be477261e": "\\frac{x}{(1+x)(1-2x)}.", "3dbb8ada49b88deec3a86efa4ca46265": "A_\\textrm{i-1}", "3dbbb1e5b013b939163288c4f21f2752": "4 t", "3dbbfc4607e5e9a56eb373234fd1c451": "(2ax + b)^2 = b^2 - 4ac", "3dbc12621cb4bf3b3bd1f6d0e97a28c7": "\\vec{\\textbf v}_b\\,", "3dbc4cec512a18226bf322c8f13b2b6f": "[a\\;\\|\\;M_1]_h\\;\\|\\; [M]_m \\rightarrow [[b\\;\\|\\;M_1]_h\\;\\|\\;M]_m", "3dbc71823d35c14293e70c75d054ffdb": "(1 +2^{2^{n-1}})^{2} \\equiv 2^{1+2^{n-1}}", "3dbc7481cc81e15bdd845bee42cb4d66": "k_a = \\frac{X_{max}}{X_{rms}}", "3dbca3c0e7873113b9867b1fb342bb66": "x>m", "3dbcc8d324519a532c0e746ea1786221": "x \\mapsto x + I", "3dbd31e9500cda43876880d7942992d6": "\\Phi=L\\cdot I", "3dbd7693931dfcb4c3e6a08c2505c37e": "\\Phi=1/\\sqrt{2}(\\phi_{cl}(\\vec{r}-\\vec{R(t)})+\\sum_n q_n(t)\\beta(\\vec{r}))e^{i \\zeta(t)}, \\sigma=\\sigma_{cl}(\\vec{r}-\\vec{R(t)})+\\sum_n q_n(t)\\alpha(\\vec{r}). ", "3dbe2eb8603af647075821331dcfada0": " J_\\nu^{(2)}(x;q) = \\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty} (x/2)^\\nu {}_0\\phi_1(;q^{\\nu+1};q,-x^2q^{\\nu +1}/4) ", "3dbe59c7c1c32f7f8f0226cb30fbb0be": "\\nabla_\\beta\\, \\tau^{\\alpha \\beta} = 0", "3dbe7f0eca1140e614e06d899f364255": "A_{f}(x)", "3dbe8527721886033d74b9874a0621af": "V_n(R) = (2\\pi) V_{n-2}(R) \\int_0^R (1 - (r/R)^2)^{(n-2)/2}\\,r\\,dr.", "3dbf18f56cad204a60d7e34640875237": "D = \\{(x_1,\\cdots,x_n) \\in \\Bbb Z^n : x_1+x_2+\\cdots+x_n=0\\}", "3dbf5ff0d52bc73eefef794ad5cb9ab7": "x_1,x_2,x_4", "3dbf6da5cdb5cac535aea96556998e4f": "\\ COP = 1+\\frac{T_L}{T_H - T_L}", "3dbf91689b7cb1b68e45aca9145258d0": "\\begin{align}\n\\alpha &= \\textstyle{\\frac{1}{2}}\\left(-p+\\sqrt{p^2 - 4q}\\right)\\\\\n\\beta &= \\textstyle{\\frac{1}{2}}\\left(-p-\\sqrt{p^2 - 4q}\\right)\n\\end{align}", "3dbfc7336daa3b87ca1ab3f9716b8fd1": "e_b(k,0) = e_f(k,0) = x(k)\\,\\!", "3dbff6918966a78488b595d7c8f32980": "[X(t),U(t)]", "3dc04700164547ee2ed7a36d270e4e92": "(rs)x = r(sx)", "3dc06f9c413067aa0a3af96865633c0e": "m_2 = c^{d_Q}\\text{ (mod }q\\text{)}", "3dc0a56dd393dd1b3b9e4799a23a0e0c": "\\Pr[|Q_{h}(D)-Q_{h}(\\widehat{D})|\\geq \\alpha/2\\,\\!", "3dc0d6680830e22400ba973762f5b4cc": "[a]p\\,\\!", "3dc12a5e27cdd665cccc8be02aab073f": "\\delta R_{ab}/\\delta g_{cd}", "3dc15171ffc8634c41a879349b4d8528": "C_i = c_i \\cdot N_{\\rm A}", "3dc157ae32f4db6e1a11e92037447a44": "f: X \\rightarrow X'", "3dc1aa24220dc2c008523010bb746e85": "-j \\frac{\\sqrt 2}{2}", "3dc21ad113dd279d1cde1f3299633c4b": "\\sqrt{a_ih_i}=\\sqrt{\\frac{a_i+h_i}{\\frac{a_i+h_i}{h_ia_i}}}=\\sqrt{\\frac{a_i+h_i}{\\frac{1}{a_i}+\\frac{1}{h_i}}}=\\sqrt{a_{i+1}h_{i+1}}", "3dc21e1bb56b3308e5cb9a66690f218b": "fl(x \\cdot y)=fl(fl(x_1*y_1)+fl(x_2*y_2))", "3dc239f740bdf9b1eebe40fb2933cfa9": "{\\rho}_{s}=0.45(\\frac{A_g}{A_{ch}}-1)\\frac{f'_c}{f_{yt}}", "3dc23ba4bd5c5fe1c2a9666dac0e1530": "[gL^3(t_s-t_\\infty)]/v^2T", "3dc2459e27a6326641336bbf8837e03d": " w_{1} \\neq 0 ", "3dc295bd1cbc31d7ebbea1b3fed8668f": "R_B(x)= \\sum_{k=0}^\\infty r_k(B) x^k", "3dc2bbe579a687e4865bc2f0db2f60ff": "E_d = \\frac{P}{Q_d}\\times\\frac{dQ_d}{dP}", "3dc2c2138b431bf718ef642b8157d504": "\\int_V\\varphi\\circ F(x) \\psi(x)\\,dx = \\int_U\\varphi(x) \\psi \\left (F^{-1}(x) \\right ) \\left |\\det dF^{-1}(x) \\right |\\,dx.", "3dc336399110db67d2dac9e00aead0bf": "\\therefore (x/a)^2 + (y/a)^2 = 1,", "3dc36693737fc6fa9e3850c210651e3c": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {(U\\otimes V)A(U\\otimes V)^\\dagger} \\right) = \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A} \\right),", "3dc36eb5e5a839fa0bd89ed0e830e0a7": " D=h-h'=\\frac{1}{\\frac{1}{\\lambda}-\\frac{1}{\\lambda'}} ", "3dc39cb9cf0de507ef24a524b9d9f245": "E(z) = 1", "3dc3be8a77a2fed351e1aabc5818c8b2": "f_d=f(1+\\frac{v}{c})", "3dc43724a6b9d3cddfb7bcead5519657": "Var(Y|X=x)", "3dc46dfc0894f98e29862cb640fdc009": "\nT(u)T(v)=T(v)T(u).\n", "3dc4f4a7c76d2008dd3f95c88ba0f375": " \\sigma^1 \\otimes \\sigma^1 + \\sigma^2 \\otimes \\sigma^2 = \\exp(2 p) \\, \\left( dx \\otimes dx + dy \\otimes dy \\right). ", "3dc54d439202b1433cdedc7a95d79c7e": " \\int_{-r}^r \\pi y^2 \\,dx = \\int_{-r}^r \\pi(r^2 - x^2) \\,dx.", "3dc59fbef7808500855b6842d4f69264": "baa = b,\tbab = c,\tbac = a,\tbba = a,\tbbb = b,\tbbc = c,\tbca = c,\tbcb = a,\tbcc = b,", "3dc5d25ef708b9d3c559f8884747f7e9": "M_1 ={1\\over 2m}M \\supset M.", "3dc5da72ecca950704f414676a66c1b9": " = l", "3dc62ae366ceffa5dedf7c6a21bf5d2e": "e^{-i\\langle x,\\zeta\\rangle}", "3dc6d51a11418af9e8c0cde642bccce2": "B^T=VMV^{-1}", "3dc7571880ae265cf09520c2b6da13b8": "\\Delta_r G^\\circ = -RT(\\ln K_{eq}) = -2.303RT(\\log_{10} K_{eq})", "3dc76311dfb639c00f25af4c6548dbcc": "U(t+\\bigtriangleup t) =\\int_0^\\infty U(t+\\bigtriangleup t,w)dw.\\quad (2.b)", "3dc767291b4fc70cdfeef9071dd87c3d": "r_{k} = (1 + r)^{k}-1", "3dc7767f813a2761b237417e57c8cdaa": "\\displaystyle{\\|Y w\\|_{(1)} \\le C\\|L w\\|_{(0)} + C^\\prime \\|w\\|_{(1)}.}", "3dc7902ab8404ae07055aa2125089c8f": "\n\\begin{align}\n\\lambda = \\frac{h}{\\sqrt{2mE}}, \\qquad \\lambda[\\textrm{A}]=\\sqrt{\\frac{150}{E[\\textrm{eV}]}}\n\\end{align}\n", "3dc798f6e312ad811249f202c5d5f3db": "\\log_2\\left(M\\right)\\le m\\le\\log_2 \\left(N_x\\right)", "3dc7a66dce48552c02a0343ebcadc79c": "T = \\sqrt{\\left(\\frac{wS^2}{8d}\\right)^2 + \\left(\\frac{wS}{2}\\right)^2}", "3dc7c2639c40d2eafb9b41e4c1b5a4c0": " 2\\pi \\times \\mathit{integer}", "3dc80311678b115f6feb86377c8352a9": " \\Delta Y_t=Y_t-Y_{t-1}", "3dc812de83c2a01894f647284d30f8fd": "\\{A, B, C, D\\} \\in Q, R", "3dc815ec76a126f37c94f707312125be": "\\lambda_1, \\lambda_2, \\lambda_3", "3dc8374077f1bbc14a210181788f89b6": " \\alpha_0 ", "3dc84c5cd09b49b58c00660ef5734f76": " \\lambda_{\\pm} = {1\\over 2}\\left( a+b\\right) \\pm {1\\over 2}\\sqrt{ \\left(a-b\\right)^2+4c^2}. ", "3dc84d071ecf9c807823aef8a312db33": "{{S}_4 = 1 + \\varphi }", "3dc86c20ffd9db6c4ab49f69fc402f50": "{\\alpha\\choose n} = \\prod_{k=1}^n \\frac{\\alpha-k+1}k = \\frac{\\alpha(\\alpha-1)\\cdots(\\alpha-n+1)}{n!}", "3dc906c36ad90365e22e2ee82276f2e7": "\\delta = \\frac{\\mbox{difference between a resonance frequency and that of a reference substance}}{\\mbox{operating frequency of the spectrometer}}", "3dc915bd8b2365b914b657923ae7a418": "LFL_{mix}", "3dc932ba11d070b15b7b87084d7e4ab0": " \\frac{\\|p-\\hat{p}\\|}{\\|p\\|}\\leq \\Lambda_n(T)\\frac{\\|u-\\hat{u}\\|}{\\|u\\|}", "3dc93da5c94b22a09a15bd9f7d2b3866": "\\operatorname{E}[nS^2] = (n-1)\\sigma^2\\text{ and }\\operatorname{Var}(nS^2)=2(n-1)\\sigma^4. ", "3dc940b6300cd062df7d07398ebafe6d": "y \\in Q", "3dc9577a2dc1db29fd210291411d643f": "\\mathcal{T}\\,\\!", "3dc99e217d0c51f53d14b8fcef6964e1": "d_p\\equiv d \\bmod (p-1)", "3dca25933ce76a43647010054ba54785": "M\\ddot{X} = - \\nabla U(X) - \\gamma M\\dot{X} + \\sqrt{2 \\gamma k_B T M} R(t)", "3dca2dd6a73e84784b398b94d567e0dc": "S_1,\\dots,S_p", "3dca79984a41d1406fccf7e2d69dec9d": "2^{-p} \\le 1 - \\frac{y}{x} \\le 2^{-q} ", "3dcaaa1a72c57aa5ff14b74214db7d6e": "\\mathbf{e}_i", "3dcab918c41f1a27f13acbc6f5a2df7f": " |x| = 1 ", "3dcabef358a7531a56eefc27f20e55f7": "\\langle T_v\\exp_p(v), T_v\\exp_p(w)\\rangle", "3dcbc8af810c140ecdd72792f91d3ba9": "\\mathbf{\\hat{d}}_\\mathrm{s} = \\mathbf{R} \\; \\mathbf{\\hat{d}}_\\mathrm{i},", "3dcc235a930b9c19afc51d5848141d2f": "\\Psi(\\mathbf{q}) ", "3dcc2925ead197b8514f11ae4c64061f": "\\lambda |B|", "3dcd5377b93afa95bceea09746ae66bb": " h_{ii}=(H)_{ii} ", "3dcd5f7d85a40be8543cd49c752d862b": "*(R_q),C_{pq}^-,*(R_p),C_{pq}^+,*(R_q)", "3dcda8c3442a004496aeca51328d29ca": " \\Pr[X_N > t] \\leq \\exp\\left(\\frac{-t^2}{2 N}\\right).", "3dcdbbacb6c4e1b85c310b28caea1417": "v_\\mathrm i = Vu(\\kappa t-x)\\,\\! ", "3dcdd37dfe651128b7a749f6cbacb036": "\nv \\ \\stackrel{\\mathrm{def}}{=}\\ \\rho \\cos(\\phi) \\ \\stackrel{\\mathrm{def}}{=}\\ - \\frac{1}{\\omega} \n\\left( \\frac{dP}{d\\zeta} \\right)\n", "3dcddb207019896a9571aaeed16a466c": " X = \\left[ \\begin{matrix} t+z & x-iy \\\\ x+iy & t-z \\end{matrix} \\right]. ", "3dce34c0cd034700cdefa856d103817b": "\\{x\\}", "3dce42e7882bec13e6b5b2deb61ba130": "A\\subseteq \\mathbb{R}", "3dce4f849d7108b1dbee2ce39ec4a0ba": "(\\dot{x}^i,\\ddot{x}^i)", "3dced1553c70cc116b876bb99c3d91f2": " 1,t,t^2,t^3, \\cdots ", "3dd02ed358e3b47b5e469fe6fc4575a1": "A \\cdot \\neg B \\cdot \\neg C", "3dd06d4d987de3a4de535983feabf9ba": "{\\color{Blue}~2.27}", "3dd07a00f4ba66d9ede72937ab582992": "\\frac{\\sigma^2}{2\\theta}", "3dd098b3f3ebe312f80cfeeaf96cc31c": " q \\in Q_N ", "3dd0dbf9007df0724a3b259cfd86f6d9": "A=A_o \\cos[(2\\pi/\\lambda)(x- ct) + \\varphi]\\,", "3dd0eb6ce438ddd13eea8d40102abb14": "a_{11} = 0", "3dd12d5116f172fac92643443ffe79bf": "x \\ge y", "3dd18283d333337bb4d7b3002b697026": " M^{-1} = \n\\left[\n\\begin{array}{ccc|c}\n & & & \\\\\n & R^T & & -R^T T \\\\\n & & & \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n", "3dd1d0fc420e17ff68bfb83453202131": " b_t ", "3dd1fb43bfcb099523954b949b91c8d8": "\nr = \\frac{TP+FN}{TP+TN+FP+FN} = \\frac{\\textit{Positives}}{N}\n", "3dd22ebf1fd9cb958cc73b2a8cc8a603": "TR^{-1} \\pmod{N}. \\, ", "3dd2a85e1d6512f87f7ed32fc1fad5e7": "\\left(H_{0}-E_{0}\\right)-\\left(H_{1}-E_{1}\\right)=E\\left(\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}-1\\right)", "3dd2c9e7d9c60184046762737e459058": "\\{p,s\\}", "3dd2d9c86dfeb1da21625dea4f576e87": "q\\frac{dL}{dq} = \\frac {L^2-M}{12}", "3dd31392ac6bba03c08197907eb12eba": "\\hat f \\circ T= \\hat f \\, ", "3dd317bb5816c37c44b059aa31c23fc0": "\\boldsymbol{F}", "3dd3491b179086451f43ddb619236c52": "\\bar{\\lambda}_\\text{C} \\approx 386.159\\;268\\;00 \\times 10^{-15} \\mbox{m}", "3dd352a4673f3c29eb14ca609691671f": "T \\in W", "3dd368e4da344739b49e83a40db6fdc3": "{s,p,\\gamma,r}", "3dd384a962800d8a6a467828e6ca4b53": "M_{k}^{\\prime} = M_{k}", "3dd38dad52330c090b23ff587e0086cf": "V_\\mathrm {rms} = \\frac{V_\\mathrm {peak}}{2}", "3dd3ae0884437eeee5db65ffe9ea29b6": "\\delta_p", "3dd3b1d86b72ccc28e5a0ea241511dad": "\\tilde{h}:V\\otimes W\\to Z", "3dd3b8a8a9b693fe0fc722a69837f1d8": "U(q,1)\\begin{pmatrix}z & 0 \\\\ 0 & z^* \\end{pmatrix} = U(q z , z^*) \\thicksim U((z^*)^{-1} q z , 1).", "3dd45f5263da813d7a568323236d06d1": "\\frac{(2\\lambda+1)^2}{2(4\\lambda+1)} \\frac{ g_2^2\\big(3g_2^2-4g_1g_3+g_4\\big)}{g_4\\big(g_1^2-g_2\\big)^2} - 3,", "3dd4becc796f408ad3f00869cbcfe963": "\\left(\\sigma^2 + \\tau^2\\right) d\\sigma \\, d\\tau \\, dz", "3dd4d13956762b009596cc7e0b6af1ca": "\\mathbf B = \\begin{bmatrix}\n-2 & 2 & 1\\\\\n1 & 3 & 2\\\\\n1 & -2 & 0\\end{bmatrix}\n", "3dd549e191406dff17d69db2783c8ba4": "\\liminf_{n \\to \\infty}n^2\\left |\\xi-\\frac{m}{n}\\right |,", "3dd55ef8a9175060008fb7b7d0010e28": "\\exists p: \\mathcal{B}\\mathcal{B}p \\wedge \\neg\\mathcal{B}p ", "3dd56a5596e0ea2397ed0ebfd9e00bc5": "\\overline{Z}_{1}", "3dd581dd4d40accd8c5e2dba8e7a0faa": "\\Sigma = P\\,\\Lambda^{1/2}\\,P^{-1}", "3dd59d2a98004ebb3acf4a180a553ce4": "f_{s+1}.", "3dd5a3f98cac41666d67bcb5789e3e50": "g(k)", "3dd6149fb9654e640d45ddf756c287fb": "e \\in L'", "3dd6249e5fa759f47d97b025d982e3a2": "D_{CSD}=V_{DS}t_g", "3dd62a780519bcb1375164a986a09350": "\\left(a_1 b_2 + a_2 b_1 + \\frac{a_3 u_1}{b_1^2+b_2^2} - \\frac{a_4 u_2}{b_1^2+b_2^2}\\right)^2+\\,", "3dd648774d25ace07e5775d648cb0df5": "r_t(i)", "3dd670ab778c02b04d0c39fff07baf1a": "[\\mathbf{L}, H] = 0, [L^2, H] = 0", "3dd67575fd5b0ca965a6df85f06c0e4a": "u_{l} = \\frac {1} {\\sqrt {N}} \\sum_{q} Q_{q} e^{ i q a l }", "3dd6780046002c90a40aa3d641727489": "\\scriptstyle (P_k)_{k \\in K}", "3dd6a6b4c320de04c25e74f6454e574a": "\\mathbf{A}_{ij} : \\mathcal{H}_j \\rightarrow \\mathcal{H} _i", "3dd6eb8a56aa13f5af5704c6f95bd115": "\\left(R - \\sqrt{x^2 + y^2}\\right)^2 + z^2 < r^2", "3dd6ec684a3698a1ac7fcffab99987c9": "q = q_0 \\cos(\\omega t + \\phi)\\,\\!", "3dd6f6bee4b60c287cb97292e3fb6ba5": "C_{II} = 0", "3dd6feb3b24bf9b85ab273c62ca33c25": "\\Phi^\\vee", "3dd737a2530b24b19ec75c71770f7463": "\\scriptstyle \\sin\\,", "3dd74784de75eb556af66ab2f6499a98": "\\alpha_J = (R_i - R_f) - \\beta_{iM} \\cdot (R_M - R_f)", "3dd74e0da96ef99f700ff9c765bf6451": "Td_0 = 1", "3dd76f7d8b4b64c1c1f697f030a27c17": " \\Omega = 2\\pi - \\arg \\prod_{j=1}^{n} \\left(\n \\left[ s_{j-1} s_j \\right]\\left[ s_{j} s_{j+1} \\right] -\n \\left[ s_{j-1} s_{j+1} \\right] +\n i\\left[ s_{j-1} s_j s_{j+1} \\right]\n \\right)\n", "3dd7800d01aeeae5ce65ce9bf3a6ba93": "|N(S)| \\leq d|S|\\,", "3dd7b7a7f9de7ae50a23efaf57f94504": " \\nabla_X(\\lambda v)=\\lambda \\nabla_Xv", "3dd7cb7890c1da00e4197f08f137dde7": "y(i,j) = \n\\begin{cases} \n 1, & x(i,j) > \\bar x \\\\\n 0, & x(i,j) \\le \\bar x \n\\end{cases}", "3dd7d5f092fe93d6278b3de751ad5bf3": "A \\supset B", "3dd892663879a65b2cf48241afec73df": "\\omega_j=\\exp \\left(\\tfrac{2\\pi i j}{n}\\right)", "3dd8dca7208d892e99d826db80724a52": "\ng_n = \\sum_{n_1=0}^n n - n_1 + 1 = \\frac{(n+1)(n+2)}{2}\n", "3dd957ffe2b5ee2e17858f92eddd5c9a": " (a^2+b^2)(c^2+d^2)= (ac-bd)^2+(ad+bc)^2. \\,", "3dd96c0a5a2afbd6d003cf48b022cd2d": "W^0/P,", "3dd9897b3282fe96cac54faeb90d5dc0": "UAV=D", "3dd991e6387551ca31b742597756ecee": "\\rho_\\text{EtOH}(A_w)", "3dd9bd06178a69b381ac459bb6289e94": "\\forall p\\,\\phi", "3dd9d672fdc338aaf9921494cd283c0b": "GW_{g, n}^{X, A} \\in H_d(\\bar M_{g, n} \\times X^n, \\mathbb{Q}).", "3dd9dff8157f6cc5ecb9248487518d03": " \\textbf{p} = [T(t)]^{-1}\\textbf{P}(t) = \n\\begin{Bmatrix} \\textbf{p} \\\\ 1\\end{Bmatrix}=\\begin{bmatrix} A(t)^T & -A(t)^T\\textbf{d}(t) \\\\ 0 & 1\\end{bmatrix}\n\\begin{Bmatrix} \\textbf{P}(t) \\\\ 1\\end{Bmatrix}.", "3dd9ecf1403ac8e6af7463340c6e13b3": "\\Psi\\colon G \\to \\mathrm{Aut}(G)\\,", "3dda08770f60af43fa2e8072b9eb8461": "\\vert{\\Psi_0}\\rangle = (1 + C)\\vert{\\Phi_0}\\rangle", "3dda125a7161aa63cc8469489fae7876": "z = \\frac {R(t_0)}{R(t_e)} - 1 \\ . ", "3dda1fb20cc2a4a05e75b922f7afc957": "x(t+\\tau)=a \\cdot b^{\\frac{t+\\tau}{\\tau}} = a \\cdot b^{\\frac{t}{\\tau}} \\cdot b^{\\frac{\\tau}{\\tau}} = x(t)\\cdot b\\, .", "3dda736b77e923b1fd6763273325a6e5": "f(x)= \\int_0^\\infty \\chi_{f(x)>r} \\, dr", "3ddb1782ac5f86bfc6e29eb7e37ee85f": "L(\\theta,\\widehat{\\theta})", "3ddb5dcafbd9ded902c89fac273d6391": " \n \\begin{align}\n \\alpha &= \\frac{1}{2h} \\Big( f(t_0+h, p(t_0+h)) - f(t_0, p(t_0)) \\Big), \\\\\n \\beta &= f(t_0, p(t_0)), \\\\\n \\gamma &= y_0. \n \\end{align} \n", "3ddb6ab991d7fc0d7c98a0de7bfcc894": " \\mathbf{\\hat X} ", "3ddbe96f8a8b7c2d47fc71cd49f5fc2d": " (x_1, x_2).", "3ddc55be85d364b1635cc09fcecb49c9": " \\oint_S \\mathbf{B} \\cdot {\\rm d}\\mathbf{l}= \\mu_0 \\oint_S \\mathbf{J} \\cdot {\\rm d}\\mathbf{A} + \\mu_0 \\oint_S \\mathbf{J} _{\\rm d} \\cdot {\\rm d}\\mathbf{A}, \\,\\!", "3ddc57aecc604c4003042350d4ca44c3": "\\tfrac{1}{3} \\pi r^2 h = \\tfrac{1}{3} \\pi r^2 (2r) = (\\tfrac{2}{3} \\pi r^3) \\times 1,", "3ddc6917ec41991a3d79f18591fa5285": " \\rho_1 v_{n1} \\mathbf{v_{t1}} - \\frac{\\mathbf{B_{t1}}B_{n1}}{\\mu_0}= \\rho_2 v_{n2} \\mathbf{v_{t2}} - \\frac{\\mathbf{B_{t2}}B_{n2}}{\\mu_0},", "3ddc8626805a82969d6b00bc74d2f60d": "A_k/A_{k-1}", "3ddca1afde69660647b49589f62a1079": "t_r' =", "3ddca70c19067406693dd8ccd36cefc9": "3\\uparrow^m n", "3ddcae4cbb9a5cd085911831cae571f6": " \\left( \\sum_{n=0}^\\infty p(n) x^n \\right) \\cdot \\left(\\prod_{n=1}^\\infty (1-x^n)\\right)\n = 1 ", "3ddcd1773bd77388b51d6c00abfd5fcd": "J(\\mathbf X,t)\\,\\!", "3ddcdf404af466223de21c043ceb7fab": "q_{ij}", "3ddcef5b0bc5e4ce155d6a2a763b2a77": "\\ [M]\\in H^{BM}_{top}(M) ", "3ddd52bba35a9e8cbcf4864762098b73": "\\mathfrak h = \\mathfrak k\\oplus\\mathfrak m,", "3ddd5c71549933b6c5b61d69cda9dc4c": "\\mathrm{error}\\bigl(x(t_0 + T)\\bigr) = \\left(\\frac{T^2}{2\\Delta t^2} + \\frac{T}{2\\Delta t}\\right) O(\\Delta t^4)", "3ddda17bd0536538df1835656d8a4faa": "{\\pi} r^2\\left ( -D_c \\frac{dC}{dx} \\right )_x = {\\pi}r^2\\left ( -D_c \\frac{dC}{dx} \\right )_{x+{\\Delta}x}+\\left ( 2{\\pi}r{\\Delta}x \\right )\\left ( k_1C \\right )", "3dddade35bc2ded239387b59f381f62f": "\n{\\mathbf l}^{'t} \\left( \\sum_i x_i {\\mathbf T}_i \\right) {\\mathbf l}^{''} = 0\n", "3dde53aa4c6730b4a6931a958da8f639": "R_G\\neq 0", "3dde61390c74accaa059c883c119fb0a": "P_i=\\infty_1", "3ddede2b81d96b04c16738433f733422": "T(x) = 5x", "3ddf1d38e4810f976d3c0248ebfc13d2": "a(n)", "3ddf3bfaf005f479c34a671e5acf4b35": "\\begin{matrix} {4 \\choose 4}{3 \\choose 1}^4{36 \\choose 1} \\end{matrix}", "3ddf60a7f889507f7b311f60d9fb2925": "\\langle u,u'\\rangle^k = \\langle u^k , k u^{k-1} u' \\rangle \\quad (u \\ne 0) ", "3ddf9c1d8bfbc96f294f1658ead6e929": "s^2 = 0 \\,", "3de0292d09a775d02b0957fa3fd4a1b0": "0 \\times 3 = 0.", "3de079adc3007dd50b55738af8f6951e": "76^2", "3de0a77544f221e36acc594dfb9a935f": "\n\\mu(A)=\\begin{cases}\n\\vert A \\vert & \\text{if } A \\text{ is finite}\\\\\n+\\infty & \\text{if } A \\text{ is infinite}\n\\end{cases}\n", "3de0a87d337b1d2db63d7545652bc025": " Z/pZ \\oplus Z/pZ \\oplus Z/pZ ", "3de10d8e72124502fd38d875d5a289c7": "d(\\langle \\mu, \\xi \\rangle) = \\iota_{\\rho(\\xi)} \\omega", "3de16c0a2c4a67137ebb7d8f412c75c2": "D \\subseteq P", "3de1b5f7ed623f0c4a46b7d21208930c": " g(x,y) \\,", "3de1b63b33348f1c6599b64fec0fb4b8": "\\vdash \\Psi \\rightarrow (\\Box \\Psi \\rightarrow P)", "3de1f3d674e2ee46449100345a7d0658": "xR", "3de20ecef4072afb951b9c3c65600265": " S(x, y) =\n \\sum_{i=\\lfloor x \\rfloor - a + 1}^{\\lfloor x \\rfloor + a}\n \\sum_{j=\\lfloor y \\rfloor - a + 1}^{\\lfloor y \\rfloor + a}\n s_{i\\,j} L(x - i) L(y - j).", "3de246bde65fcd102e451612661440f2": "0 < b < a", "3de24c8edaad3621b0884f7abdc8732b": "f(y^*)=\\sum_{i=1}^n\\left(\\frac{y_i^*-y_i}{\\sigma_i}\\right)^2", "3de2812fe84931ee75a611ca96fce764": "\\nu^3", "3de2b1efbc4eb0a420105063253646b3": "\\alpha T = \\frac{\\Pi_{WW}(0)}{M_W^2} - \\frac{\\Pi_{ZZ}(0)}{M_Z^2}", "3de2e5c303df77f42dd4f17a8fe3f85d": "\\textstyle I=\\frac{1}{12} m \\ell^2", "3de3010f18c7f68091538c78f7b95f89": "\\frac{d\\sigma}{d\\Omega}", "3de32d30e739c5c4dd5cb3408e4a599d": "H_*(X)", "3de34ba465980f468012617e2c5fedd6": "d_{max} = f^2R", "3de38ce494c9e9fce906ad266d12c8c1": "\\nu(U) + \\nu(V) = \\nu(U \\cup V) + \\nu(U \\cap V)", "3de3a49c5abfd376b5475f9551817abf": "r \\approx z \\left( 1 + \\frac{1}{2} \\frac{\\left(x - x^\\prime \\right)^2 + y^{\\prime 2}}{z^2} \\right)", "3de3c39c5435f5f454a71c43d2220a5c": "N_{R_1} \\times N_{R_2} \\times N_{R_3}", "3de3d122fad069602cb6de65954da79f": "(a, b)^* = (a, -b).\\,", "3de407178e444032a217640005c3a5e5": "P_{xy}", "3de43312e3ccef502e0345b8b0c04e55": "\\beta\\times 10^{11}", "3de495674180c8ccf0d445763cac891b": "(\\cdot \\cdot )_\\infty", "3de4bd5f093bcff6c05eff3ab075854a": "\\text{Composition}\\ (wt%)", "3de4bd979217ae9f8ef06b510dfb6718": " \\mathcal{B} = \\mathfrak{P}(\\mathfrak{P}_{\\ge 1}(\\mathcal{Z})).", "3de4bf9bb8fc70e250666dba1f3ca45a": "p_0 = 1", "3de50e187918d8af16365101b48a3b5c": "\\operatorname{Reg}_K", "3de57c027c92343fd8501ce5468b124c": "p({\\tfrac{1}{2}}) = 1", "3de59b9492a5f90a10fe92438dea09ad": "c_{i,0}\\neq 0", "3de5b301343350b9224d8f9f8cd8c067": "\n\\frac{1}{t_{1/2e}} = \\frac{1}{t_{1/2p}} + \\frac{1}{t_{1/2b}}", "3de5e323c9a8be969cca1d9ca0526d4b": "\\langle O(n^2), O(n^2) \\rangle ", "3de653f543a0dfbee3723972db3028cd": "\nV \\ \\stackrel{\\mathrm{def}}{=}\\ \n \\left\\{\\begin{matrix}\n & e &\\qquad \\mbox{(axis)} \\\\\n & Q &\\qquad \\mbox{(the general equation of the conic)}\n \\end{matrix} \\right.\n", "3de65589482eb8effed01c4f900418ee": "N^2\\left(\\frac{\\partial^4 z}{\\partial x^4} + \\frac{\\partial^4 z}{\\partial x^2 \\partial y^2} + \\frac{\\partial^4 z}{\\partial y^4}\\right) + \\frac{\\partial^2 z}{\\partial t^2} = 0 ", "3de67e54ffeb405c8f70fa93e5aecbac": "|\\overline{AC}|\\cdot |\\overline{BD}|=|\\overline{AB}|\\cdot |\\overline{CD}|+|\\overline{BC}|\\cdot |\\overline{AD}|", "3de6dc171b781e97d5910c19dd846f57": "\\mathbf F\\,\\!", "3de7ba7c448599a9336d6392581445de": "M \\otimes_R N", "3de7c63a42252ea2df712de168623207": "\\boldsymbol{F}^{-T}", "3de7dd32c85e56635d9d7848ad1cc0b3": "\\tau(mn) = \\tau(m)\\tau(n)", "3de80bcec963fc3dc71de1158604ab54": " \\omega_r ", "3de827e91102c3e41ceedbe5983090f2": "\n d\\boldsymbol{\\sigma}:\\frac{\\partial f}{\\partial \\boldsymbol{\\sigma}} = 0 \\quad \\text{and} \\quad d\\boldsymbol{\\sigma}:d\\boldsymbol{\\varepsilon}_p = 0 \\,.\n ", "3de8444f17fba6fbf895ef909ada9e33": " A_{m}(\\omega, \\gamma)", "3de8711169170d06f8a9c7a9d3b1f12a": "e^{j \\omega t} = \\underbrace{\\cos(\\omega t)}_{R(t)} + j\\cdot \\underbrace{\\sin(\\omega t)}_{I(t)}", "3de8dd923cdea62e122832149e664735": "H(x,y) = \\begin{pmatrix}f_{xx}(x,y) &f_{xy}(x,y)\\\\f_{yx}(x,y) &f_{yy}(x,y)\\end{pmatrix}", "3de8fda3fa9f0a254177ce2dbbd85d9c": "\\mathbf{p} = m \\frac{d\\mathbf{r}}{dt}", "3de90564c61daf602b582735803fed9c": "a+bi", "3de94fe8d3b4fa9b61ec5b1a7eaca3dc": "n=\\{n_1,\\cdots,n_k\\}", "3dea36fa8e630d4f03ee075e48809c80": " Z = \\pi^{-1}(Y' \\oplus R) = \\pi^{-1} (Y \\oplus N \\oplus R), ", "3dea7c52b1b256047417ae6ccb213cb5": "\\Delta_i=\\exp(-\\zeta^\\prime_i(0))", "3dea8cfd976db28643d4a5666beacf24": " W_V ", "3deaaad508336d2c7849f103563533af": "f = n_D n_T \\langle\\sigma v\\rangle = \\frac{1}{4}n_e^2 \\langle\\sigma v\\rangle ", "3deaf4b58ff40493c4b4eb3f5c7cd972": "J=\n\\begin{bmatrix}\n0 & I_n \\\\\n-I_n & 0 \\\\\n\\end{bmatrix}", "3deb38190c5b4a13b57b5253785ed0c3": "K_*(X) = MU_*(X)\\otimes_{MU_*}K_*,", "3deb4bb442c6b0e51b32ddbc304632ec": "{N_Q+q_0(t_P-t_Q)-k_0(x_M-x_U)}", "3debd4747a67bb0ded3b8c3bcaeb00cb": "X\\to \\langle aZb\\rangle Y", "3dec364739d92c7918a5f1555f2ad016": "\\alpha,\\beta,\\gamma,\\delta", "3dec91a867677b3a8032905dc3c9e5d2": "Doppler = 180,000^o/s = 720 \\left( \\frac{75 \\times 10^9}{3 \\times 10^8} \\right) = 720 \\left( \\frac{Velocity \\times Transmit \\ Frequency}{C} \\right)", "3decdaf7ca2b9a7685bc9068e712366b": " E = \\frac{\\hbar^2}{2m}(k_x^2 + k_y^2 + k_z^2) ", "3decfb3f79eb8f28ab8df8769533bbdf": " 0 \\leq i \\leq d ", "3ded0b552177d486b8abc721e8dba8b9": "\\lambda+\\delta\\lambda", "3ded2184a3e467984dba5788f82cc430": "a,", "3ded841415054ebb3b2f1ab1935ee853": " \\widehat{\\mathbf{S}} = (\\widehat{S_x}, \\widehat{S_y}, \\widehat{S_z}) ", "3ded9104b2afc8571b7704aded4fe113": "\\mbox{gl dim } A = \\dim A", "3dedc4671ffe3ec879f0c773626ca550": "Q(X)=\\sum_{ij} a_{ij}X_iX_j=0\\,", "3dedc605ecb0c998943524f929177f1d": "x_1+\\cdots+x_n=y_1+\\cdots+y_n", "3dedd28b27706358fd3837dabbcecd7f": "I(\\nu,T) ~A ~d\\nu ~d\\Omega", "3deeb742dbc93f29145eaca79436d8b6": "y(a)= \\alpha \\ \\text{and} \\ y(b) = \\beta", "3deebd7e15000ee5cb9928b46dcc5707": " uv", "3deec5e79624dd8f13b8b94b4b7562b0": "\\mathcal{I}(\\theta)", "3def6223bbff67f428255847d996b54b": "MRP = MPP \\times \\text{Price}\\,\\!", "3def6d54b7a8e203ca85d9939621bfe0": "\\frac{Y}{1-e^{-Y}}", "3defad9350ff2f77012c6840d130576e": "w_i^+=w_i+x_i\\Theta(\\sigma_i\\tau)\\Theta(\\tau^A\\tau^B)", "3defb9cfe09b6cd288ac399720504243": "(p,q)\\sim (r,s) \\leftrightarrow ps=qr", "3defd3c89a3be31b495a42b314735fac": "f(u^n)= \\left[ f(u) \\right]^n ", "3df071d3d2f2464a1c9b11e2ce44ee6f": "m = (p, u, v)", "3df088a8e0ae490c8ea6aca0f0f0feca": "^\\ast", "3df09532ff71e7a824f46a7a20dee631": "\n \\begin{align}\n \\limsup_{n\\to\\infty} L_{\\hat{X}^n_{DUDE}}\\left( X^n,Z^n \\right) = \n \\lim_{n\\to\\infty}\\min_{\\hat{X}^n\\in\\mathcal{D}_n}\\mathbf{E} \\left[L_{\\hat{X}^n}\\left( X^n,Z^n\n \\right)\\right]\\,,\\,\\text{ almost surely}\\,.\n \\end{align}\n ", "3df0b6999532ef8ac426275f1bce8834": "N = nN_A\\,", "3df0c826860f58218bcf7868e2f596df": " \\frac{dI}{dt} = (R_0 S/N - 1) \\nu I ", "3df0dc440b02482d58096d246ba5b9e9": "{\\bar{S}}_8", "3df0f5cf2cc0f8dceb1aa0b4cce8e324": "S_F=\\int d^4x\\,\\sqrt{-g} \\left( \\frac1{12}F_{\\mu\\nu\\rho}F^{\\mu\\nu\\rho} - \\frac14\\mu^2 A_{\\mu\\nu}A^{\\mu\\nu} \\right)\\;", "3df13a805134c97512cd9b3c2583e7ed": "\\phi(E_i)\\subseteq F_i\\,", "3df155317af75d0096cc009307a67e94": "\\varepsilon_2 = \\frac{1}{E}((1+\\nu)\\sigma_2-\\nu(\\sigma_1+\\sigma_2+\\sigma_3))", "3df1b1ca23dcf82696872232afe15c28": " A = \\oplus _{k = 1} ^m M_{n_k},", "3df1b2777be5250905ef524bae07df33": "r_{xx}", "3df1db69f777cab7e43f4360cc98dad3": "\\left(\\frac{\\gamma + 1}{2 \\gamma}\\right) T", "3df20aaa9b238d9b2634abb25fbfac4c": "\\bigstar\\bigstar\\bigstar \\mathbf S", "3df2207426474d5a80aedfc4659a305d": "\\textstyle\\frac{9}{10}\\log_2{10}\\approx 2.99", "3df2569b0f87919f757e46a3f61754d1": "\\mu(AB)=\\mu(A'B').\\,", "3df2c5250c3d55fafa3e4b56673aa06c": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} = \n \\cfrac{2hE}{(1-\\nu^2)}~\\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} u^0_{1,1} \\\\ 0 \\\\ 0 \\end{bmatrix}\n", "3df2d46a833b49d99805e649c915f712": " \\frac{1}{Z_{\\text{GSE}(n)}} e^{- n \\mathrm{tr} H^2} \\, ", "3df2e54f39e57716611d43758243cc7d": "|\\mathcal{Z}|", "3df34d4e6fb29fbddb65ce555c83959c": "\\dot{M}_{i,j} = W_{i,j(i)} M_{i,j} ", "3df3a70c6eb8705add458637fc2e405f": "S \\to aSSb", "3df3c5e22ad6462b20c1d9c45a8e2883": "\\mathbf{y}_i\\in \\mathbb{Z}_{\\geq0}^4", "3df420d40239968c5156fc6bd0d88ed2": "\\sum_{k=1}^{k=N} \\cos (-2\\pi\\frac{n(k-1)}{N})/N = 0,0,0...,1 \\text{ sequence with period } N ", "3df461a25ef188a3d43f182e57f46d4a": " \\operatorname{build-param-lists}[\\lambda N.S, D, V, R] \\equiv \\operatorname{build-param-lists}[S, D, V, R] ", "3df48cbffad1c8ab6789d32e4aef9522": "\\exp_c", "3df4c5286dab944f31cb28c6b9ae7e4b": " \\frac {Q}{T}", "3df4e6ca35cc71464d7075731d6efe38": "\\bigstar\\bigstar \\mathbf S", "3df4f49d652ffee1584d73eb143c977c": "\\lim_{x \\to x_\\pm} \\|y(x)\\| \\rightarrow \\infty", "3df55ae8b5349d12d434108ec7b8fc35": " \\begin{cases}\n \\frac{ \\partial L_1 }{\\partial w} = 0\\quad \\to \\quad w = \\sum\\limits_{i = 1}^N \\alpha _i y_i \\phi (x_i ) ,\\\\\n \\frac{\\partial L_1 }{\\partial b} = 0\\quad \\to \\quad \\sum\\limits_{i = 1}^N \\alpha _i y_i = 0 ,\\\\\n \\frac{\\partial L_1 }{\\partial \\xi _i } = 0\\quad \\to \\quad 0 \\le \\alpha _i \\le c,\\;i = 1, \\ldots ,N .\n \\end{cases} ", "3df57177ebbac1a51fb1776c01f0797b": "\\theta = \\arctan\\left({-(-4) \\over 6}\\right)", "3df5cfe3e375e492efa601c6b0706dd2": "\\{2, 4, 5\\}", "3df5fb26ff7305260c4d922c2da5375e": "(0,0,-z '-2z", "3df62e848ab340805078bca0c275e95f": "x(t)=e^{it^3}", "3df663c6e9a6ac8878f573d92d8215ca": "\\tau_{\\mathrm{sen}} = R_{\\mathrm{sen}} C_{\\mathrm{sen}} = \\frac{d^2 \\rho C_p}{\\lambda}", "3df692afaf7ec004832b4574d112b295": "X_n \\ \\xrightarrow{p}\\ X \\quad\\Rightarrow\\quad g(X_n)\\ \\xrightarrow{p}\\ g(X);", "3df6a7439ca8106cf94e28d1774b8572": "\nI^\\text{filtered}(x) = \\frac{1}{W_p} \\sum_{x_i \\in \\Omega} I(x_i)f_r(\\|I(x_i)-I(x)\\|)g_s(\\|x_i-x\\|),\n", "3df6e42d12a047d947009cc8cf747a4d": "\\Psi_{fRep} : fRep \\to \\mathbb{R}/R\\mathbb{Z}, \\; (x, f) \\mapsto d(x) + f", "3df7127aefcf502324d5fa35f9cd4db4": "\\sin(2\\pi t/28)", "3df82548710ab596aa5dc65b5e06478b": "u_a(t) = (m(t) + i\\cdot \\widehat{m}(t)) \\cdot e^{i(\\omega t + \\phi)}", "3df844f40744a6bc5ba33946e7baa454": "\\scriptstyle \\prod g^{c_i y_i} \\;=\\; 1", "3df853e03ec6d6796ed3acb4ab4d3605": " \\zeta_K(s) \\,", "3df854f16d5a389c7d137ce771666471": "t = \\frac{h}{V_{Y0}}.", "3df942f8f4dfb8760b2b949dd0b31f0c": " \\hat n_{21}=2^{1/2}|\\hat\\alpha_{LO}|\\hat q_{\\theta}", "3df99411974f6ba32ebaac66290393e2": "\\frac{\\nu \\tau^2}{\\nu+2}", "3df998fb10a8595db6d13b1ef6324d06": "\\vec{e}_{ix}\\vec{e}_{iy}", "3df9ee5b9ccdd696e929a697c98a828a": "e^{2\\pi i\\theta}", "3dfa244ce94b8bd724e779a91f8ded8b": "\\int_0^1\\frac{x^8(1-x)^8(25+816x^2)}{6328}\\,dx =\\frac{911}{5\\,261\\,111\\,856} = 0.000\\,000\\,173\\ldots,", "3dfa6707294c388a2489d9fcdb83cd81": "f'_t", "3dfa86e0be2a37e0ddcfb791aaf95c5b": "p_0(\\xi), p_1(\\xi), p_2(\\xi),\\ldots, p_m(\\xi). \\,\\! ", "3dfab29f682bae55ad0f5d26f791d220": "T = 0.20 P_{pre} d ", "3dfabf1de5fd3ea850e80d7723809e48": "y(t)-r\\geq 0", "3dfad80952443e25f64a00204529eae8": "\\frac {n - 1} {n_\\mathrm{obs} - 1} = \\frac {\\rho} {\\rho_\\mathrm{obs}} \\,.", "3dfb22bd0a6aaa24598f5dd676cebdd1": "\\displaystyle{H^\\varepsilon f(z) - {i(1-\\varepsilon)\\over \\pi} f(z)={1\\over \\pi i} \\int_{|\\zeta -z|\\ge \\delta} {f(\\zeta)-f(z)\\over \\zeta -z} \\, d\\zeta.}", "3dfb4ad0be6989b6f8c0a0db1d82fad2": "\\varphi/m", "3dfb95364dd45003091a344e83232203": "S = {{\\sqrt{\\frac{1}{2}\\left(\\frac{1}{2}+1\\right)}}}\\ \\hbar = \\frac{\\sqrt{3}}{2}\\hbar .", "3dfbef97d8faac539cb10bb9615a11f8": "gx(f)=x(g^{-1}f)", "3dfbfb11151db918cb23974ffe76c604": "\\Lambda^c = \\mathbb{L}\\setminus\\Lambda", "3dfc9dfdddbcf47740137915ccef80e7": " \\nabla_{\\vec{e}_0} \\vec{e}_0 = \\frac{m}{r^2} \\, \\vec{e}_2 ", "3dfcdc59cbf1234a217c94ed8334d048": "R (u) ", "3dfcecb3b914ac70de97c9d478730daa": "r = \\pi/4t = \\pi/4\\arcsin(1/\\sqrt{N}) \\approx \\pi\\sqrt{N}/4", "3dfd0dde70a58be34518659519d71069": "\nl_{n}= \\sum_{i}^{n} k_{i} =2m\n", "3dfda650a3ceb24da6f8fa9b8d51eb40": "Q : F^2 \\mapsto F", "3dfdc5a8d1436c67422090db3f6b72eb": "\\hat{b}_2 \\,\\hat{b}_3 = \\hat{b}_3 \\,\\hat{b}_2", "3dfe9d79ac37933933c93f4f0ed179c3": " {\\rm det}\\, e^A e^B e^{-A} e^{-B} = \\exp {\\rm Tr} (AB-BA). ", "3dfea538d7f8e31c2fdc2e7edd84b635": "m,\\, x,\\, y,\\, N,", "3dfeac0bb23b4ba32dc7d056e8a0998a": "L_{xx}", "3dff0eee36e2e84e1830861e9042932f": "\\tan (\\arccsc x) = \\frac{1}{\\sqrt{x^2-1}}", "3dff998a3387bd3504393c1277394e5d": "\\displaystyle e^{-\\pi\\left(a^2x^2+b^2y^2\\right)}", "3dffb2b62c4cccf873452e06bef2b679": "\\delta_k=\\pm 1", "3dffc9f206c3d8304145139899df29f1": "n \\in \\mathbb{Z}", "3e0016172a0b4451689d816f3431df7d": " F_c= m \\frac{V_t^2}{r} ", "3e004bd12c223e5e548674083df53b02": "v\\left(\\sum_{i=-m}^\\infty a_iT^i\\right) = -m", "3e006b07e12d2e1b8c9bb92342b960db": "\\ z_g ", "3e00cea7a8b5634a17c3333c7253d936": "(a,b) = \\{x \\mid a < x < b\\}", "3e00dbfc700b63310c08fd8dd4c1c906": "{f_o >> \\beta y}", "3e00f9a1e18c7251df05848cdc0b416b": "G\\,", "3e01b5a397d45dce9eb71a91e9dbe9df": "p=(R_1, \\ldots, R_n)", "3e01e0c7ae627c1f17ac924a5db08cf4": " \\ M^{reg} ", "3e02122d5e5b246c7c58b409e5785905": "b_{k+1}=\\frac{Ab_{k}}{\\|Ab_{k}\\|}=\\frac{A^{k+1}b_{0}}{\\|A^{k+1}b_{0}\\|}", "3e024f83ae89bae75995dd7fe7529d11": "\nL(\\beta) = \\prod_{i:C_i=1}\\frac{\\theta_i}{\\sum_{j:Y_j\\ge Y_i}\\theta_j},\n", "3e029476154c883aec0913d908d631cb": "\\frac{2^n}{n!}.", "3e02b8b2666e6d5460e4ce2432e01c81": "\n\\begin{align}\nNX|n\\rangle &= (XN+[N,X])|n\\rangle\\\\\n&= (XN + cX)|n\\rangle\\\\\n&= XN|n\\rangle + cX|n\\rangle\\\\\n&= Xn|n\\rangle + cX|n\\rangle\\\\\n&= (n+c)X|n\\rangle.\n\\end{align}\n", "3e02d6517eb685987d6183011de83cb0": "(X^{st})^{st} = \\begin{bmatrix}A & -B \\\\ -C & D\\end{bmatrix}.", "3e03064c4da4374323be69a935f1d1ec": "\n(\\hat{\\theta}-\\theta_0)/{\\rm SE}(\\hat{\\theta})\n", "3e032043efcae009eaad6a85db9288f3": "\\Delta x \\to 0", "3e03eea3361d03157b9a40e45126935b": "\\gamma_{ri}=\\beta_{ri}-\\alpha_{ri}", "3e0461a1f29d719a2a64ad9157963e92": "(\\mathbb{R}, +) \\cong (\\mathbb{R}^+, \\times)", "3e04dc539b05223268b2b7e75e988cc6": "T_\\alpha = \\inf\\{ 0 < t \\mid X_t=\\alpha \\} \\sim IG(\\tfrac\\alpha\\nu, \\tfrac {\\alpha^2} {\\sigma^2}).\\,", "3e04fdce8f479ac23c5e3bafbbd3dd00": "h(t-\\tau)\\,", "3e0546fb8bc9f932543c8e7bd8c20fbe": "\\psi(u)=\\psi(v\\div a)", "3e05dc391b05bf4f675c3561a99aa949": "\nK \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{M}{2}\n", "3e05ed3f9f2e60a9bbd16cab1c710ee8": "x^4-17=2y^2", "3e0659a08933262ecb6835f2af645cdb": "X_j(t)=p_j+x_j(t)", "3e069cd595d87e17d113c309115282f4": " (x_1,y_1) + (x_2,y_2) = \\left( \\frac{x_1 y_2 + y_1 x_2}{1 + dx_1 x_2 y_1 y_2}, \\frac{y_1 y_2 - x_1 x_2}{1 - dx_1 x_2 y_1 y_2} \\right) \\, ", "3e06e7c558cf1f2c5223cec7d0a350bb": " P \\ = \\ F \\times (P/F) \\ = F \\times \\ { 1 \\over (1+i)^n } \\ = \\ \\frac{\\ 100}{1.05} \\ = \\ 95.24", "3e06f4c2e6efd04d9d2feee5eeae2480": "K^a=\\left(\\frac{\\omega}{c}, \\mathbf{k}\\right)", "3e07550cc6c292b3a3b61daf573f0389": "B_{ij} = F_{ij} \\cdot \\epsilon_j + \\sum_{k=1}^{N_s}((1-\\epsilon_k) \\cdot F_{ik} \\cdot B_{kj})", "3e07719e4bd98a93e2a262a9bcd87e59": "n_{\\rm film} > n_{\\rm air}", "3e07bfcb29e1e8caabcf214903e4fcba": "(x * y) * z", "3e07d60dbe6adba93bfdfa2e9582ba38": "\\omega \\ll \\Omega_{i} \\ll \\Omega_{e}", "3e0817b7b126fcd73eab055140df721e": "\\Gamma_0(N)", "3e0825abff98a710a3f21ce7a787e7c9": "[\\mathrm{ad}^k_{\\mathbf{f}}\\mathbf{\\mathbf{g}}] = \\begin{bmatrix} \\mathbf{f} & \\cdots & j & \\cdots & \\mathbf{[\\mathbf{f}, \\mathbf{g}]} \\end{bmatrix}. ", "3e083254ecbc6491846cc78cfedb1057": "\\begin{align} \n& \\hat{E}^2 = c^2\\hat{p}^2 + (mc^2)^2 \\\\\n& \\hat{E}^2\\Psi = c^2\\hat{p}^2\\Psi + (mc^2)^2\\Psi \\\\\n\\end{align}\\,\\!", "3e08357a8594053f514c1edc684aaced": "\\frac{R(x,s)\\; :\\ R(x,\\textit{do}(a,s))}{R(x,\\textit{do}(a,s))}", "3e08a8ec0bd19447fcb3365a367cbef6": "p \\in \\Pi_n", "3e08c6ab93b5d51cf44a0f03ee47b318": "(L_{n+1}',R_{n+1}')=\\mathrm H^{-1}(L_{n+1},R_{n+1})", "3e08dd140a6d297ddda252d145bdbf68": "\\sigma_y^2(n\\tau_0, M) = \\text{AVAR}(n\\tau_0, M) = \\frac{1}{2n(M-1)} \\sum_{i=0}^{\\frac{M-1}{n}-1}(\\bar{y}_{ni+n}-\\bar{y}_{ni})^2", "3e091f7eec5804e2019ec8a9b64361df": "\\langle\\beta_{2,i}\\rangle", "3e0932cee4764199540062bdf0299c3c": "\\frac{U}{Nk} = 9T \\left({T\\over T_D}\\right)^3\\int_0^{T_D/T} {x^3\\over e^x-1}\\, dx = 3T D_3 \\left({T_D\\over T}\\right)\\,,", "3e093392a5bab2ade484f12b95ba78cd": "p_U \\colon Y_U \\to U", "3e094e382cb1d56c17f7d63601c1a476": " n = N ", "3e095cb4ffe300ad00c7a402ef79b158": "m(R) = M", "3e09689fadf669a1efc75558c3d10495": "\\{\\; \\rho\\sigma \\;:\\; \\rho\\in R \\wedge \\sigma \\in S\\;\\}", "3e09e31b0a5b8530699b72173a8b6b52": "x_i \\neq 0", "3e09f231ac0c19192d1d29fa650c450b": "\\langle p|J^0(0)|p\\rangle =\\lim_{p'\\rightarrow p}\\langle p'|J^0(0)|p\\rangle ", "3e09f7c09c862dc5f505b703f6344c13": "A_{k+1}^{-1}= A_k^{-1}+ \\left( 1-A_k^{-1}A\\right) u_k \\otimes {v_k^* \\left(1-A A_k^{-1} \\right) \\over v_k^* A\\left(1-A_k^{-1}A \\right) u_k}.", "3e0a0358a47ed0ebeee44f325b606a16": "H(x,p;R(t))", "3e0a1218ab358f59f14d221bee008b57": "p V = n R T.\\,", "3e0a30cdef39ffe9902b7d68e98ee9c4": "F_n = \\frac{\\varphi^n-\\psi^n}{\\varphi-\\psi} = \\frac{\\varphi^n-\\psi^n}{\\sqrt 5}", "3e0a928a4caeeed81248c2c8422a5878": "x_{21}", "3e0a9a894d707fbd6fb3828279d4f21b": "\\frac{a}{\\text{∆}(APB)}+\\frac{c}{\\text{∆}(CPD)}=\\frac{b}{\\text{∆}(BPC)}+\\frac{d}{\\text{∆}(DPA)}", "3e0b17c62ce2bf3fc2d2b62922fe52d8": " \\frac{dx}{dt}=rx-x^2. \\, ", "3e0b1acb281fad827954c1e0e06a1b03": "\\ \\ \\ \\ \\tau(p^{j+1})=\\tau(p)\\tau(p^j)-p^{11}\\tau(p^{j-1})\\ (j=1,2,3,\\dots),", "3e0b378392b3f634968079524bdb2920": " a^n b^n ", "3e0b3c23d462534b1f5a8d92c794cd68": " {D =\\ } {RT \\over\\ 6\\pi rN_{A}\\eta},", "3e0b6090cc684db9bbba6f89ca174fe6": "\\omega^2 = {1 \\over R_x R_2 C_x C_2}", "3e0b7f22183957ef3f24dde562cf471a": "B_{i}^{*} (z_{i}) = z_{i} u_{i}.", "3e0b9210d4d2644047ebeed3362578dd": "G=(\\Sigma_V, \\Sigma_A, V, A, s, t, \\ell_V, \\ell_A)", "3e0ba9ac9183a8c8ff6a90656a3d3eb4": " y \\neq z", "3e0c35e895a54daebbbbe86f4ec46a1a": "\\Delta E = B \\mu_B \\Delta m_l", "3e0c7abd104b3a1be7ba24f29c7d3db4": "00g_0,", "3e0c92c243c65690463f05f36ed7859d": "h=f*g", "3e0ca2d9d814cc16876dedab42ec0f75": "I_{B} = \\frac{I_{C} (= I_{E} = I_{R2})}{h_{FE(min)}}", "3e0ca4b22ae8adbaab0638924b9ad8f7": "\\xi \\in K", "3e0ccb474211b642e162bf7ae5448325": "\\mathbf{P}_{1}", "3e0d193ac55252b31faa73df7b341f01": "K_{cs} = \\frac{Dividend_{Payment/Share}} {Price_{Market}} + Growth_{rate}.\\,", "3e0d2313c09858f67f2f8573646a34a0": "\n\\mathit{L_C} = \\frac{V_{\\rm body}}{A_{\\rm surface}}\n", "3e0d28675d7636bf56dc2f6f7e7a8c20": "E\\left\\{ {{\\left| \\left\\langle Y,x \\right\\rangle \\right|}^{2}} \\right\\}=\\left\\langle Kx,x \\right\\rangle =\\sum\\limits_{n=0}^{N-1}{\\sum\\limits_{m=0}^{N-1}{R\\left[ n,m \\right]x\\left[ n \\right]{{x}^{*}}\\left[ m \\right]}}", "3e0d2c620d2a3e2d963156ecce33679d": "\\mathcal{S}[g] =\\frac{c^4}{16 \\pi G}\\int_{\\mathcal{M}} R \\sqrt{-g} \\, \\mathrm{d}^4 x", "3e0d691f3a530e6c7e079636f20c111b": "x_0", "3e0debf6752a55d18299700fddea73b8": "V(L)", "3e0e59efdb4a24817faf2e06ea8b4f77": "\\frac{\\part\\lambda}{\\part a} = \\frac{1}{2}\\left ( 1 \\pm \\frac{a - d}{{\\rm gap}(A)} \\right ),\\qquad \\frac{\\part\\lambda}{\\part b} = \\frac{\\pm c}{{\\rm gap}(A)}", "3e0e948e60782f9f4b62f87c0c98ec64": "\\displaystyle \\varepsilon_i", "3e0ec3356e56c08db34845c1ba39dbe8": "\\scriptstyle f_\\mathrm{la}", "3e0ed2327ba27938f68bc8f997ea19c7": "F = h_2/2^n", "3e0efd71573979b41e96d827673a462f": "L_n (\\pi_1 (X))", "3e0f2b963991e7f679e86d3506de26b5": "\\ell = \\frac{k_{\\rm B}T}{\\sqrt 2 \\pi d^2 p}", "3e103176d0a1bb4d7b771dd506b02db1": " \\vec{u}_{P} ", "3e1033e1fcceed038d592cac507e235a": "(M,\\varphi:M\\to \\mathbb{R})", "3e103a1238cfc1124de70dbbbb362136": " y(t) = \\pm\\big(\\tfrac23t\\big)^{3/2}. ", "3e1043cadbf0328c6b85937dd26b3aba": " \\displaystyle{\\mu(z)=\\overline{\\mu(\\overline{z})},}", "3e1046f47cb3e338d4a09411aa04bb2e": "\\forall x . P(x)", "3e109a8ac21de31472a6b6c36657e639": "v=v_1", "3e10bfdcea42c6b022b6dde19711266e": "u_1=(1,0,0,0)", "3e113f80766c9436beaa6fb5ebb4ea8f": "C_1 > 0", "3e1183615394e19012bf8ef53db56cb4": "U_0=A", "3e120134663380ad43ece7ca1e600fac": "|{\\Phi^A_i}\\rangle", "3e123fc66a6a78509d6705e9a8769046": "g_{uc}(g_{uc}^{-1}(\\{ \\langle A \\rangle, \\langle bb \\rangle \\})) = g_{uc}(\\{ \\langle a \\rangle\\}) = \\{ \\langle A \\rangle \\} \\neq \\{ \\langle A \\rangle, \\langle bb \\rangle \\}", "3e128c9e77a8e83ab5c1c297133799aa": "p = -vi \\,", "3e12b0f5f21873e9b15031e3c799a487": " C_i ", "3e13237fa116b0a9b25471b1ede4379c": "|K| <\\infty,\\ char K \\ne 2", "3e1326dafc2a84a41fc15894f7bfff86": "P(X\\in E) = \\int_{x\\in E} f(x)\\,dx\\,.", "3e132c22441e737eb7445feaa3fec2f4": "\n\\begin{array}{rcl}\n(J^\\alpha f)(t) &= &\\frac{1}{\\Gamma(\\alpha)}\\mathcal L^{-1}\\left\\{\\left(\\mathcal L\\{p\\}\\right)(\\mathcal L\\{f\\})\\right\\}\\\\\n&=&\\frac{1}{\\Gamma(\\alpha)}(p*f)\\\\\n&=&\\frac{1}{\\Gamma(\\alpha)}\\int_0^t p(t-\\tau)f(\\tau)\\,d\\tau\\\\\n&=&\\frac{1}{\\Gamma(\\alpha)}\\int_0^t(t-\\tau)^{\\alpha-1}f(\\tau)\\,d\\tau\\\\\n\\end{array}\n", "3e133ed48ec2265cfd8e70fc63fce1db": " y(x) = y_{D}(x) + y_{R_{1}}(x) + \\cdots + y_{R_{h}}(x) + y_{C_{1}}(x) + \\cdots + y_{C_{k}}(x) ", "3e13528c3282d1844e07d20d569e6593": "x\\in \\partial D", "3e13af06cf4ce5eb72abc6bddb112621": "f:V \\to S^3", "3e14172d616a78b56cea60049f4b4bfa": " ln(\\gamma_2^\\infty )", "3e1423b5c8d80b81ce2ec7dd25c07011": "\\Pi V = n R T i", "3e1445f2ab1f48287e641f5e0936a484": "\\begin{matrix}p(x)&=&a_n x^n+a_{n-1}x^{n-1}+\\ldots+a_1 x+a_0\\\\\n&=&a_n(x-r_1)(x-r_2)\\ldots (x-r_n).\\end{matrix}", "3e147d6d7b503f03db057102ac9f9c7f": "\\mu_{eff} = constant\\sqrt{T \\chi}", "3e14a6468b4d724fb37931aaed130e47": " \\leq 2g", "3e14e339b62b101c016699448267e691": " p(x) = \\sum_{i=0}^n c_i x^i. ", "3e153ab8b4545077edb41a3389bf4d31": "kn-l", "3e156db418dd377a36a1449a7fec84b8": "d_{A\\cup B}= max(d_A,d_B)", "3e15872851f1d84be5b61cce5635b884": "j_i", "3e15cd7564c5ba5024772f95bd21839c": "\\displaystyle C(x)", "3e16db1051a379df66933493a0caa5de": " \\frac{1}{2} + \\epsilon", "3e1738ce65314e125ae8e76511a61a46": "u_i^0 = u_0(x_i)", "3e175fef3ade6f7f375d1956cbd03b79": " b_{ij} ", "3e17b9c915f0147bd6c77d7684b17958": "\\zeta_f(z) = \\exp \\left( \\sum_{n=1}^\\infty L(f^n) \\frac{z^n}{n} \\right), ", "3e17f9b284d91b0fa51c83bb5cc148ab": "i<\\alpha", "3e1800805970ff7bb99f31976ab5c796": " k = 2*\\pi/ b ", "3e181e6512d3ab629d17872fd99a4945": "\\sum_{m=0}^n P(m)^2=P(n+2)^2-P(n-1)^2-P(n-3)^2", "3e183ad2822294422ceaa02ad4b973c9": " \\mathbf{b} \\prec_w \\mathbf{a} ", "3e183badae0017a76065ab538d4f9c31": "p;q\\ ", "3e188c4dcae553fe239faaa466d9e5bd": "\\left(\\sum_{i=1}^{n}x_{i}\\right)\\left(\\sum_{i=1}^{n}\\bar{x}_{i}\\right)=\\sum_{i=1}^{n}x_{i}\\bar{x}_{i}+\\sum_{i 0} \\; t^{-\\theta} K(x, t; X_0, X_1), \\ 0 \\le \\theta \\le 1.", "3e19d739b97073faac3e4d235eda2341": "p=\\frac{2\\pi Zr_0}{k}", "3e19f2e8c913cf76c695ad4c978de74d": "\\left( \\begin{smallmatrix} 2 & 4 \\\\ 1 & -3 \\\\ \\end{smallmatrix} \\right)", "3e1a34f811bbedaea9a02be5286aa1ba": "\\zeta^*(s)= \\Gamma(s/2)\\pi^{-s/2}\\prod_p \\frac{1}{1-p^{-s}}", "3e1a7a4b882395db171037a9477a3959": "\\gcd(a,b)=\\frac{a\\cdot b}{\\operatorname{lcm}(a,b)}", "3e1a9889ae1ff98676fe651620be8d04": "s(i)", "3e1ae9a076b534f5674df047a0265333": "l^\\infty(R)=\\{(x_n)_n\\subseteq R:sup_n||x_n||<\\infty\\}", "3e1af2e9bd1d8682a67f6d7b4c00057d": "\\frac{H^\\mathrm{ig}-H}{RT} = \\int_V^\\infty \\left[ T \\left(\\frac{\\partial Z}{\\partial T}\\right)_V \\right] \\frac{\\mathrm{d}V}{V} + 1 - Z", "3e1b3fc8706a390fd51afbdb7fdd5c55": "f(m+h) = f(m) + f'(m)\\cdot h + f''(m)\\cdot\\frac{h^2}{2!} + f'''(m)\\cdot\\frac{h^3}{3!} + \\dots ", "3e1bb6ee9079ff3b2f7ce41ea00fccf1": "\\Sigma^{0}_m", "3e1bb7ced5e47b36d9d2fe055ce4f073": "\nS \\cdot \\begin{bmatrix} 0 & 0 & 0 & 1 \\\\ 1 & 1 & 1 & 0 \\\\ 1 & 1 & 1 & 1 \\end{bmatrix} \\equiv \\begin{bmatrix} 0 & 0 & 0 & 0 \\end{bmatrix} \\pmod{2}", "3e1bddefd4c46ba56123b7051c9a750d": " \\|\\pi(F)\\|\\le \\int_G |F(g)| \\, dg \\equiv \\|F\\|_1,", "3e1c5062ae7aa688577091cc4009d8dd": "a>1,", "3e1c6a8a05101cd098109a17edeed2ac": " = \\lambda_1^k\\left\\{a_1\\mathbf{u}_1 + a_2\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^k\\mathbf{u}_2 + a_3\\left(\\frac{\\lambda_3}{\\lambda_1}\\right)^k\\mathbf{u}_3 + ... + a_n\\left(\\frac{\\lambda_n}{\\lambda_1}\\right)^k\\mathbf{u}_n\\right\\} .", "3e1d50165f72415ed73b42ddf285f12b": "F_0(a, b) = \\ln(e^{a} + e^{b})", "3e1d9fa7965f7730316a60aeb6913750": "\\Delta P_{\\mu}\\Delta x_{\\mu}=\\Delta (mc\\,U_{\\mu})\\Delta x_{\\mu}\\ge\\frac{\\hbar}{2}", "3e1da97c0cf39e4c74bce2df214b5a22": "\\alpha \\in [0,\\;1]", "3e1da9ecce45d9668eddc7cc98250383": "k_1=b", "3e1ddb867b429a7bb4091b916c4aa317": "e=m c^2\\,", "3e1de9cf0fa0e128a6931dd7fd5473cd": "A =\n\\begin{bmatrix}\n2 & 0 & 0 \\\\\n0 & 3 & 4 \\\\\n0 & 4 & 9\n\\end{bmatrix}", "3e1e06c8ae67f327b2771d751ce2891a": "\n p_mf_m=c^{-1}p(y|m,n_m)p(m|n_m)p(n_m), \\, \n", "3e1e0833a42a86f57c37784b8b744424": " B\\,\\!", "3e1e393e01de45faefecb00f004197f2": "r_h\\,", "3e1e7bfedca3038776869d652ff96622": "E = h \\nu = \\frac{hc}{\\lambda} \\,\\! ", "3e1e7f2e1559ca3f7e71ea68ff4ab203": "\\ h_i(x) = 0", "3e1ebf1f7a2fd789f955d47f67a3a959": "\\Gamma(\\tfrac14) = \\sqrt \\frac{(2 \\pi)^{3/2}}{AGM(\\sqrt 2, 1)}.", "3e1eed35fbb605ccf7723610c1003279": "2^{12_{dec}}", "3e1fa0410133bb1872012c7d7566e0f0": "\\bar{x}(t') = \\frac{m_{1} \\cdot x_{1}(t')+m_{2} \\cdot x_{2}(t')}{m_{1}+m_{2}}", "3e20128a85719261146fce0bcbcc1c13": "\\lambda = \\ln (2|\\mathcal U|)", "3e202caf20e4e9d2773d57daa98cdf8a": "\\frac{d\\Phi}{dt}\\approx \\frac{dU}{dt}", "3e202e4734a11f876e992be3c942bcae": "\n F(\\mu+n\\sigma) - F(\\mu-n\\sigma) = \\Phi(n)-\\Phi(-n) = \\mathrm{erf}\\left(\\frac{n}{\\sqrt{2}}\\right),\n ", "3e20f6c937b213fa929288c87eb6933d": " 2^\\alpha || m ", "3e21129dada1b94a6ea230555efaed35": "1-2\\delta", "3e2132d95429aa553553ea36813ce3cd": "c_k=\\mathrm{min}\\{r|e_r=e_k^{\\,}\\}", "3e21336ca79f52847dc3148204db288d": "O_{ji}= \\langle f_j| O | e_i \\rangle ", "3e21457002a617f85d984c88ba56b91a": "f'(c) = 0", "3e21673ce6c9b09f9ec50b7237248576": "\\sin(x)", "3e21746bc68a23bce3d6183117058449": "\\hat{H} = H(\\hat{x}, \\hat{p})", "3e219b62390ed5f82064f06dc4155e67": "\n\\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{2c\\,e(m+2p+2)(d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{(p+1) (2 p+1)(2 c\\,d-b\\,e)^2}\\,+\\,\n \\frac{(d+e\\,x)^{m+1}(b+2 c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^p}{(2 p+1)(2 c\\,d-b\\,e)}\\,+\\,\n \\frac{2c\\,e^2(m+2p+2)(m+2 p+3)}{(p+1) (2 p+1)(2 c\\,d-b\\,e)^2} \\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^{p+1}dx\n", "3e219cc804513cd20e893ab1590ba912": " U =\n \\begin{bmatrix}\n 0 & 3 \\\\\n 0 & 0\n \\end{bmatrix}.", "3e21c0c319dfef45d2ec7d2fae0c48d0": "\\varphi_i^{-1}(U)", "3e2227620a244a212ce77e33d66ff89e": "\\text{B} \\in \\text{APX} \\implies \\text{A} \\in \\text{APX}", "3e22365fa43e1171e809c35f926fca40": "(X(x,y),Y(x,y)):=(x^{q^{2}}, y^{q^{2}}) + \\bar{q}(x, y)", "3e22e23c69367026bfbc7fc6c5c9e275": "\\angle AP_1K", "3e232b917930664fcd9a74c9efb5890c": " (b, a) = (a, b)^{-1} ", "3e2362cf6f1f8eeca9e5bd345e4acea3": "P \\sim_b Q\\,\\!", "3e23f630235d2cd480dbab7ae3892709": "p(\\sigma^2|\\nu_0,\\sigma_0^2) = \\frac{(\\sigma_0^2\\frac{\\nu_0}{2})^{\\frac{\\nu_0}{2}}}{\\Gamma\\left(\\frac{\\nu_0}{2} \\right)}~\\frac{\\exp\\left[ \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right]}{(\\sigma^2)^{1+\\frac{\\nu_0}{2}}} \\propto \\frac{\\exp\\left[ \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right]}{(\\sigma^2)^{1+\\frac{\\nu_0}{2}}}", "3e243bf49429d12111558cac4bb4fc46": "x(z) = \\sum_{n=0}^{N-1} x_n z^n.", "3e24c7910fc572fdf684bdc5ab29fed6": "\\Phi_{S}=\\frac{0}{0}", "3e24dbb65e6ce5a8e94e6814f0fe495e": "|\\psi\\rangle = \\alpha_{00}|00\\rangle + \\alpha_{01}|01\\rangle + \\alpha_{10}|10\\rangle + \\alpha_{11}|11\\rangle", "3e24f8dc572760f931d0ef7ef3f0f9e8": "B(p,q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p+q)}", "3e250ea1e17744372a4607a5eedb6bfa": " \\frac{2}{\\sqrt{3}}\\leq \\kappa \\leq 2.", "3e253e56209bca000ba70dbd5a4085be": "V_\\mathrm {dc}=V_\\mathrm {av}=\\frac{3V_\\mathrm {LLpeak}}{\\pi} \\cos (\\alpha + \\mu)", "3e2552e44c127ed3a14eb8f871582796": "H(2n,q^2)(n\\geq 2)", "3e2591ddc433a19ed03510acf428fa4b": "\\lceil x\\rceil", "3e26469ef8de737f7aa6c26897e2f8a4": "n = \\frac{3f}{2 \\pi r^2}\\,\\!", "3e26a743a35e7a2188bff03b6ec409fd": " K(k)=\\frac{F'(k,0)}{F(k,0)}. ", "3e26fa119cd7e78b5fa80cbe250dc1b1": "Z = 1 - 1728C\\,", "3e270c640df865849967e15243048a2c": "\\exp(\\nu) = \\sum_{n=0}^\\infty {\\nu^{*n} \\over n!}", "3e27567d7b1bb41bcaf104f94393c334": " \\frac{\\exp \\left( -\\frac{1}{2} ( \\vec{x} - \\vec{\\mu})^\\top \\Sigma^{-1}\\cdot(\\vec{x} - \\vec{\\mu}) \\right)} {(2\\pi)^{N/2} \\left|\\Sigma\\right|^{1/2}}", "3e2786bb0662425fc505d4870e6c12f9": "D = \\begin{bmatrix}b_{0,0} \\end{bmatrix}", "3e278a7af91424618653a038f0a80f3e": "R(X_1, \\ldots, X_{n-1},0) = \\tilde{Q}(\\sigma_{1,n-1}, \\ldots, \\sigma_{n-1,n-1}) = P(X_1, \\ldots,X_{n-1},0)", "3e27a78a91f0740a8b0847419031a546": "\\hat{z} \\in C^{0} ([0, \\hat{T}]; X);", "3e27c256690a866f1fc3bbc1a73bd4ca": "H_1(z)=\\frac{1+z^{-1}+z^{-3}}{1-z^{-2}-z^{-3}},\\,", "3e282ee9e49784953f1e31596c3ae80f": "2.5 \\times 10^{15}", "3e286fa0132b8c9dd316ac4848fe2d52": "\\boldsymbol{r}_0=\\lVert\\boldsymbol{r}_0\\rVert_2\\boldsymbol{v}_1", "3e28ce18f2bbd85f724fd91a8938e607": "\\scriptstyle \\text{curry}(f) \\colon X \\to (Y \\to Z) ", "3e28e3a89366868fe737457c9c053ff2": "L_\\beta", "3e29003e81acf2d51c164859d9fdce14": "\n\\rho_{\\alpha\\beta}(\\omega) = \\frac{1}{\\mathcal{Z}}\\sum_m 2\\pi \\delta(\\xi_\\alpha-\\omega)\n\\delta_{\\xi_\\alpha,\\xi_\\beta}\\langle m |\\psi_\\alpha\\psi_\\beta^\\dagger\\mathrm{e}^{-\\beta (H-\\mu N)}|m \\rangle\n\\left(1 - \\zeta \\mathrm{e}^{-\\beta \\xi_\\alpha}\\right),\n", "3e2903167a3c3c25f18000810dee3f13": "p^{*} := \\frac{n p}{n - p}.", "3e29b1bfcc947fab670c0149d1d4dbdd": "g\\mathrm{Tr}^U_{X,Y}(f)=\\mathrm{Tr}^U_{X,Y'}((g\\otimes U)f)", "3e29b4af2648f0d841e7acd09fb69b11": "\\lambda^{G}_{H}", "3e29e10eb629b875589303aeb1582f68": "\\operatorname{cl}_G(A)", "3e2a2c0023c58f44b294a848d8252e9e": "ALG(\\sigma)", "3e2a63c119c5ff740d88e46df75f1aeb": "(\\Phi,D)\\,", "3e2abae802d64da4d91c7905f43a0697": " \\left(\\mathbf{ab}\\right)\n\\!\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\times \n\\end{array}\\!\\!\\!\n\\left(\\mathbf{ab}\\right)=\\left(\\mathbf{a}\\times\\mathbf{a}\\right)\\left(\\mathbf{b}\\times\\mathbf{b}\\right)= 0", "3e2bbbb038cd48a787d90e347e6fa377": " \\int X(x) e^{- a x } a^x\\, dx,", "3e2bf5a4114a11df39878d4c73594311": "p\\left(z=\\eta\\right)=-\\sigma\\eta_{xx}.\\,", "3e2bfe4cd647dfd9f8fcac0d78f2e565": "f(n)=0", "3e2c669cda1904552e478989d62a0f3c": "V_{nm} = \\langle V\\phi_m, \\phi_n \\rangle", "3e2c889c5adc4bb26e080b33d7170360": "R_{tot} = R_c + R_{ch} = R_c + \\frac{L}{W C \\mu (V_{gs} - V_{ds})}", "3e2ca0149262467ef9d0b1dbc8653ff8": "r_{\\pi} +( \\beta +1 ) R_{\\text{E}}\\,", "3e2ca17fb4789d327e5f9ffeb2ca093d": "\\text{primpart}(\\text{gcd}_{F[X]}(q_1,q_2))=\\text{gcd}_{R[X]}(\\text{primpart}(q_1),\\text{primpart}(q_2)).", "3e2cb4cdd100776dfbc9522ca7e3ce59": "A \\sim B \\wedge B \\sim C \\Rightarrow A \\sim C", "3e2ccdeb4854913e9aea58fd0a2ed4fd": "E_r = \\sqrt{ (m_0 c^2)^2 + (pc)^2 } \\,\\!", "3e2d1da924bdf6aa6dbc71a3b424eb83": "V(F)=\\cup_{i=1}^{e}W(T_i)", "3e2d20ea0f47ae5c1fc65e4d65dfdb25": "c_0 = \\sqrt{2 \\pi}\\,", "3e2d320dfa58efe4b7df85cca4e39d79": "\\mathbf{E} \\big[ (\\delta u)^{2} \\big] = \\mathbf{E} \\big[ \\| u \\|_{H}^{2} \\big] + \\mathbf{E} \\big[ \\| \\mathrm{D} u \\|_{H \\otimes H}^{2} \\big].", "3e2d5f072b97c8e233603d8e30576dd3": " \\rho \\,", "3e2d61792bea821e2811e0899541ac92": "{}- \\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{r}_B ) ", "3e2d61e90d821f69d7f589e475f25b5f": " \\frac{d}{dt} \\frac{\\partial T}{\\partial \\dot{q}_j} -\\frac{\\partial T}{\\partial q_j} = Q_j, \\quad j=1,\\ldots,m.", "3e2d6816705c11c5ec61eaf20ee39df5": " \\left[ \\widehat{V}_a , \\widehat{J}_b \\right] = i \\hbar \\varepsilon_{abc} \\widehat{V}_c ", "3e2d7fd83d950906bd62ec1457d74335": " (x_1,x_2,x_3) ", "3e2d888e4400aa3b025baf621aeda807": "\n\\alpha = \\alpha_{\\mathrm{max}} \n\\sqrt{1- \\left( \\frac{2}{f_{0}} \\right)^{2} \\epsilon^{2}}\n", "3e2df6e841e57d28d9435292d53fc4bc": "\\alpha_v = 2\\arctan\\frac{v}{2f} = 2\\arctan\\frac{24}{2 \\times 50}\\approx 27.0^\\circ", "3e2e4b39a72a979ff28134206af37614": " \\exists \\xi\\in[x,y]\\ f'(\\xi) = \\frac{f(x)-f(y)}{x-y} ", "3e2e679715125e26f8ae12ee3aa46665": "p \\text{ and } q ", "3e2e6812601ebc21e1a168f9792c15ac": "F_2\\,", "3e2e714b5cfa131e59e49d2a21ae7707": "\\langle L \\rangle", "3e2eab72baaba79d3814b2eb16ca434c": "k \\alpha", "3e2efda8f87a8fa724afa6fc1dbe3d29": " u \\left ( x \\right ) = x_1 + \\sqrt{x_2} ", "3e2f2c83e374a6ac5d4e05dac212fd6e": "Pv = -v", "3e2f384df4d4e682026b1500659f5cd4": "I^{F} (x) = \\sup_{y \\in X} \\big[ F(y) - I(y) \\big] - \\big[ F(x) - I(x) \\big].", "3e2fd2514d1e02330d0925edcfaa47e4": "(1-\\cos\\phi)", "3e300a5202146ddc11a3fdbe2ea62a14": "\\text{Var}\\left(y_d\\right)=\\sigma^2", "3e3028e6d5db05d881467bc8d0d1fe1a": "\\frac{W_n}{n} = \\frac{W_m}{m}", "3e30473d3159e8ce6c1b5b276ef02df9": "\\epsilon = \\epsilon_0 + \\frac{C}{T-T_c}.", "3e3055efe0be80596772e3e6c9f68fa9": "\\begin{align}\n H_0 = &+1\\\\\n H_1 = \\frac{1}{\\sqrt2}\n &\\begin{pmatrix}\\begin{array}{rr}\n 1 & 1\\\\\n 1 & -1\n \\end{array}\\end{pmatrix}\n\\end{align}", "3e3061c4528c3a1d8c1a51f6319af058": "\\frac{250}{47\\frac{5}{6}} \\approx 5.226", "3e30804e09e4d5dc1b9d79004c2fd828": "J_{the} = A_{FN} \\cdot E_{FN}^2 \\cdot e^{-B_{FN}/E_{FN}}.", "3e30a3095864fcb9d4bff21f4f11ce72": " r=0 ", "3e30d8a610a768459d04872536338b38": "B=\\mu_0 H", "3e30e57ae92c1d3f4636a26d0e0ee66d": " \\bar V_t = \\sum_{i=0}^{\\infty} \\gamma^i r_{t+i} ", "3e319ab4e7146d73e613b3f656cd71c2": "|Pr[A_1(Y(W)) = f(W)] - Pr[A_2() = f(W)]| \\leq \\epsilon ", "3e31bd6e7faf78940270496b8b928ef7": "\\Omega\\, \\in \\mathcal{A}", "3e31bfcccd81ede69f4e63d63ee92017": "\\rho_q =\\frac{d Q}{d V}\\,,\\quad", "3e31d69373230dad6a7f0a41ac89207b": "\\nu(i)", "3e31ece8b1f5fb9b45389c9462fe1eff": "v_g = v - \\lambda\\frac{dv}{d\\lambda}.", "3e31fe8c91842ffb8f6e2d7ec49fc498": "Z^1(G)=Z^2(G)", "3e321d4ecfe074df2d391b0b208e74fc": "F_n=\\frac{v}{(g\\frac{A}{B})^{0.5}}", "3e323e41186ae91fd6b9b55fd005a433": "\\phi (x, t) = \\frac{N(x, t)}{a \\Delta x}", "3e324c8b454a081a15aca215d9d0be96": "A^*(\\mathbb{P}^n) = \\mathbb{Z}[\\omega]/(\\omega^{n + 1})", "3e3262a0d2d5adbe6915ebf070ccd950": "\\mathbf{E}' = \\gamma \\mathbf{v} \\times \\mathbf{B}", "3e326301adb80b61115f23948344336f": "r_s(T_L)", "3e32f23a11571f76ab3f995fc6d239c9": "\\Gamma \\; \\vdash \\; x \\; : \\; \\sigma", "3e32f3a289c2a66628c7343e57ef390f": " p_1^{n_1} < p_2^{n_2}<... \\alpha(k), |x_m - x_n| < 1/k", "3e3b1e7228995f33903ed7f4a1a6fba8": " \\mathbf{x}(i) ", "3e3b267cb0796b788ca635696d39c543": "G = E/(2(1+\\nu))", "3e3b32ca896a1e4d20b6b75563b95f95": "G(T,P)\\,", "3e3b3feccf8c08c6e653b8c589c0dfe2": "\\mathbf{L}_{M^+} ", "3e3bc7c0d934bd8129789e2853428a6a": "Y = \\lim_{\\nu_2 \\to \\infty} \\nu_1 X", "3e3c0832db32060fe86176e92e7da171": "d\\mathcal{F}(\\Omega_0;V) = \\langle \\nabla \\mathcal{F}, V \\rangle_{\\partial \\Omega_0}", "3e3c87604a8c6b2b73ea07e713ad4137": "H^1_0(\\Omega).", "3e3c9559eaeb9394bef30cb6b1a5c92f": "\\epsilon\\rightarrow 0^+", "3e3c9bd57bba32cd7859f2e8830abe55": "T(n, t) = e^{-t} I_n(t)\\,", "3e3cef7748db3f689474b6d40661f2bc": "rn", "3e3d5b7d26b17aee409d6d2f12ed7edc": "G_{\\mu\\nu} = \\frac{8\\pi G}{c^4} T_{\\mu\\nu}.", "3e3d693e5be9d115e3edf0ea404dee14": "(a,b)\\in I_D", "3e3d79e9ce76f6f9705f8d5fa30531f1": "\\textstyle ((1,0), (3,2), (17,12), (99,70), \\ldots)", "3e3dbe7db7e3e0b708b48038a0ede742": "\\displaystyle abc=(a+b-c)(a-b+c)(-a+b+c)\\quad\\text{(Lehmus)}", "3e3dec84b3d588317eeb4009085afb4a": "\\mathbf{a} \\cdot \\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3.", "3e3e0783d9b706fb1524787f878f5620": " \\; 0 \\log_2 0 = 0", "3e3e0c720ac47a2e581c6d75f3f09129": "\\mbox{M} = \\frac{100%}{n}\\sum_{t=1}^n \\left| \\frac{A_t-F_t}{A_t}\\right|, ", "3e3ea6f1249369bccf54048d38fab3b3": "P_b = B(E,m) = \\frac{\\frac{E^m}{m!}} { \\sum_{i=0}^m \\frac{E^i}{i!}} ", "3e3eecb6dc9c59549d8df0bfe7be2816": "Q_C = \\tfrac{Pa}{L}", "3e3f5b166619665ece895ee4e630e9d2": "\\eta = \\sinh^{-1}\\frac{w}{c} = \\tanh^{-1}\\frac{v}{c} = \\pm \\cosh^{-1}\\gamma ", "3e3fa65a5d67fdea3ad83e9add1cdb8c": "L(y_i,\\hat{y})", "3e3fcb1279b6d4ededcadf77d72804f8": "(S \\cup T)^* \\subseteq S^* \\cup T^*", "3e3fe769290f9a170b3dfb1a85034a4f": " a\\in\\mathfrak{A}", "3e401e9401f7b89c254c702c0f805c10": "N_d", "3e402016518c5fb8b8f9aa756e88fbbd": " A(m, n) =\n\\begin{cases}\nn+1 & \\mbox{if } m = 0 \\\\\nA(m-1, 1) & \\mbox{if } m > 0 \\mbox{ and } n = 0 \\\\\nA(m-1, A(m, n-1)) & \\mbox{if } m > 0 \\mbox{ and } n > 0.\n\\end{cases}\n", "3e404b5f7625f7bc6ebadb361cbeaeb2": "\\scriptstyle L-1", "3e4055831d68e5ab7bed0465e0f0105a": " u' = - i u \\quad ", "3e408329c5ae324398fc0ec9d6bacbcc": "x^2 \\cdot 2^{-1} = \\frac{\\sqrt{5}}{2}", "3e40c3aea1be25ae0c97aa67c5abba6b": "\\Psi_0=\\Psi_1=\\Psi_3=\\Psi_4=0\\,,\\quad \\Psi_2=-\\frac{M(u)}{r^3}\\,,", "3e40e7cd393a0a063d34beaef66548e1": "f(z)=c\\sin(z)", "3e414ab7b5e2f2c7c7fe313cc0a7ad3c": " \\left(\\frac{1}{\\sigma_0^2} + \\frac{n}{\\sigma^2}\\right)^{-1}", "3e415bf2c42bdeaa9a67561358bf6375": "R(\\pi)\\phi(x) \\phi(-x) = \\begin{cases}\\phi(-x) R(\\pi)\\phi(x) & \\text{ for integral spins}, \\\\ -\\phi(-x) R(\\pi)\\phi(x) & \\text{ for half-integral spins}.\\end{cases}", "3e4160c23cc8f6630fa4e66a574c2beb": "OTF(\\nu) = \\frac{2}{\\pi} \\left(\\arccos(\\nu)-\\nu \\sqrt{1-\\nu^2}\\right)", "3e41c91b71f435a2315d8adc4f28c205": "A = - i(U + 1)(U - 1)^{-1} ,\\,", "3e41ecdc10a727a11750feb219083cf4": "q \\equiv 3 \\bmod 4", "3e42ce16e59aa2a1abe28acbeff49d5b": " w \\wedge u \\wedge v = (a u + b v + x) \\wedge u \\wedge v = x \\wedge u \\wedge v.", "3e42e99cf65f3d87f05e007b37199579": "R(S)=1", "3e42f57166e814c0863511460a36d646": "\\sin(x^2)\\ ", "3e43907dbde540c116216ea50372e683": "M_{K_{m,n}}(x) = n! L_n^{(m-n)}(x^2). \\, ", "3e439b71b5c0898b541336f6e433b617": "n\\neq c", "3e43aa0c5ef4d7a45df24a33024deb30": "\\mathrm{error}\\bigl(x(t_0 + 2\\Delta t)\\bigr) = 2\\mathrm{error}\\bigl(x(t_0 + \\Delta t)\\bigr) + O(\\Delta t^4) = 3\\,O(\\Delta t^4)", "3e43acbcf6bdf7263ae3b4582eab66fc": "\\langle e_m, e_n\\rangle = \\begin{cases}1&\\mbox{if}\\ m=n\\\\\n0&\\mbox{if}\\ m \\not= n.\\end{cases}", "3e43e682f227662018057299b380f9fb": "\\vec{v}' = \\hat q \\cdot \\hat v \\cdot \\hat q^{-1}", "3e44107170a520582ade522fa73c1d15": "xy", "3e4422098653bad871405c0018dd3906": "\n u_s^{\\mathrm{botface}}(x,z) = -\\left(z + h + \\tfrac{f}{2}\\right)~\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x}\n ", "3e44fbf0fd0db8d038b25ac658a10ade": "|\\psi_{-}\\rangle", "3e4539eab95d57f154ac8f56f4e7ef32": "a\\cdot G_x(t,f) + b\\cdot G_y(t,f)\\,", "3e455472f31a889a6b9ab7084cbd1798": "A, B", "3e455c3789a5488f024502b023b7787f": "\\frac{3x^2 + 12x + 11}{(x+1)(x+3)} = \\frac{12 -24 +11}{(-1)(1)} = \\frac{-1}{(-1)} = +1 = B.", "3e4582ab6097b5dc6c2f666b7dc3db7e": " \\scriptstyle \\mathrm{PV} = \\frac{FV}{(1 + i)^n} \\,", "3e458a3c60fe6e3d4fb7e026e08e66ae": "\\zeta = \\sum_{k=1}^s |\\mathbf{a}_k|^2/n.", "3e458c0e26a1485bc7691c38ec001f2b": "z_{match} = - j (1.52),\\!", "3e45a9713ac30846eda62028297912cd": "\\operatorname{MU}_*(S) \\simeq L", "3e45d0689c7c4921fee92c060c38097e": " v(x) = c_1 x + c_2 \\;", "3e468221d2dadd51c6a97270164b405d": " \\mathbf \\zeta_{rel}= \\frac{\\mathbf Z}{\\sqrt {1-\\mathbf Z^2/c^2}}", "3e47193aa83221428b8d24ef479fd4a1": "\\mathfrak{f}(\\chi)", "3e471d82661b563d716058db5fade093": "\\sigma(\\omega) = \\frac{\\sigma_0}{1 + i\\omega\\tau}= \\frac{\\sigma_0}{1 + \\omega^2\\tau^2}- i\\omega\\tau\\frac{\\sigma_0}{1 + \\omega^2\\tau^2}.", "3e47af8fe6818ee115dc4a66bdac834e": "x_ix_{i+1}", "3e47d9fddfa684ebad82f3674efda6d3": " C_{NPSH} = \\frac{g NPSH}{n^2 D^2} ", "3e47e5ecf6b0b9c8715a3fa57d7d9b3d": "\\Pi_k^p", "3e47fb8e75f862c6285226de3a44ceeb": "C_{4,3/4}", "3e482c62fff7aff2d0aa7c51455f5ecc": " \\frac{b}{d}", "3e483ca08ad66515deb0ba705d1fe6b5": "(M_1,d_1)", "3e48c1d2a98d528aa66b4a5dc42b245b": "\\begin{matrix} \\frac{cosine \\;of \\;latitude \\;A} {cosine \\;of \\;latitude \\;B} \\end{matrix}", "3e48f050a2d6c94c6224d1bc9865dcca": "p'(a' \\otimes b') = \\|a'\\| \\|b'\\|.", "3e48f0f7b3d0b3c9a9901cda13033a63": "\\mathrm{d}\\ln X_t=\\left(\\mu-\\frac{1}{2}\\sigma^2\\right)\\mathrm{d}t+\\sigma\\mathrm{d}W_t,", "3e48feaf7b90c4311bfa2dc6645ae48c": "y=\\frac{x^2}{x^2+\\nu},", "3e4918050ac863db8806b68c669ea22d": "P_{\\!n,\\theta+r_n^{-1}h_n}", "3e492763871740cd3cbc96657e6e7320": "I\\ddot \\theta + \\Gamma\\dot \\theta + \\mu \\theta = \\tau\\,", "3e4944423b4496c023547354eafc94fb": " \\frac{d\\phi}{dt} ", "3e498a82e4b074e4ab648b8c458246f4": "\\frac{1}{2\\pi} \\int_{0}^{2\\pi} (\\pi-s) f(t-s) ds", "3e49f5dd134db958991de1db78602a9b": "p_z=p(x,y)=q_z(x,y)", "3e4a3f95f08f35cf6407cd1fae89385e": "ds^2 = dt^2 \\left(1-\\frac{2m}{r}\\right) - dr^2 \\left(1 - \\frac{2m}{r}\\right)^{-1} - r^2 (d\\theta^2 + \\sin^2 \\theta \\, d\\phi'^2) ,\n", "3e4a5d1c170a3e316025832abad1f084": "x \\in (a,b]", "3e4aab28a574eaec1f0d112d5659c934": "R = 0", "3e4b229d78bb6f63ed86d1dffea8973f": "N_{eq}", "3e4b8fa14b80e8b3db69a5a3f44bc141": "f_{a}(x) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}} \\in \\mathbb F_p ", "3e4b99466a218ddc70799c1095380469": "\n \\hat{f}_h(x) = \\frac{1}{n}\\sum_{i=1}^n K_h (x - x_i) \\quad = \\frac{1}{nh} \\sum_{i=1}^n K\\Big(\\frac{x-x_i}{h}\\Big),\n ", "3e4bb710d2d25ee0fc94cf927582fdd5": "(\\textrm{Id}_k)_p(x)=\\frac{1}{1-p^kx}.", "3e4bca80ca411244f60e7718f545f9fb": "f(x,y)=\\text{Im}(e^{x+iy})", "3e4c0ebea9334f510c8c594c95ee1c7f": "y>x", "3e4c1aa5be2d066d087e75b0562e4d5f": " H(i,j)=\\sum_{r=1}^{\\infty} rP(i,j,r) ", "3e4c357349b428269bfd178eff6ba05d": "\\scriptstyle 1/2W = T.", "3e4c3dadfdfdaf29ee8b4a481a184200": " \\begin{align} \n\\mathbf{A} & = (A_0, \\, A_1, \\, A_2, \\, A_3) \\\\\n& = A_0\\mathbf{e}^0 + A_1 \\mathbf{e}^1 + A_2 \\mathbf{e}^2 + A_3 \\mathbf{e}^3 \\\\\n& = A_0\\mathbf{e}^0 + A_i \\mathbf{e}^i \\\\\n& = A_\\alpha\\mathbf{e}^\\alpha\\\\\n\\end{align}", "3e4c852ad5965596f02af152cf219d4b": "I_A \\leq I_B \\leq I_C", "3e4cac68d713fe4fa0140e830ce932b5": "\n\\frac{N_i}{N} = \n\\frac{1}{Z} \n\\exp \\left[\n-\\frac{p_{i, x}^2 + p_{i, y}^2 + p_{i, z}^2}{2mkT}\n\\right]", "3e4ce8ca3d781c5095bffdba1d2c07cc": "v_s = \\sqrt{v^2 + 2\\omega r v \\cos\\phi \\sin\\theta \\sin A_z + (\\omega r \\cos\\theta)^2},", "3e4d01881438e4681d213e6e5e0e134c": "\nh(B) \\approx h(\\beta) + \\nabla h(\\beta)^T \\cdot (B-\\beta)\n", "3e4d0780385bbf3dbed18d8b2810e64d": "C_J = \\kappa \\varepsilon_0 \\frac{A}{w(v_R)} \\ , ", "3e4e073b7ac7b906785ee251ade6a444": "x_2 \\searrow x_1", "3e4e363b645a2db4ce3699e5e9b827ce": "P_{a\\le p\\le b} (t) = \\int\\limits_a^b d p \\, |\\Phi(p,t)|^2 ", "3e4ea350dcc6ff850fc64965f2639926": "\\psi(a) ", "3e4ec903c1c2f9df325e4cbebd256569": "\\lim_{x\\to 0}\\left( \\frac{p}{x}\\right) = k_{\\rm H}", "3e4ef917f31b91e948445f537ce1d7f8": "Eq.2\\;\\frac{d\\theta}{dx}=\\frac{M}{EI}", "3e4f1bc89175ea4fb62403590750726f": "x_1 = \\frac{\\omega^{-k}}{2} (y_0 - y_1), \\, ", "3e4f2e58e520904c1d28c253991e2980": "\\omega^3 = \\omega \\omega^2 = \\omega \\overline{\\omega} =1, \\;\\;\\ \\overline{\\omega} = \\omega^2", "3e4f4c43096dbba761c0e5782459c953": " M^0_1 = \\sum_{i=1}^N e Z_i \\langle \\Psi | z_i | \\Psi \\rangle.\n", "3e4f580699feb76f1d52ab256ae4a9f8": "S \\subset X", "3e4f71f4510ccc185fdb11ac1ff622df": "2^{n/4+O(1)}", "3e4fa01b440750f38257522ad76618c5": "\\xi^1,\\dots,\\xi^N", "3e4fd8d60ca033d726e9b3ed25256407": "\\mathrm{tolerance} = 1-R_{j}^2,\\quad \\mathrm{VIF} = \\frac{1}{\\mathrm{tolerance}},", "3e500918b438cdd863b53746018f598b": "\n\\begin{align}\n& V_{\\text{obs, r}}=A\\,d\\,\\sin\\left(2l\\right) \\\\\n& V_{\\text{obs, t}}=A\\,d\\,\\cos\\left(2l\\right)+B\\,d \\\\\n\\end{align}\n", "3e503c239cb3e4e450570ab33359e729": "F(x; \\alpha, \\beta) = \\frac{\\Gamma\\left(\\alpha,\\frac{\\beta}{x}\\right)}{\\Gamma(\\alpha)} = Q\\left(\\alpha, \\frac{\\beta}{x}\\right)\\!", "3e50d6a4d1c5e4185e4abbb5791db4c0": "(a_{13}+b_{13})-c_{13}", "3e50ed43fe3b91a73360288900ed9a83": " (\\Delta t)_2 = \\int_R^{R_2} dt \\approx \\frac{m+R_2}{R_2} \\, \\sqrt{R_2^2-R^2} = \\sqrt{R_2^2-R^2} + m \\, \\sqrt{1-(R/R_2)^2} ", "3e5113b1f179071957585c46c2121f94": "deg(Q(X)) \\le {deg(P (X)) + deg(E(X))} \\le {e + k - 1}", "3e517c57c2024e39483af85b5a6c9c1f": "\\left[\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right]", "3e51882a4a287ef0ade318ce4a3b90c6": "\\tau = 2/3", "3e51bbd93db11ce2f07fafca9d2de564": "u\\Rightarrow\\Rightarrow v\\,", "3e51d5d638764f9752b762423bb476ea": " \n \\pi^+ \\rightarrow \\mu^+ + \\nu_{\\mu} .\n", "3e524909623de491645d9a6a192ad252": "q(x) = p_{n-1}(x)", "3e5290db964e4f632b865f721d26f2fb": "C_V = {\\partial u\\over\\partial T} = -{\\varepsilon\\over2} {1\\over \\sinh^2\\left({\\varepsilon\\over 2kT}\\right)}\\left(-{\\varepsilon\\over 2kT^2}\\right) = k \\left({\\varepsilon\\over 2 k T}\\right)^2 {1\\over \\sinh^2\\left({\\varepsilon\\over 2kT}\\right)}.", "3e52b1e94c77ba5102be2a3042ae5c88": "\\begin{bmatrix} \\dfrac{\\Delta \\mathbf{[z]}}{z_{22}} & \\dfrac{z_{12}}{z_{22}} \\\\ \\dfrac{-z_{21}}{z_{22}} & \\dfrac{1}{z_{22}} \\end{bmatrix}", "3e52ff5f92b695f0ffd8f568eaefca66": "\\int\\frac{dx}{s^7}\n=-\\frac{1}{a^6}\\left[\\frac{x}{s}-\\frac{2}{3}\\frac{x^3}{s^3}+\\frac{1}{5}\\frac{x^5}{s^5}\\right]", "3e5314e9fd31509fdeb83faa0f729ba2": "[x]", "3e531b6e89919cfae194c8e70c6edc1d": "\\gcd(N,cv)=c", "3e5333a4566e81a60f437bca6c168c70": " dW = -PdV ", "3e533a73e2aea8cf434d0b7a8f709846": "\nG(t) = \\int\\limits_0^t \\varphi(\\theta^1(t) - \\theta^2(t))dt\n", "3e534e7c60347a431d11c62029082a35": " \\vec{G} = \\vec{k}_{f}\\ - \\vec{k}_{i}", "3e53a2eb4706fdb49c01a57df6b6ee1d": "f(e_0)=e_1", "3e53db5aace42d2526c03afa9997517d": "W_\\text{eff}=\\frac{1}{2}\\ C \\cdot\\ ( V_\\text{max}^2 - V_\\text{min}^2 )", "3e53f5d1c0867bea755241b5ef1ff140": "e_L^C", "3e544b660ad1078ee35ec5785fd6d88c": "N \\approx \\frac { v_{\\mathrm N} - v_{\\mathrm F} } { 2 c } \\,.", "3e548bdafb1632d617f34645884d31a6": "(e, f, e): (A, e) \\rightarrow (A, e)", "3e5491e523aedca00b9369eac2d8660f": "\\,U(t)\\psi_{\\alpha}(x) = \\chi(E_{\\alpha}(t))\\psi_{\\alpha}(x)", "3e54a02dbea9158ae1a97b8bed4c4a1a": "R_w", "3e54b053393b6054430766b5ee984c0c": "f : D^m \\to D", "3e54c67a6e0af7118cb8022980eef795": "q_{1/2}=(\\log_e 2)^{-\\frac{1}{\\alpha}}.", "3e55813196b5928ea9dba42eba0f897b": "\\frac{P_1}{T_1}=\\frac{P_2}{T_2} \\qquad \\mathrm{or} \\qquad {P_1}{T_2}={P_2}{T_1}.", "3e55abdb6fb8ef442f3a6546cffb06a1": "|E(z)| > |E(\\bar z)|", "3e55c67dc75b376ff0aca959198bbb0d": "X/A", "3e55e702c1edfd452ad2362ed67913db": "[\\hat{a}_{j}, \\hat{a}_{j}^\\dagger] = 1", "3e567490829307609746d4ba72f26a88": "h_{\\times} = -\\frac{1}{R}\\, \\frac{G^2}{c^4}\\, \\frac{4 m_1 m_2}{r}\\, (\\cos{\\theta})\\sin \\left[2\\omega(t-R)\\right].", "3e569ec876ea1274ef09177a8c5e9b1d": "P = \\frac{\\hbar c^6}{15360 \\pi G^2 M^2} \\;", "3e56b3e3a4ab12ce94f8f414335a0bf8": " n(x)\\in U ", "3e570802bba496d5c0335445a3d38f20": "(r_n)_{n\\in\\N}", "3e575891115a795d3c566df4a1695c80": "\nI \\approx \\frac{V_{in}}{R}\n", "3e5792dc2ebee3146b8813ab243f4de7": "\\mathrm{Re} \\, (k) < 0.", "3e57b7ed2f78ebf223a095dbf9d19312": "(M f)(e^{i\\theta})=\\sup_{00", "3e5ba621ad0e185797aac2671100ac27": "H^\\pi_n(X;U) := H_n(U \\otimes_{\\mathbf{Z}[\\pi]} C_*({\\tilde X})) = 0", "3e5bb1f16ccf7347ff09e843c3766bc1": "A + h \\nu \\rightarrow A^*", "3e5c2dc0bfb850c5dc0d5eb798740103": "ID = \\sum_{i=1}^n h_i^2", "3e5c695dc0f87fd3c684f46dd5bf22f7": "(2\\nu)", "3e5cfe6a8d778e5e60f1a350cfe7770d": "\n\\cong\n\\coprod_{x\\in X} g^{-1}[\\{f(x)\\}]", "3e5d2ba00c25c3e5b9843b6deec048e3": "{\\rm Tr} \\left[ \\bold{F}\\wedge\\bold{A}-\\frac{1}{3}\\bold{A}\\wedge\\bold{A}\\wedge\\bold{A}\\right].", "3e5d3d91f673829b503ef25dec952b25": "in\\;m.P", "3e5d545974ef5fb34489f4143c7447c3": "nY", "3e5daa58533be3648eef11e505520fcb": "M_X(t)", "3e5dc8a9e58fac43ec3377c25606be6b": "b_{0}", "3e5e14869dbe9f5f35a8a2a69d522695": "\\sum_{i=1}^n \\mathrm{Cauchy}(a_i,\\gamma_i) \\sim \\mathrm{Cauchy}\\left(\\sum_{i=1}^n a_i, \\sum_{i=1}^n \\gamma_i\\right) \\qquad -\\infty0 ", "3e5e153cac9b4c0785301c09a4270664": "\\varphi\\ \\stackrel{\\mathrm{def}}{=}\\ -{\\pi \\ln(ky)\\over \\ln(W)},", "3e5e20db0ec84e6f4b7c2a6e60757f1e": "S_\\mathrm{BET} = 156 \\mathrm{m}^2/\\mathrm{g}", "3e5e3bfc7b0ff86f859475e4388b1c66": "\\displaystyle{(f,v)=(u,v)_{(1)}.}", "3e5e4b34dc43c61747292fd1abb25905": "(x_1, \\dots, x_s)", "3e5ebaefdb287d6bf3176c96ed6b25c4": " t( x,w ) = \\frac{ \\arcsin( \\sqrt{ w \\, x } ) - \\sqrt{ w \\, x \\ ( 1 - w \\, x ) } }{ \\sqrt{ 2 \\mu } \\, w^{3/2} } ", "3e5ecb1f323bd1782b9b94a1a290f1b4": " \\sum_{i=1}^m x_{ij} \\leq 1 \\qquad j=1, \\ldots, n", "3e5f00d592a086935b2be836bf20fb86": "\\operatorname{cr}(G) \\geq e - 3n.\\,", "3e5f26bad30f6df4ec55bdbc94a97902": "\\epsilon_0", "3e5f5dbcd8379402ccc553ca3b5bb2cf": "h = 2 \\frac{\\operatorname{IQR}(x)}{n^{1/3}},", "3e5f81f5bf4c1308c58849fc1dd8ac39": "\\nu M", "3e5f92fbd6562468f82fe2689b67865f": "\\nu = M + (2 e - \\frac{1}{4} e^3) \\sin M + \\frac{5}{4} e^2 \\sin 2 M + \\frac{13}{12} e^3 \\sin 3 M + ...", "3e5fd1d955d10d198a3137dffd9e029d": "a_{i_1,i_2,...,i_k}", "3e60510d98422ae5c3c7eca6cf6dc047": "\\scriptstyle =(2.6\\pm1.9)\\times10^{-43}", "3e60530ce65be5e7d1980d328719ae3b": "v^{\\odot k} = \\underbrace{v \\odot v \\odot \\cdots \\odot v}_{k\\text{ times}}=\\underbrace{v \\otimes v \\otimes \\cdots \\otimes v}_{k\\text{ times}}=v^{\\otimes k}.", "3e6057573555a9d77cdb25e1744eab9c": "2cr_te^{-aT}", "3e606306c7237f6add8be1e6681f583a": "d\\mu_B=-\\frac{n_A}{n_B}d\\mu_A", "3e6098d9a9256a1aca94b5a814e61fe8": "\\bold{p}\\rightarrow -\\bold{p}", "3e609caa7b497478a84342b1d5c4a6a9": "\\psi_g(t)", "3e6107281525ac32aad8aefb7b309a55": "X \\leq_T A", "3e615e0725af4dd8a5a01cae67cffa53": "\n\\begin{align}\n0 = 2 f(x_n) f'(x_n) &+ \\big(2 [f'(x_n)]^2 - f(x_n) f''(x_n) \\big) (a - x_n) \\\\\n&+ \\left( \\frac{f'(x_n) f'''(\\xi)} {3} - \\frac{f''(x_n) f''(\\eta)} {2} \\right) (a - x_n)^3.\n\\end{align}\n", "3e618706eb74f541b4121055bf175825": " H_2 ", "3e619fe6c573bff7feb92a646be259dd": "|Q(x_i)-f(x_i)|<|P(x_i)-f(x_i)|.", "3e61c79b91ca744095a42335f499bcca": "e_1 = (1, 0, 0, \\ldots, 0)", "3e620d7b3ef3052dff469eca6dc07ec7": "P_2=|010\\rangle\\langle010|+|101\\rangle\\langle101|", "3e624cd0f7dc2ae10fb0ac1fafd5bab1": "C=(S,T)", "3e62e70ced70e908ace40a932bfd1ac0": "\\scriptstyle x \\, > \\, y", "3e632759d40799e3b0072107fe5e1a47": "f_j[n]", "3e6360f3f19b61f645ce7949c912c2b0": "k_{0}^{2}= k_{t}^{2}+\\beta ^{2}\\Rightarrow \\left ( \\frac{\\beta }{k_{0}} \\right )^{2}=1-\\left ( \\frac{k_{t}}{k_{0}}^{2} \\right )", "3e63bf1a38d519df6a47a0453c356d48": "(X,\\tau)", "3e63e4b03a6ec614d927f28295669b00": "\\tan a = \\sin b \\cdot \\tan A", "3e640222db9a079fda7df5cf3cbb5e58": "\n\\begin{bmatrix} 0 & z & -y \\\\ -z & 0 & x \\\\ y & -x & 0 \\end{bmatrix}\n\\lrarr\n\\frac{1}{K}\n\\begin{bmatrix}\n w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\\\\n 2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\\\\n 2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2\n\\end{bmatrix} ,\n", "3e6444c75e0ba8dfb161417c8434ddc3": "S_{t+\\ell} T_B", "3e6eea6c4dcbe4d41318601875002ad3": "s\\geq 2", "3e6f27541beb836c8ffb95ef5241a922": "10^{10^{26}}", "3e6f3ba89a1a6e2dd18e0940b804d27b": "p(f_i|c)\\ ", "3e6f5087277e1e1410df8991901091a1": "\\mathcal V(K)=K^n", "3e704286660b399c80c6c577bc64d2a0": "-M <\\lambda_i < M", "3e7052945a9d9ac123a557dd6cd6a0eb": " E_\\text{k} \\approx m c^2 \\left(1 + \\frac{1}{2} v^2/c^2 + \\frac{3}{8} v^4/c^4\\right) - m c^2 = \\frac{1}{2} m v^2 + \\frac{3}{8} m v^4/c^2 ", "3e705b8d8c9cdeacc8fb2307e0abe741": "\\mu + 2 \\beta \\lambda/ \\gamma^2", "3e70927f0117b38fb7321f1f7d18f78b": "\\Omega = \\begin{cases} \\tan\\tfrac{\\pi\\alpha}{2} & \\text{if }\\alpha \\ne 1 ,\\\\\n -\\tfrac{2}{\\pi}\\log|t| & \\text{if }\\alpha = 1. \\end{cases}", "3e70fd9da23ea10fb87ae701e631660f": "W(\\boldsymbol{\\varepsilon})=a_{ikjh}(\\boldsymbol{x})\\varepsilon_{ik}\\varepsilon_{ik}", "3e71357440ef70934dde0a9c47540466": "x=f^{-1}(-a)\\,", "3e7178a6913f619667290e10f1a71816": "\\text{light}=\\int_{r_0}^\\infty L(r) N(r)\\,dr", "3e71ae46bbe18f6a5657d15e0b607b8d": "H\\ge\\exp{\\{(\\ln T)^{\\varepsilon}\\}}", "3e72013e37861b8b4fb5e425a0ddc515": "P_2(\\cos\\theta)=0 \\,", "3e723d06ee958cdd74fdafe64a842127": "\\Delta \\setminus K", "3e726ee04b1189351b87ceb656e235f5": "P = \\frac{2\\gamma}{r}", "3e729a9bdc3af9f41af3f47075ce1362": "{\\lVert Ax-b\\rVert\\over \\lVert A\\rVert \\lVert x\\rVert n \\epsilon} \\leq O(1)", "3e72db6fc3f3d79c6581d3c88d02d83c": " F(t) = \\left(1-\\frac{\\rho_V}{\\rho_L}\\right)R^2\\frac{dR}{dt} ", "3e72dee25d72897efe9eaf3ac468ae7c": "-\\frac{1}{x+x^2}\\,", "3e7303f4951f416dd628e406ac316401": "T_b=\\epsilon T\\,", "3e737d3628bb10f9d3c808b6dd590caf": "\\,i \\Delta = \\Delta_+ - \\Delta_-", "3e73bb726a69cb6143b2ef4a330f38bf": "\\textstyle -\\sum_{i,j} p_{i,j}\\cdot \\log(p_{i,j}/p_{b})", "3e73ca6b1046ffa34c3e0ac5f26149f1": "\n F_m(x) = F_{m-1}(x) + \\gamma_m h_m(x), \\quad\n \\gamma_m = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L(y_i, F_{m-1}(x_i) + \\gamma h_m(x_i)).\n ", "3e73ebc1e4ec95d37300ff039e72716c": "e d \\equiv 1\\pmod{(p-1)(q-1)}", "3e7408f681a902abd18108bbb659673d": "Sp_2(\\mathbb{F}_q) = \\left \\{ g \\in GL_{2n}(\\mathbb{F}_q) | ^tgJg = J \\right \\}.", "3e7435e3c54e13500c68f9a4de112df7": "(X_t)_{t \\geq 0}", "3e7444c0282f8698221674ac5515002b": "\n\\sigma_x = \\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n", "3e7444f0c424cb31461a1abaf99a88f4": "\\mathbb P (V)", "3e7462e2545dd834492ed0b509385377": "\\sqrt{(2\\pi)^N |\\Sigma|}", "3e7464b0c87bd25e8b0f8c457826f5e9": "\\scriptstyle\\mathcal M", "3e74fe04d0877e70ca36474edee38461": "A_n, BC_n, D_n,", "3e754c1c23fc26dc02d92bfebc3a0f2e": "\\|Df^nv\\| \\le c\\lambda^n\\|v\\|", "3e757374c9ceb2a15ec77607900e17a8": "\nU_{LJ}(r_{ij}) =\n\\begin{cases}\n4\\epsilon_{LJ} \\left (\\left (\\frac{\\sigma_{ij}}{r_{ij}}\\right )^{12} - \\left (\\frac{\\sigma_{ij}}{r_{ij}}\\right )^6 \\right ) + \\epsilon_{LJ}\n& r_{ij} < 2^{1/6} \\sigma_{ij} \\\\\n0 & r_{ij} > 2^{1/6} \\sigma_{ij}\n\\end{cases}\n", "3e758aa4b1fcd755d654f5abd9534f3e": "p_n(z)={z \\choose n}= \\frac{z(z-1)\\cdots(z-n+1)}{n!}", "3e75ad4492a99d943b53fc9ab0ba0b5d": "(\\varepsilon,e)", "3e764224165c2511e9af2262a1eb71db": "(1,0,0,0,0,0,0,1)^T = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}^{\\otimes 3} + \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}^{\\otimes 3}", "3e7659b618eae986401b6d54575cf772": "B_3(f,g)=\\left[\\begin{matrix}u_1v_0-u_0 v_1 & u_2 v_0-u_0 v_2 & u_3 v_0-u_0 v_3\\\\u_2 v_0-u_0 v_2 & u_2v_1-u_1v_2+u_3v_0-u_0v_3 & u_3 v_1-u_1v_3\\\\u_3v_0-u_0v_3 & u_3v_1-u_1v_3 & u_3v_2-u_2v_3\\end{matrix}\\right].", "3e766e1aaa544d243758e2ea2b8a38fd": "S = \\left\\{z \\in \\mathbb{C} : \\mu(\\{x : |f(x) - z| < \\varepsilon\\}) > 0\\ \\text{for all}\\ \\varepsilon > 0\\right\\}", "3e766fbe3b33ab6d30d85bb54ed61061": "\\mathrm{COVV}(X,Y)= Tr(\\Sigma_{XY}\\Sigma_{YX}) \\, .", "3e76aa884b96cda46b7423b6fe4aab60": "\\lambda(g)f(x)=f(g^{-1}x),\\,\\,\\rho(g)f(x)=f(xg).", "3e76aaed285d633fa9ef0943b58c5a42": "a^2 - b^2 = ab - b^2 \\,", "3e76c3c33af16b11cee4e32614e3b1aa": "\\frac{1}{2}\\ln(2)", "3e76ea5c22ca680a4730211cca819996": " v_\\mathrm{per} = \\sqrt{ \\tfrac{(1+e)\\mu}{(1-e)a} } \\,", "3e76ee1db46a388fd20225cd8d7d83cc": "x = 1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7} + \\frac{1}{8} \\cdots", "3e7705666b059313789aa8e964322b9b": "x_3 \\in X", "3e7708374838cd3d14326fa2b54b00ae": "\\forall X \\exists R ( R \\;\\mbox{well-orders}\\; X).", "3e770869a9e4cfa6933efdd654102e2f": "A \\rtimes H", "3e7745a76d6ac3b1763bf01fbeced5be": "c+d-2.", "3e77542f0cea738997b49ceb31fd0ca1": "w(z):=e^{-z^2}\\operatorname{erfc}(-iz) = \\operatorname{erfcx}(-iz)\n =e^{-z^2}\\left(1+\\frac{2i}{\\sqrt{\\pi}}\\int_0^z e^{t^2}\\text{d}t\\right).", "3e77f753a8058a64814ea17d57bdb06f": "\\nabla_{\\dot\\gamma} X = 0 ", "3e77f7b43c7de9ed711970f30da22bff": "f\\times g", "3e7811662f2739bd5faff12a8dc42b42": "\\mu(H, \\cdot) = P(X \\in H | \\mathcal{G})", "3e782d7628712ea46f3948a2722fd022": "2 C_1", "3e785b87b4a262329569b764a80480d3": " I_s = -|I_c|\\sin(\\phi) = |I_c|\\sin(\\phi+\\pi)", "3e78705f714ea55f7e727a3f56354d2d": "\\exists x\\,\\forall y\\,(P(y) \\leftrightarrow y = x).", "3e787bcc0b8a6b167a98d8b35c63ba1c": "R_n := H^0(V, K^n),\\ ", "3e7902023061c202d1ffcc4c5df9a9de": "\n \\sigma_1^2 + \\sigma_2^2 - \\cfrac{2~R}{1+R}~\\sigma_1\\sigma_2 = (\\sigma_1^y)^2 \n ", "3e7922eda0e5c6a7e74005348a90c289": "[F , G]^{IJ} = [P^+ F + P^- F , P^+ G + P^- F]^{IJ}", "3e79a859985e29cdc7fda6b5a0a880ae": "\n\\begin{bmatrix}\n 0.771281 & -0.633718 & 0.059391 \\\\\n 0.613092 & 0.714610 & -0.336824 \\\\\n 0.171010 & 0.296198 & 0.939693 \n\\end{bmatrix}\n", "3e79a91ccd6f3ab846bbd259f8176da0": "\\mathfrak{g}(C)\\simeq\\mathfrak{g}(C_1)\\oplus\\mathfrak{g}(C_2),", "3e79df0ce95d1d28a226eb1e3c37bd78": "2Rr = \\frac{abc}{a+b+c}", "3e7a2fd93de00b9806dad656a1e76e1a": "x=b\\cdot(a^2+b^2)\\cdot (a^2-b^2)", "3e7ad40feaba2a6c6ecbcc62561d0369": "\\langle f, P^* g\\rangle_{L^2(\\Omega)} = \\langle P f, g\\rangle_{L^2(\\Omega)}", "3e7ad5435b872d20e4987e4f8117bad6": "x=\\alpha", "3e7adc840c2a925371d7827fc9b74f66": "\\mathbf{b}_i=\\dfrac{1}{\\left|\\dfrac{\\partial\\mathbf{r}}{\\partial q_i}\\right|}\\dfrac{\\partial\\mathbf{r}}{\\partial q_i}=\\frac{1}{h_i}\\dfrac{\\partial\\mathbf{r}}{\\partial q_i}", "3e7af1eb5e55bf84c668a71edde1fa94": "x \\in \\Omega\\,", "3e7af21642d03567b1da697c156589c4": "\\arg(\\gamma)", "3e7c2138d950d40c922aca8fc9f33ca7": "\\kappa=\\frac{|r'(t) \\times r''(t)|}{|r'(t)|^3}", "3e7c608e2af953cf9944c02afd3952b1": "\\mathbf{\\Omega} = (-\\kappa E, 0, \\delta)", "3e7c83bef6af5bebd03b8c93601bd124": "\\tau_{ij}=\\mu\\left(\\frac{\\partial v_i}{\\partial x_j}+\\frac{\\partial v_j}{\\partial x_i} \\right)", "3e7c902f73f3694fde44017c752ab312": "1 \\Rightarrow (2, 1, 0)", "3e7cc13b1e22dbfc8b9c971d3ef2997a": "\\frac{\\textrm{d}[\\textrm{CO}_2]}{\\textrm{d}t}= -k_1[\\textrm{CO}_2] + k_{-1}[\\textrm{H}^+][\\textrm{HCO}_3^-], ", "3e7cce96708f4adaa6209150367908a0": " A = L L^{*}. \\, ", "3e7d13d99205aa8948c738b1aea95a70": "bs(x) := d^{-1}(d(x) + f_x)", "3e7e8200cb665e38132415ee23c92bd5": "F = \\frac{\\sum_x {}^sE_{x,t}^c {}^sm_{x,t}}{\\sum_x {}^sE_{x,t}^c} \\left/\n\\frac{\\sum_xE_{x,t}^c {}^sm_{x,t}}{\\sum_x E_{x,t}^c}\\right.", "3e7e9af26aa55305964550c8ae21e179": "|c_{33}| \\ge |c_{31}| + |c_{32}|", "3e7ecd51d8f92e6de457dbc1f8f21412": "C_1=\\{(0,0),(-1,-1),(0,-1),(1,-1)\\}", "3e7efcbaff6fb69ad99a0de687d3f936": "f: X \\to S, f':X' \\to S', g':X' \\to X, g:S' \\to S", "3e7f167e7d94a74af8f62e7b2e1f984a": "\\alpha_{t_i}", "3e7f4cccb3444cf6b3b1a8ab80420224": "T = b H", "3e7f4f7c9cf5309ba7c14af8a6b23d8d": "\\textstyle (X'',Y'')", "3e7f71efec13dea576abc82ac98d4fd1": "(10\\uparrow \\uparrow)", "3e7f8e16548e07520088e85728494072": "(\\lnot \\phi \\to \\lnot \\psi) \\to (\\psi \\to \\phi).", "3e7fa32674c74ac1bbf4a5ae45427124": "\\mu (C_k[s_0,s_1,\\ldots,s_n]) = \\frac{1}{Z_n(V)} \\exp(-\\beta H_n (C_k[s_0,s_1,\\ldots,s_n]))", "3e7fbe04193182635455b803278c9ea3": "f(x) = (x-a)^n", "3e7fdd32fb0afb452a01e2ea0d7d3714": "V(I) = \\{ \\mathfrak{m} \\in X \\mid I \\subset \\mathfrak{m} \\}", "3e80380050b5d8735c8382b700e6b5b4": "{\\mathbf{k}}_{\\rm THz}", "3e8039bb44d459ccd7de4dcdd8a54e0d": " f(z) = \\int |g|^{pz} |h|^{q(1-z)}.", "3e8089e5d83ed82f98b10e26f57cf969": "\\pi : H_1 P_n \\to \\Bbb Z", "3e80b8f3b360e6f20fbcf5fbfabb287c": "181^2", "3e81092cda98dcc908e302155e05dd5b": "A=\\{a_1,\\ldots,a_n\\}", "3e810e14c68de0b433a80a6133fcf389": "\\mathfrak{P}^{116}", "3e813265bbdc68b90fd3249f4658fcea": "n - \\sum_{i=1}^{k-1} 2^{2^i}", "3e816e46d06d439681b9523afb566ba5": "t=\\tau", "3e820f80d66fa367730d2780d33414e5": "\\mathbb Q[\\mathbb Z]/\\Delta K", "3e824a2f1a3f1e4da9aa869fbe691a9a": "K_\\alpha^{-1} (u, u') = K_\\alpha^* (u, u') = K_{-\\alpha} (u', u) ", "3e826890be7ba5fc6eb1f87114122eb6": "2H_n \\, ", "3e826e7aeb0c332abda05cc4fa3e2fa2": "E[R]=R_{1}P_{1} + R_{2}P_{2} + R_{3}P_{3} = 10*0.5 + 20*0.25 + (-10)*0.25 = 7.5.", "3e82865eb964a9604dc6178a48c780dd": "{1 \\over 3} \\cdot {3 \\over 12} = {1 \\over 12}", "3e837c16fb3f6892dbb1c28962b6a262": "n 1/2", "3e9a39962ec9d31d2e7babfc737b3b28": "p_i(T_p)", "3e9a8a83a91d81cc6f04bb09f2b387eb": "C,\\theta", "3e9aa026875667391aefdde0e3fde15f": "|\\psi(t)\\rangle", "3e9aaeff0a5f0d0ac69757ccaabb36a4": "v_f^2 = v_i^2 + 2av_it + 2a\\Delta d - 2av_it\\,\\!", "3e9aeb52e6b17509b6557d58ee5372e2": "F_{1..n} = \\log_s {m\\over{m-n}}", "3e9b01ec59dae29c2d07f26fdf9d5528": "\\|f+g\\|_p \\le \\|f\\|_p + \\|g\\|_p", "3e9b1c71167bb607b65a20c175be1890": "\\sum_{k=0}^n {n \\choose k}\\frac {(-1)^k}{s-k} = \n\\frac{n!}{s(s-1)(s-2)\\cdots(s-n)} = \n\\frac{\\Gamma(n+1)\\Gamma(s-n)}{\\Gamma(s+1)}= \nB(n+1,s-n)", "3e9b8f4465b86a2da6c2f25f6aa7e5fa": "\\sigma(A)\\sigma(B) \\geq \\frac{1}{2}\\langle i[\\hat{A}, \\hat{B}] \\rangle", "3e9b9e79170eee76ce954ab0f8716b34": "\\pi_n=\\pi_n(\\alpha,\\beta,N)=\\frac{N-1}{n}\\frac{2n+\\alpha+\\beta+1}{\\alpha+\\beta+1}\n \t\t\n\\frac{\\Gamma(\\beta+1,n+\\alpha+1,n+\\alpha+\\beta+1)}{\\Gamma(\\alpha+1,\\alpha+\\beta+1,n+\\beta+1,n+1)}/\\binom{N+\\alpha+\\beta+n}{n}", "3e9c02d270e69843e837e1d07830d2de": " \\textbf{f}_1 \\cdot \\textbf{h} \\pmod q ", "3e9c6d48756111a93788eca57f6d1790": "[\\mathcal{L}_X, \\mathcal{L}_Y] T", "3e9c75d701e7660f84ccf1bd846325a2": "\n\\begin{align}\n3\\varphi^3 - 5\\varphi^2 + 4 & = 3(\\varphi^2 + \\varphi) - 5\\varphi^2 + 4 \\\\\n& = 3[(\\varphi + 1) + \\varphi] - 5(\\varphi + 1) + 4 \\\\\n& = \\varphi + 2 \\approx 3.618.\n\\end{align}\n", "3e9c809700bfd13ba9415d7778e17199": " B_t = \\sum_{i=1} ^ {\\infin} (1+r)^{-i} PB_{t+i} ", "3e9c93fd304c544f73f32b500ac70908": "\\sigma _1", "3e9cab1116549d87a0d338c1a8575527": "\\nabla f\\left( p \\right)=\\lambda \\, \\nabla g\\left( p \\right) \\qquad \\Rightarrow \\qquad \\nabla f\\left( p \\right)-\\lambda \\, \\nabla g\\left( p \\right)\\,\\,=\\,\\,0.", "3e9cd20d3c395bfab2a53d497dbc9a00": "f(0)\\oplus f(1)=1", "3e9cdb0a5ca19decc668d34290ae4fa5": "\\frac{dy(t)}{dt}=x(t)-x(t)*z(t)+c*y(t)+u", "3e9ce7e0f8eaf56b546372d15ae13f97": "\\boldsymbol{N}", "3e9d1844b611d1620fac6556c231dbdb": "T^{m-r}\\times\\mathbb R^r", "3e9d769a705a1f64c8b99e4b0a26a78e": "D = -\\sqrt{2} - \\sqrt{3}.", "3e9d8d432a33d956203f567cc6cc8b84": "\\forall z(\\phi(z,t_1,\\dots,t_n)\\rightarrow\\chi^\\ast)", "3e9d929b18d9e4819e7f8cb5c60fb6e3": "KC(\\mathcal{S})", "3e9db9a6a4ba767717101e3c273c07a7": "C_2 =0", "3e9dcff0d26f4092f734d9bb074962cf": "\\operatorname{Log}(-i) = -\\frac{\\pi i}{2} \\ne \\frac{\\pi i}{2} + \\pi i = \\operatorname{Log} i + \\operatorname{Log}(-1).", "3e9dd414bdd6768431588b021dacf73b": "\\mathbf{A}^\\mathrm{T} \\,\\!", "3e9ddc684b27365099f8cf6d7a4729ce": "\\{S_k\\}", "3e9e4dfdd9307df0a7283ec89e1bc14c": "\\sum_{n=0}^\\infty a_{\\sigma(n)} = \\sum_{n=0}^\\infty a_n.", "3e9e4ff9a201d35b237effd6f0519c89": "\\omega \\in \\Omega_{Z,[t_l,\nt_u]}", "3e9ebcd28cfba2e3a4a60090cf9a5140": "a=\\min\\left(|X-Y|,|Y-X|\\right) ", "3e9ebec2761830f3b6fabdc1743c6b40": "\\operatorname{rank}(C_1\\mid C_2) = \\operatorname{rank}(C_1) + \\operatorname{rank}(C_2)\\,", "3e9efffc82b49703cbfdc7956448ba44": "\nx_2 = \\ell \\left ( \\sin \\theta_1 + \\frac{1}{2} \\sin \\theta_2 \\right ),\n", "3e9f1e8a55c82b362080e493edef15ba": "\\nabla \\theta", "3e9f3b2399d2263e87e4457ce3dd0a34": "x_n \\rightarrow 0", "3e9f9d221c8763c5aadf525020fd8a10": "A_N(t, z+h) = \\exp\\left[i \\gamma |A|^2 h \\right] A(t, z), ", "3e9f9f53c159d16439fe8378a8f74d73": "\\!t_n", "3e9fa573f5baf959d106fc055b2c12b3": "E_{2,1}(z) = \\cosh(\\sqrt{z}).", "3e9fd7fa816ee5414e4d11ab9dbe04f8": " BB^* = I_n\\, ", "3e9fe5cdbe3a69c75211740f4ee8e8a3": "\n\\dot{\\mathbf{q}} \\equiv \\frac{d\\mathbf{q}}{dt}\n", "3e9ffc0741e6f668b8a62b2af91d3e1a": "p_{1} =\\varepsilon _{1}\\hat{P}+p;~~p_{2}=\\varepsilon _{2}\\hat{P}-p~.", "3ea070dd8e074a26252ca3bcbd02282b": "\\ w[n]=x[n]-a_1 w[n-1]-a_2 w[n-2].", "3ea129ff7d2685e6660f7ced3c7fc717": " e_n = \\frac{1}{n} \\sum_{j=1}^n (-1)^{j-1} e_{n-j} p_j \\,.", "3ea14a60c8dd056871bcdfed0964e04c": "\\boldsymbol\\Sigma ", "3ea15108d0561d958bd1329f9bcfa3bb": "\\gamma_i = \\frac{\\partial \\Phi}{\\partial X_i}", "3ea15d8e54ad06fc223d9cb18b6a8243": "(-)^n", "3ea17f8c4617e07ea21c6736b69af5f9": "L^{norm}=I-D^{-1/2}AD^{-1/2} = D^{-1/2} L D^{-1/2}.", "3ea1b01fd520709ffd1d97da2ce5750f": "\\Omega_{\\perp}", "3ea20fc629c1c85506d578acc7a5e373": "G^u{}_v = R^u{}_v - \\frac{1}{2} R I^u{}_v = 8 \\pi T^u{}_v", "3ea221c33d04b0f35282b41dd1d687c4": "\\, b_k", "3ea224b29a0e386320503ed35c9b01b3": " S_N f(x) := \\sum_{-N \\leq n \\leq N} \\hat f(n) e^{\\frac{2i\\pi n x}{L}}\n= \\frac{1}{2} a_0 + \\sum_{n=1}^N \\left( a_n \\cos\\left(\\frac{2\\pi nx}{L}\\right) + b_n \\sin\\left(\\frac{2\\pi nx}{L}\\right) \\right),", "3ea236a95fe1ef656603c31c6e7dfaab": "\\operatorname{index}\\, T = \\dim\\ker T - \\dim\\operatorname{coker}\\, T.", "3ea26ca612b266a08c02b20e6e6456af": "y=Ax", "3ea292c030d6b77f3fb6729763f8f710": "\\phi (x) = y", "3ea2d29992dda38b10e319292b5598a0": "p^\\omega s_1 s_2 \\cdots s_n q^\\omega", "3ea323560c269cfa82b0b15edd48f67d": "\\mathrm{res}", "3ea32405a6717c007316b394e6e6da4a": "\\scriptstyle{t_{r_i}}", "3ea33a2495444d5fcb4face745bb60a0": "c_{ijk\\ell} = c_{ij\\ell k}\\,", "3ea37c16a0b4bb817b875792e6072ca7": "\\Gamma_{\\alpha}=\\{z:\\arg z\\in [\\pi-\\alpha,\\pi+\\alpha]\\}", "3ea38bc64928c6af311950ed1f5e98bc": "H_n=\\begin{cases}1&\\mbox{if }n=0;\\\\H_{n-1}+2P_{n-1}&\\mbox{otherwise.}\\end{cases}", "3ea3b6400237caff42733f669fb1a33a": "V_n(R) = V_{n-1}(R) \\cdot R \\cdot \\frac{\\Gamma(\\frac{n + 1}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{n}{2} + 1)}.", "3ea3d2a453f3e8013fb7f9bfa7724379": "N=\\mathcal{T}(Q)", "3ea3d34a2e4d3425f1dd8965e9f093fd": "\\lim_{x \\to c} \\, f(x)^n = L_1^n \\qquad \\text{ if }n \\text{ is a positive integer}", "3ea435acd56ee225bc776f3a5b20b7f1": "m \\geq 4, r \\geq 6, s \\geq 3", "3ea44e5956bf3f06002dd06bf7bc68e2": "\\displaystyle M(\\alpha\\beta, n)=\\sum_{[i,j]=n}(i,j)M(\\alpha,i)M(\\beta,j)", "3ea4561cf445e74640798251d46b8cd9": " d_i\\left(X^{(1)} \\right), d_i\\left( X^{(2)} \\right), \\ldots, d_i\\left( X^{(r)} \\right) ", "3ea48ccc43ed183e685d8aa06a3cad0d": "H_\\mathrm{norm} = -(\\ln{T_\\max)}^{-1} \\sum_{t=1}^{T_\\max} P(t) \\ln{P(t)}", "3ea495bae170eb891f7a3fb1e8753137": "c_1 \\geq c_2 \\geq \\cdots \\geq c_k", "3ea4dfade6724b1b1f058aa261bb4d0e": "c_{k\\ell m}=g_{mp} {c_{k\\ell}}^p\\ ", "3ea4e534bd5ab708cd12bb6a655cc046": "\\theta_C", "3ea52409a3082a2cc8b00241394e917c": "y = 2x + 20 - 10", "3ea548633bcb035423f4ee6233e0369f": "F_{edgeness}=\\frac{|\\{p | Mag(p) > T\\}|}{N}", "3ea5f516cca26e9ab49b8da97871171d": "\n\\begin{align}\nV(\\xi, \\eta) & = \\frac{-\\mu_{1}}{a\\left( \\cosh \\xi - \\cos \\eta \\right)} - \\frac{\\mu_{2}}{a\\left( \\cosh \\xi + \\cos \\eta \\right)} \\\\[8pt]\n& = \\frac{-\\mu_{1} \\left( \\cosh \\xi + \\cos \\eta \\right) - \\mu_{2} \\left( \\cosh \\xi - \\cos \\eta \\right)}{a\\left( \\cosh^{2} \\xi - \\cos^{2} \\eta \\right)},\n\\end{align}\n", "3ea6237bf6cb46e4e76c281385c96397": "\\nu < k-t-1", "3ea629b9c46d52bdefd7cc236d00b92c": "\\overline{D}_{\\dot{\\alpha}}X=0", "3ea64b4627e61a9dfd1a33bba31a7a8d": "P_n(A_n)=1", "3ea64e11b59699c8a00fb4adf5dbf610": "L = \\left\\{a+bi+cj+dk \\in \\mathbb{H} \\mid a,b,c,d \\in \\mathbb{Z}\\right\\}", "3ea658d8a8fa652c57446b5346fc2e41": "\\quad u/n=t", "3ea67b3e08626dc599f56ec58f9ff252": "S\\subseteq T", "3ea6801f03af3dc78ea96b96e454b612": " X\\to {X}^{t}.", "3ea68d3fd751f96d847f151a131b2c60": "\ns^{(k+1)}(z)=a(z)\\cdot(\\uparrow 2)(s^{(k)}(z))+b(z)\\cdot(\\uparrow 2)(d^{(k)}(z))\n", "3ea690205e57a5f994c31044fa2ff28f": " \\kappa = i \\kappa_\\mathrm I \\ ", "3ea7123cb1c8b44ad77bd786e3a51832": "(\\lambda \\lambda \\lambda \\lambda 1 (\\lambda 5 5 (\\lambda \\lambda 3 5 6 (\\lambda 1 (\\lambda \\lambda 6 1 2) 3)) (\\lambda \\lambda 5 (\\lambda 1 4 3))) (3 1)) (\\lambda \\lambda 1 (\\lambda \\lambda 2) 2) (\\lambda 1)) (\\lambda \\lambda 1)) 2)", "3ea75ac333a32e03a16ec405cfca09cf": " X = [0,1] \\cup [2,3] \\cup [4,5] \\cup \\dotsb \\cup [2k, 2k+1] \\cup \\dotsb", "3ea762a86e77572636c3fe942d347403": " \\mathbf{y}' = \\frac{1}{x'_{3}} \\, \\tilde{\\mathbf{x}}'\n", "3ea7c9d969a0355213e347af6144c9d5": "r_{d_n}", "3ea7e2b031c6a1d0c70c836f8fa3d2b9": "\\bar{e}=e \\cdot \\hat{x}", "3ea7fefa1f6fc0d907e126d1ee5acc34": "\\Delta>0", "3ea82573881f99e5cfea7b1194785d0b": " [2]P=-P ", "3ea84aa6d0c9d46c1b956397657f4f4e": " \\delta\\ \\mathbf{r}", "3ea860a2b35f38d05cfd671b90956721": "z^w = e^{w \\log z}", "3ea86ba94f37270d9de716c05aa52b83": "\\{m(x_1)/x_1,\\dots,m(x_n)/x_n\\}.", "3ea8948d4563dd912c1be55fd29cbdb6": "|\\Psi^\\pm\\rangle = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |1\\rangle_B \\pm |1\\rangle_A \\otimes |0\\rangle_B)", "3ea8c413bb22680c54da31899242fc9d": " L'(n,k) = (-1)^n {n-1 \\choose k-1} \\frac{n!}{k!}.", "3ea96eee4245436b9cfcf58b2de988bf": "|\\alpha\\rangle=\\widehat{D}(\\alpha)|0\\rangle", "3ea987eb03d0be2f948f76e89e49be5b": " D[n] = [[\\_, \\_, (g\\ m\\ p\\ n)], [\\_, \\_, (g\\ q\\ p\\ n)]]", "3ea9e11a7c2870abf74ec6294272dea4": "\\dot{g}\\ =\\ 0\\,", "3eaa295229374ce7caee19de5fca9b2e": "\n \\exists y \\left( \\forall x \\left( p(x,y) \\right) \\right) \\vdash \\forall x \\left( \\exists y \\left( p(x,y) \\right) \\right)\n ", "3eaa3e2eefd1fcd78cba6e83adde0fa9": "\\frac{\\partial\\eta}{\\partial \\phi} = \\frac{1}{2}\\sin\\left(2\\left(\\phi-\\theta\\right)\\right) + h\\sin\\phi = 0. \\,", "3eaa4390f03644a9a0a12e83ed918015": "\\textstyle n = 1 \\cdot n", "3eaa77286e649674ec14fa78c0e517f4": "s = \\sqrt{3/2}\\,l", "3eaa97e0c3804aa4c035de90d3a330b5": "\\circ:{\\mathbb Z}[x]\\times{\\mathbb Z}[x]\\rightarrow{\\mathbb Z}[x]", "3eab0d98d709609dc2a9b312f284846f": "\n\\phi(u) \\phi(v)=\\phi(u + v)\n", "3eab12b617e5acd7bb85fd44ade1ce22": "\\eta=0", "3eab724096e3dfabe01effe483092727": "u \\in F^\\times", "3eab7e5ab5233491b2d062cb55abdb7e": "F(C)", "3eab8d590785afe0cae651f7276b0352": " \\pi_{j} : \\prod_{i \\in I} X_i \\to X_{j},", "3eac718a8c0320ce235b93c2dbcad2f9": " \\begin{align}\n&\\lim_{\\alpha = \\beta \\to 0} G_X = 0 \\\\\n&\\lim_{\\alpha = \\beta \\to \\infty} G_X =\\tfrac{1}{2}\n\\end{align}", "3eacb79610989990bfbe6fde24a8a638": "\\scriptstyle{DTFT}\\displaystyle \\{x_N\\}(f) = \\frac{1}{N} \\sum_{k=-\\infty}^{\\infty} \\left(\\scriptstyle{DFT}\\displaystyle\\{x_N\\}[k]\\right)\\cdot \\delta\\left(f-k/N\\right)", "3eace0401ec57774ce25f5e5629243c5": "f\\circ r(x)=x", "3eacfa1bdcfa487b236bd97b73bb2238": "z=\\Phi(x)", "3ead491aef6b6cf4557deabbd4b313cc": "A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1} \\cdots A_n B_n A_n^{-1} B_n^{-1} C^2 = 1", "3eadba528b8e2f45bad93846b858c705": "\\sqrt{yz}", "3eae63594a41739e87141e8333d15f73": "uw", "3eaf7357273bdb2566075541a27baea6": "\\bold{j}(x)=D\\nabla \\mu", "3eafb25f270a96461713d601ec9796e2": "\\alpha < \\gamma \\,", "3eafe486bb60fbbd8eef94054969cdfc": "\\scriptstyle{\\bar{V}}", "3eb075e01cf0d26c9355862f2d0c0759": "\\textstyle\\sum_\\alpha c_\\alpha X^\\alpha", "3eb0bf9e9367e4c61aa964cbde879aa0": "K_n^M(F)/2 \\cong H_{\\acute{e}t}^n(F, \\mathbb{Z}/2\\mathbb{Z})", "3eb0c25fad94880a23053e45454832ac": " \\int\\!\\!\\!\\!\\int_S \\mathbf{j}_2(\\mathbf{r},t) \\cdot d\\mathbf{S} = ", "3eb0cf2af2f3e5b269bd4d598c1b5273": "{3 \\choose 2, 0, 1} = \\frac{3!}{2!\\cdot 0!\\cdot 1!} = \\frac{6}{2 \\cdot 1 \\cdot 1} = 3", "3eb10ce93baee00eb019d09b0a446343": "\\scriptstyle\\sqrt{n}(\\hat{F}_n-F)", "3eb15584eed6ca336d39c8e4a42cc478": "\n{u}(x,z) = \\frac{1}{\\sqrt{{q}_x(z)}} \\exp\\left(-i k \\frac{x^2}{2 {q}_x(z)}\\right).\n", "3eb1ecba3ca9935c6857b950dd26f96d": "z = t \\pm \\pi k i,\\ \\ t \\in [-\\pi k, \\pi k],", "3eb22962bab7069c8891c8f59cec454a": "[\\![x]\\!]", "3eb22ebd4505bd95f841bb43d3ffac60": " f^\\prime(x_{n-1}) \\approx \\frac{f(x_{n-1}) - f(x_{n-2})}{x_{n-1} - x_{n-2}}", "3eb25d5d059459437a0372acb418d678": "F(t) = (1-t)^d F(t) (1 - t)^{-d}", "3eb271ae8dd6b39833b49f2ae1ba8000": "\\frac{\\partial}{\\partial X} \\sin(X) = \\cos(X),", "3eb2eb373c726294e9e46bed955ab5b8": "AA^*\\leq\\lambda^2 BB^*", "3eb33487086370de4cba739f1848c69b": "\\displaystyle{\\widehat{f}(t)={1\\over \\sqrt{2\\pi}}\\int_{-\\infty}^\\infty f(x) e^{-itx} \\, dx.}", "3eb390e562b02891a62d62be96386f95": "\\Rrightarrow 1 \\text{ rad} = \\frac{180^\\circ}{\\pi}", "3eb3acc408c48fa418778881880ac812": "D = \\varepsilon E \\;", "3eb3b6e335db2adbf54d92a1ea19e48e": "(x_i\\,,y_i\\,)", "3eb3badac105903397e7731addc39a5d": "\\bar M_{g, n}^{J, \\nu}(X, A).", "3eb3d54009b115e37c5b1e89696f81d6": "f_{x_0}(x) = f_W(x-x_0)", "3eb4379e36f582e17525784ddfb0d347": "Em = \\tfrac{18}{25} + \\tfrac{9}{25} + \\tfrac{12}{25} + 0", "3eb4f08a88f425ab474d6b80c4681a31": "p_2(x)", "3eb4f0d1876baeb9ec0f0f982c869a97": "(I_n)_{ij} = \\delta_{ij}. \\,", "3eb53cff474c7c58ae069c162d75184c": "\\mathfrak c^{\\aleph_0} = \\mathfrak c,", "3eb54e47f3de0035b8a02f0bebb87d20": "\\tau \\propto \\frac{\\partial u}{\\partial y}", "3eb5a79d4428594241ced882d818436a": "h(x)=\\overline{f(x)},", "3eb5d5a5356b7bd5f5b17faee657fcbc": "m(n) = \\Omega(n^{1/3}2^n)", "3eb5e4f3a4e2ed00557c12d95f4f1c94": "y = \\sin (6x) + 2", "3eb6a9154f8321acce1f83a1de2d5d99": "\\frac{d \\rho_{ee}}{dt} = -\\gamma \\rho_{ee} + \\frac{i}{2}(\\Omega \\bar \\rho_{ge} - \\Omega^*\\bar \\rho_{eg})", "3eb6b8f04c2d6d641ca874cf8827c8df": "N(\\mathfrak p)", "3eb6ef5c556ff967e5fc3cc335caacc9": "\\displaystyle z=\\exp(\\mu/kT)", "3eb6f6f488696b9626ecf60693d35274": "\\operatorname{SU}(4) \\cong \\operatorname{Spin}(6).", "3eb6f7c799414ae12c126c5e4c55ff75": "\\forall\\beta.\\beta", "3eb71cf93b7f99a681fbc62abd375a36": "(\\mathbf{\\mu}=I\\mathbf{A})", "3eb732c5b5e3b80c277aea3c99e8cdea": " \\mathbf{D}^{t+1} = \\text{diag}\\left((G+\\lambda \\mathbf{I})^{-1} \\widetilde{G} \\boldsymbol{\\alpha}^t \\right)", "3eb738feeac1937c4842dd65165d5be4": "(AB' \\parallel A'B \\ \\and \\ BC' \\parallel B'C) \\Rightarrow CA' \\parallel C'A.", "3eb756b679ab73db2a4ce80d5f56d122": "\\gamma=1/2", "3eb7d9342f1b3213b8cb1492a4764d17": " s^{k+1} - s^k - s^{k-1} - \\dots - s - 1 ", "3eb7e77c5cf80f8643583f9f6108ed57": "\\begin{pmatrix} \\lambda & 0 \\\\ 0 & \\lambda^{-1} \\end{pmatrix}", "3eb7ec97aa0d55aff6b9a37f64259b0f": "\\langle m|H=\\langle m|E_m", "3eb82850cc4bab843da1d3dbc72587c6": "\\text{price}_{\\text{today}}", "3eb872485ed7cdea54d0779366fac43a": "\\frac{3+\\sqrt{21}}{2}", "3eb88a98ec08fc243ddb0bbe5b7013a5": "\\mathbf{w}_X^* = \\arg\\min_{\\mathbf{w}} \\left\\{ \\sum_{i=1}^N (x_i - \\langle\\mathbf{w}, \\mathbf{z}_i \\rangle)^2 \\right\\} ", "3eb8c4a7dfa2974000b45102e49998a5": "\\lim_{x \\to 0^+} \\log_a x = -\\infty", "3eb8ff4abf4429b4333d87809d311913": "\\frac {\\partial M_y(t)} {\\partial t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _y - \\frac {M_y(t)} {T_2}", "3eb91d041b640dc4ebc2580e5884a149": "\\mathbf{c} \\cdot \\mathbf{a} \\times \\mathbf{b} = c_i\\mathbf{e}_i \\cdot a_j\\mathbf{e}_j \\times b_k\\mathbf{e}_k = \\varepsilon_{ijk} c_i a_j b_k ", "3eb9297c4a844cb0fd2341eaef3ad87a": "== Summary ==", "3eb9df0c9ed5c329111215d1d950ead0": "\\theta = 2\\pi \\left( \\frac{x}{\\lambda} - \\frac{t}{T} \\right) = k x - \\omega t,", "3eb9e3b93160bc23e6fc99118e31e178": "\\frac{1}{AB}=\\int^1_0 \\frac{du}{\\left[uA +(1-u)B\\right]^2}", "3eba57f0734afafd8ceabc9bedb69de1": " \\partial p/\\partial r = 0 ", "3eba5d5c9af5c1ea2a58da397cecb222": "\\sin\\theta=0", "3eba69d8d9499c458b43930e994bfe9a": "1 + z = \\sqrt{\\frac{1+\\frac{v}{c}}{1-\\frac{v}{c}}}", "3eba73fc575e61ec55e0a641fa07425e": " A_i ", "3ebb0ad16576e139a69702d14d4918f9": "\\alpha'", "3ebb5beaf0fc88621da27e4405b01f25": "a=\\frac{2}{5}\\text{ and }-14\\le u<14", "3ebb633167719ac9820a8d38fd29c14b": "\\|u\\|_{L^{p^*}(\\mathbf{R}^n)}\\leq C \\|Du\\|_{L^{p}(\\mathbf{R}^n)}", "3ebb79135bacc1cc5cee5c9d7c8a4e69": "\\Delta\\ Unemployment", "3ebb9639b8e00ef3fb7d81c8adaeb05f": "[r,Kr]", "3ebc3d70109c965492085192fe6de16d": "\\left \\{ (x,t) \\in \\mathbb{R}^{n+1} : \\lVert x \\rVert \\leq t \\right \\} ", "3ebc540705708eb0cdc19718a7f8d124": " x_{n+1} = A x_n + b, \\, ", "3ebc7308aab9ada456bdbf7649c4f8f0": "S_{T}", "3ebc805c5e0f5b2f6f3fd44541192be0": "\n J \\approx 1 + \\mathrm{tr}(\\boldsymbol{\\varepsilon}) ~;~~ \\boldsymbol{B} \\approx \\boldsymbol{\\mathit{1}} + 2\\boldsymbol{\\varepsilon}\n ", "3ebc973316649a50a1467ae1a32730aa": "e(\\varphi) = \\frac12g^{ij}h_{\\alpha\\beta}\\frac{\\partial\\varphi^\\alpha}{\\partial x^i}\\frac{\\partial\\varphi^\\beta}{\\partial x^j}.", "3ebcbed06f9700b7b760c3f9ac462987": " \\frac{a - b}{c} = \\frac{\\sin\\left(\\frac{\\alpha - \\beta}{2}\\right)}{\\cos\\left(\\frac{\\gamma}{2}\\right)}. ", "3ebceda01a1d8276b9438a99ed8effa9": "\\mathfrak{so}^*_6(\\mathbf H)", "3ebd1affbb0fe58cdfd7f52313c23ace": " \\Box = \\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\nabla^2.", "3ebd459227451734583a91f475cb76e1": "a_{k,k}=z_k+w_k", "3ebd8000f198ce033d0db24f005d9e06": "P_{amb}", "3ebe270bff6d2940c72cb48c3712e833": "\\mu = \\delta \\, d", "3ebe51c77e1787b51c2d4c7acfe3abba": "{\\mathcal P}:=\\C\\cup \\{\\infty\\}, \\infty \\notin \\C", "3ebeb0e9c9752233c6a57ecaf929a58e": "\\mathrm{False \\ discovery \\ rate \\ (FDR) = \\frac{Median \\ (90^{th} \\ percentile) \\ of \\ \\# \\ of \\ falsely \\ called \\ genes}{Number \\ of \\ genes \\ called \\ significant}}", "3ebf54c4819e5e87b9ac1703d98becbe": "\n\\hat g = \\,\\,{k \\over {T^2 }}", "3ebf9a2f071a071822fd13e4f974ba58": "M^{f}", "3ec056076060f086ed013868a7370297": "\\Delta E_2 ", "3ec0633992fc4e14d82327b7ceb067f7": "x^{-1} \\cdot (x \\cdot y) ", "3ec0b924fdcc1a8727cffa33f4032807": "{\\color{Blue}R}", "3ec118978f72f8a1ac6955a4870c0bc9": "(h'_e)_{\\alpha \\alpha} = U_{\\alpha \\gamma}^{-1} (x) (h_e)_{\\gamma \\sigma} U_{\\sigma \\alpha} (x) = [U_{\\sigma \\alpha} (x) U_{\\alpha \\gamma}^{-1} (x)] (h_e)_{\\gamma \\sigma} = \\delta_{\\sigma \\gamma} (h_e)_{\\gamma \\sigma} = (h_e)_{\\gamma \\gamma}", "3ec12e9d836703ad2e101a19f541633f": " [M]_{B}^{C} [M]_{C}^{B} = [M]_{B}^{B} = \\mathrm{Id} ", "3ec14027e0ffdbd14c982bc5d038f9d6": "n_x = 1", "3ec1610acb906aa6c80aa87d74a347b4": "\\exp\\left[~it\\mu\\!-\\!|c t|^{3/2}~\\right]", "3ec16e44afbf4bd609800af6bca23208": "\\lambda = (1,0,0,0)", "3ec1c18d11c71ab9f3f4bbc0b362179b": "\\frac{\\partial^2 w}{\\partial x^2}\\bigg|_{x = L} = 0 \\quad ; \\quad \\frac{\\partial^3 w}{\\partial x^3}\\bigg|_{x = L} = 0 \\qquad \\mbox{(free end)}\\,", "3ec26e156e66f2ff023158ca551eaa0f": "(5.c)\\quad \\gamma_{,\\,z}=2\\,\\rho\\,\\psi_{,\\,\\rho}\\psi_{,\\,z}\\,,", "3ec284660a56b0a2a389290a03d33cab": "\\varphi(h)(n)=hn", "3ec2cfeaf5363f558648f21c695f8f04": " pv ", "3ec321a9a32ae955e5c207cdf03c44c6": "\\omega=\\lambda_{mn}", "3ec32c2a5a192c546415977fdbd32632": " \\mathbf{E} ( \\mathbf{r}, t ) = \\frac{1}{r} \\mathbf{E}_0 \\cos( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} + \\phi_0 ) ", "3ec35934f1b64d5e3fa89a3ce313779c": "\\varphi_p \\simeq \\lambda y.Q(p,y)", "3ec366a17d137474d9518ba1f714d516": " \\operatorname{E}[XY] = \\iint xy \\, j(x,y)\\,\\mathrm{d}x\\,\\mathrm{d}y.", "3ec3856588410d5a9b34ab556e267c62": "{u}=v+\\overline{u}", "3ec386a68a9746e4415fef475838578e": "(-1, 1) = \\{y : -1 < y < 1\\}", "3ec38966103cb6ef43907190a6974642": "(0,\\ 0,\\ \\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm1)", "3ec3d7cc901c6618d3de89c86feda60c": "\\partial F_0 = f_0 ", "3ec43049d6017ea635e9bd1d5f7e421a": "\n P_1 \\oplus P_3 \\oplus C_1 = K_2.\n", "3ec45ba56eb3e8e878a5ae8159a2764a": "|CA|", "3ec49e2dd595976b765029d8386693c6": "u(x) = \\mathbf{E}^{x} \\left[ g \\big( X_{\\tau_{D}} \\big) \\cdot \\chi_{\\{ \\tau_{D} < + \\infty \\}} \\right] + \\mathbf{E}^{X} \\left[ \\int_{0}^{\\tau_{D}} f(X_{t}) \\, \\mathrm{d} t \\right]", "3ec4c0379a90a46585a5b40231d83a53": "\\mathrm{Mg}^\\times_\\mathrm{surface}", "3ec4eb4438f28f054f5a80c7157981d4": "\\hat{K}(s)", "3ec4f4c69404c36a4deee37ef673dbb1": "k = N_2 k_1 + k_2", "3ec4f974f0e50e9fe71dc4b3e00b26c8": "(d,e)\\subseteq(a,b)", "3ec531b5dd3fd9230e3e753c41b11fcf": " \\omega_p = E_{\\beta} - E_{\\alpha} ", "3ec59cb685288acf8f4e9750f6e1661d": "s_k = x_{k+1} - x_k\\,\\!", "3ec5a40c6ffb3931a51317ee0d8f6bac": "\\scriptstyle{\\lambda^+_i, \\lambda^-_i}", "3ec5ecdd97660810bca11bf68cb554b7": "F_{O_2loop}(t)=\\frac{(Q_{dump}+V_{O_2})*F_{O_2feed}-V_{O_2}}{Q_{dump}}+(F_{O_2loop}^{start}-\\frac{(Q_{dump}+V_{O_2})*F_{O_2feed}-V_{O_2}}{Q_{dump}})*e^{-\\frac{Q_{dump}}{V_{loop}}t}", "3ec603e0fc6a93cd0d3fb8854a219c12": "b_{ij} = -1", "3ec6271745ecd648e6fa6f32d9399d3b": "\\cfrac{9 Pmf - 4 Pmf}{3} = 1", "3ec6589e49bab3f348ea7911f1589cb5": "R\\ ;\\ \\overset{\\alpha}{\\rightarrow}\\quad {\\subseteq}\\quad \\overset{\\alpha}{\\rightarrow}\\ ;\\ R", "3ec69172b1ac4ea0e64bc417075409b8": "S^n = \\left\\{ x \\in \\mathbb{R}^{n+1} : \\|x\\| = r\\right\\}.", "3ec6972d44305f694cc7ce6535041671": "\\|Lv\\| \\le M \\|v\\|,\\,", "3ec7122626586a9a4462755e4b66ea40": " Q ", "3ec784417c8b4df63901a74775b331c2": "E(\\alpha)", "3ec78d9e6136c7209dc11b10251c63a3": "{\\varphi^4 = \\varphi^2 - \\varphi^3}", "3ec7ec62075cc2d7d2972f9b46810804": "T\\cap S_i", "3ec7f311859f7eef5aa65808e3e3d195": "\\langle R, \\mathcal{E}(\\sigma)\\rangle = \\langle \\mathcal{E}^{\\dagger}(R), \\sigma\\rangle \\geq \\langle Q,\\sigma\\rangle", "3ec8db2c200da2b4eacb495d60ed935e": "\\left(\\dfrac{K}{N}\\right)\\geq \\left(\\dfrac{(\\dfrac{n}{m})S - (n-k)S}{(\\dfrac{n}{m}) S}\\right)", "3ec8f64951b7fe1968e914d6dbad9d9a": "\\psi \\left(q_k, \\frac{\\partial S}{\\partial q_k} \\right)", "3eca04d36e30e7254ac67d89e335b8c9": "x_4=x_1+(x_3-x_2)", "3eca1043dcb7bc91cde47c664d86d861": "P = {1 \\over 2}V_p I_p \\cos \\theta = V_{rms}I_{rms} \\cos \\theta \\,", "3ecac405eaa711c3411f6df022467d39": "G^0(X) ", "3ecaf2b5f6fd65b8392c300e8c69148b": "\\lambda=|\\vec{B}|\\,A = BA", "3ecb1eed43724ae3d227d474de123e5c": "\\theta=x\\,dy+y\\,dz+z\\,dx.", "3ecb27b3d18eef619e2d8addfc7ba832": " \\ln(z) = \\left\\{ \\ln(r) + (\\varphi + 2\\pi k)i \\;|\\; k \\in \\mathbb{Z}\\right\\}", "3ecb3ac4b282d5004550a9087b9b90b4": "C_2=\\{(-1,0),(0,0),(-1,-1),(0,-1)\\}", "3ecb3c47bddbbde9207e5140db745cd7": "\\! F_M(x).", "3ecb5a205cf6abade1e126ce6e39cc65": "\\omega_{\\Psi}", "3ecbac51a9d8d7e24f118f04fee0c9fa": "\\frac{P-P_c}{P_c}", "3ecc31ce9608593321a55737af90dd8d": "(r ',\\theta ',z')", "3ecc5a81920adf012e62a2eb7368fd46": " d = \\det(B) ", "3ecc76c230b33e6bbda174777a2e4faf": "2+\\frac{\\sqrt{2}}{2}+\\frac{\\sqrt{6}}{2}", "3eccb8ff63eb064472ed3028b6795795": "{\\mathit{He}}_n^{(m)}(x)=\\frac{n!}{(n-m)!}\\cdot{\\mathit{He}}_{n-m}(x)=m!\\cdot{n \\choose m}\\cdot{\\mathit{He}}_{n-m}(x),\\,\\!", "3ecd01d0fa5c8b92f524c931932545d6": "\n\\begin{align}\n&F_x = -\\frac{\\partial u }{\\partial x} = J_2\\ \\frac{x}{r^7} \\left(6z^2 - \\frac{3}{2} (x^2 + y^2)\\right) \\\\\n&F_y = -\\frac{\\partial u }{\\partial y} = J_2\\ \\frac{y}{r^7} \\left(6z^2 - \\frac{3}{2} (x^2 + y^2)\\right) \\\\\n&F_z = -\\frac{\\partial u }{\\partial z} = J_2\\ \\frac{z}{r^7} \\left(3z^2 - \\frac{9}{2} (x^2 + y^2)\\right)\n\\end{align}\n", "3ecd2a043ab961437fd8dae07616196b": "Y_{4}=\\{X_{7},X_{8}\\}", "3ecd83279b59b658aaf94df1d8975667": "\\!\\mathcal A", "3ecdbbc9a64a9bfc57883ae306bf51cd": "\\Delta T", "3ecdcfcb45871b3a60b9e816c7529109": "H(s) = \\frac{ \\frac{\\omega_0}{Q}s}{ s^2 + \\frac{ \\omega_0 }{Q}s + \\omega_0^2 }", "3ecde7eb78719c9a416650a96a8b7d9f": "\nf(x) = \\frac{2\\mu}{\\hbar^2} \\left(E - V(x) \\right) - \\frac{l(l+1)}{x^2}\n", "3ece176bb5aa64bdf83ac65a6e26cacc": "\\delta\\hat x_{BA}[\\hat\\mathcal{F}]", "3ece50f7e834c671a8dc382b1dd1e643": "a|b\\,\\!", "3ece53a56865890140b3f78c1620ad83": "\\text{Base amperes }=\\frac{\\text{base kva}}{\\text{base kv}_{L-L}}", "3ece92ae6b475c81b4e90d570fa16f37": "0 < \\alpha < \\pi", "3ecec62ee0260556f8699bd2ded20b2f": "\\int \\frac{d^{2n}\\xi }{(2\\pi \\hbar )^{n}}\\hat{B}(\\xi )Tr[\\hat{B}(\\xi )\\hat{f}] =\\hat{f},", "3ecfa2c45b0d5fdc5e5c2b39a028e110": "G=(N,A=A_{1}\\times\\ldots\\times A_{N}, u: A \\rightarrow \\reals^N) ", "3ecfa949eaf2b1cbbbe06dd060970cc0": "z \\bar z - z \\bar \\gamma - \\bar z \\gamma + \\gamma \\bar \\gamma - r^2 = 0.", "3ed005b4e5317110e23d35726e73c24e": " f(x;b,\\eta) = b e^{-bx} e^{-\\eta e^{-bx}}\\left[1 + \\eta\\left(1 - e^{-bx}\\right)\\right] \\text{ for }x \\geq 0. \\,", "3ed026a66772698636f31b355734f643": "(x)_n = \\sum_{k=0}^n s(n,k) x^k,", "3ed02a2dc7966a40107dc253b1291208": "\\scriptstyle \\mathbf{v}", "3ed038b62dedb6714018d0ca210029d3": "p= \\operatorname{P}(T \\ge t) = \\operatorname{P}(T > t) =1 - F_T(t).", "3ed048da0828903cf8797ff89e844772": "s | (s \\uparrow 2)", "3ed06026f8718974a3e8598c8db2ddc2": "B(v,w) = v^\\mathsf{T} G w .", "3ed0640b5ebe6294dd7a506effd4b080": "b^{(a-1)/2}\\equiv +1 \\pmod a\\;", "3ed0f2277cb35b8bfa83c5ec0bd7b457": " \\Delta_h f(x) = f(x-h) - f(x)", "3ed0f9b408e06024f98ff278724de7a8": "\\vartheta(z; \\tau) = \\prod_{m=1}^\\infty\n\\left( 1 - \\exp(2m \\pi i \\tau)\\right)\n\\left( 1 + \\exp((2m-1) \\pi i \\tau + 2 \\pi i z)\\right)\n\\left( 1 + \\exp((2m-1) \\pi i \\tau -2 \\pi i z)\\right).\n", "3ed1099c2dd87d19cacf0e78277c724e": "\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}=-\\gamma \\mu_0 \\mathbf{m} \\times \\mathbf{H_{eff}} + \\frac{\\alpha}{m} \\left( \\mathbf{m} \\times \\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}\\right)", "3ed12c526f6e1283b6231bd176a22931": "L(M) = \\{ w\\in\\Sigma^{*}: M \\text{ accepts } w \\}", "3ed13cf2cad8f138320d1622e8f6c884": "\n\\omega_\\mathrm{c} = {1 \\over \\tau} = {1 \\over R C}\n", "3ed18e24ea05f6a1c8e4e6badeaa4f7b": "\\phi(a)=\\operatorname{id}", "3ed195be19d4640445d9819703aef40a": "\\boldsymbol{\\Sigma}^0_1", "3ed1a402987cff8db10f317b38f0b2d4": "\\mathop{\\rm el}(P)", "3ed21ab2f881f8676b3aef6e86fd0322": " P(X_s \\in B \\mid X_t = x) = E[u_s(x) \\mid X_t = x]", "3ed2c0c90b5bc02eb87e824d28c75abc": "g=2h^0\\otimes h^0 -g^R", "3ed2f4fe062465f634ba066daca3896b": " \\mathrm{d}f = \\frac{\\partial f}{\\partial x^1} \\,\\mathrm{d}x^1 + \\frac{\\partial f}{\\partial x^2} \\,\\mathrm{d}x^2 + \\cdots +\\frac{\\partial f}{\\partial x^n} \\,\\mathrm{d}x^n. ", "3ed34b2d6e9f0f79b053f00c3186cedc": " \\mbox{If } x \\in F, \\ y \\in H, \\ \\mbox{and } x \\le y \\mbox{ then } y \\in F.", "3ed3bcb42f743be71c312a0a6e13eb94": "L(x) = \\sum_{i} l_i x^{p^i}, l_i \\in \\mathrm{GF}(p^m).", "3ed404206bbcaba6cb25217344369f88": "E = \\int_v T_{00} dV \\qquad P^i = \\int_V T_{0i} dV ", "3ed43f151bd174f1dee01ad8b044bdda": "[X]_t=\\sum_{00", "3ee2a72a1d4b6a8e055358dce6987d3d": "\n\\overline{\\mathbf{\\rho}}=\\frac{1}{N}\\sum_{n=1}^N z_n.\n", "3ee2f1beb45d87a4119bdb2e5ed9441f": "w=\\frac{K}{nK_n}\\mathrm{cd}^{-1}\\left(\\frac{\\pm j}{\\epsilon},\\frac{1}{L_n}\\right)+\\frac{mK}{n}", "3ee2f2386356311fd1dce15a9b2d0964": " \\exists x \\left( F \\left( x \\right) \\land \\forall y \\left( F \\left( y \\right) \\rightarrow x=y \\right) \\right) ", "3ee350a2ecac961403b8b10f277a5761": " y^m(\\mathbf{x}) ", "3ee35440430b8900977293aa0167b91d": "(x,y)\\mapsto x^{-1}y", "3ee3731e4f840daa834aed5437d6e0af": " \\partial L/\\partial T = \\alpha L \\,\\!", "3ee3937343b7f3e4b054aff11a3ac7db": "\\inf\\{ |f - v| \\mid f \\in d(x) \\}", "3ee3a6eb17bda14b63bebfb34f4df03b": "x\\mapsto x^2", "3ee413ef8b04fe79024a02a35ce21ab2": "L_{0}=L'\\cdot\\gamma. \\qquad \\qquad \\text{(3)}", "3ee4208fcc458147cff4b0c8dbd120bb": "2|\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|(\\mathbf{X^{\\rm T}A^{\\rm T}X})^{-1}\\mathbf{X^{\\rm T}A^{\\rm T}}", "3ee4559a6b98fd2248ea716bce220ba3": "\\delta = \\delta(k,\\lambda,\\epsilon)>0", "3ee4699560155c7c9badfbc7a85345d7": "\\begin{bmatrix}1&2&1\\\\-2&-3&1\\\\3&5&0\\end{bmatrix}R_2\\rightarrow 2r_1 + r_2 \\begin{bmatrix}1&2&1\\\\0&1&3\\\\3&5&0\\end{bmatrix} R_3 \\rightarrow -3r_1 + r_3 \\begin{bmatrix}1&2&1\\\\0&1&3\\\\0&-1&-3\\end{bmatrix} R_3 \\rightarrow r_2 + r_3 \\begin{bmatrix}1&2&1\\\\0&1&3\\\\0&0&0\\end{bmatrix} R_1 \\rightarrow -2r_2 + r_1 \\begin{bmatrix}1&0&-5\\\\0&1&3\\\\0&0&0\\end{bmatrix}", "3ee47725e2886583969aef2158d2ff00": " \n\\frac{\\partial \\Gamma (s,x) }{\\partial x} = - \\frac{x^{s-1}}{e^x}\n", "3ee4ed9566ade3ec101b0edc669195f1": "\n\\chi=\\chi_0 (-1)^n\n", "3ee501507f1b8739f2b65ef300c76188": "\\left(\\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "3ee56e9eb94e7d9c8ec9a885c6fa96a6": "T = M/\\Gamma", "3ee64f16b9df12d9097302f25fdd5cc7": "L(s,a)=\\sum^\\infty_{n=1}\\frac{a(n)}{n^s}=\\prod_p\\biggl(1-\\frac{a(p)}{p^s}\\biggr)^{-1},", "3ee692a199c125547910ce579e53df0e": "a > 0", "3ee69ebeaee8676262d1c00fe705d77b": "A_j=H(A_{j-1}||auth_{j-1})", "3ee6a9ef4b48941f78904ffd4a58c397": "\n\\left( -\\frac{\\hbar^2}{2m} \\nabla^2 + V \\right) \\psi = E \\psi~,\n", "3ee6c74f8568ed1ee4150a6459e94128": "FWER = P_{all}(V \\ge 1) ", "3ee7000afe4753ed5943e8b5a9d66cd2": "(q \\and p) \\or (\\neg q \\and r)", "3ee733a76b82ca37c2dca5527818c362": " \\mathbf{s}^T \\nabla f(\\mathbf{x}_k)", "3ee7d85ccec2fc22f428c9fe030f6ffd": "T f(x)=\\int_Y K(x,y)f(y)\\,dy.", "3ee7e0009583c2954c72d9cbe49ff06f": "x e^x /(e^x - 1) = n", "3ee7f0fc80cc65f6f4e7383a39a87bb1": "(3)\\quad F_{ab}=A_{b\\,;\\,a}-A_{a\\,;\\,b}=A_{b\\,,\\,a}-A_{a\\,,\\,b}", "3ee87a0ac483856638118ad2b07e5c1b": "\n{\\rm E} \\left(\n T \\cdot \\frac{\\partial}{\\partial\\theta} \\ln f(X;\\theta)\n\\right)\n=\n\\int\n t(x)\n \\left[\n \\frac{\\partial}{\\partial\\theta} f(x;\\theta)\n \\right]\n\\, dx\n=\n\\frac{\\partial}{\\partial\\theta}\n\\left[\n \\int t(x)f(x;\\theta)\\,dx\n\\right]\n=\n\\psi^\\prime(\\theta)\n", "3ee8817595431f6dc1b96e56b3e45d82": "\\varphi (t, \\omega) ( \\mathcal{A} (\\omega) ) = \\mathcal{A} (\\vartheta_{t} \\omega)", "3ee8ac906cc9197023874496351e6cd0": "\\tau=\\prod_{i=1}^{K}\\sigma_i", "3ee8b66c28ad8ebb8e9cfad1eefbe9ea": "s/a", "3ee8c94400c12dc087d07b8c4102bc5e": "\nE_n= {p^2 \\over 2m} = {n^2 h^2 \\over 8mL^2}\n", "3ee8f44275768f8ad0d668ef9a9a3e1d": "\\widetilde{\\mathcal{O}_P} / \\mathcal{O}_P", "3ee91cd8c0965cf28a04c7392e2aa5d6": "\\vec v_i", "3ee929992e4786fd792501c239fd9ab3": "\\operatorname{isnil} \\equiv \\lambda l.l\\ (\\lambda h.\\lambda t.\\operatorname{false})\\ \\operatorname{true}", "3ee95ed0e52eddcb28fa1fde4f525db1": "\\eta_0 = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} \\approx 377 \\Omega", "3ee977eb82a175821217936070d4120d": " r = -e (-ae+r \\cos \\theta) - a\\,\\!", "3ee9a4e9b2f98de1d68840f604a74cf8": "B_i = 100b_{i-1} + b_i", "3ee9a64e7dfc371d25f10e14370ac96f": "\\frac{(7+2+1+11+6+5+3+4)^2}{8} + \\frac{(7+6+10+7+3+11+4)^2}{7} ", "3eea1213244e1aac2f713e20f2b4117e": "[L'_{ij},H']=0 \\,\\!", "3eea35c9c2a1ff5a6ed98811dfd10312": "\\textstyle(x\\pm1, y, z)", "3eea597caffa25da034b9501ef632de7": "\\tfrac{x^2-x+1}{5x^2+3}", "3eea8d68caf3f77deed5bc50567ae225": "\\ \\vec\\mathrm{M}_{hull/G} ", "3eea9a08ab9523df341848fbd6dcf4d9": "\\,\\mathcal{M}(x_1,x_2)\\,", "3eead35c07b49a661204cd45652ede3b": "(f-g)^*(x)", "3eeadfda1f542d8bcd7b493e964cbb44": "\\tbinom{n}{k}", "3eeb06c1fd9b17459f1f30c79587c820": " \\{b_i\\},\\;\\{f_i\\} ", "3eeb113ab847cc95627dce646b98839d": "\\scriptstyle a \\in R \\setminus \\{0\\}", "3eeb1636786afc6090f4c3b3a9d90907": "\\mathbf{\\rho_{S}}(t)", "3eeb41d9f9a123dc04a0ffaffd6c9076": "\\theta_t = ", "3eebc7c4562a24e8b8a2b9dad36e57c5": " DF(T) = \\frac{1}{( 1 + \\frac{r}{365} )^{ 365T } } ", "3eebf8727ad163ad604a7c90c5cfac64": "\n\\begin{align}\ng_1(p) & = 0 && \\text{these mean the point satisfies all constraints} \\\\\ng_2(p)& =0 \\\\\n& \\ \\ \\vdots \\\\\ng_M(p) &= 0 \\\\\n & \\\\\n\\nabla f(p) - \\sum_{k=1}^M {\\lambda_k \\, \\nabla g_k (p)} & = 0 && \\text{this means the point is a stationary point}. \\\\\n\\end{align}\n", "3eec3bed9028447a4c7f36187e77420a": "\\operatorname{IsZero} = \\lambda n.n\\ (\\lambda x.\\operatorname{false})\\ \\operatorname{true}", "3eec556b590cb66a86c756d3a8445911": "\\hat\\psi(0) = 0", "3eedc635bf57691aee7de14df431dc30": "R(g)", "3eedd99426b836b1581c4cdac12d92e9": "g^{(2)}(\\mathbf{r}_{1},\\mathbf{r}_{2})", "3eee0580641ef7cff6f41c346bdbd080": "(\\mathbb{C},~ 0\\! \\rightarrow\\! 1)", "3eee29b544f2b0515a075264f8aa2c55": " E_a = \\sqrt{(m_k c^2)^2 + (m_t c^2)^2 + 2 (m_t c^2) (m_k c^2)} ", "3eee30b03f0fa44bd572ed099a4e1659": " (a_2,b_2,c_2)", "3eee799801c342165ebf9d6cf413073b": " |\\phi\\rangle = \\sum_i c_i |f_i\\rangle \\quad \\text{with } c_i = \\langle f_i | \\phi \\rangle.\\, ", "3eeed24ade2a4ed8145421dff262e568": "\\frac{32}{516} = 0.0620", "3eef99db95fd98711c012ade5c8b2b1c": "\\mu_r = \\frac{\\mu}{\\mu_0},", "3eefd7a4589168a60f7a1b5167056028": "V_{\\rm pp}", "3ef0406d0b5121f37be8dfaaa4e8e8a8": "c_2\\in\\mathrm{up}(c_1)", "3ef07a6625f535909c3766aef13d0def": "\n\\begin{align}\n(f+g)(x) & = f(x)+g(x) & \\text{(pointwise addition)} \\\\\n(f\\cdot g)(x) & = f(x) \\cdot g(x) & \\text{(pointwise multiplication)} \\\\\n(\\lambda f)(x) & = \\lambda \\cdot f(x) & \\text{(pointwise multiplication by a scalar)}\n\\end{align}\n", "3ef08e4c322ae56e10155c01eab3e8d4": " t = \\frac{\\bar {X}_1 - \\bar{X}_2}{s_{X_1X_2} \\cdot \\sqrt{\\frac{2}{n}}} ", "3ef0a64fd3c432ac031cfe44dde5f1f4": "A=\\tfrac{1}{2}h(b_1+b_2)", "3ef0ba8cfa4ca7d1d38b2a87da54768a": "\\frac{N_H}{A(V)} \\approx 1.8 \\times 10^{21}~\\mbox{atoms}~\\mbox{cm}^{-2}~\\mbox{mag}^{-1} ", "3ef0f70458f9cce4880273b6703b8f37": "g\\left(x\\right)=G'\\left(x\\right)=0", "3ef10bdfd85114772f6f67ca08b30ed8": "\n\\Pr \\{X_{ni}=1\\}=\\frac{\\exp({\\beta_n} - {\\delta_i})}{1 + \\exp({\\beta_n} - {\\delta_i})},\n", "3ef14a4fd429f381a9bf8bd799c6fcca": "T_0\\in\\mathcal{F}", "3ef1638630a55bbff26e8e345e8efa52": " 256/81 r^2 h ", "3ef191b1457e059f71a682fa8efce377": "\\displaystyle f(x)=(a_1x_1+a_2x_2)^2=(b_1x_1+b_2x_2)^2", "3ef1d113a65de4f2f4aee097085c35dd": "\\scriptstyle \\frac{1}{\\Gamma(k) \\theta^k} x^{k \\,-\\, 1} e^{-\\frac{x}{\\theta}}", "3ef1d5a593eb2e1d89f31c404e5fc51a": "\nm_n=E(z^n)=\\int_\\Gamma P(\\theta)z^n d\\theta\\,\n", "3ef23f2e2f177b8905a0071e6bd983d6": "\\pi_{21}", "3ef2455ccdcae45c3b1beff216169f4e": "\\textstyle{ax^2 + bx + c = 0, a \\ne 0}", "3ef294a7aeaf5c7ee0c7f549876b9f25": "F'(x_1,\\dots,x_n)", "3ef2b89e2bf8a9ccd0e218f488b9c59c": "n = \\sqrt{\\mu \\epsilon}", "3ef2c1fec1d40aa8f6da4d588acf6af2": "\\frac{x}{r^2-z^2}\\,", "3ef2f377eac451fe50877be073bc0d0e": "\n\\max_{f(X_1,X_2)} I(X_2,X_2;Y_3) = \\frac{1}{2} \\log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \\sqrt{ \\beta c^2_{31} c^2_{32} P_1 P_2})\n", "3ef34548a427669be37667c6ed254e07": "T = 2\\pi\\sqrt{I/\\kappa}\\,", "3ef36b2bf6b7c08a1d2f3b7e1f15224d": "\\alpha=\\frac {|w|}{(2 c_r Q_r)} \\quad (1)", "3ef3913aca48b60922c6fa53296bfe4c": "\\hbar \\gamma = 0.13\\,\\mathrm{meV}", "3ef3bc48d9d0c4b5a697d9fb0e4090be": "L'=A_L(t) L", "3ef3be39c0ce6db13e25b74b1c003dd1": "\\hat{W}", "3ef3cadf29d48026f675ae9b1c31cc7b": "z=f(y)", "3ef3d7a43d12f1976d4b106166593b2d": "X \\sim \\text{Wilcoxon}(m,n)\\,\\!", "3ef3e12311e597f146bee25fac2fa1f4": "\\beta l= n\\pi", "3ef41cf44c14ef8084844b1aa1b527e3": "x=a \\sin({m\\theta}) \\cos({\\theta})", "3ef433130a1d9727b86dea72fe500e3e": "\\int_{Y} f(y) \\, \\mathrm{d} \\mu (y) = \\int_{X} \\int_{\\pi^{-1} (x)} f(y) \\, \\mathrm{d} \\mu_{x} (y) \\mathrm{d} \\nu (x).", "3ef435635c174bc897c1244f62c21624": "[F_{\\lambda}]+R[R] =G[G]+B[B]", "3ef44505e12106903accf514dc7a6351": "p(\\mathbf{y}|\\mathbf{X},\\boldsymbol\\beta,\\sigma)", "3ef4806b537a243e3709e5404c8ba294": " y \\to x \\or y = 1 ,", "3ef4991dfc1c0bc68bf5987cd46fa781": "\\mbox{epi}_S f = \\{ (x, \\mu) \\, : \\, x \\in \\mathbb{R}^n,\\, \\mu \\in \\mathbb{R},\\, \\mu > f(x) \\} \\subseteq \\mathbb{R}^{n+1}.", "3ef4fc4d191558bc1fc5b5b950daec45": " N(\\mu,\\sigma^2)", "3ef52a06b3c31d892589b3b742b8ffb2": "\\deg(2x(1+2x)) = \\deg(2x) = 1", "3ef5dc9e5618ad9ba29b276cf301727f": "-(\\beta r_O - R_E) \\frac{r_E + R_E}{r_\\pi + r_E + 2R_E} ", "3ef5f912006e23851fbb58e2e01694a1": "\\prod_p\\left(1+\\frac{1}{p(p-1)}\\right)=\\frac{\\zeta(2)\\zeta(3)}{\\zeta(6)} = \\frac{315}{2\\pi^4}\\zeta(3),", "3ef602bea68de5883a3de03bd3cd1fa0": "a_{15}+c_{15}", "3ef6190c2f19b15f82d699cc8a36334e": "F_p", "3ef63695ee9318db5cf2b2fa1cf1078a": "=\\int d\\mathbf{1} f(\\mathbf{0},\\mathbf{1})", "3ef6a56415cb78029347dae3c912b1ca": " \\gamma(\\mathbf{u}) = \\frac{1}{\\sqrt{1- \\frac{\\mathbf{u}\\cdot\\mathbf{u}}{c^2}}}", "3ef6b0fee8ed6fc7db8a5a561c41096f": "i=1,2,3..N", "3ef6c81a916ee10830767f8edf26a837": "\\ L_m", "3ef6fcc6981824413af304579127e7c2": "\\left(1-\\tfrac{2}{n}\\right)\\pi", "3ef72ef683bfc30cb6ba272c692c65c1": "\\boldsymbol{\\mathsf{a}}", "3ef74ef738b09bf317d69f1a53947758": "\\left\\lbrace c_i^* \\right\\rbrace", "3ef79175f27e661c69300b089b55f6b6": " e^{v\\hat{a}^\\dagger - v^*\\hat{a}} e^{u\\hat{a}^\\dagger - u^*\\hat{a}} = e^{(v+u)\\hat{a}^\\dagger -(v^*+u^*)\\hat{a}} e^{(vu^*-uv^*)/2},", "3ef7924fc09d2c4ace5234c54f0017ae": "D(c,d,e,f)", "3ef7a79d5f9e9a9f5cb1a6e7718e734c": "B_{2}= \\langle b_{2}^{*}, b_{2}^{*} \\rangle =\n\\begin{bmatrix}\\frac{-4}{3}\\\\\\frac{-1}{3}\\\\\\frac{5}{3}\\end{bmatrix} \\begin{bmatrix}\\frac{-4}{3}\\\\\\frac{-1}{3}\\\\\\frac{5}{3}\\end{bmatrix}= \\frac{14}{3}.", "3ef82b8750aa3794483ba01666a8379b": "\\partial_t\\bold u+\\bold u\\cdot\\nabla\\bold u=-\\frac{1}{m}\\nabla\\left(\\frac{1}{\\sqrt{\\rho}}\\hat H\\sqrt{\\rho}\\right)=-\\frac{1}{\\rho}\\nabla\\cdot\\bold p_Q-\\frac{1}{m}\\nabla U,", "3ef850c38d65e8bfe8b1dfd51a955637": "\nS_{2} = V_{0} sin(\\omega t - \\pi)\n", "3ef88e46d5de9de042bd255e3e7254ed": " y_{n+1}^{(2)} = y_{n+1}^{(1)} + \\tau_{n+1}^{(1)} ", "3ef8bc2b1a3facbb0aa57fcc33b2f0d2": "H_n(X) = \\ker \\partial_n / \\mathrm{im}(\\partial_{n+1})", "3ef926b433478991e68bba18970bf940": "10 \\times 9 \\times 8 \\times 7 \\times 6 \\times 2^{5} = 967,680", "3ef92874260c03b54bdc8960e7587e8a": "y\\in \\mathbb{R}^N", "3ef92abfbf65ede9030fd9918e41d4c1": "g^i {}_k=\\delta^i {}_k\\ ", "3ef965b758e7e0c5e47b30df0c3461c5": "j_r \\leq j_s", "3ef988dad6363d669e1b6963fcba1abf": "a = \\frac{8(\\pi-3)}{3\\pi(4-\\pi)} \\approx 0.140012.", "3ef9b9444fdb80ead50059eb26b80fde": "q=\\sqrt{a^2+d^2-2ad\\cos{A}}=\\sqrt{b^2+c^2-2bc\\cos{C}}.", "3ef9bcdbbb978867b48cee010428d6c1": "u(x, 0)=\\delta(x).", "3ef9ee95e23f9ea9f31ea497a82f052a": " 10^{15} ", "3ef9f73e9b97f6648670424855b66567": "A_{i,j}=\\langle e_i, Ae_j \\rangle", "3efa08ddb4817e3bb451c0660a882c50": " \\frac {d\\varepsilon(t)} {dt} = \\frac { \\frac {E_2} {\\eta} \\left ( \\frac {\\eta} {E_2}\\frac {d\\sigma(t)} {dt} + \\sigma(t) - E_1 \\varepsilon(t) \\right )}{E_1 + E_2} ", "3efa09e91fb4af68d252b69be09c838f": "\\left ( M_t \\right )_{0 \\le t < \\infty}", "3efa0c6121c72dc0b1dba8bd722df534": "{\\color{Blue}~6.13}", "3efa5a1f19b7c2b2db6df7790536d159": "\\omega_\\infin=\\frac{\\omega_c}{\\sqrt{1-(mm')^2}}.", "3efa7a24bb8dc4f8a1e650772660ebae": " K=\\pi\\dot{N}\\dot{G}^3/3 \\,\\!", "3efa80b568c1676da3e11762d436bec4": "{t_{k}}_{(i)} = \\mathbf{x}_{(i)} \\cdot \\mathbf{w}_{(k)}", "3efa91244682a8ef2cdce51af157d1b3": "\\frac{k}{-i}=j", "3efaa0b617bada0a447cc8bb523ed5fb": "\\left (\\zeta(3)\\right )^2-\\tfrac{4}{3}\\zeta(6)", "3efb22d643c8cd956a3bc2ff2dd6964a": "dq = \\rho(\\boldsymbol{r'})\\,dV'.", "3efbc51921433f095bfb36314942e370": "\\exists x \\phi", "3efbf43dfa8cc069bcc243830373d5ae": "\\scriptstyle F_\\sigma", "3efcac745acb6edfdef962f58505fcb6": "\\dot{a}, \\ddot{a}, \\acute{a}, \\grave{a} \\!", "3efd0b191cba3648942ef4bcf59171cc": "g \\otimes {\\Bbb C}", "3efd7d17bcec22f959ecb8f5df4b1148": " I(t)", "3efd9a725c59adafaaeab781575dd763": "\\frac{K_{sp}}{x^x y^y} = {\\left(S_0\\right)}^{(x+y)}\\,", "3efdb98f4d23ddfae6427fd687df4821": "x\\mapsto P(x).", "3efdd4464ac7c9bfb60f1eacbbcb2ac0": "\\frac{(N-2K)(N-1)^\\frac{1}{2}(N-2n)}{[nK(N-K)(N-n)]^\\frac{1}{2}(N-2)}", "3efdd9c18905f96158e0166297d44268": " k-2 ", "3efe6c5aede420ae0bd008f5f9f1befd": "\\Delta w'''=0", "3efeda71e237425a28a6adab6449009a": " \\frac{d\\ln k}{dT} = \\frac{a-bT}{RT^2} ", "3efee3fa7d6d0a0777aa250929c331fb": "\\phi_n\\in\\mathcal{C}^\\infty[0,b]", "3efef5d9fb602d128f0f99696bb9f051": "\\scriptstyle \\sum_{n=1}^\\infty \\frac{x^n}{n^2}", "3effa2917c2cb75f3fcbea8ca307feef": "\\alpha V_0", "3effa67d9b8d6e62c790654d9f391fd9": "\n\\delta x_{i} = \\frac{\\epsilon}{2} \\left[ 2 p_{i} x_{s} - x_{i} p_{s} - \\delta_{is} \\left( \\mathbf{r} \\cdot \\mathbf{p} \\right) \\right]\n", "3effa837df344d2333e20d50dfe3ccd2": "\\Delta N_k = N_k \\otimes 1 + e^{-\\lambda P_0} \\otimes N_k + i \\lambda \\varepsilon_{klm} P_l \\otimes R_m .", "3effcee5578337d0d37071b73e448e0d": " u^L_{i + 1/2} = u_i + 0.5 \\phi \\left( r_i \\right) \\left( u_{i+1} - u_i \\right),\n u^R_{i + 1/2} = u_{i+1} - 0.5 \\phi \\left( r_{i+1} \\right) \\left( u_{i+2} - u_{i+1} \\right),", "3f000ef0728fb83c0d669ac050e178f5": "x^{(n)} = {(x + n - 1)}_n ,", "3f00b07870522fd9f0d5ca51243f9106": "{\\beta}=0", "3f00d217c08928f5776d040a28ad0882": "\n\\lambda_1 x'^2 + \\lambda_2 y'^2 + \\frac{\\det A_Q}{\\det A_{33}} = 0\n", "3f016ca60249fb56c77a10d2edce2e75": "E = {1 \\over 2} \\left |\\partial \\left (\\Phi e^{iqAx} \\right) \\right |^2 = {1 \\over 2} q^2\\Phi^2 A^2.", "3f017a2d5231040a6f554b32913cd648": "\n\\mathit{RD} = \\frac{\\rho_\\mathrm{substance}}{\\rho_\\mathrm{reference}}\\,\n", "3f018dd08747f6e00a8aeac34a7b2f7b": "\\begin{align}\nP&=\\frac{2V^2}{X}\\sin\\left(\\frac{\\delta}{2}\\right)\\\\\nQ&=\\frac{2V^2}{X}\\left[1-\\cos\\left(\\frac{\\delta}{2}\\right)\\right]\n\\end{align}", "3f018f31eb22244075c4ccf66caa3779": "\\alpha_{1 \\dots K}", "3f019cc47c9ed427c4716f137ec9c613": "e^{i(\\mathbf {k}_{i}\\cdot\\mathbf {r} -\n\\omega_{i}t)}", "3f01b50c85d53c28833663bc4bd07865": "V^*(s) = \\sup \\limits_\\pi V^{\\pi}(s).", "3f01eb0f2fc40d2e04917771acd00e11": "\\tfrac{1}{3}\\scriptstyle{\\left(1+\\sqrt[3]{19-3\\sqrt{33}}+\\sqrt[3]{19+3\\sqrt{33}}\\right) \\approx 1.83929}", "3f0257ab9c6f67281416bb01c422b5db": "\\Gamma_n.", "3f02b48917934f9fc498495924865e1f": " d\\eta+\\tfrac{1}{2}[\\eta,\\eta]=0. ", "3f02f2ac61a6c54ace6d534cad372822": " [A,B] := \\mathcal{L}_A B = - \\mathcal{L}_B A.", "3f03396e6f011d5198e110d836c1ca4d": "\nd(e_{i_1...i_p}) := \\sum _{j=1}^{p}(-1)^{j-1}x_{i_j}e_{i_1...\\widehat{i_j}...i_p}.\n", "3f033dcd45a079427bc80771ef253ea6": "\nJ_G \\left(\\mathbf{x}^{(0)}\\right) = \\begin{bmatrix}\n 3 & 0 & 0\\\\\n 0 & 2 & 0\\\\\n 0 & 0 & 20\n\\end{bmatrix}\n", "3f0346d32c9306461f8dbbc3d9d2c119": " f(x, \\beta) = \\sum_{j = 1}^{m} \\beta_j \\phi_j(x)", "3f036dbdda72c7c020a7afe7c252373a": "s' = \\sum_{i}X{_i^?}", "3f038456fb9d77026eb35ac802550668": " k_{f_2} ", "3f03acdbd3d49d286e819fd28b8023a7": " t_2 ", "3f03b3152e7e12916883c6607242e89d": "P(\\mathbf{x})", "3f03cdff73a9c80ff34a4e1ba5827d83": " g(x; t)", "3f03e54f5d2d7f21c64aa86548652e96": "M_{xy}(t) = M_{xy}(0) e^{-t/T_2} \\,", "3f03ee055310178b295375bf6bb7c4fc": "\\left(\\frac{\\partial U}{\\partial N_i}\\right)_{S,V,\\{N_{j \\ne i}\\}}=\\mu_i", "3f042417eb1c92f0b6edc26ac4bbc52f": " \\boldsymbol{\\nabla} \\left( \\frac{\\partial\\varphi}{\\partial t} + \\tfrac12 \\boldsymbol{u} \\cdot \\boldsymbol{u} + \\frac p\\rho \\right) = \\boldsymbol{0}. ", "3f0476a04be10592bdbb6a6795bccbd2": "=\\beta^{s}[sb/(t+sb)]{_2\\text{F}_1}(s+1,(t/b)+s;(t/b)+s+1;1-\\beta), ", "3f04a28aae695d52ded38386e0e86297": "e^{iz \\sin(\\phi)} = \\sum_{n=-\\infty}^\\infty J_n(z) e^{in\\phi},\\!", "3f04a9c5a25464d0fc9ba421e58642b5": " X ~\\sim \\Gamma(k_1,\\theta) \\mbox{ and } Y \\sim \\Gamma(k_2,\\theta) \\,", "3f04d56ac313934f1f3ef74c5794ba9d": "L=s(s\\otimes s)^T+n(s\\otimes n)^T+s(n\\otimes s)^T+n(n\\otimes n)^T,", "3f04e6f0e396704b7fdb4714876ba70b": "H\\Phi_0(x)=E_0\\Phi_0(x)", "3f04ff676d858ced50e01b186a2c5610": "g_p < Cp^{\\frac{1}{4}+\\epsilon}.", "3f054ae4ae88129eee443d07f91785bc": "p^* \\leq p' \\leq p", "3f05d95267d9e1d7d663b8430a82d5c0": " J_i = \\sigma_{ij} E_j \\,\\, \\rightleftharpoons \\,\\, E_i = \\rho_{ij} J_j . \\,\\!", "3f05eaf7777352530dee2987f1d6b81d": "\\beta_k = m_{k+1}/\\Delta_k", "3f06801aa94559b1795fb4c761fcea9e": "A^{(12)}", "3f0689a426807165d97b27db519aed50": "e_i+e_j", "3f06b09f390b4146646c9099cc542a58": "\n\\begin{align}\n\\Delta_i &= \\frac{1}{60 dx} (y_{i+3} - 9y_{i+2} + 45y_{i+1} - 45y_{i-1} + 9y_{i-2} - y_{i-3})\n\\end{align}\n", "3f06b84d620a7ece90b640a003c5a72f": "S_{+}=S_x+i\\cdot S_y", "3f06b89589b4c1403cf0a6d92747cc90": "\\mathfrak{H}", "3f06bd74c2f6ff131b69094583229e8b": "\\varphi*\\psi^*", "3f06c79ea3dbfdac6e8c23421719aae0": "\\forall xyz\\, [ C(x;yz)\\rightarrow C(x;zy) ],", "3f0737dc33be850b1635893f439181f0": "\\Delta E_{int} = E(A_{1}, A_{2}, .., A_{N}) - \\sum_{i=1}^{N} E(A_{i})", "3f073a14113c88b4cda0c57d2e5a5e44": "\\sigma_{a \\theta v}( R ) = \\{\\ t : t \\in R,\\ t(a) \\ \\theta \\ v \\ \\}", "3f07652a3975d015718f2b8898bd9cc3": "\\mathcal{L}=-\\frac{1}{4g^2} \\operatorname{Tr}[F^{\\mu\\nu}F_{\\mu\\nu}]+{1\\over 2g^2} \\operatorname{Tr}[BB]-{1\\over g^2} \\operatorname{Tr}[BG]-{\\xi\\over g^2} \\operatorname{Tr}[\\partial^\\mu b D_\\mu c]", "3f076e94b1ef479c3196ca8ed87a5cfb": "\\sigma_{n,d}(t), \\; d=1\\ldots D", "3f0780ea8a827e0977b50b7eb7766d0f": "A[i,i] + A[r,s]\\le A[i,s] + A[r,i]", "3f078b10e09ea619eafd3c3c62427d06": " n = t | m - T |^{ - 2} a m^b ", "3f078fae8073427b258f1530c3132d57": "x = g(y) \\in A_0", "3f07b835210c57dd11b839310010a9ff": " \\log \\left( \\frac{\\Pr(\\text{Sense}_A | \\text{Collocation}_i)}{ \\Pr( \\text{Sense}_B | \\text{Collocation}_i)} \\right) ", "3f07cbbb57b5eb1b7e34ffd9dc6fc5d9": " (\\tilde{\\nu}) ", "3f080b2905f303b4cb42b6eb5ba4c058": "[\\mathfrak{g},I]\\subseteq I,", "3f081a24e1d122f48d704c5a2eb80b70": "\\left\\{\\, \\left\\{\\, w \\in X \\mid f(x)=f(w) \\,\\right\\} \\mid x \\in X \\,\\right\\}.", "3f0839072d6f6477aec69bf3e990f170": " x_1=0", "3f08dcc317957cd8305972b6b59e1d89": "D_{2}\\,\\!", "3f0956382ea1ae49bab735231f37870d": " \\left(\\mathbf{A}+\\mathbf{B}\\right)\\times\\mathbf{C}=\\mathbf{A}\\times\\mathbf{C}+\\mathbf{B}\\times\\mathbf{C} ", "3f09c1f36201d2811881ace61b69e81d": "|L_i|+\\frac12|L_{i+1}|+\\frac14|L_{i+2}|+\\cdots+\\frac1{2^j}|L_{i+j}|+\\cdots,", "3f0a2ebf4416e1f7408b9f2c981bb99c": "P \\underline{\\land} Q", "3f0a33a658444e04461bde39aef3886e": "P_H", "3f0a5433414aeebb827e0298a4a60cf1": "\\dot{x}(t)=Ax(t)+Bu(t)\\, ", "3f0ab79479f9fbefdd29a676a31d8948": "\np= {nh \\over 2L}\n", "3f0ac07f2faeaaadb1137fcea4bb5789": "c=yxxzyyzxyzzyx", "3f0ae9f91ca07a650fa0e01b320e43c6": "cov(w_i, z_i) = az(1 - z) - (a + b)z^2(1 - z) = z[1 - z][a - (a + b)z]", "3f0b1fd0416fd0bbb17f35c101dd266b": "p_3=\\frac{100}{5}\\left(3-\\frac{1}{2}\\right)=50.", "3f0b2400da58aaa4e88e438e1529a7d6": "\\mathbf{A}^{-1}=\\mathbf{Q}\\mathbf{\\Lambda}^{-1}\\mathbf{Q}^{-1}", "3f0bbff909caaa2178bf035d30e90966": "f^{n_{k+1}-n_k}", "3f0bc45083916a201bf3583b322cc03e": "y''-4y'+5y=0", "3f0bcd69ba81a710411390023ae0ae4d": "\\omega^\\alpha = \\alpha", "3f0c1d4ed83c977a5bb1ae71d10c726e": "\\begin{align}\n \\lim_{x \\to a} \\frac{(x - a) P_1(x)}{P_2(x)} &= \\lim_{x \\to 1} \\frac{(x - 1) (\\gamma - (1 + \\alpha + \\beta)x)}{x(1 - x)} = \\lim_{x \\to 1} \\frac{-(\\gamma - (1 + \\alpha + \\beta)x)}{x} = 1 + \\alpha + \\beta - \\gamma \\\\ \n\\lim_{x \\to a} \\frac{(x - a)^2 P_0(x)}{P_2(x)} &= \\lim_{x \\to 1} \\frac{(x - 1)^2 (-\\alpha\\beta)}{x(1 - x)} = \\lim_{x \\to 1} \\frac{(x - 1) \\alpha \\beta}{x} = 0\n\\end{align}", "3f0c4e01e01e22b5a33f376b8f93fb91": " \\lim_{n \\to \\infty} \\left( 1 + {a \\over {1 + n}} \\right)^n = e^a ", "3f0cc6ee15ff326bf98ec19ec70f3b12": "v_{1,1} = 10", "3f0cdb32d28c385975ddbc70a909f955": "f = \\mathrm{d} N/\\mathrm{d} t \\,\\!", "3f0ce4c42641c623128af889449bff4a": " u \\geq u_{mf} ", "3f0cff38f57e1647e300b9a3c40d7600": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = 1 + \\frac{\\sin^2\\theta}{\\cos^2\\theta}\n = \\frac{\\cos^2\\theta + \\sin^2\\theta}{\\cos^2\\theta}\n = \\frac{1}{\\cos^2\\theta}\n = \\sec^2\\theta \\, .\n", "3f0da95fe7c8a793f0e774c0b20a276b": " \\tau \\subseteq\\gamma \\subseteq \\sigma", "3f0e9d384aa88a9dd6c2d51ebd28518b": "\\scriptstyle \\operatorname{Var}[ X] = \\frac{\\alpha}{\\beta^2}", "3f0ec860b1ca0a584000db91f9ef76ef": "I(\\nu, T)", "3f0f1152af7fe421b469f0139b8db722": "w(\\cdot)=w(z_{0},\\Omega,\\cdot)", "3f0f655c6c7d060020fddfa516177c30": "X : T \\times \\Omega \\to \\mathbb{R}^{n}", "3f0fa5ccc7bc3d0dcf179a7ae5a3fb8c": "M \\to_G M'", "3f0fac87d626947b0caf9caae1898021": "ug = g^{-1}u", "3f10c942491c83052ff09064fd6c0bcd": "|\\lambda|", "3f10ca40ea1d3e18ccdb24aca019bd88": "\\sigma_{ij}=\\frac{\\partial W}{\\partial\\varepsilon_{ij}}", "3f10d77c34cf9e83c82b81dfb2f75275": "C_n\\supset C_{n+1}", "3f10dac112fa0e27ce452ed2bed3d8f9": "a_{i},a_{j}", "3f10ff2270c3b3e1d78be602e52bcaa1": "\\Xi(\\alpha)", "3f1122789d75049c2d0f342a462b2bcd": "d(a \\cdot b) = (da) \\cdot b + (-1)^{|a|}a \\cdot (db)", "3f1173bd709f187732c8b2d9d696d1de": " A_r f(e^{2 \\pi i x}) = \\sum _{k \\in \\mathbf{Z}} f_k r^{|k|} e^{2 \\pi i k x}.", "3f119239f61c1f0702d8abc290c80749": "s_{zj}=1", "3f11b18001455e13fb6efb6182b545b0": "\\exp\\colon \\mathfrak g \\to G", "3f1213230b327d0516acf599a75c94c2": "\\mathbf{\\Sigma}^1_n", "3f1226c019a954d37d6af70896412af8": "{{\\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{A}} \\over {\\mathbf{U}_n^T \\mathbf{A} + V_p C_p}}", "3f12f733f4001a38f8534bfc6cbcf0ec": "A^{2-} + H^+ \\rightleftharpoons HA^-; K_1=\\frac{[HA^-]}{[H^+][A^{2-}]}", "3f1301252568868f653c97507ab5749f": "\\alpha=1/9", "3f1323fc56796cd586903cc6eef4a3a3": " [M]_{C}^{B} ", "3f132fdfd41d1919fba964b4342bb588": "\\hat{d}^{\\dagger 2}", "3f13d61a04f0ba2ff55533a116162568": "{}^{14}_8 \\text{O}_6 \\rightarrow {}^{14}_7 \\text{N}^*_7 + \\beta^+ + \\nu_\\text{e}", "3f13e6a83e0c56a5cc30fd662b920408": " \\sigma_\\theta = A + \\dfrac{B}{r^2} \\ ", "3f13ee243c4f617a8066e13e72ee0dea": "\\lfloor \\nu-1 \\rfloor", "3f140192d223c6cdd8f8698aa19bc0bb": "P_{\\,a\\,CO_2}", "3f140d93ded8a9e9890627c14071ed9c": "\ng^{\\alpha\\kappa} = \\frac{1}{6} \\varepsilon^{\\alpha\\beta\\gamma\\delta} \\varepsilon^{\\kappa\\lambda\\mu\\nu} g_{\\beta\\lambda} g_{\\gamma\\mu} g_{\\delta\\nu} / \\det(g)\n\\,.", "3f146f4b6a4dddebb434187e8eb5e0b2": "\\mathit{alg}(A_2,B_2)", "3f14c7377b0bf6d55e30f12231a163b8": " \\qquad \\rightarrow \n\\alpha^2 |0 \\rangle_A |0\\rangle_B |A_0\\rangle_C + \\beta^2 \n|1\\rangle_A |0\\rangle_B |A_1\\rangle_C+ {\\sqrt 2} \\alpha \\beta |\\Phi \\rangle_{ABC}. ", "3f156cb35686098b0f26343abe629361": "\\left\\{{3'\\atop5'}\\right\\}", "3f157fdf9fd4254fa305e9664f71536d": "f(x,q_{k+1}) \\leq r_{k+1}\\,\\!", "3f1589131dec66ffb43e6279d34f9ef0": "\n\\wp(z) = e_{3} + \\frac{e_{1} - e_{3}}{\\mathrm{sn}^{2}\\,w}\n= e_{2} + \\left( e_{1} - e_{3} \\right) \\frac{\\mathrm{dn}^{2}\\,w}{\\mathrm{sn}^{2}\\,w}\n= e_{1} + \\left( e_{1} - e_{3} \\right) \\frac{\\mathrm{cn}^{2}\\,w}{\\mathrm{sn}^{2}\\,w}\n", "3f15a2c9fd1ff87ad641dec335be3fc5": "\\lim\\limits_{n\\rightarrow\\infty}\\prod_{i=0}^na_i,", "3f15f3003c71b488c76f88794c41a6a1": "[1.25] = 1", "3f1600cebbabb9fb2d727d48c5641240": "\\sqrt{b^2 - 4 a c} = \\sqrt{200^2 + 4 \\times 1 \\times 0.000015} = 200.00000015...", "3f160aa2de21e182429d74a9ff14ba32": " A[x] ", "3f1652093ee26b00fa4de934122b1e8d": " \\xi = \\begin{Bmatrix} \\omega \\\\ q \\times \\omega \\end{Bmatrix}.", "3f167543abc37fa0e83df4e0a377bf41": "Z_{2,t}", "3f16863e3666ec060787d37a205cf73f": "\\mathbf{f} = \\mu (\\mathbf{v} \\times \\mathbf{H}) - \\frac{\\partial\\mathbf{A}}{\\partial t}-\\nabla \\phi ", "3f175b32c340bbb2f6f3f07fb652e019": "m \\setminus S=\\{u\\in M \\;\\vert\\; mu\\in S \\}.", "3f181a42b55f7f0186759559f3977b5e": "2^{32} \\times 2^{32}", "3f18265b7d847b0a6680a91da7e388d1": "|\\alpha|=\\alpha_1+\\ldots+\\alpha_n", "3f184454d26ac9e2127c98bb5753056e": "Q_{dump}=P_{amb}*K_{bellows}*K_E*V_{O_2}", "3f185050943228a7f0f2524785187da7": "f(x;k,\\lambda, \\theta)={k \\over \\lambda} \\left({x - \\theta \\over \\lambda}\\right)^{k-1} e^{-({x-\\theta \\over \\lambda})^k}\\,", "3f18607481d2abd5ec7295f36c9aac83": "T_\\mathrm{e} = (L-1)T", "3f18b2dd3df3bc1944971731659c3f19": "E_{i+1}=E_{0}+R\\,", "3f1919adb573f60dcde30932fcc1dcfe": "(\\rho, \\phi, z)", "3f194c646d3b86bf09308765dce77f72": "\\operatorname{tr} \\mathbf{M}_{\\mathbf{Y}} (\\theta)", "3f19735449cd785715ebda6df18527e8": "\\Delta w = \\mu \\,", "3f19bc240d28a05b23495097f9bd0610": "F(x) = \\frac{1}{2}|x|", "3f19bc6d155554c53ab9e29a3be02873": "\\mathbb{P}(x \\in X) = 1 - \\mathbb{P}(x \\not\\in X).", "3f19bda9ec9f0ac154febd4183a45e67": "\\bar{x} = (x_0, \\dots, x_{k-1})", "3f1a12f77030f4e4a809f148b45fd041": "T/T_D", "3f1a23483f836fd37220e70ed49b2e43": "[u][v][w]", "3f1a6f1de6f5e977fe196c9b6bde90e4": "\\cos\\psi=\\cos(\\beta_g)\\cos(\\beta_e)\\cos(\\alpha_g-\\alpha_e)+\\sin(\\beta_g)\\sin(\\beta_e)", "3f1a93837e40d67c276dc6606b46a556": "\\psi_a", "3f1ac17ca5f5b85d806b61833ea1634e": "f(x) g'(x)", "3f1aebf448aae512be1aeecd958d5ac6": "\n\\int\\limits_{-a}^{a}dx\\phi _{m}^{\\mathrm{even}}(x)\\phi _{n}^{\\mathrm{odd}\n}(x)=0.\n", "3f1af597039bfe6c136295a8e29a3bc2": " \\left [ 1+2\\sum_{n=1}^\\infty \\frac{\\cos(n\\theta)}{\\cosh(n\\pi)} \\right ]^{-2} + \\left [1+2\\sum_{n=1}^\\infty \\frac{\\cosh(n\\theta)}{\\cosh(n\\pi)} \\right ]^{-2} = \\frac {2 \\Gamma^4 \\left ( \\frac{3}{4} \\right )}{\\pi} ", "3f1b0b1030cc194ec16808465bb621c8": " \\sigma_i(x_1,\\ldots,x_{i-1},x_i, x_{i+1},\\ldots, x_n)=\n(x_1,\\ldots, x_{i-1}, x_{i+1}, x_{i+1}^{-1}x_i x_{i+1}, x_{i+2},\\ldots,x_n).\n", "3f1b397029a64f03dd9bfdfcb40c72c0": "O(\\varepsilon \\sqrt{n})", "3f1b41d43462ad81dec5bc117ea13631": "\\mathbf{k} / \\mathbf{N}", "3f1b9c85b1179ae427b43ea6e09304dd": "\\mathbf{A} = \\begin{bmatrix} T( \\vec e_1 ) & T( \\vec e_2 ) & \\cdots & T( \\vec e_n ) \\end{bmatrix}", "3f1ba7d98f4276d4cf6f9d9e2b523e75": "\\Psi_{0} = \\psi_{1s} (1) \\psi_{2s} (2) - \\psi_{2s} (1) \\psi_{1s} (2)", "3f1bb974576c9e82834cfbffe4126324": " 2c_t r_t ", "3f1bc087b35167ff1ed15b5e5ba5c911": "\\nabla_X v=dv(\\bar{X})+\\eta(\\bar{X})\\cdot v", "3f1bd67e56743345295592449514d30a": "\\left[\\frac{1} {4}\\left(P_1^\\alpha+P_2^\\alpha+P_3^\\alpha+P_4^\\alpha\\right)\\right]^{\\frac{1} {\\alpha}}", "3f1bf9d25d81baf65565dd8c6394ac69": "\\textstyle \\alpha_{transition}", "3f1c0a3b6caeb56159595e48211f09e4": " R_n = \\left\\{ x = \\sum_{i=r}^{n-1} b_i x^i | b_i=0, \\ldots, p-1 \\text{ for } r 0", "3f2b6fcb0cda6f5b8eed9d9ef76a5b96": "s_{Tx}= \\lVert v_{T}\\rVert \\cdot \\sin(\\theta_{AOB})\\,", "3f2bc0e0df742981a60aa4a0208b6850": "\\mathbf{K} =\\boldsymbol{\\Upsilon}^T \\boldsymbol{\\Upsilon} ", "3f2bd36c5450d041b5389cb88feec451": "\\! \\rho = \\rho_mc^2", "3f2c55d49d3f67534e3f6e669670e469": "10^{10^{10^{10^{10^{4.829}}}}}", "3f2cc38d86712479b984ed9b93ec6d53": "\\begin{align}\n\\frac{d^2y}{dt^2}&=\n\\tfrac12\\left(\\frac{1}{y}+\\frac{1}{y-1}+\\frac{1}{y-t}\\right)\\left( \\frac{dy}{dt} \\right)^2\n-\\left(\\frac{1}{t}+\\frac{1}{t-1}+\\frac{1}{y-t}\\right)\\frac{dy}{dt} \\\\&\\quad +\n\\frac{y(y-1)(y-t)}{t^2(t-1)^2}\n\\left(\\alpha+\\beta\\frac{t}{y^2}+\\gamma\\frac{t-1}{(y-1)^2}+\\delta\\frac{t(t-1)}{(y-t)^2}\\right)\\\\ \n\\end{align}", "3f2d10d614bd6a4d20cdf608c452def7": "S(\\boldsymbol \\beta^s+\\alpha\\Delta) < S(\\boldsymbol \\beta^s)", "3f2d25f82de3d5242ed2e398b7997685": "\\frac{dU}{dt} =-\\frac{U}{RC}+\\eta \\left( t\\right),\\;\\;\n\\left\\langle \\eta \\left( t\\right) \\eta \\left( t^{\\prime }\\right)\\right\\rangle = \\frac{2k_{B}T}{RC^{2}}\\delta \\left(t-t^{\\prime }\\right).", "3f2da9ee439646cfb26b82067dbd1675": "\\nu_e = 2.91 \\times 10^{-6} n_e\\,\\ln\\Lambda\\,T_e^{-3/2} \\mbox{s}^{-1}", "3f2dae004b957e2489abd538ba837193": "{\\mathbf S}={\\mathbf P}\\times_{\\kappa}\\Delta_n\\,", "3f2e5a7817c44ccc52e7a187b260cda7": "\\begin{matrix} \\frac{m}{s} \\end{matrix}s = m", "3f2eb0f0a892994afc6d082cbec60114": "d \\theta = \\frac{h}{r \\sqrt{(n(r))^{2} r^{2} - h^{2}}} d r", "3f2ec123a0db5eaa791005512cad3bc7": "F : \\text{Hom}_R(M,N) \\to \\text{Hom}_S(_SM,N)", "3f2f1f7e39a7cc42228933decb9d830d": "R[t] = \\left\\{ a_n t^n + a_{n-1} t^{n -1} + \\dots + a_1 t + a_0 \\mid n \\ge 0, a_j \\in R \\right\\}", "3f2f27b3e68beae5afd2af97afdde7d7": "\\displaystyle\\gamma^\\mu\\gamma^\\nu\\gamma^\\lambda = \\eta^{\\mu\\nu}\\gamma^\\lambda + \\eta^{\\nu\\lambda}\\gamma^\\mu - \\eta^{\\mu\\lambda}\\gamma^\\nu - i\\epsilon^{\\sigma\\mu\\nu\\lambda}\\gamma_\\sigma\\gamma^5", "3f2f835d6d1c51ed0205328b1477b3e9": "x_i \\prec x_{i+1}", "3f2ff59d56a730e391665c7fb5d22872": " i|f(x_{i}) = y_{i}", "3f300e4402b1cad698c370864f44da09": "\\rho_i =m_i n_i", "3f30a282c8ca96e0bd9445aebfea19f8": "\\tfrac{\\alpha}{\\beta} X \\sim \\beta^{'}(\\tfrac{\\alpha}{2},\\tfrac{\\beta}{2})\\,", "3f30fc35f5abc1833045adc165c8db46": " \\Omega \\left( \\big[ \\rho (R) \\big] ; \\mu , T \\right)", "3f315c77151d3c496399208f8531cab2": "R(s,s) \\leq (1 + o(1))\\frac{4^{s-1}}{\\sqrt{\\pi s}}.", "3f31771943e5380e2b6d1e6abd20507d": "\nr^\\ell\\,\n\\begin{pmatrix}\n Y_\\ell^{m} \\\\\n Y_\\ell^{-m}\n\\end{pmatrix}\n=\n\\left[\\frac{2\\ell+1}{4\\pi}\\right]^{1/2} \\bar{\\Pi}^m_\\ell(z) \n\\begin{pmatrix}\n(-1)^m (A_m + i B_m) \\\\\n\\qquad (A_m - i B_m) \\\\\n\\end{pmatrix} ,\n\\qquad m > 0.\n", "3f31e55416cabbe4a3ad926788cd840a": "\\mathrm{DAOE} = \\frac{\\mbox{NOPAT (t) - WACC x NOA (t-1)}}{\\mbox{WACC}} + {\\mbox{NOA}} - {\\mbox{BVD}}", "3f320bdaa5d378e44cea8ef596189613": "\\Phi_{\\alpha,\\beta}(u)", "3f3231b436a2a4a253410a8d3d6fc9e7": "\\mathcal{F}\\mathcal{F}^{-1} = \\mathcal{F}^{-1}\\mathcal{F} = \\operatorname{Id}_{\\mathcal{S}'(\\mathbb{R}^n)}.", "3f32330466f5b0d8f1feac4a3272f081": "3ts + 9t + 5s", "3f3249170042103ed3d41035b865a30a": " X_1,X_2,\\dots : \\Omega \\to \\mathbb{R}, \\,", "3f32986679b7c833dc8292709a71dc02": "\\mathbf{E} = \\mu \\mathbf{v} \\times \\mathbf{H} - \\frac{\\partial\\mathbf{A}}{\\partial t}-\\nabla \\phi ", "3f32d4965341acb96d75fa236db5060a": "N_{nl} = \\left[\\left(\\frac{2Z}{na_0}\\right)^3 \\cdot \\frac{(n-l-1)!}{2n[(n+l)!]^3}\\right]^{1 \\over 2}", "3f32f8129fc9058223b180103e610cab": "N_{C} > N_{V}", "3f339a8608a8f408715b452e1517dc9e": " V(x):= \\sum_{i\\in I} v_i(x),", "3f33e26f195be9803a18ca735093b208": "\\beta\\ = f_7(\\eta,\\phi),\\,", "3f341d9dfd4cb5f0a33e10ace184e396": "PVx", "3f3423ca353c0ce9c26d2438d02740e3": " r_1, \\ldots, r_n ", "3f343169a67ebe8024d6b3e79e14ec6a": "\\frac{I}{I_S}", "3f3544e4912032fcd7b73b0cc3005e6c": "\\theta(z;q)=\\theta\\left(\\frac{q}{z};q\\right)=-z\\theta\\left(\\frac{1}{z};q\\right). ", "3f358c0a422e57f05fc5dd98209b8c16": "r(f,D)=\\inf_{z\\in \\partial D}\\frac{|f(z)|}{\\|q(z)\\|},", "3f3599d7b9dd2ff243a7ad321b80c63f": "X=(X_t,\\ t\\in T)", "3f359d3da035ad6875d623b3349fb8c5": " x \\sim y \\iff x\\le y \\land y \\le x", "3f35f195b49890ed844bcda21ae2b7fe": "x\\wedge (x\\vee y) = x = (y\\vee x)\\wedge x ", "3f364b2b9e53163e3b511a0c44e4db12": "\\Omega_n(k)", "3f365e0da35c27a8786efc067599c011": " \\mathbf{a} = \\mathbf{b} ", "3f3667bf1e43e797e244ead38de1a95d": "\\dim(V_\\Lambda) = {\\prod_{\\alpha \\in \\Delta^{+}}(\\Lambda+\\rho,\\alpha) \\over \\prod_{\\alpha \\in \\Delta^{+}}(\\rho,\\alpha)}", "3f36ad913ad8ffae1bc263421ba4e18a": "f(b)-f(a)", "3f36f75b0a54e03fce03fa52b1c47d85": "\\psi(\\varepsilon_{\\Omega+1})", "3f3703165a6dda071c531c0323454ad6": "|\\alpha|=q^{(d-1)/2 }.", "3f370b86de9043589be9278661078907": "\\mathbf{N}(s) = \\frac{\\mathbf{T}'(s)}{\\|\\mathbf{T}'(s)\\|}.", "3f37134d5de24cc387701d98bce48737": "f(tx_1+(1-t)x_2)\\leq t f(x_1)+(1-t)f(x_2).", "3f37b071bf4bb583900152db68346700": "\\mathbf{w}^T \\mathbf{x}", "3f37ddfd153b66756912467ed014f3a9": "\\hat{\\mathbf{\\imath}}", "3f383d5180fa32e67a5b70b6cea80123": " \\int\\!\\!\\!\\!\\int_{S_1} \\mathbf{j}_1(\\mathbf{r},t) \\cdot d\\mathbf{S}_1 = \\int\\!\\!\\!\\!\\int_{S_2} \\mathbf{j}_2(\\mathbf{r},t) \\cdot d\\mathbf{S}_2 ", "3f38512460aa79316fbe6bdca6fd8c8c": "f_A: \\varnothing \\rightarrow A.", "3f3904b98f4bf3451af4102cf830634c": " \\rho_X(t) = \\sup \\Bigl\\{ \\frac{\\|x + y \\| + \\|x - y\\|}{2} - 1 \\,:\\, \\|x\\| = 1, \\; \\|y\\| = t \\Bigr\\}. ", "3f3921cf989260090dfda03fff68de77": " U^* ", "3f397536237b1e9b2ad792d96cb70841": "\\frac{\\partial M_r}{\\partial c}=\\frac{(c)_r(c+1-\\gamma)_r}{\\left( (c+1-\\alpha )_{r} \\right)^{2}} \\sum_{k=0}^{r-1} \\left( \\frac{1}{c+k}+\\frac{1}{c+1-\\gamma +k}-\\frac{2}{c+1-\\alpha +k} \\right) ", "3f397decebb6d86fe4a24a2a71f0d282": "( I_p- C_a ) / C_d = 0.71 N_x + 0.71 N_z ", "3f39c87b2893a8716a49ab99d8068b6c": "S_0 = 2", "3f39ce3d7ee0c2455874c20472e3e9dc": "\\int_{x=-\\infty}^{\\infty} \\int_{y=-\\infty}^{\\infty} g(x, y, t) \\, dx \\, dy = 1 ", "3f39e4137dc1a9234df7ff80c2c90421": "{\\rm Shi}(x) = \\int_0^x\\frac{\\sinh t}{t}\\,dt = {\\rm shi}(x).", "3f39f6db2bbe9a25dab1c79b8a6ddcb4": "\\hat V_s", "3f3a4ebf25dd67d77244c7ddf824cc84": "d \\Xi = - U d \\frac {1} {T} - V d \\frac{P}{T} + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "3f3a6696c8ec61a28b1c8695965e0d55": "G(x, z; \\lambda) = \\sum_{i=1}^n \\frac{e_i (x) f_i^*(z)}{\\lambda - \\lambda_i}.", "3f3a9a2821cbb55ccf58298b2aaf6fb1": "\\int\\limits_0^{1}\\! \\frac{\\ln\\ln\\frac{1}{x}}{1-x+x^2}\\,dx = \n\\int\\limits_1^{\\infty}\\! \\frac{\\ln\\ln{x}}{1-x+x^2}\\,dx = \n\\frac{2\\pi}{\\sqrt{3}}\\ln \\biggl\\{ \\frac{\\sqrt[6]{32\\pi^5\n}}{\\Gamma{(1/6)}} \\biggr\\}\n", "3f3a9cea851bf5cbf872039457588d89": "2s(t)\\cdot \\cos(2\\pi f_0 t),\\,", "3f3abcc7b09fc62c0fa24f5bad1cde2a": "O_{/\\sim_f}", "3f3aed56462dbd3be386556b492b0b9f": "1/i\\omega", "3f3b30221d32a40b5122e8a5691a8775": "U_G^2\\subset U", "3f3b451a9e346ba922bd1b937165c8c4": "\\mathbf{R} \\to [-1,1] : x \\mapsto \\sin(x)", "3f3b69bcc8056a2141263b21169071df": "z_{n+1} = \\gamma z_n \\left(1 - z_n\\right),", "3f3b830f9848eb32aa4e526e47603c7b": "f_i^*", "3f3bc71ea378dfafd73c27fe0dc60a3c": "d_{n-1}, d_{n-2},\\ldots, d_1, d_0", "3f3c11aa5c8de6da6d8765e21f16b6d1": "51^2", "3f3c3043613e89140e76c4b543d4fa88": "a^2+b^2+c^2", "3f3c97da047bc5c9855dcd56738c0cc7": "\\scriptstyle a \\,+\\, b \\;=\\; 0", "3f3cc1976c326a82f414bd9a07c34388": " \\langle H \\rangle = 8E_1 + \\langle V_{ee} \\rangle = 8E_1 + \\Bigg(\\frac{e^2}{4\\pi\\epsilon_0}\\Bigg) \\Bigg(\\frac{8}{\\pi a^3}\\Bigg)^2 \\int \\frac{e^{-4(r_1 + r_2)/a}} {|\\vec{r_1} - \\vec{r_2}|}\\, d^3\\vec{r}_1 \\, d^3\\vec{r}_2 ", "3f3d12c2fa0c29277fdb9b29fc6f8590": "\\zeta_m^c\\alpha", "3f3d3c9309aced97898aa68b2512d9b7": "1.5 < z < 7", "3f3d5118e374c670258e6e2b2cfb1b0c": "t = 1", "3f3d891883ab6c6e93959d34b01da6ec": "T^{i\\alpha }_{j\\beta }", "3f3df5633daa5077fd2d534643f9b439": "\\varepsilon_\\alpha=\\alpha", "3f3e2e9875946d5ab9f722a6ea38def9": "f^{*} \\in H_k", "3f3ea5c35c882f390a10e928b150311d": "\nH=\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 1 & 0\\\\\n1 & 1 & 0 & 1\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 0\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 1\n\\end{array}\n\\right] .\n", "3f3ec768778846336f6eb62cdc8fca12": "V^{\\bot\\bot}", "3f3eda6b1484bf0715f9a26ab4e17a0e": "u = \\sqrt{a^2-x^2}", "3f3edcb68afc069a5e120f2175ebf977": "R(x, \\xi) = \\sum_\\alpha {1 \\over \\alpha!} \\left({\\partial \\over \\partial \\xi}\\right)^\\alpha P(x, \\xi) \\left({\\partial \\over \\partial x}\\right)^\\alpha Q(x, \\xi).", "3f3f7f477d8890dff9c78958e4ff2762": "\\mathfrak{G}\\{\\mathcal{B}\\} = \\mathcal{E} + \\mathcal{B} + (\\mathcal{B} \\times \\mathcal{B}) + (\\mathcal{B} \\times \\mathcal{B} \\times \\mathcal{B}) + \\cdots.", "3f3f8a616b22c7984d07cf747753a0b6": "\\mathbf{R}^\\infty", "3f3fd945bdb1c8411318fd2b617da1ef": "\\theta = Ae^{-\\alpha t} \\cos{(\\omega t + \\phi)}\\,", "3f3ffa84a0350f48a03554a2fcb70499": " f = -kT \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\\log Z_N ", "3f4004da3384913bb26e5147e8db0a09": " H(m) \\, \\equiv \\, x r + s k \\pmod{p-1}.", "3f4011813e62686da2fffe1a23859ce7": "\\mathbf{X}+ \\Delta \\mathbf{X}=(X_I+\\Delta X_I)\\mathbf I_I\\,\\!", "3f4081939609aad5fcf7ba6826176bb3": "\\Gamma(x) = \\prod (1- x \\, \\alpha^{j_i})", "3f409763ed2b1e4071ee5d7c97f8c8b5": "m_D", "3f40d4662d540419aacb9817be07f7c9": "[A]_\\infty =\\frac{k_b}{k_f+k_b}[A]_0;[B]_\\infty =\\frac{k_f}{k_f+k_b}[A]_0 ", "3f41757766564061adf5f614d848ccf9": "F\\widehat M/\\widehat H\\to \\widehat M", "3f41fba3d7f3013787deadc1045a1e42": "[S_-] = \\begin{bmatrix}\n\\langle+|S_-|+\\rangle & \\langle+|S_-|-\\rangle \\\\\n\\langle-|S_-|+\\rangle & \\langle-|S_-|-\\rangle \\end{bmatrix}\n=\n\\hbar \\cdot\n\\begin{bmatrix}\n0 & 0 \\\\\n1 & 0 \\end{bmatrix}\n", "3f422e30705391fb9d8963069cd4cf94": " a^{-s_i} \\leq p_i", "3f426566a1f6347bf2d42612138de7aa": "\nf = \\frac{v}{\\lambda}.\n", "3f426c915f25ba67e92196e5264740f7": "\\theta/(\\theta+J-1)", "3f42b47d9675e6f6fb87c4026d9c50d5": "\n E=\n \n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_{\\mathit l \\mathit l^{\\prime}}^2 }\n\\;\\mathcal J_0 \\left ( \\cos \\theta \\; k \\right) \\;\\mathcal J_0 \\left ( \\sin \\theta \\;k \\right) \\;\\mathcal J_0 \\left ( k{r_{12}\\over r_{\\mathit l \\mathit l^{\\prime}}} \\right)\n", "3f4312413252068bd7b87760247e1ce2": "\\mu_5^{'}=15\\sigma^5\\sqrt{\\pi/2}\\,\\,L_{5/2}(-\\nu^2/2\\sigma^2)", "3f43acef559c6e470c11af96070cf9fa": " \\mathfrak{sp}(S_+)\\oplus\\mathfrak{sp}(S_-) ", "3f43de0a748b79ef37a76c858924a0ee": "\\mathfrak g := \\mathfrak k \\oplus\\mathfrak m", "3f4427f558c1d122f30106b8dff95919": "N(0,R)", "3f442a3ece2af34d5771d935d5484e25": "\\alpha\\colon F^\\%\\to F", "3f4464385ad24daf2e957df1c1eda172": "\n\\mathbf{r} = [r_{d,s} \\quad r_{d,r}]^T \n = [h_{d,s} \\quad h_{d,r} h_{r,s}]^T x_{s} + \\left[1 \\quad \\sqrt{|h_{d,r}|^2+1} \\right]^T n_{d}\n = \\mathbf{h} x_{s} + \\mathbf{q} n_{d}\n", "3f447000656ae652a6443aabf1655e4e": "\\frac{1}{2} \\left[(v-1)-\\frac{2k+(v-1)(\\lambda-\\mu)}{\\sqrt{(\\lambda-\\mu)^2 + 4(k-\\mu)}}\\right]", "3f44826ffbee2634a7a9330b39df50c9": "W^{24}=F_4/Spin(8)", "3f44f4a18e003d7ec43086be244e2b86": "f\\colon M \\to N", "3f453d81f6f897ab9a029854fcb09815": "dE = \\frac{\\kappa}{8\\pi}\\,dA+\\Omega\\, dJ+\\Phi\\, dQ,", "3f455973f9cc710a333dcbde4a9db57f": "(-n,m+1), (-n,m), (-n-1,m-1), (-n-1,m)", "3f456c10181cd82fdd654afe646971f6": "\\bar{R}^ 2=\\overline{z}\\,\\overline{z^*}=\\left(\\frac{1}{N}\\sum_{n=1}^N \\cos\\theta_n\\right)^2+\\left(\\frac{1}{N}\\sum_{n=1}^N \\sin\\theta_n\\right)^2", "3f45e884af4591a95babce0e3ad39f3d": " r_{\\mathrm{e}} \\ ", "3f45fe0df6f216254a93755863d214b6": "\nH^\\text{RWA}=\\hbar\\omega_0|\\text{e}\\rangle\\langle\\text{e}|\n-\\hbar\\Omega e^{-i\\omega_Lt}|\\text{e}\\rangle\\langle\\text{g}|\n-\\hbar\\Omega^*e^{i\\omega_Lt}|\\text{g}\\rangle\\langle\\text{e}|.\n", "3f4608aa9772d07a1404c8b842307211": " \n\\leq B + Vp(t) + \\sum_{i=1}^KQ_i(t)y_i(t) \n", "3f461ff1432e43ae32ce1d6d788639cf": "\\text{Prim}=V\\cup\\Sigma", "3f462775a3502f1fac396189e6e54460": " \\frac{\\Gamma(\\tfrac{1}{20})\\Gamma(\\tfrac{9}{20})}{\\Gamma(\\tfrac{3}{20})\\Gamma(\\tfrac{7}{20})} = \\frac{\\sqrt[4]{5}\\left(1+\\sqrt{5}\\right)}{2}", "3f46730487026f3f6ccb204af769a37a": "\\mbox{CR} = \\frac { \\tfrac{\\pi}{4} b^2 s + V_c } {V_c}", "3f46cc05bf6167b9fbf756d92e64ddca": "\\tilde{g}(u) = 1 - g(1-u)", "3f46e9173682e2b3c6a38b916f1f8080": " \\mathcal{A}\\left\\{x(t)\\right\\}\\ \\stackrel{\\text{def}}{=}\\ \\int_{t-a}^{t+a} x(\\lambda) \\, \\operatorname{d} \\lambda. ", "3f47176f95ad16d6850fb5df8f3ba475": "\\frac{L}{c}\\ 0.25", "3f4755d60cda55408da80cf494b9ddb0": "\nA_3^2 = (A_1 \\cos\\theta_1 + A_2 \\cos \\theta_2)^2 + (A_1 \\sin\\theta_1 + A_2 \\sin\\theta_2)^2,\n", "3f47642401741824e953e9b1a8d7931b": "\\int\\frac{\\sin ax\\;\\mathrm{d}x}{\\cos ax - \\sin ax} = -\\frac{x}{2} - \\frac{1}{2a}\\ln\\left|\\sin ax - \\cos ax\\right|+C", "3f4774c63553897d77a06dc9aedf023b": "\\,f_0", "3f478df156e0ad96658894154c667044": "\\frac{1}{3 \\cdot 11}", "3f47da0380f0fe6f0862c6deb3b6e7fd": "i = 2, \\dots, n", "3f47ed030e660d2001a903f6ffbb2dda": " \\frac{d}{dt} \\frac{\\partial T}{\\partial \\dot{q}_j} -\\frac{\\partial T}{\\partial q_j} = -\\frac{\\partial V}{\\partial q_j}, \\quad j=1,\\ldots,m.", "3f47f5f595f8a67bf825ffca6fdbbca6": "0 \\cdot x^{-1}", "3f47fdda504e0cf42c906012cd01c0f9": "\\nabla B_z", "3f4812a686d41802983cd215100a316b": "\\beta \\twoheadrightarrow \\gamma", "3f485003f9fd34a28e1b50a66642af42": "\\Delta u = \\frac{1}{2} \\log \\left( \\frac{y}{z} \\right)", "3f48eb48a1526140c4961ca1a684519d": "(\\bar{x} \\vee z)", "3f48fca1c472393bd377191711563e09": "AdjFactor", "3f490c7e9c6b6323ee218de61b6e666f": "\\left|G_n(z) \\right||^{2}=Z_{DP}^{2}\\frac{\\hbar \\omega _{q}}{2V\\rho c^{2}} (N_{q}+\\frac{1}{2}\\pm \\frac{1}{2})\\delta _{k', k \\pm q} \\; \\; (15)", "3f4cb07449f6b1ec2347878340d03434": "f = \\frac{1}{2\\pi\\sqrt{LC}} \\,", "3f4ceb7dde91b5b4f7fd61ac62f6a325": "x \\notin S", "3f4cfb938c1a581695fc9602d61a8744": "A_T \\cong \\bigoplus_{p\\in P} A_{T_p}.\\;", "3f4d3e22ff1ab144c565d07424d207f5": "\\zeta(a,a,a)=\\tfrac{1}{6}(\\zeta(a))^{3}+\\tfrac{1}{3}\\zeta(3a)-\\tfrac{1}{2}\\zeta(a)\\zeta(2a)", "3f4db81e384fe93d7c576d82ce499973": " \\frac{\\Delta m}{A}\\ = \\frac{N_q \\rho_q}{\\pi Z f_L}\\tan^{-1} \\left [ Z\\tan \\left ( \\pi \\frac{f_U-f_L}{f_U} \\right ) \\right ] ", "3f4e3f7f5a9893bd74ac6c6aba6e2b15": "\\Delta\\theta = \\theta_1 - \\theta_2", "3f4e700d3c35a13579a789ef8d62a0f8": " L_u=10 \\log_{10}\\left(\\frac{{u_{\\mathrm{{rms}}}}^2}{{u_{\\mathrm{ref}}}^2}\\right)", "3f4e7a267815dfb483be12a1dff4137b": "\\scriptstyle \\frac{6}{\\alpha}", "3f4e93612bfd8225c8a50fb7349f82ba": "\\mathcal F \\otimes \\mathcal L^{\\otimes m}", "3f4e97ff50ecdb567d49e0c00fd6325d": "A^{- 1}", "3f4ed24dda8ad4804270a7d0f466b5f5": "L^2(dx)", "3f4efe77720ce2e04b2a6731abf5076d": " \\text{score}(D,Q) = \\sum_{i=1}^{n} \\text{IDF}(q_i) \\cdot \\frac{f(q_i, D) \\cdot (k_1 + 1)}{f(q_i, D) + k_1 \\cdot (1 - b + b \\cdot \\frac{|D|}{\\text{avgdl}})},", "3f4f21e7bf20b89e871cb369aa608e5c": "\nG^{\\mathrm{T}}(\\mathbf{x} t|\\mathbf{x}' t') = \\int_\\mathbf{k} d \\mathbf{k} \\int \\frac{\\mathrm{d}\\omega}{2\\pi} G^{\\mathrm{T}}(\\mathbf{k},\\omega) \\mathrm{e}^{\\mathrm{i} \\mathbf{k}\\cdot(\\mathbf{x} -\\mathbf{x} ')-\\mathrm{i}\\omega(t-t')}.\n", "3f4fa9221199bb30c9b386b8b8a008fb": "C B = \\omega B C", "3f4fc2f0b0475de112a57d272c9987f3": " \\begin{align} \n\\boldsymbol{\\tau} & = \\frac{{\\rm d}\\bold{L}}{{\\rm d}t} = \\frac{{\\rm d}(\\bold{I}\\cdot\\boldsymbol{\\omega})}{{\\rm d}t} \\\\\n& = \\frac{{\\rm d}\\bold{I}}{{\\rm d}t}\\cdot\\boldsymbol{\\omega} + \\bold{I}\\cdot\\boldsymbol{\\alpha} \\\\\n\\end{align} \\,\\!", "3f4fd60d3fc979e33bf3cd3b242abec3": "\\,{}_{t|k}q_x", "3f50430dfc1af40dce69fe31c29b714f": "\n\\ln Q = N \\ln {(V-Nb')} + \\frac{N^2 a'}{V kT} - N \\ln {(\\Lambda^3)} -\\ln {N!} \n", "3f5048dfd219d606df3350c2bcc02eff": "\\lim_{x \\to a}(f(x)\\cdot g(x)) = \n\\lim_{x \\to a}f(x)\\cdot \\lim_{x \\to a}g(x),", "3f506e791baca523b4278783f0b49f58": "d_\\mu(A,B) = 1 - J_\\mu(A,B) = {{\\mu(A \\triangle B)} \\over {\\mu(A \\cup B)}}", "3f509dc121febf6f6f0497baae5ac536": "V_r = \\sqrt{\\frac {\\mu}{p}} \\cdot e \\cdot \\sin \\theta", "3f50b67154ac2bb596fb67ee940c5efe": "\\scriptstyle M'", "3f50ced4341ed8fda29fb3c78fb239df": " v_t = v_s + \\frac{\\sigma_s^2}{v_s}", "3f50f8f9f72802ef2b2330afcfcc27c6": "\\bigoplus_{H \\in A} K e_H ", "3f51737e9afdeaa1a6fc95b9c41ab2f9": "F(s) = (s-s_0)\\int_0^\\infty e^{-(s-s_0)t}\\beta(t)\\,dt,\\quad \\beta(u)=\\int_0^u e^{-s_0t}f(t)\\,dt.", "3f51856358b2251c3715b46a7da3eabe": "|(F/E)(z)|,|(F^{\\#}/E)(z)| \\leq C_F(\\operatorname{Im}(z))^{(-1/2)}, \\forall z \\in \\mathbb{C}^+", "3f51a78af7f9e6f1136df0c0b990b4ca": "\\frac{v_{\\text{L}} \\left( t \\right)}{i_{\\text{L}} \\left( t \\right)} = \\frac{\\omega I_p L \\cos(\\omega t)}{I_p \\sin \\left( \\omega t \\right)} = \\frac{\\omega L \\sin \\left( \\omega t + \\frac{\\pi}{2}\\right)}{\\sin(\\omega t)}", "3f5222fea62942ee271a376d30e32e47": "\\frac{1_2}{11_2}", "3f5246a7558d6d6433eb7411249598b5": "f^n(U) \\cap U = \\varnothing.\\,", "3f527fbf06f7f594ab8dda009d9fec2b": "\\scriptstyle f \\colon (X \\times Y) \\to Z ", "3f5293940bb68ab028c4c1435bfd1069": "O/U", "3f52acb40343caa210b246b3775978b5": " 2V \\approx \\sum_{s,t=1}^{3N-6} F_{st} S_s\\, S_t. ", "3f53680dea2058ca1f9974598cb28551": "\\{y_{t}\\}", "3f536a6ce94cbf599b4ca88ccceb8079": " W_B T_D \\ge 1 ", "3f5414d6d48a97a3076d5d027245a792": "Z = \\frac{Z_{L}Z_{C}}{Z_{L} + Z_{C}}", "3f543428ce442a7699ffaea46825920f": "\\{f(x_i) \\}_{i=1}^\\infty", "3f5439fd6ed873f0210e3884b315cef2": "\\textstyle l \\leq n-k-\\log_q (n-l)+2", "3f54d66368725a330178a9190c83c5d3": "\\frac{\\sin (x - x_i)}{x - x_i}.", "3f54e91822e4953ba7ff3e9a0e4d479b": "\\eta_{sp} = \\eta_r - 1 = \\frac{t - t_0}{t_0}. \\,", "3f5526b7e71cebe36a0afc1f0d7cc2bc": "\n x(1-x)y''+\\left\\{ \\gamma -(1+\\alpha +\\beta )x \\right\\}y'-\\alpha \\beta y=0\n", "3f553c67f0e600bf9e4f1abb66d4e328": "\\rho(x,\\theta)=-\\log(f(x,\\theta))", "3f55fefc5052648430817df569eb9f94": "(x_1 + x_2)^n = \\sum_{k_1+k_2=n} {n \\choose k_1, k_2} x_1^{k_1} x_2^{k_2};\\ \\ k_1, k_2, n \\in \\mathbb{N}_0", "3f565deefcb538e05e35356f41e81314": "\\nabla^2 \\nabla^2 w = 0", "3f570c1479aa9b91dc6d268b09806d45": "\\sigma = \\mathbf{P} \\cdot \\mathbf{n}", "3f570c7a2ff5f3d2e89d3287ba4470f2": " \\mathrm{MA} = \\frac{F_B}{F_A} = \\frac{v_A}{v_B}. ", "3f575f9bf91e1526d45a0d6721106b33": "\\hat{\\mu}_{ij}", "3f57dbecc3c0f8d8139cbfccadf2bf51": "\\mathcal{N'}_k(x), k = 1, . . . , k'_{max} ", "3f57f99f81daf9d21e635a2d750cfa2f": "r'_2", "3f58053b6da266ab8069565a4aec95b0": " \\bar{n}_i = \\frac{1}{e^{(\\epsilon_i-\\mu) / k T} + 1} ", "3f580a8375107f7ad9372442df9a82c7": "\\Delta m_{H}^{2} = - \\frac{\\left|\\lambda_{f} \\right|^2}{8\\pi^2} [\\Lambda_{\\mathrm{UV}}^2+ ...].", "3f5833edc1b5b2f9b0b46ee85a7ee165": "\\mathbf{H}\\,\\!", "3f5862bf3016736bc39c1b468b9adc47": "{d \\over dt}\\left\\{ A \\right\\} =- k_+ \\left\\{ A \\right\\} \\left\\{B \\right\\} + k_{-} \\left\\{B \\right\\}^2 \\,", "3f58904a7e1724e8a6dffd6899c2598d": "-\\mathrm{tr}\\varepsilon", "3f58f66916935aec43a26d473d76c688": " 371=3^3+7^3+1^3", "3f59820dbfd16481a76b1f50a7558cb9": "W = \\frac{\\sqrt{2m}}{h} \\frac{e^2}{2 4\\pi \\epsilon_0 (l+1)} - \\frac{h(l+1)}{r\\sqrt{2m}}", "3f59a389883d0b415b014353ef4f5a73": "\\varnothing \\setminus A = \\varnothing\\,\\!", "3f5a509b1f3dd8cd9cf958f808d9b769": "e^{TB}", "3f5a92b4a8bb563380a152aab7e91a5e": " U = P_{11} \\cdot U_{11} + (1-P_{11}) \\cdot U_{21} + P_{12} \\cdot U_{12} + (1-P_{12}) \\cdot U_{22} ", "3f5b4794a1e1acbce7044db9b7525541": "\\operatorname{ess\\inf}", "3f5b502d113129b38fe6fb2fa44f3b2e": "R_1 = R_2 = \\cdots = R_n = R_{\\text{f}}", "3f5b550d2a484dcd8e795a9a42e22ee1": "Q=(X_2 : Y_2 : Z_2)", "3f5b5f0932dc2ad1decb84a0bd9c7766": "\\displaystyle p^{\\prime}=-\\frac{\\partial H}{\\partial q} = 2pq+b", "3f5ba7571ebd13d3a2b81f68d99cd943": " f''(x_n) > 0 \\,\\!", "3f5bb8cade2f66d8000a8e4f93852dd6": "\\nabla \\left( \\frac{f}{g} \\right) = \\frac{1}{g} \\det \\begin{bmatrix} \\nabla f & \\nabla g \\\\ f & g \\end{bmatrix} \n \\left( \\det {\\begin{bmatrix} g & \\nabla g \\\\ 1 & 1 \\end{bmatrix}}\\right)^{-1} ", "3f5c2191d5128f36ec6b726c77df3b06": "{n \\choose k}", "3f5c2fd0fe15a2c9d158e83c4e0cb326": "p^2+2pq+q^2", "3f5c42c11a3b06fe109f39e995330ee0": "V(r) = \\mu \\omega^2 r^2 /2", "3f5c97fd1138dd5830ae5b2414a024fd": "\\mu S+\\lambda S=(\\mu+\\lambda)S", "3f5ce17f4e4e7b73ca22d4095e0eb896": "\\nu = \\frac{{\\bar{R}} T}{P}", "3f5d167417396282c2eab925251d8f96": "\\tau_{(Q)} = {\\lim_{\\epsilon\\to0}{\\left[ \\frac {ln{I_{{(Q)}_{[\\epsilon]}}}} {ln{\\epsilon}} \\right ]}} ", "3f5d5b50bd1638ea63a624e876158554": "n\\cdot 1=1+1+\\ldots+1", "3f5d97abc8b6ca85794534bacadea847": "a_1^2\\cdot10^4", "3f5def1490354d34085b326574af94bf": "\\mathbf{A}\\frac{\\partial \\mathbf{U}}{\\partial x}\\mathbf{B}", "3f5df5cd114d150a4c98e3ad50ad2439": "\\operatorname{var} = \\frac{\\mu(1-\\mu)}{1 + \\nu}= \\frac{(n-s)s}{(1+n) n^2} ", "3f5e72601b4d3d95c11ff68a0908d73d": "f(z) = u(x,y) + i v(x,y)", "3f5ecd9efd366662aeeeedda5f3e666c": " q_2 =\\frac{Aq_1+B}{Cq_1+D}", "3f5ee07fd456bce3c43141898cb255d2": "\nI(R) = I_e \\cdot e^{-7.67 \\left( \\sqrt[4]{\\frac R {R_e}} - 1 \\right)}\n", "3f5ef1278dad11bbf06579c24b8aaa81": "\\mathbb Z[p_1,\\ldots,p_n].", "3f5ef1297199a0ef393d40e7246259e6": "f(V_j)\\subseteq W_{i+j}", "3f5f2acfe50e5dea259b875269441cd5": "n_i \\;", "3f5f5725cd14c17c75f0c5f9545b6da4": "\\begin{align}\nB &= \\left \\{\\alpha \\in R \\ : \\ p_1(\\alpha), \\cdots, p_k(\\alpha) \\in \\mathbf{F}_q \\right \\} \\\\\n&= \\{u\\in R \\ : \\ u^q=u\\}\n\\end{align}", "3f5f8d18270a043ff046e0d40501549f": "\n p_H = \\rho_0 \\chi U_s^2 = \\frac{\\rho C_0^2 \\chi}{(1 - s\\chi)^2} \\,.\n ", "3f5f8e4b34e5ecf11eb97366d9ab4782": "\\bold E =-\\frac {\\bold p}{3 \\varepsilon_0} \\ .", "3f601e5809812743583b972abf34a4c0": "(n,q,\\pi,G)", "3f60297885dd727b0548504d4f8e7f6f": "g^{\\alpha\\gamma}", "3f602a2477fd5155abe5400d1930ca41": " \\bar{x}_{j}=\\frac{1}{N}\\sum_{i=1}^{N}x_{ij},\\quad j=1,\\ldots,K. ", "3f60d1751447e81be80c601ccc883335": " \\langle f \\rangle_h = \\int f(x) e^{-h(x)} \\, d\\mu(x) \\Big/ \\int e^{-h(x)} \\, d\\mu(x). ", "3f60d9ecc2da5c1d8897552d1ba9913a": "L_\\text{max} = T_\\text{max}*M", "3f615f5811cf113840298d736bf07816": "g'(t)=\\mu\\cdot(1+\\varepsilon\\cdot (e^{-t}-1))^{-1}. \\,", "3f6205ec8ad11e92b5ba6793bb281a75": "\n\\mu_n = \\sum_{j=0}^n {n \\choose j} (-1) ^{n-j} \\mu'_j \\mu'^{n-j},\n", "3f623423efda6b5fde2dfd0ae3a3b6dd": "C = \\frac{\\int x g(x) \\; dx}{\\int g(x) \\; dx}", "3f62621fdba933a2ccb4591fb9bd6a02": "ProPoints = \\max \\left\\{ \\mathrm{round} \\left( \\frac{protein}{10.9375} + \\frac{carbohydrates}{9.2105} + \\frac{fat}{3.8889} + \\frac{fiber}{35} \\right) , 0 \\right\\} ", "3f6275e179d9b405c005d305e2265a01": "N = \\frac{ C * M + ((p * h / 4) - I)} { M}", "3f627c48e434ae231f2711d1d3093e01": "E(\\alpha_i)= 0", "3f628b3536f93cf748e1e9d9439de3fd": "\n \\mathbf{u}\\times\\mathbf{v} = \\varepsilon_{ijk}~\\hat{u}_j~\\hat{v}_k~\\mathbf{e}_i\n", "3f630258d864796afc93806edfd06363": "Z(X,t)=\\sum_{n=0}^\\infty [X^{(n)}]t^n", "3f63361400ac1013077223e948fb2a1b": "\\varepsilon(n)", "3f633f19fb5e8b2ab536083d5e8d10e8": "\n\\Delta(t) = L(t+1) - L(t) \\leq \\frac{1}{2}\\sum_{i=1}^ky_i(t)^2 + \\sum_{i=1}^K Q_i(t)y_i(t)\n", "3f63410aa429aa0c83d3cf373dfbf564": "y^2 = f(x) = a^2(x)", "3f63412cd0af4a0b306c05d8f089cb49": "B \\cup C", "3f63749a643944d48516b03f907eba02": "[\\cdot,\\cdot,\\cdot] : R \\times R \\times R \\to R", "3f63904278f7dacf27b7d3c746aa3191": "v_2 = 6", "3f639ece8a4d86408759afd0eaaf4c34": "x_1,x_2 \\in \\mathrm{GF}(p^m),", "3f63c201a3c0ae78a6eeecf47926b05b": "x_{t-1}", "3f6467e6caa97d0a16d48cb13f55bb22": "\\sum_{k=1}^{m}a_{k}=0,", "3f646ec48f4130702574274e761b6104": "\\mathrm{Cov}[X] = \\nabla \\nabla^{\\rm T} A(\\boldsymbol\\theta)\\, ,", "3f648139f520db10e3866386a6ef77e3": "F_{Dn}", "3f64d24752811f35d35848ebc538fa4d": " \\mathrm{III}_T(t) = \\sum_{n=-\\infty}^{+\\infty} c_n e^{i 2 \\pi n t/T} \\ ", "3f64db5a990809d60be801c17286dede": "\\sigma_\\mathrm{n} = \\frac{1}{2} ( \\sigma_x + \\sigma_y ) + \\frac{1}{2} ( \\sigma_x - \\sigma_y )\\cos 2\\theta + \\tau_{xy} \\sin 2\\theta\\,\\!", "3f64ddf54210e375c7cd7f0673124ed7": "\\exists x \\phi \\in \\Phi", "3f652479f5f87baddc433231f81b47f6": " R(t) = U(t)(\\rho_{A} \\otimes \\sigma_{0})U^{\\dagger}(t)", "3f6529bbed4ac9355634fabd3869ba2d": "2 \\pi k", "3f65340083e0ead66be0735c7734f4d8": "{\\gamma}_{ij}", "3f659aba13e88b2c281041306604bc8d": " s_n = s_{n-1} - \\frac{f(s_{n-1})}{ (s_{n-1}-p_n)(s_{n-1}-q_n)(s_{n-1}-r_n) }. ", "3f6604d3ef29b9703c27429ae97c8110": "T^{n}", "3f661161e91ce35af9ac3ea09c552fb7": "\\Delta(X)", "3f661d658d23ed6f1468f60c11777aeb": "a\\cdot\\cos E=a\\cdot\\varepsilon+r\\cdot\\cos \\theta.", "3f666db0e69f4c3864ed25d25d0cf5b9": "P = (x_1:y_1:1)", "3f66fbb28c7deec74517843fbea75904": "\\scriptstyle 1+\\log_2n", "3f672acdc83bf75a12cef666a474094f": "S_I = - \\sum p_i \\ln p_i", "3f6740fbc9944fe1a114410244c8fee6": " I_x(a+1,b) = I_x(a,b)-\\frac{x^a(1-x)^b}{a B(a,b)} \\, ", "3f6756eac7ca69145f47410584129542": "v ", "3f6773bb2e6f4d086e406b3b47de9fb1": "Z_{\\mathrm {in}} = \\frac {V_\\mathrm T}{I_\\mathrm T} = Z_0 \\frac {(1+\\mathit \\Gamma)\\cosh(\\gamma x) + (1-\\mathit \\Gamma)\\sinh(\\gamma x)}{(1-\\mathit \\Gamma)\\cosh(\\gamma x) + (1+\\mathit \\Gamma)\\sinh(\\gamma x)}", "3f67801afb9a09cfd4615b38ca894b6d": "\\ I(r,d)=\\eta I_b \\left(1-\\frac{1}{Q}\\right)", "3f678ced65ceefa8dd855c9a398a96b6": "\\langle f, K_x \\rangle = \\left \\langle \\sum_{i=1}^\\infty a_i K_{x_i}, K_x \\right \\rangle= \\sum_{i=1}^\\infty a_i K (x_i, x) = f(x).", "3f67df92992eed5b70a6f2aa86061560": "s_j^{\\eta + 1} = \\frac{v_j}{\\sum_i m_{ij}^{(2\\eta+1)}}", "3f681215d5dc017fc9a08f337f66372c": " \\displaystyle \\mathrm{Tr}\\left[ S_t^2 \\right] = \\sum_{j,k} \\left| \\langle \\phi_j | \\phi_k \\rangle \\right|^{2t} = \\frac{n^2 t! (d-1)!}{(t+d-1)!} ", "3f68149c738d50ce7bfd1548d8ded6a8": "{}_1Q_2 = mc_v \\left( {T_2 - T_1 } \\right)", "3f6869ab4ddff686ccbbeeda1abe72af": "\\nabla\\cdot\\nabla", "3f688fcd621117a5075c0b7b33d4a48c": "t_1^{-1}t_2", "3f68c8845e7cdfe91ac1d43ed28b1c3b": "\\scriptstyle \\gamma^0", "3f68f83670302a31a9d59eb6de0a8f57": "\\pm\\sqrt{\\frac {3} {7}}", "3f6912ba5b2409c85c4e536d2ac54998": "S={A,B,C,D}", "3f692c304a7d1b801e10aafae313da1c": "\\begin{bmatrix} k-1 \\\\[5pt] -\\dfrac{1}{\\theta} \\end{bmatrix} ", "3f698c3c8448d32e70fee88917b6089b": "U = \\sum_{n_x}\\sum_{n_y}\\sum_{n_z}E_n\\,\\bar{N}(E_n)\\,.", "3f69af43481af321ad215f50efcb5f8b": " \\mu = m ", "3f69e428ded39b0a341100d0e15dce63": "a\\ne b = c \\ne d, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "3f6a741ca616b2bd52132279dbf8d892": "\\scriptstyle \\theta \\;=\\; \\frac{2}{3}\\pi", "3f6a80fb91269e5613d507554d425ae4": " \\nabla_u R ", "3f6a90a7678ee1f6b8bd13e13d035175": "T\\left(\\frac{\\partial S}{\\partial x}\\right)_y\\!dx +\n T\\left(\\frac{\\partial S}{\\partial y}\\right)_x\\!dy = \\left(\\frac{\\partial U}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial U}{\\partial y}\\right)_x\\!dy + P\\left(\\frac{\\partial V}{\\partial x}\\right)_y\\!dx +\n P\\left(\\frac{\\partial V}{\\partial y}\\right)_x\\!dy", "3f6aa245a5d53ccd5b575294da8b3336": "{\\overline {\\mathcal{M}}}_{g,n}", "3f6ac423596b8e9b433874d0a044df09": " V_{0} = \\frac{E_{0}}{e}", "3f6ac5d62c7ea1904e8edc7dcbaf9522": "\n\\begin{align}\n\\dot{q}_1=q_1 k_{11}+q_2 k_{12}+\\cdots+q_n k_{1n}+u_1(t) \\\\\n\\dot{q}_2=q_1 k_{21}+q_2 k_{22}+\\cdots+q_n k_{2n}+u_2(t) \\\\\n\\vdots\\\\\n\\dot{q}_n=q_1 k_{n1}+q_2 k_{n2}+\\cdots+q_n k_{nn}+u_n(t)\n\\end{align}\n", "3f6b25a3e53c284e29b9b09dd9e73049": "C^m(\\bar \\Omega)", "3f6b5e5bb6885e221f33a6283048bafe": "x_t = a \\cdot x_{t-1}", "3f6b9922903e999f910c04f212744ac6": "I_{\\text{D}} = I_{\\text{S}} \\left( e^{\\frac{V_{\\text{D}}}{V_{\\text{T}}}} - 1 \\right).", "3f6be2150a4a429f5097aed46784910d": " \\nabla r\\cdot\\nabla\\theta\\times\\nabla\\zeta > 0", "3f6c22a74c4a47d1bdb86c55de7fdf91": "\\textrm{Ker} \\{\\textrm{Sp}(2n,\\mathbb{Z})\\rightarrow \\textrm{Sp}(2n,\\mathbb{Z}/k\\mathbb{Z}) \\}", "3f6c2d2468907198d017d27ac35ea3c2": "\\mathbb{S}^\\lambda V", "3f6c36a55022c208012863a5218d6a7b": "\\cong \\!\\,", "3f6c484efdb10a336806fa62b1774cac": "\\xi^1, \\xi^2, \\xi^3", "3f6ce49954e9ceacf858a4e02ff1e257": "f = \\gamma \\left( 1 - \\frac{v}{c} \\right) f^\\prime = \\gamma \\left( 1 - \\beta \\right) f^\\prime = f^\\prime \\sqrt{\\frac{1-\\beta}{1+\\beta}}", "3f6ce7b6145d19141aefa59e5a6f4a14": "\\begin{align}\n y[n] &= b_0 x[n] + b_1 x[n-1] + \\cdots + b_N x[n-N] \\\\\n &= \\sum_{i=0}^{N} b_i x[n-i]\n\\end{align}", "3f6d0d01104a8a7773e2736a303c662f": "D_{\\mathrm F} = \\infty", "3f6d1939b892c5dce1a5c8ab9a566e39": "\\frac{ds}{dt} = \\frac{1}{r}", "3f6d3db76a136e388f9eafefc501223b": "e_2 = e_3 = -1", "3f6d44d14d064ade52b0fc57283c7ebd": " -K V ", "3f6d84ce6571b9a7193fad92d85874f4": "U(f) = \\sum_{t=1}^{T}w_t e^{-2\\pi ift}.", "3f6da1df6e7238ef789a946b7a95a009": " \\mathbf{P}(t) = \\Pi\\ ", "3f6dadeefef58e118a72361c1941c924": "\n\\frac{p(A|X,S,h,\\Theta)}{p(A|X,S,h,\\Theta_{bg})} = \\frac{p(A|h,\\Theta)}{p(A|h,\\Theta_{bg})}\n", "3f6dccbf35487f15f443956a812ed566": "i_\\$ = \\rho_\\$ + E(\\pi_\\$) + \\rho_\\$E(\\pi_\\$) \\approx \\rho_\\$ + E(\\pi_\\$)", "3f6dd83c294996b5e12d2c0fb31d900c": "a z\\infty = \\begin{cases} z\\infty & \\text{if }a > 0, \\\\ -z\\infty & \\text{if }a < 0. \\end{cases} ", "3f6e57392bc54a5d6d16c21381e22c7b": " y^2\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)E(z,s) = s(s-1)E(z,s), ", "3f6e9f52917d2d13165d7b98acbb3914": "\\scriptstyle \\{|j \\rangle_B\\}", "3f6ea4f47e2b44bb73a456eedb85189f": "Z = \\frac{\\lambda_X X}{\\lambda_Y Y}", "3f6ed0f6ef9f6f0cea19f636a76908a3": " a_2 := S p_2 = B^* A^{-1} B p_2 = B^* p_1 ", "3f6ee055baf317c9f10fa163b7b1eb26": "TC(A)", "3f6eee6451717c871694e95cc1109eb8": "\\rho = Ve^{-B^{2}} + p", "3f6ef78c925d5c8ab279aa1a1b293c53": " \\theta \\left( \\xi - \\sum_{i=1}^{\\lfloor{k}\\rfloor} {\\ln(U_i)} \\right) \\sim \\Gamma (k, \\theta)", "3f6efb641413674a3829fd756c9d7217": "\\left|\\begin{array}{cccc}\\chi_1\\left(0\\right)&\\chi_2\\left(0\\right)&\\chi_3\\left(0\\right)&\\chi_4\\left(0\\right)\\\\\n\\chi_1'\\left(0\\right)&\\chi_2'\\left(0\\right)&\\chi_3'\\left(0\\right)&\\chi_4'\\left(0\\right)\\\\\n\\Omega_1\\left(1\\right)&\\Omega_2\\left(1\\right)&\\Omega_3\\left(1\\right)&\\Omega_4\\left(1\\right)\\\\\n\\chi_1''\\left(1\\right)+\\alpha^2\\chi_1\\left(1\\right)&\\chi_2''\\left(1\\right)+\\alpha^2\\chi_2\\left(1\\right)&\\chi_3''\\left(1\\right)+\\alpha^2\\chi_3\\left(1\\right)&\\chi_4''\\left(1\\right)+\\alpha^2\\chi_4\\left(1\\right)\\end{array}\\right|=0\n", "3f6f26193e3dc68602f1f21a8a45d0ec": "\\partial x/\\partial \\theta", "3f6f51e0b56ec444c8f2ab2bdf31e65e": "\\mathrm{2\\ AmF_3\\ +\\ 3\\ Ba\\ \\longrightarrow \\ 2\\ Am\\ +\\ 3\\ BaF_2}", "3f6f8570b9d4819cf38d43b71b90ff94": "(c,\\vec v)", "3f6fb1a1c6a20f822b078662ca2f5f64": "\\operatorname{Li}(n) = \\int_2^n \\frac{dt}{\\ln t}.", "3f70264c63f64e93ec90fe6e478c0d73": " F_{2} = \\frac{4\\pi^2 m_{2}r_{2}}{T^2} ", "3f70869265f69e665039f61b58c82aeb": " \\sigma_{log} = \\sqrt{\\ln\\!\\left(1 + \\!\\left(\\frac{s.d.}{m}\\right)^2 \\right)} ", "3f70f50d472f834bda55d82996fb80fa": "X^{\\prime}=r\\left(1 - \\frac{X}{K}\\right)X - c(t)X, ", "3f70f9273bcaedcf4ac21272177145c6": "\\begin{align}4x + 2(-2x + 4) &= 12 \\\\\n4x - 4x + 8 &= 12 \\\\\n8 &= 12 \\end{align}", "3f7140dff88a304aa309b5155d1c3d8d": " \\alpha=X'(\\eta), \\beta=T'(\\eta) ", "3f714172ed3efacc252dd03aebcd1828": "C_D = 2 = C_{D_{max}} ", "3f719019e6a7704ebdbbcc63ec13fb79": "\\mu\\sqrt{n}", "3f71f788c8084f58727e69b3621e0d3b": "T = \\left\\vert\\frac{2\\pi m}{e B}\\right\\vert", "3f725f4e2c93f7bcedfef289e4248e08": "S_t=e^{at}\\ \\sum_{k=0}^{n-1}\\ \\frac{t^k}{k!}\\ (z-a)^k ~.", "3f72644506633fea10e40d761802885e": "\\frac{\\operatorname{Re}(z^m (z-a)^{-n})}{\\operatorname{Im}(z^m (z-a)^{-n})} = const.", "3f72a62f6495d94b35da996839dd3d54": "U\\equiv\\left\\langle E \\right\\rangle = \\sum_{r}P_{r}E_{r}= -\\frac{\\partial \\log Z}{\\partial \\beta}\\,", "3f732a9eeafb8ffb5015d3a4eda0146e": "A \\preceq_{(0)} B", "3f73351152b4b2f7dc216c9d52ff8918": "\\varphi_{i,j}", "3f7398d3a98aab8c6901cc7de4e64f39": "T_B = \\frac{1}{11.9..}\\frac{N_A^2h^2}{MR}\\left( \\frac{N_A}{V_m}\\right)^{2/3}. ", "3f73b23d23b2f4952fd91e56c8d789fb": "\\lambda_- ", "3f73d1bcdc5b55d615ddea055822866a": " gh = f", "3f742daf2b1b08cffe4c988a0bf40449": " \\begin{align} \\mathcal{G}(\\Pi_\\beta) &= \\displaystyle \\sum_\\alpha \\Pi_\\alpha \\left| \\langle \\psi_\\alpha | \\psi_\\beta \\rangle \\right|^2 \\\\\n &= \\displaystyle \\Pi_\\beta + \\frac{1}{d+1} \\sum_{\\alpha \\neq \\beta} \\Pi_\\alpha \\\\\n &= \\displaystyle \\frac{d}{d+1} \\Pi_\\beta + \\frac{1}{d+1} \\Pi_\\beta + \\frac{1}{d+1} \\sum_{\\alpha \\neq \\beta} \\Pi_\\alpha \\\\\n &= \\displaystyle \\frac{d}{d+1} \\Pi_\\beta + \\frac{d}{d+1}\\sum_\\alpha \\frac{1}{d}\\Pi_\\alpha \\\\\n &= \\displaystyle \\frac{d}{d+1} \\left( \\Pi_\\beta + I \\right) \\end{align}", "3f74692b4856a85f0328ef2c482d4d3e": " \\zeta_S(a) = \\operatorname{tr}\\, S^{-a} \\,, ", "3f746d2efbefe77ae9ce6dd1537c64b9": "f_c=\\tfrac{1}{2\\pi C(R+S)}", "3f74b34724cc962008e2a35aa73acc8a": "\np(V) = \\frac{K_0}{K_0'} \\left[\\left(\\frac{V}{V_0}\\right)^{-K_0'} - 1\\right] \\,.\n", "3f7510e9d7d2fb875069546dd739d733": "\\text{Muscle force} = \\text{Total force} \\cdot cos \\Phi", "3f7519969e15f08505e8e25d09a4a141": "\\displaystyle -i\\sqrt{\\frac{\\pi}{2}}\\cdot \\frac{(-i\\omega)^{n-1}}{(n-1)!}\\sgn(\\omega)", "3f7537c0f823d977fa707e46c30126ae": "_3^4", "3f75913b721eddaac1c54a31d32bf867": " n \\geq \\left(\\frac{z_{\\alpha}-\\Phi^{-1}(1-\\beta)}{\\mu^{*}/\\sigma}\\right)^2 ", "3f7596a414866c34195fe40e0e1f680b": "\n\\begin{align}\n\\sin x&=2\\sin\\frac{x}{2}\\cos\\frac{x}{2}\\\\[8 pt]\n&=2t\\cos^2\\frac{x}{2}\\\\[8 pt]\n&=\\frac{2t}{\\sec^2\\frac{x}{2}}\\\\[8 pt]\n&=\\frac{2t}{1+t^2}.\n\\end{align}\n", "3f75af55a3c1e2d86a20e3e74ef8ad5c": "\n\nEe = \\frac{1}{2}K (2 cm)^2 \\!\n\n", "3f7610d82257508c6df2c5e739ad64f0": "[J,A] = 0;", "3f761c83a6ac48ee441fa3b9cb48519c": "1 \\text{ V/mil} = 3.94\\times 10^{4} \\text{ V/m}", "3f769ff2efd03c0c25df5adbc070c086": " \\gamma_1^\\infty = \\gamma_2^\\infty > exp(2) \\approx 7.38", "3f76a1c4a864e9216f39bcf16cf7e3ce": "\\forall \\phi", "3f779884561b33251c4c9b19793d886d": " \\overrightarrow{F}= - \\overrightarrow{\\nabla}U= \\begin{cases} \\frac{V \\chi}{2 \\mu_0} \\overrightarrow{\\nabla}\\left| \\overrightarrow{B} \\right|^2 & \\qquad \\text{in a weak magnetic field} \\\\ \\frac{1}{2} \\overrightarrow{\\nabla} \\left( \\overrightarrow{m}_{sat} \\cdot \\overrightarrow{B} \\right) & \\qquad \\text{in a strong magnetic field} \\end{cases} ", "3f77bd9f878d9040cbd56696a0c58a06": "\\frac{P}{\\dot{P}}", "3f77c69c6338be56e4abfa8bf6866ca5": "g_z", "3f780c3a5cb52f563f1c0dd94a99e390": "\\Delta T ", "3f784b875f74a9e33cb30034fe22ac3b": "\ne = v \\;|\\; (e\\; e) \\;|\\; x \\quad\\quad v = \\lambda x.e \\quad\\quad C = \\left[\\,\\right] \\;|\\; (C\\; e) \\;|\\; (v\\; C)\n", "3f78ef2bf6601b9c230309331afd2209": "\\neg\\exists x\\, \\neg\\phi", "3f78fa423258bd25ed3a51ee90ff8a65": "f(\\boldsymbol{x}) = f(\\boldsymbol{a}) + L(\\boldsymbol{x}-\\boldsymbol{a}) + h(\\boldsymbol{x})|\\mathbf{x}-\\mathbf{a}|,\n\\qquad \\lim_{\\boldsymbol{x}\\to\\boldsymbol{a}}h(\\boldsymbol{x})=0. ", "3f790939e7f973eb6fda39ba95c17df0": "x_{\\infty}", "3f791087255fe8fa853c68d11a2576ff": "p = 0.1", "3f79959c5ae0efce61c95a0cd0f3b408": " (x^2+y^2+z^2 + R^2 - r^2)^2 = 4R^2(x^2+y^2). \\,\\!", "3f7997d462006e6d6a2f94fd4a03e429": "x \\in (-\\infty, +\\infty) \\,\\;(\\xi = 0)", "3f7a1af099d683e9ffb8af5b33f2c5cb": "\n\\begin{align}\n\\chi_{k-1\\mid k-1}^{0} & = \\textbf{x}_{k-1\\mid k-1}^{a} \\\\[6pt]\n\\chi_{k-1\\mid k-1}^{i} & =\\textbf{x}_{k-1\\mid k-1}^{a} + \\left ( \\sqrt{ (L + \\lambda) \\textbf{P}_{k-1\\mid k-1}^{a} } \\right )_{i}, \\qquad i = 1,\\ldots,L \\\\[6pt]\n\\chi_{k-1\\mid k-1}^{i} & = \\textbf{x}_{k-1\\mid k-1}^{a} - \\left ( \\sqrt{ (L + \\lambda) \\textbf{P}_{k-1\\mid k-1}^{a} } \\right )_{i-L}, \\qquad i = L+1,\\dots{}2L\n\\end{align}\n", "3f7a2b87156da0063016a477f6e4c990": " n\\log\\left(\\frac{n}{e}\\right)+1 \\leq \\log n! \\leq (n+1)\\log\\left( \\frac{n+1}{e} \\right) + 1.", "3f7a5570b6c1765ac48b3c7b2cb7a308": "\\scriptstyle { x \\in S_*: x > 0 }", "3f7a84060f3839e838b322797631f2e5": "x^{16} + x^{15} + x^2 + 1", "3f7aa38a55cac2645a817fdfbf7514a4": "\\vert \\phi \\rangle", "3f7ad237629a60bcf0134aa76baca6e3": "\\textstyle e^{0{.}3/n}\\approx 1+\\frac{0{.}3}{n}", "3f7ae704ba08fae7fbb99b6942a489be": "\\theta(0)=0\\,\\!", "3f7aef8d2fae1e19b96dd7db3a5fa135": "\\ (\\lambda - 1)^2", "3f7af033ada57201cab2eaa9c39279c0": "\\operatorname{probit}(\\Phi(z))=z.", "3f7b165050239c2b0f38f4ca4a74ee77": "|w|", "3f7b4093474bf6547f71310d3221c91b": "\t\\mathrm{v}=(A^T A)^{-1}A^T b", "3f7b4b15362a4f0470a6c1e3a2ca4f8b": "d \\in D", "3f7b4c58361a98c87aee007b7058a5f6": " y_2", "3f7b4fa72b3c99d9e9678675993b0788": "\\frac{AB}{CD}=\\frac{IA\\cdot IB}{IC\\cdot ID},\\quad\\quad \\frac{BC}{DA}=\\frac{IB\\cdot IC}{ID\\cdot IA}.", "3f7b52c613d4b1eeae78481e65ec4c28": "r= \\frac{d[A]}{dt}", "3f7b6d31f5de60352789d2795e2bfa75": "L = K[\\pi]", "3f7b921448181e0972cff0e9253d60b0": "L\\equiv D^n+a_1D^{n-1}+\\cdots +a_{n-1}D+a_n", "3f7bad1fafe7e0664376bda0ec2ce528": "X \\,\\sim Gam(r, \\lambda) ", "3f7be74303aa786449877dcc8ec66f54": "\\hat{\\mathbf{p}}", "3f7c0ddf0ed7c83958baae25e6069c65": "x_{ir}", "3f7c98dd1d9f31b57846efb4cb4cf226": "\\theta (0 \\leqslant \\theta \\leqslant \\pi)", "3f7cc0e643b12a5ba9e2553832cdb3f6": "\\tau_ 1", "3f7d031cfd0370ea16a02125fe569681": "f = a^3b^{-2}", "3f7d0edd3838424de4d559b37b31db6c": "y = (a - b) \\sin(t)\\ - b \\sin(t ((a / b) - 1)), k = a/b ", "3f7d1f8f05b87f222731d22eaf66f1e2": "\\mathbf{r}=\\boldsymbol{\\alpha}+\\boldsymbol{\\beta}sc_1+\\boldsymbol{\\delta}s^2c_2", "3f7d5d9b46dd08600b5704b4294d28c6": "\\Delta\\sigma=2\\arcsin\\left(\\frac C2\\right).", "3f7d6b488b92b073e086a34edefed1f7": " 1 + 2 + \\cdots + n = \\frac{1}{2}\\left(B_0 n^2+2B_1 n^1\\right) = \\frac{1}{2}\\left(n^2+n\\right).", "3f7e315dba9320228936bf3cfea05401": " i_C : x \\mapsto \\left\\lbrace \n\\begin{align}\n& 0 & \\mbox{if } x \\in C\\\\\n& + \\infty & \\mbox{if } x \\notin C \n\\end{align} \\right.\n", "3f7e376fad611f7440c6c5efc3c797e2": "M_x= \\frac{248}{3}", "3f7ea9b5fed280cf7688e123772deb64": "\\bar{F}=\\frac{1}{n} \\sum F(x_i).", "3f7ec4769fc4c0d7bb5db1a4ec0cd553": "\\nu_1 \\ge \\nu_2", "3f7ed2a52e0097daa7c2ddbbbfd56b22": " 1 \\le j \\le n ", "3f7f2e05339e0e2342784dd8b4af4efe": "\\frac{59!\\times 20!\\times 3^{19}}{2\\times 5!^{12}} \\approx 2.20\\times 10^{82}", "3f7f63ddc7411525f2fe9c3caf14a844": "(n-1)(n-2)/2", "3f7fa919f056f9965ac9aa0c928bb776": " y\\ f\\ x = f\\ (y\\ f)\\ x ", "3f7feb659a498a724335f9551e483023": " u(x) = c_{1} + c_{2}x + c_{3}x^2 + \\cdots + c_{k}x^{k-1} ", "3f7ff30daaed19c3e5f555f84ab87a29": "f(t,0) = f(0,n) = 1", "3f80078ca3bca51e18e2b71750a0f8d3": "\\phi'(x,z;y)", "3f8027a50e2a93d0ef165471b22b7d4f": "{} + x_n \\epsilon_n", "3f8038dac63204782c6a8cf758dde280": "\\displaystyle{\\mathbf{n}=(-\\dot{y},\\dot{x}).}", "3f8061f5a525889df9a834066365630b": "p_{ij} = {\\part \\phi_i \\over \\part Q_j} = \\left({\\part \\phi_i \\over \\part Q_j} \\right)_{Q_1,...,Q_{j-1}, Q_{j+1},...,Q_n},", "3f80a25a36ff589babebe30e302e20cc": "\\pi^{(n)}(k_1, \\ldots, k_{n-1}, k_n) := \\pi ( \\pi^{(n-1)}(k_1, \\ldots, k_{n-1}) , k_n) \\,.", "3f80c6022d0410c31c345860ed2195c4": "\\operatorname{tr} \\left( \\gamma^\\mu \\gamma^\\sigma \\gamma^\\nu \\gamma^\\rho \\right) \\,", "3f80e4b5f5869bf3e93e182ebef2943f": "\\rho = \\rho{}(p)", "3f81144441637249630a8338796445b9": "\\scriptstyle P_j", "3f8120da913a874e87e5d87623e52a62": "V\\left(\\phi\\right)=-a \\phi^\\alpha + b\\phi^\\beta", "3f812510bc9e676638c42c408769e87f": "p\\in H(p,C)\\subseteq X\\backslash C", "3f81772677c183110c18bedb44d72783": "C_{MP,C}", "3f819e358820e7c1d95a18129bd886ab": "\\scriptstyle E(X)", "3f81cdd2bea3b743adb19a156aa4107b": "v(x_1) = au(x_1) + b\\!", "3f81fc1241cff04abce9f21f630956c7": "\\Omega^6", "3f8217c918515d8a4b08e8b040e96c90": "\n\\textrm{response} = \\textrm{constant}\n", "3f823fdb4df65db70012cc187ab6ada0": "(c + \\gamma)_{1 - \\gamma} = (c + \\gamma)(c + \\gamma + 1) \\cdots c.", "3f827fb0d256b88210ce442f6c64a519": "\\varphi(G)", "3f82c7552e198537447a20fa3343249d": " -\\triangle u = |u|^{p-1}u,\\text{ with }u\\mid_{\\partial \\Omega} = 0.", "3f82c7dddf2e87b4139fed5a1be97ef6": "u_2 = 0", "3f82eaf24c5ae900d9e76ea13a5bb9a9": "(\\mu_R)^c", "3f835ded21dfeeb894f2247d328d3e10": " u(x_j, t_n). ", "3f836bdd9f4d80bc6d030efeb06c0e7f": "\n\\mathcal A f = \\lambda f\n", "3f83870068e2c77a495c6e3681484cfd": "\\frac{p_2 \\cdot X_2}{N_2} = \\frac{p_1 \\cdot (X_1-R)}{N_1} \\,", "3f838aa6a17756ab157089b24af3ae49": "\\nu_D", "3f83b77af70b9fdd404853d53bb318b6": "\\overrightarrow{s}", "3f83cbfa32f3ef2d937aa70b95827e9b": "a^k x a^m = a^{k-m}x", "3f83d00f3c257e809926f13c7dcb8dff": "X_n \\sim \\mathcal{N}(\\mu, 1).", "3f83d2d6fcf496271434ef6e68b0b811": "{\\mathbf z}^T=(0,\\ldots,0,1,0,\\ldots,0)", "3f83d4ee148e31a3eda4996906dea157": "+S_x \\otimes I", "3f8431e9f1cf6411c16e9a29c1b865a9": "V'(\\Phi)=R", "3f84444ce3f3846f19ca9f595bf66330": "(x_i,y_i+1)", "3f8482833290779f9db716f5678d96f2": " {{\\partial}A\\over{\\partial}t} + {{\\partial}\\over{\\partial}x}(Au) = 0", "3f84b642f462c2044acc970b88bf36c7": "\\langle R \\rangle ^3 - \\langle R \\rangle _0 ^3 = \\frac {8 \\gamma c_{\\infty}v^2D} {9R_g T} t ", "3f84fbf15dacc5a90daff5833e6dd366": "F_2()\\,", "3f84fee7cb771ed099b4d04bea0bbdbf": "f = f_0 + \\beta y", "3f85384dc96e5ffac1a4555381954b83": "\\frac{1+x}{1+x+x^2}=1-x^2+x^3-x^5+x^6-x^8+\\cdots;", "3f853c7cb6f06a41b7ed4b4e001343cc": " \\overline \\sigma < \\sigma_0 ", "3f8563fa433242f09ffff1051e840f95": "\\left[ N\\right] :\\left( \\mathbb{Z}\n_{2}\\right) ^{2}\\rightarrow\\left[ \\Pi\\right] ", "3f85ad758ea5a0012c81602a749e547a": "\\sigma_{1},\\sigma_{2}= \\frac{\\sigma_{x} + \\sigma_{y}}{2} \\pm \\sqrt{\\left (\\frac{\\sigma_{x} - \\sigma_{y}}{2}\\right)^2 + \\tau_{xy}^2}\\,\\!", "3f85bb1599045c87c91a0ee496d2554d": "\n\\widehat{\\boldsymbol \\theta}_{JS} = \n\\left( 1 - \\frac{(m-2) \\sigma^2}{\\|{\\mathbf y}\\|^2} \\right) {\\mathbf y}.\n", "3f85c69e10cda86104611494fca3d8bb": "\\{ \\, ; \\, \\}", "3f86267552a01f6275fbedd8b503cded": "q=0.72", "3f86c8bf39e3fe06cc6ec5c4eb9be5dd": "T^{-1/p} + T^{-1/p^2} + T^{-1/p^3} + \\cdots", "3f87f42ad781ea6d733bc54063288f2d": "\\pi(V)\\subset B\\,", "3f88488a688abfb9e803e1035ff3c28c": " \\vdots ", "3f88feb701b80f1ca807a539d21f895d": "r_{I1}\\,\\!", "3f8978640863c6f2153bbf972c89f11f": "\\left\\{\\mathcal{F}_{t}\\right\\}_{t\\geq 0}", "3f89b1d3aa89af5e4a8b18b9d196fd73": "x_1,\\dots,x_{k-1}", "3f89b739d2ffcd0e1feee153cbf9ec16": " \\mathbf{D_{jj'}} ", "3f89db20de8089586f12eda5b57f2d48": "\n\\begin{align}\nm_{12}/b &= \\sqrt{1 + k^2\\sin^2\\sigma_2}\\, \\cos\\sigma_1 \\sin\\sigma_2\n - \\sqrt{1 + k^2\\sin^2\\sigma_1}\\, \\sin\\sigma_1 \\cos\\sigma_2 \\\\\n &\\quad - \\cos\\sigma_1 \\cos\\sigma_2 \\bigl(J(\\sigma_2) - J(\\sigma_1)\\bigr),\\\\\nM_{12} &= \\cos\\sigma_1 \\cos\\sigma_2\n+ \\frac{\\sqrt{1 + k^2\\sin^2\\sigma_2}}{\\sqrt{1 + k^2\\sin^2\\sigma_1}}\n\\sin\\sigma_1 \\sin\\sigma_2 \\\\\n&\\quad - \\frac{\\sin\\sigma_1 \\cos\\sigma_2\n\\bigl(J(\\sigma_2) - J(\\sigma_1)\\bigr)}\n{\\sqrt{1 + k^2\\sin^2\\sigma_1}},\n\\end{align}\n", "3f8a371a5f64d2a5f93158898367f204": "f(e^{i\\theta})=\\sum_{n\\in \\mathbf{Z}} a_n e^{in\\theta},", "3f8ade60980687790de83d02247f6bc9": "\\sigma_{\\text{strike}}^2", "3f8b13133fc80a4e77ceb3f9a478dd6e": " \\sum \\alpha_nb_n", "3f8b620182ef97f210f06d01e08365a0": "S_t,", "3f8b792f5bdaece2bebd5c861f924fb0": " a=0,\\dots d+1 ", "3f8bd7dcba607a633a56e8cd487b6fa8": " f' = f {\\sqrt{1-v^2/c^2}\\over (1-v/c)}{(1-u/c)\\over \\sqrt{1-u^2/c^2}}.\n\\,", "3f8bf2ed12cbf41ce69d930e474d10aa": "1 + {n \\sigma^2_\\text{Treatment}} / \n{\\sigma^2_\\text{Error}}", "3f8bf6f436d531e08550715c8bc9093e": "Q(X) = \\begin{cases}\nX_0^2+X_1^2+X_2^2+X_3^2\\\\\nX_0^2+X_1^2+X_2^2-X_3^2\\\\\nX_0^2+X_1^2-X_2^2-X_3^2\n\\end{cases}\n", "3f8c1859fa2a9f30ffb1001769e8d1f9": "\\frac{1}{M_r} \\frac{\\partial M_r}{\\partial c}= \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta +k }- \\frac{2}{c + 1 + k}\\right).", "3f8c1e7f93668bfb689df9e863f9d03f": "B \\in A", "3f8c39c364b95de5100eca268f54d896": "\\operatorname{d} \\operatorname{tr} (X) = \\operatorname{tr}(\\operatorname{d\\!} X)", "3f8c500772453df9f2464c09c112a96a": "(-1)^m \\lceil 2^{m-1}\\rceil ", "3f8c5f1272b170a6d118637525a7db65": "\\{i_{1}, i_{2}, \\ldots, i_{l}\\}", "3f8c98d832fb0a0239388c59435bb2d8": "\\varepsilon([X,Y]) = \\varepsilon(X)\\varepsilon(Y)", "3f8cb4ce5977c5d11bd7ed850d182de5": "a_i-\\varepsilon", "3f8cd217334f62ffd971f6d7284e1371": " \\Vert T\\Vert_{L^2\\to L^2} \\le\\sqrt{\\alpha\\beta}.", "3f8ce7c47355c28af7abcfe1a07e3223": "\\ddot{y}", "3f8d13d46e2b4a423ed60475c2fd7917": " 0 \\leq p \\le 1 - 1/q ", "3f8d1e2c88c565add40d9053316676a0": "(- \\sigma_1 \\sigma_2) \\, \\{a_1 + a_2\\sigma_1\\sigma_2\\}\n=a_2 - a_1\\sigma_1\\sigma_2", "3f8d2ae9356c399ffe2be5ddfa6f1f2e": " X = \\sum x_{ kj } ", "3f8d2b6a66f220b13496875e3d170ab0": "G(\\tau)=-\\langle \\mathcal{T}_\\tau \\psi(\\tau)\\psi^*(0) \\rangle ", "3f8d2fe428be485662ed2fe5c862fa55": "t=x^{-\\alpha}", "3f8d4c7b428bf9fb438531b9de3bc05e": " 101_2 \\rightarrow 10_2 \\rightarrow 1 ", "3f8e1dc14e9815eac9c41cd3741968ca": "\nA=\\sum^t_{i=j}\\Phi(x_i)\\Phi(x_i)^T\n", "3f8e4517e78151376be1ee9c51340552": " \\|[A_0,A_1]\\| \\le 2 \\|A_0\\| \\|A_1\\| \\le 2", "3f8e47d1475d047ed6bd497fc62bea11": "\\sum_{k=1}^n q(k-1;d) = n - d + d \\left (\\frac {d-1} {d} \\right )^n.", "3f8ebe43d6e5659f2ba9280a4915810c": "\\sum_{k=1}^\\infty \\frac{(-1)^{k+1}x^k}{(k-1)!} = x \\exp(-x)", "3f8ebfc2777c7c4b6f3f6d75d207e0a5": "\\Box \\exists{x} Ax", "3f8ed4047bb240d944ee72757f5cd4cc": "\\mathfrak{P}(\\mathfrak{D}_0)) = \\frac{1}{|\\mathfrak{D}_0|} \\sum_{d \\in \\mathfrak{D}_0}\\mathfrak{P}(\\{d\\})", "3f8f518e757c07ff7bb03b4efb0abc7c": "\\frac{\\beta}{\\alpha}", "3f8f885dbd7e4c531e8d5e3817e1e14d": "\\begin{align}{ \\frac{x_1}{\\alpha} \\frac{x_2}{\\alpha} \\cdots \\frac{x_n}{\\alpha} } &\\le { e^{\\frac{x_1}{\\alpha} - 1} e^{\\frac{x_2}{\\alpha} - 1} \\cdots e^{\\frac{x_n}{\\alpha} - 1} }\\\\\n& = \\exp \\Bigl( \\frac{x_1}{\\alpha} - 1 + \\frac{x_2}{\\alpha} - 1 + \\cdots + \\frac{x_n}{\\alpha} - 1 \\Bigr), \\qquad (*)\n\\end{align}", "3f9004fbc0e6b336cbd454ca5167f64f": "dc/d\\lambda", "3f901cda74b75c628ad04d258b268b0d": "D = 1-x", "3f903d3d53bf0a42b4fdb389f85bf3c0": "\\mathrm{IMG}f := \\frac{\\pi_1 (X, t)}{\\bigcap_{n\\in\\mathbb{N}}\\mathrm{Ker}\\,\\digamma^n}", "3f904e2e1162af7d7d2cee6bcea85292": "D(q;\\Delta t) =B(q)+T(q)I(q)[1-f(q;\\Delta t)] \\,", "3f908922594e7f0242a82e16906d1243": "\\| \\mathbf{M} \\| = \\| \\mathbf{M}^* \\mathbf{M} \\|^\\frac{1}{2}", "3f90af5e1452e15647b558161fc5ff3b": "f: X\\rightarrow Y", "3f90d63037db1f03824a182de5bcab85": " \\mathbf{I} = \\mathbf{ii} + \\mathbf{jj} + \\mathbf{kk} ", "3f9120dabd860474ca8bc9fe595a44c9": "[e_i,f_j] = \\delta_{ij}\\alpha_i^\\vee ", "3f912de8d9b16c643c912b195d3792c2": "24 = 2mn ", "3f9166a2bff1caba19de389f5fa8dc61": "G = \\frac{4 \\pi A}{\\lambda^2} e_A ", "3f917d770423c7884700a651787f2ab5": " c_{1} \\, ", "3f918c1e2ff8db12c9fba5c48dfcb160": "2\\pi \\nu", "3f91d5ed7b32ff9afaab5f13eff89663": "f(x)=(a_1x_1+a_2x_2+a_3x_3+\\cdots)^n=(b_1x_1+b_2x_2+b_3x_3+\\cdots)^n=(c_1x_1+c_2x_2+c_3x_3+\\cdots)^n=\\cdots.", "3f91e1b23db9830c308bdcfe297e7f07": " f^{(0)}=5.92714 - \\frac{6.09648}{T_r} - 1.28862 \\cdot \\ln T_r + 0.169347 \\cdot T_r^6 ", "3f91ec3b88fff86ea427650817b31f97": "\\,R(P)=G(P)/H(P)", "3f92519d28aa87ec3a168cc8e5b28730": "\\neg \\exists x \\neg \\phi(x)", "3f926ded67972842e35093e364f5bd73": "\\mathbf{v}_i = \\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i + \\mathbf{V}_R", "3f927d954f84b8524731e4cb8f5476c7": "\\int_a^a f(x) \\, dx = 0. ", "3f9303aceb2e9b41406a2b2eddf48838": "\\,\\!\\ ^{n}a", "3f93689ba974e54b0c29853258a63ca5": "f(x)=3x+2", "3f939bca6725d79a2a7f048059736707": "A_1({\\nu})", "3f93a598c30516b54adddf0f277bcd4d": "GW_{0, 3}^{X, 0}(a, b, c) = \\int_X a \\smile b \\smile c;", "3f93fa9916a1d0892d49b51e48434725": "\\mathcal{F}^W_t", "3f9408834cd4ff066e7e8165e50cb307": "(\\Sigma, \\text{Prim}, S, \\triangleleft)", "3f9457a6ad1c3df41b5a73e9400efb2d": "Q_{\\rm ts} = \\frac{Q_{\\rm ms} \\cdot Q_{\\rm es}}{Q_{\\rm ms} + Q_{\\rm es}}", "3f94cb495ea5019febf6b11044fa89b2": " m_k ", "3f94efda8a55c60c2aac1d80b2b32929": "\\mathrm{SNR}=\\frac{3 \\times 2^{2n}}{1+4P_e \\times (2^{2n} - 1)} \\frac{m_m(t)^2}{m_p(t)^2}", "3f94f9fe2881050cfea24c9fc7102452": "\\sum_{n=1}^\\infty a_{\\sigma(n)} = M.", "3f95749768c1dedfca83677b7d7aef67": "\\{ n \\} ", "3f9576a232e180b1975fc244652e0547": "\nu^1(x_{1}^{1},x_{2}^{1}) \\geq u_{0}^{1}\n", "3f9579ab41e117704691376b31184c23": "V_3''", "3f9589c597ae50e67a84bed500e489bb": "IJ \\subset K", "3f959b22eede325c34f46af03b284509": "2\\pi/q", "3f95bb0c1bbf5cc479e28ead3cac42fb": "K/BB = \\frac{K}{BB}", "3f95cdfa71a5e6397b46b92b6203bb04": "z_i(\\mathbf{x})", "3f95d76ca1bfda291d88c3fbfc9e2245": " \\operatorname{tr} (\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma) = 4 \\left( \\eta^{\\rho \\sigma} \\eta^{\\mu \\nu} - \\eta^{\\nu \\sigma} \\eta^{\\mu \\rho} + \\eta^{\\mu \\sigma} \\eta^{\\nu \\rho} \\right) \\,", "3f9604fd0c3d44d4ef23a15c63edc761": "\\frac{L^{\\alpha}}{1-\\left(\\frac{L}{H}\\right)^{\\alpha}} \\cdot \\frac{\\alpha * (L^{k-\\alpha}-H^{k-\\alpha})}{(\\alpha-k)}, \\alpha \\neq j ", "3f96128272876149748bb377900b8d43": "\n\\exp\\begin{pmatrix}\n. & . & . & . & . & . & . & . & . & . \\\\\n-4 & . & . & . & . & . & . & . & . & . \\\\\n. & -3 & . & . & . & . & . & . & . & . \\\\\n. & . & -2 & . & . & . & . & . & . & . \\\\\n. & . & . & -1 & . & . & . & . & . & . \\\\\n. & . & . & . & 0 & . & . & . & . & . \\\\\n. & . & . & . & . & 1 & . & . & . & . \\\\\n. & . & . & . & . & . & 2 & . & . & . \\\\\n. & . & . & . & . & . & . & 3 & . & . \\\\\n. & . & . & . & . & . & . & . & 4 & .\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & . & . & . & . & . & . & . & . & . \\\\\n-4 & 1 & . & . & . & . & . & . & . & . \\\\\n6 & -3 & 1 & . & . & . & . & . & . & . \\\\\n-4 & 3 & -2 & 1 & . & . & . & . & . & . \\\\\n1 & -1 & 1 & -1 & 1 & . & . & . & . & . \\\\\n. & . & . & . & . & 1 & . & . & . & . \\\\\n. & . & . & . & . & 1 & 1 & . & . & . \\\\\n. & . & . & . & . & 1 & 2 & 1 & . & . \\\\\n. & . & . & . & . & 1 & 3 & 3 & 1 & . \\\\\n. & . & . & . & . & 1 & 4 & 6 & 4 & 1\n\\end{pmatrix} ", "3f9650ded7ca83a35702de501b50a1d5": "\\tilde{f} \\circ \\otimes = f.", "3f966dd07191d28d45a60af414cd7b7c": "k_{j}", "3f966fd9872d705ad04b405d4728df23": " V_o \\ = \\ v(t)|_{t=0}. \\, ", "3f96b40aa9f994f4ef56653a55c75c0c": "\\left(\\left(\\dfrac{\\delta- \\left(\\dfrac{\\lambda}{d}\\right)}{1-\\left(\\dfrac{\\lambda}{d}\\right)}\\right)\\right)^2", "3f96ba47201d5ff665e67537b018f77e": "\\overline{B}_0", "3f96df51785f123c3484bdc33be8d058": "\\psi(s+1)= -\\gamma + \\int_0^1 \\frac {1-x^s}{1-x} dx", "3f96e233261fe6fed403e2082b573bfa": "f_o = \\frac {f_s} {\\gamma}. \\,", "3f96f28148515487dd810480bcc33722": "\\Delta_i", "3f96f6acd549a811b199bd50a794ba75": "\\mathbf{p} \\Psi(\\mathbf{r}) \\ \\stackrel{\\text{def}}{=}\\ \\lang \\mathbf{r} |\\mathbf{p}|\\Psi\\rang = - i \\hbar \\nabla \\Psi(\\mathbf{r})", "3f97824fc9150e24f3655217545ba5c9": " X=X_1\\cup X_2", "3f97e585027448bb10a38297753ff145": "(t,R)", "3f97eec21560ff04b8ae6a17617ff289": "\\begin{bmatrix} \\dfrac{-\\Delta \\mathbf{[h]}}{h_{21}} & \\dfrac{-h_{11}}{h_{21}} \\\\ \\dfrac{-h_{22}}{h_{21}} & \\dfrac{-1}{h_{21}} \\end{bmatrix}", "3f980a9cd7cdf34438b3a10de183a165": "T_g", "3f9814905e2c9fe81fde649badc73ea5": "\\Delta\\lambda\\,\\!", "3f981b129bfa48aa762f05bf2ce290d1": " \\sigma^1 = g(r) \\, dr", "3f982f94c67b48e3c2cfcfdf43c7ea69": "\\mathbf{e}_i(\\mathbf{r}) =\\lim_{\\epsilon \\rightarrow 0} \\frac{\\mathbf{r}\\left(x^1,\\ \\dots,\\ x^i+\\epsilon,\\ \\dots ,\\ x^n \\right) - \\mathbf{r}\\left(x^1,\\ \\dots,\\ x^i,\\ \\dots ,\\ x^n \\right)}{\\epsilon }\\ ,", "3f986237763ed5a36770e840c00dd5d2": "\\mathcal{B}(\\overline{\\mathbb{R}})", "3f9869709be92e3f1c25f81cbe3c7798": "-7\\le x,y \\le 4", "3f9899171bb50e7a0a927070076d74fb": "SVR", "3f9918fa0ed1a80eedd55b36ea353bdb": "X_1 X_2 X_3 - 2 X_1 X_2 - 2 X_1 X_3 - 2 X_2 X_3 \\,", "3f9922f7dd9130e3ac3c80a2137476bf": "\\Sigma^f_{ij}= \\sum_k^n A_{ik} \\left(\\sigma^2_k \\right)^x A_{jk}.", "3f99284ec279a20e1e4136eae7afedff": "y' = 1 1 \\ldots 1 ", "3f993261d036264fb8317700b87cf2e6": " max(S-K,0) ", "3f9944fa0b7c3a343aa4fa38ef910601": "y= \\int^x F(\\lambda) \\, d\\lambda + C \\,\\!", "3f995fb2d852d777a922996038487451": "-a(e-1)\\,", "3f99b5f43777abf9f4fa4b2714fdad0e": "\\frac{dr}{dt_r}=\\beta = -\\sqrt{\\frac{2M}{r}}. \\,", "3f99b766e013738dfb535d13dcfb9ab6": "{\\mathbb L}_{y^2}(L)\\equiv{\\Big\\langle\\Big\\langle} L,\\partial_{yy}+\\frac{2}{x-y}\\partial_y+\\frac{2}{(x-y)^2}{\\Big\\rangle\\Big\\rangle}.", "3f99f9b5db6a72925a1fddbe46e2ea75": "\n\\Pr \\{X_{ni}=1\\} =\\frac{e^{{\\beta_n} - {\\delta_i}}}{1 + e^{{\\beta_n} - {\\delta_i}}},\n", "3f9a53a97e8356b00fd5d00d372e4fe7": " P=a_0x^n+a_1x_{n-1}+ \\cdots +a_n.", "3f9a915a6b69b606230016ae069b425f": "E[uu'] = \\sigma^2 I_n", "3f9aa7be54bb9ecf4c42b11cd6f505e8": "T_{par}=\\frac{\\sqrt{2}}{3}\\sqrt{r^3\\over{\\mu}}", "3f9b0b4f9a351e213e59bc9953f3226d": " g_c ", "3f9b145013cab8141100c71321865864": "\\sigma_{put,25}", "3f9b3dd0f599894e1cf701abaae2f8ad": "F(Y) = \\operatorname{Hom}_X(x, Y);", "3f9b88993b3b4bb1b31d98dc39347dd3": "170MeV g^{-1} cm^{2}", "3f9ba6a7989aa46e84a5984e95c1208c": "\\beta(p_{2})", "3f9bbceb1852f02987d6ed303063b95d": " {n \\choose k} = 2\\times{n-1 \\choose k-1} + {n-1 \\choose k}.", "3f9bc1a6b80a2a07583979c81516d8d6": "\\,\\! R_x(t_1,t_2) = R_x(t_1-t_2).", "3f9bc4275cc478faa1f225fe8e755b33": "T_\\mathrm{n}\\,", "3f9be97b27116c5c0e417ff9e4f7322a": "\\operatorname{Spec} K[x]_{(x-c)}", "3f9c16394f7ae6f04c51bb7b1f31c865": "\\begin{align}1/7\\ & = 0.142857142857142857\\ldots \\\\[6pt]\n\n & = 0.14 + 0.0028 + 0.000056 + 0.00000112 + 0.0000000224 + 0.000000000448 + 0.00000000000896 + \\cdots \\\\[6pt]\n\n & = \\frac{14}{100} + \\frac{28}{100^2} + \\frac{56}{100^3} + \\frac{112}{100^4} + \\frac{224}{100^5} + \\cdots + \\frac{7\\times2^N}{100^N} + \\cdots \\\\[6pt]\n\n & = \\left( \\frac{7}{50} + \\frac{7}{50^2} + \\frac{7}{50^3} + \\frac{7}{50^4} + \\frac{7}{50^5} + \\cdots + \\frac{7}{50^N} + \\cdots \\right) \\\\[6pt]\n\n & = \\sum_{k=1}^\\infty \\frac{7}{50^k} \\end{align}", "3f9c199d8a7a222d179007129579dcd1": "j \\in \\{1, 2, \\dots, n\\}", "3f9c3803b31014472237e70be4b42c46": "\\frac{d}{d t}\\left(\\frac{m \\dot{\\vec{x}}[t]} {\\sqrt {1 - \\frac{v^2 [t]}{c^2}}}\\right) = q \\vec{E}[\\vec{x}[t],t] + q \\dot{\\vec{x}}[t] \\times \\vec{B} [\\vec{x}[t],t] ", "3f9c6c626c404b7b2015f9597ca65f85": "P_r/P_t", "3f9cb32d917db02d8779e9bbe3b5fb1e": "x^7 + ax^3 + bx^2 + cx + 1 = 0", "3f9cc529e14e4f1b72367184ce2f6ea6": "\\mathrm{Sp}(6,\\mathbb C)", "3f9d5a788ce43ac91446e51a29b86284": "V=10 \\sqrt{Y}", "3f9d699000751a0f69229ea4b703a221": "s_g(x_d - x)", "3f9d8769f633e6f03f5edfd2bd339872": "= [P^+ F, P^+ G]^{IJ} + [P^- F , P^- G]^{IJ} .", "3f9d876ad2ab69d94b0b14f3aef8e524": "2k_BT\\approx0.052\\,eV", "3f9dd42790ab8533f00af517804108be": "a\\,|000\\rangle + b\\,|001\\rangle + c\\,|010\\rangle + d\\,|011\\rangle + e\\,|100\\rangle + f\\,|101\\rangle + g\\,|110\\rangle + h\\,|111\\rangle", "3f9ddca033cba2f1c8ffacb296ef418b": "f^l \\circ f \\le \\mbox {id}\\qquad\\mbox{(left counit)}", "3f9df02019add25b3eeb9b566d0fb2fa": "\n\\left(\\frac{\\alpha}{\\mathfrak{p} }\\right)_n \\equiv \\alpha^{\\frac{\\mathrm{N} \\mathfrak{p} -1}{n}}\\pmod{\\mathfrak{p}}.\n", "3f9df9336d464ece6989396d54e649e3": "-L", "3f9e4f733b9cef12e22fd5888812180a": "p_n(x)\\mapsto np_{n-1}(x)", "3f9e61298234820cecc519ae382f1bb3": "y=u+z", "3f9e78e18632c0d7895dac8addbf2575": "C_{Di} = \\frac{L^2}{\\frac{1}{4} \\rho_0^2 V_e^4 S^2 \\pi e AR} ", "3f9e7def4045cb39d494d5206fd47d3f": "C < r \\times B ", "3f9e9bb652e0492fca97634937817990": "VP = V+S =", "3f9eec5bbcf1cad0d812a4d164016210": "C(X_1,X_2,\\ldots,X_n) = \\left[\\sum_{i=1}^n H(X_i)\\right] - H(X_1, X_2, \\ldots, X_n)", "3f9f315ffc4e745707281961238f7e1d": "\\frac{1}{2} mv^2 = eV_{ion}", "3f9f7d7693a8d70aca19550f5379e8ca": "\\frac {d\\epsilon_{Total}} {dt} = \\frac {d\\epsilon_{D}} {dt} + \\frac {d\\epsilon_{S}} {dt} = \\frac {\\sigma} {\\eta} + \\frac {1} {E} \\frac {d\\sigma} {dt}", "3f9f8f62cf7e0e7bd569633c88f4ea53": "g(x)/h(x)", "3f9fea165f63a71c42823a27d9b9ad6d": "T(v_{i_1} \\otimes v_{k_1} \\otimes \\cdots \\otimes v_{k_N}) = \\sum_{i'_1,\\ell_1, \\dots \\ell_N} T_{i_1 k_1 \\dots k_N}^{i'_1 \\ell_1 \\dots \\ell_N} v_{i'_1} \\otimes v_{\\ell_1} \\otimes \\cdots \\otimes v_{\\ell_N}", "3fa05944b995d9b1e6c2609b564e3fa5": "H_{N,q,s}=\\sum_{i=1}^N \\frac{1}{(i+q)^s}", "3fa08006a602dd939386a95a6b62c179": "\\delta_e \\approx 16 \\lambda", "3fa0b1b7c9cf8dc037ed55e51e86850f": "\\rho^{*}", "3fa0b9612a5e5d29a0cdd50d5593e214": "{\\tilde{A}}_7", "3fa133d3eaff5df4df6c745bfdd04307": "= \\frac{i \\Psi^\\prime}{z \\lambda} \\int_{-\\frac{a}{2}}^{\\frac{a}{2}}\\int_{-\\infty}^{\\infty} e^{-ik\\left[z+\\frac{ \\left(x - x^\\prime \\right)^2 + y^{\\prime 2}}{2z}\\right]} \\,dy^\\prime \\,dx^\\prime", "3fa141e946b1a9afa18066ffd2053cc5": " \\{a_1,\\ldots,a_n\\} \\mapsto \\langle \\langle a_1, a_2, ... , a_n \\rangle \\rangle \n = \\langle 1, a_1 \\rangle \\otimes \\langle 1, a_2 \\rangle \\otimes ... \\otimes \\langle 1, a_n \\rangle \\ ,", "3fa14675cd05df48c68ce2c9ead6154e": "w_1, w_2 ... w_N,", "3fa1840156a104ccdf387f12826fccaa": "\\sum M_D=0=-5*1+\\sqrt{3}*F_{AB} \\Rightarrow F_{AB}=\\frac{5}{\\sqrt{3} }", "3fa1a0a69644647d12b974b65cfc7c86": "r_0^* \\leftarrow b^*-x_0^*\\, A^T ", "3fa1e65cf28b755ea17c0133cff9c1c2": "\\scriptstyle{f_\\mathrm{rest}\\sqrt{\\left({1 + v/c}\\right)/\\left({1 - v/c}\\right)}}", "3fa20a38621415556591010ddb85e5b2": "u=\\alpha s + \\beta n", "3fa25944ba48a7ebd83d0c67952bcaf6": "x=x'+a", "3fa2817ba9cbe43e85b21d5f6c840b89": "Ax_n\\to y\\in Y", "3fa2b92b5604cf7258415520e8b1bb46": "\nG^TG = I\n", "3fa2ce70fe51b7d785c23f323b7fa70f": "\\left(-\\frac{3}{2}\\right) + \\left(-\\frac{1}{2}\\right) + \\frac{1}{2} = \\frac{3\\left(-\\frac{3}{2} + \\frac{1}{2}\\right)}{2} = -\\frac{3}{2}.", "3fa2de0c524900bf7754fab06c3031d9": "\nh_{\\tau} = a\\sqrt{\\frac{\\sigma^{2} - \\tau^{2}}{1 - \\tau^{2}}}\n", "3fa361f0da61d316e8f27bb9dfd662b0": " \\mbox{lim}_{t \\ \\infty} x(t) = 0 ", "3fa364a42ff781492a5213b4541b9bb9": "\\min_b SS_\\text{err}(b) \\Rightarrow \\min_b \\sum_i (y_i - X_ib)^2\\,", "3fa378a8a0fa788ac4dc3bc239e0c7e1": "\\text{PAM}_n(i,j) = log \\frac{f(i)M^n(i,j)}{f(i)f(j)} = log \\frac{f(j)M^n(j,i)}{f(j)f(i)} = \\text{PAM}_n(j,i) ", "3fa385f334dd6c9bbfaee0d5d17b70b0": "[(\\gamma_2)_\\mu (p_2-\\tilde{A}_2)^\\mu+m_2 + \\tilde{S}_2]\\Psi=0.", "3fa39012f711f7a067f006e1d08d56b8": "\\forall k>0 \\; \\exists n_0 \\; \\forall n>n_0 \\ |f(n)| \\ge k\\cdot |g(n)|", "3fa3ebcf83db3de145cfeaf2018efcc5": "\\scriptstyle A_1=a_1+a_2", "3fa44071bd09708fb2c53cd5c7db270d": "P=\\mu\\otimes \\nu", "3fa509a2c82a0a87ecce1abeb09a27a8": "V_\\text{Hall}", "3fa50f604f23436cb2c1a04f8878f47e": "V_{pp\\pi}", "3fa52e6e515977fe5d11e21297d90fca": " \\Gamma^\\lambda {}_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu + \\ddot x^\\lambda = 0\\ .", "3fa54b9ecef4ad0246b5ae50b779c3e4": "m = N + 1 - a", "3fa54eaf39ced2d3f1fe0a12b2d03b19": "\\mathbf{C^\\infty}", "3fa593e8caf3b2fa12b4e722dd8131d0": "\\lambda=\\frac{C_{\\rm Tip}}{C_{\\rm Root}}", "3fa5bd510ccda012a974661a3830b473": "H^i(V; \\bold Z)=0,\\text{ for }i>n \\, ", "3fa5d2d35a22d36a6da812db96df0d08": "A_\\varepsilon(y_1,\\ldots,y_N)=\n\\begin{cases}\n1-\\frac{1}{\\mu}\\left(\\frac{1}{N}\\sum_{i=1}^{N}y_{i}^{1-\\varepsilon}\\right)^{1/(1-\\varepsilon)}\n& \\mbox{for}\\ \\varepsilon \\in \\left[0,1\\right)\\cup\\left(1,+\\infty\\right) \\\\\n1-\\frac{1}{\\mu}\\left(\\prod_{i=1}^{N}y_{i}\\right)^{1/N}\n& \\mbox{for}\\ \\varepsilon=1,\n\\end{cases}\n", "3fa605d7dc75720ce963a2611edadc7c": "a(n)=e^{\\int_0^n \\delta_t\\, dt}\\ ,", "3fa669519ec9168212f0177967d49357": "e^{\\hat{n}\\theta}", "3fa6966d97af34e8de3860bdbe52f5e2": " \\Delta = \\frac{2\\pi t }{\\lambda} C ( \\sigma_{1} - \\sigma_{2}) ", "3fa710e5bfa96b37009e07f1e8475ffc": "L_1\\,", "3fa7598db55ff51e66af36b4878d15a0": "s(\\alpha)", "3fa78b25c3f90b3a2aeb627023a34470": "x_i = a + \\frac{i}{n}(b - a)", "3fa7a689b2f900e6700343f657a4aaca": "A\\star B", "3fa7de2674f277c38e25387b27c1755a": "d_{xz} = N_2^c \\frac{xz}{r^2} = -\\frac{1}{\\sqrt{2}} \\left(Y_2^1 - Y_2^{-1}\\right)", "3fa7e1a1dfbecee86d79349d205d82b9": "\\textstyle R_0", "3fa7e1b0183c724fea77b1ae010637c6": "\\frac{\\partial u}{\\partial t} = \\frac{k}{c_p\\rho} \\left(\\frac{\\partial^2u}{\\partial x^2}\\right) ", "3fa7e1e4287b666b62bcf0b23728e764": "\\mathit{P}", "3fa7fffcc7c5977e52e79acbb4cb5cd4": "\\mathrm{Re}_x \\approx 5 \\times 10^5", "3fa8110e288354058a8825ee0cb79347": "v_i(t|s)", "3fa81c7c28d5fbb91b2143e462a79077": "\\scriptstyle F = f(G) ", "3fa826b5674633a448cc2d2d4c5570d5": "\\wedge^m_n = \\vartriangle^{m-1}_n", "3fa85307a4f5701ac29827e6ff7f345a": "\\ln \\gamma_i = \\frac{\\partial(\\frac{G^{ex}}{W_wRT})}{\\partial m_i}\n=\\frac{z_i^2}{2}f' +2\\sum_j \\lambda_{ij}m_j +\\frac{z_i^2}{2}\\sum_j\\sum_k \\lambda'_{jk} m_jm_k \n+ 3\\sum_j\\sum_k \\mu_{ijk} m_jm_k+ \\cdots\n", "3fa8a8dc2af920bc5356c86ea4bb3cc3": " d^2 F_x = \\frac{I I' ds ds'}{r^2} \\left[\\left[2\\alpha_1 cos\\epsilon + 2\\alpha_2 cos(rds) cos(rds')\\right]cos(rx) - \\beta_0 cos(rds')cos(xds)-\\beta_0 cos(rds)cos(xds')\\right] ", "3fa8a92d7bcab37d036f73679e9368f1": "\\lim_{n\\rightarrow\\infty}\n\\,n\\left(\\,\\left|\\frac{a_{n+1}}{a_n}\\right|-1\\right)<-1", "3fa8ad05ebb163ce8b84997f25a91e77": "C_{OA}", "3fa8b6dc8e855cdc645cc65bdcb53fe4": "\\mathrm{OPT_B}(f(x)) \\le \\alpha \\mathrm{OPT_A}(x)", "3fa8b8e50a4a9754340d48c8cb6d78ae": "\\widehat{f'}(n) = in \\hat{f}(n)", "3fa8e9f273b36f8eddfcaa068d66e54f": "{\\Delta v}_t \\approx \\overline {\\Delta v}_{t-16} = \\frac{v_t - v_{t-16}}{16} \\,.", "3fa944d51846576ad1a2705fa88b35e4": "N_S = \\frac{1}{2}\\int_{-\\infty}^{+\\infty} D(E)f(E-U_{SF})\\,dE ", "3fa98ef7ca2b7a4307ae28789fbc7365": "\\dot{x}_k=dx_k/d\\sigma", "3fa98f428a755724a576949d765d0772": "\\mathbf {F}_{i}^{(T)} = m_i \\mathbf {a}_i,", "3fa99b53c75186d396ce5ff688299be7": "(S,\\mathcal{S},\\lambda)", "3faa030ff2ad01a29e996aaf663b53fb": "Cut\\ Hose\\ Length\\ = Hose\\ Assembly\\ Overall\\ Length - C1 - C2", "3faa63399f227cd2d9603c83b9193b6c": "\\sigma_0(n) = \\sigma_0(n + 1)", "3faac37b17e658443e8909745d8c00db": "\\hat{\\mathrm{ch}}(f_*([E]))=f_*(\\hat{\\mathrm{ch}}(E)\\widehat{\\mathrm{Td}}^R(T_{X/Y}))", "3fab0b6f442feec7c9954a0fef47a57d": " \\nabla_t \\mathbf{v}:= \\partial_t \\mathbf{v} + (\\mathbf{v} \\cdot \\nabla) \\mathbf{v}", "3fab801157978f2bb8ca94a5aa52626d": "\\Gamma^i_{jk}", "3fab9c72075c8a27f1f59b1d6a0319fe": "2 \\Gamma ^a_{bc} = (h^a{}_{b,c}+h^a{}_{c,b}-h_{bc,}{}^a)", "3fabb16cfd7e94c514adb7d675ccd522": "(\\hbar=m=e=4 \\pi\\varepsilon_0 =1)", "3fac367508337bff023173a57de793d5": "\n\\begin{align}\nd\\Phi & = (dM)\\wedge\\Theta+M\\wedge d\\Theta \\\\\n& =(dM)\\wedge\\Theta \\\\\n& =(dM)M^{-1}\\wedge\\Phi.\n\\end{align}\n", "3fac904df7095c889e89db3415fbf4fd": "i^2 = -1", "3facb753a522eb28f84e6153d198ffb7": "M + 1", "3facbf6b3b67341ca75558103adfcc8b": "x = \\frac{{F_0}}{{k}}[1 - cos{(\\omega t)}]", "3fadfc04f550b99f300712d6f2698562": "X_1^2 + \\cdots + X_n^2\\ \\sim\\ \\chi_n^2.", "3fae01509105ae89e0bf63e3d9347636": "(\\alpha, \\beta)", "3fae243b655024a54580e28e85969f94": "\\sum_{n=1}^\\infty {S_n(s)\\over n!} x^n = {1 \\over 2} e^x \\left [ 1-e^{-2sx} \\cos\\left (2x \\sqrt{s(1-s)}\\right )\\right ] .", "3fae2baf9158c7fab57cea2c261cd907": "\\frac{x^2}{a^2}-\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1", "3fae47c93f439d001bd935de063f105c": "X_1, X_2, \\dots, X_n", "3fae4d76de9ec57c22915e212362d6c9": "\\vec E={-q\\over 4\\pi \\varepsilon c^2_\\circ}{d^2\\ \\over dt^2}\\left(\\vec e_{r'}\\right)=-q10^{-7}{d^2\\ \\over dt^2}\\left(\\vec e_{r'}\\right)\\, ", "3fae794206186b6b21408eb4ef019b58": " \\binom{n}{r-k} ", "3fae988840fa7f6b1ea5ff32f51a84a9": "\\{f_i\\in \\mathrm{Hom}(P,R) \\mid i\\in I\\}", "3faf2e20f478e97f10936a730e964efc": "T = |P| ", "3faf33c909f1373ef2f22704882895af": "\\langle \\alpha \\vert \\alpha \\rangle = 1", "3faf5014288f479b2a726ebd99f869d3": " I_sr = \\begin{cases}\n+1, & \\text{if } \\displaystyle\\lim_{x\\uparrow s}r(x)=-\\infty \\;\\land\\; \\lim_{x\\downarrow s}r(x)=+\\infty, \\\\\n-1, & \\text{if } \\displaystyle\\lim_{x\\uparrow s}r(x)=+\\infty \\;\\land\\; \\lim_{x\\downarrow s}r(x)=-\\infty, \\\\\n0, & \\text{otherwise.}\n\\end{cases}", "3fafa0fab965ded9742c827afb689949": "c_{n/2-1}", "3faff5918b0d502873b30690bba2eab3": "q(0) = D(0)S_C = \\sqrt{\\frac{2\\hbar}{\\rho}} \\ ", "3fb02711b693c92328804e2d01ef874f": "\\vec M(t)", "3fb02e4bed8871920b24ea79cdda9424": "\\nabla\\times v", "3fb060c49ad58a692dbe7d16dcdf1ccc": "x^{2} + bx + c", "3fb06df4cb7689a06dc2c2b2d7816162": "H\\psi_E = E\\psi_E \\,", "3fb09a9979bb2f6357f6746edb7f27b0": "4 \\pi k_{\\rm C} \\epsilon_0", "3fb112227e646febea6e53ee4972aea5": " \\textit{false}", "3fb135a7176ed52348a35197cfe07588": "O(N^{1/3})", "3fb13a90e985fc6c1c61cc81ec15b3dd": " \\boldsymbol{\\mu} = -\\frac{g_J \\mu_\\mathrm{B} }{\\hbar}\\boldsymbol{J}", "3fb13b96074f79e72e84df7db7c8abcd": "\\omega = 2/(\\lambda_{min}(A)+\\lambda_{max}(A))", "3fb149f6b8e0e7ebcfafdd781b9e33ea": "(2)\\quad R_{ab}-\\frac{1}{2}Rg_{ab}=8\\pi T_{ab}\\,,", "3fb168d843662dcd9dcd0d51af990f48": " return: H_2(\\tilde{w}, \\tilde{s}) ", "3fb1bb4f8ac7cf225173bfc8df4b425b": "U = \\frac 1 2 V \\sigma \\epsilon = \\frac 1 2 V E \\epsilon^2", "3fb216620e0dd96d9b88f6ad6fac1066": "\\sigma^2\\left(1 - \\frac{2}{\\pi}\\right) ", "3fb2ef932e16a129c5a3079866c3959b": " A + Ex = E + L + I ", "3fb344f7bb3cedd8175eb6df661112c7": " \\!\\ \\sqrt[ni]{x} = \\cos(\\ln \\sqrt[n]{x} ) - i \\sin(\\ln \\sqrt[n]{x} ).", "3fb37e0df4287fb0c1138709a5f43bb6": "K_{\\mathrm{max}} =\\ q_eV_0", "3fb38a38d8b2cd21e2a87d48e2ae38bd": " g(s)=\\sum_{n=1}^{\\infty} \\frac{a(n)}{n^{s} }=s\\int_{0}^{\\infty} A(x)x^{-(s+1) } dx. ", "3fb3c59f67839b458334767ac377c369": "p_1^{-1}(U)", "3fb3cfc6b993ef9a017b8f7e4a2f2640": "n(x_3) =\n\\begin{cases}\nn_A & \\mbox{if } x_3<0 \\\\\nn_B & \\mbox{if } x_3>0 \\\\\n\\end{cases}\n", "3fb4023d0a41028063d6d3e3c0d9a66b": "\\lceil\\pi\\rceil=4", "3fb425f0b9b40b8b2e997135ea260fab": " \\frac{H(X)}{\\log_2 a} \\leq \\mathbb{E}S < \\frac{H(X)}{\\log_2 a} +1 ", "3fb465fb29802f1c27eaad7efa510d24": "a=2^{-i}", "3fb49929db9ebe477f1137b4b6650164": "d(\\phi,x) = x \\ \\,", "3fb4fda2d69d8ff55a26bb8e6ebbadfa": " \\frac{dI_1}{dt} = a E_1 - (\\nu +\\mu ) I_1 ", "3fb50087a4e6cbec3530b534b29f396f": " {D \\rho \\over Dt} = {- \\rho \\left(\\nabla \\cdot \\mathbf{v} \\right)}. ", "3fb50756ec797c83a7d1eb95911a80df": "s = \\frac{a+b+c+d}{2}.", "3fb57a1849fd27de679d7dc70c71a70c": "\n \\hat{\\boldsymbol\\beta} = (\\mathbf{X}^{\\rm T}\\mathbf{X})^{-1} \\mathbf{X}^{\\rm T}\\mathbf{y}\n = \\big(\\, \\tfrac{1}{n}{\\textstyle\\sum} \\mathbf{x}_i \\mathbf{x}^{\\rm T}_i \\,\\big)^{-1}\n \\big(\\, \\tfrac{1}{n}{\\textstyle\\sum} \\mathbf{x}_i y_i \\,\\big).\n ", "3fb5ac21972bff3ec0391c87b1adf65f": "P(L) = \\frac{-(l+l^{-1})}{m} P(L_1)P(L_2)", "3fb5c2b6d6da4712bb1fe2653905d4f0": "\\sum_{k=0}^{n} ar^k = \\frac{a(1-r^{n+1})}{1-r}.", "3fb5c522e1a3b95d75e7141c8c698aac": " |\\bold{k}| 0 ", "3fcd1c0b4a33bb444d1a96a0f73f4731": "C(\\Psi, \\mathbb{C}^{2 \\times 2})", "3fcd5fca303d5e04af3d66c3f8fe9520": "\\boldsymbol{f}", "3fcdaec8ebd56402ad383828a23366cc": "\n\\begin{array}{*{20}c}\n q' + q = p \\\\\n q + p = p'\n\\end{array} \\to \\left[ {\\begin{array}{*{20}c}\n q & q' \\\\\n p & p'\n\\end{array}} \\right]\n", "3fcde2d70b098584502a5a03ede656f5": "\\begin{align}\n\\delta f_k &= 2^{ \\frac {1}{n} } * \\delta f_{k-1}\n\\\\ &= \\left ( {2^{ \\frac {1}{n} }} \\right )^{k} * \\delta f_{\\mathrm{min}}\n\\end{align}", "3fce4c7eb5c9991d07cbc422b8e5b544": "f^{(2)}<0", "3fce9a5153d9d1481bef6399f7ecc1ba": "b = \\left ({\\text{YDS} \\over \\text{ATT}} - 3 \\right ) \\times .25", "3fcf0e6884cec222068b0aaa026f790d": " O^* ", "3fcf1c5d7ba027ad52ab82e4b8cf5d24": "S= \\int_{\\mathbf{A}}^{\\mathbf{B}} \\mathbf{p} \\cdot d\\mathbf{s}=\\int_{\\mathbf{A}}^{\\mathbf{B}} \\nabla S \\cdot d\\mathbf{s}=S(\\mathbf{B})-S(\\mathbf{A})", "3fcf727ab54658042f01afc4bd3583fd": "\\boldsymbol{\\tau}=\\mathbf{m}_2 \\times \\mathbf{B}_1.", "3fcfa68bac30344d59032d17fffc1f98": "c=(2,0)", "3fcfe0a28a116c5f1e23b602001235c8": "\\mathbb{R}^3\\times\\mathbb{R}", "3fcff132717cffbb7e5d1859506bfeab": "M\\left(|\\uparrow \\rangle \\otimes |O_{\\downarrow} \\rangle \\right) = 0", "3fd0191687aecb6e3ff3014113126a7d": " \\tau_A ", "3fd18ae73d3e5998c70507a92147245d": "P = a \\cdot (D - b)^c", "3fd1eb0cd64804d3c17f0b328f7d2c18": "", "3fd1ed89f7bb3a23b9132966dda7e372": "\\delta t=0.18\\pm0.69\\ (\\mathrm{stat.}) \\pm2.17\\ (\\mathrm{sys.})", "3fd1ee70e9e0275d924522c14bd08734": "\\sigma \\subseteq V", "3fd2a9381567cd287861387e244f90c6": "\\left( R \\cdot V \\right)^{\\alpha},", "3fd2ee28475bfb03c9ede8d13ec7389f": " G_{ab} = 8 \\pi T_{ab} \\ ", "3fd335c707e2afb3c9593a77d78e6608": " I_{L}(E)=\\frac{I_{0}}{\\pi}\\frac{\\Gamma /2}{(E-E_{b})^2+(\\Gamma /2)^2} ", "3fd3d0b6c246c382b5622fd161702e4a": "\\bar{v}<\\bar{x}(1 - \\bar{x}).", "3fd3fa20921f81e5938c38f79486b478": "z_{n+1}' = \\frac{d}{dc} f_c^{n+1}(z_0) = 2\\cdot{}f_c^n(z)\\cdot\\frac{d}{dc} f_c^n(z_0) + 1 = 2 \\cdot z_n \\cdot z_n' +1.", "3fd4447b8955fa6bd803da7943b17126": "y'' - 2xy' + 2n\\,y = 0~.", "3fd449831cacdf54a3f20f277941c5e9": "\\hat{\\mathbf{B}}", "3fd44e9cf35ecc39e48d894c87fd0231": "\\dot x = -x.", "3fd45bdbe595d065b9ad1ca531e57f60": "v(d):= \\min_{s\\in S(d)} f(d,s)\\ , \\ d \\in D", "3fd49508c3cef44c3f84f05b00209c42": "p_{k,n+1}=p_{k,n}=p_k", "3fd4967fea2cd78ed95455372c9e7ee7": "p = \\frac {A_{1}}{A_{2}} = \\frac {\\pi5^2}{\\pi1^2} = 25", "3fd4de0b8f8d0a6e91f320bb441e31de": "S_{1,t}", "3fd50bf52b5a02115707ec5755926e9e": "a_1, \\dots, a_i \\in F", "3fd51b742a5c96506bec55eee0166222": "P-P*R+P*R*R-P*R*R*R+\\cdots", "3fd5cae831a97a02f4fb12ec2678d4ae": " \\{x_{n_j}\\}", "3fd5fe219a51cf7dc99124f7f35c8718": "\\hat{P}^2", "3fd6a29557fb92afa699013b75e98e72": " \\frac{H_t}{ H_d }= H_d+(1-H_d)(1+1.7 exp (-0.415 D)-0.6exp(-0.011D))", "3fd6c8358d5698f7599022e744fa846b": "\nC_{4,3/4}=\n\\begin{bmatrix}\nc_1&c_2&c_3&0\\\\\n-c_2^*&c_1^*&0&c_3\\\\\n-c_3^*&0&c_1^*&-c_2\\\\\n0&-c_3^*&c_2^*&c_1\n\\end{bmatrix},\n", "3fd6f9990a1b3643ec18bfea5ddef788": "a = b \\times c", "3fd70eaa8500431de2720891f751e8c1": "RS = \\frac{\\text{EMA}(U,n)}{\\text{EMA}(D,n)}", "3fd71b1def76de57e741bc2ec11d9758": "\\sum_{i=1}^t \\lambda_i = 1", "3fd71ba06446910f74f3b447d6887cc6": "\n\\begin{align}\ns'_0& = x_0\\\\\ns''_0& = x_0\\\\\ns'_{t}& = \\alpha x_{t} + (1-\\alpha)s'_{t-1}\\\\\ns''_{t}& = \\alpha s'_{t} + (1-\\alpha)s''_{t-1}\\\\\nF_{t+m}& = a_t + mb_t,\n\\end{align}\n", "3fd76a3ac1c12b6ea1c38828bfec1450": "|S\\cup\\Gamma(S)|\\geq \\sum_{i=0}^{r+1}{n\\choose i}.", "3fd790f0429cfa93255cf399f6385d5b": "N \\max \\left(\\frac{S_{k_i, t_i} - S_{k_i, t_0}}{S_{k_i, t_0}}, \\ 0\\right)", "3fd7c953a1b91939200d0a319ede75ef": "{\\tan \\phi}={{(1+\\rho)}\\over{(1-\\rho)}}\\cot(kx).", "3fd83d0167b82c6b7d17f6e1507c0dd3": "A=\\frac{1}{2}\\langle R(\\theta,\\theta)\\#,\\#\\rangle + \\operatorname{Ric}(\\theta,\\#) \\, ", "3fd85467364eee3746840df1977c0932": "\\frac{1}{n}\\sum_{i=1}^n x_ix'_i\\ \\xrightarrow{p}\\ \\operatorname{E}[x_ix_i']=M_{xx}, \\qquad \n \\frac{1}{n}\\sum_{i=1}^n x_i\\varepsilon_i\\ \\xrightarrow{p}\\ \\operatorname{E}[x_i\\varepsilon_i]=0", "3fd883efeef9ca17f34b0bae096d4412": "\\omega\\cdot2+1", "3fd8cc287a519d02216f0380d99a2b0a": "\\mathcal X_1,\\cdots,\\mathcal X_n", "3fd8eaedb619fad0827fd7819c0c5164": "\\vec{y}", "3fd93ba2e95477b3837ccb2a4f42ba96": "\\gamma(x,y)=\\gamma_s(y-x).", "3fd982c72fc4d1cbef8ea440d547718b": "\\mathcal{B} = h^2\\nu/10", "3fd98cfdbfe0844301211bdbcfd4000e": "\n \\left(\\begin{matrix} m {\\boldsymbol 1} & - m [{\\bold c}]\\\\\n m [{\\bold c}] & {\\bold I}_{\\rm cm} - m [{\\bold c}][{\\bold c}]\\end{matrix}\\right),\n", "3fd9ad7c95b4f6060149d45354f760f9": "k = \\lambda^{2}, \\lambda^{-2}", "3fd9d684141c1bfb9e03fcfbd3d66e54": " \\frac{dX}{dt} = A + X^2Y -(B+1)X ", "3fd9fe6341a9dc4e27ecca4649be6b4d": "\n\\frac{\\partial {\\mathbf{z}^{\\pm}}}{\\partial t}\\mp\\left(\\mathbf {B}_0\\cdot{\\mathbf \\nabla}\\right){\\mathbf z^{\\pm}} + \\left({\\mathbf z^{\\mp}}\\cdot{\\mathbf \\nabla}\\right){\\mathbf z^{\\pm}} = -{\\mathbf \\nabla}p \n+ \\nu_+ \\nabla^2 \\mathbf{z}^{\\pm} + \\nu_- \\nabla^2 \\mathbf{z}^{\\mp} \n ", "3fd9feb18f2a5cf23e43825c0ca2c32c": "F_0:S^0\\rightarrow T^0", "3fda1b8fa54aeea2ce5e95bd7f12836a": "\\tau\\ \\ \\stackrel{\\mathrm{def}}{=}\\ s ", "3fdaf68ecac25afec5ddeda59de419d2": "e^{\\lambda M^*} \\left[e^{-\\bar\\lambda M}T e^{\\bar\\lambda N}\\right] e^{-\\lambda N^*} = U(\\lambda) T V(\\lambda)^{-1}", "3fdbc042a05b7af77236f2113202eb66": "f^{-1}(U) \\to f^{-1}(V)", "3fdbd327fd63b1a5178971473822eaa0": "\\scriptstyle\\mathbf J = -\\sigma\\boldsymbol\\nabla V", "3fdc49f7b58c347c0fab4fcca63003d8": "\\delta_S[g] = \\int_S g(\\mathbf{s})\\,d\\sigma(\\mathbf{s})", "3fdc582d1533af95be8d3b61ecdd488b": "\\mathfrak{g}_T", "3fdc7b7e1cec96fa92ddd040324e7661": "f_2=x+x^7", "3fdc940c97b7a3a7950b9e2f7dabe933": " \\displaystyle {T_2} ={T_1} r^{\\gamma-1} ", "3fdcf13127f0633853ad710adb805f66": "N = \\left\\lfloor\\frac{1}{\\epsilon}\\right\\rfloor", "3fdd2edc33a15816c40292e096d28479": " \\mathbf{v} = \\mu \\left( \\mathbf{E} + \\mathbf{v \\times B} \\right), \\ ", "3fdd39fad7b41fadeec4d563e6255a38": "\\Delta z = \\frac{a}{a + b} - z ", "3fdd7a6737ef200910632deac4912d0c": "X_{1 \\dots m}", "3fdd897ae6dafd48bc0cf5fe3dc02fd0": "l\\equiv \\lnot x", "3fdde81fac586f5899b171eecf1d70d2": "\\scriptstyle \\boldsymbol\\omega_p", "3fddfb402c7a305b7afa213b93476918": "F+G", "3fde521e1b433562017b5c2869b66f81": "\\prod_{i\\in I} X_i", "3fdeb6df29f5e1f301ef8ebdd7612949": " [[a,b],c] = [a,[b,c]]+ [[a,c],b]. \\, ", "3fded16fc5b1c33e7e2fe90db6cf51eb": "\\forall g \\in G\\;\\; |f_k(g)|=1", "3fdf4021d8e42d9bfe61dd3787086087": "\\|Tf\\|_{p,w} \\le N_p\\|f\\|_p,", "3fdf5ec1e24007d810aa0117273503a9": "Z = X(t-1)", "3fdf710b432d84ef7be46379f599eeff": "\\partial_{\\nu} {J^{\\nu}}_{\\text{bound}} = 0 \\,", "3fdf89ccea43facf1a781f8773191c3c": " u(1)=1. ", "3fdfdf59de7b83f31cc5ad0d7ffabd40": "f^r(n) = \\sup \\{m \\in \\mathbb{N} | f(m)\\le n\\}", "3fe015cd0306e2bbc7c90939c2710cbf": "S = \\frac{p+q+r+s}{2},", "3fe062b90b16fb749f29c4ca5d119bd6": "a_B = \\frac{\\lambda_0}{2\\pi \\alpha} \\ ", "3fe069f52ca0d2d88911bee829669d9e": "[e_0,a_0,i_0,\\Omega_0,\\omega_0,M_0+n\\delta t]", "3fe099c7285248dd8f13c298f0b0aa8c": "\\Gamma_1(N) g \\Gamma_1(N)", "3fe0b7a27cd5c30d4116bd4a5603ed28": "\\phi\\colon G \\to G", "3fe0c7e62e6a0ab1e632915b76022e0e": "\\partial_\\mu\\left[f^\\mu-\\left[\\frac{\\partial}{\\partial (\\partial_\\mu\\phi)}\\mathcal{L}\\right]Q[\\phi]-2\\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\partial_\\nu \\phi)}\\mathcal{L}\\right]\\partial_\\nu Q[\\phi]+\\partial_\\nu\\left[\\left[\\frac{\\partial}{\\partial (\\partial_\\mu \\partial_\\nu \\phi)}\\mathcal{L}\\right] Q[\\phi]\\right]-\\,\\cdots\\right]\\approx 0.", "3fe0f69f53ee81821f72d141c74e1014": "H^{-i}(j_x^! IC_p) ", "3fe0fc1b4bf8653f2e56ff1c92dd285f": "Fdr(z) = \\frac{{{p_0}{F_0}\\left( z \\right)}}{{F\\left( z \\right)}} ", "3fe19cdd01141db53628d2a8c15bdac4": "X=z_k", "3fe2050b0d0adafe6f47b98d34be19cf": "X_{n+1},", "3fe23d21f48c8db0b5f7ce32eb7e0f82": "\\langle c-\\mu|\\Sigma^+|c-\\mu\\rangle", "3fe24a4a70e54fd3645e292baa39d67a": "\\varphi \\rightarrow \\chi\\,\\!", "3fe2ced8b7577080dfdb970ff620ecb1": " D(\\theta) \\equiv \\frac{ K(\\theta) }{C(\\theta)} \\equiv \\frac{\\partial h}{ \\partial \\theta} ", "3fe2f80518081b4f4bb907e2d346f3ee": "\\kappa + \\mu = \\max\\{\\kappa, \\mu\\}\\,.", "3fe2ff4d7a6fffba8d4523e432b65005": " V_6 = \\frac{\\pi^3 r^6 }{6} ", "3fe2ff95ffcb9501064a8312814422ba": "\\cos x + i \\sin x \\,", "3fe30d02cb55b30156df7358db29fdd4": " [a,a,x] = [x,a,a] = x \\ \\forall \\ a,x \\in H. ", "3fe311693ab3b0693c57f8666eafe690": "5:3", "3fe31d06f0f4db402d30cb5f8cb65fde": "\\left(J+1\\right)^{\\mathrm{th}}", "3fe3b391ba03e52329429c2f4d371d18": "y=x, y=-x;", "3fe3ccd2dca3afb356bf80224169afea": "V = \\oplus_{1 \\le \\alpha \\le N} S .", "3fe3de84275c2edcd95efff999764322": "RF", "3fe3edb5163642f86ee67c1992e57168": "W_h=\\eta\\Beta^{k}_{max}", "3fe4a9a26f054fd1fe68b4edf9ad70b4": "\\mathbf{F}' = q \\mathbf{E}' = q \\gamma \\mathbf{v} \\times \\mathbf{B}.", "3fe580c60ff13f17c4e50d300ea305e0": "\n\\begin{bmatrix}\nn+1\\\\\nk\n\\end{bmatrix}_q\n=\n\\begin{bmatrix}\nn\\\\\nk\n\\end{bmatrix}_q\n+\nq^{n-k+1}\n\\begin{bmatrix}\nn\\\\\nk-1\n\\end{bmatrix}_q.\n", "3fe5dc51345db9c924139d2ed80560ec": "{\\mathfrak H}", "3fe601d1a70ca968adcb84714df23122": "O_n = P_{n-1} + P_n.", "3fe6233cc32f8e5f39c81ba28626a268": "\\hat\\ell(\\theta\\,|\\,x_1,\\ldots,x_n) ", "3fe65ad7723c142607a1611274dea9ed": " cr +cs = a^2 +b^2 \\ , ", "3fe69d7b8a736d111c92870a34df1b14": "p(z) = a + c_k (z-z_0)^k + c_{k+1} (z-z_0)^{k+1} + \\ldots + c_n (z-z_0)^n.", "3fe6f3c534ff766d95125127d19403da": " 0.6 H_s ", "3fe75f22b2c013877d45cb996a9fa3c3": "(qw)^3 = (wq)^3 = (w^3)(q^3) = (-1)p =~-p", "3fe773303432273509f1b46636d01695": "\n\\hat z(\\mathbf{s}_0 ) = \\hat m(\\mathbf{s}_0 ) + \\hat e(\\mathbf{s}_0 )= \\sum\\limits_{k = 0}^p {\\hat \\beta _k \\cdot q_k (\\mathbf{s}_0 )} + \\sum\\limits_{i = 1}^n \\lambda_i \\cdot e(\\mathbf{s}_i )\n", "3fe7a97f8e028c7e1ed0d9fb00dfda81": " \\{\\operatorname{cl}(E_{\\alpha}) : \\alpha\\in A \\}", "3fe7c4098944a2d23c31aa18108f2821": "\n\\begin{align}\n \\pm \\sqrt{-i} = (i)\\cdot (\\pm\\frac{1}{\\sqrt{2}}(1 + i)) \\\\\n & = \\pm\\frac{1}{\\sqrt{2}}(1i + i^{2})\\\\\n & = \\pm\\frac{\\sqrt{2}}{2}(i - 1)\\\\\n\\end{align}\n", "3fe7eb9f972e2fcc91f165e235b5fd83": "-1/\\xi \\le x\\le 1/\\xi", "3fe878822bfb2e4f28a6fdd2ca0c01cb": "T_f(x_0,\\dots,x_n)=\n\\begin{pmatrix}\nf[x_0] & f[x_0,x_1] & f[x_0,x_1,x_2] & \\ldots & f[x_0,\\dots,x_n] \\\\\n0 & f[x_1] & f[x_1,x_2] & \\ldots & f[x_1,\\dots,x_n] \\\\\n\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n0 & \\ldots & 0 & 0 & f[x_n]\n\\end{pmatrix}", "3fe8a0d9c04a8f44a461e49bc34bc5b4": " 2(\\Gamma^\\lambda {}_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu + \\ddot x^\\lambda) = U^\\lambda {d \\over ds} \\ln |U_\\nu U^\\nu| \\ .", "3fe8e48c9d46117891bd281f2f299bb7": "\\mu_T", "3fe8ebdc83767d1b0fc4c101465c77a3": "E \\varepsilon = [constant] = \\sigma", "3fe976990a02be7e83da1760cf56f32b": " x^{*}(\\alpha) = \\left[ \\sum_{i=1}^{n} \\alpha_i \\Sigma_i^{-1} \\right]^{-1} \\times \\left[ \\sum_{i=1}^{n} \\alpha_i \\Sigma_i^{-1} \\mu_i \\right],\n", "3fe9a4e5086a06f5467fcdb639c0e52f": "Q_\\mathrm{solar} = \\eta_\\mathrm{Optics} I C A ", "3fe9aa72900e951c398ffbb50f6c9cb0": "\\vartheta(z;\\tau)", "3fe9c711a81bf1fd8d8127e1a66cd61f": " T_1 = \\frac{t - 1}{A-C}\\sum_{j=1}^t \\left(R_j^2 - rC\\right) ", "3fe9ed39ec32f4149ae8c379e2c60f64": "s(t).\\,", "3fea10bac3193dcc2bdd96ca5a4179be": "J(t)=\\left.\\frac{\\partial\\gamma_\\tau(t)}{\\partial \\tau}\\right|_{\\tau=0}", "3feb0654ea1bdeaaf7a6b96a51cc6726": "\\langle r \\rangle", "3feb136fc5833d11557fefdbaad1033e": "\\nabla(D(r)\\cdot\\nabla)\\Phi(\\vec{r},t)-v\\mu_a(\\vec{r})\\Phi(\\vec{r},t)+vS(\\vec{r},t)=\\frac{\\partial\\Phi(\\vec{r},t)}{\\partial t}", "3feb61780810c31daeb5692936b066a5": "{\\tilde{D}}_{2n}", "3feb7182996b28ec380caf7e9c92fec1": "\\scriptstyle|\\zeta|^{nq}\\leq \\|a\\|_p^q \\frac{|\\zeta|^{qn}}{|\\zeta|^q-1}", "3feb99a24eafcde7bc4abe88da805b92": "t^2 (t - 1)", "3feba33f741f888786c03f4285d28f96": "f \\in k[x]", "3febb8bd0c9811ea90e63a82ddda24b0": "(\\partial \\langle P \\rangle/\\partial V)_{\\mu,T} = 0", "3febfd9caa822cb7a7e3b81c0b9c5a5f": "K = \\frac{|\\tan \\theta|}{4} \\cdot \\left| a^2 + c^2 - b^2 - d^2 \\right|.", "3fec7a196c6d94df9ef16c6890e1e451": "\\sqrt[25]{5}", "3feced7a2dae068aab2feca41c377196": "L/K", "3fed0ab64e29047904ac78d4b3cb3a20": "dG = -SdT + VdP + \\mathcal{E}dZ\\ , ", "3fed9937736d82e0b093f1247d2b1f12": "\\lambda(y_i)= \\frac {L(y_i|\\theta_0)}{L(y_i|\\theta_1)}", "3fedaafb8c970dc4956f4f4b50024808": "t=16", "3fee068627eb04b9c256ad9a0faf4ad4": "x = (2,3,1)", "3fee0af75aad2b30cc6ad20c83b7e5cc": "y' = x A_{1 2} + y A_{2 2} + b_{2},\\,", "3fee56de3355af7ff6081898f26611ea": "\nx = A \\sin(t - K) + b \\ ,\n", "3fee5f9226bfe9f89d1031ac5a73394d": "\\textrm{hacoversin}(\\theta) := \\frac {\\textrm{coversin}(\\theta)} {2} = \\frac{1 - \\sin (\\theta)}{2} \\,", "3fee78da7c12e743bebb7fbdec321471": "r_{12}", "3feea48811f21a4dd46e3b523fa300ee": "-1 \\in \\mathrm{Spin}(n)", "3feeb76d49a0a3838e7ffb0d35c7509d": " P(no~disease~WHOIFPI) = 0.997", "3feef34989779a10c360175d92a31e1e": "|u|_{k,\\alpha;\\Omega} = |u|_{k;\\Omega} + [u]_{k,\\alpha;\\Omega} = \\sum_{|\\beta| \\leq k} \\sup_{x\\in \\Omega} |D^\\beta u(x)| + \\sup_{\\stackrel{x,y\\in \\Omega}{|\\beta| = k}} \\frac{|D^\\beta u(x) - D^\\beta u(y)|}{|x-y|^\\alpha}.", "3fef82997b24469cffdaec636e5dfeb4": "X(x)", "3fefc20f8211d9fa7f72f6587a091e33": " 0 = v t_g \\sin \\theta - \\frac{1} {2} g t_g^2 ", "3fefc9b6cffe8c8d3c86c19c0f81786a": "f_0 > B/2", "3ff015141acdebb042e4a23b819852ae": " \\operatorname{const} = \\lambda u.x ", "3ff035efc8e790c787a961c64c5e71f7": "\\varphi=0(\\textrm{mod}2\\pi)", "3ff077ddd8f13eb63891d54e9b675d72": "g_k=\\Gamma(1-k\\xi)", "3ff0a4fcb6cb05e040eadaee4571a7c6": "\\,\\overline{a}_x =\\int_0^\\infty v^{t} [1-F_T(t)]\\,dt= \\int_0^\\infty v^{t} \\,_tp_x\\,dt\\,", "3ff0c69d2de5fe9b0fda2e5e48f6a74c": "[\\overline{u}\\;\\|\\;v'\\;\\|\\;N\\;\\|\\;[u\\;\\|\\;v\\;\\|\\;M]_h]_m \\rightarrow [w\\;\\|\\;M]_h \\;\\|\\;[w'\\;\\|\\;N]_m", "3ff0da237f77c6fbb030004eef48fae2": "{| \\psi_A \\rangle} \\in \\mathcal{H}_K", "3ff0ef40d225f783027ee7f2f4269f76": "d_1\\cdots d_r", "3ff1161b4225a1e2e9225b5c732826ba": " S^{'} \\subseteq S", "3ff117ac4ddb6b71f645819ec17ffc71": "\n\\tan(\\phi_{a}) = \n\\tan(\\phi_{b}) = \\frac{1}{2\\omega_{k}}\n", "3ff1372a001a795e28e245b33a1bec84": "\n\\begin{align}\ns\n&=p_1 p_2 \\cdots p_m \\\\\n&=q_1 q_2 \\cdots q_n.\n\\end{align}\n", "3ff1439f531ad7f87767d7f9ae85be03": "V(0) = 0 \\,", "3ff16dc35c85d96b71827824d9956abc": " Z_1,Z_2,\\cdots ", "3ff1d165319a93ce61aa1910c2d6a86d": "\\frac{\\part^2 f}{\\part x \\, \\part y}, \\; \\frac{\\part^2 f}{\\part x \\, \\part z}, \\text{ and }\\frac{\\part^2 f}{\\part y \\, \\part z}.", "3ff28cb8d869a3a94b1a62b6a2b5161d": " \\langle \\psi(t)| \\psi(t) \\rangle = \\langle \\psi(t_0)|U^{\\dagger}(t,t_0)U(t,t_0)| \\psi(t_0) \\rangle = \\langle \\psi(t_0) | \\psi(t_0) \\rangle.", "3ff2960e625af6d70de019453806d306": "T_1(x)", "3ff2a9594cf1a4288763cab1d01e61c6": "\\Rightarrow\\delta Q_0=T_0\\frac{\\delta Q}{T}", "3ff303d7e6c61bfbf986db5a63b420df": "K = \\lim_{r\\to 0^+}12\\frac{\\pi r^2-A(r)}{\\pi r^4 } ", "3ff315c1fd09adaac4e573e14697164e": "g = \\exp(\\langle \\phi \\rangle)", "3ff323ff0544bed08b9b72b4f238bc96": "r_A (d) = \\left| A_d \\right| - \\frac{\\omega(d)}{d} X.", "3ff33c704f369a4f08a811560dcc8d48": "\\boldsymbol{\\mathcal{A}}(\\mathbf{r}, t) = [\\mathcal{A}_0(\\mathbf{r}, t),\\mathcal{A}_1(\\mathbf{r}, t),\\mathcal{A}_2(\\mathbf{r}, t),\\mathcal{A}_3(\\mathbf{r}, t)] ", "3ff35274fa5973b715f57669303c1f5a": " V[\\hat\\beta_{OLS}] = V[ (\\mathbb{X}'\\mathbb{X})^{-1} \\mathbb{X}'\\mathbb{Y}] = (\\mathbb{X}'\\mathbb{X})^{-1} \\mathbb{X}' \\Sigma \\mathbb{X} (\\mathbb{X}'\\mathbb{X})^{-1}", "3ff36b33e437a73e9d2e15f1fcc8bdbd": "\\mathbb{P}(n \\leq n^*|s+b) \\leq \\alpha", "3ff3a8fe7337be16d12060a3452b4043": "\\mathbb{R}^6", "3ff3d245a809d8c6d878ce7918644ef5": "\\mathbf{E} \\left[ \\big| Y_{t} - \\hat{Y}_{t} \\big|^{2} \\right] = \\inf_{Y \\in K} \\mathbf{E} \\left[ \\big| Y_{t} - \\hat{Y} \\big|^{2} \\right]. \\qquad \\mbox{(M)}", "3ff3daefc3db72e327ed8bf65d634b98": "(-\\infty,x]", "3ff47b026619a1fb178a0803b0108440": "Y(t)=e^{(t-t_0)A}\\ Y_0+\\int_{t_0}^t e^{(t-x)A}\\ F(x)\\ dx ~.", "3ff4affc49e53568b0793ebc32d1395b": "i > j = 2, 3, \\dots .", "3ff51e364fc0732146cd47ff39d2592b": "S = \\{z \\in \\mathbb{C} : 0 \\leq \\mathrm{Re} z \\leq 1\\}", "3ff5421a79ba50d3ece73090f6f67445": "K_i\\phi", "3ff54ecca5e4c92d79d0d499dae4b946": "PV_{3} = \\frac{$35}{(1.05)^{3}} = $30.23 \\, ", "3ff572bd2bd22f06c74e45b1d273b8c2": "k_{f_{n+1}}", "3ff610ff393c9349645354f30a21f2c2": "\\kappa(\\theta)=\\lambda^{-1}\\log\\int e^{\\theta z}\\cdot \\nu_\\lambda\\, (dz)", "3ff6d7988bcc08301d6e8ebff9c815cd": "G = (U, V, E)", "3ff6f4fe5e0008522d4bdef67d8bd015": "\\forall (u, v) \\in E \\ f(u,v) = - f(v,u)", "3ff6fdcaf3a4a0686d88b1647f6186f3": "J = \\int F dt", "3ff7019aed63b5625d534203a122cb56": "\\uparrow, \\downarrow, \\updownarrow \\!", "3ff70c137dec31eecf134c8a811a350d": "t_0=1 \\colon", "3ff772df4c9c639958e7f29b14c39bf2": "\\Delta \\tfrac{W}{L}", "3ff7743b92e699171ee87234aea3b1bc": "u^a\\,", "3ff7b4388c04100ee3395a74ef786b7c": " |n_1\\rang |n_2\\rang \\pm |n_2\\rang |n_1\\rang ", "3ff7d504f6d73a93e550d54ae167ee83": "t \\equiv c \\ | \\ x \\ | \\ f (t_{1}, ..., t_{n})", "3ff822e32a0cfb96f40934da278850da": "f=f^+-f^-", "3ff854688acdb70e6c8034772225ff24": "\nS_m=\\int{\\mathcal L}({\\hat g}_{\\mu\\nu},f^\\alpha,f^\\alpha_{|\\mu},...)\\sqrt{-{\\hat g}}d^4x,\n", "3ff85ad4547be49a605e3cf13244ab50": "\\frac{dx}{dt}=x", "3ff8957be3e23cc08e70788786b89b72": "\\frac{K-P}{P}=\\pm e^{-C}e^{-kt}", "3ff94001c6399bfee97a7d4241d77b98": "p_0 \\,", "3ff95a8d2991d70e09d13bb949c5cab0": "\\begin{align}\n &z_1 = z_1^+ - z_1^-\\\\\n &z_1^+,\\, z_1^- \\ge 0\n\\end{align}", "3ff95ca1412114d6535c2ccb7577c854": "\\frac{\\partial E}{\\partial \\hat{h}_i} = -2(e[n])(s[n-i])", "3ff9c517af80e07a08ab23953327acde": "{C}({1+1/A_v})", "3ffa1abda8018ba9caf1877423300d82": " \\boldsymbol{\\Omega} ", "3ffa1c6cf990b64dbcb9b1ecd16f5ea9": " \\mathbf{x}_\\mathrm{B} = \\sum_{j=1}^3 x_j \\mathbf{u}_j \\ . ", "3ffa729af511c1e05086279acb4203d1": " F^{-1}( y ), y \\in [0,1] ", "3ffab0d67694db9b511c3f792040a345": " {x - \\mu} \\over \\sigma ", "3ffabcd5e7b2421d1c09d52de5c72b79": "|{\\mathbf E} (t)|^2", "3ffaece6134aa1bab3f34b475fcb2e44": "P = \\frac{x}{47} \\times \\frac{x-1}{46}= \\frac{x^2-x}{2,162}.", "3ffb513229c1cdcd1a7468dd0c22d3af": "\\dot{e}", "3ffb6611736473fb77a74e497ebeae54": "\\operatorname{Var}(\\mathbf{X}_{ij}) = n(v_{ij}^2+v_{ii}v_{jj})", "3ffba2e2bf4bdd6b56ad9ee3a1710fa0": "P_+P_+=P_+ \\quad P_-P_-=P_- \\quad P_+P_-=P_-P_+=0.", "3ffbf78677e8432a4393a04d088471c9": "3^{(F_n-1)/2}", "3ffbf9877c864f2bd78a275d03cdeae4": " \\mathbb{R}^3 ", "3ffc165e1929366a1acaa34a913aac67": "\\phi [Z:=\\neg Z]", "3ffc4183fd8aca3479a6b62932bd1e2f": "\n w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0\n ", "3ffc555d1acecc257740ef17dc3878fc": "\\frac{b}{a^2+b^2}.", "3ffc5a7fc59c047f3416d3347c0d0e93": "\nf(x, \\mu, \\sigma) = \\frac{1}{\\sigma} \\phi\\left(\\frac{x-\\mu}{\\sigma}\\right).\n", "3ffc8dda737bdc4b8b8104e90e4c14c6": "A\\rightarrow\\neg C", "3ffc9730206d53e84fbf0f134999677e": " \\xi = \\sqrt{\\frac{\\hbar^2}{2 m |\\alpha|}}. ", "3ffcd86d6a26df8410ee6347d228fa17": "\\partial^2 = 0", "3ffd11c5c0a70eef97778f46479cf25c": " X = f_1({\\bold x})\\,\\frac{\\partial}{\\partial x_1} + \\cdots + f_n({\\bold x})\\,\\frac{\\partial}{\\partial x_n} ", "3ffd32bc0d40e8aa73da77567d1ce5d7": "\n\\alpha_{H_{n-i} A^{i-} }= {{[H^+ ]^{n-i} \\displaystyle \\prod_{j=0}^{i}K_j} \\over { \\displaystyle \\sum_{i=0}^n \\Big[ [H^+ ]^{n-i} \\displaystyle \\prod_{j=0}^{i}K_j} \\Big] }\n\n", "3ffd600804158eea0a466416a9a85852": "y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \\, ", "3ffd75ee0f4a49770e076803333b3e91": " ({}^\\rho \\mathcal{I}^\\alpha_{a+}f)(x) = \\frac{\\rho^{1- \\alpha }}{\\Gamma({\\alpha})} \\int^x_a \\frac{\\tau^{\\rho-1} f(\\tau) }{(x^\\rho - \\tau^\\rho)^{1-\\alpha}}\\, d\\tau, ", "3ffdbaefb810dda28bfceba685d01afb": "q_2(x) = 0", "3ffdd8ba230254c75304bfd68f8dd739": "b^{\\log_b n + \\log_b m} = nm", "3ffe1792b2d396813d4a009c1df06d0b": "\\theta: H^d (M; o(M)) \\to \\mathbb R", "3ffe4b50edd734ea6bc19a4239b3a9c4": "X_k = \\frac{1}{2} (x_0 + (-1)^k x_{N-1}) \n + \\sum_{n=1}^{N-2} x_n \\cos \\left[\\frac{\\pi}{N-1} n k \\right] \\quad \\quad k = 0, \\dots, N-1.", "3ffe9998347f89569ef4bb99378d8cbc": " \\eta_{ij} = 0 \\,\\!", "3ffed5b562e91d4dc5954653893c10e0": "\\frac{-1}{4 \\pi r}", "3ffee467bf804b1df67ca7b923462af6": "Tf(\\omega)=\\int_C f(\\Omega^k)", "3fff11844de6beab9d82f988671752ef": "(P\\phi)(v)=P(v)\\phi(v).\\,", "3fff417ae9df4e26746a0ed7dd93fb24": "F_{-n}(x)=(-1)^{n-1}F_{n}(x),\\,L_{-n}(x)=(-1)^nL_{n}(x).", "3fffb008fd84b7664de5346e36ef8518": "A(z) = \\frac{1}{1 - B(z)}", "3fffd65593cbd3b43559640d3b228096": "((a^2 + b^2) + (a^2 + b^2)(\\lambda\\ - 1)^2 + (a^2 + b^2)\\lambda^2)^2 = 2((a^2 + b^2)^2 + ((a^2 + b^2)(\\lambda\\ - 1)^2)^2 + ((a^2 + b^2)\\lambda^2)^2)\\,", "3ffff759492da109105b98a98df8a5c2": "\\mu_\\mathrm{sat}^{(1)} = \\rho (V_\\mathrm{S}^{(1)})^{2}", "400045923b75d51ad5ae1e8050c5c4c9": "{\\text{MPS}}=\\frac{\\text{Skill Variety + Task Identity + Task Significance } }{\\text{3} }{\\text{ x Autonomy x Feedback}}", "4000c816c3a05c25557a24a920973235": "a' ^1\\Sigma_u^-", "4001153699eaf5fafb6df0840fdb51ae": "R_\\text{sym}", "400115911159df638b254b8f29dc36d6": "\\vec{\\mathcal{P}}(x) \\left| \\psi \\right\\rangle = 0", "40012be1c03c48bffc291476efad4d73": "\\sigma^2 = \\int_{0}^\\infty (t- \\bar t)^2 \\cdot E(t)\\, dt", "4001474f69292978dbd9dbddec8572b1": "AI=I_w-I_o", "4001661a2431306c6159158d1ba79ac5": "=\\partial^\\mu\\phi\\left(x^\\nu\\partial_\\nu\\phi+\\phi\\right)-x^\\mu\\left(\\frac{1}{2}\\partial^\\nu\\phi\\partial_\\nu\\phi-\\lambda\\phi^4\\right).", "4001720b48e0686fe7f429e770f74992": "p^\\alpha = \\left(p_0, p_1, p_2, p_3 \\right) = \\left(m c, p_x, p_y, p_z \\right) \\, ,", "40018494eec46e8ac3f510faa2195ff1": "\\ln(1 + x) = x - \\frac{1}{2}x^2 + \\frac{1}{3}x^3 - \\frac{1}{4}x^4 + \\cdots.", "400186d83d2f4f10e9fb4cbb57020abb": " I_i", "400191c8262bd72e788fb2fdc39fc92e": "\\frac{\\sigma(m)}{m} < \\frac{\\sigma(n)}{n}", "40028e34080bf49f12583428942bba50": "\\scriptstyle\\overline{x}", "4002d5cb0ffb1c097a3d3a31a5fbe69f": "\\; \\varrho_{A_1\\ldots A_m} = \\sum_{i=1}^N p_i \\varrho_1^i \\otimes \\ldots \\otimes \\varrho_k^i.", "40030fd3bab5d2a4716e5eba63d063b9": "I=I_\\circ e^{j\\omega t}", "40034f43793695ad7a0cdd0b68f0ab1c": "\n \\cfrac{\\Gamma \\vdash \\Delta}{\\Gamma \\vdash A, \\Delta} \\quad (\\mathit{WR})\n ", "400383831b25d3b819a598d1da2b6fb4": "xy \\equiv zw \\rightarrow zw \\equiv xy\\,", "4003f6840c8b20edd45c2d30042cf2d0": "\\mathbf{r}_B (t) = vt\\ \\left( \\cos ( \\theta - \\omega t), \\ \\sin ( \\theta - \\omega t)\\right) \\ , ", "40044d2abe97e2bcaf73bff8a124580f": "Y^i(X_j) = \\delta^i_j,", "40053f04e294f94d6fc6f7505e8aa969": "M(\\infty, \\dots, \\infty)", "40054d8994cfe0d3ad7956b08c872eef": "A\\otimes B_1\\otimes B_2\\otimes\\ldots\\otimes B_8\\otimes B_1\\otimes B_2\\otimes\\ldots", "40055591b41f989f28d86ef722ad9e91": "|s,m\\rangle = \\sum_{m_1+m_2=m}C_{m_1m_2m}^{s_1s_2s}|s_1m_1\\rangle|s_2m_2\\rangle", "400594579365b4b5bbc0caf76050473b": " {(3/2)^2 \\over 2} = {9/4 \\over 2} = {9 \\over 8}, ", "40067e45785caad09ff9654c4a29ddf3": "C_{\\alpha\\beta} = g_{\\alpha\\beta}", "400699324d36f1573735891ef0e948de": "N \\subseteq F", "4006a369640946df83a4b72b57d65639": "1_{\\{X \\leq \\mathbb{E}[X]\\}}", "4006f92c823da759b7ceb0a0d20cdbb0": "\\scriptstyle\\|A\\|", "4006feb4e2cf41df37d42dc89eb6db3c": "1, 2, 3, 4, 6, 8, 9, 12, \\ldots", "40077120da83c3ee780c31984144b199": "\\mathbf{I}\\;\\colon\\left(\\mathbf{ab}\\right) = \\left(\\mathbf{I}\\cdot\\mathbf{a}\\right)\\cdot\\mathbf{b} = \\mathbf{a}\\cdot\\mathbf{b} = \\mathrm{tr}\\left(\\mathbf{ab}\\right)", "400801ca880e85ec19e5f2b8249fbf45": "m(j)", "40084c8af4a4fcbaed7fa86e09fe0684": "\n{\\partial \\chi \\over \\partial t} = \\left ( a_0 + a {\\partial \\xi \\over \\partial x} \\right ) (K-\\chi) - k_1(\\beta+\\chi)\\chi^n + \nD {\\partial^2 \\xi \\over \\partial x^2}\n", "400875627656fc9cb5c449264c84c30c": "T_{ij}^k = \\theta^k(\\nabla_{\\mathbf e_i}\\mathbf e_j - \\nabla_{\\mathbf e_j}\\mathbf e_i - [\\mathbf e_i,\\mathbf e_j])", "4008b97c8e3265c9df0f40c2513ae78c": " \\textstyle P=A+q \\cdot P+I\\,\\ ", "4008f475d916f04bcb101e3627200f90": "k_{et}", "4008f7343a934a6ce7d9985ffc5ce6ed": "P_i^{(j)} (t_i) = P_{i+1}^{(j)} (t_i)", "4009076c72d3cd9be9fe9828346f478f": "M = \\frac{1}{1-\\int_{X_1}^{X_2} \\alpha\\, dx}", "40091d08e1e5758a08c690ff5b0c0ac1": "c = \\frac\\lambda2\\left(1-\\frac\\lambda2\\right).", "40095abcb4df0c69f51ae6e6e6ac190a": "1 - k_i\\in U(R)", "4009813b77eada8752c989bad7882e2e": "P=KQ-\\frac{K}{URR}Q^2 \\qquad \\mbox{(4)} \\!", "40099fbfa4b1cf8353969f0055c1e72b": "\np=-A/\\rho^\\alpha\n", "4009c59f0237615372e936825eeb73fe": "\\frac{x_0 + \\cdots + x_n}{y_0 + \\cdots +y_n}", "4009e40e70ea98ac6820a8031c6e5447": "\\angle PBN + \\angle NMP = 180^\\circ", "4009f6475cc7a16a24a47afeb32babf0": "R=\\exp(A)=\\sum_{n=0}^\\infty \\frac{A^n}{n!}.", "400a18fa2a729e21085672ca1a67a8a5": "\\frac{dT}{T}", "400a1f18fc6bab8bee903d153b56a841": " b s [1 - (\\sqrt{t}/(3-t))(\\sqrt{b} - \\sqrt{t})] = t ", "400a227133dcfee9736004e9b2b2325c": "r \\in \\mathbb{Z}/n\\mathbb{Z}", "400a407574ddbb0bef15d416365774da": "\n \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = D\\mathbf{f}(\\mathbf{v})[\\mathbf{u}] \n = \\left[\\frac{d }{d \\alpha}~\\mathbf{f}(\\mathbf{v} + \\alpha~\\mathbf{u})\\right]_{\\alpha = 0}\n", "400a5cba3370607fb514c548de0a1d89": "\\chi^{(\\lambda)}", "400ad317484d87c2da142632e0f03d1a": "x \\in \\mathbb{R}^{K}", "400b9b9a77e8f030663d5c996908ce7a": "\\begin{matrix}\n\\times & c & d \\\\\na & ac & ad \\\\\nb & bc & bd\n\\end{matrix}", "400bc1e1d53015d4cee0bcd6fe866ef2": " M_v = (-2.43\\pm0.12) (\\log_{10}(P) - 1) - (4.05 \\pm 0.02) \\, ", "400bec776dfa60c21b96d29782e12fd7": "\\mathrm{Stk} = \\frac{u_{0} V_{s}}{d g}", "400bf07c63ad31a5077272a02ee2ed41": " \\alpha/2 ", "400c0d780654e8cdfa316896ff5955cc": "X = (X_1, X_2, \\ldots, X_n) ", "400c2befe63987d867259bd5553471f3": "\\ T \\,", "400c94d8a7eb7771e8d156102df0c00c": "\\int_S {\\mathbf v}\\cdot \\,d{\\mathbf {S}}.", "400c9b6b938b2fa851cdb2a48de1586d": " \\theta_i(x) \\geq \\theta_j(x), \\forall~ i < j ", "400ccd6eb94d2b3d505d5a4b0335df30": "r = \\sqrt{x^2+a^2}", "400cf6e187f73497d338d2b1749528c2": " P^{1-\\gamma}T^{\\gamma}= \\operatorname{constant}", "400d217d7ef5612663b3831c71a46b39": "w_{t+1} = \\frac{aw_t+b}{cw_t+d}.", "400d2c1ae8a642017f8a486e8b7497a4": " X(t_i) = X(t_{i-1}) + \\theta \\Delta G_i + \\sigma \\sqrt{\\Delta G_i}Z_i.", "400d6a57c7cfc26d6584f6d0c150c362": "F_0\\colon X \\rightarrow Y", "400e1b5c42a805128acc154cdeb3c25b": "N=\\left|V\\right|", "400e3bba519c8f2fb0a71778b6a567d2": "\\langle f|g\\rangle = \\langle\\Psi|(\\hat{A}-\\langle \\hat{A}\\rangle)(\\hat{B}-\\langle \\hat{B}\\rangle)\\Psi\\rangle ", "400e523bcc0d86258f8df4a23973cb93": "\\hat{\\alpha},\\hat{\\beta}", "400e5a1d6ea0ce71de015e01c18ce9c4": "W^u(f,p) =\\{q\\in X: f^{-n}(q)\\to p \\mbox{ as } n\\to \\infty \\}.", "400e807b86ad76b22f70622d0719d101": "\\ \\alpha \\ ", "400eba478f6a296a3beb153fdd377f0d": "\n \\boldsymbol{\\sigma} = -p~\\boldsymbol{\\mathit{1}} + \n 2~\\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{B} ~;~~ \\cfrac{\\partial W}{\\partial I_1} = \\sum_{i=1}^n i~C_i~(I_1-3)^{i-1} ~.\n ", "400f2ea1f76eaa9eb3f4fd96c11b3313": "\\mathbf{f} = \\mathbf{P}^{-1}\\boldsymbol{\\varepsilon}", "400f366496f8e810c953e97b831f9cbb": " V_t = V_i + V_r \\, ", "400f5ea531645d6c69d84f0dbc5aadcd": " H_j ", "400f72a24afcf4b111b6eb455b16225b": "\\! w\\approx -1", "400f769d3ed9341282b1ae3c518f1c71": "\\sin\\left(\\frac{\\pi}{3}\\right)", "4010ab8072d6a8ad83cd3d85bde9ade5": " \\mathfrak{I} \\;\\mathfrak{p}\\; \\mathfrak{I}^{-1} = - \\mathfrak{p}, ", "4010b1d9ca9da8496718c134f6ef8bef": " N = \\sum_{i=1}^n n_i ", "4010cab7f4c3e2f5c0df31bb413000dd": " t_k = \\frac{(3 + 2\\sqrt{2})^k + (3 - 2\\sqrt{2})^k - 2}{4}. ", "40110fe8f085240e0b310accddf38b44": "\\forall i\\,\\!", "40112a428dcf629c5f7da353d8ab3cb0": "\\scriptstyle R \\ll \\omega L", "40116706154c911a7cf0da9dcfa5221b": " \\rho[\\phi,\\psi] = [\\rho(\\phi),\\rho(\\psi)] .", "4011823c10b31d7ce894d3c37cb9e50f": "\nk^2 = 1 - \\frac{m \\mu}{L_1^2}\n", "40119a74a37ecb5baf0a5d13c9142efc": " \\ldots \\left(\\frac{10}{11}\\right)\\left(\\frac{6}{7}\\right)\\left(\\frac{4}{5}\\right)\\left(\\frac{2}{3}\\right)\\left(\\frac{1}{2}\\right)\\zeta(1) = 1 ", "4011a27d0f7bc25a9cc1c4702ffc7612": "\\mathbb {RP}^\\infty", "4011bd6e9b005992c2d903e80e12f0fd": "\\ddot{y}^i=\\frac{\\partial^2 y^i}{\\partial x^j\\partial x^k}(0)\\dot{x}^j\\dot{x}^k+\\frac{\\partial y^i}{\\partial x^j}(0)\\ddot{x}^k.", "4011cc1a66b251433482edd2722f7bc7": "C_\\nu(x) = \\operatorname{Re}\\, \\chi_\\nu (e^{ix})", "4011dfbeaf9d2d84302c8ad30599a131": "\\mathrm{Var} \\left[ \\tau_{r} \\right] = (2/3) r^{4}.", "40120d32fa7c55cb9755b442a0b9ced4": "R_s/R=1", "401228e0085baac605138ff6b7340a4e": " H(A) = \\sum_{w_i > 0} w_i h(a_i) = \\sum_{w_i > 0} w_i \\log_2{1 \\over w_i} = - \\sum_{w_i > 0} w_i \\log_2{w_i}. ", "4012399be226733f7cfd7dadae7a8f6c": " I(t) = \\int_0^\\infty i(a,t) \\, da ", "401339d299bb7adb14bc4397a3473fac": " \\rho_f(r) = \\frac{Q}{\\sigma^3\\sqrt{2\\pi}^3}\\,e^{-r^2/(2\\sigma^2)},", "40135cf2ac430cf1b01b901f5c49d892": "D^2\\phi=\\Omega\\wedge\\phi", "4013c9da3db19220508a2e39af8ae699": " \\phi(1) = 1 \\ ", "4013d38504ad1d017748a8205abfc886": "\n\\begin{bmatrix}\nf_{x1} \\\\\nf_{y1} \\\\\nm_{z1} \\\\\nf_{x2} \\\\\nf_{y2} \\\\\nm_{z2} \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nk_{11} & k_{12} & k_{13} & k_{14} & k_{15} & k_{16} \\\\\nk_{21} & k_{22} & k_{23} & k_{24} & k_{25} & k_{26} \\\\\nk_{31} & k_{32} & k_{33} & k_{34} & k_{35} & k_{36} \\\\\nk_{41} & k_{42} & k_{43} & k_{44} & k_{45} & k_{46} \\\\\nk_{51} & k_{52} & k_{53} & k_{54} & k_{55} & k_{56} \\\\\nk_{61} & k_{62} & k_{63} & k_{64} & k_{65} & k_{66} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{x1} \\\\\nu_{y1} \\\\\n\\theta_{z1} \\\\\nu_{x2} \\\\\nu_{y2} \\\\\n\\theta_{z2} \\\\\n\\end{bmatrix}\n", "4013da765b69dee1f73e1e6293bc17ea": "U^{-1}=e^{-\\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\theta} = \\cos \\theta - \\beta \\mathbf{\\alpha} \\cdot \\hat{p} \\sin \\theta \\,,", "4013f1d711650b1d061c0f06bb89357f": "a_{33}|a_{11}\\cdot a_{22}", "4014112c3875da54328765d157b19e5f": "I = \\frac{2 m r^2}{5}\\,\\!", "40143f4238936ea024aea02a1cb45417": "\\{a< b< c\\}", "401441cbc02434def3ab79b4e3d4cafd": "(m',", "401444486a255181256baf88c9e1c51f": "\\mathfrak m", "401489314e9a30ce2b4ef0c460f5d428": "\\scriptstyle\\sqrt[3]{X/k}", "4014e5292c12a34cf4be2453365cb99d": " S_t = S_0 \\exp\\left\\{ \\left(r - \\delta - \\frac{\\sigma^2}{2}\\right) t + \\sigma B_t \\right\\} ", "4014fb60298bd6403c2dcd354fa614e1": "\\ln\\mathbf{B}=\\sum_{i} {x_i \\ln\\mathbf{B}_i}", "40153d226601fd5d4ac194315394a2ef": "H_{\\epsilon_0}\\,\\!", "4015afafa59d2b91fc7da76999a68e80": "\\mathfrak g/\\mathfrak h", "4015d4114623119d8bcbd90bcfe61e67": "\\begin{align}\n\\mathcal{H} & = T + V \\\\\n& = \\frac{\\mathbf{p}^2}{2m} + V \\\\ \n& = \\frac{(\\mathbf{P}-e\\mathbf{A})^2}{2m} + e\\phi\n\\end{align}", "4016a9d58af4fed63f8a86de57ec844d": "\n\\begin{align}\n\\textbf{k}-\\textbf{k}_0 = \\textbf{G}_\\textrm{hkl}, (1)\n\\end{align}\n", "4016da56af8ad47ebf691a6065a2a656": "P^D(\\tilde t_i)", "4016ed23cb29d4aa77d34a38168b53a2": "\\ C=\\frac{P_tG_tA_r}{(4\\pi)^2R^4}\\frac{\\pi}{4}(R\\theta)(R\\phi)(c\\tau/2)\\eta", "4017465df69b834e6e7fad57552dca0f": "\\!\\tau_b", "4017c1bee8aea8cb310f8e3c8531d0cd": "n+\\nu ,\\, \\boldsymbol\\Psi + \\sum_{i=1}^n (\\mathbf{x_i} - \\boldsymbol\\mu) (\\mathbf{x_i} - \\boldsymbol\\mu)^T ", "4017c50070990db8d71743c89ff6dd02": "\\mathrm{MV(t)}=K_p\\left(\\,{e(t)} + \\frac{1}{T_i}\\int_{0}^{t}{e(\\tau)}\\,{d\\tau} + T_d\\frac{d}{dt}e(t)\\right)", "40180db9cb78c03624b838451d3f9dc9": "h=0.7", "401852d722f71e46e48315fa56ee4ab0": "\\delta \\mathcal{L} = -\\frac{(\\partial_{\\mu} A^{\\mu})^2}{2 \\xi}", "40185cdf316dfe6c3becc867b55a6ea2": "\\phi\\mid\\psi ", "4018c6c7daf315da632495eab665678c": "X\\! ", "40190c9d70b5142510e3b8ea9cdbc1a1": "\n\\left(\\frac{1001}{9907}\\right) \n=\\left(\\frac{7}{9907}\\right) \\left(\\frac{11}{9907}\\right) \\left(\\frac{13}{9907}\\right). \n", "401915af30d02cbda45a4df58fcbf863": "A = \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1\\mathit{n}} \\\\ a_{21} & a_{22} & \\cdots & a_{2\\mathit{n}} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{\\mathit{n}1} & a_{\\mathit{n}2} & \\ldots & a_{\\mathit{nn}} \\end{bmatrix}", "40192b7d5391bfd15561186562bddce1": " < \\Omega ", "40195e33fa115deb7293b769cfd60537": "E^{(2)}_{lm} = \\frac{1}{l(l+1)}\\int \\mathbf{E}\\cdot \\mathbf{\\Phi}^*_{lm}\\,\\mathrm{d}\\Omega", "40196854cddba54633c29bc6b052054e": "\\overline{\\bigcap_{i \\in I} A_{i}}\\equiv\\bigcup_{i \\in I} \\overline{A_{i}}", "40197186e3ea8833b5e5a4a8bdfb3888": "\\hat{H}'_0", "401984f61d48668c65a944ed2d60a3fa": "x_0 = \\alpha \\sin(\\chi/\\alpha) \\sinh(t/\\alpha) \\cosh\\xi,", "4019bea21cbe71d02bf363ce7a780a8d": "\\begin{align}L(e) &= 0,\\\\\nL(g^{-1}) &= L(g)\\\\\nL(g_1 g_2) &\\leq L(g_1) + L(g_2), \\quad\\forall g_1, g_2 \\in G.\n\\end{align}", "401a85ba1e1b955977a5240f275978cb": "r(0)=f(0)", "401a8a49d3286cf9d9c81285c28c4e03": "(1+2\\epsilon)|I_1|+\\tau|E|<|I_2|", "401a8db945d17d5b901425fd5b09dd03": "T_k(\\beta)", "401abc3a4b8b14ad11ffd3b134d22542": "\\scriptstyle \\chi(S^2)=F-E+V=2", "401afd9a84d089e4e5146a288d47994e": " \\; f \\;", "401b093606e37133ff2542fccc8a1314": "t\\isin I, a\\le t\\le b\\,", "401b798587e13cc2a599a514bb9debfc": " r_m = \\frac{r_2 + r_1 }{2} \\quad (13)", "401b7cd182bff1ad7d45defe13adc23c": "(i - 1)", "401bc24d57343121560bdbeaa212045e": "\n\\bar \\theta = \n\n\\left.\n\\begin{cases}\n\\arctan \\left( \\frac{\\bar s}{ \\bar c} \\right) & \\bar s > 0 ,\\ \\bar c > 0 \\\\\n \\arctan \\left( \\frac{\\bar s}{ \\bar c} \\right) + 180^\\circ & \\bar c < 0 \\\\\n\\arctan \\left (\\frac{\\bar s}{\\bar c}\n\\right)+360^\\circ & \\bar s <0 ,\\ \\bar c >0 \n\\end{cases}\n\\right\\}\n\n= \\arctan \\left( \\frac{0.086}{0.986} \\right) \n\n= \\arctan (0.087) = 5^\\circ.\n\n", "401bc38dca591d2ad01b15d345f9ebb3": "\\bar{u}(x)", "401c369c96ccb7e4e1a467093170be46": "\\mu_L", "401c8216906c939c3ed21c02b6a1768b": "z_b = 2C_RD", "401cdf03a9bf818e78837e749c120212": "P_i = D_K(C_i) \\oplus P_{i-1} \\oplus C_{i-1}, P_0 \\oplus C_0 = IV", "401d1c26e544a176cafc8eda4fa7b27d": "\\gamma^*:=f\\circ\\gamma", "401d60abc3efbcca8f3504522c693f1c": "\\theta(0)\\equiv \\theta(1)", "401d71f2a2494cae4070283a9702a825": "E = \\sigma A \\, ", "401d8955481e3f9ca27483d98b79cd5c": " f_-(z)={1\\over2i\\pi} \\int_\\Gamma {\\phi(t)-\\phi(z)\\over{t-z}}\\, dt\n\\quad z\\in\\Gamma ", "401dee2f98eb0f74a8d226fa5b219661": "\\frac{j_e}{j_i} = \\sqrt{\\frac{m_i}{m_e}}.", "401e56534354acce09108d887817be94": "{\\rm Space\\;Savings} = 1 - \\frac{\\rm Compressed\\;Size}{\\rm Uncompressed\\;Size}", "401e5eed79235bbac2e931e9acbf95cd": " \\frac{\\Delta F}{V} = \\left( \\frac{A^2}{4} \\right) \\left[ \\left( \\frac{\\partial^2 f}{\\partial c^2} \\right) + 2\\, \\kappa\\, \\beta^2 \\right].", "401e7b4a1bae69912d6651f38d79b426": "\\sqrt \\pi", "401ee43cd8c99e73953f7b7c43925ba1": "\\mathbf{j}_s = -\\frac{n_se_s^2}{m}\\mathbf{A} + \\frac{n_se_s\\hbar}{m}\\mathbf{\\nabla}\\phi", "401f4b048380cda4063130bb0a9de136": "m_1+m_2+m_3=0\\,", "401f92d9c336926f6097ca30ad618847": "\np \\propto \\dfrac{1}{r} \\,\n", "401f9d1df3dca5731b1286746fc76f46": "H(Tx,Ty)\\leq L d(x,y),", "401fd1847eabb61b6b4a2b6df8676d14": "y^{n-1}", "401fff421c4c3e54f164cb7c4c424f14": "\n\\mathbf{Q}_2= \n\\begin{pmatrix}\n0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\\\\n1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\\\\n0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 1 \\\\\n1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\\\\n0 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\\\\n0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0\n\\end{pmatrix},\n", "4020108ac7c60a892d74e535e5a8db2a": "{i}^2 = -1 \\,", "402016e0af25a92e4f948ced2197f3e4": "\\sigma(n)x_\\mathrm{m} \\\\\n0 & \\text{otherwise}.\n\\end{cases}", "4023f20d236c6890a79b14112c0a781f": "f_u=f(1-\\frac{v}{c})", "40240d0e8702e026d3842ce82f6a235f": "Y_W(c, \\tau) = \\frac{1}{\\sqrt{c}}\\cdot\\int_{-\\infty}^{\\infty} y(k) \\cdot \\Psi\\left(\\frac{t - \\tau}{c}\\right)\\, dt ", "40242488498f8cd4d476201484a83b70": "\\log{f(z)-f(\\zeta)\\over z -\\zeta} = -\\sum_{m,n\\ge 0} d_{mn} z^n \\zeta^n,", "40247661ea3cb6c621f461ab9529a615": "\n\\tan\\left({\\textstyle\\frac{\\pi}{4}}\\right) = \\tan\\left({\\textstyle\\frac{5\\pi}{4}}\\right)\n= \\tan\\left({\\textstyle\\frac{-3\\pi}{4}}\\right) = \\tan\\left({\\textstyle\\frac{(2n+1)\\pi}{4}}\\right) = \\cdots = 1.\n", "4024c6ac637f2e7c04a8d3f9eecc369a": "SU(4)", "4024db3ea0f725dbda04f3d140d03173": " {} = p_{12} z_0 - p_{02} z_1 + p_{01} z_2 . \\,\\! ", "4025fcaf8a7f959aa761ce80d960467d": "\n\n\\mathbf{S} \\approx \\mathbf{L}_{\\rm ISCO} + \\mathbf{S}_1.\n\n", "402607e731fa51e6cdfaadad1e31e619": "S_4 = {27 \\over 25} \\approx 133.2 \\ \\hbox{cents}", "402611c0a67cc17e45ee420bffd14874": "\\mathrm{d}U=\\delta Q-\\delta W\\,", "402621c22ef3cec41aee4342cb4b7829": "\\mu = r\\lambda", "40262e9558717ce5a3184ed2bcb35e92": "A^{k-2}", "40263058fc4c4c024bc7adda409fa3be": "\\mathcal{L} = \\frac{1}{2} \\partial_x \\psi\\, \\partial_t \\psi \n+ \\left( \\partial_x \\psi \\right)^3 \n- \\frac{1}{2} \\left( \\partial_x^2 \\psi \\right)^2 \\quad \\quad \\quad \\quad (1) \\,", "40267e50bf963abb25e73bbe2dd48cd1": "\n\\mathrm{CNR} = \\frac{C}{N}\n", "4026a275320a476b7ba6c0061bb619c7": "O(N^2D^2)", "4026bef01407f82813b3f2829094294a": " y_1' + u' = q_0 + q_1 \\cdot (y_1 + u) + q_2 \\cdot (y_1 + u)^2,", "4026bf46bb5fb404f2358986022c2057": "U_n=(R-2Q)U_{n-2}-Q^2U_{n-4}=(a^2+b^2)U_{n-2}-a^2b^2U_{n-4}", "4026cf632332be69a1723ff1370bfd34": " \\mathrm{div}(F)= \\sum_{0 \\leq k < n}(P+k\\cdot Q) - \\sum_{0 \\leq k < n} (k\\cdot Q). ", "4026fc10cba9e08b059f9d8c4b4e0050": "\\sum_{j=1}^{\\infin} M_j = K", "402720918fd3b2f4f611288d75a7bc70": "Position = -0.5", "402748ed1d05e0f1b927b92369e42dbf": "\\left[\\frac{\\partial D}{\\partial n}\\right] _{\\text{x,t}} = -P_{\\text{x,t}}", "4027703d2632291aef5fca2fb8a9a896": "\\left[ dpa \\right]", "4027a8fd12c408f265fabe46a90df231": "U[\\sigma f] \\to bU[\\sigma]", "4027aad4edde6705847380a18e49f78c": " \\phi=1+\\frac{1}{\\phi}", "4027e08f5de8567ce483705703eb6eb2": "S_b < C_S.\\,", "4027e0a4f72457502ab52cee9037629e": "f(x) - f(\\bar{x}) \\not\\in -(C \\backslash \\{0\\})", "4028740252037531cf99b976a5ae7644": " C = \\pi^{-n/4} (2\\pi)^{-n/2}", "40289fbe3e39ad61f6474e5ae4a5a41f": "\\int_{\\mathfrak{H}^2} f_1\\overline{f_2} \\,dA = \\int_{-\\infty}^\\infty \\tilde{f}_1 \\overline{\\tilde{f}_2} \n\\,{\\lambda \\pi\\over 2} \\tanh({\\pi\\lambda\\over 2})\\, d\\lambda.", "4028da662e10e617fb9fa064a61bc0e8": "\\mu_e = \\frac{\\varepsilon_r\\varepsilon_0\\zeta}{\\eta}", "4028ded576e05984cddd94244a122c1a": "\\Phi = \\int_0^z \\left[ \\frac{Gm}{(a+z)^2} \\right] dz", "4028f048b2ae7f3b96b0791a9b23489b": " \\beta \\left(\\left[X,Y\\right],Z\\right)=\\beta \\left(Z \\triangleleft Y,X \\right) ", "4028f97511fbf2e2558f1c045d662a07": "\nf \\ll \\frac{k_B T}{h}\n", "40291fa87a84c783adab3bfdf51addc6": " \\left [ \\S n \\in \\mathbb{N} \\quad n^2 \\leq 4 \\right ] = \\left\\{0, 1, 2\\right\\}", "40299983c6c1ab8a5de7bbdca45ca25d": "\\beth_\\alpha \\ge \\aleph_\\alpha", "4029a72f9b278079663b5ddb9266f0e6": "\\left |\\varphi\\left(f(M^{-1}(z))\\right) \\right|=\\left|\\frac{f(z_1)-f(M^{-1}(z))}{1-\\overline{f(z_1)}f(M^{-1}(z))}\\right| \\le |z|.", "4029d551874e16f153007d0942b2f467": "\np_y(y) = p_x(\\phi^{-1}(y)) ~ \\left|\\det \\left[ D\\phi ^{-1}(y) \\right] \\right|. ", "4029e477924928abbc7c474d741243b8": "D(a)", "402a1768516004d1c34c3f1ed31f3057": "\\tilde{S}_t = e^{-rt} S_t", "402a2296f596ae46d612545e1a490bda": "z_1 \\dots z_m ", "402a6357c57b9ba2cac6d87a8f6bf636": "t \\rightarrow \\infin", "402a9d6930244d4303a7c44869bdd076": "y/x", "402ac479a72d8bda83474beacabac956": "{\\bar{BP}}_3", "402b2e7f0b46144b065d645b6710c722": "\\pi_i(S^2)= \\pi_i(S^3)\\oplus \\pi_{i-1}(S^1) . \\,\\!", "402c05600b8336e431a8629395aa4343": "\\frac{d [S]}{d t} = k_1[A][B] - k_f[E][S] + k_r[ES]", "402c66fec51df82d0f50da4e0b3f5310": "\\sigma: V \\longrightarrow T", "402d59c60cf72140a640af2190572c5f": "1+\\cot^2 y = \\csc^2 y\\,\\!", "402d6d52e9a0a09d3c3b9bf2d4bba600": " c_{2n} = \\frac{s_n}{s_{2n}} = 2 \\frac{s_{2n}}{S_{2n}} , ", "402de3ea4b2ea0477c1dd154746ca1af": "\\exist y[My \\and \\varnothing \\in y \\and \\forall z(z \\in y \\rightarrow \\exist x [x \\in y \\and \\forall w (w \\in x \\leftrightarrow [w = z \\or w \\in z])] )].", "402decc208e54ddcb345c8e720968824": "\n\\tau_0(z) = b_0 + z,\\quad \\tau_1(z) = \\frac{a_1}{b_1 + z},\\quad \n\\tau_2(z) = \\frac{a_2}{b_2 + z},\\quad \\tau_3(z) = \\frac{a_3}{b_3 + z},\\quad\\cdots\\,\n", "402e2229fd08649ffebe1aee46e9c8e6": "E>0\\,\\!", "402e536ddbb9918fcc61c19eeb2f477a": "u(t,x)\\in C^2((0,T)\\times\\mathcal{M})", "402e57e581ac5c7a73324edb7e7996f3": "\\Delta_2^{\\prime}F(J)^{observed} = \\bar \\nu [R(J) ] - \\bar \\nu [P(J) ] ", "402ec18da099488bb57bd8b6c105c134": "2h^2=\\cos\\frac{\\pi}{n}-\\cos\\frac{2\\pi}{n}.", "402f1ada090f47e2da2a29b395ca0882": "\\mathbf{A}=A_{1}\\otimes\\cdots\\otimes A_{n}", "402fd76f15d427e7c75880914d597bd5": " C_{(\\pm)} \\Gamma_\\text{chir} C_{(\\pm)}^{-1} = \\beta_{d+2} \\Gamma_\\text{chir}^T\n~~~~ \\beta_d= (-)^{d(d-1)/2} ", "403009a2d28adf64681f052f4bfecf3c": "\\prod_p (a,b)_p=1", "40300d269b0fa7c42a1648ac86a02cad": " d_\\text{f} = d - \\frac{\\beta}{\\nu}\\,\\!", "40301f4f4463f025d8ecfc711ca554f2": "p_\\text{H}^2", "4030342260df6e95cc05eee20a79fdec": " E_{kin} = \\frac{1}{2} m v^2 ", "40309644804849154add8faab329bd32": "\\exists (a,b)\\in D", "4030c863dbab2951104bde46a75af04b": "-\\chi(t) = \\beta {\\operatorname{d}A(t)\\over\\operatorname{d}t} \n\\theta(t) . ", "4030d2d1b029cf02d1f6b9e69e80126c": "{d \\over dt}\\left\\{ A \\right\\} =-\\alpha k_+ \\left\\{ A \\right\\}^\\alpha \\left\\{B \\right\\}^\\beta +\\alpha k_{-} \\left\\{S \\right\\}^\\sigma\\left\\{T \\right\\}^\\tau \\,", "403156ea3e4eb52e0ab51c30091342e4": "p-m\\ge 0\\,\\!", "4031571fd4ccc392de47740aad8017a2": "\n\\tilde{S}_{1} =M_{1}-m_{1}-\\frac{i}{2}G\\gamma _{2}\\cdot \\partial \\mathcal{L}\n,", "403178a47b4912272bf2232f863754b5": "x_k[n]", "4031adfdfe40954438ce071c210e00da": " q = ( v \\times y_1 ) = (2 \\times 2 ) = 4.0\\text{ m}^2/s", "4031bfcf2cb79b7b3c5d6c15fa940d62": "a q", "4031c236d5a8c3a119e0f8e43e9b0f61": " y = c_1, ~~~~ t-z = c_2, ~~~~ t^2-x^2-z^2 = c_3. ", "403231c78f9b927838679e9295869815": "\\gamma_0^2 = {+1}", "4032527ff4116a2127a92b34c925ee6c": "\\neg (p \\land q) \\vdash (\\neg p \\lor \\neg q)", "40327dca00bdeb932c922b83c7742646": "R := \\operatorname{Hom}_{\\mathcal{L}} (I,I)", "4032ac45a5d6b88042ec1add7ac645ef": " \\mathbb{R} ^ \\mathrm{N} ", "4032e86dae5c2ee647825430af141e22": "\\boldsymbol x=\\{x_1,\\ldots,x_m\\}", "4032fb7eeb4b1f911ac7ecc1abbc86d3": " c_n = 0\\, ", "40338332bbae73e7570329d95c2ff158": "f(x)=\\frac{1}{b-a}\\,\\operatorname{rect}\\left(\\frac{x-\\left(\\frac{a+b}{2}\\right)}{b-a}\\right) .", "4033d0d85fda3d21c921ea17b6a156ba": "J_k(I)\\,", "4033d595444f7fb0f40ec163fa738a3b": " ...HC... \\ ", "40349019cf7e8c35bb069ef4001377d7": " V(a,c) = \\underset{s \\in \\mathbf{R}}{\\operatorname{argmax}} \\ (W(s) - c(s-a)^2), ", "403495e5e9e7870f5f69815fd03d0457": "I^\\pm[S] = I^\\pm[\\overline{S}]", "40354637a58a42d221d55737e428c6b3": " C=ENC_{k_n}(ENC_{k_{n-1}}(...(ENC_{k_1}(P))...))", "40354c2e11729c3b02a03016245ea1e3": "f(A)=\\sum_{S\\subseteq A}(-1)^{\\left|A\\right|-\\left|S\\right|}g(S)\\qquad(**)", "4036197c98002a4f482d3a9f60d95b1b": "256^{\\,\\!256^{256^{257}}}", "403660b00f512b5ca3d9af07478caf9e": "\n \\rho_0 c^2_{33} = B_{3333}, \\qquad \\rho_0 c^2_{31} = B_{1313}, \\qquad \\rho_0 c^2_{32} = B_{2323},\n ", "40369eb3627fac5005189e8164108424": "\\left(\\frac{dn_1}{dt}\\right)_\\mathrm{spon}=A_{21} n_2", "4036decc8d7bd20081a285545fea445b": "X=\\textstyle{\\frac 32}U", "4037375406929e1b4ed5510af3f57e61": "b = 2", "403789203bf5512ad91bf53251074613": "C'_{Op} == [C_{Op} | P]", "4037b0bfc5dd28d532bfcaeb0114e9d4": "2^{2 \\times 8} + 2^{6 \\times 8} = 281474976776192", "4037bbc1b8187437497881c338beb8b1": "(\\varphi, \\theta) = \\left(2 \\arctan\\left(\\frac{1}{R}\\right), \\Theta\\right).", "4037c4968d3347918252558e8a668e6f": "p^L = gc", "4037e1c381ffca7cc72318eb3221e169": " \\oint_C w'(z)\\,dz =\\oint_C (v_x-iv_y)(dx+idy)= \\oint_C (v_x\\,dx+v_y\\,dy)+i\\oint_C(v_x\\,dy-v_y\\,dx)=\\oint_C \\mathbf{v}\\,{ds} +i\\oint_C(v_x\\,dy-v_y\\,dx).", "40383225f6f7f7172102718d49bf21cb": " V = \\int_0^H r^2 \\pi \\, dh ", "4038b9988290d9cf537ce49b84559255": "\\boldsymbol{\\omega}_1", "4038ef82386ffc158cad48dbe1c74aa9": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 7.072977\\log_e(T+273.15) - \\frac {4094.935} {T+273.15} + 59.41150 + 1.092121 \\times 10^{-5} (T+273.15)^2", "40392fd1620c587c1b9e4a85f7364e7c": "f=2^{(p-69)/12} \\times 440\\,\\text{Hz}", "40393294d896f24e78609f612d33729f": "3\\Delta^2(a_n) + 2\\Delta(a_n) + 7a_n = 0", "40393b357a4c4c2eddceb07cf9cd5b35": "Y(t)=N_x \\times \\delta t \\times \\sigma (E,t )[1-\\omega_X] \\exp(-t\\cos \\frac{\\theta}{\\lambda}) \\times I(t)\\times T\\times\\frac{d(\\Omega)}{4\\pi}", "403995b729be9f9ce705315565cf6f48": "\\scriptstyle r =k(T) [A][B]", "403a77ee510e0100f2fee1de539af103": "f: N \\to M", "403a8f9f79730ab2b7b2eef20d188627": "\\langle x , y \\rangle = \\sum_{i=1}^{k} x_i y_i\\ \\bmod\\ 2\\,.", "403a99dc2c71c66d13e16a7aa23f0caf": "H(Y|X)\\,=\\,H(X,Y)-H(X) \\, .", "403ab1839df5257a03edd63fb81ab155": "\n T_{11} = \\cfrac{\\sigma_{11}}{\\lambda} =\n 2C_1\\left(\\lambda - \\cfrac{1}{\\lambda^5}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right] = T_{22}~.\n ", "403ac6b497729ecf1f1fba1869a09d61": "\\frac{a-b}{a+b}\\,\\!", "403b083697ecbb3a5bdcd0a5229bd3bf": "(Ax, x) \\le \\lambda_k.", "403b369aeaded6494746f8a3ab714dac": "\n K_{\\rm I} = \\frac{1}{2\\sqrt{\\pi a}}\\int_{-a}^a F_y(x)\\,\\sqrt{\\frac{a+x}{a-x}}\\,{\\rm d}x\\,,\\,\\,\n K_{\\rm II} = -\\frac{1}{2\\sqrt{\\pi a}}\\left(\\frac{\\kappa -1}{\\kappa+1}\\right)\\int_{-a}^a F_y(x)\\,{\\rm d}x, \\,.\n ", "403c21d0f2125e7b682d0eb5d45665cc": "\\mathbf{T_{02}}=\\mathbf{T_{01}}", "403c4fe767746b6081b313ba1c32f626": "\\Vert \\cdot \\Vert_{l^2}", "403c5b1c7ffccef6b2182d2fa493455a": "x \\in [0, 0.5]", "403c71422f65569d5489c7ecd7ec1824": "T(t,\\sigma) = C e^{A(t-\\sigma)} B", "403ccee7ef8d0258280d1db26e71ea99": "(G, s)", "403cf5a0cd7d1291f4e859ccbc24270f": "E_{CMI}", "403d1b79f7d4a4c6ed9ddc49c7b1169a": " |\\epsilon| /|\\lambda - \\lambda_{\\mathrm{closest~ to~} \\lambda}| ", "403d20e0acf815b37ced8939966930c2": "\\sigma_{\\bar{x}}^2 = \\sigma^2/n", "403d389846f7823b2107f3a335f17290": "K(z,w)", "403d77ee42af7b7c5b0e14fb03480674": "g = \\tfrac{1}{24}(p+2)(p-3)(p-5).", "403d9af68c6c869095fce37670fa5b22": " |n_1, n_2; S\\rang \\equiv \\mbox{constant} \\times \\bigg( |n_1\\rang |n_2\\rang + |n_2\\rang |n_1\\rang \\bigg) ", "403df0c8227ff3fd556b65da287e115e": "\\mathbf{F} = - G \\frac{m_1 m_2}{r^2}\\mathbf{\\hat{r}}", "403e147a89a1128739c272dc055029c8": "\\frac{d^{|n+1|}}{dx^{|n+1|}}\\delta(x-a)\\,", "403e6f7dddce11ff6a8cad1e42ddaff6": "_{nominal}\\alpha", "403f218c77409bbb592bec3de66cf2f9": "\n\\Psi(t)=U(t,t_0) \\Psi(t_0) \\ \n", "403f251ff2681ce10e7ae16d0184091e": "\n(x,y)\\mapsto (x-\\lambda a, y)\n", "403f5ddff204efa2256688293c384cb4": "Q+k(2mP)", "403fd7001e72bbcc6f5e7a4c384d9508": "\\frac{dz}{dz'}=\n\\frac{-1 \\cdot -1 \\cdot \\mathit S \\cdot {far} \\cdot \\mathit{near}}\n {\\left( z'\\left(\\mathit{far} - \\mathit{near}\\right) - {far} \\cdot S \\right)^2}\n\\cdot \\left(\\mathit{far} - \\mathit{near}\\right)\n", "403ff2b0efb15d760c041336618b6154": "~U_{21} = U(\\mathbf{r}_2 - \\mathbf{r}_1) ", "40400c764bc85a7eb8440aece4dcd350": "p(y,A)", "404011d86812ce7758cee9119423a8af": "\\frac{d\\psi_S}{dt} = \\frac{3}{2}\\left[\\frac{Gm}{a^3 (1-e^2)^{3/2}}\\right]_S\\left[\\frac{(C-A)}{C}\\frac{\\cos\\epsilon}{\\omega}\\right]_E", "4040613a5e16cd0497cbc6bb22dc9096": " S \\subsetneq N ", "4040e59b2702c56010075143dedb7450": "\\mbox{D}(y)=21(y-8)\\mbox{ mod }26", "4041886f9e7cb846b656335f57d7c36d": "s_\\theta = s_\\zeta = +1", "4041acb9584f5bcb613744802568f554": "F(f_1 + f_2)", "4041d1f1391edaa67ca5291dc8c8e732": " \\left( \\begin{matrix} \\rho\\\\ u \\end{matrix}\\right)_t +\\left( \\begin{matrix} u&\\rho\\\\ \\frac{c^2 }{\\rho }&u \\end{matrix}\\right) \\left( \\begin{matrix} \\rho\\\\ u \\end{matrix}\\right)_x=\\left( \\begin{matrix} 0\\\\ 0 \\end{matrix}\\right) ", "4041d5dab0f0228c9133dc54850d0401": " f : K \\to L ", "40424d5af0ab04825477a1e075558691": "\\scriptstyle |\\phi\\rangle \\rightarrow |\\phi_n\\rang ", "404267f5a461d7b72ea5327883f2a146": "\n\\begin{alignat}{2}\nJ_i & = -\\mathbf\\Sigma_i \\nabla v_i \\\\\nJ_e & = -\\mathbf\\Sigma_e \\nabla v_e.\n\\end{alignat}\n", "404269a44724a23564fe68ce464cecb6": "\\sum_{j=0}^{\\sqrt{n}-1}{|\\operatorname{intersect}(j/\\sqrt{n})|} \\leq n", "4042cff29b3df2863b2b1bbd64ae3187": " x(s,t) = {3 t - {1\\over 3} (s^2 + s t + t^2)^2 \\over t (s^2 + s t + t^2) - 3} ", "4042dc00bb1f6218a00111943b901922": "\\lambda\\to 0", "4042eca3462a0ac9f8f5ebd152c35d24": "K_{\\rm Ic}", "404340e643244e0aa4305b5a3d2c4568": "c\\ne 0", "40434cf4d06c20cffa3e681e6310e240": "N(t) = N_0\\,e^{-t/ \\tau} =N_0\\,2^{-t/t_{1/2}}. \\,\\!", "4043b29051a1e820f2a6e33faa9c73f4": "\n\\begin{align}\n\\sigma_0(24) & = \\prod_{i=1}^{2} (a_i+1) \\\\\n& = (3 + 1)(1 + 1) = 4 \\times 2 = 8.\n\\end{align}\n", "4044325c556109c3d5f89b4514dd9ee4": "CE = \\%C + 0.33 \\left( %Si \\right)", "40443dbbe89840d5432d6578accd6284": "\\sum_{v\\in S\\cup T} y(v)", "40443f41a65f558f94805dd4558bc13e": "K_1 = \\frac{1}{cot(A)-cot(\\alpha)}", "40445061fbf852ef40d5df02f646ac9b": "P_c(z) \\,", "40446d33139acdf18858f200f1229bda": "y|X\\ \\sim\\ \\mathcal{N}(X\\beta,\\, \\sigma^2I)", "4044a49c9969aca66a50dc82d6170d7f": "K_2 \\times V_c \\times (\\tfrac {4d}{3D})^2", "4045152e04e670d44afea37bf5f84526": "TR^{-1} \\pmod{N}", "40454fbb9a742b2005a1dce01fcd3de6": "\\mathrm{^{238}_{\\ 92}U\\ \\xrightarrow [6 \\beta^-]{+\\ 15 n} \\ ^{253}_{\\ 98}Cf\\ \\xrightarrow{\\beta^-} \\ ^{253}_{\\ 99}Es}", "4045ab0599780ea4a181c125160684ad": "W_u", "4045d65f5447c3eeb4c2635f3d2345ec": "\\omega=\\sqrt{\\omega_0^2-\\gamma^2}", "40466ccc977ac515591cf41ae5ef9d6c": "\n | \\psi \\rangle = \\int_X \\Psi (x)| x\\rangle\\; d\\nu (x)\\; ,\n", "40469bf876939f2c42c93b76f906a503": "\\frac{1}{V}\\frac{dV}{d\\xi}=\\frac{1}{\\xi}(-1+U+(n+1)^{-1}V)", "40470f0daafa893301573660b8f92586": "\\sqrt{1-x^2} \\,\\!", "40471b8630be258da6f5c29aa49ff508": "(\\mathcal{L}f)(s) = \\int_{0^-}^\\infty e^{-st}f(t)\\,dt", "4047386ab7115fcd58f871a285084df2": " 3 ", "40473e6db12cbbf5871e594cde0b0e2a": "\\mathbf{R} = \\arg\\min_\\boldsymbol{\\Omega} \\|\\mathbf{A}\\boldsymbol{\\Omega} - \\mathbf{B}\\|_F \\quad\\mathrm{subject\\ to}\\quad \\boldsymbol{\\Omega}^T\\boldsymbol{\\Omega} = \\mathbf{I}", "40476864467ab58dae190e83e14d4901": "SS^{-1}", "40476b3ec37a21292d8e359213a7aabb": "P' = UPU^{-1} = \\sqrt{AA^*} = W \\Sigma W^*.", "4047a7232699a30c3cce63aade37de4c": "\\scriptstyle<2\\times10^{-29}", "4047ea39d96ef067fca8db1bd92164bc": "S_i = k(T_a+T_{rx})B{\\cdot}\\frac{S_o}{N_o}", "40483013c20617c5255ac4e300e76d47": " \\frac {du_f}{dt}=\\frac {dU} {dt} \\cos(\\beta+\\psi)-U\\frac {d(\\beta+\\psi)} {dt} \\sin(\\beta+\\psi) ", "40484787359d1ce029106305e2a195c7": "\\alpha=a_{t,j_t}/\\beta", "40488a2e1f19d48ccd1d818209957a9e": "f=\\sum_{k=0}^m \\frac1{k!}f_{a_1\\ldots a_k}(z)c^{a_1}\\cdots c^{a_k}", "40488dd17057853ff4702d1f89be6844": "\n\\frac{d}{dt} w_{k} = \\frac{\\partial K}{\\partial J_{k}} \\equiv \\nu_{k}(\\mathbf{J})\n", "40489cdad099bef51993883e6f40d626": "\\displaystyle{|a_2|\\le 2.}", "4048a8f9aa0144b80689c56cadcc8b2b": "\n f^{*}(\\cdot) = \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) = \\sum_{i = 1}^n \\alpha_i k(\\cdot, x_i),\n", "4048fb6f8aef95283d027ac75572bc95": "y^2=2a(x-\\tfrac{3}{2}a)", "40492a154fdcbfaa10ff662b3cedad78": "\\widehat P/\\widehat H\\to \\widehat M", "4049434557a0a3169aa76082fa817b35": "I_C=\\beta I_B+(\\beta+1)I_{CBO}\\,", "404954d3fbdf6a31e1054b4ff1704d70": "V =S \\otimes \\mathbf{C}[\\Lambda]=\\bigoplus_{\\alpha \\in \\Lambda} S \\otimes e_\\alpha.", "40497b49ef28bc7e42d3915596a0ad53": "232250619601 = 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 31 \\cdot 37 \\cdot 73 \\cdot 163\\,", "4049c18cd589d602feb5e4f95f5ec7e5": "t+dt", "404a51aa268cd2216be622a2bc9dea10": "X \\to X', Y \\to Y'", "404afb411a08f9da84e94ea81371df9e": "x,y\\ge 0", "404b1d7b3731cc87e1192a6dbdfbf6d2": "\\frac{| \\partial A |}{| A |} = \\frac{2}{\\lfloor N / 2 \\rfloor} \\to 0 \\mbox{ as } N \\to \\infty.", "404b309a59d0c66e159af86e086177d3": " \\sum_{i=1}^J [\\alpha p_{0i}+\\mu_i(x_i)(1-p_{ii})]=\\sum_{i=1}^J[\\frac{\\alpha p_{0i}}{\\lambda_i}\\mu_i(x_i)+\\lambda_i p_{i0}]+\\sum_{i=1}^J\\sum_{j\\ne i}\\frac{\\lambda_i}{\\lambda_j}p_{ij}\\mu_j(x_j). \\qquad (4) ", "404b3f40d6ba3319854d22c45a53c4f8": "\\scriptstyle f '_n", "404b7ef35d457d0505d4c8003c1bf600": "{}_3F_2 (a,b, -n;c, 1+a+b-c-n;1)= \\frac{(c-a)_n(c-b)_n}{(c)_n(c-a-b)_n}.", "404b9130539f37491ccc1c0e32277de0": "\\mathbf{\\Sigma}^0_\\alpha \\cup \\mathbf{\\Pi}^0_\\alpha \\subseteq \\mathbf{\\Delta}^0_{\\alpha+1}", "404c22bf0567dd84aa36a2740ec24c24": "T_qQ", "404ced5f619974dd288bb70a0eaeb010": "\\sum_{k=1}^K \\dot{M}_k \\hat{S}_k ", "404d01312495e1c37e3ea8a6df178865": "k_1, \\dots, k_m", "404d064f83685f5e55ee98b3014da26d": "\\vec{\\pi} = (\\pi_1, \\pi_2, \\pi_3, \\pi_4)", "404d80758e99020c7797037f81fefc5b": "u,v\\in\\left(\n\\mathbb{Z}_{2}\\right) ^{2}", "404d851681f3b9b5b803a96d7572854a": "\\textstyle f", "404d923f17708e03510d9f3d0ce1bfeb": "W_{2 n-1}(q)", "404dab242fa56882541c2b26d2f9ffaa": "{}^IE^1_{p,q} = H^{II}_q(C_{p,\\bull}).", "404def72022153c7b9245b61406c8d9a": " x^4 = 5, \\ ", "404e15c97202552de8c530ba922f3145": "\\left[\\frac{a}{b}\\right] = 1.", "404e186eafeb28d72e1c5617cb4368f4": "A,B\\in \\mathrm{RAT}(N)", "404e56a0cfa2602f54f787acc7a1dd3e": "\\mathrm{tr}\\left(\\left(\\begin{array}{cccccc}\n0 & 1 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right)\\cdot\\left(\\begin{array}{cccccc}\n1 & x_{12} & x_{13} & 0 & 0 & 0\\\\\nx_{12} & 1 & x_{23} & 0 & 0 & 0\\\\\nx_{13} & x_{23} & 1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & s_{1} & 0 & 0\\\\\n0 & 0 & 0 & 0 & s_{2} & 0\\\\\n0 & 0 & 0 & 0 & 0 & s_{3}\\end{array}\\right)\\right)=x_{12} + s_{1}=-0.1", "404e6f1ed71326a7821835cacfbbdd17": "\\ \\ \\ \\ f'(x)", "404ea599f26ee20968ab25a0ee5fed5a": " \\overline{A}_M: \\overline{M}\\to E^*,\\;\\; c\\mapsto \\left(h\\mapsto \\int_c h \\right),", "404eb32ade19fe758af8d3fd094607c1": "\\left(\\sqrt{\\frac{2}{5}},\\ \\frac{-2}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "404eccd7c7d5674e386a3b7c24c2e5d6": "{\\mathbf r}_1", "404f4fc922de87020ace89b2e1bd7a18": "0 \\rightarrow M \\rightarrow I_*, \\ \\ 0 \\rightarrow M' \\rightarrow I'_*,", "404f9574c324f9f3e1c1abebaf741d04": "\\text{Trail}_\\text{bicycle} = \\frac{R_w \\cos(A_h) - O_f}{\\sin(A_h)}", "404fb4bc0c48b485b9bac7374504324f": "\\pi_* A", "404fcf124f37c53d5b597901354328a4": "\\frac {\\mathrm dy}{\\mathrm dx} = \\frac {\\mathrm dy} {\\mathrm du} \\cdot\\frac {\\mathrm du}{\\mathrm dx}.", "404ff27771a37b27cbe6a6146a3bfc0a": "b^{-1}\\,\\bmod\\,n", "40502acde535df03678be248b5a80e5e": "\\mu_{03} = M_{03} - 3 \\bar{y} M_{02} + 2 \\bar{y}^2 M_{01}. ", "4050c0423454da302580b92d8ca17ca5": "\\varphi\\colon\\operatorname{St}(A)\\to\\mathrm{GL}(A),", "40517671c9fbe0ca8822970b44bc0d47": "A:H\\to H,", "4051cd03907764d580e5e2957244a44d": " \\begin{align} x_0 & = \\sqrt2 \\\\\n y_1 & = \\sqrt[4]2 \\\\\n p_0 & = 2+\\sqrt2\n \\end{align}\n", "4051ecc4282ccf950c41c94951c8c694": "\\mathrm{d} f(x) (y - x) \\leq f(y) - f(x);", "405207115306d529de306d96bdc49dc3": "C=(P,L,I)", "4052607ec388efd44190030cc29152ac": "\\operatorname{E} \\left [ \\frac{R(n)}{S(n)} \\right ]", "40529299add8e5389d4619a428b48dd2": "1\\le j\\le 3", "4052fd4bd263c4f9ac8d51b242d380f0": "[c,L_n]=0", "405382e7b8ad0783f22667bb5897d45e": "dR\\bar{y} = dR'\\bar{y}", "405387cc1dcfa672398dc22b663c9761": "\\Delta I_{L_{Off}}", "405390acf7f6911a932ee0f43cd2c4c1": " \\hat{\\mathbf{x}}_r(t) ", "4053ae577582d06f6ac4bfe2370fec8e": "\\rm Succinate + Q \\rightarrow Fumarate + QH_2 \\! ", "405443697652c4bad7aa6efe4f7d5f7e": "\\Delta{}H_{\\mathrm{vap}}", "405452f4658a2f6e5ffd68821b15e033": "xy\\,", "4054700c21aa35c739426e61a6dab75e": "\\mathbf{E} = e^{-i \\omega t} \\sum_{l,m} \\sqrt{l(l+1)} \\left[ a_E(l,m) \\mathbf{E}_{l,m}^{(E)} + a_M(l,m) \\mathbf{E}_{l,m}^{(M)} \\right]", "40548d4ebfa88b5b7d60c76ed1aaf75e": "\\gamma_s = \\frac{(G_s+e)\\gamma_w}{1+e}", "40553485aa1b1bf1e9022e942c1f4d32": "\\scriptstyle\\{\\varnothing, \\Omega\\}", "40554aa5074dcb265c996ccd96bea793": " \\mathbf{F} = \\{F_1, F_2, \\ldots, F_m\\} ", "405571007c15fa919a0579a6b9e7796e": "XY = X\\cdot Y", "40559a6f4e5072ebe5362b0aa1dbb297": "A^\\alpha B_\\beta{}^\\gamma C_{\\gamma\\delta} + D^\\alpha{}_\\beta{} E_\\delta \\nrightarrow A^\\lambda B_\\beta{}^\\gamma C_{\\mu\\delta} + D^\\alpha{}_\\beta{} E_\\delta \\,.", "4055c045b092d4a57c7df342b6044a25": "|A-B| < |B-C| \\Rightarrow |f(A) - f(B)| < |f(B) - f(C)|. \\,", "40564d0ee97e0081b99f0df76e825375": " \\mathbf{u} \\times \\mathbf{v}= u^j v^k\\epsilon^i{}_{jk} \\mathbf{e}_i ", "4056915d7c85d0f41aa4337391a5d877": "\\epsilon_{ii}", "4056f7803b0e2550343401e094d0406c": "\\log_{10}0.012=\\log_{10}(10^{-2}\\times 1.2)=-2+\\log_{10}1.2\\approx-2+0.079181=-1.920819", "4057197ce44f87ce8ce3624875fdcaeb": "R_0 = \\beta\\tau = {\\beta\\over\\mu}", "405736ca7ee09d966b64b8f36685c1d2": "\\boldsymbol{X}_{i}", "40573c0dedad9625155d1aa81c7b0893": "\\psi_{-|E|}(x) = C_1 e^{\\sqrt{2m|E|/\\hbar^2}\\,x} + C_2 e^{-\\sqrt{2m|E|/\\hbar^2}\\,x}.\\,", "40575e34d77356cf44e3105f3f8f257e": "\\left(A - \\lambda I\\right)^k {\\bold v} = 0,", "4057a85bce65ac3f040f48ef8a769ec4": "C_D(\\{x\\})", "4057d37dac63459e0c860327fb4b39c2": "\\mathbf{mn} = e^{\\mathbf{B}},", "4057da4b24eb7618b510742f641de2ab": "\\left|\\mathbf{p}\\right|", "405839e87123b81832d77f366879241d": "\nA_{CFG} = \\{\\langle G,w \\rangle \\mid G \\text{ is a CFG that generates string } w \\}\n", "40597e9090ccd2bd97ed66c0b35ee7ab": "s_o", "405999b7f9afd1a709f9cfb021bcc020": " E_n ", "40599ed918bf9edbad28d99f31a9e0a0": "E=\\sum_{j=1}^n \\frac 1 2 mv_j^2\\,,", "4059f1387dba2de5ab1d6b659c3aed44": "\n\\sum_{i=0}^n J_i = I. \\qquad (5)\n", "405a2b97f5b05c863f31e96a49dfd724": "g^{\\nu\\xi}\\,", "405a2fe5b47e67c3ec5cd7d16772017d": "\\mathbf{P}\\left ( \\frac{1}{m} \\sum X_i \\ge q\\right )\\le \\inf_{t>0} \\frac{E \\left[\\prod e^{t X_i}\\right]}{e^{tmq}} = \\inf_{t>0} \\left[\\frac{ E\\left[e^{tX_i} \\right] }{e^{tq}}\\right]^m.", "405a37fb43abb450ee131ec5cd66da57": "~N_1+N_2=N~", "405a4b358cbb4bbe2fc4fbb615087efc": "lift(B \\Rightarrow 1) = \\frac{P(1|B)}{P(1)} = \\frac{P(B \\and 1)}{P(B)P(1)}", "405a6658162863e7926d224a0bcb59ab": "\\forall \\varepsilon > 0 \\; \\exists c \\; \\forall x < c :\\; |f(x) - L| < \\varepsilon", "405a84b805dea7224d08aa47b4979980": "a_i,b_i,a\\in{\\mathbb Q}(x,y)", "405ac8a30ec5a45f485ac3e3388603ed": "\\text{DOR} = \\frac{26/3}{12/48} = 34.666\\ldots", "405ad9ab94c707227303a232696ff4a4": "(...P_{-2},P_{-1},P_0,P_1,P_2,...)", "405b07fe5fd93dd1a84f81bcbc6e50ed": "n\\times s", "405b0a1bbf19e3fcf95572b8bcbd12bf": "\\frac{d}{dt}\\mbox{det}\\left(A\\left(t\\right)\\right)=\\left(\\nabla\\mbox{det}\\left(A\\left(t\\right)\\right)\\right):\\left(\\frac{d}{dt}A\\left(t\\right)\\right)=\\mbox{tr}\\left(\\mbox{adj}\\left(A\\left(t\\right)\\right)\\frac{d}{dt}A\\left(t\\right)\\right)\n", "405b745ec342f856a8bac6209880f7c7": "R_{DT}", "405b91c230e565d7b28280a951875bc7": " \nN\\left(U^\\left(n\\right)\\right)-M\\left(U^\\left({n-1}\\right)\\right)+L\\left(U^\\left({n-2}\\right)\\right)=0\n", "405ba8d0ac2bd1091ea71c43f6ab9492": "g(X')", "405bb3765aebb42e8654543241add837": "h/\\lambda", "405c2b657d5cf192f7ad574d7de93ca6": "\nU^{-1}(t,t_0)=U(t_0,t)\n", "405c6dfaa58c861bc45af3d988d9a629": "(15)\\qquad T_{ab}=\\frac{1}{4\\pi}\\,\\Big(\\, F_{ac}F_b^c -\\frac{1}{4}g_{ab}F_{cd}F^{cd} \\Big)\\,,", "405c7ecd309a6b679e9e7c7456e031b2": "i\\not\\in J", "405cd6f9518d03dd6081a257ab9eab11": "H_{(1-X)} = \\frac{1}{\\operatorname{E}[\\frac{1}{(1-X)}]} = \\frac{\\beta - 1}{\\alpha + \\beta-1}\\text{ if } \\beta > 1, \\& \\alpha> 0. ", "405ce73a36ff9aa303a24d46cf6cc249": "x_N * y\\ =\\ \\scriptstyle \\text{DTFT}^{-1} \\displaystyle \\left[\\scriptstyle \\text{DTFT} \\displaystyle \\{x_N\\}\\cdot \\scriptstyle \\text{DTFT} \\displaystyle \\{y\\}\\right]\\ =\\ \\scriptstyle \\text{DFT}^{-1} \\displaystyle \\left[\\scriptstyle \\text{DFT} \\displaystyle \\{x_N\\}\\cdot \\scriptstyle \\text{DFT} \\displaystyle \\{y_N\\}\\right].", "405d4fb002ece3be60d98fda36b884f7": " \\max_{\\nu=m+1,\\dots,m+n} \\left | \\sum_{j=1}^n b_j z_j^\\nu \\right |, ", "405d83bab6bd2d1c507a50bb9dcaf99d": "G={\\left({P \\over S}\\right)_{ant} \\over \\left({P \\over S}\\right)_{iso}}\\,\\!", "405de147cffd11d08948513a4f5dab6a": "\\mathcal{r} F = \\underline{0}", "405df3562f5e4ca221fe2b9b58f931d5": "\\left\\{\\mathcal{F}_t\\right\\}_{t\\geq 0}", "405e25266c6c9a59cc82db27dcf9c4f3": "\\left| S_{2m+1} - L \\right| = S_{2m+1} - L \\leq S_{2m+1} - S_{2m+2} = a_{(2m+1)+1} ", "405ef23cecf3deef0b284f3dde18bf0f": "D \\theta^1\\theta^2\\bar\\theta^1\\bar\\theta^2", "405f5aacf401067162db98e7856c5a86": "\\partial_i f = \\frac{\\partial f}{\\partial\\xi^i}:=\\frac{\\partial\\bar{f}}{\\partial\\xi^i}", "405f62f818cc0e903f81b8d2d978da86": " C_{\\alpha \\beta}\\,\\!", "405f72ebbe32fd0ed2531a861d93c63c": "c > \\sqrt{N}", "405f80c20c49cae26f91da874c8bc97e": "\\lim_{x \\to -\\infty}{f(x)}", "405fb0582007b16c79f03ba595aede3b": "H_k=B_k^{-1}", "406002ff11913852d9099a567a797682": "(N-1)^{th}", "406030a9d3e81ce9bb6b542669dae657": "\n= \\sum_i \\langle (M_i \\otimes I)(\\rho \\otimes \\omega)(M_i \\otimes I), \\; I \\otimes \\Psi_i ^*(O)\\rangle \n", "4060c1c725a917282e54a15e194de032": "p(z).", "4060dc285a034da3b65ff77a61b8ac72": "c_1 \\in C_1", "4060f630f1d4f287f7160e469c880e86": "p_1,\\, p_2,\\, p_3,", "406107c75bc2096e19359d9ce753d70d": "\n\\delta(x-vt)\\frac{\\mbox{d}}{\\mbox{d}t}\\left[m\\frac{\\mbox{d}w(vt,t)}{\\mbox{d}t}\\right]=\\delta(x-vt)m\\frac{\\mbox{d}^2w(vt,t)}{\\mbox{d}t^2}\\ .\n", "406122e0cf7323bb606f7016bb473d08": "\\scriptstyle\\lfloor {n/2}\\rfloor", "406128b698a2271526b355d6a2661b44": "\\lim_{k \\to \\infty}P(k)= p3^{k-1} \\, ", "406188588dcf415ba4dd12b19349c041": " \\alpha= \\frac{U_x\\Delta t}{4 \\Delta x}", "4061bdfc342c4354b9d153bdc64841ec": "E_{K_{MAC}}(m_1' \\oplus t \\oplus t) = E_{K_{MAC}}(m_1')", "4061ce96abf0a84f439ade145eb5064a": " S(\\omega)= \\left|\\mathcal{F}[W(t)](\\omega)\\right|^2= \\frac{S_0}{\\omega^2} ", "4061e293039932e6eaa77456724a510b": "m_{2}\\rightarrow \\infty", "4061ed0b1134bec4a02e386111f0163e": "T_n(x)^2 - (x^2-1) U_{n-1}(x)^2 = 1 \\,\\!", "4062b8247bb3c44f78d631b54c41599f": "P_{y0}", "40634545c66c447381fae0432e03c0e7": " \\sum_{i=1}^n \\sum_{j=1}^{n_i} \\widehat\\varepsilon_{ij} = 0 \\,", "4063615b49f554ef298181418cde241f": "(V\\cup\\Sigma)^{*}", "406381251d2d4637ebfac7f24cc49c6f": "-s", "4063e5195ffdc48f9039d241e632795d": "(\\mathcal{T},\\mathcal{F})", "4063f3d37161c4bdbdb87350c674162e": "\nRE_{\\hat g} \\,\\, = \\,\\,{{\\sigma _g \\,} \\over {\\hat g}}\\,\\,\\, \\approx \\,\\,\\,\\sqrt {\\,\\,\\left( {{{\\sigma _L } \\over L}} \\right)^2 \\,\\,\\, + \\,\\,\\,\\,4\\left( {{{\\sigma _T } \\over T}} \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {{\\theta \\over 2}} \\right)^4 \\left( {{{\\sigma _\\theta } \\over \\theta }} \\right)^2 \\,}\n{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(18)}}", "406455f2fe4221c77360582a8b9c50dc": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{7}{\\sqrt{3}},\\ \\pm1\\right)", "4064a1f9119370c5a9b66022cbd9329b": "~A_0 U~", "40653a500d162d172a6a52b07080d179": "G_\\mathrm{srgb}", "406566d18fbf5a3bd3ff106348e28ba1": "(Df)_x E^s_x = E^s_{f(x)}\\text{ and }(Df)_x E^u_x = E^u_{f(x)} \\text{ for all }x\\in \\Lambda,", "40657852fe447e13b7ca9e0a4b4881f4": "\\int_0^{\\theta}\\log(1+\\sin x)\\,dx=2G-2\\text{Cl}_2\\left(\\frac{\\pi}{2}+\\theta\\right)-\\theta\\log 2", "4065d0a5b63bac3357b5cdf9ab0f4137": "\\{f\\}_i", "4065e5ec5bd3491ac86ed988d0d9dc60": " Q(x) = \\frac{x}{1 - e^{-x}}=\\sum_{i=0}^\\infty \\frac{(-1)^iB_{i}}{i!}x^{i} = 1 +x/2+x^2/12-x^4/720+\\cdots", "4066495d6a829dcfaf9d2947726801e7": "\n\\mathrm{SNR} = \\frac{\\sigma^2_\\mathrm{signal}}{\\sigma^2_\\mathrm{noise}}.\n", "406656fe4cc0c5fc368956687510fa9b": " \\delta W = \\mathbf{F}\\cdot\\mathbf{v}\\delta t,", "4066c7063b60a8e5b2fcacb164574c4c": "10^{-21}", "4066d87b4c122746d1b62d17fdb86b45": "E_1(x)=x-1/2\\,", "406700e37452b321d97032ef743d169b": "\\frac{J_\\alpha(x)}{\\left( \\frac{x}{2}\\right)^\\alpha}= \\frac{e^{-t}}{\\Gamma(\\alpha+1)} \\sum_{k=0}^\\infty \\frac{L_k^{(\\alpha)}\\left( \\frac{x^2}{4 t}\\right)}{{k+ \\alpha \\choose k}} \\frac{t^k}{k!}.", "40676546a3cfe27128d5a2b8492b9599": "\\prod_{i=1}^n G_i \\;=\\; G_1 \\times G_2 \\times \\cdots \\times G_n", "406772acba175878e3c3d687f4646a7d": "\\left(\\frac{-1}{\\sqrt{10}},\\ \\sqrt{\\frac{3}{2}},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "40679003cfbea58224cd11a8e4865d44": " \\hat{x}_T(\\omega) = \\frac{1}{\\sqrt{T}} \\int_0^T x(t) e^{-i\\omega t}\\, dt.", "40679f4bf81d35e9f26711064d40ca3c": "\\langle\\mu|\\xi\\rangle = i\\,^t\\mu^* C \\xi", "4067b0b029c0e910b7a30454cd754aae": "\\hat {a(t)} = \\hat {a}e^{-j\\omega t} \\ ", "4067c89e87b0d95af39aee2531f5234e": "\\alpha \\geq 0 ", "4068234306e78e5559da6a56a1bcfa01": "{n} \\geq {q}", "4068937013d8973ef8673aee34e4a62b": "f_1, \\dots, f_m", "4068a5ba1364d79e9812a29e1977bb4e": "\\frac{2}{\\lambda^2}\\bigg(\\frac{1}{1+2\\lambda}-\\frac{\\Gamma(\\lambda+1)^2}{\\Gamma(2\\lambda+2)}\\bigg),\\,\\,\\lambda > -1/2", "4069178f2236eec2280bc148596595c6": "c/2(\\textrm{erfc}^{-1}(1/2))^2\\,", "40691cc713638762990785a5d36d1126": "\n \\tau = \\sigma~\\tan(\\phi) + c\n ", "40692e816cb05a3cbece3b13385942c9": "\nH_{abc}+H_{bca}+H_{cab}=0.\n", "40693048ddfed251843fd7603106213e": "\\psi:\\Sigma^*\\to M\\,", "406936b9b35e1bee91e34649b520ca0d": "\n\\left( \\frac{\\partial Q_{m}}{\\partial p_{n}}\\right)_{\\mathbf{q}, \\mathbf{p}} = -\\left( \\frac{\\partial q_{n}}{\\partial P_{m}}\\right)_{\\mathbf{Q}, \\mathbf{P}}\n", "406976451e849b704a5243efd63bb98a": "\n\\sigma_t = \\frac{8 \\pi}{3} \\left(\\frac{\\alpha \\lambda_c}{2\\pi}\\right)^2\n", "406a06432ddc4709b277e01dd802d329": "VAG(x^3 -7x + 7,(2,4)) ", "406a17ca99ab800c7687039748853cef": "\n\\begin{align}\nX_{k_1,k_2} &=&\n \\sum_{n_1=0}^{N_1-1}\n\\left( \\sum_{n_2=0}^{N_2-1}\n x_{n_1,n_2} \n\\cos \\left[\\frac{\\pi}{N_2} \\left(n_2+\\frac{1}{2}\\right) k_2 \\right]\\right)\n\\cos \\left[\\frac{\\pi}{N_1} \\left(n_1+\\frac{1}{2}\\right) k_1 \\right]\\\\ &=&\n \\sum_{n_1=0}^{N_1-1}\n \\sum_{n_2=0}^{N_2-1}\n x_{n_1,n_2} \n\\cos \\left[\\frac{\\pi}{N_1} \\left(n_1+\\frac{1}{2}\\right) k_1 \\right]\n\\cos \\left[\\frac{\\pi}{N_2} \\left(n_2+\\frac{1}{2}\\right) k_2 \\right] .\n\\end{align}\n", "406a561b8378805df2cd719d1eec061b": " S_{s-th} = ln(10) {kT \\over q}(1+{C_d \\over C_{ox}}) ", "406a64c35b8d2059effdcb7c95588706": "P_{net}", "406aebd5618d428bbab677f44861741b": "\\mbox{Transformity} = \\frac{\\mbox{emergy input}}{\\mbox{exergy output}} ", "406b2898d6915ad45fd0c33910ba1a88": "\\psi=\\begin{pmatrix}u \\\\ v \\end{pmatrix} = \\begin{pmatrix} u^1 \\\\ u^2 \\\\ v^1 \\\\ v^2 \\end{pmatrix} ", "406b29ee433abc831658b659945bdab0": "2N", "406b2a6ded4c862ae4618b040847589a": "H(x) \\ \\stackrel{\\mathrm{def}}{=} \\begin{cases} 0, & x < 0; \\\\ 1, & x \\geq 0; \\end{cases}", "406b2fcaac3315545edeeab59337c26c": "\\sigma > 0, \\quad \\sigma \\notin \\mathbb{ N} \\ ", "406b307cbc0ed9d16923a2612449c517": "\\scriptstyle |\\lambda| \\; \\le \\; \\max_i \\sum_j |A_{ij}|.", "406b5a4ff7c29e79d517e22627ffa5f6": "\\tau = \\tfrac{\\tau_N}{2}", "406b60b7dc497e5a62528ddfc4ace885": "S(q)=\\frac{1}{(q)_{\\infty}}\\sum_{n=1}^{\\infty} \\frac{q^n \\prod_{m=1}^{n-1}(1-q^m)}{1-q^n}", "406b788ee1fe9183993a4a4cfb197814": "L(1) = \\frac{\\pi}{\\left(2-\\left(\\frac{2}{q}\\right)\\right)\\!\\sqrt q}\\sum_{n=1}^\\frac{q-1}{2}\\left(\\frac{n}{q}\\right) > 0.", "406b903451e4e051a0129bb69df63bff": "\\langle p_1-p_2,p_3-p_4\\rangle=\\#\\{p_1,p_2\\}\\cap\\{p_3,p_4\\}", "406bd84a8b3f1ef65f3be92ee0234adc": "h=1.0.", "406c1b6579a91e89c5492cacf29b4e21": "=1234+166x+94x^2\\,\\!", "406c337fcb16cffaa55770fb1fe9f778": "\\sigma(s_i) = s_{i+1} \\quad\\mbox{for }0\\leq i 0", "40745d13340648dda73eeaf653ebdaaa": " K_{i1j} ", "4074e2e840aa64689074159a5b8f2cdb": " \\theta_0\\,", "40753097dcb8d0f5b948ac682cc0b490": "\\lim_{p \\rightarrow 0+} p \\log p = 0", "407543affa65062c724599121fee9260": "\\Phi : L(H_1) \\rightarrow L(H_2) ", "407565eb74df80c9df78e5f015a480aa": "\\pm \\varepsilon", "40759916c7f41da40d1c85fecc1b584f": "\\displaystyle{Tf(w)={1\\over 2\\pi}\\int_{\\partial\\Omega}\\partial_n (\\log|z-w|) f(z)={1\\over 2}\\Re (Hf)(w),}", "40759ce4f486294ebda7d6b3696c6aa1": "c_4(n)\\,=\\,\\sqrt{\\frac{2}{n-1}}\\,\\,\\,\\frac{\\Gamma\\left(\\frac{n}{2}\\right)}{\\Gamma\\left(\\frac{n-1}{2}\\right)}\n \\, = \\, 1 - \\frac{1}{4n} - \\frac{7}{32n^2} - \\frac{19}{128n^3} + O(n^{-4})", "4075c9d6c1d8aaff4bdec69da28b8b97": "\\gamma(s,z) = z^s \\, \\Gamma(s) \\, \\gamma^*(s,z)", "4075e73f1c052bc71e0e66d56b0651f0": "0 \\leq i \\leq k", "4075f0e31b6714a03616f91a371a8674": "a,\\; b,\\;c,\\text{ and }d", "40769d773131bb163f6dd6263fd6ebad": " Loss_{min} = 20 \\ log_{10} \\left ( \\sqrt{ \\rho - 1 } + \\sqrt{\\rho } \\quad \\right ) \\, \n\\quad \\text{where} \\quad \\rho = \\frac {\\max [ Z_S, Z_{Load} ]}{\\min [ Z_S, Z_{Load} ] } \\, \n", "4076acbdd844449eb8b4c09585a7fc73": "|x_\\nu| < |\\rho|.", "4076cf7a090341b0dcf8164bf485bbd8": "y_{n+1} = y_n + h \\sum_{i=1}^s b_i k_i\\,", "4077217b3ef08ed1943464c4e3edb624": " D_\\gamma(f) = \\frac{d}{dt}(f \\circ \\gamma)(t=0) = (f \\circ \\gamma)'(0)", "4077222bc347cd74de946326bd654d3e": "\\begin{matrix}\\left({\\left\\lfloor{\\frac{y}{12}}\\right\\rfloor+y \\bmod 12+\\left\\lfloor{\\frac{y \\bmod 12}{4}}\\right\\rfloor}\\right) \\bmod 7+\\rm{anchor}=\\rm{Doomsday}\\end{matrix}", "40775b8fd4f7a41d3c6f49778de7f875": "\\begin{bmatrix} g_{11} & g_{12} \\\\ g_{21} & g_{22} \\end{bmatrix}", "4077769cee76acd48104415fc1a45c3a": "4n-1", "4077c20b7314f87a124389501d5ff6ff": "|\\psi_2\\rangle", "4077f1bca8e6e9d6b30e40fed739161c": "\\theta=\\Theta e^{-\\kappa t/2m}\\cos\\left ( \\omega \\right )\\,\\!", "4078637fa1b8ee6d1ef7c458b233e99b": "\\mbox{FFD} = f \\left( 1 + \\frac{ (n-1) d}{n R_2} \\right), ", "4078c9d751061536f4cc54d86caf4590": "score(X, Y)", "4078e7609a53847880b4047bbd6c7750": "Z^0_0", "4078f00be178dd9652ae7909ac2a1b3a": "L_\\text{ice}(T) = (2834.1 - 0.29 T - 0.004 T^2)~\\text{J/g}.", "40796441baeb370f1de52f690fca6bdb": "A_{}^{}", "407985daa02813b3d79a0b02dcc24039": "\\pi\\circ\\sigma", "4079a85f81832057711b58923600a912": "a_k = \\frac{2}{\\pi} \\int_0^\\pi f(\\cos \\theta) \\cos(k \\theta)\\, d\\theta", "4079ca97d24c43cbf808bb448fcb972f": "\\mathbf{E}\\,\\mathrm{e}^{i(kz-\\omega t)} = E\\, \\mathbf{\\hat p}\\,\\mathrm{e}^{i(kz-\\omega t)} = E (\\cos\\theta\\, \\mathbf{\\hat f} + \\sin\\theta\\, \\mathbf{\\hat s})\\mathrm{e}^{i(kz-\\omega t)},", "4079fadc2693045f25228c872b386182": "\\{\\mathcal{F}_t\\}_{t\\geq 0}", "407a48419547661c47bf5de683bc4bd3": " \\frac{w_\\mathrm{photoreceptor}}{{(f/\\#)}_\\mathrm{microlens}} \\ge \\frac{w_\\mathrm{pixel}}{{(f/\\#)}_\\mathrm{objective}}", "407a62a59cb751d4b21881472bedcbea": "\\ \\frac{dM^2}{M^2} = \\frac{1 + \\gamma M^2}{1 - M^2}\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)\\frac{dT_0}{T_0} ", "407b4932083b1e260096c7303baef1f8": " f:[0,\\pi/2] \\times [0,2 \\pi] \\rightarrow \\mathbb{R}P^2. ", "407b64c54acff749d0ae789e162de4dd": "\\mathfrak{g} \\rightarrow \\mathrm{L}(\\Gamma(\\mathfrak{g}))", "407bd9252787af19943803951ea4e9d0": " t_1, t_2, \\ldots, t_n ", "407be614120997ea1bc1f83b85a1f857": "P,Q \\in z'", "407bec515942dec9c41665eaaea993de": "2P_{3/2}", "407c3d155f6a8cf6f82048309e93d329": "n_B(a+b)-n_B(a-b)=2n_B^\\prime(a)b+\\cdots", "407c4b48cee3fd2aaa9240f4ad165f26": " \\nabla\\times\\vec v = -y^2\\mathbf{i} - 3x\\mathbf{k} ", "407c66628859ebf3cc1a54968e3ac68f": "\\frac{dv^{0}}{d\\mu}+\\frac{v^{0}}{2v_{0}}\\frac{\\partial\ng_{ij}}{\\partial x^{0}}v^{i}v^{j}=0", "407c9d83ee0a7559d3b7964c4d7100de": "r_{12} = a\\sec\\alpha\\left|\\tan^{-1}\\sinh\\left(\\frac{y_1}{R}\\right)-\\tan^{-1}\\sinh\\left(\\frac{y_2}{R}\\right)\\right|.", "407d374c8b339627753c4ebd0d26fc5d": " \\hbar \\omega_k^\\prime ", "407d98f0278ab9ff009e9ff9e6d5466d": "w_{n}", "407d99d12e74bbd3d5be95eeeb386853": " w \\in \\mathbb{M} ", "407dbd7c5ce46c8cf120bf6a876800d4": "\\bigcap_{n=1}^\\infty C_n=\\varnothing", "407dd06c43056b35e7a1fca2f47c8716": "x_5", "407df9db53e6af7ab35ddae3490cda1d": "\\exists j (N \\equiv \\pm r^j \\pmod{B})", "407e262d6c6c1c40e3bbb4ad31e31a19": "\\bar{\\textbf{c}}(t) \\approx \\textbf{m}(t)\\bar{d}(t)", "407e4604112dd8c676690a3fe84d26b7": "v_0 \\rightarrow v", "407ea0092e65f84e486532d5ad247fd3": "\\mathcal{C}_{-n}", "407eae591172863d39f7551928bc8142": "M \\in SU(p, q, R)", "407eb5ed8e225387fdb02ed6fbeb955c": "\\omega_{\\text{r}}", "407f0feda6dc98be3f09134a20d34d4b": "P(H|D)", "407f73b6611a11c102c6c769a18b7b95": "y = \\sum_{r=0}^\\infty a_r x^{r + c}", "407f751cd010122d5e0f5f2348a746ce": "(-1)^S", "407f975ea33026d32673db6f8f8e96f5": "\\zeta_{ 4}", "407fa0605d06eb5ed7469ab7f0137b6e": "G(z) = \\sum_{n \\ge 1} \\left(\\sum_{G\\in \\operatorname{Cl}(S_n)} \\frac{c_G}{|G|}\\right) g(z)^n. ", "407fb6c3240d02d7f94c71cf9e124d29": "\\Vert \\phi-v_0\\Vert_X<\\delta\\,", "407fb95062393cd18149325506704670": "s_n^2 := \\sum_{k=1}^n \\sigma_k^2 .", "407fe54b07f4ba2a73043539313c69dc": "T_a T_b = \\frac{1}{2n}\\delta_{ab}I_n + \\frac{1}{2}\\sum_{c=1}^{n^2 -1}{(if_{abc} + d_{abc}) T_c} \\,", "408012ab88873a75c1a111da0b1f443a": "\nu = A \\cos ax \\sin by \\sin cz,\n", "40804f79aaf47defecbfe1492f411de3": "\\pi(g,\\gamma)F_w= (\\overline{\\alpha} +\\overline{\\beta}w)^{-\\frac{1}{2}} F_{gw}=\\frac{1}{\\overline{\\gamma}} \\left(1+{\\overline{\\beta}\\over \\overline{\\alpha}}w\\right)^{-1/2}F_{gw},", "4080959eac498a51f901b117ac72a8cf": "r=Ae^{{-E_a}/{kT}}", "4080c8743336eaf26d54ac36c4f77795": "\n\\dot{\\sigma} = \\dot{e}_1\n= a_{11} e_1 + A_{12} \\mathbf{e}_2 - v( e_1 )\n= a_{11} e_1 + A_{12} \\mathbf{e}_2 - v( \\sigma )\n", "4080e3b4ec9913ae32f7cc30b0c58c68": "\\mathcal{AL}", "40812614762644a6d46341afc07e1a1a": "\\mathrm{SU}(8)/\\{\\pm I\\}\\,", "40814da1db3c7cd315ddbabf2286013a": "H(X)=-\\sum_{n=0}^{N-1} P_n\\ln P_n", "4081752df02e749c4b96f996c26d4602": "\\xi\\propto |T-T_c|^{-\\nu}\\,,", "4081c9235f69fd9558756e45cbcb216c": "\\exists x \\forall y A_D(x; y)", "40821d1ef0ee54cc65524cc0e4ae2ac6": "\\epsilon^{ijkl}", "40821e9c0a198e71b3991fb574c30d37": "10_1", "40826e12790bc13f97cf515a0d776a4e": "U_e = \\int {\\frac{Y A_0 \\Delta l} {l_0}}\\, dl = \\frac {Y A_0 {\\Delta l}^2} {2 l_0}", "4082c8b8623fdecb3dba4d6e7258483b": "\\widehat\\sigma^2 = \\frac{1}{n} \\sum_{i=1}^{n} (\\hat{x}_{i} - \\bar{x})^2. ", "40833e43a4dbfa9ef64a50b1bbe80727": "\\operatorname{Fl}(\\mathcal{C}),\\quad u\\mapsto |u|=", "40834dbf628d4f7ebfc19a61527551da": "\n\\frac{1}{T} = \\frac{\\partial S}{\\partial E} = k_{\\rm B} \\frac{\\partial \\log \\Sigma}{\\partial E} = k_{\\rm B} \\frac{1}{\\Sigma}\\,\\frac{\\partial \\Sigma}{\\partial E} .\n", "408378e8bc55170258126d10000c53d9": "\\frac{\\partial f}{\\partial y}", "4083c45f0942d3253c84a54225d4e552": "\\vec{J}(\\vec{r},t)=\\int_{4\\pi}\\hat{s}L(\\vec{r},\\hat{s},t)d\\Omega (\\frac{W}{m^2})", "40842b9f01ddcd922aeab0c546ccabd4": "\\alpha_{\\rm L}=\\frac{k_{\\rm C}}{\\alpha_{\\rm B}c^2}", "408430fb12a72d9362086a0e38e77670": "\\Delta t = \\frac{1}{\\sqrt{1-\\tfrac{v^2}{c^2}}} \\Delta\\tau .\\ ", "408437b592c5eab425bce0d0d6133053": " \\bar r(t_2)=\\bar r_2.", "4084605f8de45ac29581d80b8b74916b": "i,j p[Y,X]", "409ced13f14b82b41189d5f725168b7b": "a_e", "409d179cbbf4a8e593a2b37414713c8e": "r_i\\in\\{1,2\\}", "409d48e3cbbf12d37c4de3f1b6797be3": "N = 26 + 24(N_1 + N_2) + 48N_3.\\,", "409de92ca6587d19d6c406eb7456be3a": "Lc(z)=\\int_0^z\\pi x\\log(\\sin \\pi x)\\,dx=z\\log(\\sin \\pi z)-\\int_0^z\\log(\\sin \\pi x)\\,dx=", "409df7ca3a24ec6da299dc0a52c50c95": " h_{rgb} = \\mathrm{atan2}\\left( \\sqrt{3} \\cdot (G - B), 2 \\cdot R - G - B \\right) ", "409e2740edf304efeeafc7e60b6f8221": "z\\le x\\Rightarrow y,", "409e449d409b664e02e495031c719cd8": "E_{l}^{k} = E_m^{(0)} + \\Delta \\epsilon_l + \\frac{1}{N_l^k} \\left\\langle \\Psi_{m}^{(0)} \\left| \\left(V_{l}^{k}\\right)^{+}\n\\left[\\hat{\\mathcal{H}}_v , V_{l}^{k} \\right] \\right| \\Psi_{m}^{(0)} \\right\\rangle\n", "409e4bfc114bbfcbd919a76546d9ee53": "F,G,H,L,M,N,I,J,K", "409e5e0748cb140207cc960c65f6eb74": "\\widetilde M = M//G", "409ea924be7366292cd1142ba3312f49": "P_1=(2,\\sqrt{3})", "409ebf4bbb9302a79354491c03dad46a": " \\scriptstyle \\zeta = 1 \\,", "409f870d4da5b512316ee2bda84a1ef4": "\\psi(\\Omega^{\\psi(\\psi(0))})", "40a041b8828a732244711019c655271a": "p=23", "40a08aabeeafeb3aee2dfec7d52aa1c0": "V_U = V_L \\,", "40a093e03789a7cea67e389b63690f15": "\\sigma_2^{(t+1)} = \\frac{\\sum_{i=1}^n T_{2,i}^{(t)} (\\mathbf{x}_i - \\boldsymbol{\\mu}_2^{(t+1)}) (\\mathbf{x}_i - \\boldsymbol{\\mu}_2^{(t+1)})^\\top }{\\sum_{i=1}^n T_{2,i}^{(t)}} ", "40a14666f9bf9ce620fb59f8d088c6a8": " f(x;\\mu,\\lambda) = \\left[\\frac{\\lambda}{2 \\pi x^3}\\right]^{1/2} \\exp{\\frac{-\\lambda (x-\\mu)^2}{2 \\mu^2 x}}", "40a1ad5c82fc4b5b18f2d23811b56bbb": "y_{12}", "40a1dd12fcccff7f9d34439991168f19": "d\\colon E \\to E \\wedge E", "40a1e944b0ea0487386ff00a02d998b4": " (x, y) \\mapsto (\\alpha x +\\lambda y^n, \\beta y)", "40a25d80a034b64291d0719d4a1d1399": "F(X) \\rightarrow \\prod_{\\alpha\\in A} F(X_\\alpha) {{{} \\atop \\longrightarrow}\\atop{\\longrightarrow \\atop {}}} \\prod_{\\alpha,\\beta \\in A} F(X_\\alpha\\times_X X_\\beta)", "40a2ca53ca45b750462b60b381b47bb8": "\\vec{u}\\in \\mathcal{D}(A)", "40a2cb0f5f34223969144502a96b8cfa": "X_o", "40a2de939afdf8711d8afa84978b12af": "C_1=nC,\\,", "40a2ed1d63302ece2fa16911cfddb780": "{{5(n-1)^2 + 5(n-1) + 2} \\over 2}.", "40a2f2e3c11b370ff3e52f693c3b7f10": " \\frac{((3!)!)!}{3!} = k \\cdot n!, ", "40a3214f2ab5b5bff4a0cf04d29f9d85": "\ng=\\left(\\frac{f}{8}\\right) \\frac{4}{3}\\pi n^3 \n = \\frac{4\\pi f}{3} \\left(\\frac{Lp}{h}\\right)^3\n", "40a32869e502b6ab947f35ad911a3c4a": "f d\\mathbf{S}.", "40a3329566ae896de05f73e752f367f1": "\\frac{\\partial}{\\partial \\rho}", "40a3724832b37cd4640162d15c108330": "f^2 = {R^2 \\over 1 - R^2}", "40a3b8e5b9ae7f1b015c52e5bb2bf05c": " \\sigma_{x}\\sigma_{p} \\geq \\frac{\\hbar}{2}, ", "40a3bfc67363cee2745a307ef29e0c5b": "\\mu \\ll 1", "40a410dffd65244ae58ef65a761b4f74": "E[\\hat{\\beta}_{FD}]=\\beta", "40a45420e4019fd642126435c0832e7a": "A = \\begin{pmatrix}\n0&1&0\\\\\n0&0&1\\\\\n0&0&0\\end{pmatrix}\n", "40a4f8fa8107b0390be037f7a01d27e5": "\\nu^\\nu n^{-n} \\left( n - \\nu \\right)^{n - \\nu} {n \\choose \\nu}.", "40a528c3cf44b52d83db6a79f12b9e72": "\\theta_{k,n}(z) = \\sum_{\\gamma\\in\\Gamma^*} (cz+d)^{-2k}\\exp\\left(2\\pi i n\\frac{az+b}{cz+d}\\right)", "40a568d1fb7792772f7dd94d53522090": "\nr_{\\mathrm{outer}} \\approx r_{\\mathrm{inner}} \\approx 3 r_{s}\n", "40a69242158ea61dd5f3b8c170d720b2": "Gm^2", "40a71bb7c367c531dd2b9159c2516788": "\\,\\{F^{uv}+iF^{uv}\\} = e^{i\\theta}\\{F^{uv}+iF^{uv}\\} ", "40a728d10c02b7465b8ca2d112a92ce9": "\\begin{align}\n1_F &= \\varepsilon F\\circ F\\eta\\\\\n1_G &= G\\varepsilon \\circ \\eta G\n\\end{align}", "40a730b9ee4cd0b4d53577bcc7f92170": "~g=2Gh~", "40a792ee0069893d72b632e26af75515": "m = (f^{1}(2) - f^{1}(1)) / (2 - 1) = (x_{2}^{1} - x_{1}{1}) / (2 - 1) = (9 - 2) / (2 - 1) = 7 / 1 = 7", "40a7cc239a91f65bc8aa3aabab6c28d4": " \\dot{\\textbf{x}}(t) = \\textbf{a}\\,[\\,\\textbf{x}(t),\\textbf{u}(t),t\\,],", "40a7e4bb11dae4dc868316a68a5cfc43": "\\hat{r} = \\frac{\\mathbf{r}}{|\\mathbf{r}|}. ", "40a7e9f83f62996ac022e6a4ad2e21a8": "L: \\Sigma^{*} \\mapsto \\mathbb{N}_0", "40a7ec129c1b41c23700a6ee52a3d764": "V\\;", "40a7ef91ef12e0e06de277e4d31b0157": "\\left({24 \\over 200}\\right)", "40a826518659d850c6c7d716358e69c6": "{}_{\nx_{N-1}=-\\frac{a}{2b}\\sqrt{\\left(\\frac{c}{b}\\right)^{N-1}}{}_{N-1}F_{N-2}\n\\begin{bmatrix}\n\\frac{N+1}{2N},\\frac{N+3}{2N},\\cdots,\\frac{N-2}{N},\\frac{N-1}{N},\\frac{N+1}{N},\\frac{N+2}{N},\\cdots,\\frac{3N-3}{2N},\\frac{3N-1}{2N};\\\\[8pt]\n\n\\frac{N+1}{2N-4},\\frac{N+3}{2N-4},\\cdots,\\frac{N-4}{N-2},\\frac{N-3}{N-2},\\frac{N-1}{N-2},\\frac{N}{N-2},\\cdots,\\frac{3N-5}{2N-4},\\frac{3}{2};\\\\[8pt]\n-\\frac{a^2c^{N-2}}{4b^N\\left(N-2\\right)^{N-2}}\n\\end{bmatrix}\n-\\sqrt{\\frac{c}{b}}{\\rm{i}}{}_{N-1}F_{N-2}\n\\begin{bmatrix}\n\\frac{1}{2N},\\frac{3}{2N},\\cdots,\\frac{N-4}{2N},\\frac{N-2}{2N},\\frac{N+2}{2N},\\frac{N+4}{2N},\\cdots,\\frac{2N-3}{2N},\\frac{2N-1}{2N};\\\\[8pt]\n\n\\frac{3}{2N-4},\\frac{5}{2N-4},\\cdots,\\frac{2N-3}{2N-4};\\\\[8pt]\n-\\frac{a^2c^{N-2}}{4b^N\\left(N-2\\right)^{N-2}}\n\\end{bmatrix}\n}", "40a82aa1612699e7afa5c2172289d5ed": "\\eta\\colon S^3 \\to S^2", "40a84006a762ee7589709eeee3d4e7c2": "E_l^k", "40a89676f79aa2749e0eb7803371fd99": "H_2O^+ + H_2O \\longrightarrow H_3O^+ + OH", "40a8bb1c2154788cac9120d0aed47892": "\\varepsilon_F=(k^2_F+m^2)^{1/2}", "40a90030a6b28ecc09e565a42948409c": "\\int_{0}^{+\\infty} e^{-x} f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)", "40a90eb52f2ab9c7b380b2a25f80d9b5": "D(n,b) = \\left\\lfloor \\log_{b}\\left(2^{2^{\\overset{n}{}}}+1\\right)+1 \\right\\rfloor \\approx \\lfloor 2^{n}\\,\\log_{b}2+1 \\rfloor ", "40a92c79d5691a77c8b90f1f191764ad": "\\mathbf{ x}(1) = [u(1)\\, u(2)\\, u(3)]=[85\\, 80\\, 89]", "40a960305bbc3b961b520af73a9519c8": "\\Lambda(B_1),\\ldots,\\Lambda(B_n),", "40a9a0434711b2c13b7f0f00121c3164": " \\dot x = u(t), \\quad x(t_0) = x_0 ", "40a9df94cefe23de6dd100a51986abbf": "\\mathcal{D}[A]a\\equiv AaA^\\dagger-\\frac{1}{2}\\left(A^\\dagger Aa+aA^\\dagger A\\right)\\,.", "40aa096058f805bbda81c77c69342895": "\\Pi_A", "40aa2227f8ab9f9737e2ce467090bb9c": "m=1", "40aa52dd884a0e370ed6b3ccd1461796": "g(x, y) \\equiv f(u(x, y)),", "40aa77d8de35beb2cf5f3045c07dd904": "\\log_{10} \\mbox{ year}", "40aa98cd31a46a57ff47ad9b80986793": "\\scriptstyle{1/(R_{\\text{t}}\\,C_{\\text{dc}})}", "40ab5cc3f2664afa2dbf0eefd3f0a638": " \\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\frac{\\partial f_1}{\\partial \\boldsymbol{F}_2}:\\left(\\frac{\\partial \\boldsymbol{F}_2}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} \\right) ", "40ab618367e77af4a788d0cdaddf4f6e": "g^{-1}(V_i)", "40ab8e5e99e42e63ac1d60dbb6f2502a": "T=\\begin{pmatrix}T_{00}&T_{01}\\\\T_{10}&T_{11}\\end{pmatrix}", "40aba56a522d77f990de84ad8817318d": " m\\left( q\\right) =\\operatorname{E} \\left( e^{q\\int_0^t Z_s \\, ds}\\right)\n,\\qquad q\\in \\mathbb{R}, ", "40abc68c810c57b79a539c680ab24acb": "P(n \\mid N) = \\frac{1}{c} ", "40abe0134d47c9fc2c0f0b1304706eff": "x \\beta = \\alpha - E\\,", "40ac15ce1e17e5a0262c80cdf7c4662f": " f^{*} = \\frac{b - 1}{2b} \\! ", "40acb945f39e353f87d06b6055acd9e7": "\\omega=\\frac{-1+\\sqrt{-3}}{2}, ", "40ace8930e1afc844b0822a5f1dac3c2": "C_{V,m} = \\left(\\frac{\\partial C}{\\partial n}\\right)_V", "40ad1511faaf7b5b86d8994b97199dfe": "K^{-n}(X)=\\widetilde{K}^{-n}(X_+)", "40ad2e89aa430897e69d667226bdf37b": "\\phi_{1,m}", "40ad4412417656111c219177e35c67e6": " \\int_{\\mathbf{R}} \\psi_{n_1, k_1}(t) \\psi_{n_2, k_2}(t) \\, d t = \\delta_{n_1, n_2} \\delta_{k_1, k_2}, ", "40ad6644ed1d1922b0eca8fd7a2692d3": "w=Az^n, \\,", "40ad68321dd6f3634d52330190fa996a": "m \\leftarrow \\frac{a+b}{2} = \\frac{0+2}{2} = 1", "40adcddaa1c2f5d47cca294959700282": " \\mathbf{v} ", "40ae264b3e39253259ffc5c98d6d8bf0": "y(t).\\ ", "40ae87bee7fbcdbeb2f891f06dbc4aa4": "b_m ( \\boldsymbol{R_p+R_{\\ell}}) = e^{i\\boldsymbol{k \\cdot R_{\\ell}}}b_m ( \\boldsymbol{R_p}) \\ , ", "40aef2fa5b2db2a620826a2b591ec242": "\\rho' : (a, b) \\rightarrow (d, d)", "40af1de26d2b37ba2a4d7c237152ea62": "\\mathfrak{sp}_1 \\oplus \\mathfrak{sp}_1\\oplus \\mathfrak{sp}_1", "40af5fc078289fef413f109a5d733ac7": "\\Zeta", "40b017ff9ff17c8a7773429646d6c74a": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \\left( \\mathbf{\\hat k}\\cdot \\mathbf{\\hat r}\\right)^2\n { \\exp \\left ( i\\mathbf{k}\\cdot \\mathbf{r}\\right ) \\over k^2 +m^2 } = \n\\int_0^{\\infty} {k^2 dk \\over \\left ( 2 \\pi \\right )^2 } \\int_{-1}^{1} du u^2 {\\exp\\left( ikru \\right) \\over k^2 + m^2}\n", "40b05281d5336938be041817a84a5a29": "\\begin{align}\n(1+x)(1+y) &= 1+x+y+xy &&\\approx 1+x+y\\\\\n\\frac{1}{1+x} &= 1-x+x^2-x^3+\\cdots &&\\approx 1-x\n\\end{align}", "40b063806a815266b91997c4e29ca5a4": "(G, G^+) = \\varinjlim (H_k, H_k^+) , \\quad \\mbox{where} \\quad (H, H_k^+) = (\\mathbb{Z}^{n_k}, \\mathbb{Z}^{n_k}_+).", "40b0735a269f0273c59a67cee9b33fff": "X = -2\\sum_{i=1}^k \\log_e(p_i) \\sim \\chi^2(2k) .", "40b0a1cc8a8d0a366f3b6cfd115491f3": "\\frac{13+10}{2} - 10=1.5", "40b0aaabb1a28660d9cfd74b6f754ddb": "\nT_{ij} = \\frac{{cT_i dT_j }}\n{{C_{ij}^b }}\n", "40b0cec66dbf43d16f2bfca73cb02dc4": "\\beta_{nl} = k_0 n_2 I = k_0 n_2 \\frac{|E|^2}{2 \\eta_0 / n} = k_0 n_2 n \\frac{|A_m|^2}{2 \\eta_0} |a|^2", "40b0d51c19d5e80e4e67d021d359d30c": "\n\\begin{bmatrix} 0 & \\tan \\frac{\\theta}{2} \\\\ -\\tan \\frac{\\theta}{2} & 0 \\end{bmatrix}\n\\lrarr\n\\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{bmatrix} .\n", "40b0ed64379a2e072ba44a5e9f5abf50": "\\frac{x^2-1}{x-1} = \\frac{(x+1)(x-1)}{x-1} = x+1", "40b136445985b3ab52376234ac90de3d": "\\ y[n]", "40b169be9023b6aec9500933242b24d9": "\\varphi(x_1,x_2,x_3,x_4,x_5) =\n\\varphi(x_4,x_5,x_3,x_1,x_2) +\n\\varphi(x_4,x_2,x_5,x_1,x_3) +\n\\varphi(x_1,x_4,x_5,x_2,x_3),", "40b16f75fffbba8af9c4184dd383a742": "\\Phi_{10}=\\Phi_{20}=\\Phi_{11}=\\Phi_{12}=\\Phi_{22}=\\Lambda=0\\,,\\quad \\Phi_{00}=\\frac{M(v)_{\\,,\\,v}}{r^2}\\;.", "40b1717258aefa7cc734fafd9060b810": "\\mu(x+y) + \\mu(x) + \\mu(y) \\equiv \\lambda(x,y) \\pmod 2 \\; \\forall \\,x,y \\in H_k(M;\\mathbb{Z}_2)", "40b1b713aa24ea913c6e1a7af81e4d0c": " \n\\begin{bmatrix}\n a & b \\\\\n c & d \n\\end{bmatrix}\n= \n\\begin{bmatrix}\n 1 & 0 \\\\\n \\frac{-1}{\\lambda R} & 1 \n\\end{bmatrix}.\n", "40b1db68b783ee5426e1ae385aee116f": "\\nabla h = \\left(\n{\\frac{\\partial h}{\\partial x}},\n{\\frac{\\partial h}{\\partial y}},\n{\\frac{\\partial h}{\\partial z}}\n\\right) = \n{\\frac{\\partial h}{\\partial x}}\\mathbf{i} + \n{\\frac{\\partial h}{\\partial y}}\\mathbf{j} + \n{\\frac{\\partial h}{\\partial z}}\\mathbf{k}", "40b1ec66d227fe02940284db3086c0a1": "\n\\sum_{\\delta\\mid n}\\mu(\\delta)=\n\\sum_{\\delta\\mid n}\\lambda\\left(\\frac{n}{\\delta}\\right)|\\mu(\\delta)|=\n\\begin{cases}\n&1\\mbox{ if } n=1\\\\\n&0\\mbox{ if } n\\ne1.\n\\end{cases}\n", "40b2758ef75276096dc9e11d08d000fa": "~~~\\oplus~~~", "40b28f2198764edf4f750c143a41b45f": "f = c T^{k/n} + \\cdots", "40b300f6aacf9894ff1d369195efd0ab": "\na_z = \\left({\\partial\\over\\partial t} + u{\\partial\\over\\partial x}\\right)^2h = \n{\\partial^2h\\over\\partial t^2} + 2u {\\partial^2h\\over\\partial t \\partial x} + u^2 {\\partial^2h\\over\\partial x^2},\\,\n", "40b325fdc8250456d2732728053ad2a9": "F_{23}=\\sin^2{(\\alpha)} \\sin{(\\theta)} d\\theta\\wedge d\\phi", "40b332ea096cdcb2dbabb3af1f77bdf0": " j^{th} ", "40b361b52ccee2dab1c718eb50b608ad": "\\rho_{ABC}", "40b362f1322f29d5c919e5e61684ca08": "\\int_D\\langle {\\nabla u} , {\\nabla (v - u)}\\rangle \\mathrm{d}x \\geq 0\\qquad\\forall v \\in K, ", "40b3c299a13534469c33cf1346eef625": "\\sum r_i g_i", "40b3ca6e45d391744c24b66133b8c148": "\\cup : H^* (M; \\mathbb R)\\otimes H^{d-*}(M, o(M)) \\to H^d(M, o(M)) \\simeq \\mathbb R", "40b3d279f254503cf82dd5888b2a394a": "\\langle \\sigma_z\\rangle = -D ", "40b4035401f04531f98f27c00805fe43": " A : G \\times M \\to M ", "40b4c1d904984beef8cb384f467c5ff2": " \\int_{a}^{b}( \\alpha f(x) + \\beta g(x))\\,dx = \\alpha \\int_{a}^{b}f(x)\\,dx + \\beta \\int_{a}^{b}g(x)\\,dx. ", "40b4f2f3612245ee78320ee6ed83451e": "f(x_1,\\ldots, x_n)=g(x_1,\\ldots, x_n)+ih(x_1,\\ldots, x_n),", "40b5037f0dd4d90b51c5422a706836b4": "R_{isol}", "40b53fce48388f21539af5638b25a0df": "\\mu = A_1 \\cdot T \\cdot \\left[1 + A_2 \\cdot e^{B/RT}] \\cdot [1 + C \\cdot e^{D/RT} \\right],", "40b5537916e3d0e9f16411d507224270": " H_0 = -\\frac{1}{2} \\nabla_{r_1}^2 - \\frac{1}{2} \\nabla_{r_2}^2 - \\frac{Z}{r_1} - \\frac{Z}{r_2} ", "40b5815cca4f21acf774fa99c44f944e": " \\Gamma \\, ", "40b58bd195eddf9354b39eb3338c7b85": "C_{abcd}\\, k^bk^d=0= {^*C}_{abcd}\\, k^bk^d", "40b59973714ef8457a3afe013a3e86c0": "\n\\gamma^2 = \\left( \\begin{array}{cccc}\n0 & 0 & 0 & -i \\\\\n0 & 0 & i & 0 \\\\\n0 & i & 0 & 0 \\\\\n-i & 0 & 0 & 0\n\\end{array} \\right),\n\\; \\; \\; \\; \\gamma^3 = \\left( \\begin{array}{cccc}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0\n\\end{array} \\right).\n", "40b600c318c1304793e8c64aacad434a": "q_1, q_2 \\in Q", "40b66f217c447ea178c25b2fab62736e": "\\bar{x}^{\\iota}", "40b67e69a2b51b3d09e2c9b3ecb58c2a": " x_i \\sim x_j ", "40b68de19d78c0ccd2af60164b0f995c": "\\sum_{i=1}^n \\tfrac12r(\\varphi_i)^2\\,\\Delta\\varphi.", "40b6cbb5312e4f6b82c51e314a577e27": "T \\sin \\varphi = \\lambda gs,\\,", "40b73d31938b8b388e17d2309bc32ee5": "LB = (2W - X + Y)\\sqrt{8}", "40b770bae76085dac3e32a4150b98f4b": " \\left(\\int_{x \\in J} |p(x)|^p\\,\\mathrm{d}x\\right)^{1/p} \\leq e^{\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J} \\left( \\frac{C \\,\\, \\textrm{mes} J}{\\textrm{mes} E} \\right)^{n-1+\\frac{1}{p}} \\left(\\int_{x \\in E} |p(x)|^p\\,\\mathrm{d}x\n\\right)^{1/p}, ", "40b7a91b3b2d4d7d62eedc64d851bb67": "\\Delta g_{FA}", "40b801ad4226c9c740338e843b1e417a": "\\chi(x_{\\sigma(1)},\\dots,x_{\\sigma(r)})=\\text{sign}(\\sigma)\\chi(x_1,\\dots,x_r)", "40b80bed326323a4ce3750d91d89c3a2": "\\frac{P}{Q_i}=A_i + O((x-\\lambda_i)^{\\nu_i})\\qquad ", "40b813fdcd55a1b7576cc55293d723f4": "\\lambda = L + (1-C) f \\sin \\alpha \\left\\{ \\sigma + C \\sin \\sigma \\left[\\cos (2 \\sigma_m) + C \\cos \\sigma (-1 + 2 \\cos^2 (2 \\sigma_m)) \\right]\\right\\} \\, ", "40b85027598d87611b1c8d5d11e46812": "n+1", "40b86574e9495f7f31b70fe7ff06f1b4": " \\mathbf{s}(x) ", "40b89fa0fa1510e56fc5309f740d7faa": " \\lim_{x \\to c} \\frac{f(x)}{g(x)} = \\lim_{x \\to c} \\frac{1/g(x)}{1/f(x)} \\! ", "40b8f9557f841c9652b1f10dcc8cef84": " DGS_\\gamma ", "40b91f7d3f1f1891d5b781d5dd878f2d": "|E\\cup E'|>1", "40b947131cbee40db2c94dd5e19199c9": "\\alpha(u+v) = e^{i\\pi E(u,v)}\\alpha(u)\\alpha(v)\\ ", "40b9579b90931ea3e89d0bf35cdb3d83": "b_n \\,", "40b97eb8b68f2c75f8e7fe2f56cd243c": " t \\!", "40b9813b8f3f9a42ee22efa4b8ed88a7": "f(t,n)", "40b98e8b3d9a732a62d3fbbf643103d7": "\\beta_w = -\\frac{dV_w}{dp}\\frac{1}{V_w}", "40b9a6f43f303ef5e6ff841f48591b4e": "\n V^* \\cong\n \\biggl(\\bigoplus_{\\alpha\\in A}F\\biggr)^* \\cong\n \\prod_{\\alpha\\in A}F^* \\cong\n \\prod_{\\alpha\\in A}F \\cong\n F^A\n ", "40ba2104e0b6721d1d9b80dbe01e7e04": " \\text{where} \\quad A= \\sqrt{2/\\pi} (\\sigma_1+\\sigma_2)^{-1}", "40ba47b620ee482c03351a99623ea473": " \\mathbf{U} \\equiv \\frac{d \\mathbf{R}}{d \\tau} = \\gamma \\frac{d \\mathbf{R}}{dt}\\,,", "40ba55cd3c58225334c65204b80c6ca3": "\\varphi (n)", "40ba8aaf58a05868f6a92c153204a92f": "\\varphi\\left(\\mathbb{E}\\left[X|\\mathfrak{G}\\right]\\right) \\leq \\mathbb{E}\\left[\\varphi(X)|\\mathfrak{G}\\right].", "40ba939aea08ff27006272db3f9d57f1": "\\mathbf{H_{eff}}\\left(t-\\delta t\\right)=\\mathbf{H_{eff}}\\left(t\\right)-\\delta t \\frac{\\mathrm{d}\\mathbf{H_{eff}}}{\\mathrm{d}\\mathbf{m}}\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}+\\dots", "40bb2d082eb81371cdacfd316c950b3a": "T_m \\mapsto (T_n)^{n/m}", "40bbd446f98e312f66a266382ce9dcb0": "QT_{LC} = {QT + 0.154(1 - RR)}", "40bc2acf691fce096c9c4ae098522f75": "\\lambda_B = \\frac{e^2}{4\\pi \\varepsilon_0 \\varepsilon_r \\ k_B T},", "40bc6df3d29f3300fdda8708755a590b": " d(p,q) = \\inf L(\\gamma)", "40bcd959849d18010ad2ed7684119b30": "f(n) \\in \\Theta(g(n))", "40bd062aa8885de732a9003bea4d38fc": " h\\, = \\Delta \\theta\\,", "40bd8331d1a17db069d94bae7e23ad29": "\\mathcal A_k,\\mathcal B_k", "40bdd90f8b5a5f0d14bbf8f59e64a421": " G =\\left(\\frac{1}{2r_{1}}+\\frac{1}{2r_2}-\\frac{1}{R}\\right)\\cdot\\left(\\frac{1}{\\epsilon_{op}}-\\frac{1}{\\epsilon_s}\\right)\\cdot(\\Delta e)^2 ", "40be0cab0a0264428a8a12fe7f360a46": "A_{\\mathrm{right}}^{-1} = A^T \\left(A A^T\\right)^{-1}", "40be4455145d0a79da884ca552618aff": "w_\\Gamma:= \\frac{m(\\Gamma)}{(2\\pi)^{2n}n!}\\int_{C_n(H)} \\bigwedge_{j=1}^n\\mathrm{d}\\phi(u_j,u_{t1(j)})\\wedge\\mathrm{d}\\phi(u_j,u_{t2(j)})", "40be630cd67daabb6f49805b09ccea2a": "X^*_{c}", "40be6ddd39966623ab92c279f417d4d1": "1 = \\text{id}_S\\left(x\\right) = x", "40be899b021bef3a37b116acb0eae179": "p-1 \\equiv -1 \\pmod p.", "40be8cda4d92f79402724fe5c8d12cb4": "v^i[\\mathbf{f}] = \\sum_{k=1}^n g^{ik}[\\mathbf{f}]a_k[\\mathbf{f}].", "40beb0b454d2772c892b582b1e104e93": " N_B = N_{A0} - N_A = N_{A0} - N_{A0}e^{-{\\lambda}t} = N_{A0} \\left ( 1 - e^{-{\\lambda}t} \\right ). ", "40bed7cf9b3d4bb3a3d7a7e3eb18c5eb": "Person", "40bf0c9d93034e3516e981dec0742696": "\\begin{bmatrix}M\\end{bmatrix}^{-1}", "40bf5d59357d8bd2b4f62f18b170246c": "b = 2a", "40bfd02ea6203d86a39e7638ac20d3ce": "k=2,\\dots,K", "40bffe78e4ad4d37448ac5af838052e8": "\\textstyle k = 49,581", "40c013d00f630c22feb78751ed25fa71": "[K_0] \\mathbf{x}_{0i} = \\lambda_{0i} [M_0] \\mathbf{x}_{0i}. \\qquad (1)", "40c01ef67b28dc2279094c0b829381c9": "C: \\dot{\\bigcup}_{\\lambda\\in\\Lambda} M(\\lambda)\\times M(\\lambda) \\to A", "40c062537f29974e9094472fb3b8e3c3": "d(x,Y)=\\inf \\{ d(x,y) | y \\in Y \\}\\ ", "40c0c8282467e2241f53b002c7a44634": "\\sum_{k=a}^{n-1} 2^k = 2^n - \\sum_{k=0}^{a-1} 2^k - 1", "40c139d816ac6f5a01620c7025e87d1e": " \\mathbf{B}\\cdot\\nabla\\alpha=0 ", "40c176fa124cd9791ecada5bf6f3e726": "E_0 = E_1 + E_2 \\,\\! ", "40c1810e21c33e71286ec55836be0e39": " SubCipher_1=DEC_{b_1}(k_{b_1},s)", "40c1be3abb81bd0570688e6be36b9602": "\\gamma'_x(t) = V(\\gamma_x(t)) \\qquad ( t \\in (-\\epsilon, +\\epsilon) \\subset \\mathbf{R}).", "40c1c037b732857934e4794ea684c05d": "f(x)=x^2\\sin\\tfrac{1}{x},\\ f(0)=0,~x_n=\\tfrac{1}{(n+\\frac12)\\pi},\\ y_n=x_{n+1}", "40c20227fc1ddf44f28b0015e5161e2b": "\\Delta E^*_{CMC} = \\sqrt{ \\left( \\frac{L^*_2-L^*_1}{l S_L} \\right)^2 + \\left( \\frac{C^*_2-C^*_1}{c S_C} \\right)^2 + \\left( \\frac{\\Delta H^*_{ab}}{S_H} \\right)^2 }", "40c215e3f7abfbfeff336fa67bb84911": "\\Phi_n(x)", "40c22bb74ef18fef1869b28e9a8a702b": "\\mu \\in [0,1]", "40c253c0b14f1aa9702df4208c9b7c59": "w(i, j)", "40c2690fdba1a3d17aa9729561b57a93": "F^P_i", "40c29a997875799447ddfb06ebb51bcf": "\n\\begin{align}\nq_r = \\sum_{Aj} d_{Aj}& \\big( q^A_{rj} \\big) \\\\\ns_i = \\sum_{Aj} d_{Aj}& \\big( M_A \\delta_{ij} \\big) =0 \\\\\ns_{i+3} = \\sum_{Aj} d_{Aj}& \\big( M_A \\sum_k \\epsilon_{ikj} R^0_{Ak} \\big)=0 \\\\\n\\end{align}\n", "40c2b62f2110e38dfb7c1e0fbf5cbf10": "\nP = \\int_S (\\mathbf{E} \\times \\mathbf{H}) \\cdot \\mathbf{dA}. \\,\n", "40c2f708c41ed2b4f79641879122ee0a": "\nL(z) = c_0 + \\sum_{n=1}^\\infty c_n \\prod_{k=1}^n (z - \\beta_k)\n", "40c30884e102762304b23c432ff7ad22": "\\_ \\ast Q ", "40c3c3b2195e1c427f0fd939b1ec8a97": " p^2 \\le \\Gamma(S )^2 \\le 2 N p^2 ", "40c41e223d94d10cfa58d5c233c7f0d3": "L\\sin \\theta = {mv^2\\over r}", "40c42642043a98e69c9fea5b07ae8e98": "R(\\theta ) = R(0) + G \\sin^2 \\theta + F ( \\tan^2 \\theta - \\sin^2 \\theta )", "40c44fb7a72eae60bc87cd585c74c4c4": "A \\otimes_R B = B[x] / f(x)", "40c47a780ced9484e9a196ac27132f33": "\\sum_{j}{e_{jk}} = q_{k}\\,", "40c4f2d1e34e680eb427ac6003e0fc4e": "\n\\frac{6\\pi k^{2}}{TL^{2}c^{2}} ~,\n", "40c51f7782cd3405a54547dfadbc5b0a": "\n M_3 = 262.5 + R_a (x-10) + R_b (x-25) - 25 x \n = -675 + R_a (30 - 0.6 x) - M_c (1 - 0.04 x) + 12.5 x\\,.\n ", "40c5422eb6a76b7447ef5d927da62e1e": "\nv(t + \\Delta t)\n = v(t)\n + \\frac{1}{12}\\Bigl(5a(t + \\Delta t) + 8a(t) - a(t - \\Delta t)\\Bigr)\\Delta t\n + O(\\Delta t^3)\n", "40c56eb37b72a9677c6b41de309371c5": " 0.164 H_s^2 ", "40c571a0d4ec0769972c6bbda616c6bd": "\nb^{2}(-P^{2},m_{1}^{2},m_{2}^{2})", "40c58c3dafa053251b919e7b75a1e151": "(J^3_0f)(x)=-x-\\frac{x^2}{2}-\\frac{x^3}{3}", "40c61250c11a99a3f6a5599848c26a47": "X=\\bigcup U_j", "40c71d8afa86881899e32478bfe829e8": "\\mathrm{d} H = \\delta Q +V\\mathrm{d}p-\\delta W^\\prime.", "40c7366df1e39d2071b303da328cf89d": "(P \\or (Q \\leftrightarrow R)) \\leftrightarrow ((P \\or Q) \\leftrightarrow (P \\or R))", "40c7715ea1900b7dc0b191e70d2eedf1": "\nM_f(x_1,\\dots,x_{n\\cdot k}) =\n M_f(M_f(x_1,\\dots,x_{k}),\n M_f(x_{k+1},\\dots,x_{2\\cdot k}),\n \\dots,\n M_f(x_{(n-1)\\cdot k + 1},\\dots,x_{n\\cdot k}))\n", "40c7b62d993b90e5164e97b62152e668": " y=-2x+x^2 ", "40c7d039100b18e55343524021aeb358": "m \\in \\mathcal{N}", "40c8391d05ec4030dba1bb4e87a7caff": "F_\\leftrightarrow(x,y) = 1 - |x-y|", "40c879f6865a2087bbecb909e503bc28": "\\hat{h}_\\gamma \\Psi [\\eta] = h_\\gamma \\Psi [\\eta]", "40c8ba7ff65e4d5bd9657654c95f8334": "x v \\alpha(x)^{-1}\\in V", "40c8be24880d641f42d2b0aec2a22992": "\\begin{align}\n 0&=\\operatorname{adj}(\\varphi I_n-A^\\mathrm{tr})\\cdot((\\varphi I_n-A^\\mathrm{tr})\\cdot E)\\\\\n &= (\\operatorname{adj}(\\varphi I_n-A^\\mathrm{tr})\\cdot(\\varphi I_n-A^\\mathrm{tr}))\\cdot E\\\\\n &= (\\det(\\varphi I_n-A^\\mathrm{tr})I_n)\\cdot E\\\\\n &= (p(\\varphi)I_n)\\cdot E;\n\\end{align}", "40c906dd30da0d052fc34a39222f3ee2": "d(\\gamma(t_1),\\gamma(t_2))=|t_1-t_2|.\\,", "40c95ddc362ddd0e9fecef2ae1973fac": "t =", "40c96957f318f4a2a314fa1b8b77e5f4": "4.00= \\left ( \\frac{q_\\mathrm{max}^2}{32.2} \\right )^\\frac{1}{3}", "40c98f86a7e3cd314e5afb61ecdb2742": "\\mathbf v = \\mathbf V + \\mathbf v'", "40c9ab079c215c7a299ccd1e487b4338": "h\\colon F\\times [-1,1]\\to M", "40c9d961bba47048ce1f815b152f4f3c": "\\triangledown _{t}^{2}=\\triangledown ^{2}-\\frac{\\partial^2 }{\\partial z^2} \\ \\ \\ and \\ \\ \\ k_{t}^{2}=k^{2}-k_{z}^{2}", "40ca07e3e74681798c7390625eec9ea2": "A+A^*", "40ca36ce2d937df9ff1d3276a6ed53e9": " \\varepsilon_6 < 2^{-95} < 10^{-28}. \\, ", "40ca4678cc4f4dd39a2e66a5d72691e6": " \\sum_{i=1}^J \\alpha p_{0i}= \\sum_{i=1}^J\\lambda_i-\\sum _{i=1}^J\\sum_{j=1}^J\\lambda_j p_{ji}=\\sum_{i=1}^J\\lambda_i-\\sum_{j=1}^J\\lambda_j(1-p_{j0})=\\sum_{i=1}^J\\lambda_ip_{i0} ", "40ca65b8a332230321e2ef2b0db26247": "S_L = \\begin{matrix}\\frac{3}{10}\\end{matrix}S_L(1) + \\begin{matrix}\\frac{7}{10}\\end{matrix}S_L(2)", "40cad3986b06268ed6ddf5a8e9154218": "P(h(a) = h(b)) = \\frac{1}{m}.", "40cb4567c191beb7451d995dce728a31": "H_0^-(x)", "40cb558a32433608aa31f84ab9cb4279": " (T_l f)\\left(\\frac a b\\right) = f\\left(\\frac{la}{b}\\right) + \\sum_{k=0}^{l-1} f\\left({\\frac{a+kb}{lb}}\\right) - \\sum_{k=0}^{l-1} f\\left(\\frac k l \\right) \\ . ", "40cb8239e66820231e42be60cda62717": "X_{j(i)}", "40cbbf5790a23b12fc8863afb089edee": "\\mathbf{e}_{12} + \\mathbf{e}_{34} = \\mathbf{e}_{1} \\wedge \\mathbf{e}_{2} + \\mathbf{e}_{3} \\wedge \\mathbf{e}_{4}.", "40cbfadb9891d48916e4be0e611bd20c": "\\Delta\\mu = \\frac{\\beta}{1-\\beta^2}eE_0l_se^{z/l_s},", "40cc0e5ac4a364b9c54acf466aaca97e": "\\lfloor x\\rfloor - \\sum_{i}\\left\\lfloor\\frac{x}{p_i}\\right\\rfloor + \\sum_{i \\overline{AD} \\ . ", "40d0a587192fb7dbd0f8ee2f2e46aa47": "\\nabla^2 \\Phi = 4 \\pi G \\rho |\\Psi|^2 = 4\\pi G\\langle\\rho\\rangle", "40d0eae782ad235ad6910443da301082": "APY = \\left(1 + \\frac {i_\\text{nom}} {N} \\right)^N -1", "40d113cff97ce136572b39ce3245c721": "\\tilde{2}", "40d13ebf04b34c9547dcfd0fc362d410": "\n\\bar{f}_{k_j}(s)=\\bar{\\psi}_{k_jk_j+1}(s)\\bar{\\psi}_{k_j+1k_j}(s).\n", "40d160c455d1e2bcec20599b21e90107": "\\Sigma^\\ast", "40d18c6b79f11af155007ecae3e4408c": "\\sqrt{1 + 1 + 1 + 1 + 1 +1} = \\sqrt{6} = 2.4495,", "40d18fcf8fcecd7fcd25904518b8bbfb": "\\, (x_{min}, y_{min}, x_{max}, y_{max})", "40d1c3fa1d7386ef4166b7039bdd18f9": "\\nu(i)=\\sup_{\\tau>0}\\frac{\n \\left\\langle\\sum_{t=0}^{\\tau-1}\\beta^t R[Z(t)]\\right\\rangle_{Z(0)=i}}{\n \\left\\langle\\sum_{t=0}^{\\tau-1}\\beta^t \\right\\rangle_{Z(0)=i}}\n ", "40d1cbfc123a7e313d5305c35945b138": " v( \\emptyset ) = 0 ", "40d1d96e16f6ce4ba92b0f4a272550c0": "\n\\varphi(t;\\alpha,\\beta,\\lambda,\\mu) = \n[1+\\lambda^{\\alpha}|t|^{\\alpha} \\omega - i \\mu t]^{-1}\n", "40d23dfd8fffd792fa2439952ace0e16": "\\pi ^\\pi", "40d33ebb2dc78fc6e2020c31be34f9f6": "\n\tZ_t = \\sum_{i:x_t\\notin x_i}D_t(i) + \\sum_{i:x_t\\in x_i}D_t(i)e^{-y_i\\alpha_i h_t(\\boldsymbol{x_i})}\n", "40d358d448f2e9c7402d69fb16f06929": "\\displaystyle ax \\equiv 1 \\pmod y,", "40d36df13d7b0d9006d93d0e090eb813": " H(s) = A \\prod \\frac{(s - x_n)^{a_n}}{(s - y_n)^{b_n}} ", "40d429398660db7acf1dccc3b9a57e86": "m \\ne \\; n\\,", "40d45691b7359927923be0288024891a": "\\textstyle {N}_i", "40d46d1c2daf1ebdd58f5e90674f6527": "-j", "40d481ac8c18239e6dd8dcc5069e8540": " (l_A a_B + l_B) l_B + (1 + r) l_A a_B ", "40d49943afe20d39d0136f7356c80197": " \\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x) ", "40d4a61be935ebe606f86f6ff81e2f98": "r=\\frac{N_{AA}}{(N_{BB}+2N_B)}", "40d4d9f9d7b60a09d29a82f38f0dc613": "\\scriptstyle{t_{r_O}}", "40d5034bffb2e330732e514eff323759": " \\hat{x} = \\frac{\\sigma_x^2 - m^2}{(\\sigma_x^2 - m^2)+\\sigma_w^2} y + \\frac{\\sigma_w^2}{(\\sigma_x^2 - m^2)+\\sigma_w^2} m.", "40d5248c64ea5664b17e2c5ba1950b27": " \\arcsin(x)+\\arccos(x)=\\pi/2\\;", "40d557321f9f2913a18e94503a8b92fc": "W = (A^TA)^{-1}A^T.", "40d5b738aadc01419292d931d228124b": "j_\\mu^3 = \\frac12(\\overline U_{iL}\\gamma_\\mu U_{iL} - \\overline D_{iL}\\gamma_\\mu D_{iL} + \\overline \\nu_{iL}\\gamma_\\mu \\nu_{iL} - \\overline l_{iL}\\gamma_\\mu l_{iL})", "40d62df2336c804fdfaedc9efdea1d84": "\\| \\cdots \\|", "40d717dcb410c938f4387463d00fbe39": " F_s= \\frac{G \\cdot m \\cdot M_s} {d_s^2}", "40d792bb48c94975132f90f23094286c": "A f(x)\\sqrt{\\alpha(x)} \\exp\\left(-\\int\\alpha(x)\\,dx\\right)", "40d924b688814cb117cc0da07479d43a": "\n\\begin{align}\n \\mathbf{v}_{\\mathrm{rot}} &= \\mathbf{v} \\cos\\theta + ([\\mathbf{k}]_\\times \\mathbf{v}) \\sin\\theta \n + \\mathbf{k} (\\mathbf{k}^\\mathsf{T} \\mathbf{v}) (1 - \\cos\\theta) \\\\\n &= \\mathbf{v} \\cos\\theta + [\\mathbf{k}]_\\times \\mathbf{v} \\sin\\theta \n + \\mathbf{k} \\mathbf{k}^\\mathsf{T} \\mathbf{v} (1 - \\cos\\theta).\n\\end{align}\n", "40d9477e326ad4881499e8d11e6a4ed9": "(\\mathbb{Z}/2\\mathbb{Z})", "40d96545e04b4a2d55b0f5530042bfa1": "i^{th}\\,", "40d9a61b1c5290f6b6c855792672cc53": "a_0 a_n \\ne 0", "40da23ac90097d0aa1b5e84792dda1a5": "2\\, i\\, \\mathbf{k}_0\\, \\cdot \\nabla E_0 + 2\\, i\\, \\omega_0\\, \\mu_0\\, \\varepsilon_0\\, \\frac{\\partial E_0}{\\partial t} - \\left( k_0^2 - \\omega_0^2\\, \\mu_0\\, \\varepsilon_0 \\right)\\, E_0 = 0.", "40da3a88dc7e8b645053845b59d35d9d": "\\sum_{a_i > 0} a_i (f(x_i) - t_i) = \\sum_{a_i < 0} (-a_i) (f(x_i) - t_i), \\quad \\forall f \\in \\mathcal{F}", "40da675a5e8b9107433dc902014c5b4e": "h << l", "40da79f1594f8a0b61797548a4f73c1b": "\\mathfrak{so}_5 \\cong \\mathfrak{sp}_2 ", "40daabd83fc46e692e68d170ab94d650": "\n\\Vert T\\Vert^2_{L^2\\to L^2}\\le\n\\sup_{x\\in X}\\int_Y|K(x,y)| \\, dy\n\\cdot\n\\sup_{y\\in Y}\\int_X|K(x,y)| \\, dx.\n", "40db6bfbeedea40b37e8b6a6c5ce1892": "p(x) = C(x - x_1)(x - x_2)\\cdots(x - x_n), ", "40db999fecca5312d0a82431ac339900": " NX = X - M \\ ", "40dbcb33b179daee4fcce57bc38466ad": "\\scriptstyle \\sin\\theta = \\sin\\left(\\theta + 2\\pi k \\right)", "40dc0b1c78e982448682c45bb5e86770": "\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}} ", "40dc391d9945875c2221f2baf89cf51d": "L_{total} = \\rho V_\\infty \\int_{tip}^{tip} \\Gamma_{(y)} \\operatorname{d}y", "40dc6e251700b4ced2fab619ca184dd1": "\\nu_E", "40dc73c0b085f1a317f38cac7f24acab": "x\\in\\mathrm{supp}\\,T,", "40dc7b03ed3e84f3a81eaff5ee559a0f": "p_{n}(0)=0;", "40dcaaff818834299ef38b0cf7d900ad": "\\rho=v^{\\lambda}p_{\\lambda}-L.", "40dcba645bd712704ed0f67ebff9d39a": "\\lim_{a\\to\\infty} I(a) ", "40dd1eb91b60872d1743fcc12f7c790c": "\\mathbf{V}_\\mathrm{quad} = \\frac{\\lambda d^2 Cos[2 \\phi]}{4 \\pi \\epsilon_0 s^2} ", "40dd3b264095a76c98c3570548495333": "\\phi \\land \\psi", "40dd66875168b0322c73d20dfe564fe5": "(X_1 \\times \\cdots \\times X_j \\times \\cdots \\times X_k)", "40dd7cde7322d5bfb815479170bd4fae": "n = d_k^{d_k} + d_{k-1}^{d_{k-1}} + \\dots + d_2^{d_2} + d_1^{d_1}\\,.", "40dd7d718377e611c887fd58098ceaa2": "P_\\nu = m_0 U_\\mu ", "40dd96223f7c8bb2d942577de3374352": "E_i = l_i + l_A a_i", "40ddac7cff89c1a83dbcd69ba021b1bf": " V_{OUT}(t) = -V_{fb}(t) = - \\frac{1}{\\frac{1}{fC_s}C_{fb}} \\int V_s(t)dt \\, ", "40ddba83c22acc2acaddff12c66d7adf": "R.", "40ddcdb7c3398723c8c7a7e53d3e879c": "F = \\frac{L}{4\\pi r^2} \\,", "40dde0a1bcf28fb79ec2552efb8f01b4": "\\mathrm{for\\ acids\\ with\\ } \\big(pH - pK_a\\big) > 1,\\ log\\ D_{acids} \\cong log\\ P + pK_a - pH", "40de134122653473bf2e98e9d2f95a41": "C(1,\\dots,1,u,1,\\dots,1)=u ", "40deff01e19e8325240aac7f6e4a85fd": "\\displaystyle M_3(\\tau) = q^{47/168}F_3(q) + R_{7,3}(\\tau)", "40df23c75595890cd1d765ec0d714441": "g_t(x)", "40df885312c68d4eadb483b1aa650184": "y = m x + c", "40dfb615dc3689ab06bd498f02c15528": "\\left(\\frac {1}{e}\\right)^e", "40dfec24d93b2ab92feedd8122d0f6ec": " \\phi(y) \\ge V(y) ", "40dff08820a77391c33567102a70bd8d": "\\frac{2i}{z}", "40dff8be7b1aeb0db8d893c49e56d68e": "z\\frac{d^2w}{dz^2} + (b-z)\\frac{dw}{dz} - aw = 0", "40e0321f0193c8c13888baedfcfebda4": "\\{x \\mid x \\not\\in x\\}", "40e04e880c70508977a7ad59c516fb52": "{\\mathbf P}=[ {\\mathbf I} \\; | \\; {\\mathbf 0} ]", "40e08561a200f03ac98e8eed1803f2d7": "s^2= \\eta_{\\mu\\nu} x^\\mu x^\\nu\\ .", "40e0f41d1463da3cbedaa0661e1bd451": "\\nu = \\frac{1}{2}\\,", "40e14e0b934223304f61fd035e125dc1": "\\overline{Y} = \\frac{\\sum_i \\overline{Y}_i}{a} = \\frac{\\overline{Y}_1 + \\overline{Y}_2 + \\overline{Y}_3}{a} = \\frac{5 + 9 + 10}{3} = 8", "40e1ca351d24b1a3a5d532cfbc2c9f25": "\\scriptstyle\\sum\\limits_{n=0}^\\infty \\left|a_n\\right| = \\infty.", "40e2274bc7298fffe1253667db387a4c": "E=\\frac{1}{1+(r/R_0)^6}\\!", "40e2592464ee75634e9417bca990f4eb": "(2n + 1)^2-2", "40e3beb1e23cd04a85bc72750020ba8e": "\n\\begin{bmatrix}\n 1 & \\lambda D \\\\\n 0 & 1 \n\\end{bmatrix}.\n", "40e3c656321bc032bbfef600bdfb5fb7": "\\frac{\\$\\ \\mbox{60,000}}{\\$\\ \\mbox{300,000}}=0.20=20\\%", "40e3f250de286455c28dae05cd6e548e": "\\alpha.\\,", "40e43f29a709d1e4d89cfb682ec50e30": "\\mathcal{H}_\\tau", "40e47ab0beb2bad4746372c72a7b0f01": "\\scriptstyle V_{in} \\;=\\; V", "40e493ff8039b1543e2b1ee33dc6d177": "\\begin{align}h_{jk} & = \nh\\left(\\tfrac{\\partial}{\\partial\\theta_j}, \\tfrac{\\partial}{\\partial\\theta_k}\\right) \\\\\n & = \\frac{1}{4} \\mathrm{E}\\left[\n\\frac{\\partial\\log p}{\\partial\\theta_j}\n\\frac{\\partial\\log p}{\\partial\\theta_k}\n\\right] \n + \\mathrm{E}\\left[\n\\frac{\\partial\\alpha}{\\partial\\theta_j}\n\\frac{\\partial\\alpha}{\\partial\\theta_k}\n\\right]\n - \\mathrm{E}\\left[ \\frac{\\partial\\alpha}{\\partial\\theta_j} \\right]\n\\mathrm{E}\\left[ \\frac{\\partial\\alpha}{\\partial\\theta_k} \\right] \\\\\n& - \\frac{i}{2}\\mathrm{E}\\left[\n\\frac{\\partial\\log p}{\\partial\\theta_j}\n\\frac{\\partial\\alpha}{\\partial\\theta_k}\n- \n\\frac{\\partial\\alpha}{\\partial\\theta_j}\n\\frac{\\partial\\log p}{\\partial\\theta_k}\n\\right]\n\\end{align}", "40e5ddc66c6995b164f8e0d7c0eea567": "\\langle x,y\\rangle=1", "40e66515ba2e1ecc9b3f3942244dead0": "m_{\\text{r}}", "40e666e6456ae45e7ac9de8a72566437": "D(p||q))\\geq f\\sum p\\frac{q}{p}=f(1)", "40e6a4567fc1fc12f502941587a9d787": "f(x*s)=f(x)*s", "40e6b7591517e93e6db7813993a8aa7a": "\\left(Y_{[r_1:n]}, \\cdots, Y_{[r_k:n]} \\right)", "40e6c7792bd33eb7b8b66c9d95651c4a": "F(b) - F(a)", "40e6e54df6c94948f36cfd06f0e8d4f6": "\\scriptstyle (\\partial f/\\partial x_i)(0)=0,\\;(i=1,\\dots, n)", "40e6f19b2e153e2b44d05b4b6b5d54a8": "1 + 1 =", "40e6fa4d5776f233830bad4ddf4429ef": "\\Gamma = D|q|^2 \\qquad(4)", "40e6fb2c27640f5c09b82dc5c7ebef3b": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, conc(\\langle b \\rangle, S, \\langle b \\rangle), \\langle b \\rangle), \\langle a \\rangle)", "40e7494929982245823ba78325ac2963": "-\\infty = M^-_\\infty < M^+_\\infty = +\\infty; ", "40e79470fd5669576a654b720988e0db": "(\\lfloor i\\varphi\\rfloor, \\lfloor i\\varphi^2\\rfloor)", "40e79fe29fd9edc5a592f682dc39b652": " ZZ=Z\\cdot Z ", "40e7a911592bf13448ce366a4930e2d4": "u=u_x +iu_y", "40e7bf331e87fd935530564fee98fb4b": "\\nabla_{\\bold{v}}(h\\circ g)(\\bold{p}) = h'(g(\\bold{p})) \\nabla_{\\bold{v}} g (\\bold{p})", "40e7e2e9525c9dca49598c217ecb5041": "p<2", "40e7fee5d023356a9ace309456168985": "\\displaystyle f'(x_{0}) \\approx \\displaystyle \\frac{-\\frac{11}{6}f(x_{0}) + 3f(x_{+1}) -\\frac{3}{2}f(x_{+2}) +\\frac{1}{3}f(x_{+3}) }{h_{x}} + O\\left(h_{x}^3 \\right), ", "40e82ab5c31559a7c111dd349467f605": "P_{\\mathbf{k}}", "40e8552bd8fbfec47013ad27b4e74f71": " \\frac{dm_V}{dt} = \\rho_V\\frac{dV}{dt} = \\rho_V\\frac{d(4\\pi R^3/3)}{dt} = 4\\pi\\rho_VR^2\\frac{dR}{dt} ", "40e89fe34fef92bd40eca47169ab2e7a": " \\mathbf{d}_i^{[k]} = (1-\\alpha_{k,i}) \\mathbf{d}_{i-1}^{[k-1]} + \\alpha_{k,i} \\mathbf{d}_i^{[k-1]}; \\qquad k=1,\\dots,n; \\quad i=\\ell-n+k,\\dots,\\ell ", "40e8a15eea01d9ec2ef3471eceb59cb4": "{\\mathrm{m}}", "40e8ed83ff4709e58859d56c91d5ab40": "g(\\vec{r}, \\vec{r}') = p(\\vec{r},\\vec{r}') V^2 \\frac{N-1}{N}", "40e9020da7cd472433a88c36c0b8a1ce": "\\sum_{j=1}^{20}n(j)M^n(j,j)", "40e92586c8d5cf63fc1ac27a53ce8469": "A\\in\\mathbb{M}_n (\\mathbb{C})", "40ea0336807f13225ffb59b62505d741": "D \\triangleq \\{J\\subset I : |J|<\\infty \\}", "40ea1a326879b0337aeaefbee64e5238": "\nJ_\\mu |0\\rangle = k_\\mu |\\pi\\rangle \n\\,,", "40ea2cb22fd54873007245ee51f60659": " x \\left( t \\right) =A \\cos \\left( \\sqrt{k \\over m}t \\right).", "40ea471596fcb0a09cfe3d5bfe6e4f7c": "\\Gamma\\ \\vdash\\ e:\\sigma", "40ea5cb9e4f6fa2d06315db0650dc313": " x, y ", "40eaadea3b5c3a73186968c6b911c95f": "\\psi(1) = \\omega^{\\omega^2}", "40eacf8e3905ec0c998b913d45dd741a": "f_n = \\frac{(-1)^n }{2\\pi i} \n\\int_\\gamma \n\\frac {\\phi(s)}{\\Gamma(-s)} \\frac{n!}{s(s-1)\\cdots (s-n)}\\, \\mathrm{d}s", "40eb38468ed85aa87159a0f25f0905fb": "J_i={C_i} {j_i}", "40eb69efe7283ecc4f091c7906bf4076": "b \\left ( \\alpha\\, \\right ) = \\mbox{systematic brute forcing on copies of the program with the same address space}", "40eb824bb677b222a160aed3a9c28517": "y^i = y^i(x^1,x^2,\\dots,x^n),\\quad i=1,2,\\dots,n", "40ebb6f8be8903d3c8442d6b78035506": "p_i =\\frac{\\partial L}{\\partial\\dot q_i},", "40ec0a5374afed7b3edd58946d625776": "(m s) r = r (s m)", "40ec54cd9639a96cd3fb5446fabfa9b3": "\\scriptstyle \\partial_a", "40ec8a6e972df057e83345a0a4fce562": "\\sum_{r=1}^{m+n}{x'_{r,s}}=1", "40ecae15b2bc3539de4180598d437655": " A \\underbrace{ A^{T}(AA^{T})^{-1} }_{ A^{-1}_\\text{right} } = I_{m} ", "40ecbaa26d22c1a0a4ab8103dea95ce9": "\\hat{B}", "40ecccbf7d18240e13d0f04154827644": "\n\\begin{pmatrix} x \\\\ y \\\\ 1 \\end{pmatrix}\n", "40ecd813ceb9b34e8e67d2bb305fd315": "g(r)=\\frac {exp(-qr)} {(2 \\pi \\Lambda r)}", "40ed8abd6ae8af3300aa8a8b08d22c47": "\\mathbb{Z}^{2g}", "40ee163d44d7899d334d4b1a58fc575e": "\\Gamma^i{}_{jk}=\\Gamma^i{}_{kj}.\\ ", "40eec9c6e8fcea72226b552067324870": "f_\\triangle :K \\rightarrow L", "40eef8459c4b764fa270f5867633bd59": " \\forall n\\!\\in\\!\\mathbb{N}\\; \\bigl( Q(n) \\rightarrow P(n) \\bigr) ", "40ef151b045963fb8e7df9f0b5ae3538": "\\theta=x", "40ef5a56d29f0095e273d9327c83c1d1": "\\frac{a}{2}[\\text{1 0 -1}] \\rightarrow \\frac{a}{6}[\\text{1 1 -2}] + \\frac{a}{6}[\\text{2 -1 -1}]", "40efc92613fa65ea9b1c2372465497b9": "cd/m^2", "40f017d69c2531f9156da420ab837b67": "\\hat{x}_1 = x_1(1+\\delta_1)", "40f070efcaabdef370595e254201c0fe": "L(X^*_{\\sigma(X^*, X)}, Y_{\\sigma(Y^*, Y)})", "40f0a95883e5148219999b4f6c82a251": "\\forall z", "40f0b72630416e8c53b6e5b33136223d": "\\sigma \\in (0,\\infty) \\,", "40f0d4dadb6383b2d3c57a9f60b6d0c6": "a=K_{IC}^2/\\pi\\sigma_Y^2", "40f10b7080b8b6942c1b851107481c06": "z_1+ z_2 = z_2 + z_1,", "40f12d113a33f204afc6a1b20efacdbd": "g(h) = \\frac{1}{1 + e^{-2 \\beta h}} \\! ", "40f1310d256c7380f4eacf5e3da0e3ae": "x^5 + cx + d", "40f1496f848ee133b6c5ae43069f008e": "{\\mathcal B}", "40f178625f6b0282787354fbf216770c": "y_{n+\\frac{1}{2}}=y_n+\\frac{h}{2}f(t_n,y_n)", "40f189528f587d4208d1c94434b20bf9": "dx/dt", "40f19e44153f0893ab039549fe556a53": "\\mathrm{Im}\\,\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)", "40f1c6afd34297556b412a75bcd1ffbd": "L = \\pm|\\mathbf{r}||\\mathbf{p}_{\\perp}|", "40f1e0dba6ff8005d7b9ce1f10f1e959": " r_{\\text{o}} = \\frac{V_{\\text{A}}}{I_{\\text{C}}}", "40f20013604bb77b29d32572911be9f1": "= \\left(i\\dot \\omega-\\omega^2 \\right) R e^{i\\theta} ", "40f22f02b381e495f5856d7adc41de21": "\\operatorname{div}(\\operatorname{curl}(\\mathbf{F}))=0", "40f23c2676a1c22d964c9bafd6f98d8b": "\\{1, \\dots, k\\}", "40f256dbc2262275d25df03d9996cdd7": "S_\\mathrm{BET}", "40f28d91a2c6da4b40e96b24849fce98": "\\vec{S}", "40f299034528b65c04c76758139f55aa": " q_{S} = |A_{S}|", "40f30a22b35d834d80f4750658fc08f9": "\\scriptstyle (N,d)\\neq(1,1) ", "40f34cf30b086ab3d66c79b6fe69f3d3": " P( S_j > y_j ) = \\frac{r_{j+1}}{\\overline{r}_j}", "40f3ba82c1f87a4eb6fd06467649e02e": "\\gamma_2 \\to \\infty", "40f413b1fe30eb5a015695dcf1ca13a6": "\\exist r \\in p (a \\in r \\and \\forall x \\in r (x = a)) \\and \\exist s \\in p (a \\in s \\and b \\in s \\and \\forall x \\in s (x = a \\or x = b))", "40f4256800031b76d60839f026c5bd6c": "p_{\\alpha+1}", "40f4a4b42720ffdda94d928d44dad62e": "\\frac{4 \\pi m}{\\sqrt{1 - \\frac{2m}{r c^2}}} ", "40f4ed6571112cb3115a75df0bab68e2": "n_s \\sim s^{-\\tau}(1+\\text{const} \\times s^{-\\Omega})\\,\\!", "40f57124e82cac681cc004ab1727e7ae": "\\scriptstyle\\boldsymbol{x}_i", "40f57d57598af67e9a8b73c5c7f3f49e": "M_{ts}=t(\\lambda_s),\\,\\,N_{st}= (-1)^{\\ell(t)}\\cdot t(\\psi_s),", "40f58220104d9d17c8e6ebb7a7e3d50e": "m_E=-A-2B", "40f5c5e21397577eb433006b27991fbf": "(f*g)(x)=\\int f(x-y)g(y) dy = \\int h(x,y) dy", "40f5e5122f8907f2257e9ed2ccf014f4": " n \\in \\mathbf{N}\\,\\!", "40f61f782b50b44aa520ac4ed0dbbe6a": "x^3 + x^2 + 1", "40f62c63c17c5c1ef2953fc743121859": " \n\\left[\n\\begin{array}{cc}\n {[-4,-3]} & {[-2,2]}\\\\\n {[-2,2]} & {[-4,-3]}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\nx_1\\\\\nx_2\n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array}{c}\n{[-8,8]}\\\\\n{[-8,8]}\n\\end{array}\n\\right]\n", "40f64f3adf5fbcab81c903a79e1b66ca": "x^{24} + x^{22} + x^{20} + x^{19} + x^{18} + x^{16} + x^{14} + x^{13} + x^{11} + x^{10} + x^8 + x^7 + x^6 + x^3 + x + 1", "40f665ea91ee0e75576229f01ea942ee": " (\\lambda x.f\\ (x\\ x))\\ \\operatorname{get-lambda}[x, x = \\lambda q.f\\ (q\\ q)] ", "40f6988bab391d5991b0d88985f762d1": "f=(c_0+2mc_1+c_2m^2)x^2+(d_0+3md_1+3m^2d_2+d_3m^3)x^3+\\dots.\\,", "40f6a3b6cb22eb95ff51504baa0e1fae": " \\bigcup_{i=1}^m\\psi_i (V) \\subseteq V, ", "40f6b23148fd3f5cb6fd708d247cc365": "\n\\begin{align}\n\\Pr(Y_i=1) &= \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{1 + \\sum_{k=1}^{K-1} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\\\\n\\Pr(Y_i=2) &= \\frac{e^{\\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i}}{1 + \\sum_{k=1}^{K-1} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\\\\n\\cdots & \\cdots \\\\\n\\Pr(Y_i=K-1) &= \\frac{e^{\\boldsymbol\\beta_{K-1} \\cdot \\mathbf{X}_i}}{1 + \\sum_{k=1}^{K-1} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\\\\n\\end{align}\n", "40f6d994e0cf983c0f9e544a07e88217": " A \\mapsto \\psi(A) = \\bigcup_{i=1}^m \\psi_i(A) ", "40f705c78f3fb00390bc77be2dbcd326": "\\mathcal{A} = \\mathcal{B} \\times \\mathcal{C}", "40f71521d258eb29c6866702ef65a804": "\\tfrac{p+1}2", "40f720eff3474e94604a231c2e576461": "\\nu \\in Y", "40f73157f8aca276cee6284f67d508e3": " \\delta^i_k ", "40f7898b64fe4527000acf70e5133569": "ct_{LB}", "40f79775461db3743ff60250bb1d49b0": "\\frac{\\partial^2V(x)}{\\partial x^2} = \\gamma^2 V(x)", "40f838b4ac0e3f424a4119543f3fbf05": "A_x(\\eta,\\tau) = \\int_{-\\infty}^\\infty x(t+\\tau /2)x^*(t-\\tau /2) e^{-j2\\pi t\\eta}\\, dt,", "40f841ef358e47e931e738c5c63ca11b": "H_b = \\frac{n J_{ex} S_F S_{AF}}{2 M_F t_F}", "40f84bca5316bab94dff9b10093e105a": "\\frac{\\partial F}{\\partial s} {\\rm d}\\Sigma", "40f86bf516bf17ce346815ded57b15fc": " N_4 ", "40f8d395a7018aa47a2f38152551d292": "\\sin(180^\\circ-\\alpha) = \\sin(\\alpha)", "40f9bfa0270a215a24f438df3f82bf4f": "s^2\\mathcal{L}\\{f(t)\\}-sf(0)-f'(0)+4\\mathcal{L}\\{f(t)\\}=\\mathcal{L}\\{\\phi(t)\\}", "40f9c4295f0c86400619c94f607a78a6": "Pwf = \\cfrac{2 Pmf}{3} = \\tfrac{6}{15} = \\tfrac{2}{5}", "40f9dd98263b91ddb875d4e48e89f1b3": "(\\lambda x.y)", "40f9e5edc269e70b609a5975e949fdbd": "\\ M_{pitch}= D_{pitch} \\times drag \\times 1 ", "40fa06926fa24ab2fb7192715ab2d033": "\\pi_* R \\to \\operatorname{MU}_*(R)", "40fa14706b1be5b70ac84876c0efafb3": "\\begin{align}\nd(T^n(x_0), T^m(x_0)) &\\le \\frac{d(T(T^n(x_0)), T^n(x_0)) + d(T(T^m(x_0)),T^m(x_0))}{1-q}, \\\\\n&= \\frac{d(T^n(T(x_0)), T^n(x_0)) + d(T^m(T(x_0)), T^m(x_0))}{1-q} \\\\\n&\\le \\frac{q^n d(T(x_0), x_0) + q^m d(T(x_0), x_0)}{1-q} \\\\\n&= \\frac {q^n + q^m} {1-q} d(T(x_0) ,x_0) \n\\end{align}", "40fa5045097ad7770b74961a2f27dd1f": "H=H_0,H_1,H_2,\\ldots, H_k=G", "40fad460dd45fde8735172a730fd0a03": "\\Pr_{h \\in H} \\left[ h(x_1)=y_1 \\land \\cdots \\land h(x_k)=y_k \\right] = m^{-k}", "40fae8ea73f9c785a9903eb9135d8948": "2|\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|(\\mathbf{X}^{-1})^{\\rm T}", "40fb4f39a0aebabaf6dbc2fd3e82e0c9": "\\mathbf{Z}/4,", "40fb57cd5146fa03c69710fb43647ead": "(a + b) + c = a + (b + c)", "40fb64fec5fe45afe68a6cae37081561": "D(p)=(-1)^{\\frac{1}{2}n(n-1)}\\frac{1}{a_n}R(p,p').\\,", "40fb8606b55bb191c797f63571e9d46e": "\\scriptstyle s.e.(\\hat\\beta)", "40fbd72692855518988f2fe9bd0b46ea": "\nY=Y_0 e^{[\\nu+\\alpha(\\eta-\\nu)]t}=Y_0 e^{\\delta \\,t}.\n", "40fbdeff8bcaff1dbc63cbd5259c8e00": "r_p", "40fbf7074569907f7f152cb6ca2403aa": "c_n/c_{n-1}", "40fc34bff4a51743cc48a0639bf45c2c": "C(x,y)", "40fc6088830aad25a1d13fc4fa2692e4": "\\Delta\\Theta \\approx \\Lambda/d", "40fc8a6e6d554b8ef25903e1141ae0ae": "\n \\boldsymbol{F} = \\begin{bmatrix} 1 & \\gamma & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\n ", "40fcb782ad0d6b4bdd2a36a99f200628": "D \\subset \\Sigma", "40fcbdc1bce74575467761e364ea192f": "X \\in L^{\\infty}_T", "40fd050563a8323f9211783e926420bb": "\n\\begin{align}\n\\frac{x}{R} &= \\int_{0}^{\\lambda'} \\frac{H-S^2}{\\left(1+S^2\\right)^{1/2}}d\\lambda' - \\frac{S}{\\left(1+S^2\\right)^{1/2}}\\ln\\tan\\left(\\frac{\\pi}{4}+\\frac{\\phi'}{2}\\right) \\\\\n\\frac{y}{R} &= \\left(H+1\\right) \\int_{0}^{\\lambda'} \\frac{S}{\\left(1+S^2\\right)^{1/2}}d\\lambda' + \\frac{1}{\\left(1+S^2\\right)^{1/2}}\\ln\\tan\\left(\\frac{\\pi}{4}+\\frac{\\phi'}{2}\\right) \\\\\nS &= \\left(P_{2}/P_{1}\\right) \\sin i \\cos \\lambda' \\\\\nH &= 1 - \\left(P_{2}/P_{1}\\right) \\cos i \\\\\n\\tan\\lambda' &= \\cos i \\tan \\lambda_{t} + \\sin i \\tan \\phi / \\cos \\lambda_{t} \\\\\n\\sin\\phi' &= \\cos i \\sin \\phi - \\sin i \\cos \\phi \\sin \\lambda_{t} \\\\\n\\lambda_{t} &= \\lambda + \\left(P_{2}/P_{1}\\right) \\lambda'. \\\\\n\\phi &= \\text{geodetic (or geographic) latitude.} \\\\\n\\lambda &= \\text{geodetic (or geographic) longitude.} \\\\\nP_{2} &= \\text{time required for revolution of satellite.} \\\\\nP_{1} &= \\text{length of Earth rotation.} \\\\\ni &= \\text{angle of inclination.} \\\\\nR &= \\text{radius of Earth.} \\\\\nx,y &= \\text{rectangular map coordinates.}\n\\end{align}\n", "40fd23f4296e800d6079983d79d9f438": "\\{2^i \\mod{9}\\ |\\ i \\in \\mathbb{N}\\} = \\{2,4,8,7,5,1\\}.", "40fd6586a299a1958aed7b9f4ec72a97": " \\frac{\\partial}{\\partial z_i} \\left(\\alpha f+\\beta g\\right)= \\alpha\\frac{\\partial f}{\\partial z_i} + \\beta\\frac{\\partial g}{\\partial z_i},\\quad \\frac{\\partial}{\\partial\\bar{z}_i} \\left(\\alpha f+\\beta g\\right) = \\alpha\\frac{\\partial f}{\\partial\\bar{z}_i} + \\beta\\frac{\\partial g}{\\partial\\bar{z}_i}", "40fd75ebe13271830daf3df4a7067003": "\\Lambda(x)=\\alpha^{3}+\\alpha^{1}x.", "40fdb5e9cae9a76b2e166640ef46363c": "\\scriptstyle C_S", "40fdbe5714e48c83bd42198936216df3": "1\\le i\\le m", "40fdc2b1875ee7f8db992755f59fc347": "\\int_0^\\infty e^{n(\\ln y -y)} dy \\sim \\sqrt{\\frac{2\\pi}{n}} e^{-n}", "40fdc9e2a8b37ebb2dcbda82d2e26e63": " \\Upsilon_i := \\Omega^{-1} \\partial_i \\Omega\\, . ", "40fdd727431ed6c50d1ee12d838653a9": "\\gcd(p,q)", "40fdfc219213371a1a8682dc624df079": "2 \\le a \\le (b+1)/2", "40fe29f8498ed35a3d430123f0f35dd4": "(A_{i,j})_{i,j \\ge 1}", "40fe9ad4949331a12f5f19b477133924": "hg", "40fec5273e77d8661b94b97f3e58b3dd": "X^2\\backslash\\left\\{ y~\\backepsilon~y\\succcurlyeq x\\right\\}=\\left\\{ y~\\backepsilon~x\\succ y\\right\\}", "40fec549d6f77f087015606dfa81b917": "V = E_f d \\Rightarrow E_f=\\frac{V}{d} ", "40ff67ce3ea734efa48f3882df3cb5d4": "\\tan\\delta'=\\frac {\\sin \\delta}{\\gamma(\\cos \\delta +\\beta)} \\approx \\frac {\\sin \\delta}{\\cos \\delta +\\beta} = \\frac{\\sin \\delta}{\\cos \\delta \\cdot \\left(1+\\frac{\\beta}{\\cos \\delta}\\right)} \\approx \\tan \\delta \\cdot \\left(1-\\frac{\\beta}{\\cos \\delta}\\right)", "40ffaf0e0eb28873346765012975b247": "\n\\begin{align}\n& {} \\quad \\frac{a_1+a_2+a_3+a_4}{4} \\\\[8pt]\n& {} \\ge \\sqrt{\\frac{a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4}{6}} \\\\[8pt]\n& {} \\ge \\sqrt[3]{\\frac{a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4}{4}} \\\\[8pt]\n& {} \\ge \\sqrt[4]{a_1a_2a_3a_4}.\n\\end{align}\n", "41002dcd457f75ea1b7f5a43cb394d93": "{x^2 \\over a^2} + {y^2 \\over b^2} - {z^2 \\over c^2} = 1 \\,", "4100334037ba581822ce50166592a778": "E_R", "41008db73f3ffe6ef659b776481e7bde": "6^6", "410138918876984467a609c7c2de3832": "Angle\\ of\\ Bank \\approx \\frac{TAS(kt)}{10} + 7", "41016fb113f9b7b8583e9ec681950f33": "\\mathcal{T}=1+\\sum_R\\hat{\\mathcal{T}}_R", "410183dc076802dd1a24e9f0aa3ee4eb": "\\mathcal{B}_r = ( \\lfloor nr \\rfloor)_{n\\geq 1}", "4101a25a8ac54d5e8c1b9dd194ea3606": "2^{\\varnothing} = \\{\\varnothing\\}\\, .", "4101be4c871086d3106b1918f649362e": "\\text{if both } A,B \\neq \\emptyset \\text{ then } A \\times B \\subseteq C \\times D \\iff A \\subseteq C \\and B \\subseteq D.", "41020db1d819bdd5c7d40c6e9b94e22f": "\n \\rho~\\dot{\\eta} \\ge \\cfrac{1}{T}\\left(\\rho~\\dot{e}-\\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v}\\right) + \n \\cfrac{1}{T^2}~\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T \n \\qquad \\implies \\qquad\n \\rho~\\dot{\\eta}~T \\ge \\rho~\\dot{e}-\\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} + \n \\cfrac{\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T}{T}. \n ", "410242ab969eaff299d20a647929af39": "\n\\left( \\begin{smallmatrix} 0\\\\ 0\\\\ 0 \\end{smallmatrix}\n\\begin{smallmatrix} 0\\\\ 0\\\\ 1 \\end{smallmatrix}\n\\begin{smallmatrix} 0\\\\ 1\\\\ 0 \\end{smallmatrix}\n\\begin{smallmatrix}1 \\\\ 0 \\\\ 0 \\end{smallmatrix}\n \\begin{smallmatrix} 1 \\\\ 1\\\\ 0 \\end{smallmatrix}\n\\right) \\qquad \\Longleftrightarrow \\qquad \\begin{matrix} 2 & 1 \\\\ 1 & 1 \\end{matrix} \n", "41025d55474603bbfea41844b5c7c4ef": "V^+_i", "410268fccd6bde1a9e6f1bd11016355e": " \n\\mathbf{h}(n) = \\left[h_0(n), h_1(n), \\dots, h_{p-1}(n)\\right]^T,\\quad \\mathbf{h}(n) \\in \\mathbb{C}^p\n", "41027e5e55f4d64cdf0b10c65b2af705": "{{T}_{s}}-{{T}_{sat}}", "41033268fa8e1d98eb0b80709675f55b": "\\mathbf{\\dot{P}} = - \\frac{\\partial K}{\\partial \\mathbf{Q}}\\,,\\quad \\mathbf{\\dot{Q}} = + \\frac{\\partial K}{\\partial \\mathbf{P}} \\,,", "4103904aabe71e958b60d7876616a7d8": "\\nabla_1^2", "4103b187a95fb2166ad5b5aa0b882bb2": "V_\\text{cap}(t)", "4103c5877f9db8a92937d3f3ee8fe2c2": "\\displaystyle{f(z_n)-f(z)=\\int_{\\partial\\Omega} (K(w,z_n) + K(z_n,w) - K(w,z) - K(z,w))\\varphi(w) \\, |dw| =\\int_{|w-z|\\ge \\delta} + \\int_{|w-z|\\le \\delta}.}", "41044d17a0de70bf465c1f5b65ae3330": "\\left\\vert \\langle \\psi_{x+} \\vert \\psi \\rangle \\right\\vert ^2", "410456641a738ca74ada067c45d94873": "\\sum_{r} \\tilde{P}_{r}\\log\\left(\\frac{\\tilde{P}_{r}}{P_{r}}\\right)\\geq \\sum_{r}\\left(\\tilde{P}_{r} - P_{r}\\right) = 0 \\,", "410491212e6c7d0cc8400ca4b882b276": "\\left(\\begin{bmatrix}y_0\\\\y_1\\\\y_2\\\\\\cdots\\\\y_{m-1}\\end{bmatrix},\\begin{bmatrix}y_m\\\\y_{m+1}\\\\y_{m+2}\\\\\\cdots\\\\y_{2m-1}\\end{bmatrix},\\cdots,\\begin{bmatrix}y_{n-m}\\\\y_{n-m+1}\\\\y_{n-m+2}\\\\\\cdots\\\\y_{n-1}\\end{bmatrix}\\right)", "41049ed90095b396a5fac0c0800101f8": "-\\infty x_0", "410c34c5080aaabc42cc19e899d2f1ee": "~\\left(\\frac{n_2}{n_1}\\right)_{\\!T}=\n\\exp\\!\\left(\\frac{\\hbar \\omega_{\\rm Z}}{k_{\\rm B}T}\\right)~~~~,~~~\n", "410c4b817dec11481851e3117127576f": "x_{n+1}=x_{n}-\\frac{f(x_n)-\\alpha}{f'(x_n)} \\left(\\frac{f(x_n)}{\\alpha}\\right).", "410c58b51a431240781f82ebf11ab0df": " i\\,T_2:=\\frac{T-T^*}{2}\\,),", "410c7131933333e052ec295230219e96": "P(R_n)", "410c7656de83503155c3125c67aaafad": "\n\\operatorname{ec}(G) = t_w(G) \\prod_{v\\in V} \\bigl(\\deg(v)-1\\bigr)!.\n", "410ccd5df3851918320ce41291b18026": " V(f,\\Omega):=\\sup\\left\\{\\int_\\Omega f(x)\\mathrm{div}\\phi(x)\\,\\mathrm{d}x\\colon \\phi\\in C_c^1(\\Omega,\\mathbb{R}^n),\\ \\Vert \\phi\\Vert_{L^\\infty(\\Omega)}\\le 1\\right\\}, ", "410cf93adebe1c2f3f10855bdc875cfe": "7^4", "410d169cffa4da5c716592a855b17a85": "\\bar x_X = \\bar x_A + \\delta\\cdot\\frac{n_B}{n_X}", "410d1cc34ee0d087d96054e9622fab09": "G(\\mathbf{v})", "410d21d6b793c690557bab7ea75fbd28": "\\mathbf{\\tilde{U}}_{S}^{\\dagger}", "410d78a5567ae04a1b7d4f1cd1c0ddac": "\n\\vartheta(\\theta,\\tau)=\\sum_{n=-\\infty}^\\infty (w^2)^n q^{n^2}\n \\text{ where } w \\equiv e^{i\\pi \\theta}", "410da07b81b19249400a6ef053aab950": "\\scriptstyle\\mathbf{R}^n ", "410dc5b5ad9d7a634b86aa257202b8b8": "(a_1b_4 + a_2b_3 - a_3b_2 + a_4b_1)^2\\,", "410de1ee3d12850e0141bd666d4ea1f9": " \\psi(x) = - \\psi_1(0, x) ", "410e13c69d07004fc948b086ba77e433": "\nh_{n, m} = \\rm {max}_{z=1, \\ldots ,8}\\sum_{i=-1}^{1}\\sum_{j=-1}^{1}g_{ij}^{(z)}\\cdot f_{n+i,m+j}\n", "410e36624dd6f3359681c02144d3356f": "Q=I-P", "410ea1dceb75794787dcf4470fc992ed": "a=\\eta a_{max}", "410ec60955241b2904c0a3a82c08d49f": "D \\cdot x > 0", "410f18fa45e9456a0b37d4f26231e7aa": "\\pi=\\textstyle \\cfrac{4}{1+\\textstyle \\frac{1^2}{2+\\textstyle \\frac{3^2}{2+\\textstyle \\frac{5^2}{2+\\textstyle \\frac{7^2}{2+\\textstyle \\frac{9^2}{2+\\ddots}}}}}}\n=3+\\textstyle \\frac{1^2}{6+\\textstyle \\frac{3^2}{6+\\textstyle \\frac{5^2}{6+\\textstyle \\frac{7^2}{6+\\textstyle \\frac{9^2}{6+\\ddots}}}}}\n=\\textstyle \\cfrac{4}{1+\\textstyle \\frac{1^2}{3+\\textstyle \\frac{2^2}{5+\\textstyle \\frac{3^2}{7+\\textstyle \\frac{4^2}{9+\\ddots}}}}}", "410f6e16f70ccba7815b813d023f0aac": "9\\times 10^{16}", "410fa2477a907ac9283403c78d8204df": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_\\text{f} + \\frac{\\partial \\mathbf{D}} {\\partial t}", "41100a97e02ab609997dc4eb969eacf9": "\\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 1 & 1\\end{pmatrix}\n\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;\n\\begin{pmatrix}1 & 0 & 0 \\\\ 1 & 1 & 1\\end{pmatrix}", "41102d8989fbc417e9aa42caff399d01": "x_B \\overset {d}{=} (x_A + z)", "411035f064db077896b4658752bfe32c": " \\left[(\\mathbf{AB})^\\mathrm{T}\\right]_{ij} = \\left(\\mathbf{AB}\\right)_{ji} = \\sum_k \\left(\\mathbf{A}\\right)_{jk}\\left(\\mathbf{B}\\right)_{ki} = \\sum_k \\left(\\mathbf{A}^\\mathrm{T}\\right)_{kj}\\left(\\mathbf{B}^\\mathrm{T}\\right)_{ik} = \\sum_k \\left(\\mathbf{B}^\\mathrm{T}\\right)_{ik}\\left(\\mathbf{A}^\\mathrm{T}\\right)_{kj} = \\left[\\left(\\mathbf{A}^\\mathrm{T}\\right) \\left(\\mathbf{B}^\\mathrm{T}\\right)\\right]_{ij} ", "41104c3ff33f630c3dafa7a2079a6f4d": "0 \\le k \\le n", "411055fe99631c3be4528051522cabba": "\\delta(q,a)=\\delta_a(q)", "411069363473c2a3fb4b2d85e2580a43": " \\mu(E) = \\inf \\{\\mu(U): E \\subseteq U, U \\mbox{ open}\\} ", "4110dcded9e28c5cb10f3b78fca86093": " \\begin{align} \n\\mathbf{A} & = (A_t, \\, A_x, \\, A_y, \\, A_z) \\\\\n& = A_t \\mathbf{e}_t + A_x \\mathbf{e}_x + A_y \\mathbf{e}_y + A_z \\mathbf{e}_z \\\\\n\\end{align}", "411193d018b597a177357e8c355b862a": "\\rho: M \\to M \\otimes C", "4111ddc6d8ef3e6566d6616512d6dffd": "\\sum_{k=0}^n {n \\choose k}^2 = {2n \\choose n}.", "411277a4089b0cf6898d790a188d0d30": " E_1, E_2, ... , E_{n-1}, E_n \\vdash S, ", "4112cb742204b9f55ddab92f29e06cb9": " L \\in X \\to \\operatorname{sink-test}[L, X] = L ", "4113d0ed339546c597139341f840bb2c": "p_4 = A\\rightarrow a", "4113fc37e42032174b9b49387b8b7125": "|a(v, w)| \\le \\gamma \\|v\\|\\,\\|w\\|", "4113feecb501085342aadddacb004321": "(a)^{(\\alpha )}_\\kappa=\\prod_{i=1}^m \\prod_{j=1}^{\\kappa_i}\n\\left(a-\\frac{i-1}{\\alpha}+j-1\\right).\n", "411407f717c0404c61e459121c8c7192": "\\vdash A \\to A", "4114c7d8a6c5b8facb1b81a0f35a213d": "g_n", "4114d8204ca379a65697648a7615efc4": " \\ TimeStep = 1/4 ", "41150a334bed9f65d38382e5420009de": "\\left(\\prod_{x \\in X_{}} x^{r(x)}\\right).\\left(\\prod_{q \\in P_\\Delta} f^{t(q)}_{q}\\right) = 1", "41152577b4fc11f508c0df2144a934a5": "\\delta w = -V\\sum_{ij}\\sigma_{ij}d\\varepsilon_{ij}", "4115bbb29610136892710201e6ae565b": "\\begin{align}\n &\\sinh(\\operatorname{arcosh}\\,x) = \\sqrt{x^{2} - 1} \\quad \\text{for} \\quad |x| > 1 \\\\\n &\\sinh(\\operatorname{artanh}\\,x) = \\frac{x}{\\sqrt{1-x^{2}}} \\quad \\text{for} \\quad -1 < x < 1 \\\\\n &\\cosh(\\operatorname{arsinh}\\,x) = \\sqrt{1+x^{2}} \\\\\n &\\cosh(\\operatorname{artanh}\\,x) = \\frac{1}{\\sqrt{1-x^{2}}} \\quad \\text{for} \\quad -1 < x < 1 \\\\\n &\\tanh(\\operatorname{arsinh}\\,x) = \\frac{x}{\\sqrt{1+x^{2}}} \\\\\n &\\tanh(\\operatorname{arcosh}\\,x) = \\frac{\\sqrt{x^{2} - 1}}{x} \\quad \\text{for} \\quad |x| > 1\n\\end{align}", "4115c63b5073274e1be4752aeb98d4c3": " L(y) = 0 \\,", "4115ec06521af93c6b94901d7c4cf79d": "f: \\mathbf{R}^{d} \\to \\mathbf{C}", "4115fde4a42803e10dc76e1ede9f8082": "t, s, y", "4116236c4c1a0c0ff5b6efab58241155": "\\arctan{\\left(\\frac{Z_o\\Omega}{R(1+R)+Z_o^2\\Omega^2}\\right)}", "41162e6dc6c46cb4a122c67d978603a5": "\\scriptstyle \\frac{2^n}{m}", "41163e6d17666ebb24f941a9e1a3f06a": " C' = \\frac{L}{R_0^2}", "4116b22a939db81fdf2a25237a8d99f3": "\\ H_{in}", "41171aae01ab9d1acc50198bb83760be": "\nm(\\phi) =\\int_0^\\phi M(\\phi) d\\phi\n= a(1 - e^2)\\int_0^\\phi \\left (1 - e^2 \\sin^2 \\phi \\right )^{-3/2} d\\phi\n", "4117495ec386bc7981333615a776552e": "k_C=\\frac{-C_m}{\\sqrt{(C_1+C_m)(C_2+C_m)}}.", "41182f996eb1d3f1c90f3bbaee5c97fa": "(1 - \\epsilon) N", "411875962e2d73217ddfab4d16343ec9": "W^{1,p}(R^n)", "4119c66b14483a654ff8404a913acf9e": "E_\\gamma", "4119e1a3cfd14715a7110cff2c2f4be6": "\\beta_i(t)", "4119facab40a83137f315f85cf918bad": "\\gcd(n, s_1, \\ldots, s_k) = 1", "411a015017b9697be549fb51f71aabfd": "K_1\\sqcup K_2", "411a396841e50b6e05dbd1cc74f32037": "(\\hat{\\mathbf{S}^T}\\mathbf{P}^T\\hat{\\mathbf{M}})(\\hat{\\mathbf{M}^T}\\operatorname{diag}(\\mathbf{P}\\mathbf{1})\\hat{\\mathbf{M}})^{-1}", "411a468a1968e9e79e733ce6604cb960": " I(\\omega,T) = \\frac{\\hbar \\omega^3 }{4 \\pi^3 c^2}~\\frac{1}{e^{\\frac{\\hbar \\omega}{k_\\text{B} T}}-1} ", "411a6481d5a81df089fd6bb5c8c221e9": "S \\times [0,1]", "411a80cff0a5988aaae184e3c7007d63": " \\underline u_i = \\min \\{u_i : K(p)u = Q(p), p\\in {\\mathbf p}\\} ", "411a92c2646d9233f37e8a54f19578f7": "\\nu(\\gamma\\delta,z)=\\nu(\\gamma,\\delta z)\\nu(\\delta,z)", "411aa3292f3eb4184ce0e5d3bf1e7ae2": "{}^{5}i = i^{\\left({}^{4}i\\right)}", "411b3ff08a437e3f81b4944272ece3de": "k=\\pm 1.", "411b602b469d266166271ae8c8e677f4": "\\mathbf{EMF} = \\oint_{path} \\mathbf{E} \\cdot {\\rm d}l = -\\oint_{path} \\frac { \\partial \\mathbf{A} } { \\partial t } \\cdot {\\rm d}l = -\\frac { \\partial } { \\partial t } \\oint_{path} \\mathbf{A}\\cdot {\\rm d}l = -\\frac { \\partial } { \\partial t } \\int_{surface} \\mathbf{B}\\cdot {\\rm d}s ", "411b7adeeabaf231289a068e182eea38": "C' \\to L = \\frac{1}{\\omega_c \\,\\omega_c'\\,C'}", "411b89e8eea5aa33c3cfc74cdc2bf91a": "i_J", "411b8cbde39c64f4a85d4d478bcd7745": "F_D", "411bf02d226779cfe1c6420f236e9bf8": "\\Psi(x) = \\Lambda(x) \\, \\Gamma(x)", "411c5541fda493fc9c6dadc9c2c74773": "\\cos \\theta_s = 0 \\,", "411c64cc5444a06ad662f08bc73e0f00": "\n\\left(\n 1 - \\sum_{i=1}^p \\phi_i L^i\n\\right)\n\\left(\n 1-L\n\\right)^d\nX_t\n=\n\\delta + \\left(\n 1 + \\sum_{i=1}^q \\theta_i L^i\n\\right) \\varepsilon_t \\,\n", "411c8ac943d9ddb364afd36e330d8387": " \\frac{\\partial V}{\\partial t} + \\frac{1}{2} S^2\\frac{\\partial^2 V}{\\partial S^2} + S\\frac{\\partial V}{\\partial S} - V = 0. ", "411d263c018f2116a76dd568bd025971": "\\ln(\\beta)+\\gamma+1\\!", "411dadf0f4a9413ac4e07f4c2e8e96f2": "N_{i} = {^{(4)}g_{0i}}\\,\\!", "411db98b8b0f199b34067a43e70f91c2": "\\langle L^{op}, \\ge \\rangle", "411dd49540ea0084c1d796a9be8738e8": "{m^+}_2 = [13.045, 0.622]", "411dde2c7cc146efdbfe0c4272cdbca4": "\\displaystyle{L(a^{m+1})=2L(a^m)L(a)+L(a^2)L(a^{m-1})-2L(a)^2L(a^{m-1}),}", "411e2157947200ae70a943eb31dd95c1": "\\partial_u\\,\\!", "411ef71929b61bf85a0aeb397e06c1b6": "(f+g)(x) = \\sum_i \\left (f_i+g_i \\right )x^i = \\sum_i{f_i x^i} + \\sum_i{g_i x^i} = f(x) + g(x)", "411f0b91f11e5ce6e2eafaff91c2d903": "1 \\equiv \\begin{pmatrix} 1 & 0 \\\\ 0 & 1\\end{pmatrix} \\qquad i \\equiv \\begin{pmatrix} i & 0 \\\\ 0 & i\\end{pmatrix} \\qquad \\varepsilon \\equiv \\begin{pmatrix} 0 & 1 \\\\ 1 & 0\\end{pmatrix} \\qquad i_0 \\equiv \\begin{pmatrix} 0 & i \\\\ i & 0\\end{pmatrix}.", "411f3ad591206c59dde8019db156fffa": "\\chi(\\mathcal{F})", "411fb77fb9f7152cf89d22ef8624ce76": "ordinal(date) = week(date) * 7 + weekday(date) - (weekday(year(date), 1, 4) + 3)", "411fba450672ab5aa4a183dae0b00e12": "\n\\left[ - \\frac{\\hbar^2}{2\\mu} \\left({1 \\over r^2}{\\partial \\over \\partial r}\\left(r^2 {\\partial R(r)\\over \\partial r}\\right) - {l(l+1)R(r)\\over r^2} \\right) + V(r)R(r) \\right]= E R(r),\n", "411fe5d936104765000404c6f9f5a959": " \\alpha c ", "411ff08efd6e00aecb07d20f0971cf5e": "\\mathrm{idx}_A \\lambda\\,", "41201e9afb79eb6557e84746828770d3": "\\left\\vert T_{\\delta}^{\\mathbf{p}^{n}}\\right\\vert \\leq2^{n\\left[ H\\left(\n\\mathbf{p}\\right) +\\delta\\right] }.\n", "41205f83d7ba3f4e29856e1b3ccdb77c": " \\mathbb{G}_n = \\sqrt{n} (\\mathbb{P}_n - P) \\rightsquigarrow \\mathbb{G}, \\quad ", "412061a9eccd8570cd3273ede6b41c25": "((A \\rightarrow B) \\rightarrow B) \\rightarrow ((B \\rightarrow A) \\rightarrow A)", "4120f94b599b1e2817c80004a3b4cddd": "\\theta_u(v) = g(u,\\pi_* v)\\,", "41217bef4fe4e59a445d8c650cc8e989": "I=(I_{ij})", "4121a5a7a258a913a794849bf3549bf6": "\\scriptstyle > 0", "4122143d3f932f2b8084355ca89dd51b": " ik ", "412230ee2f1eb48efc06e67107e39bc6": "\\mathbf{x}\\cdot\\mathbf{y} = \\sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\\cdots+x_ny_n", "41235074fc390e4f63e4e1932c1c56f4": "e^{\\mu T}", "4123798cc3df925f31387d06c62ebba4": "(S^{*}, {\\overline{S}}^{*})", "4123be7bdb4bf6270e3e713401294384": " d_1 = \\frac{\\ln(F/K) + (\\sigma^2/2)T}{\\sigma\\sqrt{T}} ", "4123eff2165b813faa6d51f8e77d0f59": "N^\\epsilon", "41241a95720cc70400aad6ea8a0e2ff3": "Q(x)\\leq \\tfrac{1}{2}e^{-\\frac{x^2}{2}}, \\qquad x>0", "41241af47d6987a8e27420fb26c39cea": "s_1 + s_2 + \\cdots + s_n = {n \\choose 2}.", "41242e4a25e1979ce6ec2ccb6bb1235d": "\\mathbf{E}^n", "41247590b1787770992c387859839f98": "\\varphi(x,0) = x;", "4124ac440d521d03844cb2f6318c5562": "\\mbox{Diff} \\to \\mbox{Top}", "4124bc0a9335c27f086f24ba207a4912": "aa", "4124c6308b09007bbfb3531b1fb3bfdf": "R=\\sqrt{(z-z_0)^2+\\rho^2+\\rho_0^2-2\\rho\\rho_0\\cos(\\varphi-\\varphi_0)}.\\,", "4124ce2be28b2746deafa4d7af975a99": "\\mu = \\frac{q}{m^*}\\overline{\\tau}", "4124da4e04ba0eab3195240844879e91": " { 1 \\over s } ", "412559aaba4ab67f09a25adf7aef3a83": "ds^{2} = -\\left(1-\\frac{2GM}{r} \\right) dt^2 + \\left(1-\\frac{2GM}{r}\\right)^{-1}dr^2+ r^2 d\\Omega^2 ", "412566367c67448b599d1b7666f8ccfc": "mn", "41257ae775fad4a04e5ffd2267230249": "\\frac{32450625}{59056400}", "4125a310a4f101cbb590074dff527994": "\\left[ J_z , \\widehat{T}_{k}^{q} \\right] = \\hbar q \\widehat{T}_{k}^{q} ", "4125bd2fbbad206a8399b09375f2b4a1": "\\int_{\\mathbb{Q}_{\\infty}} \\chi_{\\infty} (a{x_{\\infty}}^2 + bx_{\\infty})dx_{\\infty} \\prod_p \\int_{\\mathbb{Q}_{p}} \\chi_{p} (a{x_p}^2 + bx_p)dx_{p} = 1", "4125c0dec3984a24ebba29ed259ebd0f": " ^{14}\\text{NO}_3^- + ^{15}\\text{NO}_3^- \\rightarrow {^{14}}\\text{N}^{15}\\text{NO}, ", "412605349c4b4139f11b63df04cd9305": "\\pi = 4 \\sum_{k = 0}^{\\infty} \\frac{1}{(16^k)(8k+1)} - 2 \\sum_{k = 0}^{\\infty} \\frac{1}{(16^k)(8k+4)} - \\sum_{k = 0}^{\\infty} \\frac{1}{(16^k)(8k+5)}\n- \\sum_{k = 0}^{\\infty} \\frac{1}{(16^k)(8k+6)}. \\!", "4126b86258a2fb01b9770ef38afc15fb": "\\mathrm{2\\ SO_4^{2-}\\ \\xrightarrow \\ \\ S_2O_8^{2-}\\ +\\ 2\\ e^- \\ \\ \\ \\ E^0 = 2.01 V}", "4126c9c05e762267788f865a718fcd53": "(\\Delta E \\sim 1/r^4)", "4127638260b0f5433a2427f8a6ad2ea8": "w(n) = 1 + \\sum_{l = 1}^P a_l \\cos \\left ( \\frac{l 2 \\pi n}{N-1} \\right)", "4127d7baea6943b2640e83a88675e950": "\\sigma_{0,\\,0.5}", "4127fdb3bea43e93496a2ef2dbd78a36": "H(u)(t)", "41288c3e60b12758f0422e0048b1d84c": "(T \\times S) (x,y) = (T(x), S(y))", "41289014a7c8cafcc9ee703fbbb18c10": "\\Uparrow, \\Downarrow, \\Updownarrow \\!", "4128b5ab8707b629bdbe0aa451a9eb85": "\\mathbf{x}=x^i\\mathbf{e}_i\\,,\\quad \\mathbf{x}=x_i\\mathbf{e}^i", "41290cfb82e8c865a2b7c807cdb2bd26": "g\\in\\widehat{G}", "4129127bdd6463eef92fa70283117107": "{\\mathfrak k}_n", "41292116f68219788801ae4226442228": "\\operatorname{deg}(pq) = \\operatorname{deg}(p) + \\operatorname{deg}(q).", "41293e5897e5abf110ade9bd8086e37c": "f(T_1,T_3) = \\frac{g(T_3)}{g(T_1)} = \\frac{q_3}{q_1}.", "4129a6d9fc747767d1f7c1a3f8601ea1": "\n\\frac{d}{d\\omega}Q(\\omega)<0,\n", "412a025714214529d1bab208e0b066c1": "f : \\mathbb{N} \\to \\mathbb{N}", "412a1b6a7870925f18067b49140e8fda": "A \\mapsto A-B--B+A++AA+B-", "412a3f47eb5ec15c5267e725eaec990d": "p_{k}\\sim k^{-\\tau}", "412a75e7ae9fa56fbd8c17657d610644": "{\\mathcal A}_p", "412a891c481d5edd6324d0cbe35b7408": "\\Delta S_\\mathrm{mix}=-R\\sum _i x_i\\ln x_i", "412b68119c1473ca514926154d8c1570": "\\mathfrak{so}_3(\\mathbf K)", "412bd1def2f1b61da9169d88e6c8b9e9": "X,", "412bd6fda8082d41029ede94a123d269": "\\beta_{t,k}", "412c3b314d0f1db5f3baef2c830f2806": "\\scriptstyle{Rt=f(Rc,X)}", "412c5bc5e99db5c9de1c300416970c65": "C(u)=\\sum_{i=1}^k R_{i,n}(u)P_i", "412c8421dec2ee49f1e0f533843f2854": " AX_2=\\lambda X_2", "412cb938ba5da24cc0ec08a0b6d7f123": " T = 2 \\sqrt{D \\over A}", "412cd4beb62b7d84ca79847ac123ccf9": "z_{n+1} = f_c(z_n) \\,", "412cd676de15a706a78b56c6522aca0e": "\\left|\\mathbf{1}_F-\\mathbf{1}_G\\right|=0", "412d004c62c25be7080db2154b0e7101": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) = \\sum_{h=1}^n \\frac{\\prod_{j=1}^n \\Gamma(a_h - a_j)^* \\prod_{j=1}^m \\Gamma(1-a_h + b_j) \\; z^{a_h-1}} {\\prod_{j=n+1}^p \\Gamma(1-a_h + a_j) \\prod_{j=m+1}^q \\Gamma(a_h - b_j)} \\times\n", "412d494c1f332c4619ad60c9c404b160": "\n\\left( -\\frac{1}{2} \\nabla^2 + V(\\textbf{x}) \\right) \\psi_{+} = E_{+} \\psi_{+}\n", "412d6e2ea252cd5d31a2e45cc8a7d89c": "\n\\hat{q}_1=\\frac{2}{N}\\sum\\limits_{n=1}^{N/2}m_A(x_n), \\quad\n\\hat{q}_2=\\frac{2}{N}\\sum\\limits_{n=1+N/2}^{N}m_B(x_n)\n", "412de614dc93905c2e4351ce506f4e51": "b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \\dots {x_2}^{(1)}x_1 \\le n", "412de9434d405d4269ec652d076306d2": " [t_l, t_u] \\subset \\mathbb{T} ", "412e2835938f913c46626f8d38a42dfb": "P_1=1-R/d", "412e2f5e11a465186e069d31a35e6df2": "\\Psi(w,z) = \\bar w_1 z_1 + \\cdots + \\bar w_p z_p - \\bar w_{p+1}z_{p+1} - \\cdots - \\bar w_n z_n.", "412e5ee741e850311d2992883085aaff": "x\\in [0,1],", "412e6f7f571c8855fcec32f7a0a8bbd7": "H = \\begin{bmatrix}\nT1 & T2 & T3 \\\\\nR(A) & & \\\\\n & W(A) & \\\\\n & Com. & \\\\\nW(A) & & \\\\\nCom. & & \\\\\n & & W(A) \\\\\n & & Com. \\\\\n & & \\end{bmatrix}", "412e8433a8d74b7bba840111b487e100": "\\operatorname{im}\\, \\kappa \\oplus \\operatorname{im}\\, \\sigma", "412ea3c4a1576614219f308aaf35ec96": "\n \\mathrm{Area} = \\pi \\times (5m)^2 \\approx 78{.}54m^2", "412eb4bc0cc4b9a8bf06a1892a6cd7ed": "(\\neg Q \\to \\neg P)", "412ebfba305ef4771e67c89a42121515": "\\displaystyle{\\beta(R(a,b),R(c,d))=(R(a,b)c,d)=(R(c,d)a,b),}", "412ec04f4a8189f2ef589681618fb100": "\\frac{11}{8}", "412f1b0e0588a4d9aa8722b30bc2d09b": "\\color{blue}\\mathcal{S} \\color{blue}\\rightarrow \\color{blue}\\mathcal{I} \\color{blue}\\rightarrow \\color{blue}\\mathcal{S}", "412f25a29d3ecec67574f27b9b4ce8b0": "\\begin{matrix} {48 \\choose 3} = 17,296 \\end{matrix}", "412f44c2e98d0ef2f4d2fad7f25aa5e8": " r_2 - r_1 = 2A \\quad (3)", "412f5062e169369891aa81bb2ac683e5": "\\hat{H}(t)=\\hat{T}+\\hat{V}_{\\mathrm{ext}}(t)+\\hat{W},", "412f62db388eaf49a7d23832bc1fb203": "\\lnot\\lnot (x \\vee y) = \\lnot\\lnot x \\vee \\lnot\\lnot y \\mbox{ for all } x, y \\in H,", "412f9581aeba755a6a2d1045535d1a44": "\\textstyle 4.\\ since", "412fa675a3c21e5a843782cc4f234dc7": "v_{i+1}(t) = \\{\\ldots\\}_{\\text{eq}}", "412faddedd77f5141b57d00d43032019": "\\{a_{k,n} : k = 0, 1, \\ldots, m_n\\}", "41304f96a4bedcb2a747393cf4954dff": "\\log\\left(\\frac{R_1}{R_2}\\right)=\\log\\left(b\\right)+s\\cdot\\log\\left(\\frac{Rf_1}{Rf_2}\\right)", "4130701a0e8abffb2b1ebcc37a16214d": "T_1^{(4)}k=T_2^{(4)}k=X_1^{(4)}k=0.", "413081316d2db0667c7639f5f368f8b6": "-2 < \\alpha < -1", "4130c89f2d12c3ac81aba3adbff28685": "p(x)", "4130fa996cf48f2375c0865227def129": "B_{j_{k}}", "413100d7be973577e6793c13e2b5e832": "R^*=C \\times \\rho^*", "41311c5a2b9b79c18f0cef8ea0a16096": "\\tau(g \\circ f) = g_* \\tau(f) + \\tau(g)", "41316a507fdead5b4dab62cefe84a4d4": "\\scriptstyle \\leq2\\times10^{-43}", "41316b35bc226f3ddeca1ba7aed80b9a": "(1, 0)", "41316dadc1f68d705b0961d65ff0262a": "F(a x)=a F(x) \\,", "413187e65da512f5eac53c383ef1841c": "\\operatorname dt", "4131924946cde58ddb09d36c8a24f82e": "\nposterior (male) = \\frac{P(male) \\, p(height | male) \\, p(weight | male) \\, p(foot size | male)}{evidence}\n", "4131a2798c64ce6b31e6b3034f5d7f60": "\\exp_p\\ ", "41326549b2f762be433a349b6fca1ac1": "\\rho=\\rho_c\\theta^n", "4132888434bffdee30f326f038a467cb": "t \\in [0,\\infty)", "4132a30c54147b83f8cd04a293041e44": "\\omega_{\\mu\\nu}", "4133261164edb2e8ea58976de8c48684": "\\lambda=\\mu_1-\\frac{a-C_1} {1-2 b_2}\\!", "41333861ce6e13da218764a9462340bd": "\\lnot\\phi(\\boldsymbol{x}) \\in p(\\boldsymbol{x})", "41334c981095838c1540a9d00dea17f9": "P_{airgap}=\\frac{R_r}{s} * I_r^{2}", "4133594683e5893e36c8d06c6cef6f0c": "\\sigma_{long}", "41336548e5942f8e32f1b04f5788e2c3": " \\|f\\|^2= {1\\over \\pi} \\iint_D |f^\\prime(z)|^2 \\, dx dy = {1\\over 4\\pi}\\iint_D |\\partial_x f|^2 + |\\partial_y f|^2 \\, dx dy,", "413376234d2135321dbda74d6a689778": "\n\\bold A_x= \\left[ \n\\begin{array}{c c c c c}\n0 & 1 & 0 & 0 & 0 \\\\\n\\hat{\\gamma}H-u^2-a^2 & (3-\\gamma)u & -\\hat{\\gamma}v & -\\hat{\\gamma}w & \\hat{\\gamma} \\\\\n-uv & v & u & 0 & 0 \\\\\n-uw & w & 0 & u & 0 \\\\\nu[(\\gamma-2)H-a^2] & H-\\hat{\\gamma}u^2 & -\\hat{\\gamma}uv & -\\hat{\\gamma}uw & \\gamma u\n\\end{array}\n\\right].\n", "4133f929308e88056d4a7cb256839ae0": "dS = \\frac{1}{T}\\sum_{j}E_{j}dP_{j}=\\frac{1}{T}\\sum_{j}d\\left(E_{j}P_{j}\\right) - \\frac{1}{T}\\sum_{j}P_{j}dE_{j}= \\frac{dE + \\delta W}{T}=\\frac{\\delta Q}{T}", "41341e8cf6cd03733363db5736800814": "T =\\,", "41341f75998a13239ebfd3142ccaba60": "u_{i}^{n+1}", "4134416801c11f5749fdf90e7c48430f": "-\\nabla \\times \\mathbf{E} = \\frac{1}{c}\\frac{\\partial \\mathbf{B}} {\\partial t}", "4134b9997a996c092dee695c991acb79": "\\alpha,\\beta, \\gamma,", "4134de78c502ad72f47bd1145eb9806f": "\\mathbf s", "4134f3382133bcf0ea4683a9b3d3d10e": "c=kd/2", "41354992c111fa5fd8fcc4e53c972dc0": "\\frac{d p}{d s}=\\rho_o g", "41357e29299a87a083763ebd115fb916": " [z^{2n}][u] g(z, u),\\,", "4135dfd38576ebc8ce1905b5b4abffab": "(x,y,u,p,q)", "4136391b15cbae2b7ab4b9add480da10": "da_{i,j} = \\bar a_{i,i}a_{i+1,j}+\\bar a_{i,i+1}a_{i+1,j}+\\cdots+\\bar a_{i,j-1}a_{j,j}", "4136601c1c013de4492727dfa848c204": "\\mu (A \\cap T^{-n}B) - \\mu(A)\\mu(B)", "4136669d8137487ad4dad2e9f712d386": "A\\left(t\\right)", "41372a82e2f6dba7b9a35b6abe01f9ae": "z! = \\Pi(z)\\,", "4137358e081ce8019fdcd85c3a73f8f3": "\\operatorname{Im}\\,2\\bar{A}B", "41379fa57086b93c605f5d678bcba915": "F = \\rho u", "4137a73431ab3e213414641451148af4": "R(z) =z^2 +c", "4137f1b85c8abff14c29533cd1a7ad95": "d_{\\textit{f}}", "41383ce756f62084d6a7d11cf25b1e6f": "\\aleph_{\\alpha + 1}", "4138d072f203bd3f92bb1a26bb81d61b": " x = 0", "41391c8e94b7a3aeb924cb267a869994": " \\vec a = \\frac {d}{dt} \\vec v = \\frac {d}{dt} \\left(R\\ \\omega \\ \\hat u_\\theta \\ \\right) \\ . ", "4139cf0dfec11080b8968aa6d583720b": "K=I\\left\\{ \\forall d \\right\\}\\otimes T\\left\\{ :\\mathop{D}_{o}^{i}\\circ \\mathop{S}_{w}^{p}\\circ \\mathop{R}_{e}^{c}\\circ \\mathop{P}_{v}^{\\rho }: \\right\\}", "4139e78c6ea1113780d78574a9f35468": "\\ell(p)-KP(x)", "4139f9382d35b5ac0fd465fbb3f7ab27": "\\operatorname{cov}(w_i, z_i) + \\operatorname{E}(w_i\\,\\Delta z_i) = wz' - wz = w\\,\\Delta z\\,", "413a36c4cd8c26ed122569ee065a6491": "x \\in \\mathbb{F}_2^n", "413a7c0d3671a3a414287065dcc249c2": "\\psi_{\\bold{k}}(\\bold{r}) = u_{\\bold{k}}(\\bold{r}) e^{i\\bold{k}\\cdot\\bold{r}}", "413a9216f1d2c33eadc572f18c16d4e7": "p(n)=\\sum_k (-1)^{k-1}p(n-g_k)", "413acfaf4e7699495eeea2f118b93b53": "C = \\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|c_n(z-a)^n|} = \\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|c_n|}|z-a|", "413b1e62d098e7c18befcffc2dc438d0": " \\langle N, \\in_M, \\ldots \\rangle", "413b3ee66bbe1c90ab1aae3112da4b5e": "1^{-1} ", "413b75c3a702908183694a1170494283": "|\\psi_n\\rang", "413b8d7707a6ed20dfa90e68ff98bad2": "\\theta_m=\\frac{\\pi}{2}\\,\\frac{2m-1}{n}.", "413b949a981e3504ddeebc797c6a0e7a": " N=2^{20}, 2^{22},\n\\ldots, 2^{44}", "413beb263399c132fcb2dceea50e48c4": "X_n = ", "413c329617ab7ba7042490fbc67335a5": "\\left\\{{n+1\\atop k+1}\\right\\} = \\sum_{j=k}^n {n \\choose j} \\left\\{{ j \\atop k }\\right\\}, ", "413c411a32f157cfaf7b0ea9610addd6": "\\operatorname{E}(T) = \\left. \\frac{\\mathrm{d}}{\\mathrm{d}z} G(z) \\right|_{z=1}.", "413c54a5b5ceac70233b7903953ff386": "f_1(z)= \\frac {(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}", "413ce1ce7e514df4a17c78c247aada4c": "x_{n+1} = \\frac{\\alpha+\\sum_{i=0}^k \\beta_ix_{n-i}}{A+\\sum_{i=0}^k B_ix_{n-i}},", "413d2dc174d1f60336025890b62d384e": " u^2_\\rho /c^2 ", "413d53680e59c3ae5ed2874970ed3742": "(\\boldsymbol{x}_i,y_i)", "413d5b5e97df8bde2625d324c075e534": "P(v_1,\\ldots,v_n)", "413de94e7c4aaa04a85c12a9e95da825": " \\operatorname{E}_e\\left[f_a(o,s)\\right] = \\operatorname{E}_p\\left[f_a(o,s)\\right] \\quad \\text{ for all } a .", "413e45ac53271aeaea759cec348c8810": "P_{r} = \\mathrm{A}(\\theta,\\Phi) W\\,", "413eb375380f5ac0a7a49ac19dd0cd39": " h = z \\, ", "413ed835e9ddc147503786b35e4b1b3e": "\\mathbf{e}_3.", "413f1f1ffd124a1666ddfbc3e51b8b76": " S = 1 - \\log_e( 2 ) \\approx 0.31 ", "413f2282b22ab6a1c6a055d91aead87e": "\n\\bigoplus_{i=1}^{n} \\bold{A}_{i} = {\\rm diag}( \\bold{A}_1, \\bold{A}_2, \\bold{A}_3 \\cdots \\bold{A}_n)=\n\\begin{bmatrix}\n \\bold{A}_1 & \\boldsymbol{0} & \\cdots & \\boldsymbol{0} \\\\\n \\boldsymbol{0} & \\bold{A}_2 & \\cdots & \\boldsymbol{0} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\boldsymbol{0} & \\boldsymbol{0} & \\cdots & \\bold{A}_n \\\\\n\\end{bmatrix}\\,\\!", "413f273626102fd92747e1294c5776f1": "V_{\\text{out}} = -RC \\,\\frac{\\operatorname{d}V_{\\text{in}} }{ \\operatorname{d}t} \\, \\qquad \\text{where } V_{\\text{in}}\\text{ and } V_{\\text{out}} \\text{ are functions of time.}", "413f6779616525db74f685ada56a5f21": "\\frac{1}{c^2} + \\frac{1}{h^2} = \\frac{1}{k^2}.", "413f9f196e29a1e29457d2c2c8b717fe": "\n\\overline{\\bigcap\\limits^{\\{q\\}}X_i}=\\overline{\\overset{\\{m-q-1\\} }{\\bigcup }X_i}=\\bigcap^{\\{m-q-1\\}}\\overline{X_i}\n", "413fd04ad288892095ded30141159554": "3^{(3^3)}=3^{27}=7625597484987", "413fd3bdbec37e2ac98931864dab16a7": "\\mathbf{F}\\cdot\\mathbf{n} \\, dS,", "413fe853e30bc875b042cbdc5984e253": "\\delta(z-w) \\equiv \\sum_{n \\in \\mathbf{Z}} z^{-n-1}w^n", "413ffa3b01501ed5d3352c082a50eca5": "\\ VOP = \\frac{lgPTS}{lgFGA - lgORB + lgTO + 0.44 \\times lgFTA} ", "41402205757a625a333ea3fd3961cdd5": "\\gamma_n(1)=\\gamma_n.\\;", "414027eab45cde436eba1d458bd67f35": "\\psi_0(x) = \\pi^{-1/4} \\, \\mathrm{e}^{-\\frac{1}{2} x^2}", "41403c2ac0ee13b2ef94676e6881ba6f": "\n\\sqrt[12]2 = 1+\\cfrac{1} {12+\\cfrac{11} {2+\\cfrac{13} {36+\\cfrac{23} {2+\\cfrac{25} {60+\\cfrac{35} {2+\\cfrac{37} {84+\\cfrac{47} {2+\\ddots}}}}}}}} = 1+\\cfrac{2 \\cdot 1} {36-1 - \\cfrac{11 \\cdot 13} {108-\\cfrac{23 \\cdot 25} {180-\\cfrac{35 \\cdot 37} {252-\\cfrac{47 \\cdot 49} {324-\\ddots}}}}}.\n", "414072aa5e706ccd17c56c89352623c0": "\\ell = 2", "41410216a1ee0bf2faa41b5a1cd3a065": "\\frac{d}{dt}\\mathbf{p}(t) = q\\left(\\mathbf{E}+\\frac{\\mathbf{p}\\times\\mathbf{B}}{m}\\right) - \\frac{\\mathbf{p}(t)}{\\tau},", "41416dae685fc5a2cc5d2f12e2b1dfd3": "HY=\\Gamma(Y\\times_X TX ) \\subset TY", "4141a12e9d4ad6ccfba89a5d39b7df74": "\\Delta\\Theta", "4141debafb5104c75b764a7b26913ef3": "|\\hat{R}_n - R|_{\\mathcal{F}}", "41420b627afced3eae7e46c19a25a968": "\\Lambda(\\cdot)", "414271632a88fc8f75738c504a06fbe7": " \\mathrm{Input} = \\mathrm{Output} + \\mathrm{Accumulation} \\, ", "414271e080f7d8ba0326a0ae2ee2fd3b": "dF_X(x)=f_X(x)\\,dx", "41429d0e73384c209d1736958411e1dd": "\\delta (i)", "414322ad6f6a6e41cd97580ec949f679": "O(d^2).", "41439ebc8dca092c549a4ea362c7c70f": "[M_x(OH)_y] = \\beta_{x,-y}* [M]^x [H]^{-y} ", "4144a92eb23e722b46ad0d2ee010805f": "E_{ij}^\\text{dual} = \\sum_{a=1}^n x_{ai}\\frac{\\partial}{\\partial x_{aj}}.", "4144b22ed968bda9e78b294174c4ac7d": " f(1) = \\sum_{k\\ge 0} c_k^2 \\int_{-\\infty}^\\infty {\\rm Tr}(\\pi_{\\nu,k}(f)) (\\nu^2+k^2)\\, d\\nu.", "4144e097d2fa7a491cec2a7a4322f2bc": "AC", "4144ecf991a07fbfabecaf4960f14a71": "\\delta(0)", "4144f3905db2e471abe6ae1fdc1f6cda": "M\\cong R^r\\oplus R/(a_1)\\oplus R/(a_2)\\oplus\\cdots\\oplus R/(a_m)", "4144fe17ac210df70ec8b1a846f6f9d1": "R^\\rho{}_{\\sigma\\mu\\nu} = \\partial_\\mu\\Gamma^\\rho{}_{\\nu\\sigma}\n - \\partial_\\nu\\Gamma^\\rho{}_{\\mu\\sigma}\n + \\Gamma^\\rho{}_{\\mu\\lambda}\\Gamma^\\lambda{}_{\\nu\\sigma}\n - \\Gamma^\\rho{}_{\\nu\\lambda}\\Gamma^\\lambda{}_{\\mu\\sigma}", "41455cfe60bc9af91423dc2a98fe33cd": "F_1 = \\frac{M_0^\\mathrm{act} M_1^\\mathrm{pass}}{r^2}", "414586534af85ad6c9c9dcf7fb64a5b3": "(\\sqrt[3]{8})^3 = 8", "4145fc9e52fcd776e46d8df50a27495a": "O(e^n) = n^{O(1)}\\, ", "41462d9fcba6299fc65b1582c24768e5": " \\epsilon = \\frac {1}{\\left\\vert B \\right\\vert}", "4146597ed144486f8ac4f05ca1533bea": "p(x,y) = \\sum_{i=0}^3 \\sum_{j=0}^3 a_{ij} x^i y^j.", "414681fb294d75f46472eb907ad59b38": "\n\\langle \\theta \\rangle=\\mathrm{Arg}\\langle z \\rangle = \\mu\n", "4146c0fb38f25250cba24dfe5230e8d6": "\\operatorname{perm}\\begin{pmatrix}a&b&c \\\\ d&e&f \\\\ g&h&i \\end{pmatrix}=aei + bfg + cdh + ceg + bdi + afh.", "4146d19c953c31330b0260f5f87cf3ca": " \\nabla \\cdot \\mathbf{v}=0", "4146d7f98422c33f653c19e74a2e1629": "(x^2 + y^2)^4 - 45(x^2 + y^2)^3 - 41283(x^2 + y^2)^2 + 7950960(x^2 + y^2) + 16(x^2 - 3y^2)^3", "4146e193559cae477fc2fb6cbe207f42": "u \\in \\mathbb{R}^I", "4147288fdeeff06f7df3f90f9768a1bb": "C_0 = 1 \\quad \\mbox{and} \\quad C_{n+1}=\\frac{2(2n+1)}{n+2}C_n,", "4147387b8bb4ed297063fda1b1555b4c": "\\rho = \\text{constant} ", "41477f0e5b5180f70e842beb316e69e5": "\\mathbf{D} ", "41477f2af495da49ce2726312951b045": " V_d = \\frac{n_d-1}{ n_F - n_C }", "4148000f94fd769766365445b3df2fda": "\\mathbf{B}({\\mathbf{r}})=\\nabla\\times{\\mathbf{A}}=\\frac{\\mu_{0}}{4\\pi}\\left(\\frac{3\\mathbf{r}(\\mathbf{m}\\cdot\\mathbf{r})}{\\left | \\mathbf{r} \\right |^{5}}-\\frac{{\\mathbf{m}}}{\\left | \\mathbf{r} \\right |^{3}}\\right)", "41480431453103f4552b3fd85577c159": "\\rho(g_1 g_2) = \\rho(g_1) \\rho(g_2) , \\qquad \\text{for all }g_1,g_2 \\in G.", "4148135b1845cc2ad5363b8b3c8bc31d": " P2 ", "414838f42f72cbb1567bd21dcfbf04d9": "\n b_n(\\hat\\theta_n - a_n)\\ \\xrightarrow{d}\\ G ,\n ", "41483b7658e178c4a8f94014866b22c1": "E_6 \\supset SU(6)", "41489783b563657b1401d099185ede6a": " i \\in C ", "41489b561d485cc769bcd422933bbb87": "\n\\theta\\frac{J!}{n!(J-n)!}\n\\frac{\\Gamma(\\gamma)}{\\Gamma(J+\\gamma)}\n\\int_{y=0}^\\gamma\n\\frac{\\Gamma(n+y)}{\\Gamma(1+y)}\n\\frac{\\Gamma(J-n+\\gamma-y)}{\\Gamma(\\gamma-y)}\n\\exp(-y\\theta/\\gamma)\\,dy\n", "4148d902da3f64c40bbe5d41870187ce": "U_{\\varepsilon}(a)", "4148ed54fb1dc0c45458799fe8082b63": "\\textstyle K_{\\lambda g} = \\frac{1}{\\lambda} K_g", "41491776834800a4425c29f5ee80a11e": "{}_{-2}Y_{lm}", "4149225acb8d639faaa1e9cd48de4dcd": "Y[x,y]=\\frac{(x'^2-y'^2)x'+2xyy'}{xy'-yx'}", "4149a2eda65f94335696bdb3b68f83e1": "\\{a, b\\} = ab + ba .", "414a1b998e8448b74a3651c045bb1dc7": "\\beta(1+r)=1", "414a30be534f2b4da11808411a62888c": "(y-f(x)) ", "414ac0c3ee3363235fc652dc4d9eb2a9": "\\vec{D}", "414ae87925e1c428431791d57cb3e9e5": "\\sqrt{-1} = \\frac{\\ln(-1)}{\\pi} \\qquad\\qquad \\mathrm{e}^{i\\,\\pi} = -1", "414bbade85ca3b47930b6ab4ecfe8da7": " v \\ ", "414bf2d391acd18c85e4ae9840bcd1a9": "V \\subset \\mathbb{C}^n", "414bf9234493e8e08ecaf4ca54ea4b46": "C_{2k-1}", "414c7c6f0517f4c7f2fddf3cf653a19e": "z_i'", "414c820e8857a103383745504f3f48e1": "\\scriptstyle P_0", "414cc58d0a752504e2f4e47600fdb23a": "V(\\mathbf{x}) = -\\int_{\\mathbf{R}^3} \\frac{G}{|\\mathbf{x} - \\mathbf{y}|}\\,dm(\\mathbf{y}).", "414cf83b396e3a4f246ff637a0c500e1": "p_1 V_1^{\\gamma} = p_2 V_2^{\\gamma}\\,\\!", "414cfe30d08c211b8331f0598165f427": "y_p = \\frac {1} {4} t \\cos{t} + \\frac {1} {4} t^2 \\sin{t}. ", "414d190418d7c38a436c7183850a1770": "(x_1, y_1), \\ldots, (x_n, y_n)", "414d1acc809588064065ae04d1a48aac": "\\mathcal{M}_{+} (X)", "414decd168eeea72030482b286edf7a3": "A=\n\\begin{pmatrix}\na_{00} & a_{01} & a_{02} & a_{03} \\\\\na_{10} & a_{11} & a_{12} & a_{13} \\\\\na_{20} & a_{21} & a_{22} & a_{23} \\\\\na_{30} & a_{31} & a_{32} & a_{33} \\\\\n\\end{pmatrix}\n", "414e1843242a21ecc7fc8176f212988b": "\\frac{ \\partial Y}{ \\partial t} = \\frac{ \\partial Y}{ \\partial K} \\frac{ \\partial K}{ \\partial t} + \\frac{ \\partial Y}{ \\partial L} \\frac{ \\partial L}{ \\partial t} + \\frac{ \\partial Y}{ \\partial A} \\frac{ \\partial A}{ \\partial t} ", "414e248647f649a0a18e0d05e1c11ca1": "u'_i = 1-u_i", "414e56660e8dd9c038eae856d6cb80e0": "d_{ijk}", "414e70f8a66c2091e3813d744494f17f": "\\begin{matrix}{4 \\choose 2}{4 \\choose 1}{52 - 4r \\choose 1}\\end{matrix}", "414e8a932b21ca3f06ec8d88736a80fb": "x^{\\frac{1}{2}} \\!\\ ", "414eaf14e4aff98c2782c0fbfe15d341": "\n \\sqrt{J_2} = \\begin{cases}\n \\cfrac{1}{\\sqrt{3}}~\\sigma_t - 0.03\\sqrt{3}\\cfrac{\\rho}{\\rho_m~\\sigma_t}~I_1^2 \\\\\n -\\cfrac{1}{\\sqrt{3}}~\\sigma_c + 0.03\\sqrt{3}\\cfrac{\\rho}{\\rho_m~\\sigma_c}~I_1^2 \n \\end{cases}\n ", "414f2092b08b6facd46816e8a37d872e": "S(\\rho||\\sigma) - S(\\mathcal{E}(\\rho)||\\mathcal{E}(\\sigma)) \\geq 0", "414f3a8b4a18b59c5b84b62328231885": "\\frac{1}{c_{eq}} = \\frac{1}{c_1} + \\frac{1}{c_2} ", "414f7cb65d66348790f00452b30419e5": "1 \\leq K(\\sigma) \\leq 4", "414f7f80bb94f0ae08d299ab104c9a53": "a_n =\n\\begin{cases} \na_{(m-1)/4} & \\text{if } m = 1 \\mod 4 \\\\\na_{(m-1)/2} + 1 & \\text{if } m = 3 \\mod 4\n\\end{cases}", "414fa9001f34fc02cc2f4bda1577d95c": "message'", "414fbf484525f27cc88f8a5d4c00dc7c": "= \\gamma (t)\\cdot e^{j \\omega_0 t},", "414fd3fe1a2a4f0ba93e59e8a6ae5127": "\\mathsf{(CH_2CH_2)O+HCN}\\rightarrow\\mathsf{HOCH_2CH_2CN\\ \\xrightarrow[-H_2O]\\ CH_2\\!\\!=\\!\\!CH\\!\\!-\\!\\!CN }", "414fe2e8738b4fa747528ba8c3cbeb41": " \\langle \\psi_m | \\psi_n \\rangle = 0 ", "415023ae67de88dd35c5649116de3578": "\\begin{align}\n v_{n+1} &= v_n + g(t_n, x_n) \\, \\Delta t\\\\[0.3em]\n x_{n+1} &= x_n + f(t_n, v_{n+1}) \\, \\Delta t\n\\end{align}", "415051d72298c06f40e7239372dafd7e": " x_{ij} \\in \\{0,1\\},", "4151146d3122daf5b58e7c0673fd9bf9": "uLv = u \\left[\\frac {d}{dx} \\left( p(x) \\frac {dv}{dx} \\right) + q(x) v \\right], ", "415150424b59bc9120b54263f56ad6e6": " 1 = \\int_0^{a_M}{\\varphi(a) \\exp\\left(- \\int_0^a{\\mu(q)dq} \\right)da } ", "415164a75b4e25f690123c27cc7a3657": " \\ell^+ \\ell^+", "4151d98f8e119ac8659e52861ad628ff": " t_{k+1}", "4152460ff7a5c588fb9712b29a5f4b7c": "RPM = {Cutting Speed\\times 12 \\over \\pi \\times Diameter}", "4152752e786128c6b7e60548c4322e85": "H(\\mathbf{x},\\mathbf{v}) = \\frac{1}{2} v^2 + \\Phi", "415285b85e7351c5a00588f6b24db53e": "(x, x)", "415287ff0ad4ae32774de09424db0583": "\\mu'", "415290769594460e2e485922904f345d": "y", "4152a3d1dca8dd55b612835203ec7429": "U_t = \\int_{h_{b}}^{h_{t}} U(h) \\cdot dh = \\int_{h_{b}}^{h_{t}} \\left[ B(h) + W(h) \\right] \\cdot dh", "4152f28c5a6e9fd33201a5a319fde586": "|n_{\\nu_1}, n_{\\nu_2}, n_{\\nu_3},\\dots \\rang", "4152f5266bf1aa67f7db49f7a94f12f4": " \\mathcal{C} = \\bigcup_{\\mathit{w}=d_{min}}^{\\mathit{w_{max}}} \\mathcal{C}_w", "41531bae51c40ed90aeaf7a2b9b220d9": "\n P_{\\theta_1}=P_{\\theta_2} \\quad\\Rightarrow\\quad \\theta_1=\\theta_2 \\quad\\ \\text{for all } \\theta_1,\\theta_2\\in\\Theta.\n ", "41531f79baf77ef35e01b4b455bc4d3c": "\\begin{matrix} {2 \\choose 1}{2 \\choose 2}{3 \\choose 1}^2{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "415368b3d5f487b6bc3865f1c9906643": "V_\\pm", "4153a41ad367075723b5f4db4d1d5048": "H(X_1) \\ge 0", "4153a493376852c3b21d5c6644ac308e": "\\sum_{i=1}^p n_i \\chi_{V_i}(g)=|G|\\delta_{ge}", "4153e89d6342933c19be444d5f88d2f6": "\\mathbf{\\hat{j}}", "4153f62574549726757ca0cd79300cb5": "O(\\sum_{i=j}^{h} \\log |T_i|) = O(\\sum_{i=j}^{h} \\log 2^{2^i}) = O(2^ h)", "415401c0650095a10e01d5ca86e82bad": " C-\\text{vertex}= \\sec^2 \\left(\\frac{A}{2}\\right) :\\sec^2\\left(\\frac{B}{2}\\right):0", "415425dc0eaf63d1515f3964beedd3fb": "\nt = -RC \\times \\ln\\left(\\frac{-V_\\text{CC}}{-2 V_\\text{CC}}\\right) \n", "4155242252e21dd9f9c2bc6f6293b2f5": "M\\!\\,", "4155356c4d6bcad93aeff6fa23630d43": "\\zeta(s) = \\sum_{n=1}^{\\infty} \\frac{1}{n^s}.", "415588a6ebf6d1227bc4a2d19dd1a457": "a=a_0t^{w(a)}\\prod_{i\\ne 0}(1-a_it^i)", "41559413fff57a033b32be9edab3d25e": "\n\\rho = \\frac{a' \\Sigma _{XY} b}{\\sqrt{a' \\Sigma _{XX} a} \\sqrt{b' \\Sigma _{YY} b}}.\n", "41560db2e410ce8e83ff9d0007a8108b": "\nf(x) = g(x + y)\n\\Rightarrow\nf^\\star(p) = g^\\star(p) - p \\cdot y\n", "415678ffb9f559330ef65beb50e92c66": "\\Omega \\le \\tfrac{2\\pi}{\\tau_0}", "4156961d4fa43094a24e07b42ffd4a8a": "\\{e_1,\\cdots,e_n\\}", "415704d618d0fa83b9f1af070d083c51": "{{{5}}}", "41570b26ad7e7d44813de10f3d928d81": " r_{1} = a + bi ", "41572a23c9127d7b0c277f83ae3d9a83": " \\{a_{k \\ell, i j}\\} \\quad 1 \\leq k, i \\leq m, \\quad 1 \\leq \\ell,j \\leq n ", "415749a94b82d71988ca1a71ef3a5829": " \\frac {U} {T^2} = \\frac { \\partial \\Phi } {\\partial T}", "415754f191ca681fe3b52cbee05e9d30": "200000^n", "4157550672dc936fa54fb6fd7cac059b": "a_{A1} + a_{A2} = 1 > a_{G1} + a_{G2} = \\frac 2 n\\,", "415756dc9785a67d7edc1f51163dd009": "H_0 = \\hat{H}_{\\text{field}} +\\hat{H}_{\\text{atom}}", "41585d3777dbaf77a3a2877fde4ada95": "\\chi_G(\\lambda) = \\lambda^n", "415889ca30884f6792a5b2fa066ca315": "\\rho_0 = \\frac{\\sqrt{C - 2 n \\sin \\phi_0}}{n} ", "4158bd101b04b6c924a47f4f8dfdf626": "e^{-\\tau_{\\nu}}", "4159206871a38592f6cb3e6ec8b5189e": " p_{n+1} - p_n < p_n^\\theta, ", "415954bece090c0b2f1860b62b60aa41": "\\sup_n \\left\\vert \\sum_{i=0}^n x_i \\right\\vert < \\infty.", "415999888b15d8ba8d2140974a2e0200": "u_{tt} = u_{xx} + u_{yy}, \\; u(0,x,y) = p(x,y), \\; u_t(0,x,y) = q(x,y) ", "41599a98f6642247563e8a2953b76ddd": "H = K \\oplus L", "4159b5aafaaeb8fb6e1110732cd3bfe4": "\n\\log \\left( 1+\\frac{x}{y} \\right) = \\cfrac{x} {y+\\cfrac{1x} {2+\\cfrac{1x} {3y+\\cfrac{2x} {2+\\cfrac{2x} {5y+\\cfrac{3x} {2+\\ddots}}}}}} \n= \\cfrac{2x} {2y+x-\\cfrac{(1x)^2} {3(2y+x)-\\cfrac{(2x)^2} {5(2y+x)-\\cfrac{(3x)^2} {7(2y+x)-\\ddots}}}}\n", "415a090a00b4c2f809e8bc7434ba2888": "Y=2k(\\phi(front))=2k(\\theta-\\psi)-2k\\frac{a}{V}\\frac{d\\theta}{dt}", "415a27ca930b2cb7f01a6b0f4e73adf0": "\\max_{x\\in X}\\min_{u\\in U(x)} \\ \\{f(x,u): g(x,u)\\le b,\\forall u\\in U(x)\\}", "415a3fae48b52c0924afd36bc634ebd8": "\n(0,\\ 0,\\ 0.378951,\\ 0.925417) = (0,\\ 0,\\ \\sin\\frac{\\theta}{2},\\ \\cos\\frac{\\theta}{2})\n", "415a8cb3e41f1f25a0dacd8e54173c3e": "d+*d*", "415a8d51a272ae290e9cb13139bcfed8": " b_i > v_i ", "415a9190c3f55dde500d6a01bf0244ba": "\\nabla f=\\frac{\\partial f}{\\partial x}\\hat x + \\frac{\\partial f}{\\partial y}\\hat y", "415af845730a49b8f65bcd60dbafba7e": "\\Sigma(2,5,7) ", "415b3115fa873668d4f96d3b1576be30": "V(t)=t^{- {3 \\over 2}}(-1+t-2t^2+t^3-2t^4+t^5).", "415b346922c9b11f8f7a57e88586b902": " G_{ab}=R_{ab}-\\frac{1}{2}g_{ab}R", "415b4b6826328a9a7106d2102632c132": "n_\\eta(\\xi)=\\frac{1}{e^{\\beta\\xi}-\\eta}", "415b7804fdb8445a386d0c59f93b0806": "\\mathbf{w}=\\mathbf{p}_1^T", "415b7f770f63fbfe1f762cd28b869abf": "W=\n\\left( \\begin{array}{cccc}\n0.0 & 0 & -0.7745 & -0.8960 \\\\\n2.8669 & -4.4622 & 0.0 & 0.0 \\\\\n0.0 & 0.0 & 7.9272 & 2.4523 \\\\\n-4.0225 & 20.6505 & 0.0 & 0.0 \\\\\n0.0 & 0.0 & -9.2789 & -0.1239 \\\\\n-0.5092 & -18.4582 & 0.0 & 0.0 \\end{array} \\right)\n", "415bb6da92c50b0b94a90023f1965fbe": "\\delta_{y}(s)=(y,s').", "415bba4d871053728d259dd411255999": "a \\cdot(b \\cdot c)=(a \\cdot b) \\cdot c\\quad \\forall a,b\\in Q", "415bc8cea10a042635eb4b4925a2eccd": "AJ - JA = 0,", "415c37834be9df1dd9fe579638a985bc": "\n:\\alpha_V V = \\sum_i \\alpha_{V,i} V_i + \\sum_i \\frac{\\partial V_i^{E}}{\\partial T}\n", "415c493ae0d47b00badab7d94c53512d": " | 0 1 \\rangle \\mapsto | 0 1 \\rangle ", "415c4e23132c55bad41fec69eec3e0f6": "12\\nabla_a\\left\\{ \\xi^f\\,\\lambda^{(d}_{fb[c}\\,D^{e)b}_{gh]} \\right\\}", "415c5ad89e5e00a7263b000493212b83": "\\{ z \\in \\mathbb{C} | |1+z| < 1 \\}", "415c5c5665f1ae5f0c66fc65eca3fb72": "10^{30} a^{2+2}", "415c60683d9047cb0808248806a6dda0": "P = \\frac { \\beta_1 + \\beta_2 }{2} \\qquad Q = \\frac {\\beta_2 - \\beta_1}{2}", "415c7327bbca1526172536d9f4d8db03": "\\begin{matrix}kd\\sin\\theta\\,\\end{matrix}", "415c794d3bf1838ae6ca7dba8edf9a04": "(\\ddagger)", "415c8514cb3f987f761fe1b6e43b18ae": "f(bb)d_{bb}+f(Bb)d_{Bb}+f(BB)d_{BB} = 0.", "415c88ea5587cedb417fa9c8d83de344": "\\rfloor", "415c9581582b0a8f0f8dd58cbca696f5": " \\omega \\to \\pi^+ \\pi^- ", "415c96f9b821979d490f616eb240c264": "\n \\left.\n \\underbrace{(2222), (2223), (2233), (2333), (3333)}_{(d)}\n \\right\\}\n", "415d0a0740027b860508a7a5a5077e90": "\\pi_{X,Y}R", "415d0a2f99dca817fdffc9402f0fa177": "Y^X", "415d0b867a3b8c2f08fad3f0e2fd51aa": "\\epsilon=\\mathcal{O}\\left(N^{-1/2}\\right)", "415d0dff9f800de3cc8532ea28b56143": "\\lim_{q \\to 0} ce_r(\\omega,q)= \\cos {r \\omega}", "415d444255db51d64935766b54a26549": "f'(x)={\\rm st}\\Bigg( \\frac{\\Delta y}{\\Delta x} \\Bigg)", "415dd69e673e9db4497a6a40b4276bb3": "\\textstyle \\Lambda", "415df80d2c85fcf9939816804cc4804c": "\\gamma_1 = \\frac{\\mu_3}{\\mu_2^{3/2}} = \\frac{(a_1+8a_2)}{(a_1+4a_2)^{3/2}}", "415e492a68045142eea7e28a9dff24f0": "\np(h|n,b) =\n\n\\begin{cases}\n\\frac{1}{ \\textstyle \\prod_{f=1}^F N_f^{b_f}}, & \\mbox{if } h \\in H(b,n) \\\\\n0, & \\mbox{for other } h\n\\end{cases}\n\n", "415e7eb8d766edce211521d166075a07": "\\overline{\\mbox{SL}(2,\\mathbf{R})}", "415e8c9e45aa9ead4d9f692afdf74965": " \\vec{x}", "415fa2d811711ef9dc2ebc3d730a869e": "m(\\xi)", "415fb03d112450d28db33da398e4de6f": "(\\hbar/e)", "416002ef386d74788b67beee9e74e0bd": "x(N)", "4160112d4cc877ceca6f7aa1d401f820": "c(0,0) = 4\\pi", "41608eeaff1b1ee41bfbc263c84a2540": "g(x)=\\sum_{i=1}^{k-1} f_i(x).", "4160bdca8da432edd51e8f0a70e1c330": "\n\\begin{align}\n& \\int_{\\theta_j} P(\\theta_j;\\alpha) \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) \\, d\\theta_j = \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j \\\\\n= & \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)}\\frac{\\prod_{i=1}^K \\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)}{\\Gamma\\bigl(\\sum_{i=1}^K n_{j,(\\cdot)}^i+\\alpha_i \\bigr)} \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K n_{j,(\\cdot)}^i+\\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{n_{j,(\\cdot)}^i+\\alpha_i - 1} \\, d\\theta_j \\\\\n= & \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)}\\frac{\\prod_{i=1}^K \\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)}{\\Gamma\\bigl(\\sum_{i=1}^K n_{j,(\\cdot)}^i+\\alpha_i \\bigr)}.\n\\end{align}\n", "4160c03e051d035ae614a5de9b6c3406": "u=y", "41611c1841ef248820d39c7e964caccd": " \\frac{ | Re( \\overline{\\lambda} ) | }\n{ | Re( \\underline{\\lambda} ) | }. \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (9) ", "41613817bc67fd5261db32b169294775": "(P \\to Q) \\leftrightarrow (P \\to (P \\and Q))", "41615155c71ed9651dfe8fded3bd9359": "K_\\lambda", "4161afa84ee500175af702f010bd70f7": "\\phi_{\\lambda}^{\\mathrm{R}}(\\mathbf{k})", "41620165e62aeb3fb8d54f6f8cdbc01a": "P_b(E) \\approx K_b(A)Q\\sqrt(2\\gamma_c(A)E_b/N_0),", "41621c1d09462a570fa2509fd61e5caf": "\\mathbf{q}_3 = \\mathbf{q}_2 \\otimes \\mathbf{q}_1", "4162453155942bcbf46d64f163f111e4": "\\mathbf{v}_{min}", "4162579bb8ad04b480bdb51b631a77dd": "\\eta =({\\gamma_{\\perp}})^2 |A|^4L^2 \\propto \\gamma_{\\perp}^2I_{in}^2L^2", "4162d8d34e2c476022484e32b39b270f": "\\alpha = O\\left(\\frac{1}{\\nu}\\right)", "4162dff7802412c23e5566a193803ed1": "s'_n = s_{n+2} - \\frac{(s_{n+2}-s_{n+1})^2}{s_{n+2}-2s_{n+1}+s_n}.", "41632ed88191a6a9a3036d79c023a035": "2B \\int_{S} ds = \\mu_0 I_{enc}", "416360bebae2b88fe6223ebf016cc6a5": "b\\in I\\cap J.", "41639a6aa86b70cb5d4b12fd712851d9": "a'_{i}", "4163a2d89c8aeedc8e4c633b50f4ea54": "\\langle \\bar T T\\rangle_{ETC}", "4163fd221d25f785116e10bd2bbdad75": "P ( \\omega_s )", "416405e9a327c304403ff07d7d08d618": "\\tau_{\\max,\\min}= \\pm R = \\pm 50 \\textrm{ MPa}", "416406e325844909af29d595ea0ad070": "a_1, ..., a_r", "41642cb0a902a44b349726288c4beba1": "\nx = \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cfrac{\\ddots}{\\quad\\ddots\\quad b_{k-1} + \\cfrac{a_k}{b_k + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\ddots}}}}}}\\,\n", "4164baea3bcec9195d7eb7b27d7fc0c2": "G^i(x,\\lambda\\xi) = \\lambda^2G^i(x,\\xi),\\quad \\lambda>0.\\,", "4164c08ecb4c579b56691f0f091f576a": "\\text{pH}=6.1+\\text{Log}\\left[\\frac{24}{0.03\\times 40}\\right]", "416535b1163aa80f74f3f0f1422186fe": "[,]: L_i \\otimes L_j \\to L_{i+j}", "41659a13e234c85f1ecad01cb281cd82": " \\xi = \\sqrt{\\frac{\\hbar^2}{4 m |\\alpha|}}. ", "4165e88125a2bb3dda797c8d3e24ec52": "x(x^2+y^2)+2ay^2=0", "416653f7fe9ab913ff62286f6ad94a6f": "\\theta_0\\,", "41665a851c456bf661d36718ee1db0dc": " f(x_0) = y_0 ", "416687facba2a93ed681a6b8a6896013": "T_2/T_3 \\,", "4167056f419c5f2103ab52d270f998d7": " s_N ", "4167273f34b8610f7afb214505530f9a": "|t|\\rightarrow\\infty", "416750586235f319ff49ffb94deb7c44": "\\scriptstyle{1/{3\\sqrt{3}}}", "41675ac24d33d121299758fc53cc71a7": "\\sigma=(x_n^2)_{n=0,\\dots,M-1}", "41680437065e60aa8ec8cabefd1cd2a2": "\\neg(A\\land B)\\leftrightarrow(\\neg A\\lor\\neg B)", "41681f64163281ec7f46238d8788a499": "\\begin{cases}\\mbox{An ellipse or a circle},\\ \\mbox{if}\\ B^2\\ -\\ 4AC\\ <\\ 0 \\\\\n \\mbox{A parabola},\\ \\mbox{if}\\ B^2\\ -\\ 4AC\\ =\\ 0 \\\\\n \\mbox{A hyperbola},\\ \\mbox{if}\\ B^2\\ -\\ 4AC\\ >\\ 0\\end{cases}", "41683415f46676a8f310ce039b6cfad7": "f' = 0. \\,", "416866bcbd1c3991ff93d8b4019020fd": " F = \\tfrac{{\\displaystyle \\sum_{j=1}^k n_j\\left(\\bar y_j- \\bar y\\right)^2}/{(k-1)}} {{\\displaystyle {\\sum_{j=1}^{k}} {\\sum_{i=1}^{n_j}} \\left(y_{ij}- \\bar y_j\\right)^2}/{(n-k)}}", "416879e14dd6997e05b9ec4be9bcf9b0": "2^{1/p}>2", "41688ba6a031904f5b519422d8b09012": " N_c = 5.7 \\ ", "4168c8b9af46d0a44e2c5bc08a3c79a0": "\\varepsilon_0 \\hat{\\bold n} \\cdot \\left[ \\chi \\bold{ (r_+)}\\bold {E(r_+)}-\\chi \\bold{ (r_-)}\\bold {E(r_-)}\\right] =\\frac{1}{A_n} \\int d \\Omega_n \\ \\rho_b = 0 \\ , ", "4168d303f392f8a57c70f0adf9c69cda": "\\frac{\\partial E}{ \\partial w_{ji} } \\,", "4168e944edf19533a64cfd5505106c7b": " \\alpha(\\rho,\\sigma) \\ge 215.3^\\rho 44.7^\\sigma . ", "416906799024eca24d8ebe55436575fc": "2t\\log(\\mathcal{F})", "41690ea5b70d72068db4e8acb5ff59a9": "G=\\sum_{x}p(x)E(x)-T\\sum_xp(x)\\ln(p(x)) \\,", "416942a69e6c9f902fc5bc95daca1122": "\\forall x\\colon F(x)", "41696a73e6f9beff906d47fd1b2a538d": "\\hbar\\omega_0", "4169c438d0b07b45e66d99883a75d7a4": "\\dot{y} = \\frac{dy}{dt} \\,,", "416a10391207bc332bc87881d2701ab3": "RT=P(V_m-b)+\\frac{a}{V_m(V_m+b)T^\\frac{1}{2}}(V_m-b)", "416a1b8f76b53aa2a213967acd268d85": "e_{(0)}=\\frac{1}{\\sqrt{2+(x^3)^2}}\\left( x^3\\partial_0-\\partial_1+\\partial_2\\right)", "416a1f0d29dccbf5ed7e19a00c6242c7": "365\\tfrac{385}{1539}", "416a676c937ebb9f8a41dc4a7d1491d9": "T \\square \\square F", "416a83bd1e5b6f501e324b4a88502750": "Y_{6}^{2}(\\theta,\\varphi)={1\\over 64}\\sqrt{1365\\over \\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(33\\cos^{4}\\theta-18\\cos^{2}\\theta+1)", "416a8922654cfad9a4182a06349ac76d": "\\frac{3a^2}{8\\pi G}H^2 = \\rho a^2 - \\frac{3kc^2}{8 \\pi G}", "416a9d5222b9eb12285ca78d4666b33f": " ADC", "416adad6bf68d54d26a675a974fecb84": "\\lim_{\\Delta g\\to 0} \\left.\\frac{\\Delta U}{\\Delta g}\\right|_{c.p.}", "416afe36f00d44cf356c8939c3ea2f1d": "\\mathbf{A}^k = \\underset{k \\mathrm{\\, times}}{\\mathbf{A}\\mathbf{A}\\cdots\\mathbf{A}}", "416b2b62374e597d1781160b051ea1b4": "\\Delta E_k = F \\cdot s ", "416b6d6e48a4ce5f6da9c82bca1a8fd1": "\\sum_{n\\geq 1}\\tau(n)q^n=q\\prod_{n\\geq 1}(1-q^n)^{24}.", "416c01bfba136129063f58b44151e984": "\nI =I_0{{\\left(\\frac{\\sin \\left(\\frac{{\\pi a}}{\\lambda } \\sin\\theta\\right)}{\\frac{{\\pi a}}{\\lambda }\n\\sin\\theta}\\right)}^2}{{\\left(\\frac{\\sin \\left(\\frac{\\pi }{\\lambda } N d \\sin\\theta+\\frac{N}{2} \\phi \\right)}{\\sin\n\\left(\\frac{{\\pi d}}{\\lambda } \\sin\\theta+\\phi \\right)}\\right)}^2} \n", "416cff4fce8c34bfc29865175b5369be": "\n\\left( \\zeta + b \\right)^{l} = \\zeta^{l} + l b \\zeta^{l-1} + \\ldots + l \\zeta b^{l-1} + b^{l}\n", "416d0a01947204835ec4a384d3cd6ea6": "c \\equiv a \\times b \\pmod n. \\, ", "416d1ff9c2cc4db62117c80a3546af13": "g_\\otimes\\in \\Gamma\\left((TM\\otimes TM)^*\\right).", "416d425b2c0bbf654876e4707837121c": "z_1, z_2, z_3", "416d68f7fff72bb4e99adcf8bd1fa155": "(r_i r_j)^m", "416dec0ee8dda26d2aa7ca0d9690c52e": "[s] = \\mathrm{\\tfrac{J}{m^{3} \\cdot K}}", "416e07bc52354a26f291d90cda71c913": "z\\mapsto\\frac{az+b}{cz+d}", "416e3716dc6247f8418c502af057810a": " y(t) = \\int_{-\\infty}^{\\infty} h(t,s) x(s) ds ", "416e74576a8a460dcbaf5fb9f28cfb93": "\\delta S = \\frac{\\delta Q_{rev}}{T}, \\delta W_{rev}= p \\delta V ", "416e85f1b7d1ed53197917017bad1bd0": "e^x = \\sum_{n=0}^\\infty {x^n \\over n!} = \\lim_{n \\to \\infty}\\left(\\frac{1}{0!} + \\frac{x}{1!} + \\frac{x^2}{2!} + \\cdots + \\frac{x^n}{n!}\\right).", "416ebe19b2d5c7d1953090e1f3544547": "\n \\begin{align}\n \\frac{\\partial}{\\partial x_i} \\left( \\rho\\, v_i \\right) &= 0, \\\\\n \\rho\\, v_j\\, \\frac{\\partial v_i}{\\partial x_j} &= - \\frac{\\partial p}{\\partial x_i},\n \\end{align}\n", "416ed4f0935b738667b0d7f98b550ae7": "\\| f \\| := \\sup \\left\\{ \\left| \\int_{I} f \\right| : I \\subseteq \\mathbb{R} \\text{ is an interval} \\right\\}.", "416f18e4affb6eccb198587401680230": "\\left(\\frac{1/\\sigma_x^2}{1/\\sigma_x^2 + 1/\\sigma_d^2}\\right)^2 \\sigma_x^2 + \\left(\\frac{1/\\sigma_d^2}{1/\\sigma_x^2 + 1/\\sigma_d^2}\\right)^2 \\sigma_d^2 = \\frac{1}{1/\\sigma_x^2 + 1/\\sigma_d^2}", "416f31473b82c4991b1282b2d67dcd62": "\\bar{X_t} = \\bar{X_t} + \\langle x_t^{[m]}, w_t^{[m]} \\rangle", "416f57841bf32cff414088e9c04ec443": "e^{-e^{-(x-\\mu)/\\beta}}\\!", "416f9bfa2d2040738cc6628a38fc5fe6": "W = |\\sum_{i=1}^{N_r} [\\sgn(x_{2,i} - x_{1,i}) \\cdot R_i]|", "41700c1af3178e044f5747e9db4df838": "L \\in \\mathcal{C}", "417024bf6f09a088710a45ea9663adcc": " F_{\\alpha\\beta} = \\partial_{[\\alpha} A_{\\beta]} = \\nabla_{[\\alpha} A_{\\beta]}", "417038041ee0cef0a431b343969d606b": "Q_n = (V_n, E_n)", "4170a89776deb8c88c367139e88682e6": "\\digamma^n:\\pi_1 (X, t)\\rightarrow \\mathrm{Sym}\\,f^{-n}(t)", "4170acd6af571e8d0d59fdad999cc605": "CD", "4170c219c476b06cf1777ecf0a8f776a": "\\hat{\\textbf{y}}_{k\\mid k} = \\hat{\\textbf{y}}_{k\\mid k-1} + \\sum_{j=1}^N \\textbf{i}_{k,j}", "4170ebf1a9f2cb703cf7d753a49b1bf2": "\\alpha\\to 1", "41711bf2b34d38de98a406c26f117c57": "\\beta_0, \\ldots, \\beta_p", "4171518beb9d6b30a475153c1c71ffa0": "\\,\\sigma", "41716debf274e8bbde48ef0180298b41": "y^2 = 4x^3 + b_2 x^2 + 2b_4 x + b_6", "41717f6aed0bbaeb6c37ab07f1041895": "\n \\sqrt{n}(\\hat{\\theta} - \\theta_0) = \\Bigg[\\, {- \\frac{1}{n} \\sum_{i=1}^n \\nabla_{\\!\\theta\\theta}\\ln f(x_i|\\tilde\\theta)} \\,\\Bigg]^{-1} \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n \\nabla_{\\!\\theta}\\ln f(x_i|\\theta_0)\n ", "4172bc56e31ef5638bae0e491ebcecad": "\n D\\,\\nabla^2\\nabla^2 w^0 = -q(x, y, t) - 2\\rho h \\, \\ddot{w}^0 \\,.\n ", "417302f2e6b053153c9e8bd2fbbb2a03": "\\lambda_{c}", "4173400383df31f09e2a25b65ca3deb1": " \\operatorname{Var}(X) = \\sum_{k=0}^{\\infty} \\frac{\\lambda^k}{k!} e^{-\\lambda} (k-\\lambda)^2 = \\lambda,", "41735b06797091d82cb09f6f59fd8c5d": "f(x) = x^2 + bx |_{b=\\{1,2,3,4\\}} \\!", "417362dfcc0de4250da739fd407a8d86": "f_1 \\lor \\dots \\lor f_n \\Leftrightarrow f", "4173a25da06516000b7796bb063ba33f": " -\\frac{\\pi}{2} < y < \\frac{\\pi}{2} \\, ", "4174272007ee851d048f2ed892f387bb": "\\sum_{n=0}^\\infty b_n = B", "417439d5d976a5672162d184e34368ae": "p(X) \\equiv X=a \\or X=b", "41743ebcbe8f4c21003026584adb07e1": "\\mathbf{P}(O = j)=\\sum_{i} \\pi_i b_{i,j}", "417484eca5c3b3a74ba5cf1056113cf1": "\n\nx_i = a_{1i}x_1 + a_{2i}x_2 + \\ldots + a_{ni}x_n + d_i,\n\n", "4174a1cf29c4bbe75814a267a75420c8": "M_2(\\phi,\\tau) = \\phi_2(\\theta,\\tau)\\int s^*(u-\\dfrac{1}{2}\\tau)s(u+\\dfrac{1}{2}\\tau)e^{j\\theta u}\\, du", "4174ef56b3487130f7cff157c59e670f": "g(x)= \\sqrt{\\frac{a}{\\pi}}\\cdot e^{-a \\cdot x^2}", "417511eafda5bb6aa1e3daf31f9691de": "\n \\nabla^2\\nabla^2 w = -\\frac{q}{D} \\,.\n ", "417517d510ea0da2631421873eed0f9e": "\\operatorname{adj}(-A)=\\sum_{i=1}^nc_iA^{i-1},", "41751dfb5cab5860356f47b15b671e9e": "[z,x^{-1},y]^{x}=1", "4175c21edb56de33d2fe43552147f29d": "\\scriptstyle {\\sqrt2 - 1}", "4175cb1a402b6a3078102d323b15e622": "P(A) = Q(A)", "4175d7950537da7207c7b41983b3cf47": "\\mathcal{F}\\varphi", "4175f876759114730f63d8cc22f6436a": "\\pi_i(B^+C)", "4176309b36d20358787499ff0ec411fe": "d_1d_2^{-1}", "41763cb1ffdc54e30eaf81110b93d0b1": "\nI = \\frac{V_{in}}{R+1/j\\omega C}\n", "4176c4c06642287644a8fc00a4f925f7": "f(x)=f(a)+\\sum_{i=1}^n (x_i-a_i) g_i(x),", "4176c800d7632761b0a2534f714cef29": "\n\\rho (u ) = -{ 1 \\over {\\pi u }} \\sum\\limits_{k = 0}^\\infty \n {{( - 1)^k } \\over {k!}}\\sin (\\pi \\beta k)\\Gamma (\\beta k + 1) u^{\\beta k } \n", "41773a28fe2a5f23495ec9b6175272fe": " \n\\Omega_d = \\frac{\\Gamma_\\theta}{f_r}\n", "417748f905fcf618688a30338c8c6e10": "X_{t}", "41775329215e9210d69c7f6c70787273": "\\hat{x}_1=\\sin \\theta \\sin \\varphi _1 \\dots \\sin\\varphi _{n-3}", "41778f81a9252a96b10a6ee1ca2a57d1": " \\frac{y}{b} = \\sqrt { 1 - \\left( \\frac {x}{a}\\right) ^2 } = \\sqrt { 1 - \\cos^2 E } = \\sin E \\ , ", "4177abe8e34e1838fb0a0b7ae3b0063b": "\\mathbb{R}^n, n=1,2,3,\\dots", "4177ed0d15a6c35e70871364eba4f709": "\\lnot \\mathrm{Proof}^R_T(x,y) \\equiv \\mathrm{Proof}_T(x,y) \\to \\exists z \\leq x [ \\mathrm{Proof}_T(z,\\mathrm{neg}(y))].", "4177f404b63421bff05325b2095a8718": "Q_{g \\circ f}=g(Q_f)", "417819924fa2b42fa09669db32dda0be": "c^2 = a^2 + b^2 + 2bd.\\,", "41788fec3c7b7f424a042089e2509c3f": "\\Gamma(E)", "417894ac7c35647ca70c1d92f77108c4": "{a_0=eA/m_e c^2}", "417904fef2f447bfccb1b56c00905ad2": "{v_{o} \\cdot w^{j} (s - a_{v})\\over 100} > {c_{o} \\cdot r^{j} \\cdot a_c\\over 100}", "41799ffb0a3fc317db38e4130e16519e": "T_{initial}", "4179a4d06a00ca19d1a66ebdc1fb754a": "-RT \\left(\\frac{\\partial \\ln k_x}{\\partial P} \\right)_T = \\Delta V^{\\ddagger}", "4179a5d1f856671e151d276131c16441": "\\pi^0 \\to\\gamma\\gamma", "4179c9f61cc2c1b964499f5916396e4b": " y^{(N)}(t) = f(t, y(t), y'(t), \\ldots, y^{(N-1)}(t)) ", "417a24fccae39f408b8fc70288900cc6": "\\mathcal{L}_C=-\\frac g{\\sqrt2}\\left[\\overline u_i\\gamma^\\mu\\frac{1-\\gamma^5}2M^{CKM}_{ij}d_j+\\overline\\nu_i\\gamma^\\mu\\frac{1-\\gamma^5}2e_i\\right]W_\\mu^++h.c.", "417a393d49e2bbc33afdb9fb5b022610": "E(u)=(u(0)+u(100))/2", "417a7f4d3ab3a37150001afec51981f9": "exp[\\psi(\\theta)]=\\sum (C(v)+\\theta^iF_i)", "417abb2a3e5507c7f329365458c8dddd": "r= a \\frac{\\sin 3\\theta}{\\sin 2\\theta} = {a \\over 2} \\frac{4 \\cos^2 \\theta - 1} {\\cos \\theta} = {a \\over 2} (4 \\cos \\theta - \\sec \\theta)\\!", "417b3782769eac23fb57905386386459": "\\mbox{EBC} = \\mbox{SRM} \\times 1.97", "417ba1dd16d2be236b39fea0d12dbd41": "f(s)\\not \\le_Q f(t)", "417c1439799679218e0725fe03647986": "\\mu_1 \\dots \\mu_i", "417c59ad49efd02423ad25520360bafe": " c \\geq 12 ]", "417c9d911c37aaf5956257aec38df5db": "\\prod_{i \\in I} X_i = \\prod_{i \\in I} X", "417caaa64ac79c2cf50a00c993d05ff2": "a_n(\\mathbf{r-R}) = \\frac{V_{C}}{(2\\pi)^{3}} \\int_{BZ} d\\mathbf{k} e^{-i\\mathbf{k}\\cdot(\\mathbf{R-r})}u_{n\\mathbf{k}}", "417d4c7e196d41804f72e4fcede78c95": "\n\\begin{align}\n\\hat{H}_\\text{field} &= \\hbar \\omega_c \\hat{a}^{\\dagger}\\hat{a}\\\\\n\\hat{H}_\\text{atom} &= \\hbar \\omega_a \\frac{\\hat{\\sigma}_z}{2}\\\\\n\\hat{H}_\\text{int} &= \\frac{\\hbar \\Omega}{2} \\hat{E} \\hat{S}.\n\\end{align}\n", "417dbc0db5f3106eb022460df5bef8eb": "H(x) \\gets \\frac{N - p_f \\cdot (N - C - 1) }{N} \\cdot \\lg\\left[\\frac{N}{N - p_f \\cdot (N - C - 1)}\\right] + p_f \\cdot \\frac{N - C - 1}{N} \\cdot \\lg\\left[N/p_f\\right]", "417dc60d26e5f55293b49fb5770eeee2": "\\sigma_P(x,dF(x)) = 0.", "417dcbb41bd7a1bf22943aa95381fdce": "\\Phi(\\mathbf{r}_3)", "417e706c11324371873405c89d9ab18f": "\\textstyle P(A\\mid[x]) = \\alpha", "417e9323c23628b7d215a4285da4a5b3": "\\omega_f(x_0) = \\lim_{\\epsilon\\to 0} \\omega_f(x_0-\\epsilon,x_0+\\epsilon).", "417e980627e724a6b08b57a6620ec967": "4*k_B*T*B/R)^{1/2}", "417e9a77c6643e446f328ea2aae7d295": "Z = sL\\,", "417ea7ce370459fb533cb1de12a4b620": "R \\frac{\\mathrm{d}q}{\\mathrm{d}t} + \\frac{q}{C} = \\mathcal{E}\\,\\!", "417ea8336a401ec70a495e47bd5a098d": "F(Y) = Y \\otimes_R X \\quad \\text{for } Y \\in \\mathcal{C}", "417ef3939269d6265eba5ac6d78a1cd3": "\n64,864,800^2 \\approx 4.2 \\cdot 10^{15}\n", "417ef8ba1f557f72d6c84ee2a8163bb0": "p_i = c_i + v_i + s_i = l_A a_i + l_W l_i + s_i ", "417f21342b1f24e6829a78b73de97fe6": "X_2 = \\,\\!", "417f30e98e7ee418849198f0ebde3589": "\\ P(\\mathbf{i}x)x.", "417f827aa803f7a4d058a048d034518b": "\\mathcal{H}_N = \\mathcal{H}_K \\otimes \\mathcal{H}_M ,\n", "41800d853f240062dd0ce10d03c9561c": "\\eta>0", "418017675383af28c32d0cd77a22198e": " c = f\\,^2 / (N H)", "41808a632be95c3f6fd894a23798905b": " OI = \\frac{F_i O_2 * M_{PAW}}{P_a O_2}", "41809796e8db9ffd858e710f36a595c0": "\ni\\hbar {d\\over dt} |\\psi\\rangle = \\hat H |\\psi\\rangle\n", "41812708904111f4f0f9bae0acf82ef6": "\\mu\\,=GM", "41817efd7044ce5ffe68609ed4b34468": " (q_1,\\omega_1,q_2) ", "41823ab86043a081f8de839650d4b361": "\\displaystyle{\\alpha_m={1\\over N} \\sum_{n=1}^N{1\\over n} \\lambda_n z_n^m.}", "418277b0dd30832544cbc986a0edc422": " \\hat{p}_x = -i\\hbar\\frac{d}{dx} ", "4182be47bd837919f8db54cf5d837b8d": "\\tilde{y}", "4182d4c239870d692abd2a053adb5eb4": "p = 2^{521} - 1", "418342ff161c253ec52492ccde0d3776": "\\mathtt{Map\\ (Set\\ string)\\ int}", "4183b25e8ddb06066f7b8eb6459efb6f": " \\frac{f_o}{f_s} = \\frac{ 1 - \\frac{ \\|\\vec{v_o}\\|}{\\|\\vec{c}\\|} cos(\\theta_{co}) } { 1 - \\frac{ \\|\\vec{v_s}\\|}{\\|\\vec{c}\\|} cos(\\theta_{cs}) } \\sqrt{ \\frac{ 1-(v_s/c)^2 }{ 1-(v_o/c)^2 } } ", "41842638e5b09f8618afeca3579ead5e": " k_i = a + b m_i ", "418439cd7180163d515649d80805199c": "H_2A \\rightleftharpoons HA^- + H^+ :K_1=\\frac{[HA^-][H^+]} {[H_2A]}", "418471a992862a79fdef5a6a25cd1f18": "V(r,t) = \\frac {1}{4 \\pi} \\int_{S} \\left\\{[V] \\frac {\\partial}{\\partial n} \\left(\\frac {1}{s}\\right) - \\frac {1}{cs} \\frac {\\partial s}{\\partial n} \\left[\\frac{\\partial V}{\\partial t}\\right] - \\frac{1}{s} \\left[\\frac{\\partial V}{\\partial n} \\right] \\right\\} dS ", "418482540147d10e3c584045961997e3": "T_v = \\log_2", "418490b996a4f8bec1f15c698d010296": "F_{a}(x) = (a_{1}^{x_{1}} a_{2}^{x_{2}}\\dots a_{n}^{x_{n}})G ", "4184b2012600b8bf27f51c96d8adcbbc": "D_-f(t) \\triangleq \\liminf_{h \\to {0-}} \\frac{f(t + h) - f(t)}{h}.", "4184d489245c978a6c130d1bd5a8eaaf": "\n Q_x^{\\mathrm{core}} = \\kappa\\int_{-h}^h \\sigma_{xz}~dz = \\tfrac{\\kappa(2h+f)}{2}~C_{55}^{\\mathrm{core}}~\\cfrac{\\mathrm{d}w_s}{\\mathrm{d}x}\n ", "418529d47ebfb16abb7d3499b232ee00": "T(\\mathcal{M}, \\theta)", "4185dcb9aa453ffab7c147890e2e151d": "r = r* + r'", "41863aa6e3e6580a687f878074859582": "D_k(x)=\\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \n\\zeta^k(w) \\frac {x^w}{w} \\,dw", "4186500394f13b6e41706df5446c2a29": " D[g] = [[x, \\operatorname{false}, \\_], [o, \\operatorname{true}, p], [y, \\operatorname{true}, n]] ", "4186565f1337ef1f881f091f7a1c1e65": "B=\\prod_{i=1}^{d}[x_i,y_i]\\subseteq [0,1]^d ", "41868729cfdc544a5c0105dbbca24025": "\\int_0^\\theta \\log| \\sec(t)| \\,dt = \\Lambda(\\theta+\\pi/2)+\\theta\\log 2.", "4186d395d8b7d784a352385b7c778b10": "\\Psi [A] = \\sum_\\gamma \\Psi [\\gamma] W_\\gamma [A] ", "4186dd5201d123d4378b20c6fe6cc918": "||A|| = \\inf\\{C: ||A\\mathbf{x}||_V \\leq C||\\mathbf{x}||_U\\}", "41870e2c87281c547285c0641d88421a": "\\eta(f) \\mapsto \\sum_{s \\in \\bigcup_{S \\in \\mathfrak{P}(V)}\\sum{v_i \\in S} \\left( \\{v_i\\} \\times D_i \\right)} f(s)", "41874d1de7f7d38d604ec3430a2ecaa7": "\\langle X,C,k \\rangle", "4187f076fcfabf1bf600176ac7743970": "(Y,Y^*)", "4188b2c9c1a870605c2ff493129e485a": "E_{e'} = \\sqrt{(p_{e'}c)^2 + (m_ec^2)^2}.", "4188f1aa4660448ce82bd748c48629fd": "\\le 1000", "41896e8772250934a1de3f642f863e33": "\\left|\\int_\\text{Arc} f(z)\\,dz\\right|\\le {a\\pi \\over (a^2+1)^2} \\rightarrow 0\\ \\mathrm{as}\\ a \\rightarrow \\infty.", "41898670d560dcb3e592c58a2638ef0c": "\np = \\frac{1}{C_2} \\left(\\frac{r_0}{r}\\right)^4\n", "4189bd72964d02f23cbb89574bdc126d": "\\min\\left(x\\right)", "4189eb5dedd1f2247b771dc8de91cf2d": " 3570 = 80L ", "418a0321dc80862893aea5b8eda7bf5b": "\\Gamma^k_{ij}", "418a276cf60526b7e105d0e966621db3": "\\ \\gamma \\,", "418a603bec08787d0cf985472f0d06c9": "L(C) = \\sup_{a=t_0 < t_1 < \\cdots < t_n = b} \\sum_{i = 0}^{n - 1} d(f(t_i), f(t_{i+1}))", "418a6972602ab918e97eb6fd06afa90e": " Q_d ", "418a9ff64a267e0aea4cc85a4ebe7a39": "O(1/p(z))", "418ad60004c96bb8dbdbb616d8ec991e": "\n \\sqrt{J_2} = A + B~I_1\n ", "418b19b833e5bbb53ab9938202c83b6c": "\\int \\hat\\mu''(x)^2 \\, dx", "418baacb0b85f5c41ea00b03ee3fb862": "2^d \\alpha", "418bb0a90f747d38970a657d05c42f4b": " \\| A^* A \\| _{op} = \\| A \\| _{op}^2. ", "418c0a3dd65188837ffcca8526393fa4": "\\Omega\\ or\\ \\omega > 0", "418c4c8aa00f7630adc2882051c6a3a9": "(x \\circ y) \\circ z", "418ce4d6934b851d8c33bc1c277bb6ba": " \\mathbf{ \\hat n}\\,", "418d72a61a0f85744115c83b81803d3e": "[\\![p]\\!]_i = V(p)", "418ddcc4ac5d1150e5b90cce6b71e14c": " P ", "418e3854922a1a51068adffa5eb4ec88": "ST_x(\\varphi \\rightarrow \\psi) \\equiv ST_x(\\varphi) \\rightarrow ST_x(\\psi)", "418e45c1c8d45de6d59a01acdc635b02": "\\mathbb P(\\theta_k^n=s|\\theta_{k-1}^n=r,\\theta_k^{n-1}=t)=\\Lambda^n(s|r,t)", "418e649175dce49a08b8e7b384d4a271": "P_{y,w}(q) = \\sum_iq^i\\dim IH^{2i}_{X_y}(\\overline{X_w})", "418e9f8bd8daa2272f0434d9b5a889fa": "\nS_\\lambda(x_1,x_2,\\ldots,x_n)=\\sum_T x^T = \\sum_T x_1^{t_1}\\cdots x_n^{t_n}\n", "418ebb9e4ede4a3ae8926598af8d3ab6": "\\mathbf{g}(n)=\\mathbf{P}(n-1)\\mathbf{x}^*(n)\\left\\{\\lambda+\\mathbf{x}^{T}(n)\\mathbf{P}(n-1)\\mathbf{x}^*(n)\\right\\}^{-1}", "418ec10eba0bb67c698400c8ddc6013e": "\\pi \\sim 6 \\times 2^i \\times t_i,\\qquad\\mathrm{converging\\ as\\ i \\rightarrow \\infty}\\,", "418f61c02d46f02e1110b45d28955b6f": " \\rho(z) = z^s + \\sum_{k=0}^{s-1} a_k z^k \\quad\\text{and}\\quad \\sigma(z) = \\sum_{k=0}^s b_k z^k. ", "418f7b3d2c4f40b0016b5112eacb6add": "L[\\gamma]:=\\int_a^b F(\\gamma(t),\\dot{\\gamma}(t))\\, dt", "418fbaf7992a1d3470215959877e59eb": "\\zeta\\left(\\frac{1}{2}+it\\right) = e^{-i \\theta(t)}Z(t),", "418fceb4eb06c346b136eefc77638f21": "\\mathbf{E} ( Y_{t} \\mid \\{ X_{\\tau}, \\tau \\leq s \\} ) = Y_s, \\ \\forall\\ s \\leq t.", "418ffbb0ac6a8760b694cbd4e68625d0": "q \\mapsto u^{-1}q u ", "41900f17a7a7ba0ba680f749ee39ba95": "\\int_0^\\infty \\sin ax^n=\\frac{1}{na^{1/n}}\\Gamma(1/n)\\sin\\frac{\\pi}{2n}\\quad ,n>1", "4190251187c0e38a45e798a1c2c2d99b": "\\quad \\sqrt{(x_1-y_1)^2 + (x_2-y_2)^2 + (x_3-y_3)^2} ", "41905feb4988d09687ca5b1149318214": "\\Delta S\\,\\!", "41909202e076793e01daeda08e9de8fc": "\\textstyle \\frac{4(x^p -1)}{x-1} = y^2 \\pm pz^2", "4191090d0fded0b32e2b64534307257c": "F(0)=0", "419185f61c15870d9f8b96c78542527d": "\\operatorname{Z}(R)", "4191aa38592466ad383c01863314f779": "\n\\mathbf{M} = \\mathbf{v} \\times \\mathbf{L} - k\\mathbf{\\hat{r}}\n", "419202c9636f08ee3ba5c2aec9e0d76c": "\\tfrac{1}{50}", "41921196ee456c8e49b35be3bb9ee728": "E_{K} = \\begin{matrix} \\frac{1}{2} \\end{matrix} m \\boldsymbol{v}^2.", "41922c7551828b3441b07e1dfee25bf6": "(\\alpha - \\beta)^2 = (\\alpha + \\beta)^2 - 4\\alpha\\beta\\!", "4192b0174cc1d8ebd4f89c5294a57f5b": " R_1 = Z_0 \\coth \\left ( \\frac {\\gamma_ \\mathrm \\Pi}{2} \\right ) = Z_0 \\frac {1 + A} {1 - A} ", "419346e7b5cffce55c269716a22ce3c0": "y=t(x+1)=\\frac{t(t+2)}{1+t+t^2}\\,,", "4193513eff3272cec8af2930d7b2f73a": "\\mathbf{g}=\\frac{\\mathbf{F}}{m}=-\\frac{{\\rm d}^2\\mathbf{R}}{{\\rm d}t^2}=-GM\\frac{\\mathbf{\\hat{R}}}{|\\mathbf{R}|^2}=-\\nabla\\Phi,", "419395aea99598d9eeb6429d4609571a": " B \\in M(m,p; \\mathbb{K})\\,\\!", "4193ed879e770cef282066687a06dfb4": "\\{ M , C (\\vec{N}) \\} = 0", "41940cd0dcd6014f30ab5974ecbf524d": " H_n^{(b)}=+1+\\frac{1}{2^b}+\\frac{1}{3^b}+\\cdots", "419473299add2a6d86754eca0d5d2fd4": " T = \\frac{1}{2} m \\dot x^2, \\,", "4194c41dfca6651c7577002614d4050a": "pq =100", "4194f60aa55bc6e2a96f5ef9ae26707b": "\\lbrace 1, 3, 5, \\dots, 2^{w-1}-1 \\rbrace P", "4195828d5d3ff2a58ebfdd1cdaf661c9": " a_kz_0^k = O \\left( k^{-{\\textstyle \\frac{1}{2}}} \\right), \\qquad \\forall k \\geq 0, ", "4195d087df97ddff697866cdc787e6da": "\\overline{0}", "419788f770296c836cb42fc503831976": "\\sup_{T \\in F} \\|T (x)\\|_Y < \\infty ", "41979f5125ba28dfe7836bb7178b1bae": "(X,\\pi)\\,", "4197de86973041c8caf6c37b6d726475": " a_i^\\dagger ( n_i + n_j ) a_j ", "4198054823f5d5b8176f9b0c8dc294c0": "A:x \\mapsto 2\\sqrt{x} \\, ", "41983307c5268f803690a1c343c7d351": "\\mathcal{L}\\left\\{f(t)\\right\\}=F(s)", "41986975d435867f98f80fe6482e4da4": "j\\neq l", "41986d6df9054ab4599291d905b7a0b6": " \\sum_{i=1}^{d+2} a_i x_i=0,\\quad \\sum_{i=1}^{d+2} a_i=0,", "41988ad9ac43e77736e655c95d2fd529": "x_i+\\sum \\lfloor \\bar a_{i,j} \\rfloor x_j - \\lfloor \\bar b_i \\rfloor = \\bar b_i - \\lfloor \\bar b_i \\rfloor - \\sum ( \\bar a_{i,j} -\\lfloor \\bar a_{i,j} \\rfloor) x_j.", "41989e7849207d9b75400ce5133cd85f": "f(\\epsilon) = \\frac{1 - \\epsilon^2}{\\epsilon^3} \\cdot \\left[ \\left(3-\\epsilon^2 \\right) \\cdot \\mathrm{arcsinh} \\left(\\frac{\\epsilon}{\\sqrt{1-\\epsilon^2}} \\right) -3 \\epsilon \\right]", "4198b2be7b7c26e8f0f689c68766f75d": "M = {f \\over f-d_o}", "4198c91c9caa3461e5538eab03297599": "z_{i+1}=z_i - \\frac{P(z_i)}{P^{\\prime}(z_i)}.", "4198d1bda9f97d2a21dc8d7372c7ffc8": "\\begin{align}\n\\mathbf{x}+ d\\mathbf{x}&= \\mathbf{X}+d\\mathbf{X}+\\mathbf{u}(\\mathbf{X}+d\\mathbf{X}) \\\\\nd\\mathbf{x} &= \\mathbf{X}-\\mathbf{x}+d\\mathbf{X}+\\mathbf{u}(\\mathbf{X}+d\\mathbf{X}) \\\\\n &= d\\mathbf{X}+\\mathbf{u}(\\mathbf{X}+d\\mathbf{X})-\\mathbf{u}(\\mathbf{X}) \\\\\n &= d\\mathbf{X}+d\\mathbf{u} \\\\\n\\end{align}\n\\,\\!", "41991148576e7fc658fbd35ce83aa65e": "S^{\\sigma}(n)", "41991d0b4a169b02e78e8452a1f5cc0a": "\\sqrt{\\Delta}", "41991e706ccee3cef4ca15c21b3b8df2": "PMNB", "419930a31439db67be136f26c6893f0a": "\\int_\\Omega f(x) h(x) dx = 0\\,", "41995545c21174b81c2bbf265d5f50bc": "\\sqrt [24]{2}", "41996eb8f2a00a05130854f48b830cd8": "xy \\vee xyz \\vee \\bar{x}z \\vee \\bar{x}yz", "41998b3f19b5e12366d35551ac6b5dfb": "1 + MD( \\Box p \\rightarrow p) =", "4199cc9a6f0e0d190e757cf4b8923d71": " \\Delta = -r^2(\\partial_x^2 + \\partial_r^2).", "4199d78e2757031d2fd2bbf827839532": " S = {\\mathcal T} \\exp \\left(\\sum_{j=-\\infty}^\\infty h_j\\right) = \\mathcal T \\exp \\left(\\int dt\\, d^3 x \\, \\frac{H(\\vec x,t)}{i\\hbar}\\right).", "4199ee1096aba59a1be22bd0e6feb2c8": " C_{mV} = \\partial^2 Q/\\partial m \\partial T \\,\\!", "419a5fdddfb1e8589156c7d6f729456b": "L_{i+1} = R_i\\,", "419a7e3baf7a5b4bd53b11c64ec525ce": "c_i=a_i\\pm b_i \\mod m_i", "419ab8956089312ba2eae60e42e5a5ce": "f(x;\\mu,\\sigma_1,\\sigma_2)= A \\exp (- \\frac {(x-\\mu)^2}{2 \\sigma_1^2}) \\quad \\text{if } x< \\mu\n", "419ad638fc89f1e49ebad887afd66713": "A_1, A_2, \\dots, A_N", "419af079d7bb613d1b336cda6ee5d32b": "\n E\\{G_i x(n) G_j x(n)\\} = 0;\\qquad i\\neq j\n", "419b6e0162618c242288a543838120e3": "a \\rVert c , b \\rVert d ", "419b6fac614cbb8bf5f4f59b2e9b8e7c": " \\beta = \\arccos \\left( \\frac{W\\cos \\alpha+V}{A} \\right) = \\arccos \\left( \\frac{W\\cos \\alpha+V}{\\sqrt{W^2 + V^2 +2WV\\cos{\\alpha}}} \\right)", "419ba1bddc3912b6d032ab5216bac5e5": "p_K\\left( \\frac{1}{2} x + \\frac{1}{2} y\\right) \\le r + \\epsilon = \\frac{1}{2} p_K(x) + \\frac{1}{2} p_K(y) + \\epsilon .", "419ba5dcf8d961d749688b2418a3cfeb": "D^2F : U\\times X\\times X\\to Y", "419ba9a20a7abb1ad6b3cb1f8af77ea4": "\\scriptstyle\\mathbf x_{k\\beta}(t)", "419bc237205a04ea779f6845e500c785": "\\displaystyle{\\overline{H_\\varepsilon f} = - u^{-1} H_\\varepsilon( u \\overline{f}).}", "419c0cd8f0dc86a78c2a8f289456c7a1": "\\textstyle B", "419c13b939ca3d819b4f3419d3c2b9b8": "H_1: Y_i = N_i + X_i", "419c4ce048dc0bea2d66025181d9d551": "\\tilde{E} (\\mathbf{r},\\omega - \\omega_0) = A_m \\tilde{a} (\\omega - \\omega_0 , z) f(x,y) e^{i \\beta_0 z} ", "419c55c90d1040d9f5d7fcd9b7827c16": "\\pi = \\pi^-", "419c6a02b3795504e6e7363aa1077551": " C_k ={ 1\\over{k+1}}{{2k}\\choose {k}},\\quad k \\ge 0 ", "419c7b7b7ebf0082443f378f80f8851a": "\\angle DFE", "419cfed52ab5bc2a4b07e189bfec3df2": "\\omega\\in U", "419d0ba1b2edd87642c010746f6c1b20": "D_n(x,\\alpha) = xD_{n-1}(x,\\alpha)-\\alpha D_{n-2}(x,\\alpha) \\, ", "419d4bbcd6fea39db26bda978568df46": "1/(2\\mathcal U)", "419d6d8adc38fc7658448c0387d11bf9": "\\Phi(s,\\tilde{L})", "419d85054b5aaeab065e427da3b16a63": "E_8 \\times E_8", "419d874a81c1168a4885ee284389679a": "\\kappa_1", "419dd4424edee9a47016c609b0bbd5e8": "I_{Na}(t)=\\bar{g}_{Na} m(V_m)^3h(V_m)(V_m-E_{Na}),", "419df2067f4ba052349e7d6192b5e22d": "P = 0\\,", "419e08cf9d90eb3ec77f8bbe79a3e366": "I(rain;dark;cloud)", "419ed991964e9252f90218b1d6b7fcb9": "{(x \\ne y \\and Bxyz \\and Bx'y'z' \\and xy \\equiv x'y' \\and yz \\equiv y'z' \\and xu \\equiv x'u' \\and yu \\equiv y'u')} \\rightarrow zu \\equiv z'u'.", "419efa87c8aaffec8b21e5adcd4d44e8": "g_{k+1}(x) =|f(x)|e^{i \\theta'_{k+1}(x)}", "419f8b414a498e6306e2b3ca3edc3082": "\\lim_{n\\rightarrow\\infty} \\frac{a_n}{a_{n-1}^2}=1.", "419fc6af438f2a135908a10a0a19b07c": "H(p_1, \\cdots, p_n ) = - \\sum_i p_i \\log_2 p_i", "419fd5f68dbd0814ba151277af0986f0": "\\int_X \\! f^-(x) \\, d\\nu (x) < \\infty, ", "41a00464ae3432d1f6c362308f868f99": "-i\\hbar \\nabla", "41a007fcac1d7d8d672e7f5ec4005848": "\\Lambda_n(T)=\\max_{x\\in[a,b]} \\lambda_n(x) ", "41a01808229f413c8c9ba8ccf38a44e8": "A \\in K", "41a03acff4cc2675be1acaee84d5947a": "\\mathbb{E}(V\\mid\\theta)", "41a05b69c98663cfa5b67d3a4eef6427": " f = \\frac{64}{\\mathrm{Re}}", "41a0686235875afdfc435c3c940517f2": "d(mn)=d(m)\\times d(n)", "41a06c02d82022bb68edb2b7b4407674": "\\textstyle h \\leq \\lambda l", "41a0a5e6258c710834360d6ad48a4448": "(a, b) + (c, d) = (a + c, b + d)\\,", "41a0ca20cd14731bc8ca58670d6d8bc3": "C_n X = F_n X / \\Sigma_n", "41a0f3bf2d1b35ff2639aae5471d5aed": "\\displaystyle \\tilde{u}\\ ", "41a0fb873d0efd03684bfe424797d0ba": " X = {T \\over S} = {\\sqrt{5} / 2 \\over 8 / 5^{5/4}} = {5^{1/2} \\cdot 5^{5/4} \\over 8 \\cdot 2} = {5^{7/4} \\over 16}. ", "41a1318d14e7ac1c751aa05153774ca1": "U_{k+1}=\\frac12\\left(\\gamma_kU_k+\\frac1{\\gamma_k}(U_k^*)^{-1}\\right)", "41a13b1717ace92774c1b8a0ce6b24d1": "y(t)=g_0(\\textbf{x},u)(u(t)-a),", "41a14458b387b40724abc360838afe80": "X_\\infty", "41a1730c2d3ed988ddd49987f23fbef1": "\\tfrac{a}{ab-1}=\\tfrac{1}{b}+\\tfrac{1}{b(ab-1)}.", "41a1d5278250badaf84a7c01dfe65603": "\\left(x^\\prime\\cos \\theta\\ -\\ y^\\prime\\sin \\theta,\\ x^\\prime\\sin \\theta\\ +\\ y^\\prime\\cos \\theta\\right)", "41a1eb3f744109a4d1f5656dd6f70f3d": "\n\\cdots \n~+~ \\frac{1}{3} \\gamma_{t x x } \\frac{\\partial^3 \\textbf{A} }{ \\partial x^2 \\partial t}\n~+~ \\frac{1}{3} \\gamma_{t y y } \\frac{\\partial^3 \\textbf{A} }{ \\partial y^2 \\partial t}\n~+~ \\frac{1}{3} \\gamma_{t t x } \\frac{\\partial^3 \\textbf{A} }{ \\partial t^2 \\partial x} + \\cdots\n", "41a239768b5859079e75d2313c4e76e4": "Z_k = \\ker \\partial_k", "41a2411fd061b2d3dc85a43812bb48e3": " \\sum_{i < j} f(|x_i-x_j|)", "41a2d1771489dd07ffd5238ab11acd5e": " \\bar{x} ", "41a305f51875749658639c657707467f": " Nq ", "41a332d9e28353ba8a4b3da8d7cc1c42": "f(r)=\\frac{-k}{r^{2}}", "41a33c2e3f163e19ce1eb8be9e85341d": "(y, p) \\in \\mathbb{R}^m \\times \\mathbb{R}^{mn}", "41a3c0c88ba384bf59b5af6f64b175d9": "q^k", "41a3c5d72fc085b89c4fd10156b59cc5": "V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} )", "41a45dbaedbb94e7928f779c389bb4ca": "\n\\begin{bmatrix} \n+3 & +10 & +3 \\\\\n 0 & 0 & 0 \\\\\n-3 & -10 & -3 \n\\end{bmatrix}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\begin{bmatrix} \n+3 & 0 & -3 \\\\\n+10 & 0 & -10 \\\\\n+3 & 0 & -3 \n\\end{bmatrix}\n", "41a50ed06aca89beb4994a913df4ae73": "R=r_\\infty e^{B/T}", "41a53ceda25ea58f5b10571a7ae3531a": "P = 1 - \\left[1 - \\frac{L}{G}\\right]^N.", "41a54f465d3d21c0ddd919ff5a497075": "R[x]/(x^n)", "41a569d588c8bdaaf53185ef7507a2da": " - k x = m \\frac{d^2 x}{dt^2}. \\,", "41a5d0f4c7590a604428a82b9e480069": "\n\\frac{1}{2} W^2NcC_y = 4\\pi U_{\\infty}(1 - a)\\times\\Omega a'r^2\n", "41a5f246fa38a9c3e6de5f03223b64b2": "\n \\begin{array}{ccc}\n (e^r)^p & = & 2 \\\\\n e^{rp} & = & 2 \\\\\n \\ln e^{rp} & = & \\ln 2 \\\\\n rp & = & \\ln 2 \\\\\n p & = & \\frac{\\ln 2}{r} \\\\\n & & \\\\\n p & \\approx & \\frac{0.693147}{r}\n \\end{array}\n", "41a60a8b0050ab7223c384f691adee0c": "\\Omega(T)", "41a6434f1731e65c7e6c89cabc57ea56": "R_\\mathrm{m} = \\mathcal{M}/\\Phi_B", "41a6705b6bdc4ff606ad51284725306f": "|\\mathsf{Pad}(M_{1})| = |\\mathsf{Pad}(M_{2})|.", "41a6db9ead6cba1001a555c35d013aa2": "u:= e_1 e_2 e_n", "41a6e50ed6b1e3f2d9b3ada0d45a9f0b": "\\left|\\int_0^T e^{-x t}\\phi(t)\\, dt\\right| < \\infty", "41a792cb6780a8ec19cb27f34382d266": " V^\\alpha {}_{;\\beta}", "41a7a768a27732caa99a91232c610727": "a^n b^{n-i}", "41a7c4cc56a5c003d71cddecb0eb1e15": "e^{(c+o(1)) (\\ln n)^\\alpha (\\ln \\ln n)^{1-\\alpha}}", "41a7d88abdddba30329055b36a118979": "M>1", "41a80b166f2d9945f9d4dec2e6be5fa3": "S=s T \\;", "41a83526f06d935a8f5f2c7db3371936": "\\textstyle(x\\pm1, y\\mp1, z\\pm1)", "41a83dbd375c133032aeb46160fadc06": "e_j=1", "41a86ba51bcfd83e11cc6b6a955d94ae": "\\frac{1}{2\\pi i }\\oint_C G(x,y;\\lambda) d \\lambda = -\\sum_{i=1}^n \\langle x, e_i \\rangle \\langle f_i , y\\rangle = -\\langle x, y\\rangle = -\\delta (x-y). ", "41a8731e31e4f993829c1bc019f59e3f": "\n\\begin{align}\n B_0 &= b\\bigg(1+n+\\frac{5}{4}n^2+\\frac{5}{4}n^3 \\bigg),\n \\qquad\nB_4 = b\\bigg(\\frac{15}{16}n^2+\\frac{15}{16}n^3 \\bigg),\\\\\n B_2 &= - b\\bigg(\\frac{3}{2}n+\\frac{3}{2}n^2+\\frac{21}{16}n^3 \\bigg),\n \\qquad\n B_6 = - b\\bigg(\\frac{35}{48}n^3\\bigg).\n \\end{align}\n", "41a884927a2d4a7199478e0afc82f52a": "L(s,\\Delta)=\\sum_{n=1}^\\infty\\frac{a_n}{n^s}", "41a8f8181a67b2ba3c79bc4467c82672": "F(x_1 + \\Delta x) - F(x_1) = f(c) \\Delta x.", "41a93f39aafcbf203749a033d5ebbe77": " x(t) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty G_x(\\tau,f) e^{j2\\pi tf}\\,df \\,d\\tau", "41a9512a23cd1680c587686d2d6a7554": " p(\\textbf{x}_k\\mid \\textbf{Z}_{k-1}) = \\int p(\\textbf{x}_k \\mid \\textbf{x}_{k-1}) p(\\textbf{x}_{k-1} \\mid \\textbf{Z}_{k-1} ) \\, d\\textbf{x}_{k-1} ", "41a958f4c4fbbe17a843eb5336e25716": " v_1(\\vec r)-v_2(\\vec r) = const", "41a9c5d7fbb2d2268b5e873126904e1a": "2.8461", "41a9c6188f0fc006b4dff7f6935f9503": "\\pi^{\\#}: k[U] \\to k[\\pi^{-1}(U)]^G", "41a9d4eb925425e9a1fd26ba9d835010": "l=1\\,", "41aa052658fc8fbd3e4b96291974dc56": "\nG_{\\mu\\nu}+\\Lambda g_{\\mu\\nu}=0 \n", "41aa3b9ccb8b2c2cdd915dbfe3d298b2": " \\sum_{i=1}^k \\ln \\Gamma(\\eta_i+1) - \\ln \\Gamma\\left(\\sum_{i=1}^k\\Big(\\eta_i+1\\Big)\\right)\n", "41aa40a95ee5d4602b0eb7d16d256329": "\\! \\{(x_i, y_i)\\}_{i=1}^n,", "41aa46a13da5965fa7abfc84e1370880": "P_{\\mathcal{C}}", "41aaa8ec7c902013b006aec8306e7af3": "\\frac{\\operatorname{d}}{\\operatorname{d}x} \\left( \\rho \\frac{v^2}{2} + p \\right) =0", "41aab433b0ef53ff7e78c28838b925e7": "\\exist^{\\ge n}", "41aab95f89fe7f02651f9001cddad31b": "\\Pi = -\\tau_{ij}^{r} \\bar{S_{ij}}", "41ab539d8b689c38e381fb2741227954": "\\epsilon = 2^{-cm}", "41abdce0ffde2d55009cda6ed1462224": " G_{ab} = \\frac12 h_{ab} h^{cd} G_{cd} ", "41abeb33c6a8409e9c5e579469f1094c": "W(abcd;ef)=\\Delta(a,b,e)\\Delta(c,d,e)\\Delta(a,c,f)\\Delta(b,d,f)w(abcd;ef)\n", "41abecfd0cf3e3fe5a038cbe3fe672ef": "RC\\ln(2)", "41abed6cf526a03ac620e1838d50248d": " r_\\mathrm{corr} = r \\left[ 1 - \\frac{ 2 }{ n^2 m_x } \\left( \\frac{ 1 }{ m_x } - \\frac{ s_{ xy } }{ m_x m_y } \\right) \\left( 1 + \\frac{ 13 }{ 2n } + \\frac{ 8 }{ n m_x } \\right) \\right]", "41abfc2ee98eac57794afd332d579c10": "P^3 + 2bP^2 + (b^2 - 4d)P - c^2\\,", "41ac1af7812fffea8103d99cf8638d24": "f(t)=(f_1(t),f_2(t),\\ldots,f_n(t))", "41acffb1679d2352a7653691f9c0a100": "h(t) = \\begin{cases}\n 1-\\beta+4\\dfrac{\\beta}{\\pi},\n & t = 0 \\\\\n\n\\dfrac{\\beta}{\\sqrt{2}}\n\\left[\n\\left(1+\\dfrac{2}{\\pi}\\right)\\sin\\left(\\dfrac{\\pi}{4\\beta}\\right) +\n\\left(1-\\dfrac{2}{\\pi}\\right)\\cos\\left(\\dfrac{\\pi}{4\\beta}\\right)\n\\right],\n & t = \\pm \\dfrac{T_s}{4\\beta} \\\\\n\n\\dfrac{\\sin\\left[\\pi \\dfrac{t}{T_s}\\left(1-\\beta\\right)\\right] + 4\\beta\\dfrac{t}{T_s}\\cos\\left[\\pi\\dfrac{t}{T_s}\\left(1+\\beta\\right)\\right]}{\\pi \\dfrac{t}{T_s}\\left[1-\\left(4\\beta\\dfrac{t}{T_s} \\right)^2 \\right]},\n & \\mbox{otherwise}\n\\end{cases}", "41ad023d676a0ddb997cda57332f738f": "\\Psi [A]", "41ad3149a624eed4c878410280985d6b": "F^g = \\sqrt{\\theta^{g}}\\mathbb{I}+[1-\\sqrt{\\theta^{g}}]n_{0}\\otimes n_{0} \\,", "41ad426acbe6c695abc69d67d013a6b4": "x \\in [0; 1]\\!", "41ae05a84322a3de9bf22b0eb3fa8c6f": "\\displaystyle{\\tau(a,bc)=\\tau(ba,c),}", "41ae110a7a4e1eeb02f99841e1ac5bee": "\\scriptstyle X_L\\,", "41ae230e890004b5d28f08d3694916c1": "{{f}_{a}}(t)", "41ae9a5496fc66484f5fc8d7f2e13c5b": "B_2 \\in \\Sigma_2.", "41ae9e5aa681e6deb7e8443942bbd402": "E_a^j", "41aea28355f867d4b2f4eb181faa5d8d": "({v_0+v_i})10^{b_1E_i}", "41aec51ddd26daa17a27eec84a714a67": "1.25 + 2.50 =", "41aecdacf278b6a757b6ba2214db2e8a": "O(e^{-as}).", "41af03122550594e585239d25fd1b0e8": "\\bar{\\rho} = \\rho_A / \\Delta z,", "41af06866e2bc354dab8fc6872a1704c": "\\#X(k)=\\sum_i (-1)^i \\mathop{\\rm tr} F_q| H^i_c(\\bar X,{\\Bbb Q}_\\ell).", "41af8abe810ecb49636091588c980341": "\\log(1+y) = y - \\tfrac{1}{2}y^2 + \\tfrac{1}{3}y^3 - \\tfrac{1}{4}y^4 + ...", "41af9fd8ace77b1eb5e032cca4ad18fa": "H_3(a, b) = a^{b}\\,\\!,", "41afb3b4564c214881adac6525057521": "C \\to A \\times C", "41afdf1e57cc011850f5e6c7ac987e95": "e_j^{p} = 1 = \\omega^{p} \\,", "41b043ad650c3a051783058b29e88744": " m \\dot {\\mathbf{v} } = \\mathbf{F}_\\mathrm{rad} + \\mathbf{F}_\\mathrm{ext} = m t_0 \\ddot { \\mathbf{{v}}} + \\mathbf{F}_\\mathrm{ext} .", "41b09085070a774898b8e4aadbd52483": "{U_i}", "41b0d8331ce0bd796d9b1c7ae8556de5": "y[n]=\\pm 1", "41b0dea6f47c36d9a13738847cb6373f": "P^{(\\pm)} = {1 \\over 2} (1 \\mp i *).", "41b101c9fcc1f4d1ab90d654c068357c": " f(x,y) = \\frac{1}{2\\pi \\sigma_x \\sigma_y \\sqrt{1-\\rho^2}} \\exp\\left[ -\\frac{1}{2(1-\\rho^2)} \\left(\\frac{(x-\\mu_x)^2}{\\sigma_x^2} - \\frac{2\\rho(x-\\mu_x)(y-\\mu_y)}{\\sigma_x\\sigma_y} + \\frac{(y-\\mu_y)^2}{\\sigma_y^2}\\right) \\right] ", "41b17199d0c9ca6cdcd4a8ff3ea2a742": "1\\or 1=1", "41b239eb1b03fe9f25cca924af3adbef": "M=m", "41b24f24b487864c6ac71f5c1e923e8d": "\n\\langle A \\rangle_{\\text{mc}} = \\frac{1}{\\mathcal{N}} \\sum_{\\alpha'=1}^{\\mathcal{N}}A_{\\alpha' \\alpha'} \\approx \\frac{1}{\\mathcal{N}} \\sum_{\\alpha'=1}^{\\mathcal{N}}A = A.\n", "41b253c98af2e74971c3a318f81c9e51": "\\{\\phi , \\lnot \\phi \\}", "41b2563d9ede33eb738aa611da5fa1e1": " D_k ", "41b2bb74e9a2b88d1517591b7604cbdb": "\\displaystyle \\omega_{m,n}(x) = \\frac{e^{-x+\\pi i (m/2-n)}}{\\Gamma(1+n-m/2)}U(m/2-n,1+m,x).", "41b2d4d93ea3a42b0731b18405a28099": "f(z,w)=z^2+w^3", "41b32ae095f0845d782d4d3508740c82": " A_{2N} < \\pi < A_{2N} + D_{2N}.", "41b33024eecddb86895aa199136482f5": "\\sum_{i=0}^n", "41b351f775655abeddd6a5ac48eb5e17": " \\operatorname{build-param-lists}[g, D, V, T_8] \\and \\operatorname{build-param-lists}[m, D, V, K_8] ", "41b36b8260a56784e4502833c676de09": "m^2-N", "41b3c66c6ab7c811677dfc0e41be7944": " A_{i} = \\begin{matrix} {i_\\mathrm{o} \\over i_\\mathrm{i}} \\end{matrix} \\Big|_{R_{L}=0} ", "41b3da0809d0015b5fa4d460ce594421": " \\textbf{x}(t_0) = \\textbf{x}_0", "41b45074a777c8ca9a7eaa3ea0210b87": "\\exists x (P \\lor Q(x))", "41b5155a18f6ac77cd10a0fc82ce51de": "T=R \\cup S", "41b53d48ac80b0a43df75ceb0b30ae3d": "\\langle A\\rangle \\simeq \\frac{1}{N}\\sum_{i=1}^N A^{*}_{\\vec{r}_i}", "41b5436fe8416a32ca6cb289295c2926": "\\omega^i(\\mathrm{X}_j)=\\delta^i_j\\,", "41b5a111d99d877107d9a1ba6e0c4313": "\\left(a,b\\right)", "41b5eaf9f6798c09764149ff8709fdac": " f(z)=\\frac{z^{2}-5}{(z^2-1)(z^2+1)}=\\frac{z^{2}-5}{(z+1)(z-1)(z+i)(z-i)}", "41b60ea0ce4066867bf7af138576ac93": "n \\mapsto (-1)^n.", "41b6112a65826762f54cfdf0e532085d": "\n v_{i,jk} = A_{ij,k} ~;~~ v_{i,kj} = A_{ik,j} \n", "41b67df8c5d8bb5a8bab5d2f1f27c45e": "2^{n+1} - 1", "41b6a5d9c3a881336b4005ce6c1fe011": "g ", "41b6aa40a55401aac122741e73e10f72": "J=\\int_{0}^{\\infty}\\frac{x^{3}}{e^x - 1}\\,dx = \\frac{\\pi^{4}}{15}.", "41b6bdaa95dacfc67e994f3e66343ab7": "\\gamma = \\lim\\limits_{m \\to \\infty}\\sum_{k=1}^m{m \\choose k}\\frac{(-1)^k}{k}\\ln(\\Gamma(k+1)).", "41b6c23dae23f0be2a2d8df53f5c5029": "T = C(1-B)^{-1}A.", "41b6e942693ba22c8f653cee502040ab": "\\forall x,y,z,\\ldots \\in \\mathbb{N} : p \\vee \\neg p", "41b7231579a7d4a6ba37ef2a42daa3c5": "\\sum\\limits^g_{i=1}v_{2i-1,2i-1}v_{2i,2i} \\pmod 2.", "41b73f82646766d577d9f7e713bf921f": "\\begin{matrix} {4 \\choose 4} \\end{matrix}", "41b7c0f94fb913df5393a27db4ec339c": "2 \\le a \\le q-2.", "41b7d836e8c72c0ba6d52248f1672e6c": "\\theta = nS", "41b86966de4f9ff9640294547af1a4e7": "w(x) < \\alpha", "41b877a50f3d51aaf7abfee1f3a38631": "T^{(0,1)}\\mathbb{C}^n = \\mathrm{span}\\left(\\frac{\\partial}{\\partial \\bar{z}_1},\\dots,\\frac{\\partial}{\\partial \\bar{z}_n}\\right).", "41b8b5b6bac3e488635c79eee1447356": "\n\\begin{align}\n \\omega^2 =& \\left( gk\\, \\tanh\\, kh \\right)\\;\n \\left\\{\n 1 \n + \\frac{9 - 10\\, \\sigma^2 + 9\\, \\sigma^4}{8\\, \\sigma^4}\\, ( ka )^2\n \\right\\}\n \\\\\n &+ \\mathcal{O}\\left( (ka)^4 \\right),\n \\qquad \\text{with}\n \\\\\n \\sigma =& \\tanh\\, kh.\n\\end{align}\n", "41b8b7a86ea20380d100d4d9c9c3ff30": "while (C_k \\neq \\emptyset)", "41b93d74f13c1257dd187e2ac93690c5": "\\Phi_2(s) = \\frac{\\sqrt{2 \\alpha_2}} {(s+\\alpha_2)} \\cdot \\frac{(s-\\alpha_1)}{(s+\\alpha_1)}", "41b943ef186ff0f53229f142607f94ce": "i\\in \\{1,\\dots,n\\}\\, ,", "41b953c630e560f68dc90e0a8ef04ec9": " d(.,.) ", "41b97d4a3de2ea2285130b32bc6bb669": "(1-x+x^2)^{xy}", "41b9c09dd65d9c08ed70d6d3a303abe9": "|\\Phi^+\\rangle_{AC} \\otimes (\\alpha |0\\rangle_B + \\beta|1\\rangle_B)", "41b9ce3c13b9837a9bd1fd5fcf5f43a2": "\\frac{\\partial f_k}{\\partial x_j}", "41b9f28ddf07d10a0663142ab3166f20": "y_t = y_0 + \\sum_{i=1}^t u_i + a_0t ", "41ba07917087cd51b3d51fe43ea8451c": "\\displaystyle \\frac{2(-i)^n T_n (\\nu) \\operatorname{rect} \\left(\\displaystyle \\frac{\\nu}{2} \\right)}{\\sqrt{1 - \\nu^2}} ", "41ba91c8c0ed4326e708a7aeb91a700d": "w(t)", "41ba938ffd3d385c2d2b0744956993f9": "\\hat{\\epsilon}_i = y_i - \\hat{y}_i, (i = 1,\\dots, n)", "41baa26212044a953938e95afc4ca7f7": " \\theta = \\cos^{-1}\\left( {\\varphi \\over 2}\\right) = {\\pi \\over 5} = 36^\\circ. ", "41bae0cbcdc28efd251bef6e98e174ae": " \\nabla\\cdot(\\bold{A}{B}) = (\\nabla\\cdot\\bold{A}){B} + \\bold{A}\\cdot(\\nabla{B}) \\Rightarrow (\\nabla\\cdot\\bold{A}){B} = \\nabla\\cdot(\\bold{A}{B}) - \\bold{A}\\cdot(\\nabla{B})", "41bb3524cdbf47aa07aa96545c65b79d": "U(KM)", "41bb380331fadf572d54c5110111b955": "\\pm L\\frac {\\hat \\sigma}{\\sqrt n}", "41bb6a0191bd895fbef8561afb59f9d0": " \\mathrm{nDCG_{p}} = \\frac{DCG_{p}}{IDCG{p}}. ", "41bbf7f0306c9a88f13a6aba6a747244": " = R_\\mathrm{S}+r_{\\pi} +(\\beta+1)R_\\mathrm{E} \\ . ", "41bc0cb6a228c6343186bd830cf2ccb7": "\\vec{\\ell} = \\vec{e}_0 - \\vec{e}_1", "41bc30668fb678fbb8111ded52629f2f": "\\xi = \\frac{1}{3}\\left(\\sqrt[3]{17+3\\sqrt{33}} - \\sqrt[3]{-17+3\\sqrt{33}} - 1\\right)", "41bc472ffb9fa29e65f2f9bbdfb00c88": "\\sum_{i=0}^n{(-1)^i}.", "41bccc1a872e8c24a0fd4851c129cfc1": "\\scriptstyle i \\,=\\, 0,\\, \\ldots,\\, N", "41bd7a046446ab6359e9a20e187688ae": " R_{ij} \\ \\equiv\\ \\rho \\overline{ u'_i u'_j} ", "41bd7c8d042fa8eeb5c4bb1511e37362": "\\mathfrak{sl}(2,\\mathbb C)\\times\\mathfrak{sl}(2,\\mathbb C)\\times\\mathfrak{sl}(2,\\mathbb C)", "41bdc18f1657fb4853512e2ad9b5e766": "\\zeta(q)", "41bdc961bc1914a2f394aaeaa5b37704": "P_{t_0}", "41be2b0c2e441b964fa01420fe9a53f4": "\\Phi = \\mathcal{F}_t \\mathcal{F}^{-1}_f \\Pi", "41be4c3b82d00a5fbb63edec55329c6e": "\\frac1{f(x)}=\\frac1{f(x)-f(a)}=\\frac{x-a}{f(x)-f(a)}\\cdot\\frac{-1}{a(1-x/a)}\n \\approx\\frac{-1}{af'(a)}\\cdot\\sum_{k=0}^\\infty\\left(\\frac{x}{a}\\right)^k.\n", "41be5cb2674dd26416590151ac8d4b36": "H(u)\\in \\mathcal{D}'_{L^p}", "41be7ea775747d1248deebabf865e7a5": "\na=xy \n", "41be8266946509300440718b203173a0": " \\tilde \\nu_{J^{\\prime}\\leftrightarrow J^{\\prime\\prime},K} = F\\left( J^{\\prime},K \\right) - F\\left( J^{\\prime\\prime},K \\right) \n= 2 \\tilde B \\left( J^{\\prime\\prime} + 1 \\right) \n\\qquad J^{\\prime\\prime} = 0,1,2,...", "41bea9ec1381557f3c9952f53af3a0a1": "\\mathcal{B}(x) = \\{ B_{1/n}(x) ; n \\in \\mathbb N^* \\}", "41beb9bc114f00cf75b7af239ff26907": "\\sigma^2 \\,", "41bed6a85a5ba57107fc715b3198a450": "g(y) \\leq f(x)", "41bfff2a34ce38ff2e4fe846ef7511d0": "A = (a_{ij}) ", "41c0ba1be497139c69794588511d45dc": "1\\over k^2", "41c0ee4305973a0ad8c0ea74703f0178": "P_{\\mathrm{error}\\ 1\\to \\mathrm{any}} \\le M^\\rho \\sum_{x_1^n} Q(x_1^n) \\left(\\frac{p(y_1^n|x_2^n)}{p(y_1^n|x_1^n(1))}\\right)^{s\\rho}. ", "41c10fec9f319ccf50f0c209142eea61": "g: Y \\to \\mathbb{R} \\cup \\{+\\infty\\}", "41c13bb1949b44d611f6a97aa170a147": "\\displaystyle{\\mu v -\\mu_n v_n= \\mu(v-v_n) +(\\mu-\\mu_n)v_n.}", "41c14a4605945c01b2bb2de1475b4e41": "[a(f),a(g)]=[a^\\dagger(f),a^\\dagger(g)]=0", "41c188cf618304af8fe9c39bbf2d1323": "{\\mathbf R}_p\\,", "41c1a277e6ecf165866b6648975c449b": "cosk_n(K) := i^! i_* K.", "41c1af0627a55a4dc9e604683ce3ff75": "\\log(xy) = \\log(x) + \\log(y)", "41c1dea393b4c41d8847acbb0036c804": " \\langle \\phi \\vert \\psi\\rangle = \\int_X \\phi^*(x;\\theta) \\psi(x;\\theta) dx ", "41c22410cebbd7fbcca5407b0cc84ca9": "\\int \\operatorname{artanh} \\, x \\, dx = x \\, \\operatorname{artanh} \\, x + \\frac{\\ln\\left(\\,1-x^2\\right)}{2} + C , \\text{ for } \\vert x \\vert < 1 ", "41c23dc7a57765d023e7b2069656fd09": "2.4682", "41c2545b8a21e0380f0ef0a03dfada00": " \\land , \\lor ", "41c275bf34d7faea968b872166cdd51b": "\\psi(x + 1) = \\psi(x) + \\frac{1}{x}.", "41c282db9148d9fd0dc28288b0d32a27": "t^a(d,n):", "41c2f2136110516f7d332adc5041b0fe": "f(X)", "41c38df71cc601c22ec0b909a937e4e8": "\\mathcal{B}(X, Y; Z) = B(X, Y; Z)", "41c391db0292c8d8248616c5769d1828": "\\bold{x} = p + t\\bold{A}", "41c39d4e25741f9e1c4895d565e125bb": "\\ell/p", "41c3ab8ebc1fe1af448721146f308590": "M_{+\\infty}(x_1,\\dots,x_n) = \\lim_{p\\to\\infty} M_p(x_1,\\dots,x_n) = \\max \\{x_1,\\dots,x_n\\}", "41c3d396799ce08baaf5d62db25205a6": " \\lambda = \\sum_{i=1}^R p_i^2", "41c3dd8e533b47ca436632c5a257f341": "ds^2 = -\\left(1-\\frac{r^2}{\\alpha^2}\\right)dt^2 + \\left(1-\\frac{r^2}{\\alpha^2}\\right)^{-1}dr^2 + r^2 d\\Omega_{n-2}^2.", "41c3e5fb662cfaa36ed61c2e649e6c79": "\\tilde{\\Omega}^*", "41c3e71354f8adacf369a7c0ec4a6346": "f(\\boldsymbol{S}) = f_1(\\boldsymbol{F}_2(\\boldsymbol{S}))", "41c43e2351e78fedcbe57aa0cd1c03df": " u(x) \\, ", "41c45692e1ea2a4e975a7c63787505b4": "L_m<0.", "41c4604aa3a064a85e1924ba6c8b6b34": "\\mathbb{Q} \\left( \\zeta_p \\right)", "41c46425c7cdf3f95767cb56f6a0f2da": "\\chi = \\frac{n \\mu_0 \\mu^2}{k_BT}", "41c4af26b8b30005441e549cfd76d4c7": "D_a V_i^b = \\partial_a V_i^b - \\Gamma_{a \\;\\; i}^{\\;\\; j} V_j^b + \\Gamma^b_{ac} V_i^c", "41c535f5b08cb95e1a1029f3a81934f6": "T:l_p\\to c_0", "41c553fdd2d4d81e7ad039c511afa4e0": "\\alpha_J=\\frac{2\\mu}{r_0}-\\mathbf{v}_0\\cdot\\mathbf{v}_0-2V_0", "41c576c78943e1bf132014e4d07cff45": "N = t_1^2 - D u_1^2", "41c5bfe922324bad1512f5e114a8cd25": "(\\phi_n), (\\psi_n) \\in \\Delta", "41c5d55fda69c4a119d0ecc4113493e4": " \\gamma^\\mu\\hat{P}_\\mu = i\\hbar \\gamma^\\mu\\partial_\\mu = \\hat{P}\\!\\!\\!\\!/\\ = i\\hbar\\partial\\!\\!\\!/\\ ", "41c617352a669441775fcdfe2a4c3bb6": "\\textstyle{3\\frac{\\log(\\varphi)}{\\log (1+\\sqrt{2})}}", "41c6411ad7a4052ba20a43e66ed67142": "\\frac{\\partial \\mathbf{U}}{\\partial X_{ij}},", "41c6921a240ab6993064b908b798f055": "T\\colon \\Sigma\\times X \\to X", "41c69a1c5d69e0e43a16d7dc41f6f73a": " V \\approx \\frac {c}{n} + v \\left(1 - \\frac{1}{n^2} \\right) ", "41c6cce53fc2f6239c4a9295489c52e9": "k_2=2.148 v^{0.878} H^{-1.48}", "41c6d52c3ad33c74c14538bf0e053644": "(\\nabla I)'(\\nabla I)", "41c70d20fe29690a83d526f1df6c1456": "X_E", "41c7419604992f2b68615919940b16fa": "n_\\mathrm{A} = n_\\mathrm{A*} \\; (R_\\mathrm{A*B}=R_\\mathrm{AB} \\and R_\\mathrm{A*} = R_\\mathrm{A})", "41c78786d68ff9f9a7b7d7e74d13e98b": "a_k(\\mathbf{y}, t)=mod(\\phi_k(\\mathbf{y}, t)+\\pi,2\\pi)-\\pi", "41c7aa13b394c6fda93a3f9d1b1acbe3": "\\begin{bmatrix} -\\frac12\\mathbf{V}^{-1} \\\\[5pt] \\dfrac{n-p-1}{2} \\end{bmatrix}", "41c7b4623741cc8b20ee441382e2d46e": "\\text{CLV} = \\text{GC} \\cdot \\sum_{i=0}^n \\frac{r^i}{(1+d)^i} - \\text{M} \\cdot \\sum_{i=1}^n \\frac{r^{i-1}}{(1+d)^{i-0.5}}", "41c7d308ad5e2ed67e55ab97812a10a7": "\\exp\\left(\\frac{2\\pi i}{\\lambda} \\frac{r^2}{2f}\\right)", "41c8309032ba4371de489cfe5e8e38fc": "\\delta \\mathbf{r}\\,", "41c86902e6faa682678b807b4d7049f9": "\\omega \\,=\\, e^{2\\pi i/n} \\,=\\, \\cos\\left(\\frac{2\\pi}{n}\\right) + i\\sin\\left(\\frac{2\\pi}{n}\\right)", "41c88ef8b9190ab50c3fd8e1a4a757da": "q \\text{ and } \\beta", "41c8952e2c5dc5f70b5f32386ce97404": "d p_s(t) = -p_s(t) h(t) \\, dt.", "41c913bd573b28a1d757b17aed453b4e": " \\text{ } (5) \\text{ } V_n \\equiv P \\pmod {n} . ", "41c9d76d1e2d636fabe5b0ebf50e6416": "K_{sp} = x^x y^y \\frac{{(-1)}^{(x+y)}{(N_{AxBy(\\Delta)})}^{(x+y)}}{V^{(x+y)}}\\,", "41c9d9feabba840001b7d3f7c29eef69": " H= -{\\nabla^2\\over 2} \\, .", "41ca1a634efcd33f581330dffc8a0533": " \\prod_{p b=a} g_b = g_a \\ . ", "41ca38e939ee63f3490110898a5880cb": "T_\\text{load}", "41ca6c045a846aa0fc24228429d9eb9d": " \\approx ", "41ca746d30fc53ef240f1ab58185c38e": "N^{-1}D", "41cac6e919a82211ea799103b81339af": "v_{e}(x)", "41cb089c76b58ec793b502c35ca108e4": "^\\mathcal{(D)}", "41cb256c319cf2af6cfb24831ccf3f7f": "\\int\\!\\!\\!\\int\\!\\!\\!\\int_W y\\, dV = \\int\\!\\!\\!\\int\\!\\!\\!\\int_W z\\, dV = 0.", "41cb28f93023137532bd227769499b48": "1,277 \\cdot 20 \\cdot 253 = 6,461,620\\,", "41cb34123c21a8f1568cec2c3cc793c6": "Q = \\frac{N}{D} \\frac{F_1}{F_1} \\frac{F_2}{F_2} \\frac{F_{\\ldots}}{F_{\\ldots}}.", "41cb7a9f46a228e89cb8df9238233069": "R=\\mathrm{End}(_D V)\\,", "41cba883d0121d29e36f1475b74fc860": " { \\partial^2 p \\over \\partial x ^2 } - {1 \\over c^2} { \\partial^2 p \\over \\partial t ^2 } = 0 ", "41cbc0e388a4992e3cac7a15f6cfd85b": "[g_{ij}] = \\begin{pmatrix}\nh_1^2 & 0 & 0\\\\\n0 & h_2^2 & 0\\\\\n0 & 0 & h_3^2\n\\end{pmatrix}", "41cbc6310c398d3f87f9aa2f1815fe23": "\\mathrm{diam}(V)^d \\le C \\, \\lambda_d(V)", "41cbff4d16c05873efbc04243df10bec": "\\delta W = P\\,\\mathrm{d}V,", "41cc0153175d3a54ed19d592e5512943": "\nx^{2} +\n\\left( y - a \\cot \\sigma \\right)^{2} = \\frac{a^{2}}{\\sin^{2} \\sigma}\n", "41cc3beface906d3ef1f8a5b75d07562": "f_e(x_e)", "41cc533af49f184a11454ee7a2ad5658": "\\Delta \\Omega", "41cc7f912ec93c11b581bc1f27b1e8d2": "\\vdash p \\to q\\!", "41cceaaf7d7f654e85cb51b477c7d6d7": " \\frac{\\pi h^3}{6}. ", "41cd03e8ebb99345eac61db891a61280": "S = \\cfrac{BH^2}{6} - \\cfrac{bh^3}{6H}", "41cd347516668e7302e2eb3698d176a8": "\\epsilon (r)", "41cd52a5afb5e65fab0133470ab4414b": "\\frac{(a*(a+1))*(a+2)}{1*(2*3)}", "41cda7a98a4c1f95828ab349afcb2eb3": "\\rho(\\boldsymbol{r}) = F[\\rho] = \\int \\rho(\\boldsymbol{r}') \\delta(\\boldsymbol{r}-\\boldsymbol{r}')\\, d\\boldsymbol{r}'.", "41cdc00729df89ad61b4ed7d492e18b2": " (a,f,b)(n,i,e)(d,s,\\#)(\\#,h,\\#)", "41cdc228257fb9f7c9fea8295577f716": "\\Gamma_{ij} = \\left\\{\\begin{matrix} \n-1, & \\mbox{if } i \\ne j & \\mbox{and }R_{ij} \\le r_c \\\\ \n0, & \\mbox{if } i \\ne j & \\mbox{and }R_{ij} > r_c \\\\\n-\\sum_{j,j \\ne i}^{N} \\Gamma_{ij}, & \\mbox{if } i = j \\end{matrix}\\right.", "41cdc5e1e08a6a6e73b3c56c774b321c": " a - 2 \\cdot b.", "41cddb42dbfb6200600a9511283b2391": "\\sum_k Y_{ik} V_k(s) + Y_i^{\\text{sh}} V_i(s) = s\\frac{S_i^*}{V_i ^*(s^*)}", "41cdf04552e4309db8b71fe99a010dd2": " \\begin{align} \\hat{T}_x & = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2 }{\\partial x^2} \\\\\n\\hat{T}_y & = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2 }{\\partial y^2} \\\\\n\\hat{T}_z & = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2 }{\\partial z^2} \\\\\n\\end{align} ", "41ce988f3a91dce9bd832db987b8cfdc": "\\operatorname{F}(\\dots\\mid z)", "41cea5d53f6d56defc6bfaf5c2481d21": "\nP_{\\nu_b\\rightarrow\\nu_a}=P_{\\nu_b\\rightarrow\\nu_a}^{(0)}+P_{\\nu_b\\rightarrow\\nu_a}^{(1)}+P_{\\nu_b\\rightarrow\\nu_a}^{(2)}+\\cdots,\n", "41ceb115615fcde0cb85fd93c9c5419e": "\\sqrt{n_1 n_2/(n_1+n_2)}\\,g", "41cec20164112471b0f8c3d3fbbdd3a7": "\\exp(\\sum_{H \\subset G} x^{[G:H]}/[G:H])=\\sum_{n\\ge 0} \\frac{a_{G,n}}{n!}x^n,", "41cec27548a23e43dfad772a96116db6": " A = Q R ", "41cee7cdbd786b9c3ea8e6c1bb2b8c82": "O(N^2L)", "41cf65e9948bdff9f3e1d8ff013a0f2b": "g(x,y) \\rightarrow x, \\; g(x,y) \\rightarrow y", "41cf7bddaf980bae0408a24004227586": " V' = \\{ v_1 ,\\ldots, v_i \\} ", "41cf922e918141d085d48f85898c7144": " \\|x_1 - x_{-1}\\| \\ge t, \\quad \\|x_{\\varepsilon_1, \\ldots, \\varepsilon_k, 1} - x_{\\varepsilon_1, \\ldots, \\varepsilon_k, -1}\\| \\ge t, \\quad 1 \\le k < n.", "41cfef629beae500f9ce757e88c9089e": "2 \\times \\sqrt{2}", "41d050439a8f511c944f178b0a11f992": " r\\theta={\\frac {r \\sin \\theta }{\\cos \\theta\n }}-(1/3)\\,r\\,{\\frac { \\left(\\sin \\theta \\right) ^\n{3}}{ \\left(\\cos \\theta \\right) ^{3}}}+(1/5)\\,r\\,{\\frac {\n \\left(\\sin \\theta \\right) ^{5}}{ \\left(\\cos \n\\theta \\right) ^{5}}}-(1/7)\\,r\\,{\\frac { \\left(\\sin \\theta\n \\right) ^{7}}{ \\left(\\cos \\theta \\right) ^{\n7}}} + \\cdots", "41d081f5c63bc4e9390f7796f79331a6": "{\\mathbb P}\\biggl(\\bigcup_{i=_1}^{n+1} A_i\\biggr) \\le {\\mathbb P}\\biggl(\\bigcup_{i=_1}^n A_i\\biggr) + \\mathbb P(A_{n+1})", "41d0b1cfa7e9c16b58cce7b2f71d6565": "m_1+m_2", "41d0d7b598b0f91d8de355a8adf9e7cd": " \\mathbf{x}_{k+1} = \\mathbf{x}_k + \\alpha \\mathbf{p}_k ", "41d0d8c048a7adff739a71185a6f52d4": "T_A^1 \\longrightarrow^* T_A^2", "41d1f449b308e5b8419098873677c1f6": "(X_n,P_n)", "41d216e07a36c409bc7d6c0514df60cf": "\\{f_j\\}_{j=1}^r", "41d234a21224e7e6c5879e5270485a16": "\\ (1 0.", "41d9fefeaf849b1a39647c2b7763d0d3": "P_u", "41da006cdcad0935ade457acd8da0e53": "V_{DS} \\gg V_T", "41daa7904efdbe9c580f65b25f2ad9b4": "a \\mid b,\\, a \\mid c \\Rightarrow b=ja,\\, c=ka \\Rightarrow b-c=(j-k)a \\Rightarrow a \\mid (b-c)", "41dac26c6512e81244810cc832d0f275": "\\{X\\}_i", "41db424f5c02cc7c51f1f480735df0e4": "[a,t]", "41db5337d5bb1c0ee31a9293f21e1b28": "A_2B_1", "41db655c3086f3ec7cfe5d8be29cd259": "c_i(x)", "41db711e7e82d5128ee887ae4ac8be1b": "R_{\\alpha \\beta \\gamma}^{\\;\\;\\;\\;\\;\\; \\delta} V_\\delta = (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_\\gamma", "41db8a6bb9a3d1169561fe96a2b3fdc1": "P-E-\\frac{\\partial (\\int_0^\\infty \\! \\rho q\\,dz \\,)}{\\partial t}=-\\nabla \\cdot \\mathbf{F}\\!", "41dba35faadd4f96e24d2860d9f6628e": "f^{(n)}", "41dba44c60dae10ab31f214ac7a45fed": "\\mbox{Average point differential}=\\frac{\\mbox{Total points for}-\\mbox{Total points against}}{\\mbox{Total games played}}", "41dbe3f5e3bd28f882a43d0122f61186": " df/dk - (n+d)= 0 ", "41dbe4d7daeffb853f270dd58a9076ff": "L_\\phi\\;", "41dc08b3ad1fd08580b01cc219d17269": "\\mathbf{A} \\in \\mathbb{M}(2,2)", "41dca4e2ca7a1a89e65a927c162150b1": "T(x_i,y_i) = \\iint T(k_x,k_y) ~ e^{j((k_x/M) x_i + (k_y/M) y_i)} ~ dk_x \\, dk_y", "41dcf01ec70ceca53e9ec9fa9b49d699": " \\alpha_i^2 = \\beta^2 = I \\,, ", "41dd0534df9d2d8710c3e6910534d1dd": "\\operatorname{E}(A) = [\\operatorname{GL}(A),\\operatorname{GL}(A)]", "41dd23773235474430bc32d2e9268b69": "F'_1,\\dots,F'_t", "41dd288523f52fd16c730388bcd0f7b5": "OB\\% = \\frac{-1600}{227.1} \\times (14 + 2.5 - 6)", "41dd3eacef3107db27b48b7e39472223": " Q_{Fan} = Q_{Building}\\,\\!", "41dd4d1ccaa89a72f35b4be44021f758": "\\hbar=\\frac{h}{2 \\pi}", "41dd56f4641318378cbdcaaaf908f9aa": "O(\\text{min}(\\text{size}(t_1),\\text{size}(t_2)))", "41dd7754cef854cf3b9a188d9730639f": "258890850 = 15 \\,\\frac{75!}{5! \\, 70!}", "41dd9ca800865881bb42edd4aa10bd9c": "\\beta_1 = \\alpha_2 ", "41ddacae53c3fae52ca0a6cbbd71080d": "\\mu\\left(\\gamma \\cdot U \\cap U\\right)=0", "41de34907da08ffca81944c8655c6c86": "30^2 - 3^2 = 891", "41de3876e819b2897425299bb06a8f95": "\n\\biggl(\\sum_{i=1}^n a_i c_i\\biggr)\n\\biggl(\\sum_{j=1}^n b_j d_j\\biggr) = \n\\biggl(\\sum_{i=1}^n a_i d_i\\biggr)\n\\biggl(\\sum_{j=1}^n b_j c_j\\biggr) \n+ \\sum_{1\\le i < j \\le n} \n(a_i b_j - a_j b_i ) \n(c_i d_j - c_j d_i )\n", "41deca77eaeb0cb747123ada97512a2e": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{havercosin}(x) = \\frac{-\\sin{x}}{2}", "41dede49f800259199842ccc60cf6472": " \\varepsilon_i ", "41dee42e904f9881a40d74fbe8a30fd8": " \\left(\\mathbf{A} \\otimes \\mathbf{B}\\right)_{ij} = A_{1+\\lfloor\\frac{i-1}{p}\\rfloor,\\, 1+\\lfloor\\frac{j-1}{q}\\rfloor}B_{1+\\,(i-1)\\, \\text{mod}\\, p,\\, 1+\\,(j-1)\\, \\text{mod}\\, q} ", "41dfac9d35d0db84b1de5d6d6c7bf66b": "a_i = \\gamma_{ci} \\left [ {\\rm X}_i \\right ]/\\left [ {\\rm X}_i \\right ]^{\\ominus} \\,", "41dfbb1e22f24cfabfaf376323714b33": "\\displaystyle(a_1,...,a_m;q,p)_n=(a_1;q,p)_n\\cdots(a_m;q,p)_n", "41dfd87b606f4370214e6ca62d26347e": "\n \\begin{align}\n K_{\\rm I} & = \\frac{4P}{B}\\sqrt{\\frac{\\pi}{W}}\\left[1.6\\left(\\frac{a}{W}\\right)^{1/2} - 2.6\\left(\\frac{a}{W}\\right)^{3/2}\n + 12.3\\left(\\frac{a}{W}\\right)^{5/2} \\right.\\\\\n & \\qquad \\left.- 21.2\\left(\\frac{a}{W}\\right)^{7/2} + 21.8\\left(\\frac{a}{W}\\right)^{9/2} \\right]\n \\end{align}\n ", "41dfd9a81a3340ee8ca030e62fcf3651": "\\frac{dW}{d\\omega}\\sim \\frac{e^2}{4\\pi \\varepsilon_0c}\\left ( \\frac{\\omega \\rho}{c} \\right )^{1/3}", "41e02f298ab89d433fd122ce4e78525f": "\ne^{t\\,X} = 1 + t\\,X + \\frac{t^2\\,X^2}{2!} + \\frac{t^3\\,X^3}{3!} + \\cdots +\\frac{t^n\\,X^n}{n!} + \\cdots.\n", "41e0576d7e640121bff4451ce7351e06": "R (ESR)", "41e0a72eb9d5169b95c83595eda553d2": "\\neg (P \\wedge \\neg P)", "41e0c9b26f74e9abe6d4f24b507c07f2": "\\begin{align}h(a)=h(b)&\\iff f(a)-r\\,g(a)=f(b)-r\\,g(b)\\\\ &\\iff r\\,(g(b)-g(a))=f(b)-f(a)\\\\ &\\iff r=\\frac{f(b)-f(a)}{g(b)-g(a)}.\\end{align}", "41e1539c5a8f7943129754a4c5fa6ef6": "x_a=x_a(y_b,\\xi_j)\\,,\\; \\theta_i=\\theta_i(y_b,\\xi_j)\\;,", "41e17084e5f29f47756c503bc108b238": "x^{k+1}\\leftarrow x^k + \\alpha h_x", "41e1d920ed4bb7019581dcbb5fc45421": "\\Pr[\\xi(B_i) = k_i, 1 \\leq i \\leq n] = \\prod_i e^{-\\lambda \\|B_i\\|}\\frac{(\\lambda \\|B_i\\|)^{k_i}}{k_i!}.", "41e1f3766ad77ec049739eeaf76ebb1f": "\\bar x = \\sum_{i=1}^n x_i/n ", "41e247c1902063dc8b129bd3ed3656c6": "\n\\left ( \\alpha R, \\beta G, \\gamma B \\right ) \\rarr \\left ( \\frac{\\alpha R}{\\frac{\\alpha}{n} \\sum_i R }, \\frac{\\beta G}{\\frac{\\beta}{n} \\sum_i G }, \\frac{\\gamma B}{\\frac{\\gamma}{n} \\sum_i B } \\right )\n", "41e262b078e79f7abdc4eb69cd269b76": "\\frac{1}{m} + \\frac{\\ell}{p}", "41e27cf2b42d5c42785ca03d741467ce": " \\displaystyle{Au_n=\\mu_n u_n,}", "41e3130702248e1bb643284abbaa3827": "\\stackrel{\\mathrm{d}}{=}", "41e316c1fa1b2f436b820d1c03f8c14a": "m_a^2 + m_b^2 = 5m_c^2 = \\frac{5}{4}c^2.", "41e39f243befbf1dccbc38e43dfcad68": "7*2^{14}+1=114689 ", "41e42aad52626c8e59bf8cb5db3e3bad": " -ln(X) \\sim \\textrm{Exponential}(a)\\,", "41e42dcd08e8bcd69df9d819f50ac542": "\\iiint_D (x^2 + y^2 +z^2) \\, dx\\, dy\\, dz = \\iiint_T \\rho^2 \\ \\rho^2 \\sin \\theta \\, d\\rho\\, d\\theta\\, d\\phi,", "41e43dcd1498bee513cdfdf8a8f4c5f8": "D_a(z)", "41e46d8144dc18c2bfd33a20d632ee60": "w()", "41e47faa4deece61c4cb41fc8d9e1e70": "\\boldsymbol{\\mathsf{L}}^{\\mathrm{T}}=\\boldsymbol{\\mathsf{L}}^{-1}\\Rightarrow(\\boldsymbol{\\mathsf{L}}^{-1})_i{}^j=(\\boldsymbol{\\mathsf{L}}^{\\mathrm{T}})_i{}^j=(\\boldsymbol{\\mathsf{L}})^j{}_i=\\mathsf{L}^j{}_i", "41e4a77c051c3c453bd4ddc14c1e4d66": "P_1^2 P_2^0 \\or P_3^0 P_2^0", "41e4d0a17973251a949fd21c1223fed6": "c=s+1. \\, ", "41e4eeeeffc63436aa240249fc78b9a1": "\\frac{x}{2}", "41e50482a366cabc9c11297a4dbe682c": "\\scriptstyle \\mathcal{N}_2=\\{(0,0),(0,1),(1,0),(0,-1),(-1,0)\\} ", "41e5068a30e483bd1f18f9bcd1b188b9": "F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu \\,\\!", "41e531e38320c7fb3c64468d96526e6d": " R_{ a b } \\ \\stackrel{\\mathrm{def}}{=}\\ {R^{ s }}_{ a s b } ", "41e56bf1e5844273d6b6af2589b57ab7": "(\\forall a,b\\in M)\\ a + b = 0 \\implies a = b = 0 \\!", "41e584226130cb6a964adba31f5f3a46": "\\mbox{Total external virtual work} = \\int_{V} \\boldsymbol{\\epsilon}^{*T} \\boldsymbol{\\sigma} dV \\qquad \\mathrm{(d)} ", "41e59692852c9d8687091ce0e3a9cd4e": "\nS^{T}AS=\\begin{pmatrix}\n2 & 2 & 2 & 1 & 0 \\\\\n2 & 3 & 2 & 1 & 0 \\\\\n2 & 2 & 2 & 1 & 0 \\\\\n1 & 1 & 1 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix}.", "41e5bc4eed8b47c49abb7aeabf6d76fb": "\\begin{matrix} {4 \\choose 2}{3 \\choose 2}^2 \\end{matrix}", "41e5c02a4c9c17c40bf89d48b8b79f18": "Q_t \\cdot Ca_{O_2} = Q_s \\cdot Cv_{O_2} + (Q_t - Q_s) \\cdot Cc_{O_2}", "41e5dd34a6942c6ca860fcb6a0052e49": "k^a", "41e622d2a994cc1b6e7375207bc91c8b": " V_0 \\subsetneq V_1 \\subsetneq \\cdots \\subsetneq V_n\\ ", "41e62626d49cef5f8ecb2f913953579e": "\\tilde{k}^\\prime \\in \\tilde{K}^\\prime", "41e62bb8d321538120e9b40f1a60070e": " \\langle x_0 | \\psi \\rangle = \\int\\limits_R d x \\, \\langle x_0 | x \\rangle \\psi(x) = \\int\\limits_R d x \\, \\delta( x_0 - x ) \\psi(x) = \\psi(x_0) \\,.", "41e6385753210e37cab28be4f28a6661": "P_n=\\frac14\\binom{2n+2}{3}.", "41e671191bc0c8528869fbefc48ec0c0": "\\mathbf{}\\begin{bmatrix}\n\n4&-3&0&0 \\\\\n-3 & 10/3 &1&4/3 \\\\\n0 & 1 &5/3& -4/3 \\\\\n0 & 4/3 & -4/3&-1 \\end{bmatrix},", "41e6c60b0643d9922dd7145511c41ad3": "\nF^k_{\\mu\\nu} = \\partial_\\mu A^k_\\nu-\\partial_\\nu A^k_\\mu+g f^{klm}A^l_\\mu A^m_\\nu\n", "41e749fd79240ff098b7064215aae3e6": "\\sum_{k=0}^\\infty a_k = \\sum_{k=0}^\\infty(-1)^k(k+1),", "41e74d412fdbd8c204282d70d5ad977c": "m_1,\\ldots,m_r", "41e77579b929decc5184b8eedd2e05f1": "\\underset{x\\in \\Bbb{R}}{\\operatorname{arg\\,max}} (x(10-x)) = 5", "41e77db74c03845a860b75e7b521f36a": "\n\\begin{align}\n\\operatorname{cn}(x+y) & =\n{\\operatorname{cn}(x)\\;\\operatorname{cn}(y)\n- \\operatorname{sn}(x)\\;\\operatorname{sn}(y)\\;\\operatorname{dn}(x)\\;\\operatorname{dn}(y)\n\\over {1 - k^2 \\;\\operatorname{sn}^2 (x) \\;\\operatorname{sn}^2 (y)}}, \\\\[8pt]\n\\operatorname{sn}(x+y) & =\n{\\operatorname{sn}(x)\\;\\operatorname{cn}(y)\\;\\operatorname{dn}(y) +\n\\operatorname{sn}(y)\\;\\operatorname{cn}(x)\\;\\operatorname{dn}(x)\n\\over {1 - k^2 \\;\\operatorname{sn}^2 (x)\\; \\operatorname{sn}^2 (y)}}, \\\\[8pt]\n\\operatorname{dn}(x+y) & =\n{\\operatorname{dn}(x)\\;\\operatorname{dn}(y)\n- k^2 \\;\\operatorname{sn}(x)\\;\\operatorname{sn}(y)\\;\\operatorname{cn}(x)\\;\\operatorname{cn}(y)\n\\over {1 - k^2 \\;\\operatorname{sn}^2 (x)\\; \\operatorname{sn}^2 (y)}}.\n\\end{align}\n", "41e799d374bd8446ae15b6b9877e2912": "\\omega = \\sqrt{\\frac{g}{L}} \\left [ 1 + \\sum_{k=1}^\\infty \\frac{\\prod_{n=1}^k \\left ( 2n-1 \\right )}{\\prod_{n=1}^m \\left ( 2n \\right )} \\sin^{2n} \\Theta \\right ]\\,\\!", "41e7b2e91f00d1e2e7d0993dc74c26ec": " \\boldsymbol{\\mathsf{X}} \\cdot \\boldsymbol{\\mathsf{X}} = \\left ( c \\tau \\right )^2 \\,\\!", "41e813ba63c7549549e3c735bac17671": "l^a", "41e824d0f6f2b27dfbe4dc224a314d2e": "A \\in O_{/\\sim_{\\mathbb{F}_1}}, B\\in O_{/\\sim_{\\mathbb{F}_2}}, C\\in O_{/\\sim_{f}}", "41e82929bbe023796f15c1f59922b8e7": "\\text{Visibility}(\\text{ideal}) = \\frac{2\\sqrt{I_1I_2}}{I_1+I_2},", "41e8820b2059833362d1936788e264a5": "A\\mid B \\hbox{ and } A'\\subset A\\Rightarrow A'\\mid B.", "41e8b908e5377bca288dab37d9e6116a": "\\sigma_{x}", "41e8f5439c76f7c49473adbab97b384c": "\\displaystyle{\\overline{H_\\varepsilon f} = - H_\\varepsilon( \\overline{f}),}", "41e9274ec1866c12c012a196020bf8ea": "X[k] = \\sum_{N} x_N[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}},", "41e93e25411f0955aefb4b40771e8b02": "R_2 - R_1 = R \\left( 1 + \\frac{1}{2} \\left( \\frac{x_2^2 + y_2^2}{R^2} \\right) \\right) - R \\left( 1 + \\frac{1}{2} \\left( \\frac{x_1^2 + y_1^2}{R^2} \\right) \\right)", "41e9ab92d5bee6aeafb496ebaf72ca2a": "\\frac{d}{dt}\\langle \\mathbf{p}\\rangle = -\\langle \\nabla V \\rangle", "41e9d8d975b4b6510dd0951927b0b3db": "R_1R_3 = \\frac{R_aR_b^2R_c}{R_T^2}", "41e9fbbdcd4d906072e9d2bb75bc4365": "d < 19", "41ea0e204216040955546a7022ef0951": "\nm_\\text{H}=126\\,\\text{GeV} ,\n", "41ea10df76745ebccbc592e305d7c736": "\\theta (z^k) = k z^k,\\quad k=0,1,2,\\dots ", "41ea26c0480b9c66cc8c80650e109dce": "\\varepsilon_0 ", "41ea272e21f78b4fb9057d09f90fd55c": "x = a/c", "41ea6a434e1d1fed14b85e9d04426b36": " K(n)= e^{-(n^2-1)\\zeta^\\prime(-1)} \\cdot\nn^{\\frac{5}{12}}\\cdot(2\\pi)^{(n-1)/2}\\,=\\,\n(Ae^{-\\frac{1}{12}})^{n^2-1}\\cdot n^{\\frac{5}{12}}\\cdot (2\\pi)^{(n-1)/2}.", "41ea6bcc842d7a25fdf75faa552e9aa3": "\\mu \\sim 3 \\lambda f \\pm 100 \\mbox{GeV}", "41ea77afa2a2ae5bc2c7bec19aaf6f4c": "b_5", "41ea7fb24909280b67f008819ac4b62e": " H_i ", "41eb4ec1194402d369262919e3393115": "W^{-1} = w^{-1}W^{T}", "41eb5581ebf7954e1158fd9375b522c4": "Z_{P^n}", "41eb7b725742fe0394162f57b181cb1a": "\\frac{173205\\times117557-100000\\times161803}{200000}\\approx20906.", "41eb9125c1d5b48ca7cf3816d790f195": "D+h", "41ebad7e2da15f7d48c95fa44e9e3ee0": " \\langle \\varepsilon' | \\psi_N \\rangle = \\langle \\varepsilon' | \\left( \\frac{1}{\\|\\psi\\|}\\int d \\varepsilon | \\varepsilon \\rangle \\psi(\\varepsilon) \\right) = \\frac{1}{\\|\\psi\\|}\\int d \\varepsilon \\langle \\varepsilon' | \\varepsilon \\rangle \\psi(\\varepsilon) = \\frac{1}{\\|\\psi\\|}\\int d \\varepsilon \\delta( \\varepsilon' - \\varepsilon ) \\psi(\\varepsilon) = \\frac{\\psi(\\varepsilon')}{\\|\\psi\\|} \\,,", "41ebb374be2579773748a55be979d113": "\\lim_{x \\to -\\infty}{f(x)} = L", "41ebcd755737efd1dceffb3ab0ef2cfe": "\\frac{dp_\\alpha}{d\\tau} = q_{\\mathrm e} F_{\\alpha\\beta}v^\\beta + q_{\\mathrm m} {\\star F_{\\alpha\\beta}}v^\\beta ", "41ebf8bfdc45b4b6ec4194c8a06dd0fc": "10ms/km", "41ec486101b769e7fd4c4cf1af6c10ea": " \\operatorname{dom}\\ A = \\{\\xi \\in H_1: \\phi_\\xi: \\eta \\mapsto \\operatorname{Q}(\\xi, \\eta) \\mbox{ is bounded linear.} \\} ", "41ec911816796a5a82015a341cccc19e": "\\begin{bmatrix}c_1 \\\\ c_2 \\\\ \\cdots \\\\ c_n\\end{bmatrix} = E^{-1} \\vec y_0 = \\begin{bmatrix}\\lambda_1^{n-1} & \\lambda_2^{n-1} & \\cdots & \\lambda_n^{n-1} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\lambda_1 & \\lambda_2 & \\cdots & \\lambda_n \\\\ 1 & 1 & \\cdots & 1\\end{bmatrix}^{-1}\\,\\begin{bmatrix}y_{0}\\\\ y_{-1} \\\\ \\vdots\\\\ y_{-n+1}\\end{bmatrix}.", "41ec943f239684947c1941f8a2c58dd4": "S \\to aSSb | c", "41ecc7aa3079f473049e2e0ec0def13d": "R=K[x_1,\\dots,x_n],", "41ecdccf2cb1feccfb9cd7b2a4af9251": "\nX_{2n} = (X_n+Z_n)^2(X_n-Z_n)^2\n", "41ed4492229a1222a9618c1d032c18c0": " -1<\\gamma<1", "41ed95b31253bfee916af638aaf319cf": "H(X) = H_{\\mathrm b}(p) = -p \\log_2 p - (1 - p) \\log_2 (1 - p). \\,", "41ee7a892261127876424289a2b65d7c": "2xc_n^\\lambda(x;k) = c_{n+1}^\\lambda(x;k) + c_{n-1}^\\lambda(x;k)", "41ee8b4f24f802758d549f547b249ba5": "\\frac{\\partial}{\\partial t}\\Bigl( \\rho\\, h \\Bigr)\\, +\\, \\frac{\\partial}{\\partial x} \\Bigl( \\rho\\, h\\, \\tilde{U} \\Bigr)\\, =\\, 0\\,", "41eeb37e2945fa710cea8c5b86b3ec54": " \\frac{dr}{dt} = \\frac{1}{\\hbar} \\nabla_k E(k) ", "41ef0bce7c605896e276a036ff896fa9": "\\beta=\\beta", "41ef1cd3d16c3820719e9ace1d6191ff": "N_{\\mu\\nu...}", "41ef47593889232d0d9fd7d7c9cecbe0": "\\scriptstyle x_2 \\;=\\; kb \\,-\\, a \\;=\\; \\frac{b^2 \\,+\\, 1}{a}", "41ef59aef209183ace7c5216ef3df8e4": "j_0 = F k_0 ( C_{oxy}^{1-\\beta} C_{red}^\\beta ) ", "41ef638a8c193a86a251efc3872a9656": "\\textstyle{\\overline{\\sum_i V_i^\\perp} = \\left(\\bigcap_i V_i\\right)^\\perp}", "41ef862db8693981a745b16f6966be31": "\\displaystyle\\mathbf v=\\sum^n_{i=1}\\frac{\\partial\\mathbf r}{\\partial q^i}\\,\\dot q^i\n", "41efa849210f99c8e40f800cfb9c25c7": "\\sin 9^\\circ = \\cos 81^\\circ = \\dfrac{\\sqrt{90} + \\sqrt{18} + \\sqrt{10} + \\sqrt2 - \\sqrt{20 - 4 \\sqrt5} - \\sqrt{180 - 36 \\sqrt5}}{32}\\,\\!", "41eff5e86ba721af1dce7a9d05305008": "\\nabla^{bas}_{\\!\\phi\\,}X := [\\rho(\\phi),X]+\\rho(\\nabla_{\\!X\\,}\\phi).", "41eff700e207020e651d627f1ea963a1": "w \\models \\neg P", "41f01a9241b2c084aa5f9f6b60e90e61": "\\alpha N := \\{ \\alpha x \\mid x \\in N\\}", "41f040b9d03c6c8ce12e3425f6b7e4ed": "OPD = 2n_2d\\cos\\big(\\theta_2)", "41f06cbc8a9d8c7577528a3264ac490d": " (m_1-1)(m_2-1)/2 = 2.3 \\times 10^{18}", "41f091e9a6df9a1c2eba3cb6e3b0f706": "\\mathcal L \\{f \\}", "41f09ebb33f1e201629e8341869b4487": " \\frac{S A+ 2RA}{R} = B. ", "41f0cfb47497b36c5f02d4cc5427369a": "\\forall x \\,\\phi x \\rightarrow \\phi y", "41f11453def24d37131f0d4842ed9cfa": "\\ \\Delta G(T)=\\Delta H(T_d)(1-\\frac{T}{T_d}) - \\Delta C_p[T_d -T +Tln(\\frac{T}{T_d})]", "41f11966df7d395f3a75c1eee27a16a6": "B\\in V", "41f1908fd59a6e5045f460f3b8471742": "\\operatorname{E}_Y(Y)= \\operatorname{E}_N\\left[\\operatorname{E}_{Y|N}(Y)\\right]= \\operatorname{E}_N\\left[N \\operatorname{E}_X(X)\\right]= \\operatorname{E}_N(N)\\operatorname{E}_X(X) ,", "41f19727042a3eb191c424d1ae3961d6": "W = \\frac{d\\mathcal{R}}{dt}\\mathcal{R}^T", "41f1d516b5c7f7890fb77b969e14d33e": "w^0_{,2211} = w^0_{,1212} = w^0_{,1122}", "41f1f4efe657919e14a8efede2cbbd0a": " p,q>1 ", "41f2187c39b4d69f60c1bb895c39338f": " H_{\\text{mix}}", "41f26c91128dcab789bfe3ebfdde0eb4": ">4", "41f29d3235e4f71169aa55c55dfb849c": "x > 0", "41f2d9feba49fa2a8b10d4a7b4b5565a": " \\begin{align}\n\\bold{F} \\equiv & \\frac{1}{2}F_{\\mu\\nu} dx^{\\mu} \\wedge dx^{\\nu} \\\\\n= & B_x dy \\wedge dz + B_y dz \\wedge dx + B_z dx \\wedge dy + E_x dx \\wedge dt + E_y dy \\wedge dt + E_z dz \\wedge dt\n\\end{align}\n", "41f31cd97c8c8497f0ae733252829e41": "x_1, x_2, \\ldots \\in X^*", "41f34f7c5436070bad33dafbdbb95ca4": "\\frac{}{id:\\!\\!-~~ \\alpha ~\\vdash~ \\alpha}", "41f367cb3e78098bd19ddc70380e5c43": " P( ( X - E[ X ] )^T S^{ -1 } ( X - E[ X ] ) < k ) \\ge 1 - \\frac{ N }{ k } ", "41f372cd6ff0b3359265ca1926033cde": "T(\\mathbf{e}_j)=\\mathbf{e}_i T^i_j.\\,", "41f3a688406c49e396c853b6751f81a8": "\n h^H(c) = \\cfrac{c^2}{2R} - d^H + u^H(c)\n ", "41f3bd4c44365d248a0dcddb7b44d61c": "m_j < f(t_i) < M_j,", "41f3c4d50fd61b6f72373d90b49e2804": "\\lim_{t \\to + \\infty} \\varphi (t, \\vartheta_{-t} \\omega)", "41f432ae40f04d6f58ec6bb1e7d0dda4": "a,\\alpha,r", "41f4673aa24a2a9e518c5ac5d8ae365b": "x(20p + x) \\le c", "41f4683a18c4acbf7d80d8377b978c21": "\\vartheta_{0} = \\mathrm{id}_{\\Omega} : \\Omega \\to \\Omega", "41f4a409de91f5982583143ab9ce42a4": "\n \\begin{cases}\n 0 & \\mathrm{for\\ } x < a, \\\\\n \\frac{2(x-a)}{(b-a)(c-a)} & \\mathrm{for\\ } a \\le x \\leq c, \\\\[4pt]\n \\frac{2(b-x)}{(b-a)(b-c)} & \\mathrm{for\\ } c < x \\le b, \\\\[4pt]\n 0 & \\mathrm{for\\ } b < x.\n \\end{cases}\n ", "41f4b7eace9ecb69ba6df5728ccb3c09": "(\\Delta^nc)_0=\\sum_{k=0}^n\\binom nk (-1)^{n-k}z^k=(z-1)^n.", "41f4e9087d5369981f3319de9e57caf6": "\\mathbf{q}=\\mathbf{k}_o - \\mathbf{k}_i", "41f51383d44d2a9ed38440f3786a3046": " 0 < \\frac{b^{2n+1}I_n\\left(\\frac\\pi2\\right)}{n!} < 1, ", "41f523bc06a4d2a90132912b805ea989": "\\mathcal{L}_X T=0", "41f526af94e34fce8a895d2d62c78d38": "H_p = -J_p \\sum_{(i,j)}\\delta(s_i,s_j) \\,", "41f5982775f4b93c469e02a93768b245": "\\Phi(t,\\omega)", "41f5f92d5a5b6bf88dd948bf7375fd8b": "\\alpha \\in [0,\\infty) ", "41f624381a8136ab61962dd6b67eddcd": "LM\\times LM", "41f63f7dfba1a0d2f26adb217e95de63": "U(O)", "41f656da544256895f199b819bc575d9": "H^{ \\ell(w) }( G/B, \\, L_\\lambda )", "41f6636c7617613790840c64adcd9b23": "H(\\phi,\\pi) = \\int dx \\left[\\frac{1}{2} \\pi^2 + \\frac{1}{2} (\\partial_x \\phi)^2 + \\frac{1}{2} m^2 \\phi^2 + V(\\phi)\\right].", "41f6deffaa8e1eb3b0be8a535a75eb74": "f_{i \\pm \\frac{1}{2}} =f \\left( x_{i \\pm \\frac{1}{2}}, t \\right) ", "41f70e86673c457e336bbdba511a6340": "\\scriptstyle z' \\,=\\, f'(a_0,\\,a_1,\\,\\dots,\\,a_n)", "41f73ea9f83daac7688d598ac8bf81d1": "t\\dot\\gamma(t)", "41f751ce4c66708efe3197befe864824": " mgh = \\frac{1}{2}kx_{max}^2 - mgx_{max}\\ ; \\ F_{max} = k x_{max} ", "41f78a82168115dbf288e3278994582a": "\\begin{matrix}\\frac{3}{10}\\end{matrix}", "41f819ef00998c4f7bde7cdf5a900e9d": "a/b = c/d", "41f872f77919caa199acc49c09fd5037": "f_i(x) \\leq 0", "41f89219ed2dfb89418728ae6dfb126f": " G_\\mathrm{pixel} \\simeq \\frac{w_\\mathrm{photoreceptor}}{2{(f/\\#)}_\\mathrm{microlens}} ", "41f8a7eaf23d60206f16954f60ae7536": "\n\\delta=\n\\ln\\frac{\\sin(\\gamma)}{\\gamma}\n", "41f8cd580477381af2c8ba587ce010d5": "E_\\text{F}\\ll \\Delta_0", "41f8ef5ba843ecc3499e1a3bcc1c8669": "\\frac{\\sin\\beta}{\\sin\\phi} = \\sqrt{1-e^2\\cos^2\\beta},", "41f90eda1ba6eeca71ed54977ac0f68f": "\\dot{\\theta} = (1-\\cos\\theta) + (1 + \\cos\\theta)\\overline{g}(0,y,0)", "41f92e218e378a501c21b2da6937ce38": " C_P = \\frac{\\gamma n R}{\\gamma - 1} \\qquad \\mbox{and} \\qquad C_V = \\frac{n R}{\\gamma - 1}", "41f932d18b94ec6f6cb6a7a2ae126d9e": "\\mathrm{CLASSICAL \\;DYNAMICS}", "41f9340d248ff999edc8a7b1f7545a35": "\\beta_2 = -1", "41f9793cfc6227c702ff7941b264f226": "-\\sqrt{\\frac{8}{35}}\\!\\,", "41f98c12d87238b9c12d75ffeb0e0eed": "{(1-\\rho^2+2\\rho\\cos \\alpha_1)(4\\cos \\alpha_1-4\\rho) -2\\rho(2\\cos \\alpha_1- \\rho)(-2\\rho+2\\cos \\alpha_1) = 0}", "41f9aaa560643fd0a7d9c5c3bbd891ec": "fRep_{red}", "41fa2cb7b5351789db925cab029df3da": "f(g^a h^c, g^b h^d, \\tau)", "41fa4b6573e98fd9448444ed46c11211": "\\begin{bmatrix} ^\\diagdown k_{r\\diagdown} \\end{bmatrix}", "41fa630660a20f74dba2658ba8082273": " t_1 = \\sum x_i ", "41fa77e0028bbf0ddf760692206d7b25": "D/2f", "41fa9f5032bb011688f8571e1fcf1935": "\\int_0^T e^{-st}dg(t)", "41faa7645aefda374ac536727d850d8a": "\\vert vP(s)Q_\\text{accept} - \\eta \\vert \\ge \\delta", "41fb35e06be16aba39c31f56c941ad2c": "\\zeta(s) = \\frac{e^{(\\log(2\\pi)-1-\\gamma/2)s}}{2(s-1)\\Gamma(1+s/2)} \\prod_\\rho \\left(1 - \\frac{s}{\\rho} \\right) e^{s/\\rho},\\!", "41fba7fb25b0947f521700cce942073b": "\\rho \\colon X \\to Y", "41fbb4a531573e932f51c5a14c584c4b": "\\textrm{shortestPath}(i,j,k+1) = \\min(\\textrm{shortestPath}(i,j,k),\\,\\textrm{shortestPath}(i,k+1,k) + \\textrm{shortestPath}(k+1,j,k))", "41fc45edf2f7296ca050499312f0a70e": "A\\mid B", "41fc6642be00af16f5e391a597b7f907": "\\mathbf{Q}(\\sqrt{D})", "41fc9089cca10372c624b27ee9f7582a": "R[t]", "41fd28fa5f594332b4161c93c4acef5b": "\\frac{W_{n + 1}}{W_n} \\to 1", "41fd50738d08efeea0ab82810c1dfa6e": "\\frac1 {n^2} \\le \\frac{1}{n-1} - \\frac{1}{n}, \\quad n \\ge 2,", "41fd51ef085c6ca628cb803a20a9e013": "E(\\max_{i>1} v_i~|~v_ij. ", "42047194a8a972ad57431ac5ab095661": "\\log L(\\theta^{(t-1)};\\mathbf{x},\\mathbf{Z})", "4204780b7edc79e3bb0872e0d93f4369": "{V_a}", "420486f49c226c1695d9d3b480914b9d": "\\begin{matrix} \\frac{cosine \\;of \\;latitude} {1} \\end{matrix}", "420508b9c3fab37ed4cd33df4d77be30": "\n\\operatorname{maj}(w) = \\sum_{w(i)>w(i+1)} i.\n", "42050b737b8451eeb7686426e19c5e7a": "\n [f^*(\\varphi),\\, v] = [\\varphi,\\, f(v)],\n ", "42054935a0ef2a6dcf79e6853e688bde": "M(a,b,c)", "42055b08f32006b1da09ec1b5d0cadfd": "y \\succ^p_v x", "4205a84e990e7a859723cf620779a3eb": "\\Psi=e^{iS/{\\hbar}} ", "4205b4c3fab61881691d0148b96a1394": "A\\cup \\lbrace p \\rbrace", "4206102b8997626801b7fbcc6a77690e": "F_{\\mathrm{kf}}", "42063198f1e08f61ac764ba7640aa806": "\\Gamma(1-\\beta\\,t)\\, e^{\\mu\\,t}\\!", "4206483fa50f38cdae1c6c133bae8a10": "\\Delta\\subseteq J", "420665b81ca35bd75e4e12e80c1acc97": "Engine Displacement = Cylinder Volume * Number of Cylinders", "420683d4f3582ab42e86c2cfc0f29729": " P = (1-\\dfrac{d}{a})(1-\\dfrac{d}{A}) Q \\,", "42076b179515c1c32d726751c0209751": "f(n) \\in o(n)", "42076d0fd0b036333dc81ef14522b759": "\\frac{d}{dt}\\boldsymbol{f}=\\frac{df_x}{dt}\\hat{\\boldsymbol{\\imath}}+\\frac{d\\hat{\\boldsymbol{\\imath}}}{dt}f_x+\\frac{df_y}{dt}\\hat{\\boldsymbol{\\jmath}}+\\frac{d\\hat{\\boldsymbol{\\jmath}}}{dt}f_y+\\frac{df_z}{dt}\\hat{\\boldsymbol{k}}+\\frac{d\\hat{\\boldsymbol{k}}}{dt}f_z", "42078c6ccb6542d201ef98817610131b": "k_{t+1}=Ak^a_t - c_t \\geq 0", "4207974ac63edb01da8c2386f109ce0e": "x \\in H \\to f_y(x) = \\langle x, y \\rangle", "4207c064b87b43e7f812bcdaadde3a51": " F = \\; \\beta V, ..........(31) ", "4207d2294a05eb360487fa383334f26e": " A = - \\left ( \\frac{\\partial G }{\\partial \\xi} \\right )_{P,T}", "4207e280cc7f9cc83a06b6e18f64deb6": "r= \\theta^{1/2}", "4208345698bf930f3e7e706bd4b0828f": "~\\frac{s}{U}\\gg 1~", "4208856af6d6766145696aecc7cefe65": "\\mathrm{d}U = T\\,\\mathrm{d}S - \\delta W\\,", "4208bffa384a9c8e87cb561d6e802200": "{\\mathbf v} ", "42090de62834d5167049b15998ddfff0": " w=i {1+z\\over 1-z},\\,\\, z={w-i\\over w+i}.", "42093afa0830a0f560c170f635fbadc5": "\\cos\\frac{\\pi}{6}=\\cos 30^\\circ=\\tfrac{1}{2}\\sqrt3\\,", "4209511ee0f67a1e98784e2c8c98b8b2": "M\\in X' ", "42096a61f4e7f80f2f15c9853d487336": "\\xi = \\frac{g}{f}\\nabla^2 Z ", "4209f0b0c6818b2d093883e843d672d7": "\\textstyle l = 121", "420a06e74c16d59856c83d5fd72cac2f": "M/G", "420a07d6475fdd41bc8152ec0918afe0": " Z = e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}", "420a3ae6382189ee2b5493bf7620684b": " \nH_n(\\underline q, \\underline x)=\\sum_{i", "420bdc3142f7e8fa5303aa10457cfc5c": "\\mathbf{B}(t) = (1 - t)[(1 - t) \\mathbf P_0 + t \\mathbf P_1] + t [(1 - t) \\mathbf P_1 + t \\mathbf P_2] \\mbox{ , } t \\in [0,1]", "420c13da7b03b0fa7ab57d3bf3c31da1": "Eq \\; 1", "420c7a0614a0620a22351329df2af597": "G[\\text{true}] \\land H[\\text{false}]", "420c9298eaec7e30bd5883f11134331b": " \\mu \\in \\mathbb R, \\sigma, \\gamma > 0", "420c98a9e3cae9960cef021c405fa9b4": "p_0 \\equiv -\\frac{\\partial}{\\partial q_0} L_d\\left( t_0, t_1, q_0, q_1 \\right)", "420d0d22590b301605229403ea1c9a80": "\n{\\rm Dec}(y, z) = h(f^{-1}(y)) \\oplus z\n", "420d84f8ca907d9062c9801848e70812": "\\phi(w,x,y,z) = \\oint_\\gamma f\\left((w+ix)+(iy+z)\\zeta,(iy-z)+(w-ix)\\zeta,\\zeta\\right)\\,d\\zeta", "420d85d914e3a02caad85033df0dc92c": "\\lim_{k\\to\\infty} \\int f_k \\, \\mathrm{d}\\mu = \\int f \\, \\mathrm{d}\\mu. ", "420dd467ba48ae02f1f26d0b9c07d5d0": "\\mathcal F_n ", "420dec7b4c98ba3ff833536232a514a3": " \\operatorname{inc}\\ g = \\operatorname{value}\\ (g\\ f) ", "420df5ad9cd6ea61554d64185501dbb1": " f(x) = \\sec x + \\tan x. \\, ", "420e8f8616887550c6617bcaf0fa72f2": " \\sum_{k=1}^n \\frac{1}{k^s} = \\frac{1}{n^{s-1}} + s \\int_1^n \\frac{\\lfloor x\\rfloor}{x^{s+1}} dx \\qquad \\text{with }\\quad s \\in \\R \\setminus \\{1\\} ", "420ec53045cecf769d1fbcc18d12e0b3": "z_{t,d,n}", "420f19f4a73c758780d03dfe22c9dad0": "Pr(X),", "420f35ec7a39574c4af6e5b20e09edf9": "T^*Y", "420f37492ac9d74833064926dfad0063": " C_H = [H^+]+ \\beta_1 [A^{3-}][H^+]+ 2\\beta_2 [A^{3-}][H^+]^2+ 3\\beta_3 [A^{3-}][H^+]^3 -K_w[H]^{-1}", "420f94543d74a8b529cc295c863a6597": "n=2048, k=1751, t=27", "420fab68ee9f600906809bcaee2ba768": "=\\left({1 \\over 2}(u+v)(u'+v')\\right) - \\left({1 \\over 2}(u-v)(u'-v')\\right)", "4210aaba3b6069b93ca31d53d7e88466": "\\begin{align} Z\\{x[n-k]\\} &= \\sum_{n=0}^{\\infty} x[n-k]z^{-n}\\\\\n&= \\sum_{j=-k}^{\\infty} x[j]z^{-(j+k)}&& j = n-k \\\\\n&= \\sum_{j=-k}^{\\infty} x[j]z^{-j}z^{-k} \\\\\n&= z^{-k}\\sum_{j=-k}^{\\infty}x[j]z^{-j}\\\\\n&= z^{-k}\\sum_{j=0}^{\\infty}x[j]z^{-j} && x[\\beta] = 0, \\beta < 0\\\\\n&= z^{-k}X(z)\\end{align} ", "4210cdc3bb47e0fc8ca49dc71c170814": "\\lbrace \\theta,\\sigma^2 \\rbrace", "421100bf280d4f724a0ee6ffff8c324a": "\\frac{\\partial K(z,w)}{\\partial w} = \nc(w) K(z,w)+\\frac{zb(w)}{w} \\frac{\\partial K(z,w)}{\\partial z}", "4211158a41caee945d208dd3ed0c6d19": "H_{x,3} = \\frac{1}{2}\\sum_{n=1}^{\\infin}(-1)^{n+1}(n+1)(n+2)x^n\\zeta(n+3).", "4211362ecbd7ee9545c0791248de2dc0": "\n \\boldsymbol{\\sigma} \\equiv\n \\begin{bmatrix}\n \\sigma_{11} & \\sigma_{12} & \\sigma_{13} \\\\\n \\sigma_{12} & \\sigma_{22} & \\sigma_{23} \\\\\n \\sigma_{13} & \\sigma_{23} & \\sigma_{33}\n \\end{bmatrix} =\n \\begin{bmatrix} 0 & 0 & \\mu~\\cfrac{\\partial u_3}{\\partial x_1} \\\\\n 0 & 0 & \\mu~\\cfrac{\\partial u_3}{\\partial x_2} \\\\\n \\mu~\\cfrac{\\partial u_3}{\\partial x_1} & \\mu~\\cfrac{\\partial u_3}{\\partial x_2} & 0 \\end{bmatrix}\n ", "421142e4059581f6d0a5b39a236e1b8d": " \\mathbf{H}_{m} ", "42114b7df194d960f06122cecb1a5c79": " p(x) = a_0 + \\sum_{k=1}^K a_k \\cos(kx) + \\sum_{k=1}^K b_k \\sin(kx). \\, ", "4211e8461a51330e70a35badc38efe2d": "a_1^3.\\,", "4212bfc6a0c0b2afce7427f1e9204fe6": "K < 1 \\,", "4212dc426ce445b64fb1239fd7f79686": "\\quad\n\\beta^{2} =\n\\begin{pmatrix}\n0&0&0&-1&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\\\\\n1&0&0&0&0\\\\\n0&0&0&0&0\n\\end{pmatrix}\n", "4212e442db35a38073ae30a8e7251611": "\\scriptstyle f,", "4213525a0aa5786541e15342bd2586b4": "\n\\mathcal{S}_{i0}=\\frac{i}{\\sqrt{2}}\\gamma _{5i}(\\gamma _{i}\\cdot\np_{i}+m_{i})=0, \n", "421362170a6365edd5293b879cc622a5": "c \\in (a,b)", "42137c92997490c6e66bfc55c3221ff5": "|G| = C_G(t)C_G(z) \\sum_x\\frac{a(x)}{C_G(x)}", "42138b08983106528b12741e851ba77a": "\\textstyle \\Upsilon_n", "42140771e880f0333fcabb6d162bff6a": "K(\\!(T_m)\\!) \\to K(\\!(T_n)\\!)", "421454d760842082e27f6f4d41c0c598": "T = s , \\; Z = Z_0, \\; R = s ", "4214594ab331618ab5b276c672124c73": "", "4214cbbe214fa2ebebbde3c731b427f4": "\\alpha=\\beta", "4214d1ec6bdcb33b2ecd2b74317bcf9b": " v = - \\frac{1}{4 \\eta} \\frac{\\Delta P}{\\Delta x} (R^2 - r^2) ", "42150ba76a96a7ba9367225baafd4da1": "r = \\min_{v \\in V} \\epsilon(v)", "421526c4b843788e8e5676299770adc8": "LR+", "42153b666686a277e676279db0d57083": "\\scriptstyle \\operatorname{sinc} \\left(t/T\\right).", "421570724060381e95985593de9d77c9": "\\theta_2=0", "42157ef5dada021bbcbd77a9ac932a85": " [P_\\mathrm{c} - P_\\mathrm{i}] - \\sigma[\\pi_\\mathrm{c} - \\pi_\\mathrm{i}] ", "4215a330f75c4c2660cab1e34e95b79a": "C^{-1}=\\Pi =\\partial_{\\tilde{\\mu}\\tilde{\\mu}} G(\\tilde{\\mu})", "42162bbf9ba8cdf0ac9f079b5e3b7108": "\\bold u", "421683d0b1c1a545a2b70727adad6f6c": "\\int e^x \\cos (x) \\,dx = e^x \\cos (x) + e^x \\sin (x) - \\int e^x \\cos (x) \\,dx. ", "4216867c9f384feb306c899fd3065b68": "h_{\\text{in}}(G) = \\min_{0 < |S|\\le \\frac{n}{2}} \\frac{|\\partial_{\\text{in}}(S)|}{|S|},", "4216f6a3ba0b1de221f3b43313ba9c0e": "L_\\mathrm o", "4216f783cb93740da56f8daf5c184f89": "D_2(P \\| Q) = \\log \\Big\\langle \\frac{p_i}{q_i} \\Big\\rangle \\, ", "4216f9abca917b1ad527315f881c6fd7": "(X_0, X_1)_{\\theta, q} \\subset (X_0, X_1)_{\\theta, r}.\\,", "421704f4a8ec05ae485ed468e99e4f4e": "g(\\cdot) = \\int \\cdot \\; d \\mu", "421712f7a534f93fdae495fd29f89ea6": "u(r, t) = R(r)T(t).", "4217751e1f2dbdefe9f3e0036ab65834": " \\max_{-1 \\leq x \\leq 1} |P^{(k)}(x)| \\leq \\frac{n^2 (n^2 - 1^2) (n^2 - 2^2) \\cdots (n^2 - (k-1)^2)}{1 \\cdot 3 \\cdot 5 \\cdots (2k-1)} \\max_{-1 \\leq x \\leq 1} |P(x)|. ", "42178c82181d027117588aedac0bf524": " y(t_{n+1}) - y(t_n) = \\int_{t_n}^{t_{n+1}} f(t,y(t)) \\,\\mathrm{d}t. ", "42182c52ef9d0135fab1eaf1b25a07d6": "\\frac1n \\log \\frac{a(n,k,X)}{j(n,X)} \\quad \\text{and} \\quad \\frac{1}{n} \\log\\frac{j(n,X)}{a(n,X)}", "421839508cb21915f301145e735ffe57": "V = I R", "42185645556cd0c0c33d85555680bde0": "\\mathbf{IC}_{expected} = \\frac{\\displaystyle\\sum_{i=1}^{c}{f_i}^2}{1/c}.", "42185b22e713df5114c9f6432045d587": " k \\geq 1 ", "42186cb438d05e0ef57d4cdee1d8a650": " \\sum_{n=0} ^ {\\infty} \\frac {f^{(n)}(a)}{n!} \\, (x-a)^{n}", "4218b7ed5149721ed631a24e6250b50f": "P(x) = \\sum_{i=0}^n{ a_i x^i} =a_0 + a_1 x+ a_2 x^2 + \\cdots + a_n x^n,", "4218cb73de01241970d2b4dda6a8705b": "\\int_a^b \\frac{f(t)}{(t-x)^2}\\, dt = \\frac{d}{dx} \\int_{a}^{b} \\frac{f(t)}{t-x} \\,dt.", "4218ecf619c24c6d1c0e5bdaa8ae2a26": "((\\operatorname{trace}_{V}(T))^M)_{\\ell'_1 \\dots \\ell'_N }^{\\ell_1 \\dots \\ell_N} ", "42191eb17b3eb272a72a68a03e98ace7": "S = \\{i_{1}, \\ldots, i_{r}\\}", "4219d820368d9573e026b8097e4a1134": " \\Gamma (t) ", "421a2047062aaaf028baa29074f386be": "\\mathbf\\Theta", "421a33eed039fd73ddca86d41bbb24e5": "\\ell^q\\xrightarrow{\\kappa_q}(\\ell^p)^*\\xrightarrow{(\\kappa_q^*)^{-1}}", "421a6b1e8b864e31d503a4bfe0cc8be4": "j^{\\star} = \\sigma T^4,", "421a966db17187201b4c83a080ca1ed1": "J_n(k\\rho)", "421a9c8c784a242d1a3f6d743172310d": "\\mathbf{A}=\\left(\\begin{matrix} L(\\phi ) & L(\\gamma)\\\\\\phi & \\gamma\\\\ \\frac{\\partial {\\phi }}{\\partial n}&\\frac{\\partial {\\gamma}}{\\partial n}\\\\ {\\gamma}&0\\\\ \\end{matrix} \\right),\\quad \\mathbf{b}=\\left(\\begin{matrix} f\\\\g\\\\h\\\\0\\\\ \\end{matrix} \\right),\\quad \\phi = \\phi\\left( x_i,x_j\\right),\\quad \\gamma = \\gamma_k\\left(X_i\\right).\\qquad (7)", "421ad0f1eeb80c0a6dbd2a6b77bb5c38": "\\left.\n\\begin{align}\n x\\left(\\theta\\right) &= \\plusmn a\\cos^{\\frac{2}{n}} \\theta \\\\\n y\\left(\\theta\\right) &= \\plusmn b\\sin^{\\frac{2}{n}} \\theta\n\\end{align} \\right\\} \\qquad 0 \\le \\theta < \\frac{\\pi}{2} ", "421ae39e43b6f1477ca739c15fe724c0": "m = \\frac{ 21 - 15}{3 - 4} = \\frac{6}{-1} = -6.", "421b04b10fd532cb98c579ced68ac424": " rQ_D l_A a_D ", "421b07b670e1c5d9e6eb572773fe7231": "\\alpha_{\\rm B}=\\alpha\\cdot k_2=k_{\\rm B}", "421b70a60e8f7a4f7e9ccfd2f47b6ab6": "{ }^\\lambda M\\subseteq M \\,.", "421b99a4f122d2902c28e3935577d41e": "\n\\begin{align}\n\\min&\\|f(x)-z^{ideal}\\|\\\\\n\\text{s.t. }&x\\in X\n\\end{align}\n", "421b9fc4472160ce30d9a9057e93e72a": "g_i(x)=0", "421c2807756de96e48e40cf14b58442f": "C = A+B \\quad\\mbox{with}\\quad A = \\frac{1}{2}(C + C^\\dagger) \\quad\\mbox{and}\\quad B = \\frac{1}{2}(C - C^\\dagger).", "421c46e1e41362e835a33515ccdb9d12": "\\Omega \\to \\mathbb{R}^n", "421c5ad4909a98d2c4e9a43ad929c9cd": "e^{(2)}_i = -a^*_{i+1}", "421cb100ef53a99cc90c6f0bd83b1dcd": " \\scriptstyle E_{\\rm C} ", "421cea0090f903db71b36286a642a44e": "\\phi=\\textrm{inf}\\{\\ell\\in [0,T)\\}", "421d1b5313c186e7b107299a5dc08ce4": "N(x_0)", "421d2200240cde1a1a7c8caa0fe531e6": "R_0(*, *)", "421d4be32422d56d1964ab31c79d8d6b": "x_1,...,x_n", "421d7077cf63cbe36b7ac9488cb6c0d7": "\\omega = \\sup\\{G<1\\} <+\\infty", "421d85cb2757d77900fa4a3eabad3735": "E_K(P)=C \\Leftrightarrow E_{\\overline{K}}(\\overline{P})=\\overline{C}", "421e29600ec0ec08bda21ba1d4b371f7": "X=\\cup_{s\\in S} f_s(X)", "421e4ff1a916738db8ae0f49a881cd0a": "ds^2=-dt^2/U(t) + 4l^2U(t)(d\\psi+ \\cos\\theta d\\phi)^2+(t^2+l^2)(d\\theta^2+(\\sin\\theta)^2d\\phi^2)", "421e72ada4b768bbe485e89911aa9a3c": "\\frac{P \\or P}{\\therefore P}", "421e767ccfc58016a6ce709df049f60d": "-\\frac{(b-a)^7}{1935360}\\,f^{(6)}(\\xi)", "421e90e317ae736f6f5fb5529f694823": "(\\partial_\\mu\\varphi,\\partial_\\mu\\varphi)", "421ea3bd98aab0436aa1012320aadc8c": "\\operatorname{ev}_z f = \\int_D f(\\zeta)\\overline{\\eta_z(\\zeta)}\\,d\\mu(\\zeta).", "421ea9da84e9c362b743d49f3f22a8cf": " -qe^{-q \\tau} \\Phi(-d_1) - e^{-q \\tau} \\phi(d_1) \\frac{2(r-q) \\tau - d_2 \\sigma \\sqrt{\\tau}}{2\\tau \\sigma \\sqrt{\\tau}} \\, ", "421eab98c31a1f29e7c0d38e3be4fbbf": "\\sum_{n=1}^N \\lambda_n(A)=\\operatorname{Tr}(A).", "421eb4c46f972247f493f0eafcca4314": "\\sum_i \\langle T_f u_i, u_i \\rangle = f(I) \\leq \\|f\\|,", "421eb585092bb59551df7ec393c95263": "\\arccot (1/x) = \\tfrac{3}{2}\\pi - \\arccot x = \\pi + \\arctan x,\\text{ if }x < 0 \\,", "421f17d71cab8d8b30f7e93fb3387f0c": "K_2=\\cos(\\phi_m)N\\frac{\\pi}{180}\\,\\!", "421f657cc0426bd74281ec830546901f": "h,g,f\\colon\\quad V \\to U \\to T \\to S", "421f693f9ea0d06713ed70be5ab8a7af": "p^{-N}", "421f8647eb04742d96431ab0564115e7": "\\min_{w\\in\\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n (y_i- \\langle w,x_i\\rangle)^2+ \\|w\\|_1, ", "421f9ae48c3ef68b352d381cb4f14da9": "x:=y+1\\,\\!", "421fba5ef53644895521645370f97540": "[\\hat H, a^\\dagger ] = \\hbar \\omega \\, a^\\dagger .", "421fe56bf19bfc400f99313d818b7c8b": "j_1, j_2, \\ldots, j_n \\in S .", "42209fd1b98a48bdf1b6ac97dccf200c": " t^2 + 2\\beta t - 1 = 0 . \\,\\! ", "4220b4ef42d37e5177bd5cf2e3b81ed8": "2^{-\\mathrm{depth}(\\ell)}", "4220e3512f04d1de6049ab3aed6167b8": "0=d^2C_p", "422110099e9b432b1d033fdf832bc3cc": "Q(x) = \\frac{1}{\\pi} \\int_0^{\\frac{\\pi}{2}} \\exp \\left( - \\frac{x^2}{2 \\sin^2 \\theta} \\right) d\\theta.", "422161ae4ddc343ae580e4bcc414d8a5": " f(Z_t)-f(0) ", "42217049a06e499c99e6eb69a9bf145f": "V = a^3 \\cdot 4\\sqrt{5+2\\sqrt{5}} \\approx 12.3107 \\cdot a^3", "42217d5a0609f5cc6187e1cfb0844a3b": "\\omega \\in \\Omega", "42217e9aa6ec3cffdb8696fb4d97feb7": "\\mathbf{R}^{2m}", "4222896ed4d83b5bb433634f148a6bde": "\\Omega=\\oplus_n\\Omega^n,\\ {\\rm d}:\\Omega^n\\to\\Omega^{n+1}", "4222dba475c82ad5a00abceb75161817": "S(t)=\\int_0^{a_M}{s(t,a)da}", "4222dcf6d1f1958cfc225253bb95a950": " \\bar h^i = \\mathcal{M}^i_jh^j ", "4222f87965a05f129adede6e70b3b2bd": "u_{\\mathbf 1^n} \\mapsto q^{-n(n-1)}e_n \\, ", "42231a2d4aacfb20d98d06af2e7d3084": "t={}_*u", "42232c05857615513bd80ff0af8f4501": "m = 0.9832\\,", "42232ce11d3ebe3d81d3cbc3810b4c22": "dU = TdS-pdV+\\sum_{i=1}^k\\mu_idN_i", "4223a05733bb3c3b518b7f8a510f109d": "\nx = a \\ \\frac{\\sinh \\tau}{\\cosh \\tau - \\cos \\sigma}\n", "4223c59a9388e2a0e67fcb5f911dd830": "0 \\subset \\mathbf{Z}/p \\subset \\mathbf{Z}/p^2 \\subset \\mathbf{Z}/p^3 \\subset \\cdots \\subset \\mathbf{Z}(p^\\infty)", "4223e650f073f5a04d4341dc8f408875": "E_\\mathrm{p,e} = \\frac{1}{2}kx^2\\,\\!", "42244c820b8cba35f6046a703f3b5ed0": "-K+S_{T}", "4224b199199d716fc7e39732c1fb6bab": "x_\\text{median}=\\frac{\\ln(1+\\sqrt{p})}{\\beta}", "4224e11d8962b7462861f27d053c35df": "K_{a,b}", "42250244826653fc7374695993e401a8": "J_{12}", "422526ed15b3aeb16436a480dfc9544a": "ab\\,\\bmod\\,n = ((a\\,\\bmod\\,n)\\,(b\\,\\bmod\\,n))\\,\\bmod\\,n", "4225555efc97bb78cf143343e98320f5": " \\psi^{(\\alpha)} ", "42257b7bf339d6e1ded77061935d426c": "\\scriptstyle g (x_1,\\dots, x_m)", "4225e8596e01a20bba87ec2b561b5037": "(\\mathbb{C}\\otimes\\mathbb{O})P^2", "422603b1c298c4f06bf840c306cd2d71": "X,\\,Y", "422610a8ce1574130967d0e1d699c700": "\nQ_L(\\mathbf{p}) = \\sum_\\mathbf{k} F^\\dagger(\\mathbf{k})\n\\left [ c_2(\\mathbf{p}/2-\\mathbf{k})a_2(\\mathbf{p}/2+\\mathbf{k})\n+ c_1(\\mathbf{p}/2+\\mathbf{k})a_1(\\mathbf{p}/2-\\mathbf{k}), \\right ].\n", "42264ced561289c0a8e1bd84882e76a6": "\nu_t + 2\\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. \\,\n", "4226974f180ae559cea06c42bf210e51": " \\forall X \\exist Y \\forall uvw[(u,v,w) \\in Y \\leftrightarrow (v,w,u) \\in X],", "4226bc923e1b6c3fb0055f83d02a69d7": "\\delta < \\kappa", "4226cf82f3774696d7f6fcc95424b409": "\\cos(x \\pm y) = \\cos(x) \\cos(y) \\mp \\sin(x) \\sin(y)\\,", "42270139d00452fc1d1e4096096c35b6": "Prob_z[Accept] = 1", "42270811afeb495f50e3536efbea6b6c": "R_{\\mu\\nu}=0", "42271f96fcfdd216f4139ecdf14f34f7": "C = -\\sqrt{2} + \\sqrt{3}", "42274288bcc6c7c5583ddd2030a1ff8f": " U_k ", "4227692f2cd656a924e1e362b498c6b2": "\\log|X-O| + \\log|O-Y| = c'\\,. ", "42277df7668fd3babc119581ff1bebd4": "(-\\infty, p_1), [p_1,p_1], (p_1, p_2), [p_2, p_2], ..., (p_{m-1}, p_m), [p_m, p_m], (p_m, +\\infty)", "42286ae6ee1f1e5afd15716fca23ea24": "L= \\infty", "42287a3a68ff489b5308b8c3770ae03a": "Y=(Y_i :i=0,1,2,...,N)", "42288c8074267b623b4905aedde4a2fe": "(a,b)\\in A\\times B", "422899891e4182f7848719014025558f": "\\varphi = {1 + \\sqrt{5} \\over 2}.", "42289dd97f39a9b0dd17cb08e04f7246": "\\begin{bmatrix}\\cos(\\phi)&0&-\\sin(\\phi)\\\\\n0& 1& 0\\\\\n\\sin(\\phi)& 0& \\cos(\\phi)\\end{bmatrix}\n", "42289eec22d139db2162f142be21e71c": "\\frac{4}{n} = \\frac{1}{n} + \\frac{1}{(n-2)/3+1} + \\frac{1}{n((n-2)/3+1)}.", "4228c42305d2a4d2d3e0f68e8e5b4546": " \\left(r, \\theta, \\phi, t \\right)", "4228c607ae4b625e7b05efec97ae5933": "O(n).", "4228e059dd09a931647ecef64e8cc431": "\\phi_i: M_{i, [\\epsilon_i, \\infty)} \\rightarrow M_{[\\epsilon_i, \\infty)},", "4228e76530bd6e75fb045ac93eba8444": "(X_0, Y_0), (x, y), (X_1, Y_1)", "422905755732b728717f843bae87b635": "\\lambda^{-1}\\mathbf{R}_{x}^{-1}(n-1)", "422906f0b92c3712f68f0a7a5deaee8f": "\\mu_r \\mu_0 = \\mu", "4229304d36f78df27a09b0db938eade6": "\\phi(A)=A/N", "422a2511a2ca6ecf55d6bc8e3ed36330": "\\gamma = \\frac{1}{\\sqrt{2}}", "422a77a910ea6cb39eb82a948b89fca8": " a \\ = \\ \\Omega^2 \\vec{r} \\ + \\ 2 (\\vec{\\Omega} \\times \\vec{v}) ", "422a9f993b06b0a101ab323852b5e281": "W_c(t) = \\int\\limits_{t_0}^{t} \\Phi(t_0,\\tau)B(\\tau)B^T(\\tau)\\Phi^T(t_0,\\tau) d\\tau", "422ac7076efbd4d800ef899a70207057": "\\frac{\\pi}{\\sqrt{12}}", "422ae55fb14fd7456358aac4933296f1": "\\bold{X}(\\bold{u})", "422af929b9e4e3e6aab9bfa46b06206f": " \\sum_{P \\in E}{c_P P} = \\sum_{P \\in E}{d_P P}", "422b249337fcb06cf6b77da2416ffdc9": "[1+\\cos((\\Delta \\omega_1 - \\Delta \\omega_2)t)]", "422b2fff9a42ba21ac3951944005367b": "(1-g", "422b46f3c0508d67c0058bb5be451b72": " v_n ", "422b4907511857d9891691015c2e2865": "\\frac{d\\mathbf{T}}{ds} + \\mathbf{G} = \\mathbf{0}.\\,", "422b5fec1ce05d5cb6818f22bac4c661": "\\left(\\frac{-1}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "422bcdc4e7f05bf2a1eea51f4535192e": "\\Delta _r G=\\left (\\frac{\\partial G}{\\partial \\xi}\\right )_{p,T}", "422bdfe1e69bcc8f17d2e52b7f5f5480": " \\lambda(E) = \\frac {\\hbar} {\\sqrt{2mE}} ", "422c0597c0b3481ca784532f331261b1": "y_1=x^3", "422c2f5a9ba98a7a1e02b10a526387a3": " f\\sim g \\iff (f-g) \\in o(g) ", "422c45e0fbeca1dfc733d0be23409aaf": "\n[{\\mathbf x}^']_{\\times} \\left( \\sum_i x_i {\\mathbf T}_i \\right) [{\\mathbf x}^{''}]_{\\times} = {\\mathbf 0}_{3 \\times 3}\n", "422caf3757f024878096a25b901b1095": "(m, n)", "422d2cdf3cd4915945dd7cbd92c4ef34": "\\gamma = 1.4", "422d605ad5b47e52b7275a53d9d87499": "\\rightharpoonup \\rightharpoondown \\leftharpoonup \\leftharpoondown \\upharpoonleft \\upharpoonright \\downharpoonleft \\downharpoonright \\rightleftharpoons \\leftrightharpoons \\,\\!", "422d66cb41c8fa09bc47b806b8295394": "\nA_p(t) = \\begin{cases} 0 & {\\rm if}~p~{\\rm~is~not~modified~at~time}~t\\\\ t - {\\rm modification~time~of}~p &\n{\\rm otherwise} \\end{cases}\n", "422e4f9a401c56a0a35adffdb68960b3": " R = \\left ( \\frac{P_t \\ G_t \\ A_r \\ \\sigma F \\ D^2}{16 \\ \\pi^2 \\ K_b \\ T \\ B \\ N} \\right)^{-4} ", "422e7bb6db5389da99c1559e4dbfa872": "\\sum_{k \\in \\mathbf{Z}} \\hat{f}(k) e^{ikx} = \\sum_{k \\in \\mathbf{Z}}\\frac{1}{2\\pi} \\left (\\int_0 ^{2 \\pi} f(t) e^{-ikt} dt \\right) e^{ikx},", "422ea94f4fe57a3a943fd47c68f4d14f": "g(1) = 1", "422eae3029aa965e84f7ee45ce441966": "\\phi(x,u_1,\\dots,u_n)", "422ecdcf5ec81c7d27f6d63928a3ce4d": "\\left|\\alpha-\\beta\\right|<\\left|\\alpha-\\alpha_i\\right|\\text{ for }i=2,\\dots,n \\, ", "422f7f531f19aecc9b03104d73d8672b": "Q(x'; x^t)=\\mathcal{N}(x^t;\\sigma^2 I) \\,", "422f9a0b971be65cb09d4f92e67852f9": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = \\left(x - 5\\right)^{2} + y^{2} \\leq 25 \\\\\n g_{2}\\left(x,y\\right) & = \\left(x - 8\\right)^{2} + \\left(y + 3\\right)^{2} \\geq 7.7 \\\\\n\\end{cases}\n", "422fb1fc9319e3d037553b838206673f": "p(\\mathbf{\\theta}|\\mathbf{\\alpha})", "422fd3780d9931f4eed2c370c88c29f1": "\\tilde x", "42301145b3e24e354cae97c004e67893": "{\\epsilon\\over k} \\ne T_D\\,,", "423055c263194e91c4d9e4bea3a66cec": " g(\\theta) = \\sqrt{ \\xi{(\\theta)} \\left[ 1+r^2\\right] - 2},", "423079780f802f5c8c86945ae9fcb828": "V_n = \\frac{\\pi^\\frac{n}{2}}{\\Gamma(\\frac{n}{2} + 1)}", "42307db75999f93b646971b135c8bdd7": "\\frac{\\zeta(s)\\zeta(s-a)\\zeta(s-b)\\zeta(s-a-b)}{\\zeta(2s-a-b)}=\\sum_{n=1}^{\\infty} \\frac{\\sigma_a(n)\\sigma_b(n)}{n^s}", "4230f24e9ac675c0a086a5ec7d67d89a": "[\\theta^i]", "42314878ab26f82899a7537ea07c77e8": " d = km = \\frac{zp}{\\pi} = z\\frac{m_n}{\\cos\\beta } ", "4231c28b3895de11d61eccac5b2d8b56": "m_{\\rm u} = \\frac{m_{\\rm e}}{A_{\\rm r}({\\rm e})} = \\frac{2R_\\infty h}{A_{\\rm r}({\\rm e})c\\alpha^2}", "4231c73dbb976ba6ef67fa95ee284557": " \\Delta U=Q+W", "4231c98ed0aae5f834bb2f0c22791a8b": "\\Omega_4(t) =\\frac{1}{12} \\int_0^t dt_1 \\int_0^{t_{1}}d t_2 \\int_0^{t_{2}} dt_3 \\int_0^{t_{3}} dt_4 \\ (\\left[\\left[\\left[A_1,A_2\\right],\nA_3\\right],A_4\\right]", "4231d05381d62b65e3e4ac70c9b85508": "\\textstyle (M_{\\mathrm f} R^k)", "4231eeb3c33ebe207f916b021ea63a39": "\\boldsymbol{f}(s) \\sim \\frac{4\\pi\\mu}{\\ln(\\ell/a)} \\frac{\\partial \\boldsymbol{X}}{\\partial t} \\cdot \\Bigl( \\mathbf{I} - \\textstyle\\frac{1}{2} \\boldsymbol{X}'\\boldsymbol{X}' \\Bigr) ", "4231f787e3bdf4fb9b9eb05a0ac237ad": "\nC_{0i} = \\frac{\n{\\left( {3m^2 - 7 - 20i^2 } \\right)/4}}\n{{m\\left( {m^2 - 4} \\right)/3}}\n", "42320b32d2f89b158b33e835a4e22e9c": " C_x = \\frac { s^2 / m - 1 } { nm - 1 } ", "42322c9705afe36bc0e5e4db12ec8c0d": "\\displaystyle{a^*=-a +2(a,1)1,\\,\\,\\, L(a)b = ab,\\,\\,\\, R(a)b=ba.}", "423276db9c54c7318624c67c92616255": "A_r = A_l ; \\, ", "423280733888f42310074907f5220438": "u(\\mu,\\eta) = F(\\mu) + G(\\eta)\\,", "4232a36385f99fdac35551e117b8d4da": "T \\perp Y(0), Y(1) \\,|\\, p(X).", "4232be8f740ae76285e26d1ddcaaa61a": "\\omega _-", "4232d7590954fd1a8b89242efc1b2bca": "s^{\\overline{k}}", "4232e43653694037d868fd6cd8f96db7": "\\mathbf{S} - \\mathbf{1}\\mu_s^T", "4232fc41e139c1199c69746418c6fd5a": "\\scriptstyle f \\,\\mapsto\\, f(\\zeta)", "42332eaff5cdd6a7791c9e8824833971": " \\int_\\Omega uD^\\alpha\\varphi\\;dx=(-1)^{|\\alpha|}\\int_\\Omega \\varphi v \\;dx, \\ \\ \\ \\ \\varphi\\in C_c^\\infty(\\Omega),", "4233454e25722e3832cd2d5a1f06385e": "(\\triangleleft) \\subseteq \\text{Tp}(\\text{Prim}) \\times \\Sigma", "4233720ad3b4d96cb8d23d5c02013814": "E_n = E_0 + \\frac{\\hbar^2 \\pi^2}{2 m L^2} n^2. \\,", "4233915643ff88749b8d19c6c8d4ca0f": "2^1 + 1 - 1", "423464424571f93b8be2ffa2a88811ba": "7\\varepsilon", "42346877fc30b240f1fae1b082273235": " Q_2 = - \\frac{R}{p} \\frac{\\partial \\vec{V_g}}{\\partial y} \\cdot \\vec{\\nabla} T ", "423498a1cffd732c78946c982287e4e6": "A f (x) = \\lim_{t \\downarrow 0} \\frac{\\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}.", "4234a8f81f2d37b23cdda2bc01efc205": "f_X(t),f_Y(t),f_Z(t)", "4234c196c9414da6264e289abfed563b": "c_0,\\, c_{\\pm 1},\\, c_{\\pm 2},\\ldots", "4234e90f366f468d1b41ece043d82491": "\\cdots \\subset M_2 \\subset M_1 \\subset M.", "4234f19c98db67d2618606ef2205c4e8": "\\mu =1", "42351c1624426b2621036cc3841dedbd": "\n s_{1j} X_1 + s_{2j} X_2 \\ldots + s_{Nj} X_{N} \\xrightarrow{k_j} \\ r_{1j} X_{1} + \\ r_{2j} X_{2} + \\ldots + r_{Nj} X_{N},\n", "42352a48287c6f024f47234f56119cd7": "S_n=\\binom mn \\frac{(m-n)!}{m!}=\\frac1{n!}", "4235f61d6dbb94ac9d0e8a23c8636972": "\\mathbf{X}\\boldsymbol{\\beta}=\\mu\\,\\!", "4235fab09620c6635836629502e0baa7": "\\Gamma \\in \\mathbb{Z}", "42369228d853faa8d33fb38849012823": "\\frac{1}{2 \\mathrm{NA_o}} = \\frac{m-1}{m}\\, N. ", "4236ebc40ae8c0700eeadaa7073bf746": "\\sum_{n=1}^\\infty\\frac{1}{n^s} = \\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}},", "4237e9e7b9e7cd77c0a762586bbb9b64": "p_w(\\theta)\\,d\\theta=p_w(z)\\,dz", "42380e9b5e03cb0d3846a6e505ba54ae": "Aged\\ dependency\\ ratio\\ = \\frac{number\\ of\\ people\\ aged\\ 65\\ and\\ over} {number\\ of\\ people\\ aged\\ 15-64} \\times 100 ", "42382bb0a7ece15b9a81f8c536b7f57b": "|T|>t_{1-\\alpha/2}\\,\\!", "423879a1326300565dc8b94498d0c3e2": "x_2\\cdots x_p = wy_2\\cdots y_q", "42389ae8194eb8268ee5b6126f549feb": "r_{\\pi} = \\frac{v_\\mathrm{be}}{i_\\mathrm{b}}\\Bigg |_{v_\\mathrm{ce}=0} = \\frac{\\beta_0}{g_m} = \\frac{V_\\mathrm{T}}{I_\\mathrm{B}} \\,", "4238fa560b5f7a577347dfdb4472fd75": "x > \\mu", "42390dd36c71b4596c845a7e15e46514": "1 \\times 2 \\times \\cdots \\times n", "42396b539c2582f325208d05851284c9": "\\frac{\\sin x}{x} = \\prod_{n = 1}^\\infty\\cos\\left(\\frac{x}{2^n}\\right)", "4239714140254dcf8519745a3907b637": "\\mathbf{B} = \\begin{pmatrix}0\\\\0\\\\B\\end{pmatrix}", "4239fb8d34a8149cfcb167ad637d4c31": " \\hat{f}_2^{(i)} = [S_2 \\sum_{\\alpha = 0}^{i-1}(S_1 S_2)^\\alpha(I-S_1)]Y ", "423a25216a5a58aa1e92697d478cf130": "EU(n)", "423a618cefb15a65c03dabd3d06b4095": "{\\bold \\ f_T}", "423ae41944ef88942a0e82e30fe3050b": " A = \\arcsin\\left( \\frac{20\\sin 40^\\circ}{24} \\right) \\approx 32.39^\\circ. ", "423b3076bd449cb329a49ef3cf0c8f5f": "t=1,..,T", "423bb6aa7af3991740d7452810972198": " \\ J1: ((A)A) = . ", "423bd9001de05c695d4641fa59faedcf": "{\\mathbf\\mu}", "423c4246d42cb83b56bb4a562feecf4e": "x_n = \\frac{1}{1 + \\sum_{j=1}^{n-1}{y_j}},", "423cabdda4bd7f2b50866526cc3270eb": "\\|x\\|^2 = \\langle x,\\ x\\rangle", "423cad92e05c265be04516596a681dc4": "F(x)=C", "423cd85fa0e3b52c00104fccb0e3274f": "\\xi=\\frac{1}{2}\\left(\\frac{R_I}{R}-\\frac{R}{R_I}\\right)\\sin\\beta", "423d4026802d1575b297a04bfd45f5fd": "(I \\otimes \\Phi) (\\rho) \\geq 0.", "423d48019009bb6066ecb96ef1d99fb6": "(\\omega+dd'\\phi)^m = e^f\\omega^m", "423d539d50d53702456418faf9512318": " \\hbar \\frac{\\partial \\alpha_k}{\\partial t} (t)= i\\left[ H , \\alpha_k \\right] = 2(i \\gamma_k m - \\sigma_{kl}p^l) = 2i(p_k-\\alpha_kH) \\,\\!\\;", "423dc2594c95ba90a01b240cfd2dee8e": "tf'(r_k) \\equiv -(f(r_k)/p^{k})\\,\\bmod{p^m}\\,", "423e4d3560660bce0e4915c550827759": "\\frac{d\\theta}{dt}=q", "423f44c7099f81521012bf36f2526741": "E_{external}=E_{image}+E_{con}", "423f51d7af6441c7bf88b2b16ecc40ad": "y(x)=Cx+f(C),\\,", "424032d17fe9233ff7ec535ba70e2af5": " \\hat{T}T\\sigma=\\sigma,", "42404e04b8046f18f2c573d400c8ed4d": "\\pi r^4/3", "4240654bc9f092543772c2fba0c2a15b": " g''(x) = \\frac{-f''(g(x))}{[f'(g(x))]^3}", "424071db8d03ced6a4233e3938af17aa": " \\sum_i d(\\mathbf{x_i}, \\hat{\\mathbf{x_i}})^2 + d(\\mathbf{x_i}', \\hat{\\mathbf{x_i}}')^2", "4240c6b27740f946f6ba796c6e0f264c": "\n f(x) = \\frac{2^{-p}\\chi^{2(p+1)}}{\\Gamma(p+1)-\\Gamma(p+1,\\,\\tfrac{1}{2}\\chi^2)}\\ \\cdot\\ \n \\frac{x}{c^2} \\bigg( 1 - \\frac{x^2}{c^2} \\bigg)^p\n \\exp\\bigg\\{ -\\frac12 \\chi^2\\Big(1-\\frac{x^2}{c^2}\\Big) \\bigg\\},\n \\qquad 0 \\leq x \\leq c,\n ", "4240f97cb3fc46b6b65bdbe5bd6cacc2": " C_n(x;\\beta|q) = \\frac{(\\beta;q)_n}{(q;q)_n}e^{in\\theta} {}_2\\phi_1(q^{-n},\\beta;\\beta^{-1}q^{1-n};q,q\\beta^{-1}e^{-2i\\theta})", "42410eb756342ca33a2e925202392e04": "p(a \\otimes b) = \\|a\\| \\|b\\|", "4241377f1936f6fa8bc0c0df8981a01b": "T = U_i^{-1}", "4241d2825add997ed44df971d74b4ebb": "\\sqrt{\\frac{49}{40}} \\frac{C_8}{\\sqrt{C_6 C_{10}}} = 1", "42425d3d94b61e88991f418166763286": "\\sqrt{\\frac{15}{28}}\\!\\,", "42426060bc84f769ebe1f7e30fba2bee": "R / (I_1 \\cap I_2 \\cap \\ldots \\cap I_m) \\rightarrow R/I_1 \\oplus R/I_2 \\oplus \\cdots \\oplus R/I_m", "4242968b9f08bce37f3e6a702db37498": "\\mu_r", "4242ce112f58724b3f954822a191f02e": " 2 \\alpha ", "4242fcbb95742d9b5654fe46e3dd6f02": " \\prod_{p} \\Big(1 - \\frac{2p-1}{p^3}\\Big) = 0.428249... ", "424337321692880e719282d54f0961fb": "(A_x\\ ,\\ A_y)", "42435b034a40d2bbafbb4e231d2d1572": "(r)\\,", "424397b904c9c5540570f104f05d8cd0": "\\scriptstyle \\dot{Q}", "4243ef1854af4be7d80eff10cc6fdb6a": "\\zeta(2,1,2,1,3) = \\zeta(\\{2,1\\}^2,3)", "42440a0c938d3acf499c5581a4cfc363": "F \\vdash _{A} f", "424466f2f0ef87dcaffe8509acffcbc4": "R=(2.9, 0.0)", "4244b6ed51ac7d486ae1341f42cddb37": "\\mathcal{I}_i \\to \\mathcal{I}_j", "4244db87e0d870494772e64543ff2224": "x^*\\in D(p,m)", "42451e0536bea249b25f035a8d6ab1b6": "v_h=v \\cos \\theta,\\quad v_v=v \\sin \\theta \\;", "42452f72bc892cd40cf7f828934dadea": "n \\geqslant 2\\qquad \\,", "424579421929254ae98810939babd5c1": "x = \\cos(t)(R + r \\cos(u)),", "4245ea83cef6ef0cad36f24e5bd4fd4c": "V_C = - \\frac{1}{2} \\omega^2 \\Delta d^2 = - \\frac{G M }{2 d^3}\\Delta d^2 \\,", "4245f68a1240d913119295016f9c3412": "\\mathbf{B}(\\mathbf{r}, t) = \\frac{\\mu_0}{4 \\pi} \\int \\left[\\frac{\\mathbf{J}(\\mathbf{r}', t_r)}{|\\mathbf{r}-\\mathbf{r}'|^3} + \\frac{1}{|\\mathbf{r}-\\mathbf{r}'|^2 c}\\frac{\\partial \\mathbf{J}(\\mathbf{r}', t_r)}{\\partial t} \\right] \\times (\\mathbf{r}-\\mathbf{r}') \\mathrm{d}^3 \\mathbf{r}'", "424612c979d0809a7517ece53f38a0e6": "\n\\operatorname{Ran}(J)=\\operatorname{Ker}(J), \\qquad J[X,Y]=J[JX,Y]+J[X,JY],\n", "42463f2883cabf63d27a03b249332c84": " Var( \\theta ) \\sim\\ \\frac{ 2 } { N } ( 1 - \\frac{ 1 } { n } ) ", "4246a0548540a48a8ea39f853a2b9ace": "-m\\bar{\\psi}\\psi\\;=\\;-m(\\bar{\\psi}_L\\psi_R+\\bar{\\psi}_R\\psi_L)", "4246fabaf9ac9b6d2f3d04cc38465499": "\\displaystyle {\\mathbf v}_{xt}+({\\mathbf v}_x{\\mathbf v}_t){\\mathbf v}=0", "4247058e2656d8dcbc21313ac9dfbb3b": "\\nabla g =0", "42470fd9b656409ce2b1977fba3d86bd": "\\sigma' = \\sigma - u,", "424748e4fdc07a74270ff36f1012a60c": "\\eta_0=\\omega_{\\rm s}/\\omega_{\\rm p}~~", "42474ba4cd75852bbf87a1954ca7785b": "Z[J_{ij}]", "42475f74c69e4091869250c8d2ce6e71": "\\begin{pmatrix}0 & 1\\\\-1 & 0\\end{pmatrix}", "4247d05c3bce7cd2af5e40dde937e9a5": "\\tau = \\{\\, G \\subseteq X : \\forall x,y\\in X\\ \\ x\\in G\\ \\land\\ x\\le y\\ \\rightarrow\\ y \\in G,\\}", "4247daad28d2cb7225d4ad1a039a1421": " Ff\\circ \\alpha = \\beta \\circ f", "4248115885d575be034437709117d2c5": "\\mathcal{G''}", "4248d0080a18612dc265195baefdd388": "\n\\begin{align}\n -\\rho \\mathbf{U}\\cdot\\nabla\\mathbf{u} &= -\\nabla p\\, +\\, \\mu \\nabla^2 \\mathbf{u},\n \\\\\n \\nabla\\cdot\\mathbf{u} &= 0,\n\\end{align}\n", "4248d18cad5c1ae84574f134213cd0d9": "I_i = \\frac{3}{2}^+ \\Rightarrow I_f = \\frac{5}{2}^+ \\Rightarrow \\Delta I = 1", "4248e38141d6a88a4ccc65e2123aa9d3": " F = x, G = x, V = q, E = f\\ (q\\ q) ", "42494764a8867339a8824ecbbfa29153": "\\frac{dI_\\nu}{ds}=j_\\nu\\rho-\\kappa_\\nu\\rho I_\\nu.", "424978768395f28dae62d44669428a81": "Z = Z + \\alpha X Y", "42498d45018eff9051fca2ef0250a5da": "{dQ_g \\over dt} = F_g (C_{art} - {{Q_g} \\over {P_g V_g}}) + R_{ing}", "42498deb9039b1dbf2f4c8547b88be11": "\\left(1-\\frac{2M}{r}\\right) \\dot{t}^2 - \\frac{M}{r} \\dot{t}^2 = 1", "4249a7004723ec5b4b5364f54c8b76be": "\n\\frac{1}{r} = \\frac{1}{b} \\cosh\\ \\left(\\frac{\\theta_0 - \\theta_2}{\\lambda} \\right)\n", "4249b20ea57fc3834bb6581bb1e819ff": "(x - \\alpha) ^{u+v}", "4249ccbc0cb371052f4f7e920a2901aa": "\\frac{d(g_i^{-1}Z_i)}{dx_i} = \\left(\\sum_{j=1}^{r_i} \\frac{(-j)T^{(i)}_j}{x_i^{j+1}}+\\frac{M^{(i)}}{x_i}\\right)(g_i^{-1}Z_i)", "424a07997eed927b9dffa7499b8cabd1": " \\gamma(0)=p", "424a9a54d706390729a8a3043b09b773": "\\mathcal{M}_R", "424a9c12e671c287d7928a0d4e212200": " \\Delta_i = \\omega_i - \\omega_l ", "424adfc996c50364089ec8b060b3dd29": "T:c_0\\to l_p", "424b2ee07ae26961428bcdab0230ce90": "Y \\sim N(0, \\sigma^2)", "424b35f0e7977ddd829eb729b504b8c2": "\nWC(\\theta;\\theta_0,\\gamma)=\\sum_{n=-\\infty}^\\infty \\frac{\\gamma}{\\pi(\\gamma^2+(\\theta+2\\pi n-\\theta_0)^2)}\n=\\frac{1}{2\\pi}\\,\\,\\frac{\\sinh\\gamma}{\\cosh\\gamma-\\cos(\\theta-\\theta_0)}\n", "424b48e2d0826957423fc319cf34c2f0": "10\\uparrow\\uparrow\\uparrow\\uparrow 8=(10 \\uparrow \\uparrow\\uparrow)^8 1=", "424bc0e7a9e47280be820b752a9cad42": " |i,\\epsilon_i\\rang = |i \\rang|\\epsilon_i\\rang ", "424bc99244f29fa18df1dccdba983b6e": "\n\\begin{align}\n \\Phi_{p_g}(\\omega) &= \\frac{\\sigma_w^2}{2VL_w}\\frac{0.8\\left(\\frac{2\\pi L_w}{4b}\\right)^{\\frac{1}{3}}}{1+ \\left(\\frac{4b\\omega}{\\pi V}\\right)^2} \\\\\n \\Phi_{q_g}(\\omega) &= \\frac{\\pm \\left( \\frac{\\omega}{V} \\right)^2}{1+ \\left( \\frac{4b\\omega}{\\pi V} \\right)^2} \\Phi_{w_g}(\\omega) \\\\\n \\Phi_{r_g}(\\omega) &= \\frac{\\mp \\left( \\frac{\\omega}{V} \\right)^2}{1+ \\left( \\frac{3b\\omega}{\\pi V} \\right)^2} \\Phi_{v_g}(\\omega)\n\\end{align}\n", "424c148f7df8a6e5d7f8a91927b0d29a": "d_2 \\;", "424cd1d59f4b1bde57904af5f50af637": "|\\{p\\}\\ \\mathrm{out}\\rangle", "424cea87b4a73f2266e458cb5c72fe12": "\\left(\\nabla^2 + k^2\\right)\\psi = 0", "424d1ee78a6310f7e501ecd2188211c4": "t = 1/r\\,", "424d228915a4babe67d493fcd27fc5eb": " \\ a = -1.47\\left[1 + 0.146{\\ e^{-2.9\\times {10^{-5}}He}}\\right] ", "424d265b35ffacee2ca484cd2bb1b934": "F < (\\frac{1}{2})^n", "424dab175dcc43b5049c820103f6d26e": "\nrx+sy = 1.\n", "424daddab8de03b31cc137752d2f8ac6": "\\frac{u_{i-1}-u_i}{\\Delta x}\\ f", "424df179dcc5585e2aa5c7089b00d96a": "\\frac{w_i}{n_i} + c\\sqrt{\\frac{\\ln t}{n_i}}", "424e3ec3b1aeff7fbc0e896787be47b4": "v\\left(x_0\\right)", "424e47f295054ebaad19bb6f78b9ab63": "S_y = \\left[y \\cdot x, y \\cdot Mx, y \\cdot M^2x \\ldots\\right]", "424e5439044d99934936695432e87f67": " V(\\mathbf x) \\to \\infty ", "424e59e7cb8e49738de96375fba71243": "\\hat{\\alpha}\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t} = \\alpha \\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}", "424ea1a6505faa6be86a5f960379fbcf": " 0 = x r \\cos \\varphi + y r \\sin \\varphi - z \\sqrt{R^2-r^2} \\,\\!", "424ed889d2d217646268a59b3f4f9600": "\\dot{v}_4 = -({1 \\over {C_4 R_6}} + {1 \\over {C_4 R_2}}) v_4 + {1 \\over {C_4 R_2}} v_3 + { 1 \\over {C_4 R_6}} v_1", "424ee0b9fb64762c201fc1a8445551d3": "(U_{\\alpha}, \\varphi_{\\alpha})", "424f00a638b08b9dbbbe9fecdbef0411": "\\scriptstyle W(m,n)=w(r)", "424f1b9605a49cc87b5089482e926e8d": "\\scriptstyle{ \\text{i}=\\sqrt{-1}} ", "424f32383af25170a18d6a6741214ea7": "\\pi_1(X,x)", "424f559ac74869a54d2a49934b1d6935": "\\lambda F_1 + \\mu F_2", "424fbd16c370c049836c35be01facb49": "\\frac{\\text{d}T(r)}{\\text{d}r}\\ =\\ -\\frac{3 \\kappa(r) \\rho(r) L(r)}{(4 \\pi r^2)(16 \\sigma) T^3(r)}", "424fc320d21943d444a2cd2e058d2677": "(x+\\sqrt{-m}y)(x-\\sqrt{-m}y)", "4250992fa634c94a4e24b2dab70415f9": "\\omega(s)\\sim s^{-\\rho}L(s^{-1}),\\quad\\rm{as\\ }s\\to 0", "4250a8fd3047d00d29e3dc0ea8e75a28": "k = 2\\pi f/v", "4250d722539f66405872216a82eef3a7": "\\cos(\\phi)r=-10M ", "4250d9e85e47a0762179119666ef20e5": "d\\Omega=\\sin(\\theta)\\,d\\theta\\,d\\phi.", "4250e3f207a1af353d50c389162941fe": "o(f(n))", "4251f0ad3b1eccf575175b3ee4844aaa": "\\sum F_x=0=R_{Ax}", "425227eb53eb4743a69e265f53338719": "\\mathbf{x}^{(k+1)}", "42522eb29157c8d54e0826a2b5437e25": "\\ M - m = - 2.5 \\log_{10}(F_1/F_2) \\,.", "4252942d1bf1d9ea96db94698a1165ca": "\\partial^2u/\\partial x^2+\\partial^2u/\\partial y^2+k^2u=0", "4252cb9880b07af1fe0e1e30105dec76": "D_n(z) = (n! \\sqrt{\\pi})^{1/2} \\psi_n(z/\\sqrt{2}) = \\pi^{-1/4} \\sqrt{2} \\mathrm{e}^{z^2/4} \\frac{d^n}{dz^n} \\mathrm{e}^{-z^2}", "42533db4801c065d27aa64de5d9e6523": "\\mathbf Z_2", "425346384351ead86814aaba2c146077": " \\widehat{\\mathit{G}} ", "4253506e0812d7950567b33c04620a43": "(0,\\varphi,1)", "425396a9bf798abc57d1d1a370f7fe1f": "\\left(\\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ -2\\sqrt{2/5},\\ 0,\\ 0,\\ 0\\right)", "4253ae7949b0d6fdd5e94ef1ef77d2c5": "\\partial_t + \\gamma ", "4253de1326cf0c8929e12faae452ee04": "{\\rm Aff}({\\Bbb R}^n)", "4253e68b337d7957392c4c4216184edb": " \\Big(2(xz+wy) , 2(yz-wx) , 1-2(x^2+y^2)\\Big) , \\,\\!", "425432e2e6441af660e3f6853186922e": "\\begin{align}\n \\mu &= \\sum_n n\\cdot\\Pr(N=n\\mid M=m,K=k) \\\\&\n = \\sum_n n\\cdot [n\\ge m]\\frac {m-1}n \\cdot \\frac {\\binom{m-2}{k-2}}{\\binom{n-1}{k-1}} \\\\&\n = \\frac{m-1}1 \\cdot \\frac{\\binom{m-2}{k-2}}1\\sum_{n=m}^\\infty \\frac 1{\\binom{n-1}{k-1}}\\\\&\n = \\frac{m-1}1 \\cdot \\frac{\\binom{m-2}{k-2}}1 \\cdot \\frac{k-1}{k-2} \\cdot \\frac {1}{\\binom{m-2}{k-2}}\\\\&\n = \\frac{m-1}1 \\cdot \\frac{k-1}{k-2}\n\\end{align}", "4254a198f563646e8c474436a364c015": "a\\le b", "4254e666ab48d42ae78d66a315f59ce0": " \\tau,", "425541079435bba2bcdebcd1ec38d03d": "\\phi_x = \\frac {F x} {2 E I} (2L - x)", "42554d81c358d321457ae531802c830f": "FPT", "42556a93b0ddd9a1e1b8c170c8356144": " F\\left( J \\right) = \\tilde B J \\left( J+1 \\right) \\qquad J = 0,1,2,...", "42557970c726972bb1ff49a1d0b910e9": "\n\\partial_\\mu\\partial^\\mu \\langle \\phi(x) \\phi(y)\\rangle =0\n", "42558176045ca7bf8fff2819a7294c39": "2q^2 = p^2, \\, ", "42558428b9f01a9b749f5eb75083e64c": " \\Delta[x,z] = \\sum_{x < y < z} [x,y] \\otimes [y,z] \\ . ", "42560dd5f3d263238eed92625fa58f9a": "\\{f,g\\} = \\sum_{i=1}^{N} \\left( \n\\frac{\\partial f}{\\partial q_{i}} \\frac{\\partial g}{\\partial p_{i}} - \\frac{\\partial f}{\\partial p_{i}} \\frac{\\partial g}{\\partial q_{i}}\\right).", "42563ba88d1dbdd6b4849664802858ab": "\\sigma= \\sigma_0e^{-(T_0/T)^{1/(d+1)}}", "42566d3b0c5eb96fdd0536ba77a7d93d": " j = 1, \\ \\dots \\ , \\ n\\ ", "42568c0e4c8257d0edffec7fc9a2cd7e": "h\\in F", "4257b521592098cc025fa1049834dda9": "\\scriptstyle \\mathcal{C} \\;\\times\\; \\mathcal{C}", "4257c0633ca9666cf8a791beeb913225": " [\\widetilde \\tau,\\widetilde \\tau']=\\widetilde {[\\tau,\\tau']}.", "4257c2fc29594bf3603ea34b8321a408": "Ih_p < \\epsilon", "4257efbf2323def703a3148bb8b7e95d": "k\\partial_k g_\\alpha(k) = \\beta_\\alpha(g_1,\\cdots,g_N)", "425801eab1cf896cb55b2f975e11330d": "T_s", "42581eedd132d2be595aced7b4353c0b": "\n\\boldsymbol{\\hat \\varphi}\n=-\\sin (\\varphi) \\boldsymbol{\\hat{\\imath}} + \\cos (\\varphi) \\boldsymbol{\\hat{\\jmath}}\n", "42583aa38fc96475df5e465901aac1f0": "\\begin{pmatrix}0 & -1 \\\\1 & 0 \\end{pmatrix}", "42584db47fbaf437a4e7f04a93195bb3": "\n\\frac{d \\vec x}{dt} = \\vec v(\\vec x, t)\n", "42585dfd51eb79bead788950faced471": "\\frac{N_\\nu}t = \\Phi_o \\frac{\\lambda}{hc}", "425864d6a963fe06e6fa31364cfed1c9": "\nu_{xx}=xu_{yy}. \\,\n", "425875033afc430568b566139ec56a9d": "v_1 =1.28, v_2=2.54", "42587df18003ea4f059f7cc38aebabe0": " {\\mathrm MinN}(L, D+1 , n) \\le M*{\\mathrm MinN}(L,1,n^M)", "425963845ee523dd2d6291ee7540ccbd": "\\begin{align}\n\\pi_{(\\frac{1}{2},0) \\oplus (0,\\frac{1}{2})}(J_i) &= \\frac{1}{2}\\biggl(\\begin{matrix}\n\\sigma_i&0\\\\ 0&\\sigma_i\\\\\n\\end{matrix}\\biggr),\\\\\n\\pi_{(\\frac{1}{2},0) \\oplus (0,\\frac{1}{2})}(K_i) &= \\frac{i}{2}\n\\biggl(\\begin{matrix}\n\\sigma_i&0\\\\ 0&-\\sigma_i\\\\\n\\end{matrix}\\biggr)\\\\\n\\end{align}.", "4259808a0db5175e7f8935688547e1b7": "\\ O[s(t)] = (G V - \\frac{3}{4} D_3 V^3) \\cos(\\omega t) - (D_3 \\frac{V^3}{4}) \\cos(3 \\omega t)", "42599104f351380f01a3e6d5d305036f": "R(r)=Aj_l\\left(\\sqrt{2m_0(E-V_0)\\over\\hbar^2}r\\right),\\qquad r0}(1-e^{-\\alpha})},", "425a41b39ecc41c83b92bc78a14cd81e": " \\sum_{i=1}^n g(x_i) ", "425a78d383c3e5e5f9f095de1d5c5115": "|z|", "425a9f8bc4863fac698d2a3b60bdb333": "\\mathrm{Spec}(R)", "425aca3b35f4f8643f7c72258af3a116": "\\delta(q_s, \\sigma) \\to q_t", "425ada0c9121053550c6c58b9c583217": "H_A \\oplus H^\\perp_A", "425b2f972aa1a988dc90aaba236792b1": " e_i = y_i - \\widehat{y}_i ", "425c213c42d56c81945b459b30f02246": "\\subseteq \\!\\,", "425c3fa4142598de0fcc3d93255cdeed": " R(S) = \\frac{1}{|S|^{2}}\\sum_{f_{i},f_{j}\\in S}I(f_{i};f_{j})", "425c897890ed8666658ce3b6f5c74c00": " t(s_2) = C m(s_1,s_2) + D M(s_1,s_2) ", "425cca4f3a2a3be2cd1b2912b878fac3": "S_1,...,S_n", "425ccf9be28a9b7f7dadfe1a9874217d": "r\\approx0.4", "425cfc631bb9d6c046767c6f3f4a2554": "-N_\\text{s}/N_\\text{a}", "425d0f5f0e8f3c4b786221f15472d302": "\\gcd(p,q)=\\gcd(a_1p+b_1q,a_2p+b_2q)", "425d13b5524b7a16fe71f32f1925d9aa": "K_1=\\frac{[\\mathrm{HCrO_4^-}]}{[\\mathrm{CrO_4^{2+}}][\\mathrm{H^+}]}", "425d1d2ce186e71298004f8c43b8b037": "\\Box\\mathbf A = \\mu_0 \\mathbf J", "425d28f57e9000159acb2eb54e0d8d71": "|l|\\leq n", "425d39f88f46610f917191c8fdd0ab2a": "H_4(a, b) = a\\uparrow\\uparrow{b}\\,\\!,", "425d7601e762562fcc18671cd1e049fe": "q = \\frac{\\cos \\left( \\frac{1}{3} \\arccos \\left( \\sqrt{\\alpha} \\, \\right) \\right)}{\\sqrt{\\alpha}}\\!", "425dcadd60bc24012e9937e9a7cc92a2": "s_i = \\frac{M \\pm 1}{2}", "425e247beeb2a7d65139b26948c35ad9": "Q_x", "425e3cc283318b6f9c183e39fe9f26dc": "\n\\frac{dM}{dD}\\frac{dD}{dt} + 2.3026 \\; atanh \\left( \\frac{D-4.9}{3} \\right) = 5.2 - 0.45 \\; ln \\left[\\frac{ \\Phi (t - \\tau) }{4.8118~\\times~10^{-10}} \\right] \\;\\;\n", "425e484dfc0b078a48c3111e0221432b": "R = \\sum_{a\\in\\Sigma} V(a) R_a", "425e7db50e20c299ed6928508ffbd226": "S \\rightarrow aSb", "425eaab3ad1df1ecff057051f2f09022": "4 \\rightarrow \\infty", "425ef1a4253b81d145840a14469c37c2": "m_\\mathrm{interior}=\\frac{4}{3}\\pi r^3 \\rho", "425f32afca39072bd74f3777c2a4db75": "V(u)=\\Sigma E(e) \\mod |V(G)|", "425f3cdac65a2e883da34e209f7e819b": "\\textstyle Q=Q_1R", "425f7c0e7289f79fb78b743a3be54349": "\\psi_1(\\alpha) = \\frac{d^2\\ln\\Gamma(\\alpha)}{d\\alpha^2}= \\frac{d \\, \\psi(\\alpha)}{d\\alpha}.", "425f9a63f0c6a5e82addf6ffa3a41768": " C(x_1, x_2, \\dots , x_n) \\in [ \\min(x_1, x_2, \\dots , x_n) , \\max(x_1, x_2, \\dots , x_n) ] ", "425f9cd83439afc5f8edd8b5a5946b10": "E_x\\,", "425fa7e4f953742aaa9dae3c161b09ee": "P (X)", "425fb9061ecae36e549e8c414e9f9790": " T(h,a) + T(ah,\\frac{1}{a}) = \\frac{1}{2} \\left(\\Phi(h) + \\Phi(ah)\\right) + \\Phi(h)\\Phi(ah) - \\frac{1}{2} \\quad \\mbox{if} \\quad a < 0 ", "425ffa0651e0cfefc96090471390586d": "a_i(w) = 1", "425ffae0345990882490ad7de949a192": "V(+\\infty)=V(a_0, b_0, c_0, d_0, \\dots)\\,", "426012dedf23faa0db42e274a9720ed8": "F_{ii}", "426059b5aca3f076a9bf8f10fbe49fe3": "\\langle f_j|O| \\psi \\rangle = \\sum_{i=1}^{n} c_i \\langle f_j| O | e_i \\rangle = \\sum_{i=1}^{n} \\langle f_j| e_i \\rangle \\langle f_i | h \\rangle = \\langle f_j | h \\rangle, \\quad \\forall j ", "426060e376331d25673fce0bd5cf7a64": " \\mathcal{F}_n", "42609d717a42de1d8e7a61d184de3f14": "\n\\frac{6}{\\pi^2}<\\frac{\\phi(n)\\sigma(n)}{n^2}<1.\\;\n", "4260a648198f2bd6bd26b96844e63c98": " G(0) = \\sum_{i\\omega}(i\\omega-\\xi)^{-1}", "4260af3412073f93c02bd191e149d7fc": "\nD=\\frac\n{d}\n{\\sqrt\n{\\hat{V}(d)}\n} = \n\\frac\n{\\hat{k} -\n\\frac{S}{a_1}\n}\n{\\sqrt\n{[e_1S+e_2S(S-1)]}\n}\n", "4261033040b69bd23fb9c25c52c57d91": "\n\\mathrm{A}_{x}\\mathrm{B}_{y} \\rightleftharpoons x\\mathrm{A} + y\\mathrm{B}\n", "426177b49818756709a60be608b07942": "\\ell_V\\colon V\\rightarrow\\Sigma_V", "426198648cca5ec4076dae349d676763": "P(c|d)", "4261996d6da033a98eeb6288b867b368": "Q_P", "426213a42e5e4d585efc277d7dbd49c2": "{\\rm MSe}\\,\\,\\, = \\,\\,\\,\\sigma ^2 + \\,\\,\\,\\beta ^2", "426221597d87f213baee77cfe9dbfd44": "\\mu = \\mu_0(1+\\chi_v)\\,", "42622b5756c4a2a3b6b6f9a565c93088": "G_{\\mathrm{F}} / (\\hbar c)^3", "4262302f1cf914294417243d2a81f9c2": "t_9\\ ", "426231f80e2bde80c9f4c0467de11bf8": " C \\,", "42628747b1fc30b325fb760a6cf34dbe": "|\\psi(t)\\rang = U(t)|\\psi(0)\\rang = c_+ e^{\\frac{-iE_{+}t}{\\hbar}} |+\\rang + c_- e^{\\frac{-iE_{-}t}{\\hbar}} |-\\rang,", "4262b224a3b1ae639b1e789ad6fd575e": "I_n = \\frac{2x^n\\sqrt{(ax+b)^3}}{a(2n+3)}-\\frac{2nb}{a(2n+3)}I_{n-1}\\,\\!", "4262c8db51dec07c4f89ed53455d8a0c": "\\mathbf{U} \\circ \\frac{\\partial \\mathbf{V}}{\\partial x} + \\frac{\\partial \\mathbf{U}}{\\partial x} \\circ \\mathbf{V}", "4262d04543c9bdf2b038fd3527d3175c": "r_<", "4262e2b481cfd3e8519e58c7bdcb1636": "X^1", "4263bcc7bc168895facd5b60450f16ec": "\\mbox{E}(x)=(ax+b)\\mod{m},", "4263c559de37f0088aac94410aeb7e79": "\n J_1\\left( x\\right)\n=\n{a_1 \\over L_B} {1 \\over 2 \\pi r} \\delta^2\\left( r \\right)\n", "4263e4ac50dff06f5ade5b13e0e12a23": " H' = \\frac{1}{r_{12}} ", "42641517673fd335d2fa1fe0048d9188": "\\sum_{i=1}^n \\psi(x_i) = 0", "42645086efff9e3970aa5e168a67faba": "\\beta_m", "42645e73a70ff88cd2889d7ddc9fd2c9": "(\\boldsymbol{\\alpha},S)", "42646d1fc4929621c32c8796e859ed91": "\\tilde \\nu = \\tilde \\nu_{vib} \\pm BJ(J+1) ", "4264d66f1f31e05416ee71f358deca15": "\\frac{11\\cdot\\pi}{3(\\sqrt{6}-\\sqrt{2})}", "426506953ad812039e25466b599bab88": " u_0(r_0) ", "426563fe91285460e65a52736712e400": "\\theta > 0\\,", "426574d91cdd5785d24489b352861d0a": " \\{X_{\\mp},Y_{\\pm},Z_{\\pm}\\}_{\\pm} := [[X_{\\mp},Y_{\\pm}],Z_{\\pm}].", "42659dc079baeee930b3de4ecaa364b2": "\\operatorname{E}(L(\\delta_1(X)))\\leq \\operatorname{E}(L(\\delta(X)))\\,\\!", "4265a578b691b6bdbaba672436f2b87b": "(B, \\cdot) ", "4265d0e91da1efc6d3bc2367c63f66f3": "x,y,\\ldots", "4265da5ea4a11e6656af2deb8be3352e": "\\scriptstyle{\\mathrm{R}^- =_{\\mathrm{def}} \\mathrm{R} \\setminus \\{\\emptyset\\}}", "4265e169ff76bf88ee34a23918d80b0d": "\\mu_0\\mathbf{J}=\\nabla\\times\\mathbf{B}", "426676e7a78610f0cd3e9886380fe7b5": "J(f_c)", "4266ad6ad7a5261b7656cb989465fef4": " \\sum_{i=1}^m{a_{ij} x_i} \\cdot y_j + e_j t_j \\cdot y_j \\ge g_j \\cdot y_j ", "42670becb84f0d3ae7dce1df514461ab": "\\rho_{\\mathrm{free}} = \\nabla\\cdot \\mathbf{D}", "426713dbbfd20aeb04b3301bff118ee0": "f'(x-):=\\lim_{h \\to 0^-}\\frac{f(x+h)-f(x)}{h}", "42674790ff9dfedc782f38508e41f0e0": "\\cot \\theta", "4267578ce1fd83763caa4213d9334bb8": "I + \\operatorname{Ann}(M).", "4267e2b5aec24534f74ed5e976473db1": "\\beta\\ ", "4267ec256fd7daff57d1d4e413225299": "\\Phi _E", "4267f11548214449cddfbf21b72cbd52": "log (t_r')", "42680cbe259cb41b1a4a81c009e11218": "\\left((-1)^{n-i} L_{n-i}^{(\\alpha)}\\right)_{i=0}^n", "4268298cff5b424660d2fe7e2d4922ba": "x^*=(2x_0)^{1/2} -x_0", "426836267ccaca343958f1b10bafaa00": "\\mathbf{U}_i \\equiv \\{\\mathbf{u}_i,\\mathbf{u}_{i+1}\\dots,\\mathbf{u}_{N-1}\\}", "426887048c4eab17cdc9bf7763e433be": "{{f}_{M}}=\\sum\\limits_{m=0}^{\\infty }{{{\\theta }_{T}}\\left( \\left\\langle f,{{g}_{m}} \\right\\rangle \\right){{g}_{m}}}", "4268f5c4138b3ecc5ce07ea8bce6b1ce": " \\mathit{m} ", "4269646ed27ef0e6011b0b4ec662240d": "G_a(z) = \\ln(1+z)/z ", "4269e319af3366a1399e8aa10d64c95d": "(\\nabla_uR)(v,w)+(\\nabla_vR)(w,u)+(\\nabla_w R)(u,v) = 0.", "426a7cf5c66c0627ce6d667a9cd5098e": "\\frac{dK_\\nu}{dz}=-(\\alpha_\\nu+\\sigma_\\nu)H_\\nu", "426a8e2e827b548650b1042ea1ba8007": "2\\bar B", "426aca80e0eca511cc5fc9e00c0ed633": "\\frac{11\\pi}6\\!", "426b8d1528c9f8cc6f37829cb13be224": "(x_1 \\vee \\neg x_3) \\wedge (\\neg x_2 \\vee x_3) \\wedge (\\neg x_1 \\vee \\neg x_2)", "426b8db34c98c1bd1863eca9e63dadda": "z=x+iy=R(\\cos \\theta +i \\sin \\theta)=Re^{i\\theta}\\ ,", "426bfa2fa3d5e1ae9b2bd14b000cdb9c": "~[v]_\\beta=Q[v]_{\\beta'}", "426bfa35c85fd1dd962c06247ddccbda": " {\\mathbf{}}S(T) = F.", "426c60fa511716c83df6014075068e3e": "\n\\begin{align}\n& \\sum_{i=0}^{n } a_{i } x^{i+1} + \\sum_{i=0}^n a_i x^i \\\\\n& {} = \\sum_{i=1}^{n+1} a_{i-1} x^{i } + \\sum_{i=0}^n a_i x^i \\\\\n& {} = \\sum_{i=1}^{n } a_{i-1} x^{i } + \\sum_{i=1}^n a_i x^i + a_0x^0 + a_{n}x^{n+1} \\\\\n& {} = \\sum_{i=1}^{n } (a_{i-1} + a_i)x^{i } + a_0x^0 + a_{n}x^{n+1} \\\\\n& {} = \\sum_{i=1}^{n } (a_{i-1} + a_i)x^{i } + x^0 + x^{n+1}\n\\end{align}\n", "426c64a08f24ef217c8cd22014d9809d": "D^{KL} \\,", "426d105c75535de96935633770a656ed": "F_r = 0 \\,", "426d202ae80a25037c7c5421a06b4df8": "f(x_0),", "426dc4c4b2459465fde5b99bb8d452e5": "(r/a)<1", "426e0f651496b00d1e5905edcfe109f9": " \\operatorname{Var}(8+2K, 44-K , 48(22-K)/11, 44+k) \\, ", "426e39351f24bce1629ec659f62e5297": "\n\\begin{align}\nP(X_{(k)}=x)&=P(X_{(k)}\\leq x)-P(X_{(k)}< x) ,\\\\\n&=\\sum_{j=0}^{n-k}{n\\choose j}\\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\\right) ,\\\\\n&=\\sum_{j=0}^{n-k}{n\\choose j}\\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\\right).\n\\end{align}\n", "426e6b541860f7dcc58c5c6bc0c4ded9": " u(x) = \\frac{x^{1-\\gamma}}{1-\\gamma}. ", "426e898a3c4c14f8f02b329969bd35a2": "\\mathbb{N}^k", "426ecefe566218aa579cfb7ecaac4307": "A_{rx}", "426f02a6baf8ac47a4056090952dc546": "\\scriptstyle \\delta t_{\\text{clock,rec}}", "426f1523dd3c28d5df25c8c96783d748": "X \\in \\mathbb{B}", "426f1c678ae461e6015109e8b731573d": "\\Lambda=\\Lambda(x,t)", "426f2e4f780739b7213179900f20c6c6": "f(\\partial V)", "426f2e5e517deb5e5e11a12b6d904af1": "(V(x_1) \\wedge T(x_2)) \\rightarrow (R(x_1,y_1) \\wedge R(x_2,y_2) \\wedge H(y_1, y_2) \\wedge H(y_2, y_1))", "426fc3e8ee31359f18dda5d84a6a4b8e": "x^{1-\\varepsilon} \\ll N(x) \\ll \\frac{x\\log\\log x}{\\log x}", "42700b0757e42c141bee4446a455785f": "h\\in\\mathcal{O}", "42700ff4611cae507eb4336661174a57": "V_{eff}^{1-loop}=\\dfrac{1}{64\\pi^2}\\operatorname{str}\\bigg[M^4\\ln\\Big(\\dfrac{M^2}{\\Lambda^2}\\Big)\\bigg] = \n\\dfrac{1}{64\\pi^2}\\operatorname{tr}\\bigg[m_{B}^4\\ln\\Big(\\dfrac{m_{B}^2}{\\Lambda^2}\\Big)-\nm_{F}^4\\ln\\Big(\\dfrac{m_{F}^2}{\\Lambda^2}\\Big)\\bigg]", "427043c895f8a91ca3185d426f96eb85": "c_{2,0}(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta) = \\alpha \\widehat{x} \\beta \\gamma y \\delta", "427072a5e2d063dbbe9e8d00fd8bf955": "a_{1}+b_{1}", "4270a93c48c370b5bc829b09ed05b6ff": "A\\subset O", "4270ab6db5eb134158ab54f4495d43f6": "\\frac{2b}{3}", "42718852c4dcd56b34c9be92f8b92dde": "\\epsilon_{\\mathrm r}", "4271aa52ec6bcd80f1803cd23ba626cc": "XX''\\,", "4271c168a17eb762ef87b45c8bb5e9f3": " F = \\frac{\\left|q_1q_2\\right|}{4 \\pi \\varepsilon_0 r^2}\\!", "42726af7114dfa5c4c9a182f988d4617": "\\deg(u) \\leq g", "4272714e96f92e02f839845402d534d9": "m_{\\rm star}=m_{\\rm sun}-2.5\\log_{10}\\left({ L_{\\rm star} \\over L_{\\odot} } \\cdot \\left(\\frac{ d_{\\rm sun} }{ d_{\\rm star} }\\right)^2\\right)", "4272e3c4602ee5fc8ed11ac77ca7e3c7": "(B+B_R) / 2 + (B-B_R) / 2 = B", "4272f0b94b28db0a68a7826d56cb21ae": "F_i = J^{i-1}M/J^iM", "427361721966c6806fa4f1922a8a2703": "z \\succeq x ", "4273622c979699f63c31fa30a6bf4e74": "\\operatorname{Majority}(x,y,z) = (x \\wedge y) \\vee (x \\wedge z) \\vee (y \\wedge z);", "42736a2e33fab9f1f7d875f70ff9f0ca": "\\mathbf{F}_{\\mathrm{Centripetal}} = -m \\omega_S^2 R \\mathbf{u}_R \\ . ", "427380e783dcae4467a8257e38fda6fc": "\n \\boldsymbol\\mu = \\begin{pmatrix} \\mu_x \\\\ \\mu_y \\end{pmatrix}, \\quad\n \\boldsymbol\\Sigma = \\begin{pmatrix} \\sigma_x^2 & \\rho \\sigma_x \\sigma_y \\\\\n \\rho \\sigma_x \\sigma_y & \\sigma_y^2 \\end{pmatrix}.\n ", "42738b36b7c30f037210f8f87bfac1b5": "C:[0,1]^d\\rightarrow [0,1]", "4273bc13a84a118feb5adc1b09b0815e": "\nR_{\\mu\\nu\\alpha\\beta}=\\frac{-1}{\\alpha^2}(g_{\\mu\\alpha}g_{\\nu\\beta}-g_{\\mu \\beta}g_{\\nu\\alpha}) \n", "42745366749fe47237d5f4e4fa348016": " \\frac{23.976}{29.97} = \\frac{4}{5}", "427459df51f0ec2b101b12f7385619c4": " I_{cont} \\sim \\frac{4 \\pi a M_A D_{AB}}{RT} \\left( P_{A \\infty} - P_{AS}\\right)", "4274b85d2e83caf14ea2d4ccd513b57d": "\\,\\alpha_1 < \\alpha_2 < \\alpha_3 <\\alpha_4\\,", "42750293fa9f998abaedab82b55dd25a": "\\begin{bmatrix}\nT_1 & T_2 \\\\\n& Read(A) \\\\\nRead(B) & \\\\\n &Write(C) \\\\\nWrite(C) & \\\\\nCommit & \\\\\n& Commit \\end{bmatrix} \\Longleftrightarrow\n\\begin{bmatrix}\nT_1 & T_2 \\\\\n& Read(A) \\\\\nRead(B) & \\\\\n& Write(C) \\\\\n & \\\\\nCommit & \\\\\n& Commit\\\\\n\\end{bmatrix}\n", "427548822282a314fcb35a7bf7c6c0a2": "\\mathbf{n}_i\\,\\!", "42756e5f249fa6daa01fc021abfd36fd": "\\scriptstyle{t_0 \\rightarrow t_1}", "42765a2a343bda8f5fb4853a62978638": "\\Omega_{-}(k)^2={ {\\omega^2+\\Omega^2-\\sqrt\n { {(\\omega^2-\\Omega^2)}\n^2+4{g}\\omega^2\\Omega^2 }\\over 2 }},", "42765fc804444b95666b67061c3dd27a": " \\Delta Cd ", "4276d8265b276f1ed33c5b28e2a57f60": "\\mathrm{RMS}\n =\\sqrt{\\frac{1}{n}\\sum\\limits_{n}{{{x}^{2}}(t)}}\n = \\sqrt{\\frac{1}{n^2}\\sum\\limits_{n}{{{\\left| X(f) \\right|}^{2}}}}\n = \\sqrt{\\sum\\limits_{n}{{{ \\left|\\frac{X(f)}{n}\\right| ^ 2 }}}}.\n ", "4276e72a30f3a220e774f0ad3d48f35f": "H=\\frac{1}{2}m \\omega^2 x^2 + \\frac{p^2}{2m}.", "4276e96010a080f95d71d9a4d26cf4cd": "\\begin{matrix}\n\\left\\{\n\\begin{matrix}\nx &=& l \\sin \\varphi\\\\\ny &=& - l \\cos \\varphi - a \\cos \\nu t\n\\end{matrix} \\right.\n\\end{matrix}", "42773cf30789f182d9af7dd133b6f347": "\\textstyle \\begin{pmatrix}1\\\\0\\end{pmatrix} ", "4277439fd7d2263ea6a873f013411770": "d \\tau = \\int_P \\sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2}", "42775e7af2e8fd2a4e11096cd5fa5f5c": " p_0 = \\exp( -a m^b ) ", "4277824228304f33c67a98cf24bdec88": " \\frac{k_v(V)}{k_p(P)} = \\frac{P}{V} \\quad\\forall P, \\forall V", "4277a241d7cb93d14a7b3a3f52ada7a2": "F = \\lim_{\\alpha \\to 0} \\alpha E_{\\alpha, 0}(1).", "4277b48b94bcb3890a8901775ace0b75": " \\mathbf{K}_x = \\left( k(x_1, x), \\dots, k(x_n, x) \\right)^T ", "4277b7828d281c4cfb703c0f2fbd2c9c": "\\begin{pmatrix}3/4 & 5/4 \\\\ 5/4 & 3/4 \\end{pmatrix},\\ \n\\begin{pmatrix}-3/4 & -5/4 \\\\ -5/4 & -3/4\\end{pmatrix}, \\ \n\\begin{pmatrix}-5/4 & -3/4\\\\ -3/4 & -5/4 \\end{pmatrix}", "4277cedba9b41951e69e4730a65c749c": "\\{(z,w)\\in \\mathbf{C}^2;~|z|^2+|w|^{2/p}<1\\} (p>0,\\neq 1)", "4277dc3352c83fe5d634faec1da0d7e1": "\\vec r_B-\\vec r_A= \\underbrace{\\vec r_{Bi}-\\vec r_{Ai}}_{initial\\;separation} + \\underbrace{(\\vec v_B-\\vec v_A ) t}_{relative\\;velocity}", "427815c285fab26818e0f641f85820c4": "t_2s_2\\overline{t_2s_2}^{-1} = (1\\ 2\\ 3).", "4278ad1da641aa231db44b20babcac2f": "Re(s)=\\gamma", "4278bb3cac8742331e864984a49ec42b": "\\sum_{n=- \\infty}^ \\infty e^{in \\theta} \\left ( f_n + (2-n) \\bar{f}_{2-n} + (1-n) \\bar{g}_{1-n} \\right ) = 0", "42793a63c36c6bb296abcc3c7ad6a325": "\\alpha(n)=d(n)-\\mathbf{x}^{T}(n)\\mathbf{w}_{n-1}", "427989bd61f58f43f9c243188271d372": "\n\\frac{d^{2}u}{d\\varphi^{2}} + u = -\\frac{1}{mh^{2}u^{2}} F(1/u)\n", "42798e799120ab732c0a0a1d358b514f": "\\zeta(12) = 1 + \\frac{1}{2^{12}} + \\frac{1}{3^{12}} + \\cdots = \\frac{691\\pi^{12}}{638512875} = 1.000246\\dots\\!", "4279f5102f315301646f7ddc6dcdc1c1": "\\prod_{i=1}^n R_i", "427baa4ed6f4cb4e56c083bf73393d88": "V_f \\subset \\C^2", "427bce8408862fb69fefaf2922b09e68": "I \\xrightarrow{a} G", "427be496f24da888388e3613e694eebd": "n/2 = 2^{m_1}+l_1", "427c4d5401cc63c8b185ca1c3c36c667": "h(F) \\in \\{X, Y \\wedge Y\\}_{\\mathbb{Z}_2}", "427c9bb02b992298de24c36227f5c426": "v_{e,th} = \\sqrt{\\frac{k_B T_{\\mathrm{e}}}{m_e}}", "427caada6f03e5c817794e852e48445f": "f^{-1}(p)=\\{x_1,x_2,\\ldots,x_n\\} \\,.", "427ce3e31142b1e5cc412b0d739d3806": " {dz \\over dt} = iz,\\ dt = {dz \\over iz}. ", "427d0b93beb87bbdf08977ab8f57cb4b": "P(a,b)\\in C(K)", "427d2031ba00b8a64c150d7f2b78078e": " \\sum_{j=1}^{n} w_j = 1", "427d7be0aa8c242d6b826adbe7a56cf2": " \\frac{4^n}{2n} \\le \\binom{2n}{n}.\\ ", "427e45d5f79651519b772059a85d8786": "\\int_{\\frac{-a}{2}}^{\\frac{a}{2}} x^2\\sin^2 {\\frac{n\\pi x}{a}}\\;\\mathrm{d}x = \\frac{a^3(n^2\\pi^2-6(-1)^n)}{24n^2\\pi^2} = \\frac{a^3}{24} (1-6\\frac{(-1)^n}{n^2\\pi^2}) \\qquad\\mbox{(for }n=1,2,3,...\\mbox{)}\\,\\!", "427e492ca9dbad83779c310d9601d261": "\\log (g)", "427ec0c8f3cf7d2b05214c77425e2544": "f = {1\\over 2 \\pi} \\sqrt {k\\over m} ", "427eec08b15dcd292049f434469ed10e": "\\Phi_A(\\mathrm{id}_A)=u", "427ef152422a895bff2e5792f8ee35ba": "O(\\min(n^2,s \\log n))", "427f0ce315efb1031323ac4be4355220": "X \\sim \\textrm{Cauchy}(x_0,\\gamma_0)\\,", "427f456381bc6093528df90c0bb30514": " \\Delta M = M_\\Sigma - M_\\mathrm{nuc} \\,\\!", "427f85b579fb3accc6304370b8ab8559": " A(c)=1/c", "428004a9d09fff43535cf195e84de395": " y_{n+1} = 2y_n - y_{n-1} + \\tfrac{1}{12} h^2 (f_{n+1} + 10f_n + f_{n-1}). ", "4280414b9d17666128b04745843e4586": "H_1 (\\tilde P_n, \\tilde \\partial)", "428052357623a49382e2d11825ef2a77": "\n\\int_S \\liminf_{n\\to\\infty} f_n\\,d\\mu\n\\le \\liminf_{n\\to\\infty} \\int_S f_n\\,d\\mu\n\\le \\limsup_{n\\to\\infty} \\int_S f_n\\,d\\mu\n\\le \\int_S \\limsup_{n\\to\\infty} f_n\\,d\\mu\\,.\n", "42807b57c6b27260c80d33fd0ffee98e": "H\\psi =E\\psi", "4280b31c897fb2fb76ecc4f56f491d41": "\\mathbb{STUVWXYZ} \\!", "4280e23702b54ecd32e5c9cb684628e0": "V_s = \\frac{2}{9}\\frac{r^2 g (\\rho_p - \\rho_f)}{\\mu}", "42812c91c7891158488c2069336fd459": "(-1)^n (n+1)^{-(n+1)} \\int_0^\\infty u^n e^{-u} du\\, . ", "42823004aad3a03694f0c776a844be65": "A^{j}", "428241b038ec98d05a10e53565bcf093": " n \\ge n_0 ", "42829db18c6a0f9101f9b7596f5826aa": "\\{(\\mathbf{E}_1, \\mathbf{E}_2, \\mathbf{E}_3), O\\}", "4282c1db24572fe84f6ff25486dab0d8": "\\vec{F} = -\\nabla \\Phi. \\,", "42834911cde57eec1d23a38f0950a835": "A + \\operatorname{core}B = \\operatorname{core}(A + B)", "4283528710c2408b5c0b5877c15e204a": "\\sum_{j=1}^n X^{j} (\\varphi(x)) \\frac{\\partial}{\\partial x_{j}}(f \\circ \\varphi^{-1}) \\Big|_{\\varphi(x)},", "428352ec0825d1a93981d0a548d9a686": "(4)\\; h_j=\\frac{y_1 \\sqrt{1+8 F r_1^2} - 3 y_1}{2}", "42837fd53e6ca2b990a8b28f62cc4452": " (E_{d/s}) = (E_{u/p} + Z) = (6.04 + 2) = 8.04ft \\,\\!", "42838abb9d69312263267c3334202666": "k < 1", "42838c85e0c7dd0c3f6aac8ff72f84d4": "(n \\pi r^2)/A", "4284146e97e783dc07ff79444ee3c257": "\\mathbf{Q}/\\mathbf{Z}", "4284e008aa30f9c0dc6b39f70cf09b3d": "\\beta_T^{max} > 1", "428535d8ab51652628acc8a2bae500bf": "R_{n,m}", "42854b79fb3116cb56fbb02f5634f12f": "\\nabla_i = \\nabla_{{\\mathbf e}_i}", "428564f86f05b5edb8c58c2a54002803": "\\prec", "42858d0b30b9df1112d4c1191168c0d5": "n = 2p", "4285a0ae284727fd01300c055e230912": "O_{p2}", "4285b32250b485131cec465c330cd697": " \\omega\\mathit{l}_{H} /V = ( \\delta C_{mgH} /\\delta\\alpha_{H}) \\times ( \\rho S \\mathit{l} /2 ) \\times \\mathit{l}_{H} \\times V \\times \\omega \\,", "42862cd764299fd0b6c90e19a8ed27c1": "\\mathrm{d} \\mu = V_m\\mathrm{d}p - S_m\\mathrm{d}T.", "42863d40708846d88d537a26e8f86a79": "\\log_a y = \\gamma x + \\log_a \\lambda. ", "4286a99a269cd12eeded6f1abe3f3e65": "\\sigma_0 \\otimes \\sigma_3 \\otimes \\sigma_2 ", "4286ad6d707e220da5188c2f0c64195f": "\\tbinom{7}{5}", "4286edd25d16ec16ac858a5c5c67192e": "g_2 \\leftarrow g_1^w rem P", "428717d06486b6fb38a98bc56f5ff032": "f'(\\gamma_1)= k\\,", "4287225044ced9259bc4f9fb930a6686": "\nc_{gz,n}=H \\frac{\\partial \\sigma}{\\partial \\alpha_n}\n", "42875ce703f01bb45abdeb1f43390e8b": "\\mu (\\mathbf{v} \\times \\mathbf{H})", "428796a2ae1911df789e1fb551c69299": "X \\setminus A", "42879d137cfaa004a48cd22d6ffc197e": "c1 = 1/R1", "4287bf2312c48ad9e0a83c29c6b4ddb4": "\\,Q_\\textrm{F} \\subseteq Q", "428831443ca49f6c825e543411625266": "z_8=\\chi_{\\psi_{8,8}}(z_8,\\rho_{\\psi_{1,8}}(z_1))=\\chi_0(z_8,\\rho_{12}(z_1))=0+\\sin x_1=\\sin x_1", "428840485770a0f6475e1ddf312eeb46": " V_1 = Z_{11} I_1 + Z_{12} I_2 = \n Z_{11} I_1 - Z_{12}{Z_{21} \\over Z_{22}} \\, I_1 ", "4288b0a255087014fe241c886c0df74c": "\\theta_\\mathrm{i} = \\theta_\\mathrm{r} ", "4288ffbe770182d77c9e936200f051b0": "\\mathbf{P}^5;", "4289124f34a8372f36b5d6af42ed8e46": "H_{(1)} \\ldots H_{(R)}", "428919def88cb8dbd3d779539791bdc9": "f(x_{i_0})\\geq f(x_i)", "42895700c3617b9ac11f59781d56068c": "\\left( \\frac{1}{\\sqrt{10}},\\ -\\sqrt{\\frac{3}{2}},\\ 0,\\ 0 \\right)", "4289a1c1852c084b8dffc5cc794f3b04": "O(\\log n/\\log \\log n)", "4289e66f636249650c5ccada8abf5e96": "\\vartheta_{s}^{-1} = \\vartheta_{-s}", "428a00cd13af7fbb4cfe7fa245a03953": "|j\\rangle \\mapsto \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} \\omega^{jk} |k\\rangle. ", "428a01868a1d22adcdcba746c11415c6": "\\mathbf{p}_{n} = [H f(\\mathbf{x}_n)]^{-1} \\nabla f(\\mathbf{x}_n)", "428a36113e2bbdc7fd8b21855700cdcb": " r(t.i) = q_i ", "428a51414ee4bcc75c9c61c52cf28fdb": "F(L) \\propto L^{\\alpha}", "428a83559a50255edc8cb465c8be20eb": "\\frac{H^2}{H_0^2} = \\Omega_R a^{-4} + \\Omega_M a^{-3} + \\Omega_k a^{-2} + \\Omega_{\\Lambda}.", "428aaef3f4ff324c66d0abea024a2df5": "\\overline{y}=\\frac 1n \\sum_{i=1}^n y_i .", "428ae802fe02892664816d06f3e7fa67": "\n\\begin{align}\n y[n] &= \\sum_{k=0}^{RM-1} x[n-k] \\\\\n &= y[n-1] + x[n] - x[n-RM].\n\\end{align}\n", "428b3d6b45ca2203c082cfbbf6a3b97d": " S_k ", "428b98823d98729f1382d7d9edd3b8f0": "\\dot x=X(t,x)", "428bbbbf3602db05ba542cd0d8d93b84": "-(x^2-2 x-5) (x^2+x-1)^2 (x^3-4 x^2-9 x+10) (x^4+2 x^3-7 x^2-18 x-9)^2", "428bbd1149c4c45a9515ba0fd2486ea1": "u^2(y) = c_1^2u^2(x_1) + \\cdots + c_N^2u^2(x_N),", "428bcfff6ea13478352aa949dbb2fb15": "(f\\cdot g)(x) = f(x)\\cdot g(x)", "428bdd9ca9b0a3712aaf034de18e70a0": "M_s", "428c0c5253dfd2d1e3ce9ea12a4e847a": "{d \\tau}^{2} = -(dx^0)^2 +4(x^3)(dx^0)(dx^2)-2(dx^1)(dx^2)-2(x^3)(dx^2)^2-(dx^3)^2.", "428c6dba5f5b5cf9dd531817315832ef": "{\\hat c}", "428c7167741c9567659d222725b71f57": "\\frac{\\partial n}{\\partial t}= - \\nabla \\cdot \\mathbf{J} +W \\, ,", "428c7787fcd739cbf2885f9f989d9547": " y \\mapsto \\langle x \\mid A y \\rangle ", "428cd1a51116eefe1e2b738fcf34a617": "\n \\nabla^2 \\varphi = \\cfrac{1}{\\sqrt{g}}~\\frac{\\partial }{\\partial q^i}([\\boldsymbol{\\nabla} \\varphi]^i~\\sqrt{g})\n", "428cec6e73e9cfb8254856535d562f50": "[L,R]=LR-RL=0", "428cffbea2a7376ebb9c24a7fbbe20e6": "B+C+\\cdots+Z+\\frac{Z}{3} = \\frac13 A", "428d08381d3f660a8adc5d6fc2feb613": "\\mathrm{Ek}=\\frac{\\nu}{2D^2\\Omega\\sin\\varphi}", "428d3aae7828bc459b04729ecba7ebcb": "(\\mathbb{Z}/n\\mathbb{Z})^*", "428ec57caa250f2095e6a1b6d64c6019": " \\operatorname{drop-param}[(g\\ m\\ p\\ n), D, V, \\_] ", "428f03f836d3d7e87fadd65b8aadc4ef": "\\gamma = \\kappa/m \\,\\!", "428f0ea6e7a6ccd7edf3ef13ff830d12": "\\mathbf{}V(t), W(t)", "428f29685657558987af61cdf594849c": "{T_{2lm}}", "428f486b344da7de5ebd215604d52688": "AB = e^{\\ln(A)+\\ln(B)}. \\, ", "428ff68fab5fc5356a8f978c20f943d7": " \n \\lim_{N\\rightarrow\\infty} G_\\mathrm{bin}(x;p,N)\n = \\lim_{N\\rightarrow\\infty} \\Big[ 1 + \\frac{\\lambda(x-1)}{N} \\Big]^{N} \n = \\mathrm{e}^{\\lambda(x-1)}\n = \\sum_{k=0}^{\\infty} \\left[ \\frac{\\mathrm{e}^{-\\lambda}\\lambda^k}{k!} \\right] x^k\n", "4290b332f231c0a84fafb7983653151a": "2x^2+((c_{1}+b_{2})+(c_{1}-b_{2}))x-((c_{1}+b_{2})(c_{1}-b_{2})-(c_{1}-b_{2})^2))=0", "4291133bdfc56466e7cfcd8555856e06": "\\tilde{P_{n}}(x)", "42914f74e375c3bbe178ff3c413b01a0": " (0.00, -0.25); ", "42916f591570277d9d6da3b8e8c78ca0": " \\frac{3a+b}{4} ", "42918c5d5aafc33ffa63d0ec5c35cbdb": "pH=\\frac{nF}{RT}(E^0-E)", "4291f1694b6a32af215abadcdde682bd": "X = v^1[\\mathbf{f}]X_1+v^2[\\mathbf{f}]X_2+\\dots+v^n[\\mathbf{f}]X_n = \\mathbf{f}\\begin{bmatrix}v^1[\\mathbf{f}]\\\\v^2[\\mathbf{f}]\\\\ \\vdots\\\\ v^n[\\mathbf{f}]\\end{bmatrix} = \\mathbf{f}v[\\mathbf{f}]\\,", "42927e311f355b1d2ed291d3dde5738f": "f(t_1,\\ldots,t_n)", "4292c762b77071a2bfe6ce28ef0ef18d": "l_{ij} = \\frac{\\Big(\\frac{\\mathrm{tf}_{ij}}{\\max_i(\\mathrm{tf}_{ij})}\\Big) + 1}{2}", "4292eb7f941d429a605a02c8c7b3da03": "A\\cap(B+C)\\neq(A\\cap B)+(A\\cap C)", "42931c3e2894e602c6f06a7cc4538242": " \\lambda x.f\\ (x\\ x) ", "42931ee4b7772c540d2d22391189104f": "299 792 458 \\ ", "42931f65eacea7c399fe91139e751571": "V_{B}=40.6\\ L", "429326391fbb383a4c43366f3f8d6620": "\n\\frac{\\partial^2}{{\\partial t}^2} I =\n\\frac{1}{LC} \\frac{\\partial^2}{{\\partial x}^2} I\n", "42933a3b7bd28e752893aa42c61bb1a2": "\\epsilon_{it} \\sim N(0,\\sigma_i) \\,", "429381c1e0f97fe30b3fd54ac5ae736b": "\\mathbf{u}_\\mathrm{t}(s) = \\left[ x'(s), \\ y'(s) \\right] \\ , ", "4293e5a16dc08222834ac1fd247626ba": " | \\sin{(\\mathbf{v} , \\mathbf{u})} | \\leq 1 ", "4293f1cf87f52dd7246dd15434849215": "\n\\begin{array}{c|cc}\n0 & 1/2 & 0 \\\\\n1 & 1/2 & 0 \\\\\n\\hline\n & 1/2 & 1/2\\\\\n\\end{array}\n", "4294517f52afc39ea47b9d107776ea3e": "J''", "429468d4401928eabfff616a5c777745": "u \\in [0,\\ 2n\\pi)\\ \\left(n \\in \\mathbb{R}\\right),", "42948aa5e91f8265ef082ed1f8113f56": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n \\frac{(A\\,b-a\\,B) x^{m-n+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^{q+1}}{b\\,n (b\\,c-a\\,d) (p+1)}\\,-\\,\n \\frac{1}{b\\,n(b\\,c-a\\,d)(p+1)}\\,\\cdot\n", "4294f68f4297d43065503857fabbf241": "K(R)", "4295776faa6c72d89d642baa2531b050": " V_1=V_2=\\cdots=V_N ", "4295dffb61289dee0057ec75c0153c47": "O(p)", "429637cee602ccd01d4cf1276f7365f6": "a \\to b \\to a", "429693c80b24ce50ba874a2096e6d28d": "\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2} = 1", "4296db5192210af040f03966ac1bdda3": "i : \\tilde{F} \\to F_{S}", "4296e1de708fd8fb1954089d439b3162": "\\frac{W_q^{(H)}-W_q}{W_q}=\\frac{1-\\rho^2}{\\rho^2+\\lambda^2\\sigma_B^2}", "429738d226da830e86939d6dc833fb84": "h(y) = y(1-y)", "4297971aea2d28d336bcfba8a42fe4bc": "N_{A}", "42983b05e2f2cc22822e30beb7bdd668": "CO", "4298620cb8ef54c466953618a0b767bb": "\\int_{[s, t)} \\Delta(\\mathrm{d} z) \\leq \\delta(t) - \\delta(s).", "429877f02fda8592c4f6248605b5c1c1": "V = \\frac{3 \\sqrt{3}}{2}a^2 \\times h", "4298a0278a03afc589dd27b456d0a6dd": "\\left(\\frac {d}{dz}\\right)_q p_n(z) = \n\\frac{p_n(qz)-p_n(z)} {qz-z} = p_{n-1}(z)", "4298a7bf46865ef1cd18992dd7588d50": "v(t) \\le\\int_a^t\\alpha(s)\\beta(s)\\exp\\biggl({-}\\int_a^s\\beta(r)\\,\\mathrm{d}r\\biggr)\\mathrm{d}s.", "4298c963de29f9d9c209ed92268eea79": "f\\left( \\frac{x_1+x_2}{2} \\right) \\le \\frac{f(x_1)+f(x_2)}{2}", "42996a504f68a5b87938eeae914c6397": "\n \\begin{align}\n M_{xx} & = \\int_A z~\\sigma_{xx}~\\mathrm{d}A = \\int_A z~E~\\varepsilon_{xx}~\\mathrm{d}A = \n -\\int_A z^2~E~\\frac{\\partial \\varphi}{\\partial x}~\\mathrm{d}A = -EI~\\frac{\\partial \\varphi}{\\partial x} \\\\\n Q_{x} & = \\int_A \\sigma_{xz}~\\mathrm{d}A = \\int_A 2G~\\varepsilon_{xz}~\\mathrm{d}A = \n \\int_A \\kappa~G~\\left(-\\varphi + \\frac{\\partial w}{\\partial x}\\right)~\\mathrm{d}A = \\kappa~AG~\\left(-\\varphi + \\frac{\\partial w}{\\partial x}\\right)\n \\end{align}\n", "4299818cea37c057f80e7e33a38b49ae": "E_\\text{k} = m \\gamma c^2 - m c^2 = \\frac{m c^2}{\\sqrt{1 - v^2/c^2}} - m c^2", "429982942d1e845d89eb46eb13e9de1a": "L \\sim \\frac{R}{Q \\beta^2}", "4299c3eeaa05619cb78c493c5d403d5c": "1 + c_1(L) + c_1(L)^2 / 2", "429a2f264432234e0d0f0e2c5b6836a0": "H^i(U,-)", "429a41765cf19a00d872eb7f345ba5c3": "L_{\\eta}(p)=\\{0.n_1n_2n_3 \\ldots \\vert 0\\le n_k

\\eta \\}", "429a5feba051e91eb2c6f84db4c24f48": "nw(X)\\triangleq\\,", "429a88b6f166eb16664cf81cf3350231": "\\frac{dU}{dt}=0.", "429b2b08f7aa687d0c7c64cb2f720cd6": "\\zeta(2n)=\\sum^{\\infty}_{k=1} \\frac{1}{k^{2n}}=(-1)^{n+1} \\frac{B_{2n} (2\\pi)^{2n}}{2(2n)!} ", "429b8a6894ed61e730f06503f40df8fa": " \\Phi_{123}= \\Phi = x_1 \\otimes x_2 \\otimes x_3 \\in \\mathcal{A \\otimes A \\otimes A}", "429bd6d5ad9a364fafd76ff5a1909e81": " \\operatorname{max}(a,b) ", "429bff525275b7afd29bcd2b26edb945": "x+(-1)\\cdot x=1\\cdot x+(-1)\\cdot x=(1+(-1))\\cdot x=0 \\cdot x=0", "429cacdd1e26157a183aea82e7c327a6": " _{E_y}\\, \\sum_{j=0}^\\infty a_j := \\sum_{i=0}^\\infty \\frac{1}{(1+y)^{i+1}} \\sum_{j=0}^i {i \\choose j} y^{j+1} a_j .", "429cccbe23b32040c34c1d068db58da8": "\\iint_R f(x,y) \\, dA = \\int_a^b \\int_0^{r(\\varphi)} f(r,\\varphi)\\,r\\,dr\\,d\\varphi.", "429cec525f0497c9420e9e89be734190": "P_0(L)", "429cee78cde8836eecf89c7c60c5e762": "rel_{i} \\in \\{0,1\\}", "429cf1a02da039e39e2a3db1f48062eb": " R\\# H ", "429d672dcd6ad2394c74285ca092a7ac": "f(w) = we^w", "429d6d007a9a946392eb350df1eb1ccd": "f(x)=\\cos (\\arcsin(x)) = \\sqrt{1-x^2}", "429d9ed9d3859a1103c31b62c5258cd2": "N_S < N", "429df40ac244f9925380988a3072cfd8": "A \\approx B \\,", "429ef4296db80564fe0e368448f581c2": " I_\\mathrm{D}", "429ef42dbe18d2505bd33a5d13af850a": "h = \\frac{-b}{2a}; \\ \\ k = \\frac{4ac - b^2}{4a}", "429f21778df1f83d2011c2e271bb7931": " P_M ", "429f3e9b4adb66f9c7c887672bd73a07": "\\partial F_1\\wedge\\bar{\\partial} F_1\\wedge\\dots \\wedge \\partial F_k\\wedge \\bar{\\partial} F_k \\not= 0.", "429f5088a8d36954c2b47f59f2559ee9": "m_{\\mathit{eff}}", "429fb9fbe696ab7e76a25c66ff622631": " \\mathbf{X}\\mathbf{X}^T ", "429fc78517f2ff2b8fb78b117e83630b": "(q_1, q_2, \\dots, q_n)", "429fe5254b6d9d1bef16fbf4cefe7745": "A \\rightarrow BC", "42a00798cbd59672fff495bbd04fb966": "={\\mu_\\delta}^2 + {\\mu_\\alpha}^2 \\cdot \\cos^2 \\delta \\ ,", "42a018fb0efc6779738b64acc4ead07d": " c\\left(\\| \\nabla I\\| \\right) = \\frac{1}{1 + \\left(\\frac{\\|\\nabla I\\|}{K}\\right)^2} ", "42a0190770c49e955cf868d137de9dc6": "\\tau(G) = O \\left (\\nu_k^n n^{-1} \\log n \\right ),", "42a0e55180f66514cae5856bfc1d21f8": "y|V|", "42a0f4d36580871f3d7c22bc6003e2f0": "\\text{Bin size}=2\\, \\text{IQR}(x) n^{-1/3} \\;", "42a10a5190edab0b67ffda1d83e8c734": "q \\ge 2", "42a12557608b8d3db97ae56fa60a030e": "\\!\\, a ", "42a16aa65c5ed53d9d49b8bf6dd0a074": "\\operatorname{Ad}_g", "42a1830a0ba7a6249f9f25fb2a635d3c": "\\gamma(1),\\dots,\\gamma(n),\\gamma(n+1),\\dots,\\gamma(N)", "42a21c1501145576685f0c5b9d648756": "D_{NE}", "42a21e14eafaa7344c00ff19c8b1ead6": "\\zeta\\in A", "42a23ade3e523897b6caeeaf27063942": "\\sum_k A_{ik}B_{kj} \\neq \\sum_k B_{ik}A_{kj} ", "42a2490254127b19c8f81f4de73a14bf": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix} = \n \\begin{bmatrix} C_{11} & C_{12} & 0 & 0 & 0 \\\\ C_{12} & C_{22} & 0 & 0 & 0 \\\\\n 0 & 0 & C_{44} & 0 & 0 \\\\\n 0 & 0 & 0 & C_{55} & 0 \\\\ 0 & 0 & 0 & 0 & C_{66}\\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{23} \\\\ \\varepsilon_{31} \\\\ \\varepsilon_{12}\\end{bmatrix}\n", "42a264e00b0a10e0dd179c577e930d31": "\\mathbf{x}\\cdot\\alpha = s", "42a27a1ea42ae0aa33df47dd576a0987": "R = f_s \\log_2(M)", "42a29c3b73767daec74bc9a3738d0792": "\\chi_{\\text{mass}}", "42a29eba07c93ab3a5a88a07771fcf49": "\\mathbf{{\\Sigma}})", "42a2a3672bc59906b085709d99776aa6": "a ^1\\Pi_g", "42a2af91d500f25f363e54ff16b7c01c": "\\hat{f}_i \\, \\hat{f}_j = -\\hat{f}_j \\, \\hat{f}_i ", "42a2bb04d3b2d328eaf02706cd47b75b": " B", "42a2f2ea04ee5535700a2ed92b2b1d15": "H.", "42a327d20b536bdedcbd4898e3818461": "\\mathcal{I} \\models \\phi", "42a339fb7ceacbe528528b5b78a5cf1e": "\\Delta^2(p_1) \\le 0 ", "42a33d7bf569ad856d7f2cf576754637": "\\mathcal{H}_{bath}=\\sum_{\\alpha}\\left(\\frac{P_{\\alpha}^2}{2m_{\\alpha}} +\\frac{1}{2} m \\omega_{\\alpha}^2 Q_{\\alpha}^2\\right)", "42a34377720e94ea8e20b67279d5bece": " C^S=\\{y\\in C|Ty=y, \\, y\\in S\\}=\\bigcap_{T\\in S} C^T \\,", "42a3932740a0422baf0c14ea3cfbefbe": "\\scriptstyle 0=V_0 1", "42b24b3d913ca221ec70957f8a02a91e": "x^2 + bx + c \\;=\\; (x-h)^2 + k,\\quad\\text{where}\\quad h = -\\frac{b}{2} \\quad\\text{and}\\quad k = c - \\frac{b^2}{4}.", "42b28bfe7229bb2fff9f0bb2a1ffac13": "rr'", "42b2918c8b03e16710a63fce2469daf3": "J_- |j,m_{min}\\rangle = 0", "42b2cad036b5e1da7a49489c397c585f": "y=3,z=1", "42b2d98201e1aede05e81592f49f38ad": " \\vec u = (u_1, \\ldots, u_s) ", "42b2dfdc2a8af063323bbb65b7700f89": "F_e (E)", "42b36112d7c50a7d64a57f5e368d4818": "y(x) \\approx c_1Q^{-\\frac{1}{4}}(x)\\exp\\left[\\frac{1}{\\epsilon}\\int_{x_0}^x\\sqrt{Q(t)}dt\\right] + c_2Q^{-\\frac{1}{4}}(x)\\exp\\left[-\\frac{1}{\\epsilon}\\int_{x_0}^x\\sqrt{Q(t)}dt\\right].", "42b3957d60c34cdf32132c3db6ddaad2": "\\gamma < 1", "42b3c59298b8f6d12abca277daa50cf2": "{m \\choose r}_1 = {m \\choose r}", "42b40cf9c958a8a8749c8e34147edbf1": "Z_L=0", "42b4106cfd01b7c5e08a8b190219e405": "f(t)\\,e^{-g(t)}", "42b4171eb3e5ea7c3ef82b52359b1096": "(i=0,1,2...n)", "42b438e9a4e4eb2d7e6e18f24639617b": "\\mathcal{R} = \\frac{\\mathcal{F}}{\\Phi}", "42b4596479b02d24532a4a7ed8235234": "f\\in K", "42b4dcb8961adab875336c29af8736ed": "K \\hookrightarrow G", "42b5219f49a4917674fdd084c418b78c": "f:A \\rightarrow B", "42b526a6a18f18a9fbcfd93403bf0ab5": "\\varphi_V^{-1}\\circ\\varphi_U : (U\\cap V)\\times\\mathbf{R}^k\\to (U\\cap V)\\times\\mathbf{R}^k", "42b580e354ba651668b0090843d77ad3": "k^{\\Dagger } = \\kappa\\nu ", "42b5a035a292b1f18b536e70a73b6b2f": "\ny = \\sqrt{\\varepsilon /2}w\\left( z\\right) \\sinh \\xi \\sin \\eta.\n", "42b6511786ba4489de6bd871487de13b": "d \\colon V \\to \\mathbb{R}^n\\;", "42b65ba9cd07af0a2b5901a7a68770e9": "\\alpha_{1}", "42b6605e9d98f4ce1fc40ea647e6f135": " \\mathbf{\\hat T} (\\varepsilon) ", "42b673362fd1ed31deffd6ebe7040e37": "\\Psi (z)= E_0 e^{inkz}= E_0 e^{i(1-\\delta)kz} e^{-\\beta kz}", "42b6b9613342ef3088047b4584901560": " H = -J \\sum_{\\langle i,j\\rangle} s_i s_{j} - h \\sum_i s_i", "42b6cef2476f3483e477ad25b435520f": "G = 6.6742\\times10^{-11}\\, m^3kg^{-1}sec^{-2}", "42b700f23168ec58d770cba98d08ae92": "j_1", "42b72e420b2a3b6ffc2015593e1b1728": " \\text{Wright} {{=}} -61.1 + (1.798 \\times \\text{EU}) - (0.001594 \\times \\text{EU}^2) + (0.0000007713 \\times \\text{EU}^3)", "42b7d5d18b54a844cbd1961801e54fb2": "D^\\sharp_n(z)=z^n D_n(1/z)=d_n+d_{n-1}z+d_{n-2} z^2+\\cdots+d_{n-1}z^{n-1} + d_0 z^n ", "42b7dd498894491c2a1374fe32e68db5": "\n\\tan \\theta\n= \\frac {F} {W} \n= \\frac {\\rho_M}{\\rho_E} \\frac {V_M}{V_E} {\\left( \\frac {r_E}{d} \\right)}^2\n", "42b7eb4fd4761fa8ee58565b47594b4c": "\\,\\frac{a + bi}{c + di} = \\frac{\\left(a + bi\\right) \\cdot \\left(c - di\\right)}{\\left (c + di\\right) \\cdot \\left (c - di\\right)} = \\left({ac + bd \\over c^2 + d^2}\\right) + \\left( {bc - ad \\over c^2 + d^2} \\right)i. ", "42b7fa117615f02a4d47823ab2862a10": "x_{k}", "42b8415b8901be649989b2b6b57f1229": "\\chi_{P}", "42b8796fd22c20a937b494a5743339f1": "X_i\\in A", "42b8f712f5e70fb630e6db9a20188ad0": "\\exists f \\in \\mathbf{P}^{(1)} \\, \\forall i \\in \\mathrm{Domain}(\\nu_1) : \\nu_1(i) = \\nu_2 \\circ f(i).", "42b925429c9e7dada6cbcb0651570ddb": " g_J = g_L\\frac{J(J+1) + L(L+1) - S(S+1)}{2J(J+1)} + g_S\\frac{J(J+1) - L(L+1) + S(S+1)}{2J(J+1)}", "42b93f611dc41b0c2d3e3014ee09b8db": "C_4, C_5, \\bar{C_4} (=K_2+K_2)", "42b941ea3d8a2c5cccacd55107c0fcae": " R_i = 0", "42b944966ae4bad603883f95baadd56c": "L=\\{x | g^w=x \\}", "42b9572a8668eafee9099a4c47909527": "R_{nom-load} = 2 \\times R_{full-load} = 3.33\\,\\Omega", "42b981e987699cbf30cd6100c465ba74": "\\scriptstyle{\\binom{t}{k}}", "42b9e077b45a9c78b04e3000a699350c": "L^1(\\mathbb{R}^d)", "42ba2915c5cf8c5793f5c5c13ace7ef1": "\n \\mathrm{\\Beta}(x,y) = \\int_0^1t^{x-1}(1-t)^{y-1}\\,\\mathrm{d}t\n\\!", "42ba3e7113c9674be4f9aecc7ecf65e7": "\\ln\\left(F/K\\right) = \\ln(S/K)+rT.", "42ba64a88489bb4128ecedeba2b69f3b": " E_n = {R_\\mathrm{E} \\over 2 n^2 } ", "42ba98fb135676b5d07b5b59688443af": " n a + m b = 0 \\;", "42bab0ff9ff3b461c2c66bc98009c005": "r=5", "42bab7bc054d10bfde2d7d7e254128d1": "T_G(-1,-1)", "42bac2b9b9b283974022fc287cca9459": "\\displaystyle C(zt^r B(t))=\\sum_{n\\ge0}\\Phi_n^{(r)}(z)t^n", "42bb294ed604f763aa29ed2af26f3e5a": "dH=TdS+VdP\\,", "42bb64641588f86739c59462c5b0f177": "D_n - \\sigma_o", "42bbdbd81f8930045cf6d40f541b8225": "[M] \\in \\mathfrak{N}_n", "42bbf4a3d793e0ecaccf52ca4b448186": "SN(d_1)", "42bbf551737f3b512cbda5f027d554ad": "f_\\text{step}", "42bc068c0b89d89b4e8448bc936c9ec7": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} x_1 \\\\ y_1 \\\\ z_1 \\end{pmatrix}, \\begin{pmatrix} x_2 \\\\ y_2 \\\\ z_2 \\end{pmatrix}, \\begin{pmatrix} x_3 \\\\ y_3 \\\\ z_3 \\end{pmatrix}", "42bcac7f0103e123655c26a41b3e9d9b": "\n f(x) = \\frac{2 \\beta}{x} \\exp\\left\\{ -\\beta \\left[ \\log \\left(\\frac{x}{\\sigma} \\right) \\right]^2 \\right\\}\n \\cdot \\log\\left( \\frac{x}{\\sigma} \\right), \\qquad x \\geq \\sigma > 0.\n", "42bcc0905a3b9d0a17d064a8fbd0a83d": " h_{HPF}(t) = \\delta(t) - 2B_H \\, \\mathrm{sinc}\\left(2B_H t\\right)", "42bd85f314d68b590bf6f98bbb14b0ea": "\\vec y_n = \\vec C^n\\, \\vec y_0 = c_1\\,\\lambda_1^n\\,\\vec e_1 + c_2\\,\\lambda_2^n\\,\\vec e_2 + \\cdots + c_n\\,\\lambda_n^n\\,\\vec e_n", "42bdac611ae54b2566349f12c26f1a48": "J-1= 0", "42bdd857dbe2062b93b3814963e2f585": "\n \\begin{align}\n \\boldsymbol{\\nabla}\\boldsymbol{S} & = \\left[\\cfrac{\\partial S^{ij}}{\\partial q^k} + \\Gamma^i_{kl}~S^{lj} + \\Gamma^j_{kl}~S^{il}\\right]~\\mathbf{b}_i\\otimes\\mathbf{b}_j\\otimes\\mathbf{b}^k \\\\[8pt]\n & = \\left[\\cfrac{\\partial S^i_{~j}}{\\partial q^k} + \\Gamma^i_{kl}~S^l_{~j} - \\Gamma^l_{kj}~S^i_{~l}\\right]~\\mathbf{b}_i\\otimes\\mathbf{b}^j\\otimes\\mathbf{b}^k \\\\[8pt]\n & = \\left[\\cfrac{\\partial S_i^{~j}}{\\partial q^k} - \\Gamma^l_{ik}~S_l^{~j} + \\Gamma^j_{kl}~S_i^{~l}\\right]~\\mathbf{b}^i\\otimes\\mathbf{b}_j\\otimes\\mathbf{b}^k\n \\end{align}\n ", "42bde18032cda2bf191c4009f66d44ad": "\\max \\sum_{t=0}^T b^t \\ln(c_t)", "42bdedc2708e3d6ce0bbf8c0b6c49cac": "1 - P(M)=0", "42be5c284e21d52eee4ba43f32e8238f": "\\epsilon 4", "42bfaf8f94e4ad2e9c6d85a7371f0975": " f_{*}^{\\delta}(e)=\\sup_{a\\in\\mathbb{R}^{n}}\\frac{1}{m(T_{e}^{\\delta}(a))}\\int_{T_{e}^{\\delta}(a)}|f(y)|dm(y)", "42bfc212be38d7a84898becb6eb63e21": "\\sum_{i=1}^d h(a_i) \\geq \\sum_{i=1}^d h(b_i)", "42c0817b0291038612eadd6d31dac777": "N\\rtimes_{\\phi}H", "42c0947577fdf1ccaf14449864718947": "VdP", "42c153ec9cf8f6aa59c3df635bd5b186": "P_k^{(\\alpha,\\beta)}(x)", "42c15e35f3676ceda899a16175c9968f": "-\\frac{V_{dd}}{2}", "42c16d6edca347f3263fd3e34b697f97": "v_i\\!", "42c18635b62fb15166dfbf1a217ad9d1": "\\Pr(Heads) = p_f", "42c1a3173ca7fc1525f2d0215be079a8": " E_\\text{k} = \\begin{matrix} \\frac 1 2 \\end{matrix} mv^2,\\,", "42c1c5a48891a0772a06b30195c530b2": "\\log_2{(x^{-1/2})}", "42c203440c5863dd34f0a1f2a2df6140": "\\mathrm{CouponFactor} = \\frac{1}{\\mathrm{Freq}}", "42c24376babf7124694da4db89d3355f": " (f_g \\cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_{g*h}(x) ,", "42c24457eac360997cb3eb9786b8780d": "\\sum_{x=1}^\\infty (-1)^{x+1}\\,\\mathrm{sinc}(x) = \\mathrm{sinc}(1) - \\mathrm{sinc}(2) + \\mathrm{sinc}(3) - \\mathrm{sinc}(4) +\\cdots = \\frac{1}{2}", "42c261be3018d1fbfb9a5216e449a475": "\\,{}_tp_x = {l_{x+t} \\over l_x}", "42c26b1ee153bf907fe34d48572c36c2": "x \\leq 0", "42c2766fa8380264f1d74ebe75fc6213": "L(s)", "42c317b67e6cc057a7804181151900b5": "\\Sigma_2 \\not \\subseteq \\mathsf{P/poly}", "42c35913b10c36a49b292e73047217b3": "P(x_1\\mid x_2,x_3)", "42c366629f1534045cb25919aa2a379f": " F : \\mathbb{R}^2 \\times \\mathbb{R}^3 \\to \\mathbb{R}", "42c417503da42f7375c0e08afe64d3f2": "\\hat{\\rho}(\\mathbf{r}) = \\sum_{i=1}^{N}\\sum_{s_{i}}\\ \\delta(\\mathbf{r}-\\mathbf{r}_{i}).", "42c457fe03d1be76efd43f9f8558fd53": "\\left\\{\\mathcal{B} f\\right\\}(-s)", "42c4b4fb93a9b714d3274db3d3d41cbe": "P(\\mathbb{R})", "42c511df6f19c153067019b33479584c": "19249 \\cdot 2^{13018586} + 1", "42c5576146469119b7d629add9983a98": " g(a;p) =\\sum_{n=0}^{p-1}e^{2{\\pi}ian^2/p}=\\sum_{n=0}^{p-1}\\zeta_p^{an^2}, \n\\quad \\zeta_p=e^{2{\\pi}i/p}. ", "42c567a304f27e8888b56730c15a1859": "\\text{GF}(2) = \\{0,1\\}\\,", "42c5995042c1f3403283b530d0b2fc35": "a^2(u^2-v^2) = 1,\\,", "42c620b923a4049cf306540797be2b33": "\n \\oint_C f(z)dz = 0,\n", "42c69d0c46edc37929a1b40348f6dce9": " * ", "42c6f512d8028f8ff04164d3410e0dc8": "A\\subseteq \\mathbb{R}^n", "42c71acdf3f7609dd83bc75179d1d840": "\n\\begin{pmatrix} {a_0 } \\\\ {a_2 } \\\\\\end{pmatrix}_j=\n{1\\over70}\\begin{pmatrix} 34 & -10\\\\ -10 & 5 \\end{pmatrix} \n\\begin{pmatrix} 1&1&1&1&1\\\\4&1&0&1&4\\\\\\end{pmatrix}\n\\begin{pmatrix}y_{j-2} \\\\ y_{j-1} \\\\ y_j \\\\ y_{j+1} \\\\ y_{j+2} \\end{pmatrix}\n", "42c732618c2daad93bcad90d519adc4f": "\\epsilon^{\\text{v}}", "42c74ecef1903955e36db5091e2c5835": "\\langle P \\rangle = \\frac{\\langle \\rho \\rangle}{\\bar m}kT", "42c7a3aef29290737ba76c9f743c0dc7": " U_n(x) =2\\sum_{j\\,\\, \\text{odd}}^n T_j(x) ", "42c8ae79f7204efa2edda674b6a709b7": "s_{fin}", "42c8bc22e67a32ac27ff810353bbf4e3": "\\eta^*", "42c8d2e471fe149457eb04adfe323ae3": "A_c", "42c8f15669e197613b474a1b880b5a2b": "\\begin{align}\ny & = xs \\\\\ndy & = x\\,ds.\n\\end{align}", "42c904eb6efe0e68fd9a56a8abaf887b": " N_j ", "42c9343e9e28b05dbcd5f9036ac1d75c": "(b^{kn}-1) = (b^n-1) \\sum _{r=0}^{k-1} b^{rn}", "42c95e2782e182b9e2ea0460169c8bf9": "X \\twoheadrightarrow Y", "42c9aca3471dfa088544df2c34c5507d": "Z_D=Z_T=1", "42c9d760aa084e12b2ec295c1e09de10": " \\hat p_t = \\frac{1}{K}\\,\\sum_{i=1}^K 1(X_i \\ge t) W(X_i),\\,\\quad \\quad X_i \\sim f_*", "42ca554a49d1f96a5c7cc59d2c4c76f2": " a_n = a_{n-1} + \\operatorname{gcd}(n,a_{n-1}), \\quad a_1 = 7, ", "42ca8c25ce1e44462e2232bcfa9244a8": "\\cos\\frac{\\pi}{8}=\\cos 22.5^\\circ=\\tfrac{1}{2}\\sqrt{2+\\sqrt{2}}\\,", "42caa5c266596745005a46068f84e7ec": "0 \\mid 0", "42caf224c6b7b72bd92bc2462f9259b4": "x^2 = A", "42caff2830099e528908ef14e927abba": " (X - \\mathbb{E}(X))^2 ", "42cb9fda2d65b2e0d47c1da2b31414e6": "\\mathbf{\\hat{r}^{\\prime}}", "42cbccdc3b7580bf0d934068420fca81": " \\langle g_1, g_2, \\ldots, \\rangle ", "42cc001bade80447090d00d45565c96d": " \\widehat{\\mathbb{Z}} = \\varprojlim \\,\\mathbb{Z}/n\\mathbb{Z}. ", "42cc7a08707db28c616f602422bf6cec": "\\mathbf{N} = \\nu\\left(1 , \\hat{\\mathbf{n}} \\right)", "42cddf19db7f3148b9c611b478d7e5ec": "\\vec k = (k_x,k_y,k_z)", "42ce133d3e8e5e8a5952d75df9bdd1a6": "\\{\\phi_j, H^*\\}_{PB} \\approx 0", "42ce35e1a01e7e41ec8986fdd5141f2d": "\\frac{\\partial u}{\\partial \\varphi} = \\frac{\\partial u}{\\partial x}\\frac{\\partial x}{\\partial \\varphi} + \\frac{\\partial u}{\\partial y}\\frac{\\partial y}{\\partial \\varphi},", "42ce43e007a252db86d27401df33730e": "\\operatorname{Aut}(X)", "42ce60d146b5084feb2aa9f56b176182": "H(|f|^2)+H(|\\hat{f}|^2)\\ge \\log(e/2)", "42ce656539fd5dab09846f8124b4f8cd": " T=\\frac{Z}{\\sqrt{V/\\nu}} = Z \\sqrt{\\frac{\\nu}{V}} ,", "42ce67f788c4285e74e1f391db17fd22": "R+2\\delta\\le1", "42cec26ee0abc453870add63b91dadb7": "p_i^*Ap_j=r_i^*M^{-1}r_j=0", "42cf322c1f704192d1c368cd36166fac": " \\frac{1}{\\pi} = 12 \\sum^\\infty_{k=0} \\frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}} ", "42cf7676456d0b6ce5b278493e23d2b9": "X\\subseteq A", "42cfca95684a2e82830d3101d8826dd9": "c \\in \\mathbb{R}", "42cfdf4ef9cebbd81a97e6292f414fe1": "\\vec{v}(t + \\Delta t) = \\vec{v}\\left(t + \\tfrac12\\,\\Delta t\\right) + \\tfrac12\\,\\vec{a}(t + \\Delta t)\\Delta t\\,", "42cff713d6a3a48eece38b1905e244a3": "\\textstyle \\mathbb{R}^+_0", "42d02522c3c178d6ab59162c2c45e2a3": " x \\oplus y = - \\log(e^{-x}+e^{-y}) \\ , ", "42d029caf1f5db26cca04546f5321941": " \\left(\\frac{\\ldots\\cdot10\\cdot6\\cdot4\\cdot2\\cdot1}{\\ldots\\cdot11\\cdot7\\cdot5\\cdot3\\cdot2}\\right)\\zeta(1) = 1 ", "42d0683ec48151fe37b2322dee7e75bc": " \\int_0^1 \\int{1\\over (k^2 + m^2 + v 2p \\cdot k + v p^2)^2} dk dv = \\int_0^1 \\int {1\\over (k'^2 + m^2 + v(1-v)p^2)^2} dk' dv", "42d0beab3581b116a01b6983201bc8b2": "\n\\begin{alignat}{2}\n U(\\xi) & = \\int{\\int{\\log(p(\\theta | y,\\xi))p(\\theta, y | \\xi)d\\theta}dy} - \\int{\\log(p(\\theta))p(\\theta)d\\theta} \\\\\n & = \\int{\\int{\\log(p(y | \\theta,\\xi))p(\\theta, y | \\xi)dy}d\\theta} - \\int{\\log(p(y| \\xi))p(y| \\xi)dy} ,\n\\end{alignat}\n\\, ", "42d0f315298d72a8aaab4933c74bd8a4": "w^{f}(f^{*}) \\leq \\sum_{e \\in E} \\left( a_e \\cdot \\left( (f_e^{*})^2 + (f_e)^{2}/4 \\right) \\right) + \\sum_{e \\in E} f_e^{*} \\cdot b_e", "42d1dd07ecfec2b22623749c45a5db73": " \\lambda x.\\lambda y.\\operatorname{drop-formal}[D, p\\ x\\ y, F] ", "42d258bf91e865326ac3797b803efa33": "x^5-5s^3x^2+15s^5 ", "42d273ccc8918185e03ccdf3fa73ad0c": "R/xR", "42d29533e73a1c57dc1db19a953c314b": "\\{L(u,v): u, v \\in X\\}", "42d2bfc7e80a193c025ad4a77686c4a4": "z_{cr} \\,", "42d2c606ef4423b5da920c51077d988b": " L(n,k) = \\frac{n!(n-1)!}{k!(k-1)!}\\cdot\\frac{1}{(n-k)!} = \\left (\\frac{n!}{k!} \\right )^2\\frac{k}{n(n-k)!}", "42d2ca3300571c28bd2ce16201e5b73f": " K \\rightleftharpoons R", "42d38b7782813a96220896cc03be3edc": "\\Theta \\, \\theta \\, \\vartheta \\,", "42d3bd3e3fdc0cc0e75659fea200a6cc": "\nG_\\mathrm{dB} = 10 \\log_{10} \\bigg(\\frac{1000~\\mathrm{W}}{1~\\mathrm{W}}\\bigg) \\equiv 30~\\mathrm{dB} \\,\n", "42d41e07715a1fd44e1fa07eabcafd7a": "a \\propto t^{1/2}", "42d42fe87fb42629843e64e98dad5078": "M^\\frac{1}{2}L^\\frac{3}{2}T^{-1}", "42d445b963a4f2fd57132855dc00a7aa": "\\{X_\\alpha\\}", "42d46ce3e82d48d3309a914f47a53e43": "\\scriptstyle \\tan \\delta", "42d486ef7fa8ace2afe73f268fd945f7": "s(a, b) \\in [0, 1]", "42d4d0eafae42d97f6eb7bd162be80ca": "d \\big( x, T(x) \\big) \\leq f(x) - f \\big( T(x) \\big).", "42d51368f5f8ebf4e0aecd44fd0ef51f": "\\mathcal{Z}\\left[J_{ij}\\right] = \\operatorname{Tr}_{S}e^{\\left(-\\beta H\\right)}", "42d58d551fccc645698a335639d739a7": "X^n(i)", "42d5a28e63aa65d3f16ce9e01f5c4e65": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^m (A\\,b-2 a\\,B-(b\\,B-2 A\\,c) x)\\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{(p+1)\\left(b^2-4 a\\,c\\right) }\\,+\\,\n \\frac{1}{(p+1)\\left(b^2-4 a\\,c\\right) }\\,\\cdot\n", "42d5e4109233d6d5b0a7452517e81469": "\\tau_k", "42d5e7699192350b68c522e156a25fa6": "Z_{i+1} = \\{x\\in G \\mid \\forall y\\in G:[x,y] \\in Z_i \\}", "42d60e9781874f93e8ecd16bfd24392c": "c^Tx", "42d637811b18b53904db8ac6585dd1d4": "f_{\\psi(\\Omega^\\omega)}(n)", "42d664af461399ac108cb9a74f2d8b8c": "n>10^{78}", "42d677404d3bac38e5c7f3ac80ffebf2": "\\begin{align}\n\n \\eta(s) &= \\frac{1}{2}+\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\frac{1}{n^s}-\\frac{1}{(n+1)^s}\\right], \\Re(s)>-1 \\\\\n \\Rightarrow \\eta'(s) &= (1-2^{1-s})\\zeta'(s)+2^{1-s} (\\ln 2) \\zeta(s) \\\\\n &= -\\frac{1}{2} \\sum_{n=1}^\\infty (-1)^{n-1}\\left[\\frac{\\ln n}{n^s}-\\frac{\\ln (n+1)}{(n+1)^s}\\right], \\Re(s)>-1\n\\end{align}", "42d6b3677c8329c1262508d6aee781a4": "\\sigma_i^2 = 1", "42d71fd4a695a468d72f54b64ce139f3": "\\omega/k", "42d7645528fb5171a397367f7a14ef35": "p_M(\\lambda) := \\sum_{S \\subseteq E} (-1)^{|S|}\\lambda^{r(M)-r(S)},", "42d7909c3c9ab4f52135530b063f015f": "p_0=\\frac{m+1}{2m+1}", "42d7ed5db723c7cfe380b3ecca7fef28": "H_{N,q,s}", "42d80a0f264365babfc8be93e9dd65e6": "f \\sim g_1 + g_2", "42d81ac2d6858686513cf98bee0107e4": "\\log p(\\mathbf{X}|\\boldsymbol\\theta^{(t)})", "42d82d88d7c86551c0635dd7e9039d1b": "S_k(Tr(g^b))=\\left(Tr(g^{b(k-1)}),Tr(g^{bk}),Tr(g^{b(k+1)})\\right)\\in GF(p^2)^3", "42d848639589ece33cdbdc740c0bfc71": "A=\\begin{bmatrix}\n\\alpha_{x} & \\gamma & u_{0}\\\\\n0 & \\alpha_{y} & v_{0}\\\\\n0 & 0 & 1\\end{bmatrix}", "42d85b80f8b8925fb205486945ff02d7": "\\omega_h", "42d88c2b2e91d3494352e1fe0d636be8": "E(S_{T}) - K > 0", "42d899b2b891443d7c7c13729bbe5e04": "L^{rigid} \\left( t + \\frac{\\Delta t}{2} \\right) = L^{nonrigid} \\left( t + \\frac{\\Delta t}{2} \\right)", "42d8d7aa5e807e449327a2f90a92292d": " \\gamma'(0) \\longmapsto D_\\gamma ", "42d9aac217d396be22069c287119d33e": "{T_H}", "42da71679baf10996e6ed67749e422dc": "K_{ij} ", "42daa1f22a8327edb0d350bb7f55f4c8": "\\, a \\mapsto nan^{-1} .", "42dadb0d7b16b0beb66a792d3743e500": "ln \\mathbf{L(W)} ={1 \\over N}\\sum_{i}^{M} \\sum_{t}^{N}ln(1-tanh(w^T_i x_t )^2) + ln |\\mathbf{W}|", "42daeb36545b1c357d8015d1fdc8b730": "\\scriptstyle \\vec A", "42db170c4fc2dd52bf077c1ffa77c6cf": "\\tilde{\\mathbf{y}}(t)=\\mathbf{z}(t)-\\mathbf{H}(t)\\hat{\\mathbf{x}}(t)", "42db38f56fb1451eb5e839ace4b50745": "(0,\\infty)", "42db730eb348ae45f9cecf4f572b15ac": "\\theta_{(ab)c\\cdot d} \\circ ((\\theta_{ab \\cdot c},1_d) \\circ ((\\theta_{a\\cdot b},1_c),1_d))", "42db745cba7e3bdea67fbb9aec7722e6": "(Tf)(n)=\\sum_{d\\mid n} f(d)\\mu(n/d)=\\sum_{d\\mid n} f(n/d)\\mu(d)", "42db8d208909083cfa38b9924b715c37": "X := \\{1/n\\,;\\ n \\in \\mathbb{N}\\} \\quad \\mbox{and} \\quad Y := \\{-1/n\\,;\\ n \\in \\mathbb{N}\\}", "42db95cbcb835cea7470ae648f0e1036": "z_4 = x_4 y_1 + x_3 y_2 - x_2 y_3 + x_1 y_4 + u_4 y_5 + u_3 y_6 - u_2 y_7 + u_1 y_8", "42dbfcb5afe9ce2165016af027e4a8e7": "\\Bigl\\langle x_{n} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = k_{B} T \\quad \\mbox{for all } n", "42dc0f7831571fa1726a834b199506af": "(1-F_{IS})(1-F_{ST}) = 1-F_{IT}, \\, ", "42dc496c06e02542a810130e8fdebbb0": "\\dim_\\text{box}(S) = n - \\lim_{r \\to 0} \\frac{\\log \\text{vol}(S_r)}{\\log r},", "42dc8d2796ea1904fdef27ca9c1dfe31": " E = f\\ (x\\ x) ", "42dcca2920bd08c1a3afcec57e4c76e7": "H_{ext}(t)", "42dd3c7cee2ab91c3012ed2b5506ac69": "d=\\frac{h}{\\sin 20^\\circ}", "42dd9a9de5a92d54dafd48c74a6c6f5d": "\\,I^g(t)\\,", "42dda6727b03a7442e9326c47a4bf8dd": "-{dy \\over dx}\\sqrt{1-\\cos^2 y} =1", "42de3c91cbeb5fe08df9c35a25cacbea": " y_2 = y_1 = 2.0m", "42de4b694bd30794d2df61269f6eb58d": "d=(1-{1 \\over q})n, |C| \\le 2qn ", "42de4f7792289a6d354dbdafa39ff05e": "\\,\\eta_k=\\beta_k\\Sigma_{xx}^{1/2}\\quad\\forall\\; k", "42dea4d96746850559d7f11553fa3c4e": "h \\in GF(p^6)", "42dea8e0a735ed5ffb6c5d36ab900063": "X={1\\over 2}\\begin{pmatrix}1 & i\\\\ i & -1\\end{pmatrix}", "42dec5642ebd5f959fe63db4d5c734b1": "L'\\leq L", "42df1be8751a393b110e448907100446": "\\rho_{X^*}(t) = \\sup \\{ t \\varepsilon / 2 - \\delta_X(\\varepsilon) : \\varepsilon \\in [0, 2]\\}, \\quad t \\ge 0,", "42df528cde9d69ece1ed8798dd4639b9": "I = 1.1 \\times I_\\mathrm{o} \\times 0.76^{(AM^{0.618})} \\,", "42df73d2a81d26770d093388a21c5f71": "\n\\frac{1}{2}\\sum^n_{i=1}F^Q_id^Q_i + \\sum^n_{i=1}F^P_id^Q_i = \\frac{1}{2}\\int_\\Omega \\sigma^Q_{ij}\\epsilon^Q_{ij}\\,d\\Omega + \\int_\\Omega \\sigma^P_{ij}\\epsilon^Q_{ij}\\,d\\Omega\n", "42df81224b33b15da6e7b0fe2a9fea30": "H\\psi(x)=\\left[-\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}+V(x)\\right]\\psi(x)=E\\psi(x)", "42df8c1b817fb9373e1abce1ddf41cca": " \\left\\vert{ \\sum_{n=a}^b e(f(n)) }\\right\\vert \\ll \\frac{1}{\\sqrt \\lambda} \\max_{\\alpha \\le \\gamma \\le \\beta} \\left\\vert{ \\sum_{\\nu=\\alpha}^\\gamma e(g(\\nu)) }\\right\\vert \\ . ", "42dff1c271f8e6ca5288ebea4476264a": "\\lim_{\\varepsilon\\downarrow 0}\\int_{-\\varepsilon}^\\infty.", "42e025ad459e39c3ff9867b8d7aaea5e": "\\displaystyle{g=\\begin{pmatrix} A & B \\\\ C & D\\end{pmatrix}.}", "42e02d9dd51e339d42bc7d25b4c2cd0d": "Df(p)", "42e0831065be4907f096c62573797ee1": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 54.9\\cdot 2.42)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 28.6\\cdot R_{\\bigodot}\n\\end{align}", "42e0fe6b5c2edd06f715efcee3d331e7": "\\textstyle \\big( (0,1), \\mathcal{F}, m \\big) ", "42e11437f74caebdf6794c9d0428a878": "Z_0 = \\sqrt \\frac{L}{C} = \\sqrt \\frac{R}{G}", "42e131b61b513bea69fbed82d6fb0737": "\\pm \\frac{c\\,PRF}{4\\,f}", "42e13529d2bf198150e0c1f64ae8ab79": " T_n (x_m) = \\cos\\left(\\frac{\\pi m n}{N}+n\\pi\\right)=(-1)^n \\cos\\left(\\frac{\\pi m n}{N}\\right)", "42e159a532281f14b7f65bf1a09e4c76": "x^TAx", "42e16fa01acc0ce4d9f5d4f568957c39": "df(x_0,\\ldots,x_p)= \\sum_i(-1)^if(x_0,\\ldots,x_{i-1},x_{i+1},\\ldots,x_p).", "42e17e5f9213a2d8c39c640e873bfba4": "(u-t)", "42e233c254866ec2cf8ce2a7128f0d00": "x=a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3 + \\ddots}}}.", "42e24f7c3995f7f7fbba3ae4ba015076": "T_b=\\frac{I_{\\lambda}\\lambda^4}{2kc}", "42e27792d81a97f90ad3e1e7f42d13a6": "\\|\\mathbf{p}\\|=n", "42e28e909eec8a508c92dd49129a97b4": "\nw(n,g)=\\frac{(n+g-1)!}{n!(g-1)!}.\n", "42e2a4d6e4650cff4d9305d0cd6bb900": " G_1 \\cdot G_2 \\cdot G_3 \\cdots ", "42e2e1b3fc56f448d53d7b0c38b61736": "k_{0}=0.9996", "42e2f78a959c5a03d080067c8d9478b5": "G(\\theta,\\phi) = E_{antenna} \\cdot D(\\theta,\\phi)", "42e31250b4104984edab22773ad3cc2a": "P_M", "42e363d05a2dfe70da471b5900915ded": "\\int_a^b f(x)dx=\\tilde{\\mathsf{I}}(f)", "42e39af311474da27db3c73c902f4c37": "Q_\\min", "42e3b7fb814558970f853c533e2fe627": "\\mathbf{F}\\cdot\\mathbf{v} = q\\mathbf{E}\\cdot\\mathbf{v}", "42e41bdb1eabf8bc7fcc3c9176ca6b0f": "\\begin{align}\n \\ln p_{n,\\theta+\\delta\\theta} \n &\\approx \\ln p_{n,\\theta} + \\delta\\theta'\\frac{\\partial \\ln p_{n,\\theta}}{\\partial\\theta} + \\frac12 \\delta\\theta' \\frac{\\partial^2 \\ln p_{n,\\theta}}{\\partial\\theta\\,\\partial\\theta'} \\delta\\theta \\\\\n &= \\ln p_{n,\\theta} + \\delta\\theta' \\sum_{i=1}^n\\frac{\\partial \\ln f(x_i,\\theta)}{\\partial\\theta} + \\frac12 \\delta\\theta' \\bigg[\\sum_{i=1}^n\\frac{\\partial^2 \\ln f(x_i,\\theta)}{\\partial\\theta\\,\\partial\\theta'} \\bigg]\\delta\\theta .\n \\end{align}", "42e442427dd01817afbe5065bd3ecb0d": "\\left(3\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ 0\\right) \\pm \\left(0,\\ \\sqrt{\\frac{2}{3}},\\ \\frac{5}{\\sqrt{3}},\\ \\pm1\\right)", "42e45c9db91899d64d7b454e02259a20": "\\begin{align}\n\\left[ \\sigma_\\mathrm{n} - \\tfrac{1}{2} ( \\sigma_x + \\sigma_y )\\right]^2 + \\tau_\\mathrm{n}^2 &= \\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2 \\\\\n(\\sigma_\\mathrm{n} - \\sigma_\\mathrm{avg})^2 + \\tau_\\mathrm{n}^2 &= R^2 \\end{align}\\,\\!", "42e45e9f9a62b3587badaae5d331c6e5": "\\scriptstyle{\\pi(\\mathbf{n})}", "42e4dca336a6fa1b6ff190a34d936bf4": "\\scriptstyle\\lfloor \\scriptstyle{\\sqrt{\\varphi(r)}\\log(n)} \\scriptstyle\\rfloor", "42e51bf81669a3a398475dbd17bf4f42": " \\left\\{{n \\atop k}\\right\\} \\sim \\frac{k^n}{k!}.", "42e549abbc25a1bb5d4c2c44b9ca128a": "\n\\log p(\\mathbf{X}|\\boldsymbol\\theta) = \\log p(\\mathbf{X},\\mathbf{Z}|\\boldsymbol\\theta) - \\log p(\\mathbf{Z}|\\mathbf{X},\\boldsymbol\\theta) \\,.\n", "42e552816fe69a4eb00fa094a19a0cd4": "\n\\mathcal{G}(\\mathbf{x} ,\\tau|\\mathbf{x} ',\\tau') = \\langle T\\psi(\\mathbf{x} ,\\tau)\\bar\\psi(\\mathbf{x} ',\\tau')\\rangle.\n", "42e5587d8110bb93d917283811bcd299": "X \\in \\mathbb{R}^{m\\times n}", "42e5631b157b906708cc65a03019219b": "H: X_i = N_i", "42e5b6ed3c9c97c4da864c4b1e7da27c": "0.\\overline{85714}\\overline{2}", "42e5ba7c99481d3847d5b405274f3d01": "g_{11} = \\left. \\frac{I_1}{V_1} \\right|_{I_2=0} ", "42e5e597999e9369a1555d8f7bec5187": "\\vec{v}\\in V, a\\in k.", "42e612acd7819f9f8912715fb953ebae": "\\rho\\, V\\, \\dot{u}", "42e618b189ae2392d3004445ffd18817": "{\\rm d}s=\\frac{1}{T}{\\rm d}u-\\sum_{i \\geq 1}\\frac{\\mu_i}{T} {\\rm d} n_i", "42e63188369e2120cdff9107a1253df9": "\n \\begin{align}\n J_2^0 = & \\cfrac{1}{6}\\left[(a_2+a_3)\\sigma_{11}^2+(a_1+a_3)\\sigma_{22}^2-2a_3\\sigma_1\\sigma_2\\right]+ a_6\\sigma_{12}^2 \\\\\n J_3^0 = & \\cfrac{1}{27}\\left[(b_1+b_2)\\sigma_{11}^3 +(b_3+b_4)\\sigma_{22}^3 \\right]\n -\\cfrac{1}{9}\\left[b_1\\sigma_{11}+b_4\\sigma_{22}\\right]\\sigma_{11}\\sigma_{22} \n + \\cfrac{1}{3}\\left[b_5\\sigma_{22}+(2b_{10}-b_5)\\sigma_{11}\\right]\\sigma_{12}^2 \n \\end{align}\n ", "42e6986f0f6de7b4acfee1adfae8be7c": "H(u,x,y)=a(u) \\, (x^2-y^2) + 2 \\, b(u) \\, xy + c(u) \\, (x^2+y^2)", "42e6993d1b86cf0732a2236355508115": " t\\in[0,1]. ", "42e6f1ba20121a1435c8f9469d6dccdf": "\\neg Z \\vee X_k \\cdots \\vee X_n", "42e708609a4d8364e79ad670a2d4fca1": "d(fg) = f\\,dg+g\\,df.", "42e7290095e37d13e0fc111531589f71": "M(\\alpha,n)={1\\over n}\\sum_{d\\,|\\,n}\\mu\\left({n \\over d}\\right)\\alpha^d,", "42e7417975690fd4a293f2dbdc1f041f": "\\gcd\\left(N,10^m-1\\right)=\\gcd\\left(N_c,10^m-1\\right),", "42e749686eab3e7dae99bdbb0075f72a": " Y = K^\\alpha L^{\\beta} ", "42e7c952a66973130c722cdf0dd225ee": "\\theta'=\\theta", "42e7ef7e6b7569286742eb6de74d79fe": "\\lim_{h \\to 0} \\sup_{0 \\leq t \\leq 1 - h} \\frac{| B_{t+ h} - B_{t} |}{\\sqrt{2 h \\log (1 / h)}} = 1.", "42e8112f12c3930bf5633bcbc56e6197": " \\mu_0 \\dot \\gamma \\gg \\tau^* ", "42e813e208e06ae693f1dba86e3079fb": "\n\\begin{align}\n\\sum & \\Big( Y_1Y_2 \\ln P(\\varepsilon_1>-X_1\\beta_1,\\varepsilon_2>-X_2\\beta_2) \\\\[4pt]\n& {}\\quad{}+(1-Y_1)Y_2\\ln P(\\varepsilon_1<-X_1\\beta_1,\\varepsilon_2>-X_2\\beta_2) \\\\[4pt]\n& {}\\quad{}+Y_1(1-Y_2)\\ln P(\\varepsilon_1>-X_1\\beta_1,\\varepsilon_2<-X_2\\beta_2) \\\\[4pt]\n& {}\\quad{}+(1-Y_1)(1-Y_2)\\ln P(\\varepsilon_1<-X_1\\beta_1,\\varepsilon_2<-X_2\\beta_2) \\Big).\n\\end{align}\n", "42e86252cf3fd7a463c1dc75401e55dc": "\\displaystyle \\frac{d^n f(x)}{dx^n}\\,", "42e8cf882581008f2b1f5e9f2e0fc3ab": " \\max \\{\\, d(x,y) : x \\in \\mathcal{A},\\, y \\in \\mathcal{B}\\,\\}. ", "42e902294c137c036f17920ff2da55af": "\\frac{\\varphi^2}{\\xi}", "42e95b7b5af08d4bf4a361245a27f52b": "i=1,\\ldots,p", "42e9bf3d728ab138c22f6d425493f3e3": " \\dot{\\hat{\\mathbf{x}}}_r(t) = F_r(t)\\hat{\\mathbf{x}}_r(t) + K_r(t) {\\mathbf{y}}(t),\\hat{\\mathbf{x}}_r(0)={\\mathbf{x}}_r(0),", "42e9cce9f5fb1835194d5645c61043cd": "Q_{acc} := (-|1\\rangle, |0\\rangle)", "42e9e3d7aaeef3774b71987f5987a95e": "\\scriptstyle\\approx", "42e9eadeed47c76d458aaac8fa9b98ce": "~\\Phi_{12}(x) = x^4 - x^2 + 1", "42ea3df58a9fc2afb48f5c281581aca8": " \\mathcal I (\\mathcal V(I)) = \\sqrt I .", "42ea4a5e41513172702ec7c5e6e204b6": "\\textstyle{\\sum_{k=0}^\\infty u_k}", "42ea8e10cdb20d0afab59502eaa38737": "\\zeta_\\Gamma(s)=\\prod_p(1-N(p)^{-s})^{-1},", "42ea97fcf5bd5801d0400b3a2647d978": "\\frac{3}{3+2\\sqrt{5}} = \\frac{3}{3+2\\sqrt{5}} \\cdot \\frac{3-2\\sqrt{5}}{3-2\\sqrt{5}} = \\frac{3(3-2\\sqrt{5})}{{3}^2 - (2\\sqrt{5})^2} = \\frac{ 3 (3 - 2\\sqrt{5} ) }{ 9 - 20 } = - \\frac{ 9-6 \\sqrt{5} }{11}", "42eaac65c621b9e94e5ed35c69ee5644": "\\Delta{v} = \\int_{t_0}^{t_1} {\\frac {|T|} {m}}\\, dt", "42eab255bbbf9d4364b981a27adfb952": "\\alpha(\\lambda)=\\left(\\frac{I_0(\\lambda)}{I(\\lambda)}-1\\right)\\frac{1-R_\\text{eff}(\\lambda)}{d}", "42eabf956269b2d3a5172960c7fab347": "F(\\mathbf{x})", "42eba6d0477ada578c95923e4b61ac9c": " \nR_z = \\frac {R_a R_b + R_a R_c + R_b R_c} {R_c} \\qquad\nR_x = \\frac {R_a R_b + R_a R_c + R_b R_c} {R_b} \\qquad\nR_y = \\frac {R_a R_b + R_a R_c + R_b R_c} {R_a}. \\qquad \\, ", "42ec44e5327ac24048c62b892dee1ce9": " \\Delta \\phi = 2 \\pi \\, \\Delta \\omega \\approx -\\pi \\, \\sqrt{\\frac{m^3}{r^5}}", "42ecd8c3ba556953c48dc64867518449": "x_i \\in GF\\left( {q^N } \\right)", "42ecf5062178826e3f3cc606cfb56b97": " E_{n+1} ", "42ed69b274801e1b208a9babc92427f5": "(\\mathbf{r}, t)", "42ed9773f0b1117491459c0b9ee05b7e": "F_{}^c + (mg - ma_G) = 0", "42ee086bf0d0e65e7e44c7a29a21613d": "\\frac{m^2}{\\kappa}", "42ee3996b32a40f8df010fab6aab23ba": "\\int x^2\\,dm = I_{y} \\ne I_{x} ", "42ee78a1681e8595b2d8dddcce170f75": "H = X \\left(X^\\top X \\right)^{-1} X^\\top.", "42ee805b682d26c920c7f775d3fa4f32": "AB \\parallel CD", "42ee9c8fe820cccbc68d562d2f407008": " \\scriptstyle \\phi(\\varepsilon_n | \\Omega) ", "42eec9d04674e9591cf762df9918e372": "\\pmod{m}.", "42eed56da80c18140da3dd870a39aebe": "J \\approx\\beta a b^3", "42eeeaf1aed7ab4ce84d459257df20e9": "v=n_1\\cdots n_{N_G}", "42eeeb147f9adb28a8d1c73b1fe995ff": "C_0 exp(RF/\\alpha)", "42ef17c319fb94c3fd5c5cff974451a9": "\n|\\eta(s)| = \\lim_{n\\to\\infty} |\\eta_{2n}(s)| = \\lim_{n\\to\\infty} |R_n(\\frac{1}{{(1+x)}^s},0,1)|\n = |\\int_0^1 \\frac{dx}{{(1+x)}^s}| = |\\frac{2^{1-s}-1}{1-s}| = |\\frac{1-1}{-it}| = 0.\n", "42ef59ca2b7fa075637f3a733ebec3b4": "\\,\\!\\alpha", "42f03189fefd7c0897c45054d472eb94": "\\varepsilon: L\\rightarrow L", "42f07682f58ba2452b0656b913d92df4": "\\tau\\in[0,1] ", "42f08818433c456f5ee544973c0e0af0": "\\mu=24", "42f088f563df0e0d160188ea07d64837": "q \\,\\ ", "42f16b30d624ba89d30408c6fd5ea3b6": "c_i + c_j = x\\cdot g_{i+j} = c_{i+j}", "42f189523fca9924e5037e9d747e9fa2": "xy \\le zu \\leftrightarrow \\forall v ( zv \\equiv uv \\rightarrow \\exists w ( xw \\equiv yw \\and yw \\equiv uv ) ).", "42f1b0d4d4c127c2dcff79163a97b310": "y_2=\\frac{2}{x}+\\frac{3}{x^2}e^{-2x}.", "42f1b95b116733e04b49cb6c41454c9d": "\\sum_{n=0}^t f(2n) + \\sum_{n=0}^t f(2n+1) = \\sum_{n=0}^{2t+1} f(n)", "42f1cd35faf2ac94d87f0b72849996a4": "\\hat{x}\\ ,\\ \\hat{y}\\ ,\\ \\hat{z}", "42f240f68fd52e0190d8374a53366edb": "y^p - y = f(x)", "42f2584543deb2f0a70258ef0eaad5c1": "S \\circ \\theta : E \\to F_{S}", "42f26bd87c30e9b54d0bea6854f4163d": "p^3 \\; (1-p)^9", "42f28c463929e4be3594148f50b693ac": "\\,\\!x\\ne 1+1+1", "42f2adf3668ae00b33b5fc2c9b4c510b": " \\mathbf{A}\\cdot\\mathbf{B}=\\mathbf{B}\\cdot\\mathbf{A} ", "42f2f47f5f10d4c13bc8d09b576be685": "Z_o = 50\\Omega\\,", "42f395c03fcc57a733bc8ce075448093": "V_\\text{o} = \\frac{A}{1 + j\\omega R_\\text{i}C_\\text{M} }V_\\text{i} \\,", "42f3c7366b65626551b9707138751c27": "\\lfloor \\frac{y}{12} \\rfloor", "42f3e792bfef7ba7472fc7d53c63fea9": "Q_{total} = {2 \\over 3} \\pi P \\mu \\omega \\left (R_{shoulder}^3 - R_{pin}^3 \\right ) ", "42f40c3c3d2f2f5fa55ac13489552529": "\\frac{a}{c} < \\frac{b}{d}", "42f42ff029938578c8da4c7e8d9e61bd": "W : [x_1, x_2] \\to \\mathbb{R}", "42f4602848d723f36827297865672f6c": "\\psi_\\sigma", "42f4f6443a08ba343fa936aa907d97dd": "E_1, E_2, E_3, \\dots", "42f5898b2dc1c839e1fab6555425353d": "f({\\mathbf{x}})", "42f59e8e1e91fc30e79c921ee7924985": " G = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0 \\end{bmatrix} ", "42f5be15b568b97fd4c4a065fecfabe9": "d = 2a", "42f5dd20aa7049cf9100e2a31c1cfd20": "\\sigma \\gg \\omega\\epsilon_0\\epsilon_r", "42f623098b0a79946db7f93aaebdae24": " a = \\gamma /( \\sqrt 2 \\sigma) ", "42f630d7b904eed1ea903688b99cb130": "\n \\overset{\\circ}{\\boldsymbol{\\tau}} = \\phi_{*}[\\dot{\\boldsymbol{S}}]\n", "42f6739a6e72c142ced1073e09ff9fd5": "\\mbox{DTIME}\\left(n\\right)", "42f6a32cb279fc19b44ac5c269aa11b9": "G_{A}", "42f6f1421745b4149c7090a05a7ec724": "L_2(2) \\cong S_3 \\twoheadrightarrow S_2", "42f72949dfe19422e3adbe4e962e449b": " \\epsilon H = p(t)(q(t+\\epsilon) - q(t)) - \\epsilon L \\,", "42f74e2621bd79c69fc9ea30f41c7a12": "H_n(x+y)=\\sum_{k=0}^n{n \\choose k}H_{k}(x) (2y)^{(n-k)}= 2^{-\\frac n 2}\\cdot\\sum_{k=0}^n {n \\choose k} H_{n-k}\\left(x\\sqrt 2\\right) H_k\\left(y\\sqrt 2\\right).", "42f758bb2e680343d15ed49d27e26239": "V_H = -\\frac{IB}{nte}", "42f769a3dfad3614684953637f816a8a": "Q= \\mathbb R^4", "42f7a1a7599cfca7aecd90244f0b6c81": "(Bu|u) \\ge c\\|u\\|^2", "42f7b54f92ceb6f2e5a17ea6c41a135c": "B\\geq 0", "42f89345e1ad8b2c436ab964117e9e8b": "\\tfrac{2}{5}", "42f8a6e3d2e5d5203eda223bed186783": "K*l", "42f8e89fd9802061294d66b5fd40f890": " \\partial_\\mu A^\\mu = f \\,", "42f8efc0c14ebe873a7409d328da6d24": "\nW_0(x)^r = \\sum_{n=r}^\\infty \\frac{-r(-n)^{n-r-1}}{(n-r)!}\\ x^n,\n", "42f8f9197123920452125ae79744fb52": "\\;\\delta\\ne\\pm 90^\\circ\\; \\rightarrow \\;\\cos \\delta \\ne 0", "42f9092b84a648f8bf4f9e5d633175b3": "p^{2}\\Psi\n=p_{\\perp }^{2}\\Psi ", "42f9290e63acff88833442f771200499": "\\mathbf{A} = \\left[\\begin{array}{ccc}\n -1 + 2q_1^2 + 2q_4^2 & 2(q_1q_2 - q_3q_4) & 2(q_1q_3 + q_2q_4)\\\\\n 2(q_1q_2 + q_3q_4) & -1 + 2q_2^2 + 2q_4^2 & 2(q_2q_3 - q_1q_4)\\\\\n 2(q_1q_3 - q_2q_4) & 2(q_1q_4 + q_2q_3) & -1 + 2q_3^2 + 2q_4^2\n\\end{array} \\right]", "42f95a9bd9958a0adc1510f82d94d7ac": "\\mathrm{Si}", "42f96a06f6db4695fda65f09e8e7c642": "r_1=1,\\,r_2=1,\\,\\ldots,\\,r_k=1", "42f97af643caecbbb9fd201afecc65c1": "z\\cap z' =\\{P\\}", "42f98d9e2da48d68d8a3619b879b38fd": "d_1 = d_2 = a = \\sqrt {c^2+b^2}", "42f9c4656525c46d7bbb388827ebc73c": "S, p, \\{N_i\\}", "42f9c6550d1bb5e563a413f422916cec": "C=C\\left(Y\\right)", "42f9c9429dc2a968922d935433501316": "x_1,x_2,\\dots,x_{2g-1},x_{2g}", "42fa516e4231059f016655e9096064bd": "(a;q)_\\infty^{-1} = \\sum_{k=0}^\\infty \\left(\\prod_{j=1}^k \\frac{1}{1-q^j} \\right) a^k\n = \\sum_{k=0}^\\infty \\frac{a^k}{(q;q)_k}", "42fa767ad71c4453c987afe3b05f4687": "\\Phi(\\Omega) = \\int_M \\Omega \\wedge * \\Omega,", "42fac8eae629330090e8e6f8b254ef62": "z^2U_{r}(z)''+p(z)zU_{r}(z)'+q(z)U_{r}(z)=I(r)z^r", "42fb1a9ca93f3a633b6bdde6110ccca5": "L^{B/A}\\otimes_BC\\rightarrow L^{C/A}\\rightarrow L^{C/B}\\rightarrow (L^{B/A}\\otimes_BC)[1]", "42fb24cc49aa06ca05986e5b319ed561": "\\frac{4x^2-8x+16}{x(x^2-4x+8)}=\\frac{A}{x}+\\frac{Bx+C}{x^2-4x+8}", "42fb518748920af58749740ebb7a1d60": "\\Delta_k(x)", "42fb98605400c40bbd85ba0b3989983f": "\n \\begin{align}\n \\rho_0 \\frac{\\partial \\mathbf{v}}{\\partial t} + \\nabla p & = 0 \\qquad \\text{(Momentum balance)} \\\\\n \\frac{\\partial p}{\\partial t} + \\kappa~\\nabla \\cdot \\mathbf{v} & = 0 \\qquad \\text{(Mass balance)}\n \\end{align}\n", "42fbbe2ee775763078d8e764ade226ef": "U\nGR=8\\log{0.25\\over L_{b}}\\Sigma_{n}({L_{n}^2\\omega_{n}\\over p_{n}^2}),\\!", "42fc2e38e4b9e59651b4489ef3943272": "\\big. R = \\frac{\\, \\Delta T}{\\dot{Q}}, \\quad", "42fc4df7d63401d452575758b149df48": "\\Pr(D=d) = {q^{100d} \\over 1 + q^{100} + q^{200} + \\cdots + q^{900}},", "42fcb19d4e63782c9f2fb419786ad364": "\\begin{smallmatrix}R = \\sqrt[3]{\\frac{GMT^2}{4\\pi^2}}\\end{smallmatrix}", "42fcc839358936ebcfefd54b38d837b8": " \\mathbf{P} \\left( \\left| \\sum_{j=1}^n X_j - \\frac{A_3 t^2}{3A_2} \\right|\\geq \\sqrt{2A_2} \\, t \\left[ 1 + \\frac{A_4 t^2}{6 A_2^2} \\right] \\right ) < 2 \\exp (- t^2), \\qquad \\text{for } 0 < t \\leq \\frac{5 \\sqrt{2A_2}}{4L}. ", "42fce56dc4d418a6f44573682a76b8dd": "x_t = (x_{t1},\\dots,x_{tN})", "42fd6cc7db0795269d96b51bc83e6025": "U \\subset \\mathbb{C}^n", "42fed25477fa25085e9232263c4cce42": "W_1 = \\frac{1}{8} \\, \\left( C_{abcd} + i \\, {{}^\\star C}_{abcd} \\right) \\, C^{abcd}", "42ff00518a5b8b7ee007a814c9238eab": "[0, 1] \\sqcup \\{*\\}", "42ff07a985f56b3261c3fc7901436374": "H\\cap H^x=\\{1\\}", "42ff0916a5a7f3b74d4adbff3b4843aa": "z_\\mathrm{R}", "42ff30982e4f9f442a6646a46602e6f3": "Y(f) = H(f)X(f)", "42ff412d68f812565e6bd2ef4bc83a18": " (n+1)(n+2)~r^n~\\cos(n\\theta \\,", "42ff9a95ba300257dad3ea516942299a": "\\alpha_1 = 1", "43005f04c8a54541866cb42bf5651bae": "B=\\mathbb{C}", "43006471d6cfd6ea4a7dd87ecb4544d6": "r_ir_j", "430093a69224ab618b34f0956255f6ea": "F_{n+1}(a, b) = (x \\to F_n(x, x))^{\\log_2(b)}(a)", "4300981bdf91256a0673af1a99b51022": "\\scriptstyle C^2~=~ \\alpha/ \\tau_{_0} .", "4300b4e770badbcfab84c9e7757baafd": "\\mathbf{Min}", "4300d6ecfa7ce6a97da8fc3254132894": "\n\\begin{array}{lcl}\nK,N &=& \\text{as above} \\\\\n\\theta_{i=1 \\dots K}, \\phi_{i=1 \\dots K}, \\boldsymbol\\phi &=& \\text{as above} \\\\\nz_{i=1 \\dots N}, x_{i=1 \\dots N}, F(x|\\theta) &=& \\text{as above} \\\\\n\\alpha &=& \\text{shared hyperparameter for component parameters} \\\\\n\\beta &=& \\text{shared hyperparameter for mixture weights} \\\\\nH(\\theta|\\alpha) &=& \\text{prior probability distribution of component parameters, parametrized on } \\alpha \\\\\n\\theta_{i=1 \\dots K} &\\sim& H(\\alpha) \\\\\n\\boldsymbol\\phi &\\sim& \\operatorname{Symmetric-Dirichlet}_K(\\beta) \\\\\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& F(\\theta_{z_i})\n\\end{array}\n", "43014f0226b487e924ee18952582d71c": "\\Phi = \\begin{cases} \\tan\\tfrac{\\pi\\alpha}{2} & \\text{if }\\alpha \\ne 1 \\\\ -\\tfrac{2}{\\pi}\\log|t| & \\text{if }\\alpha = 1 \\end{cases}", "4301a04cfc6303d0b8a7e623665657fa": "\\psi^{(-2)}\\left(\\frac12\\right)=\\frac14\\ln\\pi+\\frac32\\ln A+\\frac5{24}\\ln2", "4301c7c85c4b2c568c6c7416f1620cb2": "U \\mapsto \\varinjlim_{V\\supseteq f(U)}\\mathcal{G}(V).", "4302218711ebf367e492e8a4f7a264b5": "p_{1H}", "43026866b6c626ead468e5715cce274b": "A_n^{-1}=a_n;\\quad A_n^\\alpha=\\sum_{k=0}^n A_k^{\\alpha-1}", "4302b5144815c49bbf5982ea366263eb": "(M^\\ast)^\\ast=M", "4302fde67d4b84111be436172c8dc323": " \\vdash \\ \\ \\forall a f(a) \\rightarrow \\ f(c) ", "43030564ddcef841ca6007330abb1ebb": "({\\mathbf e}_1,\\dots,{\\mathbf e}_n,{\\mathbf f}_1,\\dots,{\\mathbf f}_n)\\,", "4303714385e924868ac1a96da16b2755": "\\Delta \\lambda=\\left[\\frac{2 \\delta n_0 \\eta}{\\pi}\\right]\\lambda_B", "4303f52bcf9bb5e89a14ff385d124515": "SL(2,3)", "43046ce883a756f54a64c9f332a7e0ac": "10^{-11}", "4304b24523b23f1b2139a6daf1628bef": "m^*(t)", "4305760248a2e7a8c9855fbab881aaa4": "f,g:S^n\\to S^n \\,", "4305b68906218db8d55720176de38fc5": "\\gamma v t' = \\gamma^2 v t - x \\left ( \\frac{\\beta^2}{1-\\beta^2} \\right )", "4305bbd91a7fa9e6d7e17369eb228fb0": "s^2 + 2 s \\zeta \\omega_n + \\omega_n^2", "4305d4ea6fd8d8ec9855227108ab890c": "m^a", "4305fe42040a0689e95c96abb12d71df": "{172,200}\\over {49\\choose 6}", "4305fe6117d86798fc089181f778d2c5": "\\chi^{(1)}", "43065f0ae6dc0d7ff9780b1414abeae4": "E_{ij} x^k_1=0", "430686b160700d6b27d5c98521a58127": "T = \\sqrt{\\frac{\\phi^2}{\\sqrt{(r-1)(c-1)}}} .", "43071ce13a9bac66323cc7d79022fdae": "\\epsilon_z=\\frac{\\partial u_z}{\\partial z}\\,\\!", "430740a774d85ef8ec107e2c799e9bb5": "ky = \\int k \\frac{dy}{dx} dx. \\quad \\mbox{(2)}", "430778e35d3fc1202c9bcbd89498c01b": "\\mathrm{N}_G(S)=\\{ g \\in G \\mid gS=Sg \\}", "43078bbe67ed8d55801deec58a70f40b": "v_{2}", "43079b0ee2cc4ec1f2c83753636a6e07": "\\hat{L} = L(\\hat{x}, \\hat{p})", "43079db630a62d4c07231da3434b32e2": "\\!4\\pi r^2 = A(r).", "43081c08da109f43aecd6a974b5a7799": "\\int_2^5 x^2\\, dx = F(5) - F(2) = \\frac{5^3}{3} - \\frac{2^3}{3} = \\frac{125}{3} - \\frac{8}{3} = \\frac{117}{3} = 39.", "430830b29a47a24fbadc077e6db2f54f": "{\\bar{PP}}_3", "430833c92e6e391ad4b5e64d15779273": "a_1y_1+\\cdots+a_ny_n", "43083475b9b82bc5fda46a5e7a9f20f4": "\\Omega=1\\otimes \\delta_1", "4308ab9029f243263018c2f86fea512a": "\\hat{X}_U", "4308d8233e31278820ab26ad7656ec7a": "H, C_a", "43094b526d2ed86cea0661e4de60319e": "\\hat{H}\\,", "430987c00a3ed16df8647f9029018352": "1\\text{pg} = 978 \\text{Mb}", "43098e4de7fd050c09e1811c124dc106": " \\oplus_n f_*\\omega_Y^{\\otimes n}", "4309e3cdc6b99daef0e52db1a9673e93": "f(\\boldsymbol\\mu,\\boldsymbol\\Lambda|\\boldsymbol\\mu_0,\\lambda,\\mathbf{W},\\nu) = \\mathcal{N}(\\boldsymbol\\mu|\\boldsymbol\\mu_0,(\\lambda\\boldsymbol\\Lambda)^{-1})\\ \\mathcal{W}(\\boldsymbol\\Lambda|\\mathbf{W},\\nu)", "4309e8ed159447a1e30c4c0a94a421b5": "D_k\\subseteq C_{n-1}", "430a398d12b26ece2acf48a49de28ef8": "\\frac{2}{\\pi} = \\frac{ \\sqrt{2} }{ 2 } \\cdot \\frac{ \\sqrt{2 + \\sqrt{2}} }{ 2 } \\cdot \\frac{ \\sqrt{2 + \\sqrt{2 + \\sqrt{2}}} }{ 2 } \\cdots", "430a62e250d15141ce0a79ebf5e0bf5c": "\\cos \\left( \\frac{\\arccos (1/2)}{3} \\right) = \\frac{1}{2} \\left( 2\\cos \\left( \\frac{\\pi}{9} \\right) \\right)", "430b8b2e000bda3dd9591bcdf7c54f90": " \\frac{\\delta d}{d}= \\frac{2 \\gamma_{1-l}}{3 p_{l} q_{m} t r} ", "430ba36cdc2cc066cf582bb8d24153aa": "v_{\\mathrm{c}}", "430bddb2e223b30980ef4384f45b38a7": "\\lambda\\to +\\infty", "430be459db8937d595ecb646f89dc14d": "3, 6, 9, 12, 15, 18;\\,\\!", "430c0613a292687463694919346ce296": "R_D(f)=\\frac{f}{6.8966888496476\\cdot10^{-5}}\\cdot\\sqrt{\\frac{h(f)}{(f^2+79919.29)\\,(f^2+1345600)}}", "430c5b41b5c35c3a65d715c4c93f2959": "\\Phi(\\mathbf{u}) = \\left(1+\\frac{u}{2}\\right)e^{-u^{2}/4}.", "430c60e5ba351ff1ed57629a8c2cb21a": "\\frac{\\omega^2}{k^2}=c^2\\,\n\\frac{v_s^2+v_A^2}{c^2+v_A^2}", "430c7b4abd6835a818f547a7325fcf47": " {T_{ij} = G \\cdot S _{ij}} ", "430cb5f5897ca11a4b06984e052f020b": "f^0=1", "430cf705cbf3052387f811d7f3ab5e72": "\n\\frac{1}{v^2}\\frac{\\partial^2 u}{\\partial t^2}=\\frac{\\partial^2 u}{\\partial x^2}. \\,\n", "430d4c14957f5b6a0437ca682a602389": "1 = \\sum_{a = \\alpha}^\\beta e^{-ra}\\ell(a)b(a)", "430de64befdfdf00fdb550447dfd5290": " F(N,V,T) = - k_{B}T ln Q ", "430dec90bb979f33397402a7cb3e8cb2": " L^{*}_{\\left(p-k\\right)} = V_{\\left(p-k\\right)} \\Lambda_{\\left(p-k\\right)}^{1/2} ", "430e1dec9018f5a70dbbba9b5f5502d9": "\\mathbf{l}_b - \\mathbf{l}_a", "430e29afa613dc4a6068d69ce86a0eb2": "\n\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n \\cos \\theta & \\sin \\theta \\\\\n -\\sin \\theta & \\cos \\theta\n\\end{bmatrix}.\n", "430e5017cac3e7bf66c0ed47fa838331": "f '\\,", "430e76ececc831b8318095edba7a1e3e": " \\alpha\\le 1 ", "430e7de2484db3b5a632379437c6cef7": "i\\in I", "430e8da4e02b4e00fd80c147b2acbda9": "W=\\int_{1}^{2} F\\,dx.", "430f6d3ca9ce6469181715692292d629": "T = \\frac{h_ah_b}{2 \\sin \\gamma}.", "430fb980efafbd8d27b7769ca9436beb": "\\mu_B = \\frac{e\\hbar}{2m_0} = 0.5e\\nu_BS_B = \\frac{ea_B^2}{\\sqrt{L_BC_B}}. \\ ", "430ff6a52811529a0234a3d626d2252a": "U=e/C", "43102484a5b7611cb943c2a999a06382": " \\nabla\\cdot(\\mathbf{A}+\\mathbf{B})=\\nabla\\cdot\\mathbf{A}+\\nabla\\cdot\\mathbf{B} ", "43105c3278b7f6c71315f66dfc52248a": "R=\\frac{1}{\\sqrt{-K}}.", "4310a525cb0ae87541042bfefea51c79": "B: \\mathbb{R}^n \\times \\mathbb{R} \\mapsto \\mathbb{R}^{n \\times m}", "4310e70cbff5d8686999c03bd1bf86cc": "\\mu^2(w,\\overline{w}) \\;\n\\frac {\\partial w}{\\partial z}\n\\frac {\\partial \\overline {w}} {\\partial \\overline {z}} = \n\\lambda^2 (z, \\overline {z})\n", "4310f8dcf80cc3c39378fe0abd5f5c00": "\nS(x,y) \\approx \\sum_u \\sum_v w(u,v) \\, \\left( I_x(u,v)x + I_y(u,v)y \\right)^2,\n", "43112cb7b67b4062a12b144acb4d2249": "\\textrm{throughput\\,accounting\\,ratio} = \\frac{\\textrm{return}}{\\textrm{factory\\,hours}}", "4311310af470b387cf43501b46d05882": "\\mathcal{F}_\\alpha(f)", "431139622c7b3d6bc7677f69208dc610": " ^y \\sqrt{y + 1}_s ", "43119cdfaf20e7246d37b0506cbb9d82": "\\chi_U(x) = \\begin{cases}1 & x \\in U \\\\ 0 & x \\not\\in U\\end{cases}", "4311b31e44f158d1adffc82c260fdcfc": "Y_3 = D\\cdot E-F\\cdot A ", "4311ccbf49993e5058af84a198da22f9": "\\textit{great}", "4311efd9f1af840feea0e81a69a9289f": "\\theta = \\arccos \\left( \\frac{\\text{adjacent side}}{\\text{hypotenuse}} \\right)", "4312617a2a6296816a2f26de8d4e6ca4": "f+g\\in C(E)", "4312859dd325068696ecd635ec6c78bd": "\\vec{x}(t + \\Delta t)", "4313323bd73a2ecce4b168a05a7aaf80": " I_n \\!", "43135a99f00446e7c99d1831be8ae0e6": "\\psi(x) = \\rho(x)\\, e^{i\\theta(x)},", "4313d6d9ea0b421ae7eb3cc258d7273d": "i_c=2i_1|\\cos(\\pi\\frac{\\Phi_a}{\\Phi_0})|.", "4313db48baaa49e63ab3371df7522a1d": "\\lim_{\\Delta x\\to 0} \\frac{\\Delta f\\Delta g}{\\Delta x} = \\lim_{\\Delta x\\to 0} \\left ( \\frac{\\Delta f}{\\Delta x}\\frac{\\Delta g}{\\Delta x}\\Delta x \\right ) = \\lim_{\\Delta x\\to 0}{\\frac{\\Delta f}{\\Delta x}} \\cdot \\lim_{\\Delta x\\to 0}{\\frac{\\Delta g}{\\Delta x}} \\cdot \\lim_{\\Delta x\\to 0}{\\Delta x}= f'(x_0) g'(x_0) \\cdot 0 = 0", "431440bd24c5fe60649759ebb8b2c063": "y(t) = F(x(t))", "43149a5b5875036c0c4ed3304bae28ad": "\\mathbf{A}_i", "43149cdc7e8c6c770600a4ff12d9eaa2": "H\\mathbf{x}", "4314c919044bf4d484a945ff8faeadb3": "r\\left(r-1\\right) + p\\left(0\\right)r + q\\left(0\\right) = I(r)", "4314e751a3f2b2f87e21b60b74b34880": " \\Delta \\omega \\equiv \\omega _1 - \\omega _2 ", "431545e638e927e611f5ed86c421eaf8": "H_{\\omega^{\\omega + 1} + 1}(1) - 1", "4315b1e737eec4c6fa12ab93f152406d": " p \\rightarrow \\Box \\Diamond p", "4315e3c2464d6b08b3fe47d91faee2d1": "\n\\begin{array}{lrclr}\n\\max\\limits_{x_{t}} & E[Q_{t+1}(W_{t+1},\\xi_{[t+1]})|\\xi_{[t]}] & \\\\\n\\text{subject to} & W_{t+1} &=& \\sum_{i=1}^{n}\\xi_{i,t+1}x_{i,t} \\\\\n &\\sum_{i=1}^{n}x_{i,t}&=&W_{t}\\\\\n\t\t & x_{t} &\\geq& 0\n\\end{array}\n", "4315eb9c09ab2ad39b94e23ee36ab4a2": " \\displaystyle{ds^2=(1+t\\kappa(\\theta))^2 d\\theta^2 + dt^2,}", "43164fef62d5beaff5afab848918041d": "p_0(x)=1 ;", "431662e5cd278ef815f5186a1d6f4bab": "k'_x,k'_y,k'_z\\,", "4316e7e875431aba8ad0ad8de2d5bcd4": "sim(d_k,q) = \\frac{\\sum _{j=1}^n \\sum _{i=1}^n w_{i,k}*w_{j,q}*t_i \\cdot t_j }{\\sqrt{\\sum _{i=1}^n w_{i,k}^2}*\\sqrt{\\sum _{i=1}^n w_{i,q}^2}}", "431743e416fed49d44bfa05347185e42": "{{{1}}}", "4317c3bb4cbb367b00fe6818a14176de": "\\zeta_4", "4317cbcd52e85551b6ece28817e5dbc8": "Y_{10}^{-7}(\\theta,\\varphi)={3\\over 512}\\sqrt{85085\\over \\pi}\\cdot e^{-7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot(19\\cos^{3}\\theta-3\\cos\\theta)", "43186e53fe7ace814dad956dd2eb4411": "\\,L_n = F_{n-1}+F_{n+1}=F_n+2F_{n-1}", "431886999e1101c53861a9300593e894": "\\scriptstyle \\gamma_{\\mathrm{ls}} - \\gamma_{\\mathrm{sa}} ", "4318897026c7d116f38dfc7ffea27b86": " y + \\Delta{y} = (u + \\Delta{u}) + (v + \\Delta{v}) = u + v + \\Delta{u} + \\Delta{v} = y + \\Delta{u} + \\Delta{v}. \\, ", "4318cdab0619dd3ca60ada970f8549e2": " \\operatorname{E}[\\log(X_i)] = \\psi(\\alpha_i)-\\psi(\\alpha_0)", "43190c3957bc75ba04279c24812fab04": " \\{a_1, a_2, a_3, \\dots , a_n\\} ", "43194b2b4829b5bbb8055b72dcf6c8a7": "\\exists \\;xP(x)", "43194dec961e250596d6ad93226afd42": "x_1=a(e(x_1,x_2),f(x_1,x_2))", "431964f61ac2ec6f02b9b7cf6fda0235": "\\underbrace{{{d}_{p}}}_{\\begin{smallmatrix}\n \\text{width of } \\\\\n \\text{electric field}\n \\\\\n \\text{within p-side}\n\\end{smallmatrix}}{{N}_{A}}=\\underbrace{{{d}_{n}}}_{\\begin{smallmatrix}\n \\text{width of } \\\\\n \\text{electric field}\n \\\\\n \\text{within n-side}\n\\end{smallmatrix}}{{N}_{D}}", "4319d236cadfabf379292ff560626d61": "\nF(r) = Ar^{-3} + Br^{-2}\n", "4319f478cadf3bc9ec76cc692d399057": " M_{\\psi\\psi}\\ddot{\\psi} +\n C_{\\psi\\psi}\\dot{\\psi} +\n K_{\\psi\\psi}\\psi +\n M_{\\psi\\theta}\\ddot{\\theta_r} +\n C_{\\psi\\theta}\\dot{\\theta_r} +\n K_{\\psi\\theta}\\theta_r =\n M_{\\psi}\\mbox{,}\n", "431a1548ace55a9edeaa6681e0c6d820": "P(\\overline{\\theta})", "431a5bdb75e1d9bdee0419e0445c0935": "U_1,\\ldots,U_n", "431a829b7ef80de5e7bff15618a0bc58": " F=-kx, \\ ", "431aa7fd4559114f94cb46ee1abe6398": "n_{\\text{max}} = N", "431ab893c69e1cd04b3a19f98a54c515": "\\begin{matrix}{7 \\choose 5} = 21\\end{matrix}", "431af19cbd26c07c356c1ebb714819e4": "x \\neq -x", "431b16568729e705176877f04eddb999": "G(A,B)\\subseteq G(A',B')", "431b4d0e262f67394a098f9b56fbf797": "\\pi = \\partial_t\\phi", "431b80ab5a7a5cb11191e4f71c5e74a7": "U^\\dagger", "431b8e3cc3c7accd52cfdc6788a2034b": "(\\omega_i = \\Delta(C_\\text{in}(y_i'), y_i) < {d \\over 2})", "431b92ae6fa1e6603937bba55f593c19": "\\frac{P \\to Q, R \\to S, \\neg Q \\or \\neg S}{\\therefore \\neg P \\or \\neg R}", "431b93021f49c7ed46e7e47b3dc23f77": "Z^V", "431b9b80dd2618fff715346f3925741a": "M\\preceq N", "431baacfc3f9e8da1fb48b6cc6c39434": " (\\operatorname{arccsc} x)' = -{1 \\over |x|\\sqrt{x^2 - 1}} \\,", "431baed00254a242d429539339bb7df6": "\n\\pi = 16 \\tan^{-1} \\cfrac{1}{5} - 4 \\tan^{-1} \\cfrac{1}{239} \n= \\cfrac{16} {5+\\cfrac{1^2} {15+\\cfrac{2^2} {25+\\cfrac{3^2} {35+\\ddots}}}} \n- \\cfrac{4} {239+\\cfrac{1^2} {717+\\cfrac{2^2} {1195+\\cfrac{3^2} {1673+\\ddots}}}}.\n", "431bce03e969d4c366d21b1b889c2b52": "GL(d,p)", "431c235f007d242f7f4df089324577b5": " \\operatorname{dim}_{\\mathrm{Haus}}(X) =\\sup_{i\\in I} \\operatorname{dim}_{\\mathrm{Haus}}(X_i).", "431c2e8f0c48eb190501ab5b5f3fc9aa": "X^{p}", "431c2ea2b1ff0a9a12223850e2ea4de8": "\\Box A=\\{x\\in F;\\,\\forall y\\in F\\,(x\\,R\\,y\\to y\\in A)\\}", "431c301b0bcc5355f7ec5dcf9a1cd79d": "\\mathbf{v}=a_1 \\mathbf{u}_1 + \\cdots + a_m \\mathbf{u}_m + b_1 \\mathbf{w}_1 +\\cdots + b_n \\mathbf{w}_n", "431c6a54f56fd7a10ce989e0c9f64045": "C^\\infty\\ ", "431cfe6421d7f816acd91e323005be0d": " Y_i = \\begin{cases} 1 & \\text{if }Y_i^{1\\ast} > Y_i^{0\\ast}, \\\\\n0 &\\text{otherwise.} \\end{cases} ", "431d07fb971f427bca0bdfd7b700f8f6": "\\omega(\\alpha,\\beta,\\gamma) = \\sin\\! \\beta \\,,", "431d1e7631fc46eacefcdf4c8d64f2ac": "V[u + \\varepsilon v]", "431d2cce90421fab746de4617cb22849": "{\\mathbf T}_1, \\; {\\mathbf T}_2, \\; {\\mathbf T}_3", "431ddded08e4a06724eb2788d15d9aba": "\n \\left|A\\right|_{ij} = a_{ij} - r_i^j\\, \\bigl(A^{ij}\\bigr)^{-1}\\, c_j^i .\n", "431ddf58209b8889980c97ccda32632c": " e^{(a)} = e^{(a)}_\\mu dx^\\mu ", "431e354bb62fc85ea0605f3aeba7a1e3": " U_{\\mathrm{max}} = B_0", "431e4a5314cd67d4a9c1d9197606232f": " x_{n+3} - 5x_{n+2} - 5x_{n+1} + x_n = 0, \\, ", "431e674b634c18c8a27fd3427dda11c7": "\\underline{+\\mathit{3} 20 \\mathit{6}}", "431e79c533203a79d6847ac79e123fca": "\\mathbf{F}=\\frac{d\\boldsymbol{p}}{dt}", "431e9f20aa29ca8fda5ca839c3220edd": "(a+b)^* = (a^*b)^*a^*,\\,", "431eac28141016644e412bd834f969da": "\\scriptstyle f_s.\\,", "431ed3830136d4256a1cd78789b74e7a": "y_a=y(a)", "431ed9c66d4b1402b83e1ac5030fec6f": "Z = X/Y", "431efed5dde170e51c67208344f394fa": "\\int\\frac{\\sinh^m ax}{\\cosh^n ax} dx = \\frac{\\sinh^{m+1} ax}{a(n-1)\\cosh^{n-1} ax} + \\frac{m-n+2}{n-1}\\int\\frac{\\sinh^m ax}{\\cosh^{n-2} ax} dx \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "431f1663c5de189a593a50afcd6ae1b9": "\\alpha c=1", "431f215ba65f4c986813430360d9b48e": "\\boldsymbol{\\lambda \\Gamma} =0 ", "431f235275799dcf20a9518f0af77d5c": "\\mathbb{Z}[x,y,A,B]", "431f521c8c734f7561b974880b8d7f63": "\\scriptstyle I_{o_{\\text{lim}}}=\\frac{V_i\\, T}{2L}D\\left(1-D\\right)=\\frac{I_{o_{\\text{lim}}}}{2\\left|I_o\\right|}D\\left(1-D\\right)", "431f597aa59aefe0e064aad01e6b3cba": "\\Delta\\tau_s = - \\frac{\\omega^2}{2c^2} \\sum_{i=1}^{k} R_i^2 \\cos^2 \\phi_i \\Delta t_i", "431f9d1a112063ef97ce547cead130cc": "f_i e_i", "431fe4ae2c04a432bb9842ec7a01fed5": "\\bar X \\sim \\mathcal{N}\\Big(\\mu,\\, \\frac{\\sigma^2}{n} \\Big)", "43204f7d193f3e5d17a61135118f0cb3": "P(t)=P_0 e^{rt}-\\frac{M_a}{r}(e^{rt}-1).", "432107e712a56e128ffa0d4962fa2a27": "r_{\\theta} = r_z = f_r = 0", "43213046bce9c8e26bb2b1b80241dd4d": "\\lambda^{2} ( \\{ 0 \\} \\times A ) = \\lambda ( \\{ 0 \\} ) \\cdot \\lambda (A) = 0", "432160fd2b098790af454d0bc7ac7d91": "F =6 \\, \\pi \\, \\eta \\, R \\, v ", "432178a82689de8beb1dddf007b326a9": "\\frac1k+\\frac1k=\\frac2{k+1}+\\frac2{k(k+1)}", "432191adb1593f671652962acfc85492": "\\xi(\\alpha)", "43219e3fa59d5031e1580152b6e9b334": "A = ar", "4321c17da7614b4016673a3675399dd5": "R=-e^{-2 k^{2} h^{2} sin^2A}", "4321cbda617e59fa05d3e00a5e956a8c": "2^{8 \\times 8} = 18446744073709551616", "4321d3362479bb6d6b2e9199695c25e5": "\\langle U_{g} [e], [e] \\rangle_f = \\int_{\\widehat{G}} \\xi(g) d\\mu(\\xi).", "432214e0d24e8b84c7f4984049b0895c": " X = R_0 \\cos(\\Phi), \\; Y=R_0 \\sin(\\Phi)", "4322d1422d4ea7577267a76b435bf475": " C_1, C_2, \\cdots, C_l ", "4322e5d2210b42c0c6d28a9a635247a1": "\\ F \\sim 1 ", "43234e57379aa5177edcd86509150572": "\\Delta E \\Delta t \\ge \\frac { \\hbar } {2 } ", "432385a4eaac20a73b9c914b1517985e": "\\forall n (n \\in \\mathbf{N} \\iff ([n = \\empty \\,\\,\\or\\,\\, \\exist k ( n = k \\cup \\{k\\} )] \\,\\,\\and\\,\\, \\forall m \\in n[m = \\empty \\,\\,\\or\\,\\, \\exist k \\in n ( m = k \\cup \\{k\\} )])).", "4324248018471a61336d0f61f7f7baf1": "J_M", "43242d3a9f36452e7e133da89b3112df": "T^2 M = T(TM).\\,", "43243e6dc978caf4af7eb1c76320af4b": "\\mathbb{R}^{2n+1}", "432444dd18700f01eeef92106569cdff": "d\\theta + \\frac{1}{2}[\\theta,\\theta]=0.", "43246d6641003301a0fa6ad27bcf3a46": "G_{in} = \\frac{G_\\infty \\tanh(L) + G_L}{1+(G_L / G_\\infty )\\tanh(L)}", "4324dcf1fb8bf66d753be2d9a262a0a8": "0\\leq\\beta\\leq 1,\\;\\alpha\\geq 0", "4324e467a87551e80004ccf71634ef2c": "\\scriptstyle(I_1 \\,\\cap\\, \\cdots \\,\\cap\\, I_k) \\neq (I_1 I_2\\cdots I_k) ", "43253b82028ddd05679877443b7d512a": "k = 23\\,", "4325a5554a679c841bcd5b4f86fc9d22": " COP_{heating}=\\frac{T_{hot}}{T_{hot}-T_{cold}} ", "4325dafdb36b226de6aef09183e6dabb": " V_G = V_{ch} + E \\ t_{ins} = V_{ch} + \\frac {Q t_{ins}}{\\kappa \\epsilon_0}, ", "432653ebad07b4ff3110caf02f5c0041": "(\\mathbb{Z}/3\\mathbb{Z})^*", "43266965f7f3d404b033b1393fd2620c": "\\displaystyle{\\pi_\\pm(g)F_\\pm(z)= (-\\overline{b} z + \\overline{d})^{-1\\pm 1/2} F_\\pm\\left({\\overline{a} z -\\overline{c}\\over -\\overline{b} z + \\overline{d}}\\right).}", "43268bc286c600b6c315c71cd0499fd1": "elongatedness = \\frac{length}{width} = \\frac{area}{(2 d)^2}", "43272a6ad66216213fdbf00dc802a751": "x_i = -\\lambda_{i, n+1}x_{n+1} - \\cdots - \\lambda_{i, m}x_m, \\mbox{ for } 1\\leq i \\leq n", "432774234b892323d3242c5988e02120": "\\Delta I_{\\text{L}_{\\text{Off}}}", "432774675edfaae198a39df8ef8b881e": "{\\hbar}\\,\\coth(\\hbar\\omega/2k_\\mathrm{B}T)", "432790857172cff907f3ed16fff21688": "\\hat{H} |\\psi_1\\rangle = E |\\psi_1\\rangle", "432793f77ad85e52e700605f2d4ffc84": "\\pi_{s}", "4327a4c370480739558cf85ba582a9f9": " \\text{If } |\\Delta P| = \\mathit{ \\iota}: \\quad \\frac{\\Delta F(P)}{\\Delta P}=\\frac{dF(P)}{dP}=F'(P)=G(P);\\,\\!", "4327de41ad41c4f73fdc498a7d2aee17": " m \\ge n ", "4327e50f818e01ccef4de679190e63cd": "\\scriptstyle \\sqrt[4]{g\\sigma/\\rho}", "4327eb1f8a1b5e6fe1a70f9b7980044d": "\\Omega_{\\mu \\nu} ^{ab} = R_{\\mu \\nu} ^{ab}", "43285a3e865b3c268046fdb156cd73b8": "U_\\textrm{eff}(\\tilde{a})=\\frac{\\Omega\\tilde{a}^2}{2}\\;", "4328b26d6aae3fbf4ff9a7ab18f5912a": "{1 \\over T}=a+b\\,\\ln(R)+c\\,(\\ln(R))^3", "4328d3071ec84c33888bb2504dbd041e": "\n\\begin{alignat}{3}\n& S^{\\mathrm{core}}\\gamma - D^{\\mathrm{face}} \\cfrac{\\mathrm{d}^3 w}{\\mathrm{d} x^3} = Q &\\quad\\quad& (1)\\\\\n& D^{\\mathrm{beam}}\\left(\\cfrac{\\mathrm{d} \\gamma}{\\mathrm{d} x}+\\vartheta\\right) - \\left(D^{\\mathrm{beam}}+D^{\\mathrm{face}}\\right)\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} = M &\\quad\\quad& (2)\\,\n\\end{alignat}\n", "4328ec599dd67a5f18c292a71708c425": "Ind_{Cent(g)}^G(\\chi)=kG\\otimes_{kCent(g)}X", "43293202109299984abef585911fb80b": "L = \\mathbf Q(\\zeta_p),", "4329418f79551518be2cdc67369b042b": "R= \\{ 1,0 \\} ^ {|r|} ", "43296ec5bbee6d6e0f1c213c5662e8df": "H^{2N-i}(X,\\mathbf{Z}/n\\mathbf{Z})", "432979cfc179778d23cce15af9c90afc": " \\tilde \\nu_{J^{\\prime}\\leftrightarrow J^{\\prime\\prime},K} = F\\left( J^{\\prime},K \\right) - F\\left( J^{\\prime\\prime},K \\right) \n= 2 \\left( \\tilde B - 2D_{JK}K^2 \\right) \n\\left( J^{\\prime\\prime} + 1 \\right)\n-4D_J\\left(J^{\\prime\\prime}+1\\right)^3 \\qquad J^{\\prime\\prime} = 0,1,2,...", "432985da3bba3d488dbff3c8bf5ef3f0": "(b/a)", "432a0b0ed48a04ab26bece60767439ce": "c_f : V\\times V \\to R^+", "432aa147207af3ef122a9fd3e59564a0": " \\arcsec z = \\arccos {(1/z)} \n= \\frac {\\pi} {2} - \\left( z^{-1} + \\left( \\frac {1} {2} \\right) \\frac {z^{-3}} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {z^{-5}} {5} + \\cdots\\ \\right) \n= \\frac {\\pi} {2} - \\sum_{n=0}^\\infty \\frac {\\binom{2n} n z^{-(2n+1)}} {4^n (2n+1)}; \\qquad | z | \\ge 1 ", "432aa323ce11730668a268dfcca6ca01": "\\Pi_{ij}=-C_{1}\\frac{\\epsilon}{k}\\left (R_{ij}-\\frac{2}{3}k\\delta_{ij}\\right )-C_{2}\\left (P_{ij}-\\frac{2}{3}P\\delta_{ij}\\right )", "432ad4ce9dc0d1563cf93ab7e55cde2d": " H = \\langle h_1, h_2, \\ldots, h_s \\rangle ", "432afc896361fe64b90997ef6c2ba04f": "\n \\sigma_{rr} = \\frac{E}{1-\\nu^2}\\left[\\varepsilon_{rr} + \\nu\\varepsilon_{\\theta\\theta}\\right] ~;~~\n \\sigma_{\\theta\\theta} = \\frac{E}{1-\\nu^2}\\left[\\varepsilon_{\\theta\\theta} + \\nu\\varepsilon_{rr}\\right] ~;~~\n \\sigma_{r\\theta} = 0 \\,.\n", "432b014ce8e9acc5ee275e02dfddfb59": "\\begin{align} \nA^k &= (PDP^{-1})^k = (PDP^{-1}) \\cdot (PDP^{-1}) \\cdots (PDP^{-1}) \\\\\n&= PD(P^{-1}P) D (P^{-1}P) \\cdots (P^{-1}P) D P^{-1} \\\\\n&= PD^kP^{-1} \\end{align} ", "432b03c528621bac10ee06de1f9e9764": "f(q) = q^2", "432b12ccd669a1939f37582a774f04d9": "e_q(z)=\n\\sum_{n=0}^\\infty \\frac{z^n}{[n]_q!} = \n\\sum_{n=0}^\\infty \\frac{z^n (1-q)^n}{(q;q)_n} = \n\\sum_{n=0}^\\infty z^n\\frac{(1-q)^n}{(1-q^n)(1-q^{n-1}) \\cdots (1-q)}", "432b33ff0711b4140c18e92f509685be": "K=\\frac{k}{2\\pi}", "432b4cfb979f14a140b7205d6bc5dc56": "\\lambda \\geq 0", "432b565676bfd7de35cc25329641df45": "\\sigma_e=\\omega\\epsilon''=2\\pi f\\epsilon_0\\epsilon_r''\\,", "432b91fc8148bb15e7ac77077d4a2745": "{2N \\choose k} p^k q^{2N-k} ", "432ba40f3e3cf2a2c8498de92135157e": "t = Ls", "432c09ae1bc761ab37390971b5549c6a": " \\lim_{t \\to 0}\\frac{o(t)}{t} = 0, \\, ", "432c1df69e11aba7c5c5070e7578609f": "\\cup ", "432c2973439e3fb06ce342a082e92c68": "S = \\{\\} \\!", "432c52ce01cc0ff8c7f20a1d14e45715": "\\int_0^\\infty x^{2n+1}e^{-\\frac{x^2}{a^2}}\\,dx = \\frac{n!}{2} a^{2n+2}", "432cc4cc166c188e5d8039a9f41777da": "\n\\rho=m_1\\,\n", "432d8b36e11970cccf8353c224cc3242": " \\mathbf{PSF_{lens}(x,y) *\n PSF_{sensor}(x,y) *} ", "432da3b7f0c6456a474ad991f5add370": "\\sum_{t=0}^n g(t){n \\choose t}r^t = 0 .", "432dbf32f7edf10320195f495da6034e": "\\tfrac{x^3-y^3}{x-y},", "432dc1fedec8720816f16fb3ffbcb65d": "c_F(a,b)=\\begin{cases}\n \\frac{1}{2|b|}, & \\mbox{if } |a|<|b| \\\\\n 0, & \\mbox{if } |a|>|b|\n\\end{cases}", "432e51de1a60ab56070211aad8c2d8c1": "a^3/T^2 = M", "432e78eb00bdd82a71b20a347f4d1145": "\\ M_\\mathrm{filtered}(t) = M(t) * h(t)", "432e96ccefa82ffdd1d91631f9486dfd": " T_C ", "432eac9b03ec56705b3f472fe90ba15f": "\\alpha \\in (0,2)", "432ecb824303bd35b04faf652fbe1ac0": "\\mathrm{d} X_t = \\mu(X_t)\\,\\mathrm{d} t + \\sigma(X_t)\\,\\mathrm{d} W_t", "432f3c863de0864cea68263f2b2a23b5": "\\frac {1}{(2\\pi i )^2} \\int_L^* \\frac{d{\\tau}_1}{{\\tau}_1 - t}\\ \\int_L^*\\ g(\\tau)\\frac{d \\tau}{\\tau-\\tau_1} = \\frac{1}{4} g(t) \\ , ", "432f44ac9e1d2cf71d195285f6def769": " \\theta: C_K/{N_{L/K}(C_L)} \\to \\text{Gal}(L/K)^{\\text{ab}}. ", "432fd833b3854cbe78020a43a28950ef": "n=6\\,", "433006d89673dd1a21c4b40eba5275a1": "c_{i_1i_2..i_N}", "43300a94b82f6fb5ec15c0e5c22c1dfc": "u_{\\star}=\\sqrt{\\frac{\\tau_b}{\\rho}}", "43305721d46514c1ca5d24923ecc3f74": "T(s, x)", "43306d7579ad69625c3f4be1b6d205a7": "\na_{0}=\\left( \\frac{\\alpha D_{\\alpha }\\hbar ^{\\alpha }}{Ze^{2}}\\right) ^{1/(\\alpha -1)}. \n", "4330acd8840edc74e925fb1d73b230fa": "\\operatorname{core}(K)", "4330f7da39c4783285c2676fc51210ae": "\\Omega\\left(E\\right)=\\sum_{Y}\\Omega_{Y}\\left(E\\right)\\,", "43310662a6470c0e0d463b2b3545dddc": "H(p, q) = \\frac{1}{2m}p^i p^j g_{ij}(q) + V(q)", "43314a62d4213a98a6d1d55643f8ac1a": " {\\rm Tr}\\Bigl(f\\Bigl(\\sum_{k=1}^nA_k^*X_kA_k\\Bigr)\\Bigr)\\leq {\\rm Tr}\\Bigl(\\sum_{k=1}^n A_k^*f(X_k)A_k\\Bigr),", "4331bfd09bb5e3d404ee4cf642593f0a": "\\! \\chi", "433201beb820fbc1d6fe84c444ecfeb4": "C_v = F \\sqrt{\\dfrac{SG}{\\Delta P}}", "4332152409e2c3b52a5e8d897afa74a8": "U = -\\mathbf{M}\\cdot\\mathbf{B} = -(M_x B_x + M_y B_y + M_z B_z).", "43323f8f3c3361c9d458b40c7fcb8716": "\\theta_k^1", "4332d0b368f83fa9a2338c34ebec5ad5": "\\mathbf{e}_i,~~ i=1,2,3", "4332d32574a7ebee502e3bd8b13c302d": " F_m = {Q v I \\over 2 \\pi \\epsilon _0 c^2 R} ", "4332ef3a7ad0c013510db7ff70bc3f5d": "f''(z) = -\\frac{1}{z^2}.\\,", "4332f9dfaded36017584c59ba77ef467": "\\;p(t) = P r^t - A \\sum_{k=0}^{t-1} r^k", "43330b575a0033b67110f476c234202a": "x'_{r,s}", "433343b476537f8772a57b93ff4c0db9": "R \\sim \\text{Rice}\\left(0,\\sigma\\right)", "43335eb5208e530058180d7ed75e3b25": "D(A,I) = \\{ (x,y) \\in A \\times A : x-y \\in I \\} \\ . ", "4333799d3e2c634a0fa707efa81f828e": "(E_h, x_h)", "4333b9f5c6117159857dba38af33c01a": " \\text{LAM} = \\frac{\\sum_{v=v_\\min}^{N}vP(v)}{\\sum_{v=1}^{N}vP(v)},", "4333cc2cf985092907255eb6e04b45b6": "AX + XA^H + Q = 0", "433406f0449e60fd72bf8d64f69b39e6": "Pr(x|S) = K", "43342fb3a672db36c64c5ed7f4f9245d": "|U(S)| \\geq d(1-2\\varepsilon)|S|\\,", "4334672bc743775678e922172fbab883": "\\mu_s - a\\mathbf{R}\\mu_m", "4334ad380044a075ef1f42eceff6f427": " \\gamma = 0 ", "4334ad85d878583d0bc1bc42b4e805aa": "\\omega^i{}_{k\\ell}=\\frac{1}{2}g^{im} \\left(\ng_{mk,\\ell} + g_{m\\ell,k} - g_{k\\ell,m} +\nc_{mk\\ell}+c_{m\\ell k} - c_{k\\ell m} \n\\right)\\,", "43354c3a6612976d1a412e88194a4d43": "h^{ab}\\equiv g^{ab} + u^{a}u^{b}", "43357167ef025e036ed73d348f08b2f4": "Z(J_0(S))", "4335808ce5c1844557dba0fc3d3afb7d": "\\varphi_h(n) = \\beta^{-1}(\\gamma(h)\\beta(n)\\gamma(h^{-1})).", "4335fbc48ad94e38a3ffb2ef0bbaa081": "X = \\{ \\circ, \\bullet \\}", "43360f1996c5deba15c366efea5033ae": "\\cos(\\omega t).", "4336a72200806e350631e82d1d69a1a0": "\\mathfrak c=\\aleph_{\\aleph_1}", "43372adeed844bffe779c90b1532e43a": "G_{TX}", "4337e766c705701100975f146cda71ca": "I = \\frac{\\pi r^4}{8}", "433844f911136425a263c4e96736ba85": " x^5+\\frac{10}{13}x+\\frac{3}{13}\t", "433892e49d7b2e0c764d92fc7f13e303": " \\widehat{\\boldsymbol{\\beta}}_{ols} ", "43389715d9c6eead85dc7939197a0aeb": "v = a \\cosh \\tfrac{x_2}{a} - a \\cosh \\tfrac{x_1}{a}.\\,", "4339020b72e4ee1663ed43245170d573": "\\omega(\\xi\\cdot v,w)+\\omega(v,\\xi\\cdot w)=0", "4339076864ed03c556af0cf65cc642ad": "\\,(4 + 9 + 1 + 3)^3 = 4{,}913", "4339b14e4219ebd5ae1f04b372acb619": "\\gamma_s(h)=\\gamma_s(-h)", "4339c6e881ef2e0ff298ee71825639be": "X_{n-1}\\cap X_n.", "4339d4e58efa779e533ffd1d0508203d": "\\widehat{\\sigma}_\\varepsilon", "433a72446f2504ed075525160b1a2398": " g_{\\rm effective} \\approx g_{\\rm gravitational} + g_{\\rm centrifugal}\n= g_{\\rm gravitational} - |g_{\\rm centrifugal}|\\ .", "433a8b15672fb17c3b833a56d250a71c": "\\frac{\\delta}{\\delta g(x)}\\left(\\frac{\\delta S(g)}{\\delta g(y)} S^\\dagger(g)\\right)=0 \\mbox{ for } x\\le y. ", "433a8b7584fbe72e20c65da145a76956": "B = B_oM ^ {3/4}\\,", "433abdf2916010e01f931c294d0959cc": "e = y - X \\beta = y - X(X^TX)^{-1}X^Ty = [I - X(X^TX)^{-1}X^T]y = My. \\, ", "433abfeeff7039817c1b8c30410294be": "\n\\mathbf{\\hat{v}}=(\\sin \\theta, ~0, ~\\cos \\theta).\n", "433b0e312fc1e5e8e00f8582ce81409c": " S_1 = {{5\\over4} \\div {6\\over5}} = {25 \\over 24} \\approx 70.672 \\ \\hbox{cents} ", "433b311f824db983698eaa0e6f2ecb59": "P(T \\le t_0 + t | T > t_0) = \\frac{P(t_0 < T \\le t_0 + t)}{P(T > t_0)} = \\frac{F(t_0 + t) - F(t_0)}{S(t_0)}.", "433b317833735503df44a8bc2f2d5ef6": "\n\\frac{{\\partial \\ln N!}}\n{{\\partial N}} \\approx \\ln N\n", "433b584d70f9e35cac6cf26eaa0abe3e": "\\phi (0)\\,= \\phi_0", "433baca653ce567711347934e67cbf57": "\\pi_1(s\\!:\\!\\sigma,t\\!:\\!\\tau) = s\\!:\\!\\sigma", "433bc0a25fab4e2c0dff98defeeb144a": "\\tau \\to 0", "433be533f4cf7465eb9c21c1b3d1f0ef": "\\Phi(||\\mathbb{P}_n - P||_{\\mathcal{F}}) \\leq \\mathbb{E}_{Y} \\Phi \\left(\\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n [f(X_i) - f(Y_i)] \\right|\\right|_{\\mathcal{F}} \\right)", "433c13ef573990edfe7a38395dd68d0b": "x\\le y\\ ", "433c2293b0b36a58b7b59e93b6d378ad": "\\|\\tau_h f - f\\|_p=o(1).", "433c76f0c978f72bb32632a8dba6fe5a": "\\,c(u,v)", "433c9ae5a913e92f8e208febce210509": "v=\\frac{0.6581y}{y-0.15735x+0.2424}", "433cbc21e011d6eecbe991fc3cb5085d": "\\mathbf{U}_0 = U_0\\left(\\cos\\theta\\, \\hat{\\mathbf{r}} - \\sin\\theta \\,\\hat{\\mathbf{\\theta}}\\right) = U_0 \\left(\\mathbf{Y}_{10} + \\mathbf{\\Psi}_{10}\\right) ", "433cf38238f71dcc3682a9b6e6c67872": "\\nabla_{\\bold{u}}{f}(\\bold{x}) = \\nabla f(\\bold{x}) \\cdot \\bold{u}", "433d5dd21c317e94ab1bf41944cfc61e": "\\rho = \\left({a\\over 2}\\right)\\frac{\\cos(\\pi/p)}{\\sin(\\pi/h)}", "433d8d950ac1c8498faacb081e02f195": " \\mathrm{Input} - \\mathrm{Output} = \\mathrm{0} \\, ", "433dbbe420fcbf961ed3f1bd93fbd886": "\\mathbf r(0) + s\\mathbf T(0) + \\frac{s^2\\kappa(0)}{2}\\mathbf N(0)+ o(s^2).", "433df45af0fc52bce7b68093d483c441": "\\text{Viscosity}\\ (\\eta_o)\\ (kPa\\cdot s)", "433e2ad258b8ad2dfb51219c42a48d0c": "9, 8, 7, 6", "433e34b2cc18fb9e13084ac611c39660": "G\\left(\\epsilon\\right)=G_0\\,\\, e^{-\\left(\\frac{4\\pi\\epsilon}{\\lambda}\\right)^2}", "433e4d7720b5407bf7ae8ffdd7386512": "\\, ta(s)=\\sigma ", "433e53f50d647e039aac4cb769c6ab10": " F", "433e57b789c792b162942fd1905c1a91": "\\frac{1}{h(y)} \\frac{dy}{dx} = g(x),", "433e8454b78b277d5959aca595ec33a8": "\\lim_{x \\to+\\infty} {1 \\over x^b} \\log_a x = 0 \\quad \\mbox{if } b > 0", "433eb9e03028fbbab166862003809266": "\n\\mu_j p_i (j) = v_i q_ j (i),\\quad i,j=0,\\ldots,n. \\quad(10)\n", "433ebaa24beb3f98118dbf604f6de9ec": "i^{\\ast}_{\\mathrm F_{SO}(M)}(V\\mathrm FM) \\to \\mathrm F_{SO}(M)", "433f267f0853820af0228dcd90d4b88c": " S = - k_B \\sum p_i \\ln p_i ", "433f3d881c3d4c84a01803c58724b6df": "\\Gamma(t)=\\frac{\\Gamma(t+n)}{t(t+1)\\cdots(t+n-1)},", "433f8a09d990b1130c95f8f50474c2e5": "\\beta\\leq\\alpha", "433faa27cbc2842084b8ae3f45adcc14": "\\limsup_{n\\to\\infty} E_n = \\bigcap_{n=1}^{\\infty} \\bigcup_{k=n}^{\\infty} E_k.", "433fb0ea15912e851def008fc11108b6": " \\prod_{p} (1-p^{-s}) = \\sum_{n=1}^{\\infty} \\frac{\\mu (n)}{n^{s}} = \\frac{1}{\\zeta(s)} ", "434083a41f7b755fb323bba1fd16c36c": "\n \\sigma^{*} = \\sigma^{*}_{\\rm initial} - D~\\left(\\sigma^{*}_{\\rm initial} - \\sigma^{*}_{\\rm fracture}\\right)\n ", "4341410cb3c8755c7a5536a2f2235c95": "{P}^{2}-2Q\\, ", "43418a37d7c7d924e64c451faeaa1bfa": "d_1 = \\frac{1}{\\sigma\\sqrt{T - t}}\\left[\\ln\\left(\\frac{F}{K}\\right) + \\frac{1}{2}\\sigma^2(T - t)\\right]", "4341e7c8b39a400418f72b5eb3876f09": "g_n = p_{n + 1} - p_n.\\ ", "4342117d214f26621b3d71c98babfac6": " Z(\\lambda_1,\\dotsc, \\lambda_m) = \\int m(x)\\exp\\left[\\lambda_1 f_1(x) + \\dotsb + \\lambda_m f_m(x)\\right]dx.", "4342227d904eaf84678d66ae197122d9": "\\rho(\\vec x)=x_1x_2+x_3x_4", "4342257f830aea2acfebc64ec6bfd3f6": "\n\\eta= \\left[ 2\\, a\\, \\cos \\left( \\frac{k_1 - k_2}{2} x - \\frac{\\omega_1 - \\omega_2}{2} t \\right) \\right]\\;\n \\cdot\\;\n \\sin \\left( \\frac{k_1 + k_2}{2} x - \\frac{\\omega_1 + \\omega_2}{2} t \\right).\n", "4342281a2550d18657a098ad5676d551": "s_1=0", "4342e3c41dde2619d2e1984279a5492e": "\\beta_1=\\frac{1}{2}n-\\frac{2}{3}n^2+\\frac{37}{96}n^3,\\,\\,\\,\\beta_2=\\frac{1}{48}n^2+\\frac{1}{15}n^3,\\,\\,\\,\\beta_3=\\frac{17}{480}n^3,", "4342fdf886b6067777f46bb7ffbd8363": "g(x) = x^{10} + x^8 + x^5 + x^4 + x^2 + x + 1", "434307f5e92eb8722e89a4744de2ee31": "Y'-A\\ Y=F(t)", "4343383c4a9fba0c9fa76142b2c2e5f0": "X'_w", "4343591f3708a94b9ad0324e8eceae5f": "\n\\begin{align}\nu &= u^{0} + u^{1} + u^{2} + \\cdots + u^{N} \\\\\n &=\\alpha + \\beta x + \\gamma x^{2}/2 - \\frac{1}{2} L^{-1} (u^{0}+u^{1}+u^{2}+\\cdots+u^{N}) \\frac{\\mathrm{d}^{2}}{\\mathrm{d}x^{2}}(u^{0} + u^{1} + u^{2} + \\cdots + u^{N})\n\\end{align}\n", "43438195714e6718508b140fe638cf34": "(x-1)(x-1)(x+1)(x+i)(x-i) = 0", "4343e0b96af1e1e9f4dfe886a574e0d8": "\\Psi \\Psi ^* \\Delta V", "43441fd1142acb6b1e051e9b526e4f14": "m\\ge\\tbinom{k}{2}", "43442166aae4ed4cac9b5a79d37ddb6f": "\\langle x,x\\rangle", "4344409e8a3ec8b22ba51358fdbaf819": "p_{1},p_{2},\\ldots.", "43446ea9226d534dd75e880e980ef392": "V_0 ", "43448602d43c75cf6c61733e081ce33d": "\\begin{align} 2\\cdot R_*\n & = \\frac{(240\\cdot 2.44\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 126\\cdot R_{\\bigodot}\n\\end{align}", "43449f72a0242afaaa0e47627c5a32f1": " a \\kappa \\gg 1", "4344f5e892cda0f2d4643e594ad4e57d": "\\textstyle \\varepsilon", "43456920baba049127fd7004a859112c": " \\pi(G)^\\prime = P \\rho(G)^{\\prime\\prime}P. ", "43459dc48911e746db1eef663b122b66": " \\int e^{h S} d\\mu_j(S) \\neq 0 , \\, \\forall h \\in \\mathbb{H}_{+}:=\\{ z \\in \\mathbb{C} | \\Re{z}>0 \\} ", "4345ba30686773656d69686a008fa69a": "\\,x = r \\cdot \\sin(t)", "43469f22bc99f0dd6896864b1eb732ee": "\nB_0 = \\mu_0 M\n", "4346cde261008a8023651713ad1becc6": "u_1=0", "4346ed5c863f7446ef2f1fcc6e202cd5": "\\frac{\\partial S}{\\partial t} = U(S)\\frac{\\partial S}{\\partial x}", "4347269783c220187cbaa91f811b09db": " |e| = \\sqrt{-g}", "434796c9a88dc16c1c3c76ff1d1f2b56": "\\mathbf{B}' = \\frac{\\mathbf{E} \\times \\mathbf{v}}{c^2} \\,,", "4347ac66502b0e6b41ee5986d91df978": "\\varepsilon = \\frac{c}{a}", "4347f4020d29d5d80c94d2e086eb77d9": "\\mathbf{V}_i = \\vec{\\omega}\\times(\\mathbf{X}_i-\\mathbf{d}) + \\mathbf{v},", "43483929d61ed55ba34e80ade43be5db": "\\left ( \\frac{ax}{b} \\right )^3 + \\left ( \\frac {ax}{b} \\right )^2 = \\frac {ca^2}{b^3}.", "43488198de39ad9088df135b83e8407e": " ds^2 = g^R_{ij} dx^i dx^j, \\, ", "4348882c6303b6ee7ec41aeb6807c054": "c_{3,1}(\\widehat{a}, w(\\widehat{\\epsilon}, \\widehat{b}c), \\widehat{d})", "4348a0077c250409b6135cc3d0d4298c": "\\sum_{k=-\\infty}^{\\infty} \\hat s(\\nu + k/T) = \\sum_{n=-\\infty}^{\\infty} T\\cdot s(nT)\\ e^{-i 2\\pi n T \\nu} \\equiv \\mathcal{F}\\left \\{ \\sum_{n=-\\infty}^{\\infty} T\\cdot s(nT)\\ \\delta(t-nT)\\right \\},", "4348c530a7f465a7ce4762a5fb355cbb": "U =", "4348f63aaa1c8a05e30a47a0ec3e2a49": "\\textstyle p=\\infty", "4348fc0e294a481010c6135652428c25": "\\textstyle\\mathbf{IPC}+\\bigvee_{i=0}^n\\bigl(\\bigwedge_{j d ", "43587328d31942b0372a271576c788b2": " a\\in U", "4358745e8c555f3b0273dcdb98ea1f4f": "H^2_+(R)", "43591fae33070fc5a867d9142a3bcc83": "\nL=\\frac{1}{2}\nmr^2\\left(\n \\dot{\\theta}^2+\\sin^2\\theta\\ \\dot{\\phi}^2\n\\right)\n+ mgr\\cos\\theta.\n", "43592c58b70500ce471f32239ce30380": " F_2 \\ ", "43593872a2fb3637877e9fc9dbd97dbc": "\\mathfrak{p}\\cap S=\\empty", "435945d223a60758021ac59891c5c4ce": "\n \\begin{align}\n y &= (x+1)^2 \\\\\n &= x^2+2x+1\n \\end{align}\n", "435962c7554b56de955be618f218dfc4": "{1 \\over 2}+{1 \\over 3}+{1 \\over 5}+{1 \\over 7}+{1 \\over 11}+{1 \\over 13}+\\cdots \\rightarrow \\infty.", "4359b132f74e7e7f69dc25205236f374": "\\forall i\\in \\mathbb{N}, p_i\\notin P \\;", "4359d9690b364936dfa099a2e5897683": "[-2^{n+1}+1,-2^n]", "4359e8c76a62bca6e1297fb880029705": "P=X^k-1", "435a10ee11de20758dcaaa6cd8049051": "j\\not\\in A_i", "435a2b21f9cbacac10b1de3d6f7a5b3d": "\\tbinom{10}{5}", "435a6f5768936a3b6247294bc63d29eb": " V_{1} = 5 ", "435aa34c59ea6d0e85a1b156edec6c25": "\n\\frac{d \\vec v}{d t} = \\frac{\\partial \\vec v}{\\partial t} + (\\vec v \\cdot \\vec \\nabla) \\vec v = - \\frac{1}{\\rho} \\vec \\nabla p + \\vec B + \\frac{\\vec \\nabla \\cdot \\tau}{\\rho} \n", "435b2707a4f3bd11723590cf6e6be5f8": "X\\times Y = \\{\\,(x,y)\\mid x\\in X \\ \\and \\ y\\in Y\\,\\}.", "435b28048282e4b5aaf8a54e2a9db76c": "\n\\begin{align}\nI(X;Y) & {} = \\sum_y p(y) \\sum_x p(x|y) \\log_2 \\frac{p(x|y)}{p(x)} \\\\\n& {} = \\sum_y p(y) \\; D_{\\mathrm{KL}}(p(x|y)\\|p(x)) \\\\\n& {} = \\mathbb{E}_Y\\{D_{\\mathrm{KL}}(p(x|y)\\|p(x))\\}.\n\\end{align}\n", "435b3db3e1df105e38de5ffbf8708b21": "\\mathbf{u}(t) \\in \\mathbb{R}^p", "435b6395887baec548ecc17b4d10e061": "\\{a_1,a_2 , \\ldots,a_n\\}", "435b83049171958c0671e7691155ff1f": "\\alpha \\in \\mathbb R", "435c04f0e4c697ac11125c876c7a399c": "C=\\left(\\begin{matrix} \\frac{\\sigma}{\\epsilon} & 0 & 0 \\\\ 0 & \\frac{\\sigma}{\\epsilon} & 0 \\\\ 0 & 0 & 0 \\end{matrix}\\right).", "435c09fb8f3e8e0990fb61f1697ced87": "0 = t_0 < t_1 < \\ldots < t_M = T ", "435c2689c20de871f9a2f77149f1f801": "m^{e^d} \\equiv m^{e d} \\equiv m^{(e d - 1)}m \\equiv m^{k(p-1)(q-1)}m \\equiv 1^{k(q-1)}m\\equiv m \\pmod{p}", "435c656a37e24ee97c5ca25ad8f9f242": "\\int \\arcsin(x)\\, \\mathrm{d}x = x \\arcsin x + \\sqrt{1-x^2}+C ", "435c86b1fc9939d9df40fed8efbbabfa": " \nA= \n\\begin{bmatrix}\n0 & a_{1 2}& a_{1 3} & \\cdots & a_{1 n} \\\\\n0 & 0 & a_{2 3} & \\cdots & a_{2 n} \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n0 & 0 & & \\ddots & a_{n-1 n}\\\\\n0 & 0 & \\cdots & \\cdots & 0 \n\\end{bmatrix}. \n", "435cac50066aa2453236f1eea410c264": "\\eta'(s) = \\sum_{n=1}^\\infty \\frac{(-1)^n\\ln n}{n^s} = 2^{1-s}\\ln 2 \\zeta(s)+(1-2^{1-s})\\zeta'(s)", "435cba37337ba4671866cd99a90ab681": "\\operatorname{mean}(v) = [1/n, 1/n,\\dots,1/n]\\cdot v.", "435cf334b547287410a35ec275ce9f08": "Y_{2}^{-2}(\\theta,\\varphi)={1\\over 4}\\sqrt{15\\over 2\\pi} \\, \\sin^{2}\\theta \\, e^{-2i\\varphi}", "435d4a093724f621127518c8f4f97f61": "c (x, y, z) = \\frac1{R} = \\frac{4 A}{| x - y | | y - z | | z - x |},", "435d6d10a8cdf0c1d03daeac9b8b1cd3": "\\frac{n}{2\\omega}, n = ...-1,0,1,...", "435d88ba2186a88093c85c59d2039ce9": "I_1 \\otimes \\cdots \\otimes I_r \\cong J_1 \\otimes \\cdots \\otimes J_s.\\,", "435dd5465e092d6886d6580f153ce851": "\n[x] \\subset [y] \\Rightarrow C([x])\\subset C([y])\n", "435df06e76ee7df09f0b7996fd1b5968": "\\forall (f_1,f_2) \\in \\mathbb{R}(Z)^2, \\delta f_1 f_2 = f_1 \\delta f_2 + f_2 \\delta f_1", "435df676bac289f78f72a1c7afd4fd37": "V(G)", "435e185262da0230dbf9581e1a7e7fc5": "\n\\begin{align}\nx 0 \\!", "4387176b2d10a70cea731537599d27f5": "\\cos\\varphi = \\frac{1 - t^2}{1 + t^2},", "43879b4b90d86bdaf11f5dfa18570617": "\\Theta^{s}(\\mu,a)", "4387a845364488ce7a2ee06758f6d1f6": "(4, 2), (4, 3)", "4387ee46dd8e59b8b0aa0aa3c041d3f0": "b \\in \\mathbb{R}", "438811f95fe74c5bb5fa3b6d05f7349c": "\\nabla\\cdot \\mathbf{A} = \\frac{1}{r^2}{\\partial \\over \\partial r}\\left( r^2 A_r \\right) + \\frac{1}{r \\sin\\theta}{\\partial \\over \\partial\\theta} \\left( \\sin\\theta A_\\theta \\right) + \\frac{1}{r \\sin \\theta} {\\partial A_\\varphi \\over \\partial \\varphi},", "43889ba8299c956dd7c890b6fca00166": "\\begin{align}\nU_J& = \\delta_{Ji}x_i-X_J =x_J - X_J\\\\\n\\frac{\\partial U_J}{\\partial x_k} &= \\delta_{Jk}-\\frac{\\partial X_J}{\\partial x_k}\\\\\n\\end{align}\n", "4388faf5475f9430540f8b3109cc8cb5": "n! \\simeq n^n e^{-n}\\sqrt{2 \\pi n}\\qquad \\text{as } n \\to \\infty.", "438915a56743844b25741dd76447e08d": " 2 {^{15}}\\text{NO}_3^- \\rightarrow {^{15}}\\text{N}_2\\text{O} , ", "43892b44cd868f1e8b98ccfef8d995a2": "= {1 \\over 2} \\epsilon_{MN}^{\\;\\;\\;\\;\\;\\;\\; IJ} (F^{MK} {1 \\over 2} \\epsilon_{OPK}^{\\;\\;\\;\\;\\;\\;\\;\\;\\; N} G^{OP} - {1 \\over 2} \\epsilon_{OP}^{\\;\\;\\;\\;\\;\\; MK} G^{OP} F_K^{\\;\\;\\; N})", "4389439239b95ce093919ed86357439b": "\\{n_k\\}_{k\\in \\mathbb{N}}", "43894e7983db60e16afa765595fd01c5": "\\log (\\hat p_{x-1} / \\hat p_x)", "4389d81e99f3f2b21a8137c06b278acd": "B_3(2, M, r, \\mu) = \\frac{2M + MF(Mr) - \\sum_{n=1}^{M-1} (M-n)\\left [ 2F(nr) - F((M+n)r) + F((M-n)r)\\right ]}{M^{\\mu+2} \\left [ F(r) + 2\\right ]}", "438a26e62fdf3eae7ddecbce5c437a15": "\\mathcal{P} = \\{P_1, P_2, P_3, \\dots , P_n\\}", "438a44cd32b393b58ca8e1d01011bc2d": "f \\in L(G-P_{i_1} - \\dots\n- P_{i_{n-d}})", "438ae99d9135beb510dcf8477e6132f7": " Z_N(K,L) = 2^N(\\cosh K \\cosh L)^N \\sum_{P \\subset \\Lambda} v^r w^s ", "438b7bee8975c4b7356205b092448cd6": "(D_n*f)(x)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(y)D_n(x-y)\\,dy=\\sum_{k=-n}^n \\hat{f}(k)e^{ikx},", "438b9f6c2544fc1671000c4837eda9ae": "c_1 = 0.48 + 1.574 \\; \\omega - 0.176 \\; \\omega^2", "438c5d5501bc528c0dd1b47045512b46": "C_{|x|}(x)=0", "438c73eccabfa7928b5acc2cf6907a78": " (U, \\varphi)", "438c91f13d5e6edf51e7781e94ef1670": "Y[x,y]= y+x'\\frac{x'^2+y'^2}{x'y''-x''y'}", "438cd5bfd1144efe6789954ef087be38": "(S,<_S)", "438cfa326742d86dc226ac284908cae7": "(\\pm 1,\\pm 1,0,0)", "438d0332c16377848245dad502d0f24a": " i\\hbar\\frac{d}{dt}A_I(t)=\\left[A_I(t),H_0\\right].\\;", "438d2472cb67aac1af94dd61c4713623": "V_v = - {\\hbar \\alpha \\omega\\over 16\\pi\\varepsilon_0 Z^3}= -\\frac{C_v}{Z^3},", "438d5977f9ca74c642f04b3ffddf4eae": "dX^2", "438d6b71dc866094a2d040947a4f8933": "\\mathbf{\\left(J^TWJ\\right)\\boldsymbol \\Delta \\beta=J^TW \\boldsymbol\\Delta y}.\\,", "438d6f1e949309aa6c349c82ceb581e9": "\\pi/4=\\frac{3}{4} \\cdot \\frac{5}{4} \\cdot \\frac{7}{8} \\cdot \\frac{11}{12} \\cdot \\frac{13}{12} \\cdot\\frac{17}{16}\\cdots", "438dd69c58de3015bd06649b9ea9410b": "Y_{5}^{-3}(\\theta,\\varphi)={1\\over 32}\\sqrt{385\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(9\\cos^{2}\\theta-1)", "438e021a80a1a2f8b0850a91914c1dbe": "\\gamma=j/j_c", "438e1965ea5f94a5dc67016729667cf8": "f(m, x_1, \\ldots, x_n) = \\prod\\limits_{i=0}^mg(i, x_1, \\ldots, x_n)", "438e7fc1842e9f82ff595ffc3dc1ede4": "m = \\frac{d}{p}\\;", "438e90dd1525ab6045a04332b2d709b4": "= L_x^2+L_y^2+L_z^2", "438ea854da11f62959dee259767d598e": "\\mathbb{E}\\bigl[X_n^-1_{\\{X_n^->c\\}}\\,|\\,\\mathcal G\\bigr]<\\varepsilon\n\\qquad\\text{for all }n\\in\\mathbb{N},\\,\\text{almost surely}.", "438eba9408c540bfbab94abf7060c8ef": "\\textstyle{\\frac {\\log(5)} {\\log(1+\\varphi)}}", "438ecb3d3e49414040603e8d1259c056": "\\hat{x}(t)", "438eff0654cafa6cd2c4adeed4c5f8b1": "a\\sqrt {r}", "438f26a158ffa7395a4c1f2f02bb4193": "\\left\\{\nX_{i},Y_{i},Z_{i}\\right\\}", "438fc8b5b864f48105789c9210941613": "B_{\\lambda}(T) = B_{\\nu}(T) \\times \\frac{d\\nu}{d\\lambda}", "438ff3edde2e0d640f0de231f66a98d8": "\n\\Gamma_k[\\Phi,\\bar{\\Phi}] = \\sum\\limits_{\\alpha=1}^{\\infty} g_\\alpha(k) P_\\alpha[\\Phi,\\bar{\\Phi}] .\n", "43906e906b07f691cc9adb1aca5e4a4f": "x = \\frac { \\begin{vmatrix} {\\color{red}j} & b & c \\\\ {\\color{red}k} & e & f \\\\ {\\color{red}l} & h & i \\end{vmatrix} } { \\begin{vmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{vmatrix} }, \\quad y = \\frac { \\begin{vmatrix} a & {\\color{red}j} & c \\\\ d & {\\color{red}k} & f \\\\ g & {\\color{red}l} & i \\end{vmatrix} } { \\begin{vmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{vmatrix} },\\text{ and }z = \\frac { \\begin{vmatrix} a & b & {\\color{red}j} \\\\ d & e & {\\color{red}k} \\\\ g & h & {\\color{red}l} \\end{vmatrix} } { \\begin{vmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{vmatrix} }.", "439070b671565c9df642ba9749641b6a": "\n\\frac{\\delta V(r)}{\\delta \\rho(r')} = \\frac{1}{4\\pi\\epsilon_0|r-r'|}.\n", "439084a0b76f333ffa286f0b9d49f245": "\\prod_{i \\in I} X_i. ", "43912874936d1a7e191a49301991f5c6": "Y_m \\leq X_{m-1}", "439141e645575469e6f778785ac86781": "|C\\cap D|", "439150bac8038c8307539801114fd51f": "i \\ge n > 0", "439157c030a128c41189e941da1d7819": "\\tilde{a}^{(1)} , ... , \\tilde{a}^{(m/n)}", "439188e9ee2186e69c263a9203bcab73": "\\xi \\,", "4391b6b190902fb70503fefef52373f2": "(r_{i-1}).", "4391bb58d5f5c603d3186270e402b6c8": " \nf(x; \\nu, \\tau^2)=\n\\frac{(\\tau^2\\nu/2)^{\\nu/2}}{\\Gamma(\\nu/2)}~\n\\frac{\\exp\\left[ \\frac{-\\nu \\tau^2}{2 x}\\right]}{x^{1+\\nu/2}}\n", "43920c98f809fd28343b8e1e1599619d": "\\; \\{|\\phi_{A_1}\\rangle, \\ldots, |\\phi_{ A_m}\\rangle\\}", "43927bfc6bd3dc787712c3b20bc9711c": "|\\psi_{10}\\rangle = |1\\rangle", "4393102620f7750d259e3f050f32ba0b": "Tx", "439354dec55f14b5607d5a3f79daca8a": "(x-x_0)F'_x(x_0,y_0) + (y-y_0)F'_y(x_0,y_0),", "43935fc95055b5db72ce4e4d12cbb9cf": "A \\triangle B = (A \\cup B) - (A \\cap B)", "4393d38fd21c83f64d0fbac8a6c25d7e": "U_\\text{Pot} = -2 U_\\text{Kin}, \\, ", "4393e8867b39a8ec29d9c15198e3dcaf": " G_z ", "4393efe0889b5c6ea2530305553fd82c": "y_n = \\sum_{i=0}^{M-1} b_i x_{n-i}", "4394302e0e690df9b9365fd4766c8760": " \\frac{\\partial W}{\\partial t} + U \\frac{\\partial W}{\\partial X} + W \\frac{\\partial W}{\\partial Z}\\ = -\\frac{1}{\\rho_o}\\frac{\\partial p_d}{\\partial Z} + v \\left(\\frac{\\partial^2 W}{\\partial X^2} + \\frac{\\partial^2 W}{\\partial Z^2}\\right)\\ - g \\left(\\beta_{s}\\nabla{S} - \\beta_{T}\\nabla{T}\\right)", "43944993265b80a2f39b52b8c851eecd": "\\underline{\\underline{\\boldsymbol{A}_3}}", "4394536f59d42aef6d2b2a230831b47a": "x^2+y^2=h^2", "4394909a444cc3cf801e592e836e00b7": "{\\mathfrak g}=\\bigoplus_{i\\in{\\mathbb Z}} {\\mathfrak g}_i", "4394bd2bfb592102846b31c44ed3acec": "\\partial^n", "4394cb11feaf78579ec28c6b35f429e6": "\n\\ell^{\\prime\\prime}(\\beta) = -\\sum_j \\sum_{\\ell=0}^{m-1} \\left(\\frac{\\sum_{i:Y_i\\ge t_j}\\theta_iX_iX_i^\\prime - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_iX_iX_i^\\prime}{\\phi_{j,\\ell,m}} - \\frac{Z_{j,\\ell,m}\\times Z_{j,\\ell,m}^\\prime}{\\phi_{j,\\ell,m}^2}\\right),\n", "43955cbe13e9e93dd2aad3230b147426": " \\int_{-\\infty}^{\\infty}{\\left|h(t)\\right|\\,\\mathord{\\operatorname{d}}t} = \\| h \\|_{1} < \\infty", "4395937765ac6bd9de5802bc1ed541a8": "O(t)", "4395ff4ce664570576717514006cea0c": "\\lambda=-2.\\;\\!", "4396383eff49da5c463f675ef545d1d3": "W=\\int_0^t\\boldsymbol{F}\\cdot\\boldsymbol{v}dt =\\int_0^tkx v_x dt = \\frac{1}{2}kx^2. ", "439647557a23d06df2ccccca94e88da0": "E_{tgu} = 0.5 \\cdot [\\tfrac {(9.1 \\cdot 823) + (2.75 \\cdot 1585)} { 1000 } ]^2 / 4.54 =", "439710b7907b94348bd4570f53c1ff63": "B(t).", "4397246af10345a78b0cd8fef38709c0": " M(t,0) \\simeq (-t)^{\\beta}\\mbox{ for }t \\uparrow 0 ", "43972dcca0fa6083b4c4dfc26074f084": " \\ K_1 = \\ {16 \\over Re} + {16 He \\over 6{Re^2}}", "4397741edeb695be97170cb9abda269c": "\\left. \\frac{\\mathrm{\\partial}}{\\mathrm{\\partial}t} p_\\mathbf{k} \\right |_{\\mathrm{coll}}", "4397d133ad437f2b2d212091371c0c43": "\\frac{1}{\\Delta x}\\left(\\left.\\frac{\\partial y}{\\partial x}\\right|^{x+\\Delta x}-\\left.\\frac{\\partial y}{\\partial x}\\right|^x\\right)=\\frac{\\mu}{T}\\frac{\\partial^2 y}{\\partial t^2}", "439817acd176d1f9b0bdefa3c4616ee9": " D(10,2), \\; D(8,0.6), \\; D(2,3), \\; \\text{and} \\; D(-11,3). ", "439845c82c7c5b1821716fa96a8a1f2f": " P_c=2\\gamma\\left (\\frac{1}{R_A}-\\frac{1}{R_B}\\right)\\!", "43984b6058634f78ac1d8de866e994b1": "\nA \\| \\mathbf{v} \\|^{2} \\leq 0 \\leq B \\| \\mathbf{v} \\|^{2} ;\n", "43987ec533766b2766ad4526d3388b62": "\n\\Delta(\\mathbf{p}^{\\prime},\\mathbf{p}) =\n\\delta(\\mathbf{p}^{\\prime}-\\mathbf{p})\n\\sum_\\mathbf{k} \\left| F(\\mathbf{k}) \\right|^2\n\\left[ n_a( \\mathbf{p}/2-\\mathbf{k})\n+ n_c(\\mathbf{p}/2+\\mathbf{k})\n\\right] ", "4398b670a1788d0afb73003a87c56b48": "\\scriptstyle\\varphi = (1+\\sqrt{5})/2", "439936a0b49c3c6b4a0d6f917beb8f3a": "a^n + b^n = c^n", "43995818a34d8c1a60afc4a17d32cc99": "T(n) = 1*0.1 + 2*0.1 + 3*0.3 + 4*0.1 + 5*0.4 = 3.6", "4399a9cfc7fa68930f42f77e35c594c9": "\\delta(x)=x-\\operatorname{E}[X|\\theta=0].", "439a005871b327a0b5ad7782a703a5e8": "\\dot{W}_s=2 \\pi T \\dot{n}", "439a1fce2c984863c05dc70e26605835": " t = d^2\\log n ", "439a3ca759b688561a1bcd274271c6f2": "S_0 = 100", "439a401165fcc782847c7fb023eb4488": "\\mathrm{NPV}(r)", "439a4fcbffd999b4ab5097680bd8c449": "\\scriptstyle A_1, A_2,\\dots,A_d", "439a72d463938adef0263bd9628f4fb7": "\\gamma_K= sA -(n+\\delta)\\ ,", "439a911ccc390c1bfc6cf9f907f4df09": "+\\{g_1,\\cdots, g_{N-1},\\{f_1,\\cdots,f_{N-1},~g_N\\}\\}, ", "439ab73baea9c5d2b8da07439d4f7c1a": "\\vec{x}_S(s)", "439abeac317ae61446deb50ddabeef67": "\\neg(p \\lor q)", "439b7c94427f2b10454b7cae05bada92": "[2^{k},k,2^{k-1}]_2", "439bcd35756cfd253eae0a80d1a6ea4c": " Q(a,b)= \\int_M \\alpha \\wedge \\beta", "439c090355150533ef18949ac32ae6ac": "h_{3} = -\\frac{1}{8\\beta^{2}} \\left[ h_{1}h_{2} \\frac{J^{\\prime\\prime}(u_{0})}{2} + h_{1}^{3} \\frac{J^{\\prime\\prime\\prime}(u_{0})}{24} \\right]", "439c1736a7a91820bbefd402fb1bcb2a": "\\phi_i = \\frac{V_i}{V_{\\rm mix}} ", "439c24777f3e714f89e4df81f386d659": "m_X(H) = \\left|X^H\\right|", "439c682675942a172b257664e4d6946b": "[2,\\infty)", "439cb6dfa3e5877a5f2ba79f61674036": " n\\bot ", "439cc9aa4eab6ca4286da538f374ddaf": "v_e=\\frac{Z}{n}", "439cd29bab6df75d6c48f84c52a9f415": "| \\psi \\rangle = \\frac{|0 \\rangle +e^{i \\phi} |1 \\rangle}{\\sqrt{2}},", "439cfe342ab8f11676898bcd4aad41c8": "T_{11}=2 \\eta_0 \\lambda {\\dot \\gamma}^2 \\left(1 -\\exp\\left(-\\frac t \\lambda\\right)\\left(1+\\frac t \\lambda \\right)\\right)", "439d473dc564959b7538b39b0d1c8975": "f^c(x) = m^b(x)\\int_{\\omega} s(\\lambda,x)\\rho^c(\\lambda)d\\lambda", "439d65ca64fc37a78ebc9d7f34a8f99e": "3 \\cdot 5", "439d9beea9c8f1e8ef144e46285aeef5": "1\\leq k \\leq q", "439da3e8e008287f8ddf2c55db13a9ae": "\\ 0<\\varphi n1 \\end{cases} ", "43a6ee062b6d1d7e47407f429393eaea": "2.0003", "43a6f6d7d48b2901aaba5108e58b0bf3": " R =\\frac{[{1 \\over {2}}( V_{r3}^2 - V_{r2}^2) + {1\\over {2}}( U_2^2 - U_3^2)]}{( U_2\\,V_w2- U_1\\,V_w1)}", "43a7091e4c598fdd4e47f93b479c700c": "5x+3x=8x\\,\\!", "43a71dc32633814b423b36533d733a3f": "p\\circ g:\\mathbf{R}^k\\to E_x.", "43a79000b0b7a4b9c13a3a797dfad642": "f_*\\,", "43a79880496cfec15d8b711cc6ba2b44": "\\varepsilon \\mu = \\frac{1}{v^2}.", "43a8af73f7dac8a3b18dba08d4e1ad6d": "\\alpha \\mathbb{Q}", "43a8f8db3aa91b6d4b54bfb73b60cca3": "x=s", "43a927fc84524e82f3c61ed3af325891": "\\int_0^t H^2 d[M] <\\infty,", "43a929d85771e02fcc0902602d2089ab": "y^2-x^q=0", "43a95e6e330850382f9d99b441ce47a5": "\\frac{n(q)}{N}", "43a9750dbff4a8bac167d11e3a68bd47": "\\partial_{\\mu} A^{\\mu} = 0", "43a99692d9d32cc67134baed85e354c8": "\\frac{\\Delta f}{f} \\simeq -\\frac{v\\cos\\theta}{c}.", "43a9cf822d22e64284b18b2af4b860f5": "c=\\pi\\cdot{2r},\\,\\!", "43aa076969bce92649374eb67f78002b": "\\|A\\|_p", "43aa8af2ceec86ed56ec7dea37a1ee85": "q^\\alpha=\\|q\\|^\\alpha e^{\\hat{n}\\alpha\\theta} = \\|q\\|^\\alpha \\left(\\cos(\\alpha\\theta) + \\hat{n} \\sin(\\alpha\\theta)\\right).", "43aad869d2322ba9d33bd5ad88ddd386": "f(z)=\\frac{1}{z}", "43ab69d9b33de3653eca6e0201a8cbeb": "A(w)\\Psi(zg(w))", "43ab8116ec63bc440d62822fca059549": " b \\geq n + q.", "43abcd11c7f6f0859f5ef216686e9d6a": " u_{w} > 0 ", "43abcda566fb4172d380cb44d20f0237": "\n\\begin{align}\n\\operatorname{var}(Y(t)) &= E(\\operatorname{var}(Y(t)|N(t))) + \\operatorname{var}(E(Y(t)|N(t))) \\\\\n&= E(N(t)\\operatorname{var}(D)) + \\operatorname{var}(N(t)E(D)) \\\\\n&= \\operatorname{var}(D)E(N(t)) + E(D)^2 \\operatorname{var}(N(t)) \\\\\n&= \\operatorname{var}(D)\\lambda t + E(D)^2\\lambda t \\\\\n&= \\lambda t(\\operatorname{var}(D) + E(D)^2) \\\\\n&= \\lambda t E(D^2).\n\\end{align}\n", "43abe5eb03c546692e3d8162763a05a7": "\\theta\\ =\\ \\frac{\\pi}{4}", "43ac7c76421ed2307dcfdd8773898922": "2^{32} \\approx 4.3 \\times 10^9 ", "43aca3a9a1c856e9c1b7c2b63790084e": "\\langle a, b\\rangle", "43ad0e54d81d0bf274cc9ff009732243": "C ", "43ad28496d5b7b2a20e567136b7662a8": "J\\left(S\\right)", "43adb1db65d44520d1ef531b2bc1c3da": "-lg( \\frac{I} {I_0}) =\\epsilon^*cd; \\epsilon^*c=\\epsilon_{graphene}", "43adde95117658801ab6ac13a1fc6c35": "\\frac{1}{\\gamma\\cdot(1+\\beta)}\n=\\frac{\\sqrt{1-\\beta^2}}{1+\\beta}\n=\\frac{\\sqrt{(1+\\beta)(1-\\beta)}}{1+\\beta}\n=\\sqrt{\\frac{1-\\beta}{1+\\beta}}", "43ade64600b36d835a549a67c7e37338": "(TM\\otimes TM)^*\\cong T^*M\\otimes T^*M,", "43ae19f8cdcf3704358bbef08270c8d0": "\\mbox{newspaper } \\mathbf{\\{ \\operatorname{<}, \\operatorname{m} \\}} \\mbox{ bed}", "43ae6bbbc5c4de149ae8c05f9644a6b0": "O(A_1:A_2|B) = \\Lambda(A_1:A_2|B) \\cdot O(A_1:A_2).", "43ae97f46536940b6d8b524217706bcd": " +\\frac{1} {4 \\pi} \\iint\\limits_S\\left(\\mathbf{U}\\mathbf{n} \\cdot\\nabla \\frac{1}{R}\\right) dS_Q", "43ae9eed30951890a710f86d7d377f59": "\\left(\\frac{\\partial H}{\\partial T} \\right)_p=C_p", "43aecbec909c80e5f9fa14bd5b21b521": "\\ N", "43af12815a169742aa36ca7b484a7959": " \\frac{\\pi^3}{32} = \\sum_{n=1}^\\infty\\frac{-1^{n+1}}{(-1+2n)^3} = \\frac{1}{1^3} - \\frac{1}{3^3} + \\frac{1}{5^3} - \\frac{1}{7^3} + \\cdots ", "43af3e84ca91ace827bf855e16dbdd00": "P_{y,w}(q) = \\sum_{i} q^i \\dim(\\operatorname{Ext}^{\\ell(w)-\\ell(y)-2i}(M_y,L_w))", "43af401d5640207df004d81b1cc4a4ce": "RC = \\Delta_T \\left( \\frac{\\alpha}{1 - \\alpha} \\right)", "43afc2e242876990f6bf778f2a2278d7": "L\\,", "43afd222522b1f8242b2833400ddd0ce": "K_{sp} = \\frac{{(-xN_{AxBy(\\Delta)})}^x {(-yN_{AxBy(\\Delta)})}^y}{V^{(x+y)}}\\,", "43aff364fec2640209461ed9c0203c48": "38^2", "43b00b63e0c6878e768c2d9383d2c8e3": "r_{Mi}", "43b055f159ce5af56d3ba742e3879a4c": "D = \\{ \\langle 0 \\rangle, \\langle 1 \\rangle, \\langle 2 \\rangle, \\langle 3 \\rangle, \\langle 4 \\rangle, \\langle 5 \\rangle, \\langle 6 \\rangle, \\langle 7 \\rangle, \\langle 8 \\rangle, \\langle 9 \\rangle \\}", "43b0604b9e36174716ce7e02ebfd8fc1": "T_K=\\cfrac{\\frac{\\Delta H}{R}}{-\\ln \\left ( \\frac{{t_1}e^ \\left ( \\frac{-\\Delta H}{RT_1}\\right ) + {t_2}e^ \\left ( \\frac{-\\Delta H}{RT_2}\\right ) + \\cdots + {t_n}e^ \\left ( \\frac{-\\Delta H}{RT_n}\\right )}{{t_1} + {t_2} + \\cdots + {t_n}} \\right )}", "43b07ebf916b3f87f084bb515af8b618": "\\Gamma^*:=\\{f\\circ \\gamma:\\gamma\\in\\Gamma\\}", "43b09b3d78f53b0aa642b5a71d2ace94": "\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}}", "43b14e861f4d1412f1d1fe713e42a904": "N(d_-) < N(m) < N(d_+) = \\Delta.", "43b18140a0dbaf578b2ad531ee453225": "\\mu = A_HT\\cdot e^{Q_L/RT},", "43b1b993eabf98a406bbb3cf99ff3d82": "b^n = \\underbrace{b \\times \\cdots \\times b}_n", "43b24ecd760a4ff455dcfff994af93f3": "0 < \\Re(s) < \\frac12.", "43b25f1925293021cc6e1253b6d1b1b3": "\\dot{u} = u^2 + \\Delta I", "43b27e23c5d46c9e57e0db136398d435": "\\sqrt{2\\log{p}}", "43b283d8b1ef0254bb2aff06f739483c": "\\mathrm{SO}(10)\\cdot\\mathrm{SO}(2)\\,", "43b32ee6d3ebf0f5aed6fb0e0bb467ef": "\\varepsilon_n", "43b36d42e7f8e60be58ba4356b6af40c": "sw", "43b38b3d05adaa315cd4d57bbd6772a9": "\\vec V_r = \\vec V_t - \\vec V_m", "43b3a5b9ddb616da24dcee750c3a30ed": " N_{A}=-D_{AB} \\frac{1}{RT} \\frac{dP_{A}}{dx} ", "43b3c902fdac12792c86061cae0a3693": "\\frac{d}{dx}(u + v + w + \\dots)=\\frac{du}{dx}+\\frac{dv}{dx}+\\frac{dw}{dx}+\\cdots", "43b3ddf8423f162d8d53b4a4e7618ba9": "\\chi_Q(\\mathbf{z},\\mathbf{z}^*)= \\operatorname{tr}(\\rho e^{i\\mathbf{z}^*\\cdot\\widehat{\\mathbf{a}}^{\\dagger}}e^{i\\mathbf{z}\\cdot\\widehat{\\mathbf{a}}})", "43b4c87bf089a51e9187bb5be0444356": "\\sqrt N", "43b4cbf4a3048a24caba029194d1a54c": " \\mathbf{MTF_{sensor}(\\xi,\\eta) \\cdot\nMTF_{transmission}(\\xi,\\eta) \\cdot} ", "43b5296f42b8ff8639a4a05c77644f52": "\\sum_{n=0}^\\infty |w^n s(2^n x)| \\le 1/2 \\sum_{n=0}^\\infty |w|^n = \\frac{1}{2} \\cdot \\frac{1}{1-|w|}", "43b5563d202e9460d5f7a941ae8ced3d": "B_{i+1}", "43b5a59b12b58901cdde950a7d86a972": "\nJ_{\\parallel} = \\frac {n_e\\ e^2}{m_e\\ \\nu} \\ \\frac {E}{1+\\beta^2} \\qquad \\text{and} \\qquad J_{\\perp} = \\frac {-n_e\\ e^2}{m_e\\ \\nu} \\ \\frac {\\beta\\ E}{1+\\beta^2}\n", "43b5ce95797f244b404856b843c0871f": "\\prod_{n=1}^\\infty n^{{1/2}^n} = \\sqrt {1 \\sqrt {2 \\sqrt{3 \\cdots}}} = 1^{1/2} \\; 2^{1/4} \\; 3^{1/8} \\cdots ", "43b614d6e847ef8a3656ab808815018c": "x = 2 + 3t = 2 + 3(3 + 4s) = 11 + 12s", "43b66ad3cb47f185d15c0dc58231a950": "T_{2lm}", "43b67336d9e46a47f4a14a9709e5762c": "\nf v_{term} = m (1 - \\bar{\\nu} \\rho) g \\ \\stackrel{\\mathrm{def}}{=}\\ m_{b} g\n", "43b6a0803bfcd74d1823151a629f8358": " e^{x} = 1 + {x \\over 1!} + {x^{2} \\over 2!} + {x^{3} \\over 3!} + \\cdots = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}", "43b6cf93536b683ff1ea3be5a35708b3": "S=1", "43b6fcc626cbd70a0439d76547a5029e": "(G \\times \\{e'\\})/N", "43b7232130677ca0d250caa837ff3c1e": "\\begin{align}\n \\left(A_x \\frac{\\partial B_x}{\\partial x} + A_y \\frac{\\partial B_x}{\\partial y} + A_z \\frac{\\partial B_x}{\\partial z}\\right) &\\hat{\\mathbf{x}} \\\\\n+ \\left(A_x \\frac{\\partial B_y}{\\partial x} + A_y \\frac{\\partial B_y}{\\partial y} + A_z \\frac{\\partial B_y}{\\partial z}\\right) &\\hat{\\mathbf{y}} \\\\\n+ \\left(A_x \\frac{\\partial B_z}{\\partial x} + A_y \\frac{\\partial B_z}{\\partial y} + A_z \\frac{\\partial B_z}{\\partial z}\\right) &\\hat{\\mathbf{z}}\n\\end{align}", "43b7343b21eee72cb4525d6e1e66fcd8": "4x^3y-4xy^3\\,", "43b78eed6b394a9337b90c2a4cbd7f55": "\\sqrt n", "43b792467151e87f63eed79dd41c2bac": "\\sigma=\\pm 1", "43b7b2dd1652a5d6ff172493b94111a1": "\\gamma^*= sA -(n+\\delta)\\ ,", "43b7c9dfd7a8090a36df8fa5e414d2cb": " 0<\\alpha\\le 1 ", "43b86df66a838a5215368900155d93c0": "B(x,y) = \\begin{bmatrix}\nx & y \\\\\n\\pm ty & \\pm x \\end{bmatrix}.", "43b8d310c2aa7946d2806a95c12bb207": "c(v,u)=0", "43b8d4d38da341f8dc506dc1ba2735ed": "\\frac {d M_z(t)} {d t} = i \\frac{\\gamma}{2} \\left ( M_{xy} (t) \\overline{B_{xy} (t)} - \n\\overline {M_{xy}} (t) B_{xy} (t) \\right )\n- \\frac {M_z - M_0} {T_1}", "43b90f554cfe4a3c2b3b1e64a4e2f856": " D=a \\cdot S+b \\cdot \\frac {\\Delta P}{\\Delta t}-c \\cdot\\frac {\\Delta K}{\\Delta t}+d \\,\\ ", "43b94a79683c0994e623f8726a1a7fb5": "\\mathrm{^{238}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{239}_{\\ 92}U\\ \\xrightarrow[23.5 \\ min]{\\beta^-} \\ ^{239}_{\\ 93}Np\\ \\xrightarrow[2.3565 \\ d]{\\beta^-} \\ ^{239}_{\\ 94}Pu}", "43b968a0f944c27759feeb40af57a2b9": "u(z)=\\sum_{k=0}^\\infty A_kz^{k+r}, \\qquad (A_0 \\neq 0)", "43b97d54aae699e2799e0d0c6b109469": " R_{;l} = 2 R^m {}_{l;m},\\,\\!", "43b996c7ce8a862a8231502f82f7bf40": "P(X \\le k) \\approx \\Phi \\left( \\frac{k-n p}{\\sqrt{n p (1-p)}} \\right)", "43baa0127365021b9132a712417a84d5": "\nd_A = \\frac{r(\\chi)}{1+z}\n", "43bab48540ec8d17a00e5df7345ea146": "-i\\, \\Omega\\, F = \\nu \\frac{\\text{d}^2 F}{\\text{d} z^2}", "43bb3610100f675c84461c2384b82dd6": "\n\\rho_0(\\mathbf{k},\\omega) = \\frac{1}{\\mathcal{Z}}\\,2\\pi\\delta(\\xi_\\mathbf{k} - \\omega) \\sum_{\\alpha'}\\langle\\alpha' |\\psi_\\mathbf{k}\\psi_\\mathbf{k}^\\dagger|\\alpha' \\rangle(1-\\zeta \\mathrm{e}^{-\\beta\\xi_\\mathbf{k}})\\mathrm{e}^{-\\beta E_{\\alpha'}}.\n", "43bb3c73bbd343968091f236a8133a81": "\\Phi_V", "43bb5b2a1f7e3992f8283faff2075b2f": "\nP_{\\infty} + \\frac{1}{2}\\rho v_u^2 = P_{D+} + \\frac{1}{2}\\rho v_D^2\n", "43bb9123003cb64aa07c25c092b6640a": "+ \\sum_\\text{torsions} \\sum_n \\frac{1}{2} V_n [1+\\cos(n \\omega- \\gamma)]", "43bbb627730367a9dbb7137ff4b01577": "O\\left(\\sqrt{n}\\right)", "43bc91fe7207cae764693405746397f5": "Z_W", "43bcba452fcdee8a6038b5caaef16147": " P = \\frac{2}{3}\\frac{q^2}{m^2 c^3} |\\dot \\mathbf{p}|^2.", "43bcc2674a0407ee14073df30e0646b6": "\\mathbf{n} / \\mathbf{N} = (n_1/N_1, \\dots, n_d/N_d)", "43bcd0448fc2e28f31e3821a7ea595c7": " c_v = 0, -1", "43bcd7be31e620375c01e02db1b810e4": "\\displaystyle{2(a,b)(c,d)=(ac,bd) + (ad,bc).}", "43bd17406ad985de014698c083f57712": "C_\\nu(x) = -C_\\nu(1-x)", "43bd1f974bb34238df714a10ea6e8037": "*\\colon \\Omega^k(M) \\overset{\\sim}{\\to} \\Omega^{n-k}(M)", "43bd99981235f032e229dc954a6913e6": "\\scriptstyle v_i", "43bd9d88ed87a4dccaee61a65fd6e448": "\\scriptstyle \\leq3.4\\times10^{-16}", "43be27cd3f618e017deb69d4fdcce2f8": "\\alpha\\in(0,1]", "43be48353cc060115d1f53c25da374a7": "p[i,j]", "43be6639a660c44f6fd4159c23ac7002": " \\hat{\\mathbf{Z}} = \\varprojlim \\mathbf{Z}/n\\mathbf{Z}. ", "43bebf102fb56753bb46d7407d0da112": "(S \\otimes S)(R) = R", "43bef0c986d509e6bc4afad67adce753": "\\alpha^2 \\,", "43bf4d987a60e18be04bdf1b2ed1f41e": "|g(x)| \\leq \\alpha", "43bf4ee12517cd44434cdae555ec119b": " r^2 = (x-\\xi)^2 + (y-\\eta)^2 + (z-\\zeta)^2. \\,", "43c00a57058cc3cc2aaa065321f25d10": "\\sqcup \\in \\Sigma", "43c0385e67b99d53c259a89917b38f38": "\\sum_{n=1}^\\infty \\frac {a_n}{e^{zn}-1}= \\sum_{m=1}^\\infty b_m e^{-mz}", "43c049c5d594350ac6613232059a2c8c": "e^{aj} \\mapsto e^{(a-b)j/2}", "43c05d8c45974588cd41c31eb0c27ad0": "L^p(S, \\mu) \\equiv \\mathcal{L}^p(S, \\mu) / N", "43c07a3d7f9fa8c8550340e128808383": " \\gcd( f(x), x^n - 1) ", "43c0bdf83254eaefc03e939f5d47ae39": "\\mathcal{O}_X \\to R\\pi_{*}\\mathcal{O}_E", "43c0fe2b0d3048e7a93686ce412c05d7": "L=1,\\ldots,P", "43c103a99d948e1489f9e7a931afd030": " r=\\lim_{n\\to \\infty} \\sum_{i=0}^n \\frac{a_i}{10^i}", "43c13728adb3a3608996ec6c75c8ab3d": "r = \\sqrt[3]{\\frac\\mu{\\omega^2}}", "43c16a1a0330a42a97fa0c65c2ff7d8c": "\\sqrt{-1 \\cdot -1}=\\sqrt{1}=-1,", "43c178e6885b2480b4f1b3cc21c48843": "\\ Y = F(AK,L)", "43c1838588c2ecd3e8184e7c568cd85c": "\\pi^k = \\sum_{n=1}^\\infty \\frac{1}{n^k} \\left(\\frac{a}{q^n-1} + \\frac{b}{q^{2n}-1} + \\frac{c}{q^{4n}-1}\\right)", "43c1bec13d86d392f8683952d9bdac48": "d:\\Omega^n(M)\\rightarrow\\Omega^{n+1}(M)", "43c206664401724a9303635b795bcd93": "B*C/A", "43c266f4e2bd4582105f264faf320599": "a_0, a_1, a_2", "43c29005dde0e8ebb8fe56be4cdd77fd": "\\| P_t - P_s \\|_{op} = 1, \\quad \\mbox{for all} \\quad t \\neq s .", "43c2ecef52a7b2613112e59b3678bf2f": "\\sum_{\\alpha\\geq0}c_\\alpha(z-a)^\\alpha := \\sum_{\\alpha_1\\geq0,\\ldots,\\alpha_n\\geq0}c_{\\alpha_1,\\ldots,\\alpha_n}(z_1-a_1)^{\\alpha_1}\\ldots(z_n-a_n)^{\\alpha_n}", "43c30309e5b93488e2779ac95cda28c4": "D_\\varepsilon(A,B)", "43c34bc1ccce30e4bf3f628dc3a88254": "\\Delta T = T_m-T", "43c381a072b214058bf7dee1546e0f9a": "(a^2-4(x^2+y^2))^3=108a^2x^2(x^2+y^2).", "43c3852367527c6fe8bcceb558cb7f0b": "g_{\\gamma_n}", "43c3f21c38b6ad3a649a12418dcc9432": "\\sqrt[3]{u}", "43c41a9f6b0123183b754b81ba53bed9": "|Q_{P}(h)-\\widehat{Q_{s}}(h)|\\leq\\epsilon /2\\}\\,\\!", "43c42ae75b51773cddcd47e209a76774": "g_Y=({F_A}A/{Y})*g_A+({rK}/{Y})*g_K+({wL}/{Y})*g_L", "43c44505ffc25d624972d9c4af511fe4": "\\cos c = \\cot A \\cdot \\cot B", "43c47aa1c27100a2c1bb57eb6999b46f": "{Y}_{n}", "43c4af326353e21afec8bb7be962bb25": "H[1]=\\bot", "43c4b8638475a8de5858f3f2e35f7de5": " I(Q,t) \\propto S(Q) + \\int \\cos (\\omega t) \\, S(Q,\\omega)\\, dt ", "43c5359cf50b0b3f2be3e198732b2352": "\n \\left( B \\or C \\right) , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) , \\lnot C \\vdash \\lnot A\n ", "43c568aeee158206bcf1f91172d827d4": "d_n:= D(\\ h_{n-1},t_{n-1},y(t_{n-1}\\ )\\ y_{n-1} := \\left[ \\Phi(\\ h_{n-1},t_{n-1},y(t_{n-1})\\ ) - E(\\ h_{n-1},t_{n-1},y(t_{n-1})\\ ) \\right]\\ y_{n-1} ", "43c56da4d6fe4a4a695619653f061272": " \\max \\left ( \\frac{OPT}{f(y)}, \\frac{f(y)}{OPT} \\right ),", "43c59baba30f13b04af229358961bb98": "\\Pr[X \\ge x] \\ge 1-k/q", "43c5bb49c1e0c3079678f418d5306e27": "\\ln{\\frac{\\Gamma(\\alpha/2)}{2\\beta^{\\frac{1}{2}}}} - \\frac{\\alpha - 1}{2} \\psi\\left(\\frac{\\alpha}{2}\\right) + \\frac{\\alpha}{2}", "43c5f4ed6d0ff50ff4cd6b7a37115cc7": "\\frac{T3}{T2}= \\left(\\frac{V2}{V3}\\right)^{\\gamma-1}", "43c61c918aeb5f0d6db175931c209ece": "f(x)=a(x-x_0)^2+x_0", "43c628eb9d91b85d7213fe058641fcf2": " a = 4/3 ", "43c6320fbaf4b2d4fd5445b09d6d2436": " x^{(k+1)}_i = \\frac{1}{a_{ii}} \\left(b_i -\\sum_{j\\ne i}a_{ij}x^{(k)}_j\\right),\\quad i=1,2,\\ldots,n. ", "43c636f1e359e3bf9963c974878466a2": "\\gamma^2 = \\begin{pmatrix} 0 & -\\sigma^2 \\\\ \\sigma^2 & 0 \\end{pmatrix}, \\quad \\gamma^3 = \\begin{pmatrix} -i\\sigma^1 & 0 \\\\ 0 & -i\\sigma^1 \\end{pmatrix}, \\quad \\gamma^5 = \\begin{pmatrix} \\sigma^2 & 0 \\\\ 0 & -\\sigma^2 \\end{pmatrix}.", "43c6d71baf37a26e75767b637e03d915": "X\\in\\mathcal{X}", "43c712f7eaeee6f3f498a2148cc136da": "\\overline F", "43c71df67ca9db7ba36192ec8d59a6d0": "V - E + F = 2.\\,", "43c723921b8467bbe2cc012ce76b632f": " \\int_{t^n}^{t^{n+1} } f( q( t, x_{i-1/2} ) )\\, dt ", "43c7ba04a0d9614b2741ee316f494401": "\\langle\\alpha^\\mu_\\tau(A)B\\rangle", "43c7baa3e3256753246d2ab62b119365": "f*g=fg+\\sum_{k=1}^\\infty \\hbar^kB_k(f\\otimes g),", "43c7ea30701b66c65b385405e285bcb4": "\\epsilon x", "43c82b631f4bcd3f8e1cbbaa286aebce": "\n\\begin{array}{l}\n\\text{Im}\\langle u, v \\rangle = \\frac{1}{2}\\left(\\|u+iv\\|^2 - \\|u\\|^2 - \\|v\\|^2\\right), \\\\[3pt]\n\\text{Im}\\langle u, v \\rangle = \\frac{1}{2}\\left(\\|u\\|^2 + \\|v\\|^2 - \\|u-iv\\|^2\\right), \\\\[3pt]\n\\text{Im}\\langle u, v \\rangle = \\frac{1}{4}\\left(\\|u+iv\\|^2 - \\|u-iv\\|^2\\right).\n\\end{array}", "43c82f70f15dabf419ff06a4f3570fbf": "T_c=\\left(\\frac{N}{\\zeta(\\alpha)}\\right)^{1/\\alpha}\\frac{E_c}{k}", "43c84076c2dbe57a5150a4e6f655b428": " v {{\\mapsto}} \\varphi (x, v)", "43c840d37463a5457d023d05561522aa": "s_{0}(t)=x_{1}(t)", "43c8439e384b75b779c6dc0b1a337936": "\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}\\ = \\frac {n R}{P}\\ = \\left(\\frac{V P}{T}\\right)\\left(\\frac{1}{P}\\right) = \\frac{V}{T}", "43c86a04f8cc9dc2715a2ccda0b450a6": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "43c87442b26d684a7e3bef8b1e099b55": "S_d", "43c8e52d66d34045390844fbf4a4172a": "f(\\tfrac{H - E}{\\omega})", "43c8f43f141dcbd10c20d71b2b108b68": "\n\\tau = -p 1 + C_1 A + C_2 A^2 + C_3 A_u + C_4 A_l,\n", "43c90ae42650e07b1e7d4c7f554cbf81": " \\lambda = \\frac{h}{m c} \\ ", "43c967a8358c208265cf301a862a979e": "\nC_{3i} = \\frac{{ - \\left( {3m^2 - 7} \\right)i + 20i^3 }}\n{{m\\left( {m^2 - 1} \\right)\\left( {3m^4 - 39m^2 + 108} \\right)/420}}.", "43c97232ec007b2a37a971ed52a1f41a": "\\mathbb{E}\\bigl[|X_{t+1}-X_t|\\,\\big\\vert\\,{\\mathcal F}_t\\bigr]\\le c", "43c98a64bcde4857b095743482e04281": "t+1", "43c9a8662327a9357ccc0fea038988fb": "\\lambda_{1}= 0.0971028 + 0.995786i \\,", "43c9e7bad10941b56570f2e19a0c0910": "e_{ij}=\\frac12 \\left( \\frac {\\partial v_i}{\\partial x_j} + \\frac {\\partial v_j}{\\partial x_i} \\right)", "43ca41265388c683aaf76d6680d852ce": "{\\mathrm {Spin}}(n)\\,", "43ca5ea6b1a3b6ab0d5ec4e2dea6223c": "{n_\\mathrm{D}}", "43cae6581bd66f6cbcbda2241a369594": "z=\\exp(\\mu/kT)", "43cb2dcc1ecd3dcdbeaa124648ed941b": "\\log (1+z)", "43cb5013ec145c21fa5b8ed7d490d200": " k = \\frac{y}{x}\\,", "43cb5938ca67631d77848c7bdf365812": " K = 1 + \\alpha \\cdot \\Delta T ", "43cb69cbcabb39c2e8bd082f7c0955a3": "(n=6)\\,\\!", "43cb83331eb3a275e4bcefcfbba894d3": "\\tfrac{q-2}{q}", "43cbe457445ba8f7f51ac875c03097d7": "\\mathbf{L}_i=\n\\begin{bmatrix}\n \\gamma_i^{p^0} &\\gamma_i^{p^1} &\\cdots &\\gamma_i^{p^{m_i-1}}\\\\\n \\gamma_i^{p^1} &\\gamma_i^{p^2} &\\cdots &\\gamma_i^{p^{0}}\\\\\n \\vdots & \\vdots & \\ddots & \\vdots\\\\\n \\gamma_i^{p^{m_i-1}} &\\gamma_i^{p^0} &\\cdots &\\gamma_i^{p^{m_i-2}}\\\\\n\\end{bmatrix}\n", "43cbee7877348ebf90420a9525f2ec6f": "\\alpha\\cdot X = \\begin{bmatrix}\n\\alpha\\,X_{00} & \\alpha\\,X_{01}\\\\\n\\hat\\alpha\\,X_{10} & \\hat\\alpha\\,X_{11}\n\\end{bmatrix}", "43cbf29dcdd53d1f6ac05959e9c3a57c": "\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}", "43cbf7728694626ff5fe0f44292336f1": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 1}{4 \\choose 3}{32 \\choose 1} \\end{matrix}", "43cbf8d80084c6c6adb8914ec6df4363": "\\begin{matrix}\\frac1{256}\\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\\,", "43cc20f4dd826cad6613738f0786fab5": "V \\ge 10 LC_{50}", "43cd1f01fc40ff198193084a874be8ab": "x\\neq 0", "43cd27e3ea2c78ba13549af95ad90e81": " y_{n+5} - \\tfrac{300}{137} y_{n+4} + \\tfrac{300}{137} y_{n+3} - \\tfrac{200}{137} y_{n+2} + \\tfrac{75}{137} y_{n+1} - \\tfrac{12}{137} y_n = \\tfrac{60}{137} h f(t_{n+5}, y_{n+5}); ", "43cdb00a1e970d4ae09e4e7e8cd5c70e": "x^2\\equiv y^2\\quad(\\hbox{mod }N),\\qquad x\\not\\equiv\\pm y\\quad(\\hbox{mod }N).", "43cdf6b45cf362c290fd4468ff1b9f82": "h'\\left(6\\right)=\\left\\lfloor\\frac{6}{11}\\right\\rfloor\\mod 11=0", "43ce1648b8537ce2bd95e53154b41c5f": "\\sqrt n \\,W_{2n-2}", "43ce46c7090a28494d3ced3c97ac3f91": "B \\subseteq C\\,\\!", "43cec7767f9a36be4bada1fc91c0f910": "p(a,x_1,\\ldots,x_k)=0\\,", "43cf00fc93b678ca0dd3558084165a67": "\\,\\!\\alpha \\rho^2", "43cf049d7d151cacd73d34f27d53e3fd": "\nA(\\mathbf{r}) = \\sum_j m_j \\frac{A_j}{\\rho_j} W(| \\mathbf{r}-\\mathbf{r}_{j} |,h),\n", "43cf8e54134f77f1e6f7fb60d9573209": "KK(A,B)\\times KK(B,C)\\rightarrow KK(A,C)", "43cfa56db75d5f84488fb7d80d3175db": "X_1,\\dots,X_n.", "43d0146dabb78f225f803b9819c0d7ef": "\\langle \\Phi\\rangle", "43d050bc6c68a5127730e99be7490f6f": "(1-x^2)y''-2xy'+\\nu(\\nu+1)y=0", "43d058987713f696c08436506d514a28": "\\Psi(r)\\propto \\frac{e^{ik r}}{4 \\pi r} \\int\\!\\!\\!\\int_\\mathrm{aperture} E_{inc}(x',y') e^{-ik ( \\bold{r}' \\cdot \\bold{\\hat{r}} ) } \\, dx' \\,dy',", "43d0f5f1ef5eea6bc46898f4a45c9e61": "p^k:\\mathcal{K}^k \\to \\mathcal{Q}^{k-1}", "43d11bf4badba513b998c09d1f8f1b46": "Im[\\Delta R M (t)] = -i \\frac {dR} {dT} \\sum_{m=-M}^M (\\Delta T(m/\\tau + f) - \\Delta T (m/\\tau - f))exp(i2\\pi m t /\\tau)", "43d12832f1be674ff25a9281e429ab2c": "[b^{3.141},b^{3.142}]", "43d168da748587b30fe75c546fa8d2b1": "\\scriptstyle p_n(x,y)\\,", "43d17f08f40a35aee5d6db96c3249620": " \\langle x^{2n}\\rangle={\\int x^{2n} e^{-a x^2/2} \\over \\int e^{-a x^2/2} } = 1 \\cdot 3 \\cdot 5 ... \\cdot (2n-1) {1\\over a^n} ", "43d1b96e8b99e9b99e3f476e155dbc1d": "\\mathit{\\eta}_\\ast", "43d1c9ea177989f8c141c8fa3155444c": "\\tau_{\\leq 0} A \\to A \\to \\tau_{\\geq 1} A \\to.\\ ", "43d1c9f10c2ae156360107403962d989": "\\mu \\Delta t", "43d259a4aabed9413c1878b909bcc273": "\\scriptstyle Q_d", "43d26bebcf4d65db972038e8d3497447": "H_d(e^{ j \\omega T}) \\ ", "43d284a9bd92d2cfbd30f209e25f467e": "G(q) = \\sum_{n=0}^\\infty \\frac {q^{n^2}} {(q;q)_n} = \n\\frac {1}{(q;q^5)_\\infty (q^4; q^5)_\\infty}\n\t=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\\cdots\n", "43d29e3f1e686a8c75944d0a5ceac278": "\\frac{\\pi}{4} = \\frac{3}{4} + \\frac{1}{3^3-3} - \\frac{1}{5^3-5} + \\frac{1}{7^3-7} - \\cdots ", "43d3552a50e1e145ef199c96f396df65": "\\underline{\\mathsf{f}}(B)=\\lambda B,", "43d38be2c6a7d01e494ed5d08cd28905": "\\overline{I}=O(\\left(P_S\\left(n\\right)/n\\right)\\log\\left(n\\right))", "43d3c3cb9e53edfdf49d7b0cff1eb914": "t = d (D^T D)^{-\\frac{1}{2}}.", "43d3cacd15403ecdb4247191aec8a0fe": "\\Delta(n)\\leq\\frac{\\exp{\\left(c\\sqrt{\\log n}\\right)}}{n^{8/7}},", "43d40e234d53591c5077917cbe6506a4": "1/k_1", "43d4397aebda79edd524dd4a503c7c6b": "\n W = C_1~\\sum_{i=1}^5 \\alpha_i~\\beta^{i-1}~(I_1^i-3^i)\n", "43d45d51896c940b0e372643069269bb": " X \\pm ( \\frac{ 0.484 }{ \\alpha } - 1 ) | X - \\theta | ", "43d45df0c5a7076d83a1ea6e271e83a2": "[K_n,K_m]=0", "43d47060f929346aade8fc006ed29ab2": " \\|{R}\\|^2 ", "43d476269e1c7fc30cdc4f9c83ec86f4": "R = \\frac{\\Delta\\textrm{H \\,\\, Rotor}}{\\Delta\\textrm{H \\,\\, Stage}}", "43d497f8de34da6eef45e5c38bc49ada": "|\\Gamma_{m}|\\,\\!", "43d4a0d4b46c783a3e306f590544326a": "|\\mathbf{r}(t)| = \\sqrt{\\mathbf{r}(t) \\bullet \\mathbf{r}(t)}=\\sqrt {x(t)^2 + y(t)^2 } = r\\, \\sqrt{\\cos^2(t) + \\sin^2(t)} = r", "43d540defd7940fa846d1d92d77139d2": "W \\ge W_{critical, N_r}", "43d5c39393bf1350c63cc58aa6b26515": "\\mathbf{G} := \\begin{pmatrix}\nI_k | -A^T \\\\\n\\end{pmatrix}", "43d5dbebd8d704a0163d64ed5b41eca2": "n>468", "43d5ef27b4e3d7b62a8259efd60694f2": " Y^m_\\ell ( {\\theta,\\varphi} )", "43d5fbefb0851d5fcfe33f72dd1b5c0d": "\\operatorname{var}(\\widehat{D}) = \\frac {\\widehat{p}_m(1 - \\widehat{p}_m)}{n_m} + \\frac {\\widehat{p}_f(1 - \\widehat{p}_f)}{n_f} ,", "43d61d4a7cebbbcc6684e25af215148b": " y^e(\\mathbf{x})=y^m(\\mathbf{x})+\\delta(\\mathbf{x})+\\varepsilon ", "43d6655e5c099e5da959d8c0fcc8a90d": " D_k \\Leftarrow D_{k-1} \\wedge (\\forall z_1...z_{(n-1)m+1})(\\exists z_{(n-1)m+2}...z_{nm+1}) B_n \\Leftarrow D_{k-1} \\wedge (\\forall z_{a^n_1}...z_{a^n_k})(\\exists y_1...y_m) \\phi(z_{a^n_1}...z_{a^n_k}, y_1...y_m) ", "43d684092f9af9eb330eb7fae575fa7e": "q=\\exp(-\\pi K'/K)", "43d6850c4197b48c7d7bdfc888aeae9a": " \\mathbf{p} = \\hbar \\mathbf{k}, \\quad E = \\hbar \\omega", "43d6cb96a9f4507788a245690f1f8c74": "A(z) = B(z) + C(z)", "43d6e4cfca1b0fe9c9ac93f42a7823df": "p\\in X_j", "43d7243ddd2036f02a758711c261a9c2": "e^{x_1y}, e^{x_2y}, e^{\\gamma x_1/x_2}.", "43d72950bee23a5b6c71bad1ea66f2e9": "s^2 = r_1^2 + r_2^2.\\, ", "43d77b27b9cb137fc3eed3184e1a1bb6": " b_n = \\Delta p \\times r_i \\times ( b_i - h_i ) - h_t", "43d77c1d5a5a39ea7dc07705123af0b9": "h_i(x) = 0", "43d7ac86ecbb86ce022f16832b4c4586": "a_n(q), \\, b_n(q)", "43d7e0e62d6c27f2ba9ed2f50d66f815": "T_{o+}^{TE}=F cos(\\frac{m\\pi }{a}y)e^{-jk_{xo}(x-w)} \\ \\ \\ \\ \\ \\ \\ \\ \\ \t(23) ", "43d8f9ac6550b54ae93088c37c5a302a": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\chi + (\\tilde{u}_{1}- E)\\chi + i\\frac{\\hbar^2}{2m}[2 \\mathbf{\\tau}_{12}\\nabla + \\nabla\\mathbf{\\tau}_{12}]\\chi = 0 ", "43d94ee65d463253b3dae3af42dd0856": "v(t)=v(t_0)+\\int_{t_0}^{t}i(\\tau)d\\tau,", "43d95372a661beb0370246597ae09617": " \\mathbb P ", "43d9866e907b58c2946d9fe985cb73e0": "h \\circ f", "43d9edab1fcd25af93f7619fe541538c": "{\\sum P}", "43daa669f6a1e0b879d9f9d758d5ec91": "(a+b\\omega) \\cdot (c+d\\omega)=(ac-bd)+(bc+ad-bd)\\omega. \\,\\!", "43dad3858ccfac26e627e1cc30683d6b": "T_A = T_B \\,, T_B=T_C \\Rightarrow T_A=T_C\\,\\!", "43db1d1afe0373d27d74452d1bd8a922": "\\mathtt{function}\\,(x)\\ \\mathtt{return}\\ e\\ \\mathtt{end}", "43db47f1031813fefeca8b708ba7db4a": "\\scriptstyle\\nu", "43db5c6fa607e3f682071b5b83af9ede": "C^o = \\left \\{y\\in X^*: \\langle y , x \\rangle \\leq 0 \\quad \\forall x\\in C \\right \\}.", "43db7482213b1d1d617d2c37e613a8d7": "L_{ab} = L_{ba} = L_{ac} = L_{cb} = L_{bc} = L_{ca} = - \\frac{1}{2} L_{mr}", "43db7573040f75a7e52420950ba4dfbd": "\\,^{249}_{98}\\mathrm{Cf} + \\,^{50}_{22}\\mathrm{Ti} \\to \\,^{299}_{120}\\mathrm{Ubn} ^{*} ", "43dbed25e20dd72015b6dca239d9f782": "\\begin{matrix}P_{\\alpha\\rightarrow\\beta}=\\delta_{\\alpha\\beta} & - & 4{\\displaystyle \\sum_{i>j}{\\rm Re}(U_{\\alpha i}^{*}U_{\\beta i}U_{\\alpha j}U_{\\beta j}^{*}})\\sin^{2}(\\frac{\\Delta m_{ij}^{2}L}{4E})\\\\ & + & {\\displaystyle 2\\sum_{i>j}{\\rm Im}(U_{\\alpha i}^{*}U_{\\beta i}U_{\\alpha j}U_{\\beta j}^{*})\\sin(}\\frac{\\Delta m_{ij}^{2}L}{2E}),\\end{matrix}", "43dbee3e67599a46e790a9efb92bbd93": "\\sin x = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots .", "43dc1826f85efb35668741d61b7b56fa": "\\mu^{op} = \\mu \\circ \\sigma_{H,H}", "43dc69908f079b9e4efa55829b9b98f6": "\\operatorname{tr} (\\gamma^\\mu) = 0 ", "43dcafab76a1dcf62930ebde3d2ae3b6": "\\chi_G(\\lambda)", "43dd4cfcf50667c971847f39ddad5c47": " (x^2 - ny^2)^2 = (-1)^2 \\, ", "43ddc4d4624db695ae2d952858a98ec6": "L(n_v' )", "43ddf92d03834b706dbdca7622033a47": "\\log X_j", "43ddff50195690337dccab030f1b0427": "C,C',\\dots", "43de08c3a5fbe2ec0c485224fbe5d505": "0 \\le \\varphi \\le \\mathbf{1}_V \\ \\text{and} \\ \\int_S |\\mathbf{1}_A - \\varphi| \\, \\mathrm{d}\\mu < \\varepsilon", "43de24cc0f34764d6caa79865b45d8e3": "\\operatorname{E}[\\mathrm{d}N(t)]=\\mathrm{d}t\\operatorname{Tr}[c^\\dagger c\\rho_I(t)]\\,,", "43de94a1c10eaaed877b3c3ebc8513d6": "S \\subseteq X", "43dec338db8d64917494b9f6ef5f323b": "{\\mathcal O}_K", "43deca923b896ba7c192b619b22b3f32": "\\ell = 2 a ", "43df3e95f577391cb2e2015f9c102ee4": "log(500)-1=1.69", "43e020c6c9c9e911704cb12e2577fbd0": "a_i + a_j", "43e1488444499b1c5191465181f5056a": "a+bi = \\varepsilon\\left(m + ni \\right)^2, \\quad \\varepsilon\\in\\{\\pm 1, \\pm i\\}.", "43e19677c8003f21bc5d7925bd48d588": "\\vdash \\Box\\Psi \\rightarrow \\Box(\\Box \\Psi \\rightarrow P)", "43e1fa0f3ece7632c5b4db6fd58383e8": "M \\cap B", "43e20d0bfa2c44979da76db8f4fcd7c5": " (x,y,z) = (4 \\cos \\vartheta, -3+5 \\sin \\vartheta, 3 \\cos \\vartheta) . \\,\\!", "43e221fe1d6202a8d2b544016afbea1b": "lastBlock \\leftarrow 0", "43e2caa0289134cf69b643bc5a3abb81": " \\left( \\lim_{t \\to \\infty} T^t p \\right)_i = s \\cdot p ", "43e3031664860f781612883beeae1f1a": "N + 1 ", "43e308f158154a4f67a36d90a3361e41": "\\beth_{\\omega}", "43e3807f264b1a83eef55f06970d80d1": "w:k\\to j", "43e3819b37fbf87d85d481b78a831155": "\\bold x", "43e38b64d01930195475b53455ca43c3": "A(x)={\\left(\\frac{x_{n+2}\\,x_n-(x_{n+1})^2}{x_{n+2}-2\\,x_{n+1}+x_n}\\right)}_{n\\in\\Z^*},", "43e3e60689663da94aeb47aa435c5d8f": " M_t^{\\tau_k} \\to M_t ", "43e4422cd7fa7b662ef4894ce248dc25": "F(x,\\xi)", "43e453ac5f94dd3f9787100df1ddf06e": " \\lim_{x\\uparrow a}\\, f(x)", "43e46aa49b2cd009b11ecf8f12e75ddd": "\\sqrt{hPkA_c}\\theta_b\\frac{\\sinh {mL} + (h/mk) \\cosh {mL}}{\\cosh {mL} + (h/mk) \\sinh {mL}}", "43e46c529498ce45ebe6bc094c53dd76": "\\sqrt[13]{3}", "43e5064b24ef8e5aa92df09a5ed73295": "\n\\begin{align}\ng_{\\mu \\nu} & = [S1] \\times \\operatorname{diag}(-1,+1,+1,+1) \\\\[6pt]\n{R^\\mu}_{a \\beta \\gamma} & = [S2] \\times (\\Gamma^\\mu_{a \\gamma,\\beta}-\\Gamma^\\mu_{a \\beta,\\gamma}+\\Gamma^\\mu_{\\sigma \\beta}\\Gamma^\\sigma_{\\gamma a}-\\Gamma^\\mu_{\\sigma \\gamma}\\Gamma^\\sigma_{\\beta a}) \\\\[6pt]\nG_{\\mu \\nu} & = [S3] \\times {8 \\pi G \\over c^4} T_{\\mu \\nu}\n\\end{align}\n", "43e53dca883cead3ef119d6591ea2567": "I-B=(I-B)(I-B)'", "43e559228859208794c0175852f410a2": "E_2^{pq} = ({\\rm R}^p G \\circ{\\rm R}^q F)(A) \\Longrightarrow {\\rm R}^{p+q} (G\\circ F)(A).", "43e580ff516f923faf938841e692a017": "|x|_B := B(x,x)", "43e583dbbb8f260e5a1b77b5437f3860": "\n\\lambda_j = \\begin{cases}\n-\\frac{4}{h^2} \\sin(\\frac{\\pi (j-1))}{2n})^2 & \\mbox{ if j is odd.}\\\\\n-\\frac{4}{h^2} \\sin(\\frac{\\pi j}{2n})^2 & \\mbox{ if j is even.}\n\\end{cases}\n", "43e612802c5d12dba67e8b6df7659335": "\\lim_{n\\to\\infty} g_n\\xi\\to\\infty", "43e6374391aba4215a1ed4e114563ecf": "\nP(v)=\\sum^{p}_{k=0}A_k(v,\\dots,v)\n", "43e69c83a5f28c45240806f8c87a3b18": " \\angle CAB + \\angle CBA < 2 \\text{ right angles} \\Rightarrow \\angle aAB + \\angle bBA > 2 \\text{ right angles} ", "43e6e985a249808123f7e6c44b3ab91c": "\\mathbf{\\dot A} = \\boldsymbol{\\hat \\rho} (\\dot A_\\rho - A_\\theta \\dot\\theta - A_\\phi \\dot\\phi \\sin\\theta)\n + \\boldsymbol{\\hat\\theta} (\\dot A_\\theta + A_\\rho \\dot\\theta - A_\\phi \\dot\\phi \\cos\\theta)\n + \\boldsymbol{\\hat\\phi} (\\dot A_\\phi + A_\\rho \\dot\\phi \\sin\\theta + A_\\theta \\dot\\phi \\cos\\theta)", "43e7756ac074612e90b534e00df04a2f": "P(\\Delta,\\Omega_{\\perp})", "43e795a4d68c1cad4139040487876bb7": "0=f(x)^2+f(-x)^2\\,", "43e7f8299cae86f27603ed246e7ee914": "\\|\\check{A}^k\\|", "43e8034a309bd110ec95aca46fa64ce4": "\\scriptstyle l_0", "43e9d9d4c6d933e3eedec1e3911b164f": "\\omega_1 = \\omega_2", "43ea1aa03905e76d87e12c2e5994269f": "L_{[\\omega]}^i", "43ea2cec8df0998403ce0830ba7f7262": "{d}=[d_1, d_2, \\dots , d_D]", "43ea615fcb8492536f2868099be1140a": "G_{IC}", "43ea788837bf9c84d4545e715a5ad511": " N = 1 + \\prod_{p\\in S} p. ", "43ea832c9b930d9a8b77d5c9e27b544b": "\\kappa_0\\subset\\kappa_1\\subset\\kappa_2\\dots", "43eac3b6082eaa4ef98eaeede2d5c407": "v(t) \\rightarrow 0", "43eae44c5cfcdda3a1a7f76def7b9621": " f_X(x|\\boldsymbol \\theta) = h(x) \\exp\\left(\\sum_{i=1}^s \\eta_i({\\boldsymbol \\theta}) T_i(x) - A({\\boldsymbol \\theta}) \\right)", "43eb2446552a0b03074bd8d6dfd3f074": "\\mathcal{M} := (K^{-1} A)^T", "43eb71a23a70323bd35b02abaaec675b": "2 \\leq n < \\aleph_0", "43eb73c9443a853a4578423090846b55": "\\phi_x = -\\phi_y = \\pi/4", "43eb7c693df2c0eb3cfbc9b57fc237f8": " {2n \\choose n - 4} ", "43ec2541a824085637bb7f78845e5ea2": " f_{\\#}\\left(\\partial \\beta \\right) = \\partial f_{\\#}\\left(\\beta \\right)", "43ec3e5dee6e706af7766fffea512721": "=", "43ec6d4fba30529f8fe869a064b3f043": "f(\\beta, h)=-\\lim_{L\\to \\infty} \\frac{1}{\\beta L} \\ln (Z(\\beta))=-\\frac{1}{\\beta} \\ln\\left(e^{\\beta J} \\cosh \\beta h+\\sqrt{e^{2\\beta J}(\\sinh\\beta h)^2+e^{-2\\beta J}}\\right)\n", "43ecc3105dede104933eb6c04529d588": "\\frac{d\\psi_2}{dx}(L/2) = \\frac{d\\psi_3}{dx}(L/2) \\,\\!", "43ece7b9e366b74972613daf53d430a3": "d\\nu(\\sqrt n x)", "43ed2a17a081c4bb1fb1379f99c2aaef": "\\Delta_2(e_i) = k_i^{-1} \\otimes e_i + e_i \\otimes 1", "43ed875c0d58b311c250dfbdfe9e7dbd": "O\\left( 2^{n_1} \\sum_{i>1} n_i\\right).", "43edca8502a2ca85a089b99c3a23d531": "P(T) = T^3+a_{2,1}T^2+a_{4,2}T+a_{6,3}.\\ ", "43ee5068fa95675ccff9fb1656e18b3d": "\\mathrm{adj}(\\mathbf{A}) = \\det(\\mathbf{A}) \\mathbf{A}^{-1} \\,", "43ee68042afd02287254c4cba486630a": " \\ell(ab)=\\ell(a)+\\ell(b),\\quad \\ell(a)\\ell(1-a)=0. \\, ", "43ee680acb462f441214a2776b11646a": "\\begin{align}\n\\left | \\frac{1}{2 \\pi i} \\oint_C \\frac{f(z)}{z-a} \\,dz - f(a) \\right |\n&= \\left | \\frac{1}{2 \\pi i} \\oint_C \\frac{f(z)-f(a)}{z-a} \\,dz \\right |\\\\[.5em]\n&= \\left | \\frac{1}{2\\pi i}\\int_0^{2\\pi}\\left(\\frac{f(z(t))-f(a)}{\\varepsilon\\cdot e^{i\\cdot t}}\\cdot\\varepsilon\\cdot e^{t\\cdot i}i\\right )\\,dt\\right |\\\\\n&\\leq \\frac{1}{2 \\pi} \\int_0^{2\\pi} \\frac{ |f(z(t)) - f(a)| } {\\varepsilon} \\,\\varepsilon\\,dt\\\\[.5em]\n&\\leq \\max_{|z-a|=\\varepsilon}|f(z) - f(a)|\n\\xrightarrow[\\varepsilon\\to 0]{} 0.\n\\end{align}", "43ef38f8181bc0f0e67fc889e201e1e9": "i\\mathfrak{m}", "43ef3fb43063beb36ced447d1ddce03c": "\\chi(D) = \\chi(0) +\\tfrac{1}{2} D . (D - K) \\,", "43ef5f813379c23655e63c9cd7040af5": "S_\\lambda= \\sum_\\mu K_{\\lambda\\mu}m_\\mu.\\ ", "43ef9994543a05a02eff254c5c42ec64": "\n\\sin 2\\theta_{\\mathrm{eq}} \\ \\stackrel{\\mathrm{def}}{=}\\ \\left( \\frac{2}{f_{0}} \\right) \\epsilon\n", "43efa26c93db8e1d36aecb8ca144ff11": "f(x) = \\sum_{n=-\\infty}^{\\infty} F_n \\,e^{i \\omega_n x}.", "43f04ca8f1623bd18a3133b6f89702d9": "v_1+v_2", "43f057132082d4dc125b7aae8e2b9e39": "x_i=x^{*}_{i}- \\nu_i", "43f05e5751f717c70e9693609e9fcbc5": "{r \\over a}.", "43f06480a7c32f918e736c6bd8272d08": "\\frac{m(m+r-18)!r!}{(r-17)!(m+r-1)!}.", "43f0d88d3a0c4abeffbfe5b6006187f4": " {\\pi\\over 4} = 5 \\arctan \\left({1\\over 7}\\right) + 2 \\arctan \\left({3\\over 79}\\right) \\; ,", "43f0f6eed598e2146122a17a92687147": "\\mathbf E=\\frac{1}{2}(\\mathbf C - \\mathbf I)\\qquad \\text{or} \\qquad E_{KL}=\\frac{1}{2}\\left( \\frac{\\partial x_j}{\\partial X_K}\\frac{\\partial x_j}{\\partial X_L}-\\delta_{KL}\\right)\\,\\!", "43f10d9b491d798ea2510f4a95d74c47": "y=e^{-3x}\\left(2/3 e^{3x} + \\kappa\\right). \\,", "43f14311cb6919595c7795f6055c2493": "X_u \\perp\\!\\!\\!\\perp X_v \\, | \\, X_Z", "43f153264e57f495c19d0987713a96d7": "L \\cup P", "43f1ffa81814da2fa6c0d4ca4ad42690": "\n \\ \\ a_2\n", "43f2a13f8e007b9a59400c9acef6768d": " P(X_1^n) ", "43f3013f9549476f07705065905423ba": "H(a,b) < G(a,b)", "43f36150b9f6dbda30b31f86c25b2504": "r(\\phi)\\,", "43f3c811d2a149410ccecd79b8611e4c": "f(q) = a q + b, \\quad a, b \\in \\mathbb{H}. ", "43f3e4403d829c4178451a86ceda1f68": "x_1 = -0.504083008\\ldots, x_2= -1.573498473\\ldots, x_3= -2.610720868\\ldots, x_4= -3.635293366\\ldots, \\ldots", "43f3e9823e0ff6481ae83e7f4f148a85": " \\ n ", "43f44afd878f18bd88eb9404b2411c7c": "\\epsilon_\\mathrm{thermal} = \\alpha_L \\Delta T", "43f47709ce8d20dd2421821faaa635bd": "\\varepsilon_{r} \\le 25", "43f49f8773ce2ce7f9c20d9fabb26cf6": "1/\\mu = 1/{m_e^*} + 1/{m_h^*}", "43f4d27518824a0f1929736f143a6bae": "\\tfrac{c}{f}", "43f5174349aaa3f1e61b844563835044": "R_n=K[x_0, \\ldots, x_n]", "43f51beb6b4d88d9149dc90c789c6f06": "\\theta_r ", "43f52e28dd57ee8fe2cbf8a94b4a5c8c": "A + C = 0 ", "43f55da32e9adede800d19fb2e9f3809": "\\tfrac{1}{2}(\\tfrac{1}{2} + 1) = \\tfrac{3}{4}", "43f58ad80feaf61bdc1add06d54b0f93": "\\overrightarrow{l}", "43f60ee026772343ec3174ff28be2766": " y\\ f = x ", "43f637079bb8426fd72b46bf0854dd83": "{{{\\hat{\\mathbf{K}}}}_{i}}=\\left( \\begin{matrix}\n {{K}_{xx}} & {{K}_{xy}} & {{K}_{xz}} \\\\\n {{K}_{yx}} & {{K}_{yy}} & {{K}_{yz}} \\\\\n {{K}_{zx}} & {{K}_{zy}} & {{K}_{zz}} \\\\\n\\end{matrix} \\right)", "43f68a3b14cad7fc8e9f51efbf05ab72": "f\\colon A\\to B", "43f68d9dc3b1c0ab44642e0a25d2a859": "R_{\\hat{m}\\hat{n}\\hat{i}\\hat{j}}", "43f690f575298976066f4e0c1a7678f0": "1 \\le i \\le k-1", "43f69194896fd5ae9adab026e00e7864": "H =-p_3=-\\sqrt{n^2-p_1^2-p_2^2}", "43f696210389bda4bb644d26b408d6d2": "t:G\\to M: (p,q)\\mapsto p", "43f6cd7ef65a5e1a95c15ec7a371c7e2": "\\gamma = \\frac{C_p}{C_V} = -\\frac{dp/p}{dV/V}\\,\\!", "43f6eda70886ce181c16200cf50b49a8": " \\theta = \\tan^{-1}{\\left(\\frac{v^2\\pm\\sqrt{v^4-g(gx^2+2yv^2)}}{gx}\\right)} ", "43f74701809acb96e142c655ceeee89b": "\\sigma_a(\\zeta_p) = \\zeta_p^a", "43f7dbef1f138a7c0f72eda469ce46d3": "\\mathcal{F}f\\in L^1(\\mathbb{R}^n)", "43f8267390c51d59d7fdc4d5ae55d7bf": "\\forall z [ z \\in x \\Leftrightarrow z \\in y] \\land \\forall w [x \\in w \\Leftrightarrow y \\in w].", "43f82eea93bd745de5d8edc06c9d58b9": "N\\triangleleft G", "43f84c89ddc17c61083b9f807e6480b3": "\\rho(\\boldsymbol{r'})", "43f86ce3e8f129afce8d82f8418a7e7a": " \\delta/2\\pi = n/2 \\,\\!", "43f882224a4acebfcb2f8590094c03df": "\\lnot F[\\text{true}] \\land G[\\text{false}]", "43f91c29dfa7d0f5568b2a40a884ddb4": "\\mathbf{T}(s+\\Delta s)-\\mathbf{T}(s)+\\mathbf{G}\\Delta s \\approx \\mathbf{0}.\\,", "43f9386f074abf9896865e479624b64e": "X_{i,1},\\dots,X_{i,p}", "43f93d6ea1df4efb546b902ad4ec8a24": "\\gamma = \\alpha + c \\beta\\ ", "43f9790d274288c195a8a00f94334ed8": "X_{t_{i}}", "43fa00a168f7e58d65b9cf5296149efe": "\\|x\\|_1 \\leq \\sqrt{n}\\|x\\|_2", "43fa1e80d123bbaf13c58eeed930ec3e": "\\sqrt{x^2 + \\epsilon^2}", "43faafd025aa631c7ac577059ec70b55": "C(y) \\rightarrow B(y,z)", "43fb0140f94c60076a99efcbb805e6a3": "\\textit{hate}", "43fb6168a28315288155f053dc9f8fb2": "\\mathfrak{so}(p, q).", "43fb67cf02f31437eb9bc8568bdcb12a": "x\\in[0,1)", "43fb829fc35ea5404ed4b7463b64922d": "E[\\varepsilon|X]=0", "43fbb67cd1b266e8b33bcbfb43175844": "\\frac{en}{e} = \\frac{n}{1}", "43fbce84634c2163b24dae2bc40a4237": "\\!m_\\mathrm{e}", "43fbcfd790d5b674599d76bd2ba9f386": "\\frac{E_s}{N_0} =\\frac{E_b}{N_0}\\log_2 M ", "43fc394a8513b6a249d93e225ebb910e": "u^Tv=\\langle u,v\\rangle", "43fccda7173d689045fdb334f85c38e4": "\\scriptstyle (n-2,2)", "43fcfb26919f1ba186e028f73f5e5be7": "M M^* = I", "43fd02b3cadab31fe21106daecf9adef": "\\tau = r_m c_m \\ ", "43fd1c8dd2410c452fb0ce8aa4278ac5": "C\\ell_{1,3}(\\mathbb{R})", "43fd37fdc62ce3a7e68ae5e6154cda67": " X = \\frac{1}{\\beta}\\ln \\left(\\frac{1-p}{1-p^U}\\right).", "43fdf55f19ace6e199dcba71e40cf30e": " \\nu_L = \\mu_L/\\rho_L ", "43feb824f6d33606db323860c322eaae": "\\mathrm{efficiency}(X) = -\\sum_{i=1}^n \\frac{p(x_i) \\log_b (p(x_i))}{\\log_b (n)}", "43febe275630d169f218c9d195a17326": "P\\cdot R\\cdot [1-T]/n", "43feca9f46fb750f596fe0304233579d": "X_i^s", "43fed6f2df44fce230dcd06956bb4ff7": "\\mathop{\\rm el}(F)", "43ff60b06ec708d30c5d840793994d8f": "\\begin{align}\\operatorname{MSE}(S^2_{n-1})&= \\frac{1}{n} \\left(\\mu_4-\\frac{n-3}{n-1}\\sigma^4\\right) \\\\\n&=\\frac{1}{n} \\left(\\gamma_2+\\frac{2n}{n-1}\\right)\\sigma^4,\\end{align}", "43ff7451f075c236bcd319ddab50c1e5": "\\mu^\\circ_{solid} = \\mu^\\circ_{solution}\\,", "43ff8daa1aff9dc882efaee1033db3a0": "I(\\mathbf{y};\\mathbf{s})", "440029808aef9ffb7ad57a08ff91d386": "\\mathcal{S}_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int \\mathbf{p} \\cdot d\\mathbf{q}", "44013eb5cafa598c2a5894d86e75f73b": "g = g_n = (g_n g_{n-1}^{-1})(g_{n-1}g_{n-2}^{-1})\\cdots(g_{2}g_{1}^{-1})(g_{1}g_{0}), \\qquad \\Pi(g) \\equiv \\Pi_U(g_{n}g_{n-1}^{-1})\\Pi_U(g_{n-1}g_{n-2}^{-1})\\cdots\\Pi_U(g_{2}g_{1}^{-1})\\Pi_U(g_{1}g_{0}), \\quad g_0 = 1", "440146378809f5f21e4a7bd23a9cbc0b": "{A}_{9}^{(1)}", "44018380b04e05eddce754fb634afb61": "M(x)=\\exp{\\int a(x)\\,dx}", "44018b3463c704de0d7e83514b74e302": "\n\\bar y=\\frac{1}{T}\\int^T_0f(t)\\,dt.\n", "4401afd1bb84dbcc0183f8b2f52dce48": "\\mathbb{R}^2", "4401ba386c6f171c1cded171195b8ea9": "b_i = 0\\,", "4401c6afe87ae8df6be3e1deb6512738": "f (x) < f (x')", "4401ce57e7e4e8a3fcc29976ecd0adb0": "A \\sub \\mathbb{R}", "4401e4c0e13315db8e8b9a2431b68a80": "A-E[A]\\in T_p^{(e)}", "44021454b43ecd143998e35f93640b67": "cos({\\alpha})=1", "44022d2cc9efc4fb6fb7dabc28d647da": "\\frac{\\partial \\mathbf{U}}{\\partial x} + \\frac{\\partial \\mathbf{V}}{\\partial x}", "44023be4170d31a5c15e92d95de9d31e": "I_R", "44025c27ced2d1622cbab1115b2ca507": "\n\\begin{array}{l}\nr_0=a\\\\\nr_1=b\\\\\n\\ldots\\\\\nr_{i+1}=r_{i-1}-q_i r_i \\quad \\text {and} \\quad 0\\le r_{i+1} < |r_i|\\\\\n\\ldots\n\\end{array}\n", "44026c57d14cec148cb8b5ef07dcc910": " (\\kappa+2)~r^2~\\sin\\theta \\,", "4403007ea989b88a20a6d90bad013b8d": "s_y", "44030b0dd8fc61f140ba4d165e7db22e": "(\\hat{\\bold{e}}_x, \\hat{\\bold{e}}_y, \\hat{\\bold{e}}_z)", "440378a15f9d87d535552b924269b19a": "\\ldots d_5d_3d_1.d_{-1}d_{-3}\\ldots", "44043bab1214acc0b4c024031b600239": " \\hat{H} = \\sum_{n=1}^N \\hat{T}_n + V ", "44045a38b85601caa4edf06f361c4129": "A_1 \\to A_2 \\to ... \\to A_n \\to A_1", "440482f5a4549c5a5175fb807e5cded2": "r_3 = (Y \\to S, \\emptyset, \\{X\\})", "4404ef57b107f13ac22b8ce77c1502e4": "\\begin{matrix}\n\\frac{1}{15}{\\pi}{\\sqrt{25-10\\sqrt{5}}}+\\frac{2}{3}\\ln(5) \\\\ \n+\\frac{{1}+\\sqrt{5}}{3}\\ln\\left(\\frac{1}{2}\\sqrt{10-2\\sqrt{5}}\\right) \\\\ \n+\\frac{{1}-\\sqrt{5}}{3}\\ln\\left(\\frac{1}{2}\\sqrt{10+2\\sqrt{5}}\\right)\n\\end{matrix}", "44051107e0a478ca72bcf775b6e2d8b1": "\\int\\frac{dx}{x^2 - a^2} = \\frac{1}{2a}\\ln\\left|\\frac{x-a}{x+a}\\right| +C \\quad\\text{and}\\quad\n\\int\\frac{dx}{x^2 + a^2} = \\frac{1}{a}\\arctan\\left(\\frac{x}{a}\\right) +C.", "44051db81b897beebc78660823e73548": " \\prod_{i=1}^m Q(\\beta_i z) \\ ", "440557c28fcbc6bee1c9c4bfc065436e": "\\sum_{ab}W_{ab}(t)\\mu_{ab}(t) = 12", "440558a8daaa6919cafbf2f2b65ee7f3": " \\quad 2", "4405a717fba7da0256a33327b7bb9164": "\\sim_S \\;= \\{(s,t)\\in M\\times M \\,\\vert\\; S \\ / \\ s = S \\ / \\ t \\}.", "4405e244441357aad053aade4a9f02e1": "X^n = X \\times \\ldots \\times X", "4406031ad950dcd31c4193db40c50582": "n! \\sim \\sqrt{2 \\pi n}\\left(\\frac{n}{e}\\right)^n", "4406636306394cf59b2c5b3a835daa7a": "n=n\\left(x_1,x_2,x_3\\right) \\ ", "440684e04a91ea7b622e0bf8140d6759": " F(r) \\le 1 ", "4406889885935b038c991f6787f3c264": "\\mathbf{q} = (q_1,q_2,\\cdots q_N) ", "44068fc5041f8ce6e1f8b571a876bc46": " \\Lambda^\\alpha {}_\\beta \\Mu^\\beta {}_\\gamma = \\Nu^\\alpha {}_\\gamma ", "4406cd4453a50e7339de7a51d9f152bc": "m(A) = \\sum_{B \\mid B \\subseteq A} (-1)^{|A-B|}\\operatorname{bel}(B) \\, ", "4406e9e77174626e08b94586746f11eb": "\\displaystyle q = e^{2\\pi i \\tau},\\quad a = e^{2\\pi i u},\\quad b = e^{2\\pi i v}", "4406fd1150c8d7e6af928571b168f4ea": "X = \\{X_n\\}_{n \\in \\mathbb{N}}", "440703e7c5acb5fe81dee6b7982aabbc": "\\Theta \\subset \\mathbb{F}", "4407415124d08d5c5d76ea40efa9068e": "\\tfrac{j}{i+j} < \\rho", "44077df2d98881eb89886ad6b0d0334d": "M\\times D", "440781baa85477ba39087962f2b0bd96": "f = \\left( \\frac{1 + \\frac{v_\\text{r}} {c}} {1 + \\frac{v_\\text{s}} {c}} \\right) f_0 = \\left( 1 + \\frac{v_\\text{r}}{c} \\right) \\left( \\frac{1}{1 + \\frac{v_\\text{s}} {c}} \\right) f_0 \\,", "4407876f3bb9d64bf4af7b9e0ca66584": "T(F_\\theta) = \\theta \\, . ", "4407aa424b0fce1d1de3c4a77f621fb6": "n_C", "440813791894738695ac033a7fd1f783": "\\mathbf{F} = m\\mathbf{A} = \\left(\\gamma {\\mathbf{f}\\cdot\\mathbf{u} \\over c},\\gamma{\\mathbf f}\\right)", "4408979bfb805880dfc50174a4a53618": "0 = \\partial_\\mu j_{d+1}^\\mu", "4408b1f9402a626c056204d19209ab67": "d_c=\\sqrt \\frac{V}{100\\pi\\, RT_{60}} \\approx \\sqrt \\frac{A}{50} \\,", "4408b4fc42fe1ad03bd331faa4450ae5": "G = \\langle x,y \\rangle\\,", "4408ba6b44bba64077e9982d61838203": "\\begin{array}{lcl}\n\\frac{T_2}{T_1} &= & \\bigg( \\frac{1+\\frac{\\gamma -1}{2}M_1^2}{1+\\frac{\\gamma -1}{2}M_2^2} \\bigg) \\\\\n\\frac{p_2}{p_1} &= & \\bigg( \\frac{1+\\frac{\\gamma -1}{2}M_1^2}{1+\\frac{\\gamma -1}{2}M_2^2} \\bigg)^{\\gamma/(\\gamma-1)} \\\\\n\\frac{\\rho_2}{\\rho_1} &= &\\bigg( \\frac{1+\\frac{\\gamma -1}{2}M_1^2}{1+\\frac{\\gamma -1}{2}M_2^2} \\bigg)^{1/(\\gamma-1)}.\n\\end{array}", "440929dbae04ffaf035dabb9b59ef47c": "\\|x\\| = \\sqrt{\\langle x,x \\rangle},", "4409310c6f9780b70f086e9f14621ee4": "E=J(T,n)", "4409317080d1ad3c26c931e02b58f264": "M'=p.(\\overline{c}M'+\\overline{t}M')", "44094b22b4e37c8166c0a55e3df31575": "\\bigg\\{ \\Pr(h_1 \\And h_2) \\bigg\\}", "44096458d16d97fd79c7f7921866ea08": "U_p", "440967987e62ccd5dc3a7d834800d5ab": "\\nabla f = \\frac{\\partial f}{\\partial x_1}\\widehat{x}_1+\\frac{\\partial f}{\\partial x_2}\\widehat{x}_2+\\frac{\\partial f}{\\partial x_3}\\widehat{x}_3", "4409a9f88213f644d13e0d6136fa02f0": "(\\hat{x}, \\hat{y}; \\hat{t}) = \\operatorname{argmaxlocal}_{(x, y; t)}(\\operatorname{det} H L(x, y; t))", "4409ac38a6bcf5fb037b281ae4387f1c": "\\int_{x_0}^{\\max_x \\in X} f_{\\theta_1}(x_1) f_{\\theta_0}(x_0) \\, dx_1", "4409ee7ebd470553f0c3c20013adb3ec": "\\lambda = 2\\pi/k", "440a09e9e88bd8eab05f8430445a410c": "=\\oint _{ C }^{ }{ \\varphi \\left( -\\dot { y } dx+\\dot { x } dy \\right) }. ", "440a4fdc83990d54dfe0815ae59909e7": "[ \\tfrac{mg}{Ld}]", "440a77b4bfa27f1b75fbe38e25453ef0": "\\operatorname{Arg}\\left(z^n\\right) \\equiv n \\operatorname{Arg}(z) \\pmod {(-\\pi,\\pi]} .", "440a7c246147a4ec5977c38b5f29e166": "H(P)\\leq \\log_2 n ", "440a8c60e3595683365d8182ecad32ca": "\n\tT_d(\\theta,\\phi) = T_d([hkl])\n", "440afc29e8cafe18ff8102cf45f21b0d": "\\epsilon = \\frac{1}{2} - 2\\delta", "440b4a066e86d8f46acb855a175d8357": "(\\forall x, P \\Rightarrow wlp(S,Q)) \\ \\Leftrightarrow\\ (\\forall x, sp(S,P) \\Rightarrow Q)", "440ba9d5957efd6dc9bf0c1cfe03a022": "\\bar R = \\frac {\\sum_{i=1}^m \\left ( R_{max} - R_{min} \\right )}{m}", "440bb0ebabb6fed0508ad1e300e65301": "T=2\\pi\\sqrt{a^3\\over{\\mu}}", "440bcb2225cd249b09bb29454f83249d": "x=4", "440c0c8456263d3c9e04d531ec33fbeb": "\n\\partial A_\\mu / \\partial x_\\mu = 0\n", "440c145190b4d74a8da992e206c872d7": "x = \\left( a + \\sqrt{a^2-n} \\right)^{(p+1)/2} = \\left( 2 + \\sqrt{-6}\\right)^7 .", "440c3bfa2721780faaabc7b99e6408d0": "F_{1}=F_{load}\\frac{Sin(\\beta )}{Sin(\\alpha+\\beta)} \\,", "440c3f39c6973f9cb4452bf9cf7ae3d2": "D^{\\pm}(\\mathcal{S})", "440c53daacc0a8c02a444ca148f87cc0": "\\partial u_k / \\partial X_l", "440c5b1400d4578fcd0e4415272deb77": "\\boldsymbol{F} = \\boldsymbol{R}", "440c5fc76ff8291b27525cc2475255ba": "\\vec a_g = - \\hat r ~ G ~ \\frac{M}{(R \\pm \\Delta r)^2}", "440c719ccc55b3339619d2c6e16b4381": "P^*r", "440cb92a476a1c57fc28105634f3a49b": "\\nabla d = e", "440cbb8266396b353b45f30cba96f9d9": "y = Pe^{it}", "440d94ad67cc75b022f749eda784ae86": " \\text{subject to }\\ ", "440dafd915aa1501c84c74a7b2d24ce2": " f(z) = u(z) + i \\cdot v(z) ", "440de441916a938a420e78e5c5b9e15b": "bc-ad=1", "440e21968b5535ee98c8e9a5d131a4cd": "\\left(\\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ -2\\sqrt{2/5},\\ 0,\\ 0,\\ 0\\right)", "440e4bdc83d0e9f015938c266290f566": "-0.05916h/n", "440e5100084134161eac400876af21d5": "\\scriptstyle f\\in C^1(\\Omega)", "440e659d6c9028d7867794d23892422a": "\\dot{q}_m", "440e73d493e6eea4afd6dea6cee0cdad": "\\sqrt{\\pi n}-O(1)", "440e99a05247ca7a8cce224b2aa13e20": "(\\vec \\omega \\times)", "440ea3556e7912c560d95615401c36b3": "\\scriptstyle\\operatorname{Out}(F_n)", "440f2ad9af4d9c926dd351bd0c553e90": "(V,\\xi)", "440f85021978aa5176cde3b04d5e4679": "\nV(\\boldsymbol{y}) = -2k_e Q \n \\oint_\\ell\n \\ln \\vert \\boldsymbol{x} - \\boldsymbol{y} \\vert\n \\; dx .\n", "440f992366ccc4a9a7cc1bcba9a8215e": "\n \\begin{align}\n &\\sqrt{ gk\\, \\tanh\\left( kh \\right)}\\,\n \\\\[1.2ex]\n &=\\sqrt{\\frac{2\\pi g}{\\lambda}\\tanh\\left(\\frac{2\\pi h}{\\lambda}\\right)}\\,\n \\end{align}\n ", "440fc2ba44e05e7ee4da97d9695188f3": "Y_{4}^{-2}(\\theta,\\varphi)={3\\over 8}\\sqrt{5\\over 2\\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(7\\cos^{2}\\theta-1)\n= \\frac{3}{8} \\sqrt{\\frac{5}{2 \\pi}} \\cdot \\frac{(x - i y)^2 \\cdot (7 z^2 - r^2)}{r^4}", "440fe6f15b4da33fd4960feb7d8738cd": "\\frac{1}{r^2} P^0_1(\\sin\\theta) = \\frac{1}{r^2} \\sin\\theta", "441026d0ddab0fa615498a12ca165421": "\\scriptstyle x \\;=\\; (x_1,\\, \\dots,\\, x_n)", "441057eb6591e4e602818c8ad0be8327": "\\frac{\\Delta f^{*}}{f_f}=\\frac{N_S}{\\pi Z_q}\\frac{\\kappa _S^{*}}\\omega", "44106b2f78b539d367f6a266a16782c5": "y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \\cdots + a_my_{t-m} + b_px_{t-p} + \\cdots + b_qx_{t-q} + \\mathrm{residual}_t.", "44106c70f22005a813b2390cf01367f1": "\\displaystyle Wg(1^2,d) = \\frac{1}{d^2-1}", "44106e2be55a9907f6f72a976032088b": " P = K \\rho^{1 + \\frac{1}{n}}\\, ", "44107bff3c71ac8b4741a3167fc4cb3e": "l_1z=0", "4410aefd8afaa753739e31379e6eb1f0": " f \\ ", "4410bc7937576e27290cdacfce9a3de1": "S_z\\left|s,m_s\\right\\rangle=\\hbar m_s\\left|s,m_s\\right\\rangle.", "4410f972d0f4a359494ec820dadd3d68": "s(x) \\equiv \\sum_{c_i=1} \\frac{1}{x - L_i} \\mod g(x)", "44115ff4dcc1c9787a263b4f101050c8": "m = (3, -7)", "4411649e49fc97a45a4da0b73ee83f92": "\\begin{matrix} -\\tfrac {1}{3} \\le p_1 \\le 0, \\\\ \\ 0 \\le p_2 \\le \\tfrac{2}{3},\\\\ \\frac{2}{3} \\le p_3 \\le 1.\\end{matrix}", "4411742adaae0e545215c224692a00e0": "\\kappa=\\lim_{N\\rightarrow\\infty}\\left(Z_{N}\\right)^{1/N}.", "4411897e5dea0e4e35a3ce2e41e74fc5": " \\frac{1}{2}\\mathrm{O_2(g)} + 2\\mathrm{e'} + {V}^{\\bullet\\bullet}_o \\longrightarrow {O}^{\\times}_o ", "4411aff9fe2be278345a92ad7be929fa": "(gate7\\vee \\overline{gate2})\\wedge (gate7\\vee \\overline{gate4})\\wedge (gate2\\vee \\overline{gate7}\\vee gate4)", "4411d15c31aba1a3316d4a05790c046b": " \\det(\\mathbf{R}) = 1 ", "44122a0422f28ddb2b8dc2303297d5ca": "(X)_k", "441236bfecec9e83047231a066a4deb7": "\\delta(t-s)", "4412bd4f768cd74270ab62d0a73bc19e": "\\mathbf{\\hat{e}}_i \\cdot \\mathbf{\\hat{e}}_j = \\delta_{ij} ", "44136e598613d4a3886e865ddf8fd454": "Z' = i \\omega L' + \\frac{1}{i \\omega C'}", "4413f8837e07fc9ff2efb39a7dbceabc": " u_n =\\sum_{m=0}^{N-1} a_m \\cos\\left(\\frac{m\\pi}{N}(N+n+\\frac{1}{2}) \\right) ", "44140efec9e2f8eeeda92c0b92462760": "df(\\Omega)=0, \\, ", "44142e3f213d3cee1e372d8a5031819e": "R\\cong xR", "4414c05d0cd093879fab33e82923c130": "\nP = \\left\\langle \\psi_e' \\psi_v' \\psi_s' \\right| \\boldsymbol{\\mu} \\left| \\psi_e \\psi_v \\psi_s \\right\\rangle = \\int \\psi_e'^* \\psi_v'^* \\psi_s'^* (\\boldsymbol{\\mu}_e + \\boldsymbol{\\mu}_N) \\psi_e \\psi_v \\psi_s \\,d\\tau \n", "4414ec99ee71c92912e1193d4a474dce": " \\dot S_i =\\dot Q_L\\left(\\frac{1}{T_a}-\\frac{1}{T_L}\\right).", "441504cfbb247fa2cb8a660211937af9": "a_n(x)y^n+\\cdots+a_0(x)=0,", "441514afe0a32da1dcb498fca1c0978c": "\\frac{\\int_a^xg(t)dt}{F(b)-F(a)} ", "441577f8f46f01229ac1f9899af7403a": " \\tan \\gamma\\,_n = \\sinh (a+nb)", "44157ac061c3b30c8c041704f364586f": " \\left( \\nabla \\left( d \\phi\\right) \\right) _{ij} ^\\gamma= \\frac{\\partial ^2 \\phi ^\\gamma}{\\partial x^i \\partial x^j}- ^M \\Gamma ^k{}_{ij} \\frac{\\partial \\phi ^\\gamma}{\\partial x^k} + ^N \\Gamma ^{\\gamma}{}_{\\alpha \\beta} \\frac{\\partial \\phi ^\\alpha}{\\partial x^i}\\frac{\\partial \\phi ^\\beta}{\\partial x^j}.", "44158979c31f73645139ff034476da7c": " c = \\sqrt{a^2 + b^2}. \\,", "4415974a98e1bb30ed75d33242efcd63": "T \\not\\models c \\vee f", "4415a31d93c547370ae702bf569d4dce": "\\int\\frac{x^2}{R}\\;dx = \\frac{2ax-3b}{4a^2}R+\\frac{3b^2-4ac}{8a^2}\\int\\frac{dx}{R}", "4415a32a4c884a11eae10f958ffea826": " K^{\\pm} ", "4415a4b4ef0badf722d4c55b4c26f159": "(A\\land B)\\to A", "4415c02d7fe91b9601cdd75559e3e09d": "(ct)^2 - (x^2+y^2+z^2) = (ct')^2 - (x'^2+y'^2+z'^2) =0", "4415c9989ef0a80e5be980e0272dabd6": "T_d^l(\\theta,\\phi)", "4415caf8c223060b94bacc6ec1485403": "u(\\nu,T) = \\pi I(\\nu,T)", "4415fcc406cb5be63662be7af25ea110": "\\frac{dy}{dx} = \\frac{1}{2y}", "44163590957ce5b1de8bb6a0568cc309": "\\begin{pmatrix} 7 \\\\ 4 \\end{pmatrix}, \\begin{pmatrix} 11 \\\\ 15 \\end{pmatrix} \\to \\begin{pmatrix} H \\\\ E \\end{pmatrix}, \\begin{pmatrix} L \\\\ P \\end{pmatrix} \\to HELP", "44168bf7645e5280c9425d3bb015d269": " \\ell_B=\\sqrt{\\hbar/eB} ", "4416bee55a8106fe0eeb7c94b37873e9": "f(x) \\exp(-i\\beta_{in} x)", "4416ea4dd095ac109320cdd4e6c6b934": "\\scriptstyle \\tilde{X}(t)=X(t)-Y(t) ", "4416efa8078f1097913a0693b68b4055": " \\Pi(x) = \\sum_{n \\leq x} \\Lambda(n) \\int_n^x \\frac{dt}{t \\log^2 t} + \\frac{1}{\\log x} \\sum_{n \\leq x} \\Lambda(n) = \\int_2^x \\frac{\\psi(t)\\, dt}{t \\log^2 t} + \\frac{\\psi(x)}{\\log x}. ", "44170724e0784d103ee24ff6b9f6ef6b": "\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\cfrac{1}{h_1 h_2 h_3}~\\frac{\\partial }{\\partial q^i}(h_1 h_2 h_3~v^i)\n", "4417314ed42403c2cbf232d7049d92b2": " \\exp\\{-sT\\},", "44173e90ad1d497ed8c7b37e42bcb496": "\\,_1F_1(a;b-1;z)-\\,_1F_1(a+1;b;z) = \\frac{(a-b+1)z}{b(b-1)}\\,_1F_1(a+1;b+1;z)", "44178e55aceff3f9f63b45a99c894daf": "(\\nu x)0 \\equiv 0", "441813ae6dda24cda92c1aa519a6746f": "C_{\\tilde{Y}}", "4418256f004bc65d3293de96a4ed752d": "\\frac{d ^{2}i(t)}{dt^{2}} + \\frac{1}{LC} i(t) = 0.\\,", "44184116362c48ea8c70c4671e4d92f3": "\\left[ \\begin{matrix} B & AB \\end{matrix} \\right] = \\left[ \\begin{matrix} \\left[ \\begin{matrix} 0 \\\\ \\frac{1}{m} \\end{matrix} \\right] & \\left[ \\begin{matrix} 0 & 1 \\\\ -\\frac{k_2}{m} & -\\frac{k_1}{m} \\end{matrix} \\right] \\left[ \\begin{matrix} 0 \\\\ \\frac{1}{m} \\end{matrix} \\right] \\end{matrix} \\right] = \\left[ \\begin{matrix} 0 & \\frac{1}{m} \\\\ \\frac{1}{m} & -\\frac{k_1}{m^2} \\end{matrix} \\right]", "441842e42deb5ad16f9d7956e47fd10c": "m_{12} = \\sin(\\sqrt K s_{12})/\\sqrt K, \\quad\nM_{12} = \\cos(\\sqrt K s_{12}).", "4418bad4713b4d723c1b407c4f6a222d": "\\Delta M_J=0,\\pm 1", "4418f953c40031ce04d2284371c5b3e1": " c ' ", "441973e35294179b77468901118c7a84": "(0, +\\infty)", "441987f14eada31a8f8b8e9dc8e1e79d": "\\lim_{n\\to\\infty} c_n = L", "441992cc467b8ff1533963412c42208d": "\\| v \\|_{H} \\leq c \\| v \\|_{V};", "4419a84ba56522419b54e4c433f467b3": "2 \\div x", "4419f8aef3d22bcbb3a18e600246387e": " q_{solar} ", "441a53f90585fda0cbb6fecb36f3496c": "J_1^2", "441a6f33934d45b0a8077aadb6616004": "DP_{S}^{D}", "441a88b31bff5756ccfe59618586ddef": "\\frac{\\widehat{\\theta}-\\theta_0}{\\operatorname{se}(\\hat\\theta)}", "441a9c21da05f1ec7fdb937a93a2722b": "\\frac{(2V)^{1/4}}{\\sqrt{\\pi}}", "441aa59750388f84a787ceb45a3a80e0": "\\hat{W}^{I}(z,x)=E\\left[ \\hat{\\beta} (S,z)\\mid X_{1}=x\\right]", "441aa812ccfd975423905a975b211ed0": "T ", "441ad832103d117ef07d0af7bccbe175": " a^6 + b^6 = (a^2 + b^2)(a^4 - a^2 b^2 + b^4),\\,\\!", "441aea047b0a67471bbfa7a6523e610c": "x_n e^{i 2 \\pi n\\ell/N} \\,", "441afa2ce7d3102d2abd35ffe795ef32": "F^*_{i+\\frac{1}{2}} =\\frac{1}{2} \\left\\{\n\\left[ F \\left(u^R_{i + \\frac{1}{2}} \\right) + F \\left(u^L_{i + \\frac{1}{2}} \\right) \\right]\n- a_{i + \\frac{1}{2} } \\left[u^R_{i + \\frac{1}{2}} - u^L_{i + \\frac{1}{2}} \\right] \\right\\}. ", "441b06e79dd1159820cec3dd5be5f5f9": "s_i(\\theta_i) \\in \\arg\\max_{s'_i \\in S_i} \\sum_{\\theta_{-i}} \\ p(\\theta_{-i} | \\theta_i) \\ u_i\\left(s'_i, s_{-i}(\\theta_{-i}),\\theta_i \\right)", "441b0dc3c98c6120f140628caa61381c": "\\mbox{LOP1}=260", "441b186b00f14fff60efb8a62fab6000": "e^+e^+", "441b2dcd2a6edf47c9d078566ee01dff": "\\left( \\begin{matrix} x_1 \\\\ x_2 \\end{matrix} \\right) = \\left( \\begin{matrix} n\\pi \\\\ 0 \\end{matrix} \\right)", "441b37f35e280cd81a78e41ee688c52a": "(1 + i) = ((1 - i) - (1 - i)^2)", "441b8f61449dc62db20481c7235956b1": " \\rho^{(2)}= \\rho^{(1)}+\\phi (\\rho_\\mathrm{fluid}^{(2)} -\\rho_\\mathrm{fluid}^{(1)})", "441bc2e62e3bf4c1c84debf23b76386a": "\\forall N>\\lambda_\\varepsilon\\ ,\\left\\|\\sum\\limits_{n=1}^N a_n-A\\right\\| < \\frac{\\varepsilon}{2}", "441c46ff6d63c40571aa84b17003ed34": "[ t_0 .. t_1 ]", "441c508bf7750d2267bfceb3c7d22eef": "\\mathbb{E}\\left[g(X_1,\\dots,X_d)\\right]=\\int_{[0,1]^d}g(F_1^{-1}(u_1),\\dots,F_d^{-1}(u_d))c(u_1,\\dots,u_d) \\, du_1\\cdots du_d.", "441c513e50b159846be9cdb0f352adff": " m^{}_{}", "441c5cce7e24195f8232aa20549a4d38": "a_0, a_1, \\ldots, a_n", "441c5f26ba9436f38d4281fbc6aa0792": " (I-A \\otimes A)\\operatorname{vec}(X) = \\operatorname{vec}(Q) ", "441c9435f2f3c8ffe016c3c82d4840f9": "\\operatorname{cr}(G) \\geq \\frac{e^3}{33.75 n^2}.\\,", "441cacd3ca7a832f02fbc345711499f0": "\ny = a \\ \\cosh \\mu \\ \\cos \\nu \\ \\sin \\phi\n", "441cb31c2eaa300aecb0593acf88d1be": "w_\\textrm{eff} = w + t \\frac{1 + \\frac{1}{\\varepsilon_{r}}}{2 \\pi} \\mathrm{ln}\\left( \\frac{4 e}{\\sqrt{\\left( \\frac{t}{h}\\right)^{2} + \\left( \\frac{1}{\\pi} \\frac{1}{\\frac{w}{t} + \\frac{11}{10}}\\right)^{2}}}\\right).", "441d6e8ee3cd2e6d48e74eb6e584e9f0": "\\sum_{S \\in \\mathcal S} w(S) \\cdot x_S", "441d7e982f6b610aa6c8cd954dbd5d4a": "r = r_w + r_xi + r_yj + r_zk = \\cos \\left( \\frac{\\theta}{2} \\right) + \\sin \\left( \\frac{\\theta}{2} \\right) \\cdot \\vec{a}", "441db0552b7bbd77baf74397f3aa787d": "A_{\\mathcal{B},\\varepsilon}", "441dc50cb01740a6379222578f63fa0f": " \\cot\\left( \\pi/2-\\theta\\right) = \\tan \\theta", "441e737c82d8d6cafe22c766ea7bcd9f": " g(\\mu)=g_{obs} ", "441eadd2195b5fbf6efadc1dc5632caf": "\n\\nabla^2 \\Phi = \n\\frac{1}{a^2 \\left( \\sigma^2 - \\tau^2 \\right)}\n\\left\\{\n\\frac{\\sqrt{\\sigma^2 -1}}{\\sigma}\n\\frac{\\partial}{\\partial \\sigma} \\left[ \n\\left( \\sigma\\sqrt{\\sigma^2 - 1} \\right) \\frac{\\partial \\Phi}{\\partial \\sigma}\n\\right] + \n\\frac{\\sqrt{1 - \\tau^2}}{\\tau}\n\\frac{\\partial}{\\partial \\tau} \\left[ \n\\left( \\tau\\sqrt{1 - \\tau^2} \\right) \\frac{\\partial \\Phi}{\\partial \\tau}\n\\right]\n\\right\\}\n+ \\frac{1}{a^2 \\sigma^2 \\tau^2 }\n\\frac{\\partial^2 \\Phi}{\\partial \\phi^2}\n", "441eb22ae1992b94965ce735b068c8a3": " A_{FB} = \\frac {A_0} { \\tau_1 \\tau_2 }", "441edd87ee2285625f88d2a99eb3a61b": "(S_{(\\Delta x, \\Delta_y)} f)(x, y) = f(x-\\Delta x, y - \\Delta y)", "441f57b97b094227512a94d724d21216": "f(z) = \\cdots + \\left ( {1 \\over 3!} \\right ) z^{-3} + \\left ( {1 \\over 2!} \\right ) z^{-2} + 2z^{-1} + 2 + \\left ( {1 \\over 2!} \\right ) z + \\left ( {1 \\over 3!} \\right ) z^2 + \\left ( {1 \\over 4!} \\right ) z^3 + \\cdots", "441f9041ed79a60b421bc5ebd5662002": "\\scriptstyle{1 + \\log_k\\left(\\frac{k+1}{2}\\right)}", "441fba9af6b4281505569958b031b4ff": "\\sum_i \\sum_j A_{ij} B_{ij} = \\mathrm{tr} (A^{\\rm T} B).", "441fbe4795dcf97cb1988c20c223697a": "\\boldsymbol{d}", "44205199a590b0814d94563791c8841b": "\\dot{x}_\\alpha = \\frac{\\mathrm{d}x_\\alpha}{\\mathrm{d}t} = v_\\alpha", "442095639b102aaf5717f8877a9ae398": " \\|\\pi_f(x) \\xi\\|^2 = \\langle \\pi_f(x) \\xi \\mid \\pi_f(x) \\xi \\rangle = \\langle \\xi \\mid \\pi_f(x^*) \\pi_f(x) \\xi \\rangle = \\langle \\xi \\mid \\pi_f(x^* x) \\xi \\rangle= f(x^* x) > 0, ", "4420c12d3885162a10ac0f8edc23a872": "\\psi\\left(t\\right)", "4420c453897a3b5430ad1df7749f6387": "=|r'(t)|. \\,", "44212533bfc826c4edc12e9ea38bbb62": "K_{\\mathrm h}=\\frac{\\rm{[H_2CO_3]}}{\\rm{[CO_2(aq)]}}=1.70\\times 10^{-3}", "44218e8ebc8fe25517df07be90e20356": "d(x, y) \\leq \\max(d(x,z),d(y,z))", "4421b2b1a5bb0b8270422fe4c0dd13a0": "Q_j^*(H_j) = \\frac{1}{Z} e^{\\mathbb{E}_{-j}\\{\\ln P(H,V)\\}} ", "4421feaf2a4d4584ac574dbcb16e626c": "q=2^{2 h+1}", "4422059759891c3f190285016862b52b": " d_1 = \\frac{\\ln\\frac{S}{K} + (r-q+\\sigma^{2}/2)T}{\\sigma\\sqrt{T}},\\,d_2 = d_1-\\sigma\\sqrt{T}. \\,", "442224e62ee22de93a69cd6ed8cc863c": "\\tilde p = E(p/y) = \\tau(y) y", "44222abeee5aca181d3bbc43b67b07b5": "\\begin{matrix}\n| k \\rangle = b_k^\\dagger | 0 \\rangle.\n\\end{matrix}", "44228bc27949ad08e306aa89e14ea494": "\n\\pi = \\cfrac{4} {1+\\cfrac{1^2} {3+\\cfrac{2^2} {5+\\cfrac{3^2} {7+\\ddots}}}}\n= 4 - 1 + \\frac{1}{6} - \\frac{1}{34} + \\frac {16}{3145} - \\frac{4}{4551} + \\frac{1}{6601} - \\frac{1}{38341} +- \\cdots\n", "442298e044dc294ff19240f74842bac4": "X = \\{\\neg A\\}", "4422d46ffa3ddf3120ef5efaca232c07": "P(x)= x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1 \\,,", "4422eb92138df111a231b498b7c31dd4": "X_8", "44236313a0e159037e6b7a781b439b34": "\\alpha \\rightarrow \\infty ", "44236a443b7d7b067f8c21bd33b7202f": "\\sigma \\leftarrow 0", "442385b6670e6cfcd27125bad357f0d8": "\n\\ | H(e^{j \\omega}) | = \\sqrt{\\Re\\{H(e^{j \\omega})\\}^2 + \\Im\\{H(e^{j \\omega})\\}^2} \\,\n", "442390b3feade18d2c8d14f90c8ce633": " X\\subseteq\\operatorname{cl}(X) ", "4423bfba0e0a99a9ebdad0bebe47c4c9": " I = \\frac{1}{2}\\sum_{i=1}^N c_i z_i^2 ", "4423ddf51f0f6794306fa839f38e5aea": "\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\Gamma_0(N)", "4423f73dc6f2417de9c4c5e98371ccc4": "O({(log(n))}^3/{n^2})", "44243bac47cfe5b696300108583dde14": "g^{efghcda}", "44245becde3ca6028ca30d4fc2aa5997": "(-\\Delta)^{\\alpha/2} \\delta(x)", "442462ef679ff778b70fb5eb73385d26": "x_1,\\dots, x_n", "4424a249772edc5e29b780f3a4579109": "g\\in G,f\\in L^\\infty (G)\\,", "4424ecc6c7452092a6174b91ef369c03": "L = \\lim_{n\\rightarrow\\infty} \\left| \\frac{a_{n+1}}{a_{n}}\\right| < 1", "44251950153d6510fd82ae332f72f8cb": "B \\triangleq A A^T. \\, ", "44253f2c7dd4a4614a470be807396695": "\\{\\psi_k\\}", "4425d9227fd674f37517bab29d8f9af6": "F_k = (R + B^T P_k B)^{-1} B^T P_k A \\,", "4425f0acb0948391155c9766ddf54f89": "K = K_0 \\dots K_m", "44260e2dbf466c5f65bc37632f30df5c": "S_F", "44263988699f75daab6e02cb668272d6": "\\int x^{n-1}\\,dx = \\tfrac{1}{n} x^n + C \\qquad n \\neq 0.", "442663edc101e331a713f3dec6c991c0": "\\displaystyle{\\begin{pmatrix} p& q \\\\ q & r\\end{pmatrix}}", "4426a1062081b68bf927582bca128a22": "\\operatorname{Hol}(G)=\\{(x^i,\\sigma^j)\\}", "4426b0e8fb01536a88c60616a7181d9f": "R_x", "4426c88436d17a78946909a00f159a91": "H_n,P_n \\mbox{ with } n\\, ", "4426cc185a771dc8c86833f0dcdd6a5b": "w_{t,d,n}", "4427488fb690a786ff43492a4949a683": "x \\gamma_0", "4427801af8ca0a16da52be1ca7fe1f31": "(q, a, q_1 \\wedge q_2)", "4427ced457d388abe46c316339e3fafa": "L(L-2)=8", "442800833b9e17af8b4c4b11d7dd4c0e": "\\mathrm{d}^4 x \\;", "442824ee2725ca25652ad13e788ceab0": "\nW_s = \\frac{\\epsilon}{2}\\iiint_V |\\vec{E}|^2 dV = \\frac{\\mu}{2}\\iiint_V |\\vec{H}|^2 dV.\n", "442826c1641af88ef4f07c31320ff3fa": "x_2 - x_1", "44282e4fc9928a7b2b7669ebf7c76b9e": " g_k(x) = \\left\\{\\begin{matrix} 1 & \\mbox{if } x = a_j, j\\leq k \\\\ 0 & \\mbox{otherwise} \\end{matrix} \\right. ", "442830b98d43cdb1104f123e727ea47d": "\n\\varphi = 1 - \\exp {(- \\beta u)}. \n", "4428522c663f03b962a94595efe2bc2f": "g:= x^{q^{n}}-x \\bmod f", "4428798c6ab7287d2b11c8883e14f521": " Y \\,\\!", "442894f623bff654cbe097a5f0fd55c1": "u-u_n", "4428adaf19d4a9737137ed32c515015e": " a_n(0) = \\frac{(h(x),X_n)}{(X_n,X_n)} ", "4428ba5703dfd17119b2e469ef8d1684": " \\{ u , v \\}", "442953735b7d346fc2701b284c36ab58": "Q(w)", "442974e8c7c8779bf63e22e338b031c1": "\\ u^2x^2+v^2y^2+w^2z^2+2vwyz-2wuzx+2uvxy=0", "44297d0eef641fbe08a218792ef95300": "(1 - i) = ((1 + i) - (1 + i)^2)", "4429e846cf46a089b21859b14e203635": "\\mathrm{sinc}\\left(\\frac{t}{T}\\right)", "4429ef33c358577fa6c4ff7ac2dc7db4": "I_{eff}^2= \\frac{1}{T}\\int\\limits_{0}^{T}i^2\\,dt", "442a1ad12e29d54646c7279b98e12203": "\n\\mathrm{Ek} = \\frac{\\nu}{\\Omega L^2}.\n", "442aec4859569ce295e53429137a593e": "\\mathbf{w} = \\sum_i \\alpha_i y_i \\mathbf{x}_i.", "442b6927c1ee0070c169a151ff50734c": "g_4=-x+3x^2;", "442bb703ecd79792f40d6a3b0cfefa71": "C (B, \\succeq) \\subset B", "442bdc2d144dc662603559705726dfdf": "M \\stackrel{f}{\\to} N \\stackrel{\\omega}{\\to} T^*N \\stackrel{(Df)^*}{\\longrightarrow} T^*M.", "442bec4816687d2a6299b70439613dcd": "x^{-1} = \\frac{\\bar x}{N(x)}.", "442c3d6be108fdac1e97cba09ea47d4a": "x = (bg-cf)\\,", "442c5c12b31c926480e1a5174863523a": "r = \\overline{x}.\\,", "442cbaea309f62781dcfb471c9783b67": "(m_1,\\ldots,m_{\\ell})", "442ccf81d896a9e6a9e26e56e5e006f6": " g^{ij} = \\mathbf{e^i} \\cdot \\mathbf{e^j} ", "442cd830d23fcb63a4ee23dbfdbb52db": "F r_1^2=\\frac{x^2}{2}+\\frac{x}{2}", "442d0661e010c3c09654d7f4d6a5573c": "\\mathbf{K}_{k} = \\mathbf{P}_{k\\mid k-1}\\mathbf{H}_{k}^T\\bigl(\\mathbf{H}_{k}\\mathbf{P}_{k\\mid k-1}\\mathbf{H}_{k}^T+\\mathbf{R}_{k}\\bigr)^{-1} ", "442d1011742e43687916f3c2d0ac981e": " \\left(\\frac{1-t^2}{1+t^2}, \\frac{2t}{1+t^2}\\right) ", "442d3d93f0a58e73ee672da6ff2152d2": "\\theta:F\\Rightarrow G", "442d698f2081d132f766b68f91ed6b0f": "\\{ (x_i\\mid x_i) \\mid 1\\leq i \\leq k \\} \\cup \\{ (0\\mid y_j) \\mid 1\\leq j \\leq l \\} ", "442da22afd180df68e1e8bc44ef812a9": " F_2 = \\_ ", "442ddd8a96508eab316c203c5bbb1b72": "(\\lambda_k +\\mu_k) p_k(t)=\\lambda_{k-1} p_{k-1}(t)+\\mu_{k+1} p_{k+1}(t) \\, ", "442e161de802fa51296660ed1e90cf9c": "\\omega = \\frac{v}{r}", "442ea3aa8d734e3b0e3d22545fad0a1b": "x^{-1} = \\frac {x^*}{\\|x\\|^2}.", "442eb998deb028f1e7647738fd304874": "I(q)=\\sum_{i=1}^{N}\\sum_{j=1}^{N}f_i(q)f_j(q)\\frac{\\sin(q r_{ij})}{q r_{ij}},", "442ee4644ac77c66ecd2eb3e245062a0": "\\textstyle(x,y)", "442ef7a4d1e0fe9c792c0746ce6454ab": "\\frac{d}{d t}\\left(\\frac 1 2 \\left(\\frac{d y}{d t}\\right)^2\\right) = \\frac{d}{d t}\\left(A \\frac 3 5 y^{5/3}\\right).", "442f1888c0977939fcde0aa75e506645": "\\mathbf{IC} = \\frac{\\displaystyle\\sum_{j=1}^{N}[a_j=b_j]}{N/c},", "442f87a9f56706f159c6724308bc1fc2": "\\forall p: \\forall q: \\mathcal{B}(( \\mathcal{B}p \\wedge \\mathcal{B}( p \\to q)) \\to \\mathcal{B} q )", "442fcb5eb9bd5bd481ee47e12965aed4": "z_{c}\\begin{bmatrix}\nu\\\\\nv\\\\\n1\\end{bmatrix}=A \\begin{bmatrix}\nR & T\\end{bmatrix}\\begin{bmatrix}\nx_{w}\\\\\ny_{w}\\\\\nz_{w}\\\\\n1\\end{bmatrix}", "442fff68b5d2804a06707e42138a3de3": "\\hat H(t)\\psi_n(x,t) = E_n(t)\\psi_n(x,t)", "44300631c3ce9656ad8fa84d2cc629bc": "~N~", "443009cd1c175fb4394b22822415a6b6": "\\lambda = \\frac{\\rho \\pi ^{\\frac{D}{2}}}{\\Gamma (\\frac{D}{2} +1)} ", "443013514c74344a2ec88907aea39944": "H^{0}(G,A)=A^{G},\\,", "44301d3f8a97f773bea2275b2c946cdf": "\n\\Delta \\hat{z}\\ =\\ -J_2\\ \\frac{3\\ \\sin i\\ \\cos i}{\\mu p^2}\\left[\\hat{g}\\int\\limits_{0}^{2\\pi}\\frac{p}{r}\\ \\sin u\\ \\cos u \\ du\n+\\ \\hat{h}\\int\\limits_{0}^{2\\pi}\\frac{p}{r}\\ \\sin^2 u\\ du \\right]\\quad \\times \\ \\hat{z}\n", "443023e32358fdd35d25c5984d1933d7": "\\cfrac{[I]}{[I]+K_i} ", "44305cc86f7f65e43b7134740aa0ed1b": "\\mathcal{I}(\\mathcal{D}(A)) \\cong A", "4430669845e7744573c02aac86d4ddb1": "F(x_1,\\ldots,x_n)=\\min_{i\\in\\{1,\\ldots,n\\}}F_i(x_i),\\qquad (x_1,\\ldots,x_n)\\in{\\mathbb R}^n.", "44307775bf599a3d4fd34b4f59265cb3": "X\\in[A]^{\\omega}", "4430790c54f77770dde787684d79c8b7": "\\Sigma r (Q_0^n + n Q_0^{n-1} \\Delta Q) = 0", "44308062aaff7bb02a7325f8997b3898": "R \\equiv \\pm r \\pmod p", "4430a00a56930abbfdb2e1e7854cf943": "i = 0, 1, \\ldots, m\n", "4430a6ae1c231f41d77568be704ac585": "-\\rho_\\infty V_\\infty \\Gamma", "4430ebe1fe1673bb83acfbff6380bf4b": "\\langle jm|T^k_q|j'm'\\rangle =\\langle j||T^k||j'\\rangle C^{jm}_{kqj'm'}", "4430fbce9555f6e82ee04f8847ad205a": "a_n \\geq a_{n+1} > 0", "44313e25806d9f85bc8042843747be4d": " z(\\infty) > 0 \\iff a(1-Q)-b >0 \\iff (1-Q) > b/a .", "443159b5616c072f9a4a0455a3fe3853": "m_{\\text{r}}=\\frac{m_{\\text{h}}^* m_{\\text{e}}^*}{m_{\\text{h}}^* + m_{\\text{e}}^*}", "44317adde6323d5722b7ba8cfb0c3659": "\\mu_{a}", "443271c6fd9b6167bd36ca0276cff959": "\\Pr(X \\leq \\mu - \\sigma) \\leq \\frac{ 1 }{ 2 }. ", "4432f320b2b2340ce91dbfa43e33ea68": "y_{11}", "443342a157545731a7820dcd0aa4c4a1": " -\\nabla^2 u = f", "44338d693783745445ddb7386217032c": "\n\\partial_{t} e =\nDe_0\\nabla^2 \\left( \\frac{\\delta F}{\\delta e} \\right) -\n{\\mathbf{\\nabla}} \\cdot{\\mathbf {q}}_e({\\mathbf r},t).\n", "4433b8f20646bec69b25988e6ed9d3c4": "E = K + U = \\frac{1}{2} k A^2.", "4433b905a1d73b9df5d94c647a7f5d19": "\n\\eta_{2n}(s) = \\frac{1}{n^{it}} R_n(\\frac{1}{{(1+x)}^s},0,1),\n", "4433d927e1ff083341d6501130bbf0e5": "\\Omega = \\sum_{i=1}^n X_i X^i.", "44342c6f8d33ed27209f8c7bd4da9950": "\\frac{1}{\\zeta(s)}\\sum_{n=0,n\\ne s-1}^\\infty \\frac{\\zeta(s-n)}{n!}\\,t^n=\\frac{\\operatorname{Li}_s(e^t)-\\Phi(s,t)}{\\zeta(s)}", "44343b9517f8c04829f91581e553f673": "\\textrm{e}^{i kL_{(uv)}}", "4434548bfe59709f785b9830ac7a0d43": "\\operatorname{Cl}_{2m+1}\\left(\\pi\\right)=-\\eta(2m+1)=-\\left(\\frac{2^{2m}-1}{2^{2m}}\\right)\\zeta(2m+1)", "4435092f27678e3bfeeb7579948c8c55": "\\scriptstyle \\text{curry}(f) \\colon X \\to (Y \\to (Z \\to N)) ", "44351e336f99c656d1e5343171e192d8": "\\mathrm{nullity}(T)+\\mathrm{rank}(T)=\\dim(V).", "44353510b70feedd72f01a0eb447e2c6": "\n\\overline{\\phi(\\boldsymbol{x},t)} = \\displaystyle{\n\\int_{-\\infty}^{\\infty}} \\int_{-\\infty}^{\\infty} \\phi(\\boldsymbol{r},t^{\\prime}) G(\\boldsymbol{x}-\\boldsymbol{r},t - t^{\\prime}) dt^{\\prime} d \\boldsymbol{r},\n", "44354ad73ce125eb9423da52ae3964bb": "y(idx(i))=x(i)", "443576bf6048e73cb7f3f983f406b0e8": "r = (g^k", "443684ba4ebba97e4f2d1c81f4e6ce93": "L_\\mathrm v = E_\\mathrm v \\times R", "44368c251f29a8864a16ad63976a180e": " ~\\Upsilon_v ", "44369e819f51079cc12aeb715e06d61b": "\nG(a,0,c) = \\sum_{n=0}^{c-1} \\left(\\frac{n}{c}\\right) e^{2\\pi i a n/c}.\n", "4436ad1e51fae1186fd023c62c3c5316": " \\scriptstyle \\overleftarrow{s} ", "4436da78d48e6e9e59496fa740e2d2da": " \\mathcal{B}(\\mathcal{H}^{12})\n\\rightarrow \\mathcal{B}(\\mathcal{H}^{12})", "4437780bd4b5ad0a6176ce3373d19d51": "\\mathrm{NA} \\approx a/2", "4437ad3f9f01d9923a42e66dc7a8e67f": " \\mathcal{A}_f = \\mathcal{O} / \\langle 3x^2 - y^2, xy \\rangle = \\langle 1, x, y, x^2 \\rangle . ", "4437cfc21a675fe153565de39dc8dbd5": " \\operatorname{erf} \\left( \\frac {z}{\\sqrt{2}} \\right)", "4437e925bc675be9460e804836bdd11e": "\nE(\\alpha) =\n\\begin{pmatrix}\nm^2+n^2-p^2-q^2&2np-2mq &2mp+2nq \\\\\n2mq+2np &m^2-n^2+p^2-q^2&2pq-2mn \\\\\n2nq-2mp &2mn+2pq &m^2-n^2-p^2+q^2\\\\\n\\end{pmatrix},", "443823ce96fe77ace7efaed3230ee628": "\\int_{-\\infty}^\\infty \\left| f(t) \\right|^2\\, dt = \\int_{-\\infty}^\\infty ESD(\\omega)\\, d\\omega.", "44382ecfdbb701e3989338a021d92334": "(x + 1)^{3}p(\\frac{1}{x+1}) = 64x^3-64x^2-128x+64", "44383a51c5813fbd408160d86a747cb4": "\\operatorname{E}[\\ln|\\mathbf{X}|] = \\sum_{i=1}^p \\psi\\left(\\tfrac{1}{2}(n+1-i)\\right) + p\\ln(2) + \\ln|\\mathbf{V}|", "443861ee83350f28605cdfb1fbae9119": " \\nabla^2 E_u = \\frac{\\partial^2 E_u}{\\partial x^2} + \\frac{\\partial^2 E_u}{\\partial y^2} + \\frac{\\partial^2 E_u}{\\partial z^2} ", "4438eaa78a59937cd6a5d2508521d58e": "\\pi \\approx \\sqrt{10} \\approx 3.162", "443900e3b1b1c94d09de9e44abc326b7": "(1-x)^{-n}=\\sum_{k=0}^\\infty{-n \\choose k} \\cdot(-x)^k", "4439218b06cac3fc0c25d781fcec1c72": "\\sum_i T_{\\mathrm{amortized}}(o_i)", "44392694c32e23ec406e314af48101e7": "\\{P\\}\\ C\\ \\{Q\\}", "443938ef093707d738712c6ff48d9742": "\n\\begin{align}\n\\frac{d}{dt}\\langle x\\rangle =& \\frac{1}{i\\hbar}\\langle [x,H]\\rangle + \\left\\langle \\frac{\\partial x}{\\partial t}\\right\\rangle \\\\\n=& \\frac{1}{i\\hbar}\\langle [x,\\frac{p^2}{2m} + V(x,t)]\\rangle + 0 = \\frac{1}{i\\hbar}\\langle [x,\\frac{p^2}{2m}]\\rangle \\\\\n=& \\frac{1}{i\\hbar}\\langle [x,\\frac{p^2}{2m}]\\rangle = \\frac{1}{i\\hbar 2 m}\\langle [x,p] \\frac{d}{dp} p^2\\rangle \\\\\n=& \\frac{1}{i\\hbar 2 m}\\langle i \\hbar 2 p\\rangle = \\frac{1}{m}\\langle p\\rangle\n\\end{align}\n", "44399e294a1383c3552cad5e443487a0": "E=K\\cdot M\\in GF(p^2)", "4439aade2a68af93fd6e04c3fd032da8": "{d^n \\over dx^n} f(g(x))=(f\\circ g)^{(n)}(x)=\\sum_{\\pi\\in\\Pi} f^{(\\left|\\pi\\right|)}(g(x))\\cdot\\prod_{B\\in\\pi}g^{(\\left|B\\right|)}(x)", "4439feccb5b07c3a7b2d3720fda76ada": "\\sin(a n)", "443a805ec01610c96ea2a7e2f40476e0": " (1+\\chi_v) ", "443b0e1c2eb4d3024f78c6067bf28e0e": "\\mathrm{graph}(S)\\subset \\mathrm{graph}(S').", "443b4a8f9b73875d916163465a3a3d7b": " X \\sim \\textrm{Kumaraswamy}(1,1)\\,", "443b6d2a430fc165bfc98a4879a7707c": "x^{ 8 }+x^{ 6 }+x^{ 5 }+x^{ 4 }+1", "443b81512c75460fbad839086ef4428a": "\\textit{VendingMachine} = \\textit{coin} \\rightarrow \\textit{choc} \\rightarrow \\textit{STOP}", "443ba342f243a2501ab2ae45954bedde": "\\Phi(0) = 0.5 = 50\\%", "443c004b4233e581c4806afb89c62d27": "\\overline{y}_{st} ~=~ \\frac{1}{n} \\sum_{i=1}^{n} y_{ist}", "443c64c71bce8a119712a6b047bb9276": "\\Theta\\equiv\\Theta(\\lambda_{1},\\dots,\\lambda_{k})", "443cbe5791d78b42f7bf39e8b238791d": "\\scriptstyle \\mathrm{V}/\\sqrt{\\mathrm{Hz}}", "443ce159872391a25b0f22b38041de11": "\\rho(X) > \\mathbb{E}[-X]", "443ceeb453b4b61ed7e31ceaf3a7b8df": "\\scriptstyle{H^2(M,\\mathbb{Z})}", "443d174765d564e982492350513e2c43": "\\hat{I_z}", "443d5085fe54ea2cdb57a372da59eb67": "{\\dfrac{l}{M}}", "443d5d9b41ef778b1101f36ee9e6a288": "(x+y)^p \\equiv x^p+y^p \\pmod{p}.\\,", "443d81075fe3460abe6d037d9d76d81a": " \\Phi^{1}(\\Phi^0) = 0 ", "443d96b8382326bf5a88c8efa68ea421": "\nH_{E}=E_0 \\,z\n", "443daa20d8f2187280d83da72db98da7": "MA = \\frac{F_B}{F_A} = n,\\!", "443db4a17d1b4f94c55e0fd3077f8907": "\\mathbf{Q}(\\sqrt{3}, \\sqrt{7})", "443dfe158877bb74fb521ba34708c772": "dx^a \\wedge dx^a = 0 \\,\\!", "443e89f0bb6e3ec4690b5398dc169430": "(f,v)=(Lu,v) + (u,v) = (u_x,v_x) + (u_y,v_y) +(u,v) =((\\Delta+I)u,v) +(\\partial_{n} u,v)_{\\partial\\Omega} \n=(f,v) + (\\partial_{n} u,v)_{\\partial\\Omega}.", "443ea66fa64436e9d3649a567adb4b23": "\\pi_k(W,M)", "443ede66bd97eaea40f592f11e6f59c3": "\\frac{C_6 \\alpha_1}{4 C_9} = 1", "443f2f4590b36dc21bddb03f42e8cee6": "\\hat{\\tilde{H}}^\\dagger W_\\gamma [A] = - \\epsilon_{ijk} \\hat{F}^k_{ab} {\\delta \\over \\delta A_a^i} \\; {\\delta \\over \\delta A_b^j} W_\\gamma [A]", "443f6b471e36831627134a330ce81b58": " BC = M_b - M_v\\!\\,", "443f8e6a952f84f568ea76c7453e045a": "\\begin{pmatrix}\ne(x,y) & f(x,y) \\\\\nf(x,y) & g(x,y)\\end{pmatrix}", "443fadf4c784cc848fc334a9d8c7109d": "\n\\frac{\\mathrm d L}{\\mathrm d t} = \\int | \\nabla \\vec v \\cdot \\mathrm d \\vec s |\n", "443fcc40026320e40c1c01541c1270cd": "\\tfrac{3}{7}\\scriptstyle{\\sqrt{6(2+\\sqrt{2})}}", "443fe048d482800218960b6a187dab0e": "\\forall \\epsilon > 0 \\,\\, \\alpha_n(\\epsilon) \\leq C \\exp(-c n \\epsilon^2)", "443feae336300fe9580442748c9162c7": "\\frac{S^\\mathrm{ig}-S}{R} = \\int_V^\\infty \\left[ T \\left(\\frac{\\partial Z}{\\partial T}\\right)_V - 1 + Z\\right] \\frac{\\mathrm{d}V}{V} - \\ln Z", "443feea37e8fb17b5c6d465bfe6345f8": "F_r={\\frac {1}{z} \\times \\frac {G}{g} \\times r \\times (\\pi \\times \\frac {n}{30})^2 }", "4440060840d6774c05373aa0d71e757f": " g \\in {C^{\\infty}}(M) ", "4440e1362ab63109ce3d94ac9d1f6ffc": "1024n", "4440ead51039a515fc35fd702d6c819f": "\\phi_S \\,\\!", "4441b63b0485cc8f96351bb1ba3f7919": "| \\nu \\rang", "4441ce3b72757ed1be56d94964d2df47": "\\mathrm{SNR}", "44420820a54f48df70b87237ff538c23": "t_i =(\\alpha-1)w_i", "444264610f1df851430fd013f13d4523": "v_\\alpha^{t+1} = f(s_\\alpha^t, v_\\alpha^t, v_{\\alpha-1}^t, \\ldots)", "4442ad1daccf1e58f03205291870e305": "\\bigwedge_{\\gamma\\in\\Gamma} \\gamma \\rightarrow \\bigvee_{\\delta\\in\\Delta}\\delta", "4442d82c252ea19451befa611722a454": "{1\\over 2\\pi i}\\oint_{C} {f'(z) \\over f(z)}\\, dz", "4442f4e10d43960fc1c1dec3813e4dff": "E(X_1\\cdots X_n)=\\sum_\\pi\\prod_{B\\in\\pi}\\kappa(X_i : i \\in B).", "4443004b476a4541c6e9f97ec84681c0": "D_{KL}(P\\|Q) \\ge \\mu'_1(P) \\theta - \\Psi_Q(\\theta),", "44434deccd72b30b1fddb940637b299f": "1+\\cos x = 1+(1-\\sin^2x)^{1/2}.", "444350f4c3129e0eaa9a8057cfcf9594": " \\begin{matrix} u = -\\phi (y) \\quad (2) \\end{matrix} ", "444379927d947f44f19bbe5b0c9e5f16": "x = 3(t^2 - 3) = 3t^2 - 9\\,", "44439598187ae86e27ada4a4087f2442": "7 \\ \\mathrm{N} = 7 \\ \\mathrm{N} (4 \\times \\frac{7}{4} = 7)", "4443b6fe3cd4e436e532e7722e2da934": " \\Delta E \\Delta t \\ge h ", "44440f8d03b5ca836582fef1667c051e": "\\pi_x", "444460d12ffee70366a0edbd01b21949": "1_{d_{2}}", "4444855bf5764ffa252f5912004c3733": "2\\pi-l\\ ", "44451394182c0ababe62a2fb1d044f50": "\\tilde{f} = f + I_{\\mathrm{constraints}}", "44452b67025d320d99ce63027e6ebccd": " | \\psi_i \\rangle ", "44454315ae8e7e6f2e1bfa1495a5aefe": "\\frac{I_{3}}{\\sigma_{xx} \\sigma_{zz} - \\sigma^2_{xz}}", "4445589f5846fc414eefc49308d3c160": "M\\ni q:=\\exp_p(v)", "4445665d1ef668d076bc1e9135086cbd": "a,b,c ,\\ldots", "4445cfb7dd4299b175af949ac84c149a": "u_\\lambda(x):=u(\\lambda x)", "4445f39eebb8fd688e42b4b55469e7a2": " \\langle\\tilde v_i , \\tilde u_j\\rangle = \\delta_{i,j} ,", "4446a1f285e7982ad18f460e3c00aaf6": "q \\mapsto u q v,", "4446e8414bee37e562227a9333fefa53": "\\langle x \\rangle = \\sum_{i=1}^N \\langle S_i \\rangle.", "444706e7d28d9c92192161a81264b16b": "~~\\Pr_{h \\in H} \\left[ h(x_1)=y_1 \\land \\cdots \\land h(x_k)=y_k \\right] \\le \\mu / m^k", "44470e4119b6595a634f06332e1aaccb": "\\triangledown ^{2}A_{z}+k^{2}A_{z}=0 \\ \\ \\ \\ \\ \\ (12) ", "4447150c411978f0f02a2efb776b4566": "\n \\bar{\\Psi}^\\dagger \\rightarrow \n \\begin{pmatrix}\n \\psi_{22}^* & -\\psi_{21}^* \\\\ -\\psi_{12}^* & \\psi_{11}^*\n\\end{pmatrix}\n", "4447785d1c59193582025a2a90facfa2": "\\begin{align}\n H_2 = \\frac{1}{2}\n &\\begin{pmatrix}\\begin{array}{rrrr}\n 1 & 1 & 1 & 1\\\\\n 1 & -1 & 1 & -1\\\\\n 1 & 1 & -1 & -1\\\\\n 1 & -1 & -1 & 1\n \\end{array}\\end{pmatrix}\\\\\n H_3 = \\frac{1}{2^{\\frac{3}{2}}}\n &\\begin{pmatrix}\\begin{array}{rrrrrrrr}\n 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\\\\n 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\\\\n 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\\\ \n 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\\\\n 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\\\\n 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\\\\n 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\n \\end{array}\\end{pmatrix}\\\\\n (H_n)_{i,j} = \\frac{1}{2^{\\frac{n}{2}}} &(-1)^{i \\cdot j}\n\\end{align}", "44477e1c3ce07af446e1cf4ba5aade8c": "\\scriptstyle P=-\\rho g z+\\text{Const.},", "4447b19c040065972248c3b5cc25c918": "\\frac{2e^2}{h} \\simeq 77.41\\; \\mu S", "4447d435d27934267c7f6dc73bb1a109": "\\tau_A = \\frac{a}{v_A}", "444804f8b7259ba2ec21e196da7590b9": "(s, t; 0, 0, \\ldots, 0)", "444820856f50dd6216d4bab0c8a68c74": "y_i = \\alpha y_{i-1} + \\alpha (x_{i} - x_{i-1}) \\qquad \\text{where} \\qquad \\alpha \\triangleq \\frac{RC}{RC + \\Delta_T}", "4448686cc38f40a638dd04cdd1ef6c27": "\\begin{align}\n\\nabla \\times (\\nabla \\times \\mathbf{f}) & = \\nabla (\\nabla \\cdot \\mathbf{f} ) - (\\nabla \\cdot \\nabla) \\mathbf{f} \\\\\n& = \\nabla (\\nabla \\cdot \\mathbf{f} ) - \\nabla^2 \\mathbf{f},\\\\\n\\end{align} ", "444902e2d05aad831710729bd0def2aa": "\\mathbf{x}^\\top_i[M]\\mathbf{x}_i = 1 \\Rightarrow \\epsilon_{ii}=-\\frac{1}{2}\\mathbf{x}^\\top_{0i}[\\delta M]\\mathbf{x}_{0i}.", "444915f8cce81cdd9f0c6cddd657cb3d": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-2}{\\sqrt{3}},\\ \\pm4\\right)", "444932009a64aa6bcb1795d7ba0ecc85": "T \\to H^P_X", "444935f3fe150544973fdcbacca93ed9": "\\{_{\\mu\\nu\\alpha}\\}= -\\frac12(\\partial_\\mu g_{\\nu\\alpha} + \\partial_\\alpha\ng_{\\nu\\mu}-\\partial_\\nu g_{\\mu\\alpha}), ", "444a0eb2c483df2d0b3900d2517e4568": "{3003 \\choose 1} = {78 \\choose 2} = {15 \\choose 5} = {14 \\choose 6}", "444a38303ecd86e89b366b917372ed3e": "{q^2 \\over g} \\left({{y_2 - y_1} \\over {y_1y_2}} \\right) = {1 \\over 2} ({y_2 - y_1})({y_2 + y_1})", "444a70155d06b642e967636d9a4cc35a": "\\|L\\|_{X,Y} \\leq C \\|L\\|_{X_0,Y_0}^{1-\\theta} \\; \\|L\\|_{X_1,Y_1}^{\\theta}", "444aeee548b477ec24b343360d25ea5b": "\\hat{\\sigma}_x", "444afbd3de43d1dfbfeef9090487d21c": "\\frac{1}{1}", "444b0341dc9a6e08755b80a2d8ea519a": "(x*y)*z\\ne x*(y*z)\\qquad\\mbox{for some }x,y,z\\in S.", "444b21174ba45f5c4ab2d0955316674c": "y_k \\not = c_k", "444b3643f218abe71f8ac305d5230756": "g(U)\\cap U=\\varnothing", "444b40ae884ed5feb5cd4ff5052ac7f0": "G_i^\\ell", "444b7a05c550932eac4dcff3190481db": " \nR^m_{\\ell}(\\mathbf{r}) \\equiv \\sqrt{\\frac{4\\pi}{2\\ell+1}}\\; r^\\ell Y^m_{\\ell}(\\theta,\\varphi), \n", "444b8955ea2b5e689858d49aa07afb0c": "\\dim V =n", "444bb585d5ee7d58d97fff6030b83797": "\\theta = 2\\,\\arccos(w)\\,", "444be00a8548671764378653e5f2661e": " \\bigoplus_{1 \\leq \\ell \\leq \\omega} L^2_{\\mu_\\ell}(\\mathbb{R}, \\mathbf{H}_\\ell). ", "444c1937270b31d0b474922ab5577594": "[ x]", "444c2ba0d2fa3f2f4a5bc9a5c5d9dd5e": "\\int_{-1}^1 f(x)\\,dx = \\int_{-1}^1 \\omega(x) g(x)\\,dx \\approx \\sum_{i=1}^n w_i' g(x_i).", "444c428716081c825fff7665fe61d2e9": "\\approx g_J\\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)} ", "444c63064c0598b81a511fe2489e39bc": "\\mathfrak{f}_4", "444cb99631a7fe2ffe62777cbf079d3e": "\\displaystyle{\\mathfrak{g}=\\mathfrak{k}\\oplus\\mathfrak{p},}", "444cfe9265da26a396bbec1e96d87e72": "V_c(R)", "444d2cec7d53d51b85b667fad901c271": "A_1B_1", "444dafe50b9d47352a95e5c2b2e5284a": " e = \\sqrt{1- (b/a)^2} ", "444ded531df49dd5e9d60e42154722ce": " t \\,\\!", "444e4e98b4cf77a691e00fb602836bc8": "\\alpha = m + ni + pj + qk", "444e7893073c60ca3872e26367dffe64": "g(n+1) = \\sum_{i = 0}^{n} g(i)^{n-i}", "444ea73084d60cf0735e2e84c14f408b": "(s_i(t_i),t_i) \\in C_i", "444ec499e1cd708a98675413a6c52361": " \\mathrm{p}K_{\\mathrm{a}}^{\\mathrm{HH}}(\\mathrm{pH}) =\n \\mathrm{pH} - \\frac{\\Delta G^{\\mathrm{prot}}(\\mathrm{pH})}{\\mathrm{RT} \\ln10} ]\n ", "444f01253fef4a358befc1747375a9db": "I = \\epsilon \\sigma \\left ( T_\\mathrm{ext}^4 - T_\\mathrm{sys}^4 \\right ) \\,", "444f143822aeb794db14013007182b59": "\\frac{d \\mathbf{L}}{dt} = 0", "444f36ac10e870aa1de1bf6641a06fde": "y_j = \\sum_{k=1}^j y_k^{j+1}", "444f478ba119f3f734dd33f83a0032a1": "M(H)=\\chi_0H+N_eH^3+\\varepsilon(H^3)", "444f4a709e0835a7541984d4701046fa": "\\lambda = i^\\mu(1+i)^\\nu\\pi_1^{\\alpha_1}\\pi_2^{\\alpha_2}\\pi_3^{\\alpha_3} \\dots", "444f6c5522d44d40088ac9b556fd071f": "H^q(X, \\textstyle\\bigwedge^p\\Omega_X)", "444f849799a15f8c636d64ea9aad4ac6": "A \\in \\left\\{b,B,W\\right\\}\\,", "444f9297fe015dcd210b7f8fbe93f062": " |\\alpha\\rangle = |\\alpha/\\sqrt{2} \\rangle_1 \\otimes |\\alpha/\\sqrt{2} \\rangle_2 ", "444fba6f601cfa4583026f9fe79019b3": " \\langle fg\\rangle_h - \\langle f \\rangle_h \\langle g \\rangle_h \\geq 0. ", "44501d0a9377cd8b1f526b482219897f": " f(\\cdot,y): M \\to \\mathbf{R} ", "4450a3f4e6ac447c4b517d9185fe7640": "p_1(x),\\,\\ldots,\\,p_k(x)", "4450d37e40f1c30f8eef4aac9adbcecf": "\\{X,Y,Z\\}", "4450ef9acc0c855edd3bf786a81b7c0c": "\\int_\\Omega \\overline{\\psi_m}(q)\\psi_n(q)\\,dq = \\delta_{mn}", "445100fa212affc6fc281defabae6570": "m_{min} = -j", "44510e5d93296c70f7f3443638a9312b": "g_A(t) = \\log \\left(\\sum_i e^{t \\cdot A_i}\\right) . ", "445166771b603e4ccb0c7ea8d4767ef4": "\\beta=\\frac{\\rho_a\\sigma_a}{2k}", "445174d697d5072e7edeb58f17d35328": " \\sum_{n=-\\infty}^\\infty | x[n] |^2 = \\frac{1}{2\\pi} \\int_{-\\pi}^\\pi | X(e^{i\\phi}) |^2 d\\phi ", "445181fcfe92254945ef7d7d10c45733": "V < 0", "4451c967b29dfb2fa8325a28b09b40cd": "\\operatorname{Tr}_B ^*", "4451cfcebd176426acba13d2a15b5c61": " \\left(\\frac{P}{S}\\right)_\\text{iso} = \\frac{\\tfrac{1}{2} R_\\text{series} I_0^{\\,2}}{4\\pi r^2}.", "4451f4a130da218c3ab08417c272ab9e": "\\Sigma_k\\rm{P}", "4452183e9a6099dd81e58474f9c7b493": "f(z) = 1/z = z^*/\\mid z \\mid^2 \\text{where} \\mid z \\mid^2 = z z^* ", "445218ff6c61c8af21b96713c81e09a6": "n+1 \\over 2", "445236d0776695148a600577a8eaf73e": "\\left(\\bar\\psi m \\psi\\right)_B = Z_0 \\, \\bar\\psi m \\psi", "44523aeff9c163a247fec2a2a682cb0a": "n<50", "445358b39fc503d9b5e328a037ac82b6": "\n2\\pi \\hbar (n+\\frac{1}{2})=\\oint\npdx=4\\int\\limits_{0}^{x_{m}}pdx=4\\int\\limits_{0}^{x_{m}}D_{\\alpha\n}^{-1/\\alpha }(E-q^{2}|x|^{\\beta })^{1/\\alpha }dx, \n", "44535f17b3ee0f661272a5b8df504600": "\\{f_{n}\\}", "4453b0cfe614b847cd6c7bbe1b5f897b": "\\mathbb{Z}^n \\oplus \\mathbb{Z}_{k_1} \\oplus \\cdots \\oplus \\mathbb{Z}_{k_u},", "445421455282447d9f98837fd25ddf27": " \\and D[x] = [F_5, S_5, A_5]::\\_ ", "44543343a7efcf78b2936502264dc222": "-1 < f'(x_\\star) < 0 ", "44544a50c6134c41fe6f25e811255874": "(X,\\Omega)", "4454819cdbea833ab66e5c90cec38907": " \\omega_r * R = u ", "4454c15b8ae23c1313a93ca268840355": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right) = \\frac{1}{2 \\pi i} \\int_L \\frac{\\prod_{j=1}^m \\Gamma(b_j - s) \\prod_{j=1}^n \\Gamma(1 - a_j +s)} {\\prod_{j=m+1}^q \\Gamma(1 - b_j + s) \\prod_{j=n+1}^p \\Gamma(a_j - s)} \\,z^s \\,ds,\n", "4454f13395936f3666f424504395e70f": "f = \\sum_t \\alpha_t h_t", "4454f3eca6905bdccb9cdec03ad552f8": "d=3,5,7,\\ldots", "445527d8c8ec12c22269c4e93f19c515": "t - N", "445571169848a7b3cf0749a5c8b3b554": "(21)\\quad ds^2=-\\Big(1-\\frac{2M}{r}+\\frac{Q^2}{r^2} \\Big)\\,dt^2+\\Big(1-\\frac{2M}{r}+\\frac{Q^2}{r^2} \\Big)^{-1}dr^2+r^2d\\theta^2+r^2\\sin^2\\theta\\, d\\phi^2\\,.", "44559426f544198165f61b6e853db816": " F_{\\text{viscosity, fast}} = - \\eta 2 \\pi r \\Delta x \\left . \\frac{dv}{dr} \\right \\vert_r ", "4455ba47e0048860a344103d99ce7141": "\\varphi(n) = (p-1)(q-1) = n + 1 - (p+q), \\, ", "4455c87c0e35a6793d1fecb2fbc9249b": "C > 1", "4455d2d21cc32bf99c19491608cfce46": "RMS_{AC}", "44561613b431bec7a2e196c18b899011": "\\scriptstyle I_{\\mathrm{Bob}}", "44562ad8a0ce5ae2f234019f471faff8": "\np(X^o,x^m|h,n) = p_{fg}(z)p_{bg}(x_{bg})\\,\n", "44565a671258f34be98e406f74cf3218": " \\left(\\Gamma_{q^2}(x)\\Gamma_{q^2}(1-x)\\right)^{-1}=\\frac{q^{2x(1-x)}}{(q^{-2};q^{-2})^3_\\infty(q^2-1)}\\vartheta_4\\left(\\frac{1}{2i}(1-2x)\\log q,\\frac{1}{q}\\right). ", "4456661524835318dc878ccfcf5eff63": " \\displaystyle{\\iint_{\\Omega\\cup\\Omega^c, \\,\\, |z-w| >\\varepsilon} \\nabla N(w-z) \\cdot \\nabla D(\\varphi)(z) \\,dx\\, dy=\\int_{|z-w| =\\varepsilon} N(z-w)\\,\\partial_n D(\\varphi)(z)=0,}", "4457600d415965cee1d1b93da276dcad": " p \\approx 1000(SG-1)/4 ", "445778099ce6bd9d38884fb946eafd91": "d_{i(i+1)}", "44577ebaa3780ff2faadbdc2017bdc2c": "z=-a", "44578706596e227a2c4967718d467f66": " \\theta_i \\, \\sim \\, \\mathrm{Dir}(\\alpha) ", "4457a45a90031466a5e4dae44540061e": " \\forall a \\forall b\\; a \\wedge b = b \\wedge a ", "4457c6155b9ee9b6b177cd69a98cbf98": "\\,4^2 = 2^4", "4457c923741fe412dad454f437693491": "F_{14}=377 \\text{ and } F_{15}=610.", "445818e8db6e30058f98fef035b3758e": "x(k)\\,\\!", "44583e0ba36e952c640b89d0c4648b86": "\\lbrace a, \\theta, b, c\\rbrace", "4458479fdd682acd796bd43a652e139b": " \\varepsilon_x = \\frac {1}{E} \\left [ \\sigma_x - \\nu \\left ( \\sigma_y + \\sigma_z \\right ) \\right ] ", "4458d1731e84989f1f549a69865511eb": "\\scriptstyle M \\,=\\, H \\,+\\, P", "445962217d877d03b7734b9e91929819": "N=\\frac{a-c}{\\sqrt{Fb}}-1", "44599e4d7eda34e5972985d36f2142e8": "F_i(K_i,L_{i-1})=FL(KL_i, FO(KO_i, KI_i, L_{i-1}))\\,", "4459cd104bd24ef785489afc1c4a8099": "\\Phi_{12}:=\\frac{1}{2}R_{ab}\\bar{m}^a n^b\\,, \\quad\\; \\Phi_{21}:=\\frac{1}{2}R_{ab}m^a n^b=\\overline{\\Phi_{12}}\\,.", "4459cd4c3d1d1cf2441fb18fe3278212": "\\textstyle \\lfloor 33/2 \\rfloor = 16", "445a0927646e3cba475fa69236c73cea": "T_1 = 19 ^{\\circ}C\\!", "445a5a3ab574fdeec11f7faf4a3d98f0": "\\nearrow, \\swarrow, \\nwarrow, \\searrow \\!", "445a5f9bec32db6d1e764f606bddf4a7": "\n\\det(\\bold{U}_d) \\equiv 1 \\mod 2,\n", "445b53421f63a7a03f7bc5e725535e5c": "\\bar V_0", "445bf73595d29646b7d0e9be8f9386f1": "\\mathrm{Log}: \\left({\\mathbb C}\\backslash\\{0\\}\\right)^n \\to \\mathbb R^n", "445c1f91642eba3644bfe0f7e924cb93": "\\mathbf{J}=\\mathfrak{m} a (-\\nabla \\mu + (\\mbox{external force per gram particle}))\\, . ", "445ca9501d6512d14fa44df387da39c9": " \\textbf{f}_q = 5 + 9X +6X^2+16X^3 + 4X^4 +15X^5 +16X^6+22X^7+20X^8+18X^9+30X^{10} \\pmod {32} ", "445cb03c535e3c6cda5b2239d9bb807c": "\\sigma_D^2", "445d650ff24bb6dda3d3f570c41b5143": "k'= \\textrm{dn} K\\,", "445d6e72482827f7d6785f80d38dd744": "{^{n}a} := \\begin{cases} 1 &\\text{if }n=0 \\\\ a^{\\left[^{(n-1)}a\\right]} &\\text{if }n>0 \\end{cases} ", "445d765b5995dd7cc2c7693ff9a31c7c": "\\operatorname{Out}(A_6)=C_2 \\times C_2", "445d7d754403821d7dfcabc0a2ca15ba": "\n \\int x^{m-n} \\left(a\\,B (m-n+1)+(b\\,B (m+n\\,p+1)-A\\,c (m+n (2 p+1)+1)) x^n\\right) \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^pdx\n", "445d830265701a1b64e9a4f91e71d897": " \\scriptstyle e \\ = \\ \\sqrt{4 \\pi \\alpha} \\ \\approx \\ ", "445d96655a5dfc05fb09830bc940dd3e": " U_x ", "445da86c03200b3ff374f0f2640eb386": "\n\\begin{pmatrix}\nq\\\\ p\n\\end{pmatrix}\n\\mapsto\n\\begin{pmatrix}\n q + \\tau c_i \\frac{\\partial T}{\\partial p}(p)\\\\\n p\n\\end{pmatrix}\n", "445db7ba53cafb8f5dc244008d03829d": "S_{xx}''(x_0)", "445dffb4af6c02dc443e3375dc2dc49b": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, w z \\right) =\nw^{b_q} \\sum_{h=0}^{\\infty} \\frac{(w - 1)^h}{h!} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ b_1, \\dots, b_{q-1}, b_q+h \\end{matrix} \\; \\right| \\, z \\right), \\quad m < q,\n", "445e3b4db58518e7761e6027e70d90f4": "\\boldsymbol{\\sigma}= \\sigma_{ij} = \\left[{\\begin{matrix} \\mathbf{T}^{(\\mathbf{e}_1)} \\\\\n\\mathbf{T}^{(\\mathbf{e}_2)} \\\\\n\\mathbf{T}^{(\\mathbf{e}_3)} \\\\\n\\end{matrix}}\\right] =\n\\left[{\\begin{matrix}\n\\sigma _{11} & \\sigma _{12} & \\sigma _{13} \\\\\n\\sigma _{21} & \\sigma _{22} & \\sigma _{23} \\\\\n\\sigma _{31} & \\sigma _{32} & \\sigma _{33} \\\\\n\\end{matrix}}\\right] \\equiv \\left[{\\begin{matrix}\n\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\n\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\n\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\n\\end{matrix}}\\right] \\equiv \\left[{\\begin{matrix}\n\\sigma _x & \\tau _{xy} & \\tau _{xz} \\\\\n\\tau _{yx} & \\sigma _y & \\tau _{yz} \\\\\n\\tau _{zx} & \\tau _{zy} & \\sigma _z \\\\\n\\end{matrix}}\\right],", "445e4d6118f2b3329aa0afc15020cabc": "\\mathcal{E}(u_*,u_*)\\leq \\mathcal{E}(u,u)", "445e785ab49fb1f1e9f4b85d1c16dde1": "G(\\omega) = \\frac{1}{6 \\pi a \\alpha(\\omega)}", "445e85707990c9c0c497efe342359525": "\n\\frac{\\partial L}{\\partial t} = [ L, A ],\n", "445e87c2c52e81e5521ac84e493a9cf6": "v_r, v_\\theta", "445e8b8402b1b25bba7eaaae5fcc6a42": "EL(\\Gamma_0)", "445e8c7e93ef06246dc9395008ccb8bb": "\\alpha^x = \\beta", "445f49fdba060960b8210c54c5ecaa82": "y'=y, \\quad y(0)=1, ", "445fac47d5bac5c4c89d89809191bea3": "H(X_1,X_2)=H(X_1)+H(X_2)-I(X_1;X_2)", "445ffffc81bc9b65920c290fb48bbab3": "[Rp_!k_X,k] \\cong [k_X,p^!k] . \\,\\!", "446011ae82eff2cea40e74ae0f0d1780": " \\varepsilon\\left[\\Psi\\right] = \\frac{\\left\\langle\\Psi|\\hat{H}|\\Psi\\right\\rangle}{\\left\\langle\\Psi \\mid \\Psi\\right\\rangle}.", "446016541b88142a88bd7b8234000699": "g(x_1,x_2,x_3)", "446025de7adccf23e5bebc46a501f27c": "E_n = \\left({1\\over2}+n\\right)\\hbar\\omega_k \\quad\\quad\\quad n=0,1,2,3 ......", "44604e0c97b2985e250c067b39e9d90d": "\\zeta=\\frac{Ab+B}{Cb+D}\n\\quad \\text{ and } \\quad\n\\theta=\\frac{Ac+B}{Cc+D},", "446055dd890cf291dee2fc43e6a8e09f": "x\\ge \\mu", "446063b713aea44eeec8754c7131b39d": "\\nabla^a F_{ab} u^b", "44606c2c9b989ebcbf6a09cfe61cf7b9": "\\scriptstyle I_1 \\,\\cap\\, I_2 \\,\\cap\\, \\ldots \\,\\cap\\, I_m", "4460882675ac548f25541612ce273d56": "c \\ge 2e^{-\\gamma}\\approx1.1229\\ldots", "4460b479ca4eeb572e9924d90fc0d43b": "\\Sigma,", "4460cba505d816ccef288683a5e1c25c": "Q(x,y) = L(x,y) (y - p(x))", "4460e233a2165eea977d3348dd5c8039": "s(t) = \\int_0^t \\ dt^{\\prime} \\ v(t^{\\prime}) \\ , ", "44610d0b43e70e754ae2c34036a13217": "P_{Rx_{dBm}}\\;=10\\log_{10} \\frac{P_{Rx}}{1mW}", "4461142eca08dd0f408686118d0cc62c": "I_{{(Q)}_{[\\epsilon]}} = \\sum_{i=1}^{N_\\epsilon} {P_{[i,\\epsilon]}^Q} = ", "44615593404b40db0957df8547b48242": " k\\ln\\lambda -\\ln k", "446163f1f79d1193edf215c1f18ef8ca": "\\left(\\nabla^2 - \\frac{1}{c^2}\\partial_t^2\\right)\\psi = \\kappa^2\\psi.", "44617f354fbe3f4ab89d5ccafb037ceb": "A=10+\\sqrt{30(10+3\\sqrt{5}+\\sqrt{75+30\\sqrt{5}}})a^2 \\approx 39.306...a^2", "4461b8b1e25b09f39acad8e4b5dd12ba": "EQ(x,y)", "4462a11885ff6160d89f0bf6d06933fa": "X^0 = \\{*\\}", "4462be6592efaf7a5b08b0ef428a54be": " Y_{i+1, 1} = 40,014 \\times Y_{i,1}\\pmod {2,147,483,563}", "4462f28cc1618d9412ae879e6e7735fe": "2\\epsilon=C_3=v_{\\infty}^2\\,\\!", "446301e8699132bde22be0befd7fa574": "f(x)=\\mathcal{F}(\\mathcal{F}^{-1}f)(x)=\\mathcal{F}^{-1}(\\mathcal{F}f)(x)", "44631ddeb4b55fd9a1fcf9eacb5e3678": "\\frac{1}{2}\\frac{\\partial f}{\\partial a_{ik}}", "44637763a9920c16cc4eda83ed64ed4b": " X(x) Y(y) \\ ", "44638b4966111e013c858a9cc270bcf8": "ABV = (\\mathrm{Starting~SG} - \\mathrm{Final~SG})/.736", "4464322b62a1c5926c312abb8cc0a7bb": "L= \\sigma A T^4", "446467777fae673b327abffa3115313e": "\\pi_{\\text{contactName, contactPhoneNumber}}( \\text{addressBook} )", "446499f0344ef2e865ae8785c3699cfb": "P, R", "44650017af57711e0568370f096ec250": "\\Rightarrow h \\approx \\dfrac{\\sum_{x} F'(x)[G(x)-F(x)]}{\\sum_{x} F'(x)^{2}}\\,", "44653ea1aeadcba39e9326ba245d19ea": "g=\\gcd \\left (f^*, x^{q^i}-x \\right )", "44654b79093ba76ed9a67a3a04079b26": " \\mathbf{a} \\wedge \\mathbf{b} = \\frac{1}{2}(\\mathbf{ab} - \\mathbf{ba}) = -\\mathbf{b} \\wedge \\mathbf{a}", "4465607209b9255459ace55b147b9d69": "a_n=n!-1", "44657199fe80b8d47ad14ca11b37f33a": "F_0 = \\frac{M_1^\\mathrm{act} M_0^\\mathrm{pass}}{r^2}", "4465c3be8a09438a69bdd33929dad774": "f_\\lambda \\ \\hat{\\lambda}\\ =\\ -J_2\\ \\frac{1}{r^4}\\ 3\\ \\sin\\lambda\\ (\\sin i \\ \\cos u \\ \\hat{t}\\ +\\ \\cos i \\ \\hat{z}) =\\ -J_2\\ \\frac{1}{r^4}\\ 3\\ \\sin i \\ \\sin u\\ (\\sin i \\ \\cos u \\ \\hat{t}\\ +\\ \\cos i \\ \\hat{z})\\,", "4465fc369812450f61976ced7e841832": "0.5 < L < 0.544 \\,\\!", "446602a6e642829ae87483262b56e8d3": "\n \\frac{\\partial^4 w}{\\partial x^4} + 2 \\frac{\\partial^4 w}{\\partial x^2\\partial y^2}\n + \\frac{\\partial^4 w}{\\partial y^4} = 0 \\,.\n", "44663396eff4e40b48f79a17e5b09678": "-f\\in {\\mathcal F}", "4466a2514b35b5318a9618c67fee7723": "\\,\\log(f(x))\\,", "4466c694ac8c8910904e3f752fc61a44": "\\dim_T X \\leq \\mathrm{Cdim} X \\leq \\dim_H X", "4466f25986a36b0f8b3ba140708a33b3": "A/{\\mathfrak{p}_i} \\subset B/{I_i} \\subset K_i", "44670cca3d599f552c947500bef16edd": "F(x_1,x_2,\\ldots)= \\frac{1}{|G|} \\sum_{g \\in G, \\omega} x^\\omega |(Y^X)_{\\omega,g}|.", "44670ec0191963737ebc930b32d41ace": "m-i", "44673f1856c8ebc026f40ef45a7bccb0": "U(g), \\; g \\in G", "4467993d2c59003c2175bf485c9f5141": "K(\\overrightarrow{D},A)", "4467d10214e430109696be69f61898a5": "m_j=1", "446808091d2ed6a6311d1e155e436da0": "\\beta = (X^TX)^{-1}X^Ty \\, ", "44681913af81ba0d0b3fe067eed109b1": "\\Rightarrow b-d=\\frac{c^2-e^2}{b+d}\\, .", "44683061664ba7fa1c7e3b94ed691530": "\n{d\\over{dt}}\n{{\\partial L}\\over{\\partial \\vec \\omega}} = {{\\partial L}\\over{\\partial \\vec \\omega}} \\times \\vec \\omega + {{\\partial L}\\over{\\partial \\vec v}} \\times \\vec v, \\quad {d\\over{dt}}\n{{\\partial L}\\over{\\partial \\vec v}} = {{\\partial L}\\over{\\partial \\vec v}} \\times \\vec \\omega,\n", "44684ba4c3e8fcc48e7fb2f0bace5214": "0 + 1 + 2 + \\cdots + n = \\frac{n(n + 1)}{2}\\,.", "446866e12758aa9af313c79f41567473": "(l,m)", "44688f1fa0e40a02531aa88a17b7b6b8": "n \\geq 0", "4468bb28fc16649bb30e8dd0635e5305": "s(v_j, v_k)", "4468ef63012809c01bed184205bf6d7d": "\\frac{dP^S}{dQ^S} > \\left|\\frac{dP^D}{dQ^D}\\right|.", "44690acbbab60f3ff1d0de1a7621eedd": "S_{\\gamma}(x)=\\operatorname{prox}_{\\gamma \\|\\cdot\\|_1}(x)", "4469914be3ab668fe7e0495d89c36e3c": "\\left(a, z\\right)> \\left(c, x\\right)", "446998ccf8d357405c32e5a789dcf1df": " v_1(\\vec r)", "4469a92a9c4db1d36e6e988c40637978": "\\boldsymbol{p}_{k-1}", "4469b666ca7c18a7e23ebb7fa3c7a824": "\\,V(W)-V(\\Omega^{-1})", "4469bee41faf6bb7b13e8031d6a6072f": "\\left(\\{\\exp(j_2x)x^{j_1-1}\\}^{(i_1-1)}\\Big|_{x=(i_2-1)\\alpha}\\right).", "446a28c351565faeca09f20cede69d4e": "\\frac{-39161}{(1+0.10)^2}", "446a77e8d91e3587b6bd5afcbafbd643": "E > 2", "446aa80b35fd9da5e4aab4eb2b90573f": "\\omega_c = qB/mc", "446af7c85bbb790e1aadee2dc89dbfb0": "p^2/(p^2+pq)=p", "446b0c5dafd3719881606f8bdf1dc636": "\nT_{ij} = K_i K_j T_i T_j f(C_{ij} )\n", "446b285b57711ef9eb3ab664470b1d07": "\\mathbf x_i(t + \\Delta t) = \\hat{\\mathbf x}_i(t + \\Delta t) + \\sum_{k=1}^n \\lambda_k \\frac{\\partial\\sigma_k(t)}{\\partial \\mathbf x_i}\\left(\\Delta t\\right)^2m_i^{-1}, \\quad i=1 \\dots N", "446b303fe2fd192d3e978e91acc9e2f1": "\n \\underline{\\underline{\\mathsf{S}}} = \n \\begin{bmatrix}\n S_{11} & S_{12} & S_{13} & 0 & 0 & 0 \\\\\nS_{12} & S_{22} & S_{23} & 0 & 0 & 0 \\\\\nS_{13} & S_{23} & S_{33} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & S_{44} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & S_{55} & 0\\\\\n0 & 0 & 0 & 0 & 0 & S_{66} \\end{bmatrix} \n ", "446b3e4e1a45c419c0a40a32c0e778d8": "\\sigma_{11}", "446b845ea342ddf6e00d1a95b442517b": "\n\\left[\\frac{q}{p}\\right]_3 =1 \\mbox{ if and only if } \\left[\\frac{(\\frac{LM'+L'M}{LM'-L'M})}{q}\\right]_3 =1. \n\n", "446b8cbf8a13a6a558205f4f53f9df3f": " \\frac{\\partial^2}{\\partial x^2} F(x) = \\delta(x) ", "446c0e82e767b4c405224dc3a0da78d6": "\\eta_\\mathrm{max}", "446c56fbbd0ca5edf52f7578de66421d": " T(\\epsilon) | \\psi \\rangle = \\int dx T(\\epsilon) | x \\rangle \\langle x | \\psi \\rangle ", "446c58202db086ba166e42b54ff3a2e4": "\\prod_{i=0}^np_i < p", "446c5b884cc341a2d00cc4401e2ce6bd": "\\sum_m\\sum_n a_{m-n} \\lambda_m\\overline{\\lambda_n} =\\int_0^{2\\pi} \\left|\\sum_{n=}^N \\lambda_n e^{-in\\theta}\\right|^2 \\, d\\mu(\\theta) \\ge 0.", "446c788f303476638196aac4298b5fc2": " \\mathbf{x} \\sim \\tau(\\mathbf{y}'_{1}, \\mathbf{y}'_{2}, \\mathbf{C}_{1}, \\mathbf{C}_{2}) ", "446c79e9346bc4f0c6ca61a8d1bab288": "x \\geq \\mathbf{0}", "446c7fc1fa7950df679b566731977537": "H_v[1 \\dots \\rho_v - 1]", "446cd184ab001a427ed6071021602d7d": "\n\\beta_{cr} \\approx (2 + s)\n", "446cf33c6f2461d8d23ff06ee9be08fe": "\\Gamma: Y\\to J^1Y ", "446d19f3fb846fb9aa1374a8f784e18e": "F - L_{X+c}(F) = \\frac{\\mu_X}{\\mu_X + c} ( F - L_X(F))\\,", "446e2b75835e644cf3bd3034a944ea54": "X_1,\\dots,X_n:(\\Omega, \\mathcal{A})\\rightarrow(\\mathcal{X},\\Sigma)", "446e3947f53036747137093de61dd154": "F=F_1\\ast F_2", "446e5f23818524b980af84b4c4166986": "\\mathbf{u} = \\left[U(z)+u'(x,z,t), 0 ,w'(x,z,t)\\right], \\, ", "446e9e8cb3046bc99ce0c5baa7d9cc0a": "\\frac{1}{v_2}\\frac{dx_2}{ds_2}=\\frac{1}{v_1}\\frac{dx_1}{ds_1}", "446ec2ad2356bcfd7ed47b77c8de1c0f": "n - 2^k", "446ed4b0e83c4cbe82a4686c1c0f2bf0": "\\left(x',y',z'\\right)", "446f1b92370fa6bc6282069d202879cd": "(1-R-\\varepsilon)H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k \\cdot N", "446f8787a749baf0f0ad0bc2d867dd65": " \\operatorname{def}[F_1] = \\operatorname{false} ", "446f91fcd969543e977b7a8855aa5461": "[20 a_1 + a_2] a_2\\cdot10^2 ", "446fb0669b799252da8d494929dc8f4b": "\\displaystyle{F(z)=\\sum_{n\\ge 0} a_n z^n,}", "446fb5271f2dc2154319848844f65eb0": "N_{B(f)} = N_{B(\\Delta)}\\,", "446fe30b98e385ae849e5ee4f2ad290d": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} I_{lm} r^{l}\n\\sqrt{\\frac{4\\pi }{2l+1}} \nY_{lm}(\\theta, \\phi) \n", "446ff424e6ee4fbe3c8658c23f5e060c": "c_q(n)=\n\\sum_{\\stackrel{1\\le a\\le q}{ \\gcd(a,q)=1}}\ne^{2 \\pi i \\tfrac{a}{q} n}\n.\n", "44702e1ac69a9b256667f29e1016a5c7": "\\liminf\\frac{p_{n+1}-p_n}{\\sqrt{\\log p_n}(\\log\\log p_n)^2}<\\infty.", "447057564ad9a2b7b33814f0abcfc94a": "U(\\lambda) = e^{\\lambda M^* - \\bar\\lambda M}", "4470804fd5003639b151a329e42422f5": "x0} m_\\alpha \\, \\coth \\alpha \\, A_\\alpha,", "4476595fb49cb0c382cb6d5cd92271b5": "c^2<4km", "44767266e4d50d38d0a276b749cf7a20": "\\left(c'\\right)^+", "44771d48ba681f02dcdc602eeb8f76cf": "\n\\frac{\\sqrt{114}+10}{14} = 1+\\frac{\\sqrt{114}-4}{14} = 1+\\frac{114-16}{14(\\sqrt{114}+4)} = 1+\\frac{1}{\\frac{\\sqrt{114}+4}{7}}.\n", "44772634fe1954c6678782289779d944": " e \\equiv \\frac{h^2A}{G M}. ", "44772f04f97a372af551685eca3d83eb": "G_2/Z_2", "447733b6b6c1e85cc5ca64821d75efa5": "r_\\pi \\geq r_{\\pi '} \\, ", "447735279a512ab0d1abec924b553d82": "[g,x]=g^{-1} x^{-1} g x", "447743500b245f33104611a91cb6df43": "\nV = V_0 + \\sum_{i=1}^N \\sum_{\\alpha=1}^3 \\Big(\\frac{\\partial V}{\\partial \\rho_{i\\alpha}}\\Big)_0\\; \\rho_{i\\alpha} + \\frac{1}{2} \\sum_{i,j=1}^N \\sum_{\\alpha,\\beta=1}^3 \\Big(\n\\frac{\\partial^2 V}{\\partial \\rho_{i\\alpha}\\partial\\rho_{j\\beta}}\\Big)_0 \\;\\rho_{i\\alpha}\\rho_{j\\beta} + \\cdots,\n", "4477b60b5efb5e83beef5a884a05fa80": "T_{JMAX}", "4477bd8278a1118a3b44da4ca3143ee6": "W^\\mathrm{adiabatic,\\,quasi-static}_{A\\to O} = -W^\\mathrm{adiabatic,\\, quasi-static}_{O\\to A}\\,.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)", "4477f13a1ff33879dedf3f1cf412181c": " \\begin{align} \n& \\Psi(x_b,y_b,z_b,t) \\\\\n& \\frac{\\partial}{\\partial x}\\Psi(x_b,y_b,z_b,t) \\quad \\frac{\\partial}{\\partial y}\\Psi(x_b,y_b,z_b,t) \\quad \\frac{\\partial}{\\partial z}\\Psi(x_b,y_b,z_b,t) \n\\end{align} \\,\\!", "4477f71fb24c294ee13e259999735391": "F(d, k) = F(\\alpha \\cdot d, k)", "4478c2376dbeffa35018103c341049ae": "a = \\frac{4r}{\\sqrt{3}}.", "4479141bd61fbb874d4ffb4dfce0303c": "W^{\\prime} W=I", "44796a5e31dfee98d7265b0973687ad3": "\\forall x_1 \\ldots \\forall x_n \\; P(f(x_1,\\ldots,x_n))", "447974c411a2f1a72b37026f687897f4": "Z(G) = \\{z \\in G \\mid \\forall g\\in G, zg = gz \\}", "4479943cb43a365d834f5a3bf9840384": "\\ln u(x)=\\sum_{i=1}^L {\\alpha_{i}}\\ln x_i", "447a1f28b46ad8c819705ff03bb7e281": " \\frac{\\mathrm{d} \\mathbf{u}_{\\theta}}{\\mathrm{d}t} = -\\frac {\\mathrm{d} \\theta} {\\mathrm{d}t} \\mathbf{u}_{\\rho} \\ . ", "447a24ace058d90a2f23cde5c15e7691": "E_{average}", "447a4bdb2b963f78d15776ba0a8a9cf7": "\\{ \\,(2,11) \\}", "447abb72c5f2b9e2960fca7de0a9a45d": "n\\sim 1/\\alpha", "447b029af2b070ff56ef7396d7c1b310": "t_1 = \\alpha \\sin(\\tau), t_2 = \\alpha \\cos(\\tau),", "447b61651151de11f309582c606c6201": "\\mu(\\xi) \\sim \\xi", "447b94850c5c0949db4fc5ee8b93df2b": "(x_i-1, y_i+1)", "447bb3c4ef26fb47b4ad48e5c596ae4e": "J_2^0, J_3^0", "447bb6af0c02c8ddef7dc3095be90bb0": "d \\left( f(s), f(t) \\right) \\leq \\int_{s}^{t} m(\\tau) \\, \\mathrm{d} \\tau \\mbox{ for all } [s, t] \\subseteq I", "447bbaaabd85d9eb98f3fb3c6a307f49": "\\int f(x)dx = 1/M", "447bc5e28320e7717f121d9c72421ded": " \\phi_h = H = \\langle h| ", "447bd2bb66c2e20abf7ee1098d4cfb59": " {}_RW_P = \\int\\limits_R^P {pdV} = p\\left( {V_P - V_R } \\right) ", "447be54c82d591fc378032906c1e1bca": "\\ {F_{vert}} = {F_H} \\times sin(\\theta) ", "447c5622a903ebeafa082da86c5c8d25": "\\neg \\Box x", "447c650d217dfd01766bebe7f76c1e73": " s_1-s_2 =-s_3+s_4 ", "447c66ee06befc4d72fe95ac418e942b": "D_L", "447cb9a34dc1b414a7d5e373370b7a01": "S=\\mathrm{End}(E(R_R))\\,", "447d58c25e09cd678116872d54036092": "h(x) = h(y)", "447db215f81fb340596a45be1c1afa97": "E(r_0) = - \\frac{M z^2 e^2 }{4 \\pi \\epsilon_0 r_0}\\left(1-\\frac{1}{n}\\right)", "447e3415ff532b47ddb5db27db5345a1": "{}^{x}a \\approx \\begin{cases}\n\\log_a({}^{x+1}a) & x \\le -1 \\\\\n1 + \\frac{2\\ln(a)}{1 \\;+\\; \\ln(a)}x - \\frac{1 \\;-\\; \\ln(a)}{1 \\;+\\; \\ln(a)}x^2 & -1 < x \\le 0 \\\\\na^{\\left({}^{x-1}a\\right)} & 0 < x\n\\end{cases}", "447e6eb4dd661dc9f15dafdc8f9899b0": "F = dA", "447e74317a1397a0344e3d5f703ba874": "\\left( \\begin{matrix} n \\\\ k \\end{matrix} \\right)", "447e76f0ef6500e4d15f506588078c1c": "Physics \\geq good", "447ee0e55e097f2c29b7b092228b3cae": "A=A_o \\cos[2\\pi(x/\\lambda- t/T) + \\varphi]\\,", "447f719088c98c2a07251aac69e3e5db": "\\sqrt{\\scriptstyle{s(s-a)(s-b)(s-c)}}", "447f7847c63ec165224cc4f378a3dd91": "\\mathbf{1}_A\\colon X \\to \\{0, 1\\},", "447f8b07b0c95b96d8699748e92aa1fd": "\\frac{1}{1-(1+x)y}=1+(1+x)y+(1+x)^2y^2+\\dots,", "448003cfa0a8714f2327a176397f5a17": "k_2/Q", "448026ddf507e6d6dd98e230906e52d1": "z_{t+k}", "4480292f8a56a1789a8af984d7205536": "|\\mu| (X \\setminus K_{\\varepsilon}) < \\varepsilon.", "4480b0a04ee481ecb0ba3017058648ec": " \\deg P_n = n~, \\quad \\langle P_m, \\, P_n \\rangle = 0 \\quad \\text{for} \\quad m \\neq n~.", "4480ba6063f228328bba525b1c17d600": "\\mathbf{O(log|A|+m)}", "448166e4bc0962ab745c758cca34378a": "\\varphi\\circ f", "44817eda957c6f11962ecb5c899942e9": "\\Delta\\lambda = \\frac{ \\lambda_0^2}{2n\\ell \\cos\\theta + \\lambda_0 } \\approx \\frac{ \\lambda_0^2}{2n\\ell \\cos\\theta } ", "4482323e0dc0764cba2b6cb73c4ffac2": "\\text{DOL} = \\frac{\\%\\text{ change in Operating Income}}{\\% \\text{ change in Sales}}", "448234dc5f55be1b97dcd71179b16bd3": " A^r\\to I\\otimes A^r\\xrightarrow{\\eta^r}(A^r\\otimes A)\\otimes A^r\\to A^r\\otimes (A\\otimes A^r)\\xrightarrow{\\epsilon^r} A^r\\otimes I\\to A^r", "4482d273a2ee4e8bd96b9417f0e04dc5": "A_{\\beta,s}x=\\sum_{i=1}^r\\sum_{m=0}^{\\min(s,k_i-1)}\\binom{s}{m}(\\lambda_i-\\beta)^{s-m}A_{\\lambda_i,m}x_i", "44831a3b6a5e56887b7190c15a74167f": "\\left\\{\\begin{array}{ll}1 & m = n = 1\\\\ 2 & \\text{otherwise}\\end{array}\\right.", "448339189fa6df8cea9729084485d323": "Z_{i\\Pi}=\\frac{1}{\\sqrt{1-\\omega^2}}", "448399050edd6a4a5192c8c98209d445": "\\mathcal{Y} = \\mathcal{F}(\\mathbf{x}) ", "4483ca060008c479431b1cdca9f41f23": " 8x = 2\\sin(\\theta) \\cos(\\theta) + 2\\theta = \\sin(2\\theta) + 2\\theta \\,", "44844dfe862d9111d8d1f3a3c7cdbc02": "\\Box \\phi = 4 \\pi \\, \\rho", "44844ead77e21f2abc2faaa08146386c": "\\|f*g\\|_s\\leq \\|f\\|_r\\|g\\|_p", "44847ee33dd7de5ce47bcc2db73fb840": "T\\mathcal M", "44848ce1714f154c9430462e41bc8264": "k < {j \\over r} \\text{ and } {j + 1 \\over r} < k + 1 \\text{ and } m < {j \\over s} \\text{ and } {j + 1 \\over s} < m + 1 \\,", "44849938bc7ed8699c37c76a27fd8169": "k=1,...,n", "4484b35246e3d2d116089ba9fd381a61": " r, \\theta \\,", "4484c8dc60579fb0fef651d19732a76c": "\\lim_{\\Delta g\\to 0}{\\left.\\frac{\\Delta U}{\\Delta g}\\right|_{c.p.}}", "4484cc67456880c539dc69d1c63c5159": "\\beta_N^{rev}", "4484fb785ab5e9ba71f968cedd808ca9": " \\Gamma_a ~,~ \\Gamma_{a_1 a_2} ~,~ \\Gamma_{a_1 \\dots a_5} ", "448547db844c2b17cfd406a6d3f07a52": "(\\mathcal{O}_k/\\mathfrak{p}) ^\\times = \\mathcal{O}_k /\\mathfrak{p}- \\{0\\}.", "44856c879fe95c80ccbf9cb83466fae3": "(a(bc))d", "44857f618e4d3e2a931abb4fd2d14d3e": "\n\\overline{n_\\mu ^k}= 1/i^k\\frac{\\partial ^kC_\\mu (s,t)}{\\partial s^k}|_{s=0}.\n", "44862498ee59bc6612cdaabb79d3d00c": "s=\\tfrac{1}{2}(a+b+c)=\\tfrac{1}{2}(7+4+5)=8", "448636030f3c770187284d5b700a2d87": "\n \\cfrac{\\partial W}{\\partial I_1} = C_1 \n ", "44869f90be7b0d72004352cfac7151c5": " g(1)=0 ", "4487224631ad8f9729b00a4cae9bfd62": "c_{V,W}^{-1}(w\\otimes v):=v_{(0)}\\otimes S^{-1}(v_{(-1)})\\boldsymbol{.}w.", "4487a883deb0f21f6c3e5e29e64d961f": "\\textstyle\\vec{M}_W", "44880fe0724459fc9f82eb111d63a12b": "T_D=\\int\\bar{\\sigma}d\\bar{\\epsilon}^e=\\cfrac{1}{6G}\\bar{\\sigma}^2", "44881b458c040621c68d62584f04ee4d": "b \\in F", "44883f7165dadd9c144b86e323675aa9": "\nPoss(a,s)\\wedge\\gamma_{F}^{+}(\\overrightarrow{x},a,s)\\rightarrow F(\\overrightarrow{x},do(a,s))\n", "448851c11cd8ea76df6902ecc93d21f1": "d\\Omega", "4488818c86fcc4d4eb2225b4f29bcda0": "u_x'=d\\tilde\\phi / dz, u_z'=-i\\alpha\\tilde\\phi", "44889b853673f058fd885f65b13ba83d": " \\left(1/2,\\frac{+ \\sqrt{3}}{2}\\right) \\;\\; \\mathrm{and} \\;\\; \\left(1/2,\\frac{-\\sqrt{3}}{2}\\right) ", "4488b09592ea8e82e1fd50811c3e73f2": "\\frac{\\varphi^{n+1} - \\varphi^{n}}{\\Delta t} = F( \\varphi^{n+1} ),", "4488ba5f7e82e2d8c136b559d95283d5": "=\\,", "4488c56035cc333b99ddf4c55a52b28f": "f_i^{(j)}(\\alpha_{n_t+1})+\\dots+f_i^{(j)}(\\alpha_{n_{t+1}})", "4488df7ff1f69ce287b5ad9befd7c98f": "\\vec{v}_E = \\frac{\\vec{E}\\times\\vec{B}}{B^2}", "448907e9fd7b3766a28e4605b21233b7": "x_{k+1}=a_{k+1}", "448918bdd3e7a641824cedfcd1aa90a9": "S(x)=\\int_0^x \\sin(t^2)\\,\\mathrm{d}t=\\sum_{n=0}^{\\infin}(-1)^n\\frac{x^{4n+3}}{(2n+1)!(4n+3)}", "44892aedbcf909d35736f385cdc15d5a": "f(\\boldsymbol{x}) = \\sum_{i=1}^{n} x_{i}^{2}.\\quad", "4489358dd4e46126f4e3cc627f3c8cd6": "W=\\frac{\\tilde{W}}{\\frac{dz}{d\\zeta}} =\\frac{\\tilde{W}}{1-\\frac{1}{\\zeta^2}}.", "4489ab1fe27100c15d3bfc337cccb836": "\\left(\n\\frac{\\left(15 + 7\\sqrt{5}\\right)^2 \\pi}{12\\left(25+10\\sqrt{5}\\right)^{\\frac{3}{2}}}\n\\right)^{\\frac{1}{3}} \\approx 0.910", "448a24713372bbe4f5df28a80c0e93e2": "Z_0=Tr[\\rho_0]", "448a2fec77b67a42064f5d1686989790": "\\gamma_{n} \\mbox{ also } \\in H", "448a7bb6e3836ee6e272733959140297": "\nE =\n{a_1 a_2 \\over 4 \\pi r } \\exp \\left ( -m r \\right )\n", "448aaa9bdf63e8c52b88faa551f419bd": "K^{\\times}/(K^{\\times})^n,\\,\\!", "448ab8c778492f6438db70761cc42ee0": "p_i = (x_i,y_i)", "448b00ba1e578040dbf6d856682aea1c": "\\arctan (-x) = - \\arctan x \\!", "448b205eef89ae6b003079ba8755be4e": "\\epsilon_{w}", "448b45ff65721bb39b7a38df07b35471": " \\Big(\\sum _{x\\in X}f(x)g(x)\\mu(x)\\Big)\\Big(\\sum _{x\\in X}\\mu(x)\\Big) \\ge \\Big(\\sum _{x\\in X}f(x)\\mu(x)\\Big)\\Big(\\sum _{x\\in X}g(x)\\mu(x)\\Big).", "448b6a1b9936f0481f65fa86b0079310": "u(t_k)=u(t_{k-1})+K_p\\left[\\left(1+\\dfrac{\\Delta t}{T_i}+\\dfrac{T_d}{\\Delta t}\\right)e(t_k)+\\left(-1-\\dfrac{2T_d}{\\Delta t}\\right)e(t_{k-1})+\\dfrac{T_d}{\\Delta t}e(t_{k-2})\\right]", "448bada5d951d4fe877894838454f1d4": "\\frac{2 D^2}{\\lambda}", "448c19c0188e27c61e01e4ffc8741cac": "\\cos{\\theta} = (F - Fb) / I\\sigma\\ ", "448c2949232623499df308f28764c1c9": " \\alpha(t) = \\Omega t - \\arcsin\\left(\\frac{E}{R} \\sin(\\Omega t) \\right) ", "448c2cb4c0298f490b74cba41a65cd0b": "\\mathcal{D} := \\mathbf{P} \\mathbf{D}^t", "448c44956613fcf38b32a85682c471a5": "\\lambda \\in \\R", "448c8e0e178a213f20c1c1c0d528946d": " \\lVert \\hat{y} \\rVert_\\infty \\leq 5ip^{1/m} \\rbrace ", "448c9a4530b583a4b3566dcbe443fc52": "(\\pm\\sqrt{2},1)", "448ca66c27c5283478b51b50d52a1492": " f_A : \\lbrace 0, ... , d-1 \\rbrace ^m \\longrightarrow \\mathbb{Z}_q^n ", "448ca9e40f83e7558ee436f37d1d8d3c": "\n J =\n \\begin{bmatrix}\n 2 & 4 & 1 & 3 & 2\\\\\n -1 & -2 & 1 & 0 & 5\\\\\n 1 & 6 & 2 & 2 & 2\\\\\n 3 & 6 & 2 & 5 & 1\n \\end{bmatrix}\n", "448cad941486f48cfc023c4367a41a80": "V_c\\;", "448ccba051721096b69d387ff77a89bf": "\n\\mathbf{F}_{\\mathrm{centripetal}} = -m \\mathbf{\\Omega \\ \\times} \\left( \\mathbf{\\Omega \\times x_B }\\right) \\ ", "448cd24ac21910d37503325724ccd132": "\\exp(x), \\tan(x), \\ln(x), \\Gamma(x)", "448cef424de02e1b38faf8f88ba9ce34": "p = qr + 1\\;", "448d3fae4e8c6305d8ec922341204513": "\\sum_k {n\\choose k} x^k = (1+x)^n.", "448da86345eaaf7ad52e9b95a4e37619": "\n\\int_a^{x_0-\\delta} e^{n f(x) } \\, dx + \\int_{x_0 + \\delta}^b e^{n f(x) } \\, dx\n", "448de91a67a83bad50364543cd136d99": "-15\\, X^4+3\\, X^2-9,", "448e070bc1e79c8d475cdab8fd9893a3": "3:2\\ ", "448e0eaf4395a0df4f49a120ee29caa9": "h_i=H(X_i)", "448e1a0554e7a44653db21090441fea3": "r = 1", "448e474aff351858062426f7aa14d6cb": "f^*\\theta(E) = \\theta(f^*E) = \\theta(\\lambda_1 \\oplus \\cdots \\oplus \\lambda_n) = \\theta(\\lambda_1) \\cdots \\theta(\\lambda_n) = w(\\lambda_1) \\cdots w(\\lambda_n) = w(f^*E) = f^* w(E).", "448e7c413cd561da48872151e89cfec3": "0.0456\\pm0.0016", "448e7da39e2ce21f673a7b91edee1d4f": "\\overset{+}{|}\\quad \\overset{+}{|}", "448e9645416b0a125996cf921b98941a": "1.\\overline{1}", "448ebe8d62ab6c47b51366696cc6b439": "\\rho ' = \\rho \\sqrt{\\frac{z - 1}{z}}", "448ed946a7dea322d43cdf3a9f078d09": "\\frac{\\partial}{\\partial t} \\psi^*(x, p) = \\left[- \\frac{\\partial H(x, p)}{\\partial p} \\frac{\\partial}{\\partial x} + \\frac{\\partial H(x, p)}{\\partial x} \\frac{\\partial}{\\partial p} \\right] \\psi^*(x, p).", "448edfccf5e821e08bad6a147947c276": "\\mathit{\\Alpha\\Beta\\Gamma\\Delta\\Epsilon\\Zeta\\Eta\\Theta} \\!", "448f0c088996fc2a2d8e0d8127ba7b3c": "\\hat{\\mathbf n}(t')=\\mathbf R/R,", "448f217e4982e64d6d82a5b7bd6b4927": "X-Y \\sim \\mathrm{Logistic}(0,\\beta) \\,", "448f95f7b57a8c0364a001a605ba8a0d": " \\zeta_G ( \\alpha ) = p(N-1) + (N-1)(1-p)2^{-\\alpha}. ", "448f9d5603c2a7500035080fd6fb4e2c": "\nK(\\overrightarrow{D},A):=\\sum_{\\{a_{1},...,a_{S}|\\sum_{i=1}^{S}a_{i}=A\\}}\n\\prod_{i=1}^{S}\\frac{\\overline{s}\\left( n_{i},a_{i}\\right) \\overline{s}\n\\left( a_{i},1\\right) }{\\overline{s}\\left( n_{i},1\\right) }\n", "448fad5b5bcc85fbfa00a9bce988f863": "E \\,\\!", "4490067ac24a32e7d029c279d71164b2": "\\mathbb{R}^{3}", "44901c42de22a50e20ee09fff5bdb751": " \\textbf{f} = -1+X+X^2-X^4+X^6+X^9-X^{10} ", "44902348db61d6acfc68cea967ce48c6": "m(C)=(b_1-a_1)(b_2-a_2) \\cdots(b_n-a_n).", "44902f48d281c9786ac3a2fc04f2f231": " W_2=X_1^2+2X_2", "449059d3bbbe9d34315338c78dbc917e": "C_D= \\frac{2\\sum_{i=1}^S x_i y_i }{(D_x + D_y) XY }", "44907c4437e4c9adf2ca59d35107b12d": "\\left\\{\\begin{pmatrix} 1 & b \\\\ 0 & 1 \\end{pmatrix}\\mid b\\in\\mathbf C\\right\\};", "44908bc6c497f97c80ab70e686a40c86": "A = {x_2 \\over x_1 } \\bigg|_{\\theta_1 = 0} \\qquad B = {x_2 \\over \\theta_1 } \\bigg|_{x_1 = 0},", "4490bfca6c14cf734806ba625ef60baf": "3x+3", "4490f1b7b8e5aa02cc740394296d462f": "= 160", "4491184e91097003f1c74478537f952f": "\\det(\\mathbf{A}) = a(ei-fh)-b(id-fg)+c(dh-eg).", "449126ca16bc394254e4fcffd9dc2f4b": "\\mathbf{F}_{\\mathrm{body}} = \\int\\limits_{V} \\mathbf{f}(\\mathbf{r}) \\mathrm{d} V \\,,", "44919013a19ca08a9ce77272993aa167": "\\mathcal{F}_{\\alpha+\\beta} = \\mathcal{F}_\\alpha \\circ \\mathcal{F}_\\beta = \\mathcal{F}_\\beta \\circ \\mathcal{F}_\\alpha.", "4491ca82aa7853b35a0e1fe8aab1b96e": "\nJ \\equiv \n\\frac{\\partial (\\mathbf{Q})}{\\partial (\\mathbf{q})}\n\\left/\n\\frac{\\partial (\\mathbf{p})}{\\partial (\\mathbf{P})}\n\\right.\n", "4491de600581664065366698ca8c60c9": "v_b=\\frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b}", "44922f43804b6d9ac89a3a8b56811299": "(g_1,g_2,g_3)", "4492553eec2a3fa5bc20a8790ce9634e": "2^{km}\\,\\!", "4492623ab9985c9c7a4fbee5a2c5d113": "\\|f\\|_p=\\left(\\int_D \\left|f\\right|^p\\,d\\mu\\right)^{1/p}", "4492b71a20488fdf8ecd3de10badae55": "N_{L/\\mathbf{Q}}\\,", "4492b95f9238e472a1584ef6af87aa4e": " P_n(x) = c_n \\, \\det \\begin{bmatrix}\nm_0 & m_1 & m_2 &\\cdots & m_n \\\\\nm_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\\n&&\\cdots&& \\\\\nm_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\\n1 & x & x^2 & \\cdots & x^{n}\n\\end{bmatrix}~,", "44933cfa1e329d165d7a2d38aa7453b2": " \\vec w \\propto (\\Sigma_{y=0}+\\Sigma_{y=1})^{-1}(\\vec \\mu_{y=1} - \\vec \\mu_{y=0}) ", "44934c9b837c0163bd600c70ed785671": " (y_{1}, y_{2}) \\, ", "44939288011406871726bb6f1bc696f9": "\\langle x, x' \\rangle", "4493a0bdeab4aa7e40bb0340ea5ca48a": "\\mathrm{conn}(x)", "4493bc19e404ae546d6a7cb7bdd29004": "_{s.13 \\,}\\!", "4494197f3fe6211559a074ec23c75f85": "\\scriptstyle R\\left( z \\right) \\;=\\; R_0 \\,+\\, A_k \\cos \\left( kz \\right)", "449493360f77f674d9dbb171814a06a4": "1+\\lceil\\Delta/2\\rceil", "44949a66de446da2bcfea3c887d8cf04": " dq_n=-Kp_n q_n[\\lambda + (1-2 \\lambda )q_n ] ", "4495884fcc40afb40b84972e9192e75f": "\\xi_d^2\\left(\\frac{d^2\\theta}{d\\zeta^2}\\right)+\\sin{\\theta}\\cos{\\theta}=0", "4495a9c9688e54cd9991609d75be282e": " \\rho \\in C^\\infty(K; \\mathbb R)", "4495dd15d750f045da4379cf596347c3": "\\tfrac{2}{n} \\times x_i^2", "4495f2b3587333aa6db195e240758deb": " k^{-5/3} ", "44960e0a146e6fbf053407de8df15858": "(C^{\\textbf{.}}(\\mathcal U, \\mathcal F), \\delta)", "4496376b2a06a5874940a69f2ba96c2a": " \\cos\\theta\\ obs=\\ f \\cos\\theta\\!_1 +(1-f)cos\\theta\\!_2 ", "44968a79e11517b6f8fb4b3a104f2712": " \\frac{d\\sigma}{d\\Omega} =\\left(\\frac{ Z_1 Z_2 e^2}{8\\pi\\epsilon_0 m v_0^2}\\right)^2 \\csc^4{\\left(\\frac{\\Theta}{2}\\right)}. ", "4496a140844a5caf6aa5b1a5226a9ebd": "\\lim_{x \\to 0^+} x \\ln x = 0.", "4496a6c85064d97617fcd736da99bab6": " |\\tau_n| < \\varepsilon h ", "4496c1f61e552ddf62e27f01cd978988": "\\begin{align}\n\\int\\frac{dx}{{a^2+x^2}} &= \\int\\frac{a\\sec^2(\\theta)\\,d\\theta}{{a^2+a^2\\tan^2(\\theta)}} \\\\\n&= \\int\\frac{a\\sec^2(\\theta)\\,d\\theta}{{a^2(1+\\tan^2(\\theta))}} \\\\\n&= \\int \\frac{a\\sec^2(\\theta)\\,d\\theta}{{a^2\\sec^2(\\theta)}} \\\\\n&= \\int \\frac{d\\theta}{a} \\\\\n&= \\tfrac{\\theta}{a}+C \\\\\n&= \\tfrac{1}{a} \\arctan \\left(\\tfrac{x}{a}\\right)+C\n\\end{align}", "4497058c2a21bd9bac1ce904a1d60287": " \\dot{m_w} ", "4497266d6e3b6cd93da49cbd2350311b": "= 1/5 \\times (1/5 + 2/5 + 1/40) = 1/5 \\times (1/5 + 1/3 + 1/15 + 1/40) = 1/25 + 1/15 + 1/75 + 1/200", "44976b9113672040161793e2a720b2d6": "\\left\\{f\\left(x\\right)\\mid x\\in S\\right\\}", "44977db9771367b0828eeb1c1f4bfe52": "V(x) =\n\\begin{cases}\n0 & x < -a\\\\\nV(x) & -a < x < a\\\\\nV_0 & a < x\n\\end{cases},\n", "4497917eebae9fb5b1d80700f66c78bb": " = (\\lambda z.z\\ a\\ b)\\ (\\lambda x.\\lambda y.x) ", "44979879ac622c2ecea2ec2730284297": "g{(x)}", "4497cb254f0479f904dcb25ea4f588bd": " \\alpha\\, , \\beta ", "4497e587736fe02d412b60524526e51d": "\\partial_t q = (s_r+ i s_i) \\Delta^2 q + (d_r+ i d_i)\n\\Delta q + l_r q + (c_r + i c_i)|q|^2 q + (q_r + i q_i) |q|^4\nq.", "44983166ca42292222adda5b68004321": "\\frac{1}{\\Omega}", "4498ad9dcfb56920a6a7a2429986d94d": "A \\log x", "4498dc73248c3d9485e24c7d93536386": "\\textstyle \\frac{1 + \\varepsilon}{n}", "4498fe9e99783f8f87ca69bcd95e6528": "E^2 = c^2 \\mathbf{p}\\cdot\\mathbf{p} + (mc^2)^2 \\,.", "449919bb696698bc10874054d432135d": "\\frac{u(x+h) - u(x)}{h} \\approx u'(x)", "4499ae0a38d2ab17e44ce668d3bed02e": " v_g= \\frac {2\\pi\\Delta f}{\\Delta k} =\\frac {\\Delta \\omega}{\\Delta k}\\ , ", "4499c1c34b898e2f767871be3ab2274b": "\\ i_4^2 = \\cdots = i_7^2 = +1 .", "4499e3bc509612cf7349c8892b9c9bc3": "\nMax_{S^{1}\\wedge\\cdots\\wedge S^{T-1}}\\left[P\\left(S^{1}\\wedge\\cdots\\wedge S^{T-1}|S^{T}\\wedge O^{0}\\wedge\\cdots\\wedge O^{T}\\wedge\\pi\\right)\\right]\n", "449a33812f5a519d0d4007bce1edb8fa": "\\rho_1,\\ldots,\\rho_N", "449a48d23763b84c6e8492e36bcc0092": "r[k]=\\sum_{n=-\\infty}^{\\infty}h[n]s[k-n]+n[k]", "449a825dc6b98ab9231117ffa693a36e": "\\mathbb{THE}", "449afb3cd5dba72221f7755167c19441": "Ae^{i\\omega t}\\left( 1+\\frac{\\beta}{2}(e^{i\\Omega t} - e^{-i\\Omega t})\\right) = A\\left( e^{i\\omega t}+\\frac{\\beta}{2}e^{i(\\omega+\\Omega) t}-\\frac{\\beta}{2}e^{i(\\omega-\\Omega) t}\\right) .", "449b20015c0a6726c4a96a71d58344a0": "P_m(t)dt=0\\,", "449b39a45faa8fa544308d13e2d927b5": "P=\\Re (L\\oplus \\bar{L})", "449b6f459d5181c4e1bf39ead8eb9670": "y_{ij}=1", "449bd97ad17b6d127f6d87176f46dc39": "= (0.8 \\cdot 1 \\cdot 1.78 + 0{,}1 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.0961) = 1 \\cdot 1.78", "449beea8a145a296b8b399041eafad2b": "\\operatorname{not2}\\ \\operatorname{true} = (\\lambda p.p\\ (\\lambda a.\\lambda b. b) (\\lambda a.\\lambda b. a)) (\\lambda a.\\lambda b. a) = (\\lambda a.\\lambda b. a) (\\lambda a.\\lambda b. b) (\\lambda a.\\lambda b. a) = (\\lambda b. (\\lambda a.\\lambda b. b))\\ (\\lambda a.\\lambda b. a) = \\lambda a.\\lambda b.b = \\operatorname{false} ", "449c570fd467a1f6d90f49c3c2180e9d": "X \\leftarrow Y \\rightarrow Z", "449c7b7a1a9089abd602e559afd383f2": "\\mathcal{M}^\\prime := \\mathcal{M} + \\mathcal{D}", "449cb408accc71570ff34624d06044db": "\\nu_\\perp", "449cece5eed7055d051cd03f3ab00f47": " J = S = 3/2 ", "449d018ca52bb065d3d13406e1d3d617": " \\partial_t^k h = F\\left(x,t,\\partial_t^j\\,\\partial_x^\\alpha h \\right),\\text{ where }jR", "44c48de9c23848e9d8c8536773313bb5": "GF(2^s)", "44c49c793546b0688b2fbe6db7851b95": "G_1:=\\bigsqcup_{\\alpha,\\beta}U_{\\alpha\\beta}", "44c4d54268754aaff1134bc163e24712": "\ny_t=\\nu +A_1y_{t-1}+\\dots+A_p y_{t-p}+u_t\n", "44c51e9aa58f8958330a46cf49f6e980": "\n u_r(r) = -z\\phi(r) \\quad \\text{and} \\quad u_\\theta(r) = 0 \\,.\n", "44c54d948ab3a851efa760f120e04835": "M = M_0 \\supset M_1 \\supset M_2 \\supset \\cdots", "44c5dd4ef255bf9bad161f77cc1f6915": "x^6", "44c67a74ca9252585ee75aeb79763c64": "{\\omega}^0", "44c6f17733f441189a3880a1ac356f14": "\\mathfrak{P}^{114}", "44c76253231e3c0ae1359d3d06178b73": "q = \\int_{\\phi=0}^{2\\pi} \\int_{\\theta=0}^{\\pi/2} I \\cos \\theta \\sin \\theta d\\theta d\\phi", "44c7c5fb0fcfa95fa7dd7d8c3a665ad0": "CAGR \\approx AR - \\tfrac{1}{2}k\\sigma^2", "44c832eed48c877e481801a7d2e5877a": "\\left(1-\\frac{2}{2^s}\\right)\\zeta(s) = \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n^s} = \\frac{1}{1^s} - \\frac{1}{2^s} + \\frac{1}{3^s} - \\cdots.", "44c841fc1efbf79653541ebb7695e6b6": "\\{\\varphi(x,t):t\\in\\R\\}", "44c8b0285fd47b807e9d7ab62c272091": "x^0 > y^0", "44c8b21d23d726d6e972dcbb77ccbeb0": " \\int_{-\\infty}^\\infty \\frac{- \\ln f(x)}{1 + x^2} \\, dx < \\infty ", "44c8cebc7fded9a350d33ec9a7e74fe1": " Spin(12,\\mathbb C)", "44c97eed8218f7a041242083247912bb": "\\mathrm{C_nH_aO_bN_c\\ +\\ dCr_2O_7^{2-}\\ +\\ (8d\\ +\\ c)H^+ \\rightarrow nCO_2\\ +\\ \\frac {a + 8d - 3c}{2}H_2O\\ +\\ cNH_4^+\\ + \\ 2dCr^{3+}}", "44c9a86d519b9a3c76d6ca56568aca0f": "F_r", "44c9ab27e4aa9f4ecafd7f4aa074b451": "1/(1-t)^{\\chi(X)}", "44c9ead4157ab04738510b07f9542531": "= \\operatorname{tr} (\\gamma^0 \\gamma^0 \\gamma^5)", "44cac4401937c1d938d86574f49fd465": "\\limsup_{y\\to x} f(y)\\le f(x)\\,", "44cb3d7ac21a34d09355975fb14bea87": "\\begin{align} \ny_{1} &=b_0 \\left( \\frac{(\\beta )_{\\alpha -\\beta }(\\beta +1-\\gamma )_{\\alpha -\\beta }}{(\\beta +1-\\alpha )_{\\alpha -\\beta -1}(1)_{\\alpha -\\beta }}s^{\\alpha -\\beta }+\\frac{(\\beta )_{\\alpha -\\beta +1}(\\beta +1-\\gamma )_{\\alpha -\\beta +1}}{(\\beta +1-\\alpha )_{\\alpha -\\beta -1}(1)(1)_{\\alpha -\\beta +1}}s^{\\alpha -\\beta +1}+ \\cdots \\right) \\\\ \n &=\\frac{b_0}{(\\beta +1-\\alpha )_{\\alpha -\\beta -1}}\\sum_{r=\\alpha -\\beta }^{\\infty} \\frac{(\\beta )_{r}(\\beta +1-\\gamma )_r }{(1)_r (1)_{r+\\beta -\\alpha }}s^r \n\\end{align}", "44cb4f3d5ef2563979d08f3bfc9c5531": "\\frac{\\pi}{2} \\ (90^\\circ)", "44cc5e95f5aeac02772e559041e30ed4": "\\scriptstyle m \\;>\\; 0", "44ccb114bcedb26d1c68c7d0152065f0": "g(x_1,\\ldots,x_k)\\,", "44cd43692dc93427bbb04b480978536a": "\n X_n\\ \\xrightarrow{L^r}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{p}\\ X\n ", "44cd6b8a3bc7147d7fb3f76fbcf51b39": "\\mathbf{s}\\in \\mathbb{Z}^n_q", "44cd6c33d23c1dc985a71ba42be09461": "\\scriptstyle F\\colon R \\to (0,1)", "44cdcf3d0753b2733be50ec08022337d": "\\scriptstyle \\leq6\\times10^{-20}", "44cdfefb17386bb2d4dbb457b98254c9": "y \\in C'(B',\\succeq)", "44cdff0df2419584c7c567908129ccaa": "M_{\\mathrm{Pl}_{3+1+\\delta}}", "44ce205dae463d3ec784b1738560aff4": " (a/l),(\\omega_0/\\nu) \\ll 1 ", "44ce6e2da4c7c61fc3738d360fe1e45a": "[u\\,v\\,1]^T", "44cea66428cfeac6be5e61d17e644177": " \\| \\mathbf{U} \\|^2 = c^2 \\,", "44cefae6ffea35d1e7ebbbcf49280cad": "\\frac{({\\alpha}^2-{\\beta}^2)(2n+1+\\alpha+\\beta)}{2(n+1)(2n+\\alpha+\\beta)(n+1+\\alpha+\\beta)}", "44cf218ee35ec26df04185d2fef0a4e7": "\\Pi^0_\\alpha", "44cfc713e99dda5591d510a8e438fb50": "\\psi_{\\mathbf{k}}(\\mathbf{r})=\\frac{1}{(\\sqrt{2\\pi})^3} e^{i \\mathbf{k}\\cdot\\mathbf{r}}", "44cff7036fc92c4797c92f9a79ebd78a": "X\\subset P^4", "44cffe3084a6455723b8708fb7735e3b": "\\zeta(x,y,t);", "44d0534e40cb51b6ddf4af453822a6f4": "\\scriptstyle 2 \\pi", "44d07903079612b8b3a08eb66ee0d96f": "-K_F", "44d0dc437936b13f7cea2f77053806bd": "\\N", "44d11a3d5fea92b5910ad7e6beea1eda": " \\hat{p} = - i \\hbar { d \\over dx }\\,. ", "44d1334c06785418a1b47aac8d8ea55d": "\\textbf{for } j\\leftarrow 0 \\text{ to } J-1", "44d18b4875d3ae09c0f4c383a8b8717b": "\\hat{H} (N)", "44d1be9c1b1934a4b7b227faa7bb47e0": "p_Z(z|a) = \\frac{1}{\\pi^2(z^2-1)} \\ln \\left(z^2\\right).", "44d1e71631b0a465d3dd46e6c0abff57": "\\Omega = \\left[-1, 1\\right]", "44d2061e952db06705570bb4546605b5": "\n P(q|M_d) = K_q \\prod_{t \\in V} P(t|M_d)^{tf_{t,d}}\n", "44d21af66b0874d9b45905ea79807cb3": "n=5", "44d21eb24abf5b8ce627a91fb280d7b1": "\\overline{A}", "44d299a41b6a8d93dc25442b69925919": "f(\\mathcal{O},z)", "44d2fab528bff77633e8e12101612278": "( G1 + G1 ) /2 = G1", "44d374a5c74f65ceda02d2117930da56": "\n+:E\\times E\\to E \\qquad , \\qquad \\lambda:E\\to E", "44d37fb472a3aa2bb392c9bca40262d9": "\ns = \\sqrt{\\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\overline{x})^2}\\,,\n", "44d3a971e004b0775f98ef76c75eee16": "C\\ell(E) = \\coprod_{x\\in M} C\\ell(E_x,g_x)", "44d448f08c05aa149f61085d2c21c6e6": "R_{MMO}=\\frac{n}{1\\cdot n}=1.", "44d4532b5548273506da477c3e6cc415": "\\frac{4}{n}=\\frac{1}{(n-1)/2}+\\frac{1}{(n+1)/2}-\\frac{1}{n(n-1)(n+1)/4}.", "44d4aec0f81a8fe848464b70249c23e7": "d/2 + 1 + \\epsilon", "44d4c236902c10e34844c9c92f21cac4": "f^*(x^*) = \\sup_{x\\in I}(x^*x-f(x)),\\quad x^*\\in I^*", "44d4ea255d39bd423ed53367bec87374": "(d_1,e_1) + (d_2,e_2) = ((-1)^{e_1e_2}d_1d_2, e_1+e_2) ", "44d55d0785e986b888447221a082aa62": "\\textstyle \\operatorname{mes}", "44d58b4ebdf2d003b714a1cfe2b39f3a": "\\psi(\\Omega^2)", "44d5a062812c6dd8235aea8e910ab44d": "\\frac 1N \\sum_{n=0}^{N-1} x_n^2", "44d5ba1fe5f06f3b846116ca6bd7b85c": "f\\colon \\overline V \\to W", "44d5d11acc2810d1e7b1b5d96765cd8b": "\n\\begin{alignat}{4}\n&\\text{(Q1)}&\\qquad \\cos C&=-\\cos A\\,\\cos B, \n&\\qquad\\qquad \n&\\text{(Q6)}&\\qquad \\tan B&=-\\cos a\\,\\tan C,\\\\\n&\\text{(Q2)}& \\sin A&=\\sin a\\,\\sin C, \n&&\\text{(Q7)}& \\tan A&=-\\cos b\\,\\tan C,\\\\\n&\\text{(Q3)}& \\sin B&=\\sin b\\,\\sin C, \n&&\\text{(Q8)}& \\cos a&=\\sin b\\,\\cos A,\\\\\n&\\text{(Q4)}& \\tan A&=\\tan a\\,\\sin B, \n&&\\text{(Q9)}& \\cos b&=\\sin a\\,\\cos B,\\\\\n&\\text{(Q5)}& \\tan B&=\\tan b\\,\\sin A, \n&&\\text{(Q10)}& \\cos C&=-\\cot a\\,\\cot b.\n\\end{alignat}\n", "44d5ea907dbe09ff4647ea7d9139bd65": "(5)\\qquad T_{ab}^{(EM)} = \\frac{1}{4\\pi}\\Big(F_{ac}F_b^{\\;\\;c}-\\frac{1}{4}g_{ab}F_{cd}F^{cd} \\Big)\\;.", "44d5f9d8fd444edd80a985c1666aaae2": "1 \\in R", "44d67dc7f3e729e571a640868034e048": "D_n(x)\\propto\\int_0^\\infty{\\rm d}t\\frac{t^{n}}{\\exp(t)-1} = \\Gamma(n + 1) \\zeta(n + 1). \\quad [\\Re \\, n > 0]", "44d69d26606239e6ba4271a5226bbb30": "\\int_0^{2\\pi}\\int_0^a re^{-r^2}\\,dr\\,d\\theta < I^2(a) < \\int_0^{2\\pi}\\int_0^{a\\sqrt{2}} re^{-r^2}\\,dr\\,d\\theta.", "44d69ec03885d648cf09f4635f6d410f": "\\left.\\frac{dy}{dx}\\right|_{x=a} = \\frac{dy}{dx}(a).", "44d710e6f01e8a978084d590af011dae": "MOD (fsw) = 33\\mathrm{~feet} \\times \\left [\\left ({ppO_2\\over FO_2} \\right ) - 1\\right ]", "44d73e88a40af54b9a6ce8d0df23bd25": "Z_e = \\{ a \\in \\ker w : ea = 0 \\}", "44d7ca15ae8632622e1a35d608b93a51": " P^{\\mu }, J^{\\mu \\nu }, M ", "44d85dd8721bb622642beb16abcc6493": "\\sigma_{\\rm e}(\\omega)", "44d893a47cf22356d9d43edbc4332ffc": "[\\mathcal{L}_X,\\mathcal{L}_Y]\\alpha:=\n\\mathcal{L}_X\\mathcal{L}_Y\\alpha-\\mathcal{L}_Y\\mathcal{L}_X\\alpha=\\mathcal{L}_{[X,Y]}\\alpha", "44d8a559c8d0537df45ca8bb1e6e1aec": "\n\\mu_{\\text{min}} = \\lambda_{\\text{min}}\\left( \\sum_k \\mathbb{E}\\, \\mathbf{X}_k \\right) \\quad \\text{and} \\quad\n\\mu_{\\text{max}} = \\lambda_{\\text{max}}\\left( \\sum_k \\mathbb{E}\\, \\mathbf{X}_k \\right).\n", "44d8d7710c80dc86a12e772e72c60865": "O(n \\log(n))", "44d8df65ce3ae36152641e0f97cb2a3d": "H_{2n+1}(x) = 2(-4)^{n}\\,n!\\,x\\,L_{n}^{(1/2)}(x^2)", "44d916052f6cbf956cc30fcc0b31ff4c": "\\max\\{(S-K), 0\\}", "44d939ac7128e839de6b65fdd3b6fe8c": "X_i\\sim F_\\theta", "44d939becf809d49a0d4cc77db3dfdef": "\\Delta=\\{[i2^{k},(i+1)2^{k}]\\times[j2^{k},(j+1)2^{k}]: i,j,k\\in\\mathbb{Z}\\},", "44d95cc325dea9c64634a51052582ccf": "\\underline{\\lnot \\psi \\quad \\quad \\quad}\\,\\!", "44d95d587a9952b4cbaa7b920955b35f": "W/T(V)", "44d98b7b3b1b9255123077d653e04c11": "\\lambda_i = \\sigma_i \\pm j \\omega_i", "44d9c27a7aaea0bcd0b906baae73e2a4": "\\frac{1}{1-x}=1+x+x^2+x^3+x^4+\\cdots", "44d9d3d52a5ec73c615af08b900adc7b": "Y_i = \\beta_0 + \\beta_1 \\phi_1(X_{i1}) + \\cdots + \\beta_p \\phi_p(X_{ip}) + \\varepsilon_i \\qquad i = 1, \\ldots, n ", "44d9e88b1bad3b77e878b11d639b9b1c": "\\frac{\\rm d}{{\\rm d}t}x(t)=f(t,x(t),x_t),", "44da237f1a3f7cc1e435afa0e57f837d": "A_{21}g_2 = B_{12}g_1F(\\nu)\\,", "44da59cddc21d03cb770e17e646b19de": "b_3=B_3(a_1,a_2,a_3)=a_3+3a_2 a_1 + a_1^3,", "44dac5d9269e8876157c26d5b5359fdf": "{k -2ix -\\frac{1}{k}} = 0", "44db05ff7c492e15ab9e8640a4ecb03c": "d_{1,-1}^{1} = \\frac{1-\\cos \\theta}{2}", "44db0f8bf8d8e6065ba297a47a52972d": "\\begin{matrix} {1 \\choose 1}{11 \\choose 1}{4 \\choose 2} \\end{matrix}", "44db80234982e992d498d41afb29a1e0": "\\theta(t) = \\theta_0\\cos\\left(\\sqrt{g\\over \\ell\\,}\\,t\\right) \\quad\\quad\\quad\\quad \\theta_0 \\ll 1.", "44db8f49ca5db2f7484a78f7456db66e": " \\dot{u}^2 = f(u)", "44db91fdc241c4cecf1ede40eef5701f": "z = \\frac{ U - m_U }{ \\sigma_U }, \\, ", "44db9c58344a009da1973475be3f3acb": "I(\\omega) = |E(\\omega)|^2", "44dbb15468988cd05c801262408cab4e": "H_0:\\theta=\\theta_0", "44dcf94111022d47380806e99c1fd638": "b\\in\\operatorname{cl}(Y\\cup \\{a\\}) \\setminus \\operatorname{cl}(Y)", "44dd454ebaaa58a80fe43f4a19faf2df": "2\\sqrt{2}", "44dd4af199f5a3f32a61b47d55fadeb3": "\\cosh c=\\cosh a\\cosh b\\,.", "44dd54f97506a2a5423fca5f6ac4af2c": "\\vdots\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots", "44dd99437efaf8b65b4c668fceed52b2": " X\\in \\mathbf{L}_{M^+} ", "44ddf6e825ef5a1ea521e708af7deb73": "\\gamma \\,", "44de39aec0074594c428b7dac8ba2f6b": "\\{\\pm 1, \\pm 7\\}\\cong \\mathrm{C}_2 \\times \\mathrm{C}_2,", "44de3da6ded8b2c09c59a42db2aca5f2": "\\mathbf{p+k}", "44de5ae6fb6bd491a2beabeea9b01df3": "\\sqrt{9.2345} \\approx 3.0391 - \\frac{0.0391^2}{2 \\times 3.0391} \\approx 3.0388", "44de952291b385eca7fec14fea9ba18f": "\\scriptstyle \\frac{L_\\ast}{L_{\\odot}}=100^{(M_{V_\\odot}-M_{V_\\ast})/5}", "44dec32262b35de961110de44fc96500": "Q_n \\quad = \\quad \\sum_{i=0}^n w_if(x_i) \\quad \\approx \\quad \\int_a^b f(x)\\,\\mbox{d}x", "44df278adecf4f5cfdc17b4afb907857": "\ny_k = b\\ \\sinh \\mu _k\n", "44dfadae8a9db304d7a3c40e61a3fecc": "-\\mu_{\\kappa}", "44dfc7bcc7e6b7336012c4243d02ccb8": "\\textstyle {\\mathrm{Cov}}(X_i,X_j) = - n p_i p_j~~(i\\neq j)", "44dfd1a74fbf4dabb61680cd1d469e47": "\\frac{|ax(\\gamma(t))+by(\\gamma(t))+c|}{\\sqrt{a^2+b^2}}", "44dfe0a7f34aae0e6719485d5a32e5bd": "x_{up}^{(syn)}(t)", "44e031d541ba789f52a738821b6dc363": "\\sigma[1] \\sigma[2] \\ldots \\sigma[L] \\in W^*", "44e0880b3814c6dbaaec24fd25a0a3fd": "\nQ = \\frac{F}{BW} \n", "44e0c0e0f807483e633f4fc264ad7f0c": "\\mathbf{P}_{X|z} \\propto \\Pi^{-\\top}P_Z\\odot \\pi_z", "44e0cd35ef8b3fd754aba831a0bfe65b": "W_C = U(B)-U(A),", "44e0e263c30d1d3c988e5028c98b0bd2": "m_{rocket}(t) \\frac{\\mathrm{d}\\mathbf{V}}{\\mathrm{d}t} = -\\left(\\mathbf{v_{rel}}(t) \\frac{\\mathrm{d}m_{gas}}{\\mathrm{d}t} + F_{other}(t)\\right).", "44e14dfa294e02219c768c9e14680507": "X_{d}", "44e24fba045fec2e3a57af05c97fa929": " \\mathbf{R} ", "44e26792f46f094ff3c35a5a1d0058d6": "\\langle x,y \\mid x^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1}\\rangle.\\,\\!", "44e28499dc4cea28a487d67ef96f0bf3": "\\mathbf{s}(u_0)=\\mathbf{d}_0, \\dots,\n\\mathbf{s}(u_{p-1})=\\mathbf{d}_{p-1}", "44e2be20bddbea41af4cdc1aae7b3319": "\\log \\nu = 3", "44e30b696ab1630db0ddafb17c7b8ac8": "B ^3\\Pi_g", "44e31170b212ac8faf6cd57962307cd8": "T_{\\rm wc}\\,\\!", "44e3487f38b383c2b8847e87e2b806b0": "\\mathrm{Re_2S_7 \\ \\xrightarrow{600^oC}\\ 2ReS_2 + 3S }", "44e3eafd9a1b91d351c3fff0f3d04368": "H= {1\\over 2m} |{(qA+\\nabla)\\psi|^2},", "44e4cfe4f17877ccda34929a81f3223e": " \\| \\log(R) \\|_F = \\sqrt{2} | \\theta | ", "44e4e5fea1322117e602c6564133c985": "CVR=", "44e4fd871f6b67292765854884b9e523": "R=|R'|=1-T=\\frac{(k_1-k_2)^2}{(k_1+k_2)^2}", "44e512073f9e16125cf024d8582112db": "9746347772161 = 7 \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19 \\cdot 31 \\cdot 37 \\cdot 41 \\cdot 641\\,", "44e5542dcdfba22e4c48daf2ef37e3e2": " (\\lambda x.x\\ x)\\ \\operatorname{get-lambda}[x, x = \\lambda x.\\lambda f.f\\ (x\\ x\\ f)] ", "44e55c17daea4393d0a2731e70fe89bc": "\\mathit{\\bar{q}_i}", "44e5901b6880be4d9aca7ec737964207": " \\Delta h_o = \\Delta h_od - \\Delta h_ou = (h_d + \\frac{1}{2} C_s^2) - (h_u + \\frac{1}{2} C_s^2) = \\frac{1}{2} (C_s^2 - C_u^2) ", "44e5e831140e1e1bde834a5374754394": "\\begin{bmatrix}\nV_x\\\\[10pt]\nV_y\n\\end{bmatrix} \n=\n\\begin{bmatrix}\n\\sum_i I_x(q_i)^2 & \\sum_i I_x(q_i)I_y(q_i) \\\\[10pt]\n\\sum_i I_y(q_i)I_x(q_i) & \\sum_i I_y(q_i)^2 \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\n-\\sum_i I_x(q_i)I_t(q_i) \\\\[10pt]\n-\\sum_i I_y(q_i)I_t(q_i)\n\\end{bmatrix}\n", "44e6a3468c72d11ad43b82118160abf9": "S=\\sqrt{K_{sp}}", "44e6be22e29922a14a539f6e6e205f8f": " MI = \\sum { MOC (i,j) * (i-j)^2 }\\,\\!", "44e6c68088a197fb0af54b8075660579": "\\scriptstyle J \\cdot J^T = k I", "44e719a1f53ab5cd75cfb2ed7d19859d": "t_{i,j}", "44e733ea02feb918ca06bac98de17898": "PE =\\dfrac{BOD\\ load\\ from\\ industry\\ \\left [\\dfrac{kg}{day}\\right ]}{0.054\\ \\left [\\dfrac{kg}{inhab \\cdot day}\\right ]}", "44e74d68ce94bee0c8e82d00f455ea98": "\\operatorname{tr}(A) = a_{11} + a_{22} + \\dots + a_{nn}=\\sum_{i=1}^{n} a_{ii}", "44e7644a2534d008c4df29ad400b3703": "ua\\equiv vb", "44e7704e8aa2526c26c527d697b4daab": " + 0(x-1)^3 + 0(x-1)^4 + \\cdots \\,", "44e7743b575c3a715e71bc6605bb5e38": "r_{3}=(g_{12}-g_{21})/(2sin\\Theta)", "44e7794ebacc892be2d8b69eeb272008": "N-m", "44e7cc10bedc669e8c7dc5bee98d43a9": "\\scriptstyle r(\\boldsymbol{r}_i,\\, \\boldsymbol{r}_{\\text{rec}}) / c \\,+\\, (t_i - t_{\\text{rec}}) \\;=\\; 0 ", "44e821fa50a1e52c1866ce0ef920e1c3": "~f_c", "44e838ccefc2f9e30ab41bedff99b2fd": "Y \\to X", "44e8456bab27183932e657b5207d059f": " m = 167 + 1 - 25 = 143", "44e86ff44fc2e2df4f9e0c20158e26a7": "\\left(-2\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-5}{\\sqrt{3}},\\ \\pm1\\right)", "44e88036a98c00114384ad1bdc44b641": "\n\\frac{ ( \\widehat{ \\theta}-\\theta_0 )^2 }{\\operatorname{var}(\\hat \\theta )}\n", "44e9084b77214cbdf5d974bf68c2e215": " \\Psi = e^{i(\\bold{k}\\cdot\\bold{r}-\\omega t)} \\,\\!", "44e957360b28fa0e65487d8cd74d5a20": " Y(u,v) = r \\, (1 + \\cos v) \\, \\sin u, ", "44e9731ac917e23e3bdb1aae976dac8e": "F_m(x) = F_{m-1}(x) - \\gamma_m \\sum_{i=1}^n \\nabla_f L(y_i, F_{m-1}(x_i)),", "44e97e71fe32f8e65a80698a4dd27aff": "|\\psi_{00}\\rangle = |0\\rangle", "44e9cd42bda5de453846d415eaed7a9f": "C_{v \\times 1}=\\begin{bmatrix}s_{c+v}\\\\\ns_{c+v+1}\\\\\n\\vdots\\\\\ns_{c+2v-1}\\end{bmatrix}.\n", "44ea292e8362a478124472e049a9da13": " c\\log_b(x)+d\\log_b(y) = \\log_b(x^c y^d) \\!\\, ", "44ea3b4b4f287c7f96603d7935e779cb": "\\scriptstyle \\| z \\| \\;<\\; 1,", "44ea46b6dc911f0f73e5c3ab5e43feff": "h_n = \\int P_n^2(x) W(x) dx", "44ea52229d9aa739d8925d1418330bd0": "\\ln(\\frac{\\nu}{2}) + \\psi(\\frac{\\nu}{2}) = \\sum_{i=1}^n \\ln(x_i) - n \\ln(\\tau^2) ,", "44eae11261318da046a59f5c64a1595e": "6912a(a-2)^3-j(4a-9)", "44eb5b459dc3c5c16fde2a70711997e3": " \\frac{1}{\\pi} = \\frac{2\\sqrt{2}}{9801} \\sum^\\infty_{k=0} \\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}.", "44eb75de91d33b54a6b8ae56499184d8": "\\hat{a}_i \\,\\hat{a}_j^\\dagger", "44eb776569284d3a3a6c8296aea95206": "(A\\land B)\\lor (A\\land C)=0\\lor 0=0", "44eb8d781884b992b1af39d2c9e27bfc": "H_1, H_2, ..., H_7", "44eb8eef838834e5ef820d743b6a253a": "\\bar{r}", "44ec1fbabb1f745dd2c62f51dd663ca7": "\n \\mathcal{E}_{ijk} = J~\\varepsilon_{ijk} = \\sqrt{g}~\\varepsilon_{ijk}\n", "44ec2d73daae07d10b91aa4bbb202791": "\\quad\\sum_{i=1}^n\\frac{x_i^{n-1}}{\\Pi_i(x_1,\\ldots,x_n)}=1", "44ed26ea61a285d143983c55b54ab278": "(x-4) (x-1)^{10} x^{21} (x+1)^{11} (x+3) (x^2-13) (x^6-26 x^4+3 x^3+169 x^2-39 x-45)^4", "44ed276b4d58b11523517524e34cd9ea": "\n\\begin{align}\n\\lfloor x+n \\rfloor &= \\lfloor x \\rfloor+n,\\\\\n\\lceil x+n \\rceil &= \\lceil x \\rceil+n,\\\\\n\\{ x+n \\} &= \\{ x \\}.\n\\end{align}\n", "44ed50edb705bb63be2dab07a50f5f4c": "\\begin{bmatrix}\n 1-\\frac{z_1}{f} \\quad &\\lambda z_0-\\frac{\\lambda z_0 z_1}{f}+\\lambda z_1 \\\\\n -\\frac{1}{\\lambda f} \\quad &1-\\frac{z_0}{f}\n \\end{bmatrix}\n", "44ed65b70994c77f8a1465db55ab2895": " M(x,y) \\frac{dy}{dx} + N(x,y) = 0 \\,\\!", "44ed90cb0a2a4bd882391e43f1763ebf": "\n\\sum_{i=1}^{n}x_{it}(\\xi_{[t]}) = W_t,\n", "44edf686c787665b50d22fba8a62deb0": "= \\frac{C_{\\text{max}, \\text{ss}} - C_{\\text{min}, \\text{ss}}}{C_{\\text{av}, \\text{ss}}} \\cdot 100", "44ee0bb5ac9894a7d5d55cece190e918": "r_1,\\ldots,r_d \\in R", "44ee1adf9881fc16a7707bff45ecc669": "\\Delta_K\\equiv 0\\text{ or }1 \\pmod 4.", "44ee31882aee2c6e83e7ba0b1977e8f1": "\\Phi_{2^h}(x) = x^{2^{h-1}}+1", "44ee33534130ac29a89339bcd79e97d8": "t'_1 = TR\\,e^{3ik\\ell/\\cos\\theta}", "44ee573bc14c79b18327b563a877206a": " \\det \\begin{bmatrix} \n 0 & d(AB)^2 & d(AC)^2 & 1 \\\\\n d(AB)^2 & 0 & d(BC)^2 & 1 \\\\\n d(AC)^2 & d(BC)^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 0\n\\end{bmatrix} = 0. ", "44ee6369de57efd91668f1ecf55bc160": "\n \\ddot{w} = W(x_1,x_2)\\frac{d^2F}{dt^2} \\,.\n", "44eeb752ced905d087b9ef6cbd31859d": "U_\\beta", "44eeeb93c9cb197ab98fdba10f15cc32": "\\varphi:M\\to \\mathbb{R}", "44eeebeca2de2a7fe5443e84393cce25": "\n \\begin{align}\n \\sigma_{rr} & = \\frac{2C_1\\cos\\theta}{r} + \\frac{2C_3\\sin\\theta}{r} \\\\\n \\sigma_{r\\theta} & = 0 \\\\\n \\sigma_{\\theta\\theta} & = 0\n \\end{align}\n", "44eefb77d66766cf6e09da97c5157042": " \\to\\{\\mbox{vector fields on }U\\} \\;", "44ef20bfcf13985decdce36adb9b1490": "C = w(T-l)\\,\\!", "44ef808b440ab02282b1d2a9eb09d5cb": "\nf_x(x^*,u^*)=\\begin{bmatrix} \\frac{\\partial f_1}{\\partial x_1}|_{x=x^*,u=u^*} & \\cdots & \\frac{\\partial f_1}{\\partial x_n}|_{x=x^*,u=u^*} \\\\\n\\vdots & \\ddots & \\vdots \\\\ \\frac{\\partial f_n}{\\partial x_1}|_{x=x^*,u=u^*} &\n\\ldots & \\frac{\\partial f_n}{\\partial x_n}|_{x=x^*,u=u^*}\n\\end{bmatrix}\n", "44efab4c7437a72dc2389425558ada20": "c_1 \\begin{bmatrix} 1 \\\\ 0 \\\\ 2 \\end{bmatrix} + c_2 \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} c_1 \\\\ c_2 \\\\ 2c_1 \\end{bmatrix}\\,", "44efad3c685510d71951c9126b175026": "\\ln (1+x) \\;=\\; x \\,-\\, \\frac{x^2}{2} \\,+\\, \\frac{x^3}{3} \\,-\\, \\frac{x^4}{4} \\,+\\, \\cdots.", "44eff5a1b77a1fff5e7f7d29d935dc8b": " |A\\rangle = \\frac{1}{\\sqrt2} ( |H\\rangle - |V\\rangle ) ", "44f018db84c2d0eace91dca9541cd502": "n^m", "44f0d410cb8096b514a7d5d297de9d0b": "\\mathbf{w_p} \\leftarrow", "44f0de0437e2e5848f03f4c2d7b3b8ce": " \\left( \\begin{array}{c}\n\\partial_t u\\\\ \\partial_t v\n\\end{array} \\right) =\n\\left(\\begin{array}{cc} D_u &0\\\\0&D_v\n\\end{array}\\right)\n\\left( \\begin{array}{c} \\partial_{xx} u\\\\ \\partial_{xx} v\n\\end{array}\\right) + \\left(\\begin{array}{c} F(u,v)\\\\G(u,v)\n\\end{array}\\right)\n", "44f13c5781d790432a2bcc912b4b32e8": "\\boldsymbol{\\phi}", "44f160de2c8196b25b282b0c7d2e6573": " \\lim_{n \\to \\infty} \\int_X f \\circ T^n \\cdot g d \\mu = \\int_X f d \\mu \\cdot \\int_X g d \\mu.", "44f16605c2c570eacc4baa414d7a3671": "A+t", "44f190d907a4b043c648220b279b2c00": "A_1, \\ldots, A_n \\vdash B", "44f1ca6f7af615f765123521fde646e7": "\\delta\\in \\left[0,1 \\right]", "44f2385dde661dc22dbd13f9903cdcd7": " \\phi(\\omega) \\ \\stackrel{\\mathrm{def}}{=}\\ \\arg \\left\\{ H(i \\omega) \\right\\} \\ ", "44f251356c5cbb066f1630057e1a984d": "\\lrcorner", "44f2783280a512e190fc32b942bd6c3b": "N(t)\\,=\\,\\frac{1}{|| \\gamma'(t)||}\\cdot\\begin{pmatrix} -2\\cos(2t) \\\\ -3\\sin(3t)\n\\end{pmatrix}", "44f2886de47a6ed1bd8d3b95c3125295": "f(1)", "44f2d680d9b45877a6689bf1eae5f7f8": "\\arg \\min \\limits _{(x,y)} {D(f(x, y)), D(f(x + 1, y)), D(f(x, y + 1))}", "44f33d42a3fda644b9fec9f58b9cf961": "P \\rightarrow_{b}Q \\mbox{implies } P \\circ R \\rightarrow_{b} Q \\circ R", "44f3d26322e82b318283626d0cd0ba00": "Y(n,n-b) = \\sum_{a}^{} \\, p(n,n-a)q(n-a,n-b)", "44f3d2ba28e5a5e34b9584e72fd9b88c": "H_1 (X;\\mathbb Q)", "44f43b29421ed603943d42d28470e06d": "j = 1,\\ldots,n", "44f49329b05b4c4a255c3dffe60fa340": "\n\\sum_{\\stackrel{1\\le k\\le n}{ \\gcd(k,n)=1}} \\gcd(k-1,n)\n=\\varphi(n)d(n).\n", "44f4d26b8e8120adc233ccfaea8272f3": "\n s^2 \\ \\sim\\ \\frac{\\sigma^2}{n-1} \\cdot \\chi^2_{n-1}, \\qquad\n \\hat\\sigma^2 \\ \\sim\\ \\frac{\\sigma^2}{n} \\cdot \\chi^2_{n-1}\\ .\n ", "44f4e611ba10c34dba2553eb0438c677": "\n|g| = I_1\\, I_2\\, I_3\\, \\sin^2 \\beta \\quad \\hbox{and}\\quad g^{ij} = (\\mathbf{g}^{-1})_{ij}.\n", "44f52868db27abebe095f0b1bd3b7782": "\n \\begin{align}\n \\varepsilon_{rr} & = \\cfrac{\\partial u_r}{\\partial r} \\\\\n \\varepsilon_{\\theta\\theta} & = \\cfrac{1}{r}\\left(\\cfrac{\\partial u_\\theta}{\\partial \\theta} + u_r\\right) \\\\\n \\varepsilon_{zz} & = \\cfrac{\\partial u_z}{\\partial z} \\\\\n \\varepsilon_{r\\theta} & = \\cfrac{1}{2}\\left(\\cfrac{1}{r}\\cfrac{\\partial u_r}{\\partial \\theta} + \\cfrac{\\partial u_\\theta}{\\partial r}- \\cfrac{u_\\theta}{r}\\right) \\\\\n \\varepsilon_{\\theta z} & = \\cfrac{1}{2}\\left(\\cfrac{\\partial u_\\theta}{\\partial z} + \\cfrac{1}{r}\\cfrac{\\partial u_z}{\\partial \\theta}\\right) \\\\\n \\varepsilon_{zr} & = \\cfrac{1}{2}\\left(\\cfrac{\\partial u_r}{\\partial z} + \\cfrac{\\partial u_z}{\\partial r}\\right) \n \\end{align}\n ", "44f567510afa8e0482d3b1201c06d48c": "Mg + 2H_2O \\longrightarrow Mg(OH)_2 + H_2", "44f57c1c2cee2365c8be7857f7b4d7ab": " \\displaystyle{\\mu_n=\\lambda_n^2}", "44f5e2ff9d0bbb303aabab764b807cfa": "4 \\pi", "44f5f80baa7657032de8876f904c803d": "W_{H,i} ", "44f64d9f3d8f62db89d6b6e2f0f5640d": "q\\frac{dM}{dq} = \\frac {LM-N}{3}", "44f66dd8cadba68ac129e1b0a9069d77": "f_\\text{elong} = \\sqrt{\\frac{i_2}{i_1}}", "44f6ca65726960ea7bc062f9d483d322": "\nR_2 = R_1 \\cdot \\frac{1} {({\\frac{V_\\mathrm{in}}{V_\\mathrm{out}}-1})}\n", "44f6e1fc827529a61c725b7fdb100962": "g_{F} = g_{obs} - g_\\lambda + \\delta g_F", "44f768063e0068f463ce6bf11b534381": "\\mathit{3} 2 \\mathit{6} 4\\, ", "44f77a2db7605ea6e5265a56f0f817a1": "\\left\\langle \\delta, \\varphi \\right\\rangle = \\varphi(0)", "44f7d61d513e3cf7426e0bb96b055ba7": "\\mathbb{R} \\times \\{0\\}", "44f90035c7a29ae54cd989e6e22cfaf7": "B^{ij} = e^i \\wedge e^j", "44f92f289f86180de786deaf99d817f1": "\\,\\mathrm{slog}_b(z)", "44f93e8a9c8a3999ce529030ffe6d11c": "\\frac{d}{dx} \\left [ \\prod_{i=1}^k f_i(x) \\right ]\n = \\sum_{i=1}^k \\left(\\frac{d}{dx} f_i(x) \\prod_{j\\ne i} f_j(x) \\right)\n= \\left( \\prod_{i=1}^k f_i(x) \\right) \\left( \\sum_{i=1}^k \\frac{f'_i(x)}{f_i(x)} \\right).", "44f9469717737f95aab811859e967ea3": "\\,i^{(m)}", "44f960ee8ae3388c3d720f76f10d64a5": "W^+_t := \\frac{1}{\\sqrt{1 - \\tau}} | W_{\\tau + t (1-\\tau)} |, \\quad t \\in [0,1].", "44f9618e1e7d4ffbfe5dea1f147824d0": "\n \\begin{array}{lll}\n (2-5)^2 = (-3)^2 = 9 && (5-5)^2 = 0^2 = 0 \\\\\n (4-5)^2 = (-1)^2 = 1 && (5-5)^2 = 0^2 = 0 \\\\\n (4-5)^2 = (-1)^2 = 1 && (7-5)^2 = 2^2 = 4 \\\\\n (4-5)^2 = (-1)^2 = 1 && (9-5)^2 = 4^2 = 16. \\\\\n \\end{array}\n ", "44f96b42feb81a91769ab067483bdbe3": " P(x) = \\sum_{i=0}^n a_ix^i", "44f9fde3b2e08808eb831a4f4d7c85f6": "y_j = 0", "44fa4cf700e05064f6584a29fd1ae9f0": "F_h=0", "44fa523e5ef11564d8d7ef42e4ad3f12": "on(box,t)", "44fa782969ee3a6b140c82078ebafddb": "K_m(R1,K_b(R0,message),B)\\longrightarrow(K_b(R0,message),B)", "44fafd163480013e83bccdd3a409d017": "M_1 \\to M_2", "44fb17e3413a3ebd47403775fa121e51": " ~c^2 \\!-\\! s^2=1~.", "44fb2ab81909ec90333e37f60f07e6b7": "\\det(A+\\delta A-\\mu I)=0", "44fb4f4bdc385721806fd5ab9abb7d4c": "\n\\begin{array}{ll}\n\\min & f_j(x)\\\\\n\\text{s.t. }&x \\in X\\\\\n &f_i(x)\\leq \\epsilon_j \\text{ for }i\\in\\{1,\\ldots,k\\}\\setminus\\{j\\},\n\\end{array}\n", "44fbd638d7d6fb3322492f0d89ee00f6": "\\frac{1}{2} \\left[(v-1)+\\frac{2k+(v-1)(\\lambda-\\mu)}{\\sqrt{(\\lambda-\\mu)^2 + 4(k-\\mu)}}\\right]", "44fc6b19195aa0a6dfd11b685ab5c94b": "\\xi \\in \\mathbb{R}^d", "44fc8eb674befd4ccc5165f59723df1e": "\\mathbb{F} = \\mathbb {C}", "44fcdc382078ad5717dc48fe024cebe8": "Y_{10}^{-6}(\\theta,\\varphi)={3\\over 1024}\\sqrt{5005\\over \\pi}\\cdot e^{-6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot(323\\cos^{4}\\theta-102\\cos^{2}\\theta+3)", "44fd0abf44508a2819955793e7eedf35": " cl ", "44fd5b9c0545126e0e739f1e627ab19f": "\\alpha_i \\,", "44fdb5b607b007da81adebc859a86d65": "|R_i|= n_i", "44fde7fc6acf2edfa0fc04046dba9994": "\\mathbf{P}=\\sum\\limits _{m}p_{m}\\hat{\\mathbf{e}}_{m}", "44fe0aa34f189138950fe4d1564a1757": "m^*_\\text{conductivity} = 3 \\Bigg[\\frac{1}{m_x^*} + \\frac{1}{m_y^*} + \\frac{1}{m_z^*}\\Bigg]^{-1} ", "44fe28671ffe22e992e7bd3429d4a83d": "\\left\\{f(\\cdot;\\boldsymbol{\\theta})\\right\\}", "44fe88e45abbb38570140111a91424bd": "p(\\gamma[1] \\gamma[2] \\ldots \\gamma[L])", "44fe8da2cae83d186ea71e425d06a590": "LR-", "44fe98c4abebed595a7f086057475d0f": "|\\psi_1 \\rangle", "44fedf5757c45cf5c8d96706980f704f": "\\operatorname{ran}(T)", "44fee0d6e0c1c5193df1f1ac375cc44a": "V(S_{T,L})", "44fee3dcda7fdb8b38b7ef4d282dd4e7": "\\boldsymbol{y}_t", "44ff4147348be5fefdd8bdba30b0f674": "\\textstyle n_0", "44ff7253bac7f3c28be02829bfbf7e6c": "s'_n = s_{n+k}", "44ff94241fc2962f2676c34b058e2425": " \\frac{4!\\times 6!\\times 2^5}{4} = 138,240.", "44ffd64ea0d3bc5acd5c4f67a62ffee3": "\\varphi_{\\alpha}(\\varphi_{\\beta}(\\gamma)) = \\varphi_{\\beta}(\\gamma) \\,.", "44ffe67fcb337d84ca5bc1997370e0f5": "Q \\rightarrow K", "44ffe963eae1a0b79c84e74b38dff96a": " {\\overrightarrow{V} = \\overrightarrow{V_g} + \\overrightarrow{V_a}} ", "45001065d2b967b37edf052405de2cbc": "\\partial_\\mu F^{\\mu\\nu}=4\\pi j^\\nu", "45003fe6df863071a068d68c290825ac": "\\text{pHad}:\\{0,1\\}^k\\to\\{0,1\\}^{2^{k-1}}", "450050c7781fec803ac4597b5a33aecf": " \\mathfrak{g}^{*} ", "45006157c748a110f173cbebbfbe1822": " C_{P} ", "4500bf2e4b04ca8207b6c349658c762b": "A = [a_1:a_2:\\ldots:a_n]", "4500c076f432d4d393906ada187e89b6": "-\\frac{1}{d^2-1}\\le\\lambda\\le 1", "4500ea2dd68d90b2cc5e8ffbbee78057": "x\\in R^{p+1}, y\\in R^q", "450133c6e44e8661d6eb881550042474": "\\, \\left( \\tilde{t}_\\text{r} + b - t_i \\right) c", "4501a191eb2fc6b864e9c92966b58108": "\\beta+\\gamma<\\alpha.", "4502097e70466089dca4c29a68b8bee2": "A_{TM}", "450239a8ac7cc00348ad2cc06b2cfeb4": "\\mathcal A\\subseteq\\mathcal B", "45024e76bc8c7fe95372daa42fc20073": "F(\\pi_1,\\pi_2,\\ldots,\\pi_p)=0\\, ", "450272ad29cfef9cfa04b8f8344fde01": "\n\\delta \\phi(x) = \\Omega(x) \\phi(x) \n", "45035c42026ae620791aed5ece7532bc": "\\text{median}=\\tfrac {1}{\\sqrt{2}}", "4503745ef62a9eeecfdbcf2df21e794e": "N = p q", "45038f299d96fb90ff5d6ef93c1d8713": "p_{r+1}(x)=(x-a_{r,r})p_r(x)-a_{r,r-1}p_{r-1}(x)", "4503c689f53d11bfe2782f231e4dd202": "\\begin{pmatrix} A&B \\end{pmatrix}\\begin{pmatrix}x\\\\y\\end{pmatrix} = \\begin{pmatrix}C\\end{pmatrix}.", "45041ffe312b8916968422e184584d4f": "\\forall N \\in V: \\exists \\alpha,\\beta \\in (V\\cup\\Sigma)^*: S \\stackrel{*}{\\Rightarrow} \\alpha{N}\\beta", "45042ae24a30cb8f1d1f0964b03f85ec": "\\{\\theta_i^{}(t^{}_{n},k^{}_j),j=1,\\dots,J\\}", "4504380b0ef5f4375cfedc5019560e2f": "1, \\begin{matrix} \\frac{1}{6} \\end{matrix}, \\begin{matrix} \\frac{1}{30} \\end{matrix}, \\begin{matrix} \\frac{1}{140} \\end{matrix},", "450455fbc0b34f41add0f1041eaf22f9": "\\left|T_w(x) - T_{w,n}(x)\\right| = \\left|\\sum_{k=n+1}^\\infty w^k s(2^k x)\\right| = \\left|w^{n+1} \\sum_{k=0}^\\infty w^k s(2^{k+n+1} x)\\right| \\le \\frac{|w|^{n+1}}{2} \\cdot \\frac{1}{1-|w|}", "45045ddc39f748208d12990482e55a21": "\\gamma_m", "4504a5e6667c75521c0b59ea623c7c8d": " P = \\{x\\in\\mathbb{R}^d : 0 \\le x_i \\le 1; 1 \\le i \\le d\\} ", "4504ca3ed7aadd42d9c381384f246915": "-\\omega", "45050633a9c0fa3a95e907e3e40c79f1": "i \\in \\{1, 2, 3\\}", "450550cc5238a589a0de3b59d03749dc": "\\textstyle{(\\frac{7}{27})^n}", "450554ffc4205685c89a48adf079774d": "\\Gamma,\\ x:\\tau \\vdash e:\\tau'", "45056eb65a41f42eda18b97cb7eba62f": "\\phi_R = \\angle H_R(s) = \\tan^{-1}\\left(-\\frac{\\omega L}{R}\\right)", "45058b4c5f2f540d88bb83421d8b44e2": "z=f\\,(x,y)", "450598d3207caff62015c521a370c381": "\\bigcup{}_{i=1}^n", "4505a7b44b3cef509205fc43387c46c5": " \\Pr(\\left|q - E[q]\\right| \\ge \\frac{\\lambda}{m}) \\le 2\\exp(-2\\lambda^2/m) ", "4505ee0c4b3b1d5f34434a947473d33f": "\\vec{u}_1^T", "4505f9a477ed26ff02c56b472f42a3e5": "|S_1|=12", "45060e8e32bc224833ecc41cd2667817": "\\ ee = |F_+ - F_-|", "45062ba13d3b80fccdc285c6e6d068b5": " F = \\frac12 \\times \\rho \\times S \\times C \\times V ^ 2 ", "4506ebeba979846ab73079d79884c42a": "\\boldsymbol{r}'\\in\\Omega", "4506fb2761c70db92b345ba0b501b741": "\\scriptstyle{m}", "45071dea86bc433b4cbca541e19cd97b": "\\pi_{i + j} A", "450728c833bba305291f6a741168e463": "M(x) = O(x^{\\frac{1}{2}+\\varepsilon})", "45074174d8476ac5639fef7830c1c037": "\\frac{\\partial}{\\partial s}(\\rho u r_o^{n})+{\\frac{\\partial}{\\partial y}}(\\rho v r_o^{n})=0", "4507f8811b9f049a9750a7e66d489ec1": "0.500 \\pm 0.006 M_\\odot", "450802a795db3704366c22ce125a2375": "E\\in\\mathcal{F}", "4508287d373657c64e0c1e68405740ed": " g\\exp(F)=\\exp(F')\\exp(I'), \\qquad g:(\\exp(F)\\sigma_0,v)\\to (\\exp(F')\\sigma_0,\\exp(I')v). ", "4508ab5eb2d54d8606103378c1de3e76": "\\Delta\\rho(\\mathbf{r})", "45090e6ae867a01f44374e2f31af7f80": "\\left(\n\\frac{\\pi}{3\\sqrt{3}}\n\\right)^{\\frac{1}{3}} \\approx 0.846 ", "4509199a6754f4ce36c077019f980e56": "S\\subset D_R", "4509233dffb31ed8e490b35bec753554": " \\scriptstyle x \\,>\\, \\log(1 \\,+\\, x)", "45092ed107a3a95561f8e1287aeb6223": "\\hat{a}, \\widehat{a}, \\vec{a} \\!", "45094e6a2b12cabd11fff1f1aa315c82": "\\tau' : \\pi'^{-1} (U') \\to U' \\times X'", "45098c4478eb3e35fc7df5083d6c868f": "d = gh + p = (4/3)(1) + 2 = 10/3", "45099d0e661b044c878c4301529a883c": "\\left.\\frac{\\partial^{|I|} \\sigma^{\\alpha}}{\\partial x^{I}}\\right|_{p} = \\left.\\frac{\\partial^{|I|} \\eta^{\\alpha}}{\\partial x^{I}}\\right|_{p}, \\quad 0 \\leq |I| \\leq r. ", "450a1dcf91e19cac30648709ddde99ab": "\\tau\\, \\sqrt{\\frac{g}{h}} = 9.80,", "450a30c3ad7ef332b66a86412b645c7c": "E[\\delta_{b,\\bar{b}}]\\,", "450a4ac9f9aa7a4cce58b56080d79ac4": " \\mathbf{y}_{1} ", "450a9a013c895660063ca96f40f3517e": "x^2+(b_{2}-(b_{2}-c_{10}))x+b_{2}(b_{2}-c_{10})=0", "450aba4ae5d2880002af61e596dd34bd": " D_{KL}(f_i\\|f_j)\\leq \\beta", "450ade48519f714929c3124eb0bea4e0": "\\scriptstyle u(t)", "450ae0a6e22fbf75cfa130ee19209dd9": "2 r \\Omega \\dot\\theta '", "450b086a425416b5225fa9eff481bd00": " \\binom{n-1}{2} ", "450b1646ae685a5ab0d6b8f051c0ad88": "\\omega_f(U) = \\sup_{x\\in U} f(x) - \\inf_{x\\in U}f(x).", "450b21dd82e5b92d5993e1e4e9788863": "p(e_i |w,b,\\log \\zeta ,\\mathbb{M}) = \\sqrt {\\frac{\\zeta }{{2\\pi }}} \\exp \\left( { - \\frac{{\\zeta e_i^2 }}{2}} \\right) .", "450b5985117235eab18802297a8361a1": "\\mathbf{N} \\!\\,", "450b5ce26095f09729e5e9dfd2d35efa": "(I, f, m, g)", "450b64b911312cb5e3325df331872743": "\\textstyle\\binom nk", "450c96b7481e37f5b52ff7d8eb9169cf": "\\mathcal{N}(\\tilde{\\mathbf{x}}|{\\boldsymbol\\mu_0}', {\\boldsymbol\\Sigma_0}' +\\boldsymbol\\Sigma)", "450caa68f1beacfe50539f312b099f71": "x=\\frac{-b \\pm \\sqrt {b^2-4ac}}{2a},", "450cb7a4e1d74a5a10776c0bf763ae24": "\\frac{dx'}{dt'}=\\frac{ \\frac{dx}{dt} - v }{ 1 - \\frac{dx}{dt} \\frac{v}{c^2} }", "450cd0c402e1572f20eaae803f543887": "x>\\mu", "450cf669e0ab67099be6e661157e3bbb": "\\frac{d}{ds}=\\frac{d}{d\\varphi}\\cdot\\frac{d\\varphi}{ds}=\\frac{d}{d\\varphi}\\cdot\\frac{1}{r}=\\frac{1}{r}\\cdot\\frac{d}{d\\varphi}", "450d1ed6d0d153c3ff9fe4ebcb549387": "G^*", "450d2dc6fe8d5f78058c817a8dea51ec": "Y\\,\\!", "450d3cd4d767bd49740d3120184c32dc": "F(s) = \\int_0^{\\infty} f(t) e^{-st} \\,dt,", "450d60be4184ccde1efbdbe124ecf19e": "m_x v_0 = m_x v_1 \\cos \\theta_1 + m_y v_2 \\cos \\theta_2 \\,\\!", "450d7de84b5295a9eeaea99ee00ba7ef": "g(x)=\\begin{cases}\\frac{1-\\varepsilon}{\\mu(B)}\\frac{\\|f\\|_\\infty}{f(x)}&\\text{if }x\\in B,\\\\0&\\text{otherwise.}\\end{cases}", "450dc57c413831d07f4e5651cec33e8d": "2a|L| + O(lgq(x))", "450de2c286da7e62250b3c4842ca5b08": "x_i - x_j", "450e0b4130461ce1ff172cf4846b26df": "\\mathbb P(V)", "450e1bd2306037afb40528de2709006d": "P_1\\,\\!", "450e425104a361b83004b490e72d7bf1": "s_{p-2} \\equiv 0 \\pmod{M_p}", "450e4a7c000cb91bc2c34fcc080f2a4a": "\\mathrm{Gal}(\\bar{\\mathbf{Q}}/\\mathbf{Q})", "450ed50810575dcc820f6fe957837305": "q(x) = \\left(x_1^2+\\cdots + x_k^2\\right)-\\left(x_{k+1}^2+\\cdots + x_n^2\\right), \\, \\quad k < n .", "450ed8064a221fd88bc27d5b95b0a503": "g_k\\equiv\\nabla f(x_k)", "450ee26dc78ba085c301690f078c6ff4": "L_0(s) = G(s)P_0(s)", "450f6c862a36bb238c100140664b8ff6": "X, y, l", "450f7985a6915de2dbd56dab9c3ba140": " \\mathfrak{I}^*\\,", "450fd46a1197670f326fa277dc0a49c5": "=\\sum_{\\boldsymbol{R_{\\ell}}} \\ \\sum_{\\boldsymbol{R_n}} b^* ( \\boldsymbol{R_n})\\ \\int d^3 r \\ \\varphi^* (\\boldsymbol{r-R_n})H_{\\mathrm{at}}(\\boldsymbol{r-R_{\\ell}}) \\psi (\\boldsymbol{r}) \\ + \\sum_{\\boldsymbol{R_n}} b^*( \\boldsymbol{R_n})\\ \\int d^3 r \\ \\varphi^* (\\boldsymbol{r-R_n})\\Delta U (\\boldsymbol{r}) \\psi (\\boldsymbol{r}) \\ .", "450fdbfe745a057b0b612aa0bf7c4511": "\\pi f^2(x) dx", "4511491bffbcd8d8cfb5a4e1e1a40052": "K(\\mathbf{w}, \\mathbf{J})", "45114abb4d7d63bb19b38b94026e66a7": "R_\\Delta", "4511aed63a0ee4630fd9a89b45a612e2": "E_n^{(5)}=\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}E_{nk_4}E_{nk_5}}-\\frac{V_{nk_5}V_{k_5k_4}V_{k_4n}}{E_{nk_4}^2E_{nk_5}}\\frac{|V_{nk_2}|^2}{E_{nk_2}}-\\frac{V_{nk_5}V_{k_5k_2}V_{k_2n}}{E_{nk_2}E_{nk_5}^2}\\frac{|V_{nk_2}|^2}{E_{nk_2}}-\\frac{|V_{nk_5}|^2}{E_{nk_5}^2}\\frac{V_{nk_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}}", "4511afbce56666c8fedbe0e0203d1002": "iR_1R_2", "4511d7779da933fc9f3d22368d6134d4": "X \\sim \\operatorname{Log-\\mathcal{N}}(\\mu, \\sigma^2)", "4511e85e5572b225a1a8dc2ddf847382": "\\textstyle w", "45123fc25dd86394003b3dd64c416966": "l_a l^a=n_a n^a=m_a m^a=\\bar{m}_a \\bar{m}^a=0\\,;", "451264b462e9a5dadb456f5a2298e19f": "T\\vdash_{\\mathcal{S}}\\alpha(\\psi,\\vec{\\chi})", "45127bf7b6d2b4f91d4f5759fc8551cb": " S_BS_C = S^2 - a^2S_A\\quad\\quad S_AS_C = S^2 - b^2S_B\\quad\\quad S_AS_B = S^2 - c^2S_C \\, ", "4512a5a1d0d9d6b9e25f744ceae16f0d": "\nc(\\kappa,\\beta)=2\\pi\\sum_{j=0}^\\infty\\frac{\\Gamma(j+\\frac{1}{2})}{\\Gamma(j+1)}\\beta^{2j}(\\frac{1}{2}\\kappa)^{-2j-\\frac{1}{2}}{I}_{2j+\\frac{1}{2}}(\\kappa)\n", "4512dfd97a63a764eef44cd467ed44b5": "t \\{x_1 \\mapsto t_1, \\ldots, x_k \\mapsto t_k \\}", "45133ef916c82ec4b9bc9b78eeb8618d": "g(\\tau)=A+B \\exp(-\\Gamma\\tau)\\cos(2\\pi\\upsilon_o)+C \\exp(-2\\Gamma \\tau)\\,\\qquad (1)", "45135656af86acd990e63256c7a8facd": " \\det(A) = \\varepsilon\\det(U).", "451391d32138562be198d8c1b4f7522c": "\\varepsilon(P_0) = 0\\,", "45139b0bdb64f8b0d51dc74cf353d422": "x=-\\phi (3.9711308)^{-1} \\, ", "45141df91552a1e0559405c85ce4c9e3": "R_{in} = \\frac{-g_m}{\\omega ^ 2 C_1 C_2}", "45141f4ab0bcbf3735b76d4ab5c6edb6": "\\sigma(\\pi) = \\prod_{c\\in\\pi} (-1)^{|c|-1}", "45142869d764f5210c2d82acbc666d8b": " k = {v_{o} \\cdot w^{j} (s - a_{v})\\over 100} - {c_{o} \\cdot r^{j} \\cdot a_c\\over 100}", "45143c3d32c6a9e1ea390ac6c8037dab": "1-\\alpha'", "4514870f61d39ebab3431dc5fd10b22b": " (x \\to (y \\to z)) \\to ((x \\to y) \\to (x \\to z)) = 1 ,", "4514daad32aa91263cfd99f22878145a": "\\sum_{k=0}^{\\infty} \\frac{1}{(2k + 1)^2} = \\frac{\\pi^2}{2^3} = \\frac{\\pi^2}{8}", "4514dc6d5d88ce3150bcbd6f0c9f8b0a": "\\gamma \\in R^{1\\times (n-q)}", "4514f5661af5b4a4e70a8c01bde37363": "\\varepsilon=n^2=1+\\frac{0.69616630\\lambda^2}{\\lambda^2-0.0684043^2}+\\frac{0.4079426\\lambda^2}{\\lambda^2-0.11624140^2}+\\frac{0.8974794\\lambda^2}{\\lambda^2-9.896161^2},", "45157a09c47191a3f2913d0cdf7eb538": "E_c = \\frac{\\sigma_\\infty}{\\epsilon_c} = \\frac{\\sigma_f}{f\\epsilon_f + \\left(1-f\\right)\\epsilon_m} = \\left(\\frac{f}{E_f} + \\frac{1-f}{E_m}\\right)^{-1}", "45159d7b030f0eed88b6e173ecc44bad": "\\|k - j\\|", "4515bd9318f247956df4eca891e3a701": "\n(x^3-3xy^2-3xz^2)^2+(y^3 - 3 y x^2 + y z^2)^2+(z^3 - 3 z x^2 + z y^2)^2 = (x^2+y^2+z^2)^3", "4515c39e96ca6cb1a0dac77429b0cd9c": "F[u] = \\int_{\\Omega} f(u(x)) \\, \\mathrm{d} x.", "4516252b601fe49618f67bfcae0a1705": " a_i = \\frac{1}{N}\\sum_{j=1}^N y_{i,j}", "45166e7a815864e9295e4949948fdc89": "|a+b| \\le |a| + |b| ", "45168116294eafcc9d84d757772efebe": "\\, ax_2 + by_2 + cz_2 + d = 0", "4516e1ccb3e3eff572ed7407ab8aecad": "(p_n)", "4516fd1bfdf309e1defbf2c5fd0a5abd": "\\max_{j\\neq i} b_j > b_i ", "45170999670fa26244dc4b2c2a39fd9e": "x^n = n! \\gamma_n(x)", "451735f5dd22ce11c1528e8438cfbd7b": " \\frac{R_1^2}{\\varepsilon_1} + \\frac{R_2^2}{\\varepsilon_2} + \\frac{R_3^2}{\\varepsilon_3} = 1. ", "4517d8865556cfbd3e0491127c3e49a8": "\\textstyle\\frac {B}{C}", "45180fe211b1d4bafc20c54091f2faea": " \nm_1 + n_1 + m_2 + n_2 + \\cdots + m_p. \\, \n", "45182fd1a4758af957bb13a530294a17": "I_C = \\beta I_B = \\frac { \\beta (V_{CC} - V_{be})}{R_B+ ( \\beta+1) R_E} \\approx \\frac {(V_{CC} - V_{be})}{R_E}", "45184e4dbf30201c9a5d65ec066ab24c": "\\psi \\varphi \\subseteq \\chi ", "451868c0838307dce6cb305dbc095066": "h_{fw} \\,", "451877a8bd1a732318e4f32b5201fba6": "z_j=1", "451890d81e9ad58529efed629a59526d": "\\sum_{g\\in G}\\chi_{V_i}(g)^*\\chi_{V_j}(g)=\\sum_{k}|C_k|\\chi_{V_i}(C_k)^*\\chi_{V_j}(C_k)=|G|\\delta_{ij}.", "4519081374cc7d02265778633fcc32c6": "\\{\\varepsilon_n\\}", "4519523fa01f578ad0479d605721564e": "L.O.A", "45195dc11e498cc6dd73a802528e4b52": "\\chi^2_{\\alpha,\\nu}", "4519797eff88e8f1cb9ef2a38f8838db": "\\lambda \\geq 0 ", "45198d4d0b9cd0f53ba8c72b47c2f958": " W_0(x) ", "4519930829bf3424d439138d4638096f": "\\cdots\\to\\widetilde{K}(SX)\\to\\widetilde{K}(SA)\\to\\widetilde{K}(X/A)\\to\\widetilde{K}(X)\\to\\widetilde{K}(A)", "4519c30e1098d4d0e57464d98a0fb32b": "\\operatorname{Dir}(\\boldsymbol\\alpha)", "451a31b27cfb68764967e75a8f870a19": "\\frac{\\pi\\sigma^2}{4}", "451a3631f6e7378cc22e996953be0324": " F(x)G(x) = \\int_0^x f(t) G(t) dt + \\int_0^x F(t)g(t) dt \\;.", "451a99147b103101e04e840812fe3d6c": "U_1\\left(x,y\\right)=\\alpha", "451a9e5d3b524304ceabf43933a480d5": "\\mathcal O(e^{Lt_n}\\Delta t^2)", "451b20d17a2e1c3458d485879fefdcdc": "\n\\mathbf{e}^{(1)} = \\begin{bmatrix} \\rho_0 \\\\ -a \\end{bmatrix}, \\quad \n\\mathbf{e}^{(2)} = \\begin{bmatrix} \\rho_0 \\\\ a \\end{bmatrix}.\n", "451b2e83815a10c9b002dd2d812032ba": "10 \\log_{10} 2 \\approx 3", "451b482714c519a102b69703e9ac2ce9": "\\frac{d^3}{dx^3}[x^4]=24x", "451b80e3152898d24674c36bfbbbc161": "\\ MU_x ", "451bc4403796c75db82702b2d0c0a8e8": "\\phi^{-1}:V\\to U", "451c1bb141a5910cb8fe4ece108beeae": "|z^6+z^5+z^4+z^3+ ", "451c1d33a3203ced18773409015b9d5f": "(ax+b)\\mod(26)", "451c41201b9021f82e83ed17314bfdaa": "u(t,x)", "451cbed0b94393e3e1eac424d9bab334": "\n\\begin{pmatrix}\nT_\\mathrm{n}+\n \\frac{E_{1}(\\mathbf{R})+E_{2}(\\mathbf{R})}{2} & 0 \\\\\n0 & T_\\mathrm{n} +\n \\frac{E_{1}(\\mathbf{R})+E_{2}(\\mathbf{R})}{2}\n\\end{pmatrix}\n\\tilde{\\boldsymbol{\\Phi}}(\\mathbf{R})\n+\n\\tfrac{E_{2}(\\mathbf{R})-E_{1}(\\mathbf{R})}{2}\n\\begin{pmatrix}\n\\cos2\\gamma\n & \\sin2\\gamma \\\\\n\\sin2\\gamma &\n-\\cos2\\gamma\n\\end{pmatrix}\n\\tilde{\\boldsymbol{\\Phi}}(\\mathbf{R})\n= E \\tilde{\\boldsymbol{\\Phi}}(\\mathbf{R}).\n", "451cd2259af4c5c6e19acb8ca02fbf00": "\\hat{k}\\,", "451cd412fd2b09e29f0f38d12cc23871": "u_{2,2}^{0} = \\delta_{2}^{0}(2) = 6", "451d321408142cf75b4fde761d55d48f": "(VH V^{-1}- H) f(e^{i\\varphi}) = \\frac{1}{\\pi} \\int \\left[{g^\\prime(\\theta) e^{ig(\\theta)} \\over e^{ig(\\theta)} - e^{ig(\\varphi)}} - {e^{i\\theta} \\over e^{i\\theta} - e^{i\\varphi}}\\right]\\,f(e^{i\\theta})\\, d\\theta.", "451d32ff53deabf5334fbf9269eba557": "\\phi_x(0)=x", "451d394b1812be3674879543bbd5d8f1": "Hw = S", "451db9ddc940ad35bedd5522de8d0025": "\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon} = \\boldsymbol{\\nabla} \\mathbf{w}", "451ddfb717d4089cbe12d0c4f4e17da2": "\\varphi=\\frac{1+\\sqrt 5}{2}", "451e411a7ae02e9b2a415c53d55204f5": "\\left|\\widehat{f}(n)\\right|\\le {\\| f^{(p)}\\|_{L_1}\\over |n|^p}", "451edf7672d7febefcbfb4295fa277e4": " \\begin{align}\nI &= S_0 \\\\\np &= \\frac{\\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0} \\\\\n2\\psi &= \\mathrm{atan} \\frac{S_2}{S_1}\\\\\n2\\chi &= \\mathrm{atan} \\frac{S_3}{\\sqrt{S_1^2+S_2^2}}\\\\\n\\end{align} ", "451f0c4653a7ba8354e7bab7ca961c9e": "(2 + \\sqrt{3})^t + (2 - \\sqrt{3})^t", "451f86dac825711e39923ec6034c7c3b": " J_{z} = \\begin{pmatrix}x+y & 0 \\\\ 0 & x-y\\end{pmatrix} ,", "451f91bb19afdacdfdc153113aa99ff6": "V= \\operatorname{Var}(\\tilde{Y_k}- \\widehat{Y_k}), ", "451f9492aa838538b6d439d83f1e7105": "(\\hbar m_s)", "451fb14c73b0655860d3593004f3d99b": "\ne^{-{\\rm i}\\epsilon V(x)} e^{{\\rm i}\\frac{\\dot{x}^2}{2}\\epsilon}\n\\,", "451fc16574f191c66ac148e071314fcd": "\\phi=\\frac{1 + \\sqrt{5}}{2}=1.6180339887499...", "451fe406734497ecdef48dadf7721579": "\\phi(e, s) = \\phi(\\phi(e, s_1s_2...s_{t-1}), s_t)\\, .", "45200361d32640e4db933f4fca78c742": "V_{max}=\\sqrt{\\frac{E_a + 3}{0.0007d}}", "452037034fa2ddef89be2a4ae98ac8b4": "p_\\beta", "452069cf4677c579e0e731d2b37aa421": "F_n(x) \\ge 0", "4520c3aadbe17de9ff8a7d8f12667bee": "\\lambda=M,M+1,\\dots,L-1", "4520fc0773e23fad4b5891078f89b5a9": "\n \\mu | x_1,\\ldots,x_n\\ \\sim\\ \\mathcal{N}\\left( \\frac{\\frac{\\sigma^2}{n}\\mu_0 + \\sigma_0^2\\bar{x}}{\\frac{\\sigma^2}{n}+\\sigma_0^2},\\ \\left( \\frac{n}{\\sigma^2} + \\frac{1}{\\sigma_0^2} \\right)^{\\!-1} \\right)\n ", "45210da832f9626829457a65e9e7c4d0": "uv", "45211c5f24b46b4401c537de0bfa3b0f": "V_{exh}\\,", "45213cb3e7b8b7d7b3da2092c9f79212": "\\int_X (\\ell^i * \\ell^j)_L \\smile \\ell^k = GW_{0, 3}^{X, L}(\\ell^i, \\ell^j, \\ell^k) = \\delta(i + j + k, 5),", "452167cb6acbb984713a6e0bacee1c8b": "C_9", "4521ee426822d78c9da9bb38da7a3ee5": "|a|<1;\\Re(s)<0.", "4522387ba9a4438ea85099aaf1d52e76": "\nT=\\left[\n\\begin{array}{cccc}\n1&0&0&\\vdots\\\\0&1&0&\\vdots\\\\0&0&1&\\vdots\\\\\\cdots&\\cdots&\\cdots&\\ddots\\end{array}\n\\right]\n", "4522517fb6e280cc3e651bf3ed6fdaa3": "\\overline{X}X", "452294f01eb6bf99f5452ba4afa0e876": "g_2=-x;", "4522f978d963fb046f672af2c61620ca": "\\mathbf{P}=\\{ A \\subseteq H : A \\text{ is an orthonormal set of common eigenvectors for } \\mathcal{F}\\},", "452311712f21a1c9564829eb57f3987c": " x_u ( n + N_{\\text{ZC}} )= x_u(n) ", "45237e751694f56bbe81c3083a341abb": "c' = c \\circ \\phi\\;", "45239d246d11820e0c1bc2ec66ac9b05": "\\text{Area}=\\tfrac{k^2 csr(s^2-r^2)}{2} \\, ", "4523cd39a207172f1360a8ce705f08ab": " e ", "4524348f642f4ac039038659f7c2b3fe": "f: V^n \\to K , ", "45245940d39317f2d6558a682775f6b5": "\t\\min \\, \\max_{i,j}\t c_{ij}x_{ij}", "452460c795d4067d19f67483671c0ad0": "\\phi-1=|z^+z^-|f^\\phi+m\\left(\\frac{2pq}{p+q}\\right)B^\\phi_{MX}\n+m^2\\left[2\\frac{(pq)^{3/2}}{p+q}\\right]C^\\phi_{MX}.\n", "45246299cdd3fcd71292ab20896f760c": "~ A ~", "45248bbb287fa8c367a96cd2cf101f19": "\\frac{1}{1 -r}\\;=\\;\\frac{1}{1 -\\frac{1}{4}}\\;=\\;\\frac{4}{3}.", "45248da92476dc1d59f62ecbbc498838": " \\gamma = \\mathrm{surface \\ tension \\ of \\ subphase \\ with \\ monolayer} ", "452494d0cdd46a3270747c9bbdac5f9a": "\\operatorname{Log}\\colon \\mathbb{C}^\\times \\to S", "45249b12ccbe04ef993b0bf3353a92a5": " S^* = \\frac{S - S_B}{S_T - S_B}", "45255cc83b511cc4689f6d606b0d55ab": "p_i = \\frac{n_i}{N}", "452568f4afaf2d152a52630b752d1af0": "\\operatorname{Pr}_i(\\{t \\geq 0 : X_t = i \\} \\text{ is unbounded}) = 1", "452588c725b7efd42c267503329e89d7": "\\scriptstyle a^2 \\,+\\, b^2", "4525a58909703fc08c5c5047ec6786bd": "p_n(x) = \\left(\\sum_{k=0}^\\infty {c_k \\over k!} D^k\\right) x^n,", "45262d48f4314a65c3031ffe70cfd170": "g\\ \\varphi^3", "4526be1b03fc815e415a575cebd60153": "N_{\\mathbf{v}}=N \\, ", "4526cf1f3b96a8fb8052cbd117f9c85e": "e^{q(D,\\widehat{D})\\epsilon n/2}\\,\\!", "452724465819fa2f0958cb2aa89af111": "\\operatorname{Tor}^R_{i+1}(M_1, k) \\simeq \\operatorname{Tor}^R_i(M, k)", "45275576ae616a97f553399a6e39d5b1": "\\frac{1}{30} + \\frac{1}{45} + \\frac{1}{90}= \\frac{1}{15}", "45276fc7c7b31f9c5d3181de50b688a0": "N(x) = \\det(\\phi(x))", "452793d3fe477e61e630ad15bd612365": "|L|=n\\,", "4527de842249348eb271e5baafc91ee7": "f'(x) = \\lim_{h \\to 0}{f(x+h) - f(x)\\over{h}}", "4527fdb29005bad581f511f2c0d3d270": "F=\\mathbb{R}", "452827fe91c29aabf6dfffd71a135e67": "\\scriptstyle SL", "45282ea9a281c982fa2e60e977442bce": "\\Sigma R", "4528891a2a0a0cb1edccc54136cf0838": "pAa_1, a_iAa_{i+1}, a_nAq", "4529b37cd2209598997913b46b81770e": "\\mathbf{\\dot q}", "452a6de1b5faf77628c553194ea4473b": "\\displaystyle\\frac{\\deg(V)}{\\hbox{rank}(V)} \\le \\frac{\\deg(W)}{\\hbox{rank}(W)}", "452ab03b9aaaf10e989d8a03ac518ebc": "-V_{nn}\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_3}V_{k_3n}}{E_{nk_3}^2E_{nk_4}E_{nk_5}}-V_{nn}\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_2}V_{k_2n}}{E_{nk_2}E_{nk_4}^2E_{nk_5}}-V_{nn}\\frac{V_{nk_5}V_{k_5k_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}E_{nk_5}^2}+V_{nn}\\frac{|V_{nk_5}|^2}{E_{nk_5}^2}\\frac{|V_{nk_3}|^2}{E_{nk_3}^2}+2V_{nn}\\frac{|V_{nk_5}|^2}{E_{nk_5}^3}\\frac{|V_{nk_2}|^2}{E_{nk_2}}", "452af2591642bfbb26b87056564d8f2b": " D_m^{s}=O(m^{-1/2+\\epsilon}) ", "452b1b103efdfc2b0cf16cced983ded2": "\\mathbf{z}=(0,0,1)", "452b74593394862872d6eaaa896dc4bc": "\\{p_i,p_j\\} = \\{q_i,q_j\\} = 0", "452bc9136dc540bdd4fea32cff5c3309": "\\delta(k)/k\\,", "452c2199962642ed1053174408f27cd4": "1\\ \\mathrm{C} = 1\\ \\mathrm{F} \\times 1\\ \\mathrm{V}", "452c423a84fa88dc0e4a845f96c357d0": "\\frac {\\Delta V} {V} = \\left(1+\\frac{\\Delta L}{L} \\right)^{1-2\\nu}-1", "452c6fcd429631ac05288e80c7080a93": "\n\\frac{mv^{2}}{r} = F(r)\n", "452ceee64d94ae0613dca3bcf3eb3322": "1/N", "452d231f4c2669c558a31a0fcac90e1b": "MC_1", "452d46f86d6d19c3535fe58c2bdcf1b3": "\\psi\\otimes Y", "452d533a079d47feffb29ac3722e6d6d": "\\phi(R)", "452d5f5b9f628485122be093261f5276": "\n S (\\boldsymbol{\\psi}(u)) = \\sum_{i=1}^n u_i^2.\n", "452d925d47bde464cbff094f3771d527": "\\lambda = ln(2)/t_{1/2}", "452d9b1a5b2f5ee12c8297d46d4cac50": "f E_{ch} \\ge P_{loss}", "452e19640ca8bc264980bacc76469a33": "= d(V^{2}u_{1}) - V^{2}u_{2}dx - u_{2}d(V^{2}x) \\,", "452e4f1329536adeb4e84a3e349eafc1": "\\in, \\notin \\not\\in, \\ni, \\not\\ni \\!", "452e60a793e9ab6bd6aa0e29e4fcd91d": "\\mathbf{Z}_2", "452e9432e7b8020fbc602520363d0f01": " a_{ij}", "452efe7fbe47e4322049c25dd8b58526": "f,g:X \\to (Y,0)", "452f7615b4d48413e300131a9ed9d9ea": "[C]\\cup [C]", "452fc60536a6afe053ec76a6e288ef7c": "(x \\diamondsuit y)_n = \\sum_{j=1}^{n-1} {n \\choose j} x_j y_{n-j}", "452fd0f9c621eea2c235af32a04dfa0f": "\\liminf_{n\\to\\infty} d(n)=2.", "452feebc2ac6d9ecd0b6c3ce5397fb30": "U \\subset \\mathbb C", "4530c7430cf3f7c04b4059d2fed5ef3c": " t\\ ", "45310f6a83db5ca256fc1775650f57b3": "b \\uparrow X \\times Y + Z", "453118d6e9d94ea5dd07d3c5c837c1c3": "\\mathbf L_{ij} = \\left(\\mathbf A_{ij} - \\sum_{k=1}^{j-1} \\mathbf L_{ik} \\mathbf D_k \\mathbf L_{jk}^\\mathrm T\\right) \\mathbf D_j^{-1}", "4531373cf28cfec9bf3491cc8d959665": "\\begin{align}\n\\Phi_{105}(x) = & \\; x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\\\\n& {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\\\\n& {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1\n\\end{align}", "4531aa9caf0df5cd66e484b39b890724": "E_m=Q/A\\epsilon_0=qN_Aw/A\\epsilon_0, \\,", "4531b6467fafd5eaf383243287f30da3": "2^{13}", "4531fb01a435b327bc4fbab2cb85321f": "\\check{H}(X;A)", "45322e784b4905f186c7e9189bc3917b": "= -\\operatorname{tr} \\left(\\gamma^5 \\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^5 \\right) \\,", "45327242dec5afc498e718c1870e4b52": "e^2+f^2 = zx", "45327c672a6dd2232d5f7f7283c116fd": "Y\\,\\sim\\,\\textrm{Normal}(\\mu,\\sigma^2)", "45328c27f03c3a1f1fcb775f47d3c14b": "\\{P(f(x)), R(c), \\neg P(f(c)) \\vee \\neg R(c), \\forall x . Q(x)\\}", "4532adc79efead827dfff199c555051c": "A_H (t)=e^{i H_{ S}~ t / \\hbar} A_S e^{-i H_{ S}~ t / \\hbar}", "4532dbb1f28103845b535ca9cf88df19": " \\phi_k = \\int \\phi(x) e^{-ikx} dx, \\ \\ \\pi_k = \\int \\pi(x) e^{-ikx} dx. ", "4532f891c765300f5891b970a87a04f5": "divergences\\left(P\\right)", "4532fcef8c951afaa2561f08eaca8ed1": "\\gamma^4", "453371a5d0f700acab05aff1312cc82a": "f '(c)=g '(c)+r=0+r=\\frac{f(b)-f(a)}{b-a}", "4533994eabd730fba1c2356904700162": "\\Delta t_{p}", "45339cc1d8f85adde49d84e4b68a7cdc": "=(-1)^{f(0)}\\frac{1}{2}\\left(|0\\rangle + (-1)^{f(0)\\oplus f(1)}|1\\rangle\\right)(|0\\rangle - |1\\rangle).", "4533b4d0d88644ad0a865049f48a393a": "v_p = \\langle v\\rangle \\sqrt{\\pi}/2", "4533c2bc42d0907035905201c2a5ff25": "R_\\mathrm{K} = \\frac{h}{e^2}.", "4533ce51d3b55d46522906bf09e201d9": "\\left( \\rho^{\\prime} \\right)^{2}", "4533e079d6a53b814f9aee9fe3c0fd74": "\\partial_t P(b, t|x, t_0)=\\lambda P(a, t|x, t_0)-\\mu P(b, t|x, t_0).", "4533e45fa4f20972683b4efe81de743e": "\\begin{align}\n \\mathbf{u}(\\mathbf{X},t) & = \\mathbf{x}(\\mathbf{X},t) - \\mathbf{X} \\\\\n \\nabla_\\mathbf{X}\\mathbf{u} & = \\nabla_\\mathbf{X}\\mathbf{x} - \\mathbf{I} \\\\\n \\nabla_\\mathbf{X}\\mathbf{u} & = \\mathbf{F} - \\mathbf{I} \\\\\n\\end{align}\n", "4533f79b7bf08e1b7bbccb944a184a72": "\nd\\log {\\widehat X}_i(t) = \\sum_{k=1}^n \\mathbf{g}_k\\,\\mathbf{1}_{\\{{\\hat r}_t(i)=k\\}}\\,dt\n+ \\sum_{k=1}^n\\mathbf{\\sigma}_k \\mathbf{1}_{\\{{\\hat r}_t(i)=k\\}}\\,dW_i(t),\n", "4533f985c4325ec7463f497e136a3f03": " C_i \\, ", "45344bc711bbd00bdcb5dffe6e5764a4": "\\frac{x}{1 + x}", "4534552d2481ee5b49a05237e03d1e2b": "\\lim_{t \\to \\infty}x(t) = 0,\\quad \\lim_{t \\to \\infty}y(t) = c.\\ ", "45349fa244c7cf3bf475f6360448d8e8": "p_i(x)", "4534aca03b4072f00f9d7cc3b3d116b4": "\\sqrt{\\operatorname{deg}(v_i)}", "4534d4c09305846f162227a32f2bf8b0": "f(0,0) = p(0,0) = a_{00}", "4534ffea59d0a382a225c4383da32435": " -P \\, dV = \\alpha P \\, dV + \\alpha V \\, dP,", "45359fbb5b9ed29f4929ff2765710a38": "\\mathbb C\\setminus\\gamma[0,2\\pi]", "4535b925082650af8d1acffe7fab1318": " c_{k'}(t) =\\delta_{k,k'} - \\frac{i}{\\hbar} \\int_0^t dt' \\;\\lang k'|H_1(t')|k\\rang \\, e^{-i(E_k - E_{k'})t'/\\hbar} ", "4535d5a448e94ad12deeb987abc4f841": "\\qquad w_j\\,x_j \\ \\le w_i", "45360956dc2863e368c365b64a9b2523": "(G, \\Omega)", "4536adf4aaf0eac7790620671dffe8e1": "T = \\frac{4}{3} \\sqrt{\\sigma (\\sigma - m_a)(\\sigma - m_b)(\\sigma - m_c)}.", "4536f1aa4c3b8f3bfbdd1927c228d416": "\\int \\int x.6.{x^4 \\over 4!} = 30.{x^7 \\over 7!}", "45373c9e203cb1c5819f3b8edf8e909d": "S(\\mathbf{q}) = \\frac{1}{N_c N_p} \\left \\langle \\sum_{\\alpha \\beta = 1}^{N_c} \\sum_{jk = 1}^{N_p} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_{\\alpha j} - \\mathbf{R}_{\\beta k})} \\right \\rangle = \\frac{1}{N_c N_p} \\left \\langle \\sum_{\\alpha = 1}^{N_c} \\sum_{jk = 1}^{N_p} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_{\\alpha j} - \\mathbf{R}_{\\alpha k})} \\right \\rangle + \\frac{1}{N_c N_p} \\left \\langle \\sum_{\\alpha \\neq \\beta = 1}^{N_c} \\sum_{jk = 1}^{N_p} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_{\\alpha j} - \\mathbf{R}_{\\beta k})} \\right \\rangle", "45376556d5853e6292be76cbaa314ed9": "{\\mathcal{X}}", "45377032d8d6ae8e3713c5e476d0b4fd": "\nZ_\\mathrm{source} = \\frac{Z_\\mathrm{load}}{DF} \\,\n", "453787f80e414bb9b13782d999032864": "\\phi(x,X). \\, ", "4537bcd209ef037430900fa9b0711512": "a+be^{i\\rho}=re^{i\\theta}+ce^{i\\psi}.\\,", "4538680a0ad4e23936e1969e13e85bba": "\\Delta: \\mathcal{Y} \\times \\mathcal{Y} \\to \\mathbb{R}_+", "45388dd891c483def59557d475d22caf": "\\frac{\\sin u'}{\\sin U'} = \\frac{\\sin u}{\\sin U}", "45389b50cfa61d6f4b1a0a351a9e2ce1": "KE=\\frac{v^2}{2g}", "4538ff1f7e51ee30b240c8bcdf697440": "c_n = \\prod_{i=1}^n p_{i}", "45391f792c45795aeff1f1e8c8457907": " f(rm) = rf(m)\\text{ for all }m \\in M\\text{ and }r \\in R. \\, ", "4539249eae51f7d2859ba410006bdd92": "\\sum_{k=0}B_k z^k= \\int_0^\\infty e^{-t} \\frac{t z}{e^{t z}-1}d t= \\sum_{k=1}\\frac z{(k z+1)^2}.", "453958d8465f2429dc41b22f010c89dd": "\\liminf", "453984f300626a847250a6a876f308ce": "\\mathcal{C}(R)", "4539bb0bed41bd50ed0fdc09517ce070": "V_n(V)/GL^+(V)", "453a0af218a6dc0cb6af8c137f37e990": "\nu^{+1}_{+1}(\\mathbf{p}) = \\sqrt{ {E + p_3} \\over 2 E}\n\\left( \\begin{array}{c}\n1 \\\\\n{{p_1 + i p_2} \\over {E + p_3}} \\\\\n0 \\\\\n0\n\\end{array} \\right),\n", "453a396d32f0fa7b0bc8732844668f8f": " V_{3} = 20 ", "453a3b7d225a5205a95ded340c2944e4": " \\mathbf{A} = \\mathbf{X} \\, \\mathbf{Y}^{T} \\, (\\mathbf{Y} \\, \\mathbf{Y}^{T})^{-1} .", "453a5394573dc8c6bd1a66a5172d80ed": "\\xi_\\alpha", "453a5bb7c7639eb7c40452c5a9754992": " \\frac{1}{p!} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} a^{\\nu_1 \\dots \\nu_p} = a^{\\lbrack \\mu_1 \\dots \\mu_p \\rbrack} ,", "453a9096e6cf1d1872c6e7ebc94a5b76": "\\approx_{Y}\\ ", "453b344c6a04639f31e4fabc08c5fe72": "O(n,1)/\\mathbb{Z}_2,\\ ", "453b3477d5ab65a2038e7405c8e8e8ae": "G^{3j}= \\frac{i}{\\hbar}\\int_0^\\infty \\left\\langle\\left[\\mathcal{I}(t),\\mathcal{F}^j(0)\\right]\\right\\rangle \\, t \\, dt", "453b36ab6ec891f7636c10eb21df506e": "\\langle u,v\\rangle =\\overline{\\langle v,u\\rangle}.", "453bc540e6292c2f879583681a16ea7c": "f_\\beta (t) = e^{ -t^\\beta }", "453bdaa91335d075168a2378a9cf04d4": "c: X \\rightarrow Y", "453bf26c2102189b86792cb285bbdba0": "\\varphi (t) = \\omega_0 t - k z = \\omega_0 t - k_0 z [n + n_2 I(t)]", "453c3427d0259f155e096800033c58c4": "B_0(x) = \\pm \\sqrt{ 2m \\left( E - V(x) \\right) },", "453c46415e51a2e172488dabdc1fb6c0": "T_n = \\frac{1}{f_n} = \\frac{2\\pi}{\\omega_n} = 2\\pi \\sqrt{\\frac{I}{\\kappa}}\\,", "453cb0635325a6ad2493c61bc04b16d3": "~N_2~", "453cc0fc37793752f9da95479e02f88f": "\\lim _{t\\to 0}\\frac{\\sin t}{t}=1", "453dafa55d18cda4e9af72b043c165c7": "y = x^m. \\,", "453e24fa7c4234529f0543126aa3b0b3": "\\bar y_n = \\bar y_{n-1} + \\frac{y_n - \\bar y_{n-1}}{n} \\!", "453e3edb64b35a059f4d244af674eedb": " d=a+b ", "453ecf11232b8bc2989d5d318f0031fa": "\\tan \\phi = \\frac{x}{y} = \\frac{\\sinh b}{\\sinh a ~\\cosh b} = \\frac{\\tanh b}{\\sinh a}", "453f6f822986c426b2e501893d2cd7a5": "\\begin{matrix} 1 \\\\ 11 \\\\ 21 \\\\ 1211 \\\\ 111221 \\\\ 312211 \\\\ \\vdots \\end{matrix}", "453f977a0667a82c3c5cedaf106201b9": "\\hat{z}(\\tau) = (1 - \\theta) z_{-}(t) + \\theta z_{+}(t)", "453fe76b31c5c9a996081a0f39713406": "\\frac{m_{T}}{m}=\\frac{p}{mv}=\\frac{E}{mc^{2}}=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "45400e2e539157869629e2da9798fb09": "\\beta=2\\alpha-1", "45406bcb1d25e20881332277a6481235": " \\chi^2 = \\sum_{i=1}^{n} \\frac{(O_i - E_i)^2}{E_i}", "45409668895006d6f941e6365dbf4dea": "\\{R_a, R_b, R_c\\}", "4540a20c3cba8fea7604e8f6c6664780": "j(x)=y \\leftrightarrow J(x,y,p) \\,.", "4541068a23d24421c8de44f6f6e55143": "\nW_- = \\sum_{i:h_t(x_i)=-y_i} D_t(i)\n", "45413d643d45158c4bd040abb9f98c44": "x\\in [0;\\infty)", "4541417744f94242b0b5a8b98dec1bc8": "K_0 \\to \nL_0", "454160c0760eb27d0ac5ebee7d6609f0": "I_\\mathrm{min} = 0", "454276fc323ebab121ce6958b0ca6c87": " fx(n_1,n_2,\\ldots,n_m)= \\frac{1}{N_1 \\cdots N_m} \\sum_{K_1=0}^{N_1-1} \\cdots \\sum_{K_m}^{N_m-1} Fx(K_1,K_2, \\ldots ,K_m) e^{j \\frac{2 \\pi}{N_1} n_1 K_1 +j \\frac{2 \\pi}{N_2} n_2 K_2\\cdots+j \\frac{2 \\pi}{N_m} n_m K_m} ", "4542fc1fd2c7a438353a29905afaf42b": "y_2(t),\\,", "454318c1963c885ca11125bd5b32362d": "\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}", "4543282c5ce34f678a068f7496c268c1": "\nA(x,y) = \\psi_x(x)\\psi_y(y)\n\\,", "45433b40e08009716306e8db5c493867": "\\pi_v", "45434f863e1cba8873eeacfa015a8bb3": "2\\pi R^2\\sin\\theta \\,d\\theta ", "45435af6f43904349eef1d7114d5700a": "Z=X\\cup Y", "454376f53f765a1e595087af04889e8d": "\\begin{align}\n j &\\equiv \\cos{\\left( \\frac{\\pi}{2}\\right)} + j\\sin{\\left( \\frac{\\pi}{2}\\right)} \\equiv e^{j \\frac{\\pi}{2}} \\\\\n \\frac{1}{j} \\equiv -j &\\equiv \\cos{\\left(-\\frac{\\pi}{2}\\right)} + j\\sin{\\left(-\\frac{\\pi}{2}\\right)} \\equiv e^{j(-\\frac{\\pi}{2})}\n\\end{align}", "454387146f72f7a6f36343bada80f6e0": "\\tau_{xy}=0\\,\\!", "4544a263804cd53390750d7bba8e4f24": "D2=d2", "4544bf28051a8c5f5f9878932a1d1e52": "M^{\\tfrac{1}{2}}", "45451562138ba490135ed5206b2ba629": "\\phi\\leftrightarrow\\psi", "4545183c96b04ac0d7d0f0020ca19646": "-\\infty\\,", "45451cab5f789c6698fa22c9ef96ad46": "\\int^{\\infty}_{0} f_k e^{-x}\\,dx = \\int^{\\infty}_{0} \\left ([(-1)^{n}(n!)]^{k+1}e^{-x}x^k + \\cdots \\right ) dx", "45453346b8f8bb6dbda09601d8bc083f": "V_\\mathrm i = V_x e^{\\gamma x}\\,\\!", "454570f7c07599490f75580c39e0ad3e": "\\{ X_1,\\ldots,X_n \\}", "454596bd5f228f698c36662205f6cf42": "Q_a^{(c)}(t) - Q_b^{(c)}(t)", "45459934a386946e6f76a32587aac2d3": "1 + 2", "4545baf1867498e8502fef3b0ebb9197": "U = \\prod_{i} (q_i-\\gamma_i)^{\\beta_{i}}", "4545f7ff60c329a1dd799523fa317cb6": "\\tilde M", "45466580b018663e919117f09e30572e": "p(x) = a_0 + a_1x + a_2x^2 + \\ldots + a_nx^n, \\,\\!", "4546c8382315d1905ff791862b16e70b": "\\frac{x_1-x}{x_1-x_0}", "4546f40ba5b67abcd448752954aa0528": "D\\,= \\frac{\\Gamma A}{\\delta x}", "454763c598a4137a329f69628522d07c": "(a,b)\\times (-R,R)", "454779d921f54fdb635c7542977dd5ff": "\\{w,v\\}", "45481d14c8666ec08291deeed61282bb": " y = d Y \\,", "4548877eb14745adc1ee4c422efe9327": "\n\\frac{x^{2}}{a^{2} \\cos^{2} \\nu} - \\frac{y^{2}}{a^{2} \\sin^{2} \\nu} = \\cosh^{2} \\mu - \\sinh^{2} \\mu = 1\n", "45495a3243f0d233d51c1ab91da4c070": "\\frac {\\partial}{\\partial x} \\left( \\frac { \\partial f }{ \\partial y} \\right) =\n \\frac {\\partial}{\\partial y} \\left( \\frac { \\partial f }{ \\partial x} \\right).", "4549828cdc33b97697b25c1356a96e7a": "\\bar R^2 = {1-{SS_\\text{err}/df_e \\over SS_\\text{tot}/df_t}}", "4549daa821cc7effdf086e1008eaf615": "(N+1) \\times (N+1)", "454a29627358bafcf864237214307e9a": "\\log_b 2 = \\frac{\\ln 2}{\\ln b}.", "454a388c35fe0aaa1efa70a60a82af71": "\\rho = \\frac{R_L - Z_0}{R_L + Z_0}", "454ae434c669bfa868a4d76fa55eda17": "\\textstyle b_{1} = p(t_x, a_{(-1,1)}, a_{(0,1)}, a_{(1,1)}, a_{(2,1)})", "454ae8a645cf255f3c04510ac85ceaae": "\nR_{g}^{2} = \\lambda_{x}^{2} + \\lambda_{y}^{2} + \\lambda_{z}^{2}\n", "454af09b80286b9ecc32d42a2b10399f": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi\\epsilon_0} \\int \\frac{q \\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t_r'))}{|\\mathbf{r} - \\mathbf{r}'|} d^3\\mathbf{r}'\n", "454b127b6fcd79fd3a4c601f473a0c89": "X_{vi}", "454b8465fe96179885e80559019f125c": "n \\mapsto f(n,k)", "454b9af28eacefe610f725b13026db1b": "\\phi(U_i)", "454bbadb30c345cf83e2cc5ab5b42150": " dm_v/dt = dm_L/dt ", "454c112778e4e3ce156aebf2eea22f22": "p_{1}X_{1} + p_{2}X_{2} \\leq m", "454c31304dc6a37f874037f883eb2ac6": "\\cos \\theta = a/r,", "454c92a5a96edab36dc68f807a82b349": "\n \\begin{align}\n K_{\\rm I} & = \\frac{P}{B}\\sqrt{\\frac{\\pi}{W}}\\left[16.7\\left(\\frac{a}{W}\\right)^{1/2} - 104.7\\left(\\frac{a}{W}\\right)^{3/2}\n + 369.9\\left(\\frac{a}{W}\\right)^{5/2} \\right.\\\\\n & \\qquad \\left.- 573.8\\left(\\frac{a}{W}\\right)^{7/2} + 360.5\\left(\\frac{a}{W}\\right)^{9/2} \\right]\n \\end{align}\n ", "454ce8c4e3dac8d8a5ffb1bc4b388041": "\\circ_r", "454cf2e4244b147d41d00f0b2e76f8a1": " a = \\alpha + \\gamma\\,T^{-1.5}", "454d2df39913e9b3fdd1a425e501f0b7": "u(t) \\le \\alpha(t) + \\int_{[a,t)}\\alpha(s)\\exp\\bigl(\\mu(I_{s,t})\\bigr)\\,\\mu(\\mathrm{d}s)", "454df989723aa2b474225314c6f9c55c": "\\log |\\mathfrak p|", "454e5033de7f6e6a8b8be5f8f56f3e52": "\\forall a\\forall b\\forall C\\forall D\\; \\lnot a=b\\and aC\\and bC \\and aD\\and bD\\rightarrow C=D", "454f0331ea3c600199aa29fdc0188d96": " D_t(x,y) ", "454f324eb56df198424be8a85daed89a": "p(X = x|\\Theta) = p(X_b = x_b)p(X_s = x_s|X_b,\\Theta), \\,", "454f3bafe15b366bf4358838c1597e3b": "( -0.75, -1.25);", "454f4daf41dfd81ae173eb3d87e08ec4": "p(\\textbf{x}_0,\\dots,\\textbf{x}_k\\mid \\textbf{z}_1,\\dots,\\textbf{z}_k) = p(\\textbf{x}_0)\\prod_{i=1}^k p(\\textbf{z}_i\\mid \\textbf{x}_i)p(\\textbf{x}_i\\mid \\textbf{x}_{i-1})", "454f95931440ab6ea2c24de73ed6d510": "\\lim_{R \\rightarrow \\infty} \\sum_{t \\geq t_0} \\mu(A,\\rho(t))(\\mathbb{R}_{> R}) = 0 ", "454fa21f99a67fc5d90ac01eaaa4ab20": "\\rho_{m,t}=\\textrm{Corr}\\left(m,t\\right); \\, ", "45504f8aea7362ed3f1059979d17e886": "(A+uv^T)^{-1} = A^{-1} - {A^{-1}uv^T A^{-1} \\over 1 + v^T A^{-1}u}.", "455083d8f144cf0f2084127f46028dd9": "\\sqrt{\\det\\left((du_i X_i)\\cdot (du_j X_j)\\right)_{i,j=1\\dots k}} = \\sqrt{\\det(X_i\\cdot X_j)_{i,j=1\\dots k}}\\; du_1\\,du_2\\,\\cdots\\,du_k.", "45509a6fd1f62c2bb3634661e9a620e6": " O =\n \\begin{bmatrix}\n {1\\over \\eta} & {1\\over \\eta} \\left({ -{1\\over 2} - {\\sqrt{ 5} \\over 2} }\\right) \\\\ {1\\over \\eta} \\left({ {1\\over 2} + {\\sqrt{ 5} \\over 2} }\\right) & {1\\over \\eta}\n\\end{bmatrix}. \n ", "4550c80ca5447b60deb402c2bd2a2905": "X^2 + X b = c", "45518cce83f5dd7ee434f0e922d47988": "L/I, I", "4551eace59f05165c752585d86d399e3": "\\sqrt[12]{2}:1", "4551ef4a8b694f782cc6b242496ca876": "\\forall\\ x,\\ *Rx\\rightarrow Bx", "4551f7876f9948222164becb15b82d33": "x^2 y+y^3", "4551fa7dd273cff784aa16fefc60aacb": "\\gamma_{\\mathrm{SV}}", "45520007cb5a0f17d24bd05b0e73f9a4": "\\overline{\\alpha_{i}}=\\alpha_{i}", "455270197d12088a5c95ba1645e32030": "A = l\\cdot (\\epsilon_{X} c_{X} + \\epsilon_{Y} c_{Y} )", "45528b5467c262fd3182bbba90e33f72": "a_{1}<\\cdots1", "455a3dbe3f3c75a810fe7104f2040cfa": "a_3b_2", "455acd596b80f263805ba2441e675bc9": "H_{1,I}^{\\text{RWA}}", "455b27f2ee2dacdd6bb8e95e6ec57480": "{n\\choose g(m)}", "455b9b59712ac7b2c295fc0c0d3d95de": "-i.", "455bb86dd0dee82c32f833b31d25da13": "B = \\{v_i\\}_{i\\in I}", "455bc648881197c37b0591e52804add6": "\\chi_-", "455be6ac04333f9489dcc9f439485ee1": "G\\times G\\times G ...", "455c7be6e73d897741fe1cb47004a1a8": "\\Gamma_k^{(i)}", "455c7ced0c09ab81ee3386a0fc0af025": " Kz_t + Lz_x = \\nabla S(z), ", "455c86b37bd502b6909a4e7fc5e714c1": "\\geqslant, \\ngeqslant, \\eqslantgtr \\!", "455ca4710c1cde5ed0841bc78e6b6217": "P = P_0 \\cap f^{-1}(Y) ", "455cdb5093ebc386a782cc649f307ebe": "\n\\mu(x)=1~\\mathrm{if}~|x|\\gg 1,\n", "455ce12d5f15a18b8ef5143e7a417465": " \\bar{U_{j}^2}(T) = 9 \\hbar ^2 T^2 / m k_{B} \\Theta_{D} ", "455d01587e51aaaa9a243db575c69f5c": "I_J - \\ ", "455d9aa382b10c834ea2c7d0ac39ffdb": "-2(\\mathbf{J}^{T} [\\mathbf{y} - \\mathbf{f}(\\boldsymbol \\beta) ] )^T", "455dc56de294d361d071c4b6e3ceceeb": "V(\\sigma,\\tau,z)=\\sum_{m,n} A_{mn} S_{mn} T_{mn} Z_m\\,", "455dd3c9684c76a222d20f5d9a61fe5e": " = 5 \\times 5 + 2 \\times 2 ", "455de122f52c36a239c965eb41073e2b": "\\tilde{y}(\\bold{x}; \\bold{\\alpha}) = \\sum_{j=1}^{p} \\alpha_j y_j(\\bold{x})", "455e67fef131cded4f7251dfc312b581": "M' \\to M", "455e698e36420fe7030991cb1b23fcd2": "QC_x", "455ee4c7976fb843e25080ef741ae2de": "g(f_i)_{i \\in m} = g", "455ee7596f5804b20d25c5d81c9707ad": " i,j = 1\\dots 3 \\,.", "455eeac1d3b678ebf590f5d886af5678": "H_{\\epsilon I}, U_{\\epsilon I}", "455eeee3a2d4233b26735897f76e11d9": "S \\rightarrow A: \\{B, K_{AB}, N_A, N_B\\}_{K_{AS}}, \\{A, K_{AB}\\}_{K_{BS}}", "455f2ff67770e671f51430768523503f": "|\\varphi'(z)| = \\lim_{k\\to\\infty} |z'_{kr}|/(|z_{kr} - z^*|^{2}\\alpha^{k}), \\, ", "455f76236696e61ebab12ad74b29f521": "(n_{d/2+1}, \\ldots, n_d)", "455f85fcdbc89405bad4c5ce95fe21de": "\\begin{align}\n f \\left( \\sum_{i=1}^nw_ix_i^p \\right) &\\leq \\sum_{i=1}^nw_if(x_i^p) \\\\\n \\sqrt[\\frac{p}{q}]{\\sum_{i=1}^nw_ix_i^p} &\\leq \\sum_{i=1}^nw_ix_i^q\n\\end{align}", "455f931b42d953890beaf815b7e3aaf4": "\\sigma_x(y) = \\frac {-\\sigma_my}{c}", "4560432fe79ce0213349b931ead5db1a": "\\delta t=0.3\\pm4.0_{stat.}\\pm9.0_{sys.}", "45606906fb3b891690adc12c67cb2e97": "\\lambda_{a'}=m^2_{a'}/2E", "4560df4dfad07b7a5042b3f05566f039": "\\operatorname{corr}", "456157068b887de576e162d283a820d4": "\\mathbf{r}_i \\,", "456165afac46ad867be86da2b19765b4": "\\Pr(A|B_1)={99 \\over 100}", "4561ebf8d347f2d2591f56084660004a": "\\frac {p}{q}= \\frac{ad+cb}{ab+cd}", "4562127c332f29602588ab77a45d07bf": "\\Delta I_\\text{obj}/I_\\text{obj}", "456231865654d94e268c3a5f846190cb": "(x,h)", "45623d0670c589f6d911126edd583c47": "X(t_0)", "4562522e9263939d66aaab17f9a90b9b": "mk", "45681c9ce23ff27380461ede38a2c6f7": "\\frac{1_{10}}{3_{10}}", "4568e25526450b20abdef1b8144224c4": "2\\mathbf{A}\\mathbf{x}", "45692a631a414c1b96bc2582770e057d": "t_1,\\ldots t_r", "4569455ea9196f7d5fc0b8ff35fbe084": "\\bar{D}_{T,R}", "45699ce9974d7959a185c27bd9bc22dc": "W(n-1,x-1)", "4569a422d2356871d69b235c2e889317": "\\textstyle{\\frac {\\log(13)} {\\log(3)}}", "4569e4826f4bb7b4373734b54d7fe26d": "B_5(x)=x^5-\\frac{5}{2}x^4+\\frac{5}{3}x^3-\\frac{1}{6}x\\,", "4569ec211fba0ae560f0d8445f74ded9": "\\begin{align}\n(Y', C_B, C_R) &=& ( 16, 128, 128 ) + ( 219 \\cdot Y, 224 \\cdot P_B, 224 \\cdot P_R)\\\\\n\\end{align}", "456a362bb58f0fa31f54a554c7a9125b": "\\gamma_A", "456a6dc109c638464fce0a06a7bc1c66": "P = 0.08326 + 0.00925 + \\frac{2 \\times 6}{1081} + (0.00278 \\times 2) \\approx 0.1092", "456ae14af8b6c357c926df7b029d068a": "VCA(64x^3+192x^2+80x+8,(2,4)) ", "456ba8541f1238a9c27d85036597cad8": "\\bar\\lambda", "456bd2eb161a01cd389304bbb8ec807c": "\\frac{2s^2\\sqrt{3}}{5}", "456bd562e7de5e36e52864d4f5f1f58d": "\\; (u_{ij})_{ij}", "456bef2804b54ddbbc8aae6f74054ade": "L-y'\\frac{\\part L}{\\part y'}=C \\, .", "456bf5617761682f2932bfcb71cce82e": "\n\\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c})=\n\\mathbf{b}\\cdot(\\mathbf{c}\\times \\mathbf{a})=\n\\mathbf{c}\\cdot(\\mathbf{a}\\times \\mathbf{b})\n", "456c1646910587ac7b884a9b76480a60": " -\\int [\\int w(x)\\ \\, dx] dx ", "456c38664d21f6388cad790059c14f91": "S = -R\\Sigma_{i}p_{i}ln(p_{i})", "456caa33744ed0f8cb0ddc80e712cfa6": "\\mathrm{SNR} = \\frac{ | {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{-1/2}s) |^2 }\n { {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h) }\n \\leq\n \\frac{ \\left[\n \t\t\t{(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h)\n \t\t\\right]\n \t\t\\left[\n \t\t\t{(R_v^{-1/2}s)}^\\mathrm{H} (R_v^{-1/2}s)\n \t\t\\right] }\n { {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h) }.\n ", "456d1ab26b7333c7a6d912dce492455c": "e \\in P_i - Q_i", "456d468e75868c64b4d30084c9434929": "R_fI", "456d57f1a5fe99606d0a8dda23e74b7a": "b_{N_p}", "456d8795d36843124cf2ed825cad32a2": "S \\in N", "456dd26286abcd02588342f63b160420": "\\left(\\!\\!{n-1\\choose k}\\!\\!\\right).", "456e4758e2c221aaf675d213a642b2e7": "E_{yz,x^2-y^2} = \\frac{3}{2} m n (l^2 - m^2) V_{dd\\sigma} -\nm n [1 + 2(l^2 - m^2)] V_{dd\\pi} + m n [1 + (l^2 - m^2) / 2] V_{dd\\delta}", "456eb26915e39f3a4a31c9d11bed78ba": "W = \\{ W_t, t \\geq 0 \\}", "456ec23939f97aad5e9ce699ba4733e4": "N_\\mathrm{D}", "456eecafffd96a4f1f0b765e7ef7c13a": "\\vec a_P = \\vec a_C + \\vec \\alpha \\times (\\vec r_P-\\vec r_C) + \\vec \\omega \\times (\\vec v_P-\\vec v_C)", "456f57f3c3128e00812868bc730e7740": "\n\\Pi(n,\\phi,k) = \n\\int_0^{\\phi} \\frac{\\mathrm{d} \\theta} {(1 - n \\sin^2 \\theta) \\sqrt{1 - k^2 \\sin^2 \\theta}} = \n\\sin \\phi \\,F_D^{(3)}(\\tfrac 1 2, 1, \\tfrac 1 2, \\tfrac 1 2, \\tfrac 3 2; n \\sin^2 \\phi, \\sin^2 \\phi, k^2 \\sin^2 \\phi), \\quad |\\real \\,\\phi| < \\frac \\pi 2 ~.\n", "456f584ab0cfff9cc1eda9359d4b2d0e": "R=(0,w)\\times(0,h)", "456fd56513725576c70f4114fb2438e3": "[M_{\\mu\\nu},P_\\rho]=\\eta_{\\nu\\rho}P_\\mu-\\eta_{\\mu\\rho}P_\\nu", "456fe5b5758e37a683d3cd53c1d67106": "\\Lambda = \\Lambda _0 \\oplus \\Lambda _1 \\oplus \\cdots", "45700d0c72bdff57aee6045e6ba8768b": "^{\\;}q(\\xi ,\\tau)^{i}", "457056bce74a317d3dda896b68a2b0e8": "\\theta\\approx KP", "4570592d55978dd39018974cf11a48e0": " \\langle \\Omega | T(\\psi(x)\\bar{\\psi}(0))| \\Omega \\rangle = \\int \\frac{d^4q}{(2\\pi)^4}\\frac{i Z_2 e^{-i p\\cdot x}}{p\\!\\!\\!/-m_r+i\\epsilon} ", "45705bd6852bfe7e3d13fd84371aa2c7": "W = - {p_1} {V_1} \\ln \\left( \\frac {p_2} {p_1}\\ \\right)", "457071eca8d589165919382f9bd9a5de": "M(l)_n = M_{n+l}", "457077e50ed8f07c08ac766f3b886913": "n = 2k,", "4570ab041913315299940332c2088bdb": "P_{ik}(0;t)=P_k(t)", "4570b6a72c14971e8bd183e1969b981e": "\nM^{ \\downarrow Y} (X,Y) = \\left[ {\\begin{array}{*{20}c}\n {\\mu _2 } \\\\\n {\\Sigma _{22} } \\\\\n\\end{array}} \\right]\n", "4570bc4ddbeb9ad1ddceb981a0d38e0e": "\n\\begin{align}\n& 1 \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\frac{1}{3} \\,+\\, \\frac{1}{4} \\;\\;+\\;\\; \\frac{1}{5} \\,+\\, \\frac{1}{6} \\,+\\, \\frac{1}{7} \\,+\\, \\frac{1}{8} \\;\\;+\\;\\; \\frac{1}{9} \\,+\\, \\cdots \\\\[12pt]\n>\\;\\;\\; & 1 \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\frac{1}{4} \\,+\\, \\frac{1}{4} \\;\\;+\\;\\; \\frac{1}{8} \\,+\\, \\frac{1}{8} \\,+\\, \\frac{1}{8} \\,+\\, \\frac{1}{8} \\;\\;+\\;\\; \\frac{1}{16} \\,+\\, \\cdots.\n\\end{align}\n", "4570ecf6fae9d15789d95c660b5f135b": "\n\\max_{u(t)\\in \\Omega}\\int_0^{\\infty} e^{-\\rho t} \\varphi\\left(x(t), u(t)\\right)dt", "45710f1be5a3e9293bbfc49958f89944": "\\mathbb{Q}(\\sqrt{d})", "45713a9b01b5a4611de71637c38a2abb": "S = \\sum_{j=1}^n (\\mathbf{x}_j-\\overline{\\mathbf{x}})(\\mathbf{x}_j-\\overline{\\mathbf{x}})^T = \\left( \\sum_{j=1}^n \\mathbf{x}_j \\mathbf{x}_j^T \\right) - n \\overline{\\mathbf{x}} \\overline{\\mathbf{x}}^T ", "4571406625bae3181af78d084d79852a": "v=Q^m u", "457155cf6c0573697b62dff8b2865f54": "\\mathbf{A \\cdot B} = A^*(\\mathbf{B}) = A{_\\nu}B^{\\nu}. ", "45717ab26d402976cfb1972dd58b0459": "p_{ij} = 0", "4571ad89402560cfdd4700609be3abe7": " n! = D^nx^n \\;", "4571c2a77e88dbc2af827b5a5682547f": "q_1,\\dots,q_{k-1}", "4571cee12bec07f2389a3f3d5284b993": "\\mathbf{p}_k \\in \\mathbb R^n", "45723dcf16fd2e14b0ca112fcd69ea63": "T = 0, \\; Z = 0, \\; X = R_0, Y=0", "45727008716240af397b62db5710ae58": "\\frac{\\mathrm{d}q}{\\mathrm{d}t}\\,", "4572ba84d2913e317f19e26588554c94": "\\begin{matrix} {3 \\choose 2}{11 \\choose 1}{4 \\choose 2}{40 \\choose 1} \\end{matrix}", "4572bdd2352615a7d464db01053df8f0": "\n\\begin{bmatrix}\n2 & 3 & 1 & 1 \\\\\n1 & 2 & 3 & 1 \\\\\n1 & 1 & 2 & 3 \\\\\n3 & 1 & 1 & 2\n\\end{bmatrix}\n", "4572c6f0e00dc073057c72fd52150bca": "ds^2= \\left(1+\\frac{Gm}{2c^2 r_1}\\right)^{4}(dx^2+dy^2+dz^2) -c^2 dt^2 \\left(1-\\frac{Gm}{2c^2 r_1}\\right)^{2}/\\left(1+\\frac{Gm}{2c^2 r_1}\\right)^{2}", "4572d4a0912107042bcfbfe4cfef431e": "\\lim_{q\\to 1}[n]_q = n", "4573016191089f51cd2af9ca4b307e76": "\\nabla \\nabla X", "45730c1c3f3813577a6ee7d6e061e0da": "\\psi^{\\dagger} ( i \\hbar \\gamma^\\mu \\partial_\\mu \\psi + m c ) ( -i \\hbar \\gamma^\\nu \\partial_\\nu \\psi + m c ) \\psi = 0 \\,.", "457383f53075fe1183a7f4ed7ce121f8": "x \\not\\in L^c \\Rightarrow \\mathrm{Pr}[A^c\\,\\mathrm{accepts}\\,x] < 1/2", "4573c7d49e4925bd1408cd9d15ca7985": " \\and D[o] = [F_{10}, S_{10}, A_{10}]::[F_9, S_9, A_9]::L ", "4573cb0a1b738b8313a7d2dc8dbbbb0f": "f(y|\\theta^*)", "4573cea0cb2ce73952f01ecfe66c8a7e": " h_k = h_{d-k}. ", "4573fd274f04987cba3ce3fdf0c3a6bc": "u=m_1^2 + m_4^2 - 2p_1 \\cdot p_4 \\,", "45745b88be30cec589600c28ffd388e3": "\\mathbf F_{ph} = \\mathbf F_p + \\mathbf F_h", "4574991ec7a65697a6a6abee9cb89da8": "\\Delta T = (T - T_o)", "4574a3c6f682c999ec93a624ad0916df": "\n S(4,3) =\n \\left\\{ \n \\underbrace{(1111), (1112), (1113)}_{(a)},\n \\underbrace{(1122), (1123), (1133)}_{(b)},\n \\underbrace{(1222), (1223), (1233), (1333)}_{(c)},\n \\right.\n", "457533b7cf0774c18db6b000e938f9fe": "\\mathcal{L}_Y\\alpha=i_Yd\\alpha+di_Y\\alpha.", "4575380fc5ece02d93fef0c0bc018d7b": "\\scriptstyle [0,\\,\\frac{1}{2}]", "4575770f23e77f96d9f139404df71b4f": "C_1 = ", "4575ba181cd178b2a9841f50fb638641": "\\mathit{A} = (x^{a_i})", "4575d29d3c4b3a2b8de6c9281681b64c": "g:\\mathcal{X}\\rightarrow\\mathcal{Y}", "4575e8ab8affb562a075413d2c517fb0": "(12, 35, 37)", "45762d39f1f199e4e0288ceeec5d7c1e": "\ng ( \\ln \\tau_D ) = { 1 \\over \\pi }\n{ ( \\tau_D / \\tau )^{\\alpha\\beta} \\sin (\\beta\\theta) \\over\n( ( \\tau_D / \\tau )^{2\\alpha} + 2 ( \\tau_D / \\tau )^{\\alpha} \\cos (\\pi\\alpha) + 1 )^{\\beta/2} }\n", "457646b73dd0cf91db5db62984722cca": "g_{\\mu 5}", "45764ab59f085b5a89701b7fad9a301c": "\\arctan\\sqrt{2}=54.7^\\circ", "45764ca6b3f493815d6598b04b7350c0": "\\underline{\\mathbf{X}}(\\ell)", "457672c5a81f381263d3dce409affbd0": "\\dot{\\dot{M_{c}}=\\dot{M_{T}m_{C}}} \\,", "4576747181b1a70f2e750045604438b8": " -\\frac{\\lambda}{\\mu} -\\frac12\\ln\\lambda", "4576a33de9fcee4d4c5687b96873d975": "E_n = \\hbar \\omega( n + \\frac{N}{2})\\quad\\hbox{with}\\quad n=0,1,\\ldots,\\infty,\n", "4576b0361252068ae8aa60c8a198e4d8": "= \\int (2\\pi)^{-\\frac{p}{2}}|\\boldsymbol\\Sigma/n|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}n(\\overline{\\mathbf x}-\\boldsymbol\\mu)'(\\boldsymbol\\Sigma^{-1}-2 i \\theta \\boldsymbol\\Sigma^{-1})(\\overline{\\mathbf x}-\\boldsymbol\\mu) }\\,dx_{1}...dx_{p},", "4576d9abbe391f8bf5940d8b3c24ae45": "\\mu_s=\\tan(\\theta)", "4576e42524aca223de32e6734ec79474": "\\sec x = \\sum^{\\infty}_{n=0} \\frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\\quad\\text{ for }|x| < \\frac{\\pi}{2}\\!", "45777a635b313fb23f20d36096171e7f": "\\scriptstyle O(\\sqrt{|V|}\\cdot|E|)", "45779599ec121f1d46b5a98505752dbb": "f(V\\ \\backslash\\ \\{z_0\\})", "4578e0de4723d3aa25bb80ca3fa6bab8": "f''(x) = \\lim_{h \\to 0} \\frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.", "4578fe3bc53e601d6c2f12990d46e741": "{\\rm Tr} ", "4579265533f6baf3a6b1c54d2cfc672e": "\\delta_1 + \\delta_2.", "45797dcde7f24aa42fa92ac4834767f6": " x, y \\in \\mathcal{E}\\ ", "45798512f35302da7c4bc74c4464945e": "= \\operatorname{E}_X[\\operatorname{Var}[Y\\mid X]] + \\left(\\operatorname{E}_X\\left[\\operatorname{E}[Y\\mid X]^2] - \\operatorname{E}_X[\\operatorname{E}[Y\\mid X]\\right]^2\\right)", "45798cfc72e640486460edf4506873e7": "A = 2 \\cot \\frac{\\pi}{8} a^2 = 2(1+\\sqrt{2})a^2 \\simeq 4.828427125\\,a^2.", "457994903ca5bd0762bcb86ee94728e8": "Scenario \\quad I: \\qquad d_B = {\\left ( {\\frac {149,597,871 km}{696,000 km}} \\right )} {\\left ( 5.6 AU \\right )} = 1,204 R_{\\odot}", "4579a50422f24b15badda8f7a69a3da7": "\nF_C", "4579c51849b179193831f13c65d3ccc9": "M_{BC} = -4\\frac{2EI}{L}d_1 -2\\frac{2EI}{L}d_2 - q\\frac{L^2}{12} = -11.569", "4579d4d3208d407cce4a1cc80a52ab82": "\\beta < \\varphi_{\\beta}(0)", "457a18312fb5689b4264fb1e331fe625": "\\scriptstyle \\kappa", "457a337a4d322e0fd28862fd8ce5b49f": "\\lim_{x\\to\\infty} x^n e^{-x}\n=\\lim_{x\\to\\infty}{\\frac{x^n}{e^x}}\n=\\lim_{x\\to\\infty}{\\frac{nx^{n-1}}{e^x}}\n=n\\lim_{x\\to\\infty}{\\frac{x^{n-1}}{e^x}}.", "457a34b3b3d42146f2635449154190dd": "u_1 = -v_1", "457aca125357079728f9af59de77b1d5": "{\\rm sign} S_{xx}''(x_0)", "457b09251e8d02a253d893a6b14ff2b5": "\n \\mathbf{u}\\times\\mathbf{v} = \\epsilon_{ijk}{u}_j{v}_k\\mathbf{e}_i\n", "457b7ccf77774f6c4da42f600abaa9ed": "+ \\left( \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}} - u_{l}^{\\alpha}\\frac{\\partial \\rho^{l}}{\\partial u^{k}} \\right)\\theta^{k}.\\,", "457ba70ac1e2ed17a9f7f5bfe0739321": "\\sum G_i", "457bd2fb6775f9190b20cdf93aa40e18": "c_{it}", "457bd9be7cd0ff53e0d93d7d75608359": "t = t + \\Delta t", "457c68c909d6ea2823cc6ac422d00837": " n\\overline p - n\\underline p = n(\\overline p - \\underline p) = n\\Delta p ", "457c6e24afcff231f949f27b60e8f485": "\\frac{d^{2}u}{d\\theta^{2}}+u=-\\frac{Z_{1}Z_{2}e^{2}}{4\\pi\\epsilon_{0}mv_{0}^{2}b^{2}}=-\\kappa,", "457ce84a7864b9d06750b32c328555ad": " t \\in [0,T]", "457d2a70c037fe060c220ab290fe7b56": "x_{t=1 \\dots T}, y_{t=1 \\dots T}, F(y|\\theta)", "457d63e2f04a4aa660f04b13ee668573": "y_i \\,=\\, a_0 + a_1 x_i + a_2 x_i^2 + \\cdots + a_m x_i^m + \\varepsilon_i\\ (i = 1, 2, \\dots , n) ", "457d855c1b77f7ce02e01fa761e6877f": "\n\\begin{bmatrix} x' \\\\ y' \\end{bmatrix} = \\begin{bmatrix} s_x & 0 \\\\ 0 & s_y \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix}\n", "457d8a217890fe13754841cb454bc8aa": " \\left| e^{(\\lambda-\\lambda_0)(S(x) - M)} \\right| \\leqslant 1 ", "457d8e9c1ba69df5935eaab850570045": "\\prod_{m=1}^n\\sum_{i=0}^{m-1}X^i=1(1+X)(1+X+X^2)\\cdots(1+X+X^2+\\cdots+X^{n-1})", "457e45a2a7289162fd5dc7fff40b5daf": "\\scriptstyle c(u, v)", "457e86ab622e7137af5650f369ee6797": "\\Gamma(z+1) = z!", "457eaaa6086e6de7b4b960ec5142fcad": "\n \\mathcal{M}^K := -D\\nabla^2 w^K\n", "457ebf766fe3ace1aa14c32701d034b1": "\\dot{u} = -\\delta v - \\frac{u}{T}", "457ef38bf34f325cfde73c593c02456b": "\\mathbf{OPD} = 2 \\pi r n_{eff}", "457efb8546cc3b4306aff55f8a79dcd9": " \\theta_{cs} = 180^{\\circ}", "457f044a2117c0e4752e4051cfd09346": " f_i (x) = x^T Q_i x + 2g_i^T x + d_i \\le 0,0 \\le i \\le k. \\, ", "457f2edfda86e7d7c8b4eca29f5ab115": "P_A(X)=\\sum_{k=0}^{A-1}\\left({{A+k-1}\\atop{A-1}}\\right)2^{-k}X^k", "457f3f62d7633b489705ca9ff9fa63f3": "p(x) \\propto \\left(1-\\frac{x}{a_1}\\right)^{-\\nu (a_1-a)} \\left(1-\\frac{x}{a_2}\\right)^{ \\nu (a_2-a)}", "457f4cb2c31985f91e4fec8bf026c5f0": "f(-2)=-1, f(-1)=0, f(1)=0, f(2)=1", "457f80bc18196fb2698a5774d989e03b": "\\widehat{sl(2)}", "457f9a08afcaf3c951236d8d003fcecc": "\\nu_s", "457fbabf7c3edb9853beff5c70769170": " h = \\frac{\\sin\\alpha\\,\\sin\\beta}{\\sin(\\beta-\\alpha)} \\,l = \\frac{\\tan\\alpha\\,\\tan\\beta}{\\tan\\beta-\\tan\\alpha} \\,l", "45800a2dc4ee591e080aa066abd93dd3": "x^2 - 2\\operatorname{Re}z_j\\,x + |z_j|^2 = (x-z_j)(x-\\overline{z_j})", "45800a307af960641eccac1452d8410c": "\\left|y\\right\\rangle \\left|f(x_0)\\right\\rangle", "45800d2a97c9abf18b033078ee1e2e2a": "\\begin{align}\n \\mathrm{Gi}(x) &{}= \\mathrm{Bi}(x) \\int_x^\\infty \\mathrm{Ai}(t) \\, dt + \\mathrm{Ai}(x) \\int_0^x \\mathrm{Bi}(t) \\, dt, \\\\\n \\mathrm{Hi}(x) &{}= \\mathrm{Bi}(x) \\int_{-\\infty}^x \\mathrm{Ai}(t) \\, dt - \\mathrm{Ai}(x) \\int_{-\\infty}^x \\mathrm{Bi}(t) \\, dt. \\end{align}\n", "458037d1a205831028ef39c6ed8ead5d": "S(T) = \\int_{\\lambda _1}^{\\lambda _2}\\frac{c_1}{\\lambda^5\\left[\\exp\\left(\\frac{c_2}{\\lambda T}\\right)-1\\right]} d\\lambda", "4580939e81df0efe0d6efd308714c782": "\\mathrm{soc}(R_R)", "45809f37257195d36f2655a428f115e4": " U(\\mathbf{P}_{\\phi})= \\begin{bmatrix}\n e^{i\\phi} & 0 \\\\\n0 & 1 \\end{bmatrix}=\\begin{bmatrix} e^{i\\phi/2} & 0\\\\\n0 & e^{-i\\phi/2}\\end{bmatrix} \\text{(global phase ignored)}=e^{i\\frac{\\phi}{2} \\hat{\\sigma}_z}", "4580c2740ab6d9222ef06d7c6865583e": "tau", "4580c63b1dedfae659e6c74d5d3a8088": "\\frac{2(N_0-N)}{N_0 \\sdot f_{av}}", "4580eb762f9fa4aa141ca9e89f8c057a": "\nV_{ij}(r_{ij}) = V_{pair}(r_{ij}) - D \\sqrt{\\rho_i}\n", "4581a173603090271dbbe474b69eae1a": "\n\\operatorname{cov}(X,Y)^2 = \\operatorname{E}\\left[\n \\big(X - \\operatorname{E}(X)\\big)\n \\big(X^\\mathrm{'} - \\operatorname{E}(X^\\mathrm{'})\\big)\n \\big(Y - \\operatorname{E}(Y)\\big)\n \\big(Y^\\mathrm{'} - \\operatorname{E}(Y^\\mathrm{'})\\big)\n \\right]\n", "4581d03f2412fa6bfeb050b70caf9dfb": "C_f(z) = \\sum_{n=-\\infty}^\\infty f(a+nW)\\frac{\\sin[\\pi(z-a-nW/W)]}{[\\pi(z-a-nW/W)]}", "4581d578d1b325da5f79611ad749d67f": "\\ddot{x_0} \\ll \\ddot{x_1}", "45820aff098004386fff82fb3d6e4585": "\\Delta=v_n^2", "45828103b503be7d46738e2682a84492": " B_{ij}=O_{ij}^2 \\text{ for } i,j=1,\\dots,n. \\, ", "458325956ff29ebb9cbc06b3932b87c3": "\\sum \\beta_i=0", "4583469bc2ceb03c3b94c53294ec5048": "\\textstyle Pr(a_i \\le a^*) = \\frac{\\frac{1}{2+3/A}+A}{2A} = \\frac{A}{4A^{2}+6A}+\\frac{1}{2}.", "45836a5cddef0f8b434a204d2a37b648": " M_i ", "4583d48b1ac511d71b2b1b6630708420": " C^J_{E_1} = \\varepsilon^{2}_S C^S_{E_1} ", "45840d99e5d425f80962b35d2a2a2689": "{\\bold u}({\\bold u}\\cdot D\\nabla^2\\psi)", "458415f3cfdb98b1b4e4cde563ee4854": "|p s-q r|=1,", "45842d17b68ec7efa70ab591866b72ad": "\\frac{r-1}{r}\\cdot\\frac{n^2}{2} = \\left( 1-\\frac{1}{r} \\right) \\cdot\\frac{n^2}{2}.", "4584367255d80cb25c1c79d96366ca67": "\\mathbf{\\hat{r}} = \\sin \\theta \\cos \\varphi\\mathbf{\\hat{x}} + \\sin \\theta \\sin \\varphi\\mathbf{\\hat{y}} + \\cos \\theta\\mathbf{\\hat{z}}", "4584ec4a53555c1230561504c7b3e66e": "x^{\\prime}=\\gamma x^{*},\\quad y^{\\prime}=y,\\quad z^{\\prime}=z,\\quad t^{\\prime}=t", "458516e8624feaa2811829f8a102eafd": "x^{-1}(xy) = y = (yx)x^{-1}", "45851f9d2adf67e09a1953e4052a5b20": " \\Pr(Y=1 | X=x) = x'\\beta . ", "458587a434fbc40bde563ed6fc46a292": "\\displaystyle h(y) = x", "4585a5549b2bdf71b2e2ba395b0f5ace": "\\ V_\\mathrm{ov}= V_\\mathrm{GS}-V_\\mathrm{th}", "4585ad583efde3ef5b9ae9b7a1cfbeee": "Q = E^{-1}(D)", "4585d9e4510321f0c6fe7565c98c720c": "\\operatorname{Tor}^R_1(M, R_1) \\to P_1 \\otimes R_1 \\to P_0 \\otimes R_1 \\to M \\otimes R_1 \\to 0", "4585dd092f2575453a737e4ac5e54a9c": "\\phi=\\frac{1}{K-\\omega} f.", "4585e1ce878d8b48dbae86081a8fdf74": "J^\\alpha = J_\\alpha ", "45871e62b72ae398e296ecff8a13960b": "\\left(b, r, u\\right)\\succsim \\left(c, q, u\\right)", "45874d8d0c5e1611547e9cdf23f79840": "g^{\\mu}=\\bar{\\mathsf{h}}(e^{\\mu})", "45877b8b52533261490f16e8d7d2247b": "\n {\\mathrm{d}p} = - \\frac{\\gamma p}{V} {\\mathrm{d}V} \n", "4587a9c1f8d3a5ddf27bc343c807f4fc": "\\hat{D}", "4587c931607b0ed51e5684ff168bc68a": "a^{(n-1)/2}\\not\\equiv x\\pmod n", "4587dec12e6cbbd07430d7377faa650b": " f\\left(d\\right)=1- \\sigma^2/d^2 ", "4587f8983d5eb9d9eff54221c151403f": "\\left\\{ \\mathbf{P}_{i}\\right\\}", "4588364fa0111d76f05d231e2aa9f554": "LBA=((C \\times HPC) + H ) \\times SPT + S -1 ", "4588577ff19f4a42506e0e8e5e06e081": "\nE \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, x \\right) \n= \\sum_{h=1}^{p} \\frac{\\prod_{j=1}^{p} \\Gamma (a_j - a_h)^*} \n{\\prod_{j=1}^{q} \\Gamma (b_j - a_h)} \\Gamma (a_h) \\; x^{a_h}\n\\;_{q+1}F_{p-1} \\!\\left( \\left. \\begin{matrix} a_h, 1 + a_h - b_1, \\dots, 1 + a_h - b_q \\\\ 1 + a_h - a_1, \\dots, *, \\dots, 1 + a_h - a_p \\end{matrix} \\; \\right| \\, (-1)^{p-q} \\;x \\right).\n", "458884bbb36093a7f2499659f202fe1a": " M = \\begin{bmatrix} q_{01} & 0 \\\\ 0 & q_{01} \\\\ -q_{12} & q_{02} \\\\ q_{31} & q_{03} \\end{bmatrix} ", "45889af224329c4b90939e4b79550259": "\\textstyle \\mu / \\rho ", "4588b81535764f42d767e9af54e80d42": "~ G ~", "4588cd827c6bea767515800392b8d053": " \\psi_1(x)=\n\\begin{cases}\nA_1 e^{ik_1 x} + B_1 e^{-ik_1 x}, & x<0, \\\\\nA_2 e^{ik_2 x} + B_2 e^{-ik_2 x}, & 0L, \n\\end{cases} ", "45895f4b502b760d2770ad6badaffa96": "V_n(R) = \\int_0^{2\\pi} \\int_0^R V_{n-2}(\\sqrt{R^2 - r^2}) \\,r\\,dr\\,d\\theta,", "45896141d42e2732de39984f669046d4": "\\left[2^r-1, 2^r-r-1,3 \\right]_2", "458962b5b9b5f85c20ae6abdb0834df2": "\\{(x_1,x_2,x_3,x_4) \\in \\mathbb R^4 \\,:\\, -1 \\leq x_i \\leq 1 \\}", "4589ad699b1bfd5aaf4b846118262463": " \\vec{x}_{P} ", "4589fb61afcc0bdedcecfa1ab625614f": "\n S(4,3) =\n \\left\\{ \n \\underbrace{\n\t (1111), \n\t (1112), \n\t (1122), \n\t (1222), \n\t (2222)\n }_{(\\alpha)},\n \\underbrace{\n\t (111{\\color{Red}\\underset{=}{3}}),\n\t (112{\\color{Red}\\underset{=}{3}}), \n\t (122{\\color{Red}\\underset{=}{3}}), \n\t (222{\\color{Red}\\underset{=}{3}}) \n }_{(\\beta)},\n \\right.\n", "458a468653bfae753e6441c2eae7b785": "f^{m+n}\\ x = f^m (f^n x) ", "458a573b78371b24b5dd7c27888db2fd": "\\textrm{rcon}(i) = x^{(i-1)}", "458a5f1f1208d5a805c44b3b179ff59c": "\\sqrt{21\\over{8}}\\cos(\\theta)(5\\sin^2(\\phi)-1)\\cos(\\phi)", "458a7c2627e824c5885518193e5eb85b": "\\frac{\\partial \\phi(t, t_0)}{\\partial t} = A(t)\\phi(t, t_0)", "458a9216b81aa45a64505a870735d44a": "2^k-1", "458a9807ec30ce8c511f86b1420e2580": "*[F,G]^{IJ} = [*F,G]^{IJ}", "458a9fc6670721f038b7b9c618691c62": "( 1-e^{-\\alpha t}) \\cdot u(t) \\ ", "458ab09c4b82128298bd42f19fd5679e": "H=\\{e\\}", "458af27cbccec0c6025f4effbfdfd045": "\\mathrm{ROE} = \\frac{\\mbox{Net income}}{\\mbox{Sales}}\\times\\frac{\\mbox{Sales}}{\\mbox{Total Assets}}\\times\\frac{\\mbox{Total Assets}}{\\mbox{Average Shareholder Equity}}", "458b6dd300f5ce0396b32f4f301a23d4": "y_i'", "458b7f69bbf471ab8af20986f975f8ee": "2/(j+1)", "458ba0cceb126da99821b9769758eac5": "\\frac{\\Delta f^{*}}{f_f}\\approx \\frac i{\\pi Z_q}\\frac{-\\omega ^2u_0m_{%\n\\mathrm{F}}}{i\\omega u_0}=-\\frac{2\\,f}{Z_q}m_{\\mathrm{F}}", "458bff4c2a07a9e1b21d12a1489b5c78": "{S}", "458c3aad8f6d4f29154b969639fdce67": "CP^\\infty", "458c478490c2ddd8f243c93206fed95b": " = 1- 16\\sqrt{\\tfrac23}\\;\\pi^3 \\left(\\Gamma(\\tfrac{1}{24})\\Gamma(\\tfrac{5}{24})\\Gamma(\\tfrac{7}{24})\\Gamma(\\tfrac{11}{24})\\right)^{-1}", "458c54c7724febe86a253000f631fc3b": "\\|x+y\\|\\le\\|x\\|+\\|y\\|", "458c88c3b1026027eaebaafa40f2d3fb": "\\scriptstyle -r", "458cd94ddf481ed45a0ef4902a29c3d0": "a_1=b_1,\\text{ }a_2=b_2,\\text{ }\\ldots,\\text{ }a_n=b_n.", "458cf60f95cd5000293402e4655db6c6": "c_\\mathrm{g}=(gh)^{1/2}", "458d39cc4cd47f988ab4103cd90d138c": "x\\in\\{18,36,54,72,90\\}\\,", "458d763e5a46f98e6a3a68fae9dad9f0": " \\omega^2 ", "458d7eed9fdb7e9dd7dbcd81cf542556": "A_\\bar p", "458dbf29f4513413ccdaf198fadaac5c": "\\mathbf {F}_{i}^{(T)}", "458e09c5024f904f0d835e85d46e4781": "\\,x^2 = Ny^2 + 1,", "458e1e1b2ea2e4558161c94c1ee9d41b": "\\beta/\\alpha = r^* := \\sqrt{2}s / (2^T-2s)", "458e335a4e034abf70a4fa43d9e857a6": " P \\psi = \\lambda \\psi ", "458e84450ff7c72b926bfb542a4645c5": "\\langle\\Psi_{motion}\\vert", "458f2d07d9bcebd68cab23e6f4cc38c6": "x,y \\in E", "458fdc58ad57a0df600555473d9b92af": "I(X;Y|Z) = \\int_\\Omega \\log\n \\frac {d \\mathfrak P(\\omega|X,Z)\\, d\\mathfrak P(\\omega|Y,Z)}\n {d \\mathfrak P(\\omega|Z)\\, d\\mathfrak P(\\omega|X,Y,Z)}\n d \\mathfrak P(\\omega),\n ", "458fe0af0bf1f43416bc538924b4e41e": "e_h =\\ e\\ \\sin \\omega", "45900923a68bfc82bb35708f632fe2f8": "X_1 \\times \\cdots \\times X_n", "45900d062f140dc769f94625f52d7038": "\\hat{X}^n:\\mathcal{Z}^n\\to\\mathcal{X}^n", "459011a8a45cdfd2c7d80562dd13e95a": "x^5 - x^4 - x + 1 = 0\\,", "459069e258e9ea4020787cb85fcb818a": " \\frac {V_1 - V_O } {R_2} = j\\omega C_2 V_O \\ , ", "4590c89fccd95b12e9ab014704233325": "{\\mathbf Q}={\\mathbf P}\\times_{\\mathfrak m}L^2({\\mathbb R}^n)\\,", "459126067f6c1883d0893a482563d32e": "f_n=1/c\\left( \\frac{V^e_{n-1}-V^e_{n}}{R_{n-1}/2+R_{n}/2} + \\frac{V^e_{n+1}-V^e_{n}}{R_{n+1}/2+R_{n}/2} + ... \\right) ", "4591593fbc3aaecde961ea3018acdb16": "S_{2n}", "45919237ec05c1c52ee701c132dcf579": "\\nu \\propto \\alpha p_{\\mathrm{gas}}", "4591cd658a4d65606a1637eb5db6314c": "S_{21}\\,", "4591e8e025528a9c8e7fa38a44ff9b48": "N \\geq 0", "4591fa75a187a3269015b45a058b2486": "M_V = 0.03 + 5 \\cdot (1 +\\log_{10}{0.129}) = +0.6.", "45920710f4c04e57669caa7f3b1d2034": " \\lambda = \\frac{v}{f} ", "459237a2bbbfc440067d789527a9a932": "D_1 = [P] + [Q] - 2 [O]", "45928a933ee53d863edeb938ef177d0d": "J_1^m", "4592a196734f10b3b07b52dd3b64aca3": "T+I", "4592ab606f6c683b57239750d5e96016": "O (|V| |E|)", "4592b19004051545ab528dfb4576529b": "A \\cap B = \\{ a^n b^n c^n \\mid n \\geq 0\\}", "4592b62faf3a349c3721d99b6f95acd3": " D_{ijkl} = D_{ijlk} ", "459340a8c87009c6ba72c6955d154553": "\\begin{align}1-\\left(\\tfrac{1}{10}\\right)^n\\end{align}.", "459344f932cf95ee4956afec3aba463b": "\\begin{smallmatrix}M_{\\bigodot}\\end{smallmatrix}", "4593a283e1928e751dba8ea1d4307151": "-\\frac{{\\hbar}^2}{2m}\\frac{{\\partial}^2}{\\partial x^2}\\psi(x,t) + V(x)\\psi(x,t) = i \\hbar \\frac{\\partial}{\\partial t} \\psi(x,t).", "4593ab5d765651d2a79b758becca1f10": "\\operatorname{Ber}(X) = \\operatorname{Ber}(JX) = \\det(C-DB^{-1}A)\\det(-B)^{-1}.", "4593b9de09cbe6148983e42f5571e28f": "\\begin{align}\nf(1) & = 1 \\ \\text{, and} \\\\\nf(x+1) &= x f(x),\n\\end{align}", "4593e45abf7bc4e996fa5ec663ea5dfd": "\nG^{\\mathrm{A}}(\\mathbf{x} t|\\mathbf{x} 't') = -\\mathrm{i}\\langle[\\psi(\\mathbf{x} ,t),\\bar\\psi(\\mathbf{x} ',t')]\\rangle\\Theta(t'-t),\n", "4594888d184a0c9cf2068721d5002e60": "e_0\\;", "4594894efdadf7531a66bc56e5cb8288": "\n\\begin{align}\n& \\left(\\frac{2D^{\\mathrm{face}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} - \\left(1+\\frac{2D^{\\mathrm{face}}}{D^{\\mathrm{beam}}}\\right)\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} = \\frac{M}{D^{\\mathrm{beam}}}- \\cfrac{q}{S^{\\mathrm{core}}} \\\\\n& \\left(\\frac{D^{\\mathrm{beam}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^3 w_s}{\\mathrm{d} x^3} - \\left(1+\\frac{D^{\\mathrm{beam}}}{2D^{\\mathrm{face}}}\\right)\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x} = -\\left(\\cfrac{D^{\\mathrm{beam}}}{2D^{\\mathrm{face}}}\\right)\\frac{Q}{S^{\\mathrm{core}}}\\,\n\\end{align}\n", "4594d3826b1cf70bda2b0f6b570f3597": "\\alpha = a + \\sqrt D\\text{ to }\\alpha' = a - \\sqrt D\\, ", "4595046a3bca574e3624a0e3cdee3316": "\\sum_{k=0}^\\infty {s \\choose k} (-x)^k = (1-x)^s,", "4595238bf990ea08150935cccb52392e": "\\omega = L_{-2}|0\\rangle", "459561e8357a3e41d1c6ae1fea1b7511": "\\lambda = h/Mc ", "459605a1924f55967f2382cee7d78c40": " P_2 = P + 2 ", "4596198a6069857c973ee736c813e880": "SF\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\n1\n\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!|}}\\;I\n", "45962307984e1bd48d6c2d2c7dc8ebd1": "\\alpha \\in \\Omega^1", "45962b07080523d431939f3a381b42ef": "\\gamma = \\sinh \\left ( \\frac{ \\beta }{ 2n } \\right )", "45968e8357bf55ae010b7a6e8e33403c": " \\arctan(x)+\\arccot(x)=\\pi/2.\\;", "4596e866f4b735d65802e7f2a1f437a6": "2^{2/12} = \\sqrt[6]{2}", "4596e8cfba9bdb43864b54df442e4b6a": "\\textstyle\\binom{2n}{n}", "45974ac2b6c66d738b5a9ce690d17205": "{x}, {\\lambda}, {v}, {s} \\ge {0}\\,", "45977140194b0f0d90b1c4a183e403f8": "t\\mapsto e^{it}", "459784b699eec32159b0652ba8a36380": " H[(X,Y)]=H(X|Y)+H(Y)=H(Y|X)+H(X).", "45978db474d99795d442e4503fbd684a": "k= 2\\pi/\\lambda", "459791da628dc26e8caa329d70ef20a1": "\n\\overline{S}(z)=\\sqrt{\\ln(1/\\overline{R}^2)}=\\sqrt{-2\\ln(\\overline{R})}\\,\n", "459798b033e6e9b9b8be90ecf3185905": " \\scriptstyle\\sigma:(n)\\to(n) ", "4597dcad13df09c8239cc5f3c9657d8b": "S_x(t,f) \\approx S_y(t- \\frac{1}{3} f,f) \\, ", "4597ee83e8e6894f6fad5b6392b093de": "\n\\dot \\epsilon_{sh} = k_{sh} \\dot h\n", "4598435632df1b16da639b4fb6112a95": "{1, \\frac{2}{3}, \\frac{3}{4}, \\frac{5}{7}, \\frac{8}{11}, \\frac{13}{18}, \\frac{21}{29}, \\frac{34}{47}, \\frac{55}{76}, \\frac{89}{123}}, \\dots \\dots [0; 1, 2, 1, 1, 1, 1, 1, 1,\\dots].", "459873d8a8847fde4ef3774a48de9816": "\nK_a = { {[H^+][A^-]} \\over {[HA]} } = {{x^2} \\over {F - x}}\n", "45987650ca74c524d06e516bb04b8b76": "\\langle \\rho_{VU}S,\\varphi\\rangle = \\langle S, E_{VU}\\varphi\\rangle", "45989e1e34718dcf5bafb11e11a9414a": "S_{10:1}=1688", "4598f056a77245fda1a87f3d21d67a05": "2^p \\equiv 1 \\pmod{M_p}", "45993f49111274824efd23a742f4f086": " y= C - \\frac{1}{2}\\ A_x\\left( \n\\frac{(y'(0)+\\sqrt{{y'(0)}^2+1})\\ (1-\\frac{x}{A_x}) ^{1 - \\frac{V_t}{V_d}} }{1-\\frac{V_t}{V_d}} -\n\\frac{ (1-\\frac{x}{A_x}) ^{1 + \\frac{V_t}{V_d}} }{ (y'(0)+\\sqrt{{y'(0)}^2+1})\\ (1 + \\frac{V_t}{V_d}) } \n\\right)", "4599459fb341858650217dd30913f3a9": " \\xi = \\frac{x-y}{\\ln x - \\ln y} ", "4599539173edd66f96cf8273a6714c32": "j(n,x):=p\\left(x_0^{n-1} \\right).", "4599553834fba1c59fe72dc2aa666131": "\\left \\lfloor \\frac{a}{b} \\right \\rfloor \\quad \\left \\lceil \\frac{c}{d} \\right \\rceil", "45996c18026d39c682b5087644450cfd": "{\\mathbf{f}}", "459a3e9731baa3d7d17dbeb755ac117f": "\\forall i\\in N,\\ \\forall {a_{-i}\\in A_{-i}}", "459a7f866cea0cfbe686be8c6a6872ed": "H_{Head}", "459ad8e1c11d0a8a135770829fc0534f": "K(z,w)= \\sum_{n=0}^\\infty p_n(z) w^n", "459affd8fffe55beedd715dcdee7ef92": "\\cup_{u \\in H} u(G)", "459b16399fb3c7e73f1c5b0964c0f436": "\\beta=- ", "459b254b25d9d679b7d800ae218b36b3": "D(p||q)", "459b8154ed777c621a02b0ee7ae08f54": "\\displaystyle{v_0=0,\\,\\, v_i=m_i^{-1} X_i,}", "459c4b9b88e41998cb1a07b086f50390": "r_i^2", "459cac3ec81a0c0b227372ae9be2331e": " 0 I(q)", "45c1e3d3c386f368e15109125c4e5826": " \n\\left|\\frac{N(x_i)}{n}-p(x_i)\\right| < \\frac{\\varepsilon}{\\|\\mathcal{X}\\|}.\n", "45c1e660b585742fb3f5db4fd1c88677": "\\mathit{2754}62\\,", "45c2b1c22ca8121f49ecad9c00634cd1": "\nAcc = \\frac{TP+TN}{TP+TN+FP+FN} = \\frac{\\textit{Corrects}}{N}\n", "45c2e59d47ca09d1ae295c8cc4051e62": " \\frac{\\mathrm{d}\\varepsilon}{\\mathrm{d}t} = \\frac{C\\sigma^m}{d^b} e^\\frac{-Q}{kT}", "45c2e7b8c91b98f3acf1b204466cd97f": " (R_x,R_y) ", "45c32d1078c9abd333d427cbedad3d65": "\\frac{z}{\\ln(1-z)} = \\sum_{n=0}^{\\infty}C_nz^n, \\quad |z|<1,", "45c35ec7eb0b82312ea5b4d13164e14b": "x\\to \\lambda x", "45c3b51d9b08a63f11dc1a5e10b1a2ca": "j=0 \\dots k", "45c3eaa90c8ad7e92970fbf887ff494c": "r(x) = f(x)", "45c470c7bb64d72179fa96a0068dbfc3": " \n\\Sigma\\ F = ma = m \\ddot{x} = m \\frac{d^2x}{dt^2}. \n", "45c48cce2e2d7fbdea1afc51c7c6ad26": "9", "45c50b11c6a6b2893f6a89e91e2613b2": "G_n(\\mathfrak{A},\\mathfrak{B})", "45c510addfc702e81982e91cf9dee794": "M_R = e^{M_B}.", "45c53c58020c70cbfcda4ab77af99972": "(K,\\,E)", "45c593ad3821b2d7d9bb4c658ddb6dd0": "p _{v \\cap w}", "45c61f2caf5a5eb642b3adc200c2e20a": "f(x)=1 \\oplus x", "45c6345d3ba8c010f6a55ce3578793bd": " E(Q)0\\end{cases}\n", "45d455c9f169585031a67735a7be81d9": "gd_{\\mathbb{Q}}D_5=2", "45d4bf87d928aada05c4b8df8e5f057e": "H_{i}", "45d4da41c44767731e41336c8112f3f9": "\nh(z) =\n\\begin{cases}\n(z - a)^2 f(z) & z \\ne a ,\\\\\n0 & z = a .\n\\end{cases}\n", "45d5ddc9fc9705d9583e544700affa9c": "w(z)=\\exp(-z^2)(1+2i/\\sqrt{\\pi}\\int_0^z\\exp(t^2)\\text{d}t)", "45d5eec8d1a5b0936fc4d485c308871e": " \\Delta(x) = \\prod_{i0}p_k(X)t^k=\\sum_{k>0}\\sum_{i=1}^\\infty(X_it)^k=\\sum_{i=1}^\\infty\\frac{X_it}{1-X_it}=\\frac{tE'(-t)}{E(-t)}=\\frac{tH'(t)}{H(t)}", "45dcedf5769fcdd8bac4d702aa4a6b50": " \\epsilon \\rightarrow 0 ", "45dd0c605f7af3fa8bb76c9ebac8907b": "e^{in \\theta}", "45dd21fb32cb47b8cd70784051f33156": " \\sigma_2 \\,\\!", "45dd575f1178a76f1d3b7b5697c561cc": " \\phi(x)", "45dd80ab5fafd46f6fa8ee6dd2f63052": " \\partial_t^+ ", "45dd9c3ddc7b15b67830cdb2f95b40ed": "\nr_i = -\\frac{a_{i+1}B_{i-1}}{B_{i+1}}.\\,\n", "45ddae6a51415255e417e2e1243a53f4": "(P \\to Q) \\vdash (\\neg P \\or Q)", "45ddcc9126dc1489abffca11ff0bfb27": "\\alpha < \\omega^{\\mathrm{CK}}_1", "45dde8d9b56f441ba26c0d04ef57fb39": "\\tfrac{1}{8}", "45ddf7b3e3eed8d610c91e25778747f0": "\\mathbf{f}\\longrightarrow \\mathbf{f'}", "45de39779f20d28ee2536da25f69594e": "\\alpha\\to -\\infty", "45de864e41f69378f8c8820e8f807636": " \\mathit ROE = \\frac{\\mathit E} {\\mathit BV} ", "45dee754752ad53cf433ccf16ef4f878": "-u = \\frac{-u^\\prime + v}{1 + (-u^\\prime) \\frac{v}{c^2}},", "45df08553ca5ff9232a28caf8cd0900f": "\\nu_t", "45df18c90c71ea2066f8596159e11288": "x+y", "45dfb7c14c164ea140bb496f73e822a5": "P[M 0 & 0 < \\lambda < \\frac{1}{\\overline{x}}, \\\\[8pt] = 0 & \\lambda = \\frac{1}{\\overline{x}}, \\\\[8pt] < 0 & \\lambda > \\frac{1}{\\overline{x}}. \\end{cases} ", "45e0e43545a3c569674b919d1d3ffbbf": "\\textstyle{\\frac{1}{2}}(d+1)(d+2) - 1 = \\textstyle{\\frac{1}{2}}(d^2 + 3d).", "45e13ce58b5c43fb0159159025e4cd89": "\\log(I_n(w))", "45e19209d5086196524e13ac48a85a5c": "z_5=\\chi_{\\psi_{5,5}}(z_5,\\rho_{\\psi_{1,5}}(z_1))=\\chi_1(z_5,\\rho_1(z_1))=1\\times x_1=x_1", "45e1aa3dca16984aed6df4ff0101f9fa": " \\hat{E} = E \\,\\!", "45e1bd58310b866d12d98cdba13ef7c6": "\n\\sqrt{\\frac{\\pi}{2}}\\frac{1}{\\Gamma^2\\left(\\frac34\\right)}=\\sum_{k=-\\infty}^\\infty\\frac{\\vartheta_4(ik\\pi,e^{-\\pi})}{e^{2\\pi k^2}}\n", "45e1beb7ab8a962df5563e5c4f1981eb": "\\frac{H_{AB}B}{H_{AB}A}=\\frac{r_B}{r_A}", "45e1e5a8796fdf5a125ab0f7bb4cf2b2": "C.C", "45e212d40c3ce349f97d08709be9ffc3": "T_k(\\cos\\theta) = \\cos(k\\theta)", "45e25d853762f6e10d1b86f4bc309b82": "(\\mathbf{E}_1, \\hat{O} \\mathbf{E}_2)", "45e2d80689e37687e6e0ad6de7a0b6f7": "\\mathbf{r}(\\theta)", "45e35d3fd0743e2dee76ec37274d5959": "\\frac{dy}{dx} + p(x)y = q(x)", "45e38903e6b47ca983a5106c8ebe9eca": "l=[log_2 k]", "45e46989e3704bc2ba0899724acdca5c": "H_2O", "45e4a1d7bd45b939d0927f95e3d73727": "C(n) = Y", "45e4bc7a9f6be09af084db749e0fb564": "\\mathcal{E}(j)=\\{\\alpha\\in\\mathbf{H}\\,:\\,[\\mathbf{Q}(\\alpha): \\mathbf{Q}]=2\\}", "45e4f6b3b347b66d04981ffbaa466a95": "\\mathbb{R}^2 \\rarr \\mathbb{R}", "45e57f16b4e33bd3f425b9f2be47197e": "\\displaystyle{k(z,w)=h(z,F(z,w)),}", "45e614fb68b466c6cad1a4862989a0bd": "p=\\hbar k", "45e6255a6677ea33c07ccb205fde4df5": " \\operatorname{get-lambda}[F, F = E] ", "45e6480794a5fa556f58506f2d11fdf2": "U = \\frac{m}{2} \\left ( x \\right )^2 = \\frac{m \\left( \\omega A \\right )^2}{2} \\cos^2(\\omega t + \\phi)\\,\\!", "45e6793c2d8eb410afd9fe128fdf3ca5": "\n2\\eta p_{\\eta}^{2} - mk - mE\\eta = \\Gamma\n", "45e6c5641b403280ddb4911e54fa22c1": "(\\phi_n\\psi_n) \\in \\Delta", "45e6d04a81232399c4511f2303eafcd9": "\\,\\Gamma(n)=(n-1)!\\,", "45e6d79b6201f2be4cf97010c4e2d0b4": "\\beta_a=\\omega_a\\circ h", "45e6f86bcef0e3e2be4453b02539dc69": "\\left\\lceil \\log_2 \\frac 1 \\frac 1 6 \\right\\rceil + 1", "45e71d36cf9403c6e9a03a92b4a31da3": "\\scriptstyle \\boldsymbol{r}_{\\text{rec,ECEF}}", "45e7565e72fd5f254ab504edc55618cd": " w = 2.46\\left( \\frac{(\\rho_p-\\rho_f)gr}{\\rho_f}\\right)^{\\frac{1}{2}} .", "45e79fc0f3ee0174762a61fc77725438": "\\scriptstyle A \\or B \\or C", "45e7b21fa86b9fcc77da139a93149b0f": "\\Pi(k)=A +k^\\alpha", "45e88c70050aa9129b6f9da6cbfcfda9": "\n\\begin{bmatrix}\n1 & 0 & \\dots & 0 & -1\\\\\n0 & 1 & \\dots & 0 & -1\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\dots & 1 & -1\n\\end{bmatrix}.\n", "45e89844b9c93f37337c475c46e0bbc0": "(10)_2 ", "45e8f20e7232460528a5d1764df597f3": "\\Delta Q", "45e8f61bd36f9c8eb1dc58937fc807ef": "\\boldsymbol{w}_i=\\begin{cases}\n\\boldsymbol{r}_0 & \\text{if }i=1\\text{,}\\\\\n\\boldsymbol{Av}_{i-1}-\\sum_{j=1}^{i-1}(\\boldsymbol{v}_j^\\mathrm{T}\\boldsymbol{Av}_{i-1})\\boldsymbol{v}_j & \\text{if }i>1\\text{.}\n\\end{cases}", "45e979ad012ce6a7802b9681dceff75e": "\\chi^{(\\lambda)} (R)\\equiv \\operatorname{Tr}\\left(\\Gamma^{(\\lambda)}(R)\\right).", "45e99f914818042a05f7b3c22fa8645d": " \\operatorname{P}(E) = \\operatorname{Tr}(S E) ", "45e9a5488d9f9a9468f49074b0067dc0": "p_c=p_{\\text{non-wetting phase}}-p_{\\text{wetting phase.}}", "45e9cea39d6c7e7896ed560d22e0f679": "\n\\frac{1} {6} D = \\mathcal{A} \\psi_a(1)\\psi_b(2)\\psi_c(3).\n", "45e9ec9001275d7ef270fd8b936c8cf8": " p(x_1,x_2,\\ldots,x_n)=0,\\, ", "45e9ed3c7c4e844cf08ac1003a828603": "\\mbox{Free}(\\phi) = \\emptyset", "45e9f558152648a8ff2cae82862f55af": " \\Gamma \\tau \\geq \\hbar ", "45ea008048f8ff77d6b0a44f6b737049": "i; j \\in V", "45ea11c1e28a5519b26ea10e666fa450": "\\phi(\\nu_1 , \\nu_2) = \\frac{1}{n-1} y^T(\\nu_1) . y(\\nu_2)", "45ea1b351c4e4dcd063d905acf656513": "n-1\\choose k", "45ea2fb7074d5ba03ca6cb4c83463c60": "(\\downarrow 2)(c(z))=\\sum_{k\\in\\Z}c_{2k}z^{-k}", "45ea752c8d4f01e4f2c283727057df96": "m_\\mathrm{HCl} = \\left(\\frac{90.0 \\mbox{ g }\\mathrm{FeCl_3}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{FeCl_3}}{162 \\mbox{ g }\\mathrm{FeCl_3}}\\right)\\left(\\frac{6 \\mbox{ mol }\\mathrm{HCl}}{2 \\mbox{ mol }\\mathrm{FeCl_3}}\\right)\\left(\\frac{36.5 \\mbox{ g }\\mathrm{HCl}}{1 \\mbox{ mol }\\mathrm{HCl}}\\right) = 60.8 \\mbox{ g}", "45ea89f91bb35f82f173e0f2bbd05ecc": "T_0 = \\dfrac{1}{\\sqrt{\\beta_2 L_{NL}}}", "45eaf34bde3c8bc0293b8bcd487cb81f": "E=\\frac{q^2}{2gy^2}+y", "45eaf3d1bf2ac74afaf6fdcbeb9d346f": "f(0) = 0,", "45eb289d09f12c6dd10f32f409ea790a": " H \\psi (\\mathbf{x},t) = i \\hbar \\frac{\\partial\\psi}{\\partial t} (\\mathbf{x},t) \\,\\!", "45eb613060a6f168247e683db96f2f30": "L=L_0 -\\sum_{\\alpha>0} m_\\alpha\\, (\\coth \\alpha -1) A_\\alpha,", "45eb866e8a55195a7c9c63e4cac7e9d8": "\\frac{\\partial\\mathbf{E}}{\\partial t}, \\quad \\frac{\\partial\\mathbf{B}}{\\partial t}.", "45ebc31817b565cf1f0713156c2f6762": "\\lim_{x\\to\\infty}x^N=\\begin{cases} \\infty, & N > 0 \\\\ 1, & N = 0 \\\\ 0, & N < 0 \\end{cases}", "45ec56a593b36e5a859495d958b35d5e": "Q[\\mathcal{L}]\\approx\\partial_\\mu f^\\mu", "45ec73c4d2d8d0e6089adab77b16ab9b": "\\left\\{x\\right\\} \\overset{\\mathrm{def.}}{=} \\left\\{y : y = x\\right\\}", "45ece92f8e4ad11c9f33fbcb32b8525f": "V = \\frac{m}{2} \\left( \\frac{g}{L_a} x_a^2 + \\frac{g}{L_b} x_b^2 + \\frac{k}{m} (x_b - x_a)^2 \\right).", "45ecf7a581ef6ed71702b6a1c97654cd": "\\frac{q_C}{q_H} = f(T_H,T_C)", "45ecff579d1158403f3248d223730627": "\\scriptstyle \\frac{T}{N}", "45ed0a58e7678c028aeec9ee85df80b4": "A_n = \\frac{dE(r)}{d\\sqrt[n]{\\mu_n}} = \\frac{1}{n} \\frac{dE(r)}{d\\mu_n} ", "45ed121abb5ff3a8bd300f6b5dac4078": "\\int_A P_A~dA = 1", "45ed1e52f22d984da29d707e1a65fdf0": "\\{1,2,3,4\\}", "45ed6431743174bad12fe0da2b98d208": "\\phi_n(\\kappa)\n= \\frac{1}{4\\pi^2\\kappa^2} \\int_0^\\infty \\frac{\\sin(\\kappa R)}{\\kappa R} \\frac{\\partial}{\\partial R} \\left[R^2\\frac{\\partial D_n(R)}{\\partial R}\\right]\\,dR", "45ede4ebec518d467363718330957dc1": "-\\triangle", "45ee0ae98f416a1e4a6815fdec6a4c5c": "b = 6 u,\\ ", "45ee2ad84065c4c0da69be54bf67c78f": "x \\in \\mathbb{Q}", "45ee2ead140cb5a5e9cc773994e11be6": " \\begin{align}\nT^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s ; \\gamma \\delta} - T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s ; \\delta \\gamma} = \\, & - R^{\\alpha_1}{}_{\\rho \\gamma \\delta} T^{\\rho \\alpha_2 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} - \\cdots - R^{\\alpha_r}{}_{\\rho \\gamma \\delta} T^{\\alpha_1 \\cdots \\alpha_{r-1} \\rho}{}_{\\beta_1 \\cdots \\beta_s} \\\\\n& + \\, R^\\sigma{}_{\\beta_1 \\gamma \\delta} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\sigma \\beta_2 \\cdots \\beta_s} + \\cdots + R^\\sigma{}_{\\beta_s \\gamma \\delta} T^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_{s-1} \\sigma} \\,\n\\end{align}", "45ee3e83a10b5b8bd47e214fed86335b": "d(n) = (a + (n-1)b) (a-b)^{n-1}\\,.", "45ee56f16598f424f82f08b792321f34": "\\mathcal{H}_B", "45ee60725f14f3a3b68d3232e91aaed6": "\\textstyle \\exp(x+y) = \\sum c_n", "45ee85d5f6b3bbcbe52db2a287aabc2c": "K/N", "45ee94fd300b503d058c6faf197a73de": "\\Delta E \\Delta t \\approx {h \\over 2 \\pi}", "45eea293a9444630130192915e38a3c4": "\\operatorname{Li}_2(z) +\\operatorname{Li}_2(\\frac{1}{z}) = - \\frac{\\pi^2}{6} - \\frac{1}{2}\\ln^2(-z)", "45eec7a38ef78b9027513c7191e2cd19": "S = \\cdots (1+h_{+3})(1+h_{+2})(1+h_{+1})(1+h_0)(1+h_{-1})(1+h_{-2})\\cdots", "45ef28ed2051d754d8d4599a211030fd": "B_n(s)=(s+1)\\prod_{k=1}^{\\frac{n-1}{2}} \\left[s^2-2s\\cos\\left(\\frac{2k+n-1}{2n}\\,\\pi\\right)+1\\right]\\qquad\\mathrm{n = odd}.", "45ef49c292a6b4d01797397e3923b998": "e^{\\psi^{(-2)}(x)}\\,", "45ef5217d7b4bd7a0de822e88ac7807a": "f(n) = 2l+1", "45ef54a6947203932e5b350f1a94f815": "D_\\mathrm{r} = \\frac{k_\\mathrm{B} T}{8\\pi\\,\\eta\\,r^3}", "45ef5babe175121f296088e2f93f65ce": "m_{containing liquid}", "45ef83d0f66fb0fef4045672dfb61b52": "Y_0(t,t_1) = A(t_1)\\, e^{+it} + A^\\ast(t_1)\\, e^{-it},", "45efbbe73e795856c599956444962554": "\\|f\\|_\\infty=\\lim_{p\\to\\infty}\\|f\\|_p", "45efc6493fcd4ef5b027232d8841351f": "\\rho(q,t)=\\rho_{0}(q)\\sigma(q,t)", "45efd5d3bd29713328e5aa2c6c0e0e1f": " \\mathbf{n}=\\frac{\\mathbf{t}'}{\\kappa}, \\quad \\mathbf{b}=\\mathbf{t}\\times\\mathbf{n},", "45f04609d926b24f17bd3dd5605ab3ef": "\\scriptstyle{-j\\leq m\\leq j}", "45f06e5cd73333459e5c2c1dadf75416": "N-b", "45f08c0d475a9ba19808f485cabe3d3f": "\\displaystyle \\tau(s)", "45f0cef6fb4c09c8f9526c310ec4dfd1": "\\mathrm{Ra}_{x} = \\frac{g \\beta} {\\nu \\alpha} (T_s - T_\\infin) x^3 = \\mathrm{Gr}_{x}\\mathrm{Pr} ", "45f16b29406816a4cf31ea0fcec4a131": "D_i\\,\\!", "45f19b8ef3a0efe13ca3e2d994605f1a": " E_1, E_2, ... , E_{n-1}, E_n, H \\vdash H. ", "45f1fcfb6c4c2a01e2dbaad6aebdee62": "\\int LG(x,s) f(s) \\, ds = \\int \\delta(x-s)f(s) \\, ds = f(x).", "45f230ddd83def617f951a72f479ccd3": "\\mathbf{x} = \\mathbf{x}(\\xi^1,\\xi^2,\\xi^3)", "45f23b5b31554886adfdc50117edf586": "\\overline{\\mathrm{Nu}}_L \\ = 0.54\\, \\mathrm{Ra}_L^{1/4} \\, \\quad 10^4 \\le \\mathrm{Ra}_L \\le 10^7", "45f288f16763d2d74050b18abc16ab6b": "\\int\\frac{\\mathrm{d}x}{(\\cos x + \\sin x)^n} = \\frac{1}{n-1}\\left(\\frac{\\sin x - \\cos x}{(\\cos x + \\sin x)^{n - 1}} - 2(n - 2)\\int\\frac{\\mathrm{d}x}{(\\cos x + \\sin x)^{n-2}} \\right)", "45f2a31e47b4a70603487f20bdb19d38": " G \\mapsto G_{\\delta \\sigma}. ", "45f2b8c4ad4344b23a254a58fe3b95d8": "t_i(x) = x_i", "45f309f69d5345a7b10fb2acad5f338a": "I = C_S(V_{\\text{IN}}-V_{\\text{OUT}})f\\ ", "45f30e0b60e2d10ccf536298317bc479": "\\frac{c_{p}}{c_{v}}=\\frac{\\beta_{T}}{\\beta_{S}}\\,", "45f32efd8f007b52b7d9c5f294b06fb5": "f_v^{\\otimes |V|}.", "45f376f4f4096b31267d8a4d8a943977": "\\sqrt{8} \\rho^3 \\sin 3 \\theta", "45f38f42f9126d40302e963afe83ec87": "X(T) \\to X(T_0)", "45f3dbdcd82c0445204626aae30306e8": "F_2(x)=x \\,", "45f3fa53524817cd24a3540003985cf0": "q=\\sqrt{f^2+h^2}", "45f42628a10ec9dad597597a4c66e07f": "t^{\\ast/**}=\\frac{2\\|\\mathbf h_0\\|}{1\\pm\\sqrt{1-2\\alpha}}", "45f429bdfb0119f4350121ef32f6e4ad": "68^2", "45f42f2d39b083682d92f5f84704e15c": "\n \\|\\mathbf x_{n+1}-\\mathbf x^*\\|\n \\le \\theta^{2^n}\\|\\mathbf x_{n+1}-\\mathbf x_n\\|\n \\le\\frac{\\theta^{2^n}}{2^n}\\|\\mathbf h_0\\|.\n", "45f48647d3c65e04378b475d4b18f75d": "\nD = \\varepsilon^{2}_1 \\varepsilon^{3}_2 -\\varepsilon^{1}_1 \\varepsilon^{3}_2 + \\varepsilon^{1}_1 \\varepsilon^{2}_2\n", "45f495bee5354efa62b9c1ae1908f61f": "\\textstyle\\prod_{i=0}^{2n} P_i(q^{-s})^{(-1)^{i+1}} = \\frac{P_1(T)\\dotsb P_{2n-1}(T)}{P_0(T)\\dotsb P_{2n}(T)},", "45f4a55280ea1464071de00c386ebf21": "\\quad R_{\\mu\\nu}={R^\\alpha}_{\\mu\\alpha\\nu}.\\,", "45f4a9d16d2dc097b0e4cd3a97d73061": "x^3+2x^2 +y^2+2xy+x+y", "45f4f5f10cdc5e5f70676e57bc6ce06d": "C_n = (0,\\frac{1}{n}) ", "45f579908175856d6292707e446f1798": "(\\mathbb{Z}/n\\mathbb{Z})^\\times\\cong (\\mathbb{Z}/{p_1^{k_1}}\\mathbb{Z})^\\times \\times (\\mathbb{Z}/{p_2^{k_2}}\\mathbb{Z})^\\times \\times (\\mathbb{Z}/{p_3^{k_3}}\\mathbb{Z})^\\times \\dots\\;.", "45f5a3489ad14db90f0272deabdb769a": "Q_x = f(P_x)", "45f62ba991a63f9b3fa8d402e9c6cc45": "\\vert \\epsilon_0 \\vert \\leq {1 \\over 17} \\approx 0.059 \\,.", "45f6500ce9589cbb16700f14f1f93a3e": "\\ {^{n} a}", "45f67903eed4b1829280feaf94fb4902": "H(\\vec{x},\\vec{x}')=H(\\vec{x}',\\vec{x})\\,\\!", "45f67b7280254c1bcea83c101b8e5916": "\\Gamma_{max}", "45f72849ec7702577fb358fcb4c49cad": " \\left| \\psi (t) \\right\\rangle", "45f75d5e79bcb781d64753e56e742b0c": "x^{q^2}", "45f7623f41aaf66861fcabb9498665ae": "Z \\geq0", "45f763cab47328ceff903f307a0cd45a": "W = \\oint PdV = \\oint (dQ-dU)= \\oint (TdS-dU)\n\\quad\\quad\\quad\\quad(2)", "45f774a4a1564a63ea6357c515d55d28": "P(E_j) = 2^{-n(I(X;Y)-3\\epsilon)}", "45f7a420e715e8597e6bd9105bedd435": "(x+dx, y+dy, z+dz),\\;", "45f7a6b525ade511ee5ecce23206b55a": "\\mathcal{K}_{n}", "45f7b345e443b064612a87071fd3472f": "T - \\lambda", "45f7e128b5e7d6feb956871f03dbc47f": "\\overline{T}_1=A-\\frac{Bk_1}{k^2L}(1-e^{-kL})", "45f824cdc5519badb5ba4a1cfb6a189c": "\n\\int \\psi_0(x) \\int_{x(0)=x} e^{{\\rm i}S(x,\\dot{x})} Dx\n\\,", "45f8271abb1de913edad75d433ffe87c": "\n J_i := \\lim_{\\epsilon\\rightarrow 0} \\int_{\\Gamma_\\epsilon} \\left(W n_i - n_j\\sigma_{jk}~\\cfrac{\\partial u_k}{\\partial x_i}\\right) d\\Gamma\n ", "45f8461d31c425abc52919d4f16b1aed": "AB^* + M \\to C + M", "45f86b380462dcd3dc4cedcce957c669": "\\Psi_k(X)=\\gamma_k(X)=X\\circ B_k", "45f870d9b3b3bfcfc41c4c98a04b79a0": " \\kappa = \\frac{1}{r} \\frac{dp}{dr} , ", "45f91ad59c2e0ae9a10e1e33da7caa41": "P_{s} =\\text{the probability that the base is not paired.}", "45f92ca823531db6212284f08565e59d": "\n \\dot{\\boldsymbol{\\xi}}\\, =\\, \\frac{\\partial \\boldsymbol{\\xi}}{\\partial t}\\, =\\, \\boldsymbol{u}(\\boldsymbol{\\xi},t),\n", "45f9aa22c767b70f4b8c048660174ba1": "\\rho:\\pi(M)\\rightarrow GL(E)", "45f9aaa40c5119325997bfd6809c9381": "\\operatorname{int}", "45f9c728ee544ec8941f277821435bd7": "\\frac{\\tan^2 x -2 \\tan x+1}{\\tan x} = 0,", "45f9d3f85b02097ae1c496b50b43e591": "\\nabla^{a} T_{abcd} = 0", "45f9f174b15b32e7bacbb8e7fc62c444": "{\\lambda}_r = - r \\left( \\frac{r-1}{2} Q'' + L' \\right)", "45f9fb0f36d6044670e9d1bc32f762db": "\\Sigma\\left|a_n\\right|^2 = \\Sigma\\left|b_n\\right|^2\\,", "45fa5352faa1685d85640ac78e7533a9": "\\langle\\psi|\\psi\\rangle = \\int_R\\psi^*\\psi \\mathrm{d}x^1\\wedge\\mathrm{d}x^2\\wedge\\mathrm{d}x^3 ", "45fa658eb3887187210fe65c80d3e2fe": "\\exists x \\varphi", "45fa781adc9679e33d98807fbf6420f5": "\\operatorname{var}[\\ln (1-X)] = \\operatorname{E}[\\ln^2 (1-X)] - (\\operatorname{E}[\\ln (1-X)])^2 = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta) ", "45faf2050c22e06fab19f60cafce160e": "\\mu(B_{r} (x)) \\leq C_{0} r", "45fb0ac610bec0401c0f738cfb3c1dc8": "Li_{s}(z)", "45fb30929e8bfceeedfde3e49dd5709c": "O(2n+1) = SO(2n+1) \\times \\{\\pm I\\}", "45fb66e05780da371098a6bd8ca8316a": "C_\\text{out} \\circ C_\\text{in}", "45fb73663bb2cf15ba5881258ffd3748": "\\gamma\\in M", "45fc08ca5f847c1f90ca4adcdc5ba2e9": "\\begin{matrix} {2 \\choose 2}{3 \\choose 2}^2{40 \\choose 1} \\end{matrix}", "45fc2e536377e39b7ff64bfe1e58b9bb": "\\sum_s X_s(z)\\Phi(X_sv_i,z_i) = (z-z_i)^{-1}\\Phi(\\sum_s X_s^2v_i,z_i) + (k+g){\\partial\\over \\partial z_i} \\Phi(v_i,z_i) +O(z-z_i)", "45fc509e3f3b5448992db3ab75e5ceb0": "m=O(n)", "45fc824b9634a52e2114157dc1a0b7d9": "\\text{VaR}(X)\\leq \\text{CVaR}(X)\\leq\\text{EVaR}(X).\\,", "45fcb6a23459e8d00462a20db77806c9": "||\\bigstar \\bigstar |\\bigstar", "45fd29ca5202ef7c0bf0551db59b607d": "\\sigma \\approx 0.42 \\lambda N \\ ,", "45fd397d9be6821a81d8a2446198e8ad": "\\textstyle X", "45fd910d65bca35e09dc7ca57298316f": " \\text{FVU} = \\frac{\\text{MSE}(f)}{\\text{var}[Y]},", "45fd99a0d5337dfd6163f800e6d25e3e": "\nF L \\sin \\theta = k_\\theta \\theta \n", "45fdeaa6d9dbaa08edf323c226ec82e7": "p=2r", "45fe0cde04b30acfe697ecbbc811834b": "\\,\\alpha, \\beta", "45fe265fd6baa5830dbc2ae676ec7542": "\\langle\\ \\rangle \\!\\,", "45fe428c0a45d5965c8f588396afaf02": "\\mathbf{\\nabla} \\cdot \\mathbf{E} = \\mathbf{\\nabla} \\cdot \\left (- \\mathbf{\\nabla} V_\\mathbf{E} \\right ) = -\\nabla^2 V_\\mathbf{E} = \\rho / \\varepsilon_0, \\, ", "45fe495c3dcf46b140dcade68e5f8be2": "\\omega_1'=\\lambda\\omega_1", "45fecbec9284899fda4438d839e7417c": "\\mathrm{CINT}_x(a_{-1}, a_0, a_1, a_2)", "45fed1dd80038060bb3261494bb942ab": "\\eta_\\mathrm{m}\\approx 0.545", "45ff06b3059d7e713909ee4dc6e3ec12": "\\mathbf{Z}_p \\!\\,", "45ff34ad0868d9b51ed21acd78bc4441": "\\mathfrak{P}^{24}", "45ff405f4e139155255ab0e549b60bb7": "\\Gamma(\\tfrac16) \\approx 5.5663160017802352043", "45ff481797da4382e5bc8fb68ba0e9bd": "k=\\frac {F} {\\delta} ", "45ff5652478ee25831d0d98efc824348": "8t = 200 \\quad", "45ffe9312c39c91dddb2deee0dd1ceb8": "\\hat{\\mathbf{r}}", "460005ea21a13ca2a258536ce3ad54ce": "\\xi (m) \\Delta m= 0.086 (1/m) \\exp[- (\\log(m)-\\log(0.22))^2/(2 \\times 0.57^2)]", "460031d2d4b3d4bbc399d11dc1b27e60": "\\boldsymbol J\\times\\boldsymbol B\\ ,", "460035ede82c66ec5cfdfd87038f3e33": "V=\\frac{5a^3\\cos{36^\\circ}}{\\tan^2{36^\\circ}}", "4600542ab3e797bbd2cfbd1369fb6fde": "p \\vdash q\\,\\!", "4600a78238164ef220e7b424bc559e5d": " q E_r/{m r} = {\\Omega_c}^2/4 ", "4600f164887b14aa87808b16ffa119d7": "\\mu ^\\circ + RT\\ln \\frac{f}\n{{f^\\circ }} = \\mu ^\\circ + RT\\ln \\frac{P}\n{{P^\\circ }} + \\int_{P^\\circ }^P {\\Phi dP}", "460113ed33324e15fec132443671c152": "\\frac{q(q-1)}{4}", "460149ead252ead2228304b65fa17f7d": "1/X", "460170f9584c1f69d9347c73b2b63d04": "\\xi^b\\,", "46017e9f67b85a663032e53a1c96af13": "(13,14,15)", "4601f68ba3d0da05cb2a60a26eea713f": " = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left[\\frac{\\mathrm{d}s}{\\mathrm{d}t} \\left( x'(s), \\ y'(s) \\right) \\right]\\ ", "460203e7c82771b2eab05ed36ba60974": "k >0", "4602324162675673200469a1b2abdfa9": "k*\\,\\!", "460258755d8e816de811272cc4a10dfb": "var_{01}(p) = var((x+1)^{deg(p)}p(\\frac{1}{x+1}))", "460258c40738db6e93b92bbf6ed37945": " Q = \\sum_{i}^{}{q_i} ", "4602770d07710f132998d20cb20cbcff": "\\Xi(g)=\\int_Ka(kg)^\\rho dk,", "46031dcedc903274edfbf0410642f630": "10^{(7\\cdot2^{103})}", "46032a779b23743157bf8e1db335bda7": "\\begin{align}\n(a+b+c+d)(x+y+z+w)&=(a+(b+c+d))(x+y+z+w) \\\\\n&=a(x+y+z+w)+(b+c+d)(x+y+z+w) \\\\\n&=a(x+y+z+w)+(b+(c+d))(x+y+z+w) \\\\\n&=a(x+y+z+w)+b(x+y+z+w) \\\\\n&\\qquad +(c+d)(x+y+z+w) \\\\\n&=a(x+y+z+w)+b(x+y+z+w) \\\\\n&\\qquad +c(x+y+z+w)+d(x+y+z+w) \\\\\n&=ax+ay+az+aw+bx+by+bz+bw \\\\\n&\\qquad +cx+cy+cz+cw+dx+dy+dz+dw.\n\\end{align}", "4603a6240776deb787c63aeea501bec1": "\n\\left.\\begin{align}\nP(X)&=p(X)\\cdot(X-s_\\lambda)+P(s_\\lambda)\\\\\nH^{(\\lambda)}(X)&=h(X)\\cdot(X-s_\\lambda)+H^{(\\lambda)}(s_\\lambda)\\\\\n\\end{align}\\right\\}\n\\implies H^{(\\lambda+1)}(z)=h(z)-\\frac{H^{(\\lambda)}(s_\\lambda)}{P(s_\\lambda)}p(z). \n", "4603a987caadbda6f87a10c006fd83d6": "i = 4 ", "4603cd923115ceb8a83af97afdfc10b0": "\\operatorname{Li}_a(z) =\\sum_{k=1}^{\\infty}\\frac{z^k}{k^a}.", "460407740aacfc4048ec35575b0b5a9d": "n=1,2,3,...", "46043a35f458f1ebf924085b788f843c": "e^{\\pi \\rm{i} \\tau}\\,", "46044e1a9b281a81d52c67030b695b69": "S_j\\xrightarrow{d} S", "4604a67c557f015478d6fc4ac6e8bc33": "S_{exp}", "4604a923e74fb0288a8b58f955110b87": "{V_m}-{E_{ion}}", "4604b2e29fc45b8fc07bc440e852e554": "Q_{u,v}", "4604c97858ba357a8878f470a281903d": " \\scriptstyle \\mu - E_{\\rm V}\\gg kT", "46059306720d950af9890090a9e62799": "\\forall x_i,y_j: x_i \\not\\to y_j", "46061056a4bd9d89c1741844fbc46f64": "\n [\\boldsymbol{\\nabla}\\psi^i(\\mathbf{x})]\\cdot\\mathbf{c} = \\cfrac{\\partial \\psi^i}{\\partial q^j}~c^j = c^i\n ", "460629745266ac5e663303008259f8b9": "T_{i}", "46062e6d183d38c919527bfde2f14bca": "e^{u^2}=\\frac{1}{\\sqrt{4\\pi}} \\int_{-\\infty}^\\infty e^{-uy} e^{-y^2/4}\\;dy.", "46063c04e802b41de78d12601e5a90fa": "\\mathcal{T} = \\{A \\subseteq X \\mid A=\\varnothing \\mbox{ or } X \\setminus A \\mbox{ is finite} \\}", "46064cd1f35d41e16fd8ec536914dced": "u(x + iy) = u(x, y) = \\frac{1}{\\pi}\\int_{-\\infty}^\\infty f(s)\\frac{y}{(x - s)^2 + y^2}\\,ds", "4606aad4865c200d513201d5d0dabc6a": "f^-:=(-f)\\vee 0", "4606ef00fba164facd5985d4f7724bd0": "c_0 = 1", "4607918a79d3364d96dea8415fd617d2": "V(r)=\\frac{6}{\\pi}r^2+O(r\\exp(-c(\\log r)^{3/5}(\\log\\log r^2)^{-1/5}))", "4607b813c54168d3e75e95c68c781f2f": " \\mathcal{B}(\\mathbb R) ", "4607bea71cd009261ec21da06a7d3745": "{\\overline{z}\\over z}= \\left({\\overline{z}\\over |z|}\\right)^2,", "4607fbfa3a36a0354753076abe01d981": " L(w, r),", "460801263f0e471d4ac8cff704033cf7": "h_f = \\frac{L}{D} (aV + bV^2)", "46082961737a1ca72676690f0181f60d": " \\mu = E(y) ", "460896ffd4a724cb00ed4c828d153268": "\\begin{align} T(x,y) = f(a,b) & +(x-a)\\, f_x(a,b) +(y-b)\\, f_y(a,b) \\\\\n&+\\frac{1}{2!}\\left[ (x-a)^2\\,f_{xx}(a,b) + 2(x-a)(y-b)\\,f_{xy}(a,b) +(y-b)^2\\, f_{yy}(a,b) \\right]+\n\\cdots\\,,\\end{align}", "46098d05d32b8a1b4ebc5d6782686611": "\\varphi(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\varphi_{sr}(\\mathbf{r}) + \\varphi_{\\ell r}(\\mathbf{r})", "4609c1ed8e4c5c30725b03dbf3329128": "\\ xy", "4609cf60f96c3597c71fc2a8137604ae": "B_n(f,g)", "460a1940ceddf45878d2e095af31128a": "e\\,\\!", "460a1e917288e43976f9d378c51fcf53": " Q = { \\omega_0 \\over 2 \\alpha } = \\frac{ \\sqrt{ R_1 R_2 C_1 C_2 } }{ C_2 \\left( R_1 + R_2 \\right) }\n\\qquad", "460a59c6c18c617400fbec48c0e8b70b": "(1,1)_{0}", "460a61f45c7a20e87ab97e393a5dedbd": "\\sigma_x=\\biggl( \\begin{matrix}\n 0&1\\\\1&0\n \\end{matrix} \\biggr);", "460af46d445360649de4b30c54b09827": "A_1 , A_2, \\ldots, A_n \\in \\mathbb{B}_b(S)", "460b08a0b0a76b0dd5844ee724470982": "\\mathbb{W} = \\mathbb{R}^q", "460b2e1dd01142e029417887798f7c85": "\n\\begin{align}\n\\iint_D \\frac{\\partial L}{\\partial y}\\, dA\n& =\\int_a^b\\,\\int_{g_1(x)}^{g_2(x)} \\frac{\\partial L}{\\partial y} (x,y)\\,dy\\,dx \\\\\n& = \\int_a^b \\Big\\{L(x,g_2(x)) - L(x,g_1(x)) \\Big\\} \\, dx.\\qquad\\mathrm{(3)}\n\\end{align}\n", "460b3507fcc63a76050a26ea065b271b": "F = C \\times E ", "460b5131ef31f40f3173a63bb6449048": "\\frac{d}{dt} \\langle Q \\rangle \\,", "460bd5bf164eb13fb0aa653027357c3d": "\\mathfrak{D}_{K/Q}", "460bd92670bd833cd9226d2f7c284acf": "\\tfrac{p}{p+q}-\\tfrac{q}{p+q}=\\tfrac{p-q}{p+q}", "460c107cf8e65658ecf93902ba114221": "\\boldsymbol{j}=\\frac{\\text{d}\\boldsymbol{a}}{\\text{d}t}=\\frac{\\text{d}^2\\boldsymbol{v}}{\\text{d}t^2}=\\frac{\\text{d}^3\\boldsymbol{s}}{\\text{d}t^3}", "460c1deb72fde59df658876d243fc5c2": "\\langle\\mid r\\mid \\rangle_\\text{free}/n)", "460c620accfc67ca25373e2e3e127a73": "\\left\\{1/4,1/2,1\\right\\}", "460c64f837200f4380090511e19dbbc3": "A \\otimes_\\alpha B", "460c65bb981e5428c4bfa91872610540": " x = \\frac{2 \\pi r} {\\lambda}.", "460c75ed9de421baed8248579f26a9e1": "\\frac{\\partial u}{\\partial t} = -\\left (-\\frac{\\partial^2}{\\partial x^2} \\right)^{\\frac{1}{2}}u(t,x)", "460c96503fdd805984dc9e2d82db25b5": "R \\times R \\stackrel{a}\\to R", "460c9a20a30bf3f58bf8e2204fe76e47": "\\mu \\mathbf I", "460cdfd33fa371303b7134d82831d23f": "E_{x,xy} = \\sqrt{3} l^2 m V_{pd\\sigma} + m (1 - 2 l^2) V_{pd\\pi}", "460ce54abefb574cbfd3d6ec1b916a27": "R_{3,0} = -4+35 r^3-60 r^2+30 r", "460d72fda15ad3de8955f11d4455a250": "w_j/(x-x_j)", "460d969ced7c0687debad2db812d7998": "[y_\\nu] := y_\\nu, \\qquad \\nu \\in \\{ 0,\\ldots,k\\}", "460da26522907aff7f2c777db0cc24e2": "K_{++} \\ \\stackrel{\\mathrm{def}}{=}\\ \\{ x \\in K : \\langle x,\\,x \\rangle > 0 \\}", "460dc08fbc89baf8cd85b6957b0140a5": "S_C(n)\\leq 2^n", "460e1f6b855ab0de62094097375e55ed": "=\\partial_x^2 + \\partial_y^2 + \\partial_z^2", "460e3439009274f129ef73678e2e5a83": "p(x_i ,y_i |w,b,\\log \\zeta ,\\mathbb{M}) \\propto p(e_i |w,b,\\log \\zeta ,\\mathbb{M}) .", "460e82f1276ead06c25632f9a8d1634a": " s^2 + 2 \\alpha s + {\\omega'_0}^2 = 0 ", "460ed9ae3c954a985712dc074464a825": "\\partial\\Omega_a", "460f2edac58efd25e12eb10730a50d16": "A_{12} < A_{21}/2 ", "460fac49d8258002cc2bfdb13f05991a": "\\mathcal{P}_{i} \\equiv p_{\\perp }-\\frac{i}{2}\\Sigma _{i}\\cdot \\partial \n\\mathcal{G}\\Sigma _{i}, i=1,2.\n", "460fac74c4a2aa25bfc668d42943bf8c": "\\varphi_e(e) = (F \\circ h)(e)", "460fbc9b3accb4d8bcb194d11662b226": "\\lambda=c/f_s\\,", "461034648438ba7efd19df2ca58395ab": "\\Sigma \\chi(r)e(r/p) = G(\\chi)", "46105d7173d882b3ecb2afa34f0650b9": "f({\\bold x}) = \\sum_{|\\alpha|\\le m} \\frac{D^\\alpha f({\\bold y})}{\\alpha!}\\cdot ({\\bold x}-{\\bold y})^{\\alpha}+\\sum_{|\\alpha|=m} R_\\alpha({\\bold x},{\\bold y})\\frac{({\\bold x}-{\\bold y})^\\alpha}{\\alpha!}", "46105f787987625639ebd664a797bdba": "\\scriptstyle\\boldsymbol{x}\\in A", "4610a1f3e573a52aebc71fc10c48104d": "x,y > 0", "4610a5f6178d40a44ad1281e772f8f9f": "\\Lambda_{boost}", "4610d678a2ce2be9f927e73cc69697fe": "\\overline{a}_n - \\overline{b}_n = \\overline{(a - b)}_n", "461102d8aabda19be29ba1b1c04e38ab": "p-q=w(+\\infty)-w(-\\infty)", "461112f687fd4bc6a753d4493559dd14": "\\xi(s) = \\frac{1}{2}\\pi^{-s/2}s(s-1)\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s).\\!", "461124cc5cef22fa6d4897f0b3db741f": "\\alpha_V=\\frac{1}{V}\\left(\\frac{\\part V}{\\part T}\\right)_P", "4611265e8c95e34b98b0bc8d66b0ce69": "\n p(r,\\theta) = R(r)~Q(\\theta)\n ", "461193f0bca15ddfae568f814f6f092a": "= \\arctan \\frac{1}{5} + \\arctan \\frac{1}{5}", "4611b1ca14f5e415020732105564dd4b": "\\alpha\\approx0", "4611db7848c5ab68ad0322fae20c2923": "\\ddot{s}_{\\overline{n|}i}", "4611e33110d34085bcbf0667ed348a9d": "a\\#{}_Wb", "4611f5c793bfd4cf9bd1d02a93327d9e": "V_\\mathrm{T}", "46125f7561e75d933fc68aded3c4ac98": " {x^2 \\over a^2} + {y^2 \\over b^2} - {z^2 \\over c^2}= -1", "4612a565e1a6602e8aa9c00450b4b7ed": "u(x,t)=\\frac{1}{\\sqrt{4\\pi kt}} \\int_{-\\infty}^{\\infty} \\exp\\left(-\\frac{(x-y)^2}{4kt}\\right)g(y)\\,dy ", "4612d23c507c9d6e0cdbd54481e95c86": "\\frac{1}{2} \\frac{1}{q2^{q}}", "461330f8284b05936525b6d11e43f828": "c(n,k)=\\left[{n \\atop k}\\right]=|s(n,k)|=(-1)^{n-k} s(n,k)", "46139d04de6951e97d3cd7cab5870da2": "b \\equiv c^{2^{M-i-1}}", "4613d4c7f5e6e2a6519f4355efb9a10d": "q_5", "4613f69f906216e967ab180bc7313e14": "\\Omega_{ij}/{4\\pi}", "4614230799bbcca47943cd96553475ef": "x_{n}\\to x", "461458c3eeb3b635c744d51956d9780a": "\\mathbf{\\hat{p}} =-i \\hbar \\mathbf{\\nabla}", "46146dcfe9131cf9cc12fb148e245ad1": "m' = m + \\frac{\\delta}{n}", "461478535346a353a638f061f8d6aecd": " n \\times k_1 ", "46148ca2f18b057124ae5ec285afdb55": "\\lim_{n \\to \\infty} \\frac{n!}{\\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n} = 1.", "4614f0ba58649ae5d183a579b4adec48": " i^i_{cap}", "461529060bf35c2ef3aba87ad95d1797": " \\mathcal{D}_\\eta ", "4615c37a9df8fd27f9ce358c3993938d": "\\mathcal{O}_m", "4615c4b372268116f274e77d9b96880d": "S \\otimes_R S \\equiv\\,\\,{} I \\,{} \\oplus S \\otimes_R R.\\, ", "4615d27208ffb6348b4e32e2afef586b": "\\begin{align}\\Pi_0(x) &= \\operatorname{Li}(x) - \\sum_\\rho \\operatorname{Li}(x^\\rho) -\\log(2) \\\\ &\\ \\ \\ \\ +\\int_x^\\infty\\frac{dt}{t(t^2-1)\\log(t)}\\end{align}", "46161f184809b41ebb412c90ce4b76ca": " \\mathcal{L} = { {\\mid \\mathbf{E} \\mid^2} \\over {8\\pi\\omega} } \\left ( \\mid \\langle R | \\phi\\rangle \\mid^2 - \\mid \\langle L | \\phi\\rangle \\mid^2 \\right ) = { 1 \\over \\omega } \\mathcal{E}_c \\left ( \\mid \\phi_R \\mid^2 - \\mid \\phi_L \\mid^2 \\right ) ", "46165af899076ab03cbdd35f02f86e7b": "\n p(x,t)=\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\left(-\\frac{(x-x_0)^2}{4Dt}\\right).\n", "4616829a1d4bdce61c6a9f48f32f185f": "\\int_{-1}^1 C_n^{(\\alpha)}(x)C_m^{(\\alpha)}(x)(1-x^2)^{\\alpha-\\frac{1}{2}}\\,dx = 0.", "46169fb2ccc42abb7c71b55f3fd8c17a": "(v^*_1, \\ldots, v^*_n).", "461732aed82bf9d95c38c5aee044cc40": "(fg)(x) = f(x)g(x) = 2x(x + 1) = 2x^2 + 2x\\,", "4617674f464fa675d437cda2eb41f0d6": " \\limsup_{n\\to\\infty} |f-f_n| = 0.", "4617691cf05758d0cc50073edc4ddcbe": "\\{x^k\\}", "46176cb2327d23509f81558a8418ceeb": "\\scriptstyle\\mathbf{y}", "4617821242387b6eaf532d862f35bdf8": "i_\\text{i} = C{d \\over dt}(v_\\text{i} + Av_\\text{i}) \\,", "46188382d8ad754d5e2350335f100c2e": "\\frac{\\partial H}{\\partial q_k} =- \\dot{p}_k \\,, \\quad \\frac{\\partial H}{\\partial p_k} = \\dot{q}_k \\,, \\quad \\frac{\\partial H}{\\partial \\sigma} = - {\\partial L \\over \\partial \\sigma} \\,.", "46189f1df56b18845912caf7de0082e0": " ds^2=dr^2+ r^2 d \\theta\\ ^2 +dz^2 ", "46193c23b6ee012eed2306fdf56c2b53": "\\phi_{sl,m}=\\frac{M_{s}}{M_{sl}}", "46194f8e8440203927c68b66c059f11a": "\\alpha^2 = \\frac{\\text{transient inertial force}}{\\text{viscous force}} = \\frac{ \\rho \\omega U}{\\mu U R^{-2} } = \\frac{ \\omega R^{2} }{\\mu \\rho^{-1} } = \\frac{ \\omega R^{2} }{\\nu} \\, ,", "461964fc5f246a500755badc2821cbf0": "\n\\theta\n =\\frac{t^*}{t^{**}}\n =\\frac{1-\\sqrt{1-2\\alpha}}{1+\\sqrt{1-2\\alpha}}.\n", "4619d7b1908baaae899c9d0bdd61242e": "3 \\times 4 = 3 + 3 + 3 + 3 = 12", "4619f33fd6eaee547ef84dbe3d008ef5": "\\sum_{e \\in \\delta^+(v)} \\phi(e) = \\sum_{e \\in \\delta^-(v)} \\phi(e),", "461a2e3152ede72dde162d618b18fc34": "\\frac{\\partial \\eta}{\\partial t} = -\\frac{1}{\\varepsilon_o}\\nabla\\cdot\\mathbf{q_s}+\\sigma", "461a60588f8186e634fba6f3e6e58aaa": "D = \\dfrac{Eh_e^3}{12(1-\\nu^2)}", "461a9621abf6cd6f74c537497cd86e2f": "x,y,z\\in M", "461abc4a8dbf34c1b61284802dfa159f": "\\scriptstyle P(s), \\ Q(s) ", "461ac3b5f1d1ff2d80d75accb9e9c2ac": "g^{-1}\\circ f^{-1}\\circ g^{-1}", "461ac82720cda9ed9f959fc9b9215236": "E_k = \\begin{matrix} \\frac{1}{2} \\end{matrix} mv^2 ", "461b0c311542a942fb56816dee32e51f": "z_{ij}", "461b1990fe86af962cd15a16a26dceb8": "BG", "461b4da6cd344f704485cbb5fbbda8db": "\\frac{1}{q} = \\frac{1}{p}-\\frac{k-\\ell}{n},", "461b63f825444a6c5b22c5149444c0cf": "\\ g ", "461bc1a06d32ef5821d2d65b756da67e": "1 \\leq i,j,k \\leq d", "461c078e00f89d2c187026d6d24a2469": "{L\\over{\\delta}_2}=\\sqrt{{\\rho}VL\\over {\\mu}} =\\sqrt{{VL\\over\\nu}}=\\sqrt{N_R} \\,\\!", "461c552410bed0a602dc676904c76c4d": "x \\to \\infty ", "461c69c33d54a199f61f85556c8cd1de": "\n\\sum_{k=0}^{n-1} d_k 2^k\n", "461cf77cdd552deddd32c05d26f8d5bc": "y(t)-r> 0", "461d9769faed59c505ab872c22e2c399": " V = 0 ", "461dca8e295fcb9f4a1f13d78b6647b4": "\\left\\langle f', \\varphi\\right\\rangle = \\int_{\\mathbf{R}}{}{f'\\varphi \\,dx} = \\left[ f(x) \\varphi(x) \\right]_{-\\infty}^\\infty - \\int_{\\mathbf{R}}{}{f\\varphi' \\,dx} = -\\left\\langle f, \\varphi' \\right\\rangle", "461e395bf5097d9b5d74846f3b52c28d": "\\mu = 2(C_1+C_2)", "461e70ced09be4593280fab4068a6196": "\\beta_i(X)=\\beta_{n-i}(X) ", "461e9dc1923fceca8ed66aa4fad9a6aa": "\n D_\\mathrm{KL}( X_1 \\,\\|\\, X_2 ) = \\frac{(\\mu_1 - \\mu_2)^2}{2\\sigma_2^2} \\,+\\, \\frac12\\left(\\, \\frac{\\sigma_1^2}{\\sigma_2^2} - 1 - \\ln\\frac{\\sigma_1^2}{\\sigma_2^2} \\,\\right)\\ .\n ", "461eb9ef4dce852f53ce4cb911002d07": "\\!\\mu_1(v_1)", "461ef98391eda2c1cf115d4382415009": "f(z) - f(\\mu) = (z - \\mu)g(z). \\,", "461f0598fb29e2c43ffcdb792ce58037": "w = r e^{i\\theta}", "461f35ff73a4b91100c4b9ac523c2d6d": " b{\\left( \\frac{ap-1}{a+1} \\right)}^{\\tfrac{1}{a}}", "461f51320be15a19a40e77f573d21757": "\\gamma \\geq \\max_i |q_{ii}|.", "461fdd7337f7d79e4ebc1f59f3bdab10": "|r_1 - r_2|", "461fddbf6f8295659fdae44419c0f0cf": "(2)~~~~~\n dx =\n \\left(\\frac{\\partial x}{\\partial u}\\right)_v du\n +\\left(\\frac{\\partial x}{\\partial v}\\right)_u dv\n", "461fffd5b246efcb0acea67a44861ad2": "\\!\\phi", "46201292c27e564cd0f38082690bbe6f": " \\langle u,v \\rangle = \\langle Tu,Tv \\rangle \\, .", "46202a239cd73b38e9b4367d47afe954": "\\rho^E_\\text{P}=\\frac{E_\\text{P}}{l_\\text{P}^3}=\\frac{c^7}{\\hbar G^2} ", "462037e89b03a3d639736943362662c9": "\\sin\\delta = \\sin b \\sin 27^\\circ.4 + \\cos b \\cos 27^\\circ.4 \\cos (l - 123^\\circ)", "4620483f3d00106d86e1331ba0d22bbd": "\\left\\{ \\Lambda_{m}\\right\\} ", "46204dece62d867983b3027833930e18": "p(e^{iw})p(e^{-iw})=|p(e^{iw})|^2", "4620742bb0ea594830515adf89b41309": "m:I\\to TI", "4620d5034ecabe8ff03fecd2bf6c1449": "[n^2 (I) - n^2] \\approx 2 n n_2 \\frac{|A_m|^2 |a(x,z)|^2 }{2 \\eta_0 / n} = n^2 n_2 \\frac{|A_m|^2 |a(x,z)|^2 }{\\eta_0}", "462109d292d9710533f97b69e547e7d3": "10(1/5!)\\pi^5 = (1/12)\\pi^5 ", "462114619c4855aa7e479b98581ee615": " \\omega_0 = 2 \\pi f_0 = \\frac{1}{\\sqrt{R_1R_2C_1C_2}}\\,", "4621148ef7f8ea355e42e81909de6569": "m\\geq\\lambda(VCDIM(H)\\log(m/VCDIM(H))/\\alpha^{2})\\,\\!", "462114bb12b1f27f568f89abc726457b": "\\left( a,b \\right) = \\left\\{ x \\in X:a N", "4622c9f1b4016b3624254db219ea5415": "{d|n}", "4623009c5a80a2d98d54e26eb62bcc05": " u_4 =0.44341 ", "46237e53bb0dee7e3b52c916c0f0eb34": "1\\tfrac{7}{8},\\text{ }2\\tfrac{11}{12},\\text{ }3 \\tfrac{15}{16},\\text{ }4\\tfrac{19}{20},\\ldots", "46238cff458560111e19a1bac5ff5379": "\\Delta I_{\\alpha+2} = -I_\\alpha.\\ ", "4623c4857c4bdd1647ea1335ab43e038": "\\phi_{3}", "46242cd9d089e7eded890367459e56e3": "\\langle\\Phi|", "46244eb09d44df6046c5cc81c396de63": "\\operatorname{trunc}(x,n) = \\frac{\\lfloor 10^n \\cdot x \\rfloor}{10^n}.", "4624502c030e24be68d42ee3775f2e7a": "\\Psi(x,f(x)) = (\\phi(x), g(\\phi(x)))", "46247495569b77bf2667f0f8507f9761": "t_2 - t_1,", "4624962ef79abe8d70cd7e509ebbb1e9": "\\mathbf{Ma} = \\mathbf{F}_{ext} - \\mathbf{F}_{int}", "4624976fe3befb7f14c41acc64d9f6f6": "\\Psi_S(r_1,r_2)= \\frac{1}{\\sqrt{2}}[\\Phi_a(r_1) \\Phi_b(r_2) + \\Phi_b(r_1) \\Phi_a(r_2)]", "4624b168ca3c638d67d35a92715e54cf": "P_{AG}(\\nu,\\kappa,\\pi) = \\left[\\pi_G\\left(\\pi_A + \\pi_G + (\\pi_C + \\pi_T)e^{-\\beta\\nu}\\right) - \\pi_Ge^{-(1 + (\\pi_A + \\pi_G)(\\kappa - 1.0))\\beta\\nu}\\right] /\\left(\\pi_A + \\pi_G\\right) ", "4624bc0140d5c6f77840c118f617df4b": "\\dots\\ ", "462512f6695c25cf4c3497c1c4d28b1c": "\\log\\ e^*_i\\ = ", "462540dd306c83c5c54e7df6d379d68e": "\\mathcal{F}_{\\rm Bol}=\\sigma T_{\\rm eff}^4", "4625658b795695ee9fb2d2e8558ef1c5": "\\frac{a_{n+1}}{\\prod_{k=0}^n f_k} - \\frac{f_n a_n}{\\prod_{k=0}^n f_k} = \\frac{g_n}{\\prod_{k=0}^n f_k}", "46257588523fb080c1897f512bde9d8b": " T_\\alpha {}^{\\lambda \\epsilon} = T_{\\alpha \\beta \\gamma} \\, g^{\\beta \\lambda} \\, g^{\\gamma \\epsilon},", "46258411e32e50355ac836f4368dccac": "y^2+z^2=r_1^2-\\frac{(r_1^2-r_2^2+d^2)^2}{4d^2}.", "4625884e2c4216c2497a56d6f970454a": "p^2 \\leq \\Lambda'^2", "4625af529f5d28a6ab6e33fe9d5322d7": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm1\\right)", "462626bf978e93edd5b23a8371662d5d": "\\Delta \\langle 3\\rangle", "462656c2cf70bb08c003ad8094477f4c": " \\mu = s(1-s), s=\\frac{1}{2}+ir ", "46267370c6ef861d978a28a25607d83b": "x^{i+1} := \\arg\\max_{y\\in span\\{x^i,w^i,x^{i-1}\\}} \\rho(y)", "4626988b8ea038434f858c8ef728e139": "\\frac{N}{m}", "46269b23ef6d2c691312ce6fe191e62e": "D=\\frac{V_o}{V_o-V_i}", "4626b22c13dba8cb746705c1f6fa95b9": "{{\\text{ }\\!\\!\\varepsilon\\!\\!\\text{ }}_{2}}", "462705696eb33c061345f3bd370d86b6": "f:X\\to\\mathbb{C}", "46276432ac73121a7b386bf03e972dd1": " F_\\text{application} = F_\\text{model} \\times 3.44 ", "4627686bbe262458929a813bc5f1018c": " G = \\Delta x/L\\,\\!", "4627e6c76ee49aad7291067e538a0c54": "L(C;P)", "462836229f5bf94016a407e910bd06ce": "\\varphi' = \\varphi - \\frac{\\partial \\lambda}{\\partial t}", "462863172a1536230a134c2c564ad420": "\\phi(e_i) = f_{\\pi(i)}", "4628dd8205d802737a2afa53f87f8a6b": " \\det \\frac{\\partial(\\rho, \\theta, \\phi)}{\\partial(r, \\phi, h)} = \\frac{1}{\\sqrt{r^2+h^2}}", "4629287c0d6d1144b93ffea508bb75fc": "\\textstyle a a = 1 ", "462a063fd67346a5eda507906976d6a8": "\n\\frac{\\partial T (m,s,x) }{\\partial x} = -\\frac{1}{x} [T(m-1,s,x) + T(m,s,x)]\n", "462a3fc403bc46c5d31ef088f2f3d826": "\\Delta G = \\Delta H - T \\Delta S \\,\\!", "462a868105b3c6396d596c94065cdd43": "\t\n\\sum^n_{ i = 1}\t x_{ij} = 1 (j = 1,2,\\dots, n), \n", "462ab0458db8fd7ad273321a9b1beab4": "K = -e^2, \\, ", "462b1b81549b363c47596966347dee79": " R_{\\infty} = 1 - S(0)e^{-R_0(R_{\\infty} - R(0))} ", "462b27f4689085bfa91ebf7f62ade832": "l^2/D", "462b6ffa6d0db6d8c093bd6c2587c217": "\n\\sum_{k=0}^n\\frac{(k+a)!}{k!a!}=\\frac{(n+a+1)!}{n!(a+1)!}.\n", "462ba5726860c11d140d09359747d41d": " \\frac{d}{dx}\\log_a(x) = \\frac{1}{x\\ln(a)}.", "462bd5e018cc35fd1fe550a64d472de1": " H_2^{+} ", "462c04e4347490845364d7e76263e9e8": " \\mathrm{ IP } = IMC / m ", "462c65ae164540a0994826e9c1d26c40": "E(k) = C \\varepsilon^{2/3} k^{-5/3} ", "462c7272f6dd31f5e889790554d0050c": "\\Delta L_{\\mathrm T} = n \\Delta L = 6n \\ \\mathrm{dB/8ve}", "462ccb45ecd1fc6d418ed69efad32815": "\\exp_{10}^2(7.18045)", "462d0fe4dfd71f2be172d6a81325e347": "\\sum_{i=1}^\\infty a_i z^i \\text{ where } a_i = \\frac{(-1)^{n-1}}{2^nn}\\text{ for }n=\\lfloor\\log_2(i)\\rfloor+1\\text{, the unique integer with }2^{n-1}\\le i < 2^n,", "462d2d097733e87d596162b7d3dc8c4c": "\\,cov[E[Z|Y]]", "462d8c49ab47a90edc247198a92616a4": "(f\\circ f)(x) = f(f(x)) = f^2(x)", "462d8efe25a208a3792942a755f5c6d6": "\\langle Ax, x\\rangle \\ge c\\|x\\|^2", "462df25db42434e1326575ccad38fd21": "\\alpha_{11}, \\beta_{11}", "462e091611060551eb759561b7b6c40c": "N=16", "462ec655bee3ed6eee21fd8ccdb553c5": "\\rho(t)=\\frac{|1+f '^2(t)|^{3/2}}{|f ' '(t)|}.", "462ef08b31d0fdf243abdf1e57dd716a": "m' = 17", "462fad00e4cd1c1f7475764e1ab21864": "Z^1_1", "462fc2fafc63c7243859f9e3ee9dd836": "\\Phi(t)>0", "462fe006db18dd4868fbe542ffaeabb1": "N(\\mu,\\sigma^2)", "462ffca5a168505866dc51664e14919d": "s_p^2", "46303300b1cef5c917b1a73a78f8ffef": "\\dot{\\rho}+3\\frac{\\dot{a}}{a}\\left(\\rho+\\frac{P}{c^2}\\right)=0;", "463038e381e7d33c421e8023f0adf374": "\nx^{2} = \\frac{\\left( A - \\lambda \\right) \\left( A - \\mu \\right) \\left( A - \\nu \\right)}{B - A}\n", "46304a2398b2db660421d3d428ef85ce": "\\Delta^n=\\Delta^{\\mathrm{op}}(\\mathbf{n},-)", "46307cc622f71610b17486c4507a17aa": "B(x)^2 = \\sum_{p^\\nu \\le x} \\left| f(p^\\nu) \\right| ^2 p^{-\\nu}.", "4630a6d8cfb6e2badb643985f45e4ff1": "\n F_{22}~F_{33} - F_{23}^2 \\ge 0 ~;~~ F_{11}^2-F_{12}^2 \\ge 0 ~.\n ", "463111428fcc93e5236e6b20f1542275": "\\left(\\frac{d}{dz}\\right)_q e_q(z) = e_q(z)", "463125d52589658be8897e829739241c": "\\begin{bmatrix} \\Psi \\end{bmatrix}=\\begin{bmatrix} \\begin{Bmatrix} \\psi_1 \\end{Bmatrix} \\begin{Bmatrix} \\psi_2 \\end{Bmatrix} \\end{bmatrix}= \\begin{bmatrix} \\begin{Bmatrix} -0.707 \\\\ -0.707 \\end{Bmatrix}_1 \\begin{Bmatrix} 0.707 \\\\ -0.707 \\end{Bmatrix}_2 \\end{bmatrix}. ", "46313c5992d318ac46c2d74ec718b099": "{\\rm BW}_Q=\\frac{f}{Q}", "463164fecf2fb5050a81ab6a9d851dae": "\\frac{\\mathrm{d}r}{\\mathrm{d}t} = - \\frac{64}{5}\\, \\frac{G^3}{c^5}\\, \\frac{(m_1m_2)(m_1+m_2)}{r^3}\\ ", "46317f41a0feab0955c8e0915f3f600a": "\\Delta y = \\Delta G * \\frac{1}{(1 - b_C)(1 - b_T) + b_M}", "463185e00de292dc239357f263a60a11": "\\beta = (ad-bc)/c^{2}", "4631dded8d442ee51e6f5d7dffcfbc7d": "NH_3 + H^+ \\to NH_4^+", "463279d5d851b0a1d2295928c9882f8b": "\\delta \\int d\\tau = 0 \\,", "4632ce1dd177d4736418f0d73bd38085": " h(X) = h_1(X)^{e_1} \\cdots h_n(X)^{e_n}, ", "463402343529332b529a271782ce57a5": " I_\\mathrm{rms} = \\sqrt{\\frac{1}{T} \\int_{0}^{T} \\left [ I \\left ( t \\right ) \\right ]^2 \\mathrm{d} t} \\,\\!", "46343f80c701b5fa696211ca66c3450e": "dU = dQ - PdV", "463504f4b6456d767bfe42dbc1e30452": " p_5 \n", "463511570be337af1e5209ae2514b3aa": "{\\tilde{A}}_1^2", "4635ea26336c9758e0becdbebfe2c211": "v_h = v \\cos(\\theta)", "4636067d3e0f38776024dff2698d7489": "S_{mn}\\,", "4636440d2e5f370ced4eaf4dee715249": "\\gamma_2=\\tan^{-1}[\\tan\\gamma_1/\\cos\\beta]\\,", "4636b1877b2394e79f77ca4e8cafa66e": "\\pi(p;\\mathbf{e}_1, \\dots ,\\mathbf{e}_n) = p", "463760918346d893d11842812e03517e": "\\bar{N}(E_n)", "4637dd50b8585d5073f0a6a45ae90ece": "\\frac{d^2 \\theta}{d t^2} + \\theta = 0\\,", "4637e98f1ffb6114b8e57e08a20d532d": "X_n=O_p(a_n), \\,", "46381251c474e4ceee73b2b662d1d8a8": " \\quad ", "46383d43146ae2e18a2898167e0d77fd": "z = e^{ j \\omega T} \\ ", "4638b207f65267819297e5a3af01e97e": "X_1^3+ X_2^3-7", "463935ef07f9be2a38801e635eae23b5": " f_\\odot= \\frac{L_\\odot}{4\\pi R^2_\\mathrm{AU}},", "46394a50a6fbcb3fbe98e91e109cb2e4": "\n\\begin{pmatrix}\n0 && -i \\\\ i && 0 \n\\end{pmatrix}\n", "46395a1da810fb8ca91a27004a815c74": "\\operatorname{false}", "46397b651f2c83971542a96c722c7500": "\\psi(x,y) = -\\psi(y,x)", "4639b62eb806df4a98cef4b33342e661": "\\scriptstyle u \\in\\mathrm{End}(V) ", "4639d9e657902f536cadfdc23d298043": "D_n(2xa,a^2)= 2a^{n}T_n(x) \\, ", "463a3a302882fc4620e41b7a9af250cf": "\\Big( (\\mathcal{M}, s) \\models \\Phi_1 \\Rightarrow \\Phi_2 \\Big) \\Leftrightarrow \\Big( \\big((\\mathcal{M}, s) \\not\\models \\Phi_1 \\big) \\lor \\big((\\mathcal{M}, s) \\models \\Phi_2 \\big) \\Big)", "463a7829e58783eba94b35a3a4710301": "x \\in \\mathbb{N}", "463abf5e4b21953276ee94f29d38db29": "\n\\begin{align}\n(f\\circ g)''''(x) \n& = f''''(g(x))g'(x)^4 \n+ 6f'''(g(x))g''(x)g'(x)^2 \\\\[8pt]\n& {} \\quad+\\; 3f''(g(x))g''(x)^2\n+ 4f''(g(x))g'''(x)g'(x) \\\\[8pt]\n& {} \\quad+\\; f'(g(x))g''''(x).\n\\end{align}\n", "463b09d1f6b1bb88e31efa12711e34cc": "\\textrm{Beta} \\left(\\alpha_j, \\sum_{i=j+1}^K \\alpha_i \\right ),", "463b20c45aaefeafb666ec5c045b2b97": "1,024 - 34 = 990\\,", "463b3479761bc79aa3456c0a038afe24": " {52 \\choose 5} = \\frac{52\\times51\\times50\\times49\\times48}{5\\times4\\times3\\times2\\times1} = \\frac{311,875,200}{120} = \n2,598,960.", "463bd91e08e09da08bde487f4fed8349": " W_mH^k(X; \\mathbf{C}) = \\text{Im}(\\mathbb{H}^k(Y, W_{m-k}\\Omega^{\\bullet}_Y(\\log D))\\rightarrow H^k(X; \\mathbf{C})) ", "463be5af8f42d5eff646f40b04a9f65c": " (-1)^m g_m(z) w^m = \n\\sum_{n\\ge m} \\frac{(-1)^n}{n!} \ns(n,m) w^m z^n.", "463bf7250068d8f551158758bb4f25cd": "I_{\\text{C}}", "463c447a31dd7d475123f381c9bfedc8": "2n/(t-2(n/m)+1)", "463c6508baab5875dfbb0a0a1900fe38": "ab>_y ce(ab)", "463cde89668e8d4931bc2fe07f47dc93": "\\mathfrak{g}_{0}", "463ce0a6f20f87479a71702e17a588fd": "\\mathrm{Cu}^{++} + \\mathrm{Neocuproine}\\xrightarrow{\\mathrm{Oxidation}} \\mathrm{Cu}^{++} \\mathrm{neocuproine\\ complex} ", "463cf213a4027f2cd3d0bde8216dc884": "\\Bbb Q / \\Bbb Z", "463cf785f30cdd2b1af1c3d393b8d210": " \\hat{U}^{\\dagger} \\hat{U} = I ", "463d05dd746bb4a7c18e08bbdf68d5f0": "\\theta = \\operatorname{arccos}\\left(\\frac{z}{r}\\right)", "463d24215f6eec3e28649bd364c38016": " \\frac{n}{2} ", "463d830fe782da1d8d699205524afaa6": "\nv\\delta(v')\\gamma(v)=v'\\gamma(v')\\delta(v)\\,\n", "463db8c7c2bd368c1c3c7e8cc30a7837": "t \\approx \\frac{0.693147}{r}", "463dc0d666ce71931c2f31edf1cf39ba": "\\alpha\\wedge\\beta = (-1)^{kp}\\beta\\wedge\\alpha.", "463dd128856f0cb96b160e7d16d93322": "\n \\begin{align}\n u(\\theta) := & 2~r_c~(r_c^2-r_t^2)~\\cos\\theta \\\\\n v(\\theta) := & r_c~(2~r_t - r_c)\\sqrt{4~(r_c^2 - r_t^2)~\\cos^2\\theta + 5~r_t^2 - 4~r_t~r_c} \\\\\n w(\\theta) := & 4(r_c^2 - r_t^2)\\cos^2\\theta + (r_c-2~r_t)^2 \n \\end{align}\n ", "463dda2d798b04b3967b761bd03f8459": " 5\\log_{10}{d}=V+ (3.34) \\log_{10}{P} - (2.45) (V-I) + 7.52 \\,. ", "463e10b4289d71d8f76004d317ee77b5": "\\frac{3}{5}", "463e3a74bfffedc44869a1d0e50e8e86": "r=\\frac{ab}{a+b}.", "463e5ba9693c969d3ef417d5f9269175": " S_{i,j}^{t+1} =2 ", "463e980d398d0ce27ee93f71b47d820a": " H(f) \\leq C I(f) \\!", "463f0f6c03585f9fd13d682568c7fdfa": "F'(x) \\approx \\dfrac{F(x+h)-F(x)}{h}=\\dfrac{G(x)-F(x)}{h}\\,", "463f1c6b70b1a5622645eafdf0929b7c": " X \\sim \\textrm{Kumaraswamy}(a,1)\\, ", "463f2e5a8ecc0cafd059df3dc8b00a91": "\\mu_P", "463fa200ba335f28c3d3b62dc8fc3d28": " PN_1 = \\frac{N_2}{d}", "463fb7b1baf7150915ef5eb0681984e1": "f(0, -1) = 3", "463fe9849d0587ffebff261547988c75": "n(r-1)", "463fed026c8be378e20ae9886ed7f65a": " R_k(x) = \\int_a^x \\frac{f^{(k+1)} (t)}{k!} (x - t)^k \\, dt. ", "464007cfe9d2dab79bb57f414cc3f094": "\\frac13 a^2 - \\frac{5}{9} a ", "464015f8e5688c6027600022b4d3f8ac": "\\operatorname{red}_1(f,g)=f-\\frac{c}{\\operatorname{lc}(g)}\\,q\\, g.", "464021f1c882019af30758213076bf75": "\\frac{\\mathrm{d} N}{\\mathrm{d} t} = - \\lambda N ", "46402936535e425ff39f1bbacb80db7b": "Q=Z/R", "46402ce5bc801891eb70b9fc500e3209": "v\\geq\\sqrt{\\frac{2 G M}{r}}", "46403eebaba8cc670e8d972e69ac250c": "\\scriptstyle k = \\frac{a^2 \\,+\\, b^2}{ab \\,+\\, 1}", "464073319ae174552bd86ec404c4f62f": "\\scriptstyle L^{-}", "464088e03c4155733696563b1f083e4e": " \\theta\\, ", "4640b51682829767acc69f7f0b0d8d59": "\\alpha_1 ", "46412209045b16098518fe283f968199": "E = {Z_3}^2", "4641d03ff0375c94567a6219097cff83": "\\}", "4641dc47f58004f3e69cba6071104dfd": "(\\alpha_k)_k", "46420b45ef2945efadec87e9a3d5a33f": " \\rho \\left(\\frac{\\partial w}{\\partial t} + u \\frac{\\partial w}{\\partial x} + v \\frac{\\partial w}{\\partial y}+ w \\frac{\\partial w}{\\partial z}\\right) = -\\frac{\\partial p}{\\partial z} + \n\\frac{\\partial}{\\partial x}\\left(\\mu\\left(\\frac{\\partial w}{\\partial x} + \\frac{\\partial u}{\\partial z}\\right)\\right) + \n\\frac{\\partial}{\\partial y}\\left(\\mu\\left(\\frac{\\partial w}{\\partial y} + \\frac{\\partial v}{\\partial z}\\right)\\right) + \n\\frac{\\partial}{\\partial z}\\left(2 \\mu \\frac{\\partial w}{\\partial z} - \\frac{2\\mu}{3} \\nabla \\cdot \\mathbf{v}\\right) + \n\\rho g_z", "46422ec36505243aeed5a35fce049894": " \\mathbf{v} = \\boldsymbol \\Omega \\times \\mathbf r \\ , ", "464258b05ade714718006c6a8f715df2": "[x^{\\iota}]", "4642fc3dba2d3b600c2ef22bb320f580": "\\epsilon({{{\\it{T}}}}) = 1 -|\\langle{\\tilde{\\psi_{Tr}}}|{\\psi_{{Tr}}}\\rangle|^2 = 1 - 1 + \\epsilon^2 = \\epsilon^2", "46435691a1f1eb21e96829b13b901ac0": "k = \\frac{z-z'}{s-s'}", "4643578de5d620ad56c1b6c354e4f64d": "M= tM\\oplus F", "46435d319f88c68fc4654f879310a894": "\\textstyle x \\in \\mathbb{R}^n", "4644312215a1fd2a27d1cb2a5f5ad893": "w_1 + w_2 = w", "4644876b3e02a2842c65f779a307bdec": "\\rho\\sum_{i=1}^k\\frac{f_i(x)}{z_i^{nad}-z_i^{\\text{utopia}}}", "46448f8ebaa53ba1b5b2cd778f82ff17": "r_1(m_1)", "4644b86e879ea02bf71e321b8d9b0102": "(S \\cup T, F)", "4644ea991b356262a64f473e0c368e98": "{\\left \\{ a_n \\right \\}}_{n=1}^\\infty", "4645088d4a4dfb411462deb87fe6c30a": "N^*\\Sigma=\\{(x,\\xi)\\in T^*\\R^n:x\\in\\Sigma,\\,\\xi|_{T_x\\Sigma}=0\\}\\,", "46450d3b17db9e305db5cc34be92f3f4": "\\lim_{n \\rightarrow \\infty} \\left( 1 + \\dfrac{1}{n} \\right) ^n = e", "46468a4edd50301af2c2452432b6e868": "f(x) = \n\\begin{cases}\n \\frac{1}{q} &\\text{if }x\\text{ is rational, }x=\\tfrac{p}{q}\\text{ in lowest terms and } q > 0\\\\\n 0 &\\text{if }x\\text{ is irrational.}\n\\end{cases}\n", "4646a916ff44ec6f87fd670647f6601a": "y'=ay", "4646bfb56bf03207493fc1fc2568771a": " p_i^j ", "4646cc006392bb50163b14411baddbd3": "1/(2q^2),", "46470aa3c283810641f1791b82bacaca": " \\frac{d}{dx} z = 2 \\sin x \\ \\cos x + 2 \\cos x \\ (-\\sin x) = 0 \\ , ", "46471bb31a2cbb7791ecc55cdcb63d25": "\\vec{p}_\\mathrm{out} = q \\cos \\theta \\hat{z} + q \\sin \\theta \\cos \\phi\\hat{x} + q \\sin \\theta \\cos \\phi\\hat{y}", "464739654bfdcc7efa151b64d895948d": "M^- \\hookrightarrow W", "4647409980e306041574ae70413e6438": "\\varepsilon\\approx\\frac \\pi 4 \\frac {186-179}{186+179}\\approx 0.015,", "464744b154543089b584f91e92d3f58f": "\\alpha\\in(0,\\frac{\\pi}{2})", "4647a124e29eb42efe955c97efdb6288": "E[K(K-1)\\dots(K-n+1)]=n!\\boldsymbol{\\tau}(I-{T})^{-n}{T}^{n-1}\\mathbf{1}\\,.", "4647b74a9ecafbd48c8712458f273ab3": "(k^2B^2/\\mu_0\\rho)< - Rd\\Omega^2/dR\\ .", "4647f0d86edba05d82a53c2c992cf0de": " \\Delta S = S_{\\rm final} - S_{\\rm initial} \\, ", "46482436d0269b7c94cd832c09c43670": "\n f(x; \\chi, c ) = \\frac{\\chi^3}{\\sqrt{2\\pi}\\,\\Psi(\\chi) }\\ \\cdot\\ \n \\frac{x}{c^2} \\sqrt{1-\\frac{x^2}{c^2}} \\ \n \\exp\\bigg\\{ -\\frac12 \\chi^2\\Big(1-\\frac{x^2}{c^2}\\Big) \\bigg\\},\n ", "46486f5dc1208632d7e58441cab9f320": "\\left[ \\Pi\\right] =\\left\\{ \\left[ A\\right] \\ |\\ A\\in\\Pi\\right\\} ", "46488afb789019ba6314c8f43d06dc6e": " \\bold{p}(\\bold{r})=\\chi(\\bold{r})\\bold{E}(\\bold{r}) \\, ", "46489721acbbfe35694ca81dbbb44dc4": "\\lambda_k = 1 - \\mu_k", "4648afd405b0961cbb83f81cdc89b7a3": "\\mathbf{\\Psi}", "4648b0d5a8637b063d40b922d5f469d5": "h^2=pq.", "4648c8360f7a9ba887d29f72410d6b3b": "\\psi(x)=\\left(\\frac{m \\Omega}{\\pi \\hbar}\\right)^{1/4} \\exp{\\left( -\\frac{m \\Omega (x-x_0)^2}{2\\hbar}\\right)},", "4648f14c72d15bea1d4cee39214a0145": " V = \\sqrt{\\frac{Q^{2}}{2 U(\\phi_{0}) \\phi_{0}^{2}}}. ", "46494b5b8717ad2cae3ca8b83ef0eb27": "B^\\prime=-(n_\\mathrm{b}-n_\\mathrm{\\bar{b}}),", "464970895ca7d470d18948d4ea6077b1": "0 < \\left| \\sqrt{2} q_n - p_n \\right|^2 < \\frac{1}{2^{2^n}}", "4649b624d7abfefc2797c54ab1c2c2c6": "V_{VdW}=-\\frac {Aa_p} {6h} \\bigg[ 1+ \\frac {14h} {\\lambda} \\bigg]^{-1}", "464a3f6f6cd82ee3a7aea9ed4058e5b7": "q(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz \\quad\\textrm{(ternary)} ", "464a47942fa9a6b1c4f5fdf215df2587": "c = \\frac{a}{\\sin A} \\text{; } c = \\frac{b}{\\sin B}", "464a570c3e869dffb9fd14d593bc8a86": "\\psi_{3}", "464a9b06c5dead434516526cc2ec5263": " \\sigma ", "464ad5e7790ad03b489cc97d55e9521f": "a_{1} = \\frac{a_{0}}{2(l+1)}, ", "464b1631f7811de9ca690e59a83f589f": "(x^{q^{2}}, y^{q^{2}}) = \\bar{q}(x, y)", "464b2811ad104d6cf6a85bdc14117246": "x_3 = 0", "464b5b58a1e0a557943658202a58163c": "\\frac{d}{d\\theta} \\left (\\int_{a(\\theta)}^{b(\\theta)} f(x,\\theta)\\,dx \\right )= \\int_{a(\\theta)}^{b(\\theta)}f_{\\theta} (x,\\theta)\\,dx + f(b(\\theta),\\theta)b'(\\theta)-f(a(\\theta),\\theta)a'(\\theta)", "464ba62d613981de5d4c6ed8fb6ee32d": "\\ell_0=\\frac{x-x_1}{x_0-x_1}\\cdot\\frac{x-x_2}{x_0-x_2}=\\frac{x-4}{2-4}\\cdot\\frac{x-5}{2-5}=\\frac{1}{6}x^2-\\frac{3}{2}x+\\frac{10}{3}\\,\\!", "464bbb093d7caa2cd1dcf1a9e15faf3e": "x_1 + x_2 + x_3 = L", "464bbc8aca231a1b39f6a96266d19bd2": "\\pi=\\frac{27}{4Z}\\!", "464be884af4ac984f475df344a62f5d0": "\\zeta=\\zeta_{1}e_{1}+\\ldots+\\zeta_{n-1}e_{n-1}.", "464c14c39cae746502f66bd3bcd3b767": "\\log\\left(1 + e^x\\right)", "464c16eb60ddb127225e49f2b0d3e5cc": "\\begin{align}\nZ[j,\\bar\\varepsilon,\\varepsilon] &= \\exp\\left(-ig\\int d^4x \\, \\frac{\\delta}{i\\delta\\bar\\varepsilon^a(x)} f^{abc}\\partial_\\mu\\frac{i\\delta}{\\delta j^b_\\mu(x)} \\frac{i\\delta}{\\delta\\varepsilon^c(x)}\\right)\\\\\n& \\qquad \\times \\exp\\left(-ig\\int d^4xf^{abc}\\partial_\\mu\\frac{i\\delta}{\\delta j^a_\\nu(x)}\\frac{i\\delta}{\\delta j^b_\\mu(x)}\\frac{i\\delta}{\\delta j^{c\\nu}(x)}\\right)\\\\\n& \\qquad \\qquad \\times \\exp\\left(-i\\frac{g^2}{4}\\int d^4xf^{abc}f^{ars}\\frac{i\\delta}{\\delta j^b_\\mu(x)} \\frac{i\\delta}{\\delta j^c_\\nu(x)} \\frac{i\\delta}{\\delta j^{r\\mu}(x)} \\frac{i\\delta}{\\delta j^{s\\nu}(x)}\\right) \\\\\n& \\qquad \\qquad \\qquad \\times Z_0[j,\\bar\\varepsilon,\\varepsilon]\n\\end{align}", "464cac3b85003e0903c3bca68659e1c9": "\\sigma =1", "464cb80a423e61f2d346b9ea083e9d27": "f_y(0,1) = p_y(0,1) = a_{01} + 2a_{02} + 3a_{03}", "464cc2c5cf82a94c913c17848640d3c2": " I(X;Y|Z) = \\sum_{y \\in Y} p( Y=y ) D_{\\mathrm{KL}}[ p(X,Z|y) \\| p(X|Z)p(Z|y) ]. ", "464cf2b6c9a872bc33b39fc05bf13c16": "5x\\in\\{90,180,270,360,450\\}\\,", "464cfb1cb9a79a5f6549ffd7b0d9ae38": "a=\\tfrac{k^2(s^2+r^2)^2}{4}, \\, ", "464d30cd024cf9382bd9c22907f97b1f": "\\mu = \\sum\\limits_{i=1}^n p_i", "464d5e57100c7d67446720086ceeb140": "\\int \\Phi(x)\\, dx", "464d68d51a4ca9db0a18d04549f99ed0": "[f]\\notin N", "464d6dd9051b9ab48b276b3d5ba6a1db": " \\varphi = 2\\sin(3\\pi/10)=2\\sin 54^\\circ. ", "464d6ff93ab2454ee8ae6dac5b4e4832": "r_s/a_0", "464d8372058b1054718e7ad01d817d8c": " m=\\varepsilon c ", "464db4cd129ac6bdae6295b357f974ec": "S \\subset \\mathbb{R}^n", "464dfd53855810438a7523d369506d72": "\\operatorname{Rot}_{z_{n - 1}}(\\theta_n)\n = \n\\left[\n\\begin{array}{ccc|c}\n \\cos\\theta_n & -\\sin\\theta_n & 0 & 0 \\\\\n \\sin\\theta_n & \\cos\\theta_n & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n", "464e12a0f5475e615125f875f39e7b8e": "V = \\sum_{i = 1}^{\\infty} \\oplus V_i", "464e7adab6acfc647df583dedfa012c0": "\\scriptstyle\\epsilon = 1/2", "464e994278b837651cc79c23b1ca6e5c": " W = \\int_{\\mathbf{r}(t_0)=A}^{\\mathbf{r}(t_1)=B}\\mathbf{F}\\cdot d\\mathbf{r} = \\int_{t_0}^{t_1}\\mathbf{F}\\cdot \\mathbf{v}dt ,", "464ee0dfdebefce7fe85810cf0f437f5": "a\\!\\!\\!/b\\!\\!\\!/ = a \\cdot b - i a_\\mu \\sigma^{\\mu\\nu} b_\\nu ", "464eefa4b07b6584377884ab02ad3772": "\\,^{z_1 = x_1 y_1 - x_2 y_2 - x_3 y_3 - x_4 y_4 - x_5 y_5 - x_6 y_6 - x_7 y_7 - x_8 y_8 + u_1 y_9 - u_2 y_{10} - u_3 y_{11} - u_4 y_{12} - u_5 y_{13} - u_6 y_{14} - u_7 y_{15} - u_8 y_{16}}", "464f30fd4633ae7d1941627ecfb5a95f": "V = \\frac{{wS}}{{2}}", "464f34a5a7866437a556e2f396923063": "e_{i, j}", "464f4591bc90f05f9801b4b00451d09a": "\\displaystyle{\\omega(f)=\\limsup_{\\varepsilon\\rightarrow 0} |H_\\varepsilon f - T_{1-\\varepsilon}Hf|.}", "464f4dd0a1b8bce4e01b6fb0a0af287a": " c \\leq \\varphi( B) <\\infty", "464f852357d86df324ddf582f0708bb3": "m = \\partial T_L / \\partial C_L", "464fa918829b3dc6591d62790368659e": "\\nabla \\times \\mathbf{H} = \\frac{\\partial \\mathbf{D}} {\\partial t}", "4650118653b392124feee540e46e73d1": " W_{i,j(k)}=\\left[ \\begin{array}{ccc|c} 0 & -\\omega_z & \\omega_y & v_x \\\\ \\omega_z & 0 & -\\omega_x & v_y \\\\ -\\omega_y & \\omega_x & 0 & v_z \\\\ \n\\hline\n0 & 0 & 0 & 0 \\end{array}\\right]", "465031cc9cc75e6cdcf9b0980015bba0": "\\sum_{i=1}^m \\lambda_i v_i =0", "4650415004fbcc1d0383665807d35635": "B^2_\\sigma.", "465060ad1f6edb69265efd82e3115c6a": "\\omega^{\\omega^{\\varepsilon_0 + 1}} = \\omega^{(\\varepsilon_0 \\cdot \\omega)} = {(\\omega^{\\varepsilon_0})}^\\omega = \\varepsilon_0^\\omega \\,,", "46509272041d31bb09e337ebb784ce49": "\n\\lim_{x \\to \\infty} e^{\\lambda x}\\overline{F}(x) = \\infty \\quad \\mbox{for all } \\lambda>0.\\,\n", "465093fc5d014a4da5b29cd72c3f1a10": "i = 1,2,\\dots", "4650a62aefe748323eac1cd0024f7f4a": "fg = \\sum_{e\\in\\Gamma} \\sum_{e'+e''=e} c_{e'} d_{e''} T^e", "4650e759fa5e9ae277e6f79404b14196": "{\\mathbf{w},b}", "4651a3751cd15f66696eadbf4acad15b": "\n \\left[\\begin{array}{ccc|c}\n 1 & 0 & 1/16 & 0 \\\\\n 0 & 1 & 13/8 & 0\n \\end{array}\\right].\n", "465246a5a566ae7c5a393e6a34bc19b7": "y(xy)^m y^{-1} = (yx)^m yy^{-1}=(yx)^m", "4652ce646413daa866ed4c86ca4dabc3": "\\gamma_{23}", "4652dcbfe581b0abe76a06f554f0cb3d": "L_\\lambda", "465302df50edb07174a33f268d7e6c0f": "\\sigma(M) = \\sup_{g} Y(g),", "46530453cea878b8e0cfd814fa673608": " V_{\\sigma, \\sigma'} = e^{\\frac{\\beta h}{2} \\sigma} e^{\\beta J\\sigma\\sigma'} e^{\\frac{\\beta h}{2} \\sigma'}", "46533e9b03464ee8d820f50183f52a77": "\\varepsilon=\\frac{K}{2\\pi}", "465350c7dc2f52efdbc9e725c973682e": " {PB}_t ", "46537487fc213d235a305fe9e794a649": "\\Delta_{1}-\\Delta_{2}", "4653c88613870db9e01dfae4fa3d96c8": "\\left[{n\\atop k}\\right]", "4653d7baae6f808e024151939ee6a367": "\n \\phi(r) = \\cfrac{d w}{d r} = -\\frac{qr^3}{16D} + \\frac{C_1}{r} + C_2 r + C_3 r \\ln r \\,.\n", "46543b488e091ed19be07e026599b772": "i_Yd\\lambda", "4654e60f5a23fd14f8b33c749c43ae40": "e_1, e_2, ...", "4655f57428e34b6f573f527f599f5695": "x_1 h = 0", "46561ce99fb5a926dc2241051ea80817": "E_G", "46563a274c59e4ae629c46cbb2e6e6d9": " V_{n}(c)", "46565ee9dc8d6a449512f0b94f69642e": "l^n x = 0", "4656b6f4130435265c4854a2e998eca3": "\\frac{1}{e} \\leq \\left (1 - \\frac{1}{d+1} \\right)^d.", "465708e0239e986884321b2cee5a0071": "a=0.5", "46570c6f6a0b7a3349da65957f24e348": "v(x) = \\text{True}", "46571098c2431c7e2ffd45e92c64a850": "X(a^{-1}z)", "465784bc9754400b16a7db6d43978dc8": "U_n(t,t_0)=(-i)^n\\int_{t_0}^t{dt_1\\int_{t_0}^{t_1}{dt_2\\cdots\\int_{t_0}^{t_{n-1}}{dt_n\\mathcal TV(t_1)V(t_2)\\cdots V(t_n)}}}.", "465790e70ed6cd30cb9b5ee1aa40be91": " p_3 (x_1, x_2, \\dots,x_n) = x_1^3 + x_2^3 + \\cdots + x_n^3 \\, .", "4657952b0d316f6143d831ce795e4167": "g_{ij}=(1+2\\gamma U)\\delta_{ij}+O(\\epsilon^2)\\;", "4657c2f909388cba035f72ed58915437": " r_{0ij} ", "4657d5ddc3d64b84e2fc79d654439139": "c=\\frac{1}{2}\\mathrm{tr}\\,M", "46580de512db735f659bf39b394d44e5": "{\\partial \\mathbf{r} \\over \\partial y}=(0, 1, f_y(x,y))", "4658522f909f76b559bca41b4ca3d870": "E_\\text{k} =\\tfrac{1}{2} mv^2 ", "4658561736e68d644fddaad19a696e7a": "\\bigoplus_{i\\in I}A_i", "4658a912a6f0eb9c71febcb5abe6c765": "\\,a = 2m.", "4658cce022f0204e70899e92717e339a": "\\begin{align}\n x &\\equiv 2 \\pmod{3} \\\\\n x &\\equiv 3 \\pmod{4} \\\\\n x &\\equiv 1 \\pmod{5}\n\\end{align}", "4658e242c10222e7c1f4000b31314092": " \\mathbf{a} = - \\frac{v^2}{r} \\frac{\\mathbf{r}}{r} = - \\omega^2 \\mathbf{r}", "4658e45c74aade3c3af83886dc6ebb07": "\n\\langle \\langle \\phi \\rangle \\rangle = \\langle \\phi \\rangle. \\,\n", "465950666e6491bac452d41b060dd947": "\\underbrace{X_{1/T}\\left(\\frac{k}{NT}\\right)}_{X[k]} = \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}} \\quad \\quad k = 0, \\dots, N-1", "465961c9ab2365e278b0a27a16de46c6": "B_n=K\\phi^{|n|}", "4659c52c87e248b5daeee09df4e09a0a": "\nv_{k+1} = 2 \\beta v_{k+0.5} - v_{k}. \n\\,\\!", "465a2065cf48c989c53d4c42bd59e5dc": "\\overline{J^j} \\subset J^{n_j}", "465a26e08e7a98bcfc1d6f33a0e3979c": "P^2=(a^2 r^{n})^{n+1}=(a\\cdot ar^n)^{n+1}", "465a3d6d63548a430e1f61def1a1eff0": "\\int_0^{\\pi} \\frac{x \\sin x\\ dx}{1-2a\\cos x +a^2}=\\begin{cases}\n\\frac{\\pi}{a}\\ln (1+a) & \\text{if } |a|<1 \\\\ \n\\pi \\ln(1+1/a) & \\text{if } |a|>1 \n\\end{cases}", "465aa14c8604940fbd2e76a5c25a5328": "(P_1=1) \\or (P_2=2) \\to (P_{4}=1)", "465aee2ddfe71c50053c586a8c1d8124": "\\displaystyle \\ell_n(x,\\lambda) = e^{-n\\lambda}\\sum_k(1-e^\\lambda)^k\\binom{n}{k}\\binom{x}{k} =e^{-n\\lambda}{}_2F_1(-n,-x;1;1-e^\\lambda)", "465b1c7c448f8db18e2cf1e0f7affbe4": "\\exp(s-t)", "465b2e78bd56d52649b49f4ea2720254": " \\operatorname{inc}\\ (\\operatorname{inc}\\ \\operatorname{init}) = \\operatorname{value}\\ (f\\ (f\\ x)) ", "465b5e5532917bd135fce49a0b69a21e": "3000 \\le \\mathrm{Re}_D \\le 5 \\times 10^{6}", "465b99dec4d6feeecc69dc6b39df5c98": "\n B_\\delta = \\big\\{x\\in S\\ \\big|\\ x\\notin D_g:\\ \\exists y\\in S:\\ |x-y|<\\delta,\\, |g(x)-g(y)|>\\varepsilon\\big\\}.\n ", "465bad0d74fb9d3fb5331ead5795869f": "|\\ln \\zeta(s+it)-g(s)| < \\varepsilon\\quad\\mbox{for all}\\quad s\\in U.", "465bc4d83e9578b0af4e027e226a1d8d": "\\begin{bmatrix}\nc_3 c_1-s_3 s_2 s_1 &\t-s_3 c_2 &\tc_3 s_1+s_3 s_2 c_1 \\\\\ns_3 c_1+c_3 s_2 s_1 &\tc_3 c_2 &\ts_3 s_1-c_3 s_2 c_1 \\\\\n-c_2 s_1 &\ts_2 &\tc_2 c_1\n\\end{bmatrix}", "465bde25eca0600366f45fed8d94c07f": "{}+\\kappa(\\kappa(X_1\\mid Y),\\kappa(X_2\\mid Y),\\kappa(X_3\\mid Y),\\kappa(X_4\\mid Y)).\\,", "465c47d4d5e10813643704d16ac0e892": "O(n^{2.38} \\cdot |G|)", "465c486de5fbdf2023cf3d3f24ba73e3": "\\mathbf{\\hat{d}}_\\mathrm{s}", "465c993fc946a90a07365cb7cbb2a83e": "(x+y)(x-y) = x^2-y^2\\,", "465cb8b4a2bc39f0433b795292888597": "j = i+1", "465cd868687eaa8a500afec10c87c39b": "(\\mathbf{r}\\times\\mathbf{F}_1) + (\\mathbf{r}\\times\\mathbf{F}_2) + \\cdots = \\mathbf{r}\\times(\\mathbf{F}_1+\\mathbf{F}_2 + \\cdots). ", "465d1b50f67783858daaf65b729e48dd": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2\n\\end{smallmatrix}\\right ].", "465d3c56343a722e2c46532109308897": "(A_{u,v})^2\n= \\sum_{i [Cell(n-2) + \\\\ Cell(n-1) + \\\\ Cell(n+1) + \\\\ Cell(n+2)] \\times \\\\ Constant \\end{align} }\\end{cases}", "4660480bcb6f33f86be4c25fcafc5b97": "\\! w<-1 ", "46606f604935bffbbc091fbe0f3d43f3": " N_A(X,Y) = -A^2[X,Y]+A([AX,Y]+[X,AY]) -[AX,AY]. \\, ", "466070be996bec2a74b1ebaaf61e18a0": "\\,r\\cdot r + 2re + e\\cdot e \\le x", "4660aa979cf8b1b2553e912c7ee44d19": "\\forall A:P . \n(A \\Rightarrow A) \\Rightarrow (A \\Rightarrow A)", "4660e41856d7c37a210bbe38bfb2cb47": "T F T \\square F = \\{0 | 0\\} = \\star ", "466102830d09af54c4db41798f633590": "C '", "466108b8c7cc3cb296fd5529d24e357f": "\\mathbb{Z}_q^n \\times \\mathbb{T}", "466134289662771d6b927350497420b1": "\\beta H", "466157364b7564c3d6b568b6a3ba8721": "\\underline{AB}", "46622c667eebf19a059902945fb8b88e": "\\mathbb{R}/R\\mathbb{Z}", "46623945f376219d75486654e5f6e2f6": "f(z) = {e^z \\over z} + e^{1/z}.", "4662829e9f3db729442ee8ab37c82428": "n=1, 2, \\dots, ", "466290850b50d4079379d73b6ebc6933": "\\sup_{P\\in \\mathcal{P}(S,A)} \\mathbb E \\|P_n-P\\|_\\mathcal{F}\\to 0.", "4662e8d4d9f7273b123d763784e7250c": "\\mathrm{DM} = \\int_0^d{n_e\\;dl}", "466334ddbb9585cfc7f5dcd9e3e81a37": "A[1,n]", "466380a8cd6f7dc0e18451191d6d448a": "\\begin{matrix}\n\\mathbf{e}_1 & = & (1,0,0,\\ldots,0) \\\\\n\\mathbf{e}_2 & = & (0,1,0,\\ldots,0) \\\\\n& \\vdots \\\\\n\\mathbf{e}_n & = & (0,0,0,\\ldots,1).\\end{matrix}", "46638404235cb2f69e2124e265c6565c": "P(E_1 \\cup E_2 \\cup \\cdots) = \\sum_{i=1}^\\infty P(E_i).", "4663891fe5e51edbf614fba7497ebc07": "U(A)=U(O) - W^\\mathrm{adiabatic}_{O\\to A}\\,\\, \\mathrm{or}\\,\\,U(O)=U(A) - W^\\mathrm{adiabatic}_{A\\to O}\\,.", "46638b7e7f013f952bf7db280f8246d5": "\\mathcal{T} = 1 + \\sum_i \\left( | \\phi_i \\rangle - | \\tilde{\\phi}_i \\rangle \\right) \\langle p_i | ", "4663d255b692fc1626fa6789074b6e1b": "0 < \\operatorname{var}(X) < \\tfrac{-11+5 \\sqrt{5}}{2},", "466405e502f41c645ed088ed96aa489e": "0 < \\bar{n}_i < 1", "466410689fab9d9638fa0e7010479377": "(X_a,Y_a)", "46644998e396e5c3c65a36134587b3cb": "\\mu_{i}", "4665161d4c70ce6f6e53d4adc3a6237b": "\\mathbf{e}_{x,y,z}", "46653fb3089ade69988846d1d8dd1a60": "\\langle\\mathbb{N},<\\rangle", "4665666150d73ef721a5789b9d57f4b0": " \\bar{x} = f^{-1}\\left({\\frac{1}{n}\\cdot\\sum_{i=1}^n{f(x_i)}}\\right) ", "4665d4a55ca3da2574f803d1fc4b6ce7": "Z_2=\\{0,1\\}", "4665f1733bf15577f7aca3d9fc131e71": "\\ell' = v t' \\,", "46662daff4d52b802332b6f1411fa6ca": " \\min_{\\boldsymbol{\\beta}(y)} \\left|\\left|\\mathcal{C}_{{(X \\mid Y)}^{tr}} \\mathbb{E}_{Y^{tr}} [ \\boldsymbol{\\beta}(y) \\phi(y)] - \\mu_{X^{te}} \\right|\\right|_\\mathcal{H}^2 ", "466647328793c71ba3e48f1f0dddd528": "\\{ 1,~i_1, i_2, j \\}", "46668490c05a9f8ca42d14921a589abd": " x^{\\mu} ", "4666abd0fec3acc6fbe84a503423272c": "z=w", "4666b2060dee1c620cb73e9b7c160a2b": "F_k = \\frac{\\partial}{\\partial \\lambda_k} \\log Z(\\lambda_1,\\ldots, \\lambda_m).", "4666d3973d8852cd18c8b5802042f9ac": "\n w(r,t) = W(r)F(t) \\,.\n", "46670a4980b6302d5e604203c5aa3737": " \\ln (Cx) = \\int^{xy} \\frac{N(\\lambda)\\,d\\lambda}{\\lambda [N(\\lambda)-M(\\lambda)] } \\,\\!", "466795dc41202aff230dc460e5cce5bd": " c_{V,i}", "4667e778ff8ff9455c717504601e740e": "\\scriptstyle {}^{n}i \\;=\\; a+bi", "46680aa52aed0d8477115d20f90b3695": "\\Delta B_{1/2} = 2\\Delta B_h", "466820bebd7c4966f01fc0c391af0ca0": "\\chi_+^z = \\begin{bmatrix}\n 1\\\\\n 0\\\\\n \\end{bmatrix}\n", "46685c90b616499f87f0ff3f508acb20": "u_1, u_2, \\dots, u_n,\\ v_1, v_2, \\dots, v_n .", "4668650d56eaebde6702f79043d8bf37": " \\left(\\mathbf{A} - \\lambda_i \\mathbf{I}\\right)\\mathbf{v}_{i,j} = 0 \\!\\ ", "46688bbe79d73d6defc23dbe9629054a": "c_1=f'/a_0", "4668a07d29ff5584088cf5a0c912458a": " f:S^{1}\\rightarrow \\mathbb{R}^{n}", "4668dfb964cc78a1274f5e8b65df8f35": "f_c : z \\to z^2 + c. \\,", "466923f40b80a76e120219fae01e9cf6": "S_1, S_2", "4669430c16d9b1b1d54699cb07fac589": "\\gamma' = \\frac{\\gamma}{1 + \\gamma^2\\eta^2M_s^2} \\qquad \\text{and} \\qquad\\lambda = \\frac{\\gamma^2\\eta}{1 + \\gamma^2\\eta^2M_s^2}. ", "46695aa03c2071b8d10bd57ae467ff1c": "\\tilde f : aF \\rightarrow G", "46696e81370d917688844d10f245d9a3": "L^+(X,\\mu)", "4669bca04aac7310e4c0ac08a5a35812": "\\scriptstyle{(-\\frac {1}{3},\\frac{2}{3},\\frac {2}{3})}", "4669dafd3a159414e55ef1a253c520a1": " A_1=\\partial_x \\partial_y + x\\partial_x + 1= \\partial_x(\\partial_y+x), \\quad\nl_2(A_1)=1-1-0=0;", "466a2691535dba57190d947b4a23b477": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {l_2} {l_1} ", "466a8dd484691ac231c22fb188b09fac": "A^{\\alpha\\beta\\gamma \\cdots}", "466ad84eeee53885e047ba6386668223": "\n \\det[g^{ij}] = \\cfrac{1}{J^2}\n ", "466b07ba76be6b931618c21b7e965be4": "d = x- x'", "466b2977f078a017d3255d3db2864db0": "\\psi(x) = \\psi_0\\tanh\\left(\\frac{x}{\\sqrt{2}\\xi}\\right)", "466b7659b8585c65acb64b0d769f7c24": " \\mathrm{N}(n,S) = \\#\\{ m \\in S : m \\le n \\} . ", "466b99650a6fade1e3fbc30b032ff3fb": "K_2 = {{}^\\star R}_{abcd} \\, R^{abcd}", "466ba3997630bcf764a3fe2ff24e107a": "\\arccsc (1/x) = \\arcsin x \\,", "466badf51003b3123445dbe66d92ee6b": "d_{\\bar k}=\\deg T^{\\bar k}p", "466c5bae44f7dbf170b1427d05c3d5cc": "FR = {RPM \\times T \\times CL} ", "466ce95adda05b3f811436b7596a9d9a": " L^+_{u,v}:V_+\\to V_+ \\quad\\text{by} \\quad L^+_{u,v}(y) = \\{u,v,y\\}_+", "466cf6c14eff52524e94beae42f06fc1": "T^{(1,0)}\\mathbb{C}^n = \\mathrm{span}\\left(\\frac{\\partial}{\\partial z_1},\\dots,\\frac{\\partial}{\\partial z_n}\\right).", "466d2eeefcab920e372582c7b09e1549": "k_C=\\frac{C_m}{\\sqrt{(C_1+C_m)(C_2+C_m)}}.", "466d75c2f51d820aabdbc88479c406de": "(\\mathfrak{g}, \\mathfrak{h})", "466d772f66726648e3e11a5d21d4c0fc": "3 \\uparrow\\uparrow 3 = 3^{3^3} = 7,625,597,484,987", "466dc96336781311e1efc8a5e3c58832": "f^* \\colon Z^k(X') \\to Z^k(X) \\quad\\text{and}\\quad f_* \\colon Z_k(X) \\to Z_k(X') \\,\\!", "466dfa3beac3ef47374894c2b01f7bcd": "\\sum_{n=1}^{\\infty}\\frac{1}{n}e^{-ns},", "466e0850cba7a06332dd34ccea273d84": "LWt", "466e1968129efdb0804a3053e37d54f4": "\\oint\\frac{\\delta Q}{T}=0", "466e77b48cb91d6dc5d424d671077a10": "S = -\\left(\\frac{\\partial A}{\\partial T}\\right)_V\n=Nk\\left[ \\ln\\left(\\frac{(V-Nb')T^{3/2}}{N\\Phi}\\right)+\\frac{5}{2} \\right]", "466ec5088ad76dbe1cf37ad7d683ef37": "\\sin x+1=\\Omega_+(1)\\ (x\\rightarrow\\infty)", "466f317373cb7297905437961a30ca10": "\\mathbf{1}_{A} \\in \\mathcal{H}", "466f4d7a86ab68e7ab99cdc5d6ef4e2a": "\n\\mathbf{m}_{\\rm orb}=\\frac{-e}{2m_e} \\langle\\Psi \\vert\\mathbf{L} \\vert\\Psi\\rangle\n", "466f5e2807a60f4976a8ecc6461d70c6": "Q_e", "466f659bc8d0c8449551b0c140c00609": "c_1=\\frac{x-y}{\\sqrt{2}},\\quad c_2=\\frac{x+y}{\\sqrt{2}}", "466f6b7777911ef27e6cde6ebccab802": "K_{G}^{(a)} = \\sigma_x^{(a)} \\bigotimes_{b\\in \\mathrm{N}(a)} \\sigma_z^{(b)}. ", "46700b1d20c9f7cda00882982f3b3424": "x\\in \\mathrm{ker}\\left(f^k\\right)+f^k\\left(y\\right)\\subseteq \\mathrm{ker}\\left(f^k\\right) + \\mathrm{im}\\left(f^k\\right)", "46700f2e4d35b76be68c83efd3bbf366": "\\mathbf{A}^{mn}", "467072260237d8cddf4195111de23e39": "\\rho(T) = \\mathbb{K} \\setminus \\sigma(T)", "467097f2f6e31bbd06dbfa218792fb67": " \\mathbf{p} = M\\mathbf{V},\\quad \\mathbf{L} = [I_R]\\omega,", "4670fa2e9cdc7161d4f99144e873e9f8": "x = b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cfrac{a_4}{b_4 + \\ddots\\,}}}}", "467102e9d27b8fe97d947551a3d2a858": "\\scriptstyle x^2 \\,-\\, (kb)x \\,+\\, (b^2 \\,+\\, 1) \\;=\\; 0", "467130464a9e8acc8c4c2924608615f8": "R_e=\\frac{I_1(\\kappa)}{I_0(\\kappa)}\\,", "467139f4c2813566bc62c604674a427a": "\\beta = \\frac{R_o}{R_i}", "46716bbdd53c29ac0e4f42cf789d278d": " x \\ll y ", "46717c22c0e251766389a98e7738ab34": "f:\\kappa\\rightarrow\\kappa", "46717d31fdf28ee52456b4f30ba67c3b": " [16r^2y^4(a^2 - 1) + 1 - u^2]^2 - ", "4671bf394f59f7b599064f9f555d2fa3": "\\mathbf{a}_1", "4671e47d84b921f59742b606bd8b20bf": "t_\\text{S} = \\sqrt{\\frac{G e^2}{c^6 (4 \\pi \\epsilon_0)}} ", "4671fa3b23482a4537b185a2f76327b1": "f_n", "4672a5db1b8af285683bd26d3082ee4d": "\\langle \\phi_y,\\phi_{y'}\\rangle = \\delta(y-y')", "4672acdf8ae7bef37fde3107b46d5b68": "H\\left(\\frac{d^ku}{dt^k}\\right) = \\frac{d^k}{dt^k}H(u)", "4672ced82ce54d415aac30db4e3c09a5": "\\begin{align}\nx(t) \t& = \\iint X_{\\tau}(\\omega) h^{*}_{\\omega}(\\tau - t) d\\omega d\\tau \\\\\n \t& = \\iint X_{\\tau}(\\omega) h( \\tau - t ) e^{ -j \\omega \\left[ \\tau - t \\right]} d\\omega d\\tau \\\\\n\t&= \\iint M_{\\tau}(\\omega) e^{j \\phi_{\\tau}(\\omega)} h( \\tau - t ) e^{ -j \\omega \\left[ \\tau - t \\right]} d\\omega d\\tau \\\\\n\t&= \\iint M_{\\tau}(\\omega) h( \\tau - t ) e^{ j \\left[ \\phi_{\\tau}(\\omega) - \\omega \\tau+ \\omega t \\right] } d\\omega d\\tau\n\\end{align}", "46731b37fc8429834496334bed0bfc60": " m = \\gamma m_0 \\,\\!", "46734c86bb183efb74b869473e258574": "\\hat\\beta_1", "4673884e27a96852be2c8f1be170c329": " x(t) = 1 - \\mathrm{e}^{-\\zeta \\omega_0 t} \\frac{\\sin \\left( \\sqrt{1-\\zeta^2} \\ \\omega_0 t + \\varphi \\right)}{\\sin(\\varphi)},", "4673b0b606294f84fb14ab38677d7f90": " B_\\mathrm{0} ", "4673baa8c7b6807a51e1be6943ed1006": " E_{a,b} = \\frac{ N_a }{ N_b }", "4674163318188ff0de72395789e72345": "\\mathcal{F}_p", "46742feb40f136ee68aaf0b4ebb1cfda": " M_w ", "4674392a2f02314af2c0f3fb969f3d48": "\ne^z = \\frac{1}{e^{-z}};\n", "467471097774f82c08850c165995fd84": "=\\left|\\begin{array}{ccc}\nA_{x} & B_{x} & C_{x}\\\\\nA_{y} & B_{y} & C_{y}\\\\\nA_{z} & B_{z} & C_{z}\\end{array}\\right| = [\\mathbf{A, \\ B,\\ C }] ", "4674746b7e05e7a2eeab5014855a6711": "X_i = \\int^T _0 X(t)\\Phi_i(t) dt = N_i", "467479a357b1999bc15be5c3addc3653": "a_0 + a_1x^1 + a_2 x^2 + \\cdots,", "4674a55778263fcc01883d114e08eca5": "\\mathbf{A}^\\mathrm{H} \\,\\!", "4674fbcc661bf27d6bbdb67b982f0c2d": "K=0.333", "46753d0a1f303a7f526261df8aa738f8": "Y\\sim GIG(r_j,\\lambda_j;p)\\! .", "46755ecb66ace9256bfb2c0d2bae2a9b": "FY\\xrightarrow{\\;F(\\eta_Y)\\;}FGFY\\xrightarrow{\\;\\varepsilon_{FY}\\,}FY", "4675773653557201b63811ae89973c60": "A,", "4675c49e279dde24b6a4c556d49cca0d": "\\mathcal{L}=\\mathcal{L}_{0}(\\varphi)+\\mathcal{L}_{aux}=\\mathcal{L}_{0}(\\varphi)-\\frac{1}{2}(f(\\varphi),f(\\varphi))", "4675fc38382ca9e75e7ba8ec0391d3a6": "Q(x,\\xi) ", "4676703ae7f92f198c34408d4d1b8ba6": "\\sum_{k=1}^n \\frac{1}{k} - \\sum_{k=1}^{\\lfloor\\frac{n}{2}\\rfloor} \\frac{1}{k} =\n\\sum_{k=1}^n \\frac{1}{k} - 2\\sum_{k=1}^{\\lfloor\\frac{n}{2}\\rfloor} \\frac{1}{2k} =\n\\sum_{k=1\\atop k\\; \\text{even}}^n (1-2) \\frac{1}{k}\n+ \\sum_{k=1\\atop k \\;\\text{odd}}^n \\frac{1}{k} ", "467692032b9c449c5b9f9e685e0e6af7": "[0.x_1 x_2] = \\frac{x_1}{2}+\\frac{x_2}{2^2}.", "4676e51fbe1cc2e9abaefb1152b3c4a7": "\\displaystyle{ v_K = c Qv_0}", "4676e94a5923942ddfa8b2f0f80bbc3e": "\\mathop{\\rm Ham}(M,\\omega)", "46773061cd7935e247c27f15853a54e0": "\\ F_T", "46773bdac39660c996dc49dc98702be9": "\\displaystyle D_q(f(x)) = \\sum_{k=0}^{\\infty}\\frac{(q-1)^k}{(k+1)!} x^k f^{(k+1)}(x).", "46773fc039a26f559e0c8f31f77c14a7": "y = mx + b,\\,", "4677631da204a8a55f4b85e5982774b8": "(p,00111,Z) \\vdash (q,00111,Z) \\vdash (r,00111,Z)", "4677af001dc4496cbfb91ffe8b2dedeb": "W = Q_H-Q_a.", "4677c5e4d3d1a7c5f6b6fa347ff4ccac": "|\\varphi'(z)|", "4677e020c7659fe93d3ea4eae376c1ca": "6 ^ x\\,", "4678267ceb89c7213ff75e5cbb8ca74c": " [x_i \\, , p_0 ] = p_i/p_0 ", "467864f5320988b7dacfe18e444d3df2": "\\frac{dI}{ d \\Omega\\ I_{0}}=0", "467873715c59123e0a4fa3b98fc9f489": "(X_n)_{n \\in \\mathbb{N}}", "467884919f8b4fbe633b7fb8151d4f20": "\\mathrm{RO{\\cdot} + RH \\ \\xrightarrow {H-abstraction} \\ R{\\cdot} + ROH \\quad}", "4678e548a64d033c1b59e5d49efc4a88": "\\phi \\left( \\cdot \\right)", "4678ed186257071b9a4bf96fb57d3e20": "(x+a)^2+y^2 = a^2", "4679181ae330627ed1a019808c90740d": " v_{k} = constant \\ \\forall \\ k=0,...,n+1, ", "46792ba3473f6a7aacc67efde260bb89": "\\mu_{20} = M_{20} - \\bar{x} M_{10}, ", "46794c00d0bd19ead9fc8e4f1e4fc1b1": "\\displaystyle{f(z)=g(z,\\overline{z})}", "467a0e13cb3ccc93e2131fa244242e3d": "\\mbox{m}^3 {\\mbox{kg}}^{-1} \\mbox{s}^{-2}\\,", "467a1b725b906ffa9e78602bc79e015c": " \\phi(\\mathbf{x}) = \\int_{\\mathbf{x}'} d\\mathbf{x}'G(\\mathbf{x},\\mathbf{x'})\\rho(\\mathbf{x'})", "467a1c8e1232bf2ab85edf0ba09a9b75": "=>", "467a2163a3c188dd575c109b18580bab": "C=E[ss^T]", "467a3e1bd2190896a18d2b54202f2523": " \\mathbf{A}(\\mathbf{r},t) = -\\mathbf{r} \\times\\int\\limits_{0}^{1}\\mathbf{B}(u \\mathbf{r},t) u du", "467a5aea0fa0a41506c64b430c1384dd": "\\ 1 - \\sin^2(x) = \\cos^2(x) = \\sin^2\\left (\\frac{\\pi}{2} - x\\right )", "467a8f9eb31059ac14d425ed69a520c9": "f''(x)=e^{-\\frac{x^2}{2}} (x^2-1) \\nleq 0", "467b0cfe37e933c6bcb6be6417871b63": "\\vec F_{FK}", "467b113596f49b0bbf053be6a0d46a9d": " E = \\infty", "467b37936f6d9eebb6e73f0419068ed3": "{\\mathfrak p = \\mathfrak t}", "467b3ab3b836fdde7a3797f343bf33d3": " D_N^* ", "467b3c0720cf6e9f5e1ddad3f8811b77": "\\varphi_{X+Y}(t) = \\varphi_X(t)\\cdot\\varphi_Y(t),", "467b83cc50e5636aaaa2800578fef583": "y_{1,t}", "467bb3b08fd7a559ea8cfca6cfefb9ee": "\\int_{|\\xi| \\leq R} (1 - \\frac{|\\xi|^2}{R^2})^\\delta \\hat f(\\xi)e^{2\\pi i x\\cdot\\xi}\\ d\\xi", "467bcdeb8c0023d3d787dd3128b70182": "V(0,S(0))=X(0)", "467bf8472059d40b082af1f63a8898d6": "f\\colon X \\to X", "467c0fd6644d421b773858e39f8bc6c6": "(\\lambda_n^{(c)})", "467cae9d0fa2767031f35b517fe67d40": "5*2^{25}+1=167772161", "467ceb4d7f4094addb026d94597261ec": "|A|=|B|.", "467d1b521572219084cdc1ea4cc842cd": "(A - \\lambda I)^{k_1 - 1}", "467d2866e34f5ea28f935178337da0f1": "\\Gamma^\\lambda{}_{\\mu\\nu}-\\Gamma^\\lambda{}_{\\nu\\mu}\\,", "467d39b97aac7a08b5844f3fe21543bd": "m-n1", "468057cffc83966fa772f1e827086ddc": "\\scriptstyle a^2 + b^2", "46805eef14e93d9dda45c7bf51d73146": " Q_{Fan} = C_{Fan}{{\\Delta}P_{Fan}}^{n_{Fan}}\\,\\!", "4680617559a4686744523c0ab7985582": "CA=NX", "46807a7e8ecd94ad5766452be6a467a6": "{\\rm HbA1c} = {\\rm 0.017} \\times {\\rm Fructosamine} + {\\rm 1.61}", "4680969201d3b89421e93d095df993ae": "\\{u,r,\\theta,\\phi\\}", "4680abec77f98df189983c4680998c99": "\\Gamma_{ij,k}^{(\\alpha)}=\\sum\\partial_i\\partial_j\\ell^{(\\alpha)}\\partial_k\\ell^{(-\\alpha)}", "4680c3d28796bb6a0c3a8f6d0942358b": "\\left|2,1,1\\right\\rangle", "468125c15fa6083ed9450baace7253d5": "a_1\\chi_1+a_2\\chi_2 + \\cdots + a_n \\chi_n = 0 ", "46812d1d5e5067dcc568c821493e5aee": "Var := \\{ x_0, x_1, ... \\}", "4681565236a682ce902b1630628b07a2": "I = mr^2.", "46819da3bb5cd38a8ba97b29155dcdb9": "\\prod_{t\\in\\Lambda}\\lambda(\\mathrm{d}\\omega(t))", "4681a2548e7d1593a1a57a51c043faa9": "\\mathbf{H}_{\\text{Electric dipole}} \\rightarrow \\frac{-1}{Z_0}\\mathbf{E}_{\\text{Magnetic dipole}}", "4681badbf80a615c83be4771c5d0a545": " f \\in BV([a,b]) \\iff V^a_b(f) < +\\infty ", "46820074f78cdc5803e1c748d6082b87": "O(n^c(\\log n)^k)", "46821b354bf261c8aefe14f7dbeb753f": "R_{\\mu\\nu} = 0", "46827e70c06caca626e5e208ca5d4b65": "S<0", "4682a42031cda3de03064a955af7b98b": "\n\\phi_{j,\\ell,m} = \\sum_{i:Y_i\\ge t_j}\\theta_i - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_i\n", "4682b16701f06c1b6c4fbb6e35607e98": " \\sum{_{\\exists\\exists}}(\\mathbf{A},\\mathbf{b})=\\{x: Ax=b, A\\in\\mathbf{A},b\\in\\mathbf{b}\\} ", "4682c368636732a04079d9569add0376": "z\\mapsto z^3", "4682d1e2d26f648128e69120efabc2f4": "q \\Downarrow a", "4682ea748b744d6208e982d032ded00d": "Bu", "4682f602656e0722c23dcf0cc4f2246e": " \\frac{dI}{dx}=-QI \\,\\!", "4683aa8e475008ca1ea09242835c2cfa": "\\lim_{x \\to \\infty} f(x)", "4683c39d6e0931b990cc8212843f22c8": "\\hat E[Y|X]", "4683ed8c7499f07f62241196b35907b3": " \\sigma_\\text{s} = \\frac{ 2 \\pi^5}{3} \\frac{d^6}{\\lambda^4} \\left( \\frac{ n^2-1}{ n^2+2 } \\right)^2", "4683f8aa60a34eaa27bf43f05b3894b8": " A\\mathbf{v} = \\lambda B \\mathbf{v} \\quad \\quad", "468488280ae026da32d3b5d19cfe4734": "{k+\\ell \\choose \\ell}_q = {k+\\ell \\choose k}_q = \\frac{\\prod^{k+\\ell}_{j=1}(1-q^j)}{\\prod^{k}_{j=1}(1-q^j)\\prod^{\\ell}_{j=1}(1-q^j)}.", "4684daf75f5b4602211250a43a1844e6": "\\arctan(y/x)", "468533bbd8604c243f36ee704c27acc0": "\\vec{E}^*", "4685aae019319829cb1d50cbab730841": "x_2'=f_2(x_1, \\ldots, x_n)", "4685c0bc4a748ed18e916ee15de43f2c": "\\begin{align}\n \\nabla \\cdot \\mathbf{E} \\;&=\\; 0\\\\\n \\nabla \\times \\mathbf{E} \\;&=\\; -\\frac{\\partial \\mathbf{B}} {\\partial t}\\\\\n \\nabla \\cdot \\mathbf{B} \\;&=\\; 0\\\\\n \\nabla \\times \\mathbf{B} \\;&=\\; \\mu_0 \\varepsilon_0 \\frac{ \\partial \\mathbf{E}} {\\partial t}\\\\\n\\end{align}", "46860d19cd4c83ddc5473e41f6490a98": "{x_0}^2 + {x_1}^2 + {x_2}^2 + {x_3}^2 = 1", "4686534633681c639f21d743e742f2ec": "I^+(x) \\cap I^-(y)", "46867a140fba3b76712e445f602a9fd9": "f\\left(x_{\\sigma(1)},\\dots,x_{\\sigma(n)}\\right)= \\mathrm{sgn}(\\sigma) f(x_1,\\dots,x_n).", "4686992a0969a2d219ab6d5fce0bb41a": "b'_i = b_i \\oplus b_{(i+4)mod8} \\oplus b_{(i+5)mod8} \\oplus b_{(i+6)mod8} \\oplus b_{(i+7)mod8} \\oplus c_i", "46870657777e6a583043c7c794b53b88": "|\\mu_n(A) - \\mu(A)| < \\epsilon", "46873b7a7b6fad3940c61fb16bd3815a": "P_{F_i}(a_j)", "468784488ddf0ecdeac8bf132b46950c": " U_\\omega (r)= \\frac {1}{\\sqrt{2 \\pi}} \\int V(r,t) e^{i \\omega t}dt", "46878f7e804af57d3074c9ba33159c42": "\\hat{X}|x\\rangle = x|x\\rangle", "4687a8f345bc06455967a3512090e122": "\\begin{cases} \\dfrac{\\partial u}{\\partial t}(t, x) = A u (t, x), & t > 0, x \\in \\mathbf{R}^{n}; \\\\ u(0, x) = f(x), & x \\in \\mathbf{R}^{n}. \\end{cases}", "4687c4def78c049ca8611914939cba76": "{\\mathbb G}_2", "4687d282cbf45feaf5db102c9ec83f0f": " \\psi^{(0)}_\\pm(\\vec{r}_1, \\vec{r}_2) = \\frac{1}{\\sqrt{2}} [\\psi_{n_1,l_1,m_1}(\\vec{r}_1) \\psi_{n_2,l_2,m_2}(\\vec{r}_2) \\pm \\psi_{n_2,l_2,m_2}(\\vec{r}_1) \\psi_{n_1,l_1,m_1}(\\vec{r}_2)] ", "4687e9c985e8083c112d0b3792cb9c69": "{U_{MN}}", "468822a934491213451bde92782d8130": "\\operatorname{Supp}(M)", "46882c4021bc918e8c8909b618370059": " e_{x} = \\sqrt{\\frac{1+Q}{2}} ", "46889a97b2c40b741b28b67c6e6971f7": "1 + |\\{d \\in D: t \\in d\\}|", "4688a3385c7648440c82db2df13ca200": "D_x^2", "4688b5846605c1c5a76955a82fc7db0b": "\\underset{=}{A}", "46890bed01a9bf430e83d0a4a616a71d": "\\left|\\alpha - \\frac{p}{q}\\right| < \\frac{1}{\\sqrt{5}q^2}.", "468925b82f7cd1cdd2843dee24ebeb3e": "(z,z;z_3,z_4) = (z_1,z_2;z,z) = 1\\,", "46894a07e3becaca93d1cafcc06b7825": "\\frac {8\\pi^2}{g^2}", "468975b858a49e1b6ca28354a1c74fa3": "\\hat{H}_{l}=-\\frac{\\hbar^{2}}{2\\mu r^2}\\left(\\frac{\\mathrm{d}}{\\mathrm{d}r}\\left(r^{2}\\frac{\\mathrm{d}}{\\mathrm{d}r}\\right)-l(l+1)\\right) -\\frac{Ze^{2}}{r},", "46898a025a1ba704a1ae647956c58b34": " \\arcsec x = y \\, ", "468a128c7d4c9af4f9de1da8d0fa1f47": "\\mbox{Times-Interest-Earned} = \\frac {\\mbox{EBIT or EBITDA}} {\\mbox{Interest Charges}}", "468a3b94f1bcddaca897941b2406384a": "\\bold{F} \\cdot \\bold{\\delta} \\bold{r} \n= - \\bold{\\nabla} V \\cdot \\displaystyle\\sum_i {\\partial \\bold{r} \\over \\partial q_i} \\delta q_i\n= - \\displaystyle\\sum_{i,j} {\\partial V \\over \\partial r_j} {\\partial r_j \\over \\partial q_i} \\delta q_i\n= - \\displaystyle\\sum_i {\\partial V \\over \\partial q_i} \\delta q_i. ", "468ae2ab9b7405dd0ae7bc86a479d93d": "x(t) = \\cos(t),\\,", "468aef7ff2d9646451f5bd413b6f3c90": "2^in - 1, 2^in+ 1", "468af2096d096011d2f2ba0ba1974d61": "BD = R \\left(A - 2 \\right) + T \\left(kA - 2 \\right)", "468af6df9439bed901ce65836004c6e7": "H\\, =\\, 2\\, a \\qquad \\text{and} \\qquad a\\, =\\, \\frac12\\, H,", "468b05cbf1d1f7858374fe4bcc093c8e": " \\langle \\xi | \\eta \\rangle_{\\mathrm{graph}} = \\langle \\xi | \\eta \\rangle + \\langle A^* \\xi | A^* \\eta \\rangle ", "468b2ad0bde568726abc48a7a2c667df": "p \\leftarrow \\mathrm{not}~\\neg p", "468b44d1a5357801196b02531141e6c3": " \\widehat{q}|q\\rangle = q |q\\rangle", "468b9264bd8652066273542220abbb29": "f_{i}(y) = \\exp \\{-(y,\\, A_{i}\\, y)\\}.", "468bcbefe1de0bf9a4a557dd1466a39e": "\\eth f \\stackrel{\\text{def}}{=} P^{-s+1}\\partial (P^s f) ", "468bed6813a71393c20f2426a4d357ea": "f = \\frac{1}{2 K}, \\frac{3}{2 K}, \\frac{5}{2 K} ...", "468bf762369aa9f9eea21e9097d50447": "\\text{If }P\\text{ and }Q\\text{ are logically equivalent, then }K*P=K*Q", "468c509c49aaaa68645ad84f8f43a695": "-\\frac{\\bar{h_i}-h_i^\\mathrm{gas}}{R}=-\\frac{h_i^*-h_i^\\mathrm{gas}}{R}", "468c803e83f5111d5581c36a929b1ddc": "P/bD = \\sigma_N", "468c8aae4e82213f396e69981290c69c": "P(x) = |\\psi(x)|^2.", "468d317440541a3c5bdd2f36529351d9": "\\widehat{\\vec a} = (\\mathbf{X}^T \\mathbf{X})^{-1}\\; \\mathbf{X}^T \\vec y. \\,", "468d3fd83214cf86715e5d332213afb4": " \\mu (E) ", "468d55a830a61a25d60040342359376d": "\\Omega = D\\omega = d\\omega + \\omega \\wedge \\omega", "468d8891c2f6f2f9a069315077085856": "s = 2 D_\\mathrm N", "468dccd0ee802bb50979dcc782a3ddfd": "\\left(2+\\frac a2\\right)a\\arccos\\frac1{\\sqrt{-a}} \n+ \\left(1-\\frac a2\\right)\\sqrt{-(a+1)}.", "468e0439dca4a2f9f2d599cf6fa02160": "B_{\\mu\\nu} = D_\\mu B_\\nu - D_\\nu B_\\mu\\Big.", "468e47859d9381aea0991b3fb7c7fe4f": "({\\and}L_1)", "468e9150bf83c91b9203c98b795a6245": "\\phi_g", "468eab971a131f25a22d16a6d4f9915d": "P_m = 0.01 (V_h)^2\\,", "468eea0653031f73d3d38eac4bf90081": "\\phi={(n_2-1) \\over R}.", "468ef2496873ee886378cdf83b86793c": " ( \\nabla^2 + k^2 ) A = 0 ", "468f6832342fe95a954aa5051d613b66": "=\\det(\\Sigma)^{-n/2} \\exp\\left(-{1 \\over 2} \\sum_{i=1}^n \\operatorname{tr}((x_i-\\overline{x}) (x_i-\\overline{x})^\\mathrm{T} \\Sigma^{-1}) \\right)", "468f87fabd01f76e677207f4abce0b64": "\n-\\hbar^2\\frac{\\partial^2 \\psi}{\\partial t^2} +(\\hbar c)^2\\nabla^2\\psi = (mc^2)^2\\psi \\,,\n", "468f8859dbd1dba04ce6937841bb0033": "n \\left(x,t \\right)=n_0 \\mathrm{erfc} \\left( \\frac{x}{2\\sqrt{Dt}}\\right)", "468f9fd7b806d310c40bcea588da8f72": "|G|=n=p^km", "468fb01b98417d696b93e610d9a55402": "\n a_c^3 = \\cfrac{9R^2\\Delta\\gamma\\pi}{4E^*}\n ", "468fb2459fdc067a3ba271e9d8078b49": "\\bar{\\gamma}", "468fd86c9c6a5c12d168f7b4e3060f49": " p_m =1 \\Leftrightarrow m=0 ", "468fdc9b5d9af36f66c494ba3bae55ff": " N_D = \\frac{(\\epsilon_0 k T_e)^{3/2}}{q_e^3 n_e^{1/2}} ", "469010a416761c02b1b4baaa491ed207": "k(\\omega)", "46903155f9f9759aa856fa3291825ccc": "\\operatorname{str}(A^{-1} T A)=(-1)^{|i'|} (A^{-1})^{i'}_j T^j_k A^k_{i'}=(-1)^{|i'|}(-1)^{(|i'|+|j|)(|i'|+|j|)}T^j_k A^k_{i'} (A^{-1})^{i'}_j=(-1)^{|j|} T^j_j\n=\\operatorname{str}(T).", "4690699eae3a1830f7d14953e07d22f5": " \n\\begin{bmatrix}\n\\mathbf A, & \\mathbf B\n\\end{bmatrix}\n^{+} \n= \n\\begin{bmatrix}\n(\\mathbf P_B^{\\perp}\\mathbf A)^{+}\n\\\\ \n(\\mathbf P_A^{\\perp}\\mathbf B)^{+} \n\\end{bmatrix}, ", "4691752ac362d8dbb31cf450d79d5ad6": "P_n=\\frac{n(n+1)(2n+1)}{6}", "46919904a094436323ec480dc5ec22aa": "T = \\forall X. X \\to X", "4691acc035752546758ac6d38f8fb572": "\\xi,\\xi_1,\\xi_2,\\ldots", "4691c99e54893876d62afd46c91b9413": "\\mathbb{R}^n \\rightarrow \\mathbb{R}^m", "4691d04b587eab6facd4d5d6a6e9bdef": "1 {\\rm Td} = 10^{-17}\\rm V\\cdot cm^2 = 10^{-21}\\rm V\\cdot m^2 = 1\\rm zV\\cdot m^2", "4691ed9de9d8a10f17ce3485dcd364f4": "C_{00}=0", "469269105582ffef07597cce6e49bc83": "Q(x) = 1 - Q(-x) = 1 - \\Phi(x)\\,\\!,", "4692cf3c6c23a5488e664a8e7916f13e": "\\dot {Q}_m", "4692d8b8d51b6498e9cd62e84c940d8b": " \\phi_B^{-1}(x, y) := \\left( (x,y) - (x_0,y_0)\\right) \\begin{pmatrix} u_x & u_y \\\\ v_x & v_y \\end{pmatrix}^{-1} ", "4692df6b9c5cc8c45e1c0040f3af1131": "t_R \\,", "4692e55d8422219522c45b77402f4284": "\\mathrm{O}(n,F) = \\{ Q \\in \\mathrm{GL}(n, F) \\mid Q^\\mathsf{T} Q = Q Q^\\mathsf{T} = I \\}", "4692ed43236d46f26d18e0c5b5b4c0b0": "\\lambda = (\\lambda_1, \\dots, \\lambda_n)", "4692fa610159a3e491e1f6bdcd937594": "\\mathrm{Cov}(x-\\mu)=\\mathrm{Cov}(LF+\\varepsilon),\\,", "4693144b426691d3e1e5e98dff8cdba4": " \\text{The event horizon } H_e \\text{ exists if and only if } m<2", "46938f1d5761269e1606d89d8a4dd497": "\\left(\\frac{\\partial S}{\\partial V}\\right)_{T} =\\left(\\frac{\\partial P}{\\partial T}\\right)_{V}\\,", "4693b52f78cf4b1151f3b27c9593fd3b": "\nI =\n\\begin{bmatrix}\n \\frac{2}{5} m r^2 & 0 & 0 \\\\\n 0 & \\frac{2}{5} m r^2 & 0 \\\\ \n 0 & 0 & \\frac{2}{5} m r^2\n\\end{bmatrix}\n", "4693c5b8f6e9945e9dc4845cb298f0f3": " \\{y_t,t=1,\\ldots,\\infty\\}", "4693f75391950b972cf702b94c0af763": "U_{i}", "4693f7e4615f5107d25bf5cb825c9045": "\\mathfrak{s},\\mathfrak{t}\\in M(\\lambda)", "46943954fdbed5790346c3bab6fcc5c0": "N/m-1", "46944cdfae13c550f6d25d6822abd41e": "s=h(x_1,\\ldots,x_m)= h(g_{\\boldsymbol\\theta}(z_1),\\ldots, g_{\\boldsymbol\\theta}(z_m))", "46949b8678bd4198bcdb666d88e70e98": "b_i=g_i \\oplus b_{i-1}", "4694d82783e4710a4785b5a48b864147": "r_3 = \\frac{r}{2(s-c)}(s+f-r-d-e).", "4695148e88ffd816cc3fe0d2fef7c0e6": "\\sum_{k=0}^\\infty {k+n-m \\choose k} \\left[\\zeta(k+n+2)-1\\right]", "4695510eac55d6144447ea3d318dc2dc": " \\{ s_v : v \\in \\Lambda^0 \\}", "4695822c099b2a0728cc45a87a34b893": "t_m^2 - D u_m^2 \\equiv N^m", "4695d8ca1bff9fbc056be82ea1722b5c": " E_1 = \\frac{ I - I_{ min } }{ I_{ max } - I_{ min } }", "4695f35e38c8f98dcf4841833ffb69ff": "(p - p')", "4696203bfcfa20729ea6b364551194fb": "\\left(\\frac a {3^n}, \\frac{b\\sqrt 2}{3^n}, \\frac c {3^n}\\right)", "4696925bcaa21f622cd50062c4a6d8ef": "A_{n-k-1}B_1A_k", "4696aac287c68bc85ee59244bc77dc35": "\\bigcup_{a \\in A, b\\in B} (a \\star b)", "4696af08aacdc8ca12c56f08eb398d60": " J(\\omega) \\propto \\omega^s ", "4696c95ac461967ff0945475ac6cbba3": "\\leq p", "4696ed7a0ed6d0d73891eaabccde1906": " \\forall x \\in A, \\ \\exists y, \\ x \\in y ", "4696ee8b79fc123ae702de0848cec542": "{ 2\\ln\\left(2\\right) }", "46970414452653b8216cbbe48d43c8f8": "S_{n+1} = 2\\pi V_{n}", "46970968ce5f35d8cfc98a971594eb4f": "\\beta_r=\\beta_{ri}", "4697232e4c4e17201b17f8acf382948d": "b\\,M_{solute}=\\frac{m_{solute}}{m_{solvent}}=\\frac{w_{solute}}{w_{solvent}},", "469784d80e8a1b03416a998ad562cfef": "X \\subset R^n", "4698542752e65c96b22242d3374ed032": "\\int_{V^e} \\left[ \\frac{1}{2} \\epsilon^T C \\epsilon - \\sigma^T \\epsilon + \\sigma^T (\\nabla u) - \\bar{p}^T u \\right] dV - \\int_{S_\\sigma^e} \\bar{T}^T u\\ dS", "46986acdb8ef4833a29976773eccda77": "H(A|B)\\,=\\,H(AB)-H(B) \\, .", "46992524b7c6c5ca70fa17dad0cbb6aa": "\\scriptstyle {(\\bull \\downarrow \\mathbf{Set})}", "469931f9792228b3c8b2e3bd83ebb51a": " h\\in H_1\\setminus \\{w_{1,\\beta_1}^{-1},1\\} ", "46996390004cd983bdf147079c976def": "(a_n, b_n, c_n) = (a_{n-1}+b_{n-1}+c_{n-1}, \\, F_{2n-1}-b_{n-1}, \\, F_{2n})", "469990b7f2714da902adf44ebe8443d6": " m_1 = c_2, m_2 = b_1 \\cdot c_3", "4699a7c803e6327b69ec9cfe62ca62ce": "0 \\to K \\to M \\overset{x_n} \\to M(d_n) \\to C(d_n) \\to 0", "4699acea076e3edffb6e3491f7e69159": "\\cos(A - B) = \\cos(A)\\cos(B) - \\sin(A) \\sin(B)\\,", "4699cf25c93a42bacbd5558f5f76eba9": "C_{\\frac{1}{2}}", "469a42db8ced961bee4df87e34616ebb": "\\lim_{n\\to\\infty}t_n = e^x. \\, ", "469aadb17c1aa4dbac7f8a908c384cd9": "i_{C}(t) = C \\frac{dV_{C}}{dt}.\\,", "469aff9d7314886b8826392b432bd600": "dR = \\cos\\theta\\,d\\theta", "469b64869afe0670793f469ccdf3cefd": "C_{low}", "469b76224c59ae96f3c8742a630ed8d8": "\\displaystyle{Uf(x)=\\pi^{-1/2} (x+i)^{-1} f(C(x)).}", "469c49b200a7d10ef6901aaa5c56b1d0": "j_1(x)=\\frac{\\sin(x)} {x^2}- \\frac{\\cos(x)} {x}", "469c50bc59e82fb1ad34a9cfa3e6d536": "(Tf)(x)=\\int_{\\mathbb{R}^n} e^{2\\pi i \\Phi(x,\\xi)}a(x,\\xi)\\hat{f}(\\xi) \\, d\\xi ", "469cab7d42b888b0aaf55839e5f9ce63": "R\\widehat{\\mathcal S} : D(\\hat X) \\to D(X). \\, ", "469cd4a6d28a27ff9d293a9a7910974d": "\\sqrt{2}n < m < 2n", "469d3f25449fb6533981c49029b7929d": " t(z) = \\frac {2}{3 H_0 {\\Omega_0}^{1/2} (1+ z )^{3/2}} \\ , ", "469d679c79a1749c4b3062aff31b21eb": "B_{n,k}(x_1,\\dots,x_{n-k+1}) = {x_{n}^{k\\diamondsuit} \\over k!}.\\,", "469d77b1d6207a60270d146e335a7a22": "n > l", "469dbf3dd21841e839f951edb1651570": "\n\\widehat{H}_\\alpha =D_\\alpha (-\\hbar ^2\\Delta )^{\\alpha /2}+V(\\mathbf{r},t). \n", "469dea7e4d51775176cba4159222ad92": "\\rho=\\left\\vert 0\\right\\rangle \\left\\langle 0\\right\\vert\n", "469e414324a723744b7b3c033331deac": "a(n) = a(n-1)+(n-1)a(n-2) \\, ", "469e6c5ee0494c7d02f44fc2f537ce2d": "p = K\\exp(-r_d T)\\N(-d_2) - S_0\\exp(-r_f T)\\N(-d_1)", "469ee15776760c87e2607b34f972bcbd": "SM_n(gevol,endo,exo)", "469f31a3550a856faee65acbaf8f06c6": "\n\\mathbf{C}_{\\alpha \\beta} = \\mathbf{A}_{\\alpha \\gamma}\\mathbf{B}_{\\gamma \\beta}. \n", "469f397dbf4ca26f9d8751bc1d8baa40": "(I,I, \\ldots, Y_{r-1}^{-1}, I)", "469fe20a0bf4ebd326af8ddcae9898df": "\\hat{v_i}\\equiv i[\\hat{H}_0,x_i]=\\alpha_i", "469ff4d164e5cddff005f362a4e31210": "F:A\\times B^\\mathrm{op}\\times B\\rightarrow D", "46a02647273637c22a5e06f2c2cb0fce": "\\Phi_1 + \\Phi_2 + \\ldots = 0", "46a02eedb2c83e8570da8d7c490a681d": "(x'', f'')", "46a06cd164f8a0119f7ee5b785c1c801": "(t / (t -2))", "46a0800ea2d07230928becdef2678d23": "\\displaystyle{\\partial_t^m F(t,x)|_{t=0}=f_m(x).}", "46a09174e38e5c99955cd3f9e65fa9ea": "\\ n\\,", "46a0c189b99d8854e6ad858776303dda": "g_{ij} = \\left\\langle \\frac{\\partial\\vec\\Psi}{\\partial x^i} ; \\frac{\\partial\\vec\\Psi}{\\partial x^j} \\right\\rangle", "46a0c29ff15d833cd0975b4dc2b1b09d": " \\Delta A = 0 ", "46a0d8ed324f853c71fff39f76fcd7fb": "k\\lambda_{De}", "46a123363a90ba4afbbb842d90676de0": "(d)b + (d-1)b^{2} + \\cdots + 3b^{d-2} + 2b^{d-1} + b^{d}", "46a1483cbe97be2555a92bb3135cefe4": "\n\\ddot{\\theta}+\\theta\\left( \\frac{kL_{e}^{2}+\\frac{\\mu BH_k}{(B+H_k)}}{mL_{e}^{2}}\\right)=\\ddot{\\theta}+\\theta\\left( \\omega_o^2 +\\frac{\\mu BH_k}{mL_{e}^{2}(B+H_k)}\\right)=0,\n", "46a17ba154493545b11cdcaebfbd148c": "F(x, y, p) \\geq a(x)\\cdot p + b(x)", "46a1bd2b898583ae3b3f220b84e3b6d2": "J/k_{B}", "46a240f910be79c56c61b5ec9c83721c": " FF = {Q_n\\over\\sum_{k=1}^N TH_k} ", "46a241f09c3dab9207a1c584f31ce283": "\\sum_n \\mathcal{L}(n) = \\sum_{n=m}^\\infty \\frac{1}{n} = \\infty", "46a25e4a247bc560dd2411500e420ee0": "\n\\sqrt{n}\\left(B-\\beta\\right)\\,\\xrightarrow{D}\\,N\\left(0, \\Sigma \\right),\n", "46a2ef9aab6875f53470324cb56375cb": "d [x] = x \\wedge (1-x)", "46a306b766a24e9693b1482de90a7ce2": "2+\\sqrt{3}\\,", "46a33f68c1827fe7eadedb5ab2033187": "M(a;b;z)", "46a3469d8bf7b202d18c5ba91b0a6068": " \\int_0^\\infty x\\phi(x)\\Phi(bx) \\, dx = \\frac{1}{2\\sqrt{2\\pi}} \\left( 1 + \\frac{b}{\\sqrt{1+b^2}} \\right) ", "46a3dbbfd7fbada5e09c14e3d6fa0478": "\\frac{\\partial \\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y) }{\\partial a} = -(\\alpha - 1) \\sum_{i=1}^N \\frac{1}{Y_i - a} \\,+ N (\\alpha+\\beta - 1)\\frac{1}{c - a}= 0", "46a42e908af22123df81894de995e7ce": "V^{(2)} = \\frac{\\pi h_1^2}{3}(3r_1-h_1)+\\frac{\\pi h_2^2}{3}(3r_2-h_2)", "46a43a299e641ccd4be2f317fddacf35": "\n \\hat{y} = X\\hat\\beta = Py,\n ", "46a4665c2d7684406ee27ea25327d855": " \\scriptstyle \\mathbf{\\Phi}(\\mathbf{r}) ", "46a49a3051bc558c0398f5b553646e08": " \\frac{dr_i }{dt }=0", "46a4b40f7081859e0aa35f58805c5482": "\\alpha \\subseteq R", "46a4dfbc2f9ba116670acfc42f530883": "\nL = D - A.\n", "46a508a17d0f2ba5fbe427b1e9416a74": "\n\\sigma^2 = \\max \\left\\{ \\bigg\\Vert \\sum_k \\mathbb{E}\\,(\\mathbf{Z}_k\\mathbf{Z}_k^*) \\bigg\\Vert, \\bigg\\Vert \\sum_k \\mathbb{E}\\, (\\mathbf{Z}_k^*\\mathbf{Z}_k) \\bigg\\Vert \\right\\}.\n", "46a51b32ce0fe6f092f46256f6a0ac9a": "R_f(-\\tau) = R_f(\\tau)\\,", "46a52849158eb95c7c12e698337c0e42": "\\ell = {q + q^9 + q^{25} + \\cdots \\over 1 + 2q^4 + 2q^{16} + \\cdots}.", "46a52b217f1759c791d24157f97e99ed": "x(t)=x_h(t)+x_p(t)=C_1.e^{ -B.t}+C_2.e^{ -A.t} +\\frac{Q_t.e^{-At}-R_t.e^{-Bt}}{P}", "46a5b9cd6c6296b97085e7267af53d87": "0 < y < 1", "46a6c059ed35d0d779ab30b2cda9b550": "\\Theta_{n,m}(\\tau, z+b\\tau) = q^{-b^2} u^{-b}\\Theta_{n,m}(\\tau,z), \\qquad b \\in \\mathbf{Z}", "46a6c146df4bd85a41e41ba2ec02d227": "W_L \\ ", "46a6c4d715584adb3e6681ee351d1df6": "2\\pi ", "46a6e071d7c25a66cdab06423ff4b229": "\n\\widehat{B}_y \\equiv \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_y) = \\begin{pmatrix}\n\\cosh\\varphi & 0 & \\sinh\\varphi & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n\\sinh\\varphi & 0 & \\cosh\\varphi & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix} \\,,\n", "46a709a10d76468176f012efc80187fa": "\\mathbb{Q} \\left( \\zeta_p \\right)^{+}", "46a7197971546b9001eef24c5164e9ac": "\\lim_{n\\to\\infty}\\left(1+\\frac{1}{n}\\right)^n,", "46a7a7fac0ef19ce910dc99cb1a591f7": "dU=\n\\left(\\frac{\\partial U}{\\partial S}\\right)_{V,\\{N_i\\}}dS+\n\\left(\\frac{\\partial U}{\\partial V}\\right)_{S,\\{N_i\\}}dV+\n\\sum_i\\left(\\frac{\\partial U}{\\partial N_i}\\right)_{S,V,\\{N_{j \\ne i}\\}}dN_i\n", "46a7bc8e54939f44560fd00a030c4abb": "S_{D,x}", "46a7c06825bf7766c21d0fa74de2d752": "f^{low} = \\ ", "46a8d75e67cf291d982baae8ab1257d9": "n=1/c.", "46a9902590256404cb8bd01a83755517": "w(x) = \\alpha(x)", "46a998e647a67e096c392f5813865b7f": "{\\Bbb P}(V)", "46a9fb09ff9ff6f6347334c35ae7f494": "\\forall x \\lnot P(x)", "46aa024aeb03e78ec33dd21298ac3899": "\\omega^2 = g^\\prime k", "46aa0b43ac763ea6758ab12bc965d945": "\\Delta E.", "46aa1c42386eb18d044179aa752b0b5d": "\nx = b!\\,\\biggl(e - \\sum_{n = 0}^{b} \\frac{1}{n!}\\biggr)\\!\n", "46aa2956016c7355e7ff916ed010a850": "a_{ij} = \\Pi_{ijpq}\\sigma_{pq} \\,", "46aa65f77a191ae8657207ebbc15b67e": "c^k(\\ell,m,\\ell',m')=\\int d^2\\Omega \\ Y_\\ell^m(\\Omega)^* Y_{\\ell'}^{m'}(\\Omega) Y_k^{m-m'}(\\Omega)", "46aaa68abeb4605cc3264ead4505d485": "\\tfrac{4 \\times 18}{17 \\times 18}", "46aaaef3531b644439731842d8c24391": "\\left[a_0; a_1, \\,\\dots, a_n\\right]=\\frac{h_n}{k_n}.", "46aacd306494d5c21759639dfb4f7839": "x^2-t", "46aadf0d3b9dabbc896185ff6653dfd9": " W = \\bigcup_{k=0}^{\\infty} W^k . ", "46aafe4131bc141b7cf501a0afdd986e": "U^{(k)}", "46ab0a21a70e7c3e1914f50ede8db762": " y = (y_2,\\alpha_2 y_2)", "46ab1b3af3a468e42786a3755a06ea02": "\\Delta A= \\frac{V_{ac}}{1.1515V_{dc}\\delta_0\\sin\\omega t} ", "46ab1b8b5f56a4c3e742480b348533ba": "\\mu_s - \\mathbf{B}\\mu_m", "46ab2493b010bfe3e809f1ad0c6372a9": "|\\phi_{m}\\rangle = \\sum_{x \\epsilon A_{\\epsilon}^{(n)}}\\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|x_{1A}x_{2A}...x_{mA}\\rangle| x_{1B}x_{2B}...x_{mB}\\rangle\n", "46ab40009f121b1745da5e9352f752a6": " \\frac{1}{r} \\ e^{-m_{W,Z} \\ r}", "46ab585d46826afa81c3373ed72afcf4": "T\\Delta S_p", "46ab847bad68ea124ea72f0b80d31383": " \\vec{G} = \\beta g_0(T-T_{avg})\\vec{k} ", "46abd4a0762b746f4353bf8a2a42263d": "G(k) = \\frac{1-e^{-k}}{1+e^{-k}} \\mbox{ for } k\\geq 0. \\!", "46abd9754916b3993004639290984f56": " {\\delta^*}= \\int_0^\\infty {\\left(1-{\\rho(y) u(y)\\over \\rho_0 u_0}\\right) \\,\\mathrm{d}y}", "46ac0786c68c76abb4df7aa417df1ec6": "B \\to SS~~~~~~~~~\\text{bifurcation with a probability of 1}", "46ac0ef9c1e754a26738a7503f2f3885": "\\operatorname{Ext}^n_R (k, R)", "46ac1714747ed2401d2ad405974a634e": "\n\\begin{align}t \\cos{t} &= y_p'' + y_p \\\\\n&= [(A_0 t^2 + A_1 t) \\cos{t} + (B_0 t^2 + B_1 t) \\sin{t}]'' \\\\\n&\\quad + [(A_0 t^2 + A_1 t) \\cos{t} + (B_0 t^2 + B_1 t) \\sin{t}] \\\\\n&= [2A_0 \\cos{t} + 2(2A_0 t + A_1)(- \\sin{t}) + (A_0 t^2 + A_1 t)(- \\cos{t})] \\\\\n&\\quad +[2B_0 \\sin{t} + 2(2B_0 t + B_1) \\cos{t} + (B_0 t^2 + B_1 t)(- \\sin{t})] \\\\\n&\\quad +[(A_0 t^2 + A_1 t) \\cos{t} + (B_0 t^2 + B_1 t) \\sin{t}] \\\\\n&= [4B_0 t + (2A_0 + 2B_1)] \\cos{t} + [-4A_0 t + (-2A_1 + 2B_0)] \\sin{t}. \\\\\n\\end{align}", "46ac444d60f417e178a9a632d957c624": "\\top_{\\mathrm{H}_0}(a, b) = \\begin{cases}\n 0 & \\mbox{if } a=b=0 \\\\\n \\frac{ab}{a+b-ab} & \\mbox{otherwise}\n\\end{cases}", "46ac4d6e2a90e37b8b7e10af21438d77": "I_{\\text{B}} = 0", "46ad2215f730f9f817c9b74e076a9c4c": "c_7 = 3.45372\\times 10^{-4}, \\,\\!", "46ad823118ccd62b5df43f7e012cdc32": " \\alpha, \\lambda, \\rho ", "46add94142bbd96ff45c155e42ec0b90": "{}^{9}i = i^{\\left({}^{8}i\\right)}", "46adf36482babcf6e2e3716e6f5fb62c": "\na^{p-1}\\equiv 1 \\pmod p\n", "46ae15374518756d615a2c3448e3b42b": " \\Xi_v = \\exp { \\sum_{t=1}^\\infty \\frac{z^t}{t!} \\int \\left [ \\prod_{i = 1}^{t} \\psi^v_i \\right ] {\\mathcal U}^{(t)}_{1...t} d\\boldsymbol{r}_1...d\\boldsymbol{r}_t }, ", "46ae3ab099d8cd0a63da85e411c45895": "\\operatorname{E}\\left[f(X)\\right]\\approx f(\\mu_X) +\\frac{f''(\\mu_X)}{2}\\sigma_X^2", "46ae4859dc6d83278bc5e4ecea3c54fc": "(Z_1 \\vee \\cdots \\vee Z_n) \\wedge\n(\\neg Z_1 \\vee X_1) \\wedge (\\neg Z_1 \\vee Y_1) \\wedge\n\\cdots \\wedge \n(\\neg Z_n \\vee X_n) \\wedge (\\neg Z_n \\vee Y_n). ", "46ae988a550fb67ef87dbcf15e313587": "= \\operatorname{E}[\\operatorname{E}[XY\\mid Z]] - \\operatorname{E}[\\operatorname{E}[X\\mid Z]]\\operatorname{E}[\\operatorname{E}[Y\\mid Z]]", "46af2228338c63332cc80d6daec20696": "R\\ge1-H_q(\\delta)-\\epsilon", "46af4670c396a3822d70c04ce30e8e44": "v_i=-\\left(\\sum_{j=1}^N D_{ij}\\mathbf{d}_j + D_i^{(T)} \\nabla (\\ln T) \\right)\\, ;", "46af538fb401949fcb8cb336b532741d": "\\begin{align}\n V &= |V|e^{j(\\omega t + \\phi_V)} \\\\\n I &= |I|e^{j(\\omega t + \\phi_I)}\n\\end{align}", "46af9d7b48eaad680c57ff5c59950c3a": "\\!\\,\\|N\\| =\\sqrt{(b-y)^2+x^2}=m", "46b0269ea0741b0e2371e618f61f1a08": "\\left [\\begin{smallmatrix}\n1&-1&0&0&0&0 \\\\\n0&1&-1&0&0&0 \\\\\n0&0&1&-1&0&0 \\\\\n0&0&0&1&1&0 \\\\\n-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&\\frac{\\sqrt{3}}{2}\\\\\n0&0&0&1&-1&0 \\\\\n\\end{smallmatrix}\\right ]", "46b026a4b244d4264d9735dffe9c0851": " r_\\mathrm{ corr } = \\frac{ r }{ ( 1 + \\theta c_{ xy } )( 1 + \\theta c_x^2 ) } ", "46b0626b7080d37f658db00c7ff0bad0": "R_{Th} = R_{No} \\!", "46b11776c3fa86746ba6c01a81463321": " 0 \\leqslant \\left( 1 - \\frac{L}{R_1} \\right) \\left( 1 - \\frac{L}{R_2} \\right) \\leqslant 1.", "46b1e893138737552828dc10284dfc34": "y^2+by-a^5 = 0\\,,", "46b278a2095c65c96e7917ee8b4cb814": "\\sum_{-\\infty}^\\infty", "46b2e2a49ed127128d3378402381ddc4": "f'(c+):=\\lim_{h \\to 0^+}\\frac{f(c+h)-f(c)}{h}\\le0,", "46b3432c8c867638da6ea718bb46a3b6": "h^n = \\begin{pmatrix}1 & n \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix}1 & 0 \\\\ 0 & 1 \\end{pmatrix} .", "46b348799ceaa4b3d238cba17ee7d412": "\\gamma_{\\alpha \\beta} = -g_{\\alpha \\beta}.\\,", "46b36047a6dee66ee155a323b4cebb1b": " (A \\downarrow I)", "46b3646187e8c163dbc1e8e9f7f9d998": "[m,n]", "46b3c1be81df3b90f68166a180aa6745": "\\Delta \\phi=\\frac { 2 \\pi c \\Delta t }{\\lambda} ", "46b3d3db4df165ebfe3d8bcc8297ecc8": "\n\\begin{align}\n\\frac{d^2F(P)}{dP^2} & = \\frac{dF'(P)}{dP}=\\frac{F'(P_1)-F'(P_0)}{dP}, \\\\[10pt]\n& =\\ \\frac{dG(P)}{dP}=\\frac{G(P_1)-G(P_0)}{dP}, \\\\[10pt]\n& =\\frac{F(P_2)-2F(P_1)+F(P_0)}{dP^2}, \\\\[10pt]\n& =F''(P)=G'(P)=H(P)\n\\end{align}\n", "46b3f9ff2357f2c2491688efb0a8ebf7": "(\\sigma_k)_p(x)=\\frac{1}{1-(1+p^k) x+p^kx^2}.", "46b435cb0c588d996b529875636249bb": "0<|x-a|<\\delta", "46b46e2f3c6a9404f6da98132c7cbfa2": "c_{2n} = c_n^2 - 2c_n^p", "46b47abfe18f9089c212af1e3ebdfda5": "\\{(x_{1,i},x_{2,i})\\}_{i=1}^n", "46b49be158c1295d12ec44cdcd497211": "\\mathit{GL}_n \\neq SL_n \\times K^*.", "46b50b99c6eb953d5231321d4658899a": "c_{2k} = \\frac{(2\\pi i)^{2k}}{(2k-1)! \\zeta(2k)} = \\frac {-4k}{B_{2k}} = \\frac {2}{\\zeta(1-2k)}.", "46b51661b6ce2b5eef4621e6567f7b97": "x \\in [0,1]\\,", "46b5254e80d9c8e891540411c3c7a5a0": "(\\hat{n} \\cdot \\vec{\\sigma})^{2n} = I \\,", "46b5dab7c50daa3ad8adfd98e1a8d95c": "Q=Q_1^T Q_2^T=\\begin{pmatrix}\n0.8571 & 0.3943 & -0.3314 \\\\\n0.4286 & -0.9029 & 0.0343 \\\\\n-0.2857 & -0.1714 & -0.9429 \\end{pmatrix} ", "46b61e84b7ded7f5b79f3ccd5d271364": "\\{k x + l y : k, l \\in K\\}", "46b664333e24988811dacf5f3f4d9c0e": " \\mathbb{}H_*(C_f,pt)=0", "46b664b35f8134357e9e9b9f87e29f55": "(\\zeta, \\xi, \\phi)", "46b6aa4e39c50eea6e09a8831965e121": "(A, \\alpha)", "46b6cb350a54669d2be45d9f645d65d6": "g(p_{3},p_{4})", "46b6edcd719048dc0be2808bd94509c7": "\\alpha_r, \\beta_{rj} \\geq 0", "46b6fde9229d543e1453e20bcf6fd2f0": " \\Delta u = O(k^2)+O(h^2). \\, ", "46b708709f4b73238bcc6d998e626e5e": "[.,.]:\\Gamma E \\times \\Gamma E \\to \\Gamma E", "46b71b7c0649e8a278baaf511690ca9d": "M_l^d (S)", "46b748c947588603ec9b9a725c412f7d": " Q = \\int \\Phi \\mathrm{d} t ", "46b77bd5b5a0e546b055cf0dc8f8f652": "\\begin{align}\n\\operatorname{I}_{\\mathbb{P}^n}(S) &= \\{ f \\in R_+ | f = 0 \\text{ on } S \\}, \\\\\n\\operatorname{V}_{\\mathbb{P}^n}(I) &= \\{ x \\in \\mathbb{P}^n | f(x) = 0 \\text{ for all }f \\in I \\}.\n\\end{align}\n", "46b7a52c4bf88c1c45ab2adbefa317db": " \\Delta \\epsilon =\\epsilon_L-\\epsilon_R\\,", "46b7e635c37beb1bc9f46f8bd99de1d8": "v_1\\in V\\,", "46b82365df1767cf26267d6d6c854af4": "\\rho_k(x_{\\sigma(1)},\\ldots,x_{\\sigma(k)}) = \\rho_k(x_1,\\ldots,x_k)\\quad \\forall \\sigma \\in S_k, k", "46b83124ac6b0ff603ca6ecc77cb463e": " \\frac{2\\varphi}{\\log 2} \\approx 4.669", "46b8fb2476efd78740ab985e8f783baf": "\\hat{\\textbf{d}}_j", "46b96a69928a6c1b4bf821ae157f2592": " v=AD/BC ", "46b999c94f313b0585486aecb178f52c": "\n\\begin{align}\nx + 2y + 3z &= 0 \\\\\n3x + 4y + 7z &= 2 \\\\\n6x + 5y + 9z &= 11\n\\end{align}\n", "46ba3536aaecf507403ef0705a921d5a": "\\gcd", "46ba6abb379ac0de1f96327dc6b6a2db": "\\underline{\\psi \\quad \\quad}\\,\\!", "46ba87eb22088040f3ec15b18b6561e6": "H = aP^2 + bQ^2.", "46ba8f32da71c608c4ee050a0af3e474": "100\\sqrt{\\ell/g }", "46ba9552cbb9f845a77912a5a6bc5368": "P \\rightarrow Q. \\, ", "46bae8f1946c2224b823442e73f0b05c": "A(V) \\to A(U).", "46bb527d9cacb572b6076248da25418b": "k_\\pm \\in K_\\pm", "46bb6678b6c6db9736146e826db76e87": " A=\\begin{pmatrix} 0 & 1 \\\\ 1 & 1 \\end{pmatrix}, \\quad\nB=\\begin{pmatrix} 0 & 1 \\\\ -1 & 1 \\end{pmatrix}. ", "46bb6abd2bf5d0c81d379114fb3597ee": "K = \\frac{2 (\\Gamma(\\frac{n-1}{2}) )^2}{(\\Gamma(\\frac{n-2}{2}) )^2}.", "46bbb07e374972853ce49152b9bc799a": "\\scriptstyle\\mathbb{F}_p.", "46bbb752c0f72ea66e5206edb12b5e78": "H(p)", "46bbb7da22772d4f8af425ec6c8615ee": " G_x(t,f-f_0)\\,", "46bc08736051bdf64068c67785c6888e": "\\scriptstyle j", "46bcd01ac8c24d8d5b35ec70b29d33d8": "\\Lambda_f:=\\sum_{k\\geq 0}(-1)^k\\mathrm{Tr}(f_*|H_k(X,\\mathbb{Q})),", "46bce2e7d0536cf2f2d8e79eb6afe5fb": " \\sigma\\, ", "46bd460a9bb6433f95fcecfe952f8ef2": "\\int\\frac{x^2}{ax + b} \\, dx= \\frac{b^2\\ln(\\left|ax + b\\right|)}{a^3}+\\frac{ax^2 - 2bx}{2a^2} + C", "46bd55251e391ade7e3f015337ef7503": "{\\mathrm{R = \\sqrt[4]{\\frac{\\displaystyle {\\mathrm{\\lambda^2 \\, EIRP \\, G_R/T \\, \\sigma}}}{{\\mathrm{\\displaystyle 64 \\, \\pi^3 \\, k_B \\, BW \\, SNR}}}}}}", "46bd58aaa18eeda6b6f67c3cfffc2b78": " \\vec{x} = (x_1,x_2,x_3,...,x_k) ", "46bdcc3de3348a51b8c6d2f7aa0ca743": " \\sum_{w \\in S_n} q^{\\text{inv}(w)} = [n]_q ! .", "46be0fc95a3dcac0a5aa773ae79b81c5": "x\\wedge y", "46be299a62b62142ca8cd3a7cc5da704": "r^{n}", "46be5271b562fb1d90ba600aed55e942": "\\mu_G+\\mu_L", "46be55eae5c72dba63e42e237d85082d": " \\exists x ((K(x) \\land \\forall y (K(y) \\rightarrow y=x)) \\land \\lnot B(x)) ", "46be8c3731bb3a2463cf3c8a77967993": " x = x_0 + at \\,", "46bed9c7bd563d72b20fc1fcc55b0742": "\\displaystyle{h=(I-A)^{-1}T\\mu,\\,\\,\\, T^{-1}=(I-B)^{-1}\\mu}", "46bf69941a18558f34db6d4d7091642f": " S: (a_1, a_2, a_3, \\ldots) \\mapsto (0, a_1, a_2, \\ldots)", "46bf9922bc1747abcd840620fafc9031": "P_{G}", "46bfe19b8ea53b984c9efb939f0d1cc6": "a_{centrifugal} = \\cdot \\omega _{e}^{2}\\cdot \\left( r_0 + h\\right)", "46c007e653cf2bb3479fd6becbe7ea32": " R_{ij}=\\int_{v}N_{j} N_{i}dV ", "46c026f15205d674d2506ba2377565d9": "A_{arbelos}=\\frac{\\pi}{8}-\\left(\\frac{\\pi}{2}\\left(\\frac{r}{2}\\right)^2+\\frac{\\pi}{2}\\left(\\frac{1-r}{2}\\right)^2\\right)", "46c0354a216b0db6ad1bdf3d3e1ddb19": "\\textstyle x(t)=\\sum_{K=1}^q a(\\theta_k)s_k(t)+n(t)", "46c04d0e9d73c9b30ef30de52d9b2e08": "ds^{2} = -\\left(1-\\frac{2GM}{r} \\right) dv^2 + 2 dv dr + r^2 d\\Omega^2.", "46c0822c78c1eea1a47e6464638179af": "\\tfrac{e}{3} ", "46c09c99ff9703ee7321eeb8626c0eae": "\n\\kappa + \\tilde{\\kappa} =0\n", "46c0d03febe1b7424b702dea0296c0a4": " \\nu \\left(\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2}\\right)= 0 ", "46c0f27d35f08c69473bdab41682beae": "\\phi(pg) = \\phi(p)\\rho(g)", "46c14ad94556983d43c80eb661fa7eda": " \\|\\mathrm e^{tA}\\| \\leq \\mathrm e^{t\\mu(A)}\\, ", "46c15ff79849925fe72667d0daaa34c3": "R_{p}", "46c16e690aa7267dc1b9905bfaf0f317": "f - (g_1 + \\cdots + g_k) = o(g_k).", "46c16f06cced3da7dc0650aa632f20a7": "H^p(X, \\mathcal{F}(n)) = 0", "46c17abdcb9728eafac635d3ac3fb14a": "\\int_a^1 \\frac {1-x^s}{1-x} \\, dx = - \\sum_{k=1}^\\infty \\frac {1}{k} {s \\choose k} (a-1)^k,", "46c17f7a9149844c79071441a8a03a17": " J_\\varepsilon", "46c186acd76507e0bd29e41c971c70af": "x_1,x_2,x_3,...", "46c18fcbc810281fe0fc990cfb824b79": "\\alpha x", "46c1acf11322bf429f9ef3a1d5fd3a12": "L = \\frac{1}{2} mv^2= \\frac{1}{2}m \\left( \\dot{x}^2 + \\dot{y}^2 \\right)", "46c1e38b2802868606dd850a1e38f2b0": " R^1_1(\\rho) = \\rho \\,", "46c1ea69f2a69ef667de42c955b62772": "\\mathrm{Gr} = \\frac{\\beta g \\Delta T L^3}{\\nu^2}", "46c2438bf566bed75840c007b08638f1": "V = \\bigl( \\begin{smallmatrix}\\\\ 1&~\\;1\\\\ 2&-1\\end{smallmatrix} \\bigr)", "46c25c8979667227bb51900a04148efd": "A_{{yy}}", "46c25dc3acc9c669c1eb92a2d4a24f29": "l_{charge}", "46c2c18b9dd2e6e88c24bd915e8844fb": "\\log(x) + \\log(y)", "46c2e14b772c40d839bad6ba54ccf32b": "\\|R_n\\|", "46c34eee7f8ece709d24a57b494674f1": "x_1 \\stackrel{?}{=} x_2", "46c35052e92be55de7df69afbb869995": "\n\\lambda = \\frac{-3d(a-2)}{a(2a^2-6a+3)}\n", "46c3be9f3475303b4ac46db038579c59": "\\mathrm{Ta}>\\mathrm{Ta_c}", "46c3fea9feffe9bfb83ac3748cf92674": "A_{\\rm r}({\\rm X}) = A_{\\rm r}({\\rm X}^{Z+}) + ZA_{\\rm r}({\\rm e}) - E_{\\rm b}/m_{\\rm u}c^2\\,", "46c44079d64d5791226992360684a4d7": "f^2 = f(x)\\cdot f(x)", "46c493a078b62ccdce6276a0622803ca": " \\left| Q^{-1} \\sum_{x:\\, f(x)=f(x_0)} \\omega^{x y} \\right|^2\n= Q^{-2} \\left| \\sum_{b} \\omega^{(x_0 + r b) y} \\right|^2 = Q^{-2} \\left| \\sum_{b} \\omega^{ b r y} \\right|^2.\n", "46c4da8817189c719bc2a282b3a8fe65": "f(S)=\\bigg|\\bigcap_{i \\in \\underline{m} \\backslash S} A_i \\bigg\\backslash \\bigcup_{i \\in S} A_i\\bigg| \\qquad\\text{and}\\qquad f(S)=\\mathbb{P}\\bigg(\\bigcap_{i \\in \\underline{m} \\backslash S} A_i \\bigg\\backslash \\bigcup_{i \\in S} A_i\\bigg)", "46c5326fe4135c916e464b6fa4216883": "m_\\text{star}=-2.72-2.5\\,\\lg(L_\\text{star}/d_\\text{star}^2)", "46c532a6cde2bf2d5180625236ed91c5": "x^2 + y^2 + z^2 = 3xyz +4/9", "46c551908d700fef81d56f9a4696386d": " \\left(\\frac{k_x^2}{n_y^2n_z^2}+\\frac{k_y^2}{n_x^2n_z^2}+\\frac{k_z^2}{n_x^2n_y^2}\\right)(k_x^2+k_y^2+k_z^2)=0 ", "46c5678d148a2dd84f1eba3621ee5cb4": "K' = \\{x \\in K | \\langle x , x \\rangle _K = 0 \\} \\subset K", "46c592b71621a36be3162bd224cc2317": "\\text{ROA} = \\frac{\\text{Net income}}{\\text{Sales}} \\times \\frac{\\text{Sales}}{\\text{Total assets}} = \\frac{\\text{Net income}}{\\text{Total assets}}", "46c609eca573ed4bc1a858d89d4586b2": "n' = \\sum_i n'_i", "46c617bdc0efa73b741edb455d44ebd8": "\\left | a_{ii} \\right | > \\sum_{j \\ne i} {\\left | a_{ij} \\right |}. ", "46c7e9e84adbef9be97dd58124ffc0a6": "{{\\overline{P_1P_3}\\cdot \\overline{P_2P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}} =1+{{\\overline{P_1P_2}\\cdot \\overline{P_3P_4}}\\over{\\overline{P_1P_4}\\cdot \\overline{P_2P_3}}} \\ .", "46c84058c8d8f574992fb671b4a63a4f": "\\iota_\\star: \\pi_1(S) \\rightarrow \\pi_1(M)", "46c85583b3912c88af6f85e1b7519cd5": "d\\mathbf{p} = d(m\\mathbf{v})\\,+\\,(dm)\\mathbf{v} = (dt)\\mathbf{F}\\,+\\,(dm)\\mathbf{v} = d\\mathbf{J}\\,+\\,(dm)\\mathbf{v} = (dm)\\mathbf{v}_e\\,+\\,(dm)\\mathbf{v} = (dm)(\\mathbf{v}_e + \\mathbf{v})", "46c8821af0e133eec6f25a4c5cdae3ee": "\\vec{f}_0 = \\frac{1}{1-2m/r} \\, \\partial_t - \\sqrt{2m/r} \\, \\partial_r ", "46c88b00be85c604de3fecfd2ca8898b": "\\mathbb{R}^{4k}=\\mathbb{H}^k", "46c8b25a0e506f65e1508a24cc8cc5ed": "\\ \\frac {1}{2}", "46c8d3398177623605ea4b5b846345fd": "Re=\\frac{\\rho U_0 h}{\\mu}", "46c8e74298f3bd96ccd8715edb1ef88d": " \\sum_{p\\in\\{0,1\\}^d }(-1)^{d-\\|p\\|_1} I(x^p) \\,", "46c9a3b510123c4894e998d3bc8c178c": "\n\\mathcal{I}_T(\\theta)\n\\leq\n\\mathcal{I}_X(\\theta)\n", "46c9b43445391b9ce3bfc6adaad6a68a": "\\hat{F} = \\sum_{i=1}^{N}\\ \\hat{f}(i).", "46c9c5bf9f44d54eaf4e08fcc4dd4eb2": "\\,y(t)=(C+m(t))\\cos(\\omega t)\\cos(\\omega t),", "46c9dc834f6920358e961170bac929f8": "\\nabla I", "46ca98f5009540499f43ce3e9f657013": " |f(x)-g(x)| \\le C(x_b-x_a)^2 \\quad\\text{where}\\quad C = \\frac18 \\max_{y\\in[x_a,x_b]} |g''(y)|. ", "46cb2e7b1e01d9277337171aade20fde": "P_m(X)=\\frac{P(X)-P(\\alpha_m)}{X-\\alpha_m}.", "46cb3977f9de851d80594f21c7014b33": "f:X\\rightarrow X", "46cb702957ae3e53f59ccc77c7bc480c": "\nr_{CM}=\\frac{1}{N}\\sum_{i=1}^{N} \\mathbf{r}^{(i)}\n", "46cb7834f3dfecf0e40a1f35ddf13399": "\\|x\\|_\\infty=\\max\\{ |x_1|, \\dots, |x_n| \\}.", "46cbcfc9115375341f039f88daed3b66": "f: X \\rightarrow \\mathbb{R}\\,\\!", "46cd50ba5fdc6d79e682b97d8cb68934": "F(\\rho,\\sigma) = \\min_{\\{F_i\\}} F(p,q)", "46cd5b5bfd869dbab573975cccddfd11": "11^{35}\\ \\equiv\\ 70\\ \\not\\equiv\\ 1 \\pmod {71}", "46cd63702497df5bd7b0ae136967536d": "\\! \\text{MAP} = (\\text{CO} \\cdot \\text{SVR}) + \\text{CVP}. ", "46cd955b46b2eaa6766d2f61bf621fc8": "\\ v_{in}(t) ", "46cda6a96b5f8f3017056cd72464084c": " -\\infty < u < \\infty, \\, 0 < r < \\infty, \\, -\\infty < v < \\infty, -\\pi < \\theta < \\pi ", "46cdaff3676a3eb4427dac53fd36ca9b": "a \\not\\equiv 0\\pmod n", "46cdc19fbb4d0053899f2af13cbfbb57": "\\begin{pmatrix} \\int_\\gamma \\omega \\\\ \\int_\\delta \\omega \\end{pmatrix}.", "46cdc7a6723cd5f650073c07942ac70e": " (Sf)(z) = \\frac{f'''(z)}{f'(z)} - \\frac{3}{2} \\left ( \\frac{f''(z)}{f'(z)}\\right ) ^2 ", "46ce5a9bb9067de0ba752f4dbfc9d91d": " f_n : \\mathbf Z[G^{n+1}] \\to \\mathbf Z[G^n], \\quad (g_0, g_1, \\dots, g_n) \\mapsto \\sum_{i=0}^{n} (-1)^i(g_0, \\dots, \\widehat{g_i}, \\dots, g_n). ", "46ce6039423f1c0aa0c0ec766c352ec1": "\\partial{C} = S.", "46ce6ea79e8c1843f29f09b5cd077d2c": "\\kappa \\, = \\, - { 8 \\, \\pi \\, G \\over c^4 }~", "46ce6fa18846d52adc7a96c9ad66b454": "p=\\gamma^r", "46cea466dd78a99a03ccf55e48e4eca1": "\\phi_k = 0", "46cea8f505d783ec4b5b5dffbf5983c8": "P_1, P_2, ... P_n", "46cec288f93288be9cbc03bdea5977fa": "G \\ ", "46cf08886d0432dc318b38e265f5bd40": "\\tilde g", "46cf0d0885b58941655f37a64296fa5a": "\\arctan \\frac {1}{2}", "46cf11a95e203f03292816ab5f749145": "b + d = 180^\\circ ", "46cf1643884c5159ac518356d59d9a3c": " m n ", "46cf1aef8d5a918157531cdb950af3c7": " J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 ", "46cf2d8fdd2a6b9e77c17b022c20a326": "\\tfrac{dT}{dt}=\\tfrac{T_0-T}{\\tau}", "46cf559748060f02e40527630394fadb": " \\mathbf{v}_A = \\dot{\\theta} a \\mathbf{e}_A^\\perp, \\quad \\mathbf{v}_B = \\dot{\\theta} b \\mathbf{e}_B^\\perp,", "46cfd31715855cb1ba4a8e919c0db998": "f:G\\rightarrow G'", "46d005fe59651e4d9e0f1d22f5ad6f48": "\\alpha\\beta\\gamma", "46d032e78910ef433522c0050ced36a9": "\\begin{matrix}(B\\,\\mathcal{U}\\,C)(\\phi)= \\\\ (\\exists i:C(\\phi_i)\\land(\\forall j1;\\\\\n\\max_{i} td(G_i), &\\text{otherwise};\n\\end{cases}", "46d3afe9903c4cdb6c2497f4c397361f": "\\Delta M", "46d3b8f145aed86cbbd603052971e148": "\\boldsymbol S'(c)\\subseteq \\boldsymbol S(c)", "46d3c866e7f960ea12937cf1dba46127": "Q=\\int_A^B T\\,dS\n\\quad\\quad(1)", "46d3f282feff321e8ac0d6cc050e719c": "\\lambda_B = \\frac{e^2}{4\\pi \\varepsilon_r\\varepsilon_0\\ k_B T},", "46d4032c4eaf08747ea78dfc9391f3ce": "a+bi \\mapsto \\begin{pmatrix}a&b\\\\-b&a\\end{pmatrix}", "46d405ea74d2ccc94c84cf976a62a973": "A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 2 & 0 \\end{bmatrix}", "46d4485f5b9029f833539fd4c948adb4": "\\Gamma^{(\\lambda)}", "46d457bc8433dcc38fb58088e41b7767": "(x+y)^2 = x^2 +2xy + y^2,", "46d52210f17af3cbc485e5e2a78c923a": "\\textstyle \\beta", "46d575c7b6eb4a06f725f4845fb99128": " n(\\lambda) = \\sum_i (i-1)\\lambda_i = \\sum_i \\binom{\\lambda_i'}{2} ", "46d5b761bde19f002247187614bad7a3": " m = \\iint \\sigma \\mathrm{d} S ", "46d5cd99c62e3a522b1cac2fc3ee6042": "p(x_i) = p_1(x_i) - p_2(x_i)\\!\\cdot\\! E \\ = \\ f(x_i) - (-1)^i E,\\ \\ \\ \\ i =0, \\ldots, n.", "46d616ec8075538adf366fa76a736b6c": "R_{\\text{INIC}} < R_s", "46d61c407ab0005aaef06cab80245292": " a = \\frac{\\alpha}{2 \\pi} \\approx 0.0011614 ", "46d6601353493a894c9e9b34408635c8": " \\nabla^2 \\Phi(\\mathbf{r}) = -\\frac{1}{\\varepsilon_r \\varepsilon_0} \\, \\sum_{j = 1}^N q_j n_j^0 \\, \\exp\\left(- \\frac{q_j \\, \\Phi(\\mathbf{r})}{k_B T} \\right)", "46d7258738c1b0ad07c819e57d50bf2b": "~|\\alpha\\rangle~ ", "46d78bf43ac0e487fe6e0651846d0663": "\\scriptstyle h(t)", "46d79a10456c28c101e142ee2aa5c64a": "\\frac{\\sqrt{114}+10}{2}=10+\\frac{\\sqrt{114}-10}{2}=10+\\frac{14}{2(\\sqrt{114}+10)} = 10+\\frac{1}{\\frac{\\sqrt{114}+10}{7}}.", "46d7ed8dccadbe267b6acfba14ce31f1": "0.4330 + 10^{-14} < B < 0.472 \\,\\! ", "46d805a85686ce2276653f23ff20d9bf": "\\omega(v, I(v)) \\geq 0", "46d81a2c91c2198452bb3c67c034298e": "r>r_s", "46d84b11237ae884bd1f6ed3d859edc9": "n_{k\\ell}", "46d8779b124be4e14a17fb9f52c7410e": "\n \\left\\{\\begin{matrix} 4x + {\\ }y &{}= 6x\\\\6x + 3y &{}=6 y\\end{matrix}\\right.\n \\quad\\quad\\quad", "46d9089715ec3099c7346d83eb3999b0": "\\|\\hat{f}\\|_{\\infty}\\leq \\|f\\|_1", "46d9de7fa056b2b5514fe2e0032fcc12": "{3 \\choose 1, 1, 1} = \\frac{3!}{1!\\cdot 1!\\cdot 1!} = \\frac{6}{1 \\cdot 1 \\cdot 1} = 6", "46d9dfdfe8fa2fcc763de8d6ca0cdda3": "\\alpha = ( \\frac{\\beta}{N} \\sum_{i=1}^{N}|x_i-\\mu|^{\\beta})^{\\frac{1}{ \\beta}} .", "46d9e25095075f35907a2378e66ee48a": "\\delta(x) = \\frac{(n-1)!}{(2\\pi i)^n}\\int_{S^{n-1}}(x\\cdot\\xi)^{-n}\\,d\\omega_\\xi", "46da2ef1d1176644089387a75ea22e28": "\\boldsymbol{\\sigma} = \\mathsf{c}\\cdot\\boldsymbol{\\varepsilon}", "46da62b5a9b21964adc0e8e32c6960cc": "\\textstyle \\left\\{nP \\| n \\in \\left\\{0,\\ldots,q-1\\right\\} \\right\\}", "46da7a9ab200f235ce362c6afea7e565": "\\tan(\\theta)", "46daafbcf705cce3b205db16588e0161": "\\lim_{n\\to\\infty} \\int_S |f-f_n|\\,d\\mu= 0.", "46dad41ac190fbcd69d9b45ccc4e652e": "\\mathcal{G} \\rightarrow f_*f^{*}\\mathcal{G}", "46daf57cd231bef44b4d944c6f258d42": "\n\\frac{P_\\mathrm{avg}}{P_0} = \\frac{\\tau}{T} \\,\n", "46db03214759842c93d8309619ef2526": "T_{s}=", "46db589173ddb0e86d9110a4cceea9b7": "\n\\begin{align}\n\\mu = \\frac{\\ln(I)}{\\ln(I_o)},\n\\end{align}\n", "46db6c4f76300fd5213a0254f97b1d54": "u(s)=D(s)", "46db6cc1dbeb852c18775fbb6407bbf1": "[aT]_\\beta^\\gamma=a[T]_\\beta^\\gamma", "46db6cc909a8697c4d41568782cfd1d1": "FGT_2=\\frac {1} {N} \\sum_{i=1}^H (\\frac {z-y_i} {z})^2 ", "46db7479be31648b09575e18e5209e30": "A= \\pi \\times (0.23)^2 = .0529 \\pi \\approx 0.16619 ", "46db8d151bc7ffbf3209c35fa249a7a2": "\\scriptstyle X_{ip} \\sin(\\omega_r t)", "46db9e65487855de33505ed4439d8634": "\\overline{C}_{nm}", "46dbafa076e7da3bc67b2ee14e4c7eff": "y, z", "46dc890ce766f6b06b272324556fe679": "I = \\frac{bh^3}{12}", "46dcc55cf6e91f6d311f9fd81d59e5d1": " min_M \\ \\sum_k d(M(p_k),q_k) ", "46dce6d65e87d9136967397f6523f6ad": "p(0) \\,", "46dd272f974ebb818d4e2bbf20523674": "\\gamma_S", "46ddc4272067917aadcbd4bcdf062cf9": "p(x|z^i)", "46ddddc4afbfc02cb6fb0426329a873b": "Ff^2 (MgE)-C^1 Ri^1 ~ \\cdot ~ M=L/So\\ ", "46ddeef834bc894846e70b38ca3e9dac": "T(op_3, op_1 \\circ T(op_2,op_1)) = T(op_3, op_2 \\circ T(op_1,op_2))", "46de48174265cf9da54984d0bad34509": "A = 3 + 0.0391 = 3.0391\\,\\!", "46de59eaca83f2bcfc455037ee1ba4ad": "\ns=r\\theta \\,", "46df6d7f76478d3bc1e2ea753af9a8fe": " H_{j + 1, j}^{ } = H_{j, j - 1}^{ } \\equiv H_R ", "46dfd991c66ec4ee4c89d08b9bc93528": "QC_x = C_x = \\{x\\}", "46dfe23c9ff7866888f74463967f92f9": " Z^- \\subseteq (X^- \\cup Y^-)\\setminus\\{e\\}.", "46e00d6fb877fc2f16d08a0d1ebc0add": "\\zeta \\rightarrow \\frac{f(\\zeta)}{\\zeta-T}", "46e042fa7f4d10ab250b0071eaec56a7": "\\mathcal{P}_{n,k}=\\{(\\pi_1,\\pi_2,\\dots,\\pi_n)\\,:\\ \\pi_1+\\pi_2+ \\cdots + \\pi_n=k,\\ \\pi_{1}\\cdot 1+\\pi_{2}\\cdot 2+ \\cdots + \\pi_{n}\\cdot n = n \\}", "46e056d44fc761d0595c7d5ce0cbbb29": "\\phi_n \\overset{w^*}{\\rightarrow} \\phi", "46e05c540fa7e9855749d9ac72e24a99": "x'=\\frac{x+a}{1-ax}", "46e1010630e5f37af8b0dd6ae6bf696c": "z= \\frac{t^{2}+tq+q^2}{t+q},", "46e107e5d9037ea27cd4faa9b3a21c7e": "\\text{Spec }A", "46e16b547d8ab34808118e98052d284c": "\\leq N", "46e17133d0083784ed59fbef982e9773": "\\ \\begin{array}{rrcl} & t^* &=& \\sigma ^* n^*\\\\\n\\Rightarrow & Qt & = & \\sigma ^* Qn \\\\\n\\Rightarrow & Q\\sigma n &=& \\sigma ^* Qn \\\\\n\\Rightarrow & \\sigma ^* &=& Q\\sigma Q^T. \\end{array}", "46e1d3d9e9b841d973378c2481225b7d": "\\epsilon_{\\beta}", "46e20fd374270dbf1ea216200f4a3adc": "[x, y] = x y - yx", "46e2104e466885b9303b2a21ae5efc7d": "C^{\\infty}(M,\\mathbb{R})", "46e27c2aa46931e391a2d4ad6bba0e46": "\\sup_{\\| v \\| = 1} | B(u, v) | \\geq c \\| u \\|", "46e2976ef60b9e67093ead71c2ddc396": "\\scriptstyle 2\\pi", "46e31170baa48d3b6114c3b42989f12a": "-1.3235", "46e32ae4ef2508f87481db60b3294f86": "\n\\langle \\Phi_E (fg) h_1, h_2 \\rangle = \\int _X f \\cdot g \\; d \\langle E(B) h_1, h_2 \\rangle \n= \\langle \\Phi_E (f) \\Phi_E (g) h_1 , h_2 \\rangle.\n", "46e355fbbe8f6dbc98e1b181171b8e5e": "s_2 = 0", "46e35812655ca8fc1bd4dda9a1397bce": "\\uparrow\\!\\{1\\}", "46e378d36a23f86c6ec832b74d639f14": "\\scriptstyle |Z|", "46e37f30232a4bde8d29412eed02515c": "\\Omega_-", "46e3c8c5a4c1f9d056ed26f6ffef4a8b": " \\mathbf{1}_A \\mapsto \\pi(A)", "46e3f99f694c30adfdc6441cf359b64e": "auth_0,...,auth_{n-1}", "46e4901d8120a1d9c4bce4950a6ab675": "r=\\frac{1}{2-\\cos \\theta}", "46e500a26e59e0081766c152536dbf17": "\\mathbf{k} = \\nabla S", "46e50373be94b4e6f6013156624b34a2": "\\frac{1}{10^{11}\\pi}", "46e513ce064ed93e2820a8124338feca": "\\beta^{-1} = k_B T", "46e515bf14e81fa369e547af6bcdc6a0": "\\frac{2^k}{ke} ", "46e5436a27cf1a31b757b60e0c0cc2c6": "\\psi(z) \\sim \\ln z - \\sum_{k=1}^{\\infty} \\frac{B_{k}}{k z^k} ", "46e6197a2c2976006f06c12562bda39b": "\n\\begin{matrix}\n\\bold {M}\\ddot{\\bold{q}}(t)+(\\bold{C}+\\bold{G})\\dot{\\bold{q}}(t)+(\\bold{K}+\\bold{N}){\\bold{q}}(t)&=&\\bold{f}(t)\\\\\n\\end{matrix}\n", "46e669a31585ddd764d0f108c4d544b6": "a^2 + b^2 + c^2 = d^2", "46e695bff12b940a76f495a77bba7352": "\\int_{0}^{2 \\pi} e^{x \\cos \\theta} d \\theta = 2 \\pi I_{0}(x)", "46e6be2ea63d15fc00971eaa78d9d910": "-(m_0 c)^2 = -\\left(\\frac{E}{c}\\right)^2 + p^2\\,,", "46e6ebc11d7ad7c77ca20cc9bfbf33ab": "j=1,2", "46e703f73e9bc72903b72e7f53c2dead": "\\int_L f \\overline{dz} := \\overline{\\int_L \\overline{f} dz} = \\int_a^b f(\\gamma(t))\\,\\overline{\\gamma'(t)}\\,dt.", "46e70c73e726711d945ec4356425d2dd": "F \\theta^1\\theta^2", "46e7201efc0747b2a37f6bbb41128709": " z+\\bar{z}=2x ", "46e77066bc2fdcbd60009801813a91bf": "v_\\mathrm{rms} = \\sqrt {{3kT}\\over{m}}", "46e77c31259276ce7ee0d305647733ce": " |L\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ {1\\over \\sqrt{2}}\\begin{pmatrix} 1 \\\\ i \\end{pmatrix} ", "46e8015d2c6166f6b5047ae7faf9da95": "F = \\frac{1}{2} \\frac{dC}{dz} V^2", "46e81a7fc1505fc6f123d79e77d1e7ea": "\nF_r = J_3\\ \\frac{1}{r^5}\\ 2\\ \\sin i \\ \\sin u\\,\\ \\left(5\\sin^2 i \\ \\sin^2 u\\ -\\ 3\\right)\n", "46e840d4e5c91dace3f2e46ff2af02e6": " f_n y_n", "46e84fc3259d2a2e45c8a50a096dfbbe": "c^2=a^2+b^2-2ab\\cos C ,\\,", "46e8ba5b46974dec02c7988b38aa5bbd": "\\psi_s:s\\rightarrow S_R", "46e8e6cbad8c68e010aceb50b8d16577": " \\delta W = \\mathbf{F}_1\\cdot\\delta \\mathbf{r}_1 + \\mathbf{F}_2\\cdot\\delta \\mathbf{r}_2.", "46e8eebc7c21ccaf80f7a163da6a339b": "T(m,n,r)=r^{2n-m+1}e^{-r^2}\\frac{\\Gamma(\\frac 1{2}m+\\frac 1{2})}{\\Gamma(n+1)}{}_1F_1({\\textstyle\\frac 1{2}}m+{\\textstyle\\frac 1{2}};n+1;r^2).", "46e8ef53019b46abfe1419432e086212": "T\\triangleleft\\text{s}", "46e903a81b5108c35ac41ffcc48b787b": "\\, q_{i_{0}}", "46e90ff85d0beb42a07aaf825af1503a": "g_{\\ell,n}(x) = g(x - a\\ell)e^{2\\pi ibnx}, \\quad -\\infty < \\ell,n < \\infty,", "46e94bf00430bda64c93a61493c9a8f0": "f^+=g_0^+\\circ T", "46e94e40afb1a3c5dd88eb9557819cf4": "\\ U_J", "46e980d15cd27f035413c4e67dcefa4a": "\\Phi_t^n\\left(D\\right)\\left(a_0,a_1,\\ldots,a_n\\right)=\\int_{0\\leq s_1\\leq\\ldots s_n\\leq t}\\mathrm{tr}\\left(\\gamma a_0 e^{-s_1 D^2}\\left[D,a_1\\right]e^{-\\left(s_2-s_1\\right)D^2}\\ldots\\left[D,a_n\\right]e^{-\\left(t-s_n\\right)D^2}\\right)ds_1\\ldots ds_n,", "46e985194f8fec90ee2e6a4087e0c898": "b(x) \\in R", "46e9bd4291897d9337aeb18ecbc7c216": "{e} _i \\wedge {e} _j = {e} _{ij}", "46e9cd7a4c59aa7f0cf686caee355948": "\\vec{p}_v", "46e9ed793b4cb374473957d6f6ef919a": "\\rho>1\\,", "46ea5ae54dcc9f6066338fc6d1ee9cf8": "q= \\frac{V}{R} = \\frac{\\alpha m_0}{R} ", "46eaceaa65b77fdd7b4d697239d742d5": "A_n^{(c)}(t)", "46eb5ec913d8f64c64e764e96443da20": "z \\sim 6", "46eb7305226013ff2049f560ecaf9cf4": "\\lambda = 1,3,4,5", "46ec212511556f332c6f662eb7c9ff64": "\\ \\frac{[I]}{K_i} ", "46ec28eead27ff51ff9ec503d1339c52": "\\operatorname{f}_1(x) \\ge \\operatorname{wnchypg}(x;n,m_1,m_2,\\omega) \\ge \\operatorname{f}_2(x)\\,,\\,\\,\\text{for}\\,\\, \\omega > 1\\,,\\text{where}", "46ec5506113faa273ce05d6cb7fcf20d": "\n \\mathbf{x}_i(\\alpha) = \\boldsymbol{\\varphi}(\\alpha, q^j, q^k) ~,~~ i\\ne j \\ne k\n ", "46ec68cfa5952f0126dda81da8dab394": " B>1+A^2 \\,", "46ecee3630a1a9b895e16b172ce8f43d": "\n\\mathbf{P} = -\\frac{\\partial G_{3}}{\\partial \\mathbf{Q}}\n", "46ed19cdf0b2bc5e8882ac47656f92a5": "\\lambda =\\frac{\\mu }{\\rho }\\sqrt{\\frac{\\pi m}{2 k_BT}}", "46ed2df413e33dfe9b4d4485f525bcdc": "\\begin{align}\n \\frac{\\partial \\rho }{\\partial t} + \\nabla_\\alpha \\left( \\rho V^{\\alpha } \\right) &= 0 \\\\ \n & \\\\ \n \\rho \\left( \\frac{\\partial V^\\alpha}{\\partial t} + V^\\beta \\nabla _\\beta V^\\alpha \\right) &= -\\nabla ^\\alpha \\left( \\rho^2 e_\\rho \\right) \n\\end{align}", "46ed35b9e8655bedd124ba83915666b3": "\\begin{align}\n & \\operatorname{Pr}\\big(X\\in \\overline{g^{-1}(F)}\\big) \\leq \n \\operatorname{Pr}\\big(X\\in g^{-1}(F)\\cup D_g\\big) \\leq \\\\\n & \\operatorname{Pr}\\big(X \\in g^{-1}(F)\\big) + \\operatorname{Pr}(X\\in D_g) = \n \\operatorname{Pr}\\big(g(X) \\in F\\big) + 0.\n \\end{align}", "46ed4b8e6823b3604371c3a4345336c8": "p \\prec q", "46ed5c62884aafa58d03dd9ddc153b9e": " V = 0.2833~r^3.", "46ed9b21e4a28fe9f0840de9346345d2": "\\exists z[Czx\\and Czy].", "46eda33d79e45c3a5709f617b462b833": "2^{m \\times n}", "46edca9f4ff275d28027e034eafba9fb": " C_{Dmin} = (C_D)_{CL=0} = C_{D0} ", "46ee48b3e98dcda5b4ad84f26f8a1a29": "v=\\left|X\\right|=2^{n}", "46ee52e246b98b3add504a3b5e6f3a03": "E_*(X) = MU_*(X)\\otimes_{MU_*}R_*", "46eea5a721590ccc2cfbf2598616ab95": "D^{\\pm}_{\\alpha}", "46ef360e3e69e1bd3faf8afcfc2aac62": "\\frac{AB}{P_1 P_2}=\\frac{\\sin \\alpha_1 \\sin \\beta_2}{\\sin \\psi \\sin \\gamma}", "46ef41ef69b79caae581786157b18e69": "(\\mathfrak{g}, \\triangleleft)", "46efba944abaf2701f90d472fb4d3e72": " B \\cap C \\in G \\subset \\mathcal{B}", "46efddaf19434afd6e7a79653b1cf895": " \\rho = \\cos(3 \\theta) ", "46efe24c2a4d51d9fe131457fa98717a": "L={c \\over n\\, \\Delta f},", "46f0dbc766c87886664b2e1630bdaa33": "Z(j\\omega)=\\frac{1}{j\\omega C}", "46f16ded2ddd0233ac83702c93404d3f": "\\frac{\\sigma_f}{f} \\approx b\\sigma_A", "46f218c862263a226c4c26935ad25f96": "\\mu_{\\delta} (E) = \\inf \\left\\{ \\left. \\sum_{i = 1}^{\\infty} \\tau (C_{i}) \\right| C_{i} \\in \\Sigma, \\mathrm{diam} (C_{i}) \\leq \\delta, \\bigcup_{i = 1}^{\\infty} C_{i} \\supseteq E \\right\\},", "46f2f933f9448cd5bf899056e811d2f6": "x_{n+1}=x_1, x_{n+2}=x_2", "46f33cd7fc2dde4f47e9f9101afce51a": "\\lim_{n\\rightarrow\\infty}\\rho_{n}=0", "46f3489e0efa64e5859092846adadf04": "s(t,a),i(t,a),r(t,a)", "46f34a9738beb9681dcc7eb62945b84a": " \\mathbf{p} = \\boldsymbol{\\omega} \\times \\mathbf{m} \\,\\!", "46f39f1f6136e083f9d4cd667688cbf1": "\\sum_{n=0}^N \\frac{1}{F_{2^n}} = 3 - \\frac{F_{2^N-1}}{F_{2^N}}.", "46f4291c5907a00c03d0d5add1cd875c": "(\\mathcal{F}, \\mathcal{B}(X))", "46f4a7effe62645906ebde588145ca27": "P \\in \\mathcal P", "46f5364e9f655e55111649da5937c014": " \\triangle CBA ", "46f5e75f5148e5f154fb683514d70020": "\\mu_n^-(r)=\\int_{-\\infty}^r (r - x)^n\\,f(x)\\,dx,", "46f5f6fec6e1077e726680b665743821": "(NP\\backslash S)/NP\\triangleleft\\text{made}", "46f658de0871c06249368ff6036f2534": "\\mu(x) = dF(x^2)/dx", "46f659befee615f5e35666f8afa2d688": "2H_2O + 2e^- \\rightarrow H_2(g) + 2OH^- ", "46f67839ae253fd6c9bbd3e11dd1e6f6": "C_6 = \\{-U^2, -SU^2, -VU^2, -SVU^2\\},", "46f6b5d114d717b6bab3762bdad1d4ad": "p_n(z) = \\sum_{k=0}^n w_kI_k(z)", "46f6de91d96fedccb7e365ec2f0e8220": "P(\\text{success}| n_1,\\ldots,n_m, I_m)={s + c \\over n + m}, ", "46f6ec41d910477762fee43203c01138": "P_{A,B,\\Lambda}(a,b, \\lambda)", "46f71f83f0e1c4ed5f7e0f902d882013": "g'(x) = \\lim_{h \\to 0} \\frac{k(f(x+h) - f(x))}{h}", "46f734fc18d76a55485643e2a3318893": " \\left ( p_{\\mu} u^{\\mu} \\right )^2 + p_{\\mu} p^{\\mu} = { E_1^2 \\over c^4} -m^2 = \\left ( \\gamma_1^2 -1 \\right ) m^2 = \\gamma_1^2 { {\\mathbf{v}_1 \\cdot \\mathbf{v}_1 } \\over c^2 }m^2 = \\mathbf{p}_1 \\cdot \\mathbf{p}_1 ", "46f75afdd709d569e6e83a8074aefc9b": "\\liminf_{k\\to\\infty}\\frac{n_{k+1}}{n_k} > 1", "46f7913387b01205eca6035412fe2543": "x^2-y^2=(x-y)(x+y)", "46f7d5ab51076cc4df29425c759b4ea0": "E(X) = \\prod_{\\alpha_i \\in S} (X - \\alpha_i)", "46f7f885f58a1cddb0a2b71648fd485a": "-a^2 = ab", "46f82eee73102e1bf699f7f46386c29b": " \\mathcal{F} \\approx \\frac{\\pi \\sqrt{F}}{2}=\\frac{\\pi R^{1/2} }{(1-R)}. ", "46f89e9b2121cbefe72f690af0439fa0": "r_0 = 14a_0 + 11a_1 + 13a_2 + 9a_3", "46f8f6ea0f2fe73aa244cb737812ae21": "d\\mathbf F/dS\\,\\!", "46f8fa3f59a497b9b92796f83474ecde": "8x^2 + 1 = y^2", "46f93a70efdb49608648790221b5b7c2": "w_{n+1} = \\theta", "46f96a0641fd06b0170dd30533941eef": "W_1 = W_2 + \\rho K N_1", "46f9801533ec07982e3c9b031db13cbd": " P(PH~in~population) = 0.5~years * \\frac{1}{4000~per~year} = \\frac{1}{8000}", "46fa64d11a5805a055e14b55d0cc2982": "\\delta \\mathcal{S} = 0. \\,\\!", "46fa7900cc397f3a4b3fa2e72d6885e7": "a>b", "46fa852e1b7f0cae11445ccaba012914": "A_x(\\eta,\\tau)=\\int_{-\\infty}^{\\infty}x(t+\\tau /2)x^{*}(t-\\tau /2)e^{j2\\pi t\\eta}\\, dt.", "46faa58691344cfdd153e29a9e1c0f85": "1+F(x) + [F(x)]^2 + [F(x)]^3 + \\cdots = \\frac{1}{1-F(x)}", "46fadf7f045ef6dae34b517d97af0ca2": "\\tau(\\chi,\\psi)=\\sum_{x\\in F}\\chi(x)\\psi(x)", "46fb149a83e2ab75206fbdd3e0d6bfdd": "\\oint_{C} \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S \\nabla \\times \\mathbf{F} \\cdot \\mathbf{\\hat n} \\, dS. ", "46fb48af030cd757ad8555380c1b71cf": " \\tau = \\mathcal L_Vv = \\tfrac{1}{2}\\mathcal L_{[V,H]-H} J", "46fbcda5384d5c903a4ca7ffe7914810": "\nj = \\begin{cases} 1; & q < q_0 \\\\\n\t\t2; & q \\geq q_0\\end{cases}\n", "46fbf096901353c043c161cce87f08b5": "f(\\theta) = \\omega \\theta, \\quad f(\\omega) = \\omega,", "46fbf23b7eee0193d569856058343b31": "Z = \\frac{1}{i \\omega C}", "46fc1333f2e4684dd67de40df52a0df4": "\\log \\Lambda_i", "46fc4018a4e065f3c7e2aa9d13b8c886": "n \\in \\mathbb{N},", "46fc88efd0db89f022281b662e948414": "d_a", "46fca205dd7b2f52679e5ba18a5eee30": "x_{up}^{(syn)}(t) = \\mu \\frac{1}{n} \\int_{t_{0}}^{t} (q_{S}(t) - q_{F}(t)) dt,\\ \\ t \\geq t_{0}\\qquad\\qquad(2)", "46fcf0c1cb3fff58c52227844a7db97e": "E_b", "46fd018293d59c1ce9653fb7592ef7bc": "\\scriptstyle xy", "46fd4c0643508d63c32ebb6886e44c70": "\n- \\lambda < c^{2} < - \\mu < b^{2} < -\\nu < a^{2}.\n", "46fd6a9abd9a3365f2bcfbf2cd5a524f": "\\cdots \\to \\operatorname{Hom}(A, X[i]) \\to \\operatorname{Hom}(A, Y[i]) \\to \\operatorname{Hom}(A, Z[i]) \\to \\operatorname{Hom}(A, X[i + 1]) \\to \\cdots.\\ ", "46fd9df3494d6f24ad88712f48e23d7d": " e_{nit} = \\sigma \\eta_{n} X_{nit} + \\varepsilon_{nit} ", "46fdba1824123ea946c5191cbefd0b90": "f(x) = \\sum\\limits_{i=0}^n \\alpha_i \\chi_{A_i}(x)\\,", "46fe3a21358e99af31a962443452cedf": " _2F_1", "46fe71bd0de5572188b9cd3e73f1f24b": "\\operatorname{tr} (\\gamma^0 \\Gamma^\\dagger \\gamma^0)", "46fec765224862f507c449e1ab19e4cf": "B = b_{j_1}^{\\delta_1} \\ldots b_{j_L}^{\\delta_L}", "46ff061c2004e26723382571403be217": "FC_k", "470014252640e0e7ad0d7d1afba3c4e8": "n \\left( {}^{m+r}C_{r} -1\\right)\\,", "47004fb25aea311bf34ae657ebf5e15f": "D_t\\equiv R_{P_t} - R_{F_t}", "4700c5cd91d77fb31dd327af8aa67acb": "\\scriptstyle P(z):= a_n z^n+a_{n-1}z^{n-1}+\\cdots+a_1z +a_0", "4700d2af471e6bd0a66602adf96e7a31": "A = \\frac{1}{2} ab \\times \\frac{\\sqrt{3}}{2} = \\frac{\\sqrt{3}}{4}ab", "4700ff1b24d0c8f2a58339bd9710667b": "l, k", "470127e7705648852e05ab48b85c5c39": "{\\pi\\over 3}\\ {\\pi\\over 3}\\ {\\pi\\over 2}", "470164c76a624bdfe67851e79fdf4745": "A_1\\in \\mathbb{R}^{m_1,n}", "47016f8d17d7db76b1832860b4dd559b": "\\Gamma_S = - \\frac{1}{2RT} \\left( \\frac{\\partial \\gamma}{\\partial \\ln C} \\right)_{T,p}\\,,", "4701b6f9e0b481ffba6275ec471aa614": "S(A_n) = \\frac{A_{n+1}\\, A_{n-1}\\, -\\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}", "47026ac088ee0ecc880206fbaa6fef4f": "\\ \\phi ", "4702888d2b263eb1de082e4ec3afa449": "S_{0.05}", "470302bcbb844fe5e3a4402cdeaf474d": "(v,u)", "47032a6d358a1499ed3de2aa5bc7f2e0": "\\mathfrak{h}_1 = \\mathfrak{h}/\\mathfrak{n}", "47036f58ea5418b00eb3dfbd489b1e2d": " \\nabla \\rho(p) w = \\sum_{i=1}^n \\frac{\\partial \\rho (p)}{ \\partial z_j }w_j =0 ", "4703e02542d025bec93ba7a7e087f473": "\n\\int \\left(\\left(\\hat\\theta-\\theta\\right) \\sqrt{f} \\right) \\left( \\sqrt{f} \\, \\frac{\\partial \\log f}{\\partial\\theta} \\right) \\, dx = 1.\n", "4703f0ad1294c740336fb4f08707de89": "E_{j+1}", "47047d2e5994e67b237ce7fcddc9e2d0": "e \\to 0", "4704908e7d7768751382b0c4e02a72a4": "h_j=\\sum x_i w_{ji} \\,", "4704aaa4bcb6cddbc6b45d160040114e": "\\mathfrak{P}^{44a}", "4704e6d2f88c808855066142ab72270a": "\\mathbb{T}=\\mathbb{R}", "4704f424128560efabafe410440b67e2": " x\\,y=0, \\quad z=0\\,.", "470508b0474c82b62672c55e9b33a623": "cN", "470513a04e77e66686fcc29280139a92": "\n w(x,y) = \\sum_{m=1}^\\infty Y_m(y) \\sin \\frac{m\\pi x}{a} \\,.\n", "47059b6f9b06574a0eb887cccf220ffb": "\\vdots ", "4705f2f82e8e479117acc8c284b7015d": "s^2 = (x_1 - x_2)^2 + (y_1-y_2)^2 = (r_1 \\cos \\theta_1 -r_2 \\cos \\theta_2 )^2 + (r_1 \\sin \\theta_1 -r_2 \\sin \\theta_2)^2.\\, ", "47062bd1710ae449cd7e6730a0594626": "F:\\mathcal{C}\\to \\mathcal{D}", "47072fb88e59cca0aee77fa5c47bc8a6": "\n1200\n= 2^4 \\times 3^1 \\times 5^2\n= 3 \\times 2\\times 2\\times 2\\times 2 \\times 5 \\times 5\n= 5 \\times 2\\times 3\\times 2\\times 5 \\times 2 \\times 2\n=\\cdots\\text { etc.}\n\n\\!\n", "47074588651f68d7c6a9921a4d6b3d5e": "\\epsilon_X = \\epsilon_Y = \\epsilon \\,", "470747e7566be5fc19a2f5f02847051a": "y = -ix + az,", "47078c2100455a1728bf247bc8e6765c": "p_{n}/q_{n}", "4707d1e7bce73dff3556599ef7821b77": "\\scriptstyle{1/\\sqrt{2}}", "4707d6d067e31b78994a7191c491edc5": "n_s = V_v \\times c ", "47081a32c72393352fdb5b0e06801d3e": "\\Rightarrow_{r_1} AAS \\Rightarrow_{r_1} AAAA \\Rightarrow_{r_1} AAAA", "4708c0b2b64b70b94b8d535c17ed35f2": " M = \\begin{pmatrix}\n0 & . & . & . & 0 \\\\\n & & & & . \\\\\n & & & & . \\\\\nI_{n-1} & & & & . \\\\\n & & & & 0\n\\end{pmatrix}", "470926530e2f1389966144fb117ad657": "\\langle \\phi| A^\\dagger B^\\dagger | \\psi \\rangle^* = \\langle \\psi | BA |\\phi \\rangle ~.", "47093832fc0630ca3242c37c78b346fa": "x_*", "47093c677f1282735e8926a6abbb0e20": "x^5 + 2 x^3 y^2 + 9 x y^4 \\,", "47096eab8d0a045daa359a57411c04dd": "0\\leq x \\leq1 ", "47096f55e10500bc750dafbbdd7e608e": "1+z=\\frac{a_0}{a(t)}", "47099b600dd807c3e3b57af347a29ef9": " \\overline \\varepsilon = \\varepsilon(\\overline u) = \\frac{\\overline P}{\\underline EA} ", "4709b2cd4f68db447ef32ef8b0c4f083": "b^2 \\,", "4709e06282f23f70c2376918d8d75358": "(\\mathsf{i},\\emptyset)", "470a332e16c169b83ebd8ab38647207b": "A\\models Q_Ax_1x_2y_1z_1z_2z_3(\\phi,\\psi,\\theta)[a] \\iff (\\phi^{A,x_1x_2,\\bar{a}},\\psi^{A,y_1,\\bar{a}},\\theta^{A,z_1z_2z_3,\\bar{a}})\\in Q_A", "470a4aba957bd7ac6372df84d91559d2": "\\gamma'(1)", "470a791307771ccc0c37ac137e3a50be": "\n K(t;\\mu_1,\\mu_2)\\ \\stackrel{\\mathrm{def}}{=}\\ \\ln(M(t;\\mu_1,\\mu_2))\n = \\sum_{k=0}^\\infty { t^k \\over k!}\\,\\kappa_k\n ", "470ab619a14f18dcef1668d9248890f3": "\\frac{\\exp\\left(\\frac{i\\lambda t}{1-2it}\\right)}{(1-2it)^{k/2}}", "470ae8ab67640bdbeb86e37c15e6a3ff": "\\psi_{beta}(t|\\alpha ,\\beta ) =\\frac{-1}{B(\\alpha ,\\beta )T^{\\alpha +\\beta -1}} \\cdot [\\frac{\\alpha -1}{t-a}-\\frac{\\beta -1}{b-t}] \\cdot(t-a)^{\\alpha -1} \\cdot(b-t)^{\\beta -1}", "470b63d809c23b326dcdb2a704ed6112": "\n\\begin{align}\nC^0_1 &= \\sum_{i=1}^N eZ_i \\; z_i \\\\\nC^1_1 &= \\sum_{i=1}^N eZ_i \\;x_i \\\\\nS^1_1 &= \\sum_{i=1}^N eZ_i \\;y_i \\\\\nC^0_2 &= \\frac{1}{2}\\sum_{i=1}^N eZ_i\\; (3z_i^2-r_i^2)\\\\\nC^1_2 &= \\sqrt{3}\\sum_{i=1}^N eZ_i\\; z_i x_i \\\\\nC^2_2 &= \\frac{1}{3}\\sqrt{3}\\sum_{i=1}^N eZ_i\\; (x_i^2-y_i^2) \\\\\nS^1_2 &= \\sqrt{3}\\sum_{i=1}^N eZ_i\\; z_i y_i \\\\\nS^2_2 &= \\frac{2}{3}\\sqrt{3}\\sum_{i=1}^N eZ_i\\; x_iy_i \\\\\n\\end{align}\n", "470bcef5fe4f663b6390fc1d15644074": "x'=x \\cos \\theta - y \\sin \\theta\\,", "470c0bb8eae4865ca8a0546fc3406af0": "= 2\\ \\{\\gamma^\\nu, \\gamma^\\rho\\}. \\,", "470c23fe3643764cd09b0211e67e79f6": "\\alpha+\\beta=\\left(\\frac{2a_0^2}{\\xi_0}\\right)\\left(\\tau-\\tau_0\\right)+2\\ln a_0,", "470c2cbefc2d832e55de276c9f47d861": "1 < g 0 \\}.", "470e8968fe6d9a2dd626fa45ade4155d": "\\{D_1, D_2, \\ldots, D_{|V|}\\}", "470ec67a8e27c768e88c1524045df15f": "\\lambda(t) = \\lim_{dt \\rightarrow 0} \\frac{\\Pr(t \\leq T < t+dt\\,|\\,T \\geq t)}{dt} = \\frac{f(t)}{S(t)} = -\\frac{S'(t)}{S(t)}.", "470ed9955f0512da82bb821c47cf03ec": "\n C = \\frac13(a+b+c) = \\left(\\frac13 (x_a+x_b+x_c),\\;\\;\n \\frac13(y_a+y_b+y_c)\\right).", "470f08f51cf16d52ec8a82a699ca3c44": " \\int |h^\\prime(z)|^2 \\, dx dy= {1\\over 2 i}\\int_{C_1} \\overline{h}(z) h^\\prime(z)\\, dz -{1\\over 2 i}\\int_{C_2} \\overline{h}(z) h^\\prime(z)\\, dz,", "470f22b73c0f5653772644dd06eef29c": "v_{t} = \\sqrt{ \\frac{2mg}{\\rho A C_d} }. \\,", "470f552851bdb262433983d7cbe58b78": "f: S \\to \\mathbb{R}^2, (x,y) \\mapsto (x \\bmod 2, y \\bmod 2)", "470f57a34d5ba80776d33136a1486e14": "V_s\\,", "470f5953f068419d699340ca2e9494d4": "J_{\\rm e} = -A_{\\rm e} T_{\\rm e}^2 e^{-E_{\\rm barrier} / k T_{\\rm e}}", "470f71deb8cfe29679f693d04f9bc0e3": " u = \\overline{u} + u'", "470f9f8a7352f7ada98286e356f47589": "\\mathbf{\\phi}", "470fe55f95e793edc7a135d5f03f5d30": "P(A > O_j) = P(O_j | A)", "470ffded5b90037fbb6c84fb0a80a915": "{n\\choose k_1,k_2,\\ldots,k_r} ={n\\choose k_{\\sigma_1},k_{\\sigma_2},\\ldots,k_{\\sigma_r}}", "4710200fa0f74013442004e776eba7e5": "\\scriptstyle\\vec{p}", "4710412f7d6842add28e5bf64d8919aa": "Cr_\\sigma(S)", "471054471e43aa8f3bbf5cfeea5026e7": "B=[b_1,b_2,\\ldots,b_n]", "471079b7097d9f411dd1e37ce164cbfd": "e^-\\,", "4710ced0bb617502083029a68dd3131f": "\\frac{AB}{P_1 P_2}", "4710f1ed6275d8038666569eeb0100ed": "\\frac{dP\\left(t,T\\right)}{P\\left(t,T\\right)}=dX+\\frac{1}{2}(dX)^2.", "47113f86d954ab6414b71c90184e4c22": "v =\\frac V{U+V+W}= \\frac{6Y}{X + 15Y + 3Z}", "471145d7b5c0ca9eb4f58a6b00c2496b": "\n\\wp(z+y)=\\frac{1}{4}\n\\left\\{\n\\frac{\\wp'(z)-\\wp'(y)}{\\wp(z)-\\wp(y)}\n\\right\\}^2\n-\\wp(z)-\\wp(y).", "47116c323d0572b0d164385dc9f2ece4": "\\mathfrak{p} = \\sum_{i=1}^N \\, q_i \\, \\mathbf{r}_i \\, .", "471177262cddf9ef0bb6a6a116e5947c": " Q^{\\mathrm{core}} \\,", "4711b4c6f060d680dcb3605c89048df1": " \\acute{R} = - 8\\pi { G \\over { c^4 } } \\acute{T } ", "47129c979a007243c1b76d2dfa153ebe": " 2 x^2 \\frac{d^2y}{dx^2} - 3 x \\frac{dy}{dx} + y = 2 \\,. ", "4712cd2e5e56dc931b330475cbe69ca4": "|m|\\geqslant 2", "4712e111e009a21e0f908941286c173b": "\\tan\\psi=\\frac{r}{\\tfrac{dr}{d\\theta}}=\\frac{OP}{ON}=\\frac{OT}{OP}.", "4713fedb455d4246ae7ee1248dda5377": "(p_1(t) x_1^\\prime)^\\prime + q_1(t) x_1 = 0, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x_1(a)=1,\\,\\, x_1^\\prime(a)=R_1\\,", "471441a108c37e23f2e2b79cf32fc0d7": "kg/kg", "471468870263cfb074f9fd7c0d87d8d5": " \\psi(\\mathbf{r}, t) = \\frac{1}{(\\sqrt{2\\pi\\hbar})^3}\\int_\\mathrm{all \\, \\textbf{p} \\, space} A (\\mathbf{p})e^{i(\\mathbf{p}\\cdot\\mathbf{r}-E t)/\\hbar} d^3 \\mathbf{p} = \\frac{1}{\\sqrt{2\\pi}}\\int_\\mathrm{all \\, \\textbf{k} \\, space} A (\\mathbf{k})e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} d^3 \\mathbf{k} ", "471494b9f1d8c5642603d59ebdded27b": "x_{i}^{0}", "4714a3ff03d2716a7793936d5b0dfcb9": "MRRT rate = 30%( 1 - Extraction factor)", "4715046cc4ef7612be88a9a98b3fa8e8": "\\nabla_{f v}(m) = f \\nabla_v (m)", "47151928d6a35ae6596d8c9bfae4720e": " \\omega^2 = gk \\left[1+(kA)^2\\right]. ", "47158baa8e7fc084f2cb04a85915b70a": "k \\in G", "4715a41cd0b75702d6a18b6be59e83db": "\\begin{bmatrix}\n|\\mathbf{v}|^2 & -2v_x & -2v_y & -1 \\\\\n|\\mathbf{A}|^2 & -2A_x & -2A_y & -1 \\\\\n|\\mathbf{B}|^2 & -2B_x & -2B_y & -1 \\\\\n|\\mathbf{C}|^2 & -2C_x & -2C_y & -1\n\\end{bmatrix}", "4715fc9e54c1851936a22839c28ae6c7": " p_i = p_i(M) ", "4716419afd2d8f25680105aba486a7b4": "n_c = \\ell^2 N / L^2", "47178bf0e807c6137aa45423d78d2dea": "\n\\dot{q}_j = \\frac{\\partial H}{\\partial p_j} + \\sum_k u_k \\frac{\\partial \\phi_k}{\\partial p_j}\n", "4717c855fbfd66ce77ffdde6f10778e1": "X \\, \\sim \\textrm{Nakagami} (m, \\Omega)", "4717d6fa84990f904ff8f4b86a94f782": " \\overline{x} - s\\sqrt{n-1} \\le x_i \\le \\overline{x} + s\\sqrt{n-1}\\qquad \\text{for }i = 1,\\dots,n. ", "47180124c814b7bde5765bcbaca66f51": "z(u) \\rightarrow z(u) + 1.", "47183817c80138818014b3d062e9a68e": "\\lambda_{a;bc}\\neq\\lambda_{a;cb}\\,", "471838a39e6c31290cc3a39ea2cc3928": "\n\\begin{align} \\\\\na + \\infty = \\infty + a & = \\infty, & a \\in \\mathbb{R} \\\\\na - \\infty = \\infty - a & = \\infty, & a \\in \\mathbb{R} \\\\\na \\cdot \\infty = \\infty \\cdot a & = \\infty, & a \\in \\mathbb{R}, a \\neq 0 \\\\\n\\infty \\cdot \\infty & = \\infty \\\\\n\\frac{a}{\\infty} & = 0, & a \\in \\mathbb{R} \\\\\n\\frac{\\infty}{a} & = \\infty, & a \\in \\mathbb{R}, a \\neq 0 \\\\\n\\frac{a}{0} & = \\infty, & a \\in \\mathbb{R}, a \\neq 0\n\\end{align}\n", "471866313c5130611a33eb74611b6497": "\n2\\omega_{p} \\frac{dA}{dt} = \n\\left( \\frac{f_{0}}{2} \\right) \\omega_{n}^{2} A - \n\\left( \\omega_{p}^{2} - \\omega_{n}^{2} \\right) B\n", "4718bbaf20562c4996b7de1dc4f8cc67": "\\vec{S}(n)", "471907b4b62ae81ae7ab2f137d01bccc": "x_k > 0", "4719630b678d496b3fece0becfc87d2a": "\\langle B_E u | v \\rangle_E", "471984fa5f1dbe0c42fc3bbc9cd62304": "\n \\frac{\\partial \\Phi_1}{\\partial t} + g\\, \\eta_1 = 0,\n", "471a77ac010e48071dd1123e0f0cc264": "S_m=-S_g=-", "471a78797a0709e09a68c618b3c12f99": "C: (t,t^2,t^3)", "471aa11aa726f1de1a3eb1d0b1d02ac9": "\\prod C", "471b1d925b7d1906afeb361534d8d95d": "x = a,\\ y = b", "471b442071176725b93cd7125a82121e": "u-\\lambda \\Delta u=f", "471bc4cca27ffbfbecfe82bdafe30b1c": "\n\\begin{align}\n \\mathbf{v}_{\\perp\\ \\mathrm{rot}} &= \\mathbf{v}_{\\perp}\\cos\\theta + \\mathbf{w}\\sin\\theta\\\\\n &= (\\mathbf{v} - (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k})\\cos\\theta\n + (\\mathbf{k} \\times \\mathbf{v})\\sin\\theta.\n\\end{align}\n", "471c0815d425822ecec5767f20c620ee": "\\hat{z} = 1/z", "471c8fa653cbf8d358be460289d6c74d": "(2k-1)!! = \\frac{(2k)!}{2^k k!} = \\frac {_{2k}P_k} {2^k} = \\frac {{(2k)}^{\\underline k}} {2^k}.", "471cfcdfa1f590913aa1564f19941f10": "\nx = \\cfrac{z^{-1}}{1 + \\cfrac{z^{-2}}{1 + \\cfrac{z^{-2}}{1 + \\ddots}}}\\,\n", "471d218d5dcaec0082f5b0306de379c0": "\\sigma_{X+Y} = \\sqrt{\\sigma_X^2+\\sigma_Y^2+2\\rho\\sigma_X \\sigma_Y},", "471d30f03a23d6e8b9b14f6ae3c55a71": "\\displaystyle\\Delta (G) = \\Delta (S)", "471d5eed9769851415c21e9a9e91e93a": " \\operatorname{cl}(X)\\subseteq X ", "471dd2e14a0b4a1b3447bf887dfc9e24": "\\,f(x) = {}^{x}a", "471e264507e6279510689ecf35dd34da": "-\\frac{\\varepsilon\\gamma_{00}}{\\kappa} = \\frac{\\varphi}{4\\pi G}", "471f2707464e04b000660dfa18e4f4dd": "a,b,c\\in\\langle\\gamma\\rangle", "471f755b7edb306d37ed832805fd0649": "f : \\mathbb{R} \\to \\mathbb{R}", "471f9482e4020b57c781c5cbad568640": "V(I(Y))", "471f9832bc89915736bf56b1a7c4f4a2": " f^{*} = 2p - 1 . \\! ", "472039370c54c769550b84483e1366bb": "a \\div b", "4720702040a9d3e6f2b53d3b9f8afa15": "f(\\mathbf{s}^K) = \\frac{P(\\mathbf{s}^K|spike)}{P(\\mathbf{s}^K)}", "47207ec4ad4687547e78ae9722f1140b": "\n\\mathbf{X}=[X_1:\\ldots:X_K]=(x_{ij})_{i,j=1}^{L,K}=\n\\begin{bmatrix}\nx_1&x_2&x_3&\\ldots&x_{K}\\\\\nx_2&x_3&x_4&\\ldots&x_{K+1}\\\\\nx_3&x_4&x_5&\\ldots&x_{K+2}\\\\\n\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\\nx_{L}&x_{L+1}&x_{L+2}&\\ldots&x_{N}\\\\\n\\end{bmatrix}\n", "47208ce982df5a73c51d55efcef95aa2": "\\operatorname{Homeo}(\\mathbf{T}^n) \\cong \\operatorname{Homeo}_0(\\mathbf{T}^n) \\rtimes \\operatorname{GL}(n,\\mathbf{Z}).", "4720a3bf8e409cf0e41cd1829e938e54": "0 \\to M' \\xrightarrow{(f,0)} M \\times_{M''} N \\to N \\to 0.\\ ", "4720e145c3d59dbe90c5f9143393dcf0": "P(u,v,w)", "4720f6db8a83724f7ea20a3d9aa51fcb": " \\left(\\rho\\cos(2\\pi u),\\rho\\sin(2\\pi u)\\right) ", "47210a3bb3d4f55ddc8872efc2fe53b4": "A\\otimes(B\\oplus C)\\equiv(A\\otimes B)\\oplus(A\\otimes C)", "472132c8fe2a57ba94538fde5f468533": "c \\eta=\\Psi\\,", "4721607c1ccb39e3444d69bedd395971": "A \\in \\mathcal{C}", "4721d36b9462c132d04058455bdcf012": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{36 \\choose 2} \\end{matrix}", "472227fdaacf4dcdb170a4f64da97fab": "a = \\log(\\alpha)", "47224cfe8c1c8ca23a9cba464815c38e": "\\left\\{ \\Pi_{\\rho_{x^{n}\\left( m\\right)\n},\\delta},I-\\Pi_{\\rho_{x^{n}\\left( m\\right) },\\delta}\\right\\} ", "4722a04ba6f73539750f096acce2d64f": "a.P \\rightarrow^a a.P", "4722a36b2350a97a3fdea4a3084262a9": "\\alpha_2\\,", "47233a819c4766f409afbf9ef48c2b4c": " \\beta = \\frac{a_{\\ell\\ell} - a_{kk}}{2 a_{k\\ell}} . ", "472414eab335ef5323fd61a1883742ec": "\n \\begin{align} m(k|\\mu,M) & = \\int_0^1 l(k|\\theta)\\pi(\\theta|\\mu, M) \\, d\\theta \\\\\n\n & = \\frac{\\Gamma(M)}\n {\\Gamma(M\\mu)\\Gamma(M(1-\\mu))}\n {n\\choose k} \n \\int_{0}^{1} \\theta^{k+M\\mu-1}(1-\\theta)^{n-k+M(1-\\mu)-1} d\\theta \\\\\n & = \\frac{\\Gamma(M)}{\\Gamma(M\\mu)\\Gamma(M(1-\\mu))}\n {n\\choose k} \n \\frac{\\Gamma(k+M\\mu)\\Gamma(n-k+M(1-\\mu))}{\\Gamma(n+M)}.\n \\end{align}\n", "47249a43e8ca4455549686e3f08278cb": " X \\sim \\operatorname{F}(n,m) ", "4724a6921c86de8a753a4e61c0f61787": "E = \\frac{1}{2} \\int_a^b \\sum_{i,j=1}^ng_{ij}(\\gamma(t))\\left({d\\over dt}x^i\\circ\\gamma(t)\\right)\\left({d\\over dt}x^j\\circ\\gamma(t)\\right)\\,dt. \\ ", "4724ad3794e73f903d4f2cecc1795b19": " {t = t_1 }\\ ", "4724cecd37807595a472a5cdb24e1a57": "n = \\Sigma_{k} |X_{i}|", "47251b9c3f9a2a360a6c80018bbc50d6": "(y'+1)^n > x' \\,", "472544a8ff09389dfd1e71b4032c22aa": "\\scriptstyle{n}", "47254fab10acce46bbb8ad877e25e101": "|arg(u_i)| \\leq \\pi - \\frac{\\pi}{n}, i = 1,...,n", "47256e296e6bb697cb3544fd2ef3ea6d": " dG = -SdT + VdP + \\sum_i \\mu_i dN_i \\,", "472599e844b210b416d7293cda474dcd": "C_N \\; ", "47259ca7590856a57c4543c2b32d8463": "= \\frac{e^4}{(k-k')^4}\\left(\\sum_{r'} \\bar{v}_{k'} \\gamma^\\mu (\\sum_{r}v_{k} \\bar{v}_{k}) \\gamma^\\nu v_{k'} \\right) \\left(\\sum_{s} \\bar{u}_{p} \\gamma_\\mu (\\sum_{s'}{u_{p'} \\bar{u}_{p'}}) \\gamma_\\nu u_p \\right) \\,", "4725b3b8eaf0c3a616a3a8c0245a0fa0": "\nc^{2} d\\tau^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d\\theta^{2} - r^{2} \\sin^{2} \\theta d\\varphi^{2} \\,\\!\n", "4725e7aa3a5cf2188c15acd439f97d08": "p_1 = -\\Delta x , q_1 = x_0 - x_{\\text{min}}\\,\\!", "472636e6dbdde46fc9aef4dc2238dc7a": " H_f = \\left\\{ ([M], x)\\ |\\ M \\ \\mbox{accepts}\\ x \\ \\mbox{in}\\ f(|x|) \\ \\mbox{steps} \\right\\}. ", "472692c2b4a3352ebd118601401e283a": "(f*g)(t) = \\sum_{s \\in G} f(s)g(s^{-1}t)", "47269c6f975e138a198f07011c4d6b4f": "d_2 = d_1 - \\sigma\\sqrt{T}", "4726a3cc408f4e1329ebf5af05ade5ab": "\\beta=v/c", "4726b467b40b40639484548551372d5b": "K_i^G(X)", "472744bae1b1d4e6d885d7de1e33ff44": "\\sin(\\theta^1(t) + \\theta^2(t))", "47284187f0ac0a4557aa1a0e1a8b0109": "\n\\frac{1}{t}\\sum_{\\tau=0}^{t-1}E[y_i(\\tau)] \\leq \\frac{E[Q_i(t)]}{t}\n", "47285c1982cc2dc7afa181b91ea9c258": "\\mathrm{Re}_x", "47288d913ef14b4c9cf12edd28d2585a": "S(z,\\zeta) = \\overline{k_z(\\zeta)},\\quad z\\in\\Omega,\\zeta\\in\\partial\\Omega.", "4729031672ed1d8fefd4988ceec7a336": " \\boldsymbol{\\sigma} ", "47292aaa366fad51a479681f8bfc959a": "p : \\mathbf{R}^n\\to T_xM", "47294810b1287ea975f4e65766db7cd6": "\\frac{\\nu}{2}-1 ", "4729a20d07720867d60e5c482a1910bd": "q(x)\\,", "4729d4e8ed9521398cfb2ed3167a1f41": " \\mu _1 (\\Sigma _{11} )^{ - 1} \n", "4729d5066064fab406138a11f052ba82": " T_{gh}G \\text{ if } X\\in T_h G", "472aa0d4422c7645a5c954147deb2f07": " g_{k+1}(x) ", "472ab080ae3d47bd97ddaa8055746b1d": "p^2 = S \\cdot q^2 \\pm 1\\!", "472b017d20a0bbc7ac543d758c5f44be": "F(I) = 1", "472b037f8ed2ae9943a16260daafd85d": "R'(q)=P(q) + P'(q)\\cdot q", "472b1c5f8512a1892943a4806166cdbe": "p^\\omega", "472b94c1c5adf57b9b142fa481847ed1": "2^{352.17}", "472b974edc8cee37719dcded0f8dc051": " \n \\xi_{p}=\n \\begin{cases}\n \\cfrac\n {\n \\sin\\left(\\frac{\\pi}{Q_s}Y_kq_1\\nu \\right)-\n e^{j\\frac{\\pi}{t}Y_k \\nu}\n \\sin\\left(\\frac{\\pi}{Q_s} Y_kq_2\\nu\\right)\n }\n {\n \\left(q_1+q_2\\right)\n \\sin\\left(\\frac{\\pi}{Q_s} Y_k\\nu\\right)\n }\n e^{j\\frac{\\pi}{Q_s}Y_k \\left(q_1-1\\right)\\nu}\n &\n q_1\\neq q_2\\\\\n \\cfrac\n {\n \\sin\\left(\\frac{\\pi}{Q_s}Y_kq_1\\nu \\right)\n }\n {\n q_1 \\sin\\left(\\frac{\\pi}{Q_s} Y_k\\nu\\right)\n }\n &\n q_1= q_2\\\\ \n \\end{cases}\n", "472c451245a0ecf27a8c15a73e3211d4": "\\begin{bmatrix}\neither&A&amount&of&commodity&B\\\\\nor,&C&amount&of&commodity&D\\\\\nor,&E&amount&of&commodity&F\\\\\nor,&G&amount&of&commodity&H\\\\\nor,&J&amount&of&commodity&K\\\\\n\\end{bmatrix}", "472c915d7025cada617304c9d1465947": "(C - C^{\\dagger})", "472cb59b878f8dff48cf6b57535c8f7c": " w e^w = \\frac{I_SR}{nV_T} e^{V_S/nV_T} e^{\\frac{-IR}{nV_T}} e^{\\frac{IR I_S}{nV_T I_S}} e^{I_SR/nV_T} ", "472d532f721fa1fce96949891b94fc2b": "\\tau=\\frac{T}{T_c}", "472db9b2e372680d2b0d73de4ac1271d": "(f_{*} (\\mu)) (B) = \\mu \\left( f^{-1} (B) \\right) \\mbox{ for } B \\in \\Sigma_{2}.", "472dbdf96f858c4d2e826179dbe99bce": " G \\equiv \\operatorname{make-call}[p, \\operatorname{FV}[S]] \\equiv \\operatorname{make-call}[p, \\{f\\}] \\equiv \\operatorname{make-call}[p, \\{\\}]\\ f \\equiv p\\ f", "472e3114f6ff014df302430a2a73a9e8": "\\top \\equiv \\bot \\rightarrow \\bot", "472ededd588d159216e9de6f05218a9d": " \\left \\langle s, t \\, \\left | \\, s^2, t^2 \\right \\rangle \\right . ", "472ee97778dcf796fa89d9ea118f27c0": "H(T,J)", "472f1e29ee82fd45d0ee23a66e177c7e": " \\!\\ S_m^2 = m(a+b) + 1 ", "472f3240c06ec877826cadc626d9f94a": "\\Delta J= \\pm 1, \\Delta F = 0, \\pm 1 ", "472f9cf6495d03ad037cf46e8173641f": "S_D =\nA - BD^{-1}C", "472fb44578beb2a8e722fb1cc1c555ac": "V\\times V", "4730105100c2bf024883abb961d85647": " \\Psi = e^{-iEt/\\hbar}\\psi(\\mathbf{r}_1,\\mathbf{r}_2\\cdots \\mathbf{r}_N) ", "47306f38e97aee0e8209a4c2802e8bb0": " Z(u,v) = - \\hbox{tanh} \\left(u - \\pi \\right) \\, r \\, \\sin v,", "4730912d94102cd38537fdb28e3b7d7a": "\\operatorname{dim} \\operatorname{gr}_I(R)", "4730be865addbb7310678106d061bde8": "\\sum_{k=1}^{k=2} \\cos (-2\\pi\\frac{n(k-1)}{2})/2 = 0,1,0,1,0,1,0,1,0...", "47314cb5a81e0ef3e41c52f00e38cb58": "\n\\left\\{\\begin{matrix}n\\\\k\\end{matrix}\\right\\}\\equiv\\begin{pmatrix}z\\\\w\\end{pmatrix}\\ \\pmod{2},\\quad\nz = n - \\left\\lceil\\displaystyle\\frac{k + 1}{2}\\right\\rceil,\\ w = \\left\\lfloor\\displaystyle\\frac{k - 1}{2}\\right\\rfloor.\n", "47315100927fae3c4c0afc2ddf85b613": "T \\cong A_4,", "4731ea0d81b37d1609e62b14a0ed933d": " v^{k}_j = \\frac{a^{k}_{jk}}{2r}", "47324182c72f51730081877bbd375690": "\n M = \\mathbf{e}\\cdot\\mathbf{M} = \\mathbf{e}\\cdot(\\mathbf{r} \\times \\mathbf{F})\n ", "4732a703569514c5db685bc796a7ebaf": "\\alpha.", "4732ef8d5d311d22eab1d4814f8fafa0": " |1/2,+1/2\\rangle\\;|1/2,+1/2\\rangle\\ (\\uparrow\\uparrow)", "4733145eddb23fb0248d02af39a8a85f": "x_n \\to \\infty", "47334951b1982a2f12e2dd653eab99a8": "R_e \\propto \\langle I \\rangle_e^{-0.83\\pm0.08}", "4733641ee0f99c67abb60d9e1c2bb958": "C^T X", "473373c81a03ab9bc6fa30cac55a06ea": "F=4L_f \\,", "47337da87230b033fa2decf0afa4e530": "e^z = e^x e^{iy} = e^x(\\cos(y)+i\\sin(y)),\\,", "4733a525eb0e19e0412e423445ff9ce2": " N(x)= \\sum_{n=0}^\\infty H(x-E_{n}) ", "47344f01c2ce6e6ad4c0b3c92e7f270e": "\\frac{M}{y_c^2} = \\frac{q^2}{gyy_c^2} + \\frac{y^2}{2y_c^2}", "47346c65efd634e6045585a21c55c727": "H(z) = \\frac{(1 - q_1 z^{-1})(1 - q_2 z^{-1})\\cdots(1 - q_M z^{-1}) } { (1 - p_1 z^{-1})(1 - p_2 z^{-1})\\cdots(1 - p_N z^{-1})}", "4734b9b9f2d8243e338f15b123095de4": "\\lceil n'/k \\rceil", "473500774f08413b1e01d40e921a8fe9": " V_{wake} = \\frac{q \\omega_o R} {2 Q_o} \\ e^{j \\omega_o t} \\ e^{-\\frac{\\omega t}{2 Q_L}} = k q \\ e^{j \\omega_o t} \\ e^{-\\frac{\\omega t}{2 Q_L}}", "473512a2e0f0e03373f8cf3e505fd2d1": "F(x,y) = (x\\sqrt{1-y^4} +y\\sqrt{1-x^4})/(1+x^2y^2)", "473530610f7f609a4d251bf7ae5b1d33": "\\mathbf{T}(-5)= 5\\,", "4735658fb8af242e5de0d8b0df07ee77": "y_{ii}", "4735724cef210211ed5c4e6e1d690719": " C_P=\\frac{k^P_{tr}}{k_p} ", "4735fda8a55e97913a51a0b00b5b1caa": " \\operatorname{build-param-lists}[\\lambda p.f\\ (p\\ p\\ f), D, V, T_1] ", "473618686352494731be1eb8aed86504": "10\\uparrow\\uparrow 6", "473659eb19d35e20dd72bde5917d3f4d": "M_{33}", "47366f6f9cdf2b30957a076a7ff0ed98": "x_1^n", "47368824033ae3afc7c57d565423db1a": "B^2 - AC,", "47368ec90cff706f57b20af0827a33ee": "\\mathbf{TIME}(f( \\left\\lfloor \\tfrac{m}{2} \\right\\rfloor )).", "4736a4458c2b9e3270260e7712b433fc": "\\Theta_{bg}\\,", "4736ad629b771c275d10f4876f812a20": " \\int_C \\mathbf{v}(\\mathbf{x}) \\cdot d\\mathbf{s} = \\int_a^b \\mathbf{v}(\\mathbf{x}(\\lambda))\\cdot\\left({\\partial \\mathbf{x} \\over \\partial \\lambda}\\right) d\\lambda", "4736bd4d2d440f396a93fe253f1b5132": "\\mu _{v,w} (p_{v \\cap w}) = \\sum _{p _{v \\setminus w} \\in A _{S(v) \\setminus S(w)}} \\alpha _{v} (p _{v}) \\prod _{u adj v _{u \\neq v}} \\mu _{u,v} (p _{u \\cap v})", "473702f38a91bed6d632b5440f67323b": "\\langle x+y, A(x+y)\\rangle =0", "473705bd896be19bef5bfc09791f0403": "\\mu_p = \\left ( \\frac{L}{t_r} \\right )\\left ( \\frac{L_t}{V} \\right )", "473741ada8fdf16c5fd17937b96bca4b": "\n\\langle \\sum_{i=0}^n (B_1^*)^i h_i, \\sum_{j=0}^n (B_1^*)^j h_j\\rangle\n= \\sum_{i j} \\langle h_i, (B_1)^i (B_1^*)^j h_j\\rangle\n= \\sum_{i j} \\langle (B_2)^j h_i, (B_2)^i h_j\\rangle\n= \\langle \\sum_{i=0}^n (B_2^*)^i h_i, \\sum_{j=0}^n (B_2^*)^j h_j\\rangle ,\n", "4737708e3dbec6d0901267c08459d54a": "[[\\phi,\\psi]]= \\tfrac12\\big([\\phi,\\psi]-[\\psi,\\phi]\\big.)", "4737970eac0c2ce5f0c9b81f8a25664f": "W = {\\left(\\sum_{i=1}^n a_i x_{(i)}\\right)^2 \\over \\sum_{i=1}^n (x_i-\\overline{x})^2}", "4737c8e33d2958efe66e40d84dd4b5b3": "V_I(f) = \\frac{1}{2\\pi}\\int_a^b \\! \\left| \\frac{d}{dt} {\\rm Arg} \\, f(t) \\right| \\,\\, dt \\, = \\frac{1}{2\\pi} \\int_a^b \\! \\left| {\\rm Im}\\left(\\frac{f'}{f}\\right) \\right| \\, dt. ", "4737de3980e0e4a8139d800c47cca626": "\\overline{a}", "4737f8fa51b18c6aa1e5c4c1de20eb06": " \\chi(t) = 2\\pi t +i\\cos(2\\pi t), 0\\leq t \\leq 1 ", "473865977935aed8a74c1dd6206bacee": "\\omega^{\\omega^\\omega} + 4", "4738f7051e4e0e18524277d1c3211e0f": "h_{t}\\,\\!", "47390320701b74a4a9acd38a4052cf6d": "\\sqrt{5}:2", "473923cf8ddf1094ce560f321453e93d": "\\lambda = \\lim_{n\\to\\infty} \\text{Prob}\\left\\{P_2(n) \\le \\sqrt{P_1(n)}\\right\\}", "47394c4675c7f608286137012f831efc": " \\Delta F = \\int_0^1 \\left\\langle \\frac{\\part U(\\lambda)}{\\part \\lambda} \\right\\rangle \\, d\\lambda ", "47394fc2e26ad4c0f6d2414702467507": "|\\mathbf{R}|=1", "4739d4d0c943b90698ab47bff8455295": "e^{i \\Delta k \\Lambda}", "4739d4d6ee7123b878c82709c795b4a5": "W^u(f,q)", "473a3e1272c3aad23e49033736b82a54": " \\Sigma\\ = \\prod_{i \\in \\mathrm{N}} \\Sigma\\ ^i ", "473a6043d4b429b1c5f3f2b4e43f171a": "(x_2,y_2)", "473a73f7ee015891ad7b42a12acb0e48": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x) = \\frac{1}{2}(f(x_-) + f(x_+)),", "473aa395fa9c4bf106074ee3549458b5": " \\tau = C_{AO}(R+1) \\int \\frac{1}{(-r_{A})}\\,df_{A} ", "473aa9c07782771a00080653eb280dfc": "\\, Q_1\\times Q_2", "473ac322c8a99c8aad3a6292bac9f6dc": " P^t = p^t \\Bbb Z_p", "473ad6de1a73288621d6b2950835774d": "m_{relative}=\\gamma (m_{rest})\\!", "473ae11a9c9f75ffa6f58d2b575a3969": " {\\Psi} ", "473b4d09a07af0f76d56d0035e860dde": "{\\pi\\over 5}\\ {\\pi\\over 3}\\ {3\\pi\\over 5}", "473b804b4d19dcf930c9a1b5f1dae895": " R_{\\text{S}} \\| \\frac{1}{g_m} = \\frac{ R_{\\text{S}} }{ g_m R_{\\text{S}} + 1 }", "473cf8bad6ba8454452dc8904b71c973": "X_n \\, \\xrightarrow{\\mathrm{a.s.}} \\, X\\,.", "473d2a260b9725c602b3b193c829aca6": "T_H", "473d39db7ab17562a08878a16181ef00": "\\langle P \\rangle = - \\frac{1}{3}\\frac{E_{GR}}{V}", "473d42a7a5728d8dc453a98dcd996b58": "e_x = \\sum_{k=1}^\\infty {}_k p_x.", "473d48665e5fa9e6b910806681c82c2d": "\\omega_3 = - \\frac{1} {2} - \\frac {\\sqrt{3}} {2} i. ", "473d571a05060f7412d0186d9c9bb0eb": "-\\left ( \\frac{\\partial u}{\\partial y} \\frac{\\partial \\theta}{\\partial x} \\right ) - \\left ( \\frac{\\partial v}{\\partial y} \\frac{\\partial \\theta}{\\partial y} \\right )", "473ddb8fe7b9a780bf3170334b6b7ba8": " \\Delta \\rho = \\int_{r_1}^{r_2} g(r) dr ", "473f22350a075845991cdc13848e43f9": " SubCipher_2=ENC_{f_2}(k_{f_2},s_1) ", "473f6f391f57cc5ef2f89618d396213c": "p(y) \\propto \\exp\\left(- \\frac{1}{b_2}\\, \\int\\frac{y - \\frac{b_1}{2\\,b_2} - a}{y^2 + \\alpha^2} \\,\\mathrm{d}y \\right). \\!", "473f711e6822c9b37c987c67e0a12070": "\\|f\\|=[f,f]^{1/2},\\ \\ \\forall f\\in V.", "473fbda5f87125f4f905dc50629a30d4": " I = \\frac{ M m }{ M \\! + \\! m } x^2 = \\mu x^2 ", "473fd2d6ddecdd43219a2ff1cc4e0324": " I(t) = I_0(e^{-\\alpha t}-e^{-\\beta t})\\ ", "4740099ecd99da0bcc0c5bb62aae8ad8": "I_a^br", "4740284b3827a618856daa83a91db9d0": "(k\\times 2^k)", "4740589390609b204191e2481765fba7": "\\mathrm{NA}\\simeq \\frac{\\lambda_0}{\\pi w_0},", "4740945732ec66f2ff51015b2b783329": "\\gamma \\in \\{0,\\dots,n=\\dim M\\},", "4740c975786f425692b72e1b746c03cc": "\\scriptstyle{\\varepsilon_2(t)}", "4740e4f34a86db67fcece4fe24580b84": "|H_{NEXT}(f)|^2", "47417ad288669777f8851b30ce688dd1": "{a} \\,\\!", "4741b6d9cc519947639935d9344a45ed": "\\mathcal{L}={1\\over 2}g(\\partial^\\mu\\Sigma_a,\\partial_\\mu\\Sigma_b)-V(\\Sigma)", "4741e729cd6cf69a19ed0c0d0c66af01": "C(u_1,u_2,\\dots,u_d)=\\mathbb{P}[X_1\\leq F_1^{-1}(u_1),X_2\\leq F_2^{-1}(u_2),\\dots,X_d\\leq F_d^{-1}(u_d)] .", "4742479bd121f023db12883e837e6cdf": "\\mathbf{a} = \\left ( \\mathbf{v} \\cdot \\nabla \\right ) \\mathbf{v}", "47424c3380a17f2bb4c5f7f91571384e": "\\mathit{DTWA} \\subsetneq \\mathit{TWA}", "4742578056dd077db0e678387e9aa7f8": "\\partial^{\\nu} = \\frac{\\partial}{\\partial x_{\\nu}} = \\left( \\frac{1}{c} \\frac{\\partial}{\\partial t}, - \\bold{\\nabla} \\right) \\,,", "47427ca4234089a98a696cefadd5b3b8": " \nf := F(\\sigma_2-\\sigma_3)^2 + G(\\sigma_3-\\sigma_1)^2 + H(\\sigma_1-\\sigma_2)^2 - 1 = 0 \\,\n ", "4742b3b1d25871cded93e3864465f1aa": "\\scriptstyle P_{\\mathrm{CO}_2}", "4742c5b442c22db310e836f42acc7183": "\\psi_{,\\zeta} \\psi^{-1} = g_{,\\zeta} g^{-1} ", "4742dda35639775a07538fd8e7b20a2e": "\\tau_{ind}\\left(\\omega\\right)", "474305102a49720cb2d52063f121c066": "\\Gamma(S)", "4743561b053f78cad42a8982eeb9b86e": "\\nabla f(x^*) = 0.", "4743a25b76162cda0fdc2a53780f93a2": "p_{ij} = \\frac{p_{j|i} + p_{i|j}}{2N}", "4743b2453de48b63eb10081df4643a02": " F = \\sum_i f^i \\otimes f_i \\in \\mathcal{A \\otimes A} ", "4743b68900cc3b509c5e64544d6af635": "Q^{\\mathrm{II}}(t)", "4743c46901f5cfe378970d49a738f18a": "\\begin{matrix}\n\\hat{N_i}|\\Psi\\rangle_\\nu = a^{\\dagger}(\\phi_i)a(\\phi_i) |\\phi_1,\\phi_2,\\cdots,\\phi_{i-1},\\phi_i,\\phi_{i+1},\\cdots,\\phi_n\\rangle_\\nu\n&=& \\sqrt{N_i} a^{\\dagger}(\\phi_i) |\\phi_1,\\phi_2,\\cdots,\\phi_{i-1},\\phi_{i+1},\\cdots,\\phi_n\\rangle_\\nu \\\\ &=& \\sqrt{N_i} \\sqrt{N_i} |\\phi_1,\\phi_2,\\cdots,\\phi_{i-1},\\phi_{i},\\phi_{i+1},\\cdots,\\phi_n\\rangle_\\nu \\\\&=& N_i|\\Psi\\rangle_\\nu\\\\\n\\end{matrix}", "4743db5898162723b5a6a0b198cd4db8": "\\mathbf{Proj}\\mathcal S", "4743ece65e8c43a591a36d94acd5275c": "Q(x) = x^3 + (2 - |a_{n-1}|) x^2 + (1 - |a_{n-1}| - |a_{n-2}| ) x - a ", "474483f7db9de3efda102912c4dfecce": "\\{p,q\\}\\mapsto p+q", "4744f4b2cf4b533861c5fcb9544dc2d2": "\\eta'", "4745123a7107f6cc4b6b60995a574316": "D = \\frac{1}{2}(WS) H \\alpha F \\rho V^2", "47452938bf1069cf08df901918479ac4": "a_{12} x_2p_1", "47454abc6923e1334c93b6951f250410": "\\frac{1}{10^n} \\ge \\left| x- \\frac{p}{q} \\right| \\ge \\frac{1}{q^{\\mu}} ", "47454b97f7b302ee470807e714b86456": "AS=A+A^2+A^3+\\cdots +A^{n+1}", "474555f6612451386af6dc1c9650f397": "{A}_{16}^{(2)}", "474575a03063dc60c0ffd89b303c3903": "F:{\\textbf{C}}^n \\to {\\textbf{C}}", "4745fc996d9cc62616695a70e8d93acc": "Ax^2 + Bxy + Cy^2 = -(Dx + Ey + F). \\,", "474640aaa73c51fc9f9e999d8ee60c18": " S_{n-1} ", "4746ab44b8f380eced3107cc89f5c30c": "\nT_\\mathrm{Kozai} = 2\\pi\\frac{\\sqrt{GM}}{Gm_2}\\frac{a_2^3}{a^{3/2}}\\left(1-e_2^2\\right)^{3/2} = \\frac{M}{m_2}\\frac{P_2^2}{P}\\left(1-e_2^2\\right)^{3/2}\n", "4746ba863a9977b6667bc089acd4d375": "\n\\left[ \\begin{array}{cc|c}\n30 & 591400 & 591700 \\\\\n5.291 & -6.130 & 46.78 \\\\\n\\end{array} \\right]\n", "474742fec64cd2b0b9cb53eaaad8e54b": " r^{-2}~\\cos\\theta \\,", "4748463b7a7fb3b4563f22ec6c83c348": "(x \\top' y) \\top (u \\top' z) = (x \\top u) \\top' (y \\top z)", "474886f6acf1d39446b40d4e3ff6c7c9": "\\left (\\frac{\\mbox{Receivables}}{\\mbox{Net Sales}}\\right)\\mbox{365 Days}", "47490ba110d4d43561fba5fe6dead32d": "Q_{out} \\,", "47493ef758385fa3b0267e6719409b03": "\\frac{\\partial^2 s}{\\partial t^2} + \\frac{\\partial^2 u}{\\partial x \\partial t} = 0", "474946bf06a05e28b6a7e29f45cf2f25": "\\vec{\\tilde{x}}", "47495192a67ba75bcbbf855ed024a76a": "x < y \\or x = y \\or y < x", "47496e71fcbf27d0023d9ba53ed47ccb": "\nG' = \n\\begin{bmatrix}\n1 & 1 & 1 & 1\\\\ \n0 & 0 & 1 & 1\\\\ \n0 & 1 & 0 & 1 \n\\end{bmatrix}.\n", "474998d16c79d6ef134ce2952d24bcf4": "K=\\frac{\\{HA\\}}{\\{A^-\\}\\{H^+\\}}", "4749ddfb5cf37991991650e14fa3911a": "\\succ_W", "474a6498f2a915771bc9ebf33e8328df": "\\frac{d}{dt}\\hat{\\mathbf{x}}(t) = \\mathbf{F}(t)\\hat{\\mathbf{x}}(t) + \\mathbf{B}(t)\\mathbf{u}(t) + \\mathbf{K}(t) (\\mathbf{z}(t)-\\mathbf{H}(t)\\hat{\\mathbf{x}}(t))", "474b5265dcb1080762476f2afc191c9e": "D_\\,\\!", "474b76efd597897089734c44442fb10b": "2\\arctan (\\sqrt{t}) - \\pi/2", "474ba9c85a0c0fa70217c7e8b15b3436": "(U,V)", "474baf1ffaf50689e51cd730df8dff3c": " \n \\sigma_{ji,j}+ F_i = \\rho\\,\\ddot{u}_i = \\rho\\,\\partial_{tt}u_i.\n\\,\\!", "474be265ff9c60c87ec898a81e7ce0d5": "(\\tfrac{\\mathrm {mol}}{\\mathrm m^3})", "474c14d1ff32e34d319a536d87936a42": "\\tilde{A} = \\hat{A} + \\sum_{i,j} | p_i \\rangle \\left( \\langle \\phi_i | \\hat{A} | \\phi_j \\rangle - \\langle \\tilde{\\phi}_i | \\hat{A} |\\tilde{\\phi}_j \\rangle \\right) \\langle p_j | ", "474c2470995c16133da2cc359f45a314": " \\mathbf{p} = m\\mathbf{u} \\,\\!", "474c24b8e7dd37ae982aaa5bec4f842e": "x = (1-t)/t", "474c64e55e0b14900201994854fdcc0c": "\\psi, \\theta", "474c6977e2942d3e5ba137fc4f35c086": "\\zeta(z,q),", "474c7346292000afb797425fb66e73b6": "rate = {-d[A_n]\\over dt} = k_n \\times [A_n] \\times f([B],[C],\\cdots) \\text{ where } n=1 \\text{ or } 2", "474c94088e3e86abe72ca8f7ec1d3c87": "\\scriptstyle T_1", "474c96f7d60f49a55f1010f9a23e4daa": "Savings + Trade Deficit = Investment + Budget Deficit.", "474cf19e42a3384a50e90ff5043de94c": "\nK^{i}_{j;i}=0,\n", "474cfad47c27e2f69fc2dedf6bd2d218": "\\omega_0 = 2 \\pi f_0 \\,", "474d4071421986bcab90b313ace4b962": "\\phi_i = \\frac{1}{4\\pi \\varepsilon _0}\\sum_{j=1 (j\\ne i)}^N \\frac{Q_j}{4\\pi \\varepsilon _0 r_{ij}}. ", "474db0cff0cd10afbc77d57df8131cbd": "\\int_U (f\\circ \\phi)|\\det D\\phi| = \\int_{\\phi(U)}f", "474de8d61afe7088c1630dc1b3083d66": "\n\\left \\{\n\\begin{array}{l}\n\\dfrac{dx}{dt} = \\mu (1-y^2)x - y, \\\\\n\\dfrac{dy}{dt} = x.\n\\end{array}\n\\right .\n", "474e1e31ca8fdd62c137cbb8fc7ae61f": "\\lim_{n\\to\\infty} \\int_S{f_n\\,d\\mu} = \\int_S{f\\,d\\mu}.", "474e5cf81283d093cf19317c95b5e92f": "D=s^2/2\\tau", "474e7de61d1bcc034ae35f492bf8ec9f": " d \\geq 2t+1 ", "474eb5936bed46cc3d9aced6da393090": " M_{-j}^{r-1}", "474edb376d8059a812833c847f477b75": "-T\\frac{\\partial^2 f}{\\partial T^2}", "474f17dceabc2c9d9b68ae586d3ee245": "\\Beta(\\alpha, \\beta) = \\lim_{\\delta \\to 0}{\\rm NonCentralBeta}(\\alpha,\\beta,\\delta)", "474f35a20fbf3a23987a56d8ea9cdce1": "\n\\varepsilon = \\frac{c}{a} = \\frac{\\sqrt{a^{2} + b^{2}}}{a} = \\sqrt{1 + \\frac{b^{2}}{a^{2}}} = \\sec \\theta .\n", "474f3dc8cf5910d33fa4582c9f222a65": "\\textstyle \\{{n \\atop x}\\}", "474f4d3f63dcc6ef96fbcbffc810b6e9": "z_B", "474faa08f2474f10556b4912fff63217": "\n \\Big(\\begin{bmatrix}y_i\\\\Y_{-i}\\end{bmatrix} M_i \\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix} \\Big)\n \\Big(\\begin{bmatrix}y_i\\\\Y_{-i}\\end{bmatrix} M \\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix} \\Big)^{\\!-1} \n ", "474fd61b8a6efaf10a6cafa3d1b383c0": "\\sum_{i=1}^{m}\\sum_{k=1}^{n} X_{ij}X_{ik}\\hat \\beta_k=\\sum_{i=1}^{m} X_{ij}y_i\\ (j=1,2,\\dots, n).", "475009a25fbc913403b64d2ae70c6d04": "R_Y(\\theta)", "47504a350d9573cd351db439426a54a0": "{\\rm E}[f(A)] \\le {\\rm E}[f(B)]\\text{ for all bounded, increasing functions } f:\\mathbb R^d\\longrightarrow\\mathbb R ", "47504d1b71a3affeb97f7894bbe3a434": "\\mathcal{S}(\\mathbf{R}\\times \\mathbf{R})", "47505723dcc7b343bbd0fb9898adff2c": "\\tfrac {-1}{k}", "47507abec5d8c7d1cc3302650b8bfc84": "(d+1)p_\\alpha - \\frac1d", "475084f30449c9eada6ce1fd9ea51f26": " \\lim_{\\Delta x \\to 0}f'(n) = \\frac {f(x+\\Delta x)-f(x)}{\\Delta x} ", "47508fad21e96d3c72438c80b9617337": "w_A(x,y) := \\sum_B x^{n-\\dim f(B)} \\sum_C (-1)^{|C-B|}y^{\\dim f(C)},", "4750a5d2f6be13099f0cbc5739bc345a": "1 \\times n \\times ( n - 1) \\times \\cdots \\times 2 \\times 1 =n!", "4750cfb216abacbd20bf3b5834657ac1": "VAS(x^3+6x^2+5x+1,x+2) ", "4750dc36ecfecf8736c9f27fcdf2e605": "\n\\frac{\\partial c}{\\partial \\tau} =\n\\frac{\\partial^{2} c}{\\partial \\zeta^{2}} + \n\\frac{\\partial c}{\\partial \\zeta} \n", "47512b76e977f6422c96c159ad2a4407": "\\,e", "475139116279debe9d07e6e9d8b1860b": " V_1\\times V_2\\to \\mathbb F,", "47513e591f0b37237751c8e0d7a346b7": "{\\sum_{n=0}}^k (n+1)(n+2) = {(k+1)(k+2)(k+3)\\over 3}", "475213965571d546bbaed5c457124e67": "\n \\begin{align}\n \\sigma & = n_1^2 \\sigma_{1} + n_2^2 \\sigma_{2} + n_3^2 \\sigma_{3} \\\\\n \\tau & = \\sqrt{(n_1\\sigma_{1})^2 + (n_2\\sigma_{2})^2 + (n_3\\sigma_{3})^2 - \\sigma^2} \\\\\n & = \\sqrt{n_1^2 n_2^2 (\\sigma_1-\\sigma_2)^2 + n_2^2 n_3^2 (\\sigma_2-\\sigma_3)^2 + \n n_3^2 n_1^2 (\\sigma_3 - \\sigma_1)^2} \n \\end{align}\n ", "475285359e9e63b6de5a858753274beb": " = \\begin{bmatrix} 8 \\\\ -8 \\end{bmatrix} = 2 \\cdot \\begin{bmatrix} 4 \\\\ -4 \\end{bmatrix}.", "475290610345914ae477c41b2179c197": " \\tau_k = \\min \\{ t : Y_t = k \\}. ", "47529bfdf1bfc008b6f592c804b706d5": "(X_1,\\dots,X_d)", "4752a55ad2a9c6296af72b25d9c2c947": "e^{-\\frac{x^2}{2}} \\,", "4752f366d34910e80a83e2900263a7db": "\\{(1,2),(2,4)\\}", "4752f7b16335e0c7018677a93fec6436": " \\tilde{\\mathbf{A}} ", "47530ea6f763cda4011f957c9f8e3de3": "2 \\rightarrow \\infty", "475312425e4e2d524df454223a4b4a12": "I_m\\rightarrow I_n", "47533e12f2fc74fbe9f750b7f233d1e8": "x^+ \\to e^{+\\beta}x^+", "4753c94ec7292e19287f87fc24e874cb": "dV/dx_3=0 \\ ", "4753d936740e119219cfb591b7d1b475": "\\scriptstyle\\mathcal{S}^D", "4754053cf0b1cc53d6669d26d61e3af7": "p = A+D = \\mathrm{tr}(\\mathbf{A}) \\,,", "47540f0b9e477e5a350badba749e227f": "\\begin{align}\nU_\\text{in} &= U_\\text{out}\\\\\n\\frac{\\partial U_\\text{in}}{\\partial n} &= \\frac{\\partial U_\\text{out}}{\\partial n} + \\mathbf{M}\\cdot\\mathbf{n}.\n\\end{align}", "4754702411b9ff421641789a6af2640b": "F_{u_x}", "4754cfbcca3c09b8b72fc4cb3e095b4f": "\\begin{bmatrix} (v_1,t_1),(v_2,t_2) \\end{bmatrix} =\\omega(v_1,v_2)", "4754db5769c907992ef049f3e46ab107": "2C", "4755a969a928088c394476df10b5efa8": "E= \\dfrac{\\mathbf{p}\\cdot\\mathbf{p}}{2m}+V \\quad \\rightarrow \\quad \\hat{E} = \\dfrac{\\hat{\\mathbf{p}}\\cdot\\hat{\\mathbf{p}}}{2m} + V ", "4755b0106b0741a58610d74a0992ecf1": " E_c = \\frac{ O - \\frac{ 1 }{ K } }{ 1 - \\frac{ 1 }{ K } } ", "4756552df96ff705c770854668c7d7c4": " v_{i-1}", "475677fe5b4425ddd911c9ea86450980": "\\dot{m} = C\\;A_2\\;\\sqrt{2\\;\\rho_1\\;\\bigg (\\frac{k}{k-1}\\bigg)\\bigg[\\frac{(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}}{1-P_2/P_1}\\bigg](P_1-P_2)}", "4756b42b258f02023d99730b1f948a3c": "\nk_H = \\lim_{\\varrho\\rightarrow 0} \\frac{\\varrho(z')}{\\varrho(z)} = \\exp(-\\beta u_0),\n", "4756c103cbf57cc371cb3caf56421400": "S_{i}-S_{i-1}", "4756f110bdb2f68e39d32e2b27b48270": "\nD_{NN}(X, Y) = \\mu_{xy} + \\frac{2\\sigma}{\\sqrt\\pi}\\operatorname{exp}\\left(-\\frac{\\mu_{xy}^2}{4\\sigma^2}\\right)-\\mu_{xy} \\operatorname{erfc} \\left(\\frac{\\mu_{xy}}{2\\sigma}\\right) ,", "4757397d8df3a9f4b74ba303ffc5579d": "\\frac{\\sqrt{-C^3}}{\\pi} = \\sum_{n=0}^{\\infty} {\\frac{(6n)!}{(3n)!(n!)^3} \\frac{A+nB}{C^{3n}}}", "475746b51ec49a5d967bfc2dd73dde60": " p_{\\mathrm{av}} = 1.1\\sigma_y \\approx 0.39 \\sigma_0", "47574ef956be44f6746f121b18ea1797": "I \\propto \\sqrt\\frac{A}{L}", "47576eaf916f98d92ebb47c45b1c2f09": "\\int\\sqrt{x^2 - a^2}\\,dx", "4757901a512e2a202308240b1bdc9dc2": "\\mathbf{X}' = \\boldsymbol{\\Lambda}(v)\\mathbf{X} .", "4757b001f042f32214bf8373e2337d0a": "\\operatorname{not}_2 = \\lambda p.p\\ (\\lambda a.\\lambda b. b)\\ (\\lambda a.\\lambda b. a) = p \\operatorname{false} \\operatorname{true}", "4757c64973f81035a6c382e9355e6399": "P(x_j\\mid x_i)=T_{ij}", "4758646a45c7aea9a74a66558e2a8668": "X \\setminus \\cup_{\\lambda \\in \\gamma_k} U_{\\lambda}", "475865ee209753887b314c36f0e50397": "f\\colon x \\mapsto c x^r \\; c,r \\in \\mathbb{R}", "475879a4f967949411189eb975273d8c": " A\\frac{\\partial v}{\\partial t}+B\\frac{\\partial v}{\\partial x}=0 ", "47588ad8480fc76e2f716efb8c4b0943": "x=\\alpha^k", "4758b1814a0e1fe8247fe8c7876e41d6": "1.7756", "4758d21e94e7c8b3145bd775fc44cd16": "\\mathfrak F", "4758e2c9bdef539f62b399ecf7f4e69f": "p_j(x)=e(x)+x\\,o(x)", "4759739fd2f5905dfc8d16a13b78624d": "x_c(\\theta(t)) = A_c \\sin(\\theta_c(t)), \\quad x_r(\\theta_r(t))\n= A_r\\cos(\\theta_r(t))", "4759bc9848b392a33ac4186ada9db21b": "\\sum_{r=0}^n a_r\\left(\\overline{\\zeta}\\right)^r = \\overline{0} = 0.", "4759d4271fe07bd8af12f2027fc09399": "\\lambda\\|\\nabla\\beta\\|_0", "475a0f6243fcd260736d78ad05c47df6": " A \\subset X ", "475a861adf27c8e2668ddcd97e68f3e9": "I(2\\omega,l)=\\frac{2\\omega^2d^2_{\\text{eff}}l^2}{n_{2\\omega}n_{\\omega}^2c^3\\epsilon_0}\\left(\\frac{\\sin{(\\Delta k l/2)}}{\\Delta k l/2}\\right)^2I^2(\\omega)", "475a92bb74d79f495492148b22f548ee": "\\Delta G^\\ominus = -nFE^\\ominus = \\Delta H^\\ominus - T\\Delta S^\\ominus\\,", "475ac829a26620d22b6ecb4a3068e9de": "x_c(\\theta)", "475b226d0aa5482450d23074001032b6": "\\mathbf{F}_\\mathrm{b} = - \\rho_f V_\\mathrm{imm} \\mathbf{g} = - \\mathbf{F}_\\mathrm{g}\\,\\!", "475b25e450d069d690c1ff41f179ab3b": "TK_R", "475b273eaa34951cccf12ff3e1fdd93f": "E \\psi(x) = -\\frac{\\hbar^2}{2m s}\\frac{\\partial}{\\partial s} (s \\frac{\\partial}{\\partial s}) \\psi(s,\\phi)-\\frac{\\hbar^2}{2m s^2}\\frac{\\partial^2}{\\partial \\phi^2}\\psi(s,\\phi) +\\frac{Q \\lambda d^2 Cos[2 \\phi]}{4 \\pi \\epsilon_0 s^2} \\psi(s,\\phi),", "475b327d7c7f7af285499e8e1d5d2815": "t(A) = a_{ij} t(e_i \\otimes e^j)", "475b78897f974bc7658f55655285a0ff": "T_i", "475b8a2648b7de040dedfb71efc05454": "x \\leftarrow \\frac{a+bx}{1+x} ", "475b9085c708e6d01d263f1acdaacbf9": "x(t) = e^{-j \\omega_0 t}\\,", "475bcc839c2c7d05b4748dd1117ff743": "\\forall x: p_i(x) \\rightarrow p_j(x)", "475c2ad010efbf818ba17a2b9305add3": "(\\mathcal{E}^{\\dagger}(R), I - \\mathcal{E}^{\\dagger}(R))", "475c7438164638799c1a871899634718": "\\sum_k 1 / k^{(1)^2} k^{(2)^2} \\cdot\\cdot\\cdot k^{(s)^2} = \\left ( \\pi^2 / 6 \\right )^s ", "475c9b707393d8da682f4fd8ca9f87a2": "E(t) = \\delta(t-\\tau)\\,", "475cb2d30ac09bf4766b9887e2f33907": " t_2 = t_1^2, \\; \\; t_3 = t_1^3, \\; \\; t_4 = t_1^4", "475ce0c4594249ff38941295ca9bf3b0": "= \\frac{dT(s)}{ds}\\cdot\\mathbf{s}_u-\\frac{T(s)}{r}\\cdot\\mathbf{n}_u=-t\\cdot\\mathbf{s}_u-n\\cdot\\mathbf{n}_u", "475ce4b06d3532859578c9c178f396fd": "p_z", "475cf29625d049b0df87af83fcad2c3b": "C = \\frac{5}{9}(F - 32).", "475d04be093849446b442d02f5340c78": "\\mathcal{O}_Z = \\mathcal{O}_U / \\mathcal{I}_Z", "475d0b11d260165bd40d278b3d8c80bf": "\\cos(x)= \\frac{e^{ix}+e^{-ix}}{2} ", "475d1caed61a178768eb8a83611488ee": " f = \\sum_k a_k 1_{S_k}", "475d37ea42ddc8f1781424639e5401fa": "i_3 = {v_3 \\over R}", "475d77ad5816d24d133cc3f47fa674ea": "K=k_1\\cdot k_2.", "475d78ef8aa862030ed61078129f0f72": "d\\mathbf X=dX\\mathbf N\\,\\!", "475dac426dfda730d2771e68197f295c": "x, y \\in C", "475db05fb3c476e5ef416916fe6262b8": " F_2", "475db129791912f5b3c3a787d0e12335": "\\hat{t}_{g}(t,\\omega)", "475dcfa87a9012b4702111f96f67fb0b": "\\frac{2\\pi}{\\lambda}", "475dde561de4520de3283160c470e247": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = u_1\n\\end{cases}", "475e2f917c1c2ce1a56d40f8a2d10562": " = Max Q \\times S =\\frac12 \\times \\rho V_0^2 \\times S", "475e3c5d61fe9a6f0f857ae853ca171a": "E_\\infty^n", "475e3c8799b74a0d172d2c9c28ee288b": "\\mathbb C^{2 n+1}", "475e5d104d75a5f3d8ad981b80a38a4c": "32 - 8 - 1 = 23", "475e6c2daf3ea83329b527df6f2c012c": "\\mathbf u' = F(\\mathbf u,\\lambda)", "475eda3f4832ace581bf5856de45901b": "(\\mathrm{C_6H_5CH_3})", "475ef946def3615f896772cbde8fadfa": "\\operatorname{erf}^{-1}(z)=\\sum_{k=0}^\\infin\\frac{c_k}{2k+1}\\left (\\frac{\\sqrt{\\pi}}{2}z\\right )^{2k+1}, \\,\\!", "475f6c117b34b4c8a8e20097b49fabf9": "\\scriptstyle \\alpha \\;=\\; 0.5", "475f8fb6c6c8c7c9bb7e6cca424b767a": "m = (m_1,\\dots,m_n)^\\top\\,", "475fa71c189c74fbe7941e80c9bb42c0": "z\\in\\bar{\\Omega}", "475fc36827f1f37f5311f0409478e8e6": "u(t) \\le \\alpha(t)\\exp\\biggl(\\int_a^t\\beta(s)\\,\\mathrm{d}s\\biggr),\\qquad t\\in I.", "475fd03b8115f3e8f365f6c9547cd7cd": "ab = 0", "47600c65cabd107c9895607e130b3ad9": " \\epsilon(t)=1- \\alpha(t) \\qquad \\qquad (7) ", "47601722f28d4893ca9b67f34bf17b4e": " X_1 \\cdots X_n", "476047ed0294fddcedcefd38fd2dd5b4": "\\rho, \\omega", "476063463fa489739e9d78066163d527": "\\mathbf{L}_\\perp ' = \\mathbf{L}_\\perp ", "4760886589ee6af144d14b4d7f93ab13": "a_c = 4\\pi^2\\frac{r}{T^2}", "47608d5ddd358ea9a2e8347a55b931d9": "- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N{\\partial \\alpha}{\\partial \\beta}} = \\operatorname{cov}[\\ln X,\\ln(1-X)] = -\\psi_1(\\alpha+\\beta) ={\\mathcal{I}}_{\\alpha, \\beta}= \\operatorname{E}\\left [- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N{\\partial \\alpha}{\\partial \\beta}} \\right] = \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}", "47609543086a3b31d5c8db3778eef08e": "\\left\\{D,D^\\dagger\\right\\}=-2i\\frac{\\partial}{\\partial t}", "47609eb15658d5af23b6f648fcbdd5fb": "J_x = -\\frac{i\\hbar}{2m}\\left(\\psi^*\\frac{\\partial \\psi}{\\partial x} - \\psi\\frac{\\partial \\psi^*}{\\partial x}\\right), \\quad J_y = -\\frac{i\\hbar}{2m}\\left(\\psi^*\\frac{\\partial \\psi}{\\partial y} - \\psi\\frac{\\partial \\psi^*}{\\partial y}\\right), \\quad J_z = -\\frac{i\\hbar}{2m}\\left(\\psi^*\\frac{\\partial \\psi}{\\partial z} - \\psi\\frac{\\partial \\psi^*}{\\partial z}\\right)\\,.", "4760f576423a19a9da4b498d94e0588f": "t \\in [0,1 + \\epsilon)", "47611267eaa3e4ad0a4d0b16bbe72c87": "\\omega=\\sum_iE_i(e)\\otimes\\theta^i", "476117c84d02c4b8bcec2a7c9b3e523d": "Pwf = \\cfrac{2 Pmf}{3} = \\tfrac{2}{3}", "47611b3469f12860e74c95275d453825": " P_B - P_d = R_d \\times (Q_a - Q_e)", "4761211cbda7f82fc281aa7b73439102": "\\frac{1}{N}", "476123ec099d9636acecd271bca79180": " \\mathbf{y}'(t) = A\\mathbf{y}(t)+\\mathbf{b}(t).", "47612540c68db56743fc7dfac1533d5c": "G/\\mathbf{E}^p(G) \\cong (\\mathbf{Z}/p)^k", "47614ecf73352193f6281c1119e74e8f": "|(\\alpha+1)^n/G| - |\\alpha^n/G|", "476159dccc8c25036688147070c276e5": "A_{t} = \\left\\{ \\omega \\in \\Omega \\left| \\lim_{s \\to t} \\big| X_{s} (\\omega) - X_{t} (\\omega) \\big| \\neq 0 \\right. \\right\\},", "4761da4d40730e44b8f32f8f0bbffa1e": "\\mathbf e^2 = \\frac{\\mathbf e_3 \\times \\mathbf e_1}{\\sqrt{g}} = \\mathbf e_2", "476235ac8d83c4791eb066e3d40ef7c5": "D_{E}", "47632a69bb7fa4d6efb8c13a44ede377": "e = C_vT", "476374e9b651dea5d569377cb66a1fdd": "s\\to 0", "4763946ead6fb195ac11b4f4c8ed73cb": "f\\colon V_1 \\times \\cdots \\times V_n \\to W\\text{,}", "4763b239e58dacb5276fe9a667de8e7e": "l_{\\mathrm{sum~of~100}}", "4763e7f87f19f3c13e7bff926e865360": "[0.1, 0.9]", "4764360e2689c701dfb8b917ba7638ac": "y_1", "47647090d93b83faa719b2c815439a44": "\nf^{h}_{\\mathbf{k}} = 1 - \\langle \\hat{a}^\\dagger_{v, \\mathbf{k}} \\hat{a}_{v, \\mathbf{k}} \\rangle = \\langle \\hat{a}_{v, \\mathbf{k}} \\hat{a}^\\dagger_{v, \\mathbf{k}} \\rangle\n", "476470b961719d30b367e73f4e9e09dd": "{\\Delta p = f(Q)}\\ ", "476491506dd04b23a638763aacb4f626": "v=q^a\\mathbf{e}_a+p_a\\mathbf{f}^a.", "47649b80630d392185e27be0225b4fa4": "D[x,\\sigma]", "4764c25228ddd50a30c7d2e3af379fc2": "h(\\mathbf{K}) = \\frac{1}{a_3}\\int_0^{a_3} dx_3 \\frac{1}{a_2}\\int_0^{a_2} dx_2 \\frac{1}{a_1}\\int_0^{a_1} dx_1 f\\left(x_1\\frac{\\mathbf{a}_{1}}{a_1}+x_2\\frac{\\mathbf{a}_{2}}{a_2}+x_3\\frac{\\mathbf{a}_{3}}{a_3} \\right)\\cdot e^{-i \\mathbf{K} \\cdot \\mathbf{r}}. ", "47652d881fa3dfa60602718a8ffc90eb": "g : \\mathbb{R}^{n} \\rightarrow \\mathbb{R}", "47653fd749f3852e6989673d1e10d75a": "P_1,P_2,P_3,P_4\\to P.", "476541757e8d15223abcd9808208926b": "||f||_{S,r,p}=\\sup_x \\left({1\\over r}\\int_x^{x+r} |f(s)|^p \\, ds\\right)^{1/p}", "4765a2cd8b275fdfa0442cdbbe3e0acc": "\\scriptstyle{\\mathcal{O}}", "4766722ff3844dd0dffcf411dab851e9": "f|_{C_\\alpha} : C_\\alpha \\to Y\\,", "4766786ab61e41a40c5f19cc8278400f": "\\left(\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over 2},\\pm{1\\over \\sqrt{2}}\\right)", "47670f31502a81e2f57bc0b6b5da1dcb": "L = \\mathbf{Q}(\\zeta_{p^n})/\\mathbf{Q}", "476749c372af5d1efbf2cbac793cce8d": "\\mathrm{im}\\,D=\\{f\\in K((X)) : [X^{-1}]f=0\\}.", "47674d91f05f5dded2fc6649dceb50f0": "y = 0 ", "4767586912cb47814686f6bdfdbf979b": "\\mathrm{Possessions} = \\mathrm{.96} * (\\mathrm{FGA} - \\mathrm{ORb} + \\mathrm{TO} + (.44*\\mathrm{FTA}))", "47678b57ae4a89a2d056fb4b6f54e280": "(x-3) (x+3) (x^2-2)^6", "47679b83c407aa690bb469a6afc6f365": "\\sum_{d_1+\\cdots+d_n=3g-3+n}\\langle \\tau_{d_1},\\ldots,\\tau_{d_n}\\rangle \\prod_{1\\le i\\le n} \\frac{(2d_i-1)!!}{\\lambda_i^{2d_i+1}}\n=\\sum_{\\Gamma\\in G_{g,n}}\\frac{2^{-|X_0|}}{|\\text{Aut} \\Gamma|}\\prod_{e\\in X_1}\\frac{2}{\\lambda(e)}\n", "4767c35e0c3a593e3b7b5752931ea945": "d_{1} = \\frac{S-m_{1}^2}{d_0} = \\frac{114-10^2}{1} = 14 \\,.", "4767fb380c941954407b8fa49eaac290": "B(\\widehat{\\theta}) = \\operatorname{E}(\\widehat{\\theta}) - \\theta", "476812c79d55719b03449c8300627f22": "\\{1,\\,i,\\,j,\\,k\\}", "4768460d213d083fece0b20900877e92": "\nE_{\\ell r} = \n\\int \\frac{d\\mathbf{k}}{\\left(2\\pi\\right)^3} \\ \\tilde{\\rho}_\\text{TOT}^*(\\mathbf{k}) \\tilde{V}(\\mathbf{k}) = \n\\int \\frac{d\\mathbf{k}}{\\left(2\\pi\\right)^3} \\tilde{L}^*(\\mathbf{k}) \\left| \\tilde{\\rho}_{uc}(\\mathbf{k})\\right|^2 \\tilde{\\Phi}(\\mathbf{k}) = \n\\frac{1}{\\Omega} \\sum_{m_1, m_2, m_3} \\left| \\tilde{\\rho}_{uc}(\\mathbf{k})\\right|^2 \\tilde{\\Phi}(\\mathbf{k})\n", "4768513bbb2679bc7b6b15e8c0b8218e": "\\textstyle{\\mathbf{s}}", "47686993a3b97796c19d1579713ba7c4": "\\pi_{ref}=1.4\\pm0.3", "47686b6147e3ce42c92aab728a774faa": "a \\leq L/2", "4768ae8be09e5d58b83242813983f827": " \\langle 0 | R\\phi(x) \\phi(-x)|0\\rangle ", "476907de52e88dad5718e06bca2a17a7": "a \\triangleleft S", "47690b2e9a7997336c50bf7914d35bdf": " \\{g_1(t), g_2(t), g_3(t), \\dots , g_n(t)\\} ", "4769473806cf6511e0f97d3bfa81d837": "(x_0,y_0),~(x_1,y_1),~(x_2,y_2)\\,", "47696fb60a127d0ce4534943691865e3": "Y_i=\\frac{Z_i-a_i}{b_i}", "47697cf4476c7ab39e92c819b20a6f7d": "h_B^{(2)}(z)", "4769c88899e448ed4969dbd5046aa56d": " n > n_0(\\varepsilon) ", "476b2a159e3547a7747bf90f6e836379": "\\gamma V^{2/3} = k(T_c - 6 \\ \\mathrm{K} - T)\\,", "476b3c546ca5d97e89b0287ff9bcad52": "f(r) dt", "476b5cceb6433de9c22f863cf1320f7e": "E(|W_\\delta(t)|) = 2\\pi\\delta t + 4\\delta^2\\sqrt{2\\pi t} +4\\pi\\delta^3/3.", "476b681b40a928af4fe38e2b066ed199": "V_x = \\frac{dx}{dt}\\,\\quad V'_x = \\frac{dx'}{dt'}", "476c6510865264f4b8ca979b09648fcb": "\\eta_M >\\eta_L", "476caf972d787ec1af3c31ba5476d726": "S_{F0} = \\frac{h}{m_0\\omega_{F0}} = 1.4196\\cdot 10^{-7}m^2 \\ ", "476cdcfd6e5e93a4ca0f78ef650df2d5": "\\alpha_J = R_i - [R_f + \\beta_{iM} \\cdot (R_M - R_f)]", "476d2a7d2c306cc23493ad29914ea298": "F:[X]^{<\\omega}\\to X", "476d8a8f849f98556b552fec17012751": "\\frac{P_{\\rm min}}{\\dot m} = \\int_1^2(\\mathrm{d}h-T_a\\mathrm{d}s).", "476ddd4870af72f1e9af0880cd089f4c": "\\lim_{x\\to 1^-}\\sum_{p>2}(-1)^{(p+1)/2}x^p=+\\infty,", "476e0b8dbd78b8dce8cae7e314708792": "y_t = at + b + e_t\\,", "476e28766807d208066fb7d0f74f8ec6": "\\dot{m}_i \\mathbf{v}_i", "476e60074907f524bbb0e0571d709960": "r(t) = -be^{-i(a/b)t}", "476e91c8a273dd8feca6f43cada06ec3": " [\\mathbf{t}]_{\\times} ", "476fa553c4035bdbd6a7de0938364757": "D=D_{K_s}(V)", "476fe8794c7b24d6185591b90f726dfc": "{1\\over 2}mv^2_{\\mathrm{max}} = q_eV_0", "4770373166f41a33ac6ecc1c49d33f64": "\\qquad \\qquad k_{p,\\mathbf{s}} = \\frac{1}{8\\pi^3}\\sum_{\\alpha}\\int c_{v,p}\\tau_p(\\mathbf{u}_{p,g}\\cdot\\mathbf{s})^2d\\kappa \\ \\ \\ \\ \\ \\mathrm{ for \\ component \\ along\\ } \\mathbf{s},", "47705ed98de7f0887c338093dc222fe8": "N_f=\\frac{2\\;(a_c^{\\frac{2-m}{2}}-a_i^{\\frac{2-m}{2}})}{(2-m)\\;C(\\Delta\\sigma Y \\sqrt{\\pi})^m}", "4770650d17cd761278909ba6c9ef456e": "\\dot{V} < 0", "47706af35824fa8491a334fc1e185377": "= b^*(0)b(0)\\sum_{\\boldsymbol{R_n}} e^{-i \\boldsymbol{k \\cdot R_n}}\\sum_{\\boldsymbol{R_{\\ell}}} e^ {i \\boldsymbol{k \\cdot R_{\\ell}}}\\ \\int d^3 r \\ \\varphi^* (\\boldsymbol{r-R_n}) \\varphi (\\boldsymbol{r-R_{\\ell}})", "477073816efaf998ae7f504e84cb8673": " a_k = \\frac{1}{\\sqrt{2\\hbar\\omega_k}}\\left(\\omega_k\\phi_k + i\\pi_k\\right), \\ \\ a_k^\\dagger = \\frac{1}{\\sqrt{2\\hbar\\omega_k}}\\left(\\omega_k\\phi_k^\\dagger - i\\pi_k^\\dagger\\right), ", "477073c6f00792c2ad04b18d3a13a954": "\\delta \\psi = -v_\\theta \\delta r,\\,", "4770cb8daca0e99c8e8d0ff6e78455e4": "\\mathbf{J}_u = k T^2\\nabla (1/T)", "47717e31a874a8483b79617d38975689": "O(c)", "4771b94a807173a19068952244c56ac9": "s\\mapsto r(s)-sr^\\prime(s)", "4771f8a16aa71f94bd0569f270d7f156": "Np\\,", "47722e87d3b302e49cf679348cc5129d": "A>0\\,", "4772618783453ff9d819d122ecbe8d8b": "\n \\partial_{\\mu} A^{\\mu} = 0\n", "477266a83bb702b970ccd9386bffdc99": "\\tfrac12 \\rho v^2", "47727a46b2f49b3aa39ee6923b1448e4": " \\tau_{crit} = \\sigma_n ' \\tan \\phi_{crit} '\\ ", "47727fbe1bf3e76a1887f35ce225aa94": "d_{32} = 6\\frac{V_p}{A_p}.", "4772946898cc2597730dd4017543fc0b": " 10,", "477e240a4f921a9c96a0df03e8218531": "1 0", "4782ec33c383cb68f1178707973b37c3": " \\frac{d}{dZ} \\Bigg([-2Z^2 + \\frac{27}{4}Z] E_1\\Bigg) = 0 ", "47830913d8c40068bc6aa2b159e17a1f": "2K_p/T_u", "47832d16470b8d481f0c073725b401c2": "\n\\begin{array}{lcl}\n \\mathbf{b}_x &= &\\mathbf{f}_x / \\mathbf{f}_w \\\\\n \\mathbf{b}_y &= &\\mathbf{f}_y / \\mathbf{f}_w \\\\\n\\end{array}.\n", "4783360b3101439e761970faf4e91ef1": " \\Gamma(s) = \\int_0^{\\infty} t^{s-1}\\,e^{-t}\\,{\\rm d}t", "4783facc9b179e24ed7ff0fb48a342c0": "y_n^* = \\underset{y \\in \\mathcal{Y}}{\\textrm{argmax}} \\left(\n \\Delta(y_n,y) + \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y) - \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y_n) - \\xi_n\\right)", "4784399b444a47a1a05a04802a9219b3": " y'^2=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x') \\,", "478469be78f3dfa3b83246274bb4059b": "q: [0,+\\infty) \\to \\mathbb{R}", "4784906a664f850c960b6166ca516314": " i= r + \\pi ", "47849e7e2911aac8ca09471833c50aa2": "\n\\begin{align}\n\\sum_{i=1}^n 2(\\hat{y}_{i}-\\bar{y})(y_{i}-\\hat{y}_i)\t& = \\sum_{i=1}^{n}2((\\bar{y}-\\hat{b}\\bar{x}+\\hat{b}x_{i})-\\bar{y})(y_{i}-\\hat{y}_{i}) \\\\\n\t\t\t\t\t& = \\sum_{i=1}^{n}2((\\bar{y}+\\hat{b}(x_{i}-\\bar{x}))-\\bar{y})(y_{i}-\\hat{y}_{i}) \\\\\n\t\t\t\t\t& = \\sum_{i=1}^{n}2(\\hat{b}(x_{i}-\\bar{x}))(y_{i}-\\hat{y}_{i}) \\\\\n\t\t\t\t\t& = \\sum_{i=1}^{n}2\\hat{b}(x_{i}-\\bar{x})(y_{i}-(\\bar{y}+\\hat{b}(x_{i}-\\bar{x}))) \\\\\n\t\t\t\t\t& = \\sum_{i=1}^{n}2\\hat{b}((y_{i}-\\bar{y})(x_{i}-\\bar{x})-\\hat{b}(x_{i}-\\bar{x})^2) .\n\\end{align}\n", "4784b3223eebffa0e67bd64001e8b19c": "m^*(B)=\\inf_{S\\supset B} m (S)", "4784d4de9b30779f8e6a00c08223d64a": "\\scriptstyle f_k", "4784fdb6b596c9b02dda7526c4dda141": "\\tan \\left( \\frac{\\pi \\Delta f}{f_f}\\right) =\\frac{-Z_{\\mathrm{F}}}{Z_q}\\tan\n\\left( k_{\\mathrm{F}}d_{\\mathrm{F}}\\right)", "47852175ca108bb434870bdf4dc5a8d3": "\\mathfrak{S}\\left(R(X,Y)Z\\right) := R(X,Y)Z + R(Y,Z)X + R(Z,X)Y.", "478536aa0f4fa122cb1cb119099fd793": "\n\\{\\mathbf{e}_1, \\mathbf{e}_2\\} = \\left\\{\n\\begin{pmatrix}\n 1 \\\\\n 0 \n\\end{pmatrix},\n\\begin{pmatrix}\n 0 \\\\\n 1 \n\\end{pmatrix}\n\\right\\}\n", "4785486872cb8735beb4cae3ee1b2404": "P_D = 1 - \\zeta\\quad", "478556a8876b25c8c65a5e9ab49b442d": " f(0)g(0)=1 \\mbox{ and ends at }\n f(1)g(1)= i. ", "4785ca71ca494cef92599e52a604bab3": "|\\alpha,n\\rangle=[{\\hat{a}^{\\dagger}]}^n|\\alpha\\rangle / \\| [{\\hat{a}^{\\dagger}]}^n|\\alpha\\rangle \\|", "47860581a4fdb116f39f01e915e9ad28": "i=1\\ldots n", "47862376dc6d7ed97ad55a4afaee31b5": "\\epsilon \\neq 0", "47862ceaf0e4e5ee1f7b98b9bc6af4cb": " \\mu(f) = \\dim_{\\mathbb{C}} \\mathcal{A}_f \\ . ", "47865b4ecbc950d25c9eea57397d0bba": "r = 0 ", "47865eddf228c516541e566edf4947bc": "t\\sqrt{2}\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}\nM\\left(\\frac{k+1}{2},\\frac{3}{2},\\frac{t^2}{2}\\right)", "478716c809f3a126bf88bd45dad7c822": "S=\\cfrac{7d}{\\rho\\,(1-\\epsilon\\,)}\\sqrt{\\dfrac{\\epsilon\\,^3\\pi\\,\\delta\\,P}{l\\eta\\,Q}}", "47874aee68274b4741fc6cacdc41e61f": "\\mathbb{R}^\\mathbb{R}", "478785e88600677692597dcd93909fbe": "\\ \\|\\underline{x}\\|_X< \\frac{2k\\|\\underline{y}\\|_Y}{r} ", "478786cb75e45f14ce5df20f10682a24": "op_3", "4787960ce66d2c33b2d765ab9bfe80af": "{\\mathcal{I}}_{\\alpha, \\beta}", "4787c84838fcb043eff7059f2922152c": "\nL_\\mathrm{B} = \\log_{10} \\bigg(\\frac{P_1}{P_0}\\bigg) \\,\n", "47886075e116083470778fdaa553ce39": "\\nabla_\\alpha e_\\beta^I = 0", "47895e3d376b2b2b243127a4e4d63496": "R\\gg a", "4789620d87b795674b2e57f3a33a67c4": "R_t = a \\phi^{-m} S_w^{-n} R_w ", "4789833d879f9643f258aac7ee2889e9": "p = \\pi\\overline{\\pi}", "4789aa3d89ef9421a40fc2c952934a3e": "f(Y \\cap Z) = f(Y) \\cap f(Z)", "4789b108d949741195ebd0c79e0bb65f": "P^*(X,Y)", "4789b538e22f4a247d1d8f2513b537d8": "\\epsilon_{\\uparrow}", "4789d9b47fa89d5a19fdfffa276cd80d": "f(z)=\\sum^\\infty_{n=0}a_nq^n\\ \\ (q=e^{2\\pi iz})", "4789fa08a270f8e0dcbbe6cecb3a9f6d": "\\mathbf{x}(k+1) = \\mathbf{A}(k) \\mathbf{x}(k) + \\mathbf{B}(k) \\mathbf{u}(k)", "478a4b3086c4ad50e167691745ca86e7": "a^{2^{ \\overset{n} {}}} + b^{2^{ \\overset{n} {}}}", "478aa4c05d1c07d1ccda7190ce037a09": "\\chi_{\\rho \\oplus \\sigma} = \\chi_\\rho + \\chi_\\sigma", "478ae6415d3fb39c3735a7446da74801": "e^{i a(\\hat{n} \\cdot \\vec{\\sigma})} e^{i b(\\hat{n}' \\cdot \\vec{\\sigma})} = I\\cos{c} + i (\\hat{n}'' \\cdot \\vec{\\sigma}) \\sin{c} = e^{i c(\\hat{n}'' \\cdot \\vec{\\sigma})},", "478b31af7de4a8e2ba9be29c4f0fb538": "M_y+(E_y-B)L=-\\frac{3}{2}\\sigma{L}-\\frac{1}{2}M_x-\\frac{1}{2}(E_x-B)L", "478b6906ac22ae659de9aca29435904c": "|k^{(0)}\\rangle", "478baac95385fa3a8f7e1e36f0a2a011": "ax^4+bx^3+cx^2+dx+e=0 \\,", "478bbe79be0745b1aa992e905c17fff8": "\\mathbb Z [A, A^{-1}]", "478be52dc90a987b179aa2d3e0f1cb61": "P_c^{ 0}(c) = c", "478c722e858da5071d633325f888b6a7": "km^2", "478c78bb28c564a4c265be435d39662b": "\\scriptstyle M \\;=\\; M_P \\,\\oplus\\, M_C", "478c981f3b36cc3836a98a4508a4efd4": " \\ z_d = (z - z_0)/(z_1 - z_0)", "478cab6ea676352fb7987b2e919d31bd": "\\alpha < \\alpha_{\\mathrm{max}}", "478ce964ea2931cfe8e39f26aac150c7": "\nJ(x)(f)=f(x),\\qquad f\\in X'.\n", "478d1454d879f8b347477be11d42fb14": "\n u \\wedge v = \\sum_{i 1 \\mbox{ for all } p \\in P", "479b1924cfc485f8457cf526d3ee1410": "a + c = b + d", "479be040a5c15716049d5aed5329af26": "B_3(t) + B_2(t)", "479c37910ecfa5673384a4ffd9f6df04": "f(x) \\approx f_n (x) = a_1 \\phi _1 (x) + a_2 \\phi _2(x) + \\cdots + a_n \\phi _n (x), \\ ", "479c49eef7c5ef1d261d08bdf734e368": "A=14", "479c6ca5c74ee8afc6800fb589e34972": "\\pi|_E : E \\to Z", "479ccb09275c22e68a1e327e84181244": "|E_{\\alpha} \\rangle", "479d304af27b44d4b7feb6cee11e3c44": "\\sup_{\\lambda\\in\\Lambda}f_\\lambda", "479d38fadc725c5fe77d308d3276bb64": "B\\subseteq X\\times Y", "479d7fff2a69993863f8550fda41aa74": "G_x(x,y)", "479d9760caa4a5e0cf26f63b7d5dcac1": "(1+n)+a=1+(n+a)", "479df2ffb3d6f1fc013f322dc6fbb2ce": " \\begin{align}\n \\min_{x,y^*} & \\sum_{i=1}^n\\left(\\frac{y_i^*-y_i}{\\sigma_i}\\right)^2 \\\\\n\\text{subject to } & F(x,y^*)=0 \\\\\n& y_\\min \\le y^*\\le y_\\max\\\\\n& x_\\min \\le x\\le x_\\max,\n\\end{align}\\,\\!\n", "479e4927911f8dab329f8ee928b65e3d": "\\begin{bmatrix} -\\frac12{\\boldsymbol\\eta_1}^{-1} \\\\[5pt] 2\\eta_2+p+1 \\end{bmatrix}", "479e965359de867cda44f166d6652ff9": "\\scriptstyle x^*=b", "479ed6059247b1a080bae989cca69bfe": "\\bar{X_t}", "479f184ac94b5cea5b063883c728f1df": "\nv_a + v_b = 8000\n", "479f18f1dea7fbb2a25eef0be5fb7790": "{P_l}^m(x) = (1-x^2)^{m/2} \\frac{d^m}{dx^m} P_l(x), \\text{ for } x \\ge 0", "479f8143f2b0bd95958e37cb1d8476fd": "\\Gamma\\vdash\\Sigma", "479fd7cf27e924e434ab5d2f0e267d14": "N(x) = \\bar x x = (x_0^2 + x_1^2 + x_2^2 + x_3^2) - (x_4^2 + x_5^2 + x_6^2 + x_7^2)", "47a008c1efb0da42b0c85252b3c6cad0": "i, k\\,", "47a0209f221aceb078afa5df82056020": "r_M", "47a04fbc4c574a006b084271afd2fc93": "\\mathfrak{a}\\oplus\\mathfrak{p}", "47a05c6bf43d747ae9fe5b302d545a75": "\\displaystyle\\chi", "47a06c8cfd2ac4bf48a052b7493d50b1": "d j = \\left( \\frac{\\partial F_1}{\\partial x}\n+\\frac{\\partial F_2}{\\partial y}\n+\\frac{\\partial F_3}{\\partial z} \\right) dx\\wedge dy\\wedge dz \n = (\\nabla\\cdot \\mathbf{F}) \\rho", "47a07be0ccb0daadef82131564be54ac": "v_p = \\sqrt{\\frac{C_k}{C_d}\\frac{T_s - T_o}{T_o}(k^*_s-k)}", "47a095fb3149a40f2368e3e32a686499": "\\psi(x)=\\partial_\\lambda \\varphi_\\lambda(x)|_{\\lambda=\\mu}", "47a11110714dc7bf2a6c2bcb238d7cfd": "H\\rightarrow G", "47a1373ec483eb9d11ee46e4da24382f": "p=h^2/\\mu\\,", "47a16b7e620843a89a1f82720b26aca7": "\\max_{j\\neq i} b_j > v_i ", "47a18c9457e77775a0f575024c7d7cff": "P_\\mathrm{tot} = \\int_{\\phi=0}^{\\phi=2\\pi}\\int_{\\theta=0}^{\\theta=\\pi}U\\sin\\theta\\,d\\theta\\,d\\phi;", "47a20784778b60750049f932b6510545": "h_\\lambda (X_0)=\\lambda = \\text{constant},", "47a236d44bd797e97c3a44c7fc9e17fc": "\\Gamma_{(y)}", "47a239366037cf884d9ef20239cc6849": "\n w^0_{,1111} + 2~w^0_{,1212} + w^0_{,2222} = 0\n ", "47a26f2f68247d1558ce977c32e90ebc": "A' = M_{11} = \\begin{pmatrix}\n-49 & -14 \\\\\n168 & -77 \\end{pmatrix}.", "47a29f216ad2844dbfe3682b354a9300": "\\langle a \\rangle ", "47a2cccb0585dc0a6720658ff0c6c473": "\\mathbf{f} = \\epsilon_0\\left[ (\\boldsymbol{\\nabla}\\cdot \\mathbf{E} )\\mathbf{E} - \\mathbf{E} \\times (\\boldsymbol{\\nabla}\\times \\mathbf{E}) \\right] + \\frac{1}{\\mu_0} \\left[(\\boldsymbol{\\nabla}\\cdot \\mathbf{B} )\\mathbf{B} - \\mathbf{B}\\times\\left(\\boldsymbol{\\nabla}\\times \\mathbf{B} \\right) \\right]\n- \\epsilon_0\\frac{\\partial}{\\partial t}\\left( \\mathbf{E}\\times \\mathbf{B}\\right)\\,", "47a2d2f05f7c00938826f686527e5154": "\\lambda= \\tfrac{1}{3} + \\sum_{k=1}^\\infty 10^{-k!}", "47a335509b82ddd622aa33987e24e746": "\\{\\ \\ , \\ \\ \\}", "47a3584321cdca882d29d66815f08e88": "\\textbf{P}_{0\\mid 0}", "47a35b885b153dcd2618914326fdf2f0": "A \\cap B = \\{y | y \\in A \\text{ and } y \\in B\\}", "47a377f6fa474d895100603f4ff439e7": "(i\\cdot dx, j\\cdot dy, k\\cdot dz)", "47a3b8e02069cc7f3b0accaec63002d7": "\\mathbf{P} = \\chi_\\text{e}\\epsilon_0\\mathbf{E}", "47a3c32442bede8f0915dd3c879f6bec": "\nX^{\\{q\\}}=\\bigcap^{\\{q\\}}X_{i}\n", "47a437ca72adc052bbb04650c4cc9054": "|z| > 1", "47a4852687655cbd2c9dd12542e36817": "\\begin{align}\n q &{}= w + \\bold{i}x + \\bold{j}y + \\bold{k}z , \\\\\n 1 &{}= w^2 + x^2 + y^2 + z^2 ,\n\\end{align}", "47a4eeb843a981b7506df8e6c091c935": "M(0,b,z)=1", "47a4f9cdddd3c092637212fcc3250578": "n=4 \\ \\ \\rightarrow l_4=0, l_{41}=0, l_{42}=0,", "47a4ff2d9125f5d4ff83dfe9110af530": " E_1,\\ldots,E_4\\ ", "47a51e703fe5a4c199a3818359055821": "\\mathbf x \\cdot \\mathbf y = \\sum_i h_i^2 x^i y^i = \\sum_i \\frac{x_i y_i}{h_i^2} = \\sum_i x^i y_i = \\sum_i x_i y^i", "47a56bb9ba5b55e85394e573ea64fd13": " \\begin{matrix} \\frac {(g_m r_\\mathrm{O}+1)R_C} {R_C+r_O} \\end{matrix} ", "47a589e0ffc8599541c1261f4adde433": "R(i) = R(-i)", "47a5a4036bacb7bcdc0ad6dfc9b30c13": "\\partial_n:C_n\\to C_{n-1},", "47a5b083da15813a8e99903078f6499d": "s_2 = 1001,", "47a5ba0791befba7d5f3c6efcdf357fd": "y_{n+1}^{(0)}=y_n+hf(t_n,y_n)", "47a5bb4c87b6a7e8f5af151f0c4bbc19": "S_{n+1}=S_nS_{n-1}", "47a5bfaeac943e2a24209d01df6a6a0a": "n_i(\\varepsilon_i) = \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}-1}", "47a7218e3fec707fac96a2a8566a84ae": "f(z) = \\sum_{i=1}^n z_i g_i(z)", "47a76ce25ec3aa6f7c5574d3c00beea5": "f(\\sqrt[3]z)\\text{ where }f(u)=e^u+e^{\\omega u}+e^{\\omega^2 u}=e^u+2e^{-u/2}\\cos(\\sqrt 3u/2),\\text{with }\\omega\\text{ a complex cube root of 1}", "47a7766cfddc928af83dbeb3d5baa796": "f'_n\\,", "47a7a58a41f509b2c60f662f0e47295c": "\\frac{1}{2} =\\frac{1}{2} \\Sigma_q + \\Sigma_g + L_q + L_g.", "47a7cd99d9ec644e33f48b6e11f42d92": "\\frac{dP}{dt}", "47a81f128e8a27cb26f7840ca8b9f8e2": "P' = \\left(\\frac{m}{n},0\\right).", "47a824cb4c2a2a360e1d015280c4e23f": "\\vec{e}_{i}", "47a82fbdf2db6ed3691184662db8c229": "\\text{return} \\colon T \\rarr W \\times T = t \\mapsto (\\epsilon, t)", "47a871abe787b2f10a56d7ab2c6ce894": "T_{xuw}(J^{r}\\pi)\\,", "47a87f2499e8c1ed40b3aaf4321f0fa0": "H = {\\partial (\\beta F) \\over \\partial h} = (1+A\\varepsilon) H + B H^3 + \\cdots", "47a9058cd052af6b3f9675807ef4373a": "M_y = \\frac{1}{a} \\frac{\\partial \\Phi}{\\partial y}", "47a90aecc89b3da1e04bf1317b3dc449": "AM\\,", "47a92e69e83f92572c3e3e380502fa6a": " a_{ij} = \\nu a_{ji} \\in \\mathbb{C}", "47a9350569ec92a8e0e2d1f8eec17eda": "d(p, A) := \\inf_{a \\in A} d(p, a)", "47a9507c7c13a4bde00246581659cde7": "\\left(\\nabla^2+ k^2 \\right) \\psi (\\mathbf{r})=0", "47a955e75c197763197749f9513c7511": "V_0 0 ", "47adb657e04836047b17595695bdb121": " \\sup_{x \\in \\omega} u(x) \\le C \\inf_{x \\in \\omega} u(x)", "47add446245aa288ff13c6029a538c37": "V_n(R) = R^n \\int_{-1}^1 V_{n-1}(\\sqrt{1 - t^2}) \\,dt = R^n V_n(1),", "47ae6d5f5cf64d7b8ea87e4c808830e7": "H_{\\mathbf{P}^n_S}^P", "47aeb35a81794d8cb98b45e6e6d335b6": " e^+e^- \\to \\pi^+ \\pi^- \\pi^+ \\pi^-,~~ \\pi^+ \\pi^- \\pi^0 \\pi^0 ", "47af0156a0c39ce1c7b9084bd8d82178": "\\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{\\cdot^{\\cdot^{\\cdot}}}}} \\,", "47af07e9dd6a019a1d90145eb4c1604e": "\\mathbf{P} \\simeq \\varepsilon_0 \\left( \\chi^{(1)} + 3 \\chi^{(3)} |\\mathbf{E}_0|^2 \\right) \\mathbf{E}_\\omega \\cos(\\omega \nt),", "47af3b5c1127e8a826b1da2eba41ba66": "f'(g(x)) g'(x) = 1.", "47af510ab71d0e1ef3e8530c70a8744e": "2.5 \\mbox{ meters} \\times 4.5 \\mbox{ meters} = 11.25 \\mbox{ square meters}", "47af66b66242d6de649b0eff645e8433": "\\{0(\\mathrm{mod}\\ {2}),\\ 0(\\mathrm{mod}\\ {3}),\\ 1(\\mathrm{mod}\\ {4}),\n\\ 5(\\mathrm{mod}\\ {6}),\\ 7(\\mathrm{mod}\\ {12})\n\\}.", "47b00f21b22ba7a733c53672bac9c5a8": "\\mathrm{SPF} = \\frac{\\int A(\\lambda) E(\\lambda)d\\lambda}{\\int A(\\lambda) E(\\lambda)/\\mathrm{MPF}(\\lambda) \\, d\\lambda},", "47b0673d990de79cf10dfdcb9d3387fb": "x \\geqslant \\mu\\,\\;(\\xi \\geqslant 0)", "47b069764df1810b55cc4943307bbded": "|j\\rangle \\mapsto \\frac{1}{\\sqrt{2^3}} \\sum_{k=0}^{2^3-1} \\omega^{jk} |k\\rangle, ", "47b09e52d4f3f664468451390610961d": "\\int_V (\\sigma_{ji,j} + F_i\\,) dV = 0\\,\\!", "47b0b782d5f654833b0c6b6db653f327": "R = 2\\sqrt{\\frac{Z_1Z_2e^2L}{E_0}}", "47b0ba393423fbaf19dda28575ed9c22": "\n\\epsilon_\\mu^2(n) \\!= \\!{1 \\over \\sqrt{2}} \\left(\n{{i n_1 n_2 \\!+\\!1 \\!+\\!n_3 \\!-\\!n_1^2} \\over {1 + n_3}},\n{{- n_1 n_2 \\!- \\!in_1^2 \\!- \\!in_3^2 \\!- \\!in_3}\n\\over {1 + n_3}},\n\\!-n_1 \\!+ \\!i n_2, 0 \\right).\n", "47b0cc888050f23f57a127f477a27634": "\\epsilon_{ist}", "47b0d47411a9ce8fb70476f15d54fc53": "(x_1, x_2, x_3)", "47b0f82391d6d9bec96792e64e1c7103": "\\epsilon_{ij}", "47b1595da363689b38cec91317c9c2e7": " u{\\partial u \\over \\partial x}+\\upsilon{\\partial u \\over \\partial y}=u_0{\\partial u_0 \\over \\partial x}+{\\nu}{\\partial^2 u\\over \\partial y^2} ", "47b2778a08c78385e4d0b58ca45b5c2d": "\\mu + \\frac{\\delta \\beta K_{\\lambda+1}(\\delta \\gamma)}{\\gamma K_\\lambda(\\delta\\gamma)}", "47b2970d696294052936a1374a897df1": "n_{L2} = n_{AlInP}^{1/2} \\cdot n_{L1}", "47b2e9d1280fad56553d17955569665d": "\\displaystyle{\\nabla D(\\varphi) = \\int_{\\partial\\Omega} \\Delta_z N(z-w)\\mathbf{n}\\varphi -\\widetilde{\\nabla}\\int_{\\partial\\Omega} \\partial_{t}N(z-\\mathbf{v}(t))\\,\\varphi \\, dt=\\widetilde{\\nabla} S(\\dot{\\varphi}).}", "47b347e544d88f98c798b1a8d8f3d970": "D_{RR}(X, X) = \\frac{2\\sigma}{\\sqrt{3}}.", "47b3f45fbc36ddc3e895834e7814ec7b": "\\Big( (P \\downarrow P) \\downarrow Q \\Big)", "47b40c1580db8277171de8402773b89f": "{\\color{Blue}~6.6}", "47b4a2ff47baa3ec10ee16353f25d6bd": "S^d \\leq (1+R)S \\leq S^u", "47b50b2aca61e72d2c5e78ab555d9b7b": "\\,\\frac{F_m-F_{tr}}{F_m-F_0}", "47b5267084e59f9fef43df721403e2a5": "\nb^{2} = -\\frac{\\Delta}{\\lambda_{2}D} = -\\frac{\\Delta}{\\lambda_{1}\\lambda_{2}^{2}}\n", "47b5adedc2f0d0f34afec34fd38fd8c5": "t_{i+1}=t_{i}+\\sqrt{(x_{i+1}-x_{i})^2+(y_{i+1}-y_{i})^2}", "47b5d799925cd23a2a746b0451724065": "\\int_0^1 E_n(t) E_m(t)\\,dt =\n(-1)^{n} 4 (2^{m+n+2}-1)\\frac{m! n!}{(m+n+2)!} B_{n+m+2}", "47b5d9e6bfe532365059ccabae703f13": "2 \\times \\sqrt{7}", "47b5e26f39b8d413a58d3e541bd0cfff": "O = \\psi_{1}+ \\psi_{2}- \\psi_{3}", "47b66ecfb09bcc92301d7f8c713ec2f1": "|\\mathbf{v}\\mathbf{w}|^2 = |\\mathbf{v}|^2|\\mathbf{w}|^2,\\,", "47b7183cdb5ff74270f4a03b2cf29a73": "\\hat{H} = \\hat{H}_{0} + \\lambda \\hat{V}.", "47b740036fb6f7cd476b95c5fb786c0c": "784=2^4 \\cdot 7^2 = (2^2)^2 \\cdot 7^2 = (2^2 \\cdot 7)^2 = 28^2. \\, ", "47b7975eda4add318c58ff4b2b76e214": "\\begin{align}\n\\pi_0 (O) &= \\mathbf Z/2\\\\\n\\pi_1 (O) &= \\mathbf Z/2\\\\\n\\pi_2 (O) &= 0\\\\\n\\pi_3 (O) &= \\mathbf Z\\\\\n\\pi_4 (O) &= 0\\\\\n\\pi_5 (O) &= 0\\\\\n\\pi_6 (O) &= 0\\\\\n\\pi_7 (O) &= \\mathbf Z\n\\end{align}", "47b7b37fc07a3ac2f7439c4ab25fc7b0": "GF(2)=\\{0,1\\}", "47b7c5856b114d9eaad2b775d9b623fd": "T(x,y) = 2\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} \\sin(nx) {\\sinh(ny) \\over \\sinh(n\\pi)}.", "47b83b5ef984df1dee47fc368e7e02e0": " U_i = \\alpha P_i + \\beta D_i + \\varepsilon_i\\, ", "47b84f8b925e12c0ad4eb03a6effa9c4": " f(x):= \\lim_{k\\to\\infty} f_k(x). \\, ", "47b869c9691a17be24a487f9584837f6": "\\vec{e}_3 = \\frac{1}{r \\, \\sin(\\theta)} \\, \\partial_\\phi ", "47b8b25c8027151ed69829625b131036": "\\boldsymbol{\\beta}", "47b8b6068f32189413ad806cf6b3972d": "L_zL_xL_z(y) = L_{zxz}(y)", "47b962b172f913ec55785937776b672a": "s_1 \\ge ... \\ge s_i \\ge s_{i+1} \\ge ... \\ge s_n", "47b9671ff1c795be21afcfafc96c37ff": "\\Psi_i\\mapsto-\\Psi_i", "47b9b73ec8fbd011a09dc6ac112b44e1": "\\langle X, \\mathcal{F} \\rangle ", "47b9d608d1f237749d7401861e6bec0a": "H_8", "47ba1d016d371dc3a11b2e908bf60f44": "I_{m,n} = \\begin{cases}\n -\\frac{\\sin^{m-1}{ax}\\cos^{n+1}{ax}}{a(m+n)}+\\frac{m-1}{m+n}I_{m-2,n} \\\\\n \\frac{\\sin^{m+1}{ax}\\cos^{n-1}{ax}}{a(m+n)}+\\frac{n-1}{m+n}I_{m,n-2} \\\\\n\\end{cases}\\,\\!", "47ba43b7b7d1ef62d91ef768a356e19e": "U(y,\\xi)", "47bab6f51feeb8a1620ee3a9e6402df9": "\\mathbb{F}_q^3 ", "47bae2d9da983541bbbd2090f8d39097": "\\textstyle{\\frac {\\log(6)} {\\log(3)}}", "47bb3e81319dbdc61a22fdcff23802f5": "P_0 + P_1 x_1 + P_2 x_1^2 + P_3 x_1^3 + \\dots + P_N x_1^N - f(x_1) = + \\varepsilon\\,", "47bb7e6dc29af9b28527739215e0f721": "S[\\mathbf{q},t] = \\int_{t_1}^{t_2}L(\\mathbf{q}, \\mathbf{\\dot{q}}, t){\\rm d}t", "47bb964433dd076db14632946639d4ae": " \\left\\{ H_n(i, j)~, \\, 1 \\leq i \\leq j \\leq n \\right\\} ", "47bbc3f231de08cc1f92c69dcea8737e": "\\delta = \\frac{n_x n_y}{n_x+n_y}\\boldsymbol{\\nu}'\\mathbf{V}^{-1}\\boldsymbol{\\nu},", "47bbe6934ea1a9576dca8da9195f1d16": "10^b", "47bc400cece3338fa67ca3a1cb9ee88b": " 0 = \\mathbf{x}_{k}^{T} \\, \\mathbf{H} \\, \\mathbf{A} \\, \\mathbf{y}_{k} ", "47bc907207e612ac1389ce92f35c3a43": " \\overline{AD}^{\\,2} = \\overline{AB}^{\\,2} + \\overline{BD}^{\\,2} \\ ,", "47bcd14471af729f5d7be1627a7390ad": "f^\\sharp", "47bcdcd7bcbf990c435227b4aa4912da": "dq", "47bd038ca6584a213a406da0e1869426": " \\mathbb C^{32} ", "47bd192c43dcef8eb8d04189299fe829": "\\,{^{n}0}", "47bd7f2569eb50400b070276bca1ff19": "\\theta_{13}", "47bd8484c1b397ee5611409289aa32b3": "T\\colon X\\to X", "47bdb0665031c324d8ab2bc864cf36c5": "(X^G)_p", "47be293f2103d7b1cf062a90c31b3888": "\\phi'_A \\, ", "47be2d293d5d74bfe5f3c7f9fe31e4e6": " u_i \\ge -[y_i - a_0 - a_1x_{i1} - a_2x_{i2} - \\cdots - a_kx_{ik}] \\,\\ \\,\\ \\text{for} \\,\\ i=1,...,n.", "47be50ae54e24d64a0ee22222863219e": "U_g f(x) = f(g^{-1}x),", "47bea40f0cdcd053a4288af8bf2ca28f": " |\\psi(t)\\rangle = a(t)e^{-i\\omega_0 t}|e;0\\rangle + \\sum_{k,s} b_{ks}(t)e^{-i\\omega_k t}|g;1_{ks}\\rangle ", "47beacc17bfc636360f6d67e908c3fc0": "2^{4 \\times 8} + 2^{4 \\times 8} = 8589934592", "47bebaa822ff8f61ee3ba21863c86b83": "f\\colon X\\rightarrow X", "47becd447984ee8cc6043a3c626f1333": "\\ell' = \\frac{\\ell}{\\gamma}", "47bf5c41dce5f5c21da2cc6d4fff43cc": " \\widehat{U} ", "47bf71a71fde83abf88af44ddd63d1dd": "\nS_i-S_j=x_{ij} \\sqrt {\\sigma_i^2 + \\sigma_j^2- 2r_{ij}\\sigma_i\\sigma_j} ,\n", "47bfa82f6551a8e51387f2380f2d30ae": "\\log^* n = \\lceil \\text{slog}_e(n) \\rceil", "47bfc10bf350b9d926e3347b634835a4": "E_k = {C\\over 2^{2/3}} \\left((N_p+N_n)^{5/3} + {5\\over 9}{(N_n-N_p)^2 \\over (N_p+N_n)^{1/3}}\\right) + O((N_n-N_p)^2).", "47bfc7d7045b013b73c3bf8d49f5beae": "\\alpha-1", "47c00a9c1f565b1545bcfc319417eccb": "\\ \\omega_{xy,R}^2", "47c00fb5fccff9f8232ea0bf096dd5ff": "\\mathcal{P}_p^r(X)", "47c06f628bfd260618190b4e9d91303e": " 1,2,3,\\dots ", "47c0dac78ee9db522d1f7e1bb5a66a02": " \\alpha, \\beta , \\kappa ", "47c0ddd405bad36242eb153eb5900539": " g_m r_O ", "47c1217b07ef7e24eec27a01272213eb": "(\\Omega, \\mathcal F),", "47c1278b3ccf5ad1986f50036dac195c": "M = \\mathbb{R}^n", "47c130200f58da6bc6cb409211a608a8": " r_{k+1}=0. ", "47c193f09d1d1e65d907ebfdf98ceb4e": "3\\uparrow\\uparrow\\uparrow n", "47c1f39b6fd2d887389cbe509105cd9b": "(1-x^2)^{3/2}\\,", "47c1f53f061b6292f03d03e6bd74fc3e": "\\tfrac{\\pi-1}{2}", "47c27c9324ebd0f0ad6c0efe92ccd280": "c_{0} \\left (\\int^{\\infty}_{0} f_k e^{-x}\\,dx\\right )+ c_1e\\left ( \\int^{\\infty}_{0}f_k e^{-x}\\,dx\\right )+\\cdots+ c_{n}e^{n} \\left (\\int^{\\infty}_{0}f_k e^{-x}\\,dx\\right ) = 0.", "47c282f3e1512adadd946637475e843c": "f[B] \\to f(x)", "47c2e60bbf9d9b8841c6782ca58a8c24": "W \\in\n(0,f_{N})", "47c3a0939d3cccff520805545e396065": "g(t)=\\log M(t),", "47c40de22783a1b81e0787f59324f72b": "\\gamma_j", "47c4259ae1d13c3ea80172e2cb9ee9bf": " y(t) = 19e^{0.85t}", "47c42dea58976890ddaef26e0f140980": "\n \\varepsilon_{x} = -z\\cfrac{\\mathrm{d}^2w}{\\mathrm{d}x^2}\n ", "47c47de445288206fa058d99dff5c9de": " u(x)=v(x)+L(x) ", "47c48e6e36bedbfdaeb2d36ff7eb32c9": "t=n-k", "47c4a73a321ab840c5aefeeda3c93405": " x[n] = \\mathcal{Z}^{-1} \\{X(z) \\}= \\frac{1}{2 \\pi j} \\oint_{C} X(z) z^{n-1} dz", "47c4ebede3dafad3fd2c8af622a2a6d8": "F_X(x;\\mu,\\sigma) = \\frac12 \\left[ 1 + \\operatorname{erf}\\!\\left(\\frac{\\ln x - \\mu}{\\sigma\\sqrt{2}}\\right) \\right] = \\frac12 \\operatorname{erfc}\\!\\left(-\\frac{\\ln x - \\mu}{\\sigma\\sqrt{2}}\\right) = \\Phi\\bigg(\\frac{\\ln x - \\mu}{\\sigma}\\bigg),", "47c4ff764cd3797a2906a618328b8d99": " f \\star e_i \\rightarrow f. ", "47c546aad42985e3b15afb4c484caa7a": "a_2=\\frac{1}{2}\\textrm{ln}\\frac{W_+(p\\wedge \\neg c)+1}{W_-(p \\wedge \\neg c)+1}", "47c57e31c2d2ea7eeb87332189994ada": "\\operatorname{supp}~f_{U}\\subseteq U\\,", "47c599077ae8346affea6dce286e8581": "\\mathbf{w} = \\mathbf{k}\\times\\mathbf{v}", "47c62e1d4dabea77aad3c14ede312dab": "L(\\theta,a)", "47c63fa5dedc6e70e9a4454caafa78cc": "\\mathcal{O}_X(U)", "47c6575510795fcb8a47eef4682b3202": "\n \\frac{d^2 y}{d t^2} \n = \\frac{\\partial^2 Y_0}{\\partial t^2} \n + \\varepsilon \\left( 2 \\frac{\\partial^2 Y_0}{\\partial t\\, \\partial t_1} + \\frac{\\partial^2 Y_1}{\\partial t^2} \\right)\n + \\mathcal{O}(\\varepsilon^2).\n", "47c6a83f04202b4528a5dbe5ece5258e": "X=\\{0,1\\}^n", "47c6ca1849e2872a8f9e51232560be14": " 0a", "47f60e8abe30c45647a876da26020a75": " {\\iiint\\limits_V {\\partial \\rho \\over \\partial t} \\,\\mathrm{d}V} = {- \\iiint\\limits_V\\left(\\nabla\\cdot\\mathbf{J}\\right) \\, \\mathrm{d}V}, ", "47f68326bc717096c494f0eaaa1dba8f": "I_{L1}=I_P\\sin\\left(\\theta-\\varphi\\right)", "47f7344666ae49808c314b5e18928862": " P \\lor Q, \\lnot P \\vdash Q ", "47f7512ac76513056a203f8620522bad": "(a,0)", "47f799dace32753740cab60b493b2863": "{E}_{10}", "47f80dcc3fe643bcd2b9e0e32472d25a": "Standard~deviation~(s.d.) \\approx \\frac{ | (Mean) - (Upper~limit) | }{2}. ", "47f85f2dbdc2d846f0524067d2ce0655": "w-c", "47f878db590c212d0826f8affce9a6bf": "\nT_2 > F_{\\alpha,k-1,bk-b-t+1}\n", "47f8cc55288df0135467122a043d2fe3": " = \\mathrm{kg{\\cdot}C^{-1}{\\cdot}s^{-1}} = \\mathrm{kg{\\cdot}A^{-1}{\\cdot}s^{-2}} = \\mathrm{N{\\cdot}s{\\cdot}C^{-1}{\\cdot}m^{-1}} ", "47f98f1f6150566e9a96a9b97e35eacb": "I_{2k}", "47f9b842caf3205af5780a78f04ce4f0": "C_e = (A^TC_Z^{-1}A + C_X^{-1})^{-1}.", "47f9c65d24fd3accc39aa8d857df2c3f": "U \\times Y", "47fa03ef5536c5573b137cec5adc2d39": "-v", "47fa19f9c9dc6b047da619987b7b7867": "z=re^{ i \\theta}", "47fac45b1174139bd8dae7d2c3c28d7f": "\\alpha\\neq\\alpha'", "47fad2c059b125d6aa0938dd6b22824a": "L_1,L_2", "47fadd88e293e8b9545d352c55a561c6": " [ \\mathbf{BB}^T ]_{ik} = \\sum_{j = 1}^R S_{ij}S_{kj} f'_j (\\mathbf{\\phi}) = [ \\mathbf{S} \\, \\mbox{diag}(f(\\mathbf{\\phi})) \\, \\mathbf{S}^T ]_{ik}. ", "47fafda19dd4c06faac1488ad0ef6ee9": " g(x) = ax^2+bx+c ", "47fb1e587da664f09a87ecf76ad114aa": "\\lim^* f_{x,H}(c)=f(c)", "47fb5903c979f9e57e7028a7c17cab1f": "X = \\{X_{1}, X_{2}, ..,, X_{k}\\}", "47fb837b31c5ac0099830338c205b216": " \\sqrt[\\infty]{x}_s = x^{1/x} ", "47fbb01c5d8a62d6f84cec91c7970edc": "\\gamma\\in[0,1]", "47fbbc9fa94d047b8ba65b66e4c64ddc": "k(A)", "47fc1bb9605758cd37a53b639f385d3f": "\n\\begin{pmatrix} X_1 \\\\ \\vdots \\\\ X_n \\\\ Y_1 \\\\ \\vdots \\\\ Y_n \\\\ Z_1 \\\\ \\vdots \\\\ Z_n \\end{pmatrix}\n= \\bar{M} \\begin{pmatrix}1 \\\\ \\vdots \\\\ 1 \\\\ 1 \\\\ \\vdots \\\\ 1 \\\\ 1 \\\\ \\vdots \\\\ 1 \\end{pmatrix}\n+ \\begin{pmatrix}\\bar{X}-\\bar{M}\\\\ \\vdots \\\\ \\bar{X}-\\bar{M} \\\\\n\\bar{Y}-\\bar{M}\\\\ \\vdots \\\\ \\bar{Y}-\\bar{M} \\\\\n\\bar{Z}-\\bar{M}\\\\ \\vdots \\\\ \\bar{Z}-\\bar{M} \\end{pmatrix}\n+ \\begin{pmatrix} X_1-\\bar{X} \\\\ \\vdots \\\\ X_n-\\bar{X} \\\\\n Y_1-\\bar{Y} \\\\ \\vdots \\\\ Y_n-\\bar{Y} \\\\\n Z_1-\\bar{Z} \\\\ \\vdots \\\\ Z_n-\\bar{Z} \\end{pmatrix}.\n", "47fc2dd771876860fd3aed5de128bf78": "W^{\\mathfrak{p}}", "47fc71f0e2a0afe947f17ee1c811219a": "\\displaystyle x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}.", "47fc948af39028040b7862bd21baf781": "x_{i.} (i=1\\ldots k)", "47fca8208c4b0683109e26724c1fa156": "\\ln n\\prod_{p\\le n}\\left(1-\\frac1p\\right)-e^{-\\gamma}", "47fcbd7cb367f5d92191350a6a0c8748": "{\\mathbb E}[XY]\\le{\\mathbb E}[X^*Y^*]", "47fcd35e896055832245d005804a42b2": " f(r)= \\int {\\frac{\\sqrt{\\frac {2M}{r}}}{ 1-{\\frac{2M}{r}}}} dr = 2M\\left( 2y+\\ln \\left({\\frac{y-1}{y+1}}\\right)\\right)\n", "47fd21a835b83f58f2f67b7e1473a370": "\\varphi_i(x,y)=const", "47fdf109dbf1c697287d226d53dceb87": "E_+ \\geq E_- ", "47fe0abc7bafef0c3d444b13f20315eb": "3x^2 - 2xy + c", "47fe3731842daada45e7dd89a9d6eb34": "\n\\Delta n \\Delta \\phi > 1\n", "47fe7079697c684647dd30a85a0e2201": "\\begin{align} 2\\cdot R_*\n & = \\frac{(170\\cdot 1.14\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 42\\cdot R_{\\bigodot}\n\\end{align}", "47ff165f590e869977b5a0502f22f29b": "h^a_{\\;\\;b}", "47ff7503b7c76848bd88339ff07df8a8": "\\hat{\\alpha}>\\hat{\\beta}", "47ffe9e6c393a799f6fee92f3022ec9f": "20712^2 \\mod 84923 = (2^5 \\cdot 3 \\cdot 5^2 \\cdot 7)^2 \\mod 84923 = 16800^2 \\mod 84923.", "48001d9cf40f37e6f7bbc4d81ba02df4": "f(0,0) = 0", "4800369a4387c7d2778df9efd705f449": "2r-H=\\frac{W^2}{4H}", "4800bc66b5ae9ddea2e175dcf0957149": "f_{k-1}(x)= q_{k-1}(x)f_k(x) - f_{k+1}(x)\\,", "4800f0d1eef4b2e0eaecfb543c83ff0f": "B_k \\mathbf{p}_k = -\\nabla f(\\mathbf{x}_k).", "48013197692ad877450b53e4b5f9ad40": "g^m(n)", "480154092db083c9ff79caf902efe2d7": "f(-1/z) = z^k f(z)\\,", "48015ca6ebed99594919fd98c181fd2a": "(a+b)^p \\equiv a^p+b^p \\mod p", "48019322b071dd6cef4524e9ed3bd21e": "s_2 = c_1e_2", "4801e3252fc9aad9d9e2e180d20a17c4": "GF = \\frac{\\frac{\\Delta R}{R}}{\\varepsilon} = \\frac{\\frac{\\Delta\\rho}{\\rho}}{\\varepsilon} + 1 + 2\\nu ", "48025b7b4e1db50d36d1b2c472902ff8": "d_0", "48026a523cf7040d50a9121b436909e5": "\\Delta G_{mix} = - T\\Delta S_{mix}\\,", "480295c820c1c6ab23290fd19613a8ae": "\\psi_{4} = 4y(x^{6} + 5Ax^{4} + 20Bx^{3} - 5A^{2}x^{2} - 4ABx - 8B^{2} - A^{3}) ", "4802ad6a7d81a4e1aee74d7056b27280": "\\boldsymbol {\\beta} \\geq \\boldsymbol{0}", "48031871c9e1a09041638f089fb152c4": " n>1", "48038b2001efa7fdbfb51f4125648945": "\\begin{matrix} {9 \\choose 1}{4 \\choose 4} \\end{matrix}", "4803a24d7bfbdf3e97f6d4e1a2c6f9d3": "\n\\begin{align}\ni_{\\alpha}=&\\sqrt2 I\\cos\\theta(t),\\\\\ni_{\\beta}=&\\sqrt2 I\\sin\\theta(t),\\\\\ni_{\\gamma}=&0,\n\\end{align}\n", "4803e464cb0d51150b9d3b5706f80ff8": " s \\in S ", "4804116ccb5c0fa448ecaae5f2eb2aba": "\\Sigma_{ij}=E[X_i X_j],\\qquad \\forall i,j \\in \\{1,\\ldots,N\\}", "48042503d1e534850520aed7f0782765": "\\delta_{P}(t)= \\left( \\frac{R_{P}(t)}{R_{std}} -1 \\right) 1000. \\qquad \\qquad (4b) ", "4804517e21a502feb71d6cb618ebc9fb": "g(v,w) = \\sum_{i,j=1}^n v^iw^jg\\left(X_i,X_j\\right) = \\sum_{i,j=1}^n v^iw^jg_{ij}[\\mathbf{f}]", "48045f083191f6bdd008ffba3e2fd2a1": "\\frac{R\\sin\\phi}{a^2} - \\frac{Z\\cos\\phi}{b^2} = 0;", "48047f6d20aca878865170c4944cb271": "f^{-k}\\,", "4804bc1c6b2c333a455bc9a8756cad73": " \\operatorname{sech}\\left( (2n+1)\\, \\frac{\\pi\\, K'(m)}{2\\, K(m)} \\right) = 2\\, \\frac{q^{n+\\tfrac12}}{1 + q^{2n+1}} ", "4804d40e0ddcf26e5983050bde48f1d2": "(u_n)_{n\\in\\mathbb{N}}", "4804e9e4c81d4dd3a90617e6f9089732": "(6,-4)", "48054aa71ac9bcc7c2412b71068fc7fb": "\\hat{\\mathbf{u}}", "48057181c1c433bdbc6b2b798038603c": "\\begin{align}\nP_2 &= [P_0(1+r\\Delta t)-M_N\\Delta t](1+r\\Delta t)-M_N\\Delta t\\\\\n&= P_0(1+r\\Delta t)^2 - M_N\\Delta t(1+r\\Delta t)-M_N\\Delta t\n\\end{align}", "4805a37b95319f7db46e2a463c535c0d": "X\\subseteq M", "4805b1e8b9ce12565aa651e5b049d8c8": "{\\Bbb R}^3", "4805e0767fd2015bb664f74968b7b45b": "\\cap\\mathcal{S}_{drs}", "480628e469317473f5b7c627a61d9955": "V:", "480653d3e98246f9344335eefb01ca09": "\\,\\,\\boldsymbol{\\sigma} = \\mathsf{H}:\\boldsymbol{\\varepsilon}\\,\\,", "48070e2d87b93a9e2e1f17fec6617dac": "V_{ad}", "48072874b13e907690f64bebaf8a7e29": " \\epsilon = \\frac{v_a^2}{2} - \\frac{GM}{r_a} =GM \\left( \\frac{2a-r_a}{2ar_a} \\right) - \\frac{GM}{r_a} ", "4807375338d850145d8dfc15913596c1": "A_\\mathrm{surf} = 4\\pi r^2 ", "480753152e6494cbf1b30dc1269090df": "(g^{\\mu\\nu}) = \\mathrm{diag}(0,1,1,1)", "480794c1e1f861fd3d676e26a2eec147": "V^{\\otimes n}=\\underbrace{V\\otimes V\\otimes \\cdots\\otimes V}_{n\\text{ times}}", "48079d4b466d129636f9f47a8283c57c": "\\lim_{h\\to 0} f(x+h)-f(x-h) = 0.", "4807cd261a993d810c3ed2956dbce805": "\n\\frac{1}{2m} \\left( \\frac{dS_{r}}{dr} \\right)^{2} + \n\\frac{L^{2}}{2m r^{2}} + U(r) = E_{\\mathrm{tot}}\n", "4807d3d41476a756744833635116cb55": "=\\pi n^2 dS \\sin^2 \\alpha \\ ", "48081f48fb2c48c325b9c35dda8cd5ae": " S_{\\Gamma_0}", "480854bd8cc7bf85a7f0fa79feae0831": "V(S_{T,L})=\\{ z\\in V(\\mathbb{H}) : 0 \\leq Re(z)\\leq \\frac{3T+1}{2}, \\; |\\sqrt{3}Im(z)-Re(z)| \\leq 3L\\}. ", "48089c56d40cdfbdff200f1870b648fc": " v = \\frac{\\text{const.} \\cdot S}{1 + \\frac{S}{K_1} + \\frac{P}{K_2}}\\ (\\ast) ", "4808b608aa5af3c55583eb67f5af3394": " \\text{variance} \\le \\frac14 (M - m)^2. ", "4808cc25c2fa830509a17fef4334c1db": "\n\\begin{bmatrix}\n1 & -1\\\\\n-1 & -1\\end{bmatrix}\n", "4808f99aa91bb172378c7eed0ecc2b47": "a^n = \\underbrace{a \\times a \\times \\cdots \\times a}_n", "480922192539746f5039cb2af6a30022": " D_a T^{b \\dots c}_{d\\dots e} = \\nabla_a T^{b \\dots c}_{d \\dots e} = T^{b\\dots c}_{d\\dots e;a}", "48093628108aabb5840c946adda5439c": "\\operatorname{E} \\left[ ( {\\theta_i} - {\\hat \\theta_i})^2 \\right].", "48096bfeaadaebbda88c7518f144e40d": "G=F_m/\\langle r_1,\\,r_2,\\,\\ldots,\\,r_k \\rangle.", "48099ce31bcd5ea8fc166fd2a54b4962": "\\alpha\\div\\beta = \\frac{\\alpha}{\\beta} = \\frac{ai}{bi} = \\frac{a}{b}", "4809a2b3f06c7078745f387fcc586d6b": "m_\\alpha", "4809e3422b7ae0a77b416c58253fe6ad": "a = x_0 < x_1< \\cdots < x_{N-1} < x_N = b", "4809f0d439c564fb50cc53e07fd715c1": "P \\circ Q = P \\circ (x^2+1) = (x^2+1)^3+(x^2+1) = x^6+3x^4+4x^2+2", "480a0922e5767dd6db7f2bb574a42bbf": "b^{b - 1} + \\sum_{d = 2}^{b - 1} db^{b - 1 - d}", "480a2f0356b0dfce5a5c0b41386ea9a0": "\n\\mathrm{Ca}=\\frac{p - p_\\mathrm{v}}{\\frac{1}{2}\\rho V^2}\n", "480a8faaf3e4ce60f5af9d39c31675a1": "Q=1/2", "480abfb907a216bb44f0246104152c86": "|G|,s = 2~mod ~4", "480ace397c1f31cfa8e82691bdca0709": " \\omega = \\frac {1} {R C} \\rightarrow f = \\frac {1} {2 \\pi R C}\\,", "480adf621a436b15654c16f155551e6f": "\\nabla^2 \\varphi' = -\\frac{\\rho}{\\varepsilon_0}", "480aef07c1e9f44610c311e552e81f08": " 95%PI = mean\\pm t_{0.975,n-1}\\sqrt{\\frac{n+1}{n}}sd,", "480aefe2a870d0b2579fd74090e12d03": "\\varphi(x_i) = \\sum_{j=1}^n a_{ij}x_j", "480b3a5a5468e955e9c0b1d0cf980ea4": "V_{mn}(r,\\theta)", "480b6632f2aa479aa68f34f5e4a943d5": "\\hat{O}' \\Psi [\\gamma] = \\int [dA] s_\\gamma [A] \\hat{O} \\Psi [A]", "480b9df834ca7d6c88085286421a520f": "\\displaystyle{}_{r+1}v_r(a_1;a_6,...a_{r+1};\\sigma,\\tau;z) = \\sum_{n=0}^\\infty\\frac{[a_1+2n;\\sigma,\\tau]}{[a_1;\\sigma,\\tau]}\\frac{[a_1,a_6,...,a_{r+1};\\sigma,\\tau]_n}{[1,1+a_1-a_6,...,1+a_1-a_{r+1};\\sigma,\\tau]_n}z^n", "480bad51b94e403b3b9453c7a9b74db7": "\\frac{\\partial U}{\\partial T}\\ ", "480c5558f9f26c62082bae0fc3613a94": "\\cos\\theta=x/r", "480cc18c3337228276453c9b1289aa1c": "T^{00} = {\\epsilon_0 \\over 2}\\left({E^2 \\over c^2} + B^2\\right),", "480cedcb24d8dcb1d513b41a4e74caca": "\\mathfrak{p}_1 \\subseteq \\mathfrak{p}_2 \\subseteq \\cdots \\subseteq \\mathfrak{p}_n", "480cee337c56e218134326ac0e7b2114": "J_{ij}=J(i-j)", "480d01a9bf7d69cd1ba43e988bace27b": "\\scriptstyle{\\pi_i(\\cdot)}", "480d77fbc95b56b9b4162be4ec347b84": "|A_m|^2 = \\frac{2 \\eta_0 |\\beta_2|}{T_0^2 n_2 k_0 n}", "480da723a99d5ca44df1b63c5e05fe3e": "W^-\\to e^- + \\bar\\nu_e~", "480dad001c40c5ed7e00e90799d7b9ee": "\n\\sqrt[n]{z}=x+\\cfrac{2x\\cdot y}{n(2z - y)-y-\\cfrac{(1^2n^2-1)y^2}{3n(2z - y)-\\cfrac{(2^2n^2-1)y^2}{5n(2z - y)-\\cfrac{(3^2n^2-1)y^2}{7n(2z - y)-\\ddots}}}}.\n", "480ddfe1bfed8a2aa532f0bfc0c2e15d": " \\scriptstyle \\phi \\in\nC_c^1(\\Omega,\\mathbb{R}^n)", "480de7b57fe2458c6b9815017450ff59": "\\forall_{x,y} (G(x, y) \\Rightarrow x \\neq y).", "480e0a9e2d6cce30bcfb79785c9f1f1e": "[b_k , b_{k'}^{\\dagger} ] = \\delta_{k,k'} , [b_k , b_{k'} ] = [b_k^{\\dagger} , b_{k'}^{\\dagger} ] = 0.", "480e5b8be0b8f3694d2ae55f5d529cc1": "\\scriptstyle\\vec{M}", "480e732a614567a475a0fac90114e29d": "I(T_M)", "480e79ed0f272bf338d0ff96a32eb4e2": "\\lim_{k \\to \\infty}\\zeta(n,k) = \\zeta(n)-1", "480e925c512aa8cef79f8162a0fef072": "\\pi r^{2}", "480ea37110e8e370c6d754dccc53786d": "\\varphi(z) = \\frac{1}{(\\varepsilon \\alpha^{\\nu(z)})}", "480eaece9b157d209fa076b30048210e": "(T_x\\ ,\\ T_y)\\ =\\ (A_x\\ ,\\ A_y+V_t t)", "480edc5ca2804d863d740bef77c4c91c": "M_{D} = M", "480f14a48edd5717686b7ae83de206e5": " {\\mathbf u}_1,{\\mathbf u}_2,{\\mathbf u}_3 ", "480f2ebc8856c867bf0185ce64905295": "\\prod_{1\\leq i2", "48168b2e8a498a53ecfc023f8733e3da": " \\sum D(n) x^n = (1-6x+x^2)^{-1/2} ", "4816a442e43aa79daac802c954948072": " \\omega _1^2 = \\omega _0^2 + 2\\alpha\\theta", "4816d3565d5e5982219591a681589a66": "\\left|s\\mathbf{I}-\\left(\\mathbf{A}-\\mathbf{B}\\mathbf{K}\\right)\\right|=\\det\\begin{bmatrix}s & -1 \\\\ 2+k_1 & s+3+k_2 \\end{bmatrix}=s^2+(3+k_2)s+(2+k_1)", "4817008367e63c2a6d4998a3d4909bde": "\\hbox{Scope}=\\tfrac{\\hbox{RawScope}}{\\hbox{MaxScope}}", "4817221f8ad582128ea2509babb37117": "\\frac{\\left(\\frac z 2 \\right)^\\alpha} {\\Gamma(\\alpha+1)} \\frac 1 {t-z}= \\sum_{n=0}O_n^{(\\alpha)}(t) J_{\\alpha+n}(z),", "4817264190d0db4053937360ac72e56e": "P^0(\\cdot)", "4817f83c3bbe226af65e5e50e66b0971": " \\xi_t'= (g_1(x^*_t), \\cdots ,g_k(x^*_t), w_{1,t}, \\cdots , w_{l,t}).", "48180b25f8ee788a069cf81a27258d38": " ((\\mathbf{\\lambda} x . A(x)) t) ", "481822ae4e5898521e83b6656e6b116d": " \\nabla \\cdot \\mathbf{J} + \\frac{\\partial (\\nabla \\cdot \\mathbf{D})}{\\partial t} = 0. ", "481837fb4aa0cd3bb43ac0817b730624": "S_1 := \\emptyset", "48183a4b442129492a391e6d3f5e62b4": "\\nabla \\times \\frac{1}{\\mu} \\nabla \\times - (\\omega^2/c^2) \\varepsilon", "48184bf7ae51a2d2ad4e2ea887adacb3": "1/2\\sqrt{3}", "48186b8db4990bc5a37e61087e5fcc43": " C^J_{v_1} + C^J_{v_2} = 1 ", "48195fd2737c691508c9b42e0a3450ab": " w \\wedge u \\wedge v = \\frac{1}{2}( w ( u \\wedge v) + ( u \\wedge v) w)\n", "481971da67018c40a4458b73c0f3f46b": "\\hat{y}_d", "481a4a9fe50543e6584752b6936db314": "\\scriptstyle x\\in M", "481a7d399fc0d2c33fa2f765e4df37e2": "\\displaystyle T=\\frac{ab}{2}", "481a847b1ac0f6d061ca99bd7787c86d": "\\lfloor\\lambda\\rfloor, \\lceil\\lambda\\rceil - 1", "481a9cf2877bcaf67155136034825bda": "\n \\begin{bmatrix} \\sigma_1 \\\\ \\sigma_2 \\\\ \\sigma_3 \\end{bmatrix} = \n \\tfrac{1}{\\sqrt{3}} \\begin{bmatrix} \\xi \\\\ \\xi \\\\ \\xi \\end{bmatrix} + \n \\sqrt{\\tfrac{2}{3}}~\\rho~\\begin{bmatrix} \\cos\\theta \\\\ \\cos\\left(\\theta-\\tfrac{2\\pi}{3}\\right) \\\\ \\cos\\left(\\theta+\\tfrac{2\\pi}{3}\\right) \\end{bmatrix}\n = \\tfrac{1}{\\sqrt{3}} \\begin{bmatrix} \\xi \\\\ \\xi \\\\ \\xi \\end{bmatrix} + \n \\sqrt{\\tfrac{2}{3}}~\\rho~\\begin{bmatrix} \\cos\\theta \\\\ -\\sin\\left(\\tfrac{\\pi}{6}-\\theta\\right) \\\\ -\\sin\\left(\\tfrac{\\pi}{6}+\\theta\\right) \\end{bmatrix} \\,.\n ", "481b4767ded19819297abf8a3394a99e": "\\mathcal{D}(e,Z)=Z", "481b6baa5c6a37abf7a5d756440bc864": "n_3\\,\\!", "481b7852b2df8c0c7a6a9b54de50cc7a": "\\scriptstyle Re\\,", "481b86ea78c34c05df59304ba9afa7da": "\\,\\!\\phi^*", "481b9cade0e63a47bacd59d415bccbab": "\\textstyle z_\\mathrm {mn}", "481b9e76259b839600274e66c18f0501": " g(E) = \\int \\delta (E(x)-E_0) \\, dx= \\int \\delta (x^2-E_0) \\, dx,", "481bc58710db1f0763d86133d71d86a6": "\\left( x_1,y_1 \\right)\\ = \\left( \\frac{-b_1m}{m^2+1},\\frac{b_1}{m^2+1} \\right)\\, ,", "481bd27c9959ec541b9af3777da53f7f": " \\lVert f_{r}-f_{1}\\rVert_{L^{p}(S^{1})}\\rightarrow 0", "481bf4a75e7d1f6e21812ccdd37d74fe": "\\frac{dx-dX}{dX}\\frac{dx+dX}{dX}=2\\varepsilon_{ij}\\frac{dX_i}{dX}\\frac{dX_j}{dX}\\,\\!", "481c167120d372dd332d943fd5152188": " \\operatorname{Re}(\\epsilon (\\mathbf{q}, \\omega)) ", "481c1b8a5611e2b654f3661795396a07": "v[\\mathbf{f}A] = A^{-1}v[\\mathbf{f}].", "481c2a90f08cb476763fee54aac2934c": "s_0 \\ \\rightarrow_R \\ s_1 \\ \\rightarrow_R \\ s_2 \\ \\rightarrow_R \\ \\ldots ", "481c62dfd8bde9942b872842a8db19b9": " \\mathbf{y}' ", "481cdeaa0672cd74cf541813308fe80b": "A = (1+ C_A)(1 - C_A)^{-1} \\,", "481d0e831da5203b7c716f8e0bad2726": "\\min(k,n-k)", "481d118bcfd26110d40e0abdbf1c808c": "(a_{1}/b_{15})=10(2/3)", "481d180c0248cb6ee7794a895ab1aed6": "D=\\begin{pmatrix}\n\\pm 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & \\pm 1 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & \\pm 1 & 0 \\\\\n0 & 0 & \\cdots & 0 & \\pm 1 \n\\end{pmatrix}", "481d53f4a8e02605e22956183c239b32": " \\delta_a(x) = \\begin{cases} 1, \\quad\\mbox{if } x=a; \\\\ 0, \\quad\\mbox{if } x\\neq a \\end{cases} ", "481dd2bd5b3b851030862f7516b5adc6": "F(w)", "481de216875716a12cc2c7fc807896f2": "\\ell_2^k", "481e03294ff5e31c7cc305525304bf22": "\\Pi^1_n \\subset \\Pi^1_{n+1}", "481e425b4309bdaf02916094f397e3e7": "x \\stackrel{*}{\\rightarrow} y", "481ecfd6b0bbe36285c218faacee1a2e": "\\frac{\\nu\\!+\\!1}{2}\\left(\\psi\\left(\\frac{\\nu\\!+\\!1}{2}\\right)\\!-\\!\\psi\\left(\\frac{\\nu}{2}\\right)\\right)\\!+\\!\\ln \\sqrt{\\nu} B\\left(\\frac{1}{2},\\frac{\\nu}{2}\\right)", "481ee2ec104b2e80b47c61a094d1fa23": "S={1\\over 16\\pi G}\\int (R-2\\Lambda)\\sqrt{-g}\\,d^4x \\, +S_m\\;", "481f980f71c9ea2abf4e28dcdcd82ac2": "F^{(loc)}_1(X)=\\frac{1}{N}\\sum_{i=1}^NF_{1,i}(X)", "481facc882fdb390cdbdfc45854e8409": "\\lambda_{\\min}(A) ", "481fd211ee902301172c9c9be363e2f3": "j_c", "481fdd98a8faa1a0d8a670dd5730dec6": "C \\in \\C", "481ff069ed1d66269509cb6b78382810": " T(h,a) + T(ah,\\frac{1}{a}) = \\frac{1}{2} \\left(\\Phi(h) + \\Phi(ah)\\right) + \\Phi(h)\\Phi(ah) \\quad \\mbox{if} \\quad a \\geq 0 ", "481ff2f9c94b487f20868ff5d19bc538": "C_t = \\beta Y_t+\\epsilon_t", "48201a4159c35a912875d8164fa537fd": "V ", "4820297153f569d55331307949eda475": "\\forall y ( \\forall x Pxy \\to Pty)", "48206f84b34fd317e794a32f4440753f": "H = \\frac{F d}{e} ", "48209354f04bf3a5f48b96f67b3c2a2b": "f_{a}(x) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}}", "4820a8f68ff260097e1b0ee3b3907d1f": "( \\cdot, \\cdot )", "4820b40b3888bf2230e6b05f2f19071b": "\\mathrm{c.c.}", "4820c7a7a659422a7890722fb3366c4e": "\\left| j_2 , m_2 \\right\\rangle \\left\\langle j_1 , m_1 \\right|", "4820f09fbba67ef3ecf6d6dd6a46f29c": "(x_0,y_0,\\lambda_0)", "4820f3814d831e0cbda9e961372f6b29": "|Q^{(i)}_c \\rangle", "4822a2cb501ec825c4a8cb0266e9f36a": "\\Pr(Y_i=0) + \\Pr(Y_i=1) = 1", "4822cb4b28e3a6dd338683a5f7bfa416": " \\models_{\\mathcal S} \\varphi\\ \\to\\ \\vdash_{\\mathcal S} \\varphi", "4822d9c7a5e060a27421fe1e73900307": "\\frac{P_c}{K}= \\frac{1}{s_c - s_w} \\cdot \\frac{I}{K}- \\frac{s_w}{s_c - s_w} \\cdot \\frac{Y}{K}", "48232394ed070e218db731656ada1a74": "\\frac{dX}{dt}=U,", "482323cd4e318fddc68101b198cc631a": "g, h \\in G", "48236080fde3bf573dd248eee0d44ccc": "{E_2 \\over E_1} = {(8Fr_1^2 + 1)^{3/2} - 4Fr_1^2 + 1 \\over 8Fr_1^2(2 + Fr_1^2)}", "482393bc07afd1e7429b6322155c223c": "S(\\alpha)=\\sum_{n=1}^N\\Lambda(n)e(\\alpha n)", "4823bd4e5bd8cba1c5c6525485d7ada9": "\\parallel_+", "48244278fe08faf4d9756378ab7b8b11": "\n~\\epsilon_t = ~\\sigma_t z_t\n", "4824b38a578b272d3874675194dd1fae": "\\operatorname{stsys_2}", "4824b40cf7bc52c23195233415561a0c": " m \\frac{\\mathrm{d}^2 \\vec{x}(t)}{\\mathrm{d}t^2} = -\\nabla V(\\vec{x}(t)) ", "4824e99bd81e56c166c976f2477fd4a8": " \\partial_y, \\; \\partial_z", "482501b76e23b973767f0a6ab39f1321": " = \\int_{-\\infty}^t G(t-\\tau,\\xi)F(\\tau,\\xi)\\, d\\tau ", "48250c0df56646448022a403c7054493": "\\sigma_1^2.\\,", "482559e94c611901a287bb6e1ae342bb": "H = {50 \\choose 2}{48 \\choose 2}{46 \\choose 2} \\div 3! = 238,360,500", "4825e9950aa0c81fc9b8fc6d343898a9": "(\\mathcal{L}_Y T)(\\alpha_1, \\alpha_2, \\ldots, X_1, X_2, \\ldots) =Y(T(\\alpha_1,\\alpha_2,\\ldots,X_1,X_2,\\ldots))", "4826253d8469711b97334e0a0ce81db8": "\\cap_{u \\in H} u^{-1}(V)", "48264c4a79650e6a54d8f28134b9e98a": "\\|x+y\\|\\le \\max \\left\\{ \\|x\\|, \\|y\\| \\right\\}", "48265c1228b3e12413ad48e2f435c431": "= {(a+b\\varepsilon)(c-d\\varepsilon) \\over (c+d\\varepsilon)(c-d\\varepsilon)}\n= {ac-ad\\varepsilon+bc\\varepsilon-bd\\varepsilon^2 \\over (c^2+cd\\varepsilon-cd\\varepsilon-d^2\\varepsilon^2)}\n= {ac-ad\\varepsilon+bc\\varepsilon-0 \\over c^2-0}", "4826e6ebf2325365280b91dff845bf7b": "1 - \\frac{(m)_n}{m^n}, \\!", "482758f8c4d7198d52ae4d1b037fb6a6": "\n\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}=\\frac{1}{b_{21}}\n\\begin{bmatrix}\nb_{11}&b_{12}&b_{13}\\\\\nb_{21}&b_{22}&b_{23}\\\\\nb_{31}&b_{32}&b_{33}\n\\end{bmatrix}\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}=\\frac{1}{0.17697}\n\\begin{bmatrix}\n0.49&0.31&0.20\\\\\n0.17697&0.81240&0.01063\\\\\n0.00&0.01&0.99\n\\end{bmatrix}\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}\n", "48278279307e568cc71a48dac5ac9f63": "\\begin{matrix}\n\\mathrm{person} & \\mathrm{year} & \\mathrm{income} & \\mathrm{age} & \\mathrm{sex}\\\\\n1 & 2001 & 1600 & 23 & 1 \\\\\n1 & 2002 & 1500 & 24 & 1 \\\\\n2 & 2001 & 1900 & 41 & 2 \\\\\n2 & 2002 & 2000 & 42 & 2 \\\\\n2 & 2003 & 2100 & 43 & 2 \\\\\n3 & 2002 & 3300 & 34 & 1\n\\end{matrix}", "4827cc7a7524f3861648270355c06559": "(R=r)", "482802ee61da1a7a35839d93c495b902": "e(x) = \\sum_{i=0}^{n-1} e_i x^i ", "48285db4c2cef35b19c4acc1d7fe2950": "\n \\begin{align} \n &\\cfrac{\\partial N_{11}}{\\partial x_1} + \\cfrac{\\partial N_{21}}{\\partial x_2} = 0 \\\\\n &\\cfrac{\\partial N_{12}}{\\partial x_1} + \\cfrac{\\partial N_{22}}{\\partial x_2} = 0\\\\\n &\\cfrac{\\partial^2 M_{11}}{\\partial x_1^2} + 2\\cfrac{\\partial^2 M_{12}}{\\partial x_1 \\partial x_2} +\n \\cfrac{\\partial^2 M_{22}}{\\partial x_2^2} = q\n \\end{align} \n", "48289d2fd06832553f78d5d23c95f95e": "\\Delta G = \\Delta G_{SV} + \\Delta G_f (1 - e^{\\beta t/\\lambda}),", "4828ddaf90ea9885c26cbb989bbb31ed": "f'(\\gamma) = 1.\\,", "48291bf4cc894525cff11afcdb95d35e": " E_N ", "482937df1b7d72752fc3f16ec8ea339a": "u_x, u_y", "48293b894cb96227e1152672dd6e461f": "\\operatorname{Aut}(\\mathfrak{g})", "4829429917dd42fb6e6421697c7c8bda": "\\operatorname{somb}(0) = 1", "4829671a647110545a62f5ba95d39242": "\\Delta \\omega_2\\,", "482996f45ec7eb6e4594db30ba6d91d6": "r = w(z) \\, ", "4829d642c55231b5d2cf68d0867eb0e1": "x^4 +", "4829dd74c7174aa5f6984c5157edb2c3": "\\, y=2\\pi x \\Rightarrow dx=dy/(2\\pi)\\,", "4829eb3fc74f3208b7a186b4e8d977ac": "(\\lambda x)^* = \\bar{\\lambda}x^*", "482a79c7f2128db8cfa90b7ff0a2704d": "\\ddot x = f(x) + \\epsilon g(t)", "482a7c7e840629ec58d8746323d8c285": "\\scriptstyle\\lfloor\\frac{n}{2}\\rfloor\\cdot\\lceil\\frac{n}{2}\\rceil", "482aa8d791923a00bae9a7b637f5ddce": "\\scriptstyle \\sum_{i=1}^M n_i = N", "482aabab578cd40d8d735edf566c4364": "\n\\begin{align}\np(\\mathbf{X}\\mid \\mu,\\tau) & = \\prod_{n=1}^N \\mathcal{N}(x_n\\mid \\mu,\\tau^{-1}) \\\\\np(\\mu\\mid \\tau) & = \\mathcal{N}(\\mu\\mid \\mu_0, (\\lambda_0 \\tau)^{-1}) \\\\\np(\\tau) & = \\operatorname{Gamma}(\\tau\\mid a_0, b_0)\n\\end{align}\n", "482ab5549628b48fe528642c9332128f": "R = V/I\\,\\!", "482abe991b1c89d9167781f834399c75": "N{{u}_{b}}={{C}_{fc}}\\left( R{{e}_{b}},P{{r}_{L}} \\right)", "482ae4beac7df6acd2121f3b048ed6be": "F: C\\rightarrow D", "482b12d67b3c2259cfc289b45011e933": "T'^{\\mu a} = \\Lambda(x)^a_{\\ b} T^{\\mu b}", "482b40e97da5f4d8ca4ebcc51b7e7006": "\\frac{2U(\\varphi_0/\\sqrt{2})}{\\varphi_0^2} \\equiv \\frac{U(|\\Phi|)}{|\\Phi|^2}\\Bigg|_\\min < m^2. ", "482b51e6d5645c06c23ecf23321fe62a": " h = \\rho \\cos \\phi \\,", "482be17ac1bb4604a21a6bf54d7d4224": "e(\\mathbb Z)=2.", "482bfb19e2c4c3f553e88b5a9cf95024": " \\beta_0 + 1 \\ ", "482c009507d65d7f7fd0c1f87b7b52a4": "A(i,j)\\in\\mathcal{L}(H_i, H_j)", "482c51a254d8c00f96e794a0545312ea": "q =e^{-\\frac{\\pi K'}{K}}\\,", "482ca053b9f3da39456bf0613aa3ff0e": "\\displaystyle{z_i=\\sum_{j=1}^N a_{ij}(x)y_j}", "482cb4e0b45de7edff7a80bfb9ff085c": "R_aR_c=R_bR_d", "482d343e3b6f906330fa5567adf169c5": "(P, Q), \\quad P \\in W(V) V_N W(V), \\quad Q \\in W(V), \\quad V = V_N \\cup V_T.", "482d5e6480a28192af590b5dc7e5ed3a": "Y_1,\\ldots,Y_n", "482d8c2212a9b0f1d83a2ee963de0379": "\\displaystyle{\\begin{pmatrix}a & b \\\\ c & d\\end{pmatrix}}", "482d8eb92bd5ea5e466a7c390b96fdfa": "\\mathrm{n_0}\\,=\\, e_{rms}\\frac{ \\pi^2}{\\sqrt{5}}\\, (2f_0\\tau)^{\\frac{5}{2}} ", "482dc54e8eac04ead69fdb24732ca079": "\\quad\\hat{x}_2=\\sin \\theta \\sin \\varphi _1 \\dots \\cos\\varphi _{n-3}", "482e0b4e2259af34220fbde527cc352a": " A^* = QAQ^T ", "482e9709ca4f9ff9f95a37c29b6f57d9": "\n\\varepsilon _{w}^{2}-m_{w}^{2}=b^{2}(w),\n", "482eda61adbefe6ded3c7abafb9eefed": "n^{\\underline k}=\\frac{n!}{(n-k)!};", "482ef35bd669d613fd08bef1b534f298": "r^2 \\dot \\theta = \\text{constant} \\,", "482f0757a421d8e5f20a06f9c09aa8d8": "C_R = \\frac{v_b - v_a}{u_a - u_b}", "482f16d756c8d1c283f01d8334a60e16": "\n \\int x^{m-n}\\left((m-n+1)(2 a\\,B-A\\,b)+(m+2n (p+1)+1) (b\\,B-2 A\\,c) x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}dx\n", "482f68948bb6d358f7e390d44dec7e61": "p(x) := x^md(x) - r(x),\\,", "482f7cbb31cf2902e5893a4f35b3c273": "y=\\ln (x)\\;(=3.68)", "482fa4f7268c4f8bfddbb6d32d6ca3a0": "x_i=0", "482fbdcab200a588cdb642530f01f0ba": "A-A=\\{a-b\\mid a,b\\in A\\} \\, ", "482fc6133f91aaf7f5132612169e64bf": "\ne^{\\psi(x)}= \\operatorname{lcm}[1,2,\\dots,\\lfloor x\\rfloor].\\;\n", "48304eaa836a406e664794c863923387": "\\phi(t, t_0) = e^{A(t - t_0)}", "4830a399a3af4702be5f9fd2a8073c30": "i,j=1,2,\\ldots,N", "4830ea27f706cf680f7362c310b13559": "|\\tau(p)| \\leq 2p^{11/2}", "4831487ef809bcc1a003de0ad8cb288d": "{{Y}_{M}}=\\sum\\limits_{m=0}^{M-1}{\\left\\langle Y,{{g}_{m}} \\right\\rangle {{g}_{m}}}", "4831768fa11820cfa9938d23c7d9331c": " -i\\Gamma", "48326fb833561e83df3dd059f75601e2": "j(\\tau) = j(1/\\tau)", "4832e3ac3b2bb2fb5cae1c8293dcc7d2": "\\dot{y} = b \\cdot \\cos E \\cdot \\dot{E}", "48332ac35802db170d3c9f5ef6aff582": " q = \\frac{V_y S_x}{I_x}", "4833387a996448f9856fc3fcec980c35": "\\text{E}_{t/b}\\left(\\eta\\right)=\\int_1^\\infin e^{-\\eta v} v^{-t/b}dv,\\ t>0.", "48334ae04f04e73635a8e49b372a6d4c": "\nh(\\tau) = -\\frac{\\nu}{4\\pi} \\sum_{\\phi_i = \\pi \\pm \\phi_S \\pm \\phi_R} \\int_{z_1}^{z_2} \\delta\\left(\\tau - \\frac{m+l}{c}\\right) \\frac{\\beta_i}{ml} \\, dz\n", "4833aa3bb0426e364bb63f2acc2e43c8": "W_{t}(n) \\leq n - t + \\sum_{n+1-t < j \\leq n} \\lceil{\\log_2\\, j}\\rceil \\quad \\text{for}\\, n \\geq t", "4833d6e359412d149e536938910ea429": "(\\exists f \\colon \\mathbb{N}\\to\\mathbb{N}) \\psi(f)", "4833fc263255b6b2e9d9267d4bd78941": " x = {^\\infty y} ", "48343c0bee7be89fd131233f22aa8949": "W_{ab}=m_{[a}\\bar{m}_{b]}-n_{[a}l_{b]}.", "4834413f1df4463674974917788cd131": "\\omega({\\mathbf e}\\cdot g) = g^*\\omega_{\\mathfrak g} + \\text{Ad}_{g^{-1}}\\omega(\\mathbf e)", "4834699536c777a67f9ab4a9cfec1de2": "\\begin{align}\nw & = \\cos v\n\\end{align}", "48348ab84f002f5f2f89f62b4ecc7410": "\\prod_{p\\in P} \\frac{1}{1-1/p}=\\prod_{p\\in P} \\sum_{k\\geq 0} \\frac{1}{p^k}=\\sum_n\\frac{1}{n}.", "4834c42b68e5c9189c41ec5b0a4aa0d7": "Y \\mapsto Z=Y/\\sigma^2", "48351aa84728ee9475c5969d2643d08d": " a = \\frac{\\sqrt{ G M a_0 }}{r} ", "48357541594fd8023274c9cf59cc9cd2": "Q = g Q_0 g^{-1}, \\, \\, \\,Q_0 g^{-1} \\dot{g} Q_0= 0.", "483582bb80e0b0cc3fc5bbe98c8496cb": "D(p||m) = \\int p(x)\\log\\frac{p(x)}{m(x)}\\,dx.", "4835903cb381a2d2d35164fc0cbd7e25": "\\dim (R/P) + \\dim (R/Q) < \\dim (R)\\ ", "4835c18abdf364bb1b99fe49798ac44f": "\\operatorname{E}( X ^ 2 ).", "4835ca38e3633ab47ed63ea3f99c6481": "(i,j)\\in S", "4835f0b009f3ed2d06be1bca9bc021c6": "h = -(A+A^{\\mathrm{T}})^{-1}b \\quad\\text{and}\\quad k = c - h^{\\mathrm{T}}A h = c - b^{\\mathrm{T}} (A+A^{\\mathrm{T}})^{-1} A (A+A^{\\mathrm{T}})^{-1}b", "4836444baf72b8caba1d23b36755bccf": " \\mathbf{\\dot{r}} = \\dot{r} \\mathbf{e}_r + r\\dot{\\theta} \\mathbf{e}_\\theta + \\dot{\\zeta} \\left( R_0 + r \\cos\\theta \\right) \\mathbf{e}_\\zeta ", "48364971181be3f2e3d6ab8b53fd8a67": "\n\\frac{1}{\\epsilon(\\vec{r})} \\nabla \\times \\nabla \\times E(\\vec{r},\\omega) = \\left( \\frac{\\omega}{c} \\right)^2 E(\\vec{r},\\omega)\n", "48365195c6a1c685646f35d3ae35c937": "\\iota : A\\rightarrow X", "483680bfaffb2bb1111e200d78ed6cd9": "i_M(V) = \\frac{en\\langle v \\rangle}{4}S_z\\exp \\left (-eV/\\mathcal{E}_p \\right )", "4836fe7194c8a6cf5a05d6f4dd80da3d": "\\begin{matrix} {3 \\choose 1}{11 \\choose 1}{4 \\choose 2} \\end{matrix}", "483706b9eb0c290772d6c2473450f22b": "\\sigma_{\\kappa(q)+1}\\ldots\\sigma_{\\kappa(q^\\prime)}", "4837396348bf2411707f190dc7d31a35": "X = \\{a, b\\}, N = \\{p, q, r\\}\\text{, then }\\left\\vert\\{(a, a, b), (a, b, a), (a, b, b), \\ (b, b, a), (b, a, b), (b, a, a)\\}\\right\\vert = \\textstyle 2!\\{{3\\atop 2}\\} = 2\\times 3 = 6", "48379de9ca074607168db4e69e49856f": "\\|\\;\\|", "4837c8153b14f5adad59c921a4a8ce4a": "?\\times 3=6", "4837da6ae91789f70b573b447acbf25e": "\nA=\\frac{kR^{k-1}\\pi^{k/2}}{\\Gamma(k/2+1)}\n", "48383df84e4cc8e9d831ec0f3aa7f9c7": "R_\\mathrm{Th} = R_\\mathrm{No} \\!", "48385b785fff38e1d10a2c01edb41822": "\\left\\{ \\mathcal{H}_{i}\\right\\} _{i\\in\\mathbb{Z}^{+}}", "4838d1db07ab451c9fda3800d2cb82ec": " x(t) \\ ", "48395bb47891a642d7100db241bcb2e2": "\\mathcal{I}(\\theta) = 1\\,.", "48396dad21457053c0e9339808f03d5e": "K_X(s,t)=\\sum_{k=1}^\\infty \\lambda_k e_k(s) e_k(t) ", "4839bec24d80776d082f8edcec4167d8": "v_p = \\frac{r_a}{r_p}v_a", "4839da374574ae9436d72ba5337d2a5e": "\\ S_{1,t} = F(t_1) m_1 (1-F(t_2)) ", "483a1a8cb1f9cb42c135305e06243110": " \\Psi(x_1,x_2,\\cdots x_N,t) = e^{-i{E t/\\hbar}}\\prod_{n=1}^N\\psi(x_n) \\, ,", "483a6290e724af064fe5212ce751af6a": "\\frac{\\mathrm{d}P}{\\mathrm{d}T} = \\frac{L}{T\\,\\Delta v}, ", "483a8f0bd3525e4f53fb7aff3902b975": "dp_e = \\frac{dm_e \\ v_e}{\\sqrt{1 - \\frac{v_e^2}{c^2}}}", "483a94f15fe65b88db39bd1982873d22": "f_k(x) = 0\\,", "483a9756e5c37f1d881a9427e2d58a1d": "\\bar{e}(C)", "483aca74816def6f738fc18156c417c5": "r^2=b^2-\\left[a\\sin\\theta\\pm\\sqrt{c^2-a^2\\cos^2\\theta}\\right]^2.", "483af8b6c59629caef61e54660ac9ca5": "E_{x}=\\frac{1}{j\\omega \\varepsilon }\\frac{\\mathrm{dL} }{\\mathrm{d} z}\\frac{\\partial T}{\\partial x}^{TM}-L\\frac{\\partial T}{\\partial y}^{TE}=\\frac{-k_{z}}{\\omega \\varepsilon }L\\frac{\\partial T }\n{\\partial x}^{TM}-L\\frac{\\partial T}{\\partial y}^{TE} \\ \\ \\ \\ \\ \\ \\ (27) ", "483ba3b21f0b441f88132998ce3b4851": " \\triangle MXX' \\sim \\triangle MYY',\\, ", "483ba75fb27d98a3db2927de40d4eb37": "H_{rrc}(f)", "483c049df42a2c29ba2e57c6fde93a1e": " \\frac{a+b}{a} = \\frac{a}{b} = \\varphi\\,.", "483c3607549c662b46221ce80fd68c58": " C_\\lambda ", "483cca7b53268fa041c09517e51edede": "\\mu_0 = ", "483cd6c7b40a3520f11540a6cad40da8": "B^{*} = \\{v^i\\}_{i \\in I},", "483ce61ab4fb916e9b2e4acadc84f0af": "\n(2) \\qquad \\Psi_T(x(T))+H(T)=0 \\,\n", "483d2e31d10f5db574e1476d3b8f2586": "b=\\lim_{x\\to x_0^+}\\frac{f(x)-f(x_0)}{x-x_0}", "483d57047f12be628a415462437a0238": " \\mathbf{s}_1 = \\mathbf{H}_1 \\mathbf{x}_1, \\mathbf{s}_2 = \\mathbf{H}_2 \\mathbf{x}_2, \\cdots, \\mathbf{s}_a = \\mathbf{H}_a \\mathbf{x}_a ", "483d6400365ddca36d24746671b6485f": " \\int e^{\\bar\\eta \\partial_\\mu D^\\mu \\eta} D\\bar\\eta D\\eta \\,", "483d7441b31ae6ff776a7a0c5c421624": "t \\equiv 1 \\pmod p ", "483d75d9ca353cc4bd93833ce5c86c7a": "\\mathbf{R} \\times 0", "483d7c5c32742745dd8b68b4e8dcc8e7": "\n\\begin{align}\\\\\nF(I) & = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n I_{\\sigma(i)}^i\\\\\n& = \\sum_{\\sigma = (1,2,\\dots,n)} \\prod_{i = 1}^n I_{i}^i\\\\\n& = 1\n\\end{align}\n", "483d98d86c51921b1d2d2a72003f782f": "\\textstyle w_i \\geq0", "483dfe801cf9bf1a886903f33e4107fa": "\\scriptstyle \\vec A \\;=\\; {d \\over dt}\\gamma(0)", "483e2c2a814ac3254f13815bdf9d6dc2": "\\frac{p\\cdot n}{n-1}", "483e347f9b1ee92c8485405bcff3a8fd": " \\Pr(T_i < {\\infty}) = \\sum_{n=1}^{\\infty} f_{ii}^{(n)} < 1. ", "483e3d82094370d3c936534dd5db8a34": "M_{e}", "483e4403e431b5fe91d1ed93d8f535c9": "\\sum_{n=1}^{3} (2nx)", "483e55cd215f29e57e556e77ddbbcffe": "\\begin{align}\n 2 + 3t &\\equiv 3 \\pmod{4} \\\\\n 3t &\\equiv 1 \\pmod{4} \\\\\n t &\\equiv (3)^{-1} \\equiv 3 \\pmod{4}\n\\end{align}", "483e6eb71342bbb53b215af4117252d7": "\\exists x\\in X : P(x) \\equiv \\{x\\in X\\} \\and P(x).", "483edc41ff27392d7bfa6aad162dd9d0": "(L_g)_{*}X ", "483f2cbc88844fc724e6318e7e9c1f85": " \\scriptstyle \\omega_1", "483f487a55f98186d38c54a908b9b347": "\\ n\\,! \\sim \\sqrt{2\\, \\pi\\, n}\\, \\left(\\frac{n}{\\mathrm{e}}\\right)^n", "483f57a01cb302c0388bd2f7ce0209d7": " \\big(P_{A\\alpha}\\gamma(\\mathbf{R})\\big)+ \\langle\\chi_2|\\big(P_{A\\alpha} \\chi_1\\big) \\rangle_{(\\mathbf{r})} = 0 ", "483f5d8c0f92ce2b8019e5706bf295e3": "\\displaystyle{W(z,w)^*=W(-\\overline{z},-\\overline{w}).}", "483fea0226a7a3f4334d661fbc1d60ec": "P(\\emptyset)=0.", "484016b224ab1e77c6716e97de61f9d1": "\\kappa(s)", "4840413ef99b6f87741855d94e155e6c": "[b]=[S][a]. ", "484080d4655e86d83184aa43b9f596ca": "\\mathfrak{z}", "4840aaa2b05725542996f359825d5215": "\\Pr(W=n)=\\frac{k}{n}\\Pr(S_n=n-k)", "48411fad1e7a4af16952f215f70a0962": "\\phi = \\tfrac{1+\\sqrt{5}}{2}", "4841380c2068d7dc5823df56d8987ff4": "B = 2*p^2 = 2*7^2 = 98", "48414f229a77c9b37e9ba9e838565f51": "\\mu_i=\\left( \\frac{\\partial U}{\\partial n_i}\\right)_{S,V}=\\left( \\frac{\\partial H}{\\partial n_i}\\right)_{S,P}=\\left( \\frac{\\partial A}{\\partial n_i}\\right)_{T,V}=\\left( \\frac{\\partial G}{\\partial n_i}\\right)_{T,P}.", "484249736cc1e9bb6dda1beed1c22a42": "u p_k \\le q_k, \\quad k = 1, 2, 3, 4\\,\\!", "48428c44bdf0f14d1b87cbd1bd5aa131": "e(t)=1, \\forall t \\in [0, 1]", "484291df12b91d9e385c9a4726d961f5": "v\\otimes w - w\\otimes v.", "48429fd2f46b31f716a19c1d2c8417fd": "y\\in A_2", "4842a149cd66d341abd51414a796a21e": "(5-\\sqrt{5})/2", "4842e9b581687849c6581b2cc71d8c0f": "\\Rightarrow \\psi = A e^{i \\alpha x} + A' e^{-i \\alpha x} \\quad \\left( \\alpha^2 = {2mE \\over \\hbar^2} \\right) \\,\\! ", "4842f02805477098d47a84fc7bd40efb": "X_K=1-X_1-\\cdots-X_{K-1}.", "48430d4a3f26ab42d0b2c3bf5868220d": "C (\\vec{N}) = \\int d^3 x C_a (x) N^a (x)", "4843466868005057f0b83dbd1ff83cc3": "\\alpha_{mk} R^ky^k(1-y)^m", "48438ede4adf12e53253946e46021688": "\\scriptstyle M+i\\Gamma", "4843a71af3304cfedca4d8c27f611a07": "T_{bV} = e_V T", "4843c75f601544cb6fc4a38c968bb7b7": " \\Delta \\theta ", "4843eb67486bd091503acc3057e7c01c": " d = \\frac{N}{P_d} = \\frac{pN}{\\pi} \\qquad \\text{spur gears}", "4843f620eed7526fb9a3891baedca8e2": " \\mathbf{D} = \\frac{1}{3}\\sum^{N}_{j=1} \\mathbf{Q}^* \\left( z^3_j - z^3_{j-1} \\right) ", "4844004749b65d1fd8a4dad8d809a4aa": "V_{100} - V_0 = kV_0\\,", "48440ef4d357a3da46b571703d9e2d7d": " U(\\omega) ", "4844619993ab00b45a7eafa926012466": "\n\\begin{align}\n \\lambda(x,y)&\n= \\arctan\\bigg[ \\sinh\\frac{x}{k_0a}\n \\sec\\frac{y}{k_0a} \\bigg],\n\\\\[1ex]\n \\phi(x,y)&= \\arcsin\\bigg[ \\mbox{sech}\\;\\frac{x}{k_0a}\n \\sin\\frac{y}{k_0a} \\bigg].\n\\end{align}\n", "48448333f9be31c490207aa4c00d06d1": "a(1)e^{\\alpha(1)}+\\cdots + a(n)e^{\\alpha(n)} = 0.", "4844fd1242e80fe68b82d2bb8a2b6f33": "\\frac{\\gamma^\\mu\\gamma^\\mu}{\\eta^{\\mu\\mu}}=I\\,", "48451fd104b70b2bf7acb52adc841286": "\\begin{bmatrix} x_1+\\bold i x_2 & x_3+\\bold i x_4 \\\\ -x_3+\\bold i x_4 & x_1-\\bold i x_2 \\end{bmatrix}.\\,\\!", "4845237d986a014f521b3a4176f7271d": "-N(-d_1) = N(d_1) - 1\\,", "484535fb522e7a8571988a29602d31b5": " \\int_0^t X_s \\, \\partial Y_s ", "4845397206b02d10e7465b26ab0f2c2f": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.4002 & 0.7076 & -0.0808 \\\\ -0.2263 & 1.1653 & 0.0457 \\\\ 0 & 0 & 0.9182\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "4845c6dc241c0c8725dd2e001295f34d": "Z = \\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}", "4846355b1c30178bcf23dc662c1e1f66": "\n\\pm\\begin{bmatrix}\n1 & 1\\\\\n1 & -1\\end{bmatrix}\n", "484674487e52ee13028a6eaf8dac3bec": "c_i = x\\cdot g_i", "4846feb7ccfacb3250334bcf2fa29dbf": "!n = \\left[ \\frac{n!}{e} \\right] , \\quad n\\geq 1", "484700ebd6c09ceeb4a48deaf98cfb40": "\\varepsilon_{ij,km}+\\varepsilon_{km,ij}-\\varepsilon_{ik,jm}-\\varepsilon_{jm,ik}=0.\\,\\!", "48473071bae3d777d93b8eeda8ffd197": "h(t)=\\lim_{\\triangle t \\to 0} \\frac{R(t)-R(t+\\triangle t)}{\\triangle t \\cdot R(t)}.", "484760fa9e5ee8c6b03a820c1dff2c0c": " X^\\mu(\\tau, \\sigma + \\pi) = X^\\mu(\\tau, \\sigma)\\ ", "48476af10e19b9f6239e17a3890b3080": "-\\mu", "484772a7d4999f2350064eaf5f9dc384": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = z_2\\\\\n\\dot{z}_2 = u_2\n\\end{cases}", "4847997ac9bd40168282c1d951eaac56": "b^n-y", "4847acef124418c705153576282ebd02": "c=2", "4847c16a18cd34ccdf66169bd09e4db5": "\\scriptstyle a_1", "4847e4aa0d3a2b381f9b5377d1612e36": "\\parallel\\boldsymbol{\\phi}_{\\mathcal{B}}(x) -\\boldsymbol{\\phi}_{\\mathcal{B}}(y)\\parallel_{_2}\\leq\\varepsilon", "4847f1cba9c97beb51ff3105ed4ee9f2": " Pu(x) = \\sum_{|\\alpha| = k} P^\\alpha(x) \\frac {\\partial^\\alpha u} {\\partial x^{\\alpha}} + \\text{lower order terms}", "4848226bdfb5c90251a916a45d708dbf": "T_5", "48482f5a96d1a236e8db42035ccf3529": "dE(u,\\psi) = \\int_\\Omega f(u(x))\\psi(x) \\,dx,", "48484e2a4d8c23d81e939a173267a4d2": "A \\rightarrow \\alpha \\in P", "4848e848a778d5a16f98ee7e94ce7af6": "\\frac{\\partial u}{\\partial n}=h\\left( x,y \\right),\\ \\ h\\left( x,y \\right)\\in \\partial \\Omega_N", "484928387c7aaab96dce937cc3b4e359": "\n\\begin{array}{lcl}\nT &=& \\frac{1}{2I_1 \\sin^2\\beta}\n \\left( (p_\\alpha- p_\\gamma\\cos\\beta)\\cos\\gamma -p_\\beta\n \\sin\\beta\\sin\\gamma \\right)^2 \\\\\n &&+ \\frac{1}{2I_2 \\sin^2\\beta}\n \\left( (p_\\alpha- p_\\gamma\\cos\\beta)\\sin\\gamma +p_\\beta\n \\sin\\beta\\cos\\gamma \\right)^2 + \\frac{p_\\gamma^2}{2I_3}. \\\\\n\\end{array}\n", "48494468cb9f6d2347e2630db2d6c9ea": "\\varphi _p", "4849da0776250e6bf169870a16d5f994": "M(n) > (4 - 15^{1/2})^{1/2}(n -1)", "4849f313bff38734138c2758d742efe6": "\\begin{align}\n(x+2)^3 &= x^3 + 3x^2(2) + 3x(2)^2 + 2^3 \\\\\n&= x^3 + 6x^2 + 12x + 8.\\end{align}", "484a0fe7c4076fa78ce76aa1563c6cca": " V = k/P ", "484a71859a62d63f33dee52a1b7dd9a8": "0 \\to \\ker T \\to V \\to W \\to \\mathrm{coker}\\,T \\to 0.", "484aaf7f3c738cc748d5b15eef099ae7": "CD_n = CD_{n-1}+10(n-1) \\, ", "484b0d90958a7fb88d18befad43f6081": "a^b = (a^{(b-1)})\\cdot a", "484b0f984494acbac74d82441a23f9b4": "u(x,t,\\theta)", "484b321475f85f91804e4a2194b6aa9c": " j = l \\pm s ", "484ba5a59215e4f78ede3e456052eb9e": "\\frac{\\mathrm{d}\\varphi}{\\mathrm{d}\\alpha} = \\int_a^b\\frac{\\partial}{\\partial \\alpha}\\,f(x,\\alpha)\\,\\mathrm{d}x", "484c362bcf9978974335e67ae6fddcf7": "\\sigma_0 / (1 -\nD_b/D)", "484c7aeeac62b884054351a844394ffc": "Fraction \\ Bound = \\frac{[L\\! \\cdot \\! R]}{[R]+[L\\! \\cdot \\! R]} =\\frac{1}{1+\\frac{K_d}{[L]}}", "484cd6eabc72dabd8a20d36bd25382fe": "\\begin{align}\n L(x)y &= xy \\\\\n R(x)y &= yx\n\\end{align}", "484cde06c5a7b50316880691e2666207": "-\\frac{(b-a)^5}{2880}\\,f^{(4)}(\\xi)", "484d4bbf463f2986d664a7cdd6c12ba5": "\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 &\\qquad \\text{ ( by Lyapunov function } V_x, \\text{ subsystem stabilized by } u_x(\\textbf{x}) \\text{ )}\\\\\n\\dot{z}_1 = f_1( \\mathbf{x}, z_1 ) + g_1( \\mathbf{x}, z_1 ) z_2\n\\end{cases}\\\\\n\\dot{z}_2 = f_2( \\mathbf{x}, z_1, z_2 ) + g_2( \\mathbf{x}, z_1, z_2 ) z_3\n\\end{cases}\\\\\n\\vdots\\\\\n\\end{cases}\\\\\n\\dot{z}_i = f_i( \\mathbf{x}, z_1, z_2, \\ldots, z_i ) + g_i( \\mathbf{x}, z_1, z_2, \\ldots, z_i ) z_{i+1}\n\\end{cases}\\\\\n\\vdots\n\\end{cases}\\\\\n\\dot{z}_{k-2} = f_{k-2}( \\mathbf{x}, z_1, z_2, \\ldots z_{k-2} ) + g_{k-2}( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-2} ) z_{k-1}\n\\end{cases}\\\\\n\\dot{z}_{k-1} = f_{k-1}( \\mathbf{x}, z_1, z_2, \\ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-2}, z_{k-1} ) z_k\n\\end{cases}\\\\\n\\dot{z}_k = f_k( \\mathbf{x}, z_1, z_2, \\ldots z_{k-1}, z_k ) + g_k( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-1}, z_k ) u\n\\end{cases}", "484d7bc19d8c3adc2a954a63fb027386": "\\pm\\infty", "484de7249402c3616f750a7f7562bac6": "C_{\\beta J}^{\\;\\;\\;\\; I} e_\\gamma^J e_I^\\beta = 0.", "484df60ddc25650d8c50e50b85007700": " \\Box A^{\\mu} = 0 ", "484e3b6348fa73c426ac2a5f6d1f31e4": "\n\\operatorname{Li}_2(1-z) + \\operatorname{Li}_2 \\left( 1-\\frac{1}{z} \\right) = - \\frac{1}{2} (\\ln z)^2 \\qquad (z \\not \\in ~]-\\infty; 0]) \\,,\n", "484e40530be10348448a1759acb51112": "\\lim_{k \\to \\infty}\\lambda^k = 0", "484e5b207946e74cefc6f4f33e2697c4": "\n\\Phi_\\mu(\\phi)=e^{i\\mu\\phi}\\,\\,\\,\\,\\mathrm{and}\\,\\,\\,\\,e^{-i\\mu\\phi}\n", "484e6e450781d3a1d1cde87125939f61": "\\exp(a D_T)", "484e8260f128d6d3f5ffefa95c56a275": "\\scriptstyle \\log_{10} P_{mmHg} = 6.61184 - \\frac {389.93} {266.00 + T} ", "484ebdc3b5d0c8d84b87d74f46fe8caf": "P(D|T,M)", "484f7b83e47c1411ee9f8af5c7201ef3": "\\left(\\frac{\\partial x}{\\partial y}\\right)_z\\left(\\frac{\\partial y}{\\partial z}\\right)_x\\left(\\frac{\\partial z}{\\partial x}\\right)_y = -1.", "4850003482d5d23428e166bf2077b7bc": "A_i,", "4850291d7e9f55ce2ec93f31f67eeb4e": "p_n(x)=\\sum_{k=1}^n S(n,k)x^k", "48503269733b483cf484a1be80de5905": "add(v, \\pi)", "48504798adfba89caa3708602510c3f5": "ax + by + c = 0,", "48505daecbdab5e0bd5a6f5adf2dd16d": "k_0=0.9996", "4850d926be9fcbc5ce2a32033fcb9f8b": " \\psi= U \\left( r - \\frac{R^2}{r} \\right) \\sin\\theta. ", "48513ac61c995ec619c123fbc9dd6ad9": "E^\\prime", "4851c9a402ac699eac86c42d1357e3cc": "\\vec{S} ", "48521573238872ab807967fc80ca767e": "K(\\vec{r})\\!", "48523a622c17635650134fa32f28bd40": "\n\\begin{align}\n\\int \\sec^3 x \\, dx &{}= \\int u\\,dv \\\\\n&{}= uv - \\int v\\,du \\\\\n&{} = \\sec x \\tan x - \\int \\sec x \\tan^2 x\\,dx \\\\\n&{}= \\sec x \\tan x - \\int \\sec x\\, (\\sec^2 x - 1)\\,dx \\\\\n&{}= \\sec x \\tan x - \\left(\\int \\sec^3 x \\, dx - \\int \\sec x\\,dx.\\right) \\\\\n&{}= \\sec x \\tan x - \\int \\sec^3 x \\, dx + \\int \\sec x\\,dx.\n\\end{align}\n", "48523bba639012d796e3c3f0de14a0ef": "f^*,g^*,h^*", "4852726fc53ce51ab6f9fac2655bc215": "T^*_C", "4852891efab777568f3578779ca56977": "\\sum_{n=0}^\\infty (-1)^n {s \\choose 2n} = 2^{s/2} \\cos \\frac{\\pi s}{4}", "48529054168cd3031bcd58b245e2b1b4": "\\frac32\\Delta", "4852afb961be1a965db296412aae804b": "(R^p f_* \\mathcal{F})^\\wedge \\to \\varprojlim_k R^p f_* \\mathcal{F}_k", "4852df711c2cef547bb95d3b35c07eb7": "Y=\\R", "4852e572d7cc3d410e936f8c13c66429": "\\mathrm{S}(\\mathrm{U}(p) \\times \\mathrm{U}(q))", "4852f875be1e422a304afa5f1e68aa8b": "\\int^c S(c, c) \\leftarrow \\coprod_{c \\in C} S(c, c) \\leftleftarrows \\coprod_{c \\to c'} S(c', c). ", "48534c30df357eea756aa757044122e5": "d = 2.9 n^{2}a_{0}\\,", "48537c666c4c8297078a2224c1e9700d": "(1 - \\beta\\lambda)^\\gamma", "485381327b4c167d60e5600567d77383": "\\mathbf{G}_{ij} = \\langle \\mu_{X^{(i)}}, \\mu_{X^{(j)}} \\rangle_\\mathcal{H} ", "4853b458a379a21f66e5926a71507039": "\\nabla\\cdot\\vec{E} + m^2 \\phi \\approx 0", "4854023733662c9c1972e9d783a4bdca": "N_\\nu", "48543239ed78244ce2a668efc8dfbb92": "A \\cap B = B \\cap A\\,\\!", "485449b6a535966fe9088fc683c84210": "\\mathrm{\\delta ^{13}C} = \\frac{\\mathrm{\\Bigl( \\frac{^{13}C}{^{12}C} \\Bigr)_{sample}} - {\\Bigl( \\frac{^{13}C}{^{12}C} \\Bigr)_{PDB}} }{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{PDB}} \\times 1000\\ ^{o}\\!/\\!_{oo}", "4854b842e07e2506052ebc87d2a61342": "\\hat{A}_2 = (-4p_2 + 7 p_1^2)/5760.", "485546df6a5f6c79be8223dbd40e250f": "\\left(0,1\\right]", "48555bdab452feed627d0e6b72c7b1af": "\\{1, \\gamma, \\gamma ^ {2m},\\ldots,\\gamma^{(n/m-1)m}\\}", "4855b9b7658b32269a7ae21e47779463": "\\,r", "4855dd1ef38fbfefe4cf6232b5209961": "\\lim_{n \\to \\infty} \\sup_{x \\in K} d_{Y} \\left( f_{n} (x), f(x) \\right) = 0.", "4856179159093b5f8f764b73d6f7d1fd": "Pm = \\tfrac{3}{5} \\tfrac{3}{5} + \\tfrac{2}{5} \\tfrac{2}{5}", "485628b9775ebb9ec3319da9844f7fe6": "\\operatorname{Var}(Y) = E(N)(\\operatorname{Var}(X) + {E(X)}^2 )= E(N){E(X^2)}.", "48569545c86637080a1fe64813c2a722": "\\Sigma^{(i)}", "4856f138a13bd06ddf74f1db0b05710f": "p(ub * x) \\in \\mathbb{Z}[x], p(0) \\neq 0", "485701a2935b7986466328d34785b6a3": "X_1 = N - a_1^2 \\cdot 10^4 \\geq 0.", "48570b2c898beb47c13d89859f856b76": "| u_{n}(0) | + \\mathrm{Var}(u_{n}) \\leq L_{\\varepsilon},", "48574888561da3083126521f8d1b6682": "m0} \\left ( \\int|f(x+ iy)|^p\\, \\mathrm{d}x \\right)^{\\frac{1}{p}}.", "4859e21a3c64ef7a93e17941181708b1": "\\omega_n=\\frac{\\pi^2 \\hbar n^2}{8 L^2 m}", "485a162642a3895364099ae6e98a7499": " f = \\sum_r D^{(r)}(f) \\cdot t^r \\ . ", "485a17fefb7d6e93defb671d3c752557": "\\frac{\\mathrm{d}^2q}{\\mathrm{d} \\tau^2} + 2 \\zeta \\frac{\\mathrm{d}q}{\\mathrm{d}\\tau} + q = \\cos(\\omega \\tau).", "485a19a827bdf9fdeacd4eac2b75c3af": "\\int_{\\mathbb{R}^n} f(x)g(x) \\, dx \\leq \\int_{\\mathbb{R}^n} f^*(x)g^*(x) \\, dx", "485a43b76c89edaecfe6f3e7c81791f8": "F(\\vec{r})=\\int_{-\\infty}^{+\\infty}\\Phi(\\vec{r},t)dt (\\frac{J}{m^2})", "485af441c1d4583e5bf90e9c9a7c7daf": "\\rm ETF_{red} + Q \\rightarrow ETF_{ox} + QH_2 \\! ", "485b1bc94c611e681ff2f7c73406f807": "SS_{R}", "485b610d5d44a2341638af6a553b1a36": " \\left(\\frac{N_1}{N_2}\\right)", "485b8a3037b80195ba7adb30c9403810": "u^\\prime = c", "485b93b2dc14f142e3e629470d40a141": "\n\\psi^\\dagger i{d\\over dt} \\psi = \\psi^\\dagger {-\\nabla^2 \\over 2m} \\psi\n\\,", "485bbd42455ebab5139817304172948d": "\\langle a \\mid a^8 = 1, a^2 \\neq 1\\rangle", "485c28c1bbd09a51424fc9997c6f1814": "\\begin{align}\n \\dot{\\mu} & = \\mu'-\\partial_\\mu F(s,\\tilde{\\mu}) \\\\ \n \\dot{\\mu'} & =-\\partial_{\\mu'} F(s,\\tilde{\\mu}) \n \\end{align}", "485c9169b8e8d2544995fdebd73032f2": "Q(4,q)", "485ca4050547013b9de9e4d1b08a02b8": "X = \\begin{pmatrix} x_{1} \\\\ \\vdots \\\\ x_{i} \\end{pmatrix}", "485caa4c80ecdbbd96de7fc9caccfc68": "\\boldsymbol{\\hat k}", "485cab92a703c4bc8b0abc4a6db3f3c5": " {^{k}a} = \\log_a \\left( {^{(k+1)}a} \\right) ", "485cbc57f7a1121da7b3fb048df06ab7": " v^{\\prime} = DF(x(t)) v - 2 \\alpha v ", "485ccca2294d8c637704eb8d944877ce": " (X_1,...,X_n) ", "485ce5e99c07ce9614fa3c0d40070488": "{F_{2n} + F_n} \\over 2", "485cf49dd53d2f05316219b73f03ba9c": "A_\\Sigma(\\Gamma\\tau) ", "485d270f22a49a94cdd9237086c31b85": "y(t) = t\\, x(t)", "485d84088dd7c5a964eefe9ed13a97fe": "\\gamma_{1}\\,", "485db1e6bc7619a99dc47c89993a810b": " \\int_{-\\infty}^{\\infty} \\exp\\left( -{1 \\over \\hbar} f\\left( q \\right) \\right ) d^nq", "485db6da28f7c9384528f24264a4ac79": "\\textstyle{\\frac {\\log(6)} {\\log(2)}}", "485e059da21b4a4ba584e81656044283": "K_5", "485e86a9b770492c4c5fb13065051ba6": "\\scriptstyle x \\;=\\; x_0,\\; y \\;=\\; y_0,\\; z \\;=\\; z_0", "485ecea3200f4fda9068a53e30b38024": " \\bar{D} = \\begin{vmatrix}\nD_{\\color{red}xx} & D_{xy} & D_{xz} \\\\\nD_{xy} & D_{\\color{red}yy} & D_{yz} \\\\\nD_{xz} & D_{yz} & D_{\\color{red}zz}\n\\end{vmatrix}", "485f153aa53b7921e8bc338a9eb1d508": "(x_k, y_k)", "485f22cd737ddb1f5625b33386528ec4": "\\mathbf{y}(t) = C \\mathbf{x}(t)", "485f325478ea938db494172fdba4116d": "u'=1", "485f789851658b6f66b1cd8bd3bb13e4": "\\cos \\theta + \\cos y = 2 \\cos \\frac{\\theta+y}2 \\cos \\frac{\\theta-y}2", "485fd3931feb48434adb1dbb401924ae": "DPW = \\left(\\frac{\\displaystyle \\pi d^2}{4S}\\right) \\exp(-2 \\sqrt{S}/d)", "485ff19984ab84454712b87b2a18146c": "{\\nabla}^2 \\varphi = f\\,", "4860a9fbba92215710e1602e9d08ba83": "\nA = R = \\begin{pmatrix} 0 & I_{w - 1} \\\\ a_{w-1} & (a_{w - 2}, \\ldots , a_0) \\end{pmatrix}\n", "4860caf86e514f8f2233daa09e6d6542": "\\mu = \\sigma/\\xi", "4860f55ba4133a9122b7c9fa67cf0cb4": "E=E^0 - \\frac{RT}{nF} \\ln \\frac{[\\text{reduced species}]}{[\\text{oxidized species}]}", "48610f4e2e4bf87c78af61b294f25f1f": "L_0(x)=2 \\,", "486165db793cb884be6c30cb7580e9e0": " h \\nu \\ll kT ", "4861897cfa0f2265b54b5b5f7a67a5cd": "\\textstyle\\mathcal{R}", "486190c2c3ecd457e71bc35096987e81": "E^x", "4861d15160b2843a8dc1a97db90530a2": "E^2 = (pc)^2 + (m_0c^2)^2\\,", "4861d4cde5a1feb2c4432218495581fb": "N_A=-N_B", "4861e4a64c4739d78d8c615daaf675b3": "\n\\rho (r)=\\frac{\\rho_0}{\\frac{r}{R_s}\\left(1~+~\\frac{r}{R_s}\\right)^2}\n", "4861e95aca88df2bbb50d5eea1994ed9": "L(X^*_{\\tau(X^*, X)}, Y)", "48621c38344abcb7daef343596663e10": "z\\to z^2", "486227f454585eff313be5a5b2b5ee93": "\\partial_g(U|\\Psi_0\\rangle)", "4862289f25b363a09965e02ba9d18419": "k_{\\nu, a}", "48623014a887be72dc5e350e26c9f6f9": "\\mathbf{x} = \\mathbf{0}", "486243174246c597360492c690ecd966": "\n\\omega = \\dot{\\varphi} = \\frac{d\\varphi}{dt}\n", "48628358879297a12bdaa8e3c138ff2a": "\\mu=\\mu_1\\otimes\\cdots\\otimes\\mu_n", "486285b7e6324ce7fe7f334ca902acf1": " 10^{20}", "4862d14ad6671df897f3ff831eaf1e9c": "p\\in T", "4862d2cf9f1ba510c3d2dacfecae21fc": " \\frac{dQ}{dt} = 4 \\pi r_{p} K (T_{0} - T_{w}) \\,", "4863045f69ff566e4c7510c733574762": "\\chi_{A} : X \\to \\mathbb{R} \\cup \\{ + \\infty \\}", "4863a7fc4a1c0d59751859f0294a38fa": "\\rho(x)(v \\otimes w) = (\\sigma(x)(v)) \\otimes (\\tau(x)(w)).", "4863d0547d54477631133e6d10d13480": "\\,^{48}_{20}\\mathrm{Ca} + \\,^{248}_{96}\\mathrm{Cm} \\to \\,^{296}_{116}\\mathrm{Lv} ^{*} \\to \\,^{293}_{116}\\mathrm{Lv} + 3\\,^{1}_{0}\\mathrm{n}", "4863dc6f0049082416f492fec60478b3": "f\\ge0", "4863e31650e799ab1dcdfcce685168d1": "H_F=\\sum _{i\\neq j} V_F^{\\text{ij}}a_{c_i}{}^{\\dagger }a_{v_j}{}^{\\dagger }a_{c_j}a_{v_i}", "4863f19756357f14988ec7661d50f2b6": "\\mathbf Y", "4863f3ca7955c9aab798866f41cc8d01": "\\beta=\\frac{1}{kT}", "486477ad09167232a66938c8c2128f51": "A_n=\\sum_{k=0}^n (-1)^k \\frac{2n}{2n-k} {2n-k\\choose k} (n-k)!", "48647a969f1144185759043fe63f59a5": " \\det(A \\otimes B) = (\\det A)^q (\\det B)^p, ", "4864b7ae8e400617d46db368c38f255a": "h^0(V) - h^1(V) = c_1(V) + \\text{rank}(V)(1-g).", "4864d7f935d7d8168dc50e206d859033": "\\varphi\\,", "48650ec18c7885e77a2fdce6c8b2cf75": "(x,\\ f(x))", "48656491cd70cdba7c1913c5b640ef05": "S_{l,t}", "486580037168f845a5fc909346fb0a30": "m\\mathbf{a}=\\mathbf{F}+\\mathbf{N}", "4865f62b618f730206107b11ff8adeb5": "\\sum_{k=2}^\\infty (-1)^k(\\zeta(k) -1) = \\frac{1}{2}.", "4866f1e60b69985fa8894f23544f4d56": "CA_{i+1} = {(x_{i+1} + iCA_i) \\over {i+1}} = {CA_i} + {{x_{i+1} - CA_i} \\over {i+1}}\\,.", "48670d59480158098f6ca9a1b611b5f9": "\\sigma(v_1,v_2,\\dots,v_n) = (v_{\\sigma 1}, v_{\\sigma 2},\\dots,v_{\\sigma n}).", "48675360fb1cf93534004afbbec7da34": "g(x,y)=c", "486795c443946e5fd693e5e6112bc1c6": "a_1(x) = b(x), b_1(x) = r_0(x)\\,.", "48685723c8b1491e690cd5ea50fdedfd": "K(n,S):=\\{x\\in K\\mid x\\in K_{\\mathfrak{p}}^n \\mathrm{\\ for\\ all\\ }\\mathfrak{p}\\not\\in S\\}.", "4868583758598fdd778a8cede92bb63c": " \\sigma_j ", "48686aaf45949fef0449ce64bc3fca9b": "B_0 y_t = c_0 + B_1 y_{t-1} + B_2 y_{t-2} + \\cdots + B_p y_{t-p} + \\epsilon_t,", "48686ad05dd83af6f2f9da861778dfd2": "f^{-1}(y)", "486884cb32b200c14b434091e71733a3": "\\vec{R}_{E}", "4869a34ca489e81884390f67171d224b": "R[t_i]", "486a0ab961973d3d341bb074bace4573": "\\left\\{ \\rho_k(x) : \\Omega_x \\to [0,1] \\right\\}_{k=1}^K", "486a2dff91c58ff83f137ce73e26ab89": "\\mathcal{F} \\mathbf{x}", "486a5dc3315461b639bca3ae7e331fea": "J(x)=\\frac{1}{12}(E(x^3))^2 + \\frac{1}{48}(kurt(x))^2", "486ab76ec2a976f3f31ceb8d89a6780d": "\\{1, e^{\\pm i\\theta} \\} = \\{1,\\ \\cos(\\theta)+i\\sin(\\theta),\\ \\cos(\\theta)-i\\sin(\\theta)\\}", "486aebc8ab02d624070b5c1212a61642": "M=\\gamma m\\,", "486b2cbf486ff67975f0027b38c34ea4": "\\overrightarrow{E}", "486b4b4dab1d024e007829994d199dca": "\\displaystyle\\varphi_i(\\mathbf r_1,\\ldots,\\mathbf r_N)=0,\\quad i=1,\\,\\ldots,\\,K", "486b9909f521cdda6ab9c5137997a715": "M < N_i", "486bc9c2e98d61c6b42ed9b5f0c90f4c": "\\partial \\phi/\\partial x", "486c36313f6d0feda47276a7ebab7ca2": "|Succ(t,T) \\cap IS(s,S)|=1", "486c7954d154cb9bf5d1eb74877e9f36": "\\frac{1}{(\\mathbf{U}q)}= \\mathbf{S.U}q - \\mathbf{V.U}q = \\mathbf{K.U}q", "486dc18d4cf98ffed215e8662842280c": "\nR \\mbox{ metres} = \\frac{L + B + 1/2G +3d + 1/3\\sqrt{S} - F}{2}\n", "486de1bfb97ee98614b0f6d29e60eb1e": "\\int\\left(\\frac{1}{P}+\\frac{1}{K-P}\\right)\\,dP=\\int k\\,dt", "486e3768460920c96f0fbe84224c1118": "\\left\\langle \\frac{\\partial S}{\\partial x_{k}}, \\varphi \\right\\rangle = - \\left\\langle S, \\frac{\\partial \\varphi}{\\partial x_{k}} \\right\\rangle", "486e73ad23481bc9d9d3e73f3235d1cf": "A+A^{\\dagger} = 0", "486ec5b8b1a4064426544d708150a619": "\\nu=1/4", "486edfb5854fa8f517e484e5cee1780d": " EGB(z;\\delta,\\sigma,c,p,q) = \\frac{e^{p(z-\\delta)/\\sigma}(1-(1-c)e^{(z-\\delta)/\\sigma})^{q-1}}{|\\sigma|B(p,q)(1+ce^{(z-\\delta)/\\sigma})^{p+q}}", "486ef2481e028a0ae3803bf2a8e718b3": "\\operatorname{E}(T)=\\frac{\\alpha}{\\beta}", "486f5e04726621b27e0f951af4927d7b": "\\pi = \\frac{NS - NF}{S} = N\\left(1-\\frac{F}{S}\\right)", "486f69a1be34291ca3e2f2d2da62cd57": " g \\,= 2\\sqrt{2m}/\\hbar \\approx 10.24624 \\; {\\rm{eV}}^{-1/2}\\; {\\rm{nm}}^{-1}. \\qquad\\qquad (5)", "486f8756029a71cb03b7482edfaf57ec": "\\rho(A) \\overset{\\underset{\\mathrm{def}}{}}{=} \\max_i(|\\lambda_i|)", "486fb8e4dd31cd0e5acdb964ac780c1d": "R=1/G", "486fbd1f1d85fa82cb200ae950586eb9": "e_H", "48703c46bc8534ad9265af8e22a1969f": "\\begin{alignat}{7}\n x = \\;&& -\\frac{1}{16}z\\,\\,\\, \\\\\ny = \\;&& -\\frac{13}8z.\n\\end{alignat}", "487052195831ea02b54d495190823aca": "\\,^{254}_{99}\\mathrm{Es} + \\,^{48}_{20}\\mathrm{Ca} \\to \\,^{302}_{119}\\mathrm{Uue} ^{*} \\to \\ \n \\ no\\ atoms", "48705a7f7d9cd605c4d843ca43b6fd06": "k_1 < k_2", "48706681a3c73da6dc0e5ac3a01f5995": "\\phi(x_j)=0", "487068a5aa1596e740823a8160468903": "E^f \\gg E^c", "4870ba1081b2f533eef0737047c09321": "f(\\mathbf{x})=\\mathcal{S}\\boxtimes_{n=1}^N \\mathbf{w}_n(x_n)=\\left(\\mathcal{A}\\boxtimes_{n=1}^N \\mathbf{U}_n\\right)\\boxtimes_{n=1}^N\\mathbf{w}_n(x_n),", "4870d0080c489b2f4da687f82219b4cd": "f_1,\\ldots, f_s", "4870e2c50f9915a0da585f889e03f6ec": "G'\\rightarrow G", "4871134243e6a333e2646c3bc7294447": "\\oint_{\\gamma} f(\\zeta)\\,d\\zeta\\, + \\oint_{\\tau^{-1}} f(\\zeta)\\,d\\zeta\\,=\\oint_{\\gamma \\tau^{-1}} f(\\zeta)\\,d\\zeta\\,=0", "48713aea6b053c91de9021b424b784a3": "\\theta \\ ", "487167d26bc615fa3b7129c6b78fb61f": "E = -\\frac12\\sum_{i,j}{w_{ij}{s_i}{s_j}}+\\sum_i{\\theta_i\\ s_i}", "4871810f371c083cf04e5778f2eb5899": "\\mathfrak{sl}(n, F)", "4871907e433f1d5af47387650bfb4626": "h=k \\log n", "4871ca4ee1010dfa5732b50427f78ba9": "V(x)=V_0[\\Theta(x)-\\Theta(x-a)]", "4871dff5a212b1e0f96f68970cb5d680": "S_\\rho(z)-\\frac{Q_n(z)}{P_n(z)}=O\\left(\\frac{1}{z^{2n}}\\right).", "487253caebdf0cd36a2dddcc3a822e57": "a=qb+r \\Rightarrow 10a=q(10b-1)+(10r+q).", "4872cc3e323c1b8dfda9273b8b7a8c88": "K(iw) =k(w)+ i a(w) \\quad (1.3)", "48732c69e575df81c081dbcccef4043d": "\\operatorname{distance}(\\mathbf{x} = \\mathbf{a} + t\\mathbf{n}, \\mathbf{p}) = \\| (\\mathbf{a}-\\mathbf{p}) - ((\\mathbf{a}-\\mathbf{p}) \\cdot \\mathbf{n})\\mathbf{n} \\|. ", "487335c241ee3855d189c57b674add37": "\nL_\\mathrm{w}=10\\, \\log_{10}\\left(\\frac{P_1}{P_0}\\right)\\ \\mathrm{dB}\n", "48734607bdf471657d19fa1a834220ea": " p(x + 1) ", "48735b60ffc23d8e68b9e7ab484b0a25": "\n\\lim_{n\\rightarrow \\infty} p(n,k) \\alpha_k^{n+1}=\\beta_k\\,\n", "4873706ed004f23246d421ac730ea231": "\\vec{v}_E = \\left( 1 + \\frac{1}{4}\\rho_L^2\\nabla^2 \\right) \\frac{\\vec{E}\\times\\vec{B}}{B^2}", "4874447e69dc0bd11ecb2b10e054705e": "f(x)=x_1^2+\\cdots+x_n^2 \\, ", "4874792a82ece5afdc3352ac649b0898": "\\scriptstyle S_x(s)", "4874801ff7c563ee4e4ec879434a8a3c": " \\mathbf{r} = x \\mathbf{i} + y \\mathbf{j} + z \\mathbf{k} ", "48748f1fe9c6fc0060ef3dd3e1da77bf": "\\alpha \\in \\omega^G \\setminus \\{ {\\omega} \\} , v[\\alpha] \\in \\{1,...,r\\}", "48749651b62f8f6842adeaf308854c8a": "aC_\\mathfrak{st}^\\lambda \\equiv \\sum_{\\mathfrak{u}\\in M(\\lambda)} r_a(\\mathfrak{u},\\mathfrak{s}) C_\\mathfrak{ut}^\\lambda \\mod A(<\\lambda)", "4874aabe8f69909785aa5576f509edf4": "\\gamma_{i}", "4874c32b68818d351f3e61941144cc65": " G_\\mathrm{objective} \\simeq \\frac{w_\\mathrm{pixel}}{2{(f/\\#)}_\\mathrm{objective}} ", "4874dd2f74e24942502dcca6782c7a51": "\\frac{1}{r_1}+\\frac{1}{r_3}=\\frac{1}{r_2}+\\frac{1}{r_4}.", "4874fb5a4a2b856bbcacf612177344dc": "\\boldsymbol{\\sigma}=\n\\left[{\\begin{matrix}\n\\sigma _{11} & \\sigma _{12} & \\sigma _{13} \\\\\n\\sigma _{21} & \\sigma _{22} & \\sigma _{23} \\\\\n\\sigma _{31} & \\sigma _{32} & \\sigma _{33} \\\\\n\\end{matrix}}\\right]\n\n\\equiv \\left[{\\begin{matrix}\n\\sigma _{xx} & \\sigma _{xy} & \\sigma _{xz} \\\\\n\\sigma _{yx} & \\sigma _{yy} & \\sigma _{yz} \\\\\n\\sigma _{zx} & \\sigma _{zy} & \\sigma _{zz} \\\\\n\\end{matrix}}\\right]\n\\equiv \\left[{\\begin{matrix}\n\\sigma _x & \\tau _{xy} & \\tau _{xz} \\\\\n\\tau _{yx} & \\sigma _y & \\tau _{yz} \\\\\n\\tau _{zx} & \\tau _{zy} & \\sigma _z \\\\\n\\end{matrix}}\\right]\n", "487541ec55bf913c757841b8ed0cb4b7": " \\rho^2 J_n'' + \\rho J_n' +(\\rho^2 - n^2)J_n =0, ", "487574770578ce3ddcafd0db4e725e44": "f_Y ", "4875ac38cbdeb8d45e86a9258b756ed0": "H(X|Y) = \\mu(\\tilde X \\,\\backslash\\, \\tilde Y),", "4875bdd0f1a97196540f42facaa15a51": "\\mathbb{R}P^\\infty", "4875ffe7779ed990f60a119d1c468137": " Q \\in A[Y_1,\\ldots,Y_n] ", "48761a4b74fcd72faa58da6fd3127d51": "(E(z),F(z))\\,", "48765974d713775c8eb9bc15247fafe8": " M' \\subseteq M ", "48769d348c40557428685ed37a10347d": "(R^r f_* \\mathcal{F})_s = \\varinjlim H^r(U, \\mathcal{F}) = H^r(X_s, \\mathcal{F}), \\quad X_s = f^{-1}(s)", "4876a7ee637783fb7e5ee9737dfef27a": " \\mathrm{FWER} \\le \\alpha\\,\\! \\,", "4876be249a4c89e5e87354433e12cd5d": "w(p,q)", "4876d7b4a60732ae55ca6376fb1094b9": "\\frac {P_{T1}}{P_{T2}} = \\frac {\\rho_1}{\\rho_2} \\lambda^{3.5}", "4876dd3fb3d4dd66983afd046ddc6982": "\\overrightarrow{I}", "4876e0186c7aac079fc3331875aa6c78": "[x,y,z]\\in H", "4876ff7a94c29dfa0d743479d76cba93": "(\\rho,\\phi,z)", "48771b4a5c77ef1b234dbed4a1f2c25b": "a(z) = {\\alpha}_0 + {\\alpha}_1z + {\\alpha}_2z^2", "48773d7391aa88d83ac384b12302a51c": " \\mathop{\\mathrm{ker}}(h) := \\{u \\in G : h(u) = e_{H}\\}\\mbox{.} \\! ", "48779bd07d690af5093bd78e93e766da": "\\partial_\\mu \\phi\\;", "4877d292a5e8b2616600094c985ea450": "A^TD - C^TB = I", "48788dd9d250767cdf0945db2714f98e": "\\mathrm F_{SO}(M)\\to M", "4878cf55e58016f2f0134df71a2e0653": "k:= k+1", "487908d4bb8a9343c17f4af43fd9e151": " P(Y \\in S) = \\int_{\\phi^{-1}(S)} p_x(x)\\,dx. ", "48791e401a5ca4bde03551cfa4b38762": "\\dot x = Ax.", "4879cb83540ff49dbbf24d16e6948689": " \n\\text{Minimize: } f(\\overline{y}_1, \\overline{y}_2, ..., \\overline{y}_K)\n", "4879e53c724e3da524623b3f56011099": " \\widehat{\\boldsymbol{\\beta}}_{k}", "487a2c55fd86df06fd859e7d5fcf86cb": "e = \\overline{ij}", "487a3ed3ee81f3ce9644b8b9ced6944a": "(q^m-1)/(q-1)", "487a8dc2446a8c939e229366d623dc7a": "K_0(K)", "487a98a594abd8a0e2823a02a1c0f119": "ax^{p^h}", "487ab2315fb06c581219784e0b318ac1": "\\begin{array}{ll}\nS_{n-1}(R) &= \\displaystyle{\\frac{n\\pi^{n/2}}{\\Gamma(\\frac{n}{2}+1)}R^{n-1}} \\\\[1 em]\nV_n(R) &= \\displaystyle{\\frac{\\pi^{n/2}}{\\Gamma(\\frac{n}{2} + 1)}}R^n\n\\end{array}", "487b2470d670e0af114a474e846fe92b": "e_2 = <0,1>", "487b2dd994d99c1c441f774186ad62d5": "A\\,\\mid\\,B", "487b8733ec2a13faeba7dd577e8de0b3": "\\begin{align}\n\\rho &= \\sqrt{x^2 + y^2} \\\\\n c &= \\arcsin\\left(\\frac{\\rho}{R}\\right)\n\\end{align}", "487c0fd7414fdb8e4b86d0d8d9780e20": "\\boldsymbol Z", "487ca9cb2d148069d7b5cd4a92440bc8": "x^{\\overline{m}}=\\overbrace{x(x+1)\\ldots(x+m-1)}^{m~\\mathrm{factors}}\\qquad\\mbox{for integer }m\\ge0,", "487cb642717cdef5b0fbbc3ac0315492": "(S,T,W,M_0)", "487cc30a7cb9079b20bc779a8c87f5f2": "r=a\\frac{\\sin(2\\theta-\\alpha)}{\\sin(\\theta-\\alpha)}", "487cf5a3ee1cd3f659745a9e6bda6730": "1 \\leq i \\leq k", "487d29dd7ef4fc0130069077cf0adbad": "X^2 = \\frac{(500,000 - 499,945)^2}{499,945} + \\frac{(10,000 - 10,055)^2}{10,055} + \\frac{(200 - 254.9)^2}{254.9} + \\frac{(60 - 5.13)^2}{5.13}", "487d5ac852374be5256101a4ab78d91a": " E_n = - \\frac{e^2}{2 a_0 n^2} ", "487d66faf84984c1d512f3d0e240fc86": "U(x, y, z) := (x^2 + y^2) \\sqrt{ 1 - \\frac{x^2}{x^2 + y^2} } = 0.", "487d7b5481c78be9c05d275cfec4ace9": " [L_1, L_2]=0.\\qquad (2)", "487dce752e8c492d12ea9c53f3a9a1fc": "Z_0 = \\sqrt { {L \\over C}}", "487dddcc13f3f731e93ec458c0498346": "(g,1)(h,1)=(h^{-1}g,0)", "487eb252ee6fdc2b69e892ae0731226d": " \\text{CSS}(C_1,C_2) ", "487ed8a2ba84ad4395211f477ef63d39": "\n\\begin{bmatrix}y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ y_6 \\\\ y_7 \\end{bmatrix} = \n\\begin{bmatrix}1 &0 &0 \\\\1 &0 &0 \\\\ 1 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 0 & 1 \\\\ 1 & 0 & 1\\end{bmatrix}\n\\begin{bmatrix}\\mu \\\\ \\tau_2 \\\\ \\tau_3 \\end{bmatrix}\n+ \n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\\\ \\epsilon_7 \\end{bmatrix}\n", "487eea91ce395c896865884e38443b65": "\\phi(x)\\geq 0", "487f01764e71960c5efb5231b19635db": "\\underline{\\underline{\\mathsf{S}}}", "487f02a70f11267555ebe4d916010a4e": "\\tilde{e}_i (b \\otimes b') = \\begin{cases} \\tilde{e}_i b \\otimes b', & \\text{if }\\phi_i(b) \\ge \\epsilon_i(b'), \\\\ b \\otimes \\tilde{e}_i b', & \\text{if }\\phi_i(b) < \\epsilon_i(b'), \\end{cases}", "487f2fa2e03723944ef8161868801a8b": "\\nabla_XZ = (\\Delta S - I)X", "487f30de817471351650c816afb307aa": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{36 \\choose 1} \\end{matrix}", "487f3c3223c7d29c33c062da6e6bcc8b": "f = 2 \\Omega \\sin{\\phi}", "487f3ec7ca8b0a745fda1b02f927ec11": "f^{-1}\\Delta", "487f68bb820ba679ae8d24ea0f79380a": "\\lambda_1~\\lambda_2~\\lambda_3=1", "487ff412c647fbf9fc16d42244794f02": "\\mathcal{S}_{2}\\psi =(\\cosh (\\Delta )\\mathbf{S}_{2}+\\sinh (\\Delta )\n\\mathbf{S}_{1})\\psi =0, \n", "488025df275721bd3598af6eace6aa61": "\n\\begin{align}\n\\left[a, a\\right] & = \\lbrace a \\rbrace \\\\\n\\left[a, b\\right] & = \\lbrace x \\vert x \\in \\mathbb{R}, a \\leq x \\leq b \\rbrace \\\\\n\\left[a, \\infty\\right] & = \\lbrace x \\vert x \\in \\mathbb{R}, a \\leq x \\rbrace \\cup \\lbrace \\infty \\rbrace \\\\\n\\left[b, a\\right] & = \\lbrace x \\vert x \\in \\mathbb{R}, b \\leq x \\rbrace \\cup \\lbrace \\infty \\rbrace \\cup \\lbrace x \\vert x \\in \\mathbb{R}, x \\leq a \\rbrace \\\\\n\\left[\\infty, a\\right] & = \\lbrace \\infty \\rbrace \\cup \\lbrace x \\vert x \\in \\mathbb{R}, x \\leq a \\rbrace \\\\\n\\left[\\infty, \\infty\\right] & = \\lbrace \\infty \\rbrace \n\\end{align}\n", "48802c3046f46da961b6ff1e88189780": "\\Delta = \\sqrt{\\frac{4}{3}\\frac{m\\, c}{H}}.", "48804a256643a9eec36e23bd61942ee1": "\\Omega < \\Omega + 1", "48807216054f5e850853fc69d4511007": "C = \\frac{pi}{T 12}", "4880f0595401b7f1d583b91e329ca8ac": "n=1, d=1", "48810fea9b4d6fc6e024f340d390bd5e": " = \\hat{a}_j^\\dagger \\,\\hat{a}_i\\, \\hat{a}_l^\\dagger\\, \\hat{a}_k + \\delta_{kl}\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij}\\hat{a}_l^\\dagger\\hat{a}_k + \\delta_{ij} \\delta_{kl} ", "48817a500a6225efb46995ca2d85af9e": "4a^3 + 27b^2 \\not= 0", "488193142155b010c64c433cf153ddfc": "\\ q_{\\text{s}} = \\frac{q}{\\sqrt{4 \\pi \\varepsilon_0}}.", "488241c84c4487e149482104a204b8e5": "x \\mapsto x^3", "48824ac9a620613c5199eb8934899ae4": "\\frac{\\zeta'(\\hat\\alpha,x_\\min)}{\\zeta(\\hat{\\alpha},x_\\min)} = -\\frac{1}{n} \\sum_{i=1}^n \\ln \\frac{x_i}{x_\\min} ", "488265c7ab36cd9b029b2858d03b7e4c": "\nE[a_i(t) - b_i(t) | Q(t)] \\leq -\\epsilon\n", "4882a72049be547c4d1a37263f9a7c03": " \\bar{F}=-ip_0\\oint_C d\\bar{z} +i \\frac{\\rho}{2} \\oint_C |v|^2\\, d\\bar{z} = \\frac{i\\rho}{2}\\oint_C |v|^2\\,d\\bar{z}.", "4882ca7a83b3114c48297e2efa28ca4d": "\\frac{4EI}{L}", "48833fc1fc277ebadfc7829f2a1fd3e8": "R_i=100-4.6 \\Delta E_{UVW}", "4883bf81758bd2f426e2b0d5c82528fa": "0 < b_i < 1", "4883d842a7b8eddc711c4f756abdcb62": " P_s = N_{interact} F_{max} = \\frac{3 F_v \\gamma\\ }{2r}. \\,\\! ", "4883f6122c998a9bb96b6c74570063ec": "\\text{number of sidereal days per orbital period}= -1+ \\text{number of solar days per orbital period}", "488412ae5eae1fd29632ee19ac660ff1": "I = \\int _0^{\\frac{\\pi}{2}} \\frac{1}{\\sqrt{a^2 \\cos^2(\\theta) + b^2 \\sin^2(\\theta)}} \\, d\\theta = \\int _0^{\\frac{\\pi}{2}}\\frac{1}{\\operatorname{AGM}(a,b)} \\, d\\theta = \\frac{\\pi}{2 \\,\\operatorname{AGM}(a,b)}", "488476b101f19d8fd018174558b17162": "\\vec{R}_{cm}=\\frac{1}{I_{tot}}\\cdot\\sum_{k=1}^{N}I_k\\cdot\\vec{r}_k", "4884f5a549e3727cf29e1ea0bebf1946": "\\mathbb{C}^\\times", "48853125813fb635a6c9f18d4a46520e": "\\bar u = \\sqrt{\\frac{8kT}{\\pi m}}", "48855c88a4a691e1916ee9c72576b590": "\\int_{0}^{T} \\sigma (X_{t}) \\circ \\mathrm{d} W_{t} = \\frac{1}{2} \\int_{0}^{T} \\sigma'(X_{t}) \\sigma(X_{t}) \\, \\mathrm{d} t + \\int_{0}^{T} \\sigma (X_{t}) \\, \\mathrm{d} W_{t}.", "488578c03d655c9b0b546829002fc015": "\\Pr(\\mathbf{sig})=\\sum_{y: Pr(\\mathbf{y}) \\le Pr(\\mathbf{x)_0}} \\Pr(\\mathbf{y})", "488583d2a9184cabcd80823cfe2a6af1": "\nd_c \\approx 0.057 \\cdot \\sqrt \\frac{V}{RT_{60}}\n", "48858dbb1fe6288164c63a20fa120014": "{ [S] \\over v } = { [S] \\over V_\\max } + { K_m \\over V_\\max } ", "4885ba1e883fdc4f4f3479fdbbbe0c8d": "x = \\cosh \\ t", "4885bd9b8fd4d920c089f8b96228a456": "f^*(\\mathcal{I}) \\mathcal{O}_\\mathfrak{X}", "4885bf13222918289d7d929f78b2ab13": " V_{p \\times p} = [\\mathbf{v}_1,...,\\mathbf{v}_p] ", "488646c6f9fe7878eff7d67fea2bfbff": "825265 = 5 \\cdot 7 \\cdot 17 \\cdot 19 \\cdot 73\\,", "48866980d51c866ad24547116f2f7954": "\n{\\scriptstyle \\sin^2\\beta}\\;\\;\\mathbf{g}^{-1}= \n", "4886c58593177539654d0536d5dcb06b": "n=|a|", "4886f5973c60e25fa1f18945c744f989": "\\begin{array}{rccc}\\sigma:&\\mathbb{Q}(\\sqrt[3]{2})&\\longrightarrow&\\mathbb{A}\\\\&a+b\\sqrt[3]{2}+c\\sqrt[3]{4}&\\mapsto&a+b\\omega\\sqrt[3]{2}+c\\omega^2\\sqrt[3]{4}\\end{array}", "48876505e2b7152bbdd0b2134480bb31": "rca(X)", "48878788ab31627a93775fd613216e91": " \\mathbb{P}^2.", "4887d42f630580ede263efda50cda971": "H ", "4887dbd64157fecf1aef9e79d0191846": "(p_0,q_0)", "4887f6cefcf8d7af4db00810bab2df87": "\\left|\\frac{1}{|B|} \\int_{B}f(y) \\, \\mathrm{d}\\lambda(y) - f(x)\\right| = \\left|\\frac{1}{|B|} \\int_{B}f(y) -f(x)\\, \\mathrm{d}\\lambda(y)\\right| \\le \\frac{1}{|B|} \\int_{B}|f(y) -f(x)|\\, \\mathrm{d}\\lambda(y).", "4888224b7e72e1d82ba94daed38ed5b2": "w(x) = \\frac{1} {(2 \\pi)} \\int_{0}^{c} \\frac {\\gamma (x')}{(x-x')} dx'", "48883259a719d0975d762e5bee6de6c5": "MA = \\frac {T}{T_j}= \\frac {nDA}{d} ", "488867cce4bcb19f6e3b2cf7d152dc22": "\\operatorname{MSPE}(L)=\\operatorname{E}\\left[\\sum_{i=1}^n\\left(y_i-\\widehat{g}(x_i)\\right)^2\\right]-\\sigma^2\\left(n-2\\operatorname{tr}\\left[L\\right]\\right).", "48889f83a4dcdb487bf3b2d38c9a36b4": " \\kappa \\,,", "4888af6409868a99f143fd4f5378e332": "\n\\begin{align}\n\\frac{e^{(\\boldsymbol\\beta_c + C) \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{(\\boldsymbol\\beta_k + C) \\cdot \\mathbf{X}_i}} &= \\frac{e^{\\boldsymbol\\beta_c \\cdot \\mathbf{X}_i} e^{C \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i} e^{C \\cdot \\mathbf{X}_i}} \\\\\n&= \\frac{e^{C \\cdot \\mathbf{X}_i} e^{\\boldsymbol\\beta_c \\cdot \\mathbf{X}_i}}{e^{C \\cdot \\mathbf{X}_i} \\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}} \\\\\n&= \\frac{e^{\\boldsymbol\\beta_c \\cdot \\mathbf{X}_i}}{\\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i}}\n\\end{align}\n", "4888dee8d232e42c0555c9ce21a35d95": "g_{ab}+l_an_b+n_al_b", "48890525543cbf37f07496744400742f": "\\langle s,r \\rangle (t) = KA^2\\Lambda\\left (\\frac{t-t_r}{T} \\right)e^{2 i \\pi f_0 (t\\,-\\,t_r)} + B'(t)", "4889416474feedbd771e6092cd26e905": "F(N)=-N/(2+N)\\;", "4889604e29c650af74b647fe259c2edf": "= \\min(5-3, 3-2, 2-1) = \\min(2, 1, 1) = 1", "488968c8363007fe20e033f70ad0b931": "n + 1", "48897ce69ceb621a3f1785d29af70437": "g(n,1/\\epsilon)\\,", "48899cae054eadb27cbdc0276f984750": "[1,\\lambda_3,\\lambda_2]'", "4889df0a074087fc1561105ce48b1ef7": "\\psi_{0}", "4889e33608215ff7ea4a810205117944": "\\begin{cases} \\mathrm{d} X_{t}^{\\varepsilon} = b(X_{t}^{\\varepsilon}) \\, \\mathrm{d} t + \\sqrt{\\varepsilon} \\, \\mathrm{d} B_{t}; \\\\ X_{0}^{\\varepsilon} = 0; \\end{cases}", "488a64c30952e430d14f8113034ad408": "\nA_\\mathrm{vdB} = 20 \\log|H(j\\omega)| = 20 \\log {1 \\over \\left|1+j{\\omega \\over {{\\omega_\\mathrm{c}}}}\\right|} ", "488a6a1081ac3bbbdeffd80604eb7cac": "\\Delta \\phi_{(M,M^{+z})} = \\Delta \\phi_{(M,M^{+z})}^{\\ominus} + \\frac{RT}{zeN_A}\\ln a_{M^{+z}} \\,\\!", "488ab29adec84b0394426c82ab48b26d": "dP", "488ab9a7c1b281111b24127e0c6adae8": "a^8 = 1\\,", "488ac022ecb00bf9006b548815dae0ac": "d^*([x],[y])=d(x,y)", "488ad2f17755e47905e20db1f2314a89": "E_{m, \\mathrm{K}_{x}\\mathrm{Na}_{1-x}\\mathrm{Cl} } = \\frac{RT}{F} \\ln{ \\left( \\frac{ P_{Na^{+}}[Na^{+}]_\\mathrm{out} + P_{K^{+}}[K^{+}]_\\mathrm{out} + P_{Cl^{-}}[Cl^{-}]_\\mathrm{in} }{ P_{Na^{+}}[Na^{+}]_\\mathrm{in} + P_{K^{+}}[K^{+}]_{\\mathrm{in}} + P_{Cl^{-}}[Cl^{-}]_\\mathrm{out} } \\right) }", "488ade1c05d40b28a2680c0827797119": "V(f(\\vec{x}),y)", "488b0c5e3f73d3d44afcb7a599bf48d5": "\\sum _x \\sinh ax = \\frac{1}{2} \\operatorname{csch} \\left(\\frac{a}{2}\\right) \\cosh \\left(\\frac{a}{2} - a x\\right) + C \\,", "488b28415d56a8b08350a3310d366ba5": "(f\\circ g)'=(f'\\circ g)\\cdot g'.", "488b57d6413215bbf9f7e8e522833446": "\\hat{B}_\\mathbf{q}", "488b6a5acfc86038a262c5684fcaf204": " \\operatorname{let} p : \\operatorname{de-lambda}[p\\ f] = \\operatorname{let} x : \\operatorname{de-lambda}[x\\ x] = \\operatorname{de-lambda}[f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x) \\operatorname{in} p ", "488ba18cd2494bae1fd90a7f41a4295e": "\\lambda(\\phi^*\\phi - v^2)^2 ~, ", "488bf91a64cf1ac00a7c9e1b860de5f6": "W\\approx\\prod_i \\frac{g_i^{N_i}}{N_i!}", "488c04ac7650ea5163fe135342bcbc3f": "q_s* = \\frac{0.05}{c_f} \\tau*^{2.5} ", "488c0f1a7d88417a207a880d74fc44e5": "\\displaystyle u_t+\\nabla^4u+\\nabla^2u+|\\nabla u|^2/2=0", "488c4e24f5b2800d034cefcf4680ad86": "Z=-jX_C=\\frac{-j}{\\omega C}\\,", "488c676d87a23e09d4fe52c6770d5342": "\\{ \\bold {n}_1, \\bold {n}_2, (\\bold {n}_1 \\times \\bold {n}_2) \\}", "488c9e8aca6f39d51cb6474921141156": " \\begin{align}\n\\left.\\frac{d}{dt} \\right|_{t=0} E[I + th] &= \\frac{d}{dt} \\big|_{t=0}\\frac{1}{2} \\int_{\\Omega} g\\left( \\| \\nabla (I+th)(x)\\|^2 \\right)\\, dx \\\\\n &= \\int_{\\Omega} g'\\left(\\| \\nabla I(x)\\|^2 \\right) \\nabla I \\cdot \\nabla h\\, dx \\\\\n &= -\\int_{\\Omega} \\mathrm{div}(g'\\left( \\| \\nabla I(x)\\|^2 \\right) \\nabla I) h\\, dx\n\\end{align} ", "488ca567693f3f612f6486347b3035ba": " MM^tM \\leq nM ", "488d5d99f560de151d255976b83552e9": "AM = \\sqrt { ( r \\cos z )^2 + 2 r + 1 } \\; - \\; r \\cos z \\,", "488d657ac6f0adf2a3957be68e091199": "S=[0,1],", "488d7fc6bf8b4eb22b60ff25886c41a5": "X_1+X_2+\\ldots+X_n \\stackrel{d}{=} c_n X+d_n ,", "488da395740e761aec775a9550438475": "H_E\\ne 0", "488db17cef2a49506ad6b31735b56ab6": "BS = \\frac{1}{N}\\sum\\limits _{t=1}^{N}\\sum\\limits _{i=1}^{R}(f_{ti}-o_{ti})^2 \\,\\!", "488dc3240db2a8b86172cb02e572d248": "\\sum_{n=0}^\\infty[({\\Bbb A}^2)^{[n]}]t^n=\\prod_{m=1}^\\infty \\frac{1}{1-{\\Bbb L}^{m+1}t^m}", "488dd6690465505956671ccc1f811d2c": "ds^2\\,=\\,-\\frac{\\rho_K^2\\Delta_K}{\\Sigma^2}\\,dt^2+\\frac{\\rho_K^2}{\\Delta_K}\\,dr^2+\\rho_K^2d\\theta^2+\\frac{\\Sigma^2\\sin^2\\theta}{\\rho_K^2}\\big( d\\phi-\\omega_K\\, dt \\big)^2\\,,", "488e17f033a5d607de1b4e81f71f2727": "\\mathbf{a} \\times (\\mathbf{b} \\times \\mathbf{c}) + \\mathbf{b} \\times (\\mathbf{c} \\times \\mathbf{a}) + \\mathbf{c} \\times (\\mathbf{a} \\times \\mathbf{b}) = \\mathbf{0}.", "488e748b47a1801976b540ee12a2c6d1": "d+ W^+\\to u ", "488e86d86e300d061c2ada3a5c822246": " H = \\sum_{i=1}^N \\left( {p_i^2 \\over 2m} + {1\\over 2} m \\omega^2 x_i^2 \\right)", "488e8a2b521da2555664639f63107152": "\\begin{align}\n PV &= 0.7 e^{-(0.1)(\\frac{3}{12})} = 0.6827 \\\\\nS_0 ' &= 40 - 0.6827 = 39.3173 \\\\\n\\\\\n d_1 &= \\frac{\\left[\\ln\\left(\\frac{39.3173}{40}\\right) + \\left(0.1 + \\frac{0.3^2}{2}\\right)(0.4167)\\right]}{0.3\\sqrt{0.4167}} \\\\\n &= 0.2231 \\\\\n\\\\\n d_2 &= 0.2231 - 0.3\\sqrt{0.4167} \\\\\n &= 0.0294 \\\\\n\\\\\nN(d_1) &= 0.5883 \\\\\nN(d_2) &= 0.5117 \\\\\n\\\\\n C &= 39.3173(0.5883) - 40e^{-0.1(0.4167)} (0.5117) \\\\\n &= 3.4997 \\\\\n &\\approx \\$ 3.50\n\\end{align}", "488ef2185fdcd2b0302218f4f17ce260": "V_s=V_r+V_f=R I+V_f", "488f26577b68065b11db1e37ffebe9ba": "\\exist x_1\\exist x_2\\exist x_3((x_1 \\not =x_2) \\and (x_1 \\not =x_3) \\and (x_2 \\not =x_3))", "488f4cf5e0260cb5d8b8bd0e2963a6e8": "{I_z \\over I_0} = e^{-kz}", "488ffc5d58557330e214d5752667f1eb": "I_1^2 + I_2^2 + I_3^2", "488ffd9134779ce67326f6b6ad0d2972": "r=\\sqrt{\\rho^2 + z^2}", "4890945f6e575639fa6fade9d48dcbea": "(\\hat{l},\\hat{r})", "4890da4c40a739d53c877b2659db2ba5": " h A {(T_{\\infty}^i - T_{0}^i )}+ \\kappa A q_{sol} = (\\rho c_{p} \\Delta x A) \\frac{(T_{1}^i - T_{0}^i )}{\\Delta x} ", "4890ed013da32430868f7d9887cc5495": "E_1(\\mathbf{R}) \\approx\nE_2(\\mathbf{R})", "48910d6d464aad17eb4bbabe693f2def": "1_e", "4891490350248a92261becf158e80a2c": "\\psi^R_{\\mathrm e}", "4891b68cff80ae6a9dd4fa269518c86e": "\\omega_0^2", "48923771b66d5db7ffa1b1f6fb64971d": "v_{eq}", "48927e4527b64805164861a132b81953": "A^T A = I \\text{ or } A A^T = I. \\,", "4892860ee217de640a163a09de256a80": "\n\\sum_\\text{perm}\\langle 0 |T\\left[\\phi(x_1)\\phi(x_2)\\right]|0\\rangle\\cdots\\langle 0 | T\\left[\\phi(x_{n-2})\\phi(x_{n-1})\\right]|0\\rangle\\phi(x_n).\n", "489337b42a7948b15ca25ef69e1df283": "\\int \\sec^3 x \\, dx = \\frac{1}{2}\\sec x \\tan x + \\frac{1}{2}\\ln|\\sec x + \\tan x| + C", "48934d1c692874294455a32012860f7a": "\n\\begin{align}\nF(A)& = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\left(\\prod_{i = 1}^n a_{\\sigma(i)}^i\\right) F(I)\\\\\n& = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n a_{\\sigma(i)}^i\n\\end{align}\n", "4893508b3e694623b422a397c5914494": "f(x+y)=f(x)f(y)", "48935522cac78317224db63ef050c3cf": "E_n^{(1)} = \\langle n^{(0)} | eEz | n^{(0)} \\rangle =0", "489375b8ee6bbf6f2452d2c97d46e5d8": "=\\left(\\frac{1}{3} + \\frac{1}{4}\\right)", "4893a065097926cd59370a4f3cf348df": "\\mathbb{Z}= \\operatorname{trace}_{V^{\\otimes N}}(\\tau^M)\n\\sim \\lambda_{max}^M ", "4893ad711469c18204f3eca642c7094b": "u\\in U", "4894031060dd09d76c9376dbc74dccd5": "q_1 * (5000-q_1-\\frac{5000-q_1-c_2}{2}-c_1)", "48941c2c25d85db76e936559040944f3": " l^2 - r^2\\sin^2 A = (x - r \\cdot \\cos A)^2", "4894ce1e5b86316b1980736891fbd0e9": "F(r) = {1 \\over 2}\\ln{1 + r \\over 1 - r} = \\operatorname{arctanh}(r).", "4894e6a0ff4801598b00092ad6c24e48": "|S_{i+1}|<=(\\dfrac{1}{2-\\alpha})|S_i|", "489516380de0c0689ae7687aa56953dd": "V_n=a^n+b^n \\,", "4895421fa5abfa2097d2f91c9c5df5f6": "\\mathrm{E_n}", "4895f8fcb3242a56118a273c423518a3": "f(b)", "48960f9e3e0d189eb25b43f9def8b29c": "\\omega^\\alpha\\!{}_\\beta\\,e_\\alpha = \\nabla e_\\beta.", "4896396d3246afd2013ef6750c4ab27a": "-4.5 \\le x,y \\le 4.5", "48965a8ecf5ac6f04f9bb764e517fd6a": "\\mathbf{\\Delta \\beta}", "489663d121da935005d14de3961a33ea": "2\\gamma(x,y)=\\text{var}(Z(x) - Z(y)) = E\\left(|(Z(x)-\\mu(x))-(Z(y) - \\mu(y))|^2\\right). ", "4896942a05edf6fcd8c2550957ffb683": " 1000009 = \\left[\\left(\\frac{4}{2}\\right)^2 + \\left(\\frac{34}{2}\\right)^2\\right] \\cdot \\left(58^2 + 7^2\\right) \\, ", "4896da05bc2750fa4512bfd8b8f0a302": "{\\varepsilon_1}^{\\varepsilon_0\\omega}", "48978b16295b0428a912be36a1d0c2a6": "\\sqrt{n}\\sigma", "4898003dbabc041e0a3a8b6a42bce500": " f:(\\theta, \\phi) \\mapsto [\\cos \\phi : \\sin \\phi : \\cot \\theta], ", "48981b10539bae266ff0d896013fd370": " \\ln 2 = \\frac{1}{2} + \\frac{1}{2\\cdot 3} + \\frac{1}{2\\cdot 3\\cdot 7} + \\frac{1}{2\\cdot 3\\cdot 7\\cdot 9}+\\cdots. ", "48987ffad2254f085d442c4c77e9f89b": "\\mathbf{T} = \\{ T[i, k] \\} ", "48988bbdea045e8da2eb4248d6cf530b": "\\epsilon({{{\\it{D}}}}) = 1 - \\prod\\limits_{n=1}^{N-1}(1-\\epsilon_n)", "4898e5c2cf85f274557f6febb1cdfc8d": "P = {Nmv_{rms}^2 \\over 3V} ", "48992f0331376c26a16afec8756cd878": "T_1:=\\frac{T+T^*}{2}", "48995a516f2c588c55c9521eb4c57040": "L^2(\\mathbb{R})=\nH^2\\left(\\mathbb{C}^-\\right) \\oplus\nH^2\\left(\\mathbb{C}^+\\right).", "48996b37150c0012b3cfbc6c54bd7be2": "\\kappa_f(g) = \\int_S f g \\, \\mathrm{d}\\mu,\\qquad g\\in L^q(\\mu).", "4899981040571187bbff7447bc6ba24a": "x\\in V(G)", "4899d7a7804266a94df601140b9e9afb": "x_k + y_k\\sqrt n = (x_1 + y_1\\sqrt n)^k.", "4899f3b67ff3351466a59cbc70d7a43b": "\\frac{3 \\sqrt{2}}{4}", "4899f81337edd7ddde6a455d63a7bfa8": "(D,V,s,R) \\models \\exists \\bar{x}.P", "4899fb44f14867ddc63aa25d835c547f": "10^6", "489a5d04972ae0fbdfff7e068332d889": "\\mathcal{E}_2", "489a787d9291a8b07d9a5d82c81c0c05": "X(\\theta,\\phi)=((2+\\cos\\theta)\\cos\\phi,(2+\\cos\\theta)\\sin\\phi,\\sin\\theta)\\,", "489a8a5178a18e16a49aa9c722b948a9": "\\vec{X} = \\vec{e}_0", "489ae78d16faa398ba65fed38ebed3a6": "CFMG", "489b06813c7144efab2440f9ec20b776": " \\frac{c}{f'} - \\frac{c}{f} = \\frac{h}{m_ec}\\left(1-\\cos \\theta \\right). \\,", "489b13f7fb9a299c3d5c385dc08d1af4": "S[A[i]+d(v),A[i]+H[i+1]-1]", "489b2886d876c7fb1e4dc831bb543b3d": "\\scriptstyle\\langle\\cdot,\\cdot\\rangle:\\mathbb{R}^n\\times\\mathbb{R}^n\\rightarrow\\mathbb{R}", "489b69976f508a0835f171d12733e0cd": "\\epsilon u j", "489c99b9ff7dc093d7527f835d801af1": "\n \\boldsymbol{F} = \\frac{\\partial \\mathbf{x}}{\\partial \\mathbf{X}} = \\boldsymbol{\\mathbf{x}} \\cdot \\nabla ~.\n ", "489c9c834856b2eb6468577f3686c9d9": "\\begin{bmatrix}\nh_n & h_{n-1} \\\\\nk_n & k_{n-1}\n\\end{bmatrix}", "489d075b27ec80953055fd483cabb861": "\\langle \\mathbf{r}_0, m | \\mathbf{r}, s_z \\rangle = \\delta_{m\\,s_z}\\delta( \\mathbf{r}_0 - \\mathbf{r} )", "489d3a67955cafe706a984724292d131": "(3) E = \\frac {N_{++} - N_{+-} - N_{-+}+ N_{--}} {N_{++} + N_{+-} + N_{-+}+ N_{--}}", "489d9a88389bda8c566559c00b3de1be": "\n\\eta(i)=\\frac{\\Gamma \\left(\\frac{1}{4}\\right)}{2 \\pi ^{3/4}},\n", "489df2f7792a9d7313001ffc04dddedd": "q_{0i}=(s_{0i},0)", "489e42a013db51e3c80646a278b76746": "c^{2} (\\mu) = \\int_{\\mathbb{R}^{2}} c^{2} (\\mu; x) \\, \\mathrm{d} \\mu (x).", "489e49ceda270533d13955870dd791f3": "n(z,\\rho)", "489e505fe3e065406eaa78746d5899a8": "\\left( s \\tilde{N}(s) - N_o \\right) + \\lambda \\tilde{N}(s) \\ = \\ 0 ", "489e73b5671e73ab5a1570a6a2196e8f": " \\mathbb{H}_{ij} = \\left\\langle \\Phi_i^{SO} | \\mathbf{H}^{el} | \\Phi_j^{SO} \\right\\rangle ", "489ea00cfb78ad71860c8d195f10d471": " (\\Phi,\\,\\,H,\\,\\,\\Phi^*)", "489eb4a7d79bac11b1a7eeada0bc6cbd": "X_U/\\ker(\\mu_U)", "489f07bffe8b7e18a6112f5022b09410": "v \\in V", "489f59c8236dc96e5dcacea8dd979895": "\\Delta E^\\ominus \\left( \\mathrm{X} \\right ) = E^{\\ominus} \\left( \\mathrm{X} \\right ) - E^{\\ominus} \\left( \\mathrm{Def} \\right ) ", "489f817d387ccc8c1f1db3efc0cec427": "f({z}) = {z}^{\\mathrm{T}}({Mz}+{q})\\,", "489f8d8c6170db74cf1b971517b2a46b": "\\frac{d}{dx}\\psi_1(0) = \\frac{d}{dx}\\psi_2(0)", "489fd1a08ee3483712baeef3ff984150": "G:=1-\\frac{2M}{r} ", "48a013ec16bc730a94888e24a5c38843": " -0.2", "48a0155672635111efe282df6fcb7ee3": " f_{\\alpha+1}(n) = f_\\alpha^n(n),\\, ", "48a0156af77d1384e79b1130311a40fe": "Q = \\frac{SS_t}{SS_e}", "48a0542bb53384962e62c651776923d1": "\\int u \\;dx = \\frac{1}{2}\\left(xu+a^2\\arcsin\\frac{x}{a}\\right) \\qquad\\mbox{(}|x|\\leq|a|\\mbox{)}", "48a0b36aba4eedbdb663f5055f14ded4": "a \\triangleright (b \\triangleleft c) = (a \\triangleright b)\\triangleleft (a\\ \\triangleright c)", "48a0c2f613c6523508a1d62347b4f2c5": "x_2=+1\\,\\!", "48a0fe3ecb273cc0b425d3f41910d80b": " \\angle OAB = \\angle OB'A' \\ \\text{ and }\\ \\angle OBA = \\angle OA'B'.", "48a11d932d42588d4354281e10e9a592": "\\displaystyle i^n \\sqrt{2\\pi} \\delta^{(n)} (\\omega)\\,", "48a125db7635fae54e6d0918b6054d99": "\\ell(\\ell+1)\\,=\\,2", "48a140cad7163769dac9c7cf13ea3341": " \\Lambda^2\\mathbb C^3 ", "48a15ec5ddf47e532f33d97e6b1686e1": "\\Pi = -\\Delta \\gamma = - \\left[ \\frac{\\Delta F}{2(t_{p} + w_{p})} \\right] \\approx - \\frac{\\Delta F}{2w_{p}}", "48a1813792f67c981f64317602b36aa7": " A \\mapsto Q_{k\\ell}^T A Q_{k\\ell} = A' . \\,\\! ", "48a19cc126534d5222ca994090ba4bbd": " Pe \\,= \\frac{F}{D} \\,= \\frac{\\rho u}{\\Gamma / \\delta x}", "48a1d7c5b084ec1dd8f767d69bd9d517": " E\\left[\\sum_{x\\in {N}}f(x)\\right]=\\int_{\\textbf{R}^d} fd\\Lambda , ", "48a284488a3ee5aef173c839c4ad35b8": "\\left(\\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "48a2afee7b8d97feee92c77638819907": "\\mathcal{RA}\\,", "48a332bdb7d47f0908f3d36301b73c7f": "mB \\Omega", "48a389f04967c1ccbab993a5f6b90c0f": "x \\in [0,1/2), y \\in [1/2,1]", "48a3aae5791be5473a17f5268507e47f": " \\mathrm{dom} \\ f = \\mathbb{R}_{++}^n ", "48a439ca8d7f4b9f738e00dd7cd5345e": " E =\n-{W\\over T}=\n a_1 a_2\\int {d^3k \\over (2 \\pi )^3 } \\; \\; D\\left ( k \\right )\\mid_{k_0=0} \\; \\exp\\left ( i \\vec k \\cdot \\left ( \\vec x_1 - \\vec x_2 \\right ) \\right )\n", "48a44052ca0eefa40e6412c914b452b5": "\\Delta H_v", "48a459c7bd1290df1a1ae5258e7eb1ba": " - i \\exp \\left( i t \\right) ", "48a49eab8099b53c734b4dcd4baa9a41": " x_{i-\\frac{1}{2}} \\ ", "48a4abb3900fef98e80a4955be2eadbd": "L_{e2} = J \\times L_e", "48a4f644b27e741d4f49a9a828089feb": "A\\cos(\\omega t + \\phi)", "48a5814cdd059bc0d0d6b4bb9f03c112": "y = kx^{a} \\,\\!", "48a5a4a5bf3eca90e2b9c478f8fd779c": "f(x) w(x)", "48a5ef0e517c9dc5ed96b30298a8f37f": "\\,(z_1,\\ldots,z_4)", "48a610099fbd2c32b20fb93077e4025a": "U_0=0,U_1=1,U_2=1,U_3=R-Q=a^2+ab+b^2", "48a64b9c34c06152c5b556e99e1e8364": "\\varphi=\\psi-\\theta", "48a65bde2de6988426360d1d31cab37f": "j = \\frac{\\Delta a}{\\Delta t}", "48a703f0deb2697df6c584dfe78d8fc5": "x = \\overline{\\varphi(z)} \\cdot z /{\\left\\Vert z \\right\\Vert}^2", "48a707521da3085d8789af3f17f6bf14": " c_m = \\frac{C}{m} = \\frac{c_{volumetric}}{\\rho}. ", "48a7a601d4478ef3aad65915dc60661a": " g(L_1 \\times L_2) \\cdot gL_3 = 0, ", "48a7bf4e43f8753b6ad9ae5c218ff479": "\\mathcal{} \\theta (F,B)", "48a7d7543b5194d1b91f43fdb1ba0b4a": " x_i= \\frac{{\\frac{{m_i}}{{M_i}}}}{{\\sum_i \\frac{{m_i}}{{M_i}}}}", "48a7e14906e6eb77e8d852ae463333d6": " (f*g) (t) \\;:= \\int\\limits_{-\\infty}^\\infty f(\\tau)\\cdot g(t-\\tau)\\,\\mathrm{d}\\tau ", "48a826ad0ccc6d886a0dbcbc2ebd0802": "\\rho \\in ", "48a8967682215cfdc70a9f34212d246b": "\\mathrm{d} X_{t} = b(X_{t}) \\, \\mathrm{d} t + \\sigma (X_{t}) \\, \\mathrm{d} B_{t}.", "48a89c1a0e42584d6d620f67521c26ff": "\\delta_t(t) = 1, \\delta_t(s) = 0", "48a8ae08434f6d945e33c0099f412498": "X/X', Y' \\in B", "48a8f93d637624e7b9d4e209987291e9": "\\lim_{\\delta\\to 1,\\lambda\\to\\infty}s^\\delta(\\lambda)", "48a96b9e1408ee215792678b008179c8": "\ne_i^{t+n} - e_i^t = NSE_i + NSD_i + IME_i + IMD_i + RGE_i + RGD_i + RIE_i + RID_i\n", "48a96e7df5c3e7b9ded693600b9a5f60": "f'_* \\mathcal{O}_X = \\mathcal{O}_{S'}", "48a97577530f5456c20e1bc243a57296": " f(z + \\omega) = f(z) + C ", "48a9855b83249d916734bbf42c5c49da": "q(x) + q(y) = \\frac{1}{2}(q(x+y) + q(x-y)).", "48a9973ebb62bcfdf2a908ff42b43879": "V\\in B", "48a9c3f6d8691c172f008ad083c9e738": "\\operatorname{dom}f = \\{x \\in X: f(x) > -\\infty\\}. \\,", "48a9f3519634136b846d2c5e86897bdd": "K = 24\\,\\frac{M_g}{M_m}", "48aa167db8144f466b69fece832db780": "\\displaystyle{f_z(x) = e^{ix^tZx/2}}", "48aa69e5baa63edd7a20844e381476f3": " F = {{K}^{1/2}} {{L}^{1/2}} ", "48aab0d2a3633716efc7c37d4ad9a29f": "G(10,3)", "48aaddd9ed0f1bf1df2332dfde7df1ad": "U = \n\\begin{bmatrix} S & D_{S^*} \\\\ 0 & -S^* \\end{bmatrix}.\n", "48aaf81ca78af93a72065922fe7c44c2": "\\varphi=[L]i", "48ab0f00ebc93ebba18d3cc0f0916faf": " \\frac{\\hat{p} + \\frac{1}{2n} z_{1-\\frac{\\alpha}{2}}^2 \\pm \\frac{1}{2n} z_{1-\\frac{\\alpha}{2}} \\sqrt{4n\\hat{p}(1 - \\hat{p})+ z_{1-\\frac{\\alpha}{2}}^2}} {1+ \\frac{1}{n} z_{1-\\frac{\\alpha}{2}}^2}.", "48ab2603c66c1546e8b3ee4147dff731": "f_c = \\frac{1}{2 \\pi \\tau} = \\frac{1}{2 \\pi R_1 C},\\,", "48ab532097bfa38669e638a282c4d317": " R \\left( R \\left( x \\right) \\right) = R \\left( x \\right) ", "48ab82517d524214b98d8d4727858218": "0\\le k\\le n", "48ab889ecec2daf079ece3a35b90cf6a": "\\frac{\\partial u}{\\partial y}+\\frac{\\partial v}{\\partial x} = \\beta(x,y)", "48abb978f304b847e929a9805fb54e1c": "\\exists \\bar{x}.P", "48ac5bb96f9c023faa769398709c6b14": "\\overline{P}(Cl_1^{\\leq}) = \\{x_1,x_4,x_5,x_6\\}", "48ac5c46d729653afa657a636e308e83": "+\\sum_{i 0", "48cd53395dc682cbaecd9ff2b28f9dc7": "1\\tfrac{3}{5}", "48cd9d9a610e2b69c6685c018975ff3b": "\\beta_j 'X ", "48ce379aeeb346252ab4004ddc428358": "J_i = \\epsilon \\sigma T^4 + (1-\\epsilon)\\sum_{j=1}^{N}{F_{ij} J_j}", "48ce3d485efb0c97f7cd25313432cd5c": "K(S)+K(x|S)=K(x)+O(1)", "48ce4338de24ac8a739789323d076164": "I = m r^2 \\,\\!", "48ce7a4a42c25041139333a268c9b56d": "\\,\\!f_n = f_d\\cos{\\alpha_n}", "48ceb1de7599d8f4d6532d3581d06a7e": "A(X)\\,=\\,k[x_1, \\dots, x_n]/I(X)", "48cebe54870c9b9a22e0f3529106bd73": "| \\beta A_{OL} \\left( f_{0dB} \\right) | = 1. \\ ", "48cecd0c7d49eaaae210cffaaf66d858": "y'' - xy = 0.", "48cee1f4bee7f67200c3a7df6aef4557": "u = \\int_y^\\infty \\frac {ds} {\\sqrt{4s^3 - g_2s -g_3}}.", "48cf1dd0f5a18887c1f059c8fe6bd457": "\\sum_{1 \\le n \\le x} a_n \\phi(n) = A(x)\\phi(x) - \\int_1^x A(u)\\phi'(u) \\, \\mathrm{d}u \\,", "48cf6d2097dd555b53ea36508a26315f": "\\psi_1 = \\psi_1(x), \\ldots , \\psi_n = \\psi_n(x) ", "48cf995bfe735fedc5187291a93ab697": "|X/G| = \\frac{1}{|G|}\\sum_{g \\in G}|X^g|.", "48cfb25f6e34cc717c4a9527003bf70a": "\n \\vartheta(G) = \\max_B \\operatorname{Tr}(BJ).\n", "48cfd1aacf3fb7e876a919f75313e775": "\\pi_i(X; G)", "48cfde06555f1aff64bd3c1f519cb82a": "x(0)=x_0\\,", "48d016427d70f95a0f0bcb1b5d988434": "\\operatorname{var}\\left[\\epsilon^T\\Lambda\\epsilon\\right]=2\\operatorname{tr}\\left[\\Lambda \\Sigma\\Lambda \\Sigma\\right] + 4\\mu^T\\Lambda\\Sigma\\Lambda\\mu", "48d09d4f2c6483cb761613171a19ee88": "\\rho(X)", "48d0f92d723e46528e0e3f9cb58f8979": "(\\mathbb{Z}/16\\mathbb{Z})^\\times \\cong \\mathrm{C}_2 \\times \\mathrm{C}_4.", "48d11eb304620840f49e97f751d5c432": "\\frac{2}{E_gm^2}\\sum_{m,\\ n} {|\\langle u_{c,0}|p_{\\ell}| u_{n,0} \\rangle |}{|\\langle u_{c,0}|p_{m}| u_{n,0} \\rangle |} \\approx 20\\mathrm{eV} \\frac{1}{mE_{g}} \\ , ", "48d13552fbe09a4256cb60b8016fd6ea": " 0.2\\times std. ", "48d16adc1fe4ef2be9ea6a3476f106e5": "\\mu=2a \\sqrt{\\frac{2}{\\pi}}", "48d18fd6b376424be27401dab6928a4a": "W_s \\circ W_t = W_{s+t},", "48d1a90bb26a490288a0b311d874baa8": "\\frac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)}.", "48d1c0807446d7fa9cd5d3a7851afb62": "0 < k < q", "48d1f567d229603139ffb8a752a30e86": "O\\left(N_x\\log_2 N_x \\right)", "48d262b8d8cb2bfb58e9070c3d253367": " c(r)=g_{\\rm total}(r) - g_{\\rm indirect}(r)-1 \\, ", "48d27c922e699c3958ec46e874acf9c8": "\\tilde{\\rho}(r) \\in \\mbox{End}(V).", "48d2992bdc9da142844b7dfb2c449899": "\\textstyle \\sigma^2 = n\\theta (1-\\theta) = 98,451\\times0.5\\times0.5 = 24,612.75", "48d2dd89a8ca6a48d28f191764d8e32a": "\\scriptstyle C'", "48d3303c573becce0137448d5f698039": "\\frac{\\partial ({{f}_{1}},{{f}_{2}},{{f}_{3}})}{\\partial ({{x}_{1}},{{x}_{2}},{{x}_{3}})}", "48d361cf768c681daa2198fb7794a962": "R \\gg \\frac{1}{2}(y_1 + y_2)", "48d39217fa3dca50da03dbaaa7bb6d1f": " \\rho = \\frac{e^{- \\beta H}}{\\operatorname{Tr}(e^{- \\beta H})}, ", "48d41f1da30cb1f7da79a40398876ee3": "\\bar Y=X.", "48d435932f742b2f45f51e80059d5b40": " SUE_t = \\frac{Q_t - E(Q_t)}{\\sigma(Q_t - E(Q_t)} ", "48d45db5a504ce697c755747c265e7f5": "X = \\bigcup_{i \\in I} \\varphi_i(U_i)", "48d45e4469e12d81c857feeab448755f": "\\mathrm{O}(n+1)", "48d474256b6b5e8c24124f286cce264b": "X\\oplus Y.", "48d48484ce9fb02589a37b5eac9a13b0": "\\scriptstyle z\\, = \\,-7", "48d49fb11cad82d82ba761091331e98f": "{ \\frac{P(R) - P_\\infty}{\\rho_L} = \\left[- \\frac{1}{r}\\left(2R\\left(\\frac{dR}{dt}\\right)^2 + R^2\\frac{d^2R}{dt^2}\\right) + \\frac{R^4}{2r^4}\\left(\\frac{dR}{dt}\\right)^2 \\right]_R^\\infty = R\\frac{d^2R}{dt^2} + \\frac{3}{2}\\left(\\frac{dR}{dt}\\right)^2 }", "48d4c79da5cc78dd3e3adcb0b0ef06b4": "P_1^1,P_2^2", "48d5a76c18350c787499366875cce704": "s[n](\\xi) = 1/(1-\\xi)^n = 1 + n \\, \\xi + \\left( \\begin{matrix} n \\\\ 2 \\end{matrix} \\right) \\, \\xi^2 + \\left( \\begin{matrix} n+1 \\\\ 3 \\end{matrix} \\right) \\, \\xi^3 + \\dots ", "48d5ad2fb831365ea7618ac0407206db": "(x+y+z)\\left( \\sum_\\text{cyclic} b^2c^2(b^2-c^2)(b^2+c^2-2a^2)x\\right) =0 ", "48d5b9335e0e5ce22d5536438bb0ecb0": "\\begin{align}\nF'(t) = & f'(t) + \\big(f''(t)(x-t) - f'(t)\\big) + \\left(\\frac{f^{(3)}(t)}{2!}(x-t)^2 - \\frac{f^{(2)}(t)}{1!}(x-t)\\right) + \\cdots \\\\\n& \\cdots + \\left( \\frac{f^{(k+1)}(t)}{k!}(x-t)^k - \\frac{f^{(k)}(t)}{(k-1)!}(x-t)^{k-1}\\right)\n= \\frac{f^{(k+1)}(t)}{k!}(x-t)^k,\n\\end{align}", "48d60a56fc1c536bd3939e47f252e50a": "\\varphi(x) = k(\\cdot, x)", "48d6169248072c3d0c7455a60741b487": "\\dot{V}(x) < 0 \\quad \\forall x \\in \\mathbb{R}^n\\setminus\\{0\\},", "48d6215903dff56238e52e8891380c8f": "blue", "48d715b8cc6438448b12d547760a9ac6": "u = \\rho_w g z_w ", "48d71c066a76db001fe53f8008af4efc": "0\\le p\\le 1", "48d7335569868a28a5c8b29877ad5d8a": " \\bold J(x,y,z) ~ = ~ \\sum_{mnp} ~ \\bold J(\\alpha_m,\\beta_n, \\gamma_p) ~ e^{j(\\alpha_m x + \\beta_n y + \\gamma_p z)} ~~~~~(2.1a) ", "48d7743a0c855a9dea995ae62080f6a6": " y_{D}(x) = c_{1}e^{r_{1}x} + c_{2}e^{r_{2}x} + \\cdots + c_{n}e^{r_{n}x} ", "48d7ff6612fefa39d973d6883c2446cf": " f(x) = \\sum_{n=0}^\\infty E_n \\frac{x^n}{n!} = \\sec x + \\tan x. ", "48d8038bf746f687f2438ea6f5044d62": "E=E^0 + \\frac{RT}{F} \\ln \\left ( a_{\\text{H}^+} + k_{\\text{H}^+,\\text{Na}^+}a_{\\text{Na}^+} \\right )", "48d82d2f2773d768dc4f3ba125fe0069": "\\frac{\\partial^2 \\mathbf{U}^{-1}}{\\partial x \\partial y} =", "48d836e2fc660a2c4ee69fe2766d1b2e": "10 - \\epsilon^2 + O(\\epsilon^3)", "48d85f3ab0f4c4c18f49223def172079": "FWER\\leq\\alpha", "48d8fedcafb2def9ab440a97dc6a6b14": "\\mathrm{r.Ann}(S)", "48d90c1d30e1aff5a62a5a12e848fe30": " \\tau = \\frac{\\pi r\\gamma}{bL} \\,\\!", "48d946c051666763d71cebc97da67bf6": "K^{-1} = 9^{-1} \\begin{pmatrix} 5 & 23 \\\\ 24 & 3 \\end{pmatrix} = 3 \\begin{pmatrix} 5 & 23 \\\\ 24 & 3 \\end{pmatrix} = \\begin{pmatrix} 15 & 17 \\\\ 20 & 9 \\end{pmatrix}", "48d94c7b599b08c7209ecaf9a0caf6c2": " \\frac{f{(x)}}{2}+\\frac{f{(x)}}{2}+\\frac{f{(-x)}}{2}-\\frac{f{(-x)}}{2}", "48d992e0c11d7e43bb8bc981fab2d4f9": "\nF_1=\\begin{bmatrix}\n0 & 1\\end{bmatrix}\n", "48d9badc337701df2c5b306c82fe361c": "R_{Free}", "48d9d32a7646f1f5a3879d7496bc4028": "y^* =x' \\beta + \\epsilon", "48da369f47643f8f37e6db6dcf73c584": "\\xi|_B(\\gamma)=\\xi(\\gamma),\\forall \\gamma \\in \\Gamma", "48da7143fccae57b65606980a8142c14": "p=\\frac{R\\,T}{V_m-b} - \\frac{a\\,\\alpha}{V_m\\, (V_m + b) + b\\, (V_m - b)}", "48dac04d48ac5d1b67a2cf7bc5186a49": "m_\\text{Atom}(z) = -m g_J \\mu_B", "48db85270cd98ed9da3a40461d63e8b4": " U_P = U_R \\Rightarrow m_P u_P = m_R u_R \\Rightarrow u_P = u_R ", "48dbdb808e83ad166f2f7bc9b21e9fb5": "\\| f \\|_{k,\\alpha}", "48dc5e7f8348c4248ebc3900a69f86a4": "(X,Y,Z,aZ^4)", "48dc6f6761158a03ef8bad1e4513c54e": "R_Y\\geq H(Y|X), \\, ", "48dc7e30437ed342b89031a12d12d321": "S_- = S_x - i S_y", "48dc97a3928bd05922cb7110e045141b": "E = \\tfrac{1}{2}mv^2 + \\tfrac{1}{4}mv^2 = \\tfrac{3}{4}mv^2", "48dcd223cb0bd99bdecc6b641d94e21a": "\\int e^{M f(\\mathbf{x})}\\, d\\mathbf{x} \\approx \\left(\\frac{2\\pi}{M}\\right)^{d/2} |H(f)(\\mathbf{x}_0)|^{-1/2} e^{M f(\\mathbf{x}_0)} \\text { as } M\\to\\infty \\,", "48dcf6d5018cad3743edee2504110598": "D_i = \\frac{e_i^2}{p \\ \\mathrm{MSE}}\\left[\\frac{h_{ii}}{(1-h_{ii})^2}\\right],", "48dd14a3cc968b8c632102a00fe6176f": " \\sqrt{n+\\sqrt{n+\\sqrt{n+\\sqrt{n+\\cdots}}}} = \\tfrac12\\left(1 +\n\\sqrt {1+4n}\\right). ", "48dd28c321c5bffd9c1bbd104815a004": "4(\\delta+1)^2", "48dd4f61bb33895f769be073124cb617": "\n\\begin{align}\nE(e^{sY}) & = \\sum_i e^{si} \\Pr(Y(t)=i) \\\\\n& = \\sum_i e^{si} \\sum_{n} \\Pr(Y(t)=i|N(t)=n)\\Pr(N(t)=n) \\\\\n& = \\sum_n \\Pr(N(t)=n) \\sum_i e^{si} \\Pr(Y(t)=i|N(t)=n) \\\\\n& = \\sum_n \\Pr(N(t)=n) \\sum_i e^{si}\\Pr(D_1 + D_2 + \\cdots + D_n=i) \\\\\n& = \\sum_n \\Pr(N(t)=n) M_D(s)^n \\\\\n& = \\sum_n \\Pr(N(t)=n) e^{n\\ln(M_D(s))} \\\\\n& = M_{N(t)}(\\ln(M_D(s)) \\\\\n& = e^{\\lambda t \\left ( M_D(s) - 1\\right ) }.\n\\end{align}\n", "48de65ab025e58c112fa17679a044ede": "\\alpha<\\kappa^{+}\\,", "48dedf7b214cd40dba3f7812c0d5040b": " s\\ln p-t\\ln q = \\ln\\left(1+\\frac{d}{q^t}\\right) =\n\\sum_{m=1}^\\infty (-1)^{m+1}\\frac{(d/q^t)^m}{m}.\n", "48dfba637705dde1c0dcf531bff3910e": "R=\\tfrac{r+1}{n}", "48dfe3878f73d9573bbb7679d19a167c": " b = S_u + {S_P}{T_P}^0 ", "48dff80387b1c89b72df09a1f684d62f": "y = k^x.\\,", "48e019d349d8423a67359d6b601b5768": "[1+\\frac{V_0^2}{4E(E-V_0)}]^{-1}", "48e0c6ceb31a646b61761c1e76717fb2": "\\lceil R^n / n \\rceil", "48e0ce8ffed2db27776a6ccfa1194396": "{\\rm 1~Rayl = 1~\\frac{Pa}{m/s} = 1~\\frac{kg}{s \\cdot m^2}}", "48e10162d61364efd9a81a1d091c0310": "\\mathbf{N}_{t+1} = \\mathbf{L}\\mathbf{N}_t", "48e190498b36f4eb83164b0f9727ffcc": "\\part", "48e1b3de3b107fa3c568fee159e588a0": "GL(R) = \\pi_1(BGL(R))", "48e1b6a9b01ab9e2bb8c61e000627d72": "\\partial/\\partial x", "48e1ba3ca7bf250bad8acdbc8242f845": "p(\\mathbf{X}|\\theta)", "48e1f375082bd8a05449eebfb0b5f643": "n < m", "48e25474885e6552d68a9a3c4e4aec52": "G \\setminus e", "48e28674f5607a07a778ee984208769d": "\nW_0 (x) = \\sum_{n=1}^\\infty \\frac{(-n)^{n-1}}{n!}\\ x^n = x - x^2 + \\frac{3}{2}x^3 - \\frac{8}{3}x^4 + \\frac{125}{24}x^5 - \\cdots\n", "48e2b8a1cf13b0dc4692ac686b19635f": "\\displaystyle (M^* F)^\\sim(\\lambda)= \\tilde{F}(2\\lambda).", "48e2bc6fbc8b0277ae2cebf0f7783a68": "q(t) = A \\cos \\omega_{p} t", "48e37256266d837864eca90f5dea385e": "\\phi(y) \\lor (\\psi(z) \\rightarrow \\rho(x))", "48e3c3f3e79547c34852f976e25ca6a5": " \\theta_E. ", "48e3d013e93b9a1d2e5aa84a63b2aaf0": "f_u=(1/T)", "48e45fb172efc37a58c12179e3b009a1": "dy=\\frac{\\partial y}{\\partial x_1}dx_1 + \\frac{\\partial y}{\\partial x_2}dx_2 + \\ldots + \\frac{\\partial y}{\\partial x_n}dx_n ", "48e4606c383f404e8acd4a8d8f349495": "c_{21} e^{j\\varphi_{21}},c_{31} e^{j\\varphi_{31}},c_{32} e^{j\\varphi_{32}}", "48e47c9575a54e404d592aa0a75759df": "0\\le\\phi < 2\\pi.", "48e49d280a060d1e46ef2154b8078718": "J_{\\pm}", "48e4b77b53431fe2a37c7ae2a9c69747": " \\mathbf{H}_m = \\sum_{k=1}^{m-1} \\mathbf{X}_k + \\sum_{k=m+1}^n \\mathbf{\\Xi}_k ", "48e53e46e0189834ee9979372f0b0c6e": "Y\\ni y\\mapsto \\lambda_y", "48e546a200ed2fed8c13e1cc0961ce26": "\n\\vartheta_{0,0}(x) = \\sum_{n=-\\infty}^\\infty q^{n^2} \\exp (2 \\pi i n x/a)\n", "48e5500b9c8605abe3207d13507a26ff": "{\\mathcal L}_M", "48e55b7a11b32355f33dbfb0cbf58697": "\\mathbf{R}\\cdot\\mathbf{x}", "48e58396899ae6a056290622fa8bb457": "f(\\mathbf{x}, \\mathbf{p}) = 0", "48e59b614d86bdceb3147da7bce1c30d": " \\mathfrak P_2(K)", "48e5e904267a2454463c7d1cdc0f7214": "(x_1^2 + x_2^2 + \\cdots + x_j^2) - (x_{j + 1}^2 + x_{j + 2}^2 + \\cdots + x_n^2)", "48e5fb475a6c808fa8a967345255a749": "E \\approx 6.3\\times 10^4\\times 10^{3M/2}\\,", "48e683502dc08e28e817cf509341f38c": "\\rm \\ S_8 + 8 AgF_2 \\xrightarrow{398K} 4 S_2F_2 + 8 AgF", "48e69d208c034a785aa51518f5f82ec6": "\\tau = \\hbar/\\Gamma ", "48e6a50d4a3f02010385d15bea49907a": "\\operatorname{succ} \\mathbin{:} {\\mathbb N} \\to {\\mathbb N}", "48e740adfc2f4a7ddadfc09ee55969a7": "|f(n)| \\ge k\\cdot|g(n)|", "48e7c35ca40baeee405902c31ccaadca": "\\displaystyle\\sigma(G)=\\min_{\\pi\\in\\Pi}\\frac{|\\partial \\pi|}{|\\pi|-1}", "48e7cf2ed8d8211968ee36d783967f9d": " \\phi", "48e7d3e55cc3316d1019fa7fc68f7456": "n_0(x) \\equiv 1", "48e7f2508c68694794d0cc4eb2c25f61": "x \\leftarrow x+lb_{computed}", "48e828756986ea127d61ca107a6842fe": "V_n(R) = \\frac{\\pi^{n/2}}{\\Gamma(\\frac{n}{2} + 1)}R^n,", "48e88cc35768668d2ae7a898a44823c5": "\\ \\displaystyle \\mathop{Opt} \\ ", "48e8ef5897ded495f80c7d2360cc328d": "\\scriptstyle F_0", "48e93485f349745961e4ba688e038f30": "\\Delta p = \\frac{2 \\sigma}{R}\\left(1-\\frac{\\delta}{R}+\\ldots\\right)", "48e9b1fb9472180b4ecea4ff24c56487": "N_{a}", "48e9ddcf92e05c5c1b562b8e04e79cef": "c=\\inf_{\\mathbf{g}\\in\\Gamma}\\max_{0\\leq t\\leq 1} I[\\mathbf{g}(t)],", "48e9e4c39b14479b5884483d36a8c128": "\\Psi_4=\\delta\\nu-\\Delta\\lambda-(\\mu+\\bar{\\mu})\\lambda-(3\\gamma-\\bar{\\gamma})\\lambda+(3\\alpha+\\bar{\\beta}+\\pi-\\bar{\\tau})\\nu\\,.", "48ea235b72026adc96639aa3e022ae2c": "\\scriptstyle\\partial A", "48ea38749115900b355333b33ec09d29": "\\displaystyle{Q(a)Q(1-a^{-1} -b^{-1})Q(b)=B(1-a,1-b).}", "48ea4b6acea0f25da39ab7e39c523199": "R_S=\\frac{v_{Bullet}^2}{g}\\, \\sin(2\\delta\\theta)\\left(\\frac{\\cos(\\alpha)\\cos(\\theta-\\alpha)+\\cos(\\theta)-\\cos(\\alpha)\\cos(\\theta-\\alpha)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "48ea5d8561239a1b30c337c7f30fd216": "=\nS_n(\\alpha,\\beta,\\gamma) \\prod_{j=1}^k\\frac{\\alpha+(n-j)\\gamma}{\\alpha+\\beta+(2n-j-1)\\gamma}.\n", "48ea9be719fa783b7063fd5c5531521a": "x^{-1}", "48ebab9778848b4cdf79afb76b24d0a0": "R_{\\mu \\nu} = K \\left(T_{\\mu \\nu} - {1 \\over 2} T g_{\\mu \\nu}\\right)", "48ebeac219d272c76ff2d2db9b7a8193": "\\operatorname{Var}\\left(\\sum_{i}^n a_iX_i\\right) = \\sum_{i=1}^na_i^2 \\operatorname{Var}(X_i) + 2\\sum_{1\\le i}\\sum_{ p-1\\!", "48f554a22e6c148a8976832fbfc44353": "[-\\nabla^2 + V(\\mathbf{x})]\\psi(\\mathbf{x}) = E\\psi(\\mathbf{x})", "48f5a1ae9ecbd335bfb113f52bee571d": "{\\scriptstyle\\frac{1}{6}} (-x^3+9x^2-18x+6) \\,", "48f5c303597c253df44b425bd365e5e5": "\\int x^m \\exp(ix^n)\\mathrm{d}x = \\int\\sum_{l=0}^\\infty\\frac{i^lx^{m+nl}}{l!}\\mathrm{d}x\n = \\sum_{l=0}^\\infty \\frac{i^l}{(m+nl+1)}\\frac{x^{m+nl+1}}{l!}", "48f6091c341fcf4963ee80dfdce642fb": "\n\\mathbf{p} = \\frac{\\partial G_{2}}{\\partial \\mathbf{q}}\n", "48f6243e22935d4a01fbc541ae6bf310": "\\tfrac{1}{N}", "48f649c1fdbe2eb96ccf26da355f4dfa": "\\det(I_\\mathit{m} + AB) = \\det (I_\\mathit{n} + BA)", "48f65e8ff05cc121f11cd76e308d1a90": "e(x)_U = \\chi_U(x).\\,", "48f690123df074525f805e3385c2ec1f": " \\Delta w_{ij}(t + 1) = \\Delta w_{ij}(t) + \\eta\\frac{\\partial C}{\\partial w_{ij}} ", "48f698d57e7b546c3a9701107ea45280": "\\ell=2", "48f717af268ba3c9ec37fb14680ee5fc": "F : \\mathcal{C} \\rightarrow \\mathcal{C} \\times \\mathcal{C}", "48f76591bbd1254b9bb84ee0227dbef3": " H_{B_1} + H_{B_2} \\geq -2\\log c.", "48f77a3d19b2be7732a98c99d851ea6a": "\\log(\\exp X\\exp Y) =\n\\sum_{n>0}\\frac {(-1)^{n-1}}{n}\n\\sum_{ \\begin{smallmatrix} {r_i + s_i > 0} \\\\ {1\\le i \\le n} \\end{smallmatrix}}\n\\frac{(\\sum_{i=1}^n (r_i+s_i))^{-1}}{r_1!s_1!\\cdots r_n!s_n!}\n[ X^{r_1} Y^{s_1} X^{r_2} Y^{s_2} \\ldots X^{r_n} Y^{s_n} ],\n", "48f7c46245878b8beeea7c8ea8a75ca5": "x, y, A, B", "48f7f51e35a3b252597dce2ac717866e": " A_{FB} = \\frac { A_{OL} } {1 + { \\beta }_{FB} A_{OL} } ", "48f839937495eff745598b3fa05c4547": "\\textstyle q^k", "48f84c926fe6267a9417aac36f15168b": "\\mathbf{\\Delta k}\\cdot (\\mathbf{a}+\\mathbf{b}+\\mathbf{c})=2\\pi (h+k+l)", "48f84e0be16873195fdca397697f974f": "\\hat{\\beta} = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\sqrt{s_1^2+s_2^2}}.", "48f872afc01162d9a6549cf6d9ee1249": " v(x,y)= \\frac{2\\pi\\kappa}{\\lambda} \\int{p(x,y,z)} \\, dz", "48f87a66f48cc9f46b8a490e6e8a404e": "\\begin{align}\n M_p(x_1,\\dots,x_n) &= \\left(\\sum_{i=1}^n w_i x_i^p \\right)^{\\frac{1}{p}} \\\\\n M_0(x_1,\\dots,x_n) &= \\prod_{i=1}^n x_i^{w_i}\n\\end{align}", "48f8e4255c3aa7838f556c23aed4ee4d": " (\\alpha \\or \\beta \\or ...) \\leftrightarrow \\lnot(\\lnot\\alpha \\and \\lnot\\beta \\and ...).", "48f8e71fa13c19db86fb22a145a716e3": "\n\\alpha=10^\\circ \\quad \\beta=20^\\circ \\quad \\gamma=30^\\circ \\quad \n", "48f98b53ee18b2229c1a59c0ba1660b3": "\\operatorname{erfc}(x)\\approx \\frac{1}{6}e^{-x^2}+\\frac{1}{2}e^{-\\frac{4}{3} x^2} \\qquad x>0 \\,.", "48f99ff0ee4b8eb04f4f031ae22bf92b": "P_B", "48f9eeb591ce47eba166bb4943da8667": "\\displaystyle v=f\\lambda", "48f9fbb19aec819ba8a6c745133d85aa": "A_t", "48fa2f8a8ba2e041440f6fb0f7b27dac": "\\ \\displaystyle g \\ ", "48fa34582b35be5173aded124ba24822": "e^x = \\sum_{n=0}^\\infty {x^n \\over n!} = \\sum_{n=0}^\\infty \\frac {x^{2n}} {(2n)!} + \\sum_{n=0}^\\infty \\frac {x^{2n+1}} {(2n+1)!} = \\cosh x + \\sinh x ", "48fa6740f68030eff4f6335eaeb7fe3d": "\n \\sigma_{11} = 2~\\left(\\lambda^2 - \\cfrac{1}{\\lambda^4}\\right)~\\cfrac{\\partial W}{\\partial I_1} = \\sigma_{22} ~.\n ", "48fab941c2c478d43ee586e365547e80": "D_2=1P_1+ 5P_2", "48fae910f038a810b519dd357c07eb2a": "n=q", "48faeef69190aed0129fe46e98d50409": " \\langle E \\rangle = \\frac{\\langle p^2 \\rangle}{2m} ", "48fb3a0914725aa00409a21091ec10eb": "(u_1...u_i)", "48fb531e7ceff1b95bda487a2223c289": " U^2 = -1 ", "48fb551d02b6cdfe751da27ec3d53dd3": "N = |V|", "48fc338ab9db95570e500ece6bbc1c72": "\n(H-\\mu N)|n \\rangle = E_n |n \\rangle,\n", "48fc461a5755de11c1ddb2988a6c9224": "U^{-1}g(x)=\\int_0^\\infty g(\\lambda) \\, {1\\over2}\\tanh \\pi \\sqrt{\\lambda}\\,d\\lambda.", "48fc6b28e68cbf3ca782a9aa59b1a37f": " x^{q^{n_i}} \\bmod f", "48fc84b2219b15d8759419512c378fd7": "\\operatorname{lcm}(1,2,\\dots, n)=e^{\\psi(n)}.", "48fca3a53dd30609e135e5478d9ebb0d": "\\sgn(g)", "48fccdacbf5426418d678fdfb290ca88": "\\pi_{XY}(R)\\bowtie\\pi_{XZ}(R)", "48fcf1577832f46b9b5a8ddadd609d40": "\\beta >1.\\,", "48fd24cd8bf9352db0f15cca8fbd7dbc": " \\pi(s) := \\arg \\max_a \\left\\{ \\sum_{s'} P_a(s,s') \\left( R_a(s,s') + \\gamma V(s') \\right) \\right\\} ", "48fd457b21043f095164b07960af244c": " A_z", "48fd47bd4fbf655615404bb21ad57a44": "\\frac{1}{\\sigma}\\left(\\frac{\\partial^{}\\mathbf{u}}{\\partial t^{}}\\ +\\ (\\mathbf{u} \\cdot \\nabla) \\mathbf{u}\\right)\\ =\\ - {\\mathbf \\nabla }p\\ +\\ \\nabla^2 \\mathbf{u}\\ +\\frac {\\sigma}{\\zeta} {Q}\\ ({\\mathbf \\nabla} \\wedge \\mathbf{B}) \\wedge\\mathbf{B}, ", "48fe4dd8859ab4aa7c401321a074001c": "\\forall x, y \\in A, f(x+y)\\leq f(x)+f(y).", "48fe8cb3907c7b446f29ef6d937ba629": "\\rho_A = |\\psi\\rangle_A \\langle\\psi|_A .", "48fefce592fdd80866caffb846a657bb": "\\kappa \\,", "48ffa18306fb81ceb7ab781639869f74": "\\scriptstyle O(N^2)", "490012701392fd9195e7c9cf483d78ff": "\\left(\\sigma^2=\\int_{-\\infty}^\\infty g(x)(x-\\mu)^2\\,dx\\right)", "4900150432ff6478a6e648f7ce4c73ff": "\\mathbf{v}= \\frac{\\hbar}{m} Im \\left(\\frac{\\nabla\\psi}{\\psi}\\right) ", "49004058a0c3e3e3acca58a363677e06": "\\mathbf{\\hat{e}_1}, \\mathbf{\\hat{e}_2}, \\mathbf{\\hat{e}_3}, \\dots , \\mathbf{\\hat{e}_n}", "49006ff7b1516332a99599b831336240": "0\\le \\theta < 2\\pi", "490086021aac9f2ccfc81d7aca709848": "\\begin{align}\\dot x&=f(x,y,t),\\\\0&=g(x,y,t).\\end{align}", "4900cd3cef7396e78571965b41cd93a4": "\\tilde U(s) = \\int_0^\\infty e^{-st} \\frac{1}{b-a}\\text{d}t = \\frac{e^{-sa}-e^{-sb}}{s(b-a)}.", "490112682661b53a0f07629b853acd41": " f\\in \\mathcal{F} ", "49012247de5bb5afaa2294bac4731e4b": "(\\phi \\to \\lnot \\chi ) \\to (\\chi \\to \\lnot \\phi )", "49013285f145a4be37442abcac843fff": "\\frac{(1+2x)(1+5x-16x^4)}{(1-x^2)(1-16x^2)}", "490160b3499a43733d181ee2a08f8c62": "G_u H/H = (G/H)_v", "4901b9fa5ceb942927bcc9b1e4fe43e3": "-\\beta r_o \\frac{r_E + R_E }{r_\\pi + 2R_E} ", "4901d4039f5b3326b8e47b8354f05ae8": "m_1(x) = x^4+x+1.", "4901e007368c64271767cbd5dd873702": "\n\\boldsymbol\\alpha = \\frac{d\\boldsymbol\\omega}{dt}\n", "49022bc4b9512c0e317fbf2bf346de2c": "\\lfloor\\frac{w}{J}\\rfloor+\\lfloor\\frac{w+1}{J}\\rfloor+\\lfloor\\frac{w+2}{J}\\rfloor...\\lfloor\\frac{w+J-1}{J}\\rfloor=w", "490233445389e0a10405dfec124b15c9": "\\widehat{T}^{(2)}_{0} = \\frac{1}{\\sqrt{6}}\\left( \\widehat{a}_{+1} \\widehat{b}_{-1} + \\widehat{a}_{-1} \\widehat{b}_{+1} + 2 \\widehat{a}_0 \\widehat{b}_0 \\right)", "4902af8f1bb8c4af8d5531ea6cc2421b": " \\delta W = \\mathbf{F}\\cdot d\\mathbf{x} = \\mathbf{F}\\cdot\\mathbf{v}dt ", "4902dc618dd5ba907fae6f3393574938": "dx + (1-d)y < y", "4903656896d642756ea20ee8809ba430": "\\overline{2}", "49039a296c9ca0c6809c700d185c2874": "\\frac{\\partial(y_1, \\ldots, y_k)}{\\partial(x_1, \\ldots, x_n)} = \\frac{\\partial(y_1, \\ldots, y_k)}{\\partial(u_1, \\ldots, u_m)} \\frac{\\partial(u_1, \\ldots, u_m)}{\\partial(x_1, \\ldots, x_n)}.", "4903ac6d393ef3292899ab1ced927ee1": "x[nK]", "4903fc16e4645f0c9a49af6365587e8d": "\\operatorname{Cl}_2\\left(\\frac{5\\pi}{6}\\right)=\n2\\pi\\log \\left( \\frac{G\\left(\\frac{7}{12}\\right)}{G\\left(\\frac{5}{12}\\right)} \\right) -2\\pi \n\\log \\Gamma\\left(\\frac{5}{12}\\right)+\\frac{5\\pi}{6}\\log \\left( \\frac{2\\pi \\sqrt{2} \n}{\\sqrt{3}+1} \\right)", "4904047fe03551bf5fbdb9993fa997f7": "Q\\;", "490482397d49260d7a4a072db85d774d": " c A(c) = 1/a = const ", "4904c9b41311a70f1b2e749bbe69f78c": "(\\hat{m}_{ij}) = \\hat{a}\\hat{b}^T", "4904f3e52deb768baeb32baddbfa8e08": "\\implies P(A \\cap B) = P(A|B)\\, P(B) = P(B|A)\\, P(A), \\!", "4904f47008f1deb450985e8189532afc": "\\sigma\\mathbf F=\\langle F,\\le,\\sigma V\\rangle", "4905134349c5d1a09f7b72b67d948de9": " \\frac{dN_2}{dt} = \\frac{r_2N_2}{K_2}\\left( K_2 - N_2 + \\alpha_{21}N_1 \\right) ", "490541b0e196b040ae6f7216b60b206a": "n-1-p", "4905a344c2f54fdfcdf81c5d0577f15d": "G = Ht_1 \\cup \\ldots \\cup Ht_n,", "4905bb0249414a7f6acd1eed19f05025": "-R=P_1+P_2", "4905f75d7482000b765111a6a0d5e5b3": " x'^\\mu = \\frac{x^\\mu-b^\\mu x^2}{1-2b\\cdot x+b^2x^2} \\,. ", "4905ff4036d6f7e21b95178fbc06ea86": "F_x = M\\;a_x = M\\;\\ddot{x}", "49062cfd96d22aea6521fa5c00f5bd4c": " k \\in \\{1,...,p\\}", "49063ff99415bc99b1bc308b75d8f156": "p = \\rho kT - \\frac{\\rho^2}{6} \\int_V \\mathrm{d} \\mathbf{r} \\, r g(r, \\rho, T) \\frac{\\mathrm{d} u(r)}{\\mathrm{d} r}", "4906714c03eebaf8c7f7011d9c1258ef": "\\left\\lceil \\frac{-1}{\\log_2(1-p)} \\right\\rceil\\!", "4906cd8e5f5e160249c2965b05affdbe": "\\textstyle (n, k)", "4906d9fb69d2c52a8dc019449ee65ebe": "\\Theta_{*}^{s}(\\mu,a)\\leq \\Theta^{*s}(\\mu,a)", "4906f7f9419ea74ba4d9820c04216149": "n=2, 6, 14, 30, 62,", "4907162b8ee6f2a8126324a782207245": "1-t/OPT\\,\\!", "490716b8aa2f37bdb2cfce4b49ccd80f": "\n\\begin{align}\n\\mathbf{A}(\\mathbf{r}) &= \\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}}\n\\left(\\mathbf{e}^{(\\mu)} a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} +\n\\bar{\\mathbf{e}}^{(\\mu)} {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right) \\\\\n\\mathbf{E}(\\mathbf{r}) &= i\\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar\\omega}{2 V\\epsilon_0}}\n\\left(\\mathbf{e}^{(\\mu)} a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} -\n\\bar{\\mathbf{e}}^{(\\mu)} {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right) \\\\\n\\mathbf{B}(\\mathbf{r}) &= i\\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}}\n\\left((\\mathbf{k}\\times\\mathbf{e}^{(\\mu)}) a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} -\n(\\mathbf{k}\\times\\bar{\\mathbf{e}}^{(\\mu)}) {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right), \\\\\n\\end{align}\n", "49072a93a1b36b5e15a8245a6126bbba": "\n\\Phi^* = \\frac{q N_d x}{E_s} \\left(x_d - \\frac{x}{2}\\right) = (\\Phi_i - V_a)\\frac{x}{x_d}\n", "49074a617d45ef501fd13de6c19bc61a": "r(O_{r}^{*})", "490764db7b51e3c698d4a2eb7e2a1ff6": "a\\cdot_Lb=\\text{rem}(a._{K[X]}b),f).", "49079a35549e072e89e3a5b0f3d687f6": "\\scriptstyle \\sqrt{3}, \\sqrt{5}, \\dots, \\sqrt{17}", "4907a07eab13106fa03005bb48cdcb35": "\\Delta x \\Delta p \\ge \\frac{\\hbar}{2} ", "4907a39c256ec3c579ef3a904ead576b": "\\mathrm{SR} = \\frac{I_\\mathrm{sat}}{C}A_2", "4907b4c80022564d2c4d051eca674db4": "\\Psi^{\\mathrm{HF}}(\\mathbf{r}_{1}\\sigma(1)\\cdots\\mathbf{r}_{N}\\sigma(N)) = \\mathcal{A}\\left(\\psi_{1}^{\\alpha}(\\mathbf{r}_{1}\\alpha_{1})\\cdots\\psi_{N_{\\alpha}}^{\\alpha}(\\mathbf{r}_{N_{\\alpha}}\\alpha_{N_{\\alpha}})\n\\psi_{N_{\\alpha}+1}^{\\beta}(\\mathbf{r}_{N_{\\alpha}+1}\\beta_{N_{\\alpha}+1})\\cdots\\psi_{N}^{\\beta}(\\mathbf{r}_{N}\\beta_{N})\\right).", "49081c55c7945ab97dbc65507306c03a": " f_i = \\frac{n_i}{N} = \\frac{n_i}{\\sum_i n_i}. ", "49084385df96d682422b993ede1db786": "\\left [\\begin{smallmatrix}2&-2\\\\-2&2\\end{smallmatrix}\\right ]", "49084bc12e0ba83aecc8ff695f8e684f": "\\bar P_e", "49085e39e6d1674d8dd05057faba1b8e": "A = \\pi r^2\\!", "49089c90dc61704dab9221123e4ef3b7": " \\bold{x} = \\bold{Tx} + \\bold{c} \\quad (3) ", "4908b8a2244679e8ca7a631ef6acc1f2": " \\begin{align}\ny_1' &= y_2, \\\\\ny_2' &= \\beta_1 + \\beta_2 y_1 + y_1^2 \\pm y_1 y_2.\n\\end{align} ", "4908fd4200e8b4fbfb9d8eb45d8cbdcd": "\n\\rho=\\sum_{x}p_{X}\\left( x\\right) \\left\\vert x\\right\\rangle \\left\\langle\nx\\right\\vert .\n", "4909f1a92af0414834e9b3d75048b5b2": "\\mathcal{N}^\\omega", "490a0c9c1b740ac8d6d3fa3b79f39432": "X \\to e^{q\\Lambda}X", "490a16d0eae0a736894b7d2abe9aa2c6": "v_{2}'=-q_{1}^{*}\\,v_{1}+i\\,\\xi\\,v_{2}", "490a3f419692e6b5fb8dcb1753a9e3f4": " f(x-) = \\lim_{y \\to x^-} f(y) ", "490a4fe41e6d797c474a2de36fbf7d97": "y'(x) = P(x)y(x) + Q(x)y^\\alpha(x)\\ ,\\ y(x_0) = y_0 := [z(x_0)]^{\\frac{1}{1-\\alpha}}.", "490a82cb5c8af7c2ec950008b717ab26": "\n\\frac{d}{d\\omega}Q(\\phi_\\omega)<0,\n", "490a8d78ec672cefb37efe9bbc0116c7": "T_{tot} = \\frac{L}{V} + t_r + \\frac{kV}{2} \\left(\\frac{1}{a_f} - \\frac{1}{a_l} \\right)", "490aa6e856ccf208a054389e47ce0d06": "Id", "490ae5511c8e1c469abd52329d7a06fe": "V_{\\text{sat}}", "490af87e003e4c9bff51e461832ac8e5": "\\frac{2 k} {(k-1) (2k-1)} \\,", "490b02edd20f21c6342b2f6335a49885": "\\kappa ^2\\, ", "490b743988bcb951760abbe42ceb8938": " {R^1}_{212} = -\\Delta p", "490b989b4c3218a77eafb3ea65ef5666": " s = s T ", "490be1704cbabda672ef3db7f635aa43": "\n \\operatorname{Pr}\\Big( \\omega \\in \\Omega : \\lim_{n \\to \\infty} X_n(\\omega) = X(\\omega) \\Big) = 1.\n ", "490c969d1c782107e4746f14f24cb493": "\\theta (L)", "490cdd466736423f5aa2a9c8e53b78d9": "\\tfrac12 k\\hat{u}^2/\\omega.", "490cfe1136bacab8b1fc85b349f1ce78": "H_0^1", "490d2be2be63988b182e21b6abe49310": "\\lambda\\Pi", "490d5fddabd27a87f645f864416e0e28": "\\psi^m(z)", "490dabda50f510e05e01b9259aff323d": " E = \\frac{m \\omega ^2 A^2}{2} .", "490dd513610b8327b7a1c59c0faa8d40": "C_e = C_X - C_{XY} C^{-1}_Y C_{YX} .", "490de91047c2542ed5489a4cb8100f31": " \\mathrm{PE}={1\\over2}kx^2", "490e1c201dbbab7f1d45bdfaf5a6223b": "H(f)_{ij}=H_f(X_i, X_j) =\\nabla_{X_i}\\nabla_{X_j} f - \\nabla_{\\nabla_{X_i}X_j} f", "490e6a64b67b63583a2bae5a69e07d92": "\n \\begin{align}\n \\mathrm{J}_\\pm & \\sum_{m_1m_2} |j_1m_1\\rangle|j_2m_2\\rangle \\langle j_1m_1j_2m_2|JM\\rangle\\\\\n & =\\hbar \\sum_{m_1m_2}\\left[ C_\\pm(j_1,m_1)|j_1 m_1\\pm 1\\rangle |j_2m_2\\rangle\n +C_\\pm(j_2,m_2)|j_1 m_1\\rangle |j_2 m_2\\pm 1\\rangle \\right]\n \\langle j_1 m_1 j_2 m_2|J M\\rangle \\\\\n &= \\hbar \\sum_{m_1m_2} |j_1m_1\\rangle|j_2m_2\\rangle \\left[\n C_\\pm(j_1,m_1\\mp 1) \\langle j_1 {m_1\\mp 1} j_2 m_2|J M\\rangle\n +C_\\pm(j_2,m_2\\mp 1) \\langle j_1 m_1 j_2 {m_2\\mp 1}|J M\\rangle \\right].\n \\end{align}\n", "490eb6c577d91df23a2b9180fa909d5d": "\nm{ \\operatorname{d^2}\\vec{x}\\over \\operatorname{d}t^2 } = \n-GMm \\Big( 1-\\beta \\Big) {\\vec{x}\\over r^3}\n+GMm \\beta\n\\Bigg \\{ {\n-{ {\\vec{x}\\cdot \\vec{v}} \\over {cr} }\n{ \\vec{x}\\over r^3 } \n -{ \\vec{v} \\over {cr^2} }\n+ { R_{\\rm s}^2 \\over {cr^4} }\n\\Big( \\vec{\\omega} \\times \\vec{x} \\Big)\n} \\Bigg \\}\n", "490ef73cf24ee1606a1801a9a4ee38ef": "S_0 ", "490efd2ea9aeb3affe78972618a5630f": " j \\in \\{ |j_1-j_2|, (|j_1-j_2|+1), \\ldots, (j_1 + j_2) \\} ", "490f2d61625fd6ad89fb09887e1f4916": "\\begin{align}\nA &= A^* = \\mbox{constant} \\\\\n\\dot{m} &= \\dot{m}^* = \\mbox{constant} \\\\\n\\end{align} ", "490f3324abd71ca0906f2ccee7cf9877": "n^{(l)} c^{(l,m)}=c^{(l,m)}n^{(m)}", "490f5e09a386c58cafc2b6cce3581818": "\\left\\lfloor m/2 \\right\\rfloor", "490f63fa88ebc3002ca9396da190ee89": "f(x) = \\begin{cases}\n 1 & \\mbox{if } x < 1,\\\\\n 2 & \\mbox{if } x = 1,\\\\\n 1/2 & \\mbox{if } x > 1,\n \\end{cases} ", "4910adba7f63fa7565a03014e257aa12": "\\oint_{R} \\mathbf{B}\\cdot\\mathbf{ds} = \\mu_0 I_{enc}", "4910f661df87967415cf97f5c03bbe41": "2n+4", "49110b1b47cb795cae1b74280f7577e9": "{{r_p}\\over{r_a}}={{1-e}\\over{1+e}}", "4911100a351148d0e6bc5bbdf99714de": "\n \\begin{bmatrix}\n 4 & 3 \\\\\n 6 & 3 \\\\\n \\end{bmatrix} =\n \\begin{bmatrix}\n l_{11} & 0 \\\\\n l_{21} & l_{22} \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n u_{11} & u_{12} \\\\\n 0 & u_{22} \\\\\n \\end{bmatrix}.\n", "49111311cf44b5512fa56c1d239d6bc4": "d_n(e_n^{\\alpha})=\\sum_{\\beta}\\deg(\\chi_n^{\\alpha\\beta})e_{n-1}^{\\beta}\\, ", "49112c1ecad7b16a83e5efcfd2be821e": "A_{{xx}} \\le A_{{yy}} \\le A_{{zz}}.", "4911357ec6d89e6aa5942f35111f92f7": "W^\\ast(s|x) = \\frac{(1-\\rho)(1-\\rho r^2)e^{-[\\lambda(1-r)+s]x}}{(1-\\rho r^2)-\\rho(1-r)^2e^{-(\\mu/r-\\lambda r)x}}", "491179e5b092c6229f2902f73356d7f4": "\\gamma=\\mathrm{sh}(\\frac{\\beta}{2n}),", "49117caebbe29871e18fb6bf32e0afa2": "{\\hat p} = p", "491181928180408d37dd9da7c36acd82": "O=2(A-I)", "4911999f354a4055e9474391dc89fca1": "B[\\vec{X}]_{\\hat{m} \\hat{n}} = q^2 \\, \\sin(\\omega u)^2 \\, \\left[ \\begin{matrix} 0&0&0\\\\0&0&-1\\\\0&1&0 \\end{matrix} \\right] ", "4911ee1bab39bdade8060e658b220c49": "\\rho_{1/s}(L^*\\setminus \\{\\mathbf{0}\\}) \\leq \\epsilon ", "49122439eee01353322a8a4abd225732": " \\langle \\rho_A v, w \\rangle = -\\langle v,\\rho_A w\\rangle", "49123cfd708416edb196691c44f238cf": "\n\\int_{t}^1 e(s) ds=\\lambda e'(t)\n", "49125144ca5d73c111c0b5b6154fe338": "\\alpha = \\left. \\frac{d\\omega}{dt} \\right |_0 = \\frac{4 \\pi L n_2 I_0}{\\lambda_0 \\tau^2}.", "4912a9e62d38c567bc6f792379dedd1e": "x^2 \\frac{d^2 y}{dx^2} + x \\frac{dy}{dx} + (x^2 - \\alpha^2)y = 0", "4912c48d681833e98245a39735a1fc87": "(r,\\theta)", "4912d5779f890410115cb3f0be5c1b53": " f(x) = \\begin{cases} \n\\frac{2(x-a)}{(b-a)(c-a)} & \\mathrm{for\\ } a \\le x \\leq c, \\\\[4pt]\n \\frac{2(b-x)}{(b-a)(b-c)} & \\mathrm{for\\ } c < x \\le b, \\\\[4pt]\n \\end{cases}", "4913087c974e86e2c49f9b5c2cb6e452": "\\rho = \\begin{pmatrix} A_{11} & A_{12} & \\dots & A_{1n} \\\\ A_{21} & A_{22} & & \\\\ \\vdots & & \\ddots & \\\\ A_{n1} & & & A_{nn} \\end{pmatrix}", "4913e34292a19cbed188f1baa57083c1": "\\int_{n\\in N(A)}f(xny) \\, dn=0", "491429bfaeea547338f41a774aaac18e": "u(\\nu,T)=\\frac{2 h\\nu^3}{c^2}\\cdot\\frac1{e^{h\\nu/k_BT}-1}", "4914834715ecdfc5f998cbf505036ba4": "p\\Psi =\\frac{\\varepsilon _{2}p_{1}-\\varepsilon _{1}p_{2}}{w}\\Psi", "4914857c5262559bd3b6355e49492a8e": "\\mathbf{u} = \\frac{1}{1+x_0}\\left(x_1, x_2, x_3\\right).", "491487eb3eaa01a938dd57cf65f3325d": " AvDev = 1 - \\frac{ 1 }{ 2N } \\frac{ K }{ K - 1 } \\sum^K_{ i = 1 }( f_i - \\frac{ N }{ K } ) ", "4914b7c984a3cda4039a51ba21c3d6f3": "\\alpha_s=-\\frac{2\\pi}{\\beta_0 \\ln(E/\\Lambda_{QCD})}", "4914d37c452f15210994212d1f59743a": "\\operatorname{div}\\,\\mathbf{F}(p) = \n\\lim_{V \\rightarrow \\{p\\}}\n\\iint_{S(V)} {\\mathbf{F}\\cdot\\mathbf{n} \\over |V| } \\; dS ", "49159f71c04dcf6e541093d497fe9e34": "\n\\hat{A} = \\left[ \\hat{S} ~ \\vdots ~ \\hat{M} ~\\vdots~ \\hat{S} \\times \\hat{M} \\right] \\left[ \\hat{s} ~\\vdots~ \\hat{m} ~\\vdots~ \\hat{s} \\times \\hat{m} \\right]^T.\n", "4915b5db293e8173ddb0d060d856ba01": "\\{u(i), \\; d(i) \\}", "4916181ac91483f487bcc60b57ef1da0": "n=\\sum_{i=1}^{k}{d_i}!\\, ,\\text{ e.g. } 145 = 1! + 4! + 5! \\, .", "49168ae9be46873bd75f50401ea042bd": "\\Delta^P_i,", "49178b9b5128fb76eeac29a8b54d6dda": "y=ux", "49178e7b2c08aac20e705a74ff233335": "A_{1n} = A_{2n} = B_{1n} = B_{2n} = 0 \\quad (n \\neq \\pm 0) ", "49178e97abb7da32f2b3c107e18af971": " F = - \\frac{m_1 m_2}{r^2} ", "4917a7dbe0036c8db72ea874a859a971": " \\frac{\\part u}{\\part n} + a u =0, \\,", "4917ade1ad9167553e3fd5d8c6f00e37": "\\mathbf{p} = \\begin{bmatrix} p(1) \\\\ p(2) \\\\ \\vdots \\\\p(L) \\end{bmatrix}", "4917b97db938b6e5924182a94a3d94be": "\n\\frac{d}{dt} \\left( \\mathbf{r} \\cdot \\mathbf{r} \\right) = \n\\frac{d}{dt} \\left( r^{2} \\right) = 2 \\left( \\mathbf{r} \\cdot \\mathbf{v} \\right)\n", "4917fbedffbf756cbb67693a3ae38b95": "D\\equiv\\frac{d}{dx}", "4918c0733518badc90c794b09927dbfc": "m = (x+y)/2", "4918d85bdba616a75dd6dd361b271f13": "\\langle x/{\\sim},y/{\\sim}\\rangle\\in S\\iff\\forall A\\in V\\,(x\\in\\Box A\\Rightarrow y\\in A),", "4918fa8bf409a00f52ad212f130fcc66": " \\sum_{i \\in I} \\varphi_i(x) = 1.", "491916ce3ac28e83dd1dfc4d61313f28": "x^{-1} \\in S", "49194e1260aa7f489bbc8ce4b7b557ae": "\\Sigma M_B = M_{BA} + M_{BA}^f + M_{BC} + M_{BC}^f = 0.2EI \\theta_A + 1.2EI \\theta_B + 0.4EI \\theta_C - 2.033 = 0", "49195ab46bfe0e163792cf161e6770ef": "C_D=\\frac{D}{\\frac{1}{2}\\rho A W^2}", "4919f80e9f5c5fc35317c1db46841cca": "\\begin{pmatrix}\\overline{a} & \\overline{b} \\\\ \\overline{c} & \\overline{d}\\end{pmatrix},\\quad \\begin{pmatrix}a & -c \\\\ -b & d\\end{pmatrix},", "491a1a3c4e7e6ab9e7e090d457e002e2": "24^{3}\\tbinom{1,312,000}{3}", "491a501815911091e0521c0c3ef11954": "T^{15/30}", "491a8d336b43e0fdc4e6dc8702330015": " \\mu_a = {{\\mu_0} \\times {(1+kH)}} ", "491ae924639811ed117573a8cc71327c": " \\frac{d}{dx}\\tanh x = 1 - \\tanh^2 x = \\operatorname{sech}^2 x = 1/\\cosh^2 x \\,", "491af1c947acff381f1da4b683e1856d": "t_k t^{(k)} {p^{(k)}}^T", "491b21b412ede9aba445718f74d5d411": "ln(x)", "491b69706201cecb0dd0e54875ce1820": "w^{(L)}_k", "491b6a05da6f2c433c1ac2245642a9cd": "\n\\sum^{k}_{i=1}d_i\\leq k(k-1)+ \\sum^n_{i=k+1} \\min(d_i,k)\n", "491b727c4e01eb7550a6a819c477adf3": "\\left(\\frac\n{7}{11}\\right) = (-1)^3 = -1.", "491b84e7b6dd89b839c7fafadfd46e93": "\\vec k\\perp\\vec B_0,\\ \\vec E_1\\perp\\vec B_0", "491bc5c70192294b84c26c6ef20c13e8": "\\mathbf{A}(t+T)=\\mathbf{A}(t)", "491bd2d9f050b931aa8acbc769041e6f": "\\Omega_{i,j}", "491be5c18d42f852270b7c08d6348a6c": "D(\\mathbf{x}, r)", "491c2896ef37d71b55cc7226c9dc911f": "M\\{|\\xi|\\geq t\\}\\leq \\frac{E[|\\xi|^p]}{t^p}", "491c64966a3a6e4313c6bdea617f052c": "[\\omega]^2", "491c7e981a7203f466d62ec64cec3b9c": "\\big|\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k+\\alpha_k\\mathbf{p}_k)\\big|\\leq c_2\\big|\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k)\\big|", "491c86e488fe5e005b356a2cf58c8ba6": "r^*", "491cd0ce2a82ec735f7c985b1d8edc63": "y = x \\tan(\\phi_1)\\;", "491cd721ff7a447cd17f6b6224e7de53": " f(t,T) = - \\frac{\\partial}{\\partial T} \\ln(P(t,T)). ", "491ce633fbb0d7296f49a93e727a1d15": "\\vec E_{||}\\frac{\\partial f}{\\partial\\vec v_{||}}+\n(\\vec E_\\perp+\\vec v\\times\\vec B)\\cdot\\frac{\\partial f}{\\partial\\vec v_\\perp}\\approx0", "491cf176e96a10d90df059b06a59c53a": "X\\oplus X\\oplus X", "491cf80e3b2036b67cecbdc270875610": "r, \\theta, z", "491d42650797e0fd2098bcaf5763dca2": "\\underline{\\mathbb Z}_X", "491e482f5f305a6341107a795ff29acf": "X_C=X_L", "491e5ee2b31f99eb594c1ae54646fe9e": "u\\in R", "491e8af50d150458a329db3d5e60df96": "i \\hbar \\, \\partial_t \\Psi = H_S \\Psi - \\frac{e \\hbar}{2mc} \\, \\hat\\sigma \\cdot \\mathbf{B} \\Psi", "491e9f1764b305caf8bfe011bd8f1784": "f \\colon M \\to X", "491eb862a641222312b2c79f3f2717a2": "\\|T\\|\\le1", "491ffa35f33bd97a7bd0dcd26a1eda43": "R_t=w_{1t}r_{1t}+w_{2t}r_{2t}+\\cdots + w_{nt}r_{nt},", "4920311f5629e4b8609782b0f62c223a": "1\\,R_{\\odot} = 6.955\\times 10^8\\,\\hbox{m} = 0.0046491", "492055327525473d3f69688b47a3582e": "(\\mathcal D,\\bullet,I_{\\mathcal D})", "492075e6358d09c8a4b5acf5cef036ab": "\n\\sum_{m}\\,\\hat{\\Theta}_{m}=\\hat{1}.\n", "4920e0a88886647b17e16520f50b9731": "X\\subset Y", "4920f6deea349fbde647d1a183e9831f": " \\ \\mbox{Valve New adopters}\\ = \\mbox{New adopters } \\cdot TimeStep ", "4921c67e8753f4e14c256a71f168c2a2": "u_1 \\in \\mathcal{U}(\\alpha, \\tilde{u})", "4921de30ea055363fe56c2a75fa6dd59": "f_e(x_e) = c_e \\textrm{e}^{i k x_e} + \\hat{c}_e \\textrm{e}^{-i k x_e}.\\,", "4921f71bdecb960f70d0c40be305e80f": "\n \\left[\\begin{array}{ccc|c}\n1 & 0 & 0 & 4 \\\\\n0 & 1 & 0 & 1 \\\\\n0 & 0 & 1 & -2 \\\\\n \\end{array}\\right],\n", "4922133d28ad9545281939c0098bde4e": "n,r,w,s", "49223fcc02e3eaac6c862479a24363bf": "\\nabla_{\\mu}(\\sqrt{g}\\,)\\equiv (\\sqrt{g}\\,)_{;\\mu}=0 \\, , \n\\quad {\\mathrm{where}} \\quad {g}=|{\\mathrm{det}}(g_{\\mu\\nu})|\\, .", "49224fa64cbf1a60d3fb50c7eec27366": "20\\alpha ", "4923bc147b66fd185b904c5867c70ea3": "\\frac{}{} I(t) = I_c \\sin (\\phi (t))", "4923cb864486cb1b27477472e38cd617": "-\\cot(\\theta)\\,", "4923d13f4694f11cb089582baf8badb8": " \\frac{1}{\\sqrt{4\\pi}} \\left(\\mathbf{B}, \\Phi_\\text{m},\\mathbf{A}\\right) ", "49245c8cac17a7f3279e494325674504": " \\textbf{b} = \\textbf{a} \\pmod 3 = -X-X^2+X^3+X^4+X^5+X^7-X^8-X^{10} \\pmod 3 ", "49248f6ebea11caecc2371d773566e3a": "|k \\rangle ", "49249d43a4881be921921e004fdc17bc": " (v-b)\\Pr(b\\ \\textrm{wins}) ", "4924ac1a45fb1044192c94bdc3f2a20e": "X(t) .", "4924d9c318625f4258085e36f1773edc": "\\nu^*=4/3", "4924f3d45197892f762de4f6f9960ffe": "\\gcd(F_m,F_n) = F_{\\gcd(m,n)}.", "4925485a81bdc4ed2745f6476b715d70": "m_{\\mathrm{N}}", "49254e9e5eda3bb5201018af813061d7": " d = {\\ln(10)Re\\over 5.02} ", "4925707d2c37696402739feee1a98c71": "\\int_{t_1}^{t_2} \\sqrt{r^2 + \\left(\\frac{dr}{d\\theta}\\right)^2}", "4925e2bdc0dcf33cfcf54affedbfd87e": "\n\\begin{align}\n& L \\psi = \\lambda \\psi, \\\\\n& \\psi_{ t } = A \\psi\n\\end{align}\n", "4925ee5356b4ace0e28f0324cf3a81de": "V_{2}", "492697cdb75f1b371e27f499e3252ddc": "z={1 \\over 2 \\mathit l_B} \\left( x + iy\\right) ", "4926bf6f9de135d5c6bc606240a87eb0": " y' = f(t,y), \\quad y(t_0) = y_0, ", "4926dcb5c82c33f9a416d0698b38afc6": "|\\beta|>|\\alpha|", "4927193268485e385f1cd8d6cf91fb56": "CPI= \\frac{\\text{updated cost}}{\\text{base period cost}} \\times 100", "4927a2797f28506f0f5054b5b515ad67": "K_\\mathrm{J} = \\frac{2e}{h}", "4927da3058f17c74b617ac3e163fd980": "\\dot{\\mathbf{x}}(t) = A(t) \\mathbf{x}(t) + B(t) \\mathbf{u}(t)", "49285b589a62dec3ba3e64e7e56ab461": "\\mathcal{T}_1:\n\\mathcal{A} \\rightarrow \\mathcal{M}", "49286f285450b86d19e83f506ac45294": " k ", "49289c2a2053a039f0ef153c0390544a": "A \\overset{\\underset{\\mathrm{def}}{}}{=} P", "4928a3aedae7e83fd8a953cd33f22035": "Z_0(x) = \\left\\{ \\overline{z} : \\phi(x,\\overline{z}) = \\max_{z \\in Z} \\phi(x,z)\\right\\}.", "4928b07fa94531846c710cb4da5abeed": "\\sin[\\alpha]", "4928f9ab0f1e6f882ec652f58b172d81": "a_{ij}(x), b_i(x), c(x)", "49290964bd7bd54301abc4351e130e00": "\\xi_k", "4929130caf8985515ffac69cf9dc82d3": "v \\mapsto (x \\mapsto f(x,v))", "4929480ff5db12dcce25631dccf84586": "ROC4 = (1-Price/Price(X4))*100;", "49298fba361df81dc90f61c8ee8d87b7": "\\mu=1", "492990a765d9f09a3790ef70cce66e01": " f\\left(X_1,X_2,\\dots,X_n\\right) = \\sum_{m_1=-\\infty}^{\\infty} \\sum_{m_2=-\\infty}^{\\infty} \\cdots \\sum_{m_n=-\\infty}^{\\infty} C_{m_1m_2...m_n}\\exp\\bigl[2\\pi i\\left( m_1X_1 + m_2X_2 + \\cdots + m_nX_n \\right) \\bigr], \\text{ for integers }m_1, m_2, \\dots, m_n", "4929db421acb4e90c48fd79075d9830f": "w : X \\in L", "492a629c26cceb6cd1e29ff7be733b8d": "p \\sim_l q", "492a79442dd55d1ccb068989dd0e2fa9": "\\Pr(\\overline{X} - \\mathrm{E}[\\overline{X}] \\geq t) \\leq \\exp \\left( - \\frac{2t^2n^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right),\\!", "492af82aed4625633c5422bcf839ae36": "\\mathbf{a}(t,\\mathbf{r}_0)", "492b0cfeadcb34f8021b83f8717c3e5c": "\\color{RedViolet}\\text{RedViolet}", "492b2ba56fd90c7258b561c0b209823b": "(u_{s} i +v_{s} j + w_{s} k) \\times (u_{t}i +v_{t}j +w_{t}k).", "492bab9ef78267a8302bdb03775a84ed": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx = \n \\frac{x^{m+1} \\left(a+b\\,x^n\\right)^{p+1}}{a (m+1)}\\,-\\,\n \\frac{b (m+n (p+1)+1)}{a (m+1)}\\int x^{m+n}\\left(a+b\\,x^n\\right)^pdx\n", "492bb8c6c4f80a1bc1920e00602f2f3f": "2K", "492bbd3bd97c6d89ff0b4f3cf5aee52d": "e^{-\\alpha n} u[n] ", "492bd490647934e982aa84daae3fb06b": " \\sqrt{Zk_\\mathrm{e} e^2 m_\\mathrm{e} r} = n \\hbar ", "492c43ca3f7bd9e86dc539ebeeadada1": "\\omega_{ab}=B_{[ab]}", "492cc604c6323950a821bca6d2490a29": "p_x=p_x(x_1\\ldots x_n)", "492d25786b87ba368269336d2325c1a1": "U = -G\\int_0^R \\frac{4 \\pi r^3 \\rho 4 \\pi r^2 \\rho}{3r}\\, dr = -\\frac{16}{15}G \\pi^2 \\rho^2 R^5", "492d269566d18e99956a568ecaff7914": "\\rho\\leq 1", "492dc0761ca969f8ba59c819e577b462": "+1 \\div 0 = +\\infty", "492e2d7f7148b8da4c33e5afc9e5cccb": "x = \\sqrt a", "492e424d88ddb203e78d204338cdbb48": "X_{1/T}", "492e99450c7a6d29d154b6246a0d74a2": "\\gamma_{p,v}", "492eda0ef5f59b6b491b7c633f6813af": "\\scriptstyle{\\Delta z}", "492fae1501516f446e189fb9c2164620": "A\\ \\xrightarrow{\\quad\\Delta\\quad}\\ A \\oplus A \\oplus A\\ \\xrightarrow{\\alpha_1\\,\\oplus\\,\\alpha_2\\,\\oplus\\,\\alpha_3}\\ B \\oplus B \\oplus B\\ \\xrightarrow{\\quad\\nabla\\quad}\\ B", "492feabaab3ff01282023085dc4ac15d": " \\left \\langle \\mathbf{x} |A| \\mathbf{y} \\right \\rangle= \\left \\langle \\mathbf{y} |{}^{t}A| \\mathbf{x} \\right \\rangle.", "492fed85d91d1a8d90165c177a7a26b4": "P_1, \\ldots, P_4 ", "49303bbce6cde0b3ac0b757405c37dbb": "\\{f(x): x \\in \\Lambda\\} ", "49307c0a865b195825cc64d23cb5b938": "\\scriptstyle t\\,\\to\\, t^*", "4930f348cff2a4dbf5788cda9b318f78": "\\ln [A]", "4931277b5f2b699fcee0e9e62894c407": "(a+b\\sqrt c)(j+k\\sqrt c)\n=(aj+c\\,bk)+(ak+bj)\\sqrt c", "493131f63852ab695c3b9ffb339e3b62": "H^*(X) \\cong H_{n-*}(X)", "4931856b828262aeb35f154d5807bf52": "A\\mathbf{x}-\\rho(\\mathbf{x})\\mathbf{x}", "4931af7302f983b583e2bb352c1e2635": "\n c=\\frac{d}{t}\n ", "4931cd2db8e2361d252725cb5a25696e": "\\overrightarrow{f(p_2)f(p_3)}", "4931fa53266d3f3c534ea09b72123292": "D_{L,r}", "4932da6957089592c09fee8f93614260": "{D} = D \\cdot \\vec {I} = D \\cdot \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\end{bmatrix},", "493319b020d40d64d4be4296aba279cc": "J^\\alpha = \\left(ic \\rho, \\mathbf{j} \\right)", "49333de6322d7cd2a964b6dd627d52aa": "\\sigma:\\Sigma\\to\\Sigma'", "493380d08fc1ce5b3dbd386d3c8ae29e": "\\begin{align}\n\\sum_{1\\le n\\le x}\\mu(n)g\\left(\\frac{x}{n}\\right)\n &= \\sum_{1\\le n\\le x} \\mu(n) \\sum_{1\\le m\\le x/n} f\\left(\\frac{x}{mn}\\right)\\\\\n &= \\sum_{1\\le n\\le x} \\mu(n) \\sum_{1\\le m\\le x/n} \\sum_{1\\le r\\le x} [r=mn] f\\left(\\frac{x}{r}\\right)\\\\\n &= \\sum_{1\\le r\\le x} f\\left(\\frac{x}{r}\\right) \\sum_{1\\le n\\le x} \\mu(n) \\sum_{1\\le m\\le x/n} [m=r/n] \\qquad\\text{rearranging the summation order}\\\\\n &= \\sum_{1\\le r\\le x} f\\left(\\frac{x}{r}\\right) \\sum_{n|r} \\mu(n) \\\\\n &= \\sum_{1\\le r\\le x} f\\left(\\frac{x}{r}\\right) i(r) \\\\\n &= f(x) \\qquad\\text{since }i(r)=0\\text{ except when }r=1\n\\end{align}", "49339d1e43cf49d17618cc8d467a720a": "AUC_{l,k}", "493481260410ddd65923bdd74380d660": "\\eta(m)", "4934b972e3c40605c92d302d9df8b178": "\\overline{x}\\langle z_1,...z_n\\rangle.P", "4934c0e53430cf214a1396476d8d3dbf": "( \\cdot , \\cdot ) : F^* \\times F^* \\rightarrow G", "4934e2059d577bd3a1257a23f93bb5f6": "d(u, v) = \\cosh^{-1}(B(u, v)).", "4935027155f19e8c83b4001d0a3b5676": "\\mathrm{Iterations}", "49351c7804b246baaeefb44c583af6c4": "u_\\theta(r) = \\begin{cases} \\Gamma r/(2 \\pi R^2) & r \\le R, \\\\ \\Gamma/(2 \\pi r) & r > R. \\end{cases}", "49354bee5d87bf4124c6c00a4cf1f8ac": "\\phi(z)=\\sum a_nz^{-n-1},\\,\\,\\, [a_m,a_n]=m\\delta_{n+m,0}I,\\,\\, Ua_nU^{-1}=a_n - \\delta_{n,0}I", "4935954f12193cb44ed0f7416ea65d44": "\\mathit{K}_2", "4935fb57a20d359af3821d255cd02ea4": "{d^2\\theta\\over dt^2}+{g\\over \\ell} \\sin\\theta=0", "49360fa0b96edae5880d8e0948442086": "\\text{Working out for convenience per-unit ohms directly, we have}", "4936581e3a36b0eed36b3e3af40c1980": "x\\Leftrightarrow y \\equiv (x\\Rightarrow y)\\wedge(y\\Rightarrow x)", "493713e64d842b22b2579d3de081b52c": "x_U = x_L = x, \\qquad y_U = +y_t, \\quad \\text{and} \\quad y_L = -y_t.", "493724ba35cfcd3b375bed49ea033f1a": "e^{-i\\epsilon H(p,q)} \\,", "49373d4751f9bd9b50a605ba247b27c9": "(\\frac{\\partial}{\\partial x_2} - \\frac{\\partial}{\\partial x_1} ) \\psi (x_1, x_2)|_{x_2=x_1+}= c \\psi (x_1=x_2)", "4937426cfdbc22ece6a8d1dc43e8d7f1": "\n \\Theta(G) \\leq R(G) = \\min_{B} \\operatorname{rank}(B),\n", "49374aef272311afe18f65c875d11ccf": "\\displaystyle{\\beta(L_m)=L_m},", "49378fd3c30dae9a5532ce0a8c4d1a85": " \\forall x \\forall z ( ( \\phi \\lor \\psi) \\rightarrow \\rho )", "49379127ef7d860d7a2407bcfcfc4f63": "\\lim_{x\\to 0^-} g'(x) \\;=\\; \\lim_{x\\to 0^-} \\frac{1}{\\sqrt[3]{x}} \\;=\\; {-\\infty}\\text{,}", "49379f172883329a9ef3d98e020b81e4": "\n\\Delta^1_{\\rm LAT}= 111132.954 - 559.822\\cos 2\\phi + 1.175\\cos 4\\phi\n", "4937f1f389a3979eab85706490662b01": "|M_{1}| = |M_{2}|", "4938f2122644aba61b0adf120acd5210": "\n K_{\\rm I}(-a; x,y) = -K_{\\rm I}(a; -x,y) \\,,\\,\\, \n K_{\\rm II}(-a; x,y) = K_{\\rm II}(a; -x,y) \\,.\n", "49390b8dd936dafde6cb645895d794b2": "e_i.v_0 = 0", "49396cace343e6c3d82d64fd9154a1c7": "L^{n-1}\\rightarrow 0", "4939c4245161a834488481e0d597ee4d": "-\\nabla \\times \\mathbf{E} = \\frac{\\partial \\mathbf{B}} {\\partial t} + \\mathbf{j}_{\\mathrm m}", "4939f31b6cc9c445fb95d80fd99612ec": "u_2 = u_1a_1-1=1.175\\cdot1-1=0.175, a_2=\\left\\lceil\\frac{1}{0.175}\\right\\rceil=6 \\, ", "4939fde11fcbd6bd2e18d263dbf5bcba": "c=\\sqrt{\\frac{e}{m}}", "493a565b7114c714eb029f9737b87bd2": " i_r = g_m v_{\\pi} \\ . ", "493a61d8eef51f690ce12341524e3776": "T(Y)=\\dfrac{1}{2N}\\sum_{i,j}|Y_{i}-Y_{j}|^{2} \\leq \\dfrac{1}{2N}\\sum_{i,j}(N\\tau)^2 = \\dfrac{N^3\\tau^2}{2} \\,\\!", "493a6af3b5f311d27b362f0c7586c697": " \\operatorname{build-param-list}[\\lambda m,p,q.(\\lambda g.\\lambda n.(n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n)))\\ \\lambda x.\\lambda o.\\lambda y.o\\ x\\ y, D, V, \\_] ", "493ad8cc39142d53f012c2952614f95a": "\\mathit{l} \\gtrsim 800", "493ae82a5426a0fa6fd05819e3b68e92": "= a_0+a_1x+a_2x^2+\\cdots", "493af5447e12ed643c4fd3233568c880": "dV ", "493b397401787732ecd620eef2d8a16e": "\\scriptstyle\\frac14\\,\\frac13\\,2", "493b650afa5a8c62b986b3982862baab": "\\dot{\\nabla}F\\dot{G}=e^iF(\\partial_iG),", "493b919568000be9e9a145c26a3744f9": "Z_2\\,", "493b92735da4cf7076a90a7eb6d0504b": "\\bold{P}(\\bold{r})= \\bold{p}(\\bold{r}) \\, ", "493bd08196c2c5ae3d0bd9514a12dbec": "\\int\\frac{\\sin ax\\;\\mathrm{d}x}{\\cos^n ax} = \\frac{1}{a(n-1)\\cos^{n-1} ax} +C\\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "493be2afe91539eae55bad6c465baf69": " \\mathbf \\zeta_{rel} ", "493c947e1d9e47cdb9bf48b523e9aef5": "\\frac{1}{\\sqrt{s}}f\\left( \\frac{t}{s} \\right)\\leftrightarrow {{P}_{V}}f\\left( \\frac{u}{s},s\\xi \\right)", "493c99735976b4e9de3a52829bd98108": "\\begin{align}\nK_n R \\sin(\\theta\\pm\\phi) &= R \\sin(\\theta) \\pm 2^{-n} R \\cos(\\theta)\\\\\nK_n R \\cos(\\theta\\pm\\phi) &= R \\cos(\\theta) \\mp 2^{-n} R \\sin(\\theta)\n\\end{align}", "493cde74a332d416593b0f50a24978ec": " \\mathbf{a} = \\frac {\\mathrm{d}^2 \\rho }{\\mathrm{d}t^2} \\mathbf{u}_{\\rho} + 2\\frac {\\mathrm{d} \\rho}{\\mathrm{d}t} \\mathbf{u}_{\\theta} \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t}-\\rho \\mathbf{u}_{\\rho}\\left( \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t}\\right)^2 + \\rho \\mathbf{u}_{\\theta} \\frac {\\mathrm{d}^2 \\theta} {\\mathrm{d}t^2} \\ , ", "493ce25cbb130ba189e49900ad8de9a8": "i\\hbar \\gamma^\\mu \\partial_\\mu + mc \\equiv i\\hbar(\\gamma^0 \\partial_0 + \\gamma^1 \\partial_1 + \\gamma^2 \\partial_2 + \\gamma^3 \\partial_3) + mc \\begin{pmatrix}1&0&0&0\\\\ 0&1&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{pmatrix} ", "493d4974a5de8158cde945ea4ec31d6e": " A=\\bigcap_{n\\geq 0}f^{n}(N), \\quad A\\subseteq\\operatorname{Int}\\, N.", "493d59a50739aa94df5bce7979796dd8": "\\vec{r}(u,v)", "493d613dcd22380ae2beadd81fc79727": "i_D \\approx i_1", "493d6f96ea208f2d2921036c58e332d1": "(DD^+D)_{ij}=D_{ij}D^+_{ij}D_{ij}=D_{ij} \\Rightarrow DD^+D = D", "493d90bf44d2bb57159c6f90bca661d0": "< 2n", "493da6813758720e4345c23940de786c": " w \\in W", "493dffdea1a3bf693d78088c5f91daec": "\\mathbf{e}_{q}\\times\\mathbf{e}_{q} = \\boldsymbol{0}", "493eab4178903c6061681d5a381eb63e": "K = \\int d^3 x K_a^i \\tilde{E}_i^a", "493ed8b9ff518b524d54cf2bc17857ab": "F_{\\rm x} = {\\dot m}{(C_{\\rm s} - C_{\\rm u})} = {\\rho AC}{(C_{\\rm s} - C_{\\rm u})}", "493ef00e71e49b84f74ee950d33101e0": " Q_{Y|X}(\\tau)=X\\beta_{\\tau}", "493f0a4b60c883f2d27577f0c557bfe8": "\\exists x_1 . . .\n\\exists x_n TC ( x_1 . . . x_n, o_1 . . . o_m)", "493f1130b8b1a81051378d034fa1b1fd": "|F_n| = |F_{n-1}| + \\varphi(n).", "493f339417afd97f4c040e14afb35835": "E_k = \\hbar \\epsilon_k", "493f3d9b9e704066221ee5448f310ccd": "y^T b", "493faa3fe807edc20e6dafbd6d1667a0": "{\\left| G \\right\\rangle} =\\prod _{(a,b)\\in E}U^{\\{ a,b\\} } {\\left| + \\right\\rangle} ^{\\otimes V}", "493fe715e50215c40037e607dd15b7f3": "\\lim_{n \\to \\infty} \\varphi^{n} = u", "493fff05dda75b8b04af70b3419fee32": "\n\\begin{matrix}\na \\vert b &=& b \\vert a\\\\\n(a \\vert b) \\vert c &=& a \\vert (b \\vert c)\\\\\na \\vert \\delta &=& \\delta\\\\\n\\partial_H(a) &=& a \\mbox{ if } a \\notin H\\\\\n\\partial_H(a) &=& \\delta \\mbox{ if } a \\in H\\\\\n\\partial_H(x + y) &=& \\partial_H(x) + \\partial_H(y)\\\\\n\\partial_H(x \\cdot y) &=& \\partial_H(x) \\cdot \\partial_H(y)\\\\\n\\end{matrix}\n", "49400ddf0a8f8ef7fbb467cff5e8ce9d": "\\alpha_1\\,\\!", "4940261535a91444fe543000e50feec1": " \\, A_{(a_0,\\;a_1,\\ldots,\\;a_n)}(x)\\,", "494045d633ae4998c6cc2e23be21ce52": "Y\\to Y'", "49405237d2929734e6c3464f9e4b142a": "\\left(\\frac{L}{h}\\right) = {\\mu \\over 2 \\pi} \\ln(D/d)= {\\mu_0 \\mu_r \\over 2 \\pi} \\ln(D/d)", "49408c6b02e523fa9b73705588ddb7c8": "||y-Hy||^2", "4940d0b977d05df5cf201e2f2c00d70f": "1/q^2", "4940ed8cc0df89dc8483d0980d66eab6": "C:\\ Y^2 = X^4 + Z^4", "49412ad58c9b9eadf6a4b03c2626ecf7": "17^{70}\\ \\equiv\\ 1 \\pmod {71}.", "494130020366aaad2e5be68db63c4225": "\n\\Pr[X>x] \\sim x^{- \\alpha}\\text{ as }x \\to \\infty,\\qquad \\alpha > 0.\\,\n", "494156570e3583a1a45a01feabc1a507": "\\sum_{n=1}^{\\infty}|a_n e^{-\\lambda_n s}|,", "49419e52f272d4a52e1405347edad027": "-I\\in SO(2n)", "49419f6b077e2952834d6cb7eb9f03bd": " e = c = G = 4\\pi \\varepsilon_0 = 1 \\, ", "4941abcb9729ce97082af336e2b23114": "t_0 < T", "49423a05653a04a06103d7801be36782": "y = a\\ln \\left[\\tan \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right) \\right]", "49427949917b5b96e2bc65e81059a900": "H^1(End(F)),", "494299febaa9d89e0cd962c846ca5fb8": "\\begin{align}\n\\Delta p &= -\\gamma \\nabla \\cdot \\hat n \\\\\n&= 2 \\gamma H \\\\\n&= \\gamma \\left(\\frac{1}{R_1} + \\frac{1}{R_2}\\right)\n\\end{align}", "4942a3570c2774a4879591643835c00a": "\\pi_1(S^1)=\\mathbb{Z}", "4942af802da3f23b19714d97006ca33f": "\\phi(1,1,0)", "4942b7b6d0a72914c2fe931854df3bee": "u_{\\gamma}=1", "4942e61a9f3d4c66ca0bb3dec3a0559b": "x_k \\ne y_k", "4942fef4c83c8df5e80627e93777a9a0": "\\hat{x} = x - t\\,", "49431bf42a633191b4981e867e2de75f": "c_{i,x}", "49433837194aa5237e6c4f8175a3e9ea": " \\Gamma_{a} ~,~ \\Gamma_\\text{chir} ~,~ \\Gamma_\\text{chir} \\Gamma_a ~,~ \\Gamma_{a_1 a_2}\n~,~ \\Gamma_\\text{chir} \\Gamma_{a_1 \\dots a_4} ~,~ \\Gamma_{a_1 \\dots a_5}", "494361ef39f73c14927e64f34d2f7d5d": "T_{0}=", "49437b6687d948c0bf295f8f60d7469a": "\\varphi(1)", "4943b4feee4f68146e0cbb15ea5b9cfe": " \\frac{1}{\\sqrt \\pi} \\frac{d^{1/2}}{dx^{1/2}}f^{-1}(x)= \\sum_{n=0}^{\\infty}\\delta (x-E_{n}) ", "4943bb87148030069f6f6586133743a3": "\\operatorname{Q}_Y(1-p)", "4943dd74268aaecf5a9c1f09b946fdce": "\\begin{align}\n\\int\\sqrt{x^2 - a^2}\\,dx &= \\int\\sqrt{a^2 \\sec^2(\\theta) - a^2} \\cdot a \\sec(\\theta)\\tan(\\theta)\\,d\\theta \\\\\n&= \\int\\sqrt{a^2 (\\sec^2(\\theta) - 1)} \\cdot a \\sec(\\theta)\\tan(\\theta)\\,d\\theta \\\\\n&= \\int\\sqrt{a^2 \\tan^2(\\theta)} \\cdot a \\sec(\\theta)\\tan(\\theta)\\,d\\theta \\\\\n&= \\int a^2 \\sec(\\theta)\\tan^2(\\theta)\\,d\\theta \\\\\n&= a^2 \\int \\sec(\\theta)(\\sec^2(\\theta) - 1)\\,d\\theta \\\\\n&= a^2 \\int (\\sec^3(\\theta) - \\sec(\\theta))\\,d\\theta.\n\\end{align}", "49443438fd3df665d83acd870a98c11f": "(\\omega\\times\\omega)^{<\\omega}", "49444d78fca6cd4fe2da953e4da85078": "d^{n + 1} d^n = 0", "494461ab1989d3e7c997baae3414dda1": "1 \\leq \\alpha \\rightarrow \\epsilon_{\\omega_{\\alpha}} = \\omega_{\\alpha} \\,.", "494473c89e70843b8d680932f5321201": " \\mathbb{F}_{q^n}[x] ", "494490534ad993917cdc04524d19d245": "\\scriptstyle(+2.2\\pm1.5)\\times10^{-9}", "4944976aa9f5b3a2dfde33b1893ab885": "F^{-1}_e(\\mathbb{E}[Y_n]) = \\boldsymbol\\beta \\cdot \\mathbf{s_n} .", "4944ce06626dc0c63f51a3d0c23935a4": "\\begin{align}\n1\\,\\mathrm{lbf} &= 0.45359237\\,\\mathrm{kg} \\times 9.80665\\,\\mathrm{\\tfrac{m}{s^2}}\\\\\n&= 4.4482216152605\\,\\mathrm{N} \\text{ (exact)}\\end{align}", "494510b546df178e88b401359cf5dc12": " v= \\tfrac12\\, w.", "494516ec6275bdcc0f6034ec750a9101": "i (a,b) =\\text{ the smaller of }i \\langle a,b\\rangle\\text{ and }i \\langle b,a\\rangle,", "494530df3e7023dd7a00d90975c4ba08": "V_{2n}=\\frac{\\pi^n}{n!}\\ ", "49455a4fd9b263c58f532c2f18673207": " \\int_{a}^{b} f(x) \\, dx \\approx \\frac{3h}{8}\\left[f(x_0) + 3f(x_1) + 3f(x_2) \n+ 2f(x_3) + 3f(x_4) + 3f(x_5) + 2f(x_6) + ... + f(x_n)\\right] .", "4945d9e483c39a92acf4139fcc3a4f83": "\\scriptstyle [0.5/M,\\ 1-0.5/M]", "4945dafbefc668116f2a5bac285214d4": "0<\\epsilon<1", "4945edf9fa4c53b9d88e1e3e3380cae9": "\\eta_p= \\frac {2 \\frac {u} {c}} {1 + ( \\frac {u} {c} )^2 }", "49464f43ee63d8599c0f695d482aae31": "10^2\\equiv 2\\,\\bmod{7^2}", "494664a716200bae107177b8077ade60": "e=0", "494693cc0b6904a5ea361bb262750138": "\\operatorname{rank}(AB) + \\operatorname{rank}(BC) \\le \\operatorname{rank}(B) + \\operatorname{rank}(ABC).", "494696128e703a6fb0216cb7a8175086": " Y^{\\mu} = ( Y^0, \\vec Y) ", "494698335cf46e07fd83ba9ba99ab9e3": " Q(x,y)", "4946a7f3b6b970578ba611742d51b3a0": "\\frac{\\left(x-\\sqrt{a^{2}-b^{2}}\\right)^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1", "4946afbc7a52b174d059afd025f7d853": "\\widehat{\\varepsilon}(\\omega) = \\varepsilon'(\\omega) + i\\varepsilon''(\\omega) = \\frac{D_0}{E_0} \\left( \\cos\\delta + i\\sin\\delta \\right). ", "4946ce2dc975364dd88aac4f81a4ed24": "\\sum_{n=k}^\\infty {n\\choose k} y^n = \\frac{y^k}{(1-y)^{k+1}}.", "4946e4d763807bf04cbf30b99cbb0088": " \\varepsilon_n ! = \\varepsilon_1 \\varepsilon_2 \\ldots \\varepsilon_n", "4946ef0c9966875563e30bbd9b585e21": "m_{w}", "494756444a148bba1e0121368e16ee02": "n_{k}=1", "49476119c6a4a92a2bf05e231e994eab": "\\left[\\hat{f}, \\hat{f} \\right]_+ = 0", "494794a52bd6e1094ac9cc9ddf8fc66f": "G_{2}(\\mathbb{F}_{3})", "4947bc393537dd893380db764ee71877": "\\frac{q_{rev}}{T}", "4947bc9d65d5f0eeb7c92e3e94eb908b": "n=p^2 q", "4947ee4ef5d9e666b55efbbe26af8b32": "N_W = L\\sqrt{{\\omega}{\\rho}\\over {\\mu}}=\\sqrt{N_S}\\,\\!", "4947fe5857393bc493de6a54d27d9cce": "\n D^*(p \\parallel q) = D(q \\parallel p).\n ", "49486ed842d6c896ccfc778b0f8d5ec2": "g:Y \\rightarrow Z", "49488310fb6fd9c364254665dd0ef6ac": " Q_j = P O_L + h_j(\\theta) O_L. ", "4948df615777d7f42e181f70835cf236": "\\nabla_\\ell R^\\ell {}_m = {1 \\over 2} \\nabla_m R, \\ ", "4949146dac7f71e660daa3870af99f6a": "\\sigma_L^D ", "4949438321e6ec675789be5c6fce72f4": "L(\\theta,\\delta(x))\\,\\!", "49495d37e401b1e0d102d28dd8a98ada": "t = D T", "4949c28ecafc26d29604d88b8968f2ba": "x = 2r\\cos t, y = 0\\,\\!", "494a4ae1d013066f5eb4e08ab18fd12c": "\\Gamma(\\mathfrak{g})", "494a62012bb364bd841fdc16007bce95": "y'(t)=\\alpha+\\frac{v\\,y(t)}{c+vt}\\,\\!", "494a9f6062772e2de183f35c8a8cef85": "\\ A=2aS.", "494abb52a56288bc68d3f75cb4e5d555": "x_4 \\in_R (0, q-1]", "494bb2cb5cc28ffb25a662d4f8f8c757": "\\sum_{i=1}^{k} \\sum_{p: s_i \\rightarrow t_i} f_p^{*} \\cdot \\sum_{e \\in p} l_e(f_e) < \\sum_{i=1}^{k} \\sum_{p: s_i \\rightarrow t_i} f_p \\cdot \\sum_{e \\in p} l_e(f_e)", "494bcabfea671875372aa43729dc0ba0": "(c \\triangleright b) \\triangleleft a = (c \\triangleleft a)\\triangleright (b \\triangleleft a)", "494bdcab645259b6ba53b62278738e90": "\\rho(\\alpha,\\beta)=\\frac{\\sum_{i=1} \\sum_{j=1}(X_0 (i,j)-\\bar{X_0})(X_1 (i+\\alpha,j+\\beta)-\\bar{X_1})}{\\sqrt{(\\sum_{i=1} \\sum_{j=1} (X_0 (i,j)-\\bar{X_0})^{2})(\\sum_{i=1} \\sum_{j=1} (X_1 (i+\\alpha,j+\\beta)-\\bar{X_1})^{2})}}", "494bec2c6129b3355791d758dea81c4d": "x0}.", "49552a350673d9b661dc18850c11186b": "\\scriptstyle I_1,\\, \\ldots,\\, I_m", "495545debc9cda1354d7f468d0d49a7e": "t_{e1} ", "495564a38a9d20f5e35e87cf0c6c4de7": " m_b\\ne m_b'", "4955c8ad0da5d1cff9bdfecd0f09e80b": "\\Pi^{g,K}_{g,L\\cap K}(W) = R(g,K)\\otimes_{R(g,L\\cap K)}W.", "495627fc0454e2314b2bc268e4083c06": "\n\\begin{align}\nH &= \\frac{1}{2}\\sum_{\\mathbf{k},\\mu=-1,1} \\hbar \\omega\n\\Big({a^\\dagger}^{(\\mu)}(\\mathbf{k})\\,a^{(\\mu)}(\\mathbf{k}) + a^{(\\mu)}(\\mathbf{k})\\,{a^\\dagger}^{(\\mu)}(\\mathbf{k})\\Big) \\\\ \n&= \\sum_{\\mathbf{k},\\mu} \\hbar \\omega \\Big({a^\\dagger}^{(\\mu)}(\\mathbf{k})a^{(\\mu)}(\\mathbf{k}) + \\frac{1}{2}\\Big) \n\\end{align}\n", "49562c41009d266de643f7a73067f78d": "T = \\frac{r_i - r_f}{\\beta_i} ", "49564bb70028164aae66b300daef6920": "| \\Gamma | = {SWR-1 \\over SWR+1}", "49565acba06e5c1607c8f847452350e8": "\\left[\\varphi_Y\\left({t \\over \\sqrt{n}}\\right)\\right]^n = \\left[ 1 - {t^2 \\over 2n} + o\\left({t^2 \\over n}\\right) \\right]^n \\, \\rightarrow \\, e^{-t^2/2}, \\quad n \\rightarrow \\infty.", "49565e389414292f8fcf95678b9d3ab6": "\\left\\{0\\right\\}", "4956bef20f18987ffffc1daaeebb4ebd": "F = S \\left[ 1 + (r - c) T \\right]", "4956ca4647a7262dd4c294e20aa5db4d": "\ns^2 = \\frac{1}{N-1}\\sum_{j=1}^k\\sum_{i=1}^{n_j}A_{ij}^2\n", "4956fbd66c9c01291875ac5fe664d1d9": "{} + b^2\\,", "495719ea89b737c3383df3afe559e751": "\\{e_{\\mu}\\}", "49573d27df3ca2c87915b1a62d684239": " \\tau(uau^*)=\\tau(a)", "49574c1e427867d31b7c1ebcb23c9a5f": "\nV_c = \\frac{\\sum_i m_i v_i}{\\sum_i m_i}\n", "4957f8f2eb29cfef8e9094fed560107f": "[5,\\infty)", "4957fc759632d78ceb4c6666707316cb": "\\bold k\\parallel \\bold B_0", "49582354bcb8236ed4e8e67532e28850": "r_O = \\frac{1 + \\lambda V_{DS}}{\\lambda I_D} = \\frac{1}{I_D}\\left(\\frac{1}{\\lambda} + V_{DS}\\right)", "49584510b537ff2d5976e9ed64ec586e": "\\frac{\\delta \\mathcal{S}}{\\delta \\phi(x)}\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]+J(x)Z[J]=0", "495860b6a0ab86247c48cbf5f8a19aa7": "\n\\mathbb{E}_{X^{n}}\\left\\{ \\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}},\\delta}\n\\rho_{X^{n}}\\right\\} \\right\\} \\geq1-\\epsilon,", "4958db71c8f54d04e0fbd5a662e95681": "S^{k+l-1} \\to S^k \\vee S^l. \\,", "49596f5b423291bf24c9657bd2be4ba3": "x_i \\in [0,1]", "495993858c17e8ccaa6f996a5bdbc08f": "E_k = \\begin{matrix} \\frac{1}{2} \\end{matrix} mv^2 \\times\\left(\\frac{1\\mbox{ ft}\\cdot\\mbox{lbf}}{7000\\mbox{ gr}\\times 32.1739\\mbox{ ft}\\mbox{/s}^2}\\right)", "4959a73fff7f013a46f9d0e436766443": "10^{10^{10,000,000}}", "4959b049535085e0d67b9411a7e816ea": "\\Delta U_{system\\,(full\\,cycle)}=0", "495a16486955696acc9359d174a43ad2": "\\operatorname{Alt}^k\\ {\\mathbb C}^n", "495a276adcd76efbf4e08212db04bf78": "deg(r) ", "4975847ce15307e162deb37de084a73d": "\\mathbf{r}\\,", "497597bb3dd67af06163fb7646d5d90a": " r_2=u-\\int \\frac{c}{\\rho}d\\rho, ", "497599680c9e53dc40c99e6e9ecb2ef0": "\\,\\Gamma(x+n)\\,", "49760403d4b226e30a8855d5507cce24": "\\operatorname E(\\exp(zX)) = \\sum_n s_n \\frac{z^n}{n!} = \\exp\\left(\\sum_n u_n \\frac{z^n}{n!} \\right)", "49762025cabf0399bdf720770ce4c055": "\\ell \\leq n", "497672f353121f0fb6ebf15e7daad921": "U = {{{\\sum_\\min^{+\\infty} {(R_r - R_\\min}) P_r}} \\over \\sqrt{{{\\sum_{-\\infty}^\\min {(R_r - R_\\min})^2 P_r}}}} = \\frac{\\mathbb{E}[(R_r - R_\\min)_+]}{\\sqrt{\\mathbb{E}[(R_r - R_\\min)_-^2]}}, ", "49767fc2a119e5d1a16974dbdd8c7648": "-{dy \\over dx}\\sqrt{1-x^2} =1", "497682f97501b935c06b25ed80858fdd": "\\int_{-\\infty}^\\infty |f(x)|^2\\,dx = 1,", "49769575ccbeac4f6fdd9dc36a1092bd": "\\int_L^*\\frac{d \\tau_1}{\\tau_1-t} = \\int_L^* \\frac {d\\tau_1}{\\tau_1-t} = \\pi\\ i \\ . ", "49769ab5f8ba5b3e65e9aaa83d56426d": "\\alpha \\not\\in free(\\ \\Gamma\\ )", "4976f81671f4a17850ec889e5674ea83": "[\\mathfrak{g'}, \\mathfrak{g'}] \\subseteq \\mathfrak{g'}.", "49776e14bb916d0bacfc71381a62fb59": "\n\\pi = \\frac{4 \\left( M(1; \\frac{1}{\\sqrt{2}}) \\right)^2} {\\displaystyle 1 - \\sum_{j=1}^\\infty 2^{j+1} c_j^2}\n,\n", "4977acf5eab965fa5f42ea69ac1d07e1": " \nt=\\tau\\left(x_i\\right)\n", "4977b4cb9606c75aeed03cd85f1c59f7": "\n\\Pr \\{X_{ni}=0\\} =\\frac{1}{1+ \\sum_{x=1}^m \\exp{{\\sum_{k=1}^x (\\beta_n} - {\\tau_{ki}})}}\n", "4977c8c7e1663e89724accc2abc17d95": "e^a=z.\\,", "4977d5fe26cc249abe33e5f244a285ea": "H(z) = \\frac{Y(z)}{X(z)}", "4977dbbc9864cb5b4cf8f4cb739a2c09": "\n\\begin{align}\nC &= \\frac{p\\times C_u + (1-p) \\times C_d}{1+r} \\\\\n&= \\frac{p\\times \\max(S_u - k, 0) + (1-p) \\times\\max(S_d -k, 0)}{1+r} \\\\\n\\end{align}\n", "497896f0c4b24b498e40657873282b84": "T: (\\operatorname{ker}T)^\\bot \\to H_2", "4978b59fe483f0799a24179fe8d0902d": "f(a) = \\frac{1}{2\\pi i} \\oint_\\gamma \\frac{f(z)}{z-a}\\, dz ", "49790c5e921428d25b4b8c0f249cf583": "\\Sigma_2^T \\Sigma_2 = \\lceil\\beta_1^2, \\dots, \\beta_r^2\\rfloor", "49793efc0211efa5d42f3d9c5e669a3a": "g(x) = (Tf)(x) = \\frac{1}{1-x} f\\left(\\frac{x}{x-1}\\right).", "49797523266383f3dd078985edb07dbc": "r = \\frac{ p }{1 + e \\cos \\theta}", "49797dcb5b188ad2062cb7ac20ccbb1a": "F^a_{ \\mu\\nu}=\\partial_{\\mu}A^{a}_{ \\nu} - \\partial_{\\nu}A^{a}_{ \\mu} + g f^{abc}A^{b}_{\\mu}A^{c}_{\\nu}", "49797f2a9881f01b7d4243613b9d047d": "(D2)", "4979941654e4bc3a1023fcaa1422bbdb": "F_{X / S} \\times 1_{S'} = F_{X \\times_S S' / S'}.", "4979e8adb950d4c42e0516333e681d70": "H_s(t) = J(t)\\vec{\\sigma}_L \\cdot \\vec{\\sigma}_R .", "497a00e51b945c9591a4049c61e21353": " I = I_0 \\left( \\frac{ 1+\\cos^2 \\theta }{2 R^2} \\right) \\left( \\frac{ 2 \\pi }{ \\lambda } \\right)^4 \\left( \\frac{ n^2-1}{ n^2+2 } \\right)^2 \\left( \\frac{d}{2} \\right)^6,", "497a075274720085438084843329c6d0": "\\circledast", "497a40d59d6ea3f5333bc5841aee1550": "D_t(i)", "497a76c294f4977589e715009e146199": "\\frac{R_1}{R_2}=b\\left(\\frac{Rf_1}{Rf_2}\\right)^s", "497a810e5fcf2dafc1d5c8c4df4bdbcd": "\\mathbb{R}^{p}", "497a82499254c05aedb241bfd5222ef7": " \\Sigma_k (x_1...x_k) < \\Sigma_k (y_1...y_k) ", "497a946937ed45850a0329dc8cccbd2a": "5F_3^2=20\\equiv -1 \\pmod {7}\\;\\;\\text{ and }\\;\\;5F_4^2=45\\equiv -4 \\pmod {7}", "497aab4a2d5b306385afa824060466a4": "\\mathbf{y}_i = \\mathbf{H}\\mathbf{p}_i + \\mathbf{n}_i.", "497abf510f372437e98ed7745caee9ce": "q \\in Q, x \\in \\Gamma", "497b1b5e121bd9563d7794c795ae231d": "A=\\ell \\sum {\\epsilon c}", "497bd22c508a8ef9afab722ef805e775": " \\begin{align}\n \\mu (B) &= \\mathbb{P} ( X \\in B ) \\, , \\\\\n \\nu (B) &= \\mathbb{E} ( Y, \\, X \\in B )\n\\end{align} ", "497bf87149fa402c097e06d2251206bc": "\\frac{d}{dx}\\log_e x=\\frac{1}{x}.", "497c0606e64ede2697a75edde832ef0f": "f(A_r)", "497c09c0a2f997fecb6663a52006aae4": "F_{n+1}(a, b) = \\exp(F_n(\\ln(a), \\ln(b)))", "497c5420f445a63f29b08e01f0d5f89b": "\\tau=f_{x}\\cdot L_{0}-f_{y}\\cdot L_{0}\\sqrt{1-\\frac{v^{2}}{c^{2}}}=L_{0}\\left(f_{x}-f_{y}\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)", "497cd3f29ed40ea7f96fadc770bdaaae": "\\beta_1 = \\tau = ( \\theta_M + \\pi - s )", "497cff443ed325c378a747ca4a8a472a": "\\mathbb{E}[f(T)]\\geq p f(\\Omega)+(1-p) f(\\varnothing)", "497d3ee279663ec9a5580f6ba317f8ce": "\\mathrm{d \\theta}", "497d45c6affe323a97d182bbf83e2fbe": "\\pi_1(\\mathbf{C}\\setminus\\{0,1\\},z_0) \\to GL(2,\\mathbf{C})", "497d86675adcd456d1a1587bf147bdda": "{{|z_1-z_2|\\cdot |z_3-z_4|}\\over{|z_1-z_4|\\cdot |z_3-z_2|}}=-{{(z_1-z_2)(z_3-z_4)}\\over{(z_1-z_4)(z_3-z_2)}}.\\,", "497da835def1bd8bbeaab74ac3af01b2": "\\mathbf{T} = T^{ij}\\mathbf{e}_{ij}", "497dcca240469f7e9e4db47b68f7fb5c": "A=1,\\ldots, m", "497e4d9138780fe64591936d2ae46203": "\\Delta G_p = \\Delta H_p - T \\Delta S_p", "497edc4adc80c81aa66fa7664a7b67df": "\n \\mathsf{I} = g^{ij}~\\mathbf{b}_i\\otimes\\mathbf{b}_j = g_{ij}~\\mathbf{b}^i\\otimes\\mathbf{b}^j = \\mathbf{b}_i\\otimes\\mathbf{b}^i = \\mathbf{b}^i\\otimes\\mathbf{b}_i\n ", "497eefaa1bcbbb7449e29644a8d70ee5": " \\omega_1, \\dots, \\omega_n ", "497eefdb3c191d09e956765872a8018a": "G+*m", "497f1af87ab39f9fba76e9242af28489": "D^{1/2}", "497f658ed3c4447cb547391da532bf72": "\\mu = \\frac{m_A m_B}{m_A + m_B}", "497fa57699a2fa1dcaa93218620baca4": "K=\\sqrt{(s-a)(s-b)(s-c)(s-d)-\\textstyle{1\\over4}(ac+bd+pq)(ac+bd-pq)}\\,", "497fc18adfc2d545ed46b10c6a4f5b07": "\\frac{3^3 2^0 + 3^2 2^1 + 3^1 2^3 + 3^0 2^4}{2^7 - 3^4} = \\frac{85}{47}", "497fd1bba42026645340eaca53444b6b": "[x,x+dx]", "497fed2c709b6a959447a5c8f329136b": "(S, \\Phi)", "4980287a958bb7d80011a4b270f3d867": "\\mathcal{G}(\\omega_n)", "4980814d37d3349a6058863db66cb9f9": "L(\\mu)=\\frac{L_m + \\delta x_{m+1}}{L_n}", "49809a21b08b0060f93a05c0052fa306": "\\mathcal{C}\\subseteq\\mathcal{S}", "4980c9c57347de598dc29ccae141c8c6": "\ng_{1}(x) = 0.0520833\\, x -0.347222\\, x^{3} + 9.25186 \\times 10^{-17} x^{4} + 0.833333\\, x^{5} -0.555556\\, x^{6}\n", "4980e0261c4b26176940f074bf0f5e85": "R-S", "498141eb55363163810cd86c88f1ec75": "K\\ni u(t)\\quad\\perp\\quad F(t,x(t),u(t))\\in K^*. \\, ", "49819e763c604c458b063fe0e7bdad77": "\n\\epsilon(t) = \\int_{t_1}^t J(t,t^\\prime) \\mbox{d} \\sigma(t^\\prime) + \\epsilon^0 (t)\n", "4981ca1f73a8b1e6c6a8d6330aa0194b": "\\left\\{ \\sup_{0 \\leq t \\leq T} B_{t} \\geq C \\right\\} = \\left\\{ \\sup_{0 \\leq t \\leq T} \\exp ( \\lambda B_{t} ) \\geq \\exp ( \\lambda C ) \\right\\}.", "4981f20c3f117a49bf9611a53b6ecfe5": " D_{jj'} \\equiv U^{A}_{jj'} = E_{j}(0)\\delta_{jj'} + \\sum_{\\alpha\\beta} D^{\\alpha\\beta}_{jj'}k_{\\alpha}k_{\\beta}, ", "498214e4c6ad2c157b4e9641aa85993a": "W=\\int_{V_i}^{V_f} P\\,dV.", "49825384b07d1001b83b3a4eb7d316ea": "\n \\underline{\\underline{\\boldsymbol{K}}} = \\underline{\\underline{\\boldsymbol{A}^T_2}}~\\underline{\\underline{\\boldsymbol{K}}}~\\underline{\\underline{\\boldsymbol{A}_2}} = \\begin{bmatrix} K_{11} & -K_{12} & 0 \\\\ -K_{21} & K_{22} & 0 \\\\\n 0 & 0 & K_{33} \\end{bmatrix}\n ", "49827dbdfc250718f1f02837c6f706a6": "\\xi_{+1}(\\hat{z}) = \\begin{pmatrix}\n1\\\\\n0\n\\end{pmatrix} \\,", "498294f52878f94fed20225ae9e04064": "(\\downarrow 2)", "4982b0610d83b504ad2b8c58c444885b": "t\\;", "4982c72fa018142a57caa509da2bc00f": "\\mathrm{D} f (u_{0}) = \\lambda \\circ \\mathrm{D} g (u_{0}). \\quad \\mbox{(L)}", "4982c95f7f7d69bb4f60b16c588e30f1": "C_{t+1} = C_t +1 ", "4983157dcea63c1ed724aa12bc988dab": " \\chi_2(x) \\sim \\mathrm{Rayleigh}(1)\\,", "498386ca3725d37a21b7a27d27c1e506": "l_{mn_m}", "4983ea6ddd2130c96067bcdf3b6710e3": "\\displaystyle{2\\int_{|w-z|\\le 2\\varepsilon} |\\log|z-w|| \\,\\, |dw|,}", "4984115064468a2b5f13aad40921bcf6": "(\\alpha \\cdot f + \\beta \\cdot g)' = \\alpha \\cdot f'+ \\beta \\cdot g'", "49841e857a6d6c76a7cfbc2001a41b8f": "t=l-m", "4984202890b99dc2ed1b418ebf6f2c07": "X_H=\\sum p_i\\partial/\\partial q^i; ", "4984736b107272dabd015c3269aef84d": "\\epsilon \\in (0,1)", "498473a0ef51daa7664ca0e63b3b1b09": "\\sigma_A, \\sigma_B\\,", "498477dfd66fa2b7908f175182bf97d4": " \\prod_{i=0}^{\\infty} (1 - x^{2^{i}}) = \\sum_{j=0}^{\\infty} (-1)^{t_j} x^{j} \\mbox{,} \\! ", "4984c522b2d9512cccd3d5f4a784d305": "\\sum_{(c)}c_{(1)}\\otimes\\left(\\sum_{(c_{(2)})}(c_{(2)})_{(1)}\\otimes (c_{(2)})_{(2)}\\right) = \\sum_{(c)}\\left( \\sum_{(c_{(1)})}(c_{(1)})_{(1)}\\otimes (c_{(1)})_{(2)}\\right) \\otimes c_{(2)}.", "498516efccbcbd55b170c6adccc2b689": "U \\otimes U", "498536355e29bb8c2d53e3763ac63f45": "F = e + \\frac{1}{\\pi} \\int_{-\\pi/2}^{\\pi/2} e^{\\pi \\tan \\theta} e^{-e^{\\pi \\tan \\theta}}\\, d\\theta.", "4986436ef853f8342a7f754ecbfef3c1": " \\Phi(r)=0", "498653922593e6565717a22b4fce48a7": "\\phi_{sl,v}=\\frac{1}{1+\\frac{SG_{s}}{\\phi_{sl,m}}\\frac{M_{l}}{M_{s}+M_{l}}}", "49866b8cb98c0bb0ebc597f97a7a57b6": "\nz = a \\ \\frac{\\sin \\sigma}{\\cosh \\tau - \\cos \\sigma}\n", "49868ff6968264ac7f4aef376b4c1c05": "a_n^{k\\diamondsuit}\\,", "4986ce4db9ce081176b1efc32b83be8e": "\\sum_{j=0}^n\\frac{C_j^\\alpha(x)}{{2\\alpha+j-1\\choose j}}\\ge 0\\qquad (x\\ge-1,\\, \\alpha\\ge 1/4).", "4986fe5df7666b264a0bdf041cc34190": "\\bigcap_{i\\in I}N_i=\\{0\\}\\,", "4988753fd6c4a6638fc4694e07d2fa18": "N \\left ( 1 + m \\right )", "49887cbba4d77818919a7892fa5e01b0": "D(\\mathbf{X}, \\mathbf{Y}) =\\int_{\\Omega} \\int_\\Omega \\sqrt{\\sum_{i=1}^2|x_i-y_i|^2} F(\nx_1, x_2)G(y_1, y_2) \\, dx_1\\, dx_2\\, dy_1\\, dy_2.", "4988e8d041e646a20ab811664b175d23": "\\lambda_{k} = \\frac{p_{k}}{p} = \\frac{r_{k}}{r} = \\frac{(1 + r)^{k} - 1}{r} = k[1 + \\frac{(k - 1)r}{2} + \\cdots]", "4988eac6ce3b93a6e3a07accc2424f35": "(\\bar x - 0.98;\\bar x + 0.98) = (250.2 - 0.98; 250.2 + 0.98) = (249.22; 251.18).\\,", "49890286f264a3b8efa363896d9d2e47": "\\varrho_o", "49890470f8d1c041df0c50bb29d53d2f": " c_i(f) \\,", "49891cea87313ffcf0f1713fb5e24bbb": "\\textbf{Ra} = \\frac{\\rho_0 g \\beta \\Delta T L^3}{\\alpha \\mu}", "498942173e9588d90adf8d4e140ea9f3": "\\{(z,w)\\in \\mathbf{C}^2;~|z|<1,~|w|<1\\}", "49896fa48a5cdf6017ef860eca5ecac9": " \\theta = \\frac{\\left \\langle i \\right \\rangle}{N} ", "49897a5f5f20f780db69e131a01cd6f0": "n\\ = 10^m \\cdot n_m + 10^{m-1} \\cdot n_{m-1} + \\cdots + 10^0 \\cdot n_0", "4989c18022a1cda164edfff4be9a4787": "N_X Z", "4989f8c8d314981b429fadc3c48671e5": "\\pi(x)={\\rm Li} (x) + O \\left(x \\, \\exp \\left( -\\frac{A(\\ln x)^{3/5}}{(\\ln \\ln x)^{1/5}} \\right) \\right).", "498a5794eadc38b9523c7840cb8c3959": "\\scriptstyle F^{-1}(y)", "498a5959728e7d5580c7f743a99782c7": "(x-x_c)^2 + (y-y_c)^2 = r^2.\\,", "498a5bf6c43192079d84ac696fdcfff9": "\\vec v \\ast \\vec e_\\text{e}=0", "498ab1ea4aab0295389f6c51484cca6c": "HF", "498ad64271372a796a081a2b8c553510": "c - \\varepsilon ", "498aec3db9b244f71ebfb93037846846": "\\frac{1}{U_{exp}}", "498b0d4684eba45ebf388bdf63369814": " Ag^+ + e^- \\leftrightharpoons Ag(s)", "498b22028030a8be5142595a1e3c60a5": "O(|E|) = O(b^d)", "498be245e5a34a4485cbe616471543b6": "\\gcd{(q,m_p)}=1", "498bfc337463ae470efd72a9f2db014f": "A^\\mu \\rightarrow -A^\\mu", "498c9ef2e8dd26c8574dad46e3a2fd93": "\\text{grant}", "498da86ecbfde6774ccc39cece806b57": "\\gamma_{ik}=\\lambda_{il}\\beta^{-1}_{lk}", "498daa24a0f3b6003c743401534bad97": "C:\\mathcal{X}^*\\rightarrow\\Sigma^*", "498dc815e56b2ad303ab9e13b9edadce": " N(\\lambda x, \\lambda y) = \\lambda^n N(x,y)\\,. ", "498ddc00b6541e71564ecefe5055b76e": "\\left[{d^2 \\over dy^2} + 2 \\left(\\frac{l+1}{y}-y\\right)\\frac{d}{dy} + 2n - 2l \\right] f(y) = 0,", "498e1b979a9ff5d48d61cc6999874bf1": "fRep_{red} := \\{ (x, f) \\in X \\times \\mathbb{R}/R\\mathbb{Z} \\mid red(d(x) + f) = x \\}", "498e3a100c33d3622cbab618a048a541": "k \\nabla^2 h = -g \\rho", "498e68058cdef998412a5c99dcfb5735": "\\scriptstyle \\mathrm{X}", "498f4e3dc2903d7f819703b4e7b5ed4a": "\\,^{z_4 = x_4 y_1 + x_3 y_2 - x_2 y_3 + x_1 y_4 + x_8 y_5 - x_7 y_6 + x_6 y_7 - x_5 y_8 + u_4 y_9 + u_3 y_{10} - u_2 y_{11} + u_1 y_{12} + u_8 y_{13} - u_7 y_{14} + u_6 y_{15} - u_5 y_{16}}", "498f83ff8c7a8ce0bb1bd9a312ea216f": "\\|x\\|_p\\leq\\|x\\|_r\\leq n^{\\left(\\frac{1}{r} - \\frac{1}{p}\\right)}\\|x\\|_p", "498f85ed750c59bd869fab0742dd8239": " \n\\begin{pmatrix}\nv_1\\\\\ni_1\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n 0 & 0 \\\\ \n 0 & 0 \n\\end{pmatrix}\n\\begin{pmatrix}\nv_2\\\\\ni_2\n\\end{pmatrix}\n\n \n \\ . ", "498f8b69a1076465908ad1e40913873b": "S_{r_2} (r_1) > 0,", "498fa6eceff6820991fd3fd4e60738a2": "p(x)=z(x)^T Q z(x) ", "498fb89a6bbe68b7b54933cb1f1e6d53": "U=\\begin{bmatrix}1&2\\\\0&1\\end{bmatrix},\\qquad L=\\begin{bmatrix}1&0\\\\2&1\\end{bmatrix}.", "498fcbaa3b07a3e9be2d8a5697608a54": "\\int_{\\mathbb{R}^n} |f(x)|\\log^+ |f(x)|\\,dx < \\infty. ", "498fd49bc29f2cec38e68fbc56dac9df": "f,g:X\\to R", "49902126fb695e2902ccb9aa5d600843": "\\phi_{ij}", "49902a762b670496ff8aa8ff55c74f4d": "S_\\psi", "49903adf361533b50ab7b44771b327c2": "2 | 560", "49909061dea4f9b3d8febc7cc6e5bc1a": "y(t)=g_0(\\textbf{x},u)u(t)+ g_1(\\textbf{x},u){\\operatorname{d}^2u\\over\\operatorname{d}t^2}+ g_2(\\textbf{x},u){\\operatorname{d}^4u\\over\\operatorname{d}t^4}+ ... + g_m(\\textbf{x},u){\\operatorname{d}^{2m}u\\over\\operatorname{d}t^{2m}},", "49909df416666a790dd19d21d3ed0e58": " \\frac{\\frac{x}{y}}{\\frac{y+z}{y}} ", "4990c6f4c1fd1efab01a67e24bdda19c": " \\mathbb{C}[x,y] ", "4990df55ee30ee886fae68c6cb5a8ee0": "^\\mathbb{D}", "4990e66f606d1535a0b76597885512f1": "a(n) = 1", "4990ec7ae10df4f5045180758a577283": "\\Phi(v)", "49910587eb7a7869eb17e5cbc0e45fe9": " \\rho(J) = \\frac{E_0 * n * (\\frac{J}{J_c})^{n-1}}{J_c} \\, ", "49911150e2863b32440d1f8c58bdcb41": "-\\nabla f(x^*) = \\sum_{i=1}^m \\mu_i \\nabla g_i(x^*) + \\sum_{j=1}^l \\lambda_j \\nabla h_j(x^*),", "4991117bf2b4bc1d73b93bfc14b4cb6c": "\\left(1-\\frac{1}{3^s}\\right)\\left(1-\\frac{1}{2^s}\\right)\\zeta(s) = 1+\\frac{1}{5^s}+\\frac{1}{7^s}+\\frac{1}{11^s}+\\frac{1}{13^s}+\\frac{1}{17^s}+ \\ldots ", "4991985b0825da19c44f111a5b77348f": " k_c= \\frac {p_0 A}{S}(C^\\kappa-1) ", "49919d7f2b4746cd1c62210464a748c2": "\\mathbf{BB}^T", "4991e1ac0475f3a09fbf9cb69c4a150c": "\\operatorname{ev}_A:(R[A])[t]\\to R[A]", "4991f719ad0b004961b1cb5478d32c68": "n > 0 ", "4991fd49485ae322d61f246371701b24": "T\\in D'(\\Omega)", "49921f2582f1748d719d775ab738da55": "f_s\\equiv \\frac{V_s}{4\\pi r_u^3/3}", "49923f1116bd8f808b1784b7e5cac69a": "s=\\tfrac{1}{2}(a+b+c)", "4992d576a5ae43d602ea6015d6b35869": "\\mathcal{L}_{St}=-\\frac{1}{4}C_{\\mu \\nu }C^{\\mu\\nu }+g_XC_{\\mu }\\mathcal{J}_X^{\\mu }-\\frac{1}{2}\\left(\\partial _{\\mu }\\sigma +M_1C_{\\mu}+M_2B_{\\mu }\\right)^2.", "4992fb3413cbe1469b3240fed67b2ab1": "\n\\left[ Q_L(\\mathbf{p}^{\\prime}),\nQ_L^\\dagger(\\mathbf{p}) \\right]\n= \\delta( \\mathbf{p}^{\\prime} - \\mathbf{p})\n(1 -{\\overline \\Delta_{21}}(\\mathbf{p},\\mathbf{p})), ", "49938d9ef18d770d37bfe958d402ad62": "\\mathbb{Z}^*_n", "499394cc12336bd428337c781ac4bb90": " - \\frac {R_2}{R_2+R_f}", "4993999d631f416e5b859f684f24635d": "f|_{x_i = b} = f (x_1, \\ldots, x_{i-1}, b, x_{i+1}, \\ldots, x_n).", "499399a99bf7ae2f699ddef1327522b9": "I.M.A.= \\frac {Radius_{Wheel}} {Radius_{Axle}} ", "4993ae94fb5a7abcc67f3b9891d650ae": "P = C V^2 f", "4993d87f5104b5627c9803d1d83bed64": "f(1)=e", "499429d156823211d4d9f77328324234": "\\textstyle \\in\\Omega\\cap O_{i}", "4994624fe930c160b8242041171f0676": "b_k=1", "49947b9ad771ee852267fee13bbeea47": "\\left( \\{ x_{i \\, 1} \\}_{i=1}^\\infty, \\ldots \\{ x_{i \\, n} \\}_{i=1}^\\infty \\right)", "4994af2d88bd9cdfead16aef82724b1f": "\\det(S)^{-n/2} \\det(B)^{n/2} \\exp\\left(-{1 \\over 2} \\operatorname{tr} (B) \\right).", "4994b04b7987bf4abe5e4873849c01b3": "H = 2a\\, \\left( 1 + \\tfrac38\\, k^2 a^2 \\right).", "499529a02f0f39e83c3d25fc57bc73a0": "\\frac{dr}{dt_r}=\\plusmn 1-\\sqrt{\\frac{2M}{r}}.\\,", "4995881bb6d24e8eec0df421ddc519f5": "\\mathbf{\\dot{p}} = -\\frac{\\partial H}{\\partial \\mathbf{q}} \\quad \\mathbf{\\dot{q}} = + \\frac{\\partial H}{\\partial \\mathbf{p}} ", "49960a8eb074a661be25543016ec1d84": " \\ a_0^* \\ = \\frac{\\lambda_{\\mathrm{p}} + \\lambda_{\\mathrm{e}}}{2\\pi\\alpha},", "49965121d64bbf39de6ef2c75f746a27": "T^{2l}\\times SO(4l), T^l \\times Sp(l), T^2 \\times E_6,", "4996a260d02ab1b32b6825e046ce5e0b": "\\mathcal{N} (X) \\neq \\emptyset", "4996e97bcba141306f65c9c53a79cb73": "\nP=\\begin{pmatrix}\n1 & x^2 & x \\\\\n0 & 2x & 2 \\\\\n3x+2 & x^2-1 & 0\n\\end{pmatrix}\n=\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 2 \\\\\n2 & -1 & 0\n\\end{pmatrix}\n\n+\\begin{pmatrix}\n0 & 0 & 1 \\\\\n0 & 2 & 0 \\\\\n3 & 0 & 0\n\\end{pmatrix}x+\\begin{pmatrix}\n0 & 1 & 0 \\\\\n0 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}x^2.\n", "4996ebb12ca3cd02fae9f02bee9e7af4": "\\boldsymbol{q}=-\\frac{k}{\\mu}\\left(\\boldsymbol{\\nabla} P -\\rho \\boldsymbol{g}\\right)", "499759fc83f26f501d62e55e31f1d066": "a\\lor\\lnot a=1", "49976baa20c04a5e8369b23e1df1f7e0": "P, Q, M", "4997a165dc813586f0d296faa4ff2fe7": "\\bar{S}=X", "4997cde8d1088e5072764d0caa1eec2d": "G = \\frac {y_2}{x_1}", "49982de0c839d85d7b1020232ee6b3b8": "\\{v_1,v_2,\\ldots,v_n\\}", "49982e621c4f58e7fcdd293400e50e21": "\\Psi^{\\rm FQHE}_{\\nu}", "499898e0a42e55663c312077caa75fed": "pg(2^h-1,(2^h+1)(2^m-1),2^m-1)\\,", "49994102fb6234f9b458a9b2f191db89": "R = {V \\over I} = {1 \\over {C_S f}}.\\ ", "49994930bf3e257f68536399eb7bc055": "\n (P_0) \\quad \\min \\limits _x \\|x\\|_0 \\qquad \\text{subject to } y = Dx\n", "49997e0df4a6e2c8c3fa81da98ede3c5": "\\forall x \\Box Fx \\rightarrow \\Box \\forall x Fx", "4999a027c86ca99932dedcb4a236d901": "M=\n\\frac{1}{4}\n\\begin{pmatrix}\na_{00}+a_{11}+a_{22}+a_{33} & +a_{10}-a_{01}-a_{32}+a_{23} & +a_{20}+a_{31}-a_{02}-a_{13} & +a_{30}-a_{21}+a_{12}-a_{03} \\\\\na_{10}-a_{01}+a_{32}-a_{23} & -a_{00}-a_{11}+a_{22}+a_{33} & +a_{30}-a_{21}-a_{12}+a_{03} & -a_{20}-a_{31}-a_{02}-a_{13} \\\\\na_{20}-a_{31}-a_{02}+a_{13} & -a_{30}-a_{21}-a_{12}-a_{03} & -a_{00}+a_{11}-a_{22}+a_{33} & +a_{10}+a_{01}-a_{32}-a_{23} \\\\\na_{30}+a_{21}-a_{12}-a_{03} & +a_{20}-a_{31}+a_{02}-a_{13} & -a_{10}-a_{01}-a_{32}-a_{23} & -a_{00}+a_{11}+a_{22}-a_{33}\n\\end{pmatrix}\n", "4999be3f936e46ea6ff387c0bea513c5": "\\bar\n\\epsilon_{sh}(t,t_0)", "4999e6e328bf1491817f6423b3fd864c": "A = 4\\varepsilon \\sigma^{12}", "499a2fb0deae85d76058b3d5a19a64f4": "F=\\int\\!\\left(\\frac{A}{2}\\phi^2+\\frac{B}{4}\\phi^4 + \\frac{\\kappa}{2}\\left(\\nabla\\phi\\right)^2\\right)~dx\\;.", "499a89fd464daf95bc6ddb7debc66cdb": "E_f=\\left\\{x\\in X: f(x)>r\\right\\} \\, ", "499b0ce62b6cb4868e295a3bef2d74aa": "\\langle s_n \\rangle^\\omega_{n=1}", "499b1a6f2fd494b5eab66c3df71a2ccf": "\\sum_{i=1}^n (x_i-\\mu)^2 = \\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2", "499b3d8e418f271cc044584d95b16fcc": "\\sec A = \\frac {1}{\\cos A} = \\frac {\\textrm{hypotenuse}} {\\textrm{adjacent}} = \\frac {h} {b}. ", "499b493526bb9b2ac726310b840b2e26": "h = \\frac{dzd\\bar{z}}{(1+|z|^2)^2}.", "499c8de4eb4579edb8462fbedc7410f3": "\n \\cfrac{i\\omega \\hat{p}}{\\kappa} = \\cfrac{\\partial \\hat{v}_r}{\\partial r} + \\cfrac{1}{r}\\left(\\cfrac{\\partial \\hat{v}_\\theta}{\\partial \\theta} + \\hat{v}_r\\right) + \\cfrac{\\partial \\hat{v}_z}{\\partial z} ~.\n ", "499d094021b419f5d4762edd73365109": "\\mbox{ASAI} = \\frac{\\sum{N_i} \\times 8760 - \\sum{U_i N_i}}{\\sum{N_i} \\times 8760}", "499d22af1d96aa302e7d6f58f98235de": "B \\rightarrow A: \\{N_B\\}_{K_{AB}}", "499d34d232c0d5e3f54d73df75da3ba3": "T_l", "499d36caf339c6e36a4e2fc4a1176ad1": "\\mathbf r(0) + \\left(s-\\frac{s^3\\kappa^2(0)}{6}\\right)\\mathbf T(0) + \\left(\\frac{s^3\\kappa(0)\\tau(0)}{6}\\right)\\mathbf B(0)+ o(s^3)", "499d48db69133f45ea7764270b957e97": " \\left( \\sum_{k=1}^n a_k^2\\right) \\left(\\sum_{k=1}^n b_k^2\\right) = \n\\sum_{i=1}^n \\sum_{j=1}^n a_i^2 b_j^2 \n= \\sum_{k=1}^n a_k^2 b_k^2 \n+ \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i^2 b_j^2 \n+ \\sum_{j=1}^{n-1} \\sum_{i=j+1}^n a_i^2 b_j^2 \\ ,", "499d7fe762a8d81befa5d5578bbcbf90": " \\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))", "499d872778d6b587ea42ae7227de6916": "\\lim_{x \\to a^{-}} f(x)=\\pm\\infty", "499d918da0ee0dc606710e9bf04bc3c3": " \\scriptstyle R", "499da2e66a75d05bf11121b674c41a82": " (ru)_{tt} -c^2 (ru)_{rr}=0; \\,", "499dbe8ab03d88a69fdd4e47eb73b630": " \\epsilon \\frac{\\mathrm{d}^ny}{\\mathrm{d}x^n} + a(x)\\frac{\\mathrm{d}^{n-1}y}{\\mathrm{d}x^{n-1}} + \\cdots + k(x)\\frac{\\mathrm{d}y}{\\mathrm{d}x} + m(x)y= 0", "499dc244c281a64c41768f4346a7bbad": "I_\\max", "499e36f5736d8be4842efa7e661d8770": "r(\\pi,\\delta)=\\operatorname{E}_{\\pi(\\theta)}[R(\\theta,\\delta)].\\,\\!", "499e4eb57fced690bb8c0f26f2af7ce4": "\\left(\\frac{\\alpha}{\\beta }\\right)_m = 1.", "499e6234cba0baa6f7adbd19683bfe48": " mk = O(n)", "499ea482f0642e2bd203a3f560d4dd97": "k_{B}", "499eb2b09943008868a9a9cd6103a9c8": "id_A", "499ed7350bbc663a3023b06af9642ac9": "\\omega_c=1", "499f31fc13d645269d5a7b3e45bd9ae2": "{{If} f > f_{st}, f=\\frac{-{sm}_{fu, 1}+m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}", "499f3f8ccd20668f493fa0f1056740e5": "\\mathbb{E}\\left[t\\right]=\\tau", "499f82bdc9d93638361283ee378190ed": " (x_0, y_0,u_0,p,q)", "499fb63c16e6c4d0e66e5162883ce0ce": " |\\psi\\rang |\\phi\\rang \\rightarrow \\sum_n c_n |\\chi_n\\rang |\\phi_n\\rang, ", "499fd2e5e361b4f3c7113d4df06d41d4": "A_m(1,6) = 1,6,21,56,126,252,462,792,1287,2002,\\ldots", "49a0000a2bbb3b94be12f244684289cd": "G = \\oplus_i C_{d_i} \\ ", "49a074bbbf9c6632f34146db06e889bc": "[\\mathcal{F}_n(f)](x) = \\frac{1}{n\\gamma_n\\sqrt{2\\pi}} \\sum_{k=-\\infty}^\\infty {\\exp{\\left({\\frac{-1}{2\\gamma_n^2} {\\left({\\frac{k}{n}-x}\\right)}^2 }\\right)} f\\left(\\frac{k}{n}\\right)}", "49a080cd433f3f196eda7715a04284bf": "2\\sqrt{n/4}", "49a0c4412a8d6dc55601e65a3d2388e1": "1-F_t = P_t = P_0\\left(1-\\frac{1}{2N}\\right)^t. ", "49a12ba4c3e7dcb2a56081b67324dc91": "\\sqrt{a^2 + b^2}", "49a1412c709a7f47d458e5000260bcc5": "E\\{x|y\\} = W y + b", "49a15b7700c32f4bf21abaa486f9acd1": "\\overline{-s}", "49a19402d8136716b96701af85a42b39": "\\scriptstyle \\pi_2(SO(3)/SO(2))=\\mathbb{Z}", "49a1c4dc1630c3dea74de442c64a04ab": "\\lambda = 2\\, \\Delta\\, K(m) = \\sqrt{\\frac{16}{3}\\frac{m}{H}}\\; K(m),", "49a20e6c31d5bd9120d9ee911522a417": " \\Omega=\\frac{1}{\\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{\\,dt}{(e^t-t)^2+\\pi^2}}-1 .", "49a220dae13dba00566a5e5f03d79de3": " \\mathit{I}_T = \\{ p \\in \\mathbf{F}[t] \\; | \\; p(T) = 0 \\} ", "49a29160855572fd011a20848ed034a6": "\\frac{b}{a}=\\varphi", "49a2c2ce4e0bccb1780c6c930f43d655": "A,B \\subseteq \\mathbb{N}", "49a343125be463347a60f280fdde63df": "\\{ -1, 1 \\}", "49a3437800de359cbf6939e5e4ca041c": " \\begin{bmatrix} x \\\\ x^2 \\end{bmatrix} ", "49a37aaf8e83a6a946d7736e54e28dd8": "\n\\nabla^{2} \\Phi = \n\\frac{4\\sqrt{S(\\lambda)}}{\\left( \\lambda - \\mu \\right) \\left( \\lambda - \\nu\\right)}\n\\frac{\\partial}{\\partial \\lambda} \\left[ \\sqrt{S(\\lambda)} \\frac{\\partial \\Phi}{\\partial \\lambda} \\right] \\ + \\ \n", "49a3887a95fc2037df7b02fd70980db9": "|E|= \\sqrt{E_x^2+E_y^2+E_z^2}", "49a3dcc17b06a56a24c50e417201abca": "\\{ V_1, \\ldots, V_e \\}", "49a429af146fbd8a9381dcb4822c26f4": "\n\\text{PWI} = 100 \\times \\max_{i=1\\dots N \\atop j=1\\dots M}\n\\left\\{\\left|\n\\frac{\\text{measured value}_{[i,j]} - \\text{average limits}_{[i,j]}} {\\text{range}_{[i,j]}/2}\n\\right|\\right\\}\n", "49a44f77a6e85e617a7dc72d3cb5d348": " g=(1,0,1,-\\frac 1 2, \\frac 1 3 , - \\frac 1 4 , \\frac 1 5 , - \\frac 1 6 , \\cdots) ", "49a4b8597f8df303ebdfa624b9578f8c": "\\begin{align}\n V_{in}(s) &= V\\frac{1}{s} \\\\\n V_L(s) &= V\\frac{sL}{R + sL}\\frac{1}{s} \\\\\n V_R(s) &= V\\frac{R}{R + sL}\\frac{1}{s}\n\\end{align}", "49a4ba94da7c96b95ff84d13317a1267": "\\det(A + \\epsilon X) - \\det(A) = \\operatorname{tr}(\\operatorname{adj}(A) X) \\epsilon + O(\\epsilon^2) = \\det(A) \\operatorname{tr}(A^{-1} X) \\epsilon + O(\\epsilon^2)", "49a4bc709626c06ee95b0d93b130198f": "b = \\frac{V_c}{3}.", "49a4dfc4489af6c652319c3e66234ef0": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} =\n -\\cfrac{2Eh^3}{3(1-\\nu^2)} \\begin{bmatrix} 1 & \\nu & 0 \\\\ \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix} \n \\begin{bmatrix} \\varphi_{1,1} \\\\ \\varphi_{2,2} \\\\ \\frac{1}{2}(\\varphi_{1,2}+\\varphi_{2,1}) \\end{bmatrix} \\,,\n", "49a5119833c43c6db31ef1d0bfdef512": "a \\times a \\times a", "49a597b15a12c0909f1e9d133047d698": "(P \\or Q) \\Leftrightarrow (Q \\or P)", "49a5a81f28fd01dce862db5de9174b5d": "e^x=1+x+\\frac{x^2}{2}+O(x^3) \\,, \\qquad\\text{as } x\\to 0 \\,, ", "49a5bd4ddaa39233bae3ecaeeff8c1ae": "f:\\mathbb{Z}\\to\\mathbb{Z}", "49a5be772245fbeaa48c17d6fb964687": "C_1(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iint \\dfrac{\\phi_1(\\theta,\\tau)}{\\phi_2(\\theta,\\tau)}M_2(\\theta,\\tau)e^{-j\\theta t-j\\tau\\omega}\\, d\\theta\\,d\\tau", "49a5f058bfc02ad1a0da10ae276d6723": "\n\\begin{array}{cccccccc}\n\\scriptstyle 2n-1&&&&&& \\\\ \n\\scriptstyle 2n-3&\\scriptstyle 2n-3&&&&& \\\\ \n\\vdots&&\\ddots&&&&\\\\\n9&\\cdots&\\cdots&9&&&&\\\\\n7&\\cdots&\\cdots&7&7&&&\\\\ \n5&\\cdots&\\cdots&5&5&5\\\\\n3&\\cdots&\\cdots&3&3&3&3\\\\\n1&\\cdots&\\cdots&1&1&1&1&1 \\\\ \\hline\n\\scriptstyle =n^2& \\scriptstyle =(n-1)^2&\\cdots& \\scriptstyle =5^2& \\scriptstyle =4^2& \\scriptstyle =3^2& \\scriptstyle =2^2& \\scriptstyle =1^2\n\\end{array} \n", "49a6048211d70a3cb538f2275a722c97": "\\scriptstyle\\frac{v^2} {R}", "49a72a3a1320cee5aefcc681c7030114": "\\langle s(x_1),\\ldots,s(x_n) \\rangle \\in R", "49a77286fe0212c6b6c95a7cb2541ead": "{{}^*C}_{abcd}", "49a7920c29a89472e96f85f89f35756c": " a \\mid (b + c)", "49a7e246f16b6659cf94fa1edcb71725": "U_{tot}(\\mathbf{r})=U_{phys}(\\mathbf{r}) +U_{bias}^{LE}(\\mathbf{Q};t) ", "49a7ef08db1d1986aa8a7a91cb3a7250": " a_{i_1}\\wedge a_{i_2}\\wedge\\cdots\\wedge a_{i_k}", "49a7fe6c1de18f788b36bb219dd3782a": "P_{40}=440(\\sqrt[12]{2})^{(40-49)}\\approx 261.626~\\text{Hz}", "49a84e0f6330bc98a38de2b6ae4e3db7": "E_{pot} \\propto h", "49a86fd4f4200bd9e1ac5761d27f6880": "\\exists !", "49a91669ae2c6fc7e7a475bdbfe0c48b": "\\frac{6}{12}", "49a92844e4ce91ce70e24ab58865b322": "\\begin{align}\n x &= \\lambda \\cos \\varphi_1\\\\\n y &= \\varphi\n\\end{align}", "49a9ad98efe9e29ca8ebb656dd358bfb": "Reaction~Time = Movement~Time + \\frac{ \\log_{2}(n) } {Processing~Speed }", "49aa20fdb2aeb89d476a6eb555ead4ec": " \\Delta E = | E_i - E_s | = \\hbar \\omega ", "49aa2b5fcc28b1e7dfe1d503c2c1df80": "\n\\delta \\varphi \\approx \\frac{ 6\\pi G(M+m) }{ c^2 A \\left( 1- e^{2} \\right)}\n", "49aa3685ed780b30f050c1da6e9ef0e3": "f_{W}(w; \\theta)", "49aa75f06143d8d5b1e564a4c2413b88": " \\dot{m_s} ", "49aa9e075f238479fdf2e00ab0f304a9": "\\int r^2\\,dm ", "49aaeaaa767c2cb3bdeae68589d9c959": "\n M_{xx} := \\int_A z~\\sigma_{xx}~\\mathrm{d}A ~;~~ Q_x := \\kappa~\\int_A \\sigma_{xz}~\\mathrm{d}A\n", "49ab0482f817997b7b95e87ede770e22": "CH_4 +h\\nu \\rightarrow CH_3 + H", "49ab07135e83b7059cf165a7634cea60": "\\mathcal{M}_R = \\Sigma^*/\\stackrel{*}{\\leftrightarrow}_R", "49ab159afc76f78f0d2c360e00f297b2": "\\mathrm{Th}_{\\Pi^0_2}(\\mathbb N)", "49ab2f3758bc9eb1e78063653b45a9e7": "x=\\pm a", "49ab705a184eafc9e10a110c95de26f8": "B = m_1 + n - N -(m_1+n+2)\\omega", "49abc0a7150343c01108dc35eb14eb2b": "{ E = \\omega } \\ ", "49abd80a62be96acdbe8c6ab3edc202b": " \\displaystyle = \\frac{D_r + D_{r+1}}{2} + \\frac{t}{h}\\frac{D_{r+1}-D_r}{2}.", "49abef69e206ff605f42b97c1323ddfc": "S^{**} \\subseteq S^*", "49ac19d7221d551810235d60dd519649": "\\Delta \\bar \\nu", "49ac27c188d7f5df1cadf8a6b68199ab": "\\left[ C \\right]=\\left\\{ \\begin{array}{*{35}l}\n \\left[ A \\right]_{0}\\left( 1+\\frac{k_{1}e^{-k_{2}t}-k_{2}e^{-k_{1}t}}{k_{2}-k_{1}} \\right)+\\left[ B \\right]_{0}\\left( 1-e^{-k_{2}t} \\right)+\\left[ C \\right]_{0} & k_{1}\\ne k_{2} \\\\\n \\left[ A \\right]_{0}\\left( 1-e^{-k_{1}t}-k_{1}te^{-k_{1}t} \\right)+\\left[ B \\right]_{0}\\left( 1-e^{-k_{1}t} \\right)+\\left[ C \\right]_{0} & \\text{otherwise} \\\\\n\\end{array} \\right.", "49ac4d65eac52c7117ce95eb3520143d": "\\dot x = \\mu \\left(x-\\frac{1}{3}x^3-y\\right)", "49ac5bae71395c575d007d8053a9c035": "M \\to R \\otimes_R M \\xrightarrow{f \\otimes \\text{id}_M} S \\otimes_R M \\xrightarrow{v} N", "49ac82ed498d63ad8609ee0a23e78298": "mc\\frac{du^\\alpha}{d\\tau} = F^{\\alpha\\beta} q u_\\beta,", "49ace31d719d9fa710ca661771d7a609": "\\ H(s)=\\frac{Y(s)}{X(s)}", "49acec383009986068575e7d98a0c88d": "y_L, y_F, y_P\\geq 0", "49acfcabaf7e26fc3b1cdb4e740a9107": " f(T) e_i (T) = (T - \\lambda_i)^{\\nu_i} e_i (T)", "49ad4f1ad172eca789a6fab5abd7c807": " x^T A^T A x + b^T x + c \\leq 0. ", "49ae7fee7d6b6b85c8088a0559b50cb0": "\\,^{z_9 = x_9 y_1 - x_{10} y_2 - x_{11} y_3 - x_{12} y_4 - x_{13} y_5 - x_{14} y_6 - x_{15} y_7 - x_{16} y_8 + x_1 y_9 - x_2 y_{10} - x_3 y_{11} - x_4 y_{12} - x_5 y_{13} - x_6 y_{14} - x_7 y_{15} - x_8 y_{16}}", "49ae952f3a71fcdfece7d54d1cd0a166": "x=e^t", "49aeff04bcfb18d062f01dd65de02956": "\\frac{d[S]}{dt}=-k_S[S]\\implies S=\\frac{1}{2}e^{-k_St}", "49af3057809d7f1433a0abb4ad100fb2": "\\lambda_\\mathrm{defect}", "49af35ed816305c233a005edc2ddf170": "\\frac{dx}{dt} = x^2 + I", "49af410d72e66d0af1f4102555d81d23": "\\int_0^\\infty \\frac{\\sin x}{x}\\,\\mathrm{d}x", "49af6f1158e982df33af964cd6cce71f": "d = 19", "49af74deb0034a7a0e6458c240d43866": "x \\equiv \\pm a^{m+1}", "49b015986c64690b283c30294c67cf9e": "\\rho_3={}", "49b039f27ad643869e20bb23d3dc2346": "\\partial_t L = \\frac{1}{2} \\nabla^2 L,", "49b07106bbfa1c64dd3f2c21d5f90b58": " {X \\over M_{pl}} H_u H_d", "49b08526df3ada0005c7de94e09665d1": " u^n_{i-1} ", "49b0e954e304a740c242c3d4a3a95b27": "U_I", "49b1d242fbf2082ed17d9292ddd7c656": "p(x) = (x-x_1)(x-x_2)\\cdots(x-x_n) \\, ", "49b1f6ff6757cd332adc5fa85f0b6ea2": "D_h = \\sqrt{2 \\times H \\times R_e}", "49b1fe6fc99c3414e0be731753ec1a40": "U_{A \\to B \\to A} = (b - a) + (a - b) = 0. \\,", "49b21081313e5891fde4b250cd614dd2": "1 > \\sqrt{2} - 1 > 0", "49b21cd5bd8ca4aa184c545ba3f78106": "a'_i = 0\\,", "49b2370201b1e30c9316892e937d4197": "\n \\varepsilon_{\\alpha\\beta} = \\frac{1}{2}(u_{\\alpha,\\beta}+u_{\\beta,\\alpha})\n - x_3\\,w_{,\\alpha\\beta} \\,.\n", "49b237f4c6721af050a3412771e4cbde": "C_\\phi (f) = f \\circ\\phi", "49b240c591f01a226082c8dc78654e3d": "\n \\sum_{J=|j_1-j_2|}^{j_1+j_2} (2J+1) = (2j_1+1)(2j_2+1).\n", "49b2e60955c9047972cf46b4d1a6d153": "x(t)=\\frac{1}{\\omega_D}\\int_0^t{\\bar{p(\\tau)}e^{-\\xi\\omega(t-\\tau)}sin(\\omega_D(t-\\tau))d\\tau}", "49b311da0ed4baee4da4dd66e37ef59c": "(x, y)", "49b355ecfcaa2ee497df4f419fc0dbed": "F_{mag dir}=(H_{mag}(R), H_{dir}(R))", "49b38490ae4e6f0c43e7d009705a0e4c": "\\Lambda=\\{m\\omega_1+n\\omega_2 \\,\\,|\\,\\, m,n\\in\\mathbb{Z} \\}", "49b3dcc45223996eb5f7e90d064cd1cd": "\\mathbf x \\times \\mathbf y =\n(x^2 y^3 - x^3 y^2) \\frac{h_2 h_3}{h_1} \\mathbf e_1 + (x^3 y^1 - x^1 y^3) \\frac{h_1 h_3}{h_2} \\mathbf e_2 + (x^1 y^2 - x^2 y^1) \\frac{h_1 h_2}{h_3} \\mathbf e_3", "49b403e0420b3deed739cf7d863e2a9e": "\\Delta p = \\Delta \\psi - 59 \\Delta pH", "49b46604df08c011218d67672099bb2e": "\\pi_1 - \\pi_0 = \\sum_{\\omega \\in S_1} \\sum_{\\alpha,\\alpha'} \\Pi(A, \\omega, \\alpha, x)\\Pi(A, \\omega, \\alpha', x) - \\sum_{\\omega \\in S_0} \\sum_{\\alpha,\\alpha'} \\Pi(A, \\omega, \\alpha, x)\\Pi(A, \\omega, \\alpha', x).", "49b4842b079f81e20edba3ab298ad46b": "R_\\text{q} = \\sqrt{ \\frac{1}{n} \\sum_{i=1}^{n} y_i^2 }", "49b49409d1fae8471fc8cbbd15a086a4": "\\,\\!1\\text{Re}(s_0)", "49c04d77a60dcfbbb4c72ab7f0ac77f2": "Q=qN_Aw \\,", "49c05f53659e02924add5a5121e9699a": "t_2^\\prime", "49c0a9fae44b6f79b714ab91589734c0": "\\scriptstyle \\leq2.4\\times10^{-15}", "49c0c377c912385dda0eec6b2d4bc31a": "N_\\uparrow/N", "49c0d1addea49732a5d468f5e1842ba9": "z_n = f_c^n(z_0). \\,", "49c1359be44f227a12c364956258ac87": "q_{t+1} = 2q_{t} + p_{t} \\mod N", "49c143a81d335300faf63496b26a00ec": "J=- D {\\partial n}/{\\partial x}", "49c1526135c646cb55a3333b42165db7": " \\phi(t,iw) = iw(\\tau - i\\int_{0}^\\tau h(s) ds) \\quad(2.11.b)", "49c182fe80e04bfffc4a1b1fb373c1af": "T_C = \\csc^2{\\left( A/2 \\right)} : \\csc^2{\\left( B/2 \\right)} : 0", "49c1bc6c3c0554e4b67ab67e0be63214": "r(\\theta )=\\frac{P(\\theta )+Q(\\theta )}{R(\\theta )}", "49c1deb5c44b57c99879fb1cdfa2965f": "100 - x = 100 - 50 = 50 ", "49c2330615642d44b007443e0fccbac7": "B(x;r)=\\{y\\in M|d(x,y)0.", "4a22fb61ef7235e14aa57c5c318888a2": "\\mathfrak{so}(n,\\mathbb C)", "4a230644d2d6885250f22147a5bf04b1": "\\varphi_{\\beta+1}(0) \\,,", "4a234d2bb95b9a3c53ebbb1788625bca": " f(s) \\ ", "4a2395114d2327b6e9c9e68c683fd350": "r = m - n + c", "4a23cdb95cceee5db0ab49ac463fe664": "f_n=\\sum_{k=1}^{2^{2n}+1}\\frac{k-1}{2^n}{\\mathbf 1}_{A_{n,k}}", "4a23f0a4db985c6d4ad40d88cf6b813e": "\\frac{s}{t} \\leq \\frac{f(a)}{f(b)} \\leq \\frac{s+1}{t}.", "4a24332efe340cf82d7b32dfcdcf7e03": "\\textbf{G}_c\\textbf{G}\\textbf{H} = K\\textbf{G}\\textbf{H}", "4a24a0d8c41c7a2ffbd336cf33660855": "\\tau_c*=0.06", "4a2538c18551084ed589a9c73be9e8ee": "\\begin{align}A = \\frac12 ( a_1[a_2 \\sin(\\theta_1) + a_3 \\sin(\\theta_1 + \\theta_2) + \\cdots + a_{n-1} \\sin(\\theta_1 + \\theta_2 + \\cdots + \\theta_{n-2})] \\\\\n{} + a_2[a_3 \\sin(\\theta_2) + a_4 \\sin(\\theta_2 + \\theta_3) + \\cdots + a_{n-1} \\sin(\\theta_2 + \\cdots + \\theta_{n-2})] \\\\\n{} + \\cdots + a_{n-2}[a_{n-1} \\sin(\\theta_{n-2})] ). \\end{align}", "4a2549aa4f93b977b181f193fae7dba1": "~u~", "4a2570c109f12b4c39284dd0a0d35916": " r \\cdot s = \\sum_{i=1}^s r = \\sum_{j=1}^r s ", "4a26199ba65f5e9fe48d6b8eb1b24f45": "[\\varnothing]_{\\text{seq}} = \\varnothing.", "4a267e17067c0b495440f6e18a4ab923": "\\int_0^\\infty \\frac{ \\sin mx}{x(x^2+a^2)}\\ dx=\\frac{\\pi}{2a^2}(1-e^{-ma})", "4a26dc771506f828bb6c0c36f33f8410": "\\gamma\\in C_\\beta", "4a26f03764c77c32b9d73ef4b62e9067": "\\zeta(X,q^{-n}T^{-1})=\\pm q^{\\frac{nE}{2}}T^E\\zeta(X,T)", "4a27320a9aa26bd775f76c29f1ee7922": "\\mathbb R^d", "4a27c8cd118e1356f6bd1e652f9ebb4c": " S = \\mu B + gA", "4a27ed341801378862873dd4c0843c10": "~r_0(C,k)~", "4a282f761c5427863993667e3e2a61d2": "\nf(x_1,\\dots,x_n) = \\sum_{j_1,\\dots,j_n = 0}^{\\infty}a_{j_1,\\dots,j_n} \\prod_{k=1}^n \\left(x_k - c_k \\right)^{j_k},\n", "4a28a7ea2d0b595bd4d593c452d46890": "\nH = \\tfrac{1}\n{2}J\\int {d^d x\\sum\\limits_\\alpha {(\\nabla \\sigma _\\alpha )^2 } } + \\ldots.", "4a2997949f18c218d4b47e9625e2eb11": "\na^{\\varphi(n)}\\equiv 1 \\pmod{n}.\n", "4a29f21b6872ecdd9040d47a5111e80f": " F(\\theta)=f(\\cos\\theta) \\,", "4a29f30708fbf97467ecbd587f9f9795": "(d\\phi/dx)^2", "4a29f32be6240f7a22ed59abbd45e49e": "\n\\begin{align}\n\\langle H^{\\mathrm{kin}} \\rangle &= \\frac{1}{2m} \\langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} \\rangle\\\\\n&= \\frac{1}{2} \\biggl(\n\\Bigl\\langle p_{x} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{x}} \\Bigr\\rangle + \n\\Bigl\\langle p_{y} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{y}} \\Bigr\\rangle +\n\\Bigl\\langle p_{z} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{z}} \\Bigr\\rangle \\biggr) = \n\\frac{3}{2} k_{B} T\n\\end{align}\n", "4a2a12ab8fa44cd682a96fba19befbae": "u_{j} \\rightharpoonup u \\mbox{ in } X \\mbox{ as } j \\to \\infty", "4a2a26e4ecbac15fc00616e88d9e77bc": "x := x+1\\,\\!", "4a2a6aff3cbfac5956faf5bdd49bbc61": "\\frac{p(X)}{q(X)},", "4a2a9f70eefcbdd359d4893d958b0ded": "\n\\begin{align}\n& \\left\\langle\\phi|H|\\phi\\right\\rangle \\\\\n& = \\left\\langle\\sum_n c_n \\psi_n |H|\\sum_m c_m\\psi_m\\right\\rangle \\\\\n& = \\sum_n\\sum_m \\left\\langle c_n \\psi_{n}|E_m|c_m\\psi_m\\right\\rangle \\\\\n& = \\sum_n\\sum_m c_n^*c_m E_m\\left\\langle\\psi_n\\mid\\psi_m\\right\\rangle \\\\\n& = \\sum_{n} |c_n|^2 E_n.\n\\end{align}\n", "4a2ab102fb8a6f39d582e24a273af874": "T \\in \\mathbb{R}", "4a2ae4e03b448626fdc0979ab01bab45": "C_n\\left(X\\right)\\rightarrow C_n\\left(Y\\right)", "4a2b078571a96f1975af6cf159b77e0b": "\\tau_{xy}", "4a2b2619849bf1e5ef44b75c0f78bb0e": "\\Pr\\left( \\sum_{i=1}^n X_i > t \\right) \\leq\n\\exp\\left( - \\frac{n\\sigma^2}{a^2} h\\left(\\frac{at}{n\\sigma^2} \\right)\\right),", "4a2c9061790d602da37a6be7a800a7b9": "\\mathbb{S}^{4n+3}", "4a2c95b157a4d1cc8ccf557f3fa6d6ee": " \\partial(u) = u ", "4a2ca2e9b855615a85967120f8121692": "(66 + \\lfloor \\frac{66}{4} \\rfloor) \\bmod 7 = (66 + 16) \\bmod 7 = 82 \\bmod 7 = 5", "4a2cf1454f88bd6bccf2f7535176726a": "z'_0 = 1", "4a2d4785adcb4ba1d24a21e4058d537e": "\\left [s(nT)\\cdot (-j)^n\\right ].", "4a2d9207b7d142dc814d1be5bbf134cf": "v_{4}", "4a2dce0d734a4b8138592f14cc85039b": "\\frac{C_{P}}{C_{V}}=\\left(\\frac{\\partial P}{\\partial V}\\right)_{S}\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}=\\frac{\\beta_{T}}{\\beta_{S}}\\,", "4a2df0791390a4825db7638bb9c5fe34": "\\scriptstyle \\phi", "4a2dfabd31e9a10a21914a5b13f8b315": "L^1([a,b])", "4a2e8cbb5a31c33093578d16804f5fb8": "2\\cdot 3\\cdot r", "4a2ea34b4da32cacbd97fb18726c7a59": "P = V(+\\infty) = V(a_0, b_0, c_0, d_0, \\dots)\\,", "4a2f250457d62ad6c95e54a09ede071a": "\\psi_L= \\frac{1-\\gamma^5}{2}\\psi, \\qquad\\psi_R= \\frac{1+\\gamma^5}{2}\\psi ", "4a2f78bf6de2a23b1a88bbcb88237ac6": "\\displaystyle v(x)", "4a2fb57bfb2167939ccad8c1ba2bc936": "\n\\begin{align}\n& \\mathbf{y}_n = \\mathbf{A}\\mathbf{x}_n+ \\mathbf{B}\\mathbf{z}_n +\\mathbf{c}+\\mathbf{e}_n \\\\\n= & \\begin{bmatrix}\n\\mathbf{A} & \\mathbf{B}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{x}_n \\\\\n\\mathbf{z}_n\\end{bmatrix} +\\mathbf{c}+\\mathbf{e}_n \\\\\n= & \\mathbf{D}\\mathbf{f}_n +\\mathbf{c}+\\mathbf{e}_n\n\\end{align}\n", "4a2fcaf9553fb909e5612132736a6d3c": "{\\mathbf{H}}({\\mathbf{r}})=-\\nabla\\psi=\\frac{1}{4\\pi}\\left(\\frac{3\\mathbf{r}(\\mathbf{m}\\cdot\\mathbf{r})}{r^{5}}-\\frac{{\\mathbf{m}}}{r^{3}}\\right).", "4a300cb7dd98bee27e74cacd860b24fd": "\\mathcal{F}\\left\\{f(t)\\right\\} = F(s=i\\omega) = F(\\omega). ", "4a30386dc6cde5805add21978a41606e": "\\hat y_t", "4a308c7b247bbc6915e3d86ab271532b": "S=dD\\,\\sin\\alpha\\!", "4a30e91183fcea619fcab6aa54567a2f": "\\omega^3=1 ", "4a313e3c8eab8d4520d977c93d585b0a": "\\begin{align}\n\\frac{k_H}{k_D} = &\\left(\\frac{s^\\ddagger_H s_D}{s^\\ddagger_D s_H} \\right) \\left(\\frac{M^\\ddagger_H M_D}{M^\\ddagger_D M_H}\\right)^{3/2}\\left(\\frac{I^\\ddagger_{AH}I^\\ddagger_{BH}I^\\ddagger_{CH}}{I^\\ddagger_{AD}I^\\ddagger_{BD}I^\\ddagger_{CD}}\\frac{I_{AD}I_{BD}I_{CD}}{I_{AH}I_{BH}I_{CH}}\\right)^{1/2}\n\\\\ &\\times \\left(\\frac{\\prod\\limits_{i=1}^{3N^\\ddagger -7}\\frac{1-e^{-u^\\ddagger_{iD}}}{1-e^{-u^\\ddagger_{iH}}}}{\\prod\\limits_{i=1}^{3N -6}\\frac{1-e^{-u_{iH}}}{1-e^{-u_{iD}}}} \\right) e^{-1/2(\\sum\\limits_{i=1}^{3N^\\ddagger-7}(u^\\ddagger_{iH}-u^\\ddagger_{iD})-\\sum\\limits_{i=1}^{3N-6}(u_{iH}-u_{iD}))}\n\\end{align}", "4a31cf2b87fdeae042b7c19eef4c9f6a": "(a,b)\\times (-R,0)", "4a31e79b198eafa8a690b3e5e106aaf7": "\n\\overline{\\mathbf{P}_{k}\\mathbf{A}} + \\overline{\\mathbf{P}_{k}\\mathbf{B}} =\n\\left( r_{\\alpha} + r_{k} \\right) + \\left( r_{\\beta} - r_{k} \\right) = r_{\\alpha} + r_{\\beta}\n", "4a31e9857e765284054c11859b2ca58c": "i\\partial_\\mu \\partial^\\mu \\frac{\\delta}{\\delta J(x)}Z[J]+im^2\\frac{\\delta}{\\delta J(x)}Z[J]-\\frac{i\\lambda}{3!}\\frac{\\delta^3}{\\delta J(x)^3}Z[J]+J(x)Z[J]=0", "4a31f037462c8f31551eb9b28bd00fc6": "\\lambda^{-1/2}", "4a3293efafbb9c25885e28053fb258e4": "\\frac{B_5}{{v_0}^4}", "4a32f3517fa983ca964dda419834594e": "T_{eq} = T_{1} + \\frac{T_{2}}{G_1} + \\frac{T_{3}}{G_1 G_2} + \\cdots", "4a331f169bcd7cbce63862ea60d9f9b8": "f\\colon B\\to A", "4a332beceb76222295472f07cfb46b05": "c=2mn+|3m^2-n^2| \\, ", "4a334df23f60de60a390fd4a8f627a23": "\\varepsilon_{-1} = \\{0, 1, \\omega, \\omega^\\omega, \\ldots \\mid \\varepsilon_0 - 1, \\omega^{\\varepsilon_0 - 1}, \\ldots\\}", "4a336405092da5e85f541141fa9cefd5": "z_\\mathrm{R} = \\frac{\\pi w_0^2}{\\lambda} ,", "4a3376b34bab7cc3732a4b7b3c10795c": "2^{-m}\\alpha_m", "4a337f84e818b3f37e34ddc4e1661e20": "\\# B \\mathbb{G}_m (\\mathbf{F}_q) = {1 \\over {q-1}}", "4a338dfadc3a184f8cd228121302aaf3": "P = I^2 R.", "4a340332d44a72a62318b1b483d98f46": "R_1 = \\frac{2a+b}{3}\\,\\!", "4a3429cf3882e8ace64c02d5e6685c51": "\\scriptstyle i \\,-\\, 1", "4a34772a6cc0662157cdd5d625c1c725": "\\hat{d} = - {1\\over2}\\ln(1-2p-q) - {1\\over4} \\ln (1-2q)", "4a348800c90fb52bc561052c5cbbc3c3": "*632", "4a348a06ecc8a50c68ea697542cba255": "M \\otimes R_1", "4a348b3dbc2c1c58d6fb00a55e2a4746": "1 + 2 + 3 + ... + 97 + 98 + 99\\, ", "4a34bef2f37b0feeff1ea38d58e7ff1f": "x = 0.67 D_0\\sqrt{\\frac{P_0}{P_1}}", "4a34cb90c288b9db78eee7353cdc0b83": "\\operatorname{sgn}(\\sigma\\tau) = \\frac{P(x_{\\sigma(\\tau(1))},\\ldots,x_{\\sigma(\\tau(n))})}{P(x_1,\\ldots,x_n)}", "4a355c5e39d7d9606082e198bf8402ab": "\\frac{a}{\\sin A} = \\frac{b}{\\sin B}", "4a35a1f72dc2f68fa3e826e6f3bbe222": "\\textstyle\\arcsin \\left({{2}\\over{23}}\\right)", "4a35ea9c6f1b741fe88622dc0266d2df": "y_1, y_2,\\ldots,y_n\\in\\Gamma^{*}", "4a35fe30e48159975c5f63e3e0f3e808": "\\lim_{n\\to\\infty} t_n = 1/2.", "4a3618d919201dd0223b7d2d06364430": "W(t)=W_{f}+K\\, (24-t)^{2}", "4a36307ccca94fc6e2c63fbe0a72c83f": "a+c+e=b+d+f.", "4a3638e72e1789a82c8551eb0c58713c": "V+\\Delta V=(L+\\Delta L)(L-\\Delta L')^2", "4a36497e0e24792bb4b87139cd4ef2f6": "\\frac{\\partial B_x}{\\partial z} = \\frac{\\partial B_z}{\\partial x},", "4a3667c34d3591fc236b879451e06394": "b(\\theta) = E\\{\\hat{\\theta}\\} - \\theta", "4a3691b3a50dc8eb017835a65f4b311e": " \\int e^{cx}\\phi(bx)^n \\, dx = \\frac{e^{\\frac{c^2}{2nb^2}}}{b\\sqrt{n(2\\pi)^{n-1}}}\\Phi \\left (\\frac{b^2xn-c }{b\\sqrt{n}} \\right ) + C, \\qquad b\\ne 0, n>0 ", "4a36bd792ddfeb55b53a74749cd6d7d4": "\\left\\{\\begin{array}{ll}n - 2 k + 2 & n \\ge 2 k\\\\ 1 & \\text{otherwise}\\end{array}\\right.", "4a36c5ee6c5753c4c5ce6065ba3e1205": "\\; \\sum_i e_i = 1", "4a36e875a8c37f3f0be56477cc8fc943": "-\\tfrac{6}{5}", "4a3703e253504c37ff33ab22a5111086": " \\nu = 1+1/\\beta \\approx 2.1 ", "4a374da98c0632cc8f93925494319e70": "k = \\det(C_\\beta').\\,", "4a37fef33bf15c1271138d07c12d746c": "V= M/G", "4a38915794b724983c3eac64743a6908": "\\textstyle A_1 \\subset \\Omega_1 ", "4a38a6387196a7bb75a5b420bf4cee24": "\n(\\mathbf{B}^\\mathrm{T})^{-1} \\mathbf{M} \\mathbf{B}^{-1}\n= \\begin{pmatrix}\n\\mathbf{G}^{-1} && \\mathbf{0} \\\\\n\\mathbf{0} && \\mathbf{N}^{-1}\n\\end{pmatrix},\n", "4a38acf82202b5f5c038ccce6a602ac9": "\\tau=0.5", "4a38c823c88f53d089b57bc18d54cb31": "O(1.7272^n)", "4a38c93ec7b5b0dc5b7d5ff76e0a8ae4": "\\vec x_j", "4a393105fa6feccb098cd48d58da7cf4": "{x \\over \\sin A}={\\mbox{chord} \\over \\sin C}\\text{ or }{x \\over \\sin B}={\\mbox{chord} \\over \\sin C}\\,\\!", "4a3953d2acd7f42580da458104b4d969": "\\begin{align}\n(1,0,0,0) &= \\frac{1}{4}(1,1,1,1) + \\frac{1}{4}(1,1,-1,-1) + \\frac{1}{2}(1,-1,0,0) \\qquad \\text{Haar DWT}\\\\\n(1,0,0,0) &= \\frac{1}{4}(1,1,1,1) + \\frac{1}{2}(1,0,-1,0) + \\frac{1}{4}(1,-1,1,-1) \\qquad \\text{DFT}\n\\end{align}", "4a39ab2d6ed06bb3021c6b1c539d5949": "x^{2t-1}-1", "4a39b9bdd10a971f81ae3c94a90fa887": "f(w,z) = w^p + z^q", "4a39d650c0021ffd7ccb41728411d463": "\\mathbf{r}(u,v) = (x(u,v),y(u,v),z(u,v))", "4a3a4fb94cb4259bf493d014fd45f24a": "\\Phi(x)=\\begin{pmatrix}\ny_1(x) & y_2(x) & \\cdots & y_n(x)\\\\\ny'_1(x) & y'_2(x)& \\cdots & y'_n(x)\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_1^{(n-2)}(x) & y_2^{(n-2)}(x) & \\cdots & y_n^{(n-2)}(x)\\\\\ny_1^{(n-1)}(x) & y_2^{(n-1)}(x) & \\cdots & y_n^{(n-1)}(x)\n\\end{pmatrix},\\qquad x\\in I,", "4a3a6e4df1a0b5b9e8ba978b6e57b478": "\\rho \\sim r^{-2}", "4a3aae0259aa4294a869dc143ffc877b": "\\lim_{x_{1},...,x_{n}\\rightarrow+\\infty}F(x_{1},...,x_{n})=1", "4a3aaedfc3012c94cb7efd403ef77aa7": "d=\\gcd(a,b,c)", "4a3b69e57a00c1869f170acbe39d475a": "\\frac{\\text{FC}}{\\text{VC}}", "4a3b93e19bb9d876c3a7a018cdc51073": "X = S", "4a3b99525bbb45e7c33fbb8481cafcc2": "P_c \\, = \\, \\left [{ 0.113 + 0.0032 * N_A - \\sum {P_{c,i}} }\\right ]^{-2}", "4a3b9bbb5e3c2cf0c024e9b6fd7d1fc8": "\\begin{align}\np &= \\frac{5ac-2b^2}{5a^2}\\\\\nq &= \\frac{25a^2d-15abc+4b^3}{25a^3}\\\\\nr &= \\frac{125a^3e-50a^2bd+15ab^2c-3b^4}{125a^4}\\\\\ns &= \\frac{3125 a^4f-625a^3 be+125a^2b^2 d-25ab^3 c+4 b^5}{3125a^5}.\n\\end{align}\n", "4a3be73e4564f54770273528ac4f587f": "\\{x_{n_k}\\}\\to x", "4a3c0d03eb305a11f4445e7ad48c9851": "\\ U = S^2|D|\\,", "4a3c138b421a0cbf1a4759918a64b4d8": "SF\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\n1\n\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;I\n", "4a3c1587acaedbfeb2601c7aa4502f96": "\\mathrm{If}\\; a \\in \\mathbb{R} \\; \\mathrm{and} \\; Z \\in \\mathcal{L} ,\\;\\mathrm{then}\\; \\rho(Z + a) = \\rho(Z) - a", "4a3c7e8567fd37365a29226f816154ba": "d(x,S) = \\inf\\{d(x,s) : s \\in S \\} ", "4a3c8c63224865ebcb842fb3afe93b53": " D_{ddm} = \\frac{1}{K_e-g}", "4a3c9d01a91152f6b8c3f14c8b0e5846": "\\phi(z,\\rho)=\\frac{\\partial n}{\\partial z}", "4a3cf3eb7a612e24821f4de362164d8a": "K_{w/o}", "4a3d36f5d65f5852349f5acb8fad05ac": "\\mathbf{S} = \\frac{c}{4\\pi}\\mathbf{E}_\\text{a}\\times\\mathbf{B}_\\text{a},", "4a3d3f5d50d5bae7b13103ef498c4b15": "\\lambda_a+\\dots+\\lambda_b = \\mathrm{tr}~T_h", "4a3d7a4abf57434a485f5ef595a8b011": "\\scriptstyle{(-E_nt/\\hbar)}", "4a3d8eaea1d0917f55d967bf620d6b39": " \\omega = dx_1 \\wedge dy_1 + \\cdots + dx_n \\wedge dy_n. \\, ", "4a3dc0a2cbffa7ef285f143f2ad2dbb7": "Y_{21} = {I_2 \\over V_1 } \\bigg|_{V_2 = 0} \\qquad Y_{22} = {I_2 \\over V_2 } \\bigg|_{V_1 = 0}", "4a3e6013caf7ecb48d14e3574c26b635": "G_\\delta", "4a3e6852d188186e97892826b27d9530": "1 \\,", "4a3ec575a9a1cbf6f2b2a88b4771c5b4": "\\{w^R | w \\in L\\} ", "4a3f05bdf09481f57a20e27b001c8631": " \n\\dot{m} \\left(t \\right) = \\frac{K}{\\Sigma} F \\left(t \\right) \\quad \n", "4a3f1a2ad4e5484ba2822c4ca33cdb41": "\\Psi_{0}(x)\\propto \\frac{e^{ip_0\\cdot|x|/\\hbar}} {|x|}", "4a3f46b7c3779c6c871699698291bbbb": "\\frac{48}{25}", "4a3f4fda40832ebc9acad5a038b07523": "f_1(z)=\\lambda z", "4a3f79feb5dd29f2c0eb8886d57656fc": "Coverage > Availability", "4a3f85f761858d900f1bc694fe93b4eb": "D=U\\cup\\left\\{ z \\in H: \\left| z \\right| \\geq 1,\\, \\mbox{Re}(z)=\\frac{-1}{2} \\right\\} \\cup \\left\\{ z \\in H: \\left| z \\right| = 1,\\, \\frac{-1}{2}<\\mbox{Re}(z)\\leq 0 \\right\\}.", "4a3f8dfe552f2cd39536462b3384a045": "(u_i, v_j),", "4a3fc52d7e6d57bcf9bad4287f7c9ad0": "f'(x)=0", "4a3fe4b344775a9d29787bee11e9d149": "\\langle N_i \\rangle", "4a3ffb76ff9f77f4bdfde85c70f5d4c4": "{6}\\over {49\\choose 6}", "4a403a55feda97155fadfd7c433f5977": "\\frac{|u_n(x)-u_n(y)|}{|x-y|^\\alpha}=\\left(\\frac{|u_n(x)-u_n(y)|}{|x-y|^\\beta}\\right)^{\\frac{\\alpha}{\\beta}}|u_n(x)-u_n(y)|^{1-\\frac{\\alpha}{\\beta}}", "4a405a21bb6105c128fe65cd0fb7fccf": "(C \\sqcup D)^{\\mathcal{I}} = C^{\\mathcal{I}} \\cup D^{\\mathcal{I}}", "4a406861ee895ed1338badc56ab5d1cb": "2 \\neq 0", "4a41376c31c1406007d347b8034288f2": "\n\\begin{cases}\n \\frac{dx_1}{dt} = x_2 \\\\\n \\frac{dx_2}{dt} = a-g \\ (0 < t <= tCut) \\\\\n \\frac{dx_2}{dt} = -g \\ (tCut < t < t_f) \\\\\n x(t_0) = [0 \\ 0] \\\\\n g = 1 \\\\\n a = 2 \\\\\n x_1(t_f) = 100 \\\\\n\\end{cases}\n", "4a415d2f7e03eced14e1ec64e7827feb": "x=d_h", "4a41612c7c34c6d76a38ad75288e8665": "A + C = B + D = \\pi = 180^{\\circ}.", "4a4165506f017afcafda09f3ffa97dce": " r= k [NO_2]^2\\, ", "4a41dde9d2d6006fc35f9e5cb19b8cdc": "j_i : M_i \\rightarrow \\bigoplus_{k \\in I} M_k", "4a421367da2d6fec6f684f857b3d86eb": "\\tilde{\\mathcal{M}}^{-1}\\colon L^2(-\\infty,\\infty) \\to L^2(0,\\infty), \\{\\tilde{\\mathcal{M}}^{-1}\\varphi\\}(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} x^{-\\frac{1}{2}-is} \\varphi(s)\\,ds. ", "4a4217328275383941d1e7d21599552d": "\\sigma_{\\mathrm 50}", "4a421eb43ebb02425d766731757f6194": " W_{cu} ", "4a4245b9615334e996de8a0e4ce9642f": "h \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc k", "4a42a8eb9fa5b92c4f2c003eb309f0cd": "(t, y) \\in R ", "4a42b305ba08a949e8ad67977ba83d13": "\\text{Declination} = - \\arcsin(\\sin(23.44)\\times \\cos(B))", "4a42b654547c3e52b489b5823a2bd4ca": " H_b(\\mathcal{S}) = - \\sum_{i=1}^n p_i \\log_b p_i, \\,\\!", "4a43404a936a6f4b26a987fe4886b96c": "\\square = \\partial_t^2 - \\nabla^2", "4a438a5622b955637dd0267362c2ad26": " \\theta_1(u;q) = 2 q^{1/4} \\sum_{n=0}^\\infty (-1)^n q^{n(n+1)} \\sin(2n+1)u ", "4a439c4988347f3f76ec113a521a3622": "\\sigma(a_1)=a_2", "4a4410a1993e373111886338849c045a": " b^2 = a^2 + c^2 - 2 a c \\cos \\beta ", "4a442b095b8819ab964d9aceda719fbc": "H_{y}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}} {k_{o}^{2}-k_{z}^{2}}[j\\frac{k_{xo}}{\\varepsilon _{r}}(A \\ e^{-jk_{x\\varepsilon }w}+B \\ e^{jk_{x\\varepsilon }w})+\\frac{k_{z}}{\\omega \\mu }\\frac{m\\pi }{a}(C \\ e^{-jk_{x\\varepsilon }w}+D \\ e^{jk_{x\\varepsilon }w})]e^{-jk_{x0}(x-w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ (43) ", "4a44616475f64b789bb2ef04565a6686": " = C_{\\delta I}^{\\;\\;\\; K} e^I_\\gamma e^\\alpha_K - C_{\\gamma J}^{\\;\\;\\; K} e^J_\\delta e^\\alpha_K ", "4a446e47fa8dbd1d9a2f99c2b21c982d": " \\psi(s) ", "4a4478e83cdac61ce467ea8f6862484a": "\\frac{1}{\\sqrt{1-x^2}}, \\,\\!", "4a44a53e597758a8b18a41d874351c95": "\\int_a^b f(x)\\,dx = F(b) - F(a).", "4a44bda42ca8a35b3954db20c4d9c4a7": " R_0 = N\\frac{\\beta}{\\nu} ", "4a44d6b6492ed83d4a1c60a9fc94f3af": "V=V^{+}\\oplus V^{-}\\,", "4a44e7ea85ff70052bfa79f20e04cb38": "\\mathbf{F}_{1^n} = \\mu_n.", "4a45233c19752294a2e58b11e96c8756": "n\\in\\mathbf N", "4a453238aebe3d65e35933eaca73a415": "d W", "4a45a66ef8afd73cee69416e224a21e8": " R, G, ", "4a45c7c6fee3db464122917d820e2656": "\\alpha =1+\\frac{{{N}_{B}}}{{{N}_{A}}}", "4a461928c6eb588ed90f8aafb19e398b": "\\mathcal J", "4a464e141df790a1062325ae90c246bc": "f(x) = \\frac {2x-1} {x+2}", "4a4697b22d423628318be0e2f945c4e0": "\\boldsymbol {F} =m\\boldsymbol{a} =m \\sum_{k=1}^{d} \\left(\\dot v_k \\ + \\sum_{j=1}^{d} \\sum_{i=1}^{d}v_j{\\Gamma^k}_{ij}\\dot q_i \\right)\\boldsymbol{e_k} \\ , ", "4a46d611eff356272343f198725ba0df": "\\mathbf{S} = \\frac{q^2}{4\\pi c}\\frac{\\sin^2(\\theta)\\, a^2\\, \\hat{\\mathbf{n}}}{c^2 R^2}.", "4a46eadc90896e9c0cdce80b4acaf141": "(M,\\varphi:M\\to\\mathbb{R})", "4a46fe70883f23b94a558fe4f8600560": "j\\in A_i", "4a4777e6f04bc9b1c8e9594792c880c5": "\\begin{align}\n \\tanh (-x) &= -\\tanh x \\\\\n \\coth (-x) &= -\\coth x \\\\\n \\operatorname{sech} (-x) &= \\operatorname{sech} x \\\\\n \\operatorname{csch} (-x) &= -\\operatorname{csch} x\n\\end{align}", "4a4786b6e4935269889a1e33336e5ecd": "C\\ell_{p,q}^0(\\mathbf{R}) \\cong C\\ell_{q,p-1}(\\mathbf{R})", "4a4795603a81f8b29a760d6a878e5ad1": "P_n=\\frac{(1+\\sqrt2)^n-(1-\\sqrt2)^n}{2\\sqrt2}.", "4a47c8663cdb2e5953ac36f8fbfcfcab": "\\textbf{H}_{4 \\times 4}", "4a47e50eea94c20350ff21eea457be45": "\\displaystyle{{1\\over \\pi}\\iint_{\\Omega,\\,\\,\\,|z-w|>\\varepsilon} {\\overline{F(z)}\\over (z-w)^2} \\,dx\\,dy = Cf(w) -{1\\over 2\\pi i} \\int_{|z-w|=\\varepsilon} {\\overline{F(z)}\\over z-w}\\, d\\overline{z}=Cf(w).}", "4a47ec627ff65c392f24799074a70cc8": " k_1 , k_2 , \\ldots, k_n ", "4a47fc0a9921324f8332ac7fe33063fd": "T = \\{1,\\ldots,n\\}\\,\\!", "4a48a5911a85be21adff4deb05a78dd2": "\\varepsilon_1 ", "4a48e1d0f90d6cc1eed70c1ce16fb0ab": " \\mathcal{L}(I)\\subseteq \\mathbb{Z}^n", "4a48f976664893fb666182a8f5596150": "\\mathrm{H}_2 \\rightarrow \\mathrm{2H}^+ + \\mathrm{2e}^-", "4a4940d429e0eec85bf6330b859e243c": " \\mathcal{I}_X, \\mathcal{I}_Y", "4a4952b0d4c566ab69abb8b64ad1a519": "n\\ge 8", "4a49744cad59508c2d80cf4653eab8d2": " RCC = \\max_{(\\Delta ,x) \\in {T}} \\left\\{ - \\left\\| x \\right\\|^2 + \\operatorname{Tr}(\\Delta ) \\right\\} ", "4a497f4408eeda6e0d86f20886efb335": " E_{ij} ", "4a4992944a5ee15a3ad2c9c7f1157c8d": "\\begin{align}\n\\sum_{n=0}^{m}\\binom{m}{n}X^n &\n=(1+X)^m=\\prod_{i=0}^{k}\\left((1+X)^{p^i}\\right)^{m_i}\\\\\n & \\equiv \\prod_{i=0}^{k}\\left(1+X^{p^i}\\right)^{m_i}\n=\\prod_{i=0}^{k}\\left(\\sum_{n_i=0}^{m_i}\\binom{m_i}{n_i}X^{n_ip^i}\\right)\\\\\n & =\\sum_{n=0}^{m}\\left(\\prod_{i=0}^{k}\\binom{m_i}{n_i}\\right)X^n\n\\text{ mod } p,\n\\end{align}", "4a49a4b37705bfcd26fdc0589b5ae0d3": "| \\psi \\rangle = \\frac {|1\\rangle + |2\\rangle + |3\\rangle + |4\\rangle + |5\\rangle + |6\\rangle} {\\sqrt{6}}", "4a49b0f175054c4605a013aa6c25bd7f": "\n\\begin{align}\nf_{X,Y}(x,y) = f_{X \\mid Y}(x \\mid y)\\mathrm{P}(Y=y)= \\mathrm{P}(Y=y \\mid X=x) f_X(x)\n\\end{align}\n", "4a49ba75b556c0a11cbe4f73cd34cc73": "\\text{CIN} = \\int_{z_\\text{bottom}}^{z_\\text{top}} g \\left(\\frac{Tv_\\text{parcel} - Tv_\\text{env}}{Tv_\\text{env}}\\right)dz", "4a49d155f7c4283d0b74225331d67696": "Z=\\{z_0,z_1,\\dots,z_n\\}", "4a4ae75d6ba2bf6258e0212dd6f3d46a": "\\mathbf{B} = \\begin{pmatrix}\n 0.9 & 0.1 \\\\\n 0.2 & 0.8\n\\end{pmatrix}\n", "4a4b021bf49f0310864c8e7c607253a7": "f_c(z) = z^2 + c\\,", "4a4b44c86ba2c9d18d40161e14bb0b6d": "(\\mathcal{A},\\partial_{\\bullet}), (\\mathcal{B},\\partial_{\\bullet}')", "4a4b48541553695b3d2adc26da3b0dec": "c_n(t)", "4a4b5678b21bfac9491c3a716c2250dc": "\nF_s=- k x. \\!\n", "4a4bb6afaa48dfd8533bdac7f66dd9d5": "m_{e}", "4a4c138df6be18f27bcc58f65184be9e": "\\mathcal{K} (e_a, [e_b, e_c])", "4a4c35b9ec469a636e7c015f213d71bf": "\\delta = \\psi(0)", "4a4c47e8f17fcf76ca304eaf6f04d09c": "P(A|X) =g\\circ X", "4a4c735e20c1568f34966b85214c0510": "b_{p^n}=O(p^{n\\theta}).\\,", "4a4cac12e1a7eb224dff04bae5f249d3": "kar(C)", "4a4cfcab5f1eef48a60fef6e09951ac5": " v = \\frac{V_\\max S}{K_m + S} ", "4a4d38519946429acadbcc23b19b11f2": "C(\\mathcal A)", "4a4d521fcb14c1c857ea14b9da9a82a1": "|Q^{(j)}_b \\rangle = \\sum_i U^{ij}_{bc} |Q^{(i)}_c \\rangle.", "4a4d7bd2c4ae0cab2e19a8384b40ec28": " \\log {\\rm det}\\, (I+ zA) ={\\rm Tr} (\\log{(I+zA)})=\\sum_{k=1}^\\infty (-1)^{k+1}\\frac{{\\rm Tr} A^k}{k}z^k", "4a4dae98cb63c0c5b613816f5cdf2fb7": "Z=\\infty ", "4a4dc16da2094b4b7520cec9d2138465": "\n\\ln \\gamma_1 = \\left( \\ln \\gamma_1^\\infty + 2 \\left( \\ln \\gamma_2^\\infty - \\ln \\gamma_1^\\infty \\right) \\Phi_1 \\right) \\Phi_2^2 \n", "4a4e4cb7767a506fd6056130dd2c340b": "R=|r|^2\\,", "4a4e8a97046be12ddd230af8434a3959": "U(x_1,x_2) = x_1^{0.5}x_2^{0.5}", "4a4efafe9e32e69573074e0f48eeee17": "\\Pi_f=1-\\exp\\left[-\\frac{g_0R}{\\eta_0h_{PR}\\left(1-\\phi_e\\right)\\frac{C_L}{C_D}}\\right]", "4a4f0d94bcf3d59e895fb724eca5c198": "E[T_j^{2+\\delta}]", "4a4f14ceca869c933f46cf6cb274584c": "M_{\\Phi} = ( \\Phi (E_{pq}) )_{pq} \\in C^{nm \\times nm}", "4a4f2214ea876c1be96a9f7b76d8e6ca": "{EL}_{gas}", "4a4f3133e6659c37dd3f8c579729a776": "\\langle f(z)\\rangle= \\oint p_w(z)f(z)\\,dz", "4a4fb5d6ebb054ac85173dc7e94288ec": "\\theta\\epsilon_{\\Omega_v+1}0 = \\psi_0(\\epsilon_{\\Omega_v+1})", "4a4fffbcb8f528b3fcce1986d590091e": "(M_1, V) \\# (M_2, V).", "4a501c0ea2ff00af2d400b09342124f5": "\\limsup_n \\frac{\\log\\log a_n}{n} > \\log 2 ", "4a509325146ff7355d48fc8db31164a4": "t,y,z", "4a50a56451e19db137ff93c200d96152": "\nH(p,q) = T(p) + V(q). \\qquad\\qquad (1)\n", "4a50dc7eb0c52375b9d3876d76bdc52b": " A x = b ", "4a50e1d35b4712059ea0f3d60587ca60": "\\varepsilon_3: \\ \\vec n_3\\cdot\\vec x=d_3 ", "4a51080962e7d023675e70440478c804": "P_\\text{P} = \\frac{E_\\text{P}}{t_\\text{P}} = \\frac{\\hbar}{t_\\text{P}^2} = \\frac{c^5}{G} ", "4a511827b4496f09be064b90b850286c": "\\sigma_x(\\tau)", "4a5137d55d23f128af32d86440608125": "v_T = \\frac{\\phi D_L}{\\delta t^\\prime}=\\frac{v\\sin\\theta}{1-\\beta\\cos\\theta}", "4a5161029b6aae35c09e2bf022581273": "\\Delta(t) = (t - 1 + t^{-1})^2, \\, ", "4a5206695eef65c765b7a7051734ed38": " (8)\\,", "4a520b4a1c9bb6c85dcd6d995a205585": "B \\to 0,", "4a521f35fb83f9a9432bb08566ff536f": "\\displaystyle x_l(p,w)", "4a5231917b34bda79bf6b606982ae664": "\\mathbf{r} \\rightarrow \\mathbf{r} + \\delta\\theta \\mathbf{n} \\times \\mathbf{r}.", "4a523e21f1de365f356b73d4f56f4b10": "\\frac{f(\\mathbf{x} + (k/\\lambda)(\\lambda\\mathbf{u})) - f(\\mathbf{x})}{k/\\lambda}\n= \\lambda\\cdot\\frac{f(\\mathbf{x} + k\\mathbf{u}) - f(\\mathbf{x})}{k}.", "4a525749b0aa8b088d23e46f4dc7ffa0": "(x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 + 6x_6 + 7x_7 + 8x_8 + 9x_9 + 10x_{10})\\mod{11} \\equiv 0. ", "4a52e5923c4acbdcadb20a34303db9bb": "g_2(\\tau)=\\frac{4\\pi^4}{3} \\left[ 1+ 240\\sum_{k=1}^\\infty \\sigma_3(k) q^{2k} \\right] ", "4a52ed9df42c3b388f6c5e03ffb32342": "\\vec{k}_{i}", "4a5353cf2e5b19da336d9299f52c66c8": "\n\\varepsilon _{w}=(w^{2}-m_{1}^{2}-m_{2}^{2})/2w, ", "4a536c845ee99034ba21c602b3316ee1": "0.09090909\\ldots \\;=\\; \\frac{09}{99} \\;=\\; \\frac{1}{11}.", "4a53ae309b976c0620a77a5318a37b1f": "\\tilde\\psi(t)", "4a53b35bb5bb1e5096cb2dce30fba9dc": "\\phi_1\\!", "4a540d6690479c669831938d4856c30e": "x y^{-1} z.", "4a54146495000b16c0bfe4a221f723d4": "\\mu_0.", "4a548727b477835feb594e0368c43cdb": "AB \\geq CD", "4a54b0a605aef111b113965bc2b294d8": " \nQ = \\left[\\begin{matrix}\n \\mathbf{T}\\\\\n \\mathbf{N}\\\\\n \\mathbf{B}\n\\end{matrix}\\right]\n", "4a55744c45eb0714a36e668f1de42e16": "\\vert \\partial^\\alpha \\mathbf{f}(x)\\vert\\le \\frac{C}{(1+\\vert x\\vert + t)^K}\\qquad", "4a55c2924767399d27833243f5129930": "\na_n = e^{ L h / 2 } u_n + L^{-1} \\left( e^{Lh/2} - I \\right) \\mathcal{N}( u_n, t_n )\n", "4a55cbbdfc642045518bc78e941c0d7b": "9\\tbinom{t}{2} + 6 \\tbinom{t}{1} + 0\\tbinom{t}{0}.\\ ", "4a55ec6bc362303470bc97d53f964684": "a^b.", "4a5624eaf6011cfd7ce55a9a238774c3": "= \\mathbf{J}_{\\text{f}} + \\mathbf{J}_{\\text{P}} +\\varepsilon_0 \\frac{\\partial \\mathbf E}{\\partial t} + \\mathbf{J}_{\\text{M}} ", "4a5637986c02139fd1877c10fefc1b5d": "\n \\underbrace{V_m \\cos \\left( \\omega_m t \\right)}_{\\mbox{Audio}} \\times \n \\underbrace{V_c \\cos \\left( \\omega_c t \\right)}_{\\mbox{Carrier}} = \n \\frac{V_m V_c}{2} \\left[ \n \\underbrace{\\cos\\left(\\left( \\omega_c + \\omega_m \\right)t\\right)}_{\\mbox{USB}} +\n \\underbrace{\\cos\\left(\\left( \\omega_c - \\omega_m \\right)t\\right)}_{\\mbox{LSB}}\n \\right]\n", "4a56c189c37b1f14dc95e32325492d0f": "Y(E)=L^{-1}/(L^{-2}+V^2_{oi})", "4a56eb3a78bd73c4213df9bdc6eaac7f": "f_n = \\chi_{[1-1/n,2)} \\in D", "4a56f5c970564bad2e167c88ccbf6422": "\\xi_{z_i}", "4a56fad7cf4411a8591fe40a430d20ad": "|0\\rangle|0\\rangle", "4a570656c5f6be4f51692fbf98ebda26": "-g\\mu_B \\bold{\\sigma}/2", "4a57293863240a7226eca3deeab82422": "\\frac{1}{\\ln(x)}", "4a573e7167523e3eb35b9fe614998663": "\\cot y=x\\,\\!", "4a576586756464ab6d5f6fba666a8006": "\nc_1 = d_1 = 1. \\ \\ \n", "4a578b88924f2cc22997a60330bb99c0": "\\mathbb{C}^n = \\mathbb{R}^n \\oplus i\\,\\mathbb{R}^n.", "4a57e732342ad8d469830ed0218dfb2a": "\\frac{d p_{\\alpha}}{d t} \\, = \\, \\Gamma^{\\beta}_{\\alpha \\gamma} \\, p_{\\beta} \\, \\frac{d x^{\\gamma}}{d t} \\, + \\, q \\, F_{\\alpha \\gamma} \\, \\frac{d x^{\\gamma}}{d t} \\,", "4a586bca6fb47271c5ac0e95218041f9": "S = S_{\\mu}(\\mu) + S_{\\nu}(\\nu) + S_{z}(z) - Et", "4a5877aab967864d0aaf0bdb756b79c8": "\\mathbb{C} [G],", "4a5877f49c7240d32a8d9c20e87b96a8": "\n\\sigma_v = \\sigma_i \\, R\n", "4a594cd182972c38ef6af7cab315ae32": "\n \\sigma_{zx}^{\\mathrm{core}} = C^{\\mathrm{core}}_{55}~\\varepsilon_{zx}^{\\mathrm{core}} = \\cfrac{C_{55}^{\\mathrm{core}}}{2}~\\gamma_{zx}^{\\mathrm{core}} = \\tfrac{2h + f}{4h}~C_{55}^{\\mathrm{core}}~\\gamma_{zx}^{\\mathrm{beam}} \n", "4a596d6658cb71713e463ff0145bff32": "\\sigma_1, \\dots, \\sigma_m", "4a59baaa02c2e35e8b08142504e1c381": "(3.12^2)", "4a59d3ab2155575a02984be630befd7c": " [2]P_1=-P_2 ", "4a59de33edaea32a023f8ba70fd1db19": "\\operatorname{Func}(M)^i", "4a59e3ea2a73f987f0d5ff38b4c657d8": "rs^2", "4a59f76b5e01a370ce9866637c895389": " n_0+n_{+}+n_{-}=n.\\ ", "4a5a0a5833987f6de4b0fbfdfb4383c1": "\ng_m = {i_\\mathrm{out} \\over v_\\mathrm{in}} \n", "4a5a265209c55c60705d27a9e2d91c5a": "\n N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 ~;~~\n M_{\\alpha\\beta} := \\int_{-h}^h x_3~\\sigma_{\\alpha\\beta}~dx_3\n", "4a5a530aca9da07e727ce583aa0e6775": "\\vartheta_{\\textrm{K}}=-1", "4a5ad1edc858c70a1a54fccd10c13b03": "a^{1^0}", "4a5ae630ed4df81cbb1a67b770348807": "\\mathbf{y}_p = e^{tA}\\int_0^t\n\\begin{bmatrix}\n 2e^u - 2ue^{2u} & -2ue^{2u} & 0 \\\\ \\\\\n-2e^u + 2(u+1)e^{2u} & 2(u+1)e^{2u} & 0 \\\\ \\\\\n 2ue^{2u} & 2ue^{2u} & 2e^u\\end{bmatrix}\\begin{bmatrix}e^{2u} \\\\0\\\\e^{2u}\\end{bmatrix}\\,du+e^{tA}\\mathbf{c}", "4a5aeef7a6f54309cef191d2dd13e6e4": "\\text{then }\\cot(\\psi) + \\cot(\\theta) + \\cot(\\phi) = \\cot(\\psi)\\cot(\\theta)\\cot(\\phi).\\,", "4a5b03459a3735c11c7e78fb488172ec": "x_1 = n_1 p_1", "4a5b3e6359ffdd6b85cb1b5065c8fd61": "\\in_o", "4a5b6a6ab9f893c481b5f45cd844ad24": "\\tau_\\nu(s_1,s_2) \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{s_1}^{s_2} \\alpha_\\nu(s)\\,ds", "4a5b7f49ce476b8f2fe2be3af6c8049d": "\n\\mathcal{Q} = \\frac{\\lambda_\\min(M)}{\\lambda_\\max(M)}\n", "4a5c07434762e7693baf6b8ac9a94342": "\\{X_1 \\le a_1\\}, \\ldots, \\{X_n \\le a_n\\}", "4a5c634db7cead7b694d24595a9683d9": "\\textstyle\\left\\langle{n\\atop k}\\right\\rangle", "4a5c7d0f755a2df93d861bc0047e0ab6": "\\mathit{g}", "4a5d05f4f1ba367b1efc07892c2d5bc2": "\\overline{AD} = 2d\\tan(\\theta_2)\\sin(\\theta_1)", "4a5d0f90b045f2f65cd184cf2261b955": "\\begin{align}\n V_{abc}\n &= \\begin{bmatrix} V_0 \\\\ V_0 \\\\ V_0 \\end{bmatrix} +\n \\begin{bmatrix} V_1 \\\\ \\alpha^2 V_1 \\\\ \\alpha V_1 \\end{bmatrix} +\n \\begin{bmatrix} V_2 \\\\ \\alpha V_2 \\\\ \\alpha^2 V_2 \\end{bmatrix} \\\\\n &= \\begin{bmatrix}1 & 1 & 1 \\\\ 1 & \\alpha^2 & \\alpha \\\\ 1 & \\alpha & \\alpha^2 \\end{bmatrix}\n \\begin{bmatrix} V_0 \\\\ V_1 \\\\ V_2 \\end{bmatrix} \\\\\n &= \\textbf{A} V_{012}\n\\end{align}", "4a5d20ce9a29e13357e2229814a7ee6a": "k = \\frac{\\frac{\\nu \\Sigma_f}{\\Sigma_a}}{1 + L^2 {B_g}^2}", "4a5d60ed264b77389f98ff28864c67e8": "\\frac{1}{\\sqrt{1-2xt+t^2}} = \\sum_{n=0}^\\infty P_n(x) t^n.\\qquad (1)", "4a5d6e629e57ff3373a90b24ea22338c": "N_p P=\\infty", "4a5da6c3a589afdad3f63c6c3e1067ea": "s = -\\frac{\\ln\\xi}{\\mu_t}", "4a5dd75f2cb30b14a10b528f2475288f": "\\operatorname{bel}(A) \\le P(A) \\le \\operatorname{pl}(A).", "4a5e05adc38b334fca502184eaf0613b": "\\gcd(a,b) = a f\\left(\\frac b a\\right),", "4a5e44d0f825d28183abf41ea0692487": "\\mathbb N_0", "4a5ed1c4c1b32863541707d91a897294": "c_\\lambda := a_\\lambda b_\\lambda = \\sum_{g\\in P_\\lambda,h\\in Q_\\lambda} \\sgn(h) e_{gh}", "4a5f01ad4c1c1f33bef22b9006b1afa9": "\\pi:P\\to Q", "4a5f743462091d4a2fb22561706d49d4": "\\mathrm{erf}(x)=\\frac{2}{\\sqrt{\\pi}}\\int_0^x e^{-t^2}\\,dt,", "4a5f7724a1a974ffb99ca99c161848b3": " r_j ", "4a5f8f70ebf114d4f85b22712f44891b": "m'_b", "4a5f9c6576f43b23458378874a87bd4b": "xy'' + (n+1)y' = y. \\qquad", "4a601d9c53ac65faf9a82af2bd6e0f3f": "{t_{\\mu}}^{\\nu} = \\frac{c^4}{16 \\pi G \\sqrt{-g}} ( (g^{\\alpha\\beta}\\sqrt{-g})_{,\\mu} (\\Gamma^{\\nu}_{\\alpha\\beta} - \\delta^{\\nu}_{\\beta} \\Gamma^{\\sigma}_{\\alpha\\sigma}) - \\delta_{\\mu}^{\\nu} g^{\\alpha\\beta} (\\Gamma^{\\sigma}_{\\alpha\\beta} \\Gamma^{\\rho}_{\\sigma\\rho} - \\Gamma^{\\rho}_{\\alpha\\sigma} \\Gamma^{\\sigma}_{\\beta\\rho})\\sqrt{-g} ) ", "4a608c6bad0d614732452e6bf0ef2340": "C :\\ Y^2= eX^4-2dX^2Z^2+ Z^4", "4a618a6feafa759e773b45e69bca5453": "= \\ln(1.23456 \\times 10^2) \\,\\!", "4a61a05cdf1642acc651a05416fe5dc9": " \\ell(a) ", "4a61eedf7a926d93bca0c6d9fc17c8f7": "U_1 \\sim \\Gamma(\\delta_1, 1)", "4a62676119092598c899bd966541e0fd": "\\mathbf{RP}^{2k} = \\mathbf{P}(\\mathbf{R}^{2k+1}),", "4a629d8ba5914db2c11dc335ba287a57": "c_F(a,0)=\\delta(a)", "4a630da7966f95f00df582a3dd33c968": "\\forall k\\geq N_1 \\Rightarrow \\|A^k\\| < (\\rho(A)+\\epsilon)^k", "4a632544cf9375dc9220ef7386507531": "\n\\sum_{i=1}^k\\lambda_i\\leq\\sum_{i=1}^k\\lambda_i'\n", "4a644393f145758cebc754a103c46806": "\\alpha(1) = \\operatorname{Cor}(z_t,z_{t+1})", "4a644f769eb856b1681e90aea1adb589": "p_{01} \\leftarrow (x+1)^{\\deg(p)}p(\\tfrac{1}{x + 1}), M_{01} \\leftarrow M(\\tfrac{1}{x + 1})", "4a6462faa4f015fd83137da6e566fcea": "b=m(n^{2}+k^{2}) \\, ", "4a64716e42dbe2356550a6b0dfe6cbd6": "\\mathcal{O}\\left(\\Delta t^{p}\\right)", "4a64905bd44777553c14e791e215e1ca": "B=\\{x \\mid \\psi\\}.", "4a64a18294dde129a530e0556f6df926": "Z \\subset X", "4a64b6752c2666a2cfd32027c0b6094e": "G_F(\\tau)=\\frac{1}{\\beta}\\sum_{i\\omega_m}G(i\\omega_m)e^{-i\\omega_m\\tau}", "4a65129c231d77e8556a0e8180e1a69e": "X = X_0 \\frac{\\ln(E_0/E_c)}{\\ln2},", "4a652b6105b33085cbe8bdc62a5727c5": "s_i(X,Y)", "4a6541d411502e04b8b86889623961a3": "i(t=0) = I_0 \\cos( \\phi ).\\, ", "4a65a1c2ed03b019dd4c38c49fb2d9fb": "C_n(H):=\\{(u_1,\\dots,u_n)\\in H^n: u_i\\ne u_j\\forall i\\ne j\\}.", "4a65a6eef425a8c95c2fed85b7d4f363": "\\pi_2\\left(\\frac{SU(4)\\times SU(2)}{[SU(3)\\times U(1)]/\\mathbb{Z}_3}\\right)=\\mathbb{Z}", "4a65c0bfb6f7f1987b896874570ed773": "P = - \\frac{\\partial F_3}{\\partial Q},", "4a65e6b492aaf3fd16c789e89c1edc44": "m < \\infty", "4a661103ca1152399ae028920f3b12e2": "{\\rho}{\\omega}V={{\\mu}{V}\\over {{\\delta}_1}^2}\\,\\!", "4a6611dd2c9f3c5949241b73261ae852": "\\,2\\cdot r\\cdot e", "4a664789158b137eec84c16b4ecb164c": " F(h) = \\pi R \\int_h^{\\infty} \\Pi(x) \\, dx = \\pi R W(h),", "4a6678a92e9ae9e06989b8ab62d7412f": " O(A_1:A_2|B \\cap C) = \\Lambda(A_1:A_2|B \\cap C) \\cdot \\Lambda(A_1:A_2|B) \\cdot O(A_1:A_2) ,", "4a6680f253013f6490de7702325ae2aa": "\\left(\\frac{5}{21}\\right) = 1\\quad\\textrm{ but }\\quad5^{(21-1)/2} \\equiv 16\\pmod{21}", "4a6697d8ea75b010ad7f69f9785057ec": "\\sqrt{5}=2.2360679774998...", "4a66c759b32c5a6f92ba1c8b0e9b730d": "u,", "4a66dda47961f7fe1658bdf1bda6595e": "R - \\beta", "4a66f32b3b79ee91142e37ee50db4177": "\\lim_{h\\rightarrow 0}\n \\frac {D_{KL}(X_{\\theta+h}\\|X_\\theta)} {h^2}\n =\\lim_{h\\rightarrow 0} \n \\frac 1 {h^2}\n \\int_{-\\infty}^\\infty \\left( \\log\\frac{\\mathrm dX_{\\theta+h}}{\\mathrm dX_\\theta} \\right)\n \\mathrm dX_{\\theta+h}\n", "4a6736b35154eafbb9627942edeb9e56": "{\\text{Distance}} \\propto {\\text{Time}^2}", "4a677ba7b937b1021ca8d888fbc8fd95": " a_1 v_1 + a_2 v_2 + \\cdots + a_n v_n. \\, ", "4a67a1777316ec37f7df3ac0d825a625": "\\mathbf{e}_\\text{x}\\times\\mathbf{e}_\\text{y} = \\mathbf{e}_\\text{z}\\,\\quad \\mathbf{e}_\\text{y}\\times\\mathbf{e}_\\text{z} = \\mathbf{e}_\\text{x}\\,\\quad \\mathbf{e}_\\text{z}\\times\\mathbf{e}_\\text{x} = \\mathbf{e}_\\text{y}", "4a67a77ec8e825b2d984f254457d8cab": "{\\bar{R}}_5", "4a67d2651f0ba03d929b2b4ce1c9e89f": "\\Phi(G)", "4a683b1cf417b2623987019ded7f3542": "\\mathit{E_g}", "4a6896af550d30a20997775d46dc499d": "10^{100}", "4a68c75a89efda684e29b974f7958080": "\\frac{\\lambda^x}{(x!)^\\nu}\\frac{1}{Z(\\lambda,\\nu)}", "4a68e090c6c749429c669e59c15b1a68": "\\nabla^{bas}_{\\!\\phi\\,}\\psi := [\\phi,\\psi]+\\nabla_{\\!\\rho(\\psi)\\,}\\phi,", "4a68e24740960f397b3d3be5c1e2b5e6": "\\textstyle 99.5%=\\frac{199}{200}", "4a6928a946131ebbed459c7c8a5bd1f5": "{81 \\choose 12} = \\frac{81!}{12! 69!} \\approx 7.072432018 \\times 10^{13}", "4a693d89558c4fda241a2f1788057305": "\n\\frac{n\\widehat{\\sigma}^2}{\\sigma^2}\\sim\\chi^2_{n-1}\n", "4a694c9eb73dd7fe10303d4d02b14d85": " A_B = \\frac{A_1 - A_{ an } }{ A_1 } ", "4a6a49899b5fc089c08b75621976e8bd": "V_j = \\frac{1}{2}\\sum_{i\\neq j} q_i \\phi(\\mathbf{r}_i)=\\frac{1}{8\\pi\\varepsilon_0}\\sum_{i\\neq j} \\frac{q_iq_j}{|\\mathbf{r}_i-\\mathbf{r}_j|}", "4a6b294afac14df3c8b419c36903958d": "\\ \\|x\\|_\\infty=\\max \\left\\{|x_1|, |x_2|, \\dotsc, |x_n|\\right\\}", "4a6b8ef439d06c23db820437e0dc8216": "\\boldsymbol{\\Sigma}^1_n", "4a6bba422057cf3d4026b7d430717ec5": " \\mathfrak c ^{\\mathfrak c} = (2^{\\aleph_0})^{\\mathfrak c} = 2^{\\mathfrak c\\times\\aleph_0} = 2^{\\mathfrak c},", "4a6bf8b8b647156bb0c99c85157f9e83": " \\frac{\\frac{1}{2}MV^2}{\\frac{1}{2}mv^2} = \\frac{m}{M} \\qquad (2)", "4a6c9d839becec307a4bf3b90399559f": " a, n, m", "4a6cc52d57986f5c3a19f1b5b13f9ad0": "k(x)", "4a6cfbb7498fd0695f0fded92c73c6c7": "Y_{6}^{5}(\\theta,\\varphi)={-3\\over 32}\\sqrt{1001\\over \\pi}\\cdot e^{5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot\\cos\\theta", "4a6d0aa1da2f0785de69058589933021": "B_d (x)\\bigcap B_d (y)", "4a6d696a0df264782fad5205ef9bea1f": "E_B", "4a6d86321cb496e70ee0fea0822e5e28": "\\scriptstyle (1, \\|a\\|_p)", "4a6d9654792cbe72ebb7ba56440342f8": "\\exp (-x^2/2 + 2xt-t^2) = \\sum_{n=0}^\\infty \\exp (-x^2/2) H_n(x) \\frac {t^n}{n!}.\\,\\!", "4a6eac58871ffa401aa191e0668de6ba": "\\begin{align} \n\\boldsymbol {\\eta} &= \\left(\\frac{\\mu}{\\sigma^2}, -\\frac{1}{2\\sigma^2} \\right)^{\\rm T} \\\\\nh(x) &= \\frac{1}{\\sqrt{2 \\pi}} \\\\\nT(x) &= \\left( x, x^2 \\right)^{\\rm T} \\\\\nA({\\boldsymbol \\eta}) &= \\frac{\\mu^2}{2 \\sigma^2} + \\ln |\\sigma| = -\\frac{\\eta_1^2}{4\\eta_2} + \\frac{1}{2}\\ln\\left|\\frac{1}{2\\eta_2} \\right|\n\\end{align}", "4a6ebb634c59aee22c14a7bdfca2c6e5": "R=\\mathbf{Z}\\left[\\textstyle{\\frac{1+i}{2}}\\right]", "4a6ef818c1391c2da32a46f6f6e31eb4": " h\\nu = g_\\mathrm{N} \\mu_\\mathrm{N} B_\\mathrm{0}", "4a6f0941288b2035cdfb74ead22a1ea2": "\\frac{\\partial \\mathcal{A}}{\\partial t}\\, +\\, \\frac{\\partial}{\\partial x} \\Bigl( (U\\, +\\, c_g)\\, \\mathcal{A} \\Bigr)\\, =\\, 0\\,", "4a6f0dabf2093c8241c7aee98cdaa828": "SampEn(m,r,\\tau)", "4a6f2e7e3c264fff697cceca3a0eea88": "\\textstyle \\beta \\leq P(A\\mid[x]) \\leq \\alpha", "4a6f36881938a2768f159e7713763677": " N_{ t + 1 } = r N_{ t } ", "4a6f465413f590b3ce858c672ace3e29": "\n\\begin{array}{lcl}\nG &\\sim& \\operatorname{DP}_1(H,\\alpha) \\\\\n\\phi_{1,\\dots,N} &\\sim& G \\\\\nx_{i=1,\\dots,N} &\\sim& F(\\phi_i)\n\\end{array}\n", "4a6f50d65c777a50e4720fbcdbc5cbf5": "F: \\Omega \\to \\mathbb{R} ", "4a6f819eb601ea5272390b8d156deeb7": "er_k", "4a6fa8881d901c2c2b371356f70ff14f": "p_3(0)\\equiv 1", "4a6fbea990c0ad65d4a3c603a1ed19ab": "\\zeta \\text{ is } \\varphi(t,\\zeta)", "4a6fe2e2705564de16440f68ec4987a4": "\\frac{\\partial(y_1, \\ldots, y_k)}{\\partial x_i} = \\frac{\\partial(y_1, \\ldots, y_k)}{\\partial(u_1, \\ldots, u_m)} \\frac{\\partial(u_1, \\ldots, u_m)}{\\partial x_i}.", "4a701e3ea497721cbdc80ead5439b673": "g^\\lambda_{\\mu,\\nu}(p)", "4a70e16f72ad225e1b21ee0c8b7364a9": "\\begin{pmatrix}\nct'\\\\ x'\\\\ y'\\\\ z'\n\\end{pmatrix} = \\begin{pmatrix}\n\\gamma & -\\beta\\gamma & 0 & 0\\\\\n-\\beta\\gamma & \\gamma & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\nct\\\\ x\\\\ y\\\\ z\n\\end{pmatrix} =\n\\begin{pmatrix}\n\\gamma ct- \\gamma\\beta x\\\\\n\\gamma x - \\beta \\gamma ct \\\\ y\\\\ z\n\\end{pmatrix}.\n", "4a70fe411faec0532264108d2e923cda": "\\partial_\\tilde{t} \\tilde{\\eta} + \\partial_\\tilde{x} \\tilde{\\eta} + \\tfrac{3}{2}\\, \\tilde{\\eta}\\, \\partial_\\tilde{x} \\tilde{\\eta} + \\tfrac{1}{6}\\, \\partial_\\tilde{x}^3 \\tilde{\\eta} = 0,", "4a711a88104f5590eb14459d063aa29a": "\\begin{matrix}{k \\choose m} = 0\\end{matrix}", "4a71489f586e3b9f75b5a5a034068162": "\\ \\displaystyle (q,\\alpha.u)\\ ", "4a714b6a0988217c3313b6949216b5bf": "20%\\le m \\le 30%", "4a71b3e18e74d2efc16564735fdebdde": "\\; \\sigma _1 = W \\sigma _3 W^* .", "4a71f9b6b6d7eead4823143b53b811bf": "f(x)=\\frac{1}{x^2+1}", "4a7282b480585d9d19bbd21cadb9acc7": "dk_Q", "4a72835fcac75794d401b71e1e644356": "\n\\int_{[a, b]} \\varepsilon_N^2(t) dt=\\sum_{k=N+1}^\\infty \\int_{[a, b]}\\int_{[a, b]} K_X(s,t) f_k(t)f_k(s) ds\\, dt\n", "4a72b704f4579e709566154f111d9618": " \\Delta\\omega_{lab} = \\omega_0 \\sqrt{\\frac{8k_BT \\ln 2 }{mc^2}} ", "4a73101311a6dc7fbb2fbe35cc6bf5e1": "4663\\,X-6150,", "4a7311bb444f25a5172615cf3f8345f9": "q:(\\mathcal{D}^{n}\\times\\mathcal{R})\\rightarrow\\mathbb{R}\\,\\!", "4a7322be42b02967f6ea2b9bc28d539c": "\\theta=\\vartheta^a\\otimes\\vartheta_a", "4a733066626baa506e679c6a0a88d896": "k = \\frac{y''}{(1+y'^2)^{3/2}}", "4a7360d9d234640926b20e9d2f729dd1": "D\\cdot D = \\partial_\\alpha \\partial^\\alpha = \\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\nabla^2 ", "4a7365b174ad58ee4e2bf66b9e0df995": "~\\hat a~", "4a73838c65ebeb4cc01703e0734c5ae6": "g(z) := b + \\int_a^z \\frac{f'(w)}{f(w)}\\,dw", "4a7387db7dd354afa325560339f57fad": "\n C_0\\geq\\frac{\\sqrt{10}+3}{6\\sqrt{2\\pi}} = 0.4097\\ldots.\n ", "4a7395d40df200614c8b07f9a307fd87": "{f_1,...,f_k}", "4a73b01a7be8554d50b98286ebe1ac50": "\\bar{\\omega} = 2 - \\sqrt{3}", "4a7466575f6c9cb6962eb4b284e24717": "\\langle\\hat{\\delta},\\phi\\rangle = \\langle\\delta,\\hat{\\phi}\\rangle", "4a747b83b1c030552cc97c0ed2e95f99": "\\int d^Dx \\left[(A_\\mu^1)^2+(A_\\mu^2)^2+(A_\\mu^4)^2+(A_\\mu^5)^2+(A_\\mu^6)^2+(A_\\mu^7)^2\\right]\\,,", "4a74aa7274eae6ede19621c9d897ae40": "\\Lambda_{A}", "4a74dcb6bab22b0988fabc0affc77410": "f = \\frac{v}{2 \\pi r}", "4a7537b82713621024d74a2c7fe5cacb": "K=\\frac{{[S]} ^\\sigma {[T]}^\\tau \\cdots } {{[A]}^\\alpha {[B]}^\\beta \\cdots }", "4a754f69237e15a109a292aaa42c2109": "\\scriptstyle \\tanh \\ a = y/x", "4a75655396137de796d191fca4d938c3": "\\begin{align}\nC_\\beta(s) &= \\exp\\left\\{s\\cdot\\begin{bmatrix}0&-k\\\\1&0\\end{bmatrix}\\right\\}\\\\\n&=\\begin{bmatrix}\\cos\\sqrt{k}s&\\sqrt{k}\\sin\\sqrt{k}s\\\\ -\\frac{1}{\\sqrt{k}}\\sin\\sqrt{k}s&\\cos\\sqrt{k}s\\end{bmatrix}.\n\\end{align}\n", "4a75b9b74348c90eab196c210df05335": "\\ddot{x}=h(x)+g(x)\\cos(\\omega t)", "4a760f47c83d192229d59763d656586f": "\\lambda_a^n+\\dots+\\lambda_b^n = \\mathrm{tr}(T_h^n)", "4a762e37f4cfe728ee222cc5772f9e8c": "\\mathbf{x}_w^{(k)}", "4a7653657ad8b3c2161c940c3c079718": "E(\\bar{x}) = \\sum_{i=1}^n {w_i \\bar{x_i}}. ", "4a767ada62cc8cca3501af23d53f5c8e": "i_0", "4a76884eae2b11d8984ca62f54794ba6": "= \\frac{PL^3}{3EI}.", "4a769b524dd2b55a2ce0c54fba99f97e": " \nU^\\alpha f(x) = \\mathbf E^x\\left[ \\int_0^\\infty e^{-\\alpha t} f(X_t) dt \\right]. \n", "4a76af09483e00c23f32519cadff4a39": "\\acute{F}^{\\mu\\nu} = {\\Lambda^\\mu}_\\alpha {\\Lambda^\\nu}_\\beta F^{\\alpha\\beta},", "4a76be6315789972cddf72427b5baad0": "y\\in B", "4a770ec23adeb568edbe559fc648984f": " T_{\\mathbf{u}}T_{\\mathbf{v}} = T_{\\mathbf{u}+\\mathbf{v}} . \\! ", "4a773cb67dfcb7e42ca7d0a776fbe9b6": "E^{p,q}_2 = \\check{H}^p(\\mathfrak{U}, \\mathcal{H}^q(X, \\mathcal{F})) \\Rightarrow H^{p+q}(X, \\mathcal{F}).", "4a77411db7d59fd6560bd10583714106": "(x_1, x_2, x_3, \\ldots, x_n) = \\underset{^\\sim}{\\mathbf x}", "4a778498f706747e3c1c76fa2730981e": " \\frac {d \\hat u_\\theta } {dt} = -\\frac {d \\theta } {dt} \\hat u_R = -\\omega \\hat u_R\\ , ", "4a7839dd875ad76937611be31f858bba": "\n\\Delta G_\\mathrm{solv} = \\sum_{i} \\sigma_{i} \\ ASA_{i}\n", "4a787b837014dc8e3841c3c5f5fd03d2": "\\rho^{[{k+2}..{N}]}=\\sum_{\\gamma}{(\\lambda^{[k+1]}_{\\gamma})^2}|{\\Phi^{[{k+2}..N]}_{\\gamma}}\\rangle\\langle{\\Phi^{[{k+2}..N]}_{\\gamma}}|=\\sum_{\\gamma}{(\\lambda^{[k+1]}_{\\gamma})^2}|{\\gamma}\\rangle\\langle{\\gamma}|.", "4a788b1cd6f7123f68b491d801114eec": "\\mathcal{A} (\\omega) = \\Omega_{D} (\\omega)", "4a78cf9d94aa5a941368ca05a4a09273": "S=\\{(a_i, b_i, c_i)|i", "4a78f915ee70a619bcda87e16e227090": "x^n= T_n\\left(\\tfrac{1}{2}\\left[x+x^{-1}\\right]\\right)+ \\tfrac{1}{2}\\left(x-x^{-1}\\right) U_{n-1}\\left(\\tfrac{1}{2}\\left[x+x^{-1}\\right]\\right)", "4a790b81347a225b5f1258157565a7a0": "\nF = {k_{octane} \\over k_{nonane}} = {{Area_{octane}/M_{octane}} \\over {Area_{nonane}/M_{nonane}}}\n", "4a795567d9025959fe6a1e0c6920a690": "((A\\to B)\\to((\\bot\\to C)\\to D))\\to((D\\to A)\\to(E\\to(F\\to A)))", "4a7a00871856c54536f59dfeb80c2a25": "\\scriptstyle\\mathbf{\\hat{\\nu}}", "4a7a029a2df8cdf65b673aca36f0bade": "p(E)=1+p_1(E)+p_2(E)+\\cdots\\in H^*(M,\\mathbf{Z}),", "4a7a1a9ab56cbb26352f813629e6cd58": "N \\gg 1", "4a7a5937526521e3370cc4adccb6bfb7": "\\|u-u_h\\|_a^2 = a(u-u_h,u-u_h) = a(u-u_h,u-v) \\le \\|u-u_h\\|_a \\cdot \\|u-v\\|_a", "4a7a9a14dbc5ac7fafc6dd430717bb63": " -\\frac{dx}{dt} = {k_f x} - {k_b [B]_t}\\,", "4a7aab7813d105716f2adbb07e6a8522": "dV =(1+r^2)^{-1/2} \\,db\\, dx\\, dy,\\,\\,\\, dA= (1+r^2)^{-1/2} \\,db\\, dx,", "4a7ab4ff99a16d751af4ed5e0f96e44b": "\\left | \\left (\\frac{1}{\\sqrt{n}} \\right ) \\left (\\sqrt{\\frac{1-p}{p}}-\\sqrt{\\frac{p}{1-p}} \\right ) \\right |<0.3", "4a7ad1b49a7f9769457279d11f44e6fa": "\n\\frac{\\partial u}{\\partial y} = \\left\\{\n\\begin{matrix} \n0 &, \\quad \\tau < \\tau_0 \\\\ \n(\\tau - \\tau_0)/ {\\mu} &, \\quad \\tau \\ge \\tau_0 \n\\end{matrix}\\right.", "4a7ad9b69787da96b35a2b97d5784258": "T u = u_{|\\partial \\Omega}.\\,", "4a7bc32017bcdbd7c0db7b1cb214ad94": "\\varepsilon_\\mathrm e\\,\\!", "4a7bd071b0cb40fd4bf836d2b772be48": "x' = \\frac{x - \\text{min}(x)}{\\text{max}(x)-\\text{min}(x)}", "4a7be5de27ae885a87a2ac9abb9f9c93": "f(\\delta_1) = f(\\delta_2) \\cap \\delta_1", "4a7be8ef69a3d4c93216c0762a6e9193": "\\int_C f(z)\\,dz=(2\\pi i)\\operatorname{Res}_{z=i}f(z)=2\\pi i{e^{-t} \\over 2i}=\\pi e^{-t}.", "4a7c0e3306598d841e07d385e35e09ab": "x^\\mu = x^{-1}, \\ldots, x^{-n}", "4a7c326040946c37bed4acbfa8533787": "Q = Q_{1}\\cup Q_{2}\\cup\\{q_{0}\\}", "4a7ca2ab1df8d6c6653388a70141a4e7": "\\int \\frac{1}{x}\\,dx = \\ln x + C.", "4a7cbd7833a9021fcc9314250b71dfb4": "H^0(X, \\mathcal O(E)) = \\Gamma (X, \\mathcal O(E))", "4a7cecc75c172bcab5419e0aa25f3702": "f\\colon\\mathbb{R}^n\\to\\mathbb{C},\\quad\\hat f\\colon\\mathbb{R}^n\\to\\mathbb{C},", "4a7cee4b137b4cde9d96390cea1626de": "\\vec\\mu ", "4a7d14e52cdf1bebba8893a8b99f2493": "p=k\\alpha_{g}", "4a7d22b39e93fbbcbe107e7a19e8bd34": "O(n^3)", "4a7d2d7c24dd407bbb95a3b3fda28769": "n\\lambda=2d\\sin\\theta, \\,", "4a7d4705cc5470f0564d9cacfd0358e2": " A=\n \\begin{bmatrix}\n 2 & 3 \\\\\n 5 & 7 \\\\\n \\end{bmatrix}\n", "4a7d6b89663c6e01dadcbb8fe0d93fec": "\\rho\\propto a^{-3(1+w)}.", "4a7dcc07fa2fad810fc60e4a358baa53": "\\mathcal{}M_*", "4a7dcfc835568000c02dd263c0d62961": " Upper~limit = m + t_{0.975,11} \\times\\sqrt{\\frac{n+1}{n}}\\times s.d. = 5.33 + 2.20\\times\\sqrt{\\frac{13}{12}} \\times 0.42 = 6.3.", "4a7dd60c57fd5b8a4b1f09b3d7555f6a": "\\exp (ikx)", "4a7ddf75bcfa56d167af13b68d373bb5": "\\rho_{\\text{free}}", "4a7e9792d3e5f7dc6f98b45ddee8b310": "D=\\frac{1}{3\\Sigma_{\\mathrm{tr}}}", "4a7ec260312781840f6261519c929bb6": "(v,t)\\cdot(v',t') =\\left (v+v',t+t'+\\tfrac{1}{2}\\omega(v,v')\\right).", "4a7f7073206aee4577b78b30132affa3": "\n\\langle p_1,\\ldots,p_n\\ \\mathrm{out}|q_1,\\ldots,q_m\\ \\mathrm{in}\\rangle=\\int\n\\prod_{i=1}^{m}\n \\left\\{\n \\mathrm{d}^4x_i\\ \n i\\frac{e^{-iq_i\\cdot x_i}}{(2\\pi)^{3/2} Z^{1/2}}\n \\left(\\Box_{x_i}+m^2\\right)\n \\right\\}\\times\n", "4a7fa9f55b020a1e34b34a9b0337edac": "\\lim_{a\\rightarrow\\infty}\\int_{-a}^a f(x)\\,\\mathrm{d}x", "4a7fd701f8e10094e066886870db8b4f": "\\{X^i\\}_{i=1}^n", "4a807fb180238af6fcd8f041cba6a145": "Y^{\\text{op}}", "4a809cfb38f54b9c7e605e6d43b68d1d": "\\frac{\\bar{P}}{N_0 W}", "4a80b45deec90ba067bc80e3c6400667": "y_1(t)\\,v''+(2y_1'(t)+p(t)y_1(t))\\,v'=r(t)", "4a80ea2eb14cca8834854564c2f95047": "\\text{CLV} = \\text{GC} \\cdot (\\frac{1+d}{1+d-r})", "4a80ed2581ce96e3d75840c907d56e9e": "\\begin{cases}\n\\alpha y(a)+\\alpha' y'(a)=0\\\\\n\\beta y(b) + \\beta' y'(b)=0.\n\\end{cases}", "4a80fbfbaa6bc5b5c026857157e1094c": "k = \\mathrm{gcd}(k,\\ell)", "4a814415dd329d17b63780f8641975f1": "-\\frac{1}{2}\\frac{\\partial}{\\partial r}\\left(-1+\\frac{2M}{r}\\right) = \\kappa.", "4a81664cb4208b56b765ccda4440072f": "h = C_pT", "4a8177d2d6715b75f3ebaf8d45b886ba": "x(t) = \\frac{1}{2} g t^2", "4a81be0068d59fe795bc9cbdc219c59f": "\\lambda_{12}=7.01559", "4a81e4617e3a649a6b9799e346d3b36d": "\n\\omega_{r}^{2} = \\frac{1}{m} \\left[ \\frac{d^{2}V}{dr^{2}} \\right]_{r=r_{\\mathrm{outer}}}\n", "4a822a67fd7c1e5538c162369f8b6bc2": "\\bold g", "4a825e14cbf5a58de211997dac717bc4": "\n\\omega_{1}/2 = \\int_{e_{1}}^{\\infty} \\frac{dz}{\\sqrt{4z^{3} - g_{2}z - g_{3}}}\n", "4a825ed7d9bf40af5ca14e0c770f57e4": "Z_{I1}\\,\\!", "4a82d0a700468fbdcbe54eef05eee9a9": "R=R(r)=\\frac{R_0-r}{\\prod_{n=1}^{\\infty} \\left( 1+C\\beta_n \\right)}.", "4a82dbc1a5b4343f38e25eac6de35f01": "\\langle \\bar{T}T\\rangle_{ETC}", "4a834202150fb21b5bf067375a504e16": "F(\\varphi)", "4a83432667491044a583e704ee3cc8ec": "\\omega_f = \\omega_g", "4a83607a8a6e2faa2ba4cde23eeb7110": "(x_1 + x_2 + \\cdots + x_m)^n\n = \\sum_{k_1,k_2,\\ldots,k_m} {n \\choose k_1, k_2, \\ldots, k_m}\n x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m}. ", "4a837d0a72adad415d99016bbcc7be0a": " n_b = Osm_b \\times TBW_b ", "4a83aaf6121c973890507fb5f68438a6": "\\zeta(q),\\ q\\in{\\mathbb R}", "4a83f921e9302ec6fcb69cb3e4aaa9c5": " \\mathbf{T} =\\sum_{i=1}^N (m_i\\Delta r_i\\mathbf{e}_i)\\times (\\alpha(\\Delta r_i\\mathbf{t}_{i}) - \\omega^2(\\Delta r_i\\mathbf{e}_{i}) + \\mathbf{A}) = (\\sum_{i=1}^N m_i\\Delta r_i^2)\\alpha\\vec{k} + (\\sum_{i=1}^N m_i\\Delta r_i\\mathbf{e}_i)\\times\\mathbf{A}, ", "4a84873adce32ff313e2ea3a83417f7b": "\\rho (z+z') - \\rho (z) = \\frac{\\partial \\rho (z)}{\\partial z} z'", "4a854576111223fb1d200b5664a1b241": "\\langle E_1(t) E_2^*(t - \\tau) \\rangle = \\langle E_{a1}(t) E_{a2}^*(t - \\tau) \\rangle + \\langle E_{a1}(t) E_{b2}^*(t - \\tau) \\rangle + \\langle E_{b1}(t) E_{a2}^*(t - \\tau) \\rangle + \\langle E_{b1}(t) E_{b2}^*(t - \\tau) \\rangle", "4a854c196e26b8053fd97651dff4c952": "y_{\\sigma(i)}", "4a858e613f09c7080257f908c8937bf2": " E_z(k_z) ", "4a8594be279f1661dbcd70264d2d5c5d": "q(t_2, t_1),", "4a85f2a2731d82814e6f3a8ddefff0cb": "\\mathcal{F}_\\alpha = \\mathcal{F}^{k}", "4a86534d79edacd1f0d1858f7837cac7": "B_\\mathrm{0} ", "4a866f3e4ce60c1fa64d4bc2100b38da": "g(x)\\equiv 1", "4a86e4e3ece28c91576ec872f356f7dc": "\\begin{align}\nI^2 &= 4 \\int_0^\\infty \\int_0^\\infty e^{-(x^2 + y^2)} dy\\,dx \\\\\n&= 4 \\int_0^\\infty \\left( \\int_0^\\infty e^{-(x^2 + y^2)} \\, dy \\right) \\, dx \\\\\n&= 4 \\int_0^\\infty \\left( \\int_0^\\infty e^{-x^2(1+s^2)} x\\,ds \\right) \\, dx \\\\\n&= 4 \\int_0^\\infty \\left( \\int_0^\\infty e^{-x^2(1 + s^2)} x \\, dx \\right) \\, ds \\\\\n&= 4 \\int_0^\\infty \\left[ \\frac{1}{-2(1+s^2)} e^{-x^2(1+s^2)} \\right]_{x=0}^{x=\\infty} \\, ds \\\\\n&= 4 \\left (\\tfrac{1}{2} \\int_0^\\infty \\frac{ds}{1+s^2} \\right ) \\\\\n&= 2 \\left [ \\arctan s \\frac{}{} \\right ]_0^\\infty \\\\\n&= \\pi\n\\end{align}", "4a86f2173099cdeb199dd27611cb49c7": "((-a\\,\\bmod\\,n) + (a\\,\\bmod\\,n))\\,\\bmod\\,n =0", "4a872af7d723205ac7de36465be292c5": "\\sin(84\\tfrac38 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}.", "4a872ba9bd690767fefd4cee19364fd9": "\\textrm{var}(x)= 1-E[\\cos(x-\\mu)]\n= 1-\\frac{I_1(\\kappa)}{I_0(\\kappa)}.", "4a87640e4a8816fe480bee7de142a1df": "\\boldsymbol \\Sigma = \\boldsymbol \\sigma \\boldsymbol \\sigma^{\\mathrm T}", "4a8772fa0512f34f0176fb0337fcc0ab": "(5+2)+1=5+(2+1)=8 \\,", "4a8785a3c9eddd86d32cbea1e7a04945": " m a = m \\frac{\\mathrm{d}v}{\\mathrm{d}t} = m g - \\frac{1}{2} \\rho v^2 A C_\\mathrm{d}", "4a87937ae49b28e3cdfb9e9844d34589": "EPR", "4a87b28d6fb3c9572f112f6eddcbcfc5": "g_i (F_i) = F_i - d_i = L_i", "4a8876824f921beea6b682ba255b93c7": "HH^T=I_n", "4a889f3af5082e208b64d25d3566a311": "\\psi(x)=\\psi(x+l) \\,", "4a88d1645f7c6790b50abbce396e850a": "\\Sigma = 2^{AP}", "4a88d9bc20ec10e266978c8eeef2e9be": " a_0 = \\frac{4 \\pi \\varepsilon_0 \\hbar^2}{m_e e^2} ", "4a88fd675f48dec4ff1cd8f8acdc6c08": "\n\\mathbb{R}^n", "4a8918c1ef52dea79522a131c842ff7a": "0=d^2C_p=dG_{p+1}=dF_{p+1}+H\\wedge G_{p-1}.", "4a893a7f046b352893dc3b1738abecbb": "M = a_0 + a_1 S + a_2 \\left[ I \\left( I + 1 \\right)-\\frac{1}{4} S^2 \\right] ,", "4a896742e0ab4376d2696e14aa380fd7": "(u|v)_E=(f|v)\\,", "4a89ab921c132c43f3603fbe6403784f": "0\\leq i \\leq n", "4a89b201328ac3923779cf0d1249cdca": "{\\mathbf e}_V={\\mathbf e}_U\\cdot h_{UV}", "4a8a08f09d37b73795649038408b5f33": "c", "4a8a24acdb0dea48d26ba9fd487438c6": "E\\to B", "4a8a3ff0bf5b0489e1e692f9d8445738": " = a_1 \\cdot T\\{g_1(t)\\} + a_2 \\cdot T\\{g_2(t)\\} + a_3 \\cdot T\\{g_3(t)\\} + \\dots + a_n \\cdot T\\{g_n(t)\\}", "4a8a7c26107fb59050b333814368e06f": "M(a_0 - a_{n+1})", "4a8a8e9560f3c668575fa094be8fc1df": "T(x) \\in \\Theta \\left( x^p\\left( 1+\\int_1^x \\frac{g(u)}{u^{p+1}}du \\right)\\right)", "4a8a98917866692dbfcc926c53b22ac1": "\\vec{s}=(s_x,s_y)^T=\\vec{0}\\,\\!", "4a8af6bbca3a81c044bfac7589b1dbae": " \\langle \\mathcal{B} \\rangle \\le 2 ", "4a8b40737d233910bc3d16eb5eca7058": "A \\oplus B = B \\oplus A", "4a8b66a50405aa5d91cde57dc4e930cd": "\\theta_{22}", "4a8ba66748c43e0ec48efa85f7928623": "W=N!\\prod \\frac{g_i^{N_i}}{N_i!}", "4a8baa8430b2d58a1cf134ac6cb02956": "\\mathbf{n}_0", "4a8bbbe711795c257e000ce5f416e40b": "m(n)", "4a8bd6e60cd08f4fa9e5f5803e4c69ab": "j_n(x) = \\sqrt{\\frac{\\pi}{2x}} J_{n+1/2}(x), \\text{ for } x \\ge 0", "4a8cc9ea9d0372702d545764be72c02f": "2^{|k|}/2^{|l|}", "4a8cd4f8ae669e55cf0fcd868e8fc55c": "E(t) = \\frac{C(t)}{\\int_0^\\infty C(t)\\ dt}", "4a8d0edb47b53e16ef441a2e6efb5679": "\n\\xi_{\\times\\times}(\\Delta\\theta) = \\langle \\gamma_\\times(\\vec{\\theta}) \\gamma_\\times(\\vec{\\theta}+\\vec{\\Delta\\theta}) \\rangle\n", "4a8d1c03e59893475119f8fa70d1f27f": " E_n^{(0)} \\ne E_k^{(0)}", "4a8dc516b81a96635e1f9ff67389ee0d": " \\delta_x(s)=(s',1)", "4a8dcba3b5f16f24d77222e3df0326af": " X_{01} = \\sqrt {{Z_{01}}^{2} - {R_{01}^{2}}}", "4a8deaa2a259419764da2c472f8498f8": "B_n = (-1)^n B^\\prime_n", "4a8e349d51257028fe89eda61eb35f13": "i_$", "4a8e38e8d64cbca0241fcdb7828d90d2": "\\hat x(\\chi) = \\chi(x).", "4a8e56fdb34aadfeee553eec6aa6b1c3": "\\uparrow\\uparrow,\\uparrow\\downarrow,\\downarrow\\uparrow,\\downarrow\\downarrow", "4a8e950d2649299ab142ea32d7f87837": " \\mathbf{F} = \\alpha_{\\rm L} q\\;\\mathbf{v} \\times \\mathbf{B}\\;. ", "4a8eee7f718786880d1a1b9427715c3b": " \\bold g(x) ", "4a8ef0dd7593b2ec76fabe2e3bc5e84f": "|P_\\mathrm{R}| = |P_\\mathrm{\\gamma}| \\,", "4a8ef3766be95af63b563e024132de7c": "V_{in} = V", "4a8f537d954131e88a29d44b73932915": "k = \\frac{2\\sigma}{ax + by + cz}", "4a8f90bbfac9b24e17f4e193519d2716": " \n \\begin{align}\n w(x,y) & = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{16 q_0}{(2m-1)(2n-1)\\pi^6 D}\\,\\left[\\frac{(2m-1)^2}{a^2}+\\frac{(2n-1)^2}{b^2}\\right]^{-2} \\,\\times\\\\\n & \\qquad \\qquad \\quad \\sin\\frac{(2m-1) \\pi x}{a}\\sin\\frac{(2n-1) \\pi y}{b} \\,.\n \\end{align}\n", "4a8fafd4735504d8242b3adaf7da4a88": "\n C_1 = -\\frac{125}{24EI}(-40145 + 100 M_c + 1632 R_a) \\quad \\text{and} \\quad\n C_2 = \\frac{25}{4EI}(-1315 + 2 M_c + 48 R_a) \\,.\n ", "4a8fd6f87d0c06db227ed8b36b5259fb": "\\exp(x) = \\lim_{n\\to\\infty}\\left(1 + \\frac{x}{n}\\right)^{n}", "4a8fdc54f0659b9b98f1a887cf422a05": "\\langle\\psi'_r|P_3|\\psi'_r\\rangle = 0", "4a906b3e5b42ece8d88762c991dcca6a": "-\\sqrt{\\frac{1}{70}}\\!\\,", "4a9071061a2a18725c6d8562389e4db7": " R_{\\rm eff} = R.", "4a9088660daa79ede755cf29577aaae0": "(f\\circ i_j)_{j \\in J}.", "4a908fa97b73526e2a6a4760a4b6f595": "\\begin{matrix} {2 \\choose 1}{3 \\choose 2}{40 \\choose 1} \\end{matrix}", "4a90bfc3f083536bd0214878a0153a93": "Z/3", "4a913b26c20075767af2bf305caddaea": "{O}^{''}_i", "4a915ce601826937379b7ea865eeb7dd": "\\mathbf{J}_1", "4a91b15e61921710331910dcd0fcc9d4": "W_3=\\beta w_2=0", "4a92156ddf09644076d289d066f609ea": " \\{ x_1, \\ldots , x_k \\} ", "4a9271d89a2558a43a5b9866c0a1cc9a": "PG(3,q)", "4a92ee06bc6a172d1cef7f4fee25e730": "i(U_k)=U_k", "4a933b6c30d5c487529ef2fa81614d38": "\\textstyle \\int\\limits_{-N}^{N} e^x\\, dx", "4a939abd97821b012d08d988dced24d6": "B_\\nu(T) \\approx \\frac{2 \\nu^2 }{c^2}\\,k_\\mathrm{B} T", "4a93b581b94834c3da9f3d904cb2872a": "f^*(x^*) = \\sup_{x \\in X} \\left \\{\\langle x^*,x \\rangle - f(x) \\right \\}.", "4a93c5d4b007af8c5757f9f2326b6900": "f\\colon W\\to W'", "4a9411c02b560c0003bec8231ceccfba": "\\models_{\\mathrm P}\\psi", "4a948b84fb48abf44a283d778f10278e": "\\int x\\,\\operatorname{arsech}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{arsech}(a\\,x)}{2}-\n \\frac{(1+a\\,x)}{2\\,a^2}\\sqrt{\\frac{1-a\\,x}{1+a\\,x}}+C", "4a948c150058032b540b4bf0f4831240": "i \\hbar ", "4a94c8ae8e721e16feec5b7a6e8bc136": "A(d)", "4a94ce2dd4003751ca60bd928f3e7ada": "d=4,5", "4a94d07e2c7baac4410c28166710ef17": "(a\\operatorname{P}b) \\Leftrightarrow (a\\operatorname{T}b) \\wedge \\neg(b\\operatorname{T}a).", "4a94eb9bf72a6ba1b34e34984063f375": "\\scriptstyle\\alpha", "4a953c8536e55dcf59d71978764bb126": "\n \\log^* n :=\n \\begin{cases}\n 0 & \\mbox{if } n \\le 1; \\\\\n 1 + \\log^*(\\log n) & \\mbox{if } n > 1\n \\end{cases}\n ", "4a95ae0297991fb5b7b43771f3625b8e": "\\omega_G(X_g) = (T_g L_g)^{-1} X_g ", "4a95daa9005b010b5cb07a3236b1a898": " V_1(q) + V_1(-q) = 2A(q^2)", "4a962b7491423d6247796582a4cb7b61": " C=C_o \\text{ at } x=0 ", "4a96421a885ea6c4a06e101d48ab1b4e": "0.123\\ldots k0123\\ldots k0123\\dots = \\dot0.123\\ldots\\dot{k}, \\, ", "4a966136faa3021ff041c52c41c427c2": "\n\\begin{align}\n& {} \\quad \\int x \\cos(x^2+1) \\,dx = \\frac{1}{2} \\int 2x \\cos(x^2+1) \\,dx \\\\\n& {} = \\frac{1}{2} \\int\\cos u\\,du = \\frac{1}{2}\\sin u + C = \\frac{1}{2}\\sin(x^2+1) + C\n\\end{align}\n", "4a969ad2f64b0bf7f084ca030598c7d6": "v_{{RMS}} = \\sqrt { 4 k_B T R \\Delta f },", "4a969bcdc5ae952d3af667a3664daa5b": "\\scriptstyle 0 \\;<\\; r_n \\;<\\; \\frac{1}{n}", "4a96c0c7d7f2e663574fe0c8f4e65683": " P_B=P_g \\Rightarrow", "4a96e666a68fee31c7d54c7122f292fb": "E^2\\left(t^{'} \\right)+E^2\\left(t^{'} -A\\left(t\\right)\\right)+2E\\left(t^{'} \\right)E\\left(t^{'} -A\\left(t\\right)\\right) \\, \\mbox{ where } \\, A\\left(t\\right)=\\left(D_V\\left(t\\right)-D_F\\right)/c \\, \\mbox{ and } \\, t^{'} = t-D_F/c", "4a97190c63e0a600ea5d65409f37d281": " 0.5 + \\frac{q_\\mathrm{trans}^2}{2(9.8)(.5^2)} = 2.20", "4a976e1a62f07985bb222de34adeeaf6": "\\frac{r''}{(1+r'^2)^{\\frac{3}{2}}} - \\frac{1}{r(z) \\sqrt{1+r'^2} } = z - \\Delta p^*", "4a97e428bffc638702da5121ad99ff59": "\\nabla_\\alpha A^{\\alpha} = 0", "4a97e6537c204b3f6644d9fdc05a3225": " p'_y", "4a98266ae0b94547dc55627a0c7e2161": " I = I_0 e^{-A\\ln 10}\\,", "4a98759a247ca8ac120329434fe067ba": "X = \\langle C_X(E_0) \\mid E_0 \\subseteq E, \\text{ and } E/E_0 \\text{ cyclic }\\rangle", "4a98a05e996a114028b7e983c44f506d": "\\mathbb{E}_{X}", "4a98a225ef44fb395b9400d366fb5ce8": "\\dot{m}", "4a98d22ace9303f47603de2f718bb820": " 1 \\rightarrow N \\rightarrow G \\rightarrow G' \\rightarrow 1 ", "4a98f8ac146248de91334c3ecbc178c3": "\n \\hat\\theta_\\mathrm{mle}\\ \\xrightarrow{p}\\ \\theta_0.\n ", "4a99322e507ef9a514fa014cd1dd1892": "\nM_{sys} = M_{single} + M_{pair}\n", "4a995565564c0e432cd7a71ac676e19a": "\\mathbf{A} \\mathbf{B} = -A_{23}B_{23} - A_{31}B_{31} - A_{12}B_{12} + (A_{12}B_{31} - A_{31}B_{12})\\mathbf{e}_{23} + (A_{23}B_{12} - A_{12}B_{23})\\mathbf{e}_{31} + (A_{31}B_{23} - A_{23}B_{31})\\mathbf{e}_{12} ", "4a9985bc855db08736e7690fbe7ef9e0": "B=A+1.914\\times \\sin(W\\times (D-2))", "4a99970c1bda96281a7a13b6104c9eca": "\\frac{d}{dx}f(x)=f(x)(1-f(x))", "4a9a0a6ee0f381fa487e6d893af7bb1a": "\\scriptstyle D=\\sqrt{n(n+1)}", "4a9a6f38f892b36287c92af9c17af7ca": "\\epsilon_r ", "4a9a835ea0a0eee0a85c531a608b3ad8": " V_\\rho ", "4a9abbd925c899f2efc26aa4d7592271": "1,\\dots,n", "4a9b479bf925a2a5ea9a15f108c0b3b5": "\\forall \\alpha,\\beta \\in \\mathbb{N}^N", "4a9b4ba2cfc4ed6bb12d65715256d5a4": "\n \\frac{1}{\\beta W}\\left[\\frac{d^4 W}{dr^4} + \\frac{2}{r}\\frac{d^3 W}{dr^3} - \\frac{1}{r^2}\\frac{d^2W}{dr^2}\n + \\frac{1}{r^3} \\frac{d W}{dr}\\right] = -\\frac{1}{F}\\cfrac{d^2 F}{d t^2} = \\omega^2\n", "4a9b5b26dfbbea25665028a22b5ad908": "\\kappa _2(\\omega, E_x) = {1/2}\\kappa_2(\\omega)exp[{-4 \\over 3}({{\\epsilon_G-\\hbar\\omega} \\over {\\hbar\\theta_x}})]", "4a9b688ad242fdf2fb0cdc76bad4fc74": "E ( f(x) + g(x) ) = \\frac{f(x) \\cdot E(f(x)) + g(x) \\cdot E(g(x))}{f(x) + g(x)} ", "4a9b7c30e1d2377ff4c44f436e0bf547": "\\,\\! \\prod_{n \\epsilon CW}I_n = \\prod_{n \\epsilon CCW}I_n", "4a9b9dfc71b6686daede588fa21f23f6": "N(s)", "4a9bdaa727626d360d74058dcdaf0dd6": "\\frac{1}{(R-x)^2}+\\frac{1}{(R+x)^2}=\\frac{1}{r^2},", "4a9c0564a7ba3524d3b87d06be07940b": "\\,p_x+q_x=1", "4a9c2fd89f1bec25a4747128ad58b83d": "dz_2", "4a9c5d77ce34d99d7c215507e3300df2": "\\scriptstyle \\rho \\frac{\\partial\\mathbf{u}}{\\partial t}", "4a9d9fc3e2292df6f6d475e2deaca8fb": "O(n ^ k \\cdot \\log n)", "4a9dd4595954a3935a3b4ab4a87a80e6": " x+ {\\rm i}y \\in \\rm (negative ~ integers)", "4a9de2e2c9458897bee64399967e4d85": "\\mathfrak{cend}_n", "4a9de40c6167eb7318d487a30f25e726": "\n\\Gamma(s, z)= \\cfrac{z^s e^{-z}}{1+z-s+ \\cfrac{s-1}{3+z-s+ \\cfrac{2(s-2)}{5+z-s+ \\cfrac{3(s-3)} {7+z-s+ \\cfrac{4(s-4)}{9+z-s+ \\ddots}}}}}\n", "4a9e3027b5db83b54b71d91657e4a61a": "dn(v,\\vartheta)=nf(v)\\frac{2\\pi\\sin \\vartheta}{4\\pi} dv d\\vartheta", "4a9e5e1fb4832c9380fd98d216cacd6d": "a \\le x_0 < x_1 < \\cdots < x_{n+1} \\le b", "4a9e767576a8017a0d02ea5d46fb1e86": "\\boldsymbol{G}_{Had}=\\begin{pmatrix} 0\\ 0\\ 0\\ 0\\ 1\\ 1\\ 1\\ 1\\\\ 0\\ 0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\\\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\end{pmatrix}", "4a9e878ad04fe19a5efedb3928003d5f": "u_0=F(y_0,x)", "4a9e8c6611a97a59076803a98fb79903": "I\\subseteq\\{1,\\dots,n\\}", "4a9e8d9b027253edc526349c36c4a0d6": " \\left( \\sum_{n=0}^\\infty p(n) x^n \\right) \\cdot \\left(\\sum_{n=0}^\\infty a_nx^n\\right) = 1 ", "4a9ee01c6e459603cbb229b25131c650": "~x(a)~", "4a9efc06ef00f6cace4cfba6a053025c": "u = \\frac{1}{L_{g}L_{f}^{n-1}h(x)}(-L_{f}^{n}h(x) + v)", "4a9f0572d6087b27d99572833215b0c6": "\\ (gh)^{\\omega} = g^{\\omega} h^{\\omega}.", "4a9f465a007f50565ecb85f48f73a1ac": "\\|\\mathbf{x}\\|_p := \\bigg( \\sum_{i=1}^n |x_i|^p \\bigg)^{1/p}.", "4a9f61de3d17c4d14f37fdf52c434500": "\n\\begin{align}\nnR & \\leq \\sum(h(Y_i)-h(Z_i)) + n \\epsilon_n \\\\\n& \\leq \\sum \\left( \\frac{1}{2} \\log(2 \\pi e (P_i + N)) - \\frac{1}{2}\\log(2 \\pi e N)\\right) + n \\epsilon_n \\\\\n& = \\sum \\frac{1}{2} \\log (1 + \\frac{P_i}{N}) + n \\epsilon_n\n\\end{align}\n", "4a9faeed0527bb80ef6e4afe1ccc7268": " \\mathbf{L}", "4a9fd32a22509b1dcd6c456f87f5cd77": "E_{\\mathbf{k}\\sigma} = \\sqrt{\\xi_{\\mathbf{k}\\sigma}^2 + |\\Delta_{\\mathbf{k}\\sigma}|^2}", "4aa00a3130f223a6600998adec795f98": "f(x,y,z)=z^6+x^4y^2+x^2y^4-3x^2y^2z^2 \\, ", "4aa047b5cbc3717a6eba50c7453d131c": "\n[x, \\dot x] = x {dx\\over dt} - {dx \\over dt} x = 1\n\\,", "4aa0c22a317b7cb65aadef09858cfe7d": "( 1 - dt/\\tau )", "4aa1d31a0f1665b69015b38d5716b280": "dt^3", "4aa1dcde48966434777f062bb75f3482": "\nA=s_1^3+s_2^3=2(x_0^3+x_1^3+x_2^3)-3(x_0^2x_1+x_1^2x_2+x_2^2x_0+x_0x_1^2+x_1x_2^2+x_2x_0^2)+12x_0x_1x_2\\,,\n", "4aa1e4d40cc94c12a5e379a36f52323a": "y_{0} = \\frac{N[\\Gamma(2m+2)]}{a[2^{2m+1}][\\Gamma(m+1)]}", "4aa280853e4825693f930f2c2a888564": "y=B-Ce^{-\\tau}\\,", "4aa2a42f0ab79df2bc4a7a3f300dbefe": " Vol_q(y, pn)", "4aa2cc509ae17b96140e17f519a98b80": "\nC_{\\beta I}^{\\;\\;\\; K} e^\\alpha_K e^\\beta_J - C_{\\beta J}^{\\;\\;\\; K} e^\\beta_I e^\\alpha_K = 0 .\n", "4aa346a42b839268a2ebba8fcc6a8a50": "\\tfrac ca\\ =\\ k", "4aa38931541ffba96bea1c1bf36df67c": " = \\frac{1}{N n (n - 1)} (\\sum_{i=1}^N \\sum_{j=1}^k n_{i j}^2 - N n) ", "4aa3b4e7102acda0f860712ed25c5071": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ -\\sqrt{\\frac{3}{2}},\\ 0,\\ 0\\right)", "4aa4081a4c1bafb15fe5c950b857934c": "\\|\\cdot\\|.", "4aa424f0d12cecae15ca607d6d50bc51": " f\\cdot g \\neq g\\cdot f", "4aa456854018a1072fada728df191444": "e^{-\\Delta E/kT}", "4aa4d74ac9a3a4aa7304a219bfbc62b1": " dx:dy:du = F_p:F_q:(pF_p + qF_q). \\,", "4aa4ee7f1323b7137a1e5c9213434b33": " s \\leftarrow \\delta_{ext}(s,t-t_l, x)", "4aa5119f2cf9bff32124faf9eb9fdd60": "\\vec{U}", "4aa53ac818744c6f9eed645d1622356a": "k \\sqrt{g h} = \\frac{2\\pi}{\\lambda} \\sqrt{g h}", "4aa551fbcfe97fbd4468f5d86a324a30": "0 < p < 1", "4aa5a802db1673b9eedf4a8f4fbb627c": "\\{P^i\\}", "4aa5bb6542e155fd2ff61e9bdde8811c": "\nZ = R + iX\n", "4aa617ba1d996b49725d09dd285d7698": "Y \\in \\mathbb R^k", "4aa6321474d6cf7e7ffff8c9d236e618": "\\lambda_{\\rm min} \\approx \\frac{1239.8 \\text{ pm}}{V\\text{ in kV}} \\,", "4aa664c3bf1619877504ebdacc2c6d89": "x\\ge121.", "4aa6a1f03132126c67b6717d669988be": "f(\\mathbf{x}+\\mathbf{y}) = f(\\mathbf{x})+f(\\mathbf{y}) \\!", "4aa6ca5c4af490591635352db4bb90f5": "n(\\vec{r})", "4aa6dad37d4cb79826b9adf27e791bb4": "\\forall x \\forall y \\forall s [(\\langle s, x \\rangle \\in F \\and \\langle s, y \\rangle \\in F) \\Rightarrow x = y])].", "4aa6f1f2420a5f57747e213997b62625": "f_i: V\\to V_i", "4aa7135909f54adaae364500bd241a90": "{q} = e^{2\\pi i\\tau}", "4aa727b6a9c65d6dc2801371efba198b": "f(0) = 2", "4aa7332331ca8ba6886affa1ca196168": "\\eta(s_n)=0", "4aa7334b1e23e5b6095c0110758dfc87": "-\\frac{1}{\\pi} \\int_r^\\infty \\frac{F'(y)}{\\sqrt{y^2-r^2}} \\, dy = \\int_r^\\infty \\int_y^\\infty \\frac{-2 y}{\\pi \\sqrt{(y^2-r^2) (s^2-y^2)}} f'(s) \\, ds dy.", "4aa74366a5f45e9cdc744ac219cf09a1": "a = b^k", "4aa814badf83765fa214a3d82aec0815": "\\operatorname{angle}(x,y) = \\arccos \\frac{|\\langle x, y \\rangle|}{\\|x\\| \\cdot \\|y\\|}.", "4aa860591041ba40b84856467cde883c": "s\\cdot \\frac{s^2}{(2^2+2)r^2}, \\qquad s\\cdot \\frac{s^2}{(2^2+2)r^2}\\cdot \\frac{s^2}{(4^2+4)r^2},\\qquad s\\cdot \\frac{s^2}{(2^2+2)r^2}\\cdot \\frac{s^2}{(4^2+4)r^2}\\cdot \\frac{s^2}{(6^2+6)r^2},\\cdots ", "4aa861124eff57dd7988faa6753e8b7e": "n_j", "4aa88e28e0285df0757eab6c2155b218": "\\mathrm{tr}( \\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a},\\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha,\\alpha^*) g_{\\Omega}(\\alpha,\\alpha^*) \\, d^2\\alpha.", "4aa8aa383de1b63e3d0c61df3d13a1da": "\n\\frac{1}{N_N^2} = \\frac{1}{N_A^2} + \\frac{1}{N_B^2} + \\cdots + \\frac{1}{N_n^2}\n", "4aa8b2e2b8db974b711e2cfad2715a1d": "M^{(-)}", "4aa8ba5a01a8a8a9a031120dd450424e": "\\ln p = -\\frac{m}{n} \\left(\\ln 2\\right)^2.", "4aa8d702b024e97aba3cb54016a6381c": " \\lambda = E", "4aa91f3a4af163de0bbfa87ec1b29ed2": " t_n ", "4aa9307a873d40ea416e3a6b4757bd91": "(E, \\mathcal E^*)", "4aa93e2b85bb1519dd2cb731ccd15086": "Q= \\int_R q |\\psi(\\mathbf r)|^2 \\, {\\rm d}^3 \\bold{r}", "4aa973612ff613805206ae7eded5c605": "\\alpha_k > 3", "4aa9a95365172a3442da7e1293e65208": "PC_x \\subseteq C_x \\subseteq QC_x.", "4aa9b79d78997f83de5e19cf60fbee7e": "\n\\begin{align}\nx_s(t) & = \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\delta\\left(\\frac{t - nT}{T}\\right) \\\\\n& {} = T \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\delta(t - nT).\n\\end{align}\n", "4aaa14d0167b2dce9ebfea0100eb7cdc": "x_{n-1}(t) - v_{n-1}(t)^2/2b_{n-1}-s_{n-1} \\ge x_n(t) - \\left[v_n(t)-v_n(t-\\tau)\\right]\\tau/2-v_n(t+\\tau)\\theta-v_n(t+\\tau)^2/2b_n", "4aaa2955d56196da8f3041d1fc0ddcc7": "e_1\\leq e_2 \\leq\\cdots\\leq e_n", "4aaa2c37c8929b009b76cb22d89d0c18": " \\mu_{H} ", "4aaa578f0d489dcca5d09ed4c6371624": " \\mathbf{p}_k^\\mathrm{T} A \\mathbf{p}_k = (\\mathbf{r}_k + \\beta_{k-1} \\mathbf{p}_{k-1})^\\mathrm{T} A \\mathbf{p}_k = \\frac{1}{\\alpha_k} \\mathbf{r}_k^\\mathrm{T} (\\mathbf{r}_k - \\mathbf{r}_{k+1}) = \\frac{1}{\\alpha_k} \\mathbf{r}_k^\\mathrm{T} \\mathbf{r}_k ", "4aaaa08ef4dbd240d6c1d03e28eb481d": "{F_l} = CQ\\sqrt {\\Delta P} ", "4aaaa6a716e18d0d25f73618086d6f3d": "g(x) = \\begin{cases} 2|x|, & \\mbox{if } x < 0 \\\\ 2x+1, & \\mbox{if } x \\ge 0. \\end{cases} ", "4aaad6f8c76cdb7c7aa2ad810736ffc8": "b_i\\not =0", "4aab47a793488ab7e50e078a1e34e6e2": "a \\times b,", "4aab99bbbaf6e8f8f46c1ca220ca29cf": "G = q = 2 k \\sin(\\theta) = \\frac{4 \\pi}{\\lambda} \\sin(\\theta). \\, ", "4aabc4008c6694206a5436122aa3350b": " \\mathbf{E} =\\frac{q}{4\\pi\\epsilon_0}\\ \\frac{\\mathbf{\\hat r}}{r^2} ", "4aabe9b66ae6d324b37768e1e8c3aade": "(e)\\text{ }P(R_{i}\\bigcup R_{j})=FALSE\\text{ for any adjacent region }R_{i}\\text{ and }R_{j}.", "4aac1070890370c776a8b4574600e3f2": "r,\\theta", "4aac34e5659b8e558e792ca0117a282d": " \\delta W = (Q_1 + Q^*_1)\\delta q_1 + \\ldots + (Q_m + Q^*_m)\\delta q_m = 0,", "4aacee8e58d860013d28f09cc0215d8a": "| X^* |", "4aad32dd5d6a253f25a0b77060339e68": "\\scriptstyle e_2 \\;=\\; -15", "4aad50798b24cebd0f55e4700d1543c2": "J'", "4aad813a7c903d793917edc6bc742455": "\\mathbb{E}[X\\,|\\,\\mathcal G]\\le\n\\liminf_{n\\to\\infty}\\mathbb{E}[X_n\\,|\\,\\mathcal G]+\\varepsilon", "4aada4ec799c74edd760d83f682ef7f9": "\\Omega(N^4)", "4aae00fe2712356ba7e06a1317e66433": "x_{it}(\\xi_{[t]})\\geq 0", "4aae38b330590617f25b6715677f7e72": "T\\approx\\rho_0\\hat T", "4aae40f4672504dc6019a3be16db71da": " f(E) = \\frac{k}{\\left(E^2-M^2\\right)^2+M^2\\Gamma^2}. ", "4aae86e6e97fb75183e6bef11cb5fe7d": "p_{m}\\nabla^{2}q_{m}-q_{m}\\nabla^{2}p_{m}=\\nabla\\cdot\\left(p_{m}\\nabla q_{m}-q_{m}\\nabla p_{m}\\right),", "4aaf2186a6cc6882e5a81ac5ca94f16f": "-100\\le x,y \\le 100", "4aaf2ab0c2a4034b86fb342271775917": " \\exists f \\forall x_1 \\cdot \\cdot \\cdot x_n", "4aaf2f25840d9eb755c8ee78cba2f022": "\\mathbf{r}_2", "4aaf6e28081a29bed745bb49ad3c0bf8": "s\\in X ", "4aafa8933806577a6cc62459a8600df4": "L(p)=P(X_1=x_1, \\ldots, X_n=x_n \\mid p)=\\prod_{i=1}^n p^{x_i}(1-p)^{1-x_i}=p^s (1-p)^{n-s}", "4aafc995d7aca27faaf8917cb5ba9381": " {\\tau} = \\frac{q}{t}", "4aaffb3a73cabce99dc5492c5a36b303": "(\\ast)", "4aafffc8ec33b64461ac3a0aeb5281e6": "\\tfrac{3}{2}\\log_2 n+\\theta", "4ab021d5e4f7c6a21d94875994ba47f9": "\\mathcal{T}\\subseteq \\mathcal{V}", "4ab02a0e49c9dfa3ab2041f59fb8a34a": "\\begin{align} r^2 &= b^2 \\sin^2E + (ae-a\\cos E)^2 \\\\\n&=a^2(1-e^2)(1-\\cos^2 E)+a^2 (e^2 -2e \\cos E +\\cos^2 E)\\\\\n&=a^2 -2a^2e \\cos E +a^2e^2 \\cos^2 E \\\\\n&=a^2 (1-e \\cos E )^2\\\\\n\\end{align} ", "4ab0bd214f3d146270b796ac85ff3aa7": "\\Theta_0=M^2 {4\\lambda \\over \\pi D_0}", "4ab116d374c7a8d2bc07c18d8761059b": "G(x) = \\frac1I\\int^b_aG(t)\\varphi(t) \\, dt.", "4ab19098754db6e46a9a79b5c5b6911a": "\\,E[Z]=0", "4ab1a475a63e9e1d4e447020183ff3a2": " y_i = f(x_i) ", "4ab1b9b0e947b0318ec7959a5378c89a": " y = \\alpha + \\beta x^2, \\,", "4ab1bc8ca7954bb27e6258fdde4c0bae": " \\hat{f_i}^{(j)} \\leftarrow \\text{Smooth}[\\lbrace y_i - \\hat{\\alpha} - \\sum_{k \\neq j} \\hat{f_k}(x_{ik}) \\rbrace_1^N ]", "4ab1d1b1f45b604be2f4aac86a50ce21": "\nT = \n\\frac{L_{1}^{2}}{2I_{1}} + \\frac{L_{2}^{2}}{2I_{2}} + \\frac{L_{3}^{2}}{2I_{3}}\n", "4ab1fc7d9c39a6f350ff5f0a9f04abff": "\\begin{align}\n\\underbrace{X_{1/T}\\left(\\frac{k}{NT}\\right)}_{X_k} &= \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}} \\quad \\quad k = 0, \\dots, N-1 \\\\\n&= \\underbrace{\\sum_{N} x_N[n]\\cdot e^{-i 2\\pi \\frac{kn}{N}},}_{DFT}\\quad \\scriptstyle {(sum\\ over\\ any\\ n-sequence\\ of\\ length\\ N)}\n\\end{align} ", "4ab227cd8c4e7f2848f043e1e98657d0": " I = \\star ( F \\wedge \\star F ) = F_{ab} \\, F^{ab} = -2 \\, \\left ( \\| \\vec{E} \\|^2 - \\|\\vec{B} \\|^2 \\right) ", "4ab2597eae3ee51e9be364f484e6e7b9": "\\sqrt{2} \\sqrt{ 2 - \\sqrt{2} }", "4ab2647fb3e96553ac7b7945ddde9492": "(w_1 + w_2)", "4ab269358da151912cc6357cde59c846": "4 N + 4", "4ab271ec31612ce1d87d1c1019c368aa": "\n \\cfrac{1}{c_0^2}\\frac{\\partial\\tilde{p}}{\\partial t} +\n \\langle\\rho\\rangle\\nabla\\cdot\\tilde{\\mathbf{v}} = 0\n ", "4ab277657f1060a9a377c23c9f869a78": "(V/N)", "4ab2794525f2bd3d6c4f0f2c4990ee13": "\\biggl( \\sum_{k=1}^n |a_k|^2\\biggr) \\biggl(\\sum_{k=1}^n |b_k|^2\\biggr) - \\biggl|\\sum_{k=1}^n a_k b_k\\biggr|^2 = \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n |a_i \\overline{b}_j - a_j \\overline{b}_i|^2", "4ab2a090a7738b0d7fe1baf50e057764": " x_S = \\prod_{i \\in S} x_i \\!", "4ab2a553b5eefe8f6db9aef12ef0d518": "\\Omega(\\vec R)", "4ab2fe41f4665d5c02aa6b185878f766": "x:S^{2k+1}\\subset M", "4ab31eaf66eff9718941431bbb854126": "n> 0", "4ab388040b12414f16331bedf239e36f": "A_a^i (x)", "4ab3995ce479759cbe7be828f8bcc4f5": "44,100 \\times 16 \\times 2 = 1,411,200\\ \\mathrm{bit/s} = 1,411.2\\ \\mathrm{kbit/s}", "4ab3a8613b208f42ce0eaa51f2cab26f": "O(1/n^{1/2})", "4ab3fc5d75286619c1c25dccd38bcc7b": "s = |S|\\,", "4ab4509f4def71e29945d14a88bce62b": "\n\\begin{array}{lcl}\n\\theta_{1,\\dots,\\infty} &\\sim& H() \\\\\n\\boldsymbol\\beta &\\sim& \\operatorname{Stick}(1,\\alpha) \\\\\nz_{1,\\dots,N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\beta) \\\\\nx_{i=1,\\dots,N} &\\sim& F(\\theta_{z_i})\n\\end{array}\n", "4ab498afec33fe36327d6e9bd09be1b3": "\\Rightarrow_{r_1} X X S \\Rightarrow_{r_1} X X X X \\Rightarrow_{r_2} Y X X X \\Rightarrow_{r_2} Y Y X X \\Rightarrow_{r_2} Y Y Y X \\Rightarrow_{r_2} Y Y Y Y", "4ab513eab13a864702d53c09e85b99e1": "\n\\phi_{RMS} = \\lambda r_e \\left( a L \\right)^{1/2} \\left[ \\langle \\delta N^2 \\rangle \\right]^{1/2},", "4ab57cfd857442a27f5bbcc18a021111": "\\psi = 1-\\phi(\\tan\\beta_2+\\tan\\alpha_1)\\,", "4ab57dde18886b633d9d21fc5b7b522c": "D(E(m_1, r_1)\\cdot E(m_2, r_2)\\mod n^2) = m_1 + m_2 \\mod n. \\, ", "4ab5acdc372744f6b7c947727c9f3e17": "a\\in\\text{cl}(F)\\,", "4ab5e49ed864d1ed0f7bc3e230adc658": "\\left(\\sqrt 2 - 1\\right) \\left(\\sqrt{\\mu/r_0} + \\sqrt{\\mu/r_f}\\right)", "4ab5e982a3441f06f4cbd06381fdb9ad": "\\int_0^{2\\pi} \\sum_{n=-1}^\\infty\\sum_{m=-1}^\\infty m\\,r^{n+m}\\,|a_n|\\,|a_m|\\,d\\theta<\\infty\\,,\n", "4ab5f00d510365670acf7998941a1ebf": " 0 = (x-R)^2 + z^2 - r^2 \\,\\!", "4ab61f79f7a8cba3cd20e302f2349e1c": "|F-P|", "4ab684f1e50e285a624354b1533347b3": " s := \\frac{a+b}{2} ", "4ab6950e0f7f3d919d474a6bd0782db5": " \\langle\\mid r\\mid \\rangle_\\text{free}", "4ab6aadd658d2b4f4499e6ab72977336": "\\varepsilon_{\\Omega+1}", "4ab6bc2f59a0c3065d1ab34cc5f0761c": "\\partial\\Phi/\\partial{z}", "4ab7114c587cba2d4b7b5fdef6a21ab6": "\\Re(a)>0\\wedge\\Re(s)>0\\wedge z<1\\vee\\Re(a)>0\\wedge\\Re(s)>1\\wedge z=1.\n", "4ab7128d4c2de06987dd7c9fba90b6f1": "S_+ = \\bigoplus_{i>0} S_i.", "4ab74a334d9664fbd3d8697a6d86c9f5": " |\\psi\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} \\psi_x \\\\ \\psi_y \\end{pmatrix} = \\begin{pmatrix} \\cos\\theta \\exp \\left ( i \\alpha_x \\right ) \\\\ \\sin\\theta \\exp \\left ( i \\alpha_y \\right ) \\end{pmatrix} ", "4ab7c146d582aa1275fdbb16a0fb4aca": "x = -{\\partial g \\over \\partial p}", "4ab82aaf7cf5dcc5b7a9e0a3db8e59f9": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "4ab842db75aba3aa831b2101c7dd28f6": "\n\\vec{X}(t+1)=\\vec{X_p}(t) - \\vec{A}.\\vec{D}\n", "4ab89bbcff08a0fe336b3f39a90f2745": "(n, d)", "4ab8b948c6de32d298a8ca7b6e502022": "\\beta \\propto k^{-4}", "4ab8ecd47346c59743013cbe1cfc2aa4": "a_i^j", "4ab98e43d56455cc8bd117539df55e21": "V_{\\rm{universe}} = 4 \\times 10^{80}{\\rm\\ m}^{3}", "4ab992fb07cf3fea9da4fd9ee7e8e657": "B^\\ast", "4ab9edf30c5a28fd8e267ba72980087c": "y = \\sqrt 2 \\sin \\left( \\theta \\right),\\,", "4aba2460e8728e5fe0273f5a342e5a5f": "Z_{\\text{eff}} = \\sqrt[2.94]{f_{1} \\times (Z_{1})^{2.94} + f_{2} \\times (Z_{2})^{2.94} + f_{3} \\times (Z_{3})^{2.94} + ...}\n", "4aba457deedcf40858ff7335ae6ab9f9": "v \\mapsto (-1)^{l(v)}", "4aba7a4847e5a9627f93a21351fe3aa3": "M^{*} A M = A \\,", "4aba866b15a65724c2eaa36cb0cccd00": "\\alpha_1\\;", "4abadda36d4b23d49a1591696483867d": "\\left| S - \\sum_{i=1}^{n} f(t_i)\\Delta_i \\right| < \\varepsilon.", "4abb84ad5616531f77319651629a92de": "(\\mathcal{F}_i, \\Sigma)", "4abb886ca18dcbc82ee1b8b1b0aa3c8e": "m:O \\rightarrow X", "4abbaa6f5e6fc37ff53962e25b98c433": "F(c,X) = X^3 - cX^2 + c^pX - 1 \\in GF(p^2)[X].", "4abbca9e53aa20017dc7db6abe7e47e9": "k_j^i", "4abbeb8a497b23707f3e3c9502200153": "\\max_{j\\neq i} b_j < v_i ", "4abc44dff73ba053f60e847288b1fd0c": "E^{(1)}_{lm} = \\frac{1}{l(l+1)}\\int \\mathbf{E}\\cdot \\mathbf{\\Psi}^*_{lm}\\,\\mathrm{d}\\Omega", "4abc4966e0882f7fa02c4fab614fb450": "\\boldsymbol{\\mu_i}", "4abc77fffbc11c43beea6ee20c8e2e3f": " \\left[ -\\frac{1}{2} \\frac{d^2}{dy^2} + \\frac{1}{2} \\frac{l(l+1)}{y^2} - \\frac{1}{y}\\right] u_l = W u_l .\n", "4abc7983e422f4ad2acacfc0edf015a9": " \\rho( \\mathbf{Frob} (\\mathfrak{P})) ", "4abcb14750e0b5f21547a6fcfe65a0f8": "U_\\mathrm{E} = q_2 \\Phi_1(\\mathbf r_2).", "4abcc92d18b8143bf789eb782d7d3c64": "\\ v = \\sqrt{\\frac{GM} {r}\\ }", "4abcea42ee269b247d2fb6116bfda3f3": " \\int_{\\partial \\Omega} h(z) \\, dz = \\int_{\\partial \\Omega} \\omega = \\iint_\\Omega d\\omega = \\iint_\\Omega (i\\partial_x-\\partial_y)h \\, dx dy= 2i \\iint_\\Omega \\partial_{\\overline{z}} h \\,dx dy.", "4abcf2a16a3b845e4d09e130de9727b1": "B\\in \\Sigma", "4abd35902a6d84e1d33c2e26b225c7ba": "e^{-d \\cdot T}", "4abd864f074f5d13f7dffa57d040f038": "L,\\ \\Mu", "4abdaf113999310d0a4a0b48bf5ac2b9": "1 < i \\leq n", "4abdb2d620180a3177c72b4413581c01": "y = \\sqrt{r^2-x^2}", "4abdc5cf9ef33298c83f3f5f1a908420": "\\dot x = \\partial x /\\partial t", "4abe184a380ec1727c51f71bea4d612b": "\\rho=|\\Psi|^2=\\Psi^*(\\mathbf{r},t)\\Psi(\\mathbf{r},t)\\,\\!", "4abe7475e4c0dd5a339e5f7dc7ad3b41": "M(n) = \\sum_{1\\le k \\le n} \\mu(k)", "4abe92c4c08129dd71e444eb54aea9ca": "Y(x)", "4abed7747bb4a19468ead134551edcf9": "x^2+y^2-z^2=u^5", "4abee4a3677efe39bbbf739574f4082f": "\\Phi \\colon H^i(B; \\mathbf{Z}_2) \\to \\tilde{H}^{i+k}(T(E); \\mathbf{Z}_2),", "4abefeb208b9bae48b91b1c5e6f614f7": "V(r_e -r_h)", "4abf1f5f74a1203f6c46e35334b2204e": "S \\subseteq V_\\alpha \\,.", "4abf5b26593aecc79d4d86fba1f45cb4": " \\Delta V", "4abfa6d8298a591e5e93790d54e7719e": "\\varphi_\\alpha(\\mathbf{x} )", "4abfa903e1d935e0fb17f5fbb8690be2": "z = c", "4abfd1aa4d4e839ce638594b6c9f493d": "Z'\\,\\!", "4abfed30a6873b2ec810346ca7e7ca07": "\n\\begin{align}\ns_N(x) &= \\frac{a_0}{2} + \\sum_{n=1}^N \\left(\\overbrace{a_n}^{A_n \\sin(\\phi_n)} \\cos(\\tfrac{2\\pi nx}{P}) + \\overbrace{b_n}^{A_n \\cos(\\phi_n)} \\sin(\\tfrac{2\\pi nx}{P})\\right)\\\\\n&= \\sum_{n=-N}^N c_n\\cdot e^{i \\tfrac{2\\pi nx}{P}},\n\\end{align}\n", "4abff80f4c0826e7128b8775f8ada84c": "\nF_n = \\frac{1} {\\sqrt{5}} (\\varphi_1^n - \\varphi_2^n).\n", "4ac05af0271c9642a424904907509f42": "f:\\mathbb{R}^m \\to \\mathbb{R}^n", "4ac061287ed0aeda0ee2c79de9dae890": "\n\\frac{\\rm d}{{\\rm d}t}x(t)=f\\left(t,x(t),\\int_{-\\infty}^0x(t+\\tau)\\cos(\\alpha\\tau+\\beta)\\,{\\rm d}\\tau\\right)\n", "4ac1896c8c448b2a8f711ec4ba18b61c": " C(r) = \\frac{4}{\\pi} \\sum_{k=0}^\\infty \\frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}}~.", "4ac199b2417126f1662a10e5a00b215a": "Y_3", "4ac1c68c12605b8dbda007af98d93cad": "\n\\frac{a_0}{2} + \\sum_{m=1}^{\\infty}\\left[a_m\\cos\\left(mx\\right)+b_m\\sin\\left(mx\\right)\\right].\n", "4ac24dede12aefd0aaf1bc5750e08c48": " A \\,\\ ", "4ac2519fb5b318e1f0da3dc700777b2a": "U{}^1_n", "4ac2662ea5eb0b232e203f5ca597cd51": "\\sigma_{(r + f)} = \\frac{MWT}{M_{r}} + \\frac{MWT}{M_{f}}", "4ac2752510696fbfc7b4bee65d8529cf": " \\hat N", "4ac291aa284b495d891545fd718aeda0": "r=|n|", "4ac29869228eca61076e8a38bbf0942d": " LL(\\alpha,\\beta) \\sim \\textrm{Dagum}(1,\\alpha,\\beta)\\,", "4ac2a8128883e59d4b412b3323d786d8": "\\mu_0\\;", "4ac2dfc8b96300812a914dfb90be9333": "-1/4 0", "4acefe457fc782056c0fb8c9cbad7e59": "g^{(n)}( \\mathbf{r}_1,t_1;\\mathbf{r}_2,t_2;\\dots;\\mathbf{r}_n,t_n)=1 ", "4acf3efdcf00c234c48456e676c9b178": " \\left( \\left\\Vert A^{-1} e \\right\\Vert / \\left\\Vert e \\right\\Vert \\right) \\cdot \\left( \\left\\Vert b \\right\\Vert / \\left\\Vert A^{-1} b \\right\\Vert \\right) .", "4acfb24e15f34e6c3d97638f770a3ab9": "(8)\\quad ds^2 =\\frac{2M(u)}{r}du^2 +ds^2(\\text{flat})=\\frac{2M(v)}{r}dv^2 +ds^2(\\text{flat})\\,,", "4ad023ef12a866e1376766871777de4f": "\\scriptstyle{\\times}", "4ad0ace8372fd67b3e3455e2c3c2fee0": "HA + B \\rightleftharpoons A^- + HB^+", "4ad0e4b826088c029af4b3d96d354e63": "\nF(r) = Ar^{-5} \n", "4ad11acf2ad53ead6104dccb13d387c3": "M=\\left(\\begin{array}{cc}\\cos k'd & \\sin(k'd)/k' \\\\ -k' \\sin k'd & \\cos k'd \\end{array}\\right)", "4ad13c3636b2e11f45e7eab1a6297cf6": "\\alpha/2v", "4ad19f8dbcb30fa00e7b5799363b7c3c": "\\Phi_{2\\lambda}=M\\varphi_\\lambda.", "4ad1d3b50ff09701360200455be4be10": "t(n+m)= t(n+m-1) + t(n+m-2),", "4ad1e4d238ef0a3a7bf84901528633cb": "A+uv^T", "4ad1f089ae10f9e0dc20eef0d8c5fc57": "C_\\mathrm{eq}= C_1 + C_2 + \\cdots + C_n", "4ad2107267165fa9e6a19afdd8634772": "\\mathcal{D}[0,1]", "4ad24e31635be044f92934de3f4f4ffd": "K=\\frac{[F_z(F_{xx}F_z-2F_xF_{xz})+F_x^2F_{zz}][F_z(F_{yy}F_z-2F_yF_{yz})+F_y^2F_{zz}]-[F_z(-F_xF_{yz}+F_{xy}F_z-F_{xz}F_y)+F_xF_yF_{zz}]^2}{F_z^2(F_x^2+F_y^2+F_z^2)^2}", "4ad25d2306ae1b3f69ff2d5ee1e3a1a3": " (y_{mt+1} - c_{t+1}) - (y_{mt} - c_t) = \\left( y_{t+1} - \\frac{y_t} {R} \\right) ", "4ad2bcc870d0364745adb9990e324d8b": "(i\\sqrt x)^2 = i^2(\\sqrt x)^2 = (-1)x = -x.", "4ad34a4a7897dd6ea1f2034e8cad7063": "\\hat{A}_a^i \\Psi (A) = A_a^i \\Psi (A)", "4ad377206188e3fa079fcc6328fd55a4": "\\csc \\theta = \\frac{2i}{e^{i\\theta} - e^{-i\\theta}} \\,", "4ad3989d651c0fd1caa1592c581a1a48": "C(u_2,v_2)-C(u_2,v_1)-C(u_1,v_2)+C(u_1,v_1) \\geq 0 ", "4ad3f65fc2a488881597b44d765d40a6": " U = \\frac{1}{\\sqrt{d}}\n\\begin{bmatrix}\n 1 & 1 & 1 \\\\\n e^{i \\phi_{10}} & e^{i \\phi_{11}} & e^{i \\phi_{12}} \\\\\n e^{i \\phi_{20}} & e^{i \\phi_{21}} & e^{i \\phi_{22}}\n\\end{bmatrix}\n", "4ad4343a9623f5307ad6269ccd13671f": "\n\\dot{z}=-D_H z\n", "4ad480ba3ee84d54d3ffe82f266c353f": "\\frac{\\partial F}{\\partial n_2}=\\left(\\sigma_2^2-2\\sigma_2\\sigma_\\mathrm{n}+\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\\right) n_2 = 0\\,\\!", "4ad4c9976c90ad07afcce2d369a7f1d5": "\\left( \\frac{\\partial}{\\partial t} + \\frac{1}{2} \\frac{\\partial^2}{\\partial x^2} \\right) p(x,t) = 0 ", "4ad4ea00c2c8f4f5e89f9ecefa11aa8d": "j_X", "4ad5b5e045ca4c76931304c42be9f86f": "\\sum_{j=1}^n x_{ij} = 1,", "4ad5fb79dc71f7ad9722d672f6f8da55": "\\mathrm{Re}(s)=0,1,2,\\dots", "4ad614f1fe1979bcc27df71930c6a169": "\\exists x.\\phi(x)", "4ad6320b967faba4524ea7ca5f73e256": " \\operatorname{tr} \\mathbf{M}_\\mathbf{Y} (\\theta) \\leq \n(\\operatorname{tr} \\mathbf{I}) \\left [ \\Pi_k \\lambda_\\max (\\operatorname{E} e^{\\theta \\mathbf{X}_k}) \\right ] =\nd e^{\\sum_k \\lambda_\\max \\left ( \\log \\operatorname{E} e^{\\theta \\mathbf{X}_k} \\right ) } ", "4ad63284d72f7c1c4f3b4036b5292200": "aI_{(X \\geq a)} \\leq X\\,", "4ad657c1e57f174975cb122f4fb119e8": "a=a'", "4ad65d4f0aaa2a8e25c6ee04dce867b5": "O \\mapsto O'", "4ad686c0a7cc5aa9ca6597f99b9d750b": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 8.907666\\log_e(T+273.15) - \\frac {7221.386} {T+273.15} + 75.52152 + 5.304278 \\times 10^{-06} (T+273.15)^2", "4ad6cb126b29b6130743254d4c5700a3": "\n\\lambda = .0055 (1 + (2 \\times10^4 \\cdot\\frac{\\varepsilon}{D} + \\frac{10^6}{Re} )^\\frac{1}{3})\n", "4ad6d02164824c7400bdff4eae84a0f3": "l=\\mu-z\\sigma,u=\\mu+z\\sigma,", "4ad6fdb57f39b241a8de601440c2a73e": "\\mathbf{I}_i ", "4ad75808758f3133dac5a9eb81fb0e7f": " \\mathbf{F}_{ext} = -\\left(\\frac{\\mathrm{d}}{\\mathrm{d}t}\\big[m_{gas}(t)\\mathbf{v_e}(t)\\big] + F_{other}(t)\\right)", "4ad7798200505572ffee7a89c3c55a40": "a\\leq s \\leq b", "4ad77a6450cc257b468a54372c03336e": " \\bold { \\nabla \\cdot p(r)}=\\bold { \\nabla \\cdot} \\left( \\chi \\bold{ (r)}\\varepsilon_0 \\bold {E(r)}\\right) =-\\rho_b \\ . ", "4ad7cd6a0ebe7cff95c16588cdd80294": "\\ell = d \\cdot \\log n + O(2^d \\cdot \\log(\\epsilon^{-1}))", "4ad7d02b125c92bf5764cae4b4ea5ca0": "\n\\mathrm {DOF} = \\frac\n{2 f ( m + 1 ) / m }\n{ ( f m ) / ( N c ) - ( N c ) / ( f m ) }\\,,\n", "4ad807af57649eaaf404b1d0b2b819dd": "\n \\int_{x_1^0}^{x_1^1} \\cdots \\int_{x_r^0}^{x_r^1}\\; \\Gamma^{(\\lambda)}(\\mathbf{x})^*_{nm} \\Gamma^{(\\mu)}(\\mathbf{x})_{n'm'}\\; \\omega(\\mathbf{x}) dx_1\\cdots dx_r \\; = \\delta_{\\lambda \\mu} \\delta_{n n'} \\delta_{m m'} \\frac{|G|}{l_\\lambda},\n", "4ad819a37631b9ffb2c75f9e47cb34e4": "\\int_{0}^{\\infty}P(r)dr = 1\\,.", "4ad85633cc204a0671053e8b7842a041": " x_1,...,x_N\\in \\mathbb{C}", "4ad8ab272157d602592c1bb75fc41116": "\\liminf_{n\\to\\infty}(p_{n+1}-p_n),", "4ad8df00873dc55fa73b62a983772f91": "\\begin{bmatrix}\nX&amount&of&commodity&Y\\end{bmatrix}", "4ad9146ab27bf6c6aaf496ab7ec3f713": "\\frac{1}{\\sqrt{ISOI_{cas}}} = \\frac{1}{\\sqrt{ISOI_{1}}} + \\frac{1}{\\sqrt{ISOI_{2}/G_{p,1}}} + . . . + \\frac{1}{\\sqrt{ISOI_{n}/G_{p,1}G_{p,2}G_{p,3}. . .G_{p,n-1}}}", "4ad92b7c3ce30c7cd8b0afaea4abeded": " \\Box A^\\alpha = \\frac{4 \\pi}{c} J^\\alpha ", "4ad955b86164fcfaa134bfa8cd772018": "\\hat{a}^\\dagger_{\\lambda,\\mathbf{k}}", "4ad966c43bfaaedd5e93c54a52f99016": "\\vec{e}", "4ada24b320f5b1f6150079cc8149dcac": "H'=eEz.", "4adabf1377e74f47fa66e30e4c1b3aa8": "\\text{SSMD}= \\frac{\\Gamma(\\frac{n-1}{2})}{\\Gamma(\\frac{n-2}{2})} \\sqrt{\\frac{2}{n-1}} \\frac{\\bar{d}_i}{s_i} ", "4adaf4fe35e082408cb4641b3020e3a5": "\n\\left[\\mathbf{b_{1}}\\mathbf{b_{2}}\\mathbf{b_{3}}\\right]^T =\n2\\pi\\left[\\mathbf{a_{1}}\\mathbf{a_{2}}\\mathbf{a_{3}}\\right]^{-1}.\n", "4adbd997467dfaec36a37fdbc7b4af9f": "=\\frac{1-\\frac{\\varepsilon+\\cos \\theta}{1+\\varepsilon\\cdot\\cos \\theta}}{1+\\frac{\\varepsilon+\\cos \\theta}{1+\\varepsilon\\cdot\\cos \\theta}}\n", "4adbe77c46d48a4ef259a93ecd478a1b": "\\phi\\!", "4adbf4530b62914679d9d8916cd38ef2": " \\mathbf{Y} = \\mathbf{X}\\boldsymbol{\\beta} + \\boldsymbol{\\varepsilon} \\;", "4adc3071f599037544ebd5ef1f2cd1ca": "<\\lambda, \\alpha> \\in \\mathbb{Z} \\, \\forall \\alpha \\in \\Phi", "4adcc1ea3446159194ba0e490d798865": "{ \\sum_{a ", "4ae0da76b1b283df2ccb8dbb4c8a3100": "\\Phi_{ij}=2\\,\\phi_i\\, \\overline{\\phi_j}", "4ae0f60ac0455ced3ccea74c14a99df6": "\\sum_{\\nu\\in\\Lambda} f(x+\\nu) = \\sum_{\\nu\\in\\Lambda}\\hat{f}(\\nu)e^{2\\pi i x\\cdot\\nu}, ", "4ae107aad4861829224d1bd532afb9ea": "L(3) \\propto\\sum_{i=1}^{2}-(i-3)+\\sum_{i=3}^{9}(i-3) =[(2+1)+(0+1+2+...+6)] =24.", "4ae12391cf8895b6734d475110aa89d4": "w(x) = \\min \\{t | x=\\sum_{i=1}^t a_i r^{n(i)}\\} ", "4ae12a2b078ddfec0b71a6e2ee4bb1b7": "\\mathbb{P} \\left( \\max_{1 \\leq k \\leq n} | S_{k} | \\geq 3 \\alpha \\right) \\leq 3 \\max_{1 \\leq k \\leq n} \\mathbb{P} \\left( | S_{k} | \\geq \\alpha \\right).", "4ae12f01f3c5aeee8f675424b1a9f29a": "m = 2", "4ae19b83c23531143f9a401c0284af34": "\\{ a_i \\}_{i=1}^n, \\{ b_i \\}_{i=1}^n ", "4ae1f5b70c0fb2b9678943c5e0dc6355": "E_4 \\approx \\frac12 (\\phi_2 + \\phi_1) (\\lambda_2 - \\lambda_1)", "4ae1fcaca27bf62397e7982b47d28193": "B_\\theta(X,Y) := -B(X,\\theta Y)", "4ae269c33786dd4ac165545cbd7f0177": "f(x)=|x| = \\begin{cases}\n x \\text{ if }x \\geq 0\\\\\n -x\\text{ if }x < 0\n\\end{cases}", "4ae2befece2b40e5c75cb99f16231386": " \\hat {S}_z ", "4ae2edfb4700231edcad13e1e9272b9b": "\\kappa\\in\\operatorname{club}(\\kappa)", "4ae34470c95b6ed76ce58efb68893e76": "\\Delta_4", "4ae3a6658c949ebc34c728f8b6ec3470": "F=-\\frac{L(P)}{4\\pi G M_{*}(P)}g_\\textrm{eff}", "4ae3c0875ed2b3729cbfae34dfef723c": "\\sigma \\gets \\mathrm{Setup}(1^k)", "4ae432e0e3619959c83e7e988ff2c9b2": "\\{2^\\frac{j}{2}\\psi(2^jx-k)\\}_{j,k\\in Z}", "4ae466d65fe4482d3a71505a047bf6bc": "\\begin{array}{cc} P_{j}(d_{j})=\\left\\{\n \\begin{array}{lll}\n 0 & \\text{if} & |d_{j}| \\leq q_{j} \\\\\n\\\\\n 1 & \\text{if} & |d_{j}| > q_{j}\\\\\n \\end{array}\n \\right.\n\\end{array}", "4ae48cf64096a99e00f8e6371dcc4b03": "\\displaystyle{g^+=\\begin{pmatrix} \\overline{a} & -\\overline{c} \\\\ -\\overline{b} & \\overline{d}\\end{pmatrix}}", "4ae4a5a4683b8dba7b5f09794762e0b7": "{}_sY_{\\ell m}(\\pi-\\theta,\\phi+\\pi) = (-1)^\\ell {}_{-s}Y_{\\ell m}(\\theta,\\phi)", "4ae4bbf1012dcb278913bdd05bec1119": "\\lambda = (2,2)", "4ae4c93086aff8bbe548d06324095af7": "c_k=1", "4ae4d5614374931b1fa6255c83b2c58e": "n_k: = 2 \\big\\lceil k \\ln g(k+2) \\big\\rceil \\,", "4ae50361c210855823bf5cbe8cd136d9": " {(f/\\#)}_\\mathrm{microlens} \\le {(f/\\#)}_\\mathrm{objective} \\times \\mathit{ff}", "4ae5382a2c51dc5cc2bde29e415d4652": "\\overline{f * g} = \\overline{f} * \\overline{g} \\!\\ ", "4ae55f1245a275d45c2f876d7edcfcd5": "P(target) = \\frac{A(target)}{\\pi (dy)^2} = \\frac{a r(target)^2 v^2}{(tp)^2 d^4}", "4ae5ef8ea571cfc6ca35ec1316807daa": "\\mathrm{_{12}^{24}Mg} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{14}^{28}Si} + \\gamma + Q", "4ae6034c5e8c83a9b1f707a76c2174cd": "b^2/a", "4ae64c38f7a7ad9444802f405654ce07": "c_p\\rho \\int_{x-\\Delta x}^{x+\\Delta x} [u(\\xi,t+\\Delta t)-u(\\xi,t-\\Delta t)]\\, d\\xi = c_p\\rho\\int_{t-\\Delta t}^{t+\\Delta t}\\int_{x-\\Delta x}^{x+\\Delta x} \\frac{\\partial u}{\\partial\\tau}\\,d\\xi \\, d\\tau", "4ae6680115fe4c8f80375d1fa3ef6a03": "F(s)=\\sum_{n=1}^\\infty \\frac{a_n}{n^s}", "4ae68d9ebb48d4b367c49c1d4b4c72ec": "\\tfrac{2}{q}", "4ae69ab62f0b21551772b1237843dfd0": "M = (Q, \\Sigma, \\Gamma, \\vdash, \\_, \\delta, s, t, r)", "4ae6a7a46fcd5939218be90b885ac1e7": "\\phi+2\\pi n", "4ae6a94d87024f7b4a69fa2981f11e42": "\\frac{x}{480}\\times \\frac{10}{11}=\\frac{4}{3}\\Rightarrow x=\\frac{480\\times 11\\times 4}{10\\times 3}=704", "4ae72c93a7a440a8404317c911cc67ea": "x+0 = x\\ ", "4ae73a8de549e43373746d351c071fff": "\n\\left(\\frac{\\alpha}{\\mathfrak{a} }\\right)_n \n\\left(\\frac{\\alpha}{\\mathfrak{b} }\\right)_n \n=\n\\left(\\frac{\\alpha}{\\mathfrak{ab} }\\right)_n. \n", "4ae75b91715120652990caab2b294d59": "\\mathbf{B}(\\mathbf{r}) = \\frac{\\mu_0 I}{4\\pi} \\int \\frac{d\\boldsymbol{\\ell} \\times d\\hat{\\mathbf{r}}}{r^2}.", "4ae78c844375546595d546513c7821d1": "\\begin{cases} u_{t}=ku_{xx} & (x, t) \\in \\mathbf{R} \\times (0, \\infty) \\\\ \nu(x,0)=g(x) & IC \\end{cases} ", "4ae7c48fc8eb99e4018edae1093c3489": "M^-_\\infty = \\liminf_{t\\to\\infty} M_t,", "4ae83b5f47f8c9a934bb12dbba7fa2d2": "\\Delta H ^{\\circ}_{\\mathrm{c}}", "4ae83e3ab02aa5db954f4f0770a707d6": "\\Delta Q_i = ie- \\ ", "4ae858a203e15c983f3b7a15b199fea8": " m_j\\left(\\beta\\frac{\\partial u_j}{\\partial t} +\\alpha\\frac{\\partial u_j}{\\partial x} \\right)+l_j b_j=0 ", "4ae88154970bfd19aaf308b7b9506fb1": " \\, f=J_\\nu(x)\\,", "4ae884acca33a25e28a10832aff9bec7": "D^{j}(G,H)=\\frac{1}{\\sqrt{2}}(\\sum_{k=1}^{\\infty}[N_{G}^{j}(k)-N_{H}^{j}(k)]^{2})^{\\frac{1}{2}}", "4ae88ec4c1b9828ab870beff144efd6e": "1+\\varepsilon", "4ae92397914731190007c5eb27234bdb": "\n\\Pi(x)= \\sum_{n\\le x}\\frac{\\Lambda(n)}{\\log n}.\\;\n", "4ae942570d746959cb9e104b2709add6": "p \\in l.Clause", "4ae96604c4acc72e09e999d81e8664a2": "\\int_{t_0}^{t} f(\\tau) d\\tau", "4ae9a49e8ad4e95d82871440b0d0450d": "u_{\\mathrm{ref}}", "4ae9a57cd612007f383a5d02c78e0b31": "\\rho_p<\\rho_f", "4ae9c45695808156c3b35a197f0ddc97": "K_b(message, R)", "4ae9dea9b5468b2cc666f4382a02cec5": "\n\\begin{matrix}\n 4s &: 0.35 \\times 1& + &0.85 \\times 14 &+& 1.00 \\times 10 &=& 22.25 &\\Rightarrow& Z_{\\mathrm{eff}}(4s)=3.75\\\\\n 3d &: 0.35 \\times 5& & &+& 1.00 \\times 18 &=& 19.75 &\\Rightarrow& Z_{\\mathrm{eff}}(3d)=6.25\\\\\n3s,3p &: 0.35 \\times 7& + &0.85 \\times 8 &+& 1.00 \\times 2 &=& 11.25 &\\Rightarrow& Z_{\\mathrm{eff}}(3s,3p)=14.75\\\\\n2s,2p &: 0.35 \\times 7& + &0.85 \\times 2 & & &=& 4.15 &\\Rightarrow& Z_{\\mathrm{eff}}(2s,2p)=21.85\\\\\n1s &: 0.30 \\times 1& & & & &=& 0.30 &\\Rightarrow& Z_{\\mathrm{eff}}(1s)=25.7\n\\end{matrix}\n", "4ae9e11f676799acf0e967b99199c429": "\n\n\\left.\n\\begin{matrix}\n(A\\cap B)\\cap C=A\\cap(B\\cap C)=A\\cap B\\cap C\\quad\n\\\\\n(A\\cup B)\\cup C=A\\cup(B\\cup C)=A\\cup B\\cup C\\quad\n\\end{matrix}\n\\right\\}\\mbox{for all sets }A,B,C.\n", "4aea236beb39cc3baed5eb7d01753c25": "L': \\{0,1\\}^N \\rightarrow \\{0,1\\}", "4aea3f4c455157f7cecda64e79c7b7c6": "\\vec{k}_{f}", "4aea514dc0549b6258ad150df982d968": "\\scriptstyle{6+8\\sqrt{3}}", "4aea9e4cdb5a567d0ac00645c2a3a839": "-1/\\log_2(1-p)", "4aead8615b1b1f9b3296b7be1411103b": "K=2r^2\\left(\\frac{1}{\\sin{A}}+\\frac{1}{\\sin{B}}\\right).", "4aeade952308d5a96adb44cd1661495a": "a_{21}=0.05", "4aeae9f483c149e7b747fed66c6ce130": "\n\\begin{align}\n\\bar{X_i} &\\sim \\mathcal{N}_p \\left(\\mu_i, \\Sigma_i/n_i \\right), \\\\\n\nA_i &\\sim W_p(\\Sigma_i, n_i - 1).\n\\end{align}\n", "4aeb5bb271efc1fe4d24e74212f2f385": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2} ", "4aeb77052462ce76ce67f6d4a2d17d2f": "G_2 = -exp(-\\frac{u^2}{2})", "4aeb8270aa41a681711e1fbf984bea2f": "W'_i, W'_{1-i}", "4aebaccdf5fae3195a629bb237e1502d": "\\displaystyle F(e^X)\\delta(e^X)", "4aebe80eed1e2c7859fa9f1dfb51d2f3": " x(t) = \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} X(\\tau, \\omega) e^{+j \\omega t} \\, d\\tau \\, d\\omega. ", "4aebf5e9385fa8717fb04599cc2261ac": "\\mathbf{CR}(i,j) = \\Theta(\\varepsilon - \\| \\vec{x}(i) - \\vec{y}(j)\\|), \\quad \\vec{x}(i),\\, \\vec{y}(i) \\in \\Bbb{R}^m, \\quad i=1, \\dots, N_x, \\ j=1, \\dots, N_y.", "4aec65fc4ba17f3e0adf171a102ba8ca": "\\!\\sinh", "4aec66b45ffd086530c82fd46a937301": "\\textstyle m(x)", "4aec9321a18cbc89a2fb6869e8bc90ac": "N = \\frac{(M+1)(C+1)}{R+1} - 1,", "4aecfd60d8193e1daf68a63cf81130af": "CH_4 + Cl_2 \\xrightarrow{uv\\ light} CH_3Cl + HCl", "4aed057479ad2b588c34e48d0c76661b": "\\scriptstyle \\mathbf{f}", "4aed1228993f37f02fb6cf21e89687ea": "v\\frac{\\partial u}{\\partial y}+ w \\frac{\\partial u}{\\partial z} = 0", "4aed6a4528d30369262628b72f17f44b": "2^2", "4aeda570b32f2bf1d938ee64a63a9787": " \\{ S_1 = (2b_0-1) + i (2b_1-1),\\; S_2 = (2b_2-1) + i (2b_3-1) \\} ", "4aedc3922d21e4ffdbdf3cb84b86e113": "\\frac{1}{2} \\int_a^b\\! u'(x)^2\\, dx = \\frac{1}{2} (u|u)_E,", "4aedcdf1249f085b09617897d51d58a8": "\\psi(x,y)", "4aedec170bb34f70f2b59fefcde29663": "K_*", "4aeeb6148884cd170cda47a62f363aa8": "\\infty = (\\omega_1,\\omega)", "4aeedbae370fd89b4516dfc1ef5bd397": "T_{Sunset} = T_{Dhuhr} + T(0.833)", "4aef1b1731453fea4e134ab46eeed81d": "\n\\nabla A(\\mathbf{r}) = \\sum_j m_j \\frac{A_j}{\\rho_j} \\nabla W(| \\mathbf{r}-\\mathbf{r}_j |,h).\n", "4aef1ca6582df1410b868e1c7dbb6d6d": "I_{m,n}=\\int \\frac{x^m dx}{(ax^2+bx+c)^n}\\,\\!", "4aef24e45fa7f9c3b7c55214a05e1de5": "\\tau = \\frac {V}{v}", "4aef7dc18c45ba145501b46e7f36a73f": "n = 52", "4aefe259dcc663e8402883dbccea2ced": "g(JX,JY)=g(X,Y) \\,", "4aeffdfd94edbf619713c9d4465272d9": "\\langle X, \\leq , \\mathcal{F} \\rangle", "4af00430b7b7544c9d935ab90a06e0bf": " \\frac{f(x_0 + i h) - f(x_0)}{i h} = f'(x_0) + \\frac{f''(\\xi)}{2!} i h, ", "4af08c92f2af8744f50667e976e12042": " \\Pr(n) = \\int_{N=n}^{N=\\infty} \\Pr(n\\mid N) \\Pr(N) \\,dN = \\int_{n}^{\\infty} \\frac{k}{N^{(\\alpha+1)}} \\,dN = \\frac{k}{{\\alpha}n^{\\alpha}}", "4af1d337c5d9b72e7066f35f2bc3f864": "A_{arith}(G,H) = \\frac{1}{73} \\sum_{j=0}^{72}A^j(G,H)", "4af24d63264cedba1a517daeadf057ee": "\\tfrac{E + 3\\lambda + R}{6}", "4af2c162da0c8fc72232aa374304e3be": "D_4 \\to G_2", "4af2c975471a1e243af1b31455412ae7": "2=\\{H,T\\} .", "4af2d4e560739d9044a8c7dc09d52721": "\\gamma_n=n\\gamma\\,\\!", "4af3078c4e3d6437a2f03f05ebfc36b7": "A [\\rho]", "4af37725ee110530616045419b0cba7e": "\\mathbf{\\Psi},\\mathbf{S}", "4af3ae49b8660cdff5cef510c4bdc54b": "{ \\partial^2 u \\over \\partial t^2 } = c(u)^2 \\nabla^2 u ", "4af3ae68b9fe4d91770734cd827d3075": "{\\tilde{A}}_1 {\\tilde{G}}_2", "4af3ccd58d0cf5f3cd7f71c2a2884d2a": " \\mathbf{p} = \\frac{d}{dt}(\\sum_{i=1}^n m_i (\\mathbf{r}_i - \\mathbf{R})) + (\\sum_{i=1}^n m_i) \\mathbf{V},", "4af42ae915b8aab79cb0717dd723f5d0": "\\pi(G)^{''} ", "4af4630a41ccddd54cebbd5415e5b181": "R = K", "4af46f397ec8a2251f6eb6402212127f": "\\sigma \\in C_p(X)", "4af49cd77128d6c4b0aeb45deb2c7c18": "T h = \\sum _{i = 1} ^n \\alpha_i \\langle h, v_i\\rangle u_i \\quad \\mbox{for all} \\quad h \\in H ,", "4af53814c80b09346654c5c18dd81cb0": "U_2\\left(x,y\\right).", "4af54cd7ed8ebbb353d99cb4456334d3": "e'' = \\frac{1}{\\rho}p',", "4af563ebf81bd9468faad3b6141938dd": "y_d=\\alpha+\\beta x_d +\\epsilon_d \\,", "4af5814ffba6e287314358d84d83b41b": "c_{k'}^{(0)}=\\delta_{k,k'}", "4af5ac2aebe3c48f86ccb23ae37be976": "q(x) = \\left(x_1^2+\\cdots + x_k^2\\right)-\\left(x_{k+1}^2+\\cdots + x_n^2\\right)", "4af5d8b50084e185447f5447f0286f7b": "E^{2}-(pc)^{2}=(mc^{2})^{2} \\,", "4af60bf026aba3687548a7fe7afd5f47": "m(t) = m_{0} - [ (m_{0}-m_{\\infty})(1 - e^{-t/\\tau_m})]\\, ", "4af6500d43bf0caf91ee3295e059d1c8": "\\Theta(\\sqrt{n\\log n})", "4af664441f0b49973541ca759fe205e9": "\nf(x) + f(-x) = 0 \\, .\n", "4af6a83fd280047413fbb33759dc4f9e": "((1+0.50)(1-0.20)(1+0.30)(1-0.40))^{1/4}-1=-0.0164=-1.64%", "4af6f317ae26bf5c87fdb176d8571c47": "\\gamma^{\\mu\\prime} = U^\\dagger \\gamma^\\mu U.", "4af70549290a19ee9d9ee5f54295c79d": "\\nabla^*", "4af7d16ee8abbb68901ca728d6d66eb5": "45^{\\circ}", "4af7f467f1d018cd497b9a79ccd2b9c3": "M>m", "4af86d67f4eb6b1d6c096f7b883ea07a": "\nD_N^*(x_1,\\ldots,x_N)\\geq\\frac{1}{2N}\n", "4af8aad0d2ca025cb3700c87a7faedf8": " = 8*10000 + 2 (10000/600) + 0.16 (600/2) = $80081 ", "4af8ba81e5b00cf2c2a46e5059d8ad31": "\\Delta(f)(a\\otimes b) = f(ab).", "4af8ba8568d83ea1e0f6c3e2c07741f1": " \\psi = 1- \\phi(\\frac{1- \\psi^'}{\\phi^'})\\,", "4af9304cf1820c73133763910dd86d2e": "\\mathrm{MA} = \\frac {F_w}{F_i} = \\frac {\\cos \\phi} { \\sin (\\theta + \\phi ) } \\, ", "4af9446cf9c5c8941186db41b5555f05": "\\operatorname{cov}[\\ln X, \\ln(1-X)] = -\\psi_1(\\alpha+\\beta)", "4af94c980a455fcde6a06781ec80ced5": "x, y \\in S", "4af95d81d353257a048604873ffe3ce7": "\\forall_A", "4af98a0e1d9380a4cf25223c03b06f18": " |\\lambda_i| ", "4af9a7373cf65b9ade1087db796b7806": "\\chi(M \\sqcup N) = \\chi(M) + \\chi(N).", "4af9b71b677de27c34e7855a03e00df9": "h \\approx 10^{-20}", "4af9cb7140eff406f917e3b6d2ad8bb3": "P'", "4af9ee652a0d4964292d22272db5032b": "(b_n)_{n\\geq0}", "4af9ee9be0903527f4e1c36d03b6d762": "V_{TB} = V_{T0} + \\gamma \\left( \\sqrt{V_{SB} + 2\\varphi_B} - \\sqrt{2\\varphi_B} \\right),", "4afa10bf5c74b67ef79482422f3b59ba": "[K(u+w,v+z)-K(u+w,v)-K(u+w,z)-K(u,v+z)-K(w,v+z)-K(u+w,v)+K(v,w)+K(u,z)].^{}_{} ", "4afa18739ca3c0fa0fd6336fe8858ec1": "L(\\hat{y}, y) = I(\\hat{y} \\ne y)", "4afa23d604f0758346883e8b2aaba2ab": "TF(a)=(TF,E_a)=(F,T^*E_a)=\\frac{1}{\\pi} \\iint_{\\mathbf C} F(z)\\overline{(T^*E_a,E_z)} e^{-|z|^2}\\, dx dy=\\frac{1}{\\pi} \\iint_{\\mathbf C} K_T(a,\\overline{z}) F(z)e^{-|z|^2}\\, dxdy.", "4afa8357a3e3128f5fab45484cc6e689": "log(a) + log(b) = log(ab)", "4afa8d07d44321521611aea9feabbe56": "r=kC_S K_1C_AC_B", "4afa9f984401c6424e4a6865c6ec19f6": "K=\\frac{\\prod_k [A_k]^{m_k}}{\\prod_j [A_j]^{n_j}}", "4afacfe9e67901cde48d50db8cd8689f": "\n\\mathcal{P}^2\\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* = \\hbar^2 j(j+1) D^j_{m'm}(\\alpha,\\beta,\\gamma)^* \\quad\\hbox{with}\\quad\n\\mathcal{P}^2= \\mathcal{P}^2_x + \\mathcal{P}_y^2+ \\mathcal{P}_z^2,\n", "4afafb3d46096809e9b85613b3dd87d0": "\\begin{align}\n N &\\approx \\mu \\pm \\sigma = 19.5 \\pm 10 \\\\\n \\mu &= (m - 1)\\frac{k - 1}{k - 2} \\\\\n \\sigma &= \\sqrt{\\frac{(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^2}}\n\\end{align}", "4afb72c08eae7b50e2dccaf27685281e": " \\sum_{k=0}^{s-1} a_k = -1 \\quad\\text{and}\\quad q \\sum_{k=0}^s k^{q-1} b_k = s^q + \\sum_{k=0}^{s-1} k^q a_k \\text{ for } q=1,\\ldots,p. ", "4afb8ee1002b9aebbe3af597873315ef": "\\alpha_i = R_i P_i R^i", "4afb9b3a856e38499b979fefc915f800": "j \\le p", "4afba450efc7095843215abe7fc164af": "\\mathbb{T}^1\\cong\\left(R_2\\oplus R_3\\oplus R_5\\oplus\\ldots \\right)\\oplus\\bigoplus_{i\\in I}{F_P}\\;", "4afc1963fd03646f41bfe4ad0dddef25": "\\!\\mathcal A \\models_X^- =\\!\\!(t_1 \\ldots t_n)", "4afc67239bd1a5779043d72844c952fb": "G_{\\phi} = 4\\pi\\left(\\frac{U_\\phi}{P_{\\mathrm{in}}}\\right)", "4afcef98258ab931355213535989b7b3": "\\mathfrak{gl}", "4afcf752e31262ec9925f039d1d1424c": "Q= \\frac {k A}{\\mu} \\left( \\frac{\\partial P}{\\partial L} \\right)", "4afcfa4c7f70a2ef7abd2933eed2dc71": "\\mathit{dr}(a-b) \\equiv \\mathit{dr}(a)-\\mathit{dr}(b) \\pmod{9}.", "4afd0e96281d420240e6145ac1d27e21": "\\bold{a}(t)=\\frac{d}{dt}\\bold{v}(t)", "4afd2e9212d6b3cc39a5012667ee7ac9": "f''(x) \\text{ is finite },\\forall x\\in I; \\,", "4afd545660c4fe03ac9bd1f526054edb": "\nY_t=\\mathbf{\\nu} +A Y_{t-1}+U_t\n", "4afd741b84d5d58edcd0aed22cfcc94c": "\n\\left(\\frac{1-\\zeta_m}{a }\\right)_m =\n\\left(\\frac{\\zeta_m}{a }\\right)_m^{\\frac{m-1}{2}}.\n", "4afde489a6d680b4d095ec4066b99110": "D\\subseteq \\mathbb{C}", "4afde9a93a7aaf50527b056724921db3": " \\ w_i ", "4afe5eefbc32d8630303f6e681ed94f5": "s_3=\\alpha^{4},", "4afeae27eaee8d4a50b457fcc2294e9b": "l_2\\equiv\\partial_x+x\\partial_y", "4afec59c16d11709cd42eeef7315f907": "\\int u\\;dx = \\frac{1}{2}\\left(xu-\\sgn x\\,\\operatorname{arcosh}\\left|\\frac{x}{a}\\right|\\right) \\qquad\\mbox{(for }|x|\\ge|a|\\mbox{)}", "4afed1ba40fab80f41b7fddb4c19b211": "\\xi\\in\\mathfrak{g}^*", "4aff0825b82619957a1f03f9ab2ef0a1": "r_\\mathrm{obs} = R_\\mathrm{E} + y_\\mathrm{obs}", "4aff2a6751c53a7e4dafe31523b5b213": "i=\\lfloor d/2\\rfloor", "4aff4116b8b30eabfad3fe4134883e28": "B \\leq_T C^{(n)}", "4aff7f6bf58a26a0efedc003f57485d0": " : \\hat{f}_2 \\, \\hat{f}_1^\\dagger \\, \\hat{f}_3 : \\,= -\\hat{f}_1^\\dagger \\,\\hat{f}_2 \\, \\hat{f}_3 = \\hat{f}_1^\\dagger \\,\\hat{f}_3 \\, \\hat{f}_2", "4b000051980d642265b286cc364daf85": "\\{\\mathcal{J}_i\\}", "4b00017198c1862ee1c2c43803fc08c8": "Z[A]=\\int \\mathcal{D}\\overline{\\psi}\\mathcal{D}\\psi e^{-\\int d^dx \\overline{\\psi}iD\\!\\!\\!\\!/\\psi}.", "4b0063321994adbab077656352a2dc75": "T = a+b\\sqrt{(A/W)^2+(A/t)^2}.", "4b0075dd4dcb8c667fa20220c38565d1": "\\nabla\\cdot \\bold{J} = 0. ", "4b0165af6af2d17a3752f87c42dfcfc1": "\\Delta_1(x-y)", "4b01692c6208d265718e5e2cd785ed7d": "\n \\begin{align}\n J_2^0 := & \\cfrac{1}{6}\\left[a_1(\\sigma_{22}-\\sigma_{33})^2+a_2(\\sigma_{33}-\\sigma_{11})^2 +a_3(\\sigma_{11}-\\sigma_{22})^2\\right] + a_4\\sigma_{23}^2 + a_5\\sigma_{31}^2 + a_6\\sigma_{12}^2 \\\\\n J_3^0 := & \\cfrac{1}{27}\\left[(b_1+b_2)\\sigma_{11}^3 +(b_3+b_4)\\sigma_{22}^3 + \\{2(b_1+b_4)-(b_2+b_3)\\}\\sigma_{33}^3\\right] \\\\\n & -\\cfrac{1}{9}\\left[(b_1\\sigma_{22}+b_2\\sigma_{33})\\sigma_{11}^2+(b_3\\sigma_{33}+b_4\\sigma_{11})\\sigma_{22}^2\n + \\{(b_1-b_2+b_4)\\sigma_{11}+(b_1-b_3+b_4)\\sigma_{22}\\}\\sigma_{33}^2\\right] \\\\\n & + \\cfrac{2}{9}(b_1+b_4)\\sigma_{11}\\sigma_{22}\\sigma_{33} + 2 b_{11}\\sigma_{12}\\sigma_{23}\\sigma_{31}\\\\\n & - \\cfrac{1}{3}\\left[\\{2b_9\\sigma_{22}-b_8\\sigma_{33}-(2b_9-b_8)\\sigma_{11}\\}\\sigma_{31}^2+\n \\{2b_{10}\\sigma_{33}-b_5\\sigma_{22}-(2b_{10}-b_5)\\sigma_{11}\\}\\sigma_{12}^2 \\right.\\\\\n & \\qquad \\qquad\\left. \\{(b_6+b_7)\\sigma_{11} - b_6\\sigma_{22}-b_7\\sigma_{33}\\}\\sigma_{23}^2\n \\right]\n \\end{align}\n ", "4b01a20295d964c2486c03f2d2389b39": "\\Delta U_{system}=Q - W", "4b01e1d7f710de6818f24f140d5528cb": "\\{2,4,6,8,10\\}\\,\\!", "4b01f93c778b61ec98f48b938a7378db": "T(x) = \\prod_{k=1}^\\infty \\frac{1 + i/\\sqrt{k}}{1 + i/\\sqrt{x+k}} \\qquad ( -1 < x < \\infty )", "4b021d84b33a045c5ecc6a519db2a1da": "\\displaystyle Q_{ab}=R_{ab}-\\frac{1}{4}g_{ab}R", "4b02301ef53181072cc65f6e89f81334": " T_y^m u\\left( x\\right) =\\sum\\limits_{k=0}^{m-1}\\sum\\limits_{\\left\\vert \\alpha\\right\\vert =k}\\frac{1}{\\alpha!}D^\\alpha u\\left( y\\right) \\left( x-y\\right)^\\alpha", "4b024f931f7db22ccde45a8914e1818e": "V_\\mathrm{T} = \\frac{k T}{q} \\, ,", "4b028d9e3139a97db204fc4a2799d362": "\\begin{cases} \nv_{t}=kv_{xx}+f, \\, w_{t}=kw_{xx} \\, & (x, t) \\in \\mathbf{R} \\times (0, \\infty) \\\\ \nv(x,0)=0,\\, w(x,0)=g(x) \\, & IC\n\\end{cases} ", "4b031381ebd19645d6a61a8322137827": "L_{X^{(k)}}\\theta_k \\equiv 0 \\pmod{\\theta_1,\\dots, \\theta_k}", "4b03629b4948ad6139fd653ca692bd6d": " f(x_0 + i h) = f(x_0) + f'(x_0)i h + \\frac{f''(\\xi)}{2!} (i h)^{2}, ", "4b03b5e7f361144e47c891388f3002d3": "K = \\frac{3}{16} \\sqrt{\\frac{2\\pi}{\\mu kT}} \\frac{Q}{n\\sigma}", "4b03bb5d09b24a1e86b932bdabfb385b": "\\begin{align}\\text { and } { }_3\\text {F}_3&\\left(1,1,1;2,2,2;-z\\right)=\\\\&\\sum_{k=0}^\\infty\\left[1/\\left(k+1\\right)^3\\right]\\left(-1\\right)^k\\left(z^k/k!\\right)\\end{align}", "4b03be2afe7ae7103ef3c0978f136862": "\\vec F_g", "4b03c88fdf322523d6ab13a1f00b5c48": "T_6", "4b03cc79d9ababd04415d6ad366b4aaa": "w \\, \\Sigma \\, \\sigma", "4b03ebed75711b53cb5b253ee195210e": "\\neg \\Box \\neg p ", "4b0458f856d5e0bdca478de600bebebc": "|\\Psi(t)\\rangle=\\sum_{n=0}^\\infty {(-i)^n\\over n!}\\left(\\prod_{k=1}^n \\int_{t_0}^t dt_k\\right) \\mathcal{T}\\left\\{\\prod_{k=1}^n e^{iH_0 t_k}Ve^{-iH_0 t_k}\\right \\}|\\Psi(t_0)\\rangle.", "4b045a3c1658621b69586eedbb65ddec": "\\sum_{c,e=1}^{n^2 -1}d_{ace}d_{bce}= \\frac{n^2-4}{n}\\delta_{ab} \\,", "4b0469c63ef2decc06824132a8b343d2": "i \\partial_\\mu \\bar{\\psi} \\gamma^\\mu + e\\bar{\\psi}\\gamma_\\mu (A^\\mu+B^\\mu) + m \\bar{\\psi} = 0 \\,", "4b04796a9b710467b30de54a21dbbb8c": "\\tau(K_{m,n})=\\min(m,n)", "4b04c46a964296f25b7689fd8a4b358b": "g\\in GF(p^6)", "4b04e6eb1335dff34dc0c5c359d124ac": " \\Phi (x,-n,a) = G_{n+1,\\,n+1}^{\\,1,\\,n+1} \\!\\left( \\left. \\begin{matrix} 0, -a, \\dots, -a \\\\ 0, 1-a, \\dots, 1-a \\end{matrix} \\; \\right| \\, -x \\right), \\qquad \\forall x, \\; n = 0,1,2,\\dots ", "4b04fbfc04f9ec2deea3c844b79b6811": "\\mathbf{S}", "4b052c430d5d48ddd1674906d746d2c9": "(L,\\cdot)", "4b05562a7a9e6d8be3d270e05a9737a4": "z := (z_1,\\ldots,z_n).", "4b060f962fa717bfa99e0763aad80857": " Q(\\alpha)", "4b061ffa87d399e3cb99cc29f3178e86": "\\delta (\\varepsilon) = \\inf \\left\\{ 1 - \\left\\| \\frac{x + y}{2} \\right\\| \\,:\\, x, y \\in S, \\| x - y \\| \\geq \\varepsilon \\right\\},", "4b0625b1940fd82e43c0befd42fe222f": " \\begin{matrix} \\frac12 \\end{matrix} \\cdot (v_1^2-v_2^2) = \\begin{matrix} \\frac12 \\end{matrix} \\cdot (v_1 - v_2) \\cdot (v_1 + v_2) = v \\cdot (v_1-v_2) ", "4b06600dea9edf87720dba840911c8bd": "\\Sigma_i^{\\rm P} = \\Sigma_k^{\\rm P}", "4b066e4f4c19de01528f4f10db9d7701": "\\,\\mathrm{slog}_b(b^z) = \\mathrm{slog}_b(z) + 1", "4b06ab12ddab9a28a9d2bb0ec6e14ecc": "\\mathsf{Ba(OH)_2 + \\ K_2CrO_4 \\longrightarrow \\ BaCrO_4 \\downarrow + 2 \\ KOH}", "4b06d8c734ce322f47f10dd16ec0ec89": "\\mathrm{E}(Y) = \\frac{1-p}{p},\n \\qquad\\mathrm{var}(Y) = \\frac{1-p}{p^2}.", "4b0774d0aa6a7a1753d24960fff2c4c8": "H=\\alpha(2)", "4b07907c2570b5fc1d894b819767b55e": "\\sum_{n=1}^\\infty E[B_H (1)(B_H (n+1)-B_H (n))] = \\infty.", "4b07cbc1850c234f9e52323b04351722": "\n a_{32} = - \\frac{\\lambda(4\\mu + A) - 2\\mu B}{2(\\lambda + \\mu)}\n ", "4b07e3c682394eb7e10e3c5be3cdfe00": "\nk^{(1)}\n=\n\\frac{AE}{L}\n\\begin{bmatrix}\n1 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n-1 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{bmatrix}\n\\rightarrow\nK^{(1)}\n=\n\\frac{AE}{L}\n\\begin{bmatrix}\n1 & 0 & -1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\end{bmatrix}\n", "4b08084ce2b7a2dc9be2f9f02e901979": " V=V_+\\oplus V_-", "4b080eb55bc4357e6f08a67804803051": "\\frac{1}{1-p^{n-1}}=1+(n-1)kt[COOH]^{n-1}", "4b0828558542f4dd2e110a0347150235": "M=(Q\\,, \\Sigma\\,, \\Gamma\\,, q_0\\,, Z_0\\,, A\\,, \\delta\\,)", "4b084ef5b05c1cf75d82816fc3fc1da5": "\n\\left[ \\frac{d^2}{dx^2} -\\frac{l(l+1)}{x^2}+\\frac{2}{\\alpha x} - \\frac{1}{4} \\right] u_l = 0,\n\\quad \\text{with } x \\ge 0. \n", "4b085d2f9fd7b84eaa696d23a5789ff8": "G(x,x')", "4b088f080de4680127c2add2a04bb73d": "\\ I(hkl) = (1-x)|F(h k l)|^{2} + x|F(-h -k -l)|^{2}", "4b088fd6e16440962f72208842363566": "R=\\oplus_{i=1}^n e_{i}R", "4b08c68c21050926dee6fb3a4eeb11b6": " W_\\text{stored} = \\frac{1}{2} C V^2 = \\frac{1}{2} \\varepsilon_{r}\\varepsilon_{0} \\frac{A}{d} V^2.", "4b091fd59916475b3da16af1560b84c8": "{v_B^2 \\over 2}+gh_B+{P_B \\over \\rho}=\\mathrm{constant} ", "4b093d8a2da1bbe0c36f01fddf4a099a": "L = P + Q", "4b096de74d17cf257cb7c03c57737e15": "h_N(g)", "4b098406b7496fea7536a209246cef42": "P_3(q)", "4b0a46f0d327f7badd55c019244251f1": "\\log\\left(\\frac{r_3}{r_1}\\right)\\log M(r_2)\\leq \n\\log\\left(\\frac{r_3}{r_2}\\right)\\log M(r_1)\n+\\log\\left(\\frac{r_2}{r_1}\\right)\\log M(r_3)", "4b0ab2a509462e68a818319023e0bb35": "~ 1- \\exp\\left( - F \\frac{\\pi r^2}{S}\\alpha L \\right) ~", "4b0adb49618167f16b285dc382fc59d0": "C_{\\rm melt}/C_{\\rm gas} = \\exp\\left[-\\beta(\\mu^{\\rm E}_{\\rm melt} - \\mu^{\\rm E}_{\\rm gas})\\right]\\,", "4b0aef503dc1f7c1995ce9dbc559d0f8": "{\\left\\| B_n(f)^{(k)} \\right\\|}_\\infty \\le \\frac{ (n)_k }{ n^k } \\left\\| f^{(k)} \\right\\|_\\infty \\text{ and } \\left\\| f^{(k)}- B_n(f)^{(k)} \\right\\|_\\infty \\to 0", "4b0b2acf99badb1dc1a816d960125df0": " \\partial_t h = \\partial_x^2 h \\, ", "4b0b5c7d03751674c27cfc367e251325": "\\cos y=x\\,\\!", "4b0b8217e7f30047a9ef37e38fa48813": " \\cdot", "4b0bab7cd1e676cf0dad8a4462f4308f": "u'\\in S_{H}(u,r)", "4b0c1005c2bb0e499399687bafd65371": " \\lim_{x \\to p^+}f(x) = L ", "4b0c82c4db6fd1a6186493bd79c015fd": "\n \\mathbf{t} = \\boldsymbol{\\sigma}^T\\cdot\\mathbf{n}\n", "4b0ca67bd5957a08f1587cb222c00e3b": "\\tilde{w}", "4b0d06ec58a57f502438604c54b35c91": "E^M_{\\rm{(abs)}} = E^M_{\\rm{(SHE)}}+(4.44 \\pm 0.02)\\ {\\mathrm V}", "4b0d1f650d9a34a05eb831a376448573": "n_{i}(t,x) = (5.585 - 3.82x + (1.7531^{-3})t - 1.36411^{-3}t\\cdot x)\\cdot 10^{14}\\cdot E_{g}(t,x)^{0.75}\\cdot T^{1.5} \\cdot e^{\\frac{-E_{g}(t,x)\\cdot q}{2\\cdot k\\cdot t}}", "4b0d34e5056cd0137c8bee819b96a317": "\\begin{vmatrix} a_{1,1} & a_{1,2} & \\dots & a_{1,n} \\\\\na_{2,1} & a_{2,2} & \\dots & a_{2,n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{n,1} & a_{n,2} & \\dots & a_{n,n} \\end{vmatrix}.\\,", "4b0d5d43b392223f49452114e38e6725": "F(\\phi,k) = \\int_0^\\phi \\frac{1}{\\sqrt{1 - k^2 \\sin^2(t)}} dt,", "4b0d86db643689c68068c64174b36f78": "{\\mathbf v}f", "4b0d9a369bb7b6fc9aec3b8962498764": "n_i^2", "4b0db10a055d139198dd8c070bf955c6": "f(x) = a^T x + b ", "4b0e2abf74a8dc1b6d3a082548c6a1c5": " \nR_q = \\frac {Z_m} {\\sqrt{\\rho - 1}} \\qquad \nR_p = Z_m \\sqrt {\\rho - 1}", "4b0e615630d5232d2f1f0ba517634fc9": "\\Delta\\mu_{i,\\mathrm{mix}}=RT\\ln x_i", "4b0e8e59644905d06d29bf804abac2f7": "c_\\mathrm{y}", "4b0e99853a62c1015b57f2e2b8683917": " \\mathrm{li}(x) = \\int_{\\mu}^x \\frac{dt}{\\ln t}, ", "4b0ec3707b367daaa315216ea8ee1294": "\\ 0 = \\sum_{j=0}^Q a_{j} z^{-j}", "4b0ef86fe076926efce6fdf94406a92f": "\n\\hat{e}_{\\nu} = \\frac{1}{\\sqrt{\\sinh^2 \\mu + \\sin^2 \\nu}}\n\\left(\n-\\cosh \\mu \\sin \\nu \\cos \\phi \\boldsymbol{\\hat{i}} - \\cosh \\mu \\sin \\nu \\sin \\phi \\boldsymbol{\\hat{j}} + \\sinh \\mu \\cos \\nu \\boldsymbol{\\hat{k}}\n\\right)\n", "4b0f102e02c54312d3a8ffc7d589ece0": "\\star du=dv", "4b0f5349e2aae3c99f0a2a6cae7c17b6": "(val(b),val(a)) \\not\\in valprefs", "4b0fb25ae7f4c122bb0f43ee9fe5ccc9": "\\tilde{\\lambda}", "4b0fcccd6c27c031629b4dc971910ac8": "(x,0)", "4b100d3a5e4f5765fdccf63e08229462": "y_k = \\sum_{i=0}^{N} x_{k-i}\\, h_i ", "4b101913a20b4a16971400e194b45c63": "{F}=\\frac{Q_1Q_2}{4\\pi\\mathrm{D}^2\\varepsilon_0\\varepsilon_r}", "4b102511c277d86289cd0ef18c9b867e": "pickup(o)", "4b107d0bec845aa33db5342643504c4d": "{a}\\times{b} = -i({a}\\wedge{b})", "4b10f4cab3aaeec5b2d211c2caa5ac74": " H_{t}(\\mu) = F_{t_{n-1}}\\left(\\frac{\\sqrt{n}(\\mu-\\bar{X})}{s}\\right) ", "4b111bb20e3ea63195cde149344c0570": "\\tfrac {1}{10} \\pi^2 - \\ln^2 \\phi \\,", "4b1139a082d7c19c19e3602651b371dd": "\\begin{bmatrix} \\alpha_1-1 \\\\ \\vdots \\\\ \\alpha_k-1 \\end{bmatrix}", "4b118390cba51b5d8c520ba679299371": "V({\\mathbf{x}}+{\\mathbf{a}})=V({\\mathbf{x}}),", "4b11980420e07b729ab4770fa874bd00": "N\\setminus\\{x\\}", "4b11ae112fbe187179115dc9b1f02f3d": "{{i}_{B1}}={{i}_{B2}}\\equiv {{i}_{B}}", "4b1235dd04cd91e939b618f066eb5c7b": "\\textstyle x^{i}a(x)", "4b123b28a162d88dbf6c41531d431951": " (-1+j0) ", "4b1243e989121da3bfdedcd2a2f191a9": "n \\geq N", "4b128502e5691f81d3cb06b2ec9e7048": "\\ C=\\frac{P_tG_tA_r}{2\\log2(4\\pi)^2R^4}\\frac{\\pi}{4}(R\\theta)(R\\phi)(c\\tau/2)\\eta", "4b12b65c9cc39063ab9a5aa593f599f4": "\\operatorname{cov}(W_{t_1}, W_{t_2}) = E\\left[(W_{t_1}-E[W_{t_1}]) \\cdot (W_{t_2}-E[W_{t_2}])\\right] = E\\left[W_{t_1} \\cdot W_{t_2} \\right].", "4b13214446cf00ff5f51f62c5fa46989": "A^+=A^{-1}\\,\\!", "4b139775398557557ebd9e4b7d5bc72a": "L \\; = \\; G_B \\; G_M \\; \\left (\\frac{h_B \\; h_M}{d^2} \\right )^2\\beta", "4b13b39751779d059b0bef6a30c0b199": "\\quad h\\;=\\;\\dfrac{\\delta y}{a\\delta\\phi\\,}=\\dfrac{y'(\\phi)}{a}", "4b143fa640c31dfa62877790cb11f675": "f(\\zeta) = \\frac{1}{2\\pi i}\\iint \\frac{\\partial f}{\\partial \\bar{z}}\\frac{dz\\wedge d\\bar{z}}{z-\\zeta},", "4b149ce3dfcb6b279dfcec375bb55fff": "\n\\mathcal{A}^{AB} = \\tilde{\\mathcal{A}}^{AB} \\mathcal{A}^A \\mathcal{A}^B\\quad\\hbox{with}\\quad\n\\tilde{\\mathcal{A}}^{AB} = \\sum_{T=1}^{C_{AB}}(-1)^\\tau \\hat{T}, \\quad C_{AB} = \\binom{N_A+N_B}{N_A} .\n", "4b14b86334d0157762fe3d487d75b517": "\\langle c, \\mathcal S\\rangle", "4b14cf468cbe5ad9453fdbcd0a90eb64": "\\star \\mathrm{d}y \\wedge\\mathrm{d}z = \\mathrm{d}t\\wedge \\mathrm{d}x", "4b14dc817ff5aaf54ac5bd25bdcd40d4": "(e^{i a x} - e^{-i a x})/(2i).", "4b14e21dbf40238a5ddad0dc846fa2d4": "\\forall\\alpha.\\alpha", "4b14e8d224401391388d31b42fd4aa34": "\\begin{pmatrix} A_1 \\\\ A_2 \\\\ \\vdots \\\\ A_N \\end{pmatrix}", "4b152567a12fdbcd2200b4dc72f4a7ca": "n \\ge 0", "4b1549503a66a875393c43b10afa00c9": " \\oint p_\\mathrm{\\varphi} \\, d \\varphi = 2 \\pi p_\\mathrm{\\varphi} = n_\\mathrm{\\varphi} h", "4b154a574bdd87fc8ebb176907cb4bb5": "{6\\choose n}{43\\choose 6-n}\\over {49\\choose 6}", "4b157eb9f285ee15274ef1919b0fc0a9": " \\begin{align}\nE \\left[X_n^2 \\right ] & = E\\left[\\int\\int f_n(x)\\,f_n(y)\\,d\\mu(x)\\,d\\mu(y)\\right ] \\\\\n& = E\\left[ \\int\\int E\\left[f_n(x)\\,f_n(y)\\right]\\,d\\mu(x)\\,d\\mu(y)\\right ],\n\\end{align}", "4b15986c4beef8828b3403ba24fecb9c": "H_{0} = \\mbox{p}K_{BH^+} + \\log \\frac{[B]}{[BH^+]}", "4b15d198e560d8e137baa7d4c55d12e5": "p_{0} = \\sqrt{2m\\left| E \\right|}", "4b16331cfc2e513b526fe91670a7d9ca": " U_8(x) = 256x^8 - 448 x^6 + 240 x^4 - 40 x^2 + 1 \\,", "4b169a07bafb14908fc1a4362874fbdc": "E_{em}/c^2", "4b16a97ba67f10ae01ffdfe969ef2fcd": "\\alpha([[x, y], z] + [[y, z], x] + [[z, x], y]) = 0", "4b1747407b6a25937bd27ef7fe383b5e": "X \\sim IG(\\mu, \\lambda).\\,\\!", "4b17b8ed18d8394441fdbae1794395aa": "L_1 = \\ln\\left(R_1\\right)", "4b17d10d709c0a6b7baa383ad551633f": "s_0(x)=1;", "4b18d4532118149f26f3923e1e256d96": "x = ( \\lambda - \\lambda_0 ) (\\cos \\phi_0) \\pi / 180^\\circ", "4b18f1159318981143a1c54880219863": "\nX ", "4b19005c6a774249650a0c16d123ab3d": "v_y = v_{yo}", "4b197c8f51505c4194260b2fd3a0aa07": " \\hat{u}(x)=hull \\ \\tilde {u}(x)=hull \\{u(x,p):c(p)\\cdot u(p)=\\int\\limits_{\\partial \\Omega}\\left(G(p)\\frac{\\partial u(p)}{\\partial n} - \\frac{\\partial G(p)}{\\partial n}u(p)\\right)dS, p\\in\\hat{p} \\}", "4b19e393e5d70909e6a1182485d5cf5b": "g\\colon V \\to V", "4b19efee50d75e19ac7a7a48be9435cc": "S^3 ", "4b19f7a53d7f49432e4e63607dd33dd9": "\\begin{matrix} {10 \\choose 1}{4 \\choose 2}{9 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "4b19fb70d068b236db096e72b6416eb6": "\\rho=(x^i)\\circ (y^i)^{-1}:{\\mathbb R}^n\\rightarrow {\\mathbb R}^n", "4b1a1240324a0b0fb8294f2e91040860": " H,K ", "4b1a270c5f3824add0ef65789765d4de": "y = 1-x", "4b1a322b248952c24caae0debf24f711": "R_N ", "4b1a3970b84c9a4bede6c978d54ea0ee": "p(D|drunk) = 1.00 ", "4b1a44836a63ad19fa009b825e4ccda3": "\\beta1", "4b1a4d23ee19c61be2991297c514bf26": "\\mathbf{H} = \\frac{1}{2}\\sum_{\\alpha}(p_{\\alpha}^{2} + \\omega^{2}_{\\alpha}q_{\\alpha}^{2} -\\frac{1}{2}\\hbar\\omega_{\\alpha})", "4b1a896e2615a512ecf796e3254aa757": "a \\mapsto ax, a \\mapsto xa: A \\to A", "4b1a95aa056f59c2e4def7f0ba99734d": "c_1(TM)", "4b1abd777fc3b7d1488f8d272c8a92e5": "x^{\\left [ 2 \\right ]} = x^2 \\log(x) - \\begin{matrix} \\frac{3}{2} \\end{matrix} \\, x^2", "4b1b417cf961b759e1330bcc0fe3e467": "k \\ne 0", "4b1ba35f3a26c92043b659cb00da4721": "P(x,y)", "4b1bbd066f853bc42c26106e7af79614": "\n \\begin{align}\n u_\\rho &= - \\frac{1}{\\rho}\\, \\frac{\\partial \\Psi}{\\partial z},\n \\\\\n u_z &= + \\frac{1}{\\rho}\\, \\frac{\\partial \\Psi}{\\partial \\rho}.\n \\end{align}\n", "4b1c179f4e1358a079b31293e81f4c04": "\\mathbf{a} = \\frac{d\\mathbf{v}}{dt} = \\frac{d^2\\mathbf{r}}{dt^2} ", "4b1c491f938e0f03982ce86761904296": "\\psi \\,\\!", "4b1c51ee50cdcdd0f6546cf79ee30dec": "4 + \\sqrt{2} + \\sqrt{6}", "4b1c6d3a9753508be9e0967488450ac3": "\\mathbf{\\hat{e_i}}", "4b1c77c0135284712bce7a5eb2050c6d": "|\\Phi^+\\rangle = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_B + |1\\rangle_A \\otimes |1\\rangle_B)", "4b1cfef0da008a1777931f2417e85e11": "\\omega_c/2", "4b1d409571e2a97721dc3a0fe4e8697b": "d_1 < d_2 < d_3 < \\cdots", "4b1dc3b3083da9c4f81752f6be3d28cc": " p_m ", "4b1dd1ab88adffad264f4135a0d4bd67": "u*100%", "4b1e195497b2ecdff26088c2d0316a23": "T(3)", "4b1e36414d5281192796917c65184a74": "u_I = 1_A", "4b1e68c60b5d02ea84f6adc8c43c61b9": "\n\\begin{align}\n{0}&{} = \\frac{\\partial}{\\partial p} \\left( \\binom{80}{49} p^{49}(1-p)^{31} \\right) \\\\[8pt]\n & {}\\propto 49p^{48}(1-p)^{31} - 31p^{49}(1-p)^{30} \\\\[8pt]\n & {}= p^{48}(1-p)^{30}\\left[ 49(1-p) - 31p \\right] \\\\[8pt]\n & {}= p^{48}(1-p)^{30}\\left[ 49 - 80p \\right]\n\\end{align}\n", "4b1e70e226a046da499c0bc784b816e5": "(a_1,a_2),(b_1,b_2)\\in F", "4b1f0c39b98078e95a89f1423a7b63bd": "K\\{y\\}", "4b1f953e7bc9cb868240b86f9abfb0c0": "\\left(\\frac{-1}{p}\\right) = \\left(\\frac{-1}{q}\\right) = -1", "4b1fb61c2622a1bae499f2822c6e4a92": "\\|x\\|_p = \\left(\\sum_{i=1}^n |x_i|^p\\right)^{1/p}", "4b1fc64af2cdc2440ccf955ee69cbe03": "T_{(p,q)}(M_1\\times M_2) \\cong T_pM_1\\oplus T_qM_2", "4b20399b07cf9806ed2379fbabc7b791": "-x+1\\,", "4b2055b1a30880eb51f807ac6189337d": " \\lambda_\\mathit{D} ", "4b2069feab1239fa55e266be10704c37": " \\text{RCF} = 1.11824396\\, \\times 10^{-5}\\, r_\\text{cm} \\, N_\\text{RPM}^2", "4b20707a777f812e64f2bdd7dce81685": "- \\beta\\,", "4b20f00542ee7872dd2f70cca3aaebd7": "1 \\leq p_0 \\leq p_1 \\leq \\infty", "4b212c88572330809b7c158d18ddf3b8": "\n\\begin{align}\n \\Phi_{u_g}(\\Omega)&=\\sigma_u^2\\frac{2 L_u}{\\pi} \\frac{1}{1+ (L_u \\Omega)^2} \\\\\n \\Phi_{v_g}(\\Omega)&=\\sigma_v^2\\frac{2 L_v}{\\pi} \\frac{1+12(L_v \\Omega)^2}{\\left(1+ 4(L_v \\Omega)^2 \\right)^2} \\\\\n \\Phi_{w_g}(\\Omega)&=\\sigma_w^2\\frac{2 L_w}{\\pi} \\frac{1+12(L_w \\Omega)^2}{\\left(1+ 4(L_v \\Omega)^2 \\right)^2}\n\\end{align}\n", "4b213a6934a74a30522011c924a9d57b": "c = 1+\\frac{A}{I}=73.6 .", "4b2165750c7cbcc296cf0cd5dfe3ba4c": "\\forall m \\forall n [(SmC", "4b276a6180590a8fb705f6936e66a39a": "\\mathsf{d}_A(\\mathbf{x}, \\mathbf{y}) \\stackrel{\\vartriangle}{=} \\Big|\\{i \\mid x_i = 0, y_i = 1\\}\\Big| + \\Big|\\{i \\mid x_i = 1, y_i = 0\\}\\Big|.", "4b27854f670287d1234d6a3767cce5f0": "\\bar{s} = \\bar{v}\\sqrt{3}", "4b27ddf225194093cd1475f8d2d75bde": "X \\Rightarrow Z", "4b27ed11d2c4d889e72c0ee878896c89": " v : 2^N \\to \\mathbb{R} ", "4b285f26d578e2c991689ab198a8f31a": "r = a \\frac {\\sin n \\theta}{\\sin (n+1) \\theta}\\!", "4b2901f143c0441eb323032bc85dcb85": "F^{\\tau \\sigma}", "4b2918089b399b99ba81603eafbb4972": "\\to \\!\\,", "4b291fe92c605fcf945a8bae2f52ba72": "{d \\over dx}\\tan y={d \\over dx}x", "4b2924a50f7f1093a05d5b00b5d9d176": "f_i(r_1,\\dots,1)", "4b29564298f78317da9bb32a02356652": "\\left(h(y), y\\right)", "4b299feffbb79b4060bfd3820d0fe63c": "\\int_{0}^{1}g_n=1", "4b29afed80cff854bb3034610994eb00": "\\overline{u_{i}}=\\frac{1}{T}\\sum\\limits_{t=1}^{T}u_{it}", "4b29e7f256c8d9ac92d42f92db46dced": " \\frac{\\partial V}{\\partial S}", "4b29f60a453b085ad6121c15543d7664": "\\begin{align}\n&\\text{Target Income Sales (in Units)} & &= \\frac{\\text{Fixed Costs}+\\text{Target Income}}{\\text{Unit Contribution}}\\\\\n&\\text{Target Income Sales (in Sales proceeds)} & &= \\frac{\\text{Fixed Costs}+\\text{Target Income}}{\\text{Contribution Margin Ratio}}\n\\end{align}", "4b2a04cbd22d1f2d243a7d6afe97ad76": " p = \\sum_{i=1}^m f_i^2 ", "4b2a66fbd4f7bfbd43c710ad42d735b2": "\\tfrac{1}{2}", "4b2aa8b5db61c1ecd2c317108746b651": "F(s_1,\\ldots,s_n)", "4b2ab9a98940644e95f0b92d088016e2": " c(z)= e^{\\frac {1}{2} \\frac{b^2(z)}{a^2(z)} - \\frac{1}{2} \\left( \\frac{\\mu_x^2}{\\sigma_x^2} + \\frac{\\mu_y^2}{\\sigma_y^2} \\right)} ", "4b2ac92a88cc39447b743704c6532326": "\\overrightarrow{A'B'}", "4b2b2f8d66355a326490f2bfc5284b16": "\\frac{c}{a} - 1 = -(120 \\pm 5) \\times 10^{-4}", "4b2b6c36a6b0f92523823e32874f4b65": "\\operatorname{dom}(f)", "4b2c05e45381ab1e6b55f01851697bf9": "(A\\land B)\\ne (B\\land A)", "4b2c1d8a5e9df6e320c57806804a2705": " \\operatorname{Var}(\\mathbf{X}) = \\operatorname{E}(\\mathbf{X X^\\top}) - \\operatorname{E}(\\mathbf{X})\\operatorname{E}(\\mathbf{X})^\\top", "4b2c206f35aa7686e789ad8f938d885b": " L_{(\\sigma)}=\\frac12[a_1T^\\mu{}_{\\nu\\mu} T_\\alpha{}^{\\nu\\alpha}+\na_2T_{\\mu\\nu\\alpha}T^{\\mu\\nu\\alpha}+a_3T_{\\mu\\nu\\alpha}T^{\\nu\\mu\\alpha} +a_4\\epsilon^{\\mu\\nu\\alpha\\beta}T^\\gamma{}_{\\mu\\gamma}\nT_{\\beta\\nu\\alpha}-\\mu\\sigma^\\mu{}_\\nu\\sigma^\\nu{}_\\mu+\n\\lambda\\sigma^\\mu{}_\\mu \\sigma^\\nu{}_\\nu]\\sqrt{-g} ", "4b2c7711be9385b6d78fd3d54e7a238d": "\\lambda \\left((1-\\mu)\\|w\\|_1+\\mu \\|w\\|_2\\right)", "4b2c81b068323d3a63004552b4c02134": "F(x,y) = f(x,y) - \\overline{f}", "4b2d44671efd87399cf057abc03d92a4": "\nV(r) = -\\frac{GMm}{r} + \\frac{ L^2 }{ 2 \\mu r^2 } - \\frac{ G(M+m) L^2 }{ c^2 \\mu r^3 }\n", "4b2d56c41b64273eeba2eb1cff727092": "dE = T dS - P dV\\,", "4b2d603f3d8f57577f4ba3f83783765e": "\\textbf{Rouse}=\\frac{\\text{Settling velocity}}{\\text{Upwards velocity from lift and drag}}=\\frac{w_s}{\\kappa u_*}", "4b2dc4cb20dd6a949bd78d8c657de1e1": "h_M\\;", "4b2df465adc05df59f4cec653dcf4e0e": "{\\Delta \\left( {{{\\partial v} \\over {\\partial T}}} \\right)_P = \\Delta \\left( {\\left( {{{\\partial P} \\over {\\partial T}}} \\right)_v } \\right) \\cdot {{dv} \\over {dP}}}", "4b2ec78a72032be00cae74c049d14b01": "\\bigwedge^{\\hbox{even}}T^*M", "4b2ee5cf838b525215f4249e123dd86e": "\\lambda\\in\\mathbb{C}", "4b2ffeba2f6277804e39825d078fc8b8": "J_x = \\begin{pmatrix}\n0 & 0 & 0 \\\\\n0 & 0 & -i \\\\\n0 & i & 0\n\\end{pmatrix}\\,\\quad J_y = \\begin{pmatrix}\n0 & 0 & i \\\\\n0 & 0 & 0 \\\\\n-i & 0 & 0\n\\end{pmatrix}\\,\\quad J_z = \\begin{pmatrix}\n0 & -i & 0 \\\\\ni & 0 & 0 \\\\\n0 & 0 & 0\n\\end{pmatrix}", "4b3088f85d4f26919c5752a1b1961b04": "\n\\operatorname{Li}_s(e^\\mu) = \\Gamma(1-s) \\left[ (-2\\pi i)^{s-1} \\sum_{k=0}^\\infty \\left(k + {\\mu \\over {2\\pi i}} \\right)^{s-1} + (2\\pi i)^{s-1} \\sum_{k=0}^\\infty \\left(k+1- {\\mu \\over {2\\pi i}} \\right)^{s-1} \\right] ,\n", "4b316f135dfd45ce392a8d0e4a2a5d61": "p_n(z) = \\sum_{k=0}^n w_k\\frac{G_n(z)}{(z-z_k)G'_n(z_k)}", "4b31e195bd8682ff8e1933fb77892769": "\\boldsymbol{\\chi} + \\mathbf{T}(x)", "4b320bcf0108f988c14b2d1b2c662d85": "\nE_{\\ell r} = \\iint d\\mathbf{r}\\, d\\mathbf{r}^\\prime\\, \\rho_\\text{TOT}(\\mathbf{r}) \\rho_{uc}(\\mathbf{r}^\\prime) \\ \\varphi_{\\ell r}(\\mathbf{r} - \\mathbf{r}^\\prime)\n", "4b32182d86c3c5234ffd463cf92536f5": "y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + \\cdots + a_my_{t-m} + \\mathrm{residual}_t.", "4b32330f9386d3e439443333d780a6e9": "R_T = R_a+R_b+R_c", "4b324fa538f13ee776a79c70087d4edd": "v_1 \\triangleq u_1 - \\dot{u}_x", "4b325f93dbe8b56c5f70c97d7ed6b450": "\\mathbb{Z}_{11}", "4b329d2d9798c4c3f8ce76e213919c12": "\\mu = e^{\\int p(x)\\, dx}", "4b32a379573c4ab142fd572d4ea5394b": "Y^h=h^*Y", "4b32b5293afa4e4935f8da0957a7ef83": "h^\\sim = t^2 g_{ij}(x,\\rho)dx^idx^j+2\\rho dt^2+2tdtd\\rho,\\, ", "4b32d1d4f4e272bcd1813ed584990070": "H(t)|\\psi(t)\\rangle=i\\hbar\\frac{\\partial |\\psi(t)\\rangle}{\\partial t}", "4b3328c8743f7532b61d7258e384cba7": "\\mathbf I_n ", "4b332a7cf79ff533d8a06e3e2ffea22a": "\\mbox{FileSize} \\approx 54 + 4 \\cdot 2^\\mbox{bpp} + \\mbox{PixelArraySize}", "4b3342bcde7e3c173559441efa5c6be9": "g_S = 2 ", "4b341e7674b0ad7cdf0d865966821bb2": "s = ek + c \\pmod{n} \\,", "4b345686aba6f28a561de991918c7000": "\\frac {L}{hr}", "4b3501e1f0d69ac786d00343bd43b7b6": "D_T = N/R", "4b353bd9d1f9695a9298709307f2842f": "\\varphi+\\theta={1 \\over 2}\\pi\\!", "4b353ce39e0280946c84b24da3413277": "\\frac{d|x|}{dx} = \\frac{x}{|x|} = \\begin{cases} -1 & x<0 \\\\ 1 & x>0. \\end{cases}", "4b3569c5ef7c78ac736405b38b2e1470": "\\lambda_\\mathrm{now}", "4b356ac18a59a36c3e004e1c75bac466": "\\displaystyle f= H\\cdot\\varphi,", "4b358a0f968ce9f7a587f4210bdecfaa": "\\left\\{ 1,\\dots,2\\lambda\n\\right\\}", "4b35984ceb77410662713097425bbd6c": " M^3", "4b359fc949e93dab2736675a8197ba5a": "E\\{Z(x_i)\\}=E\\{Z(x_0)\\}=m", "4b35b180f1f52022c949813ac5fb7325": "V_{drm}", "4b35e31d071f4ac8b56e283e70535e52": "pV = n N_A k_B T = nRT", "4b36117ebb715264559f3849b067c33d": "Duration \\ gap = duration \\ of \\ earning \\ assets \\ - \\ duration \\ of \\ paying \\ liabilities \\ \\times \\ \\frac{paying \\ liabilities}{earning \\ assets}", "4b361f42a32ee73d5009126d49d7fb47": "X_{t}=\\mu+\\Phi D_{t}+\\Pi_{p}X_{t-p}+\\cdots+\\Pi_{1}X_{t-1}+e_{t},\\quad t=1,\\dots,T", "4b37127235fc09d2939e9d40484e1e5d": "L_1, L_2 \\text{ and } M", "4b3715fe78aac67ef9d61bae442a1c24": " u_{n \\mathbf{k}} (\\mathbf{r}) = \\sum^{A}_{j'} a_{j'}(\\mathbf{k}) u_{j'0}(\\mathbf{r}) + \\sum^{B}_{\\gamma} a_{\\gamma}(\\mathbf{k}) u_{\\gamma 0}(\\mathbf{r}) ", "4b37216348eff25dc063cc5f4bf0ba7e": "\\left \\| \\cdot \\right \\|", "4b3810b871cc95880dc7a286df2b82d9": "L\\frac{\\mathrm{d}^2q}{\\mathrm{d}t^2} + q/C = \\mathcal{E}\\,\\!", "4b3817e7a262df256ae373f84c03f008": "R_g", "4b385554898250bb825ec44e1af36b8d": "GL_n", "4b3896e138f64bdbe99994b19cebac9c": " | \\psi \\rangle = \\frac{1}{\\sqrt{2}} \\bigg( | 00...0 \\rangle + |11...1 \\rangle \\bigg) ", "4b38db92c97edaff216fe319164a77c0": "{\\Gamma^i}_{ii}=\\begin{Bmatrix}\n \\,i\\,\\\\\n i\\,\\,i\n\\end{Bmatrix} = \\frac{1}{h_i}\\frac{\\partial h_i}{\\partial q_i}\\! \\ ;\\ ", "4b390d3bd23b64f925c75980a795e875": "\n X_3 = \\frac{aX_1 + bX_2 - (a+b)\\mu}{\\sqrt{a^2+b^2}} + \\mu\n ", "4b3963f48abc8b05c6c268779032dc16": "\\Pi^1_2", "4b39662630c95026cbe13aa83d4f26ba": "\\widehat{\\mathbf{z}}", "4b3a126c9632cbf487aee2a58439044c": "\\mathrm{u}(t) = \\int_{-\\infty}^{t} \\delta (s)ds", "4b3a1e5677ec2b22b07d0a7c2dae1693": "\\textstyle M(2^{l-1}+1) \\le 2^n", "4b3a4d4000da214e17ef011861f4d5a5": "u_n(t) = \\cos(2\\pi nt)+ i \\sin(2\\pi nt) \\, ", "4b3a8cd79fa08bd4a0b44bbf6e7da859": " (V_{\\mathrm{A}}-V_{\\mathrm{B}}) = -(\\mu_{\\mathrm{A}}-\\mu_{\\mathrm{B}})/e ", "4b3aa126ea5837b4414f1772d5e9d17b": "x = r \\cos\\theta", "4b3abba2d64f368a46d1a444fb2d3677": "y = \\frac{x^2-3x-2}{x^2-4}", "4b3ac55120aad7f414dd7794a9ddc31f": "r = |z| = 1", "4b3ac95f7d777b7aa2f60b3ad883c452": "\\mathbf{r}_j\\,\\!", "4b3b7aad93a564ac4c68a2796ddc8e2f": " \\tau_g = -\\frac{d\\phi}{d\\omega}", "4b3b94cb3ecc01f6e917df8c4384d8cf": "\\displaystyle{H^*=KA^*K.}", "4b3bf71f9bc6b07643ed7241fccd09a0": " i{\\partial \\psi_j(t) \\over \\partial t} = H_{0} { \\lambda^2 } { \\partial^2 \\psi_{j}(t) \\over \\partial x^2 } + H_{jj} \\psi_{j}(t) ", "4b3c782404f7b1d93ec24cb1bb8bdbc7": "{\\mathcal L}_{xy}^2: L=lk;", "4b3c865bf23e236c2fcfae9826c50a07": "(x\\cdot y)\\cdot z = x\\cdot(y\\cdot z)", "4b3c92914fedcdbcce650dbdc6e97536": "kP ", "4b3cab19c2e7fe40d39f9982c4d5fd93": " \\partial_X \\langle Y,Z \\rangle ", "4b3cd5aa19183ac527f4d6d8541dd89f": "\\beta^{'}(1,1,\\gamma,\\sigma) = \\textrm{LL}(\\gamma,\\sigma)\\,", "4b3d2c0e2f3fef1ddbd289c419e198ab": "\\mathcal{F}^\\mathbf{W}(t)", "4b3d606ca4adecb0b7fa0d2a5747adc3": "\n y_{it} = y_{-i,t}'\\gamma_i + x_{it}'\\;\\!\\beta_i + u_{it}, \\quad i=1,\\ldots,m,\n ", "4b3d96d629452e6ead0dee05cb48a947": "\nA(x,y)=-A(y,x)\n\\,", "4b3e0451d2904f7ec88ee27bc42aa93e": " \\left| x- \\frac{p}{q} \\right| > \\frac{c(x)}{q^{n}}", "4b3e3551f6d7204be7e628e09ab0e01e": "\\textstyle\\gamma = -\\tfrac{1}{2\\sigma^2}", "4b3ec686eda592681a40b1e7b20c0bce": "AOP_m(x) = \\sum_{i=0}^{m} x^i", "4b3ecf78547a65358a1cc5f8663e9b20": "\\mathcal{F}(f')(n)=\\int_{-\\pi}^\\pi f'(t)e^{-int}\\,dt=\\int_{-\\pi}^\\pi in f(t)e^{-int}\\,dt = in\\cdot\\mathcal{F}(f)(n)", "4b3ee8c25091dc36d848440ffb1a18f8": "n[\\;A\\;]", "4b3ef0cc807474c969738db1948dca55": "\\Phi^{2} = \\Phi_{A}\\Phi_{A} = (\\Phi_{M}^{2} + \\Phi_{J}^{2})", "4b3f14ea944d358331256ad1b2b6624a": "\nD^*_N(x(1),\\dots,x(N))\\leq C\\frac{(\\log N)^{s-1}}{N}\n", "4b3f31a8de42fbc9ab4eed716e503e27": "E_6, E_7, E_8,", "4b3f6e1de49afe3e4a389980a3b6e375": "\\mu_{01}", "4b401d6b7891550e8ca829ce74963b57": "\n \\mathbf{N} \\ = \\ \\frac{d\\mathbf{L}}{dt} = \\mathbf{r} \\times \\mathbf{F} \\ ,\n", "4b40710a8e44987907decf32fdfdc982": " \\mathfrak{g}\\simeq\\mathfrak{g}_0\\otimes_{\\mathbb{R}}\\mathbb{C}. ", "4b40d0ca4aa197e1722e331e677c0d67": "a_0 = 0 , \\;\\;\\;\\;\\;\\; a_1 = 0 , \\;\\;\\;\\;\\;\\; a_2 = 0 , \\;\\;\\;\\;\\;\\; a_3 = 10 , \\;\\;\\;\\;\\;\\; a_4 = -15 , \\;\\;\\;\\;\\;\\; a_5 = 6", "4b40f3149176697fdcfe5a538a81cf15": "1\\le i < m", "4b4114c8222f1a05f8764bcf5ca2dbcf": " J = \\Lambda, \\Lambda +1, \\Lambda+2, ...... ", "4b415c7f88fb8ae7ac0635d5a84a781a": "\\mathbf{T}_{C,i}", "4b4174fd7e64105381babda1a82d0620": "\\sec \\varphi + \\tan \\varphi = e^{x/a},\\,", "4b418aa1052ec2362d5f6c1b9e9c73d5": " = \\hat{a}_j^\\dagger (\\hat{a}_l^\\dagger\\, \\hat{a}_i + \\delta_{il}) \\hat{a}_k + \\delta_{kl}\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij}\\hat{a}_l^\\dagger\\hat{a}_k + \\delta_{ij} \\delta_{kl} ", "4b4229a9904462d9728b26a1c3c6f3f3": "L = \\lim_{x \\rightarrow a} f(x), ", "4b42688bffc69d3cbe59f8e7b3fc4a1d": " \\sigma \\circ \\iota_{0,1, ..., p}", "4b4271173882938c004a86942958c13c": " \\vec{R}_{cm} = (a \\mathbf{\\hat e}^1 + b \\mathbf{\\hat e}^2 + c\\mathbf{\\hat e}^3)", "4b429e0ad92d648f4dbcb9cbbc87dfcb": "\\mathbb{A}^1", "4b42ab1c6ee0279ede1a9f511966498f": "v_n(t)", "4b42fb26237859b0a84678eae488cf1e": "g = \\left(\\frac{\\alpha-1}{\\alpha-2}\\right)^{\\frac{\\alpha-1}{\\alpha}} h > h", "4b43550d932e47d8b693236e0e7caced": "z_0 = (\\langle 0\\vert z^2 \\vert 0\\rangle)^{\\frac{1}{2}} = (\\hbar/2m\\omega_z)^{\\frac{1}{2}}", "4b43751e22571530da9a989974eb0836": "|{\\Phi^{[3..N]}_{\\alpha_2}}\\rangle", "4b438f7599bf535f863f784852cb9723": "\\mathbf{(x, y)}", "4b439bb65bac1353654b5ba0f56ccd74": "M_{BA} = 0.2 \\times 40.219 + 0.4 \\times \\left( -6.937 \\right) + 6.3 = 11.57 ", "4b43b0aee35624cd95b910189b3dc231": "r", "4b43c3ebb01d843740ae84a916d74840": "\n\\frac{\\perp true}{C \\ true} \\perp_E\n", "4b43eeab196d72321cc4de0617173181": "\\varepsilon_{r} \\gg 1.", "4b43f8cf105c40da820cc9358e9105e7": "m_{Lk+i}= \\ell_i", "4b43fc0ca7643ac915a39e05ac67dc8b": "S = k[x_1, ..., x_d]", "4b441b8cd8766abcc8685d31941341a4": "d_0=1", "4b443373a302c1ef7148b36425d29054": "D_\\hat{M}f=E^N_{\\hat{M}}\\left( \\partial_N f + \\omega_N[f] \\right)", "4b4450f94f967f6ab67f03fe7cbe1d1f": " S=\\{S_{ij}\\} ", "4b4469e6b1b303bb92ac92e8ca25312a": "w\\to 0", "4b4500584dca85f4a5e22688c0b3a4a7": "\\Box A_1; \\ldots ; \\Box A_n ; \\neg \\Box B", "4b450655d811bd0d7fb3b0154008b14b": " S_{j + \\nu} + \\Lambda_1 S_{j+\\nu-1} + \\cdots + \\Lambda_{\\nu-1} S_{j+1} + \\Lambda_{\\nu} S_j = 0 \\, ", "4b45186d76c5e342a926743989938b1a": " \\lambda m.\\lambda n.\\lambda f.\\lambda x.m\\ f\\ (n\\ f\\ x) ", "4b457c7acd9cc3d75725dfc7b6a80cbd": " \\lim_{N\\to\\infty} \\frac{f(N)}{N(N+1)/2} = 1. ", "4b45818af25f54d84b7ad61dea54faee": "\\frac{f}{d}", "4b458491d07bd0289b2a822fc351db23": "-n_1 - 2n_2 - n_3 = 0 \\ ", "4b45dcc90bfdbfa7ff87e70be47654a5": "\\alpha(n)=1", "4b4639088ca953f437cf937afebe4758": "\\scriptstyle\\mathfrak{G}:{{(T\\times M)}^M}\\to\\mathbf{C}", "4b46721201524a21435dcb06ffc356b8": "\\mu=\\begin{cases}\\mu_0,&\\Pi\\leq\\Pi_0\\\\k\\Pi^{n-1}+\\tau_0\\Pi^{-1},&\\Pi\\geq\\Pi_0\\end{cases},", "4b46f2a1dd99acf8dbc9ddcb764a582f": "N \\ge 2", "4b4794dd5105db5aae9f7f954050f3a4": "\\scriptstyle O(1)", "4b479c3f43c6d5b4a9dc19d5a66834f9": "|\\Omega| \\ge \\pi/T", "4b479f1300bda1b13cd31a661f160543": " U_{ni}= \\beta z_{ni} + \\varepsilon_{ni} ", "4b47d7e1da20fb791515dae1f450d71d": "\\hat{\\mathcal{P}}_{\\rm CAS}", "4b47f7f0c7ef7c1251bc2163dffd36aa": "d_H(\\mathbf{u^1G, c_{s}}) \\leq t", "4b488ed8ebb72ac041c78b06f3f853ca": "V_{max}=\\sqrt{\\frac{E_a + E_u}{0.0007d}}", "4b48910c852424bf24a40b6f404e0b68": "\n \\sqrt{n}(T_n - q(\\theta)) \\ \\xrightarrow{d}\\ Z_\\theta + \\Delta_\\theta,\n ", "4b48ceac6a7ece249ae6c8d046b50071": "\\sum_{m=0}^{n-1}(-1)^{m}A(n,m)=\\frac{2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1} \\text{ for }n \\ge 1.", "4b48f1ce9b704175f19df5f64842e41d": "\\Gamma(\\tfrac17) \\approx 6.5480629402478244377", "4b490d5908f28edd86688789fbd70840": "a \\to b \\to 2 \\to 2 = a \\to b \\to 2 \\to (1 + 1) = a \\to b \\to (a \\to b) \\to 1 = a \\to b \\to a^b = a \\uparrow^{a^b} b", "4b49aec26070a30ef72739da5fae11fb": "\n\\begin{align}\n2R'_c L'_s & = 2 \\sqrt{\\frac{R_c}{2L_s} } \\sqrt{\\frac{L_s}{2 R_c}} \\\\\n & = \\tfrac{2}{2} \\\\\n & = 1\n\\end{align}\n", "4b49ca6aeb296b7d35151598883d8a15": "E^{\\mathrm{tot}}=E^{\\mathrm{kin}}+E^{\\mathrm{pot}}+U\\,\\,.", "4b49d8ceb2abd91218a35635e23db1e0": "K= 10\\ \\log_{10} (1.380 \\cdot 10^{-23})= -228.6 ", "4b49d8e7164422c9f4b5208277da7bb2": "d_{\\rm f} = {{2D^2}\\over{\\lambda}},", "4b49dd0b2c5a7f596167b18ee2d96c4d": "\\text{Ball}", "4b4ac6aedfb634b229921da01023f4d9": " s = \\frac{2(z-1)}{T(z+1)} ", "4b4b608b63e4573b9f01be831a0a8f30": "\\nabla^2\\phi_0=0\\,", "4b4b9606afa302771a76dc595be4de5c": "X^\\mathrm{T} Z, Z^\\mathrm{T} Z", "4b4ba5e69577f5210ba96fa47603f923": "f'(z)=k(z-z_N)^{k-1}g(z)+(z-z_N)^kg'(z)\\,\\!", "4b4ca983e2dc0136f0291345778d50a2": "p_1^{e(p_1)},\\dots,p_k^{e(p_k)}\\vdash A^1.", "4b4cae62652a60ae1ec569e96f7470b1": "g=(g_{i,j})", "4b4d226bea79e8dddca628cc795975f0": "\\displaystyle{\\lambda_k=\\sum_{n=1}^N \\alpha_n z_n^{-k}.}", "4b4d2f33c29446b48e374a6071bc4887": "l = \\lg^2 n", "4b4d88d51ed38d8563bcd29b6c5e274c": "\\mathcal{L}_X g", "4b4daa669469824194debda546e404cd": " L_{\\pm}|L_,m_L \\rangle = \\sqrt{(L \\mp m_L)(L \\pm m_L +1)} |L,m_L \\pm 1 \\rangle", "4b4dac24ca56c7f471d2c3c60efc9b42": "(\\rho uA)_e + (\\rho uA)_w = 0", "4b4e0d55d783c345695d06bdcddbd15c": "\\alpha \\,\\ ", "4b4e11d31b1abff64408e8150976e113": "\\Pr(X>t+s \\mid X>t)=\\Pr(X>s).\\,", "4b4e176268469101b53399bd8f6182c9": "q=\\alpha \\beta", "4b4e23bf7c54a08bf2ea36edbda4ebcb": "y = x^2 - x - 2", "4b4e2da9da2cee10813bc9e0120b1589": "\\frac{d \\mathbf{Q}_k}{dt} (t) = \\frac{\\hbar}{m_k} Im \\left(\\frac{(\\psi,D_k \\psi)}{(\\psi,\\psi)} \\right) (\\mathbf{Q}_1, \\mathbf{Q}_2, \\ldots, \\mathbf{Q}_N, t)", "4b4e52d18eeaff0807be26a17b67ba15": "\\lambda_1 = 0, \\lambda_2 = \\lambda", "4b4e85f02bc9d49298b8c41a80ca148b": "\\scriptstyle X_n\\xrightarrow{p}X", "4b4eda92e9b96290a42d39b426fe2411": " 0.3 < M < 0.8 ", "4b4edcad0f75921a8f89d5ee3dc7ec59": " \\begin{align} \n \\tilde{\\nabla} E(f(x) | \\theta) \n &= F^{-1}_\\theta \\nabla_{\\!\\theta} E(f(x) | \\theta) \n\\end{align}", "4b4efc2fbe82a047fc08c83ea081f1d9": "\\star", "4b4fc38829d0fca45dc9baae17e9fa8c": "\\mathbb{P} \\left( \\max_{1 \\leq k \\leq n} | S_{k} | \\geq \\alpha \\right) \\leq \\frac{27}{\\alpha^{2}} \\mathrm{Var} (S_{n}).", "4b4ff4f1f39e0a56ce04522ea211fc8b": "\\sin(\\theta) = \\sin(\\arcsin x) ", "4b50457afad22cf214a8a83a5cd0210c": "\\theta _j", "4b5089be4fa45345a310a267290f9725": "\\theta = \\frac{\\mu_1 - \\mu_2}{\\sigma},", "4b50e713adcfaf419ec453677527f2b4": "P(\\overline{\\zeta}) = \\sum_{r=0}^n a_r\\left(\\overline{\\zeta}\\right)^r", "4b50ee30f96ca2708a36912aab0298ac": "c_k=a_0b_k+a_1b_{k-1}+\\cdots+a_{k-1}b_1+a_kb_0", "4b50f0f982dc067b1a89c9a2c5e5f83c": "\\ \\displaystyle \\hat{\\alpha}(q,r_{c}):=\\max_{\\alpha \\ge 0}\\,\\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})}\\,\\varphi(q,\\alpha,u)", "4b5131fec1f20c6ec9074688d7aa0cbc": "\\mathbb{D}=\\{X\\in\\mathbb{R}^2:|X|<1\\}", "4b5175e380d2a601b3fe7a218acb59e3": "R = R(t,r)~,~~~~~~~~ R' = \\partial R / \\partial r~,~~~~~~~~ E = E(r)", "4b518b0e9c77a3beec0247413948c180": "e_i=y_i-(\\widehat{a}+\\widehat{b}x_i)\\,", "4b519a7c2b76d43a883480e5145d3766": "\np_i(\\mu)= c_{i,0} \\mu^{k-i} + c_{i,1} \\mu^{k-i-2} + c_{i,2} \\mu^{k-i-4}+\\cdots\n", "4b51b11955b3b5d505dfefd4fdeef3e2": "g_{ik} = \\eta_{ik} + \\epsilon h_{ik}", "4b51b5a1f51d4db9bfd5e3f8397894c9": "|\\phi_i \\rangle |\\phi_i \\rangle = 0 ", "4b51c4495418adae0ee25d698c3d1155": "dU =\\frac{1}{\\beta}d\\left(\\log Z+\\beta U\\right) - X\\,dx \\,", "4b520e088e1e69bb9dba4e29b72b4aad": " S^+ = E^T \\mbox{Diag} (e^+) E ", "4b5253a6b277b28dd2b5be034a802a07": "\n F(b,a) = \\langle b,a| \\varphi\\rangle\\; ,\n", "4b525a92af6f2a3986792365b5733945": " \\text{kurtosis excess}(Y) =\\text{kurtosis excess}(X)=\\frac{6[(\\alpha - \\beta)^2 (\\alpha +\\beta + 1) - \\alpha \\beta (\\alpha + \\beta + 2)]}\n{\\alpha \\beta (\\alpha + \\beta + 2) (\\alpha + \\beta + 3)} ", "4b52d3a3df34d00d1dd866bd388b8e27": "(\\mathfrak{g}, e)", "4b538df5adb7286d72218ddf048e6f94": "F_1 = \\frac{F_0}{x_0^m} \\,\\, x^m, \\, ", "4b53eccd76e0048cf6e6ae8173dc2556": "\\int_{-x}^x f(t)\\,dt = \\int_{-x}^0 f(t)\\,dt + \\int_0^x f(t)\\,dt = -x + x = 0", "4b543a2ed77eb4eb6f5dc83ce583ef05": "\\displaystyle{\\mathrm{Tr}\\, \\psi(a) = \\int a}", "4b543d29e5bab4676c94b920f385556a": "\\psi\\ {\\sim}\\ \\psi + 4\\pi k/n", "4b544e6f4484ac56e90dc7deda4d4cd6": "(x-4)(x-2)(x+2)(x^3-x^2-2x+1)^2", "4b548aed1de56d3d86b78a7635c3a252": "\\textstyle L=\\frac{k_{B}T}{\\pi r_{I}^{2}}", "4b54a427d24e846e18e04e88801aa65e": "S \\rightarrow \\epsilon", "4b5507fe8fe507d3239912f63a0040a6": " \\mathbf{B} = \\mathbf{e}_1 \\wedge \\mathbf{e}_2 + \\mathbf{e}_3 \\wedge \\mathbf{e}_4 = \\mathbf{e}_1\\mathbf{e}_2 + \\mathbf{e}_3\\mathbf{e}_4 = \\mathbf{e}_{12} + \\mathbf{e}_{34}", "4b55114a8a2f40ff730a07d3c17584b4": "\\underline{d}(A)=\\overline{d}(A)", "4b557919b17a5ce760d6895ccfa248ca": "\\operatorname{IsZero}", "4b55a9c2b565804930f25591a1675cdc": "{\\Theta}={\\epsilon}_{11}+{\\epsilon}_{22}+{\\epsilon}_{33}", "4b55aa02e08e645cf630a3a3a6649c1a": "\nL_p=20 \\log_{10}\\left(\\frac{p_{\\mathrm{rms}}}{p_{\\mathrm{ref}}}\\right)\\mbox{ dB}\n", "4b55e19aceba6eb01778f438dee3c291": "J_2\\ =\\ 1.7555\\ 10^{10}\\text{ km}^5/s^2 \\, ", "4b56b35885b82f707d65d1b631639a44": " \nC=\\lim \\inf \\max_{p^(X_1),p^(X_2),...}\\frac{1}{n}\\sum_{i=1}^nI(X_i;Y_i).\n", "4b56ba8b2d036e21ccd69e492fdc4b14": "F(u)=\\int_{-\\infty}^\\infty f(u +{t^2\\over 2}) \\, dt,", "4b56dbc6601a5ba189ac634de4c999f5": " \\theta \\le \\nu \\le \\mu ", "4b56e7d56e9106aeed75ffe91eba979f": "B = \\begin{bmatrix} \\mu & j \\kappa & 0 \\\\ -j \\kappa & \\mu & 0 \\\\ 0 & 0 & \\mu_0 \\end{bmatrix} H", "4b5708457cdcb416c0f6bf3c74e3a2b4": "f_* : \\mathbf{Ab}(X) \\to \\mathbf{Ab}(Y)", "4b571efb2eb9ca230162db0ad0610fdd": "x_1, x_2, \\cdots", "4b5741a45f893784ced71452f683ad20": "S(n) \\le (2n-1)\\Sigma(3n+3) \\,\\!", "4b5747612caa0a117a7ca81a4fb99260": "\n\\chi = \\frac{\\chi_s+2g+g^2\\chi_s^*}{1+|g|^2 - 2\\mathrm{Re}(g\\chi_s^*)}\n", "4b579848f0be08fc44e67ad94929379d": "|{k_i}\\rang", "4b57af32334c45a8afb1f4fe351f2b28": "b=\\operatorname{E}(Y)-{\\operatorname{Cov}(Y,X)\\over \\operatorname{Var}(X)} \\operatorname{E}(X)", "4b57f1754ac73c825dfad6ceeb8efe23": " i \\in \\left \\{ 1,\\dots , k \\right \\}", "4b5826571a370efba4cc1bde504d4d16": "\\exp\\left(\\sin\\left(50x\\right)\\right) + \\sin\\left(60e^y\\right) + \\sin\\left(70 \\sin x\\right)+\\sin\\left(\\sin\\left(80y\\right)\\right) - \\sin\\left(10\\left(x+y\\right)\\right) + 1/4\\left(x^2 + y^2\\right)", "4b583a22fce24f61721e6bb7ded60df9": " P b-c = \\sqrt{\\gamma}q\\,", "4b59fb43610f2a2f9da867c51d35d8c5": "q\\to 1", "4b5a0a29c70901d523aa319614effcd3": "e^{2\\pi i \\frac{k}{n}} \\qquad 0 \\le k < n.", "4b5a6586945fdfa233cd7934d0e464e6": "e(i)\\,", "4b5aa94b338ccaeac6bc952d6cc66916": " \\langle A,B,C,D,E,F \\rangle ", "4b5aea5b261a222103d60ba4b44f0dd3": " \\nabla_{e_i} v = \\langle Dv, e_i\\rangle = \\sum_k e_k \\left(\\nabla_{e_i} v^k + \\Sigma_j\\Gamma^k_{ij}(\\mathbf e)v^j\\right)", "4b5b0a3cbbfa65cef63367a039b94c9b": "z^2 - (2a - b)z + (a^2 - ab + b^2). \\,\\!", "4b5b5bb8eec1abc5758338fa126dd0f4": "B(z+\\Delta z)", "4b5b62444eacd26c3777f9fdd0b53fd9": "f(x) = x^8 + 1", "4b5bca169ca6c87c3a9861175c7add10": "q = \\tfrac12\\, \\rho\\, v^2 ", "4b5c6f3448a8ecd0c97ede2f683d94fc": "F^0=B_\\nu J(J+1) ", "4b5cef3be5693a2a7e701d4901640ba3": "df(X_t) = \\sum_{i=1}^d f_{i}(X_t)\\,dX^i_t + \\frac{1}{2}\\sum_{i,j=1}^df_{i,j}(X_t)\\,d[X^i,X^j]_t.", "4b5cff3e851fdb0063fb4ae6958c7b4e": "T_n(1)=B_n.", "4b5d143a062493e90ea6dad2626969d6": "\\displaystyle{\\text{Dom}(R)=S}", "4b5d45e4bef62afa5b8f69167562d639": "\\scriptstyle f_n(x,y) = \\frac1n \\exp(nx)\\cos(ny)", "4b5d52b6623ee4efe14674c8bb537aec": " \\overline{K} ", "4b5e2d9ed4f3043968525efac375f8c5": "\\det(A) = \\exp(\\mathrm{tr}(L)). \\,", "4b5e469f64866b705f7ceebffafef79c": "U_o=A_o e^{i \\varphi}", "4b5e6536db49308e68dab855c13fbd2f": "(1+x)^{3}p(\\frac{1+2x}{1+x}) = x^3-2x^2-x+1", "4b5eb85bafef70e37ae2a4e2e6664916": "GHG\\ Savings=HL \\left( \\frac{FI}{AFUE \\times 1000\\frac{kg}{ton}}-\\frac{EI}{COP \\times 3600\\frac{sec}{hr}}\\right)", "4b5ebcb0bc17f1bcece86938496ca3a0": "a_{\\ell m}^{(M)}=\\frac{-ik^2}{\\sqrt{\\ell(\\ell+1)}} \\int d^3\\mathbf{x'} j_\\ell(kr') Y_{\\ell m}^*(\\theta', \\phi') \\left[\\mathbf{\\nabla}\\cdot(\\mathbf{x'}\\times\\mathbf{J}(\\mathbf{x'}))-k^2\\mathbf{x'}\\cdot\\mathbf{M}(\\mathbf{x'})\\right] + Y_{\\ell m}^*(\\theta', \\phi')\\mathbf{\\nabla}\\cdot\\mathbf{M}(\\mathbf{x'})\\frac{\\partial}{\\partial r'}(r' j_\\ell(kr'))", "4b5f0722e9ed12b78f40a6e670ece5f9": "U_\\alpha\\;", "4b5f4066b69d09db05048056d4ef919e": "=e/\\sqrt{\\alpha} ", "4b5f798cf76958e037126c62a7c8f73c": "\\forall s: M(s) \\geq W(s,t)", "4b604a65be959bba169ccfb2c18907b5": "W = \\int_{t_1}^{t_2}\\mathbf{T}\\cdot\\vec{\\omega}dt = \\int_{t_1}^{t_2}\\mathbf{T}\\cdot \\mathbf{S}\\frac{d\\phi}{dt}dt = \\int_C\\mathbf{T}\\cdot \\mathbf{S} d\\phi,", "4b608c08b1ec948a51e25d8fa6ba8f46": "3x+x=10-6", "4b60e62a3ef16ecee65eecefb2de0442": "(g \\circ f)(x) = g(f(x)).", "4b60ecc15e30922e080e6ea5e95d0532": "\\theta^0", "4b60faba59e7773364b391de7b828eae": "(\\Omega, 2^\\Omega, \\mathbb{P})", "4b612d3d7c75ffa115d6bf28c215c260": "R/t", "4b613786dc4b8b57c244e30560a194b6": " A \\mapsto e^{i A} ", "4b614aa25b324bfa3afb9c429639a0b9": "\\operatorname{erfc}(x) = 1-\\operatorname{erf}(x)", "4b61bac5579fca1e0aa08306f4e4c1ee": "p>3", "4b61ea9b8ab82bf458c101b3bd3cfe21": "E_n^{(-)}=a_n^\\dagger exp(i\\nu_nt)", "4b61ecdd2674ee85d11abe5cfa3ac05a": "\\nabla_0", "4b6213ab7655441d5fa033cca1377b3b": "\\Vert X \\Vert_\\pi = \n\\inf \\sum_{\\{i\\}} \\Vert e^*_i\\Vert \\Vert e_i \\Vert", "4b6214939ad41ca03336bd00791248a0": " \\rho\\left(x,t_2 \\right) ", "4b62498a147bdf93819fee4998ef4a2b": "(a,b)\\in U_{a,b}\\times V_{a,b}\\subset N", "4b626ab3ba03820856534db008680c24": "\\sqrt 3\\,\\varphi", "4b62d97ceb4b24b6fc97b355bc0e0b42": "v = \\left ( m + 1 \\right ) f", "4b62ea1a4321a90eab36745b134b4eb0": "op_1, op_2", "4b635d773b4d04922e326a9aad7af410": "\\left(\\frac{a}{N}\\right)=-1\\!", "4b637d129b7f2df70eb5fdacea4aa587": "\\scriptstyle x^2yxyx^5", "4b63a44c1b7e645b60b25076ea23c99f": "\\ln(4\\pi\\gamma) \\, ", "4b63af7e65622ffe68739f01b17771b4": "e_m : E[m] * E[m] \\rightarrow \\mu_m(\\mathbb{F}_q)", "4b63c9d6618000462f8a0706f7a21f23": "R_{\\alpha\\beta[\\gamma\\delta;\\varepsilon]} = \\, 0", "4b64031b5b180e20a384399490185cfb": "x = r", "4b647d944101221a7e25b3cd3f747c73": "\\Delta Y/Y \\approx k - c \\Delta u.", "4b64f915ae3d08e684f99eeda4a1f07d": " \\pi^{i}=\\mathbb{E}_{\\mathbb{P}^{*}}(S^{i}/(1+r))", "4b651a415143314ab33e062d248f1804": "k = 0.791 + 4.63\\times 10^{-4}\\,T - 8.44\\times 10^{-7}\\,T^2 ", "4b65503ae110afbcb624b095b0df9655": "M \\models \\phi(m)", "4b6552823dd3893ae5d360a13bc8aa4d": "x^2+2", "4b658f5946fa1b95f7ed9eafc1271c6b": "\n \\Pr\\Bigl(\\sup_{x\\in\\mathbb R} \\bigl(F_n(x) - F(x)\\bigr) > \\varepsilon \\Bigr) \\le e^{-2n\\varepsilon^2}\\qquad \\text{for every }\\varepsilon\\geq\\sqrt{\\tfrac{1}{2n}\\ln2},\n ", "4b65eb2a5984e98b6eaed525e9029e4c": "(x_0+1,y_0+1)", "4b665bcadbe886cbf7faf638e11c3887": "2\\alpha", "4b66a6138a5bca15160434767214a934": "|\\tau-\\theta|\\sim\\sigma", "4b66dcda078066ec795cfad0c4e24652": "\\; \\{ v_i \\}", "4b67114b3dbec1d6d32b19cf90478c00": " \\omega\\vert_U = i \\partial \\bar\\partial \\rho ", "4b6728a03919c2a8806accb65926d1e1": "|11 \\rangle ", "4b67a68f82202db7514eb3c0a189900c": " F = S_0 e^{(r-q)T} - \\sum_{i=1}^N D_i e^{(r-q)(T-t_i)} \\,", "4b67d9d3b640fe7ed4a44096113d94ce": "k\\,=\\,\\frac{2\\pi}{\\lambda},\\,", "4b6802dde104e0ad28e5a194ace27854": "H^{II}_q(H^I_p(C_{\\bull,\\bull})) \\Rightarrow_q H^{p+q}(T(C_{\\bull,\\bull}))", "4b684dc53e9a1f2c173adec474c9024e": "b_k={1\\over \\sqrt{2}}\\left({Q_k\\over l_k}+i{\\Pi_{-k}\\over \\hbar/l_k}\\right)\\quad,\\quad Q_k=l_k{1\\over \\sqrt{2}}({b_k}^\\dagger+b_{-k})", "4b6850caa740d436ca3e1b8132dedd6c": "Age = -8033 ln (Fm)", "4b68666596695f8821d30e66bc86bfed": "c^TX", "4b6882aacaa71b27f2e368e78fbce656": "Q_{next} = T \\oplus Q = T\\overline{Q} + \\overline{T}Q", "4b68a7275929668aade9420002d6fc65": "C_d", "4b68a7b5fc145c666a41ba080039dac1": "Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0.", "4b6928224614edc495f90e5d83073574": "\\frac{\\partial y}{\\partial \\mathbf{X}}", "4b6955622d6143de2cf909c24bb306fe": "\\alpha'(0):=v\\in T_vT_pM\\cong T_pM", "4b6958e8ef80ff0db0196a5bab01e222": " \\frac{d}{dt} U(t,s) = P(t) U(t,s), \\qquad U(s,s) = I,", "4b696cf7fa5a75ec4766fd6fc5279f4d": "NS^- + NI^+ \\rightleftharpoons MI^{(N-i)-} + (N-i)I^+", "4b69b45c94fb0b660f0ead84729b0f64": "E_{sig}(t,\\tau) = E(t)E_{gate}(t-\\tau)", "4b69e1ecd93c8d0dc70e07f97282cba0": "\\ x^P - N = 0", "4b6a22a4e5c4a89f94d6bbb477209aad": "\\boldsymbol\\alpha(t)", "4b6a264c78ac8dfe4df971271245c5f7": "\\Sigma = \\left \\{a, b, c\\right \\}", "4b6a6426e4888b9c7962399ea0907dff": "\\textstyle{\\binom{4}{2,2}=3}", "4b6a7315c0bde4440d2027d70e742543": "x = \\tanh(\\tfrac12 \\pi \\sinh t)\\,", "4b6a7a93159b46f79adf181f3bf019d3": "\nF_{12} = F_{21} = F_{13} = F_{31} = -\\cfrac{1}{2\\sqrt{X_{\\mathrm{f}} {X}_{\\mathrm{f}}^\\prime Y_{\\mathrm{f}}Y_{\\mathrm{f}}^\\prime}}\\ ,\\ F_{23} = F_{32} = -\\cfrac{1}{2Y_{\\mathrm{f}}Y_{\\mathrm{f}}^\\prime}\n", "4b6ab860f7a7c09402f51dbdede86b5c": "(\\lnot [\\forall x P(x)]) \\to \\exists x [\\lnot P(x)].", "4b6ad068698e2441c3aa964a57332362": "\\lambda(\\Gamma)^{\\prime\\prime}=\\rho(\\Gamma)^\\prime, \\,\\, \\rho(\\Gamma)^{\\prime\\prime}=\\lambda(\\Gamma)^\\prime.", "4b6b4782075cd7411e7eb24b5a1a05d3": "f:A\\to S^n", "4b6b532cdf75e118beef612a9cedaae8": " \\ x^2-Ny^2=k_1k_2", "4b6b5c82d211a12fbb14ba83c344ce54": "\\dot{\\alpha}", "4b6b601c3f1ccb283398c387947b1b23": " |a,b,c\\rangle \\mapsto \\begin{cases} i \\cos(\\theta) |a,b,c\\rangle + \\sin(\\theta) |a,b,1-c\\rangle & \\mbox{for }a=b=1 \\\\ |a,b,c\\rangle & \\mbox{otherwise.}\\end{cases}", "4b6ba4ae88d9cd356e64003a4333e9b4": "f_{\\ell}, f_i, f_c,", "4b6bb9b5a3a5aed24315361200a219e4": "\\scriptstyle p_2", "4b6bbc6bf021d723d47331e20815591c": "\\mathbb{F}_q^n\n", "4b6c27cb9cb43088c908c3679eeab586": "\n\\dot{x}=f(x,u,t) \\qquad x(0)=x_0 \\quad t \\in [0,T] \n", "4b6c29a09189f311112bbce88e5b3cf1": "\\sum_{k=1}^n \\frac{1}{2k-1} > \\sum_{k=1}^n \\frac{1}{2k} = \\frac{1}{2}H_{n}", "4b6cca01cc91c8eaaf0466ec170d8563": "i \\hbar \\gamma^\\mu \\partial_\\mu \\psi - m c \\psi = 0 ", "4b6cccff385757e2da937321b139d167": "\\frac{\\partial au}{\\partial \\mathbf{X}} =", "4b6cd41785c6bf6b2229aaf7dd95e13b": "\\mathrm{1.\\overline{09}}", "4b6cf0b5f3ee4c899b6170b6b0ee9fe3": " i \\epsilon \\,", "4b6cf7f5b010cfe9e8eeb6f7db962332": "\\beta = \\frac{\\sigma_x}{\\sigma_y} \\sqrt{1-\\rho^2}.", "4b6d48f6db584aef80674172a8e55da9": "P_{m-1} = \\sum_{i=1}^{m-1} a_i", "4b6d4a2e70e234ec6915e147ff98331b": "\n H_y=-\\frac{1}{j\\omega\\mu} k_y k_z \\cos k_x x \\sin k_y y \\cos k_z z\n ", "4b6d5283cf1afbfa8a40e0fb79739a25": "\\text{N}+\\text{1}", "4b6d9e84e7110e77a39b39ab533e43d7": "\\Omega_{-1,-1/2}\\propto\\binom 0 1", "4b6df9a5b488c1675d6a980359731377": " 1 - \\sqrt{R}", "4b6e35872afc354273bd1d6762fc203a": "T_c \\, = \\, 50.2 - 0.16 * MW + 1.41 * T_b", "4b6e6f343aa59ccc1f2c6d2a25499053": "X \\in L^2(\\Omega,\\mathcal{F}_t,\\mathbb{P};\\mathbb{R}^m)", "4b6eae70289968024474e33bd536f5d7": "t_1<0", "4b6ed430249765f426df841eb09185a7": "\n\\forall o\\,\\neg broken(o,S_{0})\n", "4b6f9d4282460116efdc8fb7ddb70de8": "w(i)=\\inf\\{k:v(i,k)=k\\}", "4b6face4f06daf73c6914d6fc803e466": " Q_{fb}(t) = Q_s(t-T) + Q_{fb}(t-T)\\, ", "4b6fcd20ebdf87b7219ec18136a5302d": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{J} + \\frac{1}{c^2}\\frac{\\partial \\mathbf{E}} {\\partial t}", "4b6ffaf60959d31032da80204b35ae33": " = 2\\left(r \\sin(t)\\right) r \\cdot \\cos(t)+4\\sin(t) \\cos(t) = 2\\left(r^2+2\\right) \\sin(t) \\cos(t).", "4b702959ed0adacde64115ca7a66b8e5": "a^2 \\in \\mathbb R", "4b7081e07dc145bfead6d7b770e60688": "D l^a=(\\varepsilon+\\bar{\\varepsilon})l^a-\\bar{\\kappa}m^a-\\kappa\\bar{m}^a\\,,", "4b70dd9ebb71cffde8e1143f30fa9b8b": "Z^{M}_{i,j}:", "4b70e10ebcc08840680cda2d66b667ba": "\\Pr(X=n) = \\tfrac{1}{2^n}", "4b717c518399a4e9533dba5e57c8df4d": "\\bigcup_i Y_i", "4b71aacde3897a1095062505ec0d48fc": "(C)\\int\\, f d\\nu + (C)\\int g\\, d\\nu \\ge (C)\\int (f + g)\\, d\\nu.", "4b71f88e65980b54cc7257884abb18ed": "1\\ {\\rm u} = \\frac{M_{\\rm u}}{N_{\\rm A}} = 1.660 \\, 538\\, 782(83)\\times 10^{-24}\\ {\\rm g}", "4b7241dcfb7bf9afe8a1c8b693bd7a30": " k > q", "4b724b351e801f29b59ec110745653ad": "M=n t,", "4b7279d27828f9593746dc7c90dd75a8": "k_1=\\tfrac{a-b}{b(b+1)}, k_2=\\tfrac{a+1}{(b+1)(b+2)}, k_3=\\tfrac{a-b-1}{(b+2)(b+3)}, k_4=\\tfrac{a+2}{(b+3)(b+4)}", "4b729038f6e90530f43a185ac41c58cc": " \\operatorname{de-lambda}[(\\lambda F.E) L] \\equiv \\operatorname{let-combine}[\\operatorname{let} F : \\operatorname{de-lambda}[F = L] \\operatorname{in} E] ", "4b72d81f7973a555e8908c400c4da8eb": "J \\approx a b^3 \\left ( \\frac{1}{3}-0.21 \\frac{b}{a} \\left ( 1- \\frac{b^4}{12a^4} \\right ) \\right )", "4b72e14cd391ed881c934f608d4030a9": "\\operatorname{Sp}(E)", "4b732212710279df3ed3ac742fdb61c3": " \\sigma_{\\mathbf{A},i} \\sigma_{\\mathbf{B},j}, \\qquad i=1,\\ldots,r_\\mathbf{A} ,\\, j=1,\\ldots,r_\\mathbf{B}. ", "4b734c82a55ee8f6ef5cdcf197b00ce5": "{B_m}^2 = \\frac{\\nu \\Sigma_f - \\Sigma_a}{D}", "4b735633ed575f690ff39b2a091fb2e8": "V=\\left(\\frac{h}{4}\\pi\\right)\\left(r_b^2+r_b^\\frac{4}{3}r_u^\\frac{2}{3}+r_b^\\frac{2}{3}r_u^\\frac{4}{3}+r_u^2\\right)", "4b73d8e24ddf575c6060b1589d004686": "\\phi_k(x)\\,", "4b740ee47ab6abaacba7b56c7912c226": "T^{\\alpha}S_{\\alpha}=T^{\\alpha}\\eta_{\\alpha\\beta}S^{\\beta} = T_{\\alpha}\\eta^{\\alpha\\beta}S_{\\beta} = \\text{invariant scalar}", "4b7425076a9da9534246338b03381c61": "||{\\bold u}||=1", "4b7469093a0fe4d61470dceaf259be16": "100,000 \\div 1,100,000 = 0.09091", "4b74a17f54908ad9a7c0dfff7526424e": " \\Delta_i ", "4b74f57f4128561d996c5838a2826d30": "L G(x,x')=\\nabla^2 G(x,x')=\\delta(x-x').", "4b752df61c4594173ce9adfd7b62097d": "P_{\\mathcal{M}}(x, \\theta) = \\frac{1}{N} \\sum_{m \\in \\mathcal{M}} K(x, T(m, \\theta))", "4b752f9019c36a1c4719e37296059e1f": "\\left[1,x,x^2,...\\right]^\\tau", "4b7537521fb5daf115063270b43f09b5": "x_2=a_2, x_5=a_5, x_{k-1}=a_{k-1}", "4b758517dcd5af3b249c1454e9df0704": "A \\in \\mathcal{E} \\implies \\mathbb{P} (A) \\in \\{ 0, 1 \\}", "4b7589249d053cb989ec7c0a43d92df8": "(CH_3NO_2)", "4b7589b5413e9a4eb2046928cce18668": "\\Phi_{M}", "4b75c679498e3bdff69d28e83e4d2a03": "J = \\{ J_{1}, J_{2}, \\dots, J_{n} \\}", "4b75d6d7cf7ba8d48434b691a9c98d97": "G(x) = \\left( F(x) \\right)^m = \\left( \\sum_{i=1}^n x^{-|s_i|} \\right)^m\n= \\sum_{i_1=1}^n \\sum_{i_2=1}^n \\cdots \\sum_{i_m=1}^n x^{-\\left(|s_{i_1}| + |s_{i_2}| + \\cdots + |s_{i_m}|\\right)} = \\sum_{i_1=1}^n \\sum_{i_2=1}^n \\cdots \\sum_{i_m=1}^n x^{-|s_{i_1} s_{i_2}\\cdots s_{i_m}|} \n= \\sum_{\\ell=m \\cdot \\min}^{m \\cdot \\max} q_\\ell \\, x^{-\\ell} \\; .\n", "4b75ddc6863abb086dbac071003dd8c2": "D_n(x) = (x - n) \\mod {26}.", "4b76436dd1d017126b8126c578679cd3": "\\sigma(\\beta) = \\lim_{n \\to \\infty} (|x_n|^{1/n}) \\, ", "4b7646306b67f272519f48a66f46fb96": " R = Q_t \\cdots Q_2Q_1A", "4b76d1ddd6d704aa6c7268717540a824": "\\sum_i dz_i^2 = d\\Omega_{n-2}^2", "4b77998a95bfde996b0074842c183174": "X_0(N)\\ ", "4b77c77b5e2f655bc8374639f956404a": "\\lambda(t) = \\frac{f(t)}{1-F(t)} ", "4b77d69ffda0a870ee53046cdc14f637": "\\forall j~\\sum_{i=1}^M \\mu_{ij} \\leq 1", "4b77f2296b887a1e8a68b4b20917e483": "\\textstyle {(2+2+0)!\\over 2!\\times 2!\\times 0!} \\ {(1+2+1)!\\over 1!\\times 2!\\times 1!} \\ {(0+2+2)!\\over 0!\\times 2!\\times 2!}", "4b780c625724e50985a41da862c9d41d": "S^{n-1} \\cong SO(n)/SO(n-1)", "4b784130ca5bb25f150532c60c64e0fe": "\n\\begin{align}\nw_0(n)\\ &\\stackrel{\\mathrm{def}}{=}\\ w(n+\\begin{matrix} \\frac{N-1}{2}\\end{matrix})\\\\\n&= 0.54 + 0.46\\; \\cos \\left ( \\frac{2\\pi n}{N-1} \\right)\n\\end{align}\n", "4b789939202cbb258ab0af2b225e094c": "\\textstyle\\beta\\rightarrow\\infty", "4b789bc856a52ef8eb1d050ee6e14b39": " \\mu_m = \\mathrm{P}(Y = m \\mid Y \\in \\{1,m\\} ) \\,", "4b78cce2d41adc6a4bafdc9d67daf3d0": "\\bar{L}", "4b794740e3de139d1ab94b92976ac3e0": "\\scriptstyle s \\,\\in\\, \\left\\{-\\frac{1}{2},\\; 0,\\; \\frac{1}{2}\\right\\}", "4b79842e2481b50fca20656b00519dd6": " H_{2N} = \\begin{bmatrix} H_{N} \\otimes [1, 1] \\\\ I_{N} \\otimes [1, -1] \\end{bmatrix}", "4b7994e462f04157b7a0979abe2faa48": "\\exists x (x \\in V \\land G(x,y_1\\dots, y_n))", "4b79a5da1439f86c504ed0cb6acbe4d1": "\\Omega(\\alpha^{-i_k})=-\\lambda_0e_k\\alpha^{c\\cdot i_k}\\prod_{\\ell\\in\\{1,\\dots,v\\}\\setminus\\{k\\}}(\\alpha^{i_\\ell}\\alpha^{-i_k}-1).\n", "4b79b623ffd796034e783aa7d2396eee": " \\rightarrow ", "4b79e64aed96bdf71dbf44b405e58f85": " L^s_P(\\Omega; M) \\rightarrow L^s_P(\\Omega; N), \\text{ i.e. } \\operatorname{E}|\\operatorname{E}(X\\mid N)|^s \\le \\operatorname{E}|X|^s", "4b7a4c54b57c9297141ab648398f6dfc": "\\mu_B", "4b7a50c43b81758c7d5ab2cfb70a762a": "|E(H)|", "4b7a66043cdbab16fc7696482006137a": "b_i(\\mathbf{x})", "4b7a7d55a905a55944291847ebf666e9": "\\scriptstyle \\mathcal K \\;=\\; T \\frac{dS}{dT}", "4b7a93ea1de079b4da560903197624ea": "\\Delta x_i \\triangleq x_i - a", "4b7aa5966b7fc4e4c783990057d8f9ea": "w_1 a = w_2", "4b7ae512d615708777d2a846313f5b80": "U_D(d)", "4b7b024e40777f5146874349478f0969": "\n\\ R_0 \\ = 1.\n", "4b7b4da6d7f58075bc635fdaa0683ed4": "\\left( \\mathbb{I}_A \\right)_i = \\begin{cases} 1 & \\mbox{ if } x_i \\in A \\\\ 0 & \\mbox{ otherwise} \\\\ \\end{cases} ", "4b7b9d5c2e46c80f2400e62da02123b9": "\\mathbf{F}=\\mathbf{\\sigma A}\\,", "4b7b9f2bbdb8bd2d1d4799a1d208898c": "\\textstyle n = \\mathrm{LCM}(2l-1,p)", "4b7bb46f35e4dbcd1aa74537b427fc3b": "\\varepsilon^{-1}", "4b7be9db99b1ba01144f7bfa7b2cb156": "\\inf_\\alpha f_\\alpha", "4b7bf4c284bcb6913383d497c39f8890": "\\,\\overline{A}_x", "4b7bf93ac38b5ca8826a76b5a52a670c": " \nDet\\left(\\Gamma_{jk}-G\\delta_{jk}\\right)=0\n", "4b7c10e8f1a1d70821b7c1ee22732fb9": "\\mathbf{J}_{23}", "4b7c4985ad355425254c6afd6f6d2021": "\\sum_{m=0}^n (-1)^m {\\left \\langle {n\\atop m} \\right \\rangle} = 2^{n+1}(2^{n+1}-1) \\frac{B_{n+1}}{n+1},", "4b7c51a12dc277e24950016515a858fa": "D^{(\\alpha)}=\\frac{4}{1-\\alpha^2}(1-\\sum p^{\\frac{1-\\alpha}{2}} p^{\\frac{1+\\alpha}{2}})", "4b7c6ea38ba736e3f0b44962fd9a0f79": " \\text{Pr }[\\mathcal{A}^{f_{a}(x)}(p,g) \\to 1] - \\text{Pr }[\\mathcal{A}^{R} (p,g)\\to 1] ", "4b7d523e0d4c47fdc08df164f95e272f": "\\beta_q", "4b7d8cd54724e9f00707a8dd79bdc335": "\\partial B(x,r)", "4b7da8d92aaf64e0127d898f42b34764": "\\{a(f),a(g)\\}=\\{a^\\dagger(f),a^\\dagger(g)\\}=0 ", "4b7db3e246f0d38b52e149a1bf03d4cb": "=: \\!\\,", "4b7ddde109a1c0f07b98d96f6a85d4b3": "\n\\ell^{\\prime\\prime}(\\beta) = -\\sum_{i:C_i=1} \\left(\\frac{\\sum_{j:Y_j\\ge Y_i}\\theta_jX_jX_j^\\prime}{\\sum_{j:Y_j\\ge Y_i}\\theta_j} - \\frac{\\sum_{j:Y_j\\ge Y_i}\\theta_jX_j\\times \\sum_{j:Y_j\\ge Y_i}\\theta_jX_j^\\prime}{[\\sum_{j:Y_j\\ge Y_i}\\theta_j]^2}\\right).\n", "4b7df62fc9fb99b8139a67430b0aa2c5": "Cq^{k^n}", "4b7e1d56d4c0b63868355bf0c92ecc84": "S=\\{|z|=1\\}", "4b7e381081d183153374b285eda9637f": " P_m1=1-P_d(1-\\theta)-P_d'(\\theta)-P_a(1-\\theta) ", "4b7e6eb351eb209bb5f69b6b0e1492f0": "\\lambda_\\mathrm{D}", "4b7ed0e2088262a95964184c4f0f2982": "\\ S = P_t \\frac{1}{4 \\pi d^2} ", "4b7f1852268c0445e117fd145c6b210e": "\\begin{align}\n-\\frac{1}{n} \\mathbf{K} (\\mathbf{Y} - \\mathbf{K} \\mathbf{c}) + \\lambda \\mathbf{K} \\mathbf{c} & = 0, \\\\\n(\\mathbf{K} + \\lambda n \\mathbf{I}) \\mathbf{c} & = \\mathbf{Y}, \\\\\n\\mathbf{c} & = (\\mathbf{K} + \\lambda n \\mathbf{I})^{-1} \\mathbf{Y}.\n\\end{align}", "4b7f4607b3d5aa86914bb38173bf6998": "F\\left(u,x_1,x_2,\\dots,x_n,\\frac{\\partial u}{\\partial x_1},\\dots,\\frac{\\partial u}{\\partial x_n}\\right)=0", "4b7fade2bac9ee15ce92ace2aa1ecb75": "1+r= \\frac{M_2}{M_1}", "4b7fd64c89f36746cdf5c1efea08d184": "I-1\\,", "4b7fda0e5ed3e5840001a5d38ac687e9": "k=k_i", "4b802362fb7322976111adb19de9c872": "\\lambda_\\star", "4b80303cc34155c761ee19d09bc511b2": "\\beta_F", "4b808a7445d720103e794f9b0fc215f8": "u|_{\\partial\\Omega}=g.", "4b80a76d4c48a2bfb1780c27b48654b7": "V_x \\times p_{tot} = V_{tot} \\times p_x", "4b80f7902842552a628edef6d58c9e48": " -\\frac{\\hbar^2}{2 m} \\frac{d^2 R(r)}{dr^2} + [E-V_\\textrm{eff}(r)] R(r) = 0 ", "4b81022a8a3d17636670da0c3009a7b2": "\\mu = \\mu_1 + \\cdots + \\mu_s", "4b816256f8600fe7b4fae696d9d60bfa": "\\frac{j+1}{2^n}", "4b817888555b0dee05cc1bd2e8e352a3": " \\Psi(\\mathbf{r},t) = \\psi(\\mathbf{r}) e^{-iEt/\\hbar} ", "4b81b194145c9c69514754c8543748eb": " T_{\\mathrm{min}} \\leq T_{\\mathrm{max}}, ", "4b81d20a0266623ad56193114ad1ee5f": "(y_1, y_2, \\ldots, y_m) \\leftrightarrow \\begin{bmatrix} f_1(x_1, x_2, \\ldots, x_n) \\\\ f_2(x_1, x_2, \\cdots x_n) \\\\ \\vdots \\\\ f_m(x_1, x_2, \\ldots, x_n) \\end{bmatrix} \\leftrightarrow \\begin{bmatrix} f_1(x_1, x_2, \\ldots, x_n) & f_2(x_1, x_2, \\ldots, x_n) & \\cdots & f_m(x_1, x_2, \\ldots, x_n) \\end{bmatrix} ", "4b820219e58ecfa6dde2863e1d1003a4": " (\\Omega,\\mathcal{F}) ", "4b821c5cbc995caaf12557856ca503c2": "p(x)=a_kx^k+a_{k-1}x^{k-1}+\\cdots+a_1x+a_0", "4b8232f4318fc5d2a5a55b920abde441": "\\sum_{n=0}^{\\infty}\\frac{1}{(n+2)n!}=\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{8}+\\frac{1}{30}+\\frac{1}{144}+\\cdots=1.", "4b82499af6d0720d1089e9e9670e9a38": "GS_{Q_{\\varphi}}\\leq 1/n\\,\\!", "4b826b18a0151ca8981e40cbd5a88225": "\\ +\\alpha,", "4b827277625dedded7d8c51792bdcefd": "\\{n_t\\}\\,", "4b827b2097c7a3938b9485483757bb9b": "\\displaystyle{\\lambda_n=(1+n^2)^{k-1/2}\\sum_m (1+m^2+n^2)^{-k} < C_k.}", "4b827de818d22b06e2d1b4d0337523c0": "q=[q_1,...,q_n]^T \\in \\mathcal{R}^k", "4b82b329ab16a0436eddf57d7f9afd78": " {d^2 x^\\lambda \\over dt^2} =- {d x^\\nu \\over dt} {d x^\\alpha \\over dt} \\left[{\\partial^2 X^\\mu \\over \\partial x^\\nu\\partial x^\\alpha} {\\partial x^\\lambda \\over \\partial X^\\mu}\\right]+{d x^\\nu \\over dt} {d x^\\alpha \\over dt}{d x^\\lambda \\over dt} \\left[{\\partial^2 X^\\mu \\over \\partial x^\\nu\\partial x^\\alpha} {\\partial x^0 \\over \\partial X^\\mu}\\right]", "4b8326afdd972820acad8b09d9f2362e": "\\varphi=\\pm \\pi", "4b834fd4baac57743c820023298b5274": "\\begin{matrix}\n vu_n\\xrightarrow[n\\to\\infty]{} vu \\\\\n v_nu\\xrightarrow[n\\to\\infty]{} vu\n \\end{matrix}\\quad\\Longleftrightarrow\n\\quad vu\\in BV(\\Omega)", "4b837ef237685ba1027fe21a39deab17": "Y(t) = Y(0) - bt", "4b838464072a4b2196618d472baba68a": "n_1 \\sin \\left( \\theta_1 \\right) =n_2 \\sin \\left( \\theta_2 \\right),", "4b83aa6239b80cea0f9b8e948632d688": "f^{(n)}(a) = {n! \\over 2\\pi i} \\oint_\\gamma {f(z) \\over (z-a)^{n+1}}\\, \\mathrm{d}z", "4b845deb6d1d325742a5ac3a1824bd3b": "(\\{\\infty\\}\\times K), \\ \\infty \\notin K ", "4b84619e02a41d6c66a8c0c6ca45b1bb": "\\textstyle w = 0", "4b847fce631d47758601bd21002ca562": "c_1 = c_2 = \\cdots = c_r = 0", "4b8493b54deaa8bd5f957309cddbb6d6": "\n\\mathcal L= \\frac 1 2 \\part_\\mu \\varphi\\part^\\mu \\varphi - \\frac 1 2 m_0^2 \\varphi^2 +\\mathcal L_{\\mathrm{int}}\n", "4b84aee7a6fc38713b5ff6ebeb3554eb": "y_t = (1 - \\alpha) x_t + \\alpha ((1-\\alpha) x_{t-T} + \\alpha y_{t - 2T})", "4b84b59f12ec109b8bd8fa9e97dbf81a": "x=z/H", "4b84c0618dcb0f1d3d83411ef09b5c7e": "\\sum_{x=0}^{N-1} Q_n(x)Q_m(x)\\rho(x)=\\frac{1}{\\pi_n}\\delta_{m,n},", "4b84d87d6de87064118aa26cc72c61bb": "\\mathbf{v}^1", "4b84dd976c2b4171bf7395d1d50a1d0d": "k_F = \\frac{\\sqrt{2 m E_F}} {\\hbar}", "4b8501156aa2328eaa5cf03eff6c56db": "x_{pi}x_{zi}=\\xi\\,", "4b850ea6b127ae9aa16f8ea520e5838c": " H=\\frac{1}{2}\\sum_{k=-\\infty}^{\\infty}\\left[\\pi_k \\pi_k^\\dagger + \\omega_k^2\\phi_k\\phi_k^\\dagger\\right],", "4b8528fe2cb1850b88555b2450fec147": "\\{(x,y) \\mid x \\in y\\}", "4b8552b18ce79321480ae1d63608e6bd": " P(x) = \\frac{\\prod_{k=1}^n (1-x^k)}{\\prod_{(i,j)\\in \\lambda} (1-x^{h_{(i,j)}})}. ", "4b855aa68d9c71d7d1baac18142907d2": "\\,k = x,y", "4b857171f54b5c82d773afe9c2738d22": "H=-p\\log_2 p - (1-p)\\log_2(1-p)", "4b858aa1c278ac30defde602a2712ef4": "f(x_0, y_0)", "4b864f2b7ef9e34feef66f45020e2019": "\\langle Tx,y\\rangle = \\langle x,Uy\\rangle", "4b8659cf0aaa327281c38bbf4db96946": " R_0 = \\frac{R_1 \\, R_2 \\, |\\sin (\\Phi_2-\\Phi_1)|}{\\sqrt{R_1^2 - 2 \\, R_1 \\, R_2 \\, \\cos(\\Phi_2-\\Phi_1) + R_2^2}}", "4b8660b77653e08330dfbdc81da27d2f": "110.5\\pm 1", "4b867263b1f10ba9a8ed85a0894749b5": "\n\\mathbf{F} = \\frac {3 \\mu_0} {4 \\pi r^4} \\left[ (\\hat{\\mathbf{r}} \\times \\mathbf{m}_1) \\times \\mathbf{m}_2 + (\\hat{\\mathbf{r}} \\times \\mathbf{m}_2) \\times \\mathbf{m}_1 - 2 \\hat{\\mathbf{r}}(\\mathbf{m}_1 \\cdot \\mathbf{m}_2) + 5 \\hat{\\mathbf{r}} (\\hat{\\mathbf{r}} \\times \\mathbf{m}_1) \\cdot (\\hat{\\mathbf{r}} \\times \\mathbf{m}_2) \\right].\n", "4b86a415cbe77ab5b6c3104b8c0a47a9": "\n\\begin{align}\n\\int \\sec \\theta \\, d\\theta& = \\mbox{gd}^{-1}(\\theta)=\\mbox{lam}(\\theta).\n\\end{align}\n", "4b86b37d1045b8b97b500ee12079e091": "\\frac{10^4}{\\pi}", "4b86c79188ad2f660803f9cb0ea320a4": "DM/t", "4b870f8b2216a9f0c5ec693ec2ab9dc7": "\nJ = \\{x \\in M \\mid \\nabla{f(x)} + \\lambda \\nabla{g(x)} = 0 \\mbox{ or } \\lambda \\nabla{f(x)} + \\nabla{g(x)} = 0\\}.\n", "4b8740205663002a11d309923410d134": " y =\\sin(2\\pi nt)+ \\sin(2\\pi t)", "4b87519c4a544eb079ba557e9c068281": "m\\frac{d^2 y}{d t^2}+\\gamma\\frac{d y}{d t}+ky=0", "4b87a4f796b545824418afb7b307c032": "R\\left( t,s \\right)=R\\left( u+\\frac{\\tau }{2},u-\\frac{\\tau }{2} \\right)=C\\left( u,\\tau \\right)", "4b88161d180b2d6365e0a54b24783760": "\n S_k(A_n) \n = \\frac{\n \\begin{vmatrix}\n A_{n-k} & \\cdots & A_{n-1} & A_n \\\\\n \\Delta A_{n-k} & \\cdots & \\Delta A_{n-1} & \\Delta A_{n} \\\\\n \\Delta A_{n-k+1} & \\cdots & \\Delta A_{n} & \\Delta A_{n+1} \\\\\n \\vdots & & \\vdots & \\vdots \\\\\n \\Delta A_{n-1} & \\cdots & \\Delta A_{n+k-2} & \\Delta A_{n+k-1} \\\\\n \\end{vmatrix}\n }{\n \\begin{vmatrix}\n 1 & \\cdots & 1 & 1 \\\\\n \\Delta A_{n-k} & \\cdots & \\Delta A_{n-1} & \\Delta A_{n} \\\\\n \\Delta A_{n-k+1} & \\cdots & \\Delta A_{n} & \\Delta A_{n+1} \\\\\n \\vdots & & \\vdots & \\vdots \\\\\n \\Delta A_{n-1} & \\cdots & \\Delta A_{n+k-2} & \\Delta A_{n+k-1} \\\\\n \\end{vmatrix}\n },\n", "4b884d0a814d03395a17dcc976a319e5": " \\ \\textbf{b} = \\textbf{f} \\cdot \\textbf{m}\\pmod 3 ", "4b884dc0d634fdd562c3bae0f7673968": " \n2^{-n(H(X)+\\varepsilon)} \\leqslant p(x_1, x_2, \\dots , x_n) \\leqslant 2^{-n(H(X)-\\varepsilon)}\n", "4b88c4e98239c387ce32cd8e1a85e5c8": " \\bold{p}_1^\\prime = -\\bold{p}_2^\\prime = \\mu \\Delta\\bold{u} \\,\\!", "4b88f47f80273fd5788e1e20aa81c38a": "E\\,", "4b89a937c64b90c71969655d4da2883c": " |R\\rangle ", "4b89b5547547f8a46702cb6610bfb182": "EER=(88.5-(61.9*Age))+PA*((26.7*wt)+(903*ht))", "4b89e4c42fef51affcbe787f8aed3ccb": "FX = X\\;", "4b8a1981ac69174e2265550c3c78b16b": "F=\\prod_{i=1}^mF_i", "4b8a1d1529428a425f4e17858f7ad49a": "\\phi(I) = I", "4b8a7f84b6e22154b9f519e806771455": "\\arccsc x = \\frac{\\pi}{2} - \\arcsec x ", "4b8aaa1e4e39cab0991ef678f78796fe": "\\Pr\\left(\\limsup_{n\\to\\infty} E_n\\right) = 0.\\,", "4b8ac0976db0774f7b506c6b1fdce3a2": "PA' = \\left(\\frac{1}{2} \\times \\frac{M_1}{E_1 I_1} \\times L_1\\right)\\times L_1\\times \\frac{1}{3} + \\left(\\frac{1}{2} \\times \\frac{M_2}{E_2 I_2} \\times L_1\\right)\\times L_1\\times\\frac{2}{3}+ \\frac{A_1 X_1}{E_1 I_1}", "4b8b09859900081aae04b80de28c109d": "\\Lambda(\\beta)", "4b8b7c47821c48255276323e82265882": "e'^2 = e^2/(1-e^2)", "4b8c25c3b39d085a196452319556e9a1": "G_\\mathrm{net} = \\sum_i \\sum_{i=1}^{N} \\,\\!", "4b8ca7d68d0be1e60acb3ed9a99842c4": "\\hat{y} = \\arg\\max_{k \\in 1 \\ldots K} f_k(x)", "4b8cb9ee12f0403180ce43364bbc83f0": "\\bigstar \\bigstar \\bigstar | \\bigstar \\bigstar | | \\bigstar \\bigstar \\bigstar \\bigstar \\bigstar", "4b8cbd14c23149df272140199e21f36b": "r_a\\,\\!", "4b8d510f300ebef43ac6d66178112f8f": "\n \\Gamma_{ij,k}^{(D)} = -D[\\partial_i\\partial_j\\parallel\\partial_k],\n ", "4b8d7bf3c20ba241b8b56f1f345e33cd": "-\\beta\\,", "4b8d8223d4f08b1376e766e328dda60a": "a_s=2r_u\\sqrt{\\frac{1-\\cos(\\gamma_1/s)}{2}}.", "4b8da6e0d7c73258189b64655f5cb449": "(\\mathcal{H}, \\langle, \\; \\rangle)", "4b8e24c4023ab3955f05e8e97f7b1cd5": "N_{i,p}(u)w_{i}\\over{\\sum_{j=0}^{j=n} N_{j,p}(u)w_{j}}", "4b8e3193d76f4aa6a7e608fd2fc18e94": "U = [0,\\;\\;1]", "4b8e36484db7fc335867be9cc8002c1f": "\nQ_s=\\sum^n_{r=1}g_{sr}\\,F^r,\\qquad s=1,\\,\\ldots,\\,n", "4b8e4f6932f7903213c5311401cb62e2": " \\hat{n}_{\\mathbf{k}} |\\{n_{\\mathbf{k}}\\}\\rangle = \\bigg( \\sum n_{\\mathbf{k}_l} \\bigg) |\\{n_{\\mathbf{k}}\\}\\rangle ", "4b8ea3725004eccc14a96e5a3c7bb255": "\\cos^2 \\theta", "4b8eb57f64a78daabb03b391b5bf4572": "\\forall x \\left( \\phi \\to \\psi \\right) \\to \\left( \\forall x \\left( \\phi \\right) \\to \\forall x \\left( \\psi \\right) \\right)", "4b8ee797e9fe50c45c35b159b752c16c": "\\textstyle\\left(\\sum_{x}\\vec{e}_{ix}\\right)^2 = \\sum_{x}\\sum_{y}\\vec{e}_{ix}\\vec{e}_{iy}", "4b8f03e859317af7e783ad715c85ad68": "\nP(V)=\\frac{3B_0}{2}\n\\left[\\left(\\frac{V_0}{V}\\right)^\\frac{7}{3} - \n\\left(\\frac{V_0}{V}\\right)^\\frac{5}{3}\\right]\n\\left\\{1+\\frac{3}{4}\\left(B_0^\\prime-4\\right)\n\\left[\\left(\\frac{V_0}{V}\\right)^\\frac{2}{3} - 1\\right]\\right\\}.\n", "4b8f13ec6a334504ed25b0a15e9095cc": "1\\ \\mathrm{savart} = \\frac{1}{\\log_{10}{2}}\\ \\mathrm{millioctave} \\approx 3.3219\\ \\mathrm{millioctave}", "4b8f4cd0a1c668c0464f5a15e99a4d21": "\\mathrm{Tr}^{U\\otimes V}_{X,Y}(f)=\\mathrm{Tr}^U_{X,Y}(\\mathrm{Tr}^V_{X\\otimes U,Y\\otimes U}(f))", "4b8f872ec93b3473b3f7f99c9d2ab733": "\\ e^{t f}", "4b8fb0d25e59ac79ad84fea26c46353b": " T_{\\alpha \\beta} {}^\\lambda = T_{\\alpha \\beta \\gamma} \\, g^{\\gamma \\lambda} ", "4b9050b41bf3ef159342fcd8dd97a55d": "R(r) = N r^{n-1} e^{-\\zeta r}\\,", "4b909fad1869e61887c2d7524d65e0b0": "p \\vert N ", "4b90b32964a0973507fce79cfd35f18e": "P^{m}\\,\\!", "4b90e4061faea1395a7fbc3de4aa8522": "H_{\\mathrm{dR}}^{k}(T^n) \\simeq \\mathbf{R}^{n \\choose k}.", "4b9175220f5013b732c23e14893b6c3b": "a_{1}+c_{1}", "4b91e8cd816b0e259f7f034cfb1182e8": "h_r= \\epsilon \\sigma (T_{surf}^{2}+T_{surr}^{2})(T_{surf}+T_{surr})", "4b923dbba494d691b717291f208b70b8": "\\pm 2q", "4b9252b37ad69c23e66bf7a8037ebbaa": "d^2V_{prop}", "4b9272a5a6155a270dae56dc7d4df5ac": "c_e \\neq 0", "4b92abff764db809972c661b33fa5cb1": "(\\cos \\varphi, \\sin \\varphi) = \\frac{(x', y')}{(x'^2+y'^2)^{1/2}}", "4b92b57879b189299ac07c8da8eeca1e": "\\bar V_S=D\\, 0 + (1-D)(V_i-V_o)=(1-D)(V_i-V_o)", "4b92c2cf07f88eed9273e97f11706c1a": "\\displaystyle{\\|g\\|_1 \\le \\|f\\|_1.}", "4b92e2967535b1b84f8671a73a3bad64": "\\operatorname{dCov}", "4b934d7dc5d465ce35f3c7bb7bfa0b62": "H = (V_H, E_H)", "4b934f721b2f0638d0362e04fc4ec574": "(\\rho \\times \\sigma)(g,h) = \\rho(g)\\cdot \\sigma(h).", "4b93604927a6e12641351a33b82290cc": "B(\\mathbf{u})B(\\mathbf{v})=B(\\mathbf{u}\\oplus\\mathbf{v})\\mathrm{Gyr}[\\mathbf{u},\\mathbf{v}]=\\mathrm{Gyr}[\\mathbf{u},\\mathbf{v}]B(\\mathbf{v}\\oplus\\mathbf{u})", "4b9374f0e8d59772f2b3f49d670c1212": "\\mathcal{M}_{g}", "4b937ef6c55c99a87bbe824132eed947": "n'/n \\to 0", "4b93e77b4a51dd28e4ec34855fb89c78": "S^*(f;P)-S_*(f;P)\\leq(b-a)\\omega(|P|).", "4b93ffd29081e89fa3f2a1332225c586": "A_L^{(-1)}:=P_L(A P_L + P_{L^\\perp})^{-1},", "4b94099d6e6ca56194f160ba283e4ae2": "\\phi \\in \\Phi", "4b945e3ff8b125257344c874527b6077": "f(\\Theta) = X", "4b948e26387b8e70351e4bd646e2cfe2": "\nL_i = \\epsilon_{ijk} X^j P^k\n\\, ,", "4b94a8ee905caaf2c5cf837a18b1ec67": "a=\\tfrac{q^{2}(u^{2}+v^{2})^{2}}{4} \\,", "4b94b2de31ce7b4f447c733541398ec8": " \\lambda_1, \\lambda_2, \\dots ", "4b94d05f08d9dbd376e8e7802f88fdbc": "\\gamma_K > 0\\,", "4b953ff53c9fd28005d357eb831cd458": "\\omega_{ce}=\\frac{qB}{\\gamma\\cdot m_0}", "4b954bcb6154ca71b61ebf6ecd2ee0ba": "X = \\{x_1,x_2,...,x_r\\}", "4b955c3e9514ef5f65c59edbca1e810a": "\\mathbf{\\ddot{r}}_i", "4b956f93bbb55ee2532aecec51d10499": "\\textstyle K_{pub}", "4b95cac98811a1e140a8da320b44c73b": "\\frac{p(t)}{P} = 1 - \\frac{(1+i)^t-1}{(1+i)^n-1}", "4b95ed36cceef56c78ce7003f4a0c3f5": "\\frac{d^ny}{dx^n},\n\\quad\\frac{d^n f}{dx^n}(x),\n\\;\\;\\mathrm{or}\\;\\;\n\\frac{d^n}{dx^n}f(x)", "4b96566f1b91be3fe8118c13d236e56a": "\\frac{\\partial S}{\\partial \\beta_j}=2\\sum_i r_i\\frac{\\partial r_i}{\\partial \\beta_j}=0,\\ j=1,\\ldots,m", "4b9677b889a6cefedd5a7f636bf471da": "\\sin\\theta_n\\,", "4b96be83951b26fc93d156a0a60cec01": " |\\psi_\\varepsilon\\rangle ", "4b971f7da88a479709f0ee60ff7ba5b5": "d_{3/2,1/2}^{3/2} = -\\sqrt{3} \\frac{1+\\cos \\theta}{2} \\sin \\frac{\\theta}{2}", "4b972c78164d316fc3044c93bf514888": " \\hat{S}_d \\exp \\left ( i \\alpha_x -i s \\theta \\right ) |s\\rangle \\rightarrow i { \\partial \\over \\partial \\theta} \\exp \\left ( i \\alpha_x -i s \\theta \\right ) |s\\rangle = s \\left [ \\exp \\left ( i \\alpha_x -i s \\theta \\right ) |s\\rangle \\right ]. ", "4b974153d2116a053cfab3b9af5e60bd": " \\int_{-\\infty}^\\infty ({\\rm Im}\\, z)\\cdot|\\lambda-z|^{-2}\\, d\\sigma_{ij}(\\lambda) = {\\rm Im} M_{ij}(z),", "4b97c29595ff282df489ac73fe1730c0": "f(x), g(x)", "4b97dd3f1e4cb50c0145708975b0494c": " \\sum_{n=1}^\\infty \\frac{1}{10^{n!}} = \\frac {1}{10^{1!}} + \\frac{1}{10^{2!}} + \\frac{1}{10^{3!}} + \\frac{1}{10^{4!}} + \\cdots", "4b9831a114ff55b8673680d2fc4bb5b6": "\\widehat{g}: \\widehat{Y} \\to \\widehat{Z}", "4b98bfeab26263d7d9b9f68740a7db3c": "m_a=\\frac{1}{2} \\sqrt{2b^{2}+2c^{2}-a^{2}}= \\sqrt{\\frac{1}{2}(a^{2}+b^{2}+c^{2})- \\frac{3}{4}a^{2}}", "4b999243fead14d617e217c56f5e9689": "\\sigma(n) < e^\\gamma n \\log \\log n", "4b999fdb37ee6834326614c87c41ce5d": "k \\nabla^2 u=q", "4b99ce48a8aa97a3cde5f247815ce942": "Gen(w)", "4b9a4ad553d08e08cc4e424742b5a1d2": "R_{d}", "4b9a52e281c9b8cd871e4b124fec497c": "q \\in \\{0, 1\\}", "4b9a6a93a41bd0dcc31e87f44e48cfaa": "\\frac{w_1 x_1 + w_2 x_2 + \\cdots + w_n x_n}{w} \\ge \\sqrt[w]{x_1^{w_1} x_2^{w_2} \\cdots x_n^{w_n}}", "4b9a8aa13cfe3e8b7b2f2557b9d56d3e": "f(t) = \\frac{\\tau}{T} + \\sum_{n=1}^{\\infty} \\frac{2}{n\\pi} \\sin\\left(\\frac{\\pi n\\tau}{T}\\right) \\cos\\left(\\frac{2\\pi n}{T} t\\right)", "4b9ac00a669a5bba0e9d5f9484dc366b": "\\frac{\\partial \\phi}{\\partial t} +\\,\\frac{\\partial}{\\partial x}\\,J = 0\\Rightarrow\\frac{\\partial \\phi}{\\partial t} -\\frac{\\partial}{\\partial x}\\bigg(\\,D\\,\\frac{\\partial}{\\partial x}\\phi\\,\\bigg)\\,=0\\!", "4b9ae74e7b9c05f9f4cce76efdc66cf8": "a^2+b^2=h^2", "4b9b05e9ac4af933180df8ee846941d2": "\\overline{10}_{-1}", "4b9b05f3d96af4b9eb077c3231b66d5c": " \\frac{\\mathrm{d} p^1}{\\mathrm{d} \\tau} = q \\left[U_0 \\left(\\frac{-E_x}{c} \\right) + U_2 (B_z) + U_3 (-B_y) \\right]. \\,", "4b9b3a578534be7d0090e8c07c06cae2": "|\\psi(0)\\rangle=c_1(0)|1\\rangle+c_2(0)|2\\rangle+c_3(0)|3\\rangle,", "4b9b6b5b90e00bef7d794141aad0dfe7": "\\rho = \\rho (z)", "4b9b7199583064105a272450fbcda198": "\\rho^{com}_{T-1} := \\rho_{T-1}", "4b9b7b72930398d1693af0913afae2de": "u_{j}^{(t+1)} = u_j^{(t)} \\sum_{i} \\frac{d_{i}}{c_{i}}p_{ij}", "4b9bca0a86aa323765b1ca6e05d18856": "B,C", "4b9be30509fac19fa3769e69965aea21": "(B_0 k)^{-1}", "4b9c04e1c7021f8b1e3e8940f83b139f": "{\\Delta h}", "4b9c0e2048141d616ccd3a92608d3c95": "R = e^{\\Theta / a},\\,", "4b9c82a63967b8f74895fec1fa408296": "x_i = \\frac{c_i}{c} = \\frac{c_i}{\\sum c_i}", "4b9c9e91b10c227f06506a04edf63a75": "S_0 = O_X", "4b9cf8f9fa9a55671d8be4619051c1e2": " E_{nix_{nj}^m} = -\\frac{x_{nj}^m} {P_{ni}} \\int \\beta^m L_{ni}(\\beta) L_{nj}(\\beta) f(\\beta) d \\beta = - x_{nj}^m \\int \\beta^m L_{nj} (\\beta) \\frac{L_{ni} (\\beta)} {P_{ni}} f(\\beta) d \\beta ", "4b9cfe61a34f72c7f26e195c37b81610": " T_9(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x. \\,", "4b9d3f3983fcc9026cb26fbd880b5534": "\\mathcal{P}e^{i \\oint_C A_\\mu dx^\\mu} \\to g(x) \\mathcal{P}e^{i \\oint_C A_\\mu dx^\\mu} g^{-1}(x)\\,", "4b9d624924ca1cb162baa6d0cce3146d": " T_2(z)=22-Kz +22z^2 ", "4b9d9bd852e324b56814477ae5c09d18": "\\sum_{n=1}^\\infty\\frac{n}{e^n}", "4b9dd531ad4fe11db75c6c588ec66c24": "a\\,\\rho\\,b,\\quad c\\,\\rho\\,d\\Longrightarrow ac\\,\\rho\\,bd.", "4b9e0c7693cb510ce67489c3d4dee7b3": "x \\,\\!", "4b9ea7a2213b54c7a824f891f6a6752e": "(X, \\tau)", "4b9ed63aea6e541aa26643f8b65459f6": "\\alpha<\\lambda", "4b9ef13f3b8ed73ea4dfc40147287e19": "c_1(x) = \\frac{\\sinh {\\sqrt {-x}}}{\\sqrt {-x}},\\text{ for }x < 0", "4b9ef88a2dfbdcc8bd7c6b53fe3db775": "c_2\\not=0", "4b9f097c2cff9076c0c1ce13da606a73": "\\sin(\\theta) \\approx \\theta \\qquad \\mathrm{and} \\qquad \\tan(\\theta) \\approx \\theta. ", "4b9f4a43f7f993a7e7df6d56de2db5ce": " f_0 = \\frac{v}{4L}.", "4b9f8b654ae0188bd238eec7d973b56b": "g = \\begin{cases} 2\\sqrt{-\\beta_2^2\\omega_m^4-2\\gamma P \\beta_2\\omega_m^2} &;\\, -\\beta_2^2\\omega_m^2 - 2 \\gamma P \\beta_2 > 0 \\\\ 0 &;\\, -\\beta_2^2\\omega_m^2 - 2 \\gamma P \\beta_2 \\leq 0\\end{cases} ", "4b9fbf2f39c6192b3818d5ed5d1643fe": "(a\\cdot a_i)^{(n-1)/2}=a^{(n-1)/2}\\cdot a_i^{(n-1)/2}= a^{(n-1)/2}\\cdot \\left(\\frac{a_i}{n}\\right) \\not\\equiv \\left(\\frac{a}{n}\\right)\\left(\\frac{a_i}{n}\\right)\\pmod{n}.", "4ba00a615f24781e495e4a480056855c": "\\subseteq Z_8", "4ba13ef581de93c32eb9d840e65f3dc4": "\\sum_i \\alpha_{ri} A_i \\to \\sum_i \\beta_{ri} A_i", "4ba14a55b4701c821da3475f4c99e6bc": "H_{Head} = 5", "4ba14ee6e3d372aec9ba16015fef691e": "\\nu=1/3", "4ba193e8c6c2c862d3dcc4a76f0eb9f3": "D = \\{(x,y) \\colon y = 0 \\text{ or } x \\text{ and } y \\text{ are non-zero and both are squares or both are non-squares} \\}.", "4ba1ea8db29379d5d0b2d14416298468": "\\begin{matrix} {3 \\choose 2}{44 \\choose 1} \\end{matrix}", "4ba1fc34d06627ac6b92eb9b39c6950b": "\\mathbf{a}_1 = a_1 \\mathbf{\\hat b} = \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{b}| } \\frac {\\mathbf{b}} {|\\mathbf{b}|},", "4ba3518f8710c21d02625c451673a727": "C_{4,2} = 1 + 4", "4ba3ae09a2741b434e1954752ccd47bf": "k_1, k_2, n", "4ba3cc3ea8bf4c150f326273b279f1d4": " \\frac{1}{2} \\left( 1-\\frac{r_a^2}{r_p^2} \\right) v_a^2 = \\frac{GM}{r_a} - \\frac{GM}{r_p} ", "4ba4e406ee73d7d4bbc518b3cb3df09c": " \\lambda W.\\operatorname{sink}[(\\lambda V.E)\\ Y, X] ", "4ba514f3e4af3339198cd63cb9c7f9f0": "\\chi_\\lambda(e^X)={\\sum_{\\sigma \\in W} {\\rm sign}(\\sigma) e^{i\\lambda(\\sigma X)}\\over \\delta(e^X)},", "4ba51d4a1ad0fe5d7530c41f983987a6": "\\inf \\theta \\leq 131/416.", "4ba52d9f53044a411f2f8661dc1ca664": "i,j\\in\\{1,2,\\dots,n\\}.", "4ba5627b907f94ec96d83019efc372f6": " P = e^{-rT}\\Phi(-d_2). \\,", "4ba56a2899d5eebde0b7a01aae025228": "G' = (V,E_L', s, t)", "4ba5b66f0ab5de5c2e865b88864100d1": "K_n = \\mathbf C((T^{1/n}))", "4ba5d84b0936370cbe25cd8966e8be7f": " \\mathbf{v} = \\sum_{j=1}^n \\bar{v}^j\\mathbf{b}_j = \\sum_{j=1}^n \\bar{v}^j(\\mathbf{q})\\mathbf{b}_j(\\mathbf{q}) ", "4ba61a4f7275cb420b7864c1efd9444b": "|-a| = |a|", "4ba6f3344493abe1d2993a784b7d1ab8": "f^*,g^*", "4ba703ca82453cf5f57343f803fd838a": "\\mathrm{Sh} = \\frac{K L}{D} = \\frac{\\mbox{Convective mass transfer coefficient}}{\\mbox{Diffusive mass transfer coefficient}} ", "4ba70b92c6ee80fd43c6904db4bcdfc4": " {\\alpha \\over \\lambda} \\left[{1+ {x \\over \\lambda}}\\right]^{-(\\alpha+1)}", "4ba7702f7dd6aa25aebfd9139b8ee273": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{\\partial L}{\\partial \\dot{q}}", "4ba788dfa64845810118840cb4a45d36": "\\beta^+", "4ba7b00d8a4d066f3d5beb77e8fb7f98": " RSS = y^T y - y^T X(X^T X)^{-1} X^T y.", "4ba7c6b3c90d19c9865130a51198c381": "P = I", "4ba7d2891a3f82b8fd5a79a7f8fd0ba2": "\\Delta\\theta|\\alpha|=\\frac{1}{2}", "4ba83af30ce8ad61238cccea88042c15": " f^{-1} (x)=\\frac{2}{\\sqrt{4x+1} } +\\frac{1}{4\\pi } \\int\\nolimits_{-\\sqrt{x} }^{\\sqrt{x}}\\frac{dr}{\\sqrt{x-r^2} } \\left( \\frac{\\Gamma '}{\\Gamma } \\left( \\frac{1}{4} +\\frac{ir}{2} \\right) -\\ln \\pi \\right) -\\sum\\limits_{n=1}^\\infty \\frac{\\Lambda (n)}{2\\sqrt{n} } J_0 \\left( \\sqrt{x} \\ln n\\right) ", "4ba83c435aeeb8b1a84fe1bbd0c8e610": "m_{\\text{dry}}", "4ba8656fc57c29caf071ea1425c190ce": "Q'\\subset Q", "4ba9311892c9798b89983daacde2d204": "\ndu/dt=-Ax-2xy\n", "4ba99b29c3a30576e5d763e01b041883": " 3\\cdot 5^e ", "4baa1965aa57136819f1fee11e4fd781": "\n\\text{df} \\approx 7.03. \\,\n", "4baa5f8117db4196437ac514b4949d59": "e^{\\frac{1}{e}}\\color{white}...........\\color{black}", "4baa710d704ca8746ec5fb0d35470ab7": "\\phi\\Omega(T)\\psi", "4baa98287e447ba79e8a932ea24acee9": "\\exp(-v(f))", "4baaa1aa51473635701261da233194cf": "p^r {-r \\choose k} (-q)^k", "4baadc2ca856106b88d61a5c9c7cf22e": " A = False ", "4bab1f9293b82bc87535ad8ecf923503": " r - \\cos x = r \\frac{x^2}{(2^2-2)r^2} - r \\frac{x^2}{(2^2-2)r^2} \\frac{x^2}{(4^2-4)r^2} + \\cdots , ", "4bab32b0fcdfeccd2c9f9c0cdd3efb0b": "\ndf = 10. \\, \n", "4bab5eab853a1583816b8fb937b0d47e": "\\{f,z\\} = (Sf)(z)", "4bab739f9cdbf507202c3ad8b95e53e3": "\\mathrm{N} \\mathfrak{p} \\equiv 1 \\pmod{n},", "4babc6ab976afa418fc5410ddb65b104": " \\, A_{(0,\\;1,\\;0)}(x)=x\\,", "4bac0aefa87f984312db1b03df0b8420": "\\mathbf{P} [ X_t = Y_t] = 1 \\mbox{ for all } t.", "4bac113ead86453f9767c25ff43d09b9": "3n+1", "4bac19e5232625b96657ea4be9e3d1b5": "{{s}^{2}}+2\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)s\\cos z-2{{R}_{\\text{E}}}{{y}_{\\text{atm}}}-y_{\\text{atm}}^{2}+2{{R}_{\\text{E}}}{{y}_{\\text{obs}}}+y_{\\text{obs}}^{2}=0 \\,.", "4bac74a01d7d47abfc48d0fce537abe0": "d(E)=-\\frac{1}{\\pi}Im(Tr(G(x,x^\\prime,E))", "4bac76e172d5d77f75bc9370bda93d35": "d\\times(2-2^{-n})", "4bac9cb189f97b66096a3502ea564aeb": "\\tau_m = \\tau_N", "4bacb8efb5abb2a4dd1757d96954bf34": "\n\\begin{align}\n \\frac{\\partial^2 \\Phi_3}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_3}{\\partial z}\n = & - \\eta_1\\, \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi_2}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_2}{\\partial z}\n \\right) \n - \\eta_2\\, \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right) \n \\\\ &\n - 2\\, \\frac{\\partial}{\\partial t} \\left( \\mathbf{u}_1 \\cdot \\mathbf{u}_2 \\right)\n - \\tfrac12\\, \\eta_1^2\\, \n \\frac{\\partial^2}{\\partial z^2} \n \\left(\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right)\n \\\\ &\n - \\eta_1\\, \\frac{\\partial^2}{\\partial t\\, \\partial z} \\left( |\\mathbf{u}_1|^2 \\right)\n - \\tfrac12\\, \\mathbf{u}_1 \\cdot \\boldsymbol{\\nabla} \\left( |\\mathbf{u}_1|^2 \\right).\n\\end{align}\n", "4baceb2c15d8c44eeb4fdcf4f210d9a1": "t = \\min(m-1, n)", "4bacfef6cc8676483227be86baf53d43": "C_a = C\\cdot 0.9877^a", "4bad28da41bc3994c7611c0c11781761": "I(x)=1-8x-12x^2\\ln x+8x^3-x^4", "4bad73f330207688a28988598720f154": "I_0 = \\frac{\\pi r^4}{8}", "4bad7870f12bcf48d18340faf8f16413": "\\Gamma^\\dagger", "4badd22df835cf5ee6375f1680cf714f": "\\cosh c=\\cosh a\\cosh b-\\sinh a\\sinh b \\cos C,", "4bade6c43ad54f7bb1e90bb531f38593": "\\left|\\int_a^b f(x)\\, d\\rho(x)\\right|\\le V(\\rho)\\cdot \\|f\\|_\\infty", "4bade6f3e73896c2da113ac12a2edc22": "\\frac{d^2 \\psi_2}{d x^2} = -k^2 \\psi_2 .", "4bae8318da6439e01becedb0c5c10cb1": "{1\\over (4,q^n+1)}q^{n(n-1)}(q^n+1)\\prod_{i=1}^{n-1}(q^{2i}-1)", "4bae837480ba4ad6f23ac6667931a0f1": " 0 = -g -{1 \\over \\rho}{\\partial P \\over \\partial z}", "4bae88fb6fec83a827bedb33fa4d2ff7": "X \\to B(G/O)", "4bae92fca5905703e520fc16d9eadc1d": "E_0^{(0)}=\\frac{1}{2}\\hbar \\omega. \\, ", "4bae9aace31a470d564171320ab71939": "\\alpha = Gm_{Sun}", "4bae9fc92c1980c94c4b8598f81fc290": "\\bar{x}_{j}", "4baebd2938f70ae46d0f7af220ece0b0": "m_{fuel}", "4baee84af7a92177c9064ddc2fc010ce": "PP", "4baf014183ab72b3a284fd43899a8b38": " d\\langle E \\rangle = T dS + \\mu_1 d\\langle N_1 \\rangle \\ldots + \\mu_s d\\langle N_s \\rangle - \\langle p\\rangle dV .", "4baf05316fb6a7028b02785ba030ba95": "\\displaystyle 2\\pi\\delta(\\nu)", "4baf0d0ae44c13907d5a595784494793": "[M\\cdot]=\\left(\\frac{k_d[I]f}{k_t}\\right)^{1/2}", "4baf5e97cb941401e0cb8f03df6da72e": "\\sigma_{i j} = \\epsilon_0 E_i E_j + \\frac{1}\n{{\\mu _0 }}B_i B_j - \\frac{1}{2}\\bigl( {\\epsilon_0 E^2 + \\tfrac{1}\n{{\\mu _0 }}B^2 } \\bigr)\\delta _{ij} ", "4baf5fd2bacc9d6b85900a2589f066ed": "T:V\\rightarrow V", "4baf72c9cc44f582704f6e7edc0deb90": "\\mathcal{N}(0,\\sigma^2(0))", "4baf73af724cdd0d718d1be272155f38": " R_\\Delta(N_1, N_2) = \\frac{R_c(R_a+R_b)}{R_T} ", "4baf7df630729152b6a1609a42077b6c": "F(n) \\approx \\frac{9 \\times 10^{n-1}}{n!}", "4baf9ad10c409864baaa3164a20abe41": " A = \\alpha_0 + \\alpha_1 \\frac{u^2} {c^2} + \\alpha_2 \\frac{u^2_r}{c^2}+... ", "4bafb3d0c4863848a59299854e083290": "r-1", "4bafbc8a676c10e58aa2a84eaffafb3e": " {\\partial \\over \\partial \\tau} \\Psi(\\mathbf{r},\\tau) = \\frac{\\hbar}{2m} \\nabla ^2 \\Psi(\\mathbf{r},\\tau), ", "4bafc8b4397e180cc796c83bd89126e3": " X \\in X ", "4bafd4d091b21cc91bee1ce2392e4dab": " \\Delta k = \\left|\\frac{dk}{d\\lambda}\\right|\\Delta \\lambda = 2\\pi \\frac{\\Delta \\lambda}{\\lambda^2} \\ , ", "4bafdc0de2d8c6dd79ae35f75bbc43bf": "x_{10}", "4bb00c5f034cf1c00902cb20aa3ddc6d": "|A - B|^2 = (A - B)^\\top (A - B) = |A| + |B| - 2 A^\\top B", "4bb07724e06066dcb096ee4bcc0dace3": "10^2 \\equiv (-1)^2 \\equiv 1\\pmod{11},", "4bb0a019b2427f7f7145344d361d5b8c": "\\mathbb P^n_A", "4bb0e39890bda74414da75800639ebe8": "\\mathbf{x}^{(0)}", "4bb0e8dc48ccf0ab2d2a92f66e3feb0c": "\\begin{align}\n\\frac{dp_\\alpha}{dt} & = \\sum_\\beta \\nu_{\\alpha\\beta}(p_\\beta - p_\\alpha) \\\\\n\\frac{dp_\\beta}{dt} & = \\sum_\\alpha \\nu_{\\alpha\\beta}(p_\\alpha - p_\\beta) \\\\\n\\end{align}", "4bb154b47fd2c00f40f257d15e00649e": "j_*: H^q_{\\mathrm c}(U) \\to H^q_{\\mathrm c}(X)", "4bb1f95e92ecd176701169ab5f0e1a03": "A = \\frac {r}{2} \\cdot p \\cdot \\sqrt{1- \\tfrac{p^{2}}{4n^{2}r^{2}}}.", "4bb2544e6acb06643461360fd37aa53a": " \\sigma_1 = \\max \\left( \\lambda_1,\\lambda_2,\\lambda_3 \\right)\\,\\!", "4bb2b2571319fb8565ac3c27149690dc": "\\Delta R_{\\mu}\\Delta x_{\\mu}\\ge\\ell^2_{P}", "4bb318ca400f810f087e646f45a5619c": " \\int_{\\Bbb Z_p} 1 \\, {\\rm d}x = 1 ", "4bb31c4753f72e70859c5737bd3f8bbf": "Y_{11}, Y_{22}, ..., Y_{nn}", "4bb33ef4320161b3d98a49242b40b3cc": "\\mathcal{L}_YX=[Y,X].", "4bb344d0c5d2181c02d3c9157f7bd5bc": "\nr_\\pm = \\frac{1}{2}\\left(r_{s} \\pm \\sqrt{r_{s}^2 - 4r_{Q}^2}\\right).\n", "4bb36d917727ce02beb14656f3528ffa": "-\\infty \\le x_{i} \\le \\infty", "4bb39903ef84cf7b2069255c5dfc9f77": "10^{-1}\\frac{m}{s}", "4bb39d810a530083d6b07a8b8f689321": "\\; = -(\\sum_x p_x \\log p(x) + \\sum_y p_y \\log p(y))", "4bb3cce76b1f16f03fd5d3f8cc85e5c0": "\\left({12 \\over 100}\\right)", "4bb3d21e8f6a2d4d8ff0779de9b0bf53": "{\\mathrm{h}} \\ = \\frac{k} {D}\\left({0.6 + \\frac{0.387 \\mathrm{Ra}_D^{1/6}}{\\left(1 + (0.559/\\mathrm{Pr})^{9/16} \\, \\right)^{8/27} \\,}}\\right)^2", "4bb441214d6028bf725118ee0d91fcb7": "\nJ_{k} = \\left( \\frac{1}{k} \\right) \n\\int d\\theta^{\\prime}\n\\int d\\rho^{\\prime} \n\\left[ \\frac{\\sin k\\theta^{\\prime}}{\\left(\\rho^{\\prime}\\right)^{k-1}} \\right]\n\\lambda(\\rho^{\\prime}, \\theta^{\\prime}) \n", "4bb448aa2ae8a634e90a5ce05c9639da": "\\ \\|x\\|_p=\\left(|x_1|^p+|x_2|^p+\\dotsb+|x_n|^p\\right)^{\\frac{1}{p}}", "4bb4ade27c255ccb8e7a8b83142fc188": "\\forall R_0\\ldots\\forall R_m \\phi", "4bb4c8acef9c47d950ebf1679bf369a6": " g(x) = f(x) \\big/ (x-a) ", "4bb4f732775dcd3d29f43dd3c93c1ff6": "\\boldsymbol{u_{1}}, \\boldsymbol{u_{2}}, ..., \\boldsymbol{u_{n}}", "4bb55d3fd5eab06fcc2ff91ed9281ad5": " \\frac{d \\ln K_{eq}}{d {\\frac{{1 }}{{T }}}} = -\\frac{\\Delta H^\\ominus}{R}. ", "4bb576d107cfd9443d77c9b15fd724c3": "S^{2n-1}\\times S^1", "4bb583d1dc37c6b9fe2b7b5f15fa0fdb": "\n\\begin{bmatrix}\n\\boldsymbol{V}_j^{(t)}\\\\\n\\boldsymbol{V}_j'\\\\\n\\boldsymbol{V}_j^{(b)}\n\\end{bmatrix}\n", "4bb5a779dbf0d53c5d60bf7004784f05": " V(r) = \\frac{1}{2} m_0 \\omega^2 r^2. ", "4bb5ca45b9e2307bef53ccad1259d148": "\\textstyle {1 \\over 5}, {1 \\over 4}, {1 \\over 3}, {2 \\over 5}, {1 \\over 2}, {3 \\over 5}, {2 \\over 3}, {3 \\over 4}, {4 \\over 5}", "4bb6067d29e4078d822b102bb7a851e1": "f(x) = \\frac{a_0}{2} T_0(x) + \\sum_{k=1}^\\infty a_k T_k(x),", "4bb65e2f18b60ff9cd9a47efcdaac2c9": "\\begin{cases}\n y_t = \\alpha + \\beta x_t^* + \\varepsilon_t, \\\\\n x_t = x_t^* + \\eta_t,\n \\end{cases}", "4bb711ca0794258746beb95764b2ddf5": " \\mathbf{p}_1,\\ldots,\\mathbf{p}_N", "4bb7322eaf54886a0c8ebda3240e26fc": "Y=AL^{\\alpha}K^{\\beta}\\varepsilon \\,", "4bb7675e7cacdaec7bed700da5eeea4b": "MD(\\Diamond \\varphi) = 1 + MD(\\varphi)", "4bb772132f3ae93a7335f5c75b1e1cce": "\\gamma(t)", "4bb7c0a964a94bdbebf9d0fa24945427": "a^{r/2} \\not\\equiv 1 \\pmod{N}", "4bb7fc678f782e3886452e815a476b9a": "t = \\frac{2h}{c}", "4bb84a00cee697eea7ba59496b6f9c93": " ds^2 =r^{-2}(dx^2 + dy^2 + dr^2)\\ ", "4bb87b57331ec3ea667a5cbdc2eefe72": "\n\\begin{align}\nP_{(tn)} & =LB+\\frac{TN-1}{UT-1}\\Delta B \\ =UB-\\frac{UT-TN}{UT-1}\\Delta B; \\\\[10pt]\n& {} \\qquad {\\color{white}.}(P_{(1)}=LB,\\ P_{(ut)}=UB){\\color{white}.} \\\\[10pt]\nF'(P_\\tilde{a}) & =F'(LB < P < UB)=\\sum_{TN=1}^{UT=\\infty}\\frac{F'(P_{(tn)})}{UT}.\n\\end{align}\n", "4bb8d66a28b572a2838b09f62f3cf837": " \\frac{\\delta (\\rho \\epsilon)}{\\delta t}+ \\frac{\\delta (\\rho \\epsilon u_i)}{\\delta x_i} = \\frac {\\delta}{\\delta x_j}\\left[\\frac {\\mu_t}{\\sigma_\\epsilon}\\frac {\\delta \\epsilon}{\\delta x_j}\\right] + C_{1 \\epsilon} \\frac{\\epsilon}{k} 2{\\mu_t}{E_{ij}}{E_{ij}}- C_{2 \\epsilon } \\rho \\frac{\\epsilon ^2}{k}", "4bb8f22ac49c8fe9bec51dc834daf239": " \\phi \\to a_0(980) \\gamma, ~~ f_0(975) \\gamma", "4bb92fa8e242ee359d2cfb17fc4c6a20": "d{\\mathcal I}\\subset {\\mathcal I},", "4bb931c3fe9ac8a1b53bda9d87bdf3e7": " \\sum_{i=1}^n m_i(\\mathbf{r}_i -\\mathbf{R}) = 0.", "4bb9a540b2540495472fcd56d3b3dc9e": "\\Bigg(\\frac{a}{n}\\Bigg) = \\left(\\frac{a}{p_1}\\right)^{\\alpha_1}\\left(\\frac{a}{p_2}\\right)^{\\alpha_2}\\cdots \\left(\\frac{a}{p_k}\\right)^{\\alpha_k}\\mbox{ where } n=p_1^{\\alpha_1}p_2^{\\alpha_2}\\cdots p_k^{\\alpha_k}", "4bb9b5756851db7c6e80a3e0fba1d2d8": " f_x(x,\\lambda) ", "4bb9c7085108efbc6ab93e312549326f": "\n\\left\\{ C_{1}, L_{i} \\right\\} = \\left\\{ C_{1}, D_{i} \\right\\} = \n\\left\\{ C_{2}, L_{i} \\right\\} = \\left\\{ C_{2}, D_{i} \\right\\} = 0 ~.\n", "4bb9d76edaaab5eece30f86672959edb": "\nC_1 = - \\frac{F_1}{\\pi} ~;~~ C_3 = -\\frac{F_2}{\\pi}\n", "4bb9d97f46406006f52b0b79c75a58c6": "\\delta = 0", "4bb9e0f1c2eb459e9e0a883baba9752c": "\n |(j_1j_2)jm\\rangle = \\sum_{m_1=-j_1}^{j_1} \\sum_{m_2=-j_2}^{j_2}\n |j_1m_1j_2m_2\\rangle \\langle j_1j_2;m_1m_2|j_1j_2;jm\\rangle\n", "4bba3058b81878ce0c6d78e67e640055": "\\rho(1)=0", "4bba67006ec6281770c729d4ebf13b5d": "\\mathcal{L}(n) = [n \\ge m]\\frac{1}{n}", "4bbb0ac272cb4b51f52f8303667b4256": "\\frac{d\\Gamma}{dx} \\sim (3x^2-2x^3).", "4bbb224c2603162b316a16861d670f86": "f^{\\mathcal{A}}_i", "4bbb6e580a7a396ecaeeb8af81eecf6b": "\\omega_3 = -\\frac{1}{2} - \\frac{\\sqrt{3}}{2}i", "4bbb7a3465f6756fc52715202e9c3332": "\n|t - t_{0}| = {\\sqrt{\\frac{m}{2}}} \\int \\frac{|dr|}{\\sqrt{E_{\\mathrm{tot}} - U(r)}}\n", "4bbb8d4dba779845739e80fd9d6794a6": "\\ \\lambda=l_1/l_2 ", "4bbbbc1b3ab94f2d4e918ff3405405a7": "T(F):=\\arg\\min_{\\theta\\in\\Theta}\\int_{\\mathcal{X}}\\rho(x,\\theta)dF(x)", "4bbbd57214229a382a39f3800744395b": "F_i ", "4bbc50d5b51825118464f421d38b8115": "\\textstyle \\dot{x}_1(t) = \\dot{u}(t) = x_2(t)", "4bbc78ce52df2b07eaf848e9870dd3f0": "A=ULU^{-1}", "4bbcb66457dec847c76dfa1b9aeaee97": "\n\\delta \\mathcal{S} = \n\\left[ \\boldsymbol\\varepsilon \\cdot \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}}\\right]_{t_1}^{t_2} + \n\\int_{t_1}^{t_2}\\; \n\\left( \\boldsymbol\\varepsilon \\cdot \\frac{\\partial L}{\\partial \\mathbf{q}}\n- \\boldsymbol\\varepsilon \\cdot \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right)\\,dt\n", "4bbd1101492d6dddd3f35cde308139c1": "\n\\beta_\\text{max} = 0.072 \\left(\\frac{1+2^2}{2}\\right)\\frac{1}{5/2} = 0.072.\n", "4bbd4bf3030310e4cace45c010a5b90f": " g = \\dot{X}^2 X'^2 - (\\dot{X} \\cdot X')^2 ", "4bbd4c761e851785c633b9047a308e94": "\\zeta(x,y,t)\\, =\\, \\Re\\left\\{ \\eta(x,y)\\; \\text{e}^{-i\\, \\omega\\, t} \\right\\},", "4bbd7daf2ab0e080e0b870ca0b58fc26": "{\\mathbf B} = \\nabla\\times ({\\mathbf A}+ \\nabla \\psi) = \\nabla\\times{\\mathbf A}", "4bbe7f5471fea2f3edc6c6dc4b859447": "\\| h\\|_{1} := \\int_{\\mathbb{R}^{n}} h(x) \\, \\mathrm{d} x \\geq \\left( \\int_{\\mathbb{R}^{n}} f(x) \\, \\mathrm{d} x \\right)^{1 -\\lambda} \\left( \\int_{\\mathbb{R}^{n}} g(x) \\, \\mathrm{d} x \\right)^{\\lambda} =: \\| f\\|_{1}^{1 -\\lambda} \\| g\\|_{1}^{\\lambda}. \\, ", "4bbe851e7b1836eeb1d3a3af8fa4c6e5": " Ax = b. \\, ", "4bbea9ab4d3f6c23ca57f00986b4b4b0": "(s+t)|st(s+1)(t+1)", "4bbef58a14900dbe9f4259042fd28bd1": "\\empty ^{c} =U.", "4bbefe2530b009b6f0f33c7358711ea5": "z_{n-1} = \\sqrt{z_n - c} .", "4bbf1705495b5197d9550246acdbf1aa": " \\lambda_{jx,jy,jz} = \n-\\frac{4}{h_x^2} \\sin\\left(\\frac{\\pi j_x}{2(n_x + 1)}\\right)^2\n-\\frac{4}{h_y^2} \\sin\\left(\\frac{\\pi j_y}{2(n_y + 1)}\\right)^2\n-\\frac{4}{h_z^2} \\sin\\left(\\frac{\\pi j_z}{2(n_z + 1)}\\right)^2\n", "4bbf29bf684fa6ba32598679ce281e32": "\n\\langle T_v\\exp_p(v), T_v\\exp_p(w_N)\\rangle = \\left\\langle \\frac{\\partial f}{\\partial t},\\frac{\\partial f}{\\partial s}\\right\\rangle(0,1).\n", "4bbf949fac13058a60fa132f19f24ca8": "A\\subseteq B\\,", "4bbfda4ea5b1b48a46b007de1f997137": "\\log w", "4bbff79bc58fc1330a5c5414af5090be": " F_{Y}(y) = 1 - F_{X}(-\\mathrm{log}(e^{y} - 1)) \\, ", "4bbffad90db9537ee530d3b2911189d7": " R \\leq 1 - H_q(p) - \\epsilon ", "4bc01abe3edf9306950b6f2f23c7941f": "\\sqrt{a_1 + \\sqrt{a_2 + \\sqrt{a_3 + \\sqrt{a_4 + \\cdots}}}}", "4bc07e4bb539ff25c1694c609bbf8f9d": "N^4", "4bc09745ee27c36b05a67829a042897b": "\n a^3 = \\cfrac{9R^2\\gamma\\pi}{E^*}\n ", "4bc0ac9f52814c7804d9aff8fa0ce939": " S_E = \\sqrt[12]{2} = 100.000 \\ \\hbox{cents}. ", "4bc0af79f9eb36d053ffaad2fa33cf26": "Y,Z", "4bc0baa1abb5bfd12cc11089f6a06096": "\\sum_{n =-\\infty}^\\infty h[n] \\cdot \\sin(\\omega \\cdot (n - \\alpha) + \\beta)=0", "4bc0bd3c315bfef89b427ca2c29562ac": "\\begin{smallmatrix}R_\\star\\end{smallmatrix}", "4bc1102f17e6bc7a5e6d32fed8ad300b": "M\\cdot V_T =\\sum_{i} (p_i\\cdot q_i)=\\mathbf{p}^\\mathrm{T}\\mathbf{q}", "4bc14bfde80975aea69ac3ecdc50f77d": "N=3", "4bc19af34eeebdb3e311039083848c90": "\\mathbf{G} ", "4bc1bf7bf781d26966457e2905fa3a14": "\n\\sum_{k=1}^\\infty\\frac{\\mu(k)}{k}(-\\log(1-x^k))=x, \n", "4bc1e0023c76f84f4c86e716574a517f": "\\scriptstyle \\leq5\\times10^{-15}", "4bc20d72c675d260c66247919950c747": "y_p = y_{p_1} + y_{p_2}.", "4bc243e02fb81d797ff7159fcb448410": "c(\\pi, B) = \\sum_{i=0}^{T-1} w_{v_i,v_{i+1}}", "4bc25f170405b3ede8d6b42471476d66": "\nU(\\theta)=\\frac{\\partial \\log L(\\theta | x)}{\\partial \\theta}.\n", "4bc2d19f4481fb9b469fb3581155d134": "\\Gamma_Y \\circ f_* = \\Gamma_X", "4bc342d1345b5bf7c36bd7c8d59c2189": "\\varphi\\otimes\\psi", "4bc36ce3d1ade0d22473e01cc3406d23": "\\,\\! \\delta g = \\delta \\det(g_{\\mu\\nu}) = g \\, g^{\\mu\\nu} \\delta g_{\\mu\\nu}", "4bc3dddffb5433b565847cf17a35d799": "\\eta_s:=\\exp\\left(\\frac{2\\pi i}{2^s}\\right)+\\exp\\left(-\\frac{2\\pi i}{2^s}\\right)=2\\cos\\left(\\frac{2\\pi}{2^s}\\right).", "4bc3e2d2d4fb945cf939227e03f886cf": " \\mathrm{d}\\bold{F} = 0", "4bc468f103f6cf1232e2a76ec40f5ac8": "k\\ge 3", "4bc4a4d047e39f74ff33f18fbcbe60cc": " \\mathcal{L}[\\phi(x)] ", "4bc4ba4ad4325ae301e74dd5fc79fee1": "A_1 \\dots A_t.", "4bc4bd8f63ca19eb7b22df5f9b5f7a6b": "I^+(S) \\cap T", "4bc4d6aeae1036521381570a4c5cdd06": "LMTD=\\frac{\\Delta T_A - \\Delta T_B}{\\ln \\left( \\frac{\\Delta T_A}{\\Delta T_B} \\right ) }", "4bc51bfd8f6afafc72e5e0a7033ba138": " O\\left(\\frac{1}{\\sqrt{N}}\\right) ", "4bc544811b36aa09f02bb6479304d553": "B_{\\mu\\nu}", "4bc58d367e5f31bd7f0baf8be21955d7": "\\mathbf{a} + \\mathbf{b} = \\mathbf{c}", "4bc5967a7bdfc8a76909be2d62386ab6": "\\scriptstyle dZ", "4bc5a6cd2e15d14e80e1316652073055": "J_n^{-1}=J_{n-1}^{-1}+\\frac{\\Delta \\vec{x}_n-J^{-1}_{n-1} \\Delta \\vec{F}_n}{\\Delta \\vec{x}_n^T J^{-1}_{n-1}\\Delta \\vec{F}_n} (\\Delta \\vec{x}_n^T J^{-1}_{n-1})", "4bc6009ff4943ec573e933c0a217629f": "\\pi \\approx 10.011\\overline{1}111\\overline{1}000\\overline{1}011\\overline{1}1101\\overline/11111100\\overline{1}0000\\overline{1}1\\overline{1}\\overline{1}\\overline{1}\\overline{1}0\\overline{1}", "4bc61adfb3810ce69ae7cde3215ef71b": "\\bar y = \\displaystyle \\sum_{i=1}^n y_i/n", "4bc657b8c14f4d4c6420bc1860bd68c1": "\n\\eta_q(n) = \n\\begin{cases}\n0&\\;\\mbox{ if }q\\nmid n\\\\\nq&\\;\\mbox{ if }q\\mid n\\\\\n\\end{cases}\n", "4bc6719940cc369dd3203013c5828821": "U_{0,n}", "4bc6a8151c3f0825c344a823afaed27f": "k_{js}", "4bc6b9ae5e0a74ac46e595e769148768": " \\Delta t = \\Delta x/v ", "4bc72b41b794032dc80cc74837d205d1": "\\sigma : V^n\\to V^n", "4bc747012f52d0201eaa363f3b5f44b2": "u(y)>u(x)", "4bc75f36a99355ecc2ff531b5512a152": "k_2\\ll1\\,", "4bc75fee5f69670c79b6d55a6c7d562e": "\n= \\left[\\log_{10}{\\left(\\frac{N_{\\mathrm{O}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{star}} - \\log_{10}{\\left(\\frac{N_{\\mathrm{O}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{sun}}\\right] -\n\\left[\\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{star}} - \\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{sun}}\\right].\n", "4bc7b35c432f49dd01ca1a82951b247e": "-\\frac{b}{a}", "4bc7f7ba5cc4927e342ad54f3c92a038": "f\\in{\\mathbb{F}_q[X]}", "4bc8490558f5dba74f32eedea4a34c93": " b_n < \\frac{1}{c-\\varepsilon} a_n ", "4bc8f3aeb5ca3f73834ba7c6cd4098f8": "\\begin{align}\nV_0 & = k_2[ES] = \\frac{k_2 K_i [S][E]_0}{K_m K_i + K_i[S] + K_m[I]} \\\\\n& = \\frac{k_2 [E]_0 [S]}{K_m + [S] + K_m\\frac{[I]}{K_i}}\n\\end{align}\n", "4bc915ed12f457dd7884e3d5ebc3bf66": "\\lim_{z\\rightarrow 1}(1-z)\\frac{d\\log(_{p}F_{q}(a_{1},\\ldots,a_{p};b_{1},\\ldots,b_{q};z^{p}))}{dz}=\\sum_{i=1}^{p}a_{i}-\\sum_{j=1}^{q}b_{j}", "4bc921bd2e0952abaaff376d88ad8b68": "(x_0, y_0; t_0)", "4bc93896bfa4063f9816cbe9d8c73bb5": "s_i: S_n(X)\\to S_{n+1}(X)", "4bc94950bef6d8fc5d47d767b62aae45": "\\dim_k \\mathfrak{m} / \\mathfrak{m}^2 = \\dim A\\,", "4bc98a8528f8deeb992749abe2990ddf": "I_{b-}", "4bc9927952cb84c7b841459ebf468679": "{{P}_{V}}f(u,\\xi )", "4bc99ca623bdc3bd18a9338450f328f4": "\\eta_{th} = 1-\\frac{r^{1-\\gamma}(r_c^\\gamma - 1)}{\\gamma(r_c - 1)} \\,", "4bc9cd39889f651b68656e4a99c0e4c4": "a_n \\le \\ \\left | a_n \\right \\vert", "4bc9e39907536ae35d31872c72d20a3a": " f_{fast , slow} ", "4bca1f61a85dd297b0d261ee138afde4": "2 \\sqrt{n}", "4bca3fa23cddd2a5854d4e8245029eac": "\\lim_{n\\rightarrow\\infty}\\frac{N(A,n)}{n}", "4bca7930ecfdb8e3c773b71f055f5e3a": "\\textstyle2\\left(\\frac{1 + \\sqrt{5}}{2} - \\frac{1}{2}\\right) = \\sqrt{5}", "4bca8777d71cbbe8836dabfae0c1a972": " D[g] ", "4bcb0056445d48e48b89f26d81f01a8f": "X \\sim \\mathrm{GH}(-\\frac{\\nu}{2}, 0, 0, \\sqrt{\\nu}, \\mu)\\,", "4bcb09a44a31138777ef0c4c82e87670": "\n\\begin{align}\n& {} \\quad E_{Q^N}\\left[\\left.\\frac{S(T)}{N(T)}\\right| \\mathcal{F}(t)\\right] \\\\\n& = E_{Q}\\left[\\left.\\frac{M(0)}{M(T)}\\frac{N(T)}{N(0)}\\frac{S(T)}{N(T)}\\right| \\mathcal{F}(t)\\right]/ E_Q\\left[\\left.\\frac{M(0)}{M(T)}\\frac{N(T)}{N(0)}\\right| \\mathcal{F}(t)\\right] \\\\\n& = \\frac{M(t)}{N(t)}E_{Q}\\left[\\left.\\frac{S(T)}{M(T)}\\right| \\mathcal{F}(t)\\right]= \\frac{M(t)}{N(t)}\\frac{S(t)}{M(t)} = \\frac{S(t)}{N(t)}.\n\\end{align}\n", "4bcb1d8dbb68fa755fdf8b35938a4621": "G=BWB =\\coprod_{w\\in W}BwB", "4bcb7641fdc8aa274295d9f78eb87acb": "\\mathcal{L} = - \\frac{\\hbar^2}{m} \\eta^{\\mu \\nu} \\partial_{\\mu}\\psi^{*} \\partial_{\\nu}\\psi - m c^2 \\psi^{*} \\psi\\,.", "4bcb79346568af50eb006dad02962b6c": "\n\\langle\\Psi_{motion}\\vert {k_z}^2 z^2 \\vert \\Psi_{motion} \\rangle^{1/2} \\ll 1\n", "4bcb86b7f34af075f2773b4c291ed789": "\\left(\\tfrac an\\right)=0", "4bcb8a9a314e4ded9ffe9a4cbad71c15": " \\begin{alignat} {3}\n\\bar{n}_i \\ & = \\frac{ \\displaystyle \\sum_{n_i=0} ^1 n_i \\ e^{-\\beta (n_i\\epsilon_i)} \\ \\ Z_i(N-n_i)}\n { \\displaystyle \\sum_{n_i=0} ^1 e^{-\\beta (n_i\\epsilon_i)} \\qquad Z_i(N-n_i)} \\\\\n\\\\\n& = \\ \\frac { \\quad 0 \\quad \\; + e^{-\\beta\\epsilon_i}\\; Z_i(N-1)} {Z_i(N) + e^{-\\beta\\epsilon_i}\\; Z_i(N-1)} \\\\\n& = \\ \\frac {1} {[Z_i(N)/Z_i(N-1)] \\; e^{\\beta\\epsilon_i}+1} \\quad .\n\\end{alignat} ", "4bcbbce2fcf6c0575d9ee664295b439e": "\ns_{ij} = \\begin{cases}\n\\frac{q_{ij}}{\\sum_{k \\neq i} q_{ik}} & \\text{if } i \\neq j \\\\\n0 & \\text{otherwise}.\n\\end{cases}\n", "4bcc3a3061185665838018fd5d16eb15": "c=c_L \\qquad \\forall x<0", "4bcc3ef4873039a92d4feea5ab84e05a": "f = \\gamma \\left( 1 - \\frac{v}{u^\\prime} \\right) f^\\prime.", "4bcc510dc88e037ca9ad15e18b44a36b": " \\lambda = \\tfrac{h}{m v} ", "4bcc791b6aac742988d8b820f04a1f0b": "a_1=a_2=\\cdots=a_n=0 ", "4bcca248d586ac7faa3d54d2aede8ac1": "Y=\\beta_{40} +\\beta_{41}X +\\beta_{42}Mo +\\beta_{43}XMo + \\varepsilon_4", "4bccdef029dd0530a4b92b5a44c4c849": " \\left|x\\right|^r + \\left|y\\right|^s + \\left|z\\right|^t =1", "4bcce91c33542b249de0b0b0520d788b": " \\mathbf{\\Psi} = \\frac{1}{2}\\sum_{j=0}^3 S_j \\mathbf{\\sigma}_j,\\text{ where}", "4bcce97833c705d53ef9d7875139e33a": "\\boldsymbol{\\mu} = \\frac{g_s q}{2m} \\mathbf{S} ", "4bccf26b115769be1ab70e232952c555": "\\langle a\\rangle", "4bccf32342d5fb1d5f300004e9db6564": "\\mu(A_1 \\cup A_2) = \\mu(A_1) + \\mu(A_2) \\mbox{ whenever } A_1,A_2 \\in \\mathcal{A} \\mbox{ and } A_1 \\cap A_2 = \\varnothing.", "4bcd21365a349361a64bcbc55ccaab40": "\\{\\omega\\in B(k)\\}\\Leftrightarrow\\{L(k,\\omega)=1\\}\\quad\\text{and}\\quad\\{\\omega\\in H(k)\\}\\Leftrightarrow\\{L(k,\\omega)=k\\}.", "4bcd544cb5a0682cb7a492070d69f833": "\\gamma(x)=\\gamma(y)", "4bcd62def24b3a75d14c9a6e3d5819b5": "\\lambda\\ x.(x\\ x)", "4bcdfd1c1028224b735f49d44b23ed9d": "c = e^b\\,", "4bce9937331922465d5cadba26884d10": "\\theta = \\arctan \\left(\\frac{\\text{opposite}}{\\text{adjacent}} \\right) = \\arctan \\left( \\frac{\\text{rise}}{\\text{run}} \\right) = \\arctan \\left( \\frac{8}{20} \\right) \\approx 21.8^{\\circ}.", "4bcf6402eeca68620e2d5e14f51ed0a3": "\\{0(\\mathrm{mod}\\ {3}),\\ 1(\\mathrm{mod}\\ {3}),\\ 2(\\mathrm{mod}\\ {3})\\},", "4bcfa078f4cc5eb3b46135f8a2d51395": "T^\\dagger\\,", "4bcfbd17db8f305adf5a80e45cf35d76": "Q = 1/\\sqrt{2}", "4bcfd4bfc519f3a9af75a4b5b4771c28": "n \\in A \\mbox{ iff there is a finite function } \\theta \\mbox{ such that } \\varphi_n \\mbox{ extends } \\theta \\wedge c(\\theta) \\in A", "4bd01f4a6739d95ec7743e82072870d7": "\\mathcal{N}", "4bd025d0dc8bc240c1fe933be66ac4c7": "v^{i} = \\dot{u}^{i}", "4bd0abef26de01c37ecae8b4bce5dc28": " \\mathbf{E(r)} = -\\nabla \\varphi \\mathbf{(r)} .", "4bd0bcb8a35074810edf10009ba1e343": "Q_1 = BC^2", "4bd0bfd8da5936bda9933ba308e09252": "\n\\begin{align}\nF(\\dots, A^{j_1}, \\dots, A^{j_2}, \\dots)\n& = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\left(\\prod_{i = 1, i \\neq j_1, i\\neq j_2}^n a_{\\sigma(i)}^i\\right) a_{\\sigma(j_1)}^{j_1} a_{\\sigma(j_2)}^{j_2}\\\\\n\\end{align}\n", "4bd10d486c5636a6a3a9b6ff3b3dec18": "\\tilde{X} : [0, + \\infty) \\times \\Omega \\to \\mathbb{R}^{n}", "4bd1241d43b60e0e4190660b97d2f686": "C_i", "4bd14f3a049f95837ed44999212fdfa5": "{|Z_W|} = \\sqrt{2}\\frac{A_W}{\\sqrt{\\omega}}", "4bd1673cbad60ebdf8d8414bd27d2bd7": "\\sum a_{i,j} \\vec e_i", "4bd1798d4f97c8dd20ae62cbe2a59c22": " \\frac { w_1 } { ( 1 - w_2 ) } = m ", "4bd2098a28b1b509dd01c16a8eda45a1": " \\lambda_i ", "4bd22e8cb10236bda1bc8260b297f0cc": "D=i_D\\circ j^k", "4bd273dec97c3f27306d0a55d060215b": " \\mathrm{Oh} = \\frac{ \\mu}{ \\sqrt{\\rho \\sigma L }} = \\frac{\\sqrt{\\mathrm{We}}}{\\mathrm{Re}} ", "4bd28096787a5a5f1275a39a4487fdfc": "\\pi(m)=\\Phi(m,n)+n(\\mu+1)+\\frac{\\mu^2-\\mu}{2}-1-\\sum_{k=1}^\\mu\\pi\\left(\\frac{m}{p_{n+k}}\\right)", "4bd29c2d5fbeba88316908117c40fbcb": "\\mathbf{D = l_1 + l_2 + \\cdots + l_n}", "4bd2d9eeb37d2651a53ff695a1adbf5a": "\\text{then }\\tan(x) + \\tan(y) + \\tan(z) = \\tan(x)\\tan(y)\\tan(z).\\,", "4bd33b6aeba4fc389178307c435d17ea": "\\hat\\mu^*_i", "4bd33caa0fd8056add6866abbdea5d34": "[N,K,D] ", "4bd42b9f9c53c1031676402876d142a5": "xS''+xS'^2+(c-x)S'-a=0\\,", "4bd44fe37eb5d2dcc07feef301dd2e49": "\\mathbb{R}^n ", "4bd46ea3b66695f6d2efe458823525a2": "M\\left(t;\\alpha,x_\\mathrm{m}\\right) = E \\left [e^{tX} \\right ] = \\alpha(-x_\\mathrm{m} t)^\\alpha\\Gamma(-\\alpha,-x_\\mathrm{m} t)", "4bd4fbd2e32aabba593a093331aa5e7a": "\\curvearrowright \\circlearrowright \\Rsh \\downdownarrows \\leftleftarrows \\leftrightarrows \\leftarrowtail \\looparrowleft \\,\\!", "4bd51f5ec842466d40939efdac84682f": "u^{\\prime} = \\rho^{(1/2)} u \\rho^{(1/2)} = \\rho u, ", "4bd537db828a4e604740ddcff93a9bb6": "\\mathit{d_H}^R(\\mathit{p},\\mathit{q}) = \\mathit{d}_H(\\mathit{p},\\mathit{q}^R)", "4bd55f631b11523668fd41f280f91a70": "u,p \\in \\{ String \\}", "4bd6320492ea0b68acb75a631f5dc5c5": " 0.3048 \\ \\frac{\\mbox{m}}{\\mbox{ft}} ", "4bd6603d73c29c611042a969f3fa04d4": "\n \\quad (8) \\qquad G = \\frac{e^{a(t+\\Delta t)} e^{ik_m x}}{e^{at} e^{ik_m x}} = e^{a\\Delta t}\n", "4bd66f0a7b5e81253321e0ac3c726c96": "\nG(t) = G_0 - \\Sigma_{i=1}^{N} G_i [1-\\exp(-t/\\tau_i)]\n", "4bd6a4727200002be782b24b49a9bced": "\\operatorname{Var}(f) = \\left(\\frac{\\partial f}{\\partial x}\\right)^2 \\operatorname{Var}(x) + \\left(\\frac{\\partial f}{\\partial y}\\right)^2 \\operatorname{Var}(y) + 2\\frac{\\partial f}{\\partial x}.\\frac{\\partial f}{\\partial y} \\operatorname{Cov}(x,y)", "4bd700f80169cf4e417f6898df65ef93": "r=s^2", "4bd7015a512e0cbae03bdcb1454f224d": "t \\le t_{0}", "4bd77a58c054b8672712514b8dcf9171": "( A+uv^T )^{-1}=( I+wv^T )^{-1}{A^{-1}}=\\left( I-\\frac{wv^T}{1+v^Tw} \\right)A^{-1}", "4bd78219b0c54dd9d1d4435f6302d8de": "it\\rightarrow \\tau", "4bd7bf932b0f56077a122914d02249f1": " \\frac{\\partial^2 u}{\\partial t^2} = c^2 \\left(\\frac{\\partial^2 u}{\\partial x^2}+\\frac{\\partial^2 u}{\\partial y^2}\\right) \\text{ for }(x, y) \\in \\Omega \\,", "4bd824a21c63348f82d29f88ae0dd3e3": "Q = -\\lambda dT/dz \\,", "4bd8412a44a384a838792e79b3c36199": "\\Box \\Box A", "4bd855f9443d79c30fa8638f12e935c0": "\\ MU_x/P_x=MU_y/P_y ", "4bd85775a4b524bdf2ed2341ae70fbd3": "E\\{x|y\\}", "4bd863841b1b0a68545a61e4bf9a4431": "x_n = \\cos(n\\pi/N)", "4bd86477f3055d4cfc594e9df5780749": "\\gamma^0 \\Gamma^\\dagger \\gamma^0", "4bd8994a8ffd86abaddde369434739f6": " m\\equiv 1\\bmod 4", "4bd9046f983c3122aa9151ebccac2b5b": "L:=\\{(p,0): p\\in \\mathbb R\\}", "4bd92ede64faa731f0191ffea948d311": "\\partial_n", "4bd95ebb2b7e00ce927044df4132f5f8": "E(\\lambda)", "4bd97f4d0801ccf22702340c99fc56dc": "2\\pi R \\sinh \\frac{r}{R} \\,.", "4bda0cb27c6e36b99d412cdc0ffad371": "\\mathbf A = A_x\\mathbf{\\hat x} + A_y\\mathbf{\\hat y} + A_z\\mathbf{\\hat z} \n = A_\\rho\\boldsymbol{\\hat \\rho} + A_\\theta\\boldsymbol{\\hat \\theta} + A_\\phi\\boldsymbol{\\hat \\phi}", "4bda11da5b2f02e12d5e07376fd593f6": "(\\operatorname{plus}\\ 1)", "4bda165a2162ad04047321a84138cc4e": " \\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\left(\\frac{\\partial f_1}{\\partial \\boldsymbol{S}}:\\boldsymbol{T}\\right)~f_2(\\boldsymbol{S}) + f_1(\\boldsymbol{S})~\\left(\\frac{\\partial f_2}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} \\right) ", "4bda91c20785dfd3fad20eb02f1da415": " \\alpha = \\theta_G - \\lambda_o - h", "4bdb6e094fd87d7ee0997144d0006aa3": "\n~\\sigma_t^2 = K + ~\\alpha ~\\epsilon_{t-1}^2 + ~\\beta ~\\sigma_{t-1}^2 + ~\\phi ~\\epsilon_{t-1}\n", "4bdb8ead9b8841c0bb89d6cba5c4cb9b": "\\alpha=\\{\\alpha_1,\\dots,\\alpha_n\\}\\in \\N_0^n,", "4bdbd7d0adbf48c53bf1453a6c040283": "x_1=a_1,\\ldots,x_k=a_k", "4bdbe515b9760664b124cd368f9c5a03": "\\phi(g)=\\delta_{g=e}", "4bdc27888874a635a6580af912722512": " R_{12}(\\lambda)R_{13}(\\mu)R_{23}(\\nu) = R_{23}(\\nu)R_{13}(\\mu)R_{12}(\\lambda)", "4bdc97d5e89e7b7abed67dad6112e3a8": "\\tan(3\\pi/16)", "4bdce33bbba06f1e6c75847951e16485": "|\\mathbf{p}|", "4bdd1e69e4fb5d13f727831b00ef5f8a": "\\alpha\\mapsto x_+^\\alpha", "4bdd3d53fc3228565d47970ff794ecbb": "\n a^3 = \\cfrac{3R}{4E^*}\\left(F + 6\\gamma\\pi R + \\sqrt{12\\gamma\\pi R F + (6\\gamma\\pi R)^2}\\right)\n ", "4bdd5893df765e6431af5f523510ea4c": "y = \\frac{1}{2}", "4bdd5b1065f5c9e3ae19c233bfd85f98": "y=\\pm\\sqrt{3}x.", "4bddaec04c4a66c599adb4a5a263b500": "J\\left(\\frac{\\gamma}{\\delta},\\frac{\\alpha}{\\beta}\\right) = \\begin{bmatrix}\n0 & -\\frac{\\beta \\gamma}{\\delta} \\\\\n\\frac{\\alpha \\delta}{\\beta} & 0 \\\\\n\\end{bmatrix}.", "4bddb843a868694a6b23f3ed72ed92e1": "\n\\frac{|x+y|}{2} = 0 ,\n", "4bdde4542c512e4833f1b2942a0417cf": "p^{n-i}", "4bddec02ba345a4f74998ee5e63dacf1": "\\theta \\in \\Theta_0", "4bde042aaaf1d54f004a68f230d0c786": "= \\frac {15}{8} \\sqrt{\\pi}\\,", "4bde2f9b74cf76e688ad8656b63b91d4": "M^{\\alpha\\beta} = X^\\alpha P^\\beta - X^\\beta P^\\alpha = 2 X^{[\\alpha} P^{\\beta]} ", "4bde6b7d4d3b59189f800d1c723943ac": "\\frac{1}{r} \\frac{d}{dr} \\left (r \\frac{d v_z}{dr} \\right )= \\frac{1}{\\eta} \\frac{dp}{dz} - \\frac{F_z}{\\eta}", "4bdea26a92dc2edfc7954fdc4875602d": " \\chi_T(t,0) \\simeq \\begin{cases}\n\t(t)^{-\\gamma}, & \\textrm{for} \\ t \\downarrow 0 \\\\\n\t(-t)^{-\\gamma'}, & \\textrm{for} \\ t \\uparrow 0 \\end{cases}\n\t ", "4bdf0194f1646fb2d54b91ab786cc460": "(\\lnot P) \\lor (\\lnot Q)", "4bdf1041cf97bf4ed88efce3fb3ab541": "t = 1 + 1 + \\cdots + 1", "4bdf3bb49ba05f91d7030b1daaadab98": "\\sum_{i=1}^\\ell \\binom{2m}{i}\\leq N-K+1", "4bdf4604db1929a90c4d66836b7f29b4": "2 \\log q", "4bdf702088539b0e95ba48b9e5275e89": "P=100\\,\\rm atm", "4bdf80405b131acd85f63801a75eb6ac": "H^1(M,\\mathbf{O})\\to H^1(M,\\mathbf{O}^*)\\to 2\\pi i H^2(M,\\mathbb{Z}) \\to H^2(M, \\mathbf{O}).", "4bdfe6d14e8b7a869519eb14b1976d0a": "e^x = \\sum^\\infty_{n=0} {x^n\\over n!} =1 + x + {x^2 \\over 2!} + {x^3 \\over 3!} + {x^4 \\over 4!}+\\cdots\\!, -\\infty0\\\\\nY_{-a-1,b}& \\text{if }a<0\n\\end{cases}", "4bf1aef0a0f9ff98d4286d7e24ca2929": "L(\\mathbf{r},i) = \\ln(r_i) ", "4bf1d2acde98c7a3567c7fe5939234cc": "\\pi^2 = 1\\,", "4bf29352cba5cad2ae3dceb07d571c56": "p_1(x) = 4x^3+3x^2-1", "4bf29d7e2ae82e823e8ddec94d4ae826": "\\scriptstyle{\\lambda_\\infty = 3.570}", "4bf2ddb0257cd8d89fa114986202c5a9": "\\bigg\\{ \\frac{\\Pr(h_1|h_2) = \\Pr(h_1 \\And h_2)}{\\Pr(h_2)} \\bigg\\}", "4bf2f60f916e30b92f1235e0f64a0968": "2~\\mu~u_\\theta\\, ", "4bf2fabf6191fa36f442f6d71802224c": "H^*(X)", "4bf30933068847924c4e7146c01db9fb": " \\partial\\mathbf{A} = -\\mathbf{F},", "4bf347253c45a5821bf0058bd1547de0": "\\omega^i{}_{k\\ell}=\\frac{1}{2}\\eta^{im} \\left(\nc_{mk\\ell}+c_{m\\ell k} - c_{k\\ell m} \n\\right)\\,", "4bf35665e383a40a21ca3bc3d71e5aba": "\\{0,1\\}^k\\to\\{0,1\\}", "4bf35667756d57eb9a464f572a9ac9b5": "L=\\limsup_{n\\to\\infty} \\frac{\\log\\vert\\sum_{k=1}^n a_k\\vert}{|\\omega_n|}", "4bf374bd85d8a053253782df8981ac5f": "= \\sum_{j} (v_j^{T} x)^2\\lambda_j\\le\\lambda_n \\sum_{j} (v_j^{T} x)^2", "4bf37ea7901b5a87b8dc7bb4ec8f26ad": "R(N)", "4bf3d43d963b3e9c49bc7ccc19fffc68": "\n \\tau_{yz} = T(k, z, \\omega)\\,\\exp[i(k x - \\omega t)] \n ", "4bf463030bbbdf5e8d37dc555f8e3dee": "\\sigma \\dot{\\sigma} < 0", "4bf4806ae498ee25927107928d82d445": " \\left(y - \\frac{h(x)}{2} \\right)^2 + h(x) \\left(y - \\frac{h(x)}{2}\\right) = f(x)", "4bf48ffb0899881f862883592534b875": "\\qquad + \\ V^{\\alpha}_{i_{1}i_{2}}(x,u,w) \\frac{\\partial}{\\partial w^{\\alpha}_{i_{1}i_{2}}} + \\cdots \\ + \\ \\cdots + V^{\\alpha}_{i_{1}i_{2} \\cdots i_{r}}(x,u,w) \\frac{\\partial}{\\partial w^{\\alpha}_{i_{1}i_{2} \\cdots i_{r}}}\\,", "4bf4e4d3ee450b7cbe1acaf19a4dc495": " f =45 Hz", "4bf50470733835dffb12f2d510f5b580": "\\frac{|00\\rangle+|11\\rangle}{\\sqrt{2}}", "4bf50db47ad117bd2967a8b7e87b6a0b": "(k+g){\\partial\\over \\partial z_i} \\Phi(v_i,z_i) = \\lim_{z\\rightarrow z_i} \\left[\\sum_s X_s(z)\\Phi(X_sv_i,z_i) -(z-z_i)^{-1}\\Phi(\\sum_s X_s^2 v_i,z_i)\\right].", "4bf512b0de21e34d9b6488d2d17f6fe6": "\\alpha < \\omega t < 2 \\pi - \\alpha : I(\\omega t) = I_{tcr-max} \\sqrt{2} [cos(\\alpha)-cos(\\omega t)]", "4bf513ec4ca731dfb0e8ff32ddbdbdbb": "r_{t+1} \\pi_{t+1}", "4bf51c3f2ca031db6b05ad1ec3adb193": "M_{c/4}", "4bf533c29c6ccf79ef8edd5377308b3e": "\\operatorname{N}(A)=\\operatorname{Null}(A)=\\operatorname{ker}(A) = \\left\\{ \\mathbf{x}\\in K^n : A\\mathbf{x} = \\mathbf{0} \\right\\},", "4bf573b06751e99b64439669bdc83aa1": "\n \\begin{align}\n x_U &= x - y_t\\, \\sin \\theta, \\qquad &\n y_U &= y_c + y_t\\, \\cos \\theta, \\\\\n x_L &= x + y_t\\, \\sin \\theta, &\n y_L &= y_c - y_t\\, \\cos \\theta,\n \\end{align}\n", "4bf5e6e7f38061ce5bb452465c966eed": "c_j", "4bf5ef87fccca1f08b7d0973f0dc0b4d": " \\begin{align}\n& \\text{maximize} && \\mathbf{c}^\\mathrm{T} \\mathbf{x}\\\\\n& \\text{subject to} && A \\mathbf{x} = \\mathbf{b}, \\\\\n& && \\mathbf{x} \\ge \\mathbf{0}, \\\\\n& \\text{and} && \\mathbf{x} \\in \\mathbb{Z},\n\\end{align} ", "4bf5f2a151c883d30c44433a84617b7b": "{\\rm Ci}(x)= \\gamma+\\ln x+\\sum_{n=1}^{\\infty}\\frac{(-1)^{n}x^{2n}}{2n(2n)!}=\\gamma+\\ln x-\\frac{x^2}{2!\\cdot2}+\\frac{x^4}{4! \\cdot4}\\mp\\cdots", "4bf608761bd939e2d68f3935f538cccd": "\\frac{1}{1-\\lambda}", "4bf6196bb3d082ff3095d1bbd18458dc": "\\cos(x-y)=\\cos x\\cos y - \\sin x\\sin y", "4bf63bd9acf4da201df2bcdb149ff0cb": "E_{\\pm}(n) = \\hbar\\omega_c \\left(n+\\frac{1}{2}\\right) \\pm \\frac{1}{2} \\hbar\\Omega_n(\\delta),", "4bf687f05bd455c7fd64c1b8ccd8d1cc": "\\begin{align}\n H &= H_i + \\frac{100 (h^\\prime - h_i) / e_i}{(h^\\prime - h_i) / e_i + (h_{i+1} - h^\\prime) / e_{i+1}} \\\\\n e_t &= \\textstyle{\\frac{1}{4}} \\left[ \\cos\\left( \\textstyle{\\frac{\\pi}{180}}h + 2\\right) + 3.8 \\right]\n\\end{align}", "4bf68f3e7721e7410ce18dfe9f1ddb8d": "|c_{k'}(t)|^2", "4bf6ab0e704649620e08eaa0b5414d31": "\\sec", "4bf6f6eb50d297ad6dee65e3d03f75a5": "f(X) = \\sum_{n=0}^\\infty a_n X^n ", "4bf7199256f2a92c6e4420f666c52b25": "x_0^2-x_1^2-\\cdots-x_n^2 = r^2,\\quad x_0>0.", "4bf754ad16ab5daa439c4c36c0206352": "x^2+y^2-z^2=0", "4bf767f70a2b7392171755a9fc253523": "a \\in A \\cup \\{\\varepsilon\\}", "4bf781bf7f1cace4adb602b231532a18": "\n \\frac{d \\bar{\\lambda}}{dt} =\n - \\frac{1}{\\tau} \\left(\\bar{\\lambda} -\n \\frac{\\tilde{\\nu} + \\mu}{\\tilde{\\delta} + \\mu} \\right),\n \\quad\n \\frac{d \\bar{\\varepsilon}}{dt} =\n - \\frac{1}{\\tau} \\left( \\bar{\\varepsilon } -\n\\frac{\\tilde{\\eta} + \\mu}{\\tilde{\\delta}+ \\mu} \\right),\n", "4bf78b746c6427bf6ab80919b08cb94b": " M^-_\\infty = M^+_\\infty = +\\infty, ", "4bf7d37a8e58114c6370668345681b07": "\\mathbf{Gr}(r,m)", "4bf7dfc870b560ce6c7b8a48886f4ea6": "\\scriptstyle \\mu_2/\\bar{\\Gamma}^2", "4bf88664077273de841eaffb68d2e8da": "\\gamma v t' = \\gamma^2 v t + x \\left ( 1 - \\gamma^2 \\right )", "4bf8a1ac59cd4d582714e5a3d1be99e2": "dg_A", "4bf8b00ecaec552351361a2cea2434e6": "wu=wv", "4bf969ff491e88695514997f1015bab7": "f^2 = {R^2_{AB} - R^2_A \\over 1 - R^2_{AB}}", "4bf9b62fdfb422d24c6ff7786c6b8254": "0+0", "4bf9fb1e3d864dacb373b7eb1cd35512": "\\displaystyle L'C'={\\varepsilon \\mu}.", "4bfa1e2829ab6f9f9b78dbe6eef15f48": "2^{2 + 1} + 2^2", "4bfa1ff3927da8d58f05918528a9de68": "A_+ = \\left \\{ \\begin{pmatrix} e^{-y} & 0 \\\\ 0 & e^y \\end{pmatrix} \\ : \\ y > 0 \\right \\}.", "4bfa2ae5e26e3c9a6a379602fb6495f8": "R_{TE}+R_r^'\\ll{R_{TE}+X_r^'}", "4bfa85e43248aa35f9841f57e90889b8": " p_{eq} ", "4bfa8b70276015229aebe08ab6d13561": " E_i G_{i\\pm1} E_i =l E_i. ", "4bfac13e66716f6c3c17fbea92a1aeb0": "\\cos(0.01) = 0.99995", "4bfaef98eb06da2f451ba762b54ac273": "\\lambda_s(4)=6s-2+(s\\, \\bmod\\, 2).", "4bfafa82181d498cc7b23b8100eb80bc": "0.\\underbrace{999\\ldots9 }_H \\, ", "4bfb2431f1b812fbd0283c1c95c6401a": "\\rho = \\psi(\\Omega) = \\zeta_0", "4bfb63cd8e5c4526ba86ab9cd1762a1e": "\\frac{\\tan{A}+\\tan{B}+\\tan{C}+\\tan{D}}{\\cot{A}+\\cot{B}+\\cot{C}+\\cot{D}}=\\tan{A}\\tan{B}\\tan{C}\\tan{D}.", "4bfbf8d77e0126fcfc010f462c1376a3": "\\scriptstyle\\mathbf{UDV}^H \\,=\\, \\mathbf{H}", "4bfc08a18b029815a701628080aa7fe2": "H_{\\text{loc}}", "4bfc12604ae4bb0f3dcf141a7d0be158": "\\textstyle N-2", "4bfcb9952f5248e4ed4d7da25d2499d4": "A = N^{1/2} \\operatorname{diag}(a) M^{1/2}", "4bfcd6deea4143eee991088d71c22a36": "\\pi = 4\\arctan\\frac{1}{2} + 4\\arctan\\frac{1}{3}\\!", "4bfcf6fb61389e725eac90d95913770b": "H_{\\mathrm{ZOH}}(s)\\, = \\mathcal{L} \\{ h_{\\mathrm{ZOH}}(t) \\} \\,= \\frac{1 - e^{-sT}}{sT} \\ ", "4bfda7a5e193316ff354c9389edfa33a": "b = k2^{-j}", "4bfdf0f85e0df1eaaee5d4af568e17bf": "\n\\ H(e^{j \\omega}) = \\left[1 + \\alpha \\cos(\\omega K)\\right] - j \\alpha \\sin(\\omega K) \\,\n", "4bfe51105b36eba44ab0943603837ef5": "\n\\begin{matrix}\nI & \\ge & 0, \\\\\nV & \\in & \\mathbb{R}, \\\\\nL & \\in & \\mathbb{C}, \\\\\n\\end{matrix}\n", "4bfe5a7aa412c40d5699eff5d0db7289": "\\frac{\\partial \\bar{V^E}_i}{\\partial T} = R \\frac{\\partial (ln(\\gamma_i))}{\\partial P} +RT {\\partial^2\\over\\partial T\\partial P} ln(\\gamma_i)", "4bfe793d936ff36af53bc73a7f0d0a74": "(x^2-(2+\\alpha)x+1+\\sqrt{7}+\\alpha)(x^2-(2-\\alpha)x+1+\\sqrt{7}-\\alpha),", "4bfec339a9a595c62df690432f751a7e": "\n\\begin{align}\ne^{J} & {} = \\exp \\big( J_{a_1}(\\lambda_1)\\oplus J_{a_2}(\\lambda_2)\\oplus\\cdots\\oplus J_{a_n}(\\lambda_n) \\big) \\\\\n& {} = \\exp \\big( J_{a_1}(\\lambda_1) \\big) \\oplus \\exp \\big( J_{a_2}(\\lambda_2) \\big) \\oplus\\cdots\\oplus \\exp \\big( J_{a_k}(\\lambda_k) \\big).\n\\end{align}\n", "4bfee77b8ea05010ae30e7a05f7ca819": "\\log_{10}(d) = 1 + \\frac{\\mu}{5}", "4bff064360f94c8834ae1c9e2b0cd6dc": " \\begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 2 & 0 \\\\ 0 & 0 & 5 & 0 & 0 \\\\ 0 & 7 & 0 & 0 & 0 \\\\ -1 & 0 & 0 & 0 & 0 \\end{bmatrix}. ", "4bff0a6f8103ebeb9101448e2be645b6": "E^2/2", "4bff5adbdb98fe36b6367827908275ab": "A_i\\cap A_j=\\emptyset", "4bff8cde174310c3073a9605d2b35c50": "1\\cdot(p-1)\\cdot 2\\cdot (p-2)\\cdots m\\cdot (p-m)\\ \\equiv\\ 1\\cdot (-1)\\cdot 2\\cdot (-2)\\cdots m\\cdot (-m)\\ \\equiv\\ -1 \\pmod{p}.", "4bffd0a48d95467091b1d70dca77f866": "V = \\frac{1}{\\alpha} \\times \\rho \\times L^2 \\times (G - G^P)", "4c00169dcaf25edea954de7b20896191": "\\tbinom{6}{5}", "4c00328f3188119358361054292b6014": "\\beta_2 = \\frac{-1 + \\sqrt{-3 -4c}}{2}.", "4c004e4ef379da6a60f2bc4d95b3c9f0": "\\{x\\leq10\\}\\; \\mathbf{while}\\ (x<10)\\ x := x+1\\;\\{x=10\\}", "4c005ce9becc5db1342d2eb1e1e73c04": "a + 0 = a,\\,", "4c00de4acda59973500eed4e0e151c97": "\\mathsf{PP} \\not \\subseteq \\mathsf{SIZE}(n^k)", "4c0140242d9dc63ddb2e7712bb697b4c": "((x,\\emptyset),x)", "4c017af23b7bb8117e568b3c9ff8898d": "\\Delta n\\, ", "4c018a1a30313a30c7a6c2a99891ada3": "c'_n(t)", "4c01c0bbce297ed18994b614ce623516": " V[\\varphi] = \\frac{1}{2}\\iint_D \\nabla \\varphi \\cdot \\nabla \\varphi \\, dx\\, dy.\\,", "4c02129e9b88da3c9f5039c2fea860ec": "L_p\\left[\\frac{1}{2},\\sqrt{2}\\right]", "4c025f264775b78c5562d27a37347f70": "[Z,X,Y]=1", "4c0275b77b1164f13b463457d127cf00": "[X,G/PL]", "4c02eb3061ff37083d54b62f806d0bae": "k\\geq l", "4c03c2084f36f9bfcf5b6b314fdd6f14": "\n\\begin{align}\n(x_N * h)[n] \\ &\\stackrel{\\mathrm{def}}{=} \\ \\sum_{m=-\\infty}^\\infty h[m] \\cdot x_N[n-m] \\\\\n&= \\sum_{m=-\\infty}^\\infty \\left( h[m] \\cdot \\sum_{k=-\\infty}^\\infty x[n -m -kN] \\right).\n\\end{align}\n", "4c03f17d41f88ab2a3bcf65ec15536dc": "\\forall x_1\\dots\\forall x_n(\\phi(x_1,\\dots,x_n)\\leftrightarrow \\psi(x_1,\\dots,x_n))", "4c041be8b5990646f35f73ff3715a2bc": " c(n,k)=\\left[{n \\atop k}\\right]=|s(n,k)|\\,", "4c0498a73c3a03084666ea82fb69a8cc": "\\scriptstyle{r}", "4c04df18c056ac7cfee23bcde048bf91": "r_k = \\sqrt{E_s}e^{j\\phi_k} + n_k", "4c05484037b8bfa844f0a336727bbed6": "z^5", "4c05a9308576d9c90f13803ef53e54b6": "\n\\begin{align}\nu_E&=\\pm V_0 \\cos\\left(\\frac{\\pi}{4} + \\frac{\\pi}{D_E}z\\right)\\exp\\left(\\frac{\\pi}{D_E}z\\right),\\\\\nv_E&= V_0 \\sin\\left(\\frac{\\pi}{4} + \\frac{\\pi}{D_E}z\\right)\\exp\\left(\\frac{\\pi}{D_E}z\\right),\\end{align}\n", "4c05ea3396d24b5ff388bf7871280459": "c=\\alpha^2", "4c05fe4ad9b0d134fdbd7fffe2eaf6cf": "\\|f_N - f\\|_2", "4c0660cd2b0cd4a4cc1a06b5ddee65e9": "(13)\\qquad \\theta_{(l)}=\\hat{h}^{ba}\\nabla_a l_b=m^b\\bar m^a\\nabla_a l_b+\\bar m^b m^a\\nabla_a l_b =m^b\\bar \\delta l_b+\\bar m^b \\delta l_b=-(\\rho+\\bar\\rho)\\,,", "4c06d5136c1fbf711b01d03cdf9aa848": "", "4c06e138cb016d31b0091d74a2bac4d2": "T=v_1\\otimes w_1\\otimes \\alpha_1 + v_2\\otimes w_2\\otimes \\alpha_2 +\\cdots + v_N\\otimes w_N\\otimes \\alpha_N.", "4c07313cd1f452b3d1e159d3f4086a03": "\\Pi\\sigma_{Reactants}", "4c074a9ef43287193da95b879c763a4c": "\\rho^\\pi = E[R|\\pi],", "4c076e4d8a57f190d21e878463db4329": "\\frac{D \\zeta}{D t} = -v \\beta,", "4c07c860e9257951216ac71452f8d4e5": "\\oplus_p", "4c07cb3726750ce3cc7fe45ffc879f37": "\\gamma: [0, 1] \\to M", "4c0804081296b716432bf09fd924c2ec": " B_{12} = B_{21}", "4c087e270d7905bd0203764fcc9aa02b": "\\scriptstyle (1.9\\pm2.1)\\times10^{-5}", "4c08bee63e249ccb9e51fe7e8cbeac24": "dX(t)=\\Delta(t) \\ dS(t) + r(t)(X(t)-\\Delta(t)S(t)) \\ dt", "4c08ca313fe07b791593debda34cf555": "{n \\choose m}", "4c08da76b1b945f2cdb6f3ea15c519d2": "a_{22}=0.1", "4c0904b930390705b074cfe3ccdddb38": "H' = \\epsilon^{abc} \\epsilon_{ijk} \\{ A_a^i , K \\} \\{ A_b^j , K \\} \\{ A_c^k , V \\}", "4c091b74614dd5f964cb8ee77dacb3db": "d = \\gcd(x, MN - 1)", "4c0978749afffbd42aa41a4fdddb3841": "I_n(\\kappa)=\\frac{1}{\\pi}\\int_0^\\pi e^{\\kappa\\cos(x)}\\cos(nx)\\,dx.", "4c0979e0e99f4742ee48201ae52bcbee": "i \\in S \\text{ and } j \\notin S", "4c09868f86ee39ef30cdb86d9c8c0fba": "\\mathbf{E} ( \\vert X_n \\vert )< \\infty ", "4c0998a8685f9285db91dfa4c95b0a0e": "{\\mathbf{j}}_s", "4c09dea442e51ad8175f5e5a1ed186bd": "\\times \\omega", "4c0a3b024df4f14d0af78ac0c7603c28": "K_d = \\cfrac{[A^+ ] [B^- ]}{[AB]} = \\cfrac{(\\alpha c_0 )(\\alpha c_0 )}{(1-\\alpha) c_0 } = \\cfrac{\\alpha^2}{1-\\alpha} \\cdot c_0 ", "4c0a454ceee5e78c6bf902c8379288cf": " g ", "4c0aef83b2ce2fcaa507ae64f47cd018": "H_{DR}^{n-i}(M)", "4c0baaf60cb2e234d0e7db916e1f4ae5": "{}_2F_1(a,1-a;c;z) = \\Gamma(c)z^{\\tfrac{1-c}{2}}(1-z)^{\\tfrac{c-1}{2}}P_{-a}^{1-c}(1-2z)", "4c0bbbd45fc403fd70e33de4fa753fd2": "\\nabla : \\Gamma(E) \\to \\Gamma(E\\otimes T^*M)", "4c0bf7fa0e72856c46eeef134d925adf": "\\Delta\\gamma=0", "4c0c37594b89ed9488e920aa8714c16f": "P_\\text{loss} = R_\\text{i} \\cdot I^2", "4c0c4c8286856fa954072c6576befa86": "\\pi\\circ D_p=p.", "4c0c5cb7486738b9919c8768838c0c2f": "1-\\left( (1-u)^\\theta + (1-v)^\\theta - (1-u)^\\theta(1-v)^\\theta \\right)^{1/\\theta}", "4c0c670d431133c9a817724820a1b9db": "\n\\left[a_i , a_j \\right] = 0 \\quad,\\quad\n\\left[a_i^\\dagger , a_j^\\dagger \\right] = 0 \\quad,\\quad\n\\left[a_i , a_j^\\dagger \\right] = \\delta_{ij},\n", "4c0c82f492d60269448ba099206e2f28": "S(\\mathbf{X})", "4c0ccfe5b7700bc4df80758eedb9cffc": "\\Delta G = \\Delta G^0 - (T-T^0)\\Delta S^0 ", "4c0cf670b6aa7417daa08112b5b372d0": "\n\\mbox{cost}(x,c) = \\sum_{i=1}^d | x_{i} - c_{i} |\n", "4c0d94c678fc1f5fc81e3b23c5ba6154": "p(F_i \\vert C, F_j,F_k) = p(F_i \\vert C)\\,", "4c0e0103cf5b4bc3806c0bef294c4f78": " i <_\\mathcal{O} 2^i ", "4c0e4283a8748249efbf9115017e2f45": "D_1 = \\sum_{P \\in E}{c_P [P]}", "4c0e49ca972c3dace31ee96180ffad15": "\\ C _{ik}", "4c0e49e7694c0277451c53b32f174085": "Lu(x)=\\sum_{i,j=1}^n a_{ij} (x) u_{x_i x_j} + \\sum_{i=1}^n \\tilde b_i u_{x_i}(x) + c(x) u(x)", "4c0eb3066437d72c3432eba8100b4d3b": " \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} ", "4c0ef632c37519ce1e4884c67e2425a5": "n_x \\phi_y - n_y \\phi_x,\\quad n_y \\phi_z - n_z \\phi_y,\\quad n_z \\phi_x - n_x \\phi_z", "4c0f0c41fb22fe112c8989221bf68e86": "\n\\begin{align}\n\\alpha_{ij}&=\\sigma_{ij}\\sigma_{i,j+1}\\\\\n\\mu_{ij}&=\\sigma_{ij}\\sigma_{i+1,j}.\n\\end{align}\n", "4c0f0e7868801dc149ef1e87e3093db9": "g_3", "4c0f295ed1c984f0570572346167dbe6": " V^2 = \\frac{1}{288} \\det \\begin{bmatrix} \n 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 & 1 \\\\\nd_{21}^2 & 0 & d_{23}^2 & d_{24}^2 & 1 \\\\\nd_{31}^2 & d_{32}^2 & 0 & d_{34}^2 & 1 \\\\\nd_{41}^2 & d_{42}^2 & d_{43}^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 1 & 0\n\\end{bmatrix} ", "4c0f647ae3aa4f3341ab4ebb012a4848": " {d^2 x^\\mu \\over ds^2} =- \\Gamma^\\mu {}_{\\alpha \\beta}{d x^\\alpha \\over ds}{d x^\\beta \\over ds}\\ +{q \\over m} {F^{\\mu \\beta}} {d x^\\alpha \\over ds}{g_{\\alpha \\beta}}.", "4c0faa02a7654f56b8edf68be4864859": "e^{-{2\\pi i \\over N}}", "4c100ffcb9996d2d7a0d1661072eaf30": "\\mathcal A \\models_X \\vec{t_1} \\mid \\vec{t_2}", "4c104af834b2fb08ce76f28119b41f34": " \\frac{z^{-1} }{ (1 - z^{-1})^2 }", "4c1054ac7fdbb33a67fc8a051641c357": "\\text{true} \\rightarrow \\text{open}_2", "4c1055a7167a4a92b75611d1dd063a33": "a(n) = \\left\\lfloor A^{3^n}\\right\\rfloor", "4c1078fa29ea9c0a0e9e34ef9c9baddf": "p_M, p_{M-1},\\dots,p_{M-(n-1)}", "4c10dd06b38286de292a01f6530fdbb2": "G = \\int_0^1 \\int_0^1 \\frac{1}{1+x^2 y^2} \\,dx\\, dy \\!", "4c11890a8a0e5d421ed87cf8ccb92450": "kr\\rightarrow \\infty ", "4c11c584b594df50bd09effb0f8ea498": "t_{N}=1", "4c11e771dab4f7aba15f0edbc2b4f9d6": "\\epsilon-", "4c120db448ab828e33189252bf872573": "x=2ab,", "4c12238cda75cd88f22a1ae66dba5877": " f(\\cdot)\\,", "4c124235260feb14d1ab893196bb48cc": "\n\\hat{H} = \\tfrac{1}{2}\\left[ \\frac{\\mathcal{P}_x^2}{I_1}+ \\frac{\\mathcal{P}_y^2}{I_2}+\n\\frac{\\mathcal{P}_z^2}{I_3} \\right].\n", "4c124392eb78494851d81c1a2303c4e9": "\\|O_1-O_2\\|", "4c125d62d286c6906fc914126370a490": "a_1 \\circ a_2", "4c130a23bb7fe556c589c816771c12ca": "m_G(x) := \\sum_{k\\geq0} m_k x^k.", "4c130be80c22a131149d2e01ae101771": " \\mathbf{V} = \\lim_{\\Delta t\\rightarrow0}\\frac{\\Delta\\mathbf{P}}{\\Delta t} = \\frac {d \\mathbf{P}}{d t}=\\dot{\\mathbf{P}} = \\dot{x}_p\\vec{i}+\\dot{y}_P\\vec{j}+\\dot{z}_P\\vec{k}.", "4c135e9ee450b29456f68624b0e36914": "(x_n)_m=\\begin{cases}\n 0 & m < n \\\\\n 1 & m \\ge n \n\\end{cases}", "4c13678b1009ff27177e42d5ab9b6afd": "P=\\rho gh", "4c13ac6a5c2776d63bfdcde4cca9b378": "A \\simeq J_1 \\times J_2/J_1 \\times J_3/J_2 \\times ... \\times J_n/ J_{n-1} \\times A / J_n ", "4c146d8a4317adb38e271b8906111b1d": "K\\cdot j+i ", "4c147659bf25cfb7fdaf3e3b415066ce": "T^0X", "4c148a67b44e7142ed9c4aee95266854": "\\mathbf{M} =\\mathbf{L}\\mathbf{S} = \\begin{pmatrix} 1 & d \\\\ \\frac{-1}{f} & 1-\\frac{d}{f} \\end{pmatrix} ", "4c14a91d1b3afbc30def021b35fffd3e": "Amplitube = MF \\left ( \\frac{F_{dynamic}}{2k} \\right ) ", "4c14c583818213b6b453ffa3d012a49e": "d_p(f,g) = N_p(f-g) = \\|f - g\\|_p^p", "4c14d3afd488f10d39ccf2f57cb12cbe": "\n\\Pr \\left\\{ \\bigg\\Vert \\sum_k \\xi_k \\mathbf{B}_k \\bigg\\Vert \\geq t \\right\\} \\leq (d_1+d_2) \\cdot e^{-t^2/2\\sigma^2}.\n", "4c14e0f763fa464ca1d8d45c20b6343b": "~~~h_d ^d = \\sqrt{\\tfrac{2}{d(d-1)}} \\left ( h_1 ^{d-1} \\oplus (1-d)\\right )~.", "4c14f22c841d5a464aa147854b21cca7": "(S_N f)(t)", "4c15aac97648ed37262023c747eda9af": "\\Omega(E)", "4c15adf98bdb3b06a50caccf921ac1a9": "\\sqrt{\\frac{12}{35}}\\!\\,", "4c15ef2dd7e24c57cf0b03e99809e65a": "m_1 =11", "4c1610e9ddf3ef2eed1f6cc10f3741ee": "2^d", "4c16da274998815f975a6bb1ea105677": " a^2 x^2+c=y^2,", "4c16dd705ad24a0460858dec50774500": "T = G \\circ F", "4c173deadaca9d696d744f13de4ee6fe": "\\Gamma\\left (-2.5 \\right ) = (-3.5)! = \\Pi\\left (-3.5\\right ) = {2\\over -1}\\cdot{2\\over -3}\\cdot{2\\over -5} \\sqrt{\\pi} = {(-4)^3 3! \\over 6!} \\sqrt{\\pi} = -{8 \\over 15} \\sqrt{\\pi} \\approx -0.9453.", "4c173e52ba208adf908d7375f7dac5cb": "\\omega_\\alpha^{\\;\\;\\; IJ}", "4c174fef2db3cdf3804677ba9db04405": "f_k \\in FC_k", "4c17a09931c494ed2431dba82ebb4923": "\\mathfrak{g}_2^{\\mathbb C}\\times\\mathfrak{sl}(2,\\mathbb C),\\mathfrak{sl}(2,\\mathbb C)\\times\\mathfrak{so}(3,\\mathbb C)", "4c17af0d81ab2abf13261cf7ee568849": "\\text{U}_\\text{L}", "4c17c26a4ce7b5a656edb59ad69d2221": " \n\\frac{\\partial \\textbf{A} }{\\partial z } = \n~-~ \\beta_1 \\frac{\\partial \\textbf{A} }{\\partial t}\n~-~ \\frac{i}{2} \\beta_2 \\frac{\\partial^2 \\textbf{A} }{\\partial t^2}\n~+~ \\frac{1}{6} \\beta_3 \\frac{\\partial^3 \\textbf{A} }{\\partial t^3}\n~+~ \\gamma_x \\frac{\\partial \\textbf{A} }{\\partial x} \n~+~ \\gamma_y \\frac{\\partial \\textbf{A} }{\\partial y}\n", "4c17c5b6bbdd630187713060a031d8dd": "\\mathbf{\\nabla} \\cdot", "4c184c4518fcc47869de5cb78e5fdd33": "a^{-2}+b^{-2}=d^{-2}", "4c18991cb0b32fd049b8b56283b946e7": "\\textstyle \\nu ", "4c18d6286e7381ca754646f911811b72": "\n \\mathbf{x}^{(2)} =\\mathbf{x}^{(1)} P \n = \\begin{bmatrix}\n 0.9 & 0.1\n \\end{bmatrix}\n \\begin{bmatrix}\n 0.9 & 0.1 \\\\\n 0.5 & 0.5\n \\end{bmatrix}\n \n = \\begin{bmatrix}\n 0.86 & 0.14\n \\end{bmatrix} \n", "4c18fc3b3d4c3c78a262033ee1d6c088": "A=D+L+U, ", "4c18ff002a71c9ab1226abcc8645eff8": "W_{Le} = \\frac{L_eI_e^2}{2} = \\frac{\\Phi_e^2}{2L_e} = 2\\pi \\alpha W_0. \\ ", "4c190768049b322fac30b6f6ed59e96f": " u_e ", "4c1926dcf0d01637efb440cd4d168ef7": "A = 8\\sqrt{2}a^2 \\approx 11.3137085a^2", "4c19381dd1a295b392b1aca8f45882fe": "K_R = 0.299", "4c194e2fac91c659d95f658dc989f9fc": "p_{i}, q_{i}", "4c19ab902908615f2d4eed816e484128": "\\begin{bmatrix} \\dfrac{e^{\\eta_1}}{1+\\sum_{i=1}^{k-1}e^{\\eta_i}} \\\\[10pt] \\vdots \\\\[5pt] \\dfrac{e^{\\eta_{k-1}}}{1+\\sum_{i=1}^{k-1}e^{\\eta_i}} \\\\[15pt] \\dfrac{1}{1+\\sum_{i=1}^{k-1}e^{\\eta_i}} \\end{bmatrix}", "4c19ef9f713cf4cb8d1a9198135e683b": " b = \\frac{ g^2 + 1 }{ k + \\frac{ 3( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } } ", "4c19f31e0bc5d78ea86bf35874566cc3": "| n\\rangle", "4c1ab209e7822b9141267f4cda85101b": "(L/D)", "4c1ad246c0055eece1d4b96dce1a21b6": "4t^2+4t+1=8s^2+1 \\,", "4c1b1fb9bf8369ad60d0964e78a5643f": "g_{ab}\\rightarrow\\Omega^2(x)g_{ab}", "4c1b9c883e729d1b9e4bda1b1dc08aba": "E(x^k)=\\lambda^k,E(\\ln(x))=\\ln(\\lambda)-\\frac{\\gamma_E}{k}\\,", "4c1bc5217d7666230c072643d5822efa": "f : \\mathbb N\\to\\mathbb N", "4c1bca36b7f5295977772e20f2fa6013": "2^{2^i}, d", "4c1bd2d9c7b76e1218fd6fb3e6304ff7": "\\lambda(\\pi(u)v)=\\chi(u)\\lambda(v)", "4c1bf8d4726428bbe77257f4d6c6c1d2": "i_n", "4c1c0f3e2aac9c91981118af4c9ea4eb": "X = \\text{Spec} L[x_1, \\dots, x_n]/(f_1,\\dots,f_m)", "4c1ca8bcff6a058f2d08730555f713c3": " x(t) = A \\sin \\left(t \\sqrt{\\frac{k}{m}} \\right) + B \\cos \\left(t \\sqrt{\\frac{k}{m}} \\right). \\, ", "4c1cb4ad6e5104716316d3acb3e1ae02": "\\ t=0", "4c1d4e99bdfee3dddeeaa1e5d697ec8e": "\\lambda(x) = \\frac{a+b}{2} + \\frac{b-a}{\\pi}\\tan^{-1} x.", "4c1d52fafe16a3fe3471cd37d709112d": "N_n = M_n \\cap N", "4c1d98c472df395169864ad8a8bef1e5": "H^{\\otimes n}", "4c1ddf244d6d4a9e8ecd7b9b1db4bde2": "ab\\le \\Phi(a) + \\Psi(b).", "4c1e00c83479ac801b0c2358bc68fc11": "\\mathbf{z}_0 \\in\\mathbb{C}^n", "4c1e0c8e05159875a88db86fb030377e": "\\Delta l/\\Delta t = A' \\cdot e^{-\\Delta H/ ( R \\cdot T )}", "4c1e44dbd5ebe734a792b73e37608857": " (3:0:0:0) + (0:0:1:1) = (3:0:0:0) \\ ", "4c1e89c62c37263483c50b84801fc71e": " \\Gamma_\\text{chir}= \\alpha_{d+2} \\Gamma_0 \\Gamma_1 \\dots \\Gamma_{d-1} = \\gamma_\\text{chir} \\otimes \\sigma_3\n~~~~ \\alpha_d= i^{d/2-1}", "4c1e9bd0827b1ccc5dbb850be473ce11": " \\mathbf{A} = \n \\left( \\begin{array}{rrrr}\n a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m1} & a_{m2} & \\cdots & a_{mn}\n \\end{array} \\right).\n", "4c1ed562b08dd3844b0edce1d27ab479": "I_{\\perp}", "4c1f1b88e6f1743be40bb0b43d88f771": "\\lambda = \\iota\\kappa.\\,\\!", "4c1f252fd8a10bd116e15b330b00b585": "\\mathbb{Q}[y]/q_i(y)=K[x]/p_i(x)", "4c1f260c911d9f61df2bf5e96eb355df": "\\Omega_2 \\subseteq \\{ \\land, \\lor, \\rightarrow, \\leftrightarrow \\} \\,", "4c1f7a5153dd5dd73ae2261b9316afcf": "f_1(x_1)", "4c1fc9fc7971c3d9ad0ab494110041ad": "\\vec a = R \\left( \\frac {d \\omega}{dt}\\ \\hat u_\\theta + \\omega \\ \\frac {d \\hat u_\\theta}{dt} \\right)", "4c200bd1c05f12ec1a175a59859d0c0c": "G = t ", "4c205720fc2512917e17c1b7d41ede10": " u v \\, \\partial_u + \\frac{1-u^2+v^2}{2} \\partial_v ", "4c207023d5bff42b4451351cc313d2fe": "\\scriptstyle E^{0}_n", "4c207147da5498bdb6fc2d7c840910ff": "E_\\mathrm{k} = \\int \\mathbf{F} \\cdot d \\mathbf{x} = \\int \\mathbf{v} \\cdot d \\mathbf{p}", "4c20b0f43a2d5cbc83a88d86375626e9": "(y_0,v_0)", "4c20ccd706e229a9eb81524c48edcc6b": "\\mu_4 = k_4+3k_2^2 = a_1 +16a_2 + 3 (a_1+4a_2)^2", "4c20e1e0b1c6adc6d2e594329d6da34f": "d(w) = \\frac{d(w0) + d(w1)}{2}", "4c20ff91a3159d99de7f6672c50dbf6b": "x^{-4} \\cdot 2^3 = \\frac{8}{5}", "4c21075e9a31754c394310f0b9c79462": "\\scriptstyle\\R", "4c215a998099a630d32d282aef65a20b": "\\Omega^n(M)", "4c21c256007025614acccd151a2bc323": "\n \\hat f(\\nu)\n = \\int\\limits_{-\\infty}^\\infty f(t) e^{-2\\pi i\\nu t}\\,dt.\n", "4c21f33fa06b6ae485fb581cad3ba5aa": " V_1 ", "4c2254c483e6a3c2ee3494f68d6cc4f6": "\\mathfrak{so}(5,1)\\cong \\mathfrak{sl}(2,\\mathbb H)", "4c2277a96979588ac8e573727cb20260": "(1 + i_n) = (1 + i_r)(1 + p_e)\\,\\!", "4c2294ccf5eaef96f8bd39372555f3bc": "f(x_1,x_2)=x_1^a x_2^{1-a}", "4c2345aed0e4b44cfe1245d199bfe287": "b(v_1) \\cdot v_1 = \\frac{v_1^2}{2} \\Rightarrow b(v_1) = \\frac{v_1}{2}", "4c23844580967a088c8cc623173140e8": "f=\\frac{1}{2 i k}(e^{2 i \\delta_s}-1)\\approx \\delta_s/k \\approx - a_s", "4c238ae98330a7cf7cb3a1124cab0961": "0 \\leq k \\leq n", "4c246fb32e913eed26f57e2cefce7a16": " {} = a^{i}b^{j}-a^{j}b^{i} . \\,\\!", "4c24c69249f328b139c278fdc6dee277": "\\mathbb{C}[O_d(\\mathbb{R})\\backslash\nM_d(\\mathbb{R})/O_d(\\mathbb{R})]", "4c24dd77a187d7326acf989db2364711": "i=\\frac{I_{N}R}{V_P}", "4c2599f94a0b612fc2dd1421b86d7300": "F(x)=\\sum_{n=0}^\\infty a_n x^n", "4c25f7df019569c521f95778b171058d": "\n \\lim_{n\\to\\infty}\\Pr\\!\\left(\\,|\\overline{X}_n-\\mu| > \\varepsilon\\,\\right) = 0.\n ", "4c26763bb6df15605bd32757a4398fb1": "F_b = (\\rho - \\rho_o) g", "4c26ab88796432a38def9af9677dd8c5": "\\hbar /2", "4c26be4dba34efe5161dd239ed581984": " V=\\frac{1}{\\sqrt{aa^{\\dagger}}}a", "4c26e1a65f9d3aff2a21085846466b09": "{}\\quad k\\alpha^n(1\\pm \\varepsilon)", "4c271322aa610f3d49d9d6ab7eea0e0d": "S_{H-P} = \\int d^4 x \\; e \\; e^\\alpha_I e^\\beta_J \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "4c27397bf4908401134ebee38b18d60c": "\\omega_c=eB/m", "4c27452451270caafc6fb581da3933a2": "= du - u_{1}dx \\,", "4c2746bc5db8703040aca892d89a1516": "\\phi-1", "4c27514e73eb86c2ee4d963f152974ff": " R = \\omega^r\\, \\cap \\, (\\omega^r)^{-1} ", "4c276195590682e8bd15aef2b4ba3469": "z_1 = Z_1/Z_i, z_2=Z_2/Z_i, \\dots, z_{i-1}=Z_{i-1}/Z_i, z_i = Z_{i+1}/Z_i,\\dots, z_n=Z_{n+1}/Z_i.", "4c27649f76fb1e593decc19d623ed215": " \\sin(180^\\circ-x) = \\sin(x).", "4c276941c136cc636b49e6b4be288c09": "E \\left\\{ (\\hat{x} - x)^2 \\right\\}", "4c277968ece6a93355e93f3cdda0f493": "\\Phi(z)", "4c2810cf77f3d7d2fbfb4a9c39ba6fe6": "|n^{(1)}\\rangle =\\frac{V_{k_1 n}}{E_{n k_1}}|k_1^{(0)}\\rangle", "4c281a80f854051bbab92558539e8886": "\\text{DSPACE}\\left(\\left(f\\left(n\\right)\\right)^2\\right)", "4c2821e3f0635a00aabe61409df3a65a": "z_{a}", "4c29268fd694d41dc820a7606457286f": "B^c", "4c2929d860905ee602471c19e749ada8": "\\frac{1}{\\sqrt{2}}(|0\\rangle|1\\rangle + |1\\rangle|0\\rangle)", "4c2952e983122a88e7aa08fc079acb25": "K_1 \\times I", "4c2a88158d7231f7a1bf13d24bd20b6d": "n/(d+1)", "4c2a9515b1f6c08a945412d8b37237bc": "m\\rightarrow q", "4c2aa57c3f7257d5ba79b90c891a8159": "{\\part A \\over \\part z} = - {i\\beta_2 \\over 2} {\\part^2 A \\over \\part t^2} + i \\gamma | A |^2 A = [\\hat D + \\hat N]A, ", "4c2acc625d6868c0f2253c827dc2f4ee": "\\Delta t = \\frac{x \\sin{\\theta}}{c} ", "4c2ad45e104009773c7e493ce95c178a": "g_{ij}=\\langle\\partial_i,\\partial_j\\rangle=\\partial_i\\eta_j=\\partial_i\\partial_j\\psi", "4c2ad5a9e14f52954545c5de3b96e6cc": "\\displaystyle P(w|d) = \\sum_{z=1}^Z P(w|z)P(z|d)", "4c2ad76e71f2fb872bcc2ff7bd1025ad": " \\begin{align}\n\tx_0 &= 0\\\\\n\tx_i &= \\beta_i x_{i-1}+(1-\\alpha_i-\\beta_i)x_i+\\alpha_ix_{i+1}\\quad i=1,\\dots,N-1\\\\\n\tx_N &= 1\n\\end{align}", "4c2b4290a381087c2a2187cfc8965855": " \\mathbf{b}_1,\\mathbf{b}_2, \\mathbf{b}_3 \\in Z^{3}", "4c2b502c9644632762b3d132bd46b650": "C_x \\subseteq QC_x", "4c2ba40657f1928edf8f18f925f5da88": "B=\\operatorname{adj}(tI_n-A).", "4c2baba022af6b03a2e9020eab6b8555": " (L-L_{o})\\rightarrow x\\, ", "4c2bccf841ed26dfcbaf5e2c1be68d21": "\\,a\\,", "4c2bef076ee38654efdfaf0ddef9db6a": "F \\cap E_j = \\emptyset \\rightarrow F \\cap F_j = \\emptyset", "4c2bff9425f23231dc7126973908f56e": "\\sum_{n=0}^\\infty p(n)x^n = \\prod_{k=1}^\\infty \\left(\\frac {1}{1-x^k} \\right).", "4c2c108581787765c4f7cb9c9ee4b5b1": " \\dot{a} = -\\partial_aF(s,\\tilde{\\mu}) ", "4c2c35c0f6bb7e7e15ce1350110df331": "\\begin{align} \\cot \\frac{\\theta}{2} &= \\csc \\theta + \\cot \\theta \\\\ &= \\pm\\, \\sqrt{1 + \\cos \\theta \\over 1 - \\cos \\theta} \\\\[8pt] &= \\frac{\\sin \\theta}{1 - \\cos \\theta} \\\\[8pt] &= \\frac{1 + \\cos \\theta}{\\sin \\theta} \\end{align}", "4c2c4ae2527d5b369e45f4176b04943f": "(T,\\le_T)", "4c2ca116e686b3a466c9dedcc662fe54": " \\frac{1}{v \\left [ \\left ( {p_0}/{p} \\right ) -1 \\right ]} = \\frac{c-1}{v_\\mathrm{m} c} \\left ( \\frac{p}{p_0} \\right ) + \\frac{1}{v_m c}, \\qquad (1)", "4c2cca45b922a39104cbf912872b5bc4": " (\\ 0\\ ,\\ 0.001036\\ )\\,", "4c2ce80e4763b01d59fa3facb18b5df6": "K^n \\to K.", "4c2cea7dc6d85277fcee7008d421fa6d": "\\mathbf E_{\\rm emf} = - S \\boldsymbol\\nabla T", "4c2d31304fe4e7332774a525cf1f452e": "\n\\frac{\\pi }{4} N \\sin\\theta+\\frac{N}{2} \\phi = \\frac{\\pi }{2} \n", "4c2d387156f0c9d93b0d6bfe20897128": "\\forall t\\in [a,b], X_t\\in L^2(\\Omega,\\mathcal{F},\\mathrm{P}),", "4c2d5977e09e75a8eae21b940529c951": "G_1 = C", "4c2d8a9a41acd42caa7113e0e0a2bd6a": "\\log_{10}\\left(\\frac{k}{k_0}\\right) = N^+", "4c2e18f3341feb7e5659fa19e419a6d6": "\\int_{E}\\varphi=\\infty", "4c2e2d20262d3b7b011ca5ac85d2e043": "Q_1[\\mathcal{L}]\\approx\\partial_\\mu f_1^\\mu", "4c2e74df669fd4afdd24d24dee2e41c9": "\\Box\\varphi = \\left[\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\nabla^{2}\\right] \\varphi = \\frac{1}{\\epsilon_0}\\rho\\,.", "4c2ed8f73fcba3522448907ba34166ac": "f_i=i\\frac{c}{L}", "4c2f098641a30b3ffd979cea170d7eba": "v={-0.00028} \\text { m/s}\\,\\!", "4c2f1af6e088ad83d46d4a3a432f3bdd": " \\lambda(X)f(g)=-\\rho(g^{-1}Xg)f(g).", "4c2f27e389feb5e80caa24af47f821ed": "1\\times 3^{-10} + 1\\times 3^{-11} + 1\\times 3^{-12} + 1\\times 3^{-13} + 2\\times 3^{-14} + \\cdots", "4c2f33e1af2726e0a05464edc4172a8e": " GL(V) ", "4c2fa49bc7587421bea5a40efbe60974": "m(x) = \\frac{1}{h(x)}.", "4c2fa7ae8cf5b4887a61fabaf89d1705": "R=\\frac{A_{t+1}}{P_{t1}}", "4c2ff0474e94740419889a784d77777d": "{E[\\vec{X}]^a}_a", "4c302efdfbf2d0ee2dc6dab341b814d0": "81m", "4c303b9988198f6e57df24f3dc9fb417": "\\eta_k", "4c30af6e694ede68fec9bd0f72c4608a": "E^{p,q}_1=H^{p+q}(gr^W_nH)\\Rightarrow H^{p+q},", "4c30e8f59f5fa7480ae663da7ed7ae82": "\\textstyle G_1 = G_2 = G", "4c312dfe8d8a4514fab54aac8495d597": "F_{Y}(q)<0.5", "4c31469b8124483c20632117b8e31510": "\\nabla \\wedge F = \\langle \\nabla F \\rangle_{r+1} = e^i \\wedge \\partial_i F.", "4c31552ba837016639d1ba31ef834170": "\\Delta_{S^{N-1}}", "4c318042f1b3b3a504f6514b3f3e3b7d": " r_1 = \\text {fixed rate at t of currency1. A set of rates for every t are fixed at time 0. } ", "4c319cf3f5386ad9fd2edaa7e6289629": "n > w", "4c31ac8d74bcfa7d9f90f976d17698f3": "I(S;\\Omega-S)", "4c31b359f3eb4d98fad39a3a7923a697": "Q\\bar Q", "4c31f46d5acc2a3e873c66cbabea760a": "\\text{bcclabccefab}", "4c320414a2b686eccda2b76bd1b35dcc": "{{D}_{o}}", "4c321d7ab4e8b2b98018fabd4b103e7a": "1 - {1 \\over 2} + {1 \\over 3} - {1 \\over 4} + {1 \\over 5} - \\cdots =\\sum\\limits_{n=1}^\\infty {(-1)^{n+1} \\over n}", "4c323fea2caa06f4f52b922f58170358": "\\sigma_{(uv)(vw)}", "4c3275094c79a0b1ea28e45528d72e7e": "x^2-ay^2=0.", "4c32b71325d6806fb5f25adbd80cfb64": "\\{\\Omega,\\mathcal{F}\\}", "4c32dfe228cbb8a0d852ff96d28e0807": "\\mathbf{\\alpha_2=\\alpha_3=\\alpha^{-0.5}}\\,\\!", "4c3330ea91ad299505253d535beff004": "M_{sl}=\\frac{M_{s}}{\\phi_{sl}}", "4c335085024498c757ef6d8b9cbb9133": "10^{3.47\\times 10^{20}}", "4c337ceb407b0f561fbe7bd8d1561141": "2 \\left\\lfloor\\frac{N}{4}\\right\\rfloor + 1, \\ldots, 3, 1", "4c33ade531cbf24da2cbccfdc894d2e9": "(x - [[z]]) \\cup [[w]]", "4c33f04194b21279a5b273c91a98fd08": "-2\\frac{N_c^4}{N_f^2} + N_c^2 -1", "4c340bf23d447ea6f80cdd457a662b69": "I:f^\\infty=J\\cap R.", "4c349ca442dea8db5e4d2494d2dfabcb": " X = \\langle A,C,B,D,E,G,C,E,D,B,G \\rangle ", "4c34d8d5322ce51df89a9dee8e8cc5ef": "\\rho : \\mathrm O(n) \\to \\mathrm{Aut}(C\\ell_n\\mathbb R)", "4c351e6a43942dd284e5971f41937e03": "\\Delta_c", "4c3536af1bfca26aa33f531b6b833934": "\nt(i)= \\frac{1}{iM}, \\qquad \\qquad (i=1,2,\\dots,M-1), \\,\n", "4c3575723430597ca4613b3a54a22af7": " \\operatorname{p.v.} \\int_{-\\infty}^\\infty x\\,\\mathrm{d}x = \\lim_{b\\to\\infty}\\int_{-b}^b x \\, \\mathrm{d}x = 0.", "4c35c2dce2aa9e4192702ca8605fd30a": "X[k] = \\frac {1}{N[k]} \\sum_{n=0}^{N[k]-1} W[k,n] x[n] e^{\\frac{-j2 \\pi Qn}{N[k]}} ", "4c36072d4b7ec141d2ab84b8ff0afefc": "\\left(\\pm1/2,\\ 0,\\ \\pm1/2,\\ \\sqrt{1/8},\\ \\sqrt{3/8}\\right)", "4c373c6ef9b5b60fe95f24c926fc47f2": " \\nabla_H ", "4c373fecb21d8d9c8ae7fef01da9684e": "q_t(z)=\\prod_{1\\le i \\frac{K}{\\sqrt{y}}", "4c392d0b27d35dee8bc42a191dc6aa76": "\\displaystyle{\\|P(Y_{n+1}-Y_n)\\|\\le \\|P^\\perp(Y_n)\\|.}", "4c394d37dfc324b76b7e81a71117bd65": " \\gamma_{\\mu\\nu} = g_{\\mu\\nu} + n_{\\mu}n_{\\nu} ", "4c395198bb2b1902828b88a45ac94130": " \\frac {\\left(\\rho + e - \\rho \\right ) \\ d \\theta} {\\rho \\ d \\theta} = \\frac {e} {\\rho}\\ .", "4c39650b32a7f761b5ca36c469958c18": " - \\mathrm{dln} i / \\mathrm{d} (1/V) = \\; \\kappa V + B, ..........(43)", "4c397b4c29b6a0f851c5a76a5f1c0ebd": "\\imath", "4c39ea653acf9f3471bb1f6071f59312": "\\begin{align}D & = \\max\\left(\\frac{U}{P_\\mathrm{tot}/\\left(4\\pi\\right)}\\right) \\\\\n & = \\frac{\\left. U(\\theta,\\phi)\\right|_\\mathrm{max}}{\\frac{1}{4 \\pi} \\int _{0}^{2 \\pi}\\int _{0}^{\\pi}U(\\theta,\\phi) \\sin\\theta \\, d\\theta\\, d\\phi}.\\end{align}\n", "4c3a55e06b943f9c966640a810b2ff8b": "C_\\text{L} (\\alpha)=\\frac{2 L}{ \\rho U^2S} \\quad \\text{and} \\quad C_\\text{D} (\\alpha)=\\frac{2 D}{\\rho U^2S}.", "4c3a72ba20bdae842422d4e6542a420c": "p \\to (q \\to p)", "4c3a826b25b7facb2a39787a8b4e7c10": "c = 0.035 - \\frac{1}{0.5} \\times \\ln( \\frac{1300}{1371} ) = 0.14135 = 14.1% ", "4c3aa1bbc317a05e168cc0d87a7e0012": "\\nabla \\times", "4c3aa3c6616b081d69e62b7aa51eb693": "\n r_0 = l\\sqrt{N}\n ", "4c3b12d8db28d0b27d1bde045bc3bc7b": " x_1^k + x_2^k + \\cdots + x_N^k = n\\,", "4c3b25f55e8c698ab6028ab871b9b264": "H[4]=3", "4c3b51f3cadff814cdf2ab02912e9529": "R = S + E + C", "4c3b7c04c3bd2f8ffea65faac01ec300": " (\\neg A \\rightarrow A) \\rightarrow A ", "4c3b8f07209953ccf492b1c467be46bc": "a \\text{ (digit at } i\\text{ )} \\times b \\text{ (digit at } (n-i)\\text{)}.", "4c3bf0a65dbc2bfd06a9ef01b5795617": "{1 \\over s_i} + {1 \\over s_o} = {1 \\over f}", "4c3bf5745ec36cbd17e194ab7e319754": "+(k-i)\\delta_{ij}", "4c3c3472239bba7c7738186697eb39d7": " f(x) \\le f(x_0) + \\frac{1}{2} (f''(x_0) + \\varepsilon)(x-x_0)^2 ", "4c3c5baf3f8a9e0fedd4504122dc5b9b": " G_n(x) = \\int_a^x g_n(t) \\, \\mathrm{d}t,", "4c3cfbdc61080a2271db9828bfb70a21": "\\varphi_y", "4c3d05a2077fad3a4892b08dc65f5de8": " R(x)R(y) + \\theta R(xy) = R(R(x)y + xR(y)).\\, ", "4c3d238c586d47efe72ab737704f0cd7": "e_1\\in e_2", "4c3d48e1408d7ac59082c680ce3ebd9b": " \\sum_{k=0}^n \\binom{n}{k} \\left(\\frac{\\mathrm{d}}{\\mathrm{d}x}-\\sin x\\right) \\circ \\left(\\frac{\\mathrm{d}}{\\mathrm{d}x} - \\sin x + i\\right)\\circ\\cdots\\circ\\left(\\frac{\\mathrm{d}}{\\mathrm{d}x}-\\sin x+(k-1)i\\right) (\\sin x)^{n-k} = 0. ", "4c3d61bd46464b71e73bf931f8e24984": "|A| > |B|+1", "4c3daa99288012ebed7aded25a3f7e8e": "V=\\frac 1 {\\pi} - 0.142 \\cdot 0.9 ", "4c3dadd8f30074349dc60e4f1cfd86a0": "\\delta(\\alpha) = \\prod \\left({\\alpha - \\alpha^{(i)}}\\right) \\ ", "4c3e31f6ffb1342e3ea5e92693898e89": "t \\mapsto tf'(x) ", "4c3ee926d6b5c13b74e5622f1e58343e": "f(p_1) < f(p_2) < \\cdots < f(p_k) ", "4c3f028d41eb06d04ca73c25cce41ec9": "\\scriptstyle\\varphi(\\alpha) \\,", "4c3f20ace4379f05ffc3a581198c5ed6": "x=(1,0,1,0,\\ldots)", "4c3f634667212b4ab99a190c90888f37": "\\| \\nabla u \\|_0", "4c3f7baca501da327610406db69bc143": " \\mathbf W ", "4c3fd8b308c795afabea48c32688cc1b": "\\log K =\\log K^0 + \\log \\gamma_{HA} - \\log \\gamma_{H^+} - \\log \\gamma_{A^-}", "4c3fdbced0f90b7b40c1470744324797": "x \\in \\{0,1\\}^n", "4c3ff3eb7488d5151cc07aea3f58bef7": "\\left|f(t+T)-f(t)\\right|<\\varepsilon.", "4c401a3f10fde240a86a3cc1b46e5b1a": " \\tau=I \\frac{d^2\\theta}{dt^2}=h F(t)", "4c40433bbd5f88058bbca9fcdf8116d4": "\\| D_n \\| _{L^1} \\approx \\log n", "4c4048c485cdd4a83ee94a8cdb375bc4": "U_{coul} = k {{q_1\\,q_2} \\over r}={1 \\over {4 \\pi \\epsilon_0}}{{q_1 \\, q_2} \\over r}", "4c40684a5da3a1899f660aa0bd990cba": "W_j = \\frac{B^2_j} {SE^2_{B_j}}", "4c407c22152749c6d49ce4017b3e095f": "\\tau_1 \\tau_2 ", "4c40821490f5ff28c3387463548d1429": "\\mathrm{NB}(r,\\,p)", "4c4094654690885d5536c39d7e6aaa00": "J \\subset I \\Rightarrow I : J = R", "4c40fc417002a58027fd622600991cdf": "\nr_{\\mathrm{inner}} \\approx \\frac{3}{2} r_{s}\n", "4c410c9bfd6fe42cd4c5ee5623447552": "C_{00} = 0", "4c411c9cd56a0a2132317d40502e5dfd": "\\rho(A)=\\{y \\in Y : \\exists x{\\in}A~.~xRy\\}", "4c412bb1fdbe0a5483503d0746e33cfc": "z(a)=\\{i: a_i=0\\}\\quad", "4c41815d4195a83d344d06689a4d8061": " \\ddot{y} = y = 0.", "4c418c7044822dd20949a5912fea33f7": " M = (m(s,t))_{s,t \\in S}", "4c41afc4bb1b8ef83f915175b6d82b9f": "\n\\begin{align}\n\\operatorname{arcosh}(2x^2-1)=2\\operatorname{arcosh}(x)\\\\\n\\operatorname{arcosh}(8x^4-8x^2+1)=4\\operatorname{arcosh}(x)\\\\\n\\operatorname{arcosh}(2x^2+1)=2\\operatorname{arsinh}(x) \\\\\n\\operatorname{arcosh}(8x^4+8x^2+1)=4\\operatorname{arsinh}(x)\n\\end{align}", "4c41d3b06f5fb8796b964063d1f9f54c": "X_{arv}", "4c41f1cd3c2eef347229e00c2f1cebf9": "\\int_0^\\infty a(x)d\\mu", "4c4216c5c446d86f44e87db46a1a5f74": "\\operatorname E(f|\\pi_1)(x_1)= \\int_{X_2} f(x_1,x_2) \\mu(\\mathrm d x_2|x_1),", "4c421e23de54c2d4bbbce4950810d6a1": "\\langle l_2, r_2 \\rangle_w", "4c42da19cb956b967f269666ff9822cf": "x=a\\cos \\theta \\sec^3 \\frac{\\theta}{3} = a(\\cos^3 \\frac{\\theta}{3} - 3 \\cos \\frac{\\theta}{3} \\sin^2 \\frac{\\theta}{3}) \\sec^3 \\frac{\\theta}{3}", "4c4385943ab179d6851a4fd11997e7ec": "\\frac{3^3 2^0 + 3^2 2^1 + 3^1 2^4 + 3^0 2^5}{2^7 - 3^4} = \\frac{{125}}{47}", "4c438660aaee6e73b46599d20502e1d4": "n(\\omega) k_0 = \\beta (\\omega) = \\underbrace{\\beta_0}_{\\mbox{linear non dispersive}} + \\underbrace{\\beta_l (\\omega)}_{\\mbox{linear dispersive}} + \\underbrace{\\beta_{nl}}_{\\mbox{non linear}} = \\beta_0 + \\Delta \\beta (\\omega) ", "4c43b4d34b0af85c0d735f0f0eac98c9": "S_0 = S_Te^{-\\mu T}\\text{ and }\\sigma_0=\\frac{\\sqrt{\\ln\\left(1+\\left(\\frac{\\sigma_T}{S_T}\\right )^2\\right)}}{\\sqrt{T}}.", "4c43e0a7e7fb54894a3f965ed6c9564c": "E_K=\\pi k_B T", "4c44365de6293f8470de5dcff0268536": "\\Box A\\to A", "4c44411b051ef7d47189c95e9a64a319": "\\text{Drag}_\\text{induced} = \\rho V_\\infty \\int_{-s}^s \\Gamma \\sin{\\alpha_i} dy ", "4c44539004fc552cc25b7963d0a49d85": "q_s=\\frac{Q_s}{b}", "4c4480c73dc0362793f8682aedde2f24": "\\phi: X \\to Y", "4c4484d3c5b56e71b7f9446c5949b410": "D^*=(\\mu^* E)^2 \\frac{(\\delta t)^2}{16 t_0}", "4c44a3c97bdee75f65c05851bc4cdb36": "\\left[ \\begin{matrix}\ng_{11} & g_{12} & \\cdots & g_{1N} \\\\\ng_{21} & g_{22} & \\cdots & g_{2N} \\\\\n\\vdots & & \\ddots & \\vdots \\\\\ng_{N1} & g_{N2} & \\cdots & g_{NN}\n\\end{matrix}\\right] \\left[ \\begin{matrix}\nw_1 \\\\\nw_2 \\\\\n\\vdots \\\\\nw_N\n\\end{matrix} \\right] = \\left[ \\begin{matrix}\nb_1 \\\\\nb_2 \\\\\n\\vdots \\\\\nb_N\n\\end{matrix} \\right]", "4c44a503959f3fbe9387a895dd92fffd": "f_1 \\cap f_4 \\cap f_6 = \\varnothing", "4c44d79919f9882afeca0534095ab313": "Y_{4}^{2}(\\theta,\\varphi)={3\\over 8}\\sqrt{5\\over 2\\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(7\\cos^{2}\\theta-1)\n= \\frac{3}{8} \\sqrt{\\frac{5}{2 \\pi}} \\cdot \\frac{(x + i y)^2 \\cdot (7 z^2 - r^2)}{r^4}", "4c4537448ce9ce615d89e62a3064141d": "\\oplus_{j\\geq 0} I^j/I^{j+1}", "4c457af53dc891800a5b6c1b1a438428": "\\|y-x\\|\\ge\\|y-u\\|", "4c45a65237bed559518cdf0df2bb2ada": "m(\\mathbf{c})= \\sum m_i c_i", "4c45d92dac1915c35675a942ff2dc2f0": "(x^2+3x+5)\\circ(x-2)=(x-2)^2+3(x-2)+5=x^2-x+3.", "4c45df5f4da886984e77113616c17348": "f_{\\sharp} : C_n(X) \\rightarrow C_n(Y).", "4c46115b1685c3df8c3a18ebbafc24d2": "\\int_{0}^{\\Delta t} R_i(t') dt' = \\ln(1/u^\\prime)", "4c4661d51a053fe76d7de34b2dcc9069": "E \\not \\in \\operatorname{FV}[G] \\and E \\not \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] = G\\ H ", "4c46ab04f96f6e57c070a6c1782ff9ec": "8 + 8 + 8 = 24,", "4c46adafa8802f44916f3450d9c73499": " V_{\\rm w1} = 0 ", "4c46b5075cf15a8ccc38446d7b472deb": "\nG = \\left\\| {\\begin{array}{*{20}c}\n g_1 & g_2 & \\dots & g_n \\\\\n g_1^{[m]} & g_2^{[m]} & \\dots & g_n^{[m]} \\\\\n g_1^{[2 m]} & g_2^{[2 m]} & \\dots & g_n^{[2 m]} \\\\\n \\dots & \\dots & \\dots & \\dots \\\\\n g_1^{[k m]} & g_2^{[k m]} & \\dots & g_n^{[k m]}\n\\end{array}} \\right\\|,\n", "4c46bacd299b8bb55e61a278cf46399d": "u(\\mathbf{x})\n=\n\\begin{cases}\nu^+(\\mathbf{x}) &\\text{if } \\sigma(\\mathbf{x}) > 0 \\\\ \nu^-(\\mathbf{x}) &\\text{if } \\sigma(\\mathbf{x}) < 0\n\\end{cases}", "4c46cf76ddb1753826b0292dc527a090": "E\\,.", "4c473aba508b5fc552aa8464ebb4d6cb": "(x+y)^2=x^2+2xy+y^2", "4c474500d1e96addfc684c12527e0e3f": "P(t+s) = e^{Q(t+s)} = e^{tQ} e^{sQ} = P(t) P(s)", "4c474514b50a814954f9b25d6e442ecf": "\nj = j_{ion}^{sat} \n\\left( -1 + \\sqrt{m_i/2\\pi m_e}\\,e^{-e\\Phi_{sh}/k_BT_e} \\right)\n", "4c4756a2f0e9decc510c90ef1adf1f7d": "x_n^2=(x_0+n)^2.", "4c477c13974adaba0c4ba012c6d20af0": "\\sqrt{n}(\\hat{y}_0 - y_0)\\ \\xrightarrow{d}\\ \\mathcal{N}\\big(0,\\;\\sigma^2x'_0Q_{xx}^{-1}x_0\\big),", "4c47b1976670db0362b003babb4b97e8": "A_n = \\sum_{i=0}^n a_i,\\quad B_n = \\sum_{i=0}^n b_i\\quad\\text{and}\\quad C_n = \\sum_{i=0}^n c_i", "4c47bf357ae0201343196ef7f0e1799a": "\\textstyle \\prod_{i=1}^n h_i H'", "4c47d6c31cf0028d940468403b76429b": "\\Phi({-\\infty}) = 0 = 0\\%", "4c4874bf630bbb43e39d77948cce271a": "(f \\circ g)'(a) = \\lim_{x \\to a} \\frac{f(g(x)) - f(g(a))}{x - a}.", "4c48800d12b5b625a2be20f1c83071db": "V \\times \\{0,1,2,3,\\ldots, N\\}", "4c48ad1f99c0bb0d7f0dc5f0d89c390a": "\\Delta y_{it}=y_{it}-y_{it-1}=\\Delta x_{it}\\beta + \\Delta u_{it}, t=2,...T ,", "4c48b532f497debc171b78edff003c95": "L \\mathbf{v}_0 = 0 .", "4c497976a12386b022664ffaebc14132": "F = \\left( \\frac{E A_0} {L_0} \\right) \\Delta L = k x \\,", "4c4a702d231c883bb0f220c2a3e66b19": "(x+y)^{M_p} \\equiv x^{M_p} + y^{M_p} \\pmod{M_p}", "4c4a83e6c4ea7d7e784b2d855b9cdb18": "f({{v}_{2}}^{\\prime }|{{v}_{1}}^{\\prime })f({{v}_{2}}|{{v}_{1}})\\ge f({{v}_{2}}|{{v}_{1}}^{\\prime })f({{v}_{2}}^{\\prime }|{{v}_{1}})", "4c4aba4f6928f3b70bb11a6efad5d96d": "\\cap_{i \\in \\underline{m} \\backslash A} A_i", "4c4ad23a301b88ceb528508d6f35c896": "\\langle 0|\\phi|0\\rangle=0", "4c4b55f05c10dfa2ef46b31e2b632aa7": "i_1 =

_{Kin}.", "4c514f2452ebdf1f1822c69384019490": "Y_{8}^{3}(\\theta,\\varphi)={-1\\over 64}\\sqrt{19635\\over 2\\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(39\\cos^{5}\\theta-26\\cos^{3}\\theta+3\\cos\\theta)", "4c517e753a4e1b5fad6a920e8210bc24": "0 = \\frac{\\partial L}{\\partial x} - \\frac{\\mathrm{d}}{\\mathrm{d}t} \\frac{\\partial L}{\\partial \\dot x} = m g - m \\frac{\\mathrm{d} \\dot x}{\\mathrm{d} t} ", "4c5192e58ab7c3f5106f580f4f8fccc2": "p({\\tfrac{1}{2}}) = -8", "4c5219a662483ee82db890c085bbc511": " \\ln \\left(\\sum_{i=1}^{k} e^{\\eta_i}\\right) = \\ln \\left(1+\\sum_{i=1}^{k-1} e^{\\eta_i}\\right)", "4c5256dc970b0b8f69cb909152460618": "\n\\mathrm{NDTF}^2_{j\\rightarrow i}(f)=\\left| H_{ij}(f) \\right|^2\n", "4c526c4be55c84185f2e1e43a066efee": "E=\\sum_{all\\atop faces}\\epsilon\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right),", "4c526f1579e040561883a250901cf7ed": "b=2\\sin\\tfrac{\\pi}{6}=1", "4c528735b171f208c5b473350e636af3": "(\\operatorname{id}\\otimes \\Delta)\\Delta A=(\\Delta \\otimes \\operatorname{id})\\Delta A.", "4c52c7334284062ca97f7fddfffebdac": "\\Psi_b", "4c52d5772ecce1b598e3ea95379360bb": " f\\,\\!", "4c52e3d912bf4083876bf87253a31049": "\\Delta\\,T_m(x)= \\frac{k_{GT}}{x}", "4c530ac7f493b6a503edfb37611d481b": "|x| = \\sqrt{x^2}", "4c533ef38080a71a4c42021c079de9aa": "P \\le \\sum\\limits_{m \\in \\mathbb{F}_q^k \\backslash \\{ 0\\} } {{\\Pr}_{\\text{random }G} } [wt(mG) < d]", "4c5341ee384bdee6b674addece1a9c73": "A_{ij} = 1", "4c538ed7c8b787d0c301ededc91fc29e": "AA^\\mathsf{T} = I. \\,", "4c539f54d05269ec60efa8fcc16f9a81": "\\operatorname{det}(W)=\\pm w^{n/2}", "4c53cb4027ed9faea47af98fbe20f8ce": "G = A_3 \\rtimes H", "4c53d9bc4c2df14ab66154039f796250": " \\phi \\to \\pi^+ \\pi^- ", "4c544c418632412a9ae4f05dca51203b": "\\scriptstyle | \\phi_i \\rangle ", "4c546e3d7ac509d4206c272ae9d933b8": " = |\\lambda|^2 \\cdot \\| x \\|^2 + \\| \\lambda \\cdot x \\|^2 - \\overline{\\lambda}\\cdot \\langle \\lambda\\cdot x, x \\rangle - \\lambda\\cdot \\langle x, \\lambda\\cdot x \\rangle ", "4c5471fef5ffe4d82df40e930496e00a": "\\chi^2_{d_1}", "4c54764243625f71417b10104403be22": "c_1^p", "4c54cab62e1501ef781b6edf42c894b3": " y_0 = 0, \\alpha=1, \\lambda=1 ", "4c55113cd15da401ae32155b9be12cfb": "\\textstyle{\\frac{\\partial u}{\\partial x}}\\,", "4c5545526d63c7aa2e4e33bc9445a307": "\\Pi = \\gamma^o - \\gamma ", "4c556b8ae740f5cd68c9c9f2629f5783": "if\\,(s_1\\,and\\,s_2\\,\\,\\mathit{differ}\\,\\,in\\,\\,exactely\\,\\,one\\,\\,attribute)", "4c55ed12d8cd2ece68f487528e5e8373": "\\mathrm{La} = \\frac{\\mathrm{Re}^2}{\\mathrm{We}}", "4c562e77f26fb2ca3c9e97f2189e7a8d": " E_a ", "4c563c46bf334fb52bfd2f8bfd689546": "1 \\over 8", "4c5645510c6b63f6bd0c770fceea4ac1": "\\theta_{j}", "4c56d8f4a3b6d7974d9918246c3de58a": "[T^i ,T^j] = 2i \\epsilon^{ijk} T^k", "4c56dfdece5c1dc3b1534a714b1998de": "i=1,\\dots, n", "4c56ea6d90947cc1bb5e4c3b7183b2ac": "A \\cdot B \\neq C", "4c56f7f43e3fc38feb2dbccdece15d91": "S = \\{s_0,s_1,s_2,s_3,\\dots\\}", "4c570d7dbc762e49c0c2621174281b97": "\\textstyle E \\in \\{E_n\\}", "4c571d7f51e9236bf2de139b656af166": "x_1,x_2,\\cdots,x_N", "4c573816aaade8a364dceaebe3fd18ca": "ST_x(\\varphi \\vee \\psi) \\equiv ST_x(\\varphi) \\vee ST_x(\\psi)", "4c57b554ac7aa3bd876c01228483dcf9": "V.\\ ", "4c57ee0ab0ebec7cc94eeb5bf497ac02": "d^2 = \\frac{1}{q^2 \\sum_{i=1}^{m}{(g(RD_i))^2 E(s|r,r_i,RD_i) (1-E(s|r,r_i,RD_i))}}", "4c57fc4913b28f59e234ea320354dc14": "\\langle Q\\rangle = \\frac{1}{V}\\int_\\Omega Q(\\boldsymbol{r}) p(\\boldsymbol{r}) \\,d\\boldsymbol{r}", "4c5832e7fc40efe8676651491fc0c6b2": "\\ H(Y)", "4c5878f5ed3a453de3ce6f24c2782ae8": "m + v = x = p + q\\,.", "4c58cb3e1048a197b9f1ae6afe99e649": "\\mathrm{_{24}^{48}Cr} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{26}^{52}Fe} + \\gamma", "4c58f3dec430c9c835922f62d6c8bfd0": "F_{n} = x^2 + y^2", "4c5903576f5a803100648fdef3ca98e0": "v=E/B\\,", "4c5939abfa6f6206c34b8fda45a23a0a": "\\langle \\sigma_i \\sigma_j\\rangle-\\langle \\sigma_i \\rangle\\langle\\sigma_j\\rangle=C(\\beta)e^{-c(\\beta)|i-j|}", "4c5952723caca2fd5ff417d39fa62023": "\\left[P^{\\mu},W^{\\nu}\\right]=0,", "4c5976fd6dde0d04add668b8082c65a6": "X^2 \\sim \\chi^2(3)\\,", "4c5981055ef90d5d956ffd1dbddea99e": " Au_{xx} + 2Bu_{xy} + Cu_{yy} + \\text{(lower order terms)} = 0 \\,", "4c599257aab407ee7080c9d9e0356229": " \\mu_Y ", "4c599a2f54753b3815094c61a8d56157": "B^\\alpha_\\alpha = -2/R", "4c59bb2c4a44313d6a1fec2fc0bb13bb": "FO+TC", "4c59c212948d318fbde613e658d0d6ce": " e_i \\equiv \\delta_{ij} \\pmod{u_jR} ", "4c59c2a15fd7abf0a1f353d6eb845484": "\\frac{d}{dx}\\left(\\frac{1}{g(x)}\\right) = \\frac{d}{dx}\\left(\\frac{f(x)}{g(x)}\\right)", "4c5a5766dac60ad3fe085e48ae414462": "\\gamma^k", "4c5a6119885038e4ec255e6993585ad1": " \\operatorname{MSE}(\\delta) = \\mathrm{var}(\\delta) +[ \\mathrm{bias}(\\delta)]^{2}\\ ", "4c5a8bf0f6d6cb8e470e1c174b7d4089": "1+h_{j}", "4c5ac3032dad5c20f5357421c32642cd": "\\beta'=\\mathrm{\\frac{[M(H_2O)_{n-1}L] [H_2O]} {[M(H_2O)_n] [L] }}", "4c5ae7b931adf67538b1249470a844df": "\\frac{d \\ln T}{d \\ln P} = 0.4", "4c5af31a9ec8beb8ed602249d2e8866a": "J=[0,1]", "4c5b19837f530a3cd8b82f6fab5ba6e8": "\\Delta (\\nu)=(\\mathcal{R}_{21}\\mathcal{R}_{12})^{-1}(\\nu \\otimes \\nu )", "4c5b51a8745193c287dfa388fe2478d6": "P(\\mathbf{Z}\\mid \\mathbf{X})", "4c5b9f3925f3db3a3de14bb23c5a2deb": "s_C + s_B = 1.\\,", "4c5be1018a8133a69b5246530ec463a3": "A\\equiv -k \\ln Z", "4c5be320fef4c6c9ba71831e8e87e20c": "d\\lambda", "4c5be5a1e5101c87b8a23a691aa7684c": " \\frac{ \\sin ( x ) }{ x }", "4c5bfb6fd2cf12ca1c7febbdf70b13a0": "\nT(\\cdots, a_{-1}, \\hat{a}_0, a_1, \\cdots) = (\\cdots, \\hat{a}_{-1}, a_0, a_1, \\cdots)\n", "4c5bff7a35e111831dab92dafe5fc8b6": "f_*:\\tilde H_r(X)\\to\\tilde H_r(Y)", "4c5c05a57d0a2e66ef58f18c828fea16": "r=\\sqrt((2c_{6}-b_{2})b_{2})", "4c5c20a57f64fcaeed62f9ad1d7e0272": "p_f(\\vec{r})", "4c5c227677241dacd67244f889b97393": "Z_{\\mathrm {in}} = \\frac{{Z_0}^2}{Z_\\mathrm L}", "4c5c3f224df4c05249c553ccaea6e922": "\\Gamma^i{}_{k\\ell}=\\frac{1}{2}g^{im} \\left(\\frac{\\partial g_{mk}}{\\partial x^\\ell} + \\frac{\\partial g_{m\\ell}}{\\partial x^k} - \\frac{\\partial g_{k\\ell}}{\\partial x^m} \\right) = {1 \\over 2} g^{im} (g_{mk,\\ell} + g_{m\\ell,k} - g_{k\\ell,m}), \\ ", "4c5ca9600fb530c64bce1d735fe44854": "\nINT\\_MIN \\le x + y \\le INT\\_MAX \n", "4c5cd09cec480bc31fdcde8e1f493e1d": "\\omega r", "4c5d285515ec3e16705ac1d7429cba7c": "N = \\{1, 2, 3\\}\\,\\!", "4c5d46283c6ccbb462bcc24397c3ad64": "\\textbf{P}_{k\\mid k} = \\textbf{P}_{k\\mid k}^a", "4c5d7b58a8370f922e3f9a7345deb2bb": "l_0 (y) = \\frac{c_0}{y}", "4c5deb7668460e5de77a57fa5cfce4cc": "m = \\frac{\\Delta f(a)}{\\Delta a} = \\frac{f(a+h)-f(a)}{(a+h)-(a)} = \\frac{f(a+h)-f(a)}{h}.", "4c5df7ae5d839b28355c287e1b6e5a42": " \\left(\\frac{N}2f_\\mathrm{s},\\frac{N+1}2f_\\mathrm{s}\\right), ", "4c5e0730aa6150ea3601f16eb7aa9f3a": "x = X/Z", "4c5e78e12a87ceeaedecf0c141420702": "\\ln \\gamma_i^c = \\ln \\frac{\\phi_i}{x_i} + \\frac{z}{2} q_i \\ln\\frac{\\theta_i}{\\phi_i} + L_i - \\frac{\\phi_i}{x_i}\\displaystyle\\sum_{j=1}^{n} x_j L_j ", "4c5e935b4ac45154f05190fca32e6683": "C = \\begin{pmatrix}1 & 0 \\\\ 0 & -1\\end{pmatrix} .", "4c5e955192d13c1bdbfd4793a4e4f572": "\n\\begin{vmatrix}\n\\frac{\\partial U}{\\partial Y} & \\frac{\\partial U}{\\partial Z} \\\\\n\\frac{\\partial V}{\\partial Y} & \\frac{\\partial V}{\\partial Z} \\\\\n\\end{vmatrix}\n=\n\\begin{vmatrix}\nZ & Y \\\\\n0 & 1 \\\\\n\\end{vmatrix}\n= |Z| .\n", "4c5f0bcf71b50021a0aad66d2feec3ce": "p(a/2)/(p(a/2)+p(a))", "4c5fb4106aa9c7b0e44db26c7260626e": "[a] := \\{x \\in X | a \\sim x\\}", "4c6033f9c3d5650d7e063d889f02748d": " \\frac{\\partial u_i}{\\partial p_j} < 0 ", "4c605815820f819a5119632b371e0a90": "\\bigcap_{m=1}^{\\infty}\\bigcup_{n=1}^{\\infty} \\left(r_{n}-{1 \\over 2^{n+m} }, r_{n}+{1 \\over 2^{n+m}}\\right)", "4c606685fdcd9745e925fa639763d72c": "F(x;k,\\theta) = 1-\\sum_{i=0}^{k-1} \\frac{1}{i!} \\left(\\frac{x}{\\theta}\\right)^i e^{-\\frac{x}{\\theta}} = e^{-\\frac{x}{\\theta}} \\sum_{i=k}^{\\infty} \\frac{1}{i!} \\left(\\frac{x}{\\theta}\\right)^i", "4c607e21ce2a5d0d50446980d1397c1e": "(0,\\ \\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm1)", "4c6084310daf3b36acf977fc5d1f405c": "ln N! \\, = \\, NlnN - N", "4c6085370900285fa0864b486f1ab8dd": " \\sum_{i=1}^m p_{ij}x_{ij} \\leq T \\qquad i=1, \\ldots, m", "4c608e23924a2104ec591394bcfc9993": "AC = \\textstyle \\varepsilon \\sqrt { 1 + {9 \\over 4} a\\ }.", "4c60c04986d6d903cf867231f7de0a17": " \\mathbf{D(\\omega)} = \\varepsilon (\\omega) \\mathbf{E}(\\omega) , ", "4c614360da93c0a041b22e537de151eb": "U", "4c61712ac980f3b5db3cdd49113c58af": " U(n)/U(n_1)\\times\\cdots \\times U(n_k)", "4c6197e6fb05cff5552d25bd9244adc3": "P(t)=\\sum_{i\\geq 0}t^i\\text{P}^i", "4c61ca1d4275f5f577f4f486a12f587b": "(|\\mathbf{\\hat{n}}|=1)", "4c61e4d21fe78e34d60ac4c06820b38b": "\\,\\!K_X = \\Omega^n_X,", "4c622e4f6e18b5a5985a41414c16a3d1": "\\{1,2,\\dots,k\\}", "4c623f1a9c39af0c073fb942f3dd8e38": "\nD\\equiv\n\\begin{vmatrix}\n\\psi_a(1) & \\psi_a(2) & \\psi_a(3) \\\\\n\\psi_b(1) & \\psi_b(2) & \\psi_b(3) \\\\\n\\psi_c(1) & \\psi_c(2) & \\psi_c(3)\n\\end{vmatrix}.\n", "4c6254893c175a81e171c2f0ce2a8350": "\\sigma = \\sigma_0", "4c62b87ad21ec409fb6c44a63daa0ec1": "\n\\mathrm{Fr}=\\frac{N^2d}{g}.\n", "4c62ccc8418c659c49b5e1aa89924ab8": " \\kappa_t \\exp ( \\overline{\\lambda} x ) \\bold c_t ", "4c631bf379ff20c4cced8df3b0195999": "(KBrO_3)", "4c63214e5e29466c7064c23e5c423e47": " (1,1) ", "4c632e2749baba288c5b3b945790fdda": " \\tilde{H}_n = \\begin{bmatrix}\n h_{1,1} & h_{1,2} & h_{1,3} & \\cdots & h_{1,n} \\\\\n h_{2,1} & h_{2,2} & h_{2,3} & \\cdots & h_{2,n} \\\\\n 0 & h_{3,2} & h_{3,3} & \\cdots & h_{3,n} \\\\\n \\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n \\vdots & & 0 & h_{n,n-1} & h_{n,n} \\\\\n 0 & \\cdots & \\cdots & 0 & h_{n+1,n} \n\\end{bmatrix} ", "4c635e307083071dd35992aa28de377c": "\\mathrm{Cov}[\\langle \\varphi,f \\rangle, \\langle \\varphi,g \\rangle] = \\langle f,g \\rangle", "4c6363103e7e0815459bc61d13064119": "x^{(p-1)/d} \\equiv 1 \\pmod p", "4c63f2fd29fec6c20010a36949e9752d": "x = 3", "4c643b94ccc9860c6773c37d1597c532": "g(r) = -\\frac{GM(r)}{r^2}.", "4c645c1679496e925dcc69c8071f96df": "f = \\sum_{n\\geq 0} a_n X^n", "4c64632051e65405d44872c731850675": "[a,b] = \\{x \\in \\mathbb{R} \\,|\\, a \\le x \\le b\\},", "4c6477c483bfa4c4085384264b5e7623": "6:2\\ ", "4c6514fdaf243650d97e3a698b11a020": "g^\\alpha = \\gamma^{\\alpha \\beta} g_\\beta = -g^{0 \\alpha}.\\,", "4c65348ae55269589bc109e5fd16d0ed": "|\\psi^\\prime\\rangle", "4c65640ce3679c78cd8d846b596067e5": "\\displaystyle K = \\sqrt{abcd}.", "4c65be6cc3323e15e136fbb8336eb44b": "A_3 = L U_3 U_1", "4c661094846c2e3d249b9013272c368f": "D_{n-1}=\\sum_{j=1}^{n-1}\\frac{\\partial}{\\partial x_{j}}", "4c662375ff2eed50fe7952fee74bbbe0": "\\mathbf{a}^k", "4c66a6e09a4f5b41f35295a3ede6889b": "\\{0^{n + m - 2}, (\\pm \\sqrt{n m})^1\\}", "4c66c3cb95c6d5b3319bcc1938dfd932": "\\varepsilon_{n+1} = \\frac {\\varepsilon_n^2}{2 (1 + \\varepsilon_n)}", "4c6741ba856844fd4265c1f679f0bae3": "j(\\tau) = 1728{g_2^3 \\over \\Delta}", "4c674df717d020bd55b68d196f63cfcb": "a\\in A\\setminus B", "4c67637088939609ad1b70d0c943c533": "D_\\text{out}", "4c677c2231d7cb7be5b8700ad257b958": "\\bar{Z_i}", "4c685d71b663db61200159f806408eb3": "(K) \\frac{\\Box A_1; \\ldots ; \\Box A_n ; \\neg \\Box B}{A_1; \\ldots ; A_n ; \\neg B}", "4c68cbd83b84e50f3684c6f3f0b01be8": "S \\ / \\ m=\\{u\\in M \\;\\vert\\; um\\in S \\}.", "4c68d294590643f80bb97b6926994ee9": " \\mathbf w(x)= 10 (kN/m)", "4c68e2a38da9f7fb81a46d5b11af87f4": " E[X^k] \\,", "4c69cbc221eec9ea80c0165a92cbe8fe": "\\mathrm{Fr} = \\frac{v}{\\sqrt{g\\ell}}", "4c69fc6e5c382365c588386137e5426a": " |a|< 4/(5 \\cdot \\sqrt[4]{5}) \\approx 0.53499 ", "4c6a0a04384aebfa6525df46d84a00c1": "\\forall i > 1, \\alpha _i = 0", "4c6a3b862297cd9a1a8e3e79a2599b30": "H, P, N, Z, I\\,\n", "4c6a466f09627327ef66798abb203a61": "\\langle Ax, Ay\\rangle =\\langle x, y\\rangle", "4c6a68c9bbc2da4e785123f3e3b57dc9": "\n\\left[\n\\prod_{i=1}^{k-1}B(a_i,b_i)\\right]^{-1}\np_k^{b_{k-1}-1}\n\\prod_{i=1}^{k-1}\\left[\np_i^{a_i-1}\\left(\\sum_{j=i}^kp_j\\right)^{b_{i-1}-(a_i+b_i)}\\right]\n", "4c6a8429269bd7b073028ca4e677b22b": "\\left.\\frac{\\partial}{\\partial z_2}g\\right|_{z_1=\\bar{z};z_2=z}", "4c6af1b6ad5ecab1fa0de4f97bd406b9": " E_{b \\leftarrow B}[max_{\\mathrm{a}} P[A = a | B = b]]", "4c6b0b9618e9ed411cf847d626733145": " ds^2 = -\\frac{\\Delta}{\\Sigma}\\left(dt - a \\sin^2\\theta d\\phi \\right)^2 +\\frac{\\sin^2\\theta}{\\Sigma}\\Big((r^2+a^2)d\\phi - a dt\\Big)^2 + \\frac{\\Sigma}{\\Delta}dr^2 + \\Sigma d\\theta^2 ", "4c6b222ac016882926ad1ceff0ff3cae": "\\ y = 0", "4c6b66f2cea884d0b7a137cbe7393553": "E : [N] \\times [D] \\rightarrow [M]", "4c6b687b589680bcc3985b9f0851d5df": "\\{(d (q - 1) - q i)^{\\binom{d}{i} (q - 1)^i}; i = 0, \\ldots, d\\}", "4c6baa44982f97f1f6e5852b45f3f61f": "\n= 1\\,\\mathrm{k}\\Omega + \\left[ \\left( 1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega \\right) \\| 2\\,\\mathrm{k}\\Omega \\right]\n", "4c6bf8264df3a0e53e5f9103e10b9958": " 370 ", "4c6c3e4af43af2bf5542f5d81f7b46a1": "\\mu \\rightarrow \\infty", "4c6c3fab371103bec16babc95da5c240": "\\scriptstyle \\iota", "4c6d4cdc160890fca308b913e1c7350f": "ds^2 = \\frac{4}{\\left(1 + u^2 + v^2\\right)^2} \\left(du^2 + dv^2\\right).", "4c6d5176244f6c50720b440ad0cc1dcb": "\\begin{align}\n \\mathbf{V}(t) &= \\mathbf{A}e^{st} = \\mathbf{A}e^{(\\sigma + j \\omega) t} \\\\\n \\mathbf{A} &= A e^{j \\phi} \\\\\n \\Rightarrow \\mathbf{V}(t) &= A e^{j \\phi}e^{(\\sigma + j \\omega) t} \\\\\n &= A e^{\\sigma t}e^{j ( \\omega t + \\phi )}\n\\end{align}", "4c6d87e494358966921f30abd39fc5b1": "\\mathbf{\\alpha}[\\mathbf{f}] = \\begin{bmatrix}\\alpha_1[\\mathbf{f}],\\alpha_2[\\mathbf{f}],\\dots,\\alpha_n[\\mathbf{f}]\\end{bmatrix}", "4c6db31863bff1bc2ecc808c0382b46b": "\\mu_{G}^{x} (F) = \\mathbf{P}^{x} \\big[ X_{\\tau_{G}} \\in F \\big]", "4c6deb8326413eca610709e063d710cc": "R_i(t)\\frac{}{}", "4c6e0f303a34458bed6319be669e25b6": "\\Omega^p(M, V) = \\Omega^0(M, V) \\otimes_{\\Omega^0(M)} \\Omega^p(M),", "4c6e13cacb44131db3978d652092b4fd": "\\ hc\\sim 2\\cdot 10^{-23}\\ J\\ cm", "4c6e7a186a7ca2770020b4ab9966974e": "b_n=(-1)^n a_{N-1-n}", "4c6ea0f46895ec701082fe7af244484b": "\\vartheta \\left( x \\right)\\log \\left( x \\right) + \\sum\\limits_{p \\le x} {\\log \\left( p \\right)} \\vartheta \\left( {\\frac{x}{p}} \\right) = 2x\\log \\left( x \\right) + O\\left( x \\right)", "4c6ecee17286b68228434c4dafbd4c29": "= \\mbox{Arg} \\left(1 - \\left| a \\right| e^{i \\theta_a} e^{-i \\omega} \\right)", "4c6ee9827f84c6e887f58d0764d52633": "\\int_{0}^\\infty R_m(x)\\,R_n(x)\\,dx=\\frac{2}{2n+1}\\delta_{nm}", "4c6f10f9c1d2f82cdae145d607eff679": " r_2 - r_1 ", "4c6f36248f521a566c5fc6f8427f06ec": "= \\frac{7}{3}(- \\log_{10} T) + 1", "4c6f6afd80b2b31e5aff4140fc4eec40": "\\cos(\\sqrt{z})", "4c6fd18dda8e927492ba89ec72dd7682": "\\left [\n\\begin{smallmatrix}\n 1 & 3 \\\\\n 3 & 1 \\\\ \n\\end{smallmatrix}\\right ]", "4c7014995e476328470ec3f5a7e1b5f9": "T=\\int_{\\mathbf{A}}^{\\mathbf{B}} \\, dt = \\frac{1}{c} \\int_{\\mathbf{A}}^{\\mathbf{B}} \\frac{c}{v} \\frac{ds}{dt}\\, dt = \\frac{1}{c} \\int_{\\mathbf{A}}^{\\mathbf{B}} n\\, ds\\ ", "4c7030bff8513a44046272db1b39577c": "\\frac{dL}{dt}v+LMv-MLv=\\frac{d\\lambda}{dt}v.", "4c70806baff0644e2a31545c9cc33191": "\\hat{Y}(X):\\mathbb{R}^p \\to \\mathbb{R}", "4c70e1e7f43e686a3125956f2a9c9c30": "\\nabla^2 \\mathbf{B} = \\mu_0 \\epsilon_0 \\frac{\\partial^2 \\mathbf{B}}{\\partial t^2}.", "4c712832db2f6eb1c04baa836bfba84e": "\\hat{f}(t) = \\sum_{i=1}^{N} A_i e^{\\sigma_i t} \\cos(2\\pi f_i t + \\phi_i)", "4c718f570e8412cea34011b50728c048": "\\omega_f = \\omega|_U,", "4c71b7010f3fa2b4ee61ecad4ab30e9d": "1+\\varepsilon^2T_n^2(\\cos(\\theta))=1+\\varepsilon^2\\cos^2(n\\theta)=0.\\,", "4c71ec5d823230993db826e1d9f1d919": "\\frac{\\partial u_\\varepsilon (y)}{\\partial n} = 0\\text{ for }y \\in \\partial\\Omega_r", "4c71f0bd99197edaa37458aa987a90c2": "\\begin{align}\n (volts) A_\\mathrm{eq} &= A\\left(1 - \\frac{1}{n + 1}\\right) \\\\\n (volts) B_\\text{1..n} &= \\frac{A}{n} \\left(1 - \\frac{1}{n + 1}\\right) \\\\\n A - B &= 0\n\\end{align}", "4c7207d987a73d2d0de5b9f49189a91a": "\n\\begin{bmatrix}\n 0 & 12 & 53 & 93 & 146 & 53 & 73 & 166 \\\\\n 65 & 32 & 12 & 215 & 235 & 202 & 130 & 158 \\\\\n 57 & 32 & 117 & 239 & 251 & 227 & 93 & 166 \\\\\n 65 & 20 & 154 & 243 & 255 & 231 & 146 & 130 \\\\\n 97 & 53 & 117 & 227 & 247 & 210 & 117 & 146 \\\\\n 190 & 85 & 36 & 146 & 178 & 117 & 20 & 170 \\\\\n 202 & 154 & 73 & 32 & 12 & 53 & 85 & 194 \\\\\n 206 & 190 & 130 & 117 & 85 & 174 & 182 & 219 \n\\end{bmatrix}\n", "4c72b74d5d34bb8c9e315fb878c04aa5": "d_{i,j}:=\\left\\{\n\\begin{matrix} \n\\deg(v_i) & \\mbox{if}\\ i = j \\\\\n0 & \\mbox{otherwise}\n\\end{matrix}\n\\right.\n", "4c733bc49dfe52da0e4de3b9d7f4cedf": "\\{g_{m,n}: m, n \\in \\mathbb{Z}\\}", "4c73c0d4cc3d855fc2daf28cd947e837": " u \\wedge v", "4c73d13d68a05fc5760dccb320582b36": "C=\\int d^n x c\\left(x,t\\right)", "4c744bcbb50534c6a6e11292908d86f7": "\\scriptstyle\\hat\\Sigma", "4c7505c162587c6a29b40a544c485d60": "\\psi(f_e)", "4c752cf2007b9a87b3299d3b866a26a6": " \\mathrm {log}[F (x_2)] = m \\log (x_2) + b. \\, ", "4c7599f1b97f38bc98772fbe9a3e3392": "[G]_o>>>[H]_o", "4c75a9b3a716a92b39a77ae7cb78a2b6": "\\scriptstyle\\pi(K_n)\\, =\\, n", "4c7608117c93df850714acc8d5162f1d": "x^2+y^2=r^2", "4c765a24e12fb90cfd86b7ebe22e6642": "\\omega = \\mathrm{vol}_n = \\varepsilon = *(1) . \\,\\!", "4c768a64bd0c5da05efffa78c4dd1eb7": "Z(t) = \\Omega\\left(\n\\exp\\left(\\frac{3}{4}\\sqrt{\\frac{\\log t}{\\log \\log t}}\\right)\n\\right),", "4c76a11927e0a2a23e052a5410ebefc3": "\n\\partial_t P(\\mathbf{x},t\\mid \\mathbf{x_0})= \\Sigma_{j=-M}^M D_j\\partial^2_{x_j} P(\\mathbf{x},t\\mid \\mathbf{x_0}).", "4c76a1847de641c33309530a73e869e6": " \n\\begin{matrix}\n\\frac{d\\mathbf{T}}{ds} &=& & \\kappa \\mathbf{N} & \\\\\n&&&&\\\\\n\\frac{d\\mathbf{N}}{ds} &=& - \\kappa \\mathbf{T} & &+\\, \\tau \\mathbf{B}\\\\\n&&&&\\\\\n\\frac{d\\mathbf{B}}{ds} &=& & -\\tau \\mathbf{N} &\n\\end{matrix}\n", "4c76a5d221f7633a80f3bdd247c8eaa0": " I-\\kappa M = I-M= P ", "4c76f6f25233ae291cebcb05a15fe389": "I_0 = \\frac{\\Epsilon_A^2 A^2}{2 R^2} = \\frac{P_0 A}{\\lambda^2 R^2}", "4c7706de30812285690e6b80ecdd049c": "u^\\alpha", "4c772dec7ca211c060950edb6e463b2c": "r = \\frac{2K}{P} = \\sqrt{\\frac{(s-a)(s-b)(s-c)}{s}}.", "4c775a572a65d43d4332ca32f6b771b5": "(a+b)^4+(c+d)^4", "4c7765e1a2849449eac97b465a8cd436": "HD^{16}O", "4c77ce7060c4f31e04a35aa80352184c": "\\mathbf{E}\\,", "4c78df7bd5e4de7d22eb028b58410724": "x_{n+1} = \\frac{(x_n + 5/x_n)}{2}", "4c791baefb0272a077eada0c9f2facce": "(\\mathbf{x}, e_1)", "4c791ff8aa932bc841fe40d74445ca59": "\\mu \\ll m_e", "4c7920194de2d52d6810a1140ead88db": "\n \\quad (6) \\qquad e^{a\\Delta t} = 1 + \\frac{\\alpha \\Delta t}{\\Delta x^2} \\left(e^{ik_m \\Delta x} + e^{-ik_m \\Delta x} - 2\\right).\n", "4c796fbdf20375aa86e836101f2754ce": "m_3 =17", "4c79804b43b0f93ccb823fe6eca61303": "(X_i)", "4c79d4be59eea9035bf00a2f9e7527f8": " k \\approx \\sqrt{n/2} ", "4c7a15ede9844ac8dba2d7415f627b96": "4a^2x^2 + 4abx + b^2= - 4ac + b^2", "4c7a1a3e838529769d21d03c3c6a807b": "\\cos^2x=1-\\sin^2x", "4c7a9d0843e9537f4830aa5876532b7e": "\\dot{W}_S", "4c7ad13bdeb37bfea7534aee8065b155": " G \\leq S_n ", "4c7aecf1020cf64469d3d544f729db42": "(14.a)\\quad \\nabla^2 \\psi =\\,(\\nabla\\psi)^2", "4c7b679e0d3f8ac1664adbcec246e1b9": "w_{k}", "4c7bc55637d1b5ad2bd66fefe783d847": "\\varepsilon_e", "4c7bcbae97ad322481c5cf3d35d7ac9b": " W ", "4c7be0021ac4c97d3a1127dc0f08526f": "{\\color{white}\\frac{d}{dx} 2^x} = 2^x \\lim_{h\\to 0} \\frac{2^h - 1}{h}", "4c7bfdaab06155a92873ce3c8d3fb04d": "\\displaystyle \\gamma_\\mu \\gamma_\\nu = - \\gamma_\\nu \\gamma_\\mu", "4c7c1e75bd0b277c10b0555e56725e5e": "f : \\bigcup_{S \\in \\mathfrak{P}(V)}\\sum{v_i \\in S} \\left( \\{v_i\\} \\times D_i \\right) \\leftarrow \\mathbb{N} \\cup \\{\\infty\\}", "4c7c71c84e4beb77062f4a10d91ec05f": "N \\approx \\frac{1}{2\\;\\mathrm{NA_i}}", "4c7c7486c26ca92d4e22c1e79e2dabc9": "\\Phi_{ij}=\\,\\text{Tr}\\,(\\digamma_i \\,\\bar{\\digamma}_j)", "4c7c7beb1b081e02f7f94709b433e52d": "\\mathbf{H} \\!\\,", "4c7c97054a5b4b33ebca7ee6171492a6": "\\tilde\\gamma", "4c7ca0016bb397ef598964bd6016f971": " \\{C_i^c \\}_{i \\in I} ", "4c7d32b8f3163f366715b5e0ee05e7d4": "|\\det N| = \\bigg (\\prod_{i=1}^n \\|v_i\\| \\bigg) |\\det M| \\leq \\prod_{i=1}^n \\|v_i\\|.", "4c7d5a3835d08d46d246605c28956c11": "M(j,j) = 1 - \\frac{ \\sum_{i=1, i\\neq j}^{20}\\lambda m(j)A(i,j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)}", "4c7d6b78e12674bd9d43bc1cccb37b0a": "\\psi_n(\\mathbf{r})", "4c7d8f206665985953eee7b73e7f4714": "\\begin{align}\n\\left |\\operatorname{E}\\left [f(Y_n)\\right ] - \\operatorname{E}\\left [f(X) \\right ]\\right | &\\leq \\left|\\operatorname{E}\\left[ f(Y_n) \\right ]-\\operatorname{E} \\left [f(X_n) \\right ] \\right| + \\left|\\operatorname{E}\\left [f(X_n) \\right ]-\\operatorname{E}\\left [f(X) \\right] \\right| \\\\\n &\\leq K\\varepsilon + 2M \\operatorname{Pr}\\left (|Y_n-X_n|\\geq\\varepsilon\\right )+ \\left |\\operatorname{E}\\left[ f(X_n) \\right]-\\operatorname{E} \\left [f(X) \\right ]\\right|.\n \\end{align}", "4c7db8067fa75073b61d199c1a173583": "\n\\zeta_i(t) = \\frac{\\sum_{k=1}^m \\mu_{(k)}(t) \\mathbf{1}_{\\{\\mu_i(t) = \\mu_{(k)}(t)\\}}}\n{\\sum_{l=1}^m \\mu_{(l)}(t)}\n\\qquad \\text{ and }\n\\eta_i(t) = \\frac{\\sum_{k=m+1}^n \\mu_{(k)}(t) \\mathbf{1}_{\\{\\mu_i(t) = \\mu_{(k)}(t)\\}}}\n{\\sum_{l=m+1}^n \\mu_{(l)}(t)}\n", "4c7e09366f57fefed4d29862fdf36823": "\n\\{\\mathbf{e}^1, \\mathbf{e}^2\\} = \\left\\{\n\\begin{pmatrix}\n 1 & 0 \n\\end{pmatrix},\n\\begin{pmatrix}\n 0 & 1 \n\\end{pmatrix}\n\\right\\}\\text{.}\n", "4c7e2d92fb5d58a72e942cd3a279e650": " \\textbf{h} = p\\textbf{f}_q \\cdot \\textbf{g} \\pmod {32} = 8 + 25X +22X^2+20X^3 + 12X^4 +24X^5 +15X^6+19X^7+12X^8+19X^9+16X^{10} \\pmod {32} ", "4c7e94cc1fe1c1e3252a13d1e43a42b1": "f\\mapsto f'", "4c7e9711bc79c6268e0f94eccfef02d4": "\\frac{4}{9\\pi} \\frac{(\\log g)^2}{g}.", "4c7ea28bf45de1d1361ff540a6d1e2af": "\\frac{\\partial v(p,w)}{\\partial w} \\neq 0", "4c7ee58ec03d76249a7e06e8c3908af6": "\\; t_0 \\equiv t_1 \\in \\Phi", "4c7f47ef5b1d07e65ec43bef2291c95d": "\\sigma(\\bigoplus_i A_i) \\ge 1 - \\prod_{i}(1 - \\sigma A_i).", "4c7f8a1b38dd85e2768d0f5a692e47f6": "Twist = \\frac{C D^2}{L} \\times \\sqrt{\\frac{SG}{10.9}}", "4c7fbf6f57b0376fff1f6731bc6d94f0": " E = \\frac{p^2}{2m}+V(x,t)=H.", "4c80d9209341810145da138e6b7066a6": "\n EI w_B = \\dfrac{Pba^3}{6L} -\\cfrac{Pba}{6L}(L^2-b^2) = \\frac{Pba}{6L}(a^2+b^2-L^2)\n ", "4c812c2e1f944fcede5145df8b2ff758": "P(i\\xi)", "4c820f343a4a4b57b301a177e34fdd98": "\\bar{\\Phi} \\left[ \\mathbf{r} \\right]", "4c8235fdab8ac575632fc369b736e015": "\n\\frac{\\delta(v)}{v\\gamma(v)}=\\frac{\\delta(v')}{v'\\gamma(v')}.\\,\n", "4c8252a32952cef87863895246b67499": "\\nabla \\cdot \\mathbf{D} = - \\rho", "4c8255ca4cd0d9cf52fc6109885cc33b": "\\mbox{NP} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NTIME}(n^k).", "4c826ebc8c4b16be5ec3e081259f3905": " i_{\\text{E}} = I_{\\text{S}}\\left(e^{\\frac{V_{\\text{BE}}}{V_{\\text{T}}}} - e^{\\frac{V_{\\text{BC}}}{V_{\\text{T}}}}\\right) + \\frac{I_{\\text{S}}}{\\beta_F}\\left(e^{\\frac{V_{\\text{BE}}}{V_{\\text{T}}}} - 1\\right)", "4c82a646cdc9102e89800d1326bff96d": "\\frac{b}{4\\pi c^2}", "4c82fe1fa05354a64fc9ee2b5b1116a7": "\\quad\\quad\\lambda_3 = \\lambda_2^* = -1/2 - \\mathbf{i}\\sqrt{3}/2", "4c836319a40a04641c81d556d204a53c": "Z_0 = \\sqrt {{R_1}^2 + 2R_1 R_2}", "4c83bc259c8142d2d0f3e7f3b1b5e05b": "a \\in \\mathbb{R}", "4c83ced43e6b9a390f0c566492baa742": "S^-", "4c83ea36eca0294992b067d330b3608d": "\\,{l_{x + 1} \\over l_x} = p_x", "4c842669abf8662e747b085315cc391d": "P_1(n) = \\frac{n}{1} = {n+0 \\choose 1}", "4c8462c2353cd0c9b87b18355a2a68b9": " \\displaystyle \\pi=2+\\cfrac{4}{3+\\cfrac{1\\cdot3}{4+\\cfrac{3\\cdot5}{4+\\cfrac{5\\cdot7}{4+\\ddots}}}}", "4c84b6510326ad4603b9ffbe198faef6": "\\scriptstyle v_i>0 ", "4c84bd4315a5dbda5ecf71302309afe3": "v_1 = 1 ", "4c84cecfb5fd8821f1bffe66710e904b": "\\ln(z_2)=\\ln(K)+\\int\\frac{r_1(z_1)}{a-r_1(z_1)}\\,\\frac{dz_1}{z_1}", "4c84d85fd8504f28c396b418b6eeb520": "PV_m=RT\\left(1+\\frac{B^\\prime(T)}{P}+\\frac{C^\\prime(T)}{P^2}+\\frac{D^\\prime(T)}{P^3}+...\\right)", "4c85068f146f64c35f7cfc43bc87f9c3": "n^{(2)}", "4c8545a99bc22742dc458f8a1fe3cf4f": "S_1 \\equiv_3 S_2 \\Leftrightarrow Cr_\\sigma(S_1) = Cr_\\sigma(S_2)", "4c8610134c1f3582fa8670b021fe04cf": "92^2", "4c864d84bd5db5a2e203e662f697230f": "U=TS-PAx+\\mu N+mgx\\,", "4c8658a0ccaf80bb3fbe4867923b2851": "[\\hat{\\mathbf L}^2, \\hat H]=0", "4c8661281e6d75d6d46b428407418637": "e=(v_1,v_2)\\in E", "4c86bc912564e219b3445310a8e9609e": " \\prod_{k=1}^{n-1} \\tan\\left(\\frac{k\\pi}{n}\\right) = \\frac{n}{\\sin(\\pi n/2)}", "4c86f070327c9ee0537b429664a686d8": "f = SV/L", "4c871584d9b1ff4a0e20f10510a465ba": " L_1 = \\{ a^m b^n c^n : m,n \\ge 1 \\}", "4c88105872afd31de14b4f09196eff2c": "m_1+m_2=m", "4c883768b4e36536a8cc06da7cd5119c": " \\xrightarrow{\\ p\\ } ", "4c88643e8fcde94d875db029d1db32f7": "\\ V_x=f(x,y)", "4c887bd62fc535b6789d2a192e84e73a": " a \\in F -\\{0\\}", "4c8891a2e8411edd96b5469ab4403b44": " \n\\mathcal{A}_2=(p_1\\partial_x+p_2\\partial_y+p_3)(p_4\\partial_x+p_5\\partial_y+p_6).\n", "4c88cc637ab22c0ed868a8f7d2578406": "\\frac{df}{dr} = \\left( \\beta^{2} - 3 \\right) \\frac{f}{r}", "4c88e944146b75a3326db53de4d23211": "SD = D[3,2] = d_{32} = \\frac{d_v^3}{d_s^2}. ", "4c892c414132ba68ef441dfcb564117c": "T_{i_1} \\times \\dots \\times T_{i_k}", "4c8942cc7922b57e64011bb0235e3736": "\\mathit{TWA} \\subsetneq \\mathit{REG}", "4c89cf85294adf8f18cd59de1e1102e0": "\\hat A\\hat A^\\dagger -\\hat A^\\dagger\\hat A =\\hat a\\hat a^\\dagger -\\hat a^\\dagger \\hat a=1.", "4c8a11263127fe1f0f652e728bba1ba3": "\\mathrm R^*+\\mathrm{SO}_4^{-\\bullet}+\\mathrm{OH}^\\bullet\\longrightarrow n\\mathrm{CO}_2+\\dots", "4c8a35e40e1bf20eaecf9323d237d17b": " \n\\mathrm{T_n}(\\mathbf{R})_{k'k}\n\\equiv \\langle \\chi_{k'} | T_n | \\chi_k\\rangle_{(\\mathbf{r})}\n = \\delta_{k'k} T_{\\textrm{n}}\n + \\sum_{A,\\alpha}\\frac{1}{M_A} \\langle\\chi_{k'}|\\big(P_{A\\alpha}\\chi_k\\big)\\rangle_{(\\mathbf{r})} P_{A\\alpha} + \\langle\\chi_{k'}|\\big(T_\\mathrm{n}\\chi_k\\big)\\rangle_{(\\mathbf{r})}.\n", "4c8ac923edaa72e6f564f4294d4c00e5": " FNR = E\\left( {\\frac{T}{{m - R}}} \\right) = E\\left( {\\frac{{m - {m_0} - \\left( {R - V} \\right)}}{{m - R}}} \\right) ", "4c8b2c3ca4e821c713cae8f818dead5d": "HA^- \\rightleftharpoons A^{2-} + H^+ :K_2=\\frac{[A^{2-}][H^+]} {[HA^-]}", "4c8b7f657ca895fe622f25fa00a1015c": "[0,\\ 1]", "4c8b9d8e9b6cb39a39a4317cd9f9cc86": "\\mathfrak{g}_{\\pm 1}", "4c8bdc305c2eb404e09597f8e86d65f5": " H(t) \\ = \\ H_0 + W(t). ", "4c8c9b1ea775b7f6b0666ebb4110f8b3": "\\omega(x,f) = \\bigcap_{n=1}^\\infty \\overline{\\bigcup_{k=n}^\\infty \\{f^k(x)\\}}.", "4c8cf659dcd302669cec36be03e5b579": "\\mbox{percent yield} = \\frac{\\mbox{170.0 g PbO}}{\\mbox{186.6 g PbO}} \\times \\!\\, 100 = 91.12\\%", "4c8cf9860d9c7e1d3d1296a7c940642b": "x < y ", "4c8d2fc23ab8e1eadabb2c0038bd835a": "\\Phi(1.64-\\sqrt{n}/\\hat{\\sigma}_D) <0.10.", "4c8d32a73b9c3ecf327eb999a00a7816": "d^\\mathrm{th}", "4c8d36998277c37bdc266036a36eb936": "\\scriptstyle \\,{}^{(-1)}i \\;=\\; 0", "4c8d4e2aa3675e88100c1fd7aba2cf9f": " \\left(\\sigma^{\\alpha}, \\frac{\\partial^{|I|} \\sigma^{\\alpha}}{\\partial x^{|I|}}\\right) \\qquad 1 \\leq |I| \\leq r. \\,", "4c8d95c820e2c1645251be29e99c470f": "f(x_i,\\boldsymbol \\beta)= \\alpha e^{\\beta x_i}", "4c8d9f47cdf83558de6b7faa85a2e059": "P_0 = \\frac{D_1}{r}", "4c8deaf4ed92162cab93d7b3e7324785": "f(z)=\\sum_{n=0}^\\infty a_nz^n", "4c8df82a5f3b2fbc7f4f8e5a3994fa1b": " d = 20.362955 + 29.530588861 \\times N + 102.026 \\times 10^{-12} \\times N^2", "4c8e1e675c18bf82a0caf99ae6edfddf": "\\mathbf{q} = \\left[ \\mathbf{u} \\quad \\mathbf{\\Psi} \\right]^T ", "4c8e62c37be4ac429f86ddc67c836a89": "\\sqrt{1+\\alpha^2}", "4c8e67e4f49134866e468e70fb72d7f2": "\\begin{align}\n\\sigma(A)^2 & = \\langle(A-\\langle A \\rangle)^2\\rangle \\\\\n& = \\langle A^2 \\rangle - \\langle A \\rangle^2\n\\end{align}", "4c8e71e1392f844559045a957f3b4af4": " r^2R''(r) + c rR'(r) + d R(r) = 0 ", "4c8e8f995b26aeca8a77c2a01d5def1d": "\\varphi_{\\alpha} (x) = \\frac{\\alpha}{2} \\| A_{\\alpha} x \\|^{2} + \\varphi (J_{\\alpha} (x))", "4c8eb90d094c8c10ecd0fd3a7fb1e6ff": "\\mathbf{A} = \\mathbf{U} \\mathbf{V}^*", "4c8ee04af7cef74f8f9368838c67c959": "i^* \\leftrightarrows i_*=i_! \\leftrightarrows i^!", "4c8f303809b3b80c777275a8aace14c1": "Vp=Vn=0", "4c8f3f1c6f0b8bc980c35e5b1c0ff131": "\\scriptstyle x^2 - N y^2 = 1", "4c8f3f92f573fd363fc195f024038cb7": " \\begin{align} \n\\mathbf{A} & = (A^0, \\, A^1, \\, A^2, \\, A^3) \\\\\n& = A^0\\mathbf{e}_0 + A^1 \\mathbf{e}_1 + A^2 \\mathbf{e}_2 + A^3 \\mathbf{e}_3 \\\\\n& = A^0\\mathbf{e}_0 + A^i \\mathbf{e}_i \\\\\n& = A^\\alpha\\mathbf{e}_\\alpha\\\\\n\\end{align}", "4c8f7527a1365791a0220932999e4c84": " y_i \\in R ", "4c8f8fc64bc9576024acf8d5ddc4d1a2": "\n \\varepsilon\\,|\\,X\\ \\sim\\ \\mathcal{N}(0,\\, \\sigma^2I_n).\n ", "4c8fb1ffb59dc21f0de96ee53ca62660": "4\\times 10^{10}", "4c8fb7f08193726199443bd5f8be8904": "yri[ab]", "4c9014c40ed1590f63b4f813d33858de": "R_e < 1", "4c903c1c43b2e0891ce72a805a5dbf7e": "\\tilde v_{ij}", "4c90697a5847a2187daf78c18697b3fe": "[D_P] = [1 [P] - 1 [O]]", "4c90848b5ed681571624755be5d36d38": "\\overline{E_0}", "4c90a8d5f7d10982c5b119b0b3e75a23": "g\\in G,p\\in P \\Rightarrow \\exists h\\in G, h^p=g\\;", "4c90b6684177c363378d4d53c53e1417": "\\text{Ohms conversions:}", "4c90bf85c9b7c20d69c8e8c9d3f30f5b": "::=\\,", "4c90d022c50c1d065d389c9c30b40530": "s=\\{s_1,\\dots,s_n\\}", "4c90f927af458f729bd7aad71752b028": "P(E,\\Omega)<+\\infty", "4c914504ec2b6375d172609679e7e47e": "\n\\begin{align}\nI(\\theta)\n&\\propto \\cos^2 \\left [{\\frac {\\pi d \\sin \\theta}{\\lambda}}\\right]~\\mathrm{sinc}^2 \\left [ \\frac {\\pi b \\sin \\theta}{\\lambda} \\right]\n\\end{align}\n", "4c914f83353a60b46d1e6a904087e4a8": "J\\sum_{n\\ne 0} a_ne^{in\\theta} =\\sum_{n\\ne 0} i\\, \\mathrm{sign}(n) a_n e^{in\\theta},", "4c91790a26fbf35b1a18867ed8acd91e": "\\nu_{f}", "4c91b246bb2d1dadc430ab7ac1affa99": "A[1]", "4c92b86f3ec797b2f4fca442790571ab": "\n\\sum(X_i-\\mu)^2=\n\\sum(X_i-\\overline{X})^2+n(\\overline{X}-\\mu)^2 ,\n", "4c9464b909a46552a41b43c2c4d7f4b2": "\\frac{d\\mathbf{x}}{dt} = \\mathbf{f}(t,\\mathbf{x})", "4c9508dc2170cbac5fa04d2d68291f0e": "\\rho(\\theta) \\,=\\, 1 - \\cos \\theta.\\,", "4c95522a24bd695a288d05f61fe7aa82": "\\text{DOR} = \\frac{LR+}{LR-}", "4c9567a6c05ac4bd210545bcf054e352": "A,B\\subseteq X\\,", "4c956957a6420c7b7298251605196165": "C \\subset H", "4c9581e40700e75e7d27482a71c5c919": "n^2.", "4c95ac70765562c0e72cfd00d16bdbf1": "\nP_{m-\\frac12}^n(\\cosh\\eta)=\\frac{(-1)^m}{\\Gamma(m-n+\\frac12)}\\sqrt{\\frac{2}{\\pi\\sinh\\eta}}Q_{n-\\frac12}^m(\\coth\\eta)\n", "4c95f1c6ba2aeec0970652e345b00d51": "\n\\min \\sum_a {\\int_0^{v_a} {S_a \\left( x \\right)} } dx\n", "4c96a0e9c486941924113e1bb39c6aa4": "x_o", "4c96cd309c90e35ec8c2dcd0b3f08a44": "\\langle.,.\\rangle: E\\times E\\to M\\times\\R", "4c9714216d46ef0a4f969c7e636aca70": " \\sum_i C^J_i \\varepsilon^i_S = 0 ", "4c97206be603a03d430b6718a4e6a3d6": "G(z) = \\left(\\frac{p}{1 - (1-p)z}\\right)^r.", "4c97257641499119c99108e0becc178f": "(1-\\epsilon)", "4c977a36aae7ea259137781190e35ea9": "\\operatorname{E}(w_iz'_i) = \\frac{1}{n}\\sum_i w_i z'_i n_i", "4c984e5e6b255700ce8397efb693daee": "m[n,W]", "4c98cbd6b766b4f749845dcaf6952d36": "1- \\exp\\left( - F \\frac{\\pi r^2}{S}\\alpha L \\right) ,", "4c99de43c42b466a53c6d56eddb9405f": "\\tilde{\\Lambda}_k = \\textbf{H}_k^T \\textbf{S}_k^{-1} \\textbf{H}_k + \\hat{\\textbf{C}}_k^T \\hat{\\Lambda}_k \\hat{\\textbf{C}}_k", "4c99e7322e4eecdbe5d83d79599cab84": "\\beta \\ge 0\\,", "4c99eceeba03c60bffe626ddeba00586": "2N\\mu", "4c9a28717689227771d0a91d6582aad6": "\\{ K( \\alpha ) \\ : \\ \\alpha < \\beta \\}", "4c9a499d996655b297f626574da537c8": "f(x)=1/x, f(x)=\\sqrt{x}, f(x)=\\frac{ \\sqrt{1+x^3}}{x^{3/7}-\\sqrt{7} x^{1/3}}", "4c9a884899bc643cb385a9b93c081dc5": "A\\in R^{m \\times n}", "4c9a9b2bac4b736cea9dee86ac158fea": "^hP(x,y,1)=P(x,y,z),", "4c9aa4d164cf75beb1c2ef3215a2107b": "K_a = \\frac{k_{on}}{k_{off}} = \\frac{[AbAg]}{[Ab][Ag]}", "4c9b0433e00ff72f147a7682aedbbb65": "\\frac{D}{Dt}", "4c9b2d33edec8897a80f4822f47262d7": "|\\mathop{\\text{Im}} \\, s| \\leq \\beta", "4c9b529ba27269c172cef70b655a84f4": "t_2=\\tau\\ln{10}\\,", "4c9b6de8e938a37bd12b936a664415e7": "\\theta_f", "4c9baf40af3ecf92aa82cac8c3df32e3": "n = 6,7", "4c9bca7a61d109204e7b35c487607c7f": "p_i^{m_i}", "4c9bd9abd423596725bcf791e9b7cdbf": "\\bold{p}\\rightarrow \\bold{p}", "4c9bddfb49c9b19095890c8453421f12": "P(M|\\mathbf{E}) = \\frac{P(\\mathbf{E}|M)}{\\sum_m {P(\\mathbf{E}|M_m) P(M_m)}} \\cdot P(M)", "4c9bea77a84c6f58633dd59d6ddee8dd": "\\frac{\\partial f}{\\partial\\overline{z}} = 0,", "4c9c0631e333c112d910074d02b0d756": "x_0^*= x_0", "4c9c3031e7fe5acd457d5c2606a389d2": "P = K \\rho^{1+\\frac{1}{n}} ", "4c9c4102f71c560b0c392f3507cf515e": "\\xi_0(x) = 1", "4c9c87596617d4ae7469d6cb8ff133ea": "x\\in I", "4c9c8bf5a2e9ea70060998dc9648facc": "a \\mapsto O_{p'}(C_G(a))", "4c9cc904c38bd047fac3224fb514fb96": "0\\leq a_i\\leq 9", "4c9cf021f6cae094999f0f1dfb117888": "\\mathbf{\\nabla} \\times \\mathbf{E} \\neq \\mathbf{0} ", "4c9d3e92e7d932feb7aa28a110b61e4d": "z=jx_L=j \\frac{\\omega L}{Z_0}=j\\omega LY_0\\,", "4c9d52f476949883f9602ebb0a6306f7": "b^n", "4c9d73a3e93d30706546d40fa9bfa3d1": "b > \\arcsin (\\sin c\\,\\sin\\beta)", "4c9d99c9461b3ebb1238841b29aaa5e3": " 0 = f(0) = f(x-x) = f(x) + f(-x) \\ ", "4c9e2bef54c492cb4d5888a77bbcd569": "r(v|W) = (d(v,w_1), d(v,w_2),\\dots,d(v,w_k))", "4c9e368bce05a0cc4074d5625a4d92b0": "\\int_N^{M+1}f(x)\\,dx\\le\\sum_{n=N}^Mf(n)\\le f(N)+\\int_N^M f(x)\\,dx.", "4c9e4b38a1bd1693bb96251f2c0bc6c1": "a_m = 0.", "4c9e77fc43dc6cac78edd7ae18a861ed": "\\theta \\in \\mathrm{Lin} (E; G)", "4c9e938ca062c2b46f6d0055a5ba9e90": " (x_1 - x_2)(f_{(1)}(x) - f_{(2)}(x)) \\ge 0\n ", "4c9e9dab6bd3e265c3e9859db7d5b600": "\\ v_a\\ ", "4c9f0b4c353277804fabdbb72eb409f3": "\\bold{u}_0 \\in U", "4c9f3b0b2fbc83446ceae27f79119e86": "y(t)= 0 =\\left(v_{bullet}\\sin(\\delta\\theta)-\\frac{1}{2}g t \\right)t ", "4c9fb4f9b9f9e170e840724a81266c51": "\\mathbb{E}^g[X] := Y_0", "4c9fd6d5586350bbcf133d444d0a5d6b": "{}G= \\sqrt{r^2- \\tfrac{M^2}{4}}", "4ca0420b71fccfe4a8745cc65ae16ae5": "\\frac{\\partial \\phi(x,t)}{\\partial t}=\\nabla\\cdot (D \\nabla \\phi(x,t))=\\sum_{j=1}^3D_{ij} \\frac{\\partial^2 \\phi(x,t)}{\\partial x_i \\partial x_j}\\ . ", "4ca08b8f3896ab29d150ba14fcc75bdb": "=\\tau^\\alpha\\,", "4ca0abd9089b88a264967803b6ae2800": " \\partial_t q = (s_r+ i s_i) \\Delta^2 q + (d_r+ i d_i) \\Delta q + l_r q + (c_r + i\nc_i)|q|^2 q + (q_r + i q_i) |q|^4 q.", "4ca178b970c98a657f723c460a662d61": "\n\ns (t) = (1 cm) sin (wt) \\!\n\n", "4ca2044761796c7c491738907e00c5a2": "N^{\\mathrm{th}}", "4ca23617efc6a3034da57dad6fbd055c": "\\frac{\\pi}{4} = 4 \\arctan\\frac{1}{5} - \\arctan\\frac{1}{239}\\!", "4ca2d62090e217e9d047ed603d752909": "3 (1+0+0) + 7 (1+0+2) + (1+0+5) \\mod 10 = 0.\\,", "4ca2fba347e5676d9576e4e4d7a1749e": "\\mathbf{A} \\cdot \\mathbf{B} = A_{\\mu} \\eta^{\\mu \\nu} B_{\\nu} ", "4ca357cd46c6f1e56a1fbdbc8feaf23f": " \\sum_{r'} (\\;A[r',k] - r' {dA \\over dJ}[k-r']\\;)\\left(\\; B[0,r'] +(k-r'){dB\\over dJ}[r']\\;\\right)- \\sum_r A[r,k] B[0,r]\n\\, ,", "4ca3c1674b5db5b7c45509b08fc065d0": "\\mathcal{E}_\\mathrm{rms}, \\sqrt{\\langle \\mathcal{E} \\rangle} \\,\\!", "4ca3c484d19f0fa443ac03df25e87a58": "x\\backslash y = x^{\\lambda}y", "4ca3fd13379af50cede1a9dfbc9645e3": "F[y]=y^{[-1]} \\ ", "4ca407f24821d0b2db2ce90f886595f2": "w_{x_i}", "4ca411e79d2f5a08dcdb848e8e308136": "u(t)=LI_{ex}w_{ex}\\cos(w_{ex}t) + F_{Rog}\\frac{d}{dt}\\left[\\int_0^{Lp}H(l)dl\\right]+F_{Neel}\\left[\\int_0^{Lp}H(l)dl\\right]\\frac{I_{ex}^2}{2}w_{ex}\\sin(2w_{ex}t)", "4ca41e0919e5dc7515dba9178ac85de7": "\\nu(x)dx.", "4ca4d4eb857fda2b4ac697df9432a539": "v_1,\\dots,v_n", "4ca4f08b81f2a9db9771dd16e78f2581": "X(t)^*X(t)", "4ca533365b17962917a503e64ec69fde": "\\hat{\\lambda}_n = 0", "4ca577a1a48fada2282cae1708a7b9c0": "\\psi(u)", "4ca5a03b8327058b2935713d9ae372b0": "\n{EL}_{oil}=\\frac{{WI}\\times{LOE}}{{NRI}[{P_o}+({P_g}\\times{GOR})/1,000]\\times(1-{T})}\n", "4ca5a175ede0ebaa126b939d246f8662": "\\Phi_{in}", "4ca5b9be9781ecb0819ae3c4e4458e89": "\\ \\frac{Dp}{Dt} = 0.\n", "4ca5f6253d02379139e34b06aaa7925f": "\\frac{T_0}{T} = 1+\\frac{\\gamma-1}{2}M^2\\,", "4ca617bc1fdf9b46d800e467acb43792": " \\square=\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}. ", "4ca62089fe885bade590045b97591e93": "\\textstyle \\mathbb{F}_{256}", "4ca631186170dbccbf719e4eeb2a2e82": " \\frac {L}{hr} \\cdot hr = L", "4ca64684bfef6cccea2e310568ef4f52": "B_r = -2B_0\\left(\\frac{R_E}{r}\\right)^3\\cos\\theta", "4ca65c07faf8f5ad4d80fe1372c3dd73": "\\tfrac3{160}", "4ca6a29419ec2882b164816ffb0c1b3d": " A = \\frac{45}{2} \\frac{\\zeta(3)}{\\pi^3}\\approx 0.872284041.", "4ca6d5547bae9bc79cebed4daa7003d9": "E=p^2/(2m)", "4ca6e8e8ceb43a8a4efb90d5e848668c": "(a\\lor b)", "4ca6f2b99b0e551603f12d35b898c60c": "s\\left( t \\right) = \\sum_p B_p h_p(t) = \\sum_p B_p\\left( \\frac{\\alpha}{\\pi} \\right)^\\frac{1}{4} e^{-\\frac{\\alpha}{2}(t-T_p)^2 }e^{jt\\Omega_p}", "4ca7387bddca5ddecf22093b4a523fc5": "\n\\begin{align}\n\\operatorname{pmi}(x;y) + \\operatorname{pmi}(x;z|y) & {} = \\log\\frac{p(x,y)}{p(x)p(y)} + \\log\\frac{p(x,z|y)}{p(x|y)p(z|y)} \\\\ \n& {} = \\log \\left[ \\frac{p(x,y)}{p(x)p(y)} \\frac{p(x,z|y)}{p(x|y)p(z|y)} \\right] \\\\ \n& {} = \\log \\frac{p(x|y)p(y)p(x,z|y)}{p(x)p(y)p(x|y)p(z|y)} \\\\\n& {} = \\log \\frac{p(x,yz)}{p(x)p(yz)} \\\\\n& {} = \\operatorname{pmi}(x;yz)\n\\end{align}\n", "4ca74045563b9f20fa958a075aee9046": "{E(b,N)}", "4ca74dacb90aa1ef69560eacc7b6a1e3": "\\xi\\cdot v+dv(X_\\xi)=0", "4ca77b3e60b7053a390a2d1e723085e7": "c^{(l,m)}", "4ca77b5bc8558749466fb5798ea95fbd": " n = \\sqrt{ 1 + \\chi} ", "4ca78a47847c21bcbe8da8076f8b9293": "E_F = \\frac{\\hbar^2 \\pi^2}{2m L^2} n_F^2 = \\frac{\\hbar^2 \\pi^2}{2m L^2} \\left( \\frac{3 N}{\\pi} \\right)^{2/3}", "4ca79388e82f1d8d8d0ce9ee985576f8": "67\\frac{1}{5}", "4ca7bd5bd58140ac183227d174a11202": "r\\equiv|\\mathbf{r}|", "4ca802eeac8f723f6c8d6e897ed85dce": "\nI(\\theta)\n=\n -\\mathrm{E}\n \\left[\n \\frac{\\partial^2}{\\partial\\theta^2} \\log f(X;\\theta)\n \\right].\n", "4ca834b7b9443e088fe75f7472464731": "P\\ =\\ 2\\pi\\ r\\ \\sqrt{\\frac{r}{\\mu}}\\,", "4ca8698050d5f5934b7bad6e4f152e47": "\\frac{E}{c^2} = m_\\mbox{original}-m_\\mbox{final}", "4ca896f09d809509bff7b57f81fd1ca5": "\\gamma a_i \\equiv \\zeta_n^r a_p \\pmod{\\mathfrak{p}}", "4ca8a00827a674a079ef63c1d6b49eb6": "\\dot{Q}=\\frac{T_{surf}-T_{envr}}{\\left ( \\frac{1}{h_{conv}A_{surf}} \\right )}", "4ca8cb705da5b3fe43655846ddb4ebb2": "U = \\frac{1}{2} C_{ijkl} u_{ij} u_{kl}", "4ca8fa91b7487525e85442202346d0c9": " X = \\sum_{i=1}^{k} Z_i, \\quad Y = \\sum_{i=k+1}^{n+1} Z_i,", "4ca8fd98055628cfa1b5d0562901b400": "y=\\lambda a^{\\gamma x}", "4ca90a30a8542496a131ccf538056077": "\\begin{align}\nK_\\nu(x) & = \\textstyle\\frac{\\pi}{2} i^{\\nu+1} \\big(J_\\nu(ix) + i N_\\nu(ix)\\big) \\\\\n & = \\begin{cases}\n \\displaystyle \\frac{I_{-\\nu}(x) - I_\\nu(x)}{\\sin \\nu\\pi}, & \\text{for } x \\ge 0 \\text{ and } \\nu \\notin \\mathbb{Z} \\\\[10pt]\n \\displaystyle \\frac{\\pi}{2} \\lim_{\\mu \\to \\nu} \\frac{I_{-\\mu}(x) - I_\\mu(x)}{\\sin \\mu\\pi}, & \\text{for } x < 0 \\text{ and } \\nu \\in \\mathbb{Z} \\\\\n \\end{cases}\n\\end{align}", "4ca93686496644ff15f94d1d453a4837": "|x/y| = |x|/|y|.\\,", "4ca99d53e7b31cfc4531890ab8d5a7ae": "D_P = 1 [P] - 1 [O]", "4caa5cd5a30e11c909ff2b3594e2dab6": "\\alpha_2", "4caae6c84015f92031b480a4d051f3de": " \\sigma_{DC}(p) \\propto \\sigma_d (p_c - p) ^{-s} ", "4cab3092a96816cf236058a44664c86b": "\\frac{H_{k,q,s}}{H_{N,q,s}}", "4cabbd9578e4e6304ed31d8c2bdecbab": "r e^{-i \\phi}", "4cabfe8efe6a6047bf57288311cd50ca": "\\tau^*", "4caca7747783f74ea8ad2bce3c07a52a": "f \\colon T \\to A, g \\colon T \\to B", "4cacca79542a92a74c222f7651e9dc9a": "R = \\left(1.25 \\times 10^{-15} \\mathrm{m} \\right) \\times A^{1/3}", "4cace302c3b761b6f18ee4ebb75607ae": "E \\in H", "4cad1f9e2f8a76b1fdd883b125ba82f6": "O(|E|\\sqrt{|V|})", "4cad2a6f2cb910e720dbf4884bddb479": "K_{IC}=\\frac{F_{MAX}}{t \\cdot \\sqrt{w}} \\cdot Y_{MIN}", "4cad33d1149758c929812cc254ca0e63": " \\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R}) \\, dV=0, ", "4cad9556c03ab60c756a290f71fe9571": "f_n(z) = z-1+\\frac{1}{n}, \\qquad z \\in \\mathbb C", "4cada9ecbb331e3f1abd06adc61f0865": "\\scriptstyle\\varphi_\\epsilon", "4cadbaced8b0cac339c9aa594f90419b": "\\int\\frac{\\cos ax\\;\\mathrm{d}x}{\\sin ax(1+\\cos ax)} = -\\frac{1}{4a}\\tan^2\\frac{ax}{2}+\\frac{1}{2a}\\ln\\left|\\tan\\frac{ax}{2}\\right|+C", "4caeaa7057916998a639b533b0994fd8": "\\Delta v_1 = \\sqrt{ \\frac{2 \\mu}{r_0} - \\frac{\\mu}{a_1}} - \\sqrt{\\frac{\\mu}{r_0}} ", "4caefccf64f546dfc52744c41d104816": " \\stackrel{\\triangledown}{\\mathbf{A}} = \\frac{D}{Dt} \\mathbf{A} - (\\nabla \\mathbf{v})^T \\cdot \\mathbf{A} - \\mathbf{A} \\cdot (\\nabla \\mathbf{v}) ", "4caf95d22f18576bd33979ace5be60d2": "y = \\sqrt{1 + x^2}", "4caf98c8dc20b778286e3c127445156d": "\n\\mathbf{\\gamma_t}(i) =\n\\mathbf{P}(X_t=x_i | o_1, o_2, \\dots, o_T, \\mathbf{\\pi}) =\n\\frac{ \\mathbf{P}(o_1, o_2, \\dots, o_T, X_t=x_i | \\mathbf{\\pi} ) }{ \\mathbf{P}(o_1, o_2, \\dots, o_T | \\mathbf{\\pi} ) } =\n\\frac{ \\mathbf{f_{0:t}}(i) \\cdot \\mathbf{b_{t:T}}(i) }{ \\prod_{s=1}^T c_s } =\n\\mathbf{\\hat{f}_{0:t}}(i) \\cdot \\mathbf{\\hat{b}_{t:T}}(i)\n", "4cb06c51ae72087fe3a440c4d56179ef": " (20, 35); ", "4cb06cb9e73a167905e9be51e93e3d19": "\n2V_\\mathrm{harm} = \\mathbf{d}^\\mathrm{T} \\mathbf{H} \\mathbf{d}\n= \\mathbf{v}^\\mathrm{T} (\\mathbf{B}^\\mathrm{T})^{-1} \\mathbf{H} \\mathbf{B}^{-1} \\mathbf{v} = \\sum_{r, r'=1}^{3N-6} F_{r r'} q_r q_{r'},\n", "4cb07a2b319c22a7dcba71499ead24b3": "p_n(z) = \\sum_{k=0}^n z^k \\Psi_k h_k.", "4cb0b9f82c34fa027e3d4e41b755a784": "F(x) = \\frac12 {\\rm Riesz}(4 \\pi^2 x)", "4cb0f011940991c27cd25c37b361e76b": "\nN = \\left(10^{\\frac{L_N-40}{10}}\\right)^{0.30103} \\approx 2^{\\frac{L_N-40}{10}}\n", "4cb0fac6e4d8372b4624f6c6cd924eed": "\\textbf{R}_k", "4cb12084c2613da9ca0cb1977acc8a74": "\\arctan \\frac {a_n}{b_n}", "4cb2e0b2a7dacdc0bcc4025e20401085": "a \\gg b", "4cb2ee26c8b26918a477891370b3c05b": "|n^{1-s}|=|n^{-it}|=1", "4cb2f917cdfffc37f1f6958c7a30922a": "e^{{\\delta}(A+B)} = \\lim_{\\delta\\rightarrow0}[e^{{\\delta}A}e^{{\\delta}B} + {{\\it{O}}}(\\delta^2)].", "4cb3108ee9bd522ca6b99fadb346e87e": "\\mathbf{J}=\\mathbf{I}\\cdot\\boldsymbol{\\omega}", "4cb31f6f387da96be2bb6c89ab69ce9e": "{{f}_{a}}(t)=a(t){{e}^{i\\phi (t)}}", "4cb35b906816a10aa2aacee4430da9ff": "\\text{x y z}", "4cb3628a55e62b685a86b541ae70b29b": "P_{0} = \\mathbf{E}_{0} (P_{t}) ", "4cb369cf00f4db944847e0d8c3d63173": "v_{\\text{out}} = -R I_{\\text{S}} e^{\\frac{v_{\\text{in}}}{V_{\\text{T}}}}", "4cb37b00eaad9b2aaf89bf47aec3b4f9": "\\sqrt{nN}", "4cb3e96b03750fcdaeb7e327bfa5b82c": " c_1 | \\psi_1 \\rangle + c_2|\\psi_2\\rangle ", "4cb4379a637599d279ccbcecca63564d": "\n T(\\hat\\theta) = \\frac{M(\\hat\\theta)+\\frac{k}{2}-C_1}{C_2},\n ", "4cb4610995de470b0a54af3c30d0f75b": "|n\\rangle\\equiv |n(0)\\rangle", "4cb48a226b9392a85cc73aae26a5677f": "k_l", "4cb4b8a32d4582e77d5d6ad463b2b8fd": "v\\in \\mathrm{T}_p S", "4cb500ecb55e2daa86392e00b95c5804": "\\mathbf{O}P^2", "4cb502d8cf0e64d773080a11e78d1510": "\\begin{align}\n x_2 + 2x_3 + s_1 &= 3\\\\\n -x_4 + 3x_5 - s_2 &= 2\\\\\n s_1,\\, s_2 &\\ge 0\n\\end{align}", "4cb50e3afaa0c5aca981c31d34410285": "n \\wr \\wr = \\begin{cases} 1 \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\ n \\leqslant 0 \\\\ (n-2) \\wr \\wr n^{\\big[ \\text{n odd} \\big]} (4/n)^{\\big[ \\text{n even} \\big]} \\quad n > 0 \\end{cases}", "4cb5796a6eef2092af88bc2e67088886": "h=\\frac{v^{2}}{2 g}", "4cb5b1f56d83f6a086382ed050152740": "T [ R] \\subset S \\rightarrow S/L", "4cb5bf619e7c823311b9e26d2613cd07": "\\begin{align} 2\\cdot R_*\n & = \\frac{(87\\cdot 3.55\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 66\\cdot R_{\\bigodot}\n\\end{align}", "4cb5de8680545de73f1dddaf35d38192": "B_0(x) = \\pm \\sqrt{ 2m \\left( E - V(x) \\right) }", "4cb60467ca1f4ba5ad03a3755ddb1956": "\nlocation(S_{0})=(0,0)\\,\n", "4cb621ca2d7cafe1f74829fae6cbfea7": "\\frac{1}{16}", "4cb63128c5fe10995b5fe76cc163aab4": "\\gamma_1~", "4cb64649f1dc2fbb8ad119a53cfc8599": "\\frac{\\delta \\mathcal{S}}{\\delta\\phi}=\\frac{\\partial\\mathcal{L}}{\\partial\\phi}-\\partial_\\mu \\left(\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu\\phi)}\\right)-\\partial_\\nu \\partial_\\mu \\left(\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu \\partial_\\nu \\phi)}\\right)-.~.~.-\\partial_\\sigma \\partial_\\lambda ...\\partial_\\nu \\partial_\\mu \\left(\\frac{\\partial\\mathcal{L}}{\\partial(\\partial_\\mu \\partial_\\nu ...\\partial_\\lambda \\partial_\\sigma \\phi)}\\right)=0.", "4cb6b8d49cf726eacbd876b9c6a527ec": " ~z ", "4cb6ced0496f44479ec6ab682a3077b9": "x^2 = 4py.\\ ", "4cb7070263865430ae04971ffccb65d4": "2 W_{1} (\\mu, \\nu) \\leq C \\rho (\\mu, \\nu),", "4cb7364639aa18cc9e601e89d4410626": "\\scriptstyle (a_1+a_2)\\times (a_1a_2)", "4cb79b5d78cafb40f83f772fcc9964c6": "\\scriptstyle{\\sqrt{g_{00}}/c}", "4cb7a065c4a24cbde58ec24c4abd62d2": "d|\\vec{p}_1|\\,", "4cb7a7edf4d3d46a05648bdccbb82247": "\\mathrm{crd}\\ {36^\\circ}=\\mathrm{crd}\\left(\\angle\\mathrm{ADB}\\right)=\\frac{a}{b}=\\frac{2}{1+\\sqrt{5}},", "4cb7f98195ff81fb7914525ac737d0b0": "\\hat{p}_y", "4cb8d9c59bbdcab175288e3ba9f35274": "\\sum\\frac{\\kappa}{\\sin(\\theta)^3}=0", "4cb9a2fe0fc74628cd647706e01509e2": "\\scriptstyle \\vec F", "4cb9be039935373de7c02e16eaa62bcf": "\\displaystyle \\sum_n \\frac{s_n(x)}{n!}t^n = \\left(\\frac{1+t}{1-t}\\right)^x(1-t)^{-1}", "4cb9e5e9f2265be6e891bf2a4a6a27af": "\n\\textrm{response} = \\textrm{constant} + 0.5 \\mathrm{(all\\ factor\\ and\\ interaction\\ estimates)}\n", "4cba000fcd32225056b2e9793ac3ec24": "\\sum_{n=1}^\\infty \\frac{1}{10^n} = {1 \\over 10} + {1 \\over 100} + {1 \\over 1000} + \\cdots = 0.\\overline{1}", "4cba334dc278f5fc33a04039dd6d63e4": " \\Delta{y} = \\Delta{u} + \\Delta{v}. \\, ", "4cba6e786feb6493989aded4bbe088d6": " s = - {R \\over L } ", "4cba9641523f2d0f222760229114a33e": "\\boldsymbol{L}\\cdot\\boldsymbol{S}= {1\\over 2}(\\boldsymbol{J}^2 - \\boldsymbol{L}^2 - \\boldsymbol{S}^2)", "4cba973622dd4b0eceade757b1d2eb5b": "g_{obs}", "4cbacabeabf59cc1aec155de814dcea6": "\\lambda n", "4cbad299f192c69381ca1d326c4ece24": "f(1) = 555", "4cbbb1c8f630825434012c958b269023": "H(x,y)=(x-y)^2\\, ", "4cbbb763b3a9f0343f16e87429f08382": "\\frac{d^2x}{dy^2} = \\begin{cases}\n\\mbox{unbounded} & \\mbox{if } 2x^2 + 2y^2 = a^2 \\\\\n0 & \\mbox{if } x = 0 \\mbox{ and } y = 0 \\\\\n\\frac{3a^6(x^2 - y^2)}{x^3(a^2 - 2x^2 - 2y^2)^3} & \\mbox{else } \n\\end{cases}", "4cbbeca801fc98f3577ba7bcac8ab2c6": "x_B \\overset {d}{=} (x_A+y)", "4cbc4076acbf4f8fd125d2c7b7595919": "\\epsilon\\varphi(n\\tau)\\,", "4cbd4cd79d65676314a06285ac1e7587": "1, 3, 9, 18, 36, 60, 100, 150 ", "4cbd66962730ea82b25f91fe9921f319": "\\bigcup_{\\alpha<\\gamma} M_\\alpha\\in K", "4cbda50a5bfdf0df3a5d4f7d822fcb6b": "\\textstyle\\deg A_j<\\nu_j\\text{ for }j=1,\\dots,r.", "4cbe294f1f356690e3bcafe30b6a023f": "\\int\\sinh ax\\,dx = \\frac{1}{a}\\cosh ax+C\\,", "4cbe3bd312e60f7557182128d23846da": "k x ( 180 - x ).\\,", "4cbe509eafa064cc9f481c0a454297d4": "{\\color{Blue}~2.11}", "4cbebe5cd46f97bf1abe95f2034fd3a0": "y_L+F_1\\cdot y_F+P_1\\cdot y_P\\geq S_1", "4cbf153310325db7b2e4397136ea0535": "u\\notin C", "4cbf1a4248ef06a34211e1db76c0b00a": "(\\pi+20)^i=-0.999 999 999 2\\ldots -i\\cdot 0.000 039\\ldots \\approx -1", "4cbff768d99fb1b5d7df884ca7dd6c6e": " E(x,y,z)=\\frac{e^{ikz}}{i \\lambda z} \\iint_{-\\infty}^{+\\infty} E(x',y',0)e^{{ik \\over 2z}[(x-x')^2+(y-y')^2]}dx'dy' ", "4cc09c1eee8b979e9ea1ff822b3fd26c": "Cp", "4cc0c17327cf9d7785f257421b608997": "\\mathcal{C} \\subseteq \\mathcal{H}", "4cc0d03c7d577c8204d58d152375ddc9": "a + b", "4cc0d0827547850794cfa72bd21e5936": "\\phi: A \\to A / \\mathfrak{m}", "4cc122c36e127cbabf2e2f63ced568e4": "\\alpha_{x}", "4cc1393bf12b8de46fb7dfbd804893ce": " w' ", "4cc1aeb37a37dddab8023aea1f8576b0": "\\scriptstyle exp(i \\omega t) ", "4cc1ce8706fd139b4d5578ca55965d50": "D_1={1}/{n}", "4cc235eacce83cc5b664b2f9290401d3": "\\begin{pmatrix}i & 0 \\\\ 0 & -i\\end{pmatrix}", "4cc2388465e0cd1cdb55966a5b240399": "a + bi + cj + dk", "4cc24c3e9187663711009854cedf0ab3": "(r, \\theta, z) = \\left(\\frac{2 R}{1 + R^2}, \\Theta, \\frac{R^2 - 1}{R^2 + 1}\\right).", "4cc25c1a27a3149bb796bd60fd3b155c": " D_i(a^{ij}(x)D_ju)=D_t(u)", "4cc264042ce0bab820adf2232402d7a8": "A_{t}(x) = A + (x,t)", "4cc2838c94b52ac486db98dee93854c1": "\n S_{yc} = \\tfrac{1}{\\sqrt{2}}\\left[(\\sigma_1-\\sigma_2)^2+(\\sigma_2-\\sigma_3)^2+(\\sigma_3-\\sigma_1)^2\\right]^{1/2} - c_0 - c_1~(\\sigma_1+\\sigma_2+\\sigma_3) - c_2~(\\sigma_1+\\sigma_2+\\sigma_3)^2\n ", "4cc2944a049bf27152066ff5d40e87d5": "\\scriptstyle t_\\mathit{on} \\,=\\, DT", "4cc2a3f4c4385a8cf4c2883eacd11be7": " \\bold{J} = \\bold{J}_1 + \\bold{J}_2 \\,\\!", "4cc2b84b58940ba608c4294ec0489249": " C \\approx 1.44 \\cdot B \\cdot {S \\over N}.", "4cc30f65cef674bdb728fd8ea99f3c6a": "x_i = y_i + \\sum_jX_{ij}", "4cc36308defcb9830e773d351a8639e5": "\\pi_0 B(S^{-1} S) = K_0(R)", "4cc37331f0a954f21e024f79732d5df6": "\\mathcal R = \\frac{l}{\\mu_0 \\mu_r A}", "4cc41145034fb2149a6aab63d4666951": "\\frac{n^{\\underline k}}{k!}=\\frac{n(n-1)(n-2)\\cdots(n-k+1)}{k(k-1)(k-2)\\cdots1}.", "4cc48730375e998e8948cab9e8d9fc38": "p(\\mbox{types of other players}|\\mbox{type of this player})", "4cc48f59384be348996b1a9844451983": "y\\succ_i^p x", "4cc4ef1fe4ff58a7b506ea6d9d76dd51": "GF=\\frac{\\Delta R/R_G}{\\epsilon}", "4cc5497b43fce92f2c0989b1d64f0fb9": " \\tan\\psi ", "4cc58cdf6ba4c431aa8a3519296d2727": "\\begin{align}&\\lambda(n)\\mu(n) \\\\&= |\\mu(n)| \\\\&=\\mu^2(n).\\end{align}\n", "4cc5975458598a8a611c4bccd1901f12": "\\displaystyle \\hat{f} \\left(\\xi - a\\right)\\,", "4cc5d09e0a894ce255cf7f348af2313b": "U_{\\mathrm{breakdown}}=\\frac{L\\cdot p\\cdot d\\cdot E_{I}}{e\\left(\\ln(L\\cdot p\\cdot d)-\\ln\\left(\\ln\\left(1+\\gamma^{-1}\\right)\\right)\\right)}\\qquad\\qquad(15)", "4cc5ebe68e4bb37cfee28b5c01c5a8f9": "\\frac{(2n+1+\\alpha+\\beta)(2n+2+\\alpha+\\beta)}{2(n+1)(n+1+\\alpha+\\beta)}", "4cc6823965a649aa06dbd0c9cf8e4499": " \\lim_{t\\rightarrow 0} \\int_x f(x) \\rho_t(x) = f(0) \\,", "4cc70312ef326ae7be04e7f8bd2b86e6": "10{\\frac{}{}}", "4cc71265e1d7d551947f64c56663bf28": "\\sigma^2_{MC} / \\sigma^2_{IS} \\,", "4cc74a798bc0cbbb73397ff175e3296b": " 0 \\leq e_\\lambda \\leq e_\\mu \\leq 1\\quad \\mbox{ whenever } \\lambda \\leq \\mu. ", "4cc7a79ec9234271276c2c6c9345b972": "M_{11} + M_{01} + M_{10} + M_{00} = n.", "4cc7ad93d04e79dd9dadc273256edf3a": "{\\vec b}", "4cc8ac8ce7093a88e488587e4be236b2": "\\scriptstyle\\bar f(\\boldsymbol{u}(\\boldsymbol{x}))", "4cc8b8ace2ea7aac0c2e064e21c5228b": "|f(z)| \\leq C e^{\\tau|z|}, \\quad \\Im z \\geq 0", "4cc9111eb78c43eeff03147fca7e58b9": "z = \\rho e^{i\\theta}", "4cc9288a88808ac5207ba6c4d241a1fa": "\n \\mathcal{P}_\\nu(\\eta_0) = \\big\\{ P_\\theta: \\nu\\in N,\\, \\eta=\\eta_0\\big\\}.\n ", "4cc9449afba1ba87ff905bf9666a64ad": "E(x) = e_1 x^{i_1} + e_2 x^{i_2} + \\cdots \\, ", "4cc961d13a0462b881a20b60b242c755": "Q_i(p)", "4cc98856c1cc725b0ad981af977b48ad": "\\frac{1}{2\\mu}\\cdot\\frac{\\rho}{1-\\rho}.", "4cc9b74a297d010f08c9c3c584c948f4": "F^a{}_br^b \\, =\\lambda r^a", "4cca2c952a09bd04211db6422d756426": "f(n) = O(g(n) \\log^kg(n))", "4cca3c0f5da84dc7c1aa819501ebab7b": " P = \\frac{ \\sum x_i } { { N - 1 \\choose n - 1 } X } ", "4cca703bf59d20cecd2fc744b46cb938": "177147/3 = 59049", "4cca8f838dccfb31f8ca4b740d77f12b": " \\mathrm{d} f \\in \\mathfrak{g}^{**} \\cong \\mathfrak{g} ", "4ccaa859f717068b94e0273dd6c9d984": "c+tv", "4ccabcc7ab32eb4102bee46d5c30cc5b": "\\frac{\\tan \\delta' - \\tan \\delta}{\\triangle\\delta} \\approx\\frac{d}{d\\delta}\\tan \\delta = \\frac{1}{(\\cos \\delta)^2}\\; \\rightarrow \\; \\triangle\\delta\\;=\\;(\\cos \\delta)^2 \\cdot (\\tan \\delta' - \\tan \\delta)", "4ccad6b9e530b64ee9ae751b1f8aeaee": "\\scriptstyle \\leq8.7\\times10^{-23}", "4ccaf3abee4cea3be66acbd6093432c3": " p_0 ", "4ccb2078cc7bc5cadfbaae5d5f82840b": "\\nu^{-1}(\\epsilon)", "4ccb3aa9b9d82a707653fe6f0a792b74": "N(0,\\sigma^2).", "4ccb6744496ae6a5931bedbe8b1b4bae": "50 = x", "4ccb6e7e74884f3a19c123451fe1a396": "\\sigma \\sqrt{r_t}", "4ccbad2598a44967d95bb81fd7150e16": "T^{\\mu\\nu} = \\begin{pmatrix} T^{00} & T^{01} & T^{02} & T^{03} \\\\ T^{10} & T^{11} & T^{12} & T^{13} \\\\ T^{20} & T^{21} & T^{22} & T^{23} \\\\ T^{30} & T^{31} & T^{32} & T^{33} \\end{pmatrix}.", "4ccbe26ac609fbdf68a74c7947abdb01": "\\frac{\\partial \\mathbf{Y}}{\\partial \\mathbf{x}}", "4ccc01bdbe332280222849fd4eb45382": " P_{n+1}(z) = P_n(z^2) + z P_n(-z^2) ; \\, ", "4ccc2c2bfb1e5672b0ff48e26ce881b2": "V(r) = -\\frac{V_0}{1+\\exp({r-R\\over a})}", "4ccc922d6a3f5fb52357f5a6f824faaf": "\\boldsymbol\\Lambda", "4ccc9b4493c9cba0d7dad1d836e0c182": "a \\in G", "4ccccd5f97a727de0841ede3fefeaf53": "\\cos(\\phi_0)<=0\\,", "4ccd031750d17171ee880c374d2fbef2": "\\begin{pmatrix} c t' \\\\ x' \\\\ y' \\\\ z' \\end{pmatrix} = \\begin{pmatrix} \\gamma & - \\gamma \\beta & 0 & 0 \\\\ - \\gamma \\beta & \\gamma & 0 & 0\\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix}\\begin{pmatrix} c t \\\\ x \\\\ y \\\\ z \\end{pmatrix}", "4ccd56a113ec5f31768998e1f822a7c8": "e^{-2 \\pi i K} = 1.0", "4ccdbca206ee56e152accb0a1dc4c79b": " \\textit{eff} = \\frac{K^N-1}{K^S N (K-1)}.", "4cce0b0360ce1faeda8ad7faf05644f2": "\\vec{e}_3 = \\frac{ \\omega }2 \\, \\partial_\\phi - \\partial_t,", "4cce47c53b6e82feb9b20a8adb371282": "\\{x\\in X\\mid m(r)\\le\\Phi(x)\\}", "4cce7bcdcad14656e9b9f6efc220e104": "\\text{ESS} = \\sum_{i=1}^n \\left(\\hat{y}_i - \\bar{y}\\right)^2.", "4cce952d40484485b04859bb7503b9a8": "\\Lambda(X_k^{-1}) = 1 + \\Lambda_1 X_k^{-1} + \\Lambda_2 X_k^{-2} + \\cdots + \\Lambda_\\nu X_k^{-\\nu} = 0 ", "4cce996912186c16cbbc8657109cd785": "\\sum_{a} P \\left ( {a,b}{|}{A,B} \\right ) = \\sum_{a} P \\left ( {a,b}{|}{A',B} \\right ) \\equiv P \\left ( {b}{|}{B} \\right ) \\quad \\forall {b,B,A,A'}", "4ccea012807fc5986cafd6fa6c8f0e68": "V := \\mu^{-1}(\\epsilon) / U(1),", "4cceb9005b6ac3fe47a320a8d03724a1": "v_{t} = \\frac {v_{a}}{M} = \\frac {52}{81.25} = 0.64^\\circ", "4ccee1e31ce70714c518ba268da74461": "A \\to aA", "4cceefe2ae9bc345a7a55cd4e7ea5755": "e'-e", "4ccf2f2acf8a4f32e0ea5f5aa26345eb": "\\tfrac{H}{4}", "4ccf39b0c2eb848d390c8987e31e215c": "\\begin{bmatrix}~~a+bi & c+di \\\\ -c+di & a-bi \\end{bmatrix}.", "4ccf442c511d885d77ffd20bdc4b537d": "\\frac{\\partial \\mathbf{Y}}{\\partial \\mathbf{X}}", "4ccf5572795706f28c8d2954a9bf84ca": "\\mathfrak{A}", "4ccf881664ddb1b6f7e4c62f697c07c5": "\\begin{align}\\sum_{k=0}^n f_k g_k &= \\sum_{i=0}^{M-1} f_0^{(i)} G_{i}^{(i+1)}+ \\sum_{j=0}^{n-M} f^{(M)}_{j} G_{j+M}^{(M)}=\\\\\n&= \\sum_{i=0}^{M-1} \\left( -1 \\right)^i f_{n-i}^{(i)} \\tilde{G}_{n-i}^{(i+1)}+ \\left( -1 \\right) ^{M} \\sum_{j=0}^{n-M} f_j^{(M)} \\tilde{G}_j^{(M)};\\end{align}", "4ccfa4034d1dfe6043d51ac7d1504f24": "\\textbf{x}_{k} = \\textbf{F} \\textbf{x}_{k-1} + \\textbf{G}a_{k}", "4ccfa4622d71e1fef91a2ed0b2d40936": "\\phi(r)v = rv", "4ccfc4ef2fa4a239550f03682d178676": "\\zeta(s)=\\frac{P_1(T)\\cdots P_{2n-1}(T)}{P_0(T)P_2(T)\\cdots P_{2n}(T)}", "4cd0376a5166f6de9e8c477fd443ad3d": "f_i^{(j)}(x)", "4cd042ac7d8d73e1f3431a2935eb7bb6": "\\frac{dx}{dt}(X-x)+\\frac{dy}{dt}(Y-y)=0.", "4cd044dacc686fd1a129c14b88975ed8": "\\psi\\to \\exp[iQ\\phi(x)]\\psi", "4cd09af2ff53a91f8f1a7c556977aca7": " x \\and y \\to x = 1 ,", "4cd0a271031f4b8280fe37a25d1569a7": " H(\\omega) = \\frac1{M} \\sum_{k=1}^{M} S( k \\delta t).", "4cd0a6d839354a76c4a7e187653658fb": " M_a(b) = 1 ", "4cd13001a46a0eef2f4228e9646caf94": " \\int_{-\\infty}^{\\infty} \\exp\\left[ -{1 \\over \\hbar} \\left( f\\left( q_0 \\right) + {1\\over 2} \\left( q-q_0\\right)^2f^{\\prime \\prime} \\left( q-q_0\\right) + \\cdots \\right ) \\right] d^nq", "4cd1445dad7ac2f8596b68b0ddc57417": "_{s.1\\ s.2 \\,}\\!", "4cd18bb7873a880db523d977af4e92d3": "\\bar \\nu_{sub}", "4cd191751c10142222520f1b82355626": "\\mathcal{M}(S)", "4cd1d609962adfe372401b4f77642da6": " dx^i", "4cd1e7670751def31920fe773d70ef9d": "\\omega = \\sqrt{1 \\over LC}", "4cd23512c3ba9c1a9d102be950ce163e": "\nm_t \\ddot{\\vec x} = - m_g \\nabla U\n", "4cd245ee4d632965d822098463325a93": "x_m(t)", "4cd27f55301e3467d9159d6f41122b4b": "P(N)=1-\\sum_{x=0}^{n-1}\\binom{6n}{x}\\left(\\frac{1}{6}\\right)^x\\left(\\frac{5}{6}\\right)^{6n-x}\\, .", "4cd28811145aba654d51aac9ba9423d9": "n_{ij} = \\langle V\\otimes\\chi_i, \\chi_j\\rangle = \\frac{1}{|G|}\\sum_{g\\in G} V(g)\\chi_i(g)\\overline{\\chi_j(g)},", "4cd2904cdfdca4d2ac2c0046ff8b1dfe": "\\Phi_{fl} = (k_BT_e/e)\\,(1/2)\\ln(m_i/2\\pi m_e)", "4cd2b9f5b9d1d9b949d429f1ae04348d": " v_i = {1 \\over \\sqrt{1 + \\tan^2 \\gamma_i}} \\begin{bmatrix} 1 & -\\tan \\gamma_i \\\\ \\tan \\gamma_i & 1 \\end{bmatrix} \\begin{bmatrix} x_{i-1} \\\\ y_{i-1} \\end{bmatrix} ", "4cd2c28470b55851a7bd4ab97ccdf13b": " x = 1.05 ", "4cd2c7559b339a10e25dd3c5b74e00ad": " + (1-f) ", "4cd334b959928c56357137f13ea799dc": "m_{rel} = \\frac{E}{c^2}\\!", "4cd36e11bdcf96d4d8373863c0feb1af": "\\pi_1(p) = \\bigcup\\bigcap p", "4cd3a021a3a454d6f457e738c57a7186": " r_2 := b_2 - B^* A^{-1} b_1 - S x_2 = b_2 - B^* A^{-1} (b_1 - B x_2) = b_2 - B^* x_1,", "4cd3e7f693d14fe439dd26b827d06081": "x_1 F_1 + \\cdots + x_n F_n + G \\leq 0", "4cd40d3a2bd6a0e21c05920ce13485b8": "x_{n+1}-x_n=a_n(\\alpha-N(x_n))", "4cd43603313fd944b373139c6b4aafeb": " \\pi(x)\\sim \\frac{x}{\\log(x)}.", "4cd44175eb90d9d768808d82aaf75b58": "M()\\,", "4cd45c1ea83297defe39fd459b573e5f": "\\rho(x,y)=x^2+y^2", "4cd54ee571e9f519f1b0c08d40bf10d0": "\\sqrt{2} =\n\\prod_{k=0}^\\infty\n\\left(1+\\frac{1}{4k+1}\\right)\n\\left(1-\\frac{1}{4k+3}\\right)\n=\n\\left(1+\\frac{1}{1}\\right)\n\\left(1-\\frac{1}{3}\\right)\n\\left(1+\\frac{1}{5}\\right)\n\\left(1-\\frac{1}{7}\\right) \\cdots.", "4cd5d8e7de55e614b768d0437fe72c71": "\\tilde{g}(x) = g(-x) ", "4cd5dfd5bfa60c89fb7ea21917e8e902": "[\\mathfrak{p}, \\mathfrak{p}] \\subseteq \\mathfrak{k}", "4cd614c1005a4d4b8e880a4737f344e4": "\nK u = Q\n", "4cd650af7ae30f80e75536877f8c68d7": "\\text{Lie factor}=\\frac{\\text{size of effect shown in graphic}}{\\text{size of effect shown in data}}", "4cd6cecba24e94e916f7f6b69a243e8f": "q^{i}", "4cd6d93ff2b982d2b713a3c6f6bd6962": "|n|=|A|", "4cd6ee611469c0970d495394525758ad": "26^{n^2}", "4cd75f0f4c434782683039645eba6da1": "Q_s \\cdot (Cc_{O_2} - Cv_{O_2}) = Q_t \\cdot (Cc_{O_2} - Ca_{O_2})", "4cd801791767c9a7f1c06868c3b915fc": "dG = \\left(\\frac{\\partial G}{\\partial x} \\frac{\\partial x}{\\partial u} +\\frac{\\partial G}{\\partial y} \\frac{\\partial y}{\\partial u} +\\frac{\\partial G}{\\partial u} \\right) du + \\left(\\frac{\\partial G}{\\partial x} \\frac{\\partial x}{\\partial v} +\\frac{\\partial G}{\\partial y} \\frac{\\partial y}{\\partial v} +\\frac{\\partial G}{\\partial v} \\right) dv = 0.", "4cd88c9b7512fb8c323563e62933b93c": "\\operatorname{E}(A)", "4cd8b27affe064f460c883ffb99d1ea5": "( -200 - 200.00000015 ) / 2 = -200.000000075,", "4cd90fb29a679de914f5084cff8e21d3": " \\frac{\\mathrm{d} \\mathbf{p} }{\\mathrm{d} \\tau} = q \\gamma\\left( \\mathbf{E} + \\mathbf{u} \\times \\mathbf{B} \\right) \\, , ", "4cd913a41ba61cfa30d7e625596d4c59": "Gr_L", "4cd95421dc11962cf5e531239342b63e": "A= \\frac{7155}{9990} = \\frac{135 \\times 53}{135 \\times 74} = \\frac{53}{74},", "4cd997e519b656903d8abf6c7ae81e3c": "0.3872 + 0.0305i", "4cd9a45c0286e07011b30b73337230d8": "\\displaystyle{D(x+h)=D(x) + \\min_{z} 2h\\cdot(x-z) + o(|h|),}", "4cda2d265bc87be5ef701347098323c5": "\\mbox{NTIME}(t(n)) \\subseteq \\mbox{ATIME}(t(n)) \\subseteq \\mbox{DSPACE}(t(n))", "4cda353a6a4abc10045f7dc3b67abf81": "G' = (V, E \\setminus X)", "4cda414ef5791a891251eb2c0484d27a": "\\max_{x\\in X, Y\\subseteq U} \\ \\{size(Y): g(x,u) \\le b, \\forall u\\in Y\\}", "4cda45f50410926bcaf36ae2c31fb9ad": "X = 5(B - L) - 2Y_W \\,", "4cda7521d4bdd21dff0a0b6ab4beb00a": "{\\mathfrak p}", "4cda78c54138eb35a821a61853d992bc": " r_n + r_{n-1} + r_{n-2}\\, +\\cdots, \\, ", "4cdaaf2355b9cf562b737f056ef04355": "[\\phi, [\\chi, \\psi]] = [[\\phi, \\chi], \\psi] + [\\chi, [\\phi, \\psi]]", "4cdada0da6721b386b012be73e47f891": "\\sum_i \\operatorname{index}_{x_i}(v) = \\chi(M)\\,", "4cdb37df071bb0a710f9a9fc8c83594c": "= (1,1,1,0)^{\\otimes |U|} \\left(\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}^{\\otimes 3} + \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}^{\\otimes 3}\\right)^{\\otimes |V|}", "4cdb6e3ff71b0feaaf36eb6e6a8220e4": "\\begin{align}\n &\\Pr(N=n\\mid M=m,K=1) \\\\\n = {} &(n\\mid m) = \\frac{[m \\le n]}{n} \\cdot \\frac{[n < \\Omega]}{H_{\\Omega - 1} - H_{m - 1}}\n\\end{align}", "4cdb82732f5b194596db38b5595eab6a": "\n\\wp(\\omega_1/2)=e_1\\qquad\n\\wp(\\omega_2/2)=e_2\\qquad\n\\wp(\\omega_3/2)=e_3\n", "4cdc24aa9ee44968b412a55aa1e4d000": "\\mathfrak{g}_c", "4cdcb5465d267ab0f60b9c1691303221": "\\overline{\\overline V}", "4cdcd27552a247eb5e33b9e3c28f7a80": "\\frac {\\sigma(t)} {E\\epsilon_0} ", "4cdd0781dfab8dd5fbdc1166c1c278e4": "(b-a)^4", "4cdd4b19b805f263ee29a3cc7d650f0b": "\\Psi[\\phi_2,t_2] = \\int\\!\\mathcal{D}\\phi_1\\,\\,\\mathcal{S}[\\phi_2,t_2;\\phi_1,t_1]\\Psi[\\phi_1,t_1]", "4cdd8c6d570ddda1976558592a9a1462": "{\\Delta}A=b{\\Delta}\\epsilon{\\frac{[H]_0K_a[G]_0}{1+K_a[G]_0}}", "4cdda43c01c3ce4daab5a5e97d02ee15": "(a_n)_{n\\in\\mathbb{N}}^{(k)}", "4cde1a62e145592708c6f05f2ac01762": "\\left\\langle\\sqrt{2}e^{\\pm \\tfrac{3 \\pi}4 \\mathrm i}=-1\\pm\\mathrm i,Z_2\\right\\rangle", "4cde4d3e3ac938f3626116974cd1209a": "|P\\setminus P'|", "4cde592d7451c9126d99feb7c87ffcc9": "(a_0, a_0 + a_1, a_0 + a_1 + a_2, \\ldots)", "4cde5adb3208f8c9dba0341bd08c84f8": "\\displaystyle K", "4cde5f9a75c6a07140bc3b7e7387b5e1": " h_{k}(x)=p_{k}(x)+xp_{k-1}(x) ", "4cdea3a51e7548f0c61976adbc8373ae": " \\nabla(\\mathbf{A} \\cdot \\mathbf{B})= \\nabla_\\mathbf{A}(\\mathbf{A} \\cdot \\mathbf{B}) + \\nabla_\\mathbf{B} (\\mathbf{A} \\cdot \\mathbf{B}) \\ . ", "4cdec3a2e4eac75be246b249c0a3a2e8": "\nf_{GB} = \\sqrt{r_{ij}^{2} + a_{ij}^{2}e^{-D}}\n", "4cdef5422053ced2825d1f8e3bd9dd9e": "\n\\begin{bmatrix}\n\\cos{\\theta} & 0 & \\sin{\\theta} & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n- \\sin{\\theta} & 0 & \\cos{\\theta} & 0 \\\\\n0 & 0 & 0 & 1 \n\\end{bmatrix}\n", "4cdf0f751f07ac562866fe7b1213202f": " (a/l)(\\nu/\\omega_0)", "4cdf200e0c5d1c14ebdefa76530985a8": " = -\\frac{1}{\\sqrt{|g|}}\\, \\partial_i (\\sqrt{|g|}\\,\\partial^i f),", "4cdf2dcb8bed74200fa7f02cbfc4537a": "a_i = f_i x_i\\,", "4cdf48eb628a4f350163f17d9b2f39ae": "V_{w1}", "4cdfbe87a624b7aac43e7b6ccbc4e118": "\\frac{d\\omega}{dt}=2k\\frac{(a-b)}{I}\\beta-2k\\frac{(a^2+b^2)}{VI}\\omega", "4cdfd4252af18ff38e79e2d8e22a542c": "\nc^{2} = \\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} \\left( \\frac{dt}{d\\tau} \\right)^{2} - \n\\frac{1}{1 - \\frac{r_{s}}{r}} \\left( \\frac{dr}{d\\tau} \\right)^{2} - \nr^{2} \\left( \\frac{d\\theta}{d\\tau} \\right)^{2} -\nr^{2} \\sin^{2} \\theta \\, \\left( \\frac{d\\varphi}{d\\tau} \\right)^{2},\n", "4cdfdaa3ede50f0ed6c2944d289c0a54": " \\begin{cases} \\ \\ 1 \\\\ -\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i \\\\ -\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i. \\end{cases} ", "4cdff62ba8d2db7a66f04aea4bdd9c89": "\nP =I^2 R = I V = \\frac{V^2}{R}\n", "4ce0128adb659bf1ad5416c4c43ce213": "\\textrm{SINR}_k = \\frac{|\\mathbf{h}_k^H\\mathbf{w}_k|^2}{1+\\sum_{i \\neq k} |\\mathbf{h}_k^H\\mathbf{w}_i|^2}", "4ce03b90f4d8d1b8d2532bdefc4ae4a2": "\\ (1) \\quad \\rho(a,b) + \\rho(a,b') + \\rho(a',b) - \\rho(a',b') \\leq 2", "4ce0576f252c5237ff79bdbf7f73aa04": "\\begin{align}\n \\left \\|(\\Gamma\\varphi_1 - \\Gamma\\varphi_2)(t) \\right \\| &= \\left \\|\\int_{t_0}^t \\left (f(s,\\varphi_1(s))-f(s,\\varphi_2(s))\\right )ds\\right \\|\\\\\n&\\leq \\left |\\int_{t_0}^t \\left \\|f \\left (s,\\varphi_1(s)\\right )-f\\left (s,\\varphi_2(s) \\right ) \\right \\| ds \\right |\\\\\n&\\leq L \\left |\\int_{t_0}^t \\left \\|\\varphi_1(s)-\\varphi_2(s) \\right \\|ds \\right| && f \\text{ is Lipschitz-continuous} \\\\\n&\\leq L a \\left \\|\\varphi_1-\\varphi_2 \\right \\|\n\\end{align}", "4ce058e516ad2ddcb10fb6f9741ea46d": "P^{\\prime}(A,B)", "4ce0a2dc5ebb141408ea60f7dfb5a575": "\\frac{dk}{dt}", "4ce0ecbc7556133d7d504d4fcfa77239": "f(u,v) \\cdot k(u,v)", "4ce11b003693b50e99fa94feb96da0b9": "\\boldsymbol{\\hat \\varphi}", "4ce13ed8eee1f97a679396296b48ce98": "J=L+S", "4ce1715f3ca1dd71e4fb8a990412026c": "\\mathcal{P}(n)", "4ce18e49bc1e7bd4861a2dba8642c0ff": "\\boldsymbol{c'} \\in C", "4ce1997562146b2e861f67fd0fca421f": " \\sum_{i=1}^d S_{Ti} \\geq 1 ", "4ce19999a924c5e6ca7d3906e9beb231": "q_{\\pm} = \\sqrt{ \\frac{ 1 \\pm \\sqrt{ 1- \\gamma \\alpha \\left( 1- \\frac{3 \\gamma}{4\\alpha} \\right) } }{2 \\left(1- \\frac{3\\gamma}{4\\alpha}\\right) }}", "4ce1a662ec6d6c21f82a45a9cc1a0722": "0 < C_\\psi < +\\infty.", "4ce2099bbaac19d8676900777703b493": "e \\in Q_i - P_i", "4ce21ad9ea2aa12952fe587768470def": "{}_p\\Psi_q \\left[\\begin{matrix} \n( a_1 , A_1 ) & ( a_2 , A_2 ) & \\ldots & ( a_p , A_p ) \\\\ \n( b_1 , B_1 ) & ( b_2 , B_2 ) & \\ldots & ( b_q , B_q ) \\end{matrix} \n; z \\right]\n=\n\\sum_{n=0}^\\infty \\frac{\\Gamma( a_1 + A_1 n )\\cdots\\Gamma( a_p + A_p n )}{\\Gamma( b_1 + B_1 n )\\cdots\\Gamma( b_q + B_q n )} \\, \\frac {z^n} {n!}.\n", "4ce2437b789d805bd589f9b7e4182ac7": "\\! R_\\mbox{emp}(h) = \\frac{1}{m} \\sum_{i=1}^m L(h(x_i), y_i).", "4ce2a37a3d474e567ca45e881608852e": "Y_1=F_1(K,L)\\,", "4ce30d48dad9c70392ca8d3c52ee79fe": "f:[a,b]\\to\\mathbb{R}", "4ce33192699b6d3bbfc22e71bf08a65d": "M_{M_{61}}", "4ce35581839e6ebc4cb53b82bd3c0475": "\\mathcal{A}/E = (A/E, (f^{\\mathcal{A}/E}_i)_{i \\in I})", "4ce37abd181701feb2811fe2e1cc21fc": "\\theta = FL/\\kappa\\,", "4ce3a9ff26c59d6bf32329e79396d0a4": "\\frac{\\partial h}{\\partial t} + \\frac{\\partial F}{\\partial x} =0,", "4ce3cd56dcfd1f68b2e083c11415cca8": "F|_Y=f, \\ \\ \\text{and} \\ \\ \\forall x\\in X, \\ \\ \\operatorname{Re}(F(x))\\leq p(x). ", "4ce42009b29be34c8fbf75a927cee5df": "(A,B;C,D) = \\frac {AC\\cdot BD}{BC\\cdot AD}.", "4ce48667b2c03bd1e7d6cd94234cca9c": "\\bigvee_{i}\\delta(X_i)=\\delta\\left(\\bigvee_{i} X_i\\right)", "4ce51af6362e29ea3bac042186ebefc4": "x\\equiv{hc\\over\\lambda kT },", "4ce5244e7d7ae99616f99edf58257564": "f(x)=\\int_{-\\infty}^\\infty \\varphi_\\lambda(x) \\tilde{f}(\\lambda)\\,{\\lambda \\pi\\over 2} \\tanh({\\pi\\lambda\\over 2})\\, d\\lambda,", "4ce57e098333bacce0f018d15a761d84": "C_0(X)", "4ce59d7d8756eb9da9f76b4acda1d7da": "z=y-x", "4ce61f10e8b3c2b5c2a207340bcf1ce1": "\n DV(x_0) = \\begin{pmatrix}\n V & 0 \\\\\n J_3 & J_4 \n \\end{pmatrix} ", "4ce66c63e3d84134fa53078f363e79d5": "x^4-x^3-1=0", "4ce70af540b7e61eddbd3959ba78e633": "A+e^-=A^-", "4ce7ab0c87407fb014047437e901af0d": "r_1 + r_2", "4ce80b27c365b45e3249c57d395dde68": "t+\\Delta t", "4ce8ce46c7025468f3ec3980f0926d81": "p^2=-\\frac{1}{3}(r^2-a^2)", "4ce99d9d476ef0a7ec18076516bc7629": "\n\\begin{align}\n\\frac{d (\\sum_i c_i \\mathbf{v}_i)}{d t} + kL(\\sum_i c_i \\mathbf{v}_i) & = 0 \\\\\n\\sum_i \\left[ \\frac{d c_i}{d t} \\mathbf{v}_i + k c_i L \\mathbf{v}_i \\right] & = \\\\\n\\sum_i \\left[ \\frac{d c_i}{d t} \\mathbf{v}_i + k c_i \\lambda_i \\mathbf{v}_i \\right] & = \\\\\n\\frac{d c_i}{d t} + k \\lambda_i c_i & = 0, \\\\\n\\end{align}\n", "4ce99dc366b01461766fd5f0e178b264": "(x+3)^2-2(x+2)^2+(x+1)^2=2", "4ce9ba2cf5c2fb1a1219ae6e507a1d2f": "x=0.1", "4ce9ba811b6b94eb172965c9708c1994": "\\begin{align}\np(A)\\cdot v & = A^n\\cdot v+c_{n-1}A^{n-1}\\cdot v+\\cdots+c_1A\\cdot v+c_0I_n\\cdot v \\\\\n& = \\lambda^nv+c_{n-1}\\lambda^{n-1}v+\\cdots+c_1\\lambda v+c_0 v=p(\\lambda)v,\n\\end{align}", "4ce9c0ee7439161e3f569ff2dd986785": "n![z^n] \\frac{1}{1-z}\\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty\\frac{z^k}{k} =\nn! \\sum_{k=\\lfloor\\frac{n}{2}\\rfloor +1}^n \\frac{1}{k}.", "4ce9e4603a4010514b5941b114380504": "F(x):=\\sum_{k\\in A} e^{-\\sqrt{k}}\\cos(kx)\\ .", "4ce9f170cf33adf30d8d93446fca468f": "G_1 = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos(\\theta) & -\\sin(\\theta) \\\\\n0 & \\sin(\\theta) & \\cos(\\theta)\n\\end{pmatrix}", "4cea2df8299b54c64563421fb56c7ee7": "v=iR \\iff i=vG \\, ", "4cea309909cc54f304e9d39f96e66300": "u^3", "4ceaed1efeb77f5913bc28266ca0e76a": "\\dot{\\rho}_{f} = -3 H \\left( \\rho_{f} + \\frac{p_{f}}{c^2} \\right) \\,", "4ceb25851df9a722b951d1e05cddbdac": " q(x)=b(x,x) ", "4ceb6cfff7634004a4874c2e0518a38c": "B_r(p) \\subseteq \\overline{ B_r(p) }", "4cec3dcf6158ccd241be13db604a8cdc": "\\sigma\\colon G \\to \\operatorname{Aut}(G).", "4cec42d8b6de81b59ecc9de30b9a1512": "\\epsilon = \\phi_x - \\phi_y", "4cec6c6e1b3ae30b041d467eeedb0351": "\\mathbf E_{1s} = \\frac{<\\psi_{1s}|\\mathbf{\\hat{H}}_e|\\psi_{1s}>}{<\\psi_{1s}|\\psi_{1s}>}", "4cecbe0b3841cece20a603f1e6d21b26": "Q = Q_{in} - Q_{out} = W", "4ced4c2bf1d088cc684c72d605a41650": "p_3=q_4\\ ,", "4ced63dc1b305ff9a602ac80a5c29624": " \\mathrm{isotopic \\ abundance} \\ (\\mathrm{i}) = \\frac{N_\\mathrm{i}}{N_\\mathrm{tot}} \\ ,", "4ced6515de05fef28a4ead686587abc3": " \\delta_{ext} ", "4cee39f8e6e6a7985cf391e7a7f29567": "C(\\mathbb{R})", "4cee514b030881dc4d5bd46e6f3057c0": " X \\colon(\\Omega, \\mathcal F, \\mathbb P) \\rightarrow \\mathbb R ", "4cee7fef7ea3c0acf3570ad45ca71596": "g:(X, A) \\rightarrow (Y,B)", "4ceea57f5ce9bd9d5ca04ab170e9f6a0": " \\frac{1}{\\xi^2} \\frac{d}{d\\xi} \\left({\\xi^2 \\frac{d\\theta}{d\\xi}}\\right) + \\theta^n = 0 ", "4ceef7440e4f70f3d72a9f8947ba294a": "\\sqrt{k-1}\\,", "4cef0a3437c4bd1b14ac39b401ca180d": "\\nabla \\times \\left( \\nabla \\times \\mathbf{A} \\right) = \\nabla \\left( \\nabla \\cdot \\mathbf{A} \\right) - \\nabla^2 \\mathbf{A}", "4cef2550b45699bd98020178c69e7bf4": "V \\sim I", "4cef2f19f86cb1040ce2462d4ffeaa39": "dx^{a} \\,", "4cef2f30ac7d33419d00c1d93a090095": "locked", "4cef4d94b47d16ff117768d9ebd3f9c9": "\\mathcal{L}_X C^a{}_{bcd}=0", "4cef588e02208f1f690deec890f33adc": "\\frac{1}{t}Y_t = \\frac{X_t}{t} \\frac{1}{X_t} Y_t \\to \\frac{1}{\\mathbb{E}S_1}\\cdot\\mathbb{E}W_1 ", "4cef9e976273e8faa6cc29761aa33778": " \\mathbf{P} = \\frac{m\\dot{\\mathbf{r}}}{\\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2}} + e\\mathbf{A} \\,\\!", "4cefe7664c67089646bbcbcffa05ec52": "\\hat{v}_i", "4ceff2183aea75cd72f818eb36f4595e": "\\frac{dy}{dt} = iy", "4cf002bccf39dd6bb16c4793271edffa": "0 \\le \\dim X \\le n", "4cf0652a137356dddaa4e3b610af799f": "\\mu^'=F_{adhesion}/F_{preload}", "4cf0bc8f56fb177ca7ad1558c1e0b09f": "\\scriptstyle c\\,t_1=-5\\,Ly", "4cf0c153fed7dc73263d8299bc733432": " \\mathbf{J} = \\mathbf{I} + m \\left[\\left(\\mathbf{R} \\cdot \\mathbf{R}\\right) \\mathbf{E}_{3} - \\mathbf{R} \\otimes \\mathbf{R} \\right],", "4cf0cb90096f317c0de25350dadfea2c": "878.5~\\pm0.7_{\\mathrm{stat}}\\pm0.4_{\\mathrm{syst}}~s", "4cf0cbf198077ed97ca4ff63106ab0d2": "x \\in [\\mu-s,\\mu+s]\\,", "4cf1022e235ec119fa6fdd8234a15aa8": "E=vB\\,", "4cf1a5b198fad480dfe3f0fb0332191e": "\\xi>=1", "4cf1f274871f1ffe55b4904e77d2989c": "\\Phi_A:\\Omega \\to \\mathbb{R}", "4cf25ad5a6536e2f252fda9a9f9dae48": "\\textstyle \\mathbf{\\mu^*}", "4cf2b01e1f8f1627c400c2f329af9d81": "(v,x)\\in E^{p}", "4cf2e53cd44656e74461d8e8b3596a6a": "C_{ij}", "4cf30c2055518c51857693fa16fb04c5": "\\Delta x/(\\Delta t)^2 = 7.5 \\text{m}/\\text{s}^2", "4cf319d2a160510e65d85e4fc488a134": "\\tfrac{736}{232}", "4cf3493902953119c8f193d99b72e753": "p=3a\\,\\!", "4cf3504776a1a4c2dfbec08571c86969": "\\chi_{V\\otimes_A V}(g)=\\frac{1}{2}[\\chi_V(g)^2-\\chi_V(g^2)]", "4cf412af77f061fd94d65e48fe78a6da": "-\\ell j\\,", "4cf438c67a121b412a2bc4f84bd33605": "p = \\frac{\\log(\\mathrm{A} / |C_0|)}{\\log(\\mathrm{A} / \\mathrm{NPV}_{1,in})}.", "4cf43be63abeff06c4271e1867719372": "R(S)", "4cf47a4315ce886ad3a902effa97e7f5": "d=3,4", "4cf4c03681c390747d13952d7f802f49": "S_g+S_m=0", "4cf4ce22d7018b17bb43ead6c2b15344": "M = \\alpha \\cdot u v^*, \\quad \\mbox{where} \\quad \\|u \\| = \\|v\\| = 1 \\quad \\mbox{and} \\quad \\alpha \\geq 0 .", "4cf4f43abe34fa1c746ceebbb62ece13": "h(z) = c_2 (z - a)^2 + c_3 (z - a)^3 + \\cdots \\, .", "4cf51ff6679a40321baf629ce450fc98": "\n\\vec{\\theta} -\\vec{\\beta} = \\nabla_{\\vec{\\theta}} \\psi(\\vec{\\theta}) = \\sum_i { \\theta_{Ei}^2 \\over |\\vec{\\theta}-\\vec{\\theta}_i |} , ~ \n\\pi \\theta_{Ei}^2 \\equiv {4 \\pi GM_i D_{is} \\over c^2 D_s D_i } \n", "4cf53cf3f58d2d054652e0fccf37acd7": " Rate = {k_1 [L_nM-L]}", "4cf573f2a8402edf6b103845b8b11125": "\\{a_1, a_2, a_3, a_4,\\ldots, a_m\\}", "4cf63ca7146e78f370ac58cf8e13c36a": "r_u = \\frac{a}{2} \\sqrt{\\varphi \\sqrt{5}} = \\frac{a}{4} \\sqrt{10 +2\\sqrt{5}} = a\\sin\\frac{2\\pi}{5} \\approx 0.9510565163 \\cdot a", "4cf65df88e1ddb8e31dc882dccd44890": "\\nu\\ ", "4cf65e5b42144a9d2cce47ea47c3a2c1": "(6^6)^6", "4cf6631ce88e58921d5d5ba6ea1b41d1": " \\mathbf{F}(\\mathbf{x}) = -\\frac{dU^*}{d\\mathbf{x}}~.", "4cf67b80992dc091aff2c3c99ef453a2": "P \\in {\\mathcal P}", "4cf68b28d0cc1edc954f1538d5c86fd5": "\\alpha(E)", "4cf77c2d83728773b614a323491c0ffc": "\\mathbf{n}=\\frac{\\mathbf{p}_A-\\mathbf{p}_B}{\\|\\mathbf{p}_A-\\mathbf{p}_B\\|}=\\frac{n_A \\mathbf{i}-n_B\\mathbf{r}}{\\|n_A\\mathbf{i}-n_B\\mathbf{r}\\|}", "4cf78145aca33e53929ab2e00b52ae13": "\\rho^{\\text{induced}}(\\mathbf{r}) \\approx -e^2\\frac{\\partial n}{\\partial \\mu} \\phi(\\mathbf{r})", "4cf7ec441cf35ac07544d9e535ef5d80": "\\{\\alpha_i\\}_{i=1}^G", "4cf7f20e2f6a5f6c1ca18fdc6e0a9e49": " m^{p-1}\\equiv 1\\pmod p \\!", "4cf8dc38da36975f840b86c51d2a80b4": " F = \\frac{S_X^2}{S_Y^2} ", "4cf8e4e55b221a6d94f5cc89f0027caf": "\\ell^2(\\mathbb{N})", "4cf9be29621bc0d35ec16f75fbdd998a": " \\int\\limits_0^1\\int\\limits_0^1\\int\\limits_0^1 \\left(x + y + z\\right) \\, dx \\,dy \\,dz = \\int\\limits_0^1\\int\\limits_0^1 \\left(\\frac 12 + y + z\\right) \\, dy \\,dz = \\int \\limits_0^1 \\left(1 + z\\right) \\, dz = \\frac 32", "4cfa176e6b5fc08011601a2b03a5401a": "m_U\\colon U \\to \\mathbb{N}", "4cfa520a0d4d2581067f9edc55024262": "\\phi(t) = \\arctan\\left(\\frac {\\Delta Yt} {\\Delta Xt} \\right) .", "4cfa535d1a847db5cfdf4cc06993b57d": "\nI_{m, k} \n= \\mathrm{E} \\left[\n \\frac{\\partial }{\\partial \\theta_m} \\log f\\left(x; \\boldsymbol{\\theta}\\right)\n \\frac{\\partial }{\\partial \\theta_k} \\log f\\left(x; \\boldsymbol{\\theta}\\right)\n\\right] = -\\mathrm{E} \\left[\n \\frac{\\partial ^2}{\\partial \\theta_m \\partial \\theta_k} \\log f\\left(x; \\boldsymbol{\\theta}\\right)\n\\right].\n", "4cfa8451cfd2480c2eee5a382b7f14a1": "B_0=\\operatorname{adj}(-A)", "4cfac0ecd73b6696248b23c7b0cab214": "x_n = Ax_{n-1} + Bx_{n-2}", "4cfae11d322abaee853ee428207064fd": "x = \\operatorname{prox}_{\\gamma \\varphi}(x) + \\gamma\\operatorname{prox}_{\\varphi^*/\\gamma}(x/\\gamma),", "4cfb3fd2090b42939f44dd84c4262868": "\n F_{\\mu \\nu} =\n \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}\n", "4cfb495d752d85eaef526c71435f46b3": " V\\otimes V\\otimes\\cdots\\otimes V", "4cfb8b47e34f74ea8f4b5b01a57f6192": "d=2e+1", "4cfb8ee386e55a2de2446edda2b533a9": "\\Phi\\, =\\, \\frac{\\omega}{k}\\, a\\, \\frac{\\cosh\\, \\bigl( k\\, (z+h) \\bigr)}{\\sinh\\, (k\\, h)}\\, \\sin\\, ( k x\\, -\\, \\omega t),", "4cfbeefabd707a1bf4c93c41269db7a2": " \\supseteq", "4cfc0ae77d9495c2d8d3b66175981cac": "f: H \\to G", "4cfc45b7079751b1c6ba0dca3d0fc2ff": "\\tilde{\\phi}(\\mathbf{u},z) = \\tilde{\\phi}(\\mathbf{u},z=0)\\exp(\\pi i \\lambda \\mathbf{u}^2 z)", "4cfc8fef2cad4f85acd4b005b4f65ded": "F_{abs} = 100 \\cdot \\frac{AUC_{po} \\cdot D_{iv}}{AUC_{iv} \\cdot D_{po}}", "4cfc91a6b913e36d3548a0411d7d395b": "\nf(x,y) = \\begin{bmatrix} \\ -0.15 & \\ 0.28 \\ \\\\ 0.26 & \\ 0.24 \\end{bmatrix} \\begin{bmatrix} \\ x \\\\ y \\end{bmatrix} + \\begin{bmatrix} \\ 0.00 \\\\ 0.44 \\end{bmatrix}\n", "4cfdc9f55f2cb5b3e9f97288d9b70074": "(x_0-\\epsilon_0,x_0+\\epsilon_0)", "4cfee4c5bff9538a5c060a83a2eb4648": "\\mu(A)=\\mu_0(A)", "4cff08e50912e4ffbdd041548f0f9615": "S_{s}", "4cffa2695f17022bc407fb7bda0ce1ca": "p(C)", "4d004db807502ceda7d5f0bf12c363da": "\\sin 3^\\circ = \\cos 87^\\circ = \\dfrac{\\sqrt{30} + \\sqrt{10} + \\sqrt{20 + 4 \\sqrt5} - \\sqrt6 - \\sqrt2 - \\sqrt{60 + 12 \\sqrt5}}{16}\\,\\!", "4d0056580c5c59c3d552ad57a779adea": "{\\Bbb C} \\times H= {\\Bbb C} \\times {\\Bbb R} \\times {\\Bbb R}^{>0}", "4d00ad7ecb0b7327607e4201e7ff00c8": "B\\!", "4d00bdfd4e98c9d4d2377d387d98e4d3": "CIRC(T) = \\{ M ~|~ M \\mbox{ is a minimal model of } T \\}", "4d00e6c8dfafced8a5e1ba0dd76c5ffd": "\\textstyle x \\equiv y \\pmod{n}", "4d012b48ef5f3b2e38c17e231a45c37a": "f(1,y)=1 \\lor y = 1", "4d01a5f442011b2f072dde69b374b8cd": "\\scriptstyle a = \\sqrt{h(2r-h)}", "4d01a603d9c4f24d194d4e3f476580ba": "\\xi(x) \\leq 1", "4d01b1850c321a94b4858bcd41b442f1": "\\textstyle H_2: G_2 \\rightarrow \\left\\{0,1\\right\\}^n", "4d01d9811d6e9014be435d7f6fc70e94": " \\lambda g.\\lambda n.n\\ \\operatorname{drop-param}[(g\\ m\\ p\\ n), D, \\{p, q, m\\}, \\_] \\ \\operatorname{drop-param}[(g\\ q\\ p\\ n), D, \\{p, q, m\\}, \\_] ", "4d01e54e7381f38ddd4f77035cf68e46": "\\pi(x,z)=\\mathbb{P}\\left( Z=z \\,|\\, X=x \\right)", "4d023d6797e1481bc37929dc6693a24e": "\\vec e_1, \\vec e_2, \\ldots, \\vec e_n", "4d02ce641337fb9bc74ed6a12ba8aaf6": "k_{\\rm A} = 1", "4d0302638f385dc7f31f372257e96d24": "\\delta_\\varepsilon \\bold{A}=[\\varepsilon,\\bold{A}]-\\mathrm{d}\\varepsilon", "4d0323fccfb23315960f295fbec3ff48": "\n V(x;\\sigma,\\gamma)=\\int_{-\\infty}^\\infty G(x';\\sigma)L(x-x';\\gamma)\\, dx'\n", "4d032c39135d5ff760203df215a425c6": "\\alpha^*(t)", "4d032ec3c1a613c73433231077477ac9": " \\bigg(\\frac {m-1}{2}\\bigg) ( \\sigma_1 + \\sigma_2 + \\sigma_3 ) + \\bigg(\\frac{m+1}{2}\\bigg)\\sqrt{\\frac{(\\sigma_1 - \\sigma_2)^2 + (\\sigma_2 - \\sigma_3)^2 + (\\sigma_3 - \\sigma_1)^2}{2}} = S_{yc} ", "4d034384ba79ee0eaefde1f8fec3eb41": "\\Theta_a", "4d0354bf6035a9723da79be2af2cfcdf": "f(z)=\\frac{z}{(1 - z)^2}=\\sum_{n=1}^\\infty n z^n", "4d036bbbcaa77593c1f5e16241074088": "\\displaystyle{\\ddot{\\mathbf{v}}=\\kappa(t)\\,\\mathbf{n}(t),\\,\\,\\,\\,\\, \\kappa(t) =\\ddot{\\mathbf{v}}\\cdot \\mathbf{n}=\\ddot{y}\\dot {x} - \\ddot{x}\\dot{y}.}", "4d03b1dd54afc266727a82719e9697e0": "v=v_k+\\frac{p-p_k}{p_{k+1}-p_k}(v_{k+1}-v_k).", "4d041b248c1e6bc80ce691b80f284077": "\\lambda_s(n)=n\\cdot 2^{\\frac{1}{t!}\\alpha(n)^t(1+o(1))}", "4d0463d356b989819c6556dab2718200": " \\int_U f(x_a,\\theta_i) \\, d(x,\\theta) =\n\\int_{U'} f(x_a,\\theta_i) \\operatorname{Ber}_{+-}\\, \\frac{\\partial(x_a,\\theta_i)}{\\partial(y_b,\\xi_j)} \\, d(y,\\xi). ", "4d047870751e6414943ceb14872b2203": "\\tau_{cap}\\left(\\omega\\right) = -\\Gamma_{ind}(\\omega)", "4d047ccc398cd586e01d927821fcc9b3": "V(0) = \\dot{V}(0) = 0", "4d04a444e63e3c25caf1c3beba0dbc22": " k\\xi \\gg 1\\,.", "4d04cc8dabe218d38ef565ae354902ab": "\n\\mathbf{F}=\\nabla \\left(\\mathbf{m}\\cdot\\mathbf{B}\\right)\n", "4d0519f0d87133c9f284d78d6d93c27e": "f(x)=g(x)h(x)\\,\\!", "4d0533560c2e19f7780d81157543237d": " \\min_{x_1,\\ldots,x_I} \\sum_{i y_k^{j+1}) = \\frac{r_{j+1}}{r_k}", "4d167ab5c173acb7b8dc82f359b8e74c": "X \\to f(Y, Z, ...)", "4d1685679c6c8a1f5ca4c33517ff4118": " g^{H(m)} \\equiv y^r r^s \\pmod p.", "4d16ab58b09cfcd4a6d237675a232a3a": "E=E_B(1-\\frac{d}{D})", "4d16e7aa9e2628d0064b024ee9065ae2": "m_L = \\gamma^3 m", "4d17187398ef7e28a9ba198726cda8ec": " F(r,\\Omega ) ", "4d171fdd160092d7a4184b3cba9a5c92": "\\mathcal{D}(A)", "4d173328b4a730d544d8d528bf977168": "\\tilde{\\mathbf{x}}\\in \\mathbb{R}^n", "4d174d1d1869a7060090cae3e2025e12": "V_\\mathrm{out} \\le V_\\mathrm{CC}", "4d174ec7a9179c6cbe5283714f1407a2": "M = N_1\\#N_2. ", "4d17572dc8c2cb939f6861635c5f1645": " \\text{ } 2 \\text{ } H_2 O \\text{ } + \\text{ } \\frac {1}{2}O_2 \\stackrel {Metal Ion} {\\rightarrow} \\text{ } 2 \\text{ } H_2 O_2 {\\rightarrow} \\text{ }H_2 O \\text{ }+ \\text{ } (O)", "4d176419a53e248473d00a1f5054a099": "\\,\\frac{a + bi}{c + di} = \\left({ac + bd \\over c^2 + d^2}\\right) + \\left( {bc - ad \\over c^2 + d^2} \\right)i. ", "4d178352ef835937cfea33097acb26dc": "\\nabla_w Y_k", "4d17c3904c74d26e474334000864a24a": "S_\\theta \\cap s_i = \\emptyset", "4d1802e1d1150b97629fc40bcaa800d8": "L\\frac{\\mathrm{d}I}{\\mathrm{d}t}+RI=\\mathcal{E}\\,\\!", "4d184efeb7e22c5cc8975a9d736aa108": "L\\!\\times\\! K", "4d188e84a907b39da9180a6dfe9c4000": "N_X \\mathcal U : \\dots \\rightarrow \\mathcal U \\times_X \\mathcal U \\times_X \\mathcal U \\rightarrow \\mathcal U \\times_X \\mathcal U \\rightarrow \\mathcal U.", "4d18d00ed2ae223626ed5723f0ce352b": " \\frac{x_{n+1}-a}{x_n-a}=\\frac{f(x_n)-a}{x_n-a}\n\\approx \\frac{f'(a)(x_n-a)}{x_n-a}=f'(a),", "4d192b562b76b83485cafc5acfa27520": " \\displaystyle{a_3=-2\\int_0^\\infty \\alpha^2\\, dt +4\\left(\\int_0^\\infty \\alpha\\, dt\\right)^2}", "4d19a0e6f14ec21de86143ffd49683d0": "G_{ab}=8\\pi T_{ab}", "4d1a27d2c887102d12ff43207992b5ff": "G_r", "4d1a5944b45d077f66ebc6e4bc4c8687": "m(x)\\leq\\frac{f(x)}{g(x)}\\leq M(x)", "4d1aabfc7514b7ea0bb8ae96d7d93542": "0 \\log_2 0", "4d1ab97d4b1ec55f5d66f4263604a21a": "C(v,u) = 2- ((1-\\alpha)\\times\\frac{deg(v)+deg(u)}{max\\_deg(G)+max\\_deg(H)}+\\alpha \\times S(v,u))", "4d1abaddcb738a4975157dcd658b3913": "\\sim\\exp\\left(-\\frac{c}{\\epsilon}\\right)", "4d1abc61c8d9c8f74fcf89ee3fc02b2e": "\\mathrm{I\\!I} = L\\, \\text{d}u^2 + 2M\\, \\text{d}u\\, \\text{d}v + N\\, \\text{d}v^2, \\,", "4d1ac36d739efdbc2f2d11987a3b7442": "s_1, s_2, ..., s_n", "4d1b575f4c88bc48f50594892395ad14": " a_2^\\dagger | N_1, N_2, N_3, \\cdots \\rangle = \\sqrt{N_2 + 1} \\mid N_1, (N_2 + 1), N_3, \\cdots \\rangle.", "4d1b7b74aba3cfabd624e898d86b4602": "\\omega ", "4d1b83c3e46eb318a334d4c3d44eb884": "\n\\mathbf{V}_P(t) = [\\Omega]\\mathbf{P} + \\mathbf{v} - [\\Omega]\\mathbf{d}\\quad\\mbox{or}\\quad\\mathbf{V}_P(t) = \\mathbf{\\omega}\\times\\mathbf{P} + \\mathbf{v} + \\mathbf{d}\\times\\mathbf{\\omega},\n", "4d1bc43f2af7958fbf0e39f834b87d37": "\\frac{\\zeta(s)\\zeta(s-a)\\zeta(s-2a)}{\\zeta(2s-2a)} = \\sum_{n=1}^\\infty \\frac{\\sigma_a(n^2)}{n^s}", "4d1c37922f562947bc19ce5de10eb090": "\\mathfrak{so}_4", "4d1c41f7805e155b8ff117b74feb16fb": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 1}{4 \\choose 2} \\end{matrix}", "4d1cc709eaf231ae3846d65305c167b4": " (\\gamma_{B,C} \\otimes \\text{Id}) \\circ (\\text{Id} \\otimes \\gamma_{A, C}) \\circ (\\gamma_{A,B} \\otimes \\text{Id}) =\n(\\text{Id} \\otimes \\gamma_{A,B}) \\circ (\\gamma_{A,C} \\otimes \\text{Id}) \\circ (\\text{Id} \\otimes \\gamma_{B, C})\n", "4d1cef39e50b3550fefc2a5a194d76f9": "\n\\left(\\mathbf{A}+\\mathbf{B}\\right)^{-1} = \\mathbf{A}^{-1} - \\mathbf{A}^{-1}\\mathbf{B}\\left(\\mathbf{B}+\\mathbf{BA}^{-1}\\mathbf{B}\\right)^{-1}\\mathbf{BA}^{-1}.\n", "4d1cfb19597a5cce0c3c23765b7e5a12": " K_{ij} = 0 ", "4d1d1561c3470d695afe4538a2161a39": "V_{\\mathrm{m}} = R T / p \\,", "4d1d56d67316e82838da7f12281ade83": "\\varphi^{n} \\equiv \\sum_{i=0}^{n} u_{i}", "4d1d9ef7bdf0b2cc1eefa1f652583170": " \\Delta\\phi_0-\\delta\\phi_1=(2\\gamma-\\mu)\\phi_0-2\\tau\\phi_1+\\sigma\\phi_2\\,, ", "4d1e27cca366b0f3b90e1cc7659ea905": "\n\\begin{pmatrix}\n {{\\mathbf{{\\dot{x}}}}}(t) \\\\\n {\\mathbf{y}}(t)\n\\end{pmatrix}={\\mathbf{S}}({\\mathbf{p}}(t))\\begin{pmatrix}\n {\\mathbf{x}}(t) \\\\\n {\\mathbf{u}}(t)\n\\end{pmatrix},\n", "4d1e309afd5966c84b03e8323f01069a": " \\delta\\ \\mathbf{q}^T \\sum_{e} \\int_{S^e} \\mathbf{N}^T \\mathbf{T}^e \\, dS^e + \\delta\\ \\mathbf{q}^T \\sum_{e} \\int_{V^e} \\mathbf{N}^T \\mathbf{f}^e \\, dV^e ", "4d1e85035e582985e86223c97928c24b": "h\\ =\\ \\frac {2\\gamma_\\mathrm{la} \\cos\\theta}{\\rho g r}", "4d1e992024e6db042982b4b6a446ea55": " D(a) \\rightarrow P^{-1} D(a) P", "4d1f2600b425acd5d185212667bbe52b": "b\\in \\mathbb{R}", "4d1f53905e9fe52101df44dade781965": "I\\mathcal{Q}_{\\mathrm{Hur}})\\},", "4d1f596a3c38ce67afcbbff6fe9ad20f": " f(x(\\lambda),\\lambda)=0 ", "4d1f7861d5d5802ae4b389012c0ec841": "\\Delta \\Pi", "4d1f9c9370cdb685f7bc3b92d731331c": "E[\\textbf{w}_k\\textbf{w}_k^T] = \\textbf{Q}_{k}^a", "4d2031de1bfc6ff0d5bed4be06f421be": "41.1~", "4d203f686376d8bf0c0d062ccec65a15": " G: X \\longmapsto \\frac{X}{||X||} .", "4d2041402c4e3f5bd19af7a20569e661": "= \\frac{1}{\\sigma\\,\\sqrt{2}\\,\\Gamma\\!\\left(\\frac12\\right)} \\times \\lim_{m\\to\\infty} \\frac{\\Gamma(m)}{\\Gamma\\!\\left(m-\\frac12\\right) \\sqrt{m-\\frac32}} \\times \\lim_{m\\to\\infty} \\left[1 + \\frac{\\left(\\frac{x-\\lambda}{\\sigma}\\right)^2}{2\\,m-3} \\right]^{-m}", "4d2095aa58c40d65a2621e3b847eaa49": "\\epsilon > 0\\,", "4d20a9ade8adec31e4cd4ee87b8bae2b": "\\scriptstyle V\\times \\cdots \\times V \\to \\mathbf{R}.", "4d20a9fde7abad26760c3a5c57739b9c": "\nE= n\\hbar \\omega,\n\\,", "4d20e8438d9ef9d316e7160b863d1f05": "\\textstyle\\vec{W}", "4d2110bf0aa678dc1f62b1e03dc99ad1": "\\{poly,Moebius\\ transformation\\}", "4d212afa1b8bfb32c0e11cab465251fc": "\\rho \\otimes \\rho = \\left(\\rho \\wedge \\rho \\right) \\oplus \\textrm{Sym}^2 \\rho", "4d213d237de40a127413be4faef7be5b": "x_{n_k}", "4d215197671b0e346bd9568b631025db": "\\hat r", "4d215622be144f349077cc2779651492": "a_j^{\\nu_j}", "4d222d6800d4cac254e130ad779beb8a": "\\frac{\\partial f}{\\partial x}(X,Y) \\neq 0", "4d2257a3b6ba30b2a9ef536d0fecaa79": "g = \\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix} \\in GL_2( \\Bbb{R})", "4d227a94ae24906207552fddf82f1ae6": "(\\mathbf{I})\\,\\mathrm{d}(ab)=a\\,\\mathrm{d}b+b\\,\\mathrm{d}a+\\mathrm{d}a\\,\\mathrm{d}b\\,,", "4d22a09b73b3e386fa3b469d1572b077": "\\tilde q", "4d22c04cb467e72a6db19d7c65f4bed0": "\\theta_E = \\theta_S \\frac{t_E}{t_S}", "4d230412ab609946e9cc80d1d19b8a06": "R(u,v)w=\\nabla_u\\nabla_v w - \\nabla_v \\nabla_u w - \\nabla_{[u,v]} w", "4d236d6cdaa34c9bec882f0468ea8134": "|f_{2i}\\rangle", "4d23937c9179149a6f90d56723b67f9f": " B(z)=B_{0}+B_{a}\\sqrt{1-z/z_{0}}", "4d2409df2c729528eeada0b8b17a5809": "[OH^-]_0", "4d245de1dda1a1dbc2b272096c02f3b7": " f_k(x) \\leq f_{k+1}(x) \\quad \\forall k\\in \\mathbf{N}, \\, \\forall x \\in E. ", "4d2460b428753844121c95afb0b0bc87": "(a^2 + b^2)((x_0 - m)^2 + (y_0 - n)^2) = (ax_0 + by_0 + c)^2 ", "4d249022d923dc114f9e795694c130f6": " \\langle w_{i_1}(M)w_{i_2}(M) ... w_{i_k}(M),[M] \\rangle = \n\\langle w_{i_1}(N)w_{i_2}(N) ... w_{i_k}(N),[N] \\rangle \\in \\mathbf Z_2", "4d24c1ffd5bcc9541c56fcb13242bb49": " q = x + y r, \\quad r^2 = -1, \\quad r \\in \\mathbb{H} ", "4d253ac429581055e491b260b49941e1": " 0 = (\\tilde{\\mathbf{x}}')^{T} \\, \\mathbf{E} \\, \\tilde{\\mathbf{x}} = \\frac{1}{x'_{3}} (\\tilde{\\mathbf{x}}')^{T} \\, \\mathbf{E} \\, \\frac{1}{x_{3}} \\tilde{\\mathbf{x}} = (\\mathbf{y}')^{T} \\, \\mathbf{E} \\, \\mathbf{y}\n", "4d253c134a8d758c4362f0de690c916d": "K = \\frac{1}{2}mv^2", "4d25d051e7d2c454e7eea8e408bca56d": "\\|\\mathbf{u}\\|", "4d25f0f52702d037001145521817da18": "t \\gg \\tau,\\,", "4d2656c8102558722780fb12b3e0dcb1": "a \\mathrel{:} K \\to A. \\, ", "4d2666bab4bedbdc5b93d19d90368d46": " n\\lambda", "4d26e205995115e749c3ecdc2da1d535": "\\scriptstyle a / b = c", "4d26ee945768e475ab6ff11d5d07d793": "\\begin{matrix} {4 \\choose 2}{2 \\choose 1}^2 \\end{matrix}", "4d26ef2bd6d1cf160afa877eebae6309": "(x,y)=(11,5).", "4d271b57c83340d7a609aa33a9d1be3f": "\\ \\Omega", "4d271e6c6290c903ff0b31f94e2594be": "-H/\\hbar", "4d275b258f12f24857fbd61cc46a5d8b": "\\{f:[0,1]\\to X : \\ f(0)=x_0=f(1)\\}", "4d278f377d045543394ce66254db7eda": "\\mathbf{P}(X_i = 1) = \\mathbf{P}(X_i = -1) = \\tfrac{1}{2}", "4d27a0aebeeabdeeef5dcfa693063075": "f\\circ g=0", "4d27d92870fe9c3495edcc14904e2ec1": "\\omega_E=6", "4d28005318848bf8078ad383183aebbc": "A+B \\rightleftharpoons AB:\\log K_{I} =\\log \\left(\\frac{[AB]}{[A][B]} \\right)=pK_I ", "4d282cbff45288e6e541629ae26f4e33": "D=\\alpha", "4d28a45438a280238f4403ead48fda74": "\\or", "4d28f94fdba738729bcd8f581d3af45b": "\\Phi(\\tau+64)\\,", "4d290b30226ec74606b643fdda6f9f30": " k[\\Delta]\\simeq k[\\Delta']\\otimes_k k[\\Delta'']. ", "4d293aceaa7ced8cf69eca04d0dfb2e1": "H(p,\\{q\\})\\cap H(q,\\{p\\})=\\emptyset", "4d295b82057eedd23150c7ea8d2ac299": " \\mu_{Y \\mid x} = \\mathcal{C}_{Y \\mid X} \\phi(x) ", "4d298103a44ea9d9598910de439b397f": "p_{LB}", "4d299161242c3facbb23bdc7e9ddbe38": "K_\\mu = 2\\epsilon_{\\mu\\nu\\alpha\\beta} \\left( A^{\\nu a} \\partial^\\alpha A^{\\beta a} + \\frac{1}{3} f^{abc} A^{\\nu a} A^{\\alpha b} A^{\\beta c} \\right),", "4d299175b39094ee7be540fe05ea16a1": "\\begin{align}\n\\nabla \\cdot \\mathbf{E} &= 0, \\\\\n\\nabla \\cdot \\mathbf{B} &= 0, \\\\\n\\nabla \\times \\mathbf{E} &= -\\frac{\\partial\\mathbf B}{\\partial t}, \\\\\n\\nabla \\times \\mathbf{B} &= \\frac{1}{c^2} \\frac{\\partial \\mathbf E}{\\partial t}.\n\\end{align}", "4d2999c2bc58b27371cdda9fca6c72dc": " x \\ll x", "4d29adae59d361c6cf8671f215981a49": "e f \\equiv 1 \\pmod{n-1}", "4d29d8efdf5195c55b6a3d851e1b2f1f": "j=0,1,2,\\dots,n", "4d2a8bc35a0bb0a85004a32c9a39e507": "f_0=\\sqrt{f_ef_o}.", "4d2ab61b0518ae1a4c9a9d5887fd747c": "d_\\infty(X,Y)=\\mathrm{ess } \\sup_\\omega|X(\\omega)-Y(\\omega)|,", "4d2abe632a51597ebc49095102ff005f": "\n\\langle Y\\cdot v, w \\rangle = -\\langle v, Y\\cdot w \\rangle\n", "4d2ae06a35cb6e8c03ed2998ab8328c4": "\\mathfrak{f}(\\chi) = \\prod_p p^{f(\\chi,p)}", "4d2b27bb051b07ade21c0d31c5176a7d": "\\sum_{k=0}^{n}{n \\choose k}A_{k}B_{n-k}.", "4d2b5f77d3b60ed7a14f0b4eda2e74de": "T : D \\to D", "4d2bcde9f89ad9b859e8c0f2abcecbe4": " V_\\max = {{{vK_m} \\over {[S]}} + {{v[S]} \\over {[S]}}} = {{vK_m}\\over {[S]}} + v", "4d2bd4c5f0ec98d5e0c3f5da7a48d4a0": " {}-1672280820\n x^{15}+40171771630 x^{14}-756111184500\n x^{13} \\,\\!", "4d2bd59f6b651683c53693969effc92c": "X \\sim \\mathrm{F}(1, \\nu_2)\\,", "4d2be38408132eb9d31e772542d30882": " \\log|\\Lambda|>(16nd)^{200n}\\Omega(\\log\\Omega-\\log\\log A_n)(\\log B+\\log\\Omega)", "4d2be6c825cffec688e5a529ea42fb58": "T_n = \\sum_{k=0}^\\infty {k+n-1 \\choose k} \\left[\\zeta(k+n+2)-1\\right]", "4d2c358b9d2dc76831bc476c13cdfcce": "\\,\\alpha = (\\alpha_1, \\alpha_2)", "4d2ca0ade7b83f99f356e573b9c1f45f": "\\vec \\nabla \\times (\\vec v \\times \\vec \\omega ) = -\\vec \\omega (\\vec \\nabla \\cdot \\vec v) + (\\vec \\omega \\cdot \\vec \\nabla ) \\vec v - (\\vec v \\cdot \\vec \\nabla) \\vec \\omega ", "4d2cb6c047d4a16097504d16bf90b755": "\\mathbb{R}^d \\times \\{1,2,\\dots,K\\}", "4d2d4b93b615f75351fc67a4a120f1b6": "f\\left(x,y\\right)\\approx f\\left(a,b\\right)+\\frac{\\partial f}{\\partial x}\\left(a,b\\right)\\left(x-a\\right)+\\frac{\\partial f}{\\partial y}\\left(a,b\\right)\\left(y-b\\right).", "4d2d4fa98ed2df027700bb4ecedba766": " [0. x_1 \\ldots x_m] = \\sum_{k = 1}^m x_k 2^{-k}.", "4d2dfb5e002cbdff15d882e18a8bb954": "F_{tot}", "4d2dfc050e62a6c9ee4fc126c51438b4": "(\\alpha,\\beta) = \\left| \\ln \\frac{a'}{b'} \\right|.", "4d2dff6aeb665003afed154305f5f36f": "Y_n\\ \\xrightarrow{a.s.}\\ Y", "4d2e2f20bc7dd5304c4012f865b88952": "G: \\{0, 1\\}^\\ell\\to\\{0,1\\}^n", "4d2e4e6eb89c02efb18c0cf5667a6805": "F_{thrust-gate} = \\gamma b(\\Delta M) = \\gamma b(M_{unit,1} - M_{unit,2}) ", "4d2e54968c0e5fedcd404e39fa211a97": "(x_b,y_b)", "4d2e980ad911831149a1a8237398b6b2": "\\psi^{-1}\\,", "4d2ed6ac711d7064c062ced9c64052a8": "\\mathcal F_y", "4d2f43476cb91b16b90715e655747655": "\\quad s(x) = \\prod_{p \\in \\mathbb{P}} p^{e_p(x)}\\quad", "4d2f4fdb7da8ede73a2b80e04cd4a911": "x_i \\mapsto T_i.", "4d2f9146a78698133e4cf1a9a72091f0": "\\Psi[U\\mathbf{A}U^{-1}-(dU)U^{-1},U\\phi]=e^{i\\theta}\\Psi[\\mathbf{A},\\phi]", "4d2fbf6d08c94220a01e7cf1056b6faf": "f(x|\\mu,b) = \\frac{1}{2bx} \\exp \\left( -\\frac{|\\ln x-\\mu|}{b} \\right) \\,\\!", "4d3012d4f4ab920858923ae7950124d4": "\\ \\bar{n}(\\epsilon_i) > 1 ", "4d3110bbec95a62e50fed3c75f6ddf26": " C_{(-)} \\Gamma_a C_{(-)}^{-1} = - \\Gamma_a^T ", "4d311a393155ba7727b10703eea2dc0b": "k, s_1, s_2, p_1, q_1,\\ldots , p_k, q_k, r", "4d314e7bd9d5d49081cbe698399b9416": " - \\left( \\frac{a}{b} \\right) = \\frac{-a}{b} = \\frac{a}{-b} \\quad\\mbox{and}\\quad \n \\left(\\frac{a}{b}\\right)^{-1} = \\frac{b}{a} \\mbox{ if } a \\neq 0. ", "4d316e39eeb6e7df655051bf945bdc8e": "x = 3 (7l + 3) + 1 = 21l + 10", "4d3171deb98eed457706cbdd6ae752c8": "p(x[n]; A) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\exp\\left(- \\frac{1}{2 \\sigma^2} (x[n] - A)^2 \\right)", "4d32090cf4bc8b82587dd863096a49df": "\\| X_{T} \\|_{p} \\leq \\| S_{T} \\|_{p} \\leq \\frac{p}{p-1} \\| X_{T} \\|_{p}.", "4d3265d74f453d234e271fe3cff6fab9": "\\mathrm{IFTHENELSE} = \\Lambda \\alpha.\\lambda x^{\\mathsf{Boolean}}\\lambda y^{\\alpha}\\lambda z^{\\alpha}. x \\alpha y z ", "4d326740afbe9df3ebe0cec28bd82a85": " \\omega_0 = \\sqrt {\\frac{1}{LC}-\\frac{1}{(RC)^2}} ", "4d326aea5a0d62d5ede233d78abf8b44": " \\theta \\in T^*X ", "4d328fb1255a8dad44c6a589b484c8ee": "\\scriptstyle \\sum n^m ", "4d32d359fb370eb222c4c33d741dd3c3": "\\mathit{m} - 1\\}", "4d334d3ed51363593baff7c7a8e921bd": "\\zeta(s) = - \\frac{\\Gamma(1 - s)}{2 \\pi i} \\int\\frac{(-t)^{s-1}}{e^t - 1} dt ", "4d337636befcdabd3af6faa77af5b0e0": "\nJ(\\mathbf{w}) = \\frac{\\mathbf{w}^{\\text{T}}\\mathbf{S}_B^{\\phi}\\mathbf{w}}{\\mathbf{w}^{\\text{T}}\\mathbf{S}_W^{\\phi}\\mathbf{w}},\n", "4d339216b7315ec05db673fde9dcc574": "0 = \\frac{d\\mathcal{R}}{dt}\\mathcal{R}^t+\\left(\\frac{d\\mathcal{R}}{dt}\\mathcal{R}^t\\right)^t = W + W^t", "4d33fb4e487aa85e4c53565979df6979": "\\dot{q}", "4d34505936d9aa6cb532b1ece128e7dd": "(q,\\omega,q) \\in \\Delta.", "4d345c120d882a47e3d9d206dd44debf": "E\\left( t \\right )", "4d348b56359f2e352f99bcdd83cee2b1": "F(g) = \\langle g| \\phi\\rangle", "4d3498bb8d7b05e7c0070d87ef66817c": "\\mathbb S^1\\times\\mathbb R", "4d349ad61a82c1616573596c32d631ca": "\\mathbf{n} \\times (\\mathbf{H}_{scat} + \\mathbf{H}_{inc}) = \\mathbf{H}_{int}", "4d34df014f0916e2326f529aed3627da": "\\zeta_1=\\frac{1}{\\sqrt{1+\\epsilon^2}}", "4d34eb2eccaad49ff744e0e1a246fc44": " C(u_1,\\dots,u_d;\\theta) = \\psi^{[-1]}\\left(\\psi(u_1;\\theta)+\\cdots+\\psi(u_d;\\theta);\\theta\\right) \\,", "4d3528a690e34bb48cd9796f642d7200": "\nF_r = P (\\pi r_1^2 - \\pi r_2^2) = P \\pi (r_1^2 - r_2^2)\n", "4d35954095f7df3792dc77f210f9c187": "\\mathcal{F}\\{f*g\\} = \\mathcal{F}\\{f\\} \\cdot \\mathcal{F}\\{g\\}", "4d35a226a07b93f67e2d654d0dca50b8": "\\mathbf{x}\\in R^N", "4d35ca60986835f41fee1d707381435e": " y_{n+3} - \\tfrac{18}{11} y_{n+2} + \\tfrac9{11} y_{n+1} - \\tfrac2{11} y_n = \\tfrac6{11} h f(t_{n+3}, y_{n+3}); ", "4d35cbf673ad409c2817020b8eb9bf91": " E_n = - \\frac{1}{2} \\frac{m_e q_e^4}{8 h^2 \\epsilon_{0}^2} \\frac{1}{n^2} = \\frac{-6.8 \\ \\mathrm{eV}}{n^2} \\,.", "4d35f2ee95df5359429a6fe008655e1b": "\\sum_{k=0}^\\infty (-1)^k/(k+1)\\!", "4d36415ab4090200e5d54a7de261f3a4": "\\frac{\\partial\\mathbf{a}}{\\partial q} = \\sum_{i=1}^{n}\\frac{\\partial a_i}{\\partial q}\\mathbf{e}_i", "4d366f6feea3aa319b00a0ecc78cd89a": "P_{H_i} = \\frac{1}{L}", "4d36818c032210f33fab5b1f63c448f0": "\\int_{-\\infty}^\\infty h(\\tau)\\cdot x_T(t - \\tau)\\,d\\tau", "4d3692c7b91d057842b1b8801979a842": "M_{xz} = Fx_A(L-x_A)/L", "4d36a9c910ea8b072d17c87fd4433f10": "E(\\phi,k) = \\int_0^\\phi \\sqrt{1 - k^2 \\sin^2(t)}\\,dt,", "4d36df6f25eb9a208217beb6594027cf": "P_5(x)=x^2 \\,", "4d377ec6d42be7e559b2b4dd934961b8": "\n\\mathrm{DA} = \\frac{T_\\text{SL}}{\\gamma} \\left[1-\\left(\\frac{P/P_{SL}}{\\mathrm{T}/T_{SL}}\\right)^\\frac{\\Gamma R}{gM-\\Gamma R}\\right]\n", "4d37901ae7cb46d1079d624b6bd56676": "e=\\frac{R'(w)-w}{w}\\,\\!", "4d37f1f49c4527b28b9f0a16c97a7650": "\\left(\\frac{p}{q}\\right)=\\sgn\\prod_{i=1}^{\\frac{q-1}{2}}\\prod_{k=1}^{\\frac{p-1}{2}}\\left(\\frac{k}{p}-\\frac{i}{q}\\right).", "4d38e7580b80c235f7dd302443077e60": "\\frac{\\partial c}{\\partial t} = D\\nabla^2\\left(c^3-c-\\gamma\\nabla^2 c\\right),", "4d398291c555ab261fdae064f22b8c5c": "\\frac{1}{1+iZ_o \\Omega}", "4d39cb480ccd73da03629e2702215c58": "K_*(M)=ker(f_*\\colon H_*(M)\\to H_*(X))", "4d39d0ad5add4c72c9cd101158339db1": " \\operatorname{sys}^2 \\leq \\gamma_2\\;\\operatorname{area}(\\mathbb T^2).", "4d3a04fb5e8d7618f1730616460cf575": "\\tilde A := e^{\\Delta t A}", "4d3a12ddb9a2f9e39c33a30cff57e095": "\\int\\frac{dx}{\\sinh ax} = \\frac{1}{2a} \\ln\\left|\\frac{\\cosh ax - 1}{\\cosh ax + 1}\\right|+C\\,", "4d3a5ad27097bb4814ed2427f9edda85": "3x^3 - 5x^2 + 5x - 2 = 0\\,\\!", "4d3a888752eabd87d4870e1ecad7ea9d": "2\\sum_{n=0}^{N}\\left | c_{n}\\right |^{2}\\left [1-\\cos\\left(E_{n}T\\right)\\right ] < \\delta^{2}\\sum_{n=0}^{N}\\left | c_{n}\\right |^{2}<\\delta^{2}", "4d3ae5bdb528d5e6041154445a58677a": " \\begin{align}\n\\int_R^\\varepsilon {\\sqrt{z} \\over z^2+6z+8}\\,dz&=\\int_R^\\varepsilon {e^{{1\\over 2} \\mathrm{Log}(z)} \\over z^2+6z+8}\\,dz \\\\\n&=\\int_R^\\varepsilon {e^{{1\\over 2}(\\log{|z|}+i \\arg{z})} \\over z^2+6z+8}\\,dz \\\\\n& = \\int_R^\\varepsilon { e^{{1\\over 2}\\log{|z|}}e^{1/2(2\\pi i)} \\over z^2+6z+8}\\,dz\\\\\n&=\\int_R^\\varepsilon { e^{{1\\over 2}\\log{|z|}}e^{\\pi i} \\over z^2+6z+8}\\,dz \\\\\n& = \\int_R^\\varepsilon {-\\sqrt{z} \\over z^2+6z+8}\\,dz\\\\\n&=-\\int_\\varepsilon^R {-\\sqrt{z} \\over z^2+6z+8}\\,dz \\\\\n&=\\int_\\varepsilon^R {\\sqrt{z} \\over z^2+6z+8}\\,dz.\n\\end{align}", "4d3b2ac0067f2d8f283a651276b152f2": " D^s_{m'm}(0,0,2\\pi) = d^s_{m'm}(0) e^{-i m 2 \\pi} = \\delta_{m'm} (-1)^{2m}.", "4d3bbeee8def807f8579527382c2eefe": "\\begin{align}\n {{\\left( {{R}_{\\text{E}}}+{{y}_{\\text{atm}}} \\right)}^{2}} & ={{s}^{2}}+{{\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)}^{2}}-2\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)s\\cos \\left( 180{}^\\circ -z \\right) \\\\\n & ={{s}^{2}}+{{\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)}^{2}}+2\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)s\\cos z\\text{ ;} \n\\end{align}", "4d3bd3398bb78893511842d2c9b5bfb7": "F_{\\mu \\nu}=\\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}\\,", "4d3bd3d85a744ad0e9dee3c322a3758e": "\\frac{777480}{8288641}", "4d3be3fc13d6d5c9f4896d87b4118651": "{{\\partial w }\\over{\\partial t }} + u {\\frac{\\partial w}{\\partial x}} + v {\\frac{\\partial w}{\\partial y}} + w {\\frac{\\partial w}{\\partial z}} - {\\frac{u^2 + v^2}{R}}= - { { \\frac{1}{\\varrho}}{\\frac{\\partial p}{\\partial z}}} - g +2{\\Omega u \\cos \\varphi} + \\nu \\left({\\frac{\\partial^2 w}{\\partial x^2}}+{\\frac{\\partial^2 w}{\\partial y^2}}+{\\frac{\\partial^2 w}{\\partial z^2}}\\right),\\qquad(1) ", "4d3c1a3d9c042910338dfd1312f7622a": "\n Y = X\\beta + \\varepsilon, \\qquad \\mathrm{E}[\\varepsilon|X]=0,\\ \\operatorname{Var}[\\varepsilon|X]=\\Omega.\n ", "4d3c1fcd6cef3ffef523bd5b7f85aa69": "\\delta=\\sqrt{{2\\rho }\\over{(2 \\pi f) (\\mu_0 \\mu_r)}} \\approx 503\\,\\sqrt{\\frac{\\rho}{\\mu_r f}}", "4d3c4ae1c55b0cff09b17b37e0c03ae1": "Z_o", "4d3c5cdf29703a4be7dc462cb4f2facb": "\n (10)(10) - (1)(15)(7.5) + (R_b)(15) + (R_c)(40) - 50 + M_c = 0 \\,.\n ", "4d3cc408ed380e0218cbcaaa8ef12878": "F(z):=\\frac {f(z)}{g(z)}", "4d3ce728fdcd9fb078b36c5444bd4aa4": "\\sin \\theta = \\frac{y'(s)}{\\sqrt{x'(s)^2 + y'(s)^2}} = y'(s) \\ ;", "4d3d119862237f7ed834ebc161d6c449": "\nu(\\theta) \\equiv \\frac{1}{r(\\theta)} = \\frac{\\mu}{h^{2}} ( 1 + e \\cos(\\theta - \\theta_{0}))\n", "4d3d15de98632bdcc3c9fd4bcec2fe46": "\\omega'_0=(d\\omega/dk)_{k=k_0}", "4d3d7feb2eb36c308575fa5ceffa1695": " \\int_X^\\oplus H_x d \\mu(x). \\quad", "4d3ddf2a6a6a64bdd08e899f5c16593f": "t \\geq \\tau (\\omega)", "4d3de317814deb3798fae807643a891c": "g^{ij}", "4d3dfd0aed59275c6f5f4c99d504e1e6": "f(\\sqrt[4]z)\\text{ where }f(u)=\\cos(u)+\\cosh(u)", "4d3e107f30d055b55a714af796aa814e": " \\frac{(u^{k})'}{u^{k}} = \\frac {ku^{k-1}u'}{u^{k}} = k \\frac{u'}{u} ,\\! ", "4d3e2aa571178105bce6d663ca561cf7": " E \\ = \\frac {1 - \\exp[-NTU(1 + C_{r})]}{1 + C_{r}} ", "4d3e353a1f42317ce6add756bee5947d": "\\frac{1}{\\Psi_n} = \\int_0^\\infty t^n\\, d\\alpha(t)", "4d3f263c3d6a63e1093c74e8f52690e8": "(K-S)^{+}", "4d3f301c89e7ccc018107f3b4045e01f": "KE = {\\frac{1}{2}}mv^2", "4d3f3960fef82cb8ba29fff6eba54e3a": "||A\\mathbf{x}||_V \\leq C||\\mathbf{x}||_U", "4d3f7e643110dfb169547672c34f6af3": "Z_1 Y_2 Z_3 = \\begin{bmatrix}\n c_1 c_2 c_3 - s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & c_1 s_2 \\\\\n c_1 s_3 + c_2 c_3 s_1 & c_1 c_3 - c_2 s_1 s_3 & s_1 s_2 \\\\\n - c_3 s_2 & s_2 s_3 & c_2 \n\\end{bmatrix}", "4d3fb9ceae0d22b9420d5379520db2b5": "\n{{ \\mathbf r }} \\frac{d \\hat{\\mathbf r}}{dt}\n= \\hat{\\mathbf r} \\wedge \\frac{d \\mathbf r}{dt}\n", "4d3fc377d6bd3e40cfd16c0539fd4b0a": " = \\lim_{h \\to 0}\\frac{1}{h} log_e\\left(1 + \\frac{h}{5}\\right) ", "4d4075919fb3e6b54ae1b3ae0289abb5": "\\Rightarrow \\ - bq_2 + a - b(q_1+q_2) - \\frac{\\partial C_2 (q_2)}{\\partial q_2}=0,", "4d407da4ba3e54a1fe7cd8b2475e6811": "M+m", "4d40a7b25f3700d47c867f8bc1223944": "Y(t_0) = Y_0 ", "4d40c3df04a7b7f8e8f56779759d93b7": "\\; (A-4I)^2 p_4 = (A-4I) p_3. ", "4d4157dbb4270c12242b3f8ac45c92eb": "-P = \\lbrace x, -y \\rbrace", "4d415f062b1da573194e14d9774e8470": "\\gamma = C_D/C_L", "4d41671c060bd8eee3871f61444246dc": "\\mathbb{R}^2/\\mathbb{Z}^2", "4d41798ef5b06eee256afc2e7d290727": "\\sigma_y(\\tau).\\,", "4d4191921e03bed30be59ff4a98abce3": " {\\scriptstyle\\frac{1}{2}}(|k|^2+m^2)", "4d419a1f84eee61c86121ba3da1eac18": "x-x'", "4d41e2e851a31a0b5900a5d426a2302a": "(z_1, \\dots, z_n) \\in G", "4d41e511ab58f3df37fd3aaa4565eb77": "\\scriptstyle \\vec{E}", "4d426ed3e20a87721505a6c2c83f6bd0": "c_{i}-b_{i}", "4d428056835242c9aa3b8c04ffe8d5fb": " L = \\frac{1}{8 \\pi} \\, \\eta^{ab} \\, \\phi_{,a} \\, \\phi_{,b} - \\rho \\, \\phi", "4d42bf294ad48c641d476fca6fceee76": "\\psi(x) = \\langle T, \\tau_{-x}\\varphi\\rangle.", "4d43104ff458db661df33d38b2d18e7d": " \\mathbb{H}^n=\\mathrm{SO}^{+}(1,n)/\\mathrm{SO}(n).", "4d4333c7e6bbd1905cce55565a14a053": "\nF_z(x,t) = -G\\Sigma\\int_{-\\infty}^\\infty dy' \\int_{-\\infty}^{\\infty}\n{\\left[ h(x,t) - h(x',t)\\right]\\over \\left[(x-x')^2+(y-y')^2\\right]^{3/2}}dx'\n= -2\\pi G\\Sigma k h(x,t)\n", "4d43388961022dbe04f1dbed5536bdc7": "\\mathbf{Q}(\\sqrt{-1})", "4d437773520b775f56ac16532964134b": "\n\\begin{align}\n& \\frac{m_a u_a + m_b u_b - m_b C_R(u_a - u_b) - m_b v_a}{m_a} = v_a \\\\\n& \\\\\n& \\frac{m_a u_a + m_b u_b + m_b C_R(u_b - u_a)}{m_a} = v_a \\left[ 1 + \\frac{m_b}{m_a} \\right] \\\\\n& \\\\\n& \\frac{m_a u_a + m_b u_b + m_b C_R(u_b - u_a)}{m_a + m_b} = v_a \\\\\n\\end{align}\n", "4d439b1f05471b635e49d6481e1347f0": "\\beta _m \\ ", "4d43cdc85879bd45618d0401cb532e54": "|\\partial_\\mu n\\rangle", "4d442c91baeea7f1b367b388ae897013": "p[D,E] \\text{≥} p[E,D]", "4d44a46092906f40a436b852a37a5e2e": "M_{\\phi}(X) = -\\int_0^1 \\phi(p) F_X^{-1}(p) dp", "4d44e0947a8679e2b4995eb6570dfc65": "P_2 \\uparrow MB(X,Y,W)", "4d4511c2204ba5eeb477788aa86b9793": "T_A(i)", "4d454426e0e1aebda823300e004633b2": "\\mathbb{F}_q^n ", "4d454d8610fc0ed55694c82dd7b6e0a7": "\\operatorname{Ti}_2(\\tan \\theta)= \\theta\\log(\\tan \\theta) + \\frac{1}{2}\\operatorname{Cl}_2(2\\theta) +\\frac{1}{2}\\operatorname{Cl}_2(\\pi-2\\theta)", "4d45b5c0837c6e7d2f1609af9d35820a": "\\theta_{J+1} \\sim \\frac{1}{H\\left(S\\right)+J} \\left( H + \\sum_{j=1}^{J}\\delta_{\\theta_j}\\right) ", "4d45fa1215f78ecbbfc12fab5a880e11": "L = h(D_T \\cap R)", "4d4600a2ef571f80f23e062ae9df0fde": "_2^2\\text{P} = ^{15}\\text{N}_2\\text{O}", "4d460a477ad6f9ffae0dfd155ec40bb4": "\\operatorname{arsinh}(z)", "4d461c28df6dfb89b09f853d358240c6": "\\begin{align}\nE_\\text{k} &= m \\gamma v^2 - \\frac{- m c^2}{2} \\int \\gamma d (1 - v^2/c^2) \\\\\n &= m \\gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0\n\\end{align}", "4d46224c24bc5037439daf4b952587c5": "C = \\frac{W+RY}{T}.", "4d4685b2f47545ec09963aa89e752879": "\\log \\frac{K}{K_0} = \\sigma\\rho ", "4d46bdafcc642290734659ade857d400": "f(x; d_1, d_2) = \\frac{2d_1^{d_1/2} d_2^{d_2/2}}{B(d_1/2, d_2/2)} \\frac{e^{d_1 x}}{\\left(d_1 e^{2 x} + d_2\\right)^{(d_1+d_2)/2}},", "4d46f1e96c082fd35d1278126fe40afd": "\\frac{\\partial}{\\partial q}(p\\mathbf a) = \\frac{\\partial p}{\\partial q}\\mathbf a + p\\frac{\\partial \\mathbf a}{\\partial q}.", "4d47af86360dab9547a7c8661be57731": "\\partial_n''(c) = 0", "4d48440e6817c046231b2a48c4f7d1e1": "for\\,each\\,cand \\in CandS_{k+1}", "4d48718e2833cf00aeb7e9a4ef6d0b8e": " \\mathbb C^{7} ", "4d48901437083c98bdbb45d44ff2a36f": "t >0 ", "4d48974b3703223868145f0ae405b0f7": "w_\\min", "4d4918f350a9379acdd6c851dd568bd6": " 123, 124, 134, 234, 125, 135, 235, 145, 245, 345, \\ldots. ", "4d4937c0a2e7a215b86b9c0dcb814158": "\\sum_{S\\colon e \\in S} x_S \\geqslant 1 ", "4d49519441f51bf0deceed032b24502c": "Q(x)=c_1{x_1}^2+c_2{x_2}^2 \\,", "4d4984fd86d6a4b605edb9dc0431e511": "3^\\frac{10}{13}", "4d499246fd6db305899f5f19ae3e51cf": " T_pM ", "4d49a97f1c52cc1d00795a2dcf2abfc1": "p(t) = \\frac{1}{N} \\left[ X_0 + X_1 e^{2\\pi it} + \\cdots + X_{\\lfloor N/2 \\rfloor} e^{\\lfloor N/2 \\rfloor 2\\pi it} + X_{\\lfloor N/2 \\rfloor+1} e^{-\\lfloor N/2 \\rfloor 2\\pi it} + \\cdots + X_{N-1} e^{-2\\pi it} \\right]", "4d49b9b818cc200042e00be5f5db08be": "\\begin{matrix} {2 \\choose 1}{2 \\choose 2}{3 \\choose 2}{3 \\choose 1}{40 \\choose 1} \\end{matrix}", "4d49cf0834e7704a3dada995d22599db": "b_n = \\{b_{n, j}\\}_{j=0}^\\infty \\in V, \\ \\ b_{n, j} = \\delta_{n, j},", "4d49fe25dc2567910e3b192148592a3f": "\\mathbb{S}_4", "4d4a2e0042a0ddd52cfb2232f0b722e5": "i \\omega \\to \\left( \\frac{\\omega_c'}{\\omega_c}\\right) i \\omega ", "4d4a52847fa26ea5423591c5181f24b4": " E(\\lambda) =\\chi_{[\\lambda^{-1},1]}(T)", "4d4a52d95e023f4abe3af19f6a3d061c": " y = \\exp(f(X))\\exp(\\mathrm{er}). ", "4d4ad213bd4af5eee282f069c8fbe8f3": " (z_1-z_2) (z_1-z_3)(z_2-z_3)(w_1-w_2) (w_1-w_3)(w_2-w_3) ", "4d4af4dd425e315c31461c81cfcdbf55": "\\omega\\cdot v + w(c_k),", "4d4b630d2b95708be4f86452e537e17b": " {\\mathrm{d}\\rho \\over \\mathrm{d}t} = {\\partial \\rho \\over \\partial t} + {\\partial \\rho \\over \\partial x} {\\mathrm{d}x \\over \\mathrm{d}t} + {\\partial \\rho \\over \\partial y} {\\mathrm{d}y \\over \\mathrm{d}t} + {\\partial \\rho \\over \\partial z} {\\mathrm{d}z \\over \\mathrm{d}t}. ", "4d4b6b4f98345423adb79cb5907bcf5e": "\\varphi \\rightarrow \\psi\\,\\!", "4d4bb9a90634e6c99ca1182dbb6ca0e5": "f_1(x), f_2(x)", "4d4bbc0c8f8554700af094812b1b5aab": "ws =", "4d4bd41472b6c7448deb4339ea997ca0": "dx = bc \\quad \\mathrm {or} \\quad x = \\frac {bc} {d}", "4d4c1a24cddd3e8452dcf9f0b950eb4f": "\n\\begin{array}{rl}\n1. & \\varphi \\rightarrow \\psi \\\\\n2. & \\varphi \\\\\n\\hline\n\\therefore & \\psi\n\\end{array}\n", "4d4c558f60dc5d815837bf4ed57c3cc4": "\\dot Q_k", "4d4c995d8f810fb6b9b3f6e6e4550c74": "\\nabla F(x) = 0", "4d4c9ae0e4844090a209108b4cdef9af": "d(\\varphi_1-\\varphi_2) = 0", "4d4d999fa48e3cf4ad4d0f36f3f5c0a6": "\\vec{I} = \\vec{L} + \\vec{S} = \\vec{1} + \\vec{0} \\Rightarrow \\Delta I = 0,1", "4d4ddc95a0782fcc525083a4988f7bdf": "f(x,a)=0", "4d4df1c7277b856be635577a04e84ead": "O\\bigl(n \\sqrt{\\log \\log n}\\bigr)", "4d4e156de080dbb5bf95af07fc3394ba": "\\frac{130}{17} + \\frac{16}{17} \\sqrt{2}", "4d4e6d760b4f46dc60cb3e980e79f280": "\\{\\top, \\bot\\}", "4d4e93b0b376a01a62478e73578a6735": "\\frac{V_m}{ \\sqrt{2E} } = ", "4d4ea4c8b28a1506ab70309719d30d16": "\\mathbf R_d = \\mathbf R ", "4d4f6966e39628c3effbb73b842bfd6e": "g\\in B_{p',q'}^{1-\\lambda},", "4d4f6a04833ae86d041c9dbbde9d3f27": "p(s|m)", "4d4f81ea07a9ae48563d631081a51c8c": "\\mathbb{H}.", "4d4f8a9f0c4b11e7cd481f5d105e4559": "\\mathbf{B} = \\mathbf{x} \\wedge \\mathbf{y} = \\frac{1}{2}(\\mathbf{xy} - \\mathbf{yx}).", "4d4fbfa15ad65d8307df6856d7f95cf0": "\\phi_1 \\ ", "4d4fc482747723713e5ea7c0d8dcc839": "B_{\\mathrm{HT}}\\otimes_K\\mathrm{gr}H^\\ast_{\\mathrm{dR}}(X/K)\\cong B_{\\mathrm{HT}}\\otimes_{\\mathbf{Q}_p}H^\\ast_{\\mathrm{\\acute{e}t}}(X\\times_K\\overline{K},\\mathbf{Q}_p)", "4d5001f95e92a8c43b24ded086d4b989": "(u(x)-u(d))*(v(y)-v(d))", "4d5027dcb3b8de293842d39e685d147c": "\\left[J_\\pm, V_0\\right] = \\sqrt{2} V_\\pm ", "4d5099b10aba147181962b9fcf56e8f6": "Y(s)=\\int_0^1 W(s,\\xi)\\,d\\xi.", "4d50b474c7a8b790bdd18aa1ee199f42": "\\pi_1", "4d50c82ac497cd1e2437f73671c2e247": "G(C) = \\left( \\sum_{n \\in S(C)} \\prod_{r \\in R} \\frac{v_r^{n_r}}{n_r!} \\right).", "4d50f71846101534d17671dc1ca18bff": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-5}{\\sqrt{3}},\\ \\pm1\\right)", "4d51756f0c5ca1823e71f28a96c7d6c9": " SO(1,3) \\to SO(3)", "4d517b1148878e98bb5f1140952dc826": " f_{\\rho}(x) = \\int_{Y} y d\\rho(y|x), x \\in X", "4d51d3f4d5527565dff6845e610d16c0": "\\mathbf{f}(k,\\tilde{\\mathbf{x}})= \\tilde{\\mathbf{x}}", "4d52de41a7fca88a49a916656bf49784": "\\langle \\psi | \\hat{p} | \\psi \\rangle", "4d52e4d79d3b5f954fa2fdfb0cde589b": " H_0 \\lambda^2 = - { \\hbar \\over 2m } ", "4d53393ad70ae4b563c2a2bfcca47559": "z \\in B^{\\text{op}}", "4d54aeb8a21d5618ab0b238ab3e8a928": "\\Gamma(-\\tfrac32)\\,", "4d54e28201c40d3827e9c89fbc106cf5": "= \\sum_{n = -\\infty}^{\\infty}{\\left|h[n] (1 \\cdot e)^{-j \\omega n} \\right|}", "4d554a7e92878e1fcb95a1aa3d1186a4": "r_{1}", "4d5557f4e92c3866e2ba02c332758884": "H_2(M;\\mathbb{Z})", "4d556cc20f1fc88dc75ae520c387448f": "\n\\langle f|g\\rangle-\\langle g|f\\rangle = \\langle \\hat{A}\\hat{B}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle-\\langle \\hat{B}\\hat{A}\\rangle+\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle = \\langle [\\hat{A},\\hat{B}]\\rangle\n", "4d55982b7b15132744f1d280b2e497d2": "\\int_0^{\\tan \\theta}\\frac{\\tan^{-1}x}{x}\\,dx= \\tan^{-1}x\\log x \\, \\Bigg|_0^{\\tan \\theta} - \\int_0^{\\tan \\theta}\\frac{\\log x}{1+x^2}\\,dx=", "4d559dbf3e6d8a1122e229d1cebd039b": "\\|\\theta\\|^2 \\leq M\\,\\!", "4d55afaa4b835883610b0cfc5677efea": "x_{n}=\\sin^{2}(2^{n} \\theta \\pi)", "4d55c7ac3b1c17833cee35c1f7a474c0": "(\\lfloor k\\alpha_i\\rfloor)_{k,i\\ge1}", "4d561d4dbd9bde769af86ac414ab2450": " w\\,R\\,u\\Rightarrow \\exists v\\,(w\\,R\\,v \\land v\\,R\\,u)", "4d5620737b7f954164f75a4d4609ddab": "F_{\\rm s} = \\frac{1}{2 \\pi\\cdot\\sqrt{C_{\\rm ms}\\cdot M_{\\rm ms}}}", "4d56bccf26a499245c6d8a7fa09b79cd": "C_{abcd} = \\frac{C^{(1)}_{abcd}}{\\lambda}+\\frac{C^{(2)}_{abcd}}{\\lambda^2}+\\frac{C^{(3)}_{abcd}}{\\lambda^3}+\\frac{C^{(4)}_{abcd}}{\\lambda^4}+O\\left(\\frac{1}{\\lambda^5}\\right)", "4d56e283d1369a4b9833c678ae787240": "Q=-k\\frac{dT}{dz}", "4d570a75a85353464b1fbf9e79a496a9": " \\tilde{\\boldsymbol{a}}= \\ddot r \\hat{\\boldsymbol{r}} + r\\ddot\\theta\\hat{\\boldsymbol\\theta} \\ , ", "4d5718387cd9799bdc3ff81c86a11982": " ds^2=dr^2+r^2 d \\theta\\ ^2+ r^2 \\sin^2 \\theta\\ d \\phi\\ ^2 ", "4d571fa2d84dffa3aea657aa6198aa81": "\\cos(x) = {{8 \\over \\pi} \\sum_{n=1}^{\\infty} {n \\over(4n^2-1)}\\sin(2nx)} ", "4d5735584b8205efcb8fb6a60f87eb2b": " p = exp( \\frac{ -n^2 } { 2c } ) ", "4d5770695193225ac05aa8c7dfdbc5e0": "dU = T\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}dT +\\left[T\\left(\\frac{\\partial S}{\\partial V}\\right)_{T} - p\\right]dV.\\,", "4d57940f4b8591550a9c43f03e0c44e1": "\\widehat{\\varphi} : U \\to V", "4d581278dbb8c197b987f4a7fe997ab9": "\\left( r_{\\text{th}} + 1 \\right)^2/r_{\\text{th}}", "4d5837101d17ad9844d0d11e045e5ac6": "D_n(t)=\\frac{\\sin((n+\\frac{1}{2})t)}{\\sin(t/2)}.", "4d589d8f953a92f99669c7b85a0b5c16": "L_{0}=x_{2}-x_{1}", "4d597737ed5c9557232c8f82a4586347": "e^{i(2h-2)\\theta}", "4d599f6f8e6f2595e9ee45bea21f8e5f": "\n\\omega_{k} = \\frac{k\\pi}{\\zeta_{a} - \\zeta_{b}}\n", "4d59ac6e603f7fe80ac665c039f718bc": "r:\\mathbb R\\to\\mathbb R^n", "4d59efa6e6b652fd001373b096b56d55": "\\mathbf{M}_\\mathrm{A} = \\mathbf{r}_\\mathrm{AB} \\times \\mathbf{F}_\\mathrm{B}. \\,", "4d59f587531175b4434ae403429e823e": "\\mathrm{stsys}_2{}^n \\leq n!\n\\;\\mathrm{vol}_{2n}(\\mathbb{CP}^n)", "4d5a9caf82a156fa0571b4b840b4857f": "[HG]_{eq} = \\frac{K_a[G]_{eq}[H]_o}{1 + K_a[G]_{eq}}", "4d5b125575543838671b584db9e4c3f0": " (\\cos x + i\\sin x)^n = \\cos(nx) + i\\sin(nx). \\, ", "4d5b6a2b9a17203f592b9386fa8af110": "\\phi\\in[\\phi_{intf},\\phi_{load}]", "4d5bcb1620fed4cc3bb4a7f4e9e7431f": "A u = \\sum_{i = 1}^{N} \\lambda_{i} \\langle \\varphi_{i}, u \\rangle \\varphi_{i} \\mbox{ for all } u \\in H.", "4d5c40ff9b839d0ab243a3397a11f09e": "t_m=TR^m e^{im\\delta}\\,", "4d5c6ccba312b40a1077b166952c91d7": "\\frac{\\pi a b}{4}", "4d5c85ec3e4d71ee2414612dc390b3d1": "h = \\left( \\begin{array}{cc}\n1&0\\\\\n0&-1\n\\end{array}\\right)\n", "4d5d3dd35f65ee35945fac9c52facc08": "ca(\\Sigma)", "4d5d557389d8bd48eba13c498c90d004": "C \\sub Y + \\frac{1}{2^m} C", "4d5d7110d89adc8fc2e2f537cda3da30": "s(n) = x(n) + 2 \\cos(2 \\pi \\omega) s(n-1) - s(n-2)", "4d5d7bca20dfa66d87e8ffa88ae0ba04": "-\\boldsymbol{\\alpha}{S}^{-1}\\mathbf{1}", "4d5e3706d878a9ad2dba3c57b30282bf": "P = \\frac{1}{R} \\cdot \\frac{1}{n + 1} A_\\text{volts} \\left( A_\\text{farads} + B_\\text{farads} \\right)", "4d5e77a9fdab49d734a027afb6690eb6": "v_{\\text{out}} = -V_{\\text{T}} \\ln \\left( \\frac{v_{\\text{in}}}{I_{\\text{S}} \\, R} \\right)", "4d5e78257aab6b6280ac1a1bb8f2c329": "|g^{(1)}( \\tau)|", "4d5f00ce9c46e250390e26b1e25a15ff": "i \\in \\N", "4d5f9a9c0c66d9c6a2d8c9bcb870360b": "\\ g", "4d5fa76f04a387d0df6095f6993ada4b": "(x^2 + y^2)^2 = a^2 (x^2 - y^2)\\,", "4d5fca8bb972a29b41f0a90071508a15": "V/k^2", "4d5fe5a2b3ba71905db5312d1b5ad293": "H(j \\omega) \\ \\stackrel{\\mathrm{def}}{=}\\ H(s) \\Big|_{s = j \\omega} \\ ", "4d6014992914701b87935b7fd3cf72a5": "\\bar{g}^i_n", "4d60a6075988e75fd07ff2a5a150f87e": "\n \\begin{align}\n J_1 & = \\int_{A} \\cfrac{\\partial}{\\partial x_j}\\left(W \\delta_{1j} - \\sigma_{jk}~\\cfrac{\\partial u_k}{\\partial x_1}\\right) dA \\\\\n & = \\int_A \\left[\\cfrac{\\partial W}{\\partial x_1} - \n \\cfrac{\\partial\\sigma_{jk}}{\\partial x_j}~\\cfrac{\\partial u_k}{\\partial x_1} -\n \\sigma_{jk}~\\cfrac{\\partial^2 u_k}{\\partial x_1 \\partial x_j}\\right]~dA\n \\end{align}\n ", "4d60c82aff875b58ab8bc8f1285677f2": "FV(t)", "4d618d39caa486e409dd920f82ade4d5": "P(n){{=}}\\frac{\\lambda |B|^n}{n!} e^{-\\lambda |B|}, ", "4d61e73e536bb687e8894b9eee0b2379": "\\rho = \\frac{\\text{Mass}}{\\text{Volume}}\n = \\frac{\\text{Deflection} \\times \\frac{\\text{Spring Constant}}{\\text{Gravity}}}{\\text{Displacement}_\\mathrm{Water Line} \\times \\text{Area}_\\mathrm{Cylinder}}\\,\n", "4d6238806d17ee96d4ec4d283c1b65eb": "\\sigma_{\\varepsilon}", "4d626afc48a826e067643fd4d9c43a4f": " \\begin{align}\nF(x;a_1,a_2)& = P(X \\leq x)\\\\\n& = \\exp (-(a_1+a_2)) \\sum_{i=0}^{\\lfloor x\\rfloor} \\sum_{j=0}^{[i/2]} \\frac{a_1^{i-2j}a_2^j}{(i-2j)!j!}\n\\end{align}", "4d628a6973c08bb61e44fee433dac67b": "pH = -log \\sqrt {10^{-5} 0.1} = 3.00", "4d62c1d961afab54628dc40ade3c0204": "y_i = \\alpha x_i + (1 - \\alpha) y_{i-1} \\qquad \\text{where} \\qquad \\alpha \\triangleq \\frac{\\Delta_T}{RC + \\Delta_T}", "4d63c529aeed8bacb5ba2ebe673f8f1b": "\\rho = \\frac{m}{V}", "4d63ef8372598d3f6622a2f25db365d5": "[L_i, R_i]=0", "4d64be0363b20e12165891d4996e1134": "4e^2/{(\\pi}h)", "4d655974c5091a1354c5d0d42e2ff240": "\\forall u \\,\\lnot \\partial(u)=0\\rightarrow r(u)=0.", "4d65c8a48177685b73026a912b0149ec": "...,", "4d65e32baed389ecb27c9d99aa19ec12": "\n\\mathrm{ERBS}(f) = 21.4 \\cdot log_{10}(1 + 0.00437 \\cdot f)\n", "4d65ee29ac414f2a13bab8e3f09c64c6": "D_F=\\frac{\\mu_1-\\mu_2}{(\\mu_1)^2*log(t2/t1)}", "4d661460074da93d5134f21e9aa170ad": "K_1=\\frac{k_1}{k_{-1}}", "4d66328575fd25dda3315bc65dc3764b": "-\\sqrt{\\frac{3}{10}}\\!\\,", "4d667d3f7b269e3afe097112f0c0e0a4": "\n\\begin{align}\ng(t)-g(t^-) & =h(t) \\, dt \\int_{\\Delta g} \\, \\Delta g \\eta_g(\\cdot) \\, d\\Delta g + d J_g(t).\n\\end{align}\n", "4d668258b3741f5f91a06bb90109c500": "\\sin x = \\frac{a}{\\sqrt{1 + a^2}}", "4d6698d869c625549f5d7637a058c784": " \\Box G\\left(\\tilde x -\\tilde x'\\right) = \\delta\\left(\\tilde x -\\tilde x'\\right)", "4d6748d66010e284b1000390b6075aa1": "A \\in \\mathcal F", "4d67b4755477a12d7f9b11d9f9922a11": " \\textbf{c} = \\textbf{f}_p \\cdot \\textbf{b} = \\textbf{f}_p \\cdot \\textbf{f} \\cdot \\textbf{m} \\pmod p ", "4d67e34bb2eb9d9d4d9c3b96c1771f41": "h \\circ e ", "4d67f0f2999bca2bc4d4dd2b28f4f98f": "F^1 B^n = A^n", "4d682ec4eed27c53849758bc13b6e179": "ts", "4d687833a61b58a1a252ed0ac48af721": "K g_2 L", "4d68e664288b93e524433636cdf1b049": "\\int f(x)^{dx}", "4d68ee608078741da9241227c39bdf78": "L(\\theta|x) = f(x|\\theta)", "4d69c6b2bcfc3559f2790a0c8ba18b43": "\\Downarrow", "4d6a7c8bb066a788be0cf1f3b5b04568": " w(x) = (20!) \\ell_0(x) = \\sum_{i=0}^{20} d_i \\ell_i(x) \\quad\\mbox{with}\\quad d_0=(20!) ,\\, d_1=d_2= \\ldots =d_{20}=0. ", "4d6a8ceec8f9e178f9977de56489ac27": "y = f(x) = mx + b", "4d6ac2bc3b1242b3b54a30d8ae611a6c": "\n\\begin{align}\n \\nabla \\cdot \\boldsymbol{u} &= \n \\frac{1}{r\\, \\sin(\\theta)} \\frac{\\partial}{\\partial \\theta}\\Bigl( u_\\theta\\, \\sin(\\theta) \\Bigr)\n + \\frac{1}{r^2} \\frac{\\partial}{\\partial r}\\Bigl( r^2\\, u_r \\Bigr) \n \\\\\n &=\n \\frac{1}{r\\, \\sin(\\theta)} \\frac{\\partial}{\\partial \\theta} \\left( - \\frac{1}{r} \\frac{\\partial \\Psi}{\\partial r} \\right)\n + \\frac{1}{r^2} \\frac{\\partial}{\\partial r} \\left( \\frac{1}{\\sin(\\theta)} \\frac{\\partial \\Psi}{\\partial \\theta} \\right)\n = 0.\n\\end{align}\n", "4d6ae785bcb951b087250acbebfb0124": "d_{X}(s, t) = \\sqrt{\\mathbf{E} \\big[ | X_{s} - X_{t} |^{2} ]}. \\, ", "4d6b500036b7d0bc30bf303caacbae5f": " \\mathbf{E} ( \\mathbf{r} , t ) = \\mid \\mathbf{E} \\mid \\mathrm{Re} \\left \\{ \\mathbf{Q} |\\psi\\rangle \\exp \\left [ i \\left ( kz-\\omega t \\right ) \\right ] \\right \\} ", "4d6b68110a2471d62d8e4a99a840bf21": "dT/ds", "4d6b9b1865dbdb8659ee6b52bcf28e37": "\\ln{x} = \\sum_{n=1}^\\infty {1 \\over {n}} \\left( {x - 1 \\over x} \\right)^n = \\left( {x - 1 \\over x} \\right) + {1 \\over 2} \\left( {x - 1 \\over x} \\right)^2 + {1 \\over 3} \\left( {x - 1 \\over x} \\right)^3 + \\cdots \\,", "4d6c04799b2ae398e591e34b6fd11d3e": "X = \\bigcup_{\\alpha\\in A}X_\\alpha", "4d6c1d01d7beaed88efe98c76f020340": "t_2 = t_3", "4d6c379f66675645b3ffe28a15306857": "\\frac{\\partial}{\\partial x}", "4d6c8684635e25cbd88655288ce9ddc4": "\\hat{\\textbf{x}}", "4d6c8e34240aad0eea5ea586b036d561": "{\\tilde{E}}_8", "4d6ce2de1ac30af1722588b1b249caaf": "\\|f - g\\|_{L^1} = \\int_{\\mathbf{R}^n} |f(x) - g(x)| \\, \\mathrm{d}x < \\varepsilon.", "4d6d17308707b7c7ea063acfac1736f8": "\\alpha < 0", "4d6d22a9af6909354343ead57c17a88a": "q_n = b^{n!}\\,; \\quad p_n = q_n \\sum_{k=1}^n \\frac{a_k}{b^{k!}} \\; .", "4d6da6282dc1e9dfdea2b3a050133770": "+\\pi", "4d6dad2df8abad04722befa436082b8f": "MOD (m) =10\\mathrm{~ metres} \\times \\left [\\left ({ppO_2\\over FO_2} \\right ) - 1\\right ]", "4d6dbb0184aa3004644d725dadcfbe8f": "E(m) \\in \\{0,1\\}^n", "4d6dc52ee0f8ab6a40ef3759a7c0228d": "\\alpha\\omega^{\\beta'} = \\omega^{\\beta_1 + \\beta'} \\,", "4d6df8f2b72e7c62fc7930a5f0a59e7b": "\\delta_{a}^{\\mathbb{H}}(t) = \\begin{cases} \\frac{1}{\\mu(a)}, & t = a \\\\ 0, & t \\neq a \\end{cases}", "4d6ea388bcbf9a08649fe3ace5ebde4d": "d\\,\\sin\\theta_{n} = n\\lambda", "4d6ebbde4dcf02e272fcfb9e504d09c7": "g_{ij}.", "4d6ecf5bf110c21a8b6bde52170b3691": "p^* = \\underset{p \\in P}{\\arg\\min} \\operatorname{D}_{\\mathrm{KL}}(p||q)", "4d6f49632bae94083f80dd57dbfef7e9": "\\mathbf{a}^\\mathrm T \\, [\\mathbf{a}]_{\\times} = \\mathbf{0}", "4d6f5f841e7b949eb327049da581165a": "\\bold{x_0}", "4d6fd18638837678175603f47920126e": "M\\times S^2", "4d6ff000da9ba34af92220805ef90839": "\\overline{n}", "4d70881c3356deadab57073743c29dc5": "d^{\\star}\\mathbf{F}=0", "4d70889a8e3a7ba52c558e652d0b977a": "\\psi= \\frac{V}{U^2}\\,", "4d70905c5a05ce678c723dc0935cba34": "\\,n\\in\\mathbb{N}\\,", "4d70a81e494a522f44ca1df0009aedbc": " \\mathcal{C}(\\rho)\\equiv\\max(0,\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4) ", "4d70bbceffa3b37d30e58917427241db": "\\displaystyle{\\|Y u\\|_{(k+1)} \\le C\\|\\Delta_1 u\\|_{(k)} + C^\\prime \\|u\\|_{(k+1)}.}", "4d70ce55157c3115e0b3c38feb9a1aac": "\\Gamma^k{}_{ij}=\\Gamma^k{}_{ji}\\,", "4d70d2f6dd3ca7d7875508e5ed7f498d": "\n\\mathrm{FillRad}(X\\subset E) = \\inf \\left\\{ \\epsilon > 0 \\left|\n\\;\\iota_\\epsilon([X])=0\\in H_n(U_\\epsilon X) \\right. \\right\\},\n", "4d70e172aa601298bb34671435786665": "[x_l,p_m]=i \\hbar \\delta_{lm}", "4d715c0c40955f780ca4070dcbae3f77": "m_n = \\tfrac13 e^{C\\sqrt{n}+o(1)} \\quad\\text{with } C = 2.3523418721 \\ldots.", "4d7192583eac7c6b372a9ea776c65b8c": "\\exp_{\\mathbf{R}}^{-1}(\\hat{\\mathbf{R}}) =\\mathbf{R}^{\\frac{1}{2}}\\left(\\log\\mathbf{R}^{-\\frac{1}{2}}\\hat{\\mathbf{R}}\\mathbf{R}^{-\\frac{1}{2}}\\right)\\mathbf{R}^{\\frac{1}{2}}", "4d722a60e1a1bc6a3fcfeed404ffb6ed": "x^{*} M x < 0\\,", "4d722f3ece3697d85b3906cb5a79a218": " G_n = \\begin{bmatrix} I_{n} & 0 & 0 \\\\ 0 & c_n & s_n \\\\ 0 & -s_n & c_n \\end{bmatrix} ", "4d725d8a2ba77568424a2b9fbf25fa50": "\\overset{\\underset{\\mathrm{def}}{}}{=} \\!\\,", "4d72616a10477d9808687c758cc6bb0d": "r > 3r_s", "4d726bdb724df7975ec60870841a2d28": "(a_k^2 + b_k^2) / 2", "4d726efc9e49e143a98ffd4ece223860": "L_\\mathrm{v}", "4d72bf6005b083d8c31309106f2a1f1c": "\\mathbb{Z}_{4}", "4d72f62b70591b742a64eee7de3d15ac": "{\\rm tr}(\\mathbf{A}e^{x\\mathbf{A}})", "4d731a5eaf9f5e56c65ca2ec35748923": " \\sigma _i(x,y) ", "4d7324071c219d33ae5d060ec7d76008": "A \\otimes_{\\mathbb Z} A \\rightarrow A, a \\otimes a' \\mapsto a \\cdot a'", "4d73652c2c5f46445ed77564e0c2f13a": "3(1+4 \\beta^2/\\alpha^2)/(\\delta\\gamma)", "4d73bbf8b18ceaedb2bd91704fc54b74": " \\begin{bmatrix} \\mathbf{T'} \\\\ \\mathbf{N'} \\\\ \\mathbf{B'} \\end{bmatrix} = \\begin{bmatrix} 0 & \\kappa & 0 \\\\ -\\kappa & 0 & \\tau \\\\ 0 & -\\tau & 0 \\end{bmatrix} \\begin{bmatrix} \\mathbf{T} \\\\ \\mathbf{N} \\\\ \\mathbf{B} \\end{bmatrix}.", "4d73c9864bc9e9fa70f05d4cb3ea0b38": "1 + X", "4d742af4aa2f7fef6d845dbdc4b10803": "g_1(x) = \\sqrt{1-x^2}", "4d7469e216759290c2d317ca38e42b3c": "R_s^{2}\\gg{(X_s+X_m)^2}", "4d747610437efd2e9925abe961f494f8": "\\mathcal{L}V = f", "4d747f2d632a081e0e5b87d4625ddf18": "X_{i+1} = X_i(2-DX_i)", "4d74ed6d81b8f0c3751156840c5fd155": "i=L,G,\\,", "4d7511f0f9451acea4b3e60a2f593b8d": " 6 \\, ", "4d75717fc17d2a2f3fb4a0f139d57bd3": "(\\exists x \\phi) \\rightarrow \\exists y \\psi \\qquad (1)", "4d759336c62e670fab9e04051bc87ebe": "PHM_{k} = \\gamma \\cdot D_{k - 1}", "4d75b3cd949fec5872da742a4df8c73d": "I_b \\left( ( R_1 \\parallel r_E ) + r_{\\pi} \\right) + (I_x + I_b ) R_2 = 0 \\ . ", "4d75ba84d0a2ee4165386c5649f5a8c2": "E_{trans}=\\frac{1}{2}mv^2", "4d762ecd1d92c44c441ab01a84f98a6b": "[g]\\notin N", "4d77300af2e779587cc2ef6d39daeb56": "\\det(\\bold{A}_{[p]}) \\geq 0 ", "4d7781d118a16e2e6a5229ecee1aa22c": "f(\\alpha x + \\beta y) = \\alpha f(x) + \\beta f(y) \\,", "4d77a10653744ad01e2f7bf2616c6de4": " P = {2 \\over 3} \\frac{m_e r_e a^2}{c} ", "4d77d19f975d5941350846da15b3d653": "\\vec{V}=\\nabla\\phi.", "4d780a7a01906daa54805a0053a3ce16": " \\sum a_i^2 \\beta_i + a_i \\left ( 1-\\alpha_i^2 \\right ) =0", "4d780cfb6705e8d376f22a3d8c8e5bc2": "\\mathcal{E}(x_1) \\cdot \\mathcal{E}(x_2) = (g^{x_1} r_1^m)(g^{x_2} r_2^m) = g^{x_1+x_2} (r_1r_2)^m = \\mathcal{E}( x_1 + x_2 \\;\\bmod\\; m)", "4d7837671e5ff7df9c3af546e7ef0da8": "\n2 \\int_0^{\\frac{E}{F}} \\sqrt{2m(E - Fx)}\\ dx= n h\n", "4d78b2434c8418638d1eecc59eefc3e6": "x^0 = \\boldsymbol{\\varphi}(0)", "4d78b27c657a9f93590c9a151ba3404b": "F(x, y) = \\frac{1}{2} k_1 x^2 + \\frac{1}{2} k_2 y^2 + ...", "4d78bbbe66db4340627303e822422b97": "i=1,\\dots,n", "4d78f31b1bec20d4a17574c36e75bbd0": "n = \\log p ", "4d7902f59938d866f890e2faa4c61044": "\\underline{X} = X^+ \\cup X^-", "4d79524d2dc01004debabd8b83da6d7b": "\\gamma =2", "4d7976115bec98d9606d90660ee59dd8": "s\\div a", "4d79f9b671fb180b01c9e10b0382cb20": "{d_S}", "4d7a377ad34a4e0eb3f80bfebf235e55": "M \\stackrel{a}\\to M", "4d7a3cf063b5da0b2808c0ab370f4745": "[u^k] H(z, u) = [u^k] \\exp \\left( u \\log (1+z) \\right) =\n\\frac {\\left(\\log (1+z)\\right)^k}{k!}.", "4d7a5b4786264706c7417672ea754557": "L(u) < L(v)", "4d7a71ba3b96f15c95486fbfdf859bf6": "q^{O(kn)}", "4d7a7d72f044cb8bc4fac07ccfe5fa40": "x_{2}^{0}", "4d7a9fdaf368ae4ea93b42bb868332a4": "\nG(x) = \\left\\langle \\phi (x)\\phi (0) \\right\\rangle = \\int {d^2 k \\over (2\\pi )^2 } {e^{ik \\cdot x} \\over k^2 + m^2 }.\n", "4d7ac3924d532b815c723c53135ed2ac": "x\\rightarrow y, u(x) \\rightarrow \\varphi(y)", "4d7b1003d51ad8ed0ceccb6e43e5955d": "f(\\boldsymbol{S}) = f_1(\\boldsymbol{S}) + f_2(\\boldsymbol{S})", "4d7bb6c9f7357b78f74033d958dcf0e4": "x \\in Z _+^n ", "4d7bec753fd51423c9b75ce994748767": "V_y", "4d7ca5949c1c296fbcbff168672f096e": "\\nu=1/2 ", "4d7cf3b508548b73f9dc815c876b1719": "\\partial \\boldsymbol{\\psi}(\\boldsymbol{\\theta})/\\partial \\boldsymbol{\\theta}", "4d7d26293bb36c2e2dd2111440e99e6e": "c= \\frac{n(n-1)}{2}.\\, ", "4d7d27dcc7c03169ea381f440b387536": " \\underbrace{\\alpha - x_{n+1}}_{\\epsilon_{n+1}} = \\frac {- f^{\\prime\\prime} (\\xi_n)}{2 f^\\prime(x_n)} (\\underbrace{\\alpha - x_n}_{\\epsilon_{n}})^2 \\,.", "4d7d2a6a86898f9c8dafaae8daea4ff8": "\nv = \\xi\\cdot \\omega = \\xi(2 \\pi f) = \\frac{p}{Z} = \\frac{a}{\\omega} = \\sqrt{\\frac{E}{\\rho}} = \\sqrt{\\frac{P_{ac}}{Z \\cdot A}}\n", "4d7d8da62fe90ab8581db5b52a512917": "9/k^5\\,", "4d7db0949124457828dcb35919e7b560": "[\\mathbf{k}]_\\times", "4d7dca89d923b527422f0f73468fdbec": "0.189,\\ 0.167,\\ 0.187,\\ 0.183,\\ 0.186,\\ 0.182,\\ 0.181,\\ 0.184,\\ 0.181,\\ 0.177 \\,", "4d7e56e0da5f07890b76cc37b5c75dbb": " \\sum_{j=k}^n \\tbinom j k = \\tbinom {n+1}{k+1}.", "4d7e9b4daa715110f685e36bb38017ca": "\\mathbf{a}\\in \\mathbb{Z}_q^n", "4d7eba6f8d7ec4279bba313d76c639cc": "(\\gamma, \\omega) \\mapsto \\int_\\gamma \\omega", "4d7ecc65092a61bed4cbb2832fcaf0f2": "{1 \\over r^2}{\\partial \\over \\partial r}\\!\\left(r^2 {\\partial f \\over \\partial r}\\right)\n\\!+\\!{1 \\over r^2\\!\\sin\\theta}{\\partial \\over \\partial \\theta}\\!\\left(\\sin\\theta {\\partial f \\over \\partial \\theta}\\right)\n\\!+\\!{1 \\over r^2\\!\\sin^2\\theta}{\\partial^2 f \\over \\partial \\phi^2}", "4d7ef8668c84bd4e033f131000ab44c3": "{B}_{6}^{(1)}", "4d7f8ac5712caad095c5e791195709ee": "T_{1x}=T_1 \\cos(\\alpha) \\approx T.", "4d7fda311b406195c0ef8e5929746130": "{\\vert A\\vert}^2 = \\sum_i A_i^2 = \\sum_i A_i ", "4d7fdfc915e9b4277206aee6a02c1335": "y_k = x_k^2 + \\alpha x_k + \\beta \\,", "4d7febc82e11ad217b68b569be906811": "= 0 \\,", "4d801b6ececd0f2eba4d5c4a735ee47e": "0\\ 1\\ 00\\ 01\\ 10\\ 11\\ 000\\ 001 \\ldots", "4d80812d8b9c74c8b99e90ffec565651": " y=e^{ax}.", "4d80a387ea718a0d6a57cebe0a7cb2bb": "r=r_1", "4d80b1164c8c1915a61030c9413036da": "\\rho \\left[\\frac{\\partial \\mathbf{v}}{\\partial t} + (\\mathbf{v} \\cdot \\nabla) \\mathbf{v}\\right] = \\nabla \\cdot \\boldsymbol{\\sigma} + \\mathbf{f}", "4d80ce2f74ff3e7e9f2f1a8feb74d3c4": "[t_3,\nt_4]", "4d80d51244d5ec30760e8e7c8482ef80": "v=-u'/u", "4d811a777564d0dcd9badd5faca1b60f": "{1-} \\left( \\dfrac{35}{36} \\right) ^{25} \\approx .505532 ", "4d8167e94dc158e896ac370c7fd27694": "F_2(a, b) = a\\cdot b = e^{\\ln(a) + \\ln(b)}", "4d81d0fc2fe7db7a3a37521474bc26a7": "\\sup_{i} \\sum_{j=0}^{\\infty} \\vert a_{i,j} \\vert < \\infty", "4d823b5e01265151d4751948ceea76f6": "\nM=\\begin{bmatrix}\na_{11} & a_{12} & a_{13} & a_{14} \\\\\na_{21} & a_{22} &a_{23} & a_{24} \\\\\na_{31} & a_{32} &a_{33} & a_{34} \\\\\na_{41} & a_{42} &a_{43} & a_{44} \\\\\n\\end{bmatrix}\n", "4d824426a154e03f78e23a0de1e15764": "\\displaystyle u_t=\\mathbf{b}\\cdot\\mathbf{v}_x", "4d82492a72102e38b90579ccc86c04ef": "\\sigma_X^2 = 1/12.", "4d82518097a2d26b1fdb352d27dde9da": "\\mathfrak{g}_{\\lambda} = \\{X\\in\\mathfrak{g}_0: [H,X]=\\lambda(H)X\\;\\;\\forall H\\in\\mathfrak{a}_0 \\}", "4d825b5af13ac875149c9cb7864ba716": " X^2 - aY^ 2 = P (T) Z^2", "4d8264b9e506f9d06231218eaec93638": "r_{d,s}", "4d8278f57c82825b5bc3d4768e797dd0": "\\operatorname{E}[X] = \\mu.\\,\\!", "4d82f7f5641c854e06f4a33ccae381a5": "a_n = k_1 r_1^n + k_2 r_2^n + \\cdots + k_d r_d^n,", "4d83160a7c75debae65a3e2b637a5842": "T^2_{-m}(q) = (-1)^mT^2_{+m}(q)^*", "4d831e8859e9d1ef4b116f162183f4c1": "\\alpha\\leq\\lambda", "4d83362729bcdafe2046078c639378df": "\\,P_r", "4d835dc7d791ecbfeb40026485af5a74": "d_Y (f(b), f(c)) \\leq K \\cdot (d_X (b, c))^\\alpha", "4d8395a1191e86fe8020e45a82d77956": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 18.27131\\log_e(T+273.15) - \\frac {7241.251} {T+273.15} + 130.8048 + 2.633634 \\times 10^{-5} (T+273.15)^2", "4d83ec63001b42c3172c68d44212dd47": "p \\cdot (\\Sigma _i x_i^*) \\geq r", "4d840b554f65bd7ea3dd5f0cf9d20289": "S={1\\over 16\\pi G}\\int d^4 x\\sqrt{-\\eta}(aB^{\\mu\\nu|\\alpha}B_{\\mu\\nu|\\alpha} +fB_{,\\alpha}B^{,\\alpha})+S_m", "4d8431df37e34c832da285324a080354": "A_{[\\alpha\\beta]\\gamma\\cdots}", "4d845649889f813e4079b93627ac3596": "b_{HF}=\\frac{w_{HF}}{w_{H_2O}M_{HF}}=2.19.", "4d848461df1f28a754b57a6865b53c95": "\\cos\\left(\\frac{\\pi-y}{2}\\right) = \\sin \\frac{y}{2} \\Rightarrow ", "4d85185278725cdf108b5879b7e9ff96": "{\\%RSD} = \\frac{s}{\\bar{x}} \\times 100", "4d8529d81537b644a5ba23307b8868a1": "1 \\in S", "4d8575145aa121e40ca5a64162140398": " f = q+a_2q^2+a_3q^3+\\cdots\\ ", "4d8583f87351ca9c7afcbcfd2f668cdb": " (X\\circledast Y)^\\star\\cong Y^\\star\\odot X^\\star,", "4d86671ffab05d068697a51e1c57bff2": "\\mathcal{A}(M).", "4d86aa56820bb56c45f261c1b6e99afa": "1 \\leq i \\leq D", "4d86d136b6e06fec0e44d14e84a0d09b": "d \\circ d=0", "4d8768c6de9726d9ff80ea4eee505676": "\\nabla\\gamma_0 = \\partial_t - \\nabla", "4d877557ef12fadbca4e2d3d6094d4ed": "\\frac{1}{\\sqrt{2}} (|00\\rangle + |11\\rangle).", "4d87ad0971a12f7407e9036b5f198db2": "a(n) \\bmod p", "4d87b3ac492a4e0f5f839683c9af81e0": "\\mathbf{J}_\\rho ", "4d87bd7333ca9968909f5f21028efe11": "g_{i_1j_1}g_{i_2j_2}\\cdots g_{i_nj_n}A^{i_1i_2\\cdots i_n} = A_{j_1j_2\\cdots j_n}", "4d87e52b75cf1a96f057f5a85c85840f": " k_1 \\omega_1 + k_2 \\omega_2 + \\cdots + k_n \\omega_n = 0 ", "4d87fb6bc2dc8676fabdab15c468a633": "P_k", "4d88534ca00a3b4bdc33d300cb2ece6b": "\\Im(z) > 0", "4d8853d3d623eb5fba5e9b358af65ba5": "\\int \\operatorname{erf}(x)\\, dx = \\frac{e^{-x^2}}{\\sqrt{\\pi }}+x \\operatorname{erf}(x)", "4d88bc007c4614de33050c5d237dca35": "X \\subseteq \\mathcal{X}", "4d89198ed497127d9271d00e667051e7": "0.5198 + 0.1184i", "4d89673f03ce73ce4c86d7169ea6f516": "[(x_{i1}+x_{i2})/2, x_{i2}]", "4d896b2001b80734a36e343c1e382c6b": "x^{(k+1)}", "4d8982af22e2e320197c218b7bc0ecc3": " D = X - U ", "4d89855356e2307bb659ea9e0ba660d2": "\\delta W = \\sum_{i=1}^n ( \\mathbf {F}_{i} - m_i \\mathbf{a}_i )\\cdot \\delta \\mathbf r_i = 0.", "4d89ba5e3611d16252bafbfda5da8384": "x_1, x_2, \\ldots", "4d89ed7015d2bf1f97ac53a2966ae214": "\\displaystyle{ \\pi((g^t)^{-1})f_z(x)= m(g,z)^{-1/2}f_{gz}(x).}", "4d89f997b1a5c2cc96dae4e0f45e365a": "\\tau_{i,v}", "4d8a789e4f2032dc9b254c9f2ccfc4b8": "\\sum_{n=0}^\\infty \\frac{(it)^n\\lambda^n}{n!}\\Gamma(1+n/k)", "4d8add8a510431e63d8d04fcd94fb0c0": "f(b) = 0", "4d8ade7371872e340ef21354eb5fba3c": "(\\exp(M_t/2))_{t \\ge 0}", "4d8af23fe6581eea93a5c8bf2399e0a3": "R_i, 1 \\le i \\le n", "4d8b108e5ff42146f60514f329061755": "\\sigma_\\mathrm{n}=\\frac{1}{2}\\left(\\sigma_1+\\sigma_3\\right)\\,\\!", "4d8b2220a928e880f76a1ac28b156a27": "\\mathbf{r}(t)=\\langle f(t), g(t), h(t)\\rangle", "4d8b41e3b3922250631c19474bd15555": "{\\bar{Q}}_8", "4d8b639376faa223e3b36997720f5b7e": " \\forall t \\in \\mathbb{R}: \\quad U_{t} := e^{i t A} ", "4d8b6c5029997c29837d5b4de5d65682": "F(E) = F_e (E) + F_n (E)", "4d8b752ee87835258f70ce1f8da6ff7b": "|\\epsilon' v'\\rangle", "4d8bca2850e153f0a9d584ad124fdb07": "6{{ -4q^3 + 17q^2 - 20q + 6 } \\over { (q-2) (4q-5) (5q-6) }} \\text{ for }q < {6 \\over 5}", "4d8bde4a80b7e22457ccbd3cf928e504": "K(0)=\\frac{\\pi}{2}", "4d8c50d863e52cbd1250e7be6c97ce1e": "x \\mapsto 1-E_\\alpha (-x^\\alpha)", "4d8c6be15ee3b96a4a316e769a5a5053": " \\lbrace 0 , \\dots , q-1 \\rbrace ", "4d8c6f893ecdd97b19199a2de3e2283f": "E_b/N_0", "4d8c70c3638a460d64d198df77b57aad": " P = (K'_v:v\\in V)", "4d8c7444f9d422be3821d1be45c3939c": "A=2\\sqrt{3}a^2 \\approx 3.46410162a^2", "4d8ce26fc69fe4e1635cdf9f8cdb4da2": "{}^6_2 \\text{He}_4 \\rightarrow {}^6_3 \\text{Li}_3 + \\beta^- + \\bar{\\nu}_\\text{e}", "4d8cf491c43bc43b9f93203bd576af58": "C_A : Ran(A + 1) \\rightarrow Ran(A-1) \\subset \\mathcal{H} ", "4d8d09d6ca4f56725b0a1b97b6e67a9f": "Y_{8}^{-1}(\\theta,\\varphi)={3\\over 64}\\sqrt{17\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(715\\cos^{7}\\theta-1001\\cos^{5}\\theta+385\\cos^{3}\\theta-35\\cos\\theta)", "4d8d23c40c47eb8864956d2a812de9cb": "0.9167", "4d8d3ebc60396981be901173ee1bbd15": "\\beta =7", "4d8d6778fffee2ba1bbc0d35c03771db": " \\mathcal{F} \\approx \\frac{\\pi \\sqrt{F}}{2}=\\frac{\\pi R^{1/2} }{1-R} ", "4d8d748e43c09c52a4a61b0d1ff46f75": "\\dot{V}_x \\leq -W(\\mathbf{x})", "4d8dd662da57a9c33b5f72eb88a457b6": "\\dot{x} = f(x) + B(x) u, ", "4d8de8451e4b0f3c6000bd5cafd8559f": "\\vec{F} = \\mathrm{d}\\vec{p}/\\mathrm{d}t", "4d8dfa32dd7d9332ffdfd7c02ae54b08": " {dv \\over dt} = g(t,x), ", "4d8e14dd3f87804907a77c81478ff363": "\\textstyle{4^x(2^x-1)=4(2^x+1).}", "4d8e5f169c05f9a579ec14408e869f1b": "h:\\mathbb{R}^n\\to\\mathbb{R}", "4d8e61112c4929372682272bb832f83a": "\\varepsilon_d(n)", "4d8ebd13cca310eeb61c8d8e888448fb": "S_n(c)=(c_{n-1}, c_n, c_{n+1}) \\in GF(p^2)^3", "4d8f34258669fa836cc547e86f4100f5": "|f\\,'(x_0)|<1", "4d8f459e2ebd7ecf5e979c36da3c2149": "X_{t}(\\omega)", "4d8f4ded96e8c7f1a8f88a933ef16388": "P_F = 1 - A_o \\begin{cases} P_F = Probability \\ of \\ Mission \\ Failure \\\\ A_o = Operational \\ Availability \\end{cases}", "4d8f586f99e41bacb8bf6f665d88a864": "\\sum_{k=0}^\\infty a_k=0\\!", "4d8f81468111c37c2dbf8c6e0e5c9675": "[TC \\models O \\Leftrightarrow ^{R}TC \\models O]", "4d8fccf74f063a38157365615ed186cc": "P\\setminus K", "4d90d41aa0a2699a5654c355eaef47d8": "N \\in \\N,", "4d90f34cd9e2e399541171d2292958fe": "er_k = er_1", "4d90fff95a517a224882b1781cbf1439": " S= \\int {1 \\over 4} F^2 + {1 \\over 2} m^2 A^2.\\,", "4d911cebf1289f72fa9b87d2fd076d15": " D^* ", "4d91480cd93dc3e4f576bdb8efd316e5": "\\neg~(\\neg x_1 \\oplus ... \\oplus \\neg x_n)", "4d91801420082a6d612ab1e8872e2dbf": "\\mu(t) = \\underset{\\mu}{\\operatorname{arg\\,min}} \\{ F(s(t),\\mu)) \\} ", "4d91c49857e121e1752bdaf0a5f6bf43": "\\Psi(x)=x(1-(x/k)^2)^2 ; \\ |x|\\le k ", "4d9222b19ef5275f70a0340eff4a1f01": "\\underline{\\mathbf{x}}_k(\\ell) = \\mathrm{diag} \\left\\{ \\mathbf{F}\\left[ x((\\ell -k+1) N),\\dots,x((\\ell -k-1) N-1) \\right]^T \\right\\}", "4d9227329b4502c43fa4a8d3c0352a6c": "h(G) := \\min \\left\\{ \\left. \\frac{| \\partial A |}{| A |} \\right| A \\subseteq V(G), 0 < | A | \\leq \\frac{| V(G) |}{2} \\right\\}.", "4d924264fcc786b22483ea4d2705abd3": "|\\Psi\\rangle = \\sum_{\\{s\\}} \\text{Tr}[A^{s_1} A^{s_2} \\ldots A^{s_N}] |s_1 s_2 \\ldots s_N\\rangle", "4d924e095136ff0733180d883ed388f8": " X_i \\sim \\textrm{IG}(\\mu,\\lambda)\\,", "4d931b12ac70c8f0c193d67464627e02": " [ X - E( X ) ]^{ 2k } > 0 ", "4d935cbfc4910be5ad11a7ff4edebbcc": "\\epsilon_d = \\frac{\\partial Q}{\\partial I}\\frac{I}{Q} ", "4d937833844407ac9dfc320cef469557": "3x+ 2y - z \\geq 4", "4d9389db0e241b9b2d050d2e09e7980c": "-\\frac{\\lambda^2}{\\hbar^2}\\int_{t_0}^t dt_1\\int_{t_0}^{t_1} dt_2\\sum_m\\sum_n\\sum_q e^{-\\frac{i}{\\hbar}(E_n-E_m)(t_1-t_0)}\\langle m|V|n\\rangle \\langle n|V|q\\rangle e^{-\\frac{i}{\\hbar}(E_q-E_n)(t_2-t_0)}|m\\rangle\\langle q|+\\ldots", "4d93a11a59d5019b70969a5a22135884": "\\rho_t(\\mathbf x)=\\left |\\psi(\\mathbf x, t)\\right |^2 = \\left|\\frac{\\psi_0(\\mathbf x, t)}{a}\\right|^2", "4d93cc163ae0151f457b74db575c2f5d": "\\mathbf{M} = \\mathbf{T_2T_1R{\\tilde{T_1}}}", "4d93d4aea57049d0b8b293a179198d4b": "dev (B)", "4d93fd799b0d985408a2c68fbe7824a4": "\\begin{smallmatrix}{{T}_{\\odot}}_{\\rm eff}\\end{smallmatrix}", "4d943a7a46dd53cfad1fc5536208b702": "Pwo = \\cfrac{3 Pmo + 1 Pmf}{3}", "4d948cecce5a08c4abfcdd89f883d0b4": " \\chi^\\lambda(w)=\\langle s_\\lambda, p_{\\tau(w)}\\rangle ", "4d949ea2ad95253ed40d282435686816": "\\lim_{x\\to a} f'(x) = {-\\infty}\\text{,}", "4d953ca31b495c8f80e41f5491e3a3a5": "\\int_{E}\\varphi<\\infty", "4d958faa51bbf914eb18aa9645d2ca17": "q = \\min\\{\\lfloor m/p_k^{\\alpha_k}\\rfloor, \\sigma(n/p_k^{\\alpha_k})\\}", "4d95a67a0a7b5d388f15c90948890c21": "P':=P\\setminus\\{a\\}", "4d961c10cbb5740569f7b48189b87d13": "\n\\begin{align}\n\\int\\limits_{0}^{2\\pi}{\\left(\\frac{p}{r}\\right)}^2\\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right) \\sin u \\ du\n&=\\ 2\\ e_h\\ \\left(5\\ \\sin^2 i \\ \\int\\limits_{0}^{2\\pi} \\sin^4 u \\ du \\ -\n\\int\\limits_{0}^{2\\pi} \\sin^2 u \\ du \\right) \\\\\n&=\\ 2\\pi\\ e_h \\ (\\frac{15}{4} \\sin^2 i-1)\n\\end{align}\n", "4d962c69714ac398c22a0605713aebe6": " H(1-|x|) = G_{1,1}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} 1 \\\\ 0 \\end{matrix} \\; \\right| \\, x \\right), \\qquad \\forall x ", "4d9650db3531fd9c5cbc5fa77b579813": "\\stackrel{*}{\\leftrightarrow}_R", "4d9682a22a660c94814f1e489bb45b54": "\\rho^0\\rightarrow \\pi^-\\pi^+", "4d96900afd0fc7c34328f3a29d83a7a2": "f(t) = \\mathcal{L}^{-1} \\{F\\}(t) = \\mathcal{L}^{-1}_s \\{F(s)\\}(t) = \\frac{1}{2\\pi i}\\lim_{T\\to\\infty}\\int_{\\gamma-iT}^{\\gamma+iT}e^{st}F(s)\\,ds,", "4d9720e0dc5ff83a9be3606d9c98964c": " h_x'(x,y) = h'(x)h(y) ", "4d9744dd7404b803b5dad0a593ff4936": "\\sigma_b^2(f)", "4d9764407036a61a2a3c812999fb9d20": " \\chi_\\lambda(g)=(\\pi(g)1,1).", "4d978642e5d72b15472eaa2e7a569511": "\\mathcal{L} = -g_{ij} \\partial_{t} \\pi^{ij} - NH - N_{i}P^{i} - 2 \\partial_{i} ( \\pi^{ij} N_{j} - \\frac{1}{2} \\pi N^{i} + \\nabla^{i} N \\sqrt{g} )", "4d97bce2acefeac4b8f518eb83f2f915": "\\mathbf{p}(\\mathbf{r})", "4d97d84f3f171c8effa66432b0a8b013": "D^*_N \\le D_N \\le 2^s D^*_N . \\,", "4d97e39ab055f3b0b060165240e45ca2": "\\int_0^\\infty J_z(at) J_z(bt)J_z(ct) t^{1-z}\\,dt = \\frac{2^{z-1}\\Delta(a,b,c)^{2z-1}}{\\pi^{1/2}\\Gamma(z+\\tfrac 12)(abc)^z}", "4d9815359e47ac82b2b2c458396806c0": "\\displaystyle{\\dot{\\mathbf{n}}=-\\kappa \\mathbf{t},\\,\\,\\, \\ddot{\\mathbf{n}}= \\dot{\\kappa}\\mathbf{n} -\\kappa^2 \\mathbf{t}.}", "4d982b486b29c926deaee45068133523": "\\sigma_z=\\biggl( \\begin{matrix}\n 1&0\\\\0&-1\n \\end{matrix} \\biggr)", "4d983e258be88229e6c0a6b8f99fb82b": "x_5 = \\frac{1}{2} \\left(x_4 + \\frac{S}{x_4}\\right) = \\frac{1}{2} \\left(354.045 + \\frac{125348}{354.045}\\right) = 354.045.", "4d98a52e40a0ddaf51094b22ffd8c930": "\\int_0^\\pi \\frac{1}{\\delta}\\Omega(\\delta)\\,d\\delta = \\infty", "4d98d24e004c08e7441e09e57a4813b8": " G_{ab} = \\frac{8 \\pi G} {c^4} T_{ab} ", "4d991b32d458a6c2a9bbc1c00f386e16": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{\\partial T}{\\partial \\dot{q}_j} = \\sum_{i=1}^n m_i \\mathbf{\\ddot{r}}_i \\cdot \\frac {\\partial \\mathbf{r}_i}{\\partial q_j} + \\sum_{i=1}^n m_i \\mathbf{\\dot{r}}_i \\cdot \\frac {\\partial \\mathbf{\\dot{r}}_i}{\\partial q_j} = Q_j + \\frac{\\partial T}{\\partial q_j} \\ .", "4d99d99f2d319e369ff5de0dc9a38181": "\\scriptstyle(\\Omega, \\mathfrak{F},\\mathbb{P})", "4d99ef969c7b3f2f85dba58b0ee80d66": "\\mu _{i,{\\rm liq}} = \\mu _{i,{\\rm vap}}.\\,", "4d99f3ff7727f25b96696523f6cf1b47": "\\quad = \\sqrt{\\| \\vec{u} \\|^2 \\| \\vec{v} \\|^2 - (\\vec{u} \\cdot \\vec{v})^2}", "4d9a261d0b6ab058e2dfd4006c7b3234": "m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r} = m \\sum_j \\left[ \\sum_i \\left[ {\\mathrm{d} \\over \\mathrm{d}t} {\\partial \\over \\partial \\dot{q_j}} \\left( \\frac{1}{2} \\dot{r_i}^2 \\right) - {\\partial \\over \\partial q_j} \\left( \\frac{1}{2} \\dot{r_i}^2 \\right) \\right] \\right] \\delta q_j ", "4d9a54e2d18e4b1c5c43ae3cdb42d2e4": "g \\in L^{p_0}", "4d9aaa8c9c8a932453fce04577d4bdf5": "x^{i+1} := x^i + \\alpha^i T(Ax^i - \\rho(x^i) Bx^i),", "4d9abba92bea12eec03eeb186e31f31b": "\\omega \\not\\in S_0 \\cup S_1", "4d9ac0764c687995600ade52964d0b02": " \\neg (A \\and B)", "4d9adf29794dc567c0e360a3f6076b50": "\\langle x, \\ y \\rangle = \\frac{1}{4} \\left(\\|x + y \\|^2 - \\|x-y\\|^2 +i\\|x+iy\\|^2 - i\\|x-iy\\|^2\\right) \\ \\forall\\ x,y \\in V ;", "4d9b300f0b4354c2cce14b8a5035860d": "v_p\\,", "4d9b4557bca6aca1ec8ed78452007edf": "\\scriptstyle \\bar{\\Gamma}", "4d9bbb5a90fd44490c88b90e03a09016": "\\begin{matrix}\nc_0&=&g_0(\\vec z)&=&(-1)^n\\alpha_n(\\vec z)&=&(-1)^nz_1\\cdots z_n\\\\\nc_1&=&g_1(\\vec z)&=&(-1)^{n-1}\\alpha_{n-1}(\\vec z)\\\\\n&\\vdots&\\\\\nc_{n-1}&=&g_{n-1}(\\vec z)&=&-\\alpha_1(\\vec z)&=&-(z_1+z_2+\\cdots+z_n).\n\\end{matrix}\n", "4d9bbe6132cf851d5c282b599d2a8822": " x, y \\in G ", "4d9bef91a0e86014bbad8165798753bf": "\\mathfrak{so}_{4n}(\\mathbf K)", "4d9c04aa46097dcc95186a5e303f35e7": "\\sqrt{cN} \\cdot \\sum_{j\\in\\mathbb{Z}} \\exp\\left(-\\frac{\\pi c}{N}\\cdot(k+N\\cdot j)^2\\right)", "4d9cab692e6298322f23f24745289aba": "\\mathbf{z}'A' = \\mathbf{Y}", "4d9cb591e9ae02195c4274889270f081": "1 \\otimes z = b (1\\otimes z) b^{-1}", "4d9cbd2b2ee3df498e0d2354004eb3c3": "\\sum_{k=1}^n f_k'(x)", "4d9cec3995d2691f85378e4d4cf3c27c": "\\begin{align}\n(a+b)(c+d) &{} = a(c+d) + b(c+d) \\\\\n&{}= ac + ad + bc + bd\n\\end{align}", "4d9ced1df802a83e7ea6112f87b89885": "\\frac{n(n+1)}{2}.", "4d9d0f5153756c809eaceb327250e438": "x^5-5x+12=0\\,", "4d9d502f7ae4035bf5330217be5a0e70": "\n \\hat{Q}_n(\\theta) = \\frac1n \\sum_{i=1}^n \\ln f(x_i|\\theta),\n ", "4d9d58cdc5ab9b0b286b5639f4a8e040": "A_1=B_1", "4d9d8b4890e4c053cbb8b9a606d889ed": "P = \\tau \\omega", "4d9d94f1e74093f18d07955174de01f8": "\n \\dot{\\boldsymbol{\\varepsilon}} = \\mathsf{E}^{-1}~\\dot{\\boldsymbol{\\sigma}} + \\cfrac{\\boldsymbol{\\sigma}}{\\lambda}\\left[\\cfrac{||\\boldsymbol{\\sigma}||}{\\lambda}\\right]^{N-1}\\left[1 - \\cfrac{\\sigma_y}{||\\boldsymbol{\\sigma}||}\\right] \\quad \\mathrm{for}~||\\boldsymbol{\\sigma}|| \\ge \\sigma_y\n ", "4d9e06290311bcc59c6a94f5be5cd3df": "A \\subseteq L^p_d", "4d9e091288a6d59650d2fae0d78207f3": "(1 - \\frac{|n|}{N}) 1_{-N \\leq n \\leq N}", "4d9e0d52b7e7a37e91754ce8d6e94b3a": "\\frac{\\dot{W}_{pump}}{\\dot{m}}=h_2-h_1", "4d9e5fbc0d9dfa6f9ce537fa454ed1a7": "P(v_i=1|h) = \\sigma(a_i + \\sum_{j=1}^n w_{i,j} h_j)", "4d9e7fb439866aed96563133b1a20b99": "\\sum_{x=0}^{S_{\\text{rows}}}\\sum_{y=0}^{S_{\\text{cols}}} {SAD(x, y)}", "4d9e97069c6678e2c3bae8e9cda898a9": "\n \\begin{align}\n -\\cfrac{2h^3E}{3(1-\\nu^2)}& \\left(w_{,1111} + \\nu~w_{,2211} + 2(1-\\nu)~w_{,1212} + \\nu~w_{,1122} + w_{,2222}\\right) = \\\\\n & q(x,t) + 2\\rho h\\ddot{w} - \\frac{2}{3}\\rho h^3\\left(\\ddot{w}_{,11}+\\ddot{w}_{,22} + \\ddot{w}_{,33}\\right) \\,.\n \\end{align}\n", "4d9eb17282dead6c4c32e9038396aaa6": "\\mu^{2} < a^{2}", "4d9ef16dd6477800bdf557c71888e71b": "\n\\frac{{x}'^\\mu}{{x'}^2}= \\frac{x^\\mu}{x^2} - a^\\mu,\n", "4d9faf11557a578012713c3850be3ffa": "r,", "4d9fe3ffc623a8e7788c00ba5ac7c699": "\\tbinom {100} k ", "4d9ff5845b80223c9ec710871eb5108b": " \\|f-X(f)\\| \\le (L+1) \\|f-p^*\\|. ", "4da064b51a73ce5100b1fd85700dc581": "Q\\notin z", "4da07b60fdd2ca0d044e8fe99743f64b": " x = \\frac{1}{| A|}\\operatorname{adj}( A) b", "4da089a8a6f8c7e992743bab89bc86bb": "P(t)\\!", "4da09c6b944a207ac38fb3148e797355": " P(0)= E \\left({\\mathbf{x}}(0){\\mathbf{x}}'(0) \\right),", "4da132be2856929ffe7d44fa8805e866": "\\begin{matrix}{4 \\choose 3}{52 - 4r \\choose 1}\\end{matrix}", "4da19f5fa78c7bb18bb51af001769ae8": "t_1 \\longrightarrow_R t_2 \\longrightarrow_R \\ldots \\longrightarrow_R t_n", "4da1ab582d0dc9823971cfc505a97cc7": "B=127", "4da1af463c05c3a9df99acace61e6c1e": "\\log \\inf_{\\sigma_B} \\operatorname{Tr}(\\sigma_B)", "4da1fb605fd50ee49706c176a0fcba38": "\\frac{\\partial c}{\\partial t} = \\frac{\\partial c}{\\partial \\xi} \\frac{\\partial \\xi}{\\partial t} = -\\frac{\\xi}{2 t} \\frac{\\partial c}{\\partial \\xi}", "4da20702d69367db4d78761685d9f3d6": " a_2a_1 > a_3a_0 ", "4da272616e09bf0beb735f1522c7e6d9": "\nG = \\frac{Y_\\max - \\bar{Y}}{s}\n", "4da27721d6226de6869400995dc89f7c": "P(X=x) = \\prod_{C \\in \\operatorname{cl}(G)} \\phi_C (x_C) ", "4da340976f479da5f76eb445c29fa1a8": "x= \\nabla f^{\\star}(\\nabla f(x)),", "4da3474be225c9c3b5750cf38accc36a": "\\sigma_1\\sigma_2=\\sigma_3\\sqrt{-1}", "4da355d626ed1b6ea3c1e5bfba65ad1e": "c^+", "4da370593d2fc3963572dc2b6505f14e": "\\sigma = \\tfrac{M y}{I_x}", "4da375b219d015c3fd3c3980ef65f289": "\\mathbb{C}_{\\text{par}}\\left(\\mathbb{Z}_4\\right)", "4da3bbc316497b9f4d8f67c52d70a01c": "\\mathbf F'_\\mathrm{Coriolis} = -2m\\boldsymbol\\omega \\times \\mathbf v'", "4da3dc37b3687094c7e130276b086d6e": " 0 \\to L \\to M \\to N \\to 0 ", "4da3feae8435600070d382ea7510bf92": "\\mathrm{d}\\mathbf{p} = \\mathbf{p}_{\\mathrm{2}} - \\mathbf{p}_{\\mathrm{1}} = (m\\mathbf{v} + m\\mathrm{d}\\mathbf{v} + \\mathbf{v}\\mathrm{d}m - \\mathbf{u}\\mathrm{d}m) - (m\\mathbf{v}) = m\\mathrm{d}\\mathbf{v} - (\\mathbf{u} - \\mathbf{v})\\mathrm{d}m", "4da447d98d6ad352806cc6aaabd3f6c6": "\n\\frac{\\zeta(s) \\zeta(r+s-1)}{\\zeta(r)}= \n\\sum_{q=1}^\\infty \\sum_{n=1}^\\infty \n\\frac{c_q(n)}{q^r n^s}\n.\n", "4da4d8a75d87a24772d11e742518705f": "\\nabla \\cdot \\mathbf{J} = -\\frac{\\partial\\rho}{\\partial t}", "4da4f96f1524ea4c0904182a9857e387": "Z:\\mathcal{A}\\rightarrow(\\mathcal{V}^{\\mathcal{A}})^{op}", "4da5395c05195b03f774462a9f280bdd": "S_{GS}", "4da5599460d05b8bd9b03a0e80a93b7b": "1 = K_c\\left(x\\right) = c", "4da559e1b2687bd13cb4170f0e4cb281": "\\ j", "4da5865899f9bcdb970951489c217ac9": "\\Delta T - \\frac{1}{a^2} \\frac{\\partial T}{\\partial t} = - \\frac{1}{\\lambda} \\frac{\\partial Qv}{\\partial t}", "4da59b99ce2f49c0e45b0055529252c1": "j \\equiv \\varepsilon", "4da5c24065e7d7c0f7368a641e6019e5": "w_{22}", "4da5cddd113c60cb6b701e29f6fdfeb8": "0 = \\Gamma^{\\alpha}_{\\beta \\gamma} g^{\\beta \\gamma} \\!.", "4da5e65d845c2d03b3c6793e197ec5df": "\\rho = \\sum_{ijkl} p^{ij}_{kl} |i\\rangle \\langle j | \\otimes |k\\rangle \\langle l| ", "4da5f4baeb1c51efa4942f1b6eab2f1e": "C_1'", "4da60b97b777dfc6881904e9f958335b": "\\mathbf{ x}(1) = [u(1) \\,u(2)]=[85\\, 80]", "4da628409b5fe5579e6a98c742fd5ca0": "p = f \\times s", "4da66a63c49853c7bf587d21728469a7": "V_{in} = -V_{ref}\\dfrac{R_{a}}{R_{b}}\\dfrac{t_{d}}{t_{u}}", "4da696ae208aee617262d45baf9918c2": "b^2\\ = a^2 + c^2 - 2ac\\cos(\\beta)", "4da6e9b4f0e9b1ea24d1db72d68f5766": " g_n = O(\\sqrt{p_n} \\ln p_n), ", "4da6f37cb7a380f67dde08547a5cc163": "\\|\\nabla u\\| \\le \\|f\\|_{[H^1_0(\\Omega)]'}.", "4da70766709bfff81c8657bde043571f": "\\begin{align}\n \\rho \\left(\\frac{\\partial u_r}{\\partial t} + u_r \\frac{\\partial u_r}{\\partial r} + u_z \\frac{\\partial u_r}{\\partial z}\\right)\n &= -\\frac{\\partial p}{\\partial r} + \\mu \\left[\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_r}{\\partial r}\\right) +\n \\frac{\\partial^2 u_r}{\\partial z^2} - \\frac{u_r}{r^2}\\right] + \\rho g_r \\\\\n \\rho \\left(\\frac{\\partial u_z}{\\partial t} + u_r \\frac{\\partial u_z}{\\partial r} + u_z \\frac{\\partial u_z}{\\partial z}\\right)\n &= -\\frac{\\partial p}{\\partial z} + \\mu \\left[\\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_z}{\\partial r}\\right) +\n \\frac{\\partial^2 u_z}{\\partial z^2}\\right] + \\rho g_z \\\\\n \\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r u_r\\right) + \\frac{\\partial u_z}{\\partial z} &= 0.\n\\end{align}", "4da72a56739be1c09c2af8bfd5ba62bd": "\\bigcap_{i=1}^\\infty m^i = \\{0\\}", "4da7dd48f75d968f9d76eaabd2baa454": "\\frac{\\varepsilon-1}{\\varepsilon+2}= \\frac{4\\pi}{3} \\sum_j N_j \\alpha_j", "4da812429c063a3ec4ca45475668a50e": "\\,u = x^2 + 2y", "4da8292edaad79d465ee0b3d005dad57": "\\vec{v}_f = \\frac{1}{q} \\frac{\\vec{F}\\times\\vec{B}}{B^2}.", "4da88f11e9b422e3679c814b66518273": "2^{143}", "4da93d4851c9e2c18f0d85c6e324ba61": "q_{j_1 j_2} q_{j_2 j_3} \\cdots q_{j_{n-1} j_n} q_{j_n j_1} = q_{j_1 j_n} q_{j_n j_{n-1}} \\cdots q_{j_3 j_2} q_{j_2 j_1}", "4da9761b58dd418cfdbb05f3a6408daa": "\\frac{1}{T}", "4da97cc94540d8b0d833233d1e79b021": "x_{k+n} := x_{k+m} \\oplus ({x_k}^u \\mid {x_{k+1}}^l) A \\qquad \\qquad k=0,1,\\ldots", "4da9945f2551c70b7e12724bc4f72546": "x = \\cos \\frac{k\\pi}{n}\\text{ for }0 \\le k \\le n.", "4da9975f23d4e2061494fd8e3df738c2": "\n\\begin{align}\n& {} \\quad \\binom{2}{0}(w+2)^1(z+0)^0+\\binom{2}{1}(w+1)^0(z+1)^1+\\binom{2}{2}(w+0)^{-1}(z+2)^2 \\\\\n& = (w+2)+2(z+1)+\\frac{(z+2)^2}{w} \\\\\n& = \\frac{(z+w+2)^2}{w}.\n\\end{align}\n", "4daa483364f56748654c9a32997d4ab3": "L\\equiv\\partial_{xxx}+x\\partial_{xxy}+2\\partial_{xx}+2(x+1)\\partial_{xy}+\\partial_x+(x+2)\\partial_y", "4daa7065c473552f8a9b1d93f95615cd": "\nn_l n_e C_{lu} =n_u A_{ul}\n", "4dab7cc86b29fa65ab96d44a4f4f697a": " \\in x_2 \\in x_1 \\in x_0. ", "4dabab2d922d2e80fc4d8be85a41ff51": "(0, 1)", "4dac25bca00f0be7f027fca9a002d0ad": "\\frac{\\pi}{2}", "4dacbb5c31a4fb0803d042b569a0db07": "R_i(x_1,...,x_n,x) \\}\\ ", "4dace917549e1fb328af7f77c65e67fe": "G = -V^2\\left(\\frac{\\partial \\left(F/V\\right)}{\\partial V}\\right)_T", "4daceaa5286649eb4af6a69061d36c7d": "\\textstyle \\beta > 0", "4dacf668aeb74d2cb566fa2e63f16bfe": "A_n, \\mathfrak{sl}_{n+1}(\\mathbf{C}): \\mathfrak{sl}_{n+1}(\\mathbf{R})", "4dad12533be886d90bdef31d887658a7": "f(x+r) = f(x)", "4dad8fb761fbd7aacd0c6dc0dea3df17": "\n F_H=\\int (f_c+\\lambda)\\, dA+\\Delta p\\int dV \n", "4dada3e23077565dc31112cfea82b4a8": "M: A \\times A \\rightarrow A", "4dada63c04c33b155aa0cfaefe587fc3": "\\ M_{heel_{max}} << M_{pitch_{max}} ", "4dada7b8b84387f7b39e9ca71310585e": "(i\\cdot dx, j\\cdot dy)", "4dadac100f0ccdf213250a98007a5816": "\\partial_\\mu H|n\\rangle + H|\\partial_\\mu n\\rangle=\\partial_\\mu E_n|n\\rangle+E_n|\\partial_\\mu n\\rangle.", "4dae21c3503fc7868b9575da3735f905": " m/n ", "4dae5a71770a2271eebd9b80be938a13": "W \\cup E\\cup W'", "4dae78c091c196318947b179855f4f24": " \\tan^2 \\theta = \\sec^2 \\theta - 1\\!", "4dae99880873da08c5f180c0117ec695": "f(x; t) = { 1 \\over \\pi } \\left[ { t \\over x^2 + t^2 } \\right]. ", "4daee813b979ff68eeaa0f6c84039e06": "L' \\leq_{p} L", "4daf2aa0e4a9e0e5e27ee6510025c59a": "\\sim \\,", "4daf476ef0dde1db50c7d4caf269f0ba": "a_\\mu(x):= \\operatorname{E}[\\|X-x\\|], \\quad D(\\mu) := \\operatorname{E}[a_\\mu(X)], \\quad d_\\mu(x, x') := \\|x-x'\\|-a_\\mu(x)-a_\\mu(x')+D(\\mu).\n", "4db01f85e6877374f333cb61f7a0f8e6": " \\hat{p}=(0, p_1, p_1+p_2, \\ldots, p_1+p_2+\\cdots+p_n). ", "4db05299689fc52e7a6b632a462f49c6": "a_1=2=1+1", "4db0edd20807f9f9cf19334f91dd48a9": "O(n \\log m)", "4db0f1282d2566510f2921a190214bf3": " \\frac{1}{n^2}n\\sigma^2=\\frac{\\sigma^2}{n}. ", "4db0f9dda6752f29678a657a3dc81ef2": "\\widehat{\\lambda}_\\mathrm{MLE}=\\frac{1}{n}\\sum_{i=1}^n k_i. \\!", "4db10bdf037d300f5edbf5e2a4a9a200": "\n2 H_{2}O (l) \\rightarrow O_2 (g) + 4H^+ (aq) + 4 e^-\n", "4db14be315db9e209a68b21b6a6489d8": "f'(r)", "4db176d01f466694b9437225687f8966": "f_n(z)=z+\\frac{z^2}{t^n}\\Rightarrow F_n(z)\\to \\varphi (z)", "4db177bec33b9ff917b334bf5adb9424": "x \\equiv_c y", "4db17c51395d993f5f8850c926473d15": "\\mathrm{rect}() \\ ", "4db1d78c18aa759f9b2690bfe22429a1": "[nil,cons] : 1 + (A\\times List(A))\\longrightarrow List(A)", "4db1e2904ad7092b77ab25d7681e34d5": "\\mathbf A_1", "4db1eae52cd5c548befa7b7b7af1c6c1": "\\delta_\\mathcal{D}", "4db237b10e1215e46ff00e431fd8409d": "\\underset{\\text{differential}}{d\\mathbf{A}} = \\underset{\\text{derivative}}{\\frac{d\\mathbf{A}}{d\\lambda}} \\underset{\\text{differential}}{d\\lambda} ", "4db2b6396ea7267d5728ab848c293258": "\\int_0^T X_t \\, \\mathrm{d} B_t ", "4db2d53b2f4b98801b9fdf6e95e02add": "I_i \\rightarrow I_f", "4db2e0252ea9175a36aceae5968e490c": "A_0\\otimes J_3(B)_0", "4db2e206118aae223f9c84e87f94c552": "R(i,j)=1", "4db362a96bffe76ba877231887b43500": "\\mathcal{S}[\\phi]", "4db36d4342db58161485f4391f454260": "[f,P](s)=P(f\\cdot s)-f\\cdot P(s).\\,", "4db373e0ca228118db52a2a26052b015": "U(x,\\omega)= \\int u(x,t)e^{-i \\omega t} \\,dt ", "4db384b2f728b4ecd58bc0ed83ed3a25": "U = \\big\\{u, v, w, \\ldots \\big\\}", "4db391fe4a87b03bd35b251ae2f3b204": "T*", "4db3cfb4536593f52b2a2f9e0fce8d28": " v_{2} = \\left( \\frac{m_2 - m_1}{m_1 + m_2} \\right) u_{2} + \\left( \\frac{2 m_1}{m_1 + m_2} \\right) u_{1}\\,.", "4db3d8ce765da9d475acb560910d30d9": "2 \\lang n^{(0)} | n^{(2)}\\rang + \\lang n^{(1)} | n^{(1)}\\rang = 0", "4db40b740e00de569b96f136dc19ff2d": "\\textstyle\\frac{P}{Q_i}", "4db4321473af131813ed1b4e31037202": "\\lambda y.y", "4db44800afbc69ccf17ad73531f0a544": "\\mbox{grad}\\,(\\mbox{div}\\, \\vec v ) = \\nabla (\\nabla \\cdot \\vec v)", "4db4cc18142ee9e4bcc209d88746271c": " f(x) = \\sin x \\,", "4db4f53247200d6c7c7aa7db239ef8ac": "3\\pi/7", "4db4fed3bc1136411141192b1a7838e6": "{\\mathit{He}}_n(x)=e^{-D^2/2}x^n\\,\\!", "4db50089c5621aa6da1e5886ca02a61c": "u'= dx'/dt'", "4db5026515dc34bebbaf421e8cd88dce": "\\nu_t = \\left|\\frac{\\partial u}{\\partial y}\\right|l_m^2", "4db50e3471501588a9a86ba4e7c96ad7": "\\lambda^{-2}", "4db54ade3a267275362ea18150966145": "h,m\\!\\in\\!\\!\\mathcal{N}; u\\in V^+, v,v',w,w' \\in V^*", "4db5575a3331f4c37c6e531bd324b4ca": "p_{\\alpha} = \\langle\\alpha|\\widehat{p}|\\alpha\\rangle", "4db67fffcc8ba8522afcb3a62b8cc114": " D = 1 - \\sum_{ i = 1 }^K { \\frac{ n_i ( n_i - 1 ) }{ n( n - 1 ) } }", "4db71458ddb7c8fc6e5698ee91af2c3f": " r_2 = -e x + a\\,\\!", "4db7384ff3b31f8be706fdfb6522733c": "\\Delta E^{\\mathrm{tot}}=Q+W\\,\\,.", "4db7ea709295ca5ee935993952ee1fc9": "g_k=\\frac{1}{1-f_0a}\\sum_{j=1}^k \\left( a+\\frac{b\\cdot j}{k} \\right) \\cdot f_j \\cdot g_{k-j}.\\,", "4db8234c0ee983efe1c281ce10933571": "\\left| \\begin{matrix} a_{im} & a_{in} \\\\ \\,\\,a_{jm} & a_{jn} \\end{matrix} \\right|=\n\\det\\left( \\begin{matrix} a_{im} & a_{in} \\\\ \\,\\,a_{jm} & a_{jn} \\end{matrix} \\right)", "4db85009625374f294aaa635eed61769": "f^{-1}(U) \\cong \\mathrm{Spec} \\ A(U)", "4db86ad5fa146f7fc45bc3827d1020d3": "f_{yy}", "4db8b4168aa0310144cf3da96c71bf41": " \\mathbf{x}_n ", "4db950d571116cc9768bf46baadf875f": "L_{f, P_1} \\le U_{f, P_2} . \\,\\!", "4db95cc56f24f4e8fbd473ffe109ceef": "c = (x_c, y_c)", "4db95cc7e3e2a79cc0965b74fbeab0f5": "VAF = \\langle A, R, V, val, valprefs \\rangle ", "4db95e3e0b2debd92a653667ff059cff": "k_{crit}", "4db96810121d30358b5c61b155ef62d4": " M \\leq \\frac{2d}{2d-n} - 1.", "4db9a497590d724d63fb084dbdf01e76": " a_{ij} ", "4db9dd5883d6a500d314c3c0d44820f7": "x_\\text{lower} \\leq x \\leq x_\\text{upper}", "4db9f1000722fc6be364915a9debdfc3": "S: \\mathcal{H}_N \\to \\mathcal{H}_N", "4dba39327a07c130598a3e8517adf1ca": "\n\\begin{align}\n &f_o = f_r\\left[{M(N-A) + (M+1)A}\\right]\\\\\n\\Rightarrow &f_o = f_r\\left(MN+A\\right)\n\\end{align}", "4dba59ede1ecd8e2a98209e5798c4b28": " G_x(t,f) = \\int_{-1.9143+t}^{1.9143+t} e^{-\\pi(\\tau-t)^2} e^{-j2\\pi f\\tau} x(\\tau) \\, d\\tau ", "4dba88f6d0845d52b3e562e5a85d0aaf": "c_2(M)=c_2(F)", "4dbaf528e187bbf30f5c638bf60c603c": "\\pi =3.1416", "4dbafa3044d542bd594008ad5ed07ee9": "\\sigma(n)", "4dbb1ab209e7dfa036af32bf2f377666": " i\\hbar\\frac{d}{dt}|\\Psi\\rangle = \\hat{H} |\\Psi\\rangle", "4dbbbf86bd6a7163c296129b0b7dc4c5": "i=\\{i_1,...,i_k\\}", "4dbbc0e0c2e342adba0fa1f9e26a7199": "ad=bc", "4dbbcb50255a522bcd0c8f4f52621f0b": "\\textstyle \\sum_{k=1}^N k^2", "4dbc16421da3c35be5f88521053113a9": "D^\\prime(X,Y)=1-\\frac{I(X;Y)}{\\max(H(X),H(Y))}", "4dbc5c660c11f65e7a21c5952407780f": " 2q+1 ", "4dbc6a01f2ca3873e76eb97acd064ed2": "a_2 = b_1-b_2", "4dbd262b4929f7fd5d8a6a573a0e5867": "{\\color{Blue}~2.20}", "4dbd2f82caba119d2dec5d3b5bdcbada": "E_{1,\\text{thr}}", "4dbd38338e83d2ec76918d201d32259d": "\\nu = 3", "4dbd57a5cf633fb5f8f80056a0553c94": "K_M\\otimes L\\cong\\Omega^n(L)", "4dbd86d93978fdde75a483cd23790f39": "\\alpha_1,, \\ldots, \\alpha_{r + 1}, \\quad P_1, \\ldots, P_r, \\quad R_1, \\ldots, R_r, \\quad, R^1, \\ldots, R^r", "4dbd9ce9e90c6b8d7c91661ce662f803": "\\frac{d\\Phi_B}{dt} = \\frac{d}{dt}\\int_{\\Sigma(t)} \\mathbf{B}(t)\\cdot d\\mathbf{A}", "4dbd9e3aac3aea0a04581ec13de10b68": "\\|x\\|^2=\\langle x, x\\rangle.\\,", "4dbdac139403aa24255fa8d8e54cd5f9": "\\scriptstyle D_F(1\\rightarrow 7)= 4(1)-1+3-3=3", "4dbdbda06dc9ab90a3ac65928fbdd246": "\\omega(A:x)\\,\\!", "4dbde1c1c758ef6e08942ff6ce78d2c0": " \\operatorname{tr}_g(X):=\\operatorname{tr}(X^\\sharp)=\\operatorname{tr}(g^{jk}X_{ij}) = g^{ji}X_{ij} = g^{ij}X_{ij}. ", "4dbde9f93ab80bf1b8ff6f8e7b03ec92": "\\|u\\|_{L^q(U)}\\leq C \\|u\\|_{W^{k,p}(U)}", "4dbe22d333f5900a207252b277a28970": "{dt\\over d\\theta} = \\sqrt{\\ell\\over 2g}{1\\over\\sqrt{\\cos\\theta-\\cos\\theta_0}}", "4dbf0e3091be53d3b1ae56c5268ff21d": "G(S,T)", "4dbf74d8f9d651d3e74936715f138b0e": "x_{20}", "4dbfad437fa0731552f9b994025580e6": "L^2 ([0, T] \\times \\Omega; \\mathbb{R}^n)\\,", "4dbfc0e28b1dace041200c89a4be6c8f": "\\theta(a) = \\langle \\theta(a) \\cap C_G(b) \\mid b \\in B, b \\neq 1\\rangle \\subseteq \\langle \\theta(b) \\mid b \\in B, b \\neq 1\\rangle.", "4dbfe5bff9a2c059b8bff7d4f3111c84": "\\varphi(12) = 4", "4dbffebd3e22657da4147adfd3d746c1": "p_{HA}", "4dc002b6eaf0627d359b2d91ecfc4568": " q_2 = q_3", "4dc00add5f7d82018d5dab87e4907d1b": "d = d_i + v_it + a\\frac{t^2}2", "4dc029ba94cd381a4523b32fecbe4fe7": " \\sup\\nolimits_{T \\in F} \\|T\\|_{B(X,Y)} \\leq 2 \\varepsilon^{-1} m < \\infty.", "4dc0379cdb9c4c567460438c053882f7": "\\omega = e^{\\frac{2\\pi i}{d}}", "4dc0bf4b0bcf5670bce602349ccfefc5": "\\operatorname{pos}(U) = 0", "4dc0dcf274b198c4f102118804cec723": "\\int_{-W}^{W} {\\|U(f)\\|}^2 \\,dfe total power is fixed, say \n\n:\\int_{-1/2}^{1/2} {\\|U(f)\\|}^2 \\,df=1,", "4dc187391479087c10f62ad778ea634d": "2 n l \\cos\\theta", "4dc18df5c570800dda84dcbe683260b9": "(A\\mid(B\\mid C))\\mid[(D\\mid(D\\mid D))\\mid((D\\mid B)\\mid[(A\\mid D)\\mid(A\\mid D)])]", "4dc19155ff235f87023e21d224b3d079": "X = a^b.\\,", "4dc1b53e8bf0d3d4ddfee9e03997d356": "{}^*", "4dc1d345a3561d43241340e6a3508274": "-\\frac{\\beta(1-p)}{\\ln p (\\beta-t)} \\text{hypergeom}_{2,1} ", "4dc20def1d2b64c594917d5cec853360": " y_{\\beta}=x_{h(\\beta)} ", "4dc21ce657a7c68fb9995c9487f2b902": "v(t_0)=0\\,", "4dc25e187214dc986e301f3ed5a84fe4": "f:\\mathbb{N} \\longrightarrow \\mathbb{N}", "4dc25e369c00b71a319434d2e96508e5": "R^{+} \\subseteq G", "4dc2ba1be27f453c4f3bc855d837517b": "\\lceil \\log_2(b) \\rceil", "4dc2bd5738f0d32f7bc227583eeb96da": "\\mathbf{r}\\!:\\mathbb{R}\\rightarrow\\mathbb{R}^3", "4dc2c25e89a9151e37e2083765cce904": "\\sigma = c \\epsilon", "4dc2dfd1b68dbd71ac31817a7183d5a3": " -n~r^{n-1}~\\sin(n\\theta) \\,", "4dc2e29b3395d761988fe18c0c1e975a": "v = k\\frac{{[Cl_2][H_2C_2O_4]}} {{[H^+]^2[Cl^-]}}", "4dc2fdea7ad59a88ca4cfec645b145ac": " g_{a,b,c}(z) = \\frac{z^2(z-a)}{z-b} + c, \\, ", "4dc35cfe46e6166e71ce7c554f3b1d14": "V \\otimes U", "4dc3b0e5e4ecb4cc503a8133ec8b7b44": "\\{ 1 , ~j \\}", "4dc45e028db21ad15ab03eacf699d693": "\\sum_{n=1}^\\infty n^{-s}", "4dc45e5b60d43d99394f36e9c7a17229": "(U, g_{(a,k)})", "4dc4e0e2a4206761d4a43f353c5a6ea8": "\\Delta \\tau = \\sqrt{\\left(\\Delta t\\right)^2 - \\frac{\\left(\\Delta x\\right)^2}{c^2} - \\frac{\\left(\\Delta y\\right)^2}{c^2} - \\frac{\\left(\\Delta z\\right)^2}{c^2}},", "4dc53d6e2f6adca727c46d01ebb46c32": " \\int_{\\operatorname{U}(W)} (I_V \\otimes U^*) T (I_V \\otimes U) \\ d \\mu(U) ", "4dc578d54a2df096fce19bd86baa6c16": "\n f(z) = \\tfrac{1}{\\pi} e^{-\\overline{z}z} = \\tfrac{1}{\\pi} e^{-|z|^2}.\n ", "4dc57eb3b6763e49e67ac18d0df90c44": "M_n(p)=\\begin{cases}p/(p-q) + O(r^n), & q

p \\end{cases} ", "4dc58df149383cfd9c4b146978ba65c8": "\\varphi^*(u) := \\sup_{x\\in\\mathcal{X}} \\langle x,u\\rangle - \\varphi(x).", "4dc5dc015b082fdde21fc4ad1c0e46e8": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right) = \\frac{\\partial L}{\\partial \\mathbf{q}} ~.", "4dc5dc92ff4cba594b66514d438d8bd9": "\\lambda_n=\\sum_{\\rho} \\left[1- \n\\left(1-\\frac{1}{\\rho}\\right)^n\\right]", "4dc5dd94f95f16a1f99f102aae2cd4ff": "\\mathrm{BMI}", "4dc61809c342fc305ffad19f9a43f767": " \\boldsymbol { \\mathcal{P}} = {1 \\over 4\\pi c } \\mathbf{E}( \\mathbf{r}, t ) \\times \\mathbf{B}( \\mathbf{r}, t ). ", "4dc63a106fb844b6649c60da66b8a7e5": "L_x = 2 \\cdot L \\cdot \\frac{R_{B1}}{R_{B1}+R_{B2}}", "4dc63ae95bb70e87fc3537b167ecfc70": " \\neg L_1 \\or \\cdots \\or \\neg L_i \\or \\cdots \\or \\neg L_n ", "4dc65bc9ed8936adb1b9e1934f4e7945": "N_0+N_1 = b^k-1. \\, ", "4dc68ba9f47087000bc4aace8939983f": "V_{r2}", "4dc69cf977e93792d921fbbe99ad964c": "a \\mathbin{:} A ", "4dc6ac1024f2342fa9c5704341652f5d": "(1)\\qquad Q = C\\;A_2\\;\\sqrt{2\\;(P_1-P_2)/\\rho}", "4dc72ea8ca13482e8e6a18c0cadad044": " M", "4dc796f5c8a346ccd9c059aaef06692e": "r_i = r_i^{-1}", "4dc79f4a10f2db113153de91f177b098": "[a,2a]'", "4dc864548ebf2293d0a6763720d3afc0": "\\Lambda\\;", "4dc874318f9c80e89320828a94eddb17": " \\partial_{ z } = \\frac{ 1 }{ 2 } ( \\partial_{ x_1 } - i \\partial_{ x_2 } ) ", "4dc88d02a8c7d80401808362f414d711": " U = \\frac{1}{2} ( \\mathbf{E} \\cdot \\mathbf{D} + \\mathbf{H} \\cdot \\mathbf{B} ) ", "4dc8979cc7046b39034544abc1e6fa95": " Ra = \\frac{ L }{ l } ", "4dc91987d663378a29b80701763f0298": " q(z) = U (T_2(z)-T_1(z))/D = U (\\Delta\\;T(z))/D,", "4dc91cc9fb56c67ca53bc3858a48acd1": "2^n 7^m", "4dc948e9f73cbf4069777a90d43f019e": "f : S \\times R \\rightarrow S", "4dc950a5fd19dbe044c0bcecf0abcd09": "\\mathop{\\mathrm{ker}}\\,h", "4dc95256ed75a9751ce30579dc917654": "G^{\\hat{a}\\hat{b}} = 8 \\pi \\, \\left[ \\begin{matrix} \\mu &0&0&0\\\\0&p&0&0\\\\0&0&p&0\\\\0&0&0&p\\end{matrix} \\right] ", "4dc968403249171ef87746d0ac90f752": "\\alpha(s,t) + \\alpha(t,s) = 0", "4dc997eccd61981dd5ac4e89bdc7b86d": "\\begin{align}\nH_1^{\\text{RWA}}&=U^\\dagger H_{1,I}^{\\text{RWA}} U \\\\\n&=-\\hbar\\Omega e^{-i\\Delta t}e^{-i\\omega_0t}|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\Omega^*e^{i\\Delta t}|\\text{g}\\rangle\\langle\\text{e}|e^{i\\omega_0t} \\\\\n&=-\\hbar\\Omega e^{-i\\omega_Lt}|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\Omega^*e^{i\\omega_Lt}|\\text{g}\\rangle\\langle\\text{e}|.\n\\end{align}", "4dc9a68f7e28c87122599ecae90f1e70": " \\frac 1 {d^2} \\frac {\\alpha^k d^{2k}} {k!} \\left( {\\frac{w}{d}} \\right)^{k-2}.", "4dc9bb8e65001544a2cbb63cfd3f81f4": "\n\\frac{\\partial \\mathbf{Y}}{\\partial x} =\n\\begin{bmatrix}\n\\frac{\\partial y_{11}}{\\partial x} & \\frac{\\partial y_{12}}{\\partial x} & \\cdots & \\frac{\\partial y_{1n}}{\\partial x}\\\\\n\\frac{\\partial y_{21}}{\\partial x} & \\frac{\\partial y_{22}}{\\partial x} & \\cdots & \\frac{\\partial y_{2n}}{\\partial x}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial y_{m1}}{\\partial x} & \\frac{\\partial y_{m2}}{\\partial x} & \\cdots & \\frac{\\partial y_{mn}}{\\partial x}\\\\\n\\end{bmatrix}.\n", "4dc9ca6af12225df7f53ae0493d91478": "p(\\xi)=0", "4dc9d15440240ce88a718a7c2c18b30b": "\n\\operatorname{E}(T_m) = n \\log n + (m-1) n \\log\\log n + O(n), \\ \\ \n\\text{as} \\ n \\to \\infty.\n", "4dca2e0e8c262d0e4c96547f3dfc4f18": "\\mathfrak{P}^{11}", "4dca819e20b07323409d4f5563fa8c45": " \\vec{e}_0 = \\frac{1}{\\sqrt{2}} \\, \\left( \\partial_v + \\exp(-2f) \\, \\partial_u \\right) ", "4dcaf465cd6453be9874fe27417ff850": " L_2(X, \\mu)^{\\otimes n} \\to L_2(X^n, \\mu^n) ", "4dcb7521694990b637ee494256fd5c85": "h^0(X,L)-h^0(X,L^{-1}\\otimes K)=\\textrm{deg}(L)+1-g.", "4dcbaddff8c79d054b9e7000240e3531": "V=\\frac {\\mu} {r \\left(1- \\frac {1} {c} \\frac {dr} {dt} \\right)^2}", "4dcbb10afce8b2dbabb2e0bf3d9b2725": "\\sqrt{\\langle x,\\ x \\rangle} =\\|x\\|_p", "4dcbdd80a7f1f17006947bb893fcac0f": "\\left\\{\\begin{matrix}\n1+1+1 & = & 3 \\\\\n1+1+2 & = & 4 \\\\\n1+1+3 & = & 5 \\\\\n1+2+1 & = & 4 \\\\\n1+2+2 & = & 5 \\\\\n1+2+3 & = & 6 \\\\\n1+3+1 & = & 5 \\\\\n1+3+2 & = & 6 \\\\\n1+3+3 & = & 7 \\\\\n2+1+1 & = & 4 \\\\\n2+1+2 & = & 5 \\\\\n2+1+3 & = & 6 \\\\\n2+2+1 & = & 5 \\\\\n2+2+2 & = & 6 \\\\\n2+2+3 & = & 7 \\\\\n2+3+1 & = & 6 \\\\\n2+3+2 & = & 7 \\\\\n2+3+3 & = & 8 \\\\\n3+1+1 & = & 5 \\\\\n3+1+2 & = & 6 \\\\\n3+1+3 & = & 7 \\\\\n3+2+1 & = & 6 \\\\\n3+2+2 & = & 7 \\\\\n3+2+3 & = & 8 \\\\\n3+3+1 & = & 7 \\\\\n3+3+2 & = & 8 \\\\\n3+3+3 & = & 9 \n\\end{matrix}\\right\\}\n=\\left\\{\\begin{matrix}\n3 & \\mbox{with}\\ \\mbox{probability}\\ 1/27 \\\\\n4 & \\mbox{with}\\ \\mbox{probability}\\ 3/27 \\\\\n5 & \\mbox{with}\\ \\mbox{probability}\\ 6/27 \\\\\n6 & \\mbox{with}\\ \\mbox{probability}\\ 7/27 \\\\\n7 & \\mbox{with}\\ \\mbox{probability}\\ 6/27 \\\\\n8 & \\mbox{with}\\ \\mbox{probability}\\ 3/27 \\\\\n9 & \\mbox{with}\\ \\mbox{probability}\\ 1/27\n\\end{matrix}\\right\\}\n", "4dcc6f516e22e7534c1f65f1ce40b015": "w = 0.1\\ ", "4dccf908480364b14dc48cbefa7f8a88": " F'_{-}", "4dcd1ee432a60cb44641634c46d3f177": "a_k(0)", "4dcd4e107ba26a84be60d4ffa90aa83f": "\\mathbb X", "4dcd6d2f26f5ac1e011e7c45d9fed0f5": "\\mathbf D_d = \\mathbf D ", "4dcdabb109045fdc41267b5acb0bc3f7": "E_7\\cdot\\mathrm{SU}(2)\\,", "4dcdecf66ab538b5ef2b7cff082dcb93": "\\Sigma^1_1", "4dce0b09d53ead78a0d2e60dbc75deff": "b^n = x,\\,", "4dce27a5818374f008b3cf037d465303": "\\|cx\\|=|c|\\cdot\\|x\\|", "4dce2e953adbb45d22a9415977f4cf86": "\\frac{2 l}{t\\pi} - \\frac{2}{t\\pi}\\left\\{\\sqrt{l^2 - t^2} + t\\sin^{-1}\\left(\\frac{t}{l}\\right)\\right\\}+1", "4dce6e3dd096ee90904afb8681b0ebf2": " UV_{\\rm f} \\cot \\beta_{\\rm 2} ", "4dce99ec46d815d886ee6487c84b1de8": "V_D = nV_T \\ln \\left(\\frac {V_S-V_D}{R I_S}+1\\right) ", "4dceaa01eb1e5e99b27efe1ba30f008b": "x^2+y^2-1=0.\\,", "4dcec9a16d614a437152e683c3b9a429": "I_{n,m} = \\int \\frac{(ax+b)^m}{(px+q)^n} dx\\,\\!", "4dcee8da728806601b9cf5aff50b282a": "= \\phi(0) + 2 \\pi \\int_{0}^{t} f(\\tau)\\, d \\tau.", "4dcf0604367fecc40715e0a3c3b578e5": "x^3+px^2+qx=N", "4dcf62a803c0b0ab14adb7b8836f29dd": "\n\\operatorname{E}(Y | X) = \\int y f(y|x) dy = \\int y \\frac{f(x,y)}{f(x)} dy\n", "4dcff33e37d0d3a0a364bea9598e4cbc": "SU(n,q^2)", "4dd02ed747dc545ad3e0592878b8386a": "\\bar{f}=\\bar{y}.\\,", "4dd066248caa415d72305ddf9407acaf": " v_{ij}\\big ( \\mathbf{x} - \\mathbf{c}_i \\big ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{cases} \\delta_{ij} u \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) , & \\mbox{if } i \\in [1,N] \\\\ \\left ( x_{ij} - c_{ij} \\right ) u \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) , & \\mbox{if }i \\in [N+1,2N] \\end{cases} ", "4dd06f8f06bb0ebb7b86f2d95500c8f6": "F(x) = \\sum_{k=0}^{n} A_k y^k", "4dd0813978ff28b3fdacebfa5770d059": "i\\in[2,N]", "4dd0b749083df97a1da872070ac52bc3": "p\\notin F(t)(\\partial\\Omega)", "4dd0c331a5ff65c35d030d4250a388b3": "g\\in V_k", "4dd0f511ada36a38125b7ba4af49cb7e": "w=a_1a_2\\ldots\\,a_n", "4dd0fa3275b45033a382528bb0ba59eb": "\\begin{bmatrix} \\dfrac{a_{11}}{a_{21}} & \\dfrac{\\Delta \\mathbf{[a]}}{a_{21}} \\\\ \\dfrac{1}{a_{21}} & \\dfrac{a_{22}}{a_{21}} \\end{bmatrix}", "4dd1154fef64418efb13919160c939a1": "\\ \\frac{S}{C}=5.25\\times 10^4\\frac{1}{\\theta^{o2}R^2}\\frac{\\sigma}{\\sigma^o}", "4dd136f756f73633443c8cee695ae757": "\n\\begin{align}\ns(i,j,k) & {} = \\text{The value at grid point } (i,j,k)\\\\\nt(i,j,z) & {} = \\mathrm{CINT}_z\\left( s(i,j,-1), s(i,j,0), s(i,j,1), s(i,j,2)\\right) \\\\\nu(i,y,z) & {} = \\mathrm{CINT}_y\\left( t(i,-1,z), t(i,0,z), t(i,1,z), t(i,2,z)\\right) \\\\\nf(x,y,z) & {} = \\mathrm{CINT}_x\\left( u(-1,y,z), u(0,y,z), u(1,y,z), u(2,y,z)\\right)\n\\end{align}\n", "4dd1409922be63b848a0b4d0afe4c9be": " s =\n \\begin{cases}\n \\log_{10} \\log_{10} year & \\mbox{if } year > 10 \\mbox{ , corresponding to } year = 10^{10^{s}} \\\\\n 0 & \\mbox{if } 0.1 \\le year \\le 10 \\\\\n -\\log_{10} (-\\log_{10} year) & \\mbox{if } year < 0.1 \\mbox{ , corresponding to } year = 10^{-10^{-s}}\n \\end{cases}\n", "4dd16077b8ae5bf29283c27276a8f9a6": "1 \\longrightarrow C \\longrightarrow G \\longrightarrow P \\longrightarrow 1", "4dd18d64fb222387524daf028eaadd23": " \\frac{1}{3}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2} + \\frac{1}{7}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}+\\quad \\cdots", "4dd19c94a97c2c94982f146d106aaccf": "\\ell/a", "4dd259ab81b8713a4bba025f4fd00864": "\\text{a. For induction motors and 0.8 power factor synchronous motors}", "4dd27b003604a883caa34f599e81c3f8": "R:R\\times S \\to \\mathbb{R}", "4dd27ccec0b0be7f6570379083fc2ec2": " \\rho V v\\,\\!", "4dd2de2e30f64e63e778bd89a794c76a": "\\operatorname{E}[\\,\\varepsilon|X\\,] = 0.", "4dd2f4e04e8729d53e9d013b1aa6031f": "|\\nu| \\ll \\mu", "4dd30fa403410c03295b475d95960cc3": " \\equiv C_p = \\dfrac{p-p_\\infty}{q}", "4dd42286e9beeb9dd3700fd684c24b20": "{E[\\vec{X}]^a}_{a} = R_{mn} \\, X^m \\, X^n", "4dd4563acc9417bd20df3066b4309474": "X\\in D", "4dd48fca7c929f8b235ef4d67adaa5ed": "X \\rightarrow BG \\rightarrow B(G/O)", "4dd4cc70c79637dcb1d27ab76fd7920d": "\\Lambda(B) \\stackrel{\\text{a.s.}}{=} \\int_B \\lambda(x) \\, dx,", "4dd51efdbc05637e866a5b23190a25e6": " a_i (t+1) = a_i(t) + \\nu \\big [ y(t) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ] \\frac {\\rho \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} {\\sum_{i=1}^N \\rho^2 \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} ", "4dd55654b48bf1db1c33f7df082bd12a": "\\boldsymbol\\beta_k", "4dd57187e0ac7cb359fdac45d5981c7e": " C = M_{cb} + M_{wb} + M_{ct} + M_{wt} ", "4dd5a8461a38882bdccf21daa182cb5b": "K_d=\\frac{a_s}{a_m}", "4dd5a950e0b431934ef2c13686f31a74": " e_k ", "4dd65b158eb0961f02b061723d52d3d2": "\\varepsilon \\sim \\left(\\frac{\\xi_0}{\\xi}\\right)^2.", "4dd67ab14a471ecfa9c1c49a290e3f8c": " M = \\left \\lfloor 1 + \\frac{\\tau}{T - \\delta} \\right \\rfloor ", "4dd6a70a4a4630b1c3456d4124d0e83a": "H=\\int \\mathrm{d}x \\left[{1\\over 2}|\\partial_x\\psi|^2+{\\kappa \\over 2}|\\psi|^4\\right]", "4dd6ea5f6e22ade345fab6ac67e37656": "\\mu(B)=b.\\,", "4dd6f2d08c445afcfc74f4e7e989d286": "\n\\begin{align}\n\\frac{\\tan \\frac{A-B}{2 }}{\\tan \\frac{A+B}{2 } } & = \\frac{a-b}{a+b} \\\\[6pt]\n\\frac{\\tan \\frac{A-C}{2 }}{\\tan \\frac{A+C}{2 } } & = \\frac{a-c}{a+c} \\\\[6pt]\n\\frac{\\tan \\frac{B-C}{2 }}{\\tan \\frac{B+C}{2 } } & = \\frac{b-c}{b+c}\n\\end{align}\n", "4dd6fdfebefdffeeed2c829909e82f2a": "x_{ij}\\ge 0\\text{ for }i,j\\in A,T. \\, ", "4dd721d300ecbd45c311af1c39d01acc": "g_{ij}[\\mathbf{f}] = g\\left(\\frac{\\partial}{\\partial x^i},\\frac{\\partial}{\\partial x^j}\\right).", "4dd7b1f4edba89fd8f987c046bace8e5": " \\nabla(\\mathbf{A} \\cdot \\mathbf{B}) = (\\mathbf{A} \\cdot \\nabla)\\mathbf{B} + (\\mathbf{B} \\cdot \\nabla)\\mathbf{A} + \\mathbf{A} \\times (\\nabla \\times \\mathbf{B}) + \\mathbf{B} \\times (\\nabla \\times \\mathbf{A}) \\ . ", "4dd81d1744dba2cbf33f28e174f1e3f9": "\n\\langle R_d^2\\rangle =2\\tau^{(1-\\gamma)/(2c-\\gamma)}, \\qquad c=1/((1+a)).\t \n", "4dd87284d35971e7866a62cf30c4ca7e": " M_S = M \\otimes_R R_S, ", "4dd87db4f1470fb9db04933264f90a0f": "p_1(x)=x^2+4x+4\\,={(x+2)(x+2)}", "4dd8e11cc1dab19bf12f0e15829c3b91": "(\\lambda X.\\lambda Y. X\\subseteq Y)(B)(S)", "4dd90bab747a167e7975e1f8fb397a4d": "k-1\\,\\!", "4dd923cd997053979a27f4f495b52f4b": " \\alpha = 2^{-1/2}(\\langle\\alpha|\\widehat{q}|\\alpha\\rangle + i\\langle\\alpha|\\widehat{p}|\\alpha\\rangle) = 2^{-1/2}(q_{\\alpha} + ip_{\\alpha}) ", "4dd9b96d006ea90f0c09ddfaea9c1c45": "\n n!^{(k)}=\n \\left\\{\n \\begin{matrix}\n 1,\\qquad\\qquad\\ &&\\mbox{if }0\\le n 700\\, s^{-1}", "4dea07899e6618202578521903cca6c1": "\\sum_{i=1}^{p}a_{i}\\geq\\sum_{j=1}^{q}b_{j}", "4dea476160193bae97c23e9c0424774a": "a_2 = \\sum_{i=1}^{n-1} \\frac{1}{i^2}", "4dea63123badea1d9c3978c011bebf7b": "\\exists_\\pi S = \\{y\\,|\\,\\exists x\\, S(x,y)\\}", "4deae8bb8d1e8243b48be7c09f9cf614": "L(\\mathcal{G}, \\infty)", "4deb2c23d2291762fe9d67fab217daf6": " \\lim \\frac{a_n}{b_n} = c ", "4debadeac6486efacbd2c1f6b1c8b7ab": "\\lambda\\in R", "4debf6442534dc1145a890fcc9d5aeae": "\\mathbf{C_0 = } \\begin{bmatrix}\n1 & 0 & ... &0 \\\\\n1 & 0 & ... & 0 \\\\\n...\\\\ \n1 & 0 & ... &0 \\end{bmatrix}.", "4debff47c9b677d17ba8ff45a99a4808": " x \\in X, \\delta_{ext}(s, t_s, t_e, x) = (s',t_s - t_e) \\text{ if } \\delta_x(s,x)=s'.\n", "4dec006886a935e2cf4e9c196412966c": " r\\,", "4deca6eebaf2df52203c91f12f62863e": "\n \\mu =\n-e\\varphi + N\\hbar \\omega_c\n=\nN_0\\hbar \\omega_c\n", "4decd8e823557e30e906d55428dcfcc5": "{q}_x = {q}_y = {q}", "4ded152ae2ea1b33b952eb14f18e5d91": "B_E", "4ded1dbf7828b34bb75e014085ba6b46": "\\bar\\lambda_m", "4ded3dc5964ccbf7cc909ecdae42371d": "p_i = p_i(\\theta)", "4deddec7bda74a254a9929861e484e2a": "K_a =\\frac{[HG]_{eq}}{[H]_{eq}[G]_{eq}}", "4deed9137992b656d077542fc72c786d": "\\ln(1-Y)=-V_\\beta^e/V \\,\\!", "4deee30eb63fe7baac1383210d20a4cd": "s_\\alpha := x_{\\alpha-1} - x_\\alpha - l_{\\alpha-1}", "4def19a125561c6c44a570cce7108fda": "w,\\text{ }x,\\text{ }y,-z", "4def1b7517b94377cee34310e4effe0e": " m^{2n}\\equiv 1\\pmod p \\!", "4def319207422ef8964ba1f2d5f0d6eb": "\\ \\ \\preceq \\ \\ ", "4def43dde4eeeda37fc0fe4c7a88fcb5": "(M, \\omega)\\,", "4def658eadde3aa766a527b9522c3707": "R \\equiv Rb, \\; t \\equiv tb^2, c \\equiv b^2", "4defa232b135168370f1e45a82dde901": "E_{0,0} = 1 \\!", "4defec556826eccd946b23d5d3695511": "f(L)=|{\\partial V \\over \\partial L}|f(V)=3 L^{2} f(L^{3})", "4df079640db125a0afc1fea7b28568a5": "\\nu + d\\nu", "4df0889dd844f0b81379c415a7328505": "\\left(\\frac{\\partial^2U}{\\partial x\\partial y}\\right) = \\left(\\frac{\\partial T}{\\partial x}\\right)_y \\left(\\frac{\\partial S}{\\partial y}\\right)_x + T\\left(\\frac{\\partial^2 S}{\\partial x\\partial y}\\right) - \\left(\\frac{\\partial P}{\\partial x}\\right)_y \\left(\\frac{\\partial V}{\\partial y}\\right)_x - P\\left(\\frac{\\partial^2 V}{\\partial x\\partial y}\\right)", "4df099aa83fcd8fa483f27982ad16ef7": "a \\in \\mathbb{R}, q \\in \\mathbb{C}", "4df0d2be93013ec40990809dea17b637": "\\sqrt{2} = 1 + \\frac{24}{60} + \\frac{51}{60^2} + \\frac{10}{60^3} = 1.41421297.", "4df13cb5ca2f87e02710a539e3ace95f": "b \\geq (p-2+p^{-1})", "4df1633a1ffa6cb4dc44193fba142cf6": "E_{\\rm p,m} = -m\\cdot B", "4df1afbc8e618491b0a8d1e78f197fb2": "\\mu_{\\mathrm{tot}} = \\mu_{\\mathrm{int}} + \\mu_{\\mathrm{ext}} \\,", "4df1b251fde682b6617acc3af48241b6": "I_\\Delta=\\{x_i x_j: 1\\leq i < j \\leq n\\} ", "4df204cd63d4cc2d6e5ee1a31c134357": "\\Omega^k(\\mathbf{R}^3)", "4df2e22b662b59479bd483de1a0e4ed2": "x \\oplus y = \\min(x+y,1)", "4df2f4f55c768435ae86bbb695054c55": "X\\subseteq \\{0,1\\}^n", "4df31b269300cb089fcc111987c6f11b": "\\rho_{i,j}", "4df3928b70a9fd7517365edb2a71a7c8": "\\lambda=.5 \\sigma^2", "4df40f459eb96f28d70fa8aac3fd0d61": "1 + \\frac{1}{3} + \\frac{1}{3 \\cdot 4} - \\frac{1}{3 \\cdot4 \\cdot 34} = \\frac{577}{408} = 1.414\\overline{2156862745098039}.", "4df485fad7ee305cd05abdf896db151a": "P(k) = A exp(-\\pi^2k^2\\omega_0^2/2)", "4df48ff7e4e9eda2eea259268661ac28": "\\frac{A_1,\\dots,A_n}{\\Box B_1\\lor\\dots\\lor\\Box B_m}.", "4df4a037e7c8664a1620b8dc4e2cea41": "(\\mathfrak{g}, K)", "4df4c3b3bb40d7b3634440fec3e515eb": "(C\\to F)\\to F\\vdash((B\\to C)\\to F)\\to F", "4df4f0830b79beee78ca3c03e8f58b07": "\\operatorname{coni} (S)=\\Bigl\\{\\sum_{i=1}^k \\alpha_i x_i \\;\\Big|\\; x_i\\in S, \\, \\alpha_i\\in \\mathbb{R}, \\, \\alpha_i\\geq 0, i, k=1, 2, \\dots\\Bigr\\}.", "4df5936d992226021c034b241f8052d3": "p_{i}(t)\\ge 0", "4df5939b4e2160ddd8718a4757d599d1": "x_{1,i}", "4df5d11914982ac780c3833195de4a30": "n\\mapsto2n", "4df60a11f991cf324f7aeb51214eddeb": "\n\\partial \\rho / \\partial t + \\nabla \\cdot ( \\rho v) =0 \\; ,\n", "4df6306b5dedf35eefb6a8435494898a": "=\\Psi^\\prime \\sqrt{\\frac{i}{z\\lambda}} e^\\frac{-ikx^2}{2z} \\int_{-\\frac{a}{2}}^{\\frac{a}{2}}e^\\frac{ikxx^\\prime}{z} e^\\frac{-ikx^{\\prime 2}}{2z} \\,dx^\\prime", "4df64e0e2ec60e45ff84c3afd9047620": "k_E \\| k_M = \\textrm{KDF}(S\\|S_1)", "4df668fc2e7bd4d7dfad3218fdb30005": "\\sum_{n=0}^{\\infty} a_n", "4df68cb4f6758300609a81bede0d6cd5": " \\frac{u_{j}^{n+1} - u_{j}^{n}}{k} =\\frac{u_{j+1}^{n+1} - 2u_{j}^{n+1} + u_{j-1}^{n+1}}{h^2}. \\, ", "4df6cab93e6b10a2ddd5b29adc55edef": " A = P^TQ \\, ", "4df75125ef83b36b4e885136ed421352": "m_j = 1", "4df7531d34f623f00fdc340059342671": "\\chi_{1-\\alpha,h}^2", "4df79ee7b5c0e988d8922df602fceb0e": "\\,", "4df7a5dc8b677d115738b03d6867e693": "E = \\tfrac{1}{2}mv^2 + \\tfrac{1}{2}mv^2 = mv^2", "4df7be3d128e975a6e7a65d6d11856c6": "\\pm \\frac{R_1}{R_1 + R_2} V_{\\text{sat}}", "4df8f7a4254b27126fa9b50753ee4a7b": "\\frac{k-\\mathrm{\\#\\,samples\\ chosen}}{n-\\mathrm{\\#\\,samples\\ visited}}", "4df8fb0f4b41ba7cd0c8634f08d58578": "R_{in} = g_m \\cdot Z_1 \\cdot Z_2", "4df90f7fd012a1e9728332ea1dcec368": "g_{ik}\\ ", "4df93216045754ad4179e117dc3e7c9a": "\\,k", "4df987bc196c84e4c3eb2c18e9e505fb": "\\epsilon^p_{\\mathrm{xy}}", "4df987c98b75937faf36fcb15a30525d": "u_p e_p = Q_{pq} v_q e_p", "4df9e8ad8b1523bf18c904ed99f1c0b3": " h(u) = (1+u) \\log (1+u)-u ", "4df9fc5c937aaea1d864e4712007b819": "(h_1\\ ,\\ h_2\\ ,\\ h_3)", "4dfa61bdd2e461c4278b18a5c77be582": " \\bar {V}_{flow} = \\left [ \\frac{1}{T_{ab}} - \\frac{1}{T_{ba}} \\right ]\\left [ \\frac{Dist_{Sound path}}{2\\cos \\phi} \\right ]", "4dfa6662ed8027a5bfb523ea6c3a6d0e": " pH = pK_{a~H_2CO_3}+ \\log \\left ( \\frac{[HCO_3^-]}{[H_2CO_3]} \\right )", "4dfa786a392681a6c1002cbe236971bb": "a^{\\frac{p-1}{2}} \\equiv 1 \\pmod{p},", "4dfab581f7059b95943e2bd9577b4807": "\\lambda_i(\\vec{x},t) \\geq \\lambda_{i+1}(\\vec{x},t)", "4dfac071989662ff623d763080423acf": "\n\\begin{align}\n\\nabla^2 \\Phi =\n\\frac{\\left( \\cosh \\tau - \\cos\\sigma \\right)^{3}}{a^{2}\\sinh \\tau} \n& \\left[ \n\\sinh \\tau \n\\frac{\\partial}{\\partial \\sigma}\n\\left( \\frac{1}{\\cosh \\tau - \\cos\\sigma}\n\\frac{\\partial \\Phi}{\\partial \\sigma}\n\\right) \\right. \\\\[8pt]\n& {} \\quad + \n\\left. \\frac{\\partial}{\\partial \\tau}\n\\left( \\frac{\\sinh \\tau}{\\cosh \\tau - \\cos\\sigma}\n\\frac{\\partial \\Phi}{\\partial \\tau}\n\\right) + \n\\frac{1}{\\sinh \\tau \\left( \\cosh \\tau - \\cos\\sigma \\right)}\n\\frac{\\partial^2 \\Phi}{\\partial \\phi^2}\n\\right]\n\\end{align}\n", "4dfaeaa72f8c0398d137b7d6611eddd4": "D/c", "4dfb29ce0837b75515f9026e21924966": "\\scriptstyle s(t \\,+\\, \\alpha)", "4dfbae6b85617c003aaee3847f796bdd": "\\scriptstyle{E_\\theta}", "4dfbb4763a85e3a1464b647c77da9ab7": "\\sum_{K}\\tilde{u}_k(K)=\\sum_{K}\\frac{\\frac{2m}{\\hbar^2}\\frac{A}{a}}{\\frac{2mE_k}{\\hbar^2}-(k+K)^2}\\,f(k)", "4dfc301e35b3dbb991f9e44bdd8f2d52": " \\Sigma(k,i\\omega_n) \\approx \\Sigma_{imp}(i\\omega_n) ", "4dfc3fd4f57010f1ff689e7e07b02f90": "f(z)=z^l \\frac{g(z)}{h(z)},", "4dfc5bf0dcffd5d3e5db501ecb32d8da": "{z \\to \\infin},", "4dfcfd4266a5046e00972b7ca41cb87d": "\\forall i \\in \\mathbb{N}\\colon f^i(\\bot) \\sqsubseteq k", "4dfd43419ad6ecf4e8e644f230b04c8e": "{\\mathbf\\Psi}", "4dfd5502190a7f5a59f99a0a2284fc21": "\n\\begin{alignat}{2}\n Q_s & = 124 + 1.5 \\cdot P \\\\\n Q_d & = 189 - 2.25 \\cdot P \\\\\n\\\\\n Q_s & = Q_d \\\\\n\\\\\n 124 + 1.5 \\cdot P & = 189 - 2.25 \\cdot P \\\\\n (1.5 + 2.25) \\cdot P & = (189 - 124) \\\\\n P & = \\frac{189 - 124}{1.5 + 2.25} \\\\\n P & = \\frac{65}{3.75} \\\\\n P & = 17.33 \\\\\n\\end{alignat}\n", "4dfdb816dcfbd31bc8029c879f607b53": "(x(s),t(s))\\,", "4dfddd9b7d08880f2a8397bdc080153c": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "4dfe63ad7dfcf782990cf7aa0f1137de": "C_{\\mathrm x} = \\frac{\\int x S_{\\mathrm y}(x) \\; dx}{A}", "4dfe82f82030ed9c692faa62ce72f618": "\\sqrt{8} \\rho^3 \\cos 3 \\theta", "4dff3d8452694d8ec9686dfd9ef8a9f1": "\\max_{x(\\theta),t(\\theta)} \\mathbb{E}_\\theta \\left[ t(\\theta) - c\\left(x(\\theta)\\right) \\right]", "4dff486a237c128a6c5d0cfb3ac0a861": "\\bar{a}_i", "4dffefe19f7e8bb5a086b266a5a89774": " \\lfloor \\log_2(n) \\rfloor = \\lceil \\log_2(n + 1) \\rceil - 1, \\text{ if }n \\ge 1.", "4dfffafd75ed0639289b09a7da318a98": "(t-x)^{k-1}_+", "4e0021d9ee9d5f322d7b1f88b99ffd4c": "P_{L2}=V_{L2}I_{L2}=V_P I_P\\sin\\left(\\theta-\\frac{2}{3}\\pi\\right)\\sin\\left(\\theta-\\frac{2}{3}\\pi-\\varphi\\right)", "4e012e614935fb19286e7d984f92eb20": "V=m_{liquid}/\\rho_{liquid}", "4e013833c8fa7987f646fce89394b495": "\\frac{d(uvw)}{dx} = \\frac{du}{dx}vw + u\\frac{dv}{dx}w + uv\\frac{dw}{dx}\\,\\! ", "4e01717d2af0309999e8772745035791": "i = 1, \\ldots, m", "4e0184a42fe70cd35cd41039852ac8c0": " \\sum_{i=1}^n g_i E(g_i^{-1} \\sum_{g \\in G} n_g g) = \\sum_i \\sum_{h \\in H} n_{g_ih} g_ih = \\sum_{g \\in G} n_g g ", "4e018e624ea4ee7e3ceb74ce90678c61": "\\frac{\\sin \\theta}{\\theta} > \\cos \\theta\\,", "4e01b8a4f3813c2de4b1ca89ea325a82": "s(\\theta) = \\frac{1}{2} a_0 + \\sum_{k=1}^{m-1} \\mathrm{sinc}\\Bigl(\\frac{k}{m}\\Bigr)\\cdot \\left[a_{k} \\cos \\Bigl( \\frac{2 \\pi k}{T} \\theta \\Bigr) +b_k\\sin\\Bigl( \\frac{2 \\pi k}{T} \\theta \\Bigr) \\right] ,", "4e01fb789245988ff18d767dc751b8f2": "p a_r = \\frac{n^2a^2b^2}{r^2}\\left(\\frac p r - 1 - \\frac p r\\right)= -\\frac{n^2a^2}{r^2}b^2. ", "4e024b4ed96cd1948c039a2ae2bb9e9f": "pf_j = A/N = 0.237\\!", "4e027745302afe3609c7e7380530c0ef": "\\alpha_m(t_1,t_2,\\cdots,t_m) = \\left(\\frac{1}{u},t_1 - \\frac{2t_1}{u}, \\frac{t_1t_2}{u}, t_2, \\frac{t_1t_3}{u}, t_3, \\cdots, \\frac{t_1t_m}{u}, t_m \\right),", "4e028678f700a644d68896b8303180c9": " K_z = K_3 = i\\left.\\frac{\\partial \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_z)}{\\partial \\varphi}\\right|_{\\varphi=0} = i \\begin{pmatrix}\n0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n\\end{pmatrix} \\,.", "4e02fd9f901bfc9980fb39525c7204fe": "(\\overline{gate4}\\vee gate3)\\wedge (\\overline{gate3}\\vee gate4\\vee \\overline{x1})\\wedge (gate5\\vee x2)\\wedge ", "4e0307b46885b95980a1118a855caae0": "Tv_{parcel}", "4e0339b4e5cbbe6485ccc2a711f75546": "\\frac{1}{4} \\left [ (z+1)^2 - (z-1)^2 \\right ] = z.\\,", "4e03b7efdc6b8433176d56373187ed7f": "\\nabla^2 \\varphi + \\frac{\\partial}{\\partial t} \\left ( \\mathbf \\nabla \\cdot \\mathbf A \\right ) = - \\frac{\\rho}{\\varepsilon_0}", "4e0472c516f8074fde1c9d09d172f8a4": " T = \\overline{T} + T'", "4e047d2aa6d372df3232cc649fce3993": "\\tan(z) = \\cfrac{z}{1 - \\cfrac{z^2}{3 - \\cfrac{z^2}{5 - \\cfrac{z^2}{7 - {}\\ddots}}}}.", "4e0525a6f7c33bd1bafec27a93cb1038": "\\lambda_j\\ (j = 1,\\ldots,l)", "4e052f9379b18e57596e6dba4de18c16": "e^{i \\pi} +1 = 0 \\,.", "4e054fe59637f376a3b2418de4e5e7ea": "q = 2\\pi /a", "4e05721a75f799c6b4d65129da4768f0": "K[T]/(T-0) \\oplus K[T]/(T-0),", "4e057c5f964d8c987b732300a6f747f8": "ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\\,", "4e0586b8adaa1d1869aa4ddb170a5155": "1.2 \\text{ rad} = 1.2 \\cdot \\frac {200^\\text{g}} {\\pi} \\approx 76.3944^\\text{g}", "4e0594e2250cd739c28bab8c417f6afe": "\\overline{z^n} = \\overline{z}^n ", "4e059d7b4100b8eaf78c2362c176e0f3": "\\mathsf{BPP} \\subseteq \\mathsf{MA} \\subseteq \\mathsf{S}_2^P \\subseteq \\Sigma_2 \\cap \\Pi_2", "4e05e522a4e0e14c7324262acb17d5f4": "I(x)", "4e060d8c9c4cde586650df3c37fa6ca9": "\n\\frac{ma^{2}}{2} \\left( \\cosh^{2} \\xi - \\cos^{2} \\eta \\right)^{2} \\dot{\\xi}^{2} = E \\cosh^{2} \\xi + \\left( \\frac{\\mu_{1} + \\mu_{2}}{a} \\right) \\cosh \\xi - \\gamma\n", "4e061f2253d916b1f969c0657402c8dd": " (\\lambda \\lambda 1) (\\lambda \\lambda \\lambda \\lambda 1) (\\lambda \\lambda 2) (\\lambda \\lambda 1)) (\\lambda \\lambda \\lambda 1 (\\lambda 1 6 4) 2))) (4 (\\lambda 1) (\\lambda 1) (\\lambda 1) 1)) (\\lambda 1)) (\\lambda \\lambda (\\lambda 1 1) (\\lambda \\lambda \\lambda \\lambda 1 (\\lambda (\\lambda \\lambda \\lambda \\lambda 3 (\\lambda 6 (3 (\\lambda 2 (3 ", "4e064e5509c7c75ecc35523e643debd8": "(\\Box + \\mu^2) \\psi = 0,", "4e065d4025476314230ea5fcf36bdf90": "\n\\to\n\\begin{pmatrix}\n2 & 0 & 0 \\\\\n0 & 18 & 24 \\\\\n0 & -6 & -12\n\\end{pmatrix}\n\\to\n\\begin{pmatrix}\n2 & 0 & 0 \\\\\n0 & 6 & 12 \\\\\n0 & 18 & 24\n\\end{pmatrix}\n", "4e07022b366493b87f46d407216aa363": "c_{i,j} =\\operatorname{Lcoef}(e_j)\\cdot\n \\frac{\\partial e_i}{\\partial x_{i_k}}-\\operatorname{Lcoef}(e_i)\\cdot\n \\frac{\\partial^{p_1+\\cdots+p_l}e_j}\n {\\partial x_{j_1}^{p_1}\\cdots \\partial x_{j_l}^{p_l}}", "4e07088dd7d0694b13278c60e8afde6e": "\\gamma n", "4e073036670cdad8f6d376a3c4947409": "{1-r\\over 1+r} \\le \\left|z{f^\\prime(z)\\over f(z)}\\right| \\le {1+r\\over 1-r}", "4e0732e21cb10e9893405a111c5b124a": "\\pi_0(\\operatorname{Bun}_G)", "4e07bd64dd078d323c1876cf33ce50e1": "f(t-kT)", "4e07cbbfe698797ce2d87025ca1ca4ec": "HbO_2", "4e080451371b7580c98f9eafd8b2c07d": "\\bar c = \\frac {\\sum_{i=1}^m \\sum_{j=1}^n \\mbox{no. of defects for } x_{ij}}{m}", "4e084509806aa898ebcaa7dab173f4df": "\\mathbf{A} \\mathbf{A}^T, \\mathbf{B} \\mathbf{B}^T ", "4e0853fab3bf8a29713ddd23f796a7c2": "\\mathrm{SO}(n) \\subset \\mathrm{GL}(n,\\mathbb{R})", "4e086071ddb24c86099431ef6e0510d8": "0 = \\lambda_0\\le\\lambda_1\\le\\cdots\\lambda_{n-1}", "4e089fafc613665058315b227b1a367f": "\nP^{\\eta}[\\eta_t(x)\\neq\\eta_t(y)]=P[\\eta(X_t)\\neq\\eta(Y_t)]\n", "4e08c776b6e778b44ea81c63304f6966": "\\binom{n}{2}", "4e08c8e2bf4b6c87589141f12a5ee035": "n > Q(\\mathcal{N}) N", "4e08de33e4d7bbad3ba743d46c2263c1": " \\inf_{v\\in P_{m-1}}\\bigl\\Vert u^{\\left( k\\right) }-v^{\\left( k\\right) }\\bigr\\Vert_{L^{p}\\left( a,b\\right) }\\leq C\\left( m\\right) \\left( b-a\\right) ^{m-k}\\bigl\\Vert u^{\\left( m\\right) }\\bigr\\Vert_{L^{p}\\left( a,b\\right) }, ", "4e0a3c16a113bfc4fe58679f8d432adc": "R_\\mathrm{left}", "4e0ab9b3705a057d62813f5990516006": " u=Axy + Bx + Cy + D, \\, ", "4e0ad260f1b1a86ba9005104d85245ce": "-\\mathbf{J}^* \\cdot \\mathbf{E}", "4e0ae6da3fc62f1f58ff722ef52e2682": "p_1=\\frac{31}{61}", "4e0b1c39dcf165b133e556e0ec086cb8": "\\phi (L)\\, = \\phi_{L}", "4e0beae4c67e43948b5c70f2cac7e184": "a_{5}*b_{4} ", "4e0c1720b0f5cbeb6d383b4e58420211": "\\Omega^\\omega", "4e0c3ffe898c4fcf6b920c71da4fa3d2": "p= -Ap_o \\Bigg\\{(1-f)- a \\left( \\frac{\\partial f}{\\partial a} \\right) \\Bigg\\}", "4e0c7bf6c7cbcb161eccb96cb2c07b07": "\\frac {dm} {dt} = {k_d} C_b", "4e0c8e450e3d453e53dea8c92fe0d4a5": "c\\in A^*", "4e0c9e38b7a28cf2f169730a71b40404": "\\scriptstyle B(b,\\lambda)", "4e0cd1f46fa8b1b755d77f8cb9130296": " P(r)/r", "4e0cd23167ec37ac201c2376d13ebf00": "c,C>0", "4e0cf65cbacd4242326d28fce33e60fa": "\\mathit{3}\\, ", "4e0d121a3278796e876713bd69576390": " \\frac{e^{-jkr}}{r}", "4e0d6db2c3e13a1085eb0b9741f6d7df": "\\beta(g) = \\mu\\frac{\\partial g}{\\partial \\mu} = \\frac{\\partial g}{\\partial \\ln \\mu},", "4e0d7462cb0b85ff10c907e450f09897": "\\frac{{}_{(2)1}\\partial x^2}{\\partial x}=x\\,\\!", "4e0d82ff335e7c20157b4e71167ba8e7": "\\gamma_k \\subset \\Lambda", "4e0da0f239db0dfec850c2b9fdd21ef0": "(X,Y), (X_1,Y_1), \\dots, (X_n, Y_n) ", "4e0dccc6a7ac12e5003555b7c5fbc9cc": "\\pm(-1)^{-k/2}", "4e0e76aae794e3556ab5ecfb3132ad06": " 3 + \\frac{8}{60} + \\frac{29}{60^2} + \\frac{44}{60^3} = 3.14159\\ 259^+", "4e0e7d84c01be39abc6c87e9aed8387f": "E(m)=O(m/\\sqrt{b})", "4e0eea1bd7d6e46116f2b7fbbe8d4026": "=0", "4e0f218be73883a0925c12a1172e0020": "\\, [A \\ B]", "4e0f2a0d7138df461093b508583194c2": "\\color{Tan}\\text{Tan}", "4e0f6b6a7377b8225bdc2c22155e879d": "\\textstyle{\\varphi = \\frac{1+\\sqrt 5}{2}}", "4e0fcda697b86339721cd8c67e8fcbbd": "q_n=-\\frac{k_{ni}}{\\mu}\\left(\\partial_i P -\\rho g_i\\right)", "4e1042896738c58ac2bdfad9710f85c9": "\\Gamma_{\\lambda}", "4e104fc319ef031ee93e9387c66da7ff": "\\scriptstyle Y_k", "4e10835a372499af424136d175d8449e": "L(t)=\\log (t)", "4e1088736aca7b32ffcdf88a53f89b53": "H_{ph} =\\sum_k \\hbar\\omega_k \\left({b_k}^\\dagger b_k+{1\\over2}\\right)", "4e109ef9555e663e129a3e0fb0ad9c13": "E_r^{p,q}", "4e10b056c3f7ac229f239e0fdd7a3059": "\\displaystyle{{|f(z_1)-f(z_2)|\\over |f(z_1)-f(z_3)|} \\le a {|z_1-z_2|^b\\over |z_1-z_3|^b}.}", "4e10b1ae744e07c9a0949452322182f0": "x=\\epsilon' a \\sin( \\omega t/\\phi)", "4e11c396f79226bec92a10a202e5c00a": "\\operatorname{li}(x) = \\int_0^x\\frac{dt}{\\log(t)}.", "4e11d191754373deaf4638f5270f4bec": "X \\sim \\textrm{Cauchy}(x_0,\\gamma)\\,", "4e12ae97f716bf2af99408bff1752cdb": "\\Delta = S_{11}S_{22} - S_{12}S_{21}\\,", "4e12ca85bd9576293f2b8cd21a8fb906": "\\delta W=PdV", "4e13140430cb5bf88253ec0a10237efc": "v=1\\,\\mathrm{km}/\\mathrm{s}\\,\\!", "4e13c6d0dc983bd2f0dd3c099786ac27": "\\displaystyle n", "4e13c84d6f8d883d4bb5e932c45cf49a": "I_1 = 3 (\\lambda_i = \\lambda_j = 1)", "4e13fd15abbe5ccffdf3f7a4dd897297": " cx+d ", "4e1431a43d869a01eeb24afaf73bbc47": "\\displaystyle{a_1^{-1} - b_1 + b_3= (a_1^{-1} - b_1 +b_2) -b_2 + b_3=a_2^{-1} -b_2+b_3}", "4e143e84a7a5356e818cdffa5b52652c": "\\angle DEF", "4e14601634d7bdbe00ee303aec9cac38": "log I_b\\,", "4e146a20c28bddd94ca6ff4f5eea1f92": "Y = Bottom + \\frac{Top - Bottom}{1 + (\\frac{X}{EC_{50}})^{\\mathrm{-Hill coefficient}} }", "4e14a799c2c5bd8645b401f179ff151d": "\\varnothing^C = U", "4e14ca710483c876fe4c67223a54e78b": "E_{\\alpha} ", "4e14e830c2cdeff0fd5683f034ffdd56": "\\sum_{d\\,\\mid\\,m}f(d)", "4e1516658586f2a46253f8757700f6b0": "Y \\cup \\{A \\vee B\\}", "4e153395c2ddc4605179e487c63032aa": "D_{\\delta\\delta}(X, Y) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-y|\\delta(x-\\mu_x)\\delta(y-\\mu_y) \\, dx\\, dy = |\\mu_x-\\mu_y|", "4e155efba756597cf5fd9ddfe354be9a": "b^T y \\ ", "4e157d1232b31fa06e00123d1abcc73d": "\\|(D-\\tilde{\\lambda} I)^{-1}\\|_p=\\max_{\\|\\mathbf{x}\\|_p \\ne 0}\\frac{\\|(D-\\tilde{\\lambda} I)^{-1}\\mathbf{x}\\|_p}{\\|\\mathbf{x}\\|_p}", "4e158eb6b5720647b12cf6041501bbeb": " A^+ + B \\rarr A^* + B^+ ", "4e15998807d80ae6e7db425f5c9ee783": "\\textstyle a_B", "4e159c8f5c4e5341511eab065e2ee499": "{\\mathcal N}(\\vert \\alpha\\vert^2)", "4e15aa478ee9bb3e0cde9423bf281cd9": "\\color{blue}\\rightarrow \\color{blue}\\mathcal{E} \\color{blue}\\rightarrow \\color{blue}\\mathcal{I} \\color{blue}\\rightarrow \\color{blue}\\mathcal{R}", "4e15b1743b648ca0a605531ebfc3eb18": "Po(\\lambda)", "4e15b1c02c1d9b0d850e6132879cbcfa": "M_1 \\ne M_2 \\in \\mathbb{Z}^*_N", "4e15c1594e3a3611cadb51ed2572314a": "\\begin{align}\n\\ s_{ij} &= \\sigma_{ij} - \\frac{\\sigma_{kk}}{3}\\delta_{ij},\\,\\\\\n\\left[{\\begin{matrix}\ns_{11} & s_{12} & s_{13} \\\\\ns_{21} & s_{22} & s_{23} \\\\\ns_{31} & s_{32} & s_{33}\n\\end{matrix}}\\right]\n&=\\left[{\\begin{matrix}\n\\sigma_{11} & \\sigma_{12} & \\sigma_{13} \\\\\n\\sigma_{21} & \\sigma_{22} & \\sigma_{23} \\\\\n\\sigma_{31} & \\sigma_{32} & \\sigma_{33} \n\\end{matrix}}\\right]-\\left[{\\begin{matrix}\n\\pi & 0 & 0 \\\\\n0 & \\pi & 0 \\\\\n0 & 0 & \\pi\n\\end{matrix}}\\right] \\\\\n&=\\left[{\\begin{matrix}\n\\sigma_{11}-\\pi & \\sigma_{12} & \\sigma_{13} \\\\\n\\sigma_{21} & \\sigma_{22}-\\pi & \\sigma_{23} \\\\\n\\sigma_{31} & \\sigma_{32} & \\sigma_{33}-\\pi\n\\end{matrix}}\\right].\n\\end{align}", "4e15cafb27b617bbfd0095a4c601544d": "\\mathbf{j}_r", "4e1603eaa82edb73b058e9e318f7c567": "v_i={\\operatorname{d}[M\\cdot]/\\operatorname{d}t}=2k_df[I]", "4e161b9d807bb036e0ccbc55fa39fa9b": "\n\\theta_\\mu=\\mathrm{Arg}(m_1).\\,\n", "4e164057e6cc6fed37ee7d1efbed3277": "V = \\begin{bmatrix} a_1 & a_2 & \\dots & a_n \\\\ b_1 & b_2 & \\dots & b_n \\end{bmatrix} \\in\\R^{2 \\times n}", "4e16e6b5411d9e43e09a52880be556dd": "R \\ddot{R} + \\frac{3}{2} \\dot{R^2} = \\frac{1}{\\rho_l} \\left ( p_g - P_0 - P \\left ( t \\right ) - 4\\mu \\frac{\\dot{R}}{R} - \\frac{2\\gamma}{R} \\right )", "4e1750a2870058833d1412525e1acca5": " x(t_0)=x_0 ", "4e175562805e429da780b9be8bbe3e87": "a x^2 + b x = c", "4e177c2e7f7780f763dc69fa191679e7": "\\chi(n) = \\left(\\frac{n}{m}\\right),\\ ", "4e177f28b2dd382d7f35d23ce64058fd": "{A}_3", "4e17bd09bfa6d3469a8e1347d26b83bd": "\n \\mathbf{v}\\cdot\\mathbf{b}^i = v^k~\\mathbf{b}_k\\cdot\\mathbf{b}^i = v^k~\\delta^i_k = v^i ", "4e17cde503c99f4f80c9228fe611f621": "\n\\operatorname{E}\\left[ \\left. \\hat\\theta(X) - \\theta \\right| \\theta \\right]\n= \\int \\left[ \\hat\\theta(x) - \\theta \\right] \\cdot f(x ;\\theta) \\, dx = 0.\n", "4e17d83850e2e2118232bcd302074e40": "\nC_{2} = \\mathbf{D} \\cdot \\mathbf{L} = 0,\n", "4e17ef7dff3325f702712bd2c820b1d3": "R_1, \\ldots, R_n", "4e180e989d416a57da1453ed0e759f38": "\\ K_m=\\overline{\\xi'^2} \\left| \\frac{\\part \\overline{w}}{\\part z}\\right|", "4e183bb53b27b9e128c792df3590a410": " \\int \\phi(a+bx)^n \\, dx = \\frac{1}{b\\sqrt{n(2\\pi)^{n-1}}} \\Phi\\left(\\sqrt{n}(a+bx)\\right) + C ", "4e1860b5b35c91bfc20bf5325b3e1941": "\\top \\equiv (a|(a|a))", "4e18a55511d6649bc6fcad5faf4d56e8": "CTDI=\\frac{1}{nT}\\int_{-7T}^{7T}{D(z)dz}", "4e191f987f622cc085f469606e7c0f6f": "\\tfrac{9}{3} Pmf - \\tfrac{4}{3}Pmf = 1", "4e195a1292afc85e89c806bfbb15ef15": "\\mathbf J = \\mathbf L + \\mathbf S, \\, ", "4e19ffec2d27669ad6de33e144e16bb9": "h_{\\mathrm{e}}", "4e1a21157bd443fe4fb060b2978218f9": " C_B ", "4e1a5e30168d33b299860abddb22d47c": "\\varepsilon_x(f)=f(x)\\ (f\\in A)", "4e1a966aab58ca30b6134cf70ffe3cf6": "\\tilde E_6(q)", "4e1ab9454df8562a1c23c152741c7af1": "\\lambda_1+\\dots+\\lambda_k", "4e1ae34fc05518f80b54833326b15eb5": "17-12\\sqrt{2}=0.02943\\ldots", "4e1ae44fbd6ae0a3757a4fb057ddc3aa": "R : r", "4e1b0d381a8110fa6914db55792feaad": "F_m * F_n \\cong F_{m+n},", "4e1b1db51d1480629c12ad502a0acd61": "\\forall x,y\\in\\Delta: i(\\alpha(x\\otimes i(y))) = \\alpha(y\\otimes i(x))", "4e1b61b418306e3e1b3de41b44ac5c5c": " I_R = \\frac {\\frac{1}{j \\omega C}} {R + \\frac{1}{j \\omega C} }I_T ", "4e1b6ecbc3ebc9d32e96a15aa0cd0662": "\\theta_{i=1 \\dots N, j=1 \\dots V}", "4e1b8041ceb9eba5ffddc9a282a0ed74": "\\begin{pmatrix}\n\\mathbf{A}_{1} & 0 & \\cdots & 0 \\\\\n0 & \\mathbf{A}_{2} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\mathbf{A}_{n} \n\\end{pmatrix}^{-1} = \\begin{pmatrix} \\mathbf{A}_{1}^{-1} & 0 & \\cdots & 0 \\\\\n 0 & \\mathbf{A}_{2}^{-1} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\mathbf{A}_{n}^{-1} \n\\end{pmatrix}.\n", "4e1bb1c5499c3d617265b63a55b9b1f4": "\\rho_{S}(t) = \\sum_{l}\\hat{A_{l}}\\rho_{S}(0)\\hat{A}^{\\dagger}_{l}", "4e1bcee11386309d11085d84397569e0": "\\Delta z\\,\\, \\approx \\,\\,a\\,b\\,e^{b\\,\\mu } \\,\\Delta x", "4e1bd34a7a12f9fa12969413093b2a37": "n = \\frac {Z^2} {4 \\, E^2} \\!", "4e1bf1efc259c18a2314845940d69ee8": "\\left(\\frac{SU(N)_L\\times SU(N)_R}{SU(N)_\\text{diag}}\\right)", "4e1c33da0452f1917a859147d206c4a9": "[Q^\\dagger,F\\}=\\frac{db^\\dagger}{dt}", "4e1c35ed19014cd42037e43f29c033c9": "\\det(A) = \\sum_{\\pi \\in S_n} \\sigma(\\pi) \\prod_{i=1}^n A_{i, \\pi(i)}.", "4e1c51e3ca7b7fabacb142d47bbc0b86": "\\sqrt[3]{b}^2", "4e1c596925cf59a2a378b76489a2424d": "f''(x) = \\lim_{h \\to 0} \\frac{f'(x+h)-f'(x)}{h}", "4e1c92c3b8509bc1925fcf3550405fff": "\\phi (P)", "4e1c9c635c85b1c9cc93fc110ef0e102": "\n(\\operatorname{vl}_\\xi X)[f]:=\\frac{d}{dt}\\Big|_{t=0}f(x,\\xi+tX), \\qquad f\\in C^\\infty(TM).\n", "4e1cf8de909d3b4a0580e753fbaa297d": " c\\,", "4e1d1658b9277a75dd07fb64ab70dc70": "\\ell_1+\\ell_2+\\cdots + \\ell_n = \\ell ", "4e1d4871d48dc7e74c007b3351ba91f7": " |f(x) - f(y)| < \\epsilon \\;\\;\\;\\; \\forall y \\in G ", "4e1d8cddfa0d7c2d9f336c22517f104f": "(5)", "4e1dc12253da85734adbd6aceeb4c958": "\n K_{\\rm I}(-a; x,y) = K_{\\rm I}(a; -x,y) \\,,\\,\\, \n K_{\\rm II}(-a; x,y) = -K_{\\rm II}(a; -x,y) \\,.\n", "4e1ddab87f2472b52ff795fccd1a50c4": " \\operatorname{Var}(X) = \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n \\frac{1}{2}(x_i - x_j)^2. ", "4e1e011b511005bc39e5e26ca3de4d65": " h_3 (X_1, X_2, \\dots,X_n) = \\sum_{1 \\leq j \\leq k \\leq l \\leq n} X_j X_k X_l.", "4e1e2f1b4a42bf1e31bceee51a5c25ec": "x[n] = e^{i 2\\pi \\frac{1}{8} n},\\quad ", "4e1e62a57059ffed85854112eb7e4803": "\\bar F(x)", "4e1e845071171d8aaccfe24b09b709f7": "c \\in \\mathbb{N}", "4e1f14f7db7929ddcecc35b2ec7a530b": "{\\Delta}A=A-A_0\\,", "4e1f2a39d6e3781446ccd57585e6a676": "\\mathrm{ad}_{\\Omega}^0 A = A, \\qquad \\mathrm{ad}_{\\Omega}^{k+1} A = [ \\Omega, \\mathrm{ad}_{\\Omega}^k A ], ", "4e1f94d8a1991c0c863295e7c364eb16": "\\frac{1}{\\cos(\\phi)}\\left(\n\\left(\\sin(\\phi) \\frac{\\partial a^1}{\\partial y} + \\frac{\\partial a^3}{\\partial y} - \\frac{\\partial a^2}{\\partial z}\\right) \\mathbf e_1 + \n\\left(\\frac{\\partial a^1}{\\partial z} + \\sin(\\phi) \\left(\\frac{\\partial a^3}{\\partial z} - \\frac{\\partial a^1}{\\partial x}\\right) - \\frac{\\partial a^3}{\\partial x}\\right) \\mathbf e_2 + \n\\left(\\frac{\\partial a^2}{\\partial x} - \\frac{\\partial a^1}{\\partial y} - \\sin(\\phi) \\frac{\\partial a^3}{\\partial y}\\right) \\mathbf e_3\n\\right).", "4e1fb40f313447780706b2d1e3ecf5bd": "d\\mathbf{A}", "4e1ff2eb5aa114a122aa09d2012ae21a": "123456 \\times 789", "4e20088d56f6dee9f1c4dc2346b45a11": "M(256,256,3)\\approx(10\\uparrow)^{255}1.99\\times 10^{619}", "4e2030460201df50e9ddb2057c977f33": "k(C) = \\frac{tt}{\\left | C \\right \\vert}", "4e203e2becc30afbe9eb2b1efeaa48de": "\n\\sum_{n=1}^\\infty \\frac{2}{n(5n-3)} = \\frac{1}{15}{\\pi}{\\sqrt{25-10\\sqrt{5}}}+\\frac{2}{3}\\ln(5)+\\frac{{1}+\\sqrt{5}}{3}\\ln\\left(\\frac{1}{2}\\sqrt{10-2\\sqrt{5}}\\right)+\\frac{{1}-\\sqrt{5}}{3}\\ln\\left(\\frac{1}{2}\\sqrt{10+2\\sqrt{5}}\\right) \n", "4e204cc48b28eb345990d37ce5121613": "= \\int{(m \\dot{\\vec{x}}[t] \\cdot \\dot{\\delta \\vec{x}}[t] - m \\nabla \\zeta [\\vec{x} [t],t] \\cdot \\delta \\vec{x}[t]) \\, \\mathrm{d}t}.", "4e2074ba192cf10e4aa341b3e5446e42": "\\rho = 2k/n ", "4e210c3e0e65791a6b518a055a6d69c0": "f(x_\\star)=0", "4e21d0a75bbdd1aeca737be7b885f247": " D_l = \\left(\\frac {\\Gamma}{\\delta x}\\right)_l ;\\qquad D_r =\\left(\\frac {\\Gamma}{\\delta x}\\right)_r;", "4e21ec6d18d1b5a463cb8e95a43a627b": "\\scriptstyle\\in\\mathfrak{A}", "4e22019be6d73c70223797f07959b1ea": "D_1(0)", "4e229586b9fef1aa0b85fc8f64305df5": " \\mathsf{S}\\cdot \\mathsf{T} = (\\mathbf{S}, \\mathbf{V})\\cdot (\\mathbf{T}, \\mathbf{W}) = (\\mathbf{S}\\cdot\\mathbf{T},\\,\\, \\mathbf{S}\\cdot\\mathbf{W} +\\mathbf{V}\\cdot\\mathbf{T}), ", "4e22c8a6586f2aa59224720af14a4a94": " U_\\omega : a |\\omega \\rang + b |s \\rang \\mapsto (|\\omega \\rang \\, | s \\rang) \\begin{pmatrix}\n-1 & -2/\\sqrt{N} \\\\\n0 & 1 \\end{pmatrix}\\begin{pmatrix}a\\\\b\\end{pmatrix}.", "4e22e6b4a2f55dbf45743cb1a8c75a05": "1 \\gg \\varepsilon_{ii} \\gg \\varepsilon_{ii} \\cdot \\varepsilon_{jj} \\gg \\varepsilon_{11} \\cdot \\varepsilon_{22} \\cdot \\varepsilon_{33} \\,\\!", "4e22f4b4e520b77cae0d33854a537aa4": " M(0,H) \\simeq |H|^{1/ \\delta} \\operatorname{sign}(H)\\mbox{ for }H \\rightarrow 0 ", "4e22f609d9b61667c1347fc1549d4946": " M = m + 5 (1 + \\log_{10}{p})\\!\\,", "4e2313c211d2fe2c6ba6e0d2ba6d31a8": " \\mu_{A}^{\\ominus}~", "4e2343404472af723f5691178ca6c1bb": "ab^{RC} \\in \\mathcal{O}", "4e2360564988f4510189f96d45c306b9": "\\lim_{x \\to x_\\pm} \\in \\partial \\bar{\\Omega}", "4e23f505a5d063a82b21b3f26bd44a95": "\\sigma(x) = \\prod_{e_i=1} (x-L_i) ", "4e2457e16f0d18c8d0ae25a12006960e": "\\frac{1}{kk'}", "4e2502618686cea51d948c1c919e269c": "f \\in F", "4e2557b8a2ce622ebd040b6ffd84ac76": "\\mathbf{Y}=g(\\mathbf{y})", "4e255c8d48a70fda3fe1881b5fbf08d0": "\\left ( \\frac{I_0}{I} \\right ) = {k_q} * {\\tau_0} \\times [ Q ]", "4e267773f192dc1c59b3efbe4631cfc3": "x^{k+1}=2^{k+1}(x-1)\\,", "4e26e5c717e5b9e98af3b27b241bda93": "{625 \\pi \\over 4}", "4e2782063cb5940e3b898e49e549f82a": "\\Delta\\,T_f / \\Delta\\,T_m", "4e27be5d8ca3a36079c11515707709ae": " b \\leq x \\leq c,\\; 0 \\leq t \\leq d", "4e27d48f4dc9c7493ca2d7051886df57": " \\psi(q) = \\langle q|\\psi\\rangle ", "4e28132e9dd1dd1ac69bfe6c61324ab7": " \\mathrm{E}\\left(\\int_{\\mathcal{T}} \\left( X(t) - X_m(t)\\right)^2 dt\\right) = \\sum_{j>m} \\lambda_j \\rightarrow 0 \\text{ as } m \\rightarrow \\infty .", "4e282c96d7d06dea1905b355e1883cbf": " K'_{il} ", "4e289ffb31ad66b738d93b21a5ff1c5d": "\n \\mathbf{u}_{\\text{sol}} = \\mathbf{u} - \\nabla \\phi\n", "4e28b0d0627326e08a7b13c33f5ca4af": "m \\le n", "4e28ce0635befd90beac7c35c938d54a": "\\phi ( I ) = I", "4e28ceb867a17e569f125e01cbe4a6fa": " (X,T) \\sim \\mathrm{NormalGamma}(\\mu,\\lambda,\\alpha,\\beta) \\! .\n", "4e291b7a5e6cfe95ef06174b61e3c656": " q_2 = k(Aq_1 + B) \\,", "4e2924e5da59d1f265055b531b00daaa": " F_i F_i^* \\,", "4e295f1ee404cbf7d748e485e3d78068": "(B, \\beta)", "4e296f58beba17d1b127a05e34fbdd2a": "u^2-a_1u+\\frac{a_1^2}{4}=\\left(u-\\frac{a_1}{2}\\right)^2= \\tilde{u}^2", "4e2996a07badd92ef3153f6b82f45926": "P(q_1+q_2) = \\bigg(a - b(q_1+q_2)\\bigg)", "4e29a04e43a852b62a7e3d363c757f75": "\\omega^\\omega", "4e29f0acb0c065553b5cc020a90f3c4a": "\\geq 15", "4e29f17ef78f555745aa01499050a4e8": " \\mathrm{PGL}_2 ", "4e29fbf739a6adf863c8a44556a25afa": "\\sin (\\arccos x) = \\sqrt{1-x^2}", "4e29fe1c2857caac9c1e9e7dc32ba436": "\n\\vartheta(z; \\tau) = \\sum_{n=-\\infty}^\\infty \\exp (\\pi i n^2 \\tau + 2 \\pi i n z)\n= 1 + 2 \\sum_{n=1}^\\infty \\left(e^{\\pi i\\tau}\\right)^{n^2} \\cos(2\\pi n z) = \\sum_{n=-\\infty}^\\infty q^{n^2}\\eta^n\n", "4e2a066bb8404ee2495165249466f7c7": "E_G=0.392\\pm{0.065}", "4e2a07aba65d35a28a89f01a0a5e1623": "+\n \\left[\n \\left(\\frac{\\partial z}{\\partial x}\\right)_y\n \\left(\\frac{\\partial x}{\\partial v}\\right)_u\n +\n \\left(\\frac{\\partial z}{\\partial y}\\right)_x\n \\left(\\frac{\\partial y}{\\partial v}\\right)_u\n \\right]dv\n", "4e2a1762d029c12bc41eb65fb8408655": "\\bar{\\psi}(x)", "4e2a668f4cdfee608d8fef5a3e058d29": "A_1(h) = 2A_0\\!\\left(\\frac{h}{2}\\right) - A_0(h) ", "4e2a70403edef2fec8ef77db15e9d328": "\\mathbb{RCFM}_I(R)\\,", "4e2a948c4cdf154b8c7b00d607616548": "a,b,m,n\\in\\mathbb{R}", "4e2aa2fbd38ab4d565a0ef54ee16c702": "\\overline{(\\Delta x)^2}=2Dt=t\\frac{32}{81}\\frac{mu^2}{\\pi\\mu a}=t\\frac{64}{27}\\frac{\\frac{1}{2}mu^2}{3\\pi\\mu a},", "4e2ac102a77350e3a2d9f745153fc9a2": "1.7 kN\\ x\\ \\tfrac {s}{kg}", "4e2ad3b0cfb85712ddfd1709303c57e7": " \\begin{bmatrix} x' \\\\ y' \\end{bmatrix} ", "4e2b05f510f85c0dcad913a4f7a000ee": "\\textrm{mes} E\\cap (J\\setminus E_\\lambda)\\geq \\textrm{mes}E-\\textrm{mes}E_\\lambda \\geq \n\\frac{\\textrm{mes} E}{2}", "4e2b652e19b629c70395234657b63945": "u = \\frac{1}{r}", "4e2b80c71b9a6254b4f1541278a024c8": "\\nleftrightarrow", "4e2bd4c23a27885fe14483cceec8af10": "E=F(\\alpha).", "4e2bd56827cfdc3b4e1d88c8bafbd6b8": "\\dim_F \\operatorname{Der}_k(F, F) \\ge \\operatorname{tr.deg}_k F", "4e2bdd5f671833729dda83889a2d179d": "b \\equiv \\omega^2 LC \\left( \\frac{R}{\\omega L} + \\frac{G}{\\omega C} \\right). ", "4e2c144092d7e8979b4d7697b26d3cef": " m=0 ", "4e2c50ccb0dba182225dc2788ec76474": " n,m \\ge 0", "4e2cc10bf534ac35f0a4b3b96772e425": "\n\\begin{cases}\nu_t(x,t) - \\Delta u(x,t) = 0 &(x,t)\\in \\mathbf{R}^n\\times (0,\\infty)\\\\\nu(x,0) = g(x) & x\\in \\mathbf{R}^n\n\\end{cases}\n", "4e2cc952d7238ae4dce0559bf99345b6": "\\mathbf{K}_{ij} = k(\\mathbf{x}_i,\\mathbf{x}_j)", "4e2d4cbdb578d92a28042268a9e18aa5": "M^k", "4e2d726e1b72b8b64ca4b859f119dca3": "\\,s=s_1 + jm\\omega_s", "4e2d95776f9b1fa71350ed07957e875a": "\\forall p, q", "4e2dc3708d2653357b21ec1b3a60a493": "d\\mathbf{r}\\,", "4e2dcb7c1f7da07eba217ae1239eb619": "k\\;{}_0F_1(;a;z)+l z^{1-a}\\;{}_0F_1(;2-a;z)", "4e2defc1faa219e88de2fc6fef437ecb": "r = \\frac{1+i}{1+p}-1\\,\\!", "4e2dff3a372f76578f9308d2219f527d": "{{\\frac{q}{A}}_{min}}=C{{h}_{fg}}{{\\rho }_{v}}{{\\left[ \\frac{\\sigma g\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)}{{{\\left( {{\\rho }_{L}}+{{\\rho }_{v}} \\right)}^{2}}} \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{4}\\;}}", "4e2e0a7a1039c4058924b738f1d2ad40": "K_2 F", "4e2e1e72748155b67969d5798c5615d1": "\\mathcal{L}^*g = \\mathcal{L}(dg).", "4e2e24efbd20cb48c883e531e5c81240": "(T-I)v = u", "4e2e449e2ab0a81edd811746ab452f81": "\\varepsilon_i \\perp \\mathbf{x}_i", "4e2e608d510a771326314849cfb14675": "\\scriptstyle C\\, \\rightarrowtail\\, B\\,\\cup_A\\, C", "4e2f14d3e49742adfebb4a3d4304be0a": "\\scriptstyle E_{\\mathrm{Pl}}\\sim1.22\\times10^{19}", "4e2f3778b5b6d7929ff296bd5ccb9b4b": "y = k \\log_a (x).\\,", "4e2f39cd4745a921c12f54efdd308154": " \nEI + S \\, \\overset{k_{-3}}\\underset{k_3} \\rightleftharpoons \\, E + S + I \\, \\overset{k_1}\\underset{k_{-1}} \\rightleftharpoons \\, ES + I \\, \\overset{k_2} {\\longrightarrow} \\, E + P + I\n", "4e2f43ce5bd24ae3d1331ab83b2b01d3": "{0} = -{\\frac{DK}{Q^2}}+{\\frac{h}{2}}", "4e2f4ceb0880f05597be7fb3ac70e892": "f'(a)>y>f'(b)", "4e2f74a3e1155e942f6927e54ac31fc2": " \\kappa = 3-4\\nu", "4e2fc1fbb302ec691a0ff76c6827e6dd": "c_0^\\beta = l^1", "4e2fc540e25d0940a31ca5d30ba18869": "\\gamma=1/{2 \\sigma^2}", "4e301f86e95199f8f97fbb4349a9743d": "\n F=\\tfrac{4}{3} E^*R^{1/2}d^{3/2}\n ", "4e306edf5b41786aaa2b1d6cc6a8f8f0": " A \\cdot B = gA \\cdot gB ", "4e309849a67665a87adc1390d867ce57": "\\mathcal{E}^1", "4e30dcbb455f5306b1d87f8f87b1d213": "|a|<1,", "4e30e3af48dcb8fcba340de12c082bbf": "\\Theta^i = \\Gamma^i_{kj} \\theta^k\\wedge\\theta^j", "4e30e3bc31e1a2e798026074d3d5fa5f": "R_{ik\\ell m}+R_{imk\\ell}+R_{i\\ell mk}=0.\\ ", "4e31ab9ca4d89ff4bccc7df27efecdff": "\\zeta(z) = (\\det(I-zA))^{-1} \\ . ", "4e31c455de83ff2a4d75b9b3474d2f71": "j=0,\\ldots,n-1", "4e31fdb639a5dc839c1b5bcc99e8c12a": "\\varphi : \\mathbb{Z} \\to \\mathrm{Aut}(\\mathbb{Z})", "4e324d294449407093c67f56a8010299": "n \\bar p = \\frac {\\sum_{i=1}^m \\sum_{j=1}^n \\begin{cases} 1 & \\mbox{if }x_{ij}\\mbox{ defective} \\\\ 0 & \\mbox{otherwise} \\end{cases}}{m}", "4e3264f83f4a549276a2ba01f7512142": "X = \\partial / \\partial y_1", "4e33442ea5931110149538b0eb0317f7": "c_2 = b_2 - \\frac{n+2}{a_{1}n} +\\frac{a_2}{a_{1}^{2}}", "4e33c461dcd4605c37621eeed2cb62f9": " \\mathcal{O}(t) = e^{itH} \\mathcal{O} e^{-itH}.", "4e3416dfd623e6b949ad5a40271f0829": "u=F(y,x)", "4e342c4d17ba4ab91850207c623fbbb6": "\\theta_{A} \\,= \\, \\frac{N}{N_{S}} \\, = \\, \\frac{x}{1+x}", "4e343b2e11b8d18ef007612e6b44e1f7": " {}\\, p \\, d\\mathbf{S}", "4e344cb7beb629be3b52b734525feb16": "\\hat\\beta_i", "4e346a69721d322d1cd57e558870dc15": "\\Delta x = x_\\text{final} - x_\\text{initial}. \\,", "4e34e02ade53bdcedfa0fd9996baed75": "\\boldsymbol{\\mathsf{U}} =\\frac{\\mathrm{d}\\boldsymbol{\\mathsf{X}} }{\\mathrm{d} \\tau} = \\gamma \\left( c, \\mathbf{u} \\right) ", "4e34e341974768d915d2a55fa77ca8e0": "\n\\begin{array}{rcllll}\n2s & = & &(1-2+3-4+\\cdots) & + & (1-2+3-4+\\cdots) \\\\\n & = & 1 + &(-2+3-4+\\cdots) & {} + 1 - 2 & + (3-4+5\\cdots) \\\\\n & = & 0 + &(-2+3)+(3-4)+ (-4+5)+\\cdots \\\\\n2s & = & &1-1+1-1\\cdots\n\\end{array}\n", "4e3510c123315f50bfd522c304220794": "AD=EB=\\frac{1}{2}\\pi x^2", "4e351cfe3b413e258f0c4e17882222d8": " (L_1 \\cap L_2) \\subset L_3. ", "4e356a9a800ed10077d66222e64b2258": " d\\nu_t = \\nu_t(\\omega - \\theta\\nu_t)dt + \\xi \\nu_t^\\frac{3}{2}\\,dB_t \\,", "4e35713adad2a0505e143d431eca44cb": "f_{k_1}=0,", "4e35a6ada5a2c0e8cd72f45a786054b4": " \\nabla \\cdot \\mathbf{v} = \\cfrac{1}{\\prod_j h_j} \\frac{\\partial }{\\partial q^i}(v^i\\prod_{j\\ne i} h_j) ", "4e35d299c7dd2052a3f9e4cd83553b53": "\\displaystyle{M=\\sup_g \\|T_g\\| < \\infty.}", "4e365ffce95f073919200ed54a196424": "f(d_p)", "4e3660796e1ff3493b97d223bfe789a0": "\\phi'(x)=0", "4e366d45d6b879bd73354ec19f264403": "R(z;A)= \\int_0^\\infty e^{-zt}U(t) dt.", "4e367051dc59e78a5d89e3b6436601cb": " [a_j,a_k^*] = \\delta_{j,k}. ", "4e36965942e6b6806bd1c231d3c51225": "y\\in\\Gamma^*", "4e369dc5f0a9d791480f0da8f0fd923d": "C_{P} - C_{V}= V T\\frac{\\beta^{2}}{\\alpha_{T}}\\,", "4e36e5d3f9b4ea0828d05c2a9893c6db": "-\\omega^2/c^2", "4e36ee32b37336d416002f42248fdd32": "x^*_s", "4e370a41a1efe0aadbafc1301ce739b6": "E[A-E[A]]=0", "4e378581405c97d42c9d61a749115db4": "m' = 8 * (53 + 1468 * 12)", "4e3798a8e4b431af46d481a689f650ed": "f(a+t) = \\lim_{h\\to 0^+} e^{-t/h}\\sum_{j=0}^\\infty f(a+jh) \\frac{(t/h)^j}{j!}.", "4e3798f78597c39b91881202a3addc38": "\\Lambda_m =\\Lambda_m^0-(A+B\\Lambda_m^0 )\\sqrt{c} ", "4e380af2feb9b3944a818ca45ea647fb": "\\mathfrak{p}''_i", "4e3816b630e789a0c91d814815263132": "n=\\sqrt{\\epsilon_r}=\\sqrt{2-\\left ( \\frac{r}{R}\\right ) ^2}", "4e3856ae6bfa908c42867af2ca51d8c8": "m_{12}= a|\\phi_1-\\phi_2|.", "4e3871ed52b5cdb1c75f85329fb472c5": "m-1", "4e38c438642b82f5f94c7f9fe95e231d": " \\exp : g \\rightarrow G. ", "4e38d1a776477d03479bdc5b13413266": "\\scriptstyle{\\zeta}", "4e38d9b2c25be5b60fbe91e7f310d95b": "\\theta \\in (0,1)", "4e38efa57243617fa460b6ddabbe94b6": "\\zeta = \\frac{x + i y}{1 - z} = \\cot(\\tfrac{1}{2} \\phi) \\; e^{i \\theta}.", "4e394eabc5bc43adeb61161697b4c3fd": " \n\\mathbf{u} \\cdot \\mathbf{u} +\n2 \\mathbf{u} \\cdot \\mathbf{w} +\n\\mathbf{w} \\cdot \\mathbf{w} \n= \\mathbf{u} \\cdot \\mathbf{u} +\n\\mathbf{u} \\mathbf{w} + \\mathbf{w} \\mathbf{u} +\n\\mathbf{w} \\cdot \\mathbf{w}, \n ", "4e396ce634ab41951b4abf1f41985b1d": "R_1,", "4e39f994ab881893221f44959724a888": "z = 7.1 ", "4e3a5c3501b03af593ad4f0aacf170c7": "\\frac{60}{31}", "4e3a5cdfe85e44d606cede6c6177cbf6": "b_0 = 4a_0a_2 - a_1^2 -a_0a_3^2. \\,", "4e3a67b1984175d554b7df8108dc07bb": "\\frac{(\\rho+1)^2\\;\\sqrt{\\rho-2}}{(\\rho-3)\\;\\rho}\\,", "4e3ac58cea315d68e3f1a007f3d4c601": "G^{\\prime} = (\\{S, A, X\\}, \\{a\\}, S, \\{f, g, h, k, l\\}, (f|g|h|k|l)^{*}, \\{f, g, h, k, l\\})", "4e3acca45734fc215d0c00c26567e426": "\\forall x_1,\\ldots,x_n L_1 \\vee \\cdots \\vee L_m", "4e3ad36eb645b8de7d41bb03e4a7efc6": "|P|=\\frac{n}{K+1}", "4e3adad0e8d5315381598755565d0ba5": "\\textstyle |B_i|", "4e3adb4d41717b9da418ae909feffbd0": " h(M)=\\inf_E \\frac{S(E)}{\\min(V(A), V(B))}, ", "4e3af30abb03a631f2bd1a30fbba0d66": "x(x+2)=(0)x", "4e3b01f1486b2703ba79fc4efe9bb42c": "\n\\operatorname{CNOT}\\ |0,\\psi\\rangle = |0,\\psi\\rangle\n", "4e3b1630688114c5e379210c6d9725e4": "f^{m-n}\\ x = (f^{-1})^n (f^{m} x)", "4e3b6155055cffe91468251d936535a3": "\\sigma_{ij} = \\epsilon_0 E_i E_j + \\frac{1}\n{{\\mu _0 }}B_i B_j - \\frac{1}{2} \\left( \\epsilon_0 E^2 + \\frac{1}{\\mu _0}B^2 \\right)\\delta _{ij}. ", "4e3ba775920cedd9a4ddd1400931f36d": "x^p", "4e3bc7fbcc3074b44c0bda42b74f2432": "F: C_*(X \\times Y) \\rightarrow C_*(X) \\otimes C_*(Y), \\quad G: C_*(X) \\otimes C_*(Y) \\rightarrow C_*(X \\times Y)", "4e3c4bb200defd8b9d7f26cbae24bbd0": "\\Delta^n_h (fg, x) = \\sum\\limits_{k=0}^n \\binom{n}{k} \\Delta^k_h (f, x) \\Delta^{n-k}_h(g, x+kh).", "4e3c770c65db561062fd9f9dcda83034": "G_0=(V_0,E_0): V_0\\subseteq V, E_0=E\\cap(V_0\\times V_0)", "4e3c7c57088eba7d0278197887d1d415": "<^\\mathrm{d}", "4e3cc4ad9adf094d7af987af08cf44f1": " f(\\lambda^{w_1} x_1, \\ldots, \\lambda^{w_r} x_r)=\\lambda^w f(x_1,\\ldots, x_r)", "4e3cd8f256da8bec077450216e8a0192": " \n{2 a^{2n+2}\\over n!}\n\\int_0^{\\infty} { dr }\\;r^{2n+1}\\exp\\left( -a^2 r^2\\right) J_{0} \\left( kr \\right)\n=\nM\\left( n+1, 1, -{k^2 \\over 4a^2}\\right)\n . ", "4e3ce9ab6247a23bb786b74dbfeffa3c": "A \\mapsto U_A", "4e3d051a7878ac2d1e32c7f6880e54ec": "\\lim_{n \\to \\infty} t_n = -\\infty", "4e3d1d1abbebbe83950b18c09e50d720": "\\scriptstyle {d}\\vec{x}", "4e3d45d6053ed796eed5770b4a1c3d07": "\\frac{1}{\\sqrt{2^{n}}} \\sum_x e^{ix\\theta} \\left| x \\right\\rangle \\otimes \\left| \\psi \\right\\rangle", "4e3d80251677d803faaa91aefc00653f": "\\left(\\frac{1}{\\sqrt{10}},\\ -\\sqrt{\\frac{3}{2}},\\ \\sqrt{3},\\ \\pm3\\right)", "4e3dd0dbfbf4a047595c5c54a6d5498e": "J_+ = J_x + iJ_y,\\quad", "4e3e102e833cb6ebe01909998e69068a": "\\begin{align}\nP_0(n) &= P_d(0) = 1, \\\\\nP_d(n) &= P_d(n-1) + P_{d-1}(n) \\\\\n&= \\sum_{i=0}^n P_{d-1}(i) = \\sum_{i=0}^d P_i(n-1).\n\\end{align}", "4e3e31b07ad30f8012e0ecc4baa04f7c": "N=G-\\bigcup_{x\\in G-H}H^x", "4e3e666180e6dfcda588ed8df0afaf8b": "\\left| F, 0\\right\\rangle", "4e3f200bf28aaa25ac056ea60b8e324a": "D=VT", "4e3f615a3bd12983b8a0e045524766cc": "y_{ij}^2=a_7\\theta_{i+1j}+a_8\\theta_{i+2j}+b_1y_{i+1j}^2+b_2y_{i+2j}^2", "4e3f6e9b876aa756415aa173b4a673ff": " p_m = 1 - (p_u + p_d) \\,", "4e3f966113844d395b393b8694b4841f": "F_2(a, b) = a\\cdot b = a 2^{\\log_2(b)}", "4e3fe4d64f2a3652ec42b232f0d799ed": " \\partial_t u(x,t) = \\kappa(x) \\sum_{i, j} \\partial_{x_i} \\bigl( a_{i j}(x) \\partial_{x_j} u (x,t)\\bigr) ", "4e3feaed7647f12157f56efd833eff51": "\\forall x \\in \\{0,1\\}^k, \\forall y \\in \\{0,1\\}^n", "4e3ff266b8bcb1f58244fbc1bc2309dd": " \\underbrace{ \\left( \\begin{array}{c} \\\\ Q(X=s,Y=t) \\\\ \\\\ \\end{array} \\right) }_{\\mathcal{C}_{XY}^\\pi} =\n \\underbrace{ \\left( \\begin{array}{c} \\\\ P(X=s \\mid Y=t) \\\\ \\\\ \\end{array} \\right) }_{\\mathcal{C}_{X \\mid Y}} \n\\underbrace{ \\left(\n\\begin{array}{c c c}\n\\pi(Y=1) & \\dots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\dots & \\pi(Y=K) \\\\\n\\end{array} \n\\right) }_{\\mathcal{C}_{YY}^\\pi} ", "4e405247458b357eccab36a87eb6953a": " g\\to\\infty ", "4e407e914de245d4db4593de414f2473": "\\mathcal F (M)", "4e4081f921d89e551f9ddbe661b1289d": "\\prod", "4e4136abc10c3780272c2e868ca39aa8": " \\sharp E(\\mathbb{F}_{q}) = q+1-t", "4e41e1ca78807101bb997631f150d0a3": "\\frac{D}{P} + g = r", "4e41fd274077eeb3b07f13334e629aca": "1\\uparrow_K^G", "4e420f1910ea1fc70a50682a9e622b2b": "\\tau : TM\\otimes TM\\stackrel{\\cong}{\\to} TM\\otimes TM", "4e427943c3f6b8b5012ed59a874c640f": "\\phi (x)=0", "4e42929e5a6374e8f4155fa594f0ca42": "\\{a^nb^n\\,\\vert\\; n\\ge 0\\}", "4e42bb4f0ff606b3e4c99220c270ded4": "Y_{\\mathrm{f}}^\\prime", "4e42ffe6b287fd241a4a2c45ae0b2224": " R\\left(x,u,u_x,\\ldots,\\frac{d^n u}{dx^n}\\right)=0 ", "4e4311a362b5e0cc90afca0cf66aa2a2": "E\\times -iC", "4e435eb2de20d3c73578e856cee9c9ed": "\\mathrm{adj}(c \\mathbf{A}) = c^{n - 1}\\mathrm{adj}(\\mathbf{A}) ", "4e438077d73880ee7d95a2fa357d2750": "\\{x\\in\\mathbb{R}^n|Ax \\le b\\}", "4e43f8afc63ee98f528dbc5523c6efc5": "\\left|A\\cap B\\right| < \\infty.", "4e44c1d4d71a07d8326f8a1224a001d4": "\\epsilon (b_n - b_N)/b_n \\leq \\epsilon", "4e44e0031dee54291d82ba36b26b1944": "M V = P T ", "4e44e803ffd05c50c4c156913defd3fe": "\\left \\{1, 2, 3 \\right \\}", "4e4584ec4b2adbd958e5f2911efb5dde": "B' \\in \\binom{C}{B}", "4e45af8225924c107b001d67e2ba28e9": "\\Omega\\left({\\frac{\\sqrt V}{\\log V}}\\right).", "4e4604070b51ada2a540f47ee281a294": "x + y \\prime = 0,", "4e4693dc6bd277283724b2caefff2a99": " \\begin{align}\n\\tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\\\\n\\hat{y}_{i+1} &= y_i + \\tfrac12 h \\bigl( f(t_i, y_i) + f(t_{i+1},\\tilde{y}_{i+1}) \\bigr). \\\\\ny_{i+1} &= y_i + \\tfrac12 h \\bigl( f(t_i, y_i) + f(t_{i+1},\\hat{y}_{i+1}) \\bigr). \n\\end{align} ", "4e46cbf8ad3c5f04cebf46c1ac131fc2": "\\epsilon x.\\phi", "4e46e6af82546f39ea33ea6b0c17b279": "P^{\\sigma}", "4e46f0af0473dd8512e469d45c6ada64": "S_{H-P} = \\int d^4x \\; e \\; e^\\alpha_I e^\\beta_J (R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} + \\nabla_{[\\alpha} C_{\\beta]}^{\\;\\; IJ} + C_{[\\alpha}^{\\;\\;\\; IM} C_{\\beta]M}^{\\;\\;\\;\\; J})", "4e4710cc76ee3aecf24677fd259c4ed9": "4x^2 + 4x + 1", "4e474415bacccc7900fb6d5f347aaa4e": "\\hat{f}(n) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x) e^{-inx} \\, dx", "4e4750b4b129e4f16133e487aef3af22": "m=[2\\ldots\\infty]", "4e4765c17eb18fc55d2d5b328b8e2e8c": "\n x^{k+1} \n = \n x^{k} \n + \n \\lambda_k \n \\frac{b_{i} - \\langle a_{i}, x^{k} \\rangle}{\\lVert a_{i} \\rVert^2} a_{i}\n", "4e47d5ab51acdf157da151585ada2ee5": "\\Gamma^\\lambda_{\\,\\,\\mu\\nu}", "4e480f335e1582cd2e7949e42b4a7dc9": "r(\\theta)+r(\\theta+\\pi) = 1", "4e481f0e61b1607d6542a60ab5ab76e3": "A = A^T", "4e483a965cd4d9dace68b323803c402a": "0.04m", "4e4867259d132e5b83b2a87f237a8944": "\\nabla \\cdot \\mathbf{B} = 0", "4e4902a6f67c9b095b15941e08385b47": "\\mathcal{N}(\\mathcal{R}) = K \\mathcal{R}^{d+1}", "4e493dd0f1d2f6313eb420627d1c54ca": "\\text{Sym}^i V ", "4e494354584db78354fe828881153c55": " c^2 dt^2 ", "4e495da10beaff39ef731440f00cc9d4": "t\\rightarrow P_tf(x)", "4e4975d46a9078f3a91f7a47f284690c": "x^\\top y + c < 1", "4e49837a3752099d0cbe65b77559138d": "0<\\varepsilon < 0.001,", "4e49856cd2f53bf181eca43ca05519e6": "\\tfrac{2^m-1}{q}", "4e49ad584a7a82b7f4e3a9de0a37f181": "\\omega_{X}=(\\vec{b}, u, \\vec{a})\\,\\!", "4e49d96ad9a4ffd68c5c407a31091e06": "\\mathcal{S} = -\\frac{1}{2\\pi\\alpha'} \\int \\mathrm{d}^2 \\Sigma \n\\sqrt{(\\dot{X} \\cdot X')^2 - (\\dot{X})^2 (X')^2}.", "4e49edf6ed07ce1b506dbf44bb4599ed": "f_1(f_0(3)) - 2", "4e4a6435475fac645b0a616f128606d7": " \\int_{S} \\mathbf{u \\cdot T} dS = \\int_V \\boldsymbol\\epsilon : \\boldsymbol\\sigma dV - \\int_V \\mathbf u \\cdot \\mathbf f dV ", "4e4ae79834b841e84f28384113dc127a": "\nr_\\mathrm{out} \\approx {R_\\mathrm{E}} \\| {R_\\mathrm{source} \\over \\beta_0}\n", "4e4b640023822672eeb0ef0ff67138af": "\\bar K_n", "4e4b70934ae4194f6826b47fcdff1655": "\\frac{P_1}{P_2}\\,", "4e4b8220755dc6a17dc16b43818c73cb": "\\tfrac{1}{2}Q\\left[f(a) + 2f(a+Q) + 2f(a+2Q) + 2f(a+3Q)+\\cdots+f(b)\\right].", "4e4b9ab794edf7953e705d3b0f55c1ad": "\\sin a = \\sin c \\cdot \\sin A", "4e4bad6e32fff305ad5de08ba7d6bee6": "A^+ + B \\to A^{2+} + B + e^-", "4e4bc6780d0d95579382d1fa7670abac": " W_{N}=\\int_{H}T_{H}ds+\\int_{C}T_{C}ds \\,", "4e4bc9d86e806712a74586fed26a7f0e": "e(E) \\cup e(E) = e(E \\oplus E) = e(E \\otimes \\mathbf{C}) = c_r(E \\otimes \\mathbf{C}) \\in H^{2r}(X, \\mathbf{Z}).", "4e4c4278efed2ac5f472947a156bf30d": "N_{a}=10^{-23/2.5}\\times\\frac{S_{0}(V) \\times B}{h\\nu}", "4e4c4dd22b49ef97b57e186f29f41969": "\n \\begin{bmatrix}\n \\epsilon_{{\\rm xx}} \\\\ \\epsilon_{\\rm yy} \\\\ \\epsilon_{\\rm zz} \\\\ 2\\epsilon_{\\rm yz} \\\\ 2\\epsilon_{\\rm zx} \\\\ 2\\epsilon_{\\rm xy}\n \\end{bmatrix}\n = \\begin{bmatrix}\n \\tfrac{1}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & \\tfrac{1}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm zy}}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yz}}{E_{\\rm y}} & \\tfrac{1}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\tfrac{1}{G_{\\rm yz}} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm xy}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm xy}} \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\sigma_{\\rm xx} \\\\ \\sigma_{\\rm yy} \\\\ \\sigma_{\\rm zz} \\\\ \\sigma_{\\rm yz} \\\\ \\sigma_{\\rm zx} \\\\ \\sigma_{\\rm xy}\n \\end{bmatrix}\n ", "4e4c83e642db71bcdb5bcff88135c13e": "\\max_{k=1,\\ldots,n} \\frac{\\sigma_k^2}{s_n^2} \\to 0, \\quad \\text{ as } n \\to \\infty,", "4e4c9e1c6331e0e430c5ff3224a708bd": "0+1+2+5+12+29=49", "4e4cad74f2dc2eb92cc5b8fbf76de692": "a^2+b^2=c^2", "4e4d24ad261ebbdfd9ce98f7897bc657": "X^8+X^6-3 X^4-3 X^3+8 X^2+2 X-5", "4e4d8b324d86c7d06ad412601d5a3dd1": "\n0.39 \\left ( \\frac{\\mbox{total words}}{\\mbox{total sentences}} \\right ) + 11.8 \\left ( \\frac{\\mbox{total syllables}}{\\mbox{total words}} \\right ) - 15.59\n", "4e4dfb20c15c302f98afc434293efa9e": "f_\\textrm{sim}", "4e4e811ed92ab05cac7c4a6c5d0971c4": "(f\\star g)_i \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_j f^*_j\\,g_{i+j}", "4e4e8a228fb32614731c08e8a556e79b": "y = a \\sec \\varphi + \\beta.\\,", "4e4e8ca528d7e4ca6e4a3587f3ebc7f9": "X^2\\backslash\\left\\{ y~\\backepsilon~x\\succcurlyeq y\\right\\}=\\left\\{ y~\\backepsilon~y\\succ x\\right\\}", "4e4e9eab413450a29792ef6f8095bb29": "\n\\Omega_{n,\\mu\\nu} (\\mathbf R)={\\partial\\over\\partial R^\\mu}\\mathcal{A}_{n,\\nu}(\\mathbf R)-{\\partial\\over\\partial R^\\nu}\\mathcal{A}_{n,\\mu}(\\mathbf R).\n", "4e4f05418e220c019b778f15251c66f0": "\\phi: X \\to \\mathbb{R}", "4e4f101d7c0a653d365528a762b895d7": "\nU = {\\sum_n \\hbar\\omega n e^{-\\beta n\\hbar\\omega} \\over \\sum_n e^{-\\beta n \\hbar\\omega}} = {\\hbar \\omega e^{-\\beta\\hbar\\omega} \\over 1 - e^{-\\beta\\hbar\\omega}},\\;\\;\\;{\\rm where}\\;\\;\\beta = \\frac{1}{kT},\n", "4e4f1d9ba85189a5f123d1c66d6e67ab": " \\omega \\stackrel{\\mathrm{def}}{=} kc ", "4e4f1fa6bbe2dbb3627c79d14b58bed6": "\\forall x \\, \\forall y \\, [ (Mx \\and My) \\rightarrow \\exist z \\, (Mz \\and \\forall s \\, [ s \\in z \\leftrightarrow (s = x \\, \\or \\, s = y)])].", "4e4fc5d1bbf3cc226d2774b9c09b10e9": "Q=\\int\\limits_V \\rho_q(\\bold{r}) \\,dV", "4e4febf89f09768cbe0050558ee26200": "\\ell((I^n M \\cap M') / I^n M') \\le \\chi^I_{M'}(n-1) - \\chi^I_{M'}(n-k-1)", "4e4ffdb3ee5b0fb5bd74f2511348aa70": "\\operatorname{P}[E_1]", "4e503997549ed5a41de5fb1135e46383": "\\frac{\\partial\\rho{\\mathbf{u}}}{\\partial t} + \\nabla \\cdot \\left ( \\mathbf{u}\\otimes \\left ( \\rho \\mathbf{u} \\right ) \\right )+\\nabla p=0\\,\\!", "4e505b80a327d83109cebdfd62596316": "A\\in\\mathcal{L}", "4e5081dbc4a5d021e9dc8dde51c1fdd5": "L p = -\\sum \\frac{\\partial (f_i p)}{\\partial x_i} + \\frac{1}{2} \\sum (\\sigma \\sigma^\\top)_{i,j} \\frac{\\partial^2 p}{\\partial x_i \\partial x_j}", "4e50c4860803997b61191085f6f9e190": "r,s\\in {\\Bbb Z}", "4e513ddc2ef2387633abe7ffe62f6f29": " \\vec{B} ", "4e5165e46b87541ea71501369b70c599": "(1-\\epsilon)2^{m}", "4e516d990aecf43dadb1d7773a0fb32b": "B_n(x_1,\\dots,x_n) = \\det\\begin{bmatrix}x_1 & {n-1 \\choose 1} x_2 & {n-1 \\choose 2}x_3 & {n-1 \\choose 3} x_4 & {n-1 \\choose 4} x_5 & \\cdots & \\cdots & x_n \\\\ \\\\\n-1 & x_1 & {n-2 \\choose 1} x_2 & {n-2 \\choose 2} x_3 & {n-2 \\choose 3} x_4 & \\cdots & \\cdots & x_{n-1} \\\\ \\\\\n0 & -1 & x_1 & {n-3 \\choose 1} x_2 & {n-3 \\choose 2} x_3 & \\cdots & \\cdots & x_{n-2} \\\\ \\\\\n0 & 0 & -1 & x_1 & {n-4 \\choose 1} x_2 & \\cdots & \\cdots & x_{n-3} \\\\ \\\\\n0 & 0 & 0 & -1 & x_1 & \\cdots & \\cdots & x_{n-4} \\\\ \\\\\n0 & 0 & 0 & 0 & -1 & \\cdots & \\cdots & x_{n-5} \\\\ \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ \\\\\n0 & 0 & 0 & 0 & 0 & \\cdots & -1 & x_1 \\end{bmatrix}.", "4e51852cba9fae5d110e287eed2ecd60": "\n \\lim_{\\delta \\to 0} \\limsup_{n \\to \\infty} \\mu_{n}\\big( \\{ f \\in D \\;|\\; \\varpi'_{f} (\\delta) \\geq \\varepsilon \\} \\big) = 0\\text{ for all }\\varepsilon > 0.\n ", "4e51a558256fc56cdd8b7eb4864e69d2": "Sx = \\sum_{i \\in \\mathbf{N}} \\sigma_{i}^{2} \\langle x, e_{i} \\rangle e_{i}.", "4e51eeb493eb3cf2c6cd6f023eba4576": "H= \\sum_i f_i \\ln f_i \\,\\delta q_1 \\cdots \\delta p_r", "4e51efcafa7a39f1c2822a9908855af1": " s = \\alpha_2 = -3 - 4j ", "4e5228b3f1983baa8d309f2abd1b844b": "R_x \\approx R_2 \\cdot \\frac{R_3}{R_4}", "4e52554c13afe6ff6e49b6c86ffa228d": "I = \\iint_{\\Sigma} \\mathbf{J} \\cdot \\mathrm{d} \\mathbf{S}\\,,", "4e52814f6737e85bab82c57e1e6628ad": "\\mathbb{HP}^{2}", "4e52893efe38d1d90ab63cfd1e83df57": "2/m", "4e52e50c23d5306285927a5330e72646": "r_N^k(m) < cn^{\\frac{N}{k}-1}", "4e531f46ec61e79194800200934bb375": "\\, 2\\,\\pi", "4e5334aa6f5fa551b0718a2372816061": "y^2=x(x-1)(x-\\lambda)", "4e53ecc211290ee3df1ad8da5bbe21e8": "\\scriptstyle t_i", "4e542cab39c360359c08876a784c0a65": "G_\\mu^a", "4e543dadcaf202432bd68747109adb5f": "X^j = \\eta^{j0}X_0 + \\eta^{ji}X_i = - \\delta^{ji}X_i = - X_i ", "4e54496bed057655d2a89df7392510b9": "1^9 + 2^9 + 3^9 + \\cdots + n^9 = {16a^5 - 20a^4 +12a^3 - 3a^2 \\over 5};", "4e54c5dfd3f3a5a2530d76a48c4dc811": "A_i\\;=\\;min(2d_T\\;\\tan {\\frac{a_T}{2}}, 2d_R \\tan{\\frac{a_R}{2}}, w)\\;x\\;min(2d_T \\tan {\\frac{e_T}{2}}, 2d_R \\tan{\\frac{e_R}{2}}, h)", "4e54ddc71ca7c5314244229414484307": "\\frac{F_m-F_{tr}}{F_m}", "4e54f7568621e7b96abff07613f22e8b": "\\frac{dV_p}{dt}=Q=\\frac{\\Delta p}{\\mu}\\ A\\left( \\frac{1}{R_m + R} \\right)", "4e551d1bf9154cacfc0ee328f81d23f5": "\\Sigma a_n q^n", "4e55ba99e5b1d4dd6de6bd9bb5184e85": "S_c\\,\\!", "4e5648c41e4d14d81cdfc1b5ef5a242a": "\\Gamma[\\phi]=-J_i \\phi^i - E[J] \\, ", "4e566dab0ca108decc798d9bb5c53c56": "[S^1]\\in H_1(S^1)", "4e56938567a237de34be09ab5e527c2d": "Y_{8}^{7}(\\theta,\\varphi)={-3\\over 64}\\sqrt{12155\\over 2\\pi}\\cdot e^{7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot\\cos\\theta", "4e56a179e41d8e4693d53809aad596b9": "E(s\\otimes n)=E(n\\otimes s)=n.", "4e56cf5371158b874da4abd748b4a7e5": " Reaction Rate = \\eta \\cdot \\pi \\cdot ravg \\cdot (1-\\alpha) \\cdot Lavg \\cdot k1'' \\cdot Cc ", "4e56f1f99693d049eca139e2b48b9733": " \\sum_{ i \\in N } x_i = v(N) ", "4e571f049753c707396b2c1aa8771c23": " H(X) = - \\sum_{y \\isin \\mathcal{X}}p(y)\\log_2 p(y) ", "4e572e072e4828655cd64c8618f0afe9": "10^2-10^8", "4e5741fbf06063d1f9df4653ee71ba8f": "(E \\times F) \\times G = E \\times (F \\times G) = E \\times F \\times G", "4e574bc7fac244e9b44c744f9fe30516": "1 < q < 2", "4e574dc0ae89a699165837eba9e6ada1": "\nA_{ij}=\\frac{8 \\pi h \\nu^{3}}{c^{3}} B_{ij}.\n", "4e57f857ffdc11c3584395267bb5116f": " \\chi(x_1,\\dots,x_r)\\chi(y_1,\\dots,y_r)\\ge0", "4e57fc027e3a4e7feea5269a6f36d246": " i=1, 2, ... n+2 ", "4e580b582d539d5162557b361ae383aa": "h_{\\mu\\nu}\\,", "4e582826098b3988abf2548e8afef791": " A, B \\vdash_r C, \\ ", "4e585347a756c745faf123675ddd7026": " \\mathbf{E} = - \\nabla \\phi - \\frac { \\partial \\mathbf{A} } { \\partial t }", "4e58b3032c31fa65dbad7c8a9278e5af": " \\omega^2= \\left ( \\frac{kaK_1}{ka} \\right ) \\left [ (1-k^2 a^2) \\frac{T}{pa^3} - \\frac{\\Gamma^2}{4\\pi^2 a^4} \\right ] ", "4e58d73da4a859fe166c367208d3491f": " \\tau_{th} = \\frac{\\mbox{total kinetic energy}}{\\mbox{rate of energy loss}} = \\cfrac{\\frac{GM^2}{2R}}{L}", "4e58f33de2594066a1d53da0efd690d6": "V_i, W_i, i=1,\\dots,n", "4e5915e5cce003b258ffd6e9d791dd72": "\\frac{AF}{FB} \\times \\frac{BC}{CD} \\times \\frac{DR}{RA} = 1", "4e591fa20d92a6ef93312b26f104c24d": "0 \\le j_k(g) \\le \\lfloor n/k \\rfloor \\mbox{ and }\n\\sum_{k=1}^n k \\, j_k(g) \\; = n.", "4e594b62a04a7b535e77cb37c57ab208": "1/\\sqrt{n}", "4e59bc7e979b397dde0f047509640112": "t' = t_r", "4e59bca92178d54c9a7bcd7634b3d511": "\\Delta_{T}", "4e5a39bf3d375f7393338987d3132ca7": "\\hat{b}_i \\, \\hat{b}_j = \\hat{b}_j \\, \\hat{b}_i ", "4e5a86924aeba8376b7eaaafbdde58ab": "\\rho =", "4e5a8a9861c59172c7f3d94e83c0d63b": " m > 1, \\alpha = 2.3 \\pm 0.3 ", "4e5b2edeed9265ad801399553d2df262": "\n\\frac{1}{m}\\sum_{k=1}^m c_{m_1}(k) c_{m_2}(k) =\n\\begin{cases}\n\\phi(m), & \\text{if }\\;m_1=m_2=m,\\\\\n0, & \\text{otherwise.}\n\\end{cases}\n", "4e5b338970f896fa3817350cc0ebe9de": "x_{s}(t)0 ", "4e6f70928283c81e7e5c603351acfa75": "R_1^2 + \\cdots + R_n^2 = -I.", "4e6fad039cf104fe168a08cb07c86629": "f_0, f_1, f_2, \\dots", "4e70979d3169c6e6b3d62b8313afc826": "P_\\mathrm{O}", "4e70b3c409e3f1de6153872d00d91107": "\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}} = \n \\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{\\mathit{1}} + \n \\cfrac{\\partial W}{\\partial I_2}~(I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{F}^T\\cdot\\boldsymbol{F}) + \n \\cfrac{\\partial W}{\\partial I_3}~I_3~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{F}^{-T} ~.\n ", "4e70d3e80c8317915a32e783a7541cad": "m=k=n", "4e70d697c20ba0f108f19a82e37609bf": "\\bold{p} = m\\mathbf{\\dot{r}}", "4e717fae08af3899e2671273d4d74cf6": "(+,-,-,-,\\cdots)", "4e71844ba937a00913c8a955c24e4e27": "S(P) = P - \\alpha\\cdot(P-1)", "4e723c917c8c086acea0b7b8895d0e6f": "\\textstyle {a \\choose b}", "4e72bf16f0c2b9283a4a26574715363f": "\\textstyle 0 \\le r < \\textstyle 2l-1", "4e72d2d5db2092fd9be925ea86f2747a": "\\frac{d^2}{d\\lambda\\,^2}E(\\theta(\\lambda x))<0", "4e72f15c7eb03dfd0dd7338d31612453": "A \\smallsetminus B = \\varnothing", "4e72f2a4f466fbc2d987e3ba3809174d": "X_j = \\frac{\\mu_j-\\mu^{\\rm eq}_j}{T}; \\;\\; \\frac{d N_i}{d t}=\\sum_j L_{ij}X_j ", "4e732b86f445590f04aeb66145f4fc26": " \\lim_{\\alpha = \\beta \\to \\infty} \\operatorname{var}(X) = 0 ", "4e732ced3463d06de0ca9a15b6153677": "26", "4e733cf28e4ebf04b269205f61e9c486": "e^{-r(T-t)}", "4e73b61c72425e67564349ffe9196fde": "x(yz) = (xy)z\\,", "4e73bcfae4f7b92ca0836c817d912bc0": " R =\\left( \\frac{S_{1}}{S_{2}}\\right)_{T} = \\left(\\frac{S_{1}}{S_{2}} \\right)_{T_0} \\exp\\left[-\\frac{hc(E_{1}-E_{2})}{k}\\left(\\frac{1}{T}-\\frac{1}{T_{0}} \\right) \\right] ", "4e7406b05fe4c21da1d667111c4b2ada": "\\mathcal{O}(M)", "4e74945f01daf48855aff9cf811b3700": "\\mu(\\sqrt{y})=df(y)/dy", "4e74a8f79175ae956ac003af1ef9e648": "A X + X B = C,", "4e74c2bf647e41a6302afb3704896492": "R_E=c/\\sqrt {4\\pi G\\rho}", "4e75cb2fd94a1b812c9ce3c758782115": "|P-Q|_\\pi", "4e75e37f4a19c74a5a7df1eb0d8dfb80": "\\infty = \\left(\\frac{\\pi^3}{mc^2}\\right)^\\infty", "4e760f08aa3712a2a960e95e735e1bbf": " 2^{O(\\sqrt{k})}\\cdot |G|^{O(1)}", "4e764469a91460d6edc0d1562211c2fc": "p + p = \\text{d} + e^{+} + \\nu_e \\!\\ ", "4e76a33b5df72bfa5fd5dec20ad8a33d": "\\mathit{X_n}", "4e76f130f9236be40fdbc643cd450707": "z\\mapsto z^2", "4e7705e78c0b920b1ac2a8cbbb37131c": " a_1x+b_1y+c_1z-d_1=0 \\,", "4e7752cdd93bee6cd67a9a7a293bc07a": "Y \\sim \\Gamma(\\alpha,\\tfrac{\\beta}{\\alpha})", "4e775b05a032462422a0d34f0e4f543f": "[S,H]=0", "4e7787b943b03b77b24916466387ae52": " A = (a_{i,j})_{i,j=1}^n~, \\quad a_{ij} = f(x_i - x_j) ", "4e77dc76014fa7728a87ec9798d1bdfc": "\\operatorname{var}(\\epsilon)=R", "4e77e5fae51061cb25ae849ed4061992": "\\tau=-\\frac{1}{4\\pi\\epsilon_0}\\frac{p^2}{16a^3}\\sin 2\\theta", "4e77ff99deddde411c47bf7d5a18656f": "\\omega = \\begin{bmatrix} 0 & I_n \\\\ -I_n & 0 \\end{bmatrix}", "4e78660ad9418e15415f977622df7454": "x_1x+y_1y = r^2,\\!\\ ", "4e78ebed0a423003ecadc6cf0c97c843": "\\sin (\\alpha - \\beta) = \\sin \\alpha \\cos \\beta - \\cos \\alpha \\sin \\beta\\,", "4e78f5556de86068684f2e46ac85c7af": "\\{(x,y)\\in \\R^2: x=y\\}", "4e790078255c2261edb6ab18ade222e8": "G(\\rho,\\varphi) = \\sum_{m,n}\\left[ a_{m,n} Z^{m}_n(\\rho,\\varphi) + b_{m,n} Z^{-m}_n(\\rho,\\varphi) \\right],", "4e790a58786b8950da040d36e341f1b5": "\\ AG(p)", "4e798aa3f650b3c5bacc4b2b16d25587": "r=r_e-r_h, R={{m_e r_e + m_h r_h}\\over {m_e+m_h}}, {1 \\over \\mu} = {1\\over m_e}+{1 \\over m_h}, M= m_e + m_h ", "4e79910bb9416bb8c692e6728d3ffa64": "a=\\frac{q^2}{\\omega^2}", "4e799837622522e2bf05095839fdca5a": "\\frac{\\partial T}{\\partial t} = \\kappa \\nabla^2 T = \\kappa\\frac{\\partial^2 T}{\\partial^2 z} + \\kappa\\frac{\\partial^2 T}{\\partial^2 x'}", "4e79989fa65e331f748c7c450a9b23c3": "RvR", "4e79b7f263c9e004d45940e94bd2d179": "\\alpha^{p}=a", "4e79ed2fb15b6da94249d9d4688daca2": "\n\\frac{d_{min}}{b}=A_3 \\left ( e^{-\\tfrac{\\beta Q}{4RT}} \\right ) {\\left ( \\frac{D_{p0} G b^2 }{\\nu_0 k T} \\right )}^{0.25} {\\left ( \\frac{ \\gamma }{ G b } \\right )}^{0.5} {\\left ( \\frac{ G }{ H } \\right )}^{1.25}\n", "4e79edcb6e000ba71a18694703b801e0": "[f(x) - g(x)]", "4e7a425998ba0080e4ae405cd130cf00": "\\scriptstyle\\sum_{k=1}^n\\|x_k\\|", "4e7a48d1a1455dc15afe4bc736d5185c": "F_n^2 - F_{n-r}F_{n+r} = (-1)^{n-r}F_r^2.\\,", "4e7a782c45bc8d04197436919d238bde": " {a_b}^{c_b}= {a^c}_{b^c} ", "4e7a7c819e4d43b7b53c0eb2df3a2515": "B^2 - AC < 0.\\ ", "4e7a8c4aca5586fff229d80d77d3ac2a": "\\overline{\\psi}\\gamma^{\\mu}\\partial_{\\mu}\\psi", "4e7ad7a8bab6950cfdbd2a6e0a52056c": " \\sum \\lambda_i\\overline{\\lambda_j} \\Phi(g_j^{-1}g_i) = PT^*TP\\ge 0,", "4e7ad81762eca042fbd7f5b352723076": "E_{sig}(t,\\tau)", "4e7b35a880677e8ae53ca749155de7e3": "z_1=-\\frac{1}{2}+j\\frac{\\sqrt{3} }{2}, z_2=-\\frac{1}{2}-j\\frac{\\sqrt{3} }{2}", "4e7b6a7edebadcbbc7e1ae138109746b": "\\,\\omega", "4e7c73efcd62668f557b1205c7fc905c": "\\mathbf{P}^2 \\propto \\mathbf{T}^3", "4e7c870d3a0ceefab58840ddf99af3a1": "\\scriptstyle 1 \\,\\oplus\\, 1 \\;=\\; 0", "4e7c96077fe59454f3ee61d672ee7c6d": "v=f(t)", "4e7cf729d074ab1cb4ee6a1917057f69": "\\scriptstyle\\frac{E^2}{x^2} \\,-\\, P^2 \\,-\\, Q^2", "4e7d0aa5322bcf59666ab18faf7019cf": "\\left | E\\right| = {\\partial V \\over \\partial r}={1\\over e}{\\partial U(r) \\over \\partial r},", "4e7d209252dac3e735b0aedd1c27361a": " |\\psi\\rangle = \\psi_R |R\\rangle + \\psi_L |L\\rangle ", "4e7d23215f3b1c1b3888ae5c047703c5": "F=0.8\n", "4e7d9190e802b2822d5550715a303f01": " U_n(\\cos(\\vartheta)) = \\frac{\\sin((n+1)\\vartheta)}{\\sin\\vartheta} \\, ,", "4e7dbcc624d596c80f826ab200e45f75": " L(\\mu) = -\\left[\\sum_{x_i: x_i<\\mu} (x_i-\\mu)^2 \\right]^{1/3} - \\left[\\sum_{x_i: x_i>\\mu} (x_i-\\mu)^2 \\right]^{1/3}", "4e7dc31ec00b4e5290871f8514c55839": "(\\operatorname{trace}_{V}(T))_{k_1 \\dots k_N }^{\\ell_1 \\dots \\ell_N}", "4e7de877512fe449983423242e74007e": "\\sin^2(18^\\circ)+\\sin^2(30^\\circ)=\\sin^2(36^\\circ)\\;", "4e7e2c39cc0d10231388287620ecf5ff": "w(\\neg\\theta)=F_\\neg(w(\\theta)),", "4e7eb56edcb2994f578c5c0b25779adc": "x^2 + y^2 + z^2(1.5\\cos((z+a/2)/a))^2 = a\\!", "4e7f2d16c865c9f8a4305125ccad153d": "\\scriptstyle \\sum_{i \\in I}\\alpha_i F_i \\;=\\; 0", "4e7f783b9ab1fe4013d1e3f234732932": "\\varphi,", "4e7fc778781eb1bb994c309e5e539700": "\\mathbb{HP}^\\infty", "4e800eb693de2a8d3b63391edab70b30": "\\{A, B, C,...,\\}", "4e802540b3e058efed91efb928769c4d": "(X_1,X_2,\\dots,X_d) = \\left(F_1^{-1}(U_1),F_2^{-1}(U_2),\\dots,F_d^{-1}(U_d)\\right).", "4e805170d94fa8af50d20f3c4944fc86": "p = a_3 q_3 - 1", "4e805f8999d0b9cd33e91320b549bc06": " SO_2 = (\\frac{23,400}{pO_2^3 + 150 pO_2} +1)^{-1}", "4e809f956903af266966dd441797a5f7": "P \\ne O", "4e80e2b4fa125da506538fd414f62a93": "T \\approx 2\\pi \\sqrt\\frac{L}{g}\\!", "4e813bffa16edb39eb384061c00af1f1": "~|1\\rangle~", "4e819bcffd40eaceaff65de8ecab403e": "\\iota \\circ r", "4e81b9cff9623c6f97712bb4cb0c3c6f": "\\operatorname{E} \\left[N\\left(\\mu, \\sigma^2 \\right)^p \\right]=(-2 \\sigma^2)^\\frac p 2 \\cdot U\\left(-\\frac p 2, \\frac 1 2, -\\frac{\\mu^2}{2 \\sigma^2} \\right)", "4e81f62a9771f62bf23f1a7a570b2be7": " C^J_{v_2} = \\frac{-\\varepsilon^{1}_S}{\\varepsilon^{2}_S - \\varepsilon^{1}_S} ", "4e821d83269626ee05e1814f384c00fb": "A^\\alpha B_\\beta{}^\\gamma C_{\\gamma\\delta} + D_\\alpha{}_\\beta{}^\\gamma E^\\delta. ", "4e824fa77ac3434bc518b6c827eaec75": "\\mathit{C} = [\\mathit{c}_0, \\mathit{c}_1, ..., \\mathit{c}_{N-1}]", "4e8295a7cf9f8ec768534e0011453275": "\\text{length of refresh cycle} = 4/f = \\frac {4}{1.33(10^8)\\,\\text{Hz}} = 30\\, \\text{ns} \\,", "4e82aaea367ded14646bfb604b558b81": " s \\equiv s' \\cdot r^{-1} \\equiv (m')^d r^{-1} \\equiv m^d r^{ed} r^{-1} \\equiv m^d r r^{-1} \\equiv m^d\\pmod{N},", "4e82b758476e427ab9064c860de4cf8c": "q=\\frac{1}{2}(1+3w)\\left(1+K/(aH)^2\\right)", "4e82c639001b14dd2f6305ebea94324f": "\\mathrm{S\\scriptstyle ET}_{\\ge 1}", "4e82e084e7c6141c64bd00fe176bca76": "\\gamma_E", "4e830a938aa6e5ab4e4c781657eff87c": "\\gamma^6", "4e8350badc680f9cfc03a9c988b5f47b": "\\Delta n=n_{RHC}-n_{LHC} \\,", "4e83662ad14a4b6013e1c2c2c988fbb0": "\\left[y^{(1)}\\right]", "4e840f3d9995e2033d646df8a7a98807": "log n", "4e841132f5f81780eb577ceecddd2335": "\\mathbf{D} = \\frac{1}{2}\\left[\\boldsymbol\\nabla \\mathbf{v} + (\\boldsymbol\\nabla \\mathbf{v})^T\\right]", "4e8449691d449cc25fc24a36f93fee0f": "1,2,4,7,10,13(\\mathrm{mod}\\ {15});\\ 5,11,12,22,23,29(\\mathrm{mod}\\ {30})\n\\}", "4e847b083689834bbbff371fd8025d5d": "p \\gg m", "4e84da3481926f7b0ee2e86ea08806fe": "B(E)", "4e84ebe02d13ffaf203a235d080239d1": " \\langle x \\mid A y \\rangle = \\langle z \\mid y \\rangle \\quad \\forall y \\in \\operatorname{dom} A ", "4e854be466f01819d5e8e816ca1996ce": "\\sum_{v \\in V} c(v) x_v", "4e855ca20bfaea5da062031b4cb6fc45": "y_i = \\sum_{1 \\le j \\le k}(\\alpha_{ij}v_j).", "4e8589925168bbe50b8f8c1e022a1d54": "\\ \\theta_{out}(t) ", "4e86a4b817896614e299b0cc8cc75947": " \\lim_{k\\to \\infty} \\frac{|x_{k+1}-L|}{|x_k-L|^q} = \\mu \\,\\big|\\; \\mu > 0.", "4e86b5a9f28be18cdcc06f741552d03c": "B=N(\\mu,\\sigma)", "4e86f638dd35705e831e1e4e80f60303": "\n\\beta = 90^{\\circ} - (\\lambda + \\chi) \n", "4e86fdf86a47594967166ba637021300": "\\varepsilon = \\sqrt{1+\\frac{b^2}{a^2}} = \\sec\\left(\\arctan\\left(\\frac{b}{a}\\right)\\right) = \\cosh\\left(\\operatorname{arsinh}\\left(\\frac{b}{a}\\right)\\right)", "4e87593cba325eea7b19fb8cfb550386": " \\boldsymbol { \\mathcal{L} } = \\mathbf{r} \\times \\boldsymbol { \\mathcal{P} } = {1 \\over 4\\pi c } \\mathbf{r} \\times \\left [ \\mathbf{E}( \\mathbf{r}, t ) \\times \\mathbf{B}( \\mathbf{r}, t ) \\right ]. ", "4e8766b3d362b3cdfee463a5e2aabfa0": "\\rho=\\frac{-r+2M}{2r^2}\\,,\\quad \\mu=-\\frac{1}{r}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\varepsilon=\\frac{M}{2r^2}\\,,", "4e8795069105e6b18201cdc0ec755b14": "\\tilde{X}(t,x)=(1,X(t,x))", "4e88015c408c58c796879487972409c2": "r_K", "4e883defec2c075bb28ede8f7f1ee1ff": "A_0, A_1,\\dots,A_l", "4e88a1391256ab3241b4eaf1166497c5": "\\left ( 1 - \\frac{\\dot{R}}{c} \\right ) R \\ddot{R} + \\frac{3}{2} \\dot{R^2} \\left ( 1 - \\frac{\\dot{R}}{3c} \\right ) = \\left ( 1 + \\frac{\\dot{R}}{c} \\right ) \\frac{1}{\\rho_l} \\left [ p_B(R,t) - p_A(t) - P_\\infty \\right ] + \\frac{R}{\\rho_l c} \\left ( 1 - \\frac{\\dot{R}}{c} \\right ) \\frac{dp_B(R,t)}{dt}", "4e88b7be55cb676019b6751188e8c9ee": "A \\sim W_p(\\Sigma, m) \\qquad B \\sim W_p(\\Sigma, n)", "4e89c3fecda919b660b8ddca318954a9": " \\dot \\gamma \\ll 1/\\lambda ", "4e89eea72a42274be5a8b7ba018b662a": " M_{\\mathbf{\\Xi}}(\\mathbf{x}) := \\frac{1}{\\mid{\\det{\\Xi}}\\mid}\\chi_{\\mathbf{\\Xi}}(\\mathbf{x}) = \\begin{cases} \\frac{1}{\\mid{\\det{\\Xi}}\\mid} & \\mathbf{x} = \\sum_{n=1}^d{t_n \\xi_n} \\text{ for some } 0 \\le t_n < 1 \\\\ 0 & \\text{otherwise}\\end{cases}.", "4e8a30332e887dc87f2a6cdb53143b89": " f(r | H=h, T=t) = \n \\frac {\\Pr(H=h | r, N=h+t) \\, g(r)} {\\int_0^1 \\Pr(H=h |r, N=h+t) \\, g(r) \\, dr}. \\!", "4e8aa2fb97b9603a6e8c1db12c50eb06": "|B_{i+1}-B_i|", "4e8b0364a719347d9c9ef041877e4087": "\\overrightarrow{g} = G \\frac{m M}{r^2}\\ ", "4e8b081ce11d5305b8c825c0d9c193fc": "r=nb", "4e8b128bf967a1dc4d334d4a0a8308ad": "x=a/2", "4e8b181868a97adda0987084e3b0be3c": " \\and ((T_1 = [F_1, S_1, A_1]::R ", "4e8b477dc71455276adb9846baba5069": "\\mathrm{rad}(z,D) := \\frac{1}{f'(z)}\\,.", "4e8b650caad78097cff96face8264319": "\\left\\langle \\sum\\nolimits_\\alpha p_\\alpha \\partial^\\alpha S,\\varphi\\right\\rangle = \\left\\langle S,\\sum\\nolimits_\\alpha (-1)^{|\\alpha|} \\partial^\\alpha(p_\\alpha\\varphi)\\right\\rangle.", "4e8ba6936389ba539e118107dcb474e1": "\\phi_X", "4e8bb59422877195bc9fe46d16ed84c8": "d\\Gamma=\\mathbf{V}\\cdot \\mathbf{dl}=|\\mathbf{V}||d\\mathbf{l}|\\cos \\theta", "4e8ca5d3844b991975ef52077cd5a8f3": " b_j ", "4e8cdd64c2a3180f3c2ec5a9eb3eb6d3": "(2m-n,m)", "4e8cf1731e2583db7d404d73c92ee784": "F^\\sharp : S\\mapsto F^\\sharp S = S\\circ F.", "4e8d404341b35b1cee07e3fe2b277dda": "2^{2^{0}}+1", "4e8d8e1e90fd331bc9a5f8522e38d7bf": "\\| \\!\\,", "4e8d90165bfe816139cb9b3dfa3530e0": "a={\\ell \\over e^2-1 }. ", "4e8dabdf790133436d9639b18e5bb0eb": "\\log_2 |G|", "4e8df7775fd54cf20d0cd7611a0f1d7d": "\n\\begin{align}\n \\rho A_1 s_1 &= \\rho A_{1} v_{1} \\Delta t = \\Delta m,\n \\\\\n \\rho A_2 s_2 &= \\rho A_{2} v_{2} \\Delta t = \\Delta m.\n\\end{align}\n", "4e8e5864940bf33430a4aed0eaaab83c": "C_0(\\mathbb{R})", "4e8e6db066673ad7c1704fce6babfb35": "\\mbox{diff}(M)", "4e8e70435e674d1624160eb927b73efa": "\n F_a(z) = \\cfrac{16\\gamma\\pi R}{3}\\left[\\cfrac{1}{4}\\left(\\cfrac{z}{z_0}\\right)^{-8} - \\left(\\cfrac{z}{z_0}\\right)^{-2}\\right] ~;~~ \\frac{1}{R} = \\frac{1}{R_1} + \\frac{1}{R_2}\n ", "4e8eb02ee88cfb6d539da89c55028406": "\\begin{matrix} {3 \\choose 1}{11 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "4e8f14f1b90071474002bf996aa38ec1": " p\\leq k ", "4e8fafbb7678f88a98779da6585c3eee": "\\mu(\\tilde X \\cup \\tilde Y)=\\mu(\\tilde X)+\\mu(\\tilde Y \\,\\backslash\\, \\tilde X)", "4e8fb3e5ea9435d7a96b1d332ffb840d": "\\alpha \\in (0,\\infty) ", "4e9035f49ec2ccda093e469aa83d81fa": " m = \\frac {S_{yc}}{S_{yt}}; K = \\frac {m-1}{m+1}", "4e907185a615b37c0eb54d69c52b0c0c": "p(z)= \\sum_{i=0}^m a_i z^i", "4e907eab80d4406636ab540c12278ab8": "\\scriptstyle \\frac {\\partial} {\\partial x^{a_i}}", "4e91074267d4ceb30167ec9c40f3ba22": "H_z^{(i)}(z_1,z_2)=\\frac{A_z^{(i)}(z_1,z_2)}{B_z^{(i)}(z_1,z_2)}", "4e911302800e24112708987164af250e": "a \\otimes b = a + b", "4e911f2efc44e3a44f55d74c49c239cb": "x_{2}^{0} = f^{0}(2) = 6 + 2 \\times 2 = 10", "4e915a6abd3938d11afa8d44d375cd27": "x\\to\\lambda_i;", "4e9170e7c96be8cebe35f2cfc027622d": " \\int \\Phi(x)^2 \\, dx = x \\Phi(x)^2 + 2\\Phi(x)\\phi(x) - \\tfrac{1}{\\sqrt{\\pi}}\\Phi(x\\sqrt{2}) + C ", "4e91d61938ce0d4c1027ac8b501fdb32": "C(x) = C(x) \\ - \\ (d / b) \\ x^m \\ B(x).", "4e922ad26f01f0f7c0852cc7ba22d4f9": "\\theta = (A,B,\\pi)", "4e926948e8b18aa28e7ff1b7a702dd9c": "\\sqrt{5+2\\sqrt{6}\\ },", "4e92ba5ec7c57c339fa4fa6ec234547d": "\\theta_{1}", "4e92d85a0b87bae82cc4638214a796f7": "w^{n}", "4e93c0b3c48e08f5789f690651460548": " p_{\\theta} \\leq \\mathbb{P}( q(X) > q^* | 0 ) \\equiv p_0^*", "4e93cbe514f19615a3281e3032f374b7": " \n\\phi \\left( \\vec{r} \\right) = - \\frac{\\mu_0 v m}{4 \\pi} \\frac{x}{R^3}\n", "4e93e933ba1d5915b1aaec3911cb8fb9": " \\mathbb R ", "4e943e1172cb0321f7836a615a5295e8": "e^{i\\theta} = ix \\pm \\sqrt{1-x^2} \\, ", "4e947664d7596d223bfb1ddc12757f8f": "\n p(V) = \\begin{cases}\n k_1~\\xi + k_2~\\xi^2 + k_3~\\xi^3 + \\Delta p & \\qquad \\text{Compression} \\\\\n k_1~\\xi & \\qquad \\text{Tension}\n \\end{cases}\n ~;~~ \\xi := \\cfrac{V_0}{V}-1\n ", "4e9493c720ddb317cb044b6efdd86d02": "Y[\\mathrm{iso} ] = \\frac{E}{1-\\nu} ", "4e949b8578573c50de9f21f3b9447da2": "\\delta>0\\,", "4e94b1238941b3d416109de5e742fd42": "x_j=\\cos((2i-1)\\pi/2n)", "4e94c6f253efc875ce1b726059e64054": "\\boldsymbol{T}(X) = (T_1(X), \\ldots, T_n(X))^T", "4e94dd26f9b898c6db2cfc5fb067d04a": " w \\leftarrow W", "4e950df7839eac4e1183d221d2c47fcb": " \\displaystyle{\\pi(g) W(u) =W(g\\cdot u) \\pi(g), \\,\\,\\, \\pi(g)^*W(u) =W(g^{-1}\\cdot u)\\pi(g)^*}", "4e954a4d940f384583eed28d9b26e038": "\n S(t)\\equiv\\int_{-\\infty}^{x_c} p(x,t \\mid x_{0})dx,\n", "4e95c5ca85c55320cc5546b490d02796": "\nQ^* = Q_E \\left [e(T_s) \\right] + Q_H(T_s) + Q_{LW}(T_s^4) + Q_{SW}\n", "4e95dc8d0d9d46b979be7b5df93aea3f": " \\boldsymbol{\\mu} ", "4e95ec0c86191640be8ad8facd916aeb": "5.\\mu_{7,3}(p_{1}) = \\alpha_{7}(p_{1}) ", "4e968ce8777f6cdb94ff9cb24dc24bfc": "\\left| q \\right|_p := p^{-v_p(q)}. \\,", "4e96a13015532bf445c06522a55031e0": " p_4, ", "4e96b9f365c30797e8bb8bd7f98aa216": "x_n \\,", "4e96dac0a516c6f68ac57e9fca7ffa39": "\n\\mathrm{area}(D_r)= \\int_0^{2\\pi} \\Re\\bigl(f(r e^{-i\\theta})\\bigr)\\,\\Im\\bigl(-i\\,r\\,e^{-i\\theta}\\,f'(r e^{-i\\theta})\\bigr)\\,d\\theta = -\\int_0^{2\\pi} \\Im\\bigl(f(r e^{-i\\theta})\\bigr)\\,\\Re\\bigl(-i\\,r\\,e^{-i\\theta}\\,f'(r e^{-i\\theta})\\bigr)d\\theta.\n", "4e96fafdc38ec529e79fdcf1d7ddb5ab": "TT^* f_j", "4e970010620f70a10263cf51e1f7251e": "A_1x^2 + A_2y^2 + A_3z^2 + 2B_1xy + 2B_2xz + 2B_3yz = 0.", "4e9716235c351a6b01fd146b79df1dd7": " \\eta = 1 - \\frac{T_{cold}}{T_{Hot}}", "4e97463caf8eb92f59475e8a6c8bda91": "\\text{DNL} = {{V_\\text{out}(i+1) - V_\\text{out}(i)}\\over \\text{ideal LSB step width} }-1 ", "4e977401fca0c9d0169a359f5ffa8fda": " y_{n+1} = y_n + \\tfrac32 h f(t_{n}, y_{n}) - \\tfrac12 h f(t_{n-1}, y_{n-1}). ", "4e97b3b6b45639fe0cce5f1e2196de7b": "2\\pi i.", "4e97d2ba13af051a1d6a41e66ed2ca93": "\\frac {V_\\text{out}} {I_\\text{p}} = -R_\\text{f}", "4e97e4994e791d28c0290afe23408e6a": "\\sum_{i\\in C}y_i\\leq v(C)", "4e97fcbc47308aab10bcad638063ea07": "\\chi_{imp} = \\frac{C}{T + \\theta}", "4e9802705757a4e2cecff7feae1febb3": " e^x = \\cosh x + \\sinh x\\! ", "4e9814500ca6f73f154d05d606b3e197": "3.75<{\\frac{L}{a}}", "4e988beccf7ac1477ed953bda7a157a4": "|+\\rangle = \\tfrac{1}{\\sqrt{2}} \\left(1,1\\right)", "4e98d76906ca5aab8b88ed99b986c57e": "X = \\sec\\,z_\\mathrm t \\, \\left [ 1 - 0.0012 \\,(\\sec^2 z_\\mathrm t - 1) \\right ] \\,,", "4e98ea2e6e124705f5dfc4f182e9aa86": " I_m ( \\sum x - 1 ) + n - \\sum x ", "4e98eb95f4f4a5a7141d0a50fb305720": "\\; e_{\\lambda}", "4e98f89b2220f0877e3d28f592bce27d": "\n\\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\\text{ (rotation), }\\qquad\n\\begin{bmatrix}\n\\cos \\theta & \\sin \\theta \\\\\n\\sin \\theta & -\\cos \\theta \\\\\n\\end{bmatrix}\\text{ (reflection)}\n", "4e995d1761e97a79e065ad943d24b563": "x_0, x_1, x_2, \\dots", "4e998bc25e2d921343572d78708382ba": "\\begin{align}\n{\\phi}P_\\mathrm{n(max)} &= 0.80\\phi[0.85f'_\\mathrm{c}(A_\\mathrm{g} - A_\\mathrm{st}) + A_\\mathrm{st}f_\\mathrm{y}]\\\\\n\\end{align}", "4e998f030c1a9246844a4f9430487732": "\\lfloor (d+k-3)/2\\rfloor", "4e9a08595918cd6123baedddb0a209c2": "n(t) = \\sqrt{\\frac{(x(t)-a)^2+(y(t)-b)^2}{(x(t)-p)^2+(y(t)-q)^2}}", "4e9a1cf46aa47381dff15104fd2523ad": "\n\\mathbf{v} = \\sum_{k} \\langle \\mathbf{v} \\mid \\tilde{\\mathbf{e}}_{k} \\rangle \\mathbf{e}_{k}\n", "4e9a57bdc3d91df036ea29ee23253641": "\\frac{n(n+1)}{2}", "4e9a74770d0e580aae5f7714f77c275a": "{\\Pr}_{\\theta,\\phi}(u(X)<\\theta2", "4e9de9839593f75b55fa9c02c8b8c2c2": "F_V(t,T) = \\mathbb{E}_T[V(T)|\\mathcal{F}(t)].\\,", "4e9e322f6fbdaeba01a52f18acf2a6e8": " \\Delta ^i X_t = (1-L)^i X_t \\ .", "4e9e71ede3b8fbca1d98d9642c36d805": "B^n", "4e9e7f611d448dd9b16d431c6a73ff3f": "k=a_1+a_2+\\cdots+a_n", "4e9ef6232a668e001848299e945a243e": "R_{ab} \\, = R^c{}_{acb}", "4e9f3d58186307a053b2ccd103331206": "m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r} = m \\sum_j \\left[ \\sum_i \\left[ {\\mathrm{d} \\over \\mathrm{d}t} \\left( \\dot{r_i} {\\partial r_i \\over \\partial q_j} \\right) - \\dot{r_i} {\\mathrm{d} \\over \\mathrm{d}t}\\left( {\\partial r_i \\over \\partial q_j} \\right) \\right] \\right] \\delta q_j ", "4e9f6dcce0994447120d48ec079c639a": "T_\\epsilon = T\\ast\\varphi_\\epsilon", "4e9fcbede675261d22c3952db9593ca5": "\\begin{align}\n S_0 &= $40 \\\\\n X &= $40 \\\\\n\\sigma &= 30% \\; p.a. \\\\\n r &= 10% \\; p.a. \\\\\n T &= 6 \\; months = .5 \\; years\\\\\n D &= $0.70 \\\\\n\\end{align}", "4ea0325ab716ecac773b225e23fc4987": "N_t\\times N_s", "4ea0a79e4a51e1b21834984411cfd019": "\\mathbf{R_0}", "4ea0b4a909d12a122554df1dc507e8a8": "x_i\\notin\\operatorname{Conv}(S\\setminus\\{x_i\\})", "4ea0cf20e4edff758212a035dbb2f917": "CAEBD", "4ea0eda87cbd07bbfbc674979011bb65": "\\scriptstyle \\mathbf{Q}=\\mathbf{VSV}^H", "4ea12fd77dd91ec89054bbc8f0ad1190": " H(\\lambda , z) ", "4ea167f5becc529ed63f1107ff07d19c": "88^2", "4ea20257bed69f693f29ee1b1ddba179": "\\chi : D^n \\to S^n", "4ea22c1542e389d18d4edd02331fd512": "\\scriptstyle \\frac{S_{x,s}(s)}{S_x^{-}(s)}e^{\\alpha s}", "4ea254513c43c7690a6f0fe2b7ea75f8": "h_i(t)", "4ea335c1cdea7ecfbedc18e99657b02c": "\\{ z \\in G \\mid \\varphi(z) < x \\}", "4ea33ed463e82df48f6af57a8648c4de": "\\mu_3=8(k+3\\lambda)\\,", "4ea368e3c68a4ca885a49c5406e85738": "\\mathcal{L}=\\,i \\,\\bar{\\psi}_a\\partial\\!\\!\\!/\\psi^a+\\frac{\\lambda}{4N} \\,\\left [\\left(\\bar{\\psi}_a\\psi^b\\right)\\left(\\bar{\\psi}_b\\psi^a\\right)-\\left(\\bar{\\psi}_a\\gamma^5\\psi^b\\right)\\left(\\bar{\\psi}_b\\gamma^5 \\psi^a\\right)\\right]=\\,i\\,\\bar{\\psi}_{La}\\partial\\!\\!\\!/\\psi_L^a+\\,i\\,\\bar{\\psi}_{Ra}\\partial\\!\\!\\!/\\psi_R^a+\\frac{\\lambda}{N} \\,\\left(\\bar{\\psi}_{La} \\psi_R^b\\right)\\left(\\bar{\\psi}_{Rb}\\psi_L^a \\right).", "4ea3694af5b6a62063ed1aaba6cc79bb": " -\\ln(-\\eta)", "4ea420d4cc8c7b28fcca93a6963f0863": "\\mathbf{b}(\\boldsymbol{\\sigma}) = b_1(\\sigma_{-1}) \\times b_2(\\sigma_{-2}) \\times \\cdots \\times b_n(\\sigma_{-n})", "4ea48ace10c0b924aa472a1a71cf1c4e": "\\tau(k)", "4ea49b1dc51f99c10998fdbf8db40f52": "\n \\theta = \\arctan{ \\left( \\frac{dy_c}{dx} \\right)},\n", "4ea4bde987814d5bde75f1c56a0bd636": "I(X_1;\\cdots;X_{n+1}) = I(X_1;\\cdots;X_n) - I(X_1;\\cdots;X_n|X_{n+1}),", "4ea4c07cd7a71c65cdf706da9648ffde": "\\mathcal{D}(S)\\times\\mathcal{D}(S)\\rightarrow\\mathbb{Z}:(X,Y)\\mapsto X\\cdot Y", "4ea4c5d39718ab00787b6fccb9b0f9b8": "C_t = \\frac{1}{a} C_w \\phi^m S_w^n", "4ea4ea3bea29c8ccb8c32750f5ae4a86": " \\mu_1, \\mu_2, \\mu_3", "4ea4f23243c3d61d46fd3551f9968c9c": "\\frac{A}{x - 1}", "4ea4f60f9d5bbb59bea48dcf909385a5": "\\psi_k(x) = \\sqrt{2/\\pi}\\sin(k x)", "4ea50507a173e09c8b486b1ca11ce87e": "Z_{max} = R_e(1+\\frac{Q_{ms}}{Q_{es}})", "4ea550ee318387906cd61d83f0f2ed17": "\\psi=\\begin{pmatrix} u^1 \\\\ u^2 \\end{pmatrix}", "4ea56b8e06f8b61843e70592f1ea4b6b": "\\Pi(t) = \\lim_{n\\rightarrow \\infty, n\\in \\mathbb(Z)} \\frac{1}{(2t)^{2n}+1}", "4ea580884bf28ab1a9915a03b8627a21": "\\nu Z.(\\bigvee_{a\\in A}\\langle a\\rangle\\top\\wedge \\bigwedge_{a\\in A}[a]Z)", "4ea59597e98a5c91468787a30a8fc039": "\\displaystyle \\cos ( a x^2 ) ", "4ea5fb75d63146409e3b80b75e067d97": "\\hat{f}(a)=2^{n/2}(-1)^{g(a)}", "4ea62545d472dbc3682e33bb44ac5227": "\\mathbf{R} \\times 1", "4ea6acf3c1a7e17311c317e0d66050ee": "x^{\\iota}", "4ea6d9b3b73c782cf18511360f10e8d3": "T^{ab} = 0", "4ea6e55963bd031552469dbed176aaf9": "dF\\left(T,V,n_{i}\\right) = -SdT - pdV + \\sum_{i} \\mu_{i} dN_{i}", "4ea6ee004c1c3cf19267582e8285adaf": "\\operatorname{red}_1(f,G)=\\operatorname{red}_1(f,g)", "4ea73f106f3a8291576b1fd40ced7f8d": "s = j \\omega = \\left(\\sqrt{-1}\\right) 2 \\pi f\\,", "4ea764f40219110c6e7e10e0048e6d48": "\n\\frac{\\partial}{\\partial A} \\ln p(\\mathbf{x}; A)\n=\n\\frac{1}{\\sigma^2} \\left[ \\sum_{n=0}^{N-1}x[n] - N A \\right]\n", "4ea773c5b6f476d8fb01cab479fbe384": "H = -\\mu\\boldsymbol{\\sigma}\\cdot\\mathbf{B}", "4ea7dadd0cb03003eba8d1e806ea3300": "deg(P(X)) \\le {k - 1}", "4ea8805188e73c02ac0aedfe1b750890": " \nf := F \\sigma_2^2 + G \\sigma_1^2 + R_0 G(\\sigma_1-\\sigma_2)^2 - 1 = 0 \\,\n ", "4ea89f3eb2f45080f4dbb4d44167b8fb": "q(x) = \\mu \\omega^2 w(x)\\,", "4ea8a018c3ee8274d557927df8a37b91": "\\mu_{p^a}", "4ea8cc7867ea671bdb1d980a32af8bf9": " \\theta = \\pi ", "4ea8d1f45aad5f6d2e857b9367851b98": "\\mathrm{CRS}\\!\\!\\!\\Vert", "4ea930ea95f4d8a3b470eb3e1cac332c": "~\\gamma > 0~", "4ea936c932b428c960b5a8d4e5a74524": "\\left|H\\right|\\le k.", "4ea9555ea17e7102328c42f54700c00c": "0.\\dot{3}", "4ea9788a9b685b4deae62a00402d9efe": " P( X \\ge k ) \\le \\frac{ 2 }{ 9k^2 } \\quad \\text{if} \\quad \\frac{ 2 }{ \\sqrt{ 3 } } \\le k \\le \\frac{ 2N }{ 3 }. ", "4ea97cb4d4b734cc1656051d21c9191f": "j \\frac{\\sqrt 2}{2}", "4ea9bf3642f461d4138d08855bb56695": " \\{0\\}, \\{0, 1\\}, \\{0, 1, 2\\}, \\ldots ", "4ea9d0c8620a2eb8d4248c271a8a69f3": "\\dot{H}_G", "4ea9d55249a256e3abf29be85c90817e": "W = {Q_L}({T_a/T_L-1})+T_a S_i.", "4ea9d557380601f31b1a31181c6c8eee": "= \\left[ \\frac{\\partial \\phi^{\\alpha}}{\\partial x^{i}} + \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}}u_{i}^{k} - u_{l}^{\\alpha}\\left(\\frac{\\partial \\rho^{l}}{\\partial x^{i}} + \\frac{\\partial \\rho^{l}}{\\partial u^{k}}u_{i}^{k}\\right)- \\chi^{\\alpha}_{i}\\right]\\, dx^{i} + \\left[ \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}_{i}} - u_{l}^{\\alpha}\\frac{\\partial \\rho^{l}}{\\partial u^{k}_{i}}\\right]\\, du^{k}_{i} + \\,", "4ea9f15d5b0b0982f8ee8852d61679f5": "W_{n}", "4eaa4a6b58335a5141615da0ffe00efe": " \\lambda_2 \\; ", "4eaaf4ebdcffb2c760e73b6f7925a463": "\\gamma \\ll 1", "4eab666f60981f9cd214c5081f33b866": "f_1 \\Rightarrow f", "4eabd53f4608d03d9f2d31ca66748755": "S_{RBB}(n+1)= 2 S_{RBB}(n) + S_{RRB}(n) + 1 = 3 S_{RBB}(n) + 1 \\, ", "4eacabc4122cc8a49a2fa5ed6533d4f2": "P(v)", "4ead33ffcfcb713792a55dfd1ca6f7ee": "\\binom{p}{2}=\\binom{b}{2}\\ell.", "4ead3e52e99e2e17268e993e99ad34da": "\\Phi(G)=\\left\\langle a^p\\right\\rangle", "4ead4cce4c9ac74c48297bcd1202d7c7": "\\ {\\partial \\rho \\over \\partial t} + \\nabla \\cdot (\\rho \\mathbf{u}) = 0 ", "4ead4e86be84c3806bb5a12486b3a619": "V^K", "4ead658bb8c609d6a94342428d358b28": "\\phi = \\arg z = \\,", "4ead8fe28a0667cfc831814bebc2eabd": "g\\phi_0", "4eae2b3b006564dddcea292c2c7ce140": "FRS^{(m)}_q[n,k]", "4eae8dc134a74664d10ebfc4033c3633": "0 < S < 3 \\,\\!", "4eaebc8f33e4c2efff0f2b7e5065b92e": "\\tau_{E}\\colon TE\\to E\\,", "4eaeef76bb2b600145030f2860bc73d8": "\\omega=f(x,y,z)", "4eaf2cdc2dae3a7db5d257b526923b39": "2n + 3 = 2\\cdot(n + 3) - 3", "4eaf36868a593bbcb3a113331031d034": "1+\\sqrt{2}\\,", "4eaf7a167fc14058ea350e834fb8921b": " \\mathcal{P}_2'(\\omega)=2a_{20}\\omega+a_{11}\\ne 0,", "4eaf8fa3bfd6e064381f2433b8d065fc": "U_{[ij]k\\dots}=\\frac{1}{2}(U_{ijk\\dots}-U_{jik\\dots})", "4eafcbefffc5b0a5ce045ce6e7f16332": " \\int_{Y^{-1}(B)} X(\\omega) \\ d \\operatorname{P}(\\omega) = \\int_{B} \\operatorname{E}(X \\mid Y)(u) \\ d \\operatorname{Q} (u),", "4eb03d69608b58bf4bbdbd50e8ebdc4d": "i\\in \\{1,\\dots,n\\}\\, .", "4eb07ef66e6d60c598f0e8d5c39d5d78": "H(Z) = 1", "4eb08e749c8c447c2e5bf385fa9f07e5": "\\mathcal{C}_{m}, \\mathcal{S}_{m}", "4eb09746dc7814ad0ff472e5d3cee933": "\nZ = \\frac{p}{U}\n", "4eb0ea669146654f8dbc80cc50661110": "\\Psi(r)=\\exp(i\\mathbf{k\\cdot r})", "4eb104027f77206ebd4b1a866e1ba2e6": "U = {N^{\\prime}\\varepsilon\\over2} + {N^{\\prime}\\varepsilon\\over e^{\\varepsilon/kT}-1}.", "4eb1be669ac23806b4742a2d709089bf": "\\sum_q \\frac{A_q}{\\omega_2-\\omega_q+i\\Gamma_q}", "4eb1be8867a9cf196ba0d7af00c94845": "\\sqrt{\\frac{\\pi}{2}}", "4eb274a6654cb56297fd5e42fea60c76": "r \\circ \\iota = id_A,", "4eb27fbdfe0c5454601edcb51f7a5c64": "\\hat{x}_{\\mathrm{fl}}(t_j)", "4eb31ce1b945e5be6d178e88b11c7404": "{\\mathbf P}=\\varepsilon_0\\chi_{\\text{e}}{\\mathbf E},", "4eb35445c8c9ae0f8a18406aeb237659": "\\Lambda_{C}=\\frac{-\\rho{g}{V^{1/3}}((\\frac{1-cos(\\theta_{a})}{sin(\\theta_{a})})(3+(\\frac{1-cos(\\theta_{a})}{sin(\\theta_{a})})^2))^{2/3}}{(36\\pi)^{1/3}\\gamma cos(\\theta_{a,0}+w-90)}", "4eb362397757e3cc303832999709d5c4": "h[x]=\\alpha x", "4eb3659657ca1b5991415ce22182c01e": "e^{i(\\theta + \\Delta\\theta)} = e^{i\\theta} \\times e^{i\\Delta\\theta}", "4eb36618d813d777ca2d83fe9ac467fd": "\\left(a, q, u\\right)\\succsim \\left(b, p, u\\right)", "4eb38dbd01b2a2165eac74fd0d70d696": "(\\% \\mbox{ change in }Q)/(\\%\\mbox{ change in }P)", "4eb38f447c3f58e3aeb01b1d71142c8d": "O(n^{3.5} L)", "4eb3931b3b6985d386471e0412b563d5": "\\phi(\\xi)", "4eb3994d2134ab3e1a72ec36cb90e16a": "1/T\\ ", "4eb3a04a634df98076838b482be74687": " r_1=u+2\\sqrt{\\rho}, ", "4eb3a4ed0ff56c84b7cd967184928fd8": "\\langle \\psi_1| \\psi_2 \\rangle = \\int \\psi_1(x)^\\ast \\psi_2(x) \\, \\mathrm{d}x", "4eb3aef86c79733112578d9747391a29": "\nF(k_x,k_y)=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty\nf(x,y)\\,e^{-2\\pi i(xk_x+yk_y)}\\,dxdy.\n", "4eb3bf39f5c6e5b1349359759d814feb": " \\mathbf{p} = \\hbar \\mathbf{k} \\ . ", "4eb3ca4acbfaa11d3e840f27ab5907d5": "\\scriptstyle \\Gamma \\,\\times\\, \\mathbb{R}^1", "4eb3cbdf14010fe6a649682db50685e2": "[-\\infty,+\\infty[=\\mathbb{R}\\cup\\{-\\infty\\}", "4eb438c06a84115d7ee37a2339ff3b3a": "\n\\begin{cases}\n\\frac{4}{t\\pi} &:\\ 0 \\le x \\le \\frac{t}{2}, \\ 0 \\le \\theta \\le \\frac{\\pi}{2}\\\\\n0 &: \\text{elsewhere.}\n\\end{cases}\n", "4eb48cf72677e4b27d5eae39c65bcb29": "{\\rm Riesz}(x) = x \\left(\\frac{6}{\\pi^2} + \\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^2}\\left(\\exp\\left(-\\frac{x}{n^2}\\right) - 1\\right)\\right)", "4eb49c720af5157a68475d5eb2eec25c": " q_2 ", "4eb4c81b4a8c1d0261c2bb3f79896ffe": " \\Pr(X>40 \\mid X > 30) = \\frac{ \\Pr(X>40 , X > 30) }{ \\Pr(X > 30) }= \\frac{ \\Pr(X>40) }{ \\Pr(X > 30) } ", "4eb4cff14d148a3cdb3af39d9ef0ca49": "\\boldsymbol D(\\boldsymbol r,\\ t) = \\varepsilon_0 \\boldsymbol E(\\boldsymbol r,\\ t)\\, ", "4eb4e7d7bfdcd5c00e7121d48bc17331": "\\{ |\\omega\\rang, |s\\rang \\}", "4eb4f72a6110c53aea48680eaf2d3e07": "R_N(x)=O(x^{-2N+1} e^{-x^2})", "4eb501e1c0ef4146fe23e89149304468": "-2.7917", "4eb5204402ae237595862b7efc1003df": "P(\\theta) = P(\\theta_0) + \\Delta\\theta^jP_j(\\theta_0) + \\cdots", "4eb56c0fb4895e0a890c626e551e9855": "\\mathbf{F}^\\beta\\ \\mathbf{C}^\\beta\\ = \\mathbf{S} \\mathbf{C}^\\beta\\ \\mathbf{\\epsilon}^\\beta\\ ", "4eb586b87ab26d8ed4f04fe6d2c4fc35": "\\widehat{\\boldsymbol \\theta}_{LS} = {\\mathbf y}", "4eb63772a49d95a92bb86ef1f9011c88": "\n\\begin{matrix}\nQ = D - C A^{-1} B\n\\end{matrix}\n", "4eb63b61bd5981ecc25065fa914c7d82": "kT \\ll \\hbar \\omega_c", "4eb6948b93f7318fe6abbdbefbdaa917": "\\textstyle |B|", "4eb6c9f8781d48c0961f567d73368929": "\\tau_{net}=\\left(T_1 - T_2 \\right)r - \\tau_{friction} = I \\alpha ", "4eb6f6dc89b4d2bd7289d3d7171397c1": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi(x,t) = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2}\\Psi(x,t) + V(x,t)\\Psi(x,t) ", "4eb710e9365179d1d4ca4048a9dfed47": "f \\in F_n", "4eb715bcf342d7ea18fdbfe3e2912417": "\\arccos x = \\arcsin \\sqrt{1-x^2},\\text{ if }0 \\leq x \\leq 1 ", "4eb74fcdea1a20b2951477220ff7c334": "L=\\frac {AS^2} {400 + 3.5S}", "4eb75bf7d6cf3c12bb6510af19696156": " v \\leftarrow v -1", "4eb77caaa7e78a4114517665bca53e34": "\\scriptstyle{z = A / 2\\sigma^2}", "4eb7c82c8488189e2e39c460767f4e2f": "U_y = ((A_x^2 + A_y^2)(C_x - B_x) + (B_x^2 + B_y^2)(A_x - C_x) + (C_x^2 + C_y^2)(B_x - A_x)) / D", "4eb81e7c30ad06560a2a4df8cb29391e": "(m_1n_2-m_2n_1,\\,l_2n_1-l_1n_2,\\,l_1m_2-l_2m_1).\\,", "4eb837f74208c9066c08397313f50afa": " v^*(S) = v(N) - v( N \\setminus S ), \\forall~ S \\subseteq N.\\, ", "4eb84a3951d55c10543e081ab1340dfd": "(\\lambda+\\mu)(1+cM^{-1/2})", "4eb89522f88d070b514eb1863d2a4d16": "\\kappa\\theta", "4eb8d7c6f76c8b3909d4b428bde418cc": "\\scriptstyle{\\nabla_{\\dot{\\gamma}}=0}", "4eb90c100ef9d0d7708497368ed0f57a": "C_b", "4eb97cd2be20dd806fd6803f851a7f22": "\\Phi_m =A_5e^{im\\phi}+A_6e^{-im\\phi}\\,", "4eb9819821a89a9923d752cb12bfaff8": "(X,Y)\\,", "4eb9b69c9021201159ee3c16b635b4e1": " \\langle \\psi \\vert A \\vert \\psi \\rangle. ", "4eb9da2b3683eb13b85cc5a83850ea2f": "\\quad\\frac{\\chi_p \\left( E \\right)}{4\\pi}\\int_0^{\\infty} dE^{\\prime}\\nu_p \\left( E^{\\prime} \\right) \\Sigma_f \\left(\\mathbf{r}, E^{\\prime}, t \\right) \\phi \\left( \\mathbf{r}, E^{\\prime}, t \\right) + \\sum_{i=1}^N \\frac{\\chi_{di}\\left( E \\right)}{4\\pi} \\lambda_i C_i \\left( \\mathbf{r}, t \\right)+\\quad", "4eb9e47ad2861380d5cc0f7d759980d3": "df(p) \\colon T_pM \\to {\\mathbf R}.", "4eb9f4e751855aaac8adf79b27b0d402": "\na_{1,2}=\\frac{3\\,\\pm\\,\\sqrt{1+8\\alpha}}{2}\\ ,\\ \\ \\ \\ \\ b_{1,2}=\\frac{3\\,\\mp\\,\\sqrt{1+8\\alpha}}{2}\\ ,\\ \\ \\ \\ \\ c=2\\ .\n", "4eba421042d890ca32d0ad4d10c8cbab": "(R^2/r_i,\\theta_i,\\phi_i)\\,", "4eba65767b707ff137f6e131c0151c25": "\\ -\\frac{d[A]}{dt} = k[A][B]", "4ebada6a2af2bcba53ded1d7b414f081": "FP", "4ebc092ed9177d47b05131df22e7ab83": " n \\times L", "4ebc0b51c90b8daa2df8d53c975306c9": "E=B+S-C+R", "4ebc11e0aa3d36b4f91fab45433aa0ef": "m_1 \\bold{a}_1 = - m_2 \\bold{a}_2. \\!\\,", "4ebc3a776914901119ad7d28c5039994": "\\|\\ \\|_{HS}", "4ebc9ce3f44f9f80537831872191b08e": "m_\\mathrm{3D}^*", "4ebd652b26df2d9dc945f1178bd039fa": "\\frac{\\Gamma(n+k)\\Gamma(\\alpha+n)\\Gamma(\\beta+k)}{\\Gamma(n)\\Gamma(\\alpha)\\Gamma(\\beta)}", "4ebd994d087d5ad02b0b2a1d8b43d9e1": "\\scriptstyle S_{eff_{\\ast}}= S_{eff_{\\odot}} + aT_{\\ast} + bT_{\\ast}^2 + cT_{\\ast}^3 + dT_{\\ast}^4", "4ebde23b4f9e1d45a4df7d0fdf7d7132": "\\frac{\\partial^2}{\\partial t \\partial z} \\Phi' \\, + \\, N^2 w' = \\frac{\\kappa J'}{H}", "4ebe3e75d80bf11cd007361f2119c110": "\\left\\vert \\xi \\right\\vert_{B,z}:=\\sqrt{\\sum_{i,j=1}^n g_{ij}(z) \\xi_i \\bar{\\xi}_j }.", "4ebe638878a004174a74c107a87280cf": "\\mathbf{M} = \\chi_m\\mathbf{H}", "4ebe764210c37f981ad5acd8612905b8": " \\rho(\\theta|y) \\propto \\rho(y|\\theta) \\rho(\\theta|\\alpha, \\beta) ,", "4ebf0438388bf9af8eee5d68b06ba6b0": "R=\\eta\\frac{q}{hf}\\approx\\eta\\frac{\\lambda_{(\\mu m)}}{1.23985(\\mu m\\times W/A)}", "4ebf1ade6d20d9e0212a5c76c9edeb32": "\\Sigma^{\\infty -2} \\mathbb{C}\\mathbf{P}^\\infty", "4ebf494435b0becc21780b1443625384": "\nu'(x) + 2u(x) + 5\\int_{0}^{x}u(t)\\,dt = \n\\left\\{ \\begin{array}{ll}\n 1, \\qquad x \\geq 0\\\\\n 0, \\qquad x < 0 \\end{array} \n\\right. \\qquad \\text{with} \\qquad u(0)=0.\n", "4ebfa12ea954f0eb5774699eee1376a3": "N_n=n_e n_p \\beta_{n}(T_e)=n_e^2 \\beta_{n}(T_e)", "4ebfb5604c12e0956ef8d085cb4a3c06": "\\chi = \\frac{m_\\ell}{m_g} \\sqrt{\\frac{\\rho_g}{\\rho_\\ell}}", "4ec022a58d9766f1fee27e83c249cadc": "Y\\to \\Sigma\\to X", "4ec0c95fd77540a61547a16e2f9e5889": "g(x) = -f(x)", "4ec1390500114090d3c5dbacdca59a69": "Z_2^n", "4ec14e01f2f0de69ec5865445206b290": " r=f(\\theta) ", "4ec19c359a3520555084d5bcee493d9b": "d \\,=\\, \\frac{M}{5 \\cdot V_S}", "4ec19c39effc7975e4f81498426e2967": "\\mathbb{Q}[y]", "4ec2e233aadd109bda3e8e588f11b020": "p(\\phi)", "4ec323d8275e62226c485796a8b18459": "\\tan\\left(\\frac{\\pi}{2} - A\\right) = \\cot(A)", "4ec32a23dec814ad73bcd9d96541c383": "\\sum_{n=2}^\\infty \\frac{\\sin(n x)}{\\ln n}", "4ec33da3ede3351844ed9bee4a21b549": "\\sqrt{\\frac{5}{126}}\\!\\,", "4ec3483cbc0372be6b7a873afe4c59f3": "\\bold{F}_{12} = m_1 \\bold{a}_1. \\!\\,", "4ec36417f47615de995f86473735f0c8": "\\Xi(x).", "4ec3674bd3037441218a2972f27613a4": "(\\mathbf{L} = \\mathbf{r} \\times m\\mathbf{v})", "4ec37f131a1fdc9e1d49d9213b468f97": " M \\to \\infty ", "4ec39dfed0f0db6e8f0fc70177a902d2": " x\\in F", "4ec41ddb3e7b015d094bffee466c38ae": "{n \\choose 1}_q=1+q+q^2+\\cdots+q^{n-1}", "4ec440e907e4ff9431cf8bea23dc6caa": "(a;q)_{-n} = \\frac{(-q/a)^n q^{n(n-1)/2}} {(q/a;q)_n}.", "4ec4453738fd342c9b207ef2c2d130d9": " \\and D[q] = [F_7, S_7, A_7]::[F_6, S_6, A_6]::K_5 ", "4ec46919a1bf39ce295c87e8d28e9eff": " M \\hookrightarrow W \\quad\\mbox{and}\\quad N \\hookrightarrow W", "4ec477c2c1b7c4e2be7d4ccef0ac90b8": "\\mathbf{A \\cdot B} = \\begin{pmatrix} A_0 & A_1 & A_2 & A_3 \\end{pmatrix} \\begin{pmatrix} B^0 \\\\ B^1 \\\\ B^2 \\\\ B^3 \\end{pmatrix} = \\begin{pmatrix} B_0 & B_1 & B_2 & B_3 \\end{pmatrix} \\begin{pmatrix} A^0 \\\\ A^1 \\\\ A^2 \\\\ A^3 \\end{pmatrix} ", "4ec4be6a61399bfc4e1fa9ac64e483ca": "f(x) = \\begin{cases}\n 3\\delta-6\\delta^2 & : 0\\leq\\delta\\leq 5/16 \\\\\n 45/128 & : 5/16\\leq \\delta\\leq45/128 \\\\\n \\delta & :45/128 \\leq \\delta\\leq 1/2\n\\end{cases}\n", "4ec5097604d004c50e087e82fb1b4bd3": "a \\approx S/2.414", "4ec53474ef1e57c27344f7ff2666cfc6": "\\langle R(u,v)w,z\\rangle=\\frac 16 \\left.\\frac{\\partial^2}{\\partial s\\partial t}\n\\left(K(u+sz,v+tw)-K(u+sw,v+tz)\\right)\\right|_{(s,t)=(0,0)}", "4ec5b859ef7936aaae387b06835f744b": "\\mathbf{g}_{M} = \\mathbf{D}\\times\\mathbf{B}", "4ec5caef6904a573a37d1dc1a02003eb": "\\scriptstyle \\Bbb{Z}/p\\Bbb{Z} \\,\\oplus\\, \\Bbb{Z}/q\\Bbb{Z}", "4ec5f77804e52aef83d9534947f2a08f": "\\begin{matrix} {52 \\choose 5} \\end{matrix}", "4ec617e526ec35b2a7072fec2865bde9": " \\left( J \\otimes J \\right)(\\eta) = \\eta ", "4ec62f4ef3e8dbfca0e96f7f655be72b": "u \\to v", "4ec764be64c9db0d9663f90990fa600f": "\\mathbf{h}\\,\\!", "4ec773410df80cab9da32f979fe9775e": "\\mathbf{\\Sigma}^0_1", "4ec78b9c5dc2847bc7a2e707c5136cbb": "\\mathrm{A} = \\frac{\\rho_1 - \\rho_2} {\\rho_1 + \\rho_2} ", "4ec796f42b04641df257c693e1da586c": "d_{ACDA(s)}", "4ec7c40d360521deb20df0123a7f9486": "\\mathbb{H} \\rightarrow \\mathbb{C}", "4ec8075f9fb8403f51d062941e835208": " x = x_0 ", "4ec81807770b335cf05355d10eb3d78d": "\\int_{-\\infty}^{\\infty} f(x) \\overline{g(x)} \\,{\\rm d}x = \\int_{-\\infty}^\\infty \\hat{f}(\\xi) \\overline{\\hat{g}(\\xi)} \\,d\\xi,", "4ec830be14e5b644646b4f0c10b4e8b3": "O(\\ell)", "4ec87031bcf1dec3352ce7cf94d041be": "(22)\\quad\\quad \\frac{p_{2}}{p_{1}} = \\frac{\\frac{\\rho_2}{\\rho_1} (\\gamma+1) - (\\gamma-1)} {(\\gamma+1) - \\frac{\\rho_2}{\\rho_1} (\\gamma-1)}.", "4ec8b7576b950755e64d86bb608162eb": "K_{Ic}", "4ec91d3ed8d948c334eb51e929209402": "P_{1} \\land \\dots \\land P_{m}", "4ec93aa089c700beceff1ee60c870b5f": "\\mathrm{net} = \\sum_{i=1}^{n}w_ix_i", "4ec9750242fa5da0939f09497ad8674e": "\\frac{d}{dx} x_+^\\alpha = \\alpha x_+^{\\alpha-1}", "4ec975c46fab950da027b4d31908c145": "TV \\to saw ~|~ met ~|~ ...", "4ec97bec99b0a2212b2f46fe33974f3d": "(N,T,F,P,S)", "4ec9b3ac08182c55e40d9c2ff0b898b8": "\\partial_zX^\\mu-i\\overline{\\theta_L}\\Gamma^\\mu\\partial_z\\theta_L - i \\overline{\\theta_R} \\Gamma^\\mu\\partial_z\\theta_R", "4eca1f27137609136f5ea96540020356": "\\mathbf{s}", "4eca4b827e3040dca53f6e750d5500cc": "\\frac{1}{2}\\left(\\alpha+\\beta+\\gamma \\right)_{\\tau\\tau}=\\alpha_\\tau \\beta_\\tau +\\alpha_\\tau \\gamma_\\tau+\\beta_\\tau \\gamma_\\tau.", "4ecac3662d225f909191fa312b6b598e": "\\dim V_k(\\mathbb H^n) = 4nk - k(2k-1).", "4ecae54427925b40a7f9cca4aa816297": "\\frac{\\partial ^2C_i (q_i)}{\\partial q_i \\cdot \\partial q_j}=0,\\forall j", "4ecafdd68ab5ed1f7707ce8d5c949e4a": "\n\\begin{array}{lcl}\n\\boldsymbol\\alpha &\\sim& \\text{A Dirichlet hyperprior, either a constant or a random variable} \\\\\n\\boldsymbol\\beta &\\sim& \\text{A Dirichlet hyperprior, either a constant or a random variable} \\\\\n\\boldsymbol\\theta_{d=1 \\dots M} &\\sim& \\operatorname{Dirichlet}_K(\\boldsymbol\\alpha) \\\\\n\\boldsymbol\\phi_{k=1 \\dots K} &\\sim& \\operatorname{Dirichlet}_V(\\boldsymbol\\beta) \\\\\nz_{d=1 \\dots M,n=1 \\dots N_d} &\\sim& \\operatorname{Categorical}_K(\\boldsymbol\\theta_d) \\\\\nw_{d=1 \\dots M,n=1 \\dots N_d} &\\sim& \\operatorname{Categorical}_V(\\boldsymbol\\phi_{z_{dn}}) \\\\\n\\end{array}\n", "4ecb21a91c905de38a355175a96af7dc": " T=\\frac {1}{2}(\\frac{1}{\\rho c_0} \\frac {d \\rho c_0}{dz}) =\\gamma \\quad (2.8.c)", "4ecb4771427d34330a349ac04a883c5e": "\\mu=p+q", "4ecb7343445ee0635daaf88d730cf582": "[\\xi_1,\\xi_2]= \\delta^*(\\xi_1 \\otimes \\xi_2)", "4ecb760d5d5671eb67fa8e7b3e31b096": "(-\\Delta)^m u(x) = f(x)", "4ecbb85b37a54a05eb5817315f4fe750": "\\sigma_{11},\\sigma_{22},\\sigma_{33}", "4ecbfd9a597c033893c056e543d0f3c5": "3n-6", "4ecbfef4c80cb80971b42c1b4631db1a": "t \\ge 0 \\ ", "4ecc02e42529e37d01a198358f2a2131": "\\sup", "4ecc8055dbfb4f280d42e2d73ec48041": "\np = \\frac{x^3 - y^3}{x - y} \\qquad \\left(x = y + 1\\right).\n", "4ecca89d7fec01ff0f1850370595c561": "f=u_1P(D)y_1+u_2P(D)y_2+\\cdots+u_nP(D)y_n+u'_1y^{(n-1)}_1+u'_2y^{(n-1)}_2+\\cdots+u'_ny^{(n-1)}_n.", "4eccd78ca181913a9892ab4c0ffbb16d": "\\operatorname{recc}(A) = \\{0\\}", "4ecce110bf03f51e7e5ac03a4caab9bd": "\\frac{\\partial u}{\\partial t} + \\frac{1}{2}\\frac{\\partial}{\\partial x}\\big(u^2\\big) = 0.", "4ecd31f80bee203a5eb328aaef737cfd": "\\mid e \\rangle", "4ecd58dd547309754b7ff90d6c7f3cdc": "y_0=A+B", "4ecd7c735526fcb1cadfbec4964ad6f0": "\\epsilon=1-\\frac{\\omega_p^2}{\\omega^2+i\\gamma\\omega}", "4ecdcf36d50261c4cbbfbeca58cbc0e7": "\n\\lim_{s \\rightarrow k+1} \\left[ {\\zeta(s-k) \\over k!} \\,\\mu^k + \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} \\right] = {\\mu^k \\over k!} \\left[ \\,\\sum_{h=1}^k {1 \\over h} - \\ln(-\\mu) \\right] ,\n", "4ece6f59e09db9841328327c5a09200e": "N_i ", "4ece8723087ecc89d2062495d17b24c4": "\n\\sum_A M_A \\mathbf{R}_A = \\mathbf{0}\n", "4ece883ad24ffca79c13ba078ecb6c5e": " \\left\\{\\boldsymbol{\\beta}^\\delta, \\sigma_\\delta, \\omega_k^\\delta, k=1,\\ldots,d\\right\\} ", "4eceb3903f1f5fea127f5f8295d2f03a": "u\\big|_{t=0}=f(x),", "4ecf6e2d8fb85ebc5034267cc28a4421": "R_{abcd}= \\, S_{abcd}+E_{abcd}+C_{abcd}.", "4ecfa0aaf54a5c42f44736f700aef792": "\\,{}^{x}a \\approx x+1", "4ecfee55c33038eec746c6a9cd126cd0": "\\boldsymbol{\\varepsilon}_{\\mathrm{vp}}", "4ecff42ec760ee59d1a26dd8551b0e48": " \\Pi = 81,000 - 60,750 ", "4ecffd489c1935f93b287646905a861c": "a_k \\approx \\frac{2}{N} \\sum_{n=0}^{N-1} f(\\cos[(n+0.5)\\pi/N]) \\cos[(n+0.5) k \\pi/N] ", "4ed0295c0318dca9a47851c1deeb0eae": "\\Delta_k =\\frac{y_{k+1}-y_k}{x_{k+1}-x_k}", "4ed030de4147265313a70dfbfd0ae640": "\n\\mathbf{g} = \\nabla W = \\mathrm{grad}\\ W = \\frac{\\partial W}{\\partial X}\\mathbf{i}\n+\\frac{\\partial W}{\\partial Y}\\mathbf{j}+\\frac{\\partial W}{\\partial Z}\\mathbf{k}\n", "4ed04411683b583b518a9867fb73d42d": "O_{7}", "4ed0a28a30c8d28640f8b0e8644262b8": " U(h) = \\int_h^{\\infty} F(h') \\, dh'.", "4ed0e1e6979f48d422735989af99081a": "X = X_1 \\amalg X_2 \\amalg \\cdots \\amalg X_n", "4ed0fc4d6d810e26402c8ee1b5e17974": "0 \\equiv t_l \\equiv t_k^2 + D u_k^2", "4ed106146b3379b54984fa5090aae213": "\n\\frac{d^{2}u}{d\\theta^{2}} + u = -\\frac{m}{L^{2}} \\frac{d}{du} V(1/u) = -\\frac{km}{L^{2}}\n", "4ed15d8aca846b6abb346f426b55e215": "Q = G_{1}^T * G_{2}^T\n", "4ed17a610f67dac0450044accd0e26c1": "\\pi = \\lim_{k\\to\\infty} 2^k \\underbrace{\\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\cdots}}}}}}_{k\\ \\mathrm{square}\\ \\mathrm{roots}}", "4ed1f7e1706c2b7609117cd915d4c9e6": " u_1 = y_0 \\sum_{k=0}^\\infty \\left (\\lambda-\\lambda_0^* \\right )^k X^{(2k+1)}.", "4ed1f7ed82d1a75d0382b8403d02fbc5": "'T'", "4ed263faf5d263c20bae97c39893db80": "P_n(k,\\rho)=I_n(|k|\\rho)\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,K_n(|k|\\rho)\\,", "4ed2a32be31b6e3a35f6d87d1e340bfc": "D^\\prime", "4ed2da42683cba6696afacb0b9b14906": "\\frac{\\part \\phi(\\mathbf{x}, u)}{\\part u} = 0.", "4ed37305beb0d7728954b1f191c10014": "x_{\\mathrm{min}}", "4ed38b174b3244a105bdaf70ce1e6c41": "\\scriptstyle\\varphi(\\mathbb{E}\\{ X \\} )", "4ed39f693fee3c0d5a3737aa62375531": "(Ba_{0.9}Nd_{0.1}CuO_{2+x})_m/(CaCuO_2)_n", "4ed3a8e00fc386eaf5297cbf64acc57d": "B_k(t)={3 \\choose k} \\cdot t^k \\cdot (1-t)^{3-k}", "4ed421e09e7c3954c2e6dfcd90d5d37b": "\n\\sigma_1 = \\sigma_x =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n", "4ed4462f9be4faa304b92d8af788984f": "\nF(p) = \n\\left\\{\n\\begin{array}{ll}\n-M p_c \\sqrt{(\\phi - \\phi^m)[2(1 - \\alpha)\\phi + \\alpha]}, & \\phi \\in [0,1], \\\\\n+\\infty, & \\phi \\notin [0,1],\n\\end{array}\n\\right.\n", "4ed491a532eff520d5eeeb24c0b6bff4": "\\sigma_x, \\sigma_y, \\sigma_z", "4ed491a6238fafc7798cdb433eb18f1b": "f \\in H_k", "4ed4e1e348026f00a8c9e9abc338f841": " Q\\ {=}\\ H^2\\ ", "4ed50da12e157a725e4da8ae56ae3a29": "n_{equilibrium}=\\xi \\nu_i+n_{initial}", "4ed51122c4067564a3d6bffb0af05885": "R_n = \\frac{e^2}{4 k T_0\\,\\Delta f}.", "4ed5f70cd61014f6617882e59d8b4bb3": "\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\left(\\frac{\\partial^2 u}{\\partial r^2}+\\frac {1}{r}\\frac{\\partial u}{\\partial r}+\\frac{1}{r^2}\\frac{\\partial^2 u}{\\partial \\theta^2}\\right) \\text{ for } 0 \\le r < a, 0 \\le \\theta \\le 2\\pi\\,", "4ed651efeb1acaf09504cb3c0feac504": "T_r = \\frac{T}{T_c}", "4ed6a54a86c76f144dd38a106df99600": "|{\\psi_0}\\rangle=Q|{\\psi_{gr}}\\rangle", "4ed6b9606d53cb4c372e0eea1c1d20b4": " \\psi= 1+\\frac {v_r}{g}\\sqrt{\\frac {k}{m}} ", "4ed76ada43760143c4b6a697345ffa85": " P(IV)(\\sigma, \\sigma, 1, \\alpha) = P(I)(\\sigma, \\alpha),", "4ed7b34d4bceeb0d2cdd838eb1cb82e2": "c_i(x) = 0", "4ed7b84737265a409a423d532febdf2c": "\\,\\! z=z^+-z^-", "4ed7fa875053bed5e41e7c075c129710": "\\xi^a\\,", "4ed7fe159e30a6b3ddc15a7b15a2bc6d": "V(x) = \\sum_K \\tilde{V}(K)\\cdot e^{i\\cdot K \\cdot x}", "4ed80959b70c14dd35494062aa3eff82": "T_{g*h}", "4ed84ed652edcf40edc66775e84da0b5": "\\|\\cdot\\|_\\infty\\ ", "4ed89707f8de166997364c33f0056cfa": " \\frac{d}{dx}a^x = \\ln(a)a^x.", "4ed91b0c64b1c2458baecc06d65c87a6": "\\mathcal{N}(x)", "4ed93394c4b1fd9e611753de270848d6": "W = W(\\lambda_1,\\lambda_2)", "4ed944b4533bf23fc6d51ed47cadfbd4": "3x^5", "4ed9de336dcd5dc80f1a478cd1c47ed4": "(D^+D)^*_{ij} = \\overline{(D^+D)_{ji}} = \\overline{D^+_{ji}D_{ji}} = (D^+_{ji}D_{ji})^* = D^+_{ji}D_{ji} = D^+_{ij}D_{ij} \\Rightarrow (D^+D)^* = D^+D", "4eda18a0b0f956bbc630e04399e7563f": "\\operatorname{ht}", "4edb808d97c1be12d062b016266e1216": "\\delta:Q\\times\\Sigma \\times Q\\to \\{0,1\\}", "4edb9003e7a03be76f46724a773b0cea": " (\\lambda p.\\operatorname{de-let}[\\operatorname{let} q : q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p])\\ \\lambda f.\\lambda x.f\\ (x\\ x) ", "4edbad2aac1196ffda325c64fc32778c": "\\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\frac{m^2\\,n}{3^m\\left(m\\,3^n+n\\,3^m\\right)}", "4edbba03505531b8152cf287a40e6ab3": "X*", "4edc2b7deeff81296a6949ab5486999e": " E(a) ", "4edc4f9a465bfbce7008fdffb67f9f70": "\\sqrt{g h}", "4edc63d160dcb4a28645187e303e0e11": "\\tfrac{-1+\\sqrt{5}}{2}=\\tfrac{1}{\\phi}", "4edc63da2028a0c78147edf594891eeb": "{\\mathbf{C}_i}^n ", "4edc6891d023cb3f49ce6a51f125f3f5": "\n \\hat\\varepsilon = y - X\\hat\\beta = My = M\\varepsilon.\n ", "4edc933d28bfe3f5effe94bf892dad38": "\\ge", "4edce9be3d1432204012303b1022358c": "M_1(\\phi,\\tau) = \\phi_1(\\theta,\\tau)\\int s^*(u-\\dfrac{1}{2}\\tau)s(u+\\dfrac{1}{2}\\tau)e^{j\\theta u}\\, du", "4edd6a7354fafe7a20b52f640bf21b07": "( \\lfloor ns \\rfloor)", "4edd77a3b910403b24476122d8d217da": "\\mathrm{N} = \\frac {(\\delta \\mathrm{v})^2} { 12 } \\mathrm{W} \\,\\!", "4ede279c1ebb5ee2b9b2751636f47bfb": " \\zeta = \\varphi_H - \\delta ", "4ede99ff99a72be58b4d18fe11058016": "z_0 = P_c^p(z_0)", "4edebc3e5cf9be501fdfa502720d2c89": "-\\sqrt{\\frac{1}{2}}\\!\\,", "4eded9da5b9dcdfb6d9ca06ff3507197": "(W,W_R)", "4edef268684b7fc6dc2bb061d6613b34": "\\frac{\\partial C(\\mathbf{w}_{n})}{\\partial w_{n}(k)}=\\sum_{i=0}^{n}\\,2\\lambda^{n-i}e(i)\\,\\frac{\\partial e(i)}{\\partial w_{n}(k)}=\\sum_{i=0}^{n}\\,2\\lambda^{n-i}e(i)\\,x(i-k)=0 \\qquad k=0,1,\\cdots,p", "4edf3ec974e0590e3329e2f8446fb2f0": "\\sigma_0 \\otimes \\sigma_3 \\otimes \\sigma_3 ", "4edf73aacce8357f18b812cfdcd6b222": "\\left\\vert0\\right\\rangle", "4edf8ef63b82f7820f09ab3fd1f24c65": "\\scriptstyle x \\mapsto \\eta( v , x ) .", "4edf92a53082cfe0bd3cbe0cbe3bbeb4": "\\mathcal{V}(x) = \\mathcal{V}(0) + x .", "4ee00781cb859334550683e73c474194": " g(\\gamma) ", "4ee015ef232264e402487e32704527bd": " P_i's ", "4ee0e202900b84dbb1afc325e6e551a6": "(1/2)^2+y^2 = 1", "4ee140748c07dd31eb292f98edfdb563": "UI = mgv\\,", "4ee1617a3d94f767ce4f988078c12e85": "\\mathrm{CuFeS_2 + 4 \\ O_2 \\longrightarrow Cu^{\\,2+} + Fe^{\\,2+} + 2 \\ SO_4^{\\,2-}}", "4ee18123968ebbe81a20edaa7151f1ed": " \\int_0^{2\\pi} \\Bigl( \\sup_{0 \\le r < 1} |F(r \\mathrm{e}^{\\mathrm{i} \\theta})| \\Bigr)^p \\, \\mathrm{d}\\theta \\le C^2 \\, \\sup_{0 \\le r < 1} \\int_0^{2\\pi} |F(r \\mathrm{e}^{\\mathrm{i} \\theta})|^p \\, \\mathrm{d}\\theta.", "4ee18ec559dc0bf17f541f282f57e6fd": "\\widehat f", "4ee19da6a356b73591037bab2b2ed3bd": "((x,y \\cup \\{y\\}),z \\cup \\{z\\})", "4ee1e9b35b3f41899be2fc0e55c8257d": " <\\lambda>_{trajectories} ", "4ee20852979c5c0bef29e4f48502c857": "\\{(a,a),(b,b),(c,c),(b,c),(c,b)\\}", "4ee21db54dc933fc7a4ad7f780f6f5d6": "E(m) = 0\\ \\ \\ \\ \\text{ if } m\\leq b", "4ee26a7e72811f7f820df6eca5da0fac": "\\begin{align} \\ln\\frac{10}{9} &\n= \\frac{1}{10} + \\frac{1}{200} + \\frac{1}{3\\ 000} + \\frac{1}{40\\ 000} + \\frac{1}{500\\ 000} + \\cdots \\\\ &\n= \\sum_{k=1}^{\\infty} \\frac{1}{10^k \\cdot k} = \\frac{1}{10} \\sum_{k=0}^{\\infty}\\left[ \\frac{1}{10^k} \\left( \\frac{1}{k+1} \\right) \\right] \\\\ &\n= \\frac{1}{10} P\\left(1, 10, 1, (1) \\right)\n\\end{align}", "4ee28b6a15c2402c011806b97a0e72d2": "n=\\lfloor{x}\\rfloor", "4ee29ddeb724fca12d1003a803c63c28": "\\alpha = \\left( 2\\pi\\, \\mathrm{Re} \\, \\mathrm{Sr} \\right)^{1/2}\\, .", "4ee2a28dfab4282e6ab3dfc2a330313d": "0 < \\delta < \\alpha ", "4ee2b1ffcccb8400e609bdc211150431": "I(X;Y;Z)=I(X;Y|Z)-I(X;Y)", "4ee2b9eaf19ac6d7cc114c475dd09a68": "\\left\\langle\\log\\left(\\tilde{P}_{r}\\right)\\right\\rangle\\geq \\left\\langle\\log\\left(P_{r}\\right)\\right\\rangle\\,", "4ee2bc32f441d8cf94a0b07cf980689b": "d < \\frac{N^{ \\frac{1}{4}}}{3} \\approx 5.7828", "4ee2c7ea1de36bd1e59a31efe8490d10": "\\{[SU(4)\\times SU(2)_L\\times SU(2)_R]/\\mathbb{Z}_2\\}\\rtimes\\mathbb{Z}_2", "4ee3a215c5385884459a140f8191f96e": "\\begin{align}\n Q &{}= (I - A)^{-1}(I + A) \\\\\n A &{}= (Q - I)(Q + I)^{-1}\n\\end{align}", "4ee3e225bbd13544bd86934504b8927b": "\\left(\\frac{\\Delta}{p}\\right)", "4ee3fc61912eee9ae7d47100d92b69c1": "X \\cup \\{A \\wedge B\\}", "4ee46974823ed1e66ad20e37c0c25c38": "i(d+d^*)\\tau", "4ee48210d147d83fef3830f958268be9": "x<10 \\land x\\leq10", "4ee53b7e4fa66101b01b777c02a4614f": "2^{89}-1", "4ee5ab973f661ce91c32453d8b43ed04": " \\Omega_{n+1} = G_n \\begin{bmatrix} \\Omega_n & 0 \\\\ 0 & 1 \\end{bmatrix}. ", "4ee5d8d6f128bf9e43ef7b33e6a6974c": "List(A)", "4ee64aea91571a28474a3e2701aaa9df": "\\scriptstyle P \\,=\\, V I \\,=\\, \\frac{1}{R}V^2", "4ee67ac122262a066d1c5a955196bfce": "\nn^{-1}\\sum_{i=1}^n\\sum_{j=1}^mp_jZ_j = n^{-1}\\sum_{i=1}^n\\mu = \\mu,\n", "4ee6b6f25bc7b80cadfe5159760c3e49": "b = \\lceil\\log_2(M)\\rceil", "4ee7274a7b7ad95c04b70b0a414d9b05": "\\bar{k}>1", "4ee7ea2c7d0a10c9e3300d52f913d662": " F(\\textbf{z})\\,", "4ee7fc95d151e378d044e061cac8fd5a": "\\sqrt{E_b} \\phi(t)", "4ee80adae8cfb6c6cf3578d4c435f1e4": "\\mathrm{A}_3 \\cong \\mathrm{D}_3,", "4ee815e647cfdac74c7168c74e11b54f": "mr^2", "4ee821da2bcf528928048fa7fc2cf11a": "B_i (p,q)", "4ee82f890855a8f23d58dd9a4a9215a7": " ~ = \\exp \\left[ \\begin{matrix} 0& Re(\\alpha) & Im(\\alpha) & 0 \\\\\n Re(\\alpha) & 0 & 0 & -Re(\\alpha) \\\\\n -Im(\\alpha) & 0 & 0 & Im(\\alpha) \\\\\n 0 & Re(\\alpha) & Im(\\alpha) & 0 \\end{matrix} \\right] ~ . ", "4ee864dbe2d03cece0ae1386423702a2": "\\mu \\leq 0", "4ee8840dd0d400349faab4b5b161e7e0": "{\\Theta}(n\\log n)", "4ee8acba2ee1c83cfa689239c82447fb": "p \\in \\mathbb{N}", "4ee969b3940ef6f6f63be7914fbc97f2": "y \\in [q]^n ", "4ee981142009e1f1f9cccc3510274e74": "(S,A,O,T,\\Omega,R)", "4ee9a963304c8faec8b6b03e4bec9141": "(\\operatorname{cont}_F)_* \\mathcal{O}_X = \\mathcal{O}_Z", "4ee9b3ac6254024608114076987ef2e5": "(\\lambda I - A)\\mathbf{v} = 0\\,", "4eea1e25209c52cc71bc5cc6420108f2": "S^7", "4eea81fdb4f596c0589c304e9405a4d0": "\nL(\\theta;A,B)=\\frac{(A+B)!}{A!B!}\\theta^A(1-\\theta)^B,", "4eeadc3a642c991ff540c8b673b1bc3d": "\\scriptstyle \\Psi(x,y)", "4eeaefec6afbba1bd00a93fb3c35a503": "-\\frac{\\mu}{R^2}\\mathbf{\\hat{r}}", "4eeaf64eda1710e9d281bd0ab34da640": "{v}\\,", "4eeaf9bcad28332cd616660f9e38ab52": "\\Bigg[\\frac{-1}{\\pi}\\Bigg]=(-1)^{\\frac{a-1}{2}},\\;\\;\\; \\Bigg[\\frac{2}{\\pi}\\Bigg]=i^{-\\frac{b}{2}}.", "4eeb7bcab72d105b4dce8e6b908269e3": "\\int_{-\\infty}^\\infty \\delta(\\alpha x)\\,dx\n=\\int_{-\\infty}^\\infty \\delta(u)\\,\\frac{du}{|\\alpha|}\n=\\frac{1}{|\\alpha|}", "4eeb844f6b5fd1251d2239e54527c49d": "\\sigma (x) \\ne \\sigma (y)", "4eeb9c6932a9990e936836e8cb2c0f5a": " \\vec{e}_3 = \\frac{1}{r \\sin\\theta} \\, \\partial_\\phi ", "4eec61093122021106a499ba7745ee41": " Y = R \\cos \\epsilon \\sin \\lambda ", "4eecd91a90ff18b9ad6747154f46299a": " I_{xz} =\\cdots ", "4eecfa61d4fa1d5458b3b2c3dec0b3c5": "S^1\\hookrightarrow S^3\\rightarrow S^2. \\,\\!", "4eed55c5eace1192b75dc15f109084cb": "dU=\\delta Q-\\delta W + \\sum_i \\mu_i dN_i\\,", "4eed6e58f79ddb7bfd75605688489262": " \\epsilon = \\frac{ a - a_0}{a_0} ", "4eedaccd473cbeec5005704febec53b1": "HoldsAt(f,t) \\leftarrow\n[Happens(a,t_1) \\wedge Initiates(a,f,t_1) \n\\wedge (t_1 \\theta_i > 0, i=1,\\dots,k; a > 0. \\quad (2)\n", "4ef92610f293eb606984d1644953470c": " k(t) = s.f(k) - nk ", "4ef9c528946b3412b39b41f98890d8b4": "g_0^T(x^*-x^{(k)} ) + f_0(x^{(k)}) - f_{\\rm{best}}^{(k)} \\leq 0", "4ef9cff2047e04349f6399910a5e1592": "p_{0}", "4ef9e4b2ddebe4d0658b5de72590c900": "\\pm\\frac{1}{\\sqrt{\\csc^2 \\theta - 1}}\\! ", "4ef9fb21d0c83e9205d03adf1af5ed17": "U^{T} U = I \\,", "4ef9fd28b1ef4e4055db23423c674634": "N\\trianglelefteq M", "4efa937d89fcba9e6354031696a1fb36": "\\delta^\\dagger_Z \\circ \\delta_X = \\varepsilon_Z \\circ \\varepsilon_X^\\dagger", "4efacb9e75a8b5cec7d943f5b5649f9b": "\\omega = e^{-2 \\pi i / N}", "4efb42006128cf83191f70d56d0d09be": "X '^2 - Y'^2 = P (T ') Z'^ 2", "4efba94a97a3dd55821a6e961504ce2f": "c_2 \\sin ks=c_3 \\cos ks", "4efbb198a68a02679970a53aaf256481": "g_0^+(T(\\omega))=f^+(\\omega)=+\\infty", "4efc1d90641daac789d450385802f787": "-1/\\tau", "4efc2e2074623dade080dc37b2b4dc06": " z = x - \\frac{ \\gamma }{ 6 } (x^2 - 1) + \\frac{ x }{ 72 } [ 2 \\gamma^2 (4 x^2 - 7) - 3 \\kappa (x^2 - 3) ] + \\cdots ", "4efc341d1634cd75794a17613a876dcf": "P_{i,j,k}\\,", "4efc38301bf9aae505634db5286ef968": "\\widehat{\\pi}", "4efc832e151e3342556732c5c3461c50": "\\alpha_2 \\,\\! ", "4efd392f3352075b1255d7e4171352ac": "F_N = \\dot{m}_{air} (V_{j} - V)", "4efd8022a81f8598b3b119c0f9fe9e57": "\\mbox{s}\\,", "4efda7099f2688ea6f761a28f29ce092": "a \\le b \\iff a = \\gcd(a,b).\\;", "4efdd40490e67c9af869b46b5c8ce0ba": "\n\\sigma_N = \\sigma_0 \\left( 1 + \\frac{r l_0} D \\right)^{1/r}\n", "4efe0f7170af0e93980409e6efb5944a": "C_p - \\lambda", "4efe241f58b7472662372b12074fd121": "a>c>b", "4efeb2bb74224d77d70417195f517803": "\\psi_t = \\frac{i \\hbar}{2m} \\Delta \\psi", "4eff036795fb8fac36eb95c854fa7127": "\n\\begin{bmatrix}\n 1 & 0 \\\\\n \\frac{-1}{\\lambda R_A} & 1 \n\\end{bmatrix}.\n", "4eff19ae60812cc5265fa8d9c964b41b": "\\Delta(\\alpha) = \\alpha \\otimes \\alpha - \\mu \\beta \\otimes \\beta^*", "4eff22b504bb15cee615b4429d81c3f9": "\\ M^{reg} \\subset M ", "4eff2d2aac7b3c5014e3be4f4faa6eb4": "{\\rm d}(ab)=a({\\rm d}b)+({\\rm d}a)b,\\ \\forall a,b\\in A", "4eff9aaa7987cc60035241439d8d9d54": "C_o = C_a \\alpha_a + C_b \\alpha_b \\left(1 - \\alpha_a\\right)", "4efffb16441bfe3e639c42be38cbc3d4": "= \\text{decimal } 946", "4f000e3b41970f18b7a57c9f6ec0794b": "R^2 + 2Rh + h^2 = R^2 + d^2 \\,\\!", "4f0097cdbf07cad6c8201b925cb32467": " t_1 = -\\tau\\;\\ln\\left(1-0.1\\right) = -\\tau \\; \\ln\\left(0.9\\right) = -\\tau\\;\\ln\\left(\\frac{9}{10}\\right) = \\tau\\;\\ln\\left(\\frac{10}{9}\\right) = \\tau({\\ln 10}-{\\ln 9})", "4f00adf9b32ad823965a718d6e434040": "b_{i}+c_{i}", "4f02418b2c80b7fbf8fee93fcb2bf52e": "p(x) = g(x)\\cdot q(x)", "4f02ab07737df0c87b2043fd6778ff5c": "p(x) = 64x^3 - 28x +7", "4f030f8682dbf32b1a22efc66d535402": "\\varphi_{Z_n} =\\varphi_{\\sum_{i=1}^n {Y_i \\over \\sqrt{n}}}\\left(t\\right) = \\varphi_{Y_1} \\left(t / \\sqrt{n} \\right) \\cdot \\varphi_{Y_2} \\left(t / \\sqrt{n} \\right)\\cdot \\ldots \\cdot \\varphi_{Y_n} \\left(t / \\sqrt{n} \\right) = \\left[\\varphi_Y\\left({t \\over \\sqrt{n}}\\right)\\right]^n", "4f03166ab140e3910ce58187ee6318a8": "r(t+\\tau) = r(t),\\, \\theta(t+\\tau) = \\theta(t)", "4f033c24a27b26d3a204f120d9bbf595": "f \\circ g = f \\circ h", "4f0365eb15fbbf2512d0d5fad5b90e4e": "A = B \\!", "4f03c003264e0699da75f41946babe2d": "R= \\frac{\\omega_s-\\omega_c}{\\omega_a-\\omega_c}.", "4f03d8f73534f2027f2a9bd2be39cf2a": "p(\\alpha^i) = b_1\\alpha^{ik_1} + b_2\\alpha^{ik_2} + \\cdots + b_{d-1}\\alpha^{ik_{d-1}} = 0.", "4f0467b01a7a8cba2a0db3c5cbb79bb3": "\\lim_{n\\to\\infty} a_n = a,\\; \\lim_{n\\to\\infty} b_n = b \\implies b \\in \\Gamma(a) ", "4f046d81392d084e71ba4af501057428": "(\\hat{x}\\ ,\\ \\hat{y})", "4f0509f6383891270de327d5f11b7609": " \\begin{pmatrix} \n1 & 0 \\\\\n0 & 1 \\end{pmatrix},\\quad\n\\begin{pmatrix} \n1 & 0 \\\\\n1 & 1 \\end{pmatrix},\\quad\n\\begin{pmatrix} \n1 & 1 \\\\\n0 & 1 \\end{pmatrix} ", "4f0557e053d258078e170c2fa54c35c6": "\\theta \\in", "4f056a05ed6ff06f67852f77b9e46025": "S_+|s,m\\rangle=\\hbar \\sqrt{s(s+1)-m(m+1)}|s,m+1\\rangle", "4f0595158e880c453003f0f4466fe9c6": "|k'\\rang", "4f05d4e4c95627cd5c4174da3a83d921": " a'_{h\\ell} = a'_{\\ell h} = c a_{h\\ell} + s a_{hk} \\,\\! ", "4f05eb7aea9e54a0afeca949b83e3c09": "V( \\neg A,0) \\Leftrightarrow V(A,1)", "4f05f7a1918edeed9906c7fc0e56dc45": " \\mathrm{ind}\\,T := \\dim \\ker T - \\mathrm{codim}\\,\\mathrm{ran}\\,T ", "4f06207468eacde22ab7358c39090b2c": "G(s)=\\prod_{k=1}^nG_k(s)\\ =\\ \\frac{(s)_{\\uparrow n}}{n!},", "4f0636cbffa06563b8e25cff93f980e8": "\\bar{5}_{-3}\\rightarrow (\\bar{3},1)_{-\\frac{2}{3}}\\oplus (1,2)_{-\\frac{1}{2}}", "4f0639933359941527922cdf21cf1822": "f(\\mathcal{R}^K _{\\theta}) = \\mathcal{R}^K _{2 \\theta}", "4f06b8cc531f3371bd148ab93adeeb91": " y_1, \\ldots, y_{s-1}", "4f06bbe6b63e7a922df184dff46e8e95": "\nV_{in} \\approx \\frac{I}{j\\omega C} = V_C\n", "4f07579486042c059d7258b610c56735": "\n\\rho = \\sqrt{x^2 + y^2}\n", "4f0785ef75422402cd644ee7a6e05e70": "\\dot{x}_3=dx_3/dx_3=1", "4f07ce2df639a2c21e32fa971af0d8ce": "x_r(\\theta)", "4f07e36a97e9cd26e54c13146d61b5b3": "b\\mapsto\\|b\\|", "4f08e3dba63dc6d40b22952c7a9dac6d": "\\pi", "4f0945b4f446537d6039baa6f2c0efc6": "\\Box_a", "4f097e4830c815a092d17dbd4a75dbd4": "\\lim_{x\\rightarrow 0}\\frac{1-\\cos x }{x}=0,", "4f09864aae52b209a777ebd7df9bb3a8": "\\xi^{\\mu} = (1,0,0,0)", "4f09900a083464a042d6d3c708b480c1": "x = \\left \\langle a_0; a_1, a_2, a_3 \\right \\rangle.", "4f09c4936c18548e5fc7f72cc11e91c9": "\n\\int{ G[f] [Df] } \\equiv \\int\\limits_{-\\infty}^\\infty{ ... \\int\\limits_{-\\infty}^\\infty{ G[f] } }\\prod_x df(x)\n", "4f09ed2ac676f523a5d2eeeee1a8f43a": "T(n)=\\frac{\\mathop{He}_n(i)}{i^n}.", "4f0a30d6009b300f03b230820329e927": " \\begin{align} \n& |\\bold{J}| = \\hbar\\sqrt{j(j+1)} \\\\\n& |\\bold{J}_1| = \\hbar\\sqrt{j_1(j_1+1)} \\\\\n& |\\bold{J}_2| = \\hbar\\sqrt{j_2(j_2+1)} \\\\\n\\end{align} ", "4f0a612e63eee4334251dec4f0940f60": "\\mathcal{S}[\\phi_2,t_2;\\phi_1,t_1]=\\langle\\,\\phi_2\\,|e^{-iH(t_2-t_1)/\\hbar}|\\,\\phi_1\\,\\rangle.", "4f0aaee8ec52776142c275f03a4ba9f5": " \\mathcal{O}_X", "4f0acd49da2deda3cce73ac2cd08b5ea": "x \\vee (y \\wedge z ) \\vee x = (x \\vee y \\vee x) \\wedge (x \\vee z \\vee x).", "4f0b0acfa1f95c0ecb3184e01ce1ea2f": "log BCF=2.791-0.564 logS(S=water solubility)", "4f0b26a48bb62c8dddc5f58b6339082c": "C = 1 + \\frac{0.042(Y_0 - 30)^{1/3}}{Y_0^{1/3} - \\frac{2}{3}}", "4f0b318ebc876b631ed225e7bfb4bd21": "p=k\\alpha _{g}", "4f0b4d739da953daa83e769f9ae0334f": "{\\mathbf{}}K_i^r=H_{i+1}P_i C'_i \\left( C_i P_i C'_i+W_i \\right)^{-1},", "4f0b500ce9a8bcb17a51d366ddaa90f2": "M_1 = M_2", "4f0b81b819d27ddac47615fea473ab78": " \\overrightarrow{Dk} ", "4f0bd1448d68b66405ab479de4ddcc1a": "{\\mathcal O}_{q_0}=\\{e^{t_k f_k}\\circ e^{t_{k-1} f_{k-1}}\\circ\\cdots\\circ e^{t_1 f_1}(q_0)\\mid k\\in\\mathbb{N},\\ t_1,\\dots,t_k\\in\\mathbb{R},\\ f_1,\\dots,f_k\\in{\\mathcal F}\\}.", "4f0bf12b6343fb51817f6787768880a0": "\\int\\limits_{1}^{3}\\frac{e^3/x}{x^2}\\, dx", "4f0c2a8f9289030352bbee6fff2b65d2": "\\frac {8x}{5} = 1.6 x. \\!", "4f0cbeb855f62899f16c8876a1a4a925": "-a\\,(v_\\alpha\\,\\Delta v_\\alpha)^2\\,/\\,(2\\,\\sqrt{a\\,b}\\,s_\\alpha)^2 = -(v_\\alpha\\,\\Delta v_\\alpha)^2\\,/\\,(4\\,b\\,s_\\alpha^2)", "4f0cc6de950e7b31578d99a441c39d3a": " v_{out} = i_{in} R_L = v_s \\begin{matrix} \\frac {R_L}{R_S} \\end{matrix} ", "4f0ccbcc11956ab84bc652b3b6a73822": "\\nabla\\times\\mathbf{F}=\\mathbf{0}.", "4f0cce77f94afe71cdebf91c7f1d174c": "C_t = 0.9 Y_t", "4f0d81e3a601e1f0f895aba352866d32": " \\beta' \\colon A \\times_{C} B \\to A ", "4f0d9665b30f3cfaa78afb1d1016f416": " \\{\\,N(t) : t \\geq 0\\,\\}", "4f0dd0aae78716558ee579d2dc1796ea": "Z=\\sqrt{R_{\\mathrm{ESR}}^2 + (X_\\mathrm{C} + X_\\mathrm{L})^2}", "4f0de18862c3bad97b8047a282dac6b5": "\\frac{|q_p+q_e|}{e}", "4f0e8ac494695f0dd985eee6494891e0": "T = \\sqrt{rr_1r_2r_3}.", "4f0ed02276ca43df1b7c46fd023c4c6b": "\\frac{1}{100}\\sum_{k=1}^n\\frac{1}{k}.", "4f0f18d83f89ab5f240baddf99d06021": "DP_{*}^{S}", "4f0f26f3a4228d5239190214e3d4b38f": "\\widehat{a}(\\widehat{U}|\\alpha\\rangle) = \\widehat{U}\\widehat{a}e^{-i\\theta}|\\alpha\\rangle ", "4f0f291c3578ca1ba4cd2df296cd67ad": "H(s) = \\frac{ \\omega_0^2 }{ s^2 + \\frac{ \\omega_0 }{Q}s + \\omega_0^2 }", "4f0f34d11ccc4b1cd2d85aa4ac39293a": "A = \\begin{pmatrix}\n0&1\\\\\n0&1\n\\end{pmatrix}, \\;\\;\nB =\\begin{pmatrix}\n0&1\\\\\n0&0\n\\end{pmatrix}.\n", "4f0f4157c511b6e54f2a5862b8860df5": "OPS = \\frac{AB*(H+BB+HBP)+TB*(AB+BB+SF+HBP)}{AB*(AB+BB+SF+HBP)}", "4f0f5c1672ce04d8399f53d257509811": "\\ln(2)\\sum^{k}_{i=1}1/\\lambda_{i}", "4f0f7f00b193fc7b0b8b0facddb27720": "c\\geq 0, \\Delta c>0", "4f0ff11e6a2d0bd25c2cca16320bffea": "ex\\,", "4f10553a63e8661c07cbce861e15df84": "S_0(c)\\!\\ =(c^p, 3, c) ", "4f106ed302501e498fe65b21b746dc0d": "y_{11}-y_{21}", "4f10aef3cbe8cb8cc2deeb553fe19d48": "w_2 = bz", "4f10b06f4a40b5be6db6ee899323d39c": " \\operatorname{build-list}[B, D, V, []] \\equiv \\operatorname{build-param-lists}[B, D, V, \\_] ", "4f10bd5c18a822d187b2120b009bb3d6": "S = \\beta(1/N)", "4f10de0bc6211be6ad66686de9795bb7": "\\displaystyle{g(z)=(\\alpha z +\\beta)(\\gamma z +\\delta)^{-1}.}", "4f11686467b82bba2f83e49d98c97706": "K = h \\cdot S_b (X + \\bar{X}A) = h \\cdot S_b (xP + \\bar{X}aP) = h \\cdot S_b (x + \\bar{X}a)P = h \\cdot S_b S_a P ", "4f11bb38c73434efc1f3a5d9da779a92": "0.0199 \\approx 0.02.", "4f11d268ddc9492c20c197e672b20095": "\n \\begin{align}\n \\operatorname{E}[S^2]\n &= \\operatorname{E}\\left[ \\frac{1}{n}\\sum_{i=1}^n \\left(X_i-\\overline{X}\\right)^2 \\right]\n = \\operatorname{E}\\bigg[ \\frac{1}{n}\\sum_{i=1}^n \\big((X_i-\\mu)-(\\overline{X}-\\mu)\\big)^2 \\bigg] \\\\[8pt]\n &= \\operatorname{E}\\bigg[ \\frac{1}{n}\\sum_{i=1}^n (X_i-\\mu)^2 -\n 2(\\overline{X}-\\mu)\\frac{1}{n}\\sum_{i=1}^n (X_i-\\mu) +\n (\\overline{X}-\\mu)^2 \\bigg] \\\\[8pt]\n &= \\operatorname{E}\\bigg[ \\frac{1}{n}\\sum_{i=1}^n (X_i-\\mu)^2 - (\\overline{X}-\\mu)^2 \\bigg]\n = \\sigma^2 - \\operatorname{E}\\left[ (\\overline{X}-\\mu)^2 \\right] < \\sigma^2.\n \\end{align}\n ", "4f11e9edc08fc9f632f70fcc299c6399": "\n \\sigma_{11} = \\left(\\lambda^2 - \\cfrac{1}{\\lambda^4}\\right)\\left(\\cfrac{\\mu J_m}{J_m - I_1 + 3}\\right) = \\sigma_{22} ~.\n ", "4f122e467b20de4550617d5dead2e6a6": "S^{D-3}", "4f124633249231904e4d29e00431eceb": "b \\ge j", "4f124e5e6b7184c684cae5c2ae277441": "j\\colon X \\hookrightarrow \\mathbf{R}^n,", "4f12e0cd2fefd19f2c959eecdd072641": " g \\subset \\mathcal Q", "4f12e45f8601d1536aafd4982e7598cd": "\\Sigma_1=\\{a,b,c\\}", "4f12e96e08f31db6fa5828cd2dd44cb3": "[b_\\mathrm{in}(t),b_\\mathrm{in}^\\dagger(t^\\prime)]=\\delta(t-t^\\prime)", "4f132f9bf381aa1bb9552c0d42519299": " \\sin \\left(\\frac{\\pi}{3}\\right) = \\sum_{n=0}^\\infty \\frac {(-1)^n (\\frac {\\pi}{6})^{2n}}{(2n)!} ", "4f1343c592b39a76c08e8d736ffd3283": "\n~~~~~~~~~~~\n~+~ i \\gamma_{tx} \\frac{\\partial^2 \\textbf{A} }{\\partial t \\partial x}\n~+~ i \\gamma_{ty} \\frac{\\partial^2 \\textbf{A} }{\\partial t \\partial y}\n~-~ \\frac{i}{2} \\gamma_{xx} \\frac{\\partial^2 \\textbf{A} }{ \\partial x^2}\n~-~ \\frac{i}{2} \\gamma_{yy} \\frac{\\partial^2 \\textbf{A} }{ \\partial y^2}\n~+~ i \\gamma_{xy} \\frac{\\partial^2 \\textbf{A} }{ \\partial x \\partial y} + \\cdots\n", "4f1355467c8d89507bb920d12ccf56b2": "{\\mathbb P}\\biggl(\\bigcup_{i=_1}^{n} A_i\\biggr) \\le \\sum_{i=_1}^{n} {\\mathbb P}(A_i).", "4f13a934239569691d59d3d451cd2678": "q_{sig}", "4f13acdba680c7901830f2106fc75807": "\\sqrt[3]{\\sqrt{\\sqrt{2}}} = \\sqrt [12] {2} \\approx 1.059463094359295264561825,", "4f13b03968558c6ca97fc787aa1e04f8": "v\\in T_pM", "4f13b4d46f062f1ff1ef74eee39548b3": " 10 (0.4)(0.8)^2 = 2.56 ", "4f13ffdd1c44f11d6674b3631e9cc7a5": "t\\bar{t}", "4f144b2f36faa0646ec71d2ff3660570": "M_t = M + m + \\frac{I}{R^2}", "4f1477b140dd26cb791d2bbcc82d0482": "\\scriptstyle{\\hbar\\omega}", "4f149033378268856048139df603717e": "P_{S2} = I_o^2 R_{DSON} (1-D)", "4f149ee3e761d5d7779c785ad9eccea6": "F(x; \\ln \\sigma, 1/\\alpha, 0)", "4f1501d02a60c426c214326cb7aab350": " \\frac{\\partial\\vec{r}}{\\partial s}, \\frac{\\partial\\vec{r}}{\\partial t}, \n\\frac{\\partial^2\\vec{r}}{\\partial s^2}, \\frac{\\partial^2\\vec{r}}{\\partial s\\partial t}, \n\\frac{\\partial^2\\vec{r}}{\\partial t^2}.\n", "4f1550804a15f8e101cce96053e94213": " F(R_1, R_2, \\ldots, R_N) ", "4f156a5e76af231ddf479926d7747e76": "p_{A} = Tr_{B}(|\\phi\\rangle\\langle\\phi|)", "4f16220b69aed64a5cc3bf50f2cbe7d1": " \\left(\\mathbf{ab}\\right)\n\\!\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\times \n\\end{array}\\!\\!\\!\n\\left(\\mathbf{cd}\\right)=\\left(\\mathbf{a}\\times\\mathbf{c}\\right)\\left(\\mathbf{b}\\times \\mathbf{d}\\right)", "4f165e1f146ee9503047715f20b672b2": "\n{{\\text{weight omitted by stopping after k terms}} \\over {\\text{total weight}}} = { { \\alpha \\times \\left[ (1-\\alpha)^k +(1-\\alpha)^{k+1} +(1-\\alpha)^{k+2} + \\cdots \\right] } \\over { { \\alpha \\times \\left[ 1 + (1-\\alpha) +(1-\\alpha)^{2} + \\cdots \\right] } } }\n", "4f16a0707526bb749fdaa04fd9700865": "b_{10}-a_{10}", "4f16f3edffeeede179fcfeedd155175c": "w\\,", "4f17167b0e2af04b7d3d509a547dd0f3": " \\beta_i ", "4f17462088213322460dfb9659bbf358": "\\langle\\mathbf{p}\\rangle = q \\mathbf{E} \\tau.", "4f17c72956ce177af4b6c89c786d9e04": "\\Lambda(\\mathbf{x}) = \\frac{L\\left(\\sigma_0^2;\\mathbf{x}\\right)}{L\\left(\\sigma_1^2;\\mathbf{x}\\right)} = \n\\left(\\frac{\\sigma_0^2}{\\sigma_1^2}\\right)^{-n/2}\\exp\\left\\{-\\frac{1}{2}(\\sigma_0^{-2}-\\sigma_1^{-2})\\sum_{i=1}^n \\left(x_i-\\mu\\right)^2\\right\\}.", "4f1897e2025f38f12869af68954d408c": "de_3=R (\\cos{(\\alpha)} \\sin{(\\theta)} d\\alpha \\wedge d\\phi + \\sin{(\\alpha)} \\cos{(\\theta)} d\\theta \\wedge d\\phi)", "4f18bd54885e54d5a9a2b5c45524a0eb": "X((n_1, n_2)) = n_1 + n_2", "4f195d4a41d9e5d7b3b3ae5964fd4502": "V_\\mathrm{P}^{(1)}", "4f1986fc3f39bb9ffa3397650189afd3": "|\\cdot|_{\\ast}:\\mathbf{Q}\\to\\mathbf{R}", "4f19b36577a3a72f9672e907f9725d8e": "A_\\varepsilon(y_1,\\ldots,y_N) = A_\\varepsilon( ky_1,\\ldots,ky_N)", "4f19dd4969e0fee8b0cb2b518b7aeab3": "\\scriptstyle a:=(\\ a_0,\\ a_1,...,\\ a_{n-1})", "4f19f708a95c2a4255ba6d276b69896d": "\nP_N(\\overline{R})=N^2\\overline{R}\\int_0^\\infty J_0(N\\overline{R}\\,t)J_0(t)^Nt\\,dt\n", "4f19f7185889e5e72b6d66360f936cac": "\\theta \\colon \\mathcal{N} (X) \\rightarrow L_{n} (\\pi_1 (X))", "4f19ff9e7d40f76206b53ba5d71c546a": "f^\\phi=\\frac{f'-\\frac{f}{I}}{2}", "4f1a34ce563314409fe98546fd5bad28": "\\gamma=0.577\\ldots", "4f1a3b3280ce10ec58ba780525bc07de": "\\sharp", "4f1a5800d82864935c8bb0d932226e50": "m(x)=\\inf\\frac{f'(\\xi)}{g'(\\xi)}", "4f1a980acea710d9aa6163f970c0e665": " A-\\lambda B", "4f1ab2576a421dfd162595ec3f7ead06": "pp", "4f2caf0aeb90cead4d11f9f6107ab3b4": "V[[z]]", "4f2cd886891cc26c6ea7cf33029ee73e": "\n(q+1-p) \\, {\\rho \\over 2 \\gamma} = \\sum_{j=1}^p a_j - \\sum_{j=1}^q b_j,\n", "4f2e0851aba8c15f5a84f39a17c36dd4": "H_T = \\sum_R W_R \\cdot D_{T,R}\\ ", "4f2e201ab662e57edbfd02205d812b47": "1+x+3x^4", "4f2ea3cb507d627750d379f512dfe4e7": "D(f) = Q f'' + L f'\\,", "4f2eab7b64b7a8e6de43b328a4ad0125": "\\hat{B_i} = \\frac{\\hat{M}}{\\hat{M}+n_i}", "4f2ed631d8d28e1f367c36c131fe5541": "\\operatorname{sign}(p_{i-1}(\\xi))= -\\operatorname{sign}(p_{i+1}(\\xi))", "4f2edbbbc29afab3d18e4cacb49c0ebc": "\\left[E_x(y+\\Delta y)-E_x(y)\\right]\\,\\Delta x-[E_y(x+\\Delta x)-E_y(x)]\\Delta y = 2L'\\,\\Delta l\\frac{\\partial{I}}{\\partial{t}}", "4f2eddaedeffb4e2d8027786cdbaa488": "\\sum_{i=1}^n x_i k_i = \\sum_{i=1}^n x_i", "4f2f41a81f8b7b0c9c0ca205a6d6719a": " \\emptyset = X_{-1} \\subset X_0 \\subset X_1 \\ldots \\subset X_n = X ", "4f2f5b324ddae12b4a2bd9704fdaf8dd": "\\ln(x) = \\lim_{n \\rightarrow \\infty} n(x^{1/n} - 1).", "4f2f623f85bfe8a3ecec1417e8f25d44": "\\langle e^\\frac{-ikx^2}{2z}|e^\\frac{-ikx^2}{2z} \\rangle=e^\\frac{-ikx^2}{2z} (e^\\frac{-ikx^2}{2z})^*=e^\\frac{-ikx^2}{2z} e^\\frac{+ikx^2}{2z}=e^0=1", "4f2f8bf929bb4912859c3efbd5032869": " \\int\\limits_\\Omega \\mathbf\\varphi_i\\partial_{x_i} f + f\\partial_{x_i}\\mathbf\\varphi_i=0", "4f2fac053f87a952441dc3513d3725ff": " \n\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\, X_1^{\\sigma(1)-1} \\cdots X_n^{\\sigma(n)-1} =\\prod_{1\\le ip(y_1^n|x_1^n(1)) ", "4f346b7fea27af2732828a4e8456d1ff": "\\frac{300}{F}", "4f3485c4f2b713f6021b641a823ad6f2": "\\frac{R}{n} = \\tan V,", "4f348792dbc1b51392888783c3c4b17e": "g_{mbF}=g_{mb}+\\frac{C_{GB}}{C_T}g_m", "4f349c80880c365199d862b52b94519d": " [X,Y]_p \\in E_p", "4f350664a1094dc3bec4b0f867ab49e1": " P = {{e^2 a^2} \\over {6 \\pi \\varepsilon_0 c^3}}. ", "4f35067b9952413812e05bb9b5ab8bc3": "M \\times [0,1]", "4f350866f2adbf4bb0d0213c214d8f0a": "X^2 = \\frac{(N-k)\\ln(S_p^2) - \\sum_{i=1}^k(n_i - 1)\\ln(S_i^2)}{1 + \\frac{1}{3(k-1)}\\left(\\sum_{i=1}^k(\\frac{1}{n_i-1}) - \\frac{1}{N-k}\\right)}\n", "4f351e7b9f9de0c5451f1b398f513693": "\\frac{11}{d}-\\frac{1}{2}", "4f35bc50b88a3a5e15bbaffd0fd13cc6": " E_v ", "4f35ce79b17eda793ac101849c65291c": " (a,b) := (-1)^{\\left|a\\right|} [[\\Delta,L_{a}],L_{b}]1 , ", "4f360eabcb477fe3087752141cb9e165": " 0 \\rightarrow K \\rightarrow X \\rightarrow P^\\prime \\rightarrow 0, ", "4f364e9a9553b0d2f2a8d068c649667f": "\\inf\\, \\{ x \\in \\mathbb{R} : 0 < x < 1 \\} = 0.", "4f367b09e77e1803fc7b9c1f63fff57b": "\\scriptstyle A(a,\\lambda)", "4f367f71dc082a549c93c630eeff1182": " aB = a \\;\\big\\lrcorner\\; B + a \\wedge B", "4f36a5ee136d42dc9d8ff5179cd2f369": " \\nu=1-\\frac{n}{2pn\\pm 1},", "4f36c84606e565e1a220987ec2d6cd2a": "(X-\\alpha)", "4f37783823fc5bcb18cec232f7563904": "s(t)", "4f37a64a1cf6e88e1a08a6800a80061a": "\\mathrm{Tr}", "4f37c4468cb664bea70ab09afeec871a": "\\frac{d^2W}{d\\Omega }=\\int_{-\\infty }^{\\infty }\\frac{d^2P}{d\\Omega }dt=c\\varepsilon _0\\int_{-\\infty }^{\\infty }\\left | R\\vec{E}(t) \\right |^2dt", "4f385202faf30bc2b01195621e18ac9a": "\\displaystyle{V =(r^2+s^2)^{-1/2}Y = a\\partial_x + b\\partial_y,}", "4f3876fd1579f83e1703220089dc7f32": " D_{\\mathrm{KL}}(k_p,\\theta_p; k_q, \\theta_q) = (k_p-k_q)\\psi(k_p) - \\log\\Gamma(k_p) + \\log\\Gamma(k_q) + k_q(\\log \\theta_q - \\log \\theta_p) + k_p\\frac{\\theta_p - \\theta_q}{\\theta_q} ", "4f39278b5c98c7b969fc2aa395c0c611": "\\sum_{k=0}^\\infty \\frac{(-1)^k(2k)!z^{2k+1}}{2^{2k}(k!)^2(2k+1)}=\\operatorname{arsinh} {z}, |z|\\le1\\,\\!", "4f3929b3637e236d6520d3189fcb1d6a": "\\scriptstyle \\bar{C} \\,", "4f394a726009aa67393fcee7c2ac0a39": "f(xr,ys)=r^Jf(x,y)s", "4f39d741cd16946df4f48f2865ef6cb0": "g_2(x) = \\sum_{k \\geq 1} (-1)^k \\frac{\\sin(k \\pi / 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2.", "4f39d92eed7666b4906cbdb90a754cf6": "p_{n}", "4f39e1ba336f4636fbab4192e89c1a08": "\n\\sum_{\\delta\\mid n}\\varphi(\\delta)d\\left(\\frac{n}{\\delta}\\right)=\n\\sigma(n).\n", "4f39f774aaaec0b26a96311c661c1dbf": "\n\\left[ \\mathbf{N}\\left( \\mathbf{u+v}\\right) \\right] =\\left[\n\\mathbf{N}\\left( \\mathbf{u}\\right) \\right] \\left[ \\mathbf{N}\\left(\n\\mathbf{v}\\right) \\right] ,\n", "4f3a6e8553d2921f5e630a74f0f4ae79": "\n\\sum_{n = 1}^\\infty \\frac{(-1)^{n + 1}}{n} \\;=\\; 1 \\,-\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,-\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,-\\, \\cdots\n", "4f3a7d7017ffb5b3459338a6f787a3a0": "Q_1 = \\{O_{1},O_{2}\\}", "4f3ad940d304810c65af889a713fab03": "S(q)=\\frac{2\\pi}{q^4}.", "4f3ada93ba7a60a902f1e8af1aa79766": "\\sum_{k=a}^b f(k)=\\Delta^{-1}f(b+1)-\\Delta^{-1}f(a)", "4f3ae8620d38a9dfb94f1dc87669ae6d": "\\frac{\\partial^2 \\Phi}{\\partial x_i\\, \\partial x_i} - M_i\\, M_j\\, \\frac{\\partial^2 \\Phi}{\\partial x_i\\, \\partial x_j} = 0.", "4f3bb0bb05d8cfd534a0fb022f636538": "\\mathbb{E}[X_i^?] = {2\\omega_i \\over d}", "4f3bb0c2532e427aa755712e92937b45": "\\langle x, f\\rangle := f(x) \\qquad x \\in V \\mbox{ , } f \\in V^*", "4f3bbe0aec24e949a462fdbbbe107dd7": "NEA = \\dot{V} ( U_{NH_4} + U_{TA} - U_{HCO_3} )", "4f3c723fca822864f9b689b413a744c2": "\\vec x = \\left( {x_1, x_2, \\dots, x_n } \\right)", "4f3ccb4fa968def7b7db92f0ff2da3d4": " I(y_j) ", "4f3cd91395432382ff24e803da733acc": " \\min \\Phi_k (\\bold x) = f (\\bold x) + \\mu_k ~ \\sum_{i\\in I} ~ c_i(\\bold x)^2 ", "4f3d004577eaab984d9e36712afaf94f": " \\begin{align}\n\\frac{d x}{d t} &= y+\\phi(x)-z+I, \\\\\n\\frac{d y}{d t} &= \\psi(x)-y, \\\\\n\\frac{d z}{d t} &= r[s(x-x_R)-z],\n\\end{align} ", "4f3d3803542f7bba8e99ae32982d6e76": "f_t: g^{-1} \\to \\R", "4f3d6858220dfaa0126a46913eddb235": "\\|\\Psi\\| = 1", "4f3dcfebf44545c0ec5976f5a7c5a43c": "\\sum_{k=1}^{x}f(k) = C + \\int_0^x f(t)\\,dt + \\frac{1}{2}f(x) + \\sum_{k=1}^{\\infty}\\frac{B_{2k}}{(2k)!}f^{(2k - 1)}(x)", "4f3dd799be31f3dc4395ea115dad5e7c": " \\mathrm{I}_2(3) = \\mathrm{A}_2", "4f3de01af89592d6266fcdf3436633c2": "\\overline{X}=\\frac{1}{n}\\sum_{i=1}^n(X_i)", "4f3de1927d5173f3bfb058b1f15af7ac": "\\dot{V}(\\textbf{x})", "4f3df0c42ad4239d493cabc11e20edb8": "\\operatorname{Li}_{0}(z) = {z \\over 1-z}", "4f3e7c52a971b686f63d8320e5df29f0": "\\tfrac{23}{15}-\\ln(2) = \\sum_{n=0}^{\\infty} \\frac{1575}{2(n+1)(2n+1)(4n-3)(4n-1)(4n+1)(4n+5)(4n+7)(4n+9)}", "4f3ed49fb6c62673352b8fd3da7df2f4": "t \\rightarrow\\infty", "4f3f116134f65f90d927cc2486aa014e": "R_{k,P(k)}=S_k", "4f3f165c179f2fa936c8e6b3e8492ea3": "d(X_0, Y_0) < 1 + \\varepsilon.", "4f3f230039a2c874ffa45e1a2e104490": "\\delta_{t}=\\frac{A'(t)}{A(t)}\\,", "4f3fa5dbd49878225df1ff5124ac2db6": "I_p = N \\frac{k}{q}", "4f3fafef0f114250aeb714c4020df501": "Y_{t1}(u) = Y_{t2}(u)", "4f3fced1abc8549a2e6d7c62d72dade4": "e^{\\pm i\\pi \\sum_{k=1}^{j-1}f^{\\dagger}_k f_k}=\\prod_{k=1}^{j-1}e^{\\pm i\\pi f^{\\dagger}_k f_k}=\\prod_{k=1}^{j-1}(1-2f^{\\dagger}_k f_k).", "4f3feb2f399ebbb90f1b3b418eb58531": "Z_m(z)=A_1\\,e^{imz}+A_2\\,e^{-imz}\\,", "4f401500fe8ee26f1d71210c372580f4": "w/h \\ge 0.25", "4f406a8dfa0b3de7ae2a97aa8f5ba75c": " conf_{j} ", "4f4099ea4f1c0462ce72561497670103": "I_j =\\ O_{j-1}", "4f40a0ac4889d016342309fc9f922217": " \\mathcal{G} ", "4f40e383e2e258de1320dbe22caedbca": "\\scriptstyle 4 \\sqrt{2}", "4f410cf1b4f2cd07bfaa3acc7503f304": "H(\\omega)", "4f415d744efa1c3a1773a6fb22f40f2e": "ij=-ji", "4f41734ae1984652cd625836121005c0": "\\Big|\\prod_{i} r_{ii}\\Big| = \\prod_{i} \\sigma_{i},", "4f41a079aa6e14c534852f471c2dfd96": "\\scriptstyle{\\binom{n}{k}}", "4f41d7943207465a251593ec85b3b6fe": "\\bar{\\mu},", "4f41edf6f0e23ac3cf352ce918a3d2dd": "4\\times\\begin{pmatrix}5\\\\2\\end{pmatrix}", "4f41f0d974c1d3b5a959b6e58d06c90d": "\\cos 20^\\circ\\cdot\\cos 40^\\circ\\cdot\\cos 80^\\circ=\\frac{1}{8}", "4f42714aee6b799b443585cb735b2f85": "A = \\begin{bmatrix}\\,\\,\\,2 & 3 & 5 \\\\ -4 & 2 & 3\\end{bmatrix}.", "4f4285fadc9c7b9ed946b0fbc6eb1806": " \\sum \\sup_{|z|\\le r} |1 -F\\circ f^j(z)| \\le (1+|\\lambda|^{-1}) \\sum M(r)^j <\\infty.", "4f42ce4180354c463f6cd5fe31d34172": "E_{1,2} = M_{AA}-\\frac12\\Gamma_{AA}\\pm e^{i\\alpha}\\left|{|M_{AB}|}+\\frac12{|\\Gamma_{AB}|} e^{i(\\beta+\\pi/2)}\\right|,", "4f42d4a33e63fb3245c1364b417c8e92": "\n\\mathrm i^n \\operatorname{erfc}\\, (z) = \\int_z^\\infty \\mathrm i^{n-1} \\operatorname{erfc}\\, (\\zeta)\\;\\mathrm d \\zeta.\\,\n", "4f4327886ba5eb6debde26ceea88efbc": "\\scriptstyle w(0.01) \\times v(-1000)+w(0.99) \\times v(0)=w(0.01) \\times v(-1000)", "4f438a1c935d3997d1d224bc00f8e3cb": " S(x)= e^{ia f(x) } ", "4f43b170306b64eb49f16c2b43fd4ccf": "\\theta_{ik}=\\delta_i-\\delta_k", "4f43f3a06a42570ae0da675e08e41db3": "S^{(n)}", "4f43fb30cd934ac4684d6994c923dca0": "\\displaystyle{S_0(t)=e^{-tL_0}}", "4f4405d33620b33034c5f8e04991db76": "\\exp_{10}^3(4.55997)", "4f44285aca3e236c5048306fbade2f8d": "\\mu = 1", "4f4443b02e73d82a8b701c33bd87b610": "X_i \\sim \\mathrm{Laplace}(\\mu,\\beta)\\,", "4f444488c52e842ca188cc057bc19c1d": "I=I_\\mathrm{S} \\left( e^{V_\\mathrm{D}/(n V_\\mathrm{T})}-1 \\right),\\,", "4f4486005f9f1f93bbe7b2aeb42505b7": "J_n=\\int x^n \\cos{ax} dx \\,\\!", "4f4511b8c5464631067c037394a64adc": "\\int\\frac{1}{x^2(ax+b)^2} \\, dx = -a\\left(\\frac{1}{b^2(ax+b)} + \\frac{1}{ab^2x} - \\frac{2}{b^3}\\ln\\left|\\frac{ax+b}{x}\\right|\\right) + C", "4f455e3f97b1fc1bfedb90c1d0325302": " V(x) \\ = \\ V_{max} (1 - e^{-x /\\lambda})", "4f45bf1507f5ace45ff25334e53fece4": "K'", "4f45c230e00f7ddcf3a166ffdd8002be": "(b)\\text{ }R_{i}\\text{ is a connected region},\\text{ i}=\\text{1},\\text{ 2},\\text{ }...,\\text{n}", "4f46a0c4bc647ee447ebe9b40d4990a1": "E =k_eQ/r^2,", "4f46fd9567369053ad8cab6d72e3578d": "\\|v\\| := \\sqrt{v\\cdot v} ", "4f46ff5fdd5a6c6b2e13a191e8377a5e": "z = b e^{it} + {a\\over 2} e^{2it}.", "4f47356f30365742c4ba33269bf1a9e4": " \\frac{dy}{dx} = f(x,y), ", "4f476446b0508e985486adda9afb1b8a": "\nA = A_{N} p_{N} + A_{U} p_{U}\n", "4f478c1d24495cf69949c29aadc20822": "A \\rightarrow I: \\{N_B\\}_{K_{PI}}", "4f47a427704bf9864c5fa1090db9d760": "J = \\begin{bmatrix}0&-1\\\\1&0\\end{bmatrix} . ", "4f47a937b6b95133448612ecaeeb7de4": "g_i={di}/{i}", "4f47e7cefa5ae3a2a3ffa934485f2886": " R_0 ", "4f4820c8ced97e83f45f1a16fe3dd8cc": " \\frac{P^\\prime(z)}{P(z)}= \\sum_{i=1}^n \\frac{1}{z-a_i}. ", "4f4886c04b788f881889eb8a74e291f5": " |\\langle Tx, Ty \\rangle| =|\\langle x, y \\rangle|", "4f48a5d6306e221f0b2f48441a0870df": "\\tfrac{\\log(P(n)/P(n-1))}{\\log(n/(n-1))}", "4f48ad28a12ad23aba75dce680e8258e": "[x] = {\\mathop{\\darr}x} \\cap {\\mathop{\\uarr}x}.", "4f48eb560789e7c36ce00c0644399d3d": " F_1 = \\_ ", "4f491fbeebfe9f2d0834730146462fb2": "\\Sigma^0_{\\alpha}", "4f497cd5d92a8fca94d5cd020762f961": "d_{z^2}", "4f49d987d008fd5a33a8c75c85f24d0d": "\n\\left[ \\begin{array}{ccccc}\n1 & a_0 & a_1 & a_2 & a_3 \\\\\n0 & 0 & 2 & a_4 & a_5 \\\\\n0 & 0 & 0 & 1 & a_6\n\\end{array} \\right]\n", "4f4a010b8b5c78a1170f6e4c5a04285e": "n\\rightarrow\\infty", "4f4a3bc8b33b1502b8f977e1764d6083": "\n\\mathbf{f_{0:t}} = \\mathbf{f_{0:t-1}} \\mathbf{T} \\mathbf{O_t}\n", "4f4a89c0e2acc7bb838c8ba1a90be6b7": "\\frac{c_p}{c_v}", "4f4adba26ed62abdb09eb0463d360d7e": "KO_*(X)", "4f4b2d5c1854b010bc5e7033b8a40496": "\\|Ax\\|_a \\leq \\|A\\|_b \\|x\\|_a", "4f4bd5fb128b4b1e4150d63130d29b9a": " C(x,y)= G_{\\sigma}*I(x,y) ", "4f4bda154a48bd8adb76e1d2ab28af2d": "T_{Dhuhr} = 12 + T_Z - (L/15 + T_E)", "4f4cb61e1d7320192628024e3d7c1e6e": "h=", "4f4cbf909c2808fc820ab21c679760b7": "\\textstyle{\\int_t^u}", "4f4ce858033d5d4413f85ab601bb1b8a": "\\left [ -i\\gamma^\\mu\\left ( \\partial_\\mu + ieA_\\mu \\right ) + m \\right ] \\psi = 0\\,", "4f4d4ff7c8a6d3469f9610c829a10634": "Y(t+\\Delta t) = F(Y(t))\\,", "4f4df637deee00a36a24bdd2c1884c11": " f = q \\, g + r ", "4f4dfadc18a5e8b0caf12687071ac8fe": "U=-\\mathbf{m}\\cdot\\mathbf{B}", "4f4e1c434bad9964a9d49331ec469cf9": " \\{T^a, T^b\\} = \\frac{1}{3}\\delta^{ab} + d^{abc} T^c. \\,", "4f4e57f243bab5f0c9e1ecaf46ca06dc": "\\alpha \\times \\beta", "4f4e8f907c69ca5557d06df2eef11f75": "\\begin{array}{rccc}g\\colon&\\mathbb{C}&\\longrightarrow&\\mathbb{C}\\\\ &z&\\mapsto&\\frac{f(z)-a}{\\omega}\\mbox{,}\\end{array}", "4f4ef5b2019b6495c2938837b2389c1f": "f = sum", "4f4ef89d81161b3ac3f01bf46a951e03": "y = \\mathbf{B'}x", "4f4f87126d472c3ae2298e43efde1338": "[x^3:x^2y:xy^2:y^3],", "4f4fd37066d6a0177b9f12493b731a75": "\\forall x\\in\\mathbb{R}^{+}", "4f5090bd51281c8f28d330a395a5ca7e": "3 \\times \\sqrt{3}", "4f509fa986537b588a571097578e8b0c": " Z[J] = C e^{i \\int d^d x (\\mathcal L [B(x)] + J(x) B(x))} \\left(\\det \\frac{\\delta^2\\mathcal L}{\\delta \\phi(x) \\delta\\phi(y)} [B]\\right)^{-1/2} + \\cdots ", "4f50d5e7e545793a5547321572867a64": "\\scriptstyle (\\hat{\\boldsymbol{r}}_{\\text{rec}},\\, \\hat{t}_{\\text{rec}}) \\;=\\; \\arg \\min \\phi ( \\boldsymbol{r}_{\\text{rec}},\\, t_{\\text{rec}} )", "4f515d412b8f6d8d38bde974ad3b2fd9": "\n\\vec{E}^i(\\vec{r},t) = \\frac{\\vec{J}(\\vec{r},t)}{\\sigma} + \\frac {\\partial\n\\vec{A}(\\vec{r},t)}{\\partial t} + \\nabla \\phi (\\vec{r},t)\n", "4f517cd93ece2397a0f3af187ac4188f": " {L} = n \\hbar ", "4f51ad5a4b7bba954679b4d02b5d2e16": "y>x\\ ", "4f520bd05bf84d0d2fd3ef4652ad81c5": "T_x = \\frac{3}{2}\\frac{Gm}{a^3 (1-e^2)^{3/2}}(C-A)\\sin\\epsilon\\cos\\epsilon", "4f522636d6253498a2eec5347abd0c12": " b = 0 ", "4f523082e11cf07dffcc903b2b752163": " \\vec 0 = \\vec\\mathrm{Fsail} + \\vec\\mathrm{Fhull} + \\vec\\mathrm{Fgravity}", "4f52411f01f094be5a8a0878d92c612b": "N(S,H,B) = \\sharp \\{ x \\in S : H(x) \\le B \\} . ", "4f52595ede4601adbdedbd9e8ebaffdf": "\\forall k, f_k(x) \\le f(x)", "4f526ddb3834e3e06f13e61718122bec": "g \\ge \\frac{a + cb}{2cb}", "4f529518c903e3ba2995834cb6c44648": "h: S \\rightarrow \\{1, 2, \\dots, k\\}", "4f52e06bfcbc9cc07aa460649561cb70": "\\beta(s) = \\left(\\frac{1}{\\sqrt{k}}\\sin\\sqrt{k}s, -\\frac{1}{k}\\cos\\sqrt{k}s\\right)", "4f52e0cc2e50afbc9f5c2a8c1572d7b6": "m \\in \\{0,1\\}^k", "4f536253a46d3f9ece37c90ecc869685": "\\langle j\\,m|J_+^\\dagger J_+|j\\,m\\rangle = \\langle j\\,m|J_-J_+|j\\,m\\rangle = \\langle j\\,m+1|\\alpha^*\\alpha|j\\,m+1\\rangle = |\\alpha|^2", "4f53a8fbbabd7faebce1580d1b8cda84": "\\Theta_{\\Gamma_8}(\\tau) = \\frac{1}{2}\\left(\\theta_2(q)^8 + \\theta_3(q)^8 + \\theta_4(q)^8\\right)", "4f53e3df60ce7f19cf6c7c7197f0f85e": "\\Delta(x)/x^{1/4}", "4f5423c6cc0c649471619486b2e9ec71": "\\hat{x}|\\psi\\rangle = x_0 |\\psi\\rangle.", "4f54438ffe23d70cb87884b5da7055b0": "\\{ \\tau \\leq t \\} \\in \\mathcal{F}_{t}", "4f5446877b0e5235375207f90ad8bd91": "\np(z)=\\cfrac{a_0}{z+\n\\cfrac{a_1}{z+\n\\cfrac{a_2}{z+\n\\cfrac{a_3}{z+\\ddots}}}}\n", "4f548c6bbf77ea959853a06e44ad1160": "\\theta_{Bn}", "4f54f3f6cb8d9b908356893d267d6f2d": "\\sigma_{p}=\\sqrt{\\langle \\hat{p}^{2} \\rangle-\\langle \\hat{p}\\rangle ^{2}}.", "4f55499b7851e600370b24ba9b2dabe6": "\\mathbf{X} = \\{\\mathbf{x}_i\\}_{i=1}^{n}", "4f554b7172a503045703242d2fb4c85a": "\\Delta x_{\\mathrm{meas}}", "4f55552bfc22ce7ffe3da53bd3828cb8": " = \\displaystyle{{G_a\\over4\\pi}\\lambda^2} \\,", "4f555b22d2bba3e5203e375728bc0476": "f(x) = x^2 - 2", "4f5577daa2ec5fc075dc43382b885ffa": "M\\preceq L", "4f558526ca8fe2556f3bdeede39eac78": "F[y]= \\int_a^t \\sqrt { 1 + y'^2 }\\, dt", "4f55c64e72078521798b636d3a8af734": "\\Phi(z, w) = w^{s+1}-\\sum_{i=0}^s a_iw^{s-i} - z\\sum_{j=-1}^s b_jw^{s-j}.", "4f55d8b3e6bda32c5fbaf051e39725db": "\\|\\mathbf{a \\times b}\\| = a b \\sin \\theta_{ab} \\ , ", "4f55ee073254e95c8b1f164534e54968": "f'(r)=0\\,", "4f55f95c5299065fff3b13b0bea0ddfa": "\\sin(A \\pm B) = \\sin(A) \\cos(B) \\pm \\cos(A) \\sin(B).\\,", "4f5606ec4c68a70c5ef9a2a68954d13c": "f^{(-n)}(x) = \\frac{1}{(n-1)!} \\int_a^x\\left(x-t\\right)^{n-1} f(t)\\,\\mathrm{d}t", "4f5668b9808c2e008fa4d09944c6f8e3": " \\frac{1}{2} \\left( -v \\partial_u + u \\partial_v \\right) ", "4f56881b5f28bb0329091ca8d082a17c": " \\underline u_i=u_i(p^{min}), \\ \\overline u_i=u_i(p^{max}) ", "4f56b1c6f6916041a8b544d1a7e0b664": "S=\\{1,3,4,7, \\ldots \\}", "4f57063f52045c499ba7412d57228d26": "10^{10^{10^{122}}}", "4f571e8ed02144d86f9b4d17a2ad0a8e": "a\\ \\mathrm{arsech}\\frac{x}{a}, \\,\\!", "4f5737ab4f0b6c1331c9773fddcafd6a": " w_{k} ", "4f57825966894235d758afba0bdf2d73": "\\sin x = \\mathrm{Im}\\{e^{ix}\\} ={e^{ix} - e^{-ix} \\over 2i} ", "4f57833bf8b8a46c627f2b84b8680fac": "v_v = v \\sin(\\theta)", "4f579de79ce3334d014db485ce180367": "T=30", "4f57e468ce83344c6f2614057c186014": " \\mathrm{MA} = \\frac{T_B}{T_A} = \\frac{\\omega_A}{\\omega_B}.", "4f57fc88f8e64ee0ef3ae551b6c6cb6f": "3,\\quad 3.1,\\quad 3.14,\\quad 3.141,\\quad 3.1415,\\quad \\ldots", "4f580f1198b3f31f4e879c425c7ac01c": " \\begin{array}{llll}\n\\mathbf{ab} = & a_1 b_1 \\mathbf{i i} & + a_1 b_2 \\mathbf{i j} & + a_1 b_3 \\mathbf{i k} \\\\\n&+ a_2 b_1 \\mathbf{j i} & + a_2 b_2 \\mathbf{j j} & + a_2 b_3 \\mathbf{j k}\\\\\n&+ a_3 b_1 \\mathbf{k i} & + a_3 b_2 \\mathbf{k j} & + a_3 b_3 \\mathbf{k k}\n\\end{array}", "4f581ac7950514dbbd7606d262b3b5fc": "\\frac{1}{\\kappa} = \\frac{\\int_0^{\\infty} \\kappa_{\\nu}^{-1} u(\\nu, T) d\\nu }{\\int_0^{\\infty} u(\\nu,T) d\\nu}", "4f58200f4592979c3edffd7122b8ac8d": "0=\\boldsymbol{\\psi}(0)", "4f583161acb7e316b5dae5e4655dc524": "\\ \\mathbf U(\\mathbf x,t)=U_J\\mathbf E_J", "4f58578923e8db6fac275d287e301db7": "\n\\mathcal{G}(\\mathbf{x} ,\\tau|\\mathbf{0},0) = \\frac{1}{\\mathcal{Z}}\\sum_{\\alpha,\\alpha'} \\mathrm{e}^{-\\beta E_{\\alpha'}}\n\\langle\\alpha' | \\psi(\\mathbf{x} ,\\tau)|\\alpha \\rangle\\langle\\alpha |\\bar\\psi(\\mathbf{0},0) |\\alpha' \\rangle.\n", "4f58910e3f9bb7b48f533efccf723424": "Y_{0,4} = H_e", "4f58dd2a0b8d689d145b22694901d90c": "\\Gamma\\vdash t \\mathbin{:} \\sigma", "4f590e47502dc725632e1373dd2e04f1": " \\int S = {\\rm Tr}_\\omega( S |D|^{-d} ). ", "4f594acfbaf08607a45001f07c849b77": "1/\\gamma \\,\\!", "4f5953164dd5b1c60d1f32bf94eb11bb": "X\\setminus C", "4f5953ca6f47807307f19d741f0ad596": "D f(\\mathbf{a})\\!", "4f59610ce52e08a70febea686de12c3d": "V(f,\\Omega) = \\int\\limits_\\Omega\\left|\\nabla f(x)\\right|\\mathrm{d}x", "4f599ac31644776059300d3ce2afeba9": "\\xi^2", "4f59b3fd6f5001542919c239391a3278": "0.8333", "4f5a3356d307e70fad6df1c845bdd7f3": " h(t) = \\left(1/4-1/4{e^{-4/3t}}^{n_2}\\right)(\\lambda e^{-\\lambda t})\\ ", "4f5a47bad410984ecf56bd9f341c0dca": "\n= \\sum_r \\left(\\; A[-r+k,k] + (r-k){dA \\over dJ}[r]\\; \\right)\n\\left(\\; B[0,k-r] + r {dB\\over dJ}[r-k] \\; \\right) -\n\\sum_r A[r,k]B[0,r]\n\\, .", "4f5a5a7b9a1978d710850a8e1eaaea7e": "\\displaystyle{G=KAK,}", "4f5a8fb574e314dff9ff38e53deece8e": "\\ell_{(M,\\varphi)}(\\tilde x,\\tilde y)", "4f5aebb369a84057d848d88fc498d95b": "H_{\\nu} (\\omega)=-e^{-j \\nu \\frac {\\omega - \\pi} {2}} P_{\\nu}( \\cos (\\frac {\\omega} {2}))", "4f5aff32addda8afc5692d0d02946567": " \\gamma_{ty} ", "4f5bda896e36205e3a5b8725015ff4f9": "(\\mathcal{F}, \\mathcal{F})", "4f5c03add4efc3974e9a7d6ef0d97196": "\\scriptstyle a \\;\\neq\\; b", "4f5ccfa916650ca1e9d2c562caf3342d": "m_l=-e \\vec{L}/2m", "4f5cd641f4d5c05f59616842050f0cf2": "x-\\lfloor x\\rfloor", "4f5d0329a9476a049b9ecc071b89f25a": "(\\Pi^\\top)^{-1}", "4f5d06449e949f35c4ab234ec168825b": "p_{00}^0 = 1", "4f5d3523a7fcdc8b27d4376150773c7f": " \\pi_{t+1} = \\pi_{t}^e + c (\\pi_{t} - \\pi_{t}^e) ", "4f5d46d9763225e296833c0d90635e20": "\\vec{\\ell_1}", "4f5d4e05c5385036d123df7226afa230": "k = \\lceil \\log\\log n\\rceil", "4f5d78add9198c2df88336d9cbee96a9": "L(t, \\mathbf{x}, \\mathbf{v}) = \\frac{1}{2}m \\sum_{i=1} ^{3} v_i^2 - U(\\mathbf{x}),", "4f5dd42c90ccb37177b989beb8a93980": "\\Gamma = \\int_{A} \\vec{\\omega} \\cdot \\vec{n} dA = \\oint_{c} \\vec{u} \\cdot d\\vec{s} ", "4f5dd5c5cafbe79953f146d615a8f160": "t(\\cdot,\\cdot)", "4f5e0742ae56f2ee7fded205628e2989": "Y=\\beta_{10} +\\beta_{11}X + \\varepsilon_1", "4f5e7a76abb46f9c0c43f0a7ad81a56b": " \\tau_{xy} = \\frac{f'' (0) \\rho U^{2}\\sqrt{\\nu}}{\\sqrt{Ux}}.", "4f5ea761ca538ddde100a89fe688730e": "g_{\\nu\\mu}\\;", "4f5ebb0bde1a43460b70efaead855184": "\\oint_{\\part V}\\mathbf{g}\\cdot d\\mathbf{A} = -4 \\pi GM.", "4f5edddd8614771a031dbe7499135e16": "q\\cdot(P(\\lambda(y_i)=C|H_0)+(P(\\lambda(y_i)0", "4f78782b8989961c30a1ac193ad65ed5": " e^{x}>x ", "4f78e9f49f526605a90d75c125cea3b8": "A'(W) < 0", "4f7907d42695fa783738265cbcc16e15": "\\psi_{\\mathbf{k}}(\\mathbf{r}) = \\frac{1}{\\sqrt{N}} \\sum_{\\mathbf{R}} e^{i\\mathbf{k}\\cdot\\mathbf{R}} \\phi_{\\mathbf{R}}(\\mathbf{r})", "4f798b2ab4081424db59fd62ddc94977": "r_{\\mathrm{e}} = \\frac{\\alpha \\lambda_{\\mathrm{e}}}{2\\pi} = \\alpha^2 a_0.", "4f79b4a1b04cdc811a290a90ad8a735a": "\nw_i = \\frac{1}{\\sigma_i^2}.\n", "4f79dc99bc1cb2d15cc3681f14a24e90": " = L ", "4f79fcdeacf0bc21663a8666efeef580": "y^2 + a_1 x y +a_3 y = x^3 + a_2 x^2 +a_4 x + a_6. \\, ", "4f7a31de1aa625b04680cc1d8d36f027": "F_n^{(r+1)}=\\sum_{i=0}^n F_i F_{n-i}^{(r)}", "4f7a3b602a74bd4cbce3986ef7a50f0a": "\\operatorname{ran}(T) = \\operatorname{ran}(TT^*)", "4f7a4928b5b4b6e7920d722d7fa1c5f6": "x_{\\mathrm{FOH}}(t)\\,= \\sum_{n=-\\infty}^{\\infty} x(nT) \\mathrm{tri} \\left(\\frac{t - nT}{T} \\right) \\ ", "4f7a97fa894acb537cf7fef569f1312a": " \\mathbf{x}_{k} ", "4f7abe7f8fb1e9a8947fe750dcc6a0ba": "EL(\\Gamma)\\,EL(\\Gamma\\,')=1", "4f7b015018b146e297470b341cefb200": "O(n^3) ", "4f7b06085d2d70524665a7d442766ee4": "\\hat\\sigma^2 = \\frac{ \\|\\hat{r}\\|^2}{ n - \\mathrm{tr}( 2 H - H H' ) } = \\frac{ \\|\\hat{r}\\|^2}{ n - 2 \\, \\mathrm{tr}(H) + \\mathrm{tr}(H H') } \\approx \\frac{ \\|\\hat{r}\\|^2}{ n - 1.25 \\, \\mathrm{tr}(H) + 0.5 }.", "4f7b216245081da92b9dbf7bdd51f68a": "\\operatorname{cl}(\\varnothing)=\\varnothing", "4f7b3c7494bab79f902399cdfd2aa4a3": "\n\\begin{align}\ns(t) & = \\Re\\left\\{\\nu(t) e^{j2\\pi f_c t}\\right\\} \\\\\n & = \\sum_{k=0}^{N-1}|X_k|\\cos\\left(2\\pi [f_c + k/T]t + \\arg[X_k]\\right)\n\\end{align}\n", "4f7b402d17e94c3b54735e74d0be2abd": "D_1 = A_{1,1}", "4f7b43f06d8b5ef3776a65d93fc113e3": "{A}\\,", "4f7bfc2b013bd99865b3953890964538": "\\alpha_{\\text{r}}", "4f7c41f273ca7ec8925ca445968bd4be": "\\int v \\, ds", "4f7c70750372fce8841bce7576210440": " M4 = \\frac{ \\sum_{ i = 1 }^K | X_i - m | }{ 2 \\sum_{ i = 1 }^K X_i } ", "4f7c7c5cc50e7f1621cd7e75103c3a4c": "Q_c=1/a.", "4f7c90733e959d909243d959f923b1d8": "\\zeta^{-1}", "4f7caba7bf267810cf3dca5e1f8c9c9a": "\nz^2 + n = 0 \\quad \\Leftrightarrow \\quad z = \\pm i\\sqrt{n},\\,\n", "4f7cd3774b864624fa0661655a4ba635": "\\begin{cases}\n y_t = \\alpha + \\beta'x_t^* + \\varepsilon_t, \\\\\n x_t = x_t^* + \\eta_t.\n \\end{cases}", "4f7cd767936f8790afcb2c9812028ee0": "\\operatorname{Hom}_C\\left(\\coprod_{j\\in J}X_j,Y\\right) \\cong \\prod_{j\\in J}\\operatorname{Hom}_C(X_j,Y)", "4f7cd9f09afa0ae9642cac5ea4158bc9": " \\text{(2)} \\qquad \\delta W = P \\, dV. ", "4f7ce7b97227b01806e2dc9c54166868": "while X\\ne Y", "4f7cfa6b3b021553a4da70da7ea785c5": "c({\\alpha})", "4f7d6c17d6b551674eff16606466fcab": "2^2 + 2^1 + 2^0", "4f7dad86a4650676968ed8dc3f35ac31": "(X,\\sigma(X,Y))' \\simeq Y", "4f7dae9db7155fd68f8055eaf1ee06f5": "E_{0, 0} = 510,260 * \\frac {510,000}{510,260} * \\frac {500,200}{510,260}", "4f7de9ed5a726c67d114c847d10cfdac": " \\textbf{f}_1 \\cdot \\textbf{h} = \\textbf{g} -\\textbf{f}_2 \\cdot \\textbf{h} \\pmod q", "4f7e10f02ad15cd75455913a900db12f": "\\sum_{n=1}^\\infty \\left | a_n \\right \\vert", "4f7e2824a4ad516f31a578600e61d19c": "Q_{10}=\\left( \\frac{R_2}{R_1} \\right )^{10/(T_2-T_1) }", "4f7e2d6bb2c2ba500a95fe8d86da0c41": "G_{4}", "4f7e3bcf30c7ca539e05da87c21253cc": "x_i\\;", "4f7e4cf72d50702cb953b64f3735c132": "k_{nr}a", "4f7e5b059b8262bb9ff29bf712d6f233": "\\mathbb{Z}/\\ell ", "4f7e78e9b7c7791580fd8e41a621059b": "1 + 2 + 3 + \\cdots + n = {n(n+1) \\over 2} = {n^2 + n \\over 2}", "4f7e97a3c26e6bc09c366ec94038ff0d": " = \\sigma_\\mathrm{tot}^{-1} \\int \\Delta \\vec{p} \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\mathrm{d} \\Omega ", "4f7ea4a8e14bcb60dc61b7dedd8a5e5e": "\\gamma=\\mu-\\alpha_ja", "4f7ec4b540d9373ba5c68411418c3a40": " W", "4f7eea238a994034b59d8f90dc7a6c3a": "x,y,z,t \\in \\left\\{0, -1, 1\\right\\} ", "4f7efe4c715d4b794c78311c12be4375": "\n\\vec{A}=2\\vec{a}.\\vec{r_1}-\\vec{a}\n", "4f7f0951a82b9a57687eff915c45d3cb": "\\sqrt{r \\left(\\cos \\varphi + i \\, \\sin \\varphi \\right)} = \\sqrt{r} \\left [ \\cos \\frac{\\varphi}{2} + i \\sin \\frac{\\varphi}{2} \\right ] .", "4f7f64f0705617539aba054d675a0696": "\\phi(A)", "4f7f993dc6b92cf0ad3e36c6574272ec": "[V] = [k^{\\mbox{dim}(V)}]", "4f800529019c6d564e396415ab7f4676": "P(x_1,\\dots,x_n)=\\sum_{i=0}^d x_1^i P_i(x_2,\\dots,x_n).", "4f805bfefef6a9a9b92d26fefdcfc0a5": " H_n(C) = Z_n/B_n = \\operatorname{Ker}\\, d_n/ \\operatorname{Im}\\, d_{n+1}. ", "4f8115e5cff172f50d874634c9dbed52": " \\mathbf{v} = \\sum_{k} c_{k} \\mathbf{e}_{k} ", "4f82704c80ee36d45ea4f436c1f368e4": "S_{i+1}\\colon \\Z^k \\to S", "4f83724dba796b5aa847a4d7d32c53f7": "L \\varpropto M^{3.33}", "4f83ae3bb58af84b5eb0cbb6836adb88": "f(p_{1},p_{2},p_{5})", "4f83b625885109f4f0be5e7fd12088cb": "S_N = \\sum_{n=1}^N a_n.", "4f83c5468364e10155008d7edbac585f": "\\mu(x)=\\frac{1}{\\ln^2\\left(\\frac{x}{1-x}\\right)+\\pi^2}", "4f8430ff59f2548324f8a11d6298eb6e": "\\Delta y /\\Delta x", "4f843bf234ed8ae96f9a664feea2b3f7": "\\mathrm{Sp}(n)", "4f8455d23d574ec545ac282bf4c027c9": "a_2 ", "4f84a786b4d46edad6da0aa9acbf39d3": "{P-1\\over P}{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "4f84cbbd9bfef6b11efae16431857922": "\\displaystyle (f*g)(z)= \\sum \\{(fx)(gy)\\mid z=x\\circ y \\}", "4f84d897c85713d6e295d23beb2f031c": "\\alpha \\in R", "4f84db7080b44ff7c04eba184c08e57e": "\\Diamond P", "4f853fd6200e8c38e70ccfd55a3d9340": "H=M\\,", "4f855f4817af9ac30d2ac10217497e51": "x , \\ y \\,", "4f8568b3805b6941de8447e177ed8444": "y \\times y <2\\,.", "4f85b110f6315310b1125f4f350e751d": "A \\subseteq \\{0,1\\}^{m}", "4f85d2b7f4b14610932fcaa025a06dac": "A = (a_{ij})", "4f86293d3deac430b40abcd16436dfb8": "\\varphi(Q_n)=\\sum_{m=0}^{n-1} \\binom{m}{\\lfloor m/2\\rfloor}.", "4f864eda292637c5fc8d0ee28c9a66fa": "\\rho(r) = \\rho_0 e^{-\\alpha r^n}.", "4f868e9405dfafc9bac218357580316d": "f(z) = z^{\\frac{3}{4}} (3-z)^{\\frac{1}{4}}.", "4f86eb5fe7473d783d90eb474335d147": "\n \\quad \\min \\limits _{D, X} \\{ \\|Y - DX\\|^2_F \\} \\qquad \\text{subject to } \\quad \\forall i \\;, \\|x_i\\|_0 \\le T_0.\n", "4f86f92f94ddb2dff97a11b48f5cadea": "_{P}(f) < 0", "4f874e187e3f2a43fb6a4b00ba8a7d77": "d_{3/2,-1/2}^{3/2} = \\sqrt{3} \\frac{1-\\cos \\theta}{2} \\cos \\frac{\\theta}{2}", "4f879a4a38a17caaa4d56df983d1626f": "\\frac{\\partial}{\\partial t} \\sum_\\omega \\langle \\hat{B}^\\dagger_{\\omega} \\hat{B}_{\\omega} \\rangle = -\\frac{\\partial}{\\partial t} \\sum_\\mathbf{k} f^e_\\mathbf{k}", "4f87af74f96e0d7b98a8632cedbf7860": " \\nabla ", "4f87c02ac01c4ee59a54872003d8dc74": "X, Y \\subseteq I", "4f87dfb00c3b0b6114ec283e415d0566": "M_J", "4f8846ad99149e10fccc2e7de0aac1ee": "Z(\\mathfrak{g})", "4f88492feee3e622062f83aefff5dca3": "\\scriptstyle \\mu \\;=\\; \\frac{1}{\\lambda} > 0\\,", "4f88cc10cf161a86013c359fd561a024": "\\color{Fuchsia}\\text{Fuchsia}", "4f88f6f326a22be369becda366997c82": "\\phi_X(t) = e^{ix_0(t)-\\gamma(t)}, \\!", "4f8900fad412e4bde9f4703fe11cbe81": "\\|x\\| \\le \\|x + y\\| \\le \\|x\\|", "4f891694ccfb0c3e26abe94fda87be29": "f(X)=1/X-D", "4f8939815dd5f96edd7793bca7c92d45": " F(z) = \\exp \\left( \\sum_{l\\ge 1} \\frac{f(z^l)}{l} \\right).", "4f896c625bd62e56125d2492ee669fad": "\\,\\phi(v_i)=(\\|v_i-c_i\\|^2 + a^2)^{-1/2}", "4f89710f3aac8efbaa84fdd76539b9dd": "\\mathrm{div} \\, \\mathrm{grad} \\, \\varphi\\,", "4f898d69d388764d4a751d03249e38de": "X_{\\mathrm{f}}^\\prime", "4f89c1bb6e652c948ffd44cc22307b79": "K_7", "4f89f8e92272be0a61b14baf5b544c4c": " f \\cdot g \\mbox{ and } g \\cdot f", "4f8a0664c2b7596fd50f34453509e044": "\\begin{align}\n\\mathcal{A}\\left\\{c_1 x_1(t) + c_2 x_2(t) \\right\\}\n&= \\int_{t-a}^{t+a} \\left( c_1 x_1(\\lambda) + c_2 x_2(\\lambda) \\right) \\, \\operatorname{d} \\lambda\\\\\n&= c_1 \\int_{t-a}^{t+a} x_1(\\lambda) \\, \\operatorname{d} \\lambda + c_2 \\int_{t-a}^{t+a} x_2(\\lambda) \\, \\operatorname{d} \\lambda\\\\\n&= c_1 \\mathcal{A}\\left\\{x_1(t) \\right\\} + c_2 \\mathcal{A}\\left\\{x_2(t) \\right\\},\n\\end{align}", "4f8a4535693d8cba6143c1d41061c403": "h(v_0) = log(|v_0| + 1) \\ ", "4f8af48924e52d6cd620725758525de3": "R = -\\frac{6 \\, \\Box \\phi}{\\phi^3} ", "4f8b0660ebda8088af908f83ca8e879b": "\\mathbf{T}^{(\\mathbf{n})}", "4f8b40a797d181373a9d1166cbe3989d": "3*x^4+14x^3+21x^2+10x-31680=0", "4f8b8c656a6d8f10eb3b4b345d5b4654": "z_1 = (X-(k-1)/2)^{1/2}", "4f8bcb6a71eedf794adf7f5324359284": "a_y = b_z c_x - b_x c_z \\, ", "4f8bdcdacf896808f569defdf08011f9": " {n \\choose k} = \\frac { ( n - 0 ) }1 \\times \\frac { ( n - 1 ) }2 \\times \\frac { ( n - 2 ) }3 \\times \\cdots \\times \\frac { ( n - (k - 1) ) }k,", "4f8c163630bd047bffd4c38df46fbf50": "\n\\begin{align}\nE_\\mu g(x)^T(x - \\mu) & = \\int_{R^d} \\frac{1}{\\sqrt{2 \\pi \\sigma^{2d}}} \\exp\\left(-\\frac{\\|x - \\mu\\|^2}{2 \\sigma^2} \\right) \\sum_{i=1}^d g_i(x) (x_i - \\mu_i) d^d x \\\\\n& = \\sigma^2 \\sum_{i=1}^d\\int_{R^d} \\frac{1}{\\sqrt{2 \\pi \\sigma^{2d}}} \\exp\\left(-\\frac{\\|x - \\mu\\|^2}{2 \\sigma^2} \\right) \\frac{dg_i}{dx_i} d^d x \\\\\n& = \\sigma^2 \\sum_{i=1}^d E_\\mu \\frac{dg_i}{dx_i}. \n\\end{align}\n", "4f8c219cad65760183f4a6b351a82b46": " \\mathbb{A} = \\{x_\\infty, x_2, x_3, x_5 ...\\}", "4f8c992efcfcf2e4367a87db45f74774": "L_{\\lambda} = L \\cap M_{\\lambda}", "4f8c9e7da54b8bad00f3bfae5869cc06": "X_C", "4f8cd1e5e73414c95c16193f3a568653": "\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a}^\\mathrm{T} \\mathbf{b} = \\begin{bmatrix}\n a_1 & a_2 & a_3\n\\end{bmatrix}\\begin{bmatrix} \n b_1 \\\\ b_2 \\\\ b_3\n\\end{bmatrix}.", "4f8ceec19f472f3191bcd97d0a22d0d3": "\\int_0^{\\infty} r^2 dr\\int_0^{\\pi} \\sin \\vartheta d\\vartheta \\int_0^{2 \\pi} d\\varphi\\; \\psi^*_{n\\ell m}(r,\\vartheta,\\varphi)\\psi_{n'\\ell'm'}(r,\\vartheta,\\varphi)=\\langle n,\\ell, m | n', \\ell', m' \\rangle = \\delta_{nn'} \\delta_{\\ell\\ell'} \\delta_{mm'},", "4f8cff565a529116ebc4abcb57c4b771": "(x+0y)z = xz + 0y\\ ", "4f8d4180f8be16993d55c82210d55401": "0 = (\\mathcal{D}_\\alpha - \\nabla_\\alpha) \\eta_{IJ}", "4f8da190603e22a21657bd8ce197253c": "\\scriptstyle \\left\\langle r^2 \\right\\rangle", "4f8dd6e509d2dcedf7f5f35d2e53fe0b": "x\\le x.", "4f8dd9b7cbadd667f04b516097f690d6": "a\\neq 0:", "4f8debc79a0ae38e755782ddfe1ff329": "\\mathbb{E}_{-j}\\{\\ln P(H,V)\\}", "4f8dfd7b4c3402d8f87c9e1b3bfab4ed": "\\binom{13}{10} = \\binom{13}{3} = 286,", "4f8e1995e3690535cc2de5b4726fd2ed": "\\Delta E \\Delta t \\ge \\hbar \\ , ", "4f8e6a00ff6c3846d2708b0a4250ea15": "\n s = \\sqrt {R_\\mathrm {E}^2 \\cos^2 z + 2 R_\\mathrm {E} y_\\mathrm{atm}\n + y_\\mathrm{atm}^2}\n - R_\\mathrm {E} \\cos\\, z \\,,\n", "4f8e732acd7925c8c28d0ecfe1c8c5d9": "\\{\\mathcal{F}^W_t\\}", "4f8ef245c96b697e50f0688f991ac1fb": "\n\\frac{\\mathrm{d}^n}{\\mathrm{d}x^n}\\ \\mathrm{e}^{\\alpha\\,x}\\,=\\,\\alpha^n\\,\\mathrm{e}^{\\alpha\\,x}\n", "4f8f07f51ff1d7d7edf529e2cfb76ed9": "n=\\frac{2\\pi}{T}", "4f8f289f35a3e6f58fb7e12ab88dcdb6": "-1 < f'(x_\\star) < 0", "4f8f62716b7156f8c48a4083c99064fb": "\\frac{1}{10^a p^k q^\\ell \\cdots}\\, ,", "4f8f637828cc51138a92f4e81e66b667": "\\Phi(t, p)", "4f8f6ad5280e835339e46ca48197fe5b": "\\int_{X_2} g \\, d(f_* \\mu) = \\int_{X_1} g \\circ f \\, d\\mu.", "4f8f8e2b360645646d304b436eaf2c00": "\\hat{H}_\\text{JC} = \\hat{H}_I +\\hat{H}_{II}", "4f8f9f76b2dc6f716401b15e9b52282b": "-\\frac{\\partial\\ln(c_{t+1}/c_t)}{\\partial\\ln(u'(c_{t+1})/u'(c_t))}", "4f8ff66da5332ba5105169f9aa437937": " \\operatorname{de-let}[M\\ N] ", "4f8ff6e14b699f6e372748bdb1528b05": "{\\mathbf U}^2 = \\eta_{\\nu\\mu} U^\\nu U^\\mu = -c^2 \\,,", "4f900130230a9637245003efcecbcbbc": "\\mathbf{x}=\\boldsymbol{\\mathsf{L}}^{-1}\\bar{\\mathbf{x}}=\\boldsymbol{\\mathsf{L}}^{\\mathrm{T}}\\bar{\\mathbf{x}}", "4f9061b4f01a83bfd1d4a45880f90c81": "g_{n+1}", "4f90b861ada0e69b5e5843714bc8d240": "{}^*C_{abcd}\\, k^bk^d=\\beta k_ak_c", "4f90c8d5e9173e3c280e6a6f8215503c": "1/n \\le V_2\\le 1", "4f90e33de8b2e7b19c0d9e1972ef05eb": "\\mathcal{Z(C)}", "4f9129f603ed6301f2db733af4444f17": " C_{(\\pm)} ", "4f916dee1e805aecdd7d7fd6a19cceac": "\\alpha\\ \\cdot p_2 = 1 \\,", "4f91a2993fc1dec492255bd79c2207d9": " i = 1, \\ldots, m ", "4f92752253cfb4300184ea5363238904": "\\Delta_n(R) = Spec(R[\\Delta^n])", "4f92b5a8e6755f9e82fb83d440560a08": "\\bold{p(r)} = \\varepsilon_0 \\chi(\\bold r ) \\bold {E(r)} \\ , ", "4f92f31ee89eac6c426419e78a0c7d1a": "f(z) = \\frac{1}{h(z)}", "4f931dd098f707d6de9523c562df50e9": " \\lim_{t \\to \\infty}f(t) = 0 ", "4f932e1d43adb9bb4a356b17bf5bc29e": "i\\in [1,M]", "4f937b6fac97eefadb6835d6480842e4": "\\Omega_H=(1/2M)\\tan(\\lambda/2)", "4f93a1aff8b0302fe3937e3a61bdde88": "P_{\\lambda,\\theta}(Z\\in A)=\\int_{A} \\exp[\\theta \\cdot z-\\lambda\\kappa(\\theta)]\\cdot \\nu_\\lambda\\, (dz)", "4f94a50c59b0fe7e7f94bf327f14dd74": "V=6\\pi r^3", "4f94a9b3816221b6ed47a3df9ec58335": "x=\\int^y \\frac{d\\lambda}{F(\\lambda)}+C\\,\\!", "4f94d5b5776efab8a46a472cd99a4039": "\\Phi(x,y) = (\\phi(x,y),y)", "4f9564d8da35926dc2b22694b834936c": "x=\\lambda", "4f959befdb6ba48088eb09afe1f4e9bb": "\\boldsymbol{\\mathsf{T}} = \\boldsymbol{\\mathsf{A}} \\left( \\nabla\\mathbf{v} \\right)", "4f95a9891fe02cecf5f2879b76ee2bcd": "\n\\Gamma (z) = \\frac{e^{-\\gamma z}}{z} \\prod_{n=1}^\\infty \\left[\\left(1+\\frac{z}{n}\\right)^{-1}e^{z/n}\\right]\\,\n", "4f95f309857017ae0388501bb8047615": " (s,x) ", "4f96121988fdf399a98685c512629e24": "\\mathit{v}", "4f964676942ec8302ffaf7ececffcbb2": "|\\psi\\rangle = |\\psi_1\\rangle \\otimes |\\psi_2\\rangle ", "4f9681c4b7a9313a8f00b18a67bf4513": "\n \\operatorname{prox}_f(x) :\\mathbb{R}^N \\rightarrow \\mathbb{R}^N\n", "4f96968093626696ef15ac77b5b7c509": "\\ G_{GR}(\\tau)=1+\\frac{\\langle \\delta I_G(t)\\delta I_R(t+\\tau)\\rangle }{\\langle I_G(t)\\rangle\\langle I_R(t)\\rangle}=\\frac{\\langle I_G(t)I_R(t+\\tau)\\rangle}{\\langle I_G(t)\\rangle \\langle I_R(t)\\rangle}", "4f96a03dede81bc577608217a87805a4": "({\\neg}R)", "4f96c607243ae6179d12f8b2e540bbfd": " \\left(\\frac{\\Delta S}{\\Delta t} \\right)_i=I-O=q^{ss}-q^{trans}_i ", "4f96e1690ebb73449525891ae440ed43": "\\boldsymbol{\\tau} = J~\\boldsymbol{\\sigma}", "4f96f1bdeb41aac77bb21a1babfefe10": "\\displaystyle{\\|v\\|_{(2)}^2=\\|\\Delta v\\|^2 + 2\\|v\\|_{(1)}^2,}", "4f9746ec398af61323c7a12c36db9f26": "x^2=(0,1,0)", "4f9797ddf3021a299451f690a1abd3b1": "P+Q+R=O ", "4f97d54767ae2089bb9cd8cac524dc94": "1/d(v,w)^q", "4f97f5649cdd208350358f8f5371b343": "p(x) = x^n + p_1x^{n-1}+\\cdots + p_n", "4f98007bd83803ed26ac0f07cab694c4": " H = \\frac{p_{\\sigma}^{2} + p_{\\tau}^{2}}{2m \\left( \\sigma^{2} + \\tau^{2}\\right)} + \\frac{p_{z}^{2}}{2m} + U(\\sigma, \\tau, z). ", "4f980b0fabc0a2a3c97410a7fcee8264": "\\mathfrak{P}^{108}", "4f983b37d330f5d72c3cf427f3bef598": "\\hat y^T \\hat y = y^TX(X^T X)^{-1}X^Ty ", "4f9872e1611c22b963a94eee0ea3c278": "H_* (\\nu M, \\partial \\nu M) \\to H_{*-n} M", "4f988cbe96887ce70c30c62850a559e4": "M_4\\frac{\\mathrm{d}\\vec v_s}{\\mathrm{d}t}=- \\vec \\nabla (\\mu + M_4gz).", "4f9893a4c8446a66583f6e8c3bc92ef0": "H^\\ast_{\\mathrm{dR}}(X/\\mathbf{C})\\cong H^\\ast(X(\\mathbf{C}),\\mathbf{Q})\\otimes_\\mathbf{Q}\\mathbf{C}.", "4f98b7ff0a44c952114f99f9f16e457f": "\\big. U = \\frac{k}{\\Delta x}, \\quad", "4f98d22734ea7db779a83c48d50c7676": "h_K", "4f98e732ce1ea6069feb5cc2f4307cd8": "\\begin{align}\n T_{n+1}(x) &= T_{n+1}(\\cos(\\vartheta)) \\\\\n &= \\cos((n + 1)\\vartheta) \\\\\n &= \\cos(n\\vartheta)\\cos(\\vartheta) - \\sin(n\\vartheta)\\sin(\\vartheta) \\\\\n &= T_n(\\cos(\\vartheta))\\cos(\\vartheta) - U_{n-1}(\\cos(\\vartheta))\\sin^2(\\vartheta) \\\\\n &= xT_n(x) - (1 - x^2)U_{n-1}(x). \\\\\n\\end{align}", "4f998e70e92d46de650c7b1e0d6aa6bb": "x_\\min = (x_1,\\dots,x_N)_\\min", "4f9a216a5f4bf0960b058a874e829b9f": " \\sin(i) = i\\sinh(1) \\, = {{e - 1/e} \\over 2} \\, i = {{e^2 - 1} \\over 2e} \\, i \\approx 1.17520119 \\, i... .", "4f9ab1fed825416444279e1c5a7c8fa6": "\n\\int{ G[f] [Df] } \\equiv \\int\\limits_{-\\infty}^\\infty{ ... \\int\\limits_{-\\infty}^\\infty{ G(f_1,f_2,..) } }\\prod_n df_n\n", "4f9ac3010678ce60083a3ba3133e19ae": "{\\Omega} = \\frac{\\left[\\textrm{Ca}^{2+}\\right] \\left[\\textrm{CO}_{3}^{2-}\\right]}{K_{sp}}", "4f9b0a14f72701f855cdf580b59b13b7": "X_{2\\pi}(\\omega) = \\frac{\\pi}{i} \\sum_{k=-\\infty}^{\\infty} \\left[ \\delta (\\omega - a - 2\\pi k) - \\delta ( \\omega + a + 2\\pi k) \\right]", "4f9b843c4749ef09374fe6737d5129a7": "\\mbox{ASAI} = 1 - \\frac{\\mbox{SAIDI} }{8760}", "4f9c73e3b8bcdd656027a8e776c25d94": "c_i \\neq 0", "4f9c791a45b37b45fea2c7c4a73ba33d": "\\text{n(Quadrant)}", "4f9c86f2d4ff9435dc520661cac5736a": "1, 2, 3,\\ldots\\!", "4f9cabb48fc14e6ef165e4a6016bb401": "u_n = \\frac{(1 + \\sqrt{5})^n - (1 - \\sqrt{5})^n}{2^n \\sqrt{5}}", "4f9d51e9a1d3d53b8889950c7bc6f820": "\\neg (\\neg B\\lor C)\\lor (\\neg (A\\lor B)\\lor (A\\lor C))", "4f9d8d6c88c40e98b1caab1576d4a93b": "\\Delta T_{ad}=-\\int_{H_0}^{H_1}\\Bigg(\\frac {T}{C(T,H)}\\Bigg)_H{\\Bigg(\\frac {\\partial M(T,H)}{\\partial T}\\Bigg)}_H dH", "4f9deb8a02ab413e2b4ac218d25674be": "\\tau(F)=\\sigma=\\displaystyle\\limsup_{|z|\\rightarrow\\infty}|z|^{-1}\\log|F(z)|", "4f9e37cdd76ce1f854cf0a266e1734e8": "_a\\mathbf{D}_t^{-\\alpha} f(t) = \\frac{1}{\\Gamma(\\alpha)}\\int_a^t \\Bigg(\\log\\frac{t}{\\tau}\\Bigg)^{\\alpha -1} f(\\tau)\\frac{d\\tau}{\\tau}\n", "4f9e8b3fbe14e0b9da217ed878a8e4eb": "\n\\begin{align}\n\\Gamma & = KW/(2U) \\\\\n\\Omega & = \\sqrt{w^2-\\Gamma^2}\n\\end{align}\n", "4f9ecc6fcec075207350ad3725e8d16e": "\\mathcal{U}_n", "4f9ef8b34ab10e5c3e132299013d467f": " n.(H(X)+\\epsilon) ", "4f9f451fcd514340630de9dc524b5017": "s=w_0 1_G +w_1 a +w_2 a^2\\,", "4f9f560723dc59cbf40020b72751a3f4": " p=q=1 ", "4f9f675c623932276d702e9ce02e496c": "\nu= \\frac{\\partial\\psi}{\\partial y},\\qquad\nv= -\\frac{\\partial\\psi}{\\partial x}\n", "4f9fbb8c9d58f1a0cf10acb713d7e0ee": "\\tilde f", "4f9fc0b435deb05484ce9e03e9c683c6": "i^{\\ast}_{Q}(TE) = TQ\\oplus \\mathcal M(Q),\\,", "4f9ff75e0a64e2c4c088dd622c722494": "O(Nc)", "4fa00d8929647b9d127c5421b5e1fa73": "b \\approx \\frac {fm_\\mathrm s} {N} \\,,", "4fa06ea21742611c87b2d6a45aa9e1c6": "A_0=A(h)", "4fa071386c9e1dde9f08865841723429": " d-\\chi_V(g) ", "4fa0b6caf156520601d83f9e02d7951a": "c_1, c_2, \\dots", "4fa17f9e694262a699424353e14b6de9": " (H=0) ", "4fa184c679745d4d82f3962c96a17170": " Z_\\mathrm{in} \\, ", "4fa1a0d698332ea1d54dc622699e1245": "\\mbox{End}\\, A_B \\cong A \\otimes_R S ", "4fa1f54504f9f302282925fda2796015": "V(y) = \\{v_i | \\text{ the } i^{\\text{th}} \\text{ position of } y \\text{ is a } 1\\}\\,", "4fa1f85afba276a5188b7a399a83db9d": "\n\\zeta_G(s) = \\frac{\\zeta(s)\\zeta(s-1)\\zeta(2s-2)\\zeta(2s-3)}{\\zeta(3s-3)}.\n", "4fa1fbe243e52d7225dc6c09c0a59580": "\\sqrt{x^2+c} = -x+t", "4fa21cdb0f9f82b500b0f4a61fe465f4": "\\bar{K} = \\{ i \\mid i \\not \\in W_i \\}", "4fa2497be69d34c1af338563fede8f24": "\\frac{\\partial g(\\mathbf{U})}{\\partial X_{ij}} =", "4fa2748fadc1f360eb9c01c7bd9c9f0a": "\\Beta(x,y) \\sim \\sqrt {2\\pi } \\frac{{x^{x - \\frac{1}{2}} y^{y - \\frac{1}{2}} }}{{\\left( {x + y} \\right)^{x + y - \\frac{1}{2}} }}", "4fa2e08262b3b772d0299e3d29f66493": "\\frac{\\sin A}{\\sin a} = \\frac{\\sin B}{\\sin b} = \\frac{\\sin C}{\\sin c}.", "4fa2f1d1d40bd5fe86435f83c8610293": "\\mathcal{L}=K=\\frac{1}{2}\\,\\vec{\\omega}^{T}I\\vec\\omega\\,,", "4fa2f27a558b047bfe63e951918a78a5": "S \\to \\operatorname{Spec}\\mathbb{Z}", "4fa3952ce7c97bb873232b2a34c24f1a": "\n\\frac{1}{\\sqrt{\\lambda}} = 1.8\\log[\\frac{Re}{0.135Re(\\frac{\\varepsilon}{D}) +6.5}]\n", "4fa3a77a38cbe464a09889a97c7fcced": "m + n + 1", "4fa3d789f0802611472357b21bc62080": "\\{1,..., P\\}", "4fa3e9c3249cce0dbe7b8f335876e972": "L(\\partial_t,\\nabla_x)(e^{i\\varphi(t,x)/\\varepsilon}) a_\\varepsilon(t,x) = e^{i\\varphi(t,x)/\\varepsilon} \n\\left( \\left(\\frac{i}{\\varepsilon} \\right)^2 L(\\varphi_t, \\nabla_x\\varphi)a_\\varepsilon + \\frac{2i}{\\varepsilon} V(\\partial_t,\\nabla_x)a_\\varepsilon + \\frac{i}{\\varepsilon} (a_\\varepsilon L(\\partial_t,\\nabla_x)\\varphi) + L(\\partial_t,\\nabla_x)a_\\varepsilon \\right)", "4fa43f37513bb57536dbd4c95faf2b23": "\\tilde{p}_P", "4fa46e0e4c289be2bd099a6fdbc45c90": "\\bigcup A", "4fa47247c2c197d4c0f837d1852d89ad": " C \\leq \\max_{p(x_1,x_2)} \\min \\left( I\\left( x_1; y_1, y | x_2 \\right) , I\\left( x_1, x_2 ; y \\right) \\right) ", "4fa4ece96b1b3e80fae2ea5583ff353c": "r < 100", "4fa50234be31fbfb894521f24161933c": " f'(x) = 2x. \\, ", "4fa50b2e9ad8bd27aca292b8c794e135": "(\\tfrac{17}{p})=1", "4fa51538f8ae1a412fbd7e9ca2bba067": "\\nu:[S:T] \\mapsto [S^n:S^{n-1}T:S^{n-2}T^2:\\ldots:T^n].", "4fa547b49c47ca2d63e806f6ad63be99": "M_x\\,\\!", "4fa58ca039aad561f92c319b0449daa6": " \\langle{\\varphi_k} |\\big( P_{A\\alpha} \\varphi_k\\big) \\rangle_{(\\mathbf{r})} = 0 \\quad\\textrm{for}\\quad k=1, \\, 2.\n", "4fa59dc5949cf235559ccc843f8553e6": "\n\\begin{align}\n\\arcsin x &{}= \\int_0^x \\frac {1} {\\sqrt{1 - z^2}}\\,dz,\\qquad |x| \\leq 1\\\\\n\\arccos x &{}= \\int_x^1 \\frac {1} {\\sqrt{1 - z^2}}\\,dz,\\qquad |x| \\leq 1\\\\\n\\arctan x &{}= \\int_0^x \\frac 1 {z^2 + 1}\\,dz,\\\\\n\\arccot x &{}= \\int_x^\\infty \\frac {1} {z^2 + 1}\\,dz,\\\\\n\\arcsec x &{}= \\int_1^x \\frac 1 {z \\sqrt{z^2 - 1}}\\,dz, \\qquad x \\geq 1\\\\\n\\arcsec x &{}= \\pi + \\int_x^{-1} \\frac 1 {z \\sqrt{z^2 - 1}}\\,dz, \\qquad x \\leq -1\\\\\n\\arccsc x &{}= \\int_x^\\infty \\frac {1} {z \\sqrt{z^2 - 1}}\\,dz, \\qquad x \\geq 1\\\\\n\\arccsc x &{}= \\int_{-\\infty}^x \\frac {1} {z \\sqrt{z^2 - 1}}\\,dz, \\qquad x \\leq -1\n\\end{align}", "4fa5ded7da4e3574c66bb95c5a01c0f6": "X(t,dx)", "4fa64a4a6374829058b751115aa2985c": "A(z) = \\sum_{k = 0}^\\infty \\left( \n \\sum_{l=0}^\\infty \\frac{(-1)^l(2l + 2)^k}{(2l+1)!} \\right)\nz^k. ", "4fa6683781e784d05b97d3954b27bb31": "|{\\tau^{[3..N]}_{\\alpha_1i_2}}\\rangle", "4fa668b95f1845c41ebf859a4339e906": "q^2\\rightarrow 0", "4fa6a1f05955283dc71b99d660de6964": "||\\alpha||", "4fa6e63a98d6eff5c5aa2bd762574c39": "= {1 \\over 4} [F , G]^{IJ} - {1 \\over 4} i [* F , G]^{IJ} + {1 \\over 4} i [* F , G]^{IJ} - {1 \\over 4} [F , G]^{IJ}", "4fa7006101e5f1a8a6f7994dbc8f8d44": "(x,k)", "4fa71a32d284a76bf6ee31e349b9b0f4": "\\sum_{j=0}^\\infty x^j P_k(j)= \\sum_{i=0}^k \\frac{x^i}{(1-x)^{i+1}}\\sum_{j=0}^i {i \\choose j} (-1)^{i-j} P_k(j) ", "4fa71c61882f410426b796444bab4368": "u_n=((u_n)_1,(u_n)_2,\\ldots,(u_n)_p)^T", "4fa71d007c094ac3c858919aec515277": "C_1", "4fa74a46f02f4c9e37cce7ca3193ceb2": "A rem n\\,", "4fa74e515dea2a60e9199beb30338537": "M_i = \\sum_{j} \\frac{z_j}{r_{ij}/r_0}.", "4fa74e68ed64a6ffac7b81bd1c364a89": " \\Delta z / \\cos \\theta \\approx \\Delta z\\theta ", "4fa75d2bd9f20ea417a072941ffa3c22": " p'_x", "4fa78c8d14a843ce3df8dbebdf15ef91": " X_{2i} = \\lambda_0 + \\lambda_1 X_{1i}. ", "4fa7a24c4c625f30e60c62229d8f52d8": "\\vartheta_{10}(z|q) = 2 q^{1/4}\\cos(\\pi z)\\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 + 2 \\cos(2 \\pi z)q^{2m}+q^{4m}\\right).", "4fa80db921e33628005afc5dc5503a10": "f(L_0 \\times \\{0\\}) = L_0 \\times \\{0\\}", "4fa83f763380ed4b1d8008895ff17d20": "\n \\omega_1 = \\beta_1^2 \\sqrt{\\frac{EI}{\\mu}} = \\frac{3.515}{L^2}\\sqrt{\\frac{EI}{\\mu}} ~,~~ \\dots\n", "4fa8470fa91cceed99a691345ad454e7": "\\hat{H}\\Psi(\\mathbf{r}_{1},\\mathbf{r}_{2}) = E\\Psi(\\mathbf{r}_{1},\\mathbf{r}_{2}).", "4fa872134df4df8d684835123be1c2a5": "\\mathcal{A}'-\\mathcal{B}(\\overline{\\mathbb{R}})", "4fa8afcb0baa7e6b70b961623a403e01": " p \\in P^N", "4fa8b0fceca0c253aedc33ac60d1226b": "\\sum_{i=0}^{n}a_if^{(i)}(t)=\\phi(t)", "4fa965c71f3ff280a2b828274eacedde": "\n\\overline{ \\frac{\\partial \\phi}{\\partial s} } = \\frac{\\partial \\overline{\\phi} }{\\partial s}, \\qquad s = \\boldsymbol{x}, t.\n", "4fa9da7133eca737c3f7567f9b4c7455": "\\vec{F} = - k \\vec{r}", "4faa4ba8543b06492e8b69d6f4ea24f4": "1-\\exp(-\\lambda x-\\frac{\\alpha}{\\beta}(e^{\\beta x}-1))", "4faa661f569c0818e646fd22d44863b1": "\\text{var}\\,(Y) = a[\\text{E}\\,(Y)]^p ,", "4faaaa4880961dc4f8c43cb3eaac2207": "\\frac{dz}{dz'}=\n\\frac{-1 \\cdot -1 \\cdot \\mathit S \\cdot {far} \\cdot \\mathit{near} \\cdot \\left(\\mathit{far} - \\mathit{near}\\right)}\n {\\left( \\mathit S \\cdot \\left(\\frac{-\\mathit{far} \\cdot \\mathit{near}}{z} + \\mathit{far}\\right) - {far} \\cdot S \\right)^2} =\n", "4faacebbf3aff5c3de6dad0d7bd83812": "x+y{\\sqrt{k}}", "4faaf68cd173a5bed1412825f8da2b28": "\n\\frac{\\partial \\tilde{\\nu}}{\\partial t} + u_j \\frac{\\partial \\tilde{\\nu}}{\\partial x_j} = C_{b1} [1 - f_{t2}] \\tilde{S} \\tilde{\\nu} + \\frac{1}{\\sigma} \\{ \\nabla \\cdot [(\\nu + \\tilde{\\nu}) \\nabla \\tilde{\\nu}] + C_{b2} | \\nabla \\nu |^2 \\} - \\left[C_{w1} f_w - \\frac{C_{b1}}{\\kappa^2} f_{t2}\\right] \\left( \\frac{\\tilde{\\nu}}{d} \\right)^2 + f_{t1} \\Delta U^2\n", "4fab07a182559f78cab14505298a1245": " I_1 \\;\\supseteq\\; I_2 \\;\\supseteq\\; I_3 \\;\\supseteq\\; \\cdots ", "4fab200175e18533459444c65ed71c90": "V \\otimes V", "4fab2478c8a5aa89ee0cf28e9577c0e2": "\\theta_1+\\theta_2+\\theta_3+\\theta_4=180^\\circ\\,", "4fab4ca026ffef9a33b7b187d22acb90": " K ", "4fab70b2476fd5016ba39a71580b9a84": " V_{l} = 1\\ \\mathrm{l} \\times \\frac{310\\ \\mathrm{K}}{273\\ \\mathrm{K}} \\times \\frac{100\\ \\mathrm{kPa}-0\\ \\mathrm{kPa}}{100\\ \\mathrm{kPa}-6.2\\ \\mathrm{kPa}} = 1.21\\ \\mathrm{l} ", "4fab7e8a66a1afcabc05a9c56ed5c5b7": "DO_{crit}", "4fabaca04c0c8d6189e2fbd43976783d": "a_{ji}=0", "4fabae1eca6f222dd80114f97c2edb0f": " \\dot{\\tilde{\\chi}}=-i[\\tilde{H}_{BS},\\tilde{\\chi}(0)] - \\int^t_0 dt' [\\tilde{H}_{BS}(t),[\\tilde{H}_{BS}(t'),\\tilde{\\chi}(t')]]", "4fabb24fcf0bbdb37f7bb55313d522e9": " \\sigma_1 \\otimes \\sigma_1 \\otimes \\sigma_0 ", "4fabc5721033bac822b69c4beab230ef": "d_0(z) = az", "4fac086e2de69c6595ec18131b9b94b7": "e^{-\\frac{x^2}{2}}\\cdot H_n(x) \\sim \\left(\\frac{2 n}{e}\\right)^{\\frac{n}{2}} {\\sqrt 2} \\cos \\left(x \\sqrt{2 n}- n\\frac \\pi 2 \\right)\\left(1-\\frac{x^2}{2n}\\right)^{-\\frac{1}{4}}", "4fac0be2a82fdc5e0f4565fadf22bdd3": " \\nabla(p+\\Phi)=0 \\Longrightarrow p+\\Phi = \\text{constant}.\\,", "4fac2e29ce0ea241bc8b9f043deb47f3": " y'_{2} ", "4fac9116aa4ec0b270d796328eb97dc6": "f(k)=\\sum_{K}\\frac{\\frac{2m}{\\hbar^2}\\frac{A}{a}}{\\frac{2mE_k}{\\hbar^2}-(k+K)^2}\\,f(k)", "4facaebe1a4e85989bca5e7a51d24aad": "q(x_0,x_1,\\ldots,x_n)\\,", "4facd77bab74cd5c85224f381f5ea8f5": "\\vec A=\\phi\\nabla\\psi-\\psi\\nabla\\phi", "4face491d22eacb52594e16e12b9da0c": "NP\\backslash S", "4face9271371a00bbeef6b465829bd07": "\\mathcal{f}0\\mathcal{g}^j", "4faced85fcb07d00a543cb298c0bb6b6": "f(n)=2^n", "4facfb2f6d04a3033dc676ead63967ab": "d(f,g) = \\sum\\nolimits_n 2^{-n}\\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}.", "4fad9cfb12f1f4c09d6e9851192c05f0": "{{f}_{M}}=\\sum\\limits_{m\\in {{I}_{M}}}^{{}}{\\left\\langle f,{{g}_{m}} \\right\\rangle {{g}_{m}}}", "4fadb93a736cec106b814b599d7cfaa5": "\\mathfrak{P}^{101}", "4fae0d7192ab05255990cde164d693b3": "E = m c^2", "4fae4fcc1959e041ae902487769d092e": " P = \\frac{U^2 \\cdot tan \\delta}{2\\pi f \\cdot C}", "4fae8984c6ebf1458bce5e48516ffb3e": "f_P(x) = \\sum_{n=-\\infty}^\\infty f(x + nP) = \\sum_{n=-\\infty}^\\infty f(x - nP).", "4faf203a3a3976ad55269d86a80b1eb9": "_{q=p\\ \\Leftrightarrow\\ q'p=qp'\\,}\\!", "4faf76feeebe0750c00c9fc9162b1eb6": " e(X). ", "4fb03b58a94173b939f778e3ad093adf": "\\Delta \\pi = 0 \\Rightarrow", "4fb04de2d99947f0a6b1d76ac0719534": "{ P(z) \\over P_0 } = 1 - e^{-2} \\approx 0.865\\ .", "4fb0d7ef2f35a4470a9a8d4b4c2f73d3": "-I\\in\\Gamma", "4fb108b76c2c0b1f0e429b5486017a18": "y_k = \\mathbf{h}_k^H \\sum_{i=1}^K \\hat{\\mathbf{w}}_i s_i +n_k, \\quad k=1,2, \\ldots, K.", "4fb117d5856267caa2f7b6186e9faed9": "C_V", "4fb19e7f83d73b006d283493070fca28": " a_n(q), \\, b_n(q) ", "4fb1b3a3875c496496f6693e79573ba4": "(3n^2-3n+1)", "4fb1b943e63237bbcb9bccdd2df3a8df": " \\gamma_s=2\\pi LR\\sigma=\\frac{2V}{R}\\sigma ", "4fb1d160ac9e363682840c06ceb907c7": "\\hat{H}_\\text{int}(t) = \\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{\\sigma}_{-} e^{-i(\\omega_c+\\omega_a)t}\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_{+}e^{i(\\omega_c+\\omega_a)t}\n+\\hat{a}\\hat{\\sigma}_{+}e^{i (-\\omega_c+\\omega_a) t}\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_{-}e^{-i (-\\omega_c+\\omega_a) t}\\right).", "4fb1fb9da82f1b75c4bd6b01aa822f56": "\\delta_1\\frac{\\sigma_1 - \\sigma_e}{\\sigma_1 + 2\\sigma_e}\\,+\\,\\delta_2\\frac{\\sigma_2 - \\sigma_e}{\\sigma_2 + 2\\sigma_e}\\,=\\,0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(5)", "4fb2411294c6d27f42878d958ce3ae67": "\n\\begin{align}\n& {} \\qquad ((2+2)+(2+2))+(3+3) \\\\\n& {} = ((2+2)+4)+(3+3) \\\\\n& {} = (4+4)+(3+3) \\\\\n& {} = (4+4)+6 \\\\\n& {} = 8+6 \\\\\n& {} = 14\n\\end{align}\n", "4fb26fade481bfc923724044c9dc0d7a": "\\dot n_k, \\dot m_k", "4fb2888f044575bba412aa0cc1b1e450": " 0 \\le \\varphi(r \\mathrm{e}^{\\mathrm{i} \\theta}) \\le \\frac{1}{2\\pi} \\int_0^{2\\pi} P_r\\left(\\theta- t\\right) \\varphi\\left(\\mathrm{e}^{\\mathrm{i} t}\\right) \\, \\mathrm{d} t, \\ \\ \\ r < 1.", "4fb2c3bb403f698cedc1ab3ca239c719": "(z_1, z_4; z_3, z_2) = {\\lambda\\over{\\lambda-1}}", "4fb2f1f0c2aea1a4415c26ae0e882b02": " (x \\oplus y) \\oplus z = x \\oplus (y \\oplus z),", "4fb308d770749dd35fdb2bc993e2dce1": "x>\\sigma", "4fb3197ed9303ac78f3855f1f8ec20b8": "(M^{2n}, \\omega)", "4fb3237cf984809c530bd9ab10e722b7": "\\tilde{d} > 1", "4fb362ff1ac809e4bfb865dc5f69927f": " ds^2 = -(R^2-z^2) \\, dt^2 + {dz^2\\over (R^2- z^2)} + R^2 \\, d\\Omega^2 \\,", "4fb38aa859ebda86c73a49c07686b478": "Q_{hot}=Q_{cold}+W ", "4fb3f64a4efed16176287805315390a1": "\nR_\\ell^m(r,\\theta,\\varphi) = (-1)^{(m+|m|)/2}\\; r^\\ell \\;\\Theta_{\\ell}^{|m|} (\\cos\\theta)\n e^{im\\varphi}, \\qquad -\\ell \\le m \\le \\ell,\n", "4fb437564892944d73f87c822342e2bb": "4.8", "4fb479710dfdf1c7d7ebb3dd2c746d47": "f(x) = c x_1^{a_1} x_2^{a_2} \\cdots x_n^{a_n} ", "4fb4c9847cad672e4f62fa741e65750f": "L(\\boldsymbol{q},\\ \\boldsymbol {\\dot{q}},\\ t)= T-U", "4fb53477570d6671fd344f74ff944be3": "z_{\\pm i} = -z_b + 2i(d+2z_b)\\pm(z'+z_b)", "4fb62e3fe45a42f690cb7eecb11511a7": "\\zeta(z) = \\arctan \\left( \\frac{z}{z_\\mathrm{R}} \\right) \\ .", "4fb6344681edfe890279f7e582b0e7e9": "\\star \\mathrm{d}\\eta=\\left({\\partial C \\over \\partial y} - {\\partial B \\over \\partial z}\\right)\\mathrm{d}x - \\left({\\partial C \\over \\partial x} - {\\partial A \\over \\partial z}\\right)\\mathrm{d}y+\\left({\\partial B \\over \\partial x} - {\\partial A \\over \\partial y}\\right)\\mathrm{d}z.", "4fb65749e7cb3b2bccd05bd187a7ba31": "\n r(\\theta) := \\cfrac{u(\\theta)+v(\\theta)}{w(\\theta)}\n ", "4fb6933f7f8b84ed7605116b8b62c58f": "\\left(\\frac{g}{f}\\right) =\n\\begin{cases}\n+1 \\text{ if }\\gcd(f,g)=1 \\text{ and there are } h,k \\in \\mathrm{F} [x] \\text{ such that }g-h^2 = kf \\\\\n-1 \\text{ if }\\gcd(f,g)=1 \\text{ and } g \\text{ is not a square }\\pmod{f}\\\\\n\\;\\;\\;0\\text{ if }\\gcd(f,g)\\ne 1.\n\\end{cases}\n", "4fb7bb7ee2a941c07fe01bdec3e9c6d5": "\\sigma = (\\text{left}, \\text{right},\n\\text{right})", "4fb7f9918239289d035cda5baca85666": "\n \\tau_m = \\cfrac{\\sigma_1-\\sigma_3}{2} ~;~~ \\sigma_m = \\cfrac{\\sigma_1+\\sigma_3}{2} \n ", "4fb82d778b9b307fd39cb3e5adf4e654": "s = \\sigma + i \\omega \\,", "4fb83c983a00fe4456442059a2fcbe29": "T:L^2\\to L^2", "4fb84aa35bd45fe0ddafca3d0aa65431": "g(0,s)=0", "4fb87522712533a53a75d14e07ff4b84": "\\scriptstyle t_\\frac{1}{2}", "4fb914d94a2622987dbb53edb3d6ec5e": "m=\\frac{\\Delta y}{\\Delta x}=\\tan( \\theta )", "4fb94b015f7ef899047564f990a29e1f": "f(3, 0.5) = 0", "4fb95127ec06c9cef5c597cd440165ac": "1 + 2(r^1 + r^2)", "4fb998777cf3504bb044fd07fe7f8475": "\\Delta = I_{-\\infty}^{+\\infty}\\frac{-\\mathfrak{Im}[f(x)]}{\\mathfrak{Re}[f(x)]}= I_{-\\infty}^{+\\infty}\\frac{b_0\\omega^{n-1}-b_1\\omega^{n-3}+\\cdots}{a_0\\omega^n-a_1\\omega^{n-2}+\\ldots} \\quad (24)\\,", "4fb9e102b54a3a34e7c903926e7d4e0a": "F_{R_k}(r_K)", "4fb9fecf99d04f3d00086ea8a3aade50": " Q^*Q = I ", "4fbbcd1934e8bbc6f1c431d3eb5ca476": "\\begin{bmatrix} -\\dfrac{\\nu}{2}-1 \\\\[10pt] -\\dfrac{\\nu\\sigma^2}{2} \\end{bmatrix} ", "4fbbf71742af92acd49bc5baa2e99b0c": "f / (a_{1}, a_{2} \\ldots, a_{n})", "4fbc15f92bc9e1bc4fb354047cb73f62": "\\vec{r_V}", "4fbc391332328a5fde0190be88aebb05": "\\gamma(0)=p", "4fbc6a8b05f26bef85073cd47ad21b0f": " \\min \\le \\mathrm{round}(value) \\le \\max ", "4fbc6c6bbd7808700b916291e74b574c": "y_{k,t} = c_{k} + a_{k,1}^1y_{1,t-1} + a_{k,2}^1y_{2,t-1} +\\cdots + a_{k,k}^1y_{k,t-1}+\\cdots+a_{k,1}^py_{1,t-p}+a_{k,2}^py_{2,t-p}+ \\cdots +a_{k,k}^py_{k,t-p} + e_{k,t}\\,", "4fbc95ca3c0acac4324de7c48e1e946f": " \\mathcal{C}_{XX} =\\left(\n\\begin{array}{c c c}\nP(X=1) & \\dots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\dots & P(X=K) \\\\\n\\end{array}\n\\right)\n", "4fbd30e7a31f66b47f8c5c63f91ac63c": "p(v;\\xi)", "4fbd6fbbd674f0dedc3aad194d796194": "2^m \\ge 2^n (a n + 1)", "4fbd8b0f717d6f117f2175b0ab65b82f": "{\\sqrt{\\beta} \\over C_q} e_q^{-\\beta (x-\\mu)^2} .", "4fbd921cdf2100ea050362609def6406": "{e}^{\\Theta /bT}", "4fbde91a0f3439aa309d3ab5bd0a0215": "1+\\Phi\\approx 2.6", "4fbeba786546e67ef50bdd29575d1c92": " a_m =\\frac{p_m}{N}\\left[\\frac{1}{2} (u_0 (-1)^m +u_N)+\\sum_{n=1}^{N-1} u_n T_m (x_n)\\right] ", "4fbed74ee35bd1f30bbbae8ddade2859": "\\left|V\\right|_{min, max}", "4fbf0f58fa846f48be60bcb5b0bd30be": " [F] ", "4fbf55a70848111d468888aaba9246b2": "\\prod_{n=1}^\\infty(1+p_n)", "4fbfb82106be06e6765d963ec7d3c477": "\\varphi: \\text{Mod}_R \\to \\text{Mod}_{R[S^{-1}]} \\quad M \\mapsto M[S^{-1}]", "4fbfc47aecb6f5b9a10b7a8027846789": "re^{i \\psi} = \\frac{1}{N} \\sum_{j=1}^{N} e^{i \\theta_j} ", "4fc0d3482eaebce7076e585d3ef115b8": "Q = (1 - u \\cdot v)^2,\\,", "4fc11d71d6cbcbc9bf9807d61aceb449": "L_z = m_\\ell \\hbar\\,\\!", "4fc1adf3ed859a4aa5f473e8c2474baa": "b = \\frac{\\tbinom nt}{\\tbinom kt}", "4fc1b16130a0cd050c77362ff290543e": "\n\\begin{align}\n \\tilde{s} & =\\tilde{g}(\\tilde{x},\\tilde{u})+\\tilde{\\omega}_s \\\\ \n \\\\ \n s & =g(x,u)+\\omega_s \\\\\n s' & =\\partial_x g\\cdot x'+\\partial_u g\\cdot u'+\\omega'_x \\\\ \n s'' & =\\partial_x g\\cdot x''+\\partial_u g\\cdot u''+\\omega''_x\n\\\\\n& \\vdots \\\\\n \\end{align} \\qquad \\begin{align}\n \\tilde{x} & =\\tilde{f}(\\tilde{x},\\tilde{u})+\\tilde{\\omega}_x \\\\ \n \\\\\n \\dot{x} & =f(x,u)+\\omega_x \\\\ \n \\dot{x}' & =\\partial_x f\\cdot x'+\\partial_u f\\cdot u'+\\omega'_x \\\\ \n \\dot{x}''& =\\partial_x f\\cdot x''+\\partial_u f\\cdot u''+\\omega''_x \\\\ \n & \\vdots \n \\end{align}", "4fc1da883ae6b47316a4415385ecf562": "{\\psi}^i", "4fc2472df0cef83d40a958405ab7f7e2": "\\begin{align}x^*& = (1/b)\\ln\\left[(1/s)(\\beta-1)\\right], \\\\&\\text{with } 0<\\text{F}(x^*)<1-(\\beta s)^s/\\left[(\\beta-1)(s+1)\\right]^s<0.632121,\\\\& \\beta > s+1\\\\& = 0, \\quad \\beta\\le s+1\\\\\\end{align}", "4fc2507666afc0fd70ab47b0ab2cf2c0": " \\sigma", "4fc26b2fa729df47f36408b2de684cc6": "S= \\int L\\, dx_3 ", "4fc2708754de3c8100bee9d37eb67d65": " {n \\choose k_1, k_2, \\ldots, k_m}\n = \\frac{n!}{k_1!\\, k_2! \\cdots k_m!}", "4fc27e10e30b589e9fd0fffdf87551c1": "{A}_{2}^{(1)}", "4fc2ae0ce2162c965c1ab4d47e4d83fd": "\\and_{\\gamma < \\delta}{A_{\\gamma}}", "4fc2af8b3385367f02531732989b7716": "a_i = \\sum_{1 \\leq j < i} \\lambda_j \\quad \\text{and} \\quad a'_i = \\sum_{1 \\leq j < \\pi(i)} \\lambda_{\\pi^{-1}(j)}.", "4fc2ba2d355a628c2e67d27f314b1a67": "\\Delta H_\\text{SO}= {\\beta\\over 2}(j(j+1) - l(l+1) -s(s+1))", "4fc2ca3ecc70acbbf37268058e4269b9": "k_1[S]", "4fc2dfc615fb7a8e1a86e2affea4c054": "\\alpha(t_{n}-t_{n-1}) = 1-\\exp\\left({-{ {t_{n}-t_{n-1}} \\over {W \\times 60} }}\\right)", "4fc3070af4fa1f1527262273815d7309": "\nT(m,s,z) = - \\frac{(-1)^{m-1} }{(m-2)! } \\frac{{\\rm d}^{m-2} }{{\\rm d}t^{m-2} } \\left[\\Gamma (s-t) z^{t-1}\\right]\\Big|_{t=0} + \\sum_{n=0}^{\\infty} \\frac{(-1)^n z^{s-1+n}}{n! (-s-n)^{m-1} }\n", "4fc30e9d68b477ca17ef4c02c1e2dcf3": "\nf' = f {1 \\over (1-u/c)(1-v/c)}\n\\, .", "4fc3127ece534dcb5b6dcbfb04de6488": "\\Gamma_{\\mu\\nu\\alpha}=\\{_{\\mu\\nu\\alpha}\\} + \\frac12(T_{\\nu\\mu\\alpha} +T_{\\nu\\alpha\\mu} + T_{\\mu\\nu\\alpha}). ", "4fc3707f0edeadded2657ca6a0200561": " \\psi(n)\\ =\\ H_{n-1} - \\gamma", "4fc3bb6c6699c9f7ae24ee77b66b9560": "\n\\delta = a(a-1) \\dfrac{\\partial}{\\partial a} + (l+b)\\dfrac{\\partial}{\\partial b}+ (2ac-c)\\dfrac{\\partial}{\\partial c}+(-aP+P)\\dfrac{\\partial}{\\partial P}\n", "4fc3be82861822db88aa6c6d2bf0671f": "Z(z) = z", "4fc3cb653d2319da577ccff2256969dc": "\\frac{uv}{1-\\theta (1-u)(1-v)}", "4fc3cf1d35bd61a5490c13c3e01c925f": "[n]_q! := [1]_q [2]_q \\dots [n]_q", "4fc3d1b1738d30e449f00cd425b01446": "p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\\\\naq&bdq&cq\\\\ \\end{matrix};q;q\\right]", "4fc3e3c98d13ed389817f11dc66c10a6": "X_{i}", "4fc40a022deb4a2e95f389071e78836b": "d = gh + p = (1)(1) + 1 = 2", "4fc477b8ffd0b7190601bfc12b95588a": "K(x,y)=\n\\begin{cases}\n -1 & \\text{if } x 0 ", "4fdc45f680580f0a91afd126ae632881": "\\frac{2\\lambda l}{\\lambda n - \\lambda k}\\quad=\\quad\\frac{2l}{n-k}", "4fdc60a51c8e5283616461fb3f9fafbf": "\n\\partial e^{ f(x) \\mathbf{e}_3 } = \n (\\partial f(x)) e^{ f(x) \\mathbf{e}_3 } \\mathbf{e}_3,\n", "4fdcede4b6094c31ae659035effe203b": "p_i = \\mathbb{E}\\left[\\left.\\frac{Y_i}{n_{i}}\\,\\right|\\,\\mathbf{X}_i \\right], ", "4fdd1439b2a6204d33d572695fc17524": "G_{\\nu} (\\omega)", "4fdd465688198dc0e7c227377cb5decd": "\nP(V) = \\frac{K_0}{K'_0} \\left(\\left(\\frac{V_0}{V}\\right)^{K'_0}\n - 1\\right) \\qquad (6)\n", "4fdd8216f22c9f608d830a70b00adef0": "A, B, G, H", "4fdd835664b2b30d5a5681d1b62eba49": "\n\\left[ \\eta \\right] = \\lim_{\\phi \\rightarrow 0} \\frac{\\eta - \\eta_{0}}{\\eta_{0}\\phi}\n", "4fdda62d723ceae9a16230a4ba36268e": "\\mathbf{\\mathcal{C}}_\\bullet", "4fddc31a09fd373dc92d37997828c758": "K = \\mathbf{Q}(\\sqrt d)", "4fddd3411c1848a7689fc048205549ed": "\\omega_f(\\delta)=O\\left(\\log\\frac{1}{\\delta}\\right)^{-1}.", "4fde33cc160276caa1e4cce4ad28fb23": "2rt \\approx \\pi/2", "4fde993b8152602de80c393ec7760df0": "\n\\begin{align}\nI(X;Y) = h(Y) - h(Y|X)\n&= h(Y)-h(X+Z|X)\n&= h(Y)-h(Z|X)\n\\end{align}\n\\,\\!", "4fded3a4417054306ed1990e31f6e292": " \n \\vec{F}_m = \\int _m \\vec{J} \\times \\vec{b} d^3\\vec{r}\n", "4fdeddb85f44ee6ef00c9c40c2c802fe": "ct", "4fdefb997a92c7a260b23f39af518e49": "F_{c}=\\frac{m\\ v^2}{r}", "4fdefba26320686bb2bd0579a0df421c": "\\nu", "4fdf0c9a1798bc98497f963a2b330ae7": "\\begin{align} m_1 u_1 + m_2 u_2 &= m_1 v_1 + m_2 v_2\\\\\n\\tfrac{1}{2} m_1 u_1^2 + \\tfrac{1}{2} m_2 u_2^2 &= \\tfrac{1}{2} m_1 v_1^2 + \\tfrac{1}{2} m_2 v_2^2\\,.\\end{align}", "4fdf3764aa82c0aa964653426e01f269": "D(\\hat X)", "4fdf5443ae5f06bf11f87b8b97a3365a": "L_A = \\{x \\mid \\exists y\\in A\\mbox{ s.t. }x\\le y\\}.", "4fdf8e22db576913afcacdba203513ee": "35^8 \\approxeq 17000^3", "4fdf9c08958cfe1c720bc03fd8511fab": "Tu.", "4fdf9e9786bdd79d83e2349b7c66a15d": "\\,y \\prec z", "4fdfa1e5a4f2209779dc1b55f716b6f3": "k(F) \\approx \\frac{1}{\\sqrt{1-F}}", "4fdffbd1032abdd10dc252485a4f9912": "\\frac{\\mathrm{d}p}{p} = \\frac{-g}{R \\cdot T} \\, \\mathrm{d}z.", "4fe01167dde89365a59ee3bed251eb7d": "\n\\psi(z) = \\psi_{\\rm D} e^{-\\kappa z}\n", "4fe0eddbe65860eebd79aa8e7197b023": "D=D_1N_2+D_2N_1", "4fe115fba1ad79774e90cd1a16e721b1": "\n[d(\\rho, \\rho+d\\rho)]^2 =\n \\frac{1}{4}\\mbox{tr}\\left[ d \\rho d \\rho + \\frac{1}{\\det(\\rho)}(\\mathbf{1}-\\rho)d\\rho (\\mathbf{1}-\\rho)d\\rho \\right]\n", "4fe178d70bb22197cbcef6ebcb3203c5": "I_jI_k \\subset I_{j + k}", "4fe1836f2538e6ce7b0e12c6ee636fe2": "\\displaystyle{\\hat{K}} = (x^3, y^4, z^8, x^2z^7)", "4fe19c3119735ac228eb86aa93347b27": "k=[\\vec{L}\\cdot (\\vec{E}-2\\vec{N}(\\vec{N}\\cdot \\vec{E}))]^n=[\\vec{L}\\cdot (\\vec{E}-2\\vec{N}(0\\cdot \\frac{\\sqrt{3}}{2}+1\\cdot 0.5 +0\\cdot 0))]^3=", "4fe1eb8bb2b898728925bf81dfe3619e": "\\operatorname{erf}^{-1}(z)=\\tfrac{1}{2}\\sqrt{\\pi}\\left (z+\\frac{\\pi}{12}z^3+\\frac{7\\pi^2}{480}z^5+\\frac{127\\pi^3}{40320}z^7+\\frac{4369\\pi^4}{5806080}z^9+\\frac{34807\\pi^5}{182476800}z^{11}+\\cdots\\right ).\\ ", "4fe1f90379be0eb89b0c5618d737e292": "V_v = 1723 \\cdot V_w", "4fe2331f4c915514adfa384e76c208a0": "\\mathcal{L}_{YU} = \\overline U_L G_u U_R \\phi^0 - \\overline D_L G_u U_R \\phi^- + \\overline U_L G_d D_R \\phi^+ + \\overline D_L G_d D_R \\phi^0 + hc", "4fe24a9ac5c6e11789d84094627bf9d7": "(B y + \\beta)^n - B^n y^n \\le B^n r + \\alpha", "4fe2891dc0f1964c5fbd0c5e9d484e4b": "x_{n+1} = x_n + x_n r_n", "4fe2dad93ddc92a26b0c4db55ecdd303": " \\mathbf{v}=(\\dot{x}, \\dot{y}) = (L\\cos\\theta, L\\sin\\theta)\\dot{\\theta},", "4fe2e0d78351a6bad21c9f22474a6d43": "P_{\\nu\\mu} = \\sum_{\\alpha} P_{\\nu\\mu}^\\alpha\\frac{\\partial}{\\partial x^\\alpha}.", "4fe327b5507738fe961cd0d5f5bd2776": "1-s_0", "4fe32e3d0b44c9f2cd5d6397672b638c": "C_{p,m} = \\left(\\frac{\\partial C}{\\partial n}\\right)_p", "4fe34e27bdb75bc4157b1ae4dcb89f4e": "(C_1,C_2,\\lambda_+,\\lambda_-,\\alpha)", "4fe3ab3e3c3534e3363612e10a671805": "F \\subset (P \\times T) \\cup (T \\times P)", "4fe3b2e219996e4497b61c94bbe3edcf": "q = C \\ \\frac{\\rho}{g}\\ \\sqrt{\\frac{d}{D}} u_*^3", "4fe3c7c100acd748d961d9d835bddf7e": "{f}^2", "4fe3d2897c0288b6710d9881cc37337f": "\nx = {e^{-t}\\over(1+e^{-t})^2}={1\\over2+2\\cosh t}.\n", "4fe43cb1ed7e37fddcd266cc166b969f": " G : U \\rightarrow (0, \\infty)", "4fe48b2db58faaa38674594e86c5f87f": "c \\leftarrow Enc(pk, x)", "4fe4b4d4ff75225d6c73b297618803d5": "\ndA = {\\partial A \\over \\partial x} dx + {\\partial A\\over \\partial p} dp = \\{ A,G\\} ds\n\\, .", "4fe4b89d270d30202fa2275d1ae0b607": "\\mathit{alg}_A(B)", "4fe522c78f93507862bc60c6d95a6f98": "\\Delta^{P[log]}_2", "4fe556ec9c8619ad69394e7f975f35bd": "10^8", "4fe56d4852363fa07ceac1f2b20f30c3": " \n\\alpha^*(t) = \\omega\\text{-only action that satisfies (Eq.3)-(Eq.4)} \n", "4fe573e4e0dd73d0f769753f4aeb377a": " v_1 \\neq 0 ", "4fe577b2879d380ca9129defe8017161": "M_\\lambda\\to V.", "4fe58e78fff979d6917291c522effb2c": "c_p = c_g = \\sqrt{g(h+H)}.", "4fe5a59fdd4713e26e0f144589360695": "A(x,y) \\rightarrow C(y)", "4fe5bcb12110eceda176bb9a9856b9b4": "\\deg(D_1)=6+4=10", "4fe5eeb063cf17caeddc56ef13ab6777": "\\sin \\beta\\,", "4fe6a41fb1538eeaea831e4d5b642fc2": "\\left(10^{10^{10}}\\right)^{\\left(10^{10^{10}}\\right)^{\\left(10^{10^{10}}\\right)}}.", "4fe6ffd6a422fc2373a0b414097c9e16": "z_k = \\sum_{ij} T_{ijk}x_iy_j\\,", "4fe700d06298de828b9628d969705f8d": "1/2\\rho", "4fe71363325cad6c9d52f9519c985fc0": "S(A:B | \\Lambda)", "4fe7330ba0efe67b7a60a3392cad452b": "|x| = 2.", "4fe753bf7d6965c27f863a703677c76c": "X = T^{-1/p} + X_1", "4fe7615a7bbd229c3091681027d45558": "\\{0\\}^j \\times S^{m-j-1} \\subset D^j \\times D^{m-j} = H^j", "4fe78ee304e9fd7a7d56c16db82257e3": "i \\frac{\\partial}{\\partial t} \\psi(x, t) = \\Bigl[-\\frac{\\partial^2}{\\partial x^2} + V(x) \\Bigr] \\psi(x,t), \\qquad\\qquad \\psi(t_0) = \\psi_0", "4fe7d26d93f5b9185ff936328746c9a4": "(\\alpha, \\alpha^\\vee)=2", "4fe7f7d19c5c389a9e3a1e67f2cafa78": "\n\\cos \\theta = \\frac{d_{\\mathrm{s}}^{2} + d_{\\mathrm{non}}^{2} - d_{\\mathrm{T}}^{2}}{2 d_{\\mathrm{s}} d_{\\mathrm{non}}} \\equiv C_{\\pm}.\n", "4fe852cb785065e3f1f804fe56055f0d": "\\alpha=4.90", "4fe92d20260f5a9c0d6d591b806bec65": " \\omega \\frac{\\operatorname{d}}{\\operatorname{d}\\theta} = \\frac{\\operatorname{d}}{\\operatorname{d}t}", "4fe97105758b726f9b9332448176ce14": "\\,\\!\\delta", "4fe9b01daee5f7551d18a3d8e3c37519": "\\textbf{P}_{k\\mid k}^a = E[(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k})(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k})^T] ", "4fe9c67bdaa0129ca03e2dd0c735659a": "\\textstyle{\\frac {\\log(\\scriptstyle\\varphi)}{\\log(2)}=\\frac{\\log(1+\\sqrt{5})}{\\log(2)}-1}", "4fe9e20c923592937a79cc578b6c78ee": "X^{(m)}=\\frac{1}{m}(X_1+\\cdot\\cdot\\cdot+X_m)", "4fe9f15fb53c589e023213ae35fcf78f": "O(|V|+|E|)", "4fe9f2b84c6fcb871bad782ba516076e": " f(x)= \\Phi(e,x)", "4fea3dbfde8158c293ff1c6f60a50f81": "\\mathrm{d}(U-TS+pV)=V\\mathrm{d}p-S\\mathrm{d}T+\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}N_i - \\sum_{i=1}^n X_i \\,\\mathrm{d}a_i + \\cdots", "4feab4fc1ed88d87a8f4cfdc23a8483a": "\\,\\delta_1+\\delta_2+\\delta_3\\,", "4feae835c143d70dd357a6a40e61523c": "c_d", "4feafe6e24ec6d9fcd00953bae5f114f": "\\begin{cases}\n\\overbrace{ \\begin{bmatrix} \\dot{\\mathbf{x}}\\\\ \\dot{z}_1\\\\ \\dot{z}_2 \\\\ \\vdots \\\\ \\dot{z}_{i-2} \\\\ \\dot{z}_{i-1} \\end{bmatrix} }^{\\triangleq \\, \\dot{\\mathbf{x}}_{i-1}}\n= \n\\overbrace{ \\begin{bmatrix} f_{i-2}(\\mathbf{x}_{i-2}) + g_{i-2}(\\mathbf{x}_{i-1}) z_{i-2} \\\\ 0 \\end{bmatrix} }^{\\triangleq \\, f_{i-1}(\\mathbf{x}_{i-1})}\n+\n\\overbrace{ \\begin{bmatrix} \\mathbf{0}\\\\ 1\\end{bmatrix} }^{\\triangleq \\, g_{i-1}(\\mathbf{x}_{i-1})} z_i &\\quad \\text{ ( by Lyap. func. } V_{i-1}, \\text{ subsystem stabilized by } u_{i-1}(\\textbf{x}_{i-1}) \\text{ )}\\\\\n\\dot{z}_i = u_i\n\\end{cases}", "4feb269c4e45c89a2416d28cb4132f6c": " \\frac{r}{r_s} \\left( \\frac{v}{c} \\sqrt{1 - \\frac{r_s}{r}} \\right)^2 = \\frac{1}{2} ", "4feb55da2bba429a263344f095d2d520": "K(x,y)=\\frac{1}{\\pi}\\frac{1}{(1-x\\overline{y})^2}.", "4feb58ad57969c02c58dc0e766d99fa3": "\\sqrt{-3} = \\pm\\sqrt{3}i", "4feb679e84dc6080ddf18f19c228154f": "\\Big\\|Y|_I - U_k\\Big\\|_1 \\leq \\delta", "4febe5bd8040ab76528c85952a10181b": "{\\tilde{D}}_{4}", "4febe8d410428bc78830355e0550bf0c": "x^T \\bar{S} = x^T \\frac{S}{1+r}", "4febf095fce1f5218f9c15eb7102c220": "\\mathfrak{p}_0 \\subsetneq \\cdots \\subsetneq \\mathfrak{p}_m", "4fec1dd5cb4d84db56e9dbcc86f7b60d": "\\Phi\\subseteq\\Lambda", "4fec31763f3f0697352fd0e6ea67e363": "\nX = \\frac {4 L \\tau_ y} {D \\Delta P}\n", "4fec6334b443129c6ae0c804a5378cad": "\\prod _x a^{f(x)} = a^{\\sum _x f(x)} \\,", "4feca19ba65f0f3f7e7f56b17c16bb2f": "t\\equiv\\alpha\\tau", "4fecbc62d9fc1c84cdba7c32418d0126": " \\widehat{\\mathcal{C}}_{YX}^\\pi, \\widehat{\\mathcal{C}}_{XX}^\\pi = \\boldsymbol{\\Upsilon} \\mathbf{D} \\boldsymbol{\\Upsilon}^T ", "4fecd5eee1744e4a8260b557edab16b1": "\\mathsf{d}_A(\\mathbf{x}, \\mathbf{y})", "4fece51418aad71fee71fa244e40ca66": "\\lambda=\\aleph_0", "4fecef49dbd81b058d4b59fdc43b3a77": "(x - a)^{n} \\equiv (x^{n} - a) \\pmod{n} \\qquad (1)", "4fed040e628b84ef35a28e3d0c892d1f": "256 - 13 = 243", "4fed1266f167bedc320c56e67324658a": "\\sum_{n\\ge 1} \\frac{2^{\\omega(n)}}{n^s} = \\frac{\\zeta(s)^2}{\\zeta(2s)}", "4fed293fa379425ca743fa16d98e3038": "V_{pu} = \\frac{V}{V_{base}} = \\frac{136 \\, \\mathrm{kV}}{138 \\, \\mathrm{kV}} = 0.9855 \\, \\mathrm{pu}", "4fed3e01e2f7e0b19485ff0b4f97b559": "f(F_i,k)=kF_i\\!", "4fed4ac7abc3f3777809bf8ac83d411b": "K=\\frac {\\epsilon^3}{36k(1-\\epsilon)^2}d^2 ", "4fed5e7ebd8eafe2b09501c6f76d67b6": "(\\mathbb{N}, \\mathbb{R}, G)", "4fed91e6bd95c36e93021e425d760a47": "\n \\hat F_n(t) = \\frac{ \\mbox{number of elements in the sample} \\leq t}n = \n\\frac{1}{n} \\sum_{i=1}^n \\mathbf{1}\\{x_i \\le t\\},\n ", "4fedc6a1b8ff33a311b7426b30344264": "\\delta^a_b = \\begin{cases} 1 & \\mbox{if } a = b, \\\\ 0 & \\mbox{if } a \\ne b. \\end{cases}", "4fee40592a91c2147711ea0e381c3b14": "{\\boldsymbol{S}}=\\frac{1}{2}{\\boldsymbol{\\sigma}}", "4fee55b34ff714033762c37147a234a9": "h(e)", "4fee6d094e8fb68e1ed2691ed40b8d4d": "R_{AC} = R_{DC}\\bigg(Re(M) + \\frac{(m^2-1) Re(D)}{3}\\bigg)", "4feea5f418537522e84c35f2d2278fcd": "d^P_i", "4feed09df01603ad8ce8af63a3203b39": "\\theta \\log{\\tan \\theta} - \\frac{1}{2}\\int_0^{2\\theta}\\log\\left(\\frac{2\\sin (x/2) }{2\\cos (x/2)}\\right)\\,dx=", "4feeda516f56cdfb2241a5e0501cd32f": "-1 < m(n) < 1", "4feede09296c353eef46b9d51b96586b": "Y_i = \\int^T_0 Y(t)\\Phi_i(t)dt = \\int^T_0 [N(t) + X(t)]\\Phi_i(t) = N_i + X_i", "4feef7be4ede51e732bd6d436c758369": "\n\\hat{\\beta}_j = \\bar{Y}_{\\cdot j} - \\bar{Y}_{\\cdot\\cdot}.\n", "4fef4c97ef1eb75e1100b4a0af6081ef": "X^\\bullet(\\mathrm{Res}_{L/K}T) \\cong \\mathrm{Ind}_{G_L}^{G_K} X^\\bullet(T).", "4fef501a8abb2a4ef5690a45822380ba": "F_C = \\frac{k q_1 q_2}{d^2}", "4fef89ecc8c3ae3743827fbe8be11125": "\\tau\\in \\mathbb{H}_n", "4fefa1c65d408090875627d9dc9eca1d": "\\frac{36}{38}", "4ff00e61d6a1209f8d049cc9395c212d": "\nw = \\frac{{n!}}\n{{\\prod_{i = 1}^n {n_i !} }}\n", "4ff0986b83b95a5c37c78ff6409ad863": " \\hat{H}=\\hat{H}_{\\mathrm{el}}+\\hat{H}_{\\mathrm{back}}+\\hat{H}_{\\mathrm{el-back}},\\,", "4ff09d48233d837fe05408fcc2a7c79d": "r = \\frac{1}{| \\sin \\theta| + |\\cos\\theta|}", "4ff0a1eabfe3b4ce02042e0844a653a8": " \\theta=tan^{-1}\\left(\\frac{X}{R}\\right) ", "4ff0ac3638ca31863419b2176c94d402": "|\\tilde{\\Psi}\\rangle=\\sum_i|\\tilde{\\phi}_i\\rangle c_i", "4ff0da8fc960145a64db35357f6503e9": "T := \\mathbb{N}", "4ff105cc1a6250b094c7a2030a8d1f03": "\\exists c. \\, ( \\, \\text{Cat}(c) \\land \\forall m. \\, (\\text{Mouse}(m) \\rightarrow \\text{Fears}(m,c)) \\, )", "4ff13c06ec250847c849ca6f21ba53d0": "f^{(-1)}(a) = \\int_a^a f(t)\\,\\mathrm{d}t = 0", "4ff161c8ae49bef9e028250c51b5ea52": "T_{ON}", "4ff1a285cf8c2050ecfc1730cfb70787": "\n{\\rm var\\ } T \\geq \\frac{[\\psi^\\prime(\\theta)]^2}{{\\rm var} (V)}\n=\n\\frac{[\\psi^\\prime(\\theta)]^2}{I(\\theta)}\n=\n\\left[\n \\frac{\\partial}{\\partial\\theta}\n {\\rm E} (T)\n\\right]^2\n\\frac{1}{I(\\theta)} \n", "4ff2a4668ef1d00df3b9d2f17be76665": "x\\,L\\,y", "4ff2a973a1ef891fc64c24a3689de338": "H_1 = -\\hbar\\left(\\Omega e^{-i\\omega_Lt}+\\tilde{\\Omega}e^{i\\omega_Lt}\\right)|\\text{e}\\rangle\\langle\\text{g}|\n\n -\\hbar\\left(\\tilde{\\Omega}^*e^{-i\\omega_Lt}+\\Omega^*e^{i\\omega_Lt}\\right)|\\text{g}\\rangle\\langle\\text{e}|", "4ff304237f42cb89090cbff7df277713": "(D)", "4ff314de762ea9543070c53003917e79": "\\hat{Y}_{t} = P_{K(Z, t)} \\big( X_{t} \\big) = \\mathbf{E} \\big[ X_{t} \\big | G_{t} \\big].", "4ff33519fabf368383abdc7f6688c9b6": " \\frac{dN_i}{dt} = F_{io} - F_i + V \\nu_i r_i ", "4ff3366ba4a05fcd5e07611e247cf7bf": "M^{1/2}=\\begin{bmatrix}1.2 & 0 \\\\0 & 1.7\\end{bmatrix}", "4ff33908f4c4ff5c89c34e73f4cfc6bd": " T \\subset S ", "4ff351cafd3392a29986ec0e91c42f4f": "f\\left(S\\right)", "4ff36f6d563e1783fab4a6a1b6ae5069": "z_2 = e^{i\\,\\xi_2}\\cos\\eta. ", "4ff374313fa221a11ab2208abf6fa2a2": "T_H = {1 \\over 8 \\pi M}.", "4ff3cf1c3c9cd0b7af0c89d394e3af52": "\\Kappa\\,", "4ff3d021e06365382a9ccf7debe267de": "\\sin^2(x/2) = \\frac{1-\\cos(x)}{2}", "4ff3ddbcebe785e634d6050367b95344": " \\frac{P_1}{T_1}=\\frac{P_2}{T_2}", "4ff3e53c567e66d089c657641180aa32": "k_{\\lambda}.v = q^{(\\lambda,\\nu)} v", "4ff3f6e11633f4df584e67e29c71f94c": "\\phi\\land\\psi", "4ff420a385fe4e34e88f681f5c8aa591": "P^\\mu", "4ff4b08dec3bbd82a8bc8b24716daef9": "\\Omega_*^{PL}(X), \\Omega_*^{TOP}(X)", "4ff4bf64a5e9d36f5176afc8626edada": "\\Omega_0 = \\frac {\\rho}{ \\rho_{crit}} \\ , ", "4ff4c1cfccf3a6bb729a39be4d54bc6b": "f (y_i | \\mu_i , \\sigma_i , \\nu_i , \\tau_i )", "4ff4e7dad744c67132915bda07365e9d": " i_{\\text{C}} = I_{\\text{S}}\\left(e^{\\frac{V_{\\text{BE}}}{V_{\\text{T}}}} - e^{\\frac{V_{\\text{BC}}}{V_{\\text{T}}}}\\right) - \\frac{I_{\\text{S}}}{\\beta_R}\\left(e^{\\frac{V_{\\text{BC}}}{V_{\\text{T}}}} - 1\\right)", "4ff517b9343cebc2d272499feef21181": "\\frac{1}{N}\\frac{dN}{dt}=\\frac{\\ln\\left(\\frac{1}{2}\\right)}{T_{1/2} }", "4ff51e633e388798b1bdb89b1a36c6f4": "P_{r}= \\frac{e^{-\\beta E_r}}{Z}\\,", "4ff52e442077655a9ced61a154a9fb87": "12x^2y-18y+7x^2y-3y^2+16y-8y^2-8x^2y\\,\\!", "4ff5355b3596ee398585c17a54bc2b4c": "\\mathbf{ x}(j)\\text{s}", "4ff55a68c5e9623c5e12df786021031d": "\\left(\\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ \\sqrt{1/3},\\ \\pm1\\right)", "4ff566428f50a996d5bec34b68bba274": "\\Gamma^*", "4ff5a667c2569bbb328b6a9714695864": "H_{el} = D_{ac} \\sum_{q} \\sqrt{ \\frac {\\hbar} {2 M N \\omega_{q} } } ( i e_{q} \\cdot q ) [ a_{q} e^{i q \\cdot r} - a^{\\dagger}_{q} e^{-i q \\cdot r} ]", "4ff5c350227aebf39ac3a9ad951dfc36": "\\sum x_n", "4ff60d0bb84ca6079c4805cebd1fb738": "u' = u - v \\, .", "4ff681491bdac9d1b6d6be7858d1247d": "V_1= 19.72 \\left [ \\frac{7800 d^3 \\left [ \\left ( \\frac{e_h}{d} \\right) \\sec \\theta \\right ]^{1.6}}{W_T} \\right ]^{0.5}", "4ff69288ccfec85f179df881d5d8076a": "E[G|K] = \\int^T_0 k(t)E[x(t)|K]dt = \\int^T_0 k(t)S(t)dt \\equiv \\rho", "4ff76b8abca5898ba9b4f77197401f08": "E_\\mathrm{k} = {{3}\\over{2}}nRT = \\frac{3}{2}NkT", "4ff77e32ccf996db3ca09dc4b6adbe54": "N_{p} = 0, N_{n} = N", "4ff7b61ee83385a2b7cdc089f633b838": "J(v,w) = (-w,v).\\,", "4ff7cca8fe1c4126c2d22e897e7d1bd7": "a_1,\\ldots,a_m", "4ff7d5486e424c9bc8f2f6810412c85c": " y(4) = e^4 \\approx 54.598 ", "4ff826baaad95b3b7fd12d1131922b42": "\\sum_{n \\ge 1} f(n) \\frac{x^n}{n!}.", "4ff8bde7777b0eec6f2abecd6a73d416": "FGT_1=\\frac {1} {N} \\sum_{i=1}^H (\\frac {z-y_i} {z}) ", "4ff905b0b0d75eec25dcabe1a5a362d4": "m_{\\rm e} = \\frac{2R_{\\infty}h}{c\\alpha^2}.", "4ff909170176e0e2262f00585a37f713": "\\Theta=2 \\arctan\\left(\\frac{D_f-D_i}{2 l}\\right).", "4ff947073354e48eeb22ad027593451a": " \\langle X^1 \\rangle", "4ff9608c1b28be2363957393436c5a32": "{\\color{white}.}\\qquad\n\\cos\\alpha\\,d\\sigma = d\\beta, \\quad\n\\sin\\alpha\\,d\\sigma = \\cos\\beta\\,d\\omega.", "4ff9e7263999ca0c67fcca26d7057f0a": "\\left\\{a_n\\right\\}_{n=1}^\\infty", "4ffa1a8cbdeea13a184efe1b2d9ebe97": "\\frac{\\partial \\phi(x,t)}{\\partial t}=\\nabla\\cdot (D(x) \\nabla \\phi(x,t))=D(x) \\Delta \\phi(x,t)+\\sum_{i=1}^3 \\frac{\\partial D(x)}{\\partial x_i} \\frac{\\partial \\phi(x,t)}{\\partial x_i}\\ ", "4ffa49bea15284d2f8394a1a3f935c60": "{\\mathcal L}_n=L_n+ {1\\over 2} (n+1)J_n", "4ffac2e0bc3f3534f9c40aa23c0e212b": "\n \\cfrac{\\rm{d}\\mathbf{x}_i}{\\rm{d}\\alpha} \\equiv \\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\n ", "4fface7fbe5ff2cf2d9d7b065e70ffe1": "\\gamma^5/20=O(\\epsilon^{45})", "4ffbb4da5d2ebf9b7f819c5966c5b302": "a^{\\dagger}|n\\rangle= \\sqrt{n+1} \\ |n+1\\rangle", "4ffc094458e8eb83bb5d2b13e8fb5ad4": " s_k = \\frac{(3 + 2\\sqrt{2})^k - (3 - 2\\sqrt{2})^k}{4\\sqrt{2}} ", "4ffc0ed97e1ef2c6585ae8eeb07b760d": "-x_1,y_1)", "4ffc1b3352715642443f17b8b1fa0dec": "\\Box_b", "4ffc643cb3aa0ae0d9c528220699f786": "X_i ", "4ffc6ee3f78b94c8cd66a294a0b668f9": "\n \\begin{align}\n w_x(x) & = \\frac{q_{x1}}{6bD}\\,(3ax^2 -x^3) \\\\\n \\theta_x(x) & = \\frac{q_{x2}}{2bD(1-\\nu)}\\left[x - \\frac{1}{\\nu_b}\\,\n \\left(\\frac{\\sinh(\\nu_b a)}{\\cosh[\\nu_b (x-a)]} + \\tanh[\\nu_b(x-a)]\\right)\\right]\n \\end{align}\n", "4ffc7a6401792b9c63d5c4c254d29d49": "uv\\equiv xy", "4ffc7bdd365ef4c47b785a68e13cee45": "\\frac{1}{h_1}+\\frac{1}{h_3}=\\frac{1}{h_2}+\\frac{1}{h_4}", "4ffccd07b312e4c9540113facc2881fc": "\\Phi_b(w)", "4ffd0626d8afeef97c92cfc0431ca55f": "\\vec p\\!", "4ffd3d66b7a85c366cba9cba53012107": "\\Delta_{P2}", "4ffd43545e6a0431d6355f17f7fc7fe7": "2^{8/12} = \\sqrt[3]{4}", "4ffdfd6c6eeb9193470ce464d0aa7019": "f \\frac{r}{c} = 2 \\pi^2 \\frac{\\mu N}{P} \\left( \\frac{r}{c} \\right)^2 = 2 \\pi^2 S", "4ffe2466d8f84b9010531c37953b2d69": "k = \\sqrt{k_g^2+k_n^2}", "4ffe6952894f08e696057b79ab72231d": "(A - \\lambda I)^{k_1} = 0", "4ffe6dd71c517c7f98e7b909bcb050b7": "\\ C=\\frac{P_tG^2\\lambda^2}{4^4R^2}(\\theta^o/180)^2\\sigma^o", "4ffe98209cbc370833c886c468905576": "\\mathbf{q} = c_1 e^{\\lambda_1 t} \\mathbf{v_1} + c_2 e^{\\lambda_2 t} \\mathbf{v_2} + \\cdots + c_n e^{\\lambda_n t} \\mathbf{v_n}", "4ffea5248f45e497b88c3a474d4c324e": " P_c(z)=z^2+c \\,", "4ffefdab0a8eade9952da40dd4093c56": "\\mu\\colon M\\times Q \\to Q", "4fff2e38c2290305466853b4146b7789": "\nR(t) = e^{-\\lambda t} \\, \n", "4fff54a8936b14d767c56fa0de8a8131": "\\scriptstyle 10^{-10}", "4fff5bb5515b4daa373f232cfed5ca96": "\\scriptstyle \\hbar \\;\\to\\; 0", "4fffb0c53281f7240047c12f5dbbcd50": "\\langle x,x \\rangle = 0", "4fffd0bb026f3f8a09a5ac6dbf47f396": "V_L = -V_o", "500028badeae427418860bbdb3b8256e": "x = \\frac{R_{bef} - O_{bef} \\pm H}{c}", "500031336825d0e0570cb6bd95a3b573": "{Z \\over R}", "500043758ba32c07afd5a4a8c373aa03": "\\displaystyle p(x)=x^3(x^2-2x+a)^2", "500054e194270e45790ec1cf887f8ac3": "x^{\\lambda} = x^{\\rho}", "5000e02d43c763314180f701fa5a2fae": "k \\in \\{1,2,\\ldots\\}", "500162a084125fb28a0e9387929439b3": "F\\colon X\\to X'", "5001abb10035b7b1ea716b3ca18a9278": "f(x,y) = ax^2+bxy+cy^2 \\, ", "500237e7baab8abe27e87f7f8907279a": "\\mathsf{P^{\\sharp P}} \\subseteq \\mathsf{P/poly}", "50029e30957aaf22ffed76704f9513cd": "\\Box \\psi + \\frac{\\partial{}V}{\\partial \\psi} = 0", "5002cb585821a672d48ea8c4c5bfee51": "(S,I)", "5002eeaaec17fee304f295dd3f231c96": "{\\overrightarrow{V_g}}", "50030d1868601730a459d723baf1054b": " y_{n+1} - y_n = h f(t_{n+1}, y_{n+1}) ", "50033c15af7795cd6617f117b29d6922": "Lt", "5003ae4b9892505ffd726a49f97c5488": "\n\\begin{align}\n\\sum_{k=-\\infty}^\\infty \\hat f(k)\n&= \\sum_{k=-\\infty}^\\infty \\left(\\int_{-\\infty}^{\\infty} f(x)\\ e^{-i 2\\pi k x} dx \\right)\n= \\int_{-\\infty}^{\\infty} f(x) \\underbrace{\\left(\\sum_{k=-\\infty}^\\infty e^{-i 2\\pi k x}\\right)}_{\\sum_{n=-\\infty}^\\infty \\delta(x-n)} dx \\\\\n&= \\sum_{n=-\\infty}^\\infty \\left(\\int_{-\\infty}^{\\infty} f(x)\\ \\delta(x-n)\\ dx \\right) = \\sum_{n=-\\infty}^\\infty f(n).\n\\end{align} \n", "5004228399eb7c587709363aefb9e11a": " \\textbf{P}(t) = R(t)\\textbf{e}_r + Z(t)\\vec{k}.", "50047830240dc1852f89f9fbf68c6d72": "{R\\over r} = \\tan\\frac{\\pi}{p}\\tan\\frac{\\pi}{q}=\\frac{{\\sqrt{{sin^{-2}{(\\theta/2)}}-{cos^{2}{(\\alpha/2)}}}}}{\\sin{(\\alpha/2)}}. ", "500488a181089ac3d105ad188199fc1c": " x_i , i=1,2,3,\\ldots,m ", "50048a4cba995a2f356ec6520a79c95d": "B=\\ast_{v\\in V} A_v/{\\rm ncl}\\{\\alpha_e(g)=\\omega_e(g), \\text{ where }e\\in E^+T, g\\in G_e\\}", "500490e5d5b0b773243c0abcbdef2f75": " k=k(\\rho) ", "5004b6a7c019e59477d14bb92712a13b": "(\\overline{z})", "5004f3b31fbdc56b9e0e2b027deabcbc": "\\text{Hom}_{\\Gamma(F)}((c_1,x_1),(c_2,x_2))", "50053104d029b8e2347ca36dadeac513": "1\\ \\mathrm{savart} = \\frac{1.2}{\\log_{10}{2}}\\ \\mathrm{cent} \\approx 3.9863\\ \\mathrm{cent}", "5005ec80c62ee1d3674f108a1747276f": "\\alpha, \\beta, \\gamma, \\delta, \\epsilon, \\zeta", "500648a0429fac81b606f850596d06a3": "\\frac{2R}{\\sqrt(3)}", "500651ec4cb74fe99c36a73b796f8d33": " \\hat{U}^{\\dagger} \\approx I + i\\hat{H}^{\\dagger}\\tau ", "50066b5c8c1d7664b9fcf23996dadbfd": "\\rho*\\lambda^2/(4*pi*)^3 *d^4", "50067a17563fdd21bb24fefeaf69adec": "\\underline{\\mathbb C}", "5006833d96fcdf40d1fff83ed4bfb7c1": "\n\\sigma=\\int \\rho \\, ds\n", "500683545127d15956ce81a5e1d2843f": "\\alpha_i = P", "5006e5de1f4de566d0b46444cc3a3194": "t_{n+s}", "5006f12bfe1fed33896c3c489ab0903c": "\\overline{\\epsilon}", "50073557790b8f63ec7ead04c7b06b6a": "X_1 + X_2", "50078b88a3f3dcd6db51b902a8bb64c7": "\\lim_{x\\to c}{f(x)} = \\lim_{x\\to c}g(x) = 0", "5008ae39f2451cc6f1ff02885b01c61e": "\\frac {V_\\mathrm i}{V_{x1}} = \\frac{\\delta Z+\\frac{Z_0 /\\delta Y}{Z_0+1/\\delta Y}}{\\frac{Z_0 /\\delta Y}{Z_0+1/\\delta Y}} \n= 1 + \\frac {\\delta Z}{Z_0} + \\delta Z \\delta Y", "50090cf4fbe9c7fe31da69f7b5e0f462": "P(x\\rightarrow x')", "50091983f91f4f97d392a9f3e22334f5": " \\acute{R}_{\\alpha \\beta} = 8\\pi { G \\over { c^4 } } \\left ( { A \\over 2 } \\acute{T}_{\\alpha \\beta} + {B \\over 2} \\acute{T} g_{\\alpha \\beta} \\right ) ", "50091a4916fcf87929d778327ae9d861": "\\ \\delta_{d}", "50095ad1e32d6ae68d6095f9b865b048": "\\beta_1 = .703", "500963121b9349006d0bc01acb34e5b8": " q^*=\\frac{f_D q_{0,D} + f_A q_{0,A}}{f_D + f_A} ", "50096c3d83dd73868e3050d3b4e56f15": " \\cot \\theta\\! ", "500981a9f2ba2d74607c782f974f1d4a": "G(z) = G(z_1,\\ldots,z_d)=\\operatorname{E}\\bigl (z_1^{X_1}\\cdots z_d^{X_d}\\bigr) = \\sum_{x_1,\\ldots,x_d=0}^{\\infty}p(x_1,\\ldots,x_d)z_1^{x_1}\\cdots z_d^{x_d},", "50098c290214fa46aa26dd560fb503d0": "\\frac{\\hat{\\mu}^0_{ij}}{\\sum_{i=1}^{M+1} \\hat{\\mu}^0_{ij}}", "5009bfd282c5d47d29a011564284d709": "H_k(x)", "500a6170038a888a9a53623d944c9772": "\\aleph_{\\mu}^{\\aleph_{\\alpha}} = \\aleph_{\\mu} \\; \\aleph_{\\mu -1}^{\\aleph_{\\alpha}}.", "500a9e31ec6985067b62e7fdbd14dd15": "(P_i=a) \\and (P_j=b) \\and \\dots \\and (P_k=c) \\to (Q=d)", "500ac703e4e89281a74221abae63fdb5": " a_{32} ", "500ad73e00faa3be8ee09f91db162e85": "E = K + \\varphi ", "500b04f1190e6f9a7f9d83a92659ea01": "\\gcd(n,q+1)", "500b1fa7d066aa7fc537e05d4be59a62": "\\lim_{x \\to 0^{+}}{1\\over x} = +\\infty, \\lim_{x \\to 0^{-}}{1\\over x} = -\\infty.", "500be45b6e3fcb9987a00e080a324583": "+\\infin", "500bf4d4ee9442148e66bf39aea67935": "h^{\\mu\\nu}T_{\\mu\\nu}", "500c1146f4b1e30e161755bf342f148c": " \\mathbf{Z} = \\begin{pmatrix} 0 & 1 & 0 \\\\ -1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} ", "500c20e914962a07a2786b279eaa7948": "\\operatorname{Li}_{-2}(z) = {z \\,(1+z) \\over (1-z)^3}", "500c52f2c78fbd9de01e425e0b27b141": "\\displaystyle \\left(\\frac{i}{2\\pi}\\right)^n \\delta^{(n)} (\\xi)\\,", "500c727643b6e4034e48afeb2f6f038b": "e[n]=e_a[n]+e_f[n]\\,", "500c7958fafa02929f6f92773a0e8532": "f_* M = \\operatorname{Hom}_R(S, M)", "500ccf0fecc7d78c3f848a19bfe669dd": "\\,g(X,Y) = 0", "500ce8012397e20220226244bcb33a1f": " \\frac{\\partial\\rho(x,t)}{\\partial t}=-\\nabla\\cdot J(x,t) ", "500d75dfa1aabfd0e00d5269545e0e61": "T =\\frac{2 \\pi\\, k}{\\sqrt{g \\overline{GM}}}\\ ", "500dbda5543aee3bf02647040d0fad4e": "\\displaystyle W(x)<0", "500dbfa3d2137d8afaab70a508d47ca5": "\\text{Hg}_2^{2+} + 2\\text{e}^- \\rightleftarrows 2\\text{Hg(l)}", "500de063e52be1e706a71e7a887f95bf": "\n\\sigma(n)=\n\\frac{\\pi^2}{6}n\n\\left(\n\\frac{c_1(n)}{1}+\n\\frac{c_2(n)}{4}+\n\\frac{c_3(n)}{9}+\n\\dots\n\\right) .\n", "500e0371db6ed73455838e16e9a6625d": "X^{\\star}", "500e1d92035e4a8a2061fd40d48a7320": "\\int^{N_f}_0 {\\rm d}N = \\int^{a_c}_{a_i}\\frac{{\\rm d}a}{C(\\Delta\\sigma Y \\sqrt{\\pi a})^m } =\\frac{1}{C(\\Delta\\sigma Y \\sqrt{\\pi})^m }\\int^{a_c}_{a_i} a^{-\\frac{m}{2}}\\;{\\rm d}a ", "500e2f3221150c9d9691a4ddcb3ac3d0": " t^*=\\sigma ^* n^* ", "500e7d3cb2ca5f428df34009c2bbfdc7": "k = \\frac{2 \\pi}{\\lambda}", "500e8c8363d796a30bf6751865323b1a": "\\theta_i=\\theta_i(\\xi_j)", "500f2ab9ab140d52391a0fe1ab6dec7a": "TE = - G \\frac{M m}{r}\\ + \\frac{1}{2}\\ m v^2", "500f447167ac3adcc98ab8e5467a8c33": "\\alpha_{avg.} = \\sqrt {(\\alpha_t)(\\alpha_b)}", "500f5830aa000a1e2e8874a5ed3c879e": "\n\\min_{x\\in X_N} \\hat{g}_N(x)\n", "500fb55917e022c60030adc586462bf7": "\\begin{cases}\n\\mathbf{R} \\times TM \\to TM \\\\\n(t,v) \\longmapsto tv\n\\end{cases}", "500fdad65362df2d8ec3b93e9a9ef508": "k\\geq \\frac{2n R_L(x_0)}{\\epsilon} \\log \\left(\\frac{f(x_0)-f^*}{\\epsilon \\rho}\\right)", "501036ab0703f24b1b4f99521d0175ae": "\\overline{\\mathbf{GT}}", "50103d3a0471b73461a576ae892388b8": "\\mathrm{ker}(\\partial_2) = \\mathrm{Im}(\\partial_2) = 0", "5010684359d01486f6e73d5b355e1b34": " \\Sigma_k (x_1...x_k) = \\Sigma_k (y_1...y_k) ", "50109245064bc820b59582ece4e54438": "\n\\begin{align}\n{} & (a^2 - b^2)^2 + (b^2 - c^2)^2 + (c^2 - a^2)^2 \\geq 0 \\\\\n{} \\iff & 2(a^4+b^4+c^4) - 2(a^2 b^2+a^2c^2+b^2c^2) \\geq 0 \\\\\n{} \\iff & \\frac{4(a^4+b^4+c^4)}{3} \\geq \\frac{4(a^2 b^2+a^2c^2+b^2c^2)}{3} \\\\\n{} \\iff & \\frac{(a^4+b^4+c^4) + 2(a^2 b^2+a^2c^2+b^2c^2)}{3} \\geq 2(a^2 b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4) \\\\\n{} \\iff & \\frac{(a^2 + b^2 + c^2)^2}{3} \\geq (4\\Delta)^2,\n\\end{align}\n", "50109c3b605fad720376c802cf69ff85": "[B] = \\frac{k_1}{k_1+k_2}[A]_0 (1-e^{-(k_1+k_2)t})", "5010b7738c83764bae8ee3fd9a6b763c": "0\\cdot g(x),\\quad 1\\cdot g(x),\\quad x\\cdot g(x), \\quad (x+1) \\cdot g(x),", "5010f35cffa731a11c2be08033dc3506": "P_k=\\sum_{n \\neq k}c_n^*(t)c_n(t)", "5010f610a326b64f2528c3330338b7bf": "{| \\psi_B \\rangle} \\in \\mathcal{H}_M", "50111b65a0809f4d5099c056864d799c": "H_{1} (x|q) = 2x", "50111db1a986f904c1d3817517345d08": "f \\cdot \\mathbf{1}_A", "50111de555114ef9eafb93f4a74895dc": "y' = B y + \\beta ", "5011797d66a59436ee970611be2a73dd": "b^{n}\\equiv b\\pmod{n}", "501189c0be9ab5f18a61c2ffe3d637b3": "\\color{blue}\\mathcal{S}", "50119a13ce1600e426beca4202705daf": "s(\\pi(y))=y\\,", "50119ac63ebeb52e290443c98a1fa506": " {S_4 \\over S_2} = {{27\\over25} \\div {135\\over128}} ", "50125e654a62d2f1d3b6258c3ceaa270": "\nX_i \\sim IG(\\mu,\\lambda w_i), \\,\\,\\,\\,\\,\\, i=1,2,\\ldots,n \n", "501274943de8a15e34c34435b398a528": "(9)\\quad \\hat\\theta_{ab}=\\hat B_{(ab)}\\;,\\quad \\hat\\theta=\\hat h^{ab} \\hat B_{ab}\\;,\\quad \\hat\\sigma_{ab}=\\hat B_{(ab)}-\\frac{1}{2}\\hat\\theta \\hat h_{ab}\\;,\\quad \\hat\\omega_{ab}=\\hat B_{[ab]}\\;.", "5012927296870ec60658c33ea14c8b32": "\\begin{align}\nc^2 - a^2 & {} = b(b + 2a\\cos(\\pi - \\gamma)) \\\\\n& {} = b(b - 2a\\cos\\gamma),\n\\end{align}", "50129348a136ff8688cb15f942961d1a": "P(\\epsilon_n=\\pm1)=\\frac12", "50131a0a53c64dca317d21495f393c60": "dS = 8 \\pi M dM = d(4 \\pi M^2).", "50132fcbb739bd745e4f59ca5a37fa34": "P(x) \\nleftarrow (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\nleftarrow Q(y))", "501352e3b566c0f649f4fc869b02faef": "h_{\\,\\times}", "50136157945a410d6f3383b2e3a1b7c9": "-\\frac{\\hbar^2}{2m}\\nabla^2 \\psi(\\bold{r}) = E \\psi(\\bold{r}) ", "50136a915b92218c35659fe1fef8c08b": "I_{n} = \\int x^{ax} \\sin^n{bx} dx\\,\\!", "50139f5afc139e86616042acb3d68044": "PQ = 4x^2 + 21xy + 2x^2y + 12x + 15y^2 + 3xy^2 + 28y + 5", "50145cf6707de180f2ba6dce08bde39a": "P_r = {{P_t G_t A_r \\sigma F^4}\\over{{(4\\pi)}^2 R_t^2R_r^2}}", "5014a32fdaf7bbc601020c0cc02148b8": "(1-\\lambda)", "5014eed0753b3f0a8cce7f67d6c0333a": "n_0 = \\lfloor x \\rfloor", "501531d565a0a20e9d6b93cd3d041920": "{\\mu}", "50159e4cee736dc23dcae8b9651b54ee": "dS_t=\\mu S_t dt + \\sigma S_t ^ \\gamma dW_t", "5015a2edac49661aed88e5b85e311192": " \\hbar=1", "5015e5a0d7132bba6f02b21130ff53e2": "A(X)", "5015fb61e47f0d6631d6440860cb5571": "d(h)^2 = {D^2} \\left({H-h \\over H-h_b}\\right)^{1.6} ", "50161e042070a756e8c022679ec52809": " [R_O(x)]^2 - [R_I(x)]^2\\ \\not\\equiv \\; [R_O(x) - R_I(x)]^2", "50167ea217ca17be9aef2eb5fad71ff1": "\\{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\\}", "50169b05f287834a7dc9fa853a4fdf5e": "w_1^3, w_1 w_2, w_3", "5016ca32fd411451fb91fcbd157027e4": " l ", "501715567398b38130dea6681af4524b": "\\Pi(k_i)=\\frac{k_i}{\\sum_j k_j}", "5017b971d653b4070b8c0b4299ea6960": "\\begin{smallmatrix}R_\\odot\\end{smallmatrix}", "5017c7edac324c3c9bcbec35769cced1": "S(a,q,0)=0", "5017d32c7a3ed7ca6318f6baf100af0c": "\\gamma=1/\\sqrt{L} ", "5017f4f7501c0298beba769626aa0b6f": " \\eta: X \\setminus M \\rightarrow Y \\setminus N ", "50180362e85d1418c4681d3fe992e91b": "\\rho(N) = \\|N\\|_{op}.", "50186df2feff940402eaf1db02f4251f": "T = F^{-1}(U)", "5018bdfbb17976f82f0c6868a98286c0": "P(A \\cup B)", "501965040efe4c50edc8d109756180d9": " 0< \\operatorname{Re}(s)<1 \\!", "5019655dd08b4c73fba4b4fff9c5fb08": "x^{'}_{i} \\leftarrow x^{'}_{i} - \\delta", "5019a1c118d5b0f6705931defd264c63": "b\\stackrel{\\mathrm{def}}{=}\\left\\{0,1\\right\\}", "5019b24d28854565dadbdbfc05d7ed43": "\\rho(u)\\approx u^{-u}.\\,", "5019e9c59367cf81db67fa518937b550": "\\ a=b", "5019ef8cb51410d8ca44c6be8b89f1cd": "\\alpha = 1", "501a40f033fa30e338be60d185805fe4": "U_1(x,y)=\\alpha \\left(\\alpha x^\\rho +(1-\\alpha)y^\\rho\\right)^{\\left(1/\\rho\\right)-1} x^{\\rho-1}", "501a73fe3416f18681209472191995fa": "\\frac{W_\\alpha}{W_{TOT}} = \\frac{c-b}{a-b}", "501a96e1ba389c43c073869ee2608364": "\\mathbb{Q}(\\sqrt{p})", "501a9a2eee444fc5b85dac82700e246d": "\n\\bar\\varepsilon = \\varepsilon K / P. \n", "501aa952433c8c40f2a570b35f93acb7": "\\Sigma|b_k|", "501b0f05aff8bd242375c4c3b8720976": "b_k(X)=\\prod_{1\\le j\\le n,\\;j\\ne k}(X-z_j),\\quad k=1,\\dots,n,", "501b1ddac5b4089ffdda205c6e21e39f": "{\\omega^1}_3 = -\\left( 1 + \\frac{r \\, g'}{g} \\right) \\, \\sin(\\theta) \\, d\\phi", "501bdf36a31cde4af63f189c9d108d83": " \\Phi_{00}\\,\\hat{=}\\,0\\,,\\quad \\Phi_{10}=\\overline{\\Phi_{01}}\\,\\hat{=}\\,0", "501be9c7889dcb05ad467333a9dcc204": "P_2=(1/10)-\\epsilon", "501c3ec4aa64a9cc5e3de48bc6ab6e88": "\\vec{s}_a \\cdot \\vec{s}_b", "501c8ff9ba82eb65ed85095bb3229310": "5x \\equiv 2 \\pmod 6\\ ", "501cc797a271f079706218f4eb7634a3": "U {{\\partial v'\\over \\partial z_1} \\over {\\nu \\triangledown^2 v'}} = O \\left({r \\over a} \\right).", "501cd05d61b676738a207e471eb2f996": "\\vec r(x, \\phi) = (x, R\\cos\\phi, R\\sin\\phi). ", "501d6c7172799065fefb6a61ae05ac86": "\\sin(\\theta) = x ", "501d6f40d47df86fa66711f7bbee4459": " \\psi_{n\\ell m}(r,\\theta,\\phi) = \\sqrt {{\\left ( \\frac{2}{n a_0} \\right )}^3\\frac{(n-\\ell-1)!}{2n[(n+\\ell)!]} } e^{- r/na_0} \\left(\\frac{2r}{na_0}\\right)^{\\ell} L_{n-\\ell-1}^{2\\ell+1}\\left(\\frac{2r}{na_0}\\right) \\cdot Y_{\\ell}^{m}(\\theta, \\phi ) ", "501dc5c138ab9a3f8fd1ecd0dc2e0265": "v(x) \\leq \\varphi(x)", "501dd8a088476d1afca08bc3f7961da9": "\\begin{matrix} {3 \\choose 1}{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "501e443bce84b46333d36aa858460042": "X = 100\\cdot c_p\\cdot (T_s - T_b)/H", "501e9fe078344c03a8d2e0afed176c85": "[n]_q := \\frac{q^n-1}{q-1}=1+q+q^2+\\dots+q^{n-1}.", "501ead6aece7e17059d6349f0c12d34b": " G(\\boldsymbol{w},\\boldsymbol{\\psi}) ", "501eb78fec7f5ee6aa3ca156eb5b0631": " \\operatorname{rank}(G)=\\min\\{ |X|: X\\subseteq G, \\langle X\\rangle =G\\}.", "501f053f8490559eac06baa847314d89": " \\vec{a} \\times \\vec{b} = \\vec{c}", "501f77a0831329c572446da433950387": " \\oint_C \\Gamma(\\alpha)\\,d\\alpha = 0 ", "501f871befba633c95263877dc70674b": " { \\mathit l = \\mathit l^{\\prime} } ", "501f8adf8d433a4f7e6dd7b9e2498686": "A^T P + P A + Q = 0", "501f9ee7cac0f8fa81d2da8f2e330892": "r=\\frac{1}{T}\\left (W(-se^{-s})+s\\right )", "501fa04cd561c9b2969c18d6ce180440": "-S_x \\otimes S_z", "501fc75acdbb336d8da1ec5eabdcddb9": "\\alpha f\\in C(E)", "501fe7e72a575f0b5cae33402889467c": "y''(t_0) = {\\partial f\\over\\partial t}(t_0, y(t_0)) + {\\partial f\\over\\partial y}(t_0, y(t_0)) \\, f(t_0, y(t_0)).", "502009bd0c8dec755ad0af9e3e6fb09d": "b_{t+1}\\geq\\eta", "502055d86c88fa3a698ac7ff4646505a": "a \\in R", "50209b7fc1f8e1ae1acb6b27dd40e7d8": "\n H_{p}x(n) = \\sum_{\\tau_1=a}^{b}\\cdots\\sum_{\\tau_p=a}^{b}\n {h_{p}(\\tau_{1},.\\,.\\,,\\tau_{p})\\prod^{p}_{j=1}{x(n - \\tau_{j})}},\n", "5020a63d6218d916ba1c61fc4af6ad1e": "l(X) = \\aleph_0", "5020cb40c6036c616c1a3710f3c6a04a": "\\partial\\boldsymbol{X}/\\partial t", "5020e0741351422ec49422f36844b9ae": "\\phi_{\\rho_{n},e}(c)", "50215110a5d8340aaf45c581aa053fd1": "x = -1.84208\\dots", "5021a75e39090a2200307108b2fc0e84": "\\scriptstyle{\\mathrm{R}^- = \\emptyset}", "5021b1956ecfe067fccef70ababbdec7": "\\omega_{pe}/\\omega_{ce} = 3.21\\times10^{-3}\\,n_e^{1/2}B^{-1}", "502231a8760e8e53fef66641bc1a0121": "d>3", "502270288d8be1ca2db1c26761a88bc3": "-x_0^2 + \\sum_{i=1}^n x_i^2 = \\alpha^2", "5022802d4aa5df1d269bfbe57c4f7126": "X = E(X) + \\int_0^\\infty C_s\\,dB_s.", "502299d494f5971512e0493e1f1f60a1": "K_2 =\\mathrm{\\frac{[[Ag(NH_3)_2]^+]}{[[Ag(NH_3)]^+][NH_3]}}", "5022bf25168895ae1443a6ae5017d1b4": "\\frac{\\displaystyle\\Bigl(\\bigwedge_{i=1}^n(p_i\\to q_i)\\to p_{n+1}\\lor p_{n+2}\\Bigr)\\lor r}{\\displaystyle\\bigvee_{j=1}^{n+2}\\Bigl(\\bigwedge_{i=1}^{n}(p_i\\to q_i)\\to p_j\\Bigr)\\lor r},\\qquad n\\ge 1", "5022c48cf78732cb93fb6db98bb357cc": "y=2xc+x^2", "502320cb33fda6758e4dc94f32215263": "K(X)(\\sqrt[p]{X}) \\supset K(X)", "502328270503cdca5b5ad13fc1879a7d": "\\int_{x_1}^{x_s(t)}w_t \\, dx\\rightarrow 0", "50233c5adf44baa6c8f019e626419c08": "P_i := (x_i,x_i^2)", "50236b93792d3d714ef7930a189fe7c2": "\nQ_i(t+1)^2 = \\max[Q_i(t) + a_i(t) - b_i(t), 0]^2 \\leq (Q_i(t) + a_i(t) - b_i(t))^2 \n", "502379ed9b181b53f045e0ed4c41fd96": "\\|x+y\\|=\\|(1,1)\\|=|1|+|1|=2=\\|x\\|+\\|y\\|.", "5023909d75ddb4c83fd1d943a05fc88d": " (2^{k-1}+1):(2^{k-2}+1) ", "5023c35822ad288b42f1319039d86403": "aab\\widehat{b}ccdd", "5023d23c675d57246392080c41e46744": "x = \\frac{X}{Z^2}", "5024093c5bdd9b5f09d04b6fe313af50": "l=0,1,2,\\dots,n-1\\,", "50243eed1bacafbe03c1ef32c2c05f40": " \\frac{\\langle \\Psi(a) | H | \\Psi(a) \\rangle} {\\langle \\Psi(a) | \\Psi(a) \\rangle } = \\frac{\\int | \\Psi(X,a) | ^2 \\frac{H\\Psi(X,a)}{\\Psi(X,a)} \\, dX} { \\int | \\Psi(X,a)|^2 \\, dX}. ", "5024952028217b65152b87c8eb78bb40": "\nZ_k=\\int_{[a,b]} X_t e_k(t)\\, dt\n", "5024e507cdad895c9e3f1a2c74994a42": "\nA_{s} = \\left[ p^{2} x_{s} - p_{s} \\ \\left(\\mathbf{r} \\cdot \\mathbf{p}\\right) \\right] - mk \\left( \\frac{x_{s}}{r} \\right) = \n\\left[ \\mathbf{p} \\times \\left( \\mathbf{r} \\times \\mathbf{p} \\right) \\right]_{s} - mk \\left( \\frac{x_{s}}{r} \\right)\n", "5025599df4343f18e4156723eb28b2d6": "0 \\le x \\le 2 ", "502561559b9a3c5d70d572a11bfebc46": "\\mathit{L}", "502600b712055d2a7089d988731eda41": "\\lambda =(y_1-y_2) (x_1-x_2)^{-1}", "50260fedafa9f167ceacad16344f17f4": "O(g)", "50261e419a3841ecbcf44611a58d48a7": "\\Phi =", "50262dd8360d50b8c9ad2195dafddc48": "\\mathbf{C}_{2,1} = \\mathbf{A}_{2,1} \\mathbf{B}_{1,1} + \\mathbf{A}_{2,2} \\mathbf{B}_{2,1} ", "50265f1502ee81ec3ec4a8ed347922e1": " H^*(X;A) \\,", "50268ffe5f8b6f9edf9050011e60c0c6": "p_2: H\\rightarrow G'", "5026b8de237ace3c33c96ad8519b375e": "(f\\cdot g)'=f'\\cdot g+f\\cdot g' \\,\\! ", "5026ba86fded88324bfdf737005d63ae": "A = \\left[ \\begin{alignat}{3} 1 && 3 && 2 &\\\\ 2 && \\;\\;-4 && \\;\\;\\;\\;5 &\\end{alignat} \\,\\right]\\text{.}", "5026c15c12f11d214e6a4466e0880edc": "W_{f}", "5026d4a2fd816b729e71cf623c3237dd": "P_{0}(\\zeta)", "5026d5bf98636a67577ca7a55fdc83e8": "J(u_n) \\to \\inf\\{J(u)|u\\in V\\}.", "502707f238743da1f0b5c367781d6088": "f(xv) = xf(v)", "502739d2a39aa9006a7414534084de81": "\n\\frac{\\partial L}{\\partial x_k} -\n\\frac{d}{dx_3}\\frac{\\partial L}{\\partial \\dot x_k} = 0\n", "502760934cf9660d3ef380bce9d037d6": "\n\\mathbf{b}_{i,j} = \\frac{\\partial \\mathbf{b}_i}{\\partial q^j} = \\Gamma_{ijk}\\mathbf{b}^k \\quad \\Rightarrow \\quad\n\\mathbf{b}_{i,j} \\cdot \\mathbf{b}_k = \\Gamma_{ijk}\n", "5027cfc254a3b42212fa837f9de8eb0d": "y\\perp\\!\\!\\!\\perp\\textbf{x}\\,|\\,\\eta^T\\textbf{x},", "5027e454a430423d7bedf41a04f869cc": "H_0 (X) = \\log n = \\log |X|.\\,", "5027fc06d6ff436afd61cee0e81b5c9a": "\\hat{b}_j^{(0)} := 1", "502804b848280fd9e7609c6c28a9859c": "\\{ d, f(d), f(f(d)), f(f(f(d))), \\ldots \\}", "502823ac4e68af7806df4e539b1f245b": "A_\\mu (x) \\rightarrow A'_\\mu(x) = A_\\mu(x)+ \\partial_\\mu f(x)", "5028317ea9a149d88bfdc96a46a04778": "T_G(x,y)= x^i y^j, ", "502853c0e00479003837d268dfa0ae42": "G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji}\\,\\!", "502853cf1ddd8e233dedb9334bf06b1a": "\\Delta{G} \\ll 0", "5028820ae57dda1a75892fe177f9dbb9": "\\Phi(x|B)=M\\{\\xi\\leq x|B\\}", "50292dba9d60321a0d46e0a78b587971": "{J^{\\nu}}_{\\text{free}}=(c\\rho_{\\text{free}},\\mathbf{J}_{\\text{free}})=\\left(c \\nabla \\cdot \\mathbf{D}, - \\ \\frac{\\partial \\mathbf{D}}{\\partial t}+\\nabla\\times\\mathbf{H}\\right) \\,,", "50297902fbb15573de01edd0d3f2c4e2": "dN= \\frac{dg}{\\Phi}", "50298ba0da621d2d3d8f288a124e35a3": "\\pi = \\sum_{k = 0}^{\\infty}\\left[ \\frac{1}{16^k} \\left( \\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15} \\right) \\right].", "5029a26b1fcabf852ce63e76e427445e": "\\vartheta\\ll T", "502a02817f61f69de0c77866e35cdcfb": "z=1\\cdot z+0\\cdot a_i", "502a02f4d787c1f13110c3a41f37da9a": "\\scriptstyle m X^2 + n Y^2", "502a1ae4f991d9c3ff10a0fbeae21b19": "\\tilde L", "502aaf48788e953e00c5ce43c6aee605": "\\nabla I(\\mathbf{x'})=[I_{\\mathbf{x}},I_{\\mathbf{y}}]^{\\top}", "502add68abb7c97c54838dd5b977de47": " \\mathrm{tr} \\left(\\mathbf{R}^{-1} + (\\mathbf{P}^T \\, \\otimes\\, \\mathbf{I})^H \\mathbf{S}^{-1} (\\mathbf{P}^T \\, \\otimes\\, \\mathbf{I}) \\right)^{-1}", "502ae4250d8e16ff2d1a61535f1ce5c8": "P_i^k \\stackrel{\\mathrm{def}}{=} f(x_1,\\ldots,x_k) = x_i", "502b11c1fb9ca437a0d6599746016f87": " A \\times B = -(A \\times -B) = -(-A \\times B) = (-A \\times -B) \\,", "502b282fb3917b996c11a0fd00a90c51": "W_{in} = W_{out} + W_{fric} \\, ", "502b7679f394e0c9ceec5bd7098906b1": " R_P(t) = \\frac{ 15 {_2^1}P }{ 14 {_2^1}P + 29 \\ {_2^0}P }", "502bbe9051679eba417f4e1f73620815": "\\mathcal{S} [\\varphi] = \\int \\mathrm{d}^4x\\; {\\mathcal{L} [\\varphi (x)]\\,} ", "502c36f2cbb8cf4f5270fe8cd87ee7fb": "\\ f(v,u)=-1", "502cc6efd91bae724e166836ede1f845": "i_1>1", "502cd9271da1a105f04685322c900b9f": "V_{S-} \\le V_+,V_- \\le V_{S+}", "502ce8ad5291af121919199c944884ad": "Q (y, f (x))", "502ce8b4e15a5046dd472f7eda95c03f": "r_a=\\frac{p}{1-e}", "502cf82f9e8f15ffa64971dc34901e82": " u(x,t) = \\frac{1}{\\sqrt{2\\pi}} \\int^{\\,\\infty}_{-\\infty} A(k) ~ e^{i(kx-\\omega(k)t)} \\,dk ", "502d251d5a478627a1732afc7e9e2f3c": "\\scriptstyle \\frac{\\partial f}{\\partial t} \\;=\\; 0", "502d345334ce577f16f2776faa68eb1e": "\\Omega_2 \\subseteq \\{ \\land, \\lor, \\rightarrow, \\leftrightarrow \\}.", "502d37dc24423679a7a7ce51bde89735": "\\omega_1,\\omega_2 \\in \\Omega: \\; (X(\\omega_2) - X(\\omega_1))(Y(\\omega_2) - Y(\\omega_1)) \\geq 0", "502d55b15c2e53705f4a341f62b0ee55": " y\\ f = f\\ (y\\ f) ", "502d83b007a4562c67a3d457cb8bc34b": "\\Vert \\cdot\\Vert_{L^2}", "502d86d4080a9ca040a992da26cc274c": "Ran ( 1 - \\tilde{C} )", "502db98753038a9c71a478b5d8351c42": "n \\alpha s", "502dfdab9da6d2b38b26e9e19036bb15": "\\alpha_n\\rightarrow a.\\ ", "502e1c7d9280b92e8282e1d43a65f398": " \\frac{\\partial \\boldsymbol{F}}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\left(\\frac{\\partial \\boldsymbol{F}_1}{\\partial \\boldsymbol{S}} + \\frac{\\partial \\boldsymbol{F}_2}{\\partial \\boldsymbol{S}}\\right):\\boldsymbol{T} ", "502eb7049dd08a2c0264d480f3ca5993": "b_n =\n\\begin{cases} \nb_{(m-1)/4} & \\text{if } m = 1 \\mod 4 \\\\\n-b_{(m-1)/2} & \\text{if } m = 3 \\mod 4\n\\end{cases}", "502ed8f3eaf62558a53cbf74d009ed5f": "\\alpha A +\\beta B ... \\rightleftharpoons \\rho R+\\sigma S ...", "502f55a7dab43a9f8b31f20a9ce9ded2": "\n\\left(\n 1 - \\sum_{i=1}^{p'} \\alpha_i L^i\n\\right) X_t\n=\n\\left(\n 1 + \\sum_{i=1}^q \\theta_i L^i\n\\right) \\varepsilon_t \\,\n", "502f910ecfca890648458b4cace7b7e5": "\\Delta\\tau", "502faf775aede2326001ebe5174c855c": "c_{3,1}(\\widehat{a}, w(c_{3,1}(\\widehat{a}, w(S, \\widehat{b}c), \\widehat{d}), \\widehat{b}c), \\widehat{d})", "502fba3fbd820ac32448a9f45ed69c6a": "\n{\\Vert r \\Vert}^2 = r \\cdot v = \\frac{1}{{\\Vert{ u}\\Vert}^2} \\sum_{i 0", "50356ff125112bee79346c3db6ed66d7": "t\\to \\infty", "5035a693a2986f32ba6e53394410bda5": "\n\\{x^{a_{11}}y^{a_{12}}z^{a_{13}}, y^{a_{22}}z^{a_{23}}, z^{a_{33}}\\}\n", "5035aa6da16c75d2daf2dde9cb2f75a7": "O(m + n)", "5035ac53c41660d81c4af9d6e6cf89ec": " \\omega_0 = 2 \\pi f_0 = \\frac{1}{RC\\sqrt{mn}},\\, ", "5035d6bd1219243557c2cb2dcc2fbf62": "\n\\begin{pmatrix}\n{\\color{BrickRed}1} & {\\color{BurntOrange}2} &\n\n{\\color{Violet}3} \\\\\n{\\color{BrickRed}4} & {\\color{BurntOrange}5} &\n\n{\\color{Violet}6} \\\\\n{\\color{BrickRed}7} & {\\color{BurntOrange}8} &\n\n{\\color{Violet}9} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\color{BrickRed}a} & {\\color{BrickRed}d} \\\\\n{\\color{BurntOrange}b} & {\\color{BurntOrange}e} \\\\\n{\\color{Violet}c} & {\\color{Violet}f} \\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n{\\color{BrickRed}1} \\\\\n{\\color{BrickRed}4} \\\\\n{\\color{BrickRed}7} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\color{BrickRed}{a}} & {\\color{BrickRed}{d}} \\\\\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n{\\color{BurntOrange}2} \\\\\n{\\color{BurntOrange}5} \\\\\n{\\color{BurntOrange}8 }\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\color{BurntOrange}{b}} & {\\color{BurntOrange}\n\n{e}} \\\\\n\\end{pmatrix}+\n\\begin{pmatrix}\n{\\color{Violet}3} \\\\\n{\\color{Violet}6} \\\\\n{\\color{Violet}9} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\color{Violet}c} & {\\color{Violet}f} \\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n{\\color{BrickRed}1a} & {\\color{BrickRed}1d} \\\\\n{\\color{BrickRed}4a} & {\\color{BrickRed}4d} \\\\\n{\\color{BrickRed}7a} & {\\color{BrickRed}7d} \\\\\n\\end{pmatrix}+\n\\begin{pmatrix}\n{\\color{BurntOrange}2b} & {\\color{BurntOrange}2e}\n\n\\\\\n{\\color{BurntOrange}5b} & {\\color{BurntOrange}5e}\n\n\\\\\n{\\color{BurntOrange}8b} & {\\color{BurntOrange}8e}\n\n\\\\\n\\end{pmatrix}+\n\\begin{pmatrix}\n{\\color{Violet}3c} & {\\color{Violet}3f} \\\\\n{\\color{Violet}6c} & {\\color{Violet}6f} \\\\\n{\\color{Violet}9c} & {\\color{Violet}9f} \\\\\n\\end{pmatrix}.\n\n", "503600c50248b924ad1c8c6f30a959e3": " \\hat{\\mathbf{e}}^1", "50361299c15a18b26ef998944ad7d488": "\\mathfrak L", "50363cb7fd223f50093091451c5f09cb": "\\mathbf{r_x}", "50367b3b4a1e8d1253ac728916550601": " \\nabla_G \\phi = G^\\mathrm{T} G m - G^\\mathrm{T} d = 0 \\, ", "5036a4bf007dc9e36619c1cb25470ee0": " < ", "5037101d942a08de2b96ba87b7828365": "T = 2\\pi\\sqrt{a^3/\\mu}", "503737093c4e50e94aa2d7a039c7f7e4": "f(z) = \\frac{1}{2\\pi i}\\iint_{{C\\backslash K}} \\frac{\\partial f}{\\partial \\bar{w}}\\frac{dw\\wedge d\\bar{w}}{w-z}, ", "5037726549e7922caf13505d64d5fcfa": "T(\\rho,\\sigma) := \\frac{1}{2}||\\rho - \\sigma||_{1} = \\frac{1}{2} \\mathrm{Tr} \\left[ \\sqrt{(\\rho-\\sigma)^\\dagger (\\rho-\\sigma)} \\right] .", "503785e6c7537bf5e13b078283974e4b": "(a_i,b_i=\\langle a_i,\\mathbf{s} \\rangle/q + e_i)^m_{i=1}", "503788995a97c6bea237844b1f1cee28": "\\Rightarrow \\ q_1^* = \\frac{a + \\frac{\\partial C_2 (q_2)}{\\partial q_2} - 2*\\frac{\\partial C_1 (q_1)}{\\partial q_1}}{3}", "50380d39f33a60ecf732abb3b7666c05": "1/T_{c}", "5038c55ac470d159edc80e02e8db565d": "\\sigma_{ij,kk}+\\frac{1}{1+\\nu}\\sigma_{kk,ij}=0", "5039390e952d24a23151b431c50799e7": "\\nabla_{\\vec{e}_0} \\vec{e}_1 = \\nabla_{\\vec{e}_0} \\vec{e}_2 = \\nabla_{\\vec{e}_0} \\vec{e}_3 = 0", "503969f2bc18c2c88c14456c1011be02": "[0, m_k]", "5039a9ad969eb49e1e0860b8a3881fdf": "f_\\mathrm{obs} = f_\\mathrm{rest}\\sqrt{\\left({1 + v/c}\\right)/\\left({1 - v/c}\\right)}", "5039aaaf0abb11e54fe20b9c561d6cab": "x=\\lfloor x\\rfloor+\\{x\\}", "5039aef5a640fbfc873bc426e3f01c00": "(\\lambda - 1)(\\lambda + 5) = 0 \\,\\!", "5039e0776d7fadd8ca1a958781856a09": "e^\\gamma\\cdot\\log(10)/\\log(2)", "503a1fe7f9787aa9b7bfdaf3d4ae99b1": "\\frac{\\partial S}{\\partial t} + u \\frac{\\partial S}{\\partial x} + v \\frac{\\partial S}{\\partial y} + w \\frac{\\partial S}{\\partial z} = (E-P)S(z=0)", "503a72142ab18dbabc8accabee7e3571": "E \\in \\mathcal{E}", "503a7c69141689cddf493efed9275847": "F_\\mathrm{crit}", "503aa622d29b7591f497735f1db06b43": "\\alpha^+", "503ae83904d68768597c625768bcad0e": "q\\gamma(a,q) = \\gamma-\\sum_{j=1}^{q-1}e^{-2\\pi aij/q}\\log(1-e^{2\\pi ij/q}),", "503b3794057bb50d616d5fb9877a3af8": "l\\equiv\\partial_x+A_2", "503b4524578ccb2e813df4b6eabd3f14": "\\Delta x = l + r \\cos \\Delta \\alpha \\,", "503b5a0f2df044615d4ad35fb03b119d": "\\omega_k = \\sqrt{k^2+m^2}", "503bda3ee05888c45c84482ffa8a59e6": "\\overline{\\mathbf{Q}}_p", "503bddae5e384972b59ea674bf5de215": "\\psi(I) = k I ^a , \\,\\!", "503be0de4134f48be50669524b177263": "y_i = f(x_i, \\boldsymbol \\beta)", "503bf275ba4b2e31f71985354c24e637": "\\boldsymbol{\\hat{\\imath}}, \\boldsymbol{\\hat{\\jmath}}, \\boldsymbol{\\hat{k}}", "503c5e18af59681f6d762dda207e64c3": "U: C \\to D", "503c79c73d069d0532499d59c5543f8f": "RL(\\mathrm{dB}) = P_\\mathrm i(\\mathrm{dBm}) - P_\\mathrm r(\\mathrm{dBm})\\,", "503c7e3ab57b933187905092c76ffc5c": "\n \\begin{matrix}\n 1 & \\mbox{for }k=k_0 \\\\0 & \\mbox{otherwise }\n \\end{matrix}\n ", "503c8ad152e088d190ec8bf702fc5e52": "\\Delta \\alpha = d^* d\\alpha + dd^*\\alpha.\\,", "503cbe9a528f262b9a574fa88e2fb719": "\\frac{2.00 \\mbox{ g NaCl}}{58.44 \\mbox{ g NaCl mol}^{-1}} = 0.034 \\ \\text{mol}", "503cd1be293cfc12cb255b683fac273b": "\\sigma_\\mathrm{Total}=\\sigma_D = \\sigma_S", "503cebfe89f4dfb9b5019f15c2f10f5e": "S, T \\subseteq V", "503d0e6d8ab23fbded0a78ee5ad1dad4": "V=w^3 \\left (h/ \\left (\\pi w \\right ) -0.071 \\left (1-10^ \\left (-2h/w \\right ) \\right ) \\right )", "503e3ac4fe77b174e338e196f4a9c0fd": "\\frac{1 + {\\scriptstyle\\frac{1}{2}}z + {\\scriptstyle\\frac{1}{12}}z^2}\n{1 - {\\scriptstyle\\frac{1}{2}}z + {\\scriptstyle\\frac{1}{12}}z^2}", "503e5c1867025084050268a031a30d34": "\\sum|S_i|=n\\sum(\\dfrac{1}{(2-\\alpha)^i})=O(n)", "503e7f6efc29ab745c9f48f2607080c2": "n_1\\times n_2\\times\\ldots\\times n_d", "503ed72ddde67e8867d929cdebfbd698": "(7,4,3)", "503edd90fec23e9942a5829ce4c3b9fd": " K = \\frac {\\dot{m}}{C_{\\infty}} \\qquad(3b)", "503f0b041624474c1feb38d6d693a18e": "Cl_t, t \\in T", "503f95ea37769162d25feb731f30b734": "n(I) = n_0 + n_2 \\cdot I", "503ff59cc659604d9c9a5866cbfd6609": "|\\bigstar \\bigstar ||\\bigstar", "50402d297f27b1a6affc72418abbac79": "A(0,T) = \\frac{1}{T} \\int_{0}^{T} S(t) dt.", "50403a45f51e78e3c27cbed453930eb9": "\t\\frac{12 + 144 + 20 + 3 \\sqrt{4}}{7} + (5 \\times 11) = 9^2+0", "504051f5e8f7cdb23757b0336735bcd2": "n = [L : K]", "5040892a7ca8bc043539db025507bc72": "I_\\mathrm{AMPA}(t,V) = \\bar{g}_\\mathrm{AMPA} \\cdot [O] \\cdot (V(t)-E_\\mathrm{AMPA})", "5040955ad6ae352ad0db79b15d376807": "\\forall w_1,\\ldots,w_n \\, \\exists B \\, \\forall x \\, ( x \\in B \\Leftrightarrow \\varphi(x, w_1, \\ldots, w_n) )", "50409a5bca925cb19033183cd45beef0": "\\emptyset \\in \\mathcal{C} \\Rightarrow g(\\emptyset)=0", "5040c5f0a538a57a9d7276cf0f4c9fe4": "\\left(\\int_{\\mathbb{R}^n} |u|^{\\frac{n}{n-1}}\\right)^{\\frac{n-1}{n}} \\le n^{-1}\\omega_n^{-1/n}\\int_{\\mathbb{R}^n}|\\nabla u|", "5040c673caf80691a71fc4f6546d081f": "f:M\\to M'", "5040fdf2a584ca9a310f14df5a5eb2e6": " \\begin{bmatrix} 1 & 1 \\\\ 0 & 1 \\end{bmatrix}. ", "5040fe98098575520630585edd02f419": "u_j = \\frac {x_j - y} {\\left \\| x_j - y \\right \\|}", "50411ba0fcae2dcdc6484ba3be6ae7e8": "\\tau_N = \\tau_0 \\exp \\left(\\frac{K V}{k_B T}\\right)", "5041ac04547715941937bcc574077279": "p:S\\rightarrow \\mathrm{Spec}(R)", "5041dd12ed6c779a559444552df428a8": "T_P", "5041efc0c229bf224a9df39db3cf1e87": "g(x)=f^{(0)}_n(x) + \\epsilon f^{(1)}_n(x)", "5042165e2b5b40726ee4c5ed24a4a8d2": " x_j = v(j) ", "504245f3803475a708408e17399d4c15": "2 \\Rightarrow (1, 1, 0)", "504312dcfbe53fd888dcf3c8a2606b3d": " b^e ", "50438221bfd114b94ffbcc2f2e979e31": "\\langle\\phi^0\\rangle=v", "5043c16debc27b5000a901d60a367e4f": "\\frac{}{eval:\\!\\!-~~ (\\alpha \\rightarrow \\beta) \\times \\alpha ~\\vdash~ \\beta}", "5043e18455002e54995b54ebedba9ea5": "\\frac{1}{r^3} P^2_2(\\sin\\theta) \\cos2\\varphi = \\frac{1}{r^3} 3 \\cos^2 \\theta\\ \\cos2\\varphi", "5043fcba281f04541f950c6c39b5d398": "h_0 = y_0/2", "50440210f55e9b687278fa0ba84dce9a": " A \\subseteq B \\Rightarrow \\operatorname{cl}(A) \\subseteq \\operatorname{cl}(B) ", "50443518b3cbaf1df9c5a5d89de84695": " \\eta^*= \\frac {\\tau^*} {\\dot \\gamma^*} = ( \\frac {\\tau'}{\\dot \\gamma}+i \\frac{\\tau''}{\\dot \\gamma})=\\eta'+i\\eta'' ", "5044491d83899d1e9f57c613d8bedaed": "(j, J, \\nu)", "504464859b5280e620f7130df5248138": "\n\\sigma_{\\text{impl}}^{\\text{n}}=\\alpha\\;\n\\frac{F_0-K}{D\\left(\\zeta\\right)}\\;\n\\left\\{1+\\left[\\frac{2\\gamma_2-\\gamma_1^2}{24}\\;\\left(\\frac{\\sigma_0 C\\left(F_{\\text{mid}}\\right)}{\\alpha}\\right)^2+\\frac{\\rho\\gamma_1}{4}\\;\\frac{\\sigma_0 C\\left(F_{\\text{mid}}\\right)}{\\alpha}+\\frac{2-3\\rho^2}{24}\n\\right]\\varepsilon\\right\\}.\n", "504494692957a679f17a2782316dcf4d": "t_{crit}", "5044a0068bc09329e7bd28640d4745a3": "F[y]=\\exp \\left( \\frac{1}{t-a}\\int_a^t \\ln y\\,dt \\right) ", "504534a5ea48054164ffa344d36e7bae": "6P", "50453a0356b771c492699db2281c35b5": "(x_{i-1}\\ ,\\ y_{i-1})", "504575aa0276599812fa961de034fcb3": "\\operatorname{P}({X \\in \\mathcal{A}}) = \\operatorname{E}[I_{\\mathcal{A}}(X)]", "504587e2609b6e4eab438d6b9d3f0499": " \\scriptstyle |\\phi\\rang ", "50459ee2d05d69a67a2e56bb21f8d169": " R(\\alpha,\\beta,\\gamma) \\leftrightarrow P(\\alpha,\\gamma) \\land I(\\beta,\\gamma)", "50461a4ed4e8144398995536d573e90b": "\\! f(\\gamma')", "504631fc0e486497fd4c43cb2640b306": "\\{x : A \\ |\\ P(x)\\}", "50463dc4b64c36c5dbb08451ba808d12": " \\Delta E_{SO} = \\xi (r)\\vec L \\cdot \\vec S", "5046851ef5f5ec1c23cdb021e9e9c287": "\\omega \\ ", "5046a2e0ce1c79bf8548c893adfefc06": " F = \\{ U \\cap Y \\ | \\ U \\in N_x \\}", "5046bcf04edaabd26e158b6e49178b2e": "k \\leq 3", "5046e84a74f6ead642fdef9006714dd5": " \\dfrac{d^a}{dx^a}x^k=\\dfrac{\\Gamma(k+1)}{\\Gamma(k-a+1)}x^{k-a}\\;.", "50474239a2c5bbb253db4566704fcac7": "i = 1, 2", "5047506d94a45a73aa51fc971a0ff38d": "O(|E|)", "50476c783beb17efce42611f85596859": "\\left(\\frac{\\partial P}{\\partial T}\\right)_{V}", "504870e237c74e57166e613f242be41e": " \\bold x - \\bold x = {\\bold 0} ", "5048c8258d07bf0aa6d1710ba3fa6358": "{\\mathcal M}B(z^3)=B(z^1) {\\;} {\\mathrm {or}} {\\;} {\\mathcal M}{\\;}{\\mathrm {yellow}}\\subseteq{\\mathrm {red}}.", "5048d90b88950f08a4ac01f2b0444f99": "\\begin{cases}\nR_{i,parr} = (R_{i,b} \\cdot R_{load}) / (R_{i,b} + R_{load}) \\\\\nR_{i,a} + R_{i,parr} = R_{load} \\\\\nR_{i,parr} / (R_{i,a} + R_{i,parr}) = Ratio_i\n\\end{cases}", "50491c54dd2e5cfa2f9c1501c3f9e3ff": "w= \\frac{T_0}{\\rho \\cos^2 \\varphi}.\\,", "50497de8190ded30416425ce6a600660": "f_i\\in F", "5049991a5b2bbf9d3833fd6ec82c7a38": "h\\in G", "5049c3e3c706f9b903a9f456855c373e": "f_*(\\mu)(B)= \\mu(f^{-1}(B)) \\,", "5049e29d94c01c4094d57490e8f3772f": "L(x_i)=y_i+0+0+\\dots +0=y_i", "504a0d90fea4c8531d9f2390ac28bea2": "\\chi(\\mathbf{r}_1t_1,\\mathbf{r}_2t_2)=\\chi_{KS}(\\mathbf{r_1}t_1,\\mathbf{r}_2t_2)+\n\\chi_{KS}(\\mathbf{r_1}t_1,\\mathbf{r}_2't_2')\n\\left(\\frac{1}{|\\mathbf{r}_2'-\\mathbf{r}_1'|}+f_{xc}(\\mathbf{r}_2't_2',\\mathbf{r}_1't_1')\\right)\n\\chi(\\mathbf{r}_1't_1',\\mathbf{r}_2t_2) ", "504a76df8d4518ea2b5ac76117d8948d": "\\mathfrak{P}^{20}", "504a7e7ca6ee509e679af7520f7ca81a": "x = -3", "504a7f079733f14e3eb81f99eeb3e148": "\\operatorname{erf}(x)\\approx 1-(a_1t+a_2t^2+a_3t^3)e^{-x^2},\\quad t=\\frac{1}{1+px}", "504b07ac860d0a44e0c7d0351efd0d2c": "\\sigma = \\frac{My}{I} = E y \\frac{\\partial^2 u}{\\partial x^2}\\,", "504b50ea53699186b5972e5d3f625d93": "\nd_2 = \\frac{4MG}{15Rc^2}\n", "504b577682effd45a0d24e961150620e": "\\ b_{i}", "504b7f029949c9d3a66cfdd313eccf25": "F = S_{0}(1 - \\delta)^{n(T)}e^{rT}\\,", "504b84fcda84126c4a1bf56680ea2574": "W^u(f,p) = W^u(f^k,p).", "504b858157c6d62d0683998aca5225e0": "= \\mbox{R}(z, dt)|x, y, z\\rangle", "504bd3d2b072fe016daa4a05c7735f46": "\\sum_{i \\mathrm{\\ even}} p(n{-}g_i) = \\sum_{i \\mathrm{\\ odd}} p(n{-}g_i),", "504be999cfd72dbab09861c80b93d795": "F(\\omega) = \\mathcal{F} \\{ f(t) \\}(\\omega)", "504bf04de298583884d54391b45b517f": "U_i=U_i/\\operatorname{E}(U_i^2)", "504bf1af408d4926baf17a59bd559c07": "\\int a^x\\,dx = \\frac{a^x}{\\ln a} + C", "504c1737228dcd55dedbb5679f6f2c69": " \\and (S_7 \\implies (\\operatorname{equate}[A_7, q] \\and V[F_7] = A_7)) \\and D[F_7] = D[q] ", "504c3be7694b8c39fe8370dfa736254b": "\\frac{\\sigma_j}{\\langle\\sigma_j\\rangle} = f_j,", "504c6cff8f8b16440e45eb4bcdda8c53": "\\langle\\psi_V(t)|E_1^{(-)}(t)E_2^{(+)}(t)|\\psi_V(t)\\rangle=\\kappa\\langle 1_{\\nu_1}0_{\\nu_2}|a_1^\\dagger a_2|0_{\\nu_1}1_{\\nu_2}\\rangle exp\\lbrack i(\\nu_1-\\nu_2)t\\rbrack\\langle c|c\\rangle=\\kappa exp\\lbrack i(\\nu_1-\\nu_2)t\\rbrack\\langle c|c\\rangle", "504c6dcd86672f5e96fedb7b6d6b63f7": "\\left \\langle N,e,C_1,C_2,f \\right \\rangle", "504c7ecda335a5ce7e2083f0f8c5fda1": "C \\to \\mathbb{P}^1", "504ca902fe23506649fe2a1f90419036": "\\mu(X)= \\frac{a + 4b + c}{6}", "504caaea40613f0f74bbcbb184762dbd": "^{\\ast}", "504cf75ba6fef3ca937cf22738680685": " g(E) = 0 ", "504d2a7aa0907096fe4aa22e2b8dcf93": "\n\\big[ \\tilde{\\mathcal{A}}^{AB}, H^{(0)}\\big] \\ne 0 .\n", "504d7b1a145a4e9503472f6196ab4af9": "U=C_vT", "504d7f9286be189848e394f932f13a1a": "c_{\\alpha}", "504d81e8c8c065f2b0ee293289fa7ab8": "\\|AB\\| \\le \\|A\\|\\|B\\| ", "504d8fd1e3f395bab89dea42ecc790a5": "R_n(\\xi,x)", "504dc447c182c97d0ce1d6217d12e58c": "\\begin{bmatrix}0&0\\\\1&1\\end{bmatrix}:\\mathbf a", "504dc93782a1b2d3eff43b3533a10b98": "\\,I_{H_s}", "504de48a1dd9d61df9ed988e814450fc": "n_d =", "504e1b40d6de42decab41052d03fc3bd": "\\varphi : F \\rightarrow G\\mbox{ and }\\psi : F \\rightarrow H.", "504e48cf1c394ab8749bf6d53a5a83c4": " B(y;b,c,p,q) = GB(y;a=1,b,c,p,q). ", "504e521d229636c56784c0f205ee0d34": "w_1-T", "504e6d16c6c4b62aed9beef2dedfd07d": " m = \\frac{\\operatorname{E}^2 \\left[X^2 \\right]}\n {\\operatorname{Var} \\left[X^2 \\right]},\n", "504ee14448df55076469a92555cd1b13": "G = (S, T)", "504ee35938da240f6e0976ed1a17c079": "\\phi \\to (\\chi \\to \\phi )", "504efeb0259c69f23a46857877eea4f3": " S_{\\vartheta + \\varphi} = \\frac {S_\\vartheta S_\\varphi - S^2} {S_\\vartheta + S_\\varphi} \\quad\\quad S_{\\vartheta - \\varphi} = \\frac {S_\\vartheta S_\\varphi + S^2} {S_\\varphi - S_\\vartheta} \\, ", "504f26f618d579e299806161f1c166d9": "\n f(x;\\mu,\\sigma^2) =\\Phi(-\\frac{\\mu}{\\sigma})\\delta(x)+ \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\; e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2}}\\textrm{U}(x).\n", "504f73baa48f61f129325fc28ab3d855": "a_{2n-1}=\\sum_{i=1}^{n-1} a_i a_{2n-i-1}", "504fe1b78358e038319f7f69a1625386": "\\chi_\\text{m}^\\text{SI} = \\chi_\\text{m}^\\text{LH} = 4\\pi \\chi_\\text{m}^\\text{G}", "505048d7f4a3dccb3d81e7a296913936": "\\mathbf{\\hat{p}} = -i\\hbar\\nabla .", "50505f7f3430a1829db2a51d30550ef9": "\\alpha + \\lambda = \\bigcup_{\\beta < \\lambda} (\\alpha + \\beta)", "50508746ccd0fe0abadb158d37b3a32a": "t' = \\gamma \\left ( t - \\beta \\frac{x}{c} \\right )", "5050c04a061d774e342622e7a47955e6": "K(x, y)=1\\,", "5050e051bcee0e3a416457ee73224666": "V = V_{12} - V_{1a} - V_{1b} - V_{2a} - V_{2b} + V_{ab}", "5050f58848424497f305a1f994bcb6e3": "\\omega_1=d\\gamma_1/dt", "5050fc89e7551c0a852a62665b4b9775": "\\alpha, \\beta \\in Z^{*}(X)", "505120d0481372dfa78fcef6ef6cba51": "(\\begin{matrix} \\frac{m}{s} \\end{matrix})", "5051981c58a3fbf732bdddc084145a1b": "P_{loss}", "5051ad556e5a8ea78a2b33add44a6c56": "t:U_{\\alpha\\beta}\\to U_\\beta", "5051cba565cbfcd270786b0e94dd7b7d": "(X,.)\\times(X,.)\\to(X,.)", "5051cc475309ba8bca5bea2d8a954419": "f_{0}() = \\begin{cases}\nf_{1}(0)\\cdot f_{1}(1) & \\text{ if }S = \\forall \\\\\nf_{1}(0) * f_{1}(1) & \\text{ if }S = \\exists. \\\\\n(1-r)f_{1}(0) + rf_{1}(1) & \\text{ if }S = R.\n\\end{cases}", "50524e1c16ccf6f45b111a5f87cd1492": "\\ \\sum_{j=0}^Q a_{j} z^{-j} Y(z) = \\sum_{i=0}^P b_{i} z^{-i} X(z)", "5052b9cab30e97583be75dbd1fff0649": "x = \\left(\\sum_i \\hat n_i \\hat n_i^\\top\\right)^{-1}\\left(\\sum_i \\hat n_i \\hat n_i^\\top p_i\\right).", "5052ce246096df6cc93957e0e993f629": " t_M\\le t_r\\le t_c ", "50530e1d734d177213c45ec550f3363a": "\nK(p) = {i\\over p_0^2 - \\vec{p}^2 - m^2}\n\\,", "50533a5c37ca328c3c19f37a66e9ce1d": "\\ln\\left(\\frac {1}{2}\\right)=-\\ln 2", "505343a387de7e556a3b9d1496ac0901": "\\Gamma(0.5 + 0.5i) \\approx 0.8181639995 - 0.7633138287 i", "505370488d0641f118ac7cc9cd496922": "f(z) = e^{2 \\pi i t} z^2(z - 4)/(1 - 4z)\\ ", "5053955c3516902bfde20809343a03e8": "\\prod_{i=1}^nx_i^{w_i} \\leq \\sum_{i=1}^nw_ix_i", "5054152398e0219cd2a7af89220c3094": "s = (a+b+c)/2", "505415947b7fe2e25e6dc80f18022c8a": "e^{-2t / \\tau}", "50547ac7f5a9919145ccca8c84adfb72": "\\epsilon_{\\rm ij}", "5054bc3b9f7f0bc6e06d32ab8bd69e08": "\\scriptstyle{\\simeq 2}", "5054c10ba5e7e6bc15c72d8e557f65d4": "I(t):=\\frac 1 {2 \\pi i} \\oint_{\\partial K} \\frac{F'(z)}{F(z)+t}dz ", "50551b4bd43df80d3d59fb3cab1d5a57": " C_n(x)", "50557d4b672646a6d6333aada29ca5b7": "\\Delta z \\;\\stackrel{\\mathrm{def}}{=}\\; z' - z = \\frac{1}{3}", "5055840deee8e0eabe0c21e3044e6303": "\\#(n)\\le b^n", "5055b189dd22286d5a28c0fbf15670c5": "\\frac{2}{\\pi}\\arcsin{\\sqrt{u}}.", "5055d6a06637cbc145f3a918f4d42e14": "d_2 = d_2(K)", "5055f8dc6174012f9c316e9c1424de9d": "\\text{RCS}_\\text{Plate} = \\frac{4 \\pi A^2}{\\lambda^2},", "505656af0c88f6c13bd9dd4bd3b824f0": "t[A] = \\{ (a, v) : (a, v) \\in t, a \\in A \\}", "5056709c744d16aa947b74882bf69039": "B_{k+1} = B_k + \\frac{\\mathbf{y}_k \\mathbf{y}_k^{\\mathrm{T}}}{\\mathbf{y}_k^{\\mathrm{T}} \\mathbf{s}_k} - \\frac{B_k \\mathbf{s}_k \\mathbf{s}_k^{\\mathrm{T}} B_k }{\\mathbf{s}_k^{\\mathrm{T}} B_k \\mathbf{s}_k}.", "505681e61e71d9c28f9e4bcaf2d046de": "{\\mathbf y}_1,\\dots,{\\mathbf y}_{n_y}\\sim N_p(\\boldsymbol{\\mu},{\\mathbf V})", "5057208c07fde439315662440683c9eb": "-\\boldsymbol{\\alpha}e^{x\\Theta}\\Theta\\boldsymbol{1}", "50573d62f7b5608d36d935fe28c21ca9": "C:\\mathcal{X}\\rightarrow\\Sigma^*", "50576bdabb0fce1548889e120dce078d": "F_S^{-1}", "5057b1eb4fb5ce672c10dd335f1762ab": "{\\mathfrak c} = 2^{\\aleph_0} \\,.", "5057cf5d33462d510d9536595db28247": "A_5 \\cong PSL(2,5)", "50582aec84a4ff2a4843dacc9edb53e0": " \\vec{e}_3 = \\frac{1}{C\\left(\\frac{q^2}{\\omega^2}, \\, \\frac{2q^2}{\\omega^2}, \\, \\omega u \\right)} \\partial_y ", "50582fa15ffbca30d654205e293034d8": "\\|(\\lambda I-A)x\\|\\geq\\lambda\\|x\\|.", "50582fc25ae3606c41366381390c7498": " \\land", "5058b714039bf3344f56128dfbf5dc61": " \\Omega_x^{(k)} ", "5058c17065929b089fc2ce89d311cf72": " \\frac{d^2}{d t^2}N_2 = c_2 c_1 N_2 ", "5058edc0ff5d25b221b139ab0ad97090": "\\left|V_o\\right| = D", "5058f1af8388633f609cadb75a75dc9d": ".", "5058f323f521c14a4fba492da6b760e0": "\n \\dot{\\varepsilon}_{\\mathrm{vp}} = \\left\\langle \\cfrac{f(\\boldsymbol{\\sigma}, \\boldsymbol{q})}{\\tau} \\right\\rangle = \\begin{cases}\n \\cfrac{f(\\boldsymbol{\\sigma}, \\boldsymbol{q})}{\\tau} & \\rm{if}~f(\\boldsymbol{\\sigma}, \\boldsymbol{q}) > 0 \\\\\n 0 & \\rm{otherwise} \\\\\n \\end{cases}\n ", "50594d83f27e7d6332ca50125da3f0cf": "\\begin{bmatrix}-2\\\\0\\end{bmatrix} + \\begin{bmatrix}2 & -1 \\\\ -1 & 1\\end{bmatrix}\\begin{bmatrix}x_2 \\\\ x_3\\end{bmatrix} = \\begin{bmatrix}-1\\\\2\\end{bmatrix}", "5059a07a66618dd8b856fc0ffb31975a": " n", "5059b44a1213d92f6e6b21f356b282fb": "\\phi_D:A\\to\\hat A", "5059bc62d4cba129141a58f7a821f692": "p - 1 \\mid n - 1", "5059e5c11212701b98a20159b116e229": "z^q : \\Bbb{Z} \\times \\Bbb{Q} \\rightarrow \\Bbb{C}", "505a60bada0642e8d3013e996621e8ce": "L_e \\sim \\sigma_o^4", "505a940e9dec7ee5706a78edc0645431": " \\frac{2}{n} = \\frac{1}{2} \\frac{1}{n} +\\frac{3}{2} \\frac{1}{n} ", "505ab0e91ee65f0dd0ebbdc148d80185": " \\rho_{\\text{fluid}} ", "505ae0fb79c0551866b058a45397f4eb": "L_{p1}", "505af99402393049de3f21b239a0df02": "n(r) \\sqrt{{r'}^{2} + r^{2}} - n(r) \\frac{{r'}^{2}}{\\sqrt{{r'}^{2} + r^{2}}} = h", "505b2ca43d62627d8f2db80cd337269f": "H_\\text{KL}+H_{\\text{D}3}", "505b30474bf75fb6f9f0d876bf9fdcf1": "M(n)", "505b79816808bf1e2d779ee64ff016ce": "R=\\frac{|a_0|}{\\max(|a_0|,\\,|a_1|+|a_2|+\\cdots+|a_{n}|)}", "505bd2584d9eb7f605d48a696d86a914": " \\psi^{(CSha) }(t)=\\operatorname{sinc}(t).e^{-j2 \\pi t}", "505caf8213c6f1ba5ac67236434a5b5d": "( \\lambda x . xx)( \\lambda x . xx) \\to ( xx )[ x := \\lambda x . xx ] = ( x [ x := \\lambda x . xx ] )( x [ x := \\lambda x . xx ] ) = ( \\lambda x . xx)( \\lambda x . xx )", "505d2a7da2596eb3e9d7eb21b5d29e3d": "\\langle\\mathbf{p}(t_0+dt)\\rangle=\\left( 1 - \\frac{dt}{\\tau} \\right) \\left(\\langle\\mathbf{p}(t_0)\\rangle + q\\mathbf{E}dt\\right),", "505d5034b58dbe4dab19a0fec0ca5889": "\\mathit{d_H}^{RC}(\\mathcal{B}) \\geq \\mathit{d_{min}} ", "505d867838ccdfffa57a0bb81223280f": "x_i \\sim p(x_i|\\theta)", "505da4dfee5c8cc5d2ebb2556a4b92fe": "T_{e}^{\\delta}(a)", "505dbcab7ca1774e5b6f9599bd96c6c0": " \\lambda \\,\\!", "505dbfaa90367cde805a5b47888a7e41": "af(N)=f(aN).\\,", "505dc0a0f7f506450185fcda220867a0": "\\dim A(M) \\le n(n+1)", "505dc4e00ffcf4cea128a26ec8f1e5f8": "F''\\,", "505e1bfc34e4d249df641870b1a1242d": "u(x_1, \\cdots, x_n)", "505e48126906049eadc0b49afcbf6fb3": "\n\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2} = 0 ", "505eb02a0d992e989453c89e9ac82c75": "\\mathcal{F}^{-1}=\\mathcal{F}R=R\\mathcal{F}.", "505edba23ef64dbeefa4c2c690968379": " I_{|X|\\geq K} ", "505eeb47f6f26cb80ae4e9b9464934f0": "241_5 = 2\\times 5^2 + 4\\times 5^1 + 1\\times 5^0 = 50 + 20 + 1 = 71_{10}", "505f141ab677086c1c020b3fba7e61ce": "F_2(x)=\\frac{1}{4}+\\frac{3}{4}x^2", "505fbf39a55f4534d6de22caccc5d472": "\\tau(s,a,C) = \\tau(t,a,C).", "505fc1267b6fb64a5d72693338a156dd": "\\overline{\\mathbf{O}(2s)}", "505fcc9effe8dcb680db87ec840c2c73": "\nE(S) = UNC + REL - RES.\n", "505fdf87cf55e28830846bb55e042899": " I = \\lim_{A_i \\rightarrow 0} \\sum_i J A_i = \\int_S J {\\mathrm{d} A} \\,\\!", "505fe82ffddac7c33fed9e44e4c94d75": "I\\subset\\mathbb{R}", "50601cb74e746991efafd541f370c09c": " e_s(T)= 6.1094 \\exp \\left( \\frac{17.625T}{T+243.04} \\right).", "50602a7c94262988d712feb100b4af80": "x^{\\overline{n\\!}}", "50611a1003359b5baa96c1db60c4be2b": "Instrumented \\ Range = F_r-F_t = \\frac {Speed \\ of \\ Light}{(4 \\times Modulation \\ Frequency)}", "50613b99a0922c7df7f47e0c5034fd23": "-T^a_b l^b", "5061795158063dd7360c9629ee3f7918": " S = | \\psi \\rangle \\langle \\psi |, ", "50619dd1055db4e3b0d1ef995834fcd1": "\\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^* \\,", "50619f9dc37f261eb94912809bda03c8": "\\sin x + a\\cos 2 x \\ge 0\\,", "5061a302ce84af46a9c429940f29990a": "\n\\tan \\theta = \\frac{\\cos \\chi \\cos \\lambda - \\sin \\chi \\sin \\lambda \\cos \\eta + \\sin \\chi \\sin \\eta \\cot(15^{\\circ} \\times t)}{\\sin \\eta \\sin \\lambda + \\cos \\eta \\cot(15^{\\circ} \\times t)}\n", "5061cd40251a33a7b8b231e75873b503": "f=1/T\\,\\!", "5061e563b65831be60f9748a3cc02c59": " d = \\frac{v_0^2}{g}\\sin(2\\theta) ", "50632d0a610afd107647f1f080585117": " U_{324} \\equiv 0 \\pmod {323}. ", "50639b3ef7626e432a7e4bbef43da718": "f(x_1, x_2, \\dots, x_n) = \\sum_{k=1}^K c_k x_1^{a_{1k}} \\cdots x_n^{a_{nk}}", "506449d6a1aaa092ba75795354c2219e": "Z_P = \\cfrac{d_2^3-d_1^3}{6}", "50646728276f89f7f75b56b6bba99e55": " p^{\\mu} = m v^{\\mu} = \\left ( \\gamma m , \\gamma { m \\mathbf{v} \\over c } \\right ) = \\left ( \\gamma m , { \\mathbf{p} \\over c } \\right ) = \\left ( {E \\over c^2 } , { \\mathbf{p} \\over c } \\right ) ", "5064b395f64b39f8464d89d5b485ec01": "Q=\\int\\limits_V \\rho_q(\\bold{r}) \\,dV.", "5064be4d893af72ee82ff78a61d98a68": "g=g_*", "50654bece04c1d3b8e7aa46ccd7d703c": "(\\log |x_1|, \\log |x_2|)", "50657787f2c25bace866324d03d4d1af": "F_{N,D}({n,d},z)", "5065a9eaade77f8a0386389f8e7d3215": "\\langle X\\rangle_c \\doteq \\sum_{x\\in\\chi}x P\\{X=x|c\\}", "5065f9d4bebc60fda4aad700442bd4b4": "f^* = \\mu_Y\\circ Tf.", "5066317e9ab217aa6171440b683e5948": "=t+t'+\\tfrac{1}{2}(p q'-p' q)", "5066768903b9887836fea11f312333be": "\\frac{d}{dt}m_H(\\Sigma_t) \\geq 0.", "506695146f17bebbf9936057cd6562b5": "\\left(j_1,j_2,\\ldots, j_n\\right)", "5066d8db6892f37fdccf84dbf7959fd9": "E \\supset F", "5067205dfa5471abfca82c75e73c042c": "f(\\mathbf x)= f(\\mathbf a)+\\sum_{|\\alpha|\\leq k}\\frac{1}{\\alpha!} (D^\\alpha f) (\\mathbf a)(\\mathbf x-\\mathbf a)^\\alpha+\\sum_{|\\alpha|=k+1}\\frac{k+1}{\\alpha!}\n(\\mathbf x-\\mathbf a)^\\alpha \\int_0^1 (1-t)^k (D^\\alpha f)(\\mathbf a+t(\\mathbf x-\\mathbf a))\\,dt.", "50673f5f78260c59ebadfb7f6cbd00d7": "\\hat{\\tau}_{ij}^r = \\widehat{ \\overline{ u_i u_j }} - \\widehat{ \\overline{u}_i \\overline{u}_j }", "506744ea6a351377b44d97ae2eb85617": "\\sigma(t) = (\\sigma(t))_{1\\leq i,j \\leq n}", "5067b3f5dcc0532c24f6dfa6c852ac6e": "\\,\\operatorname{cr}^2(z_1,z_2,z_3,z_4)", "5067ed39e7331cbcdb2a51c6d712de0b": " L = 2\\bullet10^{-7} \\bullet \\ln \\left ( {D\\bullet e^{1 \\over 4} \\over r_x} \\right ) ", "5068013ead055be4b66e3ecf20c0d515": "I_3", "50687cfba4931199144f9dcfcf3c06d1": "M^{1/2}", "506883b21d9a8c72358f2dc8b7cb3cc4": "r~", "5068a5b3051074c80f7311563b8e6c97": " \\min {\\{f (x)|x \\in X, X \\subseteq S\\}} ", "5068a5effba827497b9e477b611b18d4": "\\displaystyle P=2(a+b).", "5068cd778ecf57970420cb8936041ffb": " I_m = \\mathrm{d}q_m/\\mathrm{d}t \\,\\!", "5068cdc11d913000b018fe7f52fb93ce": "u(t) = u_m(t) \\cdot \\cos(\\omega t + \\phi)", "5068fa9ca0d754c110c3336f996463a6": "\\neg a \\wedge b", "506931b0413886172b2fb85f52678ab8": "w(t) = \\frac{e^{i4\\pi t} - e^{i 2\\pi t}}{i 2\\pi t} .", "506934495acb62c277b414aadb52e7ac": "V_\\mathrm{LL}", "506935af5ba3d829d8433b95bacbeea7": "q = ( w_{1,q} ,w_{2,q} , \\dotsc ,w_{t,q} )", "506938698016a281e71eeb2ee25cc79a": "R(C^{*}) = R(C_\\text{in}) \\times R(C_\\text{out}) = (1-\\frac{\\epsilon}{2}) ( 1 - H(p) - \\frac{\\epsilon}{2} ) \\geq 1", "50699727945a321455675f2427c1e074": "u'= \\frac{f -v^2}{u} = \\frac{1}{16}(-25x^2 +176x -15 )", "5069ac652a22c9f6d555bcdf1779d3ce": "a(4q+2)=2k(4q+2)=4(2kq+k)", "5069b60533d867aa29475d4a471bd019": "\\sigma(\\mathcal A)=\\sigma(\\mathcal B)", "5069b996403054899a1e2e5cf53030c4": "A_{T+1}^{x_c} - A_{T}^{x_c} \\leq x_c (B_{T+1}^{x_c})^2", "5069c7e3aa415ad343bbc5320f4091e4": "X_i= \\sum_{k \\geq 0} \\left.\\frac{\\partial^2 s(n)}{\\partial n_i \\partial n_k}\\right|_{n=n^*} {\\rm grad} n_k \\ ,", "506a5300888b495b58b6485be9aa771b": "\\varepsilon_i: \\quad \\vec n_i\\cdot\\vec x=d_i, \\quad i=1,2, \\quad\n \\vec n_1,\\vec n_2", "506ad5480ebbbba8fc18288a504b6748": "e^{-1}", "506b2f6fd707b4047ce2b1be0dad14e6": "\\textbf{P}_{k\\mid k} = (\\textbf{I} - \\textbf{K}_k\\textbf{H}_k)\\textbf{P}_{k\\mid k-1}(\\textbf{I} - \\textbf{K}_k\\textbf{H}_k)^T + \\textbf{K}_k\\textbf{R}_k\\textbf{K}_k^T ", "506b420f8f6085c90e9e8c759e583731": " f_0 = \\frac{1}{T} ", "506b72b6c5a6ec5adbde79f9184c1495": "\\begin{align}\n\\sum_{k=0}^\\infty &(k+r)(k+r-1) A_kz^{k+r-2}-\\frac{1}{z} \\sum_{k=0}^\\infty (k+r)A_kz^{k+r-1} + \\left(\\frac{1}{z^2} - \\frac{1}{z}\\right) \\sum_{k=0}^\\infty A_kz^{k+r} \\\\\n&= \\sum_{k=0}^\\infty (k+r)(k+r-1) A_kz^{k+r-2} -\\frac{1}{z} \\sum_{k=0}^\\infty (k+r) A_kz^{k+r-1} +\\frac{1}{z^2} \\sum_{k=0}^\\infty A_kz^{k+r} -\\frac{1}{z} \\sum_{k=0}^\\infty A_kz^{k+r} \\\\\n&= \\sum_{k=0}^\\infty (k+r)(k+r-1)A_kz^{k+r-2}-\\sum_{k=0}^\\infty (k+r)A_kz^{k+r-2}+\\sum_{k=0}^\\infty A_kz^{k+r-2}-\\sum_{k=0}^\\infty A_kz^{k+r-1} \\\\\n&= \\sum_{k=0}^\\infty (k+r)(k+r-1)A_kz^{k+r-2}-\\sum_{k=0}^\\infty (k+r) A_kz^{k+r-2} + \\sum_{k=0}^\\infty A_kz^{k+r-2} - \\sum_{k-1=0}^\\infty A_{k-1}z^{k+r-2} \\\\\n&= \\sum_{k=0}^\\infty (k+r)(k+r-1)A_kz^{k+r-2}-\\sum_{k=0}^\\infty (k+r)A_kz^{k+r-2}+\\sum_{k=0}^\\infty A_kz^{k+r-2}-\\sum_{k=1}^\\infty A_{k-1}z^{k+r-2} \\\\\n&= \\left \\{ \\sum_{k=0}^{\\infty} \\left ((k+r)(k+r-1) - (k+r) + 1\\right ) A_kz^{k+r-2} \\right \\} -\\sum_{k=1}^\\infty A_{k-1}z^{k+r-2} \\\\\n&= \\left \\{ \\left ( r(r-1) - r +1 \\right ) A_0 z^{r-2} + \\sum_{k=1}^{\\infty} \\left ((k+r)(k+r-1) - (k+r) + 1\\right ) A_kz^{k+r-2} \\right \\} - \\sum_{k=1}^\\infty A_{k-1}z^{k+r-2} \\\\\n&= (r-1)^2 A_0 z^{r-2} + \\left \\{ \\sum_{k=1}^{\\infty} (k+r-1)^2 A_kz^{k+r-2} - \\sum_{k=1}^\\infty A_{k-1}z^{k+r-2} \\right \\} \\\\\n&= (r-1)^2 A_0 z^{r-2} + \\sum_{k=1}^{\\infty} \\left ( (k+r-1)^2 A_k - A_{k-1} \\right ) z^{k+r-2}\n\\end{align}", "506b8fdcfc486908f67f606b0356b26f": "V - E + F - C = 1", "506bc30ec17fe3e14c24107a366ea780": " \\sigma^2 = \\frac1n \\sum_{i=1}^n \\operatorname{Var}(X_i).", "506be6738610bda49192526accdc5b69": "\\begin{align}\n \\frac{\\partial y}{\\partial c} &= b_0 x^c \\left (\\ln(x) \\sum_{r = 0}^\\infty \\frac{c(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^r +\\sum_{r = 0}^\\infty \\frac{c(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} \\left\\{\\frac{1}{c} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta + k} - \\frac{1}{c + 1 + k} - \\frac{1}{c + \\gamma + k} \\right) \\right\\} x^r \\right )\n\\end{align}", "506c03fb2fa5ae15f245c7605ff2b1d2": "(M{,}R)", "506c09f04741ab8a7f58112f0ad8039e": "g(\\mathbf{y}) = f(\\mathbf{x})\\left\\vert \\det\\left(\\frac{d\\mathbf{x}}{d\\mathbf{y}}\\right)\\right \\vert", "506c83491c32176f27076b36c460913a": "\\mathrm{exp}(i \\theta_j g_j)", "506d17daf15fa473048a848c376446c5": "\\frac{CS_{scat}}{CS_{particle}}", "506d793fdab485d89fc6836bddc7625b": "F_{0}=P", "506d9a94ee346d633be7a272142ce02a": "\\widehat\\sigma^2 = \\sum_{i=1}^n(x_i-\\widehat{\\mu})^2/n.", "506dc05839e82cf593078d23ddc26dfc": "\\frac{c}{n}\\sum_{k=1}(p_f\\frac{n-c}{n})^{k} = \n(\\frac{c}{n})(\\frac{\\frac{p_f(n-c)}{n}}{1-\\frac{p_f(n-c)}{n}})", "506dd0c2d88a32ad19dacddd5577a2cd": "\\{Z_1,\\ldots,Z_m|S=s\\}", "506e03d0557db97341531bd0206e6c2d": "\\partial_t \\boldsymbol{q} = \\underline{\\boldsymbol{D}}\n\\Delta \\boldsymbol{q} + \\boldsymbol{R}(\\boldsymbol{q}).", "506ece1c375637635fa7e626cf9be963": "P \\and (Q \\vee R)", "506ee64aa7ebfd560f7f5c1a0e15c02c": "\\mathbf{F} = m\\,\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t} = m\\mathbf{a},", "506ef92a9914ea94db7133504b58f3c9": "\\begin{align}dy_{\\text{1}}\\ =\\ (-ky_{\\text{1}}-wy_{\\text{2}}+I_{\\text{1}})dt\\ +\\ cdW_{\\text{1}}\\\\\ndy_{\\text{2}}\\ =\\ (-ky_{\\text{2}}-wy_{\\text{1}}+I_{\\text{2}})dt\\ +\\ cdW_{\\text{2}}\\end{align},\\quad y_{\\text{1}}(0)\\ =\\ y_{\\text{2}}(0) = 0", "506f027332789e7aef73c5684468f9c1": "\n \\begin{align}\n EI\\dfrac{dw}{dx} &= \\dfrac{Pbx^2}{2L} +C_1 & &\\quad\\mathrm{(i)}\\\\\n EI w &= \\dfrac{Pbx^3}{6L} + C_1 x + C_2 & &\\quad\\mathrm{(ii)}\n \\end{align}\n ", "506f353d3ef92d51f3e57bae99367caf": "f()", "50706b04c96ff10d4203ab436667fd41": "e_1 = \\hat{x}_1 - x_1", "5070a768ac54186afac6b0e68b9b1164": "\\{u,r,\\varsigma, \\bar{\\varsigma}\\}", "5070bb6f103aee534205b8094345c967": "p(v)I(v)", "5070ddf8ff8e2b11feab0729e104047f": "\\widehat{f}(\\varrho)\\,", "507175197797ea435f771739ddf97054": "= (\\frac{k\\rho C}{\\mu H})^\\gamma", "5071c8a32062955bfc598eeddf8969b3": "\\,\\mathfrak{Re}\\left(\\text{Fourier} \\left[ \\frac{\\sin(x)^5}{x} \\right]\\right)", "50720a8b5a1b77c8f176d2d789d31a3e": "\\lim_{x \\to \\infty} f(x) = \\lim_{x \\to \\infty} {e^x f(x) \\over e^x} = \\lim_{x \\to \\infty} {e^x (f(x) + f'(x)) \\over e^x} = \\lim_{x \\to \\infty} (f(x) + f'(x))", "5072187e377f49959f69e458fac0c44c": "\\eta = \\left( \\frac{1}{a_0} \\right) \\left(\\frac{da}{dc}\\right) + \\frac{d \\ln a}{dc}", "50721b3c3c0456f8572eea7528289ed4": "\\boldsymbol{v}_{A\\text{ relative to }B} = \\boldsymbol{v} - \\boldsymbol{w}", "507270776e5a4fb8352b5379f73552b7": "A_{o}^{N+1} = 1 - \\left( (1 - A_{o}^{N} ) \\times (1 - A_{o}^{N} ) \\right) = 0.9999 \\ Up \\ Time ", "5072d47b8ef194dd20a020898a34ab55": "L_U=x_m-x_u", "5072d5ff51bc4ed9f6381c53a3757003": "L_{brems}", "50734c1a4dc12005132e10653568e2ff": "\\sigma(v_i)", "50735bd9205efc12a2ea7f02cf935256": "c_0,c_1,...,c_n", "507399bcffabb24edf69015bcce15227": "C = \\sqrt{\\cfrac{2E\\gamma}{\\pi}}", "50739a61871bdb1d12256d0c777662a4": "F_n(x)=\\frac{1}{n}\\sum_{i=1}^n I_{(-\\infty,x]}(X_i),", "5073b2cfdc658e0b2d2558f49ecd2c09": "X \\times_S \\mathbf A^1_S \\cong X", "5074b04104c7971bc4f5d1b6d11e0d9f": "B(\\mathbf{V},n) = \\frac{1}{\\left|\\mathbf{V}\\right|^\\frac{n}{2} 2^\\frac{np}{2}\\Gamma_p(\\frac{n}{2})}", "50751f4defaa5e178fc3830b294fe07d": "\\langle-,-\\rangle", "5075d5b2bf6f5c9304d80bd13ec45585": "\\begin{align}\nR &= Y' + V \\frac{1 - W_R}{V_{Max}} = Y' + \\frac{V}{0.877}\\\\\nG &= Y' - U \\frac{W_B (1 - W_B)}{U_{Max} W_G} - V \\frac{W_R (1 - W_R)}{V_{Max} W_G} = Y' - \\frac{0.232 U}{0.587} - \\frac{0.341 V}{0.587} = Y' - 0.395 U - 0.581 V\\\\\nB &= Y' + U \\frac{1 - W_B}{U_{Max}} =Y' + \\frac{U}{0.492}\n\\end{align}", "5075d8b2e55d6ed575e03e7cf98531a9": " {}_3H_3(a,b, f+1;d,e,f;1)=\n \\sum_{-\\infty}^\\infty\\frac{(a)_n(b)_n(f+1)_n}{(d)_n(e)_n(f)_n}= \\lambda\\frac{\\Gamma(d)\\Gamma(e)\\Gamma(1-a)\\Gamma(1-b)\\Gamma(d+e-a-b-2)}{\\Gamma(d-a)\\Gamma(d-b)\\Gamma(e-a)\\Gamma(e-b)} ", "5075ec0694e693a848a8e100edbff4b2": "a = e^{\\ln (a)} \\ ", "50761f06c34aceb2f15185c905a6ae50": " \\frac{d v_i}{dt} = \\frac{1}{\\tau_p} (u_i - v_i),", "50763c82ea44610ba3786b865b230fce": "c_\\varepsilon > 0", "50769f07794ae3ec617ec26a8fe9637b": "m + n \\ge \\dim B", "5076a459bfe1d2f80052e910717d86ff": " \\delta_d ", "5076b39147ce6d47613f7e1361ba64e9": "s/n^{1/3}", "5076dd42696ef206bd31239b16c81c90": " (x+F_{n-1})(y+F_{m-1})=x\\cdot y+F_{n+m-1}", "507757cb72cca9b0fd06eb2517797c52": "g(x) = f(x) + \\lambda a \\sum_{1\\leq i\\leq m} I_i(x) p_i ", "507764e028b76668b4517e5369d234b9": "1\\leq p\\leq q", "50778d1bf5a695da0fe71da3368c161d": "d_{0j} = j", "5078315af28c16b0250075f0ee3469e7": " K(2p+\\Delta)^{K-1}[2(1-p)-\\Delta]^{N-K}W-(N-K)(2p+\\Delta)^K[2(1-p)-\\Delta]^{N-K-1}W\\! ", "50787b936ba5eb923af98aba4da2f1d3": "0\\leqslant k_0 \\leqslant k_1 \\leqslant \\dots", "5078ab855c76dc0f4f6b48dc8285d776": "\nqA^\\mu(\\bar\\psi\\gamma^\\mu\\psi)\n", "5078cd850758c08fce30292335154633": " \\frac{\\partial ^4 y}{\\partial s^4} (\\bar v_i),\\frac{\\partial ^4 x}{\\partial s^4} (\\bar v_i)", "5079095ec0bec78fafe02020cbd217d2": "x(t) = x_0 + y_0t + \\frac{D}{2}t^2 + \\frac{\\phi(t)}{2\\pi\\nu_{nom}}", "507947724489ee099b11d04e5597cb12": "C_\\xi", "50795e10b3f2d4d6dcff7496b35a6dfc": "\\mathbf{A} = \\mathbf{Q}_{12} \\mathbf{Q}_{23} \\mathbf{Q}_{13}", "507a8c789f38a5498916c00527bf2af7": "\\mathfrak H", "507ad99d78fec925ecbb2a983a49348f": "M_1;M_2", "507b08a84555ba91fb03408fb2329945": "\\lambda_1 = 1 \\,\\!", "507b65fab9c6cef8f26f5891486afeb0": "V(\\lambda) = e^{\\lambda N^* - \\bar\\lambda N}", "507b6cc6d73fc44142cb905120eb851d": " \\mathcal{B}p \\to \\mathcal{B}\\bot ", "507c0666ebf59e0cf285b836bfa6c77d": "t_n \\leftarrow \\min\\{t_{ni}: i \\in D\\}", "507c6bfb3f43fc0b329c23af8d50323d": "\\{X_a, X_{a+1},\\ldots, X_b \\}.", "507c861e9f637b233ed72effb680d348": " F_{hullcourse} = \\frac12 \\times \\rho_{water} \\times S_b \\times C \\times V_b^2", "507caae04cb90a7fddec933918a9c5d9": "t_{i+1}:=t_{i-1}-qt_{i};", "507cc4ce2d57415d9bad003969f4e723": " \\sum_{u=0}^{m} p_{ju}^h = n_j ", "507cfc555b88df085cfe93f5c0743614": "N_B = (B^T\\cdot B)^{-1}\\cdot B^T \\cdot N_G", "507e4d7c8a1d6f60f599fad185415aa0": "x_{0} = n", "507e5debf4f8f492951a5cf3df4aa383": "e^{-T} e^{T}=1", "507e8b4c2a8d212be7e6dd2f5ff2cc87": "HoldsAt(f,t)", "507efa8b98f854e99741386d42547fe0": "\\{ a^n b^n c^n : n \\geq 1 \\}", "507f268da7fda569042a13ef18c5c41e": "\\left[\\hat{b}^\\dagger, \\hat{b}^\\dagger \\right]_- = 0", "507faaecde0f3ff5283113eeaceb330a": "\\sigma=\\frac{2k_\\mathrm{B}^4\\pi^5}{15c^2h^3}\\approx 5.670 400 \\times 10^{-8}\\, \\mathrm{J\\, s^{-1}m^{-2}K^{-4}}", "50801d14e85f022dfde284c8c8ad6086": "t_8\\ ", "508027bc0a7eeb23b392017d58ed60df": "\n\\begin{array}{rl}\n \\partial_t u &= d_u^2 \\Delta u + u -u^3 - v + \\kappa,\\\\\n\\tau \\partial_t v &= d_v^2 \\Delta v + u - v\n\\end{array}\n", "50802c385a060e75d3725a917c7fccc2": "P = Q", "50802c70bd90744193b8d7dc2b5a9724": "1, 2,\\ldots, 3m", "50804a0e4bfc1e44b706a10fc026ded9": "A_{\\mathrm{left}}^{-1} = \\left(A^T A\\right)^{-1} A^T", "508066d24d21a570470e26f0309fa6a6": "\\text{If }\\ell_{(N,\\psi)}(\\bar x,\\bar y)>\\ell_{(M,\\varphi)}(\\tilde x,\\tilde y)\\text{ then }d((M,\\varphi),(N,\\psi))\\ge \\min\\{\\tilde x-\\bar x,\\bar y-\\tilde y\\}.", "50807e4ee0eb9fbe1b0fa683e7a8039d": "\\pi : A \\rightarrow B(K)", "50808094ef2acc69cb5cd73b2f8b8798": "P_\\lambda(g(T)) = 0) = 1", "50809200ffc1e6cab89e44ff7334a1e2": "\\,\\mu = \\frac{v_d}{E}", "5080a8931ac9237d5b155d4442f28357": "H_n(\\mathcal{C})", "5080b3de6f98b67431e34d2f46254c18": "Z_+ \\sim ED^*(\\theta,\\lambda_1+\\ldots+\\lambda_n)", "5080c0d8fe6ad3749c647a58adbea6ae": "-\\frac{\\partial\\ln(c_{t+1}/c_t)}{\\partial\\ln(u'(c_{t+1})/u'(c_t))}=-\\left[-\\frac{1}{\\sigma}\\right]=\\frac{1}{\\sigma}.", "508105ec002977bac9421fb7544c1220": "(x,v=u*Mg(x))", "5081099a82ed04b5a25ecacc0281c375": "\\{(+,-,-,-)\\,,l^a n_a=1\\,,m^a \\bar{m}_a=-1\\}", "50812ff1da701d0f694739f02270e26e": "\n \\begin{align} \\pi(p|\\alpha,\\beta) & = \\mathrm{Beta}(\\alpha,\\beta) \\\\\n\n & = \\frac{p^{\\alpha-1} (1-p)^{\\beta-1}}\n {\\mathrm{B}(\\alpha,\\beta)} \n \\end{align}\n", "508162b2cb3a9ee92cfecf0e2e7d9a83": "d \\ll D", "50819566c650079290649e6e6fdf18c3": "\n\\begin{alignat}{2}\n\\epsilon(q,0) & = \n1 + V_q \\sum_{k,i}{\\frac{ q_i k_i \\frac{\\hbar^2}{m} \\frac{\\partial f_k}{\\partial \\mu} }{\\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m} }} = \n1 + V_q\\sum_k{\\frac{\\partial f_k}{\\partial \\mu}} = \n1 + \\frac{2 \\pi e^2}{\\epsilon q L^2} \\frac{\\partial}{\\partial \\mu} \\sum_k{f_k} \\\\\n& = 1 + \\frac{2 \\pi e^2}{\\epsilon q} \\frac{\\partial}{\\partial \\mu} \\frac{N}{L^2} =\n1 + \\frac{2 \\pi e^2}{\\epsilon q} \\frac{\\partial n}{\\partial \\mu} \\equiv\n1 + \\frac{\\kappa}{q}.\n\\end{alignat}\n", "5081da9ad7104c518a21e86f48570032": "(1+2\\epsilon)|I_1|+\\tau|E|+|I_1\\cup I_2|", "50820d4a9178ca6e8b56be186b29448a": "H^0(C, K) \\cong \\mathbb{C}^g,", "50825bb31c6dbc4c21914c582bc8a092": "m_k\\in\\mathbb{R}^n", "5082678a3c7b5543acdadd689703152e": "\\frac{dM(r)}{dr}=4 \\pi \\rho(r) r^2 \\;", "5082a56148fd7dda70cf575e0f619a3b": " T \\in \\operatorname{L}(V \\otimes W)", "5082ad5eb7ce9d46496623af6fd156ee": "\\epsilon_\\boldsymbol{q} = s\\hbar q", "50830de5f4f5fa8c4407f8052e5c053b": "\\zeta_q(s;\\rho)", "5083131f2174a71c9b094c6b2e911f1a": "-3\\le y \\le 3", "5083791efa9f742ee2efd3e26507179f": "\\Omega = 4 \\arctan \\frac {\\alpha\\beta} {2d\\sqrt{4d^2 + \\alpha^2 + \\beta^2}} ", "50845d79e365c2e96159541ce9e7ce23": "\\frac{1}{|G|}\\sum_{g\\in G}\\chi_{V\\otimes_A V}(g)", "50845f68e23dc58c15589259e6f5da1a": "U_{x_1,x_2,\\ldots,x_n}=\\{f\\in\\operatorname{Hom}(A,B)\\mid f(x_i)=0 \\mbox{ for } i=1,2,\\ldots,n\\}", "5084724fb31da6ab7202e1ca727ef8b0": "\\bar{x}_0", "508478667bed040b3724fcb6732de15a": "|w| < 1", "508496474bca865c96f6980bb251357f": "\\begin{array}{rclr}\nd_{ij} - p_i + p_j & \\geq & 0 & (i, j) \\in E \\\\\np_s - p_t & \\geq & 1 & \\\\\np_i & \\geq & 0 & i \\in V \\\\\nd_{ij} & \\geq & 0 & (i, j) \\in E\n\\end{array}", "5084e70587ede7e1781a3a06fbb50c6a": "p_{X \\times Y}(t) = p_X(t) p_Y(t).", "5084e792105aafdc500fdea1e5987d98": "\\frac{d H}{d t} = 0", "508518e3eaad19cc4abd85b6b21f8810": "2^{2^{2^2}}=65536", "508530f2f9c77706089cab33d8c46e98": " C_{i',j'},", "508561d9f1660dcbbd4daa6e652a2ecb": "{1 \\over 2} = {2 \\over 4}.\\,", "50856f8e3ccbb7457d751b44fd36d009": "\\mathbf{P}_\\mathbf{Z}^n", "508574b949520ca3e964583fa558d3e0": "\\mathcal{A} = \\mathcal{B} \\circ \\mathcal{C}", "50858a668e850a05a51e7cac94eac2ab": "\\frac{r}{2GM} = 1 + W \\left( \\frac{U^2 - V^2}{e} \\right)", "5085df92974ddc657a9178f0909a00c3": "D_a q_{bc} = 0", "50867aa8245b300cd363cf15accb2d80": " \\rho_A = \\frac {m} {A} ", "5086891b55565a6b23aacae626335840": "M_{CB}^{f}", "5086bcf04bf48b26f15c9569c02d9417": "tor= tor(\\pi^{ab})", "5086ed948e9f3f4cc6c53e8be79c48c4": "\\sigma_{\\text{max}}", "5086efd5ddd32bde19f6956c42b9c4c9": "s_{2}=8.37", "50872b3e32cd197e1284c63b8649a223": "N = 11351", "50879be422e2d8f7ef52bb84f045c1e9": "H^\\dagger H", "5087b50ed27585d99231d8c67cbb92f8": "\\text{SI} = \\frac{{\\text{channel length}}}{{\\text{downvalley length}}}", "50886a01d060663d50e20d2a13e51d5d": " \\delta W = \\left(\\sum_{j=1}^m \\mathbf{F}_j \\cdot \\frac{\\partial\\mathbf{v}_j}{\\partial\\dot{q}_1} + \\sum_{j=1}^m \\mathbf{M}_j\\cdot \\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_1}\\right) \\delta q_1 + \\ldots + \\left(\\sum_{j=1}^m \\mathbf{F}_j \\cdot \\frac{\\partial\\mathbf{v}_j}{\\partial\\dot{q}_n} + \\sum_{j=1}^m \\mathbf{M}_j\\cdot\\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_n} \\right) \\delta q_n ,", "508891a54ff4899dee1b8b463bab9ae3": "\\lceil\\log_2(13!)\\rceil=33", "508894b38e54d433a5e3ce653884a30e": " Ar= \\frac{Gr}{Re^2} ", "5088b5047bf6cec14a4a3acc3402fe48": "\\tan \\frac{\\theta}{2} = \\frac{\\sin \\theta}{1 + \\cos \\theta}.\\,", "5088d9a5cddbf26a7d60ea240041bedf": "Volume = \\frac{4}{3} \\times \\pi \\times r^3", "50892eeb2604069867e95e8a96b9a1f0": "H = \\begin{bmatrix}\nL_{xx} & L_{xy} \\\\ \nL_{xy} & L_{yy}\n\\end{bmatrix}", "5089a8252e2091b242c1a644d4dbec10": "f:X_1\\rightarrow X", "5089c3ad1f14e3026766c28db2d7c735": "30^o", "5089d041e00ddebe498df0ad5d355079": "\\lim_{r \\to 0} \\frac1{\\gamma \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\gamma(y) = f(x)", "508a17fd7d41690f698b7a6b939b49df": "\n\\mathbf{B} = \\frac{1}{159} \\begin{bmatrix} \n2 & 4 & 5 & 4 & 2 \\\\\n4 & 9 & 12 & 9 & 4 \\\\\n5 & 12 & 15 & 12 & 5 \\\\\n4 & 9 & 12 & 9 & 4 \\\\\n2 & 4 & 5 & 4 & 2\n\\end{bmatrix} * \\mathbf{A}.\n", "508a2059aeaae135b21c201afd26aa5c": "a_1/b_2,\\ldots,a_n/b_n", "508a395c135509ecbbe0f1ca92136f6d": "{\\rm Si}(x) = \\int_0^x\\frac{\\sin t}{t}\\,dt", "508a9007ead3ccc8fdb4c2d6d4494114": " [\\mathbf{k}]_\\times \\mathbf{v} = \\mathbf{k}\\times\\mathbf{v} = \n\\left[\\begin{array}{ccc}\n0 & -k_3 & k_2 \\\\\nk_3 & 0 & -k_1 \\\\\n-k_2 & k_1 & 0\n\\end{array}\\right]\\mathbf{v}\n", "508aae577751838fe1c64ebe9a84b361": "\\left\\{\\begin{array}{rl}\\mathbf{y}'(x) &= A(x)\\mathbf{y}(x)+\\mathbf{b}(x)\\\\\\mathbf y(x_0)&=\\mathbf y_0\\end{array}\\right.", "508aca6d1ecf677ff41629ca271b580f": " a \\ge b \\ge c ", "508adf2855b6dd8ac7c57392170124c5": "\nF[\\rho(\\boldsymbol{r})] = \\int f( \\boldsymbol{r}, \\rho(\\boldsymbol{r}), \\nabla\\rho(\\boldsymbol{r}), \\nabla^{(2)}\\rho(\\boldsymbol{r}), \\dots, \\nabla^{(N)}\\rho(\\boldsymbol{r}))\\, d\\boldsymbol{r},\n", "508aecc0d78098e24f4097f6c5466e2a": "\\mathbf{h} = (x_1,x_2) = (x_i,x_i+\\mathbf{h})", "508b33f36edf5c2cb328a6651309eea0": "\\frac{f}{g}=b+\\sum_{i=1}^k\\sum_{j=2}^{n_i}\\left(\\frac{c_{ij}}{p_i^{j-1}}\\right)' + \n\\sum_{i=1}^k \\frac{c_{i1}}{p_i}.", "508b5b3a0637023b2a7546aca059f1e3": "\\Big\\langle x_1,x_2,\\ldots,x_n \\Big| \\langle x_1, x_2 \\rangle^{m_{1,2}}=\\langle x_2, x_1 \\rangle^{m_{2,1}}, \\ldots , \\langle x_{n-1}, x_n \\rangle^{m_{n-1,n}}=\\langle x_{n}, x_{n-1} \\rangle^{m_{n,n-1}} \\Big\\rangle", "508b915f825790f9bec1d0fc4d4bf99d": "\\textstyle (X,\\Sigma) = \\{0,1\\}^\\infty ", "508be07dfb94b74f9a414ad6bfd8d5a4": "\\frac{\\infty}{\\infty}, \\infty^0, \\ldots", "508c3f66c7a26a8da74f7837e1d2d5e8": "\\sum_{n=0}^\\infin \\frac{f^{(n)}(a)}{n!} (x-a)^n", "508c709677088c32f4faca8ba81dfb03": "EU(n)={\\lim_{\\to}}\\;_{k\\to\\infty}F_n(\\mathbf{C}^k)", "508c8229a365afee209bc9b29d1dd1c3": " V(t) = \\dfrac{W(t)}{W(0)} \\,", "508d31070a905e89cb27e0bba08dfe53": "f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to \\Omega_{X/Y} \\to 0.", "508db201bd8f530d245cdd034076349f": "\\mathbf{a}\\cdot \\mathbf{b} = \\mathbf{a}^\\mathrm{T}\\mathbf{b} =\n\\begin{pmatrix}a_1 & a_2 & \\cdots & a_n\\end{pmatrix}\n\\begin{pmatrix}b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n\\end{pmatrix}\n= a_1b_1+a_2b_2+\\cdots+a_nb_n = \\sum_{i=1}^n a_ib_i\n", "508e58e8e00291680f294dce7722c5ca": "\ng_{w3c}(a) =\n\\begin{cases}\n ((16 a - 12) \\cdot a + 4) \\cdot a\n & \\text{if } a \\leq 0.25 \\\\\n \\sqrt{a}\n & \\text{otherwise}\n\\end{cases}\n", "508e5fcc532d80ce4ba5e3209aaca48f": "\\mathbb{R}\\subset{}^{\\ast}\\mathbb{R}", "508e6b2c2031e697c57e0ad2d58f03a9": "\\color{Green}\\text{Green}", "508e6f46f587ecbb69419df15225cc10": " x(t) = c_1\\cos\\left(\\omega t\\right) + c_2\\sin\\left(\\omega t\\right) = A\\cos\\left(\\omega t - \\varphi\\right),", "508e88c1bef3432434cf87cd29727a7e": "e^c", "508ec398e88ac1018532be76e61e8613": "\\Sigma = \\Sigma_i \\times \\Sigma_{-i}", "508eea0b799137ae7242c7d208204f32": "i \\not \\in W_i", "508eec6859f7c57d7c8361ae7e88dc70": "H_\\mathrm{D}={\\frac{\\hbar\\gamma_I\\gamma_S}{4\\pi^2 r^3_{IS}}}[1-3 \\cos^2\\theta](3I_zS_z-\\vec{I}\\cdot \\vec{S})", "508f1485098fbb50aa053b3e8f1aa366": " \n\\left. \\phi_{A}^{} \\right|_M = \\left. \\phi_{B}^{} \\right|_M \\; , \\qquad\n {\\mathbf{z}} \\cdot \\left. \\mathbf{\\nabla} \\phi_{A}^{} \\right|_M = {} - \\mathbf{z} \\cdot \\left. \\mathbf{\\nabla} \\phi_{B}^{} \\right|_M \\; .\n", "508f3c60445f1cc1558753d0d5e4f922": "D(X_t,t)", "508f4418538c0c266dd544a468d5781b": "\\mathcal{M}=\\left(\\varphi_i(\\zeta_j)\\right)_{1\\leq i,j\\leq N}", "508f66a837b8bc5d77cdfd8314b46fea": "A \\land (B\\lor C)= A\\land 1 = A", "508f890892e99e5f13951255560a5741": "m_\\ell", "508fb6362d72a9f1046b1e509a01ceb7": "FDR_{-1} = Fdr = \\frac{E[V]}{E[R]} ", "508fcb987ce7e945da5fd028ed572fbd": "(\\neg 1) \\frac{A}{\\neg \\neg A}", "509005738f74becf21c9ce003ce2f6bb": "T_{\\alpha}", "50900dadc3f6366c4b9f8273e156f59f": "[t_l,\nt_u]", "50903d0f4295bd9829c5aab31e80359e": "~~~~~T,V,\\{N_{i\\ne j}\\},\\mu_j\\,", "5090a1b577175cda2686a0b94916b1ad": "z_{j+1}", "5090c975aa1dea37ab5412d60193137e": "\\textstyle w (= |W| )", "5090d86d44b1f6b38260284937782044": "0=\\frac{h}{2\\pi}\\vec{\\nabla}{\\varphi}+2e\\vec{A}.", "50916053732dff34013d4a6b0b00df34": "K > 1", "5091740112e05f0dd9176ca2114b26a1": "\n N_{xx}(x,t) = \\int_{-h}^{h} \\sigma_{xx}~dz\n ", "50918aed688118592ef4e0b710ca4412": "\n\\left( \\frac{dr}{d\\varphi} \\right)^{2} = \\frac{r^{4}}{b^{2}} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{r^{4}}{a^{2}} + r^{2} \\right)\n", "5091fae395e7f6792d02926e7d3e12e8": "f \\in \\ker(\\alpha)", "50922a020d19416154940cfbf3cb5f7b": "\\alpha, \\beta\\in {\\Bbb C}", "50928be84b56f4e9b668afc875cc4dc9": "r_{i} = E_g = \\frac{k M m}{eV_{ion}} = 6.9\\cdot 10^{-20}M \\frac{m'}{V_{ion}} = 13.5\\cdot 10^{13} \\frac{m'}{V_{ion}} cm", "5092df8a997d36fb1c5dd0a8f9cc73dc": " f(z) = g(e^{z}), \\, ", "5092e5275e1d4cf9cf4bc2a116e225cb": " t^3 + pt + q = 0\\,. \\qquad (2)", "5092e95e6b552968619664b95bf13a6c": " {\\partial \\over \\partial t} \\rho = {1\\over 2} {\\partial^2 \\over \\partial x^2 } \\rho ~,", "509304b4c68a16b02fd2d0d7e504140e": "\\|\\cdot\\|_{\\beta} < \\|\\cdot\\|_{\\alpha}", "509322ff6a700af1f17ea1f13d3a7e64": "G_0 = 1 \\Leftrightarrow L/K ", "50932ab825ff7b7758651f3b0efa733d": "\n\\mathcal{B} = \\mathbf{L} \\times \\mathcal{A} .\n", "50933082d8cd6ea1faf7eab7e2e692d7": "\\scriptstyle n-1", "50937cfd3c2ba3cbd8555e3918c92240": "\\bigcup_{i\\in I} A_{i}", "509388528a52e98e463cad686f35bb84": "r = \\frac{a}{2 \\cos{\\theta \\over 3}}\\!", "50942a9df995f34f4c6c13122bcbdfad": "\\mathrm{vol}_M = \\left(\\frac{i}{2}\\right)^n \\det(h_{\\alpha\\bar\\beta})\\, dz^1\\wedge d\\bar z^1\\wedge \\cdots \\wedge dz^n\\wedge d\\bar z^n.", "5094328ba7df55f52e170baeb80aabab": " \\hat{R}(k)=\\frac{1}{(n-k) \\sigma^2} \\sum_{t=1}^{n-k} (X_t-\\mu)(X_{t+k}-\\mu) ", "50944c484a1b9325f4f12c3ecb4d66c8": "\n \\sigma_{ij} = \\cfrac{\\partial U}{\\partial \\epsilon_{ij}} \\quad \\implies \\quad\nc_{ijk\\ell} = \\cfrac{\\partial^2 U}{\\partial \\epsilon_{ij}\\partial \\epsilon_{k\\ell}}~.\n ", "509490e725064be666bc75b9403adc93": "\\alpha^{n/2} = -1", "5094b2238def64357fa2b3e1a9e80f1a": "\\mathrm d\\varphi_x(X)(f) = X(f \\circ \\varphi).", "50951a451d0682f76fa948ad498bc17c": " \\kappa \\ge 0, \\left \\Vert \\mu \\right \\Vert =1 \\,", "50958a696d1d89995a6029d7563d807d": "iw_x + jw_y", "5095c88a561fb96ef8f3389c9f78ce06": "\\eta=1-\\frac{1-4}{5-9}=1-\\frac{-3}{-4}=0.25", "50966a0dc5c186531f6d5f7d0918243a": "\\frac{1}{q} + \\frac{1}{p} > \\frac{1}{2}.", "509699d1cc2f6e759a031ca429002ec3": "\\mathcal{J}_{i-1,i}=\\mathcal{J}_{i,i-1}=\\sqrt{J_{i,i-1}J_{i-1,i}}=\\sqrt{b_{i-1}},\\, i=2,\\ldots,n.", "5096a63e9c41e9eaa7a12499b6f7a03b": "y_1 = \\frac{2y_g}{-1+\\sqrt{1+\\frac{8gy_g^3}{q^2}}}=\\frac{(2)(0.5)}{-1+\\sqrt{1+\\frac{8(9.8)(0.5)^3}{4^2}}} = 3.71\\text{ m}", "5096dc1b02659e39ce7d50634b5d5f2a": "\\int_{\\pi-\\beta}^\\gamma", "5096f066e5b24d22bf06fc1c836da041": "(x-1)\\Gamma(x)\\,", "50970814e54cdaa5ebc5878f3066c91a": "n > 0.1 \\mbox{ cm}^{-3}", "5097472fe68bedc9e7b642e466f14fea": " \\bold{p}_1^\\prime = -\\bold{p}_2^\\prime = \\mu \\Delta\\bold{v} = \\mu \\Delta\\bold{u} \\,\\!", "50978c3ca574881b6c1a59947dfc6f7a": "\\displaystyle 4(p^{2}+q^{2}+t^{2})=5a^{2}", "5097b06a626d0e122bc047cb3cd3dbfc": "\\mu^{2} < q^{2}", "5097f065b087d8dedb7c7570c65bf61b": "\\,w_{i}(n+1) ~ = ~ w_{i} + \\eta\\, y(\\mathbf{x}) x_{i}", "50981d6afe27a337d9e578382bd29c13": "\n -\\ln(1-p) = p + \\frac{p^2}{2} + \\frac{p^3}{3} + \\cdots.\n", "50985fa7ab2daf2cbe764ec413786b26": "\\Phi_e = \\sqrt{2\\mu_0 hc\\alpha} = 2\\alpha \\Phi_0. \\ ", "5098b4313e3abf4006f15db266aad997": "\n\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n i \\cos \\theta & i \\sin \\theta \\\\\n i \\sin \\theta & -i \\cos \\theta\n\\end{bmatrix}.\n", "50990074b29f3f8ec1e57de6f5b94248": " t(d,n) \\leq \\mathcal{O}(d^2 \\log n) ", "50991095446b9c61829f712e84066c87": "b_k \\sin (2\\pi \\nu_k t)", "5099316ab0f03b8d829f9fd1e931f418": "[x_t - x^*] = A[x_{t-1}-x^*]\\,", "5099a40a5cdffc48a2b5d907f8eda20d": "N\\,\\log(N)\\,\\log(\\log(N))", "509a0b974adf2ab2fc59cf21224e1526": "g=1", "509a4647f9bc5cc4d7ac1bc2982eb417": "\\scriptstyle T=\\theta|\\mathcal{N}| ", "509acd7520a120b7e8971bce126466cf": "\\hat{W}_T \\xrightarrow{p} W,", "509b324e93f6566b8eb2edaa2d2a6e7c": "(2l+1)", "509b4804a9c005add83418c083748303": "A = \\frac{1}{2}bh", "509b7cae2ffe8a63c523d6c0c61eaac5": " \\int DA \\int \\delta(F(A)) \\mathrm{Det}({\\partial F\\over \\partial G}) DG e^{iS} = \\int DG \\int \\delta(F(A))\\mathrm{Det}({\\partial F\\over \\partial G}) e^{iS} \\,", "509ba7fd144b179a86e8152f50bab6ca": " \\mathrm{Re} = {{\\rho N D^2} \\over {\\mu}}.", "509baefc853fa8ffb38785a06158b490": " K(x,y) = e^{-\\frac{\\|x - y\\|^2}{2\\sigma^2}}, \\sigma > 0 ", "509bd6edf0bd3e90df11cacf847a4d8d": " \\overline m_a = m_1 x_1+m_2 x_2+...+m_Nx_N", "509be748b6324f4f42f532b9aa60f91f": "H_\\alpha^{(2)}(x) = J_\\alpha(x) - i Y_\\alpha(x)", "509c75f46486b0a0b7391c0c04dd6c96": "\\Phi^v = \\{ \\alpha^v | \\alpha \\in \\Phi \\} ", "509ca0642d3e531288138b21ec3b62a6": "\\omega_r(\\lambda) d\\lambda", "509cca4f5a3a5dfe081ac421dcdd8ac8": "P_N = (\\frac {F} S - 1) \\frac {360} d", "509ce10e9512019288b53ccb19315002": "\\mathit{gl}_n = sl_n \\oplus k", "509cfce4670e7b2b110dfdc4c74b2338": "\\beta_\\mathrm{F}", "509d0b8b1599e0d0938637374a0a5fcc": "\\mathrm{PS}_2", "509d3cb630ee229299d38ba4754d5b39": "S=\\sum_{i=1}^n \\frac{2i-1}{n}\\left[\\ln( \\Phi(Y_i)) + \\ln\\left(1-\\Phi(Y_{n+1-i})\\right)\\right].", "509d5a8b73658cbb193458a84460e412": " \\tau = C_{AO} \\int \\frac{1}{(-r_{A})(1-\\delta_{A}f_{A})}\\,df_{A}", "509dac133654447036a67886c5ab3cb4": "\\textstyle Y_r", "509dda130b36c8b3dc0be62c4ddc4aa1": "c - c_0 = \\int A(\\beta) \\exp (i \\beta x)~d\\beta ", "509de9ffbf34c25e0c3a1d31f5be4c2d": "\n {\\mathbf{A}} = \\begin{bmatrix} \\mathbf{A}_{11} & \\mathbf{A}_{12} \\\\ \\mathbf{A}_{21} & \\mathbf{A}_{22} \\end{bmatrix}, \\;\n {\\mathbf{\\Psi}} = \\begin{bmatrix} \\mathbf{\\Psi}_{11} & \\mathbf{\\Psi}_{12} \\\\ \\mathbf{\\Psi}_{21} & \\mathbf{\\Psi}_{22} \\end{bmatrix}\n", "509e2f62e4e0059f8a6356b3da623aa6": "K_m(R1, K_b(R0, message, K_m(S1, A), K_x ), B)\\longrightarrow K_b(R0, message, K_m(S1, A), K_x )", "509e70a7a7bda1a0a61c000d45ca0a4a": " r_N^k(n)", "509e8727e3bdc49e38556907b8029849": "(X_n,T_n)", "509e96dd72429f096e1975b80baad65c": " x, x' \\in \\Omega ", "509eae197b7e2373b64d7c47d8415e48": " \\|\\psi\\| = |\\mu|(X).", "509f4bdae37f6e7d6179673dce5a82e4": "\\scriptstyle HDOP = \\sqrt{\\sigma_x^2 + \\sigma_y^2}", "509f97f6e31d699cf90f75766acccde5": "x = \\cos (x) ", "509fd4c1a38e3dc4474e882e20c4573f": " \\mathbf{x} = A^{-1} \\mathbf{b} = \\begin{bmatrix} -38\\\\ 29 \\end{bmatrix} ", "509feea287319e232552fbdd9e14b2ea": "\\tilde{M}\\to E^* = H_1(M,\\mathbf{R})", "509fefb00a28b0185a6537a689cc0397": "\\sqrt{x^2} = \\sqrt{4}.", "509ff86db816231af33400e8d3c07344": "ze^z\\mathrm{E_1}(z)", "50a031b6bde58c54afac35bc13582ba4": "\\Gamma_A(\\check{M})", "50a0a513a3765337bf1d769a91368efb": " B(t) = W(t) - t W(1)\\,", "50a0b3ce043806cb79709c94d26dcac5": "Y_{1},", "50a0bd4df83e42be25f9b88be28c31c5": "\\Gamma\\backslash \\{(x,0) | x \\in \\R\\}", "50a0d3e44bcb8ee66b9708e074955ed7": "[0,x]=[s\\cdot 0,x]", "50a17289ec2702a7eff5cced828f97b3": "P(X \\ge 90) = P\\left(Z \\ge \\frac{90 - 80}{5}\\right) = P(Z \\ge 2.00) = 1 - P(Z \\le 2.00) = 1 - 0.9772 = 0.0228", "50a1bba907ffe13c0c8303e1fce6fde5": "\\int_\\pi^{-\\pi}Q\\,dh = \\int_{h_o}^{-h_o}Q\\,dh = S_o\\frac{R_o^2}{R_E^2}\\int_{h_o}^{-h_o}\\cos(\\Theta)\\, dh ", "50a1df05124c3489c98c999dfc5a6619": " y(t) = \\int_{-\\infty}^{\\infty} h(t,t-\\tau) x(t-\\tau) d \\tau ", "50a1e2c14dc9391b30a71f9c853c7f5c": "\\left\\{{n+k+1 \\atop k}\\right\\} = \\sum_{j=0}^k j \\left\\{{ n+j \\atop j }\\right\\}. ", "50a1f11b26896e7fea4684e0f5869f25": "\n \\langle J_1 M_1 | J_2 M_2 \\rangle = \\delta_{J_1J_2}\\delta_{M_1M_2}.\n", "50a1f8420c1a5fee449b3365c51d0188": "(x_1|y_1|0|c\\,t_1)", "50a20ce04c291ac897290a4f2e3bc9e2": "x=a", "50a20d6a98813db3003e4b4149476659": " P(s) = a_4s^4 + a_3s^3 + a_2s^2 + a_1s + a_0 = 0 ", "50a21efd602ef38ae7ee749c7d8dd644": "F = E - T S", "50a23dec06dcd1bf92a7959f011b621a": "M^{10}\\times S^1\\,", "50a312f40b54bfb01b3494f11a4b78f7": "\\vec v.", "50a33af6d8b4357a7a2a559fbf3e013b": "{{1}\\over{4 \\pi r^2}}", "50a3b2687d4997725c1a6db31b06e626": "\\Vert u_j - u_k \\Vert_{BV}<\\varepsilon\\quad\\forall j,k\\geq N\\in\\mathbb{N} \\quad\\Rightarrow\\quad V(u_k-u,\\Omega)\\leq \\liminf_{j\\rightarrow +\\infty} V(u_k-u_j,\\Omega)\\leq\\varepsilon", "50a3c29862c951e735429b850d18305f": "\n K_0 \\left ( x \\right)\n", "50a3db54e084e6d13e1663dcbca4de8f": "\\tfrac{2}{5}\\tau", "50a40a5b697c246c10e387e10fbe2411": "H^2-2P^2=\\pm1", "50a41392b77e8f00d1f90a6a0339b81d": "\\scriptstyle m \\,\\leq\\, n", "50a41af26179aaacea98541a2b968e4f": "(x^2\\frac{\\varphi-1}{r-\\varphi}", "50e0774f76fe458dfbdf7925300f6189": " = \\lambda [(3:0:1:1) - (0:0:1:1)] + (0:0:1:1) \\ ", "50e0a92ead3cce3fe20f40280379f223": "unused).", "50e0ac82a5a3336c99bbb09d03351ac4": "\nK = H + \\frac{\\partial G_{3}}{\\partial t}\n", "50e0bd7bc01f9e390d03058e608aa08f": "\\mu(\\theta) = \\begin{bmatrix}\n \\mu_{1}(\\theta), \\mu_{2}(\\theta), \\dots , \\mu_{N}(\\theta) \\end{bmatrix}^\\mathrm{T},", "50e0ce4de8bedc5530f8246bf9e3b9c8": "\\phi = \\psi+\\delta\\psi", "50e112648a1bed360769c911b368c248": "0.\\overline{27}", "50e13037b11c8fcaa50931e72829196b": "\\mathrm{Tr}\\;G = m", "50e14faf5b07c0bcf8450afeabeb2ad1": "F = GH", "50e17d0759964fd28a8180f753a19bba": "K_s+\\overline{K_t} (s\\geq 1, t\\geq 2)", "50e17d0fbb6814ae610231a11962603d": "R_\\infty\\ = \\tfrac{RS}{R+S}", "50e195ddb278312c0b2058c6cd0efa83": "\\hat{\\sigma}^{2}_{u} = [n(T-1)-K]^{-1}\\hat{u}'\\hat{u} .", "50e1bc3d6f61b415d3b98677d48fa062": "\\mathbf{r}=(x, y, z)=(x, y, f(x,y))", "50e1be828e29bc028ff67377b9edbf54": "x^* \\in S", "50e256502afb6fb79d507185e8c21cd6": "T^m\\times \\mathbb R^k", "50e2d768aacb0f787ca9ba130e4e453b": "\\begin{matrix}4&6\\end{matrix}", "50e3160ad9189bc090310b11a0dc05d5": "\\Psi_v", "50e34e5640cd4ecb7184cf40f254d72a": "\\alpha'_j=2(-\\mathbf{r'}+r\\boldsymbol{\\omega}\\times\\mathbf{r})\\cdot\\mathbf{P}", "50e37134148a648f2da861f4431c9956": "\nf(x) = \\begin{cases}1 & \\text{if }w \\cdot x + b > 0\\\\0 & \\text{otherwise}\\end{cases}\n", "50e396aec56b16b02c53a44f6c5c17a1": " \\nabla \\cdot \\mathbf{j} + \\frac{\\partial \\rho}{\\partial t} = 0 \\rightleftharpoons \\nabla \\cdot \\mathbf{j} + \\frac{\\partial |\\Psi|^2}{\\partial t} = 0.", "50e40104a6d99210342b742249ea7234": " a_{20} = p_1p_4, ", "50e42a8a724ec277442b399444d9c42b": " \\frac{dI}{dV} \\propto\\rho_S\\left(E_F-eV\\right)\\ ,\\qquad\\qquad (4)", "50e44551c5a739be8cc0ea4da45639a0": "d_1:\\ M \\to M.", "50e4537ff5fb8b6312b304a6b4e79627": " A\\subseteq B", "50e46de6deb998f6e7851929a0e24818": " \\theta_0 = 2 \\psi_1 + 2 \\psi_2 \\,", "50e4baef1b14cb95e18b493f61164c2e": "\\oint \\vec{\\nabla}\\varphi \\cdot \\vec{\\mathrm{d}s} = 2\\pi n", "50e5053532e39f971ca8a8b4ce9ab940": "\\frac{(q;q)_{a_1+\\cdots+a_n}}{(q;q)_{a_1}\\cdots(q;q)_{a_n}}.", "50e52b94eda68a0713dae870e0e7555f": "x^{-1} \\cdot x", "50e539b57d010d2bf3bd09cf6d8f4a2d": "\\scriptstyle{\\mu_j}", "50e588cd1a1aaf959312bee2064b1bb7": " a_{mn}=\\sqrt{|mn|} \\cdot c_{mn}", "50e595fdfb0ada55be9a5dab093d1660": "\nr_m = {\\Delta V_{out} \\over \\Delta I_{in}}\n", "50e5d37d1bbe3f444bfa6fed423ddb7d": "\\textstyle p=0", "50e680402e34ad05569299011bfaad84": "[r^n-1]", "50e680a9c071e45544ef5c09eb8816a6": "\\langle p: \\{r_1,\\ldots,r_n\\} \\rangle", "50e6900dd7d88f1d19c28cdb44f97bd4": "(Sw)(v) = -\\left(\\frac{dw}{dv}\\right)^2 (Sv)(w)", "50e6d78193e00de138aebfbdb2004f9a": "(p,q,Tr(g))", "50e7257d913c5eb48d2cc7f3c90d795d": "\\Rightarrow t = 1", "50e7626b086e5a0a454c76be2f9b7852": "\\chi_A", "50e7765925c45a041b0355831b368925": "I^{[p^e]}", "50e77f0aec5835b28388da2bd5edeb91": "\\,e^{i\\Omega[\\sqrt{t^2-x^2-y^2-z^2}]}", "50e780a0f390d7c4dec2361d35bcb4e9": "r.\\mathrm{Ann}_R(S)\\,", "50e844d10682741c1ea0c023ccf9a0a3": "D = \\sum_{i=1}^{n-1} (s_i-s_{i+1})^2", "50e852157df644a993217011adc3c77b": " d=8 k+4 ", "50e886ccf2f88e9e1262cc27eee91428": "z R x", "50e8b6f29429bedb74e8764cd0385706": "\\dot{x}(t)=\\frac{\\partial}{\\partial t}x(t)", "50e8cce8929d8ccbb0b31579d1bcd09f": "T_M(d)=T_{MB}(1-\\frac{c}{zd})", "50e8dd64439e1fc36ab11bd89a4ad6dd": "I : (r) = \\frac{1}{r}(I \\cap (r))", "50e94240f8e1370b0ecaf398821ed520": "\\det \\left( 1-z\\mathcal{L}\\right)= \n\\exp \\mbox{Tr} \\log\\left( 1-z\\mathcal{L}\\right) ", "50e9618058df4f165915c5c1b05ed82f": "\\mu_{00} = M_{00},\\,\\!", "50e9658a2f1e8a9e13ad215e5fedc7de": "H^{-}", "50e9c72fe1be3b6089ac7653802d60fb": "u_1 \\geq \\dots \\geq u_\\lambda", "50e9f2442033c3c0e9d8a317a05f0b49": "(bool, int)", "50ea0500d5f794f55e99d57949fba7f7": "\\frac{11}{2}N_c>N_f>\\frac{68N_c^2}{(16+20N_c)}", "50ea2b2614150e1c1881e7bd8a6d2bc0": "\n\\mathcal{I}(\\theta) = - \\operatorname{E} \\left[\\left. \\frac{\\partial^2}{\\partial\\theta^2} \\log L(X;\\theta)\\right|\\theta \\right]\\,.\n", "50ea31ecdbc1fa5b7859aab868937c7e": " \\prod_{i=1}^n \\left( t - x_i \\right) = \\sum_{k=0}^n (-1)^{k} a_k t^{n-k},", "50ea6912d28f1a9b2337b8f7774e4a83": "[2]P_n=P_n+P_n=P_{2n}=(X_{2n}:Z_{2n})", "50ea95dd769a78bb02633eaafa891fd3": "\\phi_{\\alpha}", "50eaa9e4dcb0fc30f192f5c0af390adf": "f(A,B,C,D) =\\sum m(4,8,10,11,12,15) + d(9,14). \\,", "50eac30a3ea0bff9f9761a4213d2160b": "MMOS_m(pino(r),exo(s)) = RE", "50eb75ef386077a4009cd67b3a429207": "[X,Y](f) = X(Y(f))-Y(X(f)) \\;\\;\\text{ for all } f\\in C^\\infty(M).", "50ebb24987f4fe94f48d18de268809a0": "C_R = \\sqrt{\\frac{h}{H}}", "50ec2ca518d4d8ffc7859bd1727c8409": " q_f = \\frac{Aq_i+B}{Cq_i+D}", "50ec750d2965830362a74ffd97083c21": "P(E=GD|C=c) = (0.01 + 0.16(c-11))(0.5 - 0.09(c-11))", "50ecfc4fbcc53212fe006040ffdce37a": "\\int_{a}^{x_1} f(t) \\,dt + \\int_{x_1}^{x_1 + \\Delta x} f(t) \\,dt = \\int_a^{x_1 + \\Delta x} f(t) \\,dt. ", "50ed1948174d9693bb6c395ac2fed60f": "\\frac{\\part f_i^{eq}}{\\part t_1} +\\vec{e}_i \\nabla_1 f_i^{eq} =-\\frac{f_i^{(1)}}{\\tau} ", "50ee75f018b29fd1b2dd0bd440c6b90b": "\\mbox{Sp}(1) \\to S(\\mathbb{H}^{n+1}) \\to \\mathbb{HP}^n.", "50ee92302fd2fa411ae761ce3282b428": "\\epsilon = (\\alpha_1-\\alpha_2) \\Delta T \\,", "50eea96fc64a5dd08b3a00d3807f3d6b": "\n\\ln (1+z) = z - \\frac{z^2}{2} +\\frac{z^3}{3}\\cdots \\approx z.\n", "50eeccb595ac01f212bb63962a624fe8": "H \\approx \\frac{f^2}{N c}", "50ef17088cf65bebd61680ac5a54494e": "D_\\gamma(\\gamma(b)||\\gamma(a))=\\int\\int g_\\gamma(s)\\frac{\\mu(t)}{\\mu(s)}dsdt", "50ef1de079f77896bccc5fa1cd543ddd": "1-\\sqrt{-5}", "50efa2daa30c0101202390bfcd9ebafb": "3^6 \\equiv 1 \\pmod 7, \\,\\!", "50efb73d9834cbd81139694725778713": "~\\ell/L \\ll 1~", "50efbb55f58ae991e15b589e3fb2c26d": "\\phi_{\\gamma_n,e}", "50efbca7d5e1e19f1b0545ae6c3a06b7": "\\Delta^\\text{w}_\\text{o}\\phi = \\phi^\\text{w} - \\phi^\\text{o} = \\Delta^\\text{w}_\\text{o}\\phi^\\ominus_i + \\frac{RT}{z_iF}\\ln\\left(\\frac{a^\\text{o}_i}{a^\\text{w}_i}\\right)", "50f003332cba36a0b1b762be84f39125": "\\omega_{ce}=\\frac{eB}{m}", "50f020a4da5c445cc4c6fd8aa20db2e1": "\\text{SUBEPT}=\\text{DTIME}\\left(2^{o(k)} \\cdot \\text{poly}(n)\\right).", "50f03fa7ea932453fad25ca73fd02bc6": "L_{nl} = \\frac{2 \\eta_0}{k_0 n n_2 |A_m|^2}", "50f0c069a14aa93a811d2b94b759ce01": "(2P_{n}P_{n+1}, P_{n+1}^2 - P_{n}^2, P_{n+1}^2 + P_{n}^2=P_{2n+1}).", "50f0c8979df3444f1357520278f4d32f": "[ E_{ij}, \\det(E+(n-i)\\delta_{ij})]=0", "50f0f8ea01e511afe7b8917368a8825c": "\\ln(1+x) = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\cdots + (-1)^{n-1}\\frac{x^n}{n} + (-1)^n \\int_0^x \\frac{t^n}{1+t} \\,dt.", "50f124b36312ed6ba571053cfe34e74f": "\\text{Throughput} = \\text{Efficiency} \\times \\text{Net bit rate}\\,\\!", "50f17e5c11d610b19c0471830dc4dda1": "n \\times n", "50f18eccc959a1ec86e2805f0940bea8": "\n \\begin{bmatrix}\n 1 & 3 & 2 \\\\\n 2 & 3 & 1\n \\end{bmatrix}\n\\oplus\n \\begin{bmatrix}\n 1 & 6 \\\\\n 0 & 1\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1 & 3 & 2 & 0 & 0 \\\\\n 2 & 3 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 6 \\\\\n 0 & 0 & 0 & 0 & 1\n \\end{bmatrix}\n", "50f205321fbdf412b2cebf252e8285d2": " g_m(z) = \n\\sum_{n\\ge m} \\frac{(-1)^{n+m}}{n!} \ns(n,m) z^n", "50f227c40ee81b349b85bfe19928f6d6": "H_2(G, \\mathbf{Z}) \\cong \\bigl( H^2(G, \\mathbf{C}^\\times) \\bigr)^* ", "50f285aeb60aebef9a10e59c657f8d96": "K = \\{ x \\}", "50f298ba2254b872cbb967a1c329ebed": "\\scriptstyle U \\;\\sim\\; \\mathrm{Exponential}(\\lambda)\\,", "50f2c166995959e0289047896cf21e84": "\\rho = T_{ab} \\, X^a \\, X^b \\ge 0.", "50f2c16707de0ef29312d3d2bf8bc64e": "f(x)=\\frac{1}{x}", "50f328e5ea6672d1d9b5b0703463fb4a": "dp\\!/\\!dx", "50f331d7eb4f3094aaa783d4ea4828b0": "f(\\eta) = -\\eta^3 + 2\\, \\left( c - 1 \\right)\\, \\eta^2 + r\\, \\eta + s.\\,", "50f34ab2ebeaf922040d91fcb2f88460": "\\psi=\\psi(\\Phi)", "50f37456352784e3e72f44f3268f1e98": "\n\\begin{matrix}\n Q^{T} A = Q^{T}Q\\,R = R; \n\\end{matrix}\n", "50f389a42e1e4977366302c309fe13a4": "C_{awgn}=W\\log_2\\left(1+\\frac{\\bar{P}}{N_0 W}\\right)", "50f459b5ecb20bc2bdb8fefcef483b46": "x \\in M", "50f4c04035bd64316d2eeb13555adf2e": "k_i=\t\\textstyle \\sum_{j=1}^N A_{ij}", "50f4da86c82b63ef6fea83854cb07a16": "\\varepsilon{}_n ^ \\alpha = \\frac{1}{2} [ (1 - \\varepsilon{}_n ) + (1 + \\varepsilon{}_n ) e^{- \\pi i_n \\alpha } ]", "50f53d48198265ffe7f4ad9a3d733304": "= 2\\pi\\cdot I_{max}\\int\\limits_0^{\\pi/2}\\frac{\\sin(2\\theta)}{2}\\,\\operatorname{d}\\theta ", "50f572e63e6ed867f6e2292af2aae130": "\\ u_g ", "50f5aa1b30ea58f74ab2309f46017cb8": "\\vec v_{A|C}=\\vec v_A", "50f5d4600a5e903f4f88a4cd9114782e": "\\ P \\ ", "50f62fa201fe4805b8d076b0d19d12e0": "\n\\boldsymbol{\\Tau}_{\\boldsymbol{1}}(z) = \\tau_0\\circ\\tau_1(z) = \\tau_0(\\tau_1(z)),\\quad\n\\boldsymbol{\\Tau}_{\\boldsymbol{2}}(z) = \\tau_0\\circ\\tau_1\\circ\\tau_2(z) = \\tau_0(\\tau_1(\\tau_2(z))),\\,\n", "50f6770abb325b81dfda18b2719d750e": "X_C = 1000 \\sqrt{\\frac{50}{(1000-50)}} = 1000\\,\\times\\,0.2294\\ Ohms = 229.4\\ Ohms", "50f67f6ac25744275d2297a4ac640a63": "^{\\;}g(\\xi)", "50f683489e85531d577532223fc945cd": "\\delta(x,y)", "50f6aa1e18c734436ebf0ca0bcef0be9": " \\chi (\\mathrm{M}(a,b,c)) = \\left\\{ \\begin{matrix} |K|^n \\ \\omega( h c) & \\mbox{ if } a = b = 0 \\\\ 0 & \\mbox{ otherwise}. \\end{matrix} \\right. ", "50f714bb6f9adb434a395355201acb99": "r \\equiv s \\pmod{p^{k}}.", "50f728f73c69db9bd68b29c69346d9a6": "\\psi_1(\\psi_1(0)) = \\varepsilon_{\\Omega+\\varepsilon_{\\Omega+1}}", "50f73d662c3843fd55025b9139234ee9": "2^{8^{10^{12}}}", "50f77479e6d1b945d9ab0a64f082d459": "U'':=\\bigcup_{a\\in A_0}U'_a\\supset A", "50f77604cdb298c6c35545d6d9e69932": "[T^i_j,\\overline{Q}^k]= \\delta^k_j \\overline{Q}^i", "50f797890a480227bd5105347ed6b88c": "X = \\{\\boldsymbol{x}_1, \\dots, \\boldsymbol{x}_{\\ell}\\}", "50f7ff66bf32332b30f15bff1cfc896b": "\\Phi = \\Phi_R\\times\\frac{Int}{Int_R}\\frac{1-10^{-A_R}}{1-10^{-A}}\\frac{{n}^2}{{n_R}^2}", "50f814ce62127a3baaab42a5c2c6dccd": "IEAC = \\sum AC + { \\left( BAC - \\sum EV \\right) \\over CPI }", "50f817f5dd43e0cfefbb0439daa3036c": "r = 0.5L_x|N*I|+0.5L_y|N*J|+0.5L_z|N*K|\\,", "50f81fa36986cd3f728eb4eb40d7a0ec": "BV \\leftarrow BE \\leftarrow \\varnothing", "50f840fb31b7bacf5c4d480686da3f47": "a = 2k", "50f8524b3a1509a1c6415f32a450b863": "\\mbox{Golden rule savings rate: } s^G=\\frac{(n+d)k^G}{f(k^G)}", "50f860c3079e045fa939b3e7fb5e5880": "\\scriptstyle \\{|0\\rangle_B, |1\\rangle_B\\}", "50f87f990968b8fec1d10d352c960b03": "d_{00}={4\\lambda f \\over \\pi D_{00}}", "50f880e083bf9e03134f8024d4190d2c": "\\mathcal{F}_{\\infty}", "50f890fe6de53459132f13dfd33d40dc": "(1,1,3);\\quad\n(\\frac{1}{\\varphi}, \\frac{1}{\\varphi^2}, 2\\varphi);\\quad\n(\\varphi, \\frac{2}{\\varphi}, \\varphi^2);\\quad\n(\\varphi^2, \\frac{1}{\\varphi^2}, 2);\\quad\n(2\\varphi-1,1,2\\varphi-1).", "50f938a6225b8b9a0e3ee16f22165079": "\\mathrm{T}", "50f9dac0dc15f05271262b40d7e51bbc": "\\mathbb{R}^n\\setminus E", "50fa0d31b8d69fa882d0eb4e03804403": "\\rho'=\\alpha\\Delta T", "50fa3a3833e69ccdd06bee25e50276a6": "i \\in \\{1,\\dots,T\\}", "50fa469bc71b6cc4b65948b1a6b001ca": "z_4=0", "50fa4db5992d0829bcc4304d52867a18": "\\scriptstyle c \\;=\\; (c_1,\\, \\dots,\\, c_n)", "50faf30714bd498db82e3d829fc92258": "nw(X)\\,", "50fb36836f201e061e79eff768f84291": "g(x) = \\begin{cases}\\frac{x}{1-\\alpha} & \\text{if }0 \\leq x < 1-\\alpha\\\\ 1 & \\text{if }1-\\alpha \\leq x \\leq 1\\end{cases}.", "50fb606e449b08a7be84f692322737ae": "(3-\\sqrt{5})/2", "50fb7daf4f2d49ad05ff171e64e010a9": "m\\frac{d^{2}r}{dt^{2}} - mr \\omega^{2} = m\\frac{d^{2}r}{dt^{2}} - \\frac{L^{2}}{mr^{3}} = -\\frac{dV}{dr}", "50fb9a85dc088fb4639f3c14e04cbbf7": " [\\cdot]_{\\times} ", "50fc244f3086b21f0bedc6d6766ccd02": "\\phi_{13} : A \\otimes A \\to A \\otimes A \\otimes A", "50fc7814e2fa53848a8e8338f0b6874b": "AB^+", "50fcdc8b2c020887bf49f4e9de00cd32": "s_M=\\sum_{i=1}^m x_i", "50fd08b29d5fa12396aa912516b184f7": "\\scriptstyle {z_{n+1} = a + bz_n exp[i[k - p/(1 + \\lfloor z_n \\rfloor^2)]]} ", "50fd68590c58d099db7b908ebe667989": "\\begin{align} \\prod_{k=1}^n \\cos \\theta_k & = \\frac{1}{2^n}\\sum_{e\\in S} \\cos(e_1\\theta_1+\\cdots+e_n\\theta_n) \\\\[6pt]\n& \\text{where }S=\\{1,-1\\}^n\n\\end{align}\n", "50fd9f1f80f0c8d9e1623131d27b72de": "\\scriptstyle \\epsilon_{0} = 1/4\\pi", "50fdfa434b67c57d9172007a59cb62a8": "t > s'", "50fe182f5da084b8f8162442d5c233ac": "|\\phi(p)|^2 = \\sqrt{\\frac{2 \\ell^2}{\\pi \\hbar^2}} \\exp{\\left( -\\frac{2\\ell^2 p^2}{\\hbar^2}\\right)}", "50fe62910e07d6ba7f396016a3e3cc29": "\\sum_{k=1}^p x_k y_k - \\sum_{k=p+1}^n x_k y_k ", "50fe75fa93d49ec917847e28f3c99912": "\\mathit{E_c}", "50fe797279bb6e755c740a75b4bfa42d": " G_{0} = \\frac { R_1 R_2 } {R_F +R_1 +R_2}\\ . ", "50fea226941be44d0483ea9f3165fef0": "\\langle x\\wedge\\mathbf{v}, \\mathbf{w}\\rangle = \\langle \\mathbf{v}, i_{x^\\flat}\\mathbf{w}\\rangle", "50feb2b5ad83ddb6d16eaa618348c4fd": "f_j(x)", "50fed03cf6a77c74bdf0f083afc6f684": "ds^2=\\frac{4(dx^2+dy^2)}{(1-(x^2+y^2))^2}=\\frac{4 dz\\,d\\overline{z}}{(1-|z|^2)^2}.", "50fee62d5ea4b49c46b21ebe22f4fe15": "\n\\begin{align}\n\\Pr(Y_i = 1) &= \\Pr(Y_{i,1}^{\\ast} > Y_{i,2}^{\\ast} \\text{ and } Y_{i,1}^{\\ast} > Y_{i,3}^{\\ast}\\text{ and } \\cdots \\text{ and } Y_{i,1}^{\\ast} > Y_{i,K}^{\\ast}) \\\\\n\\Pr(Y_i = 2) &= \\Pr(Y_{i,2}^{\\ast} > Y_{i,1}^{\\ast} \\text{ and } Y_{i,2}^{\\ast} > Y_{i,3}^{\\ast}\\text{ and } \\cdots \\text{ and } Y_{i,2}^{\\ast} > Y_{i,K}^{\\ast}) \\\\\n\\cdots & \\\\\n\\Pr(Y_i = K) &= \\Pr(Y_{i,K}^{\\ast} > Y_{i,1}^{\\ast} \\text{ and } Y_{i,K}^{\\ast} > Y_{i,2}^{\\ast}\\text{ and } \\cdots \\text{ and } Y_{i,K}^{\\ast} > Y_{i,K-1}^{\\ast}) \\\\\n\\end{align}\n", "50ff364aba0bc0868a2ac51275f5cf42": "Pm", "50ff4c72b442f0d5d2af4e0e0fecf02e": "\\{h[n]\\}_{n=-N}^{N}", "50ff5fb4edc8ad47ae8d012abb915625": "a=0.6+i0.37", "50ff8d5c94c32ac2e198f6d68d2c24b1": "\\scriptstyle\\le\\, N", "50ffe3f203d34f35f5fc38614af25d65": " \\gamma_p: \\mathbb R \\to M ", "51001be8765754deac456ce278a9c19c": "\\tbinom nm", "5100943bcde858b18ec1380ba6c4f9c3": "\n\\rho = \\frac{r \\sin\\theta}{1 - \\sin\\theta}\n", "510099ac217f7d0bdeb9aceae4111cc1": "T=1/f_\\mathrm{s}\\,", "5100c743739aa324af6d359cfd01f92a": "0 = \\vec{j} [\\vec{x},t] + \\epsilon_0 \\partial_t \\vec{E} [\\vec{x},t] - {1 \\over \\mu_0} \\nabla \\times \\vec{B} [\\vec{x},t] ", "5100d39892bc34523bfda5eeb5456ad9": " 5(0.75 - f^{*}) = 1.5 + 5f^{*} \\! ", "510145a6f50a87ef494f9618e2b863cb": "\\frac{d\\Gamma}{d\\cos\\theta} \\sim 1 + \\frac{1}{3}P_{\\mu}\\cos\\theta.", "5101cf53a9051bb23b1f4a9785477517": " x \\in M ", "51024928ac639be4eca4d82cf76ad5c2": "C' \\in C(F)", "5103435cf4d25737037fcd48aad4a7de": "I = \\frac VR", "51036d64b019047e730ab56eb3c5fc4b": "{\\tilde{A}}_{4}", "51036fa42b870b4674ab27cab3a383ab": "\\sum_{n=0}^{\\infty}\\frac{T(n)x^n}{n!}=\\exp\\left(\\frac{x^2}{2}+x\\right).", "5103bfb4002e4c8824827d7910c2f64c": "\nP_\\mu (\\tau )=1-\\sum\\limits_{n=1}^\\infty P_\\mu (n,\\tau )=E_\\mu (-\\nu \\tau^\\mu ), \n", "5103c690e82da81df03f5f080d3d7b82": "\\varepsilon_{eff}\\,=\\,\\varepsilon_m\\,\\frac{2(1 - \\delta_i)\\varepsilon_m + (1 + 2\\delta_i)\\varepsilon_i}{(2 + \\delta_i)\\varepsilon_m + (1 - \\delta_i)\\varepsilon_i},\\,\\,\\,\\,\\,\\,\\,\\,(7)", "51044deaa23d747e00b7e415a6bd0c20": "0 \\le \\alpha < B^n", "5104924b1d9eb287b80883bcf30d75b9": "\\left|x- \\frac{p}{q}\\right|= \\frac{|cq - dp|}{dq} = 0\\, ,", "5104c59ba0681147bb6535128d8d2929": "g_{k,n}(z)=z+\\frac{k}{n^2}f(z).", "51054f4ca74e56500a65b1cf40be2aca": "\\mathrm{Sulfur:} \\frac {\\mathrm{mass \\ of \\ air}}{\\mathrm{mass \\ of \\ S}} = \\frac {4.773 \\times 28.96} {32.06} = 4.31", "5105db699464ec9a3c5c7b3445c29654": "Sq\\,60 \\times 18 (P9) LH", "5106367c87be78a707ac2f396d92d8de": "{u}_{2} (\\mathbf{q})", "510671b8da59797f6fc79c06b7d2a237": "F = (G_1\\cap F)\\cup(G_2\\cap F)", "51069df81ca54ff19ea19c8e92ec5152": "\\mu := \\mathcal{H}^1 \\llcorner M_2", "51071e3c3cb480f7adc40d431bc6eab3": "(f+g)^{\\mathbb C} = f^{\\mathbb C} + g^{\\mathbb C}", "51075ae5183a4dac6cdb5d6f6a3d99a8": " \\mathit{k}_\\mathit{D} ", "5107bb6c2b269426a71685d4ca610de6": "v^{i}", "5107cef4377b585f048ee432fc7d2033": " \\text{(2)} \\qquad \n w(x,y) = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\frac{a_{mn}}{\\pi^4 D}\\,\\left(\\frac{m^2}{a^2}+\\frac{n^2}{b^2}\\right)^{-2}\\,\\sin\\frac{m \\pi x}{a}\\sin\\frac{n \\pi y}{b} \\,.\n", "5107fc377b67bcf73e93b27c5070595d": "\\Delta P_i = -P_i + \\sum_{k=1}^N |V_i||V_k|(G_{ik}\\cos\\theta_{ik}+B_{ik}\\sin \\theta_{ik})", "5108217cf758724eb821511af449ec75": "\\scriptstyle \\cos(\\omega t)", "510849dfe337b325aceef65dab9f4573": "X^*_{\\sigma(X^*, X)}", "5108a486e0f8b051f0eaced343893df0": "\\mathbf{E} \\times \\mathbf{B}", "5108b5470d3e6540a92a3bf350a42254": "\\dot{v} = \\frac{\\mathrm{d}v}{\\mathrm{d}t} = a\\,\\left( 1 - \\left(\\frac{v}{v_0}\\right)^\\delta - \\left(\\frac{s^*(v,\\Delta v)}{s}\\right)^2 \\right)", "5108e42a98569ddbbeb642e0e58c492d": " e^+e^- \\to \\pi^+ \\pi^- \\pi^0 ,~~ \\pi^+ \\pi^- \\eta ", "5108e48e61ab6b68437a79a3b84c7107": "\n L = \\frac{1}{4} \\left[\n (\\rho+\\rho') \\frac{\\omega^2}{|k|} - (\\rho-\\rho') g - \\sigma k^2\n \\right] a^2 \\lambda.\n", "5108f6cf068f73282964394b597491b6": "r \\dot{A} =A(1-A)", "510916c2eb151e98a3612ab82913a493": "\n\\begin{cases}\n\\Delta u + \\lambda u = 0\\\\\nu|_{\\partial D} = 0\n\\end{cases}\n", "510922325941936dbea79b5fae1bcbd2": "\\mathcal{A} = \\mathcal{B} \\cup \\mathcal{C}", "51092fbf986090149d09c4a3bdfff17d": "\\mathbf{X}_{\\ell m}^*=Y_{\\ell m}^*\\mathbf{L}/\\sqrt{\\ell(\\ell+1)}", "510956ca423c07a21e89114ac4f240e9": "\\operatorname{E}(T - C) = (k - 1)\\sigma^2", "510972160e311f59fe68b2c67034149b": "\\left.\\frac{\\partial p}{\\partial V}\\right|_{(V,T)}", "5109a758e77174ceb57137de33ae9f00": "\\operatorname{sinc}(t):= \\frac {\\sin {\\pi t}} {\\pi t}", "5109b00f66e71cc7c7336a01209da31d": "(\\neg A\\to\\neg B)\\to(B\\to A)", "5109bec6de9a0ed2954f83d73b729b9d": "\\frac{V_{mitral} - V_{aortic}} {V_{mitral}} \\times 100%", "5109d37d34e087704c9cedc8e827c8af": "C(0)=1/2", "5109f12fa2288a0f15053be5765194d8": "t_1^*=\\lambda t_1", "510a8592e70e1f8d70c65b7920b8ccbf": "e^z-1=0\\,", "510abd4f256141de45b9f39e810c262d": "\\left( \\begin{matrix} n \\\\ 2 \\end{matrix} \\right)", "510ac2e8ad4bd2fb36effa331e786600": " X \\boldsymbol \\beta = X ( S \\mathbf y) = (X S) \\mathbf y", "510ad546c568789fae413a39e8bd5765": "\\Theta _0", "510b192bced25f48ee8e6ff0f0269f03": "j^* (\\Gamma (\\mathbf P^n, \\mathcal O(1)))", "510b1c96a2cdb992c2b513ff2dd0acef": "V^* \\times V \\to F", "510c12c5cff9a520625bff2237e6ae91": "A\\subseteq C", "510c37e21409c585fe86560539ecef97": "((x))=\\begin{cases}\nx-\\lfloor x\\rfloor - 1/2, &\\mbox{if }x\\in\\mathbb{R}\\setminus\\mathbb{Z};\\\\\n0,&\\mbox{if }x\\in\\mathbb{Z}.\n\\end{cases}", "510cbb0b5ea77e8e0f8abb817d90e176": "\\begin{align}\nx_{1}^{'} & =\\gamma\\left(x_{1}-vt_{1}\\right) & \\quad\\mathrm{and}\\quad & & x_{2}^{'} & =\\gamma\\left(x_{2}-vt_{2}\\right)\\\\\nt_{1}^{'} & =\\gamma\\left(t_{1}-vx_{1}/c^{2}\\right) & \\quad\\mathrm{and}\\quad & & t_{2}^{'} & =\\gamma\\left(t_{2}-vx_{2}/c^{2}\\right).\n\\end{align}", "510cbde153f0feee180bb0533fdf0cfc": "\\forall \\rho, \\sigma. \\;\\ \\rho \\subseteq \\sigma \\in \\Gamma \\Rightarrow \\rho \\in \\Gamma", "510cdcffba5927f7ed72230e4165bbfe": " g(x,y)=c", "510ce813236b930f0fb2afba89696bf4": "F_{\\alpha \\beta} \\, = \\, \\partial_{\\alpha} A_{\\beta} \\, - \\, \\partial_{\\beta} A_{\\alpha} \\,", "510d010cb371b359a6830280e4d83c48": "(\\;8) \\quad\\quad \\therefore w_1\\frac{dx_s}{dt}-w_2 \\frac{dx_s}{dt} + \\int_{x_1}^{x_s(t)} w_t \\, dx + \\int_{x_s(t)}^{x_2}w_t \\, dx = -\\left.f\\left(w\\right)\\right|_{x_1}^{x_2}", "510d2e5ae5d337c3e1a6162e2a94846a": "g(n) \\cdot k_1 \\leq f(n) \\leq g(n) \\cdot k_2 ", "510d320b83f4fdb0df48bc7d6f3d18c7": "(x-g)\\!\\ (x-g^{p-1})(x-g^{-p})", "510d389688568c0e883b32dd1f0700ee": "a \\left( 1 - \\frac{\\sqrt{2}}{2} \\right)", "510d6881b7a23e3d1839eddedd89b497": "P_N(r) = \\frac{D}{(N-1)!} {\\lambda}^N r^{DN-1} e^{- \\lambda r^D} ,", "510d74579ce7a90aecf4231449c88cfb": " \\mathfrak{P}(\\mathfrak{P}_{\\ge 1}(\\mathcal{Z})).", "510d7a05984cd94bd8d15773a239a6b8": "P = \\frac{E}{t} = \\frac{1}{2}A\\rho v^3.", "510d86b247afd37c3ef4234b09241dbf": "\\psi\\upharpoonright_\\alpha", "510d9d71959175c7bbf6f3c684c6e28f": "m_i/m_\\mathrm{tot}", "510de7bfbb6902caf27f95de72fa275c": "Y(t) = \\sum_{i=1}^{N(t)} D_i,", "510e2c21aff8c7cd1ff941927ca14b6b": " \\Rightarrow \\beta^2 b = \\mathrm{constant} \\ ; \\ \\alpha^2 b \\rightarrow 0 \\,\\! ", "510e38a681f6b6acb52aeccb58d6f56e": "u(x)={\\rm e}^{\\lambda x}\\int_x^1 {\\rm e}^{-\\lambda t}f(t)\\,dt ", "510e629a512038d5d9745261c849e9bd": "3. \\; \\; \\mathrm{NO}_2 + \\mathrm{O}_2 \\; \\xrightarrow{h \\nu } \\; \\mathrm{NO} + \\mathrm{O}_3 \\; ", "510ef794bfc52a164cd604afa0797116": "\\Delta S=Q_1/T_1", "510f6f543f18b9f99eb80435c75dca98": "\\textstyle p(x) = 1 + x^2 + x^5", "510f9a87836f98035ca7d93d3b31b05b": " \\ S_N = \\{85, 80, 89, 85, 80, 89, \\ldots\\} ", "510fdb5227f00fac86dfbf8b76391d89": "H \\in \\mathcal{H}", "510ff6e03bc28392b9ba08234666eb96": "\\varepsilon = 1.", "51100ccb60c9243da7dbb029e98c9b36": "\\mathbf A=\\langle A,\\wedge,\\vee,\\to,0\\rangle", "51102e879a523aaf3b6a913dd2b121dd": "\\Delta\\phi,\\Delta\\lambda", "511037e1fe40317c50e41d9d44252daf": "L \\cdot \\frac{10.67}{C^{1.85}\\quad d^{4.87}}", "511043ce7391c497b84ebf7dc43ebb1a": "i_{n-k}p", "51234918bef632bc8b40976ba5e31372": "\\kappa^*(s) \\leq \\kappa(s)", "5123d60dbe84afb38f24237204b15d92": "C_2 = (uv-vu)+(wx-xw)+(yz-zy)+...", "5123e95d99edaebd6f11304bd2f679f8": "\\nabla\\cdot(-\\nabla \\Phi) = - 4\\pi G \\rho", "51241a4c59c9f8dd9c0652cb967065f9": "\\tfrac{1}{2} (-1 \\pm \\sqrt{q})", "5124456221ed4a33597c941aa20b23f6": "\\left(\\frac{x}{a}\\right)^2 + \\left(\\frac{y}{b}\\right)^2 = 1.", "5124abde0040b8aee4d5ee5ebb095dc6": "x\\in A\\setminus B", "5124d72cb7cc5d3e8e935873765718e8": "D \\le D_\\max", "5124e152accbf8dd195676f88f84f341": "\\sum_{i=1}^kr_i=n ,", "51252b2f6c233866c2a4f691f663c80e": "\nH_{\\mathrm{pot}} = \\sum_{n=2}^{\\infty} C_{n} q^{n}\n", "512553ec562f7c9d10d1cf7d2130ecd3": "\\mathcal{I}_{\\beta, c} = -\\frac{\\operatorname{E} \\left [\\frac{X}{1-X} \\right ]}{c-a}=- \\frac{\\alpha}{(\\beta-1)(c-a)}\\text{ if }\\beta> 1", "51255ed7de47aec2c864a56e75f28f53": "\\alpha_i(\\alpha_j^\\vee) = c_{ji}", "51257d0584f24aabbef68523b35d54f8": "\\bold{X} | F_\\bold{X}", "5125ac4c8fbe46638d43f553cde7d2a8": "\\left[-\\frac{\\hbar^2}{2m}\\nabla^2+V_s(\\vec r)\\right] \\phi_i(\\vec r) = \\epsilon_i \\phi_i(\\vec r) ", "5125c47c36f1df417c7d25b8c88da09a": "T_n={n(n+1)(n+2)\\over 6} = {n^{\\overline 3}\\over 3!}", "5125c6d5c59baac38769d3e155ddaa32": "T(\\neg A)=\\Box\\neg T(A).", "5125d357e9e3e1405bfd004934934c43": "\\int \\frac{1}{h(y)} \\, dy + C_1 = \\int g(x) \\, dx + C_2,", "5126f656a7be3c9112e950fe165d03e3": "\\lim_{h \\to 0} \\frac{f(a + h) - f(a) - f'(a)h}{h} = 0.", "51270b4827dfcc9079d88c5ca34616b9": " 3=3-0", "51278d9f5be4e216a1c4eaa5f98a7bb1": "\\,\\Upsilon", "5127982bba19e3dc8cdbaf05112607f0": "2x = x + x = x\\times(1 + 1) \\equiv x\\times0 = 0 \\pmod 2", "5127b8bbfc66b19c9402a35878bf3aca": "\\text{A} \\leq_{\\text{PTAS}} \\text{B}", "5127c419d2ef7697f13aa4ad71eb5a8f": "\\det(A)\\,", "5127c878bed22f3b5af626692ac85b7d": "i\\textrm{'th}", "5127f0407db2081ab767e88bccfe775c": "\\|\\cdot\\|_k", "5128092ad6654b0601b029e7e14c3edb": "x_1, x_2, x_3, x_4", "5128418c3505722232ed0d6d6f6c001c": "\\langle x,y,z \\mid (xy)^2=x^2=y^2, zxz^{-1}=y,zyz^{-1}=xy, z^{3^k}=1\\rangle", "512848dcb51945b714163d1014d9a651": "P = \\frac{0.6 F_y \\pi d_s^2}{4}", "512881f7b3d640172b985115f71b0099": "dF_{p+1}=H\\wedge F_{p-1}", "51289fe4d4e83d08134f129d8d49c303": "\\ \\displaystyle u \\ ", "51290300cfb0231c22b8992a285ce8e9": "\\hat{\\sigma_z}", "51293c0014cc485e2d592e9ea9d1ed1b": "a_{v,t} = 1", "5129943ab118817c8419867abc206702": "r[s,t;x,y]_n+s[t,r;y,z]_n+t[r,s;z,x]_n=0,", "5129caae37406a3514a46317b39c2bc1": "\n\\mu(x, y; t, s) =\n\\int_{\\xi = -\\infty}^{\\infty} \\int_{\\eta = -\\infty}^{\\infty}\n\\begin{bmatrix}\n L_x^2(x-\\xi, y-\\eta; t) & L_x(x-\\xi, y-\\eta; t) \\, L_y(x-\\xi, y-\\eta; t) \\\\\nL_x(x-\\xi, y-\\eta; t) \\, L_y(x-\\xi, y-\\eta; t) & L_y^2(x-\\xi, y-\\eta; t) \n\\end{bmatrix}\ng(\\xi, \\eta; s) \\, d\\xi \\, d\\eta.\n", "512a1ea8e072dc69add5b350616f68fb": "\\gamma \\equiv \\frac{m_0\\omega}{\\hbar}", "512a27ae561b83c726faed4b447e146a": "f:\\mathcal{A} \\to \\mathcal{B}", "512a2c92cceddf146342eb68a401c52b": "{f \\hat{k} \\times \\overrightarrow{V} + \\nabla \\Phi} = {(f_o + \\beta y)\\hat{k} \\times (\\overrightarrow{V_g} + \\overrightarrow{V_a}) - f_o \\hat{k} \\times \\overrightarrow{V_g}} = {f_o \\hat{k} \\times \\overrightarrow{V_a} + \\beta y \\hat{k} \\times \\overrightarrow{V_g} } ", "512ac2177c9abd31a2c49e9a4d4cef95": "n\\mathrm{Li^+} + n\\mathrm{e^-} + \\mathrm{C} \\leftrightarrows \\mathrm{Li_nC}", "512b09f218c2b45e8018f1de5955a692": "\\frac{b-a}{5}", "512b172d5e33ec298899551547dc75f0": "\nR = \\frac {TP } {TP + FN}\n", "512b493381fc5b0f4df14f8221ac79be": "c'_i = \\frac{c_i}{b_i - c'_{i - 1} a_i}\\,", "512b50ccf840b8f6d6b0685cc2e520ae": "\\phi(u) = 0", "512b98db0fb881e8bf38280db1d673e0": " \\mathbf {P_{\\mu\\nu}} = \\mathbf {D_{\\mu\\nu}} \\mathbf {S_{\\mu\\nu}} ", "512baff4e93db6fd2f554cbcd3b3f0ee": "\\pi:\\mathcal{C}_g\\rightarrow\\mathcal{M}_g", "512bbda64f9fcec4f278e3dd77995045": "\\displaystyle{f^*(t)=\\sup_{0 r_c.\n \\end{cases}\n", "5143d084b2e9066684f94d4ffdd979d4": "\n\\left.\\begin{align}\n X \\perp\\!\\!\\!\\perp A \\mid B \\\\\n X \\perp\\!\\!\\!\\perp B\n\\end{align}\\right\\}\\text{ and }\n\\quad \\Rightarrow \\quad\nX \\perp\\!\\!\\!\\perp A,B\n", "5143d22e1e9adbdda3c62d6c1e9568f1": "{m \\choose r}_q = q^r {m-1 \\choose r}_q + {m-1 \\choose r-1}_q", "514453b4f1aede2f98acbb74558946d1": "t \\geq 0", "5144680ad3c256cee51f45a7e1e03ed5": "\n\\begin{align}\n\\sigma(x,y)\n&= \\operatorname{E}\\left[\\left(x - \\operatorname{E}\\left[x\\right]\\right) \\left(y - \\operatorname{E}\\left[y\\right]\\right)\\right] \\\\\n&= \\operatorname{E}\\left[x y - x \\operatorname{E}\\left[y\\right] - \\operatorname{E}\\left[x\\right] y + \\operatorname{E}\\left[x\\right] \\operatorname{E}\\left[y\\right]\\right] \\\\\n&= \\operatorname{E}\\left[x y\\right] - \\operatorname{E}\\left[x\\right] \\operatorname{E}\\left[y\\right] - \\operatorname{E}\\left[x\\right] \\operatorname{E}\\left[y\\right] + \\operatorname{E}\\left[x\\right] \\operatorname{E}\\left[y\\right] \\\\\n&= \\operatorname{E}\\left[x y\\right] - \\operatorname{E}\\left[x\\right] \\operatorname{E}\\left[y\\right].\n\\end{align}\n", "514476dae130a139c74ac0360de57503": " \\mathrm{Graff}_k(V) \\simeq \\frac{E(n)}{E(k)\\times O(n-k)} ", "51447845109c9c09bc6802fedd8fbe1a": "\\text{change open}_1 \\equiv (\\text{open}_1 \\not\\equiv \\text{open}_2)", "514479d8cb739a85b6637840c39b2353": "\\alpha:p\\Rightarrow q", "51448c1a4ff440201501e9ba7b5cc2ad": "\\|T\\| = \\sup\\{\\|Tx\\|_Y \\mid x\\in X,\\ \\|x\\|_X\\le 1\\}.", "5144b0e6436cefe4723156482bda1dc5": "\\delta \\rho(\\mathbf{r}t)=\\chi_{KS}(\\mathbf{r}t,\\mathbf{r'}t')\n\\delta V^{eff}[\\rho](\\mathbf{r'}t')", "5144b8e82739fbeb41b95ba4d75afd0d": "p(\\boldsymbol{\\theta | x})", "5144bcdec04a15630528c8c8ad1b35f3": " \\begin{align}\n\\lim_{\\beta \\to 0} \\mu = \\lim_{\\alpha \\to \\infty} \\mu = 1\\\\\n\\lim_{\\alpha\\to 0} \\mu = \\lim_{\\beta \\to \\infty} \\mu = 0\n\\end{align}", "5144c4f36bb31d354d871bbbe2841f73": " \\mathbf{x}_{k}^{T} \\, \\mathbf{H} ", "5144d52aeb003ce407e63f3631ffe4b3": "\\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_-.", "5144ea2e6372fe6895d60a8df8496b4c": "u(x)=\\prod_{i=1}^L x_i^{\\alpha_{i}}, \\quad \\sum_{i=1}^L\\alpha_{i}=1", "5144ec55d20d06e6a30ecb02d5ce5cd8": "\\sigma_{ij} \\approx \\left(\\cfrac{K}{\\sqrt{2\\pi r}}\\right)~f_{ij}(\\theta)", "5144f0062ef9b23e0d93c0d843cd8f39": "\\overrightarrow{dA}", "514524e7bb31aa33528ae1c07220e350": "h: A \\to S", "5145252618d7765b55e87afc471eef27": "L = 4\\pi R^2\\sigma T_E^4", "5145399de6a65458b87295eac7076c63": "y_c = 2/3E_c", "51456e469bd32d40279d54c025524079": "pH = pK_a \\,", "51458d48f328d7e09111c73f987683c3": "g_1(x),\\ldots,g_k(x)", "5145b8fbc4925d469850a6df000482a2": "\\frac{b^2}{p^2}=\\frac{2a}{r}+1", "5145ddebb2366a3db0fa565158ef19ba": "\\cosh (a) + j \\ \\sinh (a) = \\exp(a j) = e^{a j}", "51466bbde3d1997dbf31f2686b2805a8": "\\lim_{q\\rightarrow 1}\\frac{1-q^n}{1-q}=n", "5146acfcbfb7911de961b9770aa10e7d": "l^{(i)}_j", "5146cd02c23457ebfce5eba0ea7f7f8b": "\\mathrm{ 2 \\ (RS)_2Pb + 4 \\ NaOH + O_2 \\longrightarrow 2 \\ RS - SR + 2 \\ Na_2PbO_2 + 2 \\ H_2O}", "5146eca7b0215bd8875b12864695fdd3": "g_{t t} = -c^2 \\,", "5146fd40c377870aa31f0006e2ef4a82": "\\left(x+\\tfrac{1}{2} b\\right)^2 \\,=\\, x^2 + bx + \\tfrac{1}{4}b^2.", "514707d35a6735023df340d033747844": "\\frac{p(\\xi)}{q(\\xi)} = \\left(t\\to\\frac{f(t)}{\\xi-t}\\right)[x_1,\\dots,x_n].", "514719f72bcfc0f054167834e9b8f86d": "\n\\frac{\\partial^2}{{\\partial x}^2} V =\nL C \\frac{\\partial^2}{{\\partial t}^2} V +\n(R C + G L) \\frac{\\partial}{\\partial t} V + G R V\n", "51473a78991dc85c4dedd1c41533ab23": "\\Delta_L = \\nabla^*\\nabla", "51476421980bfb7f50595929c80f4bca": "\\displaystyle w(4,3)", "5147ab98259a22ba7c0a7bac703288dc": "\\mathbf x\\sim\\mathbf y", "5147c9e14a12126f6ebab9d4a3368a09": "\\sqrt[n]{x}_s", "51481add6ad7e73d83d8914a5f0972fd": " 2 \\alpha = (2 + h^2 \\lambda) ", "514821d3632868d369399e3065e43c55": "\\tilde{F}", "514837debd7dd88a3b9dd78dc41d3bb3": " \\sum_{ i = 1 }^n b_i^2 = 1 .", "514839fdd7c7b39885d6bb14b4fa9fe8": "(ab)x=a(bx)", "51484b73317aa6d31a8fec4f9862d8f2": "\\{N_A\\}_{A\\in\\mathcal{A}}", "514853fcfaa50888b9645ab172f7b064": " y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) \\ge 1 - \\xi_i, ~~~~\\xi_i \\ge 0", "514884be093e9ab7909b0d394e7b74d2": "T(n)", "51488657202fda87dbdb0f71589c5037": "\\mathrm{Sh} = \\mathrm{Sh}_0 + C\\, \\mathrm{Re}^{m}\\, \\mathrm{Sc}^{\\frac{1}{3}}", "5148dde867fdf182b425234d3b31552b": "\\Delta u-u^2=0\\ on\\ \\mathbb{R}^d.", "5149159f8ff3d948070686927b7ef6bc": "\\mathbf{w}\\cdot\\varphi(\\mathbf{x}) = k(\\mathbf{w'}, \\mathbf{x}).", "51498e48ce598700bbaf0f26dd323eaa": "P \\left\\{ \\begin{matrix} a & b & c & \\; \\\\ \n\\alpha & \\beta & \\gamma & z \\\\\n\\alpha' & \\beta' & \\gamma' & \\;\n\\end{matrix} \\right\\} = \n\\left(\\frac{z-a}{z-b}\\right)^\\alpha \n\\left(\\frac{z-c}{z-b}\\right)^\\gamma\nP \\left\\{ \\begin{matrix} 0 & \\infty & 1 & \\; \\\\ \n0 & \\alpha+\\beta+\\gamma & 0 & \\;\\frac{(z-a)(c-b)}{(z-b)(c-a)} \\\\\n\\alpha'-\\alpha & \\alpha+\\beta'+\\gamma & \\gamma'-\\gamma & \\;\n\\end{matrix} \\right\\}\n", "5149916f7f7ccf22b2dd5d693af26fcd": "i>k+1", "5149bdbb939b3866ec75060aed0b0f7b": " u_r = u_\\theta = 0 ", "5149ce2ff7799ed01da8519aa93672d3": "E=12n", "5149f264afdd4ce372754ca935777264": "\n{Q}=\\left[\\begin{matrix}0&\\mathbf{0}\\\\\\mathbf{S}^0&{S}\\\\\\end{matrix}\\right],\n", "5149fe187ff5cd559972341eb6e2027e": "H^n(G,M) = Z^n(G,M)/B^n(G,M).\\ ", "514a5157b0a13c4072f32fa0c2b228a1": "2^{2^n n^{-3/2+o(1)}}", "514a536fd4e632857f17009b7b086648": "0\\to M'\\to M\\to M''\\to 0", "514aa1fccc38f84a7e5e845954f820ee": "\\delta \\simeq\nc\\,\\big/\\sqrt{\\omega_{p_e}^2 - \\omega^2}. ", "514ac9bc1155e7acade44099f7f312fc": " P(X \\mid Y) ", "514b1a143f455903b58d38f215b052b2": "R_{j,\\varepsilon} f(x) =c_n\\int_{|y|\\ge \\varepsilon} f(x-y){y_j\\over |y|^{n+1}} dy", "514b6609ade736a6b84ac68ba6aff2a0": "N_i = N-n_i", "514b8098ef32ea608644167e35c31e06": "k3\\,", "514f997633c02627f18aa014b396b10c": "P = \\frac{m}{w_{t}}", "514faf5a71b4f0f67374c388f37aa0d7": "**", "514fc4076b6dd65489650f523f0561cb": "(\\mathbb{R}^2)^{\\perp} \\cong \\mathbb{R}", "514fd3510fe7b43cba86ee7f2a7a4429": "\\begin{align}\n\\hat{\\boldsymbol\\rho} &= \\frac{ x \\hat{\\mathbf x} + y \\hat{\\mathbf y}}{\\sqrt{x^2+y^2}} \\\\\n\\hat{\\boldsymbol\\phi} &= \\frac{- y \\hat{\\mathbf x} + x \\hat{\\mathbf y}}{\\sqrt{x^2+y^2}} \\\\\n\\hat{\\mathbf z} &= \\hat{\\mathbf z}\n\\end{align}", "515008884997990ab0d9777988a3e0ca": "\\varphi(t;t_0,x)", "515080ef0c702314270d2b64c8632c4a": "\nm H'_p (x,p) = p, \\qquad H'_x (x,p) = U'(x),\n", "515085cf9f7457bf045db3c0102c9298": "n > 0\\,\\!", "5150884a86c25257e9b2342970588661": "x \\in \\overline{B}(x_n, r_n).", "5150a488b6a91eb9d6620c1ee950d155": "(n+1)^3", "5150b684a78044282705c05be09c38d1": "e^{TB}=\\phi^{-1}(0) \\phi (T),\\ ", "5151081af6be315886dd5340f90c5e65": " \\sin\\left(a - b\\right) = \\sin a\\cos b - \\cos a\\sin b ", "51514fd2c4af3d38f3407e0d985f70d0": "w_{ij} = w_{ji}, \\forall i,j", "51525ebbeb141caa86bf5076c4cdd4ce": "g^{\\mu \\nu} \\,", "5152643fea171bdf67f9bb18af37cd15": "A + (A \\cdot B) = A", "5152736ad999bdb01770fa78038b00e4": "{s} = \\frac{30m}{t^2d^3l(1+l^2)}", "5152a2680b6a588c854004ae8ee91624": "ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2\\, ", "515318e66aefad6492ee6390d6ceaa42": " q _{v \\setminus w} -1 ", "51533c369ddc15f1cb3855e0b51e668b": "F(x;\\mu,\\sigma,0) = \\exp\\left\\{-\\exp \\left(-\\frac{x-\\mu}{\\sigma}\\right)\\right\\}", "5153c747353d8505f50537f0f1f629f6": "a_1,\\cdots,a_{k-1}\\,\\!", "5153f0caeb5546a647feeca6c446f032": "(\\mathbf{\\hat{e}}_1, \\mathbf{\\hat{e}}_2, \\mathbf{\\hat{e}}_3)", "515408ad4fbff085005d063b888f18ce": "\nV_{ijk}= E_{0} \\left[\n\\frac{1 + 3 \\cos\\gamma_{i} \\cos\\gamma_{j} \\cos\\gamma_{k}}{\\left( r_{ij} r_{jk} r_{ik} \\right)^3}\n\\right]\n", "5154adfeea8fc0fce4b2b3e019f4c0a4": "\\operatorname{E}(X)=\\mu+\\gamma\\beta ,", "5154bf5d81c6e6fd1e76ff4482108ee1": "-2\\Im(\\mathit \\Gamma)=\\tan(2\\beta x)", "5154c9f32731c212bd86d8ab77b34f48": "d_{mm'}", "51554e61630b8eb7e513fd9c4f6caffb": " [ Q_\\alpha , P^\\mu ] = 0 ", "5155690c5441234ce2b4c9f63a516909": "m(r,h'/h) = S(r,h) = o(T(r,h))", "51559a39317f23d8569532d80bba8ad2": "I_{n,m} = -\\frac{1}{(n-1)(bp-aq)} \\left [ \\frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+a(n+m-2)I_{m,n-1} \\right ]\\,\\!", "5155a88e4d19531e71bcdb4297247225": " p_{n+1}=p_n+K\\sin(x_n),\\; \\; x_{n+1}=x_n+p_{n+1} ", "515662524bcfc42019d386d7e58c1ebd": "x'_s\\leftarrow 0", "515693a9b866faa0b44a8b71158bb7bc": "T(x) = A x + b", "515694de1492b3bb47ffe24cc6099992": "\\mu_{2}", "5157167282544d70b8e153ed21beee85": " \\tau_\\mathcal{S} = \\inf\\{ t>0: Y_t \\notin \\mathcal{S} \\} ", "51572c709b5d3e7a5503d43c7e9319b5": " P = \\frac{q^2 a^2}{6 \\pi \\varepsilon_0 c^3} \\mbox{ (SI units)} ", "51572c8a42b550bb5f66ba299a4907db": "\\mathrm{supp}(X)", "51573a1c618533b0dd05b994b18cc713": "H^p(X, \\mathcal{F})", "51574292eac1cdccd119a10827fab56a": " \\mathbf{v}\\mathbf{w} + \\mathbf{w}\\mathbf{v} = -2 (\\mathbf{v}\\cdot \\mathbf{w}).\\!", "5157478a478ca4d88891995e7f079607": "-\\mathrm{d}\\gamma\\ = \\Gamma_1\\mathrm{d}\\mu_1\\, + \\Gamma_2\\mathrm{d}\\mu_2\\,,", "5157c730ec122aa9fd022402ac50ecbc": "0.\\overline{769230}", "51582dd4740cb2f7b3760cb2858043ef": "\\lambda=e^{\\beta \\mu}", "51582e8b0ab90f6146ec1be569965a1d": "i-j \\in \\overline n \\setminus \\left\\{0\\right\\}", "515846b1f7d392d61e021e1bc3176437": "\\dots \\longrightarrow \\textrm H^2(M;\\mathbf Z) \\stackrel {2} {\\longrightarrow} \\textrm H^2(M;\\mathbf Z) \\longrightarrow \\textrm H^2(M;\\mathbf Z_2) \\stackrel {\\beta}\\longrightarrow \\textrm H^3(M;\\mathbf Z) \\longrightarrow \\dots", "51584ac3d2fb9c3c6eb7bf5b603ef8da": "\\delta f", "515889781dcfeed2034f666b98f669c0": " k_x^2+k_y^2+k_z^2=k^2 ", "5158a7f3f6f3060478f964cf4f29c403": "\\bar 1", "5158e12edfb511b974e3449d91c59aeb": " N_1 + N_2 \\,", "5158edd70ad7a617d21a44809ec54d55": "\\dot{Q} = -k A_r \\frac{\\mathrm{d}T}{\\mathrm{d}r} = -2 k \\pi r \\ell \\frac{\\mathrm{d}T}{\\mathrm{d}r}", "51590326dddf11b9da2d396c0c760a08": "\\mathbf{U}_{C} = \\mathit{exp}\\big(\\frac{-i\\mathbf{H}_{C}t}{\\hbar}\\big)", "515912a4993142a050727fef47b9dd12": "b_3 = 3a_0 + 1a_1 + 1a_2 + 2a_3", "515950a082a48aa26476b286232bb9c3": "f_1(x)/f_2(x)\\,", "515a0ae8660891334e360e5643478711": "\\beta=-v\\alpha \\,", "515a2b3b208a03b103342dcfaac063e2": "n = 2\\eta_2+p+1,", "515a58f18c7ccd868e6b2b9daa8970a2": "\\{\\psi_n\\}", "515aab5efd30ae91804f38672cd7762e": "L^{\\,p}(X,\\mu)", "515b4953f33b904ff3e8d7265f3684ab": "\\int\\frac{x^2}{(ax + b)^n} \\, dx= \\frac{1}{a^3}\\left(-\\frac{(ax + b)^{3-n}}{(n-3)} + \\frac{2b (ax + b)^{2-n}}{(n-2)} - \\frac{b^2 (ax + b)^{1-n}}{(n - 1)}\\right) + C \\qquad\\text{(for } n\\not\\in \\{1, 2, 3\\}\\mbox{)}", "515b6808cfc89d858c418cc2349c7c2f": "s\\rightarrow T(G)", "515b755913116040cd03e9a38924abd9": "I_n=\\{x\\in I\\mid \\sigma(x)=n\\}", "515c33b6d9892dce155365b8b2f833d9": "\\displaystyle{\\pi(g,\\gamma)[zF_w](z)= (\\overline{\\alpha} +\\overline{\\beta}w)^{-3/2} zF_{gw}(z).}", "515c3e6e4bba21d2bd3de04388aafd15": "n_{i} = 0", "515c4843811af1ba5da2ea88280c2586": "2 = \\frac{p^2}{q^2}, ", "515c745fd33d6c93ef06cfe361ea3977": "[x_1 : x_2] = [\\lambda x_1 : \\lambda x_2].", "515cc2e7a337f615370afaba4daa1958": "\\scriptstyle f_1=5.033 \\mathrm {\\ Hz}", "515cf58929859094602252b52dadf6e7": "\\langle x,y \\mid x^p = y^q \\rangle", "515d0db3f3277a6495216a27c0434e65": "2^{\\mathbb N}", "515d5e4cac361b851dc310734c50326c": "(-1)^{A\\cdot X}=(-1)^A(-1)^X\\,", "515d75bbb64c3f60b2a462350a1a0297": "\\tan S = \\sqrt{\\left (\\frac{\\partial z}{\\partial x}\\right )^2 + \\left(\\frac{\\partial z}{\\partial y}\\right )^2}", "515d9c4990e82aa01e7f9f83367f01d7": "\\chi(G)=n", "515da930cabec567e4cf07e38cddab42": "A= (r^n-1)/B = 3", "515de8bce891da9980698acb3d1a8006": "\\displaystyle y = \\frac {bd}{{a^2+b^2+c^2}}", "515e0dd3e0c17c87623328f670825cf7": "\\begin{bmatrix} \\eta_1+1 \\\\[5pt] -\\dfrac{1}{\\eta_2} \\end{bmatrix} ", "515e47a71e27f6ba8a0b3214af7fa9d8": "B [U(\\$100) - U(\\$0)] > R [U(\\$100) - U(\\$0)] ", "515e95315a464a16560c81b3e768efb2": " p(x) = \\sum_{k=-K}^K c_k e^{ikx}, \\, ", "515e9c39a11963c2fbda9bff41e18920": " \\nabla _{\\mu} ", "515ead0097f203e83801be75f2aa9648": "\n\\begin{cases} \n n & \\mbox{if } k = 0 \\\\[5pt]\n \\displaystyle\\frac{n + 1}{k + 1} & \\mbox{if } 1 \\le k \\le n.\n\\end{cases}\n", "515eb0876e7f6fd55fc418699cd50da1": "n=0,\\pm 1,\\pm 2,\\dots", "515f52cb1052a6a9a65e567d7e390f91": "\\boldsymbol{N}=[0,0,1]", "515f5ae20a06e328cfdbda09a0d4eb5d": "\\mathbf{f}\\, \\mathbf{v}[\\mathbf{f}] = v = \\mathbf{f'}\\, \\mathbf{v}[\\mathbf{f'}].", "515f73d732de531e07839e4bf4536ece": "|\\det M|>1", "5160188717e914c13394e963ee9fdc37": "{\\pi}", "51609dccdc32fd2bc30100ec26e83669": " \\frac{d}{d t}\\left(\\frac{\\partial \\mathcal{L} }{\\partial\\dot{q}_j}\\right) - \\frac{\\partial \\mathcal{L}}{\\partial q_j} = 0\\,.", "5160a5f716659e5fdda9f95d6ba72a27": "{h_{\\mathrm{f}}=f\\frac{l}{d}\\frac{V^2}{2\\,g}};", "5160b05723173df94c9ba15ffb16804f": "\\{\\sigma_a, \\sigma_b\\} = 2 \\delta_{a b}\\,I.", "51611070cc09fb6421e447f3b540fe46": "\n\\begin{align}\n\\operatorname{gd}^{-1}\\,x & = \\int_0^x\\frac{\\mathrm{d}t}{\\cos t} \\\\[8pt]\n& = \\ln\\left| \\frac{1 + \\sin x}{\\cos x} \\right| = \\tfrac12\\ln \\left| \\frac{1 + \\sin x}{1 - \\sin x} \\right| \\\\[8pt]\n& = \\ln\\left| \\tan x +\\sec x \\right| = \\ln \\left| \\tan\\left(\\tfrac14\\pi + \\tfrac12x\\right) \\right| \\\\[8pt]\n& = \\mathrm{artanh}\\,(\\sin x) = \\mathrm{arsinh}\\,(\\tan x).\n\\end{align}\n", "5161a5035991ac61b0660560255023d1": "\\sigma\\in{\\mathfrak G}", "5161c0ff27080aeb17689dfe6d139139": "q=\\sqrt{\\frac{(ab+cd)(ac+bd)-2abcd(\\cos{A}+\\cos{C})}{ad+bc}}.", "5161cbea60eb3b099274a47fb7211f2e": "\\Delta{}E = W + Q ", "5161eca1fd90db4b19e437bdaba5343a": "\\rho _\\infty ", "5162152980bab490b277a567c737f189": "| (\\mathbb{Z}/n\\mathbb{Z})^\\times|=\\varphi(n).", "516228bd7ee7fcebc61d3057509bf48d": "(R_2 - R_1)/c", "51624937bb7dc2318ce2b0b00031e1a5": "n/L", "5162df21a1bcd84296d0ce2896f02a0e": "\\eta= \\left(\\frac{p}{p_0}\\right)\\left(\\frac{T_0}{T}\\right)\\, {\\rm amg}", "5162efdc466bada12a85806438b866da": "(-g)(t_{LL}^{\\mu \\nu} + \\frac{c^4\\Lambda g^{\\mu \\nu}}{8\\pi G}) = \\frac{c^4}{16\\pi G}((\\sqrt{-g}g^{\\mu \\nu}),_{\\alpha }(\\sqrt{-g}g^{\\alpha \\beta}),_{\\beta}- ", "51630ca8431ab13d0c2a60c4a5c355a8": " x^{2/3} = \\sqrt[3]{x^2} \\text{ etc.} ", "51631a7e1dfb29aa9d618f3c00874cda": "y'(\\phi)=1", "51633bbb92bc5945a76702ad2ed6a915": "X_\\delta", "5163a3f1704c756833c6134f40d62b0f": "g_m(z)", "5163aa51c95e23fa71a5c2760581e085": "2 (\\ell w + \\ell h + w h)", "5163c2b7ce135543c0f34a1d81726519": "\\text{Gain}_i(\\sigma^*,\\cdot)", "5163c9439bba3e4a231d9cf9fbfa55b1": "0 \\to \\mathcal{S}^{TOP} (\\mathbb{C} P^n) \\to \\mathcal{N}^{TOP} (\\mathbb{C} P^n) \\to L_{2n}(1) \\to 0", "5163efed9535b14956944090b231fbad": "\\eta_{D=2}=\\frac{1}{4}", "5163f0550d417d81ea76fa42d889fb8d": "C_n = I_n - \\tfrac{1}{n}\\mathbb{O} ", "51645a073b961ceb2076af7d159eab6b": "0^{(\\lambda_n)}", "516461c325bb282713e8ab0ab633bd8d": "\\exists x . \\Phi(x,y_1,\\ldots,y_n)", "516479b6ff46ea9be113d1a06c63abb4": "{X^\\mu}_{,\\nu} = 0 \\,", "5164fce2ab44114eb97bae3a534ee399": "\n \\hat{\\mathbf{b}}^1 = \\mathbf{e}_r ~;~~\\hat{\\mathbf{b}}^2 = \\mathbf{e}_\\theta ~;~~\\hat{\\mathbf{b}}^3 = \\mathbf{e}_z\n ", "5165034925e5d2ebd214731911058791": "V^{1} \\,", "5165268024282e0651b479e7e75b57c9": "1 - (1 - \\alpha_1)^n", "5165aa0087217e607b46f8384aca5025": "q = ae({i}\\times{i}) + af({i}\\times{j}) + ag({i}\\times{k}) + be({j}\\times{i}) + bf({j}\\times{j}) + bg({j}\\times{k}) + ce({k}\\times{i}) + cf({k}\\times{j}) + cg({k}\\times{k})", "5165b70a61907d7fda8822d9fb6e8297": "\\sigma_w", "51662c9d3fe57ce77fd7041fc64b36cf": "T = \\coprod_{i\\in I}U_i \\times F = \\{(i,x,y) : i\\in I, x\\in U_i, y\\in F\\}", "51663f39d2eda75b41aa5506087273ff": "a_0=v_{x\\infty}-iv_{y\\infty}\\,", "5166b497563f6533302b7b1c601699f8": "f{\\circ}id_{\\tau_1}=f", "5166c157aadf9c36e6c67fb3f29ed1bb": "H^2(\\mathbb{D},\\mathbb{C}).", "5166d32792047403c31a55c454bb0ddd": "\\eta = \\frac{P}{\\dot Q_H}.", "5166e9bdff1acc58bf9918e795510ff3": "\\begin{pmatrix}\n\\mathbf{J}_{\\mathrm e} \\\\\n\\mathbf{J}_{\\mathrm m}\n\\end{pmatrix}=\\begin{pmatrix}\n\\cos \\xi & -\\sin \\xi \\\\\n\\sin \\xi & \\cos \\xi \\\\\n\\end{pmatrix}\\begin{pmatrix}\n\\mathbf{J}_{\\mathrm e}' \\\\\n\\mathbf{J}_{\\mathrm m}'\n\\end{pmatrix}", "5166ec8d7ff692fe9a72d4a0fafc8a9b": "|a| \\ge b \\iff a \\le -b\\ ", "51671a98f1432495bb59491222f13121": "\\mathrm{FWHM} = 2 \\gamma ", "51672861d65b16ed67af41315bd87d33": "\\bold{u}\\cdot\\bold{u} = |\\bold{u}|^2 = 1", "51672f4b3abaf583cc1690063ac3e3d5": "\\xi^R_t = L_{\\exp tX}.", "51677320c5fbdf1c7cbac7de51555fed": "\\displaystyle -2\\frac{\\sin(\\pi\\alpha/2)\\Gamma(\\alpha+1)}{|\\nu|^{\\alpha+1}} ", "516791991031c38723287a06b75c5884": "\\sgn(0)^2 =0~", "5167be0d2c4b5f2d04397ee5d6ef0cf2": "\\int_{a}^{b} f(x)\\,dx = \\lim_{\\epsilon \\to 0} \\int_{a+\\epsilon}^{b} f(x)\\,dx", "5168122d20270b308c2ea9635b99b578": "x_0 \\in \\mathbf{F}_{p^2}", "51686fa6cc764b2c3fec430f022c7692": "\\mbox{median}=\\kappa\\,2^{1/\\theta}.\\,", "5168820eeaae3967e2556346c1c7fe16": "\\tilde{\\gamma}(0) = e", "5168fa53ce7749eeca0e8c1bb97c8b19": "W^{1,2}_0(\\Omega)", "51692a9ace4f907bc6ca442afc8de28b": "S({\\mathcal N},\\rho)", "51694e7b29bb2b371abb960a796c070a": "\\sigma_v = \\sigma_y = \\sqrt{3J_2} ", "516952dacf12c81080a77983fb3e28e3": "y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3.", "51695731a9aa69064e234a697a8bbc58": "f = H_p(t_{\\rm CMB})/H_p(t_0)", "5169a86701b7cf95b832179d37abfa96": "\\lambda = 0.5 \\mu m", "5169edc56a26bd1c9a195fc024137a5c": " = \\frac {A_0} {(1 + \\beta A_0) \\left(1+j \\frac {f} {(1+ \\beta A_0) f_C } \\right)} \n\\ . \n", "516a0031bdd1990ee1913e7407b27f2a": "(\\mu_{ab}^{(c)}(t))", "516a10563fb8b1524f1089f8e0ebc0bd": "k=(p+\\epsilon) n", "516a216c2d29bb9e713badd02f47c15a": "\\alpha^{1/2}", "516a470048fdf4a8a3140cfc4d3d974b": " {\\tilde H} = \\oplus_1 ^n H.", "516a54a1e32a8069424f488cb34eaac4": "\\alpha_{111} > 0", "516a7861c667ab267fb6e30b59638560": " \\scriptstyle \\;\\! \\theta", "516ae4177aae26005297e02429daab88": "k= \\, ", "516afa39ee97a9eb57d31d81deb09116": " P(x_i) = \\frac{1}{Z} \\int_{j \\ne i} \\exp(-1/2x^TAx + b^Tx)\\,dx_j", "516b5af0cea39fac91df94d700e4ff2d": "\\color{Orchid}\\text{Orchid}", "516b8f017f68123493fbadd5b1eda5c0": " U^R = \\left( \\begin{array}{ccccccc} 0 & -1 & & & & & \\\\ 1 & 0 & & & & & \\\\ & & 0 & -1 & & & \\\\ & & 1 & 0 & & & \\\\ & & & & \\ddots & & \\\\ & & & & & 0& -1\\\\ & & & & & 1 & 0 \\end{array} \\right) U^T \\left( \\begin{array}{ccccccc} 0 & 1 & & & & & \\\\ -1 & 0 & & & & & \\\\ & & 0 & 1 & & & \\\\ & & -1 & 0 & & & \\\\ & & & & \\ddots & & \\\\ & & & & & 0& 1\\\\ & & & & & -1 & 0 \\end{array} \\right)~. ", "516c0cab4994f7103a8e32f6e8a7e591": " (2 x dx ) (x^2+y^2) + x^2 (2xdx + 2y dy) - a^2 2y dy = 0 ", "516c159e209bf21f5b4f0e1c77cd8c1d": " {_2\\text{F}_1}(a,b;c;z) = \\sum_{k=0}^\\infty[(a)_k(b)_k/(c)_k]z^k/k!", "516cb5980b239552b7725584900acb17": " P\\left[ \\sum_{ i = 1 }^n \\frac{ ( X_i - \\mu_i )^2 }{ \\sigma_i^2 t_i^2 } \\ge k^2 \\right] \\le \\frac{ 1 }{ k^2 } \\sum_{ i = 1 }^n \\frac{ 1 }{ t_i^2 } ", "516cbb2f52c7555a47655178452a6c8c": "c_r", "516cc5bb4959ebd1b37225f8f93c387c": "R={R_{abc}}^d\\in {V_{abc}}^d = V^*\\otimes V^*\\otimes V^*\\otimes V,", "516ccb6e4b94c1b24484ecd8358245e5": "\n\\mathbf{W}\\ \\overset{\\underset{\\mathrm{def}}{}}{=}\\begin{bmatrix}\n58 && 26 \\\\\n26 && 52\n\\end{bmatrix}\n", "516d1146768650d435eb7208cd3e697d": "\\chi_\\text{mass} = \\chi_v/\\rho", "516d5dbc84d0be27649e2baa7c01c704": "\\textstyle \\delta_\\eta>0", "516d6e3587057b614764056128974f6a": "x^3y''-xy'+2y=0.", "516ddca186eaece4a6fff9e2c42676dc": "10\\uparrow\\uparrow 257 < \\text{mega} < 10\\uparrow\\uparrow 258", "516dfd7b27bfcf88adda34a6e5338203": "\\alpha \\in M_\\lambda[X]", "516e1efc06cab76e9a40c413d51e8266": "D_N({\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1})=0", "516e451af1993e69e406a295c44896c4": "\\sum_{i=1}^n{\\frac{2 |X_i-\\mu|}{\\beta}} \\sim \\chi^2(2n)\\,", "516e8d3306ce2af16f603f718e864440": "\\scriptstyle W", "516e8e2bea81851d22ddbd8ec1c32e67": " (\\phi \\lor \\exists x \\psi) \\rightarrow \\forall z \\rho", "516f0ae7d7eb319144fd414210095d47": " \\alpha (w)=a_1|w|^{1-\\gamma} \\quad (1.1)", "516f0d7b51271e0a304f66184caba77d": "-\\ln(-4a\\pi)/2", "516f6056046d122925995a3bfd34b928": "Z_p\\,", "516f60b7120c21242fe5a9f684a8c096": "\\frac{\\sigma(n)}{n^{1+\\varepsilon}}\\geq\\frac{\\sigma(k)}{k^{1+\\varepsilon}}", "516f6370210ff07b7c8271f9bad57c12": "\\mu = p", "516f800ffa03e16c760698e683385cf7": "dX=-d\\left(\\int_t^T f\\left(t,s\\right) d s\\right)=-A\\left(t,T\\right)dt-\\tau\\left(t,T\\right)dW\\left(t\\right),", "516f9ce5db596b124239ac37fe1fcdfb": "196884 q^1.", "516fa9f9011f81ed902b0f545221a8f5": "\\textstyle U(z)", "516fb7b9ca7f944d59ae0edda29e0b9a": "f({{v}_{1}}^{\\prime },{{v}_{2}}^{\\prime })f({{v}_{1}},{{v}_{2}})\\ge f({{v}_{1}},{{v}_{2}}^{\\prime })f({{v}_{1}}^{\\prime },{{v}_{2}})", "516ff39324a3441f1787576fd7ebbc03": "\\lambda_{\\rm peak} = \\frac{2.898\\times 10^-3 \\ \\mathrm{K} \\cdot \\mathrm{nm}}{305 \\ \\mathrm{K}} = 9.50 \\ \\mu\\mathrm{m}.", "516ffc4ffa0bf464b41459ef4b55de3c": "j+k = e_i", "517048a58ec9030c13fad442000eb860": "\\ r", "51709975e96688020260d67a05c77164": "a_{13}+b_{13}", "5170bfee58ac20ce202d70f4ce84e95b": "b: X \\to \\{0, 1\\}", "5170c235b7ed29a2c9e7c63192d2c1a2": "\\zeta_n^{s-r}\\gamma a_i \\equiv \\zeta_n^s a_p \\equiv \\gamma a_j\\pmod{\\mathfrak{p}}", "5170c81e7b8f98966d88420bedddebcb": " E_\\text{t} =\\tfrac{1}{2} mv^2 ", "5170e27ea7e4f19f674acfc021698552": "\\frac1a\\frac{ds}{d\\sigma} = \\frac{d\\lambda}{d\\omega}\n= \\sqrt{1-e^2\\cos^2\\beta}.", "5170e341c84d01f5bfae7243b3f312cb": "\\{X_i\\}", "517101e9a3431889dc46615b11fbef48": "\\mathbf{L} = m \\bold{r}\\times \\left(\\bold{\\hat{e}}_r \\frac{\\mathrm{d}^2 r}{\\mathrm{d}t^2} + r \\omega \\bold{\\hat{e}}_\\theta \\right) ", "517132379c8738e3c539964a0dfcc9c8": "\\displaystyle f(x) \\,", "51716233d71f74a9bc66aed68304b22c": "100Mcalories(meat.fac.intake)/200kcalories(calories/truckload)=500truckloads", "517170b8f1a0e8be88206a3b0ad5111b": " y = \\textstyle {3 \\over 2} {a^{1/2}}(x - a) + f(a). ", "51718398f14c2c7248fa166b1c749400": "a, b", "5171844660de430bb1e641fa864ac8c7": "S_{200}", "5171aba5e33e4a16c3617d0f8f71a846": "\\scriptstyle S_n \\mathcal{C}", "5171b2077c7fbae27e5ec513a075c7ab": "A\\cap\\overline{B}", "5171fd77b067f8fc5460863c52370f7b": "\\mathbb{Q}(\\sqrt{-a})", "517212f16cc116e22df7435e360bff4a": "\\epsilon \\circ \\nabla = \\nabla_0 \\circ (\\epsilon \\otimes \\epsilon) : (B \\otimes B) \\to K", "5172ed323caf3447b8f0f66c45bceeea": "\\lambda^4 - 2\\lambda^3 + 3\\lambda^2 - 2\\lambda + 1 = \\lambda^4 - 2\\lambda^3 + 3\\lambda^2 - 2\\lambda + 1\\,", "517318d5ca71043886de24c40fd5379d": "150\\,\\text{lb}_m \\cdot 0.249\\,\\text{g} = 37.3\\,\\text{lb}_F", "51734af7fafa296d5838e05b32434137": "\\sum_{i=0}^{n-1} i a^i = \\frac{a-na^n+(n-1)a^{n+1}}{(1-a)^2}", "51734e7c1116bbfd30555fbba4ccb1a6": "h(\\phi + \\pi) = -h(\\phi)", "51737747819a89793d32f34f662be13a": "\\; W \\pi_1 = \\pi_2 W.", "51737e4d3076b908b3855eba02151214": "e \\in V", "517397ace3898e057f84db50bdb3125e": " F(g) ={1\\over |W|}\\int_{{\\mathfrak a}^*} \\tilde{F}(\\lambda)\\Phi_\\lambda(g) |d(\\lambda)|^2 \\, d\\,\\lambda=\n\\int_{{\\mathfrak a}^*_+} \\tilde{F}(\\lambda)\\Phi_\\lambda(g) |d(\\lambda)|^2 \\, d\\,\\lambda,", "5173ace8e97670c1962dc019e575a9e2": "(x\\!:\\!\\sigma, t\\!:\\!\\tau)", "51741133cbe73714f0527becfa95b996": "(\\forall i, confidence_{i} \\in \\{0, 1, 2\\})", "51744fd85df47b3a290420121c24727a": "{-p \\choose p} = \\frac{(-1)^p}2{2p \\choose p}.", "517459a8c03b9ae7bfba08f564e73555": "r \\cdot s = s \\cdot r", "517464f9599278cd552c12bf59a3ed43": "A =\n\\pi r^2 \\cdot \\frac{\\theta}{2 \\pi} =\n\\frac{r^2 \\theta}{2}\n", "5174c675570b5f82324d4d7a04acac68": "(3/2)(p/2) = \\tfrac 3 4 p", "5174ccca1bde096616d3523e5c1d04bb": " \\mathrm{Oh} = 1/\\sqrt{\\mathrm{La}}", "5174d5fa1dc1f6a475828524aa327feb": " \\sigma \\colon \\mathbb{R}^n\\times\\mathbb{N} \\to \\mathbb{R} ", "517516abe77083ca3fe6555581902113": " r(t)=f(t,t)", "51751fe9171de28396804693c732c21a": "w:j\\to k", "51752f796f28969286c6f376646180f2": "n=1024, k=524, t=50", "5175553a75dacba648d15285caad2204": "T'=QTQ^T", "5175868f076487ffc37b460a9d887954": "n_{1s}", "51766fef5922d5aae1a0caeb7a55d0b7": "d\\tan(x)/dx = 1/\\cos^2(x)", "51768253e60de783e10baa54cd8c2685": " \\int_G |f(x)|^2 \\ d \\mu(x) = \\int_{\\widehat{G}} |\\widehat{f}(\\chi)|^2 \\ d \\nu(\\chi).", "5177015aae17a652afa8b3aa229f6481": "T^{\\hat{\\rho}}_{\\hat{\\mu}\\hat{\\nu}} = 0", "517710901afdec24682e2d20e3b01fed": " f(x)=\\sum_{k=0}^\\infty \\frac{\\phi(k)}{k!}(-x)^k \\!", "51771c9b1c91de4cda4dcd470d710482": "k_1, ... ,k_k", "51773024f1b5d2ecb4db66318e64f944": "\\forall \\rho, \\sigma. \\;\\ \\rho \\subseteq \\sigma \\in \\Delta \\Rightarrow \\rho \\in \\Delta", "51773753a224abc62d367d7ac892aad8": "A(t_n) = e^{-i t H_\\mathcal{S}} A e^{+i t H_\\mathcal{S}}.", "51775ece7879b5085798df6ba819b78d": "X_0 = T_1 + T_2 D \\approx \\frac{1}{D} \\,", "51776f21de5d14a6c7d10ea6d1c1a064": "\\ E_{-} - E_{+} = \\frac{2(CB^2 - J_{ex})}{1-B^4} ", "5177ad57ee6a389e80e64e0fcb0e70aa": " A^* = -N \\ln \\left[ 1 - A / N \\right] / k", "5177b1e4eb5096725263592dade1f0e9": "I(X_1;X_2)", "5177cb7ae94d23fc958f10b65fdf2aa1": "\\begin{align}\n\\delta \\psi_i &= \\delta\\lambda D_i c \\\\\n\\delta A_\\mu &= \\delta\\lambda D_\\mu c \\\\\n\\delta c &= - \\delta\\lambda \\tfrac{g}{2} [c, c] \\\\\n\\delta \\bar{c} &= i \\delta\\lambda B \\\\\n\\delta B &= 0\n\\end{align}", "51781ff025523e1c6d17366f27dd0d27": "A\\to(B\\to A)", "51783f538a01c61aa6933657a7afe6ae": "C\\ell_{0,3}(\\mathbb{R}) = \\mathbb{H} \\otimes \\mathbb{D}", "51785d24df3ef00943eac29632a30ccd": "q(\\tilde{x},\\tilde{u}\\mid m)", "5178de312c453fa004e115aea63e99ae": "\\sqrt{\\mathcal{I}(p)}= \\frac{\\sqrt{n}}{\\sqrt{p(1-p)}}.", "517921f924a219ec0ec90920a4a9b906": "f(s)", "51792718155d512c39f51b5ef2815e3f": " p(X, A|\\theta_{bg}) ", "51792a3bb060c8f3207f9fba42bb4fce": "S^1 a S^1", "51796328fea925abdad00edfa3757300": "\\mathbf{C}(Km',c)\\cdot Tm", "51798fc385177d8ebd7a15798a1d1422": "\\frac{\\omega - \\omega_0}{\\omega_0}\\thickapprox \\frac{\\Delta W_m - \\Delta W_e}{W_m + W_e}\\,", "51799a337d476be0f2a3110dcd06f342": "\\{\\sqrt{h_i(\\vec{x},t)}\\}", "517a4b4d28865df776b5463d82f0246e": " \\frac {df}{dE} = \\frac {1}{m_w^*} \\frac {dk}{dE} \\tan(\\frac {k l_w} {2}) + \\frac {k} {m_w^*} \\sec^2(\\frac {k l_w} {2}) \\times \\frac {l_w} {2} \\frac {dk} {dE} - \\frac {1}{m_b^*} \\frac {d \\kappa} {dE} \\quad \\quad (9-1)", "517a7561a2ac81a82f958fb41ede6790": " \\bigg[\\int_X^\\oplus A_x d\\mu(x)\\bigg]' = \\int_X^\\oplus A'_x d\\mu(x). ", "517a7856b76700f3f50ef11ab8dae825": "\\operatorname{Im}\\,(z) = \\tfrac{1}{2i}(z-\\bar{z}). \\,", "517a879a1aaa8e95d39fbe9977cc7bac": "\\delta^{\\beta_1}\\gamma_1 + \\ldots + \\delta^{\\beta_k}\\gamma_k", "517ad580e87e3ff39765a4969851abaa": "\\ C_i ", "517ad71800c3d599736012184a7519d8": "\\scriptstyle f^\\ast(n)=\\lfloor\\sqrt{n-1}\\rfloor", "517aef2b93b5a04fd0af9d2e1cf04c9f": "{\\mathbf{}}\\tau^2(t)=\\tau(t).", "517b531ee5cec5e4c6cd32285960ccc8": "\\gamma : I \\to \\mathbb{R}^2", "517bbbcdf9187d4cf2072170b6fd4f0f": "E z \\Big) \\leq 2e^{-2z^2}.\n ", "51968e891904f72d46b1ed0731e8aa66": "z=z(t)", "5196e256ea8160ade83844712066bfcc": "A \\leftrightarrow B", "519743b46b61207f77da30a1ee0ba0c1": "\\eta = K\\dot{\\gamma}^{n - 1}", "51975788eac31a0e84f8b0c3168c92f0": "\\Lambda = {L(a|X=x) \\over L(b|X=x)} = {P(X=x|a) \\over P(X=x|b)}", "5197815716b7e72a342042df4db8b31f": "\n\\mu_i = \\frac{b^*_i A b_i}{b^*_i b_i}.\n", "5197c3e003ed636ed0d12bdb14990f1c": "\\epsilon_r\\approx10-100", "51982da463c49e88ce0a3e979cca6ebe": " d(d-2) ", "5198581373d42bdb3d805e97e8b4eaa7": "\nv=\\sqrt{\\frac{B}{\\rho_0}}, \\,\n", "5198616d7cf5af2d0667c9610eaf57f1": "\\textstyle [x]", "51986a016e994aa649c5e0da841e43ec": " \\chi(n) = \\left(\\frac{n}{q}\\right),", "5198a8aeb57bd4881d94796aeff6f427": "\\frac{\\partial Y}{\\partial \\beta} ", "5198e27b7cfc0dfe56a51b109a7c3b56": "P(S|N,n,s) \\propto {1 \\over S(N-S)} {S \\choose s}{N-S \\choose n-s}\n\\propto {S!(N-S)! \\over S(N-S)(S-s)!(N-S-[n-s])!}\n", "5198e4ec29abe2c43798d4bd85fa504e": "\\nabla \\times ( \\nabla \\phi ) = \\vec{0}", "51990fc081c7a08e238f5b16ccf70597": "A_{\\text{c}}", "51992e8dc57d167c8e3a3923e3bc2879": "Bird(Bee)", "519938a7ed5ebe85b5c69bc283e611ce": "\\tilde f \\colon f^{*}E \\to E", "5199445f717e040375b89a7426ecd2e3": "p\\geq (b/a)z+(c/a)y", "51997e99050065257036f2f3f8f395af": " H_{x} = x \\sum_{k=1}^\\infty \\frac{1}{k(x+k)}\\, .", "51999f8192d36af3a519560759a0ef40": "\\,\\!E[n]", "5199d8f02e908b63410e9252b2a587fe": "\\frac1{b_n}\\sum_{k=1}^n b_k x_k = S_n - \\frac1{b_n}\\sum_{k=1}^{n-1}(b_{k+1} - b_k)S_k", "519a0d64a35cc946cdd087adb35932a4": "n_1 = \\sqrt{n_0 n_S}", "519a19c563c2be177c1565ea617d0adf": " \\left( A \\right)", "519a4013999f58ffff422a807bd65129": "\\big\\{|0\\rangle_{1}\\otimes|0\\rangle_{2}\\big\\}, \\big\\{|1\\rangle_{1}\\otimes|1\\rangle_{2}\\big\\}", "519a7af745486ab60a5370cef3af34d5": "\\textstyle n=1,2,\\dots", "519a7ccae50c78c6dbc87e272307db46": "\\hat{\\mu} = \\bar{x}", "519ab1d72bb87bb6d985d81f2c198194": "*\\infty\\infty", "519adad070d17710472fe72fae71b7b1": "\n| \\Psi(0) \\rangle = \\sum_{\\alpha=1}^{D} c_{\\alpha} | E_{\\alpha} \\rangle,\n", "519af01fab1a6f18385446c1ff02df3b": "b_{41}^{8\\text{v}8\\text{v}} = -81.93 \\,\\text{meV} \\cdot \\text{nm}^3", "519b1081e4190796b8ca317c531e348b": "K_b=\\mathrm {1.8 \\times 10^{-9}} = {x \\times x \\over .20-x}", "519b546c6d48778d5a3b424119ab09a3": "\n\\begin{array}{rl}\nW_R &= 0.2126 \\\\\nW_B &= 0.0722 \\\\\n\\end{array}\n", "519b61373ba82156351e2eafddb8966f": "|\\mathbf{F}_{\\mathrm{fict}}| = m \\omega^2 R \\ . ", "519b63bab25baea386256e153b0ed2d4": "\\lim_{n\\to\\infty}\\frac{1}{n}\\log q^n(x^n) = -D_{\\mathrm{KL}}(p^*||q).", "519b6956c0dcaad1f514ccd25c669947": "f_{mt}", "519bbabcd77772f7a52eabe46c4c34b9": "\\hat{\\rho}_{retr}^{[n]}", "519bc397d6a5ee84f46f83dbe037bea6": "\\lim_{n\\to\\infty} x_n = \\infty,", "519bd86b2c97a81a3c4ea09c8e213488": "\\nu \\equiv_1 c(\\nu).", "519c0597784367cca027de054c5d693e": "\\mathbf{J}^{(e)}=\\sigma\\mathbf{E}^{(e)}.", "519c114ce4264fb1ae46fcfd7629abf6": "(f_{[\\frac{d}{2}]},\\ldots,f_{d-1})", "519c27ace812f13888d4dfab08d2952b": "\\lambda \\|B\\|", "519d2c2350a12552ff40327d0643f427": "W_{ijkl}=W_{klij}=-W_{jikl}=W_{ijlk}", "519d7ea2b7dd6f397096edbe206004c5": "\n X_n\\ \\xrightarrow{L^r}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{L^s}\\ X,\n ", "519d887345a984593835c8424c35abed": "\\partial_t \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_t \\psi )} \\right) = \\partial_t \\left( \\frac{1}{2} \\partial_x \\psi \\right) \\,", "519dd6a8093e30809331d805209d183b": "A_p = C_0 h_{max}^2 + C_1 h_{max}.", "519e33a24000a5bb8410fc88ab0183b6": " w(x) = \\frac{1}{\\sqrt{2\\pi \\langle x^2 \\rangle}} \\exp\\left(-\\frac{x^2}{2 \\langle x^2 \\rangle} \\right).", "519e6098c36a1b3149d6e05e78d78db1": "\nr^2 =\n\\left (\\vec x_1 - \\vec x_2 \\right )^2\n", "519eaca043da41602eb4d7fb4d3f0c60": " \\frac{dq(t)}{dt} = \\frac{d}{dt} \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\rho(\\mathbf{r},t) dV = \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\frac{\\partial \\rho(\\mathbf{r},t) }{\\partial t} dV .", "519ec1a74f256730e992f12a95cddb3d": "f \\in F \\subset K", "519ecb41a8c8671aecd62d00f4987601": "\n (2.2)\\quad\n E\\left(f'(Y)-Yf(Y)\\right) = 0\\text{ for all } f\\in C_b^1 \\quad \\iff \\quad Y \\text{ has standard normal distribution.}\n", "519f109026fb28bb6d9c470a2ff5d8b2": "X_3, X_4", "519f8c63a650b5993f804d9281c63058": "t = t_0", "51a00a634fd5e0ad8b398a5d65c9cd6d": "2^{N}", "51a0141161309f3751372a8b9256c967": "r\\arctan\\left(\\frac{y}{x}\\right) = \\frac{1}{1}\\cdot\\frac{ry}{x} -\\frac{1}{3}\\cdot\\frac{ry^3}{x^3} + \\frac{1}{5}\\cdot\\frac{ry^5}{x^5} - \\cdots ,\\text{ where }y/x \\leq 1. ", "51a097578f8fb7fd1e7246ee55017c4d": "C_T", "51a0b89f056ac92b48a952300c8ca265": "K = h \\cdot S_a (Y + \\bar{Y}B)", "51a0fdebbe87b4a0b9ea181a012ff1b1": "(a) \\qquad |K(x,y)| \\leq \\frac{C}{|x-y|^n}", "51a170dc6f313dde5b5d329e37cd86ab": "\\mathbf{V_C}", "51a19b8c09db28b0cbaea019913dcf6e": "H^1(\\mathbf{P}^\\infty(\\mathbf{R}); \\mathbf{Z}/2\\mathbf{Z}) = \\mathbf{Z}/2\\mathbf{Z}", "51a1eb2ad0c2bef8772e5fd83ddd45e5": "L\\ =\\ \\frac{r \\cdot R_o}{R_s - R_o}", "51a214189bb02ab2bac1a6972180c171": "(w,x,y,z_1,z_2)\\,", "51a26b4bd06323baac7bfd0f771aec86": "L(C_n)", "51a2c0a7d06fe578f33774eec2df8eb9": "\\epsilon\\; \\mbox{ and }\\; \\delta", "51a3044e979a25090e8100fc79925b35": "\\pi_1(X,x_0),", "51a31bc114b849ce30f29e6e1d9a25b0": "M_y, M_z, I_y, I_z, I_{yz}", "51a35aea7ce0fdfccc554682744e0b55": "H_2(\\mathrm{A}_n,\\mathbf{Z})=0", "51a3a7922ee67d6c4b33e041e133b08b": "f(t)=\\begin{cases}\n\\sum_{x=1}^N[g(x)]^n & t=0 \\\\\n\\sum_{x=1}^{N-t}\\left(\n\\begin{alignat}{2} &[G(x+t)-G(x-1)]^n\\\\\n&-[G(x+t)-G(x)]^n\\\\\n&-[G(x+t-1)-G(x-1)]^n\\\\\n&+[G(x+t-1)-G(x)]^n \n\\end{alignat} \\right)& t=1,2,3\\ldots,N-1.\\\\\n\\end{cases}", "51a4579bf090fc78dfbcbfca621ff925": "=N\\left(\\delta^2- O\\left(\\dfrac{\\lambda}{d}\\right)\\right)", "51a49a511b6d1f2e6add6759e25a47d3": "G \\circ H := (V, E)", "51a4c585852d944626399473610e49f0": "u=5", "51a4c7c0f5bbb6fd5d585f4c14d2ca6d": "\\angle ACB=\\angle ACD +\\angle DCB=\\angle ACD+(90^\\circ-\\angle ACD)=90^\\circ", "51a53a37b240615106406fb67e4332cb": "\\alpha \\mapsto \\varepsilon_\\alpha", "51a53ee2ab33874e48db29469637ee92": "R_x = {{R_2 \\cdot I_2 \\cdot I_3 \\cdot R_3}\\over{R_1 \\cdot I_1 \\cdot I_x}}", "51a5e4020d09516d50a173fd86e09e54": "\\varepsilon_0", "51a6103d8656748b3e0446adce14a396": "\\frac{-\\mathrm e^{-ikr}}{4 \\pi r}", "51a660780067834db63235c1776262a7": " y^2x^2 \\rightarrow xyxy ", "51a6b646822c16f4cd1471dc53837167": " (\\mathbb F_{l})^{n} ", "51a6c7277a0bad826b9c4052298b3a20": "\\bar{\\rho\\,\\!c}\\left ( \\frac{\\part T}{\\part t} \\right )\\ = \\bar{\\lambda} \\left ( \\frac{\\part^2T}{\\part t^2} + \\frac{1}{r} \\frac{\\part T}{\\part r} \\right ) + \\frac{1}{\\bar{\\sigma}} \\left ( \\frac{I}{2{\\pi}Lr} \\right )^2 - \\underbrace{ \\rho_w\\,\\!c_w \\frac{Q}{2{\\pi}Lr} \\frac{1}{r} \\frac{\\part T}{\\part r} }_{Convection}", "51a6e1251b28934fcb41d10188226399": "\\left\\{ y~\\backepsilon~(x\\succcurlyeq y)\\land\\lnot(y\\succcurlyeq x)\\right\\}", "51a783a8e7f8f9e7367d968ebe65a3be": " \\frac{d\\phi}{d\\xi}=\\theta^n\\xi^2 ", "51a784f2c155931aa5cf3919b7c65203": "M\\left(0,\\alpha,x_\\mathrm{m}\\right)=1.", "51a791171243ea679e6f7203149783d4": "q(i) = | Q_{i+\\Delta} \\setminus Q_{i-\\Delta} | / |Q_i|", "51a7ba5ca5575eda48b90612e15b90f1": " L_\\text{int} = -q\\Phi + {q\\over c} \\mathbf u \\cdot \\mathbf A, ", "51a7c9bd37c5e5f1e89c06300c6665f6": "(\\mathcal{S}_n(f))_{n\\in\\mathbb{N}}", "51a81a2c657024abc71ffe90e9b897e0": "\\mu = \n{1\\over (j+1)}\\langle(l,s),j,m_j=j|\\overrightarrow{\\mu}\\cdot \\overrightarrow{j}|(l,s),j,m_j=j\\rangle", "51a86fcd41fa96dc54edae6f149033af": "d_{1}^{k}", "51a9177cdc942d841f3cbedf3470e7f9": "\\lambda = 500\\mbox{ nm},\\, d = 1\\mbox{ cm},\\, \\Theta = 45 ^\\circ", "51a92df02bc90f06c9b992c1a48ecad4": " d = \\frac{v \\cos \\theta}{g} \\left( v \\sin \\theta + \\sqrt{(v \\sin \\theta)^2 + 2gy_0} \\right) ", "51a971fcbfe0489997f4a6044a2f1227": " T(a\\,dx\\wedge dy + b\\,dy\\wedge dz + c\\,dx\\wedge dz) = \\int_0^1 \\int_0^1 b(x,y,0)\\, dx \\, dy.", "51a976b7f93adc979571ff1c0dd53e51": "N_{Q(\\zeta_{nl})/Q(\\zeta_l)}(\\alpha_{nl}) = \\alpha_n^{F_l-1}", "51a9cfdc6f21d0191a2186f8f78b7e44": "\\lambda = (M + 102.9372 + C + 180) \\mod 360", "51a9fb520deb71f1c178355503050052": "[g,h]^{-1} = [h,g].", "51aa0a9210ad34f93ea63f8e96540dc7": " \\mathbf{[T]}^T \\mathbf{[A]} \\mathbf{[T]} ", "51aac3e007ed7055aa80fd1399a6796e": "| f |_{0,\\alpha}", "51ab0f11683e82d8f7b3a7c60eca3669": "\\tilde f_0()", "51ab116a98d8a89443735a51c147166f": "\\tilde\\epsilon", "51ab4fc7a19b838a7fbd56088270597f": "!n = (n - 1) (!(n-1) + !(n-2)).\\,", "51ab9678a5844dc2672ea9dfaba92adc": "A_+ \\equiv \\int_0^1 e(t) \\, dt ", "51ac3ff171d54e5f02cc292e96c7966e": " |\\psi\\rangle = {1\\over \\sqrt{2}} \\begin{pmatrix} 1 \\\\ \\pm i \\end{pmatrix} \\exp \\left ( i \\alpha_x \\right ) ", "51ac471555a055fd7d05325521b1a366": "\\Delta H_m", "51acab7eb93a5a6892bbf25f7ca1b2de": "dF_{O_2loop}", "51acb900b3e7c3cedadd96c3d3416382": "L_{p,w}(x) = \\frac{\\sum_{k=1}^n w_k\\cdot x_k^p}{\\sum_{k=1}^n w_k\\cdot x_k^{p-1}}.", "51acfc06ef00c9201667d31d1bf8ce14": " \\|a-b \\|_\\infty = \\max_i |a_i-b_i| ", "51acfddfb30b9491a99e258da910993c": "\\mathfrak{m}_x", "51ad1f924283e7d476623c75b1555020": " f: X \\rightarrow X", "51ad2b8ca0fbd69df88ae4231cb00fc9": "2^{127-1}", "51ad8a60063ec923b47b3eaaa708f571": "c_m(t) = c_m(0)\\exp[-\\textstyle\\int\\limits_{0}^{t}\\langle\\psi_m(t')|\\dot{\\psi_m}(t')\\rangle dt'] = c_m(0)e^{i\\gamma_m(t)},", "51ad959902df53038c7d5cab249ee1c8": "p^2 = N(p)", "51ada76d0be788f35eb37850bc508243": "Z^{32}_2", "51add596cf581cea0f4d483f901d7e8f": "[- \\frac{\\hbar^2}{2\\mu R^2} \\frac{\\partial}{\\partial R} (R^2 \\frac{\\partial}{\\partial R})+ \\frac{\\langle \\Phi_s|N^2|\\Phi_s \\rangle}{2\\mu R^2}+ E_s(R)-E]F_s(\\mathbf R) = 0 ", "51ae001d45cc34b4e1f50e77aad21011": "P(h(x)=z_1 \\land h(y)=z_2)= 1/m^2", "51ae6b464b8af942d1983e0c82132fc6": "(G_i,G_j)", "51aee7b1295945ecd22f6da7e26da686": "\n\\begin{align}\n \\mathbf{v}_{\\mathrm{rot}}&=([\\mathbf{k}]_{\\times}\\mathbf{v})\\sin\\theta\n +(\\mathbf{k}(\\mathbf{k}^\\mathsf{T} \\mathbf{v})-\\mathbf{v}(\\mathbf{k}\\cdot\\mathbf{k}))\n (1-\\cos\\theta)+\\mathbf{v}(\\mathbf{k}^\\mathsf{T} \\mathbf{k}) \\\\\n &=\\mathbf{v}+([\\mathbf{k}]_{\\times}\\mathbf{v})\\sin\\theta\n +([\\mathbf{k}]_{\\times}[\\mathbf{k}]_{\\times}\\mathbf{v})(1-\\cos\\theta).\n\\end{align}\n", "51aefc395729091e5aedd7466b38a948": "\\Sigma\\rightarrow[0,+\\infty]", "51af1bd2352010e4a524a67afe0b375e": "P = \\{1,\\ldots,k\\}", "51af5b626edf7940bf7ce4e8cd780cb7": "2y/Ec=dV/Vdp", "51af5ca1824ddad9f51e36850e953901": "\\sigma_3 = \\min \\left( \\lambda_1,\\lambda_2,\\lambda_3 \\right)\\,\\!", "51afbddd14a4eaed09c18e4ad7f060d6": " H_q(x)\\equiv_ \\text{def} -x\\cdot\\log_q{x \\over {q-1}}-(1-x)\\cdot\\log_q{(1-x)} ", "51b0451d2fd50da5af56c8c317b36f70": "t = \\frac{T}{\\sqrt{1 - \\frac{v^2}{c^2}}} \\,", "51b068de164d86eab5e4098737df89b8": "\\ C, \\ ", "51b07507354f4b620af27f3c0ab3544d": "N(\\alpha)=|N(a)|", "51b0794067f5eb28bd63e575b9245615": "e^{\\prime}\\,", "51b0fa4859fbed49be7af6b36d4a2b17": "M \\cap \\partial D^4 = \\partial M \\subset S^3", "51b10835145489c8a1178eb1e93ab4e5": "k[x_{ij}]", "51b11639a6a1ddf740457eeb641202dd": " \\chi_V= \\sum \\dim V_{\\lambda} e^{\\lambda}", "51b15e858852fda53907c48824ee8c8a": "N - C", "51b188512df1afdd1494aa9d358bb169": "\\overline{r}.", "51b1ba062bb8c16a4422ee1d5e99406b": "\\sim_{\\mathcal{B}} = \\{(x,y)\\in O \\times O : \\,\\parallel \\boldsymbol{\\phi}_{\\mathcal{B}}(x) - \\boldsymbol{\\phi}_{\\mathcal{B}}(y)\\parallel_{_2} = 0\\},", "51b1c997575d4817e907a8608c4a02ea": "\\mp", "51b212aefa7193bdc1a198c44aefaeab": "Cl^{\\le}_t", "51b21eacef2052990d3f5c1f8205cd56": "\\alpha_{nl}\\equiv\\alpha_n ", "51b287cf821749cb4f14cf2c93bb348c": "G=[G_{ij}]", "51b3110e3dcdf2471aee3a3c116c6e7d": "{\\mathcal I}", "51b328242d5061665d026ef4210eeaf1": "T^p_n (x) \\rightarrow x+n", "51b3457fb3d9546aec911fe3586a320d": "2^{n^c}", "51b365911c73bdad2e9a1c595d0f0d99": "x\\preceq y", "51b3b8f05897ee9c0bce7a5d8bbc07f6": "CS = \\int^{P_{\\mathit{max}}}_{P_{\\mathit{mkt}}} D(P)\\, dP, ", "51b3e631dbfd04f935a69e3bd3925501": "x_\\mathrm{med}", "51b3f7f83930bff4d4a3c4810934e8a5": "\n \\qquad \\frac{\\partial u}{\\partial t} + a \\frac{\\partial u}{\\partial x} = 0\n", "51b450215a1940c679e13a642a655088": "F = S \\frac {(1+i_d)} {(1+i_f)}", "51b48875c158da6a382192c4a90a67c5": "\\langle \\phi_n, \\phi_m \\rangle = \\sum_{i=0}^N w_i \\phi_n(x_i) \\overline{\\phi_m(x_i)} \\qquad\\qquad n,m = 0,\\ldots,N", "51b4c2a540e7da7cf4573eb17b7db575": "E_n(x) = \\frac{\\Gamma(n)\\left(\\Gamma\\left(\\frac{1}{n}\\right)-\\Gamma\\left(\\frac{1}{n},x^n\\right)\\right)}{\\sqrt\\pi},\n\\quad \\quad\nx>0.\\ ", "51b513fb4148d0cc7fda61927d977af6": "\\alpha_2=-2", "51b564e0808517b061ddb715d776f037": "\\frac{\\pi}{4} = 4 \\arctan\\frac{1}{5} - \\arctan\\frac{1}{239}", "51b57abe4d5f7d28d7cf0495be9274d0": "4x^2-8x+16 = A(x^2-4x+8)+(Bx+C)x", "51b58fbdb1c29733d63d79c596833683": " \\frac{dN}{dt} = r N \\left(1 - \\frac {N}{K} \\right)", "51b5a6657c1fb238a4bfc762bc1ef40f": " m_b ", "51b5c41be101c9cd3001ea5f307636bd": "W_v", "51b5d07711705d77374d54d8e47d0228": "{\\mathbb P}(X_1\\le x_1,\\ldots,X_n\\le x_n)=\\min_{i\\in\\{1,\\ldots,n\\}}{\\mathbb P}(X_i\\le x_i),\\qquad (x_1,\\ldots,x_n)\\in{\\mathbb R}^n.", "51b607ef6891c829cdce4c9a2a5f2df4": "O\\left(\\frac{n}{p}\\right)", "51b743cf0581d7b50952366fb8a7bfc4": " h=\\lambda(b-a)", "51b75fe47535ffb90bd065e73b8eab6d": "F((x,y),(a,b,c)) = x^2 + y^5 + ay + by^2 + cy^3.", "51b7aca91c5a5b70cb02855c010bf5fc": "E_n(x,\\alpha) = xE_{n-1}(x,\\alpha)-\\alpha E_{n-2}(x,\\alpha). \\, ", "51b7fe0a09c86b0256d4bffa477067d3": "\\sigma_{22}' = a_{21}^2\\sigma_{11}+a_{22}^2\\sigma_{22}+a_{23}^2\\sigma_{33}+2a_{21}a_{22}\\sigma_{12}+2a_{21}a_{23}\\sigma_{13}+2a_{22}a_{23}\\sigma_{23},", "51b8103956d6bfc89c1afee1560283e4": "[p_1]=[x]", "51b8d72cb95b7a886209f3eb2641beeb": "\\ R_f = \\frac{\\mbox{migration distance of substance}}{\\mbox{migration distance of solvent front}}", "51b8fd992ef321c9a83b0efe771c5b47": "\n\\begin{array}{l}\nx = \\frac{\\dot\\theta_c - \\omega_c}{g_v c^*} = \\frac{\\omega_r - \\dot\\theta_e - \\omega_c}{g_v c^*},\\\\\n\\dot x = \\frac{\\ddot\\theta_c}{g_v c^*},\\\\\n\\theta_r = \\omega_r t + \\Psi,\\\\\n\\theta_e = \\theta_r -\\theta_c,\\\\\n\\dot\\theta_e = \\dot\\theta_r - \\dot\\theta_c = \\omega_r - \\dot\\theta_c,\\\\\n\\frac{1}{g_v c^*}\\ddot\\theta_e - \\frac{1}{g_v c^* RC}\\dot\\theta_e - \\frac{A_cA_r}{2RC}\\sin\\theta_e = \\frac{\\omega_c - \\omega_r}{g_v c^* RC}.\n\\end{array}\n", "51b8fdeca4906065130eb2425da1d873": "f''(x)=2>0", "51b93fc7c471f407ece4e3a89f2b557a": "(R_0,\\ \\Theta_0) = \\left(\\frac{r_0}{r_0^2 - a^2},\\ \\theta_0\\right).", "51b965b57f407a255565c269409970fb": "M \\times C", "51b96f34659698087cde68ba860bb658": "\\begin{align}\n 3 x + 2 y + z &\\le 10\\\\\n 2 x + 5 y + 3 z &\\le 15\\\\\n x,\\,y,\\,z &\\ge 0\n\\end{align}", "51b9704eb85180fab9513d0486a20072": "\\Delta x_{n}={(x_{n+1}-x_{n})},\\ ", "51b977f2a14bed96c557e58f4d18b204": " H_2(G,\\mathbf{Z}) \\cong (R \\cap [F, F])/[F, R]", "51b986a35854aa0973c8c38cd8d8e352": "H_3,", "51b997f3754a48e281e781a22e870305": "f(\\lambda x_0, ..., \\lambda x_n) = 0", "51b9e46e5ada94238add1cabeef14796": "X(t)=\\epsilon W(t)", "51ba442d09d39c8db97b776c6648e006": "\\hat{\\mathcal{P}}_{\\rm CAS}\\hat{\\mathcal{H}}\\hat{\\mathcal{P}}_{\\rm CAS}\\left|\\Psi_m^{(0)}\\right\\rangle = E_m^{(0)} \\left|\\Psi_m^{(0)}\\right\\rangle", "51ba6e0fb7b06beedbf7c893be609f39": "\\begin{align}G_X(z)\n&=G_N(G_{Y_1}(z))\\\\\n&=\\exp\\biggl(\\lambda\\biggl(\\frac{\\ln(1-pz)}{\\ln(1-p)}-1\\biggr)\\biggr)\\\\\n&=\\exp\\bigl(-r(\\ln(1-pz)-\\ln(1-p))\\bigr)\\\\\n&=\\biggl(\\frac{1-p}{1-pz}\\biggr)^r,\\qquad |z|<\\frac1p,\\end{align}", "51ba717d2fa8bcc24e5a78a2f1f49cb0": " \\pm \\sqrt{-i} = e^{i(3\\pi/4)} \\,\\! ,", "51ba7ca6f607d7d54884f0b7b6951eb6": "\\xi \\in [t_0,t_0+h]", "51ba9b150f3a33908be016efc0b3aa31": "\\sum_i \\gamma_{ri} x_i = \\ln k_r^+-\\ln k_r^-=\\ln K_r ", "51baa6475210cb8f39da8ef1d82d1484": "\\left\\langle {\\eta _{i}\\left( t\\right) \\eta _{j}\\left( t^{\\prime }\\right) }\\right\\rangle =2\\lambda _{i,j}\\left( A\\right) \\delta \\left( t-t^{\\prime}\\right).", "51bb0211b7d1374882e9bdb960e9633f": "= \\begin{cases}\n\\frac{1}{T} \\left( 1 + \\frac{t}{T} \\right) & \\mbox{if } 0 \\le t < T \\\\\n\\frac{1}{T} \\left( 1 - \\frac{t}{T} \\right) & \\mbox{if } T \\le t < 2T \\\\\n0 & \\mbox{otherwise}\n\\end{cases} \\ ", "51bb810070b1ff2167de41a5a90a68c8": "\\gamma \\gamma^* = \\gamma^* \\gamma, \\ \\alpha \\gamma = \\mu \\gamma \\alpha, \\ \\alpha \\gamma^* = \\mu \\gamma^* \\alpha, \\ \\alpha \\alpha^* + \\mu \\gamma^* \\gamma = \\alpha^* \\alpha + \\mu^{-1} \\gamma^* \\gamma = I,", "51bb8273865150f69b97eaf97b0ed4ea": "y_1=01110", "51bb9ee7e66652ef2713d3cd752839d4": "\\sigma_x^2.", "51bba9e7311ad58d76d87573d4a9e088": "\\,M\\,", "51bbc5a6427691cb04232511ddae4214": "\\sigma_{\\mathcal{O}}=\\sqrt{\\langle \\hat{\\mathcal{O}}^{2} \\rangle-\\langle \\hat{\\mathcal{O}}\\rangle ^{2}},", "51bc4455204d02b2db428ec0f85bffae": "\n\\begin{align}\n\\begin{pmatrix}\nN_{t+l_1} \\\\\nN_{t+l_2} \\\\\nN_{t+l_3}\n\\end{pmatrix} &=\n\\begin{pmatrix}\nF_1 & F_2 & F_3 \\\\\nS_1 & 0 & 0 \\\\\n0 & S_2 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\nN_{t_1}\\\\\nN_{t_2}\\\\\nN_{t_3}\n\\end{pmatrix}\n\\end{align} .\n", "51bc9c376d0f820a087798a203e84e6e": "\\omega(x)=e^{-x^2}", "51bce2ec33bfdd97b8a777a30b47ce72": "x_i.", "51bd0603c47f95b3b3e03f8be4d4d97d": "v_p(n)", "51bd5832a1c35c8beb8078f111055136": "0<\\delta\\leq5/16", "51bd5e3ebf805275817ca51ba3844720": " \\Gamma = \\frac{E_\\mathrm{C}}{kT_e} ", "51bd64d9c5296a8106fa0094d3cf52b2": " \\Delta P ", "51bd83521235c4b537c36d89bfdcc503": "g : \\mathbb N^n \\to \\mathbb N", "51bd865f14bfd590b68e3a1feccd583a": "\\int\\mathrm{havercosin}(x) \\,\\mathrm{d}x = \\frac{x + \\sin{x}}{2} + C", "51bd911a06326441f7fdc3142421d588": "|\\mathbf{w}|", "51bdc054943214f91bfe30894ea7381b": " \\tau = ct ", "51bde143ff5177765948b9eccd4b4b7e": "\n\\hat\\Omega_n = n \\cdot v_{HCE}[\\hat\\beta_{OLS}]\n", "51bdeae618a05fe872a84004ff936a55": "d := \\min_{m_1,m_2\\in\\Sigma^k; m_1\\neq m_2} \\Delta[C(m_1),C(m_2)]", "51bdf4efb37b3a13d4c9cc9f01ceee56": "x_\\pm=-\\frac{5}{4} \\pm \\frac{3}{4}\\sqrt{\\frac{35}{3}},", "51bdfbc5c7ac2f782d21066f180f7fcc": " E: y^2 +xy +y = x^3 - 663204x + 206441595", "51be552c76074b8f5a296f04e7dcf59b": "\\omega_1=(1,0)", "51be6a593599b690f6b495087bee1443": "\\gamma_1=\\frac{2 \\sqrt{2} (16 -5 \\pi)}{(3 \\pi - 8)^{3/2}}", "51be959c964daa525ead529fd89dcf6a": "I(\\mathcal{C}),", "51beb0489816d6a92944a63c1884b54d": "\\eta\\,\\! (d) = \\eta\\,\\! (d_m)10^{-p(\\lambda\\,\\!)(d-d_m)^2} ", "51bebb00c1d1d720f4f9281e2e3b827c": "G_T = \\frac{4|k_{21}|^2 \\Re{(M_L)}\\Re{(M_S)}}{|(k_{11}+M_S)(k_{22}+M_L)-k_{12}k_{21}|^2}", "51bed1077e13e36e2beed99cd74a8694": "C_y", "51beddb4dae5fdcd022f0c1f0222f81c": "_{x^c}\\!", "51bf0eb8f85e65d698c818d88aa6a730": "\\cos \\omega_\\circ = -\\tan \\phi \\times \\tan \\delta ", "51bf31c7cc5dd43bef7075a7ce7729a7": "\n\\vec{I} = p \\vec{v}\n", "51bf40bc928d5a8238407459f096166f": "AI_U=\\frac{P}{PET}", "51bf9038cec75b0025bb2f315853999a": " r = \\sigma \\, \\Phi ", "51bfa465261c36e18e9b75d92c9ced25": "\\mu_n(t+s)=\\sum_{k=0}^n {n \\choose k} \\mu_k(t) \\mu_{n-k}(s).", "51c04bb14702ef6d39ad6f18a76eef14": "a_m\\mathbf{(R_n,r)}=\\frac{1}{\\sqrt{N}}\\sum_{\\mathbf{k}}{e^{\\mathbf{-ik\\cdot R_n}}\\psi_m\\mathbf{(k,r)}}=\\frac{1}{\\sqrt{N}}\\sum_{\\mathbf{k}}{e^{\\mathbf{ik\\cdot (r-R_n)}}u_m\\mathbf{(k,r)}}.", "51c0587a22beda1af3afbbb4b97cd966": "a\\ne 0", "51c059187216d3445a1789cbb44d5e3d": " \\pm 0\\; \\overset{def}{=} \\lim_{\\;x\\rightarrow 0^\\pm} x", "51c05a69a2732522f2ac6c0beaf572d6": " k = \\{u \\in K : \\partial(u) = 0\\}.", "51c06aaeb7229e5a658a056d86f636e0": " \\ln \\left(\\frac{[A]_0 - [A]_e}{[A]_t-[A]_e}\\right) = (k_f + k_b)t ", "51c0b3612e19966b0047d20228554106": "\n\\left(\\frac{p}{q}\\right)\n=\\sgn\\left(\\prod_{i=1}^{\\frac{q-1}{2}}\\prod_{k=1}^{\\frac{p-1}{2}}\\left(\\frac{k}{p}-\\frac{i}{q}\\right)\\right).", "51c1136c07cfeaef1ec652d6c6b73702": "{}^{21}_{11} \\text{Na}_{10} \\rightarrow {}^{21}_{10} \\text{Ne}_{11} + \\beta^+ + \\nu_\\text{e}", "51c13d192b7d810e23f36a451a5768ec": "f'(x)=y", "51c17cdd418cc642edf08b547f8cb43b": "\\alpha + 0 = \\alpha\\!", "51c1dcae579a192e757f4b1efa6848c3": "E(\\vec{x})=\\vec{\\mu},\\,E((\\vec{x}-\\vec{\\mu})(\\vec{x}-\\vec{\\mu})^T)=\\Sigma\\,", "51c248d686be2934e270f429cd410808": "\\sin(56\\tfrac14 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2-\\sqrt{2}}};", "51c2af9de5996776f27cfb55c3cfbf6d": "\\scriptstyle i = 1,\\ldots,4", "51c2b9df2351677059a4004a49bdb87a": "C:\\{0,1\\}^n\\to\\{0,1\\}^s", "51c2c460195cb29a06f050823e80fd50": "\\lambda^{-\\nu} J_\\nu (\\lambda z) = \\sum_{n=0}^\\infty \\frac{1}{n!} \\left(\\frac{(1-\\lambda^2)z}{2}\\right)^n J_{\\nu+n}(z) ", "51c2e0776cf83375e372094bb8d55e24": "\\scriptstyle\\nabla_{\\vec{f}_0} \\vec{f}_0 \\;=\\; 0", "51c2e91e5da4fcd8baa481977f0f1c77": "x^{2k-1}", "51c3086371b0837b3f21e238ee039a52": "1/\\sqrt{2} ", "51c334a69f37e096fd9654c117a0b517": "t \\in \\{0,1,2,3,\\ldots,N\\}", "51c338f8090d63c3aa796fa54819fc00": "P_v(n) = \\frac{x(1 - (1 + i)^{-n})}{i}", "51c3537bdf413e91974b9ac8b4cb870f": "X = \\coprod_i X_i", "51c359cbded7b9eaa20f97d53cd6cbbc": "\\mathbf{P}(t) = \\varepsilon_0( \\chi^{(1)} \\mathbf{E}(t) + \\chi^{(2)} \\mathbf{E}^2(t) + \\chi^{(3)} \\mathbf{E}^3(t) + \\ldots )\\ .", "51c392e5db9772bcbef07df80ca82a2d": "e^{i z \\cos \\theta}=J_0(z)\\, +\\, 2\\, \\sum_{n=1}^{\\infty}\\, i^n\\, J_n(z)\\, \\cos\\, (n \\theta).", "51c40be5b6976f42e10b16754b901d7a": "x^*\\in H^*_1", "51c45e4339ce78903719018fa0c5761f": "\nm \\ddot{x} + h x \\imath + k x = 0\n", "51c47bdb86d84e99c41698ae668a8f4b": "(S,s)", "51c49d1e445d77f4884f49b8624537e3": "\\scriptstyle\\boldsymbol{x}_1,\\dotsc,\\boldsymbol{x}_n", "51c4be0c1ed03cace7a4071801e26a26": "0 \\leq 1 - \\frac{1}{r} \\leq \\left[ \\frac{m}{r} \\right]_1", "51c4fbaaa95931075e64d1ab6df7527d": "B[v, u] = \\sum_{0 \\leq | \\alpha |, | \\beta | \\leq k} \\int_{\\Omega} A_{\\alpha \\beta} (x) \\mathrm{D}^{\\alpha} u(x) \\mathrm{D}^{\\beta} v(x) \\, \\mathrm{d} x", "51c53952eb005e467b8ddaf0f2febd01": "\\scriptstyle -\\frac{2\\ell\\ell}{n}", "51c5b8794cb1b81b0a05ad7f67526abb": "MPL=\\frac{\\partial F}{\\partial L}", "51c5b8b866b68709931c972f30afdaef": "{\\mathbf{B}}", "51c637cf47246411f4243fc64429c97d": "b_1,", "51c63b935fbbc9c4826654d7430cf642": "\\eta_p= \\frac {2 \\frac {v} {c}} {1 + ( \\frac {v} {c} )^2 }", "51c6cac0aa2bc2b4aa9774055c22b529": "{d \\over dx}x = 1", "51c6e9726d815bec76b9d07119ab1c9c": "|\\alpha_i\\rangle", "51c70f6f070744b585994e0678912e56": " q_m ", "51c71ab398aa778764e7aa6c73938ae1": "\\nu_{\\tau}", "51c73114896a8adbc5ba680e14bf7932": "\\textstyle1=\\binom m0", "51c7ab06299730aabf35fc63046b9163": "\\lambda = 2H_n", "51c7d50e8504b50703d358fba205c835": "R_4(x)=\\frac{x^4-28x^3+70x^2-28x+1}{(x+1)^4}\\,", "51c8016588c78dbda952a3666369ab0f": " P( X - \\mu \\ge k \\sigma ) \\le [ 1 + k^2 + \\frac{ ( k^2 - k \\theta_3 - 1 )^2 }{ \\theta_4 - \\theta_3^2 - 1 } ]^{ -1 } ", "51c8695dffb79c49f1a4d8edb58ced05": "\n\\begin{array}{c|cccc}\nc_1 & a_{11} & a_{12}& \\dots & a_{1s}\\\\\nc_2 & a_{21} & a_{22}& \\dots & a_{2s}\\\\\n\\vdots & \\vdots & \\vdots& \\ddots& \\vdots\\\\\nc_s & a_{s1} & a_{s2}& \\dots & a_{ss} \\\\\n\\hline\n & b_1 & b_2 & \\dots & b_s\\\\\n\\end{array} = \n\n\\begin{array}{c|c}\n\\mathbf{c}& A\\\\\n\\hline\n & \\mathbf{b^T} \\\\\n\\end{array}\n", "51c87fd00cd0a360618b8b71f1076fab": "\\{X_1,\\dots,X_n\\},", "51c8a2c780f7edd2851943c05fe1dfbb": "P(A) = \\sum_{i=0}^n{ a_i A^i} =a_0 I + a_1 A + a_2 A^2 + \\cdots + a_n A^n,", "51c8c108d76efd4e2dc8b95909cc14b5": "\\omega = \\kappa (V_{TBG} \\times \\frac{d V_{TBG}}{dt})", "51c8cd426bd8a6dda56d460ffd8833f6": "(n-1)/n", "51c911e325024ca4931d97801047582f": "\\ \\delta_{h}", "51c9227a870f01fb2666ab52f2fbc525": "{ {\\underbrace{a + a + a + \\cdots + a}} \\atop{b} }", "51c9548f40287fd7ef4cac1d7cae3c19": " \\mu(B)=0.", "51c965557ee228d9d36bd2f615e53511": "Y_3 = \\left (1+1\\cdot1 \\right)\\left (\\sqrt{2}\\cdot\\sqrt{2} - 2\\cdot 0 \\cdot 1 \\cdot 1\\cdot 1\\cdot1 \\right ) + 2\\cdot1 \\left (1^2 \\cdot 1^2 + 1^2 \\cdot 1^2 \\right ) = 8", "51c97faba1b0449f0ccc0f5741d124fb": "U_1(N),U_2(N)", "51c991f814afde9e78b533a55bc361b1": " \\hat{H} X_n = \\lambda_n X_n ", "51ca05775d176495e960194bbebdcc21": "\\rho = a \\sin\\theta \\, \\cos^2\\theta.", "51ca332c1ba036fc41c01518ff957ffd": "jP", "51ca4bd01b0a968bd732c0ec1d899c29": "\\displaystyle{SK^n = (K^*)^nS.}", "51caa1ae4be3075313d0834f62f52097": "(d*3)\\times(d*3)", "51caa84f13fe96ebd5a13cf4c4c05eca": "\\delta X", "51cade6fcfa3873f1c1cf741178715c9": "\\rho_{2}/\\rho_{1}<(\\gamma+1)/(\\gamma-1)\\,", "51cb4d9b42375a671c2952f3726971a7": "\\begin{align}\n\\underline{\\int_{a}^{b}} f(x) \\, dx + \\underline{\\int_{a}^{b}} g(x) \\, dx &\\leq \\underline{\\int_{a}^{b}} f(x) + g(x) \\, dx\\\\ \n\\overline{\\int_{a}^{b}} f(x) \\, dx + \\overline{\\int_{a}^{b}} g(x) \\, dx &\\geq \\overline{\\int_{a}^{b}} f(x) + g(x) \\, dx \n\\end{align}", "51cbc57ef727043d5190e47f1651f6f7": "p(D)=0.05095", "51cbd933c74ff2e6bc100cfb7dc89fa8": "\n\\begin{bmatrix}\n\\boldsymbol{I}_{3m} & \\boldsymbol{0} & \\boldsymbol{V}_1^{[2](t)}\\\\\n\\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_1^{[2](b)} & \\boldsymbol{0}\\\\\n\\boldsymbol{0} & \\boldsymbol{W}_2^{[2](t)} & \\boldsymbol{I}_m & \\boldsymbol{0} & \\boldsymbol{V}_2^{[2](t)}\\\\\n& \\boldsymbol{W}_2^{[2](b)} & \\boldsymbol{0} & \\boldsymbol{I}_{3m} & \\boldsymbol{V}_2^{[2](b)} & \\boldsymbol{0} \\\\\n& & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots\\\\\n& & & \\boldsymbol{0} & \\boldsymbol{W}_{p/2-1}^{[2](t)} & \\boldsymbol{I}_{3m} & \\boldsymbol{0} & \\boldsymbol{V}_{p/2-1}^{[2](t)}\\\\\n& & & & \\boldsymbol{W}_{p/2-1}^{[2](b)} & \\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_{p/2-1}^{[2](b)} & \\boldsymbol{0}\\\\\n& & & & & \\boldsymbol{0} & \\boldsymbol{W}_{p/2}^{[2](t)} & \\boldsymbol{I}_m & \\boldsymbol{0}\\\\\n& & & & & & \\boldsymbol{W}_{p/2}^{[2](b)} & \\boldsymbol{0} & \\boldsymbol{I}_{3m}\n\\end{bmatrix}\\text{.}\n", "51cc905c1045eac6f382faa4d6c204ad": "S = \\sum_{i = 1}^n \\left(Y_i - \\beta_0 - \\beta_1 \\phi_1(X_{i1}) - \\cdots - \\beta_p \\phi_p(X_{ip})\\right)^2 .", "51cc9fe06c99e335c5c78a2deecbb80c": "\\begin{matrix}{52 - 4r \\choose 4}\\end{matrix}", "51ccadda77022bd7336e976d5e76d455": "\\begin{bmatrix}-\\omega^2\\begin{bmatrix}M\\end{bmatrix}+\\begin{bmatrix}K\\end{bmatrix}\\end{bmatrix}\\begin{Bmatrix}X\\end{Bmatrix}e^{i\\omega t}=0.", "51ccf643003c3a99fe91cad31e4d1883": "K_a (v_e -v_i) \\approx 10^{-pH_i} v_i", "51cd67921429c05daeda4833ed27b4bd": " \\ddot{\\bold{r}} = (\\ddot r - r\\dot\\theta^2)\\hat{\\mathbf{r}} + (r \\ddot\\theta + 2 \\dot r \\dot\\theta) \\hat{\\boldsymbol\\theta} ", "51cd8205e0800ba7fd2179106189f80b": "h = H(m)", "51cd8653820dd56d0ed920efec565c8f": " y = x \\tan \\theta - \\frac{gx^2 \\sec^2 \\theta}{2v^2}", "51cd97e064794c15f89967d47074827c": "\ny_{s} = P + Q r_{s}\n", "51cda671253d0eadb67fb50bdb1018d7": "\\Pr(\\mathbb{Z},\\mathbb{W}\\mid\\boldsymbol\\alpha,\\boldsymbol\\phi) = \\prod_d \\operatorname{DirMult}(\\mathbb{Z}_d\\mid\\boldsymbol\\alpha) \\prod_{d=1}^{M} \\prod_{n=1}^{N_d} \\operatorname{F}(w_{dn}\\mid z_{dn},\\boldsymbol\\phi)", "51cdc04be76c741e4cc1f229441596d1": "\\tfrac{1}{2}\\left (\\left (\\zeta(3)\\right )^2 -\\zeta(6)\\right )", "51cdc1214e7fe1c12aad28dc0274d490": "(S(\\exists), S(\\forall), \\Sigma, \\delta, P_0, F)", "51ce08a182d3e698b88930a9c331f6bb": "2005*10^{-6}", "51ce367d9ad33f8a9f998fa64bafe52b": "\nX^{\\{1\\}}=[3,4],\n", "51ce44a87cffad6d7652f91d19abdde5": "a^\\ast_{T+1}", "51ce64d4499b690bddf2d46c0b5d40fb": "P = \\frac{\\rho RT}{\\mu}", "51cea10940d0755e9c5b34dff3c328fd": "\\scriptstyle X", "51cf73ac19b65b6df7a0be0bcbac7a64": "\n\\psi:W \\to \\varphi(U)\\times \\mathbb R^N \\quad ; \\quad \\psi(v^k e_k|_x) := (x^1,\\ldots,x^n,v^1,\\ldots,v^N)\n", "51d001ca097cacd1b24b88ed90c25797": "P_V(V_G-V_L)=\\int_{V_L}^{V_G}P\\,dV", "51d0051115d45d673acce93634f392e8": "-\\nabla \\cdot a\\mathrm{inf}", "51d0837dfd4276291e4aa177f06f1c98": "S_0=1 \\, ", "51d0a3431927ab2ad5efe75e3104a500": "Rf(\\xi) = \\int_\\xi f(\\mathbf{x})\\, d\\sigma(\\mathbf{x})", "51d0c645ee5058dd7c3dcf59527858f3": "\\alpha=\\pm 1", "51d0c9362f1032344d316f76e286bbc7": "m=r^n-1", "51d0e2bfd7732f428a4ab439f9c8459e": "{\\Gamma \\vdash e{:}1{\\to}\\tau_1}\n \\over\n {\\Gamma \\vdash \\operatorname{lift}_{\\tau_2}(e)\\;:\\;\\tau_2\\to(\\tau_1\\times\\tau_2) }\n", "51d0ed2d6fc28be31cf21f09622e4007": "\\Sigma _{XX} =\\operatorname{Cov}(X,X) = \\operatorname{E}[X X']", "51d0edb1381e228b2affcb351ac70fe6": "-\\frac{\\hbar^2}{2 m} \\frac{d^2 \\psi}{d x^2} + V(x) \\psi = E \\psi", "51d113fb66afa362ab7e76d793e634c0": "\\scriptstyle\\mathbb{E}\\{\\cdot|\\mathfrak{G} \\}", "51d13d690a90f4ee85221625b04d06dd": "\n \\nabla\\langle p \\rangle = 0 ~;~~ \\nabla\\langle \\rho \\rangle = 0 ~.\n ", "51d15a9749f40c042ebcc5fb9d0f1414": " ln \\frac{[R]_0}{[R]_0-[C]}=\\frac{k_2[A]_0}{k_1'}(1-e^{-k_1't})", "51d1655e248fe4d0363d6b50f4f359af": "\\Sigma_{e \\in T} d(e)", "51d1fafb5370410ccb725c9ca7f5848b": "x \\mapsto f(x) + g(x)", "51d1fd3cbc328040d36f91ae7ccc9cc2": "xp(x+y)", "51d2143c87bbf8b0e1136047edc1476e": "\n\\begin{align} \n u &= u_g + \\frac{\\sqrt{2}}{fd}e^{z/d}\\left [\\tau^x \\cos(z/d - \\pi/4) - \\tau^y \\sin(z/d - \\pi/4)\\right ],\n \\\\\n v &= v_g + \\frac{\\sqrt{2}}{fd}e^{z/d}\\left [\\tau^x \\sin(z/d - \\pi/4) + \\tau^y \\cos(z/d - \\pi/4)\\right ],\n \\\\\nd &= \\sqrt{2 K_m/f}.\n\n\\end{align}\n\n ", "51d232838873eccc5fd1e28030914804": "C_{30}", "51d28e4b13f0c7abdd4440a55e282296": "f_{*}:H_n\\left(X\\right) \\rightarrow H_n\\left(Y\\right)", "51d291a3a63a25131c36acaf4e6ce643": "\n\\frac{ h^2 }{ G(M+m) } = A \\left( 1 - e^2 \\right)\n", "51d2cd321641580ad2c1a72471b161b5": "\\delta_{ij} = [i=j].\\,", "51d2dbc58032774c522f9eab8d5e8b56": " r = \\cos(3\\theta)", "51d2eebb011140e35806dc58250360f6": "[x= 3; x=x+1;]x\\ \\dot{=}\\ 4", "51d34bc95bcfaf6fad776cc1f39a9a67": "\nz\\rightarrow 2 x y +z_0\n", "51d371ec5555a3024971e455b6ea5033": "\\sqrt x = x^{1/2}.", "51d37eb396541f02c5117c58cc996f22": "w = w_x + y\\theta_x", "51d393158e568400c80b6d6b63b28f7d": " A^2 = h^2+r^2", "51d3b1d079820675803f721a91e32ec8": "_k\\mathbf{b}_{l,m,n}", "51d42d44a2670dc37db6d26e2bd8c026": "\\tbinom{n}{k}\\ge k!,", "51d446a43983f20d57a5980f7cc1574b": "M(\\vec X)", "51d4503556a22d4de0d709cd640d851d": "\\operatorname{Pr}(U>u) = \\begin{cases}\n1 &:\\ u < 1,\\\\\nu^{-D} &:\\ u \\ge 1.\n\\end{cases}", "51d45ae8492630f1243e8a14b0425fe2": "Y = b(Y - tY) + I + tY ", "51d48ffd8b726b8683ae020443bffbb6": "\\textrm{d}r\\,g(r)", "51d4a3ad3828ebbf4e10fb0c0395f888": "I^{-1} = \\{r\\in q(R): rI\\subseteq R\\}", "51d4cb3572a19643a58e6ad88fe7f660": "\\left\\{ \\hat{b}, \\hat{b}^\\dagger \\right\\} = (|u|^2 + |v|^2) \\left\\{ \\hat{a}, \\hat{a}^\\dagger \\right\\}", "51d4f1e6e76b93afdd1318b895f22fee": " {0.693 \\over 12.7} ", "51d4fe2fea5fe5577392abd64ef16c4b": " \\theta _f \\approx \\lambda / d ", "51d511190b5c1c8904a8ebc980a9a3b7": "R_{a,\\theta}", "51d5166a774b20f4451c6a7a7e5bb875": "\\begin{align}\n \\ln(M_r) &= \\sum_{k = 0}^{r - 1} \\ln(c + \\alpha + k)+ \\sum_{k = 0}^{r - 1} \\ln(c + \\beta + k)- 2 \\sum_{k = 0}^{r - 1} \\ln(c + 1 + k) \\\\ \n &= \\sum_{k = 0}^{r - 1} \\left(\\ln(c + \\alpha + k) + \\ln(c + \\beta + k) -2 \\ln(c + 1 + k)\\right)\n\\end{align}", "51d52a0073b2580d706e3584c224fca8": "\\tilde{g}^{(k+1)} = (1/\\sqrt{g^{(k+1)T} P g^{(k+1)}})g^{(k+1)}", "51d592d124f57bdac4b516b68632f6be": "f^i(\\bot) \\sqsubseteq k ~\\implies~ f^{i+1}(\\bot) \\sqsubseteq f(k)", "51d5a83a635773edca531a24be614371": "f:A^n \\to \\Bbb{R}", "51d5af715ffbec041b4ab6d60b5add9a": "R \\simeq S", "51d5b3dd96451b27b0a1e93f1f4d9f06": "(\\dot p , \\dot q)", "51d5f93b7399fcc709057eefc4aec654": "\\left( \\frac{r-2}{2(r-1)} + \\varepsilon \\right)n^2", "51d62e1de1fa3f1e27c3f72a9a81163b": "\\mathbf{\\Xi} := \\left[\\xi_1 \\dots \\xi_N\\right] ", "51d6628de58416f48833287b0c7b1d48": "a=k\\cdot (2\\xi)^{-1}", "51d67f490338719237eadd769ead5d2c": "\\omega(x,f)", "51d6b253e86b1e7ac187f0c5188a8202": "E^2 = \\sum_{j = 1}^L \\left( Y_j - j a - b \\right)^2.", "51d6b76fb14101d63268ab120909787d": "\\rho < \\sigma\\frac{\\sigma+\\beta+3}{\\sigma-\\beta-1}, ", "51d6c9ac48c2fd7f767f55c1948c9a00": "\\mathcal P \\left\\{ O_1(4) O_2(2) O_3(3) O_4(1) \\right\\} = O_4(1) O_2(2) O_3(3) O_1(4) .", "51d6d3f81f3e2ececc694b7f530a28c3": "\n D := \\cfrac{2h^3E}{3(1-\\nu^2)}\n", "51d6dc397a2dbfbc5a1a06b74ecd15ad": "E \\psi = - \\frac{\\hbar^2}{2 L} \\nabla^2 \\psi+\\frac{1}{2} L \\omega ^2 Q^2 \\psi ", "51d70f27dddffd5facff982cbed876f0": " t \\uparrow 0 ", "51d72752d58cc5b57c1b8503fbafe87e": "\\Gamma(x,y)", "51d729e9a47bdc0ea79cdd49322754ff": " \\lim_{x \\to 1} \\frac{x^2-1}{x-1} = 2 ", "51d734401f86654011085caa87cc30f3": "\n \\begin{align}\n \\frac{\\partial\\kappa_y}{\\partial{x}}\\, -\\, \\frac{\\partial\\kappa_x}{\\partial{y}}\\, =\\, 0\n \\qquad &\\text{ with } \\kappa_x\\, =\\, \\frac{\\partial\\theta}{\\partial{x}} \\text{ and } \\kappa_y\\, =\\, \\frac{\\partial\\theta}{\\partial{y}},\n \\\\\n \\kappa^2\\, =\\, k^2\\, +\\, \\frac{\\nabla\\cdot\\left( c_p\\, c_g\\, \\nabla a \\right)}{c_p\\, c_g\\, a}\n \\qquad &\\text{ with } \\kappa\\, =\\, \\sqrt{\\kappa_x^2 \\, +\\, \\kappa_y^2} \\quad \\text{ and}\n \\\\\n \\nabla \\cdot \\left( \\boldsymbol{v}_g\\, E \\right)\\, =\\, 0\n \\qquad &\\text{ with } E\\, =\\, \\frac12\\, \\rho\\, g\\, a^2 \\quad \\text{and} \\quad \\boldsymbol{v}_g\\, =\\, c_g\\, \\frac{\\boldsymbol{\\kappa}}{k},\n \\end{align}\n", "51d735ded5c3c315b01e43cc71649462": "(1_A \\otimes \\varepsilon) \\circ \\delta =1_A = (\\varepsilon \\otimes 1_A) \\circ \\delta", "51d76ecce63617448f0d9c613c40d7bf": "\\sum_{n = 1}^{\\infty}M[\\xi_n] = -\\frac{1}{37}\\sum_{n = 1}^{\\infty}r_n \\to -\\infty.", "51d7914d3e267f8f0b7884522af1757e": "k = \\frac{C_S}{C_L}", "51d7cd6c66bcc2b2d4c98ddbd7160dc9": " A = u_1 \\otimes u_2 ", "51d7e59b3ff613511483734f4d8fa740": "= {1 \\over 2}(\\sqrt{2}){[\\left|1,45\\right\\rang \\left|2,45\\right\\rang + \\left|1,45\\right\\rang \\left|2,135\\right\\rang + \\left|1,135\\right\\rang \\left|2,45\\right\\rang + \\left|1,135\\right\\rang \\left|2,135\\right\\rang + \\left|1,45\\right\\rang \\left|2,45\\right\\rang - \\left|1,45\\right\\rang \\left|2,135\\right\\rang - \\left|1,135\\right\\rang \\left|2,45\\right\\rang + \\left|1,135\\right\\rang \\left|2,135\\right\\rang} ", "51d7efec6f76816cba0b77afe529b146": "\\beta(\\langle E\\rangle)=m/\\langle E\\rangle", "51d7fa84d1bf3f0e388d9c7ebc49d4ff": "= {2p_1q_1 + 2p_2q_2 \\over 2}", "51d8b44f7f45a249c5aa7b038c705f82": "\\mathcal{KL}", "51d8cf921bcbcf2a2674bbc4395b7c4c": "\n\\begin{align}\np(\\tilde{x}=i\\mid\\mathbb{X},\\boldsymbol{\\alpha}) &= \\int_{\\mathbf{p}}p(\\tilde{x}=i\\mid\\mathbf{p})\\,p(\\mathbf{p}\\mid\\mathbb{X},\\boldsymbol{\\alpha})\\,\\textrm{d}\\mathbf{p} \\\\\n&=\\, \\mathbb{E}_{\\mathbf{p}\\mid\\mathbb{X},\\boldsymbol{\\alpha}} \\left[p(\\tilde{x}=i\\mid\\mathbf{p})\\right] \\\\\n&=\\, \\mathbb{E}_{\\mathbf{p}\\mid\\mathbb{X},\\boldsymbol{\\alpha}} \\left[p_i\\right] \\\\\n&=\\, \\mathbb{E}[p_i \\mid \\mathbb{X},\\boldsymbol\\alpha]. \\\\\n\\end{align}\n", "51d8f361b87fac28e97b69d12b6a5e18": " L^* = \\sum_{j=1}^l L_j ", "51d90d28799773c9a89089d8e866ff89": "\\mathcal{W}^{-1}", "51d937f0f94f39690a5dce3a728502ee": "x_i >_i x_i^*", "51d98b1bd8ae28c7fed8a8ffb2ea49b5": "\\text{WMA}_{M+1} = { \\text{Numerator}_{M+1} \\over n + (n-1) + \\cdots + 2 + 1} \\,", "51d9d324bc2f352f1258d0a8543fbff4": "I^a= I^a(E^a)", "51d9ed642aa6abe3c9a5b0eb00404ba6": "Z_{\\mathrm {in}} = Z_0 \\frac {Z_\\mathrm L + jZ_0\\tan(\\beta l)}{Z_0 + jZ_\\mathrm L\\tan(\\beta l)}", "51da1f4a0ec9cb125fc434752ce085e3": "f(x)=\\log\\cos x, \\quad x\\in(-\\pi/2, \\pi/2)\\!", "51da3a6d1052f9f71ff8b73c23cb6827": "\n\\left(\\part^2+m^2\\right) \\varphi_{\\mathrm{in}}(x)=0\n", "51da66eba7fcbfb6afa16749ba36107b": "\\mathfrak M", "51dab0e717bee5c9fba5ba122276951c": "d(A)", "51dabb7d4495ca035767a4c60cb7ce56": "m^3", "51dbdcd4f96c50eb2023d0650de41bd5": "\\mathbb{S}^n_T=C_1 \\cup ... \\cup C_r", "51dbdf1bbdc3971708990db3426d1951": "\\mathbf{\\Phi}_i \\ ", "51dc3609f7bc6ae2b24acb02c555f542": "\\scriptstyle G \\in G(n, p)", "51dc5a68d6b360c780323546c6adfa85": "K_{il} \\oplus K_{jl} ", "51dcf82cc32e2a9de47e68518bf7d179": " \\mathbf{F}_{ext} = -\\left(m_{gas}(t) \\frac{\\mathrm{d}\\mathbf{v_e}}{\\mathrm{d}t} + \\mathbf{v_e}(t) \\frac{\\mathrm{d}m_{gas}}{\\mathrm{d}t} + F_{other}(t)\\right).", "51dd337cd58f31f11c45bfc8c497240d": "\n\\frac{1}{r} = A \\cos\\left( k\\theta + \\varepsilon \\right)\n", "51dd6295734fe43edffa0ffcc9f9dfc0": "\\mbox{Re}(\\delta) > 0", "51ddc371b7e20852c63e86dffcfd9a5b": "\\sum_m (-1)^{j-m} \\langle j m j {-m} | J 0 \\rangle = \\sqrt{2j+1} ~ \\delta_{J0}\n", "51deb695c87236bab2218b66873dc60a": "U_1\\oplus\\dots\\oplus U_n", "51decbbdc0384eaba73241ab0ae242f0": "W = (A^TC_Z^{-1}A)^{-1} A^TC_Z^{-1}", "51ded22fb64f97ac1bb836abb4efe92f": "\\{ \\mathfrak{p} \\in \\operatorname{Spec}A | A_\\mathfrak{p} \\text{ is integrally closed} \\}", "51df7416bdaab2f17926b65e50c5e7d1": "1^1 \\cdot 2^2 \\cdot 3^3", "51df7c30edcf41df41f0d0c90682b9ee": " S_n=\\langle s_1,\\ldots,s_{n-1}|\ns_i s_{i+1} s_i=s_{i+1} s_i s_{i+1}, \ns_i s_j = s_j s_i ~\\rm{for}~|i-j|\\geq 2, s_i^2=1 \\rangle. ", "51dfc63b13bef9db35947cfc6508dae4": "E_{1} =\\varepsilon _{1} \\cosh \\mathcal{G} -\\varepsilon _{2}\\sinh \n\\mathcal{G}, ", "51dfe8f5e94c5b252dacc5f023cf10e3": "\\mathrm{Tor}_n^R(A,B)=(L_nT)(A)", "51e01dcff08b66dfc7038488a385363c": "[a,a+r]", "51e0567bdbbac0f9acc6ac4febd3263f": "X_8=0", "51e05edfeae647e687088cda0add241d": "\\vec y_2", "51e0668a1da1c6c104041b67f4a72524": " \\Delta \\tau = {Gb\\over L-2r} ", "51e06ea1dd78b580864b206de1fc8bb5": "\\bar\\alpha=a_0 -a_1\\mathbf{i} -a_2\\mathbf{j} -a_3\\mathbf{k}", "51e07c5db219224f3cc20ec9a4592a23": "0< \\epsilon< 1", "51e09c5af74696f5e0221475a1bbdb1b": "\n\\epsilon_\\mu^2( p ) = {-1 \\over \\sqrt{2}} [u^{+1}_{+1}(\\mathbf{p})]^\\dagger\n\\gamma_0 \\gamma_{\\mu} u^{-1}_{-1}(\\mathbf{p}).\n", "51e0a9b9d6ac3c64e03ca30f0f83c42b": "\\mathcal{Q}^1_{\\mathrm{Hur}}(I) = \\{x \\in \\mathcal{Q}_{\\mathrm{Hur}}^1 : x \\equiv 1 (", "51e0aaa87465f09fcadd8ce799a198c0": "\\text{NP}", "51e0ac8f75ebe6523d711a5a588c89a0": "P(k,\\rho,V) = \\frac{(V\\rho)^k e^{-(V\\rho) }}{k!} . \\,\\!", "51e11932d607970240b05e4689180d84": "a \\ge 0", "51e15187e66d518035632f2cbcd7190c": "S(\\mathbf{q}) = 1 + \\rho \\int_V \\mathrm{d} \\mathbf{r} \\, \\mathrm{e}^{-i \\mathbf{q} \\mathbf{r}} g(\\mathbf{r})", "51e151cfe74ef3beaeaf7f17700f3001": "0.16 < \\frac{r}{\\lambda} < 0.2", "51e15d05fe22b530bbe21f8cdc4f966d": "t \\mapsto \\zeta(s,\\{t\\})", "51e17a3546b3f743448d40b221135631": "Head^+(X) = \\{Y|X \\Rightarrow^+ Y \\alpha \\}", "51e1c0a1aa46a9d98414113d6523ef66": "poly(n)", "51e211a6f2453ac306b0c12f9da6f936": "\\mbox{m}\\,\\mbox{s}^{-2}\\,", "51e2521599423eb9eedadffec070d639": " w + \\bold{i} x + \\bold{j} y + \\bold{k} z \\,\\!", "51e278abe4a8bca1829954759546d6ba": "f(y_1,\\dots,y_k)= \\delta^{\\nu}\\sum_{n=0}^\\infty \\frac{(1-\\delta)^{n}\n\\prod_{i=1}^k \\mu_i \\lambda_i^{-\\nu-n}}{[\\Gamma(\\nu+n)]^{k-1}\\Gamma(\\nu)n!}\n\\exp\\bigg\\{(\\nu +n)\\sum_{i=1}^k \\mu_i y_i - \\sum_{i=1}^k \\frac{1}{\\lambda_i}\\exp\\{\\mu_i y_i\\}\\bigg\\},", "51e2c3ee258f96ed8a21f784c6e424f5": " X_p \\in T_p M", "51e30ff0f3ad7f4a08fb2aea5cbc037b": "I_n", "51e3aee4bcf98f9b14fc068f188f1e35": "a=\\arccos\\left(\\frac{\\cos\\alpha+\\cos\\beta\\cos\\gamma}{\\sin\\beta\\sin\\gamma}\\right)", "51e3b077dd909dc13c5073899e9ad3ff": "~Y~", "51e45295fa3cc1dd826b5e604b731c88": "\nH=\\left(\\begin{matrix}1&0\\\\0&-1\\end{matrix}\\right).\n", "51e48ff56998073fb1262bebb9a050ea": " r_{\\rm{particle}}\\, ", "51e4e893459f6978961d020cdd088bd7": "g_h = G \\, m_\\mathrm{Earth} / \\left( r_\\mathrm{Earth} + h \\right) ^2", "51e50c31e5eea2fd2549bef7175901ec": "h = h_{b+1}", "51e518e33fc80f074b8153d089d59da8": "0 ", "51e584f6e2e568fe3433d9d3938c9ad9": "b_1 \\ldots b_m", "51e6253a0ff1c47f2839719aeb5011d8": "R(r) = \\frac{1}{r^{n+1}}", "51e62919cf69f99a5ad4254f26fa9727": "\\rho = N/V", "51e64dacbe9684d401252b7d12e20af6": "\\frac{PN^3}{ON^3}=\\frac{PN}{NA}.", "51e658eadb41992293fa4dce27988804": "\\displaystyle{R=-iR_1 + R_2,\\,\\,\\, R^*=-iR_1 - R_2,}", "51e66368f9f1779ede1126d3d065be77": "\\mathbf{\\ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}\\hat{\\mathbf{r}}_{{Earth},{Sun}}", "51e6bf68dd6f626dffde009037e93bbe": "{\\varepsilon}_{1,2}=\\frac{1}{{\\frac{1}{\\varepsilon_1}}+{\\frac{1}{\\varepsilon_2}}-1}=\\frac{\\varepsilon_1\\varepsilon_2}{\\varepsilon_1+\\varepsilon_2-\\varepsilon_1\\varepsilon_2}", "51e6cc8ea1c4efe256f6b1c57ff6cb40": "x \\in O", "51e6ddc2b69d82cad9b00be3e5b81eb3": "= - \\frac{1}{2} \\int_{0}^{T} \\frac{1}{T^{2}} \\, \\mathrm{d} t", "51e736088ec2be495f714a5b3c226b68": "\\kappa_2(A)", "51e744c8d21e715b1ecad642564da105": "\n\\mathord{\\overbrace{\n\\begin{bmatrix}\n\\dot{e}_2\\\\\n\\dot{e}_3\\\\\n\\vdots\\\\\n\\dot{e}_n\n\\end{bmatrix}\n}^{\\dot{\\mathbf{e}}_2}}\n=\nA_2\n\\mathord{\\overbrace{\n\\begin{bmatrix}\ne_2\\\\\ne_3\\\\\n\\vdots\\\\\ne_n\n\\end{bmatrix}\n}^{\\mathbf{e}_2}}\n+\nL_2 v(e_1)\n=\nA_2\n\\mathbf{e}_2\n+\nL_2 v_{\\text{eq}}\n=\nA_2\n\\mathbf{e}_2\n+\nL_2 A_{12} \\mathbf{e}_2\n= ( A_2 + L_2 A_{12} ) \\mathbf{e}_2.\n", "51e7be0ce456299476d36e5b99a774ab": "\\tfrac{OC}{OF}", "51e7d45635a030a4dc039471faa1bf78": " =144 \\times 35=(3+4+5)^2 \\cdot (2^3+3^3) ", "51e7feef0f5254f95a1e9180f5fceb0e": "C_{60} ", "51e81b861a24e3e45216489bdac6684f": "x_{n_k}\\to x \\in C(\\theta)", "51e84ae3cd70009344ae6e5ff5a7d2aa": "\\psi_n(q) = 0 \\quad\\mbox{for}\\quad q\\notin \\Omega", "51e8964ca0d9ea07724f7eb474d19d03": "\\exists r>0", "51e8dc7efa8fbd165a7ba0f0db0e76f1": "D\\Omega=0", "51e8f0e186f412ae7c88e7eff152793d": "-x \\equiv x\\Rightarrow 0", "51e8f10e6877e0025d324fbf3d8c43ec": "a\\to a = 1", "51e935a498038b58a99a8a34543f8211": "\\phi_0(\\theta) = \\sin 0\\theta = \\sin 0 = 0", "51e9485ff8026209a0577a59d9483dfd": "p_{1}x_{1} + p_{2}x_{2} = m", "51e94cf270a4d3c3f872d2351cec4ae4": "\\textstyle \\mathcal{M} = \\left\\{-1,1\\right\\}, \\mathcal{C} = \\mathbb{Z}_n", "51e9b3aa3f84816a7caca3563b1273da": "\\operatorname{DG}(n^2;s)=\\sum_{n=1}^{\\infty} \\frac{n^2}{n^s}=\\zeta(s-2)\\,,", "51ea51620a3634168f6780a47d654a82": "V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_i\\}}", "51ea97ecf76d5d5e64d6873ed9dfef87": "f_{uc}(\\langle u2 \\rangle) = \\{\\langle U \\rangle\\} \\cdot \\{\\varepsilon\\} = \\{\\langle U \\rangle\\}", "51eabbc5fd90928dc1a538f6c9050be6": " T_n(\\phi)_{k,l} = \\widehat\\phi(k-l), \\quad 0 \\leq k,l \\leq n-1,", "51eaffa14ce144c876f7425fbf104814": "a\\succeq c\\;", "51eb0f36e85cfd2ac3f420bd74a9fa1a": "\\kappa = \\frac{|y''|}{(1+y'^2)^{3/2}}", "51eb44250e84eec14fd6760818ab3ddd": "L u =x, \\qquad x\\in(0,\\pi)", "51eb4456be41cd8cfca6b27161920936": "\n\\bar{q}(x, y)=w\\phi(x,y)", "51eb89e7843ee71cde49dca915ddddc2": "\\mathit{R}_G = \\frac{ \\sqrt N\\, l }{ \\sqrt 6\\ } ", "51eb922e404d2b55480ab6c34dbf7722": "Q \\rightarrow P", "51eba9321e72b165d0789aac431c4864": " f(x) \\approx \\sum_{k=0}^\\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \\ldots ", "51ebd69c3dc2b4a3345459f57e93bcc3": "\\cos(t) = x \\,\\!", "51ed2fb2ceb26eb38db0a2057d51947c": "\\texttt{bool} \\to \\texttt{bool}", "51ee12048348bdba9e22e97e1744d5af": " \\alpha = \\frac{1+\\sqrt{5}}{2} \\approx 1.618 ", "51eea8e822e688605de85f317d98232f": "s = |\\mathbf{a}| \\cos \\theta = |\\mathbf{a}| \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{a}| \\, |\\mathbf{b}|} = \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{b}| }\\,", "51eeda3d89b3cd67de3831c759398ca7": "a = p^\\alpha u", "51eee3901704646724946b7558c926f5": "T_{a} (x) = x + a.", "51ef04014415547315c00eea5b23cf8e": " \\Delta{x} ", "51ef0a0ef5888d5f09adb96bf8d0ae26": "\\textstyle (*) \\quad \\mathbb{E} \\sup_k| M_t^{\\tau_k} | < \\infty ", "51ef33afa88a2ab600414f60f591ed00": "f :\\!\\!-~~ \\alpha ~\\vdash~ \\beta", "51ef59b186899609d2f6a200e105afa6": "f(z)=\\sum_{n=0} a_n J_{\\alpha+n}(z)\\,", "51ef9083cdff96535051fc5dd6722f55": "C_\\text{max}", "51ef946391b865bd6352c99bb0d7249e": "f(x) = y\\,\\,\\Leftrightarrow\\,\\,g(y) = x.", "51efb315720c847816fe04fee3b3ffc1": " |j(t)\\rang = e^{-iE_j t /\\hbar} |j\\rang ", "51efd10ad5ec76673dd45ad0b295f75e": "W_1, W_2, \\ldots", "51efd29b8aff1c1ee97c82d284a1c307": "\\boldsymbol{\\nabla} \\cdot (\\mathbf{a})", "51eff8601fd33e7319f6d023b5f3e929": "\\tfrac{43}{81}", "51f08730df775d4d8da941961c76e677": "v = k i", "51f08b084ca680b66561721991f49b6a": "abcd, acbd, acdb, cabd, cadb, cdab", "51f13106ba3ec0f0008bc35c72d5a614": "\\begin{align}\n\\operatorname{MSE}(S^2_a)&=\\operatorname{E}\\left(\\left(\\frac{n-1}{a} S^2_{n-1}-\\sigma^2\\right)^2 \\right) \\\\\n&=\\frac{n-1}{n a^2}[(n-1)\\gamma_2+n^2+n]\\sigma^4-\\frac{2(n-1)}{a}\\sigma^4+\\sigma^4\n\\end{align}", "51f131481794ef5ee8ec4efe57eae3d1": "\\max_{X} E_{kl}(r_{k}^{B}, r_{l}^{X})", "51f180c76c6e2cf3b352239df90bac17": "T_{se} = (\\sigma_{se} n \\bar{v})^{-1}", "51f18d720f497462cebc8a75811df53b": "\\|u\\|_{C^{k-[n/p]-1,\\gamma}(U)}\\leq C \\|u\\|_{W^{k,p}(U)},", "51f1a8fcd05b4dff2872a4cc69d7b9ee": "SL_4/\\mu_2 \\cong SO_6", "51f1ddd49fdb426297b60cb323aa93ef": "\\bar{f}(x')", "51f231e80f7b736016f1aaea66b5c61d": "\\langle\\rangle \\in traces\\left(P\\right)", "51f24d9810efdbc313439fbe9d67359b": "\\begin{array}{rcl}\n\\eta(z) & = &\\displaystyle 1+\\frac{1}{2^z}+\\frac{1}{3^z}+\\frac{1}{4^z}+\\cdots - \\frac{2}{2^z}\\left(1+\\frac{1}{2^z}+\\cdots\\right) \\\\[1em]\n & = & \\displaystyle \\left(1-\\frac{2}{2^z}\\right)\\zeta(z),\n\\end{array}", "51f25a03d41533e1cd076412864dc272": "\\Omega(x)\\,\\!", "51f2be00c5ab33386967a67a14e99758": "\\{ \\mathbb{X} ; w_1, w_2, \\dots, w_N\\}", "51f2f7302a6c4d4f6397fcfefd5d269e": " \\frac{U_{dyn}}{U_{static}}=\\frac {A\\,L}{e^\\left(AL \\right)-1}", "51f32685b48951d9b6ab6aafafced2de": "(x,y)\\in P_1, (y,z)\\in P_2.", "51f372ef9502f65506c67e3b13d1bf2b": "\\scriptstyle f\\,", "51f3a62b74f5320853592728e0368553": "p = p_0 \\left(1 - \\frac{L h}{T_0} \\right)^\\frac{g M}{R L}", "51f3d5e4aed2faedef416b21782753ba": "a \\ne 0", "51f404ef81b596cbd4b693704bb09e97": "c=\\tfrac{2*\\pi*v_{1}}{R_{0,1}} * a", "51f42262cf93984b98a44086f20806b2": "\\lambda = \\pm 1 \\,", "51f4722c9c411d3c2629039cb701e379": " \\sigma_{21} =\\frac{d \\ln (x_2/x_1) }{d \\ln MRTS_{12}}\n =\\frac{d \\ln (x_2/x_1) }{d \\ln (\\frac{df}{dx_1}/\\frac{df}{dx_2})}\n =\\frac{\\frac{d (x_2/x_1) }{x_2/x_1}}{\\frac{d (\\frac{df}{dx_1}/\\frac{df}{dx_2})}{\\frac{df}{dx_1}/\\frac{df}{dx_2}}}\n =-\\frac{\\frac{d (x_2/x_1) }{x_2/x_1}}{\\frac{d (\\frac{df}{dx_2}/\\frac{df}{dx_1})}{\\frac{df}{dx_2}/\\frac{df}{dx_1}}}\n", "51f4789bbd5a405c3da3d9727a722d11": "\\Rightarrow cos\\vartheta _{m}\\simeq \\Delta \\theta \\simeq \\frac{2\\pi }{k_{t}L}=\\frac{\\lambda _{c}}{L}", "51f495e66062709816e706b7b0fdfdb3": "a(Z)=2^{1-A}(1+Z)^A\\,p(Z)", "51f5934a7f8cd43d9d9aa058e4db4d97": "2|r_1-r_2|", "51f595c82a1f42d190879954c2af845d": "\n H^2(P, Q) = 1 - \\sqrt{\\frac{2\\sigma_1\\sigma_2}{\\sigma_1^2+\\sigma_2^2}} \\, e^{-\\frac{1}{4}\\frac{(\\mu_1-\\mu_2)^2}{\\sigma_1^2+\\sigma_2^2}}.\n ", "51f59fbc155215ec6551c20f42b8ec8c": "X_j \\sim \\operatorname{Log-\\mathcal{N}}(\\mu_j, \\sigma_j^2)", "51f5ee301b784eceff986a2ad6afd132": "\\phi^{X}(z)", "51f5f381199c5a0e5260630b9727f7ec": "\\Omega\\subset\\R^n", "51f62c2ab92d9b30402af2153dca796b": "2. \\quad c\\leq\\sum_{n}\\left|\\hat{\\varphi}(\\omega + 2\\pi n)\\right|^2\\leq C", "51f65bceb52aadada84028b586f68a56": "|g\\cap \\mathfrak o|=2", "51f771826e064641a7ff1ee02dccc16c": " \\dot{t} = \\frac{E}{x^2}, \\; \\; \\dot{y} = P, \\; \\; \\dot{z} = Q ", "51f772324f6d75a281a96d1bdfba46dc": "\\frac{\\rho_1}{\\rho_2}", "51f7881839fb747ce4d2f3196356a1fc": " \\mathbf{E} \\left[|X_i^k|\\right ] \\leq \\frac{k!}{4!} \\left(\\frac{L}{5}\\right)^{k-4}", "51f7c166364e61895aefd69f261d2f25": "R_C(x,y) = R_F(x,y,y) = \\tfrac{1}{2} \\int_0^\\infty \\frac{dt}{(t+y)\\sqrt{(t+x)}}", "51f7e8d205200870f775ed0fbca5cb5c": "\\bar{I}_1 = \\bar{I}_2", "51f8023a5b19c3be8228c77a4ad356e6": "(\\sum_{i \\in S} a_i, x/2 + \\sum_{i \\in S} b_i)", "51f814b6fbe9f6f4d6a6e9cd64b1ee2a": "\\scriptstyle \\geq1.4\\times10^{16}", "51f847d5a79d80a52e40ed5025e9f31e": "\n C(t; 0, 1) \\;:=\\;W(L(t; 0, t^2/2)).\n", "51f8631a6f2f8a068575ad957a90888f": "\\text{(*)} \\qquad f(x+h)-f(x) = \\left(\\int_0^1 Df(x+th)\\,dt\\right)\\cdot h,", "51f8a91f692fa9e108599f1242b8cffb": "|\\Omega \\cap B_r(x_0)| > A r^n ", "51f8cbe3dc2b814b593f9c1664db359e": "\\mathrm{aff}(S)", "51f9194a8cd8b7b8ffe623e6168a8e19": " V = \\begin{bmatrix} U & I - UU^* \\\\ 0 & - U^* \\end{bmatrix}\n= \\begin{bmatrix} U & D_{U^*} \\\\ 0 & - U^* \\end{bmatrix}\n.", "51f98b7e14d4dd869ca003995fc1fa1d": "h_\\mathrm{c}", "51f98eec67d719bc079b781d2c408700": "N_R>1", "51f99cae0c800679b439afcd41873bb6": "\n1 - p = (1 - w^n)^k\n", "51f9bedeb5789431f907f50dfaac97a6": "\\left| \\frac{\\partial^2 a(x,z)}{\\partial z^2} \\right| \\ll \\left|k_0 \\frac{\\partial a(x,z)}{\\partial z} \\right|", "51f9fe79e3c6d580b37196ccc4d67de0": " q=e^{2\\pi i \\tau},\\qquad u=e^{\\pi i z}", "51fa07caa3e29533352fbb4eee056fc2": "G|_{V'}", "51fa8b9de8be1d0ae9763c87ca5b8ee7": "\\min_{\\boldsymbol{w}\\in\\boldsymbol{W}, t\\in\\R}\\left\\lbrace\nt+\\frac{1}{\\alpha}\\text{E}\\left[\ng_0(\\boldsymbol{w})+\\sum_{i=1}^{m}g_i(\\boldsymbol{w})\n\\psi_i-t\n\\right]_+\n\\right\\rbrace.\\,", "51fab83a7093d924dc8c08e74a434a6a": " \\operatorname{LUB}(\\{E_i\\}) = \\bigcup_{i=1}^\\infty E_i. ", "51fb69505b0797a957613aeabbdbd5b5": "i \\ ", "51fb6cce6b003c71640b302e8d264d8a": "\\frac{f_s D}{U} = S", "51fb8046c06bdee0f94229ac486aa8f4": "\\Delta C_t = C_t - C_{t-1}", "51fbb388b54a5a577385380ba189a2cd": "1/g", "51fc855b6dcc08fc50cb269f29952bd5": " \\exists{x}{\\in}\\mathbf{X}\\, P(x) \\or Q(x) \\to\\ (\\exists{x}{\\in}\\mathbf{X}\\, P(x) \\or \\exists{x}{\\in}\\mathbf{X}\\, Q(x))", "51fc99859b8009aaf4d81f44948cd8dd": "2n +\\gamma_{ij}", "51fca04d9521ebefdd660edd8bb3979f": "\\textstyle (**) \\quad \\mathbb{E} \\sup_{s\\in[0,t]} |M_s| < \\infty ", "51fcd51329147a5f7852f0c7130960b0": "\\mu_j=\\cos\\left(\\frac{j\\pi}{n}\\right), \\eta_k=\n\\begin{cases}\n\\cos\\left(\\frac{(2k-2)\\pi}{n+1}\\right) & j\\mbox{ odd} \\\\\n\\cos\\left(\\frac{(2k-1)\\pi}{n+1}\\right) & j\\mbox{ even.}\n\\end{cases}\n", "51fd17487fe6b1c6f3cb8cb8be93fd67": "\\phi_i = \\sum_{j = 1}^{n}\\frac{1}{4\\pi\\epsilon_0}\\int_{S_j}\\frac{\\sigma_j da_j}{R_{ji}} \\mbox{ (i=1, 2..., n)},", "51fd3f968ec99b73399e86d5f31d62d9": "\n[\\xi( \\mathbf{p}^{\\prime}),\\eta^\\dagger( \\mathbf{p})]\n= {i \\over 2} \\delta( \\mathbf{p}^{\\prime} - \\mathbf{p})\n[\\overline \\Delta_{21}(\\mathbf{p},\\mathbf{p})\n-\\overline \\Delta_{12}(\\mathbf{p},\\mathbf{p})], \\quad\\quad (13)\n\n", "51fdb7b288657d080a7af5042198d934": " \\vec{\\theta} -\\vec{\\beta} = {\\theta_{E}^2 \\over |\\vec{\\theta} |}. ", "51fdc614c7a66f2cdea8e86af1c5d100": "D \\approx 4\\pi\\frac{(\\frac{180}{\\pi})^2}{\\Theta_{1d}\\Theta_{2d}} = \\frac{41253}{\\Theta_{1d}\\Theta_{2d}}", "51fe1e222ec9fb123c4b671f7fe21f70": "RR=\\frac{D_{E}/N_{E}}{D_{NE}/N_{NE}}\\,,", "51fe5f11dc90db5d1dd2759c5aae09f6": "R_{a, (\\theta + r)} = R_{a, \\theta} \\cdot R_{a,r}", "51fe90619502ea126f656f3e7a3a025a": " \\frac{t}{s}(1+\\frac{s}{\\ell}) = 1 + \\frac{d}{\\ell} \\ \\ \\Rightarrow \\ \\ \\frac{s}{t} = \\frac{1+\\frac{s}{\\ell}}{1 + \\frac{d}{\\ell}}.", "51fe990f802c5ca1dc3461c47a0da960": " \\tfrac{\\sigma(140)}{140} = \\tfrac{1+2+4+5+7+10+14+20+28+35+70+140}{140} = \\tfrac{336}{140} = \\tfrac{12}{5}.", "51ff10a9d476f3cefde0bcd3747477ff": " \\mathbf{x}_1-\\mathbf{x}_2. ", "51ff1891c49b20f849bdfc17ebcf83fb": "d_h^{\\mathrm{(el)}} = \\frac{2F}{3b\\sqrt{h}} = \\frac{e F}{g \\sqrt{h}}. ", "51ff1b6222c67a19c7d5432419c4afde": "\tB_p =\\left \\langle s,h_p\\right \\rangle \\, ", "51fff57041d4b76d8b952b1a62cbf0ae": "\\mathcal{L}\\{f'\\}=s\\mathcal{L}\\{f\\}-f(0)", "52004e13ba3f40bcaeb0ad89c1857ed5": "\\text{cont} (q) =\\frac{\\text{cont} (p)}{c}.", "52007c10cc33353f23cf349046ab6aae": "\\psi^{(-3)}(1)=\\frac14\\ln(2\\pi)+\\ln A", "52009b8f513dae7dcf13054f65a2e20d": "\\left \\langle \\nabla_{ \\partial_i }\\partial_j, \\partial_k \\right \\rangle = \\Gamma^l _{ij} g_{lk},", "5200d7ff8b2830460bbedf510f57d75c": "10ms/km ", "520111e54c7bf27de7851c05d1570779": "\\begin{pmatrix} 0 \\\\ 2 \\\\ 19 \\end{pmatrix}", "520128173129ec044792e4d3e5e240c3": "\n \\vdash \\lnot A , A\n ", "520147030f8700b6fef1a2c650d66cfb": "\\frac{d^3W}{d\\Omega d\\omega }=2c\\varepsilon _0R^2\\left | \\vec{E}(\\omega) \\right |^2=\\frac{e^2}{4\\pi\\varepsilon_0 4\\pi^2 c}\\left | \\int_{-\\infty}^{\\infty}\\frac{\\hat{n}\\times\\left [ \\left ( \\hat{n}-\\vec{\\beta } \\right )\\times\\dot{\\vec{\\beta }} \\right ]}{\\left ( 1-\\hat{n}\\cdot \\vec{\\beta } \\right )^2}e^{i\\omega(t-\\hat{n}\\cdot\\vec{r}(t)/c)}dt\\right |^2 \\qquad (9)", "520165e3dc06d2befc8854035ec89188": "0 \\leq d_1 \\leq d_2 \\leq 1", "52016f06adf5dcdea2e163215c0b2138": "\n\\mathbf{A}^* = \\underset{\\mathbf{A}}{\\operatorname{argmax}} = \\frac{\\left|\\mathbf{A}^{\\text{T}}\\mathbf{M}\\mathbf{A}\\right|}{\\left|\\mathbf{A}^{\\text{T}}\\mathbf{N}\\mathbf{A}\\right|},\n", "52018fc7fa6292cfa972c5d48b6be23e": "w=u_1s_1u_1^{-1}\\cdots u_n s_nu_{n}^{-1} \\text{ in } F(A),", "5201a10c7c54a83fc4a2e4c7c71f1ef9": "r_1 - r_2 = \\pm\\sqrt{(-\\frac{b}{a})^2-4\\frac{c}{a}} = \\pm\\sqrt{\\frac{b^2}{a^2} - \\frac{4ac}{a^2}} = \\pm\\frac{\\sqrt{b^2-4ac}}{a}", "5201a5bb9c92f2c190dfdd4dfc2bf4cb": "vp_{sat} - vp_{air}", "5202261495902e0ec9e715365172eb63": "x^3\\sin x + 3x^2\\cos x - 6x\\sin x - 6\\cos x + C. \\, ", "520248985b0cc41c3f90cf4849f8f5c6": "G = \\operatorname{make-call}[V, \\operatorname{FV}[S]] ", "52029703abb491fb660b8e51560202e8": "L_s , s_o\\,", "5202c35f06ead52cc05040511f3829fd": " E_1\\cap(E_1\\cup E_2)=E_1", "5202eb8104f6afba6bc38dcacee9242f": "U(t_0)", "52031681c336c57513bc9c19e445f3c5": "\\begin{bmatrix}b_0\\\\b_1\\\\b_2\\\\b_3\\end{bmatrix} =\n\\begin{bmatrix}\n2&3&1&1 \\\\\n1&2&3&1 \\\\\n1&1&2&3 \\\\\n3&1&1&2 \\end{bmatrix} \\begin{bmatrix}a_0\\\\a_1\\\\a_2\\\\a_3\\end{bmatrix}", "5203698bf8ed1d7a4362e34220aa53fd": "\\textbf{Y}_{k\\mid k} = \\textbf{P}_{k\\mid k}^{-1} ", "52036bb37c9a325c2f83488fb375b734": "\\ C = \\sup_{p_X(x)} I(X;Y)\\, ", "5203700b55d535623e49096273e36611": "\\sum_{n=0}^\\infty p(n)x^n = \\prod_{k=1}^\\infty \\frac{1}{(1-x^k)}", "5203bb437c12761209e8891e443a0f63": "M_{2} \\equiv \\int d\\zeta \\ \\lambda(\\zeta) \\ \\zeta^{2}", "52040a4f2486c33e5287bf2f2a045cc1": " f(z) = \\sum_{a \\in E} p_a(z)", "52041f0b67f673ec670c37c22f0a5983": "K_{\\mathrm{eff}} =\\int_m\\frac{1}{2}u^2\\,dm", "52045813c2873183e6a917db6935bd40": "e^{\\mu+\\sigma^2/2}", "52045e9d5e05294ab2aed8ae365ac180": " \\big[q(W_2-W_1) + p(W_1 - W_0)\\big ]", "520466efd0971fd107ec33d6ea37a66f": "f(x; \\sigma) = \\frac{\\sqrt{2}}{\\sigma\\sqrt{\\pi}}\\exp \\left( -\\frac{x^2}{2\\sigma^2} \\right) \\quad x>0", "52046b0e4b11f7a861a638490dfefd61": "\\|\\mathbf{p}_B\\|=n_B", "5204b063965a5378fca673b059208514": " A (r, \\theta, \\varphi)= \\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell ( a_{\\ell m} j_\\ell ( k r ) + b_{\\ell m} y_\\ell ( k r ) ) Y ^ m_\\ell ( { \\theta,\\varphi} ) .", "5204e05ee035f70162adc6a511a80544": " \\nabla s = C_p \\ln({\\overline{T}\\over T_1}) +R \\ln({\\overline{p} \\over p_1}).", "5204fde33a37283fecb530a5edc7016a": "\\ln", "52053dc3ca4218b698d115408e557a44": "(2,3)", "5205956f3ea0c1d8b62f7133f21845a8": "H^s = \\{ v \\in S' | \\hat{v} \\in L^1_\\text{loc}(R^n), \\int_{R^n} |\\xi|^{2s}| \\hat{v} (\\xi)|^2 \\, d\\xi < \\infty \\}", "5205e29e7201a0b009466036b40a34a6": "G(T,p)=H-TS", "5205fe4075d3b6e4214cb13da48f5ecb": "\\sup_w |A^* w|^*_V/|w|^*_W", "520609864df6a92d98a3028386ed0799": "F_\\mathrm{f}\\,", "52064867797313575d205702a278bfd4": "Gcrd(L_1,L_2)", "5206560a306a2e085a437fd258eb57ce": "V", "52066a559e0cc3fe9d3c871b79f34008": "N\\bar\\psi(x)\\,\\gamma^\\mu\\,\\underline{\\psi(x)\\,\\bar\\psi(x')}\\,\\gamma^\\nu\\,\\psi(x')\\,A_\\mu(x)\\,A_\\nu(x')\\;,", "520725a212bb4a17d49b3b4fad2d1733": "Rec(W,S)", "520796668ece54d20d1ba0e31542fe84": "\\exp \\left[ -\\beta (U_{\\mathrm {spring}}+ U_{\\mathrm{internal}} ) \\right]", "5207ac74c14a294ce99dedd69ee80337": "\\mathbf{y} = \\lambda \\mathbf{x}", "5207b38b7ff5f1f437141f1e79208b69": "loaded(3)", "5207fc547da80ddc95beeb718d176576": "\\frac{\\partial \\lambda_i}{\\partial M_{(k\\ell)}} = \\frac{\\partial}{\\partial M_{(k\\ell)}}\\left(\\lambda_{0i} + \\mathbf{x}^\\top_{0i} ([\\delta K] - \\lambda_{0i}[\\delta M]) \\mathbf{x}_{0i}\\right) =\n\\lambda_i x_{0i(k)} x_{0i(\\ell)}(2-\\delta_k^\\ell).", "5208748170d787e6098e6d65adce590b": "\\mathrm{in}(f)", "520880314d67ec6aa7d309bbba562e5f": "R^{**}_{P_{d_h,\\gamma_h}}(t)= \\frac{{^{d_h}_{c_h}}P^{\\gamma_h}_h(t)}{^0_{c_h}P_h(t)} ", "520906f7df4bb0a4cbad486a09f1cf0d": "\\left( \\frac{1}{x} \\frac{d}{dx} \\right)^m \\left[ \\frac{Z_\\alpha (x)}{x^\\alpha} \\right] = (-1)^m \\frac{Z_{\\alpha + m} (x)}{x^{\\alpha + m}}.", "5209a1ec2511c1c0a79ff812a19629e5": "{\\tilde{C}}_{3}", "5209cd054bae174718ec608f1ca1cbba": "\\mathrm{NPV} = \\sum_{n=0}^{N} \\frac{C_n}{(1+r)^{n}} = 0", "5209e631610730ac47738cdd54803154": "(x,\\ y)\\,", "520a19cfa014a86dac662a9f3d8caf41": "ax^2 + bx + c =0\\,", "520a375d33f835e5506c81922597148a": "R = {{-f^{(3)}(c)}\\over {6}}h^2", "520a415a6292de6d76e910cf843a5383": "\\begin{align}\n C_1 &= \\mu_M - \\sqrt{\\frac{\\sigma^2_Mn}{2}},\\\\\n C_2 &= {\\sqrt\\frac{\\sigma^2_M}{2n}},\\\\\n \\end{align}", "520ab14f3c878641a267a09aadf89171": "f(x; \\theta)", "520ab31822ea10e87af2f4728d77cce0": "\\Delta H = K A", "520ab547ec348f74e0605e4f99291386": "\\tilde u_i:= (I-P) u_i", "520ac833c273b2ba3967a68addb7698c": "\\displaystyle{}_{r+1}E_r(a_1,...a_{r+1};b_1,...,b_r;q,p;z) = \\sum_{n=0}^\\infty\\frac{(a_1,...,a_{r+1};q;p)_n}{(q,b_1,...,b_r;q,p)_n}z^n", "520aff59651e71f45181b9339f42e2f4": "\\displaystyle |\\nabla u(x)|=F(x), \\ x\\in \\Omega", "520b0eb32935112c741785ce63ebfd0e": "U_1,U_2\\in\\mathcal{U}", "520b10b4488f97d3dabf0aa23ca003e3": "a_0,a_1,\\dots,a_n,\\dots", "520b297d1825582f8719da69b09a38ce": " \\mathcal A _P ", "520b8cd8f03d73f0d24b5c2ebfb12753": "\\,B = i\\phi_2 + B_1\\mathbf{i}+ B_2\\mathbf{j}+ B_3\\mathbf{k}\\qquad ", "520bb17b1a81b621bbe93e56861413e1": "\\gamma_{p,v}\\ ", "520c77d0fbadae5ed9a7b0f63bf2ea83": "\\Sigma_i z'_i = 0", "520c8a8c2ca9b9a0b32ae0869d5a37e2": "2^{i-1}", "520cb78226415202eca7d310fd9fdc5a": "\\mathbf{y}_i \\le \\mathbf{y^*}_i", "520d3f699dd6c747cb330852babe73ce": "V_{SC} = \\frac{-Q_t+qN_S(V_{SC})+qN_D(V_{SC})-qN_0}{C_\\Sigma}", "520d9291637ecd0e840d21b9d7078e74": " A^\\alpha = (\\varphi/c, A_x,A_y,A_z)\\,, ", "520ddc28a321e30cc02c6da5622abc18": "\\sum_{i=1}^{n} c_i'(x) y_i^{(j)}(x) = 0 \\, \\mathrm{,} \\quad j = 0,\\ldots, n-2.\\quad\\quad {\\rm (iv)}", "520e913c84f9ea59122389d7f5186287": "c_{l0} = 2 \\pi \\alpha", "520ed851d53d293fd3b104752a4e9f13": " f\\colon d\\to d', g\\colon c\\to c'", "520f1bc363a52da4da458f0efad113b9": "\\vec{A}\\bot \\vec{\\Omega }", "520f712c935009aa8d9feeb42b2bca7e": "0\\le x_{1},x_{2},x_{6} \\le 10", "520f7f730c88715a9682f0623c91211b": "\\begin{array}{ccc}\n \\mathbf{M}_f(A) & \\longrightarrow & \\mathbf{M}_f(\\hat{A}) \\\\\n \\downarrow & & \\downarrow \\\\\n \\mathbf{M}(A_f) & \\longrightarrow & \\mathbf{M}(\\hat{A}_f)\n\\end{array}", "520f8cc28bf9ca0e41f7772c67c00bcf": "2^{\\aleph_{\\alpha}} = \\aleph_{G(\\alpha)}\\,", "52100913e3df06089e7b55e73f1b8975": " \\lambda\\mathrm{d}\\ell ", "52101598373d3753d35ab5b55821f2c7": "\\textstyle \\mathbb{F}_2^{16}", "521033fb482115387527e40b338c426a": " \\gamma:\\mathbb{R}^k \\times \\mathbb{R}^k \\to \\mathbb{R}^{k\\times l} ", "52104ef59d4bc26486d6f0bfafb5ae63": "\\scriptstyle \\sqsubseteq", "5210c66f6cceed5c4661242d776b6783": "x_1(0) = a_1, x_2(0) = a_2, \\,", "5210e452c28a6d0bd07d068555011cf3": "\\begin{align} \\mathbf{T}^{(\\mathbf{n})} &= \\mathbf{T}^{(\\mathbf{e}_1)}n_1 + \\mathbf{T}^{(\\mathbf{e}_2)}n_2 + \\mathbf{T}^{(\\mathbf{e}_3)}n_3 \\\\\n& = \\sum_{i=1}^3 \\mathbf{T}^{(\\mathbf{e}_i)}n_i \\\\\n&= \\left( \\sigma_{ij}\\mathbf{e}_j \\right)n_i \\\\\n&= \\sigma_{ij}n_i\\mathbf{e}_j\n\\end{align}", "521143443e8fe1a56cac974986452824": "\\frac {Y''}{Y}+\\lambda =-\\frac {X''}{X} ", "521153902faa47f2f72cf4b8a92d009b": "\\alpha(x) = \\min_{S \\in \\mathcal F, x \\not\\in S}w(S) - \\min_{S\\in\\mathcal F, x\\in S}w(S\\setminus\\{x\\}). \\, ", "52116a3492ebe194c916297bf481c177": "\\frac{d}{cos(\\theta)} = d\\left(1 + \\frac{1}{2} \\theta^2\\right)", "52119c0c8706ae41bc665ca05668d97a": "x=50\\tfrac{1}{2}-49\\tfrac{1}{2}=1", "5212463e37406b73b693fe832f7bc8c2": "2^{64}", "52135ae719e97aad6d4b6b72d3e4ee49": " G^{+}(k) + K_{+}(k)\\hat{f}_{+}(k,0) = \\hat{f}'_{-}(k,0)/K_{-}(k) - G^{-}(k). ", "521365005f157ad8a9b909c5f225857f": "\\frac{mr^{2}}{L^{2}}", "521380ad72d07c373a81b3e3575884fc": "R^qf_!(F)=R^qg_*(j_!F)", "52138c7becc9f0384cc086d258a066d2": "Z(s) = \\frac{1}{sC}, ", "52138f2db56315bc1d6e435eaf5e1051": " \n \\int_0^{2 \\pi} {d\\varphi \\over 2 \\pi} \\cos\\left( \\varphi \\right) \\exp\\left( i p \\cos\\left( \\varphi \\right) \\right)\n=\ni J_1 \\left( p \\right)\n . ", "52139563f1f27254723c66afde79f612": "{[f(x)]}^{h(x)g(\\theta)} = e^{h(x)g(\\theta)\\ln f(x)} = e^{[h(x) \\ln f(x)] g(\\theta)}", "52139f730788d937a785fb7d45cd815f": "\\left( w_{v(1)}, w_{v(2)}, \\ldots, w_{v(d)}\\right)", "5213df26ae80cdd195d101eab63a315b": "\\{z:r<|z| 3\n \\end{matrix}\\,\\!\n ", "5219e96fba41b1de6f7c242180b17981": "(14.e)\\quad \\nabla^2\\Phi =\\,2\\nabla\\psi \\nabla\\Phi\\,,", "521a0ac69b236d73fc49ccb99765a78f": "\\scriptstyle \\frac{1+\\sqrt{5}}{2}", "521a0c10e09faf612addebefedce428b": "r \\in \\mathbf{R}", "521a9daa8c1cd6293c6e22e8e8386c42": "A^T", "521ab716bcb83dd7e6b3918b1c691f5d": "\\mathbf{C}_{1,2} = \\mathbf{A}_{1,1} \\mathbf{B}_{1,2} + \\mathbf{A}_{1,2} \\mathbf{B}_{2,2} ", "521aca611cbb381c0b152f06f4cd30a4": "N_1N_2", "521afc6a3e15400073c783caeb79d62c": "L^{p_0}(\\mu_1)", "521b21af71a2258ed4ea2aa2ce024b52": "\n F^m_{~\\gamma}\\left(\\,_{(X)}\\Gamma^\\gamma_{\\mu\\rho}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} + \n \\frac{\\partial }{\\partial X^\\rho}[\\,_{(X)}\\Gamma^\\gamma_{\\alpha\\beta}] -\n \\,_{(X)}\\Gamma^\\gamma_{\\mu\\beta}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\rho} - \n \\frac{\\partial }{\\partial X^\\beta}[\\,_{(X)}\\Gamma^\\gamma_{\\alpha\\rho}]\\right) = 0\n", "521b31c0f28ed6d011929f58298b0931": "K/\\mathbb{Q}", "521b71d67e59473595eeab55cf07cd41": "\\underbrace{\\left[\\begin{array}{cc}t&c^Tx\\\\c^Tx&d^Tx\\end{array}\\right]}_{D}\\geq 0", "521bdff5cd27c36888444e92b100218e": "\\left(f^{-1}\\right)^{-1} = f .", "521c36a31c2762741cf0f8890cbe05e3": "On", "521c69d8b455e0445a2a4346be5013bf": "\\gamma=\\sinh^{-1}\\frac{Z}{k}", "521c8b98f51b79e7f6801c7d13884f88": "G = (N, T, S, P)", "521d45393c9e4bd12133de0fa4f4dc13": "u_\\lambda(p) = \\begin{pmatrix}\nu_{-1}\\\\\nu_{+1}\n\\end{pmatrix} = \\begin{pmatrix}\n\\sqrt{E-\\lambda |\\vec{p}|} \\chi_\\lambda(\\hat{p}) \\\\\n\\sqrt{E+\\lambda |\\vec{p}|} \\chi_\\lambda(\\hat{p})\n\\end{pmatrix} \\,", "521d46ba1db0fa113485ca32f4bbe3ff": "H=1", "521d8aeec9e4fc12c3f966247c0673c5": " \\det(\\lambda~\\boldsymbol{\\mathit{I}} + \\boldsymbol{A}) = \\lambda^3 + I_1(\\boldsymbol{A})~\\lambda^2 + I_2(\\boldsymbol{A})~\\lambda + I_3(\\boldsymbol{A}).", "521d92b9424de8aeb0778486b2acd374": "V_2 = K{V_1}", "521d9740446761e4ca8039a4c49762f1": "\\mathrm{spt}(5n+4) \\equiv 0 \\mod(5) ", "521ddf230f7e777cd707ddd85c672586": "f(v_1,\\dots,v_m,\\alpha_1,\\dots,\\alpha_n) = T_f(v_1\\otimes\\cdots\\otimes v_m\\otimes\\alpha_1\\otimes\\cdots\\otimes\\alpha_n)", "521df336df9f84b6615b182c8580915f": "G = G(z) = { I(z) \\over I_{in} } ", "521ea81d7eb18ae66267c863d9390af1": "?\\times 0=6.", "521ec627cb53b829386f480a4e5e412b": "L_n(p, q) := p \\star q", "521ef5e3769cab4bbbbc0cca75ffb06c": "\\lim_{\\Delta P\\rightarrow 0}\\,\\!", "521f0fbe208e8727add093aff607d79f": "L = \\{ww | w \\mbox{ is a letter}\\}", "521f18f491579ef4f49036a2505c0abe": "y(t) = e^{L t}y_0 + L^{-1} (e^{L t} - 1) \\mathcal{N}( y( t_0 ) ).", "521f22bc9cd196e8807fd440ba76e8e3": "K(\\zeta,z) = \\overline{\\eta_z(\\zeta)}", "521f49efac8a90fab33626db2fdd1e27": "X(t)-m_1=h_1.\\,", "521ffe88b1eee8fac6a42320e69346f6": "\\Phi = (\\forall x')(\\exists y') Q(x',y') \\wedge (\\forall x)(\\forall y)( Q(x,y) \\rightarrow (\\forall u)(\\exist v)(P)\\psi )", "52205d5d092c512842b6f529bcbea745": "365\\mod 7 = 1", "5220b5c1edaa7e7ba5aff3a726393f73": "x_{ht}", "5220bf0fe33faf12f4354dd307b85a13": "E(e^2) = R_s(0) - \\int_{-\\infty}^{\\infty}{g(\\tau)R_{x,s}(\\tau + \\alpha)\\,d\\tau},", "5220e5f10769c1d094c03885043c325b": "\\frac{\\partial (\\mathbf{u} \\times \\mathbf{v})}{\\partial x} =", "5221e873cd09ab479ff56110ace42913": "A_1, \\ldots, A_r", "5221ebc2bd6710dc4be205bf196f6145": "\\!E_\\mathrm{h}", "52227a23278adf7fc931536d97e5ed70": "V_-", "522359592d78569a9eac16498aa7a087": "\\pi ", "522389f557e60fbe424075ccd6388ba3": "\\lambda_k ", "52238a16af15877daea2cd57bf7260d0": "R = \\{ r_1, r_2, \\dots, r_k, \\; | \\; r_i > 0\\}", "52239af9562a42a1c49ad5c8bf240630": "l < i \\leq Lk", "52239f78722ac7887467eaf5eb3a557a": "\\partial_\\alpha {\\star F^{\\alpha\\beta}} = \\frac{4\\pi}{c} J^\\beta_{\\mathrm m}", "5223b486f4c0a3620652d1fc84ed1a56": "O = \\phi_{1}+ \\phi_{2}- \\phi_{3}", "5223b85ac9c88e2f44a1360c99a69e8b": "g(\\mathbf{A})", "5223c4d52f0e3db27ee5a64220ecb60a": "(\\nabla \\cdot \\nabla - \\frac{1}{c^2} \\frac{\\partial^2}{\\partial t^2})\n{\\mathbf E}({\\mathbf r},t)\n= \\mu_0 \\frac{\\partial^2}{\\partial t^2} {\\mathbf P}({\\mathbf r},t)", "5223ce154becc680ac4ef6a08a7ad55d": "n \\times l", "52240bb500e6e946d0922ba6c17dbfc9": "\\varepsilon_{\\alpha + 1} = \\sup\\{\\varepsilon_\\alpha + 1, \\omega^{\\varepsilon_\\alpha + 1}, \\omega^{\\omega^{\\varepsilon_\\alpha + 1}}, \\dots\\} = \\sup\\{0, 1, \\varepsilon_\\alpha, \\varepsilon_\\alpha^{\\varepsilon_\\alpha}, \\varepsilon_\\alpha^{\\varepsilon_\\alpha^{\\varepsilon_\\alpha}}, \\dots\\}", "52241331218c56e3d0650d138aa7a4c4": "K_{bch}", "52248f37b856759e618e87460f2c2d45": "c_\\mathrm x\\,", "5224de75b3a61da1f4fab5f4003e945e": "k_{ij}", "5224ecd72eb656735bbf77942183d485": "(x, y, z) = \\left(\\frac{2 \\mathrm{Re}(\\xi)}{1 + \\bar \\xi \\xi}, \\frac{2 \\mathrm{Im}(\\xi)}{1 + \\bar \\xi \\xi}, \\frac{1 - \\bar \\xi \\xi}{1 + \\bar \\xi \\xi}\\right).", "5224f0145f132a9b3d0b93e75e25ae1c": "\\prod_a^b f(x) {}^{dx}", "5224f7bd8895c889ff3bee2d98ea42d4": "\\begin{matrix} \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty t^nW_x(t,f)\\,dt\\,df=\\int_{-\\infty}^\\infty t^n|x(t)|^2\\,dt \\\\\n\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty f^nW_x(t,f)\\,dt\\,df=\\int_{-\\infty}^\\infty f^n|X(f)|^2\\,df\n\\end{matrix}", "5224fb591becd4b351f63ddb0cb7c8f2": "a \\cdot W_0^a \\cdot \\text{E} R_1^a \\cdot \\text{E}R_2^a \\cdots \\text{E}R_T^a;", "52251553cda63a1b9b599691f84600f8": "\\mathrm{^{239}_{\\ 94}Pu\\ \\xrightarrow {4(n,\\gamma)} \\ ^{243}_{\\ 94}Pu\\ \\xrightarrow [4,956 \\ h]{\\beta^-} \\ ^{243}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{244}_{\\ 95}Am\\ \\xrightarrow [10.1 \\ h]{\\beta^-} \\ ^{244}_{\\ 96}Cm}", "5225a4c2b7e86029bc63ec2fe1d51905": "I_2\\,\\!", "5225b61070f37ee50edde3384b36cee6": "\\omega^{2^{p-1}} = -1", "5225bcf4cf2b99af4b481358e2d7e997": "\\Delta^n f(0)", "5225c5891f6e2f7a4fdd075de7bd5cb7": "\n|\\Psi_n(t)\\rangle =e^{i\\gamma_n(t)}\\,\ne^{-{i\\over\\hbar}\\int_0 ^t dt'\\varepsilon_n(\\mathbf R(t'))}\\,\n| n(\\mathbf R(t))\\rangle,\n", "5225f697a2527e24659cb152e1f3528d": "\\chi(G) \\le \\Delta(G) + 1. \\,", "522608b3dcd8383b76f88cb249bbf788": "\\begin{bmatrix}\np & t\\\\\nq & u\n\\end{bmatrix},", "5226173719dac1218e970c7d2e019697": "\\tau_{\\rm D}", "52262d41543eef244534dec04eb158e2": "p \\leftarrow \\mathrm{not}~p", "52265f593e287616a243b17e148b7d5b": "U \\times_S P", "522660f1788dd2c9961fc308836e281e": "J^rQ", "5226a1903f639bb9b94571c9928f2eea": "\\frac{\\partial u}{\\partial y} = \\frac{\\partial u}{\\partial r}\\frac{y}{\\sqrt{x^2+y^2}} + \\frac{\\partial u}{\\partial \\varphi}\\frac{x}{x^2+y^2} = \\sin \\varphi \\frac{\\partial u}{\\partial r} + \\frac{1}{r} \\cos \\varphi \\frac{\\partial u}{\\partial \\varphi}.", "5226bd7215e4b205d8e6592b8da2fb4a": "A_1 = A", "5227244379276dda062be32420ec9415": "\\rho \\int_{\\Omega} \\frac{d u_i}{d t} \\, dV = \\int_{\\Omega} \\nabla_j\\sigma_i^j \\, dV + \\int_{\\Omega} f_i \\, dV", "52272774f8c5fd83e5a217ddb8d0f2ab": "[K_\\mu, \\mathcal{O}(0)]=0", "522748524ad010358705b6852b81be4c": "ds", "5227611fb113dcd4d726d08908f31422": "p > e^{e^{24}}", "522769d736da48b63a4a53df48e15960": " \\langle q|\\mathbf{\\hat Q}|\\psi\\rangle = q\\langle q|\\psi\\rangle = q\\psi(q) ", "5227ed450dd8d8d692263aadf442a040": " v = \\omega r ", "522801766af080923e0eecd569528d29": "\\Vert x \\Vert = \\Vert\\Phi(x)\\Vert", "5228050d3f43aae6ac5d2d15022eb07f": " Q_j = \\mathbf{F}\\cdot \\frac{\\partial \\mathbf{V}}{\\partial \\dot{q}_j} + \\mathbf{T}\\cdot\\frac{\\partial \\vec{\\omega}}{\\partial \\dot{q}_j}, \\quad j=1,\\ldots, m.", "522805b142300cc302f7257bcd324e50": " \\alpha = \\frac{d^2\\theta}{dt^2}. ", "52281535b924cddb0504bfc4974fac58": " C_n(A,M) := M \\otimes A^{\\otimes n} ", "522854fe808e9bbd1e42901772f7ae4d": "V\\otimes_A V.", "522855995d58050bdfd7b407648792be": "\\alpha = ln(2)", "5228b98d430958c509f355eaf01a0d40": "z =\\frac{2+sT}{2-sT}", "5228d2590cd5334a36675606c8ce0121": "\\int_{X\\times Y} |f(x,y)|\\,\\text{d}(x,y)<\\infty,", "5228fbfbc363904be3e0045a5195928f": "0x + 0y = 0xy\\ ", "52293a9053c316c092244f04db19de3f": "\\lambda = \\chi^{-1}\\epsilon\\chi,", "522972395e53d342fdf862acd04964c3": "\\scriptstyle \\begin{align}\n \\scriptstyle \\alpha &\\scriptstyle \\,-\\, \\ln \\beta \\,+\\, \\ln[\\Gamma(\\alpha)]\\\\\n \\scriptstyle &\\scriptstyle \\,+\\, (1 \\,-\\, \\alpha)\\psi(\\alpha)\n \\end{align}", "5229db0660317cfa207b81d691f10ba3": "|q\\bar{q}\\rangle = |\\bar{q}q\\rangle", "5229e2e36e715f440a74e5ca15ecd8f7": " \\xi{\\left(\\theta\\right)} ", "522a167c99bfce45d2e50112b4ebd5be": "{ \\dfrac{p_{11}/(p_{11}+p_{01})}{p_{01}/(p_{11}+p_{01})} \\bigg / \\dfrac{p_{10}/(p_{10}+p_{00})}{p_{00}/(p_{10}+p_{00})}} = \\dfrac{p_{11}p_{00}}{p_{10}p_{01}}.", "522a9d1d3f6a3c56c79053400198b1db": "\\int_0^1 \\frac{x^n (\\log x)^n}{n!}\\; dx\n= (-1)^n (n+1)^{-(n+1)}.", "522b092011390442f2c3d1296cf7e58a": "\\phi_0(\\theta) = \\sin 0 = 0", "522b1fdf4df911c05c7c8babe6309975": "S(\\phi_x)", "522b92c3a7f22de12b817e839b7773f3": "\\displaystyle{T_c(U(D(\\varphi) + S(\\psi)))=U(D(\\varphi)-S(\\psi)).}", "522bb9359fd47a03ae5652691321c7dc": "x' \\in \\mathbb{R}^n", "522bba94dbbfc07fd241675a41762bc9": "x_\\perp", "522be928b14a0107a144c6f757f1a436": "\\scriptstyle (\\cdot)", "522bf47465a17fb30aa6507ef1ca4fcd": " \\mathcal{G}(n) = f_{R_2^\\omega(m_1)}(f_{R_2^\\omega(m_2)}(\\cdots(f_{R_2^\\omega(m_k)}(3))\\cdots)) - 2", "522c302eba1208dcf98d60e81ae520d3": "\\frac{X}{1 - X} = X + X^2 + X^3 + X^4 + X^5 + \\cdots", "522c3a60e9b4c4de0f8038dac77b014a": "F(f_1)", "522c4420d0154c7372ec395c2da55160": "\\Pi_G", "522c499edbb12c4b481e8f7f575543dc": "\nk \\partial_k \\Gamma_k\\big[\\Phi, \\bar{\\Phi}\\big] = \\frac{1}{2}\\,\\mbox{STr}\\Big[\\big(\\Gamma_k^{(2)}\\big[\\Phi, \\bar{\\Phi}\\big] + \\mathcal{R}_k[\\bar{\\Phi}]\\big)^{-1} k \\partial_k \\mathcal{R}_k[\\bar{\\Phi}] \\Big] .\n", "522c5557a2495835d225eff329962dce": " \\textbf{c} = \\textbf{m} \\pmod p ", "522cce5e178678ad53a908e3dd2ca893": " \\min(a_1, \\ldots, a_n) \\le F(a_1, \\ldots, a_n) \\le \\max(a_1, \\ldots, a_n) ", "522d6868b3d9c042d2137847a6551bee": "\\Phi = BA", "522daa5bb2ec8b17d4f07514fde63e5e": "x_m - x_n", "522de8501dd8cbee065e6a37037223f2": "g_{jk}(\\theta)", "522e00a833e699d3ea4c84cdd6ee3086": "\\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}\\ ", "522e201e3e2c1deec5ec2b67253f10ff": "y \\equiv \\pm (4a)^{m+1}", "522e4753719b4f01441d999559976a75": "0\\leq\\theta\\leq 180^\\circ", "522e6f1b3c8e3bc2ca57d1b87b811089": " \\{z \\mid z \\overline{r} + \\overline{z} r = 0 \\}", "522e74f1b5b406b5ccd906b55fd00aea": "G_{ab} \\, = \\kappa T_{ab}", "522e9f4da08a1447d21adec8832fabb7": "\\frac{d}{dt} L(t) = [P(t), L(t)]", "522f1147d4411cffbdf2a42b6c9272be": " E_{(v,N)}[n] ", "522f2f72f981a093dc94891cdac8677d": "(10x_1 + 9x_2 + 8x_3 + 7x_4 + 6x_5 + 5x_6 + 4x_7 + 3x_8 + 2x_9 + x_{10})\\mod{11} \\equiv 0. ", "522f38215399245443dbf1edbf4967cb": "{{P}_{\\theta }}g(u,\\xi )={{P}_{\\theta }}f\\left( \\frac{u}{s},s\\xi \\right)", "522f3843394b95ee43e0adeebeaeb508": "Div^0 (K)/P(K)", "523013a51c7b73e47aaa1de9986b3f82": "s, h \\models e \\mapsto e'", "523099f07ee9660421c4d184d49c4ea6": "(\\lambda x . x x x) (\\lambda x . x x x)", "5230a68009175889a8e7845779886ec3": "G_n(A)", "523102b1d18a4ce483ac309b7c58b05c": " \\mathbf{G}_2 \\supset \\mathrm{O}(3) ", "523120f4c1f15bf7a8f59aafaba28fee": "Y_{1..j}", "5231aaf8f32a98b51f1a8b050e7acca2": "R^T w = \\mu", "5231cb4b7df936e7da0b248078b6ef15": " H_2(p_b)", "523243de07271305d6a92da9f0bc7261": " f(x) = \\frac{\\varphi(0) - \\varphi(x)}{x^2}.", "52326944265a60e757ab52fbde818502": "\\scriptstyle ABCD", "52328cdc4f87e7a3373eaf735ce46434": "(f^{-1}(t_{j-1}) \\times [0,1]) \\cup H^{I(j)}", "5232a252bd0e8197eb00b40bcdaa01df": "EL(\\Gamma^*)=\\frac{2\\pi}{\\log(r_2/r_1)}=EL(\\Gamma)^{-1}.", "52330ffe9baa56268c1d8b1c9306a3a4": "gg^{-1}=g^{-1}g=e", "5233365a40bd435b71893ccf310aa395": "0<\\epsilon\\leq\\phi\\leq\\pi-\\epsilon", "523343b65cb158e9a5305cd9a3b3265c": "\\left (q+1 \\right ) r", "5233b3820d2d4709695fbac0b732355b": "E_{mass}=\\frac{m R_n + \\gamma * 6.43\\left(1+0.536 * U_2 \\right)\\delta e}{\\lambda_v \\left(m + \\gamma \\right) }\n", "523434c30012afeb3d942f55d3368079": " Y=\n\\begin{bmatrix}y_{p} & y_{p+1} & \\cdots & y_{T}\\end{bmatrix} =\n\\begin{bmatrix}y_{1,p} & y_{1,p+1} & \\cdots & y_{1,T} \\\\ y_{2,p} &y_{2,p+1} & \\cdots & y_{2,T}\\\\\n\\vdots& \\vdots &\\vdots &\\vdots \\\\ y_{k,p} &y_{k,p+1} & \\cdots & y_{k,T}\\end{bmatrix} ", "523469b98f1d56779680211260c549f4": "\\left[ \\Pi^{n}\\right] ", "523471d20382665ee583044c052d89b7": "\\partial^\\mu F_{\\mu\\nu}^a+gf^{abc}A^{\\mu b}F_{\\mu\\nu}^c=0.", "52348357d5345c6b6d0234e2f8253c4f": "K_{k} = \\textbf{P}_{x_{k}z_{k}} \\textbf{P}_{z_{k}z_{k}}^{-1}", "5234b592a7e6bdad2884e8f79a6f75d8": "X^Tr = \\mu", "5234b7cdd3927dd60ca2d1a554778082": "L(v_1) = L(v_2)\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;L(v_1-v_2)=0\\text{.}", "5234b7f11bdd77e30b42638e931d16f7": "\\mathcal L=\\frac{i}{2}\\overleftrightarrow{\\overline\\Psi\\gamma^\\mu\\partial_\\mu\\Psi} - m(\\sigma)\\overline\\Psi\\Psi-U(\\sigma)+\\frac{1}{2}(\\partial_\\mu\\sigma)^2. ", "5234b912f921172bf254dd36df38f75f": "b^{(a-1)/2}\\equiv -1 \\pmod a\\;", "5234ce179481f9f661dac2d00e3e3ce4": "I_{simp} = r \\cdot B_0 \\cdot m_t", "5234effc6ede8bc1ceb5aafbb2193818": "C_{\\alpha-1}(x) - C_{\\alpha+1}(x) = \\frac{2\\alpha}{x} C_\\alpha(x)\\!", "52350949bf25d8e950c34f2b56d2f521": "\\vec {t}", "52352e70811b4c4728f3160a15828d98": "wp(a,p)\\,\\!", "5235414d3ccfd9e0c0c201b3d6c309e2": "\\frac{h^2 n^2}{8 m L^2}", "523570abcd54b034618061aa585e1f28": "S_p(x_0,\\dots,x_n) = {f^{(n)}}^{-1}(n!\\cdot f[x_0,\\dots,x_n])", "52361a3a08d7df65e2743f06ac84cb1a": "\\bold{F}=\\mathrm{d}\\bold{A}+\\bold{A}\\wedge\\bold{A}", "52363926a4a7ba998d4da53409b5438a": "\\ \\mathcal{L}_\\mathrm{int} = \\frac{e}{\\hbar}\\bar\\psi(x) \\gamma^\\mu \\psi(x) A_{\\mu}(x) = J^{\\mu}(x)A_{\\mu}(x)", "52366f6cc14fd8a2cc32ef91824a9494": " dB^2 \\rightarrow E(dB^2), ", "52367120b6eb9ec3278f1d17d26fbedf": " \\Delta \\omega = 2 \\alpha \\,", "52369f11caa0636c135bf5886594ff51": "O_i", "5236f908d43f92846645d03f6d66cbe5": "\n \\boldsymbol{\\nabla}\\cdot\\mathbf{v} = \\boldsymbol{\\nabla}^2 f = \\cfrac{\\partial^2 f}{\\partial r^2} + \n \\cfrac{1}{r}\\left(\\cfrac{1}{r}\\cfrac{\\partial^2f}{\\partial \\theta^2} + \\cfrac{\\partial f}{\\partial r} \\right)\n + \\cfrac{\\partial^2 f}{\\partial z^2}\n = \\cfrac{1}{r}\\left[\\cfrac{\\partial}{\\partial r}\\left(r\\cfrac{\\partial f}{\\partial r}\\right)\\right] + \\cfrac{1}{r^2}\\cfrac{\\partial^2f}{\\partial \\theta^2} + \\cfrac{\\partial^2 f}{\\partial z^2}\n ", "5237346aeb47ca51fb3e5164f96adcab": "\\frac{1}{\\sqrt{1+ \\frac{1}{\\varepsilon^2}}}", "523736b54c105255e91fb2c0d41c6dad": "\\frac{\\rho}{R}=lk,", "5237d38f1aa565f3bd6a45de3f24ef54": "\\Delta(c)=\\sum_i c_{(1)}^{(i)}\\otimes c_{(2)}^{(i)}.", "5237dab0f88940d92680e7e8f21aec18": "k=1,2,\\ldots,2^{2n}+1", "523857bc56e089f997229e91f5bf92ac": "w \\equiv e^{i\\pi \\theta}", "52386172135f1399e845307c12dd0ec7": "=\\frac{1}{2}\\frac{m}{L^3}v^2\\left[\\frac{y^3}{3}\\right]_0^L", "5238c0e48cf5caed3dfe7e1a96178c5b": "P(k)", "5238e4bb3cbbe394b14e9dc7866a304b": "\\bar{G}_i^{vap}", "523906f532ddd3d8153dc158a8d433a8": "\n (H_{n}x)(t) = \\int_{a}^{b}\\cdots\\int_{a}^{b}\n {h_{n}(\\tau_{1},.\\,.\\,,\\tau_{n})\\prod^{n}_{j=1}{x(t - \\tau_{j}) d\\tau_{j}}},\n", "523960cbb901992459ec34c4f721795f": "\\lim_{\\varepsilon\\rightarrow 0^+} \\int_{-\\infty}^\\infty dE\\, \\int_0^\\infty dt\\, f(E)\\exp(-iEt-\\varepsilon t)", "5239db5fedff9c061f5a6a09d24d4bd3": "T^*\\mathcal M", "523a9b15c22fc5b2d512efb403b23b21": "\\epsilon \\ ", "523accb538201dc3de730112799f29b2": "\\{m \\setminus L \\,\\vert\\; m\\in M\\}", "523ae37a0eaebb3f5bf541989d72b133": "(513 \\cdot 537)^2 \\mod 84923 = 2^{10} \\cdot 3^2 \\cdot 5^4 \\cdot 7^2 ", "523b190fd0704f237f90f235182d530b": "(x,w) \\in R", "523b8ea234bf1826c4c41645a291edb1": " \\operatorname{inc}\\ (\\operatorname{inc}\\ (\\operatorname{inc}\\ \\operatorname{const})) = \\operatorname{value}\\ (f\\ (f\\ x)) ", "523b91c6b92b1774bd745f6f50acbc11": "+ \\ln\\Gamma_p\\left(\\eta_2+\\frac{p+1}{2}\\right) =", "523ba01993bbe3e0b7e3755fdcbee52a": " V_+\\oplus \\mathfrak g_0\\oplus V_-,", "523bb99e02fe50775594d41b6a15801d": " (I_n \\otimes A + \\bar{A} \\otimes I_n) \\operatorname{vec}X = -\\operatorname{vec}Q, ", "523bf4a3a7c4e2e6d318a12012834cda": " y\\preceq x", "523c628d78ac02f3c19c3f844acd58d6": "\\begin{matrix} \\frac{6}{1326} = \\frac{1}{221} \\end{matrix}", "523c7fcd96211a2614ecbaaf62dcd649": "\n 3s=b-a, \\,", "523c8309d1199caae406f175e8692ba6": " \\sin 2 \\theta_p = - 2 \\frac{\\sqrt{ac}}{b} ,", "523cc2c677d8f8d8d57d891a522142c0": "v = \\int \\frac{dv}{dx} \\,dx", "523d67035a6d57d2110cc239f1f00804": "T \\, dS=\\delta Q\\,", "523d87cce3e8130932f10e482032cbb3": "0=\\sum^n_{i=1}\\sum^n_{j=1}{k_{ij}}", "523dbe38a5e13d0c3d4044733fdd4961": "=\\kappa_p(V)\\|(D-\\tilde{\\lambda} I)^{-1}\\|_p \\|\\mathbf{r}\\|_p.\n", "523edfaeb3fff78e69bd9ad0dd6813eb": " \\lim_{N \\rightarrow \\infty} \\left(1 - \\frac{i}{\\hbar}\\frac{\\Delta \\theta}{N} \\hat{\\mathbf{n}} \\cdot \\widehat{\\mathbf{L}} \\right)^N = \\exp\\left( - \\frac{i}{\\hbar}\\Delta \\theta \\hat{\\mathbf{n}} \\cdot \\widehat{\\mathbf{L}}\\right) = \\widehat{R} ", "523f1e56ab066fb6f5873ac9d88da299": "\n p = \\tfrac{1}{3}~I_1 ~:~~\n q = \\sqrt{3~J_2} = \\sigma_\\mathrm{eq} ~;~~\n r = 3\\left(\\tfrac{1}{2}\\,J_3\\right)^{1/3} \n ", "523fa8379fe3dcb1bbbba0a23600716c": "...\\longrightarrow H^n(E)\\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\pi_*}H^{n-k}(M)\\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!e\\wedge}H^{n+1}(M)\\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\pi^*}H^{n+1}(E)\\longrightarrow ...", "523fd42296a88bcd6ee83071828087c8": " t_c = \\sqrt{\\frac{a}{c}} ", "523fdc86cb2d8acd121f9c475f4e98f3": "\\Omega^{(1,0)}\\mathbb{C}^n = \\mathrm{span}(dz_1,\\dots,dz_n).", "52402705fc0463a1ba209a79eeb46bc1": "X_\\text{max}", "52404a7dfc803060dac6b7094f074c62": " Df = \\sum_{n \\geq 1} a_n n X^{n-1}.", "52405011aadf91d03cc35fefed641a84": "20\\sqrt 3", "52406c6585d04f58352193da51260761": "\\{\\ \\Delta P_2\\}", "524092ebc83bf32011f04778d2af2523": "\\begin{align}g(t) &= \\log(\\operatorname{E}(e^{tX})) = - \\sum_{n=1}^\\infty \\frac{1}{n}\\left(1-\\operatorname{E}(e^{tX})\\right)^n = - \\sum_{n=1}^\\infty \\frac{1}{n}\\left(-\\sum_{m=1}^\\infty \\mu'_m \\frac{t^m}{m!}\\right)^n \\\\\n&= \\mu'_1 t\n+ \\left(\\mu'_2 - {\\mu'_1}^2\\right) \\frac{t^2}{2!}\n+ \\left(\\mu'_3 - 3\\mu'_2\\mu'_1 + 2{\\mu'_1}^3\\right) \\frac{t^3}{3!}\n+ \\cdots .\n\\end{align}", "5240b9d813ef4472db5c2bb1a720416b": "\\frac{\\partial f}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\det(\\boldsymbol{A})~\\text{tr}(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})= \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T : \\boldsymbol{T}.", "5240e389cd78ed1a447a7d1ab178cb2e": "Z_\\mathrm {i T} = \\sqrt {Z^2 + \\frac{Z}{Y}}", "5240ec44fdbcac583aa50859e1a1b0d8": "\\overset{\\nwarrow}{\\ }", "524116f795224a4beb36e17750a25a14": "\\lambda=1/p", "52411b553d087933eeb8965db85da217": " \\textrm{Magnification} = \\frac{1}{(1-(\\frac{t}{n})P)}\\cdot \\frac{1}{(1-hF)} ", "524192b88faf8e91b100f2ca77edbd06": "\n\\varphi = \\text{sgn}(b)\\arccos \\left(\\tfrac{a}{\\sqrt{a^2+b^2}}\\right)\n", "5242210a33590eebfca907e3a340fdf0": "y(t) = A\\cdot x(t-T)", "524255a706d173e3d4caf2cbdd35f4c5": " k = -C_\\mathrm{w} \\frac{L}{A} \\frac{\\Delta m}{\\Delta t} \\frac{\\Delta T_\\mathrm{water}}{\\Delta T_\\mathrm{bar}}", "5242753a4b0e472c5b523860dd06caa9": " F_b= -k_{eq}\\left( \\frac{k_2+k_1}{k_2} \\right)x_1 \\,", "52427b5052f532a8031598c4f3932f2d": "S + S\\subseteq S", "5242849c754fc47ca9c3791345eebdd5": "{\\partial \\det(A) \\over \\partial A_{ij}} = \\sum_k {\\partial A_{ik} \\over \\partial A_{ij}} \\mathrm{adj}^{\\rm T}(A)_{ik} + \\sum_k A_{ik} {\\partial \\, \\mathrm{adj}^{\\rm T}(A)_{ik} \\over \\partial A_{ij}}.", "5243a7fbd350c5b21748afd90a097657": "\\color{RedOrange}\\text{RedOrange}", "5243acfe12641ad45e87cd194d13f812": "A=\\begin{bmatrix} 0 & a \\\\ -a & 0 \\end{bmatrix}.\\qquad\\operatorname{pf(A)}=a.", "5243b00828ef84c8320b2d0a7caa27b3": "{\\alpha} = 1", "5243ed6fd50d0f75255e31a7b187edd7": " \\Delta_\\mathrm{f}H^o", "52440234052c4341662044d2402dd9eb": "U/V \\ll 1", "52440c6b424e6182e5c9d6836c705e2b": "\n[\\hat{x}, \\hat{p}_x] = [\\hat{y}, \\hat{p}_y] = i\\frac{\\hbar}{2}\n", "524425e56f6dbb53c2ed8f5e5d2676a6": "q_c \\,", "52445a2d0543591cc3c85a4137319733": "\\tfrac{3K-E}{6K}", "52449a32ac24838b5a93360ffbc6459b": "(p,p^2),", "52449c8d557b66c62a5044f8bb7a89dc": "(p-v)F(p) + (q-v)F(q) + (r-v)F(r) = 0", "5244a0617c383a401e96e6f28aa33482": "\\begin{align}\\operatorname{VAR}(S)=&\\frac{2(1728 - 24 - 192)+3(144 - 16 - 48)+ 60}{18} \\\\\n&+\\frac{(24 - 48 + 24)(192 - 144 + 24)}{9 \\times 12 \\times 11 \\times 10} \\\\\n&+\\frac{(16 - 12)(48 - 12)}{2 \\times 12 \\times 11} \\\\\n&= 185.212\\end{align}", "5244b63149e063f1fb2743a5d5f54ae5": " T_8(x) = 128x^8 - 256x^6 + 160x^4 - 32x^2 + 1 \\,", "524502c69a39c6bfa05029ffb497ebd7": " \\partial_{\\mu}F^{\\mu\\nu} + 2 \\epsilon ( b_\\mu \\times F^{\\mu\\nu} ) = J^\\nu", "52458cfd8b3ba4848c282a61ab60ac89": "\\mathbf{v} (\\mathbf{r},t) = \\mathbf{v}(x,y,z,t) = [v_x(x,y,z,t), v_y(x,y,z,t), v_z(x,y,z,t)]", "524592445f689d41c2f99b3ddd938ab0": "i=1,\\ldots,n\\,", "5245a91249c886a690ce358c1ecbd673": "\\tilde b_i = b_i \\tilde b_{i - 1} - \\tilde c_{i - 1} a_i\\,", "5245c781382ca5893b3335de89aa5f33": "\\sigma (\\varepsilon)", "5245e565c86682a75e48ead04f4286e1": "TdS \\ge \\delta Q ", "5245f24bb6c423e7097370adedebd51b": "\\mathbf{a}_1 = (12, 6, -4)^T", "5245fce9001a0d02a62f15dfde6aac64": " T_r\\equiv \\frac{1}{r^2} \\frac{d}{dr}r^2\\frac{d}{dr} = \\frac{1}{r}\\frac{d^2}{dr^2}r.", "52462a82087987ecfdf5478a4f691ff0": "\\eta_Y=\\Phi_{Y,FY}(1_{FY})", "5246c93457d2e3e8f871a3209a76de6e": "C = n_\\text{c} - n_{\\mathrm{\\overline{c}}}.\\ ", "5246ebaa7ebff65c88256a46f7fbc91d": "\\mathcal{G}_\\pi", "5246f5cb8e0998c1e3f180c26ac7d780": "R[t_1, t_2, \\dots, t_n]", "524719a5e58fe3733be5df4460e5f31e": "Q = \\frac{f_r}{\\Delta f} = \\frac{\\omega_r}{\\Delta \\omega}, \\,", "524732cec29658fdf7a1db579a87c9d1": " u=\\frac{2gr^2}{9\\eta_2}(\\rho_2-\\rho_1)\\left (\\frac{3\\eta_1+3\\eta_2}{3\\eta_1+2\\eta_2}\\right)\\!", "524745c478bc90c1ab125094159a4ba4": "\\begin{pmatrix}\n1 & 4 & 0 & 0 \\\\\n0 & 4 & 1 & 0 \\\\\n0 & 0 & 3 & 4 \\\\\n0 & 0 & 0 & 3 \\\\\n\\end{pmatrix}", "5247541ef0c485ba70178cc6568ecd50": "\\hat X [k]=X(z_k),\\quad k=0, 1, ..., N-1,", "52478fa07eb97abf0ed2360042a51a4b": "\\Pr(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots, X_{n-m}=x_{n-m}) = \\sum_{r=1}^{m} f(x_n,x_{n-r},r)", "5247cf13c20ebbc892f6f70f7a08e64d": "\\theta=(J_M-1)\\nu/(1-\\nu)\\approx J_M\\nu", "524828e8c71816c9cf79d6b09a8fba40": " T_{a} = c_1 M ", "52483374b5b079cfee1644474f71e71e": "\\mathbf{x} = \\mathbf{G} \\mathbf{p} =\n\\begin{pmatrix}\n 1 & 1 & 0 & 1 \\\\\n 1 & 0 & 1 & 1 \\\\\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 1 & 1 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} =\n\\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\\\ 2 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} =\n\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} ", "52483cba4315652287020055f84b7e23": "x= \\frac {ax'+1}{x'} ", "52484230610ea808f24029bba6ec2935": "\\tilde{U}\\, =\\, U\\, +\\, \\frac{M}{\\rho\\, h}\\, =\\, U\\, +\\, \\frac{E}{\\rho\\, h\\, c_p}\\,", "524848a6ce67852022eb6fc061c36f4f": "|\\Delta_K|^{1/2}\\geq \\frac{n^n}{n!}\\left(\\frac{\\pi}{4}\\right)^{r_2} \\geq \\frac{n^n}{n!}\\left(\\frac{\\pi}{4}\\right)^{n/2}.", "524883320555bb229516c41510fec2e5": "\\begin{cases}\n\\dot{\\mathbf{x}}_1 = f_1(\\mathbf{x}_1) + g_1(\\mathbf{x}_1) z_2 &\\qquad \\text{ ( by Lyapunov function } V_1, \\text{ subsystem stabilized by } u_1(\\textbf{x}_1) \\text{ )}\\\\\n\\dot{z}_2 = u_2\n\\end{cases}", "524887c221997a0597fcfdd712883079": "\\Delta^n(i) = \\mathrm{Hom}_{\\mathbf{\\Delta}} ([i], [n])", "52489f8614eb5664db32049ef7de8b90": "\\varphi(\\vec p)=-\\vec p", "5248b1a83c26e3915f1009b186bc3b6e": "0\\le l \\le 1", "5248c8eeb1c27a53849bb64a446d4a2d": "\n\\begin{align}\nw(c_1\\mid c_1+c_2) &= w(c_1) + w(c_1 + c_2) \\\\\n& \\geq w(c_1 + c_1 + c_2) \\\\\n& = w(c_2) \\\\\n& \\geq w(C_2)\n\\end{align}\n", "5248d010824f77fb2f2aafca32adb534": "y = px - f^\\star(p)", "5248e2fc4eb7e487bd734c0e75db03c8": "M_o", "5248f40988b5d2a1b57775040cfbb91e": "\\int\\limits_0^1 {x^4(1-x)^4 \\over 1+x^2}\\,dx = {22 \\over 7} - \\pi\\!", "5249404ca2ac4473e28eae7c4c8d223e": "\n\\left( \\frac{\\partial}{\\partial_z} - A - X \\frac{1}{z-B} D^t \\right)\n", "52495e37b9abfef661063dbe28866a57": "u_1\\equiv\\,u_2", "52497eee6635c891c1b5245cd646ae3b": "v_\\alpha", "5249a15445d0f229adcc49536a287b50": "\\psi(x_1, x_2, \\dots, x_j, \\dots,x_N)", "5249e8d777e2c73fa8edc78b8d3a96c7": "\\displaystyle A_i=a_1^{2-i}a_2^{i}= b_1^{2-i}b_2^{i}", "524a0b74d1a34650e194612470584d45": "F\\left(x, y, y', y'',\\ \\cdots,\\ y^{(n)}\\right) = 0", "524a50782178998021a88b8cd4c8dcd8": "<", "524a591bc27479c5a33f42572d99e162": "|\\chi 2 \\rangle= (|1,0,0 \\rangle - |2,1,1 \\rangle)/ \\sqrt{2} \\frac {}{} ", "524a85be7abc446456a376f5772d1752": "\\textstyle\\pi = 16\\arctan \\frac15 - 4\\arctan\\frac1{239},", "524aa83f2a377964d9475bee31c62e2c": "S_G\\;", "524ae47ad4b39c5b076f6dbfefa42490": "[12,6,6]_3", "524aeda603f9b286bb6b9994ef4d04de": "\\bold{x}=A^g \\bold{b} + (I-A^gA)\\bold{w}", "524b94b10ad03a44c43c65ed5d1cc4ea": "\\textstyle y_{11} = y_{22}", "524bca5bad19bb698ed51b7a83a62d29": " {1\\over \\rho} {\\partial p \\over \\partial y}=0 ", "524bd18d02cc03619175657daa1a0061": "\\exists z \\exists w ( z \\not = w) \\vdash \\forall x (x \\not = x),", "524c225996e48858284fc809333a6307": "h:S^2 \\times S^2 \\to S^2 ", "524c7dbec4039819a01029cf834e1d9e": " \\partial_\\gamma F_{ \\alpha \\beta } + \\partial_\\alpha F_{ \\beta \\gamma } + \\partial_\\beta F_{ \\gamma \\alpha } = 0 ", "524c9fa55a156f893db7f15f938b3890": " \\left(\\frac{a}{\\alpha}\\right)_l=\\left(\\frac{\\alpha}{a}\\right)_l ", "524ca9764431452b8b43b8440118ace1": "GF(q)^n", "524ce044c402a61b400bc14273b8cdda": "N \\rightarrow \\infty", "524d0e56d041a830146652597450a69d": "2s_n = \\frac22+\\frac24+\\frac28+\\frac{2}{16}+\\cdots+\\frac{2}{2^n} = 1+\\frac12+\\frac14+\\frac18+\\cdots+\\frac{1}{2^{n-1}} = 1+s_n-\\frac{1}{2^n}.", "524d1ba83bf8f3e4b70cef8c4e354f11": "\\{s_n\\}", "524d22cdfd5f40f4116bbd0835b3c385": "\nc_{ij} = \\frac{1}{N-|i-j|} \\sum_{t=1}^{N-|i-j|} X(t) X(t+|i-j|).\n", "524d799332a7be240320bc823b9068c2": "\\max(0,x-c)", "524d8d79fb8e962a4fcd4e178a8f0764": "V(t)= \\left\\langle a( 1+\\cos(t) ), a\\sin(t), 2a\\sin\\left(\\frac{t}{2}\\right) \\right\\rangle. ", "524daa760363032363c68268031c9eca": "A = (a_{i,j})_{1 \\leq i, j \\leq n}", "524db2f5bd3d594bbdb71fe361b1ce1c": "\\int x^n e^x \\,dx,\\,\\int x^n\\sin (x) \\,dx,\\,\\int x^n\\cos (x) \\,dx\\,,", "524dcec35781adb66a2d7aca918411ff": " B = 1 - \\frac{ 2 }{ \\pi } arctan( \\theta ) ", "524dda42a28b7c38d48751fa4e2b9152": "F \\approx \\chi_a F_a + \\chi_b F_b,", "524e12a151176f4d2a3fd67a59cd568c": "w=1/2", "524e21d22389de801ebc220bd54ab55d": "\n\\nabla W_\\lambda\\chi_E(x) =\n\\mathrm{grad}W_\\lambda\\chi_E(x) =\nDW_\\lambda\\chi_E(x) = \n\\begin{pmatrix}\\frac{\\partial W_\\lambda\\chi_E(x)}{\\partial x_1}\\\\\n\\vdots\\\\\n\\frac{\\partial W_\\lambda\\chi_E(x)}{\\partial x_n}\\\\\n\\end{pmatrix} \n\\Longleftrightarrow\n\\left\\vert DW_\\lambda\\chi_E(x)\\right\\vert = \n\\sqrt{\\sum_{k=1}^n\\left|\\frac{\\partial W_\\lambda\\chi_E(x)}{\\partial x_k}\\right|^2}\n", "524ece0da9d85df730fcd7855101052d": "C_\\beta' = C_\\beta\\begin{bmatrix}0&-k\\\\1&0\\end{bmatrix}.", "524ee7b016d0bf8cb4515957321f3ccd": "dr_t = a(b-r_t)\\, dt + \\sigma \\, dW_t", "524f0b2ce05ee15d591d8a7f76180a07": "\\mathbf{x}=0", "524faa816b24b6ec59e55e7a301cc0e9": " (x'_1,\\ldots,x'_m) ", "524fd6dbf073f12b4b69a1dfdaa10f1b": "K_{G}^{(a)} {\\left| G \\right\\rangle} ={\\left| G \\right\\rangle} ", "525014f43e5863416632d255cb244780": "b \\to [k]b\\,\\!", "525044f8f722b78956a09a5988eba453": "\\mathcal{B}_{\\epsilon}(X^*_{b(X^*, X)}, Y^*_{b(X^*, X)}; Z)", "5250466c66f06eaaac856064453b3c88": " H(P) = P^3 = PPP", "5250515bfac149e7ebc1a42f8e204b79": " e^{- \\lambda}\\, ", "52509fa6bfbb753138a20f6d858b22e0": "f = f\\chi_{\\{|f|>1\\}} + f\\chi_{\\{|f| \\leq 1\\}}", "5250a4f6f8e1183f7526a8c30289f7c3": "\n\\langle k\\cdot v, w \\rangle = \\langle v, k^{-1}\\cdot w \\rangle\n", "5250e5da00e50d6c0a1e55c4302314d4": "\\gcd(f(x),a(x)+1) = \\prod_{i \\in B} p_i(x),", "5250fe4ae42a33eaafcafc4cbcb27709": " F_{diff} = f_{slow} exp (-bD_{slow}) + f_{fast} exp (-bD_{fast}) \\,", "52514c8b373bfbdbea4d6fa2f221a2be": "\\,l_0", "52519c750c8b4a388d147bb635b5bcbe": " \\acute{f}^{\\mu} = - 8\\pi { G \\over { 3 c^4 } } \\left ( {A \\over 2} \\acute{T}_{\\alpha \\beta} + {B \\over 2} \\acute{T} g_{\\alpha \\beta} \\right ) \\left ( \\delta^{\\mu}_{\\nu} + \\acute{u}^{\\mu} \\acute{u}_{\\nu} \\right ) \\acute{u}^{\\alpha} \\acute{x}^{\\nu} \\acute{u}^{\\beta} ", "5251b27f2c41cb82790a5b071da3efd9": "O(c^4) + O(a^3 b) + O(a b^3) . \\,\\!", "5252cfe51ed03761a9fa21e8eeea9844": "\\mathrm{AND} = \\lambda x^{\\mathsf{Boolean}} \\lambda y^{\\mathsf{Boolean}}{.} x\\, \\mathsf{Boolean}\\, y\\, \\mathbf{F}", "52532b72a0656d101cb9aa2729d6edc2": "\nL(x,v)\n= \\tfrac{1}{2}\\|v-b(x)\\|_x^2\n+\\tfrac{1}{2}\\operatorname{div}\\, b(x)\n- \\tfrac{1}{12}R(x),\n", "52532d4e7713be793042655d5d806b7a": "\\textstyle f(\\Omega_1) ", "52533dde878a7e5fd909a93ea0481732": "|\\mathcal{X}|", "525351d7e2d733f4bcc82a0516b0551a": "\\Pi = R-C ", "525381fb7eba229bab345d5034495d0e": "\\mathcal{P} ", "5253d557a3b8c4bd6344837b0cec3ec5": "C[-\\pi,\\pi]", "52544b36d7cb500db055e5dfca784cb5": "Q_z", "52545b8ed31af37f9814324e0f52d42b": "(p+\\epsilon)", "52546d594690c5795b789c99e35271bb": "z^4 + 2cz^2 - z + c^2 + c = 0", "525487c91b7e5a4c6152c146793b9e65": " |\\uparrow \\uparrow \\rangle, 1/ \\sqrt{2}(|\\uparrow \\downarrow \\rangle +|\\downarrow \\uparrow \\rangle) ", "52548f91c5b25021360263e6076d988a": "K(k(3))", "525570777763eb2ed278ff8aeb3ec206": "\\nabla \\times \\mathbf{H} = \\sigma \\mathbf{E} + \\epsilon \\frac{d\\mathbf{E}}{dt}", "525595469fcc0f38fb9c152aaaf81206": "\\det(A-\\lambda I) = 0", "5255fd9077313fb974a5e4e34e384352": "E[(X(t) - E[X(t)])^3] = (2 \\theta^3 \\nu^2 + 3 \\sigma^2 \\theta \\nu)t ", "52564113afce37312d20544030859de8": "(n-1)/(n+1)", "5256812080e9ffd62aa5ae99e41c3aba": "\\{2,3\\},\\;\\{1,3\\},\\;\\{1,2\\}, ", "52569337437d328768285e62926904fb": "\\varepsilon \\rightarrow \\varepsilon + i\\sigma/\\omega", "5256b5fa2db3eddb8986cdff2ae5f254": " dW(t) ", "52573af7619ee85fb5a33997b3a73999": "\\int_a^b U\\,dV+\\int_a^b V\\,dU=U(b+)V(b+)-U(a-)V(a-),", "52575d831568365e65f79dc3aa347557": "\\begin{alignat}{2}\n\\mathbf{ij} & = \\mathbf{k}, & \\mathbf{ji} & = \\mathbf{-k}, \\\\\n\\mathbf{jk} & = \\mathbf{i}, & \\mathbf{kj} & = \\mathbf{-i}, \\\\\n\\mathbf{ki} & = \\mathbf{j}, & \\mathbf{ik} & = \\mathbf{-j}, \\\\\n\\mathbf{i}^{2} & = \\mathbf{j}^{2}& = \\mathbf{k}^{2} & = -1\n\\end{alignat}", "5257765df6f2b6c709cbfa6e754712e1": "T(A,R)|_{R = T(A,R)}.", "5257b477e89e36594ec58fd6ac0b1b1a": "\\mathit{RRA}(w) =-\\frac{wu''(w)}{u'(w)}", "5257e29a7bd72f38bc5233b1f846a636": "\\left.\\right.\\omega_f(\\delta) = \\max_t \\omega_f(\\delta;t)", "5257eeee7697bfb5d3edcf6778bddef0": "\n \\displaystyle \n w(n,g) \n =\n \\sum_{k_1=0}^{n}\n \\sum_{k_2=0}^{n-k_1}\n \\cdots\n \\sum_{k_g=0}^{n-\\sum_{j=1}^{g-1} k_j}\n 1,\n", "52583c8149318bc960dd72ac73ba3df6": "P_{AC}(\\nu,\\kappa,\\pi) = \\pi_C\\left(1.0 - e^{-\\beta\\nu}\\right) ", "5258a49e9b39c92418cb20bedbfd070a": "O_K = \\mathbf{Z}[\\zeta].", "5258f89fd95758d8d08d926d3da8b9ea": "\\sqrt{\\frac{1}{42}}\\!\\,", "525924c77f462d4c65e96f5438ed6cc5": "f \\colon \\mathbb{R}^n \\to \\mathbb{R}", "52594224fa34244424208c9415de667c": "N_G(x) = \\lfloor x \\rfloor", "5259728cfce2b262155971230a498218": "\\begin{align}V(t) &= -t^5+3t^4-5t^3+8t^2-9t+12-9t^{-1}+8t^{-2}-5t^{-3}+3t^{-4}-t^{-5} \\\\[8pt]\n& = -\\frac{1}{t^5}\\left(t^5-2 t^4+t^3-2t^2+t-1\\right)\n \\left(t^5-t^4+2 t^3-t^2+2t-1\\right) \\\\[8pt]\n& = w(t) w(1/t), \\,\n\\end{align}\n", "525999285036462230e1c4305c96e866": " P:=\\mathbb R^2 \\,", "5259bb02de8b4b55d008fdff40e6abf3": "\\mathrm{tr}(\\rho^2) = \\frac{1}{2}\\left(1 +|\\vec{a}|^2 \\right) = 1 \\quad \\Leftrightarrow \\quad |\\vec{a}| = 1", "5259e10f6a693af1a45685b50d567b13": "F_e,", "525a2e6c0d8af3500e1770a5877ae3ae": "x^n = \\frac {1}{n+1}\n\\sum_{k=0}^n {n+1 \\choose k} B_k (x)\n", "525a4befc8537e9be988f973533f2010": "cx + dy", "525a863c5b3dbc2bd887ed3fee99f61e": "f={\\rm Tr}\\, Fe^R", "525a9e104aaaee5ac4e6666bf8dad02a": "\nH_{SO}=\\Delta_{SO} \\bold{L} \\otimes \\boldsymbol{\\sigma}\n", "525aa32e85f14ffbef1fdacb9c7c0895": "\\Pr(a \\leftrightarrow b,\\ c\\leftrightarrow d) \\geq \\Pr(a \\leftrightarrow b)\\Pr(c\\leftrightarrow d)", "525aa98d2f4a25cde16dc9a5ba6745ca": " i = 0, \\ldots, n ", "525aabda3f70ea3fe888a82bbe1e7bd5": "\\psi(\\Omega^{\\Omega^\\omega})", "525b32063ee4fd5d10b4f110022a9dc4": "\\sum_{\\nu=0}^{n}\\nu b_{\\nu, n}(x) = nx", "525babc5cd700521ff622bd45c0f8588": "\n\\begin{align}\n \\Psi_0 &=& D\\exp\\left[-\\sqrt{\\frac{2m}{\\hbar^2}(V_0-E)z}\\right]\n\\end{align}\n", "525bdfc9c9b66cc42e2f5662ceb6f831": "K_N(f;t)=\\frac{1}{N}\\sum_{n=0}^{N-1} S_n(f;t), \\quad N \\ge 1,", "525be25dfdf08554fe8cdcf49e1d1d7d": "\\frac{A \\text{ plano}}{B} \\text{ subducere } \\frac {Z \\text{ quadratum}}{G}\\text{ residua erit }\\frac {A \\text{ planum in } G - Z \\text{ quadrato in } B}{B\\text{ in }G}", "525be84b89b94841694be1bf79bb4b27": "\\frac{1}{\\mathrm{gcd}(2,q-1)}q^{63}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^8-1)(q^6-1)(q^2-1)", "525c00b90a7021826c3018e77af73369": "S = a^2 \\cdot 12\\sqrt{5} \\approx 26.8328 \\cdot a^2", "525c4e9e0d7674a414eb3b94cc7db5d3": "On(\\text{coin},\\text{moon})", "525c51fc76819f440112a6dfa43b9ba1": "f(x_2)=y_2", "525c98de00a8a0cba4c4b9817dcfb1ec": "\\ , \\theta=0", "525c9a10604ca67a13d1172338d29219": "P(X > 2)\\,\\!", "525d171cd5389143256c69702cf50a67": "I_\\nu(s)=I_\\nu(s_0)e^{-\\tau_\\nu(s_0,s)}+\\int_{s_0}^s B_\\nu(T(s'))\\alpha_\\nu(s')\ne^{-\\tau_\\nu(s',s)}\\,ds'", "525d9b1c09a0c5f033a274f1cfb19222": "\\scriptstyle O(n)", "525e816ab4ec3d8eadf8eb8272673a2e": "{\\mathrm {Spin}}(n)", "525ee09dfa71f69672cabbde296ee6d3": "P(1+1/{\\epsilon})=MC", "525fd6abe5f30cf4fc3b866a627983dc": "G_{\\text{Aff}}\\;", "52601dc86a0b1e0c7c07e822e6b7963b": "\\psi(x;\\theta) = \\sqrt{p(x; \\theta)} \\; e^{i\\alpha(x;\\theta)} ", "52606ea9572f0ac504bec7c8579b0be7": "(P^{(\\pm)} [F , G])^{IJ} = [F , P^{(\\pm)} G]^{IJ}", "52607dca70f6eeb3bbdfb12626cfc968": " G = \\langle B, e| e^{-1}\\alpha(c)e=\\omega(c), c\\in C\\rangle.", "52609c120a7fad603de67f830ee2238d": " d_C(z) = d_H \\int_0^z \\frac{dz'}{E(z')}", "5260acbe3975181a1fd5def02a0ff5e8": " = V F kT (kT/h)^3 \\int_0^{T_D/T} \\,{x^3 \\over e^x-1}\\, dx\\,,", "5260d75cf6e337b848d63722e3a615b4": "L = \\frac{I}{mR}.", "5260e6d3c7bc380cfa06bd5f75512b0f": "\\tau_2\\ ", "526175620869b2840482c61e718a7e81": "P_{t_m}", "5261893e95ee37bdd25fc46f3149bc20": "a^2 = -|z|^2", "52619ab6407e667be9d53154521fed19": "\\alpha^{N \\pi - 1} \\equiv 1 \\pmod{\\pi}", "5261d218c1f78cb5ae2a31764ad2ec44": "g^{j i} g_{i k}= \\delta^j{}_k\\ ", "52620b76e785c524da390d38e4cfdb1d": "\\sqrt{(u/p)}", "5262320f78ce05842be1dd5c7ba1662a": "\\beta(s) = 2^{-s} \\Phi\\left(-1,s,{{1} \\over {2}}\\right),", "526238b06ad19a0fc269626dd5762caa": "x^\\frac{\\alpha}{\\alpha+1}+y^\\frac{\\alpha}{\\alpha+1}=1.", "5262781b4293fe87d0465d70113f4e41": "\\ -2J_{ab} \\vec{s}_a \\cdot \\vec{s}_b ", "5262a327116958d4d1c47da0534959d1": "\\tau_{xy} = \\mu\\gamma_{xy}=\\mu\\left(\\frac{\\partial u_x}{\\partial y}+\\frac{\\partial u_y}{\\partial x}\\right)\\,\\!", "52632e836922392c37da2a4d0eb5dff8": "HJ_i=HL_i-x=r-x=r_1+r_2-x~", "52633b85762a4341dab6cdb9bc2d71af": " Rep(w',P) = R ", "5263436d2675d0fd21822dd6e79ffca4": "\ns_{f_1}(n) = O\\left(M(n)\\log^2n \\right), \\,\n", "52638a8461436b92e8b33ea7fef85f0f": "(\\mu_i , \\sigma_i , \\nu_i , \\tau_i )", "526395d6c60a4531efbaabbf61275f79": "[C_{\\lambda \\pm k},H]=\\Omega_{\\pm} (k) C_{\\lambda \\pm k}", "5263eb30532a0690045c32ea04e7e081": "\\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B}} {\\partial t}", "526454c62b33c0e22ad72562d6bf73b0": "f(x) = \\sum_{n=0}^\\infty a_n(x-b)^n.", "526464a96c6ac07db374279056fdc453": "W_o = W_n P V f \\frac{(T_H - T_K)}{(T_H + T_K)} ", "5264c62b71057b2227207203adb7e8b5": " n > 0 \\, ", "5264cd3994877aebe1435516c84f522e": " \\det S = e^{-\\zeta_S'(0)} \\,, ", "52655ffed1847ed61cab4c665b0d7ba8": "\\sqrt{2 \\mathbf{e}}", "52657ddc2216d3c34c18f85cf6a8a225": " \\begin{align}\nS &= (\\sqrt{y}Z\\sqrt{y} \\,- 1_{\\!N}) (\\sqrt{y}Z\\sqrt{y} \\,+ 1_{\\!N})^{-1} \\\\\n &= (\\sqrt{y}Z\\sqrt{y} \\,+ 1_{\\!N})^{-1} (\\sqrt{y}Z\\sqrt{y} \\,- 1_{\\!N}) \\\\\n\\end{align} ", "5266167c1acfedff91956654c94c89be": " 0 = \\lambda^2 - \\lambda - 6 = (\\lambda - 3)(\\lambda + 2),", "52665187ef8e41ee1607c720638a9ee0": "\\,\\mathit{I} = \\frac{1}{2} + \\log_{2} \\mathit{H}_{av}", "52668e67dc686ea52cc2ffe269992ddb": "N = \\frac{1}{\\sum_{i=1}^n p_i^2} ", "5266df3dddea4224a98331e6218bc6be": "J_m \\gg 1", "5266e908251a3e307db08b3d6ce71f34": "\\mbox{U}(1) \\to S(\\mathbb{C}^{n+1}) \\to \\mathbb{CP}^n", "52672ec7f656ebb612300e3a3768db30": "\\scriptstyle \\hat T_0=T_0", "5267d3da4ad0dadc1c14d149e84008de": "\\sigma_t\\,", "526869dcdff24444b74535e3108d6f2d": "\\mu_{i,j}", "52687d609f29a52c36616eceeb4fad11": "\\mathbf{\\mu}=\\frac{1}{2}\\int\\mathbf{r}\\times\\mathbf{J}\\,{\\rm d}V,", "52688ecce7936a5f9e1149c01bd54f84": "\\langle n|\\partial_\\nu H|n\\rangle", "526890d0fcc999a33f84fb7915c93dc3": "\\bar{r_0} = r \\cdot \\hat{r}", "5268d432efdac67f291a2e870ab7f154": " R\\in {\\rm SO(3)}", "5268e1bc2d1e35b5b313acfe12c1b508": " M = R^2 ", "526906eedcebc63130e1a0eaab2ef29c": "B_s", "5269a3e20d334c0368d00b85212039bc": "E_{F_A}", "5269f7512b756edbc6f85fc376d677e8": " 2 \\lor 3 \\iff | -i \\rangle ", "526a1bbc656caa3035c9eca58ae00cf2": "\\lambda_0\\neq 1.", "526a3e2b5407501cf73e39f5c8598a09": " f^m \\in I, \\text{ for some } m\\in \\mathbb{N}. \\, ", "526a67511cd796b7f8127d27d06a5ab5": "\\eta_p= \\frac {2\\, (\\frac {v} {v_e})} {1 + ( \\frac {v} {v_e} )^2 }", "526aa80c8b4a0d90abd9e194b9a62303": "\\mu_{\\lambda + \\rho}", "526ad3ae7889f3317347461f25126fbd": "(d-1-k)/2", "526b9188b01dbf56e232e03ed2f18cd5": "a\\uparrow\\uparrow\\uparrow b = a\\rightarrow b\\rightarrow 3", "526bf69ac13c6c536d6d202b67cfd135": "\\gamma_2 = \\infty", "526c0146f8b95cde37264f0c18b6163e": "F_1=\\left\\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\\right\\}", "526c4a2465a2347f4fc4c1841453eec7": "a=\\frac{p}{1-e^2}", "526c79eaed85050e8d06a6d9e1a15506": "T_{1A}", "526c80e64d2373fac4a2f4a2e07e465f": "\nF(r) = -\\frac{dU}{dr}\n", "526cdd7fd02ea47253da43b912a96cb8": "\\mathrm{Supp}(M)", "526d17e11a919efe4b1bbf7804956848": "\nS(L)=e^{-ih_\\text{eff}L}\\simeq 1-ih_\\text{eff}L-\\frac{1}{2}h^2_\\text{eff}L^2+\\cdots.\n", "526d2d05435c6cbb4c652ab2b238ad4f": "\\int_{-x}^x(x^2-y^2)^n\\cos(y)\\,dy.", "526d692a10afb275e322462899fc292c": "ZY^2=X^3+aZ^2X+bZ^3", "526d6bb05e3d43d9a1ce970f5701a370": "+\\tau_\\perp (t)P(t)C'(t)W^{-1}(t)\nC(t)P(t)\\tau'_\\perp (t),", "526d96322b02d659ee296406c432dcb3": "C(n,k)", "526dc70f613309513907a08227f3db89": "\\nabla=e^i\\frac{\\partial}{\\partial x^i}=e^i\\partial_i,", "526e0ad04862487476698bc66eace1a0": "\\varepsilon_{total}= P(X_1\\oplus X_2\\oplus\\cdots\\oplus X_n=1)- P(X_1\\oplus X_2\\oplus\\cdots\\oplus X_n=0)", "526eb83d57ee34d36f772683546e072a": "\\alpha:\\mathrm{Id}_{\\mathrm{Id}_A(a,b)}(p,q)", "526eb92cdd8eccf345e4a05e5baf33a1": "E[f_i y_i]=f_i^2", "526f46c00cf3c1cce39d38ad977fdef4": "\\scriptstyle 3ab \\;=\\; a^2 \\,+\\, b^2 \\,+\\, 1", "526f488b9b07f9f963083daea9db6dee": "\\frac{a}{(x-b)^n}", "526f69dec39a3277902c373b308c6b22": "f_1 \\in L^1(\\mathbb{R}^d)", "526f853230e995a9f5963211a7c02d94": "\n\\arctan(z)\n", "526f8d377a8014ed1ad5363d8afe2866": "u_z=-\\mathbf{F}_z(\\mathbf{p}(t))\\mathbf{x}(t)=-\\mathcal{F}_z\\boxtimes_{n=1}^N\\mathbf{w}_{z,n}(p_n(t))\\mathbf{x}(t).", "526fc81a1039f53d8c4af2b58332c41d": "V_0\\to\\infty", "527043a6cb92cb14691fca33164c2488": "W=\\{A\\cap G;\\,A\\in V\\}.", "5270579d4e1b91147012665718dfec6f": "f (x) = \\lambda u (x) - \\int_a^b K (x, x') f (x') \\;\\mathrm d x'", "527087090986cb4f84c31ac99f538fba": "Z_{ij}=(X^{\\mathrm{t}})_{i}^{k} z_{k} X^{k}_{j}", "527097529329584e0b672d3775888e81": "(B,\\sigma,I')", "5270a2f57a6f216068c7a0c6f2ce2c20": "x_1 ", "5270abffedc2a885f911ee92b83f42df": "f_n = E[X_n] = K - \\binom{N}{n}^{-1} \\sum_{i=1}^K {\\binom{N-N_i}{n}}", "5270ae675fac24f97e172dcd9b18fa92": "(i,j)", "52719f437757f11e19454d68529cd389": "f = \\sqrt{a^2-b^2}.", "5271c9fca76c1ce5af43132c251b476a": "f(x) = \\pm x_1 x_2 \\ldots x_n . y_1 y_2 y_3 \\ldots", "52721209ab7e76357012cb29ed64c4cf": " avg= o*(1-2^{-n}) + c*(2^{-n})\\,\\!", "5272324c00943fe2b64fb06edd8cb35a": " \\chi^2 = \\sum_{i} {(O_{i} - E_{i})^2 \\over E_{i}} .", "5272966456a67b9cac7831e6d241584f": " y_2 = -\\frac{f \\, x_2}{x_3} ", "5272c7114d5e1f80008e963b1e1e628c": " A \\in Aff/k ", "52731af77ce64c6c796601b117700276": " A \\le^+ B \\iff \\forall a \\in A\\exists b \\in B(a \\le b) ", "527322e434a933c185e111d602849c66": " AP \\cdot AQ = AR \\cdot AS \\, ", "52733fac571b024e2de7e24ddaeb533f": "x \\equiv \\alpha", "5273474ef6854fa79b6e1b1e46ea95e0": "\\sum_{|\\alpha|\\le k}\\ \\sup_{y\\in B'_\\delta(x_{2k})} |\\nabla^\\alpha s_k(y)|\\le \\frac{1}{M_k}\\left(\\frac{\\delta}{2}\\right)^k", "5273c5b0dcd57fa624fc513bf6905609": " {} = \\begin{vmatrix}x_0&y_0&x'_0&y'_0\\\\x_1&y_1&x'_1&y'_1\\\\x_2&y_2&x'_2&y'_2\\\\x_3&y_3&x'_3&y'_3\\end{vmatrix} . ", "5273e1c08cb159f358ed12b71a9d96c0": "x_1=\\lambda_1 X_1,x_2=\\lambda_2 X_2,x_3=\\lambda_3 X_3", "527401798659e6fbe4d44e2e1add34a5": "\nT = \\frac{1}{2} \\sum_{k=1}^N m_k v_k^2 = \n\\frac{1}{2} \\sum_{k=1}^N m_k \\frac{d\\mathbf{r}_k}{dt} \\cdot \\frac{d\\mathbf{r}_k}{dt}.\n", "52743153dcdf3d2fb8c03a42e64457b7": "\n\\operatorname{Li}_{s}(z) = {1 \\over \\Gamma(s)}\n\\int_0^\\infty {t^{s-1} \\over e^t/z-1} \\,\\mathrm{d}t \\,.\n", "52743315b0530122644bd861ecba5494": "C_{\\Psi}=\\frac{16}{5}\\sqrt{\\pi}.", "52747ebbc27c0c596a63913740e16a9c": "Eb^6_9", "527496c8dc0339fae0b7febd1378d040": "~~~~~\\frac{1}{T},\\frac{P}{T},\\{N_i\\}\\,", "52752d3540c15b55457390fa1acc6aac": "L=M + \\varpi = M + \\Omega + \\omega\\,", "527545ed0edda5f1dc1956c9c9c5405a": " e_l ", "52759f982599fbbea75bf0fbb1b0400c": "\\textstyle\\int_0^\\infty f(x)\\,dx", "5275b10186810e934fb8ef4538978f1a": " \\pi(a) = \\int_X^\\oplus \\pi_x(a) d \\mu(x), \\quad \\forall a \\in A. ", "52767e6438c201198670bc45c499cdbc": "\nu'(x)\n=\n\\frac{\n 0.0204 + 0.0379\\, z \n - 0.0059\\, z^{2} \n - 0.00004575\\, z^{3} \n + 6.357 \\cdot 10^{-6} z^{4} \n -1.291\\cdot 10^{-6} z^{5}\n}{\n 1 - 0.1429\\, z \n - 0.0000232\\, z^{2} \n +0.0008375\\, z^{3} \n - 0.0001558\\, z^{4} \n - 1.2849\\cdot 10^{-6} z^{5}\n},\n", "5277cc644a989c17421ded469013e720": "Z = \\frac {\\bar X-\\mu}{\\sigma/\\sqrt{n}} =\\frac {\\bar X-\\mu}{0.5} ", "5277dfc47290dac3663067f5fcbf040e": "\n\\omega^{A}=ix^{AA'}\\pi_A.\n", "5278a1a09bc9cf788979de00afbcde46": "\\sum_{i=1}^{n} a_i^k = \\sum_{j=1}^{m} b_j^k.", "5278acc7dbe8f0acfa27326627dfe85b": "X_{2c}(\\bold{r})", "5279d46175e8020024e34923a4fcd8f1": "\\tfrac{EG}{3(3G-E)}", "527a0332191df1d3a76073eb0880259d": " a_{2k-1}=\\frac{(1-x_{2k+1}x_{2k+2})}{(1-x_{2k-3}x_{2k-2})} x_{2k+0}", "527a5b91b8d491e09f7faf975398ceb3": "|\\omega|^s(v_1,\\ldots,v_n) := |\\omega(v_1,\\ldots,v_n)|^s.", "527a8243fe01ebb4951b8b0bf59a7aaf": "\nK(p) = { i \\over {p^2 - m^2 + i\\epsilon}}\n", "527aac4a93e6f24d84e8e8ed7ca49bb0": "b in A\\wedge b\\wedge A", "527ac5d1d6152373326fcd7c1f4fa572": "N = 2n + 1", "527acb137a08b95c9f663e8d60297de7": "\\mathbf{G}", "527af9c26a3bcb50535fb0533adc124d": "\\forall i\\in I\\forall j\\in J: v(\\alpha_i-\\alpha_i^*)>v(\\alpha_i^*-\\alpha_j^*).", "527affcda3f6d68350eb538fd3e37910": "z>0\\,\\!", "527bcf9db7d6277da3329d385156f749": "\n\n\\phi(e^{-2\\pi})=\\frac{e^{\\pi/12}\\Gamma\\left(\\frac14\\right)}{2\\pi^{3/4}}\n\n", "527c24ee2307df31d807ed8c6856ba0f": "\n\\varphi\\left(e^{-2\\pi} \\right) = \\frac{\\sqrt[4]{6\\pi+4\\sqrt2\\pi}}{2\\Gamma(\\frac{3}{4})}\n", "527c2eb6806dce8beae04c1140bfc67f": "\\scriptstyle\\backsim", "527c358d1ad6463cca6ffcd46af5afb8": "P(t) = e^{tQ}", "527c48b53941e3df93501cee67c2c0fb": "e^{i\\mathcal{S}[\\phi]},", "527ca7df2b5732e033103672b981ed64": "\n\\begin{cases}\n(P_1, 1) \\to (P_4, 1) \\\\\n(P_5, 0) \\to (P_4, 1) \\\\\n(P_1, 0) \\to (P_4, 2) \\\\\n(P_2, 1) \\to (P_4, 2)\n\\end{cases}\n", "527cec4ffaece9aceb5f86a2cbcd87eb": "Y_i = \\mu(x_i)", "527d1c63fb60b3777c17337c736379b9": "\\psi_3 = He^{- \\alpha x}+ Ie^{ \\alpha x} \\,\\!", "527d24165db02dd9a22b2d628a4bbfbd": "\\supseteq\\,\\!", "527d5671821a02e8baa4817018797433": "\n R^\\gamma_{\\alpha\\beta\\rho} := \n \\frac{\\partial }{\\partial X^\\rho}[\\Gamma^\\gamma_{\\alpha\\beta}] -\n \\frac{\\partial }{\\partial X^\\beta}[\\Gamma^\\gamma_{\\alpha\\rho}] +\n \\Gamma^\\gamma_{\\mu\\rho}~\\Gamma^\\mu_{\\alpha\\beta} - \n \\Gamma^\\gamma_{\\mu\\beta}~\\Gamma^\\mu_{\\alpha\\rho} = 0\n", "527d99a08c1580eacca141e7b85e64d8": "\nc^\\epsilon(x)=\\begin{cases} 1, & 0 < x < 1 \\\\ \\frac{1}{\\epsilon^2}, & 1 < x < 2\n \\end{cases}\n", "527dde896031b582848b6fa920ba34c1": "\\displaystyle{D(\\varphi)(z)= \\Re {1\\over 2\\pi i} \\int_{\\partial\\Omega} {\\varphi(w)\\over w-z}\\,dw .}", "527df1c4d9cc306d974f6bbb255da709": " L_i \\, ", "527e11a147ddd5c0448079603ddb5294": "\\hat{e}_1 = [1,0]", "527e23370645ef023bd3042d709eaf97": " E_{RF} = \\frac{2(\\varepsilon_{RF} - 1)}{2\\varepsilon_{RF} + 1} \\frac{\\vec{M}}{r_c^3}", "527e3f90a9062fddb8f305c9251cdfb9": " 1 \\,+\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,+\\, \\cdots\\, \\frac{1}{n} ", "527e5660aee5d42d192f1f65e80cebea": "V = (\\oplus_{\\alpha \\in A} T_z) \\oplus U.", "527e8047264b607f733d3501e4bd49e8": "T_Z(w) = w - 12\\frac{g(Z,w)}{g'(Z,w)}\\,", "527e95629352a237ff424ffd891ea593": " L=\\ln(L) ", "527ed6e46ed5326142af2ff50787927d": "\\textstyle 100:200 = 1:2", "527f274886a565608863ab4cc49c4413": "G_{V_1, E_1} \\cdot H_{V_2, E_2} \\rightarrow J_{(V_1 V_2), (E_2 V_1 + E_1 V_2^2)}", "527f2988485539c9ed96e6982d921242": "K = R^{-1} B^T P \\,", "527f30c774eca76683cebe45d0595f6b": "\\left[t^a,t^b\\right] = t^a t^b - t^b t^a = C^{abc} t^c.", "527f327a65a10280a2860fd23358da44": "(n^3)", "527f4523896d99016d88f07961469b2a": "\\frac{\\partial f}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\left(\\frac{\\partial f_1}{\\partial \\mathbf{v}} \\cdot \\mathbf{u} \\right)~f_2(\\mathbf{v}) + f_1(\\mathbf{v})~\\left(\\frac{\\partial f_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} \\right)", "527f4afe5341307c41b47e09c8eae9f4": " V_{\\mathbb R} \\otimes_{\\mathbb{R}} \\mathbb{C} \\to V\\,", "527fd831d10c6bac3324b334a87d3469": "Z_{i,j}", "52803daae5b3a0078e372c23da0008a3": "T_{mm} = \\begin{pmatrix}\n\\alpha_1 & \\beta_2 & & & & 0 \\\\\n\\beta_2 & \\alpha_2 & \\beta_3 & & & \\\\\n & \\beta_3 & \\alpha_3 & \\ddots & & \\\\\n & & \\ddots & \\ddots & \\beta_{m-1} & \\\\\n & & & \\beta_{m-1} & \\alpha_{m-1} & \\beta_m \\\\\n0 & & & & \\beta_m & \\alpha_m \\\\\n\\end{pmatrix}", "52804d0f18d810e5e4196f8c266be81f": " V_{\\left(p-k\\right)}^{T}\\boldsymbol{\\beta} \\neq \\mathbf{0} ", "52806c1b87e4ac85fd24adc6d8a863ee": "q^{\\prime\\prime}(s) = sq(s)+2q(s)^3\\,", "52808c52ba94ee30e8f04b8e649512cd": "A(z)", "52818459cffc65b67c21de148363e8bb": "(a_{k})_{k=1}^\\infty, \\qquad a_k = (2k-1)^2", "52818cdc22e86ac00a34e624fa38b5c0": "f(x; k,\\lambda)={\\lambda^k x^{k-1} e^{-\\lambda x} \\over \\Gamma(k)}\\quad\\mbox{for }x, \\lambda \\geq 0,", "5282690558ac5003bd6f9b3fae57d9ce": "(3,5)", "528278a3ffcd18951b156d0ce83a4609": "\\frac{\\beta }{k_{o}}=\\sqrt{\\varepsilon _{eff}}", "52828ac2205727ac199b9e339bdc2a48": "r_j", "528297fefe818beeff7271263d584a2e": "\\scriptstyle |1\\rangle_A |0\\rangle_B", "52829eba03137db057089c677c12b5a5": "\\chi_X", "5282e49cf321b04088aa6d6a62c31b14": "a\\times x\\ \\bmod\\,2^{32}", "52835f4c8b9ff123de5e1832693426c2": "1/(1 \\pm \\varepsilon)", "528384c5d4ea0d0c543cf6480c7ee690": "v_\\mathrm{o} = (v_1+v_2)+\\frac12 (v_1+v_2)^2 + \\dots", "52838c46020ae10508a2a76fcddf396a": "B = \\frac{b p}{RT}", "5283b458d516b243ab09a6f2c6c054db": "\\mathbf{x}=(x_1,x_2,\\ldots,x_n)", "528445ae4bd1915287675b86355ca89a": "\n\\begin{bmatrix}\n \\alpha - E & \\beta & 0 & 0 \\\\\n \\beta & \\alpha - E & \\beta & 0 \\\\\n 0 & \\beta & \\alpha - E & \\beta \\\\\n 0 & 0 & \\beta & \\alpha - E \\\\\n \\end{bmatrix} \\times\n\n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n c_3 \\\\\n c_4 \\\\\n\n \\end{bmatrix}= 0\n", "52845ba4b0537920fbee1deb426f0d25": "p_{n+1}-p_n \\lambda l", "528b0612292db57b0e4819f646b33730": "z = (10^m \\cdot n_m - n_m) + (10^{m-1} \\cdot n_{m-1} - n_{m-1}) + \\cdots + (10^0 \\cdot n_0 - n_0).", "528b307bfdcea1de33a99859716c799a": "\\!\\dot{\\epsilon} = Ae^\\frac{-Q}{RT} \\frac{\\sigma^n}{d^m}", "528bdef160dba40014af7cd8fca544bb": "\\omega_{\\text{c}} = \\frac{1}{R_2 C}", "528d51f39d19fc72def1e1a572a1f5bb": " B' = \\begin{bmatrix} A & I - AA^* \\\\ 0 & - A^* \\end{bmatrix}", "528e1fec23df4ec4b1993ee844570774": "\\lim_{n \\to \\infty}{n_a \\over n}=p ", "528efd31e7fab0d00b6adc0359fdc0ef": "f(\\mathbf{a} + \\mathbf{v}) \\approx f(\\mathbf{a}) + f'(\\mathbf{a})\\mathbf{v}.", "528f26efc0ff661d5314aa1dd8ae8962": "\\ \\frac{4fL^*}{D_h} = \\left(\\frac{1 - M^2}{\\gamma M^2}\\right) + \\left(\\frac{\\gamma + 1}{2\\gamma}\\right)\\ln\\left[\\frac{M^2}{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)}\\right]", "528f769726c3ea674754b496780579cb": "\\beta=\\beta_1\\cup \\beta_2\\cup \\cup\\beta_k", "528fd04e8d2fa8cb08d14a38963f3ab5": "\\sum_{i=1}^{m} \\sum_{j=1}^m |\\langle x_i , x_j \\rangle|^2 = m+\\sum_{i\\neq j} |\\langle x_i , x_j \\rangle|^2 \\geq \\frac{m^2}{n}", "528fe9e1d6dead3762a3a509d559740a": "\\gamma_j = \\frac{\\pi N_j l_{t,j}d_{c,j}^4}{64 \\rho_c}", "528fed78d4b2d0f1d2bd4f160db8ac35": "|V_0| v_i[H^+]_0 \\text{ (alkaline region)} \\\\\n= 0 & \\text{ when } v_{0^{ }} [OH^-]_0 = v_i[H^+]_0 \\text{ (equivalence point)} \\\\\n\\approx -[H^+]_i \\text{ or } -10^{-pH_i} & \\text{ when } v_{0^{ }} [OH^-]_0 < v_i[H^+]_0 \\text{ (acidic region)} \n\\end{cases} ", "529b07f106b62085ed7493d69a14ad2c": "\\displaystyle \\Delta v = -v_e \\ln (m_f / m_0)", "529c6e02d38c1e8e18fc787982ce4075": " \\textstyle\\ \\begin{align} \n&\\mbox{I}^- + \\mbox{M} \\overset{k_{init}} {\\longrightarrow} \\mbox{M}^- \\\\\n\n&\\mbox{M}^- + \\mbox{M} \\overset{k_{prop}} {\\longrightarrow} \\mbox{M}^- \n\n\\end{align} ", "529ca30eb74564461bc8e0e7d7864e95": "\\textstyle \\Delta ", "529d0440a3c6b825d6b9417c790cee12": "x_j ", "529d096ae64f29c8f403caa5f421c810": "\\lfloor mx \\rfloor=\\left\\lfloor x\\right\\rfloor + \\left\\lfloor x+\\frac{1}{m}\\right\\rfloor +\\dots+\\left\\lfloor x+\\frac{m-1}{m}\\right\\rfloor.\n", "529d18932be7620810837fcbe120dd91": "\\tau={bmep A R K\\over 2 \\pi}", "529d952baa9adf18015a9c1e5a5ebc36": "v \\in E^+_q", "529dbecaa7d54251d696c7d294a97290": "MA^2+MB^2+MC^2=GA^2+GB^2+GC^2+3MG^2.", "529e241b178319bd66799ebea50b89a1": "\\left\\{1,...,N-1\\right\\}", "529e4cadf68b783947fd38efcf7abfd9": "\\|\\cdot\\|'", "529e660973b52121f8b0e48c03a53408": "a+3 b", "529e86f40651f5724bac76141e5202f7": "g_2(\\lambda \\omega_1, \\lambda \\omega_2) = \\lambda^{-4} g_2(\\omega_1, \\omega_2)", "529eac97a8fb9f8659d72d1530b3a14f": "P_0\\left[\\frac{B}{S_0(T)} > -\\infty \\right] = 1 ", "529ed0b41d1997468c990d0d3ad35230": "F(t) = \\sum_0^\\infty \\ell(I^n / I^{n+1}) t^n", "529efcb109b2b71e03eac910a805a595": " h_j(\\mathbf{p},u) = h_j(\\mathbf{p}, v(\\mathbf{p},w)) = x_j(\\mathbf{p},w), ", "529f0f418b87e90e15cefae3cfdf9ffe": "p = 61", "529f138e0de236ba451d583a4deb9f26": "\n \\tau = \\sigma~\\tan(\\phi) + c\n ", "529f4f7b1b5a87a0fd7988dc48d34ede": "(\\partial P_i/\\partial t_j)", "529facf7573a72069d4f69bb7d00dd17": "traces\\left(a\\rightarrow b \\rightarrow STOP\\right) = \\left\\{\\langle\\rangle ,\\langle a \\rangle, \\langle a, b \\rangle \\right\\}", "529fb3b83c9b8fefed8c1e21f725341d": "D=\\gamma^\\mu\\partial_\\mu\\ \\equiv \\partial\\!\\!\\!/,", "529ff5f8f00e23218cd91171de9da568": "K_\\text{vert}=m(\\frac{2\\pi}{P})^2", "52a0499db241e01aad51423503346f2b": "{\\partial u\\over \\partial t} =\n\\alpha \\left({\\partial^2 u\\over \\partial x^2 } +\n{\\partial^2 u\\over \\partial y^2 } +\n{\\partial^2 u\\over \\partial z^2 }\\right)", "52a049a7de736ac5f3a48a8ca948a46c": "\\sin^{-1}(x)", "52a07ce46212cbc2298415c5fca6e075": "x=", "52a0a42a8e8eb8110d6ad7d8d3ad5af3": "f = \\frac {1}{ 2 \\pi \\sqrt {LC}} \\,", "52a1238731b2b1f2e6de4dfb0219168b": "\\displaystyle{\\cdots\\subset H^{2}(\\Omega) \\subset H^{1}(\\Omega) \\subset H^0(\\Omega) \\subset H^{-1}_0(\\Omega) \\subset H^{-2}_0(\\Omega) \\subset \\cdots}", "52a1514fa90b0a908969721defb716c3": "0.5 \\ge \\beta > 0.25", "52a1521657610a2cc7ac389506b03fda": "r_3 = 108 = 3 + 7 + 2\\cdot 7^2 = 213_7", "52a160e349f6a49d397d013d42c58d94": "H(t) = H_0 e^{-(R/L)t}\\,", "52a18ed12192ba3513b14663e2c464a5": "\\tan\\phi = \\cot\\alpha_0\\sin(\\lambda-\\lambda_0).", "52a1a72c60ee6f18d4c4d374fcc7be6e": "G, \\bar{G}, \\tilde{G}", "52a1b187743e5a6a6ab29cb5bcb354a0": "C = (1, 0)", "52a2091bf58d0908f33013d2cde904cf": "\nf(x_1, x_2, \\ldots)^{j_1(g)} f(x_1^2, x_2^2, \\ldots)^{j_2(g)} \\cdots f(x_1^n, x_2^n, \\ldots)^{j_n(g)}.\n", "52a22ac9ace627816429b0a5eca41dce": "{}A_{96} = 313 {584 \\over 625} ", "52a235d07434903973344d870f25977a": "\\Delta = \\int V\\operatorname{d}t", "52a282b5b160ca164522b7130e4d5b6a": "B_r = -\\frac{2B_0}{R^3}\\sin\\lambda", "52a2cfdf6a91aea224af507e8bdbd958": "\\varphi(s)= \\{ \\mathcal{M} f \\} = \\int_0^{\\infty} x^s f(x)\\,\\frac{dx}{x}.", "52a2d50a2f8ef4a2ff2336056aafab65": "M_\\odot \\frac{G}{c^2} \\approx 1.48~\\mathrm{km};\\ \\ M_\\odot \\frac{G}{c^3} \\approx 4.93~ \\mathrm{\\mu s}", "52a2e5ca79ca3d246719d8047281084d": "360^o", "52a2f261468ad891d685ebfac5157bcd": "X_{1}^\\mathrm{opt} = \\alpha W + \\beta \\mu_1", "52a38d49ba4709f10b1986d0b98ff384": "s= \\tfrac{a+b+c}{2}", "52a3e2c0c432ee69d80a028fca6ce81c": "\\vec{\\omega}", "52a3f3d140260719a0577f06f8d5ced1": "\nP = ST = \n\\begin{bmatrix}\n\\frac{2}{right-left} & 0 & 0 & 0 \\\\\n0 & \\frac{2}{top-bottom} & 0 & 0 \\\\\n0 & 0 & \\frac{2}{far-near} & 0 \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\n1 & 0 & 0 & -\\frac{left+right}{2} \\\\\n0 & 1 & 0 & -\\frac{top+bottom}{2} \\\\\n0 & 0 & 1 & -\\frac{far+near}{2} \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "52a3fca06fdfd1d2f3e1a6d1a107272f": "\\Pr(u < X \\leq v) = F(v)-F(u)\\,.", "52a466116a7900aaca4985a9baf3cee4": "k_i+1 = k_i4^{k_i}", "52a46fc8f625ab8ece200c68b64f7af7": "\\begin{matrix} \\frac{4 \\pi GM}{c^2} \\end{matrix}", "52a4911e55613fc3cc8fa7ee728b12fc": "P_{ij}(\\nu) = \\left\\{\n\\begin{array}{cc}\ne^{-\\beta\\nu}+\\pi_j\\left(1- e^{-\\beta\\nu}\\right) & \\mbox{ if } i = j \\\\\n\\pi_j\\left(1- e^{-\\beta\\nu}\\right) & \\mbox{ if } i \\neq j \n\\end{array}\n\\right.", "52a4eddc112e6169f14337963862a0b5": " k_+ \\left\\{ A \\right\\}^\\alpha \\left\\{B \\right\\}^\\beta = k_{-} \\left\\{S \\right\\}^\\sigma\\left\\{T \\right\\}^\\tau \\,", "52a4ee52c99fc527cc0d5968b0519945": "e^{i\\beta _{j}}=\\left( -1 \\right)^{N_{j}}", "52a522eb7db01651d05fc4e096b0ac33": "\\frac{1}{F(p)}H(t)=\\sum_{n=0}^\\infty a_n \\frac{t^n}{n!} H(t) .", "52a5c2c95de56709ea86341d78361cda": " \\eta(X) = \\mathrm{E}(\\delta(X) | T) = \\mathrm{E} \\left( \\left. \\frac{T^2}{2} \\,\\right|\\, T \\right) = \\frac{T^{2}}{2} = \\frac{\\log(1 + e^{-X})^{2}}{2}", "52a60e707abfd8d63ed3dfd743163623": "M_p(x_1,\\dots,x_n) \\le M_q(x_1,\\dots,x_n)", "52a60f099a8a797edf6647b13f1911a3": "S_0'^2=1\\,", "52a65e583a48d09860c0dee2004935f8": "\\sqrt{\\frac{5}{6}}\\!\\,", "52a6809db8f542d2930fa5b9e9245a56": "\\displaystyle p=\\sqrt{\\frac{e+g}{f+h}\\Big((e+g)(f+h)+4fh\\Big)},", "52a6a1d5a21a7731816824a54259a455": "(\\mathbf{A}\\otimes \\mathbf{B})^* = \\mathbf{A}^* \\otimes \\mathbf{B}^*.", "52a6e7d048981b1aa5bb6ac043709c5d": "{p + iq \\over r + is} = {p r + q s \\over r^2 + s^2} + i{q r - p s \\over r^2 + s^2}.", "52a6f899bc9672161544d2e2103b6f3d": "f^*\\left(\\frac{x^*}{a}\\right)", "52a72d76189340fa52f28994ed417f06": "N = S(t) + I(t) + R(t)", "52a78547c5ad2cf7402b2a6bc707013b": "{\\mathcal C}_n(z) = \\pi(n)\\ _0F_1(;n+1; z).", "52a7a893207023fc3f15aad1f10fe492": "\\int_0^t fd^-g=\\text{ucp-}\\lim_{\\varepsilon\\rightarrow\\infty}I^-(\\varepsilon,t,f,dg).", "52a7c599a7711d89521774e8ab32561f": "K = \\sqrt{s\\left(s-a\\right)\\left(s-b\\right)\\left(s-c\\right)}.", "52a7f3f525a273c8c46948d72613b81b": " x_j = x_i ", "52a88cbf0fc270dea3eed647a100e1f7": "(g^a \\bmod{p}, g, p)", "52a8a4f3cbc4019028e7f35f80457616": "L_* \\mathbf{x} = \\mathbf{b} - U \\mathbf{x} ", "52a8a7fa4d82138018c70f2a1531bada": "\\lim_{n\\to\\infty} {!n \\over n!} = {1 \\over e} \\approx 0.3679\\dots.", "52a95f3d398dfbe6943dcf70ba3f67bb": "X_j= - R \\frac{{\\rm \\nabla} n_j}{n_j}", "52a977d402fdebfee7ee42cdfc4491ab": "D(Y|X)=\\exp[-2F(\\theta_1)]", "52a9884404c67e2a04df5c4e20dba138": "\\{ z \\in \\mathbb{C} | \\mathrm{Re}(z) < 0 \\}", "52a9e8b1cc6eecc1f1687e53669b2b9a": "j(g, z) = \\det(g)^{-1/2} (cz+d)", "52aa0a0a84fcd61bcbca22c16cce7597": "\\cfrac{1}{1 + \\cfrac{e^{-2\\pi}}{1 + \\cfrac{e^{-4\\pi}}{1+\\dots}}} = \\left({\\sqrt{5+\\sqrt{5}\\over 2}-{\\sqrt{5}+1\\over 2}}\\right)e^{2\\pi/5} = e^{2\\pi/5}\\left({\\sqrt{\\varphi\\sqrt{5}}-\\varphi}\\right) = 0.9981360\\dots", "52aa743613df5ccc73d97534eb73436a": "\\psi(y)^{\\dagger}", "52aa8825786b82ffc2ab643c3952384a": "\n\\lim_{N\\to\\infty}a_N = \\frac{\\pi}{2K(\\alpha)},\n", "52aaaea54a4bd7b0bfaa6f583f672860": " m_n = \\int_{-\\infty}^\\infty x^n \\, d\\nu(x), \\quad n = 0,1,2,\\ldots", "52ab287a937f0441b6e3d1d9c95c85ed": "S(r)", "52ab405df63ab76c1d41d9f4749eca0c": " \\scriptstyle {R^\\alpha}_\\beta ", "52ab5d8a9cabae8524e77d4f430c2a60": "\\mathrm{CH_3OH + \\frac{3}{2} O_2 \\to 2\\ H_2O + CO_2}", "52ab699cffdac8f259ffd74d3460e350": "x=r\\,\\cos\\theta \\quad", "52ab6dbd4101ee33ec986ed66ecb7b0e": "A=\\frac{h^2}{2}", "52ab819a9c2c50576ced6abadfb01e9d": "\\Delta \\mathcal{\\delta O}", "52ab8b00e8514e2b2069d481f24fe473": "U^{(k)} = \\mu_i^{(k)}\\cdot U^{(k-1)}", "52abec3789cb517ab3be4153fc653cad": "T_{total} = T_1 + \\frac{T_2}{G_1} + \\frac{T_3}{G_1 G_2} + ...", "52ac116d463014fad341b5d15054c5d8": "\\mathbf{R}^N", "52ac2f9ae5ccc8634cd30ad6bebaa7de": " n_{\\mathrm{A}}= V * C_{\\mathrm{A}} ", "52ac3ba0678395ad49d6b0fc5c2464b3": "\\scriptstyle \\, F^{ab}", "52ac823917e800133cab01dd85c76f91": "d\\tilde V = e^{n\\varphi}dV", "52ac830126ba889a2b78bb0f4b3217d8": "G_{\\infty} = \\lim_{T \\to \\infty }G\\ ; \\ G_{0} = \\lim_{T \\to 0 }G \\ . ", "52ad2d6e4db4b06fce09caa290544883": "f(\\emptyset)=0", "52ad3b22dac7e7bf37257a9193196237": "\\operatorname{EG}(n^2;x)=\\sum _{n=0}^{\\infty} \\frac{n^2x^n}{n!}=x(x+1)e^x", "52ad412557eab3f7c519eb6103d76f8a": " \\overrightarrow{m}_{sat} ", "52ad5706fde53105e9f368ff18b70a33": "\\overline\\mathbf{x}'=\\int_{\\Delta'} \\mathbf{x}' \\, dA' = \\frac{1}{3}(\\mathbf{v}'_1 + \\mathbf{v}'_2)\n= \\frac{1}{3}(\\mathbf{v}_1 + \\mathbf{v}_2 - 2\\mathbf{v}_0)", "52ad59b81e3b79041bbf26300db6ef0c": "\\{AX,EX\\}", "52ad5f3a80cd69462cabeaed6e4e4301": "\\langle a* \\rangle p \\equiv p \\lor \\langle a \\rangle \\langle a* \\rangle p\\,\\!", "52ad86e75c886c0c397edcc7047e0b71": "{\\mathit{He}}_6(x)=x^6-15x^4+45x^2-15\\,", "52adda75541dec4a52c1231d74a46ba2": "\\lambda=\\mu^T\\left(I-H\\right)^T\\left(I-H\\right)\\mu/2", "52ae0123c4c2d03dd9cbf774037bfa61": " \\varrho(z') ", "52ae2fcd09e2c787c6fcf4b7e4d41491": "x = 1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\ddots\\,}}}} ", "52ae3898a41a8c067256ab8fa0d950bc": "\\Delta t_\\mathrm{Voigt} = \\gamma^{-2}\\Delta t = \\gamma^{-1}\\Delta t_\\mathrm{Lorentz}", "52ae3d638ea796bc8a2df96544c0a4a5": "(12)\\quad {\\hat\\omega}^2 =\\frac{1}{2}\\,k_{[a\\,;\\,b]}\\,k^{a\\,;\\,b} =g^{ca}\\,g^{db}\\,k_{[a\\,;\\,b]}\\,k_{c\\,;\\,d}\\;.", "52ae901bd900c284158d70fceebd84a5": " P = \\begin{bmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{bmatrix} ", "52aeacfc36135fef69599ae31c87a0e8": "K = R^{-1} B^T P(t) \\,", "52af0923a8a477d823af05ee0c0f99e4": "\\int_X \\! f \\, d\\mu = \\left(\\int_X \\! f \\, d\\mu_1^+ - \\int_X \\! f \\, d\\mu_1^-\\right) + i \\left(\\int_X \\! f \\, d\\mu_2^+ - \\int_X \\! f \\, d\\mu_2^-\\right) ", "52af4a9f476842a58040646565adb1b2": "n^3 u[n]", "52af82c65ae884dd5a83dbc54c7896cc": "\\gamma=\\gamma_D=2.55268-0.959456i", "52afd74d988ac7c3289fca2c4b7f4cf4": "\\frac{v^2}{2} - \\frac{G M}{r} \\,", "52afd93b02dc14eab3e68ff326f736e5": "\n\\forall w_1,\\ldots,w_n \\,[(\\forall x\\, \\exist\\, y \\phi(x, y, w_1, \\ldots, w_n)) \\Rightarrow \\forall A\\, \\exist B\\, \\forall x \\in A\\, \\exist y \\in B\\, \\phi(x, y, w_1, \\ldots, w_n)]", "52aff869f0948ac39699040634ce5a08": " f \\mapsto \\{ f,g \\} ", "52b025d1c01e2632b4997b67dde2509a": "\\alpha _{31}", "52b02c4d0eed343654c29dab8072f9ae": "\\displaystyle{\\{f,g\\}=\\sum_i {\\partial f\\over \\partial x_i}{\\partial g\\over \\partial y_i} -\n{\\partial f\\over \\partial y_i}{\\partial g\\over \\partial x_i}.}", "52b02df4fda765be0119c6b353c38d20": "\nNT = V_c \\times K_2 \\times \\tfrac {4d^2}{3D}\n", "52b0b4d0478f25b022a6086f3086a0b8": "\\lambda(\\mathfrak{A})^{\\prime\\prime}=\\rho(\\mathfrak{A})^\\prime", "52b0e7694b21279ca7c752848fe44f1a": "\\langle x_i(t)x_k(0)\\rangle=\\langle x_i(-t)x_k(0)\\rangle = \\langle x_i(0)x_k(t)\\rangle ", "52b0f7dfe2102c34a65efc56dbc02fbf": " \\lim_{m\\to\\infty} p_{\\alpha , \\beta} (\\varphi_m) = 0 ", "52b104c2cfc0cdcfe446de65525134c5": "\\Gamma(a,x)", "52b11c8c9dcdff04a4c43856c4310c20": "-dT/dz", "52b20687c563ec1991becf3a5d0707f7": " \\mathrm{osmol/L} = \\sum_i \\varphi_i \\, n_i C_i", "52b209cef0e5f25dc1d61ea98317b293": " \\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix} \\quad\\text{and}\\quad \\begin{bmatrix} 1 & 1 \\\\ 0 & 0 \\end{bmatrix} ", "52b20d6ce7e36fea3dcea2aa735abb56": "2\\cdot A_5, 2\\cdot S_5, A_5, S_5,", "52b26110cf02112294eb08444c403358": "\\ m=3 ", "52b27278af413e9987e37a0aed2702b8": "p_5 = B\\rightarrow b", "52b29069dcfabb1136290f9991a23c38": "\n \\lim_{n\\to\\infty} F_n(x) = \\Phi(x) \\equiv \\int_{-\\infty}^x \\tfrac{1}{\\sqrt{2\\pi}}e^{-\\frac{1}{2}q^2}dq\n ", "52b2d350e2e2d9e43671e59e73ee9957": "\\Delta: \\Gamma (E) \\rightarrow \\Gamma (F) ", "52b2e082492fdb6410a123ad3f45cbd9": "\n \\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}_{\\mathrm{e}}}{\\mathrm{d}t} = \\mathsf{E}^{-1}~\\cfrac{\\mathrm{d}\\boldsymbol{\\sigma}}{\\mathrm{d}t}\n ", "52b2eac7808afb2d49284b6eccd51fc8": "\\alpha = \\frac{a + d + \\sqrt{(a - d)^2 + 4bc}}{2}", "52b2ed4a47c6bf1fca0222839c64fcfa": "\\delta[n]\\,", "52b2f97e120a3bbb8f701d2f6b3dbedf": "\\vec{E}", "52b309209becaf55248c9c5a5ecb15c2": "y=x/2,\\, dy=dx/2\\,", "52b386ceef4e280ffb4233bdbf6857f9": "S_{k'k}^{Ac}=\\frac{2\\pi}{\\hbar} Z_{DP}^{2}\\frac{\\hbar \\omega _{q}}{2V\\rho c^{2}} (N_{q}+\\frac{1}{2}\\pm \\frac{1}{2})\\delta _{k', k \\pm q}\\delta [E(k')-E(k) \\pm \\hbar \\omega _{q}] \\; \\; (16)", "52b38c087696807088e799621178ef08": "L_2(4) \\cong L_2(5) \\cong A_5,", "52b3c122d53b0d50a5c4b2e11186d83d": " \\text{DWF} = \\left\\langle \\exp\\left(i \\mathbf{q}\\cdot \\mathbf{u}\\right) \\right\\rangle^2", "52b45428158c1f7129d9d51ffde8a4ad": "\\omega_{k}\\ ", "52b4805b45d3942d4e56b9fe82bcfd9d": "Z = \\frac{\\bar{X}-\\operatorname{E}[X]}{\\sigma(X)/\\sqrt{n}}.", "52b4cbde6774f5d56c581aeacc90d570": "e^2=\\frac{1}{2}", "52b4d7760e0712061055904dd3c4af48": "\\mathbb E(D_0) = \\frac{1}{\\pi} \\sqrt{-\\rho''(0)}", "52b4f2e36f154d6174ef775fc02a54ba": "\\gamma=0.380", "52b5885e7716e342ccae4b39edcd274d": "[Z]_i", "52b63a682f84640945b528518b5f0d41": "\\frac{\\sum_{k=1}^\\eta d(k)}{\\eta} \\approx \\ln \\eta + 2\\gamma - 1,", "52b67fd288c710b1b8dbdfe1ef98b9b6": "i>k", "52b721e8df04c2299d5253591ff1d57f": "\\int_E |f|\\,d\\mu < + \\infty.", "52b779d911bf0f259d040a3a3f7cbc8c": "\\tilde{d}_{min} = \\frac{c - b^T A^{-1} b}{\\mbox{trace} A}", "52b785010d38171802c8a1771686912f": " \n \\int_0^{2 \\pi} {d\\varphi \\over 2 \\pi} \\cos\\left( \\varphi \\right) \\exp\\left( i p \\cos\\left( \\varphi \\right) \\right)\n=\ni\\mathcal J_1 \\left( p \\right)\n . ", "52b7923b284559e3b36605e9d5ff1cfe": "((\\sigma_{ij} - \\bar{\\sigma}_{ij}) \\mathbf{\\hat{n}}) \\cdot \\mathbf{\\hat{t}_2} = \\nabla_{\\!S} \\gamma \\cdot \\mathbf{\\hat{t}_2}", "52b7ca945cf7da0c83db30fde2494381": "\\left\\{\\ldots, a - 2n, a - n, a, a + n, a + 2n, \\ldots \\right\\}", "52b7ce0c7b36ed951b33f452f8b52a08": "\\frac {\\partial M_z(t)} {\\partial t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _z - \\frac {M_z(t) - M_0} {T_1}", "52b7f439e709c3ec6868d20d5be4048d": "\nU(A)=\\sum\\limits_{j} P(A > O_j) D(O_j),\n", "52b843d030a5926b8b04cb802019b5c0": "\\lim_{a\\to\\infty}\\int_{-a}^a x f(x)\\,dx, \\!", "52b85436ebb7852eb1aea249755ca425": "df({\\mathbf v})", "52b85fb884756322ae57cf4bb9450358": "\\begin{align}\n v_g &= \\frac{\\partial E}{\\partial p} = \\frac{\\partial}{\\partial p} \\left( \\frac{1}{2}\\frac{p^2}{m} \\right),\\\\\n &= \\frac{p}{m},\\\\\n &= v.\n\\end{align}", "52b8d8744587922bd35115377908451e": "n=\\frac{N_{\\rm A}}{M}\\rho", "52b8e9c0c74a1588119abfacb2fd91f6": "\\oplus_{n \\in \\mathbf Z}Hom_n(A,B)", "52b8fd789fd5969e73e2b5f19000f30d": "-0.493091109\\ldots", "52b910ce9e23e5baf577a22144bef184": "\\sigma_t = \\sigma_1", "52b9594218dcbd3970ea0165d414c9ad": "C_p = 1-|\\vec{V}|^2 ", "52b95e62fb3739438332795b2d1c2da5": "V-CVR=Viewthrough Conversions/Viewthrough Visits", "52b96ea2ae0c3ba15e43318178b81906": "\\left(\\tfrac{-4}\\cdot\\right)", "52b970147c5c2476698c7d9e045a83c1": "a_6\\times (3\\rho^2-2)\\rho \\cos(\\theta)", "52b9778bbdec33530284aea51a8328a7": "\\left[ \\begin{matrix} 1 & i \\alpha \\\\ 0 & 1 \\end{matrix} \\right] ", "52b9918f6f48a2c880b9776681d4f22c": "c_{i+1}=\\left\\lfloor\\frac{1}{2}(c_i+n_i+n_{i+1})\\right\\rfloor", "52b9a4d329e0084a00796b2008877719": "\\displaystyle{z^8}", "52b9c39944eb38561abf26bbbe8c01c3": "\\varphi(h) = \\frac{f(a+h) - f(a)}{h} - f'(a)", "52ba06c4e29951fd829c8760e75e3fd0": " \\sigma = \\sigma_0 \\sin(t\\omega + \\delta) \\,", "52ba40d5899499e1678cab90b7e43432": "\\frac{d}{dx} \\left(u+v\\right) = \\frac{du}{dx} + \\frac{dv}{dx}", "52ba6d6938f0200d4d85319318d70db6": "f(3) = 5", "52ba7ed1b68aff32df689e081b695a14": "m_n = \\frac{\\Gamma(\\alpha+\\beta)\\Gamma(\\alpha+n/\\gamma)}{\\Gamma(\\alpha)\\Gamma(\\alpha+\\beta+n/\\gamma)}.", "52ba8c3486a645b03cbb93fb65dc5754": "\\partial_t \\eta = -c\\, \\eta', \\,", "52ba99ff496748c07526d767af4c9c4c": "A_i := \\operatorname{Adj}(R_i) \\setminus R", "52ba9f5307a8be434a8322f5c23caf85": "\n\\ | H(e^{j \\omega}) | = \\sqrt{(1 + \\alpha^2) + 2 \\alpha \\cos(\\omega K)} \\,\n", "52bab5b3d1faceb96eb68ab0a5628352": " T_i , i=1,2,..,m ", "52babb530a879f15456fcd43ea1534e0": "a_n>0 ", "52babd457879b7107f6a1d64a16aeb7e": "X \\in L^p(\\mathcal{F})", "52bad69bea1dcc3baabe08203fa56b0e": " Coverage \\approx \\frac{Total \\ Faults \\ Excluding \\ Operational \\ Failure}{Total \\ Faults \\ Including \\ Operational \\ Failure}", "52baf6ad14efe3dcf9616d05b3b57bfc": "T=\\frac{(C_1-C_2)^2\\left(\\frac{1.128}{1-\\frac{0.177kC_1}{C_2}}\\right)^2}{4Dse\\!f\\!\\!f\\left(\\left.\\frac{\\mathrm dC}{\\mathrm dx}\\right|_{x=0}\\right)^2}", "52bb2821b99315a3f03ced70d9a9cf92": " E[N]\\, ", "52bb2e466003779625fd957c77776675": "\\int {1 \\over x}\\,dx", "52bb2fe12448c2b1998dc3505b1307f4": "A_q (n,d)", "52bb34f2c92ac9ece023fbbf76cbf847": "E(\\vec{k})", "52bb375901fb29bfcc0cfb6136741c4b": "R \\subseteq S", "52bb585340341f78b1558e14f6171c22": "p(r,t,k) = \\operatorname{Real}\\left[p(r,k) e^{i\\omega t}\\right]", "52bb5d411d35bfa1df958b7121503581": "l_i\\equiv\\partial_x+a_i\\partial_y+b_i", "52bbe241af913dcb193cd0fbba493d29": "c_n(s)= \\sum_{a=1\\atop \\gcd(a,n)=1}^n e^{2 \\pi i \\tfrac{a}{n} s}.", "52bc3fbe585018376bfebc8bd8de52d0": " \\alpha = { {2-q} \\over {q-1}}, ~ \\lambda = {1 \\over \\lambda_q (q-1)} .", "52bc437ec17bb09c65f005c5ffb1b390": "z=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2", "52bc768258bfce12434118dc8c8c9c74": "f(X; \\theta) = \\frac{e^{-(X-\\theta)^2/2}}{\\sqrt{2 \\pi}}", "52bcc81cb978f4187c338a7f1484d7e1": " I+R\\xrightarrow{\\beta}2R ", "52bd359b58a68826c53a3947b8757362": "\\exp_1(x)=y(|x|_1)=y(|x|)", "52bd3e03cfdb3c1ea5757950994455d1": "K_{eq}^A ", "52bd691f8237ea03ca0f3168e70e0b3c": "U_{A \\to B} = (b + c) - (a + c) = b - a \\,", "52bd9a7cbed332f83ce8ccb41070199b": "q=e^{\\lambda}", "52bdf55349e265fc820299c2d105ed1e": "\\mathbf{\\Gamma}", "52be7b5f049df8950dee4ceba1d2ff93": "FDCR", "52bf1c3a76dca7550084ecaa9ce6ff11": "\n{\\rm Pr}\\Big(\\hat{f}(x)-w(x) \\le y^* \\le \\hat{f}(x)+w(x)\\Big) = 1-\\alpha,\n", "52bf524ec1429be8db452338f8b6f985": "\n\\begin{align}\n\\rho(\\mathbf{r})&=N\\sum_{{s}_{1}} \\cdots \\sum_{{s}_{N}} \\int \\ \\mathrm{d}\\mathbf{r}_2 \\ \\cdots \\int\\ \\mathrm{d}\\mathbf{r}_N \\ |\\Psi(\\mathbf{r},s_{1},\\mathbf{r}_{2},s_{2},...,\\mathbf{r}_{N},s_{N})|^2, \\\\\n&= \\langle\\Psi|\\hat{\\rho}(\\mathbf{r})|\\Psi\\rangle,\n\\end{align}\n", "52bf712fb41a2ae32b1a96340e2f8c88": "(0,-1)", "52bf93da6b51f30461142571e9735791": "(2m+n,m)", "52c0094d87467a999d8108c999743849": "\\sin\\alpha", "52c012a26e077ed8e5cc64bf2aed16f6": "K_0=\\frac{a}{\\gamma-1},\\ \\ \\ \\phi=\\frac{(1+a)^\\gamma(\\gamma-1)^{\\gamma-1}}{(a^{\\gamma-1})(\\gamma^\\gamma)}.", "52c0177385d5380bb40a1dda69ce39d1": "\\frac{ \\partial x_i w_{ji} }{ \\partial w_{ji} }=x_i \\,", "52c0374ea2cd7a1e966dabca2fe3b510": "F(\\omega)= \\int _{-\\infty}^\\omega S_{xx}(\\omega')\\, d\\omega'. ", "52c051806cfdcabe13e8338caa1358bc": " \\gamma_3 ", "52c0b75439e549a8ecdd9dc8d7901dcd": "\\sqrt[2]{x^3}", "52c10952aba10db92f77c85905376dc1": "\\langle v-u(t), F(t,x(t),u(t))\\rangle\\geq 0", "52c1199a504faa26641d4eeab9ceebff": "E\\supseteq L\\supseteq F", "52c1b756063b23b3c9a3dc0129517c79": "(T, Con, \\vdash) ", "52c1b94f399c582fdbaff85b7cacc451": " \\sum_{n=0}^{\\infty} \\frac {8^{2^n}}{2^{2^{n+2}}-1} = \n\\sum_{n=0}^{\\infty} \\cfrac {\\tfrac {1}{2^{2^n}}} {1-\\tfrac{1}{2^{2^{n+2}}}} ", "52c1e5ae8949a6ee32ac9e6306796186": " cW_{TOT} = W_B= W_{\\alpha,B} + W_{\\beta,B} = aW_\\alpha + b\\left(W_{TOT}-W_\\alpha\\right)", "52c1f4933784554a3140adc9294864d1": " IMM_{i-1}(S_{x,{i-1}}, t) ", "52c23b2ec0cf89c3de70799b58f9442b": "\n \\delta V_{\\mathrm{ext}} = \\int_{\\Omega^0} q~\\delta w^0~d\\Omega\n ", "52c280ef34265bc3adc7d243b4c2cb81": "\n r(\\vec{c}) = \\min_{\\vec{p} \\in P}\\quad (I(\\vec{p}) - I(\\vec{c}))^2 + (I(\\vec{p'}) - I(\\vec{c})) ^2\n", "52c2e3f977253f0a424c44d8366f7d67": "B(\\lambda) \\psi = \\psi", "52c2edda309684562417f66822658519": " V = \\alpha P(0,T)\\left(F N(d_1) - K N(d_2)\\right), ", "52c3012eb4382c33ebdb4ecae734537f": "\\left(1,\\tfrac{1}{2}\\right) \\oplus \\left(\\tfrac{1}{2},1 \\right)", "52c32802cff51a4495d94de4eb2c1bcd": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 21.25867 \\log_e(T+273.15) - \\frac {16726.26} {T+273.15} + 165.5099 + 1.100480 \\times 10^{-05} (T+273.15)^2", "52c32f7bad0308f5af67fd744155442c": " \\chi = 0 \\,", "52c366126f1a9db8f1f5785c63f51cbc": "D_i = K", "52c38f2823b69a03599dbebecc55c3d8": "8128 = 2^6(2^7-1) = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3", "52c39d3fd659a39ab86ff9a579275095": "\\exp_2^{i-2}(n^{O(1)})^{\\Sigma_j^{\\rm P}}", "52c3bb1c7493f4339ceb82f1a63b7b6f": "\\operatorname{Hom}_R S^{-1}(M, N) \\to \\operatorname{Hom}_{S^{-1}R} (S^{-1}M, S^{-1}N)", "52c3d09aad2f7744cce875d95e8de8a8": " CVG = -(\\frac{du}{dx} + \\frac{dv}{dy}) ", "52c4747a514a7817e3d36c51131e6f12": "\\phi = (1+\\sqrt{5})/2", "52c49cb58e85d7ed2d57b4b2f1a7277b": "\\ell_{P}\\sim 10^{-35}", "52c4ab5e993e816b71e693aa86106555": "U = \\begin{pmatrix} \\cos\\theta & \\sin\\theta \\\\ -\\sin\\theta & \\cos\\theta \\end{pmatrix}.", "52c4b3ca89a250babc6b4c93191e1e5e": "a_{(n+1)}b = (a_{(n+1)}(b-1))_{(n)}a", "52c4c6614266d9a8d191b559438544da": "\\delta( \\sigma \\otimes \\tau) = \\delta_X \\sigma \\otimes \\tau + (-1)^p \\sigma \\otimes \\delta_Y \\tau", "52c52f49a1296e87af55d2cd5b17de96": " {\\kappa}^{\\sigma} = \\frac{4F^2Cz^2D(1+3m/z^2)}{RT\\kappa}\\left(\\cosh\\frac{zF\\zeta}{2RT}-1\\right)", "52c53e63bd4e183e0f3e5d9e73ac21e4": "\nu \\ \\stackrel{\\mathrm{def}}{=}\\ \\rho \\sin(\\phi) \\ \\stackrel{\\mathrm{def}}{=}\\ P\n", "52c554230a0ad4eaec805ac3b913d598": " e^{-za}M_X(z)=\\alpha ", "52c5a77790638b5a79fb257b6ae99be3": " S_2 = {{9\\over8} \\div {16\\over15}} = {135 \\over 128} \\approx 92.179 \\ \\hbox{cents} ", "52c6270028ebf6e30aeb8cdf655f051b": "b^d", "52c63ec4bc9ad9b5bb6d2db016419c51": "P=\\left(\\begin{matrix}p_{1,1}&p_{1,2}&\\dots&p_{1,j}&\\dots\\\\\np_{2,1}&p_{2,2}&\\dots&p_{2,j}&\\dots\\\\\n\\vdots&\\vdots&\\ddots&\\vdots&\\ddots\\\\\np_{i,1}&p_{i,2}&\\dots&p_{i,j}&\\dots\\\\\n\\vdots&\\vdots&\\ddots&\\vdots&\\ddots\n\\end{matrix}\\right).", "52c6625dc3298cfafed719e29646cce6": "\\int_{-\\infty}^x \\int_{-\\infty}^z [F_B(t) - F_A(t)] \\, dt \\, dz \\geq 0", "52c68382bed84b0c4c95249553624653": "\\begin{align} &{}\\quad {\\partial^3 \\over \\partial x_1\\,\\partial x_2\\,\\partial x_3} (uv) \\\\ \\\\\n&{}= u \\cdot{\\partial^3 v \\over \\partial x_1\\,\\partial x_2\\,\\partial x_3} + {\\partial u \\over \\partial x_1}\\cdot{\\partial^2 v \\over \\partial x_2\\,\\partial x_3} + {\\partial u \\over \\partial x_2}\\cdot{\\partial^2 v \\over \\partial x_1\\,\\partial x_3} + {\\partial u \\over \\partial x_3}\\cdot{\\partial^2 v \\over \\partial x_1\\,\\partial x_2} \\\\ \\\\\n&{}\\qquad + {\\partial^2 u \\over \\partial x_1\\,\\partial x_2}\\cdot{\\partial v \\over \\partial x_3}\n+ {\\partial^2 u \\over \\partial x_1\\,\\partial x_3}\\cdot{\\partial v \\over \\partial x_2}\n+ {\\partial^2 u \\over \\partial x_2\\,\\partial x_3}\\cdot{\\partial v \\over \\partial x_1}\n+ {\\partial^3 u \\over \\partial x_1\\,\\partial x_2\\,\\partial x_3}\\cdot v. \\end{align}", "52c6be1d95da441aaf8cbb28f210ccec": "A\\in Sp(2)\\subset SU(4)", "52c6db1c472ece074843e9708a0aac1b": "H_{\\rm int}(t)\\equiv e^{{(i/\\hbar})tH_0}\\,V\\,e^{{(-i/\\hbar})tH_0}", "52c717163d614700a8e074283fb6cdd8": " I=\\frac{4\\pi e}{\\hbar}\\int_{-\\infty}^{\\infty}\\left[f\\left(E_F-eV+\\epsilon\\right)-f\\left(E_F+\\epsilon\\right)\\right]\\rho_S\\left(E_F-eV+\\epsilon\\right)\\rho_T\\left(E_F+\\epsilon\\right)\\left|M_{\\mu\\nu}\\right|^2\\,d\\epsilon \n\\ ,\\qquad\\qquad (1)", "52c7b7f511b436208d77809cc3a89575": " w \\wedge z = (w_1 \\cdot z_1, \\ldots , w_n \\cdot z_n ), ", "52c830d4eae2d3bb54216124045934cd": "c=(\\mathbf{o}-\\mathbf{c})\\cdot(\\mathbf{o}-\\mathbf{c})-r^2", "52c89b4c8d84d657ddc69bc592c51458": "\n \\mathrm{sf}(n)\n =\\prod_{0 \\le i < j \\le n} (j-i)\n ", "52c8a0e2deaae3f82ed35a8d9e35f8ca": " \\langle x, y \\mid x^3 = y^3 = (xy)^3 = 1 \\rangle ", "52c9058c7a9ba607786bb0ae9bcb4504": "\nf(w_1,w_2,\\beta)=\\sum_{i=1}^\\beta (f^\\text{pmi}(X_i^{w_1},w_2))^\\gamma\n", "52c9211146fd62c36631f943d6ccd3eb": "R_a=100-4.6 \\Delta \\bar{E}_{UVW}", "52c92dd8d32665e0afed815a16fa3707": "[p, q, r] ~\\leftrightarrow~(q \\rightarrow p) \\and (\\neg q \\rightarrow r)", "52ca65dfeea07ae8fc6d3da526163e8b": "a,b\\;", "52cab20b8bd956c74c4ff8e763b47fde": "E=hc \\bar \\nu", "52cadadb3f8b7554c662e41713d81cf1": "D=\\{x\\in G~|~{\\rm Tr}_{q^{n+2}/q}(x)=0\\}", "52cae9ff5a91247c890559a1cf289b8f": " 1 > \\lambda \\ge 0.75", "52cb3ff5dc66f2396e5d6265d3974ce2": "(4, 10)", "52cb874956ca3b3e3346cd122188365a": "p_\\beta \\in C_\\beta", "52cb881c528ca701b8c3df88a368d2df": "\\frac{\\mathrm{d} \\mathbf x}{\\mathrm{d} t} = \\left(\\frac{\\mathrm{d} x}{ \\mathrm{d}t}, \\frac{\\mathrm{d} y}{\\mathrm{d} t}, \\frac{\\mathrm{d} z}{\\mathrm{d} t}\\right)^T", "52cb9d2b97a01a874bb9e8389d9ec020": " DF(T) = \\frac{1}{( 1 + \\frac{r}{360} )^{ 360T } } ", "52cba524d806e2d15e9060d5fabe838f": "(\\alpha_1,\\alpha_2)", "52cbb83785ea23df8b02625ab91be47e": "( \\neg \\phi \\to \\neg \\neg \\psi)", "52cbd9306b1952066734911dee28066c": "Q=n^\\nwarrow\\mid n^\\searrow(~)", "52cbe62120086357d8eaf3130de59cca": "\\gamma\\approx 2", "52cbe9e99b9ff53946eaf650d989dfb8": "\\displaystyle = \\frac{D_r + D_{r+1}}{2} \\pm \\frac{t}{h}\\frac{|D_r-D_{r+1}|}{2}.", "52cbf3c25f8b838022dacef84bc3ce32": "\n \\mathrm{i}\\hbar\\frac{\\partial}{\\partial t} \\Delta \\langle \\hat{B}^\\dagger_{\\omega} \\hat{B}_{\\omega'} \\rangle\n = (\\hbar \\omega' - \\hbar\\omega) \\,\\Delta \\langle \\hat{B}^\\dagger_{\\omega} \\hat{B}_{\\omega'} \\rangle\n + \\mathrm{i} \\sum\\limits_{\\mathbf{k}} \\left[\\mathcal{F}_{\\omega'}^\\star \\Pi_{\\mathbf{k},\\omega} + \\mathcal{F}_{\\omega} \\Pi_{\\mathbf{k},\\omega'}^\\star \\right]\n", "52cc3c429e9f4848a225733db6dea506": "\\ F_+ + F_- = 1 ", "52cc62c53b796db3de3db26a660268dc": " MPGe = \\frac{total~miles~driven}{\\left [ \\frac{total~energy~of~all~fuels~consumed}{energy~of~one~gallon~of~gasoline} \\right ]} = \\frac{(total\\ miles\\ driven) \\times (energy\\ of\\ one\\ gallon\\ of\\ gasoline)} {total~energy~of~all~fuels~consumed} ", "52cc688697bfe7f0dc2b9e4f78f9ab83": "L_n^{\\alpha }(z)", "52cc797b6421db0c3cc3f081557b1e01": "Y_{N_t}", "52cc934511696ab9e2de1e6f9ed7ae85": "\\scriptstyle \\sqrt{1+\\alpha^2}.", "52cc9cebe66a01d6f7662b80f72cea6b": "\n\\tan\\gamma_1=\\cos\\beta\\tan\\gamma_2\\,\n", "52ccf88c6cadc95ec24161330936f7fc": " [K][U] = [F] ", "52cd035d324d650c3e078fa7578cd340": "\\ w_j = 1", "52cd12d538d2346fcb8083338e8f0734": "u_{\\mathbf1^n}", "52cd2912358a22f425c19822701622e3": " R_{i} = \\begin{bmatrix} \\cos \\gamma_{i} & -\\sin \\gamma_{i} \\\\ \\sin \\gamma_{i} & \\cos \\gamma _{i}\\end{bmatrix} ", "52cd3ecfacfc770fbf19c3d97f19c7c1": "p_1+p_2+p_3=p_1^2+p_2^2+p_3^2=1.", "52cdcc9f55d19e0d4d208a6b47d2b739": "|N|!", "52ce232f12a1ecd2ed2f4d2b2868f3fb": "\n \\underline{\\underline{\\mathsf{A}_\\varepsilon}} = \\begin{bmatrix} \n A_{11}^2 & A_{12}^2 & A_{13}^2 & A_{12}A_{13} & A_{11}A_{13} & A_{11}A_{12} \\\\\n A_{21}^2 & A_{22}^2 & A_{23}^2 & A_{22}A_{23} & A_{21}A_{23} & A_{21}A_{22} \\\\\n A_{31}^2 & A_{32}^2 & A_{33}^2 & A_{32}A_{33} & A_{31}A_{33} & A_{31}A_{32} \\\\\n 2A_{21}A_{31} & 2A_{22}A_{32} & 2A_{23}A_{33} & A_{22}A_{33}+A_{23}A_{32} & A_{21}A_{33}+A_{23}A_{31} & A_{21}A_{32}+A_{22}A_{31} \\\\\n 2A_{11}A_{31} & 2A_{12}A_{32} & 2A_{13}A_{33} & A_{12}A_{33}+A_{13}A_{32} & A_{11}A_{33}+A_{13}A_{31} & A_{11}A_{32}+A_{12}A_{31} \\\\\n 2A_{11}A_{21} & 2A_{12}A_{22} & 2A_{13}A_{23} & A_{12}A_{23}+A_{13}A_{22} & A_{11}A_{23}+A_{13}A_{21} & A_{11}A_{22}+A_{12}A_{21} \\end{bmatrix}\n ", "52ce566739e8bd9c213f48c7eb660eb6": "d(x,y) = 0\\,", "52ce8a123f1dbf7fe07ac9b141038c7a": "\n\\begin{align}\n\\Pr(Y_i = 1) &= \\Pr(\\max(Y_{i,1}^{\\ast},Y_{i,2}^{\\ast},\\ldots,Y_{i,K}^{\\ast})=Y_{i,1}^{\\ast}) \\\\\n\\Pr(Y_i = 2) &= \\Pr(\\max(Y_{i,1}^{\\ast},Y_{i,2}^{\\ast},\\ldots,Y_{i,K}^{\\ast})=Y_{i,2}^{\\ast}) \\\\\n\\cdots & \\\\\n\\Pr(Y_i = K) &= \\Pr(\\max(Y_{i,1}^{\\ast},Y_{i,2}^{\\ast},\\ldots,Y_{i,K}^{\\ast})=Y_{i,K}^{\\ast}) \\\\\n\\end{align}\n", "52cead0896550a13728605a763d2a484": "f(t,n) \\leq f(t+1,n)", "52cef39b3a5c398d8e9228096f9b1c0c": "F_{ab} = \\partial_a A_b - \\partial_b A_a.", "52cf3e188c430eb48de3e5b8686fef98": "GH=2GO;", "52cf43f34a0f74eb6d4a8826be460489": "C:\\, S \\to T^*", "52cf48d6ac50a96502ea19575fd813fe": "\nu= m(1+q^2)(\\ddot{q}_d + \\alpha \\dot{e}+\\frac{\\kappa}{2}r )+K_0q+K_1q^3+b\\dot{q}\n", "52cf6e28cf2571f17263406bfafd3c7a": "A3:= 26.246", "52cf82b3f27a0064521ead00e0c1c758": "\\exp_{10}^3(8.56784)", "52cffe918af83e8100d536a14530ae32": "n/\\log(n)", "52d0116666cf13cc738d0e85921faecb": "\\mathbf{A} = A_x \\mathbf{e}_x + A_y \\mathbf{e}_y + A_z \\mathbf{e}_z ", "52d01559cf93dbd068dec08eb481d525": "\\left( \\text{map}\\,f \\right) \\circ \\left( \\text{map}\\,g \\right) = \\text{map}\\,\\left( f \\circ g \\right)", "52d03579a7a4422a4344e59475fb1555": "r=\\sqrt{(d-a)^{2}+4bc}", "52d079080fbd49a0339f428e99998ee5": "\\mathrm{Rot}_G : [N] \\times [D] \\rightarrow [N] \\times [D]", "52d0c6e6608a0d45fd6bb8fd5e23d1ab": " E [d- f(u)]^2\\,", "52d15a4c60394e906e9dd4fe504ada55": "\nY_2 = \\begin{cases}\n1 & \\text{if }Y^*_2>0, \\\\\n0 & \\text{otherwise},\n\\end{cases}\n", "52d15eaa258c7d4958fede54f85feb48": "f(a,-z), f(-a,iz)\\text{ and }f(-a,-iz).\\,", "52d18b06c6be0c2812f8b93e92f4c8ac": "e^{\\rho + \\alpha i} = k. \\;", "52d1e321b7f4d74562297e86bb366bcf": "T^2_{p,m}", "52d235a4990769bdf257b0d954f4a939": "g \\colon \\bigoplus_{i \\in I} A_i \\to B", "52d273def00bfd42c6292d557a69c292": "\\Delta E_C / \\Delta E_G = .73", "52d275f608d08926f258c866cd7ad966": "\\begin{bmatrix} \\dfrac{-b_{21}}{b_{22}} & \\dfrac{-1}{b_{22}} \\\\ \\dfrac{\\Delta \\mathbf{[b]}}{b_{22}} & \\dfrac{-b_{12}}{b_{22}} \\end{bmatrix}", "52d2d0987452b9cbcdcbd76ece296533": "\\frac{\\Gamma(s+1)}{\\zeta(-2s)} = {\\mathbf M}({\\rm Riesz}(z)) ", "52d31680c2ee7727cc862e561c228aa8": "K \\cap N = \\emptyset", "52d325877551787d40fb32388d2e8aae": "\\partial_{n}:=n\\cdot\\nabla", "52d34910723a13d2047d8eece300c34f": "\\frac{x^2}{a^2}-\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0.", "52d38faa4613d25e33ccdca51fcca1d4": "x(j^{1}_{p}\\sigma) ", "52d3d709e4707fd4314e84d3e9d5edd8": "f_x=\\frac{\\partial f}{\\partial x}", "52d415cdb5045450f1a2f9be1d8b073d": "\\begin{vmatrix} V_{11}-\\Delta E_j & V_{12} & \\dots & V_{1N} \\\\\nV_{21} & V_{22}-\\Delta E_j & \\dots & V_{2N} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nV_{N1} & V_{N2} & \\dots & V_{NN}-\\Delta E_j \\end{vmatrix}.\\,", "52d4524bd16654939cb633aca44fbe3a": "\\pm 2/\\sqrt{N}", "52d486f5519d357b20ce6db33070ecad": " \\operatorname{perm}(A)=\\sum_{k=0}^{n-1} (-1)^{k}\\Sigma_k.", "52d4d6c0345834ef4ae2c597d02209d5": "\\mathbb{R}[x]", "52d4d7c4bb971fc1b34b0b219fd5b668": " t^a(d,n) \\leq O(d\\log{n}) ", "52d4fce11e59f0395730bb90f9b239cc": "[a,b]\\!", "52d50404f827bbeee59ac8a21a1378b5": "f:X\\to Y", "52d5238b64be78a117d7ebb42c2b20bd": "S^{lm}", "52d552fb58dfff8749edc7ae9a3ec8f5": "g(p) = \\Phi^{-1}(p).\\,\\!", "52d56b5868b19a4a024cbbd908475c0e": " \\mathbb{E} ( \\mathbb{E} (Y|X) ) = \\sum_x \\mathbb{E} (Y|X=x) \\mathbb{P} (X=x) = \\mathbb{E} (Y), ", "52d5b315f816d3bf4c84b5c4fd5be6fc": "10 \\uparrow\\uparrow n < (10\\uparrow)^n a < 10 \\uparrow\\uparrow (n+1)", "52d5b7ab0919fccfa5b6bf025bf2b869": "a+b \\sqrt{c} = d + e + 2 \\sqrt{de}.", "52d5f36f86a759125ccc07aa984b84f6": "\\theta = \\sin^{-1}(2u)", "52d60551c378e265be1093f65e1ffc0c": "\\upsilon = 0\\,", "52d608234ae27fb2e222701d150766fc": "{Li_{k}(1-(ab)^{-x})\\over b^x-a^{-x}}c^{xt}=\\sum_{n=0}^{\\infty}B_{n}^{(k)}(t;a,b,c){x^{n}\\over n!}", "52d624c6014b24c6f93a47ec25acbe6d": "x^i", "52d65c28ec3fd04525eb4f78e7a5e7a1": "c \\propto \\exp( - D^{-1} \\zeta^{-1} U)", "52d68a61c750a0c7f4e260f0ac9e1369": " r_\\mathrm{ corr } = r \\left[ 1 + \\frac{ 1 }{ n } \\left( \\frac{ 1 }{ m_x } - \\frac{ s_{ xy } }{ m_x m_y } \\right) + \\frac{ 1 }{ n^2 } \\left( \\frac{ 2 }{ m_x^2 } - \\frac{ s_{ xy } }{ m_x m_y } \\left[ 2 + \\frac{ 3 }{ m_x } \\right] + \\frac{ s_{ x^2 y } }{ m_x^2 m_y } \\right) \\right]", "52d719036ab4e6757643b2ce188708dc": " \\frac{ V_{geostrophic} }{ V_{cyclone} } = 1 + \\frac{ V_{cyclone} }{ V_{inertial} } > 1 ", "52d77e343a2fe90639e4271ab372b65d": "{n \\choose m}=\\frac{n!}{m!\\,(n-m)!}", "52d7c581dfa34628def8c98613163daf": "\\omega(s) = \\int_0^\\infty e^{-st}\\,dF(t).", "52d7cb8b5d546e995999273b8aaa0068": "\\vec{f}(\\vec{u})", "52d876efd59c26ac290f5d6007461936": "Spring: W = 0.000005297L^{3.110}\\!\\,", "52d89f5cd151947811603bb14aabed7f": "[1-(1-x^a)^b]\\,", "52d933a0a583cb438b5484df82887f16": "V-\\,", "52d96fed1c6f8977d8f1edcac8fba457": "b\\left(\\frac{\\frac{1}{8}n_1^2 }{n_0 + n_1}\\right)", "52d9aca052aa6797fc72549a91e7ac43": "V_{out-down} = -\\dfrac{V_{ref}}{RC}t_{d} + V_{out-up} = 0", "52d9f08faa28ed7de12ba9e2f5054442": "G = \\begin{pmatrix}\nG_{11} & G_{12}\\\\\nG_{21} & G_{22}\\end{pmatrix}", "52da082c8c7510abf429802e1998176a": " \\Gamma_\\text{chir} \\Gamma_a ~,~ \\Gamma_{a_1 a_2} ~,~ \\Gamma_\\text{chir} \\Gamma_{a_1 a_2} ~,~ \\Gamma_{a_1 a_2 a_3} ", "52da2058f1f468a0e00eccab5367e5fe": "\\Big\\vert \\frac{f(a)}{f(b)} - \\frac{\\log_2(a)}{\\log_2(b)} \\Big\\vert \\leq \\frac{1}{t}.", "52da726de354255bbaa4640f343e9fe0": "x(t) = \\frac{1}{2}a_0 + \\sum_{n=1}^\\infty\\left[a_n\\cos(2 \\pi n f_0 t) - b_n\\sin(2 \\pi n f_0 t)\\right]", "52dafac65d264dfbfdb2de015b4f9c5f": " \\omega = \\frac{\\sigma \\left ( 1- k^2 a^2 \\right ) }{2a \\mu_B} \\Phi \\left ( k a \\right ) ", "52db555f0bf09715f325d61935fda610": "{\\underset{a_+}{\\lim}G}", "52db7071d7315b926bc0c9bccb580812": "d_u (f_{n}, f_{m}) \\to 0", "52dbc3fd7d34cbe6ae695944dca0004d": "Q_i", "52dbc9a7174bedd305a26fc8dda92595": "n_i = 2", "52dca2d5f4b877f4a65b5dc80344dc34": "\n\\frac{\\displaystyle 1}{\\displaystyle 17}\n\\begin{bmatrix}\n1 & 9 & 3 & 11 \\\\\n13 & 5 & 15 & 7 \\\\\n4 & 12 & 2 & 10 \\\\\n16 & 8 & 14 & 6 \\\\\n\\end{bmatrix}\n", "52dd1c1536bcb36b9efffd4122a69db8": "(S \\times \\Sigma, (x,i)\\rightarrow (T(x,i),G(x,i)))", "52dd514a67ed6c43b539548253761720": "R_O \\approx r_O \\left( 1+ \\frac { \\beta_2 R_2} { r_{\\pi} +R_2} \\right) \\ ", "52dd9768dde621dd455d2bb1fcfc99ea": "\\frac{L}{kA}", "52ddd16d3940cbc999ddec8a092414c1": "g_{\\theta}(x_1^n)", "52ddddf7529d59248b06810f8bdf0c67": "\\frac{3^3 2^1 + 3^2 2^2 + 3^1 2^5 + 3^0 2^6}{2^7 - 3^4} = \\frac{250}{47}", "52de2336201ea80ae9bc2475034e75a3": "\\frac{n+2{\\alpha}-1}{n+1}\\,", "52de25635fd952f86c6530be23a6f311": "\\frac{\\theta}{\\theta_b}=\\frac{\\frac{\\theta_L}{\\theta_b}\\sinh {mx} + \\sinh {m(L-x)}}{\\sinh {mL}}", "52de413dcc80be8afc1ec2d86e0fbfd6": " \\hat{A} = (A, B) = A + \\epsilon B, ", "52de4781fcc78164448038a07d6dd642": "f_{M_t}", "52de56f6cc7d5f698b34ba5f226366d0": "\\sum_{n=-\\infty}^{\\infty}x[n]z^{-n} = \\sum_{n=0}^{\\infty}0.5^nz^{-n} = \\sum_{n=0}^{\\infty}\\left(\\frac{0.5}{z}\\right)^n = \\frac{1}{1 - 0.5z^{-1}}.", "52de61031b2a568f6118ce61d98e56fb": "f_n\\in C^{\\infty}([0,1])", "52de77f16c2dc56da1a6fcbb47baabdf": "\\scriptstyle{\\mathcal{A}}", "52de91f0f9c357f63c5d23a5eac71b0b": " \\mu\\ = \\frac{a+b}{2} ", "52deb3e4e34c8300cff33e0f435b2645": "\\vec{\\alpha}(\\vec{\\theta})", "52df0b4a77ae59354f084f85ce283619": "\\lim_{n\\to\\infty}\\left\\|\\mu\\left(\\displaystyle\\bigcup_{i=n}^\\infty A_i\\right)\\right\\|=0, \\quad\\quad\\quad (*)", "52df9c13c91cdc976d94604ef5599c03": "Z\\leftarrow X^{k-1}", "52e05d1c9a08480fdbe7709ff3585a83": "f_\\infty", "52e067be3d5265f0129e15a35fe616b9": "\n\\sqrt{\\frac{1+z}{1-z}}\n\\left(\n\\exp\\left(\\frac{z^2}{2}\\right)-1 \\, + \\,\n\\exp\\left(\\frac{z^4}{4}\\right)-1 \\, + \\,\n\\exp\\left(\\frac{z^6}{6}\\right)-1 \\, + \\, \\cdots\n\\right).\n", "52e0787f32967d02e6cd6ea637b72bf1": "W = \\frac{\\rho + \\lambda \\mu \\text{Var}(S)}{2(\\mu-\\lambda)} + \\mu^{-1}.", "52e081bd903f6fe8b684e8336f84bad0": "x^5-20 x^3 -160 x^2 -420 x -8928 ", "52e0ca0764a2560b3601de9dc5d72915": " formula\\ mass \\ \\times \\ true\\ percentage_{ingredient} \\ = \\ mass_{flour}\\ \\times\\ baker's\\ percentage_{ingredient}", "52e1081bf7fb8b7839d30fed1ffc05a1": "U B_1 = B_2 U. \\,", "52e12e82764a4576dad10cec7ae5d30e": "10^a", "52e15560ffd2ca045a679d084ed5dfc1": " \\begin{align}\n \\mu (B) &= \\mathbb{P} ( X \\in B ), \\\\\n \\nu (B) &= \\mathbb{P} ( X \\in B, \\, Y \\le \\tfrac{1}{3})\n\\end{align} ", "52e193e14e11e99ae1f4a8b8cc7a0a2e": "\n\\begin{align}\n& {} \\qquad \\int_{-1}^1 \\left(10t^3+17t^2-{7\\over 9}t-{17\\over 3}\\right)\\,dt \\\\[6pt]\n& = \\left[{5\\over 2}t^4 + {17\\over 3}t^3-{7\\over 18}t^2-{17\\over 3} t \\right]_{-1}^1 \\\\[6pt]\n& = \\left({5\\over 2}(1)^4+{17\\over 3}(1)^3-{7\\over 18}(1)^2-{17\\over 3}(1)\\right)-\\left({5\\over 2}(-1)^4+{17\\over 3}(-1)^3-{7\\over 18}(-1)^2-{17\\over 3}(-1)\\right) \\\\[6pt]\n& = {19\\over 9} - {19\\over 9} = 0.\n\\end{align}\n", "52e1a9aa75a29de6ffbb3174399261d6": "P\\bigg(\\bigg[G:\\bigcap_{i=1}^kG_i\\bigg]\\ \\bigg)\n=\\bigcup_{i=1}^kP([G:G_i]),", "52e1b7af79a7858d74709419f1c0fbd5": "J_Y", "52e1c23b5aa2667a846f1333328b46d7": "e_2 = (0, 1, \\ldots, 0) \\,", "52e1f78e7ab88c91d40d9939c7d13644": "0 =-1\\cdot [1+(-1)]=-1\\cdot1+(-1)\\cdot(-1)=-1+(-1)\\cdot(-1)", "52e2160832062dfb5ab838365115d9df": "\n\\mathcal{S}[\\mathbf{q}] \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int_{t_1}^{t_2} L(\\mathbf{q}(t),\\dot{\\mathbf{q}}(t),t)\\, dt \n", "52e23a285d94fe7a31fda4ae8e1546bb": "|M_{1}| \\neq |M_{2}|", "52e2476e6326080ad9c89490eddf2fc8": "[X = \\text{green}]", "52e3208c1a717a4bd18ba404fc06608b": " E = \\{ \\exp(i x) = 1 \\} ", "52e381d48d014ecc75e3ab3638d7fda7": "C_p = I^k t,", "52e3a75ecf4d526e522eb0ee05dfc19f": " g_i \\ ", "52e3b3cea70994bd647ebcc33cd9faad": "\\scriptstyle N\\geq 1 ", "52e45d9e29fe91c072928a8d9b888d0c": "r = \\cos(2\\theta), \\,", "52e47862db98463639ccc96442c744d8": "A_{\\infty}(a,b)\\prod_p A_p(a,b)=1", "52e48037ad7d3a16f8f1d86ece6eca55": "\\boldsymbol{p(x)}", "52e4ab99511d8405da42ddf6c867f152": "v_\\text{p}=0{,}17 \\cdot w^* \\cdot Re \\cdot {\\left( \\frac{d_\\text{p}}{d_\\text{T}} \\right)} ^{2{,}84} \\cdot \\frac{2r}{d_\\text{T}}\\cdot \\left(1 - \\frac{r}{r^*}\\right)", "52e55071b8d9d2348099b92b79199fd5": "\\sum_{n=1}^\\infty r^n", "52e5b9a7e910c7cba71645d359bd3a75": "(c_{n-1}b_n/c_n+c_{n+1}/c_n/b_n)/2", "52e5d07345bf4ac766c72c9789973f31": " Z[J] = e^{i \\int d^d x (\\mathcal L [B(x)] + J(x) B(x))} \\int \\mathcal D \\eta e^{\\frac i2 \\int d^d x d^d y \\frac{\\delta^2\\mathcal L}{\\delta \\phi(x) \\delta\\phi(y)} [B] \\eta(x) \\eta(y) + \\cdots}. ", "52e5d9434351e7e910ae5266209356fb": " K(x,y) \\;=\\; \\overline{K(y,x)} \\;=\\; \\langle K_y,K_x\\rangle. ", "52e61a542b3ca6ed549e718a37d8acc8": "i+4 \\rightarrow i", "52e63b9b3d9d04bdafe6c6eaddf062dc": "\\int_{x=0}^1\\left(\\int_{y=0}^1\\frac{x^2-y^2}{(x^2+y^2)^2}\\,\\text{d}y\\right)\\,\\text{d}x=\\frac{\\pi}{4}", "52e6990bccbb2fd93bf14a6b682d7b6b": " \\forall x \\, P(x) ", "52e6b9e6205352adfb2b1f8e4bd19ac8": "|000\\cdots\\rangle", "52e6d07c56b37864f19a342c6f517eca": "I_\\text{r}=I_\\text{i}N\\sigma\\,", "52e6e9acf165d70a1cc707ff7be904ce": "R= \\{(x,w): x \\in L, w \\in W(x)\\}", "52e72caa78ce4c87167448760817fc79": " \\Lambda^3\\mathbb C^6", "52e76320d66a801053b5af34a1dfc275": " {\\operatorname{d}M \\over \\operatorname{d}t} = \\frac{\\operatorname d}{\\operatorname d t} M \\bigl(t, p_1(t), \\ldots, p_n(t)\\bigr).", "52e7b327141785d9d4d83c3400d56c63": "2^{\\aleph_0} \\leq {\\mathfrak c} \\,.", "52e8160dcb37aa9a5f6b7a2107641f03": "Q(X,Y_1,Y_2,\\ldots,Y_s)", "52e854e7de3c7c03da62428fb195feec": " 4e^2 - 4ae + b^2 c = 0, ", "52e86077236cdbf7b6ff6f7d8d058aed": "\\{ H (N) , H(M) \\}", "52e86af9c968780e8ecd4a649b089b68": "x_1,~x_2,~y_1", "52e8a35b567b0337b852f3f8f115be99": "\\scriptstyle(14\\pm14)\\times10^{-11}", "52e8ab6b8b3b33d882c9a5296794b958": " S_M = \\operatorname{st} \\sum_{k=0}^{n-1} [*f](\\xi_k) (x_{k+1} - x_k) ", "52e8bce35308a8323d730551509ec013": "V\\in \\mathbb{M}_n", "52e8c1dcfcfcd18dfb1d62aa9a121df9": "x^5 - x + 1 = 0.", "52e8c3bb62f41cd75f38b85ccf8931b1": "\nQ_E = k_E L \\rho v \\left [ RH e(T_a) - e(T_s) \\right ]\n", "52e958f6898f36589ea6bbba9a8ebb0d": "\\mathbb{D}^q f(x) = \\lim_{h \\to 0} \\frac{(-1)^q}{h^q}\\sum_{0 \\le m < \\infty}(-1)^m {q \\choose m}f(x+mh).", "52e9c63dfed90979ba3d33582c0b2a34": "\\dot \\theta ' \\ne 0 \\ .", "52ea4208c2f1f5397bfb74fc28baa519": "n = p_1^{k_1} \\cdots p_r^{k_r}, ", "52ea4f62517e9467673c503858ff32d0": "\\alpha<\\beta", "52ea51e293006e66c5f7ba2d79a26882": "\\left(\\frac 7{11}\\right) = (-1)^{ \\left \\lfloor 14/11 \\right \\rfloor + \\left \\lfloor 28/11 \\right \\rfloor + \\left \\lfloor 42/11 \\right \\rfloor + \\left \\lfloor 56/11 \\right \\rfloor + \\left \\lfloor 70/11 \\right \\rfloor } = (-1)^{1 + 2 + 3 + 5 + 6} = (-1)^{17} = -1.", "52ea8ef89889ebf078d484fcc45dc0fd": "f(xy) = f(x) + f(y)\\,\\!", "52ea96fea80dd792eba13d1d81d2e395": "M_\\mbox{linear} = \\nabla_f M \\nabla_f^T = \\begin{bmatrix}1.927 & 0.047 \\\\0.047 & 0.011\\end{bmatrix}", "52eaba2968d3f59ef66db64cb414feb8": " \\pi : M \\rightarrow V ", "52eb5e15fcb13b5160ff82e6f4e4a6ea": "\\displaystyle f'''(x_{0}) \\approx \\displaystyle \\frac{-\\frac{1}{2}f(x_{-2}) + f(x_{-1}) -f(x_{+1}) +\\frac{1}{2}f(x_{+2})}{h^3_x} + O\\left(h_x^2 \\right) ", "52eb750b88521b9e1698269962b3a0ec": "w[n],\\ 0\\le n \\le N-1.", "52ebbdd0ca188291485f969e1e08c320": "4x + 2y = 14 \\,", "52ec62c099e6db0632bd89ecad4879b2": "\\sum_{j=1}^n w_j x_{ij} \\leq Wy_i,", "52eca23894ab0514e47420c8fdd3a4fa": "w_\\lambda", "52ecc5e1de5be55cfc51279599149fb5": "L = |\\mathbf{L}|= |\\mathbf{r}||\\mathbf{p}|\\sin \\theta_{r,p}", "52ece2220466c77f608484b6671bf70c": "(2n)^{\\sqrt{2n}}", "52ecf018664b19888540afd32a57b5aa": " x-y", "52ecface9fe4020dcc38a0f1864d2302": "Q \\leftrightarrow HP", "52ed187e8afad5fdea0b02f5acd3a8a2": "\\bold{u} + d\\bold{u}", "52ee1083647c9a5cc930908e80017361": "I=\\arcsin \\left({n\\sin(R) \\over n_a}\\right) ", "52ee354dd0fc3cd3aa4cc34c02ee13e7": "w(c_2)", "52ee4c6f1753b1d464b6b9f726de8d96": " \\theta_1 ", "52ee60aaaff51521c15f0e7e155d3e7e": "\\epsilon_r \\;", "52eeba1b1020b02a649260b4ee2b7f8a": "E=\\frac{G m_\\text{P}^2}{r}=\\frac{\\hbar c}{r}", "52eedb5116592e76840cf0c58bd1954a": "P(y, x_1, \\ldots x_n)=P(y, x_i)\\prod_{j=1}^n P(x_j\\mid y, x_i).", "52ef0b9f78f54387761bf5b0c0508794": "s=-3; \\quad \\Rightarrow 2x^3+3x^2-4 = (2x^2+x-1)(x+1) - 3\\!", "52ef5f75f8a087cdcceeea0eaf1fbb71": "\\mathbb{E}[X] = \\frac{x_1p_1 + x_2p_2 + \\dotsb + x_kp_k}{p_1 + p_2 + \\dotsb + p_k}\\;.", "52efc024e1ad91b5babc0f2959965956": "B\\in\\mathcal E", "52efc53c70cdc82b1939440d8b30a33c": "\ng(f(x)) = \\exp\\left[ v(x) \\dfrac{\\partial}{\\partial x} \\right] g(x).\n", "52f00cdadba9c51f5d3865882797db2e": "\\Delta g_h \\approx - \\, \\dfrac{ G \\, m_\\mathrm{Earth}}{ r_\\mathrm{Earth} ^2} \\times \\dfrac{ 2 \\,h}{r_\\mathrm{Earth}}", "52f0393b6126cf14203eb61d0ec2c4fd": "\n\\hat{\\kappa} = \\frac{\\bar{R}(p-\\bar{R}^2)}{1-\\bar{R}^2} ,\n", "52f08ed783a2b1f1e0fe9b760dacb23e": "G = K_{|G|}", "52f0c045f231795f59722125c2011b99": "\nI_K(X;Y) = K(X) - K(X|Y).\n", "52f0e84c81a3e912cdc7399987549a74": "m(\\Omega_{gp} - \\Omega (R))", "52f0eb86a893708d18dc55cab92fb1c5": "det(\\mathbf{D})", "52f10c59d89e6b3b1106aa53ac7f183e": "{}- e^4 \\left( \\frac{ (\\bar{v}_{k} \\gamma^\\mu v_{k'} )( \\bar{u}_{p'} \\gamma_\\mu u_p)}{(k-k')^2} \\right)^* \\left( \\frac{ (\\bar{v}_{k} \\gamma^\\nu u_p )( \\bar{u}_{p'} \\gamma_\\nu v_{k'}) }{(k+p)^2} \\right) \\,", "52f123e8829f42d735fbb599ea4ecea4": "\\left(-4\\sqrt{\\frac{2}{5}},\\ -2\\sqrt{\\frac{2}{3}},\\ \\frac{2}{\\sqrt{3}},\\ 0\\right)", "52f12907d1fe7c6ec06ddc35e4f61163": "P = 10^{(8{.}20417 - \\frac{1642{.}89}{78{.}32 + 230{.}300})} = 760{.}0\\ \\mathrm{mmHg}", "52f12e658e14b81d7b6006d11aa165ef": "T_i=\\frac{1}{10c} \\int [r_i(\\mathbf{r}\\cdot\\mathbf{J})-2r^2J_i] \\mathrm{d}^3x.", "52f1aa43e1dce7c3a46b28ddaf26035e": "\\mathbf{x}_h", "52f1b63c30d4d06786532147e4946484": "\n\\lim_{\\mathrm{Re}(\\mu) \\rightarrow \\infty} \\operatorname{Li}_s(-e^\\mu) = -{\\mu^s \\over \\Gamma(s+1)}\n\\qquad (s \\ne -1, -2, -3, \\ldots)\n", "52f1f39bf7a8cf596b281c60ffdbd101": "v=\\frac{R}{\\left[ 2+p\\right]^2}-\\frac{R}{\\left[ m+s\\right]^2} with m=2,3,4,5,6,...", "52f1fcb5c7b1150ee74664af68478f74": " V_{lost ICF} = V_{ICF b} - V_{ICF a} ", "52f24a9d3e6e63e45968aedc4a125ab1": "\\exists{n\\in\\mathbf{N}},|n|_{\\ast}>1", "52f2909e01fa9bd32db5a641358cef7b": "(a - b)", "52f2e04f46360220787d1ceb8473b584": "|\\psi\\left(x,y\\right)|^2", "52f2f282eb10c1b0a8422aae78071a5f": "A^k\\mathbf{v} = \\lambda^k\\mathbf{v},", "52f2f669aabeed8ed03d7150e4980497": "\n\\begin{bmatrix}y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ y_6 \\\\ y_7 \\end{bmatrix} = \n\\begin{bmatrix} 1 & w_1 & x_1 \\\\1 & w_2 & x_2 \\\\1 & w_3 & x_3 \\\\1 & w_4 & x_4 \\\\1 & w_5 & x_5 \\\\1 & w_6 & x_6 \\\\ 1& w_7 & x_7 \\end{bmatrix}\n\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\end{bmatrix}\n+ \n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\\\ \\epsilon_7 \\end{bmatrix}\n", "52f3098333618e2a6408280d8a054539": "\\sum_{n=0}^\\infty \\frac{s(n)}{n!}x^n=(1-x^2)^{-\\frac{1}{4}}\\exp\\left(\\frac{x^2}{4}\\right).", "52f31851f8373da0172d4e35f6b7a6f3": "\\langle{\\mathbb G}_1, {\\mathbb G}_2\\rangle", "52f3320fb7d4c282a0007a6686df1b47": "S(\\vec R) = k_B \\log (\\Omega(\\vec R))", "52f34b1b9b28b571e64e24ff0731adc8": "\nF_Y^{\\text{GIG}}(y|r_1,\\dots,r_j;\\lambda_1,\\dots,\\lambda_p)\\,=\\,1-K\\sum^p_{j=1}P^*_j(y)\\,e^{-\\lambda_j\\,y}\\,,~~~~(y>0)\n", "52f3986345e39c51f1ee963d1fd0a1ab": "H=A\\ and\\ B \\ \\ \\ \\ \\ \\ E=B\\ and\\ C", "52f3b97c93f9396ad3309470c9327dbf": "\\mathbf{B}(\\mathbf{c},\\mathbf{e})", "52f3c0516d4a85ff9454bb11a1425634": "A \\leq C", "52f3cc9c101db26d356240059e4ddfaf": "R(z; A) - R(w; A) = (w-z) R(z;A) R(w;A)\\,", "52f3d81a7761604a16a6e2a6208527ff": "\\forall y\\, q(y,G(F))", "52f41ae4a206cbf5037a2eb77a75adaf": "\\rho(A) ", "52f42dccedab9282b113c517340d56af": "|M_{k_1} \\cap M_{k_2}| \\leq k - 1", "52f4510ad43a34df4d2ddab5a808cca0": " d\\colon H \\to G \\! ", "52f47183229bcab50c03deca40fd1df3": " \\begin{align} \\mathbf{F} & = q \\left ( \\mathbf{v} \\times \\mathbf{B} \\right ) \\\\\n& = \\left ( \\int I \\mathrm{d} t \\right ) \\left ( \\frac{\\mathrm{d}\\mathbf{r}}{\\mathrm{d} t} \\times \\mathbf{B} \\right ) \\\\\n& = \\left ( \\int I \\mathrm{d} t \\frac{\\mathrm{d}\\mathbf{r}}{\\mathrm{d} t} \\right ) \\times \\mathbf{B} \\\\\n& = I \\left ( \\int \\mathrm{d}\\mathbf{r} \\right ) \\times \\mathbf{B} \\\\\n& = I \\left ( \\mathbf{l} \\times \\mathbf{B} \\right ),\n\\end{align}\n\\,\\!", "52f49f825aa22c8a57ff59a603306dc5": "(E_b-E_y)", "52f5614bf47e0d9cfc8e5946548ef4ec": "(x,y)\\not \\in R.", "52f5c438842c39fe9756e2a77da1454f": "s \\in G", "52f5ece6ef7241e26a59e25ef40d89e0": "\\frac{dx}{dy} = \\frac{1}{y^2}", "52f5f55af986155b05330e9ac2eeb860": "a_n = 1", "52f6050ea89161e4d794a8639198c6fd": "b' = a + 2b\\, \\pmod{2^n}\\,", "52f6264f6b4c425770bf29571111d8ca": "S_n(x)", "52f64a192a9ae1e06c1eb9cd4f2a7189": "\nx\\rightarrow x^3 - 3 x (y^2 + z^2) + x_0\n", "52f6ba834651504d7c5e8ca5facf1f28": "l(x)\\ne j.", "52f6c8752f39db61ab3e287985e895f8": "|\\mathcal P|=n^2+n, ", "52f7757f9edc4c41aeed55261565bd26": "w_{u_i}", "52f78adbf30d4354c80c5ab776a6049f": "b(n) = a(n+1)-a(n)", "52f86c083555ac00a12043e4cf964555": "m*\\phi^0", "52f86c5ee60685ce91b312f64d0d455f": "A = {X_1}^2 = 0", "52f909ff07aa4c703b5911d112df9643": "I(v) = max(|left(v) - right(v)| - 1, 0) \\ ", "52f9358e5fa29920f285fefe9ece9f23": " H \\approx H^{MF} \\equiv -J \\sum_{\\langle i,j \\rangle} (m_i m_j +m_i \\delta s_j + m_j \\delta s_i ) - h \\sum_i s_i", "52f94c738046d074584a30c2d75f611b": "\\nabla \\times (\\textbf{E} + j \\omega \\mu \\textbf{A}) = 0\\,", "52f95ff365a96e989a586cd612f5dc41": " (0\\ ,\\ 0.001036)\\,", "52f9694efc4b06b49374a0d3d342a27d": "y^{\\ast }\\left( t\\right) ", "52f98bb9749a8ee11f845c95904e124a": "Pr[h(u) = h(v)] = 1 - \\frac{\\theta(u,v)}{\\pi}", "52f99b42a211cf147a8dfa82ec98c1f0": "\\ln(f_{WC}(\\theta;\\mu,\\gamma))=c_0+2\\sum_{m=1}^\\infty c_m \\cos(m\\theta) ", "52f9bc45d1b730dbe8cd782592e4b636": "(i=1,2,3)", "52f9ca6b178a8fa4a0797d1baae3e305": "\\begin{align}\n \\mu_t ( \\ddot{r} - R \\ddot{\\theta}) & = r \\dot{\\theta}^2 + g (\\cos {\\theta} - \\mu ) \\\\\n r \\ddot{\\theta} & = - 2 \\dot{r} \\dot{\\theta} + R \\dot{\\theta}^2 - g \\sin {\\theta} \\\\\n\\end{align}\n", "52f9d5d0732ab51f599e7a9e1108a1f8": "\\operatorname{Hdg}^k(X) = H^{2k}(X, \\mathbf{Q}) \\cap H^{k,k}(X).", "52f9ff864bdb70db7492065bcafbb2eb": " v_\\mathrm{x} = i_\\mathrm{b} \\left( R_\\mathrm{S}+r_{\\pi} \\right) \\ . ", "52fa171540cc1ee5dba9b1c16bc2ef49": "(n,k,d)", "52fa43396db710fb4d41aaa05d6f3e7f": "\\|a \\wedge b\\| = \\sqrt{(\\|a\\|\\ \\|b\\|)^2 - \\|a \\cdot b\\|^2}.", "52fa469800ed0d4ec3fe4605b1938b68": "\\theta(z;q)=\\prod_{n=0}^\\infty (1-q^nz)\\left(1-q^{n+1}/z\\right)", "52fa4abcceb3dd755b03724e7b50e8b3": "V_h.\\,", "52fa7028566b19c272c7e04af172f152": "C_1 = \\begin{bmatrix} b_{1,0} & b_{2,0} & b_{3,0} & \\cdots & b_{m-1,0} & b_{m,0} \\\\\n\\end{bmatrix}", "52fa88310780f54201f4d5635763e00c": "\\left\\vert \\Phi\n_{n}^{+}\\right\\rangle", "52facecdc5eca2ab7cb9b2d837147318": "= \\cos(\\pi/3) = 1/2", "52fad273bcefc69a14cba614ffc0665c": " H(s) = \\frac { s } {s^2 + 3 s + 1 } ", "52faf4d16adc2f480d34d152a6e38f16": "\\epsilon_{ij} = \\frac{\\mathbf{x}^\\top_{0j}([\\delta K] - \\lambda_{0i}[\\delta M])\\mathbf{x}_{0i}}{\\lambda_{0i}-\\lambda_{0j}}, \\qquad i\\neq j.", "52fb28f4a8fe75c4ee7f3b9a0a4e4ae0": "a^5", "52fbaa7818f64a4bd0e7f940db6f9c04": "u_a(t) = u(t) + i\\cdot H(u)(t)", "52fc19999197309f5f8f00d4978d2304": "\\{y,w\\}", "52fc3c300be14d9af0b36c12d7489861": "- \\Box = \\nabla^2 - { 1 \\over c^2} \\frac{\\partial^2}{\\partial t^2}", "52fc48fc6a74baec60b7389cb21a3898": "\\sigma^1", "52fc86f5d305a8183f0e603e18eb11f2": "X^8-16=(X^4-4)(X^4+4)=(X^2-2)(X^2+2)(X^2-2X+2)(X^2+2X+2).", "52fd1c5cf46ba5df5df385687a110233": "\\mathbf{w} = \\varphi(v, t).", "52fd5cc32f557f6bb09674dc0792a5ca": "V(q) = (q^{-1} + q^{-3} - q^{-4})(q + q^3 - q^4) = -q^3 + q^2 - q + 3 - q^{-1} + q^{-2} - q^{-3}. \\, ", "52fd5e63df88ec0970e6e9a5a1f38954": "\\vartheta \\left( x \\right)\\log \\left( x \\right) + \\sum\\limits_{p \\le x} {\\log \\left( p \\right)}\\ \\vartheta \\left( {\\frac{x}{p}} \\right) = 2x\\log \\left( x \\right) + O\\left( x \\right)", "52fd7115db276e2212a41a88e23c78e9": "\\mathbf{D}=\\varepsilon \\mathbf{E} + \\xi \\mathbf{H} \\,,\\quad \\mathbf{B} = \\mu \\mathbf{H} + \\zeta \\mathbf{E}.", "52fd975a5dbda1e96c87aae5a0dad609": "\\Gamma(s,z) \\sim z^{s-1} e^{-z} \\, \\sum_{k=0} \\frac {\\Gamma(s)} {\\Gamma(s-k)} z^{-k}", "52fdd6b32d26f3c4dd8fd5d9cbaf1efd": "\\left(0,\\ 0,\\ \\pm1,\\ \\pm2,\\ \\pm2,\\ \\pm2 \\right)", "52fdfb48a8488d600be62eacc3fc737a": "[\\mathfrak{k}, \\mathfrak{k}] \\subseteq \\mathfrak{k}", "52fe2fd5ae1852426d0141a08e807ae6": "O(\\varepsilon^{-2})", "52fe3c282f8abe971945c3a033a38bab": "\\Bbb F_p", "52ff0eebead2e31a9bd6fba4406aa26d": " S_3 \\implies A_3 = f ", "52ff5f61cca97b8debfb047ceb154549": "\\frac{\\partial{}^2}{\\partial t^2}u(x,t) - \\frac{\\partial{}^2}{\\partial x^2} u(x,t) = 0 ", "52ff8402d680edc4af8e4e01cea77d18": " \\tau = -\\mathbf{n}\\cdot\\mathbf{b}'. ", "52ff9fe333b42bcd42a3b064dc6a5548": " p_0 = 0 =\\hbar k_0 ", "52fff2e216954a566fce7d6ecd9931d1": "{\\tilde{B}}_5", "52fff5e1ee362c2da171ed18d21e5dfb": "y\\in Y.", "53002a81d101179a990bb0b0b5c93511": "=\\left[\\mathcal{A},\\mu\\right]", "530062da946db880678c0f76ba01e8f8": "\\rho = \\frac{1}{d^2-d \\alpha}(1 - \\alpha P),", "53006af87ac6b1034bf66b9e9ac54808": "\\sum_{t_i} \\mu(t_i|m)U_R(t_i,m,a)", "5300c5a43f62caca274ce690d636ca50": "\\begin{align}\n b_{n+1}(x) &= b_{n+2}(x) = 0, \\\\[.5em]\n b_{k}(x) &= a_k + \\alpha_k(x)\\,b_{k+1}(x) + \\beta_{k+1}(x)\\,b_{k+2}(x).\n\\end{align}", "530101e95f755cddbb75c0924f4c6969": "u_y", "53018eb63770e8bb77bc2415898c5764": "\\,^{z_8 = x_8 y_1 - x_7 y_2 + x_6 y_3 + x_5 y_4 - x_4 y_5 - x_3 y_6 + x_2 y_7 + x_1 y_8 + u_8 y_9 - u_7 y_{10} + u_6 y_{11} + u_5 y_{12} - u_4 y_{13} - u_3 y_{14} + u_2 y_{15} + u_1 y_{16}}", "5301a9405ae9c18d1ef3fbde9dc101a2": " C \\in \\mathcal{R}^{n \\times n} ", "5301ad8ffddd3a971edd32bb05b0ccf3": "\\beta_2 > 0", "5301d63f7938b31319914f3cb983b114": "\\pi\\colon \\mbox{GL}(2,\\mathbf C) \\to \\mbox{Aut}(\\widehat{\\mathbf C})", "530209263a27a5f9b9293c51f4959f1c": " r = k_2 K_1 [H_2][NO]^2 \\,", "530241e8aa30422a214f463eb47e5be8": "|Q\\rangle = \\sum_j Q_j |j\\rangle\\,", "53027634fbbf3bb3c1e300a34b3b0f80": " k X \\sim \\textrm{IG}(k \\mu,k \\lambda)\\,", "5302d0d592132339d4d7eb449a82ec2f": " + E_\\mathrm{sig}E_\\mathrm{LO} \\left[\n\\cos((\\omega_\\mathrm{sig}+\\omega_\\mathrm{LO})t+\\varphi)\n+ \\cos((\\omega_\\mathrm{sig}-\\omega_\\mathrm{LO})t+\\varphi)\n\\right]\n", "530317729b1d84b198fe8e891f1ef914": "\\partial E", "530376d4cb2518b676514e9db76ba3f4": "\\phi(\\boldsymbol{a}, b) = 0", "5303ea6a9cf4b0e1008095b06b51f4fd": "S(x, y, z) = ax^2+by^2+cz^2+2uxy+2vyz+2wzx=0", "5304a17ac58c183e7cdb1628601a98d2": "\\inf \\theta \\ge 1/4", "5305725afbaa1f3a1419b7fe630f043b": "\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu\\gamma^5) = 0 \\,", "53059f7eaf85868fbaef6e5529bbfa6c": "\\Delta_\\infty u(x) = \\langle Du, D^2 u \\, Du \\rangle = \\sum_{i,j} \\frac{\\partial^2 u}{\\partial x_i \\, \\partial x_j} \\frac{\\partial u}{\\partial x_i} \\frac{\\partial u}{\\partial x_j}", "5305cd199e5e7e4f0491f61cc0d0eb66": "\\mathcal{G}-\\mathcal{H}", "5305f2a6d60a39f00670fd53856e6c5f": "g={{Ne^2}\\over{Vm\\epsilon_0\\omega^2}}", "5306055be9625e6911121b9941fc453a": "(\\mathbf{A}^*)_{ij} = \\overline{\\mathbf{A}_{ji}}", "53060dc325a692a79b624b1ddaf3d687": "\\mbox{PH} = \\bigcup_{k\\in\\mathbb{N}} \\Delta_k\\mbox{P}", "53068e4ca278f53dbdb310dba6f28f3f": "\\langle x| y \\rangle = \\langle y, x \\rangle.", "530695eec7905010c68f8ea1681608c9": "_{s.3\\ s.14 \\,}\\!", "53071f567446a502cdb740eef983a6b7": "\\hat{A} = \\hat{A}^\\dagger", "53074497b7a00dcdf7b8d9ade6e0f3e4": "\\begin{align}[]\n \\lbrack\\mathbf{b}\\rbrack &= \\lbrack\\mathbf{b}\\rbrack_2 \\cdot \\lbrack\\mathbf{b}\\rbrack_1 \\\\\n &= \\begin{bmatrix} 1 & 0 \\\\ -sC & 1 \\end{bmatrix} \\begin{bmatrix} 1 & -R \\\\ 0 & 1 \\end{bmatrix} \\\\\n &= \\begin{bmatrix} 1 & -R \\\\ -sC & 1 + sCR \\end{bmatrix}\n\\end{align}", "530764bddaca8506de7d0b2c11d3ac9a": " a \\equiv b \\pmod{m}", "5307674f963364d7d755e2228d23afc0": "\\Delta c/c", "53078a151d3f70176e2781447fe16e26": "\\delta^{\\alpha}_{\\beta}\\,", "5307b5c5d108cddc5236df466fd136c1": "\\frac{\\partial \\mathbf{u}}{\\partial x} \\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{f(g)}}{\\partial \\mathbf{g}}", "53081cd117355bb7e6f51330b536fbbc": " \\operatorname{dom} S = \\left\\{\\xi \\in H: h \\in L^2_{\\nu_\\xi}(\\mathbb{R}) \\right\\}", "5308616dd5d55ad2d9aa3f05984b350e": "u^{\\alpha} u^{\\beta} g_{\\alpha \\beta} = - c^2 \\,.", "5308aa83b85c6d727ffd99a8595cc188": "\\neg(xRy)", "5308c45be52c8c6ff5ba20285eea747c": "\\left(\\frac{\\part S}{\\part P}\\right)_T=-\\left(\\frac{\\part V}{\\part T}\\right)_P", "53091bdbcb01b9e61a93e5f29a9c5105": "F_1 \\; ", "53094cf317241f28f684f432d662bc5f": "G = (\\{S, X, Y\\}, \\{a, b, c\\}, S, M)", "5309698f6f3204b94576a23f95199068": "\n\\frac{dr}{d\\varphi} = \\frac{r^{2}}{h} \\frac{dr}{dt}\n", "53097863d068ed6695b00a5ce7c89507": "\\pi^p_s(X) = \\varinjlim_k{[\\Sigma^k X, S^{p+k}]}", "5309822484bf949583257095f5d5b2c3": "\\pi(x;q,l)< \\frac{cx}{\\varphi(q)\\ln\\frac{2x}{q}},", "53098393459427b6c6d909c0a22f614c": "\\mathbf{D}=\\mathcal{E}_0 \\mathbf{E} + \\mathbf{P}", "530985fedc31149d332792a6ca79f82e": "A_n(k)=(2-\\delta_{n0})J_n(k\\rho_0)\\,", "5309b9e6c25a5c661d5dead87d1005b9": "\\frac 1{\\Gamma(x)}= \\sum_{k=0}^\\infty {x-a\\choose k}\\sum_{j=0}^k \\frac{(-1)^{k-j}}{\\Gamma(a+j)}{k\\choose j},", "5309d4e1581e5d6480935b307758b9fc": "\\sum_{i=k+1}^\\infty {1\\over p_i} < {1\\over 2} \\qquad(1)", "530a038ed5c034b70ef7317b2106d8a8": "C_{4,2} = 5", "530a1728c6a74c8cae532640dbc14bd9": "\n\\varphi(n) < \\frac {n} {e^{ \\gamma}\\log \\log n} \n", "530a56c3c56356b278d419738ff02148": " c = \\vec w \\cdot \\frac12 (\\vec \\mu_{y=0} + \\vec \\mu_{y=1}) = \\frac{1}{2} \\vec\\mu_{y=1}^t \\Sigma^{-1} \\vec\\mu_{y=1} - \\frac{1}{2} \\vec\\mu_{y=0}^t \\Sigma^{-1} \\vec\\mu_{y=0} ", "530a926e9105c598a2d857fcf2a6d4d1": "\\Sigma Work = 1 work/(kg*stroke)*0.00121 kg= 0.00121", "530ab66470e4e760d3fcd6c96a2c84be": "1.53 \\, z", "530ac39ca0fe5f6d02b80cdbfdc4d39b": "\\widehat{\\theta}_2 = \\frac{Y_1 + Y_2 - Y_3 - Y_4 + Y_5 + Y_6 - Y_7 - Y_8}{8}.", "530b29fb3cbb5198378879d8903e0010": " f(t) = \\int \\limits_{u_1}^{u_2} K^{-1}( u,t )\\, (Tf(u))\\, du", "530b744e702a2058ad3f745861caa3be": "\\{O_{1},O_{2},\\ldots,O_{10}\\}", "530bd056a2ae7d4319b9f1e0e5cd9bb9": "\\frac{E(M)}{N} = 1 - (1 - \\frac{1}{N})^{K}", "530bec30e5cec9d69b395e4a84cc9c75": "m_j= -\\frac{5}{2} \\log_{10} \\left | \\frac {F_j}{F_j^0} \\right | \\,", "530c3ae5882f8afd34c9d0bfad8be343": "ds^2=\\left(1-\\frac{r_s}{r}\\right)^{-1}dr^2+r^2(d\\theta^2 +\\sin^2\\theta d\\phi^2)-c^2\\left(1-\\frac{r_s}{r}\\right)dt^2", "530c761aa8ac0aeffc50641ec4ba0e13": " |\\uparrow\\rangle ", "530cb3dbd5ef273c135c981fc15e3e8c": " |x\\rangle ", "530cd09a8e7e5b8bf6d4f99608f7252f": "CO_{(g)} + H_2O_{(v)} \\rightleftharpoons CO_{2(g)} + H_{2(g)}", "530ce6a4a281347195e090b93e28da01": "\\exist X \\left [\\varnothing \\in X \\and \\forall y (y \\in X \\Rightarrow S(y) \\in X)\\right ].", "530ce9d81fd8173f07306a0bd9cf6c8a": " u_{e} > 0 ", "530d02377adc0af268df3e5a7bcd6078": "{\\mathcal{M}}^3", "530d70f4358345bdd7b685e375444dba": " n,m,q,d ", "530e4f721d20fab7bbdd92ab972c5aab": "\\mathrm{var}(\\hat{\\alpha})\\geq\\frac{1}{\\operatorname{var}[\\ln X]}\\geq\\frac{1}{\\psi_1(\\hat{\\alpha}) - \\psi_1(\\hat{\\alpha} + \\hat{\\beta})}", "530e59a243f9ce6dd0f31e9b5feaaea9": "g(y) \\propto y^{-1} \\quad \\text{ for } 0\\le b^{-1}0.", "5313f920b8278d5d48778ca7ae6db3ac": "u_i^{(m)}", "531409180f643ae616c295ebfcc2f8ec": " \nh_t = h + \\frac{v^2}{2}\n", "531447d38fa533380636fbb8ac009fea": "d = c\\cdot(0-t_1) = -c\\,t_1", "53144d8d691c62f351a210b584fd830d": "\\left[ \\cos \\left( R \\tan \\theta \\right) - i \\sin \\left( R \\tan \\theta \\right) \\right] \\, ", "53165079389aa0b43f89ac2e69994621": "j = 1, \\ldots, m\\!", "53167e9bef26b4a67482fb14b0b9e572": "L_\\rho(\\Gamma_1\\cup\\Gamma_2)\\ge 1", "5316c5ca2c0aabc8b823e946ba54d065": "Y = \\lfloor X \\rfloor,", "53170bd9dfafc58a0c0648c9ab4694e9": "O(p(x)^3)", "53176e1fadd69dd38c5867588a0dc758": "10^2", "5317a3a6e54e83815c5b29d779077198": "0=2", "5317cf692e44a682f8fc2b752903baad": "W_{\\mathrm P}", "5319284bcb86f1104ad918c662e5afa7": "\\phi(w) = (w+1)^2:", "531959a4b45d1d92987ee509d7d31094": "\\boldsymbol{ a}_C =-2\\boldsymbol{\\Omega \\times v}= 2\\,\\omega\\, \\begin{pmatrix} v_n \\sin \\varphi-v_u \\cos \\varphi \\\\ -v_e \\sin \\varphi \\\\ v_e \\cos\\varphi\\end{pmatrix}\\ .", "531961fa804e80a0101292c9e6fe9e38": "R(k)", "5319fbf542d7401a1e517bf1d654d5ba": "\\varepsilon_{\\text{c}} = { \\left( -400+60f_{\\text{c}}-0.33f_{\\text{c}} ^2\\right) x 10^{-6}} ", "531a0d6d911e9297365ddc8fd0518f82": "\\mathbf{v} = (u, v, w)", "531a2daa5be5d9929910e78e4cfb9611": "\\left(\\frac an\\right)", "531a38dda2f7522a74305b3a7d643bfb": "q-r \\geq 1", "531a3be1cb33abb9686e3c22d99abd51": "\\frac{E_b}{N_0} > \\frac{2^{2R_l}-1}{2R_l} ", "531a5f878742f22ea1b2b865a1517616": "3\\pi/5", "531ab8efae71457d17834f68ac07e35d": " O(w|A|(log |A| + l)) ", "531abad10ab1538ce43d99a06a1746d7": "r=\\|\\boldsymbol{r}\\|.", "531af1f45d6032011ded3a1f5e0c1598": "\\lim_{n \\to \\infty} a_n = 0,", "531af9a946ac896f32c4d17d5ad7043a": "U = -G \\frac{m_1 M_2}{r}\\ + K", "531b0fb66663121c06062afdb66b3f65": "\\scriptstyle (0,n)", "531b398df3f18847fe33d5959a0370a1": "g^{(n)}(0)= n! ", "531be1b21560ee142c4e71f495be5966": "\\partial/\\partial\\nu", "531bffe4328fc402c793b33bbc6be747": "(x,y),\\ y>-1/2", "531c1368ce60e4983ab90c25683dff15": "\\mathfrak{f}_{4}(\\mathbf K)", "531c8a17095b261c6c09eee4b997baa7": "K_n(x) = \\left(\\frac{x}{2}\\right)^n {\\mathcal K}_n\\left(\\frac{x^2}{4}\\right).", "531ca5eaa259d564cba52db08c2d17ff": "R_2 ^ 2 = 1 - \\frac{\\sum (y - Y_r) ^ 2 }{ \\sum (y - Y_{a2})^2}", "531cb46c2ae8afe7b335eae0717fa785": "\\forall x ((x=e) \\lor (x^2=e) \\lor (x^3=e) \\lor \\ldots)", "531cb852e24430604e1ea91a46c2cbb0": "\\mathbf{e}_3(t) = \\mathbf{e}_1(t) \\times \\mathbf{e}_2(t)", "531cd3824617f48d3f4a2351d62fb74b": "\\vartriangle^{1+\\epsilon}", "531cd9114199f616b4fa04b7d723fa83": "(1)(2)(3)", "531cdfb0f95d3cfe19616038e2ff6803": "m \\times r", "531ce11c79e67c71b24c395f735c6485": "F(x,v) := \\lim_{t\\to 0+} \\frac{d(\\gamma(0),\\gamma(t))}{t},", "531d057c0d42cfe54a93fd4065652dc0": "\\max\\{d(A),d(B)\\} \\leq d(A\\cup B) \\leq \\min\\{d(A)+d(B),1\\}", "531d3416b50bb5b9833bee4ec32827eb": "\\pi(x)\\!", "531e18cf7b639a33380202bde3499869": "f(f^{n-1}(\\bot)) \\leq f(f^n(\\bot))", "531e8b3b9cb799690ec12bf6b547587d": " Q = C{\\Delta}P^n\\,\\!", "531e9c5730a9fbf5f6618b9079f22fb9": " f_4\\circ f_3\\circ f_2\\circ f_1 (z)= f(z) = \\frac{az+b}{cz+d}.", "531f16c70a47943524428ee3deee6212": "U_3", "531f46f70ae84e56e2aa2410063bcc7e": " U(\\delta) = \\frac{\\hbar}{2 e} \\left ( -I_0 \\cos \\delta - I \\, \\delta \\right ).", "531f5e79c93f8da0115c5f7c374f56e4": " \\lim_{N \\rightarrow \\infty} \\left(I - \\frac{i}{\\hbar}\\frac{\\Delta t}{N} \\cdot \\widehat{E}\\right)^N = \\exp\\left(-\\frac{i}{\\hbar}\\Delta t \\widehat{E}\\right) = \\widehat{U}(\\Delta t)", "531f87998a631f7e735fe8a54761cb61": "\\Pr(disc(\\mathcal{H},\\chi)> \\lambda) \\leq \\sum_{E \\in \\mathcal{E}} \\Pr(|\\chi(E)| > \\lambda) < 1.", "531f89a19c50f82bd9c3c5a636916b7a": "\\mathbf{a_{2}} \\times \\mathbf{a_{3}}", "531f9ce762dc934cef35651ac56bf07c": "F_{thrust-gate} = 76,142 \\quad lbs ", "531fe422b40f5b324364acf0c37bac3f": "\\vec{v}_B", "5320225d545ef40fa0ac863829702a6b": "m\\frac{d^2}{dt^2} (\\delta r)_{\\vec{k}}=-eE_{\\vec{k}}", "53203d10aab431daa7ecae00e36d92dd": "(K-S)^{+} \\ ", "53205fe870168dfd258490cd0bcdcfa8": "\\alpha=2+1/m", "532066350de2a8ac8a108d1985d5b8ce": "\\int_a^x \\!\\!\\!\\int_a^s f(y)\\,dy\\,ds = \\int_a^x f(y)(x-y)\\,dy", "5320727725e3ba3a2983d667fe82394e": "Fib(x) = \\frac{\\varphi^x - \\cos(x \\pi)\\varphi^{-x}}{\\sqrt{5}}", "5320fc4efae73048570c1f450192bfe9": "m~= p \\exp{(bp)} = \\sinh b + p \\cosh b = \\sinh b + i \\sinh a~\\cosh b + r \\cosh a~\\cosh b", "5320fcf601f59b0a438176e31325db50": "[e_a, e_b] = f^c_{ab}e_c", "53210eeae0720f0c2655797917ab98ae": " F(x,y,z,a)=0,\\,\\,{\\partial F\\over \\partial a}(x,y,z,a)=0.", "532112c388a472f504766860c5209250": "V \\,", "53212e51091b0e3f4c9c799c4fcefa5a": "M\\left(|\\uparrow \\rangle \\otimes |O_{\\uparrow} \\rangle \\right) = |\\uparrow \\rangle \\otimes |O_{\\uparrow} \\rangle", "53212f5fcbb429384a1ac4cf43d2c310": "ds = \\frac{ds'}{a}\\,", "53213d4883c48f4d3df3ddb56d345efc": "a_{2n} = \\frac{T_{2n}}{(2n)!} = 0,", "53214974d3234f88d25575088fe6fd72": "\\frac{AB}{BF} \\cdot \\frac{FO}{OC} \\cdot \\frac{CE}{EA} = 1", "53215b328a767fe215eac6161031c990": "n_i = 1", "53217c9454e7af825c3403842313d820": "f^{-1}(B) = \\{x \\in X : f(x) \\in B\\}.", "5321c2f16c9aba20cb114c07d7089e95": "\\operatorname{NB}(\\tilde{x}|k', \\frac{\\theta'}{1+\\theta'})", "5322016648b28db498c15962254e4837": "\\scriptstyle \\nu", "53220618ea0c86dc8724c4fbedf515b6": "U_\\omega", "53229ff4ec9a131fc8c235d8d5b7e945": "A^lX + B^hX \\rightleftharpoons A^hX + B^lX", "5322af1f6a695f00b1fa815f59469be8": "\\mathbf{x}^H", "532310b0d7d5aeea7f0ea29a31de48e6": "P\\left(O^{t}|S^{t}\\right)", "532310e9415eee36abae231d5197c771": "W = \\frac {400F}{DT \\times (bulk\\ density)}", "53235d44e2ff704d571c2a163fe7c853": "\\left\\langle F\\right\\rangle=\\frac{\\int \\mathcal{D}\\phi F[\\phi]e^{i\\mathcal{S}[\\phi]}}{\\int\\mathcal{D}\\phi e^{i\\mathcal{S}[\\phi]}}", "5323973d8035a28c1b225b349814958f": "\\scriptstyle x'_1", "5323bf1625a6234e261f1bb70d1593c6": "J^k_p({\\mathbb R}^n,{\\mathbb R}^m)", "532414abe190c1593dfd1c149af333e2": "C_0", "53244b32b1210bb9b483d65e3899be48": "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle + \\gamma |2\\rangle", "5324af69c7dfae5dccc9b457567ad19c": "K_8", "5324b4d3e5d5c2c84ac2a6b3884e7634": "v(S)=1", "5324e59ba0a17801880d597b7f6ffd91": "A \\vert (v - 1)", "5324e59c9b5ca1b5ccdb74a8a281fa73": " f(\\theta)= \\sum_{-\\infty}^\\infty a_n e^{in\\theta},\\,\\,\\,\\, g(\\theta)= \\sum_{-\\infty}^\\infty b_n e^{in\\theta}.", "5324fbfb1ef6203d9dfbf895e42be070": "S_\\alpha(f) = \\{x|f(x) \\leq \\alpha\\}", "53250b8e11587f5a99616fb9e8d4e0c7": "\\Sigma^{\\mathrm{(diag)}}", "532562f400589a9764d1d6a22c561d82": "\\frac{\\mu(X)}{\\mu(G)}.", "5325ac9d817c692a7769875807a2002e": "Y = \\left( \\sum_{i=1}^n X_i \\right) \\sim \\mathrm{Pois}(\\lambda)", "5325ad17a88761dcd6919f549ac64920": "\n\\mbox{U-238} \\rightarrow \\overbrace{ \\underbrace{\\mbox{Th-234}}_{\\mbox{daughter of U-238}} \\rightarrow \\underbrace{\\mbox{Pa-234m}}_{\\mbox{granddaughter of U-238}} \\rightarrow \\ldots \\rightarrow \\mbox{Pb-206} }^{\\begin{array}{c} \\mbox{decay products of U-238} \\end{array}}\n", "5325be0a82e8a1805d5343ee05a23bb5": "\\nabla\\cdot\\mathbf{B} = 0", "5325c9e78b1b442911c84ce9b03bc2be": "k'(\\tau) = \\left({\\vartheta_{01} \\over \\vartheta}\\right)^2.", "53262a4b6b71f74130121d638a1535d3": "{1\\over\\sigma}\\varphi\\left({x-\\mu \\over \\sigma}\\right).", "53264c07a09b54e9addd59fbd77bf7a7": "\\delta=0,w(x_1,x_2)=\\mathbb{I}(x_10.", "533650c5654a9e800fc81b53bde0d70f": "\\tan ( \\alpha / 2 ) = \\frac {d/2} {S_2} .", "5336584f3c0235f646a616508327c3aa": "c \\in R", "5336cac86efcf08f4bece60dd5f4a78f": "k \\varphi (N)=ed-1 0.5\\,", "5341267fe32c1e421a0353509ca292ed": "10^9 \\tfrac{photons}{mm^2s}", "53412b39f74cdf4ca740d317ea477ce4": "\\scriptstyle (k,n-k)", "5341470f3ac49b4990020daefaa32f5d": "\np_{U} = \\frac{e^{-\\Delta G/RT}}{1 + e^{-\\Delta G/RT}}\n", "534161ae83672e742972e08243e8e5e4": "\\mu (E)= \\lambda^{1} \\big( \\{ x \\in \\mathbf{R} | (x, 0) \\in E \\subseteq \\mathbf{R}^{2} \\} \\big).", "5341bcf31eabbe330041281cef0bb0dd": "\\left.\\right.E(n)=\\Delta\\,", "5342345a1fedb8ca48484e330f29dea5": "a^{\\tau}", "53429bbf048a07157395973f309b1d16": "\\displaystyle{T = R_1\\Delta^{-1}R_0,}", "5342cc286082dc870db4e4fdf2b766f9": "\\sum_{N_1 = 0}^{\\infty} \\ldots \\sum_{N_s = 0}^{\\infty} \\int \\ldots \\int \\rho \\, dp_1 \\ldots dq_n = 1.", "5342e42e15c78d4afddb7a20998e4d26": "\\{\\sqrt{\\pi}\\}", "53430f6f4201b0c788db181e7aa47aa0": "\\mathrm{R}_\\mathrm{m} = \\frac{U L}{\\eta}", "53431cbb2cec435e69f185e3215391ec": " (0, v e^{-i\\theta/2}).", "534329971d11f401046d9be595f07aff": "B_j", "53439ddbd5657d9ba0ec678bf2dec2b8": "s^2\\mid r", "53439e6b044b02319a816790427810f9": "Q_S\\cong I_A\\cdot\\tau", "5343d5d915185861ec5c57c27a31d602": "\\dot{x}_2(t) = \\ddot{x}_1(t)", "5344a90cd116d13df26c753cac69af5b": " \\_ ", "5344b79b91dbae3c43ff7045cde405c8": "a(\\sigma-\\tau)", "534571560dcb23804be02a764d0fd728": "\\nabla P_i", "5345c22148713cddb3acf9a0b53adee9": "X_n=\\left(1-\\frac{1}{50^6}\\right)^n.", "5345cb8e40db56b534e17fc726d62dd2": "K=\\frac{[A][B]}{[AB]}", "5345de2da37e1dd8beca305017d12545": "\\text{PI}=100\\frac{mass}{height^3}", "5346502e3fdc50c44efc026f336731b9": "s \\rightarrow t", "5346b9a96342a0b648884c58b52f1dbe": "\n\\lim_{n \\to \\infty} \\max_{i \\leq n} s_i(n) = 0\n", "5346d2982c4aa06e34dfe4cd1a181bb4": "p(N,M;n) = p(N,M-1;n) + p(N-1,M;n-M) \\ ", "5346ebc59b28dbb947a86394c03488bb": "\\omega(t) = \\frac{d \\phi(t)}{dt} = \\omega_0 - \\frac{2 \\pi L}{\\lambda_0} \\frac{dn(I)}{dt},", "53471bc23fccbe58270a89cb2139e07f": "i > 0", "534722094fffaf1e458d42078e94900e": "\\text{SE}_\\bar{x}\\ = \\frac{s}{\\sqrt{n}}", "53472e3ba29cb0384b976e90daeaac95": "\\sqrt{\\sum_1^k \\left(\\frac{X_i}{\\sigma_i}\\right)^2}", "53473003eda44b1934af6a820da4ddf9": "1 - (1/n) = 1 - (1/3) = 2/3 \\approx 67%", "53474669ce91720fab1d2d08954175f0": "(pk, msk) \\leftarrow Setup(1^\\lambda)", "53477f3c041eedab1dcb0c5bdcfa13ef": "k\\{x_1, \\ldots, x_n\\}\\ ", "534794e9a27ee3cc674ca6a962dc60a2": "\\frac{1}{\\sqrt{2}}", "5347cd9788b4d728fa3091b849aa8a03": "\\hbar c", "534815201dc2570ce4b48226062a818f": " \\lambda(\\Sigma(p,q,r))=-\\frac{1}{8}\\left[1-\\frac{1}{3pqr}\\left(1-p^2q^2r^2+p^2q^2+q^2r^2+p^2r^2\\right)\n-d(p,qr)-d(q,pr)-d(r,pq)\\right]", "5348d33c9dee22744c6e3909c6f04536": "\\hat{\\sigma}_z = |e \\rangle \\langle e | - |g \\rangle \\langle g |", "5348ed8265df16318ffb1666d08f0935": "T_A=3.468F{{\\lambda}^2}10^{G/10}", "5349463deb9eac6407491e9f4354c8e1": "E = Q(y_1, y_2, y_3, y_4, y_5)", "5349a3ff1299b04f76da3bccf13d441c": "\\lfloor \\ln(U) / \\ln(1-p)\\rfloor", "5349a8bf73f688a22ce5a72c33ee8101": "\\sum_{k=1}^{n} \\varphi (k) \\sim \\frac{3n^2}{\\pi^2}\\!", "5349b3b41973725706cf349f1263defe": "\n\\begin{align}\n\\frac{s}{p_1 p_2 \\cdots p_m}\n&=1 \\\\\n&=q_k \\cdots q_n,\n\\end{align}\n", "534a29530cab81d6939c2321369aff78": "f(x)=x^4", "534a2fde5e31a892dc035f6f50282d30": "W_e", "534a459c3d4eeb9f34461a7d4d1dc5f3": "P_{\\infty}", "534ab9cec88faac591cf304aaa508dc4": "N(p) = p^2", "534b00c1aa22146243557e5c5fb8fe2d": "\\Delta^2(a_n) = \\Delta(a_{n+1}) - \\Delta(a_n)", "534b967e833c847e694d1250a1f2b6b5": "T[\\rho]", "534bf8bfc0975373902636e9af68fcfd": "t_{\\nu'-p+1}\\left(\\tilde{\\mathbf{x}}|\\boldsymbol\\mu,\\frac{1}{\\nu'-p+1}{\\mathbf{V}'}^{-1}\\right)", "534c1c7b997f11ebf0e33b1c6456c1f3": "\n\\sin 45^\\circ = \\frac{\\pi}{180\\times 60}\\left( 2430 + \\frac{51}{60} + \\frac{15}{60\\times 60}\\right)\n", "534c95ebfa8543e0d9bc82db55759c01": "A_0\\cap C_i\\ne\\emptyset", "534ca3f7ebd155926214c9960625c236": "\\epsilon>0\\,", "534cbaced9031f0607eca7fcc2281d11": "\\eta_r = \\frac{\\eta}{\\eta_s}", "534cc7d7601378122f68289c05d21395": " 40.078 + (2*26.982) + (4*15.999) = 158.038 g/mol ", "534d26be0ec138557880e1d79cf11e2d": "\\mathbb R^G \\otimes \\mathbb R^G", "534d6340166ce62096991ff17b42d1b9": " x \\diamondsuit x = ( 0,\\ 2 x_1^2 \\ ,\\ 6 x_1 x_2 \\ , \\ 8 x_1 x_3 + 6 x_2^2 \\ , \\dots ) ", "534e11206cbac02aec49f18f127e101a": "\\lambda = 0.318", "534e26be6a0c3c375a8a68b40bdb89e9": "\\tbinom{9}{6}", "534e372c33ca7501d1b83389e8746b2c": "[0,t_m]", "534e701870b199f36081e620179fe744": "\\begin{align}\n {\\frac{\\delta\\rho }{\\delta t}} &= \\rho CB_\\alpha^\\alpha \\\\\n&\\\\\n\\rho\\frac{\\delta C}{\\delta t} &=\\sigma B_\\alpha^\\alpha \\\\\n\\end{align}", "534e7c886bac0b6594502c31c87f95e8": "(\\mathbf X^{\\rm T} \\mathbf X )\\hat{\\boldsymbol{\\beta}}= \\mathbf X^{\\rm T} \\mathbf y.", "534e9d2ef0fe0d506986d74a42552c35": "\\pi_x(R\\bowtie S)=R\\bowtie\\pi_{x\\cap y}(S)\\,", "534eb9832bab2a784fb71639c2ae8d91": "\\frac{\\alpha}{2\\pi}", "534ee59b851dd2c9ef6a75e007dd4ac3": "\\rho < 10 h.", "534ef77f58243f8186cdb9c352f9bd30": "\n\\text{wage} = (\\text{base salary}) \n+ (\\text{incentives})\n\\cdot (\\text{(unobserved) effort} + (\\text{(unobserved) effects}) \n+ (\\text{weight }Y)(\\text{observed exogenous effects}))\n", "534f14c49e455d0fe4d83a4fdf23c3b3": "c(\\omega)", "534f2ba770b1da31bbc6b0a28f13ac10": "\\textit{par}(h,m) \\land \\textit{par}(h,t) \\land \\textit{par}(g,m) \\land \\textit{par}(t,e) \\land \\textit{par}(n,e) \\land \\textit{fem}(h) \\land \\textit{fem}(m) \\land \\textit{fem}(n) \\land \\textit{fem}(e)", "534f665dcf0b2a9a543ff59a848b69bf": "\\; - \\sum_j p_j \\log q_j - \\left(- \\sum_j p_j \\log p_j\\right) = \\sum_j p_j \\log p_j - \\sum_j p_j \\log q_j", "534fa2191760562013a0402bbbc941d1": "h(i,x) =\n\\begin{cases}\n 1 & \\text{if } \\text{ program }i\\text{ halts on input }x, \\\\\n 0 & \\text{otherwise.}\n\\end{cases}", "534fb40799d5c0de84528ad6841432ae": "w = w_1\\cdots w_\\ell", "534fc8d4c161e698c97ba60f424828f8": " \\lambda_i \\mu_j, \\qquad i=1,\\ldots,n ,\\, j=1,\\ldots,m. ", "534fcba5581eef9d8e65951f09981b2f": "y^2 = x^5 + 1", "534fe2b418bbc0ebefba8a69eb49a67d": "f(\\beta_n | \\theta) ", "53506be0642ff589dcfbfc81091d4f80": "\\Omega^{framed}_1 \\cong \\pi_m(S^{m-1}) \\, (m \\geq 4) \\cong \\mathbb{Z}_2", "53507d4fe16fecb46a2040e2a2b21647": " I_B =\\frac \n{\n \\frac \n{V_{CC}}{1+R_1/R_2}\n - V_{be}\n}\n{( \\beta + 1)R_E + R_1 \\parallel R_2 } .", "53509ac0f4c89a5c2790637ded5fd68f": "\\lambda_\\mathrm{boundary}", "5350a70b010686b6d0aedc90ec4af0f5": " \\ddot{q} = a q - b p - c q. ", "535113aaa636844f9e20412371eec7c3": "\\gamma\\,\\!", "5351a844deeb1c710e7d664091e75aa0": "\\mathcal{Q} = \\frac{\\lambda_\\min(\\mu)}{\\lambda_\\max(\\mu)}", "5351dc4affb4731ae781bf95c0facb43": "(\\phi \\rightarrow", "5351e0b6f6a9ea594957b2268c869dac": "\\varepsilon=\\varepsilon_r'' \\cdot \\varepsilon_0", "5351ef16c7ab05e7a15bfbe02eedce6b": "\\lambda_1 > \\lambda_2", "5352926c3c4745e22281e3e77ef5a6ad": " GM= \\sqrt[n]{\\prod_{i=1}^n a_i} = \\sqrt[n]{a_1 a_2 \\cdots a_n}", "5352c494530e66f667b1d58ded3b174d": "\\left(\\frac{\\partial^2U}{\\partial X^2}\\right)_S=-T\\left(\\frac{\\partial^2S}{\\partial X^2}\\right)_U ", "535310f69e96a6300e343ad93703b8ef": "p_3(x) = -32x-64", "5353475b5ddf9f85581768cd6c362f11": "S(t) = \\left|t\\right|^3 ", "5353bb048048980d91202d2e88f20447": "0 \\in \\mathfrak{g}^*", "5353c0312ea11341b2e61a6c8450c8d5": "\\mathbb{M}_k", "5353e2c4ca7cb3a2aebbe5cb6516c5de": "\\scriptstyle poly()", "5353f4015da2ea922e3a00e2ffcbf2be": "t_n=1-\\gamma + \\sum_{k=1}^n (-1)^k {n \\choose k} \\left[ \\frac{1}{k} - \\frac {\\zeta(k+1)} {k+1} \\right]", "5353ff8ea7cd06db29ed808ed997d944": "a_2=4=2+2", "535413ffaee15ebcd2c96f57c9e43a8b": "\\mathbf{1}_A : X \\to \\lbrace 0,1 \\rbrace \\,", "53543bba00e8d3dc0e752eda3c4d4245": "\\displaystyle{C^\\infty(\\overline{\\Omega}) \\rightarrow C^\\infty(\\mathbf{R}^n)}", "53549de6b38a84be57d5b6c9be934c98": "\\Omega_i^{alm}", "5354ad059e7ca6a28129887c1209b66a": "p^2|2^{2m\\lambda}-1", "5354b0be118deb635ccd2e55a957553e": "x^{-1}=/x", "5354cae26070b677314bd6295baad79b": "H_n = F + G", "535500cf11dcf77779eaa58f700f6bff": " x^{-n} = \\frac {1} {x^n} \\text{ etc.} ", "5355232a65ec9692e124e7bad889beea": " LWC = m_w / V_c ", "535568781b58788c008bd56eaaf129c7": "\\langle\\!\\langle a_1, a_2, ... , a_n \\rangle\\!\\rangle \\cong \\langle 1, a_1 \\rangle \\otimes \\langle 1, a_2 \\rangle \\otimes ... \\otimes \\langle 1, a_n \\rangle,", "535584bf545b3df2bf8d21e8f8e457a3": "G = \n\\begin{pmatrix}\n\\uparrow & \\uparrow & & \\uparrow\\\\ \ny_1 & y_2 & \\dots & y_{2^k} \\\\ \n\\downarrow & \\downarrow & & \\downarrow\n\\end{pmatrix}\\,.", "5355d70ff46c40eaf44c3fdfa4c4cfac": "[x, s] = 0", "5355dabbb1697bd46b1ea327a35355e6": "\\textstyle{\\int_A f\\,d\\mu = \\int_{[a,b]} f\\,d\\mu}", "5355eafa13c309420f7cfa37e3f1e923": "\\sqrt[n]{A}", "5355f9ebf601ab7589fe25ce9f725904": "1 = \\int_0^\\infty c \\cdot N_0 e^{-\\lambda t}\\, dt = c \\cdot \\frac{N_0}{\\lambda}", "535635edb98a4c1f96ec5853a1edf040": "S_g ", "5356d4aef99b64da9bcfe3a393b1878e": "e_i=[0 ... 0 1 0 ... 0]", "535719cbb3b3143d7696a157ca29cae4": "J = \\Big| {\\partial x^a \\over \\partial x^{'b}} \\Big|", "53571d2a4576631d9b726a136baac2ea": "(\\overbrace{x}^\\text{abscissa}, \\overbrace{y}^\\text{ordinate})", "5357535a8fa38fd63a26669ccb40177f": "a \\pi \\varphi", "53577b71e78a812cd2ebf7dfeb7a64f1": "F\\left(I\\right)=1", "53578eec522d50bcb19017752c07121a": " W = p \\Delta V\\, ", "5357b3d3ac36ac3428af79de4d598b5a": " E =\n - a_1 a_2\\int {d^3k \\over (2 \\pi )^3 } \\; \\;\n{\\vec v_1 \\cdot \\left[ 1 - \\hat k \\hat k \\right ] \\cdot \\vec v_2 \\over \\vec k^2 + m^2 } \\; \\exp\\left ( i \\vec k \\cdot \\left ( x_1 - x_2 \\right ) \\right )\n", "5357d594ae8c2e447b614d0233978e43": " \\sum_{k=0}^\\infty a_kz_0^k = a(z_0) ", "5357e2fc68411a2aa419e954a5c10254": "\\therefore \\frac{\\Pr[A]}{\\Pr[B]}\\geq \\frac{e^{-\\alpha\\epsilon n/4}}{2^{km}e^{-\\alpha\\epsilon n/2}}=\\frac{e^{\\alpha\\epsilon n/4}}{2^{km}}.\\,\\!", "5357f84febe9162351a556bbc682ebf8": "V_b=\\frac{\\pi\\,\\Gamma\\,\\sqrt{\\rho_t\\,\\sigma_e}\\,D^2\\,T}{4\\,m} \\left [1+\\sqrt{1+\\frac{8\\,m}{\\pi\\,\\Gamma^2\\,\\rho_t\\,D^2\\,T}}\\, \\right ]", "53582e8ea40869b9f262d867139280f5": "\\bar{\\sigma}_{ij}", "535830a66d1d9f68758ae1f8046e605f": " \\hat a ", "53583a5a4d72c2d6e3484a6d4590b487": "\\begin{align}\n & \\alpha \\in [\\,\\hat\\alpha \\mp t^*_{13} s_\\alpha \\,] = [\\,{-45.4},\\ {-32.7}\\,] \\\\\n & \\beta \\in [\\,\\hat\\beta \\mp t^*_{13} s_\\beta \\,] = [\\, 57.4,\\ 65.1 \\,]\n \\end{align}", "535881b7d30508deb0a04b5144e61d60": "\\begin{align}\n x_R &= s^2 - x_P - x_Q \\\\\n y_R &= y_P + s(x_R - x_P)\n\\end{align}", "535898108312b042ec036ec9cc99e36c": "\\left\\{p,{q\\atop q}\\right\\}", "5358c5af4db03c406c0644ed7516ef98": "\\lesssim0.1\\,\\text{eV}", "5358e691eafd9d01d644fbfa9be49dfb": "\\pi_2\\left(\\frac{SU(5)}{[SU(3)\\times SU(2)\\times U(1)_Y]/\\mathbb{Z}_6}\\right)=\\mathbb{Z}", "53594ba31214dd7aa8bf11697d123715": "\\tilde{\\pi} = \\delta L / \\delta \\dot{\\varphi}", "53596a279eead6b0d224cd46e0da2757": "T_\\mathrm{v,parcel}", "5359d71d638cc0f5c759ec42c7569e6f": "\n\\begin{matrix}\n & X_1 & X_2 & Y_1 & Y_2 \\\\\nX_1 & 1 & u & \\gamma & \\gamma \\\\\nX_2 & u & 1 & \\gamma & \\gamma \\\\\nY_1 & \\gamma & \\gamma & 1 & v \\\\\nY_2 & \\gamma & \\gamma & v & 1 \\\\\n\\end{matrix}.\n", "535a14a738431c07e44733991a202051": "L_\\mathrm{initial}", "535a18e8524c4f76cf274e3d76c6553e": " y = r \\, \\cos \\theta \\, \\sin \\phi, ", "535a57fcb454de2fee7dcac76a7b69bf": "\n\\begin{align}\n\nP ::= \\, & x(y).P \\,\\,\\, \\\\\n|\\,\\,\\, & \\overline{x} \\langle y \\rangle.P \\,\\,\\, \\\\\n|\\,\\,\\, & P|P \\,\\,\\, \\\\\n|\\,\\,\\, & (\\nu x)P \\,\\,\\, \\\\\n|\\,\\,\\, & !P \\,\\,\\, \\\\\n|\\,\\,\\, & 0\n\n\\end{align}\n", "535a8702ba290abc3409ab50c005785f": "X_K = M - \\mbox{interior}(N).", "535a913326b83f6cc7f6b0a6dbecbecc": " f(x,y,z) = 1 ", "535a9798d1cc73a54bfef30c923dd732": "\\Box A\\to\\Box\\Box A", "535aaed224025eb3d04bbdc46bbc5e00": " \\oint p \\, dq =2 \\pi n(E)= 4\\int_0^a dx \\sqrt{E_n - f(x)} ", "535b2dc5b41e313e6157b252431d81c5": "P(\\left.X\\right|_{x})", "535b4cac1627c9c9b5ace4f122f49a69": "\\{x: x^T P x = 0\\}", "535ba37212328f0316f6f60f0df80ecc": "\n\\log \\frac{1+z}{1-z} = \\cfrac{2z}{1 - \\cfrac{z^2}{z^2 + 3 -\n\\cfrac{(3z)^2}{3z^2 + 5 - \\cfrac{(5z)^2}{5z^2 + 7 - \\cfrac{(7z)^2}{7z^2 + 9 - \\ddots}}}}}.\\,\n", "535ba80c09e4f0afb16f427a79541e0f": "P_\\pi \\mathbf{g} \n= \n\\begin{bmatrix}\n\\mathbf{e}_{\\pi(1)} \\\\\n\\mathbf{e}_{\\pi(2)} \\\\\n\\vdots \\\\\n\\mathbf{e}_{\\pi(n)}\n\\end{bmatrix}\n\n\\begin{bmatrix}\ng_1 \\\\\ng_2 \\\\\n\\vdots \\\\\ng_n\n\\end{bmatrix}\n=\n\\begin{bmatrix}\ng_{\\pi(1)} \\\\\ng_{\\pi(2)} \\\\\n\\vdots \\\\\ng_{\\pi(n)}\n\\end{bmatrix}.\n", "535bd6fb90f31009804fb1e8c58ac936": "p_{i;j}(f_i\\mid f_j)", "535c478876ad10563628d9c8403ca23e": "\nL = \\frac{\\pi t}{6}\\left(2^{n}+4\\right)\\left(2^{n}-1\\right).\n", "535c65899d3c236522a025eb2d0e1c04": "q_\\text{P}=1", "535c84e131aab0bcf993fa5b7f39e600": " ~n_0(\\varepsilon) = \\max\\left\\{e^4,d^{-1}(\\varepsilon) \\right\\}~ ", "535ca41562204f52700ddac2308f7a74": "\\displaystyle{S_Z=B^{1\\over 2}\\cdot T_Z.}", "535ceb54473a71f6521901e98638c233": "\\text{max:}\\operatorname{Tr}(\\rho_{AB}E_{AB})", "535cf7bbff9f65d64e4d73e882e231c4": "f,g \\colon X \\rightarrow Y", "535cfe1215e9c3f4d37ca2dff8e4f5a2": "A^{1/2} = V D^{1/2} V^{-1} = \\bigl( \\begin{smallmatrix}\\\\ 5&2\\\\ 4&7\\end{smallmatrix} \\bigr)", "535d460ba073a42bc138e050fdd7d929": " \\Pi_{k+1}^{\\rm P} := \\forall^{\\rm P} \\Sigma_k^{\\rm P} ", "535da3e73146bc662fd5a57de4d1befa": "\\frac{m_{1}\\;u_{1}}{\\sqrt{1-u_{1}^{2}/c^{2}}} +\n\\frac{m_{2}\\;u_{2}}{\\sqrt{1-u_{2}^{2}/c^{2}}} = \n\\frac{m_{1}\\;v_{1}}{\\sqrt{1-v_{1}^{2}/c^{2}}} +\n\\frac{m_{2}\\;v_{2}}{\\sqrt{1-v_{2}^{2}/c^{2}}}=p_T", "535da86c1b9c29d6476ed03954773643": "1^3+2^3+3^3+\\cdots+n^3 = \\left(1+2+3+\\cdots+n\\right)^2.", "535db509d873ac0e36695d1f59756f36": "\\hat{Y}_{t} = P_{K(Z, t)} \\big( X_{t} \\big),", "535dc2e394302677be40eaeb783bc2dc": "A\\,\\!", "535dc9ebd6961c902badce38779efeb7": "r\\ge n-\\pi+p_a+1-i", "535e1236d7bd13bd122e8a2c7df40de9": "\\boldsymbol{\\Omega \\ \\times} \\left( \\boldsymbol{\\Omega} \\mathbf{\\times r_B}\\right)", "535e488585b5f587bdaff6ed0b7571d9": "f: U\\setminus \\{a\\} \\rightarrow \\mathbb C", "535e676e2f7a8cf7f9545d8d64da8ef8": "E[(y_i-g_i)^2] = E[\\epsilon^2] + E[(f_i - E[g_i])^2] + E[(E[g_i]-g_i)^2]", "535ebe861e63bda731257d96a4287b23": "M = D(Z)", "535f03c9f2b5143d31c4f88b6250ce9e": "\\pi(\\mathcal{C}).", "535f498b323244ed27ee4a79b8a4964d": " \\delta = \\sqrt{\\frac{\\mu / \\rho}{\\Omega}} ", "535f6b140a7dbb6c6dad5c92455ae859": "e^{-wt}", "535f6e5ba54b006b58c31a0e7771ea41": "\\{n\\in\\mathbb{N} : u_n =v_n\\} \\in \\mathcal{F}", "535fb2335a185a98064e749d346c2db0": " Se^{-q \\tau} \\phi(d_1) \\sqrt{\\tau} \\frac{d_1 d_2}{\\sigma} = \\nu \\frac{d_1 d_2}{\\sigma} \\, ", "535ff3933185e60be61db6840b5c5db9": "\\rho_{eg}(t)", "53601e156f8bd55e232dea8f7d5d2e05": "\\operatorname{Var}_Y(Y) = E_N\\left[\\operatorname{Var}_{Y|N}(Y)\\right] + \\operatorname{Var}_N\\left[E_{Y|N}(Y)\\right] \n=\\operatorname{E}_N\\left[N\\operatorname{Var}_X(X)\\right] + \\operatorname{Var}_N\\left[N\\operatorname{E}_X(X)\\right] ,", "5360af35bde9ebd8f01f492dc059593c": "bc", "5361648e180c94cbab967efc9f87a82b": "\n\\frac{1}{D_{\\mathrm{rot}}} \\frac{\\partial f}{\\partial t} = \\nabla^{2} f = \n\\frac{1}{\\sin\\theta} \\frac{\\partial}{\\partial \\theta}\\left( \\sin\\theta \\frac{\\partial f}{\\partial \\theta} \\right) + \n\\frac{1}{\\sin^{2} \\theta} \\frac{\\partial^{2} f}{\\partial \\phi^{2}}\n", "5361d0af0de1c4e3771a8caef3639b7f": "v_{e}", "53625a6b5fca3acce1e4e5fbe9ff5c37": "\\mu_1,\\ldots,\\mu_n", "5362baf84f11a4b5f20e2f7e3bfacb63": "\\mbox{min}_x \\leq x \\leq \\mbox{max}_x", "536301b8628f83e2ed33f4b8c78582a9": "\\mathcal{FS'}", "53631b1b405e18d235753d32a2ac05a9": " \\partial = \\sum a_n {{d^n} \\over {dx^n} } = \\sum a_n D^n ", "53644136de95f2459646d993a90ba6d3": "n \\oplus \\lfloor n/2 \\rfloor", "53646bc153933ec7f82dff9198324aee": "h_{ab} = \\arctan \\frac{b^{*}}{a^{*}}", "5364757d11996233e2e53b1d6ceb565f": "\\color{Sepia}\\text{Sepia}", "536495c20125c6bb3eb0aebc049247cc": "\\frac{2(1+\\alpha)}{\\alpha-3}\\,\\sqrt{\\frac{\\alpha-2}{\\alpha}}\\text{ for }\\alpha>3", "5364af889df24d73081b72747d0a6773": "\\scriptstyle\\hat p", "5364fc82771b366c3db7e38687a1f8bd": "\\varphi = \\frac{1}{4\\pi\\epsilon_0}\\iiint\\bold{P}\\cdot\\nabla'\\left(\\frac{1}{|\\bold{r}-\\bold{r}'|}\\right)d^3\\bold{r'}", "536500f8e1cecea4200ac24f36fa8dd8": "dU", "53656467cef8d5feff0cb5268bda59d5": " w''(t) = \\frac{3}{2} w^2, \\quad w(0) = 4, \\quad w(1) = 1 ", "53659675d753fec5f834c457323005a7": "NH_K (M)= \\sum_{i=1}^ \\frac{\\ell}{2} (k_{(2i-1)} +_w m_{(2i-1)})\\times (k_{2i} +_w m_{2i} ) \\mod 2^{2w} ", "53659fde69cf2b506dc62b2ed1c5b4c9": " L(F(x))=\\frac{\\int_{-\\infty}^{x} t\\,f(t)\\,dt}{\\int_{-\\infty}^\\infty t\\,f(t)\\,dt} =\\frac{\\int_{-\\infty}^{x} t\\,f(t)\\,dt}{\\mu} ", "536605fbe042ea0be0fd78a6b9ade42f": "C^2\\,", "5366510de30ce1bf3f4133b600ae4fed": "\n\\Psi(1,2, \\ldots, N)\\quad\\text{with} \\quad i \\leftrightarrow (\\mathbf{r}_i, \\sigma_i),\n", "53667f716d74fd4d6f250f95e54e626e": " -1 + \\sum_{k=0}^{n}\\binom{n}{k} \\frac{2^{n-k+1}}{n-k+1}B_{k}(1) = 2^n ", "53668ce9196ab49979cdc3c6690fb63b": "\\frac{\\sqrt 2}{2}\\sqrt{N(z)}", "5366b77516ea7fa970c388f78543999b": "Re \\leq 0.5", "5366b92cc69acd650d85598546b6c5de": "\n\\begin{align}\nh(D) & = \\frac{1}{D} \\sum_{r=1}^{|D|}r\\left(\\frac{D}{r}\\right)\\\\\n & = \\frac{1}{2-\\left(\\tfrac{D}{2}\\right)} \\sum_{r=1}^{|D|/2}\\left(\\frac{D}{r}\\right).\n\\end{align}\n", "536732bc48900d47b603816b60687a39": " \\text{If }w + x + y + z = \\pi = \\text{half circle,} \\, ", "536757bcfcce7e5b8099ff260810a425": "\\sum_{f_i \\in \\mathrm{Hom}(C)} a_i f_i,", "5367630f5af4145d3f6363bd1b27c355": "\\{ A_{2 l +1} \\}_{l \\in Z}", "536777d3072ca50018b131cfc661b8ad": " \\frac{\\partial u}{\\partial t}=u\\frac{\\partial^2 u}{\\partial x^2} \\quad\\text{on the domain }-1 \\mathbb{E}[-X]", "539131546d06151a194ca1412eb30d46": "\\mathbf{\\mathit{L}}", "5391746f1ab5a77cdacef34bea17efb2": "x=c_2", "539185a708dc05e9cd2d0ef3cec1978a": "\\sqrt{n}{2n \\choose n} \\ge 2^{2n-1}", "5391a9f600b071b56794670878020208": "f_a=f_b=f_c=f", "5391ae73b2c96cc0f5ffbce39c49e0bf": "\\mathrm{Cov}(x, y) = \\int_{H} \\langle x, z \\rangle \\langle y, z \\rangle \\, \\mathrm{d} \\mathbf{P} (z)", "5391b425009306e33bc0a4ed04f2ec6f": "SF\\;\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\n1\n\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;I\n", "5391c6d96b1f68a049fd6f9df3bd983c": " L = 10 \\ m ", "5391ca3de6d7f00e5267f2419a73f0ee": "\\scriptstyle T ", "5391cfae2738931e49fc17bf63c797bd": "\\lfloor\\;\\rfloor", "5391dd8584bbefaf2e71c257bf6d5789": " Q = U \\times Ar \\times \\frac{\\Delta T(B)-\\Delta T(A)}{\\ln [ \\Delta T(B) / \\Delta T(A) ]} ", "5391e09c3e41a0aa2be567be8cb68015": "p_Z(z) = \\begin{cases}\n1/2 \\qquad & 0 < z < 1 \\\\ \n\\frac{1}{2z^2} \\qquad & z \\geq 1 \\\\\n0 \\qquad & \\mbox{otherwise} \\end{cases}", "5391fbc3973b4379b8e701f98811280a": "D_P = \\frac{1470 \\ \\text{kPa} \\cdot \\mu \\text{m}}{P_L} ", "5392603b640818998d392010364ca780": "\\Gamma^{\\alpha}_{\\sigma \\nu} \\,", "5393063fa19e0252f01248a536673d3e": " \\begin{pmatrix} H_n & 2P_n \\\\ P_n & H_n \\end{pmatrix}= \\begin{pmatrix} 1 & 2 \\\\ 1 & 1 \\end{pmatrix}^n .", "53930bfc93fe7fff211bea016b899296": "n=4k", "539325d39de7a42ff1343c857f8f53fb": "g_{21} = \\left. \\frac{ V_{2} }{ V_{1} } \\right|_{I_{2}=0} ", "539342a1748cc01c4d51e35646727df3": "y(t) = y_I + y_O - y_\\mathrm{overlap} = e\\left( {1 - e^{ - t/\\epsilon } } \\right) + e^{1 - t} - e = e\\left( {e^{ - t} - e^{ - t/\\epsilon } } \\right).\\,", "5393cf8d264a65e6ac1e8ea7d5a900b5": "\\sum_{i \\in I} \\| T(e_{i}) \\|_{H}^{2} < + \\infty.", "53940a47f01fa05b1fa540fbc4f2f8aa": "F[r]=\\frac{r^2+2r'^2-rr''}{(r^2+r'^2)^{3/2}}", "539428c639a84461c947abb43d36b47b": "\\frac{a^2b^2}{a^2+b^2}", "53942f1e56f64683e57154f6ba38e654": " W=0 ", "539437a02c5885c575788009383e96c8": "A(b)", "53944242863f19453940a9eeb79899b4": "L=0=-F\\dot{u}^2+2\\dot{u}\\dot{r}\\,,", "53944cbadda4d9648d086a737598e81c": "cos LHA = (sin Ho - sin Dec * sin B )/(cos Dec * cos B)\\,", "53948525ad8122972c7d590844eab79c": "(1+x)^n = \\sum_{k=0}^n {n \\choose k}x^k.", "5394a20e795ccc771e3daeb76beb557f": "\n\\aleph_1", "5394e7d394c7d61d17d3e88e6d350d7e": "\\vec a\\cdot \\vec n \\geq 0\\,", "539539aed329fabc0b57f9b7aef34998": "\\boldsymbol{Y}=(Y_{1},\\dots,Y_{k})", "53959e3ddcf1330904db393599d2f847": "Eq.3\\;\\frac{d^2 M}{dx^2}=w", "5395d7d27ceff6cd537e79ead4eb0337": "D^2 A = \\mu_0 J ", "5396133e6d70c4fc2148e33ed1310fe5": "z^3/\\lambda^3", "53966ae066edcf5683d8283d203369c8": "\\scriptstyle (\\Omega, \\mathcal{F}, \\operatorname{P})", "5396d7af3ef3f7276706123de55e2919": "R_{12} = \\phi_{12}(R)", "5396ecccc4b2c5a3cb79ddc227d17296": "\\exists x \\in S : \\phi(x)", "5396fc502cc396a4081db8e22996be52": "g_{ik}\\Big(\\gamma(t),\\dot\\gamma(t)\\Big)\\ddot\\gamma^i(t) + \\left(\\frac{\\partial g_{ik}}{\\partial x^j}\\Big(\\gamma(t),\\dot\\gamma(t)\\Big)\n- \\frac{1}{2}\\frac{\\partial g_{ij}}{\\partial x^k}\\Big(\\gamma(t),\\dot\\gamma(t)\\Big) \\right)\\dot\\gamma^i(t)\\dot\\gamma^j(t) = 0,\n", "53979fcfad11a178b50c44fc572ad583": "\\mathbb{F}L(v) \\cdot w = \\frac{d}{ds}|_{s=0} L(v+sw)", "5397e360b7657c915e82bd0bf197c6ab": "\nL(p) = f_D(\\mathrm{H} = 49 \\mid p) = \\binom{80}{49} p^{49}(1-p)^{31},\n", "5397e5c9baf0e79681291ec0a5f1621f": "\\sqrt{\\frac{8}{35}}\\!\\,", "5397f9134eeee6d9c396eb09e128e898": "\\nabla \\times \\vec v", "539802621c64fab5406976ff09a73dd0": "\\rho(\\mathbf{r'})", "539896cd433d91ef6e80305bb4427abd": "B_0(x) = 0", "539896d896ca1f8e8977bae6bc7f9fdb": "J = \\epsilon\\sigma T^4 + (1 - \\epsilon) H", "5398efce5c5d80e15a4fa98e6ce0abf1": "D_{KL}^{(e)}(P||Q)", "5398fd42b8a3291c5d6ce24af6b60b5d": "P_e= 10\\ \\log_{10} (p) + 10\\ \\log_{10} (g_{t}) ", "539964f67d8d023777c2256fc98d6be9": "N_{\\alpha\\beta} = N_{\\beta\\alpha}", "539985706431efda6d8a2aa1631b98dd": " H(\\xi,\\eta) = R(\\xi,\\eta)", "53999774470193e168e6e0f28edd4e88": " {\\langle \\psi |} X {| \\psi \\rangle} =\n\\lambda", "5399a12f242e6eb715a6c3a456fed761": " Y = A[\\alpha K^\\gamma + (1-\\alpha) L^\\gamma]^{\\frac{1}{\\gamma}}, ", "5399b02fd961d76c653296438ea4b28f": "(k_1,\\ldots,k_r)", "5399c355902c1058e96f97062532beff": " N(A \\cup B)", "5399e065cd489aadee9afe640fffebbe": "[H f(\\mathbf{x}_n)]", "5399fbfa7e57a3f4b82d4138a2d9f1b0": "z^3", "539a3f2c2a1307ef6ced7d1cc3cc5c1f": " \\phi_{sm}(r) = \\max \\left[ 0, \\min \\left(2 r, \\left(0.25 + 0.75 r \\right), 4 \\right) \\right] ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{sm}(r) = 4", "539a508c09293b9c47f2dd2585e60164": " f(q) = a q ", "539aa03c54b360b8991a7066373ed9ac": "P\\to M", "539bb35381a9037e319770eb1fb91f5c": "\\mathbf{T}(s)=T(s)\\cdot\\mathbf{s}_u", "539bbdfdf82b023539f4ec047cf5fd5f": "\\sum_{n=0}^\\infty q(n)x^n = \\prod_{k=1}^\\infty (1+x^k) = \\prod_{k=1}^\\infty \\left(\\frac {1}{1-x^{2k-1}} \\right).", "539be4daa7cd7a7534c961c77b9acfcf": "\n{dV \\over dt} + \\frac{1}{\\tau} V = f(t) = Ae^{j \\omega t }. \n", "539c2046cdab21a06b487828a6bf060a": "0.26 \\, (\\langle I \\rangle_e/\\mu_B) + \\log \\sigma_o", "539c675d0eae53c30bcdafb3c86fc3b2": "(c+n)", "539c9ae8dcecb08f3f06e4fe5441d437": "\n\\begin{align}\n\\sec\\theta=\\frac{1}{\\sin\\left(\\theta + \\dfrac{\\pi}{2}\\right)}\n=\\frac{1}{2\\sin\\left(\\dfrac{\\theta}{2} + \\dfrac{\\pi}{4}\\right)\n\\cos\\left(\\dfrac{\\theta}{2} + \\dfrac{\\pi}{4}\\right)}\n=\\frac{\\sec^2\\left(\\dfrac{\\theta}{2} + \\dfrac{\\pi}{4}\\right)}\n{2\\tan\\left(\\dfrac{\\theta}{2} + \\dfrac{\\pi}{4}\\right)}.\n\\end{align}\n", "539cf86ef2924417f244a2e7d6a05d59": "\\langle x, y, z\\rangle^n = r^n\\langle\\sin(n\\theta)\\cos(n\\phi),\\sin(n\\theta)\\sin(n\\phi),\\cos(n\\theta)\\rangle", "539d36d00f550ca383029134242569fd": " A_{overall} = \\sum w_i A_i ", "539da09009dbd0e516aae679ac98c32c": " \\varphi_n(t) \\ ", "539ddf0660cafdf04413a3b1f14945c1": "\\Omega(\\Gamma)=\\{\\tau\\in H: |\\tau|>1 , |\\tau +\\overline\\tau| <1\\}", "539e8c9895051e0ad97d9a48477854b5": "Y_t = \\exp\\Bigl(X_t-X_0-\\frac12[X]_t\\Bigr)\\prod_{s\\le t}(1+\\Delta X_s) \\exp \\Bigl(-\\Delta X_s+\\frac12\\Delta X_s^2\\Bigr),\\qquad t\\ge0,", "539f9f4455260d965754fe7c4a0c900d": "y_3 ", "539fa66a54d60fdbd6278ccebed13ddd": "b>0", "539ffee3894df8b003e30c8d7b8cb7e0": "{T_L}", "53a02ac7bdce706a6e956699ed48130d": "\\frac{d \\ln T}{d \\ln P}", "53a0c569255ae6f9975172c00894ce18": " f(s_1,\\dots,s_n) \\sqcup f(t_1,\\ldots,t_n) ", "53a0e60a54093b9d0cd5e42ff3215055": " O_n ", "53a0efbc4b269b4579c906f75f790030": "u(x) = \\frac{x^{1-\\rho}-1}{1-\\rho}", "53a143cf88b8b0e56bfe43af3693a512": "S(1+r)-S = (1+r)^N - 1", "53a15a1655f1c92510b80adea7bca455": "\n\\left\\{Var\\left(\\begin{bmatrix}Z(x_1)&\\cdots&Z(x_N)\\end{bmatrix}^T\\right),\nVar\\left(Z(x_0)\\right),\nCov\\left(\\begin{bmatrix}Z(x_1)&\\cdots&Z(x_N)\\end{bmatrix}^T,Z(x_0)\\right)\\right\\}", "53a15fcac791c0c792842b875c6a6a1d": "\n(R_\\mathrm{S} + R_\\mathrm{L})^2 + (X_\\mathrm{S} + X_\\mathrm{L})^2 \\,\n", "53a17cc43dcb401055e20ccf41b5f8ba": "\\epsilon[T^*]\\epsilon", "53a1823797feb9d4c7c77a4345eb42f4": " \\sigma_y = \\sigma_0 + kd^{-1/2} \\,", "53a1aa425714c0f24078dc3cb5f8b019": "a_3 = 4", "53a1b58bf6a8198fc2174e2a7a8ba534": "\\Gamma (X, \\mathcal O_X) = \\mathbb C", "53a1d2a3c72dcfd545bb5885111cef31": "\\scriptstyle \\mathbb{Z}_N", "53a21539823c0d69039ea7356cbb1f2d": "y_2 = \\left( y_1 \\left(n^2-a\\right) - x_1^2y_1^{-1}\\right)^{-1}", "53a24f4698cac78827b662dc4625589a": "O_F^{(a2,b2,c2,d2)} \\left \\{ O_F^{(a1,b1,c1,d1)}[x(t)] \\right \\} = O_F^{(a3,b3,c3,d3)}[x(t)] \\, ,", "53a30b948bdece9a2a2648b5cb8f97f5": " \\dfrac{M}{Re} \\ll 1", "53a323d4cc4d1511faf23b33a83bc640": " \\alpha=F_k(x)\\, dx^k , \\quad \\beta=G_k(x)\\, dx^k ", "53a408cd435adc3142979f4e5fbe4a33": "{1 \\over e}\\sum_{k=x}^\\infty {k^n \\over (k-x)!} = \\sum_{k=0}^n {n \\choose k} B_{k} x^{n-k}", "53a44f63623d768a850be3cdedaa8e3f": "M_i\\,", "53a466dc405427da521624f77a276ab4": "d \\Phi = \\frac {U} {T^2} d T + \\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "53a489f0f57e5af47e4883730e315499": "\\mathfrak{p}_0", "53a4f2c990342a789c195882569df905": "\\overline{x}\\in O_n(N_G(P)/C_G(P))", "53a4fd7300c03a7ba0225953b3b53ac9": "\\text{Attenuation (dB)} = 10\\times\\log_{10}\\left(\\frac{\\text{Input intensity (W)}}{\\text{Output intensity (W)}}\\right)", "53a51e897a785ce397f13d8f37cd5853": "\\frac{P Q_e t}{\\sqrt{P Q_e t}} = \\sqrt{P Q_e t}", "53a56bc893a7c98376f2f536d1f5e6f6": "11 \\times x", "53a5710d3f8ac9e37549550b619dcb58": "\\alpha=\\omega^\\alpha", "53a59542ed2279ca52652c51a57a20d7": "0 \\rightarrow A {{q \\atop \\longrightarrow} \\atop {\\longleftarrow \\atop t}} B {{r \\atop \\longrightarrow} \\atop {\\longleftarrow \\atop u}} C \\rightarrow 0.", "53a59bc6a7f0ca6599cf138d604d1197": "{^N\\!\\mathbf{\\bar{H}}}", "53a59c199e84395920d36084a0b243df": "D_2 \\cong A_1 \\times A_1", "53a626bee4e7e802e6e79b72310e6e8d": "K=-V\\frac{\\mathrm d P}{\\mathrm d V}", "53a6d79ae38497f2a3801828479b8dcc": "\\ E(r) = A \\cos (k r - \\omega t + \\phi)/r", "53a6dc2d0fa84f7828fe73656b0d5977": "\\mu^{(1)}", "53a735cf3d426c2ff4928c2e242da6e7": "\\frac{\\partial}{\\partial t} E \\Longleftrightarrow -i (\\omega - \\omega_0) \\tilde{E}", "53a759a3b540ff8a95ac91a7ff0535e6": "\\frac{l^2-1}{2}", "53a75a8568644da69a9b96517b059522": " H(x_k)=\\frac{h(x_k)}{A}", "53a75e132aa011a796c239062ec74d67": "\nQ_{l} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int d\\mathbf{r}^{\\prime} \\rho(\\mathbf{r}^{\\prime}) \n\\left( r^{\\prime} \\right)^{l} P_{l}(\\cos \\theta^{\\prime})\n", "53a79daf6df2a5b4b270e96a3750b31c": "C_l\\!", "53a8089dfa672fb1b3109d42f2d1b944": "\n B_i^n(u) = {n \\choose i} \\; u^i (1-u)^{n-i}\n", "53a858cc10cdbf823e15d66e05a9bc17": "t_3=[v_3, v_4]\\,", "53a884ec1421c2325fedb5e0ed247377": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\n = \\cfrac{E}{1-\\nu^2}\n \\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \\,.\n ", "53a8b2998d201e921e3a3190842159d3": "s\\colon H \\rightarrow G", "53a8b4a6a15727ac6a1f5ee0d26ca60d": "P \\downarrow Q", "53a909b04ce20dd7d2608a97bbc00c53": "(2,n-1)", "53a92d82a68fd713f1be1b89d40d2765": " \\begin{align} \n & 0 = v \\tau_{m} + 2 R_{0} + \\frac {(R_{0}^2 - R_{m}^2)} {v \\tau_{m}} \\\\\n & 0 = -v \\tau_{1} - 2 R_{0} - \\frac {(R_{0}^2 - R_{1}^2)} {v \\tau_{1}} \\\\\n\n\\hline \\\\\n\n & 0 = v \\tau_{m} - v \\tau_{1} + \\frac {(R_{0}^2 - R_{m}^2)} {v \\tau_{m}}\n - \\frac {(R_{0}^2 - R_{1}^2)} {v \\tau_{1}}. \n\\end{align} ", "53a96f9484e0fb67caff7ddba6fe5120": " 2m ", "53a988991061fbfe16c05fc42304ab23": "x\\in\\mathbb{R}^N", "53a9b27abf09220f4d315d3cb834e9a5": "\\nabla f(x, y)", "53a9c32c08eb2127376a3abaf75abde7": "\\hat{\\mu}_0 = 34/3=11.33", "53a9e7b569759042d6b1e5edb395e6d6": "d_1 = d_2", "53a9f3d058801aed22be22f7541d1d79": "2^{\\aleph_{\\omega_1}}<\\aleph_{(2^{\\aleph_1})^+}", "53aa588fdc40cf4d963c77c90c54412c": " \\mathcal L = i \\psi_R^\\dagger \\sigma^\\mu \\partial_\\mu \\psi_R ", "53aa8ec5e63883aeaddaf2df3950d397": "\\frac{\\partial^2\\rho}{\\partial t^2}-c^2_0\\nabla^2\\rho=\\frac{\\partial^2T_{ij}}{\\partial x_i \\partial x_j},\\quad (*)", "53aab91fdaeb10683c63a4fb54ab3cd9": "\\frac{\\omega_{\\mathrm{obs}}}{\\omega_s} = \\frac{1}{\\gamma (1 - \\beta \\cos \\theta)} \\,", "53aafe2ac2858fa015e4a02215d05561": "24^{4}\\tbinom{1,312,000}{4}", "53ab18bf53f3719775f5aab4c487d242": "PV_{2} = \\frac{-$50}{(1.05)^{2}} = -$45.35 \\, ", "53ab48021fd7b4b0a23d57a596614828": "W(\\boldsymbol{E})", "53ab57cc7bcf7614c8f7fa37b7eec545": "h^0=h^0_\\lambda dx^\\lambda", "53ab613575c60fd9b0f7144dde7fa91a": "^i_j", "53ab6c567649b1b820780a1e30dd7b65": "4\\omega_{RF}", "53ab8f207f29a5a5f46c918d39c0c5f6": "p_x(a)", "53abd198d828850775b5c5c8e67faf4d": "\\mu(A)\\leq 0 \\, \\Rightarrow \\, \\|R(A)\\|\\leq 1.", "53abe374b14efd53731c9596da595664": "\\rho_{2}=(1-P)\\sigma", "53abfb889e0e87d55654d85cb51768b4": "\\mathrm{La} = \\mathrm{Su} = \\frac{\\sigma \\rho L}{\\mu^2}", "53ac00b463829a6c258ef4c5d6f4b61a": "|e_i\\rangle", "53ac0fe52a076f995b393483335a0ca4": "\\delta \\mathbf{r}_i = \\sum_{j=1}^m \\frac {\\partial \\mathbf {r}_i} {\\partial q_j} \\delta q_j,\\quad i=1,\\ldots, n,", "53ac2507ca90a4392e50124aaec85b5f": "b_{min}", "53ac486c23ce6493f4539e6968322edb": "\n\\frac{ \\partial \\bar{u_i} }{ \\partial t }\n+ \\frac{ \\partial }{ \\partial x_j } \\left( \\overline{ u_i u_j } \\right) \n= - \\frac{1}{\\rho} \\frac{ \\partial \\overline{p} }{ \\partial x_i } \n+ \\nu \\frac{\\partial}{\\partial x_j} \\left( \\frac{ \\partial \\bar{u_i} }{ \\partial x_j } + \\frac{ \\partial \\bar{u_j} }{ \\partial x_i } \\right)\n= - \\frac{1}{\\rho} \\frac{ \\partial \\overline{p} }{ \\partial x_i } \n+ 2 \\nu \\frac{\\partial}{\\partial x_j} S_{ij},\n", "53ac79c764d37852dc8834712085a137": "\\scriptstyle y(t)\\,", "53ac95e6082b0210589364058b27b64c": "L=L(a,\\theta)", "53ac9730feb01b8e96c7ae47dee92874": "\\begin{bmatrix} -\\eta_1-1 \\\\ -\\eta_2 \\end{bmatrix} ", "53accaaac3703ed19b852e5d30143062": "G^m= G_0^m+Kt", "53acf48555ca0e1822f462b24de9459a": "\\mathbb{C}_{\\mathcal{B},\\varepsilon}", "53ad408a4019a6c2ba59d393608d4412": "\n\n\\left.\n\\begin{matrix}\n\\operatorname{gcd}(\\operatorname{gcd}(x,y),z)=\n\\operatorname{gcd}(x,\\operatorname{gcd}(y,z))=\n\\operatorname{gcd}(x,y,z)\\ \\quad\n\\\\\n\\operatorname{lcm}(\\operatorname{lcm}(x,y),z)=\n\\operatorname{lcm}(x,\\operatorname{lcm}(y,z))=\n\\operatorname{lcm}(x,y,z)\\quad\n\\end{matrix}\n\\right\\}\\mbox{ for all }x,y,z\\in\\mathbb{Z}.\n", "53ad5f824ddc69698bb8d0b3c02ec72e": " \\sum_{i=-1}^{k-1}(-1)^{d+i}\\binom{d-i-1}{d-k} f_i = \n\\sum_{i=-1}^{d-k-1}(-1)^{i}\\binom{d-i-1}{k} f_i, ", "53ad97ad4fe7d7c944ec291e2dd0c978": "O(n^2(\\log(r)\\log(q)+n))", "53ae46d90cccde6f93923ad18e3d3de7": "R_{in}", "53ae9c2aea468475a559f759b27d6bf6": "A \\cap (A \\cup B) = A\\,\\!", "53aea22cd2eff22325680224f68025ab": "\\ \\rho {Dh \\over Dt} = {D p \\over D t} + \\nabla \\cdot \\left( k \\nabla T\\right) + \\Phi ", "53aea65e194335f93ef8ad4c6cf6bba0": "\\left(\\begin{array}{rrrrrr}\n 0 & 1 & 0 & 0 & 1 & 0\\\\\n 1 & 0 & 1 & 0 & 1 & 0\\\\\n 0 & 1 & 0 & 1 & 0 & 0\\\\\n 0 & 0 & 1 & 0 & 1 & 1\\\\\n 1 & 1 & 0 & 1 & 0 & 0\\\\\n 0 & 0 & 0 & 1 & 0 & 0\\\\\n\\end{array}\\right)", "53aed1d4fc00253e807143427441d1d9": "V_N(\\vec{r}) = \\frac{-Ze^2}{r} . ", "53af32d33ae6a5c0cbbae4389f7377c8": " \\frac{d}{dt} y(t) = A(t) \\, y(t), \\quad y(0) = y_0, ", "53af382e72267991ddb884c3611906cf": "f_\\theta(x)=a(x) b_\\theta(t)", "53af49b7f7149267c7ad429e1d0bd26f": "\\displaystyle a_0", "53afb0458e61ec41b27443cce5ec9ab0": "\\varepsilon_1-\\varepsilon_2, \\varepsilon_2-\\varepsilon_3, \\ldots, \\varepsilon_{m-1}-\\varepsilon_m, \\left\\{\\begin{matrix}\n\\varepsilon_{m-1}+\\varepsilon_m& n=2m\\\\\n\\varepsilon_m & n=2m+1.\n\\end{matrix}\\right.", "53b02c486fd8439ef9dcfcbc84c8ee27": "v \\in [0,\\ 2\\pi),", "53b0993f3ae99c7e32dacda148976793": "\\Sigma^1", "53b09a69cd6b2398463e4cf8b868690e": "W^{1, p}(\\Omega).", "53b0beb0a387028dde890977f6717efe": "V_\\mathcal{H} (\\mathcal{P}) = \\frac{1}{N} \\text{tr}(\\mathbf{G}) - \\frac{1}{N^2} \\sum_{i,j=1}^N \\mathbf{G}_{ij} ", "53b0dc4784c6d50537208ecb2bba7e13": "\nV_\\mathrm{peak}=\\sqrt{2}\\ V_\\mathrm{rms}.", "53b157903fdb2eaddbadc2c8157730ef": "\\{f_\\lambda\\}_{\\lambda\\in\\Lambda}", "53b16c8be24bb8b962762d256f752326": "w(x,y) = g(x,y,\\sigma) = \\frac{1}{2\\pi \\sigma ^2} e^{ \\left (-\\frac{ x^2 + y^2}{2\\sigma ^2} \\right )}", "53b1c4ad55547bb0c15a6f68f2282179": " a_{i_1,i_2,...,i_k}", "53b1f4f3075fa0003a8d08c66611fa80": "R_A(x) = \\frac{(Ax, x)}{(x,x)}", "53b23896ce88ef93d64ea7f73220f5a5": "\n\\begin{align}\nML &= \\left[\\begin{matrix} A & B \\\\ C & D \\end{matrix}\\right]\\left[\\begin{matrix} I_p & 0 \\\\ -D^{-1}C & I_q \\end{matrix}\\right] = \\left[\\begin{matrix} A-BD^{-1}C & B \\\\ 0 & D \\end{matrix}\\right] \\\\\n&= \\left[\\begin{matrix} I_p & BD^{-1} \\\\ 0 & I_q \\end{matrix}\\right] \\left[\\begin{matrix} A-BD^{-1}C & 0 \\\\ 0 & D \\end{matrix}\\right].\n\\end{align}\n", "53b251ff70478abd7d44194d6bef42d4": " e^z ", "53b26dcbb69697c2e363e0874a8546a0": "nil : 1\\longrightarrow List(A)", "53b27782af96e46ef81598ce96b71a15": "\\tbinom{m}{n}", "53b291dae067b5c4e030523680301ef9": "x^k=\\mathbf{x}\\cdot\\mathbf{i}_k", "53b2e48e2054838dc3b86689a751b25e": "\\langle m|\\partial_\\mu n\\rangle=\\frac{\\langle m|\\partial_\\mu H | n\\rangle}{E_n-E_m},", "53b2fade4cf15910cb8f554439b963e3": "A_1\\otimes\\mathrm{End}(E_1) \\simeq A_2\\otimes\\mathrm{End}(E_2),", "53b3518754763769325ef3792d244db8": "C[0/1].", "53b352e4a9830159af45d73a1d477eee": "u[n] \\,", "53b383e63144557aa13bfa29eb151774": "D_{gb}", "53b388ff8097a860005bea9cf0372f54": "F_{i+1} = 2 - D_i.", "53b39914b8d07af965daf1f564dc77f4": "t'_{A,B}=T(\\gamma_{B,A})\\circ t_{B,A}\\circ\\gamma_{TA,B} : TA\\otimes B\\to T(A\\otimes B)", "53b3e23fdac017facd9ac38072e030db": "3x^2 + y^2 = 1 + 2x^2y^2", "53b401ec590f78698b70dd82234d85d0": "\nU_2 \\in [14V,17V]\n", "53b4428edce0e8be3f82c11045d38918": "Z=aX+bY+i", "53b490ecab576b831f61963b1c6af543": "C_R = \\frac{v}{u}", "53b4f16390135270eebc174f04078d80": "b \\leftarrow t", "53b52a54484e61bf9637b67e36eaa639": "\\int_{-\\infty}^\\infty \\mathrm{rect}(t)\\cdot e^{-i 2\\pi f t} \\, dt\n=\\frac{\\sin(\\pi f)}{\\pi f} = \\mathrm{sinc}(f),\\,", "53b54574389c007790084aec23ce19d3": "\\dot{v}_\\alpha = \\frac{\\mathrm{d}v_\\alpha}{\\mathrm{d}t} = a\\,\\left( 1 - \\left(\\frac{v_\\alpha}{v_0}\\right)^\\delta - \\left(\\frac{s^*(v_\\alpha,\\Delta v_\\alpha)}{s_\\alpha}\\right)^2 \\right)", "53b5ac5727f3f8ad9796f9697922c907": "LC_{50} \\le 1000 \\tfrac{mL}{m^3}", "53b5f10611f339a3645af95fd8d5f234": "\\partial F", "53b6244068d9bf76419eff3953cc48cb": "{{0}}", "53b64ae80e4721029ec348e5f27546e2": "\\psi^{\\dagger}\\psi", "53b6840e17dbb70a591e526660a266d3": " \\hat{\\Phi}(1) = \\Phi", "53b6b5b04b0909513d04c42c51556d2b": "1 + \\cot^2 \\theta = \\csc^2 \\theta\\,", "53b6c2c6cb3d8beb4214a2616a2f0e17": "s = -|\\mathbf{a}_1| ", "53b6ca5279fa69cb4219f8d76cda8e92": "S_{mk}^{}=\\alpha_{mk}+\\beta_{mk}+p_{m,k-1}+p_{m-1,k-1}", "53b6f13f8b631a1af5dfbb3041467e88": "(Y_{\\text{10}}) ", "53b721edfe6ffb071d2d3c0e14b180cd": " |z| < |a|", "53b76d2d423617d9336e7ccb3c8894c9": " \\text{ln(settlement)} { \\text{spores} \\over \\text{mm}^2 } = -7.47 \\times 10^{-2} \\times ERI + 6.28", "53b7a3d4bb6d91c28bd13eab145fbab4": "\\scriptstyle \\mathbf{b}_3", "53b7cd9822701a7fe2e842fa81b623c4": "x(y_1,\\cdots,y_n).P", "53b7d431983e30af0f4ec2b82146336a": "\\scriptstyle \\epsilon_{01} = p_{01} / p_{0m} \\,", "53b7f2154341dc1e7f5025741bfc5ddf": "e\\in G", "53b80021c832d431dbdde4c64be14903": " \\rho \\ d \\theta ", "53b80816ecd0d7c05162895d0ee70123": " X = \\{x_1, x_9\\}", "53b81e6fbffb884869082ec90c0f2771": " \\left(0,\\ 0,\\ 0,\\ \\pm2\\right)", "53b885e056e315ca5dbb210e035bbe9a": "(dx/(n+1),\\dots, dx/(n+1))", "53b8d05ae0d98080e11f929527a3b3c0": "\n \\Omega \\equiv \\operatorname{E}[\\,\\varepsilon\\varepsilon^\\mathsf{T}\\,|X\\,] = \\Sigma \\otimes I_R,\n ", "53b900f28104c201c78ec97277c7872a": "U(x) = \\frac{1}{1 - \\gamma}x^{1 - \\gamma}", "53b90a968dd6dae34398a506ffb86a56": "\\log T = \\log\\left(\\sum_{k=0}^\\infty {a_k \\over k!} D^k \\right) ", "53b95aadabcc7867ade6e219f5a785c9": "\\omega^2=\\Omega_c^2+k^2v_s^2", "53ba1df3f132179e28723b2564657081": "S(t)\\,", "53ba8d27712c33499100712424c2dc99": "\\vec U^{n+1}=\\vec S(\\vec U^n,\\lambda; \\delta t),", "53bae2a55f48a6abbcd7a29932ea2297": "\\mathfrak{C}\\{\\mathcal{B}\\}", "53bae5109c087448dc3b1c9607fea29a": "Z_{\\text{para}}=\\sum\\limits_{\\text{even }J}{(2J+1)e^{{-J(J+1)\\hbar ^{2}}/{2Ik_{B}T}\\;}}\\text{ ; }Z_{\\text{ortho}}=3\\sum\\limits_{\\text{odd }J}{(2J+1)e^{{-J(J+1)\\hbar ^{2}}/{2Ik_{B}T}\\;}}", "53baef77f85f3400b2312b5a48d7f963": "\\pi_k(X) \\approxeq \\pi_{k-1}(\\Omega X)", "53bb412acfc70c3507738fa00007841a": "\\displaystyle{gw={\\alpha w + \\beta\\over \\overline{\\beta}w + \\overline{\\alpha}} .}", "53bb5a2f3206cca27907e65974dd6446": " \\theta_1(x) \\leq 0 ", "53bb7d244285f7ec192cd7c208075025": "\n\\sum_{k=0}^n \\mu_i p_i (k)p_\\ell (k)=v v_i \\delta_{i \\ell}, \\quad(8)\n", "53bbc39f5b2eb14e3ada08dde0711f78": "-1 \\le u \\le 0", "53bbd9a45021043fcf7cc605e76bf972": " = \\dfrac{5}{\\sqrt{3} + 4} \\times \\dfrac{\\sqrt{3} - 4}{\\sqrt{3} - 4}\\,\\!", "53bc063428055a9f127076b99935fba5": "\\mathcal{H}_A \\otimes \\mathcal{H}_B", "53bc3068bc889e2eb1d275af42db0771": "\\gamma^{a}", "53bc5b7f3619a42a98d475ddeb4991c0": "y = (y_0, \\dots, y_{2^n-1})", "53bc9e341879ff172c152f6d59410160": " \\operatorname{Reg}(E/K) = \\det\\bigl( \\langle P_i,P_j\\rangle \\bigr)_{1\\le i,j\\le r},", "53bcb3d68a24b0d9bacf6d23b65a2f4f": "\\ (D_\\mu \\Phi)' = G D_\\mu \\Phi", "53bcfd6aefe982ef2cf97bcc3aef0335": "\\sigma_z, \\sigma_x", "53bddc5452f937a7d699d41ea6bb629c": "dP(x) = \\frac{I(x)-I(x+dx)}{I_0} = \\frac{1}{\\ell} e^{-x/\\ell} dx.", "53bdea579fa8ea3555585c58a830f137": " \\cosh y = (e^y + e^{-y})/2.", "53be02c6d86c72f2efbbbbb06e8d53ca": "x \\in \\Omega", "53be544503d9d900ce3823c0133237f1": "\n\\cos(\\psi) =\n\\frac{\\sqrt{\\cos^2(\\theta)-\\cos^2(\\Omega)}}{\\sin(\\Omega)}\n", "53be7bab5bc7819bdd3af038dd0b64e8": "\\alpha + \\beta = Desired Reserve Ratio", "53be99fe87a57c0254d9958aa4e01d1f": "\\tfrac{2 G \\nu}{1-2\\nu}", "53bed2fc8c9ba2a9e4e218f77bcd388f": "x,\\ f(x),\\ f(f(x)),\\ f(f(f(x))), \\dots", "53bee136585f84e5fa5c176d101f2fd7": "x_2 = \\frac{c}{a \\ x_1} \\approx -\\frac{c}{b} .", "53bf48356b05b851390b3dcfe40f11dd": "\\Omega_c = q B/m", "53bf4e37f0f5ae5d400afd662d42be06": "\\phi :H \\rightarrow \\operatorname{Aut}(N)", "53bf9e3d0130aae401bb08b4a66bb6f5": "[n,p]", "53bfc17512c0340211a8207a80c97fc0": "D-", "53bfd8a5aa9ecdb922f4bfd4ff7baf38": "Z^0 \\,", "53bfe23fc78307cd0f1294ee6c461d9d": "\\scriptstyle h\\left(a\\right)\\,=\\,0", "53bfede7ab7f232f72de0d4518fe092e": "\\mathrm{O}^{''}_\\mathrm{i}", "53c0351bfa5c6f86c3639c6d4b183da0": "p_1+p_2=1, 0 \\leq p_1,p_2 \\leq 1 .", "53c09dcdc7221448513249cb2e66856c": "B_\\nu(T) \\simeq \\frac{2 \\nu^2 k T}{c^2}.", "53c0cc0a96ac7a59181f226151da4ff4": "E_t(S_{t + 1} - S_t)", "53c0cfa77cf549f1d5f7ea2bc2dbff1d": "\\mathbf{X_{t}}=(1,y_{t-1},y_{t-2},\\ldots,y_{t-p})\\,", "53c0f1def75afc759131c61bd6535c02": "\\mathcal{FS} \\,.", "53c1076d1a024d074d77c3538ccf6109": " (1+\\nu)\\sigma_{ij,kk}+\\sigma_{kk,ij}=0.\\,\\!", "53c1244e388c0d55dad76f6c6bbb168f": "\\displaystyle \\sum_{n=0}^\\infty t^nM_n(x)=\\frac 1{1+t}\\exp-\\frac{xt}{1-t}\n\n", "53c1b8da87e5f2a46386b11974458949": "\\mathbf{j}_8", "53c2124918f89a26cc3c0a0086b7a725": "S\\to aSb ~|~ ab", "53c24989a69f7cf4e58c5a47c000bff4": " 3 = 7 \\times (3/7)", "53c25b554b06dcfb30e975d39cfc5024": " K = \\frac{\\Pr(D|M_1)}{\\Pr(D|M_2)}\n= \\frac{\\int \\Pr(\\theta_1|M_1)\\Pr(D|\\theta_1,M_1)\\,d\\theta_1}\n{\\int \\Pr(\\theta_2|M_2)\\Pr(D|\\theta_2,M_2)\\,d\\theta_2} .\n", "53c29428fbc300e5d1b116437d82a6d9": "x = [2,3]", "53c2f522f586612b41e96e3e657feda0": "U_\\mathrm{E}^{\\text{total}} = \\frac{1 }{4\\pi \\varepsilon _0}\\sum_{j=1}^N Q_j \\sum_{i=1}^{j-1} \\frac{Q_i}{r_{ij}}= \\frac{1}{2}\\sum_{i=1}^N Q_i\\phi_i ,", "53c3020038cab9483ff83364a1cb0c6d": "n_f", "53c31666c509fe7ef31e4ba49413edfe": "\\ A + B", "53c31ae8d29528bea55613eef78adb7d": "a=b=s=t=0", "53c325b0c9f415f10fb67f6c32ee0569": "0 = \\nabla \\cdot (D \\nabla c) - \\nabla \\cdot (\\vec{v} c) + R.", "53c3622a535eebd0c1f6e277bf104dad": "\\{\\}", "53c385c29573addf9f1ed458e2f2ce19": "\\displaystyle{K(w,z)=\\gamma^{-1} \\cdot \\exp\\,{1\\over 2a}(c z^2 + 2wz - b w^2)}", "53c3abe79fd26ae79ed63ffc0ed43ba7": "R > 0", "53c3eae0329bb86fc5b9df610b8f8f5b": "t=(p_1-p_3)^2=p_1^2 + p_3^2 - 2p_1 \\cdot p_3 \\,", "53c3f76fb1210bb2e98cee320ead5d1d": "\\mathrm{R}(t_1,t_2)", "53c3f9f776648fbdaa1a4dd171b0f0dc": "r_i^{\\eta + 1} = \\frac{u_i}{\\sum_j m_{ij}^{(2\\eta)}}", "53c3fe259d9343107ada54761e624202": "P_\\ell^{m}(\\cos \\theta)\\ \\cos (m\\phi)\\ \\ \\ \\ 0 \\le m \\le \\ell", "53c452d6c464531e47dd92dadddb0b5f": "T_n(x) = \\begin{cases} \\{ nx \\} & 0 \\leq x < 1, \\\\ 1 & x = 1,\\end{cases}", "53c4672fe5729636bacd10e08cb44d4c": "\n(A|B) = \n \\left[\\begin{array}{ccc|c}\n 1 & 1 & 2 & 3\\\\\n 1 & 1 & 1 & 1 \\\\\n 2 & 2 & 2 & 2\n \\end{array}\\right].\n", "53c49c566d4f798e307e8ba3829e76c6": "=\\left(\\frac{1}{5} + \\frac{1}{6}\\right)", "53c4bfa0a2ee0f6c8517f02b23545a7e": "e^{i(x+y)}=e^{ix}\\cdot e^{iy}", "53c4eb536f18e33dad6bec22b17bfd3e": "\nY = W \\cos \\phi \\tan \\delta \\,\n", "53c53f29d6a83deacaea80f073e73a2d": " X_t = \\sum_{k=1}^\\infty Z_k e_k(t)", "53c5525a048e5decffa84ac5bfed84f4": " V= \\pi R\\left(\\frac{-H}{12\\pi h^2} + \\frac{64CkT\\Gamma^2e^\\kappa h}{\\kappa^2}\\right) ", "53c59fff29ea0c9cdd4f5251d0568c11": " |01 \\rangle ", "53c5c43bc955b0eb316a6191b25a21e6": "{d\\mu } = dG = -SdT + VdP", "53c5e7a2e3e5dbee4e8376a86241e013": "q\\ne 0", "53c66fa31b975208c448710aedff1486": "J\\Rightarrow K", "53c693fcc34e10add127f0242f7bf5c2": "E^2 \\Psi = 0,", "53c6aecce9cd49d7ea512d709e644928": "D = D_0 \\, {\\mathrm e}^{-E_{\\mathrm A}/(RT)},", "53c6fbd9e9307682cf0a314409119713": "L_\\mathrm v", "53c73ac6204a09fec7d426e8ff33aa18": " \\int_{-\\infty}^\\infty G_x(t,f) e^{j2\\pi ktf}\\,df = e^{-\\pi (k-1)^2 t^2} x(kt) ", "53c780ab349cebb68bf8dea2cd8ab6d4": "a(t)\\propto t^{1/2}", "53c7d399d7654e166accd90819d851a1": "\\textstyle \\mathbf{Q}=\\left[ q_{jk}\\right] ", "53c7f71e86e7b609f28043e42e767b74": "\\psi^\\prime = \\frac{1}{2}\\xi_a^b \\lambda_b^a.", "53c87b6988968b7ca3abe8c661e1e2e9": "\\vdash\\varphi", "53c886f04671966dd5041bf8de800708": "\n \\mathcal{P} = \\Big\\{\\ \n f_\\theta(x) = \\tfrac{\\beta}{\\lambda} \n \\left(\\tfrac{x-\\mu}{\\lambda}\\right)^{\\beta-1}\\!\n \\exp\\!\\big(\\!-\\!\\big(\\tfrac{x-\\mu}{\\lambda}\\big)^\\beta \\big)\\,\n \\mathbf{1}_{\\{x>\\mu\\}}\n \\ \\Big|\\ \n \\lambda>0,\\, \\beta>0,\\, \\mu\\in\\mathbb{R}\n \\ \\Big\\}.\n ", "53c8b041afc4797bb65aa5605e7f686b": "I = I_{0} e^{-x/\\ell}", "53c8f2ae9071df33ff5d828ea228fc49": "\\begin{align}A \\sum_{n=0}^{\\infty} \\frac{\\ln 2^{n+1}}{\\ln F_{n}} &= A \\sum_{n=0}^{\\infty} \\frac{\\log_2 2^{n+1}}{\\log_{2}(2^{2^{n}}+1)} \\\\ &< \nA \\sum_{n=0}^{\\infty} (n+1) 2^{-n} \\\\ &= 4A.\\end{align}", "53c919b5c5c0350523f29b3c64e14fa2": " T(A \\to B) = \\Box (T(A) \\to T(B)) ", "53c96a953bfc2c93f854e29d125de613": "S \\notin W", "53c984625d1d7d84ce2265827ee1598e": " \\rightarrow (\\mathbf{\\lambda} x . x x x) (\\lambda x . x x x) (\\lambda x . x x x)", "53ca21a00451c776e4b1148ec4e4ab7b": "(bd)\\,\\!", "53ca50286db73aaf06a1cb033c277a43": "= \\sqrt{2\\pi} \\sum_m i^m e^{im\\theta_k}\\int_0^\\infty f_m(r) J_m(kr) r\\operatorname{d}\\!r.\n", "53ca7c3b330c53c9d6c6d5623486bd8f": " E = [\\rho_{sw} \\cdot S_{Basin} \\cdot (H_{Tide} - H_{Head})] \\cdot g \\cdot H_{Head}", "53ca8cace3120ef02ecaa0acd77cc123": "F(x)=\\sin x,", "53cab3eedad054ad09811da5d8a76b43": "i=1,2,\\dots,M", "53cacae6507725386353a9d0f6efe209": "\\beta_{j}^{-}=-\\min(\\beta_{j},0)", "53cad7742d8c804728a92f85ea31934a": "((x * y) * z) * (w * v)", "53cae6498027134877ea8054b7933572": "x_0 < x_1 < x_2 < \\ldots < x_n", "53cb63f888046685f5b41eaaf4359c5e": "C=\\{1,\\dots,c\\}", "53cb71f01d2a59b8fd9ec44da31b2a99": "o_{1:t}:= o_1,\\dots,o_t", "53cb9d016b437aba244d838af25e6dee": "p \\in X", "53cba3c292cd4b1992099a31bbf472b2": "[w]", "53cbd970f703f173d00d199ae6a9524d": "\n v^i\\mathbf{b}_i = S^{ij}u_j\\mathbf{b}_i = S^i_{j}u^j\\mathbf{b}_i ;\\qquad v_i\\mathbf{b}^i = S_{ij}u^i\\mathbf{b}^i = S_{i}^{j}u_j\\mathbf{b}^i\n ", "53cbe939fe9880396dd78783af9177f3": "\n\\mathbf{S}_{x_B}\\mathbf{S}_{x_l}=S_{x_B}\\mathbb{I}\\otimes\\mathbb{I}S_{x_l}\\otimes\\mathbb{I}\\mathbb{I}\\otimes\\mathbb{I}\\mathbb{I}=S_{x_B}\\otimes S_{x_l}\\otimes\\mathbb{I}\\otimes\\mathbb{I}\n", "53cbf5dd8297358a55b6315824392d2d": "h_c(t) = \\begin{cases}\n \\sum_{k=1}^N{A_ke^{s_kt}}, & t \\ge 0 \\\\\n 0, & \\mbox{otherwise}\n\\end{cases}", "53cbfe714c82e464c523ccc4b291f13e": "e^{\\pi|\\xi|^2}\\hat f\\in\\mathcal{S}'(\\R^d) ~,", "53cc6157c0f993bb6e2ec5ad7db11df2": " \\omega = \\frac{d\\theta}{dt}. ", "53ccc4f640a8c3d25f17b8b217cd15bb": "a_{\\pi}", "53ccdfcd8bbbedb0c46a7efa08469836": "P(A^c)=1-P(A)\\,", "53cd0eadb1690997c780de91db05f269": " \\langle E \\rangle_{\\mathcal{A}} = \\bigcap \\{ F \\in \\mathrm{Con}(\\mathcal{A}) | E \\subseteq F \\}", "53cd6f9f818d74de291e80aed6d4a713": "\\,{}_2F_1(3,1;1;z) = \\,{}_2F_1(1,3;1;z) = \\,{}_1F_0(3;;z)", "53cdfca9c698c7280fdb3e4bc2f06812": "* \\rightarrow * \\rightarrow * \\rightarrow *", "53ce390921927a0292ce3c78ba1ca3f6": "\\mathbf{b_i} \\cdot \\mathbf{a_j}=2\\pi\\delta_{ij}", "53ce47a5c30b64519e7568dcb644cd97": "4 \\sqrt[4]{q}", "53ceb41bbbadccc2ad3b0ade41edaf7b": " a \\uparrow \\uparrow b = { }^{b}a", "53ceba6f4dc598ce73517ab9e27f2aa3": " (\\Delta \\otimes id)R = \\Phi_{321}R_{13}\\Phi_{132}^{-1}R_{23}\\Phi_{123} ", "53ceeb08f30ce138eaf49dc9f0fb7421": "\\mathsf{L}_i{}^j \\equiv \\mathsf{L}_{ij} = \\frac{\\partial\\bar{x}_j}{\\partial x_i} ", "53cf363ff44217c3477650c27771933d": "\\bar{\\psi}(\\gamma^\\mu\\gamma^\\nu-\\gamma^\\nu\\gamma^\\mu)\\psi", "53cf53e010fcc78099cf8d23eb044bb2": "p(X_1,X_2,\\ldots,X_n)", "53cf8ba0450ad64884c0d94cf334185d": "\\text{ENF} = \\kappa M + \\left(2 - \\frac{1}{M}\\right)\\left(1 - \\kappa\\right)", "53cf995f07497db148361c88e0353ef3": "\\text{kva base} \\approx\\text{ 0.8 * horsepower rating}", "53d00ad4deb2fc05eda31f01e3b7c3b4": "\\mathrm{MD} = \\int_0^1 \\int_0^1 |F(x)-F(y)|\\,dx\\,dy .", "53d015078cb7d3348bbdbe371666577d": "\\rho_{xy}", "53d122524aec0ed7a97e338fbd10ff0f": "(\\sum_{g\\in G} a_g g) v = \\sum_{g\\in G} a_g \\rho(g)(v)", "53d12ed5f89b0ed3bde18c86dd5c79db": "P(x,y,z)=1", "53d165387a09c39b7b09ed454e4c5723": "\\mathcal B([0,\\infty))", "53d1c0a6c023f048219d1e33ac77996c": "(28)\\quad \\theta_{(n)}=\\hat{h}^{ba}\\nabla_a n_b=\\bar m^b m^a\\nabla_a n_b+m^b\\bar m^a\\nabla_a n_b=\\bar m^b \\delta n_b+m^b\\bar \\delta n_b=\\mu+\\bar\\mu\\,.", "53d1c63363ac97d72707237f5e9a26d5": "\\textstyle\\int f (\\mathcal{F}g) = \\int(\\mathcal{F}f) g", "53d1cc84936fc288d8af3baf09465545": "c=\\sqrt{\\kappa/\\rho}", "53d2006f5bf86b0e2b15065236c1e5df": "\\mathbf{A} + \\mathbf{A}^{\\rm T}", "53d2288c50858befdad0bf434de2fc8c": "\\frac{U}{L_\\bigodot} \\approx \\frac{2.3 \\times 10^{41}\\ \\mathrm{J}}{4 \\times 10^{26}\\ \\mathrm{W}} \\approx 18,000,000\\ \\mathrm{years}", "53d253da735e32cbd4cd69532c15f4e7": "e=\\hat{z}-z", "53d2afa958c9b58a6254dc8bd67bee02": "{{F}[X]}[Y]", "53d2afe071f7108035bbf2d87b69874d": "s'_n", "53d2b2b975d7447981ab47b48462bb78": "f[A] = \\{ \\, y \\in Y \\, | \\, y = f(x) \\text{ for some } x \\in A \\, \\}", "53d2ba4a8140b9065cc9d1e377e410b3": "\\,K_i", "53d2bd28c5894474c364aa07dc918f98": "\\sigma_P(\\xi) = p(x,\\xi) = \\sum_{|\\alpha|\\le k} a_\\alpha(x)\\xi^\\alpha.", "53d2fd0743a13661638e51a4f94052a0": "F_i\\ ", "53d30e336e323ea06491b29ce95d2328": " n = ( t / D )^2 / m ", "53d33985d35472234c00b13228023c51": "A=af(a^{\\dagger}a)", "53d34848919acd9cd5d566de4a31973a": "c = \\frac{1}{\\sqrt{ \\mu_0 \\varepsilon_0}} = 2.99792458 \\times 10^8 \\, \\mathrm{m~s}^{-1} ", "53d381d6024a9bdf1cc505c6c6bc89f3": "S_{z_r}", "53d39ca1156f290f02df65b50ecdcae0": " x_{1} = y , x_{2} = \\dot{y} ", "53d3d4db321b0c8cc6e1ecb6b9d3e457": " d = ", "53d48532bc54478d9bdfb1a52659ec87": "\\widehat{b}", "53d4aaaf9d6aceebb98a5ad003c6a808": "\\begin{align}\n y &= \\frac{G}{(2 - \\gamma)_{\\gamma - 2}} x^{1 - \\gamma} \\sum_{r=\\gamma - 1}^\\infty \\frac{(\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(1)_r (1)_{r + 1 - \\gamma}} x^r + \\\\\n &\\qquad \\qquad+ H \\sum_{r = 0}^\\infty \\frac{(1 - \\gamma) (\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(2 - \\gamma)_r (1)_r} \\left(\\ln(x) + \\frac{1}{\\gamma - 1} + \\sum_{k = 0}^{r - 1} \\left \\{ \\frac{1}{\\alpha + k} + \\frac{1}{\\beta + k}- \\frac{1}{1 + k} - \\frac{1}{\\gamma + k} \\right \\} \\right) x^r.\n\\end{align}", "53d4d27d487a1ca0305da6c18ef23530": "[L,L]\\subseteq L", "53d576988e4b2f945a97049b1715f596": "\\Phi(t)", "53d58ced79085145030af9d9e1745d95": "\n\\mathbf{P}(\\tau,\\mu | \\mathbf{X}) = \\text{NormalGamma}(\\mu_0, n_\\mu ,\\frac{n_\\tau}{2}, \\frac{n_\\tau}{2 \\tau_0})\n", "53d5d7bd6f3a3a13413cfdb459f22214": "N=120f/P", "53d607274fbb05858c833a6d56ed3a86": " \\phi : S_p \\rightarrow S_q '", "53d6331cbd80a1b018b9afd5b30743ba": "\\mathrm{Re}", "53d64f375d7dd6768da11a8531a18442": "z_{n}\\left( p_{n}\\right) , ", "53d685dff126caf1660ef11bb3eb3dc3": "\nds^2 = \\sum_{k=1}^d \\left( h_k \\, dq^{k} \\right)^2\n", "53d6ad95d1aef1683b45e8e92bff58ef": "\nH = {1\\over 2} P^2 + {1\\over 2} X^2 + \\epsilon X^3 ~.\n", "53d6df55d7c000532da80024684d098b": " NGD(x,y)= \\frac{ \\max \\{\\log f(x), \\log f(y)\\} - \\log f(x,y) }{ \\log N - \\min\\{\\log f(x), \\log f(y) \\}}, ", "53d717f304c1f1f1820866630018d1e9": "\\textrm{KEYBOARD-PLAYER(Jill)}", "53d71baa77f56c507c82b22c2612f0bd": "w^2 = v^{2g+2}f(1/v) \\,", "53d729770c7dd2901588445c6c4234d0": "ds=0", "53d880c8aa1783cc920b47c1e5ddb546": " \\mathbf{P} ", "53d8976158694c21d1b397a922bcafea": "\\hat v_i", "53d8cb3f2f4dc83956fec421ed584b08": "\\scriptstyle r(\\boldsymbol{r}_A,\\, \\boldsymbol{r}_B) ", "53d92d651d5ff877f5d9ce060e7c5e78": "\\hat{\\mathbf{n}} = (\\hat{n}_1,\\hat{n}_2,\\hat{n}_3)\\,,", "53d9487b45e8daf0f7980c9499afeac0": "L(x, y; t)\\ = g(x, y, t) * f(x, y)", "53d95e56113d4ab39be8e52cbb086491": "\\dot{\\omega}_{ab} = -\\frac{2\\theta}{3} \\, \\omega_{ab} -\\left (\\sigma_{am} \\, {\\omega^m}_b + \\omega_{am} \\, {\\sigma^m}_b \\right) ", "53d96be575cfc63f7649a56c222857ad": "\\text{PointsPlus} = \\max \\left\\{ \\mathrm{round} \\left( \\frac{(16 \\cdot \\text{protein}) + (19 \\cdot (\\text{carbohydrates} - \\text{sugar alcohol})) + (45 \\cdot \\text{fat}) - (14 \\cdot \\text{fiber}) +(58.05 \\cdot \\text{alcohol}) + (11.4 \\cdot \\text{sugar alcohol})}{175} \\right) , 0 \\right\\} ", "53d9ad80c648d041396162871f3af7b0": " Y \\sim \\textrm{Gumbel}(\\mu - \\beta \\log(n),\\beta) \\,", "53d9ec42df96e515a5a827073452eb82": "+(F^nG^m) \\leftrightarrow (+F^n+G^m)", "53da834a2c5aa9201e2510156b0df6cd": "V_{\\rm rms}", "53da95b21d02a1baa1a08a288b0c7d83": "(x',y') = (x+y s, y)\\,", "53dadd8b634e29537a2c7fa1f2d24f4c": "\\scriptstyle{1 -} \\tfrac{y}{h}", "53daee207d1e55397f3ae4ed4d3e3a5b": "f: (M,\\omega) \\rightarrow (N,\\omega')", "53db7c05a43bd12e6276c78f1ff3819c": "\n\\begin{align}\n\\mathbf{B} &= (\\mathbf{B}_0^{-1} + \\mathbf{X}'\\mathbf{X})^{-1} \\\\[3pt]\n\\boldsymbol\\beta\\mid\\mathbf{y}^\\ast &\\sim \\mathcal{N}(\\mathbf{B}(\\mathbf{B}_0^{-1}\\mathbf{b}_0 + \\mathbf{X}'\\mathbf{y}^\\ast), \\mathbf{B}) \\\\[3pt]\ny_i^\\ast\\mid y_i=0,\\mathbf{x}_i,\\boldsymbol\\beta &\\sim \\mathcal{N}(\\mathbf{x}'_i\\boldsymbol\\beta, 1)[y_i^\\ast < 0] \\\\[3pt]\ny_i^\\ast\\mid y_i=1,\\mathbf{x}_i,\\boldsymbol\\beta &\\sim \\mathcal{N}(\\mathbf{x}'_i\\boldsymbol\\beta, 1)[y_i^\\ast \\ge 0]\n\\end{align}\n", "53dc3e0e1917d76d090f6a93bc2c5ba2": "\\mathbf{v}_B = \\omega_SR \\mathbf{u}_{\\theta} \\ ,", "53dc498a3291ed1cd3bd2fbc335d415a": "\\left\\lfloor m + \\left(1 - \\left[ \\frac{m}{r} \\right]_1 \\right)r \\right\\rfloor = m", "53dcc8d4632bf7120dee0c4335eaf1a0": "f(x) = \\sum_{n = 0}^\\infty a_n T_n(x).", "53dcffd562613d43dad62cf358105560": "F_j(x) = \\frac{1}{\\Gamma(j+1)} \\int_0^\\infty \\frac{t^j}{\\exp(t-x) + 1}\\,dt.", "53dd4cb26524808d6881f52296a033ce": "\\ u\\in \\mathcal{U}(\\alpha,\\tilde{u})\\ ", "53ddfe9704768f365c1469e383a2edf1": "\n\\beta+ \\Gamma^{i}_{\\lambda \\beta}[g^{h\\lambda}g_{hj},\\alpha - g^{h\\lambda}g_{mj}\\Gamma^{m}_{h\\alpha}\n", "53de0ea0ca87edbf47a5a74f2b6e3728": "\\mathcal{H}(q,p)= \\frac{1}{2} \\langle p,p\\rangle_q", "53de30561bc25792f9852887d08d8561": "\\mbox{radians}\\,", "53de40b10d88dd08db766f85b9ad1f16": "\n\\mathrm{Pr}\\left( \\sqrt{\\left( X-\\mu\\right)^T \\, V^{-1} \\, \\left( X-\\mu\\right) } > t \\right) \\le \\frac{N}{t^2}\n", "53de985ec2940ec469c562672cdea16c": "[a, c)", "53df0dc0c8e87aef21bef58d2c830b0f": "\n\\log\\left(\n\\frac{Z_{\\pi^\\mathbb{G}}(T)}{Z_{\\mu}(T)}\n\\right)\n= \\log\\left(\n\\frac{\\mathbb{G}(\\mu(T))}{\\mathbb{G}(\\mu(0))}\n\\right)\n+ \\int_0^T g(t) \\, dt\n", "53df42c211726022ec76e435bb492b5d": "\\pi ab \\,\\!", "53dfa039dd969f1a80d7e5dd233bea56": "BU(n) = \\{ V \\subset \\mathcal{H} \\ : \\ \\dim V = n \\}", "53dfcfe222461d1fa7019d580f3f562f": " f ( x_1 + x_2 ) = f ( x_1) + f ( x_2 )\\ ", "53dfdab60a803b8ba9a48c0248c12fe9": "\\frac{d}{dt}x(t)=f(x(t)) \\, \\mathrm{,} \\quad x(t_0)=x_0", "53dff4fb69cadbf3cd5d2ad16d8555bd": "V_{C} = \\pi r^2 h. \\,", "53dffdbe12d7650dc739c48e1dcc0d93": "X_{\\mathcal{G'}}", "53dfff27a931973054eae24d1bbfd64c": "W_0 = \\frac{\\hbar \\omega_0}{2} = \\frac{\\hbar}{2\\sqrt{LC}}. \\ ", "53e01e9112e4b6ebd3561ae64eac71ae": "\\sqrt{{x_1}^2 + {y_1}^2} = 1", "53e0a1c8c428c1d849d4ea54a25d6a5b": "E \\begin{bmatrix} \\phi \\\\ \\chi \\end{bmatrix} = \n\\begin{bmatrix} m \\mathbf{I} & \\vec{\\sigma}\\vec{p} \\\\ \\vec{\\sigma}\\vec{p} & -m \\mathbf{I} \\end{bmatrix} \\begin{bmatrix} \\phi \\\\ \\chi \\end{bmatrix} \\,", "53e0b004324a92cb13a67488c736dfb4": "\\Gamma_i = \\Pi_1 + \\cdots + \\Pi_i - I,\\quad i=1,\\dots,p-1. \\, ", "53e0f52d0c02fd9bb13f2c1610d6aa2e": "(x \\land y) \\lor (\\neg x \\land \\neg y)", "53e12341c72fa9b36431086b1f7ddb98": "\\mu_4=E(X-E(X))^4", "53e150f9adc32ee519ead867049de443": "0 < 1 \\and \\forall x \\in N", "53e15a613766b8a94cbd778f4eee3d3b": "\\scriptstyle 1 \\,\\oplus\\, 0 \\;=\\; 1", "53e2860833fd240073ca475dd236caa6": "~(\\cos(x))^2~", "53e28c49bf301fa73808c0b834184510": "C(i,j) = 2 C(i-1, j-1) - C(i-2, j)\\,", "53e2e474d6cabbd57fa05ae237ec2f93": "P.MPL=w", "53e2e77a78a6daa358d7c1bb37ac9678": "Wr", "53e3249138245912d37253a1cca3993a": "K(t,s)=K(t-s)", "53e372755f0ac9e97149f613ef2c1c43": "W=\\{w_1, w_2,\\dots,w_n \\}", "53e38b2de2c36844b26a4cd0dd179f61": " \\mathrm{A} ", "53e3993644232b57abba4336f2c73c16": "\\lbrace \\exp(\\epsilon t) = 1 + \\epsilon t : t \\in R \\rbrace.", "53e3a8aae0e87588485e0f58264f1437": "(x_1 + 3x_2 + x_3 + 3x_4 + x_5 + 3x_6 + x_7 + 3x_8 + x_9 + 3x_{10} + x_{11} + 3x_{12} + x_{13})\\mod{10} \\equiv 0. ", "53e3c478ae9167be6bd6e12eb1310a4d": "\\textstyle (x_{i},y_{i})", "53e3c5ca4484af96189c8d8a1dc194a5": "x_k = \\tanh(\\tfrac12 \\pi \\sinh kh)", "53e3dcaa6d701fc0021bc3df7341466f": "O(A \\rightarrow B) \\rightarrow (OA \\rightarrow OB)", "53e3f6d1e01354dd0fde943e1d395c46": " M'= (M_1,M_2,M_3,M_4,M_5,M_6)", "53e40afd6c27d87e2aa08588ceadcbc8": "\\Delta G = RT \\ln{K_a} ", "53e456f7d0554101cb8a1d65da86bd59": "~U_{12} = U(\\mathbf{r}_1 - \\mathbf{r}_2) ", "53e469fc986fb9da1ed5ad0d5f6a9b4d": "|x|^{c+1}", "53e4a1a4bf9f37d38351cb1a2651154f": "x-\\log 2\\pi", "53e4c7d77dc40a3669e659a82e1bac8c": "\\Delta{\\mathit{E}}=\\mathit{E}_{in}+\\mathit{E}_{out}=\\mathit{0}", "53e4cdbfbdb63a7a67ea475461ffa5fc": "ax^2 + bx + c \\;=\\; a(x-h)^2 + k,\\quad\\text{where}\\quad h = -\\frac{b}{2a} \\quad\\text{and}\\quad k = c - \\frac{b^2}{4a}.", "53e5361ceb7a408e8987d3375f70ac76": "\\left\\{ \\gamma^5,\\gamma^\\mu \\right\\} =\\gamma^5 \\gamma^\\mu + \\gamma^\\mu \\gamma^5 = 0. \\,", "53e58e318df27bcd49d6537d1878a14b": " E = -1 / n^2", "53e5d69438cecde14b5c10011caf71cd": "\n\\mathord{\\overbrace{v_{\\text{eq}}}^{\\text{scalar}}} = \\mathord{\\overbrace{A_{12}}^{1 \\times (n-1) \\text{ vector}}} \\mathord{\\overbrace{\\mathbf{e}_2}^{(n-1) \\times 1 \\text{ vector}}}.\n", "53e5de0f4b33c4cd002cafa0a5d9846e": "\\alpha ^ \\gamma = \\beta", "53e5e6b7591fc7547c2d74695706f8a3": " 16\\pi (T_{bd} - T_{ac} \\eta^{ac} \\eta_{bd}/2) \\, = h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \\eta ^{rs}", "53e664135e7da3074d29879999a62277": "x - x = 0", "53e6e89899571bb885c71fbbf7a2d69d": "\\alpha_k = \\alpha.", "53e75747600afbc4f3c0b6550346887c": "S_0(t) = .2", "53e7989bf7b750ed44970fd0602f1b2a": " E_f = \\varepsilon\\sigma T^{4}", "53e8dbbd69cefccc211d33dfd7d05145": "F=N\\varepsilon_0+k_BT\\sum_{\\alpha}\\log\\left(\\frac{\\hbar\\omega_{\\alpha}}{k_BT}\\right).", "53e909594ec993db324a4d841c989cf7": "\n\\max\\{\\alpha: r_{c}\\ge \\max_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} R(q,u)\\}\n", "53e92018297adf00c46b587bd0405b67": "N = { (4.76 \\times 10^6) d_r C \\over L^2}", "53e938f77cec8a6d87b4b32cbfde4994": "\\Delta x_i = x_{i+\\frac{1}{2}}-x_{i-\\frac{1}{2}} ", "53e940a62400004d127657196e37b90d": "\\left( \\frac{\\partial A}{\\partial y} \\right)_x = \\left( \\frac{\\partial B}{\\partial x} \\right)_y", "53e96bf0546060d0791d27d9442b914a": "\\rho(L) \\subseteq \\mathbb{C}", "53e973336e39ad1b2dd15938db036d22": "T = b \\cdot \\log_{2}(n + 1)", "53ea0704316c0474672f184208f02067": "q^* = w - xi - yj - zk \\!\\ ,", "53ea236cfc27dc0154305e7bff3d3d9f": "\nx_{1}^{1}+x_{1}^{2} \\leq \\omega_{1}^{tot}\n", "53ea45152e71d92730890ac9c710fcd1": " \\cdots\\rightarrow 0 \\rightarrow \\tilde{H}_{n}\\left(S^k\\right) \\xrightarrow{\\partial_*}\\, \\tilde{H}_{n-1}\\left(S^{k-1}\\right) \\rightarrow 0 \\rightarrow \\cdots \\! ", "53eac5bcade613f7e4f17eaf4b9c0155": "G_T = \\frac{4|y_{21}|^2 \\Re{(Y_L)}\\Re{(Y_S)}}{|(y_{11}+Y_S)(y_{22}+Y_L)-y_{12}y_{21}|^2}", "53ead87a6ffbda57732bed20b3fff10e": "\\int f(v)\\,dv=3N", "53eb25efcb0715e368975aaf38acb561": "4.1) \\ \\mbox{Potential adopters}\\ -= \\mbox{New adopters }\\ ", "53eb69324bc67448ff78d84c2c550a01": "|\\theta|<2\\pi", "53eb77b67dd727866f61291e68086417": " d(G) = \\sum_{k \\geq 1} k \\ \\operatorname{rank}(G_k/G_{k+1}) ", "53eb970c24913ec6872749d0f83d0fa8": "I_f", "53ebe857ae23936f2e3bd380c8a7c393": "x_1 .. x_n", "53ebeffeb57ef338724c98c40cb69a1b": "x \\not\\in A", "53ecaa60b1e4aecd4443066677589a2e": "b_2=a_1^2+4a_2", "53ed01bedde279f431d927a65626f3b5": "\\| u-f\\|_p^p", "53ed6520cb9e3b93ebbe46094bb9073f": "j_0", "53ed84dd6c81b964fac95f3b55d23f0c": "\\pi_n(x)", "53ed852d43f5166c0560d0bd3f86859d": "Y_u", "53edb3df95e7a00002cf4130a8f8a765": "H(z)=\\phi(z)\\otimes I - I\\otimes \\phi(z)", "53edefd5b47193dc95ede51b75376c30": "L^{q_1}(\\mu_2)", "53edf0f1fbb5099c2cd2e6718efd3203": " a,b,c", "53ee41d8ce19d14d9bb432653a23ea55": "(\\rho u\\phi A)_e - (\\rho u\\phi A)_w = (\\Gamma A d\\phi /dx)_e - (\\Gamma A d\\phi /dx)_w", "53ee58a104ce1b33b5a4d71ae1f17841": " \\tfrac 12 n(n-1) ", "53ee83e4462e7691853fc490ce5b3501": "\\,U_0", "53eecc1ef16b8b16a26d14cba9f6dc26": "\\hat{V}=e (\\vec{L}+2\\vec{S}).\\vec{B}/2m", "53ef11dc32b97a5af5b0a79518668d0d": "\\Omega(\\log\\log n)", "53ef17e9f72e0adcdbe0267b843149a3": "\\mathcal{C}^{op}", "53ef46409844ccb65372a8ae5fff7c36": "\\Theta(n+m)", "53ef6ab6c1282a6555917f34befea39a": " p_i^m(t_1, t_2, \\dots, t_m) = t_i ", "53efbe5e3bc5680610143bf1354d3cc4": "n \\equiv 0,1,2,3", "53f01d9166032a11d992db5c80990383": "\\epsilon=\\frac{R_{sd}}{R_{sw}}", "53f0271006e751e56ae2164ef9e6d77b": "\\phi^{*}_{j} ", "53f02b470a1465380699dac650f3ee0f": "|\\psi_j\\rangle", "53f099c729651b2797960b7bde7132e1": "\\lambda = \\frac{v}{f}", "53f09b8f4284e8f03a3ca6787387ecbb": "\\mu_{\\alpha} = \\mu_{\\beta}.", "53f0a4f197bb9968c0bbdfaf02ff5107": "\\frac{\\mbox{total} \\; \\mbox{votes}}{\\mbox{total} \\; \\mbox{seats}+1}", "53f0a80bebc8e44b42246c37e74a6440": "a(u,v) = b(v)", "53f0bceb52637b82040f856185ed3a2e": "[f,g]\\ne \\overline{[g,f]}", "53f0fe131bc430277bc07a2839110f42": "\\Psi_4 = -C_{\\alpha\\beta\\gamma\\delta} n^\\alpha \\bar{m}^\\beta n^\\gamma \\bar{m}^\\delta\\ ", "53f12c2ddaf9c0ed060c1de9036f29fa": "\\frac{k+1}{k}m - 1", "53f1cc9b19f708a3a62b159534eaa8da": " PA = A ", "53f1d6b9290de593e096b681f2bb1bfb": "a \\ll L_{sd}", "53f20476f2fa9947e1b8c0ef1308e847": " F(k) = \n \\begin{cases}\n 0, & k < -1 \\\\\n 1/2, & -1 \\leq k < 1 \\\\\n 1, & k \\geq 1\n \\end{cases}\n ", "53f20957aad78fda1a450465e094ee96": "h(\\vec x,t)", "53f2222ec584d2413274c2638ba4ca7d": "w(3,3)", "53f26ca790a9e8ee30ad572d87e2b6d7": "e_{ij} = e_{ji}", "53f272e2999c895ec5eac4e3ac02b3f8": "\nF(r) = Ar^{-3} + Br + Cr^{-5} + Dr^{-7}\n", "53f28e344e4aeefa1cca874f99ce8880": "d_i=1", "53f2d08d16608bd9caaeb0d110ef0116": "T:H\\rightarrow H", "53f2e23467f7025ff0e727427f8ad0bd": "|\\psi(x)-x|<\\frac{\\sqrt x\\,\\ln^2 x}{8\\pi}", "53f37bd13a6e5da7a298fbd9a87109ad": "3 a_1^2 a_2^2.", "53f3a3845fff94a177ab33cdb38f33c0": " i,\\ j,\\ k\\geq 1", "53f3cb25d6a54b37c9e4cc16dded5469": "\\mathbf{U}_n^T", "53f3f3c8667e5b60f4b2bdbb8cbaabb7": "\\dot m = \\rho \\, A \\, v.", "53f3febd450f13090aaed329055dbd5b": "A(x_i)=O(x^{\\varphi-1})", "53f54243bf8caef2fdf5e987af62e960": "\\{\\neg, \\and, \\mbox{AX}, \\mbox{AU}, \\mbox{EU}\\}", "53f573bea29e180f6713b6cb8dd93bc1": "T=D^{-\\frac12} \\left(I-L^{norm}\\right) D^{\\frac12}", "53f5dcca73678be155fb6dd890072e50": "0 \\le \\alpha \\le 2^{160}", "53f5e64e85b675e79a961e0b205066c9": " \\lambda_n(t) ", "53f5f1495555b43aaf9a45c626352462": "\\sum_{i\\in I}A_i<\\prod_{i\\in I}B_i,", "53f5f64e462be72e133e774d2fa41005": "\n(\\mathbf{\\gamma_2})^T = \\alpha\\begin{pmatrix}0.8834 \\\\ 0.1166 \\end{pmatrix}\\circ \\begin{pmatrix}0.3763 \\\\ 0.6237 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.3324 \\\\ 0.0728\\end{pmatrix}=\\begin{pmatrix}0.8204 \\\\ 0.1796 \\end{pmatrix}\n", "53f6249ce464f0b7b87c6b07ac7c1619": " \\frac{1}{\\sqrt{f}}= 1.1364\\ldots -2 \\log_{10} \\left( \\frac {\\varepsilon}\n{D_\\mathrm{h}} + \\frac {9.287} {\\mathrm{Re} \\sqrt{f}} \\right) ", "53f68e295839f301c534b40253957ff3": "A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}", "53f691a8c6dcd2ffde6800df0382800d": "Z = (Z_t)_{t=0}^T", "53f708df300001943c8f16c88818e887": "D_i = C", "53f7332347f81ec2fad5acc8be981134": "\nN_{ss} = \\frac{N\\sqrt{Q}} {{NPSH}_R^{0.75}} \n", "53f7aa404f2ee97bc7afdef4cb0b9d1e": " \\begin{align} P(hypercalcemia~is~caused~by~cancer~in~individual) = \\\\\n \\frac {P(hypercalcemia~WHOIFPI~by~cancer)}{P(hypercalcemia~WHOIFPI)} = \\\\\n \\frac {0.0002}{0.00335} = 0.060 = 6.0% \\end{align}", "53f7e31b4bb04e95622c5ee54da1d2b0": "\nz = x + iy\\,\n", "53f81136e69e79686d94013ca69d60ec": "(a_0,a_1,\\dots,a_{2n},a_{2n+1})", "53f874a78fba58511b90143eb65fa6d3": "\\psi_{k}(\\mathbf{q})", "53f887314ba14517fdb9659edcdd94a4": "[0, t] \\times \\Omega \\to \\mathbb{X}", "53f88861c90d6a4adc745b7c249c2ffa": "\\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{\\partial \\alpha^2}= -N\\frac{\\part^2\\ln \\Beta(\\alpha,\\beta)}{\\partial \\alpha^2}<0", "53f8a638aa6852c69b9beac860358de9": "\\frac 1 p + \\frac 1 q = 1", "53f8df89c24d51b0be4679fccdf8412c": " E_0 = \\tfrac {1}{2} m_x v_0^2 \\,\\! ", "53f968ea234e2040b31079defd1b546e": "\\left\\{\\,k 2^n + 1 : n \\in\\mathbb{N}\\,\\right\\}.", "53f9dd9d5adbbfa9700c8c1d315d14ac": "S\\in \\Omega", "53f9e5bdf2df89aba5b8a644ff0fa9b1": "V = \\,", "53fa2773e33874e33f57e2daec850c02": "\\theta_{T}\\,", "53fa2c4445d0951c568115f4fbfc5146": "\\lambda^\\star > 0", "53fa2d9d3f47ff06403763ffeb27d37d": "\\mathfrak{a}_i = R e_i", "53fa5061c26c7eb0748e1a32499cd10c": "\n\\mathbf{P} = \\begin{pmatrix}\nP^0 \\\\ P^1 \\\\ P^2 \\\\ P^3 \n\\end{pmatrix} = \n\\begin{pmatrix}\nE/c \\\\ p_x \\\\ p_y \\\\ p_z \n\\end{pmatrix}\n", "53fa5807310cc08d9c0bbb40b40c0b3a": "Velocity = (-0.07,0.07)", "53fb06f7c2d281f54a74b2e8c3a123cb": "\\lambda=\\lambda_0/n", "53fb42b7cd793bb8887554ab90cdb55e": "\\cos\\frac{7\\pi}{60}=\\cos 21^\\circ=\\tfrac{1}{16}\\left[2(\\sqrt3-1)\\sqrt{5-\\sqrt5}+\\sqrt2(\\sqrt3+1)(1+\\sqrt5)\\right]\\,", "53fb437fe990ebd27aaeebcf4b245f15": "ds^2 = - \\left( 1-\\frac{2GM}{r} \\right) dt'^2 \\pm \\frac{4GM}{r} dt' dr + \\left( 1 + \\frac{2GM}{r} \\right) dr^2 + r^2 d\\Omega^2=(-dt'^2 +dr^2 + r^2 d\\Omega^2)+\\frac{2GM}{r} (dt'\\pm dr)^2", "53fb9afe9b88bfa780725671a2696e4f": "f(x)=\\sum_{j=0}^2 y_j\\cdot\\ell_j(x)\\,\\!", "53fba0d7114a19516bc902ee0f6b4df0": "P = 80\\pi^2 \\left ( \\frac {hI_0}{\\lambda} \\right )^2 \\,", "53fc19f801efb4a975e95281a5082776": "J = \\left|\\frac{\\partial\\mathbf{r}}{\\partial q_1}\\cdot\\left(\\frac{\\partial\\mathbf{r}}{\\partial q_2}\\times\\frac{\\partial\\mathbf{r}}{\\partial q_3} \\right)\\right| = \\left|\\frac{\\partial(x,y,z)}{\\partial(q_1,q_2,q_3)} \\right| = h_1 h_2 h_3", "53fc24e54923a96c39534644448fa2e8": "\\displaystyle{S_{Z_1} S_{Z_2} =\\pm S_{Z_3}.}", "53fc48c5db0dd771ede48641803fa8d9": "\\mathcal{N}_1(\\boldsymbol\\mu_1, \\boldsymbol\\Sigma_1)", "53fc5661f71d78e9a2744c58abc357ad": "\n\\left.\\begin{array}{l}\n\\mathrm{d}\\theta_b^a + \\sum_{c=1}^p\\theta_c^a\\wedge\\theta_b^c = \\Omega_b^a = -\\sum_{\\mu=p+1}^n\\theta_\\mu^a\\wedge\\theta^\\mu_b\\\\\n\\\\\n\\mathrm{d}\\theta_b^\\gamma = -\\sum_{c=1}^p\\theta_c^\\gamma\\wedge\\theta_b^c-\\sum_{\\mu=p+1}^n\\theta_\\mu^\\gamma\\wedge\\theta_b^\\mu\\\\\n\\\\\n\\mathrm{d}\\theta_\\mu^\\gamma = -\\sum_{c=1}^p\\theta_c^\\gamma\\wedge\\theta_\\mu^c-\\sum_{\\delta=p+1}^n\\theta_\\delta^\\gamma\\wedge\\theta_\\mu^\\delta\n\\end{array}\\right\\}\\,\\,\\, (2)\n", "53fcf865c4ad092e1b984bb7a8b502c1": "\n\\frac{d^2x}{dy^2}\\,\\cdot\\,y^3 + y = 0\n\\mbox{ }\\mbox{ }\\mbox{ }\\mbox{ };\n\\mbox{ }\\mbox{ }\\mbox{ }\\mbox{ }\n\\frac{d^2x}{dy^2} = -\\frac{1}{y^2}\n", "53fd17989f37694ab7d7666b021b170f": "\\psi(z)=-\\gamma+\\sum_{n=0}^{\\infty}\\frac{z-1}{(n+1)(n+z)}=-\\gamma+\\sum_{n=0}^{\\infty}\\left(\\frac{1}{n+1}-\\frac{1}{n+z}\\right)\\qquad z\\neq0,-1,-2,-3,\\ldots", "53fd1b80b810fae8e002690a2ee8a759": "\\mathit{MS}_\\text{total} = \\mathit{MS}_\\text{regression} + \\mathit{MS}_\\text{residual}.", "53fd32893fa86c6e45efa9e55812ece1": "s \\in(0,l)", "53fd5c3ac853e4b2baa463a62d2ba890": "r(\\pi,\\delta)\\,\\!", "53fd9ad8df217a11f90acd17bed6f967": "(X_s)_{s\\ge 0}", "53fdcba9692265c2546c08d1c9582039": "!P", "53fdfe004ffe0e92432e33c0e93683b5": "f(x) = \\sum_{n=0}^{\\infty} c_n x^n", "53fe96a229bac2aa1479bf523e755f3b": "76\\frac{2}{15}", "53fec25f4752fd9ebddddc7e48f01bd1": "\\lnot(\\lnot \\phi \\lor \\lnot \\psi)", "53fef4e1b2c249ab6ef55c3a3075cd49": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\n = \\cfrac{E}{1-\\nu^2}\n \\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix} \\,.\n", "53ffc035cc585bd4d543180933e93f6b": "{RgX^*\\ \\xrightarrow[]{\\tau}\\ Rg + X + h\\nu (UV photon)},", "53ffcf2b950c1592d2b91c5458f86c62": "\\sum x_i", "54000f88e3667cc25230d8bd8546544f": "\\frac{1}{nD}", "54002150c3109d908e304af66c19ecb8": "M(x)=\\frac{ax+b}{cx+d}, a,b,c,d \\in \\mathbb{N}", "540042364cabf8286390361d618b925c": "\n\\mathrm{Ta}=\\frac{4\\Omega^2 R^4}{\\nu^2}\n", "540053113fa5f59fcd90165f58c9d8aa": "\\displaystyle{([[A,B],C],D)=([A,B],[C,D])=([[C,D],B],A).}", "540084f68a8fa66fb5a3734169db6720": "s \\models_K a", "5400e6ca94d5f95f5df403c217bb388a": "\\mathbf{a_i} + (0, \\ldots, r, \\ldots, 0)", "5400f75632ae0b6ebf8eb622580541d0": "\ne_2 = \\frac{h^2}{n} \\nabla^2 P\n", "54010ee32a8ecb11e42f970d678271fb": "U(x)", "54018cb52510a5c5d890aee707bdd31f": "\\begin{bmatrix} 3& 1 \\\\ 0 & 3 \\end{bmatrix}", "5401a637d357ecd4c709585e2a19c6d2": "\\tau_{xy}, \\tau_{xz},\\tau_{yz}", "5401d0affa4c9c55982a58233d01a83b": "V \\rtimes_\\rho G", "540235acd90190279ae463bceffb7999": "\\frac{5 \\cdot 7 \\cdot 13}{3 \\cdot 11}, \\frac{11}{13}", "5402531bd9c895143f5abb1ebd122418": "D_0 = C\\ ", "5402a21ccc01c31b5aee96746e6f46fb": "(G/H_r)", "540307a01ec9f78c36f0364ca3b4a9eb": "\\frac {\\Delta V} {V} \\approx (1-2\\nu)\\frac{\\Delta L}{L}", "54030d0c9e01123a6f9738c33bceecf9": "f = \\sqrt{\\mu \\over{s}}", "54036de5d647dc7d5658e0ac5c798d21": "\\rho_{uc}(\\mathbf{r})", "5403780c4d5bc6adc316ad2d10da4885": "y=-\\sqrt{1-x^2}", "5403f2c95eb31d4cf29894853e919c36": " \\mathcal{I}(A) = A \\text{ and } I(A)=\\mathrm{Tr}(A)I ", "540410c56c832ec24dd92480dd00678e": "log BCF=3.41-0.508 logS", "5404152578733c8ec0b29689ced73577": "\\varphi:G\\rightarrow S_n.", "54054a2cf59e0ce342e3ce66c6140412": "\\mathrm{A_{3}}", "5405825457920e6fc7232ab44e631bdc": " R_s = R_{BCS} + R_{res}", "5406142dba7f4dd87086edc013fc412f": "P_2 = \\int_0^1 \\frac{2\\theta(x)}{\\pi}\\,dx = \\frac{2}{\\pi}\\int_0^1 \\cos^{-1}(x)\\,dx = \\frac{2}{\\pi}\\cdot 1 = \\frac{2}{\\pi}.", "54064367f5b4e247e8a80af100f1d820": " F = \\beta I ,", "54064c939d47bd9a92a65542ec894a4c": "\nV_2(r) = V_1(r) + \\frac{L_1^2}{2mr^2} \\left( 1 - k^2 \\right)\n", "5406d48d69ef60580fdc6830e8ed20ca": "c_i=a_i \\cdot b_i^{-1} \\mod m_i", "5406f4e7941360ee4fed56266dfd97c8": "\\scriptstyle{R_n^0}", "5406fd2a5f96d9c2885d76a8efb2e48c": "n > 2 \\cdot N \\cdot E_C(\\mathcal{N})", "54071658cfe46e3470546da4d8309efb": "\\frac{S(z)}{X(z)} \\frac{Y(z)}{S(z)} = \\frac{Y(z)}{X(z)} = \\frac{(1 - e^{-2 \\pi i \\omega}z^{-1})}{(1 - e^{+2 \\pi i \\omega} z^{-1})(1 - e^{-2 \\pi i \\omega} z^{-1})} = \\frac{1}{1 - e^{+2 \\pi i \\omega} z^{-1}}.", "5407528fb96a7f5a67c9356602e0c0ab": "(K,+,\\cdot)", "5407587c85c1af60fe8e883d3a184479": "f(x) = \\lambda e^{-\\lambda x},\\,", "540790881db97128ef9c971121419991": "I_1\\subseteq\\cdots I_{k-1}\\subseteq I_{k}\\subseteq I_{k+1}\\subseteq\\cdots", "5407a386fcf5c9f56a030b0b5a3dac37": "s_N(x) = \\sum_{n=0}^N f_n(x)", "5407e58e3a99b708c9eb67eb4d9ea539": "0, 1, 3, 15, 75, 405, 2835, 22155, 199395, 1828575, \\ldots ", "5408485ede8a57bf6f2bdcd453e1f71b": "{\\mathbf{B}}_{n,m}", "54086c122bf3b8283ed931ffc215a996": "a \\ne 0\\,", "5408e62ddf9c8737551a3c00aa42c285": " f(r) \\equiv 0 \\,\\bmod{p^k}", "54090011351f89b616322e05fe3a9c83": "e'=e+k. \\varphi (N) ", "540905b069fc44560825f7af4bfd7241": " \\xi", "54092e6c5f68e44c2103e696b1ea37cb": "f^*(x) = \\sup_{r > 0} \\Bigl( \\mu(B(x, r))^{-1} \\int_{B(x, r)} |f(y)| \\, d\\mu(y) \\Bigr).", "540930f61a93c7e33f7cd78497b45a2a": " f(x, y) = x^2 y \\, ", "54093160ec365d2bee056849ff9f9845": "\\left(f_\\alpha\\right)_\\alpha", "54093ab0ead324e036cd77f60e497a6d": "a_{(4)}b = a^{(a^{(b-1)})}", "540943f423f527803d0d749ab1effd61": "r_2 = 0.2", "54099796aa162572c26312b3e7ed05d5": "\\displaystyle{\\iint_\\Omega g (Dh) + (Dg) h \\, dx dy =\\lim_{n\\rightarrow 0}\\iint_\\Omega g (Dh_n) + (Dg) h_n \\, dx dy =0.}", "54099ce629953ef0b0a1519b654076c7": "b_{3}^{*}= b_{3}- \\mu_{3,1}b_{1}^{*}= \\begin{bmatrix}3\\\\5\\\\6\\end{bmatrix}- \\frac{14}{3}\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}=\\begin{bmatrix}\\frac{-5}{3}\\\\\\frac{1}{3}\\\\\\frac{4}{3}\\end{bmatrix}", "540a5ce84069dc2ca36a5a358f387547": "2n + o(n)", "540a6f6aba7b53cb385b8d81009305dd": "d_I\\,", "540a8efcfc3f2e7c21ff003cfa4bdaa3": "\nV_\\text{P} = E_\\text{P} = N_\\text{P} \\frac{\\mathrm{d}\\Phi}{\\mathrm{d}t}.\n", "540aac94a76438b19671717f24cf2e87": " D_\\text{wave} = - \\frac {1}{4 \\pi} \\rho U^2 \\int_0^\\ell \\int_0^\\ell S''(x_1) S''(x_2) \\ln |x_1-x_2| \\mathrm{d}x_1 \\mathrm{d}x_2 ", "540ab4b76c02bf410325f1e767e10752": "L_n:\\mathbb{K}^{n+1} \\to \\Pi_n", "540ab710c0ffff21abcbb7960f037569": "\n\\begin{align}\nF_{X,Y}(x,y)&=\\sum\\limits_{t\\le y}\\int_{s=-\\infty}^x f_{X,Y}(s,t)\\;ds\n\\end{align}\n", "540b1ccdb513be935c8dc6578792c2c1": "\\Gamma \\subseteq \\mathbb{R}", "540b6fe377ab6bc53fd0d01f58cacd67": "\\mathbf{Q}^{*T} \\mathbf{q} ", "540b7ef1fe7cb1aabc2c31581beae989": " (\\lambda x.f\\ (x\\ x))\\ (\\lambda q.\\operatorname{de-let}[f]\\ (\\operatorname{de-let}[q]\\ \\operatorname{de-let}[q])) ", "540bdf656419f2a308d7ad1571c664a0": "W_i", "540c379fa4051c9adbff8b1ed10569fe": "\\forall i\\in\\mathbb{N}\\ \\exists j\\in\\mathbb{N}\\ f(n,p^i) = p^j", "540caa2c3c2d6b31e37397957076edbc": "\n j\\;c\n", "540cd4c7fa255574158e9a20dfe3be14": " A = a_0 + a_1 \\mathbf{e}_1 + a_2 \\mathbf{e}_2 + a_3 \\mathbf{e}_3 + a_4 \\mathbf{e}_2 \\mathbf{e}_3 + a_5 \\mathbf{e}_3 \\mathbf{e}_1 + a_6 \\mathbf{e}_1 \\mathbf{e}_2 + a_7 \\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3.\\!", "540cdbf9912b07351b67d9fea1dd40aa": "\\Delta E_{CB}^{Tot}=\\widehat{H}_{int}=Z_{DP}\\cdot \\bigtriangledown u(r,t)\\sqrt{N_{q}+\\frac{1}{2}\\pm \\frac{1}{2}}=\\pm i q Z_{DP}\\cdot \\bigtriangledown u(r,t)\\sqrt{N_{q}+\\frac{1}{2}\\pm \\frac{1}{2}} \\; \\; (13)", "540cf6d5cdb3f53f9ee78e7ecaf291ed": "2^4\\cdot 3", "540cf95a62fd6e25fd23b0fdfba5dfa6": "\\int x^3 \\cos x \\, dx.\\!", "540cfa6364cb6f7acacfd371730fd202": " \\omega_\\mathrm{n}^2 = \\frac{k}{m} \\, ", "540d2705ecc6a841c1a2f76539e1c71f": "\n\\mathrm{EVM (dB)} = 10\\log_{10} \\left ( {P_\\mathrm{error} \\over P_\\mathrm{reference}} \\right )\n", "540d30374f12b30b4ff9cadea9e02e20": "q_1^*", "540d9afadf241bbffb2458a59021db97": "\\sigma_1\\Big.", "540da2341d237e5d9a2704fbf5d9ecfe": " d \\left( \\sin{\\alpha} + \\sin{\\beta} \\right) = m \\lambda", "540dd520569b724ee8847ef17740e4e9": "\\begin{align}\n t - t_0 &= \\operatorname{arctanh} \\left(\n \\frac{1}{E}\\left[s \\left(P^2 + Q^2\\right) - \\sqrt{E^2 - \\left(P^2 + Q^2\\right) x_0^2}\\right]\n \\right) +\\\\\n & \\quad\\quad \\operatorname{arctanh} \\left(\n \\frac{1}{E}\\sqrt{E^2 - (P^2+Q^2) x_0^2}\n \\right)\\\\\n x &= \\sqrt{ x_0^2 + 2s \\sqrt{E^2 - (P^2+Q^2) x_0^2} - s^2 (P^2 + Q^2) }\\\\\n y - y_0 &= Ps;\\;\\; z - z_0 = Qs\n\\end{align}", "540e02857158dfaf3a0684a454fcf957": "x_i=a_i", "540e72a1ba25c1974983015c07bf8761": "a\\sqrt {3}", "540eb4e0ed313f5aeb405e63720a83f8": "q^\\prime", "540ed6e0f46da8d0d0f9d40c155c831c": "E_L", "540ef1afe662a552672aa5a6d57e58b4": "\n\\begin{align}\nV_j &= 2\\sum_{m_j=1}^{\\infty} \\left( A_{m_j}^2+B_{m_j}^2 \\right) \\\\\n&\\approx 2\\sum_{m_j=1}^{\\infty} \\left( \\hat{A}_{m_j}^2+\\hat{B}_{m_j}^2 \\right) \\\\\n&\\approx 2\\sum_{m_j=1}^{2} \\left( \\hat{A}_{m_j}^2+\\hat{B}_{m_j}^2 \\right) \\\\\n&= 2\\left( \\hat{A}_{m_j=2}^2 + \\hat{B}_{m_j=1}^2 \\right)\n\\end{align}", "540f020c6da9ccb680408eb2f1b88042": "f \\colon [a, b] \\to \\mathbb{R}", "540f1870626340a0621cef07fb7b78c1": "\\frac{a_{rel}}{a_0} =\\sqrt{1-(v_e/c)^2} ", "540fc76411257b96f9aa0b3d559d8c69": "P(Q-Q')=r'-r", "54102f232b28ef74c94f6e458f9024ee": "10\\!+\\!18\\,=\\,28", "54104f2117737dab440b54983964d95c": "\\beta_{T}= - \\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}\\ = - \\frac{1}{V}\\left( - \\frac{V}{P}\\right)", "541097ef5bd4b3b2a2e4bb9ad6c30682": "V_{\\text{initial}}", "5410c382aede03d42f1ea0a5fb248653": "C \\cong B/f(A).", "54112ad8b52d1ca988e7608d4b7ee66c": "\\Delta H = -n_0 F_0 \\left( \\mathcal{E} - T \\frac {d\\mathcal{E}}{dT}\\right) \\ , ", "54112afb426640fbe9f7dd558a3df603": "K_I = Y \\sigma \\sqrt{\\pi a}\\,", "541195504ccf7b0970266748f80a6e3a": "\\partial X=\\{[\\gamma]|\\gamma", "5411a22a80231715e52c5f7e3ae41296": "h_O \\; ", "5411a805ae433633661a7385763847de": "\\mathrm{MA}= \\frac{ D D_{\\mathrm{EO} }}{f_O f_E} = \\frac{D}{f_E} \\times \\frac{ D_{\\mathrm{EO}}}{f_O}", "5411a9b662a7c5fcb0e8276e1c278f63": "\\|u\\|^2 = \\langle u,u \\rangle = \\left\\langle \\int_a^b v(t) dt,u \\right\\rangle = \\int_a^b \\langle v(t),u \\rangle \\,dt \\leq \\int_a^b \\| v(t) \\|\\cdot \\|u \\|\\,dt = \\|u\\| \\int_a^b \\|v(t)\\|\\,dt,", "5411b4c88e2ab01ca925e521f033ff5c": "a_n = b_n = \\frac{(-1)^n}{\\sqrt{n+1}}\\,,", "5411c220ca80a57e514caaa5ce5386cc": "\\ H_n", "5411d0c2200b38623ba2fd9e2d32be20": "\\mu_2=k\\,", "5411e18d38afd3ff0aff64f3211e6983": "\\hat{h}=\\cos i\\ \\hat{b}\\ +\\ \\sin i\\ \\hat{n}\\,", "541208eb3f818af8e7efb64967c565da": "CTDI_{100}=\\frac{1}{nT}\\int_{-50 mm}^{50 mm}{D_a(z)dz}", "54121d6fbcb150ef153c8e90da9b4192": "B\\,=\\,(b_1,b_2,\\dots,b_n)", "541228d0c342b6c02cff8c4fdd900992": "\\scriptstyle P(s_{t+1}|s_t, a_t)", "54126cbc970e3c1f2c9e96a2ec7b2bb3": "\\displaystyle u_t+u_x+uu_x-u_{xxt}=0", "5412efaa2efc5130c69f446a3b8bda3e": "\n\\operatorname{Li}_2 (z) = -\\int_0^z{\\ln (1-t) \\over t} \\,\\mathrm{d}t = -\\int_0^1{\\ln (1-zt) \\over t} \\,\\mathrm{d}t.\n", "541300d5a57a52ca4afada8a94a10e36": "d\\omega=0", "54133795d1a6e9c8f0e61fc98cccb93f": "f(\\phi/c^2)=\\exp(-\\phi/c^2)\\,", "54133e9fa2b08861e320167d4db8367c": "MSE^o=E\\left(||\\hat{x}^o-x||^2\\right) = Tr(G(x)C_wG(x)^*) + x^*(I-G(x)H)^*(I-G(x)H)x.", "5413586f5fd8c810a579de38be6f73f5": " Det(\\partial_\\mu D_\\mu) \\,", "5413652b68876ea20af48d6830ce0a75": "\n\\vec{F}_{ab}= \\frac {3 \\mu_0} {4 \\pi |r|^4} [ (\\hat r \\times \\vec{m}_a) \\times \\vec{m}_b + (\\hat r \\times \\vec{m}_b) \\times \\vec{m}_a - 2 \\hat r(\\vec{m}_a \\cdot \\vec{m}_b) + 5 \\hat r ((\\hat r \\times \\vec{m}_a) \\cdot (\\hat r \\times \\vec{m}_b)) ]\n\n", "541392507e7ad53239a11d55f22a8b23": "2^{-L}\\approx 10^{-30}", "5413a856f0f8567a83ccbf75ebc2bdea": " \\forall g \\in G: \\ \\ \\sum_{i=1}^n g(a_i) b_i = \\delta_{g,1_G}1_A.", "5414004cbfc5b408623f3da64c6d75c2": " \\mathbf F (T) := \\inf \\{\\mathbf M(T - \\partial A) + \\mathbf M(A) \\colon A\\in\\mathcal E_{m+1}\\}.", "54140f62de5c5210e66858285c58381a": "\n\\int_{1}^{v} \\frac{ \\ln t }{ 1 -t } \\mathrm{d}t = \\operatorname{Li}_2(1-v).\n", "541434859412ae28774c8e8e920e29b9": "\\begin{alignat}{2}\n dg_E & = \\quad \\ \\left(\\frac{Vf}{\\Lambda^3}\\right)\n\\frac{1}{2}~\\beta^3E^2~dE \\\\\n P_E~dE & = \\frac{1}{N}\\left(\\frac{Vf}{\\Lambda^3}\\right)\n\\frac{1}{2}~\\frac{\\beta^3E^2}{\\Phi(E)}~dE \\\\\n\\end{alignat}\n", "5414353a8672a437fb9d88e6127ac38a": " | \\vec a_t", "541451e83170fdeb734469b5fe07e04a": "\\frac{\\partial f}{\\partial t} + \\frac {\\operatorname d\\bar{q}}{\\operatorname dt} \\cdot \\frac{\\partial f}{\\partial \\bar{q}} + \\frac {\\operatorname d\\bar{p}}{\\operatorname dt} \\cdot \\frac{\\partial f}{\\partial \\bar{p}} = 0,", "5414a1192d70a2d04792eb6fbe1ce4eb": "\\mathrm{ROA} = \\frac{\\mbox{Net Income}}{\\mbox{Average Total Assets}}", "5414f96bcc008d56f4646501fe7ac173": "\\omega \\mapsto \\mathrm{dist} (x, \\mathcal{A} (\\omega))", "541500ed0e5a40378831b16368bd03ea": "-2\\pi", "5415664619ea21c48feae801f1842ce9": "x_{19}", "54158afb56cf17100f93e8f139ea0612": "c_{n-1} \\leftarrow c_{n-1} \\oplus b_{n-N+m-1}", "54158fef703ae3442f48edaccd194921": "\\alpha\\beta^{-1}=q\\,", "5415a29a657776ef80a59d56a0af63a5": "R_0(a, b)", "5415fb14bfc63d6d622340d3acd50e16": " \\sum_P f = \\sum_{i = 1}^n (u_i - u_{i-1}) f(t_i).", "5416157e60e96b2e50a7a99cc22af2c3": "\\sigma_d^2(k) = \\lambda\\sigma_d^2(k-1) + d^2(k)\\,\\!", "5416445ef3d7ebbcd4f12405c5c94327": " \\mathbf{E} = \\int_V d\\mathbf{E} = \\frac{1}{4\\pi\\varepsilon_0} \\int_V\\frac{\\rho}{r^2} \\mathbf{\\hat{r}}\\,\\mathrm{d}V = \\frac{1}{4\\pi\\varepsilon_0} \\int_V\\frac{\\rho}{r^3} \\mathbf{r}\\,\\mathrm{d}V \\,\\!", "54167e9ad2d54e49205a067bb54d6059": "\\text{Var}(X_n) = \\gamma_0 = 2 \\int_0^{\\infty} S_{xx}(\\omega) d\\omega.", "541685294f3669e7a53f46da9ca7df78": "\\begin{matrix} \\frac{1}{4} \\end{matrix} \\pi", "5416af39c6dc9f22d1ebc0fe051d07f3": " \\left [ \\begin{smallmatrix} 0 & 1 \\\\ -1 & 0 \\end{smallmatrix} \\right ] ", "541702ea538eaefe4d19552956ecd514": "\n\\begin{align}\n& {} = (1+r)^NP - p_Nc \\\\\n& {} = (1+r)^NP - \\frac{(1+r)^N-1}{(1+r)-1} c \\\\\n& {} = (1+r)^NP - \\frac{(1+r)^N-1}{r} c.\n\\end{align}\n", "541756407babcbc3e4c5869fc4d3a326": "A_{\\epsilon}^{(n)}", "541766a5f561455ca93a96e2a4177a24": "\\operatorname E\\left [ a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right ]=a_{i0}+\\sum_{j=1}^{m}a_{ij}\\operatorname E(X_{ij})-\\operatorname E(m(\\vartheta))=a_{i0}-\\left (\\sum_{j=1}^{m}a_{ij}-1 \\right )\\mu", "54176d5f31486191fd3312b52a97fd12": " \\sigma_2 \\otimes \\sigma_1 \\otimes \\sigma_0 ", "5417804f1fed622cd6293b9265752f50": "f' \\ne 0 \\!", "5417c427f3676e94a77002b9efad5e54": "\\int_{\\tau_1}^{\\tau_2} \\mathbf{F}_\\mathrm{rad} \\cdot \\mathbf{v} dt = - \\frac{\\mu_0 q^2}{6 \\pi c} \\frac{d \\mathbf{v}}{dt} \\cdot \\mathbf{v} \\bigg|_{\\tau_1}^{\\tau_2} + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2}{6 \\pi c} \\frac{d^2 \\mathbf{v}}{dt^2} \\cdot \\mathbf{v} dt = -0 + \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2}{6 \\pi c} \\mathbf{\\dot{a}} \\cdot \\mathbf{v} dt", "5417efc9960572ce790198d8159580ec": " J = 1 - \\langle \\langle \\frac{N_\\text{receive}}{N_\\text{send}} \\rangle_h \\rangle_\\text{network} = R(0)", "54180b3fc594bb668324c529459d3528": "\n\\varphi\\left(e^{-\\pi} \\right) = \\frac{\\sqrt[4]{\\pi}}{\\Gamma(\\frac{3}{4})}\n", "54183fa19ac6806596adb46246e08503": "(\\hat{x}, \\hat{y}; t) = \\operatorname{argmaxlocal}_{(x, y)} M_c(x, y; t, \\gamma^2 t)", "5418475be288ccf0c770410bca76e58a": "y_i \\succ_i^p x", "54187dd7a126f08a1bdf497352c81d71": "Bxyz \\leftrightarrow \\forall u ( ( ux \\le xy \\and uz \\le zy ) \\rightarrow u = y ).", "541896b625c97c2d4f4470c4b4f648bb": "(\\mathbb{R}, \\mathbb{N}, G)", "5418c0f7ca5ffe3cc5e4849db99aa608": "3\\tfrac{75}{100}", "5418ca81ce249a26165b151f57b25adb": "P(u,v,w) = \\sum\\limits_{h k l} \\left|F_{h k l}\\right|^2 \\;e^{-2\\pi i(hu + kv + lw)}.", "5418caa9999cf72b0211d6a554c6f4b3": " \\textbf{V}_P=\\dot{\\textbf{P}}(t) = \\dot{\\textbf{d}}(t)=\\textbf{V}_O,\\quad \\textbf{A}_P=\\ddot{\\textbf{P}}(t) = \\ddot{\\textbf{d}}(t) = \\textbf{A}_O,", "5418fa3ea31f2cc27bb86b5c260fb987": "a,b,c,\\dots\\,", "54190b70c0a943e10c324f2809267eb7": "\\Lambda=\\frac{\\pi}{\\Delta k}", "54194858ca9885fd4bb76d866b4ff0f9": "P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)} = \\frac{P(B|A)P(A)}{P(B)} \\,", "541966c0dbd7290fb85e741567b1d8f2": "2\\pi\\sin \\vartheta d\\vartheta", "54199d6c8473a3aefcb08bcb8b25df59": "L = A - A \\cap B", "5419a44e2ef134e17f0c7e57c06f90cc": "PV \\int f(x)\\,\\mathrm{d}x,\\quad \\int_L^* f(z)\\, \\mathrm{d}z,\\quad -\\!\\!\\!\\!\\!\\!\\int f(x)\\,\\mathrm{d}x,", "5419b66136147f08a75fb354e4a30996": " \\begin{align}\\hat{H} & = \\frac{\\hat{\\mathbf{p}}\\cdot\\hat{\\mathbf{p}}}{2m} + V(\\mathbf{r},t) \\\\\n& = -\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r},t) \\\\\n\\end{align}", "5419b6d589b18bd9369e8d3401c318bc": "\\int x\\,\\operatorname{arcosh}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{arcosh}(a\\,x)}{2}-\n \\frac{\\operatorname{arcosh}(a\\,x)}{4\\,a^2}-\n \\frac{x\\,\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}}{4\\,a}+C", "5419c1b102c934d35d209e1cb65191d7": " (A \\times I) \\cup (X \\times \\{0\\})", "541bd5cc0d5395eedbf3ba15fda32d01": "p_n = 1-\\frac{1}{1!}+\\frac{1}{2!}-\\frac{1}{3!}+\\cdots+\\frac{(-1)^n}{n!} = \\sum_{k = 0}^n \\frac{(-1)^k}{k!}.", "541c0cb4b7f99870582607c3061f6403": "=\\lambda\\mathbf{P}(n)\\,\\mathbf{r}_{dx}(n-1)+d(n)\\mathbf{P}(n)\\,\\mathbf{x}(n)", "541c50ea94710e52189a46655d2b5323": "1/\\tau", "541c8a2a7278e2cc698e993ebdb9e1ee": "\\mathbb{E} (e^{z X_\\alpha}) = E_\\alpha (Cz),", "541caf3b325f37838a11a2f978e1f908": "\\, x_i", "541cb7ae6140b0bb8a527326338fda79": "Z_3 = F\\cdot C-B\\cdot E ", "541ce4d0e121ec6d58abfac510805a01": "\\Theta_p=dz -\\frac{1}{2}\\left(xdy - ydx\\right).", "541d834ea60ca8adf2af76079e14cdb5": "\\Pi_4", "541d949f968c7765d5454babd1617622": "\\dot{s}_{i\\mbox{-}j}", "541da5cf4337b56c04fa08f79b19ea31": "\\epsilon, \\delta \\in [0,1], \\epsilon + \\delta = 1", "541dd8fa2722803c7c5a235165fe44d6": "X\\times 1 \\simeq 1\\times X \\simeq X", "541e393bd06b1d541ff34b177e839ae8": "H_* \\left(M \\times M, (M \\times M) \\setminus V\\right) \\to H_* (\\nu M, \\partial \\nu M) ", "541e46ce42ab7ada996c2e42919e3328": "\\Phi_{j}^{i} = \\frac{1}{2} \\frac{\\mathrm{d}}{\\mathrm{d} t} \\frac{\\partial f^{i}}{\\partial v^{j}} - \\frac{\\partial f^{i}}{\\partial u^{j}} - \\frac{1}{4} \\frac{\\partial f^{i}}{\\partial v^{k}} \\frac{\\partial f^{k}}{\\partial v^{j}}.", "541eb3aeb8f5a85c3d15183129f95656": "\\mathbf{w} = \\mathbf{G}^+ \\mathbf{b}", "541ebf52a1beef05bdcbae3f5a31d5f2": "\\int_0^{+\\infty} e^{-x^2} dx = \\sqrt{\\pi} /2", "541ed9220e888a970e83ed00ed47c999": "\\text{NOT }x = \\sum_{n=0}^{b}2^n\\left[\\left(\\left\\lfloor\\frac{x}{2^n}\\right\\rfloor \\bmod 2 + 1\\right) \\bmod 2\\right]", "541edf1b9bb4c61ccaf5e4f292100b82": "(g^a,g^b)", "541eeda287b15ded8056be9ae5d9756c": "\n\\frac{x^{2}}{a^{2} + \\lambda} + \\frac{y^{2}}{b^{2} + \\lambda} + \\frac{z^{2}}{c^{2} + \\lambda} = 1,\n", "541f0af35e7b8a3c67fe1e3f145fe272": " x=v\\cos u,\\quad y=v\\sin u,\\quad z=\\sin 2u. ", "541f4b203a0d388c00e608148b97983f": "\\rho_t(-X) = \\operatorname*{ess\\sup}_{Q \\in EMM} \\mathbb{E}^Q[X | \\mathcal{F}_t]", "541f4dbb0147fc1fbde3cffcefd9cf4c": "\\Delta (a f + b g) = a \\,\\Delta f + b \\,\\Delta g", "541f4e386a172a445c001210774c20db": "\\mathrm{ord}(\\alpha),", "541f9100a14029b74dda9ba4b833c80b": "k_p=\\omega_p/c", "541f918fd1f609364e05fc9d2c9a9f60": "\\operatorname{Res}(a,b)", "541f9bce2e30c0a0126612ca1a2d2eed": "\\Pr(0\\text{ heads}) = f(0) = \\Pr(X = 0) = {6\\choose 0}0.3^0 (1-0.3)^{6-0} \\approx 0.1176 ", "541fa2bafb1f1b5de29f1c7503744bcf": "\\textstyle a_n = x^n/n!\\,", "541fb089927096736dec2ba0ebfad202": "\\{0,\\pm1\\}", "541fd16682b68ff6711f209728edacd3": "\\ell_\\text{P} =\\sqrt\\frac{\\hbar G}{c^3} \\approx 1.616\\;199 (97) \\times 10^{-35} \\mbox{ m}", "542020e6631afaa13bee561fa19403cc": "\\rho_{S}\\,", "54204c97100b928733bd791ef7f00709": "Q_\\mathrm{absorbed} = \\alpha Q_\\mathrm{solar} ", "5420f818f7c5370d35667cb8e1f47021": "f(\\gamma X)", "5421058ad288fe36ebfbc442496305a4": "3 \\cdot 10^{-10}", "542148829d5e79ae92f83b2cf90e7c46": "{Z_0}", "5421578214ccbca3ffe66a6bf70e8c11": "\\scriptstyle v_j(U)\\subseteq v_k(U)\\!", "54219a0807574e68c17412d6da61363a": "\\alpha > \\gamma \\,", "5421ac18aa3090da59d59eb6deaa87ce": "m\\Gamma(m)=\\Gamma(m+1)", "5421c0ef7c0adf67d17d1f6e68dff1a5": "\\ln\\frac{p}{1-p}", "5421c203e94aaa7581792f744628213b": "|SB_{0}|=\\frac{|SD||SA|}{|SC|}", "5421ec5ffb7b2928f8120d66253e52f8": "\\hat{H}=1+\\frac{1}{2}(slope).", "5421ee6a306a18b6bd7c53e2f9fc499d": "w_a = \\frac{2 \\sqrt{bcs(s-a)}}{b+c} = \\sqrt{bc\\left[1- \\frac{a^{2}}{(b+c)^{2}}\\right]}", "54221ffbb91021745f0dca769cbbb679": "{h}", "5422d419f3c410167b5e5a521ab17d01": "R_{out} = \\frac {V_X} {I_X} = r_O \\left( 1+ \\beta \\frac{R_E} {R_E+r_{\\pi}/(A_v+1)} \\right) +R_E\\|\\frac {r_{\\pi}} {A_v+1} \\ .", "5422e569f6245c0e58f2659143026941": " (\\nabla_X J)X =0 \\, ", "54238d79d16a713d4b79fd786267080e": "DM\\,", "5423becdb53b0d9d30312eb8406fe4f8": "\\scriptstyle a^2 \\,+\\, b^2 \\,-\\, qab \\,-\\, q \\;=\\; 0", "5423d344ff7a5edfcc86c5138e9df5c9": "\\{a_i\\}", "542403c904c5f2e47216e5d23d89eb9a": "x \\in_{NFU} y \\equiv_{def} j(x) \\in y \\wedge y \\in V_{j(\\alpha)+1}.", "54240b3ecb06d6f401b219742388c7f0": "\\forall n \\ge 0, x_{n+1} = (2 \\cdot x_n) \\mod 1", "54242e200a1285fb4f67c9c04ea00f88": "\\mathcal{I}{{\\left( \\theta \\right)}_{m,n}}=\\frac{1}{2}\\operatorname{tr}\\left( {{\\Sigma }^{-1}}\\frac{\\partial \\Sigma }{\\partial {{\\theta }_{m}}}{{\\Sigma }^{-1}}\\frac{\\partial \\Sigma }{\\partial {{\\theta }_{n}}} \\right)", "542436b8a74947d05084468e84c8b7b7": "\\mathbf{v}=(a,b,c)", "54248b020023fa0dd84f5c73820dca50": "\\mathbb{E}\\Big(\\exp(\\theta X_t)\\Big) = (1-\\theta/\\lambda)^{-\\gamma t},\\ \\quad \\theta<\\lambda", "5424914008be9914de50a40c2f91b79e": "r_i(\\sigma_{-i})", "5424a262018c7954550f6e6fd7781ef3": "\\vec{n}_{AB}=(\\vec{x}_A-\\vec{x}_B)/r_{AB}", "5424a6869ecfff64942abd6ba68e8e34": "\n\\det{\\frac{\\partial(x, y)}{\\partial(r, \\theta)}} =\nr\n", "5424b2e4af326802aad461d3ffbbe5ed": "F_{adhesion}", "5424db09a7bc53fb24a20b14df48c262": "f_X(x)= \\begin{cases} \\frac{\\alpha x_\\mathrm{m}^\\alpha}{x^{\\alpha+1}} & x \\ge x_\\mathrm{m}, \\\\ 0 & x < x_\\mathrm{m}. \\end{cases} ", "5424e3acab46c3d6f43f9f1dfff8db00": "\\Phi (x) = \\frac{1}{2}+ \\frac{1}{2} \\operatorname{erf} \\left(x/ \\sqrt{2}\\right).", "54252862da38ea9696a9dbc0edf67f6a": "\\begin{matrix} {9 \\choose 1}{4 \\choose 4}{44 \\choose 1} \\end{matrix}", "54254c912e7bd331bc629874839bc654": " C^T_{(+)}=C_{(+)};~~~C^2_{(+)}=1 ", "54257f4c59166cb430bfcbff880d076b": "\\begin{align}\n\\sigma_{0,j}^2\n&= \\frac{1}{j}(\\sigma_{0,1}^2 + \\sigma_{1,2}^2 + \\cdots + \\sigma_{j-1,j}^2)\\\\\n&= \\frac{j-1}{j}\\cdot\\frac{1}{j-1}(\\sigma_{0,1}^2 + \\sigma_{1,2}^2 + \\cdots + \\sigma_{j-2,j-1}^2) + \\frac{1}{j}\\sigma_{j-1,j}^2\\\\\n&= \\frac{j-1}{j}\\,\\sigma_{0,j-1}^2 + \\frac{1}{j}\\sigma_{j-1,j}^2 \\\\\n\\Rightarrow \\sigma_{j-1,j}\n&=\\sqrt{j\\sigma_{0,j}^2-(j-1)\\sigma_{0,j-1}^2}\n\\end{align}\n", "5425b53625a15246febd29887c9b7529": "\n\\mu_{opt}=\\frac{E\\left[\\left|y(n)-\\hat{y}(n)\\right|^2\\right]}{E\\left[|e(n)|^2\\right]}\n", "5425e5211cf30c9a5f8bf19927691cdb": "\\sum_{1 \\le k \\le n} f(\\gamma(t_k)) ( \\gamma(t_k) - \\gamma(t_{k-1}) )", "5426107e8bad7d0ca3239da451ab6baf": "S_{YM} = \\int_M \\left|F\\right|^2 d\\mathrm{vol}_M,", "5426122b8964e67c70c450fbbc275276": "(n-2)/n", "54267141933ce28dc6809de21ed7a9b6": "\\langle \\delta_\\epsilon \\mathcal{F}\\rangle - i \\int \\epsilon \\langle \\mathcal{F} \\partial_\\mu J^\\mu \\rangle \\mathrm{d}^dx = 0", "5426ee70cd4724912ef739d67e4ea0d1": "H_{k}+\\frac {(\\Delta x_k-H_k y_k) (\\Delta x_k-H_k y_k)^T}{(\\Delta x_k-H_k y_k)^T y_k}", "5426f119d6a8fdd869c9839e87d1be8b": "\\rm HCOOH + H_2O_2 \\leftrightharpoons HCO_2OH + H_2O", "5426f4abac92b141e07b42fb6f57a3d7": "a \\rightarrow b", "5426fe60eee44ad1dc40812aed1e9451": "x\\rightarrow\\lambda x", "54270c5568e12cb70bb40a55c321a9b0": "p_\\mu=p_\\mu(\\eta_A,\\eta)", "54271c04e40ea92ff9b996c8c8c66503": " r = 4a\\cos^2 \\frac{\\theta}{2}\\,", "54271d2b7ebf60c2f40581a5de8759a3": "J(t)=\\partial\\gamma_\\tau(t)/\\partial \\tau|_{\\tau=0}.\\,", "542773d0ace2c2efa761a33a6ca39ae6": "a \\in [2,\\min(n-1,\\lfloor2(\\ln n)^2\\rfloor)]", "542782db95e548592086e23e6f517de9": "\n{\\mathrm{d} \\over \\mathrm{d}t}{\\partial{L}\\over \\partial{\\dot \\theta}} - {\\partial{L}\\over \\partial \\theta} = 0\n", "5427be9cf9ed8b8252bfe2aa4e645979": "\\chi_1\\left(z\\right)=\\sinh\\left(\\alpha z\\right),\\qquad \\chi_2\\left(z\\right)=\\cosh\\left(\\alpha z\\right)", "54281756f2cafcd3d1844b826414c39f": "\\langle\\Psi|\\frac{d}{dt}A(t)|\\Psi\\rangle = \\langle\\Psi|\\frac{\\partial A(t)}{\\partial t}|\\Psi\\rangle + \\langle\\Psi|\\frac{1}{i \\hbar}[A(t),H)]|\\Psi\\rangle,", "5428894d1f78dde7344b0540a0a8d7a1": "S(e^x)", "5429012bfd957995ecf7060540a6871c": " F= \\circ + \\frac{1}{1!}\\cdot \\circ \\times F\n+ \\frac{1}{2!}\\cdot \\circ \\times F* F\n+ \\frac{1}{3!}\\cdot \\circ \\times F* F* F * \\cdots\n= \\circ\\times\\exp(F),", "54291395e4b46c171855f17828dbae40": "y\\le x_j", "54291d4e37d28502fc32725240d6075e": "xI \\subset R.", "542960cc709af119cc5301a3f5c0ae64": " (X^{(0)},X^{(1)},X^{(2)},...)", "542a1e0bbdf4f0e6d7613540c3bc8fe3": "\\sum_{i=1}^{n}\\frac{\\alpha}{n}=\\alpha", "542ace34f8e0686514df58edff80a735": "\\vec{\\tau} = \\vec{r} \\times \\vec{F}", "542afea0e7e5ab0b3a6ebc723d92bba1": "W=-\\int_{V_i}^{V_f} P\\,dV", "542b1d2c4cd761696ad1597f8f70bd54": "\\begin{align}\n\\mathbf A &= [A_1,\\dots,A_n] = \\sum_i A_i\\mathbf e_i\\\\\n\\mathbf B &= [B_1,\\dots,B_n] = \\sum_i B_i\\mathbf e_i.\n\\end{align}\n", "542b6f5f614dbe8ed8c1688e4b925067": "\\binom{B}{A}", "542b93904a5041385900738e73443f6d": " G= \\cap_{i\\in \\alpha} G_i", "542bf67dcedeb911cac735e23c42421e": "\nR_{\\mu\\nu}=\\frac{-(n-1)}{\\alpha^2}g_{\\mu\\nu} \n", "542bfc69a25bd376f81c931a09fe87d0": "\\mu_{\\xi,\\eta}=d\\rho_{\\xi,\\eta}", "542c0b6eb2ba9fb347ceb99af069ed13": "\\scriptstyle|\\phi(x) - \\phi(y)|\\leq \\sigma(|x-y|)", "542c1a73039fc1a240544337b49269b9": "\n\\ U(h) = U(0) h ^ \\zeta\\,\n", "542c7fb5404d093fdb2d46cf5784eabe": "\\vert g,0 \\rangle", "542c9b6306d7b4a0820ace410845aad4": "\\varphi_{\\mathrm{in}}", "542cba0c1e6777fcd337b06b69c04b02": " n > 1", "542cf561b0b881918e0412503b30ed2d": "-t < \\operatorname{Perf}_s(f,r') - \\operatorname{Perf}_s(f,r) < t", "542cfac8c30427be6d486d92b8c9711d": "P(A\\mid N=n) \\, ", "542d3aa2ceb092e6c1ab0ecb0e5b693e": " x_i < x_j ", "542d725c7e8dad2e2ddfc55e2fb99b05": "\\frac{v_1^2}{2g}+y_1 = \\frac{v_2^2}{2g}+y_2", "542d9e9c603af88172f408c728513f37": "x >> 1", "542dc4bf1d588474fcbc41f22c160800": "\\sigma_v = \\rho g H = \\gamma H", "542dd77c0edf71afe68cc9c513371f6a": "G(\\alpha', \\beta')", "542df7e30698039c8df88bd8fee2b91e": "(x+iy)(x-iy)=0", "542dfe2d8ceb42a8cfb0860d7b2a6102": "\n\\frac{\\partial v}{\\partial t} + u\\frac{\\partial v}{\\partial x} + v\\frac{\\partial v}{\\partial y} =\n-\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y} + \\nu \\left( \\frac{\\partial^2 v}{\\partial x^2} +\n\\frac{\\partial^2 v}{\\partial y^2} \\right).\n", "542e0440622ea4ee58a1d879f455d421": "\\pi(n) \\approx \\frac n {\\ln n},", "542e7fef6e4cecb0ddd0a09693c3a75f": "\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}", "542e995b6fcd37d873247f6433b69c80": "\\textstyle \\sum_{v_i \\in F_a}", "542ecabcbf8315b5452e5c4507050ab4": "(x+Y) \\cap (1+\\delta) B_{X}", "542ef17814a8245cebd965747b7c7c14": "u_k = -F x_k \\,", "542f458cce0b9353ae62b0ba5447281c": "{{|\\overrightarrow{V_a}| \\over |\\overrightarrow{V_g}|}}", "542f5a352badf06b56e8016884f7dbd8": "\\frac{1}{{\\Delta}A}=\\frac{1}{b{\\Delta}\\epsilon[G]_0[H]_0K_a} +\\frac{1}{b{\\Delta}\\epsilon[H]_0}", "54304d13e92bc5ab9e3ee9a4490de5d4": "\\mathrm{j}_y", "5430512cd8a040399fef87cff0e7f49e": "N=1 \\cdot N", "54308497697265e53ed0bc8fa679b643": "\\omega_1\\to(\\alpha)^2_n", "5430d81643f8bfaa02e4608c8e590463": "\\Delta u_{it}", "543110a3213269076b94b5d94d06e510": "\\bold{v}_1 \\times \\bold{v}_2", "5431591a8ccd16f014a9fcf2490e3de9": "\\quad \\varphi_0 + \\sum_{k=1}^N \\sum_{i_1 \\neq \\cdots \\neq i_k } \\varphi_k(x_{i_1} \\ldots x_{i_k})\\ge 0 \\text{ for all }k,(x_i)_{i = 1}^k ", "5431901b9e0b26ec38baa3c642997888": "x^*(A \\circ B)y = \\mathrm{tr}(D_x^* A D_y B^T)", "5431c75ec51f0bba67291b76a3bc7697": " \\alpha =1 ", "5431cea5c82b423ab0fd9e75191d93d0": " V = \\rho_{e} / \\rho ", "54321d5f0c20fa1776cefe4536fab1ff": "\\frac{1}{T} \\int_0^T Z(t)^2 dt \\sim \\log T", "5432673744b36ad390610e0d2f1f8291": "\\lfloor \\log_2U\\rfloor+1", "5432c4c1a0a95989a8d6bb378e144d7c": "\\mathbf{k}_o - \\mathbf{k}_i = \\mathbf{\\Delta k}", "5433226ad8683f9dc51ed092260917a5": "c_3 = {1 \\over 3} c_0 + {2 \\over 3} c_1", "5433293f30d973c8ed795f08e28984e5": "x_1^q,\\ldots,x_n^q", "54332b38df35e486c6de00cb979b1aa3": "\\lambda_1 \\ge \\lambda_2 \\ge \\cdots \\ge \\lambda_{n}", "5433445aac54530a7c4e3eaadacda3c7": " \\omega_0 = { 1 \\over \\sqrt{LC}} ", "54334d941935187267bcf924006136fc": "S=\\{x^2\\mid x\\in\\Bbb F_p\\}", "543351c9b490030a094ddbe89be24868": "b_6=a_3^2+4a_6", "5433520f8c6d21b842a7f98ce1dcb519": "{\\rm ?}(x) = a_0 + 2 \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{2^{a_1 + \\cdots + a_n}}", "54336613d7ec0dc8605e140650d26237": "14+\t51+\t62+\t3+\t7+\t58+\t55+\t10\t=\t\t260", "543399a0968a64d921fedd8e5c7f2a9a": "f(n,m) = O(n^m) \\text{ as } n,m\\to\\infty\\,", "5433b8788cb9f40e635603e968ff1bbc": "V_\\mathrm{out} = V_\\mathrm{in} \\cdot\\frac {Z_2}{Z_1+Z_2}", "5433d2131a0ad223434332296feeef54": "\\frac{1}{2}(W,W) = i\\hbar\\Delta(W) . ", "543417dc545820af9037c93f0e8efa24": "(\\mathbf{a \\times b})\\mathbf {\\cdot}(\\mathbf{c}\\times \\mathbf{d}) =\\begin{vmatrix} \\mathbf{a\\cdot c} & \\mathbf{a\\cdot d} \\\\\n \\mathbf{b\\cdot c} & \\mathbf{b\\cdot d} \\end{vmatrix} \\ . ", "54345490ad3e4aba7abe7825fc1005a9": "(t_k)", "5434c75bf56c6a4a4952f1169c02d9de": " 1/2 = 1 - 1 + 1 - 1 + 1 - ...", "54352c2662b63c49c6349bbed5dc5ccc": "\\sqrt{\\frac{\\alpha_\\text{G}}{\\alpha}} \\,", "543544e228e19fd48f6555790794807e": "Z^{(\\ell)}_{\\mathbf{x}}({\\mathbf{y}})", "5435a16c6f2be5158c46cc68879b0247": "\\operatorname{Cl}_{2m+1}(\\theta) = \\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m+1}}", "5435ce350e2d3faa064f8ac969c9ab1f": "\\bar{V}\\,", "5435d13ca70a4511aee3c0712a7ad32a": "\\int_a^b k(s)\\,ds.", "54360015ae455f0e4241b655debb7247": "it\\,", "54360c7ec9d272af78f7b9bcad010b7a": "(R, +, \\cdot)", "543651de2bf1be228c03e63f49cf2e64": "\\!\\,\\phi", "5436740d835ec38c611cb15b3029fff4": "\\mathcal O(\\mathbf C^n)", "54368c628be89bf9888a1bead2883a85": "R(X) = Q_1(X)E_2(X) - Q_2(X)E_1(X)", "54368e8bf4aad91e5bdbce627c03049b": "D_\\alpha V_I", "5436ca104030a95e84b8842cae5c696a": "P= (X:Y:Z:Z^{2}) = (\\lambda^{2}X:\\lambda^{3}Y:\\lambda Z:\\lambda^{2}Z^{2}),", "5436fd3f474a5f35c684ff40dd257ec6": "k^2", "543752921733ab15a386dc87b8ac8cce": "\\partial_x\\partial_y f|_{(0,0)} =\n\\lim_{\\epsilon\\rightarrow 0} \\frac { \\partial_y f|_{(\\epsilon,0)}-\\partial_y f|_{(0,0)} } \\epsilon = 1.", "54376d20c1e918bc19ddddb1d08d6a8c": "P( \\mathcal{N}(\\mu,\\,\\sigma^2) )", "543787b6a2b837baf469e2b1784dcdac": "E_i = \\max\\{0, d_i-C_i\\}", "54378c9991aa61e210b778f8d087aaee": "\\scriptstyle(-2.1\\pm1.9)\\times10^{-10}", "5437963ae53679a02a67312796abf172": "i=0,1,\\ldots,n/m - 1\\text{ and }j=0,1,\\ldots,m-s", "5437dd1e462b54eb01242052f2e49318": "\\alpha t = \\pi/2", "543837bdd2c47bbd0166d1c08ec0aaf2": "\\lim_{x\\to x_0\\atop x\\in I}", "543841ea2daa52538c177e054f57e88a": "x_i - m'", "5438494c4717f542c5b2eedf8c60f4b7": "\\beta(n_i, \\tilde{n}_i)", "54384a31404656d0da82223bf8408228": " V\\,k_v(V) = k_{\\text{arb}} \\,\\!", "543879b4f31ee3ba3f0b8aeb729cc360": "L^2 (B)", "5438d8ab5b44df8b257f0cfff0647e6e": "\\log_b a = \\log_2 8 = 3>c", "5439159692c0a2a4b1c796c2ead6d082": "i,j=1\\dots 6", "543981615eb9bf90ce68196ba19cb4c5": "D:=\\,(\\eta,\\eta)_{K},", "5439fb59092e406e448e5c6447ab7e15": "B_\\theta = \\frac{B_0}{R^3}\\cos\\lambda", "5439ff8850597fb9b51e6bdd27702696": "r( \\text{out}, \\text{in})", "543a39cfc15b3b531257a2f377ff3dff": " 1/e_i ", "543a46f7a41761a85f606a4aa82e5f34": " m \\dot {\\mathbf{v} } = {1 \\over t_0} \\int_t^{\\infty} \\exp \\left( - {t'-t \\over t_0 }\\right ) \\, \\mathbf{F}_\\mathrm{ext}(t') \\, dt' .", "543ab605059f2bb7e7f57de592a019b2": "\\hat{S}(z) \\hat{D}(\\alpha) \\neq \\hat{D}(\\alpha) \\hat{S}(z),", "543b1b73a61252f1c86793a34d79a0de": " \\{B_j:j\\in J'\\}, \\quad J'\\subset J", "543c05a1dbe4d0161309810ec05f7b89": "\\displaystyle\\beta", "543c08a2ffefbd4f19a9e5dfe97790b2": " \\frac{\\Gamma(r+k)}{k!\\,\\Gamma(r)}\\,p^r\\,(1-p)^k \\,", "543c0c480a0a8141cf0ce84a454e9021": "\\mathbf{a}\\times\\mathbf{b} = (a_2b_3-a_3b_2)\\mathbf{i} + (a_3b_1-a_1b_3)\\mathbf{j} + (a_1b_2-a_2b_1)\\mathbf{k}", "543c13e7dd6c7d0fe58a49f572d74290": "{\\vec x_{i}(t)}", "543c581889d10f5b249cc41fa9c63667": "\\Gamma\\vdash \\sigma \\equiv \\tau", "543ca752eb327398a3295c8cd02c1ca0": "|z_1 + z_2| \\le |z_1| + |z_2|", "543d3a590685a951849f3cce5a819b6f": "\\vec z", "543d6140fb33ee43ecd35565d1f9c587": "X=X(s)", "543da453f56ec85dd9c12e6228c75df1": "A =\\ ", "543de15700c9c59815bf33ef5fc3cc10": "b_n =\n\\begin{cases} \n0 & \\text{if } m \\text{ is even} \\\\\nb_{(m-1)/2} & \\text{if } m \\text{ is odd}.\n\\end{cases}", "543de6080c54546c6f14a23428dafc0c": "T\\vdash\\phi", "543eb9c2b5a173dcae81d46f446d719d": "\\hat{x}_{n_j} \\to \\hat{x}", "543ee309d99c8ab7f4152e5f545975ec": "\\begin{align}\n\\textrm{Var}\\left(m^{\\star}\\right) & =\\textrm{Var}\\left(m\\right) - \\frac{\\left[\\textrm{Cov}\\left(m,t\\right)\\right]^2}{\\textrm{Var}\\left(t\\right)} \\\\\n& = \\left(1-\\rho_{m,t}^2\\right)\\textrm{Var}\\left(m\\right);\n\\end{align} ", "543f178109fd26253d7c5e483e569ce8": "\\Pr(R \\mid B \\cap Y) = \\Pr(R \\mid Y).\\,", "543f1e1ddd452d94a391bd74db149568": " \\alpha_{eff} = \\alpha \\cdot \\frac{\\rho}{\\rho-\\rho_w}", "543f3ab4972ddb005910d101f0d72320": "Y\\left(\\frac{a+b+c+d}{4},\\frac{e+f+g}{4}\\right).", "543f5c6df6af063cf7ca449510c5fc05": " \\mathcal{L} = \\sum_i \\tfrac{1}{2} m \\dot{x}_i^2 + \\sum_i e \\dot{x}_i A_i - e \\phi, ", "543f88e8bc17dbab36842e42b0d59e04": "S_0=\\{2\\}", "543fb8e2d85031e33fbad87ad899613e": "d^2G_\\Sigma=d^2G_S \\ ", "543fe48ea8638d75171002e71c4450c5": "y_i = x_i \\beta + z_i \\delta + u_i,\\qquad i = 1,\\dots,n", "54403373cbdcaf01653b4e59657c9f06": "\\{b_1,b_2,\\cdots b_k\\}", "544057442572d981390bc1e805ead381": "x_{i+}", "5440f735161e803ec7ce0b78d12bad2c": "log_2(M_e) \\le log_2(M_o)", "544176dd1c6b44401d2ab0c7dcff9956": " \\mu_A/\\mu_B = 0.28 ", "5441790baba9b8fec5080ed6eab5d48e": " d^3x ", "54418a03dac4a2f6ff8d6d3349f40431": "u(t) \\le \\alpha(t) + c\\int_a^t \\alpha(s)\\exp\\bigl(c(t-s)\\bigr)\\,\\mathrm{d}s,\\qquad t\\in I.", "5441cff4042ba6a030cd6f5a00037b85": "\\sigma M\\leq \\tau M", "544245fe8ccd02e447b78108a51f864d": "Z = XY", "54428b80397d1158a378eee71da9702c": "M_R = \\frac {m_1} {m_0}", "54428debf283424cb57e7a584bd2cb89": "\\ Z_{\\text{eq}} = \\frac{Z_1 Z_2}{Z_1 + Z_2}", "5442b66e9abbfafbe9a875901627330b": "f(x) = \\lambda \\exp\\left(-\\lambda x\\right)", "5443046ea67b943840392a0d9d7a678a": "\\vec x_i=(x_i,0,0)=(iD,0,0)", "54433d938b46700cc6e902262ded5846": " P(\\mathbf{X}, t) = P(\\Omega \\mathbf{\\phi} + \\Omega^{1/2} \\mathbf{\\xi}) = \\Pi (\\mathbf{\\xi}, t). ", "544386578bc9c5abf46623d45c83d129": "\\hat{ v}", "54439c342588ca6494c7e821337dda53": "e_2", "5443a5962c32b4995f15e901593f7e49": "S_R^\\delta f", "5443c00579e139b189180d4353a04127": "a_{24}+a_{21}+a_{34}+a_{31} = 34 ", "5443e5d0b1177a13131482af014962dc": "\\tilde{m}", "5443e6f3cc32ef507233bbded8829cea": "\n\\begin{align}\n Angular \\text{ } 2nd \\text{ } Moment &= \\sum_{i} \\sum_{j} p[i,j]^{2}\\\\\n Contrast &= \\sum_{n=0}^{Ng-1} n^{2} \\left \\{ \\sum_{i=1}^{Ng} \\sum_{j=1}^{Ng} p[i,j] \\right \\} \\text{, where } |i-j|=n\\\\\n Correlation &= \\frac{\\sum_{i=1}^{Ng} \\sum_{j=1}^{Ng}(ij)p[i,j] - \\mu_x \\mu_y}{\\sigma_x \\sigma_y} \\\\\n Entropy &= -\\sum_{i}\\sum_{j} p[i,j] log(p[i,j])\\\\\n\\end{align}\n", "5443f3e5409afeeb0760f235600ca999": "\n \\begin{matrix}\n {^b} \\bar a = & \\underbrace{a_{}^{a^{{}^{.\\,^{.\\,^{.\\,^a}}}}}} & \n\\\\ \n & b\\mbox{ copies of }a\n \\end{matrix}\n ", "5443f667f2e8086a3a5546f842479155": "u_{xx} + u_{yy} = 0, \\quad v_{xx} + v_{yy}=0. \\,", "5443fb186991bb31324daa2235f95b4a": "\\mbox{Doomsday} = \\mbox{Sunday} + y + \\left\\lfloor\\frac{y}{4}\\right\\rfloor = \\mbox{Sunday}+ 5\\times (y\\mod 4) + 3\\times (y\\mod 7)", "54441309fa73cc01efff993772978f0d": "x = (B_x - A_x)t + A_x\\text{ and }y = (B_y - A_y)t + A_y\\,", "54446addcbd8715668d5e1b83899d4b4": "V_{\\mathrm{int}}^{1,2,...N} = \\frac{1}{2}\\sum_{i=0}^\\N \\sum_{j=0(\\ne i)}^\\N V_{\\mathrm{int}}^{ij}(R_{ij})", "54447f8b72000be2d1408afbc4d2fa8b": "y_{2trans} = \\left ( \\frac{q_\\mathrm{trans}^2}{g} \\right )^\\frac{1}{3}", "5444bca99e9c159f48b0c9621e190fa9": "\\alpha \\leftarrow \\gamma\\cdot \\min\\{-v_i^k/(h_v)_i \\,\\,|\\,\\, (h_v)_i < 0,\\, i=1,\\ldots,m\\}", "5444ccf91d07774645cf78cf244ddf0f": " \\vec{s}(C_{-1}^{(3)}) = [-1,+1,-1,+1], ", "5444cdc1eb35c1ecdee8a2996ab74de4": "r=\\sqrt{x^2+y^2+z^2}", "5445024b79a846355ad273877d1e65fb": "\\rho_b = \\frac{M_s}{V_t}", "54460eb764f1e75d1bce6c45788bd371": "\nH^\\text{RWA}=H_0+H_1^{\\text{RWA}} = \\hbar\\omega_0|\\text{e}\\rangle\\langle\\text{e}|\n-\\hbar\\Omega e^{-i\\omega_Lt}|\\text{e}\\rangle\\langle\\text{g}|\n-\\hbar\\Omega^*e^{i\\omega_Lt}|\\text{g}\\rangle\\langle\\text{e}|.\n", "54462b637dec421949544080c46adcdc": "\n\\mathbf{M}_{\\rm orb}=\\frac{1}{V}\\sum_{j\\in V}\\mathbf{m}_{{\\rm orb},j} \\;.\n", "54463542e17dc6785a04f7ef2cd98f5c": "U(r)\\!", "544673ba87e7fbc90066cc278ce2fd40": "f_{-} (t) = \\lim_{s \\uparrow t} f(s).", "544675898085b42ac89eac5f1a20171c": "\\vec v(0)=\\dot {\\vec x}(0)=\\vec v_0", "54467e2e24f61c71b7e6aec747fa9b10": "\\phi(p^n) = p^{n-1} \\phi(p) = p^{n-1} (p - 1) = p^n - p^{n-1} = p^n \\left(1 - \\frac{1}{p}\\right),", "54468b616fffa9164c46c30dbbc00d29": " \\frac {1}{i\\omega C}", "54468f5a4ac8fd71e349250f610f069d": "E_{\\alpha'} + \\xi_\\mathbf{k}", "544761859044689f2589f7a47ff783d8": "P_{1,0} = (0, 1)", "5447e68623938d6ffabf94f479d032dc": "F: E\\to E", "5447f68a6bd575c7d490ace9c6f6d54c": "dP_{Br}/d\\omega", "5448115316079e7a00bfa8a2b5ab9d97": " \\operatorname{Var}(y_t) = \\sum_{j=1}^t \\sigma^2=t \\sigma^2 .", "54481748501d00d43742e7b8406335e1": "V = \\frac{5 + \\sqrt{5}}{24}\\,a^3 \\approx 0.3015\\,a^3.", "54484ddac5f0ab20b93ef1ddd71dd982": "x \\in \\omega^{\\omega}", "54488c75aef292b5a703ebf942f38a92": "= (-1) \\left ( \\frac{331}{3}\\right ) \\left ( \\frac{331}{5}\\right ) (-1) \\left ( \\frac{331}{7}\\right ) (-1) \\left ( \\frac{331}{23}\\right )", "54488e0d58bb9e194a8e62db6f7e8447": "\\cot^2x<\\frac{1}{x^2}<\\csc^2x", "5448c9a2880e78b4b50c7541a7996097": "\\textstyle \\mathcal{P}", "5448d1e75a733dd8f201e28c7bf965d7": "LCL=\\overline{x}-2.66\\overline{MR}", "5448d701c476541209ef780b80ed2d10": "\\rho_{ij}\\,", "54492bc54ed300df2f8e7a2d9a4dd58a": "F=\\frac{\\mu_0 H^2 A}{2} = \\frac{B^2 A}{2 \\mu_0}", "54493ad0a30763b8150cae25df8a983d": "y = r \\sin \\left ( \\theta + \\omega t \\right ) ", "5449d356912057d651ed765d84c42fa9": "\\textstyle\\rho = |i\\rang\\lang i|", "5449e44265567ad244e11f2e2cb962a3": "\\underline{\\lambda}^{(0)} = \\mathbf 0", "544a90155ee374b2679623e343d3aebd": "\nh_{LCL} = \\frac{T - T_d}{\\Gamma_d - \\Gamma_{dew}} = 125 (T - T_d)\n", "544aa0d7c6dabf96b0659eebfaa6ac98": " \\det g=\\alpha^2 ", "544ad2fa7f2d80d017f5869424e342d2": "\\dot{K}(t) = sY(t) - {\\delta}K(t)\\,", "544ad6f99e3eb432d642a3e559c119da": "\\forall u,v \\in M, N(u) \\setminus M = N(v) \\setminus M ", "544b2603d02c0a1e4a6cf43034512af5": "f(x)=g(x)+ih(x),", "544bf58fc04d4641b7c2df131d82f16b": "\\lambda \\, \\mu \\bigl( \\{ x : f^*(x) > \\lambda \\} \\bigr) \\le b_N \\, \\int |f| \\, d\\mu.", "544c1fe14083cb56d532cfdbb4bff0fb": "Ro = \\frac{U}{fL}.", "544c42c8036087f7d2af5ba0e874ba3e": "\\Omega_{\\mu \\nu}^{\\;\\;\\;\\; IJ}", "544c7999d9c7e63340b871c641ecb317": "\n\\begin{align}\n\\hat{\\beta} & = (X'X)^{-1}X'(X\\beta+Z\\delta+U) \\\\\n& =(X'X)^{-1}X'X\\beta + (X'X)^{-1}X'Z\\delta + (X'X)^{-1}X'U \\\\\n& =\\beta + (X'X)^{-1}X'Z\\delta + (X'X)^{-1}X'U.\n\\end{align}\n", "544c994bc547168b72c51b6da4335596": "\n u_x(x,y,z,t) = -z~\\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z) = w(x,t)\n", "544cc9f99e0b6f85a34d13af64bec96c": "\\ln T_{max}", "544d0c9bdfa591c4e229d4857b34190c": "\\hat{H}'_0= \\beta p_0", "544d0cf1adcb2cc195b854ddc1aeab3b": "U_2", "544df711c100b690fc2efee404bc8950": "\\alpha \\dot{\\beta}", "544e5d5060ba6d1bf40f6e7bc90b2b3f": "h\\nu", "544eab3f6cc58d9ca8e50a5a55f5775b": "-kA_c \\left.\\left ( \\frac{dT}{dx} \\right )\\right\\vert_{x+dx} = -kA_c \\left.\\left ( \\frac{dT}{dx} \\right )\\right\\vert_{x} + Ph\\left (T-T_\\infty\\right )dx.", "544eba2d4182dc05321f25337367f8dc": "\\phi_w = L_wI_w = \\phi_0 = \\frac{h}{e}. \\ ", "544f603a3bd3a3df7ff07114a792437f": "f(a) + f(b) = f \\left((a+b) \\frac{a}{a+b} \\right) + f \\left((a+b) \\frac{b}{a+b} \\right)\n\\ge \\frac{a}{a+b} f(a+b) + \\frac{b}{a+b} f(a+b) = f(a+b)", "544ffa8cd8e559ca2f9d704912751509": "z_0, z_1, \\ldots , z_n", "545016cecdc2cb69f95b3b393ed2bbce": "u(x)\\in C^\\infty_c(R^n)", "54503cce0d07e1644c0b86597e123fba": "E(e) = \\bigcap_i K_i(e)", "545043dad9110f955d085d532d019bb3": "{4 \\choose 1}^5 - 34 = 990", "5450536d7b966c452e524a65132586a7": "x \\equiv \\frac{\\mu_B B(g_J - g_I)}{h \\Delta W} \\quad \\quad \\Delta W= A \\left(I+\\frac{1}{2}\\right)", "54505ca3fa0880b43ba8b6829e2230c9": "P = \\{1, \\ldots, m\\}", "54509e04d7f47b8f43d69ba2c8db16f9": "x=y'", "5450bcf5ae3990ff658823377c3229e3": "v \\mapsto B(w, v)", "5450d9b572333c330e69c7d81bbd7747": "e_1 = 0", "5450dc59009ba70404fa1403b2438f89": " f^{(0)}_n (x) ", "54510df68e6b590d5de3921a395a1a7f": "V_x(\\mathbf{x}(t))", "545130077b95b9d09abf8815ec7f0717": "-\\log\\left(1 - e^x\\right)", "54516ce8693c1df14a6240d2efc7c23b": "I(\\mathbf{q}) = \\frac{1}{V}\\int_V\\int_V\\rho(\\mathbf{r})\\rho(\\mathbf{r}')e^{-i\\mathbf{q}(\\mathbf{r}-\\mathbf{r}')}\\text{d}\\mathbf{r}\\text{d}\\mathbf{r}', ", "545184e1807ad71eccafa37fa1146b6f": "Q_{lm}", "54518d2c02a7b3926758aee74120998c": "\\text{fmap}: (A \\rarr B) \\rarr \\mathrm{W} \\, A \\rarr \\mathrm{W} \\, B = f \\mapsto \\text{extend} \\, (f \\circ \\text{extract})", "5451d3179abdb396585d0a71d4c5c395": "\\mathbf{x}_{0i}^\\top[\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} \\mathbf{x}_{0i}^\\top[\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i \\mathbf{x}_{0i}^\\top [M_0] \\mathbf{x}_{0i}.", "5451e8411def53afb58c06535acd1b7f": "s=r+cx", "5451f07462d8495d0dedfb921b15de63": "\\sum_{i=0}^n i\\cdot i! = (n+1)! - 1", "545247559b7757180b66afdd5a71e124": "\\gamma\\colon E\\to R", "54526e1712908e32510dfa8133680ca4": " d_i \\circ d_j = d_{j-1} \\circ d_i ", "5452751fdb2fa3d7b2f8181472e9376f": "\\gamma = 1-S_L \\, .", "5452790f13a5dbd7da5c8a6ce17ef8dd": "\\mu=m_{\\mathrm{e}}/2.", "54528d54cea2e13eb50e0bc350ad6653": "P = P_1\\cdot P_2 = \\frac{l}{t}\\frac{2}{\\pi} = \\frac{2 l}{t\\pi}", "5452a3001ed55eb9d536aebffdc97efa": "M_{z,\\mathrm{eq}}", "5452e907f40c692c9ccd2e05b0dca9a8": "H_*", "5452efe63b7f989ce8d7111b75697d6b": "2 i \\beta_0 \\frac{\\partial \\tilde{a}}{\\partial z} + [\\beta^2 (\\omega) - \\beta_0^2] \\tilde{a} = 0", "54530c2474a8e467b81a6bfcb99d0935": "\\,\\!z_> (z_<)", "54530ef378791e4fe9d308dfebdf806b": "\\forall x (x \\not = x)", "5453550f0821d7b401b958d7c3777659": "\n\\int_0^1 \\sqrt{1-x^2}\\; dx\n", "54536aef9b38b0b9d0be7d9d75269398": "\nE_{DFA} = \\{\\langle A \\rangle \\mid A \\text{ is a DFA and } L \\left( A \\right)= \\empty \\}\n", "5454067d18b804fd6e7622e163b1fb8a": "(K_2)^{\\square n} = Q_n.", "54548f05510673aa223dd80e9a2f5321": "\\|I_\\alpha f\\|_{p^*} \\le C_p \\|f\\|_p,\\quad p^*=\\frac{np}{n-\\alpha p}.", "5454a08e79eff396f502fd71d11e5089": "{{\\left\\{ {{g}_{m}} \\right\\}}_{0\\le m1", "5472c66168b213d832d4781d9cc9ae15": "\ny(n) = h_{0}+\\sum_{p=1}^{P}{(H_{p}x)(n)},\n", "5472d24f54745ce2efd03861ae42a431": "\\vec{v}_\\mathrm{BA}", "5472fa8da29e9a2e08aecfd67a489266": " \\mathbf{P}(V)\\times_k \\mathbf{Gr}(r, \\mathcal E)", "547324a168e4520601b691bbbd64ca7b": "\\log g(n) < \\sqrt{\\operatorname{Li}^{-1}(n)}", "5473485fa5c80cd9d3c100790ecc7442": "E^2 - |\\vec{p} \\,|^2 c^2 = m^2 c^4", "5473507e233e8da84c6f89adc6b287ae": "-\\left ( \\frac{\\partial w}{\\partial z} \\frac{\\partial \\theta}{\\partial z} \\right ) ", "5473844146fa66ba6b94c01a87f0084c": "P_{d-1}=0", "5474382b5e2f1826b019f1b6107fa518": "H_{ns}", "5474a25011bc72b2aa2fae9d81c5dbd0": "k = \\left\\lfloor \\frac{x}{\\Delta} \\right\\rfloor", "5474ef2d56f1e96c11b9d4db46354de2": " R_{in}(fb) = \\frac {V_x} {I_x} = \\left( 1 + \\beta A_v \\right ) R_{in} \\ . ", "5474fee2c14210ce348286fc66794a10": "n,\\alpha", "547595b249eddb2c687609b114b8e554": "x^*(p, w) = \\operatorname{argmax}_{x \\in B(p, w)} u(x).", "5475d8e4dc2056a65632251bf3d91e20": "G=C_3=\\langle x\\rangle=\\{1,x,x^2\\}", "547653abc26281aad95e7b88244b3cc6": " [x,x]=0 \\quad ", "547674d1bdf22b542f169fd0d845985a": "\\sum_{n=0}^{\\infty}\\sum_{m=-\\infty}^\\infty M(m,n) z^mq^n = \\prod_{n=1}^\\infty \\frac{(1-q^n)}{(1-zq^n)(1-z^{-1}q^n)}", "5476a7354805197e3f2af3d09f976343": "n_{00}", "5476bfc833c37f0116a0b6b1fd870d34": "|n|_\\ast=|n|^c_{\\ast\\ast}", "5477054a4f15a08ccb4fce88af75dcb7": "[0,1].", "547706f6fc8be8b303f5ce20055c803f": " R^2_\\max = 1- (L(0))^{2/n} ", "547746de0f398dc7366d69538fbc6c2d": "S=\\int \\left[ \\frac{m}{2}g_{ij}\\dot{x}^i\\dot{x}^j - V(x) \\right] dt", "5477703cf16302fdce429d1d859d0884": "-i\\hbar\\mathbf{r}\\times\\nabla,", "54777064b0cd6c10ee0796d79c6d9c4f": "\\|\\mathbf{A}\\|", "5477717d13820379a038373776a2956e": "r \\approx z + \\frac{\\left(x - x^\\prime\\right)^2 + y^{\\prime 2}}{2z}", "5477f7002b3ed44594f06d730570f5aa": "f(u):=e^{-u} u^{x-1} 1_{\\R_+}", "547856b16821383ed1a8235c396707a4": "x_s=1", "54787b21b2d19bbbb2d642d8462ec017": "|z_D - \\zeta_{k(m,n)}| < \\epsilon", "5478a7b8a655ddd3cce3120f98c3fbb2": "\\scriptstyle\\vec{e}_0", "5478c48c96dcdad92a7a5c801b5f454d": "\n\\begin{bmatrix}\n0 & (0+2)/2 \\\\\n(0+4)/2 & (0+2+4+8)/4 \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0 & 1 \\\\\n2 & 3.5\\\\\n\\end{bmatrix}\n", "5478d763f1f38ecfc3b213e374a3e5e9": "Searched", "5478e36c018ea64b20ee8e7a7f40d8ea": " B(f_1,f_2)=F^*(f_1+f_2).F(f_1).F(f_2)", "547921feb1e84da43a36212c5171ac2b": "\\hat{x} = \\frac{\\sigma_{XY}}{\\sigma_X^2}(y-\\bar{y}) + \\bar{x},", "54792323e9cfbabca586391d016c07e0": " \\omega_{2n}(x) =\\omega_{0,n}(x).", "54798a4758b4c1e5a26471222dfed311": "\n\\overline{F^{*n}}(x) \\sim n\\overline{F}(x) \\quad \\mbox{as } x \\to \\infty. \n", "5479aeed8f00a2b005993ef187c6c37c": "Re \\to 0.", "5479baa6a01a377a1ad459ed62cab6ab": "I_{n} \\in M(n,n;\\mathbb{K})", "5479f0cac9363aab27668a1bca51e393": "\\operatorname{Rot}_{x_n}(\\alpha_n)\n = \n\\left[\n\\begin{array}{ccc|c}\n 1 & 0 & 0 & 0 \\\\\n 0 & \\cos\\alpha_n & -\\sin\\alpha_n & 0 \\\\\n 0 & \\sin\\alpha_n & \\cos\\alpha_n & 0 \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n", "5479ff73c3eacd7290d4eb505e6d9748": "x'_{r,r}", "547a0c554a0a69e4f5ba7897f37b470f": "u(x,y) = |x|^{4/3} - |y|^{4/3}", "547a70ed833e0a4c5a60877aebf99746": " \\int_{S} \\mathbf{u \\cdot T} dS = \\int_{S} \\mathbf{u \\cdot \\boldsymbol {\\sigma} \\cdot n} dS ", "547a7fe6ec6a3c427c6d5c659a74f8c2": "n_-", "547ab433b60b8db28ef8aafd455c9d22": "\\frac{\\mathrm{d}\\mathbf{m}\\left(t\\right)}{\\mathrm{d}t}=-\\gamma\\mu_0 \\mathbf{m}\\left(t\\right) \\times \\mathbf{H_{eff}}\\left(t-\\delta t\\right)", "547b8ae60a8f17e36a7ed06f58904555": "y = (1+n)^x \\mod n^2", "547b9a8bd6f171a86f5c020b47864ec7": "L_{hr}(e_{x},e_{y},l_{x},l_{y})", "547bda7f3f9440f1f208c9bf0ed815c9": " u = { 1 \\over r }. ", "547c03aae3aeb53dba6ccb07b8a1cd06": "E_y ", "547c0f91bf8842bd4987be63b140dc74": "\\pi/2 \\approx 1.57 ", "547c2be0d78f71d29971c4fcfc0241ae": " \\|u\\| \\le \\frac1c \\|f\\|_{V'}.", "547c42e03d73b7e0cd339275e68d3435": " \n2 \\int_{L} \\psi_{+} \\psi_{-} ~ dV = \\int_{L} ( \\phi_A^2 - \\phi_B^2 ) ~ dV = 1 - 2 \\int_R \\phi_A^2 ~ dV\n", "547c5208f09cf00c5dd6824217c39506": " \\left(\\mathbf{B}, \\Phi_\\text{m},\\mathbf{A}\\right) ", "547c6af6bfeef8f640614ffb9753dd35": "\\sigma=\\sqrt{8c_0\\epsilon\\epsilon_0RT}\\sinh\\left(\\frac{ze\\psi_0}{2k_BT}\\right)", "547c9a4e2d08950d50d2f5265668708d": " \\mu{\\left( \\frac{a}{a_0} \\right)} = \\left(1 + \\frac{a_0}{a}\\right)^{-1} ", "547ccbdbed91bd92d825adeb855709c3": "m\\leq 20\\rho [K:\\mathbb{Q}].", "547cfcd760cc8b88e99ea54527ff0850": "P(e|X)", "547d1fef8095cc20a1623978e297193d": "\\theta = \\pi/3", "547d425d2fa8543e9d0cb95c74ba3174": "\\omega _c/2", "547d4e231f3483fd8a086606a7ecdae5": "S_{40:1}=40", "547d6372b9ee628705aec8531e24beda": "\\lbrace e^{ar}u : 0 \\le a < \\pi \\rbrace", "547d930634af9e90289886176886c401": "S_i \\geq b", "547dbaea4c7eb6e2975d8cd2ccca4581": "\nP(\\mathbf{s}) ={e^{-\\beta H(\\mathbf{s})} \\over Z}\n\\qquad\nZ=\\int_{[-\\pi,\\pi]^{\\Lambda}}\\!\\prod_{j\\in \\Lambda}d\\theta_j\\;e^{-\\beta H(\\mathbf{s})}\n\\,.", "547df9652d18d17f5d95fe437fc2381e": "-b_0", "547e174a03976e7c9503978ba924733e": "{\\overline{X}_n - \\mu \\over S_n/\\sqrt{n}}.", "547e3c334f6c3b153773dcd118977c19": " M = {4\\pi\\rho (r) r^3 \\over 3 } ", "547e7cfaa27843ea4bd16dc3d62910bc": " H_\\alpha(|f|^2) + H_\\beta(|g|^2) \\ge \\frac 1 2 \\left(\\frac{\\log\\alpha}{\\alpha-1}+\\frac{\\log\\beta}{\\beta-1}\\right) - \\log 2", "547e9716f3bf162490f008021023922f": "sk(t)^{\\alpha} = sy(k(t))", "547eb00f850913b61d47cb5fd373b764": "P_{i+1}=P_i.", "547eb069fd85d303676bb6c6890b25ce": "m_1 < \\cdots < m_n", "547ebecd2abab0adb90c798e7b838d56": "\\psi_3 = He^{- \\alpha x} \\,\\!", "547fb22a5b67c09509c21a162616fbe9": "n^N", "547fbe9c68076467b8a53e115d44e740": "O(\\sum_{i=j}^{k} \\log 2^{2^i}) = O(2^k) = O(\\log n)", "54800f11a798dda54247dbeaa6345f1b": "\n\\langle F |\\psi(t)\\rangle = \\langle F | \\exp\\left({- {i \\over \\hbar } \\hat H T}\\right) |0\\rangle.\n", "548046fb53d126e2ff85cb87682ff3b1": "E_0^{(1)}=\\lambda \\left( \\frac{\\alpha}{\\pi}\\right)^\\frac{1}{2}\\int e^{-\\alpha x^2/2} x^4 e^{-\\alpha x^2/2} dx=\\lambda \\left( \\frac{\\alpha}{\\pi}\\right)^\\frac{1}{2} \\frac{\\partial^2}{\\partial \\alpha^2} \\int e^{-\\alpha x^2} dx ", "548078e8fef7e2ebdaa75af31979f9bd": "\\mathrm{d}S = \\frac{\\delta Q}{T}.", "54807db5294e1a13d81e421e244ac7ef": "\\begin{align} \\hat{x} = x \\\\\n\\hat{y} = y \\\\\n\\hat{z} = z \n\\end{align}", "54813f0f20f3b009688207d5f0684b4f": "C_l= \\frac{2 \\gamma}{V_\\infty c} \\qquad (2)", "5481592e97262249e6d108f5582071f0": "m=7", "548174f862edf8c1514670ce5d59a34b": "A^\\alpha = (\\phi, \\mathbf{A})", "5481b1469ed091636a0b922e3f2c04fc": "\n \\qquad \\qquad u_x^- = \\frac{3u_i^n - 4u_{i-1}^n + u_{i-2}^n}{2\\Delta x}\n", "5481ff188ef0a78adb0263af445bdb4f": "g(x,b) \\cdot [x]", "54821fd4e9350baca2568d58c8c9c157": "\\scriptstyle v\\,", "54825b67b131e8c66122e7c64b308b9d": "(Tf)(x) = \\int_0^x f(t)g(t) \\, \\mathrm{d} t.", "54828ab00b4dfaa2b981f6f25ca539e9": "A^{?} = A + 1", "54837d0db6a38f5e5ea68c760d04e954": "\\pi(\\theta)=1", "54839dd19ca687c0356897923c07e333": "\\frac{\\omega_p^2}{kN} f'_0 \\frac{\\mathcal P}{kv-\\omega} + \\epsilon \\delta(v-\\frac{\\omega}{k}) ", "5483ce9df82a9df099117fcbe794d9ca": "\\{ a^p b^q c^r \\ \\ | \\ \\ p = q \\ge r \\ \\ \\or \\ \\ p \\ge q \\and r = 0 \\}", "5484722448eb365ce42f4c5a44985ce8": "d (x, y) = \\sup \\{ \\rho (f(x), f(y)) | f : B \\to \\Delta \\mbox{ is holomorphic} \\}.", "5484b36fe4739e6364996256d4b3e8ce": "\\scriptstyle \\operatorname{E}\\left[X| Y\\right]", "5484c09cacfe7a0ead9c189afe889d06": " f_a (1)= g^{a_{1}} ", "5484ea9e65ef46e46393b5e7f729a1e2": "T = \\frac{1}{2} U^2 \\left(C + \\frac{md^2}{4}\\right) \\left(\\frac{2k}{rd}\\right) \\theta^2", "5484f9773f0834015029b868979f9e71": "\\vec{J} = \\vec{L} + \\vec{S}", "5485e68aca5d7407b8e28b9b5364e6f0": "\n\\exp(a D_T) = \\sum_{n=0}^\\infty \\frac{(a D_T)^n}{n!}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (7)\n", "5485f72122cb5f515a573e384075e136": "\\sum_{\\ell \\in \\mathrm{leaves}(T')} 2^{-\\mathrm{depth}(\\ell)} = 1 \\;. ", "548670187918659c7cffee56a04ca107": "\\scriptstyle \\sqrt{f}", "548691a0c250bd19ec762a5fc9f4cdfc": " \\overline{F}(x)=1-F(x)", "5486c617450874f45957a3e1ab4ec600": "\\left.\\frac{\\partial p}{\\partial T}\\right|_{(V,T)}", "5486d364cc732496569f6837d296aaf7": "\\gamma_{\\perp}", "548708662968941e72cc2dc1bf0b17ad": "e_x", "54875ef9fe8d6c4f3e08dea6321d0384": "A = \\left(\\sqrt{P}+\\epsilon\\left(t,z\\right)\\right)e^{i\\gamma Pz}", "5487646dc573ec85e591dfd7cf6e0d64": "\\scriptstyle \\mathbb{T}", "548777dfc32ea321a1c2c7774f89b2fc": "\\hbar=\\frac{h}{2\\pi}", "5487788cf715fc725d7e62323490e01e": " b = F_{2a}\\,", "5487cd151040506e6bea2b8751bde8ab": "d=\\sqrt[3]{\\frac{FD}{0.4\\sigma} } ", "5487e60e029b9026f9236a6bbb24532d": "\n\\begin{pmatrix}S(x)\\Gamma(x)\\\\ x^6\\end{pmatrix}=\n\\begin{pmatrix}\\alpha^{-7}+\\alpha^{4}x+\\alpha^{-1}x^2+\\alpha^{6}x^3+\\alpha^{-1}x^4+\\alpha^{5}x^5+\\alpha^{7}x^6+\\alpha^{-3}x^7\\\\ x^6\\end{pmatrix}=\n", "54883146acad1a89110bb36238f39266": "A_1 \\to A_2 \\to A_3 \\leftarrow A_1", "548839b88339373e284a7a2c97690fca": " (M \\varphi)(\\mathrm{e}^{\\mathrm{i} \\theta}) = \\sup_{0 \\le r < 1} \\varphi(r \\mathrm{e}^{\\mathrm{i} \\theta}). ", "548874f0c11dbb9d5f09397b1698f6d2": "x \\wedge y = x \\wedge z", "54887b37b128f4caf69ac3f684509c4d": "S(x; 0, 1)=\\frac{1}{2} - \\frac{1}{\\pi} \\arctan(\\ln x), \\ \\ x>0", "54888001fe61be85158b7673e9a2b32c": "\\cos \\sqrt z=\\lim_{n \\to \\infty}\\prod_{m=1}^n \\left(1-\\frac z{((m-\\frac{1}{2})\\pi)^2}\\right)", "5488914d82ca7ebbbdbe5bec3ebcee4d": "x - y = -1", "54889b63b16b32a8b0fe480a1affa28b": "(\\kappa-1)~r~\\ln r - r", "5488d3479df9765365e092234182a5ec": "\\frac{d u (x,t)}{d t^\\alpha}= D \\frac{\\partial }{\\partial x^\\beta} \\left(\\frac{\\partial u(x,t)}{\\partial x^\\beta}\\right), -\\infty< x < +\\infty\\,, \\quad (1)", "5488e2f5652fabcc44d481a746c3c029": " z^{1-c} \\, _2F_1(1+a-c,1+b-c;2-c;z)", "548901a71ee285de0b5892eac24c1cad": "W=\\left( \\tfrac{\\ln(\\hbox{Victims})}{\\ln(\\hbox{Monetary Losses})} \\right)^\\beta", "5489047881250c964bf395cb83e9072b": "\\left [ Z_1 (X_c-X_s) + Z_v \\left (X_c - \\frac {1}{2} \\right ) \\right ] + \\frac {24\\pi\\,\\Kappa\\,\\mu\\,_0 r_0 (r_1-r_0)^2}{3\\Kappa\\,+4\\mu\\,_0}", "548906487d1f26a8de79e5fe680c9e34": "2\\pi \\nu=\\omega", "54895c969bba5664c5108cd49c1ec521": "2^{\\mathbb{R}}", "5489831701435ee57e61322f89645176": "B27^-", "54899e6ae9735c646e8109df1af8c30a": "P=\\{p_1,p_2,...\\}\\,", "5489aa46a292d6eeda0679ad066c1809": "a^\\ell + b^\\ell = c^\\ell.\\ ", "548ab24ee010cf43d469190cd7d49186": "\nv = \\sqrt{\\frac{g k h (1 - R_a/R_s)}{c_D}}\n", "548acc2e676a614b47efe9da54ed421a": "\\scriptstyle t \\;=\\; (N \\,+\\, 1)\\tau", "548aff26c34352418a357515b19cac61": " \\alpha = \\frac{e^2}{\\hbar c \\ 4 \\pi \\varepsilon_0} \\approx \\frac{1}{137.03599908},", "548b268bc78e814ee30b44c53b04a7f9": "\\kappa(v_{ij}) = v^*_{ji}", "548b58a87054f3ade74f548a1edd54cc": "\\theta(1)=\\frac{\\alpha}{c+v}\\,\\!", "548b9cac06f2bdf48a6c28b2647af120": "c_\\sigma=0", "548bcd6a98d8c37c6c1036dcf190b797": "\n\\frac{d\\mathbf{v}}{dt} = \\frac{1}{m} \\mathbf{F} = -\\frac{\\mathbf{v}}{\\tau} + \\frac{1}{m} \\mathbf{F}_{\\mathrm{rnd}},\n", "548be04215436f4e60521769469e928b": "A_T", "548bed4189da87ba23b3899874535fc9": "\\alpha_1 = a\\omega_1+b\\omega_2\\,", "548c04eddf0a2a46099d04ade7ea6374": "W(0)", "548c35d726b06d4aee2d6d6751ef2045": "v_i ", "548cd58df3cad168be0cda40ccc5754c": "W=\\left(1-\\frac{T_2}{T_1}\\right)Q_1.", "548ce5dc90cbdb657e2b2803b8051c42": " 0 \\log 0 ", "548d237ac44864bf27040f6739ab214d": " J = \\lim_{A \\rightarrow 0} \\frac{I}{A} = \\frac{\\mathrm{d}I}{\\mathrm{d}A} \\,\\!", "548d5ed18e28d8af7366cf51aedfd9c6": "A=\\begin{bmatrix}\n1 & \\dots & 1 \\\\\na_1 & \\dots & a_n \\\\\na_1^2 & \\dots & a_n^2 \\\\\n\\vdots & \\ddots &\\vdots \\\\\na_1^{k-1} & \\dots & a_n^{k-1}\n\\end{bmatrix}", "548dbdd70e04bf27c7d92693ee84086e": "\\operatorname{Id}(A)", "548e25fd76154ae81615b35ff4a2ae10": "\\vec x(t_0)=\\vec x_0", "548e9155178e42368c35ca4126efd2c4": "FGT_\\alpha=\\frac {1} {N} \\sum_{i=1}^H (\\frac {z-y_i} {z})^\\alpha ", "548ec841eda59978ca5f47c46d9f030e": "x \\not\\in x", "548ee60298204bb0aaca08a030a7b8a4": " 0 = \\mathbf{b}_{k}^{T} \\, \\mathbf{a} ", "548f3dfaa2b8a38b9abab447f6049edf": "\\begin{align}\nS_p(-n) & = S_p(-1)-\\sum_{k=1}^{n-1}(-k)^p=S_p(-1)-\\left(\\sum_{k=1}^n(-k)^p-(-n)^p\\right) \\\\\n& = S_p(-1)-(-1)^p\\left(\\sum_{k=1}^nk^p-n^p\\right).\n\\end{align}", "548f83cd22eb943bb6d34146475055e1": "\\sqrt{A}=A", "548ff1ac4de6ae109263dd75cc25badb": "\\sqrt{\\frac{1}{5}}\\!\\,", "54909ecc2360d86aa4aa2ed69c66a4c0": "\\beta(p,n) =\\frac{1}{p}\\left(p\\log n + p -\\psi(n-p+1) +(n-p+1)\\psi(n-p+2) +\\psi(n+1) -(n+1)\\psi(n+2)\\right)", "5490c3dfe3a6daed48a280d8ec151f9d": "W =\\frac{RI^2}{2}=\\frac{HI}{2}=H_{eff}\\times I_{eff}", "5490e10c1d84b4ac51268d27ec55c579": "E_{x,y} = \\frac{\\partial x}{\\partial y} \\cdot \\frac{y}{x} = \\frac{\\partial \\ln x}{\\partial \\ln y}", "54914fe93426a74e7421021cdfc06df1": "S=\\textstyle \\sum_{i} n_id_i", "5491ef93f7a4c0fdad76dd8650e5712c": "x=x_1,\\ \\ y=y_1 \\ \\ ", "54926a43b16417b64e05769d83a07b65": "\\zeta_{MT,r}(s_1,\\dots,s_r;s_{r+1})=\\sum_{m_1,\\dots,m_r>0}\\frac{1}{ m_1^{s_1}\\cdots m_r^{s_r}(m_1+\\dots+m_r)^{s_{r+1}}}", "549278d70668e0cba6b6b64194e9f680": "\\pi^*", "5492b4b0399e74422011584218d61969": "\\mathrm{N} \\mathfrak{p} = |\\mathcal{O}_k / \\mathfrak{p}| = |(\\mathcal{O}_k / \\mathfrak{p} )^\\times| + 1 \\equiv 1 \\pmod{n}.", "5492fb9b144314688513ffc5876db122": "\\mathbf{b}^\\mathrm T \\, [\\mathbf{a}]_{\\times} \\, \\mathbf{b} = 0. ", "5493415951d5862008701552be54ae64": "p_{\\lambda}=\\frac{v_{\\lambda}}{g_{00}}; \\qquad F_{\\lambda}=\\frac{1}{2g_{00}}\\frac{\\partial g_{ij}}{\\partial\nx^{\\lambda}}v^{i}v^{j}", "54937a2ad003f5b9de792930448dee28": "\\left(\\frac{a}{p}\\right) = (-1)^n", "549391ebafd6de4ee199c741b6459544": "P \\equiv \\log_{10} \\frac {p_i} {f_B T}", "5493e8b8c75781d4f7084fa7c3a4db9b": "A \\circ B = \\bigcup_{B_x\\subseteq A} B_x", "5493ebab72bb687fe6b4f4c00280a0b4": " \\beta := r_2^* a_2 / p_2^* a_2,\\quad p_2 := r_2 - \\beta p_2. ", "5493ee81a358b0be5dd7478a0005ef32": "\\langle \\theta,\\theta \\rangle =1 ", "54941e25651d279903bd17bc3bdbaa70": "( 32=2^{10/2} )", "54947ea07316b0fb223bde56b98d92e7": " \\overrightarrow{m}(\\overrightarrow{B}) ", "5494f5f5cfc16659767f90730fd8afdb": "\\overline{X}_n \\, \\to \\, \\mu \\qquad\\textrm{for}\\qquad n \\to \\infty", "5495574e5e9f23690ed845fcd3ff524a": " A^{(k)}={a_{ij}^{(k)}} ", "5495e9341adbfe8c0d26ca6359d25971": "|X| \\leq dim(Hom(U(d),s,s))", "549600468718199886840a22b4f35b9b": "\\vec{M} = \\chi_v \\vec{H}", "54965bb4a3c7d9d653fd3fe5bcd71618": "\\tfrac{1}{n}", "54967ad6d2a249d8ca46a0ef3fe87da3": "M_{\\rho_{T}}", "5496857518305d368fd6bde7e722b528": " a=A^T*h_i ", "5496970e19af3c4f4aebaaaf5370c2dc": "r_c/2", "549780ecace68772a15fc1f32c960dfb": "xyzx = xzyx", "5497a2cca53c132d7748cb55f7e5a839": "(\\hat x,u)", "5497a342817e9e2e5be615c3909523d6": "{i,g} \\in [0, 1, ... , n]", "54982ad1f342c434710a72080618aff5": "\\lambda y", "5499c4e8af2bf16ba29381a13187ea7a": "\n\\begin{align}\n\\Omega_1(q_1,q_2,\\ldots,q_n,Q_1,Q_2,\\ldots Q_n)=0\\\\\n\\ldots\\\\\n\\Omega_m(q_1,q_2,\\ldots,q_n,Q_1,Q_2,\\ldots Q_n)=0\n\\end{align}\n", "5499e2bd619ea6746b5adf030773fb28": "u(t)= -\\int_{0}^{\\infty} \\eta(z,t)dz", "5499e44f5ddd609290aa477463a066db": " X^{\\ast }(t) =\\{x\\in X\\left( t\\right) :f(x,t)=V(t)\\}\\text{, } ", "5499ec9b4c76491e5b1d8d02d717bb17": " c_2 \\cdot s^{-1} = m'\\cdot h^y \\cdot (g^{xy})^{-1} = m'\\cdot g^{xy} \\cdot g^{-xy} = m'.", "5499f44c89e54c73b9d663434628ceca": " q=(s,t_e) \\in Q_A", "549a20cbf09292aa902bcc1bcc18174c": "s^\\prime_i\\in S_i", "549a23aaea351d98d98587b1476e0068": "\\scriptstyle{ [T,S]^\\prime = [T^\\prime , S] + [T, S^\\prime ] }", "549a2af71dee2ee054c03d8f1c63b7c2": "Y_{ij} = \\beta_{0j} + R_{ij}", "549ad6c1ec327e297a42b5bcfa0d7d34": "N_{(-)} + N_{(+)} + 1", "549b2f0cb7d89f6ffba790ffeaa6e344": "* : \\Lambda^k(V) \\rightarrow \\Lambda^{n-k}(V). \\, ", "549b610bae5752c8f194e6c279bd5b76": "G^\\infty", "549b959e968ffdf89dd4e6fd6a24a3b0": "\\psi_{jk}(x) = 2^{j/2}\\psi(2^jx-k)", "549c41ef5853f719e63e05a59fbe4488": "Y\\supseteq X", "549c620052328770daf4aa5f790780fa": "(1-\\varepsilon, \\varepsilon)", "549cb6212dd7e1580fddbf29f7febc7a": "\\cos C=\\frac{a^2+b^2-c^2}{2ab}.\\,", "549cfa7379e1307c700ea22635c412a9": "r=kC_S C_B", "549d3ac62f48f57063be403567c989b4": "\nP= \\frac{Li}{1-\\frac{1}{(1+i)^n}}=\\frac{Li}{1-e^{-n\\ln(1+i)}}\n", "549d51d5efda9952aa047aad54c74d18": "\\beta + \\epsilon", "549d9036dae121a8e09509a342fe76c4": "R \\equiv n^{\\frac{Q+1}{2}}, t\\equiv n^Q, M = S.", "549e6bb9e638679e8985cbef9a19ecaa": "(18)\\quad L=\\frac{1}{2}\\big(l_+ + l_- \\big)\\,,\\quad l_+ =\\sqrt{\\rho^2+(z+ \\sqrt{M^2-Q^2})^2}\\,,\\quad l_- =\\sqrt{\\rho^2+(z-\\sqrt{M^2-Q^2})^2}\\,.", "549ed032f86b830b0514155497d31534": "\\mathbb{N}=\\overline{\\{\\}}", "549edbfafdf794f8a8eca3628bb17423": "{}_t q_x", "549f329772ce77bf1297c4137b321219": " \\psi\\left(\\frac{1}{4}\\right) = -\\frac{\\pi}{2} - 3\\ln{2} - \\gamma", "549fa13cf3065617a0ea77921834cbaf": "\ny = \\int \\sin \\left[\\int \\kappa(s) \\,ds\\right] ds\n", "549fef5e427feebb986afd00730780b3": "\\sum_{n=1}^{\\infty} q^n \\sigma_a(n) = \\sum_{n=1}^{\\infty} \\frac{n^a q^n}{1-q^n}", "54a06627f6aa66d29fc0bb9dcc1a1312": "f_4(x) = x^{\\frac{1}{x}} \\ ", "54a10248cf7934dbffe032e119cafda8": "\\theta=\\frac{kT}{mc^2},", "54a12f6679b3b156f0289029870f5b1c": " f_n(X) \\partial_X^n + f_{n-1}(X) \\partial_X^{n-1} + \\cdots + f_1(X) \\partial_X + f_0(X).", "54a1de7d969415c5a773536ecb60a22b": "{16\\over 5}\\times{1\\over 2} = {8\\over 5}", "54a2222c19204e105351c6054d900777": "\\operatorname{Tor}^R_1(M, k) = 0 \\Rightarrow M\\text{ flat } \\Rightarrow M\\text{ free } \\Rightarrow \\operatorname{pd}_R(M) \\le 0.", "54a22bd208c52f373d4f4699215845b5": " f_Z(x)={\\Pr(Y>x)-\\Pr(X>x)\\over {\\rm E}[Y]-{\\rm E}[X]}\\,, \\qquad x\\geq 0.", "54a2308956043ff3d39715953de3535d": "\\sum_{p}^{}{\\Delta S^0_i} \\ge \\frac{\\Delta G^0 - W_{add}}{(T_H-T^0)} ", "54a24c7978f04562d18bf8762179eef1": "\\vec c", "54a25fe06f0e5419829cf9bd243381b5": "\n \\mathbf{b}_{i,j} = \\frac{\\partial \\mathbf{b}_i}{\\partial q^j} := \\Gamma_{ijk}~\\mathbf{b}^k \\quad \\Rightarrow \\quad\n \\mathbf{b}_{i,j} \\cdot \\mathbf{b}_l = \\Gamma_{ijl}\n", "54a27edb0bd6438f18636c5e8bfe1879": "p=\\sqrt{\\frac{(ac+bd)(ad+bc)-2abcd(\\cos{B}+\\cos{D})}{ab+cd}}", "54a2bf8c09ace67d3513aaa1aa7aa0f3": "sl", "54a2c574a61bf9ed239a4b57cebab812": "\\displaystyle a(w^2x^2+y^2z^2) + b(w^2y^2+x^2z^2) + c (w^2z^2+x^2y^2) = 0", "54a2d0b95ee6127643ee3bcb76e5a3f3": "a=\\sum_{i=1}^{N}a_{i}<+\\infty ", "54a2e20d66c75e71087cedbbdc117f7e": "Q(\\alpha,\\alpha^*)=\\frac{1}{\\pi} \\int \\delta^2(\\beta-\\alpha_0) e^{-|\\alpha-\\beta|^2} \\, d^2\\beta=\\frac{1}{\\pi}e^{-|\\alpha-\\alpha_0|^2}.", "54a30c2aa4939c6ea5ac28afc8e22489": "\\pi/2\\,", "54a319fc86c4ec95d1184bcc6196dbe8": " \\vec{m} ", "54a31a77c53f67edf9f679077caa41d7": " g = \\mathrm{det} \\left( g_{ab} \\right) \\ ", "54a3426864cc22d7855884bb584338aa": "t_0 = \\frac{1}{H_0} F(\\Omega_r,\\Omega_m,\\Omega_\\Lambda,\\dots) ", "54a3a9df4547cb2cb9c77f8e4d08a012": "F = \\frac{d^{2}}{l \\lambda} \\gtrsim 1", "54a3c5885c6a35c89beb2955da04917d": "\\|X\\|_{\\Psi} \\triangleq \\inf\\left\\{k\\in (0,\\infty)\\mid E[ \\Psi(|X|/k)] \\le 1 \\right\\}. ", "54a3f0c9337b69de6a376d6f2276781b": "\\mathbf A \\mathbf x = \\mathbf b", "54a43d0ded705ace12b89892f3bdcbbf": " P(I|E) \\approx P(E|I)\\cdot \\operatorname{Odds}(I) ", "54a44c328c06b26c8244db967622dbd4": " K_n ", "54a4b4043846451f003edce2410ea649": "m_{\\mathrm{QCD}} = m_\\text{p} \\ ", "54a4c495dca05df660f56b2e5570ba19": "\\dot{\\gamma}(t)", "54a4db6913ba72c5393e0f29cf161612": "N(k,n) = {n \\choose k}=\\frac{n!}{k! (n-k)!}", "54a4de57adfbd6f931539110126659f4": "X_tY_t=X_0Y_0+\\int_0^tX_{s-}\\,dY_s + \\int_0^tY_{s-}\\,dX_s+[X,Y]_t,", "54a4ebb591daa7339a5f0045386f3dcd": "\n\\Pr \\left\\{ \\bigg\\Vert \\sum_k \\mathbf{Z}_k \\bigg\\Vert \\geq t \\right\\} \\leq (d_1+d_2) \\cdot \\exp \\left( \\frac{-t^2} {\\sigma^2+Rt/3} \\right)\n", "54a50acd196b0149e51113179a9d0504": "V = V_1 + V_2 + V_3 + \\dots + V_n = \\sum_{i} V_i", "54a5284e6a81423fad2e746576956d3e": "X \\widehat{\\otimes}_\\varepsilon X", "54a567951d2ad6234d506f06f64872d4": "\\text{Im}\\left(f(0)\\right)", "54a587e054691740d4445442b0412e79": "1s\\alpha \\;\\;\\; 1s\\beta", "54a6c4c965b9d2e725dbff6dbc37e35e": "\\mathbf{R} \\to S^1", "54a6d7219afca19d3944e690dd53d021": "\\displaystyle m_n", "54a6f6ea6eda784a10943d37603128c2": "\\begin{matrix} \\frac{1}{5} \\end{matrix}", "54a70019ae3be5c7b4473f043f7621e5": "x^{32} + x^{26} + x^{23} + x^{22} + x^{16} + x^{12} + x^{11} + x^{10} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1", "54a778a590665f1dae5711f38a039480": "\\lambda^5 = \\begin{pmatrix} 0 & 0 & -i \\\\ 0 & 0 & 0 \\\\ i & 0 & 0 \\end{pmatrix}", "54a7857b848dd85bd0f35d5f3bfdd29d": "H(.,j)", "54a8adb944a1e7eb98e2f92b44c21d81": " P_\\mathrm{E} = \\frac{D}{f_E}, \\qquad P_\\mathrm{O} = \\frac{D_{\\mathrm{EO}}}{f_O}", "54a8ae9b5fdb196e1ec05ffd5a109e5a": "j \\in {1,\\dots,k}", "54a8edc39fd4125eac9843ff916d3d17": " \\begin{align} \\mathcal{A}(1^+ \\cdots n^+) &= 0, \\\\ \n \\mathcal{A}(1^+ \\cdots i^- \\cdots n^+) &= 0. \\end{align} ", "54a90fecedf4a86402e3acafc9cc897a": "+ 7 \\cdot 8^{(8+2)} + 7 \\cdot 8^{(8+1)} + 7 \\cdot 8^8 ", "54a9210c68ef01c907bb455026e664a9": "y_1= g_1^{x} \\land y_2={(g_2^a)}^{x} g_2^b", "54a993b66466b4d7e051ce15ce381795": "P = \\frac{E*G}{K^2} + \\frac{D}{K}", "54a9d05bb19c962feb2db2a41b502899": " \\mathbf{R} = (\\mathbf{R}-\\mathbf{C}) + \\mathbf{C} = \\mathbf{d} + \\mathbf{C},", "54aa083d29e60cced226f5886f4c86a6": "|x_{2,i} - x_{1,i}| = 0", "54aa702eaf4e0cc1c8f44ecfdddadbe2": " |\\mathbf{\\Omega}| = \\frac {\\mathrm{d} \\theta } {\\mathrm{d}t} = \\omega \\ , ", "54aaa51c1236ba25b13ebb15e1324a3c": "dy^2=dt^2-\\sum_idx_i^2", "54aae346d094d754963c48bf79114d6b": "x^i(t)", "54ab1fd3ada3e2daa486c000010435cc": " \\epsilon(q) = 1 + \\lambda^2/q^2", "54ababed9261ff22dfe8bf5e52f9e71c": "1/T", "54ac32099adce0247f3520b47016d425": "\\phi(x_i-x_j) \\, ", "54ac4754a20f8e7a7aff0bbd1e7d0024": "\\lang n^{(0)}|n^{(0)}\\rang = 1", "54ac552df219f340fbf2c2b928d4aa28": "\\chi(\\tau,f) = \\delta(\\tau) \\delta(f) \\, ", "54ac61d68f77901bf3fe9e4d3e8c15be": "RWA = K * 12.5 * EAD\n", "54acfa295fb6911a1810cc03956bf7fe": "\\varpi = \\Omega + \\omega", "54acfd5bd1bf1f713eeeab875cb2d505": "\\zeta(\\alpha,x_{\\mathrm{min}})", "54ad044d7c6bb2bf72cc8339aeda1632": "\\bigoplus_k \\Lambda^k(V)", "54ad2f18887cb8db95b02845fc212bc7": " \\operatorname{de-lambda}[F = \\lambda P.E] \\equiv \\operatorname{de-lambda}[F\\ P = E] ", "54ad3df1573368de56f08845ef432dae": "J=\\begin{bmatrix} \\dfrac{\\partial F_1}{\\partial x_1} & \\cdots & \\dfrac{\\partial F_1}{\\partial x_n} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\dfrac{\\partial F_m}{\\partial x_1} & \\cdots & \\dfrac{\\partial F_m}{\\partial x_n} \\end{bmatrix}. ", "54ad43b04acc48ce658a4a9e97dae0cd": "\\frac{dz}{z} \\;=\\; \\frac{dr}{r} + i\\,d\\theta \\;=\\; d[ \\ln r ] + i\\,d\\theta.", "54ad44680fb11934e60232f9a24bd43e": "f_X: X_1 \\rightarrow X_2, \\ f_Y: \nY_1 \\rightarrow Y_2", "54ad54a4c17d68fc286dfba09ff11453": "\n\\begin{align}\nc_i(t) = c_i(0) \\exp(-k \\lambda_i t).\n\\end{align}\n", "54adef7baf828d5571ac1738951183f3": "\\epsilon < 0", "54ae08cc23a369e211fbad200d833f81": "\\sqrt[3]{y}", "54ae18778af4a4366772598e6f61ef90": "U_{j+p\\,(\\operatorname{mod} q)}", "54ae9047932144bc08a92e891093ca53": "v^{-1}=0", "54aea13cd11751425efc13a8eab6f083": " f_*", "54aed778d595a13f203082bbfdd31c91": "H = \\frac{2 x_1 x_2}{x_1 + x_2}.", "54aeecd98536aaf185bf8cb90957595b": "\\frac{1}{[S]}", "54af00623ee82cce4f795aaf2f8140d2": "R_{\\text{v}i}", "54af54ed330c22bf5e197935a4e82105": "\n\\begin{align}\n\\frac{d \\phi}{d t} + kL\\phi = 0.\n\\end{align}\n", "54af828d15cb2dead42dc71c900efcd7": "\\beta(r') = 4\\pi \\sqrt{2M(r-2M)} \\sqrt{1- 2M/r' \\over 1- 2M/r}.", "54afa46338dc2c76f7c9ad3d65b1871f": "\nk_\\text{fit}(\\gamma)=\n\\frac{2\\pi}\n{T_\\text{fit}(\\gamma)}=\n", "54afbcf595a159ef4098c2c544fe3396": " f\\in C^{1,1-\\frac{3}{p}}(S^{1})", "54afec35d0492b2793fd74791114fa2b": "\\scriptstyle \\frac{\\partial f}{\\partial t} (\\gamma,\\, t^*)", "54affe27c13b5563b095228a07180cfe": "Pt3D=\n\\begin{cases}\nx1 \\\\\ny1 \\\\\nz1\n\\end{cases}", "54b03d525150a004e9bd4e6a413f33d4": "\\hat{H_0}", "54b0a89b70963d13ae685ac40d924909": "\\scriptstyle{\\vec{r}(u,v)}", "54b0ced054b620a574a36a39bc64df2c": "\\begin{align}\n(5)\\qquad S &= 16\\pi \\frac{d}{d q^2} \\left[\\Pi_{33}^{\\mathbf{new}} (q^2) - \\Pi_{3Q}^{\\mathbf{new}}(q^2)\\right]_{q^2=0}\\\\\n&= 4\\pi \\int\\frac{dm^2}{m^4}\\left[\\sigma^3_V(m^2) - \\sigma^3_A(m^2)\\right]^{\\mathbf{new}};\\\\\n(6)\\qquad T &= \\frac{16\\pi}{M^2_Z \\sin^2 2\\theta_W}\\; \\left[\\Pi_{11}^{\\mathbf{new}}(0) - \\Pi_{33}^{\\mathbf{new}}(0) \\right]\\\\\n&= \\frac{4\\pi}{M^2_Z \\sin^2 2\\theta_W}\\int_0^\\infty\\frac{dm^2}{m^2}\\left[\\sigma_V^1(m^2) + \\sigma_A^1(m^2) - \\sigma_V^3(m^2) - \\sigma_A^3(m^2)\\right]^{\\mathbf{new}},\\end{align} ", "54b0f0686a2fd7e612db22d8b6099439": "2J = 2 \\frac {f} {\\sin \\theta} = 2 \\times \\frac {90 \\text{ mm}} {\\sin 8^\\circ} = 1293 \\text { mm} \\,.", "54b10b83299204391112d2de37fcae99": "\\mathcal{O} ", "54b14afd81d49b800491d476b3d130b6": "\\sqrt{2} =a/b", "54b15b017935a882f7c52fa695116338": " P+c=P1 \\, ", "54b175304b7d7808895906cdf6dcbab0": "\\tau = \\mathrm{id} : \\pi^{-1} (V) = \\mathrm{T} V \\to V \\times V;", "54b1e2053743e81303cb43afcbd1fe49": "c_{15}+b_{15}", "54b21fb62a3a57e0694e97ab7a5142e5": " R_i = 100 \\times { price_i - maxprice \\over maxprice } ", "54b237e67048332dfd3e5aeb2799fc6b": "\\mu_\\Delta=\\frac{\\Delta B}{\\mu_{0}*\\Delta H}", "54b2addbb837ff10a9a827c40d231d53": "\n\\left(x,y\\right)\\in D^n\\times{\\mathbb R}, z\\in{\\mathbb Z}", "54b30d06f41b53501d7a96a7c8c038bd": "\\pi_i=P(X_1 = i)", "54b32a3fb9a40773cc269b92738211e9": "b = 2(x_2 - x_1)(x_1 - x_c) + 2(y_2 - y_1)(y_1 - y_c)\\,", "54b36cab0e16d05fbb191b8094d1c857": "s(\\vec{r},t)\\!", "54b3c8e822e66738f8d8c3598eaaa2c2": "X_1 Y_2 Z_3 = \\begin{bmatrix}\n c_2 c_3 & - c_2 s_3 & s_2 \\\\\n c_1 s_3 + c_3 s_1 s_2 & c_1 c_3 - s_1 s_2 s_3 & - c_2 s_1 \\\\\n s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_2 \n\\end{bmatrix}", "54b3e3094429d8320a3e5dffc7aa0e95": "P_1(A)", "54b3e3a56efbb853dbe36379dabf9d4a": "x_k+iy_k=z_k\\qquad x_k-iy_k=u_k", "54b40129e0c84b5f70d7f4faa89135b5": "w_i(p) = \\Phi^{-1}(Sq^i(\\Phi(1))) = \\Phi^{-1}(Sq^i(U)).\\,", "54b420f0bb0ae8b2176b93bdb3c623da": "\n\\lambda = \\frac{6.4}{(\\ln(Re) -\\ln(1+.001Re\\frac{\\varepsilon}{D}(1+10\\sqrt{\\varepsilon}{D})))^{2.4}}\n", "54b430ee1c385478824d4af3ad937cd3": "\\mathcal{Z}(\\beta)", "54b43e16b0fc9112fc9c0509703aee09": "\\left | f^{\\prime\\prime} (p) \\right | ", "54b465d61b8f906ecff540bcff770851": "(i-2)", "54b47c3febd347753d743d28f8b6d605": "\\scriptstyle \\vec L", "54b4c0a1f812c94687b83dcc50e88834": "(\\mathbb{Z}/p\\mathbb{Z})^*", "54b4ce4d7a0096fbc5ceed7b71a3c0d0": " \\tau_{ij} ", "54b5eeb98e958a3eedf2c255939798ac": "R=\\frac{1}{4} \\sqrt{\\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.", "54b6bddaecc15e1417e01f1aa440c7b6": " (z,r,\\phi) \\rightarrow (h(r), \\, r \\cos \\phi, \\, r \\sin \\phi ) ", "54b6fcacd170068b96d78040ed36d41f": "|\\hat{u}| \\le \\|u\\|_{L^1}", "54b7539d3a5f9ae202d8ebd1e9e79fc3": "T^n = P_H U^n \\vert_H,\\quad n\\ge 0.", "54b7769a05a6a49df3d370ae3418f558": "B_\\nu(T)", "54b788afc04b2670da5833c8340872b2": " \\boldsymbol\\mu|\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Sigma \\sim \\mathcal{N}\\left(\\boldsymbol\\mu\\Big|\\boldsymbol\\mu_0,\\frac{1}{\\lambda}\\boldsymbol\\Sigma\\right)", "54b80057bac9faf7963f1d8b3526d4b9": " \\bar{R} \\ ", "54b87e210e70550940b34ff209c999d8": "{2^{2^{65536}}} - 3", "54b8811318f29805dbdf3d71c5e80301": "(A \\or B) \\to (A \\or B)", "54b8dbecbe31b7d30d3088ca8a7cf94e": "\n\\beta^T = \\beta \\left( \\frac{293 K}{T} \\right)^{0.8}\n", "54b990bbb173fa6ea23b758fb23b1a8d": "\\exists x_{m+1}\\ldots\\exists x_n \\, f_1(x_1,\\ldots,x_n,e^{x_1},\\ldots,e^{x_n})=\\cdots= f_r(x_1,\\ldots,x_n,e^{x_1},\\ldots,e^{x_n})=0.\\,", "54b9d04fcaeb97e192a45e803388164e": "\\forall b_n \\in \\Gamma(a_n)", "54b9e5988da92d83f179d631abb64028": " \\sigma_x ", "54ba1b9ca3c516d292b7f143b13c2d99": "\\sum_{n=1}^\\infty a_n", "54ba233fa217fdbb4534581f1a115c54": "\\gcd(a,b) = \\gcd(b, a \\,\\mathrm{mod}\\, b)", "54ba93c0cb497a1a53582f03af7f10ca": "p_{12} p_{34}+p_{13}p_{42}+p_{14} p_{23} = 0 ", "54ba9f1aacb5a2f76a649a4a69eab5c6": "\nu_t - u_{xxt} + (b+1) u u_x = b u_x u_{xx} + u u_{xxx}, \\,\n", "54bae8343994565c5d6c5d1d962695b8": "g[(t,x)] = -\\beta(x) dt^{2} + g_{S}[x]", "54bb06d383102aea1bf4f974382ac1be": "\\frac{dq}{dt}=\\frac{M}{B}", "54bb0734941f2a02e17c7808399a79d8": "\\ G(f) = \\frac{H^*(f)S(f)}{ |H(f)|^2 S(f) + N(f) }", "54bb25ba9acc81ea0684a1fee2fe6265": "\\tfrac{E(1-\\nu)}{(1+\\nu)(1-2\\nu)}", "54bb46b01c97a6b27209fc30f1c27ef3": "s = j \\omega_a \\ ", "54bb6176ed155c081d52bfe462a73906": "\\begin{align}\n x &= R( \\lambda - \\lambda_0), \\qquad\n y &= R\\ln \\left[\\tan \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right) \\right]. \n\\end{align}", "54bc7524d37270474fbafbaecb0a5470": "S_{22} = {(Z_{11} + Z_0) (Z_{22} - Z_0) - Z_{12} Z_{21} \\over \\Delta}", "54bcb588508be45153edbfdf57af1b33": "M_{max} = \\frac {q L^2} {8}", "54bd11cb05f75ff584372bcd966c7e9d": "Mod:Sign^{op}\\to", "54bd5142792eba6d100bd5eec4a69172": "\\frac {\\dot {\\sigma}} {E} + \\frac {\\sigma} {\\eta}= \\dot {\\epsilon}", "54bd5bdddef0a8aae657fff5b67280a5": "\\Delta_-(x-y) = \\langle 0 | \\Phi(y) \\Phi(x) |0 \\rangle ", "54bd6fcffd99fd43f82b61f9cc0ba19c": "P=\\frac{W}{t} = \\frac{(mg)h}{t}\\ ", "54bd92fd5a2f106e49e44c527e4ae588": "\\mu(\\tau^{-1}\\sigma ) = \\mu(\\sigma)", "54bdae835cbdd9398f9780b004faa2a6": "\nZ_\\mathrm{in} (l)=Z_0 \\frac{Z_L + jZ_0\\tan(\\beta l)}{Z_0 + jZ_L\\tan(\\beta l)}\n", "54be068524a387a48c0617e5d4f38d87": "Z = \\sum_{m=1}^r Y_m", "54bec0923f3924158b8969de252037fb": " p_{i}\\times p_{j}", "54bedf895d7c6608b88041e9c33d1a3f": "V_{in}", "54bf2971a04b5ad13cf11d970cf4cd81": "q_b = \\frac{\\bold{d} \\cdot\\mathbf{\\hat{n}}}{|\\bold{s}|} ", "54c04abd874b17b4d0b0b6e3d50bff68": "F(x,f(x))=0", "54c06198f6562bed6f438f745d7002ab": "S = \\operatorname{iso} C", "54c0af9eb105dacc157cc54c61cdf6da": "\\mathbb P(Z \\leq z) = \\Phi \\left(\\frac{z-\\mu t}{\\sigma t^{1/2}} \\right) - e^{2 \\mu z /\\sigma^2} \\Phi \\left( \\frac{-z-\\mu t}{\\sigma t^{1/2}} \\right)", "54c140c747eb3346db13cbf0c0587323": "y_p = t^2 - 2 t + 2", "54c16a0ec384c629934b0bcc749c78be": "\\sigma_{yy} -\\frac{\\sigma^2_{yz}}{\\sigma_{zz}} +|\\sigma_{xy} -\\frac{\\sigma_{xz}\\sigma_{yz}}{\\sigma_{zz}}|", "54c194ce3774375fa81687db13d8b036": "| y - z | > \\varepsilon;", "54c19ca946af5c01aeae5f3bcd80b1e0": "M_x=\\{p\\in M:f(p)\\le x\\}\\ ", "54c1a3f0498dbade385bd2fb716aac66": "(\\nabla^2+k^2)\\mathbf{E}(\\mathbf{x})=-\\left[ikZ_0\\mathbf{J}(\\mathbf{x})+ikZ_0\\mathbf{\\nabla}\\times\\mathbf{M}(\\mathbf{x}))+\\frac{iZ_0}{k}\\mathbf{\\nabla}(\\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x}))\\right]", "54c1baa98c56da260e01b4ad850e2b58": "TEE_m", "54c1ee32082a0b75da1f9e1b95f86a40": "=\\sum P(D|\\sigma, T, M) P(\\sigma|T,M)", "54c23cb40b241b429c4b82c7bc753d2f": "T\\subseteq C", "54c25f6f2030049c4689b7cead23dc46": "t_{\\operatorname{ev}} = \\frac{5120 \\pi G^2 M_{\\odot}^3}{\\hbar c^4} = 6.617 \\times 10^{74} \\; \\text{s} \\;", "54c27fd5d1fa5750b4b46acfeb98a912": "1 \\over x", "54c2eb38bcf68be880998368b52911b2": "(\\mu,\\tau)", "54c360c23d74e17d00df09a32b4f4682": "\n \\psi_= \n \\begin{pmatrix}\n \\psi_{22}^* \\\\ -\\psi_{12}^* \\\\\n \\psi_{11} \\\\ \\psi_{21}\n\\end{pmatrix}\n", "54c3feec630d4b02ddafdcf92587560e": "C'", "54c45d6d95864e9c81d2604fbad60445": " y = \\begin{cases} - \\frac{1}{\\kappa} \\log \\left[ 1- \\frac{\\kappa(x-\\xi)}{\\alpha} \\right] & \\text{if } \\kappa \\neq 0 \\\\ \\frac{x-\\xi}{\\alpha} & \\text{if } \\kappa=0 \\end{cases} ", "54c482d2fc09815e5be8123fe0ff2a4b": "H^0(M,\\mathbf{K}^*)\\xrightarrow{\\phi} H^0(M,\\mathbf{K}^*/\\mathbf{O}^*)\\to H^1(M,\\mathbf{O}^*)", "54c4e8b889d5f9e555f233f1e7ecf4e0": "[M+H]^+", "54c52e7ba3e2c4422a0d443f955f07cc": "\\scriptstyle (a + b)/2 >= \\sqrt {ab}", "54c57f34364e1d015a6cbf7697a14830": "[Q_a,Q_b]=\\gamma^{[i}_{ac}\\gamma^{j]}_{cb} J_{ij}.", "54c5979cdd8d0a935afc5d99d8b47366": "\\Delta E = \\bar{E_i} - \\sum_{i\\neq r} q_{ir}\\bar{E_r} \\, ", "54c60115e4e93207987ffba519e8435f": "g\\cap \\mathcal Q=\\emptyset", "54c60a38d6af3a8ff967b8d344f05213": "u(t),", "54c6159c8576f4287c42b61e2f1ea6b0": "\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\qquad 0 < b \\leq a", "54c61e2ff426e3eaee731bd7abe6d1a6": "f_{IF}=0 ", "54c63adc309741836ad79d52a1f81088": "\\ C_e ", "54c657b7759a9c1cc4ed3b606668573d": "x=x_{1},...,x_{m}", "54c6e8079c724422c23b06a3c62ea3d6": "G = B_0\\geq B_1\\geq \\cdots \\geq B_n = 1.", "54c72927e00d8570700d7c3c428417f7": "\n g_{mm'}:=\\Bigg((m,u)\\mapsto \\bigg((m',u')=\\big(g_{1mm'}(m,u),g_{2mm'}(m,u)\\big)\\bigg)\\Bigg) \\, \n", "54c744d5a1214cd6e98d7e6eaed7703c": "\\frac{\\Pi}{RT} = \\frac{c}{M} + Bc^2 + B_3c^3...", "54c7697b50c92e3fa84f1687ffcda214": "\\pi_\\lambda(g^{-1})\\xi(x)=|cx + d|^{-1-i\\lambda} \\xi(g(x)).", "54c7c1b011f357c9044d43f9a801a4a0": "{\\boldsymbol{\\theta}_c}", "54c84a71e6de661b07867c834e5bc136": "\\pi \\approx 3.1415925335.", "54c853b386928dba9f9cd8faa6790749": " \\exists n\\colon (n=0 \\to A) \\wedge (n \\neq 0 \\to B) ", "54c887a9736f082460513c9342ab7063": "\\mathbf{R}_2^{-1} \\mathbf{R}_1=\\mathbf{PDP}^{-1}", "54c89f8dc00db293007b664e4b4e6155": "\n\\begin{align}\nM_{[\\text{1 0 1}](\\overline{1} 1 1)}=&\\frac{(-1*1)+(1*2)+(1*3)}{((-1)^2+1^2+1^2)^{1/2}(1^2+2^2+3^2)^{1/2}}*\\frac{(1*1)+(0*2)+(1*3)}{(1^2+0^2+1^2)^{1/2}(1^2+2^2+3^2)^{1/2}}=&\\frac{16}{14\\sqrt{6}}\n\\end{align}", "54c8bb327fb7e90a0e7954b19b6881ac": "Z = a \\cdot 2a \\cdot 3a \\cdot \\cdots \\cdot \\frac{p-1}2 a", "54c9472190c8b8cae4ae9c3b4567c896": "EA + T = EAT \\neq TEA = T + EA", "54c996bb35dcaac0ddda138670c4bfed": "\\frac{1}{M(1, \\sqrt{2})} = G = 0.8346268\\dots", "54c9c960b9ad8eada6472d9180fc0fb6": "\\sum_{k=1}^{m_n} |d_{k,n}| = \\mathcal{O} (n^r)\\quad\\forall r>1/2", "54ca005f6c5a481d44245db5bce67a43": "V=1", "54ca411340941dd1836ed065053f904e": " f(z) = \\sin\\frac1z ", "54ca8a7a770c680b7a106d1abbbe827b": "U_m", "54ca9b276176795f29d4269b9b8acda4": "\\ P(A,B) = A + B", "54caf732af79e185ae2d545a5178ace0": " \\ell\\beta", "54cafa3a6d69c189cf2df3978fbdd435": " 1", "54cb0ebdd982e1fa5ecd1f4b70f33720": "\\operatorname{Br}(F/k)", "54cb2060f8b089e9a5b18f6f24d1e9b7": "y=\\frac{a\\sin^2 t}{t}", "54cb8922b13ca009df24a960e0a42d4b": "v=\\left( 1,\\dots,1 \\right)", "54cbad10135018e981b94a49407660df": "m = u - u_{xx} = c \\, \\delta(x-ct)", "54cc2ebe6aebf21f9a7b0c86d5054a28": " g_2, g_3, g_2', g_3' ", "54cc3bded9ffdccd15ecb0114db6a609": "f(t,x) = c(t,x)\\left(t^k+a_{k-1}(x)t^{k-1}+\\cdots+a_0(x) \\right)", "54cc493da7132d4a039f474a54e87ca2": "S_s = \\frac{1}{V_a}\\frac{dV_w}{dh} = \\frac{1}{V_a}\\frac{dV_w}{dp}\\frac{dp}{dh}= \\frac{1}{V_a}\\frac{dV_w}{dp}\\gamma_w", "54cc8994e9bcf5f2556e96d1852f9e8f": "2L", "54cc9d3c0a9cf1c43af380d8f98b9441": "\n k_{c}(2H+c_{0})(2H^{2}-c_{0}H-2K)-2\\lambda H+k_{c}\\nabla^{2}(2H)=0\n", "54ccf0ccddf064e0bd8b57538d845285": "Q=Q_n", "54ccfdb11917628af35132a06150d31d": "t_{TOF'}= t_{TOF}+t_{Delay}\\,", "54cd86b0f6a64b0682f35353f1811eb2": "E,F\\in\\mathcal{C}", "54ce0c322d618bfb9706e653fd021da6": "F'_i", "54ce3727af79c033441abcaa9dee9713": "i<\\xi", "54ce6120fedacdf13dfbfe3b847c9528": "\\frac{\\text{number of LCSAJs executed by the test data}}{\\text{total number of LCSAJs}}", "54ce9b5dc59fee10e8506ef2fa093502": "\\begin{bmatrix} \\dfrac{y_{22}}{\\Delta \\mathbf{[y]}} & \\dfrac{-y_{12}}{\\Delta \\mathbf{]y]}} \\\\ \\dfrac{-y_{21}}{\\Delta \\mathbf{[y]}} & \\dfrac{y_{11}}{\\Delta \\mathbf{[y]}} \\end{bmatrix}", "54cf5243f2f72f5e20bb0decb3326315": " 2 \\mathbf{E} \\mathbf{E}^T \\mathbf{E} - \\operatorname{tr} ( \\mathbf{E} \\mathbf{E}^T ) \\mathbf{E} = 0 .\n", "54cf7d684e102363526c430c93eb9296": "P = \\frac {F} S - 1", "54cfd42fbaab515ff19137f8f067250f": " \\mathbf{E} \\cdot \\mathrm{d} \\mathbf{s} = |\\mathbf{E}| \\cdot |\\mathrm{d}\\mathbf{s}|\\cos(0) = E \\mathrm{d}s ", "54cff57e92c68a3aee88a480c03dda11": "\\alpha=R_b^{-1}", "54d02c15388bb4f00eeb264b3adc5655": "|f(x)|\\leq C(1+|x|)^Ne^{-a\\pi x^2}", "54d0443aaf4e89d33a8af4b1521a47a2": "\\lambda^{-\\nu} J_\\nu (\\lambda z) =\n\\sum_{n=0}^\\infty \\frac{1}{n!}\n\\left(\\frac{(1-\\lambda^2)z}{2}\\right)^n\nJ_{\\nu+n}(z),\n", "54d097bed823972c2029904d3cce543f": "\\left \\{ a_n \\right \\}, n \\geq 1", "54d10aa91076c41a768fd647d81d651c": "\\mathbf{w}_{n-1}=\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)", "54d1667dfa97ff92d7d3f7e14eff5b9d": " \\sigma \\le \\omega \\le 2 \\sigma, ", "54d16954771f79a58f4d46d34153f373": "\\psi^* \\cdot \\psi = (\\mathbf E^2 + \\mathbf B^2)/2", "54d18f650d3596489970f2661589f2a0": "T(m) = 2 \\times T\\left(\\frac{m}{2}\\right) + \\Theta(m^{2})", "54d1a471d536f5e5c53e2c58fc49582d": "[\\phi'^i(x),\\phi'^j(y)]=[\\chi'^i(x),\\chi'^j(y)]=[\\phi'^i(x),\\chi'^j(y)]=0.\\,", "54d1b9c5878d8eaf612acec2b15464b5": " \\delta = \\frac{ \\delta_{max} \\sin^2 \\gamma }{ 1 + \\cos^2 \\gamma} ", "54d1e7444d698afe6283c94eb28d8213": "\\rho_S (t)= e^{-i H_{ S} ~t / \\hbar} \\rho_S(0) e^{i H_{ S}~ t / \\hbar} ", "54d20e48a941648e634c80750823fd5a": "\\left\\langle\\psi\\left|\\mathcal{T}\\left\\{\\frac{\\delta}{\\delta\\phi}F[\\phi]\\right\\}\\right|\\psi\\right\\rangle = -i\\left\\langle\\psi\\left|\\mathcal{T}\\left\\{F[\\phi]\\frac{\\delta}{\\delta\\phi}S[\\phi]\\right\\}\\right|\\psi\\right\\rangle", "54d228e9ebb86ede7383ebfed2b8236e": " k_A= 20,5^{\\prime\\prime}", "54d231bd8ffd7bfd9a94a796e704f374": " \\operatorname{sr}(G)=\\max_{H \\leq G} \\min\\{ |X|: X \\subseteq H, \\langle X \\rangle = H \\}.", "54d266b232f758e785b8fcf7023501c1": " x_i > y _i, \\forall~ i \\in S ", "54d29af7129e32fd9addeaa48310c2c6": "t_\\frac{1}{2} \\,", "54d2a185b6210960d6fc0efd2d226f87": "\\underset{^\\sim}a", "54d2a3eecd5b6777b887ac6417a2142e": "Z_0 = \\sqrt{\\mu_0 \\over \\varepsilon_0}\\ .", "54d2bf256f5357a739c2cd5deaf09148": "r \\ge l + 2", "54d2d1c2fca3137059cf95ca9763f0fc": "T^*=\\frac{(T-T_o)}{(T_s - T_o)}", "54d2dce4ee9b743d5335eca4674532d1": "\\hat{f}:\\mathbb{Z} \\to \\mathbb{C}", "54d2ff2e612ee6994c29568546ddf2e2": " I_k = [b_{k-1}~,~b_k)", "54d3864c1b1ad291de52bf65edde84fd": "E=s(s\\otimes s)^T+n(s\\otimes n)^T+n(n\\otimes s)^T+s(n\\otimes n)^T", "54d3942bb062a130ca0b30fa2eba6934": "\\sum_{i=0}^{2\\dim X}(-1)^i\\mathrm{Tr} f|_{H^i(X)}.", "54d40485d8a9a582fc2c0798deb0a210": " \\frac{1}{2m} \\left( \\frac{\\mathrm{d}S_{r}}{\\mathrm{d}r} \\right)^{2} + U_{r}(r) + \\frac{1}{2m r^{2}} \\left[ \\left( \\frac{\\mathrm{d}S_{\\theta}}{\\mathrm{d}\\theta} \\right)^{2} + 2m U_{\\theta}(\\theta) + \\frac{\\Gamma_{\\phi}}{\\sin^{2}\\theta} \\right] = E. ", "54d408cbae5e3c2dbe03c178c1dd6e9f": "{{P}_{\\theta }}", "54d40a31430f21952c47b74feac0d6a8": "C_{XZ} = 0", "54d42105487ea7629f670e20662b8427": "\\mathsf{(CH_2CH_2)O+O_2\\ \\xrightarrow{AgNO_3}\\ HOCH_2COOH}", "54d4396caf2568980d8b4e1b4caceb63": " \\mbox{Ext}(H_i(X),\\mathbf{Z}) \\cong \\mbox{Ext}(\\mathbf{Z}^{\\beta_i(X)},\\mathbf{Z}) \\oplus \\mbox{Ext}(T_i, \\mathbf{Z}) \\cong T_i. ", "54d461736366732e1b3d4e111bf8bf10": "\\gamma_2(t)=[-\\cos(nt),-\\cos((n+1)t)],\\quad t\\in [0,\\pi],", "54d575d5a7c8c013592dc497f1402ac4": " \\omega(A^*)= 1,", "54d608c83d47dcdcb4e6369d511d3154": "\\Delta x = \\hbar / |q|", "54d6530ffa4a301200ee1f49cc3c5ae4": "j^\\star = \\sigma T^4 ~, ~~ \\sigma = \\frac{2 \\pi^5 k^4 }{15 c^2 h^3} = \\frac{\\pi^2 k^4}{60 \\hbar^3 c^2}. ", "54d6ac1ed50532c558e4e76470901045": "\n\\begin{align}\n\\sum_{k=1}^n \\frac{\\theta}{\\theta+k-1}.\n\\end{align}\n", "54d6c2b6896137795c2310ecbbdaeb81": "u'' - 2xu'=-2\\lambda u", "54d7030062d75d661865ed2d54792849": "A=(7+2\\sqrt{2}+\\sqrt{3})a^2\\approx11.5605...a^2", "54d72f2423dcd54eb3dfa5dc16ecd7e4": "SM = {X_{AC} - X_{CG}\\over c}", "54d78b8618449c8022618ec118bf756a": "\\sigma(x) = a(x)^2 + x\\cdot b(x)^2", "54d7d54c0dd6da8ff0340b925f2701d9": "c_1 \\log n \\leq r_B(n) \\leq c_2 \\log n ", "54d833490546fbbdf8d548cfb4e10ec4": "A,B \\in \\mathcal{A}", "54d85c35ca8b8e7b6c5f419e8eee4d8e": "[T^{2^{j+1}}]_{\\Phi_{j+1}}^{\\Phi_{j+1}} \\leftarrow \\left([T^{2^j}]_{\\Phi_j}^{\\Phi_{j+1}} [\\Phi_{j+1}]_{\\Phi_j}\\right)^2", "54d867d540aa0d61176093b04cbe7172": "\n V_\\tau^S = \\sum_{i=t(\\tau-1)+1}^{t\\tau} V_i \\left(1-Z\\left(\\frac{S_i-S_{i-1}}{\\sigma_{\\Delta S}}\\right)\\right) = V-V_\\tau^B \\; .\n", "54d86f12f6e21c98ac301fdadb9a7143": "(26)\\quad \\tilde{\\omega}_{ab}=\\frac{1}{2}\\,\\Big(\\rho-\\bar\\rho \\Big)\\,\\Big(m_a \\bar m_b-\\bar m_a m_b \\Big)=\\text{Im}(\\rho)\\cdot\\Big(m_a \\bar m_b-\\bar m_a m_b \\Big)\\,,", "54d899250f697cff7099caffccce18fe": "a_1,\\,a_2,\\,\\ldots,\\,a_m", "54d91584f07420e8ad6676e24eac056b": "\\cos \\lambda \\ \\hat{\\lambda}\\,", "54d91e388bcf498cd3b2ff70070d5116": "\nw(a_i, b_j) = w\\text{(match)}\n", "54d99aa41842e50f6caa6a4e4234e2a6": "(gate1\\vee x1)\\wedge (\\overline{gate1}\\vee \\overline{x1})\\wedge (\\overline{gate2}\\vee gate1)\\wedge (\\overline{gate2}\\vee x2)\\wedge ", "54d9a98543b51ca6d578cfc59b94650c": " \\sigma_j \\in \\{1,-1\\} ", "54d9b5d34b9add571f48afac1cfbd540": "\\phi_T(\\mathbf{Z}) = \\frac{\\exp({\\rm tr}(i\\mathbf{Z}'\\mathbf{M}))|\\boldsymbol\\Omega|^\\alpha}{\\Gamma_p(\\alpha)(2\\beta)^{\\alpha p}} |\\mathbf{Z}'\\boldsymbol\\Sigma\\mathbf{Z}|^\\alpha B_\\alpha\\left(\\frac{1}{2\\beta}\\mathbf{Z}'\\boldsymbol\\Sigma\\mathbf{Z}\\boldsymbol\\Omega\\right),", "54da6c60162adfbad9d30c1449c22b18": "L = \\mathbf{r}_{uu} \\cdot \\mathbf{n}, \\quad\nM = \\mathbf{r}_{uv} \\cdot \\mathbf{n}, \\quad\nN = \\mathbf{r}_{vv} \\cdot \\mathbf{n}. ", "54da8873f55ab76fa1ff3d1f718d6021": "d(x,y) \\geq d", "54db4f864f1756f8233412aadcd7ef4d": "J = \\begin{bmatrix}\nJ_1 & \\; & \\; \\\\\n\\; & \\ddots & \\; \\\\ \n\\; & \\; & J_p\\end{bmatrix}", "54db6033d6b054fc6179430ddd015d7d": "y_{n}\\in \\varphi(x_{n})", "54db783e55a8ddf8eb694467805d4661": "(\\lambda, \\mu).", "54db9bd574aa7929e1c3bde8a27b701f": "\\eta=\\sqrt[3]{\\frac{V}{V_0}}", "54dbb9cbb20285163a561b1ffa942f74": "(I + N)^{-1} = I - N + N^2 - N^3 + \\cdots,", "54dbe5ef61d2f9977c7be03737833609": "s_1", "54dc196dbe424b8a9d3103629f34bf31": "\n(\\mathbf{J^TJ})^{-1}_\\text{even}={1\\over70}\\begin{pmatrix}34 & -10\\\\ -10 & 5 \\\\\\end{pmatrix}\n \\quad \\mathrm{and} \\quad\n(\\mathbf{J^TJ})^{-1}_\\text{odd}={1\\over144}\\begin{pmatrix}130 & -34\\\\ -34 & 10 \\\\\\end{pmatrix}\n", "54dc589a3721e09f0a7eccdbfd24d4d4": "b>1", "54dc8e472f74e44ac74bdd70ec28e2e2": "S(\\vec{k},\\omega)", "54dc989c76bed1b3637c96848f3ea240": "(x : A)", "54dcb1fc80e21351858da95d5bb59bfa": "\\sigma \\in \\textrm{End}(R)", "54dcbf30a6421f6a48779b0d287dc685": "n/r", "54dd11685d504aa2eff075007900b41f": "eff", "54dd1d674526cd57885f3a8f02b4aef2": "V_0 = R_0(A-B)\\,\\!", "54dd222a14206680ec3f37119af4b9b9": "\\log^2n) \\subsetneq", "54dd5c8437377e1cf6e19a2bc060705b": " Q = \\frac {1}{\\omega_0 R C} = \\frac {\\omega_0 L}{R} ", "54dda1e395e9affa3dea96f03c76651b": "\\psi_i = c_1(\\mathcal{L}_i) \\in H^2({\\overline {\\mathcal{M}}}_{g,n},\\mathbf{Q}).", "54ddc3c7d064822eed932015d8740336": "broken", "54ddd43e2eb59d25636da586ba7c1859": "\\begin{align}\n V_{in} &= IR + L\\frac{dI}{dt} \\\\\n V_R &= V_{in} - V_L\n\\end{align}", "54de4dfbea4b5f58fe20ebeced27cdba": "R_3=\\tfrac{1}{2}(\\sigma_1 - \\sigma_2)", "54de7c861a1adbaf940cdcf7bcbe32a8": "\\scriptstyle a_{j,k}", "54deba8016ed1466f5178ec8019ba514": "P = \\text{weight density} \\times \\!\\, \\text{depth}", "54df328d4637aa2a22f1bd538a413190": "q(x,t)", "54df57f6957543d3149053321133815b": "s_k = \\omega_c e^{\\frac{j(2k+n-1)\\pi}{2n}}\\qquad\\mathrm{k = 1,2,3, \\ldots, n}.", "54df78824f5d66b32529ebffa168e935": "\n\\boldsymbol{R} = (\\boldsymbol{A}^T \\boldsymbol{S_y}^{-1} \\boldsymbol{A} +\n\t\\boldsymbol{S_{x_a}}^{-1})^{-1}\n\t\\boldsymbol{A}^T \\boldsymbol{S_y}^{-1} \\boldsymbol{A}\n", "54e043643b98b769d84482cf5d463c78": "y^{(n+1)}=y^{(n)}-[\\nabla\\phi_k(\\mathbf{y}^{(n)},t_m)]^{-1}[\\phi_k(\\mathbf{y}^{(n)},t_m)-\\phi_k(\\mathbf{y}_m,t_m)]", "54e052ff90920421bfed09a1c2a49b92": "\\max_{x\\in X,v\\in \\mathbb{R}} \\ \\{v: v\\le f(x,u), \\forall u\\in U(x)\\}", "54e08f7c0edcb132a3262cf6c9ab6116": "\\bar a_{1,1}a_{2,n}+\\bar a_{1,2}a_{3,n}+\\cdots+\\bar a_{1,n-1}a_{n,n}", "54e0a33a23fece5504b9a4c162af53f1": "\\bar{y}_i", "54e0c58cbadbbb2b5c98dd2e632db538": "k[t]", "54e114ada49689b808027201822f217d": "m/n < 1", "54e150a9f272ca6070ec9ceb0dbe6818": "\\langle(\\widehat{\\mathbf{a}}_j^{\\dagger m}\\widehat{\\mathbf{a}}_k^n)_S\\rangle=\\int W(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)\\alpha_j^m\\alpha_k^{*n} \\, d^{2N}\\mathbf{\\alpha}", "54e152738813da61f1a3fc8a6511cdd9": " \n \\boldsymbol{\\mu}_a = \\left[ \\begin{array}{cccccc}\n 0 & 0 & 0 & 0 & 2 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0\n \\end{array}\n \\right] , \\quad \\boldsymbol{\\mu}_b = \\left[ \\begin{array}{cccccc}\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0\n \\end{array}\n \\right] ", "54e15da4633059c85ec8a937148e6eb2": "T_{eq}", "54e1659e841b9d4b90aea4e1b424aeed": "\\int_0^\\infty e^{-x} g(x) dx", "54e1af64fc7c9f70740a9de75e2449fe": "\\displaystyle{g_z(t)=e^{itz}\\chi_{[0,\\infty)}(t)}", "54e1f166d7130d0a6f71fcc8a2b1ed5a": "\\xi\\gg 1", "54e2190e961b1ef1373ec391946337ec": " (\\mu_1 \\times \\mu_2)(B_1 \\times B_2) = \\mu_1(B_1) \\mu_2(B_2)", "54e2715903ee48e16cbe9094f6ad8415": " Y = \\sigma / \\left ( \\Delta L/ L \\right ) \\,\\!", "54e29df48bbdf2756f03ac98fcf8b501": "\\mathbf{X}_k \\in A", "54e2f2365dc77cd798d9f973dc1570ec": "\\phi(\\rho,z)", "54e37a52fb3d8c12b076e61a191843c3": "p^0_{ii}=v_{i}", "54e37f550d8df11c316d88ef8d3f9eda": "\\operatorname{perm} (A) = (-1)^n \\sum_{S\\subseteq\\{1,\\dots,n\\}} (-1)^{|S|} \\prod_{i=1}^n \\sum_{j\\in S} a_{ij}.", "54e3807351c7bc206103c9314e112b9d": ",", "54e39f9e6b4006b053f51caa8a9a37cd": "{\\mathcal C}_n(x)", "54e3cd701761f2ccd2ac570d4de4db05": "F = k X", "54e3d73db8fedd6aa6c0629ddd0e0456": "B : H^{\\perp} \\rightarrow H, \\quad \\mbox{and} \\quad C : H^{\\perp} \\rightarrow H^{\\perp}.", "54e3dcc2ccf7a942484d6637ecc5cf46": "\\delta\\int p\\,dq=0", "54e3facd65a57b4ba2f8553dcb84a3c1": "\\textstyle{\\frac {\\log(6)} {\\log(1+\\varphi)}}", "54e4223982264afd5907e574cddc18a0": "N > d", "54e42e0025b83a8f4603147ac27ef039": "p_j=Prob({x_j}).", "54e4582f66a531f95eef8cf97a3d1b21": "P[S] + \\mathbb{R}^m_+", "54e4967479126be8b11dc999ed39b06a": "n_{pas} = {P}\\times \\frac{3600}{T_{tot}}", "54e4af4ac4d45394d3f22fce760ef7da": "\\ \\sin(a-b)=\\sin(a) \\cos(b) - \\sin(b) \\cos(a)", "54e51c2d36a4e929a5b7509305657464": "n > 1,\\,", "54e5338bf4b8e36981cd1e63b9db4781": "Z / 2 Z", "54e5523910c70a36b153d5d554ad76d8": "{f_{em}}(t)=- {d\\Phi(t) \\over dt}", "54e59fa769df35dea2d65bb88712586e": "X^T", "54e5ad249cb81c5e5267ab6564c47193": " \\displaystyle T = {m\\overline{v^2}\\over 3 k_B}", "54e5b4e3f4a19b7b86dd73140e077d9f": "V_L(s) = \\frac{Ls}{R + Ls}V_{in}(s)", "54e5b84b2daf8900e9beb4b047f86fb9": "\\Omega =\\omega _{\\mathrm{c}}+\\mathrm{Re} \\Sigma (\\Omega )", "54e5bed5ad403dcfb75550fe49ce1638": "4\\pi \\times 10^{-7} \\text{ T}\\cdot\\text{m/A}", "54e60e5dd6401a11d558e0c105bc4e5d": "Z_{Fish}=\\frac{P_{Fish}\\times{BCF}}{H}", "54e623c6cb64e78533252352f06699e6": "U = k \\sqrt{Q_1 Q_2}", "54e6267213f6b836f594430ec949ffd0": "\\tau_\\max=\\frac{1}{2}\\left|\\sigma_\\max-\\sigma_\\min\\right|\\,\\!", "54e6881ecd86831eae4afbd32cfc11fc": " \\Omega = -PV \\,\\;", "54e6e998381e6214020c3daff3b42a39": " W^{1,1}(\\Omega) ", "54e741322de3b5566bcad935a779e9c2": "x\\in\\,S", "54e773e3af224cac0229289d74462fbc": "\\displaystyle\\delta", "54e782e5d87605bace2bf6fade3fc0ca": "F(f_2)", "54e7893aceecbfa01446f6c67710d15c": "\\left[\\frac{\\hbar^2(k+K)^2}{2m}-E_k\\right]\\cdot\\tilde{u}_k(K)+\\frac{A}{a}f(k)=0", "54e7a4ca192aab84ae999e1e52442383": "x_{n + 1} = x_n - \\frac{f(x_n)}{f'(x_n)}", "54e7b5849d226abe45ff8504fc700fc4": " d(\\vec{x},\\vec{y})=\n\\sqrt{\\sum_{i=1}^N {(x_i - y_i)^2 \\over s_{i}^2}},\n", "54e7c1f9440cfd1359381cb856f7772c": "\\gamma<\\delta", "54e83ce2095df0d6d905f9cea24eaa4f": "\\frac{j^2 \\pi^2}{L^2}", "54e86de59dda54977b42dbf7506bd57a": "\\gamma_{A,B}:A\\otimes B \\rightarrow B\\otimes A", "54e88dfc56ec7fb6df203eed490a2a42": "t_0,\\ldots,t_{n + 1}", "54e89d25c783f8588b18c6c9a988b63d": "\\rho_q(\\bold{r}) = q \\delta(\\mathbf{r} - \\mathbf{r}_0)", "54e9629005ad7768f42da6a9751b00f6": "x_1 + x_2 + x_3 + x_4 = 3", "54e9870a3ada84bf9132419164b35820": " x(S) = \\ln(S) ", "54e9cd8b7aad41df5d1281df175317f5": " \\forall n\\in\\mathbb{N} \\ \\exists A\\subseteq\\mathbb{N} \\ \\forall x\\in\\mathbb{N} \\ [x\\in A \\text{ iff } x \\leq n].", "54e9cdf52bd7f10f86c7feffe05d1a51": "\n\\alpha_{m_1-j}(1) = m-2j , \\quad j = 0 , 1 , \\dots , m_1 - 1 ,\n", "54ea41bb57835934665e27281c261014": " A_z \\frac{\\partial^2 u}{\\partial z^2}=-fv,\\,\\!", "54ea4b1f1f99bba0ce8bc43ef192c561": "(a,b,c)\\cdot(x,y,z)", "54ea6a2dbcff50ce2fac71649df7d7ba": "0<\\alpha<1\\quad\\quad\\frac{1}{(\\alpha+1)(\\alpha+2) \\cdots (\\alpha+n-1)} \\sum_{i=1}^n\\frac{x_i^{\\alpha+n-1}}{\\Pi_k(x_1,\\ldots,x_n)}>\\frac{1}{n!}\\sum_{i=1}^n x_i^{\\alpha} ", "54ea7f8757722b4596a8ab669e7a2ddb": " j:X-D\\rightarrow X ", "54ea9049d1407ffb5441918daeffd380": "(2,3,-5,0,\\dots)", "54eaab477a9dc3780bfa999d627f1726": "\n\\textstyle\\left(10^{12} + 10^2 + 1\\right)^5\n", "54eacb3c813f6219bad39c91ba3aea98": "\\sigma = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\mu)^2}, {\\rm \\ \\ where\\ \\ } \\mu = \\frac{1}{N} \\sum_{i=1}^N x_i.", "54eb00d53fbc1d349040f2fad0900ee0": "=\\{ax + b \\mid (a,n) = 1\\}", "54eb16e8eed8198efb5ee823772e9b02": "\\angle QCB = \\angle QBA = \\angle QAC.", "54eb266d2f89f7a33324f89e2826743b": " X^{\\star}", "54ebe4c6b75e81d89e62828915ec4280": "ax+by =z", "54ec111de5986a2720df26f0f5eb7229": "i_\\max", "54ec4ea82d62298781a61c5890dbdaa5": "\\frac{dTR}{dP} = 1 \\cdot f(P) + P \\cdot f'(P)", "54ed226dc4b7589be90f19ca184a2747": "\\ \\beta<1", "54ed5f5792defa4adf51d66143664947": "\\alpha\\ ", "54ed62c41c8806ff9e54d3f23cf5deac": "\\int\\sin^n {ax}\\;\\mathrm{d}x = -\\frac{\\sin^{n-1} ax\\cos ax}{na} + \\frac{n-1}{n}\\int\\sin^{n-2} ax\\;\\mathrm{d}x \\qquad\\mbox{(for }n>2\\mbox{)}\\,\\!", "54ed90806ff3b81a48838b1a467286b0": "v_z", "54edcf605eba47e52e17c46be70b6d63": "\\text{join} \\colon {A^{?}}^{?} \\to A^{?} = a \\mapsto \\begin{cases} \\text{Nothing} & \\text{if} \\ a = \\text{Nothing}\\\\ \\text{Nothing} & \\text{if} \\ a = \\text{Just} \\, \\text{Nothing}\\\\ \\text{Just} \\, a' & \\text{if} \\ a = \\text{Just} \\, \\text{Just} \\, a' \\end{cases}", "54ee615dfcdc71fa41bda38b95385591": "\\nabla \\cdot \\mathbf a = \\frac{1}{\\sqrt{g}} \\sum_i \\frac{\\partial}{\\partial q^i}\\left(\\sqrt{g} a^i\\right) = \\frac{\\partial a^1}{\\partial x} + \\frac{\\partial a^2}{\\partial y} + \\frac{\\partial a^3}{\\partial z}.", "54ee625d179d6422dcad1768bec8e02c": "S(\\mathbf{\\beta}, b) = \\sum_i | \\mathbf{x}'_i \\mathbf{\\beta} + b - y_i |", "54ee9e96ef8d65fe03a24572bd5a1b0a": "\nP(\\zeta) = A \\cos{\\omega_{k} \\zeta} + B \\sin{\\omega_{k} \\zeta}\n", "54eec762cb7576a4b28411a8bdd7b9ee": "\\chi = (1.97\\times 10^{-3})(E_{\\rm i} + E_{\\rm ea}) + 0.19.", "54ef294bde6731b194a6965dcfcda74c": "\\mathbb T \\cong \\mbox{U}(1) \\cong \\mathbb R/\\mathbb Z \\cong \\mbox{SO}(2).", "54ef775ec9cb0e375abee7bbec0271b3": "U(N)", "54efeff6813f42e70720a7c6e816d079": "\\text{Price}(\\text{complete game})\\times\\text{Price}(\\text{Red Sox win}\\mid\\text{complete game})-\\text{Price}(\\text{Red Sox win}).", "54f023fe4c8d674692fc85156dadebb1": "|\\tilde{\\chi}(\\omega)|", "54f026770cb9b7499885f80933b5eff3": "g^s = t y^{c}", "54f0b885347248056c022c09ec73e0b7": "\\scriptstyle I \\,+\\, J \\;=\\; R", "54f0bf75ded2e146926c1441a62a34b8": "f_2(x)\\,", "54f0e998ef44e263599c9632e5cd37c5": " R_1:= \\max\\Bigl\\{ 1 , \\sum_{0\\leq k 1 , I(0)> 0 \\Rightarrow \\lim_{t \\rightarrow +\\infty} \\left(S(t),I(t)\\right) = EE = \\left(\\frac{N}{R_0(1-P)},N \\left(R_0 (1-P)-1\\right)\\right). ", "54f2bfc8d9ffd013e88dbff7d4bf29e9": "\n\\psi (x,t)=\\left(-i\\frac{Et}{\\hbar }\\right)\\phi(x)\n", "54f2d182ac66f1b9d5f547dec3a69add": "H^2 = \\left(\\frac{\\dot{a}}{a}\\right)^2 = \\frac{8 \\pi G}{3}\\rho - \\frac{kc^2}{a^2}", "54f2d60b20070ce5f03ca626ff745a58": "\\tilde{\\boldsymbol{U}}\\, =\\, \\boldsymbol{U}\\, +\\, \\frac{\\boldsymbol{M}}{\\rho\\,h}.", "54f30e6c9e74523cdb50c7708a256980": "\nS_w(p) =\n\\begin{bmatrix}\n\\int w(r) (I_x(p-r))^2\\,d r & \\int w(r) I_x(p-r)I_y(p-r)\\,d r \\\\[10pt]\n\\int w(r) I_x(p-r)I_y(p-r)\\,d r & \\int w(r) (I_y(p-r))^2\\,d r\n\\end{bmatrix}\n", "54f32557c4e5146b804d0beb61127a44": "\n \\hat\\sigma_{ij} = \\frac1R\\, \\hat\\varepsilon_i^\\mathsf{T} \\hat\\varepsilon_j .\n ", "54f32b2f577d31f514e8e46c35839641": "\\lceil", "54f34e063fa67e74a2dae739a1039a0d": "\\Omega\\subset R^n", "54f3a6296e4b69ccb441161db0283707": "\\sqrt{\\sigma_{\\pi_r}^2+\\sigma_{\\pi_{ref}}^2}=\\sqrt{4.2^2+0.3^2}=4.21", "54f3c04f73e389a5d71716b89b201d9e": "ab<0", "54f3ef6f2b9c8e9eb25490d57f57b6bc": "\\frac{2\\left(-i\\beta t\\right)^{\\!\\!\\frac{\\alpha}{2}}}{\\Gamma(\\alpha)}K_{\\alpha}\\left(\\sqrt{-4i\\beta t}\\right)", "54f4085bbf8796c028e16aeb626803fd": "=\\frac{dR}{dt} \\mathbf{u}_R + \\omega R(t) \\mathbf{u}_{\\theta}, ", "54f436c3f36f4d3b23c1e89d3062b65d": "\\sum_{i=1}^{n-1}t(i)", "54f43e9a01d5c98877c183afa1313110": "\\rho'(r,\\theta,\\phi)=(R/r)^5\\rho(R^2/r,\\theta,\\phi)\\,", "54f455352227ffff31a4747ca87057da": "Z[[q]]", "54f463fe2c7fdab2bb74dfc55b7402bd": " a_2 ", "54f46e9c1362445f24651ed47387e0c1": " \\pi R^2", "54f48c2b217dfd4531cdaad8f988870f": "\\|x\\|_2 \\leq \\|x\\|_1", "54f4b4ebcb1798e0a67667b8251e0ab0": "R = {R^m}_m", "54f4d2f6f0f4d3691d55aa9f27a9240d": "\n \\begin{align}\n \\sigma_{xx} & =-\\frac{2P}{\\pi}\\frac{x^2z}{(x^2+z^2)^2} ~;~~\n \\sigma_{zz} =-\\frac{2P}{\\pi}\\frac{z^3}{(x^2+z^2)^2} \\\\\n \\sigma_{xz} & =-\\frac{2P}{\\pi}\\frac{xz^2}{(x^2+z^2)^2}\n \\end{align}\n ", "54f53aeafdf11b19b1a0e955284e3d53": "\\mathbf{s} \\in \\mathcal{D}", "54f5dd3a32560516089aaea078f19e91": "W_{\\mu}", "54f63fc7fa03d74b46766d8de50c7a36": "c/\\omega_{pi}", "54f683fd7b9bfd91b19eb32247ca8f57": " (tan \\beta_2 + tan \\alpha_1) ", "54f6b2c4bbdcc8cf0c1d98c1d2dad69f": "\\phi\\leftrightarrow\\psi\\in T", "54f702f382929093229b788ff33b7207": "\\nabla^2\\phi=0 ", "54f751c5ab3779fb863912fb63e2e458": "\nx v(x,y) - y u(x,y) + w(x,y)\n", "54f784ad81f60427de4ca52bc48c4c52": "\\mathcal{H}(T, I)", "54f7865c6e103ab90c5514c35b501e83": "f(\\mathbf{y}) < f(\\mathbf{x})", "54f79137363d1c4895ca8f081897dbe2": "\\scriptstyle \\pi/4", "54f797067443d3a9a5e3056f15bcf403": "\\ Diff_k(\\tau)=\\frac{1}{(1+\\frac{\\tau}{\\tau_{D,i}})(1+a^{-2}(\\frac{\\tau}{\\tau_{D,i}})^{1/2}}", "54f7c9c10b0fa4084ae022706b5403a4": "\n \\min_{B,R} \\|X - BR\\|_{F}^{2}\n", "54f7dd38f6e3ebe510f9dde5e197ce2d": "\\mathbf u_0", "54f7ef1990f1ea415ae1614c33249483": "0.5\\,", "54f82f03368d4d52fad841acd62d4b42": "T_{ab}=0", "54f83a4348feadb0c00e0121b2f64012": "\\lambda = c_2/(xT)", "54f8815d7776d7700237bb66462fb3ff": "\\displaystyle{\\mu_{F\\circ G^{-1}} \\circ G = {G_z\\over \\overline{G_z}} \\mu_F,\\,\\,\\, \\mu_{G^{-1}\\circ F}=\\mu_F.}", "54f8ab9fa950fa4afa5c8240ff9d91c5": "e=\\varepsilon=\\sqrt{\\frac{a^2-b^2}{a^2}}\n =\\sqrt{1-\\left(\\frac{b}{a}\\right)^2}\n =f/a", "54f8cfb7bb6948579ee81cf89d2ef264": "\\scriptstyle{r'\\over c}", "54f8d54b66385f3acd0276b28a7db57c": "\nh(\\mathbf{X})= \\frac{\\mathbf{X}}{|\\mathbf{X}|^2}.\n", "54f9047f83f9de92190994f5d5fe27a7": "R\\bar 3m", "54f926f921f4665763a621a571249c0a": "r_\\mathrm{e}=\\frac{1}{4\\pi\\epsilon_0}\\frac{e^2}{m_\\mathrm{e} c^2}", "54f96d03693c2573953c55ec6f1118ce": "\nD = {1\\over 2|E|}\n\\begin{bmatrix}\ndeg(p_1) \\\\\ndeg(p_2) \\\\\n\\vdots \\\\\ndeg(p_N)\n\\end{bmatrix}\n", "54f9954e79b25f35e371b8478f78e3c6": "d_Y\\left(f(a),f(b)\\right)=d_X(a,b).", "54f9b8c1f12fc38403943db68c6e5252": "\\zeta = {1 \\over 2R}\\sqrt{L\\over C}", "54f9e1f9b4c2e06bc3fe1e12ca194c24": "\\sin(36^\\circ)=\\frac {\\sqrt{3-\\varphi}}{2}\\text{ and }\\sin(72^\\circ)=\\frac{\\sqrt{2+\\varphi}}{2}.", "54fa119d3a7b7029c3630bd366ef1502": " v \\propto r", "54fa1a29141817fd4c1d1b11b00dc1ca": "M =\n\\begin{pmatrix} m_{1,1} \\cdots m_{1,n} \\\\ \\vdots \\ddots \\vdots \\\\ m_{t,1} \\cdots m_{t,n} \\end{pmatrix}", "54fa6be8aea33315e72e92d37cceb8f2": "S_m(n) = {1\\over{m+1}}\\sum_{k=0}^m {m+1\\choose{k}} B_k\\; n^{m+1-k}, ", "54faf70d8a7482d622ce3e4fc924aac1": "S: z\\mapsto -1/z", "54faf94c80f64432df389bd4fe23abef": "\\mathbf{p}=\\int d^3\\mathbf{x'}\\mathbf{x'}\\rho(\\mathbf{x'})", "54fafbd3aa3458a021f252ca31322463": "\\xi_j^{(n)}", "54fb18ac025cca8e5e729fbb0570f2ca": "G_a(z)", "54fb1def8be73ebbfb5bdca04711bdc1": "0 \\notin \\Phi", "54fb88cbb7db385f8fb1eb448fe31cd6": "E_{0}=500", "54fb9bd0b58f4077eb2bfe7ac10befa9": "L_{f}h(x) = \\frac{\\operatorname{d}h(x)}{\\operatorname{d}x}f(x),", "54fbf08a7266bfcfe77293f068f4c3b3": " \\pi^{\\prime k\\ell} = x^{\\prime k}\\stackrel{\\leftarrow}{\\partial}_{i} \\pi^{ij} \\partial_{j}x^{\\prime \\ell} ", "54fbf38cf649866815e0fefc46a1f6c7": "0.4", "54fbfafce39923187a3130b91b36de0a": "\\sigma_x^2 \\sigma_p^2 = \\left( \\langle x^2 \\rangle - \\langle x \\rangle^2 \\right)\\left( \\langle p^2 \\rangle - \\langle p \\rangle^2 \\right)\\ge \\left( \\langle xp \\rangle - \\langle x \\rangle \\langle p \\rangle \\right)^2 + \\frac{\\hbar^2}{4} ~.", "54fc94eb1f1c2fe901834b8499fddb04": "\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}(\\mathbf{x},t)~\\text{dV}\\right) = \n \\int_{\\Omega(t)} \n \\left(\n \\frac{\\partial \\mathbf{f}(\\mathbf{x},t)}{\\partial t} + [\\boldsymbol{\\nabla} \\mathbf{f}(\\mathbf{x},t)]\\cdot\\mathbf{v}(\\mathbf{x},t) +\n \\mathbf{f}(\\mathbf{x},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\mathbf{x},t)\\right)~\\text{dV} \n", "54fc9c9409107081b51b8407c09ba6bc": " C_2 ", "54fcbc476cff42331c92d7b852c6db97": "\\omega t+\\theta", "54fcdccb572c95bd502fc7744aeaf8df": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {c_2} {c_1} \\frac {d_1} {d_2} \\frac {m_1} {m_2}\n= \\frac {l_2} {l_1} \\frac {d_1} {d_2} \\frac {l_1} {l_2} = \\frac {d_1} {d_2} \\,,", "54fd19762c4928ff7b5b0da8e06a18da": "\n\\hat{H} = - \\tfrac{\\hbar^2}{2}\\;|g|^{-1/2}\n\\frac{\\partial}{\\partial q^i} |g|^{1/2} g^{ij} \\frac{\\partial}{\\partial q^j},\n", "54fd43f24dd9657d483655828f65d579": "\\operatorname{MSE}(\\hat{\\theta})=\\operatorname{Var}(\\hat{\\theta})+ \\left(\\operatorname{Bias}(\\hat{\\theta},\\theta)\\right)^2.", "54fd7a313b4517967e0f5ed3b16a9a14": "\\mathbf{w}_{(1)}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\left\\{ \\frac{\\mathbf{w}^T\\mathbf{X}^T \\mathbf{X w}}{\\mathbf{w}^T \\mathbf{w}} \\right\\}", "54fdbefa78bf0d5649dbb1dc3336de1b": "\\,\\cos(\\omega z)\\cos(\\phi z) = [\\frac{e^{-i\\omega z} + e^{i\\omega z}}{2}][\\frac{e^{-i\\phi z} + e^{i\\phi z}}{2}] ", "54fded9f02e9fe05bb7f960105afe6fd": " {}^3\\!P \\, ", "54fed29f820a057ee6cba9d6f02fc773": "n\\phi(n),", "54ff0e1154ea2a35a612f93a5dbd7ea9": "\\widehat{\\mu}", "54ff3c93c5b75dcc59a3c154357b0208": "(1 + x)^r \\leq 1 + rx\\!", "54ff4612127333bba2c9bd9a6caab817": " V\\,k_v(V) = P\\,k_p(P) ", "54ff69e4bb34f2c847af06ad59bcc699": "E_\\mathrm{spring} = E_\\mathrm{force}", "54ff7e03e03f851b775e0b3e36c3fe0b": " v^{k}_{k+1} = \\frac{a^{k}_{k+1,k}-\\alpha}{2r}", "54fff897f8d5bb132002da6ebfc7b77b": "f(\\cdot,\\cdot,\\cdot)", "550037413dc925e954f55c7cbbd651c6": "\\{1,\\ldots, y\\}", "550083ed2a4697be84140c4ce8829c1d": "\\mathcal D_0, \\mathcal D_1, \\mathcal D_2,\\dots\\,", "5500afbe723f7aad685e82efe44ce19a": " \\psi_1 = \\angle DVE, ", "5500f945fd90ddfe810f7ca6bdd89cee": "R_0' = - R_0 \\,\\!", "5501486e46d1f151d6fe1c4fd7176807": " \\overline G(X)", "550164428210b1570405bdda19ca61f7": "\\boldsymbol{\\varepsilon} = \\begin{bmatrix}\n0 & 0 & \\epsilon_{13} \\\\\n0 & 0 & \\epsilon_{23}\\\\\n \\epsilon_{13} & \\epsilon_{23} & 0\\end{bmatrix}", "55017c4a61b64587a89c7844dcc70fb2": "E^{\\ominus} \\left( \\mathrm{H}^{+} \\right ) = E^{\\ominus} \\left( \\mathrm{H}^{+} \\vert \\mathrm{H} \\right ) = 0", "550188f707122f5872d644ce07afe12f": "\n\\begin{align}\ng^{H(m)} & \\equiv g^{xr} g^{ks} \\\\\n& \\equiv (g^{x})^r (g^{k})^s \\\\\n& \\equiv (y)^r (r)^s \\pmod p.\\\\\n\\end{align}\n", "55018e39117cb2ff8bf6d2f4ca7eae9d": "\n T= \n \\frac{1}{2} \\iint dx\\, dy\\; \n \\left[ \n \\int_{-\\infty}^\\eta dz\\; \\rho\\, \\left| \\bold\\nabla \\Phi \\right|^2\n +\n \\int_\\eta^{+\\infty} dz\\; \\rho'\\, \\left| \\bold\\nabla \\Phi' \\right|^2\n \\right].\n", "5501901aed0c86846dba22b882171d10": "R = 0.04\n", "5501928ec9052d45f6a527e5b8dba46e": "\\frac{1}{P_0}\\prod_{i=1}^{c} \\binom{m_i}{x_i}\\omega_i^{x_i}", "5501f3f7d8f785a0e2bee18501aa5bf6": "{(Y-\\mu)}^{-2} \\sim\\,\\textrm{Levy}(0,1/\\sigma^2)", "5502082b3c812297f1ac7960770e967b": "\\cos\\theta/\\sin\\theta=cotan\\theta", "55025645121e0bb9e54a98c780112621": "\\mathcal{A} \\star \\mathcal{B} = \\bigcup_{\\alpha \\in \\mathcal{A}, \\beta \\in \\mathcal{B}} (\\alpha \\star \\beta).", "5502772d0527d60a5d854bee140d47aa": " e_a = \\frac{\\partial}{\\partial x^a} ", "5502866cdcc815683b18eaa79d138412": "AE=0.14F - 0.36C + 1.27T", "550287dacbfb17f1bf66f247a2f5da5b": "x(1-x)\\leq 1/4\\,\\!", "550296eb9c4da083dee3456ced173286": "f = gh", "55029dbd0db05f8c0ebf9e3ac6174acb": "\\tau_H={\\mu r \\over \\gamma}", "55029e4f43cf85da6e12db2f31d689ae": "\\tfrac{n}{n} = 1", "5502afed8b4ef25c78df800ffd4528a5": "M h/2", "55153d2464f5f57d0195cc516938d38f": "L_{k + 1}(x) = \\frac{1}{k + 1} \\left( (2k + 1 - x)L_k(x) - k L_{k - 1}(x)\\right). ", "55157b64df1ea55eaf295bebea835e36": "\\psi_{0}\n\\left(\\mathbf{r}\\right)", "551588bc53a18c24dc7512c1eb4aef6e": "\\mathbf{p}\\rightarrow \\mathbf{p} + m\\mathbf{v}", "5515924d1c37f6fea578003a882d3060": " \\begin{align} \nv_p &= \\sqrt{\\frac{K+(4/3)\\mu}{\\rho}} \\\\\nv_s &= \\sqrt{\\frac{\\mu}{\\rho}}.\n\\end{align}", "5515944ce4bb0674b52ba5e602efcd0a": "\\frac{\\partial^2 \\epsilon_y}{\\partial z \\partial x} = \\frac{\\partial}{\\partial y} \\left ( \\frac{\\partial \\epsilon_{yz}}{\\partial x} - \\frac{\\partial \\epsilon_{zx}}{\\partial y} + \\frac{\\partial \\epsilon_{xy}}{\\partial z}\\right)\\,\\!", "5515d6a338b4fd4fbf9a9312c904c5c5": "T_k(x)", "5515f4c69c238b61e6af0a8b530b47f5": "x=1/w", "55162ec297321638d89621984269cc93": "\\mathrm{ H_2SO_4 + \\ RbCl \\longrightarrow\\ RbHSO_4 + \\ HCl}", "55167e53b4f4c967fdb55843fc93cee0": "g(h)=\\lim_{t \\to 0} \\frac{ f(x + th) - f(x) }{ t } ", "5516eb97912eaf2b6a40c1a7932879e6": "\\gcd(x + 2^{2^{n-1}} y, F_{n}).\\!", "55171d5f317fccbb2bc46ef2dc28796d": "\\infty\\infty", "551727673a81ffe00e0306ddef437149": " \\gamma _i ", "55179be5528d99b73d597edecdc4e553": " \\frac {f(x_n)}{f^\\prime(x_n)}+\\left(\\alpha-x_n\\right) = \\frac {- f^{\\prime\\prime} (\\xi_n)}{2 f^\\prime(x_n)}\\left(\\alpha-x_n\\right)^2 ", "5517a70c1ffee462f8c48811f596fb71": " \\psi_i: \\mathbf{R}^n \\rightarrow \\mathbf{R}^n, \\quad i=1, \\ldots , m ", "551815620d60b875e6f45fe180fd0c3c": "A^{2+} + B \\to A^+ + B^+", "551835d87f4a0aae4f583695bf8c6045": "\\begin{align}\n& u_x'=\\frac{u_x+v}{1 + (v \\ u_x)/c^2}\\\\\n& u_y'=\\frac{u_y/\\gamma}{1 + (v \\ u_x)/c^2}\\\\\n& u_z'=\\frac{u_z/\\gamma}{1 + (v \\ u_x)/c^2}\n\\end{align}", "551894444bf3cb0cc8350594310e2e99": "\\mathcal{O}(n^2)", "55189bd6679920647413d7cb98944817": "p^*=p=p^2", "5518acf4b1d4a212f2ac3f19a12743e7": "\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}", "5518addeeacd363cd6b3959680191d83": "\\tilde{R} = R_f", "5518c3d07dd1169f7714a9ce925c0054": "T_{n,n}(z)\\to e^{\\frac{\\alpha}{2}}z+b\\beta ", "5518c4291598ee12e5b75f304190a181": "\nn \\geq 2^{{h(\\text{root}) \\over 2}} - 1 \\leftrightarrow \\; \\log_2{(n+1)} \\geq {h(\\text{root}) \\over 2} \\leftrightarrow \\; h(\\text{root}) \\leq 2\\log_2{(n+1)}.\n", "5518d45800163f7e4d3ee1a316e53865": "S_\\mathrm{total}", "5518de9cf2311ed47dec975948f27faf": "(x, y) \\wedge (z , w ) = (x, w )", "55193f84961c4a4b9069d67f594c783b": " 1 = \\sum_{s_z} \\int\\limits_R d^3 \\, \\mathbf{r} | \\mathbf{r},s_z\\rangle \\langle \\mathbf{r} , s_z | ", "551981b34ee4ed3f7f344aff0aca9073": "\\vec \\nabla \\cdot \\vec \\omega = 0 ", "55199f99ab8df806d898e8471e3e2e86": "(39, 5, -4)", "5519b5c897ce25833bd1fb8ce8839db8": " d\\mathbf{B} = \\alpha_{\\rm B}\\frac{I d\\mathbf{l} \\times \\mathbf{\\hat r}}{r^2}\\;,", "5519d3c021ee9108499f1d8c7a61091f": "L(v)", "5519f92f58ab04dc30709370db670e4a": " f.\\check{f}: [D \\times D^{'} \\rightarrow D^{''}]\\rightarrow [D\\rightarrow [D^{'}\\rightarrow D^{''}]", "551a12f4dfc6bd545fcb1d972c7e0385": "(x_1,y_1)\\cdots(x_n,y_n)", "551a3d7d5630a81ff21cc0b33c436cdf": "\\sigma^2_b(t)=\\sigma^2-\\sigma^2_w(t)=\\omega_1(t)\\omega_2(t)\\left[\\mu_1(t)-\\mu_2(t)\\right]^2", "551b216f1375f1f22e30b7546839c779": "\\scriptstyle \\log_2{(1+x)}", "551bea1cfb72a0dd9590d10cdfca0c4d": "n! = \\prod_{\\text{prime }p\\le n} p^{s_p(n)}, ", "551bf72d794a2e60b291694fdd8659d7": "82_{11} \\ ", "551c4941161416ea350f4bec76dbaa89": "x^n,", "551c703ea3c2d73dcdb1982678604bb3": "y^2-z^2=1,", "551c705b5fb170db6c00fbd5b2602e23": "\\mathbb{R}^{1,1}/boost", "551c8b1eded080f55f0d7d5724cbc50d": "a_0 + \\sum_{n=1}^{N}\\Bigg[{\\frac{a_n \\alpha^n}{2^{n-1}} \\sum_{k=0}^{n} {{n \\choose k} \\frac{e^{j(n-k)(\\omega t + \\phi)}e^{-jk(\\omega t + \\phi)}}{2}}\\Bigg]} \n\n=a_0 + \\sum_{n=1}^{N}\\Bigg[{\\frac{a_n \\alpha^n}{2^{n-1}} \\sum_{k=0}^{n} {{n \\choose k} \\frac{e^{j(n-2k)(\\omega t + \\phi)}}{2}}\\Bigg]}\n", "551c8d2a02c87c587ac64328926090a8": " r_1 \\neq r_2 ", "551c989ea4ff61b6cd94c985558203c1": "2\\sin x \\cos x = \\sin(2x) .", "551c99091898cc27c01c64762643f3e5": "V(r)=\\frac{6}{\\pi}r^2+O(r^{1+\\varepsilon}).", "551cfad5fa384988eed1319d77a8b907": " \\boldsymbol{\\tau} ", "551d4bdbbd636f01a8fd6d1445002bbf": " S^2 \\cdot X^2 = 1.25 \\approx 386.3 \\ \\hbox{cents} ", "551db63dbfb674bc509161871a8dc740": "\\mathcal{G}_\\infty", "551dcd500ea811cbbb52786e287beafe": "\\Psi(x,y,z)=\\psi(x)\\phi(y,z)\\,\\!", "551dea43674af6291f343a7de1788ae0": "n = \\frac{l^2-l}{2} ", "551e0b5618806d93d7c3ac06371b51b8": " \\sigma(\\tilde{\\nu}) ", "551e2e9cf3bb64c7ca4abec99c126229": "H^{q}(K(\\pi,n),G)", "551e38c639482aa419552cd85f1e384d": "\\psi(\\mathbf{r})=j_l(kr)Y_{lm}(\\theta,\\phi)", "551e40c1f53148e6d57f9850e89302cb": "\\int_0^{\\infty}x^\\alpha e^{-x} L_n^{(\\alpha)}(x)L_m^{(\\alpha)}(x)dx=\\frac{\\Gamma(n+\\alpha+1)}{n!}\\delta_{n,m}~,", "551e82a450e60bb3060cef2e8d9be036": "x_1^2", "551ede6b6a82c1585c1d90f564ed5329": " \\Omega_k ", "551f15bcf1382b935bd819a9cee6521c": "\\varphi_X(t) = 1 + it\\mu + o(t), \\quad t \\rightarrow 0.", "551f2a474ec8d25ea8efe85fc86782be": "\\frac{dP(r)}{dr}=-\\frac{G}{r^2}\\left[\\rho(r)+\\frac{P(r)}{c^2}\\right]\\left[M(r)+4\\pi r^3 \\frac{P(r)}{c^2}\\right]\\left[1-\\frac{2GM(r)}{c^2r}\\right]^{-1} \\;", "551f64a55f9c38c88838a52840d54b84": "n_j(\\mathbf{r})", "551fb42bac42e8b0d44c8896b05d6409": "f_{1}", "551fedfc612260cbfa0a9fb02593510e": "\\alpha = a[\\mathbf{f}A]\\theta[\\mathbf{f}A] = a[\\mathbf{f}]\\theta[\\mathbf{f}]", "5520437ced61a47c767d2b1f74b3b8cf": "= z", "55207eb665e39a1866225c825908a839": "H_{\\frac{1}{2},3}=8-6\\zeta(3)", "5520928338a008bc559352647e5d21e7": "t_{diverge} = 6", "5520f5b8a57b90da597254ab292a5dfd": "F(z,w) := |z|^2+|w|^2=1,", "552123ca481d7c348cc2aaa0063e61b7": "\\ u_i", "55222a2453bcdf4fb73cd8f1f3193f35": "0.\\overline{8}4615\\overline{3}", "552236e553ffaa083e7b820d53541138": " f(y) = f(x) + \\nabla f(x)^T (y-x) + 1/2 (y-x)^T \\nabla^2f(z) (y-x) ", "5522552b632ec7f87493b35145d38797": " \\Delta E \\, \\Delta t \\ge h ", "55225938da4ac18cb71b4b3d883bb5f1": "H^0(P,\\Z) \\cong \\Z, \\ H^1(P,\\Z) \\cong \\Z, \\ \\text{and} \\ H^2(P,\\Z) \\cong \\Z. ", "55226f624917656b11ceb461bca46f37": " x_1:A_1, x_2:A_2, \\ldots ", "552272f6ca68944bae94ebf5aced22f6": "\\displaystyle 4^n (5n\\ln n).", "5522812f47f719ece65b617d2d188799": "\\mathbf{E}_1", "5522d0001dc5704dea2e78d98e2a45e2": "\n\\begin{align}\n0 & = \\delta S \\\\\n & = \\int \n \\left[ \n {1 \\over 2\\kappa} \\frac{\\delta (\\sqrt{-g}R)}{\\delta g^{\\mu\\nu}} + \n \\frac{\\delta (\\sqrt{-g} \\mathcal{L}_\\mathrm{M})}{\\delta g^{\\mu\\nu}}\n \\right] \\delta g^{\\mu\\nu}\\mathrm{d}^4x \\\\\n & = \\int \n \\left[ \n {1 \\over 2\\kappa} \\left( \\frac{\\delta R}{\\delta g^{\\mu\\nu}} +\n \\frac{R}{\\sqrt{-g}} \\frac{\\delta \\sqrt{-g}}{\\delta g^{\\mu\\nu} } \n \\right) +\n \\frac{1}{\\sqrt{-g}} \\frac{\\delta (\\sqrt{-g} \\mathcal{L}_\\mathrm{M})}{\\delta g^{\\mu\\nu}} \n \\right] \\delta g^{\\mu\\nu} \\sqrt{-g}\\, \\mathrm{d}^4x.\n\\end{align}\n", "5522d19f5c33c505aacf932cf2ed361b": "1-x,", "5523a4dcf9ecc4c305a07003bc9eba91": "T_x-x", "5523b60638d2ab815f0792ab0bbf32a4": "\\begin{align}\nQ_1 &{}= \\begin{bmatrix}0 & -1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1\\end{bmatrix} &\nQ_2 &{}= \\begin{bmatrix}0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ -1 & 0 & 0\\end{bmatrix} \\\\\nQ_1 Q_2 &{}= \\begin{bmatrix}0 & -1 & 0 \\\\ 0 & 0 & 1 \\\\ -1 & 0 & 0\\end{bmatrix} &\nQ_2 Q_1 &{}= \\begin{bmatrix}0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0\\end{bmatrix}.\n\\end{align}", "5523b66fdc7a1b91c378825b4f4c0c23": "f^{458} = f^{678} = \\frac{\\sqrt{3}}{2}, \\,", "5523faef8a5000baf940637250730128": "\nf_{1,2}(y) = 0.00413682 - 0.0813801\\, y^{2} + 0.260416\\, y^{4} - 0.277778\\, y^{6}\n", "55240407d463ee06c3a1cb28fb60ca61": "\\scriptstyle{E_{3}}", "5524287c481c571d7535a86cc3e5e08b": "[G]_o", "5524317b3ea8e1ea5e9f64a9d51dd194": " = \\frac{1}{2} \\left(\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu) + \\operatorname{tr} (\\gamma^\\nu\\gamma^\\mu) \\right) \\,", "5524ac789c43659f73c987f624b36660": "\n\\frac{1}{r} = \\frac{1}{b} \\cos\\ \\left(\\frac{\\theta_2 - \\theta_0}{k} \\right)\n", "5524c8fcf986e255ac02378a149f824a": "\\displaystyle W[u,v]", "55254735a11627543a0da69d38b5c1a2": "\\,N_1", "55254bb628f4c0769a48667dadebea38": "x = \\pm 1.", "55255de692fd80d5abaac251fa6f3234": " \\theta=0.1 ", "5525698ae57aa8d327044a5034d4eddf": "Y=\\ln(W)", "55259a224ba91e1a7724f54ab1dc3dea": "\\,p", "5525ab1252de1da080eb7676dd89cfc8": "r=\\frac{{\\rm ln} \\frac{X_2}{X_1}}{\\Delta t}", "5525bec94160eb37329d8dcdc958abd8": "\\overline{op_1}' = T(\\overline{op_1}, T(op_2, op_1))", "5525f104bf11eca1463f626b8613e056": "R_aR_b=R_cR_d", "552630c360afcdb242f4c7cbbde5f744": "a\\otimes b=T(a,b,0)", "55263a421efc3fefea0c47790c662da5": "U_i U_{i+1} U_i = U_i", "5526c0e29cd37a7f0b6545387a626069": "Z_{5}", "5526f0c3688d4401eb821601eea03da4": "(x,y)=(0,-l)", "55270709e50bd521dbf7eaff8601ca24": "\\gamma_x", "55270a103777630dfed56cf07f2692dc": "39 - (4 - s) = 35 + s", "552741ff5a63944831ceb039de809b1e": " P_e^{(n)}", "55275afbd1ec69e7118c70dd4b210ff5": "M(V) = (V/V_d)^0 = 1", "55276da1f18ef621d7da349db460d6e4": "f(y^*)\\le P_{95}", "5527719a575934d2bbcbd586c1e53eca": "\\displaystyle p^2+q^2=2(m^2+n^2).", "55278400b20794d690ca7023ad6badc1": " f(r) = \\frac{h^2}{p^3} \\frac{dp}{dr} . ", "5527c44fb26c467894a5d76f2c6ccbdd": "x_n ", "552820f19c46b19d6a4d77446634f18c": "{\\alpha} = \\frac{\\tau}{I}.", "5528308b4fb684b2824e8e8bd466ff82": "\\frac{|G|}{|C_i|}\\delta_{ij}=\\sum_{k=1}^p\\chi_{V_k}(C_i)^*\\chi_{V_k}(C_j)", "5528a8298c0e927f07bb3047e9f468ac": "{{v}_{2}}^{\\prime }", "5528ab554824850c3a1628889b33539b": "4!", "5528c23562d3621c7f72493144691350": "\\begin{array}{rcl}\n\\tau & = & t\\cdot\\exp\\left(-x^{2}-1.26551223+1.00002368\\cdot t+0.37409196\\cdot t^{2}+0.09678418\\cdot t^{3}\\right.\\\\\n & & \\qquad-0.18628806\\cdot t^{4}+0.27886807\\cdot t^{5}-1.13520398\\cdot t^{6}+1.48851587\\cdot t^7\\\\\n & & \\qquad\\left.-0.82215223\\cdot t^{8}+0.17087277\\cdot t^{9}\\right)\n\\end{array}", "5528da87bff364f1ce4999aeb6fd4c0d": "\\succ\\! ", "552aba7b35d90f1582b369069c16fa76": "B_k = \\operatorname{im} \\partial_{k+1}", "552b2aafd01b198b5f59e621855f13ae": "d ", "552b47012f512546eb152e0510903a62": "\\hat{f}(\\nu)\\ \\stackrel{\\mathrm{def}}{=}\\int_{-\\infty}^{\\infty} f(x)\\ e^{-2\\pi i\\nu x}\\, dx.", "552baef1655a1dfc4211406ea53a194a": "\nP = K_0 \\ln(V_0/V). \\,\n", "552bca14cd1d7056d8f5d75f860c00d5": "10\\rightarrow (3,2)_{\\frac{1}{6}}\\oplus (\\bar{3},1)_{-\\frac{2}{3}}\\oplus (1,1)_1", "552be29885f3d93023cfddbf3b8c3429": "f(x;\\mu, k,\\theta) \\propto \\exp{\\left(\\frac{(x-\\mu)^2}{4\\theta^2}\\right)}D_{-2k-1}\\left(\\frac{|x-\\mu|}{\\theta}\\right)\\,\\!", "552c189086838cfaee41a9ea86a16a7f": "\\left ( Q = \\frac{V}{R} \\right )", "552c50ad62d7a2f6b74853a5e9318baf": "\\scriptstyle\\pi", "552c60413a8f92668230253e25df9b92": "l_.", "552c7d949b3ccad7bd7f1ad58ed7290f": "\\frac{1}{c}", "552c848dc01fb00d3d53b422827dc653": "H\\cap K\\subseteq_e M", "552cb18a7f9728fc1d0f022cbe40746e": "\\langle G,S \\rangle \\rightarrow \\langle G',S' \\rangle", "552cb4e829deafc37b95af50e53ae633": "M_{bol_{\\rm Sun}}", "552cb9af84efb4e0cee111b4e1d27d2a": "Y_{B}", "552d17ee29d2ebb5fb12814c20966723": " \\{ \\langle \\mathbf{e}_\\mu \\bar{\\mathbf{e}}_\\nu \\rangle_V \\},", "552d5f2cfddfa4ce9a098f9c6ea055e3": "g_n=\\begin{cases}f_k(x)&\\text{ if }n=0\\\\ \\frac{c^n}{k(k+1)\\cdots(k+n-1)}f_{k+n}(x)&\\text{ otherwise.}\\end{cases}", "552d7a1b27a521b3d744d5518b7d8b3a": "|p_ip_j|", "552d7dd6fded82984a8ea9f7b014e95e": "\\zeta(2n) = (-1)^{n+1}\\frac{B_{2n}(2\\pi)^{2n}}{2(2n)!}", "552d8d1b9ee762c6df2dbcd1cc15a957": "x^3=9a \\left(x^2-3y^2 \\right)", "552d9cea915b1054b83bc8f5c920f482": "G(x) = \\frac{ie^{ik|x|}}{2k}", "552dad39a6c2224e75092302a6d40bf5": "({v_0+v_i})10^{-b_1E_{i}} \\text{ vs. } v_{i^{ }}", "552db29122cf15a36ed1e84f6a6c66b0": " a_n > 0 ", "552db4179ed2ef8cbb6760b26b4bda76": " \\hat{p}_t = r_t/n_t ", "552def453e513c6c0f5e3d4e5fa94d5e": " \\qquad \\qquad \\mathrm{H}_{ph} = \\frac{1}{2}\\int (\\epsilon_\\mathrm{o}\\mathbf{e}_e^2 + \\mu_\\mathrm{o}^{-1}\\mathbf{b}_e^2)dV = \\sum_\\alpha \\hbar \\omega_{ph,\\alpha}(c_\\alpha^\\dagger c_\\alpha + \\frac{1}{2}),", "552e24805f779ec7af6358919b1ffe4c": "\\sigma_{xx} + \\sigma_{xz} + \\sigma_{xy}", "552e33eeb07f12d2e7561f8e5594dbe0": "p(c) = \\prod_{ \\text{neighboring bonds }b',b'' \\in c } A(b',b'' ) ", "552ee6c15cb5be2082504f8aec1c6001": "\\Omega_\\Lambda=0.732", "552f3e513e799fd1ef8ab5a9f2f8781a": "T \\times T", "552f552b7848bdd9b4211b2fe8a6d1f9": "X=X_1,X_2,\\dots, X_n \\, ", "552f704da38c15192fc4f66689ce8673": "p(\\mathbf{y}|m)=\\frac{p(\\boldsymbol\\beta,\\sigma|m)\\, p(\\mathbf{y}|\\mathbf{X},\\boldsymbol\\beta,\\sigma,m)}{p(\\boldsymbol\\beta,\\sigma|\\mathbf{y},\\mathbf{X},m)}", "552f750a8250812e4a005c410c7e89fb": "m_x(t)= E[x(t)]", "552fb956e0889cd9105c1192d6a81bfa": "O_3^{(\\alpha)}(t)=2\\frac {(1+\\alpha)(3+\\alpha)}{t^2}+ 8\\frac {(1+\\alpha)(2+\\alpha)(3+\\alpha)}{t^4},", "552fd46d51f7a4fe2b42ff6253be515b": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}\\mathbf{A}\\mathbf{v} + \\frac{\\partial \\mathbf{v}}{\\partial \\mathbf{x}}\\mathbf{A}^{\\rm T}\\mathbf{u}", "553014ba8d1045f0897e8448a7575c6e": "\\{F_i,F_j\\}= s_{ij}\\circ F", "553029adca93cbae0c44d9c67dd591c3": "-R + 4 \\Lambda = {8 \\pi G \\over c^4} T \\,.", "55302c7859140fd683ad0356a4dc30d8": " {v_x - v_S \\over v_\\infty - v_S} = {T - T_S \\over T_\\infty - T_S} = {c_A - c_{AS} \\over c_{A\\infty} - c_{AS}} = 1", "55309f47046dc3e1ee17b52908dee896": "\\frac{\\partial \\tau_{xz}}{\\partial x} + \\frac{\\partial \\tau_{yz}}{\\partial y} + \\frac{\\partial \\sigma_z}{\\partial z} + F_z = \\rho \\frac{\\partial^2 u_z}{\\partial t^2}\\,\\!", "5530c0a84b74f4880028461ebd6e16fc": "f(5x) = \\ln 5x = \\ln 5 + f(x)", "553105e24a7c17819b6977ac724fff5c": "[(1-x^2)y']'+\\nu(\\nu+1)y=0", "5531402b89bb44e0534e77456fd64d62": "\\; k", "553162af68afe7b42d7756f2ea7f1234": "F(\\vec{k})=\\int_{-\\infty}^{\\infty}f(\\vec{r})e^{ -i \\vec{k} \\cdot \\vec{r}}d^3 r.", "55317b465fac1fa35f6acda5970b4e8b": " \\frac{43867}{798} ", "5531ab37b5740413da918683f4724d1f": "{\\pi\\over 3}\\ {\\pi\\over 2}\\ {2\\pi\\over 3}", "5531af7b786f61be9ffd32d290b5be75": "v_{OLS}[\\hat\\beta_{OLS}]", "553203154aea58298ce1285851aa12f1": "[X+\\xi,Y+\\eta]=[X,Y]\n+\\mathcal{L}_X\\eta-\\mathcal{L}_Y\\xi\n-\\frac{1}{2}d(i(X)\\eta-i(Y)\\xi)", "553209f5ce3fd3f017c92d0c273c0bee": "\\Pr_{h \\in H}[h(f)=d|h(a)=c]={1 \\over |D|}", "5532304620085cae15b75d673d890b0c": "f_{uc}", "553295b25d2666c767ab96bda9ffacfa": "\n\\nabla \\cdot \\mathbf{E}(x) = 0, ", "5532c0d763b241b7429ed2789adfae5f": "k< 3", "5532d1da9fd79ea94174218c9f90b0c6": "c < 0", "5532d37cb2f9818cd7533dac5fabbcbe": "\\displaystyle E^n X = [\\Sigma^{-n} X, E].", "55331a030c3487b85be905e61189fa33": "x_A", "553325bb3325cd5852c8f9e860d928b4": "\\frac {d}{dx} \\left( p(x) \\frac {dy}{dx} \\right ) +\\left( q(x)+ \\lambda w(x) \\right) y(x) = 0. ", "5533485d3a12c4b11ba7e996bc6be418": "S = ", "55337a57298b5774dba0e7263a98c763": "\\sum_{k=1}^n \\beta_k e^{\\alpha_k}=0", "5533ab90ef6520cd569eeb4fa035df94": " ds^2 = d\\rho^2 + \\rho^2 \\, \\left( d\\theta^2 + \\sin(\\theta)^2 \\, d\\phi^2 \\right) ", "5533f15acb6e904f0fe1d709b3d638e6": "\\scriptstyle f_{uv}", "55340fb7a7b392dcc5c5740347c1d61b": "\\vec{\\mu}", "553457a4cc3f338b23d425ba2920e54b": "\\int\\frac{1}{x(ax^2+bx+c)} \\, dx= \\frac{1}{2c}\\ln\\left|\\frac{x^2}{ax^2+bx+c}\\right|-\\frac{b}{2c}\\int\\frac{1}{ax^2+bx+c} \\, dx + C", "55347c6fbe34555ea2ddde5c19515afc": " a 0 ", "5535160f7e0be69a7ae4bfd44cad0387": "d/dt", "553538cbef0818b7fccb965042450efa": "\\scriptstyle \\operatorname{diam}\\;U_i<\\delta", "553566b8143c3712daa441eb46c115a0": "\n\\mathbf{A}(\\mathbf{r}, t) = \\frac{\\mu_0}{4\\pi} \\int \\frac{q\\mathbf{v}_s(t_r') \\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t_r'))}{|\\mathbf{r} - \\mathbf{r}'|} d^3\\mathbf{r}'\n", "553592cd9169df95b74e88a9b0a3857c": " \\int_0^\\infty x^{s-1} \\frac{\\gamma x+\\log\\Gamma(1+x)}{x^2} \\, dx= \\frac{\\pi}{\\sin(\\pi s)}\\frac{\\zeta(2-s)}{2-s} \\!", "5535cdd4118fa4f38ed62acf80ac9ab7": " I(f) = \\sum_S |S| \\hat{f}^2(S) \\!", "5536117d8f20f07de8341fc0a018a960": "\\sum_{k=0}^n \\sin(\\theta+k\\alpha)=\\frac{\\sin\\frac{(n+1)\\alpha}{2}\\sin(\\theta+\\frac{n\\alpha}{2})}{\\sin\\frac{\\alpha}{2}}\\,\\!", "55365e5108a65b0f342ee7054e5b07b6": "\\left[\\mathrm{Ligand}\\right] \\cdot \\left[\\mathrm{Receptor}\\right]\\;\\;\\overset{K_d}{\\rightleftharpoons}\\;\\;\\left[\\text{Ligand-receptor complex}\\right] ", "55369f27f40785baa7b070c9be15c8c8": "\\scriptstyle \\mathbb{Z}/2^N\\mathbb{Z}", "5536a705c03deefea57137f55b1385a3": "\n\\begin{align} \n q_4 &= \\frac{1}{2}\\sqrt{1+A_{11}+A_{22}+A_{33}}\\\\\n q_1 &= \\frac{1}{4q_4}(A_{32}- A_{23})\\\\\n q_2 &= \\frac{1}{4q_4}(A_{13}- A_{31})\\\\\n q_3 &= \\frac{1}{4q_4}(A_{21}- A_{12})\n\\end{align}\n", "553736e2233d8ef303f14132de01dbbd": "det(gM)=det(Mg)=det(M)det(g)", "55374044d91ed57134a272c1a6ed74d3": "\n \\kappa_1=x_0,\n", "55374863d0d3fd1a99f480d258f7285f": "\\begin{align}\n \\zeta(s, q) &= \\frac{1}{s-1}\\sum_{n=0}^\\infty \\frac{(-1)^n}{n+1} \\Delta^n q^{1-s}\\\\\n &= \\frac{1}{s-1} {\\log(1 + \\Delta) \\over \\Delta} q^{1-s}\n\\end{align}", "55377b0cdd8cd6ac7b89dce1a6fc2aba": "v\\le d-1", "553783b9831dfaebe6afae6f972e5c69": "\n\\begin{align}\n(R_{pq} M)_{m,n} & =\n\\begin{cases}\nM_{m,n} & m \\ne p,q \\\\[8pt]\n\\frac{1}{\\sqrt{2}} (M_{p,n} e^{-i\\theta} - M_{q,n} e^{+i\\theta}) & m = p \\\\[8pt]\n\\frac{1}{\\sqrt{2}} (M_{p,n} e^{-i\\theta} + M_{q,n} e^{+i\\theta}) & m = q\n\\end{cases} \\\\[8pt]\n(MR_{pq}^\\dagger)_{m,n} & =\n\\begin{cases}\nM_{m,n} & n \\ne p,q \\\\\n\\frac{1}{\\sqrt{2}} (M_{m,p} e^{+i\\theta} - M_{m,q} e^{-i\\theta}) & n = p \\\\[8pt]\n\\frac{1}{\\sqrt{2}} (M_{m,p} e^{+i\\theta} + M_{m,q} e^{+i\\theta}) & n = q\n\\end{cases}\n\\end{align}\n", "5537ac54356aa7411d9a05db00900495": "\\alpha\\in\\left[0,2\\pi\\right)", "5537d6eb21bb47ac98eaaef9e1c43724": "z=a, a\\le1 ", "55380e582a6757834db54807a8183219": "\\,k=1", "55380e6db75633cff9094e76eec13c4f": "D_{K_s}", "553849ce013a4e47593ef732429f79b2": "(p_f\\frac{n-c}{n})^{i-1} (\\frac{c}{n})", "5538ad332db5216e93faba8d14b618f3": "f=\\sum_{a \\in X}f_a", "5538b7ce7a2a111af2c92279189fd35e": "\\,x_1", "5538cae159065baa9afcf9c4f959e716": "P_1 P_2 = AB \\frac{\\sin \\psi \\sin \\gamma}{\\sin \\alpha_1 \\sin \\beta_2}.", "5539180822b2e14f5ab56e69de62e78f": "\\int_a^b f(t)\\, dt = F(b)-F(a).", "55391b0015e80235236b0c8a42489c33": "f(x,y) = \\begin{cases}\n \\frac{xy(x^2 - y^2)}{x^2+y^2} & \\mbox{ for } (x, y) \\ne (0, 0)\\\\\n 0 & \\mbox{ for } (x, y) = (0, 0).\n \\end{cases}", "55391d02a3c8d840c62696c8abbd8789": "|A \\cup B \\cup C| = |A| + |B| + |C| - |A \\cap B| - |A \\cap C| - |B \\cap C| + |A \\cap B \\cap C|.", "55398d2b600191c853824eb16fad7f20": "\\displaystyle t_2 \\geqslant 3 + \\sum_{k\\geq4} (k-3) t_k.\\,\\! ", "553993b0aa36bc1968f8e64ec8fcaf1c": "\\mathcal{D}_{\\Gamma_3^*}", "5539c94112c039bdf7a46de4eeba4196": "(-1)^{s-m-w}\\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}", "5539f807a63dcc15cf2c14a0fdd0f9a8": " b\\in A", "553a369fb2839c7a1930d255d0c5a6ce": "(y-x)^T \\nabla^2f(z) (y-x) \\ge m (y-x)^T(y-x) ", "553a3a6a8505d268d6122274b85034ce": "R_a", "553b0c903ca2072324666bcb8dbb3fd8": " \\mathcal{W} = \\int_S H^2 \\, dA - 2 \\pi \\chi(S)", "553b11a7a554f260a1ed74dca65ecccd": "z=re^{i\\theta}", "553b28a9975e9bf1d5791270ef44c072": "\\mathrm{SLI} = \\sum {(S-V)^2 \\over V}", "553b53067f46b772c3e52bca7dc5d4ac": "\\exp\\left(\\Omega\\left(n^{\\frac{1}{k-1}}\\right)\\right)", "553b645c5ea5441daa08ab49c3cb984d": "{n \\choose k-1}", "553b83bbe0fbd16ea54ce31d24baa063": " I_{C, \\text{disc}}=\\int \\rho r^2 dV =\\int_0^{2\\pi} \\int_0^R \\rho r^2 (s r dr d\\theta) = 2\\pi \\rho s \\frac{R^4}{4} = \\frac{1}{2}mR^2,", "553b8f53c9a19454d2661d3f6019e726": " x_n=-\\cos\\left(\\frac{n\\pi}{N}\\right)", "553babceab10f862512484a101d7d516": "\\! i=1, 2, ..., n", "553cc4a1e55e24fd868f13f134090b93": "\\mathbf{B} = \\mathbf{X} - \\mathbf{h}\\mathbf{u}^{T} ", "553d34fbcc1d15496316597598360baf": "\\mu_{02} = M_{02} - \\bar{y} M_{01}, ", "553d82b642137a5a13a4d93c93654260": "\\min(3-1,6-0,9-0) = ", "553d86f6d3afff16dea579c516a93b9c": " W =1.5 ", "553da620e1417ad04b8a4653b6bcc026": "P_1=(1:\\sqrt{2}:1)", "553dbb526e6c93b13b991ebb3e4c9003": "\\int_0^T \\phi_t^2 \\sigma_t^2 \\, dt < \\infty", "553e11b7cf2568cfd65e19d130020d82": " C = -\\frac{dC_v}{dK} ", "553e4d9b97b13e04c2b30fe545825d05": "2^{2 + 1} + 2^2 + 1 - 1", "553e5e9dc8febc6c6fc5524d6a7815bf": "\\stackrel{*}{\\rightarrow}_R", "553e6a5512b193c82d046dd9ca25e940": "(2/3 + 1/10 + 1/2190)", "553ed7b83841cf452ec3583f492582b9": "\\mathbf{n}_{12} \\times (\\mathbf{H}_2 - \\mathbf{H}_1) = \\mathbf{j}_s ", "553edfd4274396917ed24e69f0020ec4": "\\pm 1, \\pm i", "553ee32e0c204b6a57dd148e81cfa3ae": "n^a\\partial_a=-\\partial_r\\, := \\,\\Delta \\,,", "553f4de2e952b741e14110db0917e004": "L_{R, n} \\equiv \\frac{NH}{f_0}", "553fa11ea847b26c78f9b4829fabe117": "g(m^k,m^{k-1}n,m^{k-2}n^2,\\dots,n^k)=n^{k-1}(mn-m-n)+\\frac{m^2(n-1)(m^{k-1}-n^{k-1})}{m-n}.", "553fdd3799081fe9e364b5fa0190175a": " \\langle 0 | ( R\\phi(x)\\phi(y) - \\phi(y)R\\phi(x) )|0\\rangle = 0 \\,", "554028bc0ecd549dc881b9de3f55e987": "E' = {E\\over\\sqrt{1 - v^2/c^2}}\\ ", "554038d2e9ae2c90c1d835a1cc5a0283": "(A^\\bullet, d^\\bullet)", "55409cb87e308654e308719c4f0ab990": "rel_{i}", "5540c98b45a2a2e83b612d75e1104ef4": "\\{x_i\\}_{i\\le n}", "55413ce88f5f7b7bae3ecc0df6a8b46a": "\\mathfrak{so}_3", "5541b1de1450a545b46bcc8e926683ed": "\nH= \\int_x \\psi^\\dagger(x) {\\nabla^2 \\over 2m } \\psi(x)\n\\,", "5541c2ca2f2663d3f2890a69a0ee7c27": "\n X_n\\ \\xrightarrow{d}\\ X,\\ \\ |X_n-Y_n|\\ \\xrightarrow{p}\\ 0\\ \\quad\\Rightarrow\\quad Y_n\\ \\xrightarrow{d}\\ X\n ", "5541cbf75cdc1d607728531740e88872": " x^{(2)}= \n \\begin{bmatrix}\n 0 & -1/2 \\\\\n -5/7 & 0 \\\\\n \\end{bmatrix}\n\n \\begin{bmatrix}\n 5.0 \\\\\n 8/7 \\\\\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 11/2 \\\\\n 13/7 \\\\\n \\end{bmatrix} \n= \n \\begin{bmatrix}\n 69/14 \\\\\n -12/7 \\\\\n \\end{bmatrix} \n \\approx\n \\begin{bmatrix}\n 4.929 \\\\\n -1.713 \\\\\n \\end{bmatrix} .", "55421556c3d769e03c4780b1515d94ee": "\\displaystyle{\\tau(v,w,z) - \\tau(u,w,z) + \\tau(u,v,z) - \\tau(u,v,w)=0}", "55422d96e1a275853577efd9b76935fc": "\n\\mathcal{L}[T(y_0)] = \\frac{1}{\\sqrt{2g}} \\mathcal{L} \\left [ \\frac{1}{\\sqrt{y}} \\right ] \\mathcal{L} \\left [ \\frac{ds}{dy} \\right ]\n", "5542317b73f63ae5ff2ddfc6b4dc1413": " S_{\\nu}[Jy] = 3.34 \\times 10^4 \\lambda^2 F_{\\lambda}{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1} \\AA^{-1}]} ", "55425523f58a3899adde69d3a6cd368b": "f(x_i) = y_i", "554259fb403f6e05335f9dded6a49a48": "(\\frac{V_f}{2U})", "554320551676b18a388df066e468f6e7": "{}_sY_{\\ell m} (\\theta, \\phi) = (-1)^m \\sqrt{ \\frac{(\\ell+m)! (\\ell-m)! (2\\ell+1)} {4\\pi (\\ell+s)! (\\ell-s)!} } \\sin^{2\\ell} \\left( \\frac{\\theta}{2} \\right) ", "554353fdbe962c502dfab03bbf4dc90d": "s=\\frac{1}{4}", "55439bda699d4d3e326987df441685c8": "M(f)", "55439e728f5f157f2ac6283d524560cb": "\\forall p(p \\Rightarrow Kxp)", "5543a5abea07adca3692246eab213016": "G(T,p)\\,", "5543a69c0c6bbff61fd8c1b05533628b": "\\left(2\\rightarrow3\\right)\\rightarrow2 = \\left(2^3\\right)^2 = 64", "554488a5e2df61d4b2af5ec35d9d3a7e": "n (2^n / 2 - 1)", "5544b8759c0c7ec4c2c4704155ff60ae": "Inc_{k}=[\\frac{\\sum_{i=1}^{m}Costs_{i}}{ \\sum_{i=1}^{N}E(Costs_{i})} \\cdot[\\sum_{i=1}^{N}Revenues_{i}-\\sum_{i=1}^{N}E(Costs_{i})]]-\\sum_{i=1}^{m-1}Inc_{i}.\\, ", "5544c3a2b8db3577333c28f28743b541": "\\beta_1=V_\\text{max}", "5544d733d4a5f5c737d585ed94ce0afc": " \\bigcup_{\\alpha} V_\\alpha \\! ", "5544d9949b48c70eacbba70095cf7f9c": "V_{OC} = \\frac{kT}{q} \\ln\\left(\\frac{I_{SC}}{I_{0}} + 1\\right).", "5544dca8fb0ba7f5806b95f04a5bbe48": "T_s \\approx 100 \\rightarrow q_s* = 12.1\\left(\\tau*-0.047 \\right)^{3/2}", "55450cefecc282458a9eaaf0d6b885e2": "K_i(P(R)) = K_i(R) \\to K_i(S)", "55452a6206632c567c77ad1ffaf484dd": "\n\\begin{align}\n x &= \\frac{\\chi (\\chi^2 + \\eta^2 + 1)}{\\chi^2 + \\eta^2}\n \\qquad \\text{and}\n \\\\\n y &= \\frac{\\eta (\\chi^2 + \\eta^2 - 1)}{\\chi^2 + \\eta^2}.\n\\end{align}\n", "55458b6c0109ae102844b15e3374b434": "\\int_{\\Omega} \\nabla u_{f}(x) \\cdot \\nabla v(x) \\, \\mathrm{d} x = \\int_{\\Omega} f(x) v(x) \\, \\mathrm{d} x \\mbox{ for all } v \\in H_{0}^{1} (\\Omega).", "5545cd6eeba13d781212f14933b036ea": "H = -\\int_\\Gamma f(\\theta;\\mu,\\kappa)\\,\\ln(f(\\theta;\\mu,\\kappa))\\,d\\theta\\,", "5545d0d529577ad93ef7eefe82865320": "A=2(3+\\sqrt{3})a^2\\approx9.4641...a^2", "5545db5003d54399061a50debdfd6613": " x = r \\cos \\theta, \\ y = r \\sin \\theta.\\, ", "554618367dcccf22ccbe1dd4463bd3e5": "A \\subseteq B\\,\\!", "554640255e4a69c0e753566f1e487ea8": "\nG_{2} \\equiv \\mathbf{g}(\\mathbf{q}; t) \\cdot \\mathbf{P}\n", "554668d79ad7693c7964cf78bd8b7de7": "\\kappa_r", "55468f8317d644bff89201868259c171": "H^2(M, \\mathbb{Z})", "55469a1d38b8c819953f9795bbdd0480": "K(\\pi,1)", "5546b1efa9258b5bbaac6d9162eb0c46": "\\frac{d}{dt}\\hat{\\boldsymbol{\\jmath}}(t) = \\Omega (-\\cos \\Omega t, \\ -\\sin \\Omega t)= - \\Omega \\hat{\\boldsymbol{\\imath}} \\ . ", "5546b56e0ba6e194494fa28562a0c7ae": "\n\\text{(Eq. 7)} \\qquad \\Delta(t) \\leq B + \\sum_{n=1}^N\\sum_{c=1}^NQ_n^{(c)}(t)E\\left[\\lambda_n^{(c)}(t) + \\sum_{a=1}^N\\mu_{an}^{(c)}(t) - \\sum_{b=1}^N\\mu_{nb}^{(c)}(t)|\\boldsymbol{Q}(t)\\right] \n", "5546d7c8a86fae188dd1e77752da16f4": "f(lx)=l^\\theta f(x),", "5546f396f2e4f68347720d2a6eb936fe": " \\begin{align} \\mu \\nabla^2 \\mathbf{u} -\\boldsymbol{\\nabla}p &= \\mathbf{F}\\cdot\\mathbf{\\delta}(\\mathbf{r})\\\\\n \\boldsymbol{\\nabla}\\cdot\\mathbf{u}&=0 \\\\ \n|\\mathbf{u}|, p &\\to 0 \\quad \\mbox{as} \\quad r\\to\\infty \\end{align}", "554736d750a2baee8470915abd69d439": "\\mathrm{X}", "55474b116c3407a8272102b5c4f657c6": " I(\\theta) \\propto e^{-[\\frac{2\\pi \\sigma \\sin \\theta}{\\lambda}]}", "5547827ac751cb42b1b941384a088f03": "M \\rightarrow N", "5547f495c7467c7eb7e2952ef68dc934": " x_{n+1} = \\exp(-\\alpha x^2_n)+\\beta, \\, ", "55481d34be8b280d5538912016e45a3f": "N\\left( u\\right) =X", "5548419f7a7132982a2e2b9c61b7fcf9": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_\\text{f} + \\frac{\\partial \\mathbf{D}}{\\partial t} ", "55485039bfb141cdfbf482a84da50f78": " d = 1", "554864a0fde348021930086797987667": "\\Gamma \\cong C_2 * C_3", "5548cc5dcafb4d5de493a867c7f6e828": "\\bigcup_{k < \\omega} S_{\\omega + k}", "5548e4db802c03931b6f35181a756531": "(a1,a2,a3,...)", "5548f6997a09a47107a3711f99f2565a": "x_1 x_2 \\dots x_n", "5549268f751e8e4cd83ebaa215483c25": "\n E_\\text{kin}\\, =\\, \\overline{\\int_{-h}^0 \\frac12\\, \\rho\\, \\left[\\, \\left| \\boldsymbol{U}\\, +\\, \\boldsymbol{u}_x \\right|^2\\, +\\, u_z^2\\, \\right]\\; \\text{d}z}\\,\n -\\, \\int_{-h}^0 \\frac12\\, \\rho\\, \\left| \\boldsymbol{U} \\right|^2\\; \\text{d}z\\,\n =\\, \\frac14\\, \\rho\\, \\frac{\\sigma^2}{k\\, \\tanh\\, (k\\, h)}\\,a^2, \n", "5549928ce3f9a8fc7af66e4de5f42990": "\\begin{matrix} {4 \\choose 3}{3 \\choose 1} \\end{matrix}", "5549953195c7d358079cf272f77b2db4": "\\left(\\eta_2+\\frac{p+1}{2}\\right)(p\\ln 2 + \\ln|\\mathbf{V}|)", "5549a00eb7d558230cd7820fef0ee378": "\\mathbf{g}_j^{\\text{(net)}}=\\sum_{i\\ne j}\\mathbf{g}_i =\\frac{1}{m_j}\\sum_{i\\ne j}\\mathbf{F}_i = -G\\sum_{i\\ne j}m_i\\frac{\\mathbf{\\hat{R}}_{ij}}{{|\\mathbf{R}_i-\\mathbf{R}_j}|^2}=-\\sum_{i \\ne j}\\nabla\\Phi_i", "5549f824330f1662c276361673cbc86c": "a \\vee b = a, b \\vee a = a', a \\wedge b = b", "554a11a3d513829e8ce56e1932e8160c": " X_0 = \\frac{ X_1 * M_1 + X_2 * M_2 + \\ldots + X_n * M_n }{ M_1 + M_2 + \\ldots + M_n }", "554a11ed35f55692a7896691da0defd5": "v=\\frac{dr}{dt}", "554a40c32b6c8c8fc2615235a8add095": "\n\\mathrm{P}= \\frac{V_\\mathrm{rms}^{2}}{R} = I_\\mathrm{rms}^{2} R\n", "554a89106393815a0a9a8b285a251806": " (A_1, H, I, J, A_2) = (A_1, E''', F''', G''', A_2) ", "554a9cfa1c77de16d7438d38d9742fd8": " \\frac{d\\xi}{d\\theta} |_{\\theta=0} = -\\frac{1+\\xi^2}{2}. ", "554aacec196e0f393b34e4caae305947": " d\\,", "554b57149e35d747e756bf6db2669c93": " \\lim_i m(x_i) \\leqslant \\lim_i \\frac{f'(x_i)}{g'(x_i)} \\leqslant \\lim_i M(x_i) ", "554c43dfac1e47fa94e74f4545a80d75": "\\mathbb{C} \\propto \\mathbb{H} \\text{ but } \\mathbb{H} \\not \\propto \\mathbb{C}.", "554ce201a3ef4ea5f1983897d1659ad2": "K(n)_*(X \\times Y) \\cong K(n)_*(X) \\otimes_{K(n)_*} K(n)_*(Y).", "554d4dda06dbe7046e1a13df6b76b140": "z ^\\alpha\\,", "554deb5194bec46ceb6f16a3852df072": "\n \\qquad \\qquad u_x^- = \\frac{2u_{i+1} + 3u_i - 6u_{i-1} + u_{i-2}}{6\\Delta x}\n", "554dfed852e20d2b257e35248d09b465": "H_1(C_n, \\mathbf{Z}) = \\begin{cases} 0 & n = 0\\\\\n\\mathbf{Z}/2 & n = 1\\\\\n\\mathbf{Z}/2 \\times \\mathbf{Z}/2 & n \\geq 2 \\end{cases}.", "554e0936d4c2b5b542c59fce902d3064": "\n{\\mathbf P}=\\varepsilon_0\\chi_e{\\mathbf E},\n", "554e7046abe411a2f1b0a29b1d3d21b8": "\\tilde{A}", "554eb0430df493fc0bcc038d0f96b30a": "\\frac{\\rm d}{{\\rm d}t}x(t)=-x(t-1).", "554eb614d30f236d62b247f6ce389af3": "\\langle f|", "554eb8cd379ed61bf17fda01d3c76aa5": "\\boldsymbol{\\kappa}\\, =\\, \\nabla\\theta", "554f11822b3725141c5d0bc21ad74247": "\\Delta t_{p}=\\frac{\\ln \\left( 2 \\right)}{k}", "554f534c5e3ec7fd5c23d4a43c97052d": " {^n 0} ", "554f5f6f540c1c3084dec06d72ec2f1b": "\\lfloor n^{ 1/b }\\rfloor^b=n", "555053fa8f7c21a656efb83f397b09ca": "\\ddot \\theta = {g \\over \\ell} \\sin \\theta", "55508c6d397d12250ab5ff5470e64407": "\\{p(x), x \\in [0,1]\\}", "5550a15b1182f34e1a89c0aa17cc6650": "\n\\lfloor E \\rfloor (F) \\rightarrow^\\sigma E'\n", "5550fbadf4baa5426926d063bbc6cbfa": "\\vec{\\Delta\\theta}", "5550fed7ffe2a33b1c6bfdfe8ac02e06": "(4a)^{2m+1} \\equiv 1", "55510f2c55b6868546ac32f634eb0d24": "e \\oint_{\\partial D} A\\cdot dx = e \\int_D (\\nabla \\times A) dS = e \\int_D B \\, dS.", "5551479fb8f3cc2826880136bb808205": "\\{x_i\\}_{i=1}^n \\sim P(X), \\{z_j\\}_{j=1}^m \\sim Q(Z) ", "5551506ce338fa1cdc40911f7b9684f7": "\n\\begin{align}\n \\left(\\cos x + i\\sin x\\right)^{-n} & = \\left[ \\left(\\cos x + i\\sin x\\right)^n \\right]^{-1} \\\\\n & = \\left[\\cos (nx) + i\\sin (nx)\\right]^{-1} \\\\\n & = \\cos(-nx) + i\\sin (-nx). \\qquad (*) \\\\\n\\end{align}\n", "55520e9703d1dc9448445f5436d5230f": "k_i > 0", "55523b489dfbb1db5e962ba116195124": "\\left\\|\n{\\mathcal{A}}_{i_{n}=i}\\right\\| ", "555265ecdbe5fe893723858e46236653": "\\sum_{k=1}^nA_k^*A_k=1", "555286b575cf254383bdcadba06c3c0c": "C\\ell_{p+4,q}(\\mathbf{R}) = C\\ell_{p,q+4}(\\mathbf{R})", "5552d1bc55214454ddcfc8e753b3ead2": "\\mathit{n}/\\mathit{p}", "5552d2147a4dfbba9206fbafc6894aea": "M_V = -0.01 + 5 \\cdot (1 +\\log_{10}{0.742}) = +4.3.", "5552f47d1edc311d6740c968537e1bf1": "R \\stackrel{i}\\to R", "55530e963718d32e16a51dba64229323": "A=M", "55530f612038e82acd545a9d68260983": "\nW = \\sum_{p=1}^n (-1)^{p-1}\\, \\frac{n!}{p!}.\n", "5553144dec6bd5bf2265a3168e7e00d8": "\\begin{matrix} {4 \\choose 2} \\end{matrix}", "55531d33c23884abb00ed73b50b06f91": "0 \\le x_{ij} \\le X", "555325d4709eae8beab57dfba55224e4": "{}_p\\Psi_q \\left[\\begin{matrix} \n( a_1 , A_1 ) & ( a_2 , A_2 ) & \\ldots & ( a_p , A_p ) \\\\ \n( b_1 , B_1 ) & ( b_2 , B_2 ) & \\ldots & ( b_q , B_q ) \\end{matrix} \n; z \\right]\n=\nH^{1,p}_{p,q+1} \\left[ -z \\left| \\begin{matrix}\n( 1-a_1 , A_1 ) & ( 1-a_2 , A_2 ) & \\ldots & ( 1-a_p , A_p ) \\\\\n(0,1) & (1- b_1 , B_1 ) & ( 1-b_2 , B_2 ) & \\ldots & ( 1-b_q , B_q ) \\end{matrix} \\right. \\right].\n", "555365c4d40b5623fbc1dea39a4c5d66": "\\biggl(\\int_S|f|^{1/p}\\,\\mathrm{d}\\mu\\biggr)^{\\!p}", "555370c13429d1b177297157ad36f558": "D_i^{(T)}", "5553757dc72daba7ce2d3751d820b349": " f(x,y) = \\frac{1}{(2\\pi)^2}\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} F(k_x,k_y) e^{j(k_x x + k_y y)} dk_x dk_y ", "5553a64a420325bbe0f412e187740262": "{a \\over 6}(2 + \\sqrt{2})", "5553e611309b3d3482b2d826a1e8323d": "GL(V,F) \\rightarrow PGL(V, F)", "55541d457275e1c364be4ff8e5422ae6": "1 - \\varepsilon \\le d(x, Y) \\le 1", "555441d618b0b1ebebff6e5e4a9fa682": "c=n^2+m^2", "55544560b2e349d49384a6e95d78ed92": " w(P,Q):=\\frac{G}{F}", "55551161a2ebbfb5eed0a1501043b4cf": "\\Phi_R", "55555c4373aba5deddd0023c82f9948f": "\\psi_0\\!", "55558b93bec91a896a72dfa217a332ab": "\\Delta w ~ = ~ C x\\cdot w - w\\cdot C y.", "5555b4626a93dbbe5ba884e3ab890460": "(a\\cos 3t,a\\sin 3t).", "5555c4ed3f7629cda959fb4845d2401c": "a_nx^n+a_{n-1}x^{n-1}+\\dots +a_2x^2+a_1x+a_0 = 0,", "5555ca97a764b16096d098a999419b89": "\\, \\lambda", "55564cd3547cff0f2814f313267e8374": " h = \\mathrm{det} \\left( h_{ab} \\right) \\rightarrow \\tilde{h} = \\mathrm{J}^{-2} h \\, ", "555667ffa0d9c2be184cc99e3a13e0fc": "2^{\\aleph_\\alpha} = \\aleph_{\\alpha+1}.", "5556a01456289df0c7e8b4c2cdfbeea8": "a \\succ_W b", "555742f2617665ad2e83492ca936e2fe": "K(k) = \\tfrac{\\pi}{2} \\,{}_2F_1 \\left(\\tfrac{1}{2}, \\tfrac{1}{2}; 1; k^2\\right).", "5557ed198e7488386f6b999763975aa4": "\\,O_iOO_j", "5557ffcd0e4cf9c8464a4bee1d642825": "{\\ \\choose\\ }", "55583bcdc4547a178479058d53119c5a": "(AB)_{ij}=\\sum_s A_{is}B_{sj}.\\,", "5558d48258366e9aaa9d5d37e11d2367": " (u,I,v) ", "5558fc0a0745c770eb30d314232942ab": " Lu = -\\partial_i(a^{ij}(x)\\partial_ju) + b^j(x)\\partial_ju + cu\\, ", "55590f847b49574b9cb3cefa28cd89a4": "\\lim_{\\epsilon \\to 0} \\frac{f(x_0+\\epsilon)-f(x_0)}{\\epsilon} = K.", "55593986becdcac834b1e8613cdd8022": "N_b(\\omega)=\\sum_{1\\le k\\le n}\\ 1\\!\\!1_{B(k)}\\quad\\text{and}\\quad N_b(\\omega)=\\sum_{1\\le k\\le n}\\ 1\\!\\!1_{H(k)}.", "55598950cd070bafb3378edb291ee2d5": "C \\subseteq A\\,\\!", "5559b65f7d502b4e25858a8d57b79406": " r\\theta~\\cos\\theta \\,", "5559d0dbd33748e6f4a7227c2e9e4aa3": "\\delta \\leq 1", "555a360e3d07a06642abcd12f7fc6bec": "\\psi(r, \\theta, \\phi) = R(r)Y_{lm}(\\theta,\\phi).\\,", "555a43460422b4e7688cf9079c843327": "k_{\\lambda} e_i k_{\\lambda}^{-1} = q^{(\\lambda,\\alpha_i)} e_i", "555a5aa212235379d87ce67305a9224e": "\\mathrm{Stk} \\gg 1", "555a5bb21775a78a2f386253e3e5f2fb": "(g^* q g^{\\star})^* = (g^{\\star})^* q^* g = (g^*)^{\\star} q^{\\star} g = (g^* q g^{\\star})^{\\star}.", "555a6bc8910e0ba97ad0fd0c80289667": " N(y)=375.6\\cdot 1.00185^{1.00737^{y-1000}} \\,", "555ab3d0dfdc525fefb3471630e01c26": "\\ddot{\\theta} = - \\frac {2 \\cdot H \\cdot \\dot{r}} {r^3}", "555b03b96ed40919708967fe079b33d4": " \\frac{\\sqrt2}2 \\cdot \\frac{\\sqrt{2+\\sqrt2}}2 \\cdot \\frac{\\sqrt{2+\\sqrt{2+\\sqrt2}}}2 \\cdots", "555b1a1c4eb1ade7e2d525bf7ec053fd": " r_p=\\frac{a^2}{b},", "555b8c3b61b6e44c7e8e06ad0f7bfd39": "X^2 + aXY + b Y^2 = P (T).\\,", "555ba11fe85a6fa79514175c3b100953": " \\vec{A} = \\frac{\\mu_{0}}{4\\pi} \\int{ \\frac{\\vec{J} } {r} dV} ", "555bc7b97adb6722310bcb8dc075d476": "\\vec{F}=-k \\Delta \\vec{x}", "555c26f6cd4493af8a4bd9b525bf96dd": "P_{all}^{'}(k_{i})==\\delta\\frac{k_{i}}{\\sum_{j}k_{j}}+(1-\\delta)P^{'}(i\\in Local-World)\\frac{k_{i}}{\\sum_{j}k_{j}}", "555cd1621a55204bccb1729027075f5b": "x_1,x_2,\\dots,x_n", "555d30f6975b1dfd01b7a17bb3137df8": "\\tfrac{6}{7}\\scriptstyle{\\sqrt{783+436\\sqrt{2}}}", "555d3e1da7e6b1099a479fd7be88525b": " \\frac {d^2 \\mathbf{x}_\\mathrm{A}}{dt^2}=\\mathbf{a}_\\mathrm{AB}+\\mathbf{a}_\\mathrm{B} + 2\\ \\sum_{j=1}^3 v_j \\frac{d \\mathbf{u}_j}{dt} + \\sum_{j=1}^3 x_j \\frac{d^2 \\mathbf{u}_j}{dt^2}.", "555d785c7c8ecf129c38050ce2ed8e2e": "\\hat{N}_f - N_f", "555d9d497e84e72eb97330c5b558ed8e": "T_j^{(\\mathbf n)}= \\sigma_{ij}n_i.", "555daf8db2b0f3e55485e1a6436c6709": "\\delta(k) = (2\\pi)^d \\delta_D(k_1)\\delta_D(k_2) ... \\delta_D(k_d) \\,", "555df9e0de1ae9f47776f96266110172": "\\rho(r)", "555e6f6ed2c380f74a2768e40826ae41": "\\theta = \\frac{\\operatorname{d}y}{\\operatorname{d}x}", "555e822b0fb2cd60688bdca6b60f79bf": "d(v,x)", "555eb9e6d3ec2df39f7aac22ee1a04aa": "w^3 + q - \\frac{p^3}{27w^3} = 0.", "555f656042ca4859226f36503c26507b": "\\left\\{x_1, \\frac{x_1 + x_3}{x_2},x_3 \\right\\},", "555f75cd42dd7f6ef3c8a0bd4310782d": "|\\Gamma(A)| < \\frac{2^{k}}{e},", "555fb4be1423fb299552fc65943fd450": "\\mathit{h(x)} | \\mathit{x^N} -1", "555fe3e1ea3fbf0646c70241378ea34c": "H_m = \\frac{1}{\\sqrt2} \\begin{pmatrix} H_{m-1} & H_{m-1} \\\\ H_{m-1} & -H_{m-1} \\end{pmatrix}", "55607ddf6ac5a4dfeeca51ba03ee3070": "\\sigma_F", "556090315301d6e2ca3a640548db50bf": " \\Delta S = \\left(\\frac {Q}{T_2} - \\frac {Q}{T_1}\\right)", "5561732e7389eb0de3ed0e7cdc68c212": "\\left|\\sum^{N}_{n=1}b_n\\right|\\leq M", "5561aefca27be070ec562c74af7b355e": "1 \\to \\Gamma (X, O^*_X) \\to \\Gamma (X, M^*_X) \\to \\Gamma (X, M^*_X / \\mathcal O^*_X) \\to H^1(X, \\mathcal O^*_X)", "5562054c80d0bf1bf6b730d95710ea8e": " \\nabla \\cdot \\mathbf{q} + \\frac{ \\partial u}{\\partial t} = 0", "55630bf1f77af7c91d365fb4ae1ee006": " \\sum_{i=1}^k t_iX_{ni} \\overset{D}{\\underset{n\\rightarrow\\infty}{\\rightarrow}} \\sum_{i=1}^k t_iX_i. ", "55631c6015d5b89d23cb4914626ceddc": " \\mathbf{F} \\cdot \\mathbf{\\hat{n}} = \\frac{\\mathrm{d} \\Phi_F}{\\mathrm{d} A} \\,\\!", "55632eff1efed059631447d5cd2607be": "\\scriptstyle K_{m\\rightarrow n}", "5563714ba31da5956646336def3fc7e1": " \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 1 & 1 & 0 \\end{pmatrix}^n\n \\begin{pmatrix} 3 \\\\ 0 \\\\ 2 \\end{pmatrix} =\n \\begin{pmatrix} P\\left(n\\right) \\\\ P\\left(n+1\\right) \\\\ P\\left(n+2\\right) \\end{pmatrix}\n", "55638cdf4aec6357af161ba74452ac2a": "\\cos_k(i) \\equiv \\cos_k(t), \\, ", "5563992b43e4d27d99c1407c57e83f1d": "\n\\left [\\begin{matrix}\\underbrace{8 \\cdot 2 = 16}\\\\1+6\\to 7\\end{matrix} \\right ] + \n1 + \n\\left [ 1 \\cdot 2 = 2 \\right ] + \n2 + \n\\left [ 2 \\cdot 2 = 4 \\right ] + \n8 + \n\\left [\\begin{matrix}\\underbrace{9 \\cdot 2 = 18}\\\\1+8\\to 9\\end{matrix} \\right ] + \n8 + \n\\left [\\begin{matrix}\\underbrace{7 \\cdot 2 = 14}\\\\1+4\\to 5\\end{matrix} \\right ] =\n46\n", "55640a047495fa81615fec1afab0ea09": "P_y(x) = \\int_{R^n}e^{-2\\pi i tx-2\\pi |t|y}dt = \\frac{\\Gamma((n+1)/2)}{\\pi^{(n+1)/2}}\\frac{y}{(|x|^2+y^2)^{(n+1)/2}}", "556413e643bf2580a2eb7a9bf32dc6a9": " U(\\beta) = \\sum_{i=1}^N \\frac{\\partial \\mu_{ij}}{\\partial \\beta_k} V_i^{-1} \\{ Y_i - \\mu_i(\\beta)\\} \\,\\!", "556461f9b2ef581b923321016075c8ea": "\\sqrt R", "5564812909bea1dcdd011040ce14b7a7": "\\delta_{\\epsilon} \\psi =\\partial\\!\\!\\!/ (S+P\\gamma_{5})\\epsilon ", "55648609b736bd3ffeba6e6df3275f5d": "q_\\mu^*(\\mu)", "5564bc25809a74a0c2181656b271b281": "\\cos x - 1 = -\\frac{x^2}2 + \\frac{x^4}{24} - \\frac{x^6}{720} + {O}(x^8)\\!", "556567ac24e96026fb5f3993f596b428": "1, \\frac{2}{3}, \\frac{3}{7}, \\frac{4}{17}, \\frac{5}{31}, \\frac{6}{109}, \\frac{7}{253}, \\frac{8}{97}, \\frac{9}{271}, \\dots", "5565852fe8237806f5b05a4a027d394a": "qU = \\frac{1}{2}m\\left(\\frac{d}{t}\\right)^{2}\\,", "55659e1784168e2478a7b7cbbc27cc24": "Z=\\sum_{n=0}^{\\infty } \\left ( \\frac{4}{125} \\right )^n \\frac{(11n+1)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{6} \\right )_n \\left ( \\frac{5}{6} \\right )_n} {(n!)^3}\\!", "5565f2febb597014dba29904953c8786": "\\langle j_1 j_1 j_2 J-j_1|J J\\rangle", "556694c459e26d3d97cf0263fff59d17": "C(1)=K/N", "5566cd8541b82192b4ad3e7a5688fc3a": "\\mathbf{Z}/2,", "55671e9f82c35ab6670921cde82628b2": "\\mathrm{Sper} A", "55675107d2c12da6a3a25914c33e5b8e": "\\hat{H}_R=-\\mathbf{\\Omega}\\cdot(\\hat{\\mathbf{r}}\\times\\hat{\\mathbf{p}})", "55676359758cb21ee745b95c2f489cf8": " \\mathcal T", "55677559f775a19cfb7dcb1fa4bb3786": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n{1 \\over \\vec k^2 + k_{Ds}^2}\n", "5567a61d5863f7dc9d654236b75e1c79": "x_i = c_1(L_i),", "55680843b8ade3afe4195a2ca7627be9": "\\Delta P =\\rho a_0 \\Delta v ", "556838fa00e5efe21943098c85b9907e": "\\mathit{n - w}", "55684bd6a3e1eaecaab8610df3beb924": "(\\mathcal{L}_{\\!X} f)(p) =X^a(p)(\\partial_a f)(p)", "55689d46e95186d993e3151957e8d4df": "-\\ln(2\\pi)/2", "55689f02c717ad57238b3a28fa0e226e": "\\phi(q)= q^{-1/24} \\eta(\\tau)\\,", "5568bede0a4916ff3c32cb38ab2d26cb": "U(z, \\phi) = \\frac{z - \\phi}{1 - \\bar \\phi z}", "5568e6ca1c6333bbb1b4d2868e512ea7": "W=\\int d^3rB^2/2\\mu_\\circ", "55690f4757c0f4c29ed9519804f53919": "t = (w + xi) + (y + zi) j \\ ", "55692f7252274d9d422bd7473dc4cb47": " G'' = \\frac {\\Delta T_s} {\\Delta d} \\sin \\phi = \\frac {4A}{\\pi \\omega \\Delta d^2}", "55698825c62ee82d178a496b501ae6e8": " S_\\phi", "55698f91d7141a68d5e2ef813c806261": "\n\\widehat{R_{\\alpha}[f]}(\\sigma)=\\hat{f}(\\sigma\\mathbf{n}(\\alpha))\n", "556a0b91af7b4c53d17dcc709ca79e12": "\\frac{E_1}{E_2} = \\frac{k_1}{k_2} = \\frac{c_2}{c_1} ", "556a419e31e485b48aef9d0d0fc27d76": "\\int_0^\\infty x^2 e^{-3x} \\, dx = \\frac{2}{3^3} = \\frac{2}{27}.", "556a489d7aa97be44bace0086a3bce4d": " a_z = -\\frac {8eU} {m r_0^2 \\Omega^2} \\qquad\\qquad (12) \\!", "556a840ac3a1257f9fd0be2b173dbe19": "\nL = \\operatorname{tr} \\left\\{ \\frac{1}{g^2} F_{IJ} F^{IJ} - i \\bar{\\lambda} \\Gamma^I D_I \\lambda \\right\\}\n", "556a95380a795acdd55d561664a72bb0": "\\operatorname{Bez}:\\C^n\\times\\C^n\\to \\C:(x,y)\\mapsto \\operatorname{Bez}(x,y)=x^*B_n(f,g)y.", "556ab4cc78d18d608d4a6cf3505213f0": "n\\ \\arg(z-a) = m\\ \\arg(z) + n\\ \\theta_0", "556abface6568d39a0fe03d2187d7ad3": "2-\\alpha", "556b1b44bc7fdacaae57802b6c642fac": " = \\int f(x) \\; \\nabla_{\\theta} \\pi(x \\,|\\, \\theta) \\; \\frac{\\pi(x \\,|\\, \\theta)}{\\pi(x \\,|\\, \\theta)} \\; dx ", "556ba2362413810ecf34a8c87cf844bf": "\\implies argmax_{w_i}U_i(\\frac{w_i}{\\sum_jw_j}*B)-w_i", "556bd45c85dc2f505563de5f123d4b5a": "q_1q_2", "556c11f5a6845d01dc8be58b552f6b60": "\\tau = \\frac {1}{c} \\int \\sqrt{g_{00}}\\,dx^0.", "556c1af3be161ad5d97e294ec970ba94": "\nU(t,t_0)=U(t,t_1) U(t_1,t_0)\n", "556c1f3d0836feebefe7f60bfb0f8643": " X=\\{x_1,x_2,\\ldots,x_T\\} ", "556c6d490a0af15bcc7aab1d15162f70": "max_{\\mathrm{a}}P[A = a | B = d]", "556ca7136b64f48f1fffdbecb3c981a5": "\\sigma_2 =\\sigma_\\mathrm{min} = \\tfrac{1}{2}(\\sigma_x + \\sigma_y) - \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2}\\,\\!", "556cffe2b72d6d7d588ce4f4f59ebdc8": "N\\rightarrow \\infty", "556d28b736e76daaeccf63727fd9a2ad": "\\frac{{d^3 Y}}{{dx^3 }} = \\frac{6}{{h^3 }}a_3 {\\text{ }}", "556d65d7f6170b0d74ae904d78067472": "O(k^{d})", "556db6c8d81ec8c024b8f7c6c1411f57": " \\int f_n \\leq \\int f", "556dc0a9f52121f182861aa14562c563": "(-1)^{\\text{sign}} \\times 2^{1 - \\text{exponent bias}} \\times 0.\\text{mantissa}", "556dcfa7702cfd01de3f9f8a57f51163": " f(I_1, J_2, J_3) = 0 \\,", "556dfd5b0af8e223ad0d804e189b2ec0": "r = \\frac{K}{K-1}\\cdot\\frac{N^{-1}\\sum_{n=1}^N(\\bar{x}_n-\\bar{x})^2}{s^2} - \\frac{1}{K-1},", "556e2df5ebbe864d3df68944be9ddf66": "n \\cdot \\left( {\\frac{k}{s} + d} \\right)", "556e4ae710b26a9cdcee83d92e8df9ad": "3425 \\times 11 = 3425 \\times (10+1)", "556e7948e6054c074bd3bb564f2bc585": "a^{n}", "556e8fcf27e2108c110263bd0fcefdae": "\\exp^2(x)", "556eb7c4348798c83bcbca21fb9ed27b": " h(s_1) \\times K^{(n-1)} + h(s_2) \\times K^{(n-2)} + \\cdots + h(s_{n-1}) \\times K^1 + h(s_n) \\times K^0 .", "556f21c5d3a4cf3e551f4f3cc8467d76": "p(x)=p_e(x^2)+x\\,p_o(x^2)\\;\\implies\\; q(x)=(-1)^n(p_e(x)^2-x\\,p_o(x)^2). \\, ", "556f82d6bbd7b7048043acb4574d352c": "W = N \\cdot x \\,", "55700afddabe80e78bd93b68b0899df0": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) = \\left \\{ u(x): \\ \n|u(x) - {\\tilde{u}}(x) | \\le \\alpha {\\tilde{u}}(x), \\ \\mbox{for all}\\ x \\in X \\right \\} , \\ \\ \\ \\alpha \\ge 0.\n", "55702706d27f9f706a063c226d17b8bf": "\\phi(x, D(x, z), z)\\;", "55703c39d79397205c7bb5058834fa56": "\\alpha(x)\\,", "557067a91c8ba34e8e4170b2a9ec7d07": " (\\exists y_1...y_m) \\phi(a^n_1...a^n_k, y_1...y_m) ", "5570af2f35f507b65ca99c908fbc4ce0": "\\alpha \\in \\tilde C(\\alpha,\\rho)", "5570e890438700ad62cff565d1d39e57": "\\hat{\\beta}(\\tau;aY,X)=a\\hat{\\beta}(\\tau;Y,X),", "55710afb57a827fcc3a9753bbd6d29a6": " \\nabla_{X+Y}v = \\nabla_X v + \\nabla_Y v", "557115369c6e8b2852ae954be4459b93": "\\rightarrow_M^1", "5571362161165e3179cffee275d55317": "\\mu = \\frac{1}{2} ", "557166e177f68c2e4044c2d95ffe71a9": "S+I+R=N.", "557175a99f3631ab265901af2b401ab9": "\n {\\underline P}X= \\{x \\mid [x]_P \\subseteq X\\}\n", "5571a4ad52b9e896a4755316a9739053": "\\delta_{ab}", "5572133afd0a9114c2b9528a726fe0f9": " 1/(p^{2}-a^{2})", "557234813b3ae66b63c8f4d142ca579e": "f(x_i)-Q(x_i)\\le|Q(x_i)-f(x_i)|<|P(x_i)-f(x_i)|=f(x_i)-P(x_i).", "55725b1e00902f117cf3b187fdf6a38d": "v({\\mathbf P_1}+ {\\mathbf P_2}+{\\mathbf P}_3+ {\\mathbf P}_4) = v({\\mathbf I}) = 1.", "5572654594bc1bc19904cb779b0f8be8": "\\gcd(\\operatorname{lcm}(a,b),\\operatorname{lcm}(b,c),\\operatorname{lcm}(a,c))=\\operatorname{lcm}(\\gcd(a,b),\\gcd(b,c),\\gcd(a,c)).\\;", "5572c138f4c07dfdcb480fa157059b93": "\\mathcal{S}\\in \\mathcal{R}^{I_1\\times I_2\\times \\ldots \\times I_N \\times L_1\\times L_2\\times ... \\times L_O}", "5572d9c60eac732885371745ab99f1e6": "x_1,x_2,...", "55733ccf4cf11ee3bb2e5cb815eec637": "I = \\frac{m s^2}{6}\\,\\!", "557366aa6f163ab57ada67a4eb7b9497": "\\textstyle \\zeta_{G}(0)", "557383656dac420f5fbc1c24f9d76794": "H_{B_2}", "557389a1d51cf21ce90cc1fe17dd767c": "\\displaystyle P(x)", "5573b25091c3c4ae1ec0bc8c4cfcc305": " \\frac{d\\sigma}{d\\Omega} = \\alpha^2 r_c^2 P(E_\\gamma,\\theta)^2 [P(E_\\gamma,\\theta) + P(E_\\gamma,\\theta)^{-1} -1 + \\cos^2(\\theta)]/2 ", "5573dd0843b4ab9add8e2fe7a6591890": "\\delta_{\\pi} \\,\\!", "5573e9ae0447205536292c0f07f6c3f7": "p_{b}", "557440d914ab8f476685524028d0c683": "\\cos\\theta=\\frac{\\Omega_{c}}{\\sqrt{\\Omega_{p}^2+\\Omega_{c}^2}},\\qquad \\sin\\theta=\\frac{\\Omega_{p}}{\\sqrt{\\Omega_{p}^2+\\Omega_{c}^2}}.", "557494130b22389c79d1f211d7775439": "f(y)=A_{i}", "557497a3ca61f54604140f21edef516b": "1_{A \\,+_{\\mathrm{e}}\\, B} (z) = \\mathop{\\mathrm{ess\\,sup}}_{x \\,\\in\\, \\mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),", "5574a8de2f0d1f5168bcad57790ca3e9": "| g(y) | = 1 \\mbox{ for all } y \\in Y", "5574aec62690cdfbc4b812687042e182": "h = \\frac{\\mathrm{peak\\ value\\ of\\ } m(t)}{A} = \\frac{M}{A} ", "5574b3c2aefea19d8371cebf969a46ac": "\\mathbb{E}\\Bigl[\\liminf_{n\\to\\infty}X_n\\,\\Big|\\,\\mathcal G\\Bigr]\\le\\liminf_{n\\to\\infty}\\,\\mathbb{E}[X_n|\\mathcal G]", "55752def2a138b8b730ccfc29273106a": "\\Lambda<\\limsup_{n\\to\\infty}x_n", "5575592141c896679a0b754699f93998": "\\phi,\\tilde\\phi", "557574d36c26dcfabfc6354aeb859ccd": "\\forall a\\ \\forall b\\ne0 ((\\forall t\\ g(t)=a+b\\cdot f(t)) \\Rightarrow \\forall x\\ M_f (x) = M_g (x)", "55760cca82d5ddcecd92832907b0b9fe": "\\operatorname{var}(\\widehat{\\varepsilon}_i)=\\sigma^2(1-h_{ii}).", "55762e30c8b89f1388e2712a760b4f11": "T_1,T_2,X_1,X_2,H", "55763b58a6810385e3af5d44a279e0b4": "u^{-1}(D)", "55763c663ac41f808cdfc8bf946b60ee": "W_c", "55765ac1d6d9c85a8ea95226cb611e86": "e^{-ix} = \\cos(- x) + i \\sin(- x) = \\cos x - i \\sin x \\;", "55765f42f2d3c4fadd277426675825b2": " {U(R)}^\\dagger \\widehat{V}_i U(R) = \\sum_j R_{ij} \\widehat{V}_j ", "5576e262dbcf98b466cc73681a3174fd": "\\text{dleq}(X, Y, g_1, h_1,g_2,h_2)", "5576f37cf16e3c87e26a5350c31bf2dc": " c_s ", "5577524a185defa00c326094ed7faa88": " \\omega \\to \\pm \\omega_0 ", "5577a3a29aebd8886bb9408146e142ce": "i,j = 1,\\ldots,N", "5577f4f0b65d50bcc81adad652b17e92": "\n f^\\text{pmi}(t_i,w)=\\log_2 \\frac{f^b (t_i,w)\\times m}{f^t (t_i)f^t (w)},\n", "5577f941769f0ec29a1250ae46dc8818": "\\pi_1 : X \\times \\mathbb{R}/R\\mathbb{Z} \\to X, \\; (x, f) \\mapsto x", "5578686b2e4e4f07841c58f3f9a5c2b6": "\\sqcup", "5578c582d8c8ce7b49b3133571d3d6ba": "\\displaystyle{\\lim{\\delta\\downarrow 0} {D(c(t+\\delta)) - D(c(t))\\over \\delta} =2\\min_z X(c(t))\\cdot (c(t)-z),}", "557975d4cd42183b1b565b731304774f": "(a_1, b_1) \\in E, (a_2, b_2) \\in E, \\ldots, (a_n, b_n) \\in E", "557978b2e7f40a298da7180f76ecdd3d": "\\log M(r,f) \\leq \\left(\\dfrac{R+r}{R-r}\\right)T(R,f),\\,", "5579899b8bcff8336a4150f54a66a19d": " \\nabla \\boldsymbol{\\cdot J_f} = -\\frac {\\partial \\rho_f}{\\partial t} \\ , ", "5579a53f895f4288db0c4fbebffd5c8b": "\\omega= \\frac{2\\pi\\cdot \\mathrm{RPM}}{60} ", "5579d1ef09e03078c500371e432dd0d6": "x < 10^7", "5579d2a1bfcfef54ab4f7cdfe4404e71": "\\Delta^1_2", "5579ddfc01e2d626dcac3fc506a39c6a": "\\frac{\\partial}{\\partial x}\\,f(x,t)\\,", "557ae8b1e90c6893b4e9f9956d9afc87": "\n{\\mu_t}_\\text{inner} = \\rho \\ell^2 \\left[\\left(\n \\frac{\\partial U}{\\partial y}\\right)^2 +\n \\left(\\frac{\\partial V}{\\partial x}\\right)^2\n\\right]^{1/2}\n", "557aec43aabe6406cb2b74eddae6c54c": "A= 460^2", "557b0143aeda36fd455fbf6508fe091f": "m = -(I+1/2)", "557b080a387c2a3721560adafbf564fb": " R_{xs}(\\tau + \\alpha) = \\int\\limits_{0}^{\\infty} {g(\\theta) R_{x}(\\tau - \\theta)d\\theta}.", "557c0fc945e0c21981b91ea00c785261": "\\color{SpringGreen}\\text{SpringGreen}", "557c159eb89c085d75276828336e6ebe": "h(\\mathcal{O}(y,\\psi)) = \\{h(\\psi(y,t)): t\\in\\mathbb{R}\\} = \\{\\varphi(h(y),t):t\\in\\mathbb{R}\\}= \\mathcal{O}(h(y),\\varphi)", "557c2f491c1bf368980254cc88fdfd4e": " d \\sigma^2(t) = g[ \\sigma_L ^2- \\sigma ^2(t)]\\,dt+\\xi \\sigma(t)\\,dz_2(t)", "557ca859186327b3990004180c3efa70": "\\mathcal{H}_{i}\\approx 0\\rightarrow \\mathcal{H}_{i}\\Psi =0", "557cfefad6237b5280a09b29137c786f": "z_{i,j}", "557d0b33554d181d4fd55206a2e1fdb6": "x_+ = \n\\begin{cases} \nx &:\\ x > 0 \\\\\n0 &:\\ x \\le 0.\n\\end{cases}\n", "557d6197aaeef200b68f94007ba184d3": "p_1,p_2,...,p_m", "557db03270c226517b987c3545a37aa8": "\\int f'(x)e^{f(x)}\\,dx = e^{f(x)} + C", "557dd5e5ae84c046190db15353ccd607": "\\displaystyle -\\left[\\left(\\nabla\\phi\\times \\mathbf{\\hat z}\\right)\\cdot\\nabla\\right]\\left[\\nabla^2\\phi-\\ln\\left(\\frac{n_0}{\\omega_{ci}}\\right)\\right]\n", "557ebd19a89a2fa40344c8002923c7e8": "(V,T)\\ ", "557ef34c84bab51bd6435a1fda45140c": "W(D) = -\\frac{AR}{6D}", "557f2728e29f06d863149e4dbbeba4fd": "\\mathbf{a},\\mathbf{b} \\in \\mathbb{R}^d", "557f92ee7e347d64bc116ebee8849242": "R(Q) - C(Q) - G_{min} \\ge 0,", "557fded05b5ed4a43b0eb67639d832bf": "I = X - NX", "558037448afeddf90cee789acd3c1705": " \\operatorname{equate}[A, N] \\equiv A = N \\or (\\operatorname{def}[V[N]] \\and A = V[N]) \\equiv A = N ", "558038fe9d33ddeb8cb94013bb64072b": "R = \\mathrm{clamp}(( 298 \\times C + 409 \\times E + 128) >> 8)", "558041b663c4f9f121db087b386e1304": "\\scriptstyle K \\;=\\; R^{abcd}R_{abcd}", "55806f34ebdd4658d8f8b07e286173b8": " p(\\theta) = \\frac{\\sin \\theta}{2}", "5580981b03af46a180b29826be5358e0": "u(y)", "5580af5754af315735fd8a6dfe2fa3e6": "u=x", "5580b98c1c6645069146510cbb5e14dc": "\\scriptstyle\\ \\alpha=0.05 ", "558175b28435f26f7e37511b5bff3c91": "p\\in\\mathrm{sat}(T)\\iff \\mathrm{prem}(p, T)=0", "5581c96425c717c4fc14d902193d4275": "f(x)=x^6+3x^5-5x^4-15x^3+4x^2+12x", "5581db83316cc0d9c5da69bdfbbb6b23": "f(x + y) \\le f(x) + f(y)", "5581f23eab248f2c8d255d74b56016a5": " \\nabla_{\\theta} J (\\theta) = \\sum_{k=1}^\\lambda u_k \\; \\nabla_{\\theta} \\log\\pi(x_k \\,|\\, \\theta) ", "55823aa8f47aa019e13715d43fe6b87d": "n.5", "558291273cf55bf353b80184bab2b3e3": " \\nabla_m R^m {}_l = {1 \\over 2} \\nabla_l R\\,\\!", "5582947a099621ecd019353982bfee89": " i \\exp \\left( -i t \\right) ", "558299f124632342979afa145afc8685": "RR^{-1} \\equiv 1 \\pmod {N}", "5582b5b327bee4e0fe52024b3440874b": "2-2g \\,", "5582bb2d0e5eb0cb35913da17657f722": " \\displaystyle{{dw\\over dt} = -w p_t(w)}", "558384fc9c1d2c61ec39f669200d79b7": "K \\rightarrow H", "5583c5e5dc43c65b9fb32a4118e86e74": "\\frac{27}{20}", "5583daba3b4ce4a4e030d1991512c23a": "\\ O(n \\log n)", "558413309d8bfeef99f2ecaaa2d9e11c": "\\Pr(E_n) = |\\lang \\psi_n | S \\rang|^2 = \\frac{2}{L}~{\\rm sin}^2\\left(\\frac{n \\pi S}{L}\\right)", "5584620b76c35ec37a140efb88c08566": "x=-\\frac{a(1+p)}{1-p+p^2},\\ y=px", "55848c4b2b9c6041b11c02125da13392": "\\partial^2=0", "558496f2e2d72d284a865935bfe60822": "a_{n-1}\\ldots a_0", "5584a007dea247949ad19175912f097e": "a + b\\sqrt{-1} \\mapsto a + bq.", "5584a9981c3ce79b29a1423af014321a": "\\prod_i {C_{i}^{n_i} \\over n_i!} ", "5584bc807aed027c716e5558883e0900": "R_\\infty", "5584e64e05f9f6ee9d882c8ac05d0480": "E \\exp(i u^T X)= \\exp\\left\\{-\\sum_{j=1}^m \\omega(u^Ts_j|\\alpha,1)\\gamma_j^\\alpha +i u^T \\delta)\\right\\}", "558502336e515275d5b544efc0cfb4ec": "n + p \\rightarrow d + \\gamma", "55858965579332cf5d75b58364afa285": "S_{n} = \\frac{S_{n-1}}{3} = \\frac{s}{3^{n}}\\, .", "55859200bc70e7f81dd8d705f2b94d37": "e_1~e_2", "5585a4b71c85926a36cd7b16a3c68520": "\\mathbb{E}\\left[\\mbox{ Arnold }|\\mbox{ Charles calls }\\right] = \\mathbb{E}\\left[\\mbox{ Arnold }|\\mbox{ Charles folds }\\right]", "5585e68c9753c166be128b1eb257f79b": "\\varlimsup_{N\\to\\infty} \\frac{S_N(f;t)}{\\omega(N)}=\\infty.", "55864f59dceb979e7323b8d1fc6834b5": "\\sin \\theta \\approx \\theta", "55865a4b4807148eefd1f46ec535e680": "a(n+20) \\equiv a(n) \\pmod{100},", "558672245b2e1ad71289f472e058ba86": " \\sin(\\omega t) \\cdot u(t) \\ ", "5586827d3908cdd05a8c22b03489b894": " H^k(\\Omega) = W^{k,2}(\\Omega)", "5586840fa51cd02b8c105140d9f26b97": " \\mu_0 \\vec{J} = -\\frac{d}{dr}B_z \\hat{\\theta}", "55868e8bd0b20a92d4a1d0678d142388": "{ {\\log(Bx_1/Bo_1)} \\over{\\log(Bx_2/Bo_2)} }\\ = \\ \\left({r_2\\over r_1}\\right)^a", "55869c9b5c5dcb6675f67196c2657233": "\\Theta=\\frac{D_m}{f}.\\,", "5586a1f3654d28e6494d1f1cc234c0ae": "\n\\begin{align}\n \\Psi_i(z\\leq0) &=& Fe^{qz}\\left[\\exp\\left[i\\left(\\frac{\\pi}{a}z\\pm\\delta\\right)\\right]\\pm\\exp\\left[-i\\left(\\frac{\\pi}{a}z\\pm\\delta\\right)\\right]\\right]e^{\\mp i\\delta}\n\\end{align}\n", "5586ca8c5bf87b80cfabb01c23d59ad5": "\n\\Psi(\\mathbf{x}_1,\\mathbf{x}_2) = \\frac{1}{\\sqrt{2}}\\{\\chi_1(\\mathbf{x}_1)\\chi_2(\\mathbf{x}_2) - \\chi_1(\\mathbf{x}_2)\\chi_2(\\mathbf{x}_1)\\}\n", "55870897deb9a4396456a03dcda3dd15": "(x^2+y^2)^3 = 4x^2y^2. \\,", "55878eed93158b351d4179ae378faeca": "x = 12", "55879c8f5fa294592dc206fcd6f25dcd": "\\scriptstyle X,Y,Z \\in T_pM", "5587a774c2aadc35744aa3c0a1ab03c0": "\\lim_{N \\to \\infty}\\text{area of }N\\text{-gon} = \\text{area of circle}. \\, ", "5587f06f8515f054cd460459da000e71": "\\Rightarrow \\frac{A}{\\sin \\alpha}=\\frac{B}{\\sin \\beta}=\\frac{C}{\\sin \\gamma}", "5588507724533f91abdc932880e8d94f": "|A_i|>k_i", "55886a921d79f8ff320e73559fc4d730": "ky = k \\int \\frac{dy}{dx} dx", "55887d002269188d03bb0eb3cf4c5722": "x = \\left( x_1,x_2,\\ldots,x_N \\right)", "5588832106ebb52d7353d9fd28794176": " r_1 - r_2 =2 a\\,\\!", "558893903dc432e6e0f7d791d6592b90": " d_\\varepsilon(x)=\\frac{\\mu(A\\cap B_\\varepsilon(x))}{\\mu(B_\\varepsilon(x))}", "55889863d67a29425d8c1da4d9e096b8": "y_3 = \\frac{((1+(x_1x_2)^2)(y_1y_2+2ax_1x_2)+2x_1x_2({x_1}^2+{x_2}^2))}{(1-(x_1x_2)^2)^2}", "5588a49a2373fee4a014f617d28b58ad": "\\boldsymbol{\\hat{\\rho}}, \\boldsymbol{\\hat{\\phi}}, \\mathbf{\\hat{z}}", "5588cbf4d657bc2c7b582bcc42115bd6": "{R(t) \\over C_p(t)}", "5588f7a727ffbffbaffa60ffd1bc8091": "{\\scriptstyle z}", "558902c289c67cf6b47a24acbfd99c45": "(-a,\\ 0)", "558926f5535293c9baaa71ad954b1f2f": "K=I\\left\\{ \\forall d \\right\\}\\otimes T\\{DSRP\\}", "5589278a9f26085eb8e42033d4ced0a6": "1000K 0\\}", "55899629df0a16b165a7ee8cc44c64a3": "\\frac{0.693}{r - r^2/2}", "5589b4dbd7b651e284fdf3a35b093768": "\\mathbf{Z}_n \\!\\,", "5589d72460696af322ae24f8a983887d": "n-(d-1)=n-d+1", "5589fae830636947faa14f31da7cfcff": " d U = n C_{V} \\, d T ", "558a9cd312a86ec0d04d66aa1353afbb": "A_1 A_1 A_2 A_2 \\cdots A_n A_n", "558af053d04661f8274191d24978b12b": " p(n_1) = 1 - c", "558b82486a63e39d8d3b2b20255b6759": "\\frac{dy}{dx} + 1 = 0.", "558b8a7b964b086028246551e7e41167": "\\theta\\left(x\\right) = \\cos^{-1}\\left(x\\right)", "558b9d54f7374b21e14856ba640469c3": "\\frac{2\\pi}{a}", "558bbaead86369a2cecdad2ceac0fd8f": "x[ab]", "558bd8f075d1af69d8398a51ca346eaf": "\\frac{2}{\\sqrt{\\pi}}", "558be1d4ce5683063b32e731d6b475cf": "\\scriptstyle \\omega \\;\\to\\; \\infty", "558c38f81b3c25dc377aafb78961b029": "f_o = \\frac{f_s}{\\gamma\\left(1+\\frac{v\\cos\\theta_o}{c}\\right)}", "558c564d354d4cc1919b6e9b30399dfb": "\\operatorname{dVar}(A + b\\,\\mathbf{C}\\,X) = |b|\\operatorname{dVar}(X)", "558d1d387950db466db3875fcff1236b": "\\begin{bmatrix} \\ln \\dfrac{p_1}{1-\\sum_{i=1}^{k-1}p_i} \\\\[10pt] \\vdots \\\\[5pt] \\ln \\dfrac{p_{k-1}}{1-\\sum_{i=1}^{k-1}p_i} \\\\[15pt] 0 \\end{bmatrix}", "558d2314c9322123faaa83e768448b7f": "N=\\lambda -\\frac{1}{2}", "558d4536b68ca3647d96445e6d9b01aa": "\\mu = m_{sat}^2 \\frac{2L}{W} \\frac{1}{C_i}", "558d61d206fb4d1a0cf38148028aeb5c": "H_{i+1}/H_i \\;", "558d87a788134ed9d6f209dc4c7c79da": "\\tfrac{1}{4}", "558dc917dbd86d99c8382d0ff9b58231": " x\\in {N}, ", "558dd7da20698005ed93bab6e42afe2e": "B(l,\\alpha)", "558ddcc72e2739de9e1d2835e5b850f1": "T_{JMAX}-(T_{AMB}+\\Delta T_{HS})=Q_{MAX} \\times (R_{\\theta JC}+R_{\\theta B}+R_{\\theta HA})\\,", "558df4a4be4e34c409ea1f6677015e4e": "c_\\kappa(x\\cdot c_\\kappa y)=c_\\kappa x\\cdot c_\\kappa y", "558df6fe94f7a30234852633df94f123": "h(k), h(k)+2, h(k)+6, ...", "558dfc07dd9815a8ea973eab26fc1d07": "K= \\sqrt{\\frac{(ab^2-a^2 b-ad^2+bc^2)(ab^2-a^2 b-ac^2+bd^2)}{(2(b-a))^2} - \\left(\\frac{b^2+d^2-a^2-c^2}{4}\\right)^2}.", "558e0971fdd5edc45d01d8aec96d41c0": "\\frac{n}{2}-1", "558e13906dc8f4f609144045a0d66060": " H = A + \\frac{B}{u} + C \\cdot u ", "558e930052253309048dc367c2aae71d": "\\beta^2=\\operatorname{var}[U^{(i)}]", "558f12e1c2cb89cea1089d05b336001d": "S_{xx}\\,", "558f16de522f8348430c41f82bd34047": "\\rho_{ref}", "558f29cbff6909b22e0d551cd7f8ce93": "\\overline f\\circ r_B=f", "558f4313f16344e1e56d3c1378529ce1": "(2,k-1)", "558f6e150cb69a8fbedcd20dd11893d2": "\\, TA(t_1)A(t_2)=A(t_2)A(t_1)\\!", "558f9eb7f3941597773cc5bd84c19572": "C=A\\pm B \\mod M", "558fc24d258ccd08be48ec02f0d69767": "\n\t\\left|A\\right|_{ij} |A^{il}|_{kj}^{\\,-1} = - \\left|A\\right|_{il} |A^{ij}|_{kl}^{\\,-1} \n", "558ff6136c060dc1653d3a21300ac28c": "E_2 = E_y", "55903e8144edf88a3c8feb53656d8ec0": "\\int_0^T|\\zeta(1/2+it)|^{2k}\\,dt = O(T^{1+\\varepsilon})", "559089419f5e09bcdb91a9e3c6069b1f": "\\neg (\\exists x\\in X : P(x)) \\equiv \\forall x\\in X, \\neg P(x), ", "5590cddc866498754c197ae8205b4925": "x[n-k]", "5590dc6a45afef9f5f9c890d9bcf2aae": "\\Phi(x\\vert(t,+\\infty))=\\begin{cases} 0, & \\text{if }\\Phi(x)\\le\\Phi(t)\\\\ \\displaystyle\\frac{\\Phi(x)}{1-\\Phi(t)}\\and 0.5, & \\text{if }\\Phi(t)<\\Phi(x)\\le(1+\\Phi(t))/2 \\\\ \\displaystyle\\frac{\\Phi(x)-\\Phi(t)}{1-\\Phi(t)}, & \\text{if }(1+\\Phi(t))/2\\le\\Phi(x) \\end{cases}", "559120dec425d14b9ae112737379cbd4": "(H\\ or\\ \\overline{E})", "55912e620a35e5ede1fd180f7b8c09ee": "\\mathbf{B}(\\mathbf{x})=\\frac{\\mu_0}{4\\pi}\\left[\\frac{3\\mathbf{n}(\\mathbf{n}\\cdot \\mathbf{m})-\\mathbf{m}}{|\\mathbf{x}|^3} + \\frac{8\\pi}{3}\\mathbf{m}\\delta(\\mathbf{x})\\right],", "5591610705cba6221575e4a5bc92c397": "\\delta(\\mu,\\nu) = \\frac 1 2 \\sum_x \\left| \\mu(x) - \\nu(x) \\right|\\;.", "5591b1455f53769910d2f9ecddec906a": "\\sin\\frac{\\pi}{10}=\\sin 18^\\circ=\\tfrac{1}{4}\\left(\\sqrt5-1\\right)=\\tfrac{1}{2}\\varphi^{-1}\\,", "55920315075d8a2122ff51ffb3312eeb": "\\scriptstyle \\leq5\\times10^{-20}", "559272fd50032c87ea75b055fe930f91": "Y(K, AL)", "55928c3dca46dff6e119fc9af0fd1f0c": "Q=(p-iW)b", "55928cfc16e2041b75b3d4cb8a72517d": "\\deg(1+2x) = 1", "55929108139ecd7763bc55f56f7bb6b7": "\\boldsymbol{\\hat{\\imath}} = (1, 0, 0)", "55939543258aeb8a38bd45751257d69b": "xy=0", "5593dcf105c239d179058ba54606d85b": "S_M(n)", "5593f77731ec02b60d0a574bf2680e7d": "p_n\\in X_n.", "55940b9b1504d898cc07b19fdb78be83": "A\\vec{\\alpha}_{i}=\\lambda_{i}\\vec{\\alpha}_{i}\\qquad(i=1,2,\\cdots,n).", "5594507b6e6a89e60e38c34018d5c8a4": " y^1 ", "5594de2892e98289560a47ece9081da4": "\n\\begin{align}\nI' [\\epsilon] & = \\int_{t_1 + \\epsilon T}^{t_2 + \\epsilon T} L [\\mathbf{q}'[t'], \\dot{\\mathbf{q}}' [t'], t'] \\, dt' \\\\[6pt]\n& = \\int_{t_1 + \\epsilon T}^{t_2 + \\epsilon T} L [\\phi [\\mathbf{q} [t' - \\epsilon T], \\epsilon], \\frac{\\partial \\phi}{\\partial \\mathbf{q}} [\\mathbf{q} [t' - \\epsilon T], \\epsilon] \\dot{\\mathbf{q}} [t' - \\epsilon T], t'] \\, dt'\n\\end{align}\n", "559535eb9c2ae2d3c581051fda0d5977": "\\langle K_x, K_y \\rangle_H = K(x, y) = \\langle K_x, K_y \\rangle_G. \\, ", "559570fd7cd8df79c50c4b0862bcd222": "\\lim_{h\\to0}\\left(\\frac{f(x+h)}{f(x)}\\right)^\\frac{1}{h}=\\exp\\left(\\frac{f'(x)}{f(x)}\\right)", "55965979b95ddb52a212baff2645a04c": "A=\\begin{bmatrix} a_1 & 0 & \\ldots & 0 \\\\\n0 & a_2 & \\ldots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\ldots & a_n \\end{bmatrix} ", "5596bb0e1db45f9189e82061054407f7": "(l,r)", "55971c310ce3c5985547993120e35493": "\\int_{\\mathbb{R}^n} e^{-\\pi\\varepsilon^2|\\xi|^2 + 2\\pi i x\\cdot\\xi}(\\mathcal{F}f)(\\xi)\\,d\\xi = \\int_{\\mathbb{R}^n} \\frac{1}{\\varepsilon^n}e^{-\\frac{\\pi}{\\varepsilon^2}|x - y|^2}f(y)\\,dy = (\\phi_{\\varepsilon} * f)(x),", "55971e43dec90ff265f83781c5ecbf3d": "p_\\text{amb}", "55977118165be5d765d9b40bcbd6880d": "\\hat{\\delta}", "55978067ea8325cbf176052c67cf290a": "\nF(x|d_1,d_2,\\lambda)=\\sum\\limits_{j=0}^\\infty\\left(\\frac{\\left(\\frac{1}{2}\\lambda\\right)^j}{j!}e^{-\\frac{\\lambda}{2}}\\right)I\\left(\\frac{d_1x}{d_2 + d_1x}\\bigg|\\frac{d_1}{2}+j,\\frac{d_2}{2}\\right)\n", "5597acb00e0db58bd6145285030b0def": "p^{(l,m)}=\\frac{c^{(l,m)}}{n}", "5597b4bea7d776cc3e159ce5029efba1": "\\mathfrak{g} = \\textstyle{\\sum \\mathfrak{g}_i}.", "55987a8af802ff72ec73da925ad3b380": "R_{(1)}^t=\\min{\\{\\frac{S_1^t}{S_1^0},\\frac{S_2^t}{S_2^0},...,\\frac{S_n^t}{S_1^n}\\}}, ", "559923a460677823abddafa1eb2bb8c2": "\\lambda_n(T; x) = \\sum_{j = 1}^{n + 1} \\left| l_j(x) \\right|, \\quad l_j(x) = \\prod_{\\stackrel{i = 1}{i \\ne j}}^{n + 1} \\frac{(x - t_i)}{(t_j - t_i)}.", "559932a477b3d31eec340be4c09745eb": "d_0,", "559957b794dcabc25b332dddda851acf": "\\frac{v}{c} \\cdot \\frac{180^\\circ}{\\pi} = 20,5^{\\prime\\prime}", "559997226c87fdb8d6b0c1bc601ee32e": "C^\\infty_c(U\\backslash G /U)", "5599aee3df9394ed48a4b632245b0393": " \\exp \\begin{bmatrix} 0 & \\mathbf a & c \\\\ 0 & 0_n & \\mathbf b \\\\ 0 & 0 & 0 \\end{bmatrix} = \\sum_{k=0}^\\infty \\frac{1}{k!}\\begin{bmatrix} 0 & \\mathbf a & c \\\\ 0 & 0_n & \\mathbf b \\\\ 0 & 0 & 0 \\end{bmatrix}^k = \\begin{bmatrix} 1 & \\mathbf a & c + {1\\over 2}\\mathbf a \\cdot \\mathbf b\\\\ 0 & I_n & \\mathbf b \\\\ 0 & 0 & 1 \\end{bmatrix}. ", "5599c78818bb435b2cf76214a76dfdc9": "\\int_a^b h (x) \\;\\mathrm d x \\approx \\sum_{k=1}^n w_k h (x_k)", "5599e56e92aa5383ad46f3b65160ace8": "D_h = \\frac{4A_{ch}}{P_{ch}}", "5599f54259fd79d95c4d0eb936742515": "\\mathrm{E_0}", "559aa74144173171c8992c0ee9250d3e": "\\int_{R_1<|x| 0", "559b2e2b1d57bd7270fbf0ceb46998f4": "\n\\sum_{a E_0", "55bac1963da62f36f5fac41650b9ffa2": " \\langle\\Psi| ", "55bacb52bb733dc52aa2b292eba79c96": "\\ln{R} = 0.5\\ln{L} - 2\\ln{T_{eff}} + const\n", "55bae2beffcae6605fa3d6b5848f46a3": "5^{k+1} > n \\ge 5^k,\\,", "55bb27809f8afb37241743e9644f9273": "y(x) = x^n \\, ", "55bb6c52165bdc8a9a4632cc22709268": " \\boldsymbol{U}_e=\\int_{z_b}^{z_t} \\boldsymbol{u}_e\\; dz=-\\hat{\\boldsymbol{z}}\\times\\Big(\\frac{\\boldsymbol{\\tau}_t-\\boldsymbol{\\tau}_b}{f}\\Big).", "55bba95718baf9f7e845a7c54f1dbacf": "E_p + E_k = E_{total}", "55bbac50fe5d7808401d99e1f07b46da": "\\mathfrak{L}_{NT} = \\{0, S\\}\\,", "55bbb9aec2e51c8d5e8a005af7f64845": "E[X Y] = E[X] E[Y],", "55bbcd9f2ddef6b7c769dba19a76017f": "a^ib^ic^i", "55bbeb5fefa388fc850000a090efc0eb": "[d]", "55bca1e32f48e6cea36906b70995d26d": "D_V \\left ( t \\right )", "55bcc0d306a9a4acc98e6da285bc9a86": " H(x,u,Du,D^2 u) = 0 ", "55bcd588384330a1a88de79853544592": " (\\Delta \\otimes id) \\Delta = (id \\otimes \\Delta) \\Delta", "55bce2c5f91b423b8aa77d1fe97499e0": " 2^{O(t)}\\cdot |G|^{O(1)}", "55bd2092a67b56dd16e91d8ef20cd16b": "\\tfrac{765433}{25920}", "55bd34758e434258cb53036f126489f2": "(2+i)\\cdot(3+i) = 5+5i = 5\\ \\mathrm{cis}\\ 45^\\circ.", "55bd437e8dd877cb764870701e2d41ec": "\\psi_1 = \\psi_2 = 0", "55bd51572ef0da7163452a9ddd07d773": "\\mathbf F = \\mathbf e_r F_r + \\mathbf e_z F_z + \\mathbf e_\\theta F_{\\theta},", "55bdc8984e50eb42365b0d181049ad33": "\\mathbf{k} \\cdot \\mathbf{F}_t\\left(\\mathbf{k}\\right) = 0.", "55bdd30b5b79187f7bb37e951603125c": "\\zeta = \\frac{x + i y}{1 - z},", "55bde4c2f8b1728ff70888b6bdbeba2d": " \\frac{n_{e,A}\\pi^2 u_{ph}^2 \\dot{\\gamma}_{ph,e,sp}}{\\omega_{e,g}^2\\int_\\omega\\mathrm{d}\\omega} = \\frac{n_{e,A}\\pi\\omega_{e,g} |\\boldsymbol{\\mu}_e|^2}{3\\epsilon_\\mathrm{o}\\hbar u_{ph}\\int_\\omega\\mathrm{d}\\omega}", "55bdf75c8c6ffb5bd17610cfa46ac4c8": "\\frac {\\partial \\pi (p)}{\\partial p} - y = 0", "55be05af41ba64cac7e1f9febb4b6113": " {Q}_{i}^e ", "55be1966d55383ec5061d94260982028": "b = 2mn, \\, ", "55be1e3c32c4e8e691e6241780b1fd94": "\n\\mathbf{p} \\times \\mathbf{\\epsilon^2}(\\mathbf{p})= i p_0\n\\mathbf{\\epsilon^2}(\\mathbf{p}). \\quad\\quad\\quad\\quad (4)\n", "55be71ce471e559ff28b150e8d7cc5aa": "\n \\nabla^2 w \\equiv \\frac{1}{r}\\frac{\\partial }{\\partial r}\\left(r \\frac{\\partial w}{\\partial r}\\right) +\n \\frac{1}{r^2}\\frac{\\partial^2 w}{\\partial \\theta^2} + \\frac{\\partial^2 w}{\\partial z^2} \\,.\n", "55be851fc1c2cbb03d9d308f5d4e757c": "\\sqrt{8\\over 35}", "55be87eab8cdf5c87896949e6d235ea1": "\\mathbf{b}(\\mathbf{x},t)", "55bed75161e23d1d25e1bd704c33970e": "3 + 3 = 6 ", "55beecda0c640f63c387bf5aed7c4856": "Q_1 = (\\sigma \\, z_1) + X", "55bf41606c57a4f0c633d68d7ae18d2f": "(Cv_g)_i=\\frac{k_B^4 T^3}{2\\hbar^3\\pi^2}\\int \\frac{1}{v_{p,i}^2}\\left[\\frac{x^4 exp(x)}{(exp(x)-1)^2}\\right]\\,dx ", "55bf482e70880fec03ebeb5d7ed36167": "\\gamma_y\\in\\Gamma", "55bf8e80b0532cd9ee4acb0460209296": "f'(t)=\\frac{1}{n+1}-\\frac{1}{n+1}({x_1 \\cdots x_n})^{\\frac{1}{n+1}}t^{-\\frac{n}{n+1}},\\qquad t>0.", "55bfa42c9146e4f55df771016b6e27e8": " \\cos^{-1} \\left( \\frac {-1} {\\varphi} \\right) \\approx 128.173 ^ \\circ \\ .", "55c01821827035191b2a3bd582440196": "\\hat{\\alpha}({r_{\\rm c}}),", "55c02eb4c1cdf048995dc9246cccef9e": "I_{i_1},...,I_{i_k}", "55c0e28c78ea4ad48a326b7560f3717b": "\\epsilon_{i} \\sim N(0, \\sigma^2).", "55c171036c658ad7b7f119a20617f872": "\\lim_{k \\to \\infty} f(k T + m) = \\lim_{z \\to 1} (1-z^{-1})F(z, m).", "55c1868a1ac261b7869ec8097d880d89": "\\mathrm{ind}\\,\\mathfrak{g}:=\\min\\limits_{\\xi\\in\\mathfrak{g}^*} \\mathrm{dim}\\,\\mathfrak{g}_\\xi.", "55c1a321eed16c859ef92445f896e38c": "\\! q \\ge -1", "55c1b74eb4733f67cddd33ba09eaaa26": "\\overrightarrow{dS}", "55c23c54b6b059266919d0f25fa08e35": "\\Box = \\frac{1}{c^2} \\frac{\\partial^2} {\\partial t^2}-\\nabla^2", "55c289b52e16df7f937ff2dd1540c377": "\\,\\!\\gamma(a) = \\gamma(b)", "55c28cd825e8fed20d897742f67a3f87": "q(k) = c_d(k) k^d + c_{d-1}(k) k^{d-1} + \\cdots + c_0(k)", "55c3021e2174c2ff4daf587cc10a41d0": " v_d", "55c3786d1f26530b741048e52781b3f3": "1 \\equiv ((x^l)^m)^{l^{-l}} \\equiv x^m \\pmod{n}", "55c3b3dcc2019087cced9324cb5a02af": "{\\hat O}", "55c3e78a4bd253d2e9b8e80420a3cf98": " A(c) = -\\frac{u''(c)}{u'(c)}=\\frac{1}{ac+b}", "55c43132aa29e01ad02f57c62b43813d": "\\frac{d\\mathbf{r}}{dt} = \\mathbf{v}", "55c4561a23b62e5c9b667a4633b0608b": "\n\\begin{align}\n & \\text{minimize}_{y \\in C} & f(y) + \\frac{1}{2} \\left\\| x-y \\right\\|_2^2 &\n\\end{align}\n", "55c4c24f09a6b5d1556f2c6c120cb2ed": "n_2 = 30\\ ", "55c4cdd56178ecb1b6ce2926bd1d7fde": "\\,\\eta_1", "55c57447550fdd0187b4aff3e56f4d83": "\\tan(\\psi) = \\tan(\\pi - \\theta - \\phi) = - \\tan(\\theta + \\phi) = \\frac{- \\tan\\theta - \\tan\\phi}{1 - \\tan\\theta \\tan\\phi}", "55c59b1dddcdf1819518688aaf2e85e7": "x\\times y \\in H_{p+q}(M\\times M)", "55c5f270895601ebb29ac15f4f00b829": "K_f = RMT_f^2/\\Delta H_{\\mathrm{fus}}", "55c6132903adb1ee55629014963de756": "2(a^4+b^4+c^4+d^4) = (a^2+b^2+c^2+d^2)^2", "55c61e9eaf74b0d5e8711aff5a8b8bbe": "\\nu \\in [0, \\pi]", "55c680ab30e00b6e436e8ca1d4cf15f0": "\\left(X\\left(B_1\\right),\\dots,X\\left(B_n\\right)\\right) \\sim \\mathrm{Dirichlet}\\left(\\alpha H\\left(B_1\\right),\\dots, \\alpha H\\left(B_n\\right)\\right).", "55c692109b7f81214cf9da4c070e5d8c": "x_0 = \\frac{z+1}{\\kappa(x)\\sqrt{2}}, \\, x_1=x_1,\\, \\ldots,\\, x_n=x_n,\\, x_{n+1}=\\frac{(z-1)\\kappa(x)}{\\sqrt{2}}", "55c6c8a8522bc9f723ffdd80f77c3105": " \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}", "55c6f3564b69b4d85712b780490e924d": "u < X(\\omega) \\leq v", "55c714fbe1664ddb2f5f7f161fd9ef5a": "10 \\cdot a + b,", "55c7227306a5f8dbdd9e40dea7e27d14": "\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y} \\qquad \\mbox{and} \\qquad \\frac{\\partial u}{\\partial y} = -\\frac{\\partial v}{\\partial x}\\,", "55c810259fd86d5085b836d7614e7c7c": "K_{i2} ", "55c845e6a5a00349db250537363a0cf8": " \\mathcal{Z}(z, V, T) = \\sum_{N_i} z^{N_i} Z(N_i, V, T), ", "55c8517304d894fe1dfd2a9d64a96612": "\\frac{j}{i}=k", "55c8a58aeaa3f192445968b1e35df7b1": "\\mathrm{d^3}\\mathbf{r'}", "55c8acca865ad37708143ef6bed7b630": "\n\\begin{align}\n&\\sum_{i=1}^N \\Sigma_{ij} e_j=\\lambda e_i\\\\\n\\Leftrightarrow \\quad& \\Sigma e=\\lambda e\n\\end{align}\n", "55c8add7404b4df96506116715bb6799": "k^a \\nabla_a k^b = \\kappa k^b", "55c8c14a088aa1a8437dddeec75792fc": "P_{\\pi}P_{\\pi}^{T} = I", "55c8d8ffb8427e147ce7f96b351bd98b": "-\\nabla^2\\mathbf{H}=\\sigma\\nabla\\times\\boldsymbol{E}.", "55c8f8b5e9763fcda8d5d14afcf95db0": "\ni{d \\over dt} \\psi_n = \\sum H_{nm} \\psi_m\n\\,", "55c92f12dd24601958ca0a7474a0c3d5": "-l\\pi", "55c9690dcd4a8b17a639d10860f817de": " {}_{,\\alpha}", "55c9cc1bd42a38c9fe0fbe57b10e3f29": "|\\beta\\rangle ", "55c9ef293d650727f48f56ba1e79d1db": "a,b \\in U", "55c9f2542aa258a6a2f74138b5afabf5": "Z_L^\\prime", "55ca17560cd79c582cdb22b6284dff4b": " C_{\\min} ", "55ca1a4cfb031e11a3aea45a79d5f4e2": "\nx = b!\\,\\biggl(\\frac{a}{b} - \\sum_{n = 0}^{b} \\frac{1}{n!}\\biggr)\n= a(b - 1)! - \\sum_{n = 0}^{b} \\frac{b!}{n!}\\,.\n", "55ca1c7c6321c946943fc5c33f04cdf3": "\\displaystyle{f_t(z)|_{t=0} =f_0(z).}", "55ca36f841a0edea45d82a81265807af": "\\mbox{Th}(\\mathcal{N}) = \\bigcup_{n \\in \\mathbb{N}} \\mbox{Th}_n(\\mathcal{N})", "55caacd3cc0271708f180ce716dc3fcb": "{\\bold \\mathrm V} \\rho", "55caae8cbaf95018d1893f032d3b4d34": " c \\,", "55cace600b0280707cd67607f0805ae7": "(L^*_1,a^*_1,b^*_1)", "55cb6ffaf43ef59af655997d54fa3468": "\\text{Area}=mn(m^2-n^2) \\, ", "55cba0b587b83c7af944d6a51d4bda1a": "A\\subseteq \\Omega", "55cba3c6cd4cff1e383016b1af70d1e4": "\\forall{|j\\rangle}", "55cbac537cf9b373c7d7cb4f1e40d1db": "\\epsilon_1,\\dots,\\epsilon_n", "55cbac995693e3c3c9e3c4fc593453e8": " \\delta F = \\int \\frac {\\delta F} {\\delta \\rho(x)} \\ \\delta \\rho(x) \\ dx \\ , ", "55cbd32fef48b286ca71abacc513dfc9": "\\frac{\\partial M_r}{\\partial c} = \\frac{(c + \\gamma - 1) (c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} \\left(\\frac{1}{c + \\gamma - 1} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta + k}- \\frac{1}{c + 1 + k} - \\frac{1}{c + \\gamma + k} \\right) \\right)", "55cc0ccf246e4da0a312a739ffa95b24": "\\mathcal{H}=H^* \\times F,", "55cc2455a0e0ee508a179983ca81575e": "d\\bold{u}", "55cc5b2147dcada2e5ece47a26429a5a": "\\tbinom mr_q", "55cc93bd26dd4a2bddc7bfd99ea2f17b": " A_v | \\phi \\rangle = - | \\phi \\rangle ", "55ccb7a10dbf8064dc0b32cc4d0b6acb": "{\\mathbf{}}\\hat{P}_i\\hat{S}_i", "55ccdf8965dae3c640a3d17b816e1901": " g^{\\mu \\nu} \\approx \\eta^{\\mu \\nu} - \\eta^{\\mu \\alpha} h_{\\alpha \\beta} \\eta^{\\beta \\nu} + \\eta^{\\mu \\alpha} h_{\\alpha \\beta} \\eta^{\\beta \\gamma} h_{\\gamma \\delta} \\eta^{\\delta \\nu} \\,.", "55cce11d40b636d6f585f9f6f3535074": "\\mathbf{n}(t) = \\frac{\\mathbf{r} - \\mathbf{r}_s(t)}{|\\mathbf{r} - \\mathbf{r}_s(t)|}", "55ccf93b95a02b1140b526f5dce4739c": "G = \\langle g_1, \\ldots, g_d\\rangle", "55cd34190584edb164a7bb95d3145c87": " \\mathbf{A}\\colon\\mathbf{B}=\\sum_j\\sum_i = \\left(\\mathbf{a}_i\\cdot\\mathbf{c}_j\\right)\\left(\\mathbf{b}_i\\cdot\\mathbf{d}_j\\right) ", "55cd67ba46f44d376cf16afea0c36565": "t(n)0,a_{n-2}>0, \\ldots,\\, \\Delta_{3}>0,\\Delta_1>0", "55da8ddef771284238708de95fe39dda": "u(\\cdot)", "55da9364769f2083e02d904fcd1822dc": " R(D) \\ge h(X) - h(D) \\, ", "55daa1c1fc1b401169e8e7386200bfe3": "\\exists_x \\forall_y (x \\neq y \\Rightarrow G(x, y)).", "55dadb392648c5c66379a97e4c887084": "I=\\{i_1,i_2,\\dots,i_k\\}", "55db022f61645d8b1578f94f32d93fa9": "10x=2^x", "55db112a80bc9fbf902c444c93b32cf1": "d\\vec{\\ell}_2=(dx_2,0,0)", "55db37a8d2ad543829d37f18da25c86c": "\\sum_i \\int_{S_i} (\\phi\\epsilon \\mathbf{\\nabla}\\phi) \\cdot \\mathbf{dS}= \\int_V \\epsilon (\\mathbf{\\nabla}\\phi)^2 \\, d^3 \\mathbf{r}", "55db8a11284c9da76260187aed1a8afb": "ve=mu", "55db972f914bfbd212787407e11ad2e7": "O(log n)", "55db98fc051b69c96a8f8c009aea1676": "df = \\left.\\frac{\\partial f(\\boldsymbol{x})}{\\partial x_1}\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}dx_1 + \\left.\\frac{\\partial f(\\boldsymbol{x})}{\\partial x_2}\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}dx_2 + \\cdots \\left.\\frac{\\partial f(\\boldsymbol{x})}{\\partial x_n}\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}dx_n = \\nabla f(\\boldsymbol{a}) \\cdot d\\boldsymbol{x}", "55dbbaedb858a6762913f55400bbe193": " p_{i,j} ", "55dc657d5e8c784bdfe8971c06e1f991": "(P-|P|)= 0", "55dc875df80d01f8d46b6df7d79ba600": "\n\\begin{align}\nA^3 e^{A t} =& (3/4)^3 B_{1_1} e^{3/4 t} + \\left( (3/4)^3 t + ( (3/4)^2 + (3/2) \\cdot 3/4) ) \\right) B_{1_2} e^{3/4 t}\\\\ +& B_{2_1} e^{1 t} + \\left(1^3 t + (1 + 2) \\cdot 1 \\right) B_{2_2} e^{1 t} \\\\ =& (3/4)^3 B_{1_1} e^{3/4 t}\\! + \\left( (3/4)^3 t\\! + 27/16 ) \\right) B_{1_2} e^{3/4 t}\\! + B_{2_1} e^{t}\\! + \\left(t + 3\\cdot 1\\right) B_{2_2} e^{t}\n\\end{align} \n", "55dcb3057efb19e663ae80e77d21c033": "{\\rm as}_5(1,2,3,4,5) = 1 ", "55dd2f6650c6ed21997782066d9182d4": "C(R)", "55dd3229ea2d5594d4881acf5dc20cab": " E[A_n] = \\frac{2 n}{3} + \\frac{1}{6} \\qquad \\text{and} \\qquad \\operatorname{Var}[A_n] = \\frac{8 n}{45} - \\frac{13}{180}. ", "55dd40c6420b868e2b2f7cb0720efbeb": "X=\\omega_1", "55dd4695fe331866290ac0de664a8a9a": " M^{(n)}(B_1\\times,\\dots,\\times B_n)=\\lambda^n \\prod_{i=1}^n |B_i|, ", "55dd4bcd1140b270243e02b1994cab79": "W_l^{(t+1)}", "55dd8be655369e30d132cff1fe1dba67": "| \\psi \\rangle = \\alpha |0 \\rangle + \\beta |1 \\rangle,\\,", "55dd95c8ee60d3b4ef35860cbac8785a": "G_a(z) = \\sum_{k=0}^{\\infty} a_k z^k\\!", "55ddce638b152142cfd5f5e756313a66": "a_n=2^n-1\\, .", "55dde409693a08963c694cfac7af7d05": "P(t) = -\\frac{\\partial U}{\\partial \\mathbf{x}} \\cdot \\mathbf{v} = \\mathbf{F}\\cdot\\mathbf{v}.", "55de0604e52ae26cb7d80b230c1a35e1": " \\frac{dM}{dt} = 4 \\pi r_{p} D_{v} (\\rho_{0} - \\rho_{w} ) \\,", "55de3254a4a6bf56b20f3b60c7d003ae": "c=\\alpha\\,e^{-2^k\\,r},", "55de41afa217ef70d1a6024b60aca693": "\\gamma_2=-2", "55de8ea84ce4f1cf7b3789201fa82631": "S_{l}(u) \\sim e^{-\\frac{\\sqrt{k}}{4}u^{2}}\\,", "55de946fc316c2255cfc7cd096acebc0": "EI", "55dea2519c7261c643c04e7dd10a78bd": "\\sigma_P", "55dede7b1752176e0f0b08ce77559391": " \\Im z > 0", "55df2f4a8d590b220b1754ca9e46fb41": "{^N\\mathbf{\\bar{v}}}", "55df518f81e6244d21060def2209cb49": "F^\\times\\cong\\mathbf{Z}\\oplus\\mathbf{Z}/{(q-1)}\\oplus\\mathbf{Z}_p^\\mathbf{N}", "55dfb705cf9207012ef40aecfde44060": "\\rho \\propto T^4", "55e02338729b3bbd12c60b0cfa8ca913": "\\mathcal{L}(\\overline{x},\\Sigma) \\propto \\det(\\Sigma)^{-n/2} \\exp\\left(-{1 \\over 2} \\sum_{i=1}^n (x_i-\\overline{x})^\\mathrm{T} \\Sigma^{-1} (x_i-\\overline{x})\\right),", "55e06275e7ec9a153cdf906f561c128a": "E_{k-q} - E_k = \\frac{\\hbar^2}{2m}(k^2-2\\vec{k}\\cdot\\vec{q}+q^2) - \\frac{\\hbar^2 k^2}{2m} \\simeq -\\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m}", "55e117cfe04f89eb94f8be03ded3fc59": "\\mathbf{e}_1 = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix}; \\mathbf{e}_2 = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix}; \\mathbf{e}_3 = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{pmatrix}; \\mathbf{e}_4 = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix}, ", "55e14a356e024471bd1b31704bf8b470": "\\Phi_D=", "55e1518263a89c9c2bd79cdb99ba753b": "\\langle U^* x,y\\rangle=\\overline{\\langle x,Uy\\rangle}", "55e19aefd149131dbd9017831698e956": " 0 \\le \\tau_{P} \\le t_0 ", "55e1d4ae256d2fc6afca16d14385a8ea": "\\Delta W = p \\Delta V, \\quad \\Delta Q = \\Delta U + p \\delta V\\,\\!", "55e212c6253fdad0d51ea497a94c0075": "Q_{acc}", "55e22a0eae140acafccb8e9eae7d1a0f": "\\scriptstyle n \\in \\mathbb{N}_1", "55e23baf691d95300e751be835166386": "\\cos a = (\\cos a \\,\\cos c + \\sin a \\, \\sin c \\,\\cos B) \\cos c + \\sin b \\, \\sin c \\,\\cos A", "55e24178cbbdbd5ab2c7434e67a6bd67": "S^1 \\to S^1 : s \\mapsto s^n", "55e25c69901694f5c87c3f3d122ed721": "\\tan\\frac{7\\pi}{30}=\\tan 42^\\circ=\\frac{\\sqrt3(\\sqrt5+1)-\\sqrt2\\sqrt{5+\\sqrt5}}{2}\\,", "55e25f85f6e6228b02aacbd0ead727ca": "\\Psi_m^{(0)}", "55e2bb7f1e28496ed43b1e649860107f": " {x^2 \\over a^2} + {y^2 \\over b^2} - {z^2 \\over c^2}= 1", "55e2c6b434d2e7cad8e95a688af589b6": "w_n = \\lfloor n\\theta + x\\rfloor - \\lfloor (n-1)\\theta + x \\rfloor ", "55e2ef84ff65f07fa8fff0bd243d9a27": "\\alpha_i^2=\\beta^2=I_4", "55e3168a6deed918c7e5850f33dd89c7": "\nI_{fl} = -I_{e} e^{-e V_{fl}/(k T_{e} )} + I_{ion}^{sat}\n", "55e36d065eaa8e35da0094d10d04a9af": " x^5+x^4-12x^3-21x^2+x+5", "55e3dd2342bc0bcb022ba44a1f3717d3": " u_i ", "55e47259bcdb42c0c101754d76aaec30": "\nV_s = e_1\n", "55e489e527aa56fc525fd85aff34ce30": " \\mathrm{perm}(X^tD -(n-i)\\delta_{ij}) = \\mathrm{perm}^\\text{Calculate as if all commute}_{\\text{Put all }x\\text{ on the left, with all derivations on the right}}\n( X^t D). \n ", "55e4939bc6ba47dbe5e2df81a1a19fba": "b \\ge 0, n \\ge 1", "55e4a243c1a2c55a7a5c58a38d9751ac": " \\pi(x;q,a) \\approx \\frac{\\pi(x)}{\\varphi(q)}", "55e4c2bdf3e943a813233b67bbe94c90": "\nM = \\frac{1}{10} \\begin{bmatrix}1 & 1 & 1 \\\\\n1 & 2 & 1 \\\\\n1 & 1 & 1 \\\\\n\\end{bmatrix}\n", "55e4c643e91a0ea9293b6aa0395f7636": "E(\\mathbf{r},t) = A_m a(t,z) f(x,y) e^{i(\\beta_0 z - \\omega_0 t)}", "55e4c749a7d1ec8f2f3dfff612882c90": "K(a)=f^{-1}(-\\infty, a]", "55e4fa837453f85737a42b45c32cbad8": "\n\\sum_{i=1}^n U_i^2=\\sum_{i=1}^n\\left(\\frac{X_i-\\overline{X}}{\\sigma}\\right)^2\n+ n\\left(\\frac{\\overline{X}-\\mu}{\\sigma}\\right)^2\n", "55e55c4e7d477186723f02224278f346": "D^n\\times{\\mathbb R}", "55e566660fd71364b42a370a6c25b093": " \\sum_{n=1}^\\infty \\frac{N_1(n) + N_0(n)}{2n(2n+1)} = \\gamma ", "55e59924996f27eecd1a1ca88b54b7eb": " \\Phi \\rightarrow f'(R)~~~~~\\textrm{and}~~~~ \\frac{dV}{d\\Phi}\\rightarrow\\frac{2f(R)-R f'(R)}{3},", "55e59958459ffa0fffbc8886002c1955": "\\scriptstyle \\sqrt{5+\\sqrt{5+\\sqrt{5-\\sqrt{5+\n \\sqrt{5+\\sqrt{5+\\sqrt{5-\\cdots}}}}}}}\\;=\n \\textstyle\\frac{2+\\sqrt{5}+\\sqrt{15-6\\sqrt{5}}}{2}", "55e5f59ca56c0b2e2cd782d0971dbe4b": "(T x,\\mu_x)", "55e64496a9f0ce58df3f831fb5c1ace9": " \\rho(u)=\\frac{\\sin (\\lambda-u) \\sin (\\mu+u)}{\\sin \\lambda \\sin \\mu} ", "55e64dc9c15892b6d0b04460e32226ee": "h_{wall} = {2k \\over {d_i\\ln(d_o/d_i)}}", "55e66050863b6cf1e1a07f99d6f08984": "\\sum_{i=1}^{n} \\frac{x_i}{x_{i+1}+x_{i+2}} \\geq \\frac{n}{2}", "55e687b98afb0041729dd7e960493d74": " E_{\\pm}=E_{\\pm}(\\alpha_{1},\\alpha_{2},\\alpha_{3}.....\\alpha_{k}) ", "55e69bacf6513a6aefddefa8618b24ef": " S^*: (a_1, a_2, a_3, \\ldots) \\mapsto (a_2, a_3, a_4, \\ldots)", "55e721a8bf3102162833804225c8591d": " \\sigma(\\mathfrak{G}^2 \\oplus \\mathfrak{G}^2) = 0", "55e774fe7f8507c735dc2c90bf6e6424": "\\sum_{d\\,\\mid\\,12} \\Lambda(d) = \\Lambda(1) + \\Lambda(2) + \\Lambda(3) + \\Lambda(4) + \\Lambda(6) + \\Lambda(12) ", "55e7a0db5e76a0e51696da485fba8737": "\\displaystyle a=\\frac{2r(b-r)}{b-2r}.", "55e7ac78c6f106c1eb557c82605dc52e": "a_i,", "55e8cc2aab7d0a14ee2cc308b45cc313": "(6k + 1)(12k + 1)(18k + 1)", "55e9c1f1e46b84564ec386be9ebe99f8": "F_W", "55ea03ea3728a7e8645fa52fab83a514": "T_{k,n}(z)= g_n\\left(T_{k-1,n}(z) \\right)", "55ea0b0ebcf6b80f8b020cbec32b7189": "s'>s", "55ea2510409409217659f537548998cd": "\\mathcal{I} = \\mathcal{Z} \\times \\mathfrak{G}\\{\\mathcal{Z}\\}", "55ea3e9f41465c31ee5cb511a101a1d9": "\\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_m \\end{bmatrix}", "55ea77c2998cbfdb72a8dafb87f63ed5": "X_1, X_2, Y_1,", "55eac1e9c8cf10ecc1e26203f11e63b8": "3^5", "55eae0c7b40cab9feb62296c36e7ae2b": "x^2 \\frac{d^2 y}{dx^2} + 2x \\frac{dy}{dx} + [x^2 - n(n+1)]y = 0.", "55eae8f834bfc0b457fc773e2a2f9623": "~|p\\rangle~", "55eaef9c6656645d8dcf2ed2c308a6e0": "y=2\\pi+1", "55eaffc7f85be8413b5e42d6deb4585d": "(s,t) \\in R_i", "55eb2498330f2dddf0afe99893c5cf1f": "\\rho(x,y) = \\tfrac{m}{2\\pi ab}\\, e^{-((x/a)^2+(y/b)^2)/2},", "55eb4107714fdcc2a0da5b9ba1e912b0": "\\bar{\\delta} l^a=(\\alpha+\\bar{\\beta})l^a-\\bar{\\sigma}m^a-\\rho\\bar{m}^a\\,;", "55eb5572536115adfd9c6f9932d7e1ca": "\nF_{ax} \\ \\stackrel{\\mathrm{def}}{=}\\ \\left( \\frac{4}{3} \\right) \\frac{\\xi^{2}}{2 - (S/p^{2})}\n", "55eb6c6998bf2ad53be4639392e96b18": "\\{Y_1, Y_2\\}", "55ebeb18ff9324f5fc057d961e408d23": "C:\\Sigma^k \\to \\Sigma^n", "55ebfe537499513b46c34d5496568520": "z = y^m u^r \\mod n", "55ec12d9b507eeecf6625d355b8fb6c2": "\nG \\equiv G_{1}(\\mathbf{q}, \\mathbf{Q}, t)\n", "55ec1b9d5be104fb28772568b4f2c614": "\n\\left( \\underbrace{\\int dx_? ... \\int dx_?}_{n_A\\;terms} \\underbrace{\\int dx_?...\\int dx_N}_{n_B\\;terms}\\right)\\left(\\underbrace{\\int dv_1 \\int dv_2 ...\\int dv_N}_{\\Sigma v^2 = 2E_A\\;or\\;\\Sigma v^2 = 2E_B}\\right)", "55ec564201df400aeca4243a157a1f88": "J^{0} ", "55ecd4bc49ad29fb071d76381389ac35": "10,000\\mbox{ pixels} \\times 10,000\\mbox{ pixels} \\times 48\\mbox{ bpp} \\times \\frac{1\\mbox{ byte}}{8\\mbox{ bits}} = 600\\mbox{ megabytes}", "55ed4488c41591cd6cd71def83189683": " l_\\theta = l_0 (1 + \\alpha \\theta + \\beta \\theta^2) \\,", "55ed55e12cc71c3126a5d9bb0d897a9d": "|\\text{g}\\rangle", "55edc16bc0a68955309421ff3734590a": " N = \\frac{ \\alpha X }{ 1 - X } ", "55ee232c5a7f665fcca0ee8f960f42d4": "\\mathrm{rect}(t) = \\Pi(t) = \\begin{cases}\n0 & \\mbox{if } |t| > \\frac{1}{2} \\\\\n\\frac{1}{2} & \\mbox{if } |t| = \\frac{1}{2} \\\\\n1 & \\mbox{if } |t| < \\frac{1}{2}. \\\\\n\\end{cases}", "55ee38ec45bca2fbe5ec6ade4f499733": "\\lambda\\|\\cdot\\|.", "55ee60146a9ba34fd9c27e7ba0954c72": "\\scriptstyle\\uparrow\\uparrow", "55ee79553afccd0ba82503c7d33569ea": "f(x,y,z)=x^4+y^4+z^4-1=0", "55ee7f7b9fe09c4792fc8731b3a3a633": "\\scriptstyle \\vec{\\alpha}", "55ee8b8c8ce8ab50a2c501db6a1bebcb": "x_2 = 20", "55eeb7cfc1198898141bfdc03d5cd649": "\\mu\\!\\left(E\\right)=\\mu\\!\\left(E \\cup \\varnothing\\right) = \\mu\\!\\left(E\\right) + \\mu\\!\\left(\\varnothing\\right)", "55eef13fccde3afbb9edd3e1f2fd691b": "t+1.", "55ef1182d4df904bce3ec3f19e87e158": "\\mathbf{a_i}^k", "55ef3225289edce2640895b9b7279650": "x^2 - x - 2\\!", "55ef33117b433a5404aad21b122dfc06": " T^p_nI(x) = n-x ", "55ef849e0e0b6a98f603039bf498fc0a": "\\Delta\\,G", "55efa427ea2e28a9fbd669face2e7a54": "\\displaystyle{ \\pi((g^t)^{-1})f_z(x)= \\pm m(g,z)^{-1/2}f_{gz}(x).}", "55efa77cd14cd05131916615ffb6efb1": "\\phi_{rM, \\alpha}(z) =\n\\left\\{ \\begin{array}{ll}\n\\prod_{\\ell=0}^{r-1} \\phi_{M,(\\alpha+\\ell)/r} & \\mbox{if } 0 < \\alpha \\leq 0.5 \\\\ \\\\\n\\prod_{\\ell=0}^{r-1} \\phi_{M,(1-\\alpha+\\ell)/r} & \\mbox{if } 0.5 < \\alpha < 1 \\\\ \\\\\n\\prod_{\\ell=0}^{r-1} \\phi_{M,\\ell/(2r)} & \\mbox{if } \\alpha = 0\n\\end{array} \\right.\n", "55efb7daf2915f641e77d93751fea9ec": "\\gamma \\leq \\frac{1}{2}", "55f009abf4bd290a3af6eae77814a775": "\\frac{35}{243}", "55f0534441a920789843c79bd64b1b7d": "({u}_{1},{u}_{2})\\in\\mathbb{R}^{3} ", "55f0834c27598c02721a8067eebfbcad": "\n \\frac{\\partial }{\\partial \\boldsymbol{S}}[\\boldsymbol{F}_1(\\boldsymbol{S})\\cdot\\boldsymbol{F}_2(\\boldsymbol{S})]:\\boldsymbol{T} = \n \\left(\\frac{\\partial \\boldsymbol{F}_1}{\\partial \\boldsymbol{S}}:\\boldsymbol{T}\\right)\\cdot\\boldsymbol{F}_2 + \n \\boldsymbol{F}_1\\cdot\\left(\\frac{\\partial \\boldsymbol{F}_2}{\\partial \\boldsymbol{S}}:\\boldsymbol{T}\\right)\n", "55f08c53ddeee0b86afc55f0f78ef556": " \\gamma_1\\equiv \\gamma_2 \\iff \\left.\\frac{d}{dt}\\phi\\circ\\gamma_1(t)\\right|_{t=0} = \\left.\\frac{d}{dt}\\phi\\circ\\gamma_2(t)\\right|_{t=0}", "55f0be66de306d810f603bae092643b2": "f_{in}", "55f1c9a77367f29937c2888dc3c77384": "F(t)\\sim \\frac{1}{\\Gamma(\\rho+1)}t^\\rho L(t), \\quad\\rm{as\\ }t\\to\\infty.", "55f1e978b976b3d7b3bd93bf12e9c9b3": "\\tau =\\frac{1}{\\omega }\\frac{\\operatorname{Im}D(\\omega )}{\\operatorname{Re}D(\\omega )}", "55f25dfcb89b78dd775a9f1da5547b1f": "x^{-1} A[x^{-1}]", "55f28e1ce6af0f0e56229ba92f561ef5": "c_p = \\frac{\\sigma}{k} \\qquad \\text{with} \\qquad \\sigma=\\omega - \\boldsymbol{k}\\cdot\\overline{\\boldsymbol{v}},", "55f2a2e5e52d51b7d84ccb2503185678": "x'= 1,\\quad x''=0,\\quad y'= 2t,\\quad y''=2.", "55f3219e6a3114319e6426378e91428b": "m = k \\cdot q", "55f335eb74da5b18d2db2b050d6d4bda": "[f(x+h)-f(x)]\\over h", "55f365f23f112e537762f72a27b6ecb7": "V_{ij}=\\phi(b_i,b_j)", "55f38a7400f3585f98c8b9e2559657f4": "a_{ij}=-\\overline{a_{ji}}", "55f44fe35ab2d6d04b66b32c1d902979": "A \\circ (B + C) = A \\circ B + A \\circ C.", "55f45a0c66a340729370c5abc3c336ad": "f:G\\rightarrow K_k", "55f464330b5b99ecf34c66c4103aa5a2": "dim\\,G", "55f46baa43a0764afcf2412cfa72e56f": "X_1+X_2+\\cdots+X_n, \\, ", "55f4be7619cb43294f34153e1ea57c05": "f^N v=0", "55f4df177ac3e712302a1c09eccca924": "95\\frac{3}{5}", "55f524b7f15b596ad49adcb4a53d9399": " -[R]^2 = |\\mathbf{R}|^2[I] -[\\mathbf{R}\\mathbf{R}^T]= \n\\begin{bmatrix} x^2+y^2+z^2 & 0 & 0 \\\\ 0& x^2+y^2+z^2 & 0 \\\\0& 0& x^2+y^2+z^2 \\end{bmatrix}- \\begin{bmatrix}x^2 & xy & xz \\\\ yx & y^2 & yz \\\\ zx & zy & z^2\\end{bmatrix},", "55f52bff80ae0b926663caa6dd8d9901": "g(\\Gamma^*)", "55f5503a974090479a831035b0e0bbbb": "\\operatorname{E}[\\,\\nabla_{\\!\\beta}\\, g(x_t,\\beta)\\cdot(y_t - g(x_t,\\beta))\\,]=0", "55f5b105c29f0fd552e35076eb45060d": "\\,\\!d(x,z) \\leq d(x,y) + d(y,z)", "55f5dc5fabf316a706b6f7b7796b40fb": "2x \\equiv 4 \\pmod {8}", "55f5df0ac0c9d8051d15d29f089dfc74": "\\left(\\frac{1}{p},\\frac{1}{q}\\right)", "55f5f4a99950adef9e7078c2ba17c552": "Z_n(V) = \\sum_{s_0,\\ldots,s_n \\in Q} \\exp(-\\beta H_n(C_0[s_0,s_1,\\ldots,s_n]))", "55f5f7aebb87f9e3a26eacd0cb23bbc7": "R_n(\\xi,x)=r_0\\,x\\,\\frac{\\prod_{i=1}^{n-1} (x-x_i)}{\\prod_{i=1}^{n-1} (x-x_{pi})}", "55f6051438ae18448a7117f002f8b9d7": "\n\\begin{align}\np_i &\\propto \\Omega_B\\left(E_B\\right) \\\\\n&= \\Omega_B \\left(E - E_i\\right) \\\\\n\\Rightarrow k \\ln \\Omega_B \\left( E - E_i \\right) & \\approx k \\ln \\Omega_B \\left(E\\right) - \\frac{\\partial}{\\partial E} \\left(k \\ln \\Omega_B\\left(E\\right)\\right) E_i \\\\\n& \\approx k \\ln \\Omega_B \\left(E\\right) - \\frac{\\partial S_B}{\\partial E} E_i \\\\\n& \\approx k \\ln \\Omega_B \\left(E\\right) - \\frac{E_i}{T} \\\\\n\\Rightarrow k \\ln p_i & \\propto k \\ln \\Omega_B \\left(E\\right) - \\frac{E_i}{T} \\\\\n\\Rightarrow p_i & \\propto e^{\\ln \\Omega_B \\left(E\\right) - \\frac{E_i}{kT}} \\\\\n\\Rightarrow p_i & \\propto \\Omega_B \\left(E\\right) e^{ - \\frac{E_i}{kT}} \\\\\n\\Rightarrow p_i & \\propto e^{- \\frac{E_i}{kT}}.\n\\end{align}\n", "55f61667df83c8f980c24671333c5e6f": "\nds^2 = -\\left(1-{2M\\over r}\\right)dt^2 + {1\\over 1- 2M/r} dr^2 + r^2 d\\Omega^2.", "55f633523cd59766e6d2703011cf3740": "\\scriptstyle 2\\sqrt{3}-3", "55f673fbc6fa9e57fd15be3c430c607f": " \\partial_t I(u) + \\partial_x G(u) = 0 \\quad\\text{where}\\quad I(u) = \\tfrac12 \\omega(u_x,u) ,\\, G(u) = S(u) - \\tfrac12 \\omega(u_t,u). ", "55f697695c40c8afb28a42b3aee13c61": "\n\\text{INR}= \\left(\\frac{\\text{PT}_\\text{test}}{\\text{PT}_\\text{normal}}\\right)^\\text{ISI}\n", "55f6c15bf8794ea6a2b8c163a633d1fe": "M \\to N,", "55f6d25848cd2749493088d856a95e05": "\\tau_M\\colon M \\to BO(n).", "55f6d8d675da527d42583e465bdf2dd4": "\\partial_\\phi", "55f740c5e9fa187c99ade25199c5508d": "\\{\\vert e_k \\rangle\\}", "55f7557541330e8fd0bda335bebb4e81": "\\mathbf{\\lambda}(s) = |sI - A|. \\,", "55f7d03f3d6f978444dc43a0d4ae29cf": "\\phi \\left( r_i \\right)", "55f813f777c110fba5383c53936ac38b": "Y \\Rightarrow Z.", "55f84cd7fff1b8875740c8e597f12745": "a_{12}+b_{12}+c_{12}", "55f84f27f31d5c90b87ab08b945647ed": "\\sin^{-1} e", "55f885d5da77448e2e4db25128406299": "G(\\boldsymbol{\\eta}|\\boldsymbol{\\chi},\\nu)", "55f88feecd0d15f6cec0557cfc8ecbd6": "Z_i=X", "55f8af9fe0abac136658f826c67aa6ae": "\n \\delta U(\\phi_0) = U(\\phi)-U(\\phi_0) \n = E_J (\\cos(\\phi_0)-\\cos(\\phi_0+\\delta\\phi)\\,\n", "55f8bf86762a6bec553750d2dc5783c2": "\\langle \\cdot, \\cdot \\rangle _K ", "55f8fdfd8c2752439f73e1a510865ae6": " W_i ", "55f9fe8658dfb57e6c3d788120122762": "4^n + \\sum_{i = 0}^n 2^i.", "55fa4a145ee2bb00ce5228fca4d68f03": "R^h", "55fa7afe6d49b65b1c4e0754cd32acb9": " \\rho=\\limsup_{n\\to\\infty}\\frac{n\\ln n}{n\\ln n-\\ln|f^{(n)}(z_0)|}=\\left(1-\\limsup_{n\\to\\infty}\\frac{\\ln|f^{(n)}(z_0)|}{n\\ln n}\\right)^{-1},", "55fa9cc2afcfd6e4ff1855d8e2734541": "\\nabla = \\gamma^\\mu \\partial_\\mu.", "55face20d2bbe3153049d3d06231b8fe": "\\Rightarrow \\beta\\cos\\theta (1-\\beta\\cos\\theta) = (\\beta\\sin\\theta)^2 \\Rightarrow \\beta\\cos\\theta - \\beta^2\\cos^2\\theta = \\beta^2sin^2\\theta \\Rightarrow \\cos\\theta_{max} = \\beta", "55fae1a5d720ff025b7ac2312a873b3c": "\\,f \\colon S \\times S \\rightarrow S.", "55fafab82318e961d91abed42af0df86": "\\Delta X'\\Delta X", "55fafee6372cfce68753eb36b4ad03a1": "E_\\alpha", "55fb3c47dacf2b413e47a47fe44eff1d": "GSp(4)", "55fbc435a3ecd461328a9de75cb23eec": "L_1 = 53 \\ \\mathrm{nH}\\,", "55fbc906a14bb340db8a8462be5c4970": " V= \\pi/6 \\int_0^\\infty N(d_p)d_p^3 \\,\\mathrm{d}d_p", "55fc0204a0f1570faf08081c8039da9e": "v_\\lambda(p) \\,", "55fc12eaa9c07d702da98338d88dbc8c": "\\begin{align} \\mathcal{Z}\\{x(-n)\\} &= \\sum_{n=-\\infty}^{\\infty} x(-n)z^{-n} \\\\\n&= \\sum_{m=-\\infty}^{\\infty} x(m)z^{m}\\\\\n&= \\sum_{m=-\\infty}^{\\infty} x(m){(z^{-1})}^{-m}\\\\\n&= X(z^{-1}) \\\\\n\\end{align} ", "55fc857421c795a22167e1f21586f4bf": " a_{non-relativistic} = \\frac{v^2}{r} \\rightarrow a_{relativistic} = \\frac{1}{m} \\frac{dp}{d\\tau} = \\frac{1}{m} \\gamma \\frac{d(\\gamma m v)}{dt} = \\gamma^2 \\frac{dv}{dt} = \\gamma^2 \\frac{v^2}{r} ", "55fcc3c60a3871ca2d77351052b87823": "f''(x)\\,\\!", "55fcc72ba8e89cb57e310dd993bb3830": "\\ln\\, {(-\\ln {[1-Y(t)]})}=\\ln K+n\\ln t \\,\\!", "55fd019be3ec058e3690f15184e07107": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{T}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "55fe1bd1b3149d8fa5a9b39e94d1be0c": "G\\setminus B", "55feb874c35faf03466d8ae8b7dc13fa": "\nn^2(\\lambda) = 1\n+ \\frac{B_1 \\lambda^2 }{ \\lambda^2 - C_1}\n+ \\frac{B_2 \\lambda^2 }{ \\lambda^2 - C_2}\n+ \\frac{B_3 \\lambda^2 }{ \\lambda^2 - C_3},\n", "55fecf2b5971a75f9619f2edb04ec7ad": "A = \\oint_{C} (L\\, dx + M\\, dy).", "55feea56521c3a09905004e1f0209cb8": " T_{n-1} ", "55fefb12ffc4d0ec62bce6c8cb053970": " D_1 = \\partial_\\zeta + \\frac{2\\alpha_{,\\zeta}\\lambda}{\\lambda-\\alpha} \\partial_\\lambda", "55ff74f58a6a33b76136ca7b393a2ad6": "\n\\mathrm{d}\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0&\\kappa\\cos\\alpha\\, \\mathrm{d}s&-\\kappa\\sin\\alpha\\, \\mathrm{d}s\\\\\n-\\kappa\\cos\\alpha\\, \\mathrm{d}s&0&\\tau \\, \\mathrm{d}s + \\mathrm{d}\\alpha\\\\\n\\kappa\\sin\\alpha\\, \\mathrm{d}s&-\\tau \\, \\mathrm{d}s - \\mathrm{d}\\alpha&0\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}", "55ffc934d5bbab58562f5bc348be3da4": "\\varphi(p)(\\xi) = a_0 + a_1\\cdot \\xi + \\dots + a_n\\cdot \\xi^n", "55fff20f3f8ac7c92d0d0a5b9ad0ca34": "\\Delta = b^2 - 4ac.", "55fff29dee32fe3e8754937caa36b989": "{a+b\\varepsilon \\over c+d\\varepsilon}", "560002c3e1837aee925b7ccb431a0d7e": " R_{\\rm f} ", "560006076fbd2e3528dcfd62f8ea6fe5": "\n \\frac{\\partial A^{-1}_{ji}}{\\partial A_{kl}}~T_{kl} = - A^{-1}_{jk}~T_{lk}~A^{-1}_{li} \\implies \\frac{\\partial A^{-1}_{ji}}{\\partial A_{kl}} = - A^{-1}_{li}~A^{-1}_{jk}\n ", "5600851600bef73ff593cf4e8ed548a5": "\\operatorname{trace}\\left[ {(z - \\mu )'A(z - \\mu )} \\right] = \\operatorname{trace}\\left[ {A(z - \\mu )'(z - \\mu )} \\right],", "56013444af7037d1a5489df2724ff990": "\\alpha, \\beta \\in H.", "56013cc3d3cdb0425286725f3ef06c9d": " \\frac{1}{\\lambda} = R_\\text{H} \\left( 1 - \\frac{1}{n^2} \\right) ", "56014c7301ddc9a593b191ea429c582e": "-8\\pi", "56016fe5edac265bcd561eecc390d51e": "\\vec T_{bi}", "5601836658e5c66ff4aa313a7d50f009": "\\Box = \\frac{\\partial^2}{\\partial x_1^2} - \\cdots - \\frac{\\partial^2}{\\partial x_n^2}.", "56018d66abfc595f7088f5bd327a4ed9": "h(x)=b_{i+12}s_{i+8}+s_{i+13}s_{i+20}+b_{i+95}s_{i+42}+s_{i+60}s_{i+79}+b_{i+12}b_{i+95}s_{i+94}", "5601e25aef3d8f212f1f46bf7e9f3b1a": "\\scriptstyle{ Rt= a \\times Rc + b}", "56021a31a7b947af9ac8ce6c0ce3ceec": "R = \\alpha k^{\\alpha-1} \\,", "5602246114dde94ad68d084e54257a31": "\\mathbf{w}(t) \\sim N(0,\\mathbf Q)", "5602a2cf1d726651d5afa19434f10a04": "B \\leq_c A", "5602a7dc76d38ce6a6b888b93ae944ac": "K_b = RT_b^2/1000L_v", "56033f55dcc0c2803d32ca26fa5cbff8": "V(I) \\cup V(J)\\,=\\,V(IJ);", "56038c0b503233fabc7201c451d05b44": "E \\{ (x-\\hat{x}) y^T \\} = 0 ", "5603c9cf322be1848c3ac1cd59c1bfee": "F(t,x)=\\varphi(-t;t,x).", "5603ca16d978471d390a9608e0aaf5d0": "n_0 > 0", "56043cdfc622e4b27a401f3d03940450": "\n\\dot{x_1} = x_2,\n\\dot{x_2} = -g\n", "5604432919cb964915abbf19cc13054b": "P(e) = \\tfrac{1}{M} \\sum_{X \\in S} P(e|X) \\leq \\tfrac{1}{M} \\sum_{X \\in S}\\sum_{\\widehat{X}\\neq X} P(X \\to \\widehat{X})", "56046d8a5dbcbf0488cae8deeb8ca986": "P = \\sqrt{A^*A}", "560481fe21bc416b7cb7ed70d49993e2": "\\sigma^2 V", "56049af763f2b4dd153984f10531de70": "{\\tilde{D}}_n", "5604c341e03e891a99fd816af74f0a0b": " R^a_{\\ bcd}= e^a((\\nabla_c\\nabla_d - \\nabla_d\\nabla_c - f^e_{cd}\\nabla_e)e_b)", "5604eb71ac2b4f8372310e26c497e2ca": "v_A = B/(4\\pi n_im_i)^{1/2} = 2.18\\times10^{11}\\,\\mu^{-1/2}n_i^{-1/2}B\\,\\mbox{cm/s}", "5604ffcd0f2d3948fdaff85f075c4309": "o(f) \\subset O(f)", "56051c44e923ddf43d91a55bf75e5160": "N\\times D", "560550122d32651e84cc6c6bb16b0322": " {z g^\\prime(z)\\over g(z) -g(\\zeta)} - {z\\over z-\\zeta} = \\sum_{m,n>0} m c_{mn} z^{-m} \\zeta^{-n}.", "56055f90197e31f876e2daa204c434c5": "\\textstyle\\sum_{k=0}^x p_k(n)", "56056bc055060db07ed4fb205a0d9f01": "\\{E_n, l, m\\}", "5605d06c321a52b9f7b2af2e1b0fa5b6": "G(a_n;x)=\\sum_{n=0}^\\infty a_nx^n.", "560611c82c5641350511187faafc670a": "\\displaystyle v_e", "5606ab0bc810cdc79753ea42d4e2d675": "\\operatorname{Symmetric-Dirichlet}_V(\\alpha)", "5606dd87203bfce7726238ef0cf9c05f": "R C = \\frac{1}{2 K_p K_v}", "5606fbd449f214af4a9e7c2880dedb6c": "e^{ix} = \\cos x + i\\sin x\\,", "5607170e1369c99cfa33eb9216481126": "\\Delta\\;h =\\, ", "56076079f2c46353c4a9a45ac0a74207": " \\Sigma_b ", "560775887485101d093e4f17bd400222": "b\\ ", "56079d4e46e4d10d0b8e3661c635da59": "j\\neq m_i", "5607addcb67a22de46b5c7a4d1be435b": "l=\\frac{2}{\\frac{1}{r_{a}}+\\frac{1}{r_{p}}}=\\frac{2r_{a}r_{p}}{r_{a}+r_{p}}", "5607b1c5f663a0114d77054c7f301051": "\\begin{alignat}{7}\n2x &&\\; + && y &&\\; - &&\\; z &&\\; = \\;&& 8 & \\\\\n&& && \\frac{1}{2}y &&\\; + &&\\; \\frac{1}{2}z &&\\; = \\;&& 1 & \\\\\n&& && 2y &&\\; + &&\\; z &&\\; = \\;&& 5 & \n\\end{alignat}", "56087ddb40030a50c125a0b9955aae80": "P_{i} = \\frac{1}{n(n - 1)} \\sum_{j=1}^k n_{i j} (n_{i j} - 1)", "56088366ebfdcaf6b50c9759fd43c6c7": "1-k/|E|", "5608853b12be0c0211ef3249c1058ca5": "x_{k+1}=x_{k}+\\Delta x_k", "5609b4d5f0ba82be02772fbb0c85cbae": "\\| P_n(x) \\| = \\sqrt{\\int _{- 1}^{1}(P_n(x))^2 \\,dx} = \\sqrt{\\frac{2}{2 n + 1}}.", "5609c3cd315e3b4feab90f9f1b73816e": "\\tbinom {n+k-1}k", "560a10088b29fc1d32204fee968da29d": " g(E) = \\sum_{F \\subseteq E} M(F), \\forall E \\subseteq \\mathbf{X} .", "560a7da684b2e32e88ab05be439d2061": "\\tbinom{n-1}{n-x}", "560a7f3f72c68d465c18960cc8b307a5": " 1000 \\frac{\\text{rise}}{\\text{run}}", "560a986421e9d3b3cd2c2b2cda73c7ed": "\\rho(\\psi(P_B))^a) = \\rho(\\psi(P_A))^b) = \\rho(\\psi(g)^{ab}) ", "560ad1d036e37237fe1b507ac6cdf8e3": "Y\\times_X\\overline Y\\longrightarrow Y,\\qquad (y^i, \\overline y^i)\\longmapsto y^i +\\overline y^i,", "560aebd6fdd3b157f2574ee65ea37d8e": "I(X;Y)=H(X)-H(X|Y)\\,", "560b05ea42459273beb27cde4eb2ac2b": "\\forall A: A \\cap \\varnothing = \\varnothing\\, .", "560b0e42ca35b7713bee5405409c6cfc": "\n-\\nu \\int\\limits_L^{L+\\Delta L}\\frac{dx}{x}=\\int\\limits_L^{L-\\Delta L'}\\frac{dy}{y}=\\int\\limits_L^{L-\\Delta L'}\\frac{dz}{z}.\n", "560b27dc1bf1af0260059169de4bfb41": "\n\\hat{G}(\\boldsymbol{k}) = \\exp{ \\left( -\\frac{ \\boldsymbol{k}^2 \\Delta^2 }{ 24 } \\right) }.\n", "560b2ce4614da406094d54781e53c080": "R(d/dt) w=0", "560b3c4fa00d623d4f6aad999f4f9dbd": "f(x) := \\int_0^x \\omega.", "560b5f4f360c16495b0df9f3b27ca56f": "A\\hat{\\mathbf{x}} \\approx \\mathbf{b},", "560b6768a69feab20b95d897dc8630f6": "0 \\leq y \\leq x", "560b8fed200563bd97a4f3552067bbd9": "\\mu \\rightarrow 1", "560bb6b6808a15b3804f11fffc5b6dc6": " k_2 \\ ", "560bd2dbce57ca342a08f843db26f617": "\\;_6\\psi_6", "560c1ffb93178ecfbba3f4bad6f17985": "\\frac{\\lambda-\\lambda_0}{\\lambda_0}=\\frac{v_0}{c}", "560c366aefba39572b7e078147ab0313": " ( r\\ ,\\ \\theta)", "560c37fe8dcd08c8d46722d1b5282754": "(p,q) \\in C^+ \\cup C^-", "560c39ddff015e51d8b6d7b56c1d9e95": "y(t) \\to 0", "560c8f2b45313826bcecf1ddb60962a8": "2^7\\ln(2)\\mod{1} \\approx \\frac{64}{105}+\\frac{37}{360}=0.10011100 \\cdots_{2} + 0.00011010 \\cdots_{2} = 0.1011 \\cdots_{2}\\, ,", "560c962a6f84fd0ed2da057276b22bb5": "\n \\begin{align}\n P_\\mathrm{Br} &= \\int_{\\omega_p}^\\infty d\\omega {dP_\\mathrm{Br}\\over d\\omega} = {16 \\over 3} \\left[ Z_i^2 n_i n_e r_e^3 \\right] \n \\left[m_e c^3 \\right] k_m G(y_p) \\\\\n G(y_p) &= {1 \\over 2\\sqrt{\\pi}} \\int_{y_p}^\\infty dy y^{-1/2} \\left[1-{y_p\\over y}\\right]^{1/2} E_1(y) \\\\\n y_p &= y(\\omega=\\omega_p)\n \\end{align}\n", "560cb256da35250cb5c4e6698fd79634": "\\widehat{\\otimes}", "560d3c265c1b80061caa45ea6afb575f": "u : X^*_{b} \\times Y^*_{b} \\to Z^*_{b}", "560d7b4832765d2ea8483f802411f434": " \\left[ \\begin{matrix}\n 0 & 1_N^T \\\\\n 1_N & \\Omega + \\gamma ^{ - 1} I_N\n\\end{matrix} \\right] \\left[ \\begin{matrix}\n b \\\\\n \\alpha\n\\end{matrix} \\right] = \\left[ \\begin{matrix}\n 0 \\\\\n Y\n\\end{matrix} \\right] ,", "560d941514d6c6c5a858c41c8251e344": "\\sqrt{b^2-4ac}", "560db2f72980024295c1d760480db7fd": "-\\frac{dI}{dz}=\\alpha I+\\beta I^{2} ", "560db8dbd4048f0df250c22b7fa1ab61": "x_3 = 1-x_1-x_2", "560ddd0397389de69be08ee21ab7ba39": "T_{xx}", "560e3c9487637646defc13c66e9f05ad": "2\\beta", "560e6fedc4c878e55d7427ac6cc99781": "\\nabla E_{internal} (s)= \\Bigg[ \\bigg(\\alpha\\,\\!(s) \\nabla \\left \\| \\frac{d\\bar v}{ds}(s) \\right \\Vert^2 + \\beta\\,\\!(s) \\nabla \\left \\| \\frac{d^2\\bar v}{ds^2}(s) \\right \\Vert ^2 \\bigg) /2 \\Bigg]", "560e708113c7f5d61de20f40bb18de7f": "\\scriptstyle T ", "560ec574068586bd26b594b27f1ed40d": "f\\colon A \\to B", "560f5e74c4f9ee9423a56d4890f8d890": "100(1 - \\alpha)%\\text{CI}: \\operatorname{arctanh}(\\rho) \\in [\\operatorname{arctanh}(r) \\pm z_{\\alpha/2}SE]", "560f7ed26687638dbc396c79f7ec3854": "\\mu>0 \\,", "560f8eaab0f003fdfe570103cde8da40": "\\Delta y_1^s ", "560f99a25d71b5ca090258cdcd10982e": "\\sigma(B) - \\sigma(0) = \\alpha {e^2 \\over 2 \\pi^2 \\hbar} \\left ( ln \\left ( {B_\\phi \\over B}\\right ) - \\psi \\left ({1 \\over 2} + {B_\\phi \\over B} \\right ) \\right) ", "560fcadd0375d8a4381fc0f7472e7db6": " A_0 \\subseteq A_1 \\subseteq A_2 \\subseteq \\cdots ", "56101901f11542e173d7c472973a2837": " [a x + b y, z] = a [x, z] + b [y, z], \\quad [z, a x + b y] = a[z, x] + b [z, y] ", "561023752d267dc1522c8e22eb203bda": "\\int_0^1 x^xdx = \\sum_{n=0}^\\infty \\int_0^1 \\frac{x^n(\\log x)^n}{n!} \\, dx. ", "56109db3f62a9f7d937d98cd7ff01396": "\\mathrm{distance} = a \\bigl( \\sigma - \\tfrac f2 (X + Y) \\bigr) ", "5611017c697d721ef3ffad01f5b5a143": " ~p^v_0 \\propto \\exp{( -\\beta R_{min})}, ", "56118dc364e79bfdd527772b80f10344": "X \\in \\R^p", "5611f749349be8fabf5cdd7d88a64c6b": "93 = 1011101", "5612ce3d83ebfab8a12fed810734053d": "\\|\\mathbf{w}\\|", "5612d1f169de8e118e96b5b0b5532a6e": "g(z):=u_x(x,y)-iu_y(x,y)", "561305f7bdb66b83f0fab24928433664": "\\bar\\theta(t^{}_0)=\\theta^{(m)}", "56130eb58dd75ed00eda1e5cd7e20289": "N\\subseteq_s M\\,", "56133b5a6a6abff6fb7d42f562657e14": "\\int_{-\\infty}^{+\\infty} |\\Psi(x,t)|^2 \\, dx < +\\infty", "5613456c85cfc7cf75fad62ea2b4e4d8": "\\star (e_{i_1} \\wedge e_{i_2}\\wedge \\cdots \\wedge e_{i_k})= e_{i_{k+1}} \\wedge e_{i_{k+2}} \\wedge \\cdots \\wedge e_{i_n},", "5613db0505afd5a740fc2ae6e065ab7b": "\n\\coth{t-\\ln z \\over 2} = 2 \\sum_{k = -\\infty}^\\infty {1 \\over 2 k \\pi i + t - \\ln z} \\,,\n", "5614371f803f8a78b18b27391549a107": "\\lambda_i", "561466d313a42d8a81a18f2a6c6501d3": "F\\colon V_1 \\otimes \\cdots \\otimes V_n \\to W\\text{,}", "5614775bf50ed2ec589ed9458878c7d0": "F(x^{(1)},y^{(1)},x^{(2)},y^{(2)},x^{(3)},y^{(3)})= x^{(1)}x^{(2)}x^{(3)}+\\frac{2}{3}x^{(1)}y^{(2)}y^{(3)}+\\frac{2}{3}x^{(3)}y^{(1)}y^{(2)}+\\frac{2}{3}x^{(2)}y^{(3)}y^{(1)}.", "56147d584194afd8c189496c7ccd9eaa": "\\Phi_X(f) = (Ff)(u)\\,", "5614b8c6ee83fd6eca668ddf464d3c63": " 2 (d-2)", "5614c3507bb1689e404c13eec048a707": "\\gamma^{ij}", "5614f2f428505eeff293fcebfdc6c8c3": "A_{n}", "5614f9768b5306bd0ab0205d98df83fa": "\\hat{r}=\\cos u\\ \\hat{g}\\ +\\ \\sin u\\ \\hat{h}\\,", "561568bcc4553d208395092f31d1b31c": "\\epsilon = \\frac{\\sigma}{E} + K \\left(\\frac{\\sigma}{E} \\right)^n", "561574a4dc2c0e51f214ad35ed929b51": "\\frac{EP}{EP^\\prime}=\\frac{EQ}{EQ^\\prime}; {EP}\\cdot{EQ^\\prime}={EQ}\\cdot{EP^\\prime}", "5615dca0887d3fe3627eff2aa0305d76": " \\left. - \\mathcal{M}_0(p_1 \\cdots (p_i+k) \\cdots p_n; q_1 \\cdots q_n) \\right] ", "5615e3af86afee124fc92f6e130d6aa3": "\\{\\mathbb{Q},+\\}\\;", "561610b9bd53f098e3fdae387b3126ef": " m \\lambda = 2 d \\sin \\theta \\,", "561645ff50a6301ab719340274b3a4e4": " d^2 F = \\frac{k I I' ds ds'} {r^2} \\left( \\frac{\\partial r}{\\partial s} \\frac{\\partial r}{\\partial s'} - 2r \\frac{\\partial^2 r}{\\partial s \\partial s'}\\right) ", "56166226cc2c41dc055ed5c5df3b1ecd": "dA/dT", "5616d834c505c207d51a1da062f12151": "\\sigma^2 = \\frac{\\sum_{i=1}^N w_i x_i^2 \\cdot \\sum_{i=1}^N w_i - (\\sum_{i=1}^N w_i x_i)^2}\n{(\\sum_{i=1}^N w_i)^2}", "5616e29fa5ffe97eefb9c275c1e4ee95": "v_c (\\cot\\alpha + \\cot\\beta) = c\\quad \\mbox{or}\\quad\nv_c \\sin(\\alpha + \\beta) = c \\sin\\alpha \\sin\\beta.", "5617144eb366133d7f7fe0df08ea0bce": "\\vec v = v_1 \\vec e_1 + v_2 \\vec e_2 + \\ldots + v_n \\vec e_n = \\sum v_i \\vec e_i = E [v]_E", "56172daca4f4be42464fffe2f38ae0cb": " \\{ \\bar{P}_3, P_3 \\mathbf{e}_1, P_3, \\mathbf{e}_1 P_3 \\} ", "5617355abafc82e7d1a689a0a982ed79": "\\kappa(V)=\\kappa(F)+\\kappa(W).", "56176efd245d67ed196f1244d1dfbc88": "(\\lfloor n/k \\rfloor k)", "5617dfd0950285798672ed89e1fdc027": "(\\gamma^5)^\\dagger = \\gamma^5. \\,", "56180807ac875259a5b424e79498ba7f": "\\text{mode }= \\tfrac{\\alpha-1}{\\alpha + \\beta-2} ", "56180fa7a5ffaa3c7068bcf63e0f029a": "Q(w) = \\sum_{i=1}^n Q_i(w),", "561841715d84c0cd3fc726d9ba03e5f7": "S = {c^4 \\over 16 \\pi G} \\int R \\sqrt{-g} d^4 x + S_m \\,", "561874287b450200d42959d625a9b9c9": "U(y, \\xi) = D_{KL}(p(\\theta|y,\\xi) \\| p(\\theta|\\xi)) \\, ,", "5618960e821155844e1d28cd23c363e7": "\\;_{2}F_1 \\left[\\begin{matrix}\na & b \\\\ \na+b+1/2 \\end{matrix} \n; x \\right]^2 = \\;_{3}F_2 \\left[\\begin{matrix} \n2a & 2b &a+b \\\\ \na+b+1/2 &2a+2b \\end{matrix} \n; x \\right]", "5618f36dd6866f6e4ef1241b722ceb13": "\\ell(\\theta|X,Y) = \\sum_{i=1}^m \\left( y_i \\theta' x_i - e^{\\theta' x_i}\\right)", "56190dc3726e7a5b1c883c038912e05a": "C\\ell_{0,3}(\\mathbb{R}) = \\mathbb{H} \\oplus \\mathbb{H}.", "561929dd309347f13e17079c58762aad": "\\gcd(a(x), b(x)) = \\gcd(b(x), r_0(x)) \\,.", "561959b76b3a696c83d4799923e1fcb1": "\\vec E+\\mathbf u\\times\\vec B\\approx 0", "56197f49900a2429bc140883afc4a55b": "A^T A", "5619e2d459cd63bf4f24e3c31587133f": " \\left(c-v\\right) \\rightarrow c ", "561a28d21f73356717904536d752a217": " O(n \\log n) ", "561a40a8b1796609ff94f038cad3c05e": "\\int_b^a f\\,dx = -\\int_a^b f\\,dx", "561a8bd2e84a0c113dd638ec9d07efbe": "M _{DC} ^f = \\frac{PL}{8} = \\frac{10 \\times 10}{8} = 12.5 \\mathrm{\\,kN \\,m}", "561ad1badfbf59eafecee75fa4552451": "\\hbar \\omega_c", "561b34a4e77f63b1c9f5e4a61bc636c8": "\\phi_\\alpha\\colon U_\\alpha \\to \\mathbf R^n", "561b529836673ac608786439a2087c04": " \\pi ", "561b6cfedbee140415868e12d94325d3": "W_x(t, f)= \\int_{-\\infty}^{\\infty}x(t+\\tau /2) x^*(t-\\tau /2)e^{-j2\\pi f\\tau}\\, d\\tau.", "561c772f8362bfadb2e0b7c09ef2ce5f": "\\operatorname{coversin}(\\theta)", "561ce34f14c8fb398d0765f652e64b60": " \\textbf{x}_{t-i} ", "561cf97237a1bcea9b91817905d32642": "\\Delta G^{N-D}", "561d71d5cb1bf8f99b600f5b0657a005": "R=R(a,\\theta)=E[L(a,\\theta)|\\theta]", "561d7cc7fcb6df84222b691cd2d7bf22": "\n \\sum_{k=-\\infty}^\\infty f(k;\\mu_1,\\mu_2)=1.\n ", "561dec61c672a5c305bc49f528ef0a51": "r_{\\mathrm{log}} = \\ln\\left(\\frac{V_f}{V_i}\\right)", "561e19396aa04e10060d4039692b939e": "(x\\lor x)", "561e1c1bf831c98cd1e457184a48c557": "\\Pr[\\mathcal{A}(D_{1})\\in S]\\leq\n\\exp(\\epsilon c)\\times\\Pr[\\mathcal{A}(D_{2})\\in S]\\,\\!", "561e322a3c222415f8b93c7a1c521779": "S_j[q..n_j]", "561e5723f1a418e2ae1e7eb54ad3d441": "\\aleph_0, \\aleph_1", "561e7daf6a9b3709c03f9050f4953353": "(\\mathbb{Z}/1\\,\\mathbb{Z})^\\times \\cong \\mathrm{C}_1", "561e88bf9baa2855e383fe3d582df88f": " V_P", "561ec90d02bdb500ea4b02db9a0e2b28": "H_1^{(1)}", "561f01fefe9e2bad62ba2bfae624ec79": "P(t) = F\\cdot v.", "561f25d5c3cccb06ddc662cba22e90dc": "L_\\tau", "561faa1283bf35b99e4d1f17c08efc03": "\\begin{align}\nV_{BPSK}(t) &{}= A(t) cos(\\omega_{RF}t + n\\pi); n = 0,1 \\\\\nV^2_{BPSK}(t) &{}= A^2(t) cos^2(\\omega_{RF}t + n\\pi) \\\\\nV^2_{BPSK}(t) &{}= \\frac{A^2(t)}{2}[1 + cos(2\\omega_{RF}t + n2{\\pi})]\n\\end{align}", "561fe8721590b102732b10029e7cc8c8": "\\dots,a_0,a_1,\\dots,a_{n-1},a_0,a_1,\\dots", "562027795f0d7eb77f6ca0ee29d054b4": "q\\mathbf{A}(\\mathbf{r},t)", "56203f38b7689dd5f95bad991619369b": "c_{i}-(b_{i}-a_{i})", "5620a3848054b3b3fa979f07d6184b54": "\\alpha_\\lambda E_{b \\lambda}(\\lambda,T)", "5620dad06e11777f01a7205914d79682": "p_3 = \\{ s, v_1, v_2, v_3, t \\}", "5620f4b920854b641cbccc963eed8518": "T_{a_x}A_x.", "5620f76251836ad6092326f836e9944d": "A > C > B > E > D", "56214fceb3f6a95ae4f83e8b0d9c51b2": "\\exists x,y . P(x,y) \\vee Q(f(x))", "562176735cbd873eb71856e5d3ac9917": "\\lim_{n\\to\\infty} \\frac{N_S(w,n)}{n} = \\frac{1}{b^{|w|}}", "56219de3375b737f3e4f9db47dd03a89": "\\color{Rhodamine}\\text{Rhodamine}", "5621a335a17fdb6846be86828c0d3c0e": "\\mathit{A}, \\mathit{B}", "5621d56af0ee110f737314c033f51f59": "(x+iy)^2=x^2-y^2+i(2xy)", "562202c9163fecf67f49054fc30e1b73": "\\dot{\\omega}", "562245889494a5f546e0dfcfdd55b513": " \\langle Ux, Uy \\rangle = \\overline{\\langle x, y \\rangle} = \\langle y, x \\rangle ", "56226e15cf9df4618996f770b46944e0": "\\sqrt{9} = 3 ", "56229e9f7c64653ef95989c2608228b6": "\\exp(i\\theta)", "562328175d2e9eae0907f5f7b6ea294b": " g(x,y) \\le 0, \\;\\; \\forall y \\in Y ", "5623631f9a53269c6dc1e992cb75e463": "91=13*4+1*4+3*5+10*2=52+4+15+20", "5623b5406784112bf190806361e8579f": "D=(C_1,~C_2, ...C_l )", "5623f780ec683d6972f536d532749a7b": "c_n = a^{-1} \\int_0^a F(t) e^{-\\frac{2\\pi int}{a}}\\, dt= a^{-1} \\int_{-\\infty}^\\infty f(t) e^{-\\frac{2\\pi int}{a}}\\, dt={\\sqrt{2\\pi}\\over a} \\widehat{f} \\left (\\tfrac{2\\pi n}{a} \\right).", "562424827b1818bb0a80d7ed16d26ded": "f(x+h) = \\sum_{\\alpha\\in\\mathbb{N}^n_0}^{}{\\frac{\\partial^{\\alpha}f(x)}{\\alpha !}h^\\alpha}.", "56251954823da616b6558a7aff87a09c": "A \\vec e_i = \\lambda_i \\vec e_i", "562519ae3b7be85c5a7a073956e36ad2": " a = b u ", "562523b6ddc14b10ffe33e231e484b39": "\\scriptstyle h \\,+\\, \\frac{1}{x_0} \\,+\\, O \\left( h^3 \\right) ", "562541411896847024043ec76e538f10": " = V_{molecule} ", "56257f4e46e31e815b75534d2e14df45": "\nC(d) = \\sigma^2\\frac{1}{\\Gamma(\\nu)2^{\\nu-1}}\\Bigg(\\sqrt{2\\nu}\\frac{d}{\\rho}\\Bigg)^\\nu K_\\nu\\Bigg(\\sqrt{2\\nu}\\frac{d}{\\rho}\\Bigg),\n", "56259e3eb239466e4748404140366ee7": "I_{\\text{c}} = \\beta I_{\\text{b}}", "56260856eea12dd78f5125501b841e8d": "V_x = V_0 e^{-1} ", "56262900c1c4820aa4495aba78c8ba48": "\\varphi(x)=|\\det M|\\cdot\\sum_{k\\in\\Z^d} h_k\\cdot\\varphi(M\\cdot x-k)", "56264d2f0d28df0f2b9cb8cfc78e5d29": "\\sigma_f\\sqrt{a} \\approx C", "56271472c2b53d4f0132cd93fdfc500e": "\n[C1-3] \\quad [sc] \\vdash \\textbf{skip} \\qquad [sc] \\vdash h \\;:=\\; exp \\qquad \\frac{\\vdash exp \\;:\\; low}{[low] \\vdash l \\;:=\\; exp}\n", "56279f8404a5518f6fe46719a734cde7": " T = {I\\over I_{0}} = 10^{-\\alpha\\, \\ell} = 10^{-\\varepsilon\\ell c} ", "5627c7742969a6771c25e2f5445e0598": " 2N ", "5627ebca6a733b7fc284fac3a4ee9af3": "\\mu_{pq} = \\sum_{x} \\sum_{y} (x - \\bar{x})^p(y - \\bar{y})^q f(x,y)", "5628bf1f1491a80da8cb8ff69e73d4ec": "z_A = {3 (n_c - n_d) \\over \\sqrt{n(n-1)(2n+5)/2} }", "5628c888d5360920f087ea007a665a1b": "M_{g, n}^{J, \\nu}(X, A)", "5628ccc78c9515bf67c98bb902095ec1": "\\mu\\neq 0", "5628d76f193112065dca5c33a3fe6962": " \\pi_{n}(t) = i \\hbar a_n^*(t) ", "56298c3325ba4732da0ae17c50b6b1e6": "{x^2 \\over a^2} + {y^2 \\over b^2} - {z^2 \\over c^2} = - 1 \\,", "5629902d7c52af33e8cf2aa63e8a8943": "2\\omega_{p} \\approx 2\\omega_{n}", "562993f53fb50086cab571c4d0d4250d": "\\mathcal{K}_n", "5629bcf05214d91718373c18d1f93c57": "\\Psi \\in \\Gamma(T(J^{r}\\pi))\\,", "5629f5abb6253725a21a32922461c5e0": " 4 \\int\\limits_0^\\infty f(t)\\cos\\,{2\\pi \\nu t} \\,dt.", "562a19eb26bb8b6ebb2df1a822993163": " T= \\dot m\\, w.", "562a419cfb89f6b6bbc4453a5448c2e0": "l(f)\\cdot \\|u-v\\| \\leq \\|f(u)-f(v)\\| \\leq L(f)\\cdot \\|u-v\\|,", "562ab2b06d507beda35da391e27b1c30": "pH = pK_a + \\log(1)\\,", "562ab2e0299a0b55077a3bab8f540fe0": "\\displaystyle \\operatorname{sinc}(a x)\\,", "562b59874c84aab2772e10225a1bd785": "f^\\prime (0^+) = (\\ln a) f^\\prime (0^-) \\text{ or } f^\\prime (-1^+) = f^\\prime (0^-).", "562b5ed6c912b9a0662b164526f01759": "u_i^n = u(x_i, t^n) \\text{ with } x_i = b + i\\,\\Delta x ,\\, t^n = n\\,\\Delta t \\text{ for } i = 0,\\ldots,N ,\\, n = 0,\\ldots,M,", "562bb6c4a1b231d1bc9cd8dc2670ad87": "ar_1P + \\sum_{2 \\leq i \\leq r}(b_ir_iQ) = 0.", "562c238ac02c37a26bcf98af9bbc6682": " Jf(g)=\\overline{f(g^{-1})}.", "562c7fcc03ce1d2c998247471b21064d": "S = \\frac{R-T}{DR}", "562cb13d7fa244bd949f2b792b216977": "\\leftharpoondown", "562cef0cae14e6e5a50218e62e9153bb": "r^2-a^2-b^2+c^2+2a^2\\sin^2\\theta=\\pm 2a\\sin\\theta\\sqrt{c^2-a^2\\cos^2\\theta}),\\,", "562cf8f67709e27ecb856a6d34e2d776": "| p_1 - q_1 | + | p_2 - q_2 |.", "562d2684962dde85219b4b7d3304b05c": "wx[T\\cdot S]yz", "562d66227c7fecc2bc85b0338a464c98": "v=\\frac{|a-b|}{2}.", "562d7ea3f784994d50ac25a129d49080": "\\! k_\\mu", "562dc7320e502629ae5682bd1eb712fb": " D(n) = n! \\sum_{k=0}^n \\frac{(-1)^k}{k!} \\; \\approx \\; \\frac{n!}{e}.", "562e0673ae6f1a4e480f670a22544ee6": "y = A \\sin \\omega t", "562e095f1b81d50cedbc6b920982f03c": "M=\\frac{\\pi*\\rho*d^3}{6}.", "562e09c9ac5850171320971cfcb38e46": "\\sum_{u,v}(f(u) - f(v) )^2", "562e223ec6e7599075baeaed353fc844": "y_1, y_2 \\in [x_1, x_2]", "562f1bd46bff16b84d4c326572775962": "G_{Eq} = G_1 + G_2.\\ \\,", "562f3b8c2c7b50894ebfb6201981b7b5": "x(t) = R(t) \\cos ( \\omega t + \\phi(t) ) \\,", "563005f0eaeef2adb891f8f033d4e6e8": "m_n(x_n)", "56305397fa00b82659861480d1a9a717": "\nS_{xy} = \\frac{1}{2N^{2}}\\sum_{i=1}^{N}\\sum_{j=1}^{N} (x_{i} - x_{j}) (y_{i} - y_{j})\n", "56308888cfed857b6f9c5568fa319f33": "\\mathsf{P^{\\sharp P}} \\supseteq \\mathsf{PP} \\supseteq \\mathsf{MA}", "5630d34a194e1a975072c25151d10e22": "\\Delta t\\simeq\\frac{2l\\Delta v}{c^{2}} ", "56310b88b0026a6a74c89d3dfc769f8d": "y=2px+C,", "563138cde72b08df6ab8834129a862e8": "\\Lambda_c^*(M)", "56313c71819829cec615335b8de23483": "v_{ph}", "56318290179179e639b1e3d83068f46e": "\\alpha_k (t) = (\\alpha_k (0) - c p_k H^{-1}) e^{-2 i H t / \\hbar} + c p_k H^{-1} ", "56318d644b7f675f6e89770727905303": "\n\\mathcal M=\\langle \\beta\\ \\mathrm{out}|\\mathrm T\\ \\varphi(y_1)\\ldots\\varphi(y_n)|\\alpha\\ p\\ \\mathrm{in}\\rangle\n", "5631bddcbd5baa2e81806bb54c9d79f0": "\\tfrac{g}{m^2 d}", "5631d7d1eb2b09cca20052bc593215a4": "\\mathbf{K} \\cdot \\mathbf{r} = \\left ( l_{1}\\mathbf{g}_{1} + l_{2}\\mathbf{g}_{2} + l_{3}\\mathbf{g}_{3} \\right ) \\cdot \\left (x_1\\frac{\\mathbf{a}_{1}}{a_1}+ x_2\\frac{\\mathbf{a}_{2}}{a_2} +x_3\\frac{\\mathbf{a}_{3}}{a_3} \\right ) = 2\\pi \\left( x_1\\frac{l_1}{a_1}+x_2\\frac{l_2}{a_2}+x_3\\frac{l_3}{a_3} \\right ).", "56320147bc6b8e8c0c9f0797f89a4942": "j\\!\\!j", "56320ecdee193e082a6a5c3087356b52": "U_1 \\cap U_2 \\ne \\emptyset", "563250085388c11e6b9f9a3848cfbcb3": "b_j(y_t)=P(Y_t=y_t|X_t=j)", "5632c61edb93834dab3ca2232b584612": "a(v, v) \\ge \\alpha \\|v\\|^2", "5632cf2e08f7a69e0e9e3127a393e165": " \\bar X = \\frac{1}{n} \\sum_{i=1}^n X_i. \\,\\!", "56330f0fb16623fde5c3626d71c43953": "\\begin{align}\n |\\hat{\\mathbf u}| = |\\hat{\\mathbf v}| &= 1\\\\\n \\hat{\\mathbf u} \\cdot \\hat{\\mathbf v} &= 0\\\\\n \\hat{\\mathbf u} \\times \\hat{\\mathbf v} &= \\hat{\\mathbf w}\n\\end{align}", "56334c130494101c66e8ea64975ba349": "\\textstyle \\mathcal{M} = \\left\\{0,1\\right\\}^n, \\mathcal{C} = G_1^* \\times \\left\\{0,1\\right\\}^n", "56338a13c311e13ec445b472c9a9063b": "x_1 (t)", "5633c4e34f656ace96b7eb3f79c05015": "\\phi_{\\mathbf{R}}", "5633e1135d2660e13669306c3d86fb67": "\\mathit{Vs}(a)", "563446ba1c93211cb28cfb3471c76c13": "\\textstyle{\\coprod_{i \\in I}V_i}", "56345a27216f949c2a1bd2da3cc5db98": "T_1\\odot T_2 = \\operatorname{Sym}(T_1\\otimes T_2)\\quad\\left(\\in\\operatorname{Sym}^{k_1+k_2}(V)\\right).", "56349e45e98899bebd3acef21634610d": " -\\frac{\\hbar^2}{2m s^2} \\frac{d^2\\psi}{d\\phi^2} = E\\psi \\quad (3) ", "5634d50ef04b8a998fbe42fd8fb9a46b": "\nn^2 = \\sum_{\\ell=0}^{n-1} (2 \\ell + 1),\n", "5634ec46e47bef3dded3fe2ab3e62a1d": " T = {I\\over I_{0}} = e^{-\\alpha'\\, \\ell} = e^{-\\sigma \\ell N} ", "5634f6ff3f322add2e7a060d38f25cd8": " \n\\begin{align}\nI(\\theta) \n&\\propto \\operatorname{sinc}^2 \\left [\\frac { \\pi W \\sin \\theta} {\\lambda} \\right]\\\\\n&\\propto \\operatorname{sinc}^2 \\left [\\frac {kW \\sin \\theta} {2} \\right]\n\\end{align}\n", "5634ffe42f4efe70ceb6204586e26e3e": "\\nabla^2\\phi - \\frac{1}{c^2}\\frac{\\partial^2 \\phi}{\\partial t^2} = - \\rho/ \\epsilon_0 ", "56350592a1b34919324704045f0dc08e": "(dG)_{T,P} = \\left( \\frac{\\partial G}{\\partial \\xi}\\right)_{T,P} d\\xi.\\,", "563536ba4372355f74a0c3ebc8e0e311": "\\Delta t = \\pi/\\kappa E_0", "5635397b6cb9ac08126ec1da0091c5fa": "g_\\alpha", "56361505db85702d00c1dad95d6ae83b": "V_p = 1.16 V_s + 1.36", "56362a4abf474790e7edcc01752f1d62": "\\frac{\\partial }{\\partial x^i}", "563650d0f87df6480f0bdef0f873671c": " \\frac{d \\bar \\rho_{ge}}{dt} = -\\left( \\frac{\\gamma}{2} + i\\delta \\right) \\bar \\rho_{ge} + \\frac{i}{2}\\Omega^*(\\rho_{ee} - \\rho_{gg})", "5636b87b9d836e17477ae85b57eb45c9": "p \\rightarrow \\Box\\Diamond p", "5636bd95488f62024ac83b0504fb43ca": " v =\\tanh K ", "5636d5bebe6d35c64ce13ff5e895b5f9": "\n{\\star \\bold{J}} = \\rho dx \\wedge dy \\wedge dz - j_x dt \\wedge dy \\wedge dz - j_y dt \\wedge dz \\wedge dx - j_z dt \\wedge dx \\wedge dy\n", "5636e2aa4a55aeaeeb96b7f087ef8de3": "0 \\to (V/W)^* \\to V^* \\to W^* \\to 0.", "5637281ffc356e6757b656854c971424": "\\displaystyle \\sum_{n=1}^\\infty b^{-p_n}, \\, ", "56372f8fefde387c0b9b98fcc5b245b3": "k 1\n\\end{cases}", "563c9ea42780961e414ee0466b86932b": "y_l\\,\\!", "563cc4a6184d5465c8a9f7ac5c97de7b": "y(x) = -\\frac{2a}{\\pi}\\arctan(\\cot(\\frac{\\pi}{p}x))", "563d26761804b4d6d3471ae34b6e49b8": "\\left\\lbrace x \\left(\\frac 3 2\\right)^ n \\right\\rbrace ", "563d5c44ad6af9d47ad5541d6f5d213a": "\n(1-n)^2(1+n)=\\frac{1}{1+n}(1-n^2)^2\n", "563d880e1c878d80bb57b029b4c56166": "i>0", "563da0bcb189991169a5aac7ab756564": "D f(x) = \\frac{E f(x) \\cdot f(x)}{x}", "563dbb2bc3173e6016a8224bf7ee9197": "\\mu\\Bigl(\\bigcup_{i \\in \\mathbf{N}} E_i\\Bigr) = \\sum_{i \\in \\mathbf{N}} \\mu\\!\\left(E_i\\right)", "563dfa48c6a31733fe271a7eaabb3449": " x\\vee (x\\wedge y) = x = (y\\wedge x)\\vee x ", "563e13f6e84c864c843d2a5b2af99f54": " y \\in \\lbrace 0, . . . , d-1 \\rbrace^m ", "563e1b681e65693fc7416fb70fc109a2": " P_t \\,\\!", "563e490afc58fa9aef9dba8e16f1e4b7": " \\neg (\\operatorname{def}[F_1] \\and ...) ", "563e649e438ddab1bc9f9b15f3b22ff1": "A=(3+\\sqrt{3})a^2\\approx4.73205...a^2", "563e689eb949dbd1670280ce88c1d43e": "j_z = -\\nabla^2 A/\\mu_0. ", "563e907469318213e75647324039e870": "2^{4-1} - 2^{\\frac{4}{2}-1} = 8-2 = 6", "563ea116d28e7b92b6a2a05607db3b85": "w(L)L", "563f17eb58430c03cac922e124bcc102": " \\sum_{i=k}^d a_i^{\\downarrow} \\leq \\sum_{i=k}^d b_i^{\\downarrow} \\quad \\text{for } k=1,\\dots,d,", "563f3d01197bb55668e0f7b2096e5629": "\\begin{align}\n M^{\\mathrm{core}} &= D^{\\mathrm{beam}}\\left(\\cfrac{\\mathrm{d} \\gamma_2}{\\mathrm{d} x} + \\vartheta\\right) \n = D^{\\mathrm{beam}}\\left(\\cfrac{\\mathrm{d} \\gamma}{\\mathrm{d} x} - \\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} + \\vartheta\\right) \\\\\n M^{\\mathrm{face}} &= -D^{\\mathrm{face}} \\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} \\\\\n Q^{\\mathrm{core}} &= S^{\\mathrm{core}} \\gamma \\\\\n Q^{\\mathrm{face}} &= -D^{\\mathrm{face}} \\cfrac{\\mathrm{d}^3 w}{\\mathrm{d} x^3} \n \\end{align}\\,", "563f3ff1c50516469aef8bf7d75ec2e1": "I(t) = C_p(dU_{Cp}/dt)", "563f47f75e52b429022c0b178fae05e3": "\\left.\\right. f(z)", "563f76479fb4fca233a72d643a71f960": "\\sum_{v\\in V} \\deg(v) = 2|E|", "563fba10e9cbd7acc347cbde4db61561": "\\left(\\frac{m}{Q}\\right)\\mathbf{a} = \\mathbf{E}+ \\mathbf{v} \\times \\mathbf{B}", "563fbba0367ffb9bbfd8b98dfaf6bda9": "\\ \\dot{Y} ", "56401d212cd8aff4d9c23d50b5177e5a": "\\mathrm{Sh}_D = 0.3 + \\frac{0.62\\mathrm{Re}_D^{1/2}\\mathrm{Sc}^{1/3}}{\\left[1 + (0.4/\\mathrm{Sc})^{2/3} \\, \\right]^{1/4} \\,}\\bigg[1 + \\bigg(\\frac{\\mathrm{Re}_D}{282000} \\bigg)^{5/8}\\bigg]^{4/5} \\quad\n\\mathrm{Sc}\\,\\mathrm{Re}_D \\ge 0.2 ", "56403a4ccd96687d73bfae6a104f1552": "\\left\\{ \\left. \\left( \\theta_{*} (\\mu_{\\cdot}) \\right)_{S} \\right| S \\in \\mathcal{A} (G) \\right\\}", "56403b3f4d0a810b4188c270bbc4d108": " \\, T = 0 ", "564054e8a128c1763a8853ba4955d196": "x_{*}\\ ", "56406e8fc2baed5bb7348228ace1ec48": "T_n + T_{\\lfloor \\frac{n}{2} \\rfloor},", "56407ff72906c7920d171ffc90fd3cc5": " \\Sigma^0_1 \\cup \\Pi^0_1 ", "5640c7b41623a8b5993f74f8b7224b17": "F_5=5 \\text{ and } F_6=8.", "5640fd3748944fe95721622960b672e2": "P = U D U^* = (U\\sqrt{D}) (U \\sqrt{D})^*", "56412d498e58b0b6e7351711e3c36298": "m \\operatorname{d}\\boldsymbol{\\omega}/\\operatorname{d}t \\times\\boldsymbol{r}", "56415cfea58f5b4f991b2b0ad8745530": " X_s ", "56417244188cb70bf864841bf330e5ba": "\n\\begin{align}\n \\text{div}\\,\\mathbf{v} &= 0\\\\\n \\nabla\\cdot\\mathbf{v} &= 0\n\\end{align}\n", "56417c76a9d74c85335b4e392fe51feb": "\\hat{\\rho}(\\mathbf{k},t)", "5641b9b67407e392ad4ba28d7299ccbb": "S(\\rho_{A} || I_{A}/n_A) = \\mathrm{log}(n_A)- S(\\rho_{A}), \\;", "5641c9fc5595c28dbe34661968e7a138": " \\sigma = \\sum_i q_i | i \\rangle \\langle i |", "5641d42b569608f680ddd25775d993f7": " p_1=\\lambda_1 a+\\lambda_2 b,\\; p_2=\\lambda_1 c+\\lambda_2 d, \n", "5642224099dbf42a246fe5a258c23430": "\\geq 1-m\\left(\\dfrac{n-k}{n}\\right)", "5643285e98b487dd3ae65e025e6d4304": "M = (Q, \\Sigma, \\Gamma, \\$, s, \\delta)", "56433532a11cf9e8b7b044e7f4ca3286": "A^s x_1=\\sum_{r=0}^s\\binom{s}{r}\\lambda^{s-r}x_{r+1}", "5643435ce8d9a425088e4338fa2c8932": "\\scriptstyle A > \\sqrt{5}", "5643444bb89430d321a7322c34cb0a33": "\\psi(x,t)=\\frac{1}{x^2}\\,\\psi\\left(1,\\frac{t}{x^2}\\right)", "56434baecef9df028d5612f81907d26d": "\\,b\\,", "5643d395ddfbf0d83248459812961b80": " \\frac{kg.sec.}{cm.^5}", "564409d982fc18c90fa52adce52d7816": "\\vec a_g = - \\hat r ~ G ~ \\frac{M}{R^2} \\pm \\hat r ~ G ~ \\frac{2 M }{R^2} ~ \\frac{\\Delta r}{R} - \\cdots ", "564450a149ae73525a4d7b10b542c874": "2x^2 = y^2(5y - 3)^2 \\pm 2", "5644641722081d6cdfd6232534444973": " d(x, \\Omega)=\\inf_{y\\in \\Omega}d(x, y)", "56447742dafb1cbfafc234ed9ef59f7d": "\\hat{\\varepsilon}_{t-1}", "56447c332bc3dd6b1fe5173668878d21": "\\{p,r\\}\\ ", "5644cf29a7ae4a58cdf0ce7edecdb73a": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix} = \n \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix}\n", "564585dffd3660be94540b8217bc0186": "R_\\mathrm{AB} = \\sqrt{R_\\mathrm{A} R_\\mathrm{B}}", "5645db9c3e59f53b8bbd21b2ef28d451": "{Z_0= \\rho_0 c_0}", "5646033797322a34950481c1cde227c1": "v = \\frac{s_1u_1v_2 + s_2u_2v_1 + s_3 (v_1v_2 +f)}{d} \\mod u", "56471bc4e05e48fe5977a1cefe03b5ec": "M(n)/n = 1/2", "56473d4b9abe9e9f71fb5c31839d707a": " \\scriptstyle \\phi", "564756906c2665e1babb5a8ffb2d129f": "\n\\mathbf{L} \\cdot \\mathbf{W} = \n\\alpha \\left( \\mathbf{L} \\cdot \\mathbf{A} \\right) \\mathbf{A} + \\beta \\left( \\mathbf{L} \\cdot \\mathbf{B} \\right) \\mathbf{B} = 0 ~,\n", "56476b006299787ea0b57406b03a5502": "\\Pr[Y(t) \\leq x] = F(t+x) - \\int_0^t \\left[1 - F(t+x-y)\\right]dm(y)", "5647a8ffdec75f01c3b65643410f1227": "4 \\cdot \\int_0^1 (x^2 - x^6)^{\\frac{1}{6}} dx = \\frac{ \\Gamma(\\frac{1}{6}) \\cdot \\Gamma(\\frac{1}{3})}{3 \\sqrt{\\pi}} \\approx 2.804,", "5648330fe8054a4118003985826a8734": "S_\\mathit{orw}", "56484a8ea9498d7df62292f893478a31": "{A^2}_2", "56484ae38b564867d4c337cb73d962e6": "x_n=7x_{n-1}\\,\\bmod\\,69", "56489000a15231c268bc641ef711d92e": "\nf(x,y) = \\begin{bmatrix} \\ 0.20 & \\ -0.26 \\ \\\\ 0.23 & \\ 0.22 \\end{bmatrix} \\begin{bmatrix} \\ x \\\\ y \\end{bmatrix} + \\begin{bmatrix} \\ 0.00 \\\\ 1.60 \\end{bmatrix}\n", "5648ace9adb1e858d5877f825f31dabf": "\\begin{matrix} {3 \\choose 3}{45 \\choose 1} \\end{matrix}", "5648caa31c5f54db843f35d6f0bf73b9": " dis(w,w') \\leq t ", "564901e10a365a9549bc1d5bf86c2e26": "\\sum_i v_i = \\sum_j w_j = 0", "56492e2b62222d00b582096f9e5eeaad": "w'=w\\,\\!", "5649cd8b68f9e0ade507ed3baa0a2df8": " r_{c} \\le R(q,u),\\forall u\\in \\mathcal{U}(\\alpha,\\tilde{u})", "5649dbcffbfb1a2c2d0de4656083bd46": "k_B \\in [1, n-1]", "564ab5ce67d1e7df65e5a3a114b281a9": "\\Lambda^k H", "564b0ae62121694ba021f62e02ed496c": " \\tau_1 \\cdots \\tau_m \\sigma ", "564b405a7b35b6eb30ec7e4b46ec4860": "\\frac{n!}{(n-k)!}.", "564b68f27cdcdeb31310c0fecb8ab040": "\\mathbf{B}_\\text{el}^l = -2\\mu_\\text{B}\\dfrac{\\mu_0}{4\\pi}\\dfrac{1}{r^3}\\mathbf{l}", "564b6bc92d82770b7c6df1dca2fb04f3": "K_i^G(X) = \\pi_i(B^+ \\operatorname{Coh}^G(X)).", "564bbe65cb25d59391006ed6b9e06254": "\\nabla^2 L(x, y, t)", "564bc372c5a46621143bf8b891d70ddd": "\nH(u,v)=\\int_{\\R^n}\\left(\n\\frac{1}{2}|v|^2+\\frac{1}{2}|\\nabla u|^2+\\frac{1}{2}f(u)\n\\right)\\,dx,\n", "564be2be74826ff95f9aa9f1a000e9c4": "V \\otimes\\chi_i", "564c077493540a8ddbb1cb987e5cd00d": "\\mathbf{A}:\\mathbf{B}=\\sum_{i,j} A_{ij} B_{ij} = \\mathrm{vec}(\\mathbf{A})^\\mathsf{T} \\mathrm{vec}(\\mathbf{B}) = \\mathrm{tr}(\\mathbf{A}^\\mathsf{T} \\mathbf{B}) = \\mathrm{tr}(\\mathbf{A} \\mathbf{B}^\\mathsf{T}),", "564c151b7a422a5c4756e8f6086748ce": "N=ns^T + sn^T", "564c3fcfa5722bde8d24eabdf37e75a9": "(17)\\qquad D\\rho -\\bar{\\delta}\\kappa=(\\rho^2+\\sigma\\bar{\\sigma})+(\\varepsilon+\\bar{\\varepsilon})\\rho-\\bar{\\kappa}\\tau-(3\\alpha+\\bar{\\beta}-\\pi)\\,\\kappa+\\Phi_{00}\\,\\hat{=}\\,0\\,", "564cfc3188d30b38c5c487465bfa9752": "{\\rm Pr}_r(A(x,r) = \\mbox{wrong answer}) \\le 2^{-ck(n)}.", "564d04eaf229286bf314a9bc263ae234": " \\mathrm{U} = \\mathrm{L} + \\prod_{k=1}^{n} f_k ", "564d66114dcb0ca1ce5e25c9d94bb0a0": "(26)\\quad ds^2\\,=-e^{2\\lambda(\\rho,z,\\phi)}dt^2+e^{-2\\lambda(\\rho,z,\\phi)}\\Big(d\\rho^2+dz^2+\\rho^2 d\\phi^2 \\Big)\\,,", "564dafa62fdc6af97c6474d22a0fc22f": "\n f(\\xi, \\rho, \\theta) = 0 \\, \\quad \\equiv \\quad\n f := \\bar{\\lambda}(\\theta)~\\rho + \\bar{B}~\\xi - \\sigma_c \\le 0 \n ", "564dd68b7ae1edb4437b5a0ef45cc038": "e^{\\pi\\sqrt{67}}=12^3(21^2-1)^3+744-1.337\\cdots\\times 10^{-6}\\,", "564dde0e742a16c58c014be95a39953d": "\\ddagger", "564de1b306285894ce148a75fab3bf49": "\\!\\lnot(\\exists x (\\lnot \\phi))", "564e0d8e4a1da9fd4b9523d7e026d62f": "\\,\\!-1/(4\\pi)", "564e4054799a50dbb96d30f5578363d3": "x^{K} \\,", "564e43a865f9bdcce6254d04f95231f5": "\n\\Pi_K(x,v)=\\lim_{\\delta \\to 0^+} \\frac{P_K(x+\\delta v)-x}{\\delta}.\n", "564ec67fc467bf24ccaf2892409cfaae": "\\sqrt{-1}\\Theta", "5650411e938894cbcb7d277457401b3f": "\\zeta_{k(m,n)}", "56507ed9681fd928ec35cdaad03b16ec": "u^+ = y^+", "5650d5d7f4299966676c00a3928d0730": "(L_z, R_z, B_z)", "5650dde3c9b2a8c0d1dc3bdf9e88bbb8": "\\frac{1}{|\\mathbf{x} - \\mathbf{x'}|}=[(x-x^\\prime)^2+(y-y^\\prime)^2+(z-z^\\prime)^2]^{-\\frac12}.\n", "5651231fa4fc1cad185d40b36ba45342": "\\gamma = \\sqrt{\\frac{a + \\sqrt{a^2 + b^2}}{2}}", "565163102f35b8d04029b2160037ceb3": "\\vec a_P", "5651951f1b291114676987982bb779fb": "ECD= MW+{ \\frac{P_a}{0.052*TVD} }", "5651c4ec6f487dda4628e5ca727de36f": "N^{\\prime} = 3N", "5651ca65d1e1e5a0b5237b510c88a5ff": "d_B(s,t)", "5652217740c62140c4c9fa7fbf3ad30c": "\\mathcal{B}[f],", "5652536ad5e1f6db62b3eca73d943148": "\\rho=\\sum_k\\lambda_k\\rho_k", "56525a29f57d8035a3b2a3908520dbdb": "\\frac{d\\alpha}{dt}=\\left(1+\\frac{Z_q}{mU}\\right)q+\\frac{Z_\\alpha}{mU}\\alpha", "5652e61a834fc3a9b592a743e2651c14": " - p\\left( {V_P - V_R } \\right) = U_P - U_R \\Rightarrow U_P + pV_P = U_R + pV_R ", "5652f251bfd1705dff95fcc61849d375": "b=2mn - n^2\\, ", "5652f61936ccdd67d903ece4d325bb2d": "X, Y\\in \\text{Tp}(\\text{Prim})", "56532ca0b9ea283b6f35102080b96f06": " 1,\\, 2,\\, 90,\\, 297200,\\, 116963796250,\\, 6736218287430460752, \\ldots ", "56536120797f894d80a26937ac1fb819": "\\eta_{L/K}(s)=\\int_0^s \\frac{dx}{|G_0:G_x|}.", "56537bfc3fe009ee6921cf54481ad24f": "\\{(w^{(L)}_k,x^{(L)}_k)~:~L\\in\\{1,\\ldots,P\\}\\}.", "5653def7a2201948af1f4f66b10ea1d3": "\\hat{D}^\\dagger(\\alpha) \\hat{a} \\hat{D}(\\alpha)=\\hat{a}+\\alpha", "5653ef096036b8c2e524b3f289ea43f7": " 3(x_1,y_1) = \\left( \\frac{(x_1^2+ y_1^2) - (2 y_1)^2}{4(x_1^2-1)x_1^2 - (x_1^2-y_1^2)^2}x_1, \\frac{(x_1^2+ y_1^2) - 2(x_1 )^2}{-4 (y_1^2-1)y_1^2+(x_1^2-y_1^2)^2}y_1 \\right). \\, ", "565411513f4a325756919506c069002a": "\\hat{f}(\\xi)= \\sigma C_1 \\, e^{-\\pi\\sigma^2\\xi^2}", "56542173119692da045aeba83c09688d": "(f \\star g)(t)\\ \\stackrel{\\mathrm{def}}{=} \\int_{-\\infty}^{\\infty} f^*(\\tau)\\ g(\\tau+t)\\,d\\tau,", "565452eba2a70a918b160855e5be49dc": "\n\\int \\exp\\left( \\frac i 2 x \\cdot A \\cdot x +iJ \\cdot x \\right) d^nx\n=\n\\sqrt{\\frac{(2\\pi i)^n}{\\det A}} \\exp \\left( -{i\\over 2} J \\cdot A^{-1} \\cdot J \\right)\n", "56547033c17cb578aefd68890228d9ed": "c_n={P_{2n+1}}. \\,", "565483f26782f3f9704ef1cf546c57fc": "a\\sim b\\Longleftrightarrow ab^{-1},a^{-1}b", "5654974dfa2fe14106ad411c4cb883eb": "1/2N", "565522e0f963cf58c4e7947bbb0a2a40": "\\Delta w = f, \\, ", "56556e6a7c161012d8b5821174bfe683": "b = 2 z_\\mathrm{R} = \\frac{2 \\pi w_0^2}{\\lambda}\\,.", "565583002b2c0b98f4ec2346c0b354df": "\\operatorname{MSE}(\\hat{\\theta }) =\\operatorname{trace}(\\operatorname{Var}(\\hat{\\theta }))\n+\\left\\Vert\\operatorname{Bias}(\\hat{\\theta},\\theta)\n\\right\\Vert^{2}", "5655a1d02fa8d879542929f363577e9b": "x = ka \\sin \\theta \\approx 0, 3.8317, 7.0156, 10.1735, 13.3237, 16.4706... ", "5655be78f4d14ddc76a1f4d3a41ca521": "\\mu(\\{x\\in X\\,:\\,\\,|f(x)|\\geq t\\}) \\leq {1\\over t^2} \\int_X |f|^2 \\, d\\mu.", "5655c08dbb63037207bbc019a9a9d7a0": "\\scriptstyle r(t) \\,\\dot \\theta(t)", "56566d9cc39feb48b1f6802d36f3a496": "\\frac{ \\mathbb{Z} }{ n \\mathbb{Z} }", "5656b8c107dd2d8e1ed629881a39e578": "P(R_{n+1})", "5656e10de5bb4928778517658146fa8e": "\\partial_t\\tilde{\\phi}(k,t)=-m((A + 3B\\phi_{in}^2)k^2 + \\kappa k^4)\\tilde{\\phi}(k,t)=R(k)\\tilde{\\phi}(k,t)\\;,", "5656f640bc33d8f986bbb74af68d59c0": "G \\!", "5656f7434b347940bcdc61c25f16a682": "x_B \\notin U_c", "56572e3dc10c69bda8d8136d7d14c9eb": "g(\\alpha) h(\\alpha) = \\left(\\alpha^5 + \\alpha^2\\right) \\left(\\alpha^3 + 1\\right) = \\alpha^8 + 2 \\alpha^5 + \\alpha^2 = \\left(\\alpha^7\\right) \\alpha + 2\\alpha^5 + \\alpha^2 = 2 \\alpha^5 + \\alpha^2 + 2\\alpha.", "565777c3380446a71505aac7eb3fa42b": "u(\\mathbf{r})", "56578a042b8f77fd4a8ce78f314910a6": "h_{ab;c}\\, =2g_{ab}\\psi_c+g_{ac}\\psi_b+g_{bc}\\psi_a", "56579b901552d9cb39b2e11b8abbc03e": "|U|\\times|V|", "5657b926d1c6741d3561b17da566e508": "\\mathcal{B}_s = ( \\lfloor ns \\rfloor)_{n\\geq 1}", "5657cf7333ac489cd533424ebf693487": " S_i = \\sqrt[q]{\\frac{1}{T_i} \\displaystyle\\sum_{j=1}^{T_i} {|X_j-A_i|^q} } ", "565846105a6e7eb0c53b9d0bb57ddcf2": "x=x(\\theta)", "5658543acde49744aa271e58d934a60f": "\\{\\delta \\vec x_{0i}\\}", "56586ce2cf1c63a50b4c457a6dc2557d": "B=\\left\\{\\,x\\in A : x\\not\\in f(x)\\,\\right\\}.", "5658703a87c47cb7f579c274d490afd1": "t = (x-x_k)/(x_{k+1}-x_k)", "5659e9ddebba7c320e8c074abce7a5b3": "\n\\mathcal M=-i\\sqrt{2\\omega_p}\\ \n\\int \\mathrm{d}^3x f_p(x)\\overleftrightarrow\\part_0\n\\langle \\beta\\ \\mathrm{out}|\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n\\varphi_{\\mathrm{in}}(x)-\n\\varphi_{\\mathrm{out}}(x)\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n|\\alpha\\ \\mathrm{in}\\rangle\n", "5659f2bebd67ad6b3261000c912a1765": "\\begin{vmatrix} \\Delta OD_{\\lambda1} \\\\ \\Delta OD_{\\lambda2}\\end{vmatrix} = \n\\begin{vmatrix} \\epsilon^{Hb}_{\\lambda1}d & \\epsilon^{HbO_2}_{\\lambda1}d \\\\ \\epsilon^{Hb}_{\\lambda2}d & \\epsilon^{HbO_2}_{\\lambda2}d \\end{vmatrix}\n\\begin{vmatrix} \\Delta [X]^{Hb}\\\\ \\Delta [X]^{HbO_2} \\end{vmatrix}", "5659f9b129b214a8b5368ff45a0f6f0b": "P(y\\mid x_1, \\ldots x_n)=\\frac{P(y, x_1, \\ldots x_n)}{P(x_1, \\ldots x_n)}.", "565a1ca076f2bde9140d5d36c03a9a4a": "\\tau(AB)=\\tau (BA)\\;.", "565a392311f278ce98f30593ba0b107f": "[L, I_k] \\subseteq I_{k+1}", "565a61af49ce07d3a29f08c33b2e22ad": " \\frac{ \\bigl(\\prod_{i=1}^n x_i\\bigr)^{1/n} }\n { \\bigl(\\prod_{i=1}^n (1-x_i)\\bigr)^{1/n} } \n \\le \n \\frac{ \\frac1n \\sum_{i=1}^n x_i }\n { \\frac1n \\sum_{i=1}^n (1-x_i) }\n", "565aa3997c380f34b3ed81d30610f574": "\\tfrac{q+i}{p}", "565aba3ce6c645ae2abd7322ed196abd": "\\mathcal{H}(\\mathbf{r},t) = \\varphi(\\mathbf{r},t)\\pi(\\mathbf{r},t) - \\mathcal{L}(\\mathbf{r},t)\\,.", "565b2628b0f140dd7a5c61f09b37ec3b": " U(t) = \\mathrm{T}\\exp\\left({-\\frac{i}{\\hbar} \\int_0^t H(t')\\, dt'}\\right),", "565b2e77a6cf733aba8774db4be4e99c": "(a,b) : R", "565b86bb547cab38e982114b07160a7b": "\\Sigma_a^F", "565b97cb76f4e1efa646754a184c8fea": "F = G \\frac{m_1 m_2}{r^2}=\\left(G \\frac{m_1}{r^2}\\right) m_2", "565b99674500b55cc52dad1866e6533a": "Y = h_{1}^{x_1}h_{2}^{x_2}", "565bbad4c16a0ae2ed3e2802d5337dc1": "\\mathcal{B}(X^*_{b(X^*, X)}, Y^*_{b(X^*, X)}; Z)", "565bddf24f12a2fa2ad58feb29483c55": "\\gamma=\\arccos\\left(\\frac{a^2+b^2-c^2}{2ab}\\right)", "565c00bcae5d6984df538dc0b86921ae": "\\alpha^{x_i}", "565c27341d44d27a621105bdf7a905a2": "\\mathbf{L}\\cdot\\mathbf{\\nabla}s(\\mathbf{x})=0", "565cf1475323acd149801a4afb373a27": "T =\\frac12 T_{\\lambda\n\\mu}^i dx^\\lambda\\wedge dx^\\mu\\otimes \\partial_i, \\qquad T_{\\lambda \\mu}^i = \\partial_\\lambda\\sigma_\\mu^i - \\partial_\\mu\\sigma_\\lambda^i + \\sigma_\\lambda^h\n\\Gamma_\\mu{}^i{}_h - \\sigma_\\mu^h \\Gamma_\\lambda{}^i{}_h, ", "565d0fd1976bc8fea244b526b9eae9b7": "Pr[ABr=Cr]\\leq 1/2", "565d3a9998a536e98c9904798c789b64": "er_k=\\frac{D}{\\beta_k}p_k > R(S)", "565d8fbdb36ac7cf07855a124cecf094": " \\varepsilon = \\frac{-\\pi^2}{720} \\, \\frac{\\hbar}{d^4} ", "565da2ddd426925ae3e778ce011e57b5": "-\\frac{d[A]}{dt} = k[A]^n", "565dc87839bdd60d1453222756bf4234": "a_i \\land\na_i=a_i", "565de06e6f5d75c5087812930703ec7b": "\\displaystyle\\{ \\gamma^\\mu, \\gamma^\\nu \\} = \\gamma^\\mu \\gamma^\\nu + \\gamma^\\nu \\gamma^\\mu = 2 \\eta^{\\mu \\nu} I_4. ", "565de37f9eb44e655373544fb8b864a9": "(A h)(x) = h(x) x - \\mathrm{trace}_{H} \\mathrm{D} h(x).", "565e6a3e6bc9e8381345927eca6bc156": "NP\\,\\!", "565e9140b306c3635c02cf5571c551d9": "\\mathbf{v}\\cdot\\mathbf{w}=d", "565e929f9505ff22d5d6208af3563d0e": "\\{\\theta_k,\\theta_l^\\dagger\\} = \\delta_{kl}, \\ \\ \\{\\theta_k, \\theta_l\\} = 0, \\ \\ \\{\\theta_k^\\dagger, \\theta_l^\\dagger\\} = 0. ", "565ece3c0636a045b3249e4c86933055": "\\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \\frac{R_aR_bR_c}{R_T}\\frac{R_T}{R_bR_c},", "565ef9a850daf4e02c03c6465dc4b2b6": "V=e\\int d^3x\\bar\\psi\\gamma^\\mu\\psi A_\\mu", "565f4df7fbd496bb5b3f094b4721bd55": " \\hat{f}'_{-}(k,0) = \\hat{f}'_{-}(k,0)+\\hat{f}'_{+}(k,0) = \\hat{f}'(k,0) = C(k)F'(k,0). ", "565f6159d5171acda26ab4b29e9c97e8": "\\,R_i", "565fc464f3d014d84e55890179bdfb16": "v_\\max", "565fd827573b72b964b604ea44d5e579": "I(X;Y|Z) = \\sum_{z\\in Z} \\sum_{y\\in Y} \\sum_{x\\in X}\n p_{X,Y,Z}(x,y,z) \\log \\frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.", "565fe1b42319ca19766fffececa94d67": " \\frac{1}{2m} \\left( \\frac{\\mathrm{d}S_{r}}{\\mathrm{d}r} \\right)^{2} + U_{r}(r) + \\frac{\\Gamma_{\\theta}}{2m r^{2}} = E ", "56604d8c9a6748dcbf40a0c9080b1b74": "\\theta = 2\\tan^{-1}\\left(e^{-\\eta}\\right).", "5660568efb6ab2eb02a1ee97dd59ba56": "(0,T]", "5660a1a480f9674cb2643fe2e0d0803b": "3) A \\rightarrow B : E_{KU_B}[N_A||ID_A] ", "5660ab9c74678c1e430cdf31d347558b": "(D_{L_{CO}})", "5660b345198354761f21d349a163c417": "V \\backslash M", "5660f3676945b9f502fb5ddafea7c9e7": " R_\\mathrm{ext} = V_\\mathrm{load}/I \\,\\!", "56611b8d347621492a9180c100aa6660": "\\displaystyle{T(F)T(G)=T(F\\star G),\\qquad T(F)^*= T(F^*),}", "56612bfdea3edeed39b02f452deae81b": "\\mathbf{p}=\\frac{\\mathbf{v}d}{|\\mathbf{v}|^2}", "56615d2035eb79a08a06e8a14541d027": " w \\in \\mathbb{M} ", "566162f3afaf9f5f67e7d7ca7a4b424e": "x=2", "5661b57bd4d3fe13736b6d02b43cdd10": "\nt(i)= 0, \\qquad \\qquad (i=M+1,\\dots,N). \\,\n", "5661f4478115f0cc3ffca517f5a5d5f6": "\na_{n}=a_{0}n^{\\alpha /(\\alpha -1)}. \n", "56620f44b066c09c401fc1b74d4224d0": " \\gamma = 1 ", "56620faa8e559ea10d13664b96e3df78": "\\alpha_k \\equiv \\gamma_0 \\gamma_k", "566237e790221c74c17d0c8ad5b6dd8e": " f(k;p) = \\begin{cases} p & \\text{if }k=1, \\\\[6pt]\n1-p & \\text {if }k=0.\\end{cases}", "566242b29a7e15c1f516a93f629e49a0": "F^\\mu = m A^\\mu.\\!", "5662489a163d8be5674e0384a9a52591": "A \\in V_{\\alpha+1}-V_{\\alpha}", "56624bcf5ca6697cece6b87e8ef4c9e6": " (~ | z_k| > 1 ~) ", "56626b12f53e3047994be18c527c23ea": "\\partial_z \\rightarrow \\partial_z +iA_z(z,\\overline{z})", "56626ca6526003ff28f85dd8f8918c43": " {\\mathbf{}}K_i = P_iC ^\\mathrm T _i(C_iP_iC ^\\mathrm T _i + W_i)^{-1}, ", "5662a3d66504e71e1e8e4c2138fd294a": "CF = (N,LS,SS)", "5662b0a742eb38da64b928dba9ddc563": "\n \\operatorname{E}[\\, \\hat\\beta \\,| X \\,] = \\beta, \\quad \\operatorname{E}[\\,s^2\\,|X\\,] = \\sigma^2.\n ", "5662f76696bc45b2d4fc1c95d640ef28": "a < \\Re(s) < b", "56630b25fff08d2839639b903576fcaf": " (h_1+c+b_1)\\rho_c = (c\\rho_c)+(b_1\\rho_m) ", "5663529ae4b80ef3396fb989042dc60c": "(Q,+)", "56637b70c824401b949897c7d6097030": "\\bigcup \\mathbf{M}", "56637bfdd7694e6fd054600e8da25973": " L^p", "56638793dfb9a0941e7b08acca3c6d4f": "\n \\bigsqcup_{i\\in I}A_i = A \\times I.\n ", "5663c6cd5937c33f3512ddc3e590198a": "u,v:i\\to j", "5663f30ae9a43fde593f4cd4b4de15d0": " P_o = \\sum_c \\frac{o_{cc}}{n}", "56640a0d6beac0c67531ca50c77ebf34": "\\Lambda = \\frac{l}{k}", "5664b0ab66d69cbcdee6d18809305bc6": " \\int f^2 \\, dP<\\infty", "56652e904e3594575619927c8edf7102": "Y_n(x) =\\frac{1}{\\pi} \\int_0^\\pi \\sin(x \\sin\\theta - n\\theta) \\, d\\theta - \\frac{1}{\\pi} \\int_0^\\infty \\left[ e^{n t} + (-1)^n e^{-n t} \\right] e^{-x \\sinh t} \\, dt. ", "566593ab60ab07c93116a2076ac16a5c": "v = v_\\parallel + v_\\perp", "5666d01a469a67d7721fae3fc5ef8091": "LR(157,2207) = \\tau\\left(157^{2206}\\right).", "56673e4f4afb4bb082ed3fb59bc0be71": "B^{s}=\\{x\\in E | -x \\in B\\}", "5667683d43d32b96d2fa65bfe5e59589": "H=\\text{ucp-}\\lim_{n\\rightarrow\\infty}H_n,", "56677db52f547b88b5db8f574c94f9ed": "\\textstyle \\frac{i \\pi}{2}", "5667b8963970dd3df02937a8efde7c6a": "(n, m, d, \\gamma, 1-\\varepsilon)\\,", "5667c5df0d90f25dd3f2b6120a59f266": "k^*_s-k", "5667d33cce1291608c76a9d9e12d5b76": "\\exists t_1,t_2,\\ldots,t_{|r|}", "5667eb3d8b9745f9edef26b0c2d76a7e": "\\mathbf{\\hat{v}}_1 = \\begin{pmatrix} 2\\\\1 \\end{pmatrix}. ", "5667f5bb1ae0ab6937f8c42b6a30a7a3": "p(drunk) = 0.001", "56683022575702b40c735fd9380f1754": " s_2 ", "56688a3172516f134b80c8dcf0300118": "d/q^t", "56694c47b18ba70a9328d3b0a8539cb8": "S(0;x)=x", "56694f2af5264a50b06909059929a403": "1_{10} = 0.\\overline9_{10}", "5669720d486109ddd7509ee3a6b0f455": "J_f({\\mathbf x_0}) {\\mathbf \\gamma}'(0)=0.", "56697d3bb8a3cbdd447a5f84270bf7f9": "\\theta_i = dy^{(i-1)} - y^{(i)}dx.", "5669b17ed9bd7e5f71cecf50b56b000f": " Two \\ PRF \\ Combination \\begin{cases} Pulse \\ Spacing \\ (Ambiguous \\ Range) = \\frac{1}{PRF} \\\\ \\\\ \\frac{1}{PRF_A} - \\frac{1}{PRF_B} = \\pm \\ Transmit \\ Pulse \\ Width \\end{cases}", "5669e9f6378410a36450c6f9ca44839b": "2\\sqrt{-\\frac{p}{3}}", "566a33bb61f512f8a4bada7649b59f95": " x^{(k+1)} = x^{(k)} - t \\nabla F(x^{(k)}) = x^{(k)} - t( Ax^{(k)} - b )", "566a94d4fa0cb972a67b59f9644d4a58": "1 \\leq PoS \\leq PoA", "566a99d6a0dd4930dd2ae84b2d05ce31": "5) \\ x+1=\\pm\\sqrt{3}", "566af09921fd5eb87472ac9a328e4f6d": "u = Ax.q\\,", "566af1c2a398f9016e6b8ed4a571170a": "(A)^{-k} b", "566af91f0a281bcd0fb6107d05b23bdc": "S_N(f;t_1,t_2)=\\sum_{|n_1|\\leq N,|n_2|\\leq N}\\widehat{f}(n_1,n_2)e^{i(n_1 t_1+n_2 t_2)}", "566b0cf241bce59ce9bfdec9fba86e49": " \\vec{s}(C_{0}^{(4)}) = [0,0,+1,-1], ", "566b175e677811a5703903e31d926b41": "\\psi \\to e^{\\frac{2\\pi in}N}\\psi\\;,", "566b39a034398d6dfa939864d00709a0": "x\\cdot (y \\backslash z) = x/y \\cdot z.", "566b541e6d626c149463e509d94df2e8": "\nr'_4(n)=\n\\left(\\tfrac12\\pi\\right)^2\\left(n+\\tfrac12\\right)\n\\left(\n\\frac{c_1(2n+1)}{1}+\n\\frac{c_3(2n+1)}{9}+\n\\frac{c_5(2n+1)}{25}+\n\\dots\n\\right)\n", "566b703d2e309eda8667ef5f4b2c58bf": "M_C = 10.186 \\ kN \\cdot m ", "566b85f27e5486bb63c0d15775fbd769": "f_{n}", "566bd1e4d5758aeb724c12fda3463317": "y^{\\prime\\prime}(s) = -\\frac{1}{\\alpha}\\sin\\frac{s}{\\alpha} \\ ; \\ x^{\\prime\\prime}(s) = -\\frac{1}{\\alpha}\\cos \\frac{s}{\\alpha} \\ . ", "566c3115b8bde623fee32a9e1bfe4c83": "\\eta_\\varepsilon(x) = \\frac{1}{\\varepsilon}\\ \\textrm{rect}\\left(\\frac{x}{\\varepsilon}\\right)=\n\\begin{cases}\n\\frac{1}{\\varepsilon},&-\\frac{\\varepsilon}{2} 0 \\, ", "566e0d3bd56e223647064e1b36eab7cf": "1 \\to \\mathit{SL}_n \\to \\mathit{GL}_n \\overset{\\operatorname{det}}{\\to} K^* \\to 1", "566e568d380fb5061c5d6e780e4c54fe": "{v_2}", "566e5d5004c1d97ca69a03edb92bcdc2": "^{12}", "566ee5e3b705a1f9b67b6bc587a06789": "S_{(p)}", "566f186e8ae916c62a4c6cdd1826a8db": "\\nu_{i',k}", "566f5690d68421b601f69ea3529d7b3d": "I = I_L -I_0 \\left( e^{qV/(mkT)} - 1 \\right) \\ , ", "566fa6ac38b3db1ee02bb976a208987e": "\\alpha\\,,", "566fbd24a8d835a56a3dbbbb2473573a": "\\phi(\\theta,\\tau)", "56703d22093e03c6c3dc28240671d41b": "{R_1}", "5670aafa9de36500923895c199c76a7e": "A(x)=(G*f)(x)=\\int\\limits_{\\mathbb R^n}\\! G(x-y)f(y)\\,dy", "5670d8e923759807e6cda1ab6cfa7e6b": "\\psi(\\bold{r}_1,...,\\bold{r}_j,...,\\bold{r}_k,...,\\bold{r_N})=-\\psi(\\bold{r}_1,...,\\bold{r}_k,...,\\bold{r}_j,...,\\bold{r}_N)", "5670dd306fd106e822b5462027e48f9d": "\\qquad q(a_{n-1}p^{n-1} + a_{n-2}qp^{n-2} + \\cdots + a_0q^{n-1}) = -a_np^n.", "5670e72e3f970e6d6752fd5865bd097b": "\n\\begin{align}\nx & = a(y, \\dot y, \\ldots, y^{(\\beta)}) \\\\ \nu & = b(y, \\dot y, \\ldots, y^{(\\beta + 1)})\n\\end{align}\n", "5670f49a000afcc5eaae72140a62d5e6": "\\mathbb{C} \\times \\mathbb{R} \\approx \\mathbb{R}^3", "5671210c72105189f6e42c971a1746bd": "\\mathbb{C} = \\mathbb{R}[i]/(i^2+1),", "56719d4b9068417460e5e5357d0e07a7": "[-25,7,-7,13,-13,25]^9.\\ ", "5671b51afd7bd075e8612d327dbe3d44": "\\mbox{net run rate }=\\frac{\\mbox{total runs scored}}{\\mbox{total overs faced}}-\\frac{\\mbox{total runs conceded }}{\\mbox{total overs bowled}} ", "5671fe68940ed5f5766b764648403d59": "\\mathbf{a} \\equiv a^\\alpha \\frac{\\partial}{\\partial x^\\alpha}.", "56722e97f9fc16b4cbeb3438ecce4c98": "N = 3,", "56724755c9c66963d0c92361ed7a8a24": "X \\geq Y", "56726a945f146691f96234d43e74bcaf": " H(x) = \\int_{-\\infty}^x { \\delta(s)} \\, \\mathrm{d}s ", "5672728a0488268e7e249e5438772e9e": "n_\\mathrm{obs}", "5672b31d07626d484144e1857ef025b9": "\\sigma_{\\text{isFriend = true} \\or \\text{isBusinessContact = true}}( \\text{addressBook} )", "5672d3723b7350dcf5a58fb772d900ab": "u_1 = \\begin{bmatrix}{\\ }1\\\\-\\mathbf{i}\\end{bmatrix}", "5672edfc5ac756541591e40edafbcc70": "\\mathbb{S} = \\{1,4,9,16,25,\\ldots\\}", "5672fbde012f6dfecfcc8abfd7106026": "\\dot{r}=\\frac{F}{2}\\dot{v}", "5673a874c776e8687c597a0ccbe7b89e": " \\|r_n\\| \\leq \\left( 1-\\frac{\\lambda_{\\mathrm{min}}^2(1/2(A^T + A))}{ \\lambda_{\\mathrm{max}}(A^T A)} \\right)^{n/2} \\|r_0\\|, ", "567429a0bed7d503dc9af3d8fea93a37": "\\scriptstyle 0, 1,\\ldots r-1", "5674a259f06acbcbe6e0f108f4d14060": "\\bar{\\sigma}=\\cfrac{1}{2}\\sqrt{2}[({\\sigma}_{11}-{\\sigma}_{22})^2+({\\sigma}_{22}-{\\sigma}_{33})^2+({\\sigma}_{33}-{\\sigma}_{11})^2]^{1/2}", "5674d45294c67ddb1fbea244bf99982a": "| \\mathbf s | = \\hbar/2", "5674f9ca658ea0979a9ea3d130769763": "\\displaystyle \\Delta f_s = 4s(1-s) f_s.", "56757652affcf847a48c94a6823a703e": " \\mathcal{E} = -N {{d\\Phi_B} \\over dt} ", "5675d2ebd1a329f8e5e8a09172b264ca": "\\displaystyle{(f,g)_0=((1/2-T_K)^{-1}Sf,g)}", "5675f65244655d6621269b5f9c7a55ec": "W=\\alpha M \\tilde{Q^c}\\tilde{Q}", "56762f0e9e60142b207787b3e81cfde5": "f'(\\gamma_2)= 1/k.\\,", "5676424aa6982ed74b6a10eadb93191a": "J = \\frac{4A^2t}{U}", "56765472680401499c79732468ba4340": "1.2", "567656ab0d9a28d210c233d258cd5518": "\\overline{R}^2", "5676873e6224da4da590a3c09cc20d1f": "\n\\mathrm{PL}(n)\\sim \\frac{ \\zeta(3)^{7/36}}{\\sqrt{12\\pi}}\\ \\left(\\frac{n}{2}\\right)^{-25/36} \\ \\exp\\left(3\\ \\zeta(3)^{1/3} \\left(\\frac{n}2\\right)^{2/3}+ \\zeta'(-1)\\right)\\ ,\n", "5676fb070ebc7c1792518bbc9a6ac98b": "x_1 = x_2 = \\cdots = x_{2^{k-1}}", "5677780e2d99aee44949c936b165b5cc": "Q(X,f(X),f(\\gamma X),\\cdots,f(\\gamma_{s-1}X))=0", "567796805842e36c9332e80289a77641": "l=\\nu + \\varpi\\,", "5677b77596fb22ecc1cd0b24879747e5": "\\Box \\bar{h}^{\\alpha \\beta} = -16\\pi \\tau^{\\alpha \\beta} \\,", "5677d5e99fbcb2fcc6a38f26e0f42998": "S(f \\circ g) = \\left( S(f)\\circ g\\right ) \\cdot(g')^2+S(g).", "5677efff9240d674bfe45ab684379075": " 2 \\le j \\le n-1 ", "567855a62c1a416b8f4276bb40d47805": "F_{Anchor} = \\frac{ F_{Load} }{ 2 \\cos( \\theta_V / 2 ) }", "56788a3d403ae6c179f43552bef6401f": "\\rho_0v", "5678c033d4acca219ef597eb99c5c1a1": "\\frac{{\\rm d}^2 x}{{\\rm d} t^2} = -\\omega^2 x ", "5678e99000d8069acd323bb63d6376d4": "A(T,V)=U-TS", "56795b1c17065becd97c8dfe1a113d58": "\\vec{X_p}", "567970cf514fc9d6d9c1424002c85920": "\\widehat{\\textbf{x}}_{k\\mid k}", "56797ed2bf2a2005047a6f826ddbdc62": "d\\theta/dt\\approx 1", "56798da4fd27a9c6a41dc4668c52994b": "(x,y,z)\\mapsto (x,-y,-z)", "5679baf5d99285529f154abb9e6a2d01": "m_0 = \\sum_{k=1}^N\\ m_k \\ . ", "5679f9d55979526c497bc384b69ff462": "x_{n+k+1}", "567a1254346b390fab56ed9d77aeb814": "T_{\\pi,\\lambda}", "567a2bcb0daeb2eb173e8ef8cc4e0fd3": "\n H^2(P, Q) = 1 - \\frac{2 (\\alpha \\beta)^{k/2}}{\\alpha^k + \\beta^k}.\n ", "567a3bc866cdde3ced7c3422c18856a0": " \\text{precision}=\\frac{|\\{\\text{relevant documents}\\}\\cap\\{\\text{retrieved documents}\\}|}{|\\{\\text{retrieved documents}\\}|} ", "567a6679a8d5cf5748a24f118bd230bb": "\\lambda(s)=f_s^\\prime(0)", "567a70ab347459dd103385ef4b3a4a21": "(0, 0, 0, \\dots, 1, -1).\\ ", "567a9fe13532f0f7b20a5fa087b9adf2": "[A]_{org} \\propto [A]_{aq}", "567b14f69526b66092d8ff118c615019": "BC \\ = \\ D_L\\sin\\phi \\approx \\phi D_L = v\\delta t \\sin \\theta \\Rightarrow \\phi D_L = v\\sin\\theta\\frac{\\delta t^\\prime}{1-\\beta\\cos\\theta}", "567b1e040cb75498d4bcff4d8fc4b472": "\n\\mathbf{\\bar{a}} = {\\mathbf{v} - \\mathbf{u} \\over t}\n", "567b458ec27021eace17e577c93585e6": "C^{\\infty}", "567b7b42818f4427ad2c629eb4deaa03": "\\beta \\cap C = \\varnothing", "567b7c42c98e895d7a92661830207178": "I_1,\\ldots,I_r,J_1,\\ldots,J_s", "567b86f49483b31437be7ec3c322450a": "\n s^2 = \\frac{\\hat\\varepsilon'\\hat\\varepsilon}{n-p} = \\frac{y'My}{n-p} = \\frac{S(\\hat\\beta)}{n-p},\\qquad\n \\hat\\sigma^2 = \\frac{n-p}{n}\\;s^2\n ", "567b9d20cb902c1df85fc9e0b5219a5d": "{T_{hot}}", "567c30e73dfb97527a244ee901a2296e": "(L_{n+1}, R_{n+1})", "567c44e747a779a5199bd50b492e975a": "F_{PE}", "567c996739edfa1cdbad4c55a80580df": "dn", "567d48c8b177c674119cf8e8dd820c65": "Z(t) =2 \\Re \\left(e^{i \\theta(t)}\n\\left(\\sum_{n=1}^\\infty \nQ\\left(\\frac{s}{2},\\pi i n^2 \\right) \n- \\frac{\\pi^{s/2} e^{\\pi i s/4}}\n{s \\Gamma\\left(\\frac{s}{2}\\right)}\n\\right)\\right)", "567d56b2e80cb89627c2cc1e15fa6ef0": "0 \\leq t_1 < t_2<\\cdots Z_\\rho (T)) > 0", "567f2991dd9d246f9442efca48662b50": "a\\uparrow^n b= ab", "567f3196553cc02740fec0b1c1a351e9": "d(\\lambda(n))", "567f689d2120776b5bdf0fe4aac77d4f": "0 \\to \\mathbb{Z}/2\\mathbb{Z} \\to \\mathbb{Z}/4\\mathbb{Z} \\to \\mathbb{Z}/2\\mathbb{Z} \\to 0,", "567f9e9ca5e9d2001d18d3b1ec97a995": "\\begin{pmatrix}\n1 & 1 & 0 & 0 & 1 & 0\\\\\n1 & 0 & 1 & 0 & 1 & 0\\\\\n0 & 1 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 0 & 1 & 1\\\\\n1 & 1 & 0 & 1 & 0 & 0\\\\\n0 & 0 & 0 & 1 & 0 & 0\\\\\n\\end{pmatrix}", "568025352b4908c229bb4226dca1c6b0": "M_1 \\equiv f(M_2)\\, \\bmod\\, N ", "568056230bb1cd6186f2aa13293c3bd0": " |f_k \\rangle = |f_k' \\rangle \\ ", "56805f0524a58709142f2da459a02346": "1.\\overline{6}", "568073609fca4312a46029692b29813a": "b\\in a", "5680bfda9cac96a115f84f09ca431c91": "d_n = \\sum_{i=0}^{n-1}f(y_i)", "5680ff39cd933b03ab68883a8cc8ad1a": "\\circ_0 : \\mathbf{C}(B,C)\\times\\mathbf{C}(A,B)\\to\\mathbf{C}(A,C)", "5681285b00d0b11a0dbafd99111f637c": "|f_1|+\\cdots+|f_n|\\ge\\delta", "56816e91cff5ef1eea8fd2273dea418a": "[H]_{eq} = [H]_o - [HG]_{eq}", "56816fb88cb9948db9003e51d9c15b79": " \\lim_{z\\rightarrow -\\infty} S(z)=1.\\ ", "5681824a223532c900f56c37dd6152b7": "a^{15} = x^3 \\times ([x^3]^2)^2 \\!", "5681e3e34ff11a2c47bb3cd4d4a99d8e": "\n{}_3Q_1 = mc_p \\left( {T_1 - T_3 } \\right)\n", "568216907ff8cc5b0a6f63e495a4b96f": "\\left[ \\Pi\\right] ", "568242c01a8402da7e311db37b5c001b": "\\left ( \\bold{J} + \\epsilon_0\\frac{\\partial \\bold{E}}{\\partial t} \\right ) \\cdot {\\rm d}\\bold{S}", "568271638c25a833aa6a90b6f875a884": "j \\in \\{1,\\ldots,n-1\\}", "568289faf5cb9d9a8750832b1a3ffac7": "\\frac{-1}{k_2(1+V_p/\\lambda)}", "5682ef8f613b5e929edc15bd2fa05e1c": "X=(x_2, x_3, ..., x_{k+1})", "5682ffc2a954c52739b6040ffdc274a1": "U_{t+1}", "568311de23667e3b0e19764560c0c777": "\\mathrm{Ad}_g[x,y] = [\\mathrm{Ad}_g x,\\mathrm{Ad}_g y]", "568354889c15201ffa848fa5a79003de": "H_1(2I;\\mathbf{Z})\\cong H_2(2I;\\mathbf{Z})\\cong 0.", "5683883eaf03bce4a7908b4b93311044": "E_{n_x,n_y} = \\frac{\\hbar^2 k_{n_x,n_y}^2}{2m}", "5683a46ad890ce329fd5b805af005d96": " X = \\sqrt{(\\Omega / 2 m)}\\, Y.", "5683ec517bc4a8598f11a37a0c9082ee": "\\operatorname{Cl}_{2}(2\\pi z) = 2\\pi \\log \\left( \\frac{G(1-z)}{G(z)} \\right) -2\\pi \\log \\Gamma(z)-2\\pi \\log \\left( \\frac{\\sin \\pi z}{ \\pi } \\right) ", "56840a5dcb683c8f42ec6563bc84154b": " sp(x := E, R)\\ =\\ \\exists y, x=E[x \\leftarrow y] \\wedge R[x \\leftarrow y]", "56843265e26cda576f4059860de02abd": "b_{t}", "5684d2b450ce594d6dce6549c8c73fd3": "I_{im-}(x', y')=2I_0[1-cos f (x, y) ]", "5684f66246af7be203e635ba7f0db682": "h_a", "568505ae7cd4dcda53f205f0dbbcdd91": "H(x) = \\int_{\\mathbf{R}}\\mathbf{1}_{(-\\infty,x]}(t)\\,\\delta\\{dt\\} = \\delta(-\\infty,x].", "568551131b49d9a67affd8382656cb30": "\\mu(A) = \\Pr(D \\in A).\\,", "56856274f4775b3eafa4ab6e43e4f6e4": "\\operatorname{rank}(f)_p = \\dim(\\operatorname{im}(T_p f)).", "56857f083321584f22363b221b389d81": "S^2\\times\\mathbb{R}", "56858291ad0c39e88110221ec93fa38e": "f = M_T / \\Sigma_m (V_m Z_m) ", "56858cd468b1b30dfa327f5334f8c649": "\n\\begin{matrix}\nx_0 & y_0 = [y_0] & & & \\\\\n & & [y_0,y_1] & & \\\\\nx_1 & y_1 = [y_1] & & [y_0,y_1,y_2] & \\\\\n & & [y_1,y_2] & & [y_0,y_1,y_2,y_3]\\\\\nx_2 & y_2 = [y_2] & & [y_1,y_2,y_3] & \\\\\n & & [y_2,y_3] & & \\\\\nx_3 & y_3 = [y_3] & & & \\\\\n\\end{matrix}\n", "5685ba77ca8399ac2b646f669ccc7e6e": " {d \\over dt} A(t) = {i \\over \\hbar } [H, A(t)] + e^{iHt / \\hbar} \\left(\\frac{\\partial A}{\\partial t}\\right)e^{-iHt / \\hbar} ,", "56861fbab3985f6b24329004cf4fc335": "\\liminf_{n \\to \\infty}a_n", "5686e177d1ff7b523751a7de360632f1": " = \\dfrac{5(\\sqrt{3} - 4)}{(\\sqrt{3} + 4)(\\sqrt{3} - 4)}\\,\\!", "568713ecab3f551f9b1933ba33e1fa6c": "p= x^2 + 3y^2 \\Leftrightarrow p\\equiv 1 \\pmod{3}.", "56877593d11ecdc4e3b925430f37b1cb": " y \\equiv \\sum_{i=1}^Nx_i , \\qquad P(\\theta | k, x_1, \\dots, x_N) = C(x_i) \\theta^{-N k-1} e^{-\\frac{y}{\\theta}}", "5687a068ae50c6caa22ecfb759decbd7": "\\|\\mu-\\nu\\| = |\\mu-\\nu|(X)=\\sup\\left\\{\\,\\left|\\mu(A)-\\nu(A)\\right| : A\\in \\Sigma\\,\\right\\}", "5687b048fe7e6509a3991b49c01aa34d": "1 - \\cos (\\theta)", "5687b287d27749ff9b71480bb64c6e8e": "{1 \\over 1}-{1 \\over 3}+{1 \\over 5}-{1 \\over 7}+{1 \\over 9}-{1 \\over 11}+\\cdots = {\\pi \\over 4}.", "5687b3cee8d03afd953c4be576aded12": "\\tilde{\\chi}_i^0", "5687c840bdd45525f9b3e8721ed2369c": "F(\\mathbf u,\\lambda)=0", "5687dd4e9f73ba3778a7f997ecdcb29b": "\\mathcal{O}_x", "5687dfb684b8fb5c6e8aec54105a31f7": "\\scriptstyle v \\;=", "5687fad378b6a44720d8015183dae98a": " f: \\mathbb{R}^N \\rightarrow [-\\infty,+\\infty]", "56880c8498c8aedafe07da2a7e9fab46": "\\theta:M \\to T^*M", "56883018de080c2872498e3dbbc89d13": " \\simeq 54.4 ", "568835f19f1e68847498cabc56711df8": "\\operatorname{tr}(AB)=\\operatorname{tr}(BA)", "56883cad0f8d704ac5dc908047699fa3": "K_1,\\ K_2,\\ K_3,\\ K_4", "56892919ac00ec37c2fe3d3a4e96ae76": "y=z", "568937fabda4a679ee59e9492c8a0f93": "\\displaystyle{\\alpha(z,v)=\\sup_g |g^\\prime(z)v|,}", "56896b13c909a2b9a4c59dce747ee5d6": " \\xi_y = v_xB_z", "56898537166109b3dd9ba1f3fb760c72": "d(z,m)^2\\ge \\tfrac12d(z,x)^2 + \\tfrac12d(z,y)^2 - \\tfrac14d(x,y)^2", "5689b7426f7d8397b18fa9ada3d71be5": "\\; I_i", "568a0848b9a4fe68f0b933c37086d722": "\\ \\dot{\\varepsilon} ", "568a0dd543dbbff261822ac84844ed5c": "\\exp \\left[\\sum_{k=2}^{\\infty}(-1)^k\\frac{\\zeta(k)}{k+1}z^{k+1} \\right] = \\prod_{k=1}^{\\infty}\\left(1+\\frac{z}{k}\\right)^k\\text{exp}\\left(\\frac{z^2}{2k}-z\\right)", "568a17ea832bc46ee1ed347e8c4d572e": "g_\\infty", "568a4b15784ed099919356da0d44e174": "2\\mathbb{Z}", "568a63722af49669f7b83e5ee184d8f8": "(a,0) \\circ (b,0)=(a + b,0)", "568aa7c8be28b2a1cbe9a137bd34b639": "C' = g^{r'} \\cdot {(g^x)}^{-1} ~\\bmod~ p", "568ad09d3310b73cfe6c1b40f210ec6b": "\\color{YellowGreen}\\text{YellowGreen}", "568ae8c490410ca4db2516b6b3d3c7fd": " e^{-i\\phi(x)} ", "568b1e8c9f3d362b983dc5990090bea5": "\\{0, 1, ..., 100\\}", "568b30de01fc481b7930b50175335235": "\\frac{1}{2}\\left(1 - \\{ \\frac{t}{2\\pi}\\}\\right)", "568b53374f13d0e1f176b394496e0d0c": "\\mathcal{T}(A)\\equiv_{mem} \\mathcal{T}(B)", "568c04796fea2446a9bd29d37f51e89b": "p^2-p+1=r_i^2+2r_ikq+k^2q^2-r_i-kq+1=r_i^2-r_i+1+q(2rk+k^2q-k)=q(2rk+k^2q-k)", "568c873218035fb5b41e1780a21b98e5": "A \\xrightarrow[+f]{} a", "568c9c4c57f44ba5fec3ee5ad5443c57": "((x_1,\\dots,x_n),(y_1,\\dots,y_n)):= \\sum_j x_jy_j", "568cc770cf9d971630494a9ee952fb62": "\\frac{1}{(2\\pi)^n}\\int_0^{2\\pi}\\cdots\\int_0^{2\\pi}\\prod_{1\\le j0.", "569b2751409b151864e1644fb738ad26": "\n{V_\\mathrm{out} \\over V_\\mathrm{in}} = {Z_\\mathrm{2} \\over Z_\\mathrm{1} + Z_\\mathrm{2}} = {{1 \\over j \\omega C} \\over {1 \\over j \\omega C} + R} = {1 \\over 1 + j \\omega \\ R C}\n", "569b3e3afc4820b1af8e2d9ce87943e3": "C_1 = (u^2+v^2)+(w^2+x^2)+(y^2+z^2)+... ", "569b5f9e6ed7fcbc959328f80e854cd0": " \\sigma = \\nabla \\mathbf{n} (\\nabla \\phi_U-\\nabla \\phi_L )", "569b9a3a1d653afea839109b12b810a2": "\\lambda \\in \\mathbb C\\text{ (or }\\mathbb R\\text{),}", "569cc589219f1e61715e2d46b525b84c": " {\\boldsymbol{M}_{k}} = \\left . \\frac{\\partial h}{\\partial \\boldsymbol{v} } \\right \\vert _{\\hat{\\boldsymbol{x}}_{k|k-1}} ", "569cf173c67e730b64c613288a9b66dd": "3\\frac{r^3}{a^3} \\approx \\frac{m}{M}. ", "569d324f84b13ce5d22ed53e70b3af2c": "\\Diamond\\top", "569d567871b2dc133c5285090525ec40": "\\sum_{n=0}^\\infty {b_n \\over n!}t^n", "569d6a7435d5ad28ffc198754d36f0c4": "10^-6/Rt", "569dcc6f55abb3d84acb63ae7ce5e9ce": "fem(t)= F_{Neel} i_{exc}^2(t) H(t)", "569eb61862ed3e863d03175f3220c9f0": " m = M - 5 (1-\\log_{10}{d}).\\!\\,", "569ec0f603037e6ebd4c569215756be9": " \\frac{\\frac{dV_g}{dT}}{\\frac{dC}{dT}} = \\frac{1}{F_i} = \\frac{V_g}{V_i} ", "569ef243ead826b439879a5f85d62ff3": "\n\t\\left. \\frac{\\partial x_r}{\\partial y_k} \\right|_{y=0} = \\sqrt{H_{rr}(0)} \\left[ \\delta_{r,\\, k} + \\sum_{j=r+1}^n \\delta_{j, \\, k} \\tilde{H}_{jr}(0) \\right]. \n", "569f8ac7d0701d95110164f56a588db2": "A f (x) = \\sum_{i} b_{i} (x) \\frac{\\partial f}{\\partial x_{i}} (x) + \\frac1{2} \\sum_{i, j} \\big( \\sigma \\sigma^{\\top} \\big)_{i, j} (x) \\frac{\\partial^{2} f}{\\partial x_{i}\\, \\partial x_{j}} (x).\\ ", "569f8ce1ff903ce1c9cd0555c49fc904": "f(ax)", "569fbc9bc80776ffb498e9f281c61b53": "\\frac{1 + z + {\\scriptstyle\\frac{1}{2}}z^2}{1}", "56a06d388c8ece03837ab8ce997df105": "t\\geq 0\\}", "56a07f11fd23b2d57317b882e2ca1d99": " \\mbox{Mat}_n = \\mbox{Skew}_n \\oplus \\mbox{Sym}_n , ", "56a0f0f25b69046fbbf40e997eaea401": "{\\Bbb E}(\\operatorname{cr}_H) = p^4 \\operatorname{cr}(G)", "56a111c3e65fd673e5c7036cfe083cd2": "v = \\frac{o}{(1 + o)}", "56a13923f42071a1e43ee81cac8d6c7d": "\n M_{11,11} + 2 M_{12,12} + M_{22,22} \n - q(x,t) = 2\\rho h\\ddot{w} - \\frac{2}{3}\\rho h^3\\left(\\ddot{w}_{,11}+\\ddot{w}_{,22} + \\ddot{w}_{,33}\\right) \\,.\n", "56a152c5894cfb7d4bf450093dd4b4f9": "\\mathbf{J} = \\rho \\mathbf{v} \\,\\!", "56a165d5ff1ec881fbd04acb1b3b4636": "Z=jX_L=j \\omega L\\,", "56a1bede1d1373a389b48498251d98e4": "WXY", "56a1ca9d51ab639738c11a875d75764d": "\nI\\equiv\n\\begin{bmatrix}\n1 & 0\\\\\n0 & 1\n\\end{bmatrix}\n,\\ X\\equiv\n\\begin{bmatrix}\n0 & 1\\\\\n1 & 0\n\\end{bmatrix}\n,\\ Y\\equiv\n\\begin{bmatrix}\n0 & -i\\\\\ni & 0\n\\end{bmatrix}\n,\\ Z\\equiv\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}\n.\n", "56a1e20fd8b68bf8e1f847947e5abd84": "H^1(X, F) \\cong \\mathrm{Ext}^1(\\mathcal{O}_X,F) ", "56a257f161fb43689f9bf9cd6238574a": " V=\\frac{\\partial V_i}{\\partial x_j}(x_0) ", "56a2757523a1591070fb2d196cac1627": "f_0 = \\pi\\tilde f_0\\,", "56a279ddad7339802e2f6c5f9defc7c5": "L_{x^m y^n}(x, y; t) = \\left( \\partial_{x^m y^n} L \\right)(x, y; t) .", "56a28e16bfbd47a48a76e83bda9ce8e8": "E_{tgu} = 0.5 \\cdot 140.367 / 4.54 = \\,", "56a2b4e8eb12573ea56d7b1a5066d5f6": " \\theta_0 = \\angle DOC, ", "56a2f780db278e1efcae06bc609c8de9": "P \\not\\in \\operatorname{Ass}(Q) \\subset \\operatorname{Ass}(Q')", "56a35a515647381199fc1ff5107626d0": "A = \\frac{4}{3} \\eta H^3", "56a37434ba990f5347c88595507f02fd": "\\forall\\ \\mathbf{X}, \\mathbf{Y}, \\mathbf{Z} \\ D_{**}^{(p)}(\\mathbf{X}, \\mathbf{Z}) \\le D_{**}^{(p)}(\\mathbf{X}, \\mathbf{Y}) + D_{**}^{(p)}(\\mathbf{Y}, \\mathbf{Z})", "56a37fd5856a4297cee572d8745af496": "\\dot m_{gen.}", "56a39357d7287a080c5a36df14d5d95e": "s=\\frac{\\pi r \\theta}{180},", "56a3bb3ecb27f33f189fa66bcb7854e3": "\n \\begin{align}\n q_1(x) & = \\int_{-b/2}^{b/2}q(x,y)\\,\\text{d}y ~,~~ q_2(x) = \\int_{-b/2}^{b/2}y\\,q(x,y)\\,\\text{d}y~,~~\n n_1(x) = \\int_{-b/2}^{b/2}n_x(x,y)\\,\\text{d}y \\\\\n n_2(x) & = \\int_{-b/2}^{b/2}y\\,n_x(x,y)\\,\\text{d}y ~,~~ n_3(x) = \\int_{-b/2}^{b/2}y^2\\,n_x(x,y)\\,\\text{d}y.\n \\end{align}\n", "56a3d0479229b4d145ecbb41130a8d48": "i,\\,j\\in\\{1,\\,2,\\,3,\\,4\\}", "56a3e2d2c8a48e167ae87ca513fe6f5c": " \\frac{d}{dt}\\langle \\mathbf{\\hat P}\\rangle = 0 ", "56a3fe856a4aec300d1dd114e4fc3cbc": "\\langle\\psi |P |\\psi\\rangle = \\Vert P |\\psi\\rangle\\Vert^2", "56a4085da36260ca19808fabc4c79e81": "\tH = h A \\vec I \\cdot\\vec J + \\mu_B (g_J\\vec J + g_I\\vec I ) \\cdot \\vec B ", "56a44844eb3736b325895098c2c6ade8": "\\left\\vert F(A) \\right\\vert \\leq \\left\\vert A \\right\\vert + \\left\\vert \\sigma \\right\\vert + \\aleph_0 \\,.", "56a54bca0833c21f306a9a7bf9781a52": "r_e = {\\alpha \\lambda_e \\over 2\\pi} = \\alpha^2 a_0", "56a58baff43a02241714afce9a7380cf": "{\\partial u \\over \\partial t} = a \\frac{\\partial^2 u}{\\partial x^2}", "56a5b44bdcb40b43cd791747eeabe448": "0\\rightarrow B\\rightarrow Y\\rightarrow A\\rightarrow 0", "56a5c2aaab032820bb620945d82367eb": "v_x = Q/A", "56a6608cc6f15101ef4bddb3a6901fb0": "P_r(n) = \\frac{n(n+1)(n+2)...(n+r-1)}{r!} = {n+r-1 \\choose r}", "56a68c3d92ea3beb52ba519097038ba3": "APC=\\frac{C}{Y}", "56a6a06750c25225bd057fd47c9f998b": "\\sqrt{S}", "56a6ab2b4051d2d20e9514f2a586bf85": "M=h(L_w+L_t)\\!", "56a6bea208877c8debe8b5eb9e338d71": "\\psi(\\Omega\\omega)", "56a6c52e278aa481b8c501cef00d3c76": "(p+q)^n=\\sum_{k=0}^\\infty {n \\choose k} p^k q^{n-k},", "56a6eab00050371425b41f3769df93a6": "a \\circ b", "56a71f2bd40a23c30bddb19468cb6b3f": "(n_i,k_i,d_i)_q", "56a73d8b2390e2b1296964b065d77899": "\\sigma _n", "56a758d9ce1d9179923db1df81a23ec2": "f,~g", "56a7b5bf54b0e168c9c0aff6ea81bdf0": "O(1/\\varepsilon^2)", "56a7fa075647874afe8c16297a46a55b": "(X, \\emptyset) \\hookrightarrow (E, \\emptyset) \\hookrightarrow (E, E \\setminus E_0),", "56a800258d9a0dbde9909696d5225398": "x = x_1 \\dots x_n", "56a8591c406f334ac02fefdc8116fbde": "\\vec F = M_4\\frac{\\mathrm{d}\\vec v_s}{\\mathrm{d}t}.", "56a89b75e861520711f66956b76daf3d": "\\mu\\in S(X)", "56a90c5aab30e1442960764f125a8f48": "S_{\\rm oblate} = 2\\pi a^2\\left(1+\\frac{1-e^2}{e}\\tanh^{-1}e\\right)\n\\quad\\mbox{where}\\quad e^2=1-\\frac{c^2}{a^2}\\quad(c \\theta_i>0, \\quad i=1,\\dots, k. ", "56e32710c7d9e62d2d961cdaf0f29634": " {-(g_\\mathrm{m1}r_\\mathrm{O1}+1) g_\\mathrm{m2} r_\\mathrm{O2}} ", "56e33dd24e35775c4549db5fa77cfaeb": "q_i = \\mathbf{v}\\cdot \\mathbf{e}_i = (q^j \\mathbf{e}_j)\\cdot \\mathbf{e}_i = (\\mathbf{e}_j\\cdot\\mathbf{e}_i) q^j \\, ", "56e399047b40a296074d1a6bbd3fb278": "\\mu:A_1\\otimes_{A_0}A_1 \\to A_0", "56e3d436394f79947410713ccfec0e9e": "=\\operatorname E\\left [ X_{ik} \\right ]a_{i0}+\\sum_{j=1, j\\neq k}^{m}a_{ij}\\operatorname E[X_{ik}X_{ij}]+a_{ik}\\operatorname E[X^2_{ik}]-\\operatorname E[X_{ik}m(\\vartheta)]=0", "56e3e9233339c6240008b0384575e08c": "\\theta=\\tfrac{3}{2}\\pi.", "56e4679e93780b238c8d90c7e19aed9a": " \\delta = \\varepsilon / 3 .", "56e476cd9a27d3e4f7e3febe15b3fb08": "w_2=(0,1,b,0)/\\sqrt{1+b^2}", "56e4926793604e7386e75825d1e6879d": "f(x) = 2\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{n} \\sin(nx).", "56e4ac4d4cb2e6939fb932eef2937d7d": "q\\geq0", "56e4b69ddd8f73419f5a5c18cdebf34b": "\n\\begin{align}\n \\mathbf{a}_1 &= \\langle\\mathbf{e}_1,\\mathbf{a}_1 \\rangle \\mathbf{e}_1 \\\\\n \\mathbf{a}_2 &= \\langle\\mathbf{e}_1,\\mathbf{a}_2 \\rangle \\mathbf{e}_1 \n + \\langle\\mathbf{e}_2,\\mathbf{a}_2 \\rangle \\mathbf{e}_2 \\\\\n \\mathbf{a}_3 &= \\langle\\mathbf{e}_1,\\mathbf{a}_3 \\rangle \\mathbf{e}_1 \n + \\langle\\mathbf{e}_2,\\mathbf{a}_3 \\rangle \\mathbf{e}_2 \n + \\langle\\mathbf{e}_3,\\mathbf{a}_3 \\rangle \\mathbf{e}_3 \\\\\n &\\vdots \\\\\n \\mathbf{a}_k &= \\sum_{j=1}^{k} \\langle \\mathbf{e}_j, \\mathbf{a}_k \\rangle \\mathbf{e}_j\n\\end{align}\n", "56e555a7fba0b8ac5b32b1c812c509b7": " g(x) < 0 \\,", "56e56008da71052f332552979729bf4e": " \\mathbf{A}_k ", "56e5702f6d73b48e871a299a95b00ee5": " \\begin{bmatrix} 1- \\lambda & 0\\\\ 1 & 3- \\lambda \\end{bmatrix} =0 ", "56e586010fe7b44d8214824bd75ac688": "\\varphi={d \\varphi \\over d z}=0,", "56e5bb3c7adc6052026b01bdbd141e80": "(p-1)! \\equiv -1\\pmod p \\,", "56e5c96a77bcf97bcf7111e2d583bb59": "Y=\\underset{=}{A}(X)=\\underset{=}{A}\\Bigl(\\bigcup_{k \\in \\mathbb{N}} kU\\Bigr) = \\bigcup_{k \\in \\mathbb{N}} \\underset{=}{A}(kU).", "56e62d77e65450a1e123b1e12e19de9f": "\\textstyle\\{{n\\atop x}\\}", "56e66cd4f5572a19508e62dfbd51042c": "\\mathbf{x}_j", "56e6b130eeae231310a8f199108a30c8": "\n \\sup_{\\theta\\in\\Theta} \\left\\| \\frac1n\\sum_{i=1}^n f(X_i,\\theta) - \\operatorname{E}[f(X,\\theta)] \\right\\| \\xrightarrow{\\mathrm{a.s.}} \\ 0.\n ", "56e742fb8c88993f361e6332bd6608b8": "[H,Q] = HQ - QH \\,", "56e78d1f9067e8a54ee1f109793f6778": "\\epsilon = \\frac {1}{\\left\\vert B \\right\\vert}", "56e7957c55b1726246d84cedad7f1a22": "\\scriptstyle A\\,\\geq\\,\\pi/2 + 2/\\pi\\,\\approx\\,2.207416099", "56e7d8efa19d185b4f727a5436daba16": "w_n = \\sin \\left[\\frac{\\pi}{2N} \\left(n+\\frac{1}{2}\\right) \\right]", "56e824b4b80db04599a95518be50d065": "\\frac{\\log{1000}}{\\log{5}}=4.29", "56e8324b7a4e29f4be8baa0da72d9203": "X=S^{4k+2}", "56e8720d7cb44b2b9f06c2133210ab24": "\\alpha\\in\\mathbb R^+", "56e8ec71b6f8868e9e339bc02cdf35ab": "\\mathbf{k_T}=k_T\\sin(\\theta_T)\\hat{x}+k_T\\cos(\\theta_T)\\hat{z}", "56e8fc49f34230cdeb3336f1b75dacb3": " \\sum_{A \\in \\mathcal{A}} \\frac{x(A)}{1-x(A)}.", "56e98b13a5cea053cd0c42132ad20618": "\\sec^2 A - \\tan^2 A = 1 \\ ", "56e99db17308c13a71cfc5da5a3165df": "i_t", "56e9a42f73c9b19fad9530f72bc70f7a": "n = \\dim(\\mathcal{F}) + 2 ", "56ea0e38fcf1404bc9fe91dfb9f24ea3": "\\hat H = -\\frac{1}{2} \\sum_{j=1}^{N} (J_x \\sigma_j^x \\sigma_{j+1}^x + J_y \\sigma_j^y \\sigma_{j+1}^y + J_z \\sigma_j^z \\sigma_{j+1}^z - h\\sigma_j^{z}) ", "56ea1a8a199ec8e81f62ba421e0afc93": "\\ (a,b)", "56ea2e0d0f937fe2375c9cf73e7e28c7": "\\omega_{n}(t)", "56ea448fec4e8e555a62255b46e0cf9b": "\\cos_k(i)\\equiv \\cos_k(-i). \\, ", "56ea87747e26b9ba5c7917aa225ba228": " x(U) = V'", "56eb3fef7f5735867bdb44dfd5f9d99d": "Area = \\frac{\\pi d^2}{4} \\approx 0{.}7854 \\cdot d^2. ", "56ebdda1ea2260295b2530953c6cd8d5": "bit(x,k)", "56ec0578166d96cc2cbdff96f5c7de9a": "s-a", "56ec1eb99a4de1f93b738d031030e015": "c^{ij}_k", "56ec31f15a2cd6100f0aeebd44578a33": "\\frac{d(C \\cdot V)}{dt}= -K \\cdot C + \\dot{m} \\qquad(7)", "56ec776b0c086a815d4b64561fb2fdda": "F = e n_i E = - \\frac{\\partial}{\\partial x}(m_e n_e v_e^2)", "56ece28df930bd4d153b6c74ba96c054": "b_{2}", "56ed1932685b91e5e3a8c43dddb1636a": "\\mathbf{v}(t)\\;", "56ed71ed409364218c5fba755af69cf5": "\n\\begin{align}\nz &= e^{sT} \\\\\n &= \\frac{e^{sT/2}}{e^{-sT/2}} \\\\\n &\\approx \\frac{1 + s T / 2}{1 - s T / 2}\n\\end{align}\n", "56edecf936a3cdf758fd6221bd26284d": "y[S]z", "56ee230ff8cc6b2f9b2645ae95f9edf8": "Y_{1}", "56ee68c17d1cef74bd85eb3774fdd5e8": "\\,\\ -\\sin x", "56ee8538fa029093ff0d28018422722e": "d-2", "56eea6e5983f418a4c3f86ad46d78efc": "d(x,x) = 0", "56eebb8f5e4bb47ee5f7ebe8b3f51521": " \\frac{1-p}{p\\,n}. \\,\\!", "56eef305fa2e0e29018caefe3d8bfb1b": "|a - b| \\le |a - c| + |c - b| ", "56ef1d598ded3fb2f0392e3efb0fde38": "(h^*)_k = \\overline{h_{-k}}", "56ef3db94a48c56391006029a5711dae": "r/s = r(1 - 1/r) = r - 1,", "56ef59f4df65b810f0994e66aa4dc9a3": "{\\rho v^2}/2", "56ef81e402ffc3ef3f696a0d0bdc8179": "f:V\\to W", "56ef8bbcbcde31b970db091baf813662": "r_i \\in R, m_i \\in M", "56f02649c0880b692d6b82ef5d056fd5": "b_{2}=(12/17)c_{1}", "56f0292a2c7b6968f6cbddb6ffb0a823": "\\mathrm{H}_2 + \\mathrm 1/2{O}_2 \\rightarrow \\mathrm{H}_2\\mathrm{O}", "56f061bb0a0e26f82173af119aec9073": "O_{/\\sim_{\\mathbb{F}_1}}", "56f0928a6c5833198b72e8706bffa892": " - \\frac12 \\int_V \\frac{\\partial}{\\partial x_k} \\left( \\sum_i u_i^2 \\right) \\,\\mathrm{d} V = -\\int_V \\sum_i \\frac{\\partial(u_iu_k)}{\\partial x_i} \\,\\mathrm{d} V = \\int_A u_k \\sum_i u_i n_i \\,\\mathrm{d} S.", "56f0af6a7e10af27cbb5cda75528b33c": "\\sgn \\tau\\,b_{1,\\tau(1)} \\cdots b_{n,\\tau(n)}", "56f0d49cd7ffc5b2f07b5342a5cae3e4": " ((A \\land B) \\to C) \\Leftrightarrow (A \\to (B \\to C)).", "56f0fef5da133b6591bfbef3770a433b": "\\left | \\vartriangle \\right | < 2^m < N^{ \\frac {1}{e^2}}", "56f109169172b56f4a45ee0ef9db7245": "X(f) = IFT[x(t)] = \\int_{-\\infty}^{\\infty} x(t)e^{j2 \\pi ft} \\, dt ", "56f1277e1e483b2e652a7f3a90072119": "\\{A_1, \\dots, A_m\\}", "56f134a7098ba27239271c9954e75daa": "\\neg (P \\rightarrow Q)", "56f1a69749ff1a8629e474abd88c8d3d": "\\varphi_1^2 + 4\\varphi_2 < 0", "56f1be7b036c6d84f687690fa1600384": " \\mathbb{A}", "56f203d7c03820a65e46739dc5997230": "charK\\ne2", "56f2499363b88e19dfc9b90969872d6c": "R_{\\theta} = \\left(\\begin{array}{cc}\\cos \\theta & -\\sin \\theta\\\\\\sin \\theta&\\cos \\theta\\end{array}\\right)", "56f27f206becfc7db59721ab2b25a936": "i \\neq k", "56f32ec2d05ab54d19a15b61358b7e42": "\\phi^k_n", "56f3314894bbe01d087b3ed9ab2e3d47": "{* (\\mathbf a \\wedge \\mathbf b )} = \\mathbf {a \\times b} \\,,\\quad {* (\\mathbf {a \\times b} )} = \\mathbf a \\wedge \\mathbf b . ", "56f34816d8e79eeb629a6a316c1e2c9f": "a(x-h)^2 + k.\\,\\!", "56f378f00dfee3eab0d42488bbcbf3e8": "f_{a}(5) = 4^{1^{1} 2^{0} 1^{1}} = 4^{1} = 4 \\in \\mathbb F_7 ", "56f38185cbd994f031f51deaa52fbafd": " W_q^{(\\infty)}\\thicksim \\exp\\left(\\frac{2\\alpha}{\\beta^2}\\right)", "56f39be2c27c0a756499fc1db302cfa9": "\\iota_S , S \\subset \\{0,1,...,p+q \\} ", "56f3a65f08c60c9807cbb507db0cd165": "K_{0}-K_{1}=E\\left(\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}-1\\right)", "56f3f4017ff8ec610a4b7a47ed655479": "\\begin{align}(\\boldsymbol{\\mu}_1^{(t+1)},\\sigma_1^{(t+1)})\n&= \\underset{\\boldsymbol{\\mu}_1,\\sigma_1} {\\operatorname{arg\\,max}}\\ Q(\\theta | \\theta^{(t)} ) \\\\\n&= \\underset{\\boldsymbol{\\mu}_1,\\sigma_1} {\\operatorname{arg\\,max}}\\ \\sum_{i=1}^n T_{1,i}^{(t)} \\left\\{ -\\tfrac{1}{2} \\log |\\sigma_1| -\\tfrac{1}{2}(\\mathbf{x}_i-\\boldsymbol{\\mu}_1)^\\top\\sigma_1^{-1} (\\mathbf{x}_i-\\boldsymbol{\\mu}_1) \\right\\}\n\\end{align}", "56f3f799841dd397fda232befcbd6f0b": " \\mathbf S = \\frac {1}{\\sqrt 2} \\begin{bmatrix} 0 & -i & -1 & 0 \\\\ -i & 0 & 0 & -1 \\\\ -1 & 0 & 0 & -i \\\\ 0 & -1 & -i & 0 \\end{bmatrix} ", "56f4ec62098d4078f9d52482e912c4bb": "\\Box A^\\nu - \\partial^\\nu (\\partial_\\mu A^\\mu) + m^2 A^\\nu = j^\\nu", "56f4f02f14ba669893ad6aa8efc5e1e8": "\\nabla \\times \\mathbf{E} = - \\frac{\\partial \\mathbf{B}} {\\partial t}. ", "56f578caa6f5df74287255d0510e0b99": "H_{(1)} \\ldots H_{(R-1)}", "56f5ae26d4ae468cabb11100fc50e59d": "b\\cdot d", "56f5c9950be91e26f16e0a288c2f3e0e": "\\mathcal{F}(|\\sigma|)", "56f6391f7e450b4c2b9ab3d500925370": "\\lim_{n\\to\\infty} \\mu (A \\cap T^{-n}B) = \\mu(A)\\mu(B)", "56f65ae84e25a105ff83f370ee166d65": "\\scriptstyle \\pi\\left\\langle\\rho^2\\right\\rangle", "56f669905ea0f7a6bf848b3d8c717262": " \\mathbf{a}\\times(\\mathbf{b}+\\mathbf{c}) = \\mathbf{e}_i\\varepsilon_{ijk} a_j ( b_k + c_k ) = \\mathbf{e}_i \\varepsilon_{ijk} a_j b_k + \\mathbf{e}_i \\varepsilon_{ijk} a_j c_k = \\mathbf{a}\\times\\mathbf{b} + \\mathbf{a}\\times\\mathbf{c}", "56f6bcbbfa6af655469e55a2988f83db": "\nZ \\sim \\frac{\\sum_{i=1}^k w_iZ_i}{\\sqrt{\\sum_{i=1}^k w_i^2}},\n", "56f7327f53eb1b2ecb21a4a9a88830c9": "\\mathrm{Frac}(R)", "56f77ae1ba0e1d14ec2e93e73b158ff9": "\\neg \\nu Z. \\neg \\phi [Z:=\\neg Z]", "56f78354509c5b0c1bdc4a616ba3164f": "f_E(D)=e^{-kD}\\,", "56f7c36a74faba16d05bca7c9211073b": "S^2 \\times S^3", "56f7d833756871f9053b39382b06633f": " \\begin{Bmatrix}u, v, \\phi \\end{Bmatrix} = \\begin{Bmatrix}\\hat u(y), \\hat v(y), \\hat \\phi(y) \\end{Bmatrix} e^{i(k x - \\omega t)} ", "56f7def4c5b781b1f8ea277c2cc215ea": "\n\\frac{\\partial\\psi}{\\partial t}\n+{\\bold u}\\cdot\\nabla\\psi\n+D\\nabla^2\\psi\n=0\n", "56f7dfed9e04e9cca349efab9bdcf13c": "\\text{ESR}", "56f7eeb59d528241f24910d9794dd9ad": "S\\rightarrow S/D", "56f80c0fa03bcfb4870083b12b7a6f2a": "W^{a}_{\\mu}", "56f87d989e6eb5317cf49b7e47fe4be9": "k_{B}T=\\frac{GM \\mu}{r}", "56f94f91e758780feb4c4868e512092c": "\\textstyle S_{11} = S_{22}", "56f95937acb91e4516a98e9208c19831": "(X/Y), (Y\\backslash X) \\in \\text{Tp}(\\text{Prim})", "56f972be82ca1d3b6237df555d114945": " \\operatorname{dom}(A^*) = \\operatorname{dom}(\\overline{A}) \\oplus N_{+} \\oplus N_{-} ", "56f9778804e9e9e6d9fc9882b107669f": "K = -V\\frac{dP}{dV},", "56f97eaadff0d6ebd9aed4ebdd9489e2": "3,744\\,", "56f985bee5397d212382b648c2659054": " \\rho_s ", "56f996df9019e84e9e39f73651fc0e31": "f(\\theta) = x \\left(\\hat\\theta(\\theta) \\right)", "56f9d235efe09c1e755303a7731cf9f4": "A^\\mu= \\sum_k a_k^\\mu(t)e^{i\\vec k\\vec r} +c.c.", "56f9f35dc6904249937486433be3dcc5": "y_n(x) = -(-x)^n \\left(\\frac{1}{x}\\frac{d}{dx}\\right)^n\\,\\frac{\\cos(x)}{x}.", "56fa10b1a85689b980f232673f28ceca": "I_d", "56fa12a53349814057b8b8e7178d757c": "\\begin{bmatrix}m&-v\\\\n&u\\end{bmatrix}\\begin{bmatrix}1\\\\0\\end{bmatrix} = \\begin{bmatrix}m\\\\n\\end{bmatrix}", "56fa1925fabff85fdd29b2dd507df8d8": "R_{tot} = R_D + R_{sd,self} + R_{sd,\\mathrm{He}} + R_{sd,\\mathrm{N_2}} ", "56fa4640145ee486d6436d232b11b0e9": "\\frac{1}{{49 \\choose 6}} = \\frac{1}{13,983,816}", "56fa893ba1a74fd4db5320d897807361": "\\sqrt[12]{2} = 2^{(\\frac{1}{12})}", "56fa907f2e063a950eff46d0ffd891f4": "\\exp\\!\\Big[\\; it\\mu - |c\\,t|^\\alpha\\,(1-i \\beta\\,\\mbox{sgn}(t)\\Phi) \\;\\Big],", "56fa992f036b03a7b1d44eeab7eafddf": "\nJ = \\frac{q_2 D N_c}{V_t} \\left[\\frac{2q}{E_s}( \\Phi_i - V_a) N_d\\right]^{1/2} e^{-\\Phi_B / V_t}(e^{V_a / V_t} - 1)\n", "56fae0a8550cad011b83ac7d36055045": "x=\\langle x_1,...,x_n\\rangle ", "56faebf20307e809af7aec92126586bf": "\\displaystyle (N,A_i,u_i)", "56fb3b84ea5999bf6eb15831dcb11cb3": " f(1,1)=0", "56fbd970b26fd2ee08a692a7e4bf4f38": " A_{n,k} = \\prod_{\\begin{smallmatrix} 1 \\le j \\le n \\\\ j \\neq k \\end{smallmatrix}} \\cot(a_k - a_j) ", "56fc00d9273e88858ebb90eb988e32d4": "2^{-n\\left[ H\\left( X\\right) +\\delta\\right] }\\Pi_{\\rho,\\delta}^{n} \n\\leq\\Pi_{\\rho,\\delta}^{n}\\rho^{\\otimes n}\\Pi_{\\rho,\\delta}^{n}\\leq2^{-n\\left[\nH\\left( X\\right) -\\delta\\right] }\\Pi_{\\rho,\\delta}^{n},", "56fc41a2abbc28dabdd9c39b3fe36f85": "\\mathbf{R}_1=\\frac{\\mathbf{X}_1\\mathbf{X}_1^T}{t_1}", "56fc6da30147a405c35ce86ffef27f2d": " |c_1|^2 + |c_2|^2 + |c_3|^2 + \\cdots < \\infty. \\, ", "56fc71f7c69c94d18f2521037047b78b": "\\ \\nabla_{h^*} \\mathcal{L} = s s^\\mathrm{H} h - \\lambda R_v h = 0 ", "56fca4c2e42b2681870b8d79b132ae20": "\\sum_{n=1}^\\infty \\frac{\\bar{H}_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\\zeta(\\bar{a},\\bar{b}) ", "56fcd0824b857bcddd6f76a3b1d4e441": "(-+-)\\,", "56fcd5fc72db80f3376458c9a716fac9": "J = \\frac{q D n e^{-\\Phi / V_t}\\big|_0^{x_d}}{\\int_0^{x_d} e^{-\\Phi / V_t}dx}", "56fcd8ad574e0a2d4beb0bba41e1a8ba": "\n\\langle \\varphi_{k'} | T_n | \\varphi_k \\rangle_{(\\mathbf{r})} = \\delta_{k'k} T_n.\n", "56fcdb33e791a0f898439f5c02674f4c": "H_0: \\theta \\leq \\theta_0", "56fd64bd4ed95e7f834c020c53b1b22e": "\ndA_1 = \\frac{1}{2} r^2 d\\theta_1\n", "56fe0e7a42c272075c3ae4b2bfc487d3": " t_k = \\omega_k \\sqrt[3]{-{q\\over 2}+ \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}} + \\omega_k^2 \\sqrt[3]{-{q\\over 2}- \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}}", "56fe7de4b3df2538a0ff8d627271191f": "\\rho \\in H \\otimes H", "56fe854d99695fb9268ad8fa1533e110": " \\varepsilon_0 =\\frac {1}{\\mu_0 c^2}", "56fec20a5d1887d074ab7c2bb25624dc": "\\bar{q}", "56fed743b7cc2f395a6e96b3260bb8bd": "\\cos \\theta \\sin \\varphi = {{\\sin(\\theta + \\varphi) - \\sin(\\theta - \\varphi)} \\over 2}", "56feda81055cf7212493a895fb0f74fa": "\\bar{y} = E\\{y\\}", "56ff4edfad09b6cce8f9392fd9bcef73": "PV = \\frac{C}{(1+i)^n} \\,", "56ffaa8f9e5077fa9957fc547ca56f4f": "\\frac{\\partial \\Phi(R) }{\\partial R}\\bigg|_{R=1} = - Z_0", "56ffae7e8d88ae8183a57e782db37b37": " d[x_m(i),x_m(j)] < r ", "56ffc28ee492e8333c980c5c5858c153": "\\dim_k\\operatorname{Hom}_k(A(\\lambda),A(\\mu))", "56ffdc90dbfcdf735546595c4471e583": "u_{11}", "56ffedfc22836697569d62ddbf779aef": "E = \\begin{pmatrix} 0 & 0 & 0\\\\0& C& 0\\\\ 0& 0& 0\\end{pmatrix}", "57000612c209f5766d0b804f90e45408": " A = - \\frac{8}{3} \\pi \\left \\langle \\boldsymbol{\\mu}_n \\cdot \\boldsymbol{\\mu}_e \\right \\rangle |\\Psi (0)|^2\\qquad \\mbox{(c.g.i)}", "57003dccbc9933ac17027085d43327db": "\\widehat{\\mathbf{p}} \\psi = -i\\hbar \\nabla \\psi ", "57009f31e690a1e0a1aa81baaed6d5ba": "\\det \\tilde{g} = \\det g (\\det F)^2. ", "5700a8d2ea03f44660023ab5a5f805ea": "g_{ab}g^{bc}=\\delta_a{}^c", "5700aca88d834ab2b94a663abbddc3d6": "\n\\left(\\frac{\\partial S}{\\partial p}\\right)_{T,\\{N_i\\}} =\n-\\left(\\frac{\\partial V}{\\partial T}\\right)_{p,\\{N_i\\}}\n", "5700d5b606685d79bc3b32dc22fc92a9": "\\mathcal{T}(M)", "57014155fd249d7e441cbc72b85634bd": "n=n_\\max", "57015c4a8fc79cdb293f8e00701e26be": "\\left\\{ \\frac{Q(s)}{1 - P(s)Q(s)}, Q(s)\\subset \\Omega \\right\\}", "570176489ccd94850a90228695fe63dd": "w_i n_i", "5701b36d9320b7f610220c1143c4743e": "P = \\{ P_i \\}", "57025578a150a96c481012657978c3cc": "\\frac{\\sqrt{6}-\\sqrt{2}}{4}", "57026e744f2029324ccf153ecf88e731": "dS_h\\,", "5702e0f9a744cb7c9478e79200cfd608": "\\mathcal{X}=\\ker(G)", "5703046ad1ec574671a00ba851bf304a": " (\\hat{x} \\psi)(x) = x\\psi(x) ", "5703084b908b253e2b3fe61a50aa1e40": " X_F = \\{G \\in X", "5703155395fa998b06376b2d5047ed01": "h_R = \\operatorname{Hom}(-,R) : C^{\\operatorname{op}} \\to \\mathbf{Sets}", "57031b48f0a030abc505132f43e950eb": "J = (1-\\tilde{r})A_0T^2\\exp\\left(\\frac{-\\phi}{kT}\\right)", "57031ce84bb894ff5ed7ca7104be98b1": "\\psi_1(\\Omega_2) = \\zeta_{\\Omega+1}", "57031de4685a1350255ae93c67c81eae": "y^2 +a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 \\ ", "57031faf5da42a76944dfe2218493b17": "(\\vec{s}_a + \\vec{s}_b)^2 = \\vec{s}_a^{\\;2} + \\vec{s}_b^{\\;2} + 2\\vec{s}_a \\cdot \\vec{s}_b ", "57035199102276dd4a050c07a05fa308": "A' \\subset A", "57040792f80843819614409d55d7dbbc": "\\text{sign}(a_n)", "57040e7adbb0da4fdbe612de57430854": "V = \\left(22+14\\sqrt{2}\\right) a^3 \\approx 41.7989899a^3.", "57043c155867f7ef00f8afe3624f7572": " \\sum_{i=1}^\\infty\\|x_i\\|^2 = \\left\\|\\sum_{i=1}^\\infty x_i\\right\\|^2, ", "57045a6a1df34074a889bfd9d838a85e": "Cl_r\\,\\!", "5704ade7e0b342196dc93074435fd918": "SNR_{norm}", "5704d07111f21bfb49673204afd6a893": "\n\\begin{align}\n\\int \\ln (x) \\,dx & = x \\ln (x) - \\int \\frac{x}{x} \\,dx \\\\\n& = x \\ln (x) - \\int 1 \\,dx \\\\\n& = x \\ln (x) - x + C\n\\end{align}\n", "5704fdd6532a0b93e6b7ef4cec506224": "\\langle A|H|A\\rangle=\\langle B|H|B\\rangle", "57052fbee9a860078c71a89c43ca529b": "E_r=\\frac{Z \\,I_0 \\delta \\ell}{2\\pi}\\left(\\frac{1}{r^2}-\\frac{i}{kr^3} \\right) e^{i(\\omega t-k\\,r)}\\,\\cos(\\theta)", "5705ebc4f99864170a1317a05a2da553": " \\rho \\ ", "5706202118207347966e522d838cfc00": "f(t) = \\int_{\\mathbb{R}} e^{-2 \\pi i \\xi t} d \\mu(\\xi).", "570664489e94a3ff3f45e321370a444d": "~~~\nD(\\omega)~=~\n\\frac{\\frac{1}{\\pi^2} \\frac{\\omega^2}{c^3}}\n{\\exp\\!\\left(\\frac{\\hbar\\omega}{k_{\\rm B}T}\\right)-1}\n~~~~~\n{\\rm (D2)}\n", "57066ce9fb65c418d0dd3921671cf031": "F_x'", "5706743f49aa7524c76d822b56fa16ae": "\\vec{R}_{cm}", "5706a636948f96b4a506b014cc9fd34d": "D={d\\over dx}", "5707067020d36ea1a47ee6c52291be4c": "\\boldsymbol{e}_i", "5707623b6f4b00a6a8934044fd0a9f09": "-9K_1/4", "5707dfd4846106d314748f8ae11f9ed5": "\\Pi^p_2", "5707eb24e7b2f7b03b02ca6715c55ab7": "H_{k+1}=H_{k}-\\frac{H_k y_k y_k^T H_k}{y_k^T H_k y_k}+\\frac{s_k s_k^T}{y_k^{T} s_k}.", "57080f9a52d504d8b4d285223ac65db5": "\\mu^2", "57084d8a6db17ca48d613034f10bd6dc": " G^\\mathrm{T} G m = G^\\mathrm{T} d \\, ", "5708703b264d9ee94daf9be86728b46b": "l=1,3...,n-2,n", "5708c71ac7519e333707cae006a18772": "\n\\mathbf{E}(\\mathbf{r},t) = q\\left(\\frac{\\mathbf{n}-\\boldsymbol{\\beta}}{\\gamma^2(1-\\boldsymbol{\\beta}\\cdot\\mathbf{n})^3 R^2}\\right)_{\\rm{ret}} + \\frac{q}{c}\\left(\\frac{\\mathbf{n}\\times[(\\mathbf{n}-\\boldsymbol{\\beta})\\times\\dot{\\boldsymbol{\\beta}}]}{(1-\\boldsymbol{\\beta}\\cdot\\mathbf{n})^3R}\\right)_{\\rm{ret}}\n", "5708d2f4000b2bb78a5557ef457fbd97": "\\frac{\\text{d} [{_2^1}P]}{\\text{d}t} = \\text{k}_{3(2)} (C_2 + C_3)", "5708f0a1cadb8918068c40f58d3e67a8": "\n\\phi^{\\prime} = \\left( 1 - G \\right) \\star \\phi .\n", "570925d9acdc1a28b035819a2b618a0b": "\\scriptstyle \\sum_p \\frac{1}{p}", "57094aeec7fbe2ea7bb3c4ba3b5f4a12": "x_2 = 1.000000014487979", "57095d90b108ed04af25fd34fcbdb630": "a^k \\equiv a^{k+\\lambda(n)} \\pmod n ", "57097772c07334c0cc1742062940f227": "y - y_1 = \\frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\\,", "5709d8009b3cfa7095bf591c9edb8891": "\\lambda(20)=4", "5709e8fbddf99a4f3fd2c27d6122e502": "\\left|\\alpha-\\frac{p}{q}\\right|>\\phi(q)", "570a651de6f87ca92541062fececbca6": "(f * g)[n]\\,", "570abe8f9ef811dedb171329283c1dff": "XB \\approx Y", "570b2163b99ea5905d32674f11217ad3": "\\phi(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{j} e^{i\\mathbf{k}_j\\cdot \\mathbf{r}} a_{j}. ", "570b5a8ab344292b944060c333036341": " n\\in \\mathbb N", "570b9d4abec87b882db6caa50f63eaf8": "b \\in k[X]", "570bb28c1913b82739b7fffd71d08d42": "n/e", "570bc46915d59bf7648d990329ccd562": "\n\\mathrm{var} \\left( \\hat{A}_2 \\right)\n=\n\\mathrm{var} \\left( \\frac{1}{N} \\sum_{n=0}^{N-1} x[n] \\right)\n\\overset{independence}{=}\n\\frac{1}{N^2} \\left[ \\sum_{n=0}^{N-1} \\mathrm{var} (x[n]) \\right]\n=\n\\frac{1}{N^2} \\left[ N \\sigma^2 \\right]\n=\n\\frac{\\sigma^2}{N}\n", "570c4ff80d84b040438f437b36bf2c23": " R_0 > 1 , I(0)> 0 \\Rightarrow \\lim_{t \\rightarrow +\\infty} \\left(S(t),I(t),R(t)\\right) = \\textrm{EE} = \\left(\\frac{N}{R_0},\\frac{\\mu}{\\beta}\\left(R_0-1\\right)N,\\frac{\\nu}{\\beta} \\left(R_0-1\\right)N\\right). ", "570c8b1767b2c7054dd31120158e01b9": "e_k(t)=A \\sin\\left((k-\\frac{1}{2})\\pi t\\right),\\qquad k\\geq 1", "570c90f0d1e64aadb576e71f9ef4a8fe": "(u|v)_E =(Bu|v)\\,", "570cb453cc04da86197a55f0d44cdcf9": "\\mu \\ge \\frac{L-b}{h},", "570cb894b0ed6645aef79e4a6173d570": "u'\\left(x_1\\right)=0", "570cdccf4e34385fd246bf8cc702c5eb": "\\rho _{0}\\left( q\\right) =\\frac{\\delta \\left( H\\left( q\\right) -E\\right) }{\\int dq^{\\prime }\\delta \\left( H\\left( q^{\\prime }\\right) -E\\right) },", "570d227f1b657d058fcf0b6b7de9f81e": "-2\\pi n /L", "570d50c1d07279417ed901a357e91b03": "\\epsilon_\\lambda E_{b \\lambda}(\\lambda,T)", "570d52ae13545c2b3dc8e1d2606078f7": "d^Tx>0", "570d5817eb22c6f188c86ecff0130502": "O(\\Delta t^2)", "570d6344cf214b8c1416226ee10b523a": "\\Psi_0=\\Psi_1=\\Psi_3=\\Psi_4=0\\,,\\quad \\Psi_2=-\\frac{(Mr-M)}{r^4}\\,,", "570d6e840301addf1bcb176071d17ec2": " A \\ast B = C~ . ", "570e276c0d0c71b0c257c47314d07aa8": "z = re^{i\\varphi} \\,", "570e55003d5b7cbfa69e5402a628116c": "\\Gamma(A) = \\{ [p] \\,|\\, p^* = p^2 = p \\in A \\} .", "570e627a5b7293ceb34c1217b76f42f6": "(x+c)^2=x^2+2xc+c^2", "570f1f52d590709d39af5bbb04b94d7e": "(x_{n})", "57101aaf23b0022dd0dac245a57fd7a9": "s\\left\\{\\begin{array}{l}p\\\\q\\\\q\\end{array}\\right\\}", "57112149885ca93a4c8120588126f2e3": "L_\\rho(\\gamma)=\\infty", "57112f503d5b326cc528b291f6a191a1": "F_6(x)=x^5+4x^3+3x \\,", "571169983911f2bd5497949cd0220e55": "\\alpha'\\colon A \\times_{C} B \\to B ", "5711ddd599699795d67e336d32452415": " \\zeta = \\sqrt{\\frac{1}{s} (s-a)(s-b)(s-c)} ", "5711edd77975a2afb016bc2a32b77b11": "\\mathbf{T}'(s) = k(s)\\mathbf{N}(s)", "571237b9664567e67a4bdde369f9cef3": " a \\centerdot s ", "571286850423e91db6d2802a4857bf03": "\\Phi ", "571298a81e90261111990d2d8aebb646": "\\text{Spec }L(R)", "5712f007470beef0c8eba49dd5f40d37": "d^3\\Pi_u \\Leftrightarrow a^3\\Pi_g ", "57130d631d5e28c8b9886fac21bfa9bf": "h_1, h_2, ...", "571350b91b817d74bda27758bd7bb6b4": "\\mathrm{Ker}(\\phi)\n\\subset \\pi", "571440e8f22c5b1a3de669654b77d181": "\\rho(z)=|f\\,'(z)|\\,\\rho^*\\bigl(f(z)\\bigr).", "571476abb72d67b2d8e9a182604c6807": "\\displaystyle{Pf(w)=Cf(w) +\\pi ^{-1} \\iint_{\\mathbf{C}} {f(z)\\over z} \\, dx dy = -{1\\over \\pi} \\iint_{\\mathbf{C}} f(z)\\left({1\\over z-w} -{1\\over z}\\right)\\, dx dy= -{1\\over \\pi}\\iint_{\\mathbf{C}} {f(z)w\\over z(z-w)}\\, dx dy.}", "5714c20070f9f342b6943394a23d1585": "\\kappa = \\pm3(x^2 + y^2)^{1/2}a^{-2} \\,", "5714d2bb395dbe784629b98d527dcc15": "\\|x+y\\|_p^2 + \\|x-y\\|_p^2= 2\\|x\\|_p^2 + 2\\|y\\|_p^2", "5714dbdbf76cab319668e6027f996651": "\\vec{c}_4(v)", "57150c503066f0f68452a4374ea1a8b7": "E_{t-1}(U_{t+\\tau})", "57153dc3b044102820274a29e7a6ef6e": "\n D^{\\mathrm{face}} = \\cfrac{f^3}{12}~C_{11}^{\\mathrm{face}}\n ", "57154cc3fe94ce8adec850d5e993f3a6": "=\\frac{\\text{kva base}}{\\text{0/1 X}* \\sqrt{3} *\\text{kv}_{L-L}}", "5715733e97599fe728af925dd0906a0a": "-90^\\circ - \\delta < \\phi < 90^\\circ + \\delta", "571585dc6d60612c0cb1d6cb62354f6d": "\\displaystyle \\frac{1}{\\sqrt{2 \\pi a^2}}\\cdot \\operatorname{sinc}\\left(\\frac{\\omega}{2\\pi a}\\right)", "57158877c651e39f9a4206a8d8db572c": " O", "57159b039b3052ac90ec7ed944fb84af": "|\\alpha|\\le m", "57159b410eaf719e8e3f70254041e685": "\\log_b(x^p) = p \\log_b (x) \\,", "5715e471d4718d5348734f36e7852cf6": "\\lambda \\in (0,1)", "57160846bcde097ecfc98ac3d7d0aed8": "[J_m,J_n] = i \\epsilon_{mnk} J_k ~,", "5716ce08864f13d59b1f06929f466a72": " k_{b_n}", "5716f43ae7135a872ced87a713c607e9": "n \\le 64", "571711368f449dd741aa3e9c0181c00a": "F_0 = S_0 e^{(r+u-y)T}", "5717b1b4fd7323906b0538cbb7ab8036": " u_{l,m, \\kappa , \\alpha} = \\sqrt{(m_\\kappa )} v_{l, \\kappa , \\alpha } (\\omega , q) e^{i[\\omega t - q x (l,m)]} ", "5717c2fb3dbb898deed327a7ea560fcd": "\\sum_{k=0}^{\\infty}k^n x^k = \\frac{\\sum_{m=0}^{n}A(n,m)x^{m+1}}{(1-x)^{n+1}}", "5717d5f791112ea8cc97f677c6f4a5c4": "\\scriptstyle x_1 \\cdot y_1 \\,+\\, x_1 \\cdot y_2 \\,+\\, x_3 \\cdot y_3 \\;=\\; x_1 \\cdot (-x_2 \\,-\\, x_3) \\,+\\, x_2 \\cdot (x_1 \\,-\\, x_3) \\,+\\, x_3 \\cdot (x_1 \\,+\\, x_2) \\;=\\; 0", "5717fb3763aa356c74baac1da292d3a9": "\\Delta k \\Lambda \\rightarrow \\pi", "57181008b1c3511a7bdbef8b61ba4855": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( X \\right) = \\left\\{ {\\langle \\psi_A \\otimes \\psi_B |} X {| \\psi_A\\otimes\\psi_B \\rangle} \n: {| \\psi_A \\rangle}\\in \\mathcal{H}_K, {| \\psi_B \\rangle}\\in \\mathcal{H}_M \\right\\},\n", "57182d46469b0a54fc7d6b49708e1414": "Y(z) \\sum_{p=0}^{N}z^{-p}\\alpha_{p} = X(z) \\sum_{q=0}^{M}z^{-q}\\beta_{q}", "5718314249fc23aae2bfd10ca7b76e5f": " \\|f(x)\\|' \\to + \\infty \\mbox{ as } \\|x\\| \\to +\\infty ", "571880863d1ad079eef70a01061ea961": " L\\phi_\\lambda = \\lambda\\phi_\\lambda ", "571888d486afa5ce4f962f00ba894a89": " \\begin{cases} 0 & \\textrm{if } u\\aleph_0", "57198917aa7f2e1e13fa1a5d3dec146c": "\\mathbf{L}\\cdot(\\mathbf{\\nabla}\\times\\mathbf{V}(\\mathbf{x}))=i\\nabla^2(\\mathbf{x}\\cdot\\mathbf{V}(\\mathbf{x}))-\\frac{i\\partial}{r\\partial r}(r^2\\mathbf{\\nabla}\\cdot\\mathbf{V}(\\mathbf{x}))", "5719b5352cc95d89087235ef093c0ea0": "w_{0}", "5719f5c7f5365a791788d1ede86baa9a": "f[\\phi(a)] = 0\\,", "5719fc7522d9414ce8047bef151737a2": "\\cos(\\alpha)", "571a017f06a7e6993eb386c695b3e08a": "K(y\\mid x)-K(x\\mid y)", "571a3bf5e198296c4097ab79053a7df7": "(M_1, T_1)", "571a65e8b093f756cc5e78ef981b2857": "T = r \\cdot s,", "571a66849d070fe714ef370e5261497f": "{\\Phi(x,y,p,t)}", "571a992b7fd6df7c2e1a123c66a66a1a": "\\prod(1-|\\alpha_j|^2) = \\exp\\big(\\int_0^{2\\pi}\\log(w(\\theta))d\\theta/2\\pi\\big)", "571ab29c8833da1e54226cfb8554fd52": "k^{th}", "571abf5777a41ba9dd8aa97013211a92": "(B y + \\beta)^n", "571add79afd5c3c1db3c538f17cc9038": " \\overline{i} = -i.", "571b2de40755d1dedfd225d3772b8ecf": "\\scriptstyle v = 1/\\sqrt{2} ", "571b6620b5e77e41004bed7b63079158": " A = \\frac{1}{\\sqrt{3}\\, \\sinh \\left( \\frac12 \\right)} \\approx 1.108, ", "571b77503379da3cba82dbedfc942745": " P[\\text{Suicide}|\\text{Not Protestant}]", "571bdbb439d2bb3d18f4c6c49df678c6": "g^{ab}. \\,", "571bde046abbbfa9d048e8f5e6186f6e": "H_\\Lambda(\\phi):=\\sum_{\\Delta\\cap\\Lambda\\not=\\emptyset} \\Phi_\\Delta(\\phi).", "571c01eb2d4659b9d875ef82e832f58f": " f(p,k) = \\left\\lfloor \\frac{k}{p} \\right\\rfloor + \\left\\lfloor \\frac{k}{p^2} \\right\\rfloor + \\cdots. ", "571c1f7f4f4eb2ad8ab0b0f6ee8dccf0": "\\sum_{j=1}^n a_{ij}x_j = y_i \\qquad\\mbox{ for } i=1,\\ldots,m", "571c5f87dbe106c0fe015ee304b97bd8": "c\\in\\mathcal{C}", "571c828ccd81db43e26d311122cc91a9": "\\delta v=\\frac{\\Omega d}{\\lambda} ", "571c9048f3dea9ef08956e7925766e6b": "n-k-1", "571ca3d7c7a5d375a429ff5a90bc5099": "\\cdot", "571ca76e45e2209e1a9c78841bf8b8e3": "(n-R_1)", "571cefd47535c3946062d3ec763859e6": "F = u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1}\\,", "571d2145f36b32a7030df989ca3ac4df": "([x:y:z],t)\\mapsto t", "571d2cf57d5a601ef90c20b50d3099c4": "\\alpha\\,\\!_{CET}", "571d3ff1920673a1021d5ed6ea0fdd5d": "n (n - 1)", "571d7dd12174b5c8bbcb4869bf4138d1": "[\\mu_a, \\mu_b] ", "571d88552250d633910592a13719ea0a": "S(E,V)=\\log \\oint \\sqrt{2m\\left( E-\\varphi \\left( x,V\\right) \\right) }\\,dx.", "571da674aa011f03845333f2e3859601": "\\mathbf{A}^* = \\begin{bmatrix} 3-i & 2+2i \\\\ 5 & -i \\\\ 2i & -7+13i\\end{bmatrix}", "571e197c043d7b525a57558bec10522c": "\\textstyle 1/V(s)=\\sum_{n = 0}^\\infty b[n] s^n", "571e19a3fb35a2f712cd608e89b85dc5": "0\\ ", "571e4dd89e0b4ea888410c3b878a6b74": "\\frac{L}{\\lambda _{0}}\\simeq \\frac{0.18}{\\frac{\\alpha }{k_{0}}}\\Rightarrow \\Delta \\theta \\propto \\frac{\\alpha }{k_{0}}", "571e7701bbffe80bc9289a7f41e00e74": "1/p+1/p'=1\\text{ and }1/q+1/q'=1.", "571f050001fede5fef453051be68be97": "f_2=0", "571f2dfa9a1abc7b3a34ba4acffd3f99": "K_0''=0", "571f31effe9f66224962380f9b533514": " \n\\frac{\\Delta E}{\\hbar\\omega } \\approx -\\alpha -0.015919622\\alpha^2, \n", "571f99913ed6d30cb7e2856421bda5c6": "w{}_{ij}", "571fac3dbc260e20762f7c0d49f4f7fb": "Z(\\lambda n_1,\\lambda n_2, \\cdots)=\\lambda Z(n_1,n_2,\\cdots).", "571fb7b1f0555e6962af0cf7bb6bbeb1": "\\chi^{(3)}", "571fc0c89707044e560dc3da5d6eb531": "(a, \\infty) = \\{ x \\mid a < x\\}", "571fca828a4c7cb6f243e689ccbcaa53": " \\frac{\\partial f}{\\partial t}+ v \\frac{\\partial f}{\\partial x}+ \\frac{F}{m} \\frac{\\partial f}{\\partial v} = \\frac{\\partial f}{\\partial t}\\left.{\\!\\!\\frac{}{}}\\right|_\\mathrm{collision} ", "572009b5d51e3f6f4a95a9514c8b09f1": "\\lambda X.\\lambda Y. X\\cap Y= \\emptyset", "5720607029b3244eadfe4554c19dec14": "r=\\frac{V_f - V_i}{V_i} ", "572083a9cfcfa49637d7a9212e34d017": "c_\\mathrm l\\,", "5720a0a48846ef79b7bd8fe5a0c198d3": " G=\\langle X|R\\rangle\\qquad (*)", "5720a3f2555714ede2c88c7f871b4c62": " \\mathbf{V}_{grid} = \\mathbf{I}_f \\times \\mathbf{Z}_{grid}", "5720ada7b875b3db4ca44155ce0200f4": "Y_{o}+Y_{\\varepsilon }\\frac{Y_{o}+jY_{\\varepsilon }tan(k_{x\\varepsilon }b)}{Y_{\\varepsilon }+jY_{o}tan(k_{x\\varepsilon }b)}=0 \\ \\ \\ \\ \\ \\ \\ \\ \\ (7)", "5720e01045b846d82f747b9cbf81a3da": "\nM_y =\n\\begin{bmatrix}\n0 & 0 & - {\\rm i} & 0 \\\\\n0 & 0 & 0 & - {\\rm i} \\\\\n{\\rm i} & 0 & 0 & 0 \\\\\n0 & {\\rm i} & 0 & 0 \n\\end{bmatrix}\n= {\\rm i} \\Omega\\,,\n", "5720f36e380316e5a584937f31574e27": "f(x)=\\frac{1}{x^2+1}.", "57210cf9b687bcee287c78463bde0e7c": "\\eta_{tot}<0.6", "572135be938d63266e7301f867834638": "y(t) = a\\sin(t),\\,", "572146917af2a987eda4ef43da42ab9d": "v_e \\approx 2.364 \\times 10^{-5} r \\sqrt \\rho.\\,", "5721db805cb4785d928f090a189d8b9c": " \\forall x(\\phi \\to \\psi) \\to \\exists x(\\phi) \\to \\psi ", "5721f22c43788b9edba12b85262d0efe": "f_{\\pi^{0}} = 130 \\pm 5~\\mbox{MeV}", "57224d651329d32c1a4dc6da2f16f7bc": " H(Y|X) = \n - \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^\\infty Q_{Y|X}(y|x) P_X (x) \\log_{2} (Q_{Y|X} (y|x))\\, dx\\, dy. ", "5722f6dd2c9687e57473fc95adedaac2": "\\sum_{n=-\\infty}^\\infty |c_n|^2 = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi |f(x)|^2 \\, dx,", "572301e0f4ea930f89d8f3fff793d7ae": "{C}_{\\mathrm{m}}", "572386e97e04fa560c248c4d03e70b57": "\\Delta f\\Delta t \\ge k", "5723d736b8e4248a71ea8265fe4c7f7e": "{{P}_{RAM}}=\\left[ \\text{ }\\!\\!\\Delta\\!\\!\\text{ },\\text{ }\\!\\!\\Gamma\\!\\!\\text{ },\\ldots \\text{ }\\!\\!\\Theta\\!\\!\\text{ } \\right]*\\text{ }Vector\\text{ }of\\text{ }Performance\\text{ }Counters+{{\\lambda }_{constantRAM}}", "572403115d92614225d4633ef7d21b9d": "0\\rightarrow H^3(K,\\mathbf{Q}/\\mathbf{Z}(2)) \\rightarrow H^3_{et}(P_K, \\mathbf{Q}/\\mathbf{Z}(2)) \\rightarrow \\mathbf{Q}/\\mathbf{Z}", "57240ce9472409e5433b917de0f23c94": "x_0,", "57245a1f9de50b9b9ef03abdc92d8d80": "v_2 = \\left[\n\\begin{matrix}\n 1 \\\\\n -\\varphi \\\\\n 1 \\\\\n\\end{matrix}\\right]", "57248fc8c7c74ac299b5921fca048b3b": "\\hat{\\vartheta}_N", "5724ad2e12d264eebcb82f3fd0acc3e8": "Q_x=\\frac{1}{V} \\sum_{q,j} {\\hslash \\omega \\left (\\left \\langle n \\right \\rangle-{ \\left \\langle n \\right \\rangle}^0 \\right)v_x}\\text{,}", "572596707ef4c7157c39365b6669dc35": "\n \\alpha_l = \\frac{n_l+n_k}{n_i+n_j+n_k}, \\qquad\n \\beta =\\frac{-n_k}{n_i+n_j+n_k}, \\qquad\n \\gamma = 0.\n", "572647658eab0aebc64e4d036428beb4": "c = m^2 \\, \\bmod \\, n", "57264a48d44c4e972ec38e0b6e457dd4": " \\forall r \\forall s \\forall t[r\\mu. \\end{cases}", "572d5cd46c2c9496b171c1c157fa0623": "\\{A_1,B_1,\\ldots,A_n,B_n\\}", "572e194819be6cbc93cd292253085f11": "\\int_{D} f(y) \\, G(x, \\mathrm{d} y) = \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau_{D}} f(X_{s}) \\, \\mathrm{d} s \\right]", "572e553a759a7d86c515c7869dbf48fa": "\\phi_n(x) = \\sqrt{w(x)} P_n(x)", "572e926c4d3012ce5960d6777d8e3c84": " v^{\\mu} ", "572e94824c05464eb3b3b80e121e5032": "\\vert R_i \\rangle_\\mathcal{R}", "572e9a46de398db453abcc1d6d170be0": "g^s h \\mod q= (-1)^{f_0}2^{f_1}3^{f_2}\\cdots p_r^{f_r}", "572ec2f5e5d60aa413b5c411463a4cdb": "F^i_{ab}", "572ed2c7471de82f245d849428ab214c": "\\mathrm{ROI}(t)", "572f251aa8dc98d741f14683a14369c3": "y_3= \\frac{y_1y_1-ax_1x_1}{1-dx_1x_1y_1y_1}=\\frac{y_1^2-ax_1^2}{2-ax_1^2-y_1^2}.", "572f554a8846a20a4d043bd808d9dd9a": "(x_j - x_{k+1})", "572f6b6a0c62cdca1220c4d66673c36a": "\\mathfrak{p}=\\mathfrak{g}", "572fb8d11a95090769008296a26cb823": "y_t=a_{1}y_{t-1}+\\varepsilon_t", "572fdb0dc12e1adc92751f0093ebb1d4": " \\sum_{n=1}^N \\sum_{d=1}^D \\int_0^T \\sigma_{n,d}^2(s)ds < \\infty ", "572fe87b654606439b77fe948b502490": "0 \\le 2|\\beta_j| <\\kappa", "57301d20140d336fa591f4a5803ad1f6": "\\underline{c},\\underline{c}+r\\underline{v}\\in B_r(\\underline{c}) \\subseteq (\\overline{\\underset{=}{A}(kU)})^\\circ \\subseteq \\overline{\\underset{=}{A}(kU)}", "57305c9191402b0bbcd07aa855a91c2a": "0 \\to d\\Omega^{k-1} \\,\\xrightarrow{\\mathrm{incl}}\\, \\Omega^k \\,\\xrightarrow{d}\\, d\\Omega^k\\to 0.", "5730adda7170c8c47bcda5d4f6fca7da": " {_*}x ", "5730c086641c30c55d85fd16eb4d109a": " \\begin{pmatrix} k & \\frac{d}{nk} \\\\ 0 & \\frac{1}{k} \\end{pmatrix} ", "573116a381c53ec56cf4274725c714bb": " p\\left(\\lambda\\right) := \\det\\left(\\mathbf{A} - \\lambda \\mathbf{I}\\right)= 0. \\!\\ ", "57313ef0dd71bb7fc85e820c31b75634": "H_z^{p}(z_1,z_2)=\\frac{A_z^{p}(z_1,z_2)}{B_z^{p}(z_1,z_2)}=\\frac{\\sum_{j=1}^N\\prod_{i j}A_z^{(j)}(z_1,z_2)B_z^{(j)}(z_1,z_2)}{\\prod_{i=1}^NB_z^{(i)}(z_1,z_2)}", "57314d00e34f05b7d74dae61fd9826ff": "u^R_{i - \\frac{1}{2}} = u_{i} - \\frac{\\phi \\left( r_{i} \\right)}{4} \\left[ \n\\left( 1 - \\kappa \\right) \\delta u_{i + \\frac{1}{2} } + \n\\left( 1 + \\kappa \\right) \\delta u_{i - \\frac{1}{2} } \n\\right].", "57315fa2a1f359a8e8e8a5f785643654": "r_2=\\left[\\left(1+\\frac{r_1}{n_1}\\right)^\\frac{n_1}{n_2}-1\\right]{n_2}", "573171fd7dbda18c83f1df3bbe72e23b": "(a_1, b_1) = (a_2, b_2)\\quad\\text{if and only if}\\quad a_1 = a_2\\text{ and }b_1 = b_2.\\!", "573184d93ca1fc1c233629caebc5e615": "\n \\left( B \\or C \\right) , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\vdash \\lnot A\n ", "5731aef8e505ec798ba7ff086b3da7bd": " A \\subseteq A^*.", "5731b5018324197f84533fd85731ec6d": "M[f] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\nf_0&f_1&f_2& \\cdots \\\\\nf_0^2&2f_0f_1&f_1^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "5731c4a813c7bfb2868adaed60052d72": "\\displaystyle f_s(x) = \\mathbb{E} f(x+W_s) = \\int f(x+y) \\frac1{\\sqrt{2\\pi s}} \\mathrm{e}^{-y^2/(2s)} . ", "5731ce5931e62d14e003f275107581da": "V={v_1,..., v_s}", "5731e0f0cb763bc7d965a678b75c4dba": "k=2,\\dots,n-1", "5731f508cebbcd9d6fc7d420bf946c9e": "P_r = {{P_t G_t A_r \\sigma F^4}\\over{{(4\\pi)}^2 R^4}}.", "57322f88a90b4fc9a5da1f5d01a8f6ea": "S_{i}", "573239e009b71663ac4dbbcfe81c7357": "\\theta_{(x,p)}=\\sum_{{\\mathfrak i}=1}^n p_idx^i.", "57324897581c48a00c8ceebe14efde34": "I_D= \\mu_n C_{ox}\\frac{W}{L} \\left( (V_{GS}-V_{th})V_{DS}-\\frac{V_{DS}^2}{2} \\right)", "57327680884e1fa242a47dc5224efde1": "\nU = \\int_0^t I_s V\\,dt\n= \\frac{\\Phi_0}{2\\pi} \\int_0^t I_s \\frac{d\\phi}{dt}\\,dt\n= \\frac{\\Phi_0}{2\\pi} \\int_0^\\phi I_c\\sin(\\phi) \\,d\\phi\n= \\frac{\\Phi_0 I_c}{2\\pi} (1-\\cos\\phi).\n", "57327ae22f1fbe725f8480e46bab1a3d": "c=\\omega/\\sqrt{\\mathbf{k}\\cdot\\mathbf{k}}\\,\\!", "57327e36ba10f385ebcf8501dfd72dbe": "S_i=\\sum_{n=1}^iX_n,\\quad i\\in{\\mathbb N}_0.", "5732a9f49021818bc65c0b549a947afc": " dS^2 = - dt^2 + \\cosh^2 t \\, dx^2 + R^2 \\, d\\Omega^2\\, ", "5732ae6779656cc4221fda78495cdbc4": "{\\rm Tr} (B^{1/2}A^{1/2}B^{1/2})^r\\leq {\\rm Tr} B^{r/2}A^{r/2}B^{r/2}.", "5732d2d3cf3e85d08916a5e9ac43677c": "\n\\textstyle d=1+\\left\\lfloor\\log_b{n\\choose k_1,k_2,k_3}\\right\\rfloor,\\ \\sum_{i=1}^3{k_i} = n,\\ \\left\\lfloor\\frac{n}{3}\\right\\rfloor \\le k_i \\le \\left\\lceil\\frac{n}{3}\\right\\rceil,\n", "5732d78efedc927ac0d505b0b839d142": "\\sin t", "5733424e2e4a3b2312879c4f25441055": "2(n-1)p", "57339b77b3af15427b7154f4daf8a223": "h_i", "5733a7aa188dbf7be5f71e18580c435b": "w(z,\\overline{z})=w(z),", "5733bfadd7116c456eb9129ae81c5ad9": "H_0 \\,", "5733eb0906c01b172a0c38a301dc7249": "I\\left(x,y,\\frac{dy}{dx},\\dots,\\frac{d^ky}{dx^k}\\right)", "5733fad73b195d8fed10a90406952628": "25% * 100 =", "5734561759ba07c2aa2be599ac413143": " I^\\kappa", "573484d18580a5b7edf3032f54d90f03": "c^2 = a^2 + b^2.\\,", "5734ee96f13d632296881ab4ea32d947": " f(a) \\leq g(a)", "5735330dcb8e7c7c6c31e55d9070a7b9": "\\begin{align}\n(v_1 + v_2) \\otimes w &= v_1 \\otimes w + v_2 \\otimes w;\\\\\nv \\otimes (w_1 + w_2) &= v \\otimes w_1 + v \\otimes w_2;\\\\\n cv \\otimes w &= v \\otimes cw = c(v \\otimes w).\n\\end{align}", "57356d8c007a26b5d0458de8678012b6": "\\frac{\\mbox{Expected Return}}{\\mbox{Value at Risk}}", "5735a86a733ad5d5f904c7e08be02b11": "\\psi = \\forall y (P) \\exists z ( \\Phi \\wedge [ F(y) \\vee \\neg F(z) ] )", "5735ad2d9e906de54aa0a10cb75fdead": "\\displaystyle{\\|C_h f\\|^2 \\le \\|f\\|^2,}", "57365ae22c781d1a2d697e5abba3ae87": "x = ay^2 + by + c \\,", "57365e4c7d86ce757e9328841f2d9f61": ":1{\\to}\\tau", "5736e311aa166904f99601cafc7f01bc": "s \\approx \\,", "5737c5af919d99942cfaf01096467a01": "\\le \\left\\lceil 2^{\\left(\\lg n - h\\right) - 1} \\right\\rceil = \\left\\lceil \\frac{2^{\\lg n}}{2^{h+1}}\\right\\rceil = \\left\\lceil\\frac{n}{2^{h+1}}\\right\\rceil", "5737d67cab8cd82aac46d958598e8510": " \\varphi_0( z ) = e^z; \\qquad \\varphi_1( z ) = \\frac{e^z - 1}{z}, \\qquad \\varphi_2(z) = \\frac{e^z - 1-z}{z^2}. ", "57388c39d280428b78d6dff773ab0941": "\\frac{1}{2^{n+1}}\\cdot\\delta", "573897dd1e3c5e2d560bd637e891648e": "e_c = i_c r_c + { {d \\varphi_c} \\over {dt}}", "5738a68dcfded6606ba188f9d9b0627d": "\\textstyle \\prod_{i=1}^N x_i", "5738e27b112f67a94bacd0344d81a326": "P(x)=x^3+2x^2-x-2=0\\,\\!,", "5739be8217c25474d197635c42791ae4": "Q_\\lambda=\\{ g\\in S_n : g \\text{ preserves each column of } \\lambda \\}.", "5739f0bde062b4a0457863df0fa3198d": "\n [\\mathsf{C}] = \\begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\\\\n c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\\\\nc_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\\\\nc_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\\\\nc_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\\\\nc_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212}\n \\end{bmatrix} \\equiv \\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\\\\nC_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\\\\nC_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\\\\nC_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\\\\nC_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\\\\nC_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \\end{bmatrix}\n ", "5739f95601d3308d9f7016588803ffe1": "{w}\\,", "573a015266cadd6dae0b7ca1f31b8fe9": "(x,y)\\mapsto(x,-y)", "573a132ce63e2c1bf49924d94d5f6fd3": " ds^2 = H(u,x,y) \\, du^2 + 2 \\, du \\, dv + dx^2 + dy^2", "573a1f4201ad06fbe1f63e491b4487f7": "e^{kx}", "573a2e63f6a7e2c4038edb1be6eaf938": "Y\\left( \\begin{bmatrix} p \\\\ q \\end{bmatrix},z\\right)\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} p \\\\ q \\end{bmatrix} + O(z)", "573a5369aaf0fdc0b54a574303bc7c89": " \\Phi_F = \\iint_S \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{A} \\,\\!", "573a664eddda94dadbf634b14cbec1a9": "\\displaystyle T=\\frac{\\sqrt{3}}{4}(abc)^{^{\\frac{2}{3}}}.", "573a7269cafc61d7b25116e2d72f2939": "h = r v\\cos{\\phi},\\,", "573a8520cf5cd0d9ccf2c5641404262f": "\\displaystyle P_{Y_r}(y) = \\sum_{s\\in S} P_{S_r}(s)W'(y|s) \\left[\\sum_{x\\in X} P(x)\\right]", "573b2ee121dba2a7c4672f922e2d7f1e": "T[i]=k", "573b3584cc02776a7d298219b53242e3": " \\begin{align}\n\\operatorname{nullity} \\, A &= \\operatorname{nullity} \\, H, \\\\\n\\operatorname{nullity} \\, B &= \\operatorname{nullity} \\, F, \\\\\n\\operatorname{nullity} \\, C &= \\operatorname{nullity} \\, G, \\\\\n\\operatorname{nullity} \\, D &= \\operatorname{nullity} \\, E. \n\\end{align} ", "573b89d7dbf890c3b37aac3abcc66ed2": "\\vdash_L", "573ba0599020f168a7282d53971ce705": "-6/7", "573be29ed68f81af7d43a97b71513f8a": "\\frac{p}{2^a}", "573c0e8a1b098fc1dc4c59bd155a0fd9": "\\rho v_0 h_0", "573c4949b116c395ff223a8ee3fbadd0": " \\widehat\\alpha, \\widehat\\beta \\,", "573cc23cd9ed38184498175cecd7d568": "\\scriptstyle a^2", "573cd89f183e393d3f833bb0b0e0cf92": "\n \\overset{\\diamond}{\\boldsymbol{\\sigma}} \n = \\boldsymbol{F}^{-T}\\cdot\\left[\\cfrac{d}{dt}\\left(\\boldsymbol{F}^T\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}\\right)\n \\right]\\cdot\\boldsymbol{F}^{-1}\n", "573cfb0d5740bf5d9812f08d39593885": "l(x^3)=3", "573d03ceca861a5eb4a38aa4d415077a": "T_{eq}=\\sqrt[4]{\\frac{L_\\odot (1-\\alpha)}{16 \\pi \\sigma R^2_\\mathrm{AU}}}", "573d1f55467cf430a6428d8a9c59d6d5": "\\begin{align} 2\\cdot R_*\n & = \\frac{(258\\cdot 9.56\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 530\\cdot R_{\\bigodot}\n\\end{align}", "573d39a5ba05c5d978ee9aed79e357f4": "G = \\sum_{v}{P^{+}(v)\\ln\\left({\\frac{P^{+}(v)}{P^{-}(v)}}\\right)}", "573d3b8721b7f7d6ec7bfa297098aac7": "\\mathbf{u}_k", "573db97ea7cca5be5d073fcf141a65fc": "|k-K|", "573dc3ea1896cc45f9a4d353b3a88dfb": "\\mathrm{Hom}_{C(\\mathcal A), n} (A, B) = \\Pi_{l \\in \\mathbf Z} \\mathrm{Hom}(A_l, B_{l+n})", "573dfc22663a1e11953e9bcf19a1aa8d": "S^2_{n} = \\frac{1}{n}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2", "573e176d09ef1fc31bf3533359f8f714": "u_{\\min}=\\min\\{u_i\\}", "573e981885e9c16af64df59cfe60fd67": "e^t = \\frac{(1-p)q}{(1-q)p}.", "573f0b57ed7d05de43d07b5af96f393e": "\\kappa : C_0 \\to C_0", "573f68e49bdbf5219d0555a385ffe793": "\\rho = R^2 \\quad", "573f7748cd3f161e791c9e19b69d2dd2": "E=\\sum_{j} \\frac{1}{2}(t_j-y_j)^2 \\,", "573ffdc2a6355f7fd9221d04c56acf83": "n_1\\sin\\theta_i = n_2\\sin\\theta_t", "57401a3383ec49104e4f9c3e85f07c4b": "s^2 t_0", "57402af7acc14c200aeb43e8001ecd36": "k^{q-2}\\bmod\\,q", "57403e3953159c6b322cca136221c9e6": "\\|I-\\omega A\\|<1", "5740424f6dd9a3c1b513a0f9dd5949a5": "\\lambda_k \\gneqq 0\\;\\;\\forall k", "5740565f8733f04f695e975f0c2c7c39": "\\int_{\\Omega^{\\prime}} L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, \\xi^{\\mu} \\right) d^{4}\\xi - \\int_{\\Omega} L \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) d^{4}x = 0", "5740d9331ab763dc872ddb955389b5ca": "t _1[\\alpha]=t _2[\\alpha]", "5740ff22c3f810dc565b89648f2a3d80": "R^{\\ast} \\cdot f_p \\cdot n_e \\cdot f_{\\ell} \\cdot f_i ", "574129a3130bd7e24d5c958015d07f32": " \\gamma_{xy} ", "574162208e449d336908069830566fc2": "I=\\frac{ML^2}{12}", "57416a9b5832bf661320b5e3468134c6": "u = U \\left(1 - erf \\left(- y / (4 \\nu t)^{1/2} \\right)\\right)", "5741a73d5b09e17045812f8aabff86a0": "i+3 \\rightarrow i", "574204f07a4490121cff54e33f04c10e": "\\phi(x_0) = u(x_0)", "57423a496b2e62b18531b14853c52abb": "\\mathbf{v} = a_1 \\mathbf{b}_{i_1} + a_2 \\mathbf{b}_{i_2} + \\cdots + a_n \\mathbf{b}_{i_n},", "5742430e3a789b2b8ee408df9a86e339": "\n p_\\sigma \\gets \\underbrace{(1-c_\\sigma)}_{\\!\\!\\!\\!\\!\\text{discount factor}\\!\\!\\!\\!\\!}\\, p_\\sigma \n + \\overbrace{\\sqrt{1 - (1-c_\\sigma)^2}}^{\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\text{complements for discounted variance}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!} \\underbrace{\\sqrt{\\mu_w} \n \\,C_k^{\\;-1/2} \\, \\frac{\\overbrace{m_{k+1} - m_k}^{\\!\\!\\!\\text{displacement of}\\; m\\!\\!\\!}}{\\sigma_k}}_{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{distributed as}\\; \\mathcal{N}(0,I)\\;\\text{under neutral selection}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!}\n ", "5742d0f885cee9a1a92390f78087bcd6": "\\, a ", "5742df32a3f7c46d524629c4d22a06bd": "\\mathbf{v}_B", "574319fa8b06c3552f3cfee105393fc2": " f(\\mathbf{X} ; \\nu,\\mathbf{M},\\boldsymbol\\Sigma, \\boldsymbol\\Omega) = K\n\\times \\left|\\mathbf{I}_n + \\boldsymbol\\Sigma^{-1}(\\mathbf{X} - \\mathbf{M})\\boldsymbol\\Omega^{-1}(\\mathbf{X}-\\mathbf{M})^{\\rm T}\\right|^{-\\frac{\\nu+n+p-1}{2}},\n", "57431ce0e5a435b18a66f0f175113a04": " \\left(\\frac{M}{L^3}\\right)", "574336ae0039f1f94ad5ba643d344f89": "\\sum_{i=1}^np_ix_i=W.", "574351e6a5805679163b14d178b904a9": "B=(b_1,b_2,\\ldots ,b_k)", "5743983cf9594ef6f470c9fdf80698a4": "\nd_r=\\frac{|x-y|}{\\left(\\frac{|x+y|}{2}\\right)}\\, .\n", "5743a2580bed28ac9fa521299509f17a": "j^{\\star} = \\sigma T^4", "5744397ee7d693328eb7705d9ce55f98": " \\Delta G^\\circ_{form} = ( \\Delta A - \\Delta F' )T - \\Delta A ( T \\ln T ) - \\Delta B ( T^2 ) + \\textstyle \\frac {1}{2} \\Delta C ( T^{-1} ) + 2 \\Delta D ( T^{ \\textstyle \\frac {1}{2} } ) ", "5744893df5d91896c2c03365ceae2b70": "rP_0=M_a(1-e^{-rT})\\,", "5745205bd569a97be497855f362ec79d": "\\ P(q) = C(n,q) \\cdot (1/6)^q \\cdot (5/6)^{n-q}", "57457c163eaa69c0519e3405c4a5832d": " \\boldsymbol{\\omega} = \\boldsymbol{\\omega}_\\mathbf{T} + \\boldsymbol{\\omega}_\\mathbf{N} + \\boldsymbol{\\omega}_\\mathbf{B}. ", "5745805716c8450db34d0bdad9f8b769": " \\mathbf{F} = m\\mathbf{a},\\quad \\mathbf{T}=[I_R]\\alpha + \\omega\\times[I_R]\\omega,", "574582d2dea8fa23f2f0d132f636ffee": " f_X(\\mathbf{x}|\\boldsymbol{\\theta}) = h(\\mathbf{x}) \\exp(\\boldsymbol\\theta^\\top \\mathbf{x} - A(\\boldsymbol\\theta)) \\,\\! ,", "574589f1df3bd1ef27a8d459102aa026": ": \\Omega \\to \\mathbb{R}", "57458b09631c2085aa1ea720a18a3bfd": "f_k: I \\to \\mathbb R", "57460745e3c8a7b6b51baf79bc6975b2": "\\boldsymbol{\\ddot{\\rho}}", "57466bd7f7684b8de0b8e1a571b7fced": " - n^2 u[-n - 1] \\,", "5746c395dc5613824440df52440a9506": " \\phi + 0.5", "5746ceb1177a96c521f351d91ec7baf8": "[f,a_1\\cdots a_m] = -i_{df}(a_1 \\cdots a_m)", "5746ec0d6f3083007bfa5cf7dda8cfbf": " X^{(3)}", "574705be2dfc269fe52cd6ecc9de1205": "A \\in Fin", "57472204b99b131fb183b88e51ade691": "P_n^{(\\alpha, \\beta)} (-1) = (-1)^n { n+\\beta\\choose n}.", "574785fb852c2a2b14f92e9c3bc6a368": "\\forall y \\exists x, \\mbox{saw}(x,y)", "574822f5d549b250eb4b9e2d31ae3426": " \\hat{S} \\equiv |R\\rangle \\langle R | - |L\\rangle \\langle L | = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix} ", "574841c27393c235fc427fc39d395001": "\n\\mathbf{u} = \\frac{1}{S_0}\\begin{bmatrix} S_1\\\\S_2\\\\S_3\\end{bmatrix}.\n", "574864b9a5dcd6fdadb60ee5baa729fd": "CD = { gauge_{se}\n\\over { \\left (1 + {R^2 g^2\n \\over {V_{act}^4} } \\right ) ^ \\tfrac{1}{2} } } - super\\_el", "57489985d7c8a83e9ef3a77fcd520991": " V(t)-V(\\underline{t})=\\int_{0}^{t}y^{\\ast }(s)ds. ", "5748c30df1d3e86ff452a0862103aaf8": "\\langle \\alpha', j'm'|T_{q\\pm 1}^{(k)}|\\alpha, jm\\rangle=\\text{(proportionality constant)}\\langle jk; mq\\pm 1|jk;j'm'\\rangle", "5748d8479c1e99ed54a675ccec8223f8": " \\!\\ K_n = \\frac{1}{\\sqrt{m^2 + 4}} {(S_m^{n+1} - {(m-S_m)}^{n+1})}. ", "574915f07faea3b42c3086270e42fc81": "L(1) = -\\frac{\\pi}{\\sqrt q}\\sum_{n=1}^{q-1} \\frac{n}{q} \\left(\\frac{n}{q}\\right) > 0.", "57497b774da248e2a8fc77395bfd60cd": "\\frac1{\\sqrt2}\\prod_{p\\equiv3\\!\\!\\!\\!\\!\\mod \\! 4}\\!\\! \\underset{\\!\\!\\!\\!\\!\\!\\!\\! p: \\text{ prime}}{\\left(1-\\frac1{p^2}\\right)^{-\\frac{1}{2}}}\\!\\!=\\frac\\pi4\\prod_{p\\equiv1\\!\\!\\!\\!\\!\\mod \\!4}\\!\\! \\underset{\\!\\!\\!\\! p: \\text{ prime}}{\\left(1-\\frac1{p^2}\\right)^\\frac{1}{2}}", "57498f370dd9896f056d4fafe327a862": "P(B)", "57499528b7326573dae34f03bc45257e": "\\frac {du/dt}{u} = g_u=\\rho v-\\gamma-\\alpha", "57499efe8fce30036d880e3472610c25": "70x^4-140x^3+90x^2-20x+1", "574a4a4db3f7a45fa3bc7201613205f7": "2 \\uparrow 2 + 2", "574a58eb1d0a72ac4292710633863ad3": "T = \\frac{\\sum_{i=1}^t c_i \\bar{Y}_i}{\\sqrt{\\sum_{i=1}^t \\text{MSE } c_i^2/n_i }} \\sim \\text{noncentral } t(N-t, b\\lambda) ", "574a7479df016b1656bfba2108fa4d00": "1- c_1x^1 - c_2 x^2 - \\cdots - c_dx^d.", "574ae9d923ab2d385ed45511d79d932c": "\\forall\\omega\\in\\Omega\\;N_{\\bullet}(\\omega)", "574af00d5515f8acfd49b6f59492f7b9": "\\frac{K}{P\\left(K-P\\right)}=\\frac{1}{P}+\\frac{1}{K-P}", "574b1a9abb2275c073a7050477a58ff7": "\\left| \\mathbf{A} \\right| = A", "574baa46d319acf4282807347c0acc62": "q_1^{a_1}\\,q_2^{a_2}\\cdots q_n^{a_n}. \\,", "574bc3ab951fe9febadc0e3fec18ae7d": " G' = \\frac {\\sigma_0} {\\varepsilon_0} \\cos \\delta ", "574c157a5cf02556e8384cf75a4648a0": "\\ddot {\\bar{r}} = \\operatorname{\\bar{F}}(\\bar{r},\\dot {\\bar{r}},t)", "574c35ebe35f85527febbd005bf9a9ea": "\n\\begin{cases}\nu(\\gamma z)=u(z), \\ \\ \\forall \\gamma \\in \\Gamma \\\\\ny^2 \\left (u_{xx} + u_{yy} \\right) + \\mu_{n} u = 0.\n\\end{cases}\n", "574c525153d601bf9fccf1293faf17b4": " A^\\mu", "574c8b715b234a3f1846caa1957ecad0": "T = 1 / f", "574cfe6fe9236096d13f95f1282d2de0": "f_n(x)", "574d7f7315578a32fb3b845c6c0bcc71": " (p,q) \\in \\mathbb{Z}_d \\times \\mathbb{Z}_d \\rightarrow W(p,q) ", "574d9a38f995d0b5952ad6687f9786aa": "\\hat{N}_c \\equiv \\sum_{n=1}^{N}a_n^{\\dagger}a_n", "574e067239efd3e63afcd7aea4b1e15d": "\n \\begin{align}\n -\\cfrac{2h^3E}{3(1-\\nu^2)}& \\left(w_{,1111} + 2w_{,1212} + w_{,2222}\\right) = \\\\\n & q(x,t) + 2\\rho h\\ddot{w} - \\frac{2}{3}\\rho h^3\\left(\\ddot{w}_{,11}+\\ddot{w}_{,22} + \\ddot{w}_{,33}\\right) \\,.\n \\end{align}\n", "574e9393b22800f4732d9f00d354f5e2": "= \\sum_{\\boldsymbol{R_n}} b^* ( \\boldsymbol{R_n})\\sum_{\\boldsymbol{R_{\\ell}}} b ( \\boldsymbol{R_{\\ell}})\\int d^3 r \\ \\varphi^* (\\boldsymbol{r-R_n}) \\varphi (\\boldsymbol{r-R_{\\ell}})", "574eb3831d10a9b8d8a6029d6b099b36": "j=k", "574f0e7c24b78246e9227c5e235853e0": "x^{(k)}", "574f141c592d8191ec4c69f511ba7207": "P_{4}=2,3,4,etc.", "574f3a5c807675733fcb03c2ddf54474": "\\mathbf{v} = -\\mathbf{U}_0\\quad (r\\to\\infty)", "574f42097861da2465e0ba296c834adf": "E = {\\partial^2 \\over \\partial r^2} + {\\sin{\\theta} \\over r^2} {\\partial \\over \\partial \\theta} \\left({ 1 \\over \\sin{\\theta}} {\\partial \\over \\partial \\theta}\\right)", "574f797a4e9909971844ba30f67bf0c0": "\\Omega(n)", "574f8ac5b25b5e1f6b61cc2421082307": "C(z)", "574faa771169920f30c01b64af63ae2c": "y \\in D", "575025808006f003fc63ac4ce8568f05": "A=F_{2}\\cdots F_{m+1},", "5750503fb6492210b9381781dac33f77": "aa^{-1}b", "575061f3a3aa8ff7e07d9e148299f44d": "\\mathcal{L} = \\mathcal{L} \\left( \\Alpha,\\ \\Omega,\\ \\Zeta,\\ \\Iota \\right)", "57506d2567d2c6f5eb673ff6ccb514b7": "E(\\mathbb{Z}/p\\mathbb{Z})", "57509a6a00a5f26fd1185c0594eae810": "d \\Phi_B = V dt", "5750c66f666670396ac022044ad8da22": "d(W, W') = \\lVert P_W - P_{W'} \\rVert,", "5751092c68993ab980034c4612a4b6e5": "\\mathrm{APF} = \\frac{N_\\mathrm{atoms} V_\\mathrm{atom}}{V_\\mathrm{unit cell}}", "575121a35533e8b2bc88f0fed0167500": " T_s = 288.3 ~\\mathrm{K} \\qquad T_a = 242.5 ~\\mathrm{K} ", "57513b6a0a4f54f4d2d49726ffd0df25": "\\lambda\\rho(\\lambda\\mathbf{x},\\lambda t)", "575195de3301b5b67396eb0fad8bd347": "\\eta_{\\mu \\nu }={\\rm diag}(+---)", "5751a9dcb78753821888a3690c6be690": "\\mathbf{E} \\left[ X_{t} \\big| \\mathcal{F}_{s} \\right] \\geq X_{s}.", "5751ad7239b559c79e1447b2c770ea2c": "TP_s = 1~ \\mbox{if}~g\\ >= 0.02 ", "5751db1fcf1cec4a1132468c6d9482aa": "A_3 = A_1 / P_3 = A P_1 / P_3", "57520aafb215fa7f4b0464f103776d3e": "\\prod_{\\mathbb{N}}\\mathbb{F}_p", "57520f4a1caba13984797db2f019a790": "\\psi(\\beta) = \\beta\\otimes1+1\\otimes\\beta.", "57522b91d11353ba2e82127acdc467f2": " M^{m+1} \\leq n M^{m-1}", "5752340d1cd58ffe3988ffc89ca96ab7": "\\mbox{BI} = \\sum_{k=1}^m \\sqrt{n_k}", "575287685e3b43f224cdaf8f45b30f8b": "-3q_p(2) \\equiv \\sum_{k=1}^{\\lfloor\\frac{p}{4}\\rfloor} \\frac{1}{k} \\pmod{p}.", "575320a4850b18e3fe80181f31bc8ecd": "\\sum_{i=1}^{k} x^s_i =1 \\,\\,\\sum_{i=1}^{k} x^l_i =1 ", "5753546df0aa198365c80a622280ee6e": "\\scriptstyle f_\\mathrm{s}", "57538546568516fbdf5de5713a500d28": "\\frac{d\\psi_L}{dt} = \\frac{3}{2}\\left[\\frac{Gm(1-1.5\\sin^2 i)}{a^3 (1-e^2)^{3/2}}\\right]_L\\left[\\frac{(C-A)}{C}\\frac{\\cos\\epsilon}{\\omega}\\right]_E", "5753b37864426c3750c3eb0ce74b1518": "\\scriptstyle\\square", "5753eb0350e61a4e339abee533c6e0fe": "0'=1", "5753f07ce43c0e0a344e1090d13bfc04": "\\mathbf{e}_3", "575448d52dfccd9f73d3e2b08d00f63d": "(M,Q)", "57548f8506f074e361ed5bc6f18a0520": " Q_1 = - \\frac{R}{p} \\left[ \\frac{\\partial u_g}{\\partial x} \\frac{\\partial T}{\\partial x} + \\frac{\\partial v_g}{\\partial x} \\frac{\\partial T}{\\partial y} \\right] ", "5754a7cd6c1ecb05154a038796649c1f": " V(\\mathbf{r}) = \\frac{-k}{r},", "575512db3fb57787175db1fcf13be364": "H_{\\operatorname{QB}(r)} \\rightarrow\nH_{\\operatorname{QB}(r)}.", "575518b885ab1ab3f2c53b8914012091": "\\displaystyle{[R(a,b),R(c,d)]=R(R(a,b)c,d) - R(c,R(b,a)d).}", "57552fc82e0e65340eb8898740779385": "\nP(\\bar{R},\\bar{\\theta})\\,d\\bar{R}\\,d\\bar{\\theta}=\\frac{1}{ (2\\pi I_0(k))^N}\\int_\\Gamma \\prod_{n=1}^N \\left( e^{\\kappa\\cos(\\theta_n-\\mu)} d\\theta_n\\right) = \\frac{e^{\\kappa N\\bar{R}\\cos(\\bar{\\theta}-\\mu)}}{I_0(\\kappa)^N}\\left(\\frac{1}{(2\\pi)^N}\\int_\\Gamma \\prod_{n=1}^N d\\theta_n\\right)\n", "5755354db8850a796ba53c1ab29f17d2": "a\\, \\frac{z\\, +\\, h}{h}\\, \\cos\\, \\theta\\,", "57554a99fcbdfaf4c49f71209fe1e1a8": "f: \\mathbb{N} \\to \\mathbb{Z}", "5755e87c2617d443a12ceae53b969938": "P_1, P_2, P_3, \\ldots", "57563d06d12a06ba072c2353fae275d2": "E\\simeq|\\vec p|", "5756ecfcfbd09a0fc289cd26545b4e1c": "\\{r_1x_1s_1+\\dots+r_nx_ns_n \\mid n\\in\\mathbb{N}, r_i\\in R,s_i\\in R, x_i\\in X\\}.\\,", "57571cd8b50ed87a450a582a18da5f1b": "i \\hbar \\frac{\\partial}{\\partial t}\\Psi = \\hat H \\Psi.", "57574070781adc55b140651b0c466aec": "p=11", "57576b5b8bb0d635cce007bdcf9fcf68": "\\hat{\\tilde{E}}^3_i \\sim {\\delta \\over \\delta A_3^i}", "5757f9e458dbde6c3d8d789d686ebf35": "Q_i = 1 - P_i", "575826a09f28f77f2226d62b421d035c": "P_n = \\sum_{k=1}^nk^2 = \\frac{n(n + 1)(2n + 1)}{6} = \\frac{2n^3 + 3n^2 + n}{6}.", "5758cad395f3791e9b7029dec2e0c1f4": "U(W) = \\text{log}(W+b)", "5758e3126ffd73977ec11d0c19d06692": "H(S,p)=U+pV", "5759028acdc37e7d7d8b8b58e07ddae8": " Z=\n\\int \\exp\\left[ i \\int d^4x \\left ( \\frac 1 2 \\varphi \\hat O \\varphi + J \\varphi \\right) \\right ] D\\varphi\n", "57591e24ad23bb5ea733b0d538aada3d": "2^\\kappa = \\gimel(\\kappa)", "57591ef3e9ab731707c6479f8eb25c60": " \\Omega \\mathbf{S}^{n+1}\\to \\mathbf{PS}^{n+1}\\to \\mathbf{S}^{n+1}. ", "5759bc2857db3e8fe75bd0089e010de0": "{10^o}", "5759bce546253901599831908fbf0981": " W_{ij}", "575a7bb8298eaa69d8faa35b9cb4ff60": " V(x) = \\sup_{0\\le \\tau \\le T} \\mathbb{E}_x \\left( M(X_\\tau) + \\int_0^\\tau L(X_t) dt + \\sup_{0\\le t\\le\\tau} K(X_t) \\right). ", "575b005cd211f8cf34260dd794a08fc4": "\\forall x (x \\not \\in \\varnothing);", "575b180a1be8ea32885d1d2bfbf4373d": "(g \\cdot f)(x) = g f(g^{-1} x)", "575b22aebffa7f6e66217aaa23cfd880": " \\rho_1 \\bold{v}_1 \\cdot \\bold{A}_1 = \\rho_2 \\bold{v}_2 \\cdot \\bold{A}_2 ", "575b552f37dc54e3001a0579f654e214": "q = 1-p = \\tfrac{\\alpha}{\\alpha + \\beta}", "575b611435adcb5223b5db9650e13e3d": "\\vert A(z) \\vert^2 + \\vert B(z) \\vert^2 = 2N \\, ", "575b73df1937414b2bc93b724a59f997": " A=\\{a_11", "576dd3cba3986d114a8ad71e71532dcd": "\\textstyle x_3(t) = z(t)", "576e3a420a5e7a1e7bae5a2096e49cb3": " \\langle \\cdot, \\cdot \\rangle_V", "576e8869ff0e146565303e7833872eed": "\\begin{align}\n\\hat{d}_x & = q\\hat{x}\\\\\n\\hat{d}_y & = q\\hat{y}\\\\\n\\hat{d}_z & = q\\hat{z}\n\\end{align}", "576ea9e195dbd16377b10cf4dbc335fe": " \\mu = 0,~ \\xi = {1 \\over \\alpha},~ \\sigma = {\\lambda \\over \\alpha} .", "576ee0cb9c42cf0e9adb22a22340c27a": "V \\approx \\sqrt{177.8 d_{skid} } ", "576ee7e6a028385df9dc0728c591e91e": " o(h^{n+1}) ", "576f1dacd615219d9f8bea06b26d5fdc": "c_1", "576f6c40e591c065ae3bc85a25e818db": "1 - p^{1,000,000}", "577009dad8565bd521857fe506e25f6d": " \\omega(\\lambda) = C \\prod (1 -\\lambda/\\lambda_n),", "57701048e61c16d9762aa6b1962cb60d": "0 \\le \\beta \\le \\gamma", "577091b837950f844b4b336c893e9c06": "\\ E", "5771b6bbbfeb5c83c1d26d4c2705a29e": "[79{.}5; 80{.}5]/([1{.}795; 1{.}805])^2 = [24{.}4; 25{.}0].", "5771d4fba68099b6d9a290bbb7891f90": "|2,1,0\\rangle", "57723150505c0eb01795badc268d3e42": "\\ell_i = ", "577266d274c55a1f9cd69080c60198fe": "\\phi_0\\,= \\phi |_(x=0) ", "5772c7d67040de40998f5c9bd0d2ac88": "f_m(n) := \\frac{\\Gamma(n+m)}{\\Gamma(n)}", "57735a5a5da6926a36c4582341d51960": "b+b_1/a_1+\\cdots+b_r/a_r=1/(a_1\\cdots a_r).", "57737744e0ad724dbcb9e781b78e652f": "\nX_{mn}(t) = X_{mn}(0) e^{i(E_m - E_n)t},\\quad P_{mn}(t) = P_{mn}(0) e^{i(E_m -E_n)t}\n", "57739b697b3381b64a1ed53b1baef7a1": "z = \\pm\\sqrt{R^2 - r^2};", "5773e972c8d62d96dde2ed3f86204966": "\\epsilon_\\mu^2(n)", "577421120a5448d1c445defdb35b7e54": "\\scriptstyle\\varepsilon^{-1}", "577474b8b179eb28c3e589f1e738ad0e": " U_n(x) := \\{ x_k : n \\le k < \\infty \\} \\cup \\{x\\} . ", "5774af531f4831002cc325e813b6bda6": "(P \\wedge (P \\to \\bot)) \\to \\bot", "5774b1830bec01c06aa2a50b5fb373e4": "C_x(t, f) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty A_x(\\eta,\\tau) \\Phi(\\eta,\\tau) \\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta\\, d\\tau,", "5774b967c1f2077f924c22ef55adccd6": "d_\\text{min}", "5774d12a75db6f737bcbbdbbe32e001e": "\\frac{1}{21} + \\frac{1}{42} = \\frac{1}{14}", "5774ee7f40ebba1d36b44ff8661ce6dc": "\\scriptstyle K{x\\rightarrow y}(0) = \\delta_{xy} ", "577542d6c777b61334e661a11b4a5100": "\\begin{align}\n \\theta'_0 &= \\, \\text{arcsin} \\Big( \\frac{n_0}{n_1} \\, \\sin \\theta_0 \\Big) \\\\\n \\theta_1 &= \\alpha - \\theta'_0 \\\\\n \\theta'_1 &= \\, \\text{arcsin} \\Big( \\frac{n_1}{n_2} \\, \\sin \\theta_1 \\Big) \\\\\n \\theta_2 &= \\theta'_1 - \\alpha\n\\end{align}", "577589cd24748ae7e61df575b0fc7ee2": "\\displaystyle{(X,d)} ", "57758f4bff09bda2dedd4725b91323a8": "x^i=\\sum_{k=1}^m\\alpha_k b^i_k", "5775927e75270d52926f15bcdf983882": "\\mathbb{P}(X_t \\in A |\\mathcal{F}_s) = \\mathbb{P}(X_t \\in A| X_s).", "5775bbd34d32dbe02b9660136f61647b": "g=f^r", "57764326a749584b7f5266d820a86aac": "Fun(\\mathrm{Rord},\\mathrm{Ab})\\ ", "57768df8e761a2ec1df62ca93000121a": " Z_1, Z_2, \\ldots ", "57769c81bb5762d43ab330c107dda387": "\n\\frac{\\lambda}{1+\\lambda} \\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v \\right) = \\chi \\left( C_m \\frac{\\partial v}{\\partial t} + I_\\text{ion} \\right)\n,", "5776dc1ce108d3a1e809777953d9938c": "C_{m,n}=\\int s(t)\\gamma^*(t-mT)e^{-jnt\\Omega} \\, dt.", "5777113f3b349c274641ea0c61d99381": " \\sum_{i, j} a_{i j }(x) c_i \\overline{c_j} \\geq 0 ", "57774937231ab3bd303555d086511a5a": " m = \\frac {\\log (F_1 / F_0)}{\\log(x_1 / x_0)} ", "5777727205343d9e509df720def94160": "\nk_p (\\tau _1 ,.\\,.\\, ,\\tau _p)\\!=\\!\\frac{E\\left\\{ {\\left( {y(n)\\!-\\!\\!\\!\\sum\\limits_{m = 0}^{p - 1} {\\!G_m x(n)} }\\!\\!\\right)\\! x(n-\\tau_1)\\cdots x(n-\\tau_p)}\\right\\} }{p!A^p }\n", "5777b6ada94d38293981346f73252c6a": "D(\\begin{matrix} \\frac{\\pi}{2} \\end{matrix})", "5777c808728a5006e88aba8d3de071ca": "a \\in \\Sigma \\cup\\{\\varepsilon\\}", "5777d07aa214ba5a2d9ce5d2bf6d020c": "v=\\pi u", "5777f72e36e663863828e43d77f644db": "\\lim_{a \\to \\infty}2 \\pi \\ln a = \\infty.", "5778132750728dac9ba696929897bf6f": "\\theta_j=\\frac{K_jP_j}{\\displaystyle 1+\\sum_{i=1}^n K_iP_i}", "57786a412fc7b5d1bc0c89ef250d9724": "(\\mathcal{A}, \\wedge, \\vee)", "57786c89902e504ab4fbafdeecf15e05": "10^{32}", "57789b4efa604dbd66ae4ca28b7981d0": "\\mathcal{L} = i \\hbar c \\bar \\psi {\\partial}\\!\\!\\!/\\ \\psi - mc^2 \\bar\\psi \\psi", "5778d426249de6383d809b0a2aba5599": "S=\\Delta", "5778ead1b5a920c7206c20ec2227fcea": "(f+g)(m) : = f(m) + g(m) ", "577921081023515a8659fc2188671dce": "\\vec F = q(\\vec E+\\vec v_s\\times \\vec B)", "577922f8b9090af41e18fc8b9e2652a8": "\\nabla \\times \\mathbf{H} = \\frac{1}{c} \\frac{\\partial \\mathbf{D}} {\\partial t} + \\frac{4\\pi}{c} \\mathbf{J},", "57792a82cf502828038f34d2ac5ad331": "\n f(k;\\rho) = \\frac{\\rho\\,\\rho!\\,(k-1)!}{(k+\\rho)!}\n .\n\\,", "57793796fd785284903862a5a3f6548d": " J(m-1)\\,", "57797ac2fb94c02c56dacd1681d76b58": "\\begin{align}\n uv - \\int v\\,du &= \\left[ {f'(x)P_2(x) \\over 2} \\right]_k^{k+1} - {1 \\over 2}\\int_k^{k+1} f''(x)P_2(x)\\,dx \\\\\n &= {B_2 \\over 2}(f'(k + 1) - f'(k)) - {1 \\over 2}\\int_k^{k + 1} f''(x)P_2(x)\\,dx\n\\end{align}", "577a15672dd425ffa80e187c57e9762c": "(ax+b)(cx+d)^{-1}", "577a3a61d2604cd2f4ada95203c8d6cb": "\\frac{\\alpha}{4 \\pi} \\,", "577a549c95768374130f64991ac85987": " \\mathbf{b} = \\left( a_1, a_1, a_2, \\alpha^1 a_2, \\ldots, a_N, \\alpha^{N-1} a_N \\right) ", "577a6a13dcc3ffe7c5f5bb23595c97c7": "\n\\bar{\\Pi}^0_\\ell(z)\n= \\sum_{k=0}^{\\left \\lfloor \\ell/2\\right \\rfloor} \n (-1)^k 2^{-\\ell} \\binom{\\ell}{k}\\binom{2\\ell-2k}{\\ell} \\; r^{2k}\\; z^{\\ell-2k}.\n", "577a731909bb1b659ddfcfad72aabd6a": "\\# \\operatorname{Aut}(\\mathbf{F}_q) = \\# \\mathbf{F}_q^\\times = q - 1", "577b75f55c8450f363d9e389af7497be": "\\vdash \\Lambda\\alpha. \\lambda x^\\alpha.x: \\forall\\alpha.\\alpha \\to \\alpha", "577b85de4a6029b99fb1fddef8d824ac": "c=\\log_b a", "577b995670db0eaf376ad26630c8516f": "\\frac{d}{d\\tau} = \\sum_\\alpha t^\\alpha\\frac{\\partial}{\\partial x^\\alpha}.", "577bc4ddf3f5d678ea9f5bec53ea6cb4": "\\mathbf{M}(\\mathbf{x}) = \\mathbf{m}\\delta(\\mathbf{x})", "577c051d5fea2887a9f157174f0f639e": "C_D", "577c164927b587e4eea26cd1e76bbaa2": "E \\{ (\\hat{x} - x ) y^T \\} = 0,", "577c3ee8d6969fee23d8378a29697c26": "\\left(\\gamma_0,Bn\\right)", "577cd9a69d17ae2e8e4a3063096cdc6d": "x^3 = 1 \\quad \\text{and} \\quad x^3 = 8 \\quad \\Rightarrow \\qquad x = (1)^{1/3} = 1 \\quad \\text{and} \\quad x = (8)^{1/3} = 2.", "577ce9f5522cfd35d4615f7099bfe3d8": " \n\\overline{y}_i = \\lim_{t\\rightarrow\\infty} \\frac{1}{t}\\sum_{\\tau=0}^{t-1} E[y_i(\\tau)]\n", "577d2184e2e32b7703b49794cd9480e5": "\\begin{bmatrix}\nc_3 c_2 &\t-s_3 c_1 + c_3 s_2 s_1 &\ts_3 s_1 + c_3 s_2 c_1 \\\\\ns_3 c_2 &\tc_3 c_1 + s_3 s_2 s_1 &\t-c_3 s_1 + s_3 s_2 c_1 \\\\\n-s_2 &\tc_2 s_1 &\tc_2 c_1\n\\end{bmatrix}", "577d5b4e1e1699556dd29e3f85625d55": "\\ell_j(x) = \\ell(x)\\frac{w_j}{x-x_j}", "577d68dcac8e1d790ee7f8a89e6409b6": " x_{k+1} = x_k + u_k - w_k", "577d79d1b0f51cadc9a7a041691fea5d": "(F)-(G)-(H)", "577d915728a752ae0ecf8740a8933f5a": "\\mathrm{NRMSD} = \\frac{\\mathrm{RMSD}}{x_\\max -x_\\min}", "577db0d1c355060b55089aeb9ec9f71c": "H:\\;G\\sim H\\rightarrow G\\sim H^2.", "577dc431595344113fde373754199202": "\nA + B \\rightarrow S\n", "577e6c2b227a6fdfeea6603c09a6d3bb": "{{|z_1-z_3|\\cdot |z_2-z_4|}\\over{|z_1-z_4|\\cdot |z_2-z_3|}}=+{{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}", "577eb66a72c0e300dcc4e88614423440": "\\,C_\\theta(u,v)", "577fe935420351c13d8ecc99a29c6fc5": "{s_i \\over v_i} = {(1- l_W) \\over l_W} = \\sigma ", "577ffa269b5783558df3bbfd2ed5fc96": "c_{ij} = \\frac{\\max(s_{ij},0)}{\\sum_{j} \\max(s_{ij}, 0)} ", "577ffd51f5c1bae3c92d97a6dd162158": "\\boldsymbol{\\mu} = g \\frac{-e}{2m_e} \\mathbf{L}.", "5780369f2cb6a9649a565cc4d9977618": " \\tilde{f}(\\lambda)=\\int_U f(u){\\overline{\\chi_\\lambda(u)}\\over d(\\lambda)} \\,du", "5780854e0a226f5ee1fe3089d399ab55": "\\begin{array}{rcccl}\n\\mathbf{c} &=& \\left(c_1, \\cdots, c_K \\right ) &=& \\text{number of occurrences of category }i \\\\\n\\mathbf{p} \\mid \\mathbb{X},\\boldsymbol\\alpha &\\sim& \\operatorname{Dir}(K,\\mathbf{c}+\\boldsymbol\\alpha) &=& \\operatorname{Dir} \\left (K,c_1+\\alpha_1,\\cdots,c_K+\\alpha_K \\right)\n\\end{array}", "5780d9e27a9139d38e51e73895832a8c": "\\Gamma(s,x) = x^s \\, {\\rm E}_{1-s}(x),", "5781235dd838b3f6c1ae452ed7819611": "p_{n,1} (x)", "5781458d38e3b66bdaca8f67505fd10c": "\nd\\nu_t = \\kappa(\\theta - \\nu_t)\\,dt + \\xi \\sqrt{\\nu_t}\\,dW^{\\nu}_t \\,\n", "57815f1e1941390e411e7290aaf64c0b": "{n \\choose 2}", "5781e8ae94f168ba2291b5412e2c97d6": "\\Bbb{Z}[X]", "57820ba1c8a91a6dd9b5d1c7d26d0b01": "\\mathit{3}^{k - 1}", "57827bb8c615cbdde5250a34eab0fee6": "Z_i^{(m)} = mY_i^{(m)}", "578313b0c5e35176d8664b098a1b25e1": "b_{i,j} = 1", "57832ca24bfd5e24b4020a9559e9bfb3": "\\rho \\ge 0 ", "57837bf054bd523361de2dd082fcb85d": "A = 16\\pi G^2 M^2/c^4", "57838183fcd03ac47ba72f3dc28a77c9": " X^2 = -I = - \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n \\end{pmatrix} = \\begin{pmatrix}\n -1 & \\;\\;0 \\\\\n \\;\\;0 & -1\n \\end{pmatrix}. \\ ", "578384b8fc6c307a2ac457faf991968d": "P_D = V_D (1-D) I_o", "57838d2f6b76ae0ff91e2d7bee46c756": " \nn (\\ln (n \\ln n) - 1) < p_n < n {\\ln (n \\ln n)}\n\\!", "5783a3e3bf4eb23606995a7782428123": " G_{\\infty} = - R_F \\ , ", "5783d299489dcc10bb79415957df8c7f": "\n\\frac{dr}{dt} = \\frac{\\partial H}{\\partial p_{r}} = \\frac{p_{r}}{m}\n", "57840eb01a1fc740858d92c86c7b71d1": "(\\mathbb{R},\\geq)", "5784cb742ed805ed84a5fd2447afffab": "\\vec{J}_p = \\frac{1}{2m}(\\Psi (i \\frac{h}{2\\pi}\\vec{\\nabla} -q \\vec{A})\\Psi^* +cc )", "5784e2a95cbd19efa01ee158a80c3d99": "+1=R_\\mathrm{spatial}(\\hat{z},360^\\circ) = \\exp(-2\\pi i L_z /\\hbar)", "57855d53276e8d1138b545324b94249d": "\\sum_{i=1}^n\\left(y_i-\\widehat{y}_i\\right)^2=\\sum_{i=1}^n\\left(y_i-\\sum_{j=1}^K\\widehat\\beta_j X_{ij}\\right)^2.", "57858d1dd33750bc434ed16e59b75763": "\\scriptstyle dx\\int{Vdx}", "57865094656218d8744b20d5d0695631": "\nE = 4\\pi G^{3/2}(\\mu H/k)^{5/2}M^{1/2}R^{3/2}P/T^{5/2}", "57865ecc56bb1c7bd56ab6bbfab13180": "k\\leq 1", "5786d9f8d020d7aa566daf9f6e584b4d": "\n\\operatorname{Li}_s(\\pm i) = -2^{-s} \\,\\eta(s) \\pm i \\,\\beta(s) \\,,\n", "578737ec759e2d19b6ec0a3ee59e31d7": " w+EV = e(p_0,u_1) ", "578767752b59abf002830d4c1d3a598b": "\\frac12 \\Delta m\\, v_1^2 + \\Delta m\\, g z_1 + \\Delta m\\, \\frac{p_1}{\\rho} = \\frac12 \\Delta m\\, v_2^2 + \\Delta m\\, g z_2 + \\Delta m\\, \\frac{p_2}{\\rho}.", "578787284f828c85e23474ad077ae249": " x^\\mu", "57878bc470f9784e42fb4972008ce706": "\\dfrac{k}{2n+2-k}", "57878f63030ca837af7a884d2e2aa57f": " \\leq a_\\max \\cdot d \\leq a_\\max \\cdot (\\frac{w_\\min - 1}{a_\\max}) = w_\\min - 1 < \\text{ } w_\\min ", "5787dab7cdae08bdf42df5027538c003": "e^{C}>0", "57880282c299042bc6e8932980ae32d9": "s_{m}(z) = a_m\\cdot((2\\cdot m + 1) + (2\\cdot m - 1)\\cdot z) / z^{m \\bmod 2}", "578853ebbacc2777bfd9d43b91ad6f1d": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( A+B \\right)\\subset \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( A \\right) + \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( B \\right).", "5788b0d7977a4df029e33fb5793d1c26": " \\left( {\\frac{1}{3} + \\frac{1}{5}} \\right) + \\left( {\\frac{1}{5} + \\frac{1}{7}} \\right) + \\left( {\\frac{1}{{11}} + \\frac{1}{{13}}} \\right) + \\cdots = \\sum\\limits_{ \\begin{smallmatrix} p \\text{ prime, } \\\\ p + 2 \\text { prime} \\end{smallmatrix}} {\\left( {\\frac{1}{p} + \\frac{1}{{p + 2}}} \\right)}, ", "578937dd939ffd2982e218a99654ad23": "e^x= \\lim_{n \\to \\infty}\\left (1+ \\frac{x}{n} \\right )^n.", "57894e2ca305d71ba18f24f6440cb895": "\\left(\\frac{\\partial g(T,P)}{\\partial P}\\right)_{T}=v", "57899c8f698e9288d9b6b60086848c9b": "\\, (x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2.", "5789a95d725f35d3bb36c3cc4a07a51f": "\n A_{xx} = \\int_A E~\\mathrm{d}A ~;~~ B_{xx} = \\int_A zE~\\mathrm{d}A ~;~~ D_{xx} = \\int_A z^2E~\\mathrm{d}A ~.\n ", "5789c83f3c31a23f20cabe7bc7eddd6b": "0 \\leq \\lambda_i \\leq 1 \\;\\forall\\; i \\text{ in } 1,2,3", "578a24674e82def14ae6386b087b804a": "M_z(t) = M_{z,\\mathrm{eq}} - [M_{z,\\mathrm{eq}} - M_z(0)]e^{-t/T_1}", "578a2724fab251dfdf4b7e4d292aeb66": " \\nabla:D(A)\\ni u\\to \\nabla_u\\in \\mathrm{Diff}_1(P,P)", "578a2fddb76a64e0e764f2ceaebd9f26": " \\tan^{-1} q = q - \\frac{q^3}{3} + \\frac{q^5}{5} - \\frac{q^7}{7} + \\quad \\cdots ", "578a30aa7003998909312c1d580c1c48": "e^{\\mathrm{i}\\,t/2} \n \\prod_{i= 1}^{\\infty} \\cos{\\left(\\frac{t}{3^{i}}\n \\right)}", "578a961fcbbe8c046602f9f89701b200": "\\widetilde{V} = \\bigcup_K (V^*)^K", "578a9d3fa4422eb77863cebca10780ba": "y_2(x) = v(x) y_1(x) \\;", "578af232257938068ae0466ecf64eed7": "v_j = v_{o} \\cdot w^j", "578b166249ceeb00ef374ef2269664d5": "\\, e^2 N(\\phi) ", "578b3cd24f6647230f3bff6f486aa134": " (q_1,q_2,\\ldots q_n) q = (q_1 q, q_2 q,\\ldots q_n q)", "578b6c828dffc43d98f144d690dc9160": " \\langle \\Psi_s|L_y|\\Psi_s\\rangle = \\langle L_y \\rangle = 0 ", "578b91410eccb25c2b1b7a19168446ee": "x^3 +\\frac{S}{2}x^2-\\frac{P^2}{2S}=0", "578b9bec1d941a34a8357f37254c9b1d": "\\tilde{c}(\\Pi)", "578bc8c0b60ed7c822c931a49c362352": "|\\langle \\psi|\\theta \\rangle|^2=2\\pi\\hbar\\int_{-\\infty}^{\\infty}dx\\,\\int_{-\\infty}^\\infty dp\\,P_{\\psi}(x,p)P_{\\theta}(x,p)", "578bd6b280edfbc9565ad24e4a6ee3de": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 77\\cdot 0.72)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 12\\cdot R_{\\bigodot}\n\\end{align}", "578bf0ebe94e6bc5f6f647c0415e1db9": "f(z)=\\sum_{n=0}^\\infty p_n(z) \\mathcal{L}_n(f).", "578bf47bc501adc1c8e20b039fda5cc9": "I_\\text{P}", "578c78a02933b4143f53a0b85a047a48": "(F;f_0,f_1): (W;M_0,M_1) \\to (X \\times [0,1];X \\times \\{0\\},X \\times \\{1\\})", "578c95d2a6335f4e95a80e9161136066": "\n\\frac{\\tan(\\beta d / 2)} {\\tan(\\alpha d / 2)} = - \\frac\n{(k^2 - \\beta^2)^2}\n{4 \\alpha \\beta k^2}\\ \\quad \\quad \\quad \\quad (4)\n", "578cb805d2ba13afd5818d4ebc511e54": " \\mathbf v_1 = {\\partial H \\over \\partial \\mathbf p_1} ", "578ce4e69347c268aa9de50a80277d7d": "1\\cdot(2i)^3 + 1\\cdot(2i)^2 + 0\\cdot(2i)^1 + 1\\cdot(2i)^0 = -8i - 4 + 0 + 1 = -3 - 8i", "578d77084ce1a6541015264d6e302475": "\\phi\\circ f", "578dd0fbde9a3953f0e6b96b321c8992": "d'(i,j) \\le d(i,j)", "578ded01aa413f57627f42942a0f8a6f": "F_{76.4\\%} = 1- \\left({\\frac{1 + \\sqrt{5}}{2}}\\right)^{-3} \\approx 0.763932 \\,", "578e210225632ddf0b183cd5af6ce675": "\\mathrm{TKOF}=\\frac{0.30 \\cdot 150 \\cdot 2820}{7000}=18.1", "578e3522e00fbf3505bc4d78a0b7b41d": "C_{H_2O} = V - \\frac{U_{osm}}{P_{osm}}V", "578e68868271be03dc16b026d02f4e6a": "\\beta_{1,i}\\in C_i", "578e6e0a484146ccdb22c1d53973c4e9": "X\\in B^* \\widehat{\\,\\otimes\\,}_\\pi B", "578e94722a1641f692e8ef1a3d30dc22": "\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\\in\\operatorname{SL}(2,\\mathbf{Z})", "578ea4c124848f96121be28ec5e4cea3": "v_{6}", "578eda5238abae9318ed2911395c98ba": "\\langle\\beta_{j,i}\\rangle", "578ef5454bad1563b970887a8fefe3d5": "f(\\sup D) = \\sup f(D)", "578f1ccf323d6f484fb0fc89df4db2b2": "2a|L|", "578ff0df5bcd1082fec8490abdd20a49": "2) \\ \\mbox{Imitators}=q \\cdot \\mbox{Adopters} \\cdot \\mbox{Probability that contact has not yet adopted}", "57900032e4edde7f38449333b6fff335": "\\begin{align}\nC\\ell^0(T^*M) &= \\Lambda^{\\mathrm{even}}(T^*M)\\\\\nC\\ell^1(T^*M) &= \\Lambda^{\\mathrm{odd}}(T^*M).\n\\end{align}", "579035414eb16f8dd82f50e6942c915b": "\\xi_{-1}(\\vec{p}) \n= \\frac{1}{\\sqrt{2 |\\vec{p}|(|\\vec{p}| + p_z)}} \n\\begin{pmatrix}\n-p_x+i p_y\\\\\n|\\vec{p}|+p_z\n\\end{pmatrix} \n= \n\\begin{pmatrix}\n-e^{-i\\phi}\\sin{\\frac{\\theta}{2}}\\\\\n\\cos{\\frac{\\theta}{2}}\n\\end{pmatrix}\\,", "57903be6839479650d38e992695c8c01": "\\omega_N^{3k}", "57904eae85fa563b058dcda77116315b": "\\displaystyle \\mathbf{v}_{xt}=u_{xx}\\mathbf{b}+\\mathbf{a}\\times\\mathbf{v}_x-\n2\\mathbf{v}\\times(\\mathbf{v}\\times\\mathbf{b})", "57908c819fb7c0167198daebe4b295b5": "\\theta \\left( x, y \\right) = \\mathrm{atan2}\\left(L \\left( x, y+1 \\right) - L \\left( x, y-1 \\right), L \\left( x+1, y \\right) - L \\left( x-1, y \\right) \\right)", "5790d78c34a175281e4217af0405f16a": "L_n=\\sqrt{9-\\frac{4}{{m_n}^2}}", "5790f3e12079fda88129f3e44b8d90ef": "\\displaystyle{M\\begin{pmatrix} x & 0 \\\\ 0 & x^{-1}\\end{pmatrix} = \\begin{pmatrix} x & 0\\\\ ba^{-1}(x-x^{-1}) & x^{-1}\\end{pmatrix}M,}", "579105ceddd28274761ba43cd1cf2c96": "\\infty 2", "579134e49a5bff6a5b2f7c1ba0eb7fb1": " 2\\cos\\;\\theta\\;+1 = a_{11} + a_{22} + a_{33} \\,\\! ", "579145cf2f169efeaf04783937ba5cb9": "N = 2^3 = 8", "57915000091eaf2556dbc4336e17981e": "\\bigcup A_n", "5791adbb09006387665e89cc89f62d4d": "f:{\\mathbb R}^n\\rightarrow{\\mathbb R}^m", "5791d84255c53a7198e039aab5295a1a": "\\preceq ", "57920d840d192ef9d18fb6337dfc203a": "(X,\\varnothing)", "579238eb6e194163a27974e2aa19a70e": "{17 + 20 + 22 + 21 + 2 + 6 + 10 + 14 + 3 + 16 + 12 + 7}", "57924e52b783ce0b84a8599a55bca5cc": "(P \\wedge (P \\to \\bot))", "5792668a2f38437cee035faa425ab30f": "e(E_{\\mathbf R})", "57929a0635ac1354bc828d52dc84d836": "i_*: \\pi_q(A, C) \\to \\pi_q(X, B)", "5792bc9e047c7eb3cb13788304861a9c": "\\mathbf{a}_{n}", "5793105addc4ce19cc49181ba8ebf3f6": "P_{em} = \\frac{3R_r^{'}I_r^{'2}n_r}{746sn_s}", "579313ef1308df000c7420eb7232e129": "dx = v(t) dt", "5793473f55459e01f0e6578d35447e48": "\\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\\}", "5793aa3fcc2ee32a8899151aec1537a2": "d(u, v) = \\operatorname{arcosh} (1+\\delta (u,v)).\\,", "5793d75b083f5d80e81deac1ec09d9f5": " {e_M} = {e \\over {4 \\pi \\epsilon_0}} ", "5793e8ca228f56c5650222c5fa67dbfc": "5.47 = y_1 + \\frac{10.0^2}{2(9.81)}", "579417fab323777721344301df84a122": "x_{j,0} = 1 \\,", "57943049a4e63eacb29589ced54c65eb": "(x_0+2,y_0)", "5794360dbc76be99e0ed0487fb6768cf": " \\mathbf{A}\\cdot\\mathbf{B} = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3 = C", "57944d08a9e54a174dc2882c36bb4700": "p = p^{\\star}_{\\rm A} (1-x_{\\rm B}) + p^{\\star}_{\\rm B} x_{\\rm B} = p^{\\star}_{\\rm A} + (p^{\\star}_{\\rm B}-p^{\\star}_{\\rm A}) x_{\\rm B}", "57944d279700152fc1a98671aec73900": " \\Delta : \\{\\bold{x}\\} \\times M \\to \\mathbb{R} ", "57949dba9103f1adef958f27ed34b029": "\\nabla \\times \\mathbf{B} = \\mu_0 \\epsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t} \\qquad \\quad \\ (4)", "5794e07db1d03768ec179e4b78768999": "{A}_{1}^{(1)}", "5794f06f31700aa1c1f708f22604e414": "p(r,k)=AH_0^{(1)}(kr) + \\ BH_0^{(2)}(kr)", "579534134adb3376231008766e518e93": " \\ = i \\lang 0| [\\Theta(x^0 - y^0) \\Phi(x)\\Phi(y) + \\Theta(y^0 - x^0) \\Phi(y)\\Phi(x) ] |0 \\rang. ", "579558824a980985d1acd6b8a25b8718": "\\mathcal{F}(\\mathrm{sinc}\\cdot \\mathrm{rect}) = \\mathrm{rect} * \\mathrm{sinc}.", "5795717d3933552288acde3b37c4fcb6": "K_3", "57964d666f8fa79529940b266d849ca4": "\\hat\\beta", "5796902a278de86790ed50fbeaffa53d": "f_i = {}_0F_1(;a+i;z),\\,k_i = \\tfrac{1}{(a+i)(a+i-1)}", "5796d70a014897047b313cf7ef269155": "\\Delta N > 0 ", "5796f72839536eddffc6afa790b8093c": "y_n=y(x_n)", "579700fbea236e70a9534df489d8de8d": "(x, r) = (0, r_{o})", "57970c643b2395b703b97a97f9dcfdf7": " G = \\frac { G_0 \\Delta_0 +G_1 \\Delta_1 + G_2 \\Delta_2 } {\\Delta} \\, ", "579761116d884b2ef7f1ce308e59adbb": "R=e^{{\\left( x-{y \\over 2} \\right)}^{1\\over 3}-{\\left( x+{y \\over 2} \\right)}^{1\\over 3}}", "57976dfb1b26af6bdeebb0e111e43b6b": "\\hat{V}=eB(m_l+2m_s)/2m", "579778458e56a60e8ffdf6d5e73a7c2e": "ds/dt = c", "579782565efc90d205914498ef1e35b5": "\\partial_t L(x, y, t) \\leq 0", "579797b943455461ef7ab5f4c00f078d": "\n \\mathbf{u}^0 = u^0_1\\boldsymbol{e}_1+u^0_2\\boldsymbol{e}_2 \\equiv u^0_\\alpha\\boldsymbol{e}_\\alpha\n ", "5797c6bf81e55157ed47ebbb9b3eb8be": "|j-k|>1", "5797cd9f6eb8cb75e5fbb99f7d1f7bdd": " {X_C \\over X_M} = \\text{Cycles of concentration} ={ M \\over (D + W)} = {M \\over (M - E)} = 1 + {E \\over (D + W)}", "5797e1d6c93a173f9619ea05ca6f12a1": "\\begin{align} A_{n} & \n\n= \\frac{1}{5} + \\frac{4}{5} \\sum_{k=0}^n \\left(\\frac{5}{9}\\right)^k \\mbox{giving} \\lim_{n \\rightarrow \\infty} A_n = 2 \\, .\\end{align}", "57980ce27fde6cf84f3f56562f3ccf16": "y = x - x^3/3 - \\dot x/\\mu", "57980fcd4a9b7a30b7242290e8f7f49e": "\\psi^{(0)}", "5798218882310039b72cae3643435ca1": "\n\\hat{x} |x \\, p\\rangle = x |x \\, p\\rangle, \\quad \\hat{p} |x \\, p\\rangle = p |x \\, p\\rangle , \\quad\nA(\\hat{x}, \\hat{p}) |x \\, p\\rangle = A(x,p) |x \\, p\\rangle,\n", "579849b5e5273f12d7092f565efd3fbc": "\\mathbf{a} \\times (\\mathbf{b} \\times \\mathbf{c}) = \\mathbf{b}(\\mathbf{a} \\cdot \\mathbf{c}) - \\mathbf{c}(\\mathbf{a} \\cdot \\mathbf{b}).", "5798569a88d6bdefd2bc6b143663e694": "g_J \\approx \\frac{3}{2}+\\frac{S(S+1)-L(L+1)}{2J(J+1)}.", "5798935366895fc9cadbe5714b32f3c4": "f_1(x)\\circ f_2(x)\\circ\\dots\\circ f_{2^n}(x)", "5799008f48b209d618166d0ae18b5f63": "\\textstyle P=\\begin{pmatrix}W_{1}&W\\\\W&W_{2}\\end{pmatrix} ", "57990d92273c2326a8a1fb46ce9ade9d": "x\\!:\\!\\sigma", "57990ff3de395953c006a591119ca99c": "M(R)", "57991fd029a1e4e4055e9d8011dea46b": "\\mathrm{ker}(\\partial_0) = C_0 = \\{a_1[v_1] + a_2[v_2] + a_3[v_3] | a_1,a_2,a_3 \\in \\mathbb{Z}\\} \\cong \\mathbb{Z} \\oplus \\mathbb{Z} \\oplus \\mathbb{Z}", "57995aecc4438fb5b0eda0afaa3b7e78": "\\,(a,b,c)", "579a1b78a60ea661f7a999b69f3e3468": "[0, T) \\times \\Omega \\subseteq [0, + \\infty) \\times \\mathbf{R}^{n}.", "579a6244dce41c056335fd4469b2eb27": "\nP = \\lim _{\\Delta t\\rightarrow 0} P_\\mathrm{avg} = \\lim _{\\Delta t\\rightarrow 0} \\frac{\\Delta W}{\\Delta t} = \\frac{dW}{dt}\\,.\n", "579a72ac0912a21635e5e6d4e962bbae": " = -\\frac{T_0}{c} \\int \\mathrm{d}^2 \\Sigma \\sqrt{-g} ", "579aa671458ac8d1afa13d19b0611cca": "p_{j} = \\frac{1}{N n} \\sum_{i=1}^N n_{i j},\\quad\\quad 1 = \\frac{1}{n} \\sum_{j=1}^k n_{i j} ", "579ad7e7c82890138a4bfcb9c056e88a": "\n|0,\\psi\\rangle = a|0\\rangle |0\\rangle + b|0\\rangle |1\\rangle = \\begin{bmatrix} a \\\\ b \\\\ 0 \\\\ 0 \\end{bmatrix}", "579b023086e1fd4cd74b04ddb163691b": "|\\psi\\rang ", "579b3135e25e7bc4b76cbbd1ac9ab101": "\\bigcup_nX^\\alpha_n = \\alpha", "579b991a89546fd2c10f6c9b67676ad7": "l_1 \\vee \\cdots \\vee l_n", "579bbb5c03267fd7768f92d6bbd1a947": "q_{hash}", "579bc1c8693797fe265b837670fbfe59": "x \\prec y \\implies I^-(x) \\subset I^-(y)", "579bf11ea929625365ce837a37bb9e35": "\\alpha \\not\\in free(\\Gamma)", "579c2ceb41cf5f21d496cac0374fa59c": "\\forall A\\in V\\,(x\\in\\Box A\\Rightarrow y\\in A)", "579c2d2fd3b6a6c6cad26437af9347e1": "\\frac{1}{2}\\int_{\\mathbb{R}^4} \\operatorname{Tr}[*\\bold{F}\\wedge \\bold{F}]", "579c7b95cb87d9525593722464203d0f": "\\displaystyle{U_\\beta U_\\alpha =B(\\alpha,\\beta) U_\\alpha U_\\beta.}", "579ca64d5c494b7f79be029a5eaaec4c": "\\ D=\\omega_{xy}^2/{4\\tau_D}.", "579ce1137aae9499a6fc209b4db55b8a": "WW^{T}=wI_n", "579d2d5dfeafceab6d4cd3194daf8ea0": "R = 1/k = \\frac{(x'^2+y'^2)^{3/2}}{x'y''-x''y'},", "579d3a72849b9edd488d5a1dae17fd3b": " \\begin{align} & -\\frac{\\hbar^2}{2m}\\nabla^2 \\Psi + U\\Psi = i\\hbar \\frac{\\partial \\Psi}{\\partial t} , \\\\\n& - \\frac{\\hbar^2}{2m}\\nabla^2 \\Psi^{*} + U\\Psi^{*} = - i\\hbar \\frac{\\partial \\Psi^{*}}{\\partial t} ,\\\\\n\\end{align}", "579d3f93a93b7221fbfb45ef5c65e0e9": "EqA = \\frac{H+TB+{1.5 \\cdot (BB+HBP)}+SB+SH+SF}{AB+BB+HBP+SH+SF+CS+\\frac{SB}{3}}", "579d69d744357aea67e7405fc5e2fb1e": "\\{\\phi\\}\\alpha\\{\\psi\\}", "579db47a7bc68ddd6885b84ce0d2f0d3": "R((\\xi^{-1})) = \\left\\{ \\sum_{n<\\infty} r_n \\xi^n | r_n \\in R \\right\\}.", "579df0c3bc1d361cbb79ead9aa609327": "\\mathbb{F}_p(x)", "579df268bd59d6c2c7d390828a468961": "\\mathcal{F}_{\\tau_1} \\subseteq \\mathcal{F}_{\\tau_2}", "579df8a8c7b541b98fd52a45c5880402": " K = GF(2)", "579e0167a771b50de03f5f48d90e2ee2": "W^{k,p}(\\mathbf{R}^n)\\subseteq W^{\\ell,q}(\\mathbf{R}^n)", "579e1796feef8d320b2954e0e950ac0e": " \\psi^{(0)}_0(\\vec{r}_1, \\vec{r}_2) = \\psi_1(\\vec{r_1}) \\psi_1(\\vec{r_2}) = \\frac{Z^3}{\\pi} e^{-Z(r_1 + r_2)} ", "579ead1056c66dc9d3029da0aa0785ce": "k[x_1, \\dots, x_n]/I", "579ec3d32e92321be38c163d251a32c4": "\\mathcal{H}_\\mbox{accept}=\\operatorname{span} \\{|q\\rangle : |q\\rangle \\in Q_\\mbox{acc} \\}", "579ee537155925d9aa57aeaaba3a2ee1": "A_1+A_2", "579eee98cf6a7cb30e18a676298ec2ee": "\\int_{-\\infty}^{\\infty}\\Phi(x,t)\\,dx=1,", "579f0348bd80c80fd5f3eeb54b41677a": "\\phi : A_1 \\to A_2", "579f38669d65b2e63dcf622b5b3d5dd0": "r = g^k", "579f3b56f3a9c4a241286a7722b3c213": "C(z)=\\sqrt{\\frac{\\pi}{2}}\\frac{1-i}{4} \\left[ \\operatorname{erf}\\left(\\frac{1+i}{\\sqrt{2}}z\\right) + i \\operatorname{erf}\\left(\\frac{1-i}{\\sqrt{2}}z\\right) \\right].", "579f5633f204ad020787c48334798f63": " \\Delta t \\,", "579f6774c627d6ebdb4b1f52bf4d532e": "\\frac{137}{60}", "579f72016a0e763416d4c2537a0840cc": "\\frac{1}{2}bh", "57a0d042d2245f232885df57f6daf2ba": "\\textbf{P}:(P_1, P_2, P_3, P_4)", "57a0efc60f9e21bc0694b8733f031c53": "(C,\\alpha)-\\sum_{j=0}^\\infty a_j = \\lim_{n\\to\\infty} \\sum_{j=0}^n \\frac{{n \\choose j}}{{n+\\alpha \\choose j}} a_j.", "57a0fd5fb77796d6156c7fba6f12e4c4": "\\hat{f}(s)=\\int_1^\\infty f(x)P_s(x)dx,\\qquad -1\\leq\\Re(s)\\leq 0 ", "57a101a41a0ef94115dffed5a2e6aef5": "\n\\begin{bmatrix} 1 & 3 & 2 \\\\ 2 & 7 & 4 \\\\ 1 & 5 & 2\\end{bmatrix} \\underbrace{\\sim}_{r_2-2r_1}\n\\begin{bmatrix} 1 & 3 & 2 \\\\ 0 & 1 & 0 \\\\ 1 & 5 & 2\\end{bmatrix} \\underbrace{\\sim}_{r_3-r_1}\n\\begin{bmatrix} 1 & 3 & 2 \\\\ 0 & 1 & 0 \\\\ 0 & 2 & 0\\end{bmatrix} \\underbrace{\\sim}_{r_3-2r_2}\n\\begin{bmatrix} 1 & 3 & 2 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0\\end{bmatrix} \\underbrace{\\sim}_{r_1-3r_2}\n\\begin{bmatrix} 1 & 0 & 2 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0\\end{bmatrix}.\n", "57a14139f4304a1f79e7c657fda66a26": "\\frac{E}{{\\delta}E}", "57a1538d1404b2a272eb90dcce7b9fcf": "f_{\\omega + 1}(f_\\omega(3)) - 2", "57a18ee346e5dc0f6c2fea6628138075": "D_{**}(\\mathbf{x}, \\mathbf{x}_k)", "57a1a13fb40b0f543907c83bb96ad00f": "\\scriptstyle \\tau", "57a1c046297838d02423d5b5ac5658b1": "f = \\frac{V_f}{V_f + V_m}", "57a1f446950dc1db22a566fdb08b16c5": "(c_1, t \\cdot c_2)", "57a1fdb4d32efda11af263f8ac88b4f4": "A=[\\partial f_i/\\partial x_j]", "57a226672d26dd679b6d03f1e8d3de47": "(-1, 0),", "57a24e89614cac4bebb7a429a8e4103b": "B = [\\beta_1,\\beta_2,...,\\beta_k]", "57a2531bbba269a32e85f9108526cc86": "\\sqrt{a^2-n}", "57a2c01da43f8503da43190bdc4bb801": "\\alpha E", "57a301770c8b1f1f1767f943d3c4fd54": "\\Theta = g^*\\theta", "57a34aacd8242b351c9e9a9c1004865d": "\\lambda \\rightarrow +\\infty", "57a434f5e0444a29f2248453560459d5": "f_2\\circ g", "57a4d7f6e298445e66c0d000729dbae1": "L_1=1/2", "57a4fa96992cf23f017a8bc2a6270d83": "|\\vartheta(x)-x|\\le0.006788\\frac{x}{\\ln x}", "57a53ff0b9d1f2027a7ddb3923995f78": "\\partial^\\alpha = \\partial_1^{\\alpha_1} \\partial_2^{\\alpha_2} \\ldots \\partial_n^{\\alpha_n}", "57a58204866bea3c7381f4a1dadab5de": "Z_2 \\,\\!", "57a5db81c89058953b46a6bf6a4dbfce": " {}= \\begin{vmatrix}x_0&y_0\\\\x_1&y_1\\end{vmatrix}\\begin{vmatrix}x_2&y_2\\\\x_3&y_3\\end{vmatrix}+\n\\begin{vmatrix}x_0&y_0\\\\x_2&y_2\\end{vmatrix}\\begin{vmatrix}x_3&y_3\\\\x_1&y_1\\end{vmatrix}+\n\\begin{vmatrix}x_0&y_0\\\\x_3&y_3\\end{vmatrix}\\begin{vmatrix}x_1&y_1\\\\x_2&y_2\\end{vmatrix} ", "57a62c1ec67e7e6a7187931b552a1284": "{{\\partial \\eta}\\over{\\partial t}}=M_{\\eta}[\\epsilon^{2}_{\\eta}\\nabla^{2}\\eta-f'(\\eta)]", "57a66718162676118551660ada2879d9": "n^2_e =\n 5.35583 + 4.629 \\times 10^{-7} f\n+ {0.100473 + 3.862 \\times 10^{-8} f \\over \\lambda^2 - (0.20692 - 0.89 \\times 10^{-8} f)^2 }\n+ { 100 + 2.657 \\times 10^{-5} f \\over \\lambda^2 - (11.34927 )^2 }\n- 1.5334 \\times 10^{-2} \\lambda^2 ", "57a6c6a0138316f108af20db961f3020": "H_{5} (x|q) =32x^5 - 32x^3(1-q^n) +6x(1-q^n)^2", "57a6d9a2ebd6899e3dd9a412c89407c0": "\\frac{(\\log_a x)^x}{e^{\\operatorname{li}(x)}}\\,", "57a729f55394c1c875446c6377a1227f": " 1 = \\tfrac{\\|X - \\mu\\|_\\alpha^2}{\\|X - \\mu\\|_\\alpha^2}", "57a765c6780eefe9fb1ed7f9059752ce": "n={120\\times{f}\\over{p}}", "57a7a6685877b7ae060a52681474f387": "u_{s+t}=u_s\\sigma^{\\phi_s}(u_t)", "57a7fe3e4e730f45a6662a17c01e8367": " \\Gamma= K_{12}\\,S_1\\,S_2\\,\\Gamma ", "57a8806e172d9ab789eea221453b84b8": "j>i", "57a884a816683d15fb9119da6c4969c2": "\\mathcal{M}_R.", "57a8b3c7715d33aadb1259d87e9396a9": " \\int_{L[\\mathbf p \\to \\mathbf q] \\subset \\mathbb R^n} \\nabla\\varphi\\cdot d\\mathbf{r} = \\varphi\\left(\\mathbf{q}\\right)-\\varphi\\left(\\mathbf{p}\\right) ", "57a94188a71b10e7be4188a9f3f10044": "a \\land(b \\land c) = (a \\land b)\\land c", "57a94b466b854e434f9d0c3363345305": "\n -10 + R_a + R_b - (1)(15) - V_3 = 0\n ", "57a9bea7a2fbbcc84d067f034b13b0de": "a|L|", "57a9fd33ed40ca03ba8b439b70f5e79b": "\\begin{align}\nx & = r \\cos \\theta \\\\\ny & = r \\sin\\theta \\\\\nd(x,y) & = r\\, d(r,\\theta).\n\\end{align}", "57aa0c82ebf497befb048e4e7c25405f": "0 = \\nabla \\gamma \\cdot \\mathbf{\\hat{t}}", "57aa33c7340c47dcd2322242a266af48": "\\mathbf{r}_3 = (a/4)(3\\hat{x} + 3\\hat{y} + \\hat{z})", "57aa6df4e4cac7a128fd15ba43e26f3e": "dU=TdS-PdV.\\,", "57aa6ecd2616dd50158c3db75ef2b72e": "\\scriptstyle \\{-T,\\dots,T\\}\\} ", "57aa71b6fe8d1b6836402327fca58b40": "\\mbox{If }X \\vdash a \\mbox{ then } X \\cup \\{ a \\} \\in Con", "57aa81615006190d80b998466843dad3": "\\mu = \\nu \\neq \\rho", "57aac3ca4de5b628579a77eb8dda90ea": " E(\\chi_a)", "57aae60dfe786ceebf383f85c32c2e1c": "a^2 + b^2 = ac\\cos\\beta + bc\\cos\\alpha + 2ab\\cos\\gamma.\\,", "57ab9914880ec499505a0a06e3c1a473": "V(t) = V_0 \\sin (\\Phi(t)) \\, ", "57abe7ef23866eacffaeaa71f99a0040": "x = \\gamma \\left ( x' + v t' \\right )", "57abfffacc129300d87ad38cce39d250": " E_\\ell = B\\; \\ell \\left (\\ell+1\\right )\\quad\n\\textrm{with}\\quad B \\equiv \\frac{\\hbar^2}{2I}.\n", "57ac264010e9960184683eeed2df4ff0": "D \\approx \\frac{p}{\\alpha}", "57ac759ddaae2e6bf327ced6f929a088": "\\frac{x_i-x_i}{x_j-x_i} = 0", "57ac7a320f298830565ebf901d5b726f": "\n \\vec{\\omega} \n \\;=\\; \\nabla \\times \\vec{v} \\;=\\; \n \\left(\\frac{\\partial}{\\partial x},\\frac{\\partial}{\\partial y}\\right)\\times(v_x,v_y)\n \\;=\\; \\frac{\\partial v_y}{\\partial x} - \\frac{\\partial v_x}{\\partial y}\n", "57ac89a168217bcd5cfb48e32aadf24b": "n = \\frac{m}{M}", "57ac8a325e24b83b1182435168388786": "f(\\mathbf{x})=(\\text{det}^*(2\\pi\\boldsymbol\\Sigma))^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(\\mathbf{x}-\\boldsymbol\\mu)'\\boldsymbol\\Sigma^+(\\mathbf{x}-\\boldsymbol\\mu) }", "57ad7e210cea0b65bf9286d43696f2fb": "\\mathrm {1.8 \\times 10^{-9}} \\approx {x^2 \\over .20}", "57ad92b14a7c255b4477c1d71100481c": "(1-z)^{-a} F \\left (a,c-b;c; \\tfrac{z}{z-1} \\right )", "57ada556206eb157f1b6ef018d1c5941": "n \\times p", "57adbcc1542e6edfbe28290bdf78d7c6": "d_A(z)=x/\\delta\\theta", "57adf8dab4b7b0aa56d2ddeae1d335da": "\\frac{3 \\cdot 19}{7 \\cdot 17}, \\frac{17}{19}", "57adfe13ca50cddfe008e6921eec3fc0": "\\Phi(\\xi,\\eta,\\phi)", "57ae66bd3a4abbd685f527900e6ba3dc": "p(D\\vert S)=\\prod_i p(w_i \\vert S)\\,", "57ae830b480d8578e153cb16e277f83d": " \\Delta n< 0; x, y \\in \\mathbb{R} \\}.", "57b162464e58d95ac3bdb1b21d2cd468": " \\rho_2\\ :\\ h(y) \\rightarrow f(g(y), y) ", "57b1687c960a79c75f0e1455c09a71d8": "\\operatorname{Ind}_H^G \\pi", "57b16c6355d27f54f3cbc79ee43b8565": "\\Re (s) < 0\\,", "57b1ba0419242f6e7c647ea13de57f55": "\\mathbf{U}q = \\frac{q}{\\lVert q\\rVert}.", "57b2402f46b8e9efb1d532deb16e3925": "f = \\gamma+\\partial \\gamma/ \\partial e ", "57b2d2fcee2d601628a40c64ee974337": "h(u) \\leq h(v) +1", "57b31910719b283875be1f5bbc215134": "\na_y =\n\\begin{bmatrix}b_z\\\\b_x\\end{bmatrix} \\times\n\\begin{bmatrix}c_z\\\\c_x\\end{bmatrix},\na_z =\n\\begin{bmatrix}b_x\\\\b_y\\end{bmatrix} \\times\n\\begin{bmatrix}c_x\\\\c_y\\end{bmatrix}\n", "57b31ef2a5af1098c451557d4b302514": "\\ \\ X ", "57b338998873bce99f08abcde43bdf75": " \\vec \\alpha = {\\vec \\tau \\over I} ", "57b39aac3b512948edc45e7ca719fa7c": "\\gamma = \\frac{C_{P}}{C_{V}}\\,", "57b3e9e1e63afc076ca580f05d19ab89": " \\scriptstyle d_j^k ", "57b40d7b1d8616299260b5bf6d3c6dd3": "\\hat{U}(t,t_0) = 1 - \\frac{i}{\\hbar}\\int_{t_0}^t\\hat{H}(t^\\prime)\\hat{U}(t^\\prime,t_0)dt^\\prime", "57b41f68c8e8bd6d1531910d05487591": "\\beta_{T}= 1 / P \\,", "57b48437c32fde3840c657fd00c97921": "\n \\boldsymbol{\\sigma} = J^{-1}~\\boldsymbol{F}\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{F}^T \n", "57b4f40386381c28251c2dddf72bec98": "\\nabla_{ \\partial _i} \\partial _j = \\nabla_{\\partial_j} \\partial_i.", "57b5ef768590c77fa9b87130e0b3d8fc": "\\{\\psi_{jk}\\}", "57b6278789c4f6314d00f80eaf263a1c": "X(\\omega) = ", "57b62b6227c730b8fefaca5ecc8cc313": "q = p^{v_p(q)} \\frac {r}{s},", "57b640eeca5243ff3ec22b8efa51847f": "p_s = ( s - D - l' )", "57b68aa7665ffa547089c74a4c67f657": " 16\\pi^{2} e^{8|x|} ", "57b68dbd30e523bc3cb466cfaf5fe7c8": "\\mathbf{n}_i", "57b6d47725c9688d55f450792e72fe6c": "\\int_{0}^T |\\sum_{n=1}^N\\pi_n(t)[b_n(t) + \\mathbf{\\delta}_n(t) - r(t)]| dt < \\infty ", "57b7521a7000fa4bcd6f246ac333b5f4": "2^2 - 1", "57b77b62b4a2e8b42c95baa6027601fb": " T D_T = D_{T^*} T.", "57b798af2b9670b74adb279108a43aa8": "c_P=\\frac{T}{N}\\left(\\frac{\\partial S}{\\partial T}\\right)_P\n\\quad = -\\frac{T}{N}\\,\\frac{\\partial^2 G}{\\partial T^2}", "57b7c59f41be9912c2faf1fb54b3f736": "X_a", "57b7ed3c279e02161e178f901e65b731": " F = \\bold Q(\\mu)^H ", "57b7ffff8b49b8cd4fabae591ea48524": "\\scriptstyle \\mathbb R^d_+", "57b803830bdf260938ad2c6959bd0236": "X \\times_{\\textrm{Spec} (\\mathbb{Z})} X", "57b8883b9664ed81e1485ff21e8cd509": "[a^\\dagger_i, a^\\dagger_j] = [a^{\\,}_i, a^{\\,}_j] = 0,", "57b8c11f0bc7ac460addf39fed06a9f7": "G = \\frac{1}{\\phi}", "57b8ffa62013e87a57fabf456d365a36": " \\gamma =\\frac{\\mathcal{M}_{rec}}{\\mathcal{M}^\\downarrow}", "57b93300fee5e5b0a2983ef95168399b": "K_H(M)=[P_H\\rightarrow Fred(\\mathcal H)]_{PU(\\mathcal H)}.", "57b93e6ed273d87709bded0d17f51ffc": "\\psi_k (t):= \\int_0^t \\varphi_k(s) \\, ds\\,,", "57b9557aab8988cdab83481e6d0bc710": "p_n(x;a,b,c,d|q) =\n(ab,ac,ad;q)_na^{-n}\\;_{4}\\phi_3 \\left[\\begin{matrix} \nq^{-n}&abcdq^{n-1}&ae^{i\\theta}&ae^{-i\\theta} \\\\ \nab&ac&ad \\end{matrix} \n; q,q \\right] ", "57b99649dad74a7013f144a13985dec6": "\\lim_{x \\to \\infty} \\int_a^x f(t) \\; dt ", "57b99b47c544af11f8335a6aefc39401": "\\gamma = \\arcsin \\left(\\frac{\\sin c\\,\\sin\\beta}{\\sin b}\\right)", "57b9e31c1ab801c477a7ffd505af4fc3": "a = p", "57ba0143a145c3b1c8bd391cde213e42": " \\Phi u(w-1) + (w-u)^2 = 0 \\, ", "57ba049bd52689b0a2b364db8fb74398": "\\,I^+(x) = \\{ y \\in M | x \\ll y\\}", "57ba468641e38c3711ccf50de3f8b419": "\\parallel_-", "57ba5b7c31ec24e00790f0278268ea7e": "\n\\sum_{n=N_k}^\\infty\\frac1{n\\ln(n)\\ln_2(n)\\cdots\\ln_{k-1}(n)(\\ln_k(n))^{1+\\varepsilon}}\n", "57bb13ce594138dd6fcddc101154434c": "ab<_x ab.next", "57bb2477130f761ffd9b7365c86b43e4": "\n\\begin{matrix}\nx \\vert\\vert y &=& x \\vert\\lfloor y + y \\vert\\lfloor x + x \\vert y\\\\\na \\cdot x \\vert\\lfloor y &=& a\\cdot ( x \\vert\\vert y)\\\\\na \\vert\\lfloor y &=& a \\cdot y \\\\\n(x+y) \\vert\\lfloor z &=& (x \\vert\\lfloor z) + (y \\vert\\lfloor z)\\\\\na \\cdot x \\vert b &=& (a \\vert b)\\cdot x\\\\\na \\vert b \\cdot x &=& (a \\vert b)\\cdot x\\\\\na \\cdot x \\vert b \\cdot y &=& (a\\vert b)\\cdot (x \\vert \\vert y)\\\\\n(x + y)\\vert z &=& x\\vert z + y\\vert z\\\\\nx \\vert (y + z) &=& x\\vert y + x\\vert z\n\\end{matrix}\n", "57bb36d90731ad5385574139d8a0a283": "{F}_3", "57bb936a51c053b8d6b5fcabb7114f06": " r = v - \\hat{u} ( \\hat{u} \\cdot v)", "57bbcea098e829e7c3e34a37bdf3d9ef": "[f,g]:=\\frac{\\int_\\Omega f(t)\\overline{g(t)}|g(t)|^{p-2}d\\mu(t)}{\\|g\\|_p^{p-2}},\\ \\ f,g\\in L^p(\\Omega,d\\mu)\\setminus\\{0\\},\\ \\ 1 0\\!", "57bd988e8922965516fa2dd52a8beca0": "C_t[a]= \\{x \\in S^\\mathbb{Z} : x_t = a \\}.", "57bda22e8ba58e263253da7423868c89": "\\sum_x \\left| x \\right\\rangle \\left| f(x) \\right\\rangle", "57bdccf48d71b62e7939c91a405b3c50": "S_2(c)", "57be64b731f532fc9a10ac0611d4df48": "V_n = a^{n-1}\\prod_{1\\le ix )", "57befa7a8873c1154587c71414809a6c": "\\textstyle{k - 1\\choose n-1}", "57bf2fa1f62b8693cfc7c91750729513": "\\Leftrightarrow 2^{b+3}-8 = 4y^2+4y", "57bf35bbd930013872dabe434fc0377a": "\\boldsymbol{\\hat{v}} = \\boldsymbol{v} / \\|\\boldsymbol{v}\\|", "57bf3e3df21fd88f206ac3bea9d4ba08": "\\alpha_1\\mid\\alpha_2\\mid\\cdots\\mid\\alpha_r", "57bf9b5d9ba8a8de7c99f43b33b590af": "(s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4s_1 s_ 2 s_ 3 .\\,", "57bfd7aaee7593c98e1f24ec69d11259": "y^2 = x^2 + a^2\\,", "57bfef359e0df442323cff4aaf532e34": "\\eta - \\dot\\gamma", "57c06c7e3b4c6aee2e3f11ad8006d4ee": "\\tilde{\\mathbf{E}}^+", "57c089aa9a152fe441f468b88875f941": "q^{\\prime}\\trianglelefteq p^{\\prime}.", "57c0f8cb0a0eae04ea797fba8c1e1a32": "S(1+r) = (1+r)^N + (1+r)^{N-1} .... + (1+r)", "57c115461114240b8d2d517a345364df": " x_j \\in \\{0,1\\},", "57c11d64b9164c7f7db2530e24658e9b": "d_{i0} = i", "57c11ef2904ce088f5323447a4504612": " ~ \\omega", "57c129403e1ad607e77da5356db66903": "TC_{min} = \\sqrt{2hDK} + cD. ", "57c179570786a82716db33d5c9ff63e9": "\\kappa = \\frac{\\bar{P} - \\bar{P_e}}{1 - \\bar{P_e}}", "57c183124d88e94249bde4a0ae3f25ba": "\\sum_{i =1}^{n-1} f(e_{i, i+1}).", "57c1b8b26dbfc36d49c4a5d1a32c04f0": "\\mbox{Number of Grids} \\simeq \\frac{b_{R,C}^C \\times b_{C,R}^R}{(RC)!^{RC}}", "57c1c162e2911729cc9e8f950fb03843": "\n\\begin{align}\n& \\infty + \\infty \\\\\n& \\infty - \\infty \\\\\n& \\infty \\cdot 0 \\\\\n& 0 \\cdot \\infty \\\\\n& \\frac{\\infty}{\\infty} \\\\\n& \\frac{0}{0}\n\\end{align}\n", "57c1d1855704174c9d4faa89fb8c1526": "(k=1, 2,..., n)", "57c1f3224f304b0fa32b9d3f2cdf18ca": "{\\textbf{x}_{t+1}} = A\\textbf{x}_t", "57c2218c3ac418f1d5ea9f8e1f9b460b": "\\forall y (F(y, z_1, \\dots, z_n) \\rightarrow y \\in V) \\rightarrow \\exists x \\forall y (y \\in x \\leftrightarrow F(y,z_1, \\dots, z_n) )", "57c27f4a30a1cf20998bd6e175f0f7d2": "T = \\frac{1}{4}\\sqrt{(a+(b+c)) (c-(a-b)) (c+(a-b)) (a+(b-c))}.", "57c2ae46b59aac9fd5ccd9ff659ee874": " \\lambda = \\frac{1}{\\Sigma} ", "57c2c9eae7d0528dd13551d5587786e6": "p-\\mu", "57c2d65c98e0d74dce2986822c7d56af": "\\chi(F)", "57c3175c9706669e6e13ca61a6413b20": " \\displaystyle{K} ", "57c36def878daa286bb3cd9b1a9ef018": "\\pi(x,y,z)={x+iy\\over 1-z}\\equiv u + iv.", "57c39b00c253bf810d129a3cf6d830e2": "K^8", "57c3aab474d1544539266ef798ca713f": " \\exist r \\forall y[A(y) \\rightarrow Byr] \\rightarrow \\exist x \\forall y[y \\in x \\leftrightarrow A(y)] \\,.", "57c450726eb56ce0856937ede817fc5a": "T(x) = a_0 + \\sum_{n=1}^N a_n \\cos (nx) + \\mathrm{i}\\sum_{n=1}^N b_n \\sin(nx) \\qquad (x \\in \\mathbf{R})", "57c458804c748423db73bb5653c79e8b": "EAR=(1+{APR \\over n})^n-1", "57c492471f63e0a8a78c4941b25ec5a3": "\\ln 2\\Gamma\\left(\\frac{k}{2}\\right) - \\left(1 - \\frac{k}{2}\\right)\\psi\\left(\\frac{k}{2}\\right) + \\frac{k}{2}", "57c49b3173fdfb1b15211186121d390a": "\n\\begin{array}{rcccccccc}\n\\scriptstyle 1^2 \\scriptstyle = &1&&&&&&& \\\\ \n\\scriptstyle 2^2\\scriptstyle = &1&3&&&&&&\\\\\n\\scriptstyle 3^2\\scriptstyle = &1&3&5&&&&&\\\\\n\\scriptstyle 4^2\\scriptstyle = &1&3&5&7&&&&\\\\ \n\\scriptstyle 5^2\\scriptstyle = &1&3&5&7&9&&&\\\\\n\\vdots&\\vdots&&&&&\\ddots&&\\\\\n\\scriptstyle (n-1)^2\\scriptstyle = &1&\\cdots&&&&\\cdots&\\scriptstyle 2n-3& \\\\\n\\scriptstyle n^2\\scriptstyle = &1&\\cdots&&&&\\cdots&\\scriptstyle 2n-3&\\scriptstyle 2n-1\n\\end{array} \n", "57c4b94e83212cf2e6e7ee579889e4b5": " \\begin{matrix} {\\rm GPN's} \\\\ 0 \\\\1\\\\2\\\\~\\\\~\\\\5\\\\~\\\\ 7\\\\ ~ \\\\ ~\\\\ \\vdots \\\\ ~ \\\\ ~ \\end{matrix} \n ~~~ p(n) = \\begin{vmatrix} ~~1 & -1~ & ~& ~ & ~ &~&~&~ \\\\\n ~~1 & ~1 & -1~ & ~ \\\\\n ~~0 & ~1 & ~1 & -1~ & ~ \\\\\n ~~0 & ~0 & ~1 & ~1 &-1~ & ~ \\\\\n -1 &~0 & ~0 & ~1 & ~1 &-1~ & ~ \\\\\n ~~0 & -1~ & ~0 & ~0 & ~1 & ~1 & -1~ & ~ \\\\\n -1 & ~0& -1~ & ~0 & ~0 & ~1 & ~1 & -1~ &~ \\\\ \n ~~0 & -1~ &~0& -1~ & ~0 & ~0 & ~1 & ~1 & -1~ &~ \\\\\n ~~0 & ~0 & -1~ &~0& -1~ & ~0 & ~0 & ~1 & ~1 & ~ \\\\ \n ~~ \\vdots & ~ & ~ & ~ & ~ & ~ &~ & ~ & ~ & \\ddots \\\\ \n\\end{vmatrix} _{ n \\times n} .", "57c4ba99f0a29b92e415ad427cac3cdb": " (x_{1}, x_{2}, x_{3}) ", "57c4ee3f4369fbfb64b52c04ca174bf3": "\\sum_{i=m}^n i = \\frac{n(n+1)}{2} - \\frac{m(m-1)}{2} = \\frac{(n+1-m)(n+m)}{2}", "57c5441940eb45d95c0dd78ab8f82d6e": "\\frac{10}{7}", "57c60e57fd5cd25ccb7f1f3afb00dce9": "\n\\delta_{2s}(n)=\n\\frac{\\pi^s n^{s-1}}{(s-1)!}\n\\left(\n\\frac{c_1(n)}{1^s}+\n\\frac{c_4(n)}{2^s}+ \n\\frac{c_3(n)}{3^s}+ \n\\frac{c_8(n)}{4^s}+\n\\frac{c_5(n)}{5^s}+\n\\frac{c_{12}(n)}{6^s}+\n\\frac{c_7(n)}{7^s}+\n\\frac{c_{16}(n)}{8^s}+\n\\dots\n\\right)\n", "57c642aedd4c63cb2f830d86804b2d20": "N(0,\\mathbb{E}(f-\\mathbb{E}f)^2)", "57c662a9270a759ee03ceb0ca9a2d181": "H = \\frac {\\delta^*}{\\theta}", "57c673aebefe1ef7539ed7293e3f9dba": "f(x-x)^2=f(x)^2+f(-x)^2\\,", "57c6bda1b690d2ad898a62f5bcb50bb9": "(\\mathrm{Setup}, \\mathrm{Prove}, \\mathrm{Verify})", "57c6dea3206a2b2331de3707fdf9e496": "\\mu \\in (-\\infty,\\infty) \\,", "57c6e217f6c727505217239ed862d6d8": "\ns = - {1 \\over RC }\n", "57c7931eb54a1d2ae523d564b01812c2": "(p+1)/2", "57c80921c7756e659cecd6b25d40b2b5": "E[\\rho] = \\int_{\\mathbf{R}^{n}} \\Psi(x) \\rho(x) \\, \\mathrm{d} x", "57c8e2408d5715041df4c48e97536f2d": "m_\\alpha:L\\to L\\;.", "57c902f56d2a525c7644538c20cbe3eb": "m = \\ker(\\mathrm{coker}(m))", "57c932caf0aef5a44cfaa7bb9645b6fb": " 2\\cdot2 ", "57c9380d00c6bd41e16d425bbcbfbe6b": "\\mathrm{Salt}", "57c997442544c177e63500cf8eb0478e": "M_R", "57c99c087082e210f33be1466ab5d54a": " \\varphi_X(t) = \\operatorname{E}\\left[\\exp \\left({i\\,\\operatorname{tr}(t^T\\!X)} \\right )\\right], ", "57c9da98da8ea439feaa5dc548487c1c": "F = (R + B^T P B)^{-1} B^T P A \\,", "57c9de287813ddc2c1af8edc526f75a8": "< m", "57c9dfb757fd379b80acb134abe57182": "c=\\sqrt{\\frac{g}{k}},", "57ca327fec434772d8a24ebbad2063ce": "\\int_{-\\infty}^\\infty f(x)\\delta(x)\\, dx = f(0)", "57ca7cb2a8f45d31f3e949c56cbc3ae2": "y_p^{(n)}(x)=\\sum_{i=1}^n c_i'(x)y_i^{(n-1)}(x)+\\sum_{i=1}^n c_i(x) y_i^{(n)}(x)\\, \\mathrm{.}\n\\quad\\quad{\\rm (vi)}", "57caabe52a40811000ebf65b191e8654": "\\scriptstyle i \\;=\\; 0,\\, \\ldots,\\, k-1", "57cab3de4f821ac752991becf3745e22": " P_{\\rm nd} = \\frac{P_{\\rm d}}{\\cos\\psi} ", "57cb34217398331702860bf76fd19707": "\\sigma_p^2 = \\hbar m\\omega \\left( n+\\frac{1}{2}\\right)\\, .", "57cb40c8579c36fa056bfd24649a6b64": "Q=0\\text{ when }\\cos(\\Theta)\\le 0 \\, ", "57cb7e001c0c50c81e7d76060cb28df5": "\\ V_m\\approx\\frac{\\pi}{4}(R\\theta)(R\\phi)(c\\tau/2)", "57cbb33312383251918f45d6df58d7c5": "u = \\frac{1}{2}\\left(\\varepsilon_0 \\mathbf{E}^2 + \\frac{1}{\\mu_0}\\mathbf{B}^2\\right)", "57cbbf035ac569babe45949a1156d26c": "X_E,", "57cbcc64fc691712f624cd5ff8ccbb30": "v_\\text{out} = \\alpha_2 A_1 A_2 \\cos (\\omega_1 - \\omega_2 )t - \\alpha_2 A_1 A_2 \\cos (\\omega_1 + \\omega_2 ) t + \\ldots \\,", "57cbcd04317941255d936ee5b542a5b8": "I_{OUT} >> I_{Q}", "57cbe2915d84d70bae157060f778ae9e": "c>a", "57cc0e9982c7cc8a786639edaf751a4d": " \\langle\\mid r\\mid \\rangle_\\text{free}\\sim t^{\\alpha/2})", "57cc5c92235a38fa3b2668ccbd1030d9": "n+k-1", "57cc9f8053a983269422f46ceb10f846": "\\beta=\\beta_2", "57ccb38b54a037e62e76920e4c440209": " S_\\pm | s, m_s \\rangle = \\hbar \\sqrt{s(s+1)-m_s(m_s \\pm 1)} | s, m_s \\pm 1 \\rangle ", "57cce22d3a38940a9030839127852959": "n \\geq 254", "57ccfde39f14ac7aa1b9d429523f7951": " \\mathrm{IN} + \\mathrm{PROD} = \\mathrm{OUT} + \\mathrm{ACC} ", "57cd15a46a9075747dfe10e396a9b43b": "g (\\epsilon) = \\{ \\epsilon \\}", "57cd5d5ef11eeb230e7f13b189b42f88": "(a^{k-1})a", "57cdeebb5866b7c7de8921be4aaf233c": "\\displaystyle \\overline{\\hat{f}(-\\omega)}", "57ce33508c154d811f6580f35b1ac0b7": " {\\rm Ind} (-S_{xx}''(x^0)) = 0 ", "57ce60ba808f8448ad3cf8b9180ff58f": "\\ln\\left(V\\right) = \\ln \\left(T\\right) + \\ln\\left(nR/P\\right)", "57ce7475b1ac37c50f4853c80fdd1441": "\\alpha= 1 / T \\,", "57cec4137b614c87cb4e24a3d003a3e0": "Y", "57cef15a8ff232a990afd11a48197dfe": "\\hat{d}(n) = \\sum_{k=0}^{p} w_n(k)x(n-k)=\\mathbf{w}_n^\\mathit{T} \\mathbf{x}_n", "57cf0494fc8bb753e9893316092921fc": "m = n", "57cf441201acc0a4462e4f9e5d726fdb": "k_BT", "57cffd6b5ccc0cbc6f3a09b723f7ff4a": "x^{14}", "57d00414b6ced39b5932bec375368d96": " p \\leq \\frac{1}{d+1} \\cdot \\frac{1}{e} ", "57d0493e4ea255e9ad0f73f09f1a6e41": "\\dot{\\mathbf{U}}(t)=\\mathbf{A}(t)\\mathbf{U}(t)", "57d08dff785090b7fca5c3efbcc36b97": "(x^3+x)(x^2+1)=x^5+2x^3+x", "57d0a03700812e235306b7bfcb732372": "\\begin{align}\n R &= \\begin{bmatrix}\n\\cos ( \\alpha + \\gamma ) & -\\sin (\\alpha + \\gamma) & 0 \\\\\n\\sin ( \\alpha + \\gamma ) & \\cos (\\alpha + \\gamma) & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\end{align}\n", "57d0a355dc0307a4a1052f139bc749ac": "(T,\\eta,\\mu,t)", "57d1645b24427efb274835ebf8c6ce20": "\\lambda > -1", "57d17b0fe7d3ba697d400a59a39827d4": "{}_{Q^{ab}=Q^{[ab]}+Q^{(ab)}}", "57d17e51ff3406102a534cdeccc64475": "\\scriptstyle y(t)", "57d18cb4a8e951e172c6ce2e5e6007a3": "\\gamma (at) = |a|\\gamma (t),", "57d19ecb7c7805da1aa75e70d06bda00": "E_\\text{rest} = E_0 = m c^2 \\!", "57d2081ed4c482592b78a592915b3624": "\nV_\\textrm{max} = r_{m}I\n", "57d2246d24a154661b969ab2b1acac2b": " R:= K \\cap k[x_1, \\dots, x_n] \\ .", "57d23740d0a2ad805cecbda21bb17140": "t = 2Nc \\frac {v} {f} \\,,", "57d2b78e482295f4b6c8792e3cd69df6": "L \\colon \\mathbf{B} \\to \\mathbf{C}", "57d2f68ace471bf85b3a3ea2b55cc05d": "\\mathbf{C}^{m+n}.", "57d322f84ccdf5695a65d9740023398f": " {n \\choose 0} + {n \\choose 1} + \\cdots +{n \\choose n-1} + {n \\choose n} = 2^n. ", "57d345d34c17683731ee852e550fff5c": "H \\neq 0", "57d35d3770b1324fc2f881987f68a93c": " \\frac{h_{g;k_1, \\dots, k_n}}{m!} \\prod_{i=1}^n \\frac{k_i!}{k_i^{k_i}}", "57d35ec5fe0afca2ba4272f2fbe30e48": " (g_m R_\\mathrm{E} \\gg 1) \\wedge (\\beta_0 \\gg 1) ", "57d36e6162f9f2722b0ea5d558aa879d": "\\left| \\psi\\left(t\\right) \\right\\rangle", "57d38e6e4be32339f8bf7c2d42a2f34c": "T_{i,j}", "57d392cf876a7f64436f56e2456c1e9a": " {\\hat f}^s(\\nu) = - {\\hat f}^s(-\\nu) ", "57d3bad719823314cadd9298685747d5": "\\scriptstyle Q(\\sqrt{69})", "57d4009f5b229ce50b56fa12a59efcaa": "n = 2k+1,", "57d4ba06e24722c59704f1bf158de899": "\\Delta\\lambda ", "57d4d4ebb13c567fbaade64ede61dd10": "(s+1)(s^2+0.6180s+1)(s^2+1.6180s+1)", "57d4e6b98e9a1b9a186362020a42f0ff": "\n\\displaystyle\\mbox{Radiation Pattern (in units of dB)} \\propto 10\\log_{10}\\left(\\left|2\\frac{J_1(X)}{X}\\right|^2\\right)\n", "57d4f89f0e8720a01ae175b0cbd5ff21": "\\begin{bmatrix} \n3 & 10 & 3\n\\end{bmatrix} = \\begin{bmatrix} \n1 & 3 \n\\end{bmatrix} * \\begin{bmatrix} \n3 & 1\n\\end{bmatrix}", "57d4fa3f98c331d669f7878a3dd02f4f": "\\bar{x} = \\frac{ \\sum_{i=1}^n w_i x_i}{\\sum_{i=1}^n w_i},\n", "57d4ff66e9159f0f219e2949824f116e": "j > 0 \\,\\!", "57d51a1d73f50e70aea4f664553b3c60": " x= \\cos(2\\pi nt)+ \\cos(2\\pi t)-1,", "57d5f28c85b64e0498e30b95ea57b5aa": "\\pm\\frac{\\sqrt{1 - \\sin^2 \\theta}}{\\sin \\theta}\\! ", "57d63d86813a9e780eed36b7a1a981f9": "sh_2", "57d647f4158bcfcb8c35bd519b8c0fe3": " M(\\vec X,Y) = \\left[ {\\begin{array}{*{20}c}\n {\\begin{array}{*{20}c}\n {\\mu _1 (\\Sigma _{11} )^{ - 1} } \\\\\n { - (\\Sigma _{11} )^{ - 1} } \\\\\n {\\Sigma _{21} (\\Sigma _{11} )^{ - 1} } \\\\\n\\end{array}} & {\\begin{array}{*{20}c}\n {\\mu _2 - \\mu _1 (\\Sigma _{11} )^{ - 1} \\Sigma _{12} } \\\\\n {(\\Sigma _{11} )^{ - 1} \\Sigma _{12} } \\\\\n {\\Sigma _{22} - \\Sigma _{21} (\\Sigma _{11} )^{ - 1} \\Sigma _{12} } \\\\\n\\end{array}} \\\\\n\\end{array}} \\right]\n", "57d653f7066c22535de49d15080207f7": "L(r, c) = \\int_0^1 \\! x ^ {c - 1} (1 - x)^{r-c} \\,dx \\,.", "57d68b85d6a56567958e067bcd85af9a": "10M_{e}", "57d78c0aef1351ff367159c7ff03d681": "p = \\frac{h}{h+t} \\, = \\frac{5961}{12000} \\, = 0.4968 ", "57d7d425739bd114b7a66f0ca72c02e5": "\\veebar", "57d7f0594a602e1b67f6444d01357040": "\\mathcal{H} = \\oplus _n \\mathcal{H}_n,", "57d8d1be2747c2bccf147aeca784e625": "3T/bt^2", "57d8d94266ebc5f35017b851acfd7222": "\n\\begin{align}\n\\cos\\tfrac12(\\alpha_2+\\alpha_1) \\sin\\tfrac12\\sigma_{12} &= \\sin\\tfrac12(\\phi_2-\\phi_1) \\cos\\tfrac12\\lambda_{12},\\\\\n\\sin\\tfrac12(\\alpha_2+\\alpha_1) \\sin\\tfrac12\\sigma_{12} &= \\cos\\tfrac12(\\phi_2+\\phi_1) \\sin\\tfrac12\\lambda_{12},\\\\\n\\cos\\tfrac12(\\alpha_2-\\alpha_1) \\cos\\tfrac12\\sigma_{12} &= \\cos\\tfrac12(\\phi_2-\\phi_1) \\cos\\tfrac12\\lambda_{12},\\\\\n\\sin\\tfrac12(\\alpha_2-\\alpha_1) \\cos\\tfrac12\\sigma_{12} &= \\sin\\tfrac12(\\phi_2+\\phi_1) \\sin\\tfrac12\\lambda_{12}.\n\\end{align}\n", "57d8fbc120b53059e2e2e111d8f31e55": "(\\lambda,\\,G)", "57d9bcc7febfc6ce9e3f1cd75c0d6b2b": "\\beta_i = \\rho_i y^{\\rm T}_i z\\,\\!", "57d9e8992c2a94c25fc04e527f59f57a": "A \\to B \\quad C", "57da4c75a8e8c67ec99751f5fdc3183b": "T' \\geq T", "57da59d385e5f33f5f0fde480efcd9ec": "k(\\alpha_0)= K(\\alpha_0) b \\sqrt D /P", "57da6fc69e6bb7d29779dfc03be7d56e": "\\mathrm{longitude}=\\arctan\\left(\\frac{n_y^e}{n_x^e} \\right)", "57da83a423db3b9f54dfdcb69393174b": "W \\longrightarrow \\frac{1}{(\\pi\\hbar)^3}\\exp\\left[-\\alpha^2\\left(\\mathbf{x}-\\frac{\\mathbf{p}t}{m}\\right)^2\\right]\\,.", "57da8c56ba9040a91dc8033fefec65ae": " \\beta = \\frac{t_{0}}{t_{1}} ", "57dacb11e9a3edf6dd2d7e0be20e8d9f": "\\displaystyle{|f(z)| \\le \\left[ 1+ {K_p\\pi^{1/p} \\|\\mu\\|_\\infty\\over 1 -\\|\\mu\\|_\\infty C_p}\\right]\\cdot R = CR,}", "57dae596afae68b328029661e0601228": "2^{\\aleph_\\omega}<\\aleph_{\\omega_4}", "57dae7b4b666e91d4de32c4b3f44efbf": "\\operatorname{var} (\\Beta(\\alpha, \\beta) )=\\operatorname{var} (\\Beta(\\beta, \\alpha) )", "57db617ab73162bf244f3f6bf2b4303b": "x^3 + 3x^2 + 2x", "57db7937dc3b8ce9a54c1f300833d086": " L \\cdot g^{-1} P = 0 ", "57db940563ed33ac593a26a32d750ff8": "\\vec{e}_0 = \\sqrt{2} \\omega \\, \\partial_t", "57dbb2c554e4a0a8194d5ac679d55512": "r-c < 0", "57dbddfb1f5711cead7a3305436003c7": "\\begin{align}\nW_0 &= 0\\\\\nW_k &= W_{k-1} + w_k\n\\end{align}", "57dc138466b34664c8c1bcab01c7c270": "\\dot{\\mathbf{x}} = \\mathbf{f(x)} + \\sum_{i=1}^m \\mathbf{g}_i(\\mathbf{x})u_i", "57dd1159b4eef9ffa96015a12da2a923": "\\scriptstyle Y = \\frac{X_1 / k_1}{X_2 / k_2}", "57dd3f31ce0285833367221342a97965": "P_1=\\frac{5}{8+5}", "57dd46cd2d41eaaab120f740185a064e": "D_F^q(p, q) = F(p)-F(q)-\\langle \\nabla F(q), p-q\\rangle. ", "57dd6e79534e7004596b3707e76c8e8c": "\\{\\langle \\mathbf{a_i}, \\mathbf{s} \\rangle - \\mathbf{b_i} \\} ", "57dd99d04bef24ac75f7a3289df88a2e": "N_{samples} = 2^{11} = 2048", "57ddca903e921605985f31518d4dcf07": "N \\equiv \\sqrt{\\frac{g}{\\theta}\\frac{d\\theta}{dz}}", "57dde852fb2c9773d57e1abdf275e123": "dF_\\mathrm s\\,\\!", "57ddedcb56d6a0331247945c6dd2d45c": "u^0_1,u^0_2", "57de1661ac9c1969e97ee209970ef3b2": "{\\textbf C}_X", "57de2bac8aac1aa3ef29b3d408f6d80a": "\\psi(x,t) = \\int_{-\\infty}^\\infty \\psi(x',t') K(x,t; x', t') dx'.", "57de4b1d8e61fa996c0574f594dddbaa": "4 \\pi \\mu", "57de50408be56b22a575c3fd29576b83": "r_{k+1} = r_k + tp^k = r_k - \\frac{f(r_k)}{f'(r_k)}.", "57de5eb43852bf37c1eeb091e003c40b": "Sq^n(x \\smile y) = \\sum_{i+j=n} (Sq^i x) \\smile (Sq^j y)", "57df603d28826efedb8907f427a6a407": "L(n,k) \\leq \\left\\{{n \\atop k}\\right\\} \\leq U(n,k)", "57df681ff6b3ec86d1ca6a160ebf67db": "D_3 \\cong A_3", "57df7e7c64ca1f49963f85ddcd21c29b": "x\\vee y=y", "57dfa703130378544babe60dae6541e1": " \\begin{align}\nE[X(t)|X(0) = i] &= i \\\\\nVar(X(t)|X(0) = i) &= 2i/N(1-i/N) \\frac{1-(1-2/N^2)^t}{2/N^2}\n\\end{align}", "57e003b0179e846bd44b1c6b5f7744a7": "t_{d} = \\dfrac{2^{r}}{f_{clk}}", "57e006f2776eb911c149f82f1afa2c96": "\\mu(D):=\\lim_{n\\to\\infty} \\tilde{\\mu}_n(D)^\\frac1n", "57e0ee958434c8402da3f716952157fc": " e_{n+1} = y_{n+1} - y^*_{n+1} = \\sum_{i=1}^s (b_i - b^*_i) k_i, ", "57e1089471740723b41f0fa1793d8271": "x=n=9", "57e1c4dc92c14bcaf3860731f95a01ef": "\\Phi_q(x),", "57e1ee272b767747cce13faa0fde2836": "(A*C)", "57e24d1e527c2fa6dfcb716e4467e127": " \\Leftrightarrow e^{'} + h^{\\bullet}", "57e29741eaaca87c0b700b8cdf71cca1": " \\sum_{n \\le x} f(n) \\sim \\sum_{n \\le x} g(n) ", "57e33815258b15a7860e945bd43e3691": "|\\psi_s\\rangle.", "57e3a250eb22c6b94fd58edc40b14dca": "n \\cdot \\left[{n\\atop k}\\right]", "57e3c755654d6fafe3d79e6b1c8ebdb9": "\\, I_k=0", "57e46baafa36147ddca6352b7f5a7a2a": "\\frac{b + c - a}{a}\\,:\\,\\frac{c + a - b}{b}\\,:\\,\\frac{a + b - c}{c}.", "57e4ae3ddf972911499bfee86440e85e": "D_{\\mathrm{KL}}(p || q)", "57e4c52ce434867fcb6b9111128cc9ce": "(x-3)(x-2)(x^4)(x+1)(x+2)(x^2+x-4)^2", "57e4de28fd268a246009bda9338b184f": " I_\\text{P} = q_\\text{P}/t_\\text{P} = (c^6 4 \\pi \\varepsilon_0 / G )^ \\frac{1}{2} ", "57e4deb577638d244fc7f8090d0bf568": "\n\\begin{align}\ns_1& = x_0\\\\\ns_t& = \\alpha x_{t-1} + (1-\\alpha)s_{t-1} = s_{t-1} + \\alpha (x_{t-1} - s_{t-1}), t>1 \\,\n\\end{align}\n", "57e535d20a65a94f09c236354c98d69c": "\n\\begin{align}\ny[n] = x[n] * h[n] \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{m=-\\infty}^{\\infty} h[m] \\cdot x[n-m]\n= \\sum_{m=1}^{M} h[m] \\cdot x[n-m],\n\\end{align}", "57e572965a2828ea0aec4b30c5bf99d4": "\\{X[k]\\}", "57e58f10f00a85436d4311624c0b38ff": "\\frac{v^2}{2} \\,", "57e5e61eae0aa917a60730aa6d45150c": " 0 x^3 + a_3 x^2 + ( a_2 + a_3 ) x + ( a_1 + a_2 + a_3 ) \\,", "57e5f0d51ffbe8a84c4c5184c7f5dba0": "\n\\lim_{\\varepsilon\\downarrow 0}\n\\frac{\nP(|X_t-\\phi(t)|\\leq\\varepsilon \\text{ for every }t\\in[0,T])\n}{\nP(|X_t|\\leq\\varepsilon\\text{ for every }t\\in[0,T])\n}\n=\\exp\\left(\n-\\int^T_0\\tfrac{1}{2}|\\dot{\\phi}(t)|^2 \\, dt\n\\right).\n", "57e62227c3b483842c98a46aefcb6be4": "= \\frac{\\pi}{\\sqrt{18}} \\approx 0.74.\\,\\!", "57e66be2434ed617a24b73389c03a2a3": " \n\\begin{bmatrix}\n \\mathbf{e}_1'(t) \\\\\n \\mathbf{e}_2'(t) \\\\\n \\vdots \\\\\n \\mathbf{e}_{n-1}'(t) \\\\\n \\mathbf{e}_n'(t) \\\\\n\\end{bmatrix} \n\n=\n\n\\left\\Vert \\gamma'\\left(t\\right) \\right\\Vert\n\n\\begin{bmatrix}\n 0 & \\chi_1(t) & \\cdots & 0 & 0 \\\\\n -\\chi_1(t) & 0 & \\cdots & 0 & 0 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n 0 & 0 & \\cdots & 0 & \\chi_{n-1}(t) \\\\\n 0 & 0 & \\cdots & -\\chi_{n-1}(t) & 0 \\\\\n\\end{bmatrix} \n\n\\begin{bmatrix}\n \\mathbf{e}_1(t) \\\\\n \\mathbf{e}_2(t) \\\\\n \\vdots \\\\\n \\mathbf{e}_{n-1}(t) \\\\\n \\mathbf{e}_n(t) \\\\\n\\end{bmatrix} \n", "57e66eea42664253b61d39f87dee53d1": "\\begin{cases} q_s* = \\alpha_s \\left(\\tau*-\\tau_c* \\right)^n \\\\ n = \\frac{3}{2} \\\\ \\alpha_s = 1.6 \\ln\\left(\\tau*\\right) + 9.8 \\approx 9.64 \\tau*^{0.166} \\end{cases}", "57e69a16f99c052ad5ebabd8b491d9d3": "\\vec V_3", "57e6bc5b1365c3fc81c16df2a5a0b044": "\\text{Cl}_{2m}\\left(\\frac{\\pi}{2}\\right) = 2^{2m-1}\\left[\\text{Cl}_{2m}\\left(\\frac{\\pi}{4}\\right)- \\text{Cl}_{2m}\\left(\\frac{3\\pi}{4}\\right) \\right] = \\beta(2m)", "57e6c650dfa100985083dee33d54b733": "\\Phi_{11}=\\frac{1}{2}R_{ab}l^a n^b=\\frac{1}{2}R_{ab}m^a\\bar{m}^a", "57e7407c6fe7336ef44eab1b97ffa771": "h(x) \\ge 0", "57e75138a15bc2009b090963920de692": "w(f) = \\sum_e{f_e \\cdot l_e(f_e)}", "57e76d8790a67537bf06797efcf8e755": "P = v_1 i_1 + v_2 i_2 = (-R i_2) i_1 + (R i_1) i_2 \\equiv 0", "57e78c332ab9e9e2ad1e21cf33c50e92": "f_X(x) = \\Pr(X = x) = \\Pr(\\{s \\in S: X(s) = x\\}).", "57e79a028d193a8af708ba78fab362bf": "X \\to \\alpha \\_ \\beta", "57e7bf364f60eb98ffe4dc0eeeeca841": "x(t)=a\\int_{s=0}^t \\phi(s-\\tau)\\,{\\rm d}s+C", "57e7f8c9f6130316125efa4ee0a038d8": "\\mathbf{\\tilde{H}_{S}} = \\mathbf{H_{S}} + \\Delta", "57e817736a53b7c1b45780b40fb48318": "u(i \\,\\Delta x, n\\, \\Delta t) = u_{i}^{n}\\,", "57e8457eefd64a7157a370fad251cbc7": " G^\\sigma=\\{ g\\in G: \\sigma(g) = g\\}.", "57e84f9f2f497bcf6482ca59f0b77c53": "\\begin{align}\n\\omega_f(s+t) &=\\sup_{|x-x'|=t+s} d_Y(f(x),f(x')) \\\\\n&\\leq \\sup_{|x-x'|=t+s}\\left\\{d_Y\\left( f(x), f\\left(x +t\\frac{x-x'}{|x-x'|}\\right)\\right) + d_Y\\left( f\\left(x +t\\frac{x-x'}{|x-x'|}\\right), f(x')\\right )\\right\\} \\\\\n&\\leq \\omega_f(t)+\\omega_f(s).\n\\end{align}", "57e8832c6c971cb241a146bf775130b7": "\nR=|\\langle z^n\\rangle|=0\\,\n", "57e898a65d89fede3bb9dad8b9d546de": "\\chi^*:E^{|E|-r}\\rightarrow\\{-1,0,1\\}", "57e91e946a72cc3b48cf24a8c89ce51f": " f(z)=\\frac{1}{z+1}-\\frac{1}{z-1}+\\frac{3i}{2}\\frac{1}{z+i}-\\frac{3i}{2}\\frac{1}{z-i}", "57e94391d25f8f6194f15db881960166": "\nT(a) = \\frac{1}{a} \\sum_{0 \\leq b0) ", "57f5e6254fc6a691696d51bda6239375": "1 \\leq k < n \\qquad (1)", "57f628509960490952e8ca537ac07286": "\\varphi(x)", "57f66746b266893b3acaecd301552673": "\\sum_{j=1}^m a_{ij}N_j=b_i", "57f669571c0dc40d2517e31977dea294": "\\langle I,\\le\\rangle", "57f687d3510d3fceae9e98926be2417f": "w_i = \\sum_{j=1}^p h_{ij}v_j.", "57f69c0b36168605154bf02be8601c9c": "\\Delta \\delta T = \\epsilon(\\mathcal{L}_{X}T_{\\epsilon})_0 - \\epsilon(\\mathcal{L}_{Y}T_{\\epsilon})_0 = \\epsilon(\\mathcal{L}_{X-Y}T_\\epsilon)_0", "57f6a7871c685d4c2fa9ebc2ca27d00a": "\\sqrt{\\frac{5}{28}}\\!\\,", "57f6ae6ea02d0ca70280b85af7fe588e": "\\int_{1}^{3}\\frac{e^3/x}{x^2}\\, dx", "57f6d0181f429c851e3e26bae30c429a": " A \\rightarrow \\; B", "57f6f6ddebf5b845de79718604babc27": "\\lim_{n\\to\\infty} a_n = 0.", "57f7326e979f02ea4c906dd3d32959a2": "\n \\boldsymbol{\\nabla}\\times\\boldsymbol{S} = \\boldsymbol{0} \\quad \\implies \\quad S_{mi,j} - S_{mj,i} = 0\n ", "57f73c9cbc2218f3f94068d8c466acde": "\n f(\\phi, \\psi) = Z_s(\\kappa_1, \\kappa_2, \\kappa_3) \\ \\exp [ \\kappa_1 \\cos(\\phi - \\mu) + \\kappa_2 \\cos(\\psi - \\nu) + \\kappa_3 \\sin(\\phi - \\mu) \\sin(\\psi - \\nu) ], ", "57f7514cf8f5af22ce28072d9610c9ff": "x_i = \\alpha \\sum_{j } a_{ji}\\frac{x_j}{L(j)} + \\frac{1-\\alpha}{N}, ", "57f77c3a7202b7716da9b4e637a5ca52": "\\alpha = [1,2]", "57f796520091b4b9309945e1937ea9ae": " f(z) = e^{1/z} ", "57f7a0f65050cd5b9e84ba7164f801f4": "\\tbinom{-1}{0}=1", "57f7aa722131ae511b46305dfc7225e1": "S_-=\\wedge^{\\mathrm{odd}} W", "57f7ce62e85f226b0b7c1cfb5782b589": "(A-1 I) \\begin{bmatrix}\n0 \\\\ 1 \\\\ -3 \\\\ 3 \\\\ -1\n\\end{bmatrix} = \\begin{bmatrix} \n0 & 0 & 0 & 0 & 0 \\\\\n3 & 0 & 0 & 0 & 0 \\\\\n6 & 3 & 1 & 0 & 0 \\\\\n10 & 6 & 3 & 1 & 0 \\\\\n15 & 10 & 6 & 3 & 1\n\\end{bmatrix}\\begin{bmatrix}\n0 \\\\ 1 \\\\ -3 \\\\ 3 \\\\ -1\n\\end{bmatrix} = \\begin{bmatrix}\n0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0\n\\end{bmatrix}", "57f83543b1314dccc513b7259425e461": "\\Delta_+ = \\Lambda^{even} W,\\, \\Delta_- = \\Lambda^{odd} W", "57f8656921f46757881bc86b68bb2b9c": "u_{rest}", "57f870e6614ef3202df19fa44893e634": "\\forall u \\,\\partial(u)=0 \\and p 1 = 0 \\rightarrow r(u)^p=u", "57f8c993d5bbd40a896c8b99880d6779": "\ns_{f_2}(n)=\nO\\left(M(n)\\log n \\right).\n", "57f9393d410a3b17634efa3dd53ac6a2": "b_i(x) + \\langle p_i(x)\\rangle", "57f961e6d2705c07229eda16d4fbfc9c": "f(x_1, x_2, x_3)", "57f98ec6425f081ab84ed6cd996fc796": "M_{fs}= \\langle f|e\\mathbf\\epsilon\\cdot\\mathbf r|i\\rangle,", "57f9cf2ccd5c431b996d8f8f6f53a8aa": "T_w(x) = s(x) + w T_w(2x)", "57f9dc72ddb00c426d2755d1b4b94a35": "L\\to\\infty", "57fa0a97c92f44826e35aaf440a030d9": "a_{i j}=a_{j i}^{\\theta}", "57fa6bf881791b564972656074dc35fb": "V^0_{\\textrm{rev}}", "57fa6d61d221315b0850c27b48cfbf08": "H_{2n}(x) = (-1)^{n}\\,\\frac{(2n)!}{n!}\n\\,_1F_1\\left(-n,\\frac{1}{2};x^2\\right)", "57fa73125c5fdc163ba00ba145c32a10": " \\mathcal{F}(e,T) = \\{ X \\cup \\{ e \\} | X \\in T \\} ", "57fa98269234924d97a855c33df81f7a": "\\operatorname{span}(\\mathbf{u})", "57fa9c7d53fc4222b2029a4a85147609": "S_{k'k}=\\frac{2\\pi}{\\hbar}||^{2}\\delta (E_{f}-E_{i}) \\; \\; (6)", "57fb0b806c9afd79f70e4fb78e61106b": "\\int x^m\\arccot(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arccot(a\\,x)}{m+1}+\n \\frac{a}{m+1}\\int \\frac{x^{m+1}}{a^2\\,x^2+1}\\,dx\\quad(m\\ne-1)", "57fb1bc0bc00f3967d63a4d0f7a451e9": "K= 1/\\mathit{MPS}", "57fb420d69a1bde5c631bc2953dc4530": "D(x) = x\\log x + x(2\\gamma-1) + \\Delta(x)\\ ", "57fbc7ee0cc06365d39eabde4079c0ef": "K.", "57fbcc3a855ccfa8e57c86ae416d4fef": "\\vdash P", "57fbdfd226c98715043957109515d9d5": "p_F = \\hbar k_F ", "57fc3268b4dff94e59083f74d28fdaea": "is\\_ carrying(Ball,S_{0})", "57fc4bd8a668a5a027db2ff5928138c4": "-15\\,X^4+3\\,X^2-9,", "57fc5bf219d93ed00670b9718f3a716c": "p(x,A)", "57fc5de9749c1eb22223f2467eff7278": "\\tfrac{(m-1)!+1}{m}", "57fc63b1446073c25d46f1ce524b3e6c": "i(J)=J", "57fc66b610c3f983d160dd835241169f": "(p,00111,Z) \\vdash (p,0111,AZ) \\vdash (p,111,AAZ) \\vdash (q,111,AAZ)", "57fc693388b50ca1ae46a9397b6cc02b": "e= \\left | \\mathbf{e} \\right |", "57fc8e5b84fa954fa33453b1f132b21a": "B=-3p", "57fcf6d5a562b6640aa137d9ad1a32b8": "g\\colon\\mathbb{R}\\rightarrow\\mathbb{R}^+_0", "57fd1a88a5099fc52617623ecb003e81": "\\textstyle v_1\\neq 0 ", "57fd1da4071ae674e2a458280e6b1de6": "\\mu_j=\\mu+\\tau_j", "57fd6b31f2e8211c9b21aa54a6d30fd5": "\\theta_u(v) = g_u(u,\\pi_*v)\\,", "57fd72af87308a93ff42f527548d55a0": "S^o", "57fd741b310a8417ff55e5f88657a1a8": "{\\scriptstyle Eq \\, \\alpha \\, \\Rightarrow \\alpha \\, \\rightarrow \\left[\\alpha \\right] \\rightarrow Bool}", "57fd79ba4c87dd5e435c310191eeb1f8": "]\\ ,\\ [ \\!\\,", "57fda3aa82c3c3624905ed24e0bfb841": "x_0\\in\nM", "57fda5132b015dd8bc12309ed99019de": "\\pm\\hbar k_z", "57fdd60751782d60b1d8333cd1f35924": "0=\\det(V^{-1})\\det(A+\\delta A-\\mu I)\\det(V)=\\det(\\Lambda+V^{-1}\\delta AV-\\mu I)", "57fe245992742ad14487eff51b0f7156": "\\Omega^0(M)", "57fe86c83d163b900c24f7e240d590e0": " \\delta_s = \\frac {5 \\nu} {u^*} ", "57fe9e6a9f7d23b7cebfbc012a3ced6b": "FO(KO_i, KI_i, x)", "57fed5059d74410bc95c05baa0cc1048": "\\hat{se}_{\\theta} ", "57feef50be398c2623add85a448a83b0": "\\scriptstyle{\\mathbf{H}_A(t)}", "57fef619de33d1f62a958059c21acb43": "\n B(0)\\approx \\frac{\\Phi_0}{2\\pi\\lambda^2}\\ln\\kappa,\n", "57ff3aabab41e17c73460b706672d24a": "\\gamma={\\rm exp}(-H)/{\\rm Tr}\\, {\\rm exp}(-H)", "57ff4067acbdff7c694ed13400727415": " {S_4 \\over S_3} = {{27\\over25} \\div {16\\over15}} ", "57ff56b3431e46eac587aa2184f1a616": "F_i (x) = (x_1, x_2,\\ldots, x_{i-1}, f_i(x[i]), x_{i+1}, \\ldots , x_n) \\;. ", "57ff94fe63a61c46227e8232e7564880": "dr(n)=n-9\\left\\lfloor\\frac{n-1}{9}\\right\\rfloor.", "57ffd2fde7efa146329fcc5e6ed7f404": "S_n=\\sum_{\\scriptstyle J\\subset\\{1,\\ldots,m\\}\\atop\\scriptstyle|J|=n} \\mathbb{P}\\biggl(\\bigcap_{j\\in J}A_j\\biggr),\\qquad n\\in\\{0,\\ldots,m\\},", "5800445d061fb85a2711b563800acd37": "\\mu= \\frac{1}{\\sqrt{2}} \\left (e^{\\frac{i\\pi}{4}} +e^{-\\frac{i\\pi}{4}} \\right )=1.", "5800b336e0ceebeee97fa8138e08fd78": "a,b\\in{\\mathcal F}", "5800e4ba0dfaf4cfd08059ec0af67e6f": "\\alpha_{X,n}(t)=\\sqrt{n}(F_{X,n}(t)-F(t))", "58015872ed4af3d416f6f3c157ff9102": "\\psi_0=\\sqrt{n}e^{-i\\mu t}", "58015fa723fcd952512e79e6f5386807": "b_1=-k_1(x_1-x_0)+(y_1 - y_0)=-0.3750", "580191b2653595bcaae7657422fd7b9c": "\\left( s,p\\right) ", "5801e14bdf2063cb0ca0a9b880140701": "q = k(\\cos \\theta + \\epsilon \\sin \\theta)", "5801f61fd5ee4574d82da66ce88cdc7d": "R_t,", "58025323a5f74591dc20465ecf3639e6": "y^n", "58025caf866e0327253a060594d31d61": "t\\rightarrow\\infty ", "5802c40f5f187b148f4a42bb1c2172bb": "\\mathbf{x} ~ = ~ \\sum_j a_j \\mathbf{q}_j", "5802fd694c82bbb2800ff919a5ec4891": "(\\part T/\\part P)_H", "580374417891edec0c4a69fb4e8bdad7": "\\mathbf e_j", "58038e96a654d53b073d08cf1192f35f": "\nn_i = \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}-1}.\n", "5803cf9035c46d6648720182fe924951": "\\phi\\,", "5803ec8315a5ce5b67d2dd825aeff95c": "0 \\ge x \\,\\bmod\\, y >y.", "580404ff5058b8e0be06aae88ddd384f": "H = \\langle y \\mid y^5 = 1 \\rangle.", "58040d0df25723fcb68df2062f9777ef": "1 - \\epsilon\\,", "5804151b782c1baf7920869052d1f04e": "\\nu_A", "58044924774545d0e2638e03b13cbb35": "\\eta=1", "58044b5a646cff15c38a157fe23e4812": "[H^+_{ }]_0", "5804650cfb3c97546a2ec76dfea04e82": "p(w|D,\\lambda ,\\mathbb{M}) \\propto p(D|w,\\mathbb{M})p(w|\\lambda ,\\mathbb{M})\n", "5804b3f9ec87a8862bf2f474918a161b": " \\frac{V^2}{2.g} +\\frac{p}{\\rho.g} = \\mathrm{constant}", "5804d04cc7838c36c1af010a9b21f89b": "\\quad\\frac{\\partial\\vec V}{\\partial x^i} = \\frac{\\partial v^j}{\\partial x^i} \\frac{\\partial\\vec \\Psi}{\\partial x^j} + v^j \\frac{\\partial^2 \\vec\\Psi}{\\partial x^i \\, \\partial x^j} ", "5804e3743fb33f9b44de0256624287ba": "e^{\\lambda t}", "5805bc575426c1f67c9ecf81292c2fee": "V = \\frac{1}{3}\\left(21+14\\sqrt{2}\\right)a^3 \\approx 13.5996633a^3.", "58060cdb1efaf52ff1aa21be7d4c686b": "\\hat{\\mathbf{n}}(\\theta,\\phi) = \\cos\\theta \\sin\\phi \\mathbf{e}_x + \\sin\\theta \\sin\\phi \\mathbf{e}_y + \\cos\\phi \\mathbf{e}_z", "58060fa58bd0673da6f9d56c7a1abe9a": "\\frac{\\partial f(\\mathbf{x},t)}{\\partial t} = -\\sum_{i=1}^N \\frac{\\partial}{\\partial x_i} \\left[ \\mu_i(\\mathbf{x}) f(\\mathbf{x},t) \\right] + \\sum_{i=1}^{N} \\sum_{j=1}^{N} \\frac{\\partial^2}{\\partial x_i \\, \\partial x_j} \\left[ D_{ij}(\\mathbf{x}) f(\\mathbf{x},t) \\right],", "580661f4e6d41239021308cee95b76a7": " \\frac{dx}{dy} = \\frac{1}{dy/dx}.", "5806764a31430c63a2f0cc7f9da07d26": " g(t) = \\frac1{\\sqrt{2\\pi}} \\int \\limits_{-\\frac{\\Delta f}{2}}^{\\frac{\\Delta f}{2}} 1 \\cdot e^{j 2 \\pi f t} df = \\frac1{\\sqrt{2\\pi}} \\cdot \\Delta f \\cdot \\operatorname{si} \\left( \\frac{2 \\pi t \\cdot \\Delta f}{2} \\right)", "5806b988811b0e7990e4c03cacbbf296": "c_0 = \\frac{1}{\\sqrt{\\mu_0 \\epsilon_0}}", "5807041489e7f2125897b33a27ca31ff": "\\gamma_B \\approx 1", "580714913cf36f8c6a2955b620ac2739": " I_\\nu (x) = i^{-\\nu} \\; G_{0,2}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} - \\\\ \\frac{\\nu}{2}, \\frac{-\\nu}{2} \\end{matrix} \\; \\right| \\, -\\frac{x^2}{4} \\right), \\qquad -\\pi < \\arg x \\leq 0 ", "58073c4b3c2da2d211d867a709402a8d": "ZZ_3 = Z_3^2 = 12", "58073eb3ddf26e49b945d05339aadcfe": "\n\\text{(Eq. 10)} \\qquad \\Delta(t) \\leq B + \\sum_{n=1}^N\\sum_{c=1}^NQ_n^{(c)}(t)E\\left[\\lambda_n^{(c)}(t) + \\sum_{a=1}^N\\mu_{an}^{*(c)}(t) - \\sum_{b=1}^N\\mu_{nb}^{*(c)}(t)|\\boldsymbol{Q}(t)\\right] \n", "58077d2ae5fd04b13b41caacb0a03245": "\\mathrm{Im}(\\gamma) = \\mathrm{Re}(\\tilde{k}) = k", "5807cb7b762d864e3d4d47b9d3f415c4": "(u,v)\\in D\\subset\\mathbb{R}^2", "58084af0e41516bf3464651b74793628": " \\int_0^\\infty x^2\\phi(x)\\Phi(bx) \\, dx = \\frac{1}{4} + \\frac{1}{2\\pi} \\left(\\frac{b}{1+b^2} + \\arctan(b) \\right) ", "580851688b49a0b995a18df4e5d9146e": "\n\\begin{align}\n\\delta J(y)(h)&=\\left.\\frac{d}{d\\varepsilon} J(y + \\varepsilon h)\\right|_{\\varepsilon = 0}\\\\\n&= \\left.\\frac{d}{d\\varepsilon} \\int_a^b (y + \\varepsilon h)(y^\\prime + \\varepsilon h^\\prime) \\ dx\\right|_{\\varepsilon = 0}\\\\\n&= \\left.\\frac{d}{d\\varepsilon} \\int_a^b (yy^\\prime + y\\varepsilon h^\\prime + y^\\prime\\varepsilon h + \\varepsilon^2 hh^\\prime) \\ dx\\right|_{\\varepsilon = 0}\\\\\n&= \\left.\\int_a^b \\frac{d}{d\\varepsilon} (yy^\\prime + y\\varepsilon h^\\prime + y^\\prime\\varepsilon h + \\varepsilon^2 hh^\\prime) \\ dx\\right|_{\\varepsilon = 0}\\\\\n&= \\left.\\int_a^b (yh^\\prime + y^\\prime h + 2\\varepsilon hh^\\prime) \\ dx\\right|_{\\varepsilon = 0}\\\\\n&= \\int_a^b (yh^\\prime + y^\\prime h) \\ dx\n\\end{align}\n", "58085ca76d38292316a3fa5a44304d0b": "\\{A\\}", "58087498050f7b033ea49bbecb3d8ba5": "\\delta^*:\\mathfrak{g}^* \\otimes \\mathfrak{g}^* \\to \\mathfrak{g}^*", "5808829302c573af69fc6aa7f83b41e6": "\\vec{p}", "5808881eec69892a228d46c762c163a1": "\\frac 1{M(1,\\sqrt 2)}", "58089566677ac9245c8d0a74199acec4": "({u}_{1},{u}_{2})\\in Int({I}^{2})", "58089d9b757869c69e360c3ab53d2581": "\\Sigma_{a \\in H} W(a) \\leq k \\cdot \\Sigma_{a \\in T^*} W(a)", "5808cef287031ae9202d34c576899bfb": "r_{d_1}", "5809d09282b33fcd5154956cdc57de14": " \\psi_1\\left(\\frac{1}{2}\\right) = \\frac{\\pi^2}{2}", "580a114fa59f7815a6de1738876edf20": "1-\\frac{N_0}{N}\\,", "580a7e888574f24995b3d893864431df": "\\frac{k_H}{k_T} = (\\frac{k_H}{k_D})^{1.442}", "580af4cca285f034fe38f3a943d4fd59": "\n \\begin{align}\n \\mathbf{b}_1 & = \\mathbf{e}_r = \\mathbf{b}^1 \\\\\n \\mathbf{b}_2 & = r~\\mathbf{e}_\\theta = r^2~\\mathbf{b}^2 \\\\\n \\mathbf{b}_3 & = \\mathbf{e}_z = \\mathbf{b}^3\n \\end{align}\n ", "580bd38965c126f81bbda6d2c68a6bd7": " U_{rotating} = {1 \\over 2}I\\omega^2 \\,;", "580c05de76c6519b95a35ba00f79b77b": "\\dot{\\boldsymbol{x}}(t) = \\boldsymbol{F}(\\boldsymbol{x}(t),t), \\qquad \\boldsymbol{x}(t_0)=\\boldsymbol{x}_0.", "580c0f749691c218ff371adaa3df820f": "P \\vdash (P \\or Q)", "580c16ea125f44b984526b86e1cc5a1c": "\\ M_{heel} = D_{heel} \\times drag \\times sin(\\beta) ", "580c1f7698d16322082052e2951da942": "\n\\tan \\theta = \\cos \\lambda \\tan(15^{\\circ} \\times t)\n", "580c5956f0f73d71efaecf50b47a0341": "f=1/2", "580c65cb2b36232be04c43227805bd35": "\nE + S \\, \\overset{k_{f_1}} {\\underset{k_{r_1}} {\\rightleftharpoons}} \\, ES \\, \\overset{k_{f_2}} {\\underset{k_{r_2}} {\\rightleftharpoons}} \\, E + P\n", "580c6e97e8ae24efc6f77f7e1aa5283e": "D \\mapsto \\mathcal L (D)", "580d0c3dd48ea2dcc7b2edc4b8541d5e": "\\operatorname{tr}(AB) = 0", "580d958f3f932b767647aff8bd5df1f8": "X = \\bigcup U_j, U_j = \\operatorname{Spec} R_j", "580d9a8736b994fca05ee68ded7deb1f": "X \\,", "580daa339fe88655c8aec9865c10a7b2": "n = m^k = m^{ap} = (m^a)^p", "580ddbd7c6ce37e73cbda62d5d2f9546": "\\dot \\epsilon(y,t) = \\left(\\frac{\\partial}{\\partial t}\\frac{\\partial X}{\\partial y}\\right)(y,t) = \\left(\\frac{\\partial}{\\partial y}\\frac{\\partial X}{\\partial t}\\right)(y,t) = \\frac{\\partial V}{\\partial y}(y,t) ", "580e44f6dbe4f6df9b15eaffe2b374ee": "\\bar \\Omega.", "580e4f4559a67c318509f08500e6079e": "\\hbar \\gamma \\approx 1 /, \\mathrm{meV}", "580ea9cdbf5cad0b249d678f529fac24": "[(2+3)\\times4]^2=400", "580ed65aa706930b81882d4468770cc6": "RPI = Retic Count * {Hemoglobin(observed) \\over Normal Hemoglobin}*0.5", "580f6659bca96052d544383e02359812": "\\theta_i, i = 1,\\ldots, n", "580f756646701877bdf5fe2961fafd5b": "{{\\text{H}}_{\\text{3}}}\\text{C-}\\underset{\\underset{\\text{O}}{\\mathop{\\text{ }\\!\\!|\\!\\!\\text{ }\\!\\!|\\!\\!\\text{ }}}\\,}{\\mathop{\\text{C}}}\\,\\text{-OH+HS-C}{{\\text{H}}_{\\text{3}}}\\xrightarrow{\\text{30--50 }\\!\\!~\\!\\!\\text{ }\\!\\!{}^\\circ\\!\\!\\text{ C}\\text{,-}{{\\text{H}}_{2}}O}{{\\text{H}}_{\\text{3}}}\\text{C-}\\underset{\\underset{\\text{O}}{\\mathop{\\text{ }\\!\\!|\\!\\!\\text{ }\\!\\!|\\!\\!\\text{ }}}\\,}{\\mathop{\\text{C}}}\\,\\text{-S-C}{{\\text{H}}_{\\text{3}}}", "580f825bf4c72c298c4862f78f20246d": "\\textstyle \\mathbf{X}", "580fb2c84346a342cdacb0b30042ccbe": "K_{\\rm IIc}", "580fba9d97c53b2cedce6e9e4e48e5aa": "\\displaystyle \\partial_t g_{ij}=-2 R_{ij}", "580fe0b857240605ab7df286fbd9b2de": "k\\in\\{1,2,\\dots\\}", "580fedaa0848267f5e1b4d2f488ffc1d": "\\scriptstyle v = \\frac{2\\pi \\cdot 280000 \\cdot 9,461\\cdot10^{15}\\,m}{230\\cdot 10^{6} \\cdot 365,25 \\cdot 24 \\cdot 3600} \\approx\\, 230\\,km/s", "580ff6287f284a5ddcbe153911e0bf55": "\n\\begin{align}\n P(M1|D) & {} = \\frac{P(D|M1) P(M1)}{P(D)} \\\\\n & {} = \\frac{P(D|M1) P(M1)}{P(D|M1) P(M1) + P(D|M2) P(M2)} \\\\\n & {} = \\frac{1}{1 + \\frac{P(D|M2)}{P(D|M1)} \\frac{P(M2)}{P(M1)} }\n\\end{align}\n", "58103acfced10f2271ae65462d252813": " |\\psi \\rang ", "5810ae6b00de23fb22f56d780dd729e7": "F_\\nu", "5810c925079139239436ea08c1c39b4c": "\\lambda_{max}", "5810eefc1b9d8ae3fa2a7048284c9102": "\\langle h, x \\rangle^{\\sim} = I(h) (x)", "58110411af4949add22c9ec9f5de375c": "W_p(\\mathbf{V}, n)", "581175e56cc92536f9cd4ac2c02319c7": "Q=(T, L)", "5811b4de1df4239ee10eed70d2c3f129": "\\begin{align}\nV_{n-1}(\\mathbb R^n) &\\cong \\mathrm{SO}(n)\\\\\nV_{n-1}(\\mathbb C^n) &\\cong \\mathrm{SU}(n)\n\\end{align}", "5811c448055b041b1e8395d6faaed9a9": "\\tau_{V,W}: V\\otimes W \\rightarrow W\\otimes V,", "5811da8178ef630c945003c4718275e0": "H = \\sum_i^n p_i \\log_{2}(1/p_i + 1)", "58122737659de46fe40a9a2e8c2a361c": "\\phi(z)", "5812e016c20b57f91f29d1f5087910a3": "\n \\begin{align}\n \\delta U & = \\int_{\\Omega^0} \\left[-\\frac{1}{2}~(N_{\\alpha\\beta,\\beta}~\\delta u^0_{\\alpha}+N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta})\n + \\frac{1}{2}(M_{\\alpha\\beta,\\beta}~\\delta \\varphi_{\\alpha}+M_{\\alpha\\beta,\\alpha}\\delta\\varphi_{\\beta}) - Q_{\\alpha,\\alpha}~\\delta w^0 - Q_\\alpha~\\delta\\varphi_\\alpha\\right]~d\\Omega \\\\\n & + \\int_{\\Gamma^0} \\left[\\frac{1}{2}~(n_\\beta~N_{\\alpha\\beta}~\\delta u^0_\\alpha+n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta})\n- \\frac{1}{2}(n_\\beta~M_{\\alpha\\beta}~\\delta \\varphi_{\\alpha}+n_\\alpha M_{\\alpha\\beta}\\delta\\varphi_\\beta) + n_\\alpha~Q_\\alpha~\\delta w^0\\right]~d\\Gamma\n \\end{align}\n", "581358b398443f5ea09259fd4e3a9efd": "w\\in \\Sigma^*", "58138ff155212d9f9b1fa5241e42d618": "\\boldsymbol \\tau = \\mathbf{r}\\times \\mathbf{F}", "5813bfef72f1f3dfd7f96510846248d5": "w\\leq w''\\leq w'", "58147ce92f4a05fa42303ad334c1c9a2": "( A+uv^T )^{-1}=\\left( I-\\frac{A^{-1}uv^T}{1+v^TA^{-1}u} \\right)A^{-1}= {A^{-1}}-\\frac{A^{-1}uv^TA^{-1}}{1+v^TA^{-1}u}", "58148559beffa06ae2190cd30e559efd": "a, b, c, ...", "5814b889f2b617d1873cc47c518967b0": "\\scriptstyle (xy \\,+\\, 1)x \\,\\notin\\, IJ", "5814b994886860c267db84a977aaea9b": " {\\mathbf A_{11} } ", "5815bc093117cecd79417f9d5793a903": "I(A:BC)\\geq I(A:B)", "5815bc3a2f4a2fd1461c8a81ebae460d": "\\Omega(\\log\nn)", "5815e05bbda1842a0e55be5411b942b2": "O(n \\log^2 n)", "58160779f13d40b1739feb6e19385511": "\\alpha = K(x)+c", "58165f0a18e26109e201e1af275fce32": "\\Delta_0", "5816a3210bbbab1d8dc57a0850a78101": "k_1", "5816a50d9841a256c898339036f21d3f": "\\mathbf{e}_{\\pm}\\times\\mathbf{e}_{0} = \\pm i \\mathbf{e}_{\\pm}", "5816b0df6c82e21c4725a54fcbd95a63": "y=mx+b \\;", "5817c7d0f9309e7af885c09fdd1a2b6d": "f^{0}(0) = 6 + 2 \\times 0 = 6 = x", "5817d9f17934bc89ecbfafd74d43e17b": "K=\\frac{p^2}{2m}", "5817e558f7e65f106dd460c14466d0ad": "F_{\\mu \\nu}", "5817f50ae97b848f82a0ae918341edc8": "\\int_{\\Gamma} g(x)\\,dx = \\int_{\\partial\\Omega}\\int_{-\\mu}^\\mu g(u+\\lambda N(u))\\, \\det(I-\\lambda W_u) \\,d\\lambda \\,dS_u,", "58182bb5102c93c9cad3a3b302f1725a": " f(t)=f_0 \\theta(-t) . ", "58190c186a9e236dfda5ebbeeb09a6aa": "p_0, p_1: R \\rightarrow X", "5819430adcc58e90bc6bea50d4ea408d": "(h,v,w,x,y,z)", "58196eb11a32a10ad0503d47240c7a15": "\n\\hat{\\sigma}^2(x) := \\frac{1}{N-1}\\sum_{j=1}^{N} \\left[ Q(x,\\xi^j)-\\frac{1}{N} \\sum_{j=1}^N Q(x,\\xi^j) \\right]^2\n", "581990b04ad1e8acb24adea9a84c4b2c": " f = \\sum\\nolimits_{|\\alpha|\\le m}a_\\alpha D^\\alpha(\\tau_P\\delta)", "581a12e376465cd8d0634f14bcab3884": "\\Lambda=\\mathbb{Z}_p[[\\Gamma]]", "581a25f106d8340cd093bf356afb46ad": "{\\mathbf{}}t", "581a486129759eacdea1f7742488828a": "x_1,\\dots,x_r,y_1,\\dots,y_r\\in E", "581aa255d30e025a32285e2312b2613d": "E[y_{1i}-y_{0i}]", "581ac0d61cfc3cd7e7b1e3641fd51a97": "n(\\epsilon)=\\frac{1}{e^{\\beta(\\epsilon -\\mu)}-1}.", "581b1f3c0251dff6fcbfdea6f4e0e1cf": "~b~", "581b288aea4db0d11e0e9685fa244bb7": "\\vec{g}", "581b2c581f1390b307b874f7c079d5e0": "\\mu_{\\Phi,\\Lambda}(d\\phi, d\\lambda) = f_{\\Phi,\\Lambda}(\\phi,\\lambda) \\omega_r(\\phi,\\lambda) d\\phi d\\lambda", "581b324758099e44a4653bbd8a8a11ba": "hf = \\phi + E_{k_{max}} \\,", "581b3a7e87339dc302d69b13d19209a0": "Y_{l_1, \\dots l_{n-1}} (\\theta_1, \\dots \\theta_{n-1}) = \\frac{1}{\\sqrt{2\\pi}} e^{i l_1 \\theta_1} \\prod_{j = 2}^{n-1} {}_j \\bar{P}^{l_{n-1} - 1}_{l_j} (\\theta_j)", "581b5b3ceb24568b59ec02b3acf5e1b2": "M(\\lambda)", "581c01c15cc7dd9b681dae2f0c9fe3ad": "\\int_Y^* H_{m-n}(A\\cap f^{-1}(y)) \\, dH_n(y) \\leq \\frac{v_{m-n}v_n}{v_m}(\\text{Lip }f)^n H_m(A), ", "581c47c27802ad2bf403771846164836": " f_{a}\\;(k + 1)", "581c5d7a8cb4c7e80ecf0adea159eece": "e^{k_0t}\\,", "581c8363d044aaeb47b9921e5b7f0fd7": "2^{cn} = (2^c)^n", "581cb40363c73ca512d88f5cda228774": "f(x_1, x_2, x_3)=\\bar{x_1} \\bar{x_2} \\bar{x_3} + x_1 x_2 + x_2 x_3", "581cd69478700702bb0531e11c22142f": "\\frac{P (X_0, \\ldots, X_n)} {X_0^{deg(P)}} \\mapsto P(1,X_1,\\ldots, X_n) ", "581d50a7bcffda53929b70304a18e16d": "w = x_{1}x_{2}\\cdots x_{m}", "581d62eb6df6e9926a43bedcf6d937c9": "\\mu_{ab}^{(c)}(t) = 0", "581d9f84dcbfca5b2c79552876cad0ad": "\\,\\epsilon", "581dd7e8316957398a3876c4a50c7806": "\\begin{pmatrix}\nE(x,y) & F(x,y) \\\\\nF(x,y) & G(x,y)\\end{pmatrix}", "581e061ebbfdf205ce578925e4228bbe": "\\alpha=\\alpha(2)", "581e20f5ca0dbdf5c8a0c77c9583959a": "_{q'+p\\,}\\!", "581e3176a20c9010464b66b8625dd461": " \\mathrm{Distance}( b^\\mathrm{ideal}, b^{k}_\\mathrm{Power~Method})=O \\left( \\left| \\frac{\\lambda_\\mathrm{subdominant} }{\\lambda_\\mathrm{dominant} } \\right|^k \\right), ", "581e53cbf196a0fc814c6210fda7560a": "\\hat{a}|\\alpha\\rangle=\\alpha|\\alpha\\rangle,", "581e6e58a56534230ea0f19d3db164ee": "\\ \\displaystyle Y \\ ", "581f3a1cefc3436decc693b3ac576392": "P_0(y)", "581f40c4c377d2a5f8daad92d93b4d7e": "V'':=\\bigcap_{a\\in A_0}V'_a", "581f74b0ca8ba911e0aeeafaa980af68": "\\alpha = 0010,", "581fb7c8c3b369d95df763246ef47876": "\\phi_n", "581fd81205d3b3aec61178ed9d733897": " 0.5<\\frac{a}{L}<2 ", "581ff571138dc989bebc80c50d7a47c3": "\n\\max \\{\\,\\!u(x_1,x_2)\\mbox{ } :\\mbox{ } p_1x_1+p_2x_2=m\\}", "582014a5a23117fe9ef44fa680839f10": "L[\\vec{X}]_{11} = \\frac{1-g^2}{r^2 \\, g^2}, \\; L[\\vec{X}]_{22} = L[\\vec{X}]_{33} = \\frac{-g'}{r \\, g^3}", "58202cd9c40723ae8514bba7c58adab3": "|S| \\geq 2", "58203be486eaa1546917dcb12289568f": " a = k\\cdot(m^2 - n^2) ,\\ \\, b = k\\cdot(2mn) ,\\ \\, c = k\\cdot(m^2 + n^2)", "58203d6d11d8c77bb9f044d3eb8780ec": "\\sum_{m=-\\infty}^\\infty \\sum_{n=0}^\\infty N(m,n) z^m q^n = 1 + \\sum_{n=1}^\\infty \\frac{q^{n^2}}{\\prod_{k=1}^n ( 1 - zq^k)(1-z^{-1}q^k)}", "5820493ffe16a7d1f54e0256966ce4e4": "\\left[{3\\atop 1}\\right] = 2", "582049ab488e2d63a00bfa4a6229fe7d": "f\\;(x^0,x^1,\\dots)", "58208149b43fc54e66440162e729fa62": "z_7 = x_7 y_1 - x_8 y_2 + x_5 y_3 + x_6 y_4 + x_3 y_5 - x_4 y_6 + x_1 y_7 + x_2 y_8", "5820a424ed913a3d87e03c3ce99ff5b2": "{{({{\\partial \\over \\partial t}+{\\overrightarrow{V_g} \\cdot \\nabla}})q}={D_g q \\over Dt}=0}", "5820c91795c645fa94e593d13c8990e7": " e = \\frac{\\sin \\alpha}{\\sin \\beta} ", "5820cb56cf540857f0c3fe976e3c4612": "p=\\rho_m c_s^2", "5820f34baa85452a0f1dcedc38426231": "\\mathbf{d\\hat u_R} ", "582141f08553c9b305442a259936eddb": "\\Delta F_x = \\Delta L sin({\\beta}) - \\Delta D cos({\\beta})", "58217fbc70a7149151e03c0e64bee272": "\\Phi_n(x) = \\Phi_{qr}(x^{q^{m-1}}).", "5821c0fa07c7cc86eb23f6309967044c": "\\left( i \\frac{\\textrm{d}}{\\textrm{d} x_e} + A_e(x_e) \\right)^2 + V_e(x_e) \\ ,", "5821e7adec9f0b80ed087610758f68fb": "\\mathrm d U_0 \\,=\\, T\\, \\mathrm d S\\, -\\, P\\, \\mathrm d V\\, +\\, \\sum_{j=1}^n \\mu _j \\, \\mathrm d N_j \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, (2)", "58224e5d3dd7d07a1fab59237c9d95ef": "{lb/ft^3}", "58227cdf084e4abb07c00fc90f965577": "a_i = \\frac{f_i}{p^{\\ominus}} = \\phi_i y_i \\frac{p}{p^{\\ominus}}", "5822923ac6e5dde61e7b3bf79ce3a4d9": "w(z)", "582297ef17655c6c102af8a0de4ff555": "\\beta/2", "5822d28d064942fd929216efe9f0abe4": "0/\\exp(z)", "5822ea95c300befe170cc4e68d4b6246": " ddC_{2k-1}=\\rho. \\,\\,\\,", "582328c2c420449ae5c46e9bb27b31b1": "\\frac{\\partial L}{\\partial \\dot{x}_i} = \\frac{\\partial ~}{\\partial \\dot{x}_i} \\left( \\frac{1}{2} m \\dot{\\vec{x}}^2 \\right) = \\frac{1}{2} m \\frac{\\partial ~}{\\partial \\dot{x}_i} \\left( \\dot{x}_i\\dot{x}_i \\right) = m \\dot{x}_i,\n", "5823a53c027cb4563244c3695f49f530": "\\{x, f\\}", "5823c3aa8f611da8e21629042d3e038b": "R' \\otimes_R R[t] \\simeq R'[t]", "5823d17c63fddb67e6f442431543ad8a": "\n\\quad \\ldots~, \\quad Ti_{n+1}(z) = \\int_0^z {Ti_n(t) \\over t} \\,\\mathrm{d}t \\,,\n", "58240e6379de68dbb93c3d74d37afe89": "\\mu_\\text{sig} = \\frac{\\sum_{i=1}^n (X_i - f_i)}{n} \\qquad \\qquad \\mu_\\text{bkg} = \\frac{\\sum_{i=1}^n (X_i-f_i)}{n}", "58244c0c7a2d91baef8403ef7605657e": "\\dot{\\xi}_i(t)=-\\gamma_{ik}\\Xi_k", "58246be434c8c72326c2336d9d9c3ed4": "C = c_0 + c_1 Y^d", "582488ec3df93a93e059071a60b384e0": "V_{A}", "58252cf4a2b33ac88a5e660c4bae642f": " K(n) = a_n K(n-1) - b_{n-1}c_{n-1} K(n-2) . \\, ", "5825926ccc6867eac7785f291c60520f": "n_3=0,\\, n_1\\neq0\\,\\!", "5825da15f87e27438ecc1dbd9f05cdca": "raining(x,s)", "582639ce6fa66f39405632e54a6103b5": "K=\\sqrt{xyz(x+y+z)}.", "5826a0f5d9dce8cf2e5f4a4c43830a8d": " \\bold{j}\\left(\\bold{r},t\\right) = \\left|A\\right|^2 {\\hbar \\bold{k} \\over m} = \\rho \\frac{\\bold{p}}{m} = \\rho \\bold{v}", "5827635004c625cca297eeda8d6e3c55": "\\frac{x}{r(r+z)}\\,", "58276ae39c971a1a90df19ee5cdf0637": "\\sigma_\\text{a}", "5827a07c967847e390e7ee00c8061620": "\\dot{u}", "5827e9748cdc8c4cf09f3b2b2e23f430": "(a \\otimes h)(b \\otimes k) := a(h_{(1)} \\cdot b) \\otimes h_{(2)}k", "58283aa1a8ed37c70eec928dac39dcbd": "\\tan\\tfrac{1}{4}E \n= \\sqrt{\\tan\\tfrac{1}{2}s\\, \\tan\\tfrac{1}{2}(s{-}a)\\, \n \\tan\\tfrac{1}{2}(s{-}b)\\,\\tan\\tfrac{1}{2}(s{-}c)}", "5828ba096bd7c54f312dd13702b92249": "[X_f,T]\\subset T", "5828dca1068da2808ededc74856857c8": "H_\\lambda = \\ker(T-\\lambda).\\ ", "5829080eb735fe2924aee6e6d0a5836c": "\n \\hat{\\mu} = \\overline{x} \\equiv \\frac{1}{n}\\sum_{i=1}^n x_i, \\qquad\n \\hat{\\sigma}^2 = \\frac{1}{n} \\sum_{i=1}^n (x_i - \\overline{x})^2.\n ", "58293341ef0e05ed879a07bffec85394": "\\eta_L\\leqslant\\eta_M", "58296ad5a25b593bd1e714cecb5f6e71": "\\scriptstyle\\mathfrak{S}", "582985b26915471286997b679324f17c": "p_{n+1}\\ ", "5829f6b39cd2a902e0c49d5cce8cde35": "a \\textrm{,}\\, b \\;\\in\\; \\mathbb{N}", "582a2665001a6f1bf0404376d7c2b404": "\\pi/\\sqrt{6} \\approx 1.2825.", "582a2b5e0ecf2be4e5d581f43de38ae3": "\\beta =\\sqrt{k^{2}-k_{t}^{2}}=\\sqrt{\\omega ^{2}\\mu \\varepsilon -\\omega _{c}^{2}\\mu \\varepsilon }", "582a63afccbb112b820733e53e11349d": "{a \\over 2} + {a \\over 6}(\\sqrt{2}-1)", "582a980aa32c858b0a78cb7159cd4827": "f(x)=f(x_0)+(Df(x_0))\\cdot(x-x_0)+\\frac{1}{2}(D^2f(x_0))\\cdot (x-x_0)^{\\otimes 2}+\\cdots+\\frac{D^kf(x_0)}{k!}\\cdot(x-x_0)^{\\otimes k}+\\frac{R_{k+1}(x)}{(k+1)!}\\cdot(x-x_0)^{\\otimes (k+1)}.", "582ac8819bc20c612b0e84ec18a12495": "j=0,1,...,N-1", "582adb9215297abaeb7e148de3c3efd9": "\\left\\{\\left| n \\right\\rangle\\right\\}", "582b173fc4113dc046caf2e955b411bd": "B_2 = -2\\pi \\int {\\Big( e^{-u(|\\vec{r}_1|)/(k_BT)} - 1 \\Big)} \\cdot r^2 d\\vec{r}_1 ,", "582b49e5d13f600cd05cafeff33bfb76": "L=\\sqrt{\\frac{D}{\\Sigma_a}}", "582b5bbd4d53a606c13766246ef2bfba": "\n\\begin{matrix}\n\\tau_I(\\tau) &=& \\tau\\\\\n\\tau_I(a) &=& a \\mbox{ if } a \\notin I\\\\\n\\tau_I(a) &=& \\tau \\mbox{ if } a \\in I\\\\\n\\tau_I(x + y) &=& \\tau_I(x) + \\tau_I(y)\\\\\n\\tau_I(x \\cdot y) &=& \\tau_I(x) \\cdot \\tau_I(y)\\\\\n\\partial_H(\\tau) &=& \\tau\\\\ \nx \\cdot \\tau &=& x\\\\\n\\tau \\cdot x &=& \\tau \\cdot x + x\\\\\na\\cdot(\\tau\\cdot x + y) &=& a\\cdot(\\tau\\cdot x + y) + a\\cdot x \\\\\n\\tau \\cdot x \\vert\\lfloor y &=& \\tau\\cdot ( x \\vert\\vert y)\\\\\n\\tau \\vert\\lfloor x &=& \\tau \\cdot x \\\\\n\\tau \\vert x &=& \\delta\\\\\nx \\vert \\tau &=& \\delta\\\\\n\\tau\\cdot x \\vert y &=& x \\vert y\\\\\nx \\vert \\tau\\cdot y &=& x \\vert y\n\\end{matrix}\n", "582b5bc3dffa761197a5246e888b6590": "1 / n", "582b6149c51e05995cbf2762caa693ba": "\\scriptstyle{\\langle\\psi_m|}", "582b95370b36e8e833d121f11035efb9": "y[k]", "582bf95b2e3a818a1ae87fbe6d876e5d": "ax+by+c=0 \\,", "582bfb6671268cba9854ebd4e8a005ec": "\\mathfrak{A}\\subseteq_\\text{end}\\mathfrak{B}", "582c14c707b9b852638e6f584084456a": "\\displaystyle{ {1\\over 2\\pi}\\iint |F_+(z)|^2 (1-|z|^2)^{-1/2} \\,dx dy+ {2\\over \\pi} \\iint |F^\\prime_+(z)|^2(1-|z|^2)^{\\frac{1}{2}} \\, dxdy}", "582c67ee7f67d163764fb852554e7642": "n = \\log_{\\varphi}\\left(\\frac{F_n\\sqrt{5} + \\sqrt{5F_n^2 \\pm 4}}{2}\\right)", "582d05b6f94d5ed23eb4d60d0d81566c": "0! := 1", "582d07c4e4b7eefb2bff08bf1075e1fd": "\\pi/N", "582d371bbb5ee66ceb01e7b2a814e075": "x=(-e_{n+1}+1)(y-e_{n+1})^{-1}.", "582d40f04103981a88709f9473da8fb2": "1000^{1000^{1000}}\\approx 10^{10^{3000.48}}", "582d4e4bb7c8c44c36ed499e1bbb0ac5": "b = \\frac{R^2 - r^2 - a^2}{2a}.", "582d68783a5a30deb121c52c9f2383be": "\n{\\mathbf C}\\mathbf{X}{\\mathbf C}^T\n\\sim\n\\mathcal{W}_q\\left({\\mathbf C}{\\mathbf V}{\\mathbf C}^T,m\\right).\n", "582d9e6d45c648cf34e75d55f53d37c4": "\n\\mathbf{C} \\ast \\mathbf{D} \n= \n\\left[\n\\begin{array} { c | c | c }\n\\mathbf{C}_1 \\otimes \\mathbf{D}_1 & \\mathbf{C}_2 \\otimes \\mathbf{D}_2 & \\mathbf{C}_3 \\otimes \\mathbf{D}_3\n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array} { c | c | c }\n1 & 8 & 21 \\\\\n2 & 10 & 24 \\\\\n3 & 12 & 27 \\\\\n4 & 20 & 42 \\\\\n8 & 25 & 48 \\\\\n12 & 30 & 54 \\\\\n7 & 32 & 63 \\\\\n14 & 40 & 72 \\\\\n21 & 48 & 81\n\\end{array}\n\\right].\n", "582e24f5c58b7066eb09fdea090f52e5": " q_e = \\int \\lambda_m \\mathrm{d}\\ell ", "582e591d8d1c92da28b9b970ce136265": "4y^{3} - 3y - 1/2 = 0", "582e5b03e545183dbcb8862330d74296": "|S_i|\\le\\sum_{n=1}^i|X_n|,\\quad i\\in{\\mathbb N}.", "582e66f6a27bb911246ac4d7abb993a8": " U=\\int_V u dV \\ \\rightarrow \\ \\frac{\\partial U}{\\partial t} = \\frac{\\partial}{\\partial t} \\int_V u dV = \\int_V \\frac{\\partial u}{\\partial t} dV .", "582e74a8ee946b3c8587a2a4e54570ba": "x \\neq \\frac{\\pi}{2} + k \\pi, k = 0, \\pm 1, \\pm 2, \\ldots", "582e94be14496deb9b235afeea30fa7b": "(f * g )(t) = \\int_0^t f(\\tau)\\, g(t - \\tau)\\, d\\tau\n\\ \\ \\ \\mathrm{for} \\ \\ f, g : [0, \\infty) \\to \\mathbb{R} ", "582ec1b276b9aec983abe32ce08000c9": "s_k = a_1 + \\cdots + a_k", "582ec604465d5cac8f3c386976ebd321": "b = N_A b' ", "582f23dcdb58c67554468d43fd234631": "\\eta(\\theta) = h_1 \\cos \\left( \\beta \\theta \\right)", "582f87a08b37951660a0a7fcc9cf3419": "\\alpha_1, \\ldots, \\alpha_n", "582fed697fac186614adf5d98846767e": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n \\frac{A\\,x^{m+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^q}{a (m+1)}\\,-\\,\n \\frac{1}{a (m+1)}\\,\\cdot\n", "583074d8ad1f3f6d5baef6eb396b65ec": "\\bar{x} = \\frac{1}{n} \\sum_{i=1}^n x_i", "58313d1f37df046859569f8b8507c61d": "\\ C_2^2 (3)=\\frac{17}{50}", "583169a0c2728c07788dee493b40139d": "p=19", "58316edcd4061b6a94378e6c7f946a28": "\\mathcal{C}_m, \\mathcal{S}_{m}", "583190308da18101099591b0cc51c0a3": "W_2\\in \\beta", "5831b0b251ba7ae0f63616662920f99a": "\\scriptstyle[H^+][OH^-]=10^{-14}", "5831d57fdb3af87936d0d46704bec88c": "\\textstyle{\\frac{\\partial}{\\partial x}} \\,\\;", "5831e0f7c54e23ec0dcd158182e5a491": "S(n)=S(n-1)S(n-2) \\text{ for } n \\ge 2 \\, .", "58321207811dbe8421d7b79f743d6ccc": "\n\\begin{array}{l}\n\\text{Re}\\langle u, v \\rangle = \\frac{1}{2}\\left(\\|u+v\\|^2 - \\|u\\|^2 - \\|v\\|^2\\right), \\\\[3pt]\n\\text{Re}\\langle u, v \\rangle = \\frac{1}{2}\\left(\\|u\\|^2 + \\|v\\|^2 - \\|u-v\\|^2\\right), \\\\[3pt]\n\\text{Re}\\langle u, v \\rangle = \\frac{1}{4}\\left(\\|u+v\\|^2 - \\|u-v\\|^2\\right).\n\\end{array}", "58322584f687192261414fc397bf741d": "x > x_\\text{min}", "58322a086de647f75f01a361acad9fcd": "\\Delta_2^{\\prime}F(J)^{calculated} = (2B^{\\prime\\prime}-3D^{\\prime\\prime}) \\left(2J+1\\right)-D^{\\prime\\prime}\\left(2J+1\\right)^3", "58324448d68015d00b09d129dc294f98": " \\textstyle \\nu", "5832812ba5df6e31d15203fef2a10f57": " D = \\frac{\\mu k_B T}{q} = \\frac{k_b T}{f} ", "58328756bee692b8dd87b4a3e2bb6e0a": "L \\le (\\mbox{SAT}, \\epsilon-\\mbox{UNSAT})", "58329143ebf8c41536bff883f40324eb": " s^2 F(s) \\ ", "5832e70f6b4ea16193158b2a952b1c58": "3 < 4", "58332099bd6cf98368c5531404ae0926": "t_1 > t_0", "5833c822d4ccbeaaacdba45d68f21668": "\\left | Y \\right | = \\sqrt {G^2 + B^2} \\,", "5833c83084eb823ea9e5228be440af46": "L=x_{2}-x_{1}\\,", "5833f6594052c5510f22428a472c63ca": "{}^2E\\colon \\mathbb{N}^{\\mathbb{N}} \\to \\mathbb{N}", "58341b3a32d7116d428f1c4b4a58b378": "\\scriptstyle g_{ab}", "5834f293fd7773d79324b384e2205612": "K_a\\approx\\frac{175}{198.04}(X_n+Y_n)", "583501c61594f4cde283a1a5f0d314d7": "q=1", "583510362a314c8fca1bacd95fa24a9b": "[0,1]\\times [0,1].", "583577598a9af80f5ec6850825686a21": "M = Ng\\mu_B\\langle m \\rangle = NgJ\\mu_B B_J(x)", "5835874a493798bc2b6c37af8608a902": "\\ q \\,", "5835b338a8c954b4c9cae1277a1a6ac6": "S_{(\\Delta x, \\Delta_y)}", "5835cb5fbd49597e1d52fa8438a5e137": "\\rho = \\frac{-2x_0x_{n+1}+x_1^2+x_2^2+\\cdots+x_n^2}{t^2}.", "5835fc1c830fd1f221e2f0c60d06f659": "\\{B_\\theta:\\theta\\in {\\rm pcf}(A),\\theta<\\lambda\\}", "58363e18323f26c897ec1ea783ab638d": "C_1=\\left(\\frac{1}{2}r\\left(1+r\\right),r\\sqrt{1-r}\\right)", "5836b42e1bb134122dd1f676f9763728": "a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2=0 \\,", "5836bdbe0aa95dbad8b6bd235fb6f32d": "= a C \\frac{\\sin\\frac{ka\\sin\\theta}{2}}{\\frac{ka\\sin\\theta}{2}}\\left(\\frac{e^{-iNkd\\frac{\\sin\\theta}{2}}-e^{iNkd\\frac{\\sin\\theta}{2}}}{e^{-ikd\\frac{\\sin\\theta}{2}}-e^{ikd\\frac{\\sin\\theta}{2}}}\\right)\\left(\\frac{e^{iNkd\\frac{\\sin\\theta}{2}}}{e^{ikd\\frac{\\sin\\theta}{2}}}\\right)", "5836c73a3ac240ecd1c6b43d604c95f9": "\\frac{\\partial^2 a}{\\partial x^2} + i 2 k_0 n \\frac{\\partial a}{\\partial z} + k_0^2 [n^2 (I) - n^2] a = 0", "5836d5295a0e7ececbcf7662ef00d6ad": "= \\int u \\,dx + \\int \\left(-v\\right)\\,dx", "58370fd0021dacc7cec0716796b9589e": "5x^2+6xy+5y^2+6y-5=0:\\ \\hbox{an intersection of multiplicity 3}", "58371d417906f612461a0b6e167235cb": "\\beta_1 = 0", "58380f8b1d7bd255059966e4931d5c16": "{\\star\\mathcal{D}^{\\mu\\nu}}u_\\nu= \\frac{1}{\\mu}{\\star F^{\\mu\\nu}} u_\\nu", "58382c66873d15fd36ab0d45727b8284": "\\Delta v ", "58385e3332b4fc1590ba4c143655db7b": "m_g\\,", "5838674033b767c8bd79b4739ba42b57": "\n\\sigma_{\\hat{X}}^2 = \\frac{1/\\sigma_{Z_1}^2 + 1/\\sigma_{Z_2}^2}{1/\\sigma_{Z_1}^2 + 1/\\sigma_{Z_2}^2 + 1/\\sigma_X^2} \\sigma_X^2 ,\n", "5838c5e2bec5a7eecfaf3a78af5a0bbb": " u_{\\alpha} u^{\\alpha} = -1", "5838c96aa234d6a5a496892d1e7e886c": "\\rho(q) = \\sum_{n\\ge 0} {q^{n(n+1)/2}(-q;q)_n\\over (q;q^2)_{n+1}}", "58391dbb3b000d69e3934084dd2471ee": "\\begin{align}\n\\frac{\\delta Z^i_\\alpha}{\\delta t} & = \\nabla _\\alpha \\left( CN^i \\right) \\\\[8pt]\n\\frac{\\delta N^i}{\\delta t} & = -Z^i_\\alpha \\nabla^\\alpha C\n\\end{align}", "58393bed30fc1460d495abc6bb38abde": " \\operatorname{def}[o] \\and \\operatorname{ask}[\\_] \\and p \\subset \\{p, q, m\\}) ", "5839778489277d414036fd3496bbcc63": "x \\in {\\mathbb R}", "583977c4006cb620f4d9bea7e2c231b0": "B \\to D K, D \\to K_S \\pi^+ \\pi^-", "5839e419438d08629d56528a5e116b53": " X = UU^T X VV^T = U (U^T XV) V^T = U \\Sigma V^T ", "583a2c5fe970cd75b2aab2ba48da9c8b": "r_c =3.4 R_\\odot", "583a701c6e62645365d30e3a41ec23d0": "\\hat{a}_i \\,\\hat{a}_j^\\dagger = \\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij} = \\hat{a}_j^\\dagger \\,\\hat{a}_i + \\hat{a}_i^\\bullet \\,\\hat{a}_j^{\\dagger\\bullet} =\\,\\mathopen{:}\\,\\hat{a}_i\\, \\hat{a}_j^\\dagger \\,\\mathclose{:} + \\hat{a}_i^\\bullet \\,\\hat{a}_j^{\\dagger\\bullet} ", "583a82bb9cdde24c96353b9b889e2e96": "\n \\boldsymbol{\\varepsilon} = \\boldsymbol{\\varepsilon}_{\\mathrm{e}} + \\boldsymbol{\\varepsilon}_{\\mathrm{vp}}\n ", "583a8588d5b42af592e1124e0501b4fc": " \\psi(\\vec{r}_1, \\vec{r}_2, \\cdots, t) ", "583a9a035f91696d49c1f735f496ac4a": "\n\\begin{align}\n \\Phi_{u_g}(\\Omega)&=\\sigma_u^2\\frac{2 L_u}{\\pi} \\frac{1}{ \\left(1+ (1.339 L_u \\Omega)^2 \\right)^\\frac{5}{6}} \\\\\n \\Phi_{v_g}(\\Omega)&=\\sigma_v^2\\frac{2L_v}{\\pi} \\frac{1+\\frac{8}{3}(2.678 L_v \\Omega)^2}{\\left(1+ (2.678 L_v \\Omega)^2 \\right)^{\\frac{11}{6}}} \\\\\n \\Phi_{w_g}(\\Omega)&=\\sigma_w^2\\frac{2 L_w}{\\pi} \\frac{1+\\frac{8}{3}(2.678 L_w \\Omega)^2}{\\left(1+ (2.678 L_w \\Omega)^2 \\right)^{\\frac{11}{6}}}\n\\end{align}\n", "583afdca7902aeeb53bcb5b5b1ad7fed": "p_i \\,", "583b0f1e5991205ecdd159f70f241711": "h(x)=e^{i \\, 2\\pi \\, x \\,\\xi_0}f(x),", "583b32eb1709ecefe3c3a1c11e5e443b": "Y(t) = A + { K-A \\over (1 + Q e^{-B(t - M)}) ^ {1 / \\nu} }", "583b45835dfadc748a67eda05081929e": "\n \\mathbf{M}_x = \\int_A \\mathbf{r} \\times (\\sigma_{xx} \\mathbf{e}_x + \\sigma_{xy} \\mathbf{e}_y + \\sigma_{xz} \\mathbf{e}_z)\\, dA \n \\quad \\text{where} \\quad \n \\mathbf{r} = y\\,\\mathbf{e}_y + z\\,\\mathbf{e}_z \\,.\n ", "583b4e282bd686d19bd6f118bc4a80ba": "c^{2^S} \\equiv (z^Q)^{2^S} \\equiv z^{2^SQ}\\equiv z^{p-1} \\equiv 1 \\pmod p", "583ba1126bf2a9f5ff8f6b46cab2a7fa": " \\hat r_j ", "583ba4cf7eb6f3243ca037abcb76d3a1": "\\mathcal{SN}(\\mu,\\, \\sigma^2,\\gamma)", "583bb83bc46ef807bdd5a4cbcab9dd6d": " \\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C} ", "583bce4fe47e5223e72d7c13e7bb1556": "a^{m}", "583bd42afb35fcf7905a76f594a35fec": "\\nabla \\gamma_{13} =(\\cos\\gamma_{23})^{-1}(- F_{13}\\cos\\gamma_{12} + F_{23}\\sin\\gamma_{12})", "583bf14e8d0143c93a24d107df12a436": "( -200 - 200.0000001 ) / 2 = -200.00000005,", "583c3d623e48fae06abfa9883be7b567": "\\delta_i^j = \\int\\frac{d^dx}{(2\\pi)^d}\\psi^{\\dagger j}(x)\\psi_i(x).", "583c554a0ca7a19c3226113c7f378530": " \\mathbb{Z}^n", "583cb95e0e452beaf2b206f6f05a1f3b": "\\operatorname{E}(\\theta_i|y_i) = {\\int (\\theta^{y+1} e^{-\\theta} / {y_i}!)\\,dG(\\theta) \\over {\\int (\\theta^y e^{-\\theta} / {y_i}!)\\,dG(\\theta}) }.", "583cd7a8014ed04553ea269b2816b0f4": "\\mathrm P(A_n, \\ldots , A_1) = \\mathrm P(A_n | A_{n-1}, \\ldots , A_1) \\cdot\\mathrm P( A_{n-1}, \\ldots , A_1)", "583dd69a42f10ff37504a36e9e4c0996": "\\mathcal{N}=(2,2)", "583dd8361209a438f736038b63df7d6e": "M^{\\textit{d}} = M", "583e0f166f921021c194dab9f41536eb": "C_xH_y + zO_2 \\to xCO_2 + \\frac{y}{2}H_2O", "583ef954b542813afeeb9625fbb9542a": "S_0=R_0=\\frac{1}{2}", "583f4c6f2ce887456439e77a96d3328c": "|p,\\sigma ,n \\rangle ", "583fb962c1e19e8e67fce4840c8cfc7d": "K' = K'_0", "583fd63d6332bfc1d27b8cf188f22e72": "S^{1}(n)=S(n),\\ \\ S^{m}(n)=S(S^{m-1}(n)),\\ \\text{for}\\ m\\ge2.", "58401a3e2fdcd561783cce27fa5413f2": "h^{1,1}", "5840984844824dfbf4b64aeed342f5c5": "\\mathbf R:= f\\mathbf\\varphi", "5840b885ea0c4d463e2dabb29de95cf7": "1-\\frac{\\epsilon}{2}", "5840fa8ac34bfbe6d2df6ae2d7e565ca": "4 \\pi x 10^{-7}", "5841147003bc2a52ade8f52a5e5d57b6": "\n{\\rm Var}\\left[ {\\bar x} \\right]\\,\\,\\, = \\,\\,\\,{{\\sigma ^2 } \\over n}\\,\\,\\gamma _2", "5841586768455967936d177d759d4665": "\\sum_{p=0}^{q-1}\\psi(a+p/q)=q(\\psi(qa)-\\log(q)),", "58419ca04c689c6d707530db3d8b8121": "\\displaystyle (q,p,b)\\rightarrow (-q, -p+2q^2+t,1-b).", "5841c05466c8cb47c120f7a3166a74b3": "\n \\begin{matrix}\n a\\uparrow\\uparrow b & = {\\ ^{b}a} = & \\underbrace{a^{a^{{}^{.\\,^{.\\,^{.\\,^a}}}}}} & \n = & \\underbrace{a\\uparrow (a\\uparrow(\\dots\\uparrow a))} \n\\\\ \n & & b\\mbox{ multiplied copies of }a\\uparrow\n & & b\\mbox{ multiplied copies of }a\\uparrow\n \\end{matrix}\n ", "5841cabf6fc6474f042dd6a4b2966dc9": "\\tilde{A}+\\tilde{\\delta A}", "58422930f4d3bac9edbad1eb32602969": "\\mathbf{M} = \\mathbf{T^\\prime R^\\prime}", "58423ae8e0496eb3a050239b96ca8c96": "(\\mathbb{H}\\otimes\\mathbb{O})P^2", "58424e42ef120bb5b699514da6ccfeed": "i*", "58425c2279c79c00bf5129c902508df9": "e^*_i \\in E^*, ~ v^*_j \\in e^*_i", "5842846ae417bb297dc08cdbdee78dbf": "S_0 \\subset S_1\\subset \\dots\\subset S_r", "5842a21d9587a864758a10bddd2187ad": "E_{\\mathbf{R}}=(3/2)\\sqrt{k} E_{\\mathrm{h}}", "5842cd3140a03853d0ecb4fd59a94937": "\nP = \\lim _{\\Delta t\\rightarrow 0} \\tfrac{\\Delta W(t)}{\\Delta t} = \\lim _{\\Delta t\\rightarrow 0} P_\\mathrm{avg}\\,\n", "58439218d559cd2d2bd804d5378ed20c": " \\frac {d \\hat u_R } {dt} = \\frac {d \\theta } {dt} \\hat u_\\theta \\ , ", "58439dcb6aaedd7de43fa9895f9326ff": "x_i=\\frac{n_i}{n_T}", "5843c723c9c69515f20c593db0abd3df": "x^2+y^2=a^2 \\,", "5843f344353d9f5328737936c5d9bd8c": "Q_T P_W Q_T", "58441994d05f311c4d120fb1fc794847": "\n(b_2 b_1 - c_1 a_2) x_2 + c_2 b_1 x_3 = d_2 b_1 - d_1 a_2\n\\,", "584459aaad3edca843772a38ecede3ed": "E=T_{e}P_{e}^{'}+E_{e} \\,", "5844a4cd208402221ff6291489031efe": "O(n^2/2^{2j} + n)", "5844afeb53af6ec2e0bdf90a6d10215d": " -\\hat\\chi(\\omega) = i\\omega\\beta \\int\\limits_0^\\infty \\mathrm{e}^{-i\\omega t} A(t)\\, dt.", "5844ca1067ac42468a4b08b1f266ab7a": "\\{ |k \\rangle | 0 \\le j \\le d-1 \\} ", "5844d6a076f8adb0bc97240e2d143bc3": "(\\alpha f)(c) = \\alpha f(c).", "5844f38a38b185fbc9489a9a8bdd9fc9": " \\frac{\\Delta \\alpha}{\\alpha_\\mathrm{em}}= \\left(-0.6\\pm 0.6\\right) \\times 10^{-6}.", "5844f9a5bfb16261d237c7cb4f8f5392": "k_1,\\dots , k_s", "584584f1e079d5f86e2d88e01d06162a": "\\mathrm{null} (A^\\mathrm{T})", "5845983da815ca00f514315fb1adb5da": " \\hat{x}\\psi(x) = x \\psi(x) = x_0 \\psi(x) ", "5845c09be56bd43a0395f86aaaaf7349": " \\scriptstyle \\omega", "58462793bf23541622615ba6a075e4da": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{z}{w} \\right) =\nw^{1-a_p} \\sum_{h=0}^{\\infty} \\frac{(w - 1)^h}{h!} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_{p-1}, a_p-h \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n < p.\n", "58462ded319156d774f236dfd3629e79": "\\sin(y) - \\sin(x) \\approx (y - x)\\cos(y)", "5846c632551ecb8c7998d3c2b4d069c1": " \\vec \\mu_{y=0}, \\vec \\mu_{y=1} ", "5846ecfd45b45cd21aeb58cf91ccea8a": " (\\hat{a}_0+\\mathsf{A})^* = \\hat{a}_0 - \\mathsf{A}.\\!", "5847b57e59b9273e3c1a4df6ce937190": "n_\\text{clause}^\\text{right}", "5847eb8c1227f7a82cf74abf7b67c2cf": "A^t", "58487a4fdb836a68714b2841e6a47bb0": "(8)\\qquad \\Psi_{ERN}\\,=\\,\\ln\\frac{L}{L+M}\\;,\\quad L=\\sqrt{\\rho^2+z^2}\\;,", "58489e8d89ee592cf828bc57b79994f5": "E_+ = E_- ", "5848acdc3b7af09e7dc215d9618d3446": "35 = 32 + 2 + 1 = 2^5 + 2^1 + 2^0.", "5848ecabe8566cec59049cc06308fcbe": "\\Omega_X^p", "5849ceb6097ab571b4fc8f549d860b7c": "\\frac{4\\pi}{b}", "5849ebcf66f7693c88323e49bf574dbf": "\\langle a \\rangle p\\,\\!", "5849f9b29b47ccb71c16051384361411": "p(x) = x^2 - 2", "584a400424739f90ba41eb9754aa74fb": "\\mathfrak{M}^2 \\ge 1", "584a81dbf5bf6aa737ba43567ad6307b": "n_i", "584aa49bf0952d0bd458361f3e32c3d8": "(f * h)(x) = \\int_G f(g) h(g^{-1} x) d \\mu(g)", "584ac87579a17bb36659872fea37787b": "\\neg\\;(x \\# y) \\;\\to\\; x = y", "584b9d6bbae20527580f446d896ddae8": " U_E(r) = -\\int_\\infty^r q\\mathbf{E} \\cdot \\mathrm{d} \\mathbf{s} = -\\int_\\infty^r \\frac{1}{4\\pi\\varepsilon_0}\\frac{qQ}{s^2}{\\rm d}s = \\frac{1}{4\\pi\\varepsilon_0}\\frac{qQ}{r} = k_e\\frac{qQ}{r} ", "584bee4ec6bcc88b50719c54b4e37d44": "\\|x\\|=0 ", "584bf4477385c937e91f73d101c8faf6": "\\textstyle{\\frac {\\log(8)} {\\log(4)} = \\frac{3}{2}}", "584c27e881210bec828dce27d0260261": "T(v+v') = T(v)+T(v')", "584c7a1122fc30af334648b8eaffc43f": "X \\le _{lr} Y", "584cad7bce8d34aa0bc282715bcc442a": "\\omega'_0=\\frac{\\partial \\omega(k)}{\\partial k} |_{k=k_0}", "584cf78d7f26434d53a8f77ba6880510": "\nX=\\begin{vmatrix} \n0 & x_{12} & x_{13} &\\cdots & x_{1n} \\\\ \n-x_{12} & 0 & x_{23} &\\cdots & x_{2n} \\\\\n-x_{13} & -x_{23} & 0 &\\cdots & x_{3n} \\\\\n\\vdots& \\vdots & \\vdots &\\ddots & \\vdots \\\\\n-x_{1n} & -x_{2n} & -x_{3n} &\\cdots & 0\n\\end{vmatrix},\nD=\\begin{vmatrix} \n0 & \\frac{\\partial} {\\partial x_{12}} & \\frac{\\partial} {\\partial x_{13}} &\\cdots & \\frac{\\partial}{\\partial x_{1n} } \\\\[6pt]\n-\\frac{\\partial} { \\partial x_{12} } & 0 & \\frac{\\partial} { \\partial x_{23}} &\\cdots & \\frac{\\partial}{\\partial x_{2n} } \\\\[6pt]\n-\\frac{\\partial} {\\partial x_{13} } & -\\frac{\\partial} {\\partial x_{23}} & 0 &\\cdots & \\frac{\\partial}{\\partial x_{3n} } \\\\[6pt]\n\\vdots& \\vdots & \\vdots &\\ddots & \\vdots \\\\[6pt]\n-\\frac{\\partial} {\\partial x_{1n} } & -\\frac{\\partial} {\\partial x_{2n}} & -\\frac{\\partial} {\\partial x_{3n}} &\\cdots & 0 \n\\end{vmatrix}.\n", "584d177cbb8079ab3ca2f06eb7f86d06": "s_0,t_0 ", "584d19f6c21724d88a98227dd0153b70": "\\int_0^{\\pi/2}\\sin^{2} x\\, dx=\\int_0^{\\pi/2}\\cos^{2} x\\, dx=\\pi/4", "584d2e4cb07e994f175af9fedf33e09c": "\\lambda_1 \\,", "584dcfba031f74a7d055818604bc185b": "\\epsilon^{IJKO} \\epsilon_{LMNO} = - 6 \\delta^I_{[L} \\delta^J_M \\delta^K_{N]} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Eq.3", "584ddb30d9597a4aace228e463aca5c0": "f(J,T)", "584df6687aa9e358347f4eeab612c3ce": " \\Delta Y ", "584e13da003c4e5a25361f00f36c1acd": "C_n = \\frac{1}{n+1}{2n\\choose n}", "584e780e61c39ff2720dfbb95afa3a7d": " P(\\vec{x}) = \\sum_\\alpha c_\\alpha x^\\alpha ", "584eac4be256dd5526abd36cc4479e0d": "f\\geq 0\\,", "584ef8b66ca8e3585b4ce48637aa2dc3": "\\cdots \\rightarrow R(-i)^{b_i} \\rightarrow \\cdots \\rightarrow R(-2)^{b_2} \\rightarrow R(-1)^{b_1} \\rightarrow R \\rightarrow k \\rightarrow 0.", "584f264a2480f70d1cd5bc27354759d0": "\\tfrac{1}{2} (e_i e_j + e_j e_i) = \\Bigg\\{ \\begin{matrix} -1, 0, +1 & i=j, \\\\\n 0 & i \\not = j. \\end{matrix} ", "584f4b2783134f1785b8be648fa0cd18": "f(a^s) \\leq f(b^t) \\leq f(a^{s+1}). \\, ", "584f85daa6f655c3abfddd57d114bd05": "\\dot{Q}=\\frac{T_i-T_o}{R_i+R_1+R_2+R_o}=\\frac{T_i-T_1}{R_i}=\\frac{T_i-T_2}{R_i+R_1}=\\frac{T_i-T_3}{R_i+R_1+R_2}=\\frac{T_1-T_2}{R_1}=\\frac{T_3-T_o}{R_0}", "584f94a099b3684f012712dd4fa1c2be": "d = {{2D^2}\\over{\\lambda}},", "585033e02e0b823fb5b07ec5e0b957b5": "\\text{Utility} = aW_T^a", "585036be68324b8fb5101260301117a8": "\\scriptstyle z \\,=\\, f(x,y)", "58507ce1346357e57149e8624b055301": "s \\equiv \\langle s_{\\alpha}| \\alpha < \\gamma\\rangle", "5850ba88a0ce948679cc4218d95de79f": "i0", "58840b530f14a597f336fe9c9a913f11": "\\alpha\\leq\\zeta_0", "5884476e33d07997dbcd2b0abf025355": "r_h \\approx 1.3 a", "5884861d3f668030faec4f4c317ee8c0": "\nf(q) = \\sum_{n\\ge 0} {q^{n^2}\\over (-q; q)_n^2} = {2\\over \\prod_{n>0}(1-q^n)}\\sum_{n\\in Z}{(-1)^nq^{n(3n+1)/2}\\over 1+q^n}\n", "5884ad958e34f0e03a1d70966d84f010": "\\left\\vert \\widehat{\\beta}_{LD}\\right\\vert\n>\\left\\vert \\widehat{\\beta}_{FE}\\right\\vert >\\left\\vert \\widehat{\\beta}\n_{FD}\\right\\vert ", "5884ec8e210d50a659f539bba6240b3f": "P_{cc}(\\theta;\\zeta)", "5885062fe825dc7074a9d9a38566fc0d": " \\varepsilon_i = \\frac {1}{E} \\left [ \\sigma_i(1+\\nu) - \\nu \\left ( \\sigma_x + \\sigma_y+\\sigma_z \\right ) \\right ] ", "58851446d6fc5ccfd9491f29e9c655d6": " p^v_m ", "588586eb9554f42ccf6d1f3693c372c6": "(p^2+q^2)^2", "5885f0465d40f68606593fa73c0e3678": "R(K)", "58861ca7aa1010da6af891c97d90ba43": "V(R)=c_m R^m", "588620b0ddf03b2dd2bb0b873f378887": "\\mbox{N}\\,", "588636fbea6dca4765c43a3773ae97d6": "{\\ H = U + PV }.", "588644c0d9acb57e034c9f972a0d9529": "d_\\odot", "5886711f4d4e6416b7d6d43d1b2312fe": "\\sum_{n=0}^{N-1} A e^{i \\varphi_n} = 0", "5886e20b5ad5e156dcc7b4d4bcf638e9": "H_{0} = -\\log \\left ( a_{H^+} \\frac{\\gamma_B}{\\gamma_{BH^+}} \\right )", "5886f3508819d26ac2b173094f4fdcd9": "\n\\int\\, \\tau^{i j}(t-r,\\vec{x}')\\, \\mathrm{d}^3x'\n=\n\\int\\, x'^i x'^j \\nabla_k \\nabla_l \\tau^{k l} (t-r,\\vec{x}')\\, \\mathrm{d}^3x'\n", "588743bfab258a74bbda1eafb8378077": "I = \\frac{|E|^2}{2 \\eta}", "58874aef65ae6e6fee7e6bb76d7f027a": "E(|X|)=E(|X|,|X|>K)+E(|X|,|X|0,", "588eed15adeaa1dc2e92d5f385da3cbd": "S_k,\\ldots,S_1", "588f3a0939b4ff97ebf8f116f73c13a9": "x + y = 1", "588f643fa76c40ccd7ea2864eccb36c3": "D \\approx 3.57(\\sqrt{2}+\\sqrt{70})", "588f6fad9b08f011b727f17ecb9963bc": "K = \\mathbb{Q}(\\theta)", "588f85f674c9ff8e5a03a9608aa5ebb1": " \\begin{align}\n& \\int_0^1 h_1(f_1(y)) \\, \\mathrm{d}y = \\int_0^{1/6} \\frac{3y}3 \\, \\mathrm{d}y + \\\\\n& \\quad + \\int_{1/6}^{1/3} \\frac{2-3y}3 \\, \\mathrm{d}y + \\int_{1/3}^{2/3} \\frac{ 2 - 1.5(1-y) }{ 3 } \\, \\mathrm{d}y + \\int_{2/3}^1 \\frac56 \\, \\mathrm{d}y = \\frac12 \\, ,\n\\end{align} ", "588f8909e312148506691a4d2183cff2": "d \\rho/dt", "58900ced31a4f710ac6f2564912cbd79": "(6)\\; L=220*0.5*0.1578=17.4\\;m", "58901162b0347a393c75999129e5ebfe": "d\\mu=\\frac{dxdy}{y^2}", "58903c741f63de59f29943c1749f4a1c": "\\mbox{P2 }:\\begin{cases}\nu_{xx}(x,y)+u_{yy}(x,y)=f(x,y) & \\mbox{ in } \\Omega, \\\\\nu=0 & \\mbox{ on } \\partial \\Omega,\n\\end{cases}", "589088a4d068d322ea2548425f56b09a": "i\\,,j", "5890da605c40541e6a5feb8e28470309": " P,\\quad [2]P=P+P,\\quad [3]P=[2]P+P, \\dots , [n]P=[n-1]P+P ", "5890f3689ec6498fc656614ecca033ae": "move(x,y)", "589108c3e0497effc77dd4b7babcee40": "r_1, r_2, \\omega_1, \\omega_2 \\ne 0, \\quad \\omega_1 \\ne \\omega_2.\\,", "58910c972230a869293b067e9f6d5617": "p(r_1,\\dots,r_m) = f_k(r_1,\\dots,r_m)", "589112196e7c01677f06f8650cf7a056": "q= \\begin{bmatrix}\na(p_z) & 0 & 0 \\\\\n0 & b(p_z) & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{bmatrix} p", "58911bec8ca42509e520e436ed506611": "e\\ \\dot{=}\\ e", "589136af88ba975ad129cc23bc7e9726": " \\frac {z}{\\sqrt{2}} .", "58916ed1ab52d58d5e5d79305e0583d5": "\\scriptstyle r_in_i \\,+\\, s_iN/n_i \\;=\\; 1", "5891709e4c6326b299edd5381fd51866": "Y_1 = c_0 + c_1 \\left ( Y - T \\right ) + I + G", "5892269ee8df87b5974d9011c1b5f098": "\\begin{cases} - L u(x) = f(x), & x \\in D; \\\\ \\displaystyle{\\lim_{y \\to x} u(y)} = g(x), & x \\in \\partial D. \\end{cases} \\quad \\mbox{(P1)}", "589228b8ee4815702a86981845bee76e": "(a_1, a_2, a_3, \\ldots, a_n) = (a_1, (a_2, (a_3, (\\ldots, (a_n, \\emptyset)\\ldots))))", "5892603c375ed11c2ac74ff59b69c3cd": "\n\\langle h_{\\mathrm{pot}} \\rangle = \\int_{0}^{\\infty} 4\\pi r^{2} \\rho U(r) g(r)\\, dr.\n", "5892889e285bf077785b13a95168f89c": "x=(x_1, x_2, \\dots, x_n).", "58929ee57be45b1b6f2e9b58812da08b": "1000\\times 10 = 10,000", "5892faffac1261ff1735b7b64c137ec2": " {\\eta} ", "589301d10b656fbfd4c4c0b8cc89b1d9": "\\exp h = \\lim_{N\\to\\infty} \\left(1 + \\frac{h}{N}\\right)^N. ", "58931cc0e66030a4f5397be5eb968e6f": "u \\in C^2(\\Omega) \\cap C^1(\\bar \\Omega)", "58932255dd99c54c8308a0e22e451b62": "C = \\max(0,S_1-S_2-K)", "589331e5feb8c821577e33343fd4616b": "{\\bar{U}}_5", "5893615748d04919003681e55f9ff5bf": "g^{(j^2)}=g", "5893a34850df574fb797f16673236877": "\\gamma^{n} (A) = \\frac{1}{\\sqrt{2 \\pi}^{n}} \\int_{A} \\exp \\left( - \\frac{1}{2} \\| x \\|_{\\mathbb{R}^{n}}^{2} \\right) \\, \\mathrm{d} \\lambda^{n} (x)", "5893be63f06491341da59898186a20c0": " V_t = (1 + \\Gamma_{TL} ) V_i \\, ", "589407525edb485c9511522535a1ef31": "\\textstyle \\Gamma", "58942ac186ff25424bfe34f72b636221": " k = \\|\\mathbf{k}\\| = \\sqrt{k_x^2 + k_y^2 + k_z^2} = {\\omega \\over c}", "58942c14b673787128f071da6b423b93": "|r_1| \\neq |r_2|", "589440f3f9d898bc8d6e1fda59a15898": "F_r\\ \\hat{r}\\,", "589484963af11b35a115230dcf84a8cf": "\\ x ", "5894963b5b8cbc760870387597348302": "\n\\begin{align}\ng_{ij,k} & = (\\mathbf{b}_i\\cdot\\mathbf{b}_j)_{,k} = \\mathbf{b}_{i,k}\\cdot\\mathbf{b}_j + \\mathbf{b}_i\\cdot\\mathbf{b}_{j,k}\n= \\Gamma_{ikj} + \\Gamma_{jki}\\\\\ng_{ik,j} & = (\\mathbf{b}_i\\cdot\\mathbf{b}_k)_{,j} = \\mathbf{b}_{i,j}\\cdot\\mathbf{b}_k + \\mathbf{b}_i\\cdot\\mathbf{b}_{k,j}\n= \\Gamma_{ijk} + \\Gamma_{kji}\\\\\ng_{jk,i} & = (\\mathbf{b}_j\\cdot\\mathbf{b}_k)_{,i} = \\mathbf{b}_{j,i}\\cdot\\mathbf{b}_k + \\mathbf{b}_j\\cdot\\mathbf{b}_{k,i}\n= \\Gamma_{jik} + \\Gamma_{kij}\n\\end{align}\n", "5894ab89c84e99f67d3d9d7c41e55fa9": "X\\times Y \\simeq Y\\times X", "5894ac2317dfcade29752dbd4dea9cc9": "(x_0 + 1, y_0 + 1)", "5894cf328e2e944a6534ea4b10d3727d": "\\Delta_{S^{n-1}} = \\sin^{2-n}\\phi\\frac{\\partial}{\\partial\\phi}\\sin^{n-2}\\phi\\frac{\\partial}{\\partial\\phi} + \\sin^{-2}\\phi \\Delta_{S^{n-2}}", "5894df2489a35648aded5a952becd4d2": "\\epsilon_\\lambda", "5894efe992c29c56e0fc51c08cc813dc": "\\partial_\\nu j^\\nu = 0 \\,.", "5894f0a065c32c12204cdcecb16cccb3": "dz d\\bar{z}", "58952e7294b07c4f21ab4046cdee8abe": "P \\cup \\mathit{IC} \\cup \\Delta", "5895ce043942f60d14be169849b0addb": "n^3 - (n-1)^3 = 3n(n-1)+1.\\,", "5895f209fdda7b3def05136449ed79f5": "L_{F0} = \\frac{\\mu_0}{\\lambda_0}\\cdot S_{F0} = 7.3524\\cdot 10^{-2} \\ ", "5896224ba791e18a4f6b5cc640a9883f": "\\vec{r} \\cdot \\hat{n}", "58962a0b3352a3c50a7bd3092b8def8a": " \\mathcal{B}p \\to p ", "58962c30d85a70867ff5708162f9ab37": "\\operatorname{End}_R(\\oplus_1^n U) \\to \\operatorname{M}_n(S), \\quad f \\mapsto (f_{ij})", "58966d47cba2621f34dca5377d75748d": " \n\\begin{align}\n& A_0=T_1+iT_2, \\quad A_1=-2i T_3, \\quad A_2=T_1-iT_2 \\\\[3 pt]\n& A(\\zeta)=A_0+\\zeta A_1+\\zeta^2 A_2, \\quad B(\\zeta)=\\frac{1}{2}\\frac{dA}{d\\zeta}=\\frac{1}{2}A_1+\\zeta A_2, \n\\end{align}\n", "58966dc12ae07433dc420eb1fa4ea018": "1 \\le k \\le N-1 ", "5897331c2bbb9e4a5726c82d37669d48": "\\Delta t < 0", "5897597843fd4727b5f390eb03894dcb": " r = \\sup_{x \\in [a, b]} |f(x)|.", "5897df104007f3b286b3fdc533864104": "f_Z(z) = \\int_{-\\infty}^\\infty f_Y(z-x) f_X(x) dx", "589804d51e7beb8d21ebf6558fb8de27": "\\operatorname{sconv}\\mathcal{F}", "58983152c9850d05a27f65e471f69e02": "\\frac{V_p}{A_p} = \\frac{\\frac{4}{3}\\pi (d_v/2)^3}{4\\pi (d_s/2)^2} =\n\\frac{(d_v/2)^3}{3 (d_s/2)^2} = \\frac{d_{32}}{6}", "58984f9195e21da6b544e4bd8c8aadc9": "H(X) = \\log_2(n)", "58986a2acebf58cc93c56d4f285433c9": "\n0 \\le t \\le \\frac{C_2}{\\varepsilon}\n", "589872f72fb525457b8fd0c6c7b85c2d": "A \\xrightarrow[+f]{} x \\alpha", "58987fae7b095e93905fede77fc0ea25": "\\phi_{\\lambda}^{\\mathrm{L}}(\\mathbf{k})=\\phi_{\\lambda}^{\\mathrm{R}}(\\mathbf{k})/(1 - f^\\mathrm{e}_{\\mathbf{k}} -f^\\mathrm{h}_{\\mathbf{k}})", "5898affaae9002bbe400797b0911db59": "\\rho^n\\neq1", "5898b8f05702f007832dc972941c5b36": "m_2(t),", "5899392a38ff27b18117d7a853e925d7": "n! [z^n] \\frac{1}{1-z}\n\\exp\\left( - \\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty\\frac{z^k}{k}\\right)\n\\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty \\frac{z^k}{k} =\nn! [z^n] \\frac{1}{1-z} \\sum_{m=0}^\\infty \\frac{(-1)^m}{m!} \n\\left( \\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty\\frac{z^k}{k}\\right)^{m+1}", "5899675b7e5e4e625a6b379b25238300": "\\alpha \\colon X \\to M F", "5899b985499fbd2d65ec1c51ea57c090": "E_W = \\frac{1}{2}w^T w", "589a09aa8ec9fc31592719db40b81c14": "x\\in N_G(P)", "589a5b6363d24c6dcb5d7dc4079629ca": "\\Psi_\\epsilon", "589ac001c16aee26feadac7d965d19fa": "(z_1,z_2,z_3,z_4)", "589b21f6e078bb8deb87d1bdfaee16fe": "(\\pi(v),v)", "589b47f8d6e79d3b11c644e27363858d": "\\mathrm{Ad}_{\\mathrm S\\mathrm O}\\,", "589b576a8bbb29dd560bbb12e04ac2f8": "\\frac{x^{\\alpha-1}(1-x)^{\\beta-1}} {\\mathrm{B}(\\alpha,\\beta)}\\!", "589bbb9e2f81ccc9c574d83e51acbf14": "\\Pi^m_n", "589bc0aff9936c43e27b1ce468adb494": "S_1 = I p \\cos 2\\psi \\cos 2\\chi\\,", "589bea8a4cd89bdad424cf32b44fa762": "X \\thicksim \\chi^2(\\nu)", "589c09235305e651ef351c6aa31137f6": "n^{\\wedge} A", "589c1abbdfd6bcfc59aa198eee78a65c": "j_a=\\left(-\\rho, \\vec{j}\\right)", "589c8905f35c8e618fe483fe321b8386": "|\\mathbf{\\tau}| = |q\\mathbf{r}||\\mathbf{E}|\\sin\\theta", "589ccdf7faf27df91a88dbc291b9d7bb": "-\\infty < v,x,y < \\infty, -u_0 < u < u_0 ", "589cce6de169f2035018581ae86f1bea": "\\epsilon_j", "589cdf22380d27ef1ed863a7c00023aa": "D(m,n)=\\begin{cases}1 &\\text{if }m=0\\text{ or }n=0\\\\D(m-1,n) + D(m-1,n-1) + D(m,n-1)&\\text{else}\\end{cases}", "589d07ba0d893400787fed0981f999b5": "\n\\mu = \\frac{1}{\\frac{1}{m_{1}} + \\frac{1}{m_{2}}} = \\frac{m_{1}m_{2}}{m_{1} + m_{2}}\n", "589d964de8da9d6f9d36903421243b07": " E = \\lim_{m \\to \\infty} \\sum_{i=0}^{m} \\lambda^{i} E^{(i)}.", "589d9e637b92558349f88c2705d74d03": "\\tilde G(A,B)=G(A,B)\\backslash G(B,A)^{-}", "589dc10ae502c7ca6639ab6931c83e2f": "A=(1+\\sqrt{3})a^2", "589dd003329d62462a940147cc175921": "2^{2^{cn}}", "589e6ba5fae7e8d3a111c8bfc6612b82": "\\frac{1}{p_\\theta} = \\frac{1-\\theta}{p_0} + \\frac{\\theta}{p_1}", "589e8e5e2cba8a9c90422b2ef874178b": "\\delta(u)", "589ed530e1494fe7571b1fb1c3b9ef8c": "\\sigma=\\left(\\ldots,\\sigma_{-2},\\sigma_{-1},\\sigma_{0},\n\\sigma_{1},\\sigma_{2},\\ldots \\right)", "589f91e4285ee9229251516de19591b3": "\\begin{align}\\text{Total Surface Area}&=\\pi((R_1+R_2)s+R_1^2+R_2^2)\\\\\n&=\\pi((R_1+R_2)\\sqrt{(R_1-R_2)^2+h^2}+R_1^2+R_2^2)\\end{align}", "589fbf344a0162109576180133b439dd": "k\\in S", "589fcbbdb2a37f11e2b568da5c94575f": "v=(v_1,v_2,v_3)", "589ff5d21c675d578971c0f1e4013d95": "SD(k) = \\frac{\\sqrt{1 - w^n}}{w^n}", "589ff6e63f3526be8d140fd486e9a576": "A'(x) + A(x)^2 - B(x)^2 = \\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)", "58a03c4d9d02a267915407c71ebace88": " 0\\frac{1}{2}", "58a0d43f50180ea2c46c506e745f0e8d": "\\scriptstyle K", "58a199c168bb572ecba07c05462a0a5f": "\\displaystyle u", "58a20b261df16280dc6c6ce806aee06d": "\\Delta f_{\\mathrm{actual}} = f(x+\\Delta x) - f(x)", "58a2181ec8a7977ef5033cf6c233c5df": "Y=P_x \\cdot x + P_y \\cdot y", "58a228fb4e4c68426147ae83e50822bb": "{v'}^i ", "58a26617c5ec2a1f376b7dab386943ff": " \\alpha = {\\cos^{-1}}(\\frac{r{\\cdot}\\omega^2_t}{g{\\cdot}\\cos\\beta}) ", "58a276ed838536d168f65809330378dc": "J>0", "58a283d35cdc79f528b1da93a2cff50b": "\\int f(x)\\,dx = \\int f(x+c)\\,dx", "58a29533dd8a46c9f0b9ac943c39a75d": "y(1 - y)", "58a30817797056aa330be149b092ee69": "\\vec s", "58a3d3bdd170c08ebe74ca239e4f79d9": "r > 9", "58a44844995065c4c9b1a37fe7f8790f": " w = w( x_1, x_2, t ) ", "58a489e099c15ea53a51b6540905c270": "\\scriptstyle n \\,-\\, t", "58a5167fb952a7a204175c61519a6990": "\\lambda_2=1/\\lambda\\,", "58a52e62ad77584d4d9985dbd58a6cce": "\\textrm{Volga}= \\frac{\\partial \\mathcal{V}}{\\partial \\sigma}", "58a5ba2c51919103e31597195b69b5eb": "a \\to d", "58a600713c85e7e48d6c734ff1674280": " C_\\rho(x,y) = \\max \\{ \\beta(\\pi(x,y)) | \\pi \\text { is a (simple) path}. \\}", "58a6346fef40fc986c5b1e2c7ab0b55a": "\\{x \\mapsto 3\\}", "58a63fa37e343875ee987bc12dd5745c": "y\\succ x", "58a689402d7b9e3cdbf35cc360cb40d5": "\\left (z\\frac{d}{dz}+a \\right )w = \\left (z\\frac{d}{dz}+b \\right )\\frac{dw}{dz}", "58a6a187c5e16cff3e9dfbe5b95c7274": "P=IV=I^2R=V^2/R", "58a6be94e5ac4d1269449e1b7e167b3d": "\\Vert \\boldsymbol{\\lambda} \\Vert = \\sqrt{\\lambda(\\lambda + 1)} \\, \\hbar", "58a7014430b26b69d708f85a06b1d8ae": "D=\\frac{1}{3(\\mu_a+\\mu_s')}", "58a72c1d39c3826c22f682fadf2f81db": "\nf(x+iy)=A\\times(x+iy)=Ax+i\\cdot Ay\n", "58a7ac01c8ce00f02fafc1318dbe7642": "a_n>0", "58a7da1948fcf5496744603830cc4f73": "m=\\sum_{i=0}^{k}m_ip^i", "58a7dac626854de9544e15df54857f7a": "\\varepsilon \\approx \\left|Q - \\int_a^bf(x)\\,\\mbox{d}x\\right| ,", "58a8101b334d74b79c6e292f948b4177": " \\mathrm{Q} = \\frac{{B_0}^2 d^2}{\\mu_0 \\rho \\nu \\lambda} ", "58a812f312c0ba857e688e06a3b91111": "\\prod_{i \\subseteq \\Sigma}\\mathcal{M}_i", "58a83988ba22fb5c649477c09174e4e2": "S(f)\\ \\stackrel{\\text{def}}{=}\\ \\int_{-\\infty}^{\\infty} s(t)\\ e^{-i 2\\pi f t} dt\\,", "58a8628a5ccf90c6c4b6d39593013d2f": "E\\rightarrow {\\mbox{Div}}^0(E)", "58a87d3c777647b90f9a1e070e0fad09": "x \\in (0, c)\\!", "58a897fcf6b47da768a5a65a6dcf9860": "\\omega \\rightarrow (\\Phi_x \\otimes I)(\\omega)", "58a8c546feb7d15d33e9e09d58174d2d": " U \\, L^\\infty(X, \\mu) \\, U^* = L^\\infty(Y, \\nu) ", "58a8f8e990523add42fdacfac8ac4af3": "K-P", "58a8fd9353d3c50d9c27011a5b363cdb": "\\mathrm{Factor} = \\frac{360 \\times (Y_2 - Y_1) + 30 \\times (M_2 - M_1) + (D_2 - D_1)}{360}", "58a95237ba6cfda4c07aef711c9d7401": "N_\\alpha(t)=N_0 \\exp(-t/\\tau_\\mu) (1+\\alpha A\\cos\\omega t)", "58a96c009a36501bee80a07d76dfe0e8": "b \\gg a", "58a991fdb0459e5477653164ad8e2d6d": "C_i = {{\\sqrt{e|E_i|}}\\over{\\hbar\\theta}}", "58a9930c5609bda41b1f6659c66d1ece": "k_i\\frac{dx_i}{dt} = \\frac{\\partial H}{\\partial y_i},\\qquad k_i\\frac{dy_i}{dt} = -\\frac{\\partial H}{\\partial x_i},", "58a9b4ae7ecf5bca0678993802f543e6": " \\nabla^{2}\\theta-\\frac{1}{k_{s}}\\frac{\\partial \\theta}{\\partial \\tau}=-\\frac{g}{\\kappa_{s}} ", "58a9ef5eb149bc377cae828a7f5dd73f": "\\begin{align} (A \\oplus B) \\oplus A =& A \\oplus (A \\oplus B) \\\\=& (A \\oplus A) \\oplus B \\\\=& 0 \\oplus B \\\\=& B \\end{align}", "58aa4ce0742fcb6d32ac3101ae86409f": " f(i)M^n(i,j) = f(j)M^n(j,i)", "58ab0c4597c369968ffafb05b76b44d9": "\\zeta^{\\prime}(-2) = -\\frac{\\zeta(3)}{4\\pi^2}", "58abc9cbd170f96858f2c73fe7bdce8b": "\\Psi: \\mathcal{X} \\times \\mathcal{Y} \\to \\mathbb{R}^d", "58abe63855ff6cb07efabd45be4fce98": "\n \\nabla\\times\\mathbf{v} = 0\n ", "58abff3fd7834fc51fe0b1de051aeeb5": "\\scriptstyle (F^{\\mu\\nu})", "58ac027cc441cded5d181fbd40fcc684": "\\det {\\mathfrak{T}}^{\\alpha}_{~\\beta}", "58ac85e9e89a81e59636c51dcc72f79a": "L\\cap\\overline{L}=0", "58ac91b46d6a648d69f8261df6a34dff": "\\forall|\\phi\\rangle", "58acabc9048c0cdd8ddc91697459d9be": "O(n^2 \\log \\frac1\\delta)", "58acb76fcd2335df488021591ada5005": "\\Pr(A>x) \\le \\Pr(B>x)\\text{ for all }x \\in (-\\infty,\\infty),", "58ad14a960e75a83171fc4a5fa71e9a9": "x\\in L\\iff f(x)\\in \\textrm{TQBF}.", "58ad3fb4bd1dd09d9ff530673fed54e4": "\n\\frac{\\partial c}{\\partial t} = -\\frac{\\partial J}{\\partial z}.\n", "58ad90e0685d745ad688ce92959c651d": " b_4 = f(0,0)-f(1,0)-f(0,1)+f(1,1). \\,", "58adc3a4695727a04ed7477bf1ed4785": "\\{a^m b^n c^m d^n : m, n \\geq 1 \\}", "58ae4f2c455b8d9723c73999c568a69a": "\n\\log 2=\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)-1}{n}.\n", "58aeb19b9cb0642cd8cc6b1b953d8987": "\\mu_a \\,", "58aed6da5263f20bf35fcce0f11d70e9": "\\bigcap_{n=1}^\\infty C_n", "58aef9e990e87af95c4acb74b9c7e4fe": "\\displaystyle\\frac{d}{dt} f(t) = \\lambda f(t)", "58af3934c6c9530362de2a38b6ed639a": "k \\mapsto \\begin{pmatrix}\n 0 & -1 \\\\\n 1 & 0\n\\end{pmatrix}", "58afa8fc13fc574940319c1c2bd3fcab": "n^*(a)=C \\exp\\left(- \\int_0^a{\\mu(q)dq}\\right),", "58aff241fbe4e62b882840582b2bee67": "\\gamma:\\Delta \\to \\mathbf{Top}", "58b01c474ed0aaca4fa02aa461c73261": "E(Z_j)=\\xi", "58b038c4a9976f4cbb5ec48d287629ca": "\ne\\epsilon(t)(c_{1}(t)\\langle \\Psi_{1} |x|\\Psi_{1}\\rangle + c_{0}(t)\\langle \\Psi_{1} |x|\\Psi_{0}\\rangle)=i\\hbar \\left(c_{1}'(t)\\langle \\Psi_{1} |\\Psi_{1}\\rangle + c_{0}'(t)\\langle \\Psi_{1} |\\Psi_{0}\\rangle\\right)\n", "58b0799193688b7ed1fefea97da53573": "(x,z)\\in T\\circ S", "58b090247d0a93e021f889ed2333cd88": "x=|X|", "58b0e8612d921f68e6ca7669aeae7133": "g_i\\,", "58b12ec0382f6a05f182ddb50e135fec": "{2a_{12} \\times b_{12} \\over c_{12}}=d", "58b1316b46b3135bb049153d3be914df": " A = \\bigl[ \\begin{smallmatrix}\n a&c\\\\ c&b\n\\end{smallmatrix} \\bigr]\n ", "58b16499411c45a453d90188ae43f1d5": "\\inf \\{s : \\mathrm{some}\\ s\\mathrm{-gale\\ succeeds\\ strongly\\ on\\ all\\ elements\\ of\\ } Z \\}", "58b16c2db44a1b42bf47ef165ec7e72e": "\\left| \\psi \\right\\rangle = \\begin{matrix}\\frac{1}{\\sqrt{2}}\\end{matrix} \\left( \\left|z+\\right\\rangle_A \\left|z-\\right\\rangle_B - \n\\left|z-\\right\\rangle_A \\left|z+\\right\\rangle_B \\right).", "58b1a8f11113f5d30918afac1dfd6be9": " \\frac{x_3 - x_1}{x_2 - x_1} \\cdot\\frac{t - x_2}{t - x_3} + \\frac{x_2 - x_3}{x_2 - x_1} \\cdot \\frac{t - x_1}{t - x_3} = 1 . ", "58b1e9f4b9a86690cb106e65a265a34f": "\\delta x", "58b25b4febaac29925a137be65361ff0": "p_D\\,", "58b28519cb7bc8978e3b6f1855632ffa": "\\frac{dy}{d\\mathrm{net}} = \\frac{d}{d\\mathrm{net}}\\varphi ", "58b300f02288c838532bf4df7069dee3": " H_{\\frac{3}{4}} = \\tfrac{4}{3}-3\\ln{2}+\\tfrac{\\pi}{2}", "58b308d60f4322633176d95040090e2e": "{\\hat{q}_{{\\rm w}}}({r_{\\rm w}})", "58b308f83237e648851c2e4affd313d0": "\\Phi_n(q)", "58b31b896689ace8d668f5085299990f": " \\vec{H_a} ", "58b3301d10c0796ba3f4a06e18fbca7a": "x \\in (1+\\delta) B_{X}", "58b3ce69317a1235975db89ec0726b37": "(x,y) \\in R", "58b421cc533904911912ec749ede5fd5": "\\Rightarrow_{r_3} S Y Y Y \\Rightarrow_{r_3} S S Y Y \\Rightarrow_{r_3} S S S Y \\Rightarrow_{r_3} S S S S", "58b44b99557e199d8ecb270cfebae71d": "E_0=\\{2\\}=\\{2^1\\}=\\{2^{2^0}\\}", "58b464ab604ba914ffcfd03f4c04bbba": "\\Pr[Y_i=h|x_{1,i},\\ldots,x_{k,i}] = p_{i,h},\\text{ for }i = 1, \\dots , n,", "58b473f2400cf50886ba60965561a999": "\\gamma^2 = (\\alpha +j \\beta)^2 = (R+j \\omega L)(G + j \\omega C)\\,", "58b47da1d3804d5377e19c98fe841b77": " \\forall \\epsilon\\in \\mathbb{R}^+, \\exists \\delta \\in\\mathbb{R}^+, |h| \\leq \\delta \\implies |f(x+h) - f(x)| \\leq \\varepsilon", "58b4b4a248b91e13f74041efddf3f9c8": "\\frac{\\mathrm{d}^2 \\theta}{\\mathrm{d}t^2} = - \\omega^2 \\theta \\,\\!", "58b4d47811d766d8f87be46520f2d666": "j(p)-j(q) = \\left({1 \\over p} - {1 \\over q}\\right) \\prod_{n,m=1}^{\\infty}(1-p^n q^m)^{c_{nm}}.", "58b515ba76d367047bfc4ebaac691b9a": "\\lambda>\\lambda_c", "58b5faf50992ac668c6d17c447fb56c3": "\\int uv \\,dx = u \\int v \\,dx - \\int \\left ( u' \\int v \\,dx \\right )\\,dx.\\!", "58b5ff56fd28124d675306f6cbe9d883": "\\Delta s = s_2-s_1", "58b66d2016de0699cfebb89062315061": "T \\sin \\varphi = \\int w\\ ds,\\,", "58b68d91ee79f0897e88576a5f00fee8": "g_{\\mu}=\\mathsf{h}^{-1}(e_{\\mu})", "58b6edc5b283e36dba354c816ebd476d": "G^{(0)} := G", "58b73d9a6f03fe555e81b886220abda5": "F(d,a,n) = (n/4)(2a-d+dn)(2a^2-2ad+2adn-d^2n+d^2n^2)", "58b7c5ff4c5b69764356f91f46559acd": "\\scriptstyle F^7(12) \\;=\\; F(F(F(F(F(F(F(12)))))))", "58b7ca4bacebb963dbeaa7aa62af122c": "\\mathbf{P}=\\frac{\\langle\\mathbf{d}\\rangle}{V}", "58b7cc8d47d04c71966e28090cee0d6b": " u_z ", "58b7d7142992a60200aeaa855e09ead5": "\\approx -0.9453087204829418812\\,.", "58b7fc3474b021ff11c1a0df58a54060": "n \\,\\!", "58b825b8c02464bde7ebe321aba2f32d": "\\omega_2=(0,1)", "58b86042971b08ed7c69b4b9522d37f3": "\\pi (k_1,k_2,\\ldots,k_m) = \\frac{1}{G(K)} \\prod_{i=1}^m \\left( \\frac{e_i}{\\mu_i} \\right)^{k_i}", "58b8c9b2da0e6a363ba734e86ac93951": " t_1 t_2 = 1 ", "58b8fb0c308356eafcfa327312abdf23": "\\mathbf Z[h_1(X_1,\\ldots,X_n),\\ldots,h_n(X_1,\\ldots,X_n)]", "58b964e5d584bc7b22b4988a29e629ce": "b_1,...,b_n,a", "58b9726b4b41fb798fb05c199216236d": "\\scriptstyle\\nabla\\cdot", "58b99a45e7e697ed62ea51e250478a56": "F_\\text{m}", "58b9a688661736b896174c86a468a9ed": "\\bar{\\mathbb{F}}_{q}", "58b9c0b8f74d324428597060f20aa58a": "(\\phi\\to\\psi) \\leftrightarrow ((\\phi\\lor\\psi) \\leftrightarrow \\psi)", "58ba3167eae8ccb73ae58205e2de3c20": "I_{\\frac{d_1 x}{d_1 x + d_2}} \\left(\\tfrac{d_1}{2}, \\tfrac{d_2}{2} \\right)", "58bad387e020a64354d9d952d986c6e2": "\\mathit{x^N} - 1", "58bb76cebb09cc961e95dcedf40a57b0": "\\mbox{e}^{t\\mathcal{A}}", "58bb7c9cda59b3d1cf282f8c95ab9b71": "\n\\begin{align}\n G_{u_g}(s) &= \\sigma_u \\sqrt{\\frac{2L_u}{\\pi V}} \\frac{1}{1+\\frac{L_u}{V}s} \\\\\n G_{v_g}(s) &= \\sigma_v \\sqrt{\\frac{2L_v}{\\pi V}} \\frac{1+\\frac{2\\sqrt{3}L_v}{V}s}{\\left( 1+ \\frac{2L_v}{V}s \\right)^2} \\\\\n G_{w_g}(s) &= \\sigma_w \\sqrt{\\frac{2L_w}{\\pi V}} \\frac{1+\\frac{2\\sqrt{3}L_w}{V}s}{\\left( 1+ \\frac{2L_w}{V}s \\right)^2} \\\\\n G_{p_g}(s) &= \\sigma_w \\sqrt{\\frac{0.8}{V}} \\frac{ \\left( \\frac{\\pi}{4b} \\right)^{\\frac{1}{6}} }{(2L_w)^{\\frac{1}{3}} \\left(1 + \\frac{4b}{\\pi V}s \\right)} \\\\\n G_{q_g}(s) &= \\frac{ \\pm \\frac{s}{V}}{1+\\frac{4b}{\\pi V}s} G_{w_g}(s) \\\\\n G_{r_g}(s) &= \\frac{ \\mp \\frac{s}{V}}{1+\\frac{3b}{\\pi V}s} G_{v_g}(s)\n\\end{align}\n", "58bbd4546497ea65c59688e58ef6502d": "{\\left\\langle F \\right\\rangle}_J=\\frac{\\int \\mathcal{D}\\phi F[\\phi]e^{i(\\mathcal{S}[\\phi] + \\left\\langle J,\\phi \\right\\rangle)}}{\\int\\mathcal{D}\\phi e^{i(\\mathcal{S}[\\phi] + \\left\\langle J,\\phi \\right\\rangle)}}.", "58bc7c5cb0ba61370ac0abacb46ce3f3": "\\partial^j=(\\partial^j\\theta^i)\\partial_i=g^{ij}\\partial_i", "58bcb7abfb7dfa94d27a99277f7bc579": "C = 3(A+B) = 6", "58bce9c16b106d8d52c17ce923cf5f33": " \\delta u_{i + \\frac{1}{2} } = \\left( u_{i+1} - u_{i} \\right) , \n \\delta u_{i - \\frac{1}{2} } = \\left( u_{i} - u_{i-1} \\right),", "58bd5126d3355a1d2b82afff3c69cfa8": "[n,k]_q", "58bd579657ff2d36b94c3730003616e4": "B \\rightarrow CDC: A", "58bd5e2220b34ac14e905da479439b26": " \\|\\boldsymbol{N}_{i=1,...,k}{(0,1)}\\|^2 \\sim \\chi^2_k ", "58bdad189c04f5af521f06de791be636": " \\tilde{g} ", "58be6892e6076a1868e84977fac48f7d": " pB = \\prod_{j} P_j^{e(j)} ", "58bedd6c2f7b93a78ae43d4d28e4fa71": "\n\\mathbf{v(u)}\n ", "58beeedbfbcad05d1b021f360f361c69": "= 1+(\\alpha_{-2}+\\alpha_2)\\cos \\left ( \\frac{i4 \\pi k}{N} \\right ) + (\\alpha_{-1}+\\alpha_1)\\cos \\left ( \\frac{i2 \\pi k}{N} \\right ) > 0 ", "58bef61b23efbadb560c0fe010733e34": "|\\Psi(x,t)|^2 = \\left| e^{-iE_{\\Psi}t/\\hbar}\\Psi(x,0)\\right|^2 = \\left| e^{-iE_{\\Psi}t/\\hbar}\\right|^2 \\left| \\Psi(x,0)\\right|^2 = \\left|\\Psi(x,0)\\right|^2", "58befcea884cee55ce5a36fc0051bb40": "\\mathbf \\phi_3 = \\left (\\frac{2\\alpha_3}{\\pi} \\right ) ^{3/4}e^{-\\alpha_3 r^2}", "58bf1ad5c3e833854eac3528147ebfd7": "\nM \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{a b^{4}} \\frac{1}{J_{\\alpha}^{\\prime}}\n", "58bf3ac1c0f40f758aa9bcff2a34e5eb": "(P^{(\\pm)} [F, G])^{IJ} = {1 \\over 2} ([F, G]^{IJ} \\mp i * [F,G]^{IJ})", "58bf4f1a2298b62fc09a2328ede1fa92": "\\overline{\\bigcup_{i \\in I} A_{i}}\\equiv\\bigcap_{i \\in I} \\overline{A_{i}}", "58bf521084ee198a20293b6256a3bb29": "A_1 = 2", "58bf66dbbca350fc280f775d460036cc": " \\frac{\\mathbb{P}(q_\\theta(X) \\geq q_\\theta^* |\\theta)}{\\mathbb{P}( q_\\theta(X) \\geq q_\\theta^* | 0 )} = \\alpha' .", "58bf84bc1cdfdf2e286a896415a2c2d7": "\n \\begin{align}\n B = & \\left(\\cfrac{\\sigma_t-\\sigma_c}{\\sqrt{3}(\\sigma_t+\\sigma_c)}\\right)\n \\left(\\cfrac{4\\sigma_b^2 - \\sigma_b(\\sigma_c+\\sigma_t) + \\sigma_c\\sigma_t}{4\\sigma_b^2 + 2\\sigma_b(\\sigma_t-\\sigma_c) - \\sigma_c\\sigma_t} \\right) \\\\\n C = & \\left(\\cfrac{1}{\\sqrt{3}(\\sigma_t+\\sigma_c)}\\right)\n \\left(\\cfrac{\\sigma_b(3\\sigma_t-\\sigma_c) -2\\sigma_c\\sigma_t}{4\\sigma_b^2 + 2\\sigma_b(\\sigma_t-\\sigma_c) - \\sigma_c\\sigma_t} \\right) \\\\\n A = & \\cfrac{\\sigma_c}{\\sqrt{3}} + c_1\\sigma_c -c_2\\sigma_c^2\n \\end{align}\n ", "58bfbcaf65ba8492984335e61ddd616f": "(x_n)_{n\\in\\mathbb N}", "58bfce52e8f1f9485a789e99d086c969": "y_{t}", "58bfee033bc148a2ae6bd157b3cb134c": "\\csc(x)", "58bff630b9813da8ae982ded2647066f": "\\begin{align}\n \\mbox{curl}\\,(\\mbox{grad}\\,f ) &= \\nabla \\times (\\nabla f) = 0 \\\\\n \\mbox{div}\\,(\\mbox{curl}\\,\\vec v ) &= \\nabla \\cdot \\nabla \\times \\vec{v} = 0\n\\end{align}", "58c00fd4a916a7f7e59c74da49deef1e": "S(x)=\\int_{\\frac{\\theta}{x(t)}}^{\\infty}C(w)dw", "58c099b8b3a4c31932610be15821a076": "X\\times_Z Y = \\{(x, y) \\in X \\times Y| f(x) = g(y)\\},\\,", "58c0fd22a43748990e7534a2290ccdbe": "L\\mathfrak{g}", "58c12898b837178b35db15d6f16e04d7": " \\Xi(\\mu,M,T) = \\sum_{N=0}^M Q(N,M,T)= \\sum_{N=0}^M \\binom{M}{N} (q\\lambda)^N=(1+q\\lambda)^M", "58c12b1d7ef3391742c1da5da71d59fa": "\\bigcup_{j\\in J} E_j", "58c1352831d2281e9edc31bd7197d1aa": "\\mu = Wr_f", "58c14b37001de39eb468be08ce497913": "\\frac {dv/dt}{v} = g_v=\\frac {s(1-u)} {\\sigma} -(\\delta +\\alpha + \\beta)", "58c1de2b155cd94984621de9e7d98908": " F: R^{n_x} \\times R^{n_y} \\to R^{p}", "58c1e0637f71a81606f9612157414546": "\n\\sum_{j=1}^n h(j) = n + c\\sqrt{n} + o (\\sqrt{n}) \\,\n", "58c1fbdf64791ff2025d8bc230d009c6": "n_0(\\vec r)", "58c24598232bf829c7a95e688d3e92f2": "\\vec{\\xi}", "58c24cc77cd2acf06764e63f8ac715d7": "\\begin{align}\n r^2 \\sin\\theta \\, d\\theta \\, d\\phi &\\hat{\\mathbf r} \\\\\n+ r \\sin\\theta \\, dr \\, d\\phi &\\hat{\\boldsymbol\\theta} \\\\\n+ r \\, dr \\, d\\theta &\\hat{\\boldsymbol\\phi}\n\\end{align}", "58c25005aee205b3acb787f313b33e92": "\\Delta v<0.074", "58c251b8a00055b20399885f60fd5191": "\\begin{align}\nH(Y|X)\\ &\\equiv \\sum_{x\\in\\mathcal X}\\,p(x)\\,H(Y|X=x)\\\\\n&{=}\\sum_{x\\in\\mathcal X} \\left(p(x)\\sum_{y\\in\\mathcal Y}\\,p(y|x)\\,\\log\\, \\frac{1}{p(y|x)}\\right)\\\\\n&=-\\sum_{x\\in\\mathcal X}\\sum_{y\\in\\mathcal Y}\\,p(x,y)\\,\\log\\,p(y|x)\\\\\n&=-\\sum_{x\\in\\mathcal X, y\\in\\mathcal Y}p(x,y)\\log\\,p(y|x)\\\\\n&=\\sum_{x\\in\\mathcal X, y\\in\\mathcal Y}p(x,y)\\log \\frac {p(x)} {p(x,y)}. \\\\\n\\end{align}", "58c2cc22903546c120c2a129ddd5427c": " U = \\int_V \\mathrm{d} \\mathbf{p} \\cdot \\mathbf{E} ", "58c32f77f9ee76d0a5b9b2d162eb8ecf": "(L,\\wedge,\\vee,0,1)", "58c38cd71dce34a4929187006ab70c22": "t (\\tau)", "58c3d0d0ebe063b8eb10b3dcf9c10b0c": "\\bar{X}", "58c40bb32ef0f54c1b3f1c0adcf74b86": "k[x_1, \\ldots, x_n]/I", "58c427671008d700a9bbed5734762932": "FC_G(Q)", "58c449664c6b39f70a954a8b4d970ca8": "\nI = n \\cdot A \\cdot v \\cdot e\n", "58c45402a3237408f3acaa478bc69ca6": "V_{w1} + U ", "58c47f75cae458fa3246647eedac4af9": " \\ C_{D_i} ", "58c51205c8afcbb5dfc80aeee784a324": "\n\\begin{align}\nP &= \\text{Power in Watt} & \\left[ \\frac{Nm}{s} \\right] \\\\\nn &= \\text{Revolution per second} & \\left[ \\frac{rev}{s} \\right] \\\\\nV_{stroke} &= \\text{swept volume} & \\left[ m^3 \\right]\\\\\n\\Delta p &= \\text{pressure difference over pump} & \\left[ \\frac{N}{m^2} \\right]\\\\\n\\eta_{mech,hydr} &= \\text{Mechanical/hydraulic efficiency} & \\left[\\right]\n\\end{align}\n", "58c51368498f4e638d1cce43c8f19abe": "\\mathbf{X} = \\langle X, ( R_i )_I, \\mathcal{F} \\rangle ", "58c57befbf1b14fe492dd45f2701636d": "\\left( \\frac{-7}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0 \\right)", "58c59109381e2fb59598c08f13254caf": "Bird(Condor)", "58c67f0e50aeca7205025089a47d45f4": "s_0 = a_0", "58c6858434c7e317d365491d3c1c9be9": "C(\\Omega)", "58c68ed36ca22522ec70e61ed85d2c9c": "\\theta, \\alpha", "58c695a40f15e2da621d1bb95624e410": "m_3 = [12.3, 7.6] + [0, 2.404] = [12.3, 10.004]", "58c79378852fc1a3a56c8d275290c858": "{\\displaystyle dx_1/dt=0}", "58c7d0e6b9d0f3ecd8f6cf1b5e569d4f": "\\xi = rg\\nu", "58c7ece8d2af1080dbdb3081e5607d87": "\\scriptstyle\\lang\\lambda_i|\\psi\\rang", "58c805c091de3a8dffcb5e10e1297223": "\\mu_{T,e_1}(T)=0", "58c818cbb7d8724ca2bcd6f5661c5376": " \\prod_{p} \\Big(1 - \\frac{1}{p^2(p+1)}\\Big) = 0.881513... ", "58c97f05472dab6ae97aee009fcecbd0": " \\hat{H} = \\frac{\\hat{p}^2}{2m} + V(x) = -\\frac{\\hbar^2}{2m}\\frac{d^2}{d x^2} + V(x)", "58c9ac6ad01a743d2a6956014b67747c": " ESS = (\\hat y - \\bar y)^T(\\hat y - \\bar y) = \\hat y^T \\hat y - 2\\hat y^T \\bar y + \\bar y ^T \\bar y.", "58c9e42b7a19646a0f0d3453c506c3c2": "u \\in D", "58c9ecec3198922610751d298f022cff": "\\sqrt{\\alpha} \\,", "58ca12c83b6f43a8cd46cead08292fd5": "\\frac{2m}{\\hbar^2}\\left(V(x)-E\\right) = v_1 (x - x_1)", "58ca2344fd04128ea93c570b7ef4ff5c": "R_S=R", "58ca399ba9723f2c82900f49bae643a2": "\n L = \\cfrac{1}{2~(\\tau_{23}^y)^2} ~;~~ M = \\cfrac{1}{2~(\\tau_{31}^y)^2} ~;~~ N = \\cfrac{1}{2~(\\tau_{12}^y)^2}\n ", "58ca5d255a1718c8ad663457d4de659f": "\\alpha \\neq 1 ", "58ca8276e7017c18027ec34f1133049e": "\\lambda x.((\\lambda x.x)x)", "58caf1e0db4c2a40dc8bb39bc6ff00bd": "s_4 = 1101,", "58cb33f96fa52187dc65f50e372fb61c": "+r", "58cc0f7e205d0f275b8565d6d8c853ef": "v \\in \\Lambda^0", "58cc2290df314b7b422fc8b60ac3090f": "[L_ \\lambda L] = (2\\lambda + \\partial)L. \\, ", "58cc59fe685b51ffa5361f874738049c": "\\beth_{d-1}(|\\alpha+\\omega|+\\beth_1)", "58cc6874c67b4dd97f83f6b5c775538d": "u \\geq 0", "58cc8af2543386f74a23891cd0b48aa1": "G(\\tau=0^+)", "58ccaa057ab7067c808f9bfb5e653f47": "H_S \\otimes H_{O}", "58ccac5ad4afb972c3bdfd03abb787a0": "\\Delta{[CO]}", "58cd234c62c8562352ce2bd60dd21afd": "T_{s,t}:=\\left\\{(x,y)\\in\\R_+^2:\\ \\frac{x}{s}+\\frac{y}{t}<1\\right\\}.\n", "58cd4d425d3e30c7ffa679256f0349bc": "\\frac{d}{dt}x(t)=f(x(t))", "58cd9cdd9660130a54a580c71d669975": "\\operatorname{mod}\\sigma_y^2(n\\tau_0) = \\frac{1}{2\\tau^2}\\left\\langle \\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}x_{i+2n}-2x_{i+n}+x_i\\right]^2 \\right\\rangle", "58cdf0e3dec8f1dfd79bc10cec7c6318": "2^{2^{\\cdot^{\\cdot^{\\cdot^n}}}}", "58ce409b8672f557b6755cd0ec69fd35": " \\textrm{Pref} = \\frac {1} {2} \\, { \\rho} \\, { v^2} \\,\\!", "58ce6948ac586653ee9347bbf2b40a83": "\\mathfrak{gl}_n(\\mathbb{R})", "58ce73291f2e5a3af0cb6d203a1665e6": "n, k, t", "58ce87c1fc0d8a58a03b48ce4239a1db": "v_{0}\\,\\!", "58cebf0a77ebdd6d96480ec4f860eac1": "\\forall n\\in \\mathbb N^* \\quad \\forall u\\in\\mathbb R_+ \\quad u\\leqslant n\\quad\\Rightarrow\\quad (1-u/n)^n\\leqslant e^{-u}", "58cefc9a4687893f058934b517f77edf": "\\alpha = |\\alpha|e^{\\beta i}", "58cf11a8f05aa1e34b85f019b6525f07": "f\\colon I\\to \\R", "58cf4416423319860d7e7d8b3be75f81": "\n\\begin{align}\nx & = \\int_0^L \\cos\\theta \\, ds \\\\\n & = \\int_0^L \\cos \\left[ (a s)^2 \\right] ds\n\\end{align}\n", "58cf6890fac47dee6db944ed89647625": "x^n = {x_M}^{n_M}", "58cff1a6cfb32f7af471e1c7dd0c91e8": " F_B ", "58d03baecfd35e74c2646a7dd770b36e": "F^* = \\underset{F}{\\operatorname{arg\\,min}} E_{x,y} L(y, F(x)).", "58d063f2d02f4d156d815ade96aea7f1": "x_i\\sim\\, f_{ik}(x_i)", "58d0a1186e0964ea4c47e7853c909e62": "X' = Y'/Z'\\ ", "58d0c4f0c171cf8623c946bae14700ed": "\nf(y)=\\frac{1}{2^{\\frac{1}{2}}\\Gamma(\\frac{1}{2})}y^{-\\frac{1}{2}}e^{-\\frac{y}{2}}\n", "58d0d71f1fb3d021ce23f4b683dd1a05": "H(\\sigma)=-\\sum_{A} J_A \\sigma_A ~,", "58d0e9184be0c2026017df3044a57244": "\\Phi_t\\left(D\\right)=\\left(\\Phi_t^0\\left(D\\right),\\Phi_t^2\\left(D\\right),\\Phi_t^4\\left(D\\right),\\ldots\\right)", "58d13efbd074166e241149bc14b93574": "nm^{2}", "58d16a5042bf3f90176ea2f1ae946f99": " V=\\pi LR^2 ", "58d195bcb43989762a4293322d7ebb3f": "M := \\bigcap_{n\\in\\mathbb{N}} M_n", "58d1bbf923b3ba2bf5e9f0fbf9a8d75b": "(X-x)+\\frac{dy}{dx}(Y-y)=0.", "58d20a4249e1feedcb49b315a2b3d90f": "\\sum_{n=a}^b f(n) \\sim \\int_a^b f(x)\\,dx + \\frac{f(b) - f(a)}{2} + \\sum_{k=1}^\\infty \\,\\frac{B_{2k}}{(2k)!}\\left(f^{(2k - 1)}(b) - f^{(2k - 1)}(a)\\right)", "58d20bfdedd4b82addad2689a990e5ab": "(X;A,B)", "58d265bbfb630c049bf7712c3b20d817": "\\mathbf{Y},\\mathbf{X},", "58d28c230e6a2b58c38d5c7f22ff3d18": "\\mathfrak c=\\aleph_1", "58d2b6bc36e828e3668048bfb76d2faf": "f(t) = -t^p", "58d35f9ae55328e72fcbfa1ec957f94e": "1 << k \\le N", "58d392f490a647ad2cb2896141886430": "Y_\\alpha(x) = \\frac{J_\\alpha(x) \\cos(\\alpha\\pi) - J_{-\\alpha}(x)}{\\sin(\\alpha\\pi)}.", "58d3c0aced7461fe66605e6f7b97a961": "\\mathbf e_i = \\frac{\\partial \\mathbf r}{\\partial q^i}", "58d3ceac16a16656a7d868162ca64c7b": "\\ h^\\mathrm{H} (s s^\\mathrm{H}) h = \\lambda h^\\mathrm{H} R_v h. ", "58d3f85cb6e51539ffa4e0720def4a4d": " \\omega_0 = { 1 \\over \\sqrt{LC}} ", "58d418b68e0ee3a23c63eb4815184533": "(n-k-1) \\in (n-k)", "58d41bbeca6c07a3dbe30150a31a4b2e": "\\exp(-t)\\,", "58d42f880da273ddfd00dd337be7701c": "SE(r_1)=\\frac {1} {\\sqrt{N}} ", "58d43056593a291acccd5eed72af6311": "f : S \\times S \\rightarrow S", "58d43fe6788bb1c660c5a8f2c245f428": "\\pi_1(x) : \\mathsf{T}_1,~\\pi_2(x) : \\mathsf{T}_2,~\\ldots,~\\pi_n(x) : \\mathsf{T}_n", "58d441aa9caa907368c3f858475796bd": "\\alpha = 1 + \\frac{\\tau_d}{\\tau_i}", "58d4c722dc1b5c2c9dd9099431feee43": "\\alpha_\\text{c}", "58d4f0dc7009cd9a2928f220972c3cf8": "(1)\\quad\\langle\\phi(x),\\psi(x)\\rangle = \\int_a^b\\phi(x)\\psi(x)dx = 0,\\quad{\\rm and}", "58d4f7369eeac6d96ec0debe47d0d8ae": "\n\\text{span}\\{ u_k, \\ldots, u_n \\}\n", "58d532c2bc4c64d05323cae762cfed8c": "\\lambda_2 = e^{-\\varphi}", "58d5669b258ed948c4e240298419513e": "F^6,\\ F^4 R^2,\\ F^2 R^4,\\ R^6", "58d57be9b06ccff830314812f4632ace": "y = \\phi(\\psi y)", "58d5e31e1148ded7e500a3726514e04a": "\\frac{1}{\\sigma}\\,t(x)^{\\xi+1}e^{-t(x)},", "58d69b0bbb784efa1a55a61560ac7768": "E_1(t)", "58d6cb077d7cfde0c2f079fd85589ffb": "g(y) = \\int_0^\\infty f(x)e^{-x} L_{y}(x) \\, dx ", "58d6dc0230e564b51ef8f8d3bea0941e": "284\\text{ Mpc} \\ll 14.4\\text{ Gpc}", "58d722e1a88fb3450f896dd1fbc5fde1": "\n \\boldsymbol{\\mathsf{I}}^T = \\delta_{jk}~\\delta_{il}~\\mathbf{e}_i\\otimes\\mathbf{e}_j\\otimes\\mathbf{e}_k\\otimes\\mathbf{e}_l\n", "58d77ac09d39c12c50da0e427062d867": " L^x(t) =\\int_0^t 1_{\\{x\\}}(X_s) \\, ds.", "58d78705f8c07d1328af57f31a5d4269": " m_i \\ := \\ \\min_{x \\in X} \\, u_i(x) \\quad \\mbox{and}\\quad M_i \\ := \\ \\max_{x \\in X} \\, u_i(x)", "58d7a81ed73d7ce4f82e4fefb338c921": "*:G\\times G \\rightharpoonup G.", "58d7c001952050447f7725605438a6a4": "J(u_n+c_n\\Delta_n)_i", "58d7cfae1d540bfde2c4081dc3322fe6": " X_k = \\sum_{j=1}^{k} T_j \\quad \\text{for } k \\geq 1. ", "58d819081c6fec33dc738aafa0f36ac9": " \\alpha=(\\alpha_j)_{j=1}^J ", "58d8502c848b2c88bcd33adef3d79cdb": "a\\cdot x = (-1)^{|a||x|}x\\cdot a", "58d8f1701701addb40572504612e09a6": " \\mathrm{ NH_{4}^{+} > K^{+} > Na^{+}}", "58d8f1ba2c0bb3f5e0bc40c6c0336cdb": "\\bar{x} = 1/2 ", "58d8ff7275e15096c3512d9ed526058e": "\\mathbf{H}_{\\textrm{LS-estimate}} = \\mathbf{Y} \\mathbf{P}^H(\\mathbf{P} \\mathbf{P}^H)^{-1} ", "58d907f5cb7271a6a56c82c64a08a3f0": "\\begin{align}\n\\operatorname{H}[\\mathbf{X}] &= \\tfrac{n}{2}\\ln|\\mathbf{V}| +\\tfrac{np}{2}\\ln(2) + \\ln\\left (\\Gamma_p(\\tfrac{n}{2}) \\right ) -\\tfrac{1}{2}(n-p-1) \\operatorname{E}[\\ln|\\mathbf{X}|] + \\tfrac{np}{2} \\\\\n&= \\tfrac{n}{2}\\ln|\\mathbf{V}| +\\tfrac{np}{2}\\ln(2) + \\tfrac{1}{4} p(p-1) \\ln(\\pi) + \\sum_{i=1}^p \\ln \\left (\\Gamma\\left ( \\tfrac{n}{2}+\\tfrac{1-i}{2}\\right ) \\right ) \\\\\n&\\qquad \\qquad -\\tfrac{1}{2}(n-p-1)\\left(\\sum_{i=1}^p \\psi\\left(\\tfrac{1}{2}(n+1-i)\\right) + p\\ln(2) + \\ln|\\mathbf{V}|\\right) + \\tfrac{np}{2} \\\\\n&= \\tfrac{n}{2}\\ln|\\mathbf{V}| +\\tfrac{np}{2}\\ln(2) + \\tfrac{1}{4} p(p-1) \\ln(\\pi) + \\sum_{i=1}^p \\ln \\left (\\Gamma\\left ( \\tfrac{n}{2}+\\tfrac{1-i}{2}\\right ) \\right ) \\\\\n&\\qquad \\qquad - \\left ( \\tfrac{1}{2}(n-p-1)\\sum_{i=1}^p \\psi\\left(\\tfrac{1}{2}(n+1-i)\\right) + \\tfrac{1}{2}(n-p-1)p\\ln(2) + \\tfrac{1}{2}(n-p-1)\\ln|\\mathbf{V}|\\right) + \\tfrac{np}{2} \\\\\n&= \\tfrac{p+1}{2}\\ln|\\mathbf{V}| +\\tfrac{1}{2}p(p+1)\\ln(2) + \\tfrac{1}{4}p(p-1) \\ln(\\pi) + \\sum_{i=1}^p \\ln \\left (\\Gamma\\left ( \\tfrac{n}{2}+\\tfrac{1-i}{2}\\right ) \\right ) -\\tfrac{1}{2}(n-p-1)\\sum_{i=1}^p \\psi\\left(\\tfrac{1}{2}(n+1-i)\\right) + \\tfrac{np}{2}\n\\end{align}", "58d94bfa4cd439021c46cd57edb46b4f": "\\hat H_\\varepsilon = \\hat H_0+\\varepsilon\\sigma_3 =\\begin{pmatrix} E_0+\\varepsilon & 0 \\\\ 0 & E_0-\\varepsilon \\end{pmatrix}", "58d951bfbe8a5aec19262e500b5450db": " y=e^x(-xe^{-x}-e^{-x}+C)\\ ", "58d96f403eddfbcd9b869a73d9e34444": "\\pm\\sqrt{\\sec^2 \\theta - 1}\\! ", "58d99720ab4622ceec7aa30e9fbd3275": "\nE_{ij} = \n\\underbrace{\\frac{1}{3}(\\sum_k\\partial_k v_k) \\delta_{ij}}_{\\text{rate-of-expansion tensor} D_{ij}}\n+\n\\underbrace{\\left(\\frac{1}{2}\\left(\\partial_i v_j+\\partial_j v_i\\right)-\\frac{1}{3}(\\sum_k\\partial_k v_k) \\delta_{ij}\\right)}_{\\text{rate-of-shear tensor} S_{ij}},\n", "58d9a46ad6b6e135eab433180162b78d": "\nP_0(x) = 1,\\, P_1(x) = x,\\,P_2(x) = \\frac{3x^2-1}{2},\\,\nP_3(x) = \\frac{5x^3-3x}{2},\\ldots", "58d9c3b25d8c362e1a0dce0a132a024f": "\\Omega \\, \\omega \\,", "58d9eda93e19f61954846af9ab708e3b": "\\frac{1}{2} A_j^2 = S(\\omega_j)\\, \\Delta \\omega", "58da14931c30d7b8e5bb9fe9bb968cc3": "\\hat{A}_2 = \\frac{1}{N} \\sum_{n=0}^{N-1} x[n]", "58da7cb6b077cbbf4906820de5bb99b6": "x \\leftarrow x+p", "58da80fbf6a33394938997776b71fb30": " \\frac{2 \\pi}{P},", "58da8afca1fa4cba6764607864bf9475": " (K+v)\\cap K=\\varnothing.\\,", "58daee1e6a0d3ffe0b414d09eb0c0cc1": "{_1^0}\\text{S} + {_1^1}\\text{S} \\rightarrow {_2^1}\\text{P}", "58db01091eb1b537c62219032fce66e8": "\n\\frac{\\Gamma_1 \\vdash e_1\\!:\\!\\tau_1 \\quad \\cdots \\quad \\Gamma_n \\vdash e_n\\!:\\!\\tau_n}{\\Gamma \\vdash e\\!:\\!\\tau}\n", "58db243d2e44413a29e440e12f98982b": "\\scriptstyle x' ", "58db7e28d7b770ab2c4e771ce3f9093f": " I = \\prod_{i=1}^n P_i^{a_i}", "58db80d2b3bb3c740802e0a606ed9a41": "\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n=\n\\begin{bmatrix}\n 1 & 0 & 1.13983 \\\\\n 1 & -0.39465 & -0.58060 \\\\\n 1 & 2.03211 & 0\n\\end{bmatrix}\n\\begin{bmatrix} Y' \\\\ U \\\\ V \\end{bmatrix}\n", "58dbb3478bf7ae4ff491f09941dae050": "\\Sigma_x(f)", "58dbe855ff31c21e8027af44d39586d4": "\\operatorname{div}(\\varphi \\mathbf{F}) \n= \\operatorname{grad}(\\varphi) \\cdot \\mathbf{F} \n+ \\varphi \\;\\operatorname{div}(\\mathbf{F}), ", "58dbfb58c736aa1419ac89f4709ea02c": "\n\\begin{align}\n& A=\\frac{V_{\\text{obs, r}}}{d\\,\\sin\\left(2l\\right)} \\\\\n& B=\\frac{V_{\\text{obs, t}}}{d}-A\\,\\cos\\left(2l\\right) \\\\\n\\end{align}\n", "58dc0a64fe737bbaa2af30b8ffd01dae": " L^2(\\Omega, \\mathbb{R})", "58dc1a35bb5d5f0ae7afbff6c22000b7": "m \\times n \\times n", "58dc7bef5e736c994ce4705c33f32476": "\\frac{\\sqrt{3}}{4}+2\\times10^{-4}\\leq B\\leq \\sqrt{\\frac{\\sqrt{3}-1}{2}} \\cdot \\frac{\\Gamma(\\frac{1}{3})\\Gamma(\\frac{11}{12})}{\\Gamma(\\frac{1}{4})},", "58dcb780dd588b487177bd89c5cc926b": "\\rho(L_\\omega)<1", "58dcbabb53681c2482e6205892cf44be": "\\mathbf{c}(U) = \\sup_{x \\in U} |x|", "58dcd3f9ee45985455b96938463de649": "Q(p;\\mu,s) = \\mu + s\\,\\ln\\left(\\frac{p}{1-p}\\right).", "58dcf40adb3ee4e4367c8489cc6a576c": "b^2m + c^2n = a(d^2 + mn).\\,", "58dd056fa7ab7550d7c898fb01e2b4c2": "\\dot x, \\dot y ", "58dd24bf66bde400914b43d96a02e5a0": "du^1 \\cdots du^k.", "58dd52d1311442d4742aba070e0bd198": "\\xi_{sup}(\\alpha)=\\Phi^{-1}(1-\\alpha)", "58ddf6387b8df9027a2643f21d5671d8": "\\exist x \\in y \\; \\psi(x)", "58de12a26adbf2be76d78f0959494b94": "i=1,\\ldots,m", "58de462a3440028c9a214bf3c4c6460e": "\\Psi [\\gamma] = \\int [dA] \\Psi [A] W_\\gamma [A]", "58de534f9253df9e6a1a86da1edb8cde": "(H,\\langle\\cdot,\\cdot\\rangle)", "58de9e5e7a025219a4b1669243addd56": "J_{i+1}/J_i \\,", "58ded806f0413fbe43def36d484b62a0": "V(F) \\subset {{\\mathbf{K}}}^n", "58df1ba9da37b1864ab3ea44bc096204": "f:U\\to N", "58df212c4f7d9e93e66e4f7ad28ad96b": " E (\\langle DF, u \\rangle ) = E (F \\delta (u) ),", "58df4d0680bd268ae84bbac3a7629eff": "(\\lambda+A)(\\lambda-A)^{-1}", "58df664e98cea838dbeade6f759ffb7c": " \\left \\| A \\right \\| _\\infty = \\max \\limits _{1 \\leq i \\leq m} \\sum _{j=1} ^n | a_{ij} |, ", "58df71f5e2dedd5c779d74b5de881fea": " E^n", "58df75c31540097aa6e1cb07624af906": "O(2^n \\cdot n^2)", "58df8945cd031bc284ec6f8759d99869": "sh_1", "58dfc11663720adbb5ec107cf4a5d760": "\\pi \\ominus \\sigma", "58dfca69dbedd33b1db907257d7b7db0": "\\delta \\in B ", "58dfe0ff6ebabe2344449c431f32d739": "\\{1, -1\\}", "58dffc42e4f6b5678da79aec53dfb328": "R=\\sqrt{(1-\\mu_x)^2+\\mu_y^2}", "58e02c7d3c4def4894e2e07408ba2aac": " \\begin{align} \\mathbf{a}\\mathbf{b} &= (a_1\\mathbf{e}_1 + a_2\\mathbf{e}_2)(b_1\\mathbf{e}_1 + b_2\\mathbf{e}_2) \\\\&= a_1b_1\\mathbf{e}_1\\mathbf{e}_1 + a_1b_2\\mathbf{e}_1\\mathbf{e}_2 + a_2b_1\\mathbf{e}_2\\mathbf{e}_1 + a_2b_2\\mathbf{e}_2\\mathbf{e}_2 \\\\&= a_1b_1 + a_2b_2 + (a_1b_2 - a_2b_1)\\mathbf{e}_1\\mathbf{e}_2. \\end{align}", "58e030773c782b8f92f31110069c23e6": "\\alpha\\wedge (d\\alpha)^n", "58e11303f567e0582535419acc38cda7": "\\varphi\\left(\\int_a^b f(x)\\, dx\\right) ", "58e1458e491b7143079b81f3a784ee12": "D_{ij} \\subseteq (\\frac{1}{2})D_i", "58e14fe8acd567bcdcd0747e2b556caf": "q=W^TX^Ts=W_2^TW_1^TX^Ts", "58e150daf9ef05bf552b857b79696dcb": "x\\in{\\rm cl}_X(Y)", "58e17e8dcbc648588e5c35c69017cb1e": "p = (A_1 \\to w_1, ..., A_n \\to w_n)\\in P", "58e1a73dfc82de31f48e48bc8c376af6": "1/R_{TOT}", "58e1f5f96da3bf5b4ec2af5b2f457d4f": "dx = N \\, dz. ", "58e234d5b77fb8af674f54a4118ad91a": "n, l, m_\\text{l}, m_s", "58e2509dba5f6d6b39bc16da18d570fd": "\\tanh\\left(\\frac{t}{4GM}\\right) =\n\\begin{cases}V/U & \\mbox{(in I and III)} \\\\\nU/V & \\mbox{(in II and IV)}\\end{cases}\n", "58e25190bac8c1a921a69bda318403b7": " \\sqrt{\\sigma_S^P}\\sqrt{\\sigma_L^P}\\over \\sqrt{\\sigma_L^D} ", "58e2c0dd1b40866ef94af3f3bc4a00e7": "w\\Vdash\\bot", "58e2c4e245b062eb13b55fc55e86f90d": "G\\times G \\to G : (x,y)\\mapsto xy", "58e2c9e6bdb0245fd44b7a6514c4b301": " \n\\int_E f_k \\, d\\mu \\geq \\int_{A_k}f_k \\, d\\mu \\geq (1-\\epsilon)\\int_{A_k}\\varphi\\, d\\mu.\n", "58e3f843727c2602fc5510d05fc8777d": "\nM_{0} \\equiv \\int d\\zeta \\ \\lambda(\\zeta)\n", "58e404191c95a1266c9cb7a268651a0c": "\\mathrm{Volume} = \\text{base area} \\times height ", "58e461f80e4ed562c1fa5a108bc2655b": "E_{1,1}(z) = \\sum_{k=0}^\\infty \\frac{z^k}{\\Gamma (k + 1)} = \\sum_{k=0}^\\infty \\frac{z^k}{k!} = \\exp(z).", "58e48d3c9cc24582ba67b62312cf5678": "p=(\\succ_i^p)_{i \\in N}", "58e4cad3a07297850ef8c8bb1b8e9cfd": "f_i : Y \\to X_i", "58e4d50f2915dc415656753b3d362862": "O_{n,n}", "58e4d5e9049ec2b0cf9978620c79bfcb": "\\min(A[1..n])", "58e5469b5575dfa1355ff9024f4c7e3b": "(ax^n - 1)/(ax - 1) = y^2", "58e54a7afc8529a52e5f32a67bb9c66b": "\\,n+1\\,", "58e6007cc42989592737d615771b10c1": "P=f(Q)", "58e6239b326d79007917559ff72341e3": "\\sigma_D = s \\sigma_I,\\; s = constant", "58e638317684c8fa0ae710dfcb0e6212": " {mv^2 \\over r } = {{e_M}^2 \\over r^2} ", "58e642d2cb42ea83876f9ac34f928c7a": "\\vec j.", "58e6a964ebefdc0d07b3e200ab9b4bcb": "t \\mapsto e^t -1 -t", "58e6be29d9cb8937b1d2b2766b5f6b99": "\\frac{|\\frac{q_\\bar{p}}{m_\\bar{p}}| - \\frac{q_p}{m_p}}{\\frac{q_p}{m_p}}", "58e6dbb13d615f55bfdb7b87c8b16432": "\\{e,a,b,a^{-1},b^{-1}\\}", "58e7722edcbe9efc142a78213decd7b6": "\\mathrm{S\\ m^{-1}=A^2kg^{-1}m^{-3}s^3}", "58e890ac9024a3f14830d8ae5b53a95c": "|A\\cap B|", "58e8af06b6b711169dadcf8d7f19de72": "d\\tau = dt \\sqrt{ 1 - \\left (\\frac{r \\omega}{c} \\right )^2 },", "58e8f083dd6fb7506f25b6e71fcaba61": "n^{\\underline k}=\\frac{n!}{(n-k)!}.", "58e90c5696dfbadde69926de90b18e5c": "K_\\nu(x e^{\\pi i/4})\\,", "58e910f884ebab0177bcc3a7d9317aed": "X_{k+N/4} = U_{k+N/4} - i \\left( \\omega_N^k Z_k - \\omega_N^{3k} Z'_k \\right),", "58ea40fcad103a509f587d3d36761382": "S''_{zz}(0) = P J_z P^{-1}", "58ea882af442896ce505892f104dc476": " \\left | \\mu - \\nu \\right | \\leq \\sigma. ", "58ea8f04d035c7e0ce7d77001a48618b": "\\cot ^2 x_1 + \\cot ^2 x_2 + \\cdots + \\cot ^2 x_m\n= \\frac{\\binom{2m+1}3} {\\binom{2m+1}1}= \\frac{2m(2m-1)}6.", "58eacc51663960686ccbb930a3bc5801": "\\textstyle N \\rightarrow \\infty", "58eaf56a064925c4dd0f5bde5ca29d56": "\\alpha = \\pm 1", "58eb15a89dc5cd64584efaaef9d98552": "e^t", "58eb967c4eef117b4b35e6d36f8f61c7": "\\tau(\\vec R, \\vec R^')= {2\\sum_{i=1}^N\\sum_{j=i+1}^Ns(r_i-r_j, r_i^'-r_j^')\\over N(N-1)},", "58eccc97e46baa3d48465714b469cd6d": "S(\\rho,\\sigma)", "58ece07759ecd9c7247847f38c994486": "\\mathcal{S} = [s_{j_1,\\dots,j_N}]_{I_1 \\times I_2 \\times \\cdots \\times I_N}", "58ed11ef9125a15be2173355123402ce": "\\binom{p}{i}", "58ed1ef1d30b491d4712c09691709e9d": "P(\\alpha_i)E(\\alpha_i) = y_iE(\\alpha_i)", "58edc5d2cf2ae7caefafab5c72e86ade": "\\alpha_1=\\frac{1}{2}n-\\frac{2}{3}n^2+\\frac{5}{16}n^3,\\,\\,\\,\\alpha_2=\\frac{13}{48}n^2-\\frac{3}{5}n^3,\\,\\,\\,\\alpha_3=\\frac{61}{240}n^3,", "58edd48c88e63cbaf67a63518c7957dd": "|ax + b|", "58ee022eebed13bcef0b7c67683188cc": " \\underline{\\mathsf{f}}(1) = 1 ", "58ee0366e3dd3cb74bda71d3cf971ed1": "a(z) = {\\alpha}_0 + {\\alpha}_1z", "58ee41ab86dd0b806b52844453801ac6": "Lw ", "58ee439dd783dd073f1d340d6b10febb": "g+ij", "58ee89463520f770f4b084d6281c7f38": "u_s=dx_s(t)/dt \\,", "58ef5dd024dc23bc94d0db489c0e5b8c": " f_i(X_i) = E(Y|X_i) - f_0 ", "58efeb9aabbbcad9f73eea025abd81d1": " U(R)^\\dagger \\widehat{S} U(R) = \\widehat{S}", "58f02fcc789997441f6f95b73c512409": " \\exp(a_1(s-1)+a_2(s^2-1))\\,", "58f04d72685273c08927a746deaa902b": " f(\\lambda_1, \\ldots, \\lambda_r) \n= \\mathrm{Vol}_n (\\lambda_1 K_1 + \\cdots + \\lambda_r K_r), \\qquad \\lambda_i \\geq 0, ", "58f09ea610f5334afbc974a206dd0a5d": "\\mathcal{I}_{m,n}", "58f0d015261312fe07d8076dfd5dff1c": " C_\\mathrm{M} = C_\\mathrm{gd} \\left( 1+g_\\mathrm{m} (r_\\mathrm{O} \\| R_\\mathrm{L})\\right) ", "58f0f5ce7032b6f2f4ed3169ff69bf5c": "V=f(t)=V_0 e^{-{t \\over \\tau}}", "58f19798bc2389ecc785be3f2bd24a92": " (R+UV^{T})^{-1}=R^{-1}-R^{-1}U(I+V^{T}R^{-1}U)^{-1}V^{T}R^{-1}, ", "58f1ac0f697e60c432634d1b6561f7f3": "n^{*} \\pmod{m}", "58f1d68c0759028ae24535a37236b8cf": "T^4-T_s^4", "58f1fe2c063d3f9c23b2f7b4a0b2fc97": "\\alpha_k := \\frac{\\mathbf{r}_k^\\mathrm{T} \\mathbf{z}_k}{\\mathbf{p}_k^\\mathrm{T} \\mathbf{A p}_k}", "58f24bc64cf4695567b5e7142858e4b7": "\\scriptstyle {\\rm MCG}(\\mathbf{S}^2) \\simeq {\\mathbf Z}/2{\\mathbf Z}, ", "58f2555cfb26745d39639636e0aaf37d": "aa^{-1}", "58f27a3dae9e5e0c873f0d36f578d164": "\\boldsymbol{ a}_C", "58f28be891679cbd73a8b6d758ff6ee9": "F_L = C_Lbc \\left(\\frac {1} {2}\\rho W_m^2\\right)", "58f2925acfa77bf53cdcdb3188091cc7": "H_1 = 0\\,", "58f2c25cf2ccd3d2ed156e8667539167": "g(A \\cup B) =g(A)+g(B)+\\lambda g(A)g(B)", "58f2fccd17fab4134a8d06ad84af0df9": "\\,\\alpha=\\beta=1", "58f319ece9aaccf75ceaefe3280bcd9c": "\\pi^{-1}", "58f382fbbfc1714a4d0e1a482f38ffd5": "\\frac{1-\\gamma^5}{2}", "58f38a96b67c4a6584d9ed41bf62326c": " m = (h - 1) (1 - 3 h) \\, .", "58f39769d8983e5ede73fceda7e4b7e4": "p_{01}", "58f40d5c37321eceb49335336e4104c2": "r(\\varphi)=a \\,", "58f459f347e9a5af60d930526b8221bf": "A\\models\\forall x\\phi(x) \\text{ iff every } a\\in A\\text{ is such that }A\\models\\phi[a]", "58f4c639abe790294ee7ef18c498830f": "X_1^3+X_2^3-7=e_1(X_1,X_2)^3-3e_2(X_1,X_2)e_1(X_1,X_2)-7", "58f56ac4b504d1d5caa8273d33e01552": "\\dot{\\gamma}", "58f59d2e77cdf0077c942e93c5dd9f4e": "\\lambda (S_{R}) = \\lim_{\\delta \\to 0} \\frac{\\mu \\left( \\overline{B_{R}} + \\overline{B_{\\delta}} \\right) - \\mu \\left( \\overline{B_{R}} \\right)}{\\delta}", "58f5f33f5d184a298058a9bb42e15bb2": "O(\\lg n)", "58f604bed385ccc7525ca027157fd5f9": "\\alpha\\ne0", "58f66f1b36f79d42001c592acc6a774e": "\ny=\\frac{A_0 + A_1x} {1 + B_1x + B_2x^{2}} \n", "58f70ad9e91811e71a35d06d7965f1a8": "\\delta ^n S_n(x)", "58f72fba875aebe7ef73636433483ca7": "\\frac{-i}{-k}=j", "58f75f3a5ddcef5dcac345ffe654b17a": "\\rho = \\frac{P_{atm}}{R_aT_{in}}", "58f7a84e5c24743d1930fd7392104704": "X_n, n = 1, 2, 3, \\dots", "58f7bd8d6fe93047540dfc7d35366a87": "\\mu_\\theta(\\lambda) = e^{-\\theta}\\frac{\\theta^{|\\lambda|}(f^\\lambda)^2}{(|\\lambda|!)^2},", "58f7cf890cf99250ea7a8a8009ecd5bc": "P_1^1 \\or P_2^2", "58f7f8dabe190ae58f00766af5a1c502": "f ", "58f84a9508b75720a2586c4161ce0ffe": "\\scriptstyle 1/60^{9}\\approx 9.92\\times 10^{-17} < 10^{-16}\\,", "58f88345ddcf5041fe7d1e4b20f88e9c": "\n\\frac{3G M^{2}}{5R} > 3 N k_{B} T.\n", "58f8c7fe60dfb6145b8a8a581328fa48": "\\gamma_n(\\lambda x) = \\lambda^n \\gamma_n(x)", "58f91472c5c355a105babdc8c1ac2e9a": " \\gamma_1 \\,", "58f917ce0913aba42fddf10a9f54e95a": "\\eta(t_0) = \\eta_0 = 1.48 \\times 10^{18}\\ {\\rm s}", "58f925ea0bd855ee5dba87b4f7e77ca9": "\\varepsilon_x = \\frac{\\overline {ab}-\\overline {AB}}{\\overline {AB}}\\,\\!", "58f93ead0fcf616460f1eca9b0a6947a": "\\left|\\sum_{j=1}^{N-v} a_j a_{j+v}\\right| \\le 1\\,", "58f94e720659b7b45092b9ad05f75af6": "f = 2^{(d-69)/12} \\cdot 440\\ \\mathrm{Hz} \\, ", "58f99146cb8a675fd58ddf184d5e96df": "\\textstyle {4\\choose 0,4,0}", "58f9c86409adbe1b62925a4f0e18074f": " \\begin{align} \\mu \\left(\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2}\\right) - \\frac{\\partial p}{\\partial x} + f_x &= 0 \\\\\n \\mu \\left(\\frac{\\partial^2 v}{\\partial x^2} + \\frac{\\partial^2 v}{\\partial y^2} + \\frac{\\partial^2 v}{\\partial z^2}\\right) - \\frac{\\partial p}{\\partial y} + f_y &= 0 \\\\\n \\mu \\left(\\frac{\\partial^2 w}{\\partial x^2} + \\frac{\\partial^2 w}{\\partial y^2} + \\frac{\\partial^2 w}{\\partial z^2}\\right) - \\frac{\\partial p}{\\partial z} + f_z &= 0 \\\\\n {\\partial u \\over \\partial x} + {\\partial v \\over \\partial y} + {\\partial w \\over \\partial z} &= 0 \\end{align}", "58fa0335e0440c9479814feb75dfb892": "b^{}_i", "58fa0bdb4bbce7f7625a0efeaf89a03c": "\\!\\,\\gamma", "58fa64973a8161a5e89f9b1e53ee1245": "\nD(i,j)=\\frac {1}{N-1}\\sum_{k \\ne j}^{N-1} d(i,k|j) \n", "58fa659943de0cf801827d2bb2cc1f1a": "\\overline M", "58fa69c47bb6932667a26073d43e4a18": "\\mathrm{B}_2 \\cong \\mathrm{C}_2", "58faa2f7c2250a96d5c6ac03233c76c4": "\\Delta S_i", "58fabdda01910b88881e6cfd5b09d2fb": "\\zeta(-n)=-\\frac{B_{n+1}}{n+1}", "58faea17f244fceecbfd4c8be73a4c8d": "f(x) < 0", "58fb07e3d4fa708afd0734aab363fd36": "\\xi ", "58fb55992a6431a2af69ab14e108a7fb": "h_{bc,}{}^a \\ \\stackrel{\\mathrm{def}}{=}\\ \\eta^{ar} h_{bc,r}", "58fc1031ddb066ace542eef55c11a4eb": "\\begin{align}\nu & = \\tfrac{1}{\\sqrt{2}}(x+y) \\qquad & x &= \\tfrac{1}{\\sqrt{2}}(u+v)\\\\\nv & = \\tfrac{1}{\\sqrt{2}}(x-y) \\qquad & y &= \\tfrac{1}{\\sqrt{2}}(u-v)\n\\end{align}", "58fc14f7c629eb7a223fcb9460e690a1": "\\Theta_{\\textrm{cm}}", "58fcdead6d94153a7c1e13408e93e394": "\\text{Cl}_2\\left(-\\frac{\\pi}{3}+2m\\pi \\right) =-1.01494160 \\cdots ", "58fcefff12418facd74c60c65542b53e": "\\scriptstyle I_i \\,+\\, I_j \\;=\\; R", "58fda679acdc281d2b21c90520a7b6c9": "y \\in [0.06, 0.08]", "58fdabc72d19669fa24868b5ee85a2f7": " \\Sigma = \\frac{\\sigma}{\\rho} ", "58fdc212bc2140d7b3c38ddf5803308b": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = J^{-1}~\\boldsymbol{F}\\cdot\\dot{\\boldsymbol{S}}\\cdot\\boldsymbol{F}^T\n = J^{-1}~\\boldsymbol{F}\\cdot\n \\left[\\cfrac{d}{dt}\\left(J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T}\\right)\\right]\n \\cdot\\boldsymbol{F}^T\n = J^{-1}~\\mathcal{L}_\\varphi[\\boldsymbol{\\tau}]\n", "58fdc3affe89bc93740ac201853759b6": " \\tilde{H_n}(X) = \\ker(\\partial_n) / \\mathrm{im}(\\partial_{n+1})", "58fdd7aeaf53322cba0cb0459bbd776b": "x = u", "58fdd933c9bfb864e56f2ffc6a13f7a6": "Q(t)", "58fe4d7d44bad739681189d105bb517a": " \\left\\vert A_d \\right\\vert = \\frac{w(d)}{d} X + R_d", "58fe7d849fd66ccc221361b962bf8090": "\n\\ddot{s}_{\\overline{n|}i} = (1+i) \\times s_{\\overline{n|}i} = \\frac{(1+i)^n-1}{d}\n", "58ff4a779ef416bb18762836d2284570": " x'", "58ffbf2f1a87a186918fa7af6e787364": "\\mbox{force} = \\frac {z_1z_2e^2}{4\\pi \\epsilon _0 \\epsilon _r r^2}", "58ffc11846a256310b2c92ea48b44cae": " \\lim_{n\\rightarrow\\infty}a_{n}=0 ", "58ffd1c345cc164f0a16b549e4c32675": "f(x) = \\int_0^\\infty \\frac{g(y)}{\\Gamma (y)} L_y(x) \\, dy. ", "590002f350d490697da6e744f7224c1f": "X \\equiv x^{q} _ {\\bar{t}}\\bmod \\psi_l(x)", "59002c79d85ad792843ac3a59dcc5770": "\\varprojlim:C^{I^{op}}\\rightarrow C", "5900c5d11e5f40d29b9d088a00cce781": "x^{32} + x^{31} + x^{24} + x^{22} + x^{16} + x^{14} + x^{8} + x^{7} + x^{5} + x^{3} + x + 1", "5900d9c25dc58979227e98709e6dcefe": "\\frac{v_{\\text{in}}-v_x}{Z_1}=\\frac{v_x-v_{\\text{out}}}{Z_3}+\\frac{v_x-v_{\\text{out}}}{Z_2}", "5901325520ca2b5ad71b0993fec700c8": "AB + e^- \\to AB^{-\\bullet} ", "59016efabd038e5931ce5ee04bbb9ee6": " {\\Delta}_{\\rho}", "590186dba7d3d5f4d24ef3e723677db2": "\n\\min_{\\mathbf{a}_j, \\, \\mathbf{b}_i} \\displaystyle\\sum_{i=1}^{n} \\; \\displaystyle\\sum_{j=1}^{m} \\; v_{ij} \\, d(\\mathbf{Q}(\\mathbf{a}_j, \\, \\mathbf{b}_i), \\; \\mathbf{x}_{ij})^2,\n", "5901d0ee021fc053e2cffaeb45eb02cd": "(\\pm(P_i+P_{i-1}),0)", "590203be637f3eeb9135ca708d3b0e7d": "\\mathbf{e}_{12} = i ", "59026f3d6cea9f6478a909fec4a33bd3": "\\ X ", "59027ba5800e1084eb2857775085371b": "S_n(s) = \\frac{1}{2} - \\frac{1}{4} \\left ( 1-2s+2 \\sqrt {s^2 -s} \\right )^{n} - \\frac{1}{4} \\left ( 1-2s-2 \\sqrt {s^2 -s} \\right )^{2}.", "590280d7eb016753a8997fc53b4604a8": "\\text{FLOPS} = \\text{cores} \\times \\text{clock} \\times \\frac{\\text{FLOPs}}{\\text{cycle}}", "5902b57ebd0fb0b7eec00e661d8a4d92": "\\int d\\epsilon \\, p(\\epsilon) \\frac{1}{e^{\\beta(\\epsilon-\\mu)}-1}=1", "5902bcd2085e294f52a6b0a91f7e5576": "\\frac{\\mathrm{L} \\cdot \\mathrm{atm}}{\\mathrm{mol}}", "5902cd1151eed0a460b159baa938e3e6": "F_p\\times F_p", "5902ef086d33e376004d0369e579086a": "\\frac{X_b}{X_b^0-X_b} = exp \\left ( \\frac{-\\Delta\\,G'}{RT}\\right )", "59030791bc5ede6fe8508ebd9def1aba": "\\tfrac35\\times\\tfrac33=\\tfrac9{15}", "5903204cd18d301e2d87079f5d62bd9a": "\nc_0+c_1 p\\equiv a_0+a_1 p+b_0+b_1 p \\mod p^2\n", "5903de5cd12d5bb8fece7d3ed4cdd867": "\\mathbf{\\hat p}|-k\\rangle=-k|-k\\rangle", "590417a598900374dde48c8198fcc611": "W(x,p)=F\\left(\\frac{1}{2} m \\omega^2 x^2 + \\frac{p^2}{2m}\\right)\\equiv F(u).", "59045454043dde3acbeb076919e2dd2f": "Q T_i", "5904b447dd397e4aace9fa7122547a9f": "\\Nu_{k+1}", "5904b8c721359d3e07944691aa0479b1": "\n \\begin{align}\n \\operatorname{add} &\\mathbin{:}\\ (\\mathbb{N} \\times \\mathbb{N}) \\to \\mathbb{N} \\\\\n \\operatorname{add}(0, b) &= b \\\\\n \\operatorname{add}(\\operatorname{succ}(a), b) &= \\operatorname{succ}(\\operatorname{add}(a, b)))\n \\end{align}\n ", "5904cf31214072981179d9f3e84a0bc8": " L_2 = \\, \\frac {-G_M K_M} {s^2 L_M M} ", "5904da8f9dc680d7e874ce2c330c74e4": "S(\\rho) \\; = S(U \\rho U^*).", "5905475576a21ecdafdaab879ff45aff": "i = 1", "5905be91f61730bf19c5fa68d531972e": "\n w_1 = \\frac{5}{24EI}\\left[1026125 - 39450 x + 8 x^3 + 20 M_c (-125 + 3 x) + 480 R_a (-85 + 3 x)\\right] \\,.\n ", "5905cf686db46a5b01233c1c72e5a86a": " (\\kappa-2)~r^2~\\cos\\theta \\,", "590671f3182a37d341033e9bc85a27bb": "{1 \\over N_e} = {1 \\over t} \\sum_{i=1}^t {1 \\over N_i}", "59068ed26f0b2e504d168a07fed51bdb": "y(x) = e^{-bx} \\left( \\frac{e^{bx}}{b}+ C \\right) = \\frac{1}{b} + C e^{-bx} .", "590725c996b8d8553697d9823c9526a0": "\\dim f(X) < k", "590737f80968661e84a663ea3131a192": "\\mathbf{B}(t) = \\sum_{i=0}^{n} \\mathbf{P}_i b_{i,n}(t) \\mbox{ , } t \\in [0,1]", "590783d1219b95146f3b3fba27b64e6a": "\\frac{\\partial \\ln{T_{eff}}}{\\partial \\ln{M}} \\approx 0.1", "590795dd1353b76babbebd0fafe274a1": "x_{max} = 1", "59079e29e192ba29bc12b53b71f72ef2": "n=\\tfrac12 (3^k-1)", "5907a986ab93c76abf80ae0d2a319724": "\\Omega_{\\rm radiation}=0.266/3454", "5907ecc32883c93680f95a6bba87afe3": "\\frac{A+B}{2}.", "5907fe6edbd75ff047850c5fa49eb1a2": "G^{32}", "5908b92b4aa5ce9eef73611b662133b4": " \\frac{1}{\\sigma^2}\\sum_{i=1}^n \\sum_{j=1}^{n_i} \\widehat\\varepsilon_{ij}^{\\,2} ", "590907d9de87220ac6d25bd43fc0dd79": "\\Delta(i\\omega_n)", "590943fd8e087057d2b2cd3a1f50766b": "f(x_1-\\theta,\\dots,x_n-\\theta)", "59096d1dab14260a6e8121b5dba5add4": "(b-a)\\sigma^2", "5909ad384c7e33a1df450af45c722c90": "\\int\\cot ax\\;\\mathrm{d}x = \\frac{1}{a}\\ln|\\sin ax|+C\\,\\!", "590a8845294a4c58eb8ea82976c00c0e": "{1 \\over N_e}", "590aa17389d7c75860b5a1ba7fc8d018": "\\psi_{\\bold{k}}(\\bold{r}) = \\frac{1}{\\sqrt{\\Omega}} e^{i \\bold{k} \\cdot \\bold{r}}", "590aa4fed427cf9b1f45293f044be4f4": "\\delta V_c =\\frac{1}{3} \\delta \\sum_{j} h_j O_j = 0 ", "590aaf29ff8ad06d387f6e03cdf5bde5": "[A]^{< \\kappa} := \\{X \\subseteq A| |X| < \\kappa\\} \\,.", "590b127f03f32dbb6f62778be4676640": "\\Delta E_{94}^* = \\sqrt{ \\left(\\frac{\\Delta L^*}{k_L S_L}\\right)^2 + \\left(\\frac{\\Delta C^*_{ab}}{k_C S_C}\\right)^2 + \\left(\\frac{\\Delta H^*_{ab}}{k_H S_H}\\right)^2 }", "590b584f6fce2881aabf5b19bf158455": " m = \\frac{E}{c^2}.", "590bbc572f7ec00e76ed40f4dab07dd3": " \\mathbf{Q}_p=\\operatorname{Quot}\\left(\\mathbf{Z}_p\\right)\\cong (p^{\\mathbf{N}})^{-1}\\mathbf{Z}_p.", "590bec7fd16409007ce78d2d8119bb4d": " E_k ", "590c041da36b4c7c9b09c90bfb8e84e3": "\\frac{V(t)}{V_0}=0.9", "590c31cbd0bd49e87bf199631dd95a9d": "\n\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)-1}{4^{2n}} = \\frac{13}{30}-\\frac{\\pi}{8}.\n", "590cb34b75a4e59bc7b95a9ad7187f63": " \\forall n\\; \\bigl( Q(n) \\rightarrow P(n) \\bigr) ", "590ccc6443248ce5c1e8e4a72195223c": "J_1\\,\\!", "590d70f72eac68bd3968fc792ff25850": "\\rho_t(X) \\geq \\rho_t(Y)", "590e3329b32cea941f530704fb2ad148": "(M,x) \\to (M,x)", "590ec65fa840dbecfdc6a5e87e2ca0ee": "K_{\\rm d} \\equiv 1/K_{\\rm a}", "590edac6355206f468728f89920d6827": "n,m", "590ef5a78fe7fe1b107b1db51ed640bd": "\\lim_{n \\rightarrow \\infty} \\left( \\max_{-1 \\leq x \\leq 1} | f(x) -P_n(x)| \\right) = +\\infty.", "590f535cc82fc0fdd58e43285ab06ccf": "\\delta(P,Q) = \\frac 1 2 \\|P-Q\\|_1 = \\frac 1 2 \\sum_x \\left| P(x) - Q(x) \\right|\\;.", "590f5a912de22b5e5af5fa4d463228a6": "S_N f\\left(\\frac{2\\pi}{2N}\\right) = \\frac{\\pi}{2} \\left[ \\frac{2}{N} \\operatorname{sinc}\\left(\\frac{1}{N}\\right) + \\frac{2}{N} \\operatorname{sinc}\\left(\\frac{3}{N}\\right)\n+ \\cdots + \\frac{2}{N} \\operatorname{sinc}\\left( \\frac{(N-1)}{N} \\right) \\right].", "590f8f3a042b6c003bade8be6980e674": " D_N = \\sup_{Q \\subset [0,1]^s} | \\frac{\\mbox{number of points in } Q}{N} - volume(Q)| ", "590fa84150d00de716ca4f1f7415e1d3": "f'(t')", "590feef7d6a63becf59a0040d1fd0512": "f_j ", "59102967bb2e0555a35228f6c8cb3584": "u_x = {\\partial u \\over \\partial x}", "59113ecd31ab0ea2fb094bbd6e982ced": "s = (1-c)", "59114eaa168591c948dc79c0ce38517e": "v^{\\circ}:= \\min_{d\\in D}\\max_{s\\in S(d)}f(d,s).", "59117524f55f0e8b90de8ad49c846b05": "GF(m) \\cong F_2[x]/(p(x))", "5911bf5f11ab037bbc2576a5f62d3d6f": "\\empty", "59120e6961ed798965f113943a9b82f4": "\n \\cfrac{1}{(\\sigma_3^y)^2} = \\cfrac{R_0+R_{90}}{(1+R_0)~R_{90}}~\\cfrac{1}{(\\sigma_1^y)^2} ~;~~\n \\cfrac{1}{(\\sigma_2^y)^2} = \\cfrac{R_0(1+R_{90})}{(1+R_0)~R_{90}}~\\cfrac{1}{(\\sigma_1^y)^2}\n ", "59122c0dd76ced34fdb5abfe8a17e1e6": "\nJ^k= \\begin{pmatrix} \\lambda & 1 \\\\ 0 & \\lambda \\end{pmatrix} ^k\n=\n\\begin{pmatrix} \\lambda^k & k\\lambda^{k-1} \\\\ 0 & \\lambda^k \\end{pmatrix}, \n", "59126dcb910a5c5099e39c30e641964b": "\\vec F_{mag} = q\\vec v \\times B", "5912c3ebdc5aef13d1b3d32e9a13d6da": "X_1,\\ldots X_{j-1}", "5912fc1251cd0c1e212f6dd8d19f17ef": "\\sin", "5913030ec8f0619428085932cea61e43": "\\lambda(\\vec{r})", "59130a2c6139e888d13b2a716924ea41": "c_{\\mathrm{air}} = 20.0457\\,\\mathrm{m \\cdot s^{-1}} \\sqrt{{\\vartheta}+ {273.15\\;}}", "591318aaca0309d4f2d1f89c0a71b143": "\\widehat{\\mathbf{v}} = \\frac{1}{m}\\widehat{\\mathbf{p}}", "59131960f9622eca1a496d6000701c06": "r < 2M\\,\\!", "59131de20bf750dc8257d98c6a0c3c13": " \\binom nk = \\binom{n-1}{k-1} + \\binom{n-1}k \\quad \\text{for all integers }n,k : 1\\le k\\le n-1,", "59131e4122cdfe2693b411f3aad9eb30": "(0.5)*u(x_{1})=1\\!", "5913bcb9e77bce5d89ee9fed51c157e8": "0.1=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t_1}{2}\\right)\\right]\n\\qquad0.9=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{\\sigma t_2}{2}\\right)\\right]", "5914315c8951c8754e21a5274817381a": " X(t)", "59148ee3c3a43a71a698d76fb54ca329": "h_{d,r}", "591492e5c10c3a337b015f3cf46031fe": "\\frac{d_Y(f(x),f(x'))}{d_X(x,x')}", "5914a16eea5dc8226d1122e1f05c1eda": "L \\widehat{=} 11", "5914de9fad0b0c94ab7b1b79fc089a39": "\\forall y \\in B, \\exists x \\in A \\text{ such that } y = f(x).\\ ", "59152d8e0620685440085a2d588237b8": "C = \\sum_{j=1}^{N} C_{j}", "59152ec7a43f1f2e7942f42089f41068": "(f)_{k}=f^{\\underline k}=(f-k+1)\\cdots(f-3)\\cdot(f-2)\\cdot(f-1)\\cdot f", "59156b778b007ed36eb340d7a8de187d": "i\\in\\mathbb{N}\\,\\!", "591640220b1624c65655554f2022071f": "\\vec E_1 = - {\\gamma_eT_e \\over n_{e0}e}\\nabla n_{e1} ", "59164ca5c896da807d195cde3e00c2d9": "T^{\\hat{\\mu}}_{\\hat{\\alpha}\\hat{\\dot{\\beta}}} = 2i\\sigma^{\\hat{\\mu}}_{\\hat{\\alpha}\\hat{\\dot{\\beta}}}", "59166f43123d2847a30b7c4066b5c194": "X = (X_1,\\dots , X_n), X^{(i)} = (X_1, \\dots , X_{i-1}, X_i^',X_{i+1}, \\dots , X_n).", "5916a2cb95a748c3efe7ef34185d998b": "\n\\begin{align}\n\\begin{pmatrix}\nN_{t+l_i}\\\\\nN_{t+l_a}\n\\end{pmatrix} &=\n\\begin{pmatrix}\nS_iR_i & S_aR_i \\\\\nS_i & S_a\n\\end{pmatrix}\n\\begin{pmatrix}\nN_{t_i}\\\\\nN_{t_a}\n\\end{pmatrix}\n\\end{align}.\n", "5917953485e90d9a05c77e6f54af4d65": " \\Pi = i M R T", "5917bd7ad95d96ca51efcf277c32d459": "\\tilde{\\Theta}^i = (g^{-1})^i_j\\Theta^j.", "59189044558ed4d28ca8f7e9ef624f60": " v_{1} = \\left( \\frac{m_1 - m_2}{m_1 + m_2} \\right) u_{1} + \\left( \\frac{2 m_2}{m_1 + m_2} \\right) u_{2}\\,", "5918c9ce9848db1ce2c5bc8f64ca07ad": "(b^2 + {{a^2}\\over 2})(\\pi - \\arccos {b \\over a}) + {3\\over 2} b \\sqrt {{a^2} - {b^2}}", "5918dab4723dea5ae0af5e2cc97b968a": " P = \\begin{matrix} \\frac{dE}{dt} \\end{matrix} = F \\cdot \\begin{matrix} \\frac{dx}{dt} \\end{matrix} = F \\cdot v ", "5918e44c1889e52b466d64c62031a78c": "H_{1},...,H_{m}", "59190ad6fdac83baa55e247a696fbd06": "E\\left( \\tfrac{1}{4}\\left(\\sqrt{6} + \\sqrt{2}\\right)\\right) = 2^{\\frac 1 3} 3^{-\\frac 1 4} \\pi^2 \\Gamma\\left(\\tfrac 1 3\\right)^{-3} + 2^{-\\frac {10} 3} 3^{\\frac 1 4} \\pi^{-1} \\left(\\sqrt3 - 1\\right) \\Gamma\\left(\\tfrac 1 3\\right)^3 ", "59195fc1360647edea39bd666d1c449c": "a u(1) + bu'(1) =g(1).\\,", "5919886890477977974e7cee208922d8": "(1)\\quad ds^2=-\\Big( 1-\\frac{2M}{r} \\Big) dt^2+\\Big( 1-\\frac{2M}{r} \\Big)^{-1}dr^2+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;.", "591992f3856665c17c4a8a30393b0429": "f(a,b,x)", "59199c193ecb17f2f5aed1b57dfa6c2a": " c_p - c_v = \\frac{\\alpha^2 T}{\\rho \\beta_T} ", "5919f20f390b536946b887b7bccc201d": "f(x; x_0,\\lambda,k) = \\begin{cases}\n\\frac{k}{\\lambda}\\left(\\frac{x-x_0}{\\lambda}\\right)^{k-1}e^{-((x-x_0)/\\lambda)^{k}} & x\\geq x_0 ,\\\\\n0 & x< x_0 ,\\end{cases}", "591a0104f94d032fe69dfc216b51b9d3": "x \\Rightarrow^{ac}_{p} y", "591a47620d8d4f41cef1e76c6e42eed9": " A(t,y,0,f) = f(t,y) ", "591a883fc7760a9c516a1004a523f2b9": "\\frac{\\log|z_k|}{d^k} = \\frac{\\log(N)}{d^{\\nu(z)}},", "591a8c70b8133cdcde213a6d1898b8bd": "ky = \\int k \\frac{dy}{dx} dx.", "591a8c81e92ee77a99cad05016928883": "\n{\\mbox{LIVE}}_{in}[s] = {\\mbox{GEN}}[s] \\cup ({\\mbox{LIVE}}_{out}[s] - {\\mbox{KILL}}[s])\n", "591aa9aff862807d6af16b8b809fb016": "\n\\begin{array}{lll}\n& LAH_7=\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & . & . & . & . & . & . \\\\\n2 & . & . & . & . & . & . \\\\\n. & 6 & . & . & . & . & . \\\\\n. & . &12 & . & . & . & . \\\\\n. & . & . & 20 & . & . & . \\\\\n. & . & . & . & 30 & . & . \\\\\n. & . & . & . & . & 42 & .\n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n 1 & . & . & . & . & . & . & . \\\\\n 2 & 1 & . & . & . & . & . & . \\\\\n 6 & 6 & 1 & . & . & . & . & . \\\\\n 24 & 36 & 12 & 1 & . & . & . & . \\\\\n 120 & 240 & 120 & 20 & 1 & . & . & . \\\\\n 720 & 1800 & 1200 & 300 & 30 & 1 & . & . \\\\\n 5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & . \\\\\n 40320 & 141120 & 141120 & 58800 & 11760 & 1176 & 56 & 1\n\\end{smallmatrix}\n\\right ]\n;\\quad\n\\end{array}\n", "591acaa7684335111baf85bd1cd5cab6": "\\ v = \\frac{R T}{P}", "591b2b3002e335b873be758d431300cf": "MAP \\simeq \\frac{(2 \\times DP) + SP}{3}", "591b43666513d25e80f443434c5d357f": "I\\subseteq\\mathfrak{g}", "591b570d203010329187207fcce32a2f": "H_e = \\alpha\\ M", "591b59f3ff9a0cff027a78d604440e10": "\\lim_{\\sigma\\to 0^+} F(\\sigma+i\\omega) = \\hat{f}(\\omega)", "591b612eb025eb61bba00fe002932bd5": "\\mathbf{X}\\left(\\mathbf{a},t\\right)", "591b6521856faa7fae0b139aedf2b42b": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 13.99935 \\log_e(T+273.15) - \\frac {7284.572} {T+273.15} + 105.9605 + 1.410325 \\times 10^{-5} (T+273.15)^2", "591b7bd7ad417846a1b7ae8b8724480e": "v \\Delta t/ \\Delta x", "591bcb4d07843e835e7a0302df179277": " \\scriptstyle \\operatorname{End} \\left( \\mathbb K[x] \\right) ", "591be935a68c6467d5289463da9dd538": "f\\leq g", "591c5837bb9d9be9472f35703364de23": "\\mathbf{V}_\\mathrm{quad} = \\frac{Q \\lambda d^2 Cos[2 \\phi]}{4 \\pi \\epsilon_0 s^2} ", "591c8583385faf3d5b2caf0871299eba": " K(w, r),", "591cc2d612e27973c43eefb3dc1ba20a": "\\mu(\\omega)", "591d4fab0aed42b298b16ebc810cb826": "x \\in r", "591d61efcf1d88cebd7cd6334deeb8fe": "Tr(x)", "591d644b10bbc0c090e483e7deb92098": "\\epsilon_m", "591d69fe355a004903350a2d7d40d01b": "\\mathbf{M} = \\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^*", "591d7964462ce160fd89525998cefce1": "R_{\\mu\\nu}=0.\\,", "591d87442baed8e5a39ed94707835647": "i\\mathfrak{g}", "591d9adb96623b3eee226e8a9178fb0c": "\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}:\\mathbf a", "591dbdcf5431e7ff502961f56b897e54": "\nP(x,y) = P_x (x) P_y(y)\n\\,", "591e17a23400fe0f585003097643e746": "\\gamma_0(x) = 1", "591e354f13a62d4bbd76cf4308c82373": "{\\Delta V_w} = {V_{r1}\\cos \\beta_1+V_{r2}\\cos \\beta_2}", "591e76d18aed44b6babda6d7d4faed94": "\\int_{\\overline{\\mathcal{M}}_{g,n}} \\frac{c(E^*)}{(1-k_1\\psi_1) \\cdots (1-k_n \\psi_n)}= \\int_{\\overline{\\mathcal{M}}_{1,1}} \\frac{1-\\lambda_1}{1-k_1\\psi_1} = \\left[\\int_{\\overline{\\mathcal{M}}_{1,1}} \\psi_1 \\right] k_1 - \\left[ \\int_{\\overline{\\mathcal{M}}_{1,1}} \\lambda_1\\right].", "591ebdbcdc5a1bb1f4a6f4b8f1c575e6": "\\kappa = 3d^*(d^*-2)-6\\delta^*-8\\kappa^*.\\,", "591ee258e7a674a7d73819868c7a012a": "(2^n + 1)^2 - 2", "591f0245d88ba6b250b5ff6e3a833ee7": "k_y = k \\sin \\theta \\sin \\phi \\,\\!", "591f5d03e77b23aa1624bd6c7aeca75c": "y = \\overline{x1}\\cdot x2 + x1\\cdot \\overline{x2} + \\overline{x2}\\cdot x3", "591fc38315ab52fefb69e00995c23696": "L, \\,", "591ff81e919b2ac5604999042fd1a94f": "\\frac{\\partial(R^2\\Omega)}{\\partial R}>0,", "592096b7f55f05fd11248ce899ef2dfe": "R_{\\mu\\nu\\rho\\sigma}R^{\\mu\\nu\\rho\\sigma}", "5920c09309147daf59cdc3b053206041": " \\frac{B(rc)^2}{2\\mu_0}=n m v^2\\Rightarrow", "5920e47d59a7522b2f74225ec2944fc2": "\\mathbf{r} = \\mathbf{p} + \\mathbf{q}", "5920e9c71904cc2a5bc4aef310e691f4": "\\begin{align}\\mbox{d}M &= \\mbox{d}L\\sin\\varphi+\\mbox{d}D\\cos\\varphi\\\\\n &= \\mbox{d}L(\\sin\\varphi+\\frac{\\mbox{d}D}{\\mbox{d}L}\\cos\\varphi)\\\\\n &= \\frac{1}{2}\\rho V_1^2 C_L \\frac{\\sin(\\varphi+\\beta)}{\\cos\\varphi}b\\mbox{d}r\\end{align}", "592125840f85c61d0639de7051a6c79d": "{s_3}", "59216c3d8ee98ef8c42663e21ec84d40": "X;T \\,", "59219bb2ba2331c0258c8dde8228cde1": "i(t) = I_0 \\cos(\\omega_0 t + \\phi ).\\, ", "5921f29630296ec2f9525fee17fb6222": "x_j \\in Z _+ ", "5921f57d3c736eb3c21a069b48a93956": "{13 \\choose 4}{4 \\choose 3}{4 \\choose 2}^3{4 \\choose 1} = 2,471,040", "59221f8eab53d7f22760b353b355887e": "P(B|A)", "5922442c2df80d17622fd227f03b0a9e": "(1-b(w))w", "592290e209c1bb473c211257c00f5022": "r=-b/2.", "59232007375ff7f404ee4b528d6846a1": "\\mathbf{z}_n", "592361ca577817fa1181f717bafaee30": "V_\\mathrm{cathode}", "59237a83093df17b80fed6fbe4c410bd": "f_X(0)=f_Y(0)=f_Z(0)=1/2,", "5923f7c132cb37f2299101fb448bf921": "\n \\alpha~k^4 + \\beta~k^2 + \\gamma = 0 ~;~~ \\alpha := EI ~,~~ \\beta := m\\omega^2\\left(\\cfrac{J}{m} + \\cfrac{E I}{k A G}\\right) ~,~~ \\gamma := m\\omega^2\\left(\\cfrac{\\omega^2 J}{k A G}-1\\right)\n ", "59245875416c5f40014adad228d14bfe": "\\rho_S(x) = \\sqrt{x(2-x)}.\\,", "59245aa89a0171603b13f2b17742fecf": " \\bold{X}(\\bold{u}) + t \\bold{A}(\\bold{u}) = \\bold{X}(\\bold{u} + d\\bold{u}) + t \\bold{A}(\\bold{u} + d\\bold{u}). ", "592477c055ef56edc576397445d661b6": "\\int_{-\\infty}^{-1} \\frac{\\sin(\\pi x)}{\\pi x}\\,dx.", "5924a58eb8c25a43f79ec3cee8f8fcd7": " \\frac{1}{|\\mathbf{x} - \\mathbf{x'}|} = \n\\int_0^\\infty J_0 \\biggl( k\\sqrt{R^2+{R^\\prime}^2-2RR^\\prime\\cos(\\varphi-\\varphi^\\prime)}\\biggr)\ne^{-k(z_>-z_<)}\\,dk,", "5924b2d2b986d86b83937d609ddc128c": "a= \\frac{Z_A}{2000}", "5924fa94a46b718f1adbce4762704253": "E=\\frac{1}{2}mv^{2}", "59255bdde33f7f591e910665b1042944": "Q .", "592574e09e7b8dbdcd67afad7b2d7ab0": "Y\\in V^0", "592598607602783c166be00d59c5e4b0": "(t,x_1,x_2,\\ldots,x_n)", "5925bfe224f587fc6e966553dc648e7f": "\n{S}=\\left[\\begin{matrix}-\\lambda&\\lambda&0&0&0\\\\0&-\\lambda&\\lambda&0&0\\\\0&0&-\\lambda&\\lambda&0\\\\0&0&0&-\\lambda&\\lambda\\\\0&0&0&0&-\\lambda\\\\\\end{matrix}\\right].\n", "5925e944a732039150bcf90c16cd2499": "(x+y)^n=\\sum_{k=0}^n\\binom nk x^{n-k}y^k", "59265bc542d209467f9cc35778ba50c1": "(x)_n=x(x-1)(x-2)\\cdot\\cdots\\cdot(x-n+1).", "5926808c32c852ba7a5ca2897e4ad40e": " \\pi = \\sum_{k = 0}^{\\infty}\\left[ \\frac{1}{16^k} \\left( \\frac{4}{8k + 1} - \\frac{2}{8k + 4} - \\frac{1}{8k + 5} - \\frac{1}{8k + 6} \\right) \\right]", "59269bb15c7261a4b26a12bd5b908136": " dS_t = (r_t-d_t) S_t\\,dt + \\sigma_t S_t\\,dW_t ", "5926e1e7de225d3e730602a103eda311": "\\mathbf{E} = - \\nabla \\phi - \\frac { \\partial \\mathbf{A} } { \\partial t } ", "5926f57464d4e0960db91b41a01bd9dd": "\n\\begin{align}\n\\left\\langle\\psi\\mid H\\mid \\psi\\right\\rangle & = \\sum_{\\lambda_1,\\lambda_2 \\in \\mathrm{Spec}(H)} \\left\\langle\\psi|\\psi_{\\lambda_1}\\right\\rangle \\left\\langle\\psi_{\\lambda_1}|H|\\psi_{\\lambda_2}\\right\\rangle \\left\\langle\\psi_{\\lambda_2}|\\psi\\right\\rangle \\\\\n& =\\sum_{\\lambda\\in \\mathrm{Spec}(H)}\\lambda \\left|\\left\\langle\\psi_\\lambda\\mid \\psi\\right\\rangle\\right|^2\\ge\\sum_{\\lambda \\in \\mathrm{Spec}(H)}E_0 \\left|\\left\\langle\\psi_\\lambda\\mid \\psi\\right\\rangle\\right|^2=E_0\n\\end{align}\n", "59277def546eed7bfc1ff630f97c0f62": "g(y) = \\sqrt{1/2\\pi\\,} e^{-y^2/2}.", "592840df54675f49db842c2815676785": "a_0,a_1,\\dots,a_{d-1},a_0,\\dots", "5929026241f4abf60b1b332d51952c28": "\\parallel, \\nparallel, \\shortparallel, \\nshortparallel \\!", "592932e16776a879a9592a0b051b3a7c": " + \\frac{\\partial\\mathbf{D}}{\\partial t}", "592959463379cf7359967845a3362eac": " 3\\frac{2}{3},\\ 3\\frac{3}{5},\\ 3\\frac{20}{33},\\ 3\\frac{66}{109},\\ 3\\frac{109}{180},\\ 3\\frac{720}{1189},\\ \\cdots", "5929602cf3229396bba6dc2dcc79e422": "I_{x} = \\frac{1}{12} \\sum_{i = 1}^{n} ( x_{i+1} - x_i ) ( y_{i+1} + y_i ) ( y_{i+1}^2 + y_i^2 )\\,", "5929aa83373e3e4c062d7b2b3c11933b": " r_O \\gg R_C", "5929aec118c7ac5c5c959b667c134289": "\\mathbb{Z}^k", "5929ba517864faad2f240b0bcd06a558": "\n \\mathbf{c} = \\cfrac{\\partial\\mathbf{x}}{\\partial q^i}~c^i = \\mathbf{b}_i(\\mathbf{x})~c^i ~;~~ \\mathbf{b}_i(\\mathbf{x}) := \\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\n ", "5929c5c59664903303d3c3f2ed350566": "\\textstyle C = \\{ x : (\\forall i) \\; \\langle w_i, x \\rangle \\geq 0\\}", "5929ccb9b98887bdc7bf9f340ade65ec": "D( p + x \\| p) \\geq 2 x^2", "592a555c5a3b02b8a55b5f42f2a54178": "P_n(r) \\,", "592aa55fd8061bf6081d6c7a528dd13a": "a=2(u^2-v^2),", "592aeffb47e182b871ef54803784708d": "\\theta ' = \\theta - \\Omega t \\ . ", "592b43932b940bdb74a31a0b99accdcc": "X\\sim D(\\alpha_1,\\ldots,\\alpha_k,\\alpha_{k+1})", "592ba9159dea89cbd39fdc4f0da776f3": "\\boldsymbol{.}", "592bbac2e62f2667d7f8e14bb06c2606": "N\\triangleleft\\text{boy}", "592bf4bd2f658351e4ed90e92240752d": "{\\mathbb R}[z]", "592c035ce4f3088710d95e0730d8f0df": "L \\equiv n_{\\ell} - n_{\\bar{\\ell}}", "592c9977d65233d0c2f908b8e429c1b8": "\\scriptstyle k \\;>\\; k_0", "592cc65142ca2005c35329d59c2bbc5e": " \\hat{s} ", "592ce043d8b021faf4c3ece5bbde6097": " \n\\textbf{E} ( \\textbf{x} , t) = \\textbf{ A } ( \\textbf{x} , t) \\exp ( i \\textbf{K}_0 \\textbf{x} - i \\omega_0 t )\n", "592cf6ad0582bfc7e09a0dd7347216ba": "S_E(B) = 1 - S_E(A)\\,", "592d2c9505796e97c0168896f93aca6e": "7225 -x^2", "592d365405baeeafa7ad85de295cdfc5": "i\\hbar\\frac{\\partial}{\\partial t} |\\alpha\\rangle = \\hat H | \\alpha \\rangle", "592d59f2c7230c9cc90a6f203859792c": "G(\\underline{x}) > G_0 \\Rightarrow K, < G_0 \\Rightarrow H", "592dbd008d7d41263cd71553daa8db3f": "F_i n_i\\,dS ", "592dbddc87846f7210df0a40826bde24": " I = \\int f(x(0)) e^{-u\\int_0^t V(x(t))\\, dt} g(x(t))\\, Dx ", "592dee0953dbf40e3f21f753eb04f57b": "\\frac{\\partial\\ln(c_{t+1}/c_t)}{\\partial r}", "592df568df96116ded5fd1fa7e76c86c": "E_t=\\frac{\\pi}{d}\\sqrt{\\frac{K_2}{\\epsilon_0\\Delta\\chi_e}}", "592df5e1bd089c533a5ae58634c4382b": "\\log_B N", "592e079f2565911e0cd44c72b43d2839": "x \\, \\partial_v + u \\, \\partial_x, \\; \\; y \\, \\partial_v + u \\, \\partial_y", "592e3a9b4060f2056ceeffb07307a4e9": "2^{4\\,.}\\mathrm{GL}_{4}(\\mathbb{F}_{2})", "592e4c8f0b8f5caa831dc0354e28374b": "\\{w\\}", "592e904d6a31f9471ac76f8b37b81a07": "Nt_\\text{in}(k) + t_\\text{out}(N) = N^{O(1)} ", "592ea6fdbc65239aeedd3e76259d3a78": "\\sigma_T = T^{1/\\alpha} \\sigma.\\,", "592ea71c28370641554d719362665842": "u = u(x,y)\\,", "592ec220097d3acb156118ba64b47870": "L^\\prime = -\\rho_\\infty V_\\infty\\Gamma,\\,", "592eccf03f7dfd2d721660112c97b846": "1_A", "592f6d19e30812fffe93c774fbc77b13": "D N(d_+) F", "592f702d37c229e35f169c4cb041a981": "0 \\longrightarrow \\mathcal{A} \\stackrel{\\alpha}{\\longrightarrow} \\mathcal{B} \\stackrel{\\beta}{\\longrightarrow} \\mathcal{C}\\longrightarrow 0", "592f9d24978a43652dd9c8fa1d7b09fa": "1= \\cos(3\\pi/8) A_{T,L}^{x_c} + B_{T,L}^{x_c}", "592fc4b1656ce1a258c902144bac5439": "\\begin{align}\n& U_{\\text{rot}}=RT^{2}\\left( \\frac{\\partial \\ln Z_{\\text{rot}}}{\\partial T} \\right)\\text{ ; }C_{v,\\text{ rot}}=\\left( \\frac{\\partial U_{\\text{rot}}}{\\partial T} \\right) \\\\ \n\\end{align}", "59302678102baaa28c6b76ca28a03097": " | \\vec p \\,| = \\frac{1}{c}, \\quad \\text{and} \\quad \\vec{x_0} \\quad \\text{is arbitrary}.\\,", "5930389b88f7c5420b03a96c36182ff4": "\\cos^2\\gamma + \\sin^2\\gamma = 1.\\,", "59304f0ea5bcef11021376effd70865a": "\n\\operatorname{bin}(\\operatorname{DE9IM}(a,b)) = \\operatorname{9IM}(a,b) = \\begin{bmatrix}\na^o \\cap b^o \\ne \\emptyset & a^o \\cap \\partial{b} \\ne \\emptyset & a^o \\cap b^e \\ne \\emptyset \\\\\n\\partial{a} \\cap b^o\\ne\\emptyset & \\partial{a} \\cap \\partial{b}\\ne\\emptyset & \\partial{a} \\cap b^e\\ne\\emptyset \\\\\na^e \\cap b^o\\ne\\emptyset & a^e \\cap \\partial{b}\\ne\\emptyset & a^e \\cap b^e\\ne\\emptyset\n\\end{bmatrix}\n", "5930af1e1415f979bcde1016592cb82b": " \\mathbf{x} = x_i~\\mathbf{e}_i ", "59311cee10b178e1b0939fa0c1a2394b": "\\|\\boldsymbol{z}\\| := \\sqrt{|z_1|^2 + \\cdots + |z_n|^2}= \\sqrt{z_1 \\bar z_1 + \\cdots + z_n \\bar z_n}.", "59314e7665c01da2c2131eb80ae9dfb4": "\\mathbf{e} = \\hat{\\mathbf{x}} - \\mathbf{x}", "5931b02ead5f5b6a5f67f42d000fba2f": "\n| \\nabla \\phi | \\ll V_\\infty\n", "5931d8a368ca2efa9c0599bb1c7cd1d6": " 1, 2, 3, 4, 6, 8, 12, 24. \\, ", "5931e8c5b968261d222ae75dc678cbe2": " \\int_{c(r)} R(z) dz =0.", "5931ec423e049ec900713028c6b48f35": " \\partial = 2P_3 \\partial_u + 2\\bar{P}_3 \\partial_v - \n2\\mathbf{e}_1 P_3 \\partial_{w^{\\dagger}} - 2 P_3 \\mathbf{e}_1 \\partial_w ", "5931fd33913a19138f5ffd6ca82e3bea": "\\boldsymbol{\\nabla}\\cdot(\\mathsf{C}:\\boldsymbol{\\nabla}\\mathbf{u}) = \\rho~\\ddot{\\mathbf{u}} ", "59322f026c5c745af1fb0644ab6d36b5": " \\det\\left(\\prod_{i=1}^n \\mathbf{A}_i \\right) = \\prod_{i=1}^n \\det\\left(\\mathbf{A}_i\\right)", "5932444b5b93d64ed9433a50eb4ac4e0": " Pr(\\text{data} | \\text{something else},I) ", "59325441b4b0d911f6df39d9a518a6b0": "u_t = \\frac{k}{c_p\\rho}u_{xx},", "5932771a116fe33a736a46654a5bac74": "P_\\mathrm{tot}/4\\pi", "593291a30697b13665dc112f29824e5d": "(r,s)", "5932d12f2e26af10943674b5b87ec0b2": "\\textstyle w_{i}^{t} ", "5932ee746a3d000778bcf1de85380b56": "z^3 = 0.799901291393262 - (0.107547238170383)i{\\;}{\\;}{\\mathrm {(yellow)}}.", "593321703aac6d177519395688f20c60": "((\\infty),[\\infty])\\in I", "59332e9ded5dcb3cf1872e89cd923e1f": "X \\to (p + 1) \\to (q + 1) = X \\to (X \\to p \\to (q+1)) \\to q", "593343a9f5902f214f439c7413317c4a": "\\chi=\\chi_1", "5933820c580ecc141eb5d2248749d7ef": "\\|v\\| = \\sqrt{|\\eta(v,v)|}.", "5933ffd5e5803d4a1b25df7a41739b4f": "\\int\\operatorname{arsech}(a\\,x)\\,dx=\n x\\,\\operatorname{arsech}(a\\,x)-\n \\frac{2}{a}\\,\\operatorname{arctan}\\sqrt{\\frac{1-a\\,x}{1+a\\,x}}+C", "59346553d9a037057e517d36282d7c23": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=24,\\cdots ,29", "593486d854bad277f469b23f071156ef": "F \\, = m \\ddot {r} ", "593498bd09af5260b091926381ea7df0": "g(q)", "5934dee0aea0ff538931b9ee4749bc3c": " \\|u\\| \\le \\frac1c \\|f\\|, \\, ", "59350963674661be53b6ea4363d1b26f": "e^{\\int \\frac{k}{m} \\, dt}", "593510a0cf6982eb52edf548db1caaca": "\\sum_{n=0}^\\infty a_n,\\qquad \\sum_{n=0}^\\infty b_n,", "593539efb96c5fd2b7a85824d09bf672": "Q(\\mathbf{Z})", "593602cd3f8bf804fa02da4400fae5df": "\\begin{matrix}\\mathbf{i} = -\\sigma_2 \\sigma_3 = -i \\sigma_1 \\\\\n\\mathbf{j} = -\\sigma_3 \\sigma_1 = -i \\sigma_2 \\\\\n\\mathbf{k} = -\\sigma_1 \\sigma_2 = -i \\sigma_3. \\end{matrix}", "59361311c2d6b2e30823632c44f1114b": "\\frac{1}{\\mu_1}+\\frac{1}{\\mu_2}", "593639f17e76b8fb332e2fe90913dbe6": "x^2 + y^2 + 10 = 0 ", "59363d486782d68ae92f21e10eb5f2a8": "{x\\!:\\!\\sigma \\in \\Gamma}\\over{\\Gamma \\vdash x \\Rightarrow \\sigma }", "593665655606a5dddff95db2ea371bf8": "\\epsilon=\\frac{\\int_0^\\infty \\epsilon_\\lambda (\\lambda,T) E_{b\\lambda}(\\lambda,T)\\,d\\lambda}{\\int_0^\\infty E_{b \\lambda}(\\lambda,T)\\,d\\lambda}", "593669a4f873f43c0afad258985342a7": "\\zeta(2) = \\prod_{p} \\frac{1}{1-p^{-2}}= \\frac{\\pi^2}{6}.", "5936f58e3a6317a9ffa65e0b68e12fc1": "x. \\ ", "593720b6d235eb506223f2a217b1dfb9": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 1}{4 \\choose 3} \\end{matrix}", "59372ff974cf3a628d6e4d2f2cd01484": "\\mathrm{diag}", "59373a0f5fc08d9691efa6c0a33af1d1": "{\\partial \\rho \\over \\partial t} + {\\partial (\\rho u ) \\over \\partial x} + {\\partial (\\rho v) \\over \\partial y} + {\\partial (\\rho w) \\over \\partial z} = 0.", "59376b813089fa0c5e9a68ad4370c1a0": "(1+z_x^2)z_{yy} - 2z_xz_yz_{xy} + (1+z_y^2)z_{xx}=0", "59376d3b1ae9ec4368a1f314942c404b": " \\nu(H) \\ge \\nu(H_{11}) + \\nu(H/H_{11}) ", "59377ffcc9aab6e6bb0068a19ccd75ef": "u_p(s)", "5937a43313615d2050a452ce79b47153": " J_j = - D_{ij} \\frac{\\partial C}{\\partial x_i} ", "5938e902024190c5e65dcc9eb2b69abe": " [{L_x}, {L_y}] = i \\hbar \\epsilon_{xyz} {L_z}, ", "593948313cd38dcb657fa3e3ecffe69a": "\\scriptstyle \\log_e (\\frac{760}{101.325}) - 9.113968 \\log_e(T+273.15) - \\frac {6263.383}{T+273.15} + 74.99482 + 7.411446 \\times 10^{-06} (T+273.15)^2", "59398d8f49675ed835571e68b126dd59": "t(u, y)=0", "593996b39018805102c400c1ce8d13bc": "{\\rm JSD}(P_1, P_2, \\ldots, P_n) = H\\left(\\sum_{i=1}^n \\pi_i P_i\\right) - \\sum_{i=1}^n \\pi_i H(P_i)", "5939c0bb40e211f444c176f92bb1b2b7": " = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, e^{-j \\omega t} \\, d\\tau \\, dt. ", "5939c7d2b490b07d5791c85b4c8145bd": "u = \\frac{s}{1+\\sqrt{1-s \\cdot s}} = \n\\frac{\\left(1-\\sqrt{1-s \\cdot s}\\right)s}{s \\cdot s}.", "5939e7fae83e0506c1958f70eb15836f": " \\inf_Y \\operatorname{dim}_{\\mathrm{Haus}}(Y) =\\operatorname{dim}_{\\mathrm{ind}}(X), ", "593a93c6fea0dd5dc76990863c442b28": "P(X \\le 82) = P\\left(Z \\le \\frac{82 - 80}{5/\\sqrt{3}}\\right) = P(Z \\le 0.69) = 0.7549", "593ab1e7b1d21647d3c3854dd7412a63": "A \\subseteq Q", "593add1fb869529f238df842f84a8a9d": "(\\Omega^{-1} - 1)\\rho a^2 = \\frac{-3kc^2}{8\\pi G}", "593b22908f301da01485aec8e4ace156": "\\mathbf{g} = - \\nabla \\Phi. \\,\\!", "593b551a146cd213298b0431ecd33ac7": "V = \\frac{n \\sqrt{4\\cos^2\\frac{\\pi}{2n}-1}\\sin \\frac{3\\pi}{2n} }{12\\sin^2\\frac{\\pi}{n}} \\; a^3", "593b79e486539cb0170ddcb18fb49264": "\\alpha=\\frac{1}{3}C", "593b83a2411cc92df408762a69fe8dbb": "\\int \\phi_2 ( \\nabla^2 \\phi_1 ) dV = \\int \\phi_1 ( \\nabla^2 \\phi_2 ) dV", "593b99c30765be9b0c468bedaedeb3ee": " I(z) \\propto |\\mathbf{E}_0 e^{i(\\tilde{k} z - \\omega t)}|^2", "593be0dfc4bd9f72f06f1613a3b6d2f6": "\\Delta E_{ab}^{\\,\\bullet}\\le 2.0", "593c0063023dffce4e686b0c0040e625": " _H = _H ", "593c450bb5da636773c8a7706bbb294f": "2\\Phi_{11}=D\\gamma-\\Delta\\varepsilon+\\delta\\alpha-\\bar{\\delta}\\beta-(\\tau+\\bar{\\pi})\\alpha-\\alpha\\bar{\\alpha}-(\\bar{\\tau}+\\pi)\\beta-\\beta\\bar{\\beta}+2\\alpha\\beta+(\\varepsilon+\\bar{\\varepsilon})\\gamma-(\\rho-\\bar{\\rho})\\gamma+(\\gamma+\\bar{\\gamma})\\varepsilon-(\\mu-\\bar{\\mu})\\varepsilon-\\tau\\pi+\\nu\\kappa-(\\mu\\rho-\\lambda\\sigma)\\,,", "593c580240aa9a222d66bf14670cf7fd": "t=X/X_{0}", "593c837cc227b2125a68c76491182d3d": "\\bar{P_e} = \\sum_{j=1}^k p_{j} ^2", "593cd51507b5cd38fee087fd27617db3": "T_{\\mu\\nu}= \n\\begin{pmatrix}\n\\rho_0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\n\\end{pmatrix}", "593d03c5d59c5c67ce5faa49d711358c": " H_{D}(x,R)", "593d12b199a03826d6b7c19ab7927e44": " |x| \\sgn(x) = x,", "593d13140788dc4e5ab3f458228628cd": "f \\circ h = g", "593d7100651689185dbd958f7704a83f": "( w^- , w^+ )", "593d9b96ceba775ac243cb137dd63e4b": "Z_0", "593dbc9da24347c0112f538ac294365e": "z \\mapsto z + \\beta", "593dee0746db4f9a60c2d05315778556": "\\nabla \\cdot \\mathbf{E} = \\frac {\\rho} {\\varepsilon_0}", "593e2f67c7de0a8a2142b95a845debf8": "|\\psi'\\rangle=\\alpha_0|0_S\\rangle+\\alpha_1|1_S\\rangle", "593e6fc862eb803ec4b082ae74e03609": " \\eta = \\frac{P_{IN} - P_{LOSS}}{P_{IN}}", "593e73777737d8e673a23b7a6643c1c9": "DO_{sat}", "593e7ed2683527999ed933e7b9336f54": "\\mathfrak{E} + i\\ \\mathfrak{M}", "593e9739b280f550789d65db04caa844": "\n(3.2)\\quad \n(\\mathcal{A}f)(x)= h(x) - Eh(Y), \\qquad\\text{for all }x,\n", "593e9c9bdaac46a299861994ea1d3096": "\n\\ H(z) = \\frac{Y(z)}{X(z)} = 1 + \\alpha z^{-K} = \\frac{z^K + \\alpha}{z^K} \\,\n", "593eaf43385c1a9dd3bcae32da916f18": "\\Phi_L = LI_L. \\ ", "593ec7c06a709f478a26acc47b2ed56b": "\nB(t,T) = \\frac{2(\\exp((T-t)h)-1)}{2h+(a+h)(\\exp((T-t)h)-1)}", "593ecb49b979154603541f37a0c2c426": "\\scriptstyle \\psi_i", "593ed2e30f8c081134083fb5fdf292ea": "\\langle \\phi_i | \\phi_j \\rangle = \\delta_{ij}.", "593eee0126e62aeec93975c593964100": "A_s = 4 \\pi r_s^2 = 4 \\pi \\left( \\frac{2 G M}{c^2} \\right)^2 = \\frac{16 \\pi G^2 M^2}{c^4} \\;", "593f05c413737ef41999f1ccef9fa32c": "y_{n+1} \\ge 35", "593f5e676a2e18f65b12677f94182b56": "\\frac{f(c+h)-f(c)}{h}\\ge0,", "593fca357badd67f1ebdb94edbb89a50": "\\sum_{i=0}^n D_{i}=J ", "5940051f1e7ad31be066e374abdb5e42": "[L_{ij},L_{kl}]=i [\\delta_{ik}L_{jl}-\\delta_{il}L_{jk}-\\delta_{jk}L_{il}+\\delta_{jl}L_{ik}] \\,\\!", "594068f0e909c592f0e800793befdbf9": " q \\, ", "59408875cf5d4a30c0252564e8f30fd2": "a+_Lb=a+_{K[X]}b.", "5940cb6fc3785b16c0b3bf3bfed3aa22": "d\\mathbf{x}^2 - d\\mathbf{X}^2=d\\mathbf x\\cdot d\\mathbf x-d\\mathbf X\\cdot d\\mathbf X\n\\qquad \\text{or} \\qquad\n(dx)^2 - (dX)^2=dx_jdx_j-dX_M\\,dX_M\\,\\!", "5940e95c1dd5642cf1f121917e813c38": "\\displaystyle{-Q(Q(a)b)^{-1}Q(a)c =-Q(a)^{-1}Q(b)^{-1}c.}", "5940ef0cea0e34f282715f2ddd43a5d4": "\\rho (\\mu, \\nu) := \\sup \\left\\{ \\left. \\int_{M} f(x) \\, \\mathrm{d} (\\mu - \\nu) (x) \\right| \\mbox{continuous } f : M \\to [-1, 1] \\right\\}.", "59410a59e72bc7bedcedf5c37baae159": "aS(a^{-1})", "594153d6bb22f0bd2c264b1cc05ff9e1": "{\\mathcal O}'(M)", "59415830021361e9f7021ffa1e65f72e": "\\tbinom xn", "5941888de21ac95c48c8310c3b15d568": "t \\in \\R", "59419312d51f7c8f0847685bbfe9c66e": "A\\left ( \\mathbf{r} \\right ) \\propto \\iint_\\mathrm{aperture} E_\\mathrm{inc} \\left ( \\mathbf{r}' \\right )~ \\frac{e^{ik \\left | \\mathbf{r} - \\mathbf{r}' \\right |}}{4 \\pi \\left | \\mathbf{r} - \\mathbf{r}' \\right |} \\mathrm{d}x'\\mathrm{d}y'", "594195f0fb38a1c81d47c4ac689ba396": "n(d)=\\left\\lceil \\sqrt{2d\\ln2}+\\frac{3-2\\ln2}{6}+\\frac{9-4(\\ln2)^2}{72\\sqrt{2d\\ln2}}\n-\\frac{2(\\ln2)^2}{135d}\\right\\rceil", "5941e577e4bea1837d86734be74302cc": "1 + \\frac{\\Beta(p;k+1,0)}{\\ln(1-p)}\\!", "5941f122a0f728993a02d4e095d26b4e": "u(S_t,t)=", "594296114679b6a51cf08fe7415c2cac": "-\\frac{\\Delta E_i}{T} = \\ln\\left(\\frac{1 - p_\\text{i=on}}{p_\\text{i=on}}\\right)", "5942c6c3766df9f70d25d01eff12f96e": "\\theta_{12}\\,\\!", "5942da056275d0aed4e983afca880184": "\\mathfrak{sl}_n(\\mathbf K)\\oplus\\mathfrak{sl}_n(\\mathbf K)\\text{ or }\\mathfrak{su}_n\\oplus\\mathfrak{su}_n", "59431f82dfdd6735ef438d643d750004": " |S(\\alpha)\\ll \\left(\\frac{N}{\\sqrt{q}} + N^{4/5}+\\sqrt{Nq}\\right)\\log^4 N", "594348e822bc2380a5a7756de42e82a2": "\\pi_{t} = \\beta E_{t}[\\pi_{t+1}] + \\kappa y_{t}", "594359afeb876cf753f7d808377389b6": "RR \\geq IG", "5943cddeaeb2b751011307f5788177b1": "ST_x(\\Diamond_m \\varphi) \\equiv \\exists y ( R_m(x, y) \\wedge ST_y(\\varphi))", "594441000d758101046a46231dea6aac": "\\int \\frac{x^{n}}{S} dx = \\frac{2}{a (2 n + 1)} \\left( x^{n} S - b n \\int \\frac{x^{n - 1}}{S} dx\\right)", "5944460f706b45015fbab8eff132d613": " H=\\bigoplus_{k\\ge 0} HS(L^2(C)) \\otimes L^2(R, c_k(\\nu^2 + k^2)^{1/2} d\\nu),", "59446bb74d85fc163cb62f8aea576a27": "\\sum \\limits_i w_i E(m, \\Theta_i)", "59449d7cbefcc9eae10ccd4b48c3d4ed": "\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}~\\text{dV}\\right) = \n \\int_{\\Omega(t)} \n \\left(\\frac{\\partial \\mathbf{f}}{\\partial t} + \\boldsymbol{\\nabla} \\cdot (\\mathbf{f}\\otimes\\mathbf{v})\\right)~\\text{dV} ~.\n", "5944f94c330382d9a86e2cd4f393c07f": "L_0", "5945515991a3fb5910f5e1bc45be02ed": "S_{a/\\$} = S_{a/b} S_{b/\\$}", "5945c9af286142a51004a8400f038bd9": "C^{a}_{\\ bc}=\\varepsilon^{a}_{\\ bc}", "5945e2ed57dfde609cd3efd04e0be335": "Te_2", "5946b5a53ef63442449e5abef7ef452b": " MPGe = \\frac { E_G} {E_M*E_E} = \\frac{ 33,705 } {E_M} ", "59470cd164b94b37a2ea2d13dfcf1468": "\\Sigma = \\{ \\sigma_{1}, \\sigma_{2}, \\ldots, \\sigma_{|\\Sigma|} \\}", "59476c9721bca71a050750693f4f12b4": "\\phi:X\\to Ord", "59476f692ec62af726dc0c68e24a96f0": "d\\nu/d\\mu", "59479e71d0ecf0c7906d87ba6a358878": "x_N = y_j \\Rightarrow \\alpha^{b+d} = \\beta\\alpha^{d_j} \\Rightarrow \\beta = \\alpha^{b+d-d_j} \\pmod{n} \\Rightarrow \nx \\equiv b+d-d_j \\pmod{n}", "5948135d1207320eb220283e5411c7b8": "\\Gamma(x, y)", "594822c04d096b5226b37912e987ad2d": " y\\ f = x \\and x = f\\ x", "5948974ed95f96f036168b0f7265cc13": "X(E[A])=\\sum X^{(m)}A=\\sum X^i\\partial_iA=\\sum X^ip\\partial_i\\log pA=E[X^{(e)}A]=E[X^{(e)}A]-0=E[X^{(e)}A]-E[X^{(e)}E[A]]=E[X^{(e)}(A-E[A])]=E[X^{(e)}Y^{(e)}]=\\langle X,Y\\rangle", "5948a08eb517d697c6955c4d463cf6f3": "p_1 = p", "5948cffe8d6a809752c09b6ec64e1103": "\\min\\mathcal{L}_j", "5948e5fd0775812f27fecf7a76cb7381": "\\bold{k} = 0", "59496fb440841cafe556287c5aab3337": "f,\\,", "594a056e12dc3932fbd2c84818e659aa": "\\vec{\\mathcal C}", "594a5472f103bcb4ebcfa873e1e78411": "w{}_{ijk}", "594a989b6e3cb819094c17a542565394": "N(d_+) F", "594abc76519d30918f839d630a804172": "\n\\begin{align}\n( 0.15625) m & = (0.00101_b) m = ( 2^{-3} + 2^{-5}) m = (2^{-3})m + (2^{-5})m \\\\[4pt]\n& = 2^{-3} (m + (2^{-2})m) = 2^{-3} (m + 2^{-2} (m)).\n\\end{align}\n", "594b79b497ba92d4c65c2a261b401b9c": "x(yz) = (xy)z", "594bf6fb90e2c386de6d4824f386d85b": "\\varinjlim \\left\\{2^{-i}\\mathbb{Z}\\mid i = 0, 1, 2, \\dots \\right\\}", "594c16ca0695f6665d7cf4971707adf6": "HA", "594c51087cd1ada5eedab1951908d3bc": "0=F_0(\\lambda)\\subseteq F_1(\\lambda)\\subseteq\\dots\\subseteq I(\\lambda)", "594c77c7aba6b8a98648b39762715b65": "\\epsilon_{\\sigma \\mu \\nu \\rho}=0", "594ca484f3788783b0da487714a86aad": " \\scriptstyle{\\mathrm{V} =_{\\mathrm{def}} \\{ x\\,|\\mathrm{set}(x)\\}} ", "594ce3eb3d1662b4042b6e6c8c76d350": "red \\circ d = \\mathrm{id}_X", "594ce753850ae4906e820babbe90fdf6": "\\bar v", "594ce823c9b7bbc2a689b89629692765": " Gen(w): s = SS(w), return: P = (s, H_1(w,s)), R = H_2(w,s) ", "594d0131542870cd653e036bbfe04e2d": "Pmf = \\tfrac{3}{5}", "594d2922b102fcf8138ec4c0926debde": "T = \\bigoplus\\nolimits_{i\\in I} (T_i, a_i, b_i)", "594d5bcadf95125b277b6e9f7c8c549a": "31^3+33^3+35^3+37^3+39^3+41^3 = 66^3", "594d88141684491db2627b6a3fc2f49d": "n! = \\sqrt{2 \\pi n} \\left( \\frac{n}{e} \\right)^n \\left( 1 + O \\left( \\frac{1}{n} \\right) \\right).", "594d888e5ba8004917c4098685b34ce2": " A = L_* + U ", "594dac62d7c277cbe5f41b6e96315793": " \\sigma_{\\mathrm{ess},4}(T) = \\bigcap_{K \\in K(X)} \\sigma(T+K), ", "594dca64f86b429a6ba4ab4c260b14ae": "\\Sigma^{EXP}_k", "594e875ed8ae6376846e6fb543053a20": " \\frac{\\partial \\vec{J}(\\vec{r},t)}{c\\partial t} + (\\mu_a+\\mu_s')\\vec{J}(\\vec{r},t) + \\frac{1}{3}\\nabla \\Phi(\\vec{r},t) = 0", "594f27f5927b84093d9080d8c1939efb": "(H_\\mathrm{sat} - H_0)", "594f3a6f85c281026ffddc36e3f0d2a2": " L = g_{\\mu \\nu} \\frac{d x^{\\mu}}{d s} \\frac{d x^{\\nu}}{d s} ", "594f9d18458f21bec006bd2ca1489749": "S \\cdot A \\cdot T", "594fa08144b18b483856d086a0ab3f18": "x=x_0+\\epsilon x_1 (+\\cdots)", "594ff09ed091452c93224a829c7dd263": "(u, y) \\in {{{\\mathbf{k}}}}^d \\times {{{\\mathbf{k}}}}^{n-d}", "59506c20da3f6d595d3ece0fb7929248": "s2^{n+2}+1", "595070358cadf03cfd9bc7a9a4f3d1f2": "\\bold j = \\frac{1}{2m}\\left[\\left(\\Psi^* \\bold{\\hat{p}} \\Psi - \\Psi \\bold{\\hat{p}} \\Psi^*\\right) - 2q\\bold{A} |\\Psi|^2 \\right] + \\frac{\\mu_S}{s}\\nabla\\times(\\Psi^* \\bold{S}\\Psi) \\,\\!", "595085a3e7fc5e4ef06b92703b0a1188": "\\kappa(t) = \\chi_1(t) = \\frac{\\langle \\mathbf{e}_1'(t), \\mathbf{e}_2(t) \\rangle}{\\| \\mathbf{\\gamma}^'(t) \\|}", "5950b0c4fe14fb02f239a996ff99cc04": "T(g)(x)=g(x)", "59512150ffbbf43cf38fa30504ba0d52": "10^{(10\\cdot2^{104})}", "595184b3bf6bab4050b046f8241c5262": "\\textstyle \\left\\{ x\\right\\} \\cup B", "5952244a6488314e253bd99d443ac9d0": "P(A\\mbox{ or }B) = P(A \\cup B)= P(A) + P(B).", "595264a4acd4ee624c1e6253c4701dfa": "[Ab] + [Ag] \\leftrightharpoons [AbAg]", "59530bf79d52e782e2c67b22790b14f6": "f'(x_i)", "59536d5c1b01d8e44d9e945bc9a1abf5": "\\displaystyle{H_\\varepsilon f \\rightarrow Hf}", "5953762fa91024dc842418efa5a40faa": "\\{x_k\\}", "59537dc41096b0a08721e18c3ffc2765": "d/\\lambda", "5953c9f686b5748908bb63675bdb79ac": "\\rho_{A}", "5953da754e1e65c7b27b707e20474ce5": "\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix} =\n\\begin{bmatrix} \\rho\\sin\\theta\\cos\\phi \\\\ \\rho\\sin\\theta\\sin\\phi \\\\ \\rho\\cos\\theta\\end{bmatrix}.", "5954311c48435eb220ab8b155a017bf7": "P = \\{x \\in \\mathbb{R}^n : Ax \\leq b\\}", "595455d6e64459c111f805d4b7ceb647": "g_i = 1", "59546a7ec91db4b3f9818fd641c54056": "a_n=1", "5954f937c17ceb585086434203d1bcc5": " y' = F(x,y)\\,,\\quad y_0 = y(x_0)", "59552731fce313028f63e0c2c5c1d094": "Y = g(X)\\,\\! ", "5955db299f4249bab0dd99a60617e019": "dV/d\\log\\xi=dU/d\\log\\xi=0", "59563a92333a8c721656e8d7e1cfffb4": "d(x_n, x)<\\varepsilon/2", "59565644774f21b15b87d0e55577a557": "\\left|\\sigma_{ij}- \\lambda\\delta_{ij} \\right|=\\begin{vmatrix}\n\\sigma_{11} - \\lambda & \\sigma_{12} & \\sigma_{13} \\\\\n\\sigma_{21} & \\sigma_{22} - \\lambda & \\sigma_{23} \\\\\n\\sigma_{31}& \\sigma_{32} & \\sigma_{33} - \\lambda \\\\\n\\end{vmatrix}=0\\,\\!", "59567b4fd2a097fa43ec80a2428824c7": "\\sum_{i=1}^np_ix_i\\leq W.", "5956b0d12c40acde5fc3c13225cbe6d7": "\\frac{d_Y(f(x_1),f(x_2))}{d_X(x_1,x_2)}\\le K.", "5956b0fc4edcc1a93a7270f093583b5f": " \\operatorname{E}[X]=\\sum\\limits_{i=1}^\\infty P(X\\geq i).", "5956b6153c2c8d18ab43ec6298447c5b": " \\sigma_{j,n-1} ", "5956c8f6c13474a739d3d5f84b921059": "Y_{21} = {-2 S_{21} \\over \\Delta_S} Y_0 \\,", "5957639c5464aaaf7587b2ef574c4020": "f_\\ast(\\mu)", "59577bbfe953f01945708d74191c039c": "\\Phi(\\omega)= \\prod_{k=1}^\\infty \\frac {1} {\\sqrt 2} H\\left( \\frac {\\omega} {2^k}\\right) \\Phi^{(\\infty)}(0)", "5957fb5f4f463f626853b395ab6030de": "\\begin{align}\nd(x_m, x_n) & \\leq d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + \\cdots + d(x_{n+1}, x_n) && \\text{Triangle Inequality} \\\\\n& \\leq q^{m-1}d(x_1, x_0) + q^{m-2}d(x_1, x_0) + \\cdots + q^nd(x_1, x_0) && \\text{Lemma 1}\\\\\n& = q^n d(x_1, x_0) \\sum_{k=0}^{m-n-1} q^k \\\\\n& \\leq q^n d(x_1, x_0) \\sum_{k=0}^\\infty q^k \\\\\n& = q^n d(x_1, x_0) \\left ( \\frac{1}{1-q} \\right ) && \\text{Geometric Series}\n\\end{align}", "59580851c544a3cc1368e7dd4074b727": "\\frac{S(z)}{X(z)} = \\frac{1}{1 - 2 \\cos(2 \\pi \\omega) z^{-1} + z^{-2}} = \\frac{1}{(1 - e^{+2 \\pi i \\omega} z^{-1})(1 - e^{-2 \\pi i \\omega} z^{-1})}.", "595826807c88a4ed73fe0824defee1f1": "\\left(\\tfrac{\\Delta}{q}\\right)=1", "59585dcd3d016d47693ea153234f13a6": "\\frac{1}{h_a ^2}+\\frac{1}{h_b ^2}=\\frac{1}{h_c ^2}.", "59587c676b23b0691cc670c6ec8af07d": "p_m=-\\frac{p_b \\cdot p_r}{p_b-p_r}", "5958cfa68568947ae89472d1195dde2a": "X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow {w} X[1]", "5958da509fe89d8422e5c2eba566d01e": "\\Gamma(G,S)", "59594ba7b70593e029c943317755078e": "L_{\\omega_1, \\omega}", "59596ae88921ffc0267da4e0c38a73e0": "\\Delta'=\\nabla^*\\nabla", "595977bb533275ce95cc5947a7047113": "{\\sigma}^2_{E_i}={\\sigma}^2_{E'_i}", "595995f3babf79827665c8d450fd544a": "B= \\{ ta : a\\in A, t\\in[0,1] \\}", "5959c841dca5fdd5116901f220a3a0b8": " \\tau(\\mathbf{y}'_{1}, \\mathbf{y}'_{2}, \\mathbf{C}_{1}, \\mathbf{C}_{2}) \\sim \\mathbf{T}^{-1} \\, \\tau(\\mathbf{y}'_{1}, \\mathbf{y}'_{2}, \\mathbf{C}_{1} \\, \\mathbf{T}^{-1}, \\mathbf{C}_{2} \\, \\mathbf{T}^{-1}), ", "595a5afce7c15add5572c13795c360f3": "\\overline{\\overline{A}}=A.", "595a7f705a30ce484c5c3492a370906d": " \\delta \\phi _l = \\phi_P -\\phi_L ;\\text{ and }\\delta x_l = x_{LP}; ", "595a9a02248c6e8ee485cdbcee3ece2b": "t' = \\gamma \\left ( t - \\frac{v x}{c^2} \\right )", "595ab10107f3763dfb7558c281dd4cd1": " I=(\\bigcup I)", "595ab9dd37a1c056f5d9a8c5a72d9d39": "(\\Sigma, R)", "595adc17aa48fcbfba375a7373c630bf": "\\begin{bmatrix}B & AB\\end{bmatrix}", "595ae3672a3b9237662e9be130374b41": "A=a=b=1", "595b2e19546923e0cf30ce66c3d61712": "I(x_i,y_i) = \\int\\!\\!\\int O(u,v) ~ \\mathrm{PSF}(u-x_i/M , v-y_i/M) \\, du\\, dv", "595b425616f9988728099393628eb486": "y_1=\\frac{2}{3}-\\frac{4}{3x}+\\frac{1}{x^2},", "595be37635bde62648c435d250012767": "(\\mu)\\,", "595c3dc840b253ccf85c16b9753b91f7": "A(x\\rightarrow x') = \\min\\left(1,\\frac{P(x')}{P(x)}\\frac{g(x'\\rightarrow x)}{g(x\\rightarrow x')}\\right)", "595c57f856bd32e39671b5a46937c391": "\\lambda_e\\,", "595cd12b9e095daebf7f3ad5b4dc5569": "Lclm(L_1,L_2)", "595d004e3345592f1e5b5e666e908835": "\\displaystyle J_0 (x)", "595db4e5cd9142f934df1dcca24a6ffc": " \\{ v_0, v_1, v_2, v_3 \\},\\, ", "595dbcc4134712511a4d1340d1fe59f0": "\n\\left\\{\\begin{align}\n \\pm\\cfrac{\\sigma_1 - \\sigma_2}{2} & = \\left[\\cfrac{\\sigma_1 + \\sigma_2}{2}\\right]\\sin(\\phi) + c\\cos(\\phi) \\\\\n \\pm\\cfrac{\\sigma_2 - \\sigma_3}{2} & = \\left[\\cfrac{\\sigma_2 + \\sigma_3}{2}\\right]\\sin(\\phi) + c\\cos(\\phi)\\\\\n \\pm\\cfrac{\\sigma_3 - \\sigma_1}{2} & = \\left[\\cfrac{\\sigma_3 + \\sigma_1}{2}\\right]\\sin(\\phi) + c\\cos(\\phi).\n\\end{align}\\right.\n", "595e2bce4ccaf83ed2ae60fafb08a52b": " \\operatorname{tr} \\mathbf{M}_\\mathbf{Y} (\\theta) \\leq \n\\operatorname{tr} \\left [ \\left ( \\operatorname{E} e^{\\sum_{k=1}^{n-1} \\theta \\mathbf{X}_k} \\right ) \\left( \\operatorname{E} e^{\\theta \\mathbf{X}_n} \\right ) \\right ]\n\\leq \\operatorname{tr} \\left ( \\operatorname{E} e^{\\sum_{k=1}^{n-1} \\theta \\mathbf{X}_k} \\right ) \\lambda_{\\max} ( \\operatorname{E} e^{\\theta \\mathbf{X}_n}).\n", "595e52c2361f01e1e9c3fe31d25a1572": "\\Lambda^k(\\mathbf{R}^n)", "595e6b944702c1935c31e09144d44c5b": " \\nu \\rightarrow 0", "595e7fa6fbbd5be2a7a23a5606c4a182": "{\\pi \\ln \\beta} ", "595f1ae4ac9e6b9f0df4004a62803006": "2 k \\log_2 n", "595f4f6c7b3a62401fe6cdc91246f51e": "\\phi(\\tau,z) = \\sum_{n\\ge 0} \\sum_{r^2\\le 4mn} c(n,r)e^{2\\pi i (n\\tau+rz)}.", "595f99c5686d8ed2a8f8d9d0b232bdfa": "S^{\\alpha \\beta }", "595fbe7da974a558005bb31e83ce2659": "\\mathfrak S", "595ff5c0c9b4c4ce081b5d4873197047": "q^2 \\leq M_p = 2^p-1,", "59600293d5f801371db6cfb8e5a41094": " \\tilde{\\mathbf{A}} = \\mathbf{A} - \\mathbf{x} \\mathbf{x}^* ", "596075743835e19315e1efaed32e5393": "Z_{CO}^j", "59608f63681c1743b2e646f562970c4e": "\\forall x_1\\dots\\forall x_n\\phi(f(x_1,\\dots,x_n),x_1,\\dots,x_n)", "596093e0539c4bb5b3d58f7dbabcf754": "y = 0", "5960cc465fcbfaa600fbafd6b52add44": "2x^2 + x^3", "59610a4597951c27cc26ca367d9cc05a": "\\mu\\frac{dy}{dx}+y\\frac{d\\mu}{dx}=\\mu q(x)", "5961131802aaebcead145eb639b0810f": "\nk_z=(k_0^2-k_\\rho^2)^{1/2}\n", "5961877c0c2ec677f1a2a73e965032e6": "D_n\\,", "596188c9f06f289065786ffedf09a2a2": " {\\mathbf u}(x)=\\left\\{u(x):\\frac{d}{dx}\\left( EA\\frac{du}{dx} \\right)+n=0,u(0)=0,\\frac{du(0)}{dx}EA=P,E\\in[\\underline E,\\overline E],P\\in[\\underline P,\\overline P] \\right\\} ", "5961d176f93a9d9e2c86526ad09d18b6": " g(E) \\propto E^{-1/2},", "5961dd610ed681978d476a7b465902aa": " \\mathcal{L} = { {\\mid \\mathbf{E} \\mid^2} \\over {8\\pi\\omega} } \\left ( \\mid \\langle R | \\psi\\rangle \\mid^2 - \\mid \\langle L | \\psi\\rangle \\mid^2 \\right ) = { 1 \\over \\omega } \\mathcal{E}_c \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right ) ", "59621276fbbe51f9ba2aff3ecd8ef8d4": "\\partial_t u + \\partial_x^3 u + \\partial_x f(u) = 0\\,", "596216dcb1432f23d976811c67ba8b3b": "\\phi\\left[|m|\\right]", "59621882a6dad2fbfbc714b654ff19e9": "(\\sigma_{ij}(t))_{1\\leq i,j \\leq n}", "59623fed5b2ba23258c70da89197175e": "z \\in X", "59624c28cd1fa456d280fecae31f84d1": "e \\in \\pi_0(BS)", "59627ad913f083cf29dabc588a502629": "\\beta V=\\rho f w_E \\ ", "59634ce4f9b7eeaf38501ee57c502088": "\\,k=3", "59635638b4a293d7bd7d792d5e53338c": "\\mathcal{L}_\\mathrm{M}", "59639766aa529dd75e7f13df5ea1917b": "\\sum_{n=1}^\\infty\\,\\frac{1}{n} \\;\\;=\\;\\; 1 \\,+\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,+\\, \\cdots.\\!", "596417202c16a1a3d35384fd98e21e21": " 1 - (1 - (1 - \\alpha)^{N+1}) = (1 - \\alpha)^{N+1} ", "596422528dec948bc4575a0cde683ea1": "L_\\text{c}", "59648846d60522e67ccbab1f272ac0de": "\\left\\langle i, j \\right\\rangle", "5964cfcbcfe46a83f3f38638fee35caf": "R=M-C\\sum_{n=1}^{\\infty}\\beta_n>0", "5964d38f5f2d4176585e4ef0e9adeccb": "Work\\,output = R \\times D_R", "59658838bb5a5fa112ab52338984f529": "p \\mapsto \\Re", "5965eb90b97091e4c04c3e63c8cadc82": "D_T", "5965f302a5ebb5f900ed8c03a83034bb": "\\frac{\\zeta(1-z)}{\\zeta(z)} = \\frac{2\\, \\Gamma(z)}{(2\\pi)^{z}} \\cos\\left(\\frac{\\pi z}{2}\\right),\\,\\!", "59660997d3a2d9d169bd13b97e3142f1": "\\sum_i{p_i} < 1", "59664945bb16cd433e0e8cae1ff41043": "n_{k\\ell}=\\frac{4\\left(\\ell-1\\right)}{k\\left(k+1\\right)\\left(k+\\ell\\right)\\left(k+\\ell+1\\right)\\left(k+\\ell+2\\right)}+\\frac{12\\left(\\ell-1\\right)}{k\\left(k+\\ell-1\\right)\\left(k+\\ell\\right)\\left(k+\\ell+1\\right)\\left(k+\\ell+2\\right)}.", "59668e79b1a44da1c9b9bc38c0f71795": "F = F_{PE}+F_{SE} \\quad \\mathrm{and} \\quad F_{CE}=F_{SE} \\;.", "5966aacf51c14fcbc1c5ddd582332ada": "\\dot{d}(t)", "5966dbe7e01a646ca4f73b48c3c0648f": "\\mathcal{L} v(x_1(x,t),x_2(x,t))=0", "596741b7b2dfdcca7976b61d2fbcfef8": "j_n, y_n, h_n^{(1)}, h_n^{(2)}", "59678d7abb64b6c59d80b33ba711776e": "f(0,1.25313) = 0.292579", "5967a4c43b42f466e6b9e42884f5b756": "F(x) = \n \\left\\{\\begin{matrix}\n \\textbf{c}_N\\textbf{A}_N^{-1}e^{\\textbf{A}_Nx}\\textbf{b}_N & \\text{if }x < 0\n \\\\[8pt]\n 1 + \\textbf{c}_P\\textbf{A}_P^{-1}e^{\\textbf{A}_Px}\\textbf{b}_P & \\text{if }x \\geq 0\n \\end{matrix}\\right.\n ", "5967bfc565f98d89caa3eef8b40900a0": "\\frac{-1}{\\pi\\sqrt{u^2\\!-\\!t^2}}\\frac{d}{du}", "5967e79ba0e404a3f9550c1615950c77": "g_{m,n}(x) = e^{2\\pi i m b x} g(x - n a),", "5967eff521c650a997f82f083abf639a": "n_3", "5967f64118162fe06ddce0e9204ac92d": "\\langle ! \\rangle", "5967fbc6be769cec33cef4f8982f19b8": "x_0, x_1, \\dots, x_k", "596816738e8fd3a60cc11661fae01e5a": "\n\\nabla \\cdot \\mathbf{q} =\n\\frac{\\partial q_{x}}{\\partial q_{x}} +\n\\frac{\\partial q_{y}}{\\partial q_{y}} +\n\\frac{\\partial q_{z}}{\\partial q_{z}} = 3,\n", "59683ec320302c2b2d772f9be120f07b": "(1-\\vec{\\beta}\\mathbf{\\cdot}\\vec{\\mathbf{n}})", "5968fbec44ffa2f5559400607d472fdd": "\\mu_{eff}=\\sqrt{3\\mu_a(\\mu_a+\\mu'_s)}", "596914b59c7adb6d33d5a5e72757e742": "p(\\theta|x) \\propto p(x|\\theta)p(\\theta)", "59691984f91ba088a2dfee7d91ef93a1": "(x,y,z)\\mapsto(-y,x,-z)", "5969c5ddb91c834b284062df44b1175f": "\\textstyle H_0: \\theta=0.5", "5969cf0e40f09793a2ed7bc6ccc9022c": "f(\\theta;\\mu,\\kappa)=\\frac{e^{\\kappa\\cos(\\theta-\\mu)}}{2\\pi I_0(\\kappa)}", "5969e23c6b393904f68d52ddc14c52b3": "\\{ \\Omega_x^{(k)} \\}_{k=1}^K", "5969ea43a3ad8491b18d34252484d5b1": "2/5", "5969ef5d31b57b86de4d48a0e2fec730": "i=1,2, \\ldots, n", "5969f898f831f068dbfbf517f70c5836": " xyxy = y^2x^2 ", "596a248bffdda6a06d99a222edc79aae": "-1 < \\delta < \\alpha", "596a27c6da2b59c04420f5bb7696bda6": " \\Gamma(S^{\\sigma} ) \\le (1 - 1 / N )^{N / 2} \\Gamma(S ) ", "596a899530a633dd02fd3de6815650ed": "f_{X\\mid Y=y}(x)={f_X(x) L_{X\\mid Y=y}(x) \\over {\\int_{-\\infty}^\\infty f_X(x) L_{X\\mid Y=y}(x)\\,dx}}", "596b12c481ddcc8af5647c3a9d7cd4e9": "L^1(0,\\infty)", "596b4187d7439afcc1613232d4ab9a16": "{\\mathbf{}}L_r(t)=R^{-1}(t)B'(t)S(t)G'(t),", "596b83546e6e7b14a6d287d9ad7b1bbc": " \\hat{H}(\\mathbf{k})=\\hat{C}(\\mathbf{k}) + \\rho \\hat{H}(\\mathbf{k})\\hat{C}(\\mathbf{k}) \\, ", "596b8b6a27962b4abe97314b5005687f": "\\int \\cos^2 x \\, dx = \\frac{1}{2}\\left(x + \\frac{\\sin 2x}{2} \\right) + C = \\frac{1}{2}(x + \\sin x\\cos x ) + C ", "596b94e1e26535633b9515466bcd3f89": "\\frac{\\beta }{2}", "596ba69e9b22acf05f23f1594f164c73": "x_{s(1)...s(\\lambda)}", "596bfb56e019ef0506c0016b7172c9c2": "F(BG, X)", "596c16594a3cc08513d05defa2377b47": "t_{n+1}:=t_n-\\frac{f(t_n)}{f'(t_n)}", "596c16952a5dc03981dc32ff4831de32": "l_1 = l/2", "596c73a8b4c2f4ede068abf267e06ef2": " \\int x^m \\exp(ix^n)\\mathrm{d}x = V_{n,m}(x)e^{ix^n}", "596c9539a568ef5a956d27fa3a1f9d1a": "w^2+x^2+y^2+z^2", "596d1b6fd6d101694484072f5b9efa6f": "\\langle R(u,v)w,z \\rangle=\\langle R(w,z)u,v \\rangle^{}_{}", "596d56be03b0ea269fcb621449a3fae8": "\\displaystyle w(4,3) = 15", "596d5f72f87a1eb5ad3fbd25b9d9051b": "T = T_0 + E \\, s, \\; \\; Z = Z_0 + P ", "596d62d0a81193122f479464f8b72a38": " (\\operatorname{arcsec} x)' = { 1 \\over |x|\\sqrt{x^2 - 1}} \\,", "596df98b71027f9a2505144a143fa7c1": "= (7 * 8^2) + (5 * 8^1) + (6 * 8^0)", "596e1de8688cd8d57d7234c680f0f18c": "\\langle N,M \\rangle", "596e45a46ea595ae42b942b8973ef817": "(x,y) \\overset{\\mathrm{def.}}{=} \\{\\{x\\},\\{x,y\\}\\}", "596e8f68bf2ba43257a90693961eeaee": " X = \\mathbb{F}_2^d = \\{ x_1, \\ldots, x_{2^d} \\}. ", "596e9710175ab1dac5de8e4fcbc48300": " (x_0\\lor x_1)\\land(x_0\\lor\\lnot x_1)\\land(\\lnot x_0\\lor x_1)\\land(\\lnot x_0\\lor\\lnot x_1)", "596ec02b50fad3668c17c33eb90cf2ca": "4\\pi R^2 \\sinh^2 \\frac{r}{R} \\,.", "596ed47f5d3aff318f51eca762a31d14": "j_1,..., j_{n-k}", "596f0cd53109bc9edc63420008ef8c6f": " PV \\ = \\ {A \\over r} ", "596f317f141d28a402605655d7b25130": "n_\\text{max} = 360\n", "596f50582b5ab67a21725865911af100": "U_i \\stackrel {\\phi_i}\n\\mapsto V_i", "596fada675fb3e5ebf4da3b2036ca4c2": "\\mathbb C\\otimes_\\mathbb{R}\\mathbb H", "59702d4930e36723152b084e2acf0fbe": "\\mathbb{P}\\left( X_i=x | Z^n=z^n \\right)", "597030043d12dcf9e796846357b0c5f5": "y = {1\\over 2}{\\omega^2 m_e \\over k_m^2 k_B T_e} ", "5970339496c6229a296eb6ce456bf6ff": "F_{7}", "59705e9361e0dc7aa639e805e5fd2003": "\nf(x) = g^{-1}(x)\n\\Rightarrow\nf^\\star(p) = - p \\cdot g^\\star\\left(\\frac{1}{p}\\right)\n", "597064b420d6bf091e5ecfc7b3cd711b": "W^+\\to e^+ + \\nu_e~", "5970ff38a6ec67aa7edc87a4fe720352": "{\\rm rem}(p, \\tilde p)= \\tilde p_{i-1}\\cdot p(x)- p_i\\cdot\\tilde p(x)\\left(x+\\frac{p_{i-1}}{p_i}-\\frac{\\tilde p_{i-2}}{\\tilde p_{i-1}}\\right),", "597105bace27c1263213765e93267a5b": "\\frac{d}{dr}\\pi r^2=2 \\pi r. \\,", "5971bb22a3b4a73b52483b7cbfcf09c2": "\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2} = 0.", "5971ea3ea4c2f306a365303c3484b03c": "\\bar{\\psi}\\gamma_5\\psi", "59720c2a68c90b6396b408c84dec664f": "\\scriptstyle \\mathcal{A}", "59721851fdc15f62a3e4c266e8aa76ba": "\\kappa_X = k_1\\cos^2\\theta + k_2\\sin^2\\theta.\\,", "59724bf7a25c692ad5308bde3dcbdd18": "x=\\frac{c-b}{a}", "597263250408ee55b5ca1f71c68da22f": "(\\zeta,\\xi,\\phi)", "59727c0e1d81a5db996439daf1776eba": "\\begin{align}\n\\ln(n!) &\\sim n\\ln(n) - n + \\tfrac{1}{2}\\ln(2\\pi n) +{1\\over12n} -{1\\over360n^3} +{1\\over1260n^5} -{1\\over 1680n^7} +\\cdots \\\\\n&= n\\ln(n)-n+\\tfrac{1}{2}\\ln(2\\pi n)+{1\\over(2^2\\cdot3^1)n}-{1\\over(2^3\\cdot3^2\\cdot5^1)n^3}+{1\\over(2^2\\cdot3^2\\cdot5^1\\cdot7^1)n^5}\\\\\n&\\qquad - \\frac{1}{(2^4 \\cdot3^1 \\cdot5^1\\cdot7^1)n^7} +\\cdots.\n\\end{align} ", "5973191655aaed58c9d52ad1805dc328": "h.", "5973667e7f176524443c51e43befd15a": "\\frac{1}{g} = (l+\\pi\\frac{a}{2}) \\times{} \\frac{\\rho}{\\pi{}a^2}", "5973e9f4685d6c79a856f73dd24b05a7": "{D}_{2+}", "5973edf257f14244f497b39e518da14f": "\\delta q_1 ... \\delta p_r", "597416a9bc39f4c28b258923089400f1": "(P \\to Q) \\vdash (\\neg Q \\to \\neg P)", "59742c4b8887107a6f8fb6049d71eaca": "\n-\\Bigl\\langle \\frac{dE}{dt} \\Bigr\\rangle = \n\\frac{32G^{4}m_{1}^{2}m_{2}^{2}\\left(m_{1} + m_{2}\\right)}{5c^{5} a^{5} \\left( 1 - e^{2} \\right)^{7/2}} \n\\left( 1 + \\frac{73}{24} e^{2} + \\frac{37}{96} e^{4} \\right)\n", "59748a7f9b482bf9fd16ca3494d3eb4e": " \\mathcal{C}^{[1]}", "5974afe28bfbd83683f65687087f8ffe": "\n\\frac{ds}{dy} = T_0 \\frac{\\sqrt{2g}}{\\pi}\\frac{1}{\\sqrt{y}}\n", "5974c40fd78fa04fb6abee08d25ce964": " r \\in \\Bbb{R}", "5974d1570082d94876e4d522a7318d7f": "P_{em} = F\\times{v}", "597503762c333b6bcb75cadf4d763bb3": "H =\\sum_{i=1}^N s_i^2", "597535482c47dbff2ecf7a18e58d2e40": "I(\\alpha f) = \\int_a^b \\alpha f(x)\\, dx = \\alpha\\int_a^b f(x)\\, dx = \\alpha I(f).", "59753e3e9d095e2e95fb7baa155dc699": "\\,\\!x^2 + y^2 = r^2", "59753f715367c3ddb2bf4c246552e50f": "Z = R_\\mathrm{L} + j \\omega L \\,\\!", "5975562f519f35f2419255e8359715e5": " (\\alpha,d,\\beta)", "59756d691e8914bcfdc114f06cc78ab2": "G_c = \\frac{\\pi \\sigma_f^2 a}{E}\\,", "5975ac24364b63399760d42b98e473f6": " \\left (I-\\frac {y_k \\Delta x_k^T} {y_k^T \\Delta x_k} \\right )^T H_k \\left (I-\\frac { y_k \\Delta x_k^T} {y_k^T \\Delta x_k} \\right )+\\frac \n{\\Delta x_k \\Delta x_k^T} {y_k^T \\, \\Delta x_k}", "5975c10d345fbf9057ef265fdb81b2ed": " \\partial_\\gamma F_{ \\alpha \\beta } + \\partial_\\alpha F_{ \\beta \\gamma } + \\partial_\\beta F_{ \\gamma \\alpha } = 0", "5975f6f09622b002545b0b0d6bb128d0": "\n\\hbox{New price } y \\hbox{ years later} = \\hbox{old price } \\times \\left(1+\\frac{\\hbox{inflation}}{100}\\right)^{y}\n", "5976237bb402d71ab5cf7fc6e27d515f": "\\lim_{n \\rightarrow \\infty}(\\mathrm{Id}-T)S_n = \\lim_{n \\rightarrow \\infty}\\left(\\sum_{k=0}^n T^k - \\sum_{k=0}^n T^{k+1}\\right) = \\lim_{n \\rightarrow \\infty}\\left(\\mathrm{Id} - T^{n+1}\\right) = \\mathrm{Id}.", "59763712eeaa9e44a1b8eb94fba3b0d3": "\\Sigma\\,", "59763b26b2fe8cff1831eb1d57f85c95": " Q = T_3 + \\frac{Y_\\mathrm{W}}{2}.", "5976f86a33076b681392d5d5b187dcf7": "\\scriptstyle f_{c1}", "597719672e09b40f5697a8754126c706": "(n,m) \\in E", "597722ecf1c63210e4592249f0a9ce76": "\\! \\int \\limits_0^\\infty \\!\\! \\frac{e^{-n}}{1+n} \\, dn = \\!\\! \\int \\limits_0^1 \\!\\! \\frac{1}{1-\\ln n} \\, dn =\n \\textstyle {\\tfrac 1 {1+\\tfrac 1{1+\\tfrac 1{1+\\tfrac 2{1+\\tfrac 2{1+\\tfrac 3{1+3{/\\cdots}} }}}}}} ", "5977352863792945a5f620ecb05a2668": "\\psi : N_{M_1} V \\to N_{M_2} V", "59782f201d511defe7ba5664607206cf": "F_A=\\int_0^d\\frac{1}{2}K_2\\left(\\frac{d\\theta}{dz}\\right)^2-\\frac{1}{2}\\epsilon_0\\Delta\\chi_eE^2\\sin^2{\\theta}\\,dz \\,", "59789ca688b6fb9ef57e5878625daef3": "\\omega = \\frac{\\partial u}{\\partial z}.", "5978b906fa80fe108c371a0ec460a2c3": " \\frac{dS_j}{dt} = 0 ", "5978d289c84d06466b68d8d6ed39c245": "b^{1}\\Sigma_{g}^{+}(v^'=2) \\leftarrow X^{3}\\Sigma_{g}^{-}(v^{''}=0)", "5978f44be212e6317785368795cab4c1": "z \\mapsto \\frac{z-a}{R}.", "59791bd5674bce6fb0923c83eb79c601": " \\Omega = 4\\pi \\sin^2 \\left( \\frac{\\theta}{2} \\right) = 2\\pi \\left (1 - \\cos {\\theta} \\right) ", "59794884d77ce4c0da193a263495bf4d": "a_1\\in A_1,\\ldots,a_n\\in A_n", "5979d736f599273416694946ea07817d": "X \\stackrel f \\leftarrow X' \\rightarrow Y", "597a04a071056e7393c2aaf2bc227412": "x(t)\\in R^{n}", "597a0e6ef5e6d2a5da8701dd2eb961f9": "d(x,y)>\\epsilon", "597a2de835fc06157fd2fefef9e7933b": "\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}<1.", "597aa1a1e13fad130f1eab37ddeee345": "\\operatorname{vec}\\begin{bmatrix}a & c \\\\ b & d\\end{bmatrix} = \\begin{bmatrix}a & b & c & d\\end{bmatrix}^T.", "597ac7c695675acda507a5d22c798ba7": "\\epsilon=0", "597aed3970273109b7a4f588528f7518": "|p|_{\\ast}=\\alpha<1", "597af70edc27a8066ad8993c6afbd426": "\\mathrm{^{239\\!\\,}_{\\ 94}Pu\\ +\\ ^{4}_{2}He\\ \\longrightarrow \\ ^{242}_{\\ 96}Cm\\ +\\ ^{1}_{0}n}", "597b3711da9273992f76b38ce2ba0727": "H_1(a, b) = a + b\\,\\!,", "597ba3e6ea5e1c586a69daf38a20e490": "A\\hookrightarrow B\\twoheadrightarrow C", "597beb05e8d2c92bf2d6912345b16c08": "\\zeta_1\\;", "597bf9e2e06077c7ea5fbfe2ca313fd9": "\\scriptstyle \\hat x \\hat y", "597c1a86012d48e7783e8c8d9b414ea2": "\\mathbf{z} \\mapsto e^{i\\theta}\\mathbf{z}", "597c5adee36c658de0224a04dd4163d9": " \\Gamma \\vdash A", "597c5cd214cd9b4ee09208f66a717ac8": "\\sum_i E_i E_i ^\\dagger= I. \\, ", "597c9086801c61af9ab62692765f9569": "PHM_{k} = \\gamma \\left(1 - \\alpha - \\beta - \\gamma\\right)^{k - 1}", "597c929bcc706a863f60f90ed4606f1c": "(-1)^{0+3}(280\\times1+56\\times3+168\\times4)=-1120", "597c92eff12a7bb09d7eb371d5343aa8": "q(x) = x^3 - x + 2", "597cd34e03c04e25712449f3a6da1546": "\\xi = x + X", "597cfaeae061f1ebb5daa33d0d0ddbfb": "I_t = I_0e^{-\\alpha x}", "597d896a394bccc10d565244efc33835": "d(S_1,S_2)", "597d931d841e358898e25e64180732f4": "\\left(\\frac{\\mathit{Q}_{2-3}}{{m}}\\right)=\\mathit{u}_2-\\mathit{u}_3", "597dff9afed03926f036c6480c88efff": "B(S^{-1}S)^0", "597e137806cb2dd91069f4924b4ef262": "\nC_{1i} = \\frac{\n{5\\left( {3m^4 - 18m^2 + 31} \\right)i - 28\\left( {3m^2 - 7} \\right)i^3 }}\n{{m\\left( {m^2 - 1} \\right)\\left( {3m^4 - 39m^2 + 108} \\right)/15}}\n", "597e1d2fcdecc9f6488a54fd9c622a78": "p(1/x) = 0", "597e516bc5e44b4690aeeb1e85aecac9": "D = \\frac{OH}{\\text{Mil}}\\times 1000", "597e6427f2cac03896b2b0287eaa13f8": " \\dot{m}_{fuel} ", "597e7e4a8895afd699ca9c6975dd848e": " \\langle a \\otimes h, b \\otimes g \\rangle _K = \\langle \\Phi(b^*a) h, g \\rangle _H", "597e818d990748184dbd1e012aa2e1b8": "\\hat \\beta +2n \\pi", "597e9a612d989848a37a97fa1de59437": "\\tau(\\mu)", "597ea391e3f3aba70b3385df7d85d242": "B_\\mathrm{foreign}", "597ebd1016330153a2cc8a601f132e3b": "\\boldsymbol{L} \\cdot \\boldsymbol{S}", "597f376aa13c04baad33da87a7519b8f": "I=M(height^2+width^2)/12", "597f8c56649bcffef94305ffecf7b4a9": "\n\\operatorname{E} \\left[\\left. \\frac{\\frac{\\partial^2}{\\partial\\theta^2} f(X;\\theta)}{f(X; \\theta)}\\right|\\theta \\right]\n=\n...\n=\n\\frac{\\partial^2}{\\partial\\theta^2} \\int f(x; \\theta)\\; dx\n=\n\\frac{\\partial^2}{\\partial\\theta^2} \\; 1 = 0.\n", "597f971bea0f1a20321e28c2f5a68a1b": "\\widehat\\sigma^2 = \\frac{1}{n} \\sum_{i=1}^{n} (x_{i} - \\bar{x})^2 = \\frac{1}{n}\\sum_{i=1}^n x_i^2\n -\\frac{1}{n^2}\\sum_{i=1}^n\\sum_{j=1}^n x_i x_j.", "598005f1893c79b871ccfaad8b9d4964": "H=H_A\\otimes H_\\epsilon", "5980243e92b5d09641847b9c63edc215": "E^{\\dot{\\beta}}_{\\hat{\\dot{\\alpha}}}=\\delta^{\\dot{\\beta}}_{\\dot{\\alpha}}", "598090becf97f2e865b5e30291b9ddb1": " \\Pi = 40(2025) - 30(2025) ", "598091c09ed0d240f22fee9dfc36e52e": "T_L\\rightarrow e^{i\\beta}T_L\\text{ and }(\\tau_R)^c\\rightarrow e^{i\\beta}(\\tau_R)^c.", "5980de7e4af0c5110c08c6e30195e75e": "d = \\gcd(s, MN - 1)", "59810d1c9b5b388ace8337f24f2f4353": "L_a : x \\mapsto ax", "598144aaa87fa24fd846bd1d0cc93198": "I_{n_1, k_1}", "59816fd1f76eb48925bd06d12b5effef": "L^2\\dot\\lambda-3gv=0", "598184ff0a4db86f9deff3dcbd6a8212": " \\mathbf{a} \\cdot \\mathbf{b}", "598216e7736bbc7f9ebcdbe4827bca35": "D^{-}(\\mathcal{S})", "598233cba4baf5962c999dbafef4a464": "\\displaystyle{C_\\pm f_\\pm = F_\\pm,\\,\\,\\, C_\\pm f_\\mp = 0.}", "5982750d10af613576a9147d34564f21": "\\kappa \\geq 0", "5982977d0a1d3657d02a267d7b3ad0f3": "2 K + U = 0", "5982d06bb8b568b324b27dbdf89f017f": "\\phi_{u}", "5982d6559172260608b729cb7cb643b3": " \n\\sum_{n\\le \\lambda} \\left(1-\\frac{n}{\\lambda}\\right)^\\delta \\Lambda(n)\n= - \\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \n\\frac{\\Gamma(1+\\delta)\\Gamma(s)}{\\Gamma(1+\\delta+s)} \n\\frac{\\zeta^\\prime(s)}{\\zeta(s)} \\lambda^s ds", "5982e39b774922f8e7e71f3c98b961d4": "m \\frac{d \\vec{v}}{dt} = \\vec{F} - mg \\hat{k}\\;.", "598343ad78f0453c0f8dc47557ab6575": " 1 ", "5983b1ceb0874b26ecf80b1153a9e6a2": "\n(\\mathbf{\\alpha}^{\\text{T}}\\mathbf{M}\\mathbf{\\alpha})\\mathbf{N}\\mathbf{\\alpha} = (\\mathbf{\\alpha}^{\\text{T}}\\mathbf{N}\\mathbf{\\alpha})\\mathbf{M}\\mathbf{\\alpha}.\n", "59844f1ddb3175d4f2dca841f5af6114": "\\displaystyle Wg(21,d) = \\frac{-1}{(d^2-1)(d^2-4)}", "59848bf8fda25e042374467074f2d30a": "\\begin{align}\n & a_{0}\\left ( -(c)(c-1)+(\\alpha +\\beta -1)(c)-\\alpha \\beta \\right )s^{c}+\\sum_{r=1}^{\\infty }{a_{r-1}(r+c-1)(r+c-2)s^{r+c}} -\\sum_{r=1}^{\\infty }{a_{r}(r+c)(r+c-1)x^{r+c}}+\\\\\n& \\qquad \\qquad + (2-\\gamma )\\sum_{r=1}^{\\infty }{a_{r-1}(r+c-1)s^{r+c}} +(\\alpha +\\beta -1)\\sum_{r=1}^{\\infty }{a_{r}(r+c)s^{r+c}}-\\alpha \\beta \\sum_{r=1}^{\\infty }{a_{r}s^{r+c}}=0\n\\end{align}", "5984dc145cdc4be77efdbb1b76304055": "a\\vee b = a\\vee \\varphi -1(b), b\\vee a = \\varphi -1(b)\\vee a", "59852521231d47b1302f58a3b3640ea6": "\\textstyle p \\equiv q \\equiv 3 \\mod 4", "59853815a0f4198e0ae788d5113e429f": "(1-\\varphi_k) B_{k+1}^{BFGS}+ \\varphi_k B_{k+1}^{DFP}, \\qquad \\varphi\\in[0,1]", "5985479ddc7a0d4df148c839c30a2467": "K_{sp} = [A]^x[B]^y\\,", "59855effd44c51000f07b0146ba42aa8": "P = \\frac{\\mathcal{D}_\\uparrow(E_\\mathrm{F}) - \\mathcal{D}_\\downarrow(E_\\mathrm{F})}{\\mathcal{D}_\\uparrow(E_\\mathrm{F}) + \\mathcal{D}_\\downarrow(E_\\mathrm{F})}", "5985710bf9e209c3583b1a7d0fe1dced": "(256^{\\,\\!256})^{256^{256}}=256^{256^{257}}", "598580cd1a5635697c17f1fea1c943a7": "x(p, w) = \\operatorname{argmax}_{x^* \\in B(p, w)} u(x^*)", "5985be7e7db9b34404bbb815abcab80a": "\\beta_1^{(1)} = \\beta_1^{(0)} (1-t_0) + \\beta_2^{(0)}t_0 = \\beta_1(1-t_0) + \\beta_2 t_0", "598612529e401f437940b28cbe7a9528": " h = (a-b)^2/(a+b)^2 ", "5986321f2270a75e547ad36b0b9846ed": "\\mathbb{R}^{m}", "598671d0cbf86e14f206d376b43c71ad": "\\ 32.45", "5986ad1af302893871cf7446df9ba6c1": "\\partial_u", "5986dc545618a1eba727d03eb9a8b803": "\\exists z ( z = x+1)", "5986e11848e3c17e881eae8eada61169": "\\xi (s)=\\frac{1}{2}s(s-1) \\pi^{-s/2} \\Gamma \\left(\\frac{s}{2}\\right) \\zeta(s)", "59870b196e598879a211d4f99b43b8f4": "f(x) = \\frac{I_r - I_l}{2} \\left( \\operatorname{erf}\\left(\\frac{x}{\\sqrt{2}\\sigma}\\right) + 1\\right) + I_l.", "59874096bbb59b6aa49f758c73ba157b": "K_{(k\\ell)}", "5987e77bb7ad84f7141b68dc184be0da": "100(1-\\alpha)% ", "5988344efe10585e13747755cc81fc0b": "\\beth_{d-1}(|\\alpha+\\omega|^{\\aleph_0}+\\beth_2)", "598845930c2397fcb54d1b3eeef20468": "\\ \\nu=\\Gamma-\\lambda \\ln(p')", "5988cbeda0eb706d898a0a08c03235e1": " { m \\choose {\\lfloor \\frac{m}{2} \\rfloor} }", "5988df51f0c9d130dffb67cc7c09c9fd": "\\Theta\\mathcal{B}", "5988fb990869fb54287f6c9f2a05aad3": "M \\propto L", "59895bd4ba956812dcba592b0df15aa4": "_{p\\tilde{\\leftarrow}q=p'q\\,}\\!", "59899918691dc8315e1adf9a575b9fcc": "Q^\\mu_{-(1/2)+i\\lambda}(x).", "598a4871cd4428b32eb30b7ed4f3f37f": "\\sin(\\alpha) \\approx \\tan(\\alpha) = S", "598aa294537a722f87415f1a1c6c8f50": "loaded(1)", "598aa979a44772730626a518850ad20d": "\\frac{8! \\times 3^7 \\times 12! \\times 2^{10} \\times 24!^8}{4!^{36}} \\approx 1.95 \\times 10^{160}", "598abad192559ae589567ab11427f6e4": "F \\subseteq (S \\times T) \\cup (T \\times S)", "598acfb11e348d52b42b090b0ea34451": "h(X_1, \\ldots, X_n) = \\sum_{i=1}^{n} h(X_i|X_1, \\ldots, X_{i-1}) \\leq \\sum h(X_i)", "598bf7120c0e0bc993c883c351bb19b5": "1= \\cos(3\\pi/8) A_{T,L}^{x_c} + B_{T,L}^{x_c} + \\cos(\\pi/4) E_{T,L}^{x_c}", "598c58865841a82ef2986aa54d23604f": "x y^2 = 4 a^2 (2 a -x)", "598c6751cf561c1e075d21345def5a76": "G^{*}\\,", "598c79b1cd260fde031b0a9dbf1e381c": " B1(y;b,p,q) = GB1(y;a=1,b,p,q) ,", "598c803d84b578cbb971b9860d311d75": "\\lambda = \\sum_{A \\in H_2(X)} \\lambda_A e^A,", "598d2848af2b012768069f543f7d8f6a": "\\log_2{(x)}", "598d377ac7509e8ef63c099712bc1056": " B_\\mathrm{0} = h \\nu / g_\\mathrm{e} \\mu_\\mathrm{B} ", "598d3b45492f3f7ac97f68404ef340be": "F_\\mathrm{e} = k_\\mathrm{e} \\frac{q_1q_2}{r^2}", "598d5fe1af554080760158626456fc85": "A = \\frac{\\alpha}{2\\pi} \\cdot 4 \\pi r^2 = 2 \\alpha r^2", "598dd5a0468286823101aca7c92f1103": "{{QED}}", "598e06f732d0ff45e164a136aae1eb45": "L^{(\\alpha)}_n(x) = {n+\\alpha \\choose n} M(-n,\\alpha+1,x) =\\frac{(\\alpha+1)_n} {n!} \\,_1F_1(-n,\\alpha+1,x)", "598e1d99b18b34987a6c8130e98aa107": "\\textstyle |a_i\\rang", "598e385b6f75f0ae8d590e5980fb3522": " Q_j + Q^*_j = 0, \\quad j=1, \\ldots, m,", "598e8e900c1a601f49e7bb79f1bf592f": "g_i \\cdot x := \\alpha(g_i,x)", "598f2d188e13c994d3ca15cec7cb87bc": "k_2", "598f4de3b88fe734d1d9c8567804da96": "\\mathbf{A}_q", "598f4f716ca6cb6585467486bd21d6cd": "\\zeta(3) = \\frac{8}{7} \\sum_{k=0}^\\infty \\frac{1}{(2k+1)^3}", "598f7471d2c36e9f3045ae23776bd023": "\\begin{align}\n29 &= 2^0 \\cdot 17^0 \\cdot 23^0 \\cdot 29^1 \\\\\n782 &= 2^1 \\cdot 17^1 \\cdot 23^1 \\cdot 29^0 \\\\ \n22678 &= 2^1 \\cdot 17^1 \\cdot 23^1 \\cdot 29^1 \\\\\n\\end{align}\n", "598f8289a6bce2ae31e81045c1e3d706": "\n\\vartheta_1(z) = -i \\sum_{n=-\\infty}^\\infty (-1)^n q^{(n+1/2)^2} \\exp ((2 n + 1) i z)", "598f9ee41d2fa440625e81188b5750bc": "cos(\\theta) = 1", "598fa0be8dfc49d4f2719b1d74073c74": "\\begin{align}H(X) & = \\mathrm{(i)} \\, \\mathbb{E}_x \\{I(x)\\} \\\\\n& = \\mathrm{(ii)} \\log N - D_{\\mathrm{KL}}(P(X) \\| P_U(X) )\\end{align}", "598fab70acd93c9a8892a371acedf04c": "\\sum_{n=1}^N \\left(a_n - a_{n-1}\\right) = a_N - a_{0},", "59902f93cc32b50256f9e30e9691fc67": "y = a_0 \\sum_{r = 0}^\\infty \\frac{(c + \\alpha )_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^{r + c},", "5990433c4fb84ee107c32ff41bcc0cf6": "\\begin{align}\nL_{f,P_n} &= \\sum_{k = 1}^{n} f(x_{k-1})(x_{k} - x_{k-1})\\\\\n &= \\sum_{k = 1}^{n} \\frac{k-1}{n} \\cdot \\frac{1}{n}\\\\\n &= \\frac{1}{n^2} \\sum_{k = 1}^{n} [k-1]\\\\ \n &= \\frac{1}{n^2}\\left[ \\frac{(n-1)n}{2} \\right]\n\\end{align}", "5990995f4491cb4e1e6a7ae3ef380c40": "\n\\Bigg(\\frac{p}{q}\\Bigg)_4 \\Bigg(\\frac{q}{p}\\Bigg)_4 =\\left(-1\\right)^\\frac{fg}{2}\\left(\\frac{-1}{e}\\right).\n", "5990dcbe15f826090b7cf543a7a4f3ba": "z_a= 0\\,", "59910b315c810ec603da39bd68049ff7": " p_{X} (x) > 0 ", "599165fbe95c7d128264ecc1fb648c02": "\\hat{\\alpha}_1 \\otimes (\\mathbb{I} - \\hat{\\alpha}_2) ", "5991b64126c677187a09a573d8c1d62c": "x_i \\geq 0\\text{ for }i=1,\\dots,n", "5991c864212bca4b3a9a19e28f386f59": "\n Q_{MAX} = \n { \n { T_{JMAX}-(T_{AMB}+\\Delta T_{HS}) } \\over { R_{\\theta JC}+R_{\\theta B}+R_{\\theta HA} }\n }\n", "5991cc064a48b886356af10b5640a8eb": " \\{\\mathcal{F}(t); \\; 0 \\leq t \\leq T\\}", "5991cf51d27bdff160eadbde7fe13e8f": " \\cos y = x \\, ", "599220cf045fde182192e4a88606b5cd": "\nP_{\\mu }(n)=||^{2}=\\frac{(\\mu |\\varsigma |^{2\\mu })^{n}}{n!}\n\\left( E_{\\mu }^{(n)}(-\\mu |\\varsigma |^{2\\mu })\\right) , \n", "59923e330420613ef3cd554af99dd293": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin u\\ \\cos u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^2 u\\ \\cos u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin u\\ \\cos^2 u \\ du\\ = \\\\\n&-\\hat{g}\\ 2\\ e_g\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u\\ \\cos^2 u \\ du \n+\\hat{h}\\ 2\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u\\ \\cos^2 u \\ du =\n-\\hat{g}\\ \\left(2\\pi \\frac{1}{4}\\ e_g\\right) + \\hat{h}\\ \\left(2\\pi \\frac{1}{4}\\ e_h\\right)\n\\end{align}\n", "5992ebc6ede4739be73085c6ed00c5b5": "\n H_{\\Phi}(\\mu) = \\Phi\\left(\\frac{\\sqrt{n}(\\mu-\\bar{X})}{\\sigma}\\right) ", "5992ff4299d00b2c99cc105dae1352ae": "L_n^{(\\alpha)}(x)=\\sum_{i=0}^n {\\alpha-\\beta+n \\choose n-i} L_i^{(\\beta- i)}(x);", "599369fffef1068d96f386e5b42b9533": "i = 2 \\pi \\cdot \\frac{R \\cdot B_p}{r \\cdot B_t}", "59936ac890e5c8b75f81153010e475d2": " \\sum_{n=1}^\\infty \\pm \\frac{1}{n}.", "599379952b7ccd3059b95b5b7842e4ad": "\\Gamma(\\gamma)_u^t\\circ\\Gamma(\\gamma)_s^u = \\Gamma(\\gamma)_s^t.", "5993db33aa99fe899f36efd886b3dbf3": "l(S)", "59941adc8aacb1395f5c7198b203fbcf": "\\tfrac{3K(1-\\nu)}{1+\\nu}", "59942d697d5fc71605e92f3eed5db861": "d_{iw}", "599442e0f355961e72f43aa6f96bb786": "\\Phi_{S}=0=P_{S}\\frac{z_{S}^2F^{2}}{RT}\\frac{V_{m}([\\mbox{S}]_{i} - [\\mbox{S}]_{o}\\exp(-z_{S}V_{m}F/RT))}{1 - \\exp(-z_{S}V_{m}F/RT)}", "59945ef68150ae8d230a41d2ed5fbc4d": "\\Pi_{H}(m)\\,\\!", "5994643a8515f842714b4f9b8fd9f865": "v_{Oy}=\\lVert v_O \\rVert \\cdot \\cos(\\theta_{T})", "5994b36b82ebb9315e69ade1f3fb12b1": "F = \\frac{L'I^2}{2}", "599577a487315e32f6369d2ea6584a8b": "\\gamma_2=4\\frac{(-96+40\\pi-3\\pi^2)}{(3 \\pi - 8)^2}", "5995e427a6e7de26dfbdae9119e4daf2": "(d_{1}, d_{2})", "5995ecdba18f4060e8a484e44151b04e": "|e_n\\rangle", "5995f0d642f036271fd501408fdf838b": "\\theta = [m_k^T \\mathrm{vec}(C_k)^T \\sigma_k]^T \\in \\mathbb{R}^{n+n^2+1}", "5996182c2ae4d18b41eab57353b4f9c6": "\n V_\\mathrm{g}\n = \\iint dx\\, dy\\; \\int_0^\\eta dz\\; (\\rho - \\rho') g z \n = \\frac{1}{2} (\\rho-\\rho') g \\iint dx\\, dy\\; \\eta^2,\n", "5996508223db8a13fa08affc7b7f6493": "j_1 < \\ldots < j_r", "59967d1edf5ebabb16242d059b2eca77": "U(t,s)", "5996d03730d78cb75df366abbe8ace47": "|S|\\geq\\gamma p", "5996dd36331d1295b5669de629408ec9": "\\mathbf{w}_n(x_n), (n=1 \\ldots N)", "59970bbecfe17394263245c40e33e7aa": " V_0^{(P)} = V_0^{(1)} - C \\log_{10} \\frac{B+P}{B+P^{(1)}}", "59975bf64a3e47fd6af7a238d493584a": "K_\\mp", "59975c67abb90b97880c4d315a97fffd": "D_N(x) = \\sum_{n=-N}^N e^{inx} = \\frac{\\sin\\left((N+\\tfrac12)x\\right)}{\\sin(x/2)}.", "599775a4cf7fee3b20ae719ba665714b": "\\displaystyle{\\psi(t)=1 +a_1t +a_2 t^2 + \\cdots}", "59977683574cd4bc84f081f9687fb01b": "\\gamma v t' = \\gamma^2 v t - \\gamma^2 x + x \\,", "59977c28892724e9c9937dc5d4097e8e": " V_{i+3} = a_i V_{i+2} + b_i V_{i+1} + V_i ", "59977c9182e37a61913973a39fab1a72": "SS(w)=s=w-c", "599781d0aa01e9a82d3474d2c971ead5": " H(\\tau) = H ", "5997824dbdb9272507994fe9caca81c1": "{\\bold \\ \\mu}", "59978e58379a4919a15a2e3a9f99646b": "\\sigma(\\bar y)= \\frac{1}{\\sqrt{n}}\\sigma", "59979801c5a9aec05a98240c23e9e2b4": "I(v)=\\log p(v)=E(v)+\\sum\\xi^iF_i(v)", "5997c9d33cb5c87a7d57e51701125e89": "H_{\\frac{1}{4},3}=64-27\\zeta(3)-\\pi^3", "5997d5ea7f58fd87addee874b41993e5": "\\sum M_A=0=-10*1+2*R_B \\Rightarrow R_B=5", "5997d952fbeb7afbc41cb110f34af76d": "4.66\\,\\text{slug} \\cdot 8\\,\\tfrac{\\text{ft}}{\\text{s}^2} = 37.3\\,\\text{lb}_F", "5997eb4de64d483b4f610f9cdd758784": "F(z)=\\sum_{\\gamma\\subset\\Omega:a\\to z} e^{-i\\sigma W_{\\gamma}(a,z)}x^{\\ell(\\gamma)}.", "5998045b85a494654e5510b39735fe7c": "z=\\exp(-x^2-y^2)", "599812808c7eafbf2bfab7f441cb9c8e": "\n\\begin{cases}\nY_1^* = X_1\\beta_1+\\varepsilon_1 \\\\\nY_2^* = X_2\\beta_2+\\varepsilon_2\n\\end{cases}\n", "59986322f2b8d792d51de7ed853cd0d5": " x = y/k .\\,", "5998bdae16e85b275e57e3d8ff693c10": "p=2,3,4,\\ldots", "599904cc95caa477a3608bfb492bdc24": "P = A A^+ ", "599946f01155ec0807b986774f7718a2": "Q_{\\lambda}^{\\mu}(z) = \\frac{\\sqrt{\\pi}\\ \\Gamma(\\lambda+\\mu+1)}{2^{\\lambda+1}\\Gamma(\\lambda+3/2)}\\frac{1}{z^{\\lambda+\\mu+1}}(1-z^2)^{\\mu/2} \\,_2F_1 \\left(\\frac{\\lambda+\\mu+1}{2}, \\frac{\\lambda+\\mu+2}{2}; \\lambda+\\frac{3}{2}; \\frac{1}{z^2}\\right)", "599995fb64443466b2fa23d6419834b5": "ds^2 = -f(r)^2 \\, dt^2 + g(r)^2 \\, \\left( dr^2 + r^2 \\, \\left( d\\theta^2 + \\sin(\\theta)^2 \\, d\\phi^2 \\right) \\right), ", "5999fdbb9dd823deae3536ac59d245c6": "\\frac{dt}{dU} = 0", "599a2ae53a794ec7f4b61ea7ad4c698e": "K_2\\left(\\frac{d^2\\theta}{dz^2}\\right)+\\epsilon_0\\Delta\\chi_eE^2\\sin{\\theta}\\cos{\\theta}=0", "599a5d911c68871c48c31cb42cbcf1ec": "\\mathbf{e}_x = (1,0,0),\\quad \\mathbf{e}_y = (0,1,0),\\quad \\mathbf{e}_z=(0,0,1).", "599a70f58cc2707d1f6ad150a097e7db": " c \\ ", "599a967d64a74e92780af78204e4d189": "x_1\\le x_2\\le \\dots \\le x_n.", "599a9e44a5641bd032a1db51b675c414": " \\frac{z(z+1)^2}{(z-1)^4} = \\sum_{n=1}^{\\infty} O_n z^n = z +6z^2 + 19z^3 + \\cdots .", "599aa7c9abaffcac65794f53947a0eed": "{{\\mathbf{k}}}[x_1, \\ldots, x_n]", "599ab7a43ad463ca50b44f09314f84f9": "\\alpha^{th}", "599ac8df8eadd95d375db237b99b9344": "K(u) = \\frac{35}{32}(1-u^2)^3 \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "599ad4c77fb4f0ec16a030a1fb8fc07c": " F_n(x) = \\Phi(x) + \\sum_{j=1}^\\infty \\frac{P_j(-D)}{n^{j/2}} \\Phi(x)\\,.", "599ad7c16d6d0a8398b23de07ba097f0": "\\operatorname{Var}", "599b09aa7d371e0f22b71cad95bd93b9": "a=b=1.", "599b46b57f9e3764b4d1154b6a76a8df": "\\mathcal{A}/\\mathop{\\mathrm{ker}}\\,h", "599b526f64e698adbc30bcec982bf576": "\\int\\mathbf{\\Psi}_{lm}\\cdot \\mathbf{\\Psi}^*_{l'm'}\\,\\mathrm{d}\\Omega = l(l+1)\\delta_{ll'}\\delta_{mm'}", "599ba39d43ef5e5056aa8b82d12608b2": "10^{-3}", "599bae290b1a52288c3518d91905e1ec": "\\beta(e)=\\frac{e^3}{12\\pi^2}~,", "599bbee1dbe33e25408fb6500632fb74": "\\alpha, {\\alpha\\,\\!x^2\\over{2}}, {\\alpha\\,\\!x^3\\over{3}}, \\dots, {\\alpha\\,\\!x^n\\over{n}} ", "599bcc4c7a143319e589a2b64156d2fd": "\\scriptstyle\\circ . \\times", "599bde0db52ff63f94219b6c76424a30": " \\langle \\psi |\\hat{A}|\\psi \\rangle ", "599c01e4d1403b78b79b8e74d2d6cc6a": "{\\eta_b} = \\frac{2U(2V_1\\cos\\alpha_1-U)}{V_1^2-U^2+2V_1U\\cos\\alpha_1}", "599c0df743cf7908bda82a8c295fa669": "T_G(2,1)", "599c34da1a8d7144a4bb1dfaf9614b72": "\\lnot \\lnot \\phi \\to \\phi ", "599c4b03fa42ae956ada4f830c0e40f9": "\n\\tilde{K}(p;T) = e^{-T {p^2/2}}\n\\,", "599c55e8b2309ab356f5898019a25e6e": "\\phi(x) \\equiv x^q \\text{ (mod }\\delta) \\,\\!", "599c6997e3bc8d212ef50ab5e26eaa1f": "\\underline{\\varphi \\land \\psi}\\,\\!", "599cf09e38f44d01768cde31ab05cdea": "x_{k+1} = \\frac{1}{n} \\left[{(n-1)x_k +\\frac{A}{x_k^{n-1}}}\\right]", "599cfb272f0bca04bcea45a2a567a553": "D_1 = \\tfrac{\\kappa}{2}", "599d255805284be64ad0643a71b4d87e": "\\sqrt{-1}\\times\\sqrt{-1} = -1", "599d351db17062c32eaa1607d891bdac": "A_P", "599d91704e99f8344ce16f796bd9126f": " \\langle B,E,E,B \\rangle ", "599db4d924df74d004803060d91c6f47": "e_i = |f_i - y_i|", "599db70c3aa6ee87f82a2435b7d72835": "\n\\begin{align}\n&d_i \\log_2 p + (n_ib-d_i) \\log_2 (1-p) +n_ib-n_iRb \n\\\\= &d_i(\\log_2 p +1-R) + (n_ib-d_i)(\\log_2 (1-p) + 1-R)\n\\end{align}\n", "599df34e8704e877cadcd34c0f558613": "V_{i}(t) = 1V \\cdot \\sin (\\omega \\cdot t)", "599eee53d8a046661052bb1f1a02665e": " dU = -P\\,dV \\ ,", "599ef84975dae3289bdacd5d10917aed": "\\mathrm{d}\\Phi=\\sum_i x_i\\,\\mathrm{d}y_i\\,", "599f14315331ecc706c2b56547fd5a00": "\\mathbf N=\\mathbf J-\\mathbf S", "599fd37a66b8b49f9c6f120d8eb81cda": "\\mathbb{P}_n = n^{-1} \\sum_{i = 1}^n \\delta_{X_i}", "59a032dfd25f2a18efab6f54209f860e": "\\chi(p)", "59a094ffd3c6e5ed86d52e49dd749fbc": "\\land\\oplus", "59a0aa856d7d3670c6c85ce18364c873": "X_{2j}", "59a121181a335c1b069426960cc2fbae": "s_2", "59a13bc9c432e51774624639eb2916c6": "\\alpha = \\beta = 3", "59a25c6aa4d709e9534d86e599e31b75": "C=0.149", "59a27242e62333c79a440fffdd4bff56": "\\Delta (C \\cdot V)=(-K \\cdot C+ \\dot m_{in} +\\dot m_{gen.})\\Delta t \\qquad (5)", "59a289fd8e71210cb5c6104b70258f87": "W>0", "59a29f60b5d89f96788a523486755337": "\\Delta \\delta(x)", "59a2cbc5299d121452f683406580251f": "\\frac{1}{\\kappa(t)}", "59a38e2e397aca80620a62952a7b39e9": "f\\left(\\underbrace{-2.903534, \\ldots, -2.903534}_{(n) \\text{ times}} \\right) = -39.16599n", "59a39e76142b228eab294764145148c8": " \\sum_i X_i =0 ", "59a3c5876c9b8d73938460c183a0b4b9": "\\prod_{n=1}^\\infty (1 + a_n)", "59a3ffe5b70f9e5f8a494bd0a24ef66c": "\\phi_n(n)", "59a42debfa5e6b5fd01459a139c05e32": "((b+c)+(a+c)+(a+b))\\left(\\frac{1}{b+c}+\\frac{1}{a+c}+\\frac{1}{a+b}\\right)\\geq 9", "59a431ada758e1982e58a7ad59f9898a": "\\left|I_o\\right| = \\frac{L}{TV_i}I_o", "59a4396277b38648d0d9d7c68a069193": " f_0=\\frac{1}{2\\pi}\\sqrt{\\frac{R_\\mathrm{f}+R_1}{C_1C_2R_1R_2R_\\mathrm{f}}} ", "59a4605b027ba04f0224563e21912601": "t' = t \\pm (r^* - r)\\,", "59a4fb19fc1359fcac97d0e3e8363bae": "(a_1\\otimes b_1)(a_2\\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\\otimes b_1b_2).", "59a5a790534013717842df49dcffca65": "(f^{-1}\\mathcal{G})_x \\cong \\mathcal{G}_{f(x)}", "59a5b17a85cff9bcffae7a45c47a71b7": "WHIS \\equiv ", "59a5b1b3df95caa5a453e00c1ce82cc3": " {\\alpha \\choose k+1} = {\\alpha\\choose k}\\,\\frac{\\alpha-k}{k+1}, \\qquad\\qquad(2) ", "59a5c313c3cf46dcb82c7a8fec749cee": "\n\\cong\n\\coprod_{y\\in Y} f^{-1}[\\{g(y)\\}]\n", "59a632db817e793290e2a22580faae10": "\\vec H\\!", "59a63b7217dfe16b461fab1dfebd3f08": "2^{O(k)}\\log^2 |V|", "59a64d6742566f6b547d67d9b3091253": "a>0, d>0, p>0", "59a665e57c706e17d30d92eacaaa14b2": "r = \\frac {p}{1 + e \\cdot \\cos \\theta}", "59a673a298f2fd5b3058d95e87a84d1f": "\\scriptstyle \\dot m_{0m} \\,", "59a680b67145ac6dccd193846d6fdbf2": " f(z) =\\int_0^{2\\pi} {1 + e^{-i\\theta}z\\over 1 -e^{-i\\theta}z} \\, d\\mu(\\theta).", "59a685ff8b70f9cd90251fab4eadc30a": "-1.00031", "59a6c11782730f4b1ada75bdc5e4e09d": "a_{i_1,i_2,\\dots,i_N} = \\sum_{j_1} \\sum_{j_2}\\cdots \\sum_{j_N} s_{j_1,j_2,\\dots,j_N} u^{(1)}_{i_1,j_1} u^{(2)}_{i_2,j_2} \\dots u^{(N)}_{i_N,j_N},", "59a6cc620a6dc763082c7cbe6c781b67": "\\mathbf{R} \\to \\mathbf{R} : x \\mapsto (x-1)x(x+1) = x^3 - x ", "59a6f9abfb5e23b53ba32baf864e701f": "f(x) = \\frac{a x + b}{c x + d}", "59a7a697e6e0e7be023c2053617ad47b": "\\rightarrow -", "59a7d088aaceee7aed449f9e7301a034": "E^\\lambda \\otimes E^\\mu =\\bigoplus_\\nu (E^\\nu)^{\\oplus c_{\\lambda\\mu}^\\nu}.", "59a82d72e84c0cc85827d6616016e82e": "\\scriptstyle g\\ne 0", "59a843acb49a3b71c17c55df6d0b606a": "\nu (r) = C_1 r + \\frac{C_2}{r} ,\n", "59a8ae7900924fb35ef780971bf3127c": "\\Delta_S = (1 - S_{11}) (1 - S_{22}) - S_{12} S_{21} \\,", "59a8c2420e045ef6b6a7406c8ddc8aef": "\\mu_i = C\\mu\\text{ for }i\\geq C \\, ", "59a9005cb3c6f4941e92035f4b865026": "\\rho_{x^{n}}", "59a918eff5345fd14ea2e9cff820f9fe": "M = 2^N ", "59a9527bfe7f5cb2ee953ed585fe0261": "x_\\emptyset = \\frac{x_1 + x_{-1}}{2}, \\quad x_{\\varepsilon_1, \\ldots, \\varepsilon_k} = \\frac{x_{\\varepsilon_1, \\ldots, \\varepsilon_k, 1} + x_{\\varepsilon_1, \\ldots, \\varepsilon_k, -1}} {2}, \\quad 1 \\le k < n.", "59a978f976a0ec429c9024947a106728": " 1/\\sqrt{N} ", "59a9b7bfb91fd3c2ff9eff03965c8c6a": " i \\in [t]", "59a9bb2db8118dd2b31bc49bc491134c": "C=C^*", "59a9c6d414918b3a8b79610a021192cb": "P \\in K*P", "59a9cf0ca09bf457dda3e0ed561847d5": "\\mathbf{E_1}(\\mathbf{r},t)=\\mathbf{E}_{01}\\cos(\\mathbf{k_1\\cdot r}-\\omega t + \\epsilon_1)", "59ab9d43422116f4321f3fcb19911ab6": "\\hat{Q}", "59aba3b647ba7564de10633e63073fd6": "S/\\sim", "59abf784b890a0dd2de691228814e4e6": "\\varrho(Z+A)=0", "59ace6fd8ad826784a95a48647ce24dc": "E_s = E_s^{(0)} + \\lambda A_s \\qquad \\mbox{for all}\\; s ", "59ad3f609a83e808d855bcde4c27ae83": "\\min_{\\boldsymbol{w}\\in \\boldsymbol{W}}\\rho(G(\\boldsymbol{w},\\boldsymbol{\\psi}))\\,", "59ade0fcc05738671c5605fcef685aa5": "{i} , {i \\pm 1}", "59ae1b0c3a3892395ca4c8781223b404": "\\frac{X}{Y} \\sim \\beta^{'}(\\alpha,\\beta)", "59ae3a7df470fc9b1158508bda87c6aa": "\\scriptstyle \\frac{\\sqrt{5} - 1 }{3}", "59ae6dec4cfdf63439d478dc39c7a725": "G, L(G), L(L(G)), L(L(L(G))), \\dots.\\ ", "59ae8fb48ca9d81cf75d054139432955": "\\bigl( H^2(G, \\mathbf{C}^\\times) \\bigr)^* \\cong H^2(G, \\mathbf{C}^\\times).", "59aed60b4cff7c5ee85b8a14c7812421": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_{\\text{f}} + \\frac{\\partial \\mathbf{D}} {\\partial t}", "59af144730d5c8910e2ff3249236a2a5": "a=\\frac14", "59af175bc2c6cc60851e2e2ec8800392": " \\ F(a_1, \\ldots, a_n) = \\max(a_1, \\ldots, a_n) ", "59af5b43e5527eff1e0e1a90252683cc": "\n\\overline{X} = \\frac{1}{N} \\sum_{n=1}^{N} X_n \\sim \\mathcal{N}\\left(\\mu, \\frac{1}{N}\\right).\n", "59af5f6c7cd46996c447a30de7f899b2": "E_(el)", "59afa79de0c9eec4a1a76f9b440a22f3": "\\sum_{n=0}^{\\infty}z^{2^n}", "59b02ea0e2e0fa82a5f383630d34b526": "\n (\\mathbf{b}_m\\times\\mathbf{b}_n)\\cdot\\mathbf{b}_s =\n \\varepsilon_{ipq}~\\frac{\\partial x_p}{\\partial q^m}~\\frac{\\partial x_q}{\\partial q^n}~\\frac{\\partial x_i}{\\partial q^s}\n", "59b09d53aa02638520d1715e29fc1784": "\\scriptstyle{\\mathrm{R}^- \\notin \\mathrm{R}^-}", "59b0b5c4c2b7faba680df75258c1b8ea": " \\mathrm{T} = \\mathrm{L} \\times \\mathrm{p} ", "59b0d7fb6820ab85d59a9b2b09550e18": "c - p = S - K e^{-rT}", "59b1088c4419c0e415e38e6523d5c78e": "\\frac{\\frac{1}{3}}{x - 1} + \\frac{-\\frac{1}{3}x - \\frac{2}{3}}{x^2 + x + 1}", "59b115bfb2178bc9a3eef8cc861b99d2": "f(x)=\\left\\{\\begin{matrix} a, & \\mbox{if }x=1 \\\\ a, & \\mbox{if }x=2 \\\\ c, & \\mbox{if }x=3. \\end{matrix}\\right.", "59b13a9515724a8319c0ad5c47331337": "V = (\\oplus_{\\alpha \\in A} S) \\oplus U", "59b18b360dd3d23689de7c8e84ad8c45": "A_{\\rm total} = 4 \\pi r^2", "59b1b5240b0b3ad5456274313e17d743": "V = V_1 \\cup V_2", "59b1c34482491762f44ba6ee753a650b": "T^{\\mathrm{F}}_p(x,y) = \\begin{cases}\n T_{\\mathrm{min}}(x,y) & \\text{if } p = 0 \\\\\n T_{\\mathrm{prod}}(x,y) & \\text{if } p = 1 \\\\\n T_{\\mathrm{Luk}}(x,y) & \\text{if } p = +\\infty \\\\\n \\log_p\\left(1 + \\frac{(p^x - 1)(p^y - 1)}{p - 1}\\right) & \\text{otherwise.}\n\\end{cases}", "59b21d5fed747b16b8ce4abbc31932a3": "\\det(I_n + B A) = \\det(I_p + A B)", "59b27d3d2f3c7d12164b0668e3aab101": "\\int_{-\\infty}^\\infty dE\\, \\int_0^\\infty dt\\, f(E)\\exp(-iEt)", "59b2821087d5125aeddc19aa235434ef": "|P|", "59b34e1ea0b81443000dcca762423126": " \\mathbf{F}=\\mathbf{F}_1+\\mathbf{F}_2 = \\mathbf{B}-\\mathbf{A} + \\mathbf{D}-\\mathbf{A},", "59b36e77d342a0e20d81b15e06912a11": "\\cos \\theta", "59b37ba842b742d05a920058a164dc17": "q = m[i, k] + m[k+1, j] + p_{i-1}*p_k*p_j", "59b380ad318da36936ba1269ee283b55": "\\begin{cases}\\dot{z}_1 &= z_2\\\\\n\\dot{z}_2 &= z_3\\\\\n&\\vdots\\\\\n\\dot{z}_n &= v\\end{cases}", "59b390a5330b2959b33ce78678943cd2": "\\mu_s^o(l,p+\\Pi)=\\mu_s^0(l,p)+\\int_p^{p+\\Pi}\\! V \\, \\mathrm{d}p", "59b3bd08af3b29104d0e34068ad1b4ef": "h\\,\\!", "59b3dbb3209229c10ca06989c3f9a4f5": "\n\\begin{alignat}{2}\n\\Delta V(i) &= &{} -\\tfrac{1}{2}Q_\\mathbf{u} Q_{\\mathbf{u}\\mathbf{u}}^{-1}Q_\\mathbf{u}\\\\\nV_\\mathbf{x}(i) &= Q_\\mathbf{x} & {}- Q_\\mathbf{u} Q_{\\mathbf{u}\\mathbf{u}}^{-1}Q_{\\mathbf{u}\\mathbf{x}}\\\\\nV_{\\mathbf{x}\\mathbf{x}}(i) &= Q_{\\mathbf{x}\\mathbf{x}} &{} - Q_{\\mathbf{x}\\mathbf{u}}Q_{\\mathbf{u}\\mathbf{u}}^{-1}Q_{\\mathbf{u}\\mathbf{x}}.\n\\end{alignat}\n", "59b4211d78c375780163959635f0f987": " P_f^{1-a}=P_m^{1-a}\\theta", "59b465d38ce8ba9011bc7931878a44f4": "\\dot{\\underline{x}}=(\\mathbf{A}-\\mathbf{B}\\mathbf{K})\\underline{x}; ", "59b4752b4daadc6007b19a49738fb08a": "\\Omega(2^{n/2})", "59b4f3f8f5bce9c93ed8056d5e72fe2c": "\\neg \\neg \\exists n\\;f(n)=0 \\rightarrow \\exists n\\;f(n)=0,", "59b514174bffe4ae402b3d63aad79fe0": "images", "59b519473912cc7ba3eff0e0eb6af93f": "\\delta\\omega_{nlm} = \\omega_{nlm} - \\omega_{nl}", "59b51b8e699fd437fb6beeaf6ec8067f": "(\\{H,T\\}, 2^{\\{H, T\\}}, \\mathbb{P})", "59b60c047d435357448a4027fd03cd61": "n\\,\\log_b\\varphi", "59b65f540dde3b91ea59eb6e4af08f41": "q(D,\\widehat{D})\\geq -\\alpha\\,\\!", "59b6eebfaf1ece8a869bac51bf0c3f45": " \\lambda_+ ", "59b6f8ef91ca38c12567f3b751398e2f": "g^{(2)}( \\tau)= \\frac{\\left \\langle I(t)I(t+\\tau) \\right \\rangle}{\\left \\langle I(t) \\right \\rangle^2 }", "59b73a2630172d730b040306b9ff95ec": "r \\le \\min(m,n)", "59b75e0082f412bad0409e1f09298196": " G_{\\mathbb{C}} ", "59b7ae4e575bc25fca1eecd8882126bd": " (b_n)_{n\\geq 1}", "59b84af65b5fe167dcc9deab07b0478d": "x = 0, 0.1, ..., 1", "59b9027f40eab7f2b33ca745ad47db95": "v_{(G; c)}(S)=\\sum \\limits_{e\\in A_s} c(e).", "59b95762342d6267ce6067f00bc51c33": "\\operatorname{E}[|T_N|]\\le\\sum_{n=1}^\\infty\\bigl|\\!\\operatorname{E}[X_n]\\bigr|\\underbrace{\\sum_{i=n}^\\infty\\operatorname{P}(N=i)}_{=\\,\\operatorname{P}(N\\ge n)},", "59b95a7f867ce9d4e0f0b5f86f1260ff": "x^{0}", "59b96095d76fcaa287d7a5491764db84": "(1,1), (0,1)", "59b9d256ea4395a54fb3224e879d7089": "\\bigcap_\\alpha \\overline{B}(x_\\alpha, \\mu r_\\alpha)", "59b9deb6dd27a58c0cfa3d7f86a35e8e": " 0^0 = \\frac{0}{0} ", "59ba052a384586964f34e3f145f023a8": "a_n \\to a", "59ba0fc0ba4f8ed869e814ba5a28059c": "\\,\\!\\Psi_0 = \\Psi[n_0]", "59ba11bb4f19b919571f9a71998ccaa0": "{}_{Ei^{*}}", "59ba55e4888718e3937fab9b0a9062cf": "\\alpha (x) = x^2\\text{ and }\\beta (x) = 1", "59ba79258804d132284fb0620a4c94bd": "\\mathcal{M}_\\text{FG}", "59bac4bebf60dbed56586cf5711cda88": "\\beta_k=\\frac{B_k}{\\sum_i B_i}=\\frac{1}{1+Q_k} ,", "59baf77162be82aefc8eff1da573d423": " P(E) \\, ", "59bb07b1f6d037ea962ba639de259d1f": "\\mathcal{B}(m,n) + O(m + n \\lg \\lg n / \\lg n)", "59bb1beef8a67967803087036d20bbe1": "k \\left( 1 - \\frac{1}{9k} - \\frac{z_{\\alpha/2}}{3\\sqrt{k}}\\right)^3 \\le \\mu \\le (k+1) \\left( 1 - \\frac{1}{9(k+1)} + \\frac{z_{\\alpha/2}}{3\\sqrt{k+1}}\\right)^3, ", "59bb23cf511b30ea96f8c6eaf5f34648": "a = b = c = d, \\alpha = \\zeta = 120 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ", "59bb385a7f366346e0ecfa780aaf5218": "\\lceil \\ldots \\rceil \\!\\,", "59bb3b92f0f34f8a43f91080f6f14fc4": "V'_\\mathrm{E}=V_\\mathrm{E}- \\frac{2 \\pi l^2}{3} \\bigg[ \\frac{3 \\left(D+d \\right)}{2} -l \\bigg]", "59bb5e1e09382fa3102b15b20dd20884": "\\mathbf{y} = (y_1,\\dots,y_N) \\in [q^n]^N", "59bb8828f98cba2f8f66dc65c0b39cd2": "F \\to R", "59bba05fcc4142c2b5b77231b651be8c": "\\mathbf{\\mathit{\\rho}}_{final}", "59bc424725e178d122b1876f98e0e68a": "b_n = O(\\beta^n)", "59bc535d54fd065908ad1310610411bc": "Q_s/Q_t = (Cc_{O_2} - Ca_{O_2}) / (Cc_{O_2} - Cv_{O_2})", "59bc67374a259632da7e94038a068c73": "F(f):F(Y_0)\\to F(Y_1)", "59bd51693e5cd31d2d84e6b4c3d40cef": "\\delta(p,a,\\overline{\\gamma})", "59bd7580dafc7bea1bd4a349e04eae2b": "\\mid C \\mid \\le qnd \\cdot {{q^n} \\over {Vol_q(0,e)}} \\le q^{n(1-H_q(J_q(\\delta))+o(1))}", "59bd94c8c699e015fd75652f3a32484d": " h(X|y) = -\\int_X f(x|y) \\log f(x|y) \\,dx ", "59bdb5af2f4bfe70834701a714b8bc4d": "\\begin{align} \\cos \\alpha & = & {1 \\over \\sqrt{1 + \\tan^2 \\alpha}} \\\\ \\sin \\alpha & = & {{\\tan \\alpha} \\over \\sqrt{1 + \\tan^2 \\alpha}} \\end{align} ", "59bdbe00ba57ccf13487f75d683fd73e": " T = \\sqrt{5}:2, \\ ", "59bdf0ba696e13164c5a926386f23cb0": "f_i", "59beeb22868903bb69ab8de45d91a0d9": "\\phi_{\\mathbf{R}}(\\mathbf{r}) = \\phi_{\\mathbf{R}+\\mathbf{R}'}(\\mathbf{r}+\\mathbf{R}')", "59bf05b14bf6d19412f76434f184b530": "\\mathit{E}_{in}", "59bf15e6e9cc438043f848455bc99d39": "T(a) \\colon F(a) \\to G(a)", "59bf3cb51b9f4f441d7f15bd5439ac62": "f(x)=g(x)", "59bf3e9b3981b93e05f8283fd4485e1c": "(Q, +)", "59bf4e702c0fb63c2249406250a36f2a": "I_{sp}(vac)", "59bfd65b278efd5231be544a2a4c28b4": "f(x_1,...,x_n)", "59c00bcc65ec633966f5eec26360c06b": "\\frac{\\tfrac{3}{2}}5=\\tfrac{3}{2}\\times\\tfrac{1}{5}=\\tfrac{3}{10}", "59c043b06ee2ecf4f553f6ef61ef07b3": "\\gamma_{\\nu,\\lambda}", "59c06dd6f9c71cbb817a1cca1b6a1f2a": "\\ln 2 = \\frac{2}{3} \\sum_{k\\ge 0} \\frac{1}{(2k+1)9^k}.", "59c094d3e1f5bdf4d181f5534d69ae54": " \t E_{2n} = (-1)^{n-1} (2n-1)! \\sum_{0 \\leq k_1, \\ldots, k_n \\leq 2n-1}\n\t \\left( \\begin{array} {c} K \\\\ k_1, \\ldots , k_n \\end{array} \\right)\n\t\\delta_{2n-1,\\sum (2m-1)k_m } \\left( \\frac{-1~}{1!} \\right)^{k_1} \\left( \\frac{1}{3!} \\right)^{k_2}\n\t \\cdots \\left( \\frac{(-1)^n}{(2n-1)!} \\right)^{k_n} , ", "59c117f2c4c1e6d28b6c9488bf59c53e": "\\tbinom{n}{0} + \\tbinom{n}{1} + \\tbinom{n}{2} + \\cdots + \\tbinom{n}{n} = 2^n", "59c118dda53026bf9b4b1740dbe4a6d5": "\\mathbf{w}_{new}=\\mathbf{w}_{old}-\\eta\\operatorname{E}[\\mathbf{z}(\\mathbf{w}_{old}^T \\mathbf{z})^3 ].", "59c142168fc6600eff336e2b155c6f2a": "\\varphi = \\frac{1 + \\sqrt{5}}{2} = 1.61803\\,39887\\dots", "59c14f3ed28af7dc2c694aa27138cb76": "{\\tau_{\\rm pb}}", "59c1d0a5e181729740bae174b79a1cb5": "0.0404482", "59c1e7f7b3f403daf977f9e53956d680": "E[(X(t) - E[X(t)])^4] = (3 \\sigma^4 \\nu + 12 \\sigma^2 \\theta^2 \\nu^2 + 6 \\theta^4 \\nu^3)t + (3 \\sigma^4 + 6 \\sigma^2 \\theta^2 \\nu + 3 \\theta^4 \\nu^2)t^2", "59c20944eb7deec67ee9e1c97d367231": "\\left(\\frac{1}{y}\\right)\\left(\\frac{1}{x}\\right)", "59c23cde94eb9e32f72406ee25e4037b": "64 \\times {4 \\choose 1} = 256", "59c2a3fd55fbc272c5b25797e7e47e01": "\\hat{\\textbf{x}}_{k+1\\mid k} = \\textbf{(F}_{k}-\\textbf{K}_{k}\\textbf{H}_{k})\\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_{k} \\textbf{z}_{k} ", "59c2fbf17ebd0facc1a52b936cb1daed": "\\, E_A", "59c313928bdb0ed146b7b720e0228cf4": "\\alpha \\approx 1.8", "59c334e82a54fba6b82528838f48410b": "\\{x'_1, ..., x'_n\\}", "59c3ece4ac8bde561415868eac410a57": "G = \\bigcup_{w\\in W} BwB", "59c3f981d2e92c4f8c18e791c56f770f": "\\begin{align}\np(\\mu|D, \\sigma^2, I) \\sim & N(\\bar{x}, \\sigma^2/n) \\\\\np(\\sigma^2 | D, I) \\sim & \\operatorname{Scale-inv-}\\chi^2(\\nu, s^2)\n\\end{align}", "59c402d274cd00bac775e755cbaf86e3": " F_T = GMu\\frac{d^2-(d-r)^2}{d^2(d-r)^2}", "59c40d520a398388900937d2d36c6cab": " \\frac{P(x)}{Q(x)}", "59c4188edfaf270a3e9ae91aa8bbbc4c": "T\\left(\\frac{\\partial V}{\\partial T} \\right)_P", "59c4c106a4aed8afa1938c11eb53d898": "P(Y=y \\mid X=x) P(X=x) = P(X=x\\ \\cap Y=y) = P(X=x \\mid Y=y)P(Y=y).", "59c4e42d139511b6fdc8b48d23acf600": "a, c, F, d", "59c4fc2caa341963cd412f384042fc2d": "\\mathbf{Ma}", "59c500f5a195f42741144b7c61e2f09e": "|\\pi(x)-{\\rm li}(x)|<\\frac{\\sqrt x\\,\\ln x}{8\\pi}", "59c50bf8cb0760a341e29a8e0e0dd197": "\\coprod u_\\alpha", "59c545a862ed2a4691fc09d9e6ac5ec4": "\\frac{1}{x_n-x_{n-1}}k_{n-1}\\ +\\frac{2}{x_n-x_{n-1}}k_n = 3\\ \\frac{y_n-y_{n-1}}{(x_n-x_{n-1})^2}.", "59c59e4a872a4f8b5b10a0b219a2328f": "\\Pr(N_p = n) = (1 - p)^{n-1}\\,p\\, .", "59c5b020619040921fac6346fc3d06d0": " 4AB- E^2 =0 \\,", "59c5f3fbc4b8aefec4dd69e53f961aa3": "D_{t-1}", "59c62aa140b032821a909f0c85ee0482": "\n\\gamma_\\mu^*(t) = \\frac{1}{2}\\sum_{i=1}^n \\mu_i(t) \\tau_{ii}^\\mu(t) \\geq h > 0, \n", "59c64de85fb24a67af17358d2fac8494": "x(t) = 2pt + h; \\ \\ y(t) = pt^2 + k \\, ", "59c67204d770baddcc9544c396a69e7e": "d\\# f", "59c690059009710ceffd1fe2460e185e": "f^*\\omega = \\sum_{i_1 < \\cdots < i_k} \\sum_{j_1 < \\cdots < j_k} (\\omega_{i_1\\cdots i_k}\\circ f)\\frac{\\partial(f_{i_1}, \\ldots, f_{i_k})}{\\partial(x_{j_1}, \\ldots, x_{j_k})}dx_{j_1} \\wedge \\cdots \\wedge dx_{j_k}.", "59c6b8b965a6a27c9421fc31154c2552": " \\sigma_p^2 = \\sum_i w_i^2 \\sigma_{i}^2 + \\sum_i \\sum_{j \\neq i} w_i w_j \\sigma_i \\sigma_j \\rho_{ij}, ", "59c7659018ca54ddb03acad85532a503": "Y=AK\\,", "59c7ac30347183b45d0d3dfa9a939471": "\\mathbb{CFM}_\\mathbb{N}(R)\\cong\\mathbb{CFM}_\\mathbb{N}(R)^2", "59c7b1fbf2cc6bc234da1ca64d62dad3": " \n\\left[\n\\begin{array}{cc}\n\\frac{E^{(e)}A^{(e)}}{L^{(e)}} & -\\frac{E^{(e)}A^{(e)}}{L^{(e)}} \\\\\n-\\frac{E^{(e)}A^{(e)}}{L^{(e)}} & \\frac{E^{(e)}A^{(e)}}{L^{(e)}} \\\\\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\nu^{(e)}_1 \\\\\nu^{(e)}_2\n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array}{c}\n\\int\\limits_{0}^{L^{(e)}} n N_1^{(e)}(x)dx \\\\\n\\int\\limits_{0}^{L^{(e)}} n N_2^{(e)}(x)dx\n\\end{array}\n\\right]\n", "59c7e57b1b8487d61fa683dff5cc7eae": " \\, k = 1, \\ldots, N .", "59c87dd7eb9c85caeeeb0a68ae47b3e3": "\n\\frac{ (100- \\text{Clean price})/ \\text{Years to maturity} }{\\text{Clean price}}*100.\n", "59c8c35357d390054afa2e02228afc7a": "b-\\omega(a_0)=a'_{1} p^1 + a'_{2} p^2 + ...", "59c8d84189665bda8b561222161bdb2e": "\n\\Phi(z,s,a)=\\frac{1}{2a^s}+\n\\frac{1}{z^a}\\sum_{k=1}^\\infty\n\\frac{e^{-2\\pi i(k-1)a}\\Gamma(1-s,a(-2\\pi i(k-1)-\\log(z)))}\n {(-2\\pi i(k-1)-\\log(z))^{1-s}}+\n\\frac{e^{2\\pi ika}\\Gamma(1-s,a(2\\pi ik-\\log(z)))}{(2\\pi ik-\\log(z))^{1-s}}\n", "59c8ebcb04287618d49312966d5a7167": "\ny = \\frac{1}{2}\\left(1 \\pm \\sqrt{4z + 1}\\right)\\,\n", "59c8f8d26f4c323a958c9df0fdf4a9fc": "\\begin{align} \nS(Z,X,JY)+S(JZ,X,Y)&=-\\frac12\\langle JX, \\big(-(\\nabla_{JY}J)Z-(J\\nabla_ZJ)Y+(J\\nabla_YJ)Z+(\\nabla_{JZ}J)Y\\big)\\rangle\\\\\n&=-\\frac12\\langle JX, Re\\big((1-iJ)[(1+iJ)Y,(1+iJ)Z]\\big)\\rangle.\\end{align}", "59c90e959fd21833acd7c37bdfd851c3": "25/8=3.125", "59c92cbcbf6c58b552532a4194266a2d": "\\mu_{roll} \\,\\!", "59c974ccf7f8c27839b67b4d28816424": "\\mathrm{d} Y_{t} = { \\mathrm{d} t \\choose \\mathrm{d} B_{t} } ,", "59c9c3222e1f3af349f2afe85137f80a": "\\mathfrak S_{2k}", "59c9e8927382275d5794d62b2acf1337": " Q_{m} = \\beta_{m} = \\frac{\\partial S(\\bold{q},\\boldsymbol\\alpha, t)}{\\partial \\alpha_{m}}. ", "59c9f5c134f962ef9a04f54ce34eeb91": " _H = \\sum_{i=1}^\\infty \\frac{_{L_2}_{L_2}}{\\sigma_i} ", "59ca2d6ee4f9fde1dec0b5aaa22b0d3f": "\\pi_*", "59ca351572873932cecf12e95c185507": "\\pi_Y", "59ca5f281fe9b5c07a17706876fc806e": "\\left\\{e^{\\frac{2 \\pi i}{4}},e^{-\\frac{2 \\pi i}{4}}\\right\\}=\\left\\{\\pm\\sqrt{-1} \\right\\}=\\left\\{+i, -i \\right\\}.", "59ca7af2dce9dc36043245156b7a6d42": "{n \\choose i}=\\frac{n!}{i!(n-i)!},", "59ca8c3294950af14af7520fe6a31270": "b+d+u=1\\,\\!", "59caf030d69c9199051bb3ce7adb966f": "\n\\operatorname{Var}\\left[\\hat\\theta\\right] \\, \\geq \\, \\frac{1}{\\mathcal{I}\\left(\\theta\\right)}.\n", "59cafcd156f46c01596598479de57817": " \\int_0^1\\log\\Gamma_q(x)dx=\\frac{\\zeta(2)}{\\log q}+\\log\\sqrt{\\frac{q-1}{\\sqrt[6]{q}}}+\\log(q^{-1};q^{-1})_\\infty \\quad(q>1). ", "59cb20de564423ad3945097c629bbfda": "\n\\frac{1}{r} = A + B \\cos \\left( \\frac{\\theta_2}{k} \\right).\n", "59cb6634a7b8802575944508075de06b": "P_{2n}+P_{2n+1}", "59cb85a5a9021b2e0944d0334614bdf4": "f(P)=\\frac{ \\partial f }{ \\partial x }(P)=\\frac{ \\partial f }{ \\partial y }(P)=0.", "59cb879d01956f575da2f8f82d69f94b": "\\scriptstyle v = \\frac {dw}{d\\tau}", "59cbd1f3a702bb97a761f8b7b76fedea": " {\\Delta P} = \\frac{8 \\mu l {\\dot V}}{ \\pi r^4} ", "59cc72958054cecd087f019a3eef2418": "f(X) = \\sum_{x\\in X}f(x)", "59ccdc21c1b4c9282ac51192a8497e61": "\\vec \\mu_L= \\vec L g_L \\mu_B", "59ccfcf71abf4cf4ea90e188bdc87e51": "\\hat{\\textbf{y}}_{k\\mid N} = \\textbf{z}_{k} - \\textbf{R}_{k}\\beta_{k} ", "59cd00aeecbe6c9b69ada4344c626949": "t\\ge 0\\,.", "59cd923fd21a7bd5937df121792005bc": "\\text{Asymmetrical short-circuit current = symmetrical current * X/R factor }", "59cdfb4014c35b6641d10b9fd4d40fac": "x\\in \\mathbb{Z}_q^n", "59ce4b7e259f5320b9f2fa50f4b6be3e": "\\begin{matrix} (1,2,3) & (1,3,2) \\\\ (2,1,3) & (2,3,1) \\\\ (3,1,2) & (3,2,1) \\end{matrix}", "59cf7344a790345db91b37140fb3484a": "\\hat{A}", "59cfbbfdeabdd250f784fb82c205a4d5": "E_r=(-1/8m^3c^2)\\langle nlm|p^4|nlm \\rangle", "59cfcb5da0c7e5e2767b7d8ba350c1ad": "\n y_t = \\alpha + \\beta x_t^* + \\varepsilon_t\\,, \\quad t=1,\\ldots,T,\n ", "59cfcf4def3c722ae78fc44bed73e21a": "f_\\text{interpolated}(x) = y_\\text{lower} h_{00}(t) + h m_\\text{lower} h_{10}(t) + y_\\text{upper} h_{01}(t) + h m_\\text{upper}h_{11}(t)", "59d006f052f85ef527ef9eba07f5a6a2": "A_S(t)", "59d07f20139f609ceb1734b7ba96802a": "\\tau_0+\\Delta\\tau", "59d0997a2ef69cd555f52d082a87b432": "F=\\{m \\setminus L \\,\\vert\\; m\\in L\\}", "59d1444742d62d834dfcf14d16247e89": "u_{22}", "59d14df1095b1d5aaefa4679328b36f1": " A(x_1,\\ldots,x_{2N})=(a_1,\\ldots,a_{2N}) ", "59d151fda0ccc0bb7afb8abce11054e3": "\\alpha = 4p(1-p).\\!", "59d17d09d6c746060271cb9003ce3072": "I_x, I_y", "59d18fb5a3353912b0e7471a10563c1e": "c_f(p) = \\min\\{c_f(u,v) : (u,v) \\in p\\}", "59d2b9ef1027afe24c4fbdb4e1ebcdb7": "\\mathrm{d} X_{t} = - \\nabla \\Psi (X_{t}) \\, \\mathrm{d} t + \\sqrt{2 \\beta^{-1}} \\, \\mathrm{d} B_{t},", "59d322a617495214c2e715645c0d50bd": "\\begin{align}\n \\Pi\\colon \\mathcal{C} &\\to \\mathbb{R}^3 \\\\\n \\tau &\\mapsto \\left\\{\\quad\\begin{matrix}x^1=(a+b\\cos(n\\cdot \\tau))\\cos(m\\cdot \\tau)\\\\x^2=(a+b\\cos(n\\cdot \\tau))\\sin(m\\cdot \\tau)\\\\x^3=b\\sin(n\\cdot \\tau)\\end{matrix}\\right.\n\\end{align}", "59d3a98237c1ee85d51dc87703937240": "b = b", "59d3e27a7b05086741f74a20560dbb55": "\\textrm{Ann}(M)", "59d3e4ec61e86fbf15a1544a7d7457bc": "\nR_{1} \\in [1.846 \\Omega,2.307 \\Omega]\n", "59d453150cad482f9fe151e4c4edcaa7": "\\tan[\\arcsin (x)]=\\frac{x}{\\sqrt{1 - x^2}}", "59d458a85404dcf3b9a847586edfce7d": "N_{\\rm covering}(\\varepsilon)", "59d460841d0d0a0722ec08ee9c1d4bc3": " E\\tau", "59d47d473f161e62b13c9289306c4811": "k_{LE}", "59d48272d75cfef6402cd81eed68ae3b": "G(h, F) = g\\int M^{1/2}\\mbox{d}x, \\qquad\\qquad (4)", "59d49a4bb6d4cab62597cca7023b93c6": "L^1(G)", "59d4aac6ab198ee58f6275e4b16e139e": "U_h", "59d4f33461df7d9a34a10b4d7d25199b": "\\ T_d", "59d4f3cd298f18cd2d7e86387b9a6621": "\\langle 1, n \\rangle", "59d52d35f5db7bcdbef92883a8f65edb": "\\mathbb Z_4^6 \\times \\mathbb Z_3 \\wr \\mathrm S_8 \\times \\mathbb Z_2\\wr \\mathrm S_{12}.", "59d53ec145ac22aa571ffe079d254fd3": "D \\approx XY^T", "59d58f2993036dd66525e74c36870472": "M_v = U \\begin{bmatrix} \\Sigma \\\\ 0 \\end{bmatrix} V^T .", "59d591ee369cb2cb35fdd8fa0b085461": " a(x,\\xi) ", "59d5aff897d21cc51985f317c8fcf77f": " N_{i}^{e}=\\frac{1}{2}\\eta\\left(1\\mp\\zeta\\right),\\qquad i=2,5 ", "59d5beb6418bc7fdfc154ef3f0806183": "0 - 5", "59d5fa55072ea8b91a94b907f9b0f4ef": " \\lim_{t \\to 0} \\frac{1}{t} \\int_0^t \\mathbf{F}(\\mathbf{u}(s)) \\cdot \\mathbf{u}'(s) ds = \\frac{d}{dt} \\int_0^t \\mathbf{F}(\\mathbf{x} + s\\mathbf{v}) \\cdot \\mathbf{v} ds \\bigg|_{t=0} = \\mathbf{F}(\\mathbf{x}) \\cdot \\mathbf{v} ", "59d6660902cff67c62c35ce166847fd4": "\\frac{p^{q} - 1}{p - 1} \\text{ divides } \\frac{q^{p} - 1}{q - 1}.", "59d68c5fe91dfe17a30a3ed59a73fb7f": "X_{i+1}\\cup\\beta(X_{i+1})", "59d68f4b83dec6f052e92aaa4861bc71": "NX=NX (e, Y, Y*)", "59d6b5b14acee42a2b941f2082b6b322": " \\vec{M} = \\vec{H_k} \\chi ", "59d6f9f0fa1177c86f09409f451c6c86": "\\mathcal{N}( y(\\tau) ) \\approx \\mathcal{N}( y(0) )", "59d7a8966ca494371460d7522587251c": "r_k=\\sum_{j\\ne k}\\big|a_{j,k}\\big|", "59d7d84a64e4205cb4629dddfdc4f965": "\\frac{(6V)^{1/6}}{\\sqrt{\\pi}}", "59d7da06b44e6200575fa7d923b77512": "E_0 = \\mbox{Sup}|E(t,f)|,\\,(t,f)\\in \\mathbb{R}^2", "59d7f0bb44b76842a242d3d8914f67e7": "\\{ x_k \\}", "59d822848eff5403d599e04504758df5": "\nL \\propto \\sigma^ \\gamma\n", "59d885d3ac4fb3ac3a64fd81eae94940": "\n(J^\\alpha) (J^\\beta f)(x) = \\frac{1}{\\Gamma(\\alpha) \\Gamma(\\beta)} \\int_0^x (x-s)^{\\alpha + \\beta - 1} f(s) \\left( \\int_0^1 (1-r)^{\\alpha-1} r^{\\beta-1} \\; dr \\right) ds\n", "59d88ac9411976bea5cd563fcf883bfd": "c = a \\,\\bmod\\, n. \\, ", "59d8a7f3000a1bcbdccdca37181bbf16": "\\left[ C \\right]=\\left\\{ \\begin{matrix}\n \\left[ A \\right]_{0}\\left( 1+\\frac{k_{1}e^{-k_{2}t}-k_{2}e^{-k_{1}t}}{k_{2}-k_{1}} \\right);\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,k_{1}\\ne k_{2} \\\\\n \\left[ A \\right]_{0}\\left( 1-e^{-k_{1}t}-k_{1}te^{-k_{1}t} \\right);\\,\\,\\,\\,\\,\\,\\text{otherwise} \\\\\n\\end{matrix} \\right.", "59d8c2e685f70e42564954f6c3431df4": "\\bigcup F_k", "59d8ca14d4feec51b90700ac34b9fcff": "K(f) \\colon K(A) \\to K(B)", "59d8d8903d18029ac977610252d55077": "f(x)=\\sum_{n=0}^\\infty F_n R_n(x)", "59d91b7d1b0406353107cab6f61aa4a2": "[x:y:z],", "59d91ea9ddae3ae4dc9037af7a77e9bf": "\\displaystyle{c=D_c(Q(a)a^{-1})=2Q(a,c)a^{-1} +Q(a)D_c(a^{-1}),}", "59d926908955a87e8ca44374c2dad8ab": "z(1-z)\\frac {d^2f}{dz^2} + \n\\left[c-(a+b+1)z \\right] \\frac {df}{dz} - abf = 0.", "59d93ae10cf23a8faad62488283d2d03": "b_1^2 - 4 b_2 b_0 < 0", "59d93c35e051640ecd9dc7be6bdaee92": "\\operatorname{ht}\\mathfrak{p} = \\operatorname{dim}R_{\\mathfrak{p}}", "59d97ac4294b68376e570c68696b1006": "\n\\mathbf F=\\mathbf F_\\parallel+\\mathbf F_\\perp", "59d9c9727a2c095bb5bd776e4187b67e": "\\begin{matrix}\n \\frac{1}{2} + x \\Gamma \\left( \\frac{\\nu+1}{2} \\right) \\times\\\\[0.5em]\n \\frac{\\,_2F_1 \\left ( \\frac{1}{2},\\frac{\\nu+1}{2};\\frac{3}{2};\n -\\frac{x^2}{\\nu} \\right)}\n {\\sqrt{\\pi\\nu}\\,\\Gamma \\left(\\frac{\\nu}{2}\\right)}\n \\end{matrix}", "59daa599049d079a882afada4ce84a2d": "b{\\in}B", "59dab6281820cc169c6cfc1e38e7e1bc": "|\\Psi _{BETA}(\\omega |\\alpha ,\\beta )|=|\\Psi _{BETA}(\\omega |\\beta ,\\alpha )|.", "59dac5eb9ed124d368cc93b036daf842": "R_j/r_j", "59db2f43a4407cfd2ae3d3a6c21e4547": "Z^a", "59db317963037c1004c1873d39ed6c45": "m_{\\mathrm{eff}} = m", "59db6135c46aed40d522d50626b6f8a8": "x=\\frac{-B\\pm\\sqrt{B^2-4AC}}{2A}", "59db98506c44ef5bd5ccd9b9ce99797e": "\\mbox{curry}(g) :X\\to [Y\\to Z]", "59dbfe2cae5caec695026e220abf90b2": "\\mu\\boxplus_c\\nu", "59dc007e750558e1a55875f03a107881": "\n \\frac{d}{dx}k^\\epsilon(x)\\frac{du_\\epsilon}{dx} = - \\phi^{\\epsilon}(x), 0 < x < 2 \\quad (2)\n", "59dc5761f7e2bec07320973f182d9be7": "\\log \\Gamma \\left(z\\right) = -\\gamma z -\\log z \n+ \\sum_{n=1}^\\infty \n\\left(\\frac{z}{n} - \\log \\left(1+\\frac{z}{n}\\right)\\right),", "59dd8313b31a04cd35577b592652dcd0": "Q(i,j,k,m) \\equiv d(c_i,c_j) < \\frac{m}{k+1}", "59dddbd0ba0bcdda3cc79394f20f36df": "RPM = {100 ft/min \\over \\pi \\times 10 \\, inches \\left ( \\frac{1 ft}{12 \\, inches} \\right )} = {100 \\over 2.62} = 38.2 revs/min", "59de012d598d085957e5630bb2359829": "0 \\le \\ a_n \\le \\ b_n", "59de0cd453e633fdaa0147ce03e9f3bb": "\\sqrt[3]{8^3} = \\begin{cases} \\ \\ 8 \\\\ -4+4i\\sqrt{3} \\\\ -4-4i\\sqrt{3}. \\end{cases} ", "59dea0c301cd9bce8fd82bbe2be6d143": "\\max\\left(1+\\left|\\frac{a_0}{a_n}\\right|,1+\\left|\\frac{a_1}{a_n}\\right|,\\dots 1+\\left|\\frac{a_{n-1}}{a_n}\\right|\\right)", "59deacbfcf8a17f894da56d6f8fd95af": "\n\\begin{bmatrix}\nK_{11,reduced}\\end{bmatrix}\\begin{bmatrix}\nx_{1}\n\\end{bmatrix}=\\begin{bmatrix}\nF_{1} \n\\end{bmatrix}\n", "59decd627c83f2750070d114a3ea8e84": "C_p = \\begin{cases}\n \\tan \\frac{\\pi}{2p} & \\text{for } 1 < p \\leq 2\\\\ \n \\cot \\frac{\\pi}{2p} & \\text{for } 2 < p < \\infty\n\\end{cases}", "59df1e3b547330b5ef5cc4747077fdae": "\\, t - \\varphi ", "59df23ff385495af683ca7e293c61f14": "N = (a_0 + a_1 + \\ldots + a_{n-1})^2", "59df62b117ccad48c19f27a5207ee061": " f(v_1,\\ldots, v_k)=0", "59df65515335518ec1c8861043a9a74c": "l_e = m_0\\omega_e r_e^2 = \\hbar/2, \\ ", "59df6924bc931d8d6a402c60d09daf66": "\\{x \\in A \\mid \\varphi(x)\\}", "59df72eda473cdb1341230b3d968e124": "\\coth x = \\frac{\\cosh x}{\\sinh x} = \\frac {e^x + e^{-x}} {e^x - e^{-x}} = \\frac{e^{2x} + 1} {e^{2x} - 1} = \\frac{1 + e^{-2x}} {1 - e^{-2x}}", "59dfe5543eb13082d7b9ec885ba0790a": "\\textstyle C=C\\left( \\Omega\\right) ", "59e05b42af4e28406b735739423ae6f6": "\\psi(P^k) = \\sum_{i+j=k} P^i \\otimes P^j", "59e06f631b0ea172617718292aebdbbc": "{fp={ip-bp}}", "59e09817157bb89a2e2682a32e5a234b": "\n\\begin{bmatrix}\n \\mathbf{b}_x \\\\\n \\mathbf{b}_y \\\\\n 0 \\\\\n\\end{bmatrix}=\n\\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & 1 & 0 \\\\\n 0 & 0 & 0 \\\\\n\\end{bmatrix}\\begin{bmatrix}\n \\mathbf{c}_x \\\\\n \\mathbf{c}_y \\\\\n \\mathbf{c}_z \\\\\n\\end{bmatrix}\n", "59e0aa235cb192f7c4f774bbb1c5cd07": " \\frac{n^2 - 1}{n^2 + 2} = \\frac{4 \\pi}{3} N \\alpha, ", "59e0fa039901874400053dc63d320811": " \\max\\{\\rho(X),0\\}\\geq S\\rho(Y)-kL, \\, ", "59e0febe5bef1d74c9ab5ec6454a1f23": " F_i = V_i = 0 ", "59e1213bf632189da03ce72518fb813c": "\\begin{array}{cc}\n \\begin{array}{r} \\\\ 3 \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & -12 & 0 & -42 \\\\\n & & & \\\\\n \\hline \n \\end{array}\n\\end{array}", "59e1d835709916faae79cf657edd8136": "(a,b)\\in\\mathbb{F}_{q}^2", "59e222a84f6c3fc7d28e218bc6d2e848": "0 < \\epsilon < \\frac{1}{2} - p", "59e26d51b76d026f2aa78eccbdc3f128": "L = \\pm|\\mathbf{p}||\\mathbf{r}_{\\perp}|", "59e27effa674cf3ab3bc1b4c1a0f9e21": " d \\star d h = -h_{yy} \\, dy \\wedge dx + h_{xx} \\, dx \\wedge dy = \\left( h_{xx} + h_{yy} \\right) \\, dx \\wedge dy ", "59e2a2312f6c0f1cc0c1aa63e93e9759": "f(x) = \\sum_{n=0}^\\infty a_n \\exp(-\\lambda_n x)", "59e2b216c3051557ab30d5cd7cd60d96": "\\begin{align}\n\\langle n|aa^{\\dagger}|n\\rangle&=\\langle n|\\left([a,a^{\\dagger}]+a^{\\dagger}a\\right)|n\\rangle=\\langle n|\\left(N+1\\right)|n\\rangle=n+1\\\\\\Rightarrow a^{\\dagger}|n\\rangle&=\\sqrt{n+1}|n+1\\rangle\\\\\\Rightarrow|n\\rangle&=\\frac{a^{\\dagger}}{\\sqrt{n}}|n-1\\rangle=\\frac{\\left(a^{\\dagger}\\right)^{2}}{\\sqrt{n(n-1)}}|n-2\\rangle=\\cdots=\\frac{\\left(a^{\\dagger}\\right)^{n}}{\\sqrt{n!}}|0\\rangle.\n\\end{align}", "59e2f656f3f7f7e8dea218f5ce639b7c": " \\left(\\frac{D_1}{D_2}\\right)^5", "59e333c4535c6c01b2e909d2cf1e34e2": "x\\in [-a,a]", "59e37a658e37433745ebb880b3f22ee6": "\\frac{3h}{5}", "59e3b2986324ebcfb00772c6a9bc9c73": "\\omega_2 = \\sqrt{\\frac{3 k}{m}}.", "59e44faf74791c4c5d43556129edbe8a": " M^2(B)=\\lambda|B|+(\\lambda|B|)^2. ", "59e46a9ea35aa9c928ad76ce7bb371ea": "\\textstyle I(\\mathbf{q}) \\sim \\left | \\phi(\\mathbf{q}) \\right |^2", "59e4b7e31ccd3b6275a558f5b944ad86": " p+q>d", "59e4e7c174a4d815f90b103c2b441425": " \\dot{x} = f(t, q)", "59e513838dacb18ba03defb353f096f2": " W = \\Delta T = \\int_C \\left ( \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{r} + \\boldsymbol{\\tau} \\cdot \\mathbf{n} {\\mathrm{d} \\theta} \\right ) \\,\\!", "59e5144b3b61071dcf071d3221022ad4": "K_{\\mathcal{P}_n(S)}", "59e5d14f8e2a48e624e62ea527ac5211": " INT\\_MIN = x + y ", "59e637a371f697dfe881432aa511afe5": "\\operatorname{erf}\\left(\\frac{z}{\\sqrt{2}}\\right)", "59e6924ab373ef7940fef2660088f96f": "\\ Pv = R_{\\rm specific}T ", "59e6ba77a720a5b7eb741e9868379454": " (H^\\circ_{298} - G^\\circ_T) / T = S^\\circ_T - (H_T - H_{298}) / T ", "59e6ed269715680d8e7a3d3fe05b94d5": "\\sqrt{2}.", "59e6ef4b4bdb149d092edbcaf78f480f": "{\\mathbf T}^n", "59e6f0e64f987242c475c46f53c24b3b": "G=\\mathrm{SU}(2)", "59e6f51f2b5d46b65fc169d0372d65ae": "KE_{fly}=\\frac{\\eta_{fly} mv^2} {2} ", "59e75f117cd2c3dd90dcef56f9c3b943": "\\{U_\\alpha\\}_{\\alpha\\in A}", "59e786a1c22d635f0cff1a12ccb47d76": "-dU=\\delta Q + \\delta W\\,", "59e7bf5ea790c4ff50d7369d6d09af50": "\\sum_{i=1}^n p_i = 1", "59e7cfb637f7f5b25a4bbbaba21cb671": "G_f(V,E_f)", "59e7e00733535ff931901fc15061b851": " \\operatorname{Or}(V)\\otimes \\operatorname{Vol}(V) = \\bigwedge^n V^*, \\quad \\operatorname{Vol}(V) = \\operatorname{Or}(V)\\otimes \\bigwedge^n V^*. ", "59e7f9a7413fb4c62412e2463e239a3e": " b^0 = 1\\!\\, ", "59e81a3cb73c997622a76d419a8e5ad1": "= \\gamma^0 \\gamma^{\\mu n} \\dots \\gamma^{\\mu 2} \\gamma^{\\mu 1} \\gamma^0", "59e8211ba92a6a3fec35bbfd2a3f1966": "P= A_K(t) K", "59e8216361d30d6e54df78870b2c8509": "\\psi_{n_2, k_2}", "59e83350d0115b3c9856c115c7abc1c8": "B(R) = \\{z \\in \\mathbb{R}^{2n} | \\|z \\| < R \\}, ", "59e8ebee1af37936fca6dda34c07ce9f": "k_1,", "59e8f4193488de0760711268eca92c4c": "\\sin((n + 1/2)x).\\!", "59e90425ab4a030c9b42f21afc98ef2f": "x^3 = px + q \\,", "59e91eb3221d1fbcc7389db48ca040b5": " n_{\\infty} \\cdot n_{o} = -1 ", "59e92810b0cbe7e616f81704f1e1fb77": " X_t = c + \\sum_{i=1}^p \\varphi_i X_{t-i}+ \\varepsilon^*_t \\,.", "59e92a6709a4d6c34e96c2a54c6630c8": "(u,v) = (x,y) \\begin{pmatrix}1 & 1 \\\\1 & -1\\end{pmatrix} = (x,y) S .", "59e96cdc42eae9eb2c3e8341e726bdd9": " { d p_{\\alpha} \\over { d t } } = q \\, F_{\\alpha \\beta} \\, \\frac{d x^\\beta}{d t} \\,", "59e974eb21c0eac12a323343d6df45c3": " \\mathbf{MTF_{display}(\\xi,\\eta)} ", "59e97647cc3570747febc9580c627a77": "\\bigvee_{i 0", "59ef9d888eba94e15bff3ca6721ae470": "A\\cdot A = B\\cdot B = \\rho^2", "59effdbb5be7051e375e4a2b6cee0f71": "L=\\varinjlim_{i\\in I}L_i", "59f00060e4218be8bb4e0ea0dd9dbe86": "\\alpha _i \\ge 0,{\\rm }\\beta _i \\ge 0\\;(i = 1, \\ldots ,N)", "59f08ec913e57587759308e6d9eba22b": "\\hat{N}=\\frac{k+1}{k} m = m + \\frac{m}{k}", "59f0b3f6eee0bc7ac8582ab113169bc5": " \\tfrac{\\sigma(30)}{30} = \\tfrac{1+2+3+5+6+10+15+30}{30} =\\tfrac{72}{30} = \\tfrac{12}{5}", "59f0b5f3efab857a1114b7fc4360f79a": "\\lnot\\phi \\to \\left( \\phi \\to \\psi \\right) ", "59f0c4ca1dae6c5c7af38948fefdfa03": "\\boldsymbol{r}(t)", "59f0fde8d2672113abdfa90d4c65458e": "F_v(t)=P_v(t) \\times e^{rt}. \\, ", "59f126acf2f8e2ada74aa43a8ef87c96": " (\\kappa+n-1)~r^{-n+1}~\\cos(n\\theta) \\,", "59f1346bf8c9c3e5e71053d446c28516": "\\mathrm{const} - p_2(u_n)\\Omega \\,", "59f16c6f08950077970592007ef58466": " x'_{v_{i1}j_1}=1-\\sum_{i=1}^{j_1-1} x'_{v_{i1}j} ", "59f1adb2880dc161f4871ee60a4d32e1": "\\mathbb{C}\\oplus \\mathbb{C}\\oplus\\mathbb{C}\\oplus\\mathbb{C}\\oplus\\mathbb{C}\\oplus\\mathbb{C}\\oplus\\mathbb{C}\\oplus M_2\\left(\\mathbb{C}\\right) \\oplus M_3\\left(\\mathbb{C}\\right)", "59f1c8941ee05a95496f35e4b789120c": "\\frac{PE}{\\rho g} = y", "59f1e507d3f880eb9f9bde59c202655e": "K(k) = \\frac {\\pi /2}{M(1-k,1+k)},", "59f2057c03da74c7ad4cd2a8c204eb47": "\n\\omega^2 = (1/2)\\omega_h^2\\,\\left(\n 1 \\pm \\sqrt{\n 1 - \\left( \n \\frac{\\cos\\theta}{\\omega_h^2/2 \\omega_c\\omega_p}\n \\right)^2\n }\n\\right)\n", "59f229a17f32e3d640e48b28c207f5ed": " ((\\mathbf{\\lambda} x . A(x)) t) \\rightarrow A(t)", "59f29b17fda219c3bafd8e2e0ecfd9e6": " y_i = kF_i + \\varepsilon_i. \\, ", "59f3207b850010757838dd7ff20868a8": "5 + 5 + 8 = 18", "59f3395bdda53654b582367a6ecd8d2c": " 4=7-3", "59f3807a1e4e1a1803065987ae7cf6ef": "\\mathbf{s}=(\\mathbf{s}_j)_{j\\in \\Lambda}", "59f38c1110eca5491eff9d7a96efe63e": "l(v),", "59f3a6b9f5dcd28984ac644df8caa152": "K_k (\\tilde M)", "59f3ccc00f716222da64b948a2c25fe7": "I (D) = \\mathrm {dim} H^0 (X, \\mathcal L(D))", "59f3ccfd7031d341320a0d2cfd82c2ab": "\\lambda f.(p\\ f)\\ (p\\ f)[\\lambda f.(p\\ f)\\ (p\\ f) := q\\ p] ", "59f49418eedbb02b580e2243cacdbad6": "\n \\begin{bmatrix}\n 1 & 0 & -\\bar{\\mathbf{c}}^T_D & z_B \\\\\n 0 & I & \\mathbf{D} & \\mathbf{b}\n \\end{bmatrix}\n", "59f4d3ec5c9286cac7d8991b064a6da4": "GVD\\frac{\\tau_p}{\\tau_c}\\ll \\frac{\\tau_p}{L} \\ll GVM", "59f533763392249f16ee1c9821c94c17": "\\begin{align}\n\\sup_{y\\in [-\\varepsilon,\\varepsilon]} |P_{1-\\varepsilon}(y)| &\\le \\varepsilon^{-1}. \\\\\n\\sup_{y\\notin (-\\varepsilon,\\varepsilon)} |P_{1-\\varepsilon}(y)| &\\to 0.\n\\end{align}", "59f65ef47a2dbd4fdd08672c923b33e1": "P_2 = m\\left(V + \\Delta V \\right) + \\Delta m V_e", "59f6d45fd334a446fbc998b701bf5653": "\\begin{align}\n\\boldsymbol\\eta & = \\displaystyle \\frac{p^2 - p_0^2}{p^2 + p_0^2} \\mathbf{\\hat{w}} + \\frac{2 p_0}{p^2 + p_0^2} \\mathbf{p} \\\\[1em]\n & = \\displaystyle \\frac{mk - r p_0^2}{mk} \\mathbf{\\hat{w}} + \\frac{rp_0}{mk} \\mathbf{p}\n\\end{align}", "59f7235111a13fdb15cfb9d6c305fbcf": "f_0: X_0 \\to Y_0", "59f73e2045c8cbb291c4609180109ff3": "f \\colon D \\to \\mathbb{R}", "59f744576150dc2fd32efd812d224b6e": "\\langle \\mathcal{E}_i \\rangle", "59f763163388d7d7c78e3e4ee521b9a4": "x^k-1", "59f76672181c604284bdcab7f2c4dcf8": "K = \\rho\\frac{dP}{d\\rho}.", "59f76e9202c1d5f61b47b8aab1844d81": "\\chi\\ = \\int_{0}^{\\infty}\\frac{q}{\\pi\\left(u c d x^2\\right)\\left(cos \\alpha\\ \\right)}\\left(\\exp\\frac{y^2}{2c^2x^2}\\right) dx", "59f7a4221924f40ef1740fab1b837af5": "(\\mathbf{w}^{\\text{T}}\\mathbf{S}_W\\mathbf{w})", "59f806be08df67b8fc9f600911e4bdb2": "A_{m,1} = \\left\\lfloor \\lfloor m\\varphi \\rfloor \\varphi \\right\\rfloor", "59f8083920805bc7cde9d25c5cc44e46": "\\textstyle 5.\\ After\\ EVD\\ of\\ R_{x}:", "59f81143240f94f62e93f8c0f0c14253": "\nI =\n\\begin{bmatrix}\n \\frac{1}{12} m (3r^2+h^2) & 0 & 0 \\\\\n 0 & \\frac{1}{12} m (3r^2+h^2) & 0 \\\\ \n 0 & 0 & \\frac{1}{2} m r^2\\end{bmatrix}\n", "59f82fa436cebbd597a7d93ff7ae50db": "\\sqrt{2} \\,", "59f8a4abc8e3838cb0c39147e4fcb302": "\\ \\begin{align} v^*(x^*,t) & = Q(t)v + \\dot{c}(t) - Q(t)\\dot{Q}(t)^T[x^*-c(t)]\\\\\n & = Q(t)v + \\dot{c}(t) + \\Omega(t)[x^*-c(t)], \\end{align}", "59f8c5549eb5a27bd592ed4ed9b35625": "\\int_a^b\\bar{f}\\,dx = \\int_a^bf(x)\\,dx", "59f90a0e5e4f0fab58d1d84fff78e20a": "\n\\textrm{response} = \\textrm{constant} + 0.5 \\mathrm{(all\\ effect\\ estimates\\ down\\ to\\ and\\ including\\ the\\ effect\\ of\\ interest)}\n", "59f94907d7145ada8d094722f8ff1097": "\\scriptstyle \\sqrt{\\frac{1}{n}P \\left(1 - P \\right)}", "59f96ab3abb295fa94293365c92e0b93": "\nf^\\star\\left(x^{*} \\right)\n= \\frac{1}{q}|x^{*}|^q,\\,10 \\end{cases}", "5a27b6c12c89f53acebe9ea23c3836f3": "\\left[A, B \\right]_+ \\equiv AB + BA", "5a27b756900be10b84bff4174ba218e7": "\\displaystyle a_1", "5a27d9ee23301f88b38cccb95b897929": "\\overline{\\Theta}", "5a283b072c8de33d2d5768bc049dd866": "\\Gamma \\vdash_D\\ e:\\sigma \\Rightarrow \\Gamma \\vdash_S\\ e:\\tau \\wedge \\bar{\\Gamma}(\\tau)\\sqsubseteq\\sigma", "5a283d21e328a636e0f92e9229df850d": "s(a)=p_x(a) \\cdot g(a)", "5a2848c345f79606720369f1fbc4b0b2": " G(\\mathbf{x},\\mathbf{x'})=-\\frac{1}{4\\pi}\\cdot\\frac{1}{|\\mathbf{x} - \\mathbf{x'}|},", "5a284bf37c419c5e6415f17dfd816e0e": "B^n (y+1)^n", "5a289eb3a7b3e2e148693369f50666ad": "(N+1)", "5a2940629d3e79d56f76d85293b8edb4": "| x\\rangle\\langle x |", "5a29512f5aab4b354c104f77dddb4a21": "\\nabla \\mathbf{ A + FA = 0}", "5a29730b95a3b397c3c5f66890fdb79c": "x x_+^\\alpha = x_+^{\\alpha+1}", "5a29b231160a6963677edcbca9ffeb0d": "P:V\\rightarrow R", "5a2a2837d95afe1dce5c93a2a18e6e29": "Y_{\\ell, m}^*(\\theta, \\phi) = (-1)^m Y_{\\ell, -m}(\\theta, \\phi).", "5a2a2abbbcd3f859e5addeefe84b25af": " S_z = \\hbar s_z, \\quad s_z \\in \\{ - s, -(s-1), \\dots, s - 1, s \\} \\,\\!", "5a2a5e007179dadcdd4fbac3d6afebc2": "\\bold r = c_1\\bold {n}_1 + c_2\\bold {n}_2 + \\lambda(\\bold {n}_1 \\times \\bold {n}_2)", "5a2a8dfada98e511e7f81a23e20c118c": "\\gamma_\\mathrm{LG}", "5a2a9d374f1c9d50db7f0b7db1c3c7b7": "~\\triangle~", "5a2ae2a70617558b68bce9dd77cdee32": "\\sigma = 1 / \\nu d_\\text{f}\\,\\!", "5a2b04fdd45fc67aacb102d0dcc5ca13": "L_{0,0}(\\vec{r},t)Y_{0,0}(\\hat{s})=\\frac{\\Phi(\\vec{r},t)}{4\\pi}", "5a2b736a034c322d2f3729add041b4db": "\n|p(x)| \\le 1\n", "5a2ba465a5fcb90834a769cf0ab82c13": "\\operatorname{Re}\\{ \\alpha \\} > -1/2", "5a2bfd94e155f8538bc84f4d6aaf4c85": " \\vec{F} = \\mu_0 \\chi V \\vec{H_k} \\nabla\\vec{H_a} ", "5a2c21c418b0226b1f024f08063cf033": "T_*(T^*M)_p = T_p(M) \\oplus (T_p(M))^*.", "5a2c3383f94d051584d6629f790abf66": " \\and (S_8 \\implies (\\operatorname{equate}[A_8, q] \\and V[x] = q)) \\and D[x] = D[q] ", "5a2c3479da1d514326fcf85fdc24589a": "f / N", "5a2c8d49382e519a43fb82475fe1d0a8": "\\theta^\\prime = \\arctan\\left|\\frac{y}{x}\\right|", "5a2cafee8f6cfa8663a5047f81116681": "\n \\langle m | \\hat{C}^{(k)} | n \\rangle = (E_m-E_n)^k \\langle m | \\hat{A} | n \\rangle.\n", "5a2d29f4044b484b35ed3e6f71bba9ea": "y\\ge\\max \\varphi", "5a2d6d725e90b371ed1d9e02af715643": "x,y \\in V", "5a2dd6450728518b1478cef6e80f5199": " \n \\begin{bmatrix}\n - \\left( { b - \\lambda_{+} \\over c\\eta } \\right) \\\\ { 1\\over \\eta} \n\\end{bmatrix} \n ", "5a2ddc1502670e3e5a14eb85f7357235": "x y = \\left(N+x'\\right)\\left(N+y'\\right) =N^2 + N\\left(x'+y'\\right) + x'y' = N\\left(N+x'+y'\\right)+x'y' = N\\left(x +y'\\right)+x'y'", "5a2dec38c0ddc26e1591ebb353ff578f": "V( \\neg A,1) \\Leftrightarrow V(A,0)", "5a2df890a390e35eaee72063e673c990": "J = \\begin{bmatrix}\\cos\\theta & -r\\sin\\theta \\\\ \\sin\\theta & r\\cos\\theta\\end{bmatrix}.", "5a2e2824d7b8bbaaebe578f06d8a4577": "P_m = T \\omega", "5a2e37980ca0bd8948f0849e10e62d10": "(a-b) \\mid (a^m-b^m)", "5a2e6dfae6be152fc2dc841a5dfee815": "p_A=\\frac {h \\nu}{n c}", "5a2e8609976e8921b4b3bf4339d2ff21": "\\begin{align} \\int e^{ax} d(x^{n+1}) & = x^{n+1}e^{ax} - \\int x^{n+1} d(e^{ax}) \\\\\n& = x^{n+1}e^{ax} - a \\int x^{n+1} e^{ax}dx ,\n\\end{align} \\!", "5a2ea3025a20c6642cb681e0c0659137": "r < \\varrho < R", "5a2f08ff095e2ea87ab73e14266965fc": "\\log_d b^c=\\log_d a", "5a2f3a60b4f579d5c6776db9dc4d2da9": "<-2,4>", "5a2f4a4b4d50626a7ff52016e01aab36": "r = \\frac{1}{\\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|c_n|}}", "5a2f77b11ff539fcb563d22bb8c1457a": "\\frac{\\cos(ut)+\\sin(ut)}{\\sqrt{2 \\pi}}", "5a2f86239f56928d1082c62331355dec": "\\iiint_{\\text{all space}}p(x,y,z,t)\\,\\mathrm{d}V = 1", "5a2fb1c65bf1f2c7b6ddc2a9d2ee7f78": " y' = \\frac{y}{y_c} = \\frac{7.1 ft}{2.3 ft} = 3.1", "5a307b9650c1672c0fbfc01eea9553f0": "(2t_1, t_1, 5t_2, t_2) = t_1(2, 1, 0, 0) + t_2(0, 0, 5, 1).\\,", "5a30930eef90592a66376e28d1fd03b9": "(D,V,s,R) \\models \\exists x.P", "5a30a4744c1d475830dd91e3e331fbaf": "E \\ = \\frac{q}{q_{max}}", "5a30dda317ce495ffd130030a7320d21": "\\overline{\\alpha}", "5a311ea9100ed94a7c5f01e5f7c756b3": "(2D)^2 = \\frac{p^2}{\\sin^2 \\alpha}(1+ \\cos \\alpha)^2 + p^2\n= p^2 \\cdot \\left ( \\frac{(1 + \\cos \\alpha)^2}{\\sin^2 \\alpha} + 1\\right )", "5a314e314533b12e8b73257fc5bdeb34": "\n\\hat{\\mu}_{HT} = N^{-1}\\hat{Y}_{HT} = N^{-1}\\sum_{i=1}^n \\pi_i ^{-1} Y_i.\n", "5a31712137e4caab49ece0c59707ee6c": " V_x = I_x R_{in} + \\beta v_{out} \\ , ", "5a3187dcb71345713ccf2f2f0fa8cc8d": "\\mbox{Earnings Per Share}=\\frac{\\mbox{Income from Continuing Operations - Preferred dividends}}{\\mbox{Weighted Average Common Shares}} ", "5a31a3a86161106c9bf1f389c77d95e3": "b_1, \\ldots, b_n", "5a31db5740c13fd7846b6bfbb5e9f2f9": " (\\alpha_i, y_i) ", "5a324bff0f46a6422fce9c37f0ffbe25": "f: \\mathbb{R}^n \\rightarrow \\mathbb{R} ", "5a329123fc0bf70b5ab2c1e7a29cc72c": "\\int_0^1 G^{-1}(x)dH(x)= G^{-1}(\\alpha)", "5a329560d76b2b1f2f512e28bc7d9550": "a, b, c, d = 133,59,158,134", "5a32fbdca3e830d6a75ae2cd6177a526": "H^q(X,F)", "5a3345ccfb727e7e842748bea63f5ec8": "A.\\mathrm{dp}=m\\cfrac{\\mathrm{d}^2 x}{\\mathrm{d} t^2}", "5a33826e461c85697db9c7de1a3a23cf": "z\\in \\mathcal Z", "5a34bb082daf037b3c4b14c13af6855b": "p\\,", "5a351c355220ae642054b6a62587b108": " \\lambda_\\max", "5a355524dc86c05a810c96ed2b26a471": "\\displaystyle{\\|V u\\|_{(k+1)} \\le C^{\\prime\\prime}(\\|\\Delta_1 u\\|_{(k)} + \\|u\\|_{(k+1)}).}", "5a3565cb9b60ac3554d14b71fbf192b2": "\n \\log a_{\\rm T} = -\\frac{C^g_1 (T-T_g)}{C^g_2 + (T-T_g)} = \\log\\left(\\frac{\\eta_{\\rm T}}{\\eta_{T_g}} \\right)\n ", "5a35de9aac840c727891ca49ad6adcbd": "H\\cdot F=rF \\, ", "5a35f75917cd9a30f093f027ca2fbb4e": "\\gamma_{ik}=\\gamma_{ki}", "5a362748f6bd45a217ebe92469c5ae7b": "\\bigcap_{\\alpha\\in A} \\operatorname{cl}(E_{\\alpha}) \\neq \\emptyset", "5a362d7d0a834d883e7a7e010e04f34a": " K_{g_\\varepsilon} ", "5a363e32881ec8b1f165b05e01c5e099": "c \\mathbf{v}", "5a3643c57e3675710f4fa1b4bcaa5556": "\\chi^{d}_{ij} = \\frac{\\part M_i}{\\part H_j}", "5a369e7b8fb56d02723ea29e63e59f4a": "\\zeta = 6 \\pi \\, \\eta \\, r,", "5a36b4d35a9c5dfbcf8ce58b465d9cf3": "x = a_0+\\cfrac{1}{a_1+\\cfrac{1}{a_2+\\cfrac{1}{a_3+\\cfrac{1}{\\ddots}}}}\\;", "5a37297106104974f7a7c415038ce365": " \\Pi(X)^* f(g) = -\\rho(\\pi(g^{-1}Xg))f(g)", "5a37391405bb35dd15d0385874779ea0": "P(G,k)=P(G-uv, k)- P(G/uv,k)", "5a37611a640b9288b15953a161ba726b": "* (- [F , G]^{IJ}) = *(*[*F,G]^{IJ}) = **[*F,G]^{IJ} = - [*F , G]^{IJ} .", "5a376a5cca2a385c940730010d60dfb3": "\\begin{align}\n\\sum_{r=0}^{m+n} {m+n \\choose r}x^r\n&= (1+x)^{m+n}\\\\\n&= (1+x)^m (1+x)^n \\\\\n&= \\biggl(\\sum_{i=0}^m {m\\choose i}x^i\\biggr)\n \\biggl(\\sum_{j=0}^n {n\\choose j}x^j\\biggr)\\\\\n&=\\sum_{r=0}^{m+n}\\biggl(\\sum_{k=0}^r {m\\choose k} {n\\choose r-k}\\biggr) x^r,\n\\end{align}\n", "5a37bb2583eac39f82ad81519815317f": "\\mathbf{Q}_{p}", "5a37ce2c58541d98596fc789abf52016": "f(z)=\\sum_{n \\geq 0}a_n z^n", "5a38252ad6fd88e767ac5a0240a759ae": "M^{0.30}", "5a387eeac18b43296ade5abfdc37a7ab": "A \\leq G", "5a388411e43bb7a3fe54844bdcdd9b70": " \\phi(x, y) = y - f(x) = 0 ", "5a38c5f512dde93e585b9a030f4a4723": "\n\\int_{\\mathbb{R}^n} \\exp(-x^TAx) \\, dx = \\sqrt{\\frac{\\pi^n}{\\det{A}}} \\;.\n", "5a38cf09a6ef6d68840cbd5bf2243b4c": "b_1, b_2 ... b_t", "5a39052dbb8af0f46ad53f4e44883948": "S^2_\\infty", "5a393991ae738b09b6613413c9994013": "\n\\Pi = 2 \\epsilon \\epsilon_0 \\kappa^2 \\psi_{\\rm D}^2 \\frac{e^{-\\kappa h}}{[1+(1-2p)e^{-\\kappa h}]^2}\n", "5a39c17cae385fc69a5755976bc3f1ec": "\\scriptstyle{i=\\sqrt{-1}}", "5a39c61fc4fc0442bbd810fef4eedee2": "|K| <\\infty", "5a39f53f442b708f3deb3fda6a48708b": "\\int_0^t (H^2 \\sigma^2 + |H\\mu| )ds < \\infty.", "5a3a20e5c73d52c585f0d0545a539f33": "T_x \\mu", "5a3a2d0a07ea810f73af2305661f059c": "\\textstyle \\begin{bmatrix} A & B & x \\\\ C & D & y \\end{bmatrix}", "5a3a40dcac19a7e762d54709553b556b": "\\Omega(1/n)", "5a3b32f6ca61292ec7566b98f969368f": "H_L", "5a3b4da660caec7e71ced39b658965ab": "y'(t) = f(t,y(t)), \\qquad \\qquad y(t_0)=y_0. ", "5a3b9165b81a9c08ed8875c1ecb56f30": "b'_i", "5a3bcac85c24d2a8d5d5124b7b7ce892": "D^m", "5a3bcbf1feaad51131c5e0461e1fe246": "T^{ab}{}_{;b} = 0", "5a3c6c3eefdc8d67a6216f858b09d3fa": "\\left(\\cos z + i\\sin z\\right)^w", "5a3cc5fa10db5af664d203a1a0109849": "\\frac{d u_i}{d t} + \\frac{1}{\\Delta x_i} \\left[ \nF^*_{i + 1/2} - F^*_{i - 1/2} \\right] =0. ", "5a3cca91260fa75c16455c092c418827": "Y = f(X_1,\\ldots,X_N),", "5a3cd43f8c6d54a18e7bd0a89fe941a7": "(p_i,|\\phi_i\\rangle)", "5a3d92e8b7dd015ae94bdaad3d26f142": "\\boldsymbol{S_{x_a}}", "5a3d97e6e715203c68a753811e00f6a0": "f(x)=\\sin(x)/x", "5a3db22214ab8d6d12e54267272ef8be": "dA = 0", "5a3db653933ceed2ee4a7f9e0c96a105": " \\mu (A) >0", "5a3de35be9bbbae9beea20ce3cbca720": "O(W)", "5a3de63f6414fb8f490d1fe329c2ce64": "\\dot \\gamma \\over 2", "5a3e5c59ac2d95a773dcc207cf856078": "\nu = -\\frac{\\mu }{r} \\left(1 + \\sum_{n=2}^{N_z} \\frac{\\tilde{J_n} P^0_n(\\sin\\theta) }{{(\\frac{r}{R})}^n} + \\sum_{n=2}^{N_t} \\sum_{m=1}^n \\frac{ P^m_n(\\sin\\theta) (\\tilde{C_{n}^m} \\cos m\\varphi + \\tilde{S_{n}^m} \\sin m\\varphi)}{{(\\frac{r}{R})}^n}\\right)", "5a3e972eec97851f3ebb08112d7db834": "H_6(x)=64x^6-480x^4+720x^2-120\\,", "5a3eac6e645787cb77410b381d04f008": "\\frac{dy}{dx}\\left.{\\!\\!\\frac{}{}}\\right|_{x=a} = \\frac{dy}{dx}(a).", "5a3ee628a5fb57b5116f78851c5b0a5b": "\\frac{\\sigma \\alpha}{\\alpha-1}", "5a3f79db72b11466875d33149e443f24": "z=x", "5a3fa69c88d32da2b7b0cf4deb26ede2": "\\scriptstyle \\R^+ \\;\\equiv\\; \\left(0,\\, +\\infty\\right)", "5a3fc45199cabc8533d125ef3853002b": "G = \\operatorname{E}[\\,\\nabla_{\\!\\theta}\\,g(Y_t,\\theta_0)\\,], \\qquad \n \\Omega = \\operatorname{E}[\\,g(Y_t,\\theta_0)g(Y_t,\\theta_0)'\\,]", "5a400bae130a2f327df4799accbe5035": "q(F) = \\left|{F^\\star / F^{\\star2}}\\right|", "5a401d489ea68192db08d883ae4885da": "\\nabla \\mathcal{F}", "5a402d80918bd7fa3b880ebc566ec17b": "I_M", "5a40a91b320879dcab6f4959c0e686ed": "a^+, b^-", "5a40c081bc2eca1457cf54ed9553091f": "\\lceil\\cdot\\rceil", "5a40e24c171da6b17366020b94a6d892": "0,\\ s(0),\\ s(s(0)),\\dots.", "5a416008d7518d2cb7f25c12201115ba": "\\cos \\theta + \\cos \\varphi = 2 \\cos\\left( \\frac{\\theta + \\varphi} {2} \\right) \\cos\\left( \\frac{\\theta - \\varphi}{2} \\right)", "5a41716427aab0e06132d68cedb7989f": "\\triangle DEF\\,,\\triangle GHI ", "5a42127307026cf1afd57924a4118fb5": "\\|b_i\\|\\to 0", "5a42173339a6eee874b574d2f574c419": "\\displaystyle k_4=k_1.", "5a423cba9bdf9855dcbd8bcac8e80761": "\\textrm{Abnormal\\ Return} = \\textrm{Actual\\ Return} - \\textrm{Expected\\ Return}", "5a42478aa5175bc67c07a14a50748be5": "\\mathbf{\\hat{b}_{0:5}}", "5a42c2d6faaa0086f6f46cd91cbed606": "E \n\\;\\begin{matrix} s \\\\[-6pt] \\rightrightarrows \\\\[-4pt] t \\end{matrix}\\; V", "5a42e63bb371b722e25449f5506370d8": "(\\mathfrak{X}, \\mathcal{O}_{\\mathfrak{X}})", "5a430f51a79083d5bb33dda1c612e8c9": "\\sigma_R \\approx \\sqrt{ \\sigma_V^2 \\left(\\frac{1}{I}\\right)^2 + \\sigma_I^2 \\left(\\frac{-V}{I^2}\\right)^2 }.", "5a433f043e0480851e8c351c3d8e6789": "\\mathbf{ \\hat n}", "5a434c0299264dbbead9badf6f9eb03d": "x\\in \\alpha", "5a438c68a7c15995cc28e4b4c48a17b3": "G_0 = \\frac{v_{out}}{i_{in}} = R_D\\|r_O \\approx R_D \\ ,", "5a43be8f4ba720a2943aa5b278b46136": "\\frac{\\mbox{total number of fission neutrons}}{\\mbox{number of fission neutrons from just thermal fissions}}", "5a43e90f36d8bb26c247670c09468cf5": "f(q,q^3) = \\sum_{n=0}^\\infty q^{n(n+1)/2} = \n{(q^2;q^2)_\\infty}{(-q; q)_\\infty} ", "5a44098d1d1df7ac74415809783fc589": "S={1\\over 16\\pi G}\\int d^4 x \\sqrt{-g} F(N)K^{\\mu;\\nu}K_{\\mu;\\nu}+S_m", "5a441d240a35ec20f6c2d382314c4321": "~s=I_{\\rm s}/I_{\\rm so}~", "5a4420fcfeff722d6acdacb679ae62a2": "\\frac{\\partial u_1}{\\partial t} = \\nu \\frac{\\partial^2 u_1}{\\partial z^2}.", "5a4429f880458421c4f31f5b7f00cf06": "R_c = \\frac{R_\\mathrm{bc}R_\\mathrm{ac}}{R_\\mathrm{ac} + R_\\mathrm{ab} + R_\\mathrm{bc}} ", "5a442f63e14bc33487ba84211cd5e382": "\\phi = V_v / V_T", "5a445c23f83ce50e9a56b29f088247e7": "=\n\\frac{\\zeta(\\alpha\\!+\\!1)}{\\zeta(\\alpha)}\\,\\tau^\\alpha", "5a4489eef5c292f745039a7a970edb27": "\\sqrt{h_N(\\vec{x},t)}", "5a44b4ed613434a3c1a00d228e7273ff": "Q = \\frac {KWL}H", "5a44dbaf2dc45a86a644c9e631966632": " \\vec{E} ", "5a44e1522f69a9b06396da2bcd8739d1": "d\\mathcal H_\\Delta^k(M)=0", "5a44e539dfe6c62a3c0d96ab3c8cc08a": " \\mathrm{VOL} = 0.2933~\\mathrm{RAD}^3.", "5a4558b9d03246196ef928385465cd91": "2\\times f_2-f_1", "5a458e4633ad0846ecce304bf5a7fb21": "\\scriptstyle \\sqrt{gh}", "5a45b01f813cf35329130344a58b1735": "\\frac{[E][S]}{[ES]} = \\frac{k_{r}+k_{cat}}{k_{f}} =K_{M}", "5a45b7d3df77e297f93766d03c0a32bb": "\nS = -\\log \\sum_{j=p}^{\\min(M,n)}\\frac{\\binom{M}{j}\\binom{F - M}{n - j}}{\\binom{F}{n}}\n", "5a45d7200294d23046acf6589b0432a8": " = \\sum_{k_1+k_2+\\cdots+k_{m-1}+K=n}{n\\choose k_1,k_2,\\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\\cdots x_{m-1}^{k_{m-1}}(x_m+x_{m+1})^K\n", "5a45fbe423792959ab339ec791297c69": "\\Gamma_a^{ij}", "5a46473d74ce219e8b7bef84170045ee": " \\mathbb{C} \\oplus \\{0\\}. ", "5a465191432c3cfe1f3a771950b6a74a": "q=I\\cdot t", "5a4684c6dcd752c401f77f5633764b75": "\\boldsymbol{X}", "5a46c29cbcc054d7664e6b4404c1224f": "c = 5.7", "5a46d2282b151777f3eb661bf6fa4ce3": "\\zeta = \\frac{\\tau^2\\Delta\\bar{H}^2}{\\hbar^2}", "5a46fa7950b6f2d412b361b2f1ee228e": "\\beta_i(r_m-r_f)", "5a4765849fad53aa42b2ee5c43fdee0f": "x \\in Cl_t^{\\geq}", "5a4839321fce157f997bae09bc9d0689": "x = \\left( 1, \\frac12, \\frac13, \\dots \\right),", "5a486d6b04cdb12b0adba1d0b8275d4e": " \nU^\\alpha f(x) = \\int_0^\\infty e^{-\\alpha t}P_tf(x) dt. \n", "5a48833815828cd24bd643438f444dd7": "t=t_\\mathrm{then}+\\lambda_\\mathrm{then}/c\\,.", "5a488494ab22003404351c005436b3b6": "\\mathbf{\\theta}_{x,y,z}", "5a4911186a97fe6f8eee160365cc96e0": "df(t,T)=\\xi\\left(t,T\\right)\\left(\\int_t^T\\xi\\left(t,s\\right)\\,ds\\right)dt+\\xi\\left(t,T\\right)d\\tilde{W}.", "5a492a8ddbb79f073e64cad2c6d889f6": " \\lim_{x \\to c} f(x)g(x) = \\lim_{x \\to c} \\frac{f(x)}{1/g(x)} \\! ", "5a492d0c9609494dc941babfa08b4124": "\\beta_1 \\leftarrow 0 \\, ", "5a4965200a601498bdf3ad253d514b24": "x={a \\over 2}", "5a4969514de21636dbbdfba9cd7eeb80": " \\eta(\\sum_{g\\in G} r_g g) = \\sum_{g\\in G} r_g.", "5a498adbc4631c0985c6e3c159cd48e8": "A_n = \\frac {2}{\\pi} \\int_{0}^{\\pi} \\cos (n \\theta) (dy/dx) \\; d\\theta ", "5a49e5de11884800d09b8c25d2a3fd3a": "Z=\\bar{Z_1}n_1+\\bar{Z_2}n_2.", "5a4a9e80348a65f08e6209175dc5d986": " v_1\\left(\\frac{1}{2}\\right)=v_2 \\left(\\frac{1}{2}\\right) ", "5a4ab434769b240bd52a6bd689a38a55": "Y= \\beta_0+\\sum_{j=1}^p f_j(X_{j})+\\varepsilon ", "5a4ab8d63c89fa7e0213f82ae0eeccf6": "\n\\mathbf{C} = \\int_{\\Delta} (\\mathbf{x'}+\\mathbf{v}_0)(\\mathbf{x'}+\\mathbf{v}_0)^{\\mathrm{T}} \\, dA = \\mathbf{C}' + \\frac{a}{2}(\\mathbf{v}_0\\mathbf{v}_0^{\\mathrm{T}} + \\mathbf{v}_0\\overline{\\mathbf{x}}'^{\\mathrm{T}} +\\overline{\\mathbf{x}}'\\mathbf{v}_0^{\\mathrm{T}})\n", "5a4af0b9ea2d6bc4220b8d1f92be1424": "\nH^* = H + \\sum_j c_j\\phi_j \\approx H,\n", "5a4b1dc778de1d0d3cafdefad3743c7e": " c_{2} = - \\tfrac{1}{2}i ", "5a4b323ca324f2aed7db512f9be19b19": "d : X \\to \\mathbb{R}/R\\mathbb{Z}", "5a4b3ecacdc166135c37d08e0134cb08": "\\mathcal{P}_{ac}=e\\langle a|r|c\\rangle, \\mathcal{P}_{bc}=e\\langle b|r|c\\rangle", "5a4b9d3153c212b1a18d53689493287d": "\\lim_{t\\to 0}x = a\\lim_{t\\to 0}{\\cos t \\over t}=\\infty,", "5a4bb071c83b19499f701597f0b46d9f": "\\triangleq \\!\\,", "5a4bb5529040150a9424a37bc553e5c8": "\\alpha \\to \\alpha", "5a4c1fdc5239c533bc3f8747837ba2e2": "\\varepsilon \\rightarrow 0", "5a4c6230855a6e457e32f22088002b3b": " G = S_1\\,S_2 + S_1\\,S_2\\, K_{12}\\, G ", "5a4c66fccded43d5c9d1f75888f2ad5c": "\n\\left(\\frac{E}{c} - \\boldsymbol{\\alpha}\\cdot\\mathbf{p} - \\beta mc \\right)\\left(\\frac{E}{c} + \\boldsymbol{\\alpha}\\cdot\\mathbf{p} + \\beta mc \\right)\\psi=0 \\,,\n", "5a4ca52d2f97862828a67f2bd32fa20b": "B^*", "5a4ced54c33cdaafc6aa9161498ffc3b": "\\vec{j}_{\\text{advective}} = \\vec{v} \\, c", "5a4d015e9a4ab9e27c6232a37fbbb7d1": "T = 2 t\\left(\\theta_0\\rightarrow0\\rightarrow-\\theta_0\\right),", "5a4d13159d718d4cf0d64fbac9aec77a": "\n\\sigma_z = \\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "5a4d2862f2d5917e47d560aec2ba8dd8": "{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}=\\left(\\frac{\\partial(y,z)}{\\partial(s,t)}, \\frac{\\partial(z,x)}{\\partial(s,t)}, \\frac{\\partial(x,y)}{\\partial(s,t)}\\right)", "5a4d65ee2b56d5f352013c0d1f2d7e6e": "\\lambda>-y_0", "5a4d70236c2d7d9ccba072d0b90844a1": "(x \\triangleleft y) \\triangleleft z", "5a4d81b4f7d3b2207410b3ba0c12c029": "~R", "5a4d88101670c53cad812c6d1c3a7967": "\\left\\langle \\partial^{\\alpha} S, \\varphi \\right\\rangle = (-1)^{| \\alpha |} \\left\\langle S, \\partial^{\\alpha} \\varphi \\right\\rangle \\mbox{ for all } \\varphi \\in \\mathrm{D}(U).", "5a4dbfe03861254aa7d4ad55fc3ba77e": "f^{-1}(V) \\subseteq X.", "5a4de54a517ee9f6313265e84cb48816": "(m_1|m_2|...|m_n)^{*}", "5a4e02840c5c26e2522ea605151f1469": "NM_{\\cong_{\\mathcal{B}}}(X,Y)", "5a4e33d92b8c28fca20209d71a53dd0a": "16*x^4+5152*x^3+518420*x^2+16693124=", "5a4e60d1d780464c35656fa4bd372bc6": "\\sigma_{i+1}", "5a4e669cd7e3aa94355327fa6cb77516": "\\Im \\left[ \\mathrm{Ai} ( x + iy) \\right] ", "5a4e75c2e46aab2dba7bc18b032f88bb": "\n\\,x = \\,a \\cosh \\xi \\cos \\eta,\n", "5a4ec35e17b16b22e8eae235edc716e6": "D(f) = \\left (\\frac {\\partial f_i}{\\partial x_j} \\right )_{1 \\leq i, j \\leq n}. \\,", "5a4ecbe2a8274b9a010a42eab7668eb2": "\\tfrac{x^l}{(1-x)^{l+1}} = \\sum_{p=0}^\\infty \\tbinom p l x^p\\,.", "5a5014834130049bf213ee153e543b4a": "J_{\\beta,m}", "5a50431babcbcd4a691e47399ac298c2": "\n \\mathbf{u} = u_r~\\mathbf{e}_r + u_\\theta~\\mathbf{e}_\\theta + u_\\phi~\\mathbf{e}_\\phi\n ", "5a50be2db32113b6b84d8909d878666f": "A \\cup A = A\\,\\!", "5a50c2d1f8d35a2d1797b412c65f724f": " \\log_b(x) = \\frac{\\log_k(x)}{\\log_k(b)}.\\, ", "5a50f967f42a523922b78c072be1da2f": "\\mathbf{d\\hat u_\\theta}", "5a51fe9260b15b526f8e70cb16a0af32": "\nu=\\frac{\\partial\\varphi}{\\partial x}=\\frac{\\partial\\psi}{\\partial y},\\qquad\nv=\\frac{\\partial\\varphi}{\\partial y}=-\\frac{\\partial\\psi}{\\partial x}.\n", "5a520fc02d4611e77ebf1f95de2a6641": "\n 1-F(z k;\\,k) \\leq (z e^{1-z})^{k/2}.\n ", "5a522c8d74d5fa025e93185562a88f81": "\nM(\\vec X) = \\left( {\\begin{array}{*{20}c}\n {\\bar \\mu } \\\\\n {\\bar \\Sigma } \\\\\n\\end{array}} \\right)\n", "5a5235cbd102a289a323524341112f09": " {\\pi\\lambda/2}\\cdot \\tanh(\\pi\\lambda/2) d\\lambda", "5a5271fdfaf164721106c3cec60f2865": " A_{1} ", "5a52a86eae3dc32c4412ad1a61c04364": "g(\\Delta\\rho/\\rho)^2", "5a52d3cc802ac0553c803dda925ecbad": "non(\\mathcal{N})", "5a52e50ac34a9dd830ab253132f2dfff": "\\psi =\\psi(\\mathbf r,\\sigma)\\,,", "5a530e5d4658dc6569c018c5e9608148": "(\\alpha X_1X_2^2 + \\beta X_2X_3)\\cdot(\\gamma X_2X_1 + \\delta X_1^4X_4) = \\alpha\\gamma X_1X_2^3X_1 + \\alpha\\delta X_1X_2^2X_1^4X_4 + \\beta\\gamma X_2X_3X_2X_1 + \\beta\\delta X_2X_3X_1^4X_4", "5a5365e133c70d80c20fd2faa345a149": "\\displaystyle{A(s,t)={f(s) - f(t)\\over z(s) - z(t)}.}", "5a53a083723786c8aaf501e179971f3b": "y=f(x_1,\\dots,x_n)", "5a53aa7d246bc26fdac449fdaa4ad744": "\\mathbf{y}'(t) = (y'_1(t), \\ldots, y'_n(t)).", "5a53b9b66237a2a9a24a84a4220ffbff": " \\mbox{F} ", "5a54084e2a22032c12d3c369a2a00b78": "\\displaystyle{\\partial_n D(\\varphi)(\\mathbf{v}(s) +\\lambda \\mathbf{n}(s))= D(\\dot{\\varphi}\\mathbf{t}\\cdot \\mathbf{n}(s))+ S(\\partial_t(\\dot{\\varphi}\\mathbf{n})\\cdot\\mathbf{n}(s)).} ", "5a540e90eca8383e99c3baff350d5f24": "B\\subset X", "5a544a1c14fe313c3245c3267a292b8e": "jd", "5a547646b794d8d4af629d48b9642c5e": "V(A,B) \\in \\{0,1\\}\\,\\!", "5a5509edd4c6b0232650e9b0f60f68fc": "(\\mathbf{\\hat{e}}_x, \\mathbf{\\hat{e}}_y, \\mathbf{\\hat{e}}_z)", "5a551609d3f5b3652e4e9344d7726272": " y'' = -(py'+qy). \\,", "5a55194b2f82dd49f84b038059d3d303": "y_i\\neq y_j", "5a5622836429745af2295e2f1e27b14e": "\\begin{align}\n& \\left[\n{\\partial _\\tau ^2 - {\\rho ^{-1}}{\\partial _\\rho }\\left( {\\rho {\\partial _\\rho }} \\right) - \\partial _z^2} \n\\right]\n\\psi \\left( \\tau, \\rho ,z \\right) \n= \ns \\left( \\tau, \\rho ,z \\right)\\\\\n& \\psi \\left( \\tau, \\rho ,z \\right) = 0 \\quad \\mathrm{for} \\quad \\tau < 0\n\\end{align}", "5a567ad8b4266f465269739813f35b18": "x\\in A \\implies tx\\in A\\ \\ \\ \\text{for all}\\ t>0.", "5a568ed07af31ed5fde14b37d7ba9d48": "\\rho \\colon G \\to \\operatorname{GL}", "5a56e44d8cd9d04cd6e5324e0fb4967d": "\\boldsymbol{\\nabla}^4\\boldsymbol{\\sigma} = \\boldsymbol{0}", "5a56fe3bf0f3135064d6dc15fab396c2": "67^2", "5a5703eec25070653c50fc804dc866aa": "\\nexists A \\quad:\\quad \\aleph_0 < |A| < \\mathfrak c.", "5a570d9c909c12806fac87362c6a7fe2": " j^{1}\\sigma = (u,u_{1}) = \\left(\\sigma(p), \\left. \\frac{\\partial \\sigma}{\\partial x} \\right|_{p} \\right).", "5a5719e2d91b13a2f177b14b447b69fe": "\n\\epsilon_\\mu^2(p) \\!= \\!{1 \\over \\sqrt{2}} \\left(\n{{i p_1 p_2 \\!+\\!E^2 \\!+\\!p_3 E \\!-\\!p_1^2} \\over {E(E + p_3)}},\n{{-p_1 p_2 \\! - \\!iE^2 \\! -\\!ip_3 E \\! + \\!ip_2^2 }\n\\over {E(E + p_3)}},\n{{\\!-p_1 \\!+ \\!i p_2} \\over E}, 0 \\right), \\quad (2)\n\n", "5a573bdb2489c755e6259f72d6074ca2": "{\\mathcal K}_n(x) = \\frac{1}{2} \\int_0^\\infty \\exp\\left(-t-\\frac{x}{t}\\right) \\frac{dt}{t^{n+1}}", "5a57755fdba0c5ac0b041e98bbab3e75": "\\rho _{\\alpha -} ^{i_0^\\ast } ", "5a57d01ebbdb229318cb7652016507b2": "P = D\\cdot\\sum_{i=1}^{\\infty}\\left(\\frac{1+g_i}{1+k}\\right)^{i}", "5a57ecabb9a7e37437f938df8c1b98e4": "\\scriptstyle{X|n\\rangle}", "5a580b07cd6a1ab7e4e4d0647b3010d5": "t\\in T", "5a5822263cc1d62d02e24fd723aceee4": "D^+ = \\mathrm{max} \\left[i/n- z_i \\right],", "5a587b1de487d1308f141dc80187c3f2": " {t = (n+1) \\Delta t} \\,", "5a5881890a2351368749d84f5fb8779a": "Q ", "5a58b398d371b5bde7ee3cd63db18688": "\\mathcal{A} (G)", "5a58bb2155825dd18a7efd5d6d2650e2": "x_{i} = \\frac{\\sum_{j}e_{ij}x_{j}}{\\sum_{j}e_{ij}}", "5a593235539fc0b35eed27e32dc4e5e0": " n = p_1 p_2 ... p_i < x ", "5a59919fd879932b44328817ca870bed": "f(x)=x+\\tfrac{1}{x}", "5a59975e9481e069a6df99da96d9a778": "PSL(2,\\mathbf{C})\\cong SO^+(1,3).", "5a59a58f739bcf61f1586385fe0f4c16": " k \\in \\{1,...,p\\} ", "5a59f3f4ba1e751ab956e506fd66292a": "\n\\begin{align}\n\\mathbf{v}(t) = \\frac {d\\,\\mathbf{r}(t) }{dt} &= r \\left[\\frac{d\\, \\cos(t)}{dt}, \\frac{d\\, \\sin(t)}{dt} \\right] \\\\\n &= r\\ [ -\\sin(t),\\ \\cos(t)] \\\\\n &= [-y (t), x(t)].\n\\end{align}", "5a59ffddf65fabcb2b3d898160808906": "\\scriptstyle\\operatorname{Lk}(F,X)", "5a5a058bc0e66be0b179d80a23dfbd4d": "x^2 + y^2 + z^2 = r^2.", "5a5a6065b8a41d80e493f0dc1ae7b503": "A_{r}", "5a5a8d01dd78c1209e666b9c82269f58": "\\textit{SMA}_\\mathrm{today} = \\textit{SMA}_\\mathrm{yesterday} - {p_{M-n} \\over n} + {p_{M} \\over n}", "5a5accfd80d804ecd91607cebf0e31b5": "2.6%", "5a5ae0760dc3dac91e546c0ea25586b0": "z_i", "5a5b66bb25dd9cc1cb15be0b87559c4c": "\\scriptstyle T = \\frac{1}{f}\\,", "5a5b70dc285cedec49729cb194e0598a": "+ (\\mathbf{X^{\\rm T}A^{\\rm T}X})^{-1}\\mathbf{X^{\\rm T}A^{\\rm T}})", "5a5b9810eddc221a38d1d0282c78f73e": "\\delta_{max} = \\frac {5 M_{max} L^2} {48 E I} = \\frac {5 q L^4} {384 E I}", "5a5bb09bcc834b6a9b34c88fc92955cc": "\\mathbb{D}_{12}\\times\\mathbb{Z}_2", "5a5bc01e812c7e198b5de0f0231c445f": "\\tau = \\int_{z_1}^{z_2} \\beta_e(z) \\mathrm{d}z", "5a5c28517ed8073e1ee832a5230d55a1": " x=x_1x_2+Ny_1y_2,\\ \\ y=x_1y_2+x_2y_1 \\ \\ ", "5a5c2e78a8f72216f1b15f7432ce3e91": "Q = nP", "5a5c63fec10e4c72d9f4fd23df089128": "~z(a)~", "5a5c8491a62e6045fc513f89b62177de": " \\operatorname{cl}(\\operatorname{cl}(A)) = \\operatorname{cl}(A) \\! ", "5a5cdafa137b4d22757b4cd169c837ba": "\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\frac{\\partial v^i}{\\partial q^i} + \\cfrac{g^{mi}}{2}~\\frac{\\partial g_{im}}{\\partial q^\\ell}~v^\\ell\n", "5a5ce8568c404de24eb669af1df4a68c": "P(v) = \\frac{1}{Z} \\sum_h e^{-E(v,h)}", "5a5d7f15b0194bc965f478b1db6278bf": "\n\\begin{array}{lllll}\nA_2=\\begin{pmatrix}\n 1 & 1/2 \\\\\n 1/2 & 1 \n\\end{pmatrix};\n&\nA_2^{-1}=\\begin{pmatrix}\n 4/3 & -2/3 \\\\\n -2/3 & {\\color{BrickRed}\\mathbf{4/3}}\n\\end{pmatrix};\n\n\\\\\n\\\\\n\nA_3=\\begin{pmatrix}\n 1 & 1/2 & 1/3 \\\\\n 1/2 & 1 & 2/3 \\\\\n 1/3 & 2/3 & 1 \n\\end{pmatrix};\n&\nA_3^{-1}=\\begin{pmatrix}\n 4/3 & -2/3 & \\\\\n -2/3 & 32/15 & -6/5 \\\\\n & -6/5 & {\\color{BrickRed}\\mathbf{9/5}}\n\\end{pmatrix};\n\n\\\\\n\\\\\n\nA_4=\\begin{pmatrix}\n 1 & 1/2 & 1/3 & 1/4 \\\\\n 1/2 & 1 & 2/3 & 1/2 \\\\\n 1/3 & 2/3 & 1 & 3/4 \\\\\n 1/4 & 1/2 & 3/4 & 1 \n\\end{pmatrix};\n&\nA_4^{-1}=\\begin{pmatrix}\n 4/3 & -2/3 & & \\\\\n -2/3 & 32/15 & -6/5 & \\\\\n & -6/5 & 108/35 & -12/7 \\\\\n & & -12/7 & {\\color{BrickRed}\\mathbf{16/7}}\n\\end{pmatrix}.\n\\\\\n\\end{array}\n", "5a5d9a79a6514991291d1cc7d839ab93": "\\phi \\to \\psi ", "5a5dacafc507acce05c6c799c1eb202f": "C_{\\alpha IJ} = 0", "5a5de0daed06501f5860e20e1fc58701": "w_{1}=\\frac{dc_{1}}{dz}", "5a5df0c7f0db114eb0a961380b4bcf65": "\\mathbf{J}_{tot} = ", "5a5e1741626219ec2d094890cc5a7472": "\\bar\\xi", "5a5ea9bae64c407799738cfca86c08e0": " \\frac{\\omega_s}{\\omega_a}=R,\\quad \\mbox{so} \\quad \\frac{\\omega_s}{\\omega_a}= -\\frac{N_a}{N_s}.", "5a5ed1bc2bb36babd57f6eb1dc267328": "a>b=c", "5a5fd05dbcce2bf5dbf75669f9868c8d": "z\\simeq 1", "5a5ffa3447d180eb3093d108a18edb96": "\\displaystyle m_4", "5a600386bcac58041698a13c1739c9cb": "u = \\frac{\\sin(\\omega \\Delta t/2)}{\\sin(k \\Delta x/2) }. ", "5a6049a3a0afe9389feff19d25a613da": "\\mathfrak{p}", "5a606fe88ae1a9e27d0ae86d5dc6e330": "t_1 \\leq MacD \\leq t_n,", "5a6090323b2739b025c906ff1bbf011a": " x_{k} = x_{k-1} - \\frac{2f(x_{k-1})}{w \\pm \\sqrt{w^2 - 4f(x_{k-1})f[x_{k-1}, x_{k-2}, x_{k-3}]}}. ", "5a609fdd3e319087ee37ff2f7764fa9e": "\\vec{a}_1", "5a6132643c5f3dc79042ab3740188f19": "2^{1.6 \\times 10^{18}}\\approx 10^{4.8 \\times 10^{17}}", "5a61a640f89e85da3a5ab4f4e6905f92": "R \\xrightarrow[-g]{} RB", "5a61c12c2322bd515edf878ad4d75aec": "\\Vert x \\Vert\\to \\infty", "5a62106a4ce061e40e8ededc48fce2e3": "\\omega_s = \\frac{geB}{2mc} + (1-\\gamma)\\frac{eB}{mc\\gamma}", "5a62346924caca7323c5aa8c7fa953a2": "n_\\infty", "5a629152ce9cebe4401f7d2e15d9fbe8": " F(s,T) - F(t,T) ", "5a629c8008acd1cf32db748b9268b319": "\\partial\\!\\!\\!/=\\sum_{\\mu=0,1}\\gamma^\\mu\\frac{\\partial}{\\partial x^\\mu}\\,,", "5a62a591932ce69be534960978f6b388": "\\eta_2=0", "5a62a7d5fc29356e417bbc50af5d3be4": "m \\le f(x) \\le M\\quad\\text{for all }x \\in [a,b].\\,", "5a62b87b6af47125717edcaea63aaa99": "\\rho_1(z)= z \\, ;", "5a62dab915f858ff59d02c306369c2ab": " = 1/kT", "5a62e5728e7c784236ac4d5d8d309fd5": "\\displaystyle l(s) = \\frac{1}{n}\\sum_{i=1}^n l(s_i)", "5a630825000eaa8a51181ce5d24605f6": " F_T = \\frac{2GMur}{d^3}", "5a6308ad467b93f6d48f2c147470f128": "\\left( \\sum_{ij} | m_{ij} |^2 \\right)^\\frac{1}{2}", "5a6328b254ba3af0fca2139dfc563dae": "\\gamma(\\{a_0, \\ldots, a_k\\}) = \\sum_{i \\leq k} 2^{a_i}", "5a63374eb0996b9984cb893b2cbd4e8a": "\\displaystyle \\vec F'", "5a644d0d4f780b86c33d5910cd8ee740": "N^{-1}D = \\{\\,y \\mid xy\\in D ~\\textrm{ and }~ x \\in N \\,\\}", "5a646a32398d08a357ae5cf041e626a9": "\\scriptstyle V(\\cdot,\\Omega)", "5a64b22d098e524c122a4e03d1b54733": "(1 - f^\\mathrm{e}_{\\mathbf{k}} -f^\\mathrm{h}_{\\mathbf{k}})", "5a64e916931efa7a58876c3a6fce84c1": "\n\\frac{\\partial\nu(S,t) }{\\partial t} + \\frac12\\sigma^2 S^2 \\frac{\\partial^2 u(S,t) }{\\partial\nS^2} + \\mu S \\frac{\\partial u(S,t) }{\\partial S} =0\n", "5a658042e76b36bb8008496381a93400": "A := \\{ 1, 2, \\dots, \\lfloor N / 2 \\rfloor \\}.", "5a65836744d2c5b254a832e3031307a8": " a=\\frac{1}{\\sqrt{I_1}}, \\quad b=\\frac{1}{\\sqrt{I_2}}, \\quad c=\\frac{1}{\\sqrt{I_3}}.", "5a65852c11e26cf73132303d37203c34": "h[n]\\,", "5a65b537e349f48870f09a6ab61acce0": "(A \\and B)", "5a660248cc8bdf44ed6688d220e8a83f": "\\hat{h}_E:E(\\bar{K})\\to\\mathbb{R}", "5a667e81b7c3bc35aa6afe4721595938": " \\gcd(\\tau,m)=1 ", "5a66ada27a014825f01c5bfab4836aa8": "f^{e}_{\\mathbf{k}}", "5a66db946ef15e7e4f98671c42c640bd": "\\frac{5 \\cdot \\pi}{12}", "5a66dd644faf6c63dc4b1424b7018083": "\\begin{array}{rcl}\n \\dot x &= &Ax + b x_r(\\theta_r(t)) x_c(\\theta_c(t)),\\\\\n \\dot \\theta_c &= & \\omega_c + g_v (c^{*}x) \\\\\n\\end{array}\n\\quad\nx(0) = x_0, \\quad \\theta_c(0) = \\varphi_0.\n", "5a670f0f8ed4979324643bd0a9ba32aa": " x_1 \\approx -\\frac{b}{a} .", "5a672b3584c6f591ce83fa68d11947b9": "A-A", "5a675ab60e6c5ffef382a11f6796a55d": " | \\psi' \\rang = | n \\rang ", "5a67d1c15f6958a54b96f73ad4a71e71": "SL(m,\\mathbb C)", "5a67d22b9f7cb6df095cbca9dafd30a8": "(y_1,... y_4),", "5a67eda1ef9fae1540a58ad29f76050c": "(x,y,z)\\mapsto(x,y,-z)", "5a680d68ecb4335cb915609740348497": "\n\\begin{align}\na_1 x_{n} + b_1 x_1 + c_1 x_2 & = d_1, \\\\\na_i x_{i - 1} + b_i x_i + c_i x_{i + 1} & = d_i,\\quad\\quad i = 2,\\ldots,n-1 \\\\\na_n x_{n-1} + b_n x_n + c_n x_1 & = d_n.\n\\end{align}\n", "5a683a06ba343db32e1b023ddca78af9": " n^2 - ((n-1)^2 +1) = 2 n - 2. \\quad ", "5a692cb1e7bf00e860feb74c451d5b09": "n\\times q", "5a698f190a4c3547d7a2fb57e7fdc5b3": "\\sigma_{GB} dA \\text{ (work done)} = \\gamma_{GB} dA \\text{ (energy change)}\\,\\!", "5a69a2964d156379daf10166656dc1e5": "\\Box, \\blacksquare, \\diamond, \\Diamond \\lozenge, \\blacklozenge, \\bigstar \\!", "5a6a16078f123335c64831766a104171": "~\\mathrm{add_c}~", "5a6a9cbf4b7b0aae1181c7c9277971a9": " \\and D[f] = [F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_ ", "5a6acf2ead5c5048d763fe5713ede733": "|H_R(f)| = |H_T(f)| = \\sqrt{|H(f)|}", "5a6af0f8a951732a2c69bfc2643b2691": "\\alpha=0.5, \\beta=1.0, \n\\delta=0, \\epsilon=0.6, \\zeta=0.18, \\eta=0.6", "5a6b0397a77201dfe6cb837138ecba82": "(f\\cdot g) (m) := f(m) \\cdot g(m) ", "5a6b0b492fc89a4326f2f62d0db1836e": "S = - k_{\\mathrm{B}}\\sum_i p_i \\ln p_i \\, ,", "5a6c631e029028e6ccd9d895f10d5366": "\\delta_x(U)=\n\\begin{cases}\n0 & \\mbox{if}~x\\notin U\\\\\n1 & \\mbox{if}~x\\in U\n\\end{cases}\n\\quad\\forall U\\in\\mathcal{T}\n", "5a6cfe03aad9b5f65e965f80fa888d46": "\\delta S= \\delta\\int_{\\sigma_{A}}^{\\sigma_{B}} L\\left(q_1,\\cdots,q_N,\\dot{q}_1,\\cdots,\\dot{q}_N,\\sigma\\right)\\, d\\sigma=0", "5a6d560841e551423b9eb9c3bc7135e0": "\\textstyle r \\ge \\lceil \\log_2(n+1) \\rceil + (b-1).", "5a6d7840c4076287dd3866e7cbfdbd5b": "[\\psi_1] + [\\psi_2] := [\\psi_1 \\oplus \\psi_2].", "5a6d8ba4bd255f9c864e9531ea7482f3": " \\sum_{n=0}^{\\infty} \\frac{1}{(n+a)} ", "5a6dacfbc70e062e618d3e187d9b6bb9": "\\frac{\\hbox{twelve fifths}}{\\hbox{seven octaves}}\n=\\left(\\tfrac32\\right)^{12} \\!\\!\\bigg/\\, 2^{7}\n= \\frac{3^{12}}{2^{19}}\n= \\frac{531441}{524288}\n= 1.0136432647705078125\n\\!", "5a6de2c62c700aa04040943bf7122671": " e = 0", "5a6de8a5f6fb338f33ed524195525161": "(\\forall x:A . B)", "5a6e2e97034d70f146386c5ce10eb388": "3.4~m_e = 3.4", "5a6e7177a982101ac0c56f051fe7bc4f": "M=(Q,\\ \\Sigma,\\ \\Gamma,\\ \\delta, \\ p,\\ Z, \\ F)", "5a6f12294a0383f607e7d439c95c7c6b": "\\color{Apricot}\\text{Apricot}", "5a6f29040c5006d8f8a5151a7279733f": "{h_{\\mathrm{f}}=4f\\frac{l}{d}\\frac{V^2}{2\\,g}},", "5a6fb152b0e79d61bb16fd58014ba123": "y = 1", "5a6fb559c4acb267b0412b0887898e01": "\\begin{matrix}{4 \\choose 3}{4 \\choose 1}\\end{matrix}", "5a6fd687a8b97b9cd21315ec43d0acf6": "\n\\varepsilon^*(\\omega) - \\varepsilon_\\infty = \\frac{\\varepsilon_s - \\varepsilon_\\infty}{1+(i\\omega\\tau)^{1 - \\alpha}}\n", "5a70092b4148075ec06efb969587fd79": "\\int_{\\Omega} \\frac{\\partial u}{\\partial x_i} v \\,d\\Omega = \\int_{\\Gamma} u v \\, \\nu_i \\,d\\Gamma - \\int_{\\Omega} u \\frac{\\partial v}{\\partial x_i} \\, d\\Omega,", "5a7022b56b5f3b03b0fe2520ee3d6881": "F_2\\frac{Sin(\\beta )}{Sin(\\alpha )}Cos(\\alpha )+F_2Cos(\\beta )=F_{load} \\,", "5a702493ede9d8f04ef8bd9ae9332628": "abs(\\lambda) = 1 \\,", "5a7028a55b062451d3a8cbcf105350f3": "\\hat{A}_1 = x[0]", "5a706b393bdb83023db1ecefffecb4aa": "R(t)={\\frac{k_T D} {\\int_{0}^{t} I(t) \\, dt}}", "5a708b129efbe75427b415626f03e9ed": "\\scriptstyle P_{mmHg}=10^{7.18807 - \\frac {1416.7} {211+T}}", "5a71263c7d53eb34c5356b749773d4ae": "\\mathbb{Q}[x,y,A,B]/(y^2-x^3-Ax-B)", "5a712f357dd1ba34d1bbe98b66b97541": "E(\\sqrt{1-k^2}) - K(\\sqrt{1-k^2})", "5a7138ebb6e0b916a4629c2eabff31e6": "1-x\\,", "5a7166149c74be6f5507a714fb9afd2a": "\\lambda(L(B)) \\leq \\zeta(n) ", "5a7182e0708e8416b3878212a52c380e": "u(x) = 1 - \\exp(-x/10)", "5a71aa3cd3d0f3304769a6929cd55dbf": "\\not\\to", "5a71ae634365d7c0f222f03d1961f12c": "\\mathrm{Area} = \\pi r^2.\\,", "5a71c69b458a183975eff02b4cd80e4b": "T := I", "5a71d2037f7861365b59762fcbd546ca": "\\bigg\\Downarrow", "5a722070548e2094d52dad9d10ef4611": " k_{\\rm H,pc}(T) = k_{\\rm H,pc}(T^\\ominus)\\, \\exp{ \\left[ -C \\, \\left( \\frac{1}{T}-\\frac{1}{T^\\ominus}\\right)\\right]}\\, ", "5a72502019ac121193fd352a20e4dd0c": "p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\\\\naq&bdq&cq\\\\ \\end{matrix};q;q\\right].", "5a7254664913b100a967ef6a49e6e29c": "\\scriptstyle{a_0^2}", "5a728084b2a5da314eac60f006e1fa0b": "(\\exists x P) Q", "5a72b03f8725100acc402cc431f3f268": "\\int_0^{\\sqrt{2}-1}\\! I_\\mathbf{Q}(x) \\,\\mathrm{d}x + \\int_{\\sqrt{2}-1}^1\\! I_\\mathbf{Q}(x) \\,\\mathrm{d}x = \\int_0^1\\! I_\\mathbf{Q}(x) \\,\\mathrm{d}x.", "5a72f1304af0783657605aed0e38201a": "\\Delta t", "5a7315c7f58f44345f0277de8271a4da": "AB/HR=\\frac{AB}{HR}", "5a7327a58a9fca296b73f484b2a34a1d": "X_{3}^\\mathrm{opt} = \\alpha W + \\beta \\mu_3 = \\frac{\\mu_3 - \\mu_2}{\\mu_1 - \\mu_2}X_{1}^\\mathrm{opt} + \\frac{\\mu_1 - \\mu_3}{\\mu_1 - \\mu_2}X_{2}^\\mathrm{opt}.", "5a73ae4bcc13708eb04156912ba27063": "\\widehat{P1O1Q}", "5a73c2f98f1f292e5d18c1fab09fe58f": "i_{}", "5a73d85d0b79cffff34155a7956f6bc3": "\\left[\\frac{1} {3}\\left(P_1^\\alpha+P_2^\\alpha+P_3^\\alpha\\right)\\right]^{\\frac{1} {\\alpha}}", "5a743250d41514e6fa524b8048e1191b": "A_k(n)", "5a744131add8161f3a698e43e01ec96a": "v^{i}=dx^i/d\\mu", "5a746fcec1e43e0f5dc5d20892953982": "\\, v", "5a74b79b805c1339bdc324d15be8292e": "1 + c_1 + \\cdots + c_n = 0", "5a7542d8c159f40eb15ec2d7ebcc520e": "(g-1,g,1), (-g,-1,g-1)", "5a7555bf7ce0cf8a5d89acfa6ad133b4": "\\deg(B)=b", "5a7588ac9d5eb68f159d528dac3c8f51": "O(n^{V(\\mathcal{C}) - 1})", "5a758de987fda1dd8f29a489c045b273": "E_{\\gamma} = hf\\!", "5a75c6aaa011b76f01e85096dca311d3": "T\\,", "5a75ebf7f7e3e7679cc1da9bbbff9de2": "M_p\\equiv C^{d_p} \\bmod\\ p ", "5a760cc00595d872b7329c34e9857a93": "\\textstyle {(0+4+0)!\\over 0!\\times 4!\\times 0!}", "5a761d5e6ffa49072eaeadb12af7045b": "m \\ge 1 ", "5a7672d3b9306b5c722d7c848c813127": "e^{(\\mu + \\frac{2 \\pi i k}{T})T}=e^{\\mu T}", "5a76799562397acd5455b2c3b532da75": " \\| Df(x)\\|^n\\leq K|J_f(x)| \\,", "5a76be7c57c3eeeaf085b229cabae9ab": "n_g = n-\\lambda_0\\frac{dn}{d\\lambda_0},", "5a76c5d605a43027f00b29924d3c1d6d": " \n \\frac{dx}{dt}=Px+qf(e),\\quad e=r^*x \\quad x\\in R^n,\n", "5a77048227f5a69e65cb1a95f7c464d9": "\\mathbf{s} = \\mathbf{s} + (\\mathbf{x} \\cdot \\mathbf{r})\\mathbf{x}", "5a77424415a6817abd64418d98b20efb": "\\mathcal O_F", "5a775210bb0a1d4db35228b53ff93d2a": "x=x_s(t),y=0,z=0", "5a775dadb7521193d90aa18a20c13a72": "\\beta_{n+1} \\,.", "5a776adef967af9c8ba3f2c58f1f6593": "\\mathbf{\\hat{n}}_i", "5a77adbf37c41f47056e09128f48821c": "s_{M}=\\sum_{i=1}^m x_i", "5a77ee935e0e92da10ac46b3e8fa3273": "|e|", "5a7820c93fbec440043e7e57734730b6": "a(\\cdot, \\cdot),", "5a785240640a3cfb179136209eb1ffad": "\nN\\left( u\\right) N\\left( v\\right) =\\left( -1\\right) ^{\\left( u\\odot\nv\\right) }N\\left( v\\right) N\\left( u\\right) .\n", "5a787541bbb807b77624c937b1f2e7a3": "\\! \\prod_{n=1}^\\infty \\! \\left( 1 \\! + \\! \\frac{(-1)^{n+1}}{2n-1} \\right) \\! = \\! \\left(1 \\! + \\! \\frac{1}{1}\\right) \\! \\left(1 \\! - \\! \\frac{1}{3} \\right) \\! \\left(1 \\! + \\! \\frac{1}{5} \\right) \\cdots ", "5a78b08aa614722d2143f51dae1e8793": "z\\frac{d^2w}{dz^2}+(b-z)\\frac{dw}{dz}-aw = 0.", "5a78c63e4f3441d02eb5fbd6975a441f": " m = 2^{32}", "5a78f67e0756a6227c77c888ad69e0c4": " f^\\prime(x_n)\\,", "5a791572a1d55177429cb7f4537e7446": "k * (1 + i)^{c_0} = \\prod_{n=1}^N (b_n + a_n i)^{c_n}", "5a793c7d25b0e8dd4298800156939303": "\\varepsilon(\\mathbf e_g) = 1 \\,,", "5a7944b3bef23f227fc112ef6f9f722d": " (\\partial G)_U=-(\\partial U)_G=-VC_P+PV\\left(\\frac{\\partial V}{\\partial T}\\right)_P+ST\\left(\\frac{\\partial V}{\\partial T}\\right)_P+SP\\left(\\frac{\\partial V}{\\partial P}\\right)_T", "5a7a129d96244bf06637e2a6178fe851": "13983816 = \\frac{49!}{6! \\, 43!}", "5a7acbada160555787a226a9dd9b32cc": "X^H = \\{ x\\in X \\mid h\\cdot x = x, \\forall h\\in H\\}.", "5a7ad623757de01c65fa0c95e3aefcaa": "\\lim_{\\epsilon\\to 0} \\Delta p_{2} = \\sqrt{\\sigma^2+ \\hbar^2/16\\Omega^2}", "5a7aec643c8b6a529f45797d5df82224": "\n[x_1] =[x_1]\\ \\cap \\ \\ [a]-[x_2] \n", "5a7b0688236c3e46cfb6a575a31ae306": " D = M \\frac{\\partial^2 f}{\\partial c^2} ", "5a7bba0077e13d93b37a2993319bd67a": "\\displaystyle h_n(x;q)=q^{\\binom{n}{2}}{}_2\\phi_1(q^{-n},x^{-1};0;q,-qx) = x^n{}_2\\phi_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2) = U_n^{(-1)}(x;q) ", "5a7bd5b73b05d2978936b36abc5ea203": " s *_l r(v) = f * g(v) = g * f(v) = r *_l s(v).\\, ", "5a7bdf4618a5566d161bf0d8f6d10f7d": "\\Pi_{G \\in \\Gamma}2^{S_G};", "5a7c09b9dadff168709bb9dcbd0657cb": "\\Delta E_n(x) = E_n(x+1)-E_n(x)=2(x^n-E_n(x)).\\,", "5a7c0fcac0dfafea2fabfdd370d4b77f": "(\\mathcal D,\\bullet, J)", "5a7cba52d30f81ffcf999ab361800ef7": " \\rho_i^*", "5a7cce5b244f2db809937920e1fe7ef7": "\\sigma_g", "5a7cfd4496cf65b317c3ca4e3950b2bf": " \\sqrt{ {(x_0 - x)}^2 + {(y_0 - y)}^2} \\quad (16)", "5a7d1b1c5acf3fc76f17a1ea8d10557a": " \\langle \\Psi_1,\\Psi_2,\\Psi_3 \\rangle ", "5a7d8396aafed92fec1dd6e2a8a3d639": "\\, r \\to \\infty \\,", "5a7d83d90fca862c3effdc3f76f5d9de": "\\frac{d^2B}{dr^2} = \\sum_{i=1}^{n} c_i e^{-r t_i} t_i^2 \\geq 0.", "5a7da8473239744f1c2764f202b76680": "v_{av}=\\frac{d}{\\Delta t}=\\frac{0.57}{4.5\\times10^{-3}}=127\\text {cm/s}", "5a7dcacb49e16aa7d98383ee9774bc60": "P_X=\\frac{X_{\\uparrow}-X_{\\downarrow}}{X_{\\uparrow}+X_{\\downarrow}}", "5a7dec0ee8dc2d9ac2bd8d32d1df27c8": " n=N; \\Psi(x) = \\|x\\|_1 = \\sum_{i=1}^n |x_i| ", "5a7dfef6955238ef3245dc1ce59fb740": "F_{1}Cos(\\alpha )+F_2Cos(\\beta )=F_{load} \\,", "5a7e496677d5762ecac53da132db97ea": "f^*\\colon H^*(Y)\\to H^*(X)", "5a7e9986b9982285ca8693faaec87df1": "\\mathbf{x}_0=[x_{n_1},x_{n_2},\\cdots,x_{n_j}]", "5a7ecbccc41d671ece8971429ed4337c": "\\scriptstyle \\mathbf{2}^\\mathbb{N}", "5a7eeb231f44dc74fbc34ca40bbf96a6": "\\frac {D_{P1}} {D_{P2}} = \\frac {\\rho_{P2}} {\\rho_{P1}} ", "5a7ef5baf65c45ebcf270819242a7c90": "\\tau_b=\\rho C_f \\left(\\bar{u} \\right)^2", "5a7efa90e1f639b2c5b33a475fef401d": "\\alpha^n=1", "5a7f3b0807ee9c3f3340ad08d8d95a8c": "f(x) = e^{-1/x^2}", "5a7fa14dac5c9d494aa75b18e1ad9a51": "x_w", "5a7fc542f983287f21a1a5c04dd61a5e": " E\\mathbf{x} = \\mathbf d ", "5a7fdbb23cc85e5590b64068f3c1676b": "u(x)>u(y)", "5a7fe0b552bac6c2a6af730ab856f137": "\n\\begin{align}\n \\textbf{G}_\\textrm{hkl} = h\\textbf{a}^*+k\\textbf{b}^*+l\\textbf{c}^*\n\\end{align}\n", "5a807612a715ff6fe63037833532ba3f": "\n\\mathbf{u}\\odot\\mathbf{v\\equiv}\\sum_{i=1}^{n}u_{i}\\odot v_{i},\n", "5a80cd39eeea68b98c67cca43b77ea39": "\\varphi(s)=\\sum^\\infty_{n=1}a_nn^{-s}.", "5a818075e671cd2a4b4d4273eb06aa9e": "\\pi_i (\\mathbf{RP}^n) = \\begin{cases}\n0 & i = 0\\\\\n\\mathbf{Z} & i = 1, n = 1\\\\\n\\mathbf{Z}/2\\mathbf{Z} & i = 1, n > 1\\\\\n\\pi_i (S^n) & i > 1, n > 0.\n\\end{cases}", "5a81a3934c89640c35e6e48705600f85": " c> 0 ", "5a81f69ca58c05a21e0a23658b861f07": "R_{AW} = \\frac {P_{\\mathrm{ATM}} - P_{\\mathrm{A}}}{\\dot V}", "5a823171bd500bd233774a5d5e55e8fe": "E=\\frac{p_0a_0+p_1a_1+p_2a_2+\\cdots+p_na_n}{p_0+p_1+\\cdots+p_n}.", "5a8232b6a20915781f4efd585962624b": " x_{2}^{2} < 3.", "5a8242abbfe64f4fc3094a98007954d5": "\\frac {m_2 (u_2 - u_1)}{m_1 + m_2}", "5a8267550a5f1720b562851c55e21bb9": "\\overline{\\phi}.", "5a83341087edf295a05faac035eb6188": "|N_i|", "5a8395df0279f9096e7c8a22b0ea4d53": "\n \\hat\\varphi(t) = \\frac{1}{n} \\sum_{j=1}^n e^{itx_j}\n ", "5a83cc428ed0c5ac2bffcb3e9bc43ff8": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_\\mathrm{f} + \\frac{\\partial \\mathbf{D}} {\\partial t}", "5a83fecaf9c5a88464f3c1025732fcd1": "\\displaystyle \\mathbf{u} = \\begin{pmatrix}\n \\mathbf{u}_1 \\\\\n \\mathbf{u}_2 \n\\end{pmatrix}", "5a8409437fd73edacb0bffea15b6231e": "\nr = \\left ( \\frac{\\textit{Positives}}{N} \\right ) ^2+ \\left ( \\frac{\\textit{Negatives}}{N} \\right ) ^2=f(\\textit{Positives})^2 + f(\\textit{Negatives})^2\n", "5a8433977d29be8f3dd6b4a222730d7d": "m_{k+1}", "5a8440eb4040f15f70934a97f2dfddeb": "\\mathcal{H}^A\\otimes\\mathcal{H}^B\\otimes\\mathcal{H}^C", "5a85192c7f14cc95081ca4d237e0f893": "\\Omicron(\\log\\ell)", "5a851e31d70a2435cf7c7cbd8dbdf39b": "E(\\tau,s) =\\sum_{(m,n)\\ne (0,0)}{y^s\\over|m\\tau+n|^{2s}}", "5a85afb29a18c67ef8f069dcdcd62518": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) \\ge 1 - \\xi_i \\quad 1 \\le i \\le n. \\quad\\quad(2)", "5a85c1d5c9f30b7892504075ebbac192": "\\mathbf{R}^o ", "5a85eba63b17531373c662004a73b0dd": "(\\operatorname{arcosh}\\,x)' = {\\frac {1}{\\sqrt{x^2-1}}}", "5a86017d09bfc3bcf24f1684dcb8fddc": " \\mathit{g^{(2)}}", "5a87ac185776533f4802424b753d0237": "-\\frac{\\partial \\rho(\\mathbf{x},t)}{\\partial t} = \\nabla \\cdot \\left(\\rho (\\mathbf{x},t)\\frac{\\nabla S(\\mathbf{x},t)}{m}\\right)", "5a8839683ff152f7d082aafdebee62d5": " A + \\gamma \\rarr A^*", "5a88409b9783a19024450bacb9799e1a": "R_{long} = \\frac{r_LL_1}{2 \\pi a_1^2} + \\frac{r_LL_2}{2 \\pi a_2^2}", "5a888c58a45a2cb966b2da313b350e6b": "G = G_{\\infin} \\frac {T} {1+T} ", "5a889bf314026c0e612c81b4b4009f84": "\\frac{dI_\\nu}{ds}=\\kappa_\\nu\\rho(B_\\nu-I_\\nu).", "5a8906117cfddb715cdf7892d7117864": "\\beta = \\tfrac{v}{c}", "5a890e663b21a306b749196731ac1ce7": "\\ \\displaystyle r_{w} \\le R(q,u) \\ ", "5a8912cf5363c4952a065841f9b2eff0": "(a_1,\\dots,a_n)\\le(b_1,\\dots,b_n)\\iff a_i\\le b_i\\text{ for every }i=1,\\dots,n,", "5a891a4b195cc43c8188088fceb928b3": "C_1: f_1(x,y)=x^4+y^4-1=0,", "5a891f149f126bb3b1aacd63bc7ca311": "\\,\\! max(\\pm r_0 \\pm r_1, \\pm r_0 \\pm r_1)", "5a8991740ab669b7dd296b3a21211c20": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 211\\cdot 8.17)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 370.8\\cdot R_{\\bigodot}\n\\end{align}", "5a8a23805d88f94ebd6cec0e52b64b48": "\\langle e_i, e_j \\rangle = 0 \\qquad i\\neq j. \\,", "5a8a3176f08d989c3c86dd962f256d3a": "c_{9}-a_{11}", "5a8a42b03e5e45ddd0ac8d66648514b4": "\\frac{\\Phi_M(c)}{c} \\to 1 \\ as\\ c \\to \\infty \\,", "5a8a5803de51f21be0f36fe9c67d95c5": "\nQ(\\mathbf{p})|1_\\mathbf{p}\\rangle = |0\\rangle,\n", "5a8abae873c22f253ce64da8b0e2dff1": "x+y = 2 \\pi - \\alpha - \\beta - C", "5a8af3b7e11e098a0a1804ece14bbf39": " \\Delta F = -kT \\cdot \\log \\left\\langle e ^{\\beta (U_\\text{A} - U_\\text{B})} \\right\\rangle_\\text{A} ", "5a8b15810c9c904a7a65fc2cf93ac847": "\\frac a c < \\frac{a+b}{c+d} < \\frac b d. ", "5a8b5c72ebeda16f7c4692b863050ed0": " BA = 0.00007854 \\times DBH^2 ", "5a8b7b42a1b37839f44f70aeb43788a8": "(1-\\sqrt2)^n=H_n-P_n\\sqrt{2}.", "5a8b9958e3970b44da18e497a8ed3c66": "\\mathbb{R}\\times L", "5a8c670c16edaa6268cf58b5ee9b2487": "\\sum_if_{i*}\\alpha_i\\equiv 0,", "5a8c8c7b2c9e02d72ed0b2ca932cafa6": "\\mathcal N\\models\\psi(n) \\iff n", "5a8cac4d15e739e91ac42c56aacd49d8": "w=\\frac{\\sum_i(x_i - \\bar x)(d_i - \\bar d)}{\\sum_i(x_i - \\bar x)^2}", "5a8d2197ac28a9be62c81c36966b2658": "X\\subset M", "5a8d220733b1569d2e3c6d9a2295be98": "b_3b_2b_1b_0", "5a8dadaf1ad5b3e2ea7d8c4547d240f3": "P_1 V_1 = P_2 V_2 \\,", "5a8dd0bf5870fb28ff75dbb97b000dc9": "y_1.", "5a8e602535ce293de7d714b7fc74838a": "b(k)", "5a8e6b14b02a688028c9333e1f49095a": "P(\\vec y|j)", "5a8e9059e7aaed8cc08c511753eff61a": " \\theta = p\\alpha - p\\theta\\alpha ", "5a8eab40d97b8f81718de6dce56307ee": "m[i,w]", "5a8eb724b8aa9d37f24e125b18f8d046": " (\\cdot,\\cdot) : F^*/F^{*2} \\times F^*/F^{*2} \\rightarrow \\mathop{Br}(F) ", "5a8ec34994b6a5acf86011d84495a6c6": "i=\\sqrt{-1}", "5a8ece7ccb91338b6273e8254dd9b628": "\\nu = 2 n+1", "5a8eea82e068e69e0f2b27ba9896ff3e": "\\mathrm{d}\\mathbf{\\Sigma}^2 = \\mathrm{d}r^2 + S_k(r)^2 \\, \\mathrm{d}\\mathbf{\\Omega}^2", "5a8ef17f8dbf946815a01f71cc68a7e3": " dy = \\frac{\\partial y}{\\partial x_1} dx_1 + \\cdots + \\frac{\\partial y}{\\partial x_n} dx_n, ", "5a8efd70a89567fa5d72d0b71e9af835": "B = \\sum_i {x_i\\, B_i}", "5a8f253c077a8ff4c1a89c287ebfd776": "\n\\begin{bmatrix}\n Q & E^T \\\\\n E & 0\n\\end{bmatrix} \n\\begin{bmatrix}\n \\mathbf x \\\\\n \\lambda\n\\end{bmatrix}\n= \n\\begin{bmatrix}\n -\\mathbf c \\\\\n \\mathbf d\n\\end{bmatrix}\n", "5a8f52285053c6c5522e2f39d37c06bf": "12n + 1", "5a8fd5503ec90fc265f00c587890b15f": "\\lambda x_0 + (1 - \\lambda)y \\in \\operatorname{core}(A)", "5a8fecd19002a0ae0f6592b178076227": "((ab)c)", "5a9008dee93f780985429d2e0a6478c7": "\nP(x_1,x_2) = 49s_2(x_1,x_2)^2+14s_1(x_1,x_2)s_2(x_1,x_2)+s_1(x_1,x_2)^2.\\,\n", "5a900cd72cb9a5d5162ec5946db6d6cf": "S_{e2e}(t) = (S_1 \\otimes S_2)(t)", "5a9012576b466b88abe7f707b11a9f99": " s_i^2 = am_i + bm_i^2 \\, ", "5a9060421081c9ad6a6abc5067f58b3b": "E_{kin} = \\frac{1}{2} m v^2", "5a90ea8935b47f7a9cf2d58e2904fe3f": "U(\\mathbf{v}) =\n\\min_{\\hat{x}\\in\\mathcal{X}}\\mathbf{v}^\\top \\lambda_{\\hat{x}}", "5a91440b99bc7f52a1c2bb547fca4771": "\\mu \\ll \\nu \\iff \\left( \\nu(A) = 0\\ \\Rightarrow\\ \\mu (A) = 0 \\right).", "5a915461a0e8c601c9a5031d2dfa7900": "V\\Sigma^+U^*", "5a9172fc228c8b44b4cca185cad004ce": "\\omega^2=a^2-n", "5a9199474f5377f03a0379f9711b59b6": "L = \\sqrt{S^2 + 4d^2} = \\sqrt{S^2 + 4\\left(\\frac{{WL}}{{4T}}\\right)^2}", "5a91a658c6b7140860f161b6a0bed356": " (-1)^{m}\\sum_{r=0}^m \\binom{m}{r}B_{n+r}=(-1)^{n}\\sum_{s=0}^{n}\\binom{n}{s}B_{m+s} ", "5a91b74c8e79d8317ea432a6f0be5029": "{\\phi}", "5a91c3704d9bca088294ec9143bc3f1a": "w(C_1\\mid C_2) = \\min \\{ 2w(C_1) , w(C_2) \\}. \\,", "5a92026785ae299a10120af6d871e3f6": "\\beta + \\alpha = \\alpha.", "5a923046a12953bfe2288de7c238d5b1": "n \\ge 5", "5a92662e5108e35069708f57486637b2": "w=Ae^{-\\frac{1}{2}\\beta_{ik}x_ix_k}\\, ; \\;\\;\\;\\;\\ \\beta_{ik}= -\\frac{1}{k}\\frac{\\partial^2 S}{\\partial x_i \\partial x_k}\\, ,", "5a92aa996fc50c8114eb5c4da899216d": "\n\\frac{m-p+1}{pm} X\\sim F_{p,m-p+1}\n", "5a92c3a99f9bbe44ff93d6f9e9de6da8": "\\alpha^{-1}(L_n)=L_n -{1\\over 2} J_n + {c\\over 24}\\delta_{n,0}", "5a934df6b865f5a2ad871dda0a68fae9": "\\delta = 2", "5a934e202565cfed0a57e0a35029fd0c": "\\{\\phi'^i(x),\\phi'^j(y)\\}=\\{\\chi'^i(x),\\chi'^j(y)\\}=\\{\\phi'^i(x),\\chi'^j(y)\\}=0.", "5a9360f6f62d5abd020f8fa0d162caa6": " = q \\hat{z} \\sigma_\\mathrm{tr} / \\sigma_\\mathrm{tot}", "5a93bd2e9f041c039f347c6bfd35287d": "\\operatorname{mr}(C_n)=n-2", "5a947c1241bbb6bf6bce662c3176653a": "\nQ_{LW} = \\epsilon_i \\sigma \\left [ 0.39 (1 - k_{cc}(\\phi) cc^2) T_s^4 + T_s^3 (T_s - T_a) \\right ]\n", "5a94aae03c8350f422762cd472ea04e5": "\\pm\\sqrt{1 + \\cot^2 \\theta}\\! ", "5a94be3e6ac92fd2c85133a76b7c3456": "\\displaystyle{E(z) ={1\\over \\pi z},}", "5a94bfa0ba35ecb902f842a2e558e108": "1 + 2 + 3 + 4 = 10", "5a94f2a2a4b06ec2e3dfabda3b99c781": " I(y_j) = \\begin{cases} \n 0 & \\textrm{if} \\; y_j = y_L \\\\ \n 1 & \\textrm{if} \\; y_j \\neq y_L.\n\\end{cases}", "5a9502b49f8aeb031d94f96cecee11cf": "c_2: X_2 \\rightarrow Y_2", "5a951fb5adbdfed94a0bb9ec04e0208f": "\\frac{\\mathrm{d}G}{\\mathrm{d}t} \\leq 0.", "5a9548bc5cc5ce647bca838543990ba0": "P \\rightarrow b", "5a958131bdbcb9c2d35101124ca777b5": "|\\nabla f|=1.", "5a9599b613043383f7bfb79d3d51ac09": "\\sum_{n=0}^\\infty f(n)= \\int_0^\\infty f(x) \\, dx+ \\frac 1 2 f(0)+i \\int_0^\\infty \\frac{f(i t)-f(-i t)}{e^{2\\pi t}-1} \\, dt.", "5a95ba978677c242061c38f2d6b1186b": "n = \\pm p_1^{d_1} \\cdots p_r^{d_r}", "5a95cb547355509da37c206e7428eb35": "Z_3 = Z_1^2 Z_2^2 - eX_1^2 X_2^2 ", "5a95ebca1ef0eecc9184b1897f0aa21e": "\\langle s,r \\rangle (t) = \\int_{t'\\,=\\,0}^{+\\infty} s^\\star(t')r(t+t') dt'", "5a95ec4ceeaea6b3997eeb38b2ca1e2d": " x^M + \\sum_{m = 1}^{M} P_m x^{m - 1}.", "5a968cb37f16b70bf4369f579b11a29d": " {}^\\mathrm{t} f (\\phi ) = \\phi \\circ f \\quad \\forall \\phi \\in W^* .", "5a96c4dba11a87d4df9d0de530760f3b": "(a,1) \\circ (b,0)=(a - b,1)", "5a96d8816c2c4a1f40737e46aa2e73d3": "F = Q(s_i)", "5a972fcd85c2372afe46d9c394bcfb92": "\\sigma\\to\\tau\\to\\rho", "5a977d01923be5711fb8caa62ff7a4e9": "NP/N\\triangleleft\\text{that}", "5a981daa3121cea06ce30df378c74a66": "(2^{|k_{b_1}|}+2^{|k_{f_2}|}+2^{|s_2|}", "5a982c0fbaf5d2651abe61fa2692e436": "\\Box(x\\land y)=\\Box x\\land\\Box y", "5a9835fdbf5b2adbcff24779a1aede36": "F_{q^n}", "5a987db4db8b5329f4dcab39b1599da0": "D:= \\nabla_\\ell=l^a\\nabla_a\\,,\\; \\Delta:= \\nabla_\\mathbf{n}=n^a\\nabla_a\\,, \\;\\delta := \\nabla_\\mathbf{m}=m^a\\nabla_a\\,, \\;\\bar{\\delta} := \\nabla_\\mathbf{\\bar{m}}=\\bar{m}^a\\nabla_a\\,,", "5a988e3fb2c29c65a4c58fd4bacee3c8": "\\textbf{Cl}\\,\\!", "5a98a6aedfe0234b0d5f8176eb52e243": "\\{B(x_i,r_i):i\\in I\\}", "5a98ccb74a4b6e09168b6221bb4abad2": "\n\\mathcal{L}_\\mathrm{EW} =\n\\sum_\\psi\\bar\\psi\\gamma^\\mu\n\\left(i\\partial_\\mu-g^\\prime{1\\over2}Y_\\mathrm{W}B_\\mu-g{1\\over2}\\vec\\tau_\\mathrm{L}\\vec W_\\mu\\right)\\psi", "5a98df690105da28669b946b122f5dbc": " c_+ r_+ + c_- r_- ", "5a9999b8410447452b63ca332d98923a": "\\mbox{D}(x)=a^{-1}(x-b)\\mod{m},", "5a99e52459e5d7d1e86ba353790b975d": "\nE[U] = \\max_{d\\in D} ~ \\int_X U(d,x) p(x) ~ dx\n", "5a9a01cd2749da20209f667f1ea48a4b": "r(m+N) = rm + N \\, \\forall r \\in R, m \\in M", "5a9a06162c3b26c0ca78b298e78c4945": "Q_{\\rm ms} = \\frac{2 \\pi\\cdot F_{\\rm s}\\cdot M_{\\rm ms}}{R_{\\rm ms}}", "5a9a14c68158940e274e4b1a6bfdffde": "\\lambda X \\prec \\lambda Y", "5a9a19c0301eb86808c862fc6845b330": "\\mathfrak{sa}_3(A\\otimes B)", "5a9a3022e5627028edc95133e6ba4d2a": "\\varepsilon_0 = \\omega^{\\varepsilon_0}", "5a9a4c6a04eb86353c360f4668e7e64c": "M^{-1} = \\Omega^{-1} M^T \\Omega.", "5a9a86b1d79bba9319885871c6bdc7ba": "\\ell=p", "5a9ab0fbbadfc4da7aa740f1e1227fdc": "V_{r1} = V_2", "5a9ab6286e1124b1e17879f94ca8624e": "U_{Rp}(t) = R_p I(t) = R_p C_p(dU_{Cp}/dt)", "5a9ad302713f7739f121f71a8b263bab": "y_j", "5a9b42f09182bed17c322e161ba3d0b9": "Ki=f{(Q^{min},\\ldots,Q_i,\\ldots,Q^{max})}", "5a9b4600aeba3b27fa095f0fcc7e0b83": "\\textstyle n_a", "5a9b8809c854d038f69416b793519727": " \\frac{\\partial^2 u}{\\partial \\rho^2} + \\frac{\\partial^2 u}{\\partial \\theta^2} = 0", "5a9c3a776c90c6d7bbdda55284644572": "\\mu (A \\cup B) = \\mu (A) + \\mu (B)", "5a9cae2ca6b4795715273c371cccd27f": "x_0 \\in A", "5a9cb059d2d3d16d3ab5b62f8fbcd54c": "\\hat{\\mu}_3 = 56/3=18.67", "5a9ce7e7705c5f65657bb599b08e78f9": "L_1(x)", "5a9d1a4249e9e6d8a073c632820a0312": "\\frac{dy}{dx}(5y^4 - 1) = 1", "5a9d24a058e653bd68b9011f0bc52b25": " \\sum_{x\\in\\mathbb{Z^d}} g(z-x) = 1 ", "5a9d46c9e9572da20d29e4be9ff46817": "\\begin{matrix} {3 \\choose 2}{44 \\choose 2} \\end{matrix}", "5a9d4a6519020a9a7cce25deaf82b2aa": "(A\\equiv(B\\equiv C))\\equiv((A\\equiv B)\\equiv C)", "5a9d855a7a53f28da7a91a80cbc5da80": "\\left( \\frac{\\pi}{R} \\right)^2", "5a9e9a182e7559d2a86957a31cbcb8af": "\\mathrm{We} = \\frac{\\rho\\,v^2\\,l}{\\sigma}", "5a9f13763b5079f89642a19dc319bf02": "PG(3,2^h) (h\\geq 1)", "5a9f2e1e386b09d0cfe2285ce9f2c08b": "v_j=Be_j", "5a9f9743cf7506dcf36c442ee33ccfc9": "\\mathbf{[y]}", "5aa010555d73c023a16bccef810aff6a": "r = a + b\\theta", "5aa07d3c7471494f44f6ba8783eff2af": "u_{yy} = (-v_x)_y = -(v_y)_x = -(u_x)_x.", "5aa0ac144e79f720cb55fb0836781a64": "B \\supset A", "5aa1b3252aeca268747d44ec07e25cac": "m_{H}=2m+\\frac{E_{n}}{c^{2}}", "5aa2206a30eff624d5cf92a33d36751f": "g : S \\to S", "5aa226bde0f960cf7a1618dbe0395145": "f(|X-O|) + f(|O-Y)) = c .", "5aa26ae84b72d9350ae65d563ed5be61": "f(1) = 1,", "5aa2b60a567f9179d705e786181122aa": "(S^3, T^2)", "5aa2cb61a31af3aa4f89ce7e01e20e62": " \\mathbf{r}= \\bold{r}(t) = r\\bold{\\hat{e}}_r \\,\\!", "5aa3e18ec7b34d51979b89e98b277e97": "\\textstyle \\bold x(t) = A(\\theta)s(t) + n(t)", "5aa44d4d15cdf1ba75411b9590b5d5e9": "\\scriptstyle\\mathbf{R}", "5aa4b724ae413ba0f213d68b8d0f2389": "z^2 - 1", "5aa52412c3d5a1d1fd86b97dc6efae6c": "\\beta_2.", "5aa535757446470ccd9e94bca4f063dd": "\\bigcup_{i=_1}^n E_i", "5aa562960ab958c4fb5a398984100d58": "\n\\exp\\{\\kappa \\boldsymbol{\\gamma}_1\\cdot\\mathbf{x} + \\sum_{j=2}^p \\beta_{j} (\\boldsymbol{\\gamma}_j \\cdot \\mathbf{x})^2\\}\n", "5aa5758bb714985afb9b829f3cdf8046": "k_1 > 0", "5aa58b0f87e4eee0c23db56053462dc2": "\\widetilde{X} \\sim \\mathcal{N}\\left(\\mu, \\frac{\\pi}{2N}\\right).", "5aa599c8e96ef73e01706a7d88306b0f": "\\sigma>\\sigma_0", "5aa5ba69d5164cd74740cd946ec9f155": "a=1/(1+z)", "5aa5d7542980eadf2c276d2b83e9c2f6": "\\exists b \\in A", "5aa5e08347440c786c1c0d0afaa14dbb": "J^{\\nu} = {J^{\\nu}}_{\\text{free}} + {J^{\\nu}}_{\\text{bound}} \\,,", "5aa5e9dc406c5100abd4a992a489ce65": "\n\\operatorname{Li}_s(z) = \\sum_{k = 0}^\\infty (-1)^k \\,(1-2^{1-2k}) \\,(2\\pi)^{2k} \\,{B_{2k} \\over (2k)!} \\,{[\\ln(-z)]^{s-2 k} \\over \\Gamma(s+1-2k)} ~,\n", "5aa61fde5109e53aadfda25db2576106": "P(X, \\{x\\})", "5aa65c55129d5535e2e5d5630410f245": "P = \\big\\{p, q, r, \\ldots \\big\\}", "5aa66ae2348be0b0928bfb00aeb33893": "j-1", "5aa736ca0f78a72fc5241506560ea15d": "x^n = A", "5aa7c5c1091d75f8225bf473afbc568b": "M\\otimes_R-", "5aa7cf50c91093582aa8ef523ec2248b": "\\left(1-\\frac{1}{m}\\right)^{kn};", "5aa7e74138bd2b9f68441330309ad4ac": "T_i^{(n)}=\\sigma_{ij}n_j\\,\\!", "5aa7f81c88114bca6c9d34894739158a": "\\prod_v \\overline{\\mathcal{M}}_{g_v,n_v}", "5aa8150b3efadf511e767390e69f8d91": "d_Y(f(x), f(y)) \\leq M d_X(x, y)^{\\alpha}", "5aa8292a4867088caa853784b6bcdbc7": "H_z(z_1,z_2)=\\frac{\\sum_{l_1=0}^{L_1-1}\\sum_{l_2=0}^{L_2-1}a(l_1,l_2)z_1^{-l_1}z_2^{-l_2}}{\\sum_{k_1=0}^{K_1-1}\\sum_{k_2=0}^{K_2-1}b(k_1,k_2)z_1^{-k_1}z_2^{-k_2}}=\\frac{A_z(z_1,z_2)}{B_z(z_1,z_2)}", "5aa85007dff4cab1c2af48a93deae20e": "C_\\mathrm r = C_\\mathrm m\\cdot\\frac{(20.9 - \\mathrm{reference\\,volume\\, %\\, O_2})}{(20.9 - \\mathrm {measured\\,volume\\, %\\, O_2})}", "5aa885c944bdde012c31b3bd92dfbf9a": "\nu'(x) = \\gamma x - \\frac{\\gamma^{2}}{2} \\left(\\frac{x^{4}}{4!}\\right) + \\frac{11 \\gamma^{3}}{4} \\left(\\frac{x^7}{7!}\\right) - \\frac{375 \\gamma^{4}}{8} \\left(\\frac{x^{10}}{10!}\\right)\n", "5aa893880b6c3aaa192619a88fd90957": "\\begin{smallmatrix} \\log\\ \\operatorname{g}=\\log\\ 978.0=2.99 \\end{smallmatrix}", "5aa8f9966abb495281dde46d9cd45ac0": "\nP_{\\mathrm{acoustic}} = I \\cdot A\n", "5aa949846bb0f9e7dfb9622ddbdcb738": "\\Rightarrow \\sum_{i=0}^{h-1}N_i \\equiv 0 \\pmod{b^k-1}", "5aa9683fc3e5495d63cb285af7ec8909": " F_{forward} = lift \\times sin(\\beta) -drag \\times cos(\\beta) ", "5aa99492697fc2e042bbb9ee35424dc0": "{u}_{n+1}=u_n + {\\Delta}t~\\dot{u}_{n} + \n\\begin{matrix}\\frac{1-2\\beta}{2}\\end{matrix} {\\Delta} t^2 \\ddot{u}_{n} + \n\\beta {\\Delta} t^2 \\ddot{u}_{n+1}", "5aa9b76e2e2b32fe694917689306aa9d": "(x',y',z')", "5aa9fdb5d6f746346806224e5ebad3cf": "\n\\frac{x^{2}}{\\nu^{2}} + \\frac{y^{2}}{\\nu^{2} - b^{2}} + \\frac{z^{2}}{\\nu^{2} - c^{2}} = 0\n", "5aaa37c7c9341e50e98a61db35507d74": " g(t) = \\mathcal{L}^{-1} \\{ G(s) \\} ", "5aaa769a9f8bb84e15d5b58a1cb5b7ba": "f({\\mathbf p}q)=f({\\mathbf p})q", "5aaa81a0c93fa2f0d3e89741b9bc07d0": "\\beta \\gg 1 ", "5aaa92e6a368cfa39d1f3f64f213e343": "\\mu^{-1}((-\\infty, \\epsilon))", "5aaab0930f4df5082e30a9ed7004c219": "y'(a)= \\alpha \\ \\text{and} \\ y'(b) = \\beta", "5aaac17f5bf2e7daa957caf6451aec0e": "F(h(z)) = (F(z))^n\\,\\!", "5aab37e5b529c2831dba454d178b8212": "\\overline{\\Bigg(\\frac{\\alpha}{\\pi}\\Bigg)_3}=\\Bigg(\\frac{\\overline{\\alpha}}{\\overline{\\pi}}\\Bigg)_3", "5aab3b0ea8958be46f010825c5967b30": " m = G^{-1} d \\, ", "5aab85c112c722173a2b9d57c8048ff0": " |Y_n-X_n|\\ \\xrightarrow{p}\\ 0,\\ \\ X_n\\ \\xrightarrow{d}\\ X\\ \\quad\\Rightarrow\\quad Y_n\\ \\xrightarrow{d}\\ X", "5aac0f114ceefca820aeda05ab08fa76": "\\frac{\\sum_{i=1}^n w_i \\boldsymbol{x}_i}{\\sum_{i=1}^n w_i},", "5aac217079eaa47f2bd3f1ba2425b986": "b \\in {\\mathbb B}, x,y \\in {\\mathbb R}", "5aac52781243a2dd8e82290ae9f41fb4": "\\pi\\circ F_{\\mathbf P}=\\pi_{\\mathbf P}", "5aac61dbdb82900ce7dd98265690fd58": " \\mathbf{} C_d ", "5aac6e5cb3e9c4129bb5b522264893cf": "\\mathrm{div}\\,\\mathbf{A}\\,", "5aac84ae9d6f802156bfd1b8090490c7": "\\psi_R(x)= C_r e^{i k_0 x} + C_l e^{-i k_0x}\\quad x>a", "5aac93b76b3ab41c37d87987ba002db4": "D \\setminus \\{a\\}", "5aacdc2e954fab0cba6b56c200f2ddb8": "P=\\tau\\cdot\\omega", "5aad05c8783fd12ad8a993fb42cfddc2": " S\\otimes S \\cong \\bigoplus_{j=0}^{m} \\wedge^{2j} V^*", "5aadecc793fa80ee50690e88b720df54": "\n \\qquad \\qquad c = \\left| \\frac{a\\Delta t}{\\Delta x} \\right| \\le 1 .\n", "5aae003a655ddd335ba35510b47f8269": "\\gimel(\\kappa)>\\kappa", "5aae9917c7d2972b259df81b390bc877": "\\displaystyle c_{jk} = \\sum_{m,n}\\chi_{m,n} B_{m,n,j,k}", "5aaecd34bfe0a001071a1042271e317c": "G(\\omega,p)\\approx\\frac{Z}{\\omega+\\mu-\\epsilon(p)}", "5aaedc93e1f18f4654ab05e7a8b9e3ee": "\\{f_i,f_j\\}=\\sum_k c_{ij}^k f_k", "5aaef8d38ea9914b8c36f98df8efbb06": "\\mathbf{W} = \\begin{bmatrix}\nw_{11} & w_{12} & \\cdots & w_{1n}\\\\\nw_{21} & \\ddots & & \\vdots \\\\\n\\vdots & & \\ddots & \\vdots \\\\\nw_{n1} & \\cdots & \\cdots & w_{nn} \\end{bmatrix}", "5aaf80bfd4ea1fabe5f89d128f6ca090": "\\frac{\\delta \\delta^i_j}{\\delta t},\\frac{\\delta Z_{ij}}{\\delta t},\\frac{\\delta Z^{ij}}{\\delta t},\\frac{\\delta \\varepsilon _{ijk}}{\\delta t},\\frac{\\delta \\varepsilon^{ijk}}{\\delta t}=0", "5aaf8958e4dabe25b0c38e9e3746040e": "d=\\frac{\\lambda}{2n\\sin{\\theta}}", "5aafa2fa2af905ab5f2569dbd1e6634e": "V_{st}", "5aafae53ecf04e5dab05d6a8f0e593cd": "A_{ix}", "5aafe13e291b167198c8c5773d4da316": "\\textstyle\\mu", "5aafea085e7a5051c0f70e2fdd1ba2df": "X^+", "5ab09587a0dc872cec118fc2a767891a": "v^B=\\frac{V^B}{V}", "5ab0bd99b010f81c09adcf0a7aff433e": "\\tbinom{5}{2}=5\\times\\tfrac{4}{2}=10", "5ab0c0f4c97d6b1760fc91f19120adfd": "\\left[\\underline\\theta, \\overline\\theta \\right] ", "5ab0e51ccf7fcd5fcc9b19d1ccdef247": "\\pi(n_1,n_2,\\ldots,n_m) = \\prod_{i=1}^m (1 - \\rho_i)\\rho_i^{n_i}.", "5ab0ef9bdc31ef8ce40f452710ff2d19": "F(\\eta_Y)", "5ab1073a9b9d22d6dd19dbbb89873abb": "y_{t-1}", "5ab13774581981b7694b099f037630a6": " Y(t) = \\chi_\\text{i} X(t) + \\int_{-\\infty}^t \\Phi_\\text{d} (t-\\tau) H(\\tau) \\, d\\tau, ", "5ab16afbdfe7ea5730f11167225bdcbc": "\\mathfrak{f} = \\mathfrak{f}(B/A)", "5ab27629596eabafcdd2d915ee58d8d2": "f:X\\to X", "5ab2769a52f3442bce35a28e3e023faa": "\\gamma_{\\mu\\nu}", "5ab2a8c6d35503bcfd0c72c42b38d7a8": "\\prod^{\\infty}_{n=1}(1+x^n)", "5ab2db939d398db83e063f2160d894e3": "\\operatorname{cl}:\\mathcal{P}(X) \\to \\mathcal{P}(X)", "5ab3a22bccfb0b4904931caa9e9317bd": "\\displaystyle I_o = V_o / Z_o ", "5ab3c1b0c0f12d24f2affa2d2908513b": "V_{blood}=", "5ab3ea1f918a6e7aa1663582cd75ad8f": "\\beta_x (\\alpha) =0", "5ab44c5081ed155d4981769d674aa6b5": "X_j^s", "5ab45adc3313af745eb0e6d3b49cc95d": "a = 2a' - b' \\, \\pmod{2^n}", "5ab462172790a75a6b0284322bd1ec0e": "|zsp|=\\frac b a \\cdot|zsx|.", "5ab4986e26628cfd27b6ea5005e417e5": " \\mathcal{O} ", "5ab4cdf0fda92aff4c48f74afeda41c4": "\\sum_{n=0}^{\\infty} 2^n = -1.", "5ab4eae5f59578f6db4f001a39140ab8": "s_v = \\sum_{\\lambda \\in v \\Lambda^n} s_\\lambda s_\\lambda^*", "5ab51ad8c14b6e4850f6f88bd3cb0b5e": "\\lVert\\Psi\\rVert", "5ab5433ea7a35395eaa1d6e6920d1866": "k\\neq 2", "5ab55269b479ec59df7f9361f3732541": "[20(a_1\\cdot10+a_2) + a_3] a_3", "5ab567abdc5c5f9afb9605b2b574a3e6": "Z(t) = o(t^\\epsilon);", "5ab6119b27f8ff26c57d087ab64cf4c7": "H(v_i)", "5ab6226d3a029f524238984ff4a188c4": "\\| u - u_{\\Omega} \\|_{L^{p} (\\Omega)} \\leq C \\| \\nabla u \\|_{L^{p} (\\Omega)}", "5ab695973965c078f6b46c7130efae42": "p \\rightarrow 1", "5ab6a19b73867cf3313a158baf22743f": "\\int\\frac{\\cos^n ax\\;\\mathrm{d}x}{\\sin^m ax} = -\\frac{\\cos^{n-1} ax}{a(m-1)\\sin^{m-1} ax} - \\frac{n-1}{m-1}\\int\\frac{\\cos^{n-2} ax\\;\\mathrm{d}x}{\\sin^{m-2} ax} \\qquad\\mbox{(for }m\\neq 1\\mbox{)}\\,\\!", "5ab6ddc415e50b4812aa227e35a9be59": "\\begin{align}\n v_g &= \\frac{\\partial E}{\\partial p} = \\frac{\\partial}{\\partial p} \\left( \\sqrt{p^2c^2+m^2c^4} \\right),\\\\\n &= \\frac{pc^2}{\\sqrt{p^2c^2 + m^2c^4}},\\\\\n &= \\frac{p}{m\\sqrt{\\left(\\frac{p}{mc}\\right)^2+1}},\\\\\n &= \\frac{p}{m\\gamma},\\\\\n &= \\frac{mv\\gamma}{m\\gamma},\\\\\n &= v.\n\\end{align}", "5ab70d0efc73c5e979996b6e6ff44977": "ax^2", "5ab73d5a40d630fe9e0ed7b136135b31": " \\int_{-\\infty}^\\infty \\frac{\\sin^4(\\theta)}{\\theta^4}\\,d\\theta = \\frac{2\\pi}{3} \\,\\!", "5ab75b4ae2708aab80f7c1124d15d020": "\\begin{align}\nf'_i(x_i^0)&=\\lambda \\mbox{ if } x_i^0>0\\\\\n&\\leq\\lambda\\mbox { if }x_i^0=0.\n\\end{align}\n", "5ab76814f218c993609de345a4db9989": "\\underline{Z}", "5ab76cb1991d667139669afb88421f34": "x^6+y^6+z^6 + (x^2+y^2+z^2)(x^4+y^4+z^4)=12 x^2y^2z^2", "5ab7731a47b19ef485ca5f335d703613": "Pr[\\sigma \\gets \\mathrm{Setup}(1^k), (y,\\pi) \\gets \\tilde{P}(\\sigma): \ny\\not\\in L \\land \\mathrm{Verify}(\\sigma, y, \\pi)=\\mathrm{accept}] =\\nu(k)\\;.", "5ab819d8b1ff174532d9b05a37d6ab94": "B_n = \\frac{Wo}{P V F} ", "5ab8217841c79250b20dcf734075cf47": "\\scriptstyle V_{\\rm b} - V_{\\rm c} \\approx (S_\\mathrm{A} - S_\\mathrm{B}) \\cdot (T_{\\rm h} - T_{\\rm c})", "5ab8304018b74386fe59c32287840ed9": " = A_{c} \\cos \\left( 2 \\pi f_{c} t + 2 \\pi f_{\\Delta} \\int_{0}^{t}x_{m}(\\tau) d \\tau \\right) ", "5ab8463bcd082503cf04f730d5789320": " \\vec{x} ", "5ab85aaac2d6f8a513dac63921429671": "\\sum_{n\\ge 1} n|b_n|^2 \\le 1.", "5ab86ae82865899ff54de11e3c2310d5": "\\ h[n]", "5ab873db006d4fb414c2a66c70318a6c": "P[A,B,C,D] = P[A] P[B] P[C|B,D] P[D|A,B,C].", "5ab8829a7e680725d123e81b93892e18": " P = \\sum_{j} N_j\\ \\rho_j = \\sum_{j} N_j \\alpha_j E _{local}(j)", "5ab8f0d46bee397d2cde0a6ee34a8c49": "\\varepsilon^2=0", "5ab919ddd4e080efc5629af3818bc0b0": "y_{R}\\left(t\\right)=\\int_{0}^{\\tau}u_{R}\\left(\\sigma\\right)w\\left(t-\\sigma\\right)d\\sigma", "5ab938e0eabae28df0c4254f21c20bc6": "F = \\frac{1}{2}\\frac{\\Delta V_\\mathrm{P}}{V_\\mathrm{P}} ", "5ab95dd3a8d483c83ec6b5321cd7998c": "\\delta^\\dagger \\circ (\\overline\\psi \\otimes \\psi) = \\varepsilon^\\dagger", "5ab9c147d531452092d642a5fde95249": "\\mathbf{S}(\\mathbf{p}(t)) \\approx \\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(p_n(t)),", "5ab9fd69d57bbd637e1329f567b4cbe3": "p \\leftarrow \\mathrm{not}~q", "5aba43a8e93e5ad5647297b13419fe75": "{K_{-1}}", "5aba45702d7c466d6cafff4ac0f251dd": "\\Delta S = K \\int \\Delta y \\Delta x", "5aba4e96d977818a4007d46e1f31fa5b": "\\sin^2 n\\theta = S_n (\\sin^2\\theta)\\,", "5aba568837f22b95fd9233ad8b5753e2": "\\frac{60}{bpm}(x)=W", "5abab335abcd6a2da7d09f7bf5ea8710": "\\omega _{n-1}", "5abab965cb336b82a47ba910b3c10b99": "KU_x", "5ababea3dcf1a6916de9e4da1cb51021": "c_{1}^2 > c_2", "5abaeeb804f0012803903c51496931e7": " = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-h} h^{ab} g_{\\mu \\nu} \\partial_a \\left( X^\\mu + \\omega^\\mu_{\\ \\delta} X^\\delta \\right) \\partial_b \\left( X^\\nu + \\omega^\\nu_{\\ \\delta} X^\\delta \\right) \\, ", "5abafbfd8248465bd52696689c287de2": "u = H(s)e^{s t}", "5abafdf6e97e5037e3021b865918985a": "T^2_{+2}(q) = \\frac{1}{2}(q_{xx} - q_{yy}) + iq_{xy}", "5abb2c944df8a1868a004f2fce081f27": "\\dot{V}_1 \\leq -W(\\mathbf{x}) < 0", "5abb968d1b408238c322234bdfc7d9a1": "\\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}^{-1} = \\begin{bmatrix} A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\\\ -(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1} \\end{bmatrix}", "5abbe2a816c3c2acc8f0758b3cd3da83": "AB^T:", "5abbffa08ba9a50d302141689dc5b1db": "dv = \\cos(x)\\,dx \\Rightarrow v = \\int\\cos(x)\\,dx = \\sin x", "5abc3242af0b2e0c5f7ddc626263c4bf": "F= \\int d^dx \\left[ t H^2 + \\lambda H^4 + Z (\\nabla H)^2 \\right],", "5abc75e2b29727f4aee6a57ba983179b": "\\mathfrak{n}", "5abd0c8e9f1d423cd9d5fa0edb1cc62f": " \\widehat{\\mu}_{Y\\mid x} = \\sum_{i=1}^n \\beta_i (x) \\phi(y_i) = \\boldsymbol{\\Phi} \\boldsymbol{\\beta}(x) ", "5abd0cbb5ed64a58e7e3565bd5a78e8b": "\\frac{\\pi}{4} = 44 \\arctan\\frac{1}{57} + 7 \\arctan\\frac{1}{239} - 12 \\arctan\\frac{1}{682} + 24 \\arctan\\frac{1}{12943}\\!", "5abd1073e0a6505dfe7903feb1817bf9": "|S(\\rho)-S(\\sigma)| \\le 2T \\log (d) -2T \\log 2T ", "5abd8d11917be35979dbc892703f8fa7": " u(s,\\sigma_{-i}){{0}_{{}}}{{\\Rightarrow }_{{}}}{B}'(v)-{{B}_{x}}^{\\prime }(v)<0", "5ad1c58278f22f2b7cb886f763a641dc": " \\tau ", "5ad1dadbf8d25cf30d5009a01cad7140": " {128 \\over 125} ", "5ad1ed7d9c1b5f38addda6360130fe06": " |\\frac{\\partial S}{\\partial y}| \\ll 1 ", "5ad23de377b52d898d3cc16f960e4744": "(t, t +h)", "5ad2fdca7c15e86d6f66674d8eb1a2e1": "H_k^{\\rho, \\pi}=Z_k^\\rho /(B_k^{\\rho + \\pi }\\cap Z_k^\\rho )", "5ad31ce7af2aee5bea4422c8184e341e": "\\tfrac{1}{2}(\\tfrac{1}{2} + 1) = 0.75", "5ad351203b87da7d991b976884e06c27": "\ni\\hbar\\frac{\\partial\\Psi_\\lambda(t)}{\\partial t}=H_\\lambda\\Psi_\\lambda(t)\n", "5ad36291ec7d0aaccced8e45ae2ca4fc": " A = - \\frac{2}{3} \\mu_0 \\left \\langle \\boldsymbol{\\mu}_n \\cdot \\boldsymbol{\\mu}_e \\right \\rangle |\\Psi(0)|^2 \\qquad \\mbox{(S.I.)}", "5ad373f492cc04a405f0c5b251e7815d": "x\\varepsilon\\right) \\leq \\operatorname{Pr}(A_n) \\ \\underset{n\\to\\infty}{\\rightarrow} 0,", "5af5df1611f4d691b33eb07b1b6c8924": "AFR = 1-exp(-8760/MTBF).", "5af61ae3804fb65ed1e425c2ec336a45": "l_A a_i", "5af66daae65c04cf0d40fb552003c8c4": "g_{uc}^{-1}(\\{ \\langle SSS \\rangle \\}) = \\{ \\langle sss \\rangle, \\langle \\text{sß} \\rangle, \\langle \\text{ßs} \\rangle\\} ", "5af67933d577e2d4ace7cf9a786d0b11": "D(t) \\geq ((A \\otimes S_1) \\otimes S_2)(t)", "5af6987b87cd966129819649e121899a": "\n \\begin{align}\n 0 &= \\int_0^{\\lambda} \\eta(\\xi)\\; \\text{d}\\xi \n = 2\\, \\int_0^{\\tfrac12\\lambda} \\left[ \\eta_2 + \\left( \\eta_1 - \\eta_2 \\right)\\, \n \\operatorname{cn}^2\\, \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m \\end{array} \\right) \\right]\\; \\text{d}\\xi\n \\\\\n &= 2\\, \\int_0^{\\tfrac12\\pi} \\Bigl[ \\eta_2 + \\left( \\eta_1 - \\eta_2 \\right)\\, \\cos^2\\, \\psi \\Bigr]\\, \\frac{\\text{d}\\xi}{\\text{d}\\psi}\\; \\text{d}\\psi\n = 2\\, \\Delta\\, \\int_0^{\\tfrac12\\pi} \\frac{\\eta_1 - \\left( \\eta_1 - \\eta_2 \\right)\\, \\sin^2\\, \\psi}{\\sqrt{1 - m\\, \\sin^2\\, \\psi}}\\; \\text{d}\\psi\n \\\\\n &= 2\\, \\Delta\\, \\int_0^{\\tfrac12\\pi} \\frac{\\eta_1 - m\\, \\left( \\eta_1 - \\eta_3 \\right)\\, \\sin^2\\, \\psi}{\\sqrt{1 - m\\, \\sin^2\\, \\psi}}\\; \\text{d}\\psi\n = 2\\, \\Delta\\, \\int_0^{\\tfrac12\\pi} \\left[ \\frac{\\eta_3}{\\sqrt{1 - m\\, \\sin^2\\, \\psi}} \n + \\left( \\eta_1 - \\eta_3 \\right)\\, \\sqrt{1 - m\\, \\sin^2\\, \\psi} \\right]\\; \\text{d}\\psi\n \\\\\n &= 2\\, \\Delta\\, \\Bigl[ \\eta_3\\, K(m) + \\left( \\eta_1 - \\eta_3 \\right)\\, E(m) \\Bigr]\n = 2\\, \\Delta\\, \\Bigl[ \\eta_3\\, K(m) + \\frac{H}{m}\\, E(m) \\Bigr],\n \\end{align}\n", "5af6af0ca992120637553f39e7c55922": "\\mathbb{W}_n\\rightarrow \\underline{\\mathcal{O}}^n", "5af6af5e8a10ddf530ff1d289863df78": "1/\\ln(t)", "5af6d8c57d17ff20d9f85b6b013bc007": "a \\mapsto a^*", "5af6e2340adf6011837e1aecb60d088e": "\\mathbf{h}=\\mathbf{r}\\times\\mathbf{v}", "5af6ea335ca867b66038faae7e3f6451": "A_{(\\alpha} \\left(B_{\\beta)\\gamma\\cdots} + C_{\\beta)\\gamma\\cdots} \\right) = A_{(\\alpha}B_{\\beta)\\gamma\\cdots} + A_{(\\alpha}C_{\\beta)\\gamma\\cdots}", "5af6ec335452cbab960702775cb4a67f": "\\begin{array}{ll}\n a' = a \\sqrt{1 - |F_x|/\\mu F_n}, &\n \\mbox{for } |F_x| \\le \\mu F_n \\\\\n \\xi = -sign(F_x) \\, \\mu (a-a') / R, &\n \\mbox{i.e. } |\\xi| \\le \\mu a/R \\\\\n F_x = -sign(\\xi) \\,\\mu F_n \\left( 1 - \\left( 1 + R |\\xi| / \\mu a \\right)^2 \\right)\n \\end{array}\n", "5af70d181d58006d05af5ff19c45cd15": "\\alpha = - \\frac{\\log \\frac{\\tau_{\\lambda_1}}{\\tau_{\\lambda_2}}}{\\log \\frac{\\lambda_1}{\\lambda_2}}\\,", "5af7ee7ce853bb7720282fb2b0dcd799": "E_s = R_f + \\beta_s(R_m - R_f).\\,", "5af80dfb1f60ec32e394462e59671331": "u={\\pm_1\\sqrt{\\alpha + 2 y} \\pm_2 \\sqrt{-\\left(3\\alpha + 2y \\pm_1 {2\\beta \\over \\sqrt{\\alpha + 2 y}} \\right)} \\over 2},", "5af84569dd9efcb1c858abb4da9961ed": "L=LQ", "5af8c38f8fd16c5513dae59684f63309": "1/Z_\\nu", "5af9157dd989b12f0b6aa219eaa2edfc": "\\|F\\|^2 := \\pi^{-n} \\int_{C^n} |F(z)|^2 \\exp(-|z|^2)\\,dz < \\infty,", "5af9e28d609b16eb25693f44ea9d7a8f": "\\beta_0", "5afa0edf06d98aae35764007b1d767f4": "\\hat{\\mathbf{u}}\\,\\!", "5afa318874a9ad0fe4406b72c3a204c0": "\n\\operatorname{Li}_2 \\left( \\frac{x}{1-y} \\right) + \\operatorname{Li}_2 \\left( \\frac{y}{1-x} \\right) - \\operatorname{Li}_2 \\left( \\frac{xy}{(1-x)(1-y)} \\right) = \\operatorname{Li}_2(x) + \\operatorname{Li}_2(y) + \\ln(1-x) \\ln(1-y)\n", "5afa4b7f0f628c90ffc937b8272bff4e": "\\Delta_\\mu=x_1y_2+x_2y_3+x_3y_1-x_2y_1-x_3y_2-x_1y_3", "5afaa4077688aae9a139142025824ec4": "23 = 4 \\cdot 5 + 3", "5afab6ea4b9b49350344852fc290b162": "|b|= \\frac {a}{2}|<110>|= \\frac{a}{\\sqrt 2}", "5afade25b71ece27c98a4f2786568be9": "\nW_{f}^{\\alpha}(a,b)=\\mathcal{W}^{\\alpha}[f(t)](a,b)=\\int_{-\\infty}^{+\\infty} f(t)\\psi_{\\alpha,a,b}^{\\ast}(t)\\, dt. \n", "5afaf577ba21555181adb11f01960ff7": "\\sigma = \\int \\sqrt{k(s)}\\, ds,", "5afb4fa2bd58ddbf7c90b8254463c573": "\n\\left(\\frac{a}{p}\\right) \\equiv a^{\\tfrac{p-1}{2}}\\pmod p.\n", "5afb7f9a23ab0ce79e1eb5656d8a2659": "|S_i|=n_i", "5afb8f8b122db5fb25857316448c6d54": "E(k)=R_F\\left(0,1-k^2,1\\right)-\\tfrac{1}{3}k^2 R_D\\left(0,1-k^2,1\\right)", "5afbda3a62dff1244885f7bc7816bbc5": "\\mathcal{D}_n =\\{(X_i, Y_i)\\}_{i=1}^n", "5afc2964cf4f611172a724de29dd766e": "a = \\frac{0.457235\\,R^2\\,T_c^2}{p_c}", "5afc3f56cd81c83b0a38987ad4c27f05": "S\\left(t\\right)", "5afca9efa5a184fe70ea869178fee4d4": " x_i", "5afced6d5a89b505432c26399b31c40c": "a = l/2", "5afd1ded6fe6df42f0078e6de100a0bc": "\\mathcal{D}x", "5afd738373354b379f4305c34eed404b": "e^{i\\phi}", "5afe09bd114745468c4f90f5a67bf6e9": "(A \\leftrightarrow B) \\and (B \\leftrightarrow C)", "5afe10cd3ed5f182a8d0b7815d4afbce": "\\hat{\\textbf{x}}_{k\\mid n} = \\hat{\\textbf{x}}_{k\\mid k} + \\textbf{C}_k ( \\hat{\\textbf{x}}_{k+1\\mid n} - \\hat{\\textbf{x}}_{k+1\\mid k} ) ", "5afe790e97a5dd80287ac420b2c02212": "\\scriptstyle{\\operatorname{AGM}(a,b)}", "5afedb99853677f7c79cb98679b0d9c0": " \\tfrac 12 n(n+1) ", "5aff1a8aa7f76f12a90432c0c68e201e": "\\alpha S := \\{ \\alpha s \\mid s \\in S\\}", "5aff29052429ffd40fa11864fc242487": "|f(z_0)|\\ge |f(z)|", "5b00121c0c11564a3e4f0d482902aa1a": "\\frac{m}{100} = 1 - (1 - \\frac{1}{N})^{\\frac{nN}{100}}", "5b003bc3ff0d0f7927bef497bd08d231": "(X, \\mathcal{O})", "5b0043139740da14153b99f8d3798f48": "\\mathbb{Q}(\\zeta_l)", "5b00fb6c75b8ab83e9f5992a90e8acf9": "B/{I_i}", "5b0134e3a359975ff177adcd133fe6e5": "\\mathrm{d}S_\\rho= \\rho\\,\\mathrm{d}\\varphi\\,\\mathrm{d}z.", "5b016cc58adfac503c6d2a9dd52df4b7": "L(n_v)", "5b017b1b2a0a6ee55fbe747f08f824de": "\\mathbf{M} = (m_{ij})", "5b01af34f1027df7cc853844f0463dca": " (G_\\lambda f)(x) =\\int_a^b G_\\lambda(x,y) f(y)\\, dy.", "5b01db7286c5169333f2a8e8f9de065f": "\\phi\\, (t) ", "5b02a39568d4aaae3e68a55271309166": "\\begin{align}\nZ(\\operatorname{Spin}(n,\\mathbf{C})) &= \\begin{cases}\n\\mathbf{Z}_2 & n = 2k+1\\\\\n\\mathbf{Z}_4 & n = 4k+2\\\\\n\\mathbf{Z}_2 \\oplus \\mathbf{Z}_2 & n = 4k\\\\\n\\end{cases} \\\\\nZ(\\operatorname{Spin}(p,q)) &= \\begin{cases}\n\\mathbf{Z}_2 & n = 2k+1,\\\\\n\\mathbf{Z}_2 & n = 2k, \\text{ and } p, q \\text{ odd}\\\\\n\\mathbf{Z}_4 & n = 2k, \\text{ and } p, q \\text{ even}\\\\\n\\end{cases}\n\\end{align}", "5b02b0c50b12c21897b1bce2ce4c0bc7": "F(u)=\\int^b_a u(x)d\\rho(x),", "5b02d73bec551079561894a2e03903ff": "\\mu_i, \\Sigma_i,\\ (i=1,2)", "5b02fc014d06dc7b8a8bb429df383f89": "c_{n+2}= c_{n+1} \\cdot c - c^p \\cdot c_n + c_{n-1}", "5b036de9284e100de976f93e31593e3e": " \\zeta_n^2+\\zeta_n^{-2}-2 = (\\zeta_n - \\zeta_n^{-1})^2. ", "5b03c7910c3a364599e2f9790c44bea4": "T_{\\rm wc}=35.74+0.6215 T_{\\rm a}-35.75 V^{+0.16}+0.4275 T_{\\rm a} V^{+0.16}\\,\\!", "5b03e5a24c0bd3dba9e4f4e49bc9234a": "\\frac{\\partial}{\\partial a_i} E\\{e^2[n]\\} = 2 \\sum_{j=0}^{N} R_w[j-i] a_j - 2 R_{sw}[i] \\quad i = 0,\\, \\ldots,\\, N .", "5b0430ebeb81bd68b761d031107a1718": "\\mathbf{\\ddot{r}}", "5b0440f0d20ae09fa90476b6b35ac2e0": "\\mathcal{U}(\\hat{\\beta}(q,r_{c}),\\tilde{u})\\ ", "5b04da7aaa3bd17e669a73540fdefebe": "Area = \\pi r^2 \\approx 3{.}1416 \\cdot r^2. ", "5b0529f19dcd3330b409eec2c63f7bd6": "-\\dot{X_3}", "5b058695c1f7a08be90d6ea7a9dd6077": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = 0.5\\left(x^{2} + y^{2}\\right) + \\sin\\left(x^{2} + y^{2} \\right) \\\\\n f_{2}\\left(x,y\\right) & = \\frac{\\left(3x - 2y + 4\\right)^{2}}{8} + \\frac{\\left(x - y + 1\\right)^{2}}{27} + 15 \\\\\n f_{3}\\left(x,y\\right) & = \\frac{1}{x^{2} + y^{2} + 1} - 1.1 \\exp \\left(- \\left(x^{2} + y^{2} \\right) \\right) \\\\\n\\end{cases}\n", "5b05a87e5f448761ff562d8d57595ce9": " \\hbar \\omega ", "5b05ae842c7a8123186c853f3851f0e1": "\\approx 152.308", "5b05b76c9ba571a96df2e0a9df971ebd": " \\omega_r = -2 \\omega_0 ", "5b05b91fdac8be0c93c3e73bcc3f182c": "\\Sigma_{g+1,1}", "5b062e3f383b6c40e8c2359d685a6f8e": "\\rho_G\\,", "5b06424041565516c742a3a147cedb97": "\n\\langle \\lambda, \\mu; \\ell+\\lambda, m-\\mu| \\ell m \\rangle\n= (-1)^{\\lambda+\\mu}\\binom{\\ell+\\lambda-m+\\mu}{\\lambda+\\mu}^{1/2} \\binom{\\ell+\\lambda+m-\\mu}{\\lambda-\\mu}^{1/2}\n\\binom{2\\ell+2\\lambda+1}{2\\lambda}^{-1/2}.\n", "5b06431b50b71ca83a99c413e3c137f5": "\nX = H \\tan(15^{\\circ} \\times t)\n", "5b067184cdcfaf60f7d310909209731d": "T = \\frac {2Z_\\mathrm L}{Z_0+2Z_\\mathrm L}", "5b069181b7b956b07aade7dd2951eb7f": " x - r \\cdot \\cos A = \\sqrt{l^2 - r^2\\sin^2 A}", "5b072ee80e00782e16cbec0397e9c203": "\\mathcal{S}_{\\alpha}\\to \\min", "5b0738d0aecee072d390abd8db766952": "\n \\hat\\theta^*_\\mathrm{mle} = \\hat\\theta_\\mathrm{mle} - \\hat b .\n ", "5b07522c331e1c4898247b8237fc81c8": "x''_i = \\alpha x_i + (1 - \\alpha)x'_i \\in X_i", "5b083bdbb497754a9f63852353d7c4da": "\nV(\\mathbf{r}) = \\frac{k}{r} = ku\n", "5b08f773108da50179ec6195375bcece": "V^* \\to V^*", "5b0915bd64a5cfa9158a33c2dac3c37a": " k_3 = hf(t_n+c_3h, y_n+a_{31}k_1+a_{32}k_2), \\, ", "5b093b49e65118cfc32b60b169ec570e": "(M,\\mathcal{F})", "5b093c81a05930be6f2da9cf8103fe3c": "|set(P)|", "5b094c2155e4876782aff529c61cfac4": "q(B) = \\frac{m}{T} = \\frac{m\\,dx}{T\\,dx} = \\frac{td}{\\left | B \\right \\vert}", "5b0977d648e66e9398c2695a2e31df61": "f(x) \\in K", "5b098c9e7c16aa828fa7dcce9dfa3c6b": " \\beta_{i} = \\beta_{i-1} - \\sigma_i \\gamma_i. \\quad \\gamma_i = \\arctan 2^{-i},", "5b0a68a66041c3047714908f601ec815": " \\sin{\\frac{C}{2}}=\\sqrt{\\frac{efg + fgh + ghe + hef}{(g + e)(g + f)(g + h)}},", "5b0aba1f76b976fb9e2ca6b10091a4b5": "u_6 = \\tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+ax_6^2+x_7^2+x_8^2)x_{14} - 2x_6(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +bx_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "5b0b1a688ccc273da70ce1550f4a10ae": "\\langle \\, \\rangle", "5b0b1c0dfabe1a030b82a1a9c7cec1b9": "A_{z}=A_{z-2}+4+B_{z/2}", "5b0b519dd622eb1847cb04e3ca6213ff": "\\Lambda_{\\mathrm{m}}^\\circ = \\Sigma_i \\nu_i \\lambda_i", "5b0b968a7bb38da86ed5c28af642c250": " \\dot{V} = V_T \\times f ", "5b0bc916a13dc80a3baf90135c80b68c": "\n\\frac{\\sum_i (Y_i - a - b_1X_{1i} - \\cdots - b_pX_{pi})^2}{n}\n", "5b0c0f480d928e893019ba497b009249": "\\vec{r}(t)=\\frac{\\lambda_\\text{u} K}{2\\pi\\gamma}\\sin \\omega_\\text{u}t\\cdot \\hat{x}\n+\\left ( \\bar{\\beta_z}ct+\\frac{\\lambda_\\text{u}K^2}{16\\pi\\gamma^2}\\cos(2\\omega_\\text{u}t) \\right )\\cdot \\hat{z} ", "5b0c37fbd0eb6489b594903e8f0ef418": "\ne_1 =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n", "5b0c732b828467f6522c9745bff062d5": "\\begin{smallmatrix}\\frac{b}{a}\\end{smallmatrix}", "5b0c8ecb9c63cc3c9ded2563fdd5400e": "P = \\{\\text{abcc}, \\text{efab}, \\text{bccla}\\}", "5b0d0517f9c3ff9c731df2a74dc9c2a5": "\\mathit{CV}", "5b0d72f0e39951beccc6aba886c0397f": "\n\\epsilon_t = \\frac{\\pi \\sigma_t }{2}~I\\,n\n", "5b0d73a7a665906b141bd4137fe1566c": "\\ u_y = W - u_1 + u_2,", "5b0d8a31c4ffcf0d1fc025c94144f96f": "B_i \\rightarrow A_i,", "5b0dbf1a92677ffd4454c390cbd45b2e": "N=\\frac{ln (1-P)}{ln (1-\\frac{i}{g})}", "5b0e27686ec0e534f44f0e9e2c871945": "+p_1 p_2 p_3 (1 - p_4) [ N(1-R) \\delta_4 - \\frac{Nc}{4} (\\delta_1 + \\delta_2 + \\delta_3) ]", "5b0e41f0cf4ba8fe0f641ec5b0f0e43c": " C=ENC_{k_2}(ENC_{k_1}(P)) ", "5b0e6678e56ce2942d9a5bdce16f7623": "P - Z =", "5b0f19031360aab4c1096888e9d44597": "d_q(f(x)) = f(qx) - f(x) \\,", "5b0f65ee4ff43fd6f2ef318635ee408d": "N^{1/a}", "5b0f6d40254fb7d6bffd7c5cbe914558": "q(z,\\bar{z}) = z\\bar{z} \\, ", "5b0f9fe9cf49707e128b5991ce67516b": "\\beta=\\pi/2", "5b101687d63a60d890547ff1d3296b53": "\\begin{array}{lclcl}\n\\mathbf{c} &=& (c_1, \\ldots, c_K) &=& \\text{number of occurrences of category }i = \\sum_{j=1}^N [x_j=i] \\\\\n\\mathbf{p} \\mid \\mathbb{X},\\boldsymbol\\alpha &\\sim& \\operatorname{Dir}(K,\\mathbf{c}+\\boldsymbol\\alpha) &=& \\operatorname{Dir}(K,c_1+\\alpha_1,\\ldots,c_K+\\alpha_K)\n\\end{array}\n", "5b103b59d227bc42632a916787e2f407": "(X,\\tau')\\,", "5b105cc5b7b3e88275fec3be98a1e0d0": "I = \\sqrt{TC-(TA+{\\Delta}TD) \\over RDC(1+YC)RCA}", "5b1070191e30b9ec565375fdbfe42946": "nFE^\\circ = RT \\ln K \\,", "5b10cd7f0c1f65456dc88da09cfbd9da": " 2R\\sin x^\\circ = \\frac{4\\times 2R \\times 2Rx\\times (360R - 2Rx)}{\\frac{1}{4}\\times 5 \\times (360R)^2 - 2Rx\\times (360R -2Rx)}", "5b10d478c16e6ff61202b95c3c3a8128": "\\rho_{\\text{e}} = k\\left [ \\frac{f_{\\text{c}} }{240} \\right ] \\left [ \\frac{h}{d} \\right ]^2 ", "5b113430d00ca9cb5d8100ee6740a036": " \\operatorname{E}(X \\mid X>a) = \\mu +\\sigma\\lambda(\\alpha) \\!", "5b118f776b622959bedfd7cf172cb30f": "(\\forall k\\in\\mathbb{N}):|a_k|\\leqslant Mr^{n-k}.", "5b11d0b6c0357bb811ae87c0122eed96": "dz(t')", "5b11ec6488469314cc74ddfe0c7b3f17": " \\mathbf{O}_{1} ", "5b124c13890eb9ecdcc5f426a2d66623": "\\Omega( 2^{2j} )", "5b128f0e5cefe15ff5f29cff4752bec1": "R \\gg x_1 - x_2", "5b12b0650be9194e5d15d8672594041b": " \\Delta H^{0}_{T} ", "5b12fe4f06226fac008f73ce3679d054": "\\mathrm{deg}\\ p < \\mathrm{deg}\\ q", "5b130e0baa66b9920a02b04c68a205b2": "x\\mathcal{M}y \\and y\\mathcal{M}z", "5b13314e7eabebf852553f897a0cd023": "\\lambda^3 = 1", "5b13849acc00b36f7eca63c0516faeb8": "h< j", "5b1391fe18b86771153950cca2d52384": "v=20+5\\times\\frac{40-30}{100}(35-20)=27.5", "5b1399446b090b7e7474b76b518dcd39": "\\phi_B = \\frac {F L^2} {2 E I}", "5b13bab1d00341882f5abfc3c1a4b668": "c_1 = c_2", "5b13e4f3b1e510bd1194693315118345": "p \\approx m v", "5b146f79f14a7d9abbf69befad8237b0": "\\left\\{3,{3\\atop4}\\right\\}", "5b147ce73fc396a6c329b16adb2e9975": "B_{XX}", "5b14c35126c2b1ffc252ce5382af5adc": "\\mu (A) = \\mu (A \\cap B) + \\mu (A \\setminus B).", "5b14c5839513e3233cfe0c6c9de5f05b": "u_i = p_i/q_i", "5b14c6c29f05ae2295d9aabda5892807": "\\hat{f}(n)= \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(t)e^{-int} \\, dt", "5b14f61604083b0d8f052d5994c8869f": "m_3(x) = m_6(x) = x^4+x^3+x^2+x+1,\\,", "5b15c6701685986416953580679cd909": "q_{0}\\stackrel{\\epsilon , T}{\\rightarrow}r_{0}\\stackrel{x_{1} , T}{\\rightarrow}r_{1}\\stackrel{x_{2} , T}{\\rightarrow}r_{2}\\cdots r_{m-1}\\stackrel{x_{m} , T}{\\rightarrow}r_{m}, r_{m}\\in A_{1}\\cup A_{2}, r_{0}, r_{1},\\cdots r_{m}\\in Q\n", "5b16b7712d56a970a34db2d20421d8c5": " P \\left ( {a, b}{|}{A, B} \\right ) = \\sum_{i=1}^k P \\left ( {a, b} {|} {A, B, \\lambda } \\right ) \\rho \\left (\\lambda_i \\right )", "5b1708bbcc9b6b0293eda0ed702c0033": "p_i+q_i>0", "5b176c193cc4fe0824596d668e340e39": "f(x)=P(x)e^{-a\\pi x^2}. \\, ", "5b17e02e42749746386b1c77e2cccf35": " p_i = \\frac{1}{\\mathcal Z} \\exp((N_i\\mu - E_i)/k_B T) .", "5b1804eec2a11d265a6c7a6bf418f134": "g_F= g_J\\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}+g_I\\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}", "5b1851c49d471453eb0283dd33692b3c": "\\lim_{kh \\downarrow 0} \\mathcal{S} = \\frac{3}{4}\\, \\frac{ka}{(kh)^3},", "5b1890156064fab6b13c6b88ee993d44": "\\tau_{ij} = \\frac{1}{2\\Omega} \\sum_{k,\\ell \\in \\Omega} ( x_i^{(\\ell)} - x_i^{(k)}) f_j^{(k\\ell)}", "5b18b83150c23dddb1345aeb54e4f265": "L_v(T_L)", "5b18df671e77822114dbca8fa6dc088a": " |\\Phi\\rangle = 1/{\\sqrt 2}(|0 \\rangle_A |0\\rangle_B |A_1\\rangle_C + |1\\rangle_A |0\\rangle_B |A_0\\rangle_C), ", "5b18fb9218596eb8ccd632b18f95d0ac": "\\int_0^{2 \\pi} e^{x \\cos \\theta} d \\theta = 2 \\pi I_{0}(x)", "5b190352411c043698e7ba0830ade4cf": "\\sgn(x_{2,i} - x_{1,i})", "5b191195b4c966b251d589c2d61ed5d0": "\\textbf{R}^{a}_k", "5b1955f1660b2e87e748b82653b41d28": " K = \\left\\{ (a, x) \\, \\mid \\, N(x) \\leq a \\right\\}.", "5b1981fdbb3d0c16cb990a4f996fcdd6": "\\mbox{male shoe size} = 3\\times\\mbox{last length in inches}-24", "5b19b1389b5f0295765a3402cb573505": "\\textstyle j\\neq i", "5b1a237ea166c4ee2287372cbc445e34": "c_V=\\frac{T}{N}\\left(\\frac{\\partial S}{\\partial T}\\right)_V\n\\quad = -\\frac{T}{N}\\,\\frac{\\partial^2 A}{\\partial T^2}", "5b1a2539a34d6c64057c4d44533f16ee": "\\mathrm{Cov}(S_{t}^i, S_{t}^j) = S_0^i S_0^j e^{(\\mu_i + \\mu_j) t }\\left(e^{\\rho_{i,j} \\sigma_i \\sigma_j t}-1\\right)", "5b1a4167f3f76f79cc8739715a71fb2d": "\\tfrac{BU^n-BU^{n-x}}{BU^n}", "5b1a70b2e6faf78df33bb273b6b7f259": "\\frac{W+\\frac{1}{2}T}{GC}", "5b1a83c38ed22ff4b94c27ea7937f0cf": "V_{Eq} = V_1 + V_2.\\ \\,", "5b1a877f9644b0d72b5dc0e4d1f4806c": "{\\mathbf{}}G(t),H(t)", "5b1abdb00b536c9ff5682f58d2a59f9f": "\\left( \\begin{smallmatrix} 1 & 5 \\\\ -5 & -2 \\\\ \\end{smallmatrix} \\right)", "5b1b19a0efff69a1349bdf05276219ec": "\\mathcal{L}u=\\sum_{i,j=1}^n a_{ij}(t,x)\\frac{\\partial^2 u}{\\partial x_i\\,\\partial x_j}+\\sum_{i=1}^n b_i(t,x)\\frac{\\partial u}{\\partial x_i} + c(t,x)u", "5b1b645ea4cd2d93e42cbc5eeb269703": "\\gamma = 180^\\circ - \\alpha - \\beta", "5b1ba2fe7e642d05d58d3df27466b069": "N-1", "5b1ba55c26a6bcf21865952915935cc1": "\\scriptstyle\\sqrt{X}", "5b1be7eb33dcabd1921c413ae3188698": "(c+v)", "5b1c403dae8339c95442328bb5ef93cf": "q(x) = x_1^2 - x_2^2-\\cdots - x_n^2.", "5b1ca10f2f7e07008c688e05c74ef203": "(k^{h+1} - 1)/(k-1)", "5b1d25109ab1276805370bb562346591": "{\\mathbf D}_a", "5b1ddcdddc296823f09b539559b49848": "x_1=l_1\\|r_1", "5b1ed803c32f3f7e0bd5a66fa220a963": "\\text{root }\\simeq\\text{ known square root} - \\frac{\\text{known square} - \\text{unknown square}}{2 \\times \\text{known square root}}\\,", "5b1ee11f85e1bdfebd65d8d74ddc6105": "\\scriptstyle t_\\mathit{off} \\,=\\, (1-D)T", "5b1f2e2031121970722db29c2876a916": "K_{sp} = {\\left(\\frac{N_{A(\\Delta)}}{V}\\right)}^x {\\left(\\frac{N_{B(\\Delta)}}{V}\\right)}^y\\,", "5b1f6c0f555771dce28706d4f71ed453": " E =\n - {1\\over 2} {a_1 a_2 \\over 4 \\pi r } \n\\vec v_1 \\cdot \\left[ 1 + {\\hat r} {\\hat r}\\right]\\cdot \\vec v_2\n\\; e^{ - \\omega_p r } \\left\\{ \n {2 \\over \\left( \\omega_p r \\right)^2 } \\left( e^{\\omega_p r} -1 \\right) - {2\\over \\omega_p r} \\right \\}\n.", "5b1fa401c1ad8fe5ffba80c8de4da814": " x(t)=\\sum_{m\\in\\Z}\\sum_{n\\in\\Z}\\langle x,\\,\\psi_{m,n}\\rangle\\cdot\\psi_{m,n}(t)", "5b1fde20054f89a25b6d37cbc050d984": "\\sqrt[4]{k'} = \\psi(\\tau) = \\sqrt{\\frac{\\vartheta_{01}(0; \\tau)}{\\vartheta_{00}(0; \\tau)}}", "5b1ff9d4a6ed4a6a0e0a0f6c8b6764d3": "\\lambda y + (1-\\lambda )t", "5b20026a6f6280cf148205cfbb2bae6a": "i\\cdot x", "5b207d11ca5d9f729c87aac6417b2e47": " I_j \\cap O_i = \\varnothing, \\, ", "5b20d37faf13e40394ff3883b57a1180": "z_1, z_2", "5b2108e69110d8a72a6c85ccac7f95b9": "\\psi(t,q^{\\mathrm I},q^{\\mathrm{II}})", "5b2123fbb86bad367af40c751e5938ed": "dn(v)=nf(v)dv", "5b213a9f2ed458f1551f764f9f71c1bb": "\\frac{{(1.8)OBP} + SLG}{4}", "5b2145718f8ba82d132880e9bdd345e1": "x^n \\cdot x^m = x^{m+n}", "5b21be4ad8553be3229ddebfd4651d6a": "\\,{}^{0}a = 1", "5b21c398020a369dc5c6bdc0d9e7bea6": "\\Pr j, \\hom l, \\lVert z \\rVert, \\arg z \\!", "5b22620209174b5a653644863c4dc5a2": "1.76 \\approx 10 \\log_{10}\\left( 3/2 \\right)", "5b2263721081609e964c8193ac05c937": "\\frac{\\partial \\sigma_k(t)}{\\partial \\lambda_j}", "5b226c8939a47ea2acdc4a2a32e8c5f5": "{\\mathbf{p}}(t)", "5b22c15d85730128e4666c8ddd2e9943": "\n\\mathbf{p}^T A_Q\\mathbf{x}=0\n", "5b236ab10401a215e2cc96e7498a1e06": "\\sum_i x_i = 1\\,", "5b240f4795311f65ec1ad34a7a724013": "a = 0.5\\!", "5b242eff141772e64626cd1d581b5b2c": "\\mu(T,p_i) = g(T,p_i)=g^\\mathrm{u}(T,p^u)+RT\\ln {\\frac{p_i}{p^u}}", "5b246f0c50312369e324fd1c9b9463f7": "[L_{ss}] = \\begin{bmatrix} L_{AA} & L_{AB} & L_{AC} \\\\ L_{BA} & L_{BB} & L_{BC} \\\\ L_{CA} &L_{CB} &L_{CC} \\end{bmatrix}", "5b24858b18eefcea4e0a471617433880": "[1] = x { dx\\over dt} = x(t) {(x(t+\\epsilon) - x(t)) \\over \\epsilon } \\,", "5b24a4d203ee7c03410194cf4bfc3777": "b_{1}-b_{11}", "5b24c4cdfe510ee81f4bca7bbda40037": "\\scriptstyle \\lfloor\\frac{37+3}{4}\\rfloor=10", "5b24ce6e979db29ed1adf6ecead3cf8d": "L_m=R_m(\\xi,\\xi)", "5b2549a359f50eb34a779441ee4eb056": "Pr(X k", "5b29aa68c2f289a17410b21614209151": "\\hat{H}|\\psi\\rangle=\\hat{H}(c_1|\\psi_1\\rangle+c_2|\\psi_2\\rangle)", "5b29d638b13e8b55c94c7a814da87f24": "10^{10^{100}}\\,\\!", "5b29fc5bcbe4583d5893321fb76eac75": "\\nabla_0 \\nabla_0 \\tau^{00} = \\nabla_j \\nabla_k \\tau^{jk}", "5b2a20db30d3c311d7ce09c01f51b74a": "N_B\\sim l_B^{-d_B}", "5b2a227defa1a46c6a88e34549783565": "(x-2)^n (x-1)^n x", "5b2a24e22ffd549b7ff2cd98e21838ac": "\\frac{\\partial^2}{\\partial x\\partial y}=\\frac{\\partial^2}{\\partial y\\partial x},", "5b2a4d57b27b2ea0bc32545810cd3496": "\\textstyle (1-p^2)", "5b2a6e1a779cb75dff910edae6634b84": " b_{FB} \\geq \\frac{B}{\\log_2(1+\\rho_{FB})} = \\frac{(M-1) \\log_2 \\rho_{b,m} - (M-1) \\log_2 (g-1)}{\\log_2(1+\\rho_{FB})} ", "5b2a8599b5c1cdd5e989cc7fcc29d835": "x(i)=0", "5b2aac91371f522365ef7e231d50a493": "\\omega_{\\xi}", "5b2ab2faaf902da75ce94628db7d547a": "S=NC", "5b2b25e30234a7bb677907500da12539": "\\langle X,JY\\rangle=-\\langle JX,Y\\rangle", "5b2baacbe2c93a97eee992deda4721c7": "A_{m,n}", "5b2bddfbf96bb5dca7d71ec7f6202d0a": " \\langle \\,\\cdot\\, , \\,\\cdot\\, \\rangle ", "5b2c4c79c04458a7579bad989b43e0bc": " a_1, a_2, c", "5b2c971ef2ad2a5ed5d8e0c687187b72": "{m_2}", "5b2c9c589b1079a2b27c88304c97a455": "\\scriptstyle\\sqrt 2", "5b2cceba89a6515a4d7a215febe00e59": "\\mathcal{R}^M _{\\theta} = \\Psi_M(\\mathcal{R}_{\\theta})", "5b2d88538e54eaef3a86ec20336dd139": " d = R\\left( 2\\;\\frac {\\rho_M} {\\rho_m} \\right)^{\\frac{1}{3}} \\approx 1.26 R\\left( \\frac {\\rho_M} {\\rho_m} \\right)^{\\frac{1}{3}} ", "5b2dad85d1944bacb4bf06e59805de45": "\\frac{dr}{dt}y^2\\delta m", "5b2e2ae304bef9ff9843e39f57585b79": "{{\\Delta}V_p}", "5b2e9d6d55160a8903a3298fc8a9a30a": " D_{KY} ", "5b2ea0944c9aac17d6f2d60895b11f75": "\\dot\\lambda(t) = -\\frac{\\partial H}{\\partial x} = -\\left( \\frac{u(t)}{x(t)} \\right)^2", "5b2eb772bb93b386d1eda03882b78aba": "P(k) = \\sum_{n=0}^{f\\left(k,K\\right)} C^n_{K/2} \\left(1-\\beta\\right)^{n} \\beta^{K/2-n} \\frac{(\\beta K/2)^{k-K/2-n}}{\\left(k-K/2-n\\right)!} e^{-\\beta K/2}", "5b2ed7bf046f7464e91bcb3b7ffd7593": "\n \\tfrac{\\text{No. of waves in space}}{\\text{No. of waves in time}} =\n \\frac{\\Lambda_g / \\lambda}{\\tau_g / T} =\n \\frac{\\Lambda_g}{\\tau_g} \\cdot \\frac{T}{\\lambda} =\n \\frac{c_g}{c_p}.\n", "5b2f310c41b36ac1712698545f35ac06": " \\mathcal L = -\\frac{1}{4} Z_3 F_{\\mu \\nu,r} F^{\\mu \\nu}_r + Z_2 \\bar{\\psi}_r(i \\partial - m_r )\\psi_r - \\bar{\\psi}_r\\delta m \\psi_r + Z_1 e_r \\bar{\\psi}_r \\gamma^\\mu \\psi_r A_{\\mu,r} ", "5b2f6cb9a26ecadf5a5f36205b0acf02": "\\delta t = 0\\,", "5b2fad4133892b1d420b489a258a6d8f": " f= (x+1)(x^2+1)^3(x+2)^4.", "5b2fc4b161db9c9d3d661d93a966cda0": "RT\\ln x_i = RT\\ln \\frac{{f_i }}{{f_i^{\\star} }}", "5b2ff7b2fb560654f0f02be6237b3ae0": "A = [\\mathbf{a}_1 \\dots \\mathbf{a}_n]", "5b30431f1d28afc13d2a29f549c9a1c6": "E\\{\\hat{x}\\} = \\bar{x}", "5b304628741bad1b0b9d5f2add658f9a": " \\text { Separation ratio}= \\left ( \\frac{\\text{Protein concentration in the foam}} {\\text{Protein concentration in the outlet stream}} \\right) ", "5b3097a140cbcacb73817145de55d36d": "\\mathrm{Reg}([0, T]; X) = \\bigcup_{\\varphi} \\mathrm{BV}_{\\varphi} ([0, T]; X).", "5b30a1c64e8ebed400b56b67a04477ca": "R_\\mathrm{in}=\\begin{matrix} \\frac{1}{ g_{11} } \\end{matrix} = \\begin{matrix} \\frac{v_{in}}{i_{in}}\\end{matrix} \\Big|_{i_{out}=0} ", "5b30a8af38c339f8c2da127980799eac": "\\psi_3 \\,\\!", "5b30afb13cb0247b5bebce3ed65a887c": "\\operatorname{Tr}_{L/K}(\\alpha)=\\sum_{g\\in\\operatorname{Gal}(L/K)}g(\\alpha),", "5b30d37206d33699f31d7c9c18e4f7d9": "W : \\mathbb{R} \\times \\Omega \\to X", "5b30ff85a744e7daffb69af1a5d19bf1": "\\mathbf{\\hat{e}}_2=\\mathbf{\\hat{e}}_3\\times\\mathbf{\\hat{e}}_1", "5b3117b96cb721140fe938023bd2b446": "\\cos\\left[\\frac{\\pi}{N} \\left(-n-1+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right)\\right] = \\cos\\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right)\\right]", "5b31f3a931787210e31641b53a520b90": "\\|Ax_0-Y\\|\\le\\|Ax-y\\|", "5b3211e7262e9012f51d6905848ed51a": "\\varepsilon_{Hb}", "5b32919751a2f329a11dc0dfa15c2fb5": "\\sum_j |\\beta_j|", "5b32a1c2f9ac219f719078e4ca62c78e": "\\mathcal{L} \\left\\{\\sum_{k=0} a_k J_{\\nu+k} \\right\\}(s)= \\frac{1}{\\sqrt{1+s^2}} \\sum_{k=0} \\frac{a_k}{(s+\\sqrt{1+s^2})^{\\nu+k}}", "5b32cfed947817e0d1fd4653bded99ef": "\\mathbf{w}_Y^* = \\arg\\min_{\\mathbf{w}} \\left\\{ \\sum_{i=1}^N (y_i - \\langle\\mathbf{w}, \\mathbf{z}_i \\rangle)^2 \\right\\} ", "5b33220aa68e22c11c41ba127dd9b6dd": "\n\n\\text{average delay (}w_\\text{avg}\\text{)} = \\frac{\\text{total delay experienced by }m\\text{ vehicles}} {\\text{total number of delayed vehicles}} = \\frac{TD} {m} \n\n", "5b332402ab8a6f7d2e902bdc9aaf0867": " P = P_0 e^{-\\int_{0}^{z}{M g dz/R^*T}}\\,", "5b332466a39a06e5c59970d40fb94d8c": "\\epsilon ^2 = +1", "5b333b202ec57a4e2371bd0416063b18": "\\frac{\\partial i_l}{\\partial x}=-i_m=-\\left( \\frac{V}{r_m}+c_m \\frac{\\partial V}{\\partial t}\\right)", "5b336c4b56cae9fa5058a5140f1ea752": "b^2 = bc\\cos\\alpha + ab\\cos\\gamma.\\,", "5b33823a9866de702b152dba38a22c30": " \\boldsymbol\\beta^{(t+1)} = \\boldsymbol\\beta^{(t)} + \\mathcal{I}^{-1}(\\boldsymbol\\beta^{(t)}) u(\\boldsymbol\\beta^{(t)}), ", "5b338c965a425b8bfa617092527e3b94": "\\prod_{k=2}^{n} n(k-1) = n^{n-1} (n-1)! = n^{n-2} n!.", "5b338efb67ca6e19d62fc6ba6b845343": "\\vec{F}\\,(\\vec{p}) = \\vec{F}\\,(\\vec{q})+(\\vec{p}-\\vec{q})\\frac{\\partial \\vec{F}\\,(\\vec{q})}{\\partial \\vec{p}}+(\\vec{p}-\\vec{q})^2\\frac{\\partial^2 \\vec{F}\\,(\\vec{q})}{\\partial \\vec{p}^2} +O(\\vec{p}-\\vec{q})^3\\!", "5b34583a40a56005dfab9e6c4b819a51": "\\xi_{\\rightarrow} ", "5b3504d6ac4e1ba8ef0eb44c48029590": "(x + \\alpha e_1) \\sim (y + \\alpha e_1)", "5b35108423a49d7e0aca86dc9b53b9df": "(\\lambda 1 1 1 1 (\\lambda \\lambda 1 (\\lambda \\lambda 1) 2)) (\\lambda \\lambda 2 (2 1))", "5b358b8dc2b8876b779b96cd6ea28501": "\n \\begin{align}\n & T > T_0 \\quad \\implies \\quad a_{\\rm T} > 1 \\\\\n & T < T_0 \\quad \\implies \\quad a_{\\rm T} < 1 \\\\\n & T = T_0 \\quad \\implies \\quad a_{\\rm T} = 1 \\,.\n \\end{align}\n ", "5b359129b351dd5021463ba4a8e422b9": "\\scriptstyle -T_\\mathrm{g} \\,\\le\\, t \\,<\\, 0", "5b35d3cb759abbbf1c5012962a5de470": "x_i\\in\\mathbb{R}^n ", "5b360e0f4610ac4821ae6f26df24fe46": "\\bar{\\psi}_R\\psi_R", "5b361534ec745f2531901e34bdaa6217": " \\omega=(y, t)", "5b36432d73822b81ca2ac0a262180275": "e_{y}", "5b364fb5e8ef33351052366964b1f2b6": "\\mathbf{A} \\rightarrow \\mathbf{A}+\\nabla\\psi", "5b3676a239f5b835720b549c0720f184": "V(a) = V_A(a) = -V_B(a)", "5b37264fdb2ee07a76ca32c00a45ec16": "M = \\langle X \\mid R \\rangle", "5b374c7d8de93dfc1d3d9f7f9fc49de8": "\\tilde{H}_n", "5b378d4c2b546cbb2357bc578031ba00": "P^{(k)}", "5b37975eb1b2c1aa2c0260378ab025d8": " S_m ", "5b379890f5110c0e77f8b380022b1c6a": "1/\\beta -1", "5b37b63dd47999cf132e930f17cb2d79": "\\theta_r(t)", "5b37bd8e2a9b84a220df5cead4e452c8": "f(i)\\notin g([m])", "5b37bf328f6c8a6154255892e4ca3ad7": "{n\\over 2n-1}={2\\over 3},\\, {3\\over 5},\\, {4\\over 7},\\, {5\\over 9},\\,{6\\over 11},\\cdots ", "5b37e23a5569a6c4452c648f808221bd": "\\mathbf{O_3}", "5b37f3a820c60c0381395eebce8359cd": "V^2 = \\frac{Q^2}{A_w^2}", "5b37f8aadb675032531062749a047d99": " \\; \\phi \\in \\Phi", "5b380e753aacd37c190a5776bef83e8e": "X^{-2} \\sim \\operatorname{F}(n, 1) ", "5b381d6709c05dd04f9a7e66819a671e": " \\mathcal{D} ", "5b383326b7f750fef6c84b2bd5f791b2": "U_ \\mathbf{E} = q\\,V. \\, ", "5b3842dc98f331afb1afe66345ddcc2c": "\\kappa(p)", "5b38af819abdae2d4632ea7ed6683ee0": " \\mathbf R = \\sum_{i=1}^3 \\mathbf n_i \\otimes \\mathbf N_i ~;~~\n \\mathbf F = \\sum_{i=1}^3 \\lambda_i \\mathbf n_i \\otimes \\mathbf N_i \\,\\!", "5b38b09de29b387c576360856aca8189": " = \\int f(x) \\; \\nabla_{\\theta} \\pi(x \\,|\\, \\theta) \\; dx ", "5b38f56d513de2afc9cca7b8c7885223": "\\alpha_G=\\frac{G m_e^2}{\\hbar c}=\\left( t_P \\omega_C \\right)^2 ", "5b391db359375be595144b319d781996": "r_{u,i} = \\bar{r_u} + k\\sum\\limits_{u^\\prime \\in U}simil(u,u^\\prime)(r_{u^\\prime, i}-\\bar{r_{u^\\prime}} )", "5b3927f7d0903dcdb0dd1b3d494acd50": "\\gamma^2", "5b393149f027995c65924cc11d84a644": "\\mathcal{L}(\\underline{x}) = \\log\\frac{\\sum^n_{i=1} (2S_i x_i - S_i^2)}{2\\sigma^2} \\Leftrightarrow \\Delta t \\Sigma ^n_{i = 1} S_i x_i = \\sum^n_{i=1} S(i\\Delta t)x(i\\Delta t)\\Delta t \\gtrless \\lambda_2", "5b3954e7a5379abe7b3b766d4a9e9405": "\\operatorname{E}\\left[\\epsilon^T\\Lambda\\epsilon\\right]=\\operatorname{tr}\\left[\\Lambda \\Sigma\\right] + \\mu^T\\Lambda\\mu", "5b3a4064eee74a5fcce552a3ef2ca1a4": "\\beth_{\\alpha}", "5b3a421535a27c3820e85c1ff72b0cd2": "f'(x)=nx^{n-1},\\!", "5b3a91f5da97ca2f06843f81bdff82f8": "\\textstyle \\left(\\frac{a}{n}\\right) \\Rightarrow \\left(\\frac{a}{p}\\right) = \\left(\\frac{a}{q}\\right)", "5b3ad79740e1d84eebc4c2db6c8c6acb": "\\begin{align}\n\\varepsilon &: FG \\to 1_{\\mathcal C} \\\\\n\\eta &: 1_{\\mathcal D} \\to GF\\end{align}", "5b3aecd65a2e21e9d5bfa17fd3bdffa1": "M':=f^{-1}(c+\\epsilon)", "5b3b7d1082162ff5939d106550fc70cb": "fH_i M \\otimes fH_{n-i} M \\to \\Bbb Z", "5b3be7d136a116154fe40048ea7cca31": "\\langle\\sigma, q\\rangle \\in \\tau", "5b3bec54174c94cdbdea4b89fee80015": "1966 = [35, 6]_{56}", "5b3c7290381641669a21e2347d2df00a": "T^{-n}", "5b3c806b08b7d3a7841d985d3882bb4f": "\\left(\\sqrt{2}\\right)^{\\log_{\\sqrt{2}}3}=3.", "5b3cd620c8142ea573110cf9f866394d": "\\omega_0>0", "5b3d04b360a93f5a9b448b58543816c2": "\\alpha_{m1,m2,k}", "5b3d4e47d488dfd1edd89620e0cd270c": " u_{\\alpha} u^{\\alpha} = -1 ", "5b3d5ff649f49670a9f46be9fd4eab6d": "\\left\\{{n \\atop k}\\right\\}", "5b3daa1d33ae1f552f6e494505ecdf43": "\n\\left( \\frac{\\partial Q_{m}}{\\partial q_{n}}\\right)_{\\mathbf{q}, \\mathbf{p}} = \\left( \\frac{\\partial p_{n}}{\\partial P_{m}}\\right)_{\\mathbf{Q}, \\mathbf{P}}\n", "5b3de1a4cc85f60ef977d4b16c8a47b9": "V = I Z\\,\\!", "5b3dffe2ba1529f995cfac04e7c8e3c8": "\\text{Zomma} = \\frac{\\partial \\Gamma}{\\partial \\sigma} = \\frac{\\partial \\text{vanna}}{\\partial S} = \\frac{\\partial^3 V}{\\partial S^2 \\, \\partial \\sigma}", "5b3e12c46c9acae2a488feda2c9c94dc": "C_\\text{aer}=k' v^2", "5b3e3cf050db8ad89a140ed010d2c214": "f: A \\rightarrow B", "5b3e54e819d62009c3d146a7a12b0d5c": "(i-1)", "5b3e5ba32fc9580c961cbcdd5ecc2043": "\\int_0^{2\\pi}f(x) \\, dx = 0.", "5b3e80c5c193e88f59037cc4972ce1fe": "E\\left[ x_i x_j x_k x_n\\right]", "5b3f06f01dcec871ac205eee328268e7": "x_n'=f_n(x_1, \\ldots, x_n)", "5b3f0d9146b953d8a6dddf91bab34cfc": "SSE = \\sum_{i=1}^n (X_i-\\bar{X})^2 + \\sum_{i=1}^n (Y_i-\\bar{Y})^2 + \\sum_{i=1}^n (Z_i-\\bar{Z})^2", "5b3f5db44d37c31d3a48ec2dbc2df92b": "n=p_1^{\\alpha_1}...p_k^{\\alpha_k}", "5b3fb0d47e83100f581ae52b79aaf6c9": "\\begin{align}\n \\sqrt{i} & = \\left ( \\cos\\left ( \\frac{\\pi}{2} \\right ) + i\\sin \\left (\\frac{\\pi}{2} \\right ) \\right )^{\\frac{1}{2}} \\\\\n & = \\cos\\left (\\frac{\\pi}{4} \\right ) + i\\sin\\left ( \\frac{\\pi}{4} \\right ) \\\\\n & = \\frac{1}{\\sqrt{2}} + i\\left ( \\frac{1}{\\sqrt{2}} \\right ) = \\frac{1}{\\sqrt{2}}(1+i) . \\\\\n\\end{align}\n", "5b40352341ce8c2bd13b3708ad1e08d1": "n \\sin \\theta_\\max = \\sqrt{n_\\text{core}^2 - n_\\text{clad}^2},", "5b407a08f0510a1ddd9829817830e910": "\\mathrm{Fr} = \\frac{V^2}{g d}", "5b408ee1f5b3e7829fedbd045a391373": "K_{2k+1}(M;\\mathbb{Z}_2)=ker(f_*:H_{2k+1}(M;\\mathbb{Z}_2)\\to H_{2k+1}(X;\\mathbb{Z}_2))", "5b40b739e6dc9d158516781922f9f36d": " \\triangle CDA ", "5b41525bb09f17767049a1f8eb05747e": "r_i=y_i^\\text{obs}-y_i^\\text{calc}", "5b4193133eb09bd58918410f10185028": "\\frac{ml}{ml}", "5b42657a9141f32e4c9dbeec71d9a024": "\\Phi \\square \\Phi = -4 \\pi G T ", "5b427b08798fa303174aa2ba148bd478": "(b_n)", "5b42971e8464a1b466de7d03c76c9afc": "\\Re z", "5b42f7ccc07950af78cb23637a849769": "M=\\mathbb{R}^n", "5b43795b3c8550f5a6591c51055c6024": "\\Pi_{\\rho,\\delta}^{n}\\leq I", "5b4394877c8c0bd1cb304ea5b1c8210a": "IG(A)", "5b439eb0ec21e0555fc66eb88f1149e9": "Y_{6}^{1}(\\theta,\\varphi)={-1\\over 16}\\sqrt{273\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(33\\cos^{5}\\theta-30\\cos^{3}\\theta+5\\cos\\theta)", "5b43ad2de36c627cb2bd2361393f7d2a": "true \\rightarrow loaded(1)", "5b43bb3b05a90ac4a7efe922374d849d": "\\dot Q=\\kappa \\frac{A}{L}(T_1-T_2)", "5b43c2540942ece9dd2e475a82084fc4": "g^{k}=e", "5b44216c69054cdae3923b61baea7069": "\\cos\\theta = \n\\begin{cases}\n\\frac{1}{2g} \\left[ 1 + g^2 - \\left(\\frac{1-g^2}{1-g+2g\\xi}\\right)^2\\right]&\\text{ if }g\\ne 0 \\\\\n1-2\\xi&\\text{ if }g= 0\n\\end{cases}\n", "5b4443f844a65b48ae2a881e0e69de60": "W^{T}W=wI", "5b44a1648dcd0379e774ceca31e229da": " \\scriptstyle x_{ni} ", "5b44dfa9e1ee1f854b8577c049a38ab2": "Y_0 \\approx 1/\\sqrt{S}", "5b452753518e6cc6cde9903084b1c0e3": "k^*=p", "5b4583633fc247c0d1dbe475f7c40d4b": "U=\\sum_{t=0}^{T}\\beta^{t}u(c_t)", "5b45cee086f6f1ede0020f01786770ee": "|F-G|", "5b45d89ebe7b151be7cc175ea3431c1d": "gfg^{-1}(z) = z + \\beta\\,.", "5b461352b7e1fb617e0ad69eb199dbd5": "\\scriptstyle cT", "5b4642de76ed267565bafee94b0f6b76": "\\underline{\\hat{u}}(f)", "5b466ce4e4c5573c7e70f2d3901ec65b": "\\int_{-\\infty}^\\infty w(2^j t - k) \\cdot w(2^{j'} t - k') \\, dt = 0", "5b46ab0d331c22caa392c90bf1224541": " G( z, \\mu, \\phi) = \\exp\\left( \\frac{ \\mu }{ \\phi }\\left( \\frac{ 1 }{ z } - 1 \\right) \\log\\left( 1 - \\phi z \\right) \\right)", "5b46eba26db885f32a662ff1738d583b": " \\ \\textbf{f} = 1+p\\textbf{F} ", "5b4702f4be3c43b319974ace39b62c31": "D^{(0)} f^{(0)}_n (x)=\\lambda^{(0)}_n f^{(0)}_n (x) ", "5b473d72c807ca7eb07391d08bea5bd0": "\\tfrac12(x_1 y_2 + x_2 y_3 + x_3 y_1 - x_2 y_1 - x_3 y_2 - x_1 y_3)", "5b47697ab903c52614000e809c22976a": "\\int_0^{\\pi/2}\\sin^{2p-1} x \\cos^{2q-1} x\\, dx = \\frac{\\Gamma(p)\\Gamma(q)}{2\\Gamma(p+q)}=\\frac{1}{2} B(p,q)", "5b477d3c0d4306445bb2fa01107fbbbd": " \\langle x,y+z\\rangle = \\langle x,y\\rangle + \\langle x,z\\rangle ", "5b486b33003f60e02f128ceb16a7fbe3": " \\left . \\frac{dv}{dr} \\right \\vert_{r+dr} = \\left . \\frac{dv}{dr} \\right \\vert_r + \\left . \\frac{d^2 v}{dr^2} \\right \\vert_r dr .", "5b4879dd3c45d645b265b93e4e0e51c6": "n_1\\sin\\theta_i = n_2\\sin\\theta_t \\quad", "5b487d5e66cb0fade72ac6942f708c9d": "\\begin{alignat}{17}\nf(0) &&\\; = \\;&& 0 \\;\\;\\;\\;\\;&& \\Rightarrow &&\\;\\;\\;\\;\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; 0 \\;&& + &&\\; a_0 &&\\; = \\;&& 0 & \\\\\nf(1) &&\\; = \\;&& 1 \\;\\;\\;\\;\\;&& \\Rightarrow &&\\;\\;\\;\\;\\; a_5 \\;&& + &&\\; a_4 \\;&& + &&\\; a_3 \\;&& + &&\\; a_2 \\;&& + &&\\; a_1 \\;&& + &&\\; a_0 &&\\; = \\;&& 1 &\n\\end{alignat}", "5b48a89d0b7e8b594017f09020182287": "=-\\operatorname{tr}(\\gamma^5) \\,", "5b48c0fe587f157b874b6fab12442997": "\\dfrac{5^2-1}{3}=\\dfrac{44_5}{3} = 13_5; \\,\n\\dfrac{5^4-1}{3}=\\dfrac{4444_5}{3} = 1313_5", "5b49357496f1c4752b968018468ecda4": "\\displaystyle{E(f)=H_{f^{-1}}}", "5b493f4cbb352b29766c48ee143434be": "r_\\text{out} \\triangleq \\frac{v_\\text{out}}{i_\\text{out}}", "5b494919081bfb0315eb0c4c5526cab3": "c_1=c_1^p \\equiv (a_1+b_1- a_0^{p-1}b_0-\\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1})^p\\mod p", "5b49c0fce709e88c0f482fd3f2ea0057": "a \\,\\bmod\\, n = a-\\lfloor a m\\rfloor n ", "5b49c47887c93c365c048c4d79b3a5e1": "\\mathcal{S}=\\int \\mathrm{d}^{D-1}x \\, \\mathrm{d}t \\mathcal{L} = \\int\n\\mathrm{d}^{D-1}x \\mathrm{d}t \\left[\\frac{1}{2}\\eta^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi - V(\\phi) \\right]", "5b49cbf115ea2a97a28bc1f88f5485ac": "NAE = V ( U_{NH_4} + U_{TA} - U_{HCO_3} )", "5b49d256f07a2c5c6d997e443f1c22c0": "\\bar{G}=\\{g_c:g_c(\\theta)=\\theta + c,c\\in \\Bbb{R}^1\\},", "5b4a0381d4c0396b0a1171a9bf19b3b2": " \n\\begin{align}\n T(Y) &{}= \\dfrac{1}{2N}\\sum_{i,j}|Y_{i}-Y_{j}|^{2} \\\\\n &{}= \\dfrac{1}{2N}\\sum_{i,j}(Y_{i}^2+Y_{j}^2-Y_{i} \\cdot Y_{j} - Y_{j} \\cdot Y_{i})\\\\\n &{}= \\dfrac{1}{2N}(\\sum_{i,j}Y_{i}^2+\\sum_{i,j}Y_{j}^2-\\sum_{i,j}Y_{i} \\cdot Y_{j} -\\sum_{i,j}Y_{j} \\cdot Y_{i})\\\\ \n &{}= \\dfrac{1}{2N}(\\sum_{i,j}Y_{i}^2+\\sum_{i,j}Y_{j}^2-0 -0)\\\\\n &{}= \\dfrac{1}{N}(\\sum_{i}Y_{i}^2)=\\dfrac{1}{N}(Tr(K))\\\\\n\\end{align}\n\\,\\!", "5b4a068b33c7b8acb112844c280ab023": " \\sigma_r \\ ", "5b4a10bfb29eeef22b06ef3feebd6b99": " {\\rm MCG}(\\mathbf{T}^n) \\simeq {\\rm SL}(n, {\\mathbf Z}). ", "5b4a9823793086207303f3ce8a1493c1": "\\hat{\\phi}(\\boldsymbol{k},\\omega),", "5b4ab19c42f6df5234dce498a397751e": "\\Delta U = mg \\Delta h.", "5b4abf79c6ab5f7f4f074d160cef592b": "I \\approx I_S \\cdot e^{V_D/(nV_T)}", "5b4aeed8994b68640ae14e9cac7f8b53": "\n\\frac{\\lambda+3\\lambda^2+\\alpha\\beta(1+6\\lambda+6\\lambda\\beta+7\\beta+12\\beta^2+6\\beta^3+3\\alpha\\beta+6\\alpha\\beta^2+3\\alpha\\beta^3)}{\\left(\\lambda + \\alpha\\beta(1+\\beta)\\right)^2}\n", "5b4b2845b34abb082dc45f92f75f22bc": "10\\uparrow\\uparrow\\uparrow 5=(10 \\uparrow \\uparrow)^5 1", "5b4b73155cbd11d42e46141ed9fec636": "h \\in (-\\delta,0)", "5b4bc3c665bfc45a109c4ae743278bf1": "Lu=f\\text{ in }\\Omega,", "5b4bd9aa39bf5397220b73b7be720844": "(3, 7),", "5b4c332e42bf5a56fbec53de476fdd4e": "\\Lambda(B)=E[ {N}(B) ] ", "5b4c67f1e1b618d49d050b32bc567572": "K_1 = -1 - \\frac{2}{M^3}\\cdot\\frac{B}{D}", "5b4c80880831415ff54005dcaa2ffbf0": "H^{(0)} \\equiv H^{A}+ H^{B}", "5b4cb8e7a573aea9bcac30a840aa137e": "s_{1},s_{2}...s_{n}", "5b4ccf73ee661380ab00b6a46638721d": "\\scriptstyle 1,\\dots,n", "5b4cdc4b965cb98a65eab18410d67a70": "{ X \\ge t\\ }", "5b4ce9df8284d20a4ad99a80c1f3cc80": "\\Theta_{\\Gamma_8}(\\tau) = 1 + 240\\sum_{n=1}^\\infty \\sigma_3(n) q^{2n}", "5b4cee34ac7efd5b0298ca57bc4f1850": "p_{n,k}' (x)", "5b4d06b0dc7c03e0bd677a2cd4c8daba": "\\operatorname{Tr}(A \\, \\sigma) \\geq 0", "5b4d334643f65e6471282e32a1452c00": "i_P: J^k(E) \\rightarrow F\\,", "5b4d9767a24659e13da714b5da1ef819": "F \\overset{I}{\\to} \\operatorname{End}(V) \\overset{\\operatorname{tr}}{\\to} F", "5b4dae73820984fb61eefb9138a2c200": "[n]_q!=\\frac{(q;q)_n}{(1-q)^n}", "5b4dd256d0bd27d62cf7cc4ce2ac4cf9": " (u, v) ", "5b4de6c928ced7bc0fb4f28c2311ffc6": "A = \\left\\{Z \\in \\mathcal{L}^0: \\mathbb{E}[u(Z)] \\geq \\mathbb{E}[u(0)]\\right\\}", "5b4de8b11c537768d56ff8b4d1adc7b3": " a_{n+2} = {(n-p) (n+p) \\over (n+1) (n+2) } a_n. ", "5b4df63b4b225dbaf3273c9d84360a25": " \\Pi, A \\vdash \\Lambda", "5b4e38860cbab5ac1d486c13bc95aae3": "\\vec{u}\\,\\!", "5b4ee35daf8086648eb3c98d967a573f": " H(z,\\lambda)=B\\sqrt \\pi (\\lambda)^{-1} \\int_{-\\infty}^\\infty e^{\\frac{-1}{4\\lambda}(x-z)^{2}} \\xi(1/2+ix) \\, dx ", "5b4f39c4acaf6fc0a59ff88e1ffe52e9": "\\scriptstyle \\,s_{12,0} = 0.7854002... (+2.6 \\times 10^{-6})", "5b4f56847ea605336b2c604440a4c8e0": "\\chi_{V\\otimes_S V}(g)=\\frac{1}{2}[\\chi_V(g)^2+\\chi_V(g^2)]", "5b4f74492e3107e74c89136e12bba965": "V(x,0)", "5b4f8388e34b6db79aa498c4e944f755": " \\epsilon=0.1", "5b4fa4069f5394ebb713113e7fb7ee9c": "15 * 0% = 0", "5b4fbca0800820dcc3da0cb251ccb5d6": " R(h,k) ", "5b4fd4f3480f668559d663b30aa03bea": "T_\\Lambda M = T\\Lambda\\oplus E^s\\oplus E^u", "5b500df49a25053f352c4c612ab10160": "\\beta = \\frac {\\mathrm{Cov}(r_a,r_b)}{\\mathrm{Var}(r_b)}", "5b5028294e1aff5f7382276cf41265ec": "X = \\sum_{i=1}^n X_i", "5b5048554ddb4280974d37a59476216b": "OP = 1\\,", "5b50a6da1805ef4a8b023284c8146df2": " | \\psi_{I}(t) \\rang = e^{i H_{0, S} ~t / \\hbar} | \\psi_{S}(t) \\rang ", "5b50bc702f300e951714e65d541c4d0b": "\\beta_k(z E_{11}) = E_{11} \\otimes \\Zeta_2,", "5b5118a08447ae94fd848643df82f78e": "r_2 \\sim{} N\\left(0,N_{0}/2\\right)", "5b512b495211a407205acc432d2e4a8e": "[K_i,P_0] = -i P_i ~,", "5b519e65c2f1d2f51844d5c43705aaa4": "\\vec x \\cdot \\vec y \\ge \\vec x \\cdot \\vec y_1", "5b51aaba676c5ee76802cac97c640b60": " \\sigma_r = \\dfrac{-P}{2} \\ ", "5b5212dc5fb26efb7f61e30e279b4216": "{\\left(zw\\right)}^* = w^* z^*.", "5b522ae1f928fc6fba5ec6ca50406be9": "\\mathcal L=\\sum_{k=1}^N \\left(\\frac{i}{2}\\overleftrightarrow{\\overline{\\Psi}_k\\gamma^\\mu\\partial_\\mu\\Psi_k}-(m+g\\sigma)\\overline{\\Psi}_k\\Psi_k \\right) - U(\\sigma)+\\frac{1}{2}(\\partial_\\mu\\sigma)^2.\n\\,\\,\\,\\,\\,(3)", "5b527dfa692fa6188612c21e2033f4ad": "C(\\mathbf{x}) = \\{\\mathbf{u}\\in S^2\\mid \\mathbf{u}\\cdot\\mathbf{x}=0\\}.", "5b529aaf848d0a06f394cb8d05d2aac3": "{{B}_{x}}(x)=e(x)", "5b52d7d176e00681dc419cca446f0d65": "T_w\\circ T_v=T_{vw}", "5b5300d9651f6cc5a8053ed282143ae9": " f: \\mathbb{R}^m \\rightarrow \\mathbb{R}^n", "5b530fd8eeea32bc8a6b9cb41e9f2453": "D_R (x,y,z)", "5b536a75b764aea384e7f9e85ff1dd2d": "\\mu + \\frac{\\delta\\beta}{\\gamma}", "5b5439115770c3ef5e5abca121859ed5": "~\\left|\\frac{f_{\\rm a}(x) - f(x)}{f(x)}\\right| \\ll 1.~", "5b545fd5f47a47fdab0b10189944155c": "\\begin{bmatrix}\n & & & 0\\\\\n & O(n) & & \\vdots\\\\\n & & & 0\\\\\n0 & \\cdots & 0 & 1\n\\end{bmatrix}\n", "5b54bd212df093e3d9366a9dd45ed72a": " n \\geq 2", "5b54eb8d51cc28baedf991c02643b627": "r,s,\\ldots", "5b54f22567dc4d010f8ed3de8b6600b7": "\n \\cfrac{\\Gamma, A[t/x] \\vdash \\Delta}{\\Gamma, \\forall x A \\vdash \\Delta} \\quad ({\\forall}L)\n ", "5b54f643171efd1ace40b78ef4819f60": "\\mathrm{Re}_p*=\\frac{u_* D}{\\nu}", "5b54fb2085c85d5582056f1de7f0394d": "\\langle , \\rangle", "5b5511eb34d2088c0cbc781ac131b817": "\\frac{a-b}{b}", "5b55277e746dba01deae4c6a50ac45ee": " \\left[ a; \\frac{2at}{t^2+1}; \\frac{a(t^2-1)}{t^2+1}\\right]", "5b554238f4199c5c1cdce1727ba91e7c": "\\hat{X}^n_{DUDE}", "5b5546634062a253bf09c8925f68a163": "N\\cong N'", "5b55654ee85d353c4fadfa3511f9e0c4": " \\frac{1}{c^2} u_{tt} = u_{xx} + u_{yy},", "5b55b625414060b3878f086b23476956": "< 90^\\circ", "5b55bda6e113d185de171a98bbe7c144": "\\frac{\\mathrm{d}f}{\\mathrm{d}t} = \\{f, \\mathcal{H}\\} + \\frac{\\partial f}{\\partial t}", "5b560a881881c4398921091cb502cd8e": "\n\\Delta p \\sim p_{\\text{photon}} \\sin\\theta=\\frac{h}{\\lambda} \\sin\\theta\n", "5b56660b163bbbb67e4b0269d489a292": "\\tilde{H}_i(X) \\cong 0", "5b5667045cc0c21cd309530560aab1ef": "z_0 = \\sqrt{-2 \\ln U_1} \\cos(2 \\pi U_2) = \\sqrt{-2 \\ln s} \\left(\\frac{u}{\\sqrt{s}}\\right) = u \\cdot \\sqrt{\\frac{-2 \\ln s}{s}}", "5b568e2655f8c473a6441dbfb70f21f8": "E_6\\times\\mathrm{SO}(2)", "5b56d982cb90bf698d407206bccd0491": "f(\\mathbf{r}) = \\lambda_1 f(\\mathbf{r}_1) + \\lambda_2 f(\\mathbf{r}_2) + \\lambda_3 f(\\mathbf{r}_3)", "5b56dc6e8beb6b9fa2ae7a8b771e718f": " M(\\varphi) = \\frac{a(1 - e^2)}{\\left (1 - e^2 \\sin^2 \\varphi \\right )^{3/2}},", "5b57451142f529f07f95b54ccbf802d2": "\\left( \\Phi \\cup \\{\\phi\\}\\right)", "5b57481cab463daabc086a2bc6f18265": "D_{1}\\,\\!", "5b574fdffd9b596229af22879485d99e": "\\ln \\frac{b_n}{a_n}", "5b576c170c70368cae53deba6c8756d1": "\\mathbf{Q}^*_{M \\times 1} ", "5b578ee0a46adc3307e330b04c8da8e7": "\\det S''_{zz}(z^0) \\neq 0", "5b57a51c697786fe800f024edf2e5ef7": "(N-1, y)", "5b582a2e41f0c6eb6faa0e21e76ecbf4": "x_1y_1,\\,x_1y_2,\\,x_2y_1,\\,x_2y_2,\\,x_1/x_2.", "5b5830c3b8a3676b36c07aea9c4db91d": "n_1^2+n_2^2+n_3^2=\\frac{T_1}{{\\sigma_1}^2}^2+\\frac{T_2}{{\\sigma_2}^2}^2+\\frac{T_3}{{\\sigma_3}^2}^2=1\\,\\!", "5b586a9f9b43ea1bd3c5bf0313e55f9d": "=\\frac{(0.3* 0)+(0.7 *-0.5)+(0* 1) +(0 *-1)}{0.3+0.7+0+0} ", "5b588aea822be3fa4920de6deba02ff6": "O(\\sqrt[4]{N})", "5b58b71698c13ccc06496816a953eca9": "Var_D(X) = Var(X) - Cov(X,D)Var(D)^{-1}Cov(D,X). ", "5b58cc0cefa1115cdeb54f391b25591d": "D_j", "5b5915868d944f3dbafd7eadec2ed57a": "f_{\\text{N}_2\\text{O}_4}/p^\\ominus = K \\left(f_{\\text{NO}_2}/{p^\\ominus}\\right)^2", "5b592f3faa05b11d09fd9599bb89542f": "\\frac{\\Gamma(m+\\frac{1}{2})}{\\Gamma(m)}\\left(\\frac{\\Omega}{m}\\right)^{1/2}", "5b5980dd8db8151d4fa52d98bfb4e949": "g_1 g_2\\cdots g_m\\,=\\,h_1 h_2\\cdots h_n.", "5b599546566f251684559077f6cb66b4": "x = (a_1, \\dots, a_n).", "5b59bd3c0d018303a70cbf86b82abbbb": "\nR_s = \\frac{1}{\\sigma\\delta_s} = \\sqrt{\\frac{\\pi\\mu f}{\\sigma}},\n", "5b59d3287728dcd26bbb9de4eb56d2ff": "R_1, \\, R, \\, R_2", "5b59e10034822c8645e0146fb4286bca": "0, \\pm1", "5b5a0e72f05094c21ae75edc7ccff70e": "\\forall a,b \\in A, \\;\\; a \\neq b \\Rightarrow f(a) \\neq f(b)", "5b5a0fb296c5ad6604d5edcc4bd71a8d": "\\alpha^v = \\frac{2 \\alpha}{ ( \\alpha, \\alpha) }", "5b5a699bdef172fddfc7e092bd40af85": "\\text{Gain}=10 \\log{\\frac{(\\frac{{V_\\mathrm{out}}^2}{R_\\mathrm{out}})}{(\\frac{{V_\\mathrm{in}}^2}{R_\\mathrm{in}})}}\\ \\mathrm{dB}", "5b5ace796aadba054093dbdb7fea159e": "ab^{n-1}\\,\\bmod\\,n = a\\,\\bmod\\,n", "5b5ae529f10a6832c711699d4a957bcf": "\n \\mathrm{J}_i = \\mathrm{j}_i \\otimes 1 + 1 \\otimes \\mathrm{j}_i\\quad\\mathrm{for}\\quad i = x,y,z.\n", "5b5bb20dc13446d30f98ab899f35d1e1": "w_r(e):=w(e)+r(v)-r(u)", "5b5bf8a857c528fc97f1d68cdc773c35": "P = \\frac{2q^2\\gamma^6}{3c}\\left[(\\dot\\boldsymbol\\beta)^2 - (\\boldsymbol\\beta \\times \\dot\\boldsymbol\\beta)^2\\right].", "5b5c27160389415fd17dc47f824525b7": "\\log\\mathcal{M}(P(x))\\ge \\frac{C}{D\\log D}. ", "5b5c35c0827fcd45f3fca201baa31aa3": "\\|f\\|_p = \\left( \\int |f|^p d\\mu \\right)^{1/p}", "5b5c595a30be79daa343cf4a21cc935f": "L \\subset \\mathbb{R}^2", "5b5d5020ff6d828deea7e303144e64ca": "=0, \\hbar\\omega < \\epsilon_G.", "5b5d9a047d3df1be184a5d223db25f68": " \\mu(t) ", "5b5d9c97e15a088e27138053700797d4": "O(\\log\\log N)", "5b5dab4d00c587d786ed80c936595f00": "\\lim_{x\\rightarrow c}|g(x)|=\\infty", "5b5db385e75241b9bae8f1f9f8dc84dc": " \\left [ P(P(X_1^n(i')) \\geq P(X_1^n(i))) \\right ] \\leq \\left ( \\frac{P(X_1^n(i'))}{P(X_1^n(i))} \\right ) ^s \\, ", "5b5ddf9d4a28e9afc6d41fb76c0c64a8": "\\displaystyle{e^Z=e^{Y/2} e^X e^{Y/2},}", "5b5e16784cc339003cebe7cf92eabc35": "\\alpha, \\rho, \\lambda", "5b5e5b519cca15184ff636414ec0cf80": "\\delta^{\\mu}_{K} \\,", "5b5e6d698cac85ebe8e7719264454d3f": "\\frac{\\partial \\bold{p}_\\mathrm{em}}{\\partial t} - \\bold{\\nabla}\\cdot \\sigma + \\rho \\bold{E} + \\bold{J} \\times \\bold{B} = 0 \\,", "5b5e7e11efab1513f749c1539784b55c": "[ACO]=[ACE]-[CEO] \\,", "5b5e8e97da60450c3ec1e4616cce1f91": "E_{A,B} = \\frac{\\%\\ \\rm{change}\\ \\rm{in}\\ \\rm{quantity}\\ \\rm{demanded}\\ \\rm{of}\\ \\rm{product}\\ A}{\\%\\ \\rm{change}\\ \\rm{in}\\ \\rm{price}\\ \\rm{of}\\ \\rm{product}\\ B} ", "5b5ebdad717d422c9ab2e6e3cdd04617": " CTR = {Clicks \\over Impressions}", "5b5ec5fd43893e10291e7243efc12c02": " |\\vec{\\mu}_S| = g \\mu_B \\sqrt{s(s + 1)} ", "5b5f0112171324a6f9fc5df06f6019dd": "k\\times k", "5b5f16a86fb5b804d69603f34f0c871f": "\\mu(T)\\,=\\,\\mu_0 \\exp\\left( \\frac {E}{RT} \\right) ", "5b5f232b69f9ef872120919dbc87f235": "(i,x,y,f) \\in \\delta^*", "5b5f5272c12a3b0a13cd2aaca901e99a": "E(G_{N}) := \\{ (1, 2), (2, 3), \\dots, (N - 1, N), (N, 1) \\}.", "5b5fc3074880b9b98537c3d86d37173e": "E_{l}\\,", "5b6057d96664ae064284a6ecbc0f8b52": "\nw_{j+1}=w_j-\\frac{w_j e^{w_j}-z}{e^{w_j}(w_j+1)-\\frac{(w_j+2)(w_je^{w_j}-z)}\n{2w_j+2}}\n", "5b60cd51a31a112be13892f80990e7cb": "\\beta\\alpha \\beta^{-1}=\\alpha^{-1}", "5b619a4084fcc183e7292d9e05306042": "= 2\\ (\\gamma^\\rho \\gamma^\\nu + \\gamma^\\nu \\gamma^\\rho) \\,", "5b61a1b298a0d06efa6933a97e68d763": "AR", "5b61e2e12d24c465dcc40bf97319f357": "m,P", "5b61f5df7a455126c2daba074405dc1d": "\\lim_{n\\rightarrow\\infty} diam(C_n)=\\sup\\{d(x,y): x,y\\in C_n\\}\\rightarrow 0", "5b61fd7ec7ba304ad9d61d00c61c4443": "V \\ge 500 LC_{50}", "5b62387e2a2aa97df18b34d21969d98c": "\\frac{j-i}{2}", "5b62428166a6797f28fc5caa72b20bc5": "F^* = F \\to V \\otimes V^* \\cong \\operatorname{End}(V)", "5b625e76ff795bb0cdad1705ca64126e": "I^\\alpha : L^1(a,b) \\to L^1(a,b).", "5b629977b1fea0c490917ed52974255c": "\\theta \\rightarrow \\theta+\\theta'", "5b62cfc70c160b7ce000ca0476e8815f": "\\rho(x, p) = \\psi^*(x, p) \\psi(x, p)", "5b62dc71d4b0bfe112b9f43c16ac193a": "x^{-1}.", "5b633959285d8d3cda21cbc447cb1dd2": "x - 3 = 5^2 = 25\\,", "5b63667ee9100889d383d1521748ed3c": "\\tfrac{3}{5}", "5b637d49db4b861277314162eaae895b": "\\|f\\|_1", "5b637e3bbfbe51555fab45c0d58b9e6a": "\\varepsilon=\\frac{r_\\mathrm{max}-r_\\mathrm{min}}{r_\\mathrm{max}+r_\\mathrm{min}}.", "5b63979b43fc8b346da5ceb6e881e7c7": " \\mu^X:M \\to \\mathbb R, p \\mapsto \\langle \\mu(p),X \\rangle ", "5b639eb7bf9ecd18eb1402035ebea633": "\\bar{C} \\,", "5b63bd149459f260c7ea612f612d92f5": "\\Psi = \\begin{pmatrix}\n\\psi_{1 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} \\\\\n\\psi_{2 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} \\\\\n\\psi_{3 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} \\\\\n\\psi_{4 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} \\\\\n\\end{pmatrix}\n", "5b63cce7f3b738cae5c0e8e6667a93b6": "c_0-a_0-b_0\\equiv (a_0+b_0)^p-a_0-b_0\\equiv \\binom{p}{1} a_0^{p-1}b_0+...+ \\binom{p}{1} a_0 b_0^{p-1} \\mod p^2", "5b63cd881fbbd4e94bb39acccf0d1f8b": "{w^{T}Bw}=1", "5b6449f6171472ca9eb42bea7d84023d": "\\, t_1, t_2, t_1 + t_2 \\in I(x)\\,", "5b64508cc43802532983cb42f82b1649": "d S = \\frac{1}{T}dU+\\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i", "5b6496588ba9d3c336ef5f53be6f0ad8": " q = K\\sqrt p", "5b64d6c97d500b93514b482d729a36d5": " y = y_0 \\ln (x-a) + \\sum_{n=0}^\\infty b_n(x-a)^{n+r}, \\quad b_0 \\neq 0", "5b65264f2e5c0331221f6d491f01998f": "c_1 = 2 \\pi h c^2 \\,", "5b6545c9c22e2e3b020e9ee7d0b86daf": " v(t) = \\sqrt{ \\frac{2mg}{\\rho A C_d} } \\tanh \\left(t \\sqrt{\\frac{g \\rho C_d A}{2 m}} \\right). \\,", "5b654ed7fb090be20e194961b047bf7f": " m_H", "5b65ce2155122eb7b472c72595213ec3": " N = \\frac{m}{\\mu m_\\mathrm{u}} ", "5b6629187e41ec595b7cbd59ee4d669f": "\\mathcal{B}(\\mathbb{R}) ", "5b66c6a8f95c9a525a3dc742bd47756f": "U(1)_B^2 U(1)_R", "5b66f4c11f72f750ea737443604f82d1": "\\frac{(b/d-a/c)}{\\sqrt{3}}", "5b670972495ca37bd945c9aec5bc621e": "\\frac{a_1}{b_1+\\frac{a_2}{b_2+\\ldots}}", "5b672ccf9071482e76d7c2d6995be278": "t_x=c", "5b673db15c49519fa9bdc6d028b1e44d": "\\{ x \\}, \\{ x, y \\} \\in \\mathcal{P}(X \\cup Y) ", "5b6777acc28d17102d482154e117b2dd": " \\langle u,\\ v \\rangle \\le \\|u\\| \\|v\\| \\ . ", "5b677f7cdf0e8e29bbdbe38567dffc95": "p(v_i = 1 \\mid \\textbf{H}) = \\sigma(a_i + \\sum_j h_jw_{ij})", "5b6834ce4f267eb3ce1d55da26b90d7d": "\n\\nabla^{2} V = \n\\frac{1}{a^2 \\left( \\zeta^2 + \\xi^2 \\right)}\n\\left\\{\n\\frac{\\partial}{\\partial \\zeta} \\left[ \n\\left(1+\\zeta^2\\right) \\frac{\\partial V}{\\partial \\zeta}\n\\right] + \n\\frac{\\partial}{\\partial \\xi} \\left[ \n\\left( 1 - \\xi^2 \\right) \\frac{\\partial V}{\\partial \\xi}\n\\right]\n\\right\\}\n+ \\frac{1}{a^2 \\left( 1+\\zeta^2 \\right) \\left( 1 - \\xi^{2} \\right)}\n\\frac{\\partial^2 V}{\\partial \\phi^{2}}\n", "5b6840f8565b1b37a1c67633dd4a1b20": "v=\\sqrt{2gy}", "5b68501e8cb0501b3b2fe52cde21a8b7": "f:\\; M \\mapsto {\\Bbb R}", "5b68871d42ccde5bbfab17e66e0a6ef4": "\\exists X ( \\exists x,y (Xx \\land Xy \\land (y = x + 1 \\lor x = y + 1)) \\land \\exists x \\neg Xx \\land \\forall x\\, \\forall y (Xx \\land (y = x + 1 \\lor x = y + 1) \\rightarrow Xy))", "5b68b6931217620a32218b0b9e725223": " bk = vr, \\,", "5b693c8a3466d71242c87b67135f38f5": "O(nW*10^d)", "5b693e676f253b84f0880ce49604e90b": "\\sum_{k=1}^n A_{1k} = n", "5b6953715ccf3c4cdad4cf85f59db2c7": "c(w)", "5b696b9abf13c350b73721c862e3df07": "\n\\alpha = \\frac{J}{Mc}\\,,\n", "5b69b023f117c80ccf9e6b98af29d482": "l_\\text{P} = \\sqrt{\\frac{\\hbar G}{c^3}}", "5b69c5ba4f12c94c3a9b7677d4a775ff": " X \\subseteq R^{n_x}", "5b69dc67c4e927f8633713aac9eb1ef4": "\\Lambda \\times S \\times S", "5b6a4f1e33d2d2757b4d209ce5044521": " \nINT\\_MIN = x + y \n", "5b6a9d9c249739e0c805c95a028fed6c": "\\, u_t+u_x=0 ", "5b6ab89b9bb936dbe84b9a19c057877e": "\\{ \\mu_{i} | i \\in I \\} \\subseteq \\mathbb{R}^{n}", "5b6b08e2f2b09c418ddf894654c7fc87": "{\\mathbf{A}\\otimes\\mathbf{B}} = \\begin{bmatrix}\n a_{11} b_{11} & a_{11} b_{12} & \\cdots & a_{11} b_{1q} & \n \\cdots & \\cdots & a_{1n} b_{11} & a_{1n} b_{12} & \\cdots & a_{1n} b_{1q} \\\\\n a_{11} b_{21} & a_{11} b_{22} & \\cdots & a_{11} b_{2q} & \n \\cdots & \\cdots & a_{1n} b_{21} & a_{1n} b_{22} & \\cdots & a_{1n} b_{2q} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{11} b_{p1} & a_{11} b_{p2} & \\cdots & a_{11} b_{pq} & \n \\cdots & \\cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \\cdots & a_{1n} b_{pq} \\\\\n \\vdots & \\vdots & & \\vdots & \\ddots & & \\vdots & \\vdots & & \\vdots \\\\\n \\vdots & \\vdots & & \\vdots & & \\ddots & \\vdots & \\vdots & & \\vdots \\\\\n a_{m1} b_{11} & a_{m1} b_{12} & \\cdots & a_{m1} b_{1q} & \n \\cdots & \\cdots & a_{mn} b_{11} & a_{mn} b_{12} & \\cdots & a_{mn} b_{1q} \\\\\n a_{m1} b_{21} & a_{m1} b_{22} & \\cdots & a_{m1} b_{2q} & \n \\cdots & \\cdots & a_{mn} b_{21} & a_{mn} b_{22} & \\cdots & a_{mn} b_{2q} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots & & & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m1} b_{p1} & a_{m1} b_{p2} & \\cdots & a_{m1} b_{pq} & \n \\cdots & \\cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \\cdots & a_{mn} b_{pq} \n\\end{bmatrix}. ", "5b6b86be2af852ffada524b6c732b0fe": "\\left(\\mu_x\\right)", "5b6be5c622db185f93518849f6e3f609": " {C^{\\infty}}(M) ", "5b6c9ddf13157ecdd2011de3c67efdca": "m \\rightarrow 1", "5b6cc4c5691ed8393cd60c60e03bef1d": "P_\\pi = \\begin{pmatrix}\n\\frac{\\beta}{\\alpha+\\beta} &\n\\frac{\\alpha}{\\alpha+\\beta} \\\\\n\\frac{\\beta}{\\alpha+\\beta} &\n\\frac{\\alpha}{\\alpha+\\beta} \n\\end{pmatrix}", "5b6d14d4f64379f3742bedf314a26069": "\\begin{matrix}\n {{A}_{n}}=\\frac{\\sum\\limits_{t}{d(t)\\sin (n\\omega t)}}{\\sum\\limits_{t}{IRF(t)\\sin (n\\omega t)}}=\\frac{\\omega \\tau }{1+{{\\omega }^{2}}{{\\tau }^{2}}}, & {{B}_{n}}=\\frac{\\sum\\limits_{t}{d(t)\\cos (n\\omega t)}}{\\sum\\limits_{t}{IRF\\cos (n\\omega t)}}=\\frac{1}{1+n{{\\omega }^{2}}{{\\tau }^{2}}}, & \\omega =\\frac{2\\pi }{T} \\\\\n\\end{matrix}", "5b6d345279e978dd192f193468eb38af": " \\epsilon_{t} = \\sum_{i=1}^{m} D_{t}(i)I(y_i \\ne h_{t}(x_{i})) ", "5b6d8280154ce3d01d644dea7c7c63e5": " P^{(N)}", "5b6dd573ce651261776e45e35510d009": "\\{X_{1},X_{2},\\ldots ,X_{n}\\}", "5b6dd841ac461a7ac2737c15264b6418": "i(\\theta ) = \\frac{{k^2 V^2 \\left| {m - 1} \\right|^2}}{{4\\pi ^2}}G^2 \\left( {2ka\\sin \\frac{\\theta }{2}} \\right)", "5b6e1a99b2d8b4ea90c4b5eb059ab268": "\\psi_1(x) = \\sqrt{2} \\, \\pi^{-1/4} \\, x \\, \\mathrm{e}^{-\\frac{1}{2} x^2}", "5b6e410b10b3134f85b4ee04a0bd75e1": "Y_{p}(\\Omega) = 1 + R + 2C \\sqrt{R} f_H \\cos (\\nu - 2\\pi P_H )", "5b6e7d0720adc23706e5691beae879c6": "[\\xi^i]", "5b6eab51a2d3a379901c605ac7348855": "|z| < b", "5b6eeaec46dd274c577e3280600c06f6": " \\left | \\mathbf{B} \\right | \\left | \\mathbf{r} \\right | = \\frac{m \\left | \\mathbf{v} \\right | }{ q \\sin \\theta} , \\,\\!", "5b6f08e9b7b589195ffd49ba93564057": "|f| = \\nabla_s", "5b6f5a6cf00b91f9e2ed08a784f477af": "\\neg \\Diamond \\neg p ", "5b70149db0a29280bca9f8c58b949574": "\\Delta S_{mix} = -nR(x_1\\ln x_1 + x_2\\ln x_2) = -nR[x\\ln x + (1-x) \\ln (1-x)]\\,", "5b7017478b9102a829ad235bd924b5d1": "-1.0479", "5b70803d4cca87338d2bbc32a5834a7f": "\\operatorname{ad}_x[y,z]=[\\operatorname{ad}_xy,z]+[y,\\operatorname{ad}_xz].", "5b70e22c19f34d35c42b439f89739d9a": "\\delta[\\varphi] = \\varphi(0)\\,", "5b70f92cf55cfba10e76004bc57a67db": "\\eta= \\frac{n}{n_0}", "5b714209da05dc630c26e406f2e28bcf": "\\scriptstyle \\Delta T", "5b71b51e74d0e85d289fb2955a37a13d": "Y^i:M\\to\\mathbb{R}", "5b71b6ca85a760b44eb19994bdaa8298": "F_\\text{P} = \\frac{m c^2}{\\frac{Gm}{c^2}}=\\frac{c^4}{G}.", "5b71cc5a94f633afa76257583389c328": "\\sin{\\theta\\over 2} = \\frac{\\cos(\\pi/q)}{\\sin(\\pi/p)}.", "5b71d6dacbdcea5e00e534e92b576d55": "F \\dashv G", "5b71e8e30fe5935e0593384ae13f5ddf": "\\tilde{\\theta}\\frac{d-1}{2\\bar{x}(2-d)^2}", "5b723f55a267074a8faf0e522c42e882": "x=a \\operatorname{arcsinh}(p/a)+\\frac{T_0}{E}p + \\alpha,", "5b72520ad238c4c6d9529dabf50589ba": "\\cup\\mathcal{F}", "5b738f9c5ce41227d60990ebbc919f28": "\\alpha_i \\beta = - \\beta \\alpha_i, \\quad \\alpha_i\\alpha_j = - \\alpha_j\\alpha_i \\,,", "5b73ea308d3191f24d4650a49b2c2cfe": " B_x(p,q) = \\tfrac{x^p}{p}{}_2F_1(p,1-q;p+1;x)", "5b740864f7e92caaf557c52a523d3bea": " P_k(S_x) ", "5b74163725e22bc7cad2aeccd7e4e5d0": "p = \\frac{ \\displaystyle{{a+b}\\choose{a}} \\displaystyle{{c+d}\\choose{c}} }{ \\displaystyle{{n}\\choose{a+c}} } = \\frac{(a+b)!~(c+d)!~(a+c)!~(b+d)!}{a!~~b!~~c!~~d!~~n!}", "5b742fabb82d1f7710ed5cc6dca64b28": "V_{RM}", "5b745a3b00d316263739b8e12366abc9": " P_{TA} = (P_{AO}) - (P_{ALV})", "5b74a603f39acc8662d55a90621e88c2": "\\min(c_f(A,C), c_f(C,B), c_f(B,D))=", "5b74a8375b177b4317103294ef39dd2e": "\\lambda_{13}=10.1735", "5b74cc2e3e6e44118b79cc5899d4ef23": "i\\infty", "5b74d8f75aa934fa03b35f63219a04bb": "H(x,p;R)", "5b750ecc9382a801a890ea5ad63e1136": "\\mathbf{p}(t) \\rightarrow -i\\hbar\\boldsymbol{\\nabla}", "5b75b960ffe76a0b583f8d863507747f": "q_1=e^{-\\frac{\\pi K}{K'}}.\\,", "5b75bc53e2e0f9fa791905adfacd5150": " A \\succeq C", "5b76c2141d002bee22155e22b1808da1": "\n u_x(x,y,z,t) = -z~\\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z,t) = w(x,t)\n", "5b76cf1d8d45a5110a23d48a5b70ed73": "2^{13_{dec}}", "5b76f27126afc66b22df30bfbd2ae76a": "[0,\\infty)", "5b76fd0539f29a4c77e79455dc2b586d": "E_{KIN}\\ge 0", "5b772d1ebd5788b0c7f78610addd40e0": "\\forall N \\in V, w \\in \\Sigma^*: (N, w) \\in R \\Rightarrow w \\neq", "5b7734de883f1ccf55e5da4423bd33e2": "\\sum_{e \\in p}l_e(f_e) < \\sum_{e \\in q}l_e(f_e).", "5b773c96d0f8c597988371e07a229099": "\\overline{OE}", "5b7754feec9c63ac32b83d9cdf545caf": "\\mathrm{OPL} = \\int_C n(s) \\mathrm d s,\\quad", "5b775bf24f5ddbe96375c7c209c44e0a": "\\cdots \\to \\operatorname{Ext}^i(A, X) \\to \\operatorname{Ext}^i(A, Y) \\to \\operatorname{Ext}^i(A, Z) \\to \\operatorname{Ext}^{i + 1}(A, X) \\to \\cdots.\\ ", "5b778c232439fb03410c2637d1f6e798": "\\Pi_k(1,\\ldots,n)=(-1)^{n-k}(k-1)!(n-k)!", "5b77bc5b3cb60916278226bf4fe0db8d": "\nS^{\\nu, \\lambda} (\\omega) = \\sum_\\beta \\frac{ (E_\\beta-E_\\nu) J_{\\nu \\beta} J_{\\beta \\lambda} }{E_\\beta - E_\\nu - \\hbar \\omega - \\mathrm{i} \\gamma_{\\lambda,\\nu}(\\omega)}\n", "5b77d94ebce36684dd847ca7a26efd52": "E = h\\nu \\,.", "5b77de3fb7bc3be6f6566ff7c8dc54d5": "LC_{50i}", "5b78479d507078c60d07aafa66dcea58": "\\,k_2", "5b78a64825f15955e1b291f1f690286b": "Q(Tx) = \\lambda^2 Q(x)", "5b78dc2699285236b1d1b54a99372422": "q^{th}", "5b79b59b8425af37fcc7492ac234b2fd": "y^{(n)} = r^{n}e^{rx} \\, ", "5b79c36811cce62be6a67c41558f6fb9": "K_n", "5b7a07c9f85a58e3cb107fb393bac746": "\\det (\\lambda I - \\mathbf{U})=\n(\\lambda-1)^{\\left\\lfloor \\tfrac {N+4}{4}\\right\\rfloor}\n(\\lambda+1)^{\\left\\lfloor \\tfrac {N+2}{4}\\right\\rfloor}\n(\\lambda+i)^{\\left\\lfloor \\tfrac {N+1}{4}\\right\\rfloor}\n(\\lambda-i)^{\\left\\lfloor \\tfrac {N-1}{4}\\right\\rfloor}.", "5b7a7774e0c9e0c8d085c2d5d7f75696": "m=\\gamma(\\mathbf{u})m_0", "5b7aa7f7fa92a16f1631ccf8739726aa": "W_1 \\ge W_2 \\ge W_3 > 0", "5b7af5bd72f30cd40c10f12bae5df58e": "\\ U_n", "5b7b33bde21b89906b79e9b424b58cf9": "f(x)=\\sum_{i=1}^n f_i(x_i)", "5b7b5ce3d0b03108b5b51217af6b7139": "\\rho_d = \\frac{M_s}{V_s + V_w + V_a}= \\frac{M_s}{V_t}", "5b7ba4ab8bbea0b184b76a279fc3e6ed": "\\binom{10^6}{k} \\left(10^{-6}\\right)^k(1-10^{-6})^{10^6-k}.", "5b7bf29e833d87c813188e8ef00cfd26": "a \\cdot /b = /b \\cdot a", "5b7c05a25e4cf5c2a2b0463816149033": "1-\\boldsymbol{\\alpha}e^{x\\Theta}\\boldsymbol{1}", "5b7c5aa2a2a15bd892afe579eef0906e": "P_0", "5b7c939914948191a6c87c144e2a92bf": "E_1^{p,q} = \\frac{Z_1^{p,q}}{B_1^{p,q} + Z_0^{p+1,q-1}}.", "5b7c9ec2ee25c0ff984fd00d8dbaf52f": "c_\\max = \\max_{i\\neq j} |\\langle x_i, x_j \\rangle|", "5b7cab4c70377d690416a8f3a7438d88": "0.25 \\ge \\beta > 0", "5b7cd2eea4515286364f20b1b04508f6": "u(x) = v(x) \\quad x \\in I.\\,", "5b7cf6c33c8f0c31e35be93d1c86ebe7": "\\forall L\\in \\textrm{PSPACE}, L\\leq_p \\textrm{TQBF}.", "5b7d3670873d79a9aedf4eff20a5a2c8": "\\displaystyle{((L+I)u,v)=(u,v)_{(1)}.}", "5b7d790a47a3fba46d2194132669b908": "a < c", "5b7dbe8e0c4a5aec3138fa98075e7663": "H = \\sqrt{\\mu \\cdot p}", "5b7e0085d88305cd7024625436ed9549": "L([Tully])", "5b7e1b052d9202080109a1f2f1951843": "{{\\Delta}t}", "5b7e5fe83f4e83cf28f9f5716197e548": "\\Delta_h^\\mu[f](x) = \\sum_{k=0}^N \\mu_k f(x+kh),", "5b7ed81497e918a460186cb3e66eb7aa": "(U(z), 0, 0)", "5b7eeaa095650f78cc29f797e1386cab": "\\widehat{\\widehat{\\sigma_e^2}} = \\frac{1}{n-1} \\sum_{i=1}^n (x_i-\\hat{x_i})^2.", "5b7eef887225a7dda65bd013b089915b": "A' =\\Lambda A ", "5b7f082cb29f36eef7a1dba64718c3a2": "\\displaystyle{D_c(a^{-1})= - Q(a)^{-1}c.}", "5b7f36fa75a70b7a7590dda7ecf9487c": " \\displaystyle \\forall \\ \\varepsilon > 0\\ \\exists\\ \\delta > 0", "5b7f45635027a819241a00045c8d5255": "D_{\\hat{\\alpha}}e_{\\hat{\\dot{\\alpha}}}+\\overline{D}_{\\hat{\\dot{\\alpha}}}e_{\\hat{\\alpha}} \\neq 0", "5b7f77928e874c0d389666e34e43315f": "\\mathfrak g_{+1}", "5b7fa6a537d1806c285765e430601319": "t(n) = \\Theta(n^{\\log_2 3})\\,\\!", "5b7fe131fd2553c056b4e1f03d9e898c": " i \\hbar \\frac{\\partial \\psi}{\\partial t} = \\left( - \\frac{\\hbar^2}{2m} \\nabla^2 +V \\right)\\psi \\quad", "5b7fe7663ecb907e62311118aad6f920": "Ae^{i\\omega t}\\left( 1+i\\beta\\sin(\\Omega t)\\right) ,", "5b7fe94869a658dd608c811becca82c0": "z = e^{sT}, \\ z,s \\in \\mathbb{C}", "5b7ff70752cc660aeda087923806d69e": "k_F=\\varepsilon_F=(3\\pi^2\\lambda)^{1/4}\\sigma_0", "5b80253ad8b16ecc725e90120b683fe5": "gl(n, F)", "5b802637a17424b15e5551f2f2a3d170": " \\omega^2 \\cdot \\lambda ", "5b80b792973522120776b7d8e408025a": "\\Delta t_i", "5b80db40ed41a845fa111c17dbcd1fd2": "Z_\\mathrm{in}=Z_0 \\frac{Z_L + iZ_0\\tan(\\beta l)}{Z_0 + iZ_L\\tan(\\beta l)}", "5b80e278ecc1705094f6b5bd06398caa": "M \\circ (\\mbox {Id} \\times \\eta ) = s;\\ M \\circ (\\eta \\times \\mbox {Id}) = t", "5b8169ed650a0314dce10953f7c7032a": "\\textstyle P = \\frac{1}{Z} e^{-E/(k T)}", "5b81d3ba1e55a768188e428ce144b643": "P_B(t) ", "5b81d50966ed7ca63ca9d38906280f8b": "\\mathfrak{p}_i", "5b82027c253813e6a8022ab810a11c25": "{\\rm PGI} = \\frac{1}{N} \\sum_{j=1}^{q} \\left( \\frac{z-y_j}{z} \\right)", "5b82291d74dad677b68fc6774753ac8f": "K(a \\circ b,n) \\supset K(a,n) \\circ K(b,n) \\mbox{ and } 1 \\in K(1,n), ", "5b823981ff0b51295d8bb50c31e504c3": "H_{n} = -2(n-1)H_{n-2}.\\,\\!", "5b8272736872a190e3922b91e9859116": "\nJ \\equiv\n\\frac{\\partial (\\mathbf{Q}, \\mathbf{P})}{\\partial (\\mathbf{q}, \\mathbf{P})}\n\\left/\n\\frac{\\partial (\\mathbf{q}, \\mathbf{p})}{\\partial (\\mathbf{q}, \\mathbf{P})}\n\\right.\n", "5b82871de596615e81eeb9dcb2d2f8bb": "(f(z_1), f(z_2); f(z_3), f(z_4)) = (z_1, z_2; z_3, z_4).\\ ", "5b82884b701b62210ae9bab7601a521f": "|B:A| = |B':A'|", "5b82db340c6368bd7c1d0ee5b8f29d13": "SG_{water}", "5b82f6d68147a59017c522b1d8ed5775": "h=10", "5b834f6ed5e84ab6a9b155a8a0197d56": "\n{\\beta_2=}\\frac{\\operatorname{E}[(X-{\\mu})^4]}{(\\operatorname{E}[(X-{\\mu})^2])^2} {=} \\frac{\\mu_4}{\\sigma^4}\n", "5b8377d985e40cbfc8b07da0ac6650cb": "\\left\\{\\frac{1}{\\pi_{ji}},\\pi_{ij}\\right\\}", "5b83a1cad9a9afb55f281b6f49f7b100": "A=\\text{Angle of the wind from the direction of travel} ", "5b83a775241b4443117ddc0d62e8db90": "\n\\mathrm {DOF} = \\frac {2 H s (s - f )}\n{H^2 - ( s - f )^2} \\text{ for } s < H \\,.\n", "5b83a861cfb37c7d727d2e3b8ec63e57": "R(E):={E_{m/2+1},\\dots,E_{m-1},E_m}", "5b83b0db8d42c21717e6940ad2a0df20": "\\;\nA \n= P \\frac{r^n (r-1)}{r^n-1} \n= P \\frac{(i+1)^n ((i+\\cancel{1})-\\cancel{1})}{(i+1)^n-1} \n= P \\frac{i (1 + i)^n}{(1 + i)^n-1} \n", "5b83dba310a9e641682ea07c4e9b113f": " g : [m : b : 1]_L \\mapsto [m : -1 : b] ", "5b841f520a8915e9c8d04b4da333d6b2": "{\\rm N}(\\mathfrak a)", "5b843ca6a63cf685a1e7186ebb33f68a": "\\cos(h_o)=-\\tan(\\phi)\\tan(\\delta)", "5b846c7c384f4ec2014b68e81c687cfb": " H^s\\left(\\psi_i(E) \\cap \\psi_j(E)\\right) =0, ", "5b84902efd82d90c56fceb1da17d2c94": "\\pi_{j} : \\mathbb{R}^{d} \\to \\mathbb{R}^{d - 1},", "5b8490dd187cd89ea47fd466ac0b0f90": "\nN_0 \n", "5b84cc2c1b94381850cccceb7e7e6cbc": "p(X,f(X))", "5b84e4761a3b62cca1f90dfdfc85c217": "\\mathbf{x}_0", "5b855aa497efdabf7764298695d80359": " \\frac{TV \\or radioRating} {\\Sigma TVsets} \\cdot{100}", "5b855e0f082cd8e4009d74a54aded397": "(s_1, s_3)\\in R^{j+k}", "5b85633a8a4feee7bd9761a5cfdd641b": "\\textbf{R}^n", "5b85bafc930c3be26eab272ab36d3279": "= (\\mathbb{I} - \\hat{\\alpha}_1) \\otimes \\hat{\\alpha}_2 + \\hat{\\alpha}_1 \\otimes (\\mathbb{I} - \\hat{\\alpha}_2) + (\\mathbb{I} - \\hat{\\alpha}_1) \\otimes (\\mathbb{I} - \\hat{\\alpha}_2)", "5b85c1c8b2105f3a2b65c2779d178c7f": "\\{ U_{i} | i \\in I \\}", "5b85d5ac8a0316347fbb6b56ab30f523": "f(x_i) - g(x_i) = \\sigma (-1)^i || f - g || _\\infty", "5b868d7ad215662ee4e77589bd472520": "2^{n-2}-1", "5b86992d894dca53f26c32cb4cfb2f12": "I_{\\mathrm{xx}} = \\frac{1}{5} m( b^2+c^2),\\qquad\nI_{\\mathrm{yy}} = \\frac{1}{5} m(c^2+a^2),\\qquad\nI_{\\mathrm{zz}} = \\frac{1}{5} m(a^2+b^2),", "5b86ce8d3c95aeb12d441c102fcd216a": "\n D\\left(w_{,1111} + 2w_{,1212} + w_{,2222}\\right) = -q(x,t) - 2\\rho h\\ddot{w} \\,.\n", "5b874e52edf68e9d6ddba1e5b42bfe17": " \\frac{f'(y_i)}{g'(y_i)} + \\epsilon_i \\geqslant \\sup_{x_i < \\xi < c} \\frac{f'(\\xi)}{g'(\\xi)} ", "5b87df856dadd2b131aa14fb241b4aea": "{\\underline P}X = \\emptyset", "5b88018cba917027b0dbeaf66e856b8c": "t_k = 6t_{k-1} - t_{k-2} + 2,\\text{ with }t_0 = 0\\text{ and }t_1 = 1.", "5b880e1fee9d1c5ef6c59cfcf5970ba4": "\nl=\\frac{n(n-1)}{2} = {n \\choose 2}\n", "5b88356a1aea0eaaea3ebe7f93ef3a9b": "\\textstyle C=C\\left( m,n,\\gamma\\right) ", "5b885912cf4bf8fa69c40d991d189cb6": "u\\in H^2(\\Omega)", "5b88686adf6c1c0daac6af5448877d53": "S_{k,t}(x) =\\sum_i \\alpha_i B_{i,k,t}(x)", "5b892d6c4cfa32f8ae0b115ad7e9bb8d": "\\left\\{Var_{x_i}, Var_{x_0}, Cov_{x_ix_0}\\right\\}", "5b8957d5a6f1d3a336036b4d1341073c": "\\rho:E\\to[0,\\infty)", "5b895d3da0ea890766e2b5efc6022fd0": "\nf(t) = A \\sin(2 \\pi f t)\n", "5b895d5f4dacd4b66d0c4384df25be4f": "E \\{ L(\\theta,\\widehat{\\theta}) | x \\}", "5b896b9cfcaa16025bc9d9c8cd5d0b69": "\\beta (s) \\ ", "5b89c5da8e0dcb1d7afbaeb79b72bb12": "c = G_{Newtonian} = 1", "5b8a00628880bb8e0091746b11330b09": "\\Gamma(\\tfrac13)", "5b8a269ce8e8e2306edb42373b778224": "F^{(n-1)}(s)=\\frac{s^{\\alpha+n-1}}{(\\alpha+1)(\\alpha+2) \\cdots (\\alpha+n-1)}.", "5b8a43d5b42d4f5d090c94436e8b2bb3": "\\,(m)", "5b8a9464864e4cf906db72abb7e395f7": "\\bar{\\psi}(-i\\gamma^\\mu\\partial_\\mu - m) = 0 \\,", "5b8acc3161c5c69dba1a0903f1f12a2f": "\\bar \\nu = \\bar\\nu_{sub} + (B^\\prime-B^{\\prime\\prime})J(J+1) - (D_J^\\prime -D_J^{\\prime\\prime})J^2(J+1)^2 -(D_{JK}^\\prime -D_{JK}^{\\prime\\prime})J(J+1)K^2\n", "5b8b0324c5e452a5abdc2f7839ab2349": "\\Gamma(\\tfrac12)\\,", "5b8b17d764901c8ce20f45cafa0c57e2": "X^{(\\alpha)}=X^i\\partial_i\\ell^{\\alpha}", "5b8b3d2ddc1c8b53d231bd43a2d342f0": "\\int_{-\\infty}^{\\infty} \\, f(t)\\ dt \\ = a", "5b8b3e72fdf35f6051038e3d2cfe0ae8": " D \\,\\!", "5b8b976d994961b2de9428b9a7230c13": "N_{L/K}(L^\\times)=N_{L^{\\text{ab}}/K}\\left((L^{\\text{ab}})^\\times\\right).", "5b8b9bd9d98057847f3737b1f23a79bd": "P\\equiv1(mod q)", "5b8ba0f0d4e7cea4c58a092aba47ede9": "W \\subseteq S", "5b8bcfe8fd9a840567bdcda1a6ea2315": "\n{\\left(\\frac{2}{p}\\right) \n= (-1)^{\\frac{p^2-1}{8}} \n= \\left\\{\\begin{array}{cl} +1 & \\textrm{if}\\;p \\equiv 1\\;\\textrm{ or }\\;7 \\pmod 8\\\\ -1 &\\textrm{if}\\;p \\equiv 3\\;\\textrm{ or }\\;5\\pmod 8\\end{array}\\right.}\n", "5b8be1cb9a5ce0d1bff8fb9ed152d85d": " P(\\omega) = \\frac{\\left| X(\\omega)\\right|^2}{\\Delta f} ", "5b8c11ef7642e51e6d4fe8585323730d": "H^1_{\\acute{e}t}(X,\\mathbf{G}_m) = H^1(X,\\mathcal{O}_X^\\times) = \\mathrm{Pic}(X)", "5b8c4f932e2f7a2fc104f47923dabe89": "y_1=\\varepsilon_1(x),", "5b8c69dad82fabc0969b8677b4928952": "6914_{11} \\ ", "5b8c757b32d6f3e78918ec1fb7721005": "S_0 = 0\\,", "5b8c81ce223ca74a77bedc475ed0254e": "\\Sigma^{*} = \\bigcup_{n \\in \\N} \\Sigma^{n}", "5b8cba24426404d135dd459ff38b7e8e": "K(A_i) = 1+\\textrm{INT}\\left((m-1)\\frac{A_i-A_\\textrm{min}}{A_\\textrm{max}-A_\\textrm{min}}\\right)", "5b8cbe2520a311fd67a9acfde8deedf5": "\\,25 \\times v^5", "5b8ce05b863b74370ccad33af95d82dc": "G \\circ H", "5b8d1b7520388892808ed0a1ecd15752": "\n\\sigma _{\\hat g}^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {{{ - 8 \\times 0.5 \\times \\pi ^2 } \\over {1.443^3 }}1.0338} \\right)^2 0.03^2 \\,\\, = \\,\\,0.166", "5b8d4704d0359487c91771552f40a9fe": "L^\\infty([0, 1]),", "5b8d6b4f0b026d4b40e71023634eef63": "c_V = \\frac{3 R}{2}", "5b8d923776b433676e335b3250e58b2c": "x^2-y^2", "5b8dfc0837b6fa424a65ae9bb6c4e876": "\\Pr \\Bigl[ \\bigvee_i (r_i \\notin S)\\Bigr] \\le \\sum_i \\frac{1}{2^{|x| \\cdot |r|}} = \\frac{1}{2^{|x|}} < 1.", "5b8e456804213d3aaff6be80fa2bbf57": " L =", "5b8ef15ce1d29ea25e3629dbd9d7531c": "h_{4s}", "5b8f160dcf2938eb1813dac254fb3cc2": "\\phi_{M1}-\\phi_{M2}", "5b8f82854bba480ca3717688d2217f1f": " G=\\langle P, \\mathbf{S}, \\mathbf{F}\\rangle ", "5b8fbf6ceec613660cf0f962225086e2": "I_s = V_s/Z_m = V_s/(R_s + jX_s + \\frac{(\\frac{R_s}{s} + jX_r^')(jX_m)}{\\frac{R_r^'}{s} + j(X_r^' + X_m)})", "5b8fd6c2d776540d01faf84fdbb4e931": "\\frac{q^2}{g}\\left(\\frac{1}{y_1 y_2}\\right)=\\frac{1}{2}(y_2+y_1)\\qquad\\text{where }q_1^2=y_1^2 v_1^2=y_2^2 v_2^2.", "5b8fe6cef4b945b1b1bbdee2e89d9681": "F^{\\mu\\nu} \\, = - F^{\\nu\\mu}", "5b90102e9116b884d7ed9f6f2a4a0e42": "\\|u-u_h\\| \\le \\sqrt{\\frac{\\gamma}{\\alpha}} \\|u-v\\|", "5b90358b1daa49fd8d168591d4c4a9bc": "v_{th}=\\sqrt{\\frac{3k_BT}{m}}", "5b905de8fb652e2c28062c2ed727ed0a": "\\delta_1\\leq\\delta_2\\rightarrow\\delta\\delta_1\\leq\\delta\\delta_2", "5b9130c3b46d17fe1549e5ef4e45b704": "d\\mathcal{F}(\\Omega_0;V) = \\lim_{s \\to 0}\\frac{\\mathcal{F}(\\Omega_s) - \\mathcal{F}(\\Omega_0)}{s}", "5b924589bce03be82d491a62cdb42164": "4\\uparrow\\uparrow\\uparrow\\uparrow 4 = ", "5b925f8a9306594df9442610291dc19c": "\\sigma^t\\mathcal{B} \\subseteq\\mathcal{B}", "5b92e415b6ba2808cde639bd1b766df1": " \\lambda_{upp}=\\widehat{\\lambda} \\left (1+\\frac{1.96}{\\sqrt{n}} \\right ) ", "5b938717f1c0966e5f9a248bd601383d": "[H_m,H_n]=2m\\ell\\delta_{n+m,0},", "5b93ea489e1944a45733874493c1853f": " \\rho_M ", "5b94459dbbb3139ad525d39a54050d9b": "\\begin{align}\n E_{f_1 + f_2 + f_3} &= k E_{f_1} E_{f_2} E_{f_3} \\\\\n E_{f_1 - f_2 + f_3} &= k E_{f_1} E_{f_2} E_{f_3} \\\\\n E_{f_1 + f_2 - f_3} &= k E_{f_1} E_{f_2} E_{f_3} \\\\\n E_{f_1 - f_2 - f_3} &= k E_{f_1} E_{f_2} E_{f_3}\n\\end{align}", "5b9465bf0e99d66ac4840bcd0a15aedc": "x_i\\leq y_i", "5b953176e60e2e95d238d5f8ff706f0b": "\\alpha < \\beta", "5b956315ded624574a59c84779b48cea": "f'(x_n)", "5b957a62db79b4195b9b943f4b4c5b7d": "\n\\phi = {\\frac{5\\rho_p-2\\rho}{2\\rho_p+\\rho}}-{\\frac{\\beta_p}{\\beta}}\n", "5b959dcd20903a7b7ad8aa1f7b7d6a7b": "\\mathcal{O}(n + k)", "5b96839a2e15a2a5b29833c9cf2ea512": "X=\\{x_i\\}\\,\\!", "5b9696cdfb3583b2a2fd0c9add9c1a8e": "V_{f2}\\,", "5b96d0f0ebdabcadbc677f66a82b3ec2": "S \\vdash \\neg P.", "5b97231fdd27c1cf3fdfa6dddf7cc3da": "\\mathrm{Li}_m(z)", "5b9726c5d2793fc29548ab06ce398bfc": "\\theta_r = \\tan^{-1}\\left(r_2/r_1\\right)", "5b973c165535c89c3854b926973f7bfc": " H g_1 K \\cdot K g_2 L = \\sum_{g \\in H \\backslash G} c_g H g L.", "5b9743d549396bc8460efa05115bb85d": "\\frac{3x + 5}{(1-2x)^2} = \\frac{13/2}{(1-2x)^2} + \\frac{-3/2}{(1-2x)},", "5b974ba3363d44385c3d35cc4b00758b": " t = \\arctan(y/x) \\, ", "5b97870520d81480185c7243b7c24fd7": "\\sum_{n=-\\infty}^\\infty |\\hat{f}(n)|^2 = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} |f(x)|^2 \\, dx", "5b97940cede329d40666e76d7edbe827": "|\\Lambda|^s_M = |\\Lambda|^s(TM).", "5b97ecb6e31502670eadb5ff1735ef1c": "X \\sim \\mathcal{B}e(\\alpha, \\beta)", "5b9895ca3e0f45b682ed6f6590a7cce4": "U_{22}", "5b98f444b60a3817c0a59cfd047524a5": " \\varphi(A) = {\\rm f}(\\mu(A)) ", "5b99113c0a239d788e065d61469e900c": "\\frac{13}{2 \\cdot 5}", "5b992d309bb2eb0847df98691af9637e": " P( | X - m | \\ge ks ) \\le \\frac{ 1 }{ N + 1 }. ", "5b99521b2df43d801b250c70d380d1be": "f(x)=p", "5b9953642e4fc31ace1d8d263ea79555": "c(i)", "5b99b4661e6eb5924b90337b06513011": "v_{k+1}\\in V", "5b99c43a34fb8df21249bb62c4963335": "\\operatorname{codim}(W) = \\dim(V) - \\dim(W).", "5b9a77af89d04a685b4f649da485aed3": "2^3", "5b9afc2ba2ceab6edc6f62eb1e46b1a9": "\\begin{cases}\nf: \\mathbf{S}^2\\to \\mathbf{R}\\\\\n x \\longmapsto \\langle Y_x, x\\rangle.\n\\end{cases}", "5b9b5183c3a7cbc2b023bc0e6c3ae453": "f(x) =\n\\begin{cases}\n 1, & \\text{if }x\\text{ is rational} \\\\\n 0, & \\text{if }x\\text{ is irrational}\n\\end{cases}", "5b9b5a5c2238ea117d25d93f52155004": "P(M, t) = P(K, t) + P(M(d_n), t) - P(C(d_n), t)", "5b9b87aa1c78432abe0e2a431bcc7e6d": "b = 1/B", "5b9b8b79616831a89110d9ea4d81dd29": " x_i = \\frac{\\det(A_i)}{\\det(A)} \\qquad i = 1, \\ldots, n \\, ", "5b9ba618448aa033a723931ef2e4c07f": "d\\cdot / dt", "5b9bd1e6038f26a3e5084c83432f6f23": "\n \\frac{\\partial\\tilde{\\rho}}{\\partial t} +\n \\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\nabla \\cdot\\left[\\langle\\mathbf{v}\\rangle+\\tilde{\\mathbf{v}}\\right] +\n \\nabla\\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\cdot \\left[\\langle\\mathbf{v}\\rangle+\\tilde{\\mathbf{v}}\\right]= 0\n ", "5b9bea9b46f4b0a1493582f9ef9ed51e": "p_e = \\sqrt{\\frac{(x - x_n)^2 + (y - y_n)^2}{(x_I - x_n)^2 + (y_I - y_n)^2}}", "5b9c230fd4ec2a1d0c4cd16625eba22c": "K \\equiv \\bigoplus_{\\omega \\,\\in\\, \\Omega} A_\\omega", "5b9c3bba68f32ebf7a6facdf7d9fe1ea": "\nv^i {}_{;j}=\\nabla_j v^i=\\frac{\\partial v^i}{\\partial x^j}+\\Gamma^i{}_{jk}v^k\n", "5b9c8c2030643b37d28ebe7c7ddd8efd": "\\psi_{iI}(t)", "5b9d0564f7fc7c5b93014d5eee4b8184": "x =\\sqrt {ab}", "5b9d44fab3c299421e461adbb74f5fb7": " S_3 = U_3 ", "5b9d50bdaf18702cf4c69f73df34be64": "K = {1 \\over {\\text{radix}}^n}", "5b9d728fda928ca196d7ed12a660b2c4": "\\sigma_q", "5b9d82bebb6c2a467d5aaa3cea062981": "\\qquad (x,t)\\in\\mathbb{R}^3\\times[0,\\infty).", "5b9d9657309e655697912ef0327f95ee": "(1 - \\alpha) \\nu ", "5b9df183c68b6ea2024f1ad5c2e9312b": "\\scriptstyle{0.261}", "5b9e7fc13261366c85cede6951b30215": "\nu\\ =\\ -\\int\\limits_V \\rho \\frac{G}{r}\\, dx\\,dy\\,dz\n", "5b9ee6bde995565f7f43b2dd50741deb": "\\overline{B_1(p)} = \\{p\\}", "5b9ef8f54bfc39bffd265700eed0266e": "LC(X) = \\sum_{x \\epsilon X}p(x)L(x) = \\sum_{x \\epsilon X}p(x)(\\left\\lceil \\log_2 \\frac {1}{p(x)} \\right\\rceil + 1)", "5b9f43afe36a96d5de5c933206a9ce10": "\n\\begin{array}{ccc} \\pi\\varepsilon\\varrho\\iota\\varphi\\varepsilon\\varrho\\varepsilon\\iota\\tilde\\omega\\nu & \\varepsilon\\overset{\\text{'}}\\nu\\vartheta\\varepsilon\\iota\\tilde\\omega\\nu & \\overset{\\text{`}}\\varepsilon\\xi\\eta\\kappa\\omicron\\sigma\\tau\\tilde\\omega\\nu \\\\\n\\begin{array}{|l|} \\hline \\pi\\delta\\angle' \\\\ \\pi\\varepsilon \\\\ \\pi\\varepsilon\\angle' \\\\ \\hline \\pi\\stigma \\\\ \\pi\\stigma\\angle' \\\\ \\pi\\zeta \\\\ \\hline \\end{array} & \\begin{array}{|r|r|r|} \\hline \\pi & \\mu\\alpha & \\gamma \\\\ \\pi\\alpha & \\delta & \\iota\\varepsilon \\\\ \\pi\\alpha & \\kappa\\zeta & \\kappa\\beta \\\\ \\hline \\pi\\alpha & \\nu & \\kappa\\delta \\\\ \\pi\\beta & \\iota\\gamma & \\iota\\vartheta \\\\ \\pi\\beta & \\lambda\\stigma & \\vartheta \\\\ \\hline \\end{array} & \\begin{array}{|r|r|r|r|} \\hline \\circ & \\circ & \\mu\\stigma & \\kappa\\varepsilon \\\\ \\circ & \\circ & \\mu\\stigma & \\iota\\delta \\\\ \\circ & \\circ & \\mu\\stigma & \\gamma \\\\ \\hline \\circ & \\circ & \\mu\\varepsilon & \\nu\\beta \\\\ \\circ & \\circ & \\mu\\varepsilon & \\mu \\\\ \\circ & \\circ & \\mu\\varepsilon & \\kappa\\vartheta \\\\ \\hline \\end{array}\n\\end{array}\n", "5b9f46c78ecedd32b967d48c18c6bf0c": "\\Sigma=\\{\\alpha\\}", "5b9f6362bbbc492b3c090964da859b87": "\\{S_0,S_1,\\dots,S_r\\} \\subseteq \\mathcal P(E)", "5b9f79c35b6627ed938739e966c02e4d": "a x^3 + b x^2 + c x + d = 0", "5b9f8ab198358e629905901c301f2bb2": "\\scriptstyle{1/2\\sqrt{2}}", "5b9fbfa27c1b5c2eaa6393ffcbda877c": "\\Delta S = n C_v \\ln \\frac{T}{T_0}", "5b9fe9d587ea6ae1ca0681dc4d056b2c": "\\frac{d^{2}L}{dz^{2}}+k_{z}^{2}L=0 \\ \\ \\ \\ \\ \\ (14)", "5ba090fac8acaeb5f52dedd728f7652a": " V(\\mathbf{R}_1, \\mathbf{R}_2, \\ldots, \\mathbf{R}_N)=V(\\mathbf{R}'_1, \\mathbf{R}'_2, \\ldots, \\mathbf{R}'_N)", "5ba0bcedc26507684511dfcfc13c9c90": "\\cap \\overline{\\{0,1\\}}", "5ba11fa2c696796399e35ad1c38f9ca7": "\\pi _T", "5ba12b0da7f2604fd365d59c5049fbd6": "X_{0}", "5ba18ee35fc6a1dacf66e6ae2daaff01": "(\\mathbb{F}_p) \\cong \\mathbb{Z}_2 \\oplus \\mathbb{Z}_{2^{k-1}n}", "5ba20bdfae16da976a9d25370aa9fc7c": "p=P(a + bx)\\,", "5ba2c0c79551a8bd56b4432746b5cf34": "2^{n}\\cdot\\delta", "5ba2fbf1d94ab7912162727d9754de07": "\\exp(z) = w \\,", "5ba38d212fdc4d318ed5f22177defc65": "d = 2r \\quad \\Rightarrow \\quad r = \\frac{d}{2}.", "5ba432c40ce7251226b8cabf93fec03c": " * \\ ", "5ba468cc982256888a2ec3759e6d1b6b": "a,b,c,d\\in \\mathcal {G}(p,q)", "5ba49516112485ad28369f1e84b7b5b8": "\\hat{b}=\\left(\\sum_{i=1}^{n}(x_{i}-\\bar{x})(y_{i}-\\bar{y})\\right)/\\left(\\sum_{i=1}^{n}(x_{i}-\\bar{x})^2\\right), ", "5ba4f9568f094ab1c942afa167553bb7": "\\mathbb R^n_{+} = \\{(x_1,\\ldots,x_n) \\in \\mathbb R^n : x_n \\ge 0\\}.", "5ba5871b26295dba5155aef0c7e3d692": "0 = \\frac {\\mathrm{d}}{\\mathrm{d}t} f(p,q) = \\{f,H\\}+ \\frac{\\partial f}{\\partial t} ", "5ba5a35bed3213e63339a85a0a0a50b6": "A^{\\deg(f)}", "5ba5c7159dec49b2bba767aeaa5521fc": "\\phi(\\omega)=2\\pi\\tau(\\omega)c/\\lambda(\\omega)", "5ba60d95a102a1ece059d35d6dcd3b7e": "J_i=-\\sum_{j=0}^{np-1}\\sum_{k=1}^n\\beta_k f_i^{(j)}(\\alpha_k)", "5ba6152a39480cc4669db2d2016f2e25": "=\\sum_{\\boldsymbol{R_n}} b^* ( \\boldsymbol{R_n})\\ \\int d^3 r \\ \\varphi^* (\\boldsymbol{r-R_n})H(\\boldsymbol{r}) \\psi (\\boldsymbol{r}) \\ ", "5ba6f2aac993f638a16604ce0d06c04d": "E=-\\frac{k^{2}m}{2L^{2}}", "5ba7b91fcf3be5b284d34d9a97bda2de": "^{\\;}q^{i}(\\xi,0) = \\xi^{i}", "5ba7c6d454fd2b459541d4db2aab3eb6": "t_S", "5ba96044e35edbefc6638c82fa4e37ef": "e_1(X_1) = X_1.\\,", "5ba981cd4609681610cae99de21426f9": "T^* = J_1 T' J_2^{-1}", "5ba9d16419154b1bdbeca39d99e8e809": "\\frac{d}{dx}f(x)", "5ba9e731e950cffc871e092d840ecac8": "\nh_t(x,y) = \\sum_{i=0}^{\\infty} \\exp(-\\lambda_i t) \\phi_i(x) \\phi_i(y)\n", "5baa039c91f99793477b7eb31b69297e": "\\hat{x}\\in X", "5baa22621c7ec682e63e72e6c03bf4db": "\\pi=s_0, s_1, \\ldots", "5baa546ca4a449272d879211de2073f5": "P_1 = \\frac{l}{t}.", "5baa6ff4dd2a31f910509a869fcc85fc": "\\bold {n}_2", "5baa7ad92093f00c9c7228ae48603485": "\\Gamma \\cong {\\Bbb Z}", "5baaf2dd59fb69d457e6b8607d729ea9": "\\scriptstyle 0\\le h\\le r ", "5bab3f63f7c414a9ee180cfd04e7206f": " \\mathbf{w}_n(x_n) ", "5bab5339f340f06ed59816917169af8a": "\\frac{x_i}{B}", "5bab55658e9dd16ce8d4696aa17fd3ab": "n_2 > n_1", "5babc6dd3ecf9fc3dfff6950891a83d3": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 &\\qquad \\text{ ( by Lyapunov function } V_x, \\text{ subsystem stabilized by } u_x(\\textbf{x}) \\text{ )}\\\\\n\\dot{z}_1 = f_1( \\mathbf{x}, z_1 ) + g_1( \\mathbf{x}, z_1 ) z_2\\\\\n\\dot{z}_2 = f_2( \\mathbf{x}, z_1, z_2 ) + g_2( \\mathbf{x}, z_1, z_2 ) z_3\\\\\n\\vdots\\\\\n\\dot{z}_i = f_i( \\mathbf{x}, z_1, z_2, \\ldots, z_i ) + g_i( \\mathbf{x}, z_1, z_2, \\ldots, z_i ) z_{i+1}\\\\\n\\vdots\\\\\n\\dot{z}_{k-2} = f_{k-2}( \\mathbf{x}, z_1, z_2, \\ldots z_{k-2} ) + g_{k-2}( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-2} ) z_{k-1}\\\\\n\\dot{z}_{k-1} = f_{k-1}( \\mathbf{x}, z_1, z_2, \\ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-2}, z_{k-1} ) z_k\\\\\n\\dot{z}_k = f_k( \\mathbf{x}, z_1, z_2, \\ldots z_{k-1}, z_k ) + g_k( \\mathbf{x}, z_1, z_2, \\ldots, z_{k-1}, z_k ) u\n\\end{cases}", "5babd0a4878938fc38acebcf9b4d9122": "e=\\frac{1}{H}", "5bac2bd55ffa90b044ad60f59ad55ac4": "ad-bc=1", "5bac98310b1e8d128d7636ee1d4e6fe2": "\\overline {\\Delta q} = 0.0075 \\,", "5bac9d9f9b774fa481931f0caaa4350e": "\n0 = \\delta s = \\delta \\int ds = \\delta \\int \\sqrt{g_{\\mu\\nu} \\frac{dx^{\\mu}}{d\\tau} \\frac{dx^{\\nu}}{d\\tau} } d\\tau = \\delta \\int \\sqrt{2T} d\\tau\n", "5badc5ffc84584e0afc1412baed35581": "{13 \\choose 1}{48 \\choose 1} = 624", "5badcab3ca3fd4f65707dfbc67414e8e": " {\\theta}_{\\mathrm{lower bound}} = 0 ", "5badee929f99c84d2e970498061aa0a7": "r_{f_{i}f_{j}}", "5bae77ea933a218108424cbf69e1b830": " \\sum_{m=1}^N \\left|\\sum_{n=1}^N c_{mn}\\lambda_n\\right|^2=\\|(T_f-T_z)p\\|^2 =\\|T_fp\\|^2\\le \\|p\\|^2= \\sum_{n=1}^N {1\\over n} |\\lambda_n|^2.", "5bae797112d91658e5cdc08b8682e7a0": "\\varphi_i(A,B)=AP+BQ.", "5bae8e275111820541a563ee7e5883c5": " A_{g}\\,\\,\\, + \\,\\,\\, S \\,\\,\\, \\rightleftharpoons \\,\\,\\, A_{ad}", "5bae9e54a5b9d5049ea6aa0bec5e8bc9": "C_{M}=C (1+A_v)\\,", "5baed8deef73d9ec66d6803bbcce52b3": "X(\\mathbf{a})", "5baf4613a830e96566079792978d940c": "n_d) = 1 ", "5baf4f93b86716a386f7116ca52fb12b": "|\\uparrow z \\rangle", "5baf542c2b1ac64637fa1d2f4179daf8": " \\Delta^0_{k+1} ", "5baf9f17a3acbe2779af293ad0d23b2f": "GL(m, \\mathbb R) ", "5bafbf5ebfceeacaa752edbeef8200c9": "x^3 + 7x^2 + 8x + 2", "5baffdd902a566cdd60c01b39ecd5946": "\\color{Blue}\\text{Blue}", "5bb030fe5fff74516223d66ed1d2cc6c": "\\Delta E/V_\\text{acc}", "5bb0335c0bba99c89063ce18b99e0db6": "|n\\rang", "5bb07d52338ef37c435f7315aabbd631": "\\ \\frac{a_2}{a_1} = \\sqrt{1 + \\frac{2(\\gamma - 1)}{(\\gamma + 1)^2}\\left[\\gamma M_x^2 - \\frac{1}{M_x^2} - (\\gamma - 1)\\right]},", "5bb11b6d9a174b7249c73ae20164d431": "E \\| x - \\hat{x} \\|^2", "5bb138415d95133fcfd7b270da29c895": "n - p", "5bb1a4da77675cddbb7121adb4ca8e42": "\\tau(u,v,w)", "5bb1bf6b598f82be008369eac43d8d76": "\n\\begin{align}\n\\dot{x} =&\\ f(x, u) &\\mbox{(fast subsystem)} \\\\\n\\dot{u} =&\\ \\mu g(x, u) &\\mbox{(slow subsystem)}\n\\end{align}\n", "5bb1dc4d585e02351f7a0215fc98d3fc": " t_H \n= \\begin{matrix}\\frac12\\end{matrix} \\sqrt{\\frac{4\\pi^2 a^3_H}{\\mu}}\n= \\pi \\sqrt{\\frac {(r_1 + r_2)^3}{8\\mu}} ", "5bb1de19a063bd00e2bfe32fd24de214": "\\gcd(a,b)=\\gcd(a_1,b_1)=\\cdots=\\gcd(a_N, 0)=a_N \\,.", "5bb1dfe9be26736c1a726d7de8e974e9": "\\{ w', w \\} \\in E \\setminus M", "5bb20c77bdbf2772a865a4c0d54e5abb": "\\scriptstyle \\int_{-1}^1 ", "5bb240842a77561e39c4746253f15d51": " 2X + Y \\rightarrow 3X", "5bb25deeb6bc358c6afb1f7ecd6d5803": "K\\otimes_{\\mathbb Q}L", "5bb26199733a4b04e0aad905aab54cef": " x\\cdot a^{x-1}\\cdot 0 + a^x\\cdot (1\\cdot \\log a + x\\cdot \\frac{0}{a}).", "5bb279708bd1282e72619da588e973a1": "W=", "5bb28b65ba03ff0dab5accec0d600e9d": "\n\\left\\langle e^i, e_j \\right\\rangle = \\delta^i_j.\n", "5bb2ca93787dd90858bab9e43ce28aef": "~\\gamma=\\arcsin D", "5bb3344de74837021f3091485ebac58d": "\\sum_{i=1}^k \\left(\\frac{X_i}{\\sigma_i}\\right)^2", "5bb33fd2b6885cf21c87effa8159dc71": " \\bar{R} \\approx \\bar{\\lambda}/25 ", "5bb346dec985ddf348d51d2e2c354be2": "(R)f_! \\leftrightarrows (R)f^!", "5bb389c1ae1a84356c196e8076c7dbf4": "H_1(\\pi_1(Ff), \\mathbb{Z}) = H_2(\\pi_1(Ff), \\mathbb{Z}) = 0.", "5bb38d75e4142b2804ec876db0a6f2da": " \\theta \\in \\Bbb{R}", "5bb3aa4e6d68c72a1688831abdafbd95": "\\,\\mathbf{_2F_1}(a,b;c;z)", "5bb3cc06b14e8b13fbaf2d07ae938185": "E(e^{it\\cdot X})=\\int e^{it\\cdot x}d\\mu_X(x)", "5bb414364f9e30dc43b8b644e067ebb2": "d_A Q_B = d_A d_B G = d_B d_A G = d_B Q_A", "5bb430b8af271a6c4a2322c174473ac8": "P = {{3600 \\cdot Kh } \\over t}", "5bb43946ad26d10c07e43b76ccf68c1c": "K_a = \\frac{[H^+] [A^-]}{[HA]}", "5bb446d044f1f4cb64174d6761226888": "s_A=9.91", "5bb487104b0d5e1977e69b524d9aa1ef": "\\frac{p}{q} = \\frac{a}{\\frac{ai+b}{p}\\cdot\\frac{q}{i}} + \\frac{b}{\\frac{ai+b}{p} \\cdot \\frac{q}{i} \\cdot{i}}", "5bb54fa89eaf9546c9d3dccc8e237a1b": "v_z\\left (r \\right) = \\frac{\\epsilon \\phi_0}{4 \\pi \\eta} E_z \\left [ 1 - \\frac {I_0 \\left ( \\kappa r \\right )} {I_0 \\left ( \\kappa a \\right )} \\right ]", "5bb5bdd744ca760a52421bffd5d5018f": "R_H=R_S \\cos(\\alpha)", "5bb6217a1fdda557fd50fb9f410456f1": "+ \\tau'_\\perp (t)S(t)B(t)R^{-1}(t)B'(t)S(t) \\tau_\\perp (t),", "5bb6750b2597958a8027b889589dfb0a": "c = |S^{*}|", "5bb6e913308f1c96c3fc59c616c9f425": "(x,g) \\mapsto (x,x\\cdot g)", "5bb735b38e7d17091ebda61e136be7f4": "\nf(\\mathbf{x}\\,;\\,M,Z)\\; dS^{n-1} \\;=\\; {}_{1}F_{1}({\\textstyle\\frac{1}{2}};{\\textstyle\\frac{n}{2}};Z)^{-1}\\;\\cdot\\; \\exp\\left({\\textrm{tr}\\; Z M^{T}\\mathbf{x} \\mathbf{x}^{T}M}\\right)\\; dS^{n-1}\n", "5bb77056d3860948824b4e55cfa662c7": "O(\\sum_{i=1}^{j} \\log 2^{2^i}) = O(2^ j)", "5bb776373f2fd43831520a5b0798dd5d": "\\sum_{n=0}^\\infty H_n(x) \\frac {t^n}{n!}\\,\\!", "5bb7a0ecfc1a85387953fe338bbb4f95": "0 \\leq\nt \\leq T", "5bb7fae1967b949648d2ee9ac715dc18": "k \\geq 7", "5bb842b33667522975c9e399ac1b09d4": " u= \\sum_{k=-\\infty}^{\\infty} \\hat u_n (t) e^{ik_n x} ", "5bb85c24493ecb2ca43183b33d47adf5": "-\\nabla p + \\mathbf{f} = -\\nabla p + \\nabla F = -\\nabla \\left( p - F \\right) = -\\nabla p_m.", "5bb959cbcb33f8d6fd2bd6b2a9b3e656": "a=(x'_1,\\ldots,x'_m), b=(x_1,\\ldots,x_m)", "5bb95e28f3dd5b77da381d511332ab10": " z_i := \\left\\{\\begin{matrix} x_k & \\mbox{ if } i=2k \\mbox{ is even,}\\\\ y_k & \\mbox{ if } i=2k+1 \\mbox{ is odd.} \\end{matrix}\\right.", "5bb970f58085e0f01cb50744e4257bd2": "r=1,2,\\ldots,m", "5bb9a3d34a3df5207757040cd65b3f4a": "\\tilde{\\mathbf n} = - \\hat{\\mathbf n}", "5bb9b9f050b8730e6190d05a0c7a21e5": "\n\\sum_{n=1}^\\infty \\frac1n\n", "5bba11f76d9062ae0e858c1c9d5af4fd": "\nh_\\tau = a\\sqrt{\\frac{\\sigma^2 - \\tau^2}{1 - \\tau^{2}}}\n", "5bba7c8187b627624a2610cdc29eb7e4": "x\\, =\\cos(s);\\ y\\, = \\sin(s)\\ ", "5bba854e007b2b8903c5d6ee45994a2a": "\n\\alpha_V = \\frac{1}{V}\\,\\frac{dV}{dT}\n", "5bbaa4f49da1b69dab702384b4338d84": "a_{2} = \\frac{a_{1} + \\left(\\sqrt{k}(l+\\frac{3}{2})-E_{l}\\right)a_{0}}{2(2l+3)} = \\frac{a_{0}}{2(2l+3)}\\left(\\frac{1}{2(l+1)}+\\sqrt{k}\\left(l+\\frac{3}{2}\\right)-E_{l}\\right),", "5bbafdf7ba96a0db4827a712aa77def4": " AgCl(s) \\leftrightharpoons Ag^+ + Cl^-", "5bbaffbb775c5e7112d7a71ff94ff2e8": "\\sum_{i=1}^k \\mathrm{n_i}^\\alpha\\,\\mathrm{d}\\mu_i\\,+\\sum_{i=1}^k \\mathrm{n_i}^\\beta\\,\\mathrm{d}\\mu_i\\,+\\sum_{i=1}^k \\mathrm{n_i}^\\mathrm{S}\\,\\mathrm{d}\\mu_i\\, +A\\mathrm{d}\\gamma\\, = 0\\,.", "5bbb833240d22df4e1fdb1caad5760d6": "\\psi=Ay", "5bbbb70d1be3d26f670246101a77de2b": "R_{\\text{C}}/R_{\\text{E}}", "5bbbbbd9e617f9a8aad47636e3a9a8d2": "\nQ\\equiv \\frac{V_n}{\\langle n \\rangle}-1.\n", "5bbbe7153d272975131c2068307bc1a8": "I = {q\\over t} = q{v\\over l}", "5bbc1507633eadee2ce0586ea9fcd727": "G(X,\\Phi)", "5bbc36ce9f1707226af510c5c6dc835d": "v_0 [S^1]_0 /[S^0]_{0^{ }}", "5bbcc12a7ba598f882af7fcbe77938ec": "^i", "5bbcf16ba0361f1a8943d7005a895d34": "\n \\sigma_{11} = \\frac{\\partial^2 \\varphi}{\\partial x_2^2} ~,~~\n \\sigma_{22} = \\frac{\\partial^2 \\varphi}{\\partial x_1^2} ~,~~\n \\sigma_{12} = - \\frac{\\partial^2 \\varphi}{\\partial x_1 \\partial x_2} \\,.\n ", "5bbcfa1be949021055e9f4c9e69a74ad": "\\mathfrak I_{\\Phi} := (\\mathfrak T_{\\Phi},\\beta_{\\Phi})", "5bbd9ed9177fe3e85cbd04fe48c86c84": "r^N D = \\frac{W}{2}.", "5bbdb3bfcaf28b22d221c8bbdabd9af4": " Var(X) = np(1-p) ", "5bbdc1a0297fe94f13ba5da87eb35240": " \\operatorname{sink}[(\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ (\\lambda f.\\lambda x.f\\ (x\\ x))] ", "5bbdc3dbb6ef5128b4de7bb625bc1985": "{\\rm Pr}_r(A(x,r) = \\mbox{right answer}) \\ge 1 - \\frac{1}{3m},", "5bbdd5d676ffcc8fe8c55723ba568421": " Availability = \\frac{Available \\ Time}{Total \\ Time}", "5bbdfd1f9db6244e79ac6448ae69f66c": " y_b=a ", "5bbe5637784bfd1e640b2896e06d07ac": "\\mathsf{P^{\\sharp P}} \\supseteq \\mathsf{PH} \\supseteq \\Sigma_2 \\supseteq \\mathsf{MA}", "5bbe9ba979b51fc64e4a4bf3f56dfb03": "F_{}^{gi} = mg - ma_G", "5bbea2a7950991142896e924ae253192": "f(x)=\\lim_{k\\to\\infty}\\left(\\lim_{j\\to\\infty}\\left(\\cos(k!\\pi x)^{2j}\\right)\\right)", "5bbed676e303a1fdbe337f4ca4ecaca2": "\\widehat{\\theta}(x) = E[\\theta |x]=\\int \\theta \\pi(\\theta |x)\\,d\\theta.", "5bbf6173809529dbd40570e289f2a7ad": "\\textstyle\\frac{L}{A}=\\tfrac{1}{2}v^2\\rho C_L", "5bbfe2582a7e615d58a64bafcc7f1498": "n_{2D}(T_c) = 3.03\\times 10^{14}\\;\\mathrm{m^{-2}} \\ ", "5bbff1ab41545932f81b1c40d975f853": " \\ \\frac{\\mathrm{d}}{\\mathrm{d}s}\\mathbf{u}_\\mathrm{n}(s) = \\frac{1}{\\alpha} \\left[-\\sin\\frac{s}{\\alpha} \\ , \\ \\cos\\frac{s}{\\alpha} \\right] = \\frac{1}{\\alpha}\\mathbf{u}_\\mathrm{t}(s) \\ . ", "5bc049f532dae611375fbe5d1ead82ba": "\\eta_{\\mathrm{photon}}(T) = \\frac{\\int_{\\lambda_1}^{\\lambda_2} B(\\lambda, T)\\,\\frac{\\lambda}{hcN_A} \\,d\\lambda}{\\int_{\\lambda_1}^{\\lambda_2} B(\\lambda, T)\\,d\\lambda},", "5bc15a85936db52c4574dde2ccfd43e5": "(139 / 138 )^8 \\approx 99.9995 \\text{ cents,} ", "5bc16909823b60c819fb735cd06c028a": "W=n^{-1/2}\\sum X_i", "5bc19cda5a20540a4415bd1fec72c6e6": "\\mu_4^{'}= 8\\sigma^4+8\\sigma^2\\nu^2+\\nu^4\\,", "5bc1bcb97bd1bfb580562cea5f4bb1d2": "\n\\begin{align}\n\\oint_L f(z)\\,dz & = \\int_0^{2\\pi} {1\\over e^{it}} ie^{it}\\,dt = i\\int_0^{2\\pi} e^{-it}e^{it}\\,dt \\\\\n& =i\\int_0^{2\\pi}\\,dt = i(2\\pi-0)=2\\pi i.\n\\end{align}\n", "5bc1bdb3b7d6aab4b6a1c94bf408e4d3": "f(x_i,y_j)\\,", "5bc1e2864a5d695e06c3650cbc04b9a0": "\\hat{x}=g(y)", "5bc1e7b52a8ab7e0fc9255711f9eb656": "\nI_{xx} \\alpha_{x} - \\left( I_{yy} - I_{zz} \\right)\\omega_{y} \\omega_{z} = N_{x}\n", "5bc201db2964dd2651ec058ec5d749b7": "x^6 + y^6 = x^2.\\,\\!", "5bc205ff76ef8b2981f547c85eeec261": "1-\\varepsilon_1", "5bc2bd1a3154180e30faf7dbaa318708": "~z=t/t_0~", "5bc2c2923029332c4ef7b140da8b9653": "\n\\begin{cases}\np(y|x) = \\frac{a(y)p(x|y)}{a(y)p(x|y) +\na(\\overline{y})p(x|\\overline{y})}\\\\\np(y|\\overline{x}) = \\frac{a(y)p(\\overline{x}|y)}{a(y)p(\\overline{x}|y) +\na(\\overline{y})p(\\overline{x}|\\overline{y})}\n\\end{cases}\n", "5bc2cdfa15abe31ccf78eaadf436e4ab": "V=\\frac{a^3(15+7\\sqrt5)}{20}\\,", "5bc319bcaf3a3ec9e5736ce0b42ff377": "Y dx\\leq\\delta E", "5bc346fcfa9ede50c592a82d63bad338": "\\mathbb F=", "5bc382631cca4a6374b7a90e5f7c49f3": "O(n\\tfrac{\\log n}{\\log\\log n})", "5bc3934fa88fbfa1c74bb1c60e0ce7ae": "\\scriptstyle \\sum_{i\\in I}\\alpha_i f_i \\;=\\; 0", "5bc3e5cc7d794a12a772bef5d4bb8ff8": "(k,q)", "5bc419b9fbaeab7011d0a96c110fa4da": "\\mu = B / H", "5bc4b4e57839ca34490f02ab67d92fed": "m_{n-k+2} \\cdots m_n < m_1 \\cdots m_k", "5bc4f654c778f4e89227cf5aed6e9edf": " W(b,p) = \\mathbb E[\\log_2 S(X)] = \\sum_{i=1}^m p_i \\log_2 b_i o_i ", "5bc50409ddec8cf59046638b103c8469": "W = 3n_{\\rm e}k_{\\rm B}T", "5bc574a47246f122016869b32a6aa6f0": "CL", "5bc5d7414340e5e20be2c359a2577389": "-\\frac{\\partial U}{\\partial \\mathbf{r}} = \\mathbf{F}", "5bc5dcdb9617645dab762900d717b0bd": " \\begin{align}\n\\mathbf{a} \\wedge \\mathbf{b} = (a_1b_2 - a_2b_1)\\mathbf{e}_{12} + (a_1b_3 - a_3b_1)\\mathbf{e}_{13} + (a_1b_4 - a_4b_1)\\mathbf{e}_{14} + (a_2b_3 - a_3b_2)\\mathbf{e}_{23} \\\\\n+ (a_2b_4 - a_4b_2)\\mathbf{e}_{24} + (a_3b_4 - a_4b_3)\\mathbf{e}_{34}. \\end{align}", "5bc5e5687910b1dd21867cd4894c3bbe": " \\varphi = u + \\varepsilon v ", "5bc68ea514d91c4a4592a7592b518c68": " u(x,y) = \\int\\limits_{-\\infty}^{\\infty} f( x + t(y-x) ) dt. ", "5bc6e8db74bdfc2090b93cc82b4e382a": "\\vec{\\xi}_2, \\, \\vec{\\xi}_3, \\, \\vec{\\xi}_4", "5bc6f410e4f831e3cb904ad815493d57": "Bi = M \\wr \\mathbb{Z}_2. \\, ", "5bc6f454dca7e68b818731835af54c75": "c_f", "5bc7b087e84568e6fb70c8216c6575df": "X,Y \\in \\mathcal{M}(S)", "5bc7fa4a8d2da63c42c65a169ccb24b9": " \\scriptstyle \\{U_i\\}_{i\\in I} ", "5bc8024f7af43f745e339e5b77aed171": "N_2(t) = N_2(0) \\exp{\\frac{-t}{\\tau_{21}}},", "5bc8168a90d56250cb93e05708c0f303": "b = f^{1}(1) - m = x_{1}^{1} - 7 = 2 - 7 = -5 = 6", "5bc8381a87ca75a9a34656403d33d1bf": "\n\\tan\\frac{E_{12}}2 =\n\\frac{\\sin\\tfrac12 (\\beta_2 + \\beta_1)}\n{\\cos\\tfrac12 (\\beta_2 - \\beta_1)} \\tan\\frac{\\omega_{12}}2.\n", "5bc898f3390cea72c23ff1f900b7bc8f": "\\sum_{j \\in S}\\pi_j = 1.", "5bc8a334ab79657a09e00949a52e2707": "\\mathbb C ", "5bc8c302dd05bea408eb456f44a84121": "\\operatorname{rank}\\left(\\tilde{\\mathbf{M}}\\right) = r", "5bc8cde08535355a2427107589145cbb": "x, x', u, u'", "5bc8d2aae42eb03a130bdd82a4d4293f": "\\text{TREND} = \\frac{\\sum_{i=1}^\\tilde{N} (i-\\tilde{N}/2)(RR_i - \\langle RR_i \\rangle)}{\\sum_{i=1}^\\tilde{N} (i-\\tilde{N}/2)^2},", "5bc9132acd9a5c9d30bc3f29af19940f": "I\\left(\\frac{c+d}{2},\\frac{f+g}{2}\\right)", "5bc9a688b73ef97b14ac334181ed9418": "\\scriptstyle \\frac{1}{11}", "5bc9c2f6af0260a23ddc31187684ee70": " x=\\frac{X}{Z}", "5bca3b831657500ef73e0ab3d3c65f00": "{1\\over-I(k+r)}\\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \\over (k-j)!}A_j=A_k", "5bca6f173842309b24c21f17bfffcf09": "\\frac{35 \\sqrt{\\pi}}{64}", "5bca8cd7f026559e6ea4ad1de5cc1c21": "\n e_i \\equiv l_i/L_i - 1 = \\lambda_i - 1\n ", "5bcac7eb8e1a4204c38ed28982427aa8": "\\sqrt{2} + 1", "5bcad7eec09f719316d962fefef018a9": "QE_i(\\lambda) = \\frac{J_{SCi}(\\lambda)}{q \\phi_i(\\lambda)} \\Rightarrow J_{SCi} = \\int_{0}^{\\lambda2} q \\phi_i(\\lambda) QE_i(\\lambda) \\, d \\lambda", "5bcb63c4c9abb99e88abb5ed96d86f94": "\\text{ C}(c,\\lambda/\\mu)=\\frac{\\left( \\frac{(c\\rho)^c}{c!}\\right) \\left( \\frac{1}{1-\\rho} \\right)}{\\sum_{k=0}^{c-1} \\frac{(c\\rho)^k}{k!} + \\left( \\frac{(c\\rho)^c}{c!} \\right) \\left( \\frac{1}{1-\\rho} \\right)}", "5bcb72a415585efcf397cb38fb70754f": "\\ \\lambda", "5bcb89166bd778ee2d1d9726e59f1f49": "Q=\\frac{2}{3}[(n_\\mathrm{u}-n_\\mathrm{\\bar{u}})+(n_\\mathrm{c}-n_\\mathrm{\\bar{c}})+(n_\\mathrm{t}-n_\\mathrm{\\bar{t}})]-\\frac{1}{3}[(n_\\mathrm{d}-n_\\mathrm{\\bar{d}})+(n_\\mathrm{s}-n_\\mathrm{\\bar{s}})+(n_\\mathrm{b}-n_\\mathrm{\\bar{b}})].", "5bcbae5e645065b2e058a56af49eec0a": "T^{\\mu\\nu} = {c^4 \\over 8 \\pi G} \\left( R^{\\mu \\nu}-\\frac {1}{2} g^{\\mu \\nu} R \\right) \\,.", "5bcbb607eb6d758d72e4f77eaba968ff": "r\\in \\mathbb Z^{*}_{n^{s+1}} ", "5bcbdbdb398ac3a1ea8ec3da7f956bc9": "\\{\\xi\\in B\\}=\\{\\gamma \\in \\Gamma|\\xi(\\gamma)\\in B\\}", "5bcc038be26318cc69668823d95db7c5": "a_{\\pi(1)}\\ldots a_{\\pi(n)} = (-1)^{\\left|a_{\\pi}\\right|}a_{1}\\ldots a_{n}", "5bcc0dd81aaceb09252e791aeaa0260c": "H =\n\\left[\n\\begin{array}{cc|ccc}\n1&1&1&0&0 \\\\\n0&1&0&1&0 \\\\\n1&0&0&0&1 \\\\\n\\end{array}\n\\right].", "5bcc19a570810eeb2555b070907a4906": "V \\in V", "5bccc248565c15d621634d6275749d5c": "\\textstyle (C,\\; s)", "5bccce1701434ce7920dd2df0c987df3": " \\phi(\\vec{r}) = \\frac{1}{4\\pi}\\iiint_{\\vec{r}'} \\frac{\\vec{\\nabla}_{\\vec{r}'} \\bullet \\vec{E}(\\vec{r}')}{\\|\\vec{r}-\\vec{r}'\\|}d\\tau' ", "5bccd7d2d981fdbd61f2ce7bf7521881": "\\sigma (L) = \\mathbb{C} \\setminus \\rho (L).", "5bcd84b10ebbe6221a013bb3a9866b23": "h^{\\mu\\nu}h_{\\mu\\nu}", "5bcd94bf5487c310d35ca0d9ea6517a7": "\\left(\\mathrm{Re}<10^{-5}\\right)", "5bcd9dccea3d673a42869dfa52bdc35c": "kT = \\frac{GM\\bar m}{3R}", "5bcda296bfb3a717d5e9c22b6d1bc531": "\n\\nabla \\cdot \\vec \\psi = \\frac{1}{h_{1} h_{2} h_{3}} \\frac{\\partial}{\\partial x_3} \\left(\\psi h_1 h_2\\right) = 0\n", "5bcdd7b65be3653b06a0ff7dfe24fc8c": "\\, {e_-}^2 = -1 ", "5bce79084817b22767c374544249e098": "=\\frac{8}{13}", "5bce8fc91624c0e8f982f16333bb8796": "\n{d \\tau}^{2} = dt^2 - 2f\\, (t+z,\\,x,\\,y) (dt+dz)^2-dx^2-dy^2-dz^2\n", "5bce9eb3cc2049f804ca44edaf4ddf07": "C_{0} := C_{0} ([0, T]; \\mathbb{R}^{n}) := \\{ \\text{continuous paths starting at 0} \\}", "5bcea3dea3d41ce1a341333b5484e495": "|S^{(t)}| ~+~ \\sum_{w\\in R^{(t)}} \\frac{1}{d(w)+1}. ", "5bcf07a107c14478b2dcf60b0cae991a": "(f_U', U')", "5bcf1467479bfc0e11601e9522d416a9": "\\sum_{n=0}^\\infty (-1)^n\\,a_n", "5bcf231740fba1e4552f8450b3f661d5": "\\mathrm{NPV}(i) = \\int_{t=0}^{\\infty} (1+i)^{-t} \\cdot r(t) \\, dt", "5bcf3fbd535f268234008ce37d115cb7": "-\\mathrm{arctan}(\\mathrm{Im}[H(s)] / \\mathrm{Re}[H(s)])", "5bcf5fd27743b590319bbd79a9a6209a": "+S_y \\otimes S_y", "5bcf895b75ae1dbe4f878ec8f1d37852": "\\left(x\\cos \\theta\\ +\\ y\\sin \\theta,\\ -x\\sin \\theta\\ +\\ y\\cos \\theta\\right)", "5bcf8cadae0bd5dbeb97e035deb181d4": "A \\or (B \\and (C \\or D))", "5bcfacb9d5571437ea0b8e9007c7236e": "a_n = \\sum_{k=1}^n (-1)^{k-1} {n\\choose k}2^{k(n-k)} a_{n-k}.", "5bcfbc4233f403b0c32b02876cc86c03": " n! = \\int_0^\\infty x^n e^{-x} dx.", "5bcfd48a91d9c8bdbcec5ecb2ac94601": "0\\leq j \\leq J", "5bd009dacb11d586ae698a0b56421703": "\\mathfrak{B} = (B, \\pi)", "5bd0346a4adc01c71bb20092497a9580": " \\Delta_g ", "5bd067b30f20b7c52bdbb3bb051d012c": "0 \\rightarrow A \\overset{q}{\\longrightarrow} B \\overset{r}{\\longrightarrow} C \\rightarrow 0", "5bd0825b15108cd960569affb725147c": "\\sigma_x", "5bd0df79221450f4e5565ed6a0df1a91": "F(t)\\sim \\frac{C}{\\Gamma(\\rho+1)}t^\\rho, \\quad\\rm{as\\ }t\\to\\infty.", "5bd0fdeb0173bb0c393fe15e2cc42bd6": "V \\oplus W", "5bd254124735f610753526695b9fadf4": "\\mathbf{v}_{\\perp} = \\mathbf{v} - \\mathbf{v}_{\\parallel} = \\mathbf{v} - (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k} ", "5bd270c247fcb09378c24ae6161f5106": "X \\sim {\\rm Arcsine}(-1,1) \\ \\text{then } X^2 \\sim {\\rm Arcsine}(0,1) ", "5bd29dd9607f85d3c6f47269936ccbae": " 10 \\rightarrow 5_{-2} \\oplus \\bar{5}_2.", "5bd2f0edc4d013e6bfc0738ec40993c8": " \\frac\\pi2 (f(x+0)+f(x-0)) = \\int _0^\\infty \\int_{-\\infty}^\\infty \\cos \\omega (t-x) f(t) dt d\\omega, ", "5bd300b111e3e624a922a80a3276e69c": "X_1,\\ldots, X_n", "5bd346b8f917f5d3bc73ddbc9d68b9aa": "M_1^* + M_1 \\xrightarrow{k_{11}} M_1M_1^* \\,", "5bd3637698c546a70057de45c354335b": "\\frac{dx}{dt}(T)= \\frac{dy}{dt}(T) =0, ", "5bd398c85b694cf34689adcdd338e848": "\nn \\sqrt{\\frac{2n^2}{n-1} + \\frac14} + \\frac{n}2 - 2.\n", "5bd3aab0116959296f08cbe99ff77869": " \\frac{dv}{dr} = \\frac{1}{2 \\eta} r \\frac{\\Delta P}{\\Delta x} + A \\frac{1}{r} = 0 ", "5bd3af9fa808da098e9301429254ecd5": "F\\left(\\frac{a}{2},0\\right)", "5bd3cb14b6d0c4661d8e963bf1a50ac2": "(b-a) f_1\\,", "5bd426608b226c1ae58e7db648c37a7a": "\\boldsymbol{x}_i=\\boldsymbol{x}_0+\\boldsymbol{V}_i\\boldsymbol{y}_i", "5bd48c86456fab92ada567b6d38594c5": "T(p)", "5bd4abf63df29908407bbaad2bc96c5c": "\\mathbf{F}_{g} = m g.", "5bd4b9480065f1f894ed022a3794bc95": "np=\\omega(\\log n).\\,", "5bd534c284fa3e218a0c15721e42dddc": "\\mathbf{y}=\\mathbf{Wx}", "5bd57e24ed96fd4c93d313117d0165e9": "\\mathit({c}_0, \\mathit{c}_1, ...., \\mathit{c}_{N-k}, \\mathit{c}_{N-k+1}, ..., \\mathit{c}_{N-1})", "5bd57ee6449ac6826a1fd494b870cd35": "\n\\begin{array}{lll}\n& LAG_7=\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & . & . & . & . & . & . \\\\\n1 & . & . & . & . & . & . \\\\\n. & 4 & . & . & . & . & . \\\\\n. & . & 9 & . & . & . & . \\\\\n. & . & . & 16 & . & . & . \\\\\n. & . & . & . & 25 & . & . \\\\\n. & . & . & . & . & 36 & .\n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n 1 & . & . & . & . & . & . \\\\\n 1 & 1 & . & . & . & . & . \\\\\n 2 & 4 & 1 & . & . & . & . \\\\ \n 6 & 18 & 9 & 1 & . & . & . \\\\ \n 24 & 96 & 72 & 16 & 1 & . & . \\\\ \n 120 & 600 & 600 & 200 & 25 & 1 & . \\\\ \n 720 & 4320 & 5400 & 2400 & 450 & 36 & 1 \n\\end{smallmatrix}\n\\right ]\n;\\quad\n\\end{array}\n", "5bd592365ede9821d4cf75d02c0783eb": " \\mathrm{Gal}(\\bar{\\mathbf{Q}}/\\mathbf{Q}) ", "5bd5c0f9bb962564c62456cb4a2ae5f1": "\\,a_\\text{map}", "5bd5fe0f13466a3c9b859e3b58089644": "J_0(S)", "5bd6435556f56957b323c5044facfe12": "Q'(p;s) = \\frac{s}{p(1-p)}.", "5bd6b8d8c211a048516a2adede0dbf1e": "\\Delta N_{\\nu, \\lambda\\neq \\nu}", "5bd7218274472a387baead1d85c54d61": "\\Delta t_{1,2}", "5bd728fb0d530bcae28a9d5d0e8f1546": "g_2(x,y) = x^e - C_2", "5bd73bd3021d4e9c08d39fc1cabb7200": " \\textbf{normal}\\cdot (\\textbf{light position}) + \\text{plane}_D = \\langle a, b, c, d\\rangle \\cdot \\langle L_x, L_y, L_z, L_w \\rangle + \\text{plane}_D ", "5bd7a13f65e80beedf9b7f1bb86fd2a4": "\\frac{\\partial c}{\\partial t} = \\frac{\\partial}{\\partial x}\\overbrace{\\left[ D(c)\\frac{\\partial c}{\\partial x} \\right]}^\\text{Flux}", "5bd828e096efa7047e7c47bed627fa6c": "N_W", "5bd852e2208ef1b7d11d8aaa0429e838": "\\frac {1}{c(w)} =\\frac {1}{c_r} (1-\\frac {1}{\\pi Q_r} ln |\\frac{w}{w_r}|) \\quad (1.4)", "5bd860d64d0bec35f66767f94a94a78e": "i_1 < i_2 < \\cdots < i_n", "5bd88d89be54822553824501f56c2948": "1, 3, 5, 9, 17, 33, 65, \\ldots", "5bd8acb767f95e4a01535e77fab46051": "\\Lambda(x_i)", "5bd8c77d6302af1b5b8cb86e325b77fd": "\\{ p, p \\to q\\} \\vdash q", "5bd8eba712b0e412ec1cd7b018b9aed4": "f: Z \\times W \\mapsto \\mathbb{R}", "5bd91f6a3832b634d8ad8ffc368c3968": " n\\in\\mathbb{N}", "5bd964f64352c1a2671eae5742028fa8": "v_p(\\omega_1)\\leq v_p(\\omega_2)", "5bd98902d827be005677e8c5b2ab30a7": "GM = \\sqrt{2 \\cdot 8} = 4", "5bda001a36227e48710cc5160322708b": "\\frac{\\partial \\mathbf{A}\\mathbf{u}}{\\partial x} =", "5bda05c853c5b6516f41677c6708d7fe": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ -0.37\\end{smallmatrix}", "5bda4672e0012752809784e73d76cea6": "L_{\\alpha \\beta}", "5bda595b659f80acf8a50125ba890a59": "\\sigma _b \\to \\infty", "5bda5ceb30e0edcb001707954395827e": "\\mathcal{H} = T + V , \\quad T = \\frac{p^2}{2m} , \\quad V = V(q). ", "5bdabd0c9e4beb38ad24d0a82e154305": "\\sqrt{(x-c)^2+y^2} = a - {c \\over a}x", "5bdacc749a9938caafbb4028835f37fd": "t \\mapsto (x_1,x_2)", "5bdae561c44dbde4ae37735ce95375cd": "3 \\over 5", "5bdb1345299e3589b3a6eaed01266816": "E[F]", "5bdb1f7a200c73da27f9772d957d5b28": "\\gamma_t > 0", "5bdb52ab0e43b207b5372489e5f0e4fe": "f_{\\langle X | R\\cup \\{w\\} \\rangle}", "5bdb532cf8f0cbdeacde7caa41a7fe7c": "\\eta_{ab} \\, dx^a \\, dx^b ", "5bdb8c683a54a00844f63f6e21254d40": "\\{p_4, r_4\\}", "5bdb93aeb7f08163af739185ba16ffc1": "g_{00}=0", "5bdb9ee695c936283dd56e8c3ad78612": "Y = \\begin{cases} \n 0, & \\mbox{if }Y^*>0 \\\\\n 1, & \\mbox{if }Y^*<0.\n \\end{cases}\n", "5bdbc9ec39c200cf8107c4ecf9520a63": "f(x) = x", "5bdc2218550695b0d47cbe581ae39818": "R_\\mathrm{E} / y_\\mathrm{atm}", "5bdc4b82c73b871aa4655cefc16a0fce": "P\\left(S^{t}|O^{0}\\wedge\\cdots\\wedge O^{t}\\right)", "5bdc4d105945a6dbfdff41f1c291361f": "U=-\\mathbf{m}\\cdot\\mathbf{B} ", "5bdc624b89c003bb457ff3903139f4d5": "tr(X)=r", "5bdc80d51f0b8aa7ec80a543ec97943e": "L\n= \\lim_{n\\to\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right|\n= \\lim_{n\\to\\infty} \\left| \\frac{\\frac{e^{n+1}}{n+1}}{\\frac{e^n}{n}} \\right|\n= e > 1.", "5bdc964d6e2f1a5253897b510552d62e": "\\scriptstyle p \\,\\in\\, I \\,\\cap\\, J", "5bdca719638c7cabc77065e36f9194a5": "i=1,\\dots,N", "5bdd22ec047f54f708fedab6c311f819": " \\pm \\frac{1}{\\Delta f} ", "5bdd2676b0e864f8ed97a92fdfd357b8": "r_{IS}", "5bdd4eb4b680aa9c11d23e21a5ef6c6c": " \\nabla \\cdot \\vec v = 3y + 2yz ", "5bdd602d6c2990e1f9ac65e6c98533f9": "Y= \\int_{380}^{780} I(\\lambda)\\,\\overline{y}(\\lambda)\\,d\\lambda", "5bdd68da68f4dff8d1e1c1b40a085c0e": "\na_0 + a_0a_1 + a_0a_1a_2 + \\cdots + a_0a_1a_2\\cdots a_n =\n\\frac{a_0}{1-}\n\\frac{a_1}{1+a_1-}\n\\frac{a_2}{1+a_2-}\\cdots\n\\frac{a_{n}}{1+a_n}.\\,\n", "5bdd80626775fde3df85a0f711a1387e": "\n \\theta(x) = \\cfrac{\\mathrm{d}w}{\\mathrm{d}x}\n ", "5bdd81a80ed59452a268bda476122b1f": "d \\Omega", "5bddccd1207cc86053dc06ed56cb1024": "v\\mapsto\\infty", "5bddd7539dc4bb0c3f0afeb9c76e3511": "{}_{7.5\\,Es \\cdot \\frac{calories}{Es}}", "5bdde4ee84a5e0d0195f7af3acc24991": "x^*f : x^*\\mathcal{X} \\to x^*\\mathcal{Y}", "5bde11aca786cf377eaa17a68aa32367": "\\varepsilon = \\varepsilon_1 - \\varepsilon_0", "5bde13e844d194d37957c1323d4a8a53": "\\{ f : M \\to \\mathbb R \\}", "5bde25dde48a78cf6041ef34a1f28112": "\\ddot{(\\bullet)}", "5bde44902c9321399bd1473219272f1f": " \\mathcal{B}", "5bde534fd76cf4821c92eabb358a3ce7": "\\rho'(X)", "5bde56d175ce5e28ab3863e6477fbaff": "\\displaystyle P(t) = tr[\\rho (t)d]", "5bde6b1d94f38c031b2de90b93a70221": "f_{i,j} := n_1[+n_2[+n_3[+n_4[+n_5[+n_6]]]]]", "5bde8210f614ad7f6824a4a00c40da19": "p \\times l", "5bded2744bf2ad69cd979b5dbd68768d": "N=\\binom{c_k}k+\\cdots+\\binom{c_2}2+\\binom{c_1}1.", "5bdefb675f9f4f5a3f951b7f5234de53": " \\mathrm{BIC}= \\chi^2 + k \\cdot \\ln(n). \\, ", "5bdf520eac071682a355dc8b377d398b": " q=(s,t_s,t_e) \\in Q ", "5bdf7703df08ec82a668932b936e0dc6": "\\{\\sigma, \\tau\\}", "5bdf8d4ae3d40bdb58882039b2eef66d": "|DE|+|EF|+|FD|\\leq |GH|+|HI|+|IG| ", "5bdf91bd920d5db99627a1b9d1241369": "D\\geq D_0", "5bdfacfbb5bb03c6682707c73653fff1": "\\{[x]_R \\mid x \\in A\\}", "5be06bf848f6331a812b49f2dce3dbe3": "e_2 > e_1", "5be0dd88e2f8d8ef16aae61962666bed": " y_{n+1} = y_n + \\tfrac12 h \\Big( f(t_n,y_n) + f(t_{n+1},y_{n+1}) \\Big), ", "5be0e909f7079d8158f2b7e1ae846144": "\\Gamma(1 + i) = i\\Gamma(i) \\approx 0.498 - 0.155i", "5be18ba4770851a6484984e099c82f46": " \\frac {v_L} {i_S} = A_{FB} (R_{C2}//R_L ) \\ . ", "5be200fc997f1181809643d4114994eb": "2 G c^2", "5be20e6744cc7d09e4aaa4e2a452dd49": "\n\\overline{\\bigcap^{\\{q\\}}X_i} = \\overset{\\{q\\}}{\\bigcup}\\overline{X_i}\n", "5be2134adc4a1c35591bdbc1c0cc990c": "R = R_0 \\, \\sec(\\Phi) ", "5be234cd4297a3bd4b8bf5d7ff58a1aa": "\\left|s,m_s\\right\\rangle", "5be2ed517c2cf97146ef193f73b8b6ba": "H \\simeq G", "5be3259548c21777305006118d4d4abb": "\\widehat{u}_j= (Y-Xb)_j", "5be33791551275c23ce5145fa834eb81": "\\operatorname{El}(u_n) \\equiv \\mathcal{U}_n", "5be3b731dbd91c5219cacbee6628e962": "{\\color{Blue}~2.24}", "5be4c5f4cd8a43bc3b1ab739b9415f42": "\\cos(kt)e_i(t)", "5be521af2b397fc14eb50a22a2f49736": "q\\neq 0", "5be547c5b1a30a33e271f0deb8e9e907": "Z_T\\,", "5be56cc909fba6a1be05cd3174f85a8c": "D = \\frac{\\rho_s-\\rho}{\\rho}\\frac{t}{l_c}", "5be586170791cfb4bcdef38c1e771e8f": "\\frac{dy}{dt} + f(t) y = -f(t) \\cdot A e^{-\\int f(t)\\,dt} + f(t) \\cdot A e^{-\\int f(t)\\,dt} = 0", "5be5fb9048b5ca04f3b69f6be3b53e83": "\\scriptstyle 0\\in Z^d ", "5be65069862b619b21d0baacebf41980": "Q=\\pm 1", "5be65168308c0d067939f132a56e9b33": "T(x):=\\mathbb{E}[\\tau_{x}] ", "5be67ce82240b7b137a5aa00f2039f9d": "U_{thermal} = C(T) \\cdot T.", "5be696e646cf98f7d57e7967c870bcf0": "\\begin{align}\nf_1 (T) f_2(T) &= \\left (\\frac{1}{2\\pi i}\\int_{\\Gamma_1}\\frac{f_1(\\zeta)}{\\zeta-T} d \\zeta \\right ) \\left (\\frac{1}{2 \\pi i} \\int_{\\Gamma_2}\\frac{f_2(\\omega)}{\\omega-T}\\, d \\omega \\right )\\\\\n&= \\frac{1}{(2\\pi i)^2} \\int_{\\Gamma_1} \\int_{\\Gamma_2} \\frac{f_1(\\zeta)f_2(\\omega)}{(\\zeta-T)(\\omega-T)}\\; d \\omega \\, d \\zeta \\\\\n&= \\frac{1}{(2\\pi i)^2} \\int_{\\Gamma_1} \\int_{\\Gamma_2} f_1(\\zeta) f_2 (\\omega) \\left ( \\frac{(\\zeta - T)^{-1} - (\\omega - T)^{-1}}{\\omega - \\zeta} \\right ) d \\omega \\, d \\zeta && \\text{First Resolvent Formula}\\\\\n&= \\frac{1}{(2 \\pi i)^2}\\left \\{\\left (\\int _{\\Gamma_1} \\frac{f_1(\\zeta)}{\\zeta-T}\\left[\\int_{\\Gamma_2}\\frac{f_2(\\omega)}{\\omega - \\zeta} d\\omega\\right] d \\zeta \\right )- \\left (\\int_{\\Gamma_2} \\frac{f_2(\\omega)}{\\omega-T}\\left[\\int_{\\Gamma_1}\\frac{f_1(\\zeta)}{\\omega - \\zeta}d\\zeta\\right] d \\omega\\right)\\right \\} \\\\\n&= \\frac{1}{(2 \\pi i)^2} \\int _{\\Gamma_1} \\frac{f_1(\\zeta)}{\\zeta-T}\\left[\\int_{\\Gamma_2}\\frac{f_2(\\omega)}{\\omega - \\zeta} d\\omega\\right] d \\zeta \n\\end{align}", "5be6a744fabef259f86ae700cd61ebca": "p(H1) = \\pi_1", "5be70bfaf37c7cfe4053521ebc7d7c32": "\\cos\\,{\\theta^*}= r_{f}\\,f \\,cos\\,{\\theta_\\text{Y}}+f-1", "5be715fa8ad1eea3218135d596a99e9c": "n \\geq 3", "5be733f2d7d4dd725e4e6c6a6c3d5362": "\\delta^{(a)}_{(b)}", "5be826635bfccb640e6f4b7ed2a647d2": " \\overline{F}(x_1,x_2) = \\left(1 + \\sum_{i=1}^2 \\frac{x_i-\\theta_i}{\\theta_i} \\right)^{-a}, \\qquad x_i > \\theta_i, i=1,2.\n", "5be84364961e3f5c664c2680b5a3e339": "x_1 = z_1", "5be85cbffef4b1753313b9fa68e818ff": "\\widehat{\\mathbf{Z}}", "5be8e6626646965c9cd49d44a550fef8": "k_O", "5be900bbe8a99333c1dce8f8de4052eb": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} (k-1) \\frac{z^k}{k} =\n\\frac{z}{(1-z)^2} - \\frac{1}{1-z} \\log \\frac{1}{1-z}.", "5be90d6163e072c14af80f8a92a6ad24": "\\zeta (s,q)=\\Gamma(1-s)\\frac{1}{2 \\pi i} \\int_C \\frac{z^{s-1}e^{qz}}{1-e^z}dz", "5be94c6798a4076772d8c2805ccf502a": "\\sum_{1 \\, \\ldots \\, 10} \\left( k \\mapsto f(k,n) \\right)", "5be964b538260d9fd28bb1250fd9dd4d": "x\\text{ OR }y = \\sum_{n=0}^{b}2^n\\left[\\left[\\left(\\left\\lfloor\\frac{x}{2^n}\\right\\rfloor \\bmod 2\\right) + \\left(\\left\\lfloor\\frac{y}{2^n}\\right\\rfloor \\bmod 2\\right) + \\left(\\left\\lfloor\\frac{x}{2^n}\\right\\rfloor \\bmod 2\\right)\\left(\\left\\lfloor\\frac{y}{2^n}\\right\\rfloor \\bmod 2\\right)\\bmod 2\\right]\\bmod 2\\right]", "5be984b1b068160c2509b756cf432b7e": "f(x) \\in C(R)", "5be9f26473e7bfcea7ed2f4797d02bf7": "\n\\begin{align}\n\\vec{a}_A & = \\sum_{B \\not = A} \\frac{G m_B \\vec{n}_{BA}}{r_{AB}^2} \\\\\n& {} \\quad{} + \\frac{1}{c^2} \\sum_{B \\not = A}\n \\frac{G m_B \\vec{n}_{BA}}{r_{AB}^2}\n \\left[ v_A^2+2v_B^2 - 4( \\vec{v}_A \\cdot \\vec{v}_B) - \\frac{3}{2} ( \\vec{n}_{AB} \\cdot \\vec{v}_B)^2 \\right. \\\\\n& {} \\qquad {} \\left. {} -\n 4 \\sum_{C \\not = A} \\frac{G m_C}{r_{AC}} -\n \\sum_{C \\not = B} \\frac{G m_C}{r_{BC}} + \\frac{1}{2}( (\\vec{x}_B-\\vec{x}_A) \\cdot \\vec{a}_B ) \\right] \\\\\n& {}\\quad{} + \\frac{1}{c^2} \\sum_{B \\not = A} \\frac{G m_B}{r_{AB}^2}\\left[\\vec{n}_{AB}\\cdot(4\\vec{v}_A-3\\vec{v}_B)\\right](\\vec{v}_A-\\vec{v}_B) \\\\\n& {} \\quad {} + \\frac{7}{2c^2} \\sum_{B \\not = A}{ \\frac{G m_B \\vec{a}_B }{r_{AB}}} + O (c^{-4})\n\\end{align}\n", "5bea4286a8fbb99aaffa0d6809a3d25a": " T^{\\mu \\nu} ", "5bea46034e1660ef914b6d924a33ef63": "\\nabla\\cdot\\hat{\\mathbf{E}}=0", "5bea4df5394b78df7b1b436a7a72ec86": " \\frac{d\\mathbf{q}}{dt} = \\frac{1}{2}\\left[ {\\begin{array}{c}\n 0 \\\\\n \\omega_x\\\\\n \\omega_y\\\\\n \\omega_z\n\\end{array}} \\right] \\otimes \\mathbf{q}\n", "5bea85ceafd34075ad0031ee26f8cf74": "\\mathrm{We} = \\frac{\\rho v^2 D}{\\sigma}", "5beac1516476de0446501f5b1cd841e9": " f_X(\\mathbf{x}|\\lambda,\\boldsymbol{\\theta}) = h(\\lambda,\\mathbf{x}) \\exp (\\lambda [\\boldsymbol\\theta^\\top \\mathbf{x} - A(\\boldsymbol\\theta)] ) \\,\\! .", "5beaf6b6a8ef3ce161a22e6ae16bb1d4": "P_x(1, 2) \\ge \\frac{0.67xC_x}{(\\log x)^2}.", "5beafe7ddb8ee0f219e30cbede4eecd3": "(c_0,c_1,\\dots,c_{n-1});", "5beb20294b1db2487fcf4e503ca7da2f": "\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma_\\mu = 4\\ \\eta^{\\nu \\rho} I_4. \\,", "5beb2a58c96c63235ee486ba02bc7e70": "\\gamma_{xy}= \\alpha + \\beta\\,\\!", "5beb5f07c2db1fb1e46ae130179de1ee": "(f(s) - f(s')) (g(s) - g(s')) \\geq 0", "5bebb686ab0671d772cef62f78de3e34": "12 \\div 4", "5bebc06f1d25016efc507f2b2eccf06d": "R=\\frac{|a_0|}{|a_0|+\\max\\{|a_1|,|a_2|,\\dots, |a_{n}|\\}}", "5bebccaf93687b442b02a07549863ff8": "\\mathrm{St} = \\frac{fD}{V}", "5bec2402be69b3fe26558389d407acda": "\\begin{align}\nf_{X_1^n}(x_1^n)\n &= \\prod_{i=1}^n \\left({1 \\over \\beta-\\alpha}\\right) \\mathbf{1}_{ \\{ \\alpha \\leq x_i \\leq \\beta \\} }\n = \\left({1 \\over \\beta-\\alpha}\\right)^n \\mathbf{1}_{ \\{ \\alpha \\leq x_i \\leq \\beta, \\, \\forall \\, i = 1,\\ldots,n\\}} \\\\\n &= \\left({1 \\over \\beta-\\alpha}\\right)^n \\mathbf{1}_{ \\{ \\alpha \\, \\leq \\, \\min_{1 \\leq i \\leq n}X_i \\} } \\mathbf{1}_{ \\{ \\max_{1 \\leq i \\leq n}X_i \\, \\leq \\, \\beta \\} }.\n\\end{align}", "5bec70cbe140d3ac550024e823213d04": "E\\left(\\left[X-\\sum^k_{i=0}h_iD_i\\right]^2\\right). ", "5bec8dae759a119f705e4f9662dd2aeb": "y\\left( m \\right)", "5becf843883d054ebf75a26141fbbae7": "\n\\begin{align}\nb^2 &= m^2 + d^2 - 2dm\\cos\\theta \\\\\nc^2 &= m^2 + d^2 - 2dm\\cos\\theta' \\\\\n&= m^2 + d^2 + 2dm\\cos\\theta.\\, \\end{align}\n", "5bed3cca82fcbc861841f8c2b46a8ad5": "\\dot{\\psi}<0", "5bed6203e001c42457b0775969441de8": "a_4-a_3+a_2-a_1+a_0=0,", "5bed8825d985dab1ce4bb15d2af94d00": "\n\\{\\phi_1, H\\}_{PB}+\\sum_j u_j\\{\\phi_1, \\phi_j\\}_{PB} = -\\frac{\\partial V}{\\partial x} + u_2 \\frac{q B}{c} \\approx 0\n", "5bed9b915280b04cf1dd1286ae93ca6f": "\\textstyle \\left(\\sum_{i\\in\\N} a_i X^i\\right) \\times \\left(\\sum_{i\\in\\N} b_i X^i\\right) = \\sum_{n\\in\\N} \\left(\\sum_{k=0}^n a_k b_{n-k}\\right) X^n.", "5beddbeeea1fc602a551b3e1e48fe65f": "v(\\pi)=1", "5bee291ebeeed2a75880c20a52d82cf0": "\\sigma = \\sqrt{\\sum_{i=1}^N p_i(x_i - \\mu)^2} , {\\rm \\ \\ where\\ \\ } \\mu = \\sum_{i=1}^N p_i x_i.", "5beea77ab27b7e24e1ef23c5777f3b8c": "\\big\\uparrow \\Big\\uparrow \\bigg\\uparrow \\Bigg\\uparrow \\dots \\Bigg\\Downarrow \\bigg\\Downarrow \\Big\\Downarrow \\big\\Downarrow", "5beed02e2a39b3420b5923687bf08065": "\\mu_i > \\nu_i \\quad \\forall i = 1,\\dots,n.\\,", "5bef258e9e3bc0f9b3fb819b4f85f98d": "\\dim \\pi = d_\\lambda", "5bef92b1854f9c388d11bfbb1720c05d": "x_n", "5bf003f99955312d62b682bf4e531c9e": "\\psi_{1s}(\\zeta, \\mathbf{r - R}) = \\left(\\frac{\\zeta^3}{\\pi}\\right)^{1 \\over 2} \\, e^{-\\zeta |\\mathbf{r - R}|}.", "5bf020d677a35a58ce3c0d6b845d85c5": "x, y, z \\in C", "5bf09bf8a3e6d990ae0fbdd480ff6d44": "\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = p\\ f\\ (p\\ f) \\operatorname{in} K\\ (p\\ K\\ (p\\ K)) ", "5bf118ce2051567d44d7477cdbb36ed9": "\\left[00,\\alpha\\right] = \\frac{1}{2}\\left(g_{0\\alpha|0} + g_{\\alpha 0|0} - g_{00|\\alpha}\\right)", "5bf141c0371a24906e7643489714c533": "A_{\\text{v}} \\triangleq \\frac{v_{\\text{out}} }{ v_{\\text{in}} } \\,", "5bf192608292def9b4facdc62d3e33fe": "m - m'", "5bf19b6f15327ec3e766a6375ad838e8": "k_{21}[M_1]\\sum[M_2^*] = k_{12}[M_2]\\sum[M_1^*] \\,", "5bf1b382723c6c788610624dff0acfb0": "101\\frac{1}{5}", "5bf1bdc8f90504e6af21f36844834e7a": "(1-\\delta) \\sum_{t \\geq 0} \\delta^t u_i(h_t),", "5bf1f925e98a7ffa7fcc6ee72e4ffdd4": "\\cos\\left(\\frac{\\pi}{2} - A\\right) = \\sin(A)", "5bf23a8066448c8893012744c2db4c42": "\\sum_{j=1}^n w_j x_j \\leq W,", "5bf24521be92ab2404830e3a1892a7a1": "e(k) = \\hat{x}(k) - x(k)", "5bf2b57e92348e4454e1569618e336ff": "41041 = 7 \\cdot 11 \\cdot 13 \\cdot 41\\,", "5bf2bb3f56f2d8861e5edf7ba020e7eb": "N(g)=p^2", "5bf2e1f0230e7a9a5adda12cea496acc": "p, q, r\\,", "5bf2f2d4a7a1de98af63598b1a08636a": "x^2+2xy+y^2-2x+2y-4\\ ", "5bf2fa57472758b242cc6d6edb62e660": " x \\geq y ", "5bf38a989445240aa6adf25ff7fc3180": "A \\leq A^* \\leq 2A", "5bf393e17d5cc071dd392919030eac4b": "-2\\int_{c_R}^{c^*} \\xi \\mathrm{d}c = \\int_{c=c_R}^{c=c^*} \\mathrm{d}\\left[ D(c)\\left(\\frac{\\mathrm{d}c}{\\mathrm{d}\\xi}\\right)\\right]", "5bf39d516ace413612d8c8b780ac6014": "\\sum_{i=1}^n X_i \\sim Y", "5bf3a58bbccf3b69d81300f527bb0d6b": "C-S", "5bf3d34984829abce181a7279064336d": "\n\\frac{x_1 + x_2}{2} > \\sqrt{x_1 x_2}", "5bf4535cb4f40a45a745e43938c088c2": "F(y|\\theta)", "5bf4be0f7ba369d401e22f115a43d1af": "f(\\xi,t')=-\\frac{1}{\\tau} \\int \\exp \\left[-i \\sqrt{\\xi t'} \\left(\\frac{1}{\\omega} \\sqrt{\\frac{\\xi}{t'}}+\\omega \\sqrt{\\frac{t'}{\\xi}}\\right)\\right]\\frac{d\\omega}{\\omega^2}\n", "5bf4c7ed9e759ef6cc5fbecdc6a1cdb8": "\\delta^{-1}", "5bf5178aabe04044f4f095e6b4abb404": "t[a/b]", "5bf5405caae6fd31a34a9fafb3ee56fa": "\\Delta=\\frac{D-2+\\eta}{2},", "5bf56192e86bf4893bca8c9f41f867e6": " K(n) = \\prod_{i=0}^{n-1} K_i = \\prod_{i=0}^{n-1} 1/\\sqrt{1 + 2^{-2i}} ", "5bf56d4177ce2236f0fac4a97d30533d": "K_a \\times K_b = K_w", "5bf5a8aaba118805339c3f6703ce9d9a": "w\\Vdash(A\\land B)[e]", "5bf6f67a99d2bd633662278287d15656": "ds^2=\\left(1+\\frac{1}{S r}\\right)^{-1}dr^2+r^2(d \\theta^2 + \\sin^2 \\theta d \\phi^2)+K \\left(1+\\frac{1}{S r}\\right)dt^2", "5bf710c726e30da5f36ad4ae3fd3a692": " \\Gamma^\\mu = \\gamma^\\mu F_1(q^2) + \\frac{i \\sigma^{\\mu\\nu} q_{\\nu}}{2 m} F_2(q^2) ", "5bf99a61ae982bed3ec18a4f9c990018": "E(S_n)\\,\\!", "5bf9ee02c1225a3bc6d18c4fe44a4c0e": " r=\\sqrt{-\\lambda/\\alpha}\\text{ and }\\omega= 1 + \\beta r^2. \\, ", "5bfb873beaf60675955c36ce07f368b2": "DN_c = DN_1 + DN_2 + ... + DN_n - (n - 1), \\ ", "5bfbf6105ecd3f589cffe7381d3bc00a": "\\zeta_F(s) := \\prod_{\\mathfrak{p}} \\frac{1}{1-N(\\mathfrak{p})^{-s}}", "5bfc491fa726429c35ecddc5e2ccbbee": "\\pi \\oplus \\sigma", "5bfc6b10abf1a893160eae4db5beeb85": "p(y\\|x) = p(x)p(y|x) + p(\\overline{x})p(y|\\overline{x})", "5bfc926ab4ee96c6d6a04a35b1daa0d3": "f \\circ \\hat{f} = [n]", "5bfd487241ca603fc71b60078e826c2b": "\\bar{\\ell}\\ell^\\prime", "5bfd49e9e58dde60d9858343279c93cf": "a|n\\rangle=\\sqrt{n}|n-1\\rangle,", "5bfd5c0089b46408dad0f7dfbeec806f": "Z=\\frac{V_{\\mathrm{m}}}{(V_{\\mathrm{m}})_{\\text{ideal gas}}}=\\frac{p V_{\\mathrm{m}}}{R T},", "5bfd87207f08f0c9a67cd6be145db665": "\\widetilde{R}_{\\beta\\mu}=R_{\\beta\\mu}-g_{\\beta\\mu}\\nabla^{\\alpha}\\partial_{\\alpha}\\omega-(n-2)\\nabla_{\\mu}\\partial_{\\beta}\\omega+(n-2)(\\partial_{\\mu}\\omega\\partial_{\\beta}\\omega-g_{\\beta\\mu}\\partial^{\\lambda}\\omega\\partial_{\\lambda}\\omega)", "5bfd8adacd3ba0f1ebc37015734886cd": "\\begin{align}\n B_0(x) &= 1 \\\\\n B_n'(x) &= nB_{n - 1}(x) \\text{ and } \\int_0^1 B_n(x)\\,dx = 0\\text{ for }n \\ge 1\n\\end{align}", "5bfd8e9e6b83fc8a3bd4639e8890ac78": "\\text{Gain}=10 \\log \\left( {\\frac{V_\\mathrm{out}}{V_\\mathrm{in}}} \\right)^2\\ \\mathrm{dB}", "5bfda99f0164971fea8da3e7f97a6835": "\nK_T=P\\,\n", "5bfdb4ac511a070c564e8c0bf879f624": "\\cos(\\phi_m);\\,\\!", "5bfe4054c1d818895716df6a27f3ec8f": "z_{r0}", "5bfe8b3ce5592e3ac4fa7e22a9d249c5": "R_I(x)", "5bfee278ca1148ce766549699fa8f806": " E{d/s} = E_c = 2.20\\text{ ft}\\,\\!", "5bfee4c3fde11300e1c35d86309f7b79": " ac \\otimes bd", "5bfef3f95fc69338b30fd8430287b147": "(2n-2)", "5bff250d0a4f89cd65c43a44775f8d77": "f(x) = P_2(x) + h_2(x)(x-a)^2, \\qquad \\lim_{x\\to a}h_2(x)=0.", "5bff417179b44b527c730950118e7db7": " -2 k' \\cdot p \\,", "5bff7a3e8801306038e3526a2f86cb56": "\\frac{\\Delta I}{I}\\approx\\frac{r}{R} ", "5bff891508e35b29d7c6b8b3ecfa023b": "a^\\dagger = \\frac{1}{\\sqrt{2}}(q - i p) = \\frac{1}{\\sqrt{2}}\\left( q - \\frac{d}{dq}\\right)", "5bffab625fac97a9aedea4573ffaf432": "\\mathbb{E}(S_t)= S_0e^{\\mu t},", "5c00859dbe3f598be24d7e237caaedf1": "\\nu_0 = \\frac{a}{2\\pi} \\sqrt{2D_e/m}", "5c00daaca7c15b77cb5df9d94efeca90": " \nr_{r,s} = h_{r,s} x_{s} + n_{r,s} \\quad\n", "5c00e8085e7bd93e68667a6de2f6f5be": "\\kappa\\geq\\aleph_0", "5c00ee6e5d60c64b2ae0305cec2fb186": "0 \\le a_n \\le \\lfloor q \\rfloor", "5c00f75d89771e22a8550e94824f868b": "\\ln x = \\int_1^x \\frac {1}{t} dt ", "5c010ed7a3899e7e54d46f96ca2ef79f": "\n \\gamma_k=\\sqrt{\\frac{\\|U_k^{-1}\\|_F}{\\|U_k\\|_F}}\n", "5c012fe0a1e47362eced35df3e3d75b7": " \\Delta U = n\\,c_V\\,\\Delta T", "5c015c20d649893f0be7ee1610867eac": " X \\in \\mathbb{R}^p ", "5c01687ace76d6e58968e0e2e4dacdf1": "\\sum_{i\\in A}\\sum_{j\\in T}C(i,j)x_{ij}", "5c01b59b79355b5dfdb618f6e25ed5eb": "\\mu_{B}^{\\ominus}", "5c01bdc469af90d9d5dd07ecb7bfd5aa": "\\left(0,1\\right)", "5c01c2f1bbbd019465fd909aa79d3c44": " D_{\\mathrm{KL}}(\\alpha_p,\\beta_p; \\alpha_q, \\beta_q) = (\\alpha_p-\\alpha_q)\\psi(\\alpha_p) - \\log\\Gamma(\\alpha_p) + \\log\\Gamma(\\alpha_q) + \\alpha_q(\\log \\beta_p - \\log \\beta_q) + \\alpha_p\\frac{\\beta_q-\\beta_p}{\\beta_p} ", "5c0227ba857d2fb756d8835d8d1c433e": " b_2 = \\frac{{c_2}}{{\\rho - c_2 \\cdot M_2}} \\,", "5c022ef032b33cfdef720f26f392ef20": "P=\\frac{1}{n - 2} \\left(Ric -\\frac{ R}{2 (n-1)} g\\right)\\, \\Leftrightarrow Ric=(n-2) P + J g \\, ,", "5c023a0273b661c713e2f74b16aeb7ae": "\\epsilon_x(y)=\\frac{L(y)-L}{L} = \\frac{\\theta\\,(\\rho\\, - y) - \\theta \\rho \\,}{\\theta \\rho \\,} = \\frac{-y\\theta}{\\rho \\theta} = \\frac{-y}{\\rho}", "5c0277bcbfb470fc8b5d27642a002f9f": "\\mathbb{Q} \\!\\,", "5c028ec2178204299f954c14a7fbddd8": " \\phi_< = A r \\cos \\theta \\ ,", "5c02f80dd199212c097673962f85f5c1": "\\mathbf{WH}", "5c0301ee2008ffede7a99326e33cfe4c": "\\mathfrak{p}_2", "5c03098be92702047e00bbfb00a2ee94": "1.5 \\times 2 = 3", "5c036944c7efe3cdeae690b31752ef77": "C=2\\pi r,", "5c03ae0a05937988d65c42f34c0e8645": "\\bar{X} \\pm t_{n-1,0.975} S / \\sqrt(n)", "5c03ce8713f2eccbfe3ea3e346e47a18": "E \\widehat{=} 4", "5c03dd4d50c566363b5df60c970328dd": "d > 1", "5c048fab0c948f912a9a2871e2df8c46": "\\frac {v_\\mathrm N - v} {v_\\mathrm N} = \\frac {Nc} f", "5c04abe87ea4b320f843a735c147a465": "f(1,0) = p(1,0) = a_{00} + a_{10} + a_{20} + a_{30}", "5c04b03ac4280f25cbc17818ecc9bc78": "\\int\\frac{\\mathrm{d}x}{\\sin^n ax} = \\frac{\\cos ax}{a(1-n) \\sin^{n-1} ax}+\\frac{n-2}{n-1}\\int\\frac{\\mathrm{d}x}{\\sin^{n-2}ax} \\qquad\\mbox{(for }n>1\\mbox{)}\\,\\!", "5c05727e103f97c10a575767a5e9c72d": " \n \\eta = \\sqrt{ 1 + \\left( { a - \\lambda_{-} \\over c } \\right)^2 } = \\sqrt{ 1 + \\left( { b - \\lambda_{+} \\over c } \\right)^2 }.\n ", "5c05cf7d5ae8a32e478930e9612a119d": " A_{p}E\\left( \\left( \\sum_{i=1}^{n}\\left\\vert x_{i}\\right\\vert ^{2}\\right) _{{}}^{p/2}\\right) \\leq E\\left( \\left\\vert \\sum_{i=1}^{n}x_{i}\\right\\vert ^{p}\\right) \\leq B_{p}E\\left( \\left( \\sum_{i=1}^{n}\\left\\vert x_{i}\\right\\vert ^{2}\\right) _{{}}^{p/2}\\right) ", "5c0638043fb9afee2fec6d83b2ca998b": " x_s=\\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1} , \\quad y_s=\\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}. \\ ", "5c063edd0b55ae24702f7f99131854aa": "\\alpha_k>0", "5c06445bfd0fb5c4c7daea45217f60a0": "\\langle S_{i} S_{j} \\rangle = b^2 \\delta_{ij}.", "5c064f055618ee4a39501415d0158ccf": "m = \\tan (\\theta)", "5c06e95d95c45b397c222c2e164875f3": " I_m = n \\frac{ \\sum x^2 - \\sum x } { ( \\sum x )^2 - \\sum x } ", "5c070398f68b0cca924630ba7f4b4157": " \\nabla \\times \\mathbf{E} = -\\frac{ \\partial \\mathbf{B}} {\\partial t} \\,", "5c0730259d772f77d8a64336e0433569": "_{q \\nleftarrow p=q'p}\\!", "5c07365ec3e2e585c55c0092bb7e82e9": "A_{mn\\mu\\gamma}", "5c07efeaf14586679559c4d74b4829d7": "U(\\{c(t)\\}_{t=t_1}^{t_2})=\\int_{t_1}^{t_2} e^{-\\rho (t-t_1)}u(c(t))\\,dt,", "5c07f8135afd2eb355bb6cc9db7deb90": "n^2 + n + 41, \\, ", "5c08064cfb87aba758723b012d362a65": "A_X^{(\\beta)}", "5c083544454541e22d475c8e01f8864b": " \\lambda x_0 \\ldots \\lambda x_{i-1} . (\\lambda x_i . A(x_i)) M_1 M_2 \\ldots M_n \\rightarrow\n \\lambda x_0 \\ldots \\lambda x_{i-1} . A(M_1) M_2 \\ldots M_n ", "5c08576e98df43546e0a3c719a22f7d6": "\\sum_{k=1}^\\infty k\\,X_k", "5c085eefb5eaa18a0372600576d13414": " \\lambda_{\\mathrm{p}} \\ ", "5c08887153ce04cf46c4ba0463dd5787": "H^{II}_q(M \\otimes Q_\\bull) = \\mbox{Tor}_q(N,M)", "5c0899d68d0585ee816b787f091f4720": "\n\\sigma _{\\hat g}^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {{{ - 8\\bar L\\,\\pi ^2 } \\over {\\bar T^3 }}\\alpha (\\bar \\theta )} \\right)^2 {{\\sigma _T^2 } \\over {n_T }}", "5c08c4bb5cb943e6ddf4c8995d247519": "Y_1 Z_2 Y_3 = \\begin{bmatrix}\n c_1 c_2 c_3 - s_1 s_3 & - c_1 s_2 & c_3 s_1 + c_1 c_2 s_3 \\\\\n c_3 s_2 & c_2 & s_2 s_3 \\\\\n - c_1 s_3 - c_2 c_3 s_1 & s_1 s_2 & c_1 c_3 - c_2 s_1 s_3 \n\\end{bmatrix}", "5c08cbe3b4e4444f880c207627b89969": "r_2 = (X \\to Y, \\emptyset, \\{S\\})", "5c08d577301f6c4c6d13aacb0ede6f4b": " \\operatorname{V}(J) = \\{x \\in k^n \\ |\\ f(x)=0 \\mbox{ for all } f\\in J\\}", "5c0916e37e41a099027357216c9897c9": " \\delta \\mathbf{r} =(\\delta x, \\delta y) = (L\\cos\\theta, L\\sin\\theta)\\delta\\theta.", "5c093f37279b0830d7097522efd7661c": "\\langle f,g\\rangle := \\int_{U(d)}\\bar{f(U)}g(U) dX", "5c09609a041da3e378e5cbf8b4b7e028": "N(\\lambda,h)=+\\infty", "5c096e57cd842050ead6a2a2fa55f3bb": "x= \\frac{x'+1}{x'} ", "5c09bce40d90eaf68be277b2736f5941": "\\ \\xi \\ge 0 \\text{ and } |\\eta| \\le 1. ", "5c09ceaf744f5d9bb0c19e5fea80ca97": "\\left.\\frac{\\delta E[\\psi,\\lambda]}{\\delta\\psi(x)}\\right|_{\\psi=\\psi_{\\lambda}}=0.", "5c0a0fdb755b6ede7a9bde07c31157c3": "\\sqrt{2\\pi}", "5c0a18e924f6463cbfa1feb34f7d8bd6": "e\\left( \\rho \\right) =\\frac{\\sigma }{\\rho }", "5c0a241a1557aa044952a6372af2cead": "u(x,y,t)= \\phi (x,y) \\psi (t)\\ ", "5c0a3871f52036864ecdb691abb6a5ae": "\\bold{w} = (it+ju+kv) \\mapsto \\exp(it+ju+kv) = \\cos(|\\bold{w}|) + \\sin(|\\bold{w}|)\\frac{\\bold{w}}{|\\bold{w}|}.\\,", "5c0a6a523e8be08aa2ed5dda0aa79edb": "\\operatorname{var}(k) = \\bar{k} = 2.", "5c0ac16bf6d251b3436000c3f830a5f7": "\\scriptstyle PA\\ =\\ P'B", "5c0addffd8a9aa9186470b6280421efd": "\\tau = \\operatorname{trace}_{V}(T)", "5c0b205463d793db6313ba9421f7202b": "a*b=b*a \\,", "5c0b491a945eb7053d319c26f640c012": " g(s)=\\sum_{n=1}^{\\infty} \\frac{a(n)}{n^{s}} ", "5c0b4ce2559bc3d77a325f5b74a7a830": " ( 4 x^3 + 2 x y^2) dx + ( 2 y x^2 - 2 a^2 y ) dy = 0 ", "5c0b5ea21d2003d990a37451fa6b632b": " \\{\\epsilon_{n}\\}_{n=1}^{N} ", "5c0b5f6a5004f35ffd9f7ec79752c476": "\\dim R_1 < \\dim R", "5c0b71d863d4adf01ce54149de231504": "x_{1}0\\wedge |z|<1 ", "5c0ec45e9edadb636a173931e8335e5f": "k_1 \\in B^{20l^\\prime+64}", "5c0ee2cfee392e6c42dc515a80cfd62b": "\\mathcal A \\to (S\\downarrow T)", "5c0ee61cae9cd94693b75c865a11d861": "F_\\beta = (1 + \\beta^2) \\cdot \\frac{\\mathrm{precision} \\cdot \\mathrm{recall}}{(\\beta^2 \\cdot \\mathrm{precision}) + \\mathrm{recall}}", "5c0f254b45413eb7a3331e4fdfa6bf44": "\\text{SUBEXP}=\\bigcap_{\\varepsilon>0} \\text{DTIME}\\left(2^{n^\\varepsilon}\\right)", "5c0f3c2a7815af546c8beffbbe7c144d": "X(A_n) = \\log_2 \\left(\\frac{1}{p(A_n)} \\right) = - \\log_2(p(A_n)) ", "5c0f67d42d3fb30f08437f121ce42ba6": " \\int\\limits_{0}^{L^{e}} \\left( \\frac{d}{dx}\\left( EA\\frac{du}{dx} \\right)+n \\right)v=0 ", "5c0f680881c8f9b7092f1109bfa0d0d3": "I(X,Y;Z)", "5c0f743a1fadddc23498db550780e75e": "i:A\\to A", "5c0f843caad06ad721ca1a846c30590e": "p_A = 0", "5c0fc7f4f94bd3df891900224bb64136": "\\hat{S}_M", "5c0fe3fdec85b8ab5c2f136a3a4b9d7f": "[A \\otimes B]_{ij} = \\bigoplus_{k = 1}^p [A]_{ik} \\otimes [B]_{kj} = \\max([A]_{i1} + [B]_{1j}, \\dots, [A]_{ip} + [B]_{pj})", "5c0ff3846702201873a72b385836d681": "0+j\\omega", "5c10576dcd3a44689511bba7357084fd": "\\frac{E'}{c} = \\gamma \\frac{E}{c} - \\gamma \\beta p_x", "5c10a7164a629985fd6163db9b271976": "nN", "5c10e1f7935495b5ffb0539a1cc122c1": " \\int_{\\Bbb Z_p} (1 + a)^x \\, {\\rm d}x = \\frac{\\log(1+a)}{a} ", "5c10e48a5352a4e5dc79dac5f2827ed5": "\\displaystyle{F_{g_a\\circ f}(w) =0.}", "5c11307a143ddc5fe670149d213d7ec3": "\\frac{\\mathrm{D}\\Gamma}{\\mathrm{D}t} = 0.", "5c1142ea48e5cbc2a7b1317dbf88ff19": " V_\\lambda := \\bigcup_{\\beta < \\lambda} V_\\beta .", "5c114730ef7b3ca2e2f95764edddf2ae": "\\lbrack\\mathbf g\\rbrack = \\lbrack\\mathbf h\\rbrack^{-1}", "5c115d6634ef781313d4f8819daf5be1": "x \\cdot y = 0 = y' \\cdot x", "5c11a60b89d9a1825e62708ab4c9f5c4": " U_r = U_{in}\\frac{r}{R_{in}} .", "5c11e86870bd752212d48125099fda6c": "\\sqrt{V}=0", "5c120479917edd595e184c0712cffc8e": " (x,t) ", "5c1277720ebe52751a4adf39c6e08199": "[(\\frac{1}{2},0)\\otimes V]\\otimes [(\\frac{1}{2},0)\\otimes V]", "5c12b34b378dce2108cae80d919a4ec8": "\nC(d) = \\exp(-d/V)\n", "5c12df06db9051bcf7d2bfaa7f1ded38": "= \\int \\int (1 - \\cos \\theta) \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\sin \\theta \\mathrm{d} \\theta \\mathrm{d} \\phi", "5c12e52b3057287830866d6914427a12": "\\begin{align}\n & \\begin{matrix}\n +2a+b-c=a_1 \\quad & -2a+2b+2c=b_1 \\quad & -2a+b+3c=c_1 & \\quad \\to \\left[ \\text{ }a_1,\\text{ }b_1,\\text{ }c_1 \\right]\n\\end{matrix} \\\\ \n & \\begin{matrix}\n +2a+b+c=a_2 \\quad & +2a-2b+2c=b_2 \\quad & +2a-b+3c=c_2 & \\quad \\to \\left[ \\text{ }a_2,\\text{ }b_2,\\text{ }c_2 \\right]\n\\end{matrix} \\\\ \n & \\begin{matrix}\n +2a-b+c=a_3 \\quad & +2a+2b+2c=b_3 \\quad & +2a+b+3c=c_3 & \\quad \\to \\left[ \\text{ }a_3,\\text{ }b_3,\\text{ }c_3 \\right]\n\\end{matrix} \\\\ \n &\n\\end{align}", "5c1331023176357d0730f181a48f475e": " f_0(t) \\ ", "5c133df812a9867e5a92408e49218c99": "= (\\lambda x.g\\ (x\\ x))\\ (\\lambda x.g\\ (x\\ x))", "5c1344439830c460e1c7553ea266844f": "x_{\\mathrm {base}}", "5c1355bda6ff5c6e98bf4dd0c462f4cb": "K_1 < S_T < K_2", "5c1367f504a833aefed93009a3c426e3": "v_{001}", "5c13f4f2f5751a87d889afed1bd2bf75": "\n\\begin{align}\nH & = \\sum( n_i \\ln n_i - n_i \\ln \\delta v_\\gamma) \\\\\n& = \\sum n_i \\ln n_i + \\text{constant}\n\\end{align}\n", "5c143f693d8cda4e528ee81bd23610c1": " M \\frac{d^2}{dt^2} X(t) = - \\frac{\\partial V(X)}{\\partial X} - \\int_0^{T} d t' \\alpha (t - t') (X(t) - X(t')) ", "5c144bce6a6c3751eaca0a1e85db07ac": "P(t)", "5c14be8290418fc069f3ae921a35c576": "\\cot{ \\frac{C}{2 }} = \\frac{s-c}{\\zeta }", "5c14f2c2e7ca43257bd176a8bf41ca29": "\\displaystyle \\alpha", "5c1530d7fef27a25ecfe778d5abe29d4": "A = \\frac{1}{2} bh", "5c15509371433bf692b4595327628a03": "\\varphi_{h(x)}\\,", "5c159df56f87df521e6a5f3a1a975820": " \\frac{\\overline{GB}}{\\overline{GD}} \\times \\frac{\\overline{ID}}{\\overline{IF}} \\times \\frac{\\overline{HF}}{\\overline{HC}} \\times\\frac{\\overline{GC}}{\\overline{GA}} \\times \\frac{\\overline{IA}}{\\overline{IF}} \\times \\frac{\\overline{HF}}{\\overline{HB}}=1", "5c1684666d2458b20f0bd2cb36225e45": "\ne\\epsilon(t)(c_{1}(t)x|\\Psi_{1}(x,t)\\rangle + c_{0}(t)x|\\Psi_{0}(x,t)\\rangle=i\\hbar(c_{1}'(t)|\\Psi_{1}(x,t)\\rangle + c_{0}'(t)x|\\Psi_{0}(x,t)\\rangle)\n", "5c1696ee9713936594e7d664f5ec111f": "\\{c_i\\}_{i=1}^N", "5c16a82dd5f5eefbfcce1d17f18dfd09": "{\\scriptstyle O(\\sqrt{b-a}) = O(2^{\\frac{1}{2}\\log(b-a)})}", "5c16da5a10553fb3f6c778629f78e1ea": "\\forall a,b,c \\in R", "5c16f757233856dcf311176b7410d2d5": "(0,0)", "5c16fdefa9856385795818ca527a4c1a": "\\alpha(n)", "5c171fc50367685463f9b315b4f113ce": "b^*", "5c1731bd2598656540176c4e4a645b36": "\\scriptstyle a_1,..,a_k", "5c177afd6959abb89c02c736e73b8a7c": "\\begin{align}\n S &= \\int_0^{2\\pi} \\left[\\left(\\frac{\\operatorname d\\!y}{\\operatorname d\\!t}\\right)^2 + \\left(\\frac{\\operatorname d\\!x}{\\operatorname d\\!t}\\right)^2\\right]^\\frac{1}{2} \\operatorname d\\!t \\\\\n &= \\int_0^{2\\pi} r \\sqrt{2 - 2\\cos(t)}\\, \\operatorname d\\!t \\\\\n &= \\int_0^{2\\pi} 2\\,r \\sin \\frac{t}{2}\\, \\operatorname d\\!t \\\\\n &= 8\\,r.\n\\end{align}\n", "5c1805cbb9cc0c2320f0de0891f7dfbc": "x_n \\overset{\\mathrm{w}}{\\longrightarrow} x", "5c180da5323ca523e99c784274b3ade5": "\\leq1/\\mathrm{poly}(|y|)", "5c18304b4bc5fc1ca7ac88fd154cd009": " \\qquad \\frac {pV}{T}= k ", "5c1835fea725756aa58d0e4c48f2ff35": " \\mbox{pefsu} = \\frac{\\mbox{number loaves of bread (or jugs of beer)}}{\\mbox{number of heqats of grain}}.", "5c1856990e77a0b17e50dae5bb12102b": " y_{n+s} ", "5c185a7aab85402b07d24813749765ad": "f(bb)a_{bb}+f(Bb)a_{Bb}+f(BB)a_{BB} = 0.", "5c1862c152a840b69d59507d45b54cf6": "G=\\langle a_1,b_1,\\dots, a_k,b_k| [a_1,b_1]\\cdot\\dots\\cdot [a_k,b_k]\\rangle", "5c18a1cf2dd9694bbd1e6cb078f46bfc": "{\\Delta V_w} = {V_{r1}\\cos \\beta_1}(1+\\frac{V_{r2}\\cos \\beta_2}{V_{r1}\\cos \\beta_1})", "5c18afd468d9771631553f9de413b3c6": "(1,1,8)\\rightarrow 4\\,(1,1)_0\\oplus 2\\,(1,1)_1\\oplus 2\\,(1,1)_{-1}", "5c18f608408ca97badd77ed583f4a793": "\\scriptstyle \\mathrm{Y}+\\bar{\\mathrm{A}}", "5c18f7580f22b3309b7021e96a837048": "d^ny = f^{(n)}(x)\\,(dx)^n.", "5c1902433cb62264abb40a5368779ba2": "p(X|E = e)", "5c1966e5f247505522f8a5c38bc0804a": "f(x) = {1 \\over \\sigma\\sqrt{2\\pi} }\\,e^{-(x-\\mu )^2/(2\\sigma^2)}", "5c199953e1527351ae85e12a50b44f61": "\\partial_\\mu T(\\omega)\\varphi(x) = T(\\omega)\\partial_\\mu\\varphi(x).", "5c19ac7120c8b35632d8f93779580241": "\\frac{(-1)^n}{n!}\\,", "5c19c5be78e04a53d872804c86cf842d": "\\Omega_1, \\Omega_2", "5c19f6a5d7501e22fdf0bc64dd0c69a2": "r = k_1[A]+k_2[A]^2", "5c19f936bb0149f5268b94a1a08d5cb9": "\nn(\\omega) = \\frac{1}{\\mathrm{e}^{\\beta \\omega}-\\zeta}\n", "5c1a0e79773f309823173a170de15690": "\\lambda(C)\\in\\ [0, \\infty]", "5c1a69740b0ed89fa2f85953ff068b17": "\\left(\\nabla\\Phi\\right)_i = \\nabla_i \\Phi ", "5c1aa885ccd6777eff45c3dfbc254bfb": "\\cos_k(i)", "5c1b702c4accef877c56d5824d0d646e": "\\forall \\mu : \\forall \\delta : \\exists \\epsilon < \\gamma:A_{\\mu , \\delta} = A_{\\epsilon}", "5c1c0218d6300b4518b13684a58cc2e9": " \\mathrm{sf}(4)=1! \\times 2! \\times 3! \\times 4!=288. \\,", "5c1c1312f58691fb75734da688128023": "\\sum_{\\alpha\\in[0,\\omega_1)}f(\\alpha) = \\omega_1", "5c1c1592982bdb5067a3818c56ebde94": "c_S(I)=k", "5c1c78c61e831254b691167e4e80b424": "K(x)=e^{\\zeta^\\prime(-1,x)-\\zeta^\\prime(-1)}=e^{\\frac{z-z^2}{2}+\\frac z2 \\ln (2\\pi)-\\psi^{(-2)}(z)}", "5c1cb19e4625d64a356e077e0f560b19": "I_1 = \\int _0^{\\frac{\\pi}{2}}\\frac{1}{\\sqrt{a_1^2 \\cos^2(\\theta) + b_1^2 \\sin^2(\\theta)}} \\, d \\theta", "5c1d167d490eb2ecd66b8ea31e03e9cf": "d\\tau = dt \\,\\sqrt{1 - 2m/r}", "5c1d40a1d5eabb3a1de182bca9ef080c": "d_0^{n-1}", "5c1d9403a98c374aff989175ea8ffc84": "\\mathfrak P_n", "5c1d9e7173c08184c609dca6ccb7f1fc": " H=-\\frac{1}{2}\\sum_{i,j}J(r(i,j))s_{i}s_{j} ", "5c1da75137fc9162f4d5b60b1febface": "B(\\sigma_x, -\\tau_{xy})", "5c1df23b134f12ad6a75f71da2b9a265": " \\varphi^*(\\mathrm{e}^{\\mathrm{i} \\theta}) = \\sup_{0 < \\alpha \\le \\pi} \\frac{1}{2 \\alpha} \\int_{\\theta - \\alpha}^{\\theta + \\alpha} \\varphi\\left(\\mathrm{e}^{\\mathrm{i} t}\\right) \\, \\mathrm{d}t,", "5c1e54afe58007a090537a6902334982": "\\mathcal{L}_D = \\bar{\\psi}\\left(i\\gamma^\\mu \\partial_\\mu-m\\right)\\psi", "5c1e5e56899fb8e1fb3a2bcd890f390a": "r_s", "5c1e746076115e9714248c735c9a0f55": "R_\\text{K} =\\frac{2 \\pi \\hbar}{e^2} \\,", "5c1eb16aaf2db65c72d3902894b17f82": "\\max_\\alpha\\{\\mathsf{H}(Y) - \\mathsf{H}(Y \\mid X)\\} = \\max_p\\left\\{\\mathsf{H}(Y) - \\sum_{x \\in \\{0,1\\}}\\mathsf{H}(Y \\mid X = x) \\mathsf{Prob}\\{X = x\\}\\right\\} =", "5c1ee648262bf008a19aa55e1c6f1e39": "H(\\sigma) = -J\\sum_{}\\sigma_i \\sigma_j. ", "5c1eeca2fb4dce2402786d4ecda10119": " ||x_i - x_j||_2 << \\sigma ", "5c1f1294d161e4b05ae6ea7afb770d8f": "S = \\bigoplus_{i \\geq 0} S_i", "5c1f3708d602c5fdbba38ddf9f1f71cd": " X\\in \\mathbb{B}^n", "5c1f46496131c40e5e9f3c5218a931c1": "P \\to aWb~~~~~~~~~~~\\text{probabilities of pairwise interactions between 16 possible pairs}", "5c1f6c3cc7dd0ee385052b591ee00435": "(aq^s,bq^s;q)_\\infty", "5c1f73ee272093416bbf76d46d15e170": "f \\circ h \\neq g \\circ h", "5c1f873cbdae20aeb061e5cfec0533be": "= \\nu", "5c1fc7f94103176ae3147cf910fd92e2": "\\{\\theta^{}_i(t^{}_{n-1},k^{}_j),j=1,\\dots,J\\}", "5c2007f34a16502730b3b6aee252d84a": "E_*(X) = MU_*(X)\\otimes_{MU_*}M_*", "5c200cbf4e1750e9213ad0d905a0c9c3": "\\mathbb{E}[X_k\\,|\\,\\mathcal{F}_{k-1}]\\ge X_{k-1}", "5c202c67f0b06a7e35e6aa2375089155": "\\frac{\\Delta F}{\\Delta t} = m \\frac{\\Delta a}{\\Delta t} \\implies y = m j", "5c203beeb482b1ff5fdf3b7b1ed20188": "\\beta _{j}", "5c205b0d52f99bef795e0d66ccd3eea7": "\\Lambda=(\\lambda_1,\\ldots,\\lambda_k)", "5c206864d7ecbfdefa827fff0b33b1c1": " b \\equiv c^{2^{S-S'-1}} \\pmod p", "5c218ab75332791c3705d10c90a36916": "\\textstyle 1.\\ Subspace\\ decomposition\\ by\\ performing\\ eigenvalue\\ decomposition:", "5c21bc7be2cc01ecac1bbf2d4773e428": "|\\alpha\\rangle =e^{-{|\\alpha|^2\\over2}}\\sum_{n=0}^{\\infty}{\\alpha^n\\over\\sqrt{n!}}|n\\rangle", "5c21d82806cb83eeeeedec53b03514bb": "\\overline{D}_{st} ~=~ \\frac{1}{n}\\sum_{i=1}^{n} D_{st} ~=~ D_{st}", "5c2221dbac6362794e9e3da79b62c988": " (\\boldsymbol{\\sigma} \\cdot \\mathbf{p}) \\boldsymbol{\\sigma} = \\mathbf{p} + i \\boldsymbol{\\sigma} \\times \\mathbf{p} ", "5c222c85e6cf626d297f83b6789fd68f": "I(t+\\tau)=[1-\\int_{t-r}^{t}I(t')dt'] S_i \\left \\{\\int_{-\\infty}^{t}\\alpha(t-t')[c_3E(t')-c_4I(t')+Q(t')]dt'\\right \\}", "5c224d02827aaac92a4e2e17717d031e": "{\\hat G} = SGP", "5c228bf87b2ac812327f3be6f2f1c4d6": "I(f) = \\int_\\Omega f(x)\\,dx.", "5c22976909b09313f56ced80ce444a0b": "\\sum_{x< n\\le y} a_n\\phi(n) = A(y)\\phi(y) - A(x)\\phi(x) -\\int_x^y A(u)\\phi'(u)\\,\\mathrm{d}u \\,.", "5c2423d9bb6800593fddead13d1534d3": "t_p", "5c242938e3e56b76c35571fba4b3dc74": "f, g : G \\rightarrow \\mathbb{C}\\,", "5c244e672268b8df4c746b6979173a04": "\\xi=\\Omega t/2\\ ", "5c245df761a8f24bcf7bc7bc7daa020a": "{\\hat{a}}_{i}^{\\dagger}", "5c248b0f375d8bf98051977420881240": "\\scriptstyle \\operatorname{fix}(M,Q_i)", "5c249210fc8a6a31639914bbc67e212e": "\\scriptstyle (X,\\sigma,\\tau)", "5c2551057c7f5f8e0e3ea28e60a18c84": "E(\\tau,s) = {\\pi\\over s-1} + 2\\pi(\\gamma-\\log(2)-\\log(\\sqrt{y}|\\eta(\\tau)|^2)) +O(s-1),", "5c258138148769d9b62233731e503ad6": " R_{\\alpha \\beta } \\equiv {R^{\\nu}}_{\\alpha \\nu \\beta } ", "5c25b5a78666578e263aa37a58b70a7b": " \\hat{X}=X+K(D-HX) ", "5c25b9101460437a63ff9f7ee4a025d5": "X_t=Y_t-Y_{t-1}", "5c25bcb0752bc8be4aabb43203e773a3": "t\\in [0, 1],", "5c26763b19d9d7a96e148b6657e71d9a": "t=\\frac{\\rho _Br_s^2}{6D_ebM_Bc_{A_b}} \\left[ 1-3 \\left(\\frac{r_c}{r_s}\\right)^2 + 2 \\left(\\frac{r_c}{r_s}\\right)^3 \\right]", "5c26861a90db9d2151341d6897b22798": "y_j \\geq 0 ", "5c26b0dd5d07c6f2387fcd6cb2398946": "\\langle\\vec x\\rangle \\in P^\\perp", "5c26d00b2fd1dc8c188f3bae2501ed10": " y^*_n ", "5c27237f0d95185044cfa624a332c0d6": "{\\textstyle \\alpha}", "5c278426634b528d94ea764b4e2e59ae": " \\Delta = -r^2(\\partial_x^2 + \\partial_y^2 + \\partial_r^2) + r\\partial_r.", "5c27e0c41e8d928bf7e7020c15425766": " (r_{i}, r_{i+1}).", "5c27f26688e09c5c24a9beb648f16522": "f^{\\star\\prime\\prime}(x^\\star)\\cdot f^{\\prime\\prime}(x(x^\\star))=1.", "5c2809038ceb35f20738f0bdfc06e143": "\\gcd(w_1,w_2)=1", "5c2829f96f0d408ed51a485cb954f349": "S \\subseteq \\alpha N", "5c284f9baccdca3fd081dd7a0fce9aa9": "PET = 16 \\left(\\frac{L}{12}\\right)\\left(\\frac{N}{30}\\right)\\left(\\frac{10\\, T_a}{I}\\right)^\\alpha ", "5c28759058e4649610921ce1cf31e93e": " L = p r \\sin \\theta .\\,\\!", "5c290121e6adf18c941dda48a9295594": "a^{d 2^r} - 1 = (a^{d 2^{r-1}} - 1) (a^{d 2^{r-1}} + 1)", "5c291f2fde64301c2d5c9e1e7199dc4f": "\\omega_{x\\land (y\\lor z)} \\neq \\omega_{(x \\land y)\\lor (x\\land z)}\\,\\!", "5c297e999a0058510f51247822d24e07": "\\Lambda_n(z)+\\Lambda_n(-z)= 2^{1-n}\\Lambda_n(-z^2)", "5c298e059b1933d666acfb96aa6393c2": " P(E|S_i)\\, ", "5c29cba4f293c1bdee9f3b2d6b6c1b86": "\\hat K(n)", "5c29da53962a7a06e758517a0ccc3e9e": "X(0) = i", "5c2a0c37fb12bd7ce7f9bf456fad2996": "FreqScript:=F_1", "5c2a225258145885e00064bdf4ec0673": " \\| y \\|_1 \\equiv \\begin{smallmatrix}\n \\text{MAX} \\\\\n a \\le x \\le b\n\\end{smallmatrix} \\, |y(x)| \\ + \\begin{smallmatrix}\n \\text{MAX} \\\\\n a \\le x \\le b\n\\end{smallmatrix} \\, |y\\,'(x)| \\qquad \\text{where} \\ \\ y \\in D_1(a,b) \\, .", "5c2a6040ea3ea50ebeb4667062752d0a": " \\operatorname{smootherstep}(t) = 6t^5 - 15t^4 + 10t^3 ", "5c2a70b388b96de6fe248a230ddef4ab": "\\lim_{n\\to\\infty}\\frac{f(n)-a_1 \\varphi_{1}(n)}{\\varphi_{2}(n)}=a_2 .", "5c2a829cc568ec036252193cc3f27b92": "\\mathbf{G_y}", "5c2ab065ac710dcb15a45e49c907d231": "\\begin{align}\n W_0(k) &= \\frac{\\cos\\{N \\cos^{-1}[\\beta \\cos(\\frac{\\pi k}{N})]\\}}{\\cosh[N \\cosh^{-1}(\\beta)]}\\\\\n \\beta &= \\cosh[\\frac{1}{N} \\cosh^{-1}(10^\\alpha)],\n\\end{align}", "5c2b2b2eb053da549e5dd6f70bc90102": "D = \\sum a_i V_i, \\quad a_i \\in \\mathbb{Z}", "5c2b381a67947012e4ec52726a17d022": "(\\boldsymbol\\mu,\\boldsymbol\\Sigma) \\sim \\mathrm{NIW}(\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi,\\nu)", "5c2b445f352ccd524241e31352a25c15": "\\mathbf{X, Y, \\tilde{X}, \\tilde{Y}}", "5c2b6ded4881aa23ba8b5971c747cfe2": "f: S \\to S ", "5c2b9491ed12d5056eb07ef6c68975af": "g(\\cdot,y,z)", "5c2bd2823fee7bbc6a7e5dbd1340f53a": " n \\times m ", "5c2be2e6c59b805fe4e753ed856291d0": "|a - x_{n+1}| \\leq K |a - x_n|^3", "5c2c2cc930e9b8eeebc9a1cb914e9881": "\\deg^+(v) = \\deg^-(v)", "5c2c56e5aa53cbd6b1c15929327e12a0": "\\tan \\psi = \\frac {v'} {v' \\cos \\theta - f} \\sin \\theta \\,.", "5c2c65a2753ad80211e7ef97e1d6adbe": "\\frac{\\sin A}{p_K(a)} = \\frac{\\sin B}{p_K(b)} = \\frac{\\sin C}{p_K(c)} \\,.", "5c2c855f208a441902149165aee669e1": "s_n=1+2n\\pi i/\\log(2)", "5c2cbeccc8d790fa44a8a8b0044ff488": "x_0, x_1, x_2, \\ldots", "5c2cc86726256e553f8d3b937aeb9a71": "A \\cup B = \\{2,3,4,5,6, \\dots\\}", "5c2cce6ca638db4d00261cc584c98bc4": "\\exists \\kappa. \\mathit{false} \\Leftrightarrow \\mathit{false}", "5c2ce37ee59316117b2d2fa7cb1fbe65": "A = \\left(\\frac{n}{\\left| \\alpha \\right|}\\right)^n \\cdot \\exp\\left(- \\frac {\\left| \\alpha \\right|^2}{2}\\right)", "5c2cf1410dd61cd158b9597aecce514b": "2p_{i-n} > p_i \\text{ for } i>k \\text{ where } k=\\pi(p_k)=\\pi(R_n)\\, ,", "5c2d2851417b3f29847b1ce70ae90852": "p \\colon A \\times B \\to A,", "5c2d38ec7b3d15c3288f63e3354083bf": "r_t^2", "5c2d44adeb44bf2928bd8e25c7493829": "x_u", "5c2d46883572cd4c7e17c259cf3766a4": " \\sum_{i=1}^2 \\mathrm{Bernoulli}(p) \\sim \\mathrm{Binomial}(2,p).", "5c2da74f1fa8030d8aeb5f17da700f0c": "\\big\\{\\omega \\in \\Omega \\, | \\, \\lim_{n \\to \\infty}X_n(\\omega) = X(\\omega) \\big\\} = \\Omega.", "5c2e39c37c9d62e2d060e8611838fae8": "S_{CHSH} = 4", "5c2e72987e18d224a0501497ea6b5ca1": "\\scriptstyle PC\\ =\\ P'C", "5c2ec7096eb145bd1a76dd84f332a4c7": "\n\\frac{1}{\\left| \\mathbf{x}-\\mathbf{x}^\\prime \\right|} = \\frac{1}{\\sqrt{r^2+r^{\\prime 2}-2rr'\\cos\\gamma}} = \\sum_{\\ell=0}^{\\infty} \\frac{r^{\\prime \\ell}}{r^{\\ell+1}} P_{\\ell}(\\cos \\gamma)\n", "5c2efdfa642705e99ea245af891500c6": "M(a,c,z) = \\lim_{b\\rightarrow\\infty}{}_2F_1(a,b;c;z/b)", "5c2f0f2d5b808916f2ee9449cd053777": "\\mu_{XN} = \\mathbb{E} [x^N]", "5c2f19947bcaead9573b9e083c2dd00c": "G= \\sigma \\frac{A}{\\ell}.", "5c2f7d559c5999127a7690fad40b1cef": "\\lfloor n/2\\rfloor", "5c2fe4108b1e27fddd5184355a2cfb67": "Bu=g \\text{ on } \\partial \\Omega", "5c2ff2ac8ebf5a9c8e3d3d10329d117d": "N_f>N_c+1", "5c3040d55fe601aab069c43ebbc01bf5": "\\dfrac{\\beta : Y\\backslash Z \\qquad \\alpha : X\\backslash Y}{\\beta \\alpha : X\\backslash Z}B_<", "5c3051eec476dd9d934b5d37667c728b": " \\frac{\\part v}{\\part n} + a v =0, \\,", "5c30521c7a393daa9972b95c788f09c2": " X_{ni} = x ", "5c307767fcb4be5d0abc5ce1eb2fd004": "F_1(a, b) = a + b", "5c3089ba4e5d4ec6397f9d0701969199": "K_\\alpha(x) = \\frac{\\pi}{2} i^{\\alpha+1} H_\\alpha^{(1)}(ix),", "5c308d41ec8e98c06e0b1d04cf0e6ae1": "P(t_1t_2t_3)=P(t_1)P(t_2|t_1)P(t_3|t_1t_2)", "5c309631028467f2a352ef461bf5ccaa": "\\frac{\\rho}{\\rho+1}\\;{}_2F_1(1,1; \\rho+2; e^t)\\,e^t \\,", "5c30a2a20d52a9e9f2a44bf6be493a80": "\\displaystyle{[L(Q(a)b)+L(b)Q(a)]L(b)= L(a)[L(Q(b)a) + Q(b)L(a)].}", "5c30c45ca0103eda49466a70ed640ddb": "A \\rightarrow A / I", "5c30ee7f0e7c6a394c6858ced8527269": "\\theta_{i,1 \\dots V}", "5c316a0a961f9d9fb7486c7b906d7ae0": "f(P)", "5c317637f943df83507d88a37bd1ad5d": "I_{\\mathcal C}", "5c317808e70df8052cfb882794cfb605": "W(n,l)=n+l-\\left ( \\frac{l}{l+1} \\right )", "5c31942c9f6204c4637c855a3c2e3085": "x^x.", "5c31a2fad817e4698c5e6a27fc0bf191": "x/y", "5c31eb1223103b2bc3ebac0c5ea36373": "1 \\tfrac{1}{2} + 2 \\tfrac{3}{4} =", "5c323c7dc2fc82c8f9e8f44653484546": "1,2,\\cdots,n", "5c32f0fddad563570c3d4b1c98e7c4b5": "c^2>0\\,", "5c337e63db04050377d0fd592bf3a287": "\\textstyle\\frac{Mv^2}{R}=L\\sin\\theta=\\frac{1}{2}v^2\\rho C_L A\\sin\\theta.", "5c338d3eb67586a1a55ee5d457d9c22f": "\\rho \\wedge \\rho", "5c339601a8d035cb4d5d2389f3093020": " \\begin{align}\nX(t) &= X_0 \\sin \\omega t \\\\ Y(t) &= Y_0 \\sin\\left(\\omega t-\\phi\\right).\n\\end{align}", "5c33c577d339264a52d3bbd100828175": "3^\\frac{7}{13}", "5c33ddc0e51d4cefaf2426e2db6b8f1e": "- T(\\nabla_Y\\alpha_1, \\alpha_2, \\ldots, X_1, X_2, \\ldots) \n- T(\\alpha_1, \\nabla_Y\\alpha_2, \\ldots, X_1, X_2, \\ldots) -\\ldots ", "5c340402894f54f2539f863fb0661cfa": " t_k ", "5c3439f5f1a5fefddbe56f7bd90047c2": "\na_{\\mathrm{in}}(\\mathbf{k})=i\\int \\mathrm{d}^3x f^*_k(x)\\overleftrightarrow\\partial_0\\varphi_{\\mathrm{in}}(x)\n", "5c3482524146ce5497db607f7173dda2": "\n\\vec y = \\vec f (\\vec x)\n", "5c349bb9cbca4107a69a11571c91c3c4": " c = \\frac{1}{V_m}", "5c3508e631c31dc949db7f1a9178309f": "f^{-1}\\,", "5c352803523de96903fc947f40292d94": "Q = v \\cdot \\rho \\cdot c_p \\cdot \\Delta T", "5c357a927588e123f75e2194959f89ac": " X_0,\\,0 \\le X_0 < m", "5c35a1ddb6e99206462423fbba6ea0b6": "\\mathbf{R}^+ \\to \\mathbf{R}^+ : x \\mapsto x^2", "5c35bbfd07694b8e0e543e54a3bdcc39": "RD_{\\mathrm{new/ref}} = \\frac{\\rho_\\mathrm{new}}{\\rho_\\mathrm{ref}} = \\frac{V}{V - A \\Delta x}", "5c35e7fa55dd9838840c9c465977da53": "\n \\begin{pmatrix}X \\\\ Y\\end{pmatrix} \\ \\sim\\ \n \\mathcal{N}\\Big( \\begin{bmatrix}\n \\operatorname{Re}\\,\\mu \\\\\n \\operatorname{Im}\\,\\mu\n \\end{bmatrix},\\ \n \\tfrac{1}{2}\\begin{bmatrix}\n \\operatorname{Re}\\,\\Gamma & \\operatorname{Im}\\,\\Gamma \\\\\n \\operatorname{Im}\\,\\Gamma & \\operatorname{Re}\\,\\Gamma\n \\end{bmatrix}\\Big)\n ", "5c36a1ae25789de17a321956b8169e04": " (q,\\omega_1\\omega_2,q') \\in \\Delta ", "5c36d382789513e996937fdf5cde63e4": "V=2\\frac{|C_A\\cdot C_B^*|}{|C_A|^2+|C_B|^2}. ", "5c36d3b9268acf019cd665d32f9306c7": "1/{\\sqrt{2\\pi}} ", "5c36f74f79b28d9a16edca07a096b0d7": " E_\\text{k} = 0 \\!", "5c3717083f6da4b83c64fbb1b0383fcd": "\\partial_b A_a = \\nabla_b A_a + \\Gamma^c_{ba} A_c", "5c374f21c48f87c8c9cdf0403a676348": "2 l^2 - l", "5c3756ec942ea921a36c120e4163fb0c": "\\epsilon_j^n", "5c3787c8dc500e331acbbcab722b6f14": "r_1 = l_2", "5c37c2bac8d940c00e6755a688a7b42d": "h(p, \\bar{u}) = \\arg \\min_x \\sum_i p_i x_i ", "5c37f4b36df87f95719eaffc7b00b03b": "g_{ij}[\\mathbf{f}'] = g\\left(\\frac{\\partial}{\\partial y^i},\\frac{\\partial}{\\partial y^j}\\right).", "5c383b9b06aa9bef56c22a756ade35d7": "\n\\varphi_{0} = \\frac{\\partial S}{\\partial L} = \\frac{\\partial S_{\\varphi}}{\\partial L} + \\frac{\\partial S_{r}}{\\partial L} = \\varphi - \\frac{L}{\\sqrt{2m}} \\int^{r} \\frac{dr}{r^{2} \\sqrt{E_{\\mathrm{tot}} - U(r) - \\frac{L^{2}}{2m r^{2}}}}\n", "5c393d574816a94063f41a8ee1e584f4": " \\eta =\\frac{AMA}{IMA}.", "5c3950fac4ad1138a3aca2685f090aac": "\\tau(a):=(a,T_{X,a})", "5c3958543a8b2891cb93346e0bd1dae5": "\\displaystyle x=x_0", "5c39600e0451ed3bc50c36cfb3ca163c": " r < a \\, ", "5c39a9ae31487686efb2b3dc049cb223": "\\partial f/\\partial t=0", "5c39ae3f9778e0dbc25b0c038da557e9": "\\mathbb{H} \\oplus \\mathbb{H}.", "5c39b28154d7cb38f06e842dcd295568": "{\\frac {\\cos \\left( \\alpha/2-\\beta/2 \\right) }{\\cos \\left( \\alpha/2+\\beta/2 \\right) }} =\n{\\frac {\\cot \\left( \\alpha/2 \\right) \\cot \\left( \\beta/2 \\right) +1}{\\cot \\left( \\alpha/2 \\right) \\cot \\left( \\beta/2 \\right) -1}} =\n{\\frac {\\cot \\left( \\alpha/2 \\right) +\\cot \\left( \\beta/2 \\right) +2\\,\\cot \\left( \\gamma/2 \\right) }{\\cot \\left( \\alpha/2 \\right) +\\cot \\left( \\beta/2 \\right) }} =\n{\\frac {4\\,s-a-b-2\\,c}{2\\,s-a-b}}", "5c39fc606761348e7330fb6d3be3ed80": "\\scriptstyle \\, r", "5c3a4408bcc73bb7fbd4d76a18e29401": "\\begin{cases}\\mathrm{OPT} \\leq f(x) \\leq \\rho \\mathrm{OPT},\\qquad\\mbox{if } \\rho > 1; \\\\ \\rho \\mathrm{OPT} \\leq f(x) \\leq \\mathrm{OPT},\\qquad\\mbox{if } \\rho < 1.\\end{cases}", "5c3a7ee29f1f5710053eb8154102e754": "\\mathbf{m}_2", "5c3aa49a3ecae3edcbccbc2cbb0560ff": "\\mathbb{Q}(\\sqrt{n})", "5c3ab34395b05733b0aeb3275bc61425": " \\min_x 0.5 \\|Ax-y\\|_2^2 ", "5c3ae5f73fda3036458fbee55862955c": "\\operatorname{dim} f^{-1}(s) \\le r", "5c3af99d39141a7048c83ad9d9a0f4ac": "\\frac {\\Delta V} {V} = \\left(1+\\frac{\\Delta L}{L} \\right)\\left(1-\\frac{\\Delta L'}{L} \\right)^2-1", "5c3bc7216a39c8f0899d0153ef6761d4": "Z_p", "5c3bd71661446286885e9a22b79cbcde": " \\lambda \\to +\\infty ", "5c3bdcfcac45032297cbfe9df411592f": "\\rho'' = {M_i \\rho' M_i^\\dagger \\over {\\rm tr}(M_i \\rho' M_i^\\dagger)} = {M_i M_i \\rho M_i^\\dagger M_i^\\dagger \\over {\\rm tr}(M_i M_i \\rho M_i^\\dagger M_i^\\dagger)} ", "5c3bf1f5088ed1fa912aad34fc5940af": "f_1(z)", "5c3bf7da816112c7f6cc9697b57d2146": " g_{1,2} ", "5c3c37b29b814631d8768c850fa385b4": "\\tau=x^0", "5c3c4bf4dc006a6bb3262f8cf836bb37": "A= \\exp i \\mathfrak{t}, \\,\\, P= \\exp i \\mathfrak{k}.", "5c3c5d7bb22167d6197257534e30c70a": "H=H_0+H_1", "5c3c953f256ff1e482c5b8a54c825212": "\\,L(M) = \\left\\{ x | \\{q_0,\\Upsilon_0,\\epsilon,x\\} \\rightarrow_M^* \\{q_\\textrm{F},\\Upsilon_m \\ldots \\Upsilon_1, x, \\epsilon\\} \\right\\}", "5c3cca40bc48e0cae323f2903212f9e9": "\n\\frac{\\partial(r, \\theta, h)}{\\partial(x, y, z)} =\n\\begin{pmatrix}\n\\frac{x}{\\sqrt{x^2+y^2}}&\\frac{y}{\\sqrt{x^2+y^2}}&0\\\\\n\\frac{-y}{\\sqrt{x^2+y^2}}&\\frac{x}{\\sqrt{x^2+y^2}}&0\\\\\n0&0&1\n\\end{pmatrix}\n", "5c3cd21d7c4e771642f60ecdfc812be1": "abs(\\lambda) = 0 \\,", "5c3ce3e23fb9b017a8d1086ed7c1b708": "P_{n,j}", "5c3d05cb92a63eca67c9690789aad25d": "F_m^0", "5c3d0e8651bd2c1012d1e4029d01ef9c": "\\mathbf{R} = d \\mathcal{M}\\mathbf{R} + \\frac{1-d}{N} \\mathbf{1}", "5c3d6321c3d663ffbba58682e95ecac8": "A_N = A", "5c3d769388b2f9ae6528075eb9f09561": "T_1\\cup T_2", "5c3d8b34b0bdb028639e384dcb10f092": "\n\\begin{align}\n\\mathcal{N}(x\\mid \\mu,\\sigma^2) & = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{\\frac{-(x-\\mu)^2}{2\\sigma^2}} \\\\\n\\operatorname{Gamma}(\\tau\\mid a,b) & = \\frac{1}{\\Gamma(a)} b^a \\tau^{a-1} e^{-b \\tau}\n\\end{align}\n", "5c3e0227e03a9f75e135890c0515db08": "\\langle F_\\mathrm{DEP} \\rangle = 2\\pi r^3\\varepsilon_m \\textrm{Re}\\left\\{\\frac{\\varepsilon^*_p - \\varepsilon^*_m}{\\varepsilon^*_p + 2\\varepsilon^*_m}\\right\\}\\nabla \\left|\\vec{E}_{rms}\\right|^2", "5c3e09e4ec7b7b234699b5a088c85429": "I_S", "5c3e233647dd3f5b192702eb85a67d47": "\\sqrt{{x_2}^2 + {y_2}^2} = 1", "5c3e5ee5dfe98c4eb7f3526cdd82f521": " j \\colon U \\to X ", "5c3f261225101ecf85ad64711cf9f978": " K'_{ij}=rol(K_{il} \\oplus S_i,u)", "5c3f361ec5559847838cabddd553d61b": "r^2 \\sin\\theta \\, dr \\, d\\theta \\, d\\phi", "5c3ffe864b243ed9c5e4de6c99c2e311": "Ai\\left(\\cdot\\right)", "5c4012c049865ef92ad449311e357726": " r_2^2 = (x-a e)^2 + y^2 = x^2 - 2 x a e + a^2 e^2 + (x^2-a^2)(e^2-1)=\n(e x - a)^2", "5c401f5b0fb16e85c17aea177d482887": "\\operatorname{E} (X) = \\sum_{i=1}^{n}{\\operatorname{E}(X \\mid A_i) \\operatorname{P}(A_i)}.", "5c403f5531de0dd0098c4760db865442": "\\sin (\\arccsc x) = \\frac{1}{x}", "5c40b173713418e5ad5ab919cc016444": "\n\\frac{\\partial \\sigma}{\\partial \\mathbf{x}} \\overbrace{\\left( f(\\mathbf{x},t) + B(\\mathbf{x},t) \\mathbf{u} \\right)}^{\\dot{\\mathbf{x}}} = \\mathbf{0}", "5c40b9cfeab969b5624260c9904a9cf4": "L_{y}=i\\hbar\\left(-\\cos\\phi\\frac{\\partial}{\\partial\\theta}+\\cot\\theta\\sin\\phi\\frac{\\partial}{\\partial\\phi}\\right), ", "5c40c8fcacf241e5bfa12456f026f9e9": "\\nabla\\times \\mathbf{H} = \\mathbf{J}_{\\text{f}} + \\frac{\\partial \\mathbf{D}}{\\partial t}", "5c41189f36f385c563b62b8dd0d8803f": "n_i'=c_i+n_i-2c_{i+1}", "5c4138a1b8bcc581f2a39c5ec7cccd0d": "1+\\varepsilon+\\frac{1}{2}\\varepsilon^2+\\cdots+\\frac{1}{n!}\\varepsilon^n+\\cdots", "5c414671d59a887d65f048ed9d8f4dc3": "\\frac{}{}\\Delta \\delta", "5c41526eaa7aeb780625e782b19b042c": "f_{1}(\\Delta U) = f_{0}(\\Delta U) + \\beta\\Delta F", "5c4164ce236a3e0bf9db50029fc35490": " 6=2\\cdot3=(1+\\sqrt{-5})\\cdot(1-\\sqrt{-5}).", "5c4187d42a20c3238959be5d5f7c1a38": "\\tanh(x/3)", "5c42292c7fe2e99ba5cfcbbc23bd173b": "T \\in \\mathcal{D}_m(M)", "5c422996f616b43572feb433cd0502d1": " e^{2i\\pi/n} ", "5c423a3a061f8148dd78c8374a940d41": " \\frac{\\partial}{\\partial z} = \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x} - i \\frac{\\partial}{\\partial y} \\right) \\quad,\\quad \\frac{\\partial}{\\partial\\bar{z}}= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x} + i \\frac{\\partial}{\\partial y} \\right).", "5c42750ab7403901ed5e790800713041": "a + b \\Omega", "5c42841989bc48c241f6528d63c78877": "\\displaystyle{\\int_{\\partial\\Omega} f\\,\\,=\\,\\,\\int_{|z|=R}\\partial_n u.}", "5c42f14d67103d5114a6ed232c7f31e9": "f_k e^{-x} = x^{k}[(x-1)(x-2)\\cdots(x-n)]^{k+1}e^{-x} = \\left ([x(x-1)\\cdots(x-n)]^k \\right ) \\left ((x-1)\\cdots(x-n)e^{-x}\\right )", "5c42f6a147104f51ca60354abd986ca9": "x^2y^2-b^2x^2-a^2y^2=0 \\,", "5c439a310c8a01ce788b10a3a1e8dc3a": "K=\\, -c^2", "5c43ada5196cafeb4f0f0e0c6c7edb05": "8:5^{5 \\over 4}", "5c444a89aab3e90f778efd47e731e88f": "\\sum^n_{k=1} (n-k+1)(k|\\gamma_k|^2-1/k)\\le 0", "5c449eb1706efd7d86a3dab7131d9c14": "\\scriptstyle bx-x^2", "5c44ca146ee81f5f8ca6a10263014de4": "f=f_{t}\\sqrt{1-(\\frac{f_{L}}{f_{t}})^{2}}", "5c44d11b9793293eff22f90339e15442": "\\mathbb{N} \\subset \\mathbb{Z} \\subset \\mathbb{Q} \\subset \\mathbb{R} \\subset \\mathbb{C}", "5c451c7c08cef8300aea1933a124cab8": "U\\le_{RK}V", "5c45fb44dc205b03a8ddfd049162fa84": "Y=C+I+G+NX", "5c4651d8a931f94a5c6dff20421b5ac5": "\\rho(X)f", "5c469f14123a8319c8b417d8c4ab2125": "CBADE", "5c46b8bc3c4af6d4177081540ee45b3c": "f(x_0) \\ge f(x)\\, \\forall x", "5c46e20f4a85e04357362f7077611bea": "0 \\tau_2(j) ) \\vee ( \\tau_1(i) > \\tau_1(j) \\wedge \\tau_2(i) < \\tau_2(j) )\\}|.", "5c543c92703a6a3386826db8e2a36ba2": "E_0, E_1, \\cdots \\in \\mathfrak{E}", "5c545266c3e2b3b79ddc13d3ef7a5270": "{\\rm Tr}", "5c54944b9be11151ba7e19439162f994": "\n\\psi(k)\\psi(k') - \\psi(k')\\psi(k) =0\n\\,", "5c549c0b959f428e70ad179ef303dd9e": "P_J[f](x)", "5c549e5afb0c2e578a7379057d34f335": " {X_{i+1}} ", "5c54c1ea35dbf2b40291716a897c2d7e": " \\mathbf{T}^e ", "5c55116d75ffeefb506bebee5ec23385": "f(\\theta) = 0", "5c5526e3769aa773baa1125d6248eca6": "\\overline{X} = \\{X_1,X_2,\\dots\\}", "5c5567c873a6b6bc85ff75a5d216624d": "\\zeta(z) = \\prod_{n=1}^{\\infty} \\frac{1}{1 - p_n^{-z}}", "5c559cabbab05bf4cfb77d81b65289ca": "\n\\eta^2 = \\frac{\\omega_R}{\\omega_z} = \\frac{\\mathrm{change\\, in\\, kinetic\\, energy}}{\\mathrm{quantized\\, energy\\, spacing\\, of\\, HO}}.\n", "5c55fbe65f0419b928b21454f1da33f0": "0 \\rightarrow \\pi^*E_0 \\stackrel{\\sigma(P_1)}{\\longrightarrow} \\pi^*E_1 \\stackrel{\\sigma(P_2)}{\\longrightarrow} \\ldots \\stackrel{\\sigma(P_k)}{\\longrightarrow} \\pi^*E_k \\rightarrow 0", "5c55fd6721d5e63f746b6866873cd158": " \\tan \\alpha = \\frac {\\sin L}{(1 - e^2) \\tan \\phi_2}", "5c5616195ea9c3288f2a143a8604d57b": "\\alpha \\omega", "5c562a8e3a73f5dd597b5562942f115b": " \\tau = \\mathbf{r} \\times F", "5c567f254702da4a0c05cf07b3dc450f": " \\text{LSF} \\approx \\frac{\\text{ESF}_{i+1} - \\text{ESF}_{i-1}}{2(x_{i+1} - x_i)}", "5c56f52ccf87cce53e974969932e739c": "\\widehat{\\neg \\alpha}", "5c574021ae565bb6235baee0931687dc": "|A \\cap \\{1,\\ldots,n\\}| \\geq n^\\alpha", "5c57af15dedd80346042a503ed639cf7": "|R| < \\int_a^b \\varepsilon\\; \\mathrm{d}x = \\varepsilon(b-a).", "5c57b0fb2a502d48d21fac6011a5055c": "P_{t+1} - P_t", "5c57d2b9275b8ef9959ef7ace1adb3de": "\n\\begin{align}\n6 & = 2^1(2^2-1) & & = 1+2+3, \\\\[8pt]\n28 & = 2^2(2^3-1) & & = 1+2+3+4+5+6+7 = 1^3+3^3, \\\\[8pt]\n496 & = 2^4(2^5-1) & & = 1+2+3+\\cdots+29+30+31 \\\\\n& & & = 1^3+3^3+5^3+7^3, \\\\[8pt]\n8128 & = 2^6(2^7-1) & & = 1+2+3+\\cdots+125+126+127 \\\\\n& & & = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3, \\\\[8pt]\n33550336 & = 2^{12}(2^{13}-1) & & = 1+2+3+\\cdots+8189+8190+8191 \\\\\n& & & = 1^3+3^3+5^3+\\cdots+123^3+125^3+127^3.\n\\end{align}\n", "5c585362111889aca2095234ec47a759": "\\tilde H_i(G;\\mathbf{Z})=0", "5c585f45d16ad4ac4e2b0ffc3f985eae": "\\mathbf{I_w}", "5c586b2101ae4eba55e4415f587d7f0b": "\\varepsilon_{m} = \\varepsilon_{D} + \\varepsilon_{s_2}", "5c588b4eebf136e3ba53d23ee3104b56": "\\scriptstyle \\Lambda", "5c58ce32552f2c310eceb1c731c5a9e7": "A \\cap A\\,\\!", "5c594f492afffd602f26a8189fecdb0a": "\\psi(x,t) = \\sqrt{\\frac{\\mu - V(x)}{NU_0}}", "5c597da70f1f114116d46a0117c5ea95": "M(y) > N(y)", "5c5a305212a68e67de7ae6426975e471": "\\,\\deg(G)=\\max[2\\deg(u),2g+1+2\\deg(v)]", "5c5a3c166fdfa53856b31042c85fe49c": "t_0=-2\\sqrt{\\frac{p}{3}}\\sinh\\left(\\frac{1}{3}\\operatorname{arsinh}\\left(\\frac{3q}{2p}\\sqrt{\\frac{3}{p}}\\right)\\right) \\quad \\text{if } \\quad p>0\\,.", "5c5a584f78a9df65ecdcc5cc8baaf8d4": "P(\\alpha)=\\frac{1}{\\pi}\\rho_A(\\alpha,\\alpha^*).", "5c5a6de6b6a5995691b9652d1ca04159": "O(\\frac {nd}{\\epsilon^2} )", "5c5abee6ae6776d2c596a8790f2be31e": "\\frac{\\partial |\\mathbf{X}^n|}{\\partial \\mathbf{X}} =", "5c5ad7984f7d56c8987e33fc39f5d377": "P_0(1+i)^{n} = \\sum_{k=1}^n x(1+i)^{n-k}=\\frac{x[(1+i)^n - 1]}{i}", "5c5af9ef64f04ac7c751370cb0b717a1": "(p, R, Q)", "5c5afe41da9b4223447f2257873c995a": "(3)\\,", "5c5b240b4471858b9fd75b761df9d951": "H(z) = H_0 \\sqrt{ \\sum_i \\Omega_i (1+z)^{3(1-w_i)} +\n\\Omega_k (1+z)^2}", "5c5b98805bbb060e4c7cf3c2f99db69c": "\\vec J_T = -\\frac{NDQ}{kT^2}\\nabla T", "5c5ba045bbee3ef381212c0b1122e233": "\\displaystyle m=1", "5c5be6710373a3516924cb9f00454254": "\\operatorname{Der}_k(F, F)", "5c5bffc23b85ea026845874595a7c130": "x_1 = x_2", "5c5c15274652bc9ed4d6f41233151c0a": "F(\\psi)", "5c5c5f5a7ad4ff7686fa8bc40dc3e207": "h\\in H\\,\\!", "5c5c96a67be572ef97668de6cd392214": "p(x|\\nu,\\mu,\\sigma) = \\frac{\\Gamma(\\frac{\\nu + 1}{2})}{\\Gamma(\\frac{\\nu}{2})\\sqrt{\\pi\\nu}\\sigma} \\left(1+\\frac{1}{\\nu}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2\\right)^{-\\frac{\\nu+1}{2}} ", "5c5ca72dc2cacdf240bf25622dc47f76": "H^{-1}", "5c5cac24be69f9d3063ef9557793b011": "4x(x+5)+3y(x+5)\\,", "5c5cc8dc717c9da354df51f322c158e7": "\\sigma(z) = z\\prod_{\\omega \\in \\Lambda_{*}} \\left(1-\\frac{z}{\\omega}\\right)e^{\\frac{z^2}{2\\omega^2}+\\frac{z}{\\omega}}", "5c5cd017c5a947c9a7eb996b4569db90": "\\exists x. A", "5c5ce84c744af8aed194dc24d9d154b1": "q_s + \\lVert q\\rVert\\cdot\\mathbf{U}q", "5c5d2fd8be02e42fc8633ebf73084675": "\\begin{align}\n \\mbox{Area}(B)\n &= \\iint_B \\sqrt{\\det g}\\; du_1\\; du_2 \\\\\n &= \\iint_B \\sqrt{\\det g} \\;|\\det F| \\;dv_1 \\;dv_2 \\\\\n &= \\iint_B \\sqrt{\\det \\tilde{g}} \\;dv_1 \\;dv_2.\n\\end{align}", "5c5d935765c45d9e2bb94a469fe71e29": "\\sum_{y \\in \\text{Ball}}", "5c5db02c59c3a65a28a43fc600f14a27": "V'_a", "5c5df162d8a909f58a48bd72dec2488f": "\\sum_j a_{ij}x_j=0", "5c5e3f3f2634abacd4e7b77655a6a0c5": "a_2(t) = \\frac{p(t)}{p(0)} \\ ", "5c5ea32cb3d544eb919a04b32961bb20": "f_*\\left(\\sum_i a_i\\sigma_i\\right)=\\sum_i a_i (f\\circ \\sigma_i)", "5c5eaca2548b6dab09146e818900b110": "\n\\begin{array}{rcl}\ne(A,t) & = & \\cos( 2 \\pi F_c t ) ( 1 + c(t) + g(A,t) ) \\\\\n c(t) & = & M_i \\cos ( 2 \\pi F_i t ) ~ i(t) \\\\\n & + & M_a ~ a(t) \\\\\n & + & M_d \\cos ( 2 \\pi \\int_0^t ( F_s + F_d \\cos ( 2 \\pi F_n t ) ) dt ) \\\\\ng(A,t) & = & M_n \\cos ( 2 \\pi F_n t - A ) \\\\\n\\end{array}\n", "5c5eb95dd034bc7cd4abb2705b7c4b43": "\\frac{ index_{\\rm_{new}}}{ index_{\\rm_{old}}} = \\frac{ \\sum {\\# shares} \\cdot price_{\\rm_{new}}}{ \\sum {\\# shares} \\cdot price_{\\rm{old}}}", "5c5ee2574500d209bf96a47c9b452885": "\\sum_\\rho\\frac{x^\\rho}{\\rho} = \\lim_{T \\rightarrow \\infty} S(x,T) \\ ", "5c5ef59da0ef85c4df00f3e1223724e7": "\\binom{k}{e}=\\binom{k}{n}=\\frac{k!}{e! n!}", "5c5fda6c954f02ef102e7ca710dde974": "(i_1,\\cdots,i_k)", "5c5ff85dcbe0f15a5f5422149c59c3a9": "\\tau := (T-T_c)/T_c", "5c5ff97b15b554d01fb53719f22de446": "B(\\mathbf{u}, \\mathbf{u}) \\ge c \\|\\mathbf{u}\\|^2.", "5c5ffb63ab373e34336bed4433fb1fda": " \\qquad x_{n+1} = r x_n (1-x_n) ", "5c605c59d14ddc1d2b90bb998014bc53": "-A\\cdot \\operatorname{adj}(-A)=\\operatorname{adj}(-A)\\cdot-A=\\det(-A)I_n=c_0I_n.", "5c607053267a8fb337364b1c49492e8f": "S(\\rho^{12})+S(\\rho^{23})-S(\\rho^{123})-S(\\rho^2) \\geq 2\\max\\{S(\\rho^1)-S(\\rho^{12}),S(\\rho^2)-S(\\rho^{12}), 0 \\} ", "5c607be497adcd68fa7eded3bfe642b8": "\\Box\\Diamond A\\to\\Diamond\\Box A", "5c6131d5c6c96b89b93fa5743f5caa86": "s \\geq 0", "5c61784eac825932385bd7eae8f57c7d": "Y=|X|", "5c6180db2bb0bba4eb74f2055de86de2": "\\mathcal{D}_{L^p}'", "5c61de739ac24baa3faf0b059b6406a2": " \\exists X_1 \\cdots \\exists X_k \\, F(X_1,\\dots, X_k), \\, ", "5c62362d9a1d9f63c688dc65ec43c936": "\\nabla^2 U = 0,", "5c625bf7d29711f3fc7b1b4c48ab8a36": "\\sqrt 2 \\cosh u,\\,", "5c6278b95d5aad850d860e523d402373": "\\nabla \\cdot \\mathbf{u} = 0,", "5c629b841bef2c75bde3778799dc6837": "P(\\alpha_i) \\ne y_i \\le e", "5c62fbc2afe6c7b3f941823512ca910d": "S_6", "5c63288a602d944608615ab51f9f3e9e": "X \\sim N\\left(\\nu\\cos\\theta,\\sigma^2\\right)", "5c632b1425f5f527e2524f131e0ac30b": "m \\frac{\\operatorname{d}v}{\\operatorname{d}t}= F ", "5c63dcbebf1aacc7a031eadfba75ac01": "\\mathbb{V}\\times\\mathbb{Z}_2", "5c6412f2eaaf33e91a9af85e7b40bd91": "\\scriptstyle g \\,\\circ\\, f", "5c6483192e2bb6eb6ce34fcf9d0cff87": "\\epsilon^{\\text{s}}(p,t) = \\mu^{\\text{s}} E^{\\text{s}}(p,t),", "5c64a612585cf165ba5c3da5bf8f735f": " \\operatorname{lift-choice}[\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda y.f\\ (y\\ y))] ", "5c64bd214e4684e28b5b034f9196bf0d": "d(u,v) = \\epsilon(u)", "5c64eb29907f785c03bfb6eea9d91777": "u : M \\to N", "5c651802e9a1e9783edd80697e210e49": "\\nabla^4\\varphi=0", "5c6524c1aa806e30b9234cd181c6028c": "\\Theta(n\\log(n))", "5c6593023a9b6ac9ca53e2db4390d035": " (0.00, 0.25)", "5c65998380d068293b3f6463b73c0f0a": "\\omega_{peak} = \\omega_0\\sqrt{1 - 2\\zeta^2}", "5c65beb396388181f0ccc29896ece1c7": "p:\\mathcal{F} \\to \\mathcal{F}", "5c66651d46d25251c15bdf17c6acaa02": "Y_{8}^{5}(\\theta,\\varphi)={-3\\over 64}\\sqrt{17017\\over 2\\pi}\\cdot e^{5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(5\\cos^{3}\\theta-\\cos\\theta)", "5c67c99c3932a492fa624c16d793a6bb": " r", "5c6811cea24fc764172dcab26264af28": "y_k = \\mathbf{h}_k^T \\sum_{i=1}^K s_i P_i \\mathbf{w}_i+n_k = \\mathbf{h}_k^T s_k P_k \\mathbf{w}_k +n_k, \\quad k=1,2, \\ldots, K", "5c683647e39ea4fc47397c38b31ed71b": "g(x, y; t) = \\frac {1}{2{\\pi} t}e^{-(x^2+y^2)/2t}\\, ", "5c688c2e4ba55b3409d1f89f5e6cb4a1": "\\forall \\alpha\\in\\mathbb{R} \\ : \\ \\Pr\\left[\\frac1n\\log\\frac{j(n,X)}{a(n,X)}\\geq \\frac{1}{n}\\log\\alpha \\right]\\leq \\frac{1}{\\alpha}.", "5c68b5ffd935d54c945f0be5694d15f0": "\\mathbf{t}_a = \\frac{\\mathbf{v} - \\mathbf{u} (\\mathbf{u} \\cdot \\mathbf{v})}{\\left| \\mathbf{v} - \\mathbf{u} (\\mathbf{u} \\cdot \\mathbf{v}) \\right|} = \\frac{\\mathbf{v} - \\mathbf{u} \\cos(a)}{\\sin(a)}", "5c68e0b80f1b7ce6a959e9de9a548232": " \\int_{C_{0}} F(p) \\, \\mathrm{d} \\gamma (p) = \\mathbb{E} [F]", "5c68ea6efca2e77a62421c50a7616814": " E =\n - {1\\over 2} {a_1 a_2 \\over 4 \\pi r } \n\\vec v_1 \\cdot \\left[ 1 + {\\hat r} {\\hat r}\\right]\\cdot \\vec v_2\n", "5c68ed5d697616777fa67a8db377cc84": "f(x) = a_nx^n + \\cdots + a_1x + a_0 \\in K[x]", "5c691c55e748871119fe0b5b21c10332": "S(\\wedge)", "5c691df1873fce4722f02321f3bad140": "\\hat{\\psi}", "5c692057527419f99789240605f8f716": "E[F|x^{(t)}]", "5c69b681d095e8ad7e8a95d08dc8bf48": "c_q(q) = \\phi(q).\\;\n", "5c69d0a4a7402ad5f1cd4a3d782049af": "U_\\tau(\\lambda,a,b)", "5c69d5d99746aa32a3380f96a7d477b0": "\\begin{bmatrix} \\alpha-\\frac12 \\\\ -\\beta-\\dfrac{\\lambda\\mu^2}{2} \\\\ \\lambda\\mu \\\\ -\\dfrac{\\lambda}{2}\\end{bmatrix} ", "5c6a1ea44496869b972a70c3a1da101c": "{\\mathit{He}}_2(x)=x^2-1\\,", "5c6a415a494f10234668f178cec3cbfe": "y_{i,m} = \\mathbf{x}_i^{\\rm T}\\boldsymbol\\beta_{m} + \\epsilon_{i,m}", "5c6abcb22bbd95b1e5549c8b81963d0b": "\\varepsilon \\div a = \\varepsilon", "5c6aca8639b1eeb49d8e4e79fa295544": "\\frac{V_\\mathrm{out}}{V_2 - V_1} = \\left (1 + {2 R_1 \\over R_\\mathrm{gain}} \\right ) {R_3 \\over R_2}", "5c6b07e1581741f5a9685ba15f22b7a9": " \\frac{Z_{eff}}{r} ", "5c6b4289400ec36e4d63b37be67f7d81": " A \\mathbf{p}_{k-1} = \\frac{1}{\\alpha_{k-1}} (\\mathbf{r}_{k-1} - \\mathbf{r}_k)", "5c6bd94bcaf78f5d359f61ed281073af": "\\displaystyle{a\\circ b = L_y(a)b,}", "5c6bdb3dd7a8e92d574201d32e620620": "e^{\\sqrt{\\log x}\\log\\log x}.\\,", "5c6bdda0c7b763ded10a9d582850d23a": " A \\cong \\bigoplus_{i \\in I } K(H_i),", "5c6bf3f6c6e225666f55311b861ca137": "(7) \\ n_\\varphi = \\frac{n_0 n_e}{\\sqrt{n_0^2 \\cos \\varphi + n_e^2 \\sin^2 \\varphi}}", "5c6c1c701485a37e14616ca667c3968a": " f(x)=e^{f(x-1)} \\; \\; (x>-1)", "5c6c2041b4514fb47001c3aa750ae382": "ML+L'\\rightleftharpoons ML'+L", "5c6ca67726fd54055f8f09ac1c04d145": " F,f: R^{n_x} \\times R^{n_y} \\to R", "5c6d536a6996acc7172813f7c43f7f34": "\n=\n8\\delta_{ll'}\n\\delta_{mm'}\n(n-l)!\n(n'-l)!\n\\frac{(2\\zeta)^n}{\\zeta^l}\n\\frac{(2\\zeta')^{n'}}{\\zeta'^l}\n\\int_0^\\infty\ndk k^{2l}\n\\sum_{s=0}^{\\lfloor (n-l)/2\\rfloor}\n\\frac{\\omega_s^{nl}}{(k^2+\\zeta^2)^{n+1-s}}\n\\sum_{s'=0}^{\\lfloor (n'-l)/2\\rfloor}\n\\frac{\\omega_{s'}^{n'l'}}{(k^2+\\zeta'^2)^{n'+1-s'}}\n", "5c6da7a08892b1a037231bd3435a9ecd": "J(t)", "5c6e0498344b96c2675961266c11bfae": "\\int_{0}^{x}\\phi(t)\\psi(x-t)\\,dt=0", "5c6e12fed07b994a5d5bae5503076e53": " \\dot x = g(x,t) ", "5c6e39325789ca480ce6e72b6722e61d": "q = q_1 q_2 \\cdots q_t", "5c6ec5c01ba4a92d685a142a1109853a": "\n\\mathbf{x} = x^{\\mu}(\\tau) = \n\\begin{bmatrix}\nx^0(\\tau)\\\\ x^1(\\tau) \\\\ x^2(\\tau) \\\\ x^3(\\tau) \\\\\n\\end{bmatrix}\n\n= \\begin{bmatrix}\nct \\\\ x^1(t) \\\\ x^2(t) \\\\ x^3(t) \\\\\n\\end{bmatrix}\n\n", "5c6ec915b39db29abe839942e19aaeb9": "\n _{(x)}\\Gamma^m_{ij} = \\frac{\\partial x^m}{\\partial X^\\nu}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\nu_{\\alpha\\beta} + \n \\frac{\\partial x^m}{\\partial X^\\alpha}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j} \n", "5c6edd3b02b9294987eaebdbe00f21e8": "v^{ij}", "5c6f02c6427c491a8a2c6fae09355b74": "P_{D,max} = \\frac{T_{D,max} - T_{A}}{\\theta_{DA}}", "5c6f3280a6c4f0dd5a45c05fd3d4b46d": "\\int_0^\\pi F_{\\nu+2p} (w) F_{\\nu+2s}(-w) \\, dw = 0,\\qquad p \\ne s,", "5c6f50f6dda6200941f4d3fb989f31b2": " \nN = \\begin{bmatrix} \n 5 & -3 & 2 \\\\\n 15 & -9 & 6 \\\\\n 10 & -6 & 4\n \\end{bmatrix}\n", "5c6f58e2a7494a64ef95a29bc9ec0406": "\\begin{matrix} \\frac{2}{3} \\times 2 = 1\\frac{1}{3} \\ge 1 \\end{matrix}", "5c6f937eacd3732196734c56ec527fa4": "k = 1", "5c6feb677e96b39fda2f577a31399d9f": "Initiates(a,f,t)", "5c704d18fc14beedb1747382e369a07c": " K_\\text{g}^\\prime = {2b \\sigma_\\text{l} \\sigma_\\text{f} T_m^0 \\over k \\Delta h}", "5c705c8aa989eff3b5c6023b7c3a8955": "\\frac{R^2}{2}(\\theta -sin{\\theta})", "5c7076dc44bec78ede8950625960ad97": "x^t\\,", "5c70d0078ff008b874c7b30a9a6e3663": "\\int_0^1 {\\rm Riesz}(z) z^s \\frac{dz}{z}", "5c70d1affcad8212c60bd2c5a3255710": "O(\\Delta^{\\lfloor k/2\\rfloor})", "5c70d2621b09873d7a8253337e0f80af": "\\chi_G(k)", "5c70d8ac5ee56b7d36470c823dd26b8d": "a=\\sum_{i=0}^kF_{c_i}\\;(c_i\\ge2)", "5c70fbfa07443af6d5b562098b82f2bb": "\n\\cos z\\,\n", "5c71295a5180dd07f12ce2f2c42595ea": "|W\\rangle = \\frac{1}{\\sqrt{N}}(|100...0\\rangle + |010...0\\rangle + ... + |00...01\\rangle)", "5c71565a49e9f6d169b877311f0e6c84": "\\delta: Q \\setminus F \\times \\Gamma \\rightarrow Q \\times \\Gamma \\times \\{L,R\\}", "5c715c60ac50e2d96e5876130d7f7849": "p \\mid 1", "5c71f85e02b930d92a299c84a8e188f5": "=\\sum_{i=0}^{n}\\lambda^{n-i}\\mathbf{x}(i)\\mathbf{x}^{T}(i)", "5c71fdb3a3c018b3e68ddf7953975cd9": "h_{00}=2\\alpha U\\,", "5c7227f4cf9758825ed38c7ef52d6226": "t_P", "5c72fe85bb3db56d702ddfadd708a290": "\\psi(\\Omega_2)", "5c733a0a8ee5e19d05433a981eb0838b": "\\begin{align} \\\\\nR_Y(\\theta) =\n\\begin{bmatrix}\n\\cos \\theta & 0 & -\\sin \\theta \\\\\n0 & 1 & 0 \\\\\n\\sin \\theta & 0 & \\cos \\theta\n\\end{bmatrix}\n\\end{align}\n", "5c73cf7ea1d66f9c2acba1826ec2a0b7": "N_{\\text{var}}", "5c73df593ef2baf3d1c4bf9258b48f56": "F:=\\frac{\\frac{SS_\\text{between}}{\\sigma^2}/df_\\text{between}}{\\frac{SS_\\text{within}}{\\sigma^2}/df_\\text{within}}", "5c741f40e75f341a36afe5d849ac137a": "\\sum_{n=a}^{b} \\Delta f(n) = f(b+1)-f(a)", "5c749c0f2d58e412b49d85d81ea1c16a": "G(\\mathbf{x}, \\mathbf{x'})", "5c74a0f3a0a54b0c5ed859fc57a2c3db": "6-d(v)", "5c74a2ea5605fa01539290973217f5ca": "dS=\\frac{1}{T}dU+\\frac{p}{T}dV-\\sum_{i=1}^s\\frac{\\mu_i}{T}dN_i", "5c74a391d504d9a57959be7b99db8acd": " x^3 + y^3 + 1= Dxy ", "5c74dc01570b8ccafdc231b024208df8": "x_1, x_2,\\ldots,x_n", "5c74ec4202d495baef427ee7ed28f415": "\\Delta U = 0\\,\\!", "5c7509da3c73b33e0de2d86436370f1b": "y_1=0", "5c757540f436f2008917e99f55205c1e": "O_n+O_{n-1}=\\frac{(2n+1)(2n^2+2n+3)}{3}.", "5c7577c8e61f0da0e163285a6abc2629": "h_n", "5c75fe746a9a7e25281839dabc966612": " \\exists |\\phi \\rangle \\in \\mathbb{C}^d ", "5c7639e9ce44453d047a1ea4df84a36f": "{4 \\choose 1}^5 = 1,024\\,", "5c7662e42bd88d7e585b20122f512f47": "p_{ij}^{(n)} = \\Pr(X_{k+n}=j \\mid X_{k}=i) \\,", "5c769a8894e74b446275adfef64b9307": "b= (0.11852 - 0.05478 * \\ln(PD))^2", "5c76f651eb9949c6b0188482ae8de6e4": "\\forall\\mathbf{\\rho,\\rho'}\\in\\mathit{S}", "5c773b0d4ac112ffe8c29ce251a76f62": "f_{X}(x) = \\sum_{i=1}^{n} \\alpha_i f_{Y_i}(x)", "5c77450ff387495759f2f767477457e2": "215 = (3!)^3 - 1", "5c77aaf36c3d24e829a0917469d396ff": "[K_0]\\sum_{j=1}^N \\epsilon_{ij} \\mathbf{x}_{0j} + [\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} [M_0] \\sum_{j=1}^N \\epsilon_{ij} \\mathbf{x}_{0j} + \\lambda_{0i} [\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i [M_0] \\mathbf{x}_{0i}. \\qquad (5)", "5c77ae2f5d12b06f70d957dacee8c943": "-\\infty + \\infty", "5c77ea46255d39443b598192432dd128": "c=c_\\infty\\,", "5c780b52ca1d4dfdbe245d9db7e17a66": " 1 + 2 \\to a + b + c,", "5c78650544194abfa13e98ff55d6589e": "Q(s,x)=\\frac{\\Gamma(s,x)}{\\Gamma(s)}=1-P(s,x).", "5c7897b345214b8a993902bbe7902fc0": "w(x) = \\begin{cases}\n -\\frac{Px(4x^2-3L^2)}{48EI}, & \\mbox{for } 0 \\le x \\le \\tfrac{L}{2} \\\\\n \\frac{P(x-L)(L^2-8Lx+4x^2)}{48EI}, & \\mbox{for } \\tfrac{L}{2} < x \\le L\n \\end{cases}", "5c78dc23c3166457eadbbe3c713d4c69": "H(s)=\\frac{1}{s^2 + 2\\zeta \\omega_0 s + \\omega_0^2}", "5c79396d9b856a4637200a314d3fbcf7": "\\bar{\\varepsilon}_{pqr} = \\det(\\boldsymbol{\\mathsf{L}}) \\varepsilon_{ijk} \\mathsf{L}_{ip}\\mathsf{L}_{jq}\\mathsf{L}_{kr} \\,.", "5c795495a7175ff8c71ff4ca0478dda2": " b' = 2\\pi d^3/3 ", "5c797137f93677daad57781c82759ea2": "v_{\\lambda}(m_1)", "5c79a7d12aad0aced3330965d5f95f1d": "\\varphi_n - \\varphi_{n-1} = \\frac{2\\pi}{N}", "5c79be75788127ed7754b58f09d02cb6": "\n\\begin{bmatrix} \n+1 & 0 \\\\\n 0 & -1\\\\\n\\end{bmatrix}\n\\quad \\mbox{and} \\quad \n\\begin{bmatrix} \n0 & +1 \\\\\n-1 & 0 \\\\\n\\end{bmatrix}.\n", "5c79c0643bcc5f392666e4ab26aebddf": " \\cot(z - a_1)\\cot(z - a_2) = -1 + \\cot(a_1 - a_2)\\cot(z - a_1) + \\cot(a_2 - a_1)\\cot(z - a_2). \\, ", "5c79c6a5298ca665d5c91c2c9c8072ca": " \\int_{\\theta=0}^\\pi\\int_{\\varphi=0}^{2\\pi}Y_\\ell^m \\, Y_{\\ell'}^{m'*}d\\Omega={4 \\pi \\over (2 \\ell + 1)}\\delta_{\\ell\\ell'}\\, \\delta_{mm'}.", "5c79c82de5d78a9498b612a88c5e0039": "\\displaystyle{\\mathrm{tr}\\, K^2 = \\sum \\lambda_n^2,\\,\\,\\, \\det (I-zK^2) =\\prod_{n=1}^\\infty (1-z\\lambda_n^2).}", "5c79e87821df1146f7ee672887adcbcf": "\\operatorname{E}(X)^2 \\gg \\operatorname{Var}(X)", "5c7a6205690e8fd4e73419d0b6da7a89": "\\langle\\chi|O|\\psi\\rangle", "5c7ab8eb5553e27b16a777179bdf0db7": "\\int_{-\\infty}^{\\infty} ae^{-(x+b)^2/c^2}\\,dx", "5c7ae75eedf0485a259c033207fff524": "\\Im", "5c7af19c4fc54b993294364e9f34dee3": "<\\overline{126}_H> 16_f 16_f", "5c7b0cfb9b9104eb06bfc16c66e23b0d": "\\frac {d^2w}{dz^2} + \n\\left[\\frac{\\gamma}{z}+ \\frac{\\delta}{z-1} + \\frac{\\epsilon}{z-a} \\right] \n\\frac {dw}{dz} \n+ \\frac {\\alpha \\beta z -q} {z(z-1)(z-a)} w = 0.", "5c7b4e8163a742f0427bc0b17c0dc9dc": "D_\\mathrm F", "5c7b719c15b6aa2b629468811a8bb877": "\\mathbb C^{27}", "5c7b71bf0c8819f20129efcc772cc1bf": "\\frac{\\Delta f^{*}}{f_f}=\\frac i{\\pi Z_q}Z_L", "5c7ba9064d8a0edcd5ab21b7e8e7ccfd": " T_{k \\ell} \\in \\operatorname{L}(V) ", "5c7bab3cb9175235bf9554ac41a70bbf": "s = m = 1", "5c7bd8216879c7fc6ad911727df93de8": "\\displaystyle G", "5c7bddb5bf63b270716101cd839a52e2": "1 < r < n-1", "5c7bf5f9f4e1b634eeda09dbca4ee872": "\\scriptstyle m_\\oplus", "5c7bfa654bbe7f093f436921f0a7cba3": "\\Omega(x) = S(x) \\Lambda(x) \\mod x^4 = 546 x + 732\\,", "5c7cb5eee5da6620bbf4f1df25764e64": "\\mathcal{L}[\\vec{x},t] = - \\mu [\\vec{x},t] \\zeta [\\vec{x},t] - {1 \\over 8 \\pi G} (\\nabla \\zeta [\\vec{x},t])^2 ", "5c7cb8ef3713e1ac52552cf5c1065ee6": "\\frac{ds}{d\\theta} = r \\cos^{-1} n\\theta = a \\cos^{-1+\\tfrac{1}{n}} n\\theta", "5c7cd67eaf3e19256c7fa9b77af746f9": "(\\mathcal{L}\\mu)(s) = \\int_{[0,\\infty)} e^{-st}d\\mu(t).", "5c7d5308fb9a9cde667e540340556199": " |q_0\\rangle ", "5c7d8dad268569266f749355c703cf47": "\\ \\tau", "5c7ddde75d2874271b44abb9c07ba754": " m \\ge 3", "5c7e15b1ac65be6571b63cf6aa596eba": "C_i \\subseteq A_i \\times T_i", "5c7e228bb51d360d4a56fa6206fc2152": " \\frac{h}{p}=\\frac{q}{h}\\,\\Leftrightarrow\\,h^2=pq\\,\\Leftrightarrow\\,h=\\sqrt{pq}\\qquad (h,p,q> 0)", "5c7e2970aa17f72b29f1e30bd684d01d": "\\Phi_{Y,X} : \\mathrm{hom}_{\\mathcal C}(FY,X) \\cong \\mathrm{hom}_{\\mathcal D}(Y,GX)", "5c7e2ecd97377a511b9d43a551f9716b": " \\operatorname{value}\\ v\\ I = v ", "5c7e5bdea511dfc951e392e48b2d27f7": "p=4", "5c7f2194df8be663421ba6a6dcca7050": "1, 2, \\ldots, n", "5c7f4af5356ac932a76a92151c91ad99": "\n\\begin{align}\na_{0} &= c_{0} \\\\\na_{1} &= c_{1} + b_{1} c_{0} \\\\\na_{2} &= c_{2} + b_{1}c_{1}+b_{2}c_{0} \\\\\n&\\cdots \\\\\na_{L} &= c_{L} + \\sum_{i=1}^{\\min(L,m)} b_{i} c_{L-i}.\n\\end{align}\n", "5c7f7fd4a77a3f5b7bf755cce2b4ead5": "\\epsilon=\\alpha+\\beta-\\gamma-\\delta+1", "5c7fb4fdd96fe7b97fe1c4f36e2a2bc0": "\\psi (q)", "5c8006660e737ca8449dda957909bbb9": "\n \\beta = \\cfrac{\\sigma_\\mathrm{c}}{\\sigma_\\mathrm{t}} = \\cfrac{1 - \\sqrt{3}~B}{1 + \\sqrt{3}~B} ~.\n ", "5c801f1722ac61b69cf59903775b7366": " Ax = b\\,", "5c80384063ff17e2a698183aa54b7c40": "\\frac{1}{N_\\mathbf{P}}\\mathbf{S}^T\\mathbf{P}^T\\mathbf{1}", "5c806b3843699056529ca018bf8b61dc": "T(\\psi)=\\alpha\\psi", "5c8088b15b2d474bdc13dc544d2d90f5": "Subclutter \\ Visibility = \\left( \\tfrac{Dynamic \\ Range}{CFAR \\ Detection \\ Threshold}\\right) ", "5c809dcc8991784c47dfd337e4693182": "D_P", "5c80a305a12d79e125e2adb1fb5647fd": "it/\\hbar\\,", "5c80cd23c0c290f7ec4e545d72b916c7": "d_{0,0}^{2} = \\frac{1}{2} \\left(3 \\cos^2 \\theta - 1\\right)", "5c8112f97483f70c9087efdc03dd3114": "\\zeta_D(s) = \\prod_p{1\\over 1-{D\\choose p}p^{-s}}", "5c813177d65b2a62a93e5b0ba3594b61": "q^3 + q^3 + q^2 = 2q^3 + q^2", "5c81639ed8fb93c203f1f291854fff4d": "w_i^{(t)} = \\frac{1}{\\text{max}(\\delta, \\big|y_i - X_i \\boldsymbol \\beta ^{(t)} \\big|)}.", "5c816ce2c813cbed91542ddc59e440c5": "s_0 \\to s_1 \\to \\dots \\to s_n \\to \\dots ", "5c82091d6487b66646384a71b166e520": "|R\\rangle", "5c82b19b741958f65cc4117b0b63498a": "t\\approx \\frac{m v_\\text{e} \\Delta v}{2P}", "5c82d6b6eb378667270b8086484596d2": " \\mathbf{u}[\\mathbf{x}, \\, t] = U[\\mathbf{k} \\cdot \\mathbf{x} - \\omega \\, t] \\, \\hat{\\mathbf{u}}\\,\\!", "5c82fb603b6de67c6ffec44a5558b610": "v_s=2 t f_s", "5c831c9456b7b10f46365f6f326d75fa": "d_H \\le t ", "5c832b01d152aa422e26bafa34bc015c": "\\begin{align}\n\\mathbf{k\\times j}=&-\\mathbf{i}\\\\\n\\mathbf{i\\times k}=&-\\mathbf{j}\\\\\n\\mathbf{j\\times i}=&-\\mathbf{k}\n\\end{align}", "5c8367dc4bace20a61b0f2188f680bbc": "X = \\frac{U_1/d_1}{U_2/d_2}", "5c8368fa4b3345f25dfca8ac6d13861f": "L_v^2 L_{uv} = L_x L_y (L_{xx} - L_{yy}) - (L_x^2 - L_y^2) L_{xy}, ", "5c836acfbadb385c69c88a2fb7573ed3": " [\\mathfrak h,\\mathfrak h]\\subset \\mathfrak h,\\; [\\mathfrak h,\\mathfrak m]\\subset \\mathfrak m,\\; [\\mathfrak m,\\mathfrak m]\\subset \\mathfrak h.", "5c8388a8d5ee57b79c09317824bc90ca": " \\tilde{\\delta}. ", "5c839c501b2ad2074007fe290a09e3b5": "\\Sigma_4 = \\Sigma_2", "5c83a3048cf74a501f1c34bf862f00f8": " (N,\\{ \\cdot,\\cdot \\}_{N}) ", "5c83a91b7ab0f134817435ede6e1f06a": "dQ=0\\,", "5c83cad58d465805cdd9da624844491e": " U(y, \\xi) = \\int{\\log(p(\\theta | y, \\xi))p(\\theta | y, \\xi)d\\theta} - \\int{\\log(p(\\theta))p(\\theta)d\\theta} \\, .", "5c83d08c33537609355a6703b61760c7": "\\frac{d}{d x} = \\frac{d}{d u} \\frac{d u}{d x} = \\frac{d}{d x}\\left(u\\right) \\frac{d}{d u} = \\frac{d}{d x}\\left(x^2\\right) \\frac{d}{d u} = 2 x \\frac{d}{d u}\\,", "5c84092dbffa6dffff25e9ddf87b057d": "\\mathrm{CouponFactor} = \\frac{360 \\times (Y_3 - Y_1) + 30 \\times (M_3 - M_1) + (D_3 - D_1)}{360}", "5c8422da236bb1f59e0a0919b0df722a": "(2-\\sqrt{2})(n-4)+3", "5c842c249f9086546e3dd85af230f077": " P = \\frac{R\\,T}{V_m-b} - \\frac{a}{\\sqrt{T}\\; V_m\\, (V_m+b)},", "5c84353d244939f00e4f8c99eae74874": "\\mathrm{d}^{r} \\big( \\varphi_{j} \\circ \\varphi_{i}^{-1} \\big) : \\varphi_{i} (U_{i} \\cap U_{j}) \\to \\mathrm{Lin} \\big( E_{i}^{r}; E_{j} \\big) ", "5c84369deea00784551b6ba8c7135ef3": " \\widehat{\\mu}_{S^{t+1} \\mid h^{t+1}} = \\sum_{i=1}^T \\alpha_i^t \\phi(\\widetilde{s}^t) ", "5c843ec61366c29471be181696bf6901": "\\Lambda=\\frac{m \\pi}{\\Delta k}", "5c84584fd5982496af467382b9752170": "S S^{\\dagger} = 1", "5c84bdb50841b86ed34a849b37470999": "n + \\min(k, n-k) + O(n^{1/2})", "5c8505d68e1bbc5f5795ca2064aa9cb7": "O(\\sqrt{n}log n)", "5c85289506765963880d43db3c90f51b": "x_1\\left(\\sigma\\right)", "5c853945412c3b9ebff5903555298d6d": "\\nabla_2^2", "5c855e094bdf284e55e9d16627ddd64b": "_a", "5c85d3073918365603cc53f2dce2f6ec": "\\text{X}", "5c86691b9f66694d718260613cfda02e": "\\omega_{cyc} = \\omega_{rad}/2\\pi\\,", "5c868f607f132fc1d430e0c32a5db11c": "|T| \\leq \\gamma n\\,", "5c86af151cc9c39b353988b7bbf261f5": "\\{g_n\\} \\in B", "5c86c58afc4173d7b0c756ad71988193": "\\frac {a} {b} = \\frac {c} {d}", "5c86dc4051a541da6e4956766a9facaa": "\\begin{smallmatrix}\\frac{L_{\\rm S}}{L_{\\odot}} = {\\left ( \\frac{R_{\\rm S}}{R_{\\odot}} \\right )}^2 {\\left ( \\frac{T_{\\rm S}}{T_{\\odot}} \\right )}^4\\end{smallmatrix}", "5c870e7736cbc656660ea1331f16cb4f": " -[R]^2 = |\\mathbf{R}|^2[E_3] -[\\mathbf{R}\\mathbf{R}^T]= \n\\begin{bmatrix} x^2+y^2+z^2 & 0 & 0 \\\\ 0& x^2+y^2+z^2 & 0 \\\\0& 0& x^2+y^2+z^2 \\end{bmatrix}- \\begin{bmatrix}x^2 & xy & xz \\\\ yx & y^2 & yz \\\\ zx & zy & z^2\\end{bmatrix},", "5c87281200a785a0222ac5dcfaefeb4f": " Z = e^{-\\beta J m^2 N z /2} \\left[2 \\cosh\\left(\\frac{h+m J z}{k_BT} \\right)\\right]^{N} ", "5c87345014aec504d7af13c86daaa832": "\\therefore s = 2^{64}- 1. \\, ", "5c87a0939cabe183098609f06fd67eb3": "P_u/P_n=1000", "5c87e6d79c971a39a9ea02f7fc1c43ea": "3x^2", "5c8815df4de5dd3cecd96739327fce2a": " E(X^n) = \\int_{-\\infty}^\\infty x^n\\,dg(x). ", "5c883e68608feaabff7634143498e640": "\\omega = {i\\over 2}(h-\\bar h).", "5c8841479d75ca571be030a0d708ca18": "T_H = 1089 K\\,", "5c88dc2494dcea1d65edf6c7654884f4": " \\dots \\to F_n\\otimes_{\\mathbf{Z}[G]}M\\to F_{n-1}\\otimes_{\\mathbf{Z}[G]}M \\to\\dots \\to F_0\\otimes_{\\mathbf{Z}[G]}M\\to \\mathbf Z\\otimes_{\\mathbf{Z}[G]}M.", "5c8944caf11c6d9a2469ae3103c35eea": "\\mathbf p = m \\mathbf v_{\\rm cm}", "5c899d977cb0544f60656f70335f9f56": "O(\\Delta t^4)", "5c89fb7591669b69176c292d7a492b68": "\\textbf{c}=U\\textbf{b}", "5c8a0025a793fb2e3247a8d9ec08a865": "\\mathrm{PartitionY}", "5c8a42e77fc4460318bc9685dc6bc79e": "F(s) = \\mathcal{L} \\left\\{f(t)\\right\\}=\\int_0^{\\infty} e^{-st} f(t) \\,dt ", "5c8a46ef4141fb9bc5eb6b912f0ecc88": "\\sigma^2/\\lambda", "5c8a509d025fc55fc8e854db5a7e4375": "c_F(a,b)", "5c8ac34e015883286ced7373db998d7b": "\n\\beta_{L}=\\frac{E_{L}+D-DT}{E_{L}}\\beta_{U}=\\left[1+\\frac{D}{E_{L}}(1-T)\\right]\\beta_{U} =\\beta_{U}[1+(1-T)\\phi]\n", "5c8af5b2e6492a18786d965824a35a75": "v(k)=\\frac{1}{\\hbar}\\frac{d\\mathcal{E}}{dk}=-\\frac{Aa}{\\hbar}\\sin{ak}", "5c8b29eeeeae48d1f53fb9c5e0c3207c": "\\mathbb {M}(D) = \\Sigma^* / \\equiv_D.", "5c8b5f2049149b745f5c2a7e73519d4d": "s=3", "5c8be2a3c34162ac9935fb8fa72a805d": "0=0\\cdot b", "5c8c0f198bdc8f0cb9005703b8c6056a": "\\boldsymbol{H}_i=\\begin{bmatrix}\na_1 & b_2\\\\\nb_2 & a_2 & b_3\\\\\n& \\ddots & \\ddots & \\ddots\\\\\n& & b_{i-1} & a_{i-1} & b_i\\\\\n& & & b_i & a_i\n\\end{bmatrix}\\text{.}", "5c8c175dabaf5142e3962a04132da863": " \\mathit PB = \\frac{\\mathit P} {\\mathit BV} ", "5c8c33ccdd6114fb5c04451e58094599": "\\partial_+ V_i = \\partial_+ W_i = H_i", "5c8c4a220be6cf8dca008c70937eff66": " \\frac{\\Delta Y'}{Y} ", "5c8c53b55a1f399fbd6719cb791acd34": " H = \n\\begin{matrix} \n\\operatorname{Ker}(T)^\\perp \\\\ \n\\oplus \\\\\n\\operatorname{Ker} (T). \n\\end{matrix}\n", "5c8cdc1c2beb62eb157a8cf9b6573f10": "\np_4(x) = x^4 + 14x^3 + 47x^2 - 38x - 240 \\,\n", "5c8cf009ca979bcaa252d1e9995f1f7a": "\\nabla(z) = 2z^2+1, \\, ", "5c8cf88f4c55fa978b51d455b9c78e55": "\\|f\\|_{\\alpha,\\beta}=\\sup_{x\\in\\mathbf{R}^n} \\left |x^\\alpha D^\\beta f(x) \\right |.", "5c8d238fcf333cf96053de28447de4c0": "\\stackrel{\\mathbf {E}_{\\parallel}}{}", "5c8d4364d7eb56c4dac37f5dc4f8a881": " ds^2 = dx_1^2 + dx_2^2 - c^2 dt^2, ", "5c8d709647e1cd3a441d0db3a8cfa1f0": "(v_n)", "5c8d7405da5b5d5ba5c69bd05070041d": " T_{r} = \\nu^{-1}", "5c8da3d0aee387e2aca019f3f5c9c0f9": "x_1,x_2,x_3", "5c8da6544ad0f886ce8c5442344d6977": "\\mathbb Z^2", "5c8e104a2e1e05ae0104ac962316e3fd": " \\max_{p(x_1,x_2)} \\min \\left( I\\left( x_1; y_1 | x_2 \\right) , I\\left( x_1, x_2 ; y \\right) \\right) ", "5c8e18e83a0b99910a5e36050ecd5400": "\\alpha>0,\\alpha \\neq 1", "5c8e1a14cb774efb50169a0b2c608d85": " h \\, ", "5c8e6b80eadeef4a15fb4251a8fdbed9": "\\operatorname{dom}h", "5c8ea70b922f77e5b244154d40b4d57f": "\\prod_{n=1}^\\infty n^{{3}^{-n}} = \\sqrt[3] {1 \\sqrt[3] {2 \\sqrt[3]{3 \\cdots}}} = 1^{1/3} \\; 2^{1/9} \\; 3^{1/27} \\cdots ", "5c8eb5f9bd48eab7fed7969e36d1537b": "I_{SH} = \\frac{V_{j}}{R_{SH}}", "5c8edc57d90e59503599ebe3e35d6e08": "\\langle \\phi_n \\rangle", "5c8f207a9b5822ecd0ec40f3d33ca3bb": "\\beta<2", "5c8f4703dddffc0c4b705aa42313cf63": " \\begin{matrix}\n\\gamma_{a'} &=& S &\\gamma_{a } &S^{-1}\n\\end{matrix}\n", "5c8f63fe5525da0af0b8ced6b8b6a8af": " \\mathbf{K} = \\sum_{e} \\mathbf{k}^e ", "5c8fbe6a04e16951069ed308a0aa1359": "\n\\Phi_2(b_1,b_2,c;x,y) = \\sum_{m,n=0}^\\infty \\frac{(b_1)_m (b_2)_n} {(c)_{m+n} \\,m! \\,n!} \\,x^m y^n ~,\n", "5c8fd1ad91a588eeda8b6ed0bebacea9": "L = 1/\\cos^2\\left(\\Lambda\\right)", "5c8fd50f7f0a9982a743a7c447dc298a": "d(A\\cup B,C)", "5c906412511ab016bd0f8e9790412b0d": "C_h = \\frac {h^2}{2s}", "5c90b320d3ec6c29b11a8ce4b16d6fb2": " \\equiv a^{(N-1)/q}\\pmod{p}", "5c90ba4a04b3ba4783d13ebc4a85d266": " U^*f(z) ={f(z)-f(0)\\over z}.", "5c90d2752df315dd652a1f6902d33fcd": "C_{AB}=r(v_{AB})", "5c9115ad11c8004d2ba0f5391d1fcf1c": "M_Z^{gi} = \\overrightarrow{ZG} \\times mg - \\overrightarrow{ZG} \\times ma_G - \\dot{H}_G", "5c918727ccb8aa2cb46a73097bcc4bee": "y \\leq x", "5c918a2647789ae62486a3d1c8d82237": "\n -k_0^2 +\\vec k^2 + \\omega_p^2 { \\left( k_0^2 - \\omega_p^2\\right) \\over \\left( k_0^2- \\omega_H^2 \\right) } =0\n,", "5c9190984cd24cab07d95868d266d337": "f(x) = a \\left (-\\frac{b}{2a} \\right)^2+b \\left (-\\frac{b}{2a} \\right)+c = -\\frac{(b^2-4ac)}{4a} = -\\frac{\\Delta}{4a} \\,\\!,", "5c9212e7035eb327013dbd230616b7f9": "U=Ga^2/\\mu", "5c9292d07cce087c55f2b4a745042412": "Q = UAT_lm", "5c92ab0c570b51828099f3d82571b42f": "\\sqrt{\\{0\\}}", "5c92c00b7c8f02cfeef1cdb5aeb8b733": "y(t) = h(t) * x(t)", "5c92dc8489afdfb2927e24c12a6964b2": " \\frac{a}{\\sin A} \\,=\\, \\frac{b}{\\sin B} \\,=\\, \\frac{c}{\\sin C},\\!", "5c93336e8329b899e64b60c408c59a57": "y=\\sin x\\,", "5c93394e7f620ac59b6906d154315303": "\\operatorname{Cl}_2\\left(\\frac{\\pi}{3}\\right)=3\\pi \\log\\left( \n\\frac{G\\left(\\frac{2}{3}\\right)}{ G\\left(\\frac{1}{3}\\right)} \\right)-3\\pi \\log \n\\Gamma\\left(\\frac{1}{3}\\right)+\\pi \\log \\left(\\frac{ 2\\pi }{\\sqrt{3}}\\right)", "5c937f80ea16ddd644338714b7be76f0": "x_{\\mathrm{per}}(t_{1})=x_{1}", "5c93a920e4583f4bca030d8927b642ae": "[y_i]", "5c93af6aad858ec8cefbe13afc9e71c1": "\\bar{m}^a\\partial_a=\\bar{\\Omega}\\partial_r +\\bar{\\xi}^3\\partial_{ y}+\\bar{\\xi}^4\\partial_{ z } \\, := \\,\\bar\\delta \\,.", "5c93e584b106c9b9ceb339d8ca8395cc": "2+1", "5c94066cfda1f0086d69d15ce1dd34ca": " \\ g_{\\phi, h}", "5c941565ec0354e2892ac952f118eb82": "\\mathfrak{so}(2r + 1)", "5c9457331453cc558972d493efa9bea3": "\\int_{-1}^{1} P_k ^{m} P_\\ell ^{m} dx = \\frac{2 (\\ell+m)!}{(2\\ell+1)(\\ell-m)!}\\ \\delta _{k,\\ell}", "5c9472591dcc30c43697d33bbd2ecfd8": "y^2=1/(1-x^2)", "5c9474a6cf7e655af399f868159b322e": "J_F(r,\\theta,\\phi) =\\begin{bmatrix}\n\\dfrac{\\partial x_1}{\\partial r} & \\dfrac{\\partial x_1}{\\partial \\theta} & \\dfrac{\\partial x_1}{\\partial \\phi} \\\\[3pt]\n\\dfrac{\\partial x_2}{\\partial r} & \\dfrac{\\partial x_2}{\\partial \\theta} & \\dfrac{\\partial x_2}{\\partial \\phi} \\\\[3pt]\n\\dfrac{\\partial x_3}{\\partial r} & \\dfrac{\\partial x_3}{\\partial \\theta} & \\dfrac{\\partial x_3}{\\partial \\phi} \\\\\n\\end{bmatrix}=\\begin{bmatrix}\n\t\\sin\\theta\\, \\cos\\phi & r\\, \\cos\\theta\\, \\cos\\phi & -r\\, \\sin\\theta\\, \\sin\\phi \\\\\n\t\\sin\\theta\\, \\sin\\phi & r\\, \\cos\\theta\\, \\sin\\phi & r\\, \\sin\\theta\\, \\cos\\phi \\\\\n\t\\cos\\theta & -r\\, \\sin\\theta & 0\n\\end{bmatrix}. ", "5c9488a554308412afa8671bb30787e1": "L(z) := b + \\int_a^z \\frac{1}{w}\\,dw", "5c948c2bd99a8d028b603ac61257586f": "s_{pm}=j\\cos(\\theta)\\,", "5c949a8e4f0a653999b73e6c1df9dbb6": "\\mathbf{R}^2\\setminus\\{0\\},", "5c949d22ca4d551397f436e5557259d8": " A \\times B := \\{ a \\times b : a \\geq 0 \\and a \\in A \\and b \\geq 0 \\and b \\in B \\} \\cup \\{ x \\in \\mathrm{Q} : x < 0 \\}", "5c94b37f38a875337647972ce918eedb": " H(X|Y) = \\mathbb E_Y \\{H(X|y)\\} = -\\sum_{y \\in Y} p(y) \\sum_{x \\in X} p(x|y) \\log p(x|y) = \\sum_{x,y} p(x,y) \\log \\frac{p(y)}{p(x,y)}.", "5c96043e22eeed6d6a177458e383f213": "m_{sucked}", "5c961c4e19636d13b47f7b1f6c69aa41": " D_2(\\omega) = -\\frac{2\\pi c}{\\lambda^2}D_{tot}", "5c96510f3b40bf4552b2c2fac9a9bad4": " \\displaystyle \\alpha = -\\operatorname{sgn}(a_{21})\\sqrt{\\sum_{j=2}^{n}a_{j1}^2} ", "5c96ecca9bbb3098a3e4a47f3355b8bd": "\\operatorname{var}(\\widehat{\\theta}) = \\operatorname{E}[(\\widehat{\\theta} - \\operatorname{E}(\\widehat{\\theta}) )^2]", "5c9713aae8dbb4a68f20a1d6adeda89f": "I_k(T) = \\int_{0}^{T} k(T^\\prime) \\mathrm{d}T^\\prime.", "5c97191b6fef92efbf7b79e8c6f9d152": " V \\not \\in FV[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} V] \\equiv \\operatorname{get-lambda}[V, E] ", "5c974616dcb00011810211a23a57f90f": "\\tfrac{97}{56}", "5c9790540476292e0b8eaac95409d0b4": "\\zeta\\left(1/2+it\\right)\\in O\\left(t^c\\right)", "5c97e2c52f6ea7c5946a1f183181b485": "\\operatorname{arcosh}\\left|\\frac{x}{a}\\right|", "5c97ff3d5c66f4b948d7aecfef8c7ad2": "\\hat{\\alpha} \\hat{\\beta} = \\hat{\\beta}\\hat{\\alpha} ", "5c985c203a2564477612144cf5d7748e": "\n|\\phi\\rangle=\\sum_i\\sqrt{\\lambda_i'}|i_A'\\rangle\\otimes|i_B'\\rangle\n", "5c985d6b128b346e4bf79a3f6c85be08": "\\frac{d}{dx} \\chi_+^\\alpha = \\chi_+^{\\alpha-1}", "5c986aa5ca4dd69735b8a2a68bdf4cdb": "A_a^i = \\Gamma_a^i + \\beta K_a^i", "5c98dd5e0fa6d213f1edfeab54b775cb": "\\scriptstyle -1.5(0.74)\\times10^{-8}", "5c9914ba82c9ccfcb511fa02f854278f": "ARI = \\frac{ \\sum_{ij} \\binom{n_{ij}}{2} - [\\sum_i \\binom{a_i}{2} \\sum_j \\binom{b_j}{2}] / \\binom{n}{2} }{ \\frac{1}{2} [\\sum_i \\binom{a_i}{2} + \\sum_j \\binom{b_j}{2}] - [\\sum_i \\binom{a_i}{2} \\sum_j \\binom{b_j}{2}] / \\binom{n}{2} }", "5c9939d70d1d39b6ff868a42ca20c291": "L=f", "5c994284e7e5c1e7d59fbe0c154779af": "\\mathbf{select}_q", "5c994bfe41d467b11b29988f2dae3415": "\\operatorname{dVar}(X + Y)\\leq\\operatorname{dVar}(X) + \\operatorname{dVar}(Y)", "5c9969dd033546731211d6ac8cdb5856": "I=\\mathbf{A}^{-1}\\mathbf{A}", "5c99b4ac56b0ca7e7a5de3392edceb6b": "u = || \\ \\vec{u} \\ || = \\sqrt{ (u^1)^2 + (u^2)^2 + (u^3)^2} ", "5c99bcf9110f74c0f463155ea4864881": "s_a(t)\\ \\stackrel{\\text{def}}{=}\\ s(t) + j\\cdot \\hat s(t),", "5c99c1a3ebd0ab25005ca7896556325d": "\\begin{align}\n\\Delta\\varphi &= \\varphi(\\alpha + \\Delta\\alpha) - \\varphi(\\alpha) \\\\\n&= \\int_{a + \\Delta a}^{b + \\Delta b}f(x, \\alpha + \\Delta\\alpha)\\,dx - \\int_a^b f(x, \\alpha)\\,dx \\\\\n&= \\int_{a + \\Delta a}^af(x, \\alpha + \\Delta\\alpha)dx + \\int_a^bf(x, \\alpha + \\Delta\\alpha)dx + \\int_b^{b + \\Delta b} f(x, \\alpha+\\Delta\\alpha)\\,dx - \\int_a^b f(x, \\alpha)\\,dx \\\\\n&= -\\int_a^{a + \\Delta a} f(x, \\alpha + \\Delta\\alpha) \\, dx + \\int_a^b [f(x, \\alpha + \\Delta\\alpha) - f(x,\\alpha)]\\,dx + \\int_b^{b + \\Delta b} f(x, \\alpha + \\Delta\\alpha)\\,dx\n\\end{align}", "5c9a2ef183669153a8b924398e73ce3e": "\\zeta(s,q) = \\sum_{n=0}^\\infty \\frac{1}{(q+n)^{s}}.", "5c9a4b00c37b674613d3f39a9ded3c1c": " B[u,v] = (f,v)", "5c9aea898196d0615b3f3ad8923f1e3e": "\\mathbf{\\tau} = \\sum_{i=1}^N(\\mathbf{F_i}\\times r_i), ", "5c9af1229499d358a95f087ae15867cc": "{_{k}P_r}", "5c9b46d4916e6f4264f249965d324d9d": "K = \\mathbf{Q}(\\zeta)", "5c9b8e13045e4956bd2404372b1907d2": "\\frac{V_3}{V_2}", "5c9bcb9277e7a6b5dcccf7cdb6957558": "\\scriptstyle 2b", "5c9bceb024e1a389a53de905ecd5f05c": "\\operatorname{ip}\\langle x,y\\rangle = y-x", "5c9c2c99b224cf0ef48864620fdb32e5": " | \\Psi \\rangle ", "5c9cc29abf3c4f8580a828627555a318": "{\\varphi +2\\varphi^2 +2\\varphi^3 + \\ldots = \\varphi +2(\\varphi^2 +\\varphi^3 + \\varphi^4 + \\ldots)}", "5c9ce8b5a0aa0f770ce207cc3fc06d22": "R_S=R_H \\left(1-\\tan(\\alpha)\\tan(\\delta\\theta) \\right)\\sec(\\alpha)", "5c9d14b3d1889bb1e3ffa7c1bf86b019": "h\\left(g^{-1} \\circ u\\circ g\\right)= h(g)^{-1}\\cdot h(u)\\cdot h(g) = h(g)^{-1}\\cdot e_H\\cdot h(g) = h(g)^{-1}\\cdot h(g) = e_H.", "5c9d2deca0e6c85652ef146fb97164d8": "ab=c", "5c9d50443eed2373d006944c743e5616": "{{\\left\\{ {{g}_{m}}\\left[ n \\right]=\\delta \\left[ n-m \\right] \\right\\}}_{0\\le m 0.\n", "5c9e6954f2dc1069d85ae8f8eeb5c09d": "\\scriptstyle n! \\sim \\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n", "5c9e785a62b3b462ce2626f5c08d1ba6": "|\\mathcal Z|=8 \\ .", "5c9e83fb69b053b73b89125ab4a21a3d": "\\mathrm{d} F = 0", "5c9ebc79b0064b3a4b5f9824c159016b": "\ndP = i[P,P] ds = 0\n\\, .", "5c9ee19f03676fbb6523dbff95080b74": "A_g = \\frac{1}{e^{\\mathit{y}_g}}", "5c9f1ca08f655f79e4ac49a5516b44fb": " V = - L \\frac{\\mathrm{d}I}{\\mathrm{d} t} .\\,\\!", "5c9f22ccc0a918eb7be2b2ddcbd8c6be": "SU(N_f)_L \\times SU(N_f)_R \\times U(1)_B \\times U(1)_R", "5c9f9838fe5053f4a6508220eb1a8e01": "+V_{nn}^2\\left(2\\frac{V_{nk_5}V_{k_5k_4}V_{k_4n}}{E_{nk_4}^3E_{nk_5}}+\\frac{V_{nk_5}V_{k_5k_3}V_{k_3n}}{E_{nk_3}^2E_{nk_5}^2}\\right)-V_{nn}^3\\frac{|V_{nk_5}|^2}{E_{nk_5}^4}", "5ca002b234c77f31909b1f35dcb60918": "Y_{DW}(n, m) = \\frac{1}{\\sqrt{c_0^n}}\\cdot\\sum_{k=0}^{K - 1} y(k)\\cdot\\Psi\\left[\\left(\\frac{k}{c_0^n} - m\\right)T\\right]", "5ca02ced0167748dbd2d08007d30969d": "\\,y = 226 153 980.", "5ca07e1367dc929136b028c442d713ec": "\\hat{X} =[\\hat{S}^+][\\hat{S}^-]^*\\hat{x}.", "5ca0a22392db5c1e9441bfe4dff916a4": "f(x) = \\begin{cases} 0, & \\mbox{if }x =0 \\\\ x \\sin(1/x), & \\mbox{if } x \\neq 0 \\end{cases} ", "5ca15329b44f6228a8271609a6cb643a": "\\mathrm{ker}\\left(f^k\\right) + \\mathrm{im}\\left(f^k\\right) = M", "5ca18172e9a4163c07f63aaa31c6d118": "A=\\begin{pmatrix}\n 1 & 0 & 0 & \\cdots & 0 \\\\\n 0 & 1 & 0 & \\cdots & 0 \\\\\n 0 & a_{32} & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & a_{n2} & 0 & \\cdots & 1\n\\end{pmatrix}", "5ca1c6d16f6dc9e17ba286fadfa440b3": "\\Pi_{n \\mathbin{:} {\\mathbb N}} \\operatorname{Vec}({\\mathbb R}, n)", "5ca1c7d01409371407630d6ddfe5dc25": "c^2 = \\frac{1}{\\epsilon_0 \\mu_0}", "5ca2232dd8907417b5e33cbca27797fb": "\\mathcal{k}", "5ca266288933700eba43c915a0f22993": " |f| = f^+ + f^-. \\quad ", "5ca2b0af44fc2fab920ff6fce3410dbe": "\\sigma^2_{b2}(t)", "5ca2ccbfa2abca38c618f51a7d8a3d0c": " 1 truckload =200 kcalories(200,000 calories); there are separate trucks for meat and feed;", "5ca303d39df8f978ab947b9751e95a4d": "\\omega' = \\sqrt{\\frac{k}{m}-\\left ( \\frac{b}{2m} \\right )^2 } \\,\\!", "5ca38ec56a69495c43147dc259916e37": "[SU(2)\\times U(1)]/\\mathbb{Z}_2", "5ca3c9ac384e68f1b319188b45c57cab": "(k~~l) = (k~~k+1)\\cdot(k+1~~k+2)\\cdots(l-1~~l)\\cdot(l-2~~l-1)\\cdots(k~~k+1).", "5ca3cfb27d357ebb648abc1c6cac8175": "f(x) - a^2(x) = 0", "5ca47eed2eb38dd5a6a9dac97c3981f9": "k,l= 1,\\ldots,i", "5ca4942c056de83fc558425960f7def5": "Bxyz \\or Byzx \\or Bzxy \\or \\exists a\\, (xa \\equiv ya \\and xa \\equiv za).", "5ca4a454d73cdbf90fb2520a048798d1": "\\bold{r}=\\overrightarrow{OP}.", "5ca4ccadc82f97bf4c27e99be43db6be": "f_{i-1}(r_1\\dots r) = (1-r)f_{i}(r_1,\\dots,r_{i-1},0) + rf_{i}(r_1,\\dots,r_{i-1},1)\\text{ if }S = R.", "5ca5076d5b614a403fb8c140b0fa1070": "\\theta = 2\\arcsin \\frac{1}{\\sqrt{N}} ", "5ca58383554a292cd50b9311d9e67372": "\\sum_{n=-\\infty}^\\infty q^{n(n+1)/2}z^n =\n(q;q)_\\infty \\; (-1/z;q)_\\infty \\; (-zq;q)_\\infty.", "5ca5adc6bc995f6d0fb2503d7b5de8e5": "(1 + i_\\$) = \\frac {F_t} {S_t} (1 + i_c)", "5ca6196310213d85f85115d04c16257b": "v \\mapsto -v", "5ca6364a004a467ae8a62c021b5e269d": " q_{j} = \\Pi_{ji} J_{i} \\,", "5ca65937d317c0e23f29405253c8b602": "M_{9 \\times 12} ", "5ca65eec017601db2160ef5307bdd798": " \\mathbf{F} = m \\left ( \\mathbf{v} \\times 2 \\boldsymbol{\\xi} \\right ) \\,\\!", "5ca71fd5ec5958c585cf90758d1e6842": "\\sup\\nolimits_{u\\in H}\\|u(x)\\|<\\infty,", "5ca72c07405e44014daaf2cf9639f69a": "\\mathfrak{P}^{66}", "5ca78fa4a2054a111c367d13c690e87d": "A(r) = \\frac {M v^2} {T R} \\mathrm{e} ^ { \\frac {\\delta} {T} \\frac {v^2} {2} \\left( 1-\\frac {r^2} {R^2} \\right) } ", "5ca7c82ca99f9afb6df2bf9bc9482fd8": "\\begin{matrix} \\frac{16}{15} \\end{matrix}", "5ca7ff68b0bb63090a304f152e4400f9": " b_{2k+0}=\\frac{(1-x_{2k+2}x_{2k+3})}{(1-x_{2k-2}x_{2k-1})} x_{2k-1}", "5ca824cf56d144d884870be9de0381f8": "\\Delta_1^2 - 4 \\Delta_0^3 = -27\\,a^2\\,\\Delta\\ ,", "5ca838206f0b179fe55a2c461d37e624": "x_2 \\,", "5ca83b9ae84f2c097f2e87a3ac7cf7b8": " g_{obs}=\\frac{g_0}{1+\\beta_2 g_0 \\ln \\Lambda/m} \\,, \\qquad \\qquad\\qquad (1) ", "5ca84ee6f51a023c3c1e4c73fedf5e86": "F:M \\times [0,1] \\rightarrow M ", "5ca855921dacdfbcf22abb2fbde4cce5": "x^n+y^n=z^n \\,", "5ca855f2478bb0e98d85c86a85a46530": "\\delta^{(k)}(t)", "5ca86c2fc3044d757124e8da43b898aa": "\\begin{align}\n\\Delta K = W = [\\gamma_1(v_1) - \\gamma_0(v_0)] m_0c^2.\\end{align}", "5ca882989ddfdec4f812d35937390714": "\\operatorname{Der}_R(S,M)\\cong \\operatorname{Hom}_S(\\Omega^1_{S/R},M), \\,", "5ca88d67d6ec61977d303e130f0bd675": " k = \\frac { m } { a m^{ b - 1 } - 1 } ", "5ca8ecb1ac6937078241ac7c3c969f73": "\\overline{x_{i-1}}", "5ca906adbcf6471f0502b402791dcae8": "p\\in H", "5ca91d7e843a895062eb1561b469ce97": "\n\\begin{align}\n& \\left(\\frac{D^{\\mathrm{face}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} - \\left(1+\\frac{D^{\\mathrm{face}}}{D^{\\mathrm{beam}}}\\right)\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} = \\frac{M}{D^{\\mathrm{beam}}}-\\cfrac{q}{S^{\\mathrm{core}}}-\\vartheta \\\\\n& \\left(\\frac{D^{\\mathrm{beam}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^2 \\gamma}{\\mathrm{d} x^2} - \\left(1+\\frac{D^{\\mathrm{beam}}}{D^{\\mathrm{face}}}\\right)\\gamma = -\\left(\\cfrac{D^{\\mathrm{beam}}}{D^{\\mathrm{face}}}\\right)\\frac{Q}{S^{\\mathrm{core}}}\\,\n\\end{align}\n", "5ca935dcad8bb72edc4ed6ed496c8b70": "\\vec{y}) \\equiv", "5ca942165e4d9cea507963100ea0f0e3": "\\begin{bmatrix}\n\\,\\,\\,1 & 4 & 7 \\\\\n\\,\\,\\,3 & 0 & 5 \\\\\n-1 & 9 & \\!11 \\\\\n\\end{bmatrix}", "5ca9af70861232821870eb50caffedb7": " z = d Z \\,", "5ca9b7dc52063a9f092e449848cb7829": "V_\\text{s} = \\frac{N_\\text{s}}{N_\\text{p}} V_\\text{p} ", "5ca9c5f5feba9799965605212ff84ba9": "V_\\mathrm{out} ={ P \\times K \\times Vs_\\mathrm{actual} \\over Vs_\\mathrm{ideal} } ", "5caa0f005b04c6e1f91ed458a7b68dfd": "(p,i)", "5caa3b12bd807fabcd942f934dec5ee8": " \\nu = T_{ab} \\, k^a \\, k^b ", "5caa94fd0ec1452be6ad5bb5fdfe74ed": "\\mathbf{\\bar{o}}={\\sum_{t=1}^{N}}\\mathbf{{o_t}}/N", "5caadb57a7d909845e74b1badbff8bc9": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 1}{10 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "5caadb7bc7c2e468445be8f66d669ab3": "f: V_1 \\otimes V_2 \\otimes \\cdots \\otimes V_r \\to K", "5caaff53d67b614d4052e191525c7958": "f'(z)=-m(z-z_P)^{-m-1}h(z)+(z-z_P)^{-m}h'(z)\\,\\!.", "5cab000f0940a7e50fb9960f7cb5bce8": "p_n(x) = \\left(\\sum_{k=0}^\\infty {c_k \\over k!} D^k\\right) x^n = Sx^n,", "5cab2ce7e3cacff8a2ce4cca4f44c249": "\\omega,\\omega^\\omega\\!,\\omega^{\\omega^\\omega}\\!\\!,\\ldots", "5cab5a920c89db688836dfa28a5defe2": "\\psi_i(x) \\,", "5cab6a476db095b60db5581b9126d061": " \\hat{S}_{k}^{lm}(f) = \\frac{1}{N\\Delta t}\n {\\lbrack J_{k}^{l}(f) \\rbrack}^{*} {\\lbrack J_{k}^{m}(f)\n \\rbrack}, \n", "5cab870f37c9de1d79a855d1c69dd66c": "f(g) = \\langle U_g v, v \\rangle,", "5caba21eca092e207a121dd2d22e8c09": "\\sqrt{z^*z}", "5cabe3cac7a277fb3ea6050e6df54bca": " \\ln \\Gamma(-\\eta_1-1)-(-\\eta_1-1)\\ln(-\\eta_2)", "5cac4a37efad582495470ebfc64b24b6": " H_n = \\begin{bmatrix}\n h_{1,1} & h_{1,2} & h_{1,3} & \\cdots & h_{1,n} \\\\\n h_{2,1} & h_{2,2} & h_{2,3} & \\cdots & h_{2,n} \\\\\n 0 & h_{3,2} & h_{3,3} & \\cdots & h_{3,n} \\\\\n \\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n 0 & \\cdots & 0 & h_{n,n-1} & h_{n,n} \n\\end{bmatrix}. ", "5cac4f2ed9b96ac7ae3b7e036c6bbd03": "\\alpha=1/4", "5cac7b22d5d4b679d610ca8a1e5063ff": " m - \\pi ", "5cacd920c7502d5fa9679cba608177c5": "f \\in H^{k-2}(\\Omega)", "5cad7848ce08204cde97f046f9b5a13b": "Vol_2(d,n)", "5cadcec26f332ce3d7a52bf62c4c1ad5": "k\\geq 2", "5cade24876970e4f5597dc9cf347c809": "Q = \\frac{e^\\alpha}{\\beta-\\alpha}(\\beta e^{-\\alpha}-\\alpha e^{- \\beta})", "5cade8ffe2dfd855679280b4d43405de": "\nT_q(n) = \nc_q(1) + \nc_q(2)+\n\\dots+c_q(n)\n", "5caec960fe286d7296feb4e612c55989": " \\widehat{\\mathcal{H}f}(\\xi) = -i \\, \\mathrm{sgn}(\\xi) \\hat{f}(\\xi)", "5caecd3a8be78b3f1737bebffbe6bb2e": "B_i^n(t)", "5caed6872a9af4251f33e49e4bed5a21": "\\lambda_s", "5caed75f63dc4d5d5949640a99a5235f": "\\phi(s) = g[s+\\lambda - \\lambda \\phi(s)]", "5caf5c04b60385a4d474ca0ac9ef11a8": "Y = C + I + G", "5cafdd2860187972509ec4c023e10a30": "{\\bar{K}}_4", "5cafebec0eeda1026fab874d6dd3a3e8": "\\ T' = -\\xi' \\frac{\\part \\overline{T}}{\\part z}.", "5cb004e4283104e5707df4e51d4d59c7": "\\epsilon(q,0) = 1 - V_q \\sum_k{\\frac{f_{k-q}-f_k}{E_{k-q}-E_k}}", "5cb046868a5d26fd233e505974c258f5": "f^*(\\alpha \\smile \\beta) =f^*(\\alpha) \\smile f^*(\\beta),", "5cb04e93caeafd40729bd2b2620edee9": "\\ V_{eff, RG}=\\pi^{3/2}(\\omega_{xy,G}^2+\\omega_{xy,R}^2)(\\omega_{z,G}^2+\\omega_{z,R}^2)^{1/2}/2^{3/2}", "5cb06f2084bc66d47983c6c647e86aa8": " a_{(d_1+n-1, d_2+n-2, \\dots , d_n)} (x_1, x_2, \\dots , x_n) =\n\\det \\left[ \\begin{matrix} x_1^{d_1+n-1} & x_2^{d_1+n-1} & \\dots & x_n^{d_1+n-1} \\\\\nx_1^{d_2+n-2} & x_2^{d_2+n-2} & \\dots & x_n^{d_2+n-2} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nx_1^{d_n} & x_2^{d_n} & \\dots & x_n^{d_n} \\end{matrix} \\right]\n", "5cb0ad87b16ce992a148a2999dcc5886": "\\Bigl|\\int_S fg\\,\\mathrm{d}\\mu\\Bigr| \\le \\frac{m-1}{m}|g(m)|+\\sum_{n=m+1}^\\infty|g(n)| = \\|g\\|_1-\\frac{|g(m)|}m<1.", "5cb0f0b41e5f80492274850305cd7d5d": "C_p = C_v + R", "5cb0f88ae50b42e0929776f934fbb024": "A_1\\land\\dots\\land A_n/B", "5cb0f8aad169cadcecf587c61898b46a": " s_{\\text{unbiased}}^2 = \\frac{n}{n-1} s_{\\text{biased}}^2 ", "5cb123c0489646faebb9a9f17af30554": " h_{A+b}(x)=h_A(x)+x\\cdot b, \\qquad x,b\\in \\mathbb{R}^n.", "5cb130c82066dd4f97ebae5f685c7873": "p_i=\\omega_i \\rho_i", "5cb13b05715e82112c4601b95fcca71e": "n^{3}+2", "5cb143b46cd4809c6287ee05a0b46134": "\\frac{d}{dx} \\underline{x}^{-k} = -k \\underline{x}^{-k-1}", "5cb1a347d6c8c022bb60e5a6ff0e9547": "\n\\operatorname{P}(Z \\ge \\theta \\operatorname{E}[Z])\n\\ge \\frac{(1-\\theta)^2 \\operatorname{E}[Z]^2}{\\operatorname{E}[( Z - \\theta \\operatorname{E}[Z] )^2]}\n= \\frac{(1-\\theta)^2 \\operatorname{E}[Z]^2}{\\operatorname{var} Z + (1-\\theta)^2 \\operatorname{E}[Z]^2}.\n", "5cb1a366fcb5ea1b93b752b4cbec1e11": "\\textstyle H", "5cb1d9789feb42b0cfb134949afe040e": "\\frac{1}{|A|} \\sum_{a \\in A} f(a)", "5cb1e4a6c4fd381884751725ab2ca8b1": "\\nu(G)", "5cb1fecfb9696436d9bf6a1c71652ab4": " {\\Psi} = 1 - A{\\phi}^{\\ast} ", "5cb2d7689bfe624ce61311cb2cf0b04d": "\n\\begin{align}\nE[\\sigma_y^2]\n& = E\\left[ \\frac 1n \\sum_{i=1}^n \\left(y_i - \\frac 1n \\sum_{j=1}^n y_j \\right)^2 \\right] \\\\\n& = \\frac 1n \\sum_{i=1}^n E\\left[ y_i^2 - \\frac 2n y_i \\sum_{j=1}^n y_j + \\frac{1}{n^2} \\sum_{j=1}^n y_j \\sum_{k=1}^n y_k \\right] \\\\\n& = \\frac 1n \\sum_{i=1}^n \\left[ \\frac{n-2}{n} E[y_i^2] - \\frac 2n \\sum_{j \\neq i} E[y_i y_j] + \\frac{1}{n^2} \\sum_{j=1}^n \\sum_{k \\neq j} E[y_j y_k] +\\frac{1}{n^2} \\sum_{j=1}^n E[y_j^2] \\right] \\\\\n& = \\frac 1n \\sum_{i=1}^n \\left[ \\frac{n-2}{n} (\\sigma^2+\\mu^2) - \\frac 2n (n-1) \\mu^2 + \\frac{1}{n^2} n (n-1) \\mu^2 + \\frac 1n (\\sigma^2+\\mu^2) \\right] \\\\\n& = \\frac{n-1}{n} \\sigma^2.\n\\end{align}\n", "5cb30ab506ca1eac67115b3c62fcb48f": "\\phi(\\omega) = \\phi(0) - \\tau_g \\omega \\ ", "5cb332981ac39a0f037633c353398dd9": "2^{2^7-1}-1", "5cb364997d8263227a61773a9caef3e8": "\\Delta\\mu", "5cb3bcc67658e36e02990fa4d95eb8b4": "(a \\cos t, b \\sin t)", "5cb3c9c85adb67e383260247d1c80725": "\\star\\eta=A\\,\\mathrm{d}y\\wedge \\mathrm{d}z-B\\,\\mathrm{d}x\\wedge \\mathrm{d}z+C\\,\\mathrm{d}x\\wedge \\mathrm{d}y", "5cb3fda7e4a18f453b9a3e47b138c598": "\\partial_i=\\frac{\\partial}{\\partial\\theta^i}", "5cb453477fa0f75746cc6b561f720324": "\\mathrm{HA2} = \\mathrm{MD5}\\Big(\\mathrm{A2}\\Big) = \\mathrm{MD5}\\Big( \\mathrm{method} : \\mathrm{digestURI} \\Big)", "5cb46ee42298565937b34ff1a0e0da12": "\\frac{1 - 0.4}{1 - .5} = 1.2 ", "5cb4822c201eb7a2febedac3be7f3aee": "\\mathcal{V}=\\{X_{1},X_{2},\\ldots ,X_{n}\\}", "5cb4911d7570d698ddba602ff73ab083": "-\\nabla \\times \\mathbf{E} = \\frac{\\partial \\mathbf{B}} {\\partial t} + \\mu_0\\mathbf{j}_{\\mathrm m}", "5cb4fd1ba6707d823cad2e9954238944": "+ + -", "5cb58c9888cb25f4634bb98da197f69a": " \\vert \\Im(r) \\vert \\leq 1/2+\\delta ", "5cb5a675c55142c7c427d152e4fdfc15": "\\lim_{\\Phi_{S}\\rightarrow0} V_{m}\\ne0", "5cb5c91144b0c241c920b9901e9fcbed": "i\\in G", "5cb5ed0b9fad2f9c475b203f3788f484": "{c} = \\frac{\\cos \\beta_2}{\\cos \\beta_1}", "5cb62baaade98adaed9c62dcb9b143c7": " \\mathbf{STFT} \\left \\{ x(t) \\right \\} \\equiv X(t, f) = \\int_{-\\infty}^{\\infty} x(\\tau) w(t-\\tau) e^{-j 2 \\pi f \\tau} \\, d \\tau ", "5cb69f27a461cfd1ac9988cca21a75c3": "\\lambda \\sim \\mathrm{Gamma}\\left(\\alpha + \\sum_{i=1}^n k_i, \\beta + n\\right). \\!", "5cb6feea3cdc2400ccc9d0f9c5f711b0": "T_kF", "5cb706f0508ed5303766beecce7a5226": "S((aw+b)/(cw+d))=S(w)", "5cb75713d0754d0ea77bc15844d35b4a": "-4\\pi^2 |\\xi|^2", "5cb75bb6bdbb6b887794215cf61ee161": "\\displaystyle f(x) = y", "5cb8bc304e88f06bc1431d3aef1e0a6c": "TV = \\sum_j \\left| u_{j+1} - u_j \\right| .", "5cb8f4b5c887bca212e6846c5085089f": "E_n'(x)=nE_{n-1}(x).\\,", "5cb931b181ca42bc31fefe742dfe6fde": "\\sum_{i=1}^m\\sum_{j=1}^n p_j x_{ij}", "5cb9dfcd8e606e29ed24d1827df41e1a": "-\\frac{1}{n} \\log p(X_1^n) \\to H(X) \\quad \\mbox{ as } \\quad n\\to\\infty", "5cb9f1e119955e9191d0dd9c3d64e085": "r = r_1 = r_2 = const", "5cba230c35b43a91e5b84f219d182d25": "\\rho(t)", "5cba4cc9e4950209e49bfd5f6032ab1d": "\\mathbf{x}_1,\\mathbf{x}_2,\\ldots,\\mathbf{x}_k", "5cbafec141236743efcced8ef73bc7f5": " v_2 (x) =\\sum_{n=0}^{N} u_{n+N} T_n (y_2(x)) ", "5cbb344e05cf1fdd2c2134cf48979d14": "y^{-1}zxx^{-1}y\\;\\;\\longrightarrow\\;\\;y^{-1}zy.", "5cbb497bb950990332866866aaefc0fb": "\\Lambda \\subset \\mathbb{L}", "5cbb4b87fbc646e8e978173346552d79": "D:\\Gamma^\\infty (E)\\rightarrow \\Gamma^\\infty(F)", "5cbbea13b89600569886f4d17e786c8a": "\nf_X(x; k,\\lambda) = \\sum_{i=0}^\\infty \\frac{e^{-\\lambda/2} (\\lambda/2)^i}{i!} f_{Y_{k+2i}}(x),\n", "5cbc1148ccd7a6c1da888966bce78997": "\\sigma = \\frac{Mz}{I} = -zE ~ \\frac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}.\\,", "5cbc4b51924a01977abbeaf8a4c7377a": "\n\\begin{align}\n\\operatorname {snh} (u) &=-i\\operatorname {snh} (iu) \\\\\n\\text{where } \\operatorname {snh} (u)&= \\frac{H(u)}{k^{1/2}\\Theta(u)}\n\\end{align}\n", "5cbc5a80492f7966845f49ea709e4d0d": "{\\partial^2 V_{ij}\\over\\partial{x_i}^2} = {\\gamma\\over {s_{ij}}^2} {(x_j - x_i)}^2 ", "5cbc668307564c431887a0c0a860c9a3": " \\omega_i = 1", "5cbc6a35256725ebb3f59d9ada0e3b56": "f(t,x,y,z) = F(y, \\, t-z , \\, t^2-x^2-z^2),", "5cbc90310400c4723c7954659f46d84c": "y_4", "5cbd1328ae433a94cfcd7787d5386334": "\\phi = \\phi_a", "5cbd3070079cbca14eba81105e68d8c7": " x(t) = x(t) \\int_{-\\infty}^{\\infty} w(t-\\tau) \\, d\\tau = \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, d\\tau. ", "5cbd3da90cceff1ff74a341505e0f0fa": "p_2(x_{n+1})", "5cbd3f3a0dd62a557949f480a6fdcb95": "10^{-19}", "5cbd60a555362167167ac5327562d417": " L_{\\left(p-k\\right)}^{T}\\boldsymbol{\\beta}_* = \\mathbf{0} ", "5cbd64cdff31a4115557e24af97eba4f": "\\Delta t' = \\gamma\\, \\Delta t ", "5cbdbd7e9d33757a8fc6c8ba9cd00690": "1 \\cdot P := P \\qquad P \\in \\mathcal{E}(G)", "5cbee46bd071d0aaec95397c022dcb6e": "T^{IJ}", "5cbee5f60b844f4f21f75f48ac9ae180": "(s, t)\\in [0, 1]\\times[0, 1]\\to \\gamma_s(t)\\in \\mathbb C", "5cbee7208787369bd1f6c6e1475ef460": "X''", "5cbef1f541088ce5d25ae89e72604186": "|z| < 1", "5cbf0da511c285aa27f603fa96be044a": "\\dot\\psi<0", "5cbf6daf37f82d8d19eefba684486518": "G=I/V", "5cbf9c0ea64d8ba53f2d153545fbb8a8": "\\mathcal{R}^n", "5cbfb38df72ecfe2a1ef6201bcd3e307": "x^2 + y^2 = 1,\\,", "5cc009c125362b50fba5574e36b7c6b0": " 0 = \\Delta E_1 - \\Delta E_2", "5cc0bb344c6b1dce7405f391d0f4e334": "J=\\begin{bmatrix} \\dfrac{\\partial \\Delta P}{\\partial\\theta} & \\dfrac{\\partial \\Delta P}{\\partial |V|} \\\\ \\dfrac{\\partial \\Delta Q}{\\partial \\theta}& \\dfrac{\\partial \\Delta Q}{\\partial |V|}\\end{bmatrix}", "5cc0c168ae2522863df52bd93bd23ddb": "\\frac{1}{2} \\left( 1 + \\frac{2 k h}{\\sinh\\left( 2 k h \\right)} \\right)", "5cc0c6d2c0a734084fd2654947beed2a": "\\therefore e^{i\\mathbf{K}\\cdot\\mathbf{r}} e^{i\\mathbf{K}\\cdot\\mathbf{R}}=e^{i\\mathbf{K}\\cdot\\mathbf{r}}", "5cc0efb703a5b4ddfc9df0d87392a3f8": "= 2 \\gamma^\\nu - 4 \\gamma^\\nu = -2 \\gamma^\\nu. \\,", "5cc136eeff54c03c8ee6dd0a80141834": "h = \\frac{1}{2 g} \\omega^2 r^2", "5cc158be06d924796a7047afb22d39bf": "R[e_1,\\dots,e_n,v_n]/\\langle v_n^2-\\Delta\\rangle.", "5cc182f882ed2367f6c48163c7d7701b": "(+e a, 0)", "5cc1a26b13908d3c5a0b04f75394f3bd": "\\mathbf{f}(\\mathbf{x})", "5cc1b8a6f9e3809f85515c6b550693dd": "\\phi(\\theta)=-\\arctan\\left[{1\\over2}\\sin(2\\theta)\\right]", "5cc1b97144836d6e10d9fc4373d4d3c9": "(\\mathcal{M}, \\mathcal{S})", "5cc1eb51ebdc422116d7b9b810461f28": "X_{ab}", "5cc218edb6c157759f4c1db20b071624": "\nf(A,B) =\\frac{ A \\cdot B}{{\\vert A\\vert}^2 +{ \\vert B\\vert}^2 - A \\cdot B }\n", "5cc223b6b4fefa4753d94782656430b8": "\n\\mathcal{H} = -\\frac{1}{2} J \\sum_{i,j} \\mathbf{S}_i \\cdot \\mathbf{S}_j - g \\mu_B \\sum_i \\mathbf{H} \\cdot \\mathbf{S}_i\n", "5cc28353ad6609a54646b116009592fa": "\\phi_2(v)=\\phi_1(v)=\\frac{1}{6}\\,\\!", "5cc296e3dba4d92e772bcfb90d8834db": "\\tau_\\mathrm{eff}", "5cc3579b4071fd9591c1e152fd0f683f": "\\text{Posterior probability} \\propto \\text{Prior probability} \\times \\text{Likelihood}", "5cc3ca9643fe0f316e8a949a362fa506": "\n\\begin{align}\n \\uparrow_{\\alpha} t &= \\mathbf{SYN}\\ t \\\\\n \\uparrow_{\\tau_1 \\to \\tau_2} v &= \n \\mathbf{LAM} (\\lambda S.\\ \\uparrow_{\\tau_2} (\\mathbf{app}\\ (v, \\downarrow^{\\tau_1} S))) \\\\\n \\uparrow_{\\tau_1 \\times \\tau_2} v &=\n \\mathbf{PAIR} (\\uparrow_{\\tau_1} (\\mathbf{fst}\\ v), \\uparrow_{\\tau_2} (\\mathbf{snd}\\ v)) \\\\[1ex]\n \\downarrow^{\\alpha} (\\mathbf{SYN}\\ t) &= t \\\\\n \\downarrow^{\\tau_1 \\to \\tau_2} (\\mathbf{LAM}\\ S) &=\n \\mathbf{lam}\\ (x, \\downarrow^{\\tau_2} (S\\ (\\uparrow_{\\tau_1} (\\mathbf{var}\\ x)))) \n \\text{ where } x \\text{ is fresh} \\\\\n \\downarrow^{\\tau_1 \\times \\tau_2} (\\mathbf{PAIR}\\ (S, T)) &=\n \\mathbf{pair}\\ (\\downarrow^{\\tau_1} S, \\downarrow^{\\tau_2} T)\n\\end{align}\n", "5cc3e731f48375ba75b028f23f558b20": "\\mathcal{K}_r(A,b) = \\operatorname{span} \\, \\{ b, Ab, A^2b, \\ldots, A^{r-1}b \\}. \\, ", "5cc46cd67338963119bffc949e9f4210": "\\|\\mathbf{x}\\| = \\sqrt{\\mathbf{x}\\cdot\\mathbf{x}} = \\sqrt{\\sum_{i=1}^{n}(x_i)^2}.", "5cc479f4c3f2514651e524bc02d7c423": " C_\\alpha ", "5cc492ea76c79cfcabb87e0065a89587": " pH = 6.1 + \\log_{10} \\left ( \\frac{[HCO_3^-]}{0.03 \\times pCO_2} \\right )", "5cc4c83680494902ade07d29bfdcafcc": "\\mathcal L^{\\otimes n} = j^* (\\mathcal O(1))", "5cc4ebd8772ca0b202adae9caefc4697": "t=t_0, \\, r=r_0, \\, \\theta = \\pi", "5cc508a524bfc777457399413aaa7a93": "k-2p", "5cc50a90353c9f617a3ce6e2418e8e3e": " = \\frac { A_0/(1+jf/f_C) } { 1 + \\beta A_0/(1+jf/f_C) } ", "5cc55235085bd37cb10cfa1334617057": "R_1 \\cap R_2 = (A)", "5cc5f18a6fe76aff61bfc4373624fdeb": "\\beta^y = \\beta^z =0 \\,\\!", "5cc5f4184c692a030bce64cd8b49e0a7": "\\psi \\equiv \\phi \\Leftrightarrow \\mathcal{M} \\models \\forall x_1,\\ldots,x_n (\\psi(x_1,\\ldots,x_n) \\leftrightarrow \\phi(x_1,\\ldots,x_n)).", "5cc648c449e37d80cf735a7454b3de3a": "\\rho\\colon U \\to \\R", "5cc668a1308760b28a3c0ba4923379a4": " \\Gamma_{ij}^k = \\Gamma_{ji}^k ", "5cc6a89ce43304ea2765e9ca7af837c2": "d \\circ h + h \\circ d = f_1^* - f_0^*: \\Omega^k(N) \\to \\Omega^k(M),", "5cc6b29b5f31633b4bed4515ec6f5255": "E[X^\\nu]", "5cc6c175c3a9794faffec646ded8d917": "DICE=3.00 + \\frac {13HR + 3(BB + HBP) - 2K}{IP}", "5cc6fca43ad9195bb6a4c7f7a784c32d": "\nds = \\sqrt{-g_{\\mu\\nu} dx^\\mu dx^\\nu} + (B_\\mu + \\tilde{B}_\\mu) dx^\\mu\n", "5cc75031f417116b81e880a5d47b349e": "T = - L_*^{-1} U", "5cc76e530e8b905caf5523cb8544982c": "\\gamma_0=0.001", "5cc773c6cc8791da82fe93418c98d006": "f(x) = x^2(x-1000)+1.\\!", "5cc80f18bb30f138078dfc1795f91aec": "X \\sim \\mbox{Gamma}(\\alpha, \\beta^{-1})\\,", "5cc8108bba7ca24e7fec4eef8390c74d": " \\tau(x) = \\sum_{i=0}^{\\infty} (-1)^{t_i} \\, x^i = \\frac{1}{1-x} - 2 \\sum_{i=0}^{\\infty} t_i \\, x^i", "5cc83c096e29bdc465359cf11dfdefc4": "{w_C(c)}", "5cc876a798758e7682f535b3ffe1b864": "r_0 \\sqrt{1-\\omega^2 \\, r_0^2}", "5cc9b32b9dad2bceb09ac1b6f20c661e": "j=l\\pm1/2", "5cc9bfeb8db497068c2181baa5f17ce4": " a^2+b^2 = (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 = c^2. ", "5cc9f1074162ebc633f9590ed09eb183": "f(\\lambda^3 x,\\lambda y)=5(\\lambda^3x)^3(\\lambda y)^3+(\\lambda^3x)(\\lambda y)^9-2(\\lambda y)^{12}=\\lambda^{12}f(x,y). \\, ", "5ccaadbd51e9d40620fb80df6fb0df44": "E=-\\nabla \\phi - \\frac{\\partial \\mathbf A}{\\partial t}", "5ccabbaa4a2ac5a16ad95e75bdee113e": "A=[4,5,6,7,4,]", "5ccb2e2b1242ba55effcda2e07ea12ac": "\\hat f \\in L^1(\\mu). \\, ", "5ccbac96dfbf356677d79bf689c66c3e": "\\delta_s=\\delta_{i_1}\\cdots \\delta_{i_m}", "5ccc01722c7ed24071018166de235dd2": "\\sqrt{\\alpha} ", "5ccc026539071ba908ce3c01ab3e3934": "E={RT \\over F}\\ln {a_{H^+} \\over (p_{H_2}/p^0)^{1/2}}", "5ccc2049e29d9a312957db7766942fb3": "x = \\pm y = \\pm z", "5ccc32da406c8092e96e4a752a7e409a": "\\bar{\\boldsymbol{B}}", "5cccfa1793b4aebf2e3867c4fd610f8b": "\nH = \\left(\\begin{array}{llllcllllcllll}\n1&0&1&1 &~& 0&1&0&0 &~& 1&0&1&1 \\\\\n0&1&0&0 &~& 0&1&1&1 &~& 0&1&0&0 \\\\\n0&1&1&1 &~& 0&1&0&0 &~& 1&0&1&0 \\\\\n1&1&0&0 &~& 1&0&1&1 &~& 0&0&0&1 \n\\end{array}\\right)\n", "5cccfa6f71efa12e85fffc836f509b95": "{}_pF_q(a_1,\\dots,a_p;b_1,\\dots,b_k-1,\\dots,b_q;z),", "5ccd0da7269b2f4d1fcd342b8b095b43": "{}^{j^n(\\kappa)}M \\subset M.\\!", "5ccd718215d122e046d1573f61b1f376": "\\frac{V_o}{V_i}=\\frac{-D}{1-D}", "5ccd9d94a00a34f40cd4972cc51bb7c3": "|i-j|", "5ccda4d9640cfe9cbec8ad8a206e653f": "p\\nmid xyz.\\;\\;", "5cce3513c1a3bbfd555a3fc371b4abfa": "\nT_v\\exp_p \\colon T_pM\\cong T_vT_pM\\supset T_vB_\\epsilon(0)\\longrightarrow T_{\\exp_p(v)}M.\n", "5ccf4aaa86c89e2c6faf4f5f93ba1fd3": "\\bar{j^\\alpha}", "5ccf802baa0cb853710ed6779d791fd5": "(r_i r_j)^{c_{ij}}=1", "5ccfc5f7ddd6d2d7c7268d64b855d7d3": "\\log{n \\choose k} \\sim n H\\left(\\frac{k}{n}\\right) ", "5cd00aac6a2e6f8ae4f3baca79855642": "E_T = \\hbar D /L^2", "5cd0f1145a2d1c8cec57bf91ec13c27d": " = 1 - \\frac{1}{(1 + e^{y} - 1)^{\\theta}} ", "5cd1361631ac4f78c6e924a3554ab5bb": " \\phi_i ", "5cd1a524138d40ec3b2a6cf46e208727": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\cos^2 u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin u\\ \\cos^2 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\cos^3 u \\ du\\ = \\\\\n&-\\hat{g}\\ 2\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u\\ \\cos^2 u \\ du \n+\\hat{h}\\ 2\\ e_g\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^4 u \\ du =\n-\\hat{g}\\ \\left(2\\pi \\frac{1}{4}\\ e_h\\right) + \\hat{h}\\ \\left(2\\pi \\frac{3}{4}\\ e_g\\right)\n\\end{align}\n", "5cd1b645d6485f3c0f865d68941e30d6": "\\overline{\\boldsymbol{u}} = \\frac{\\boldsymbol{M}}{\\rho\\, \\left( h + \\overline{\\eta} \\right)}\n = \\overline{\\boldsymbol{v}} \n + \\frac{\\boldsymbol{M}_w}{\\rho\\, \\left( h + \\overline{\\eta} \\right)}.", "5cd1f1fecc4752828f0da636d9f48b09": "\\{0,1\\},", "5cd22aee9b70a2d6cb7f3c15e8cdcbac": " K_\\infty = \\bigcup K_n ", "5cd242a9447e851c9f386af581db5e6d": "E_G \\varphi \\Leftrightarrow \\bigwedge_{i \\in G} K_i \\varphi,", "5cd248c4da675cd34d7487dde291e735": "{\\Delta s}", "5cd2896c73a6bbbc1668af8a266fc845": " (1-2t(n-1))", "5cd28be1ef5a9ee1155a479d3774200a": "(a_0,a_1,\\ldots,a_q)", "5cd3003835415bb28ddb8ee42562a0e9": "r = f(\\theta) \\pm \\sqrt{(f(\\theta) \\cos\\theta - a)^2 + (f(\\theta) \\sin\\theta - b)^2}", "5cd34a69c685d3c0886309e98a797b35": "Z_{TE}=R_{TE}+jX_{TE}=\\frac{jX_m(R_s+jX_s)}{R_s+j(X_s+X_m)}", "5cd375f9d18a7fda542c1c80b37430d9": " \\sqrt{(a-b)^{\\top}S^{-1}(a-b)} ", "5cd37655dee1601e8971c00e9e6e4257": "\\frac{d^2 x^i}{{d t}^2} \\approx - \\Gamma^i_{0 0} ", "5cd3912c854e5909f92fa59152215110": "m(t) = \\mathbb{E}[X_t] = \\mathbb{E}[\\mathbb{E}(X_t \\mid S_1)]. \\, ", "5cd3a0530de01903facfad52b6320a0f": "G_0(F) = \\operatorname{Gode}(F)", "5cd3b1ee07bbeb9bb6db05fdd6f8ba8f": "\\sum_{E \\subseteq \\mathbf{X}}M(E)=1.", "5cd43ab24fbe61b5e6084f0f1a00367c": "\\exp\\left(-\\int\\alpha(x)\\,dx + i \\beta_{w} x\\right)", "5cd44ac802be9a95cce6ab110341430a": "~\\Delta_{\\rm perp}=\\left(\n\\frac{\\partial ^2}{\\partial x^2}+\n\\frac{\\partial ^2}{\\partial y^2}\n\\right)\n~", "5cd493d6568ce0926d655ac186b5c566": "Y_{c}", "5cd4e259a850338d2d6a97565550ea74": "t_2^\\prime =t_2+ \\frac{D_L}{c}", "5cd4eb424d2f715e83b9f4f46a14e173": "\\beta_X \\, ", "5cd5424ed6ba8d6feb7cde8d36b4d204": "\\frac{\\beta p}{\\rho}=1+\\sum_{i=1}^{\\infty}B_{i+1}(T)\\rho^{i}", "5cd5618c9155e48f383d6ef947f8b192": "e = e(T)", "5cd56f61d784d82e7d4559e67c0222e9": "\\overline{y}(\\lambda)", "5cd59eeebd42973055324bc6df8ee750": "-i = - {e_1} {e_2} {e_3}", "5cd5ce7dd0d32f5e42a0376756e89000": " \\chi = \\begin{pmatrix}\n\\chi_1 \\\\\n\\chi_2 \\\\ \n\\end{pmatrix} ", "5cd5de7ada512afadf8e30c0224c9c30": "\\{X_1, X_2\\}", "5cd6184ea50cdf461bf67b2f051c047f": "c_{x_0}", "5cd6623d4cbf75068e5af369fa6952f0": "y(x) = f(x) \\quad \\forall x \\in \\partial\\Omega", "5cd66ab3ce5b360b713b9e97a29d3682": "R=R_0", "5cd674f504d35565a269758ba5037858": " \\eta_p = \\frac{Acutal Power (P)}{Ideal Power (P_i)} = \\frac{F_x C_u}{\\frac{1}{2} \\rho AC (C_s^2 - C_u^2)} = \\frac{C_u}{C} = \\frac{1}{1 + a}", "5cd6a457096326f4b3d7e8d1e74118a5": "\\textstyle \\sqrt{2}", "5cd6b41c230a349421ef5ed9a0a34da4": " I= \\sum_i \\frac{I_i}{N} ", "5cd6b54630c7b15f844c7906837bb4b3": "I = \\int \\left[(x - r)^2 + y^2\\right] \\, dm", "5cd731045fc3e87dd9fb0b06b53ae4d2": "\\left[{D(K_a)} ,{D(K_b)}\\right] = -i\\varepsilon_{abc}{D(J_c)}", "5cd764aebbbab086d4361f77fe1f42e6": "b_1\\,", "5cd7dce85b6b79220a127b61cb281300": "\\displaystyle{d(x,y)=\\inf_{\\gamma(0)=x,\\gamma(1)=y} \\ell(\\gamma)}", "5cd7e855a6639f8fe34a50d59d4841e1": "P^{(t)} \\ \\sim\\ \\N(-\\alpha t, \\beta^2 t )", "5cd80bf6f9d375a17eb5e3ce78591364": " \\mu_{-1}(S) = \\mu(S^{-1}) \\quad ", "5cd846775847ce9095430f21e960e957": "\\textstyle{\\prod_{i \\in I} V_i}", "5cd8750c978aa5acc7a1222e9957792e": "Z_{V,Q}(s)", "5cd8d1b1619ac5d97c7fea980efe28fa": "1 \\le i \\le N", "5cd9031fc147f94aa724f83df47577ce": "\\ \\ \\ \\ f(x)", "5cd90870eb31ceda62842aef40cbad47": " \\mathbf{x} \\leftarrow \\mathbf{x} - E\\left\\{\\mathbf{x}\\right\\} ", "5cd9185df8306047824126be5819950e": " 2k_1 + 4k_2 + \\cdots +2nk_n=2n", "5cd92cb6a01ef07037e6f301fc09f8d4": "xy \\equiv zw \\rightarrow xy \\equiv wz\\,", "5cd99f689c3ae19536aa9c730755dfbb": "\\Sigma_{ij}", "5cda29d91c55a2a5547f22f2b423f350": " \\mathrm{FWER} = 1 -\\Pr(V = 0).", "5cda31e86c39aa2bf5430c642771969a": " (3 a_2)^2 + (3 b_2)^2 = 3 \\cdot (s_1^2+t_1^2) \\, ", "5cda3716e9ec3743c8c41fe85415c053": "\\mathbf{\\tfrac{n}{m}\\tfrac{2}{m}\\tfrac{2}{m}}", "5cdaffa14f97d3aed2aa115af1eda679": "\\frac{Wins+\\frac{1}{2}Ties}{Games}", "5cdb0749efd70cb5df3f0952b84508ae": "\\chi(G)", "5cdb0ce695e84804f190256410ee4056": " d( f^n ) = n f^{n-1} df ", "5cdb400a6a612e5b9cdb389aedc7e4f2": "V_{1,2} = V_{1,3} = V_{2,3}", "5cdb995f52818415ab5901d59e798dec": "f(x,t)\\,", "5cdbbb7be4979a192f70ddbf1996a02c": "= \\int u+\\left(-v\\right) \\,dx", "5cdbbc449d467a43f08e4fd1af0b91c3": "1/\\Lambda^{3}", "5cdbecd8a98a10ba07a9d9077319f832": " M^{G} = \\lbrace x \\in M \\ | \\ \\forall g \\in G : \\ gx=x \\rbrace. ", "5cdc003471d4ff1dcc5dc795d2b56957": "\\frac{d}{dt} \\int_{\\Omega} u d\\Omega + \\int_{\\partial\\Omega} \\vec f(u) \\cdot \\vec n d\\Gamma = 0,", "5cdc188667b6ceeed7b0c966951d4a94": "\\left(\\sum_i \\hat n_i \\hat n_i^\\top\\right) x = \\sum_i \\hat n_i \\hat n_i^\\top p_i", "5cdc3f9de33a38f1dbd1d95c80795fbe": "\\gamma \\, \\equiv \\, \\mbox{Tr}(\\rho^2) \\,", "5cdc4edf9b53435a68fda50e473cd3b6": "\\mathbb{C}^n \\oplus \\cdots \\oplus \\mathbb{C}^n = \\mathbb{C}^n \\otimes \\mathbb{C}^m .", "5cdc513ed00690bcf337cf46da9d84c7": "C=\\Delta_{\\alpha<\\kappa} C_\\alpha", "5cdc98e07f441a75ff571cc0d56399a6": "c_1 = b_1[v_1,v_2] + b_2[v_2,v_3] + b_3[v_3,v_1]", "5cdd0123ccc4d862f72d7f6c1a991b1d": "\\omega_N^N = 1", "5cdd023959630c56f453b1937bff643d": "0.14076043434\\ldots", "5cdd0284168399251a70d0ff6e2f6f36": "\\omega_\\beta=2\\pi f_\\beta", "5cdd628c8233763653ce8af5b9b1ba70": "\n\\bar{V} \n = {1 \\over L} \\oint_\\ell\n V(\\boldsymbol{y}) \\; dy \\, \n = -2k_e Q\n {1 \\over L} \\oint_\\ell \\oint_\\ell\n \\ln \\vert \\boldsymbol{x} - \\boldsymbol{y} \\vert\n \\; dx \\; dy .\n", "5cdd7052807515a10cc1c51dd1bf1ae9": "g(x) \\leq f(x) \\leq h(x) \\, ", "5cdda5ca49d1d15720409038368ec55f": "S_k(r) = \\sum_{n=0}^\\infty \\frac{(-1)^n k^n r^{2n+1}}{(2n+1)!} = r - \\frac{k r^3}{6} + \\frac{k^2 r^5}{120} - \\cdots", "5cddd4ae6b40ca76e95c6cfe94671587": "\\phi\\in T", "5cde24597651457a66caf9137126a7cf": "d(t) = 390 \\sqrt{t} + 2500", "5cde4f09191ab86c781366052711c968": "(\\frac{x^3-y^2}{N}) = 1", "5cde5635eb04fe3f9f67603d776d1144": "G =", "5cde65eefa5148b09356fabaedebc347": " = G_{\\infty} \\frac {T} {1 +T} + G_0 \\frac {1}{1+T} \\ ,", "5cde67f7dd11730f7bf64e4de28ce7cc": "\\sum_{t=1}^{\\infty} a_t = \\infty ", "5cdf0dcd5bc03877159355d4854e1d70": "G = \\operatorname{Gal}(L/K)", "5cdf2bc355f90e996c5c6cca01a4e1e0": "\\varepsilon \\sim \\operatorname{Logistic}(0,1) \\, ", "5cdf3fa6b671d3b014445ab4147d638f": "\\operatorname{pf}(A) = \\frac{1}{2^n n!}\\sum_{\\sigma\\in S_{2n}}\\operatorname{sgn}(\\sigma)\\prod_{i=1}^{n}a_{\\sigma(2i-1),\\sigma(2i)}", "5cdf9cb7aa0d512e26e0137e88300c7e": " S = S_1, S_2, \\ldots, S_m ", "5cdfca22848ed835367df5aeb8ce4920": "aa^{-1}a^{-1}", "5cdfdea98a654562d2a1828ea4bd5fa5": "f(n)=f(n-1)+\\sum^{n-1}_{i=1}\\left(2-n+ni-i^2\\right)", "5cdff1aa1ed0563b0d68a64b064a1eb0": " \\beta \\gamma = \\sinh\\phi = { e^{\\phi} - e^{-\\phi} \\over 2 },", "5cdff617c2b3cfd5af4bfe52de6f0078": "C = 1 - \\frac{1}{5 + 2\\times 5 + 5 + 2\\times 5 + 10 + 0 + 3 + 4} = 0.98", "5ce03973a7fd7dc4da337756bd3f4e02": "d(x,y)=|x-y|_p.", "5ce05834e6c4259bdf393f73ef414ff6": "(\\text{sample skewness})^2 = \\frac{4}{(2+\\hat{\\nu})^2}\\bigg(\\frac{(\\hat{c}- \\hat{a})^2}{ \\text{(sample variance)}}-4(1+\\hat{\\nu})\\bigg)", "5ce08d61e2ca721a4d506ba0d011c37a": "\\|A\\|_{\\text{max}} \\le \\|A\\|_2 \\le \\sqrt{mn}\\|A\\|_{\\text{max}}", "5ce0abc7579dceb1cd50345b949e5306": "\\xi_j \\in \\mathfrak g", "5ce0af1058e3444c1f6239435bbb5be1": "\\begin{align}\\pi &= T^2M^0L^{-1}g^1\\\\\n &= gT^2/L\\end{align}", "5ce0f02f95ab276861233a4a8ae39757": "c_k(E)c_{n-k}(F)=(-1)^{n-r}", "5ce13f807d605160d054a04cbaf388aa": " |x-x_0| < \\delta ", "5ce16a78fb200155908b0368fd3680fa": "\\alpha \\in (0,1)", "5ce1eee18269fa5d0fdaabdd802680b6": "\\tan\\varphi=\\frac{dy}{dx}=\\frac{AP}{TA}=\\frac{AN}{AP}.", "5ce28618f88cc2fe9f1d20c0055e0d5a": " \\ \\mathbf{f}_p ", "5ce2cfa6ff06a3257ba5581ea4a0cca3": "{DV}/{Dt}", "5ce32679df153065849eeed9fdc33113": "0= \\sum_{i=0}^k (-1)^{k-i} e_{k-i}(x_1,\\ldots,x_k){x_j}^i \\quad\\text{for }1\\leq j\\leq k", "5ce3577abe23d24912d8ba9daca9af5d": "::=", "5ce3a765a24291c5305985c9c3073798": "Y_1\\text{ and }Y_2 \\, ", "5ce42ca670f19475a621cdc736b517cf": "(a_L)^\\alpha_{ab}", "5ce4425f0b86c74e139fd493c6d8ecd2": "\\mathbf{H}^{(E)}(\\mathbf{x})=\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell a_{\\ell m}^{(E)} h_\\ell^{(1)}(kr) \\mathbf{X}_{\\ell m}(\\theta, \\phi)", "5ce44baccbf9e20cdf566d2745ed3582": "\\bigcirc\\!\\!\\!\\!\\!-", "5ce4cf8ed147d398a1c4e4fa0cc4785f": "P_{wc}(\\theta;\\psi)", "5ce4db3e7fd0f8ce3f5057754f8005c1": "\\bar{e}=\\frac{(V_t-V_0) \\cdot \\hat{r} - V_r \\cdot \\hat{t}}{V_0}", "5ce4df026fe7f824899db646ee8ac7f6": "\\operatorname{AveP} = \\sum_{k=1}^n P(k) \\Delta r(k)", "5ce576c33c0898bd3be00180648f6840": "r_1 = $100 * 1.43 = $143", "5ce59821878e37ab00207d28d4e86556": "V(t) = V_0 \\left(1-e^{-\\frac{t}{\\tau}} \\right)", "5ce598695699af5564a087f30d6a2982": "\\color{Black}\\tfrac{2}{m}\\tfrac{2}{m}\\tfrac{2}{m}", "5ce5fbe7c48dca302e287d12830b771e": "\\tfrac{5}{18}", "5ce60970159dde2ac7df664b6f3b844e": "g'(\\beta)= -\\frac{\\psi(1/\\beta)}{\\beta^2} - \\frac{\\psi'(1/\\beta)}{\\beta^3} + \\frac{1}{\\beta^2} - \\frac{\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta} (\\log|x_i-\\mu|)^2}{\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta}} + \\frac{(\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta} \\log|x_i-\\mu|)^2}{(\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta})^2} + \\frac{\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta} \\log|x_i-\\mu|}{\\beta \\sum_{i=1}^{N} |x_i-\\mu|^{\\beta}} - \\frac{\\log(\\frac{\\beta}{N} \\sum_{i=1}^{N} |x_i-\\mu|^{\\beta} )}{\\beta^2} ,", "5ce66b653ca52fb9a3b62287464fe8f7": "[H,J_i]=0, [H,P_i]=0, [H,K_i]=i P_i.\\,\\!", "5ce68a2c490135425ece1853ef1a1311": "\\frac{|m_p-m_\\bar{p}|}{m_p}", "5ce68c32169c124db8649f98ef618b50": "d(w) < d(u)", "5ce6e9bd77a080a372fccb61a755184d": "(1)\\; L = 220 \\times y_1 \\times \\tanh\\frac{Fr_1 - 1}{22}", "5ce6eb4c8364db349ddbe5c3f42d2d13": "\\mathbf{Z} \\to H_n(M,\\mathbf{Z})", "5ce702b476d9f4207e53626e18a19b3b": "\\frac{1}{M} \\{(n,k) | N \\leq n \\leq N+M, \\,", "5ce7178dd35620c148954d00e6792072": "X=m\\frac{dU}{dt}", "5ce772cfa2a630aa96cd28d14288d28d": "\nk^{2} = \\frac{u_{2} - u_{1}}{u_{3} - u_{1}} \\approx r_{s} \\left( u_{2} - u_{1} \\right) \\ll 1\n", "5ce78ad8f76e56b955e5c0e710903545": "\\ \\tau^x", "5ce79d5577ce4e982916987ea9652149": "i_n = i_r + p_e\\,\\!", "5ce7e9694f64eb33ccfa11f18b724b7b": " |x-x_0|\\ge \\delta ", "5ce7fc0f4ad2a6c85dc16af685dd3d9b": " \\delta_i = \\mu_i - \\mu ", "5ce86ff4ea43e08fcc5780e3bbc20b72": "\\neg P \\rightarrow \\neg Q", "5ce87ced437e0b6652799f56889d1284": " n = 5 ", "5ce891be3cc249a46247c369c07d88dd": "a_ny^{(n)} + a_{(n-1)}y^{(n-1)}+...+a_1y' + a_0y = 0,", "5ce8ece59f2ee3e1b360f22ce3778ef4": "a = a\\wedge b", "5ce9a39c60f4937376a54f94df49f5f3": "{B}_{2+}", "5ce9d683d50cf6d2db1a79bc4d520c75": "\n||E(\\mathbb{F}_q)| - (q+1)| \\leq 2 \\sqrt{q}. \\, \n", "5ce9f7862cf1bbe9595d2a53fd753987": "\\mathbf{v} = \\mathbf{e}_{124} + \\mathbf{e}_{235} + \\mathbf{e}_{346} + \\mathbf{e}_{457} + \\mathbf{e}_{561} + \\mathbf{e}_{672} + \\mathbf{e}_{713}.", "5cea157a53bf7a62fad6794778804dda": " a=\\frac{u^2-1}{u^2+1}, b=-\\frac{(u-1)^2}{u^2+1}", "5cea6bc2cf84196aadaa7e034e61f068": " \\gamma\\ = \\frac{5}{3} \\approx 1.67", "5cea943eb59d49940df15d0f3c6f5715": "\n\\begin{array}{c|ccc}\n0 & 0 & 0 & 0 \\\\\n1/2 & 1/2 & 0 & 0 \\\\\n1 & -1 & 2 & 0 \\\\\n\\hline\n & 1/6 & 2/3 & 1/6 \\\\\n\\end{array}\n", "5cea9be8153caf17be67ca86c7819c81": "\\operatorname{ker}(\\operatorname{Hom}(\\partial, G)) / \\operatorname{im}(\\operatorname{Hom}(\\partial, G))", "5ceacd6ac86002408f5a7cf65ecf3288": " Gal(L/K) ", "5ceaf0796b05ea2f2b0fcfe5246915f1": "\\psi_{j,i} \\psi_{j,i}^*", "5ceb057ad10c47abfba4e57c74bd7b43": "S(\\omega)", "5ceb064ce20f9b82e5b759cf70bf2014": "\\Delta^\\text{w}_\\text{o}\\phi^\\ominus_\\text{ET} = \\left[E^\\ominus_{\\text{O}_2/\\text{R}_2}\\right]^\\text{o}_\\text{SHE} - \\left[E^\\ominus_{\\text{O}_1/\\text{R}_1}\\right]^\\text{w}_\\text{SHE}", "5ceb2a518d9a1971b734b86073022aae": "D_c", "5ceb41666b5a88eb4d561a6ddc713e3a": "3N = 3 + 3 + (3N - 6)", "5ceb6a3d1d9b604b922fa575df7b1074": "\\frac{y(s)}{phi(s)} = \\frac{0.07s^3(s+50)}{(s+0.05)(s+0.03)}", "5ceb6ddfa69cbd054af128de67bf8bb3": " f_{Dopp} = \\frac { 2 \\cdot \\nu_0 \\cdot v_s } { c } \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (3) ", "5ceb809c35c62f09e0f2ea30503ec1c0": "N = A - Z", "5cec20ff0f7281cbf0b684ae68e96a0f": "A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2 = 1.", "5cec62fce2fe27a8bd4ace1398f0496b": "\\operatorname{cl}: \\mathcal{P}(E)\\to \\mathcal{P}(E)", "5ced1597bf7809779af11dc74d3e69b4": "\\theta_2>0", "5ced1b29bf78cd11b012a50936cc8583": " HC_i(A) \\cong K^+_{i+1}(A). \\, ", "5ced629ec4360b26cb25ac03f76f9772": "C_{111}=2A+6B+2C", "5ced8407934f70be89e8a82a5e321a79": "d = L - C_s", "5cedf38fc9933594b822143ae04a0a7f": "\\textbf{Cl}=\\{Cl_t, t \\in T\\}", "5cee388e745f821df98a6adf4191f217": "f(x,y)=0\\,", "5ceeb306fc4b469ac3ea0f91bc8ba22c": "\n \\mu =\n-e\\varphi + \\epsilon_F\n", "5cef36db962a868f07f7d40f49d80fca": "F_{top} = - P_{top} \\cdot A.", "5cef57efa625faaa938b573b1764fbf0": "\\hat{t} = \\sqrt{6}\\, t,", "5cef7b0dd9f1481793534f27be618797": "\\mathbf{R}_{N \\times 1} = \\mathbf{b}_{N \\times M} \\mathbf{Q}_{M \\times 1} + \\mathbf{W}_{N \\times 1} \\qquad \\qquad \\qquad \\mathrm{(3)} ", "5cef8ba1d7343008eba635ec315c3b0f": "X\\mapsto MXM^+", "5cef9919eaed04b01dcda4873239d221": " Z_r = \\frac{2 I k_B T}{\\sigma (\\frac{h}{2 \\pi})^2} ", "5cefa15c52a48c2d525c395807ecd4da": "S = \\cfrac{BH^2}{24}", "5cefa4adae19e5be91e1744a9f19da33": " \\text{real cost (production)} = M \\cdot \\cfrac{\\int_{T_1}^{T_2} \\frac{dC}{dt} \\, dt}{\\int_{T_1}^{T_2} \\frac{dP}{dt} \\, dt}", "5cf0388d57218267fc3176a2d8f06f40": "\\lambda=\\infty.", "5cf07c6d301581fca9f8e430819152db": "=c_0 + c_1x + \\left(c_2 - {c_0 \\over 2}\\right)x^2 + \\left(c_3 - {c_1 \\over 2}\\right)x^3+\\left(c_4+{c_0 \\over 4!}-{c_2\\over 2}\\right)x^4 + \\cdots\\!", "5cf090ba09df4f5f5c5ddfbc3f004a18": "\n \\hat\\beta\\ \\sim\\ \\mathcal{N}\\big(\\beta,\\ \\sigma^2(X'X)^{-1}\\big)\n ", "5cf0ebf3609b008ee9577a57e31af39c": "P_n^{(\\alpha, \\beta)} (1) = {n+\\alpha\\choose n},", "5cf1110f2e2a8c837cacbbdb11b7ab54": "f\\left(\\frac{n}{m}\\right)=f(1)^{n/m}=e^{k(n/m)}.", "5cf1257704d7b99bd1cda918c8ac3156": "(\\lambda_1,+\\lambda_2+\\cdots+\\lambda_k)", "5cf15b4530d15cbf34b8a0d899dda89c": " |\\Psi \\rangle ", "5cf2331504972f59bbde79ffa4ae3b60": "G_{0} = G\\ \\Big |_{T \\rightarrow 0}\\ .", "5cf2a19689fb85926ef63a13582e1a26": "\\ f \\le g \\quad\\iff\\quad \\forall x: f(x) \\le g(x)", "5cf2ea58c7f1fba8f4de9635830f2b00": "\\nu >6\\!", "5cf3adccf95decbfe78450fffa0da178": "80\\frac{1}{5}", "5cf3b003cf2da30769cb021495ae30d7": "w \\approx 1298\\AA", "5cf41985235e14c9e278f85d0d97d9d9": "\\qquad\\mathcal{T}_1(A) =", "5cf4e9a5fc94c24c99fabb5e5d43fd11": "e^{{\\delta}G}=\\prod_{even \\ \\ l}e^{{\\delta}G^{[l]}}", "5cf532597e33c751fc8ff44ffa0cfefb": "H^*(\\mathbf{RP}^\\infty; \\mathbf{Z}/2\\mathbf{Z}) = \\mathbf{Z}/2\\mathbf{Z}[w_1],", "5cf55cc0de8ecbe8c81c79c0c08787e0": " {v^2 \\over 2}+gy+{P \\over \\rho}=\\mathrm{constant} ", "5cf59ba9e311f2cf8fd8c3cf5ac89ff3": "1+\\frac{1}{2}+\\cdots+\\frac{1}{N}\\geqslant\\frac{c+v}{\\alpha}\\,\\!", "5cf5e1748dfcb03d6d03159291b4a11a": "E_{12}=\\frac{1}{2}\\sqrt{2E_{11}+1}\\sqrt{2E_{22}+1}\\sin\\phi_{12}\\,\\!", "5cf65f2553fa666382b8fd0970e67265": "x_2\\in\\mathfrak{g}_{\\lambda_2}", "5cf7208a4ad28d5914a55fe28e71d5bd": "w = 32", "5cf74a09e97f7abb70c926be74f58940": "(5+i)^4 (-239+i) = -2^2(13^4)(1+i)", "5cf777d31195f771978f013291ff59fd": "M_X^c + M_X^{gi} = 0", "5cf7af7ea9eeb8a3d21ed9dd9b7ead03": " \\psi = e^{i \\Phi_\\lambda / \\hbar} ", "5cf7e7aa8f2f686321962298af0f2db4": "{\\mathcal{O}}(G)", "5cf7e8be36e09f88031a1a27cb734bf0": "{\\color{red}1}x^6+{\\color{red}6}x^5y+{\\color{red}15}x^4y^2+{\\color{red}20}x^3y^3+{\\color{red}15}x^2y^4+{\\color{red}6}xy^5+{\\color{red}1}y^6 \\,", "5cf7f746f681511c2e3d4b29488b48a2": "[\\mathbf{k}]_\\times ", "5cf876df87a71d0b199e70b0fd8b8d7f": "2^{\\log^{1-\\epsilon}(n)}", "5cf8c69960790469c7c6ddc97afbe150": "\\mathcal{S}_{drs}", "5cf8d73ed546a0f7b6d5e1ed0d1dafde": "a_n \\ge b_n \\ge 0", "5cf9141436700d639a6dd9990cc86dd6": " q \\equiv \\Box(p\\rightarrow q)", "5cf92ec37c5bf03509ca09e0209c6fed": "|E(\\mathbb{Z}/N\\mathbb{Z})| = N + 1 - \\pi - \\bar{\\pi} = N + 1 - a. \\, ", "5cf995dc69db16c777401a43806bcf43": "\\kappa =1\\, ", "5cfafdc9c38289c0c074f020348a5ea8": "g_{00} = -1 \\!", "5cfb423a4090e920c2ace78b26814d7e": "x(t) = \\sum_k [a_k \\cos (2\\pi \\nu_k t) + b_k \\sin (2\\pi \\nu_k t)]\n", "5cfb63ce2f03faefe3a6057c9c02bf0c": "h(N_m) \\leq h^*(N_m)", "5cfb73465d6539528da36715fac54be3": "\n\\begin{alignat}{7}\n c&{}={}&1+2&&{}+3+4&&{}+5+6+\\cdots \\\\\n4c&{}={}& 4&& {}+8&&{} +12+\\cdots \\\\\n-3c&{}={}&1-2&&{}+3-4&&{}+5-6+\\cdots \\\\\n\\end{alignat}\n", "5cfba9501afc1244166282e19ca1343d": "x = 10^n \\cdot y + z \\equiv z \\pmod{2^n \\mathrm{\\ or\\ } 5^n}", "5cfbcbae7e170e8746eda41f4bdb3166": "[(\\gamma_2)_\\mu (p_2-\\tilde{A}_2)^\\mu+m_2c + \\tilde{S}_2]\\Psi=0.", "5cfbee94d0714d57448ed28851b14f1e": " x^a ", "5cfbfc9563c558f390e308a4a3ecd6de": "H^p_{{\\acute{\\rm e}{\\rm t}}}(k,\\mu^{\\otimes q}_\\ell)", "5cfc35368a76be6110428289c95a3a3a": "h\\in H]\\leq 2(\\frac{em}{VCDIM(H)})^{VCDIM(H)}\\cdot e^{-\\frac{\\alpha^{2}m}{8}}\\,\\!", "5cfc383d1c2ba277742a31bbe3132172": "t\\in \\left[ 0,1 \\right]", "5cfc3dd7eae7624836a00f15f69bd25f": "n \\ge b^{l - 1}", "5cfc63453c36d80f8df0b3beaf9b40c3": "w(x) > |T_{j-1}| = 2^{2^{j-1}}", "5cfd4a795c512d4e745916eab0f489f5": "g^{(k)}", "5cfdc493e84467b03bdca786eec4141a": "s\\in\\{L,S\\}", "5cfdec6bb8fcde479646221eee1da6da": " a=(35^2-1^2)/2=612 \\,,\\ b = 70/2=35 \\,,\\ c = (35^2 + 1^2)/2=613 ", "5cfe67c1b35a4021ef3b0097adeb1e8e": "\\left \\{a, b \\right\\}", "5cfe6c6b6744672dc8b4478007e3bf82": " \\sigma_{rr} ", "5cfe813f83771954d97c8e661360bed9": "\\vdash A \\lor \\neg A", "5cfe8965e238763717b6dd1d87cd34db": "\na_n \\, = \\,\\sum_{k=1}^n \\frac{\\binom{n+1}{k-1}\\binom{n+1}{k}\\binom{n+1}{k+1}}{\\binom{n+1}{1}\\binom{n+1}{2}}\n.", "5cfed2082a373e3fed9cafd361541ee2": " p(\\theta, \\psi) \\, d \\theta d \\psi = \\frac{\\sin \\theta \\, d \\theta \\, d \\psi}{4 \\pi} ", "5cfef8165b13278a1545e27547e759e3": "\n\\begin{align}\n &(1 - t)^2 \\mathbf{P}_0 + 2(1 - t)t\\mathbf{P}_1 + t^2 \\mathbf{P}_2 \\\\\n = {} &(1 - t)^3 \\mathbf{P}_0 + (1 - t)^{2}t\\mathbf{P}_0 + 2(1 - t)^2 t\\mathbf{P}_1 \\\\\n &+ 2(1 - t)t^2 \\mathbf{P}_1 + (1 - t)t^2 \\mathbf{P}_2 + t^3 \\mathbf{P}_2 \\\\\n = {} &(1 - t)^3 \\mathbf{P}_0\n + (1 - t)^2 t \\left( \\mathbf{P}_0 + 2\\mathbf{P}_1\\right)\n + (1 - t) t^2 \\left(2\\mathbf{P}_1 + \\mathbf{P}_2\\right)\n + t^{3}\\mathbf{P}_2 \\\\\n = {} &(1 - t)^3 \\mathbf{P}_0\n + 3(1 - t)^2 t \\left( \\frac{\\mathbf{P}_0 + 2\\mathbf{P}_1}{3} \\right)\n + 3(1 - t) t^2 \\left( \\frac{2\\mathbf{P}_1 + \\mathbf{P}_2}{3} \\right)\n + t^{3}\\mathbf{P}_2\n\\end{align}\n", "5cff4dc0cfe627655a6099e48b2607e0": "L^{norm}:=I-D^{-1/2}AD^{-1/2}", "5cff9f8b93578cbb73abc27f03531188": "\\begin{array}{ccc}[a,b]&\\longrightarrow&\\mathbb{R}^2\\\\t&\\mapsto&\\bigl(f(t),g(t)\\bigr),\\end{array}", "5d0004e25da89e43ff011668b961d0f2": "\\operatorname{prob}[n_{12} | n_1] = \\frac{{n\\choose n_{11}, n_{12}, n_{22} }} {{2n \\choose n_1, n_2}} 2^{n_{12}}, ", "5d009b6b7209c840e62174ca01f32391": "u(t) = p + (x(t)-p)(1 \\pm n(t)),\\ v(t) = q + (y(t)-q)(1 \\pm n(t))", "5d00d7c912084f9cb85c0767e18e05c6": " \\mathbb{F}_q", "5d0131593386c07885c196987d18eecb": "C_{i,j} ", "5d01a443011504498c56e1f540b76888": "f(x+y) = f(x) f(y)\\text{ for all }x\\text{ and }y \\,", "5d01b75b6ee503945af6b340859a19b1": " > 0 ", "5d01eabf967eae9f3c2ddc6c1003d823": "F_Y(y) = \\operatorname{P}(g(X) \\le y) = \n\\begin{cases}\n\\operatorname{P}(X \\le g^{-1}(y)) = F_X(g^{-1}(y)), & \\text{if } g^{-1} \\text{ increasing} ,\\\\\n\\\\\n\\operatorname{P}(X \\ge g^{-1}(y)) = 1 - F_X(g^{-1}(y)), & \\text{if } g^{-1} \\text{ decreasing} .\n\\end{cases}", "5d020a68b87e7fe98d5eeb17c210ad0a": "o(\\sqrt{n})", "5d022250e1ef8945ce55d4aad010e47e": ") = ", "5d02992f386822eeae80981584331d04": "b = b_1\\ldots b_m", "5d029ebde1a858d27aea2d78c3c7d7bd": "\\varphi(v\\otimes z) = z\\varphi(v).", "5d02a2e5420f1b265cb314f14a6921c5": "[0, \\infty].", "5d02a97f0d70756843423f71b0a4921c": "\\dot{\\boldsymbol{\\sigma}}", "5d02b916404adb3ee15b59d7835dca3d": "E(\\text{Payment}~|~\\text{Player 1 wins})P(\\text{Player 1 wins}) + E(\\text{Payment}~|~\\text{Player 1 loses})P(\\text{Player 1 loses})", "5d030998623cc3d302f2593c72fbd217": "ds^2=0=-c^2dt^2+\\frac{a^2 dr^2}{1-kr^2}", "5d0326243c7623cd70e390fb29cb3191": "\\int\\limits_{-\\infty }^{\\infty }{Pf(u,\\xi )}.du={{\\left| \\hat{f}(\\xi ) \\right|}^{2}}", "5d03ab80614215e70c7875d5b134d9f4": "X_{1}^\\mathrm{opt}", "5d03d92acd0fee2524983e889346c4a6": "\\alpha_1, \\ldots, \\alpha_N, \\beta_1, \\ldots, \\beta_N", "5d03f137a3c45ec8658744c31ef040c1": "\\phi \\in \\mathrm{span}(\\phi_1, \\dots, \\phi_n)", "5d046e33e8ab919d823d1cf1c89cf287": "1\\cdot a_1,2\\cdot a_2,\\dots,n \\cdot a_n", "5d04adf6eeb0c2a74146927e94e5826b": "(\\boldsymbol\\omega=0)", "5d04f32d3d3144e908c29914682e9c0a": "\\scriptstyle \\boldsymbol\\tau", "5d04fe67aa979f98cc1e6dd8d29535e8": " \\int_c^x | \\varphi +\\mu \\theta|^2 <-{{\\rm Im}(\\mu)\\over 2\\, {\\rm Im}(\\lambda)}", "5d052a9c03e534a609de47cf9d6a69e2": "L=T-V=\\frac{m}{2}\\mathbf{\\dot{r}}\\cdot\\mathbf{\\dot{r}}+q\\mathbf{A}\\cdot\\mathbf{\\dot{r}}-q\\phi", "5d058d6cfece7753245a083f82e17ce6": "X\\mapsto X^\\triangledown", "5d06868a4b7a8f669af18ded799764c8": "Tr(g^{ab})\\in GF(p^2)", "5d06c9cc6933305f1e4aaab3a71f89ca": "\\mathbf \\rho (\\mathbf x, t)\\,\\!", "5d06fc85ead6db1d7b3399c449bf0b29": "|f| = \\exp(-v(f))", "5d07024209021254d8eb41772c5c2e09": "V_s(q,\\omega=0) \\equiv \\frac{V_q}{\\epsilon(q,\\omega=0)} = \\frac{2 \\pi e^2}{\\epsilon q L^2} \\frac{q}{q + \\kappa} = \\frac{2 \\pi e^2}{\\epsilon L^2} \\frac{1}{q + \\kappa}", "5d071b356e43f96337a1ffa8e1a8c675": "(\\mathbb{R}^{d}, \\varphi)", "5d073ea1de9f64da4bb3f13e4bd15f80": "p = 0", "5d077700bd1a42ed0b46ca02044d4239": "\\Delta x_i", "5d0782166eaf3c08bc1c4794abd53e30": "\\Pr(\\limsup_{n \\rightarrow \\infty} E_n) = 1.", "5d078dca2f4090ddef5c4ead8f82aaf8": "q= \\begin{bmatrix}\na & 0 & 0 \\\\\n0 & b & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{bmatrix} p", "5d084df7986008e3cfb04578aafcf20b": "A^{\\varepsilon} := \\{ p \\in M | \\exists q \\in A \\mathrm{\\,s.t.\\,} d(p, q) < \\varepsilon \\} = \\bigcup_{p \\in A} B_{\\varepsilon} (p)", "5d089dbf5d81fa57e793c062a38acce9": "\\mathfrak{N}_n", "5d09122140c0234a321b92caf64622f5": "\\lambda_{mn}=\\alpha_{mn}/a,", "5d0962602f9fb80675b228849ca1162d": " U_3(x) = 8x^3 - 4x \\,", "5d09697085e8b2d48446837da84789a3": "G(x)", "5d099c2f94192e3fb5e0271c0fbeb264": "H_1(G;\\mathbf{Z})=0", "5d09a3da3f1993d68e876beb6f299118": "\\frac{dx}{dt}(t)\\in F(t,x(t)), \\quad x(t_0)=x_0", "5d09d74057e38f999a928935cf7c611f": "\\mathrm{tr}(H) = \\sum_i h_{ii} = \\sum_i \\frac{\\partial\\hat{y}_i}{\\partial y_i},", "5d0a81e56c3fb2af13996d88993c791a": "Y = k^n", "5d0a81fd410ec185a422527f28f77a03": "a\\, \\psi_0 = 0", "5d0ac1e1a5e4784e5d6f678d3c5311fd": "H(X) = \\mu(\\tilde X),", "5d0ac41197d3d152bc7ad434190f95ce": "a^2 = ab \\,", "5d0b2e10f04872fd6bb13defaff21f4b": "(p, R)", "5d0b3d6ae6dd0516d87ed90eb36bfa4d": "Jm > 0", "5d0b7cb09565803a9c6a98d83a482878": "x_ny_1 + \\cdots + x_1y_n\n\\le x_{\\sigma (1)}y_1 + \\cdots + x_{\\sigma (n)}y_n\n\\le x_1y_1 + \\cdots + x_ny_n", "5d0b97850ce35b512406e8f5439e4458": "4.06% = (1.01)^4-1", "5d0bfa2bcac674bb1c988d33caa00b4a": "\\frac{1}{2}(m_0-m_1)v_\\text{e}^2", "5d0c190144bd471d4b3bad96f95c4f92": "\\displaystyle{T(x)T(x)^t=x_1^2 +\\cdots + x_N^2.}", "5d0c3d5f78f4f3c426a51a74f5edab7c": " A = \\text{ sum of inflows } = C_1 + \\cdots + C_N \\, ", "5d0c4c4c772fa8e29b4d61d9b3bd1f23": "t_r' = t - \\frac{1}{c}|\\mathbf{r} - \\mathbf{r}'|", "5d0c4f48bfd84dfc60349987a8081ef5": "{\\mathcal C}^{\\mathcal J}", "5d0ccefff2851c63310176fbe50bcda2": "m = \\rho A", "5d0ce19c0bd9a7898f26ef996d1994f3": " | f(x) - f(y) | \\leq C|x - y|^{\\alpha} ", "5d0d2fbf61949d8aa9fe6d66b748a35a": "\n \\begin{align}\n u_\\alpha(\\mathbf{x}) & = u^0_\\alpha(x_1,x_2) - x_3~\\varphi_\\alpha ~;~~\\alpha=1,2 \\\\\n u_3(\\mathbf{x}) & = w^0(x_1, x_2)\n \\end{align}\n", "5d0d8455758118160140ffb8e6d6304f": "\\Theta\\,", "5d0e4f311948565677ef94d60ac7fd1f": "\\frac{1}{\\phi} + \\frac{1}{\\phi^2} = 1 \\,.", "5d0e993f301fd16abae35a41a3b10c3e": " r = k[sucrose][H^+][H_2O]\\, ", "5d0ecd482b415d95feb725be3f30d004": "\n\\psi (t) = g(t-u) e^{-2 \\pi i t}\n", "5d0ed3471d6d759bfa324432d1ee7e60": "X_1 \\to X_2 \\to \\cdots,", "5d0f2710cc7623d5feaced4a7a5ce0a3": "f^{-1}(U_j) = \\operatorname{Spec} (R_j / I_j)", "5d0f310467c497296767f7bf03f8bc25": "S_v[1 \\dots \\rho_v]", "5d0f6bda89c87099fc207ddc53d75282": "\\scriptstyle a_1,\\, d_1", "5d0f703bae3f48b6bf4a48595c380c71": "N(t) = N_0 e^{-t/\\tau} \\,", "5d0facd6aa2e108efe2b541f412328b3": "|a|=q^{-v(a)}.", "5d0fad84151f97a98848d1c1b15104df": "\\mathrm{P}(X_1,\\ldots,X_n)", "5d0fb66c8ec3d192143dd600fb26a82f": "\\sqrt{\\lambda_2}", "5d10444ed98cbca3724651715d8dd753": "\n \\int_\\Omega \\frac{\\partial }{\\partial t}(\\rho~\\eta)~\\text{dV} \\ge\n -\\int_{\\partial \\Omega} \\rho~\\eta~(\\mathbf{v}\\cdot\\mathbf{n})~\\text{dA} - \n \\int_{\\partial \\Omega} \\cfrac{\\mathbf{q}\\cdot\\mathbf{n}}{T}~\\text{dA} + \n \\int_\\Omega \\cfrac{\\rho~s}{T}~\\text{dV}.\n ", "5d1047ad5601329962055e0e258dde62": " (\\mathcal{M}, \\alpha)", "5d106da12e9022571f3411cf4d7ef73a": "\\log_2 3 = \\frac{m}{n}.", "5d10db8fb6cab46e35ac953a1cb54097": "\\tan\\beta = \\sqrt{1-e^2} \\tan\\phi = (1-f) \\tan\\phi,", "5d11a6469bbb74456bf3684ccf73ed64": "P(X\\in E) = \\int_{x\\in E} dF(x)\\,.", "5d12058952d072d4aea849a2213f1c40": " F_\\text{m} =\\frac{l_\\text{n}}{l_\\text{m} + l_\\text{n}} W.", "5d123761ad5a81b1eaa5724821a4de86": "\\begin{align}\n\\overline {ab} &= \\sqrt{\\left(dx+\\frac{\\partial u_x}{\\partial x}dx \\right)^2 + \\left( \\frac{\\partial u_y}{\\partial x}dx \\right)^2} \\\\\n&= \\sqrt{1+2\\frac{\\partial u_x}{\\partial x}+\\left(\\frac{\\partial u_x}{\\partial x}\\right)^2 + \\left(\\frac{\\partial u_y}{\\partial x}\\right)^2}dx \\\\\n\\end{align}\\,\\!", "5d1269bd967df70a5af2ed0196924a89": "y_0 + \\sum_{i=1}^t u_i", "5d12824648a0d18a270a2bdf7011b9cc": "Q_n(x)=\\frac{n!}{1.3\\cdots(2n+1)}\\left[x^{-(n+1)}+\\frac{(n+1)(n+2)}{2(n+3)}x^{-(n+3)}+\\frac{(n+1)(n+2)(n+3)(n+4)}{2.4(2n+3)(2n+5)}x^{-(n+5)}+\\cdots\\right]", "5d12d6c0e4348e03efb69605b4782ade": " \\alpha_1,\\alpha_2, \\alpha_3, \\beta_1 ", "5d13133f7a5569c5520cb03cb18850ce": "\nC^+ = \\max_{0 \\leq \\beta \\leq 1} \\min \\{ C_1^+(\\beta), C_2^+(\\beta) \\}\n", "5d1315d69aa3716029c33372ddc7246b": "\\partial^{2}B/\\partial x^{2} = 0", "5d135cb461a61082747e49f486e89281": "m = a^2b^3 = 10^2 \\times 6^3 \\, .", "5d138df8ec951ac3f1ea47d95e475667": "n=a b", "5d1392cb323cf9250305984464ddd492": "P = D\\times\\frac{1+g}{k-g}", "5d13b6f9706824032e726d2bded92e57": "\\sigma_{es}", "5d13c06576c12e5e5b34fb02e966d961": " \\varphi\\left(\\int_{-\\infty}^\\infty g(x)f(x)\\, dx\\right) \\le \\int_{-\\infty}^\\infty \\varphi(g(x)) f(x)\\, dx. ", "5d13cd085c0107b5dfd35d166ddc59b9": "T_{surr} \\Delta S = c_p \\left ( T_{b'} - T_c \\right )", "5d13fada834e65f12e9a037a9ecc7b59": "jk=i=-kj", "5d1415a55306f607964447c4ab9aac3b": "\\mathrm{3\\ AmO_2^+\\ +\\ 4\\ H^+\\ \\longrightarrow \\ 2\\ AmO_2^{2+}\\ +\\ Am^{3+}\\ +\\ 2\\ H_2O}", "5d1461a041c399e7a8b777ae95ed2f7a": "\\psi_\\mathbf{g}", "5d14811613cbbe289cc38eb61e13e84a": "\\text{d}x = -B^{-1}C \\text{d}a", "5d14acf2904902d1b1c358bff0ddeeef": "Lu(x)=f(x).", "5d14bdd6498e5854992577621ffa1622": "df(x_0, dx): dx \\mapsto f'(x_0) dx ", "5d15161818dcc862d25888bf1bf69f0b": "x(t+T)", "5d153df4c5c066f6903a1f0ab89f5d54": "\\varphi = \\psi \\circ \\exp_p:S' \\longrightarrow \\mathbb{R}^{3}", "5d1596c2028aa57d369612cef9bd81d7": "\\beta_{i}\\,\\!", "5d15c3c8b229ef820f609651d3f0e896": "C O_2", "5d15e128667b46d522fe999f08ed8920": "al=b^2\\,\\!", "5d15ffb3fed80854696c939479789067": "\\displaystyle \\max_{l(s) \\leq \\Lambda} \\bar{e}(s) \\leq \\exp(-n\\gamma)", "5d1653fe82b79ba28a417ca0706519da": "M^d", "5d16c3dbcbea37b54413b97d4cd59052": " T = S ( \\mathrm{Id} - (\\mathrm{Id} - S^{-1} T ))\\, ", "5d17bc07d9b0f564eae04be5e1f575e3": "\\{[X] | X \\in R\\mathrm{-Mod}\\}", "5d17d2ada1ef895a62302956581605ba": "T_pM\\ ", "5d17e9733b5e96c2729ab2bb037cc4fb": " \\mbox{Mob}(S^2_\\infty) \\cong \\mbox{Conf}(B^3) \\cong \\mbox{Isom}(\\mathbf{H}^3).", "5d18043cf595c67ff6222cf7140290bb": "\\mathbf{F_G}", "5d181fdf600b2afe39524016d5b9bebf": "T(g)(x)=g(x+1)", "5d183445e6a4e3bfa315f4f5937c3b86": "(x_1, t_1), \\ldots, (x_n, t_n)", "5d18c14f477473e8d50f7a4c7c160219": "\\angle PBC = \\angle PCA = \\angle PAB", "5d18dcc2190a7ebec358dab015d58e17": "P(\\theta )=r_0 \\left[\\left(b^2-a^2\\right) \\cos \\left(\\theta +\\theta _0-2 \\varphi\n \\right)+\\left(a^2+b^2\\right) \\cos \\left(\\theta -\\theta_0\\right)\\right]", "5d19347ba5c762554c5a7140b678f51c": "A \\in \\mathbf{R}^{m \\times n}", "5d19425b6c1b02745033acb47f6db1e7": "\\lbrace x \\ :\\ q(x) = c \\rbrace ", "5d19bb40e1c09befe06dcafd33c92bc1": "L_{ab}", "5d19bd8e1b2292cbae0907ffb5680a37": "(p,0,A,p,AA)", "5d19c17c62755ff217255f8ce6c01778": "S = - k_B\\sum_i P_i \\ln P_i = k_\\mathrm{B}\\ln \\Omega\\,\\!", "5d19cbcbe3bab38dc712abf7509f2b49": "\\left\\{r,{p\\atop q}\\right\\}", "5d19ec5e88cefcf2e6b90605119e1285": " T= \\frac{\\bold{j}_\\mathrm{trans}\\cdot\\bold{n}}{\\bold{j}_\\mathrm{inc}\\cdot\\bold{n}}, \\qquad R= \\frac{\\bold{j}_\\mathrm{ref}\\cdot\\bold{n}}{\\bold{j}_\\mathrm{inc}\\cdot\\bold{n}} , ", "5d1a5397547f53d4a62d34392db6f826": "\\displaystyle{[\\mathfrak{g}_p,\\mathfrak{g}_q]\\subseteq \\mathfrak{g}_{p+q}}", "5d1a9cfb1574635c25623a08bb7ff3f0": " M = 1 ", "5d1b18b9bbc74af09865463e6fde6f56": "\\frac{98}{49} = \\frac{\\!\\!\\!\\not98}{4\\!\\!\\!\\not9} = \\frac{8}{4} = 2.", "5d1b72ebac09ca7a8e8f1006ce4d4387": "\ns_3 = y_1y_2y_3 + y_1y_2y_4 + y_1y_2y_5 + y_1y_3y_4 + y_1y_3y_5 + y_1y_4y_5 +y_2y_3y_4 + y_2y_3y_5 + y_2y_4y_5 + y_3y_4y_5\n", "5d1b797eb1a3ceddccd9ccfab9d3b814": " P_\\mathrm{total} = P_\\mathrm{gas} + P_\\mathrm{H_2 O} \\,", "5d1c12a46b696617db281bb95f92441c": "a\\in \\mathbb{R^+}", "5d1c20c7abbe150a91f9b59b88cc72b5": "s0.", "5d472b907576dd6563c1c0951b6a85ad": "C_{4,1} = 1", "5d473f4ade19fb6ae69c8f6a8a25d121": "m=1,", "5d474ca252211e72e717102bc4a1b941": "0=\\frac{d\\mathcal{R}}{dt}\\mathcal{R}^t+\\mathcal{R}\\frac{d\\mathcal{R}^t}{dt}", "5d476b8e3b45a016b4b30d19c8103f2f": "U(1)_R^3", "5d477ddf20f642ab83cdb420fff2fda8": "\\prod_{i\\in I}X_i \\to \\prod_{i\\in I}Y_i", "5d47897458243e563bba60b634c4a159": "{\\mathbf{y}}(t'), 0 \\leq t' < t", "5d47d075fb57cfa18285c0ea43e4d39b": "\\rho_{air}\\,", "5d47daf3c406e48e6c11186b8ac3a286": "Y_{8}^{-5}(\\theta,\\varphi)={3\\over 64}\\sqrt{17017\\over 2\\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(5\\cos^{3}\\theta-\\cos\\theta)", "5d4826e9e7da3892d3c282eea0997b48": "\\oint_C \\mathbf{B} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = \\frac{1}{c} \\iint_S \\left(4\\pi\\mathbf{J}+\\frac{\\partial \\mathbf{E}}{\\partial t}\\right) \\cdot \\mathrm{d}\\mathbf{S}", "5d48d5f201a1554829e60c11192be6f9": "\nAC = [C^-] A = \\begin{bmatrix}\nc_0 & C_3 & -C_2 & C_1 \\\\\n-C_3 & c_0 & C_1 & C_2 \\\\\nC_2 & -C_1 & c_0 & C_3 \\\\\n-C_1 & -C_2 & -C_3 & c_0\n\\end{bmatrix}\n\\begin{Bmatrix} A_1 \\\\ A_2 \\\\ A_3 \\\\ a_0 \\end{Bmatrix}.\n", "5d4918480f0d465f89b8b849e4cdb516": "\\textstyle \\textbf{R}^2", "5d4965704119aa5f18dfd3c58045fb51": "\\forall z\\in\\{0,1\\}^{q(n)}\\,\\Pr\\nolimits_{y\\in\\{0,1\\}^{p(n)}}(M(x,y,z)=1)\\le1/3.", "5d49bbbfa5dd7cd7707578cc0db35c90": " \\pm \\sqrt{i} = e^{i(\\pi/4)} \\,\\! ,", "5d49c91942c4b54cc523d211cd177750": "[A] = \\sum^{\\infty}_{i=1} i\\,[A]_i = \\sum^{\\infty}_{i=1}i \\, K_1 \\, K^{i-1}_m \\, p^i_A \\, [A]_0", "5d49d4dfea84c49144ee0cd2fdd58b94": "-k^2", "5d49edb732df0f7e5da745ee11b600c4": "\\alpha_G = \\frac{m_e^2}{4\\pi}", "5d4a02cae26fad8482fda79387ceed7c": "\\frac{e^{-\\frac{(u-t)^2}{4}}}{\\sqrt{4\\pi}}\\,", "5d4a224178338dd6a9d3e4be2ecdf421": "xq'_x(a,b)+yq'_y(a,b)+p_{d-1}(a,b)=0.", "5d4a4238f0518fcfbd473137b42c9cf2": "\\theta (u,u',\\xi ,\\xi ')=\\frac{1}{2\\pi }{{P}_{V}}{{\\phi }_{\\gamma (u,\\xi )}}({{u}^{'}},{{\\xi }^{'}})", "5d4a508c8303a013671a919f10821541": "P(\\vec y|\\vec x)", "5d4af8e7cbf99f2866e8d56a3549cd67": "f(x)=\\int_{\\mathbb{R}^n} e^{2\\pi ix\\cdot\\xi} \\, \\hat f(\\xi)\\,d\\xi.", "5d4b0fbd7b0862bbfc874a4f5266fc21": " H \\left| \\psi_t \\right\\rangle = i \\partial \\left| \\psi_t \\right\\rangle/\\partial t", "5d4b2ac1854b4b3521543eba9932a1a8": "C(K,\\varepsilon)", "5d4b5bf5f0eff27fa18f576087a09c51": "C_{in}^\\alpha", "5d4bfb7f4c16ead2b93ae8df0ddfd7d1": " \\operatorname{Pr}(\\lambda) = \\operatorname{Tr}(S \\operatorname{E}_A(\\lambda)).", "5d4c64d6e3337581d886b16fc690d853": " \\xi(x\\cdot g) = \\sigma(g^{-1})\\xi(x)", "5d4ca7d9e0c55ae729ca6b235dc0bcf8": "\\hbox{dom}(f)=x\\,", "5d4cd23211360d473363955d0b610b4d": " (X_0,X_1,X_2,...)", "5d4cf72ac2474014f278dac151047806": "\\tau_A", "5d4d64d2626c3f667fe4cf61692f37ee": "E_{2n-1}\\left(\\frac{p}{q}\\right) =\n(-1)^n \\frac{4(2n-1)!}{(2\\pi q)^{2n}}\n\\sum_{k=1}^q \\zeta\\left(2n,\\frac{2k-1}{2q}\\right)\n\\cos \\frac{(2k-1)\\pi p}{q}", "5d4ddd4d64a5ab11edc9289ed990b82c": "g_i(z) = \\int_0^1 \\left. \\frac{\\partial f(z)}{\\partial z_i}\\right|_{z=(t z_1, \\ldots, t z_n)} dt", "5d4ddf64048063143affe117b7fddc56": "C^{m}_k", "5d4e1528f6b4b7bc70f94c91a93729c8": "\\lambda_1, \\lambda_2, \\ldots, \\lambda_n", "5d4e17ef23822daca0da4130b5c9e3f6": " r=120", "5d4e9d39f5fc32c21a9af247a00dbae6": " 2 \\sqrt{3}\\, s^2", "5d4ecdcc65a59b26ee402e25e6298a7e": "n\\mathbb Z", "5d4f074de6b88e92d10f43b39f080557": "\\lambda_\\max = \\frac{b}{T}", "5d4f0922e066a766406f5cb175d0ab46": "\\displaystyle{R}", "5d4f320f6fe0a3e38fccb16e318c7278": "M^T \\Omega M = \\Omega\\,.", "5d4f4bd31738b49ea9ee189500619b7b": "\\sum_{P \\in C}{-c_P [P]}", "5d4f4ee65df964b5b9256eee372a6484": "\n \\phi(v) \\phi(u)=\\phi(u)\\phi(-vu)\\phi(v)\n", "5d4f8f9f3cca759463c0ca0f3a35a35f": "\nL(z) = A + X \\frac{1}{z-B} D^t \n", "5d4fa721451f2b7bea21c80836414458": "\\frac{d (r^2 \\dot \\theta)}{dt} = r (2 \\dot r \\dot \\theta + r \\ddot \\theta ) = r a_\\theta = 0. ", "5d4fba38172cb1a85938ecb09858b38a": "|\\phi_{m}^{'}\\rangle = \\frac{|\\phi_{m}\\rangle}{\\sqrt{\\langle\\phi_{m}|\\phi_{m}\\rangle}}\n", "5d4fc0dc8a7d7b75bedd89e46dabe3f1": "\\Pi_f=1-\\exp\\left[-\\frac{\\left(\\frac{V_{initial}^2}{2}-\\frac{V_i^2}{2}\\right)+\\int{g}\\,dr}{\\eta_0h_{PR}\\left(1-\\frac{D+D_e}{F}\\right)}\\right]", "5d50136edfbacee94c2afb2ec44f5e4c": "q_n = P(R_n)", "5d502dc53d1486d1f38512b4dac983d9": "L\\frac{\\mathrm{d}^2q}{\\mathrm{d}t^2} + R\\frac{\\mathrm{d}q}{\\mathrm{d}t} + \\frac{q}{C} = \\mathcal{E} \\,\\!", "5d50346f8ce5f12e7f71faca63408db3": "A f(t, x) = \\frac{\\partial f}{\\partial t} (t, x) + \\frac1{2} \\frac{\\partial^{2} f}{\\partial x^{2}} (t, x).", "5d506c9110bdf5f64cbc61d65fb8cb06": "m[0]", "5d5098ad6aac70682781f60d78de7826": "P_\\mathrm{R} = \\begin{cases}\n0.88\\cdot\\sqrt{\\Phi}+0.049\\cdot\\Phi&(\\Phi>34\\mbox{ lm})\\\\\n0.2\\cdot\\Phi&(\\Phi\\leq34\\mbox{ lm})\n\\end{cases}", "5d50bcee7eeacd873f70e5519f5d9bf8": "\\Delta G = \\epsilon_{kink} - \\epsilon_{adatom} \\qquad (1)", "5d50eda12bdc3a862627ff55652324ef": "\\exp(-iEt/\\hbar)", "5d5105c61439bf59d05f68fb7df6407d": "\\Box \\,", "5d5145fd3f5f65a128118753a9d4e6e0": "V_\\mathrm{E}= \\frac{\\pi \\big(D+d \\big)^3}{6}", "5d5154de86286af17f1a8d6ab6311801": "q = 6 \\times 10^{-4}", "5d518a20959cc6da29f27622e8519639": "2*3^2*5^{-3}*7", "5d51b7558499b6842179b7a75324fe47": "I(\\mathbf{x};\\mathbf{s})-I(\\mathbf{y};\\mathbf{s}).", "5d51ebb9d1d0deb855a65efcb3faf942": " Df= -f^{\\prime\\prime} + qf", "5d52150543e2df32ca3b8d22cadfd051": " e ^ i \\left ( p , u ^ i \\right ) ", "5d52351762e20c57d5b6d00324fe5b02": "L=[0.71, 0.71]", "5d526dbf666a9e1c32abaed95cf01eb0": "{\\rm Poi}(\\lambda)", "5d5271cc835a0bd23fa99dd7aca7f53f": "\\alpha = \\hat{\\Delta} / \\overline{\\Delta}.", "5d52d8237743e7bab927af811a1bcd38": "S^1 \\hookrightarrow S^3 \\xrightarrow{\\ p \\, } S^2, ", "5d52fbd9d7c8cb6f76d679a68deb73f7": "dA = \\prod_{i \\neq k} ds_i = \\prod_{i \\neq k} h_i \\, dq^i\\,", "5d530b5b3dc17b3859ef36fe5e7dae22": "d\\boldsymbol{\\varepsilon}_p > 0", "5d534ebc130d0f4504c583b9477a991d": "f \\in I^mM", "5d535dc38f8f9cffe6c7d42528693505": "= \\frac{ln(2)}{t_\\frac{1}{2}} = \\frac{CL}{V_\\text{d}}", "5d535e1fad0a802272b96e355d88e3e2": "P_{B}=\\frac{|C_{B}|^2}{|C_{A}|^2+|C_{B}|^2}", "5d5367463058c0ea650f3306af11627a": "f \\in M", "5d5407c8f2b58bcafe8d8fe41edc5278": " \\text{Price}(\\text{Red Sox})+\\text{Price}(\\text{Yankees})\\neq\\text{Price}(\\text{Red Sox or Yankees}) \\, ", "5d5497a2e00da6d204e5139d2dcb9802": "q^2/(q^2+pq)=q", "5d54a897971a883d4352c903f1bfd322": "\\mathcal M \\subset \\mathbb R^{n \\times n},", "5d54e290be38c0b33b84792c516ecd1d": "A_0 \\quad \\mbox{and} \\quad A_{\\infty}", "5d54f97b85795b6ff191f6042bb39078": "|B| = B_0\\left(\\frac{R_E}{r}\\right)^3 \\sqrt{1 + 3\\cos^2\\theta}", "5d557c772f22dce61e59605fc17d08c5": "\\ v_z = v_g \\cdot \\left( \\frac {z} {z_g} \\right)^ \\frac {1} {\\alpha}, 0 < z < z_g\n", "5d5634d9f3e0a3f33e07df0cf69c828b": "k = { \\omega \\over c} = { 2 \\pi \\over \\lambda }", "5d5745a24ca5ff23ae26b498d9f16f11": "V_j = \\frac{O_j (N_{1j}/N_j) (1 - N_{1j}/N_j) (N_j - O_j)}{N_j - 1}", "5d574dbc07ac105d8a8a8a76623c785b": "\\Theta=\\theta=dx^\\mu\\otimes\\partial_\\mu", "5d5783f57f9d3f42617a7d2e4da5ee17": " \\left (-\\frac {b}{2a}, -\\frac {\\Delta}{4a} \\right). ", "5d58346e04b5c8265011cf218cf95b40": "2\\le seqs \\le50", "5d584a79b58cae69e7c646ebed302eeb": "\\pi = 4\\sum_{n=0}^{\\infty} \\cfrac {(-1)^n}{2n+1} = 4\\left( \\frac{1}{1} - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} +- \\cdots\\right) \\! = \\cfrac{4}{1 + \\cfrac{1^2}{2 + \\cfrac{3^2}{2 + \\cfrac{5^2}{2 + \\ddots}}}}\\! ", "5d591c9f6d0cca55943bcf6a87cc6411": "(n-3)/n", "5d5955771a151e1e0c6344b40002bf9c": "G_{0,loc}", "5d598e6cc68e43826ef828a4df5d1526": "\\scriptstyle A \\;\\ge\\; B", "5d59a60b40cfedc929a43400c07f8c6b": "f(x)=ax^3+bx^2+cx+d", "5d59f3f5fd228cb93eb383fed11b1a0d": " \\delta \\mathbf{U} ", "5d5a1f97db0cdbc47e50d30569d5ceb9": "x' \\in H", "5d5a2af91aa1dabcf04beb31504d4a16": "\\mathbb{E}f", "5d5a42cce1193d1b4a9086a6d694ba31": "\\mathrm{deg}\\left(P_{j'}\\right)+j 1/\\tau \\\\\n\\hbar \\omega_{ce} > k_B T \\\\\n\\end{matrix}", "5d60a29aef05d382d9ea84228709bfb5": "x,y_1,\\ldots,y_n", "5d60a8cc6105c44d0914b807a359becd": "N_x = mx - p_xt = \\gamma(u)m_0(x - u_x t) ", "5d60c89165ccf285d08ef491eaccad5d": " A + B = 4,\\ ", "5d60d6cae8294695ee32600f391bbe96": "p\\lor\\neg p", "5d615561ca2d1c938a0e5ba50b0247d5": "\n\\begin{align}\n\\det(\\mathbf{A} + \\mathbf{uv}^\\mathrm{T}) &= \\det(\\mathbf{A}) \\det(\\mathbf{I} + \\mathbf{A}^{-1}\\mathbf{uv}^\\mathrm{T})\\\\\n&= \\det(\\mathbf{A}) (1 + \\mathbf{v}^\\mathrm{T} \\mathbf{A}^{-1}\\mathbf{u}).\n\\end{align}\n", "5d61671c3ce27c32a983211816c6f390": "C = 0.1813", "5d618a16a2eac6e3e190426302cee573": "\\sum_{i=1}^N X_i", "5d6199d25f3e7d5a9ad6a9660d321240": "\nP(male) = 0.5\n", "5d61b73fdcc607fc6ca6190fd0f23f00": "l(1)=\\frac{\\ln{N}}{\\ln{K}}", "5d61ef3efe787110ec2027a7ca82b1a3": "I(A_1 : \\ldots : A_N) = S(A_1) + \\ldots + S(A_N) - S(A_1 \\ldots A_N)", "5d61f92846d96514a51e346988879da0": "\n\\max_{\\mu\\in{\\mathcal M}_X}\\,\\inf_{\\lambda\\in{\\mathcal M}_Y}k(\\mu,\\lambda)\\neq\n\\inf_{\\lambda\\in{\\mathcal M}_Y}\\,\\max_{\\mu\\in{\\mathcal M}_X} k(\\mu,\\lambda)\n", "5d621b60bced68be3c44f034e3e53962": "\\mathrm D_{\\mathsf VC}", "5d6226cfcd037d44622f754211c893c0": "v_5", "5d626db198339b12fa51cb3301919826": "\\delta D-D\\delta=(\\bar{\\alpha}+\\beta-\\bar{\\pi})D+\\kappa\\Delta-(\\bar{\\rho}+\\varepsilon-\\bar{\\varepsilon})\\delta-\\sigma\\bar{\\delta}\\,,", "5d62b1ed3893803575686f68b957e5ab": "i=1, \\dots, k-1 ", "5d62ca15c9b2c142d4fbb7290c622113": "=\\mathbf{I}_1", "5d62ef77ce177f6d925e7dad8174362f": "\\partial_t^2 + 2\\gamma\\partial_t + \\omega_0^2", "5d63199f6adb8303da74cac4d856d7dc": " |t|< \\exp\\left({c\\left({ 1\\over\\varepsilon}\\right)^{1/(2n)}}\\right) ", "5d6327dc8bd7d5defd25a4cfe731d2b9": "8 \\pi \\rho = \\frac{2 M'}{R^2 \\, R'}", "5d639b697e6e03a9e1845761b37d5695": "Cylinder Volume = \\pi*(\\frac{bore}{2})^2 * Stroke", "5d639ec2f42673c52f256c8f87dd00d5": "\\int_0^\\infty \\frac{1}{e^x-1}-\\frac{2}{e^{2x}-1}\\,dx=\\ln 2.", "5d63e1c9291d0939cac1e2e33bfed053": "\\scriptstyle\\sum^s_{j=1}1/x_j+1/x_1\\cdots x_s=n", "5d63e57d7db45cd26a0516b279f63a4d": "\\rho=a", "5d642d06b49fb632ecc94bf6cfd55906": "m_r", "5d648b75570bb87d94a6c26494f4cc5e": "C_{\\mu} \\frac{k^2}{\\epsilon}", "5d64b4c7088804f4e05b693eb06e501a": "I_z(\\alpha,\\beta)", "5d65f6ee894b43054068ad13223bf731": "\\lim_{t \\to \\infty}\\phi(t)", "5d665d9b1ed45c11d501a6d152e83a8a": "r_x", "5d66e839723fd6f556c9bc51df0986e6": "m \\leq n", "5d66ebb7455c4ec7121795c6b7432cae": "\\eta=\\frac{\\Delta G}{ \\Delta H}=1-\\frac{T\\Delta S}{\\Delta H}", "5d66ffdc939c92a37399d3f05c777fa3": "\\sigma^2_ i", "5d670ad4fa65e03601c1158f6b0ab840": "\\alpha \\in V", "5d675417a41344704e2e129954fc28f4": "\\Delta(\\tau)", "5d676d4f0191495f9c3e80a987208804": "\\frac{V\\,dP = C_{P} dT}{P\\,dV = -C_{V} dT}", "5d681e79a3b08c12aa21162a4cb1fb64": "S_\\mu =\\sum_{i=1}^m X_i", "5d68605a46257e242cada8ae314f46e3": "p_N < 0 ", "5d6869fbb6ae386b1a1f609e987d3b41": "\\,^{z_7 = x_7 y_1 + x_8 y_2 + x_5 y_3 - x_6 y_4 - x_3 y_5 + x_4 y_6 + x_1 y_7 - x_2 y_8 + u_7 y_9 + u_8 y_{10} + u_5 y_{11} - u_6 y_{12} - u_3 y_{13} + u_4 y_{14} + u_1 y_{15} - u_2 y_{16}}", "5d688da0b927da5f21f8c8a861fcffd4": " \\Delta U = Q + W + W' \\, ", "5d689772d9c53d24ffca5c442ac6d561": "\\tau=\\omega_2/\\omega_1", "5d68c68c56ecf918bbf11fa08e0c0fd2": "\\emptyset\\neq u\\subseteq D", "5d695abee3fbd3b25e4f5a2964f552b7": "\\epsilon < \\frac{1}{2}\\left(\\frac{3}{7}-\\frac{1}{3}\\right)\\simeq 0.0476", "5d6966af182ece536a38d10f16547e59": "z_1,z_2", "5d696fb191135e4915621ee05f65be32": "{\\mathbf{}}\\tau_\\perp(t)", "5d697726571eaa8a23c75082e67768bc": "W\\left(0\\right) = 0\\,", "5d6977882668b8201c73e414b600c659": "F = \\frac{Nm\\overline{v_x^2}}{L}", "5d69d4f1f5dff7fba3763ba599f54f00": " \\nu \\; = \\; a \\sqrt{ \\frac {1 + \\varepsilon}{1 + \\varepsilon c^2}}", "5d69f22d4877bb6fa169afbb3e140ad7": " = 1 - e^{-y \\theta}.\\, ", "5d6a027e73bbe52ea5d299e97fd59c23": "\\Theta_1 \\, ", "5d6a294c458cdf4e073595b99867d676": "\n\\begin{align}\na_{11}x_1 + \\cdots + a_{1n}x_n &= a_{1,n+1}x_{n+1}\\\\\n&\\vdots&\\\\\na_{n-k,1}x_1 + \\cdots + a_{n-k,n}x_n &= a_{n-k,n+1}x_{n+1}.\n\\end{align}\n", "5d6a4c67d4862f9f7ee2abe81b3ff5e6": "\\begin{align}\n (a+b+c)(w+x+y+z) = {} & aw + ax + ay + az \\\\\n & {} + bw + bx + by + bz \\\\\n & {} + cw + cx + cy + cz .\n\\end{align}", "5d6a561907622d89b7a884f56dc52bef": " \\sum_n P_n = 1.", "5d6a7038e5b4a4da8f28574d5abcb0a9": "J = \\det\\boldsymbol{F}", "5d6a77bc9131223d6aa620f2dc29a2ef": "{k_1 \\over k_2} = \\frac{\\ln ([A_1]/[A_1] ^0)}{\\ln ([A_2]/[A_2]^0) }", "5d6a8f7bd4d8a07e1dce375c97b085cf": " \\rho ^A = \\operatorname{Tr}_B \\; \\rho.", "5d6ac287e806c9dbfa4d546a7787f332": " E = h\\nu \\,", "5d6adf9636447b7832df9e9fef4876f8": "i\\le \\frac{n+1}{2}", "5d6ae5f5adbd6e9a6b8daa2a166b7300": "(X,\\in)\\prec (L_\\alpha,\\in)", "5d6b14c8cb3d9cb7c44c75685ea4fc40": "MinPts", "5d6b33365dcbcb0f3e351a7357d4a692": "Fr = \\frac{v}{\\sqrt{gy}} = \\frac{q}{y\\sqrt{gy}}", "5d6b5563f9f88e566bfc0b0ad9c4e85c": "W_{g,p} ", "5d6b58d3b211a31be9722e2865c555de": "Q = S sin(\\phi)", "5d6bc4a83c4ef56c38ac747a56d3121f": "2\\rho_{,uv} - 4\\phi_{,uv} + 4\\phi_{,u}\\phi_{,v} + \\lambda^2 e^{2\\rho} = 0", "5d6bed204a05c3b042d6d012b8eac620": "r_2+km_2", "5d6c2799bc5c5c482ae6d855656deba3": "w^{f}(f^{*}) = \\sum_{e \\in E}{f^{*}_e \\cdot l_{e}(f_{e})}", "5d6c395d444c36d24cf267a8e1c675b5": " \\int_0^\\infty x^p e^{-g(x)/x} dx \\leq e^{p+1} \\int_0^\\infty x^p e^{-g'(x)} dx. \\,", "5d6c493f41efa7c730f7957f5fcfb9a9": "(a_1,\\dots,a_{\\mu})\\in \\mathbb{C}^{\\mu}", "5d6c8f1f8c65fef927aaeb3a73878f9e": " \\frac{\\rho\\sigma_X-\\rho\\sigma_H}{f}=F_X +\\frac{(%r)R_X}{100-%r} ", "5d6c94547f43f41ac115d46cfd131729": "V(X)=\\frac{1}{12}(b-a)^2", "5d6ce704e4da7ca752beb5f24df723b5": "\\langle O, \\mathbb{F} \\rangle", "5d6d2c5cbf29fed7841ac2724d436030": "F_{m}", "5d6d8b9b98a7ec61c098ce6eef837d64": "\\Psi(x, x^{1/a})\\sim x\\rho(a)\\,", "5d6dbb4659f6b8c5be012ea06240927f": "A \\in \\mathop{\\rm Ob}(C)", "5d6de27c1481c39b0d12d0a0aaaeb4b2": "(.,.)\\colon \\mathfrak{g}\\otimes\\mathfrak{g}\\to \\mathbb{R}", "5d6e1ad2611841b8fc5b30c8632c3e7c": "\\boldsymbol{\\varepsilon} =\\tfrac{1}{2} \\left[\\boldsymbol{\\nabla}\\mathbf{u}+(\\boldsymbol{\\nabla}\\mathbf{u})^T\\right]\\,\\!", "5d6e4405a97effb4dabd8280f2cd8b7e": " u_t + cu_x = \\nu u_{xx},\\qquad 0\\le x\\le L ", "5d6e71ade978254e77726df02f54d800": " \\mathcal{S} = -\\int \\frac{1}{4 \\mu_0} \\, \\mathrm{d}^4x \\, F^{\\alpha\\beta} F_{\\alpha\\beta} - \\int \\mathrm{d}^4x \\, j^{\\alpha}A_{\\alpha} ", "5d6eb188ec73029045d12ab1da9faa33": "C = f(S, \\sigma, \\cdot) \\,", "5d6eb5340c4ceaa7c5179b52cef53f7b": "\\left(\\frac{\\Delta}{\\varpi}\\right)", "5d6f0231f438b1f27203ed26e67b9228": "\\Delta x= -R_s \\log(1-\\mathbf{R}\\cdot\\mathbf{x})", "5d6f0bb8b8814a0ad19f4bfb6089d756": " J = \\frac{ A }{ A + B + C } ", "5d6fdfe179730deff2d6991a3c610ab9": "k\\cdot \\mathbb{N}=\\{k,2k,3k,\\dots\\}", "5d6fff29889e3e045fe7ca16797e5849": " S_n / (\\sigma \\sqrt n) ", "5d709d0ce9e8bf6aaf4a06b86c46d15f": "\\sin(\\frac{\\pi}{p}) \\sin(\\frac{\\pi}{r}) < \\cos(\\frac{\\pi}{q}). ", "5d70bed16862abd203750e66f8c1fdaf": "B\\subseteq\\lambda U", "5d70c952cf59126a2d4cd550f182a58b": "R^k", "5d70dad5ae2be23c28250ad28a7004e3": "X_0, \\dots, X_n", "5d7105d2988f60f453afc86232399694": "E^2 - (pc)^2 = (mc^2)^2 \\,\\!", "5d711496ddbc6c794f80155c9794632d": "b_i\\to b", "5d7123b7117c6d24ac2809724db4baf8": "\\hat{H}_{I} = \\sum_{i}\\hat{S_{i}}\\otimes\\hat{B}_{i},", "5d718ee2c5546647ad13cc1123985ee7": "D \\neq -3", "5d71a7601f65ee3b4829280e3e08eae0": "\\scriptstyle \\Omega", "5d71d2b44ea10686256b4d1ac55af1d2": "\n V(x;\\sigma,\\gamma)=\\frac{\\textrm{Re}[w(z)]}{\\sigma\\sqrt{2 \\pi}}\n", "5d71ec8934e62db791a6d6cee86112d4": "\\tfrac{1}{3}=2^0 3^{-1} 5^0,\\;\\;\\tfrac{2}{5}=2^1 3^0 5^{-1}, \\;\\; \\gcd(\\tfrac13, \\tfrac25)= 2^0 3^{-1} 5^{-1}, \\;\\;\\operatorname{lcm}(\\tfrac13, \\tfrac25) = 2^1 3^0 5^0, \\;\\;", "5d7281c7ac0462b397cb398619b0e157": "\\tilde{S} = \\frac{S}{g\\, h_c^2}.", "5d728ff05c8c655c793026a6baaa62f6": "\\max(0, \\hat{R}_m \\cdot \\hat{V})^\\alpha = \\max(0, 1-\\lambda)^{\\beta \\gamma} = \\left(\\max(0,1-\\lambda)^\\beta\\right)^\\gamma \\approx \\max(0, 1 - \\beta \\lambda)^\\gamma ", "5d72b71a4c0d7a0e82eb0de038d809f4": "\\left\\lfloor \\frac{n+1}{2} \\right\\rfloor ", "5d72c79cb96a77dc22cff264033dbf4b": "\\,\\zeta(s,\\alpha)=L(0, \\alpha,s)=\\Phi(1,s,\\alpha).", "5d72ffe60ff80ffd64377ce6673232a0": "~p=I_{\\rm p}/I_{\\rm po}~", "5d730097bea745895e43f9001b15d1a5": "\\operatorname{sn}(u)=\\frac{2\\pi}{K\\sqrt{m}}\n\\sum_{n=0}^\\infty \\frac{q^{n+1/2}}{1-q^{2n+1}} \\sin ((2n+1)v),", "5d73088e05c49ff12b6d90779cafad33": "\n\\frac{P_n + \\sqrt{D}}{Q_n}\n", "5d73733d0dbbf581ae3bef659c9b45f0": "\\hat{H}_{II} = \\dfrac{\\mu_0\\mu_\\text{N}^2}{4\\pi}\\sum_{\\alpha\\neq\\alpha^\\prime}\\dfrac{g_\\alpha g_{\\alpha^\\prime}}{R_{\\alpha\\alpha^\\prime}^3}\\left\\{\\mathbf{I}_\\alpha\\cdot\\mathbf{I}_{\\alpha^\\prime} - 3(\\mathbf{I}_\\alpha\\cdot\\hat{\\mathbf{R}}_{\\alpha\\alpha^\\prime})(\\mathbf{I}_{\\alpha^\\prime}\\cdot\\hat{\\mathbf{R}}_{\\alpha\\alpha^\\prime})\\right\\}", "5d73ab662ed61be43509e063d5261b27": "d <\\infty", "5d73ed971d528d7a092a6036afed5cb9": " \\boldsymbol{\\sigma} = \\mathsf{C}:\\boldsymbol{\\varepsilon},", "5d744495a4fa0a283334bd261548eb5e": "t \\le 0.5", "5d74a67f6a0cc925b0e4d69146fc57a4": "a*b", "5d74a769062d7575ef4d7d5ba302e705": "\\operatorname{E} \\left [ \\frac{R(n)}{S(n)} \\right ]=C n^H", "5d74b84d177c4939840390b201ee356c": "\\quad \\{ z_k \\}_{k=1}^\\infty \\subset \\bold C ", "5d74f3f8afb4c15b96ba68a7c26ef719": "G_x = \\arctan\\frac{1}{x}.", "5d760e63143620fd3cbded415cc05462": "x_0=x", "5d76123a3aba18842dd83a893df92996": "=\\frac{2\\left(L_{L}-L_{T}\\right)}{\\lambda}\\left(\\frac{1}{\\sqrt{1-v_{A}^{2}/c^{2}}}-\\frac{1}{\\sqrt{1-v_{B}^{2}/c^{2}}}\\right)", "5d76453fb9f206bb1d4b83c6abe4e125": " (a_1\\ a_2\\ a_3\\ \\ldots\\ a_k) = (a_2\\ a_3\\ \\ldots\\ a_k\\ a_1) = \\cdots = (a_k\\ a_1\\ a_2\\ \\ldots\\ a_{k-1}).\\, ", "5d765a201055134aae123c3093b2c353": " C_i ", "5d76b681d1298189cf27cd01fe7ac11b": "\\sigma_1 = 2\\lambda_1 = 2\\lambda_2.", "5d76e493c4447853faa24369fac3bd3f": "\\textstyle{\\vec e_{r'}}", "5d770eeb779b062f745352ff157e8d3f": "\\scriptstyle \\leq2.7\\times10^{-4}", "5d7738df5d90d649f98de7762c752e70": "\\{0,\\ldots,k\\}^{<\\omega}\\,", "5d7740a2992c219ff1c94893092fcbaf": "\\lfloor b_i x \\rfloor = b_i x + (\\lfloor b_i x \\rfloor - b_i x)", "5d77523f8862aab5e67557940c47f314": "J := \\rho I", "5d77b143bc3185fff27196e552914759": "S_{[8-15]}", "5d77b94c2546b1a0f31509bf1799a2cf": "Q^N", "5d78067bb643a3a2d8c4ea12d667da06": "B^{*} = \\frac{p}{r} = \\frac{p_{k}}{r_{k}}", "5d780f8ac7433176a53816076de4a671": "1.\\quad f\\star g = fg + \\mathcal O(\\hbar)", "5d78521b2834a58d1dff3d0bffff07f5": "\\bar{\\Delta}_- \\cong \\sigma_-\\otimes \\Delta_+^*.", "5d788189bc42adfa4137441e53c4dab0": " ~\\epsilon_t=\\sigma_t z_t ~", "5d7886ffe0edd0900f1ef6da5a7464e3": " \\alpha' = \\frac{4 \\pi \\kappa}{\\lambda_{0}}.", "5d789f522874d84f4cbbbb52b7b75584": " \\mathcal{E} = \\int_0^R dr Br \\omega = \\frac {R^2}{2} B \\omega \\ , ", "5d78a5a899a19de7023eca4391234c84": "W_{in}", "5d78b6e528214e09b08d5c927111b3f8": "H_{\\text{cc}}=\\sum _{i>j} V_{\\text{cc}}^{\\text{ij}}a_{c_i}{}^{\\dagger }a_{c_j}{}^{\\dagger }a_{c_j}a_{c_i}", "5d78cf469924667befda13bcd3a4abf9": "A^{(0)}", "5d78d506f88c212cafff48a22d9e5365": "C_0=\\log x^{a-c}\\,", "5d78f576cf1013a1b976d3293ef62a87": "q=\\frac{a^3 \\omega^2}{GM}\\,\\!", "5d790761de42cde1c17c1f6ebd066290": "\\{a_n\\} \\mapsto \\{b_n\\}", "5d792d9a6b7a440839e231fcd3751509": "\\begin{align}\n \\operatorname{Re}(-3.5 + 2i) &= -3.5 \\\\\n \\operatorname{Im}(-3.5 + 2i) &= 2\n\\end{align}", "5d793a340bc086d79752627fcc78b46e": "\\mathcal{F}_\\omega^{-1}", "5d796c5e83938f40e6e88e259bec5071": "\nd=\\frac{\\pi}{2}\\epsilon", "5d797b74db439c2314eeea60754616bc": "w^-=(y^\\mu)^-(u_\\mu)^-\\;", "5d799a70375638f162a532995b8c930f": "D + w w^{T}", "5d79e34c63b0798f53cca199e8d7ca4b": "e = e^1 = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, \\dots],", "5d79ed44c3406d0fb886f6835262f8ed": "D_{\\mathrm{KL}}(X_1,X_2) = D_{\\mathrm{KL}}(X_2,X_1),\\text{ if }h(X_1) = h(X_2),\\text{ for (skewed) }\\alpha \\neq \\beta", "5d7a1543dae8d8765421a65f5695a8aa": "\n G(k) = \\sum^{\\infty}_{m=0}\\frac{(ik)^m}{m!}\\mu_{m}.\n", "5d7a99271bad94afd38ca10d4dbbafca": "\\frac{1}{19}=0.\\overline{052631578947368421}.", "5d7b6f4960c0291a1c72494a585a949c": " \n= B + C + VE[P(\\alpha^*(t), \\omega(t))] + \\sum_{i=1}^KE[Q_i(t)]E[Y_i(\\alpha^*(t), \\omega(t))] \n", "5d7b9adcbe1c629ec722529dd12e5129": "-2", "5d7bfc4d421c11ab835910f18192a346": "i = 1,2,\\dots,n-1", "5d7c04cee8021cb77db14d7514ebddc7": "y = g^x", "5d7c0e6dca4a38faf800856d42741321": "m = 0.5", "5d7c2f53346b09e709daa6f554a24b08": "\\chi_\\text{m}^\\text{SI} = 4\\pi \\chi_\\text{m}^\\text{G}", "5d7c50a502fad9954c9b97f82d800c9f": "\\mathbf{B}", "5d7cbd9b406863238abbe752fe4d3a77": " Mf(x)=\\sup_{r>0}\\frac{1}{|B(x,r)|}\\int_{B(x,r)}|f(x)|dx", "5d7cf0335dbbd08abdb4e3125ce3c663": " \\neg (\\operatorname{def}[F] \\and \\operatorname{ask}[S] \\and FV[A] \\subset V) \\to \\operatorname{drop-params}[E\\ P, D, V, R] \\equiv \\operatorname{drop-params}[E, D, V, [F, S, A]::R]\\ \\operatorname{drop-params}[P, D, V, \\_] ", "5d7d067d63c694bef0c2c808e0a01900": "F^{\\prime}(X_n)", "5d7d0b062dc64a852a6e8ed5f87219f7": "\\begin{align}\nU_{f,P_n} - L_{f,P_n} &< \\epsilon\n\\end{align}", "5d7d9438ee0c84cebda6bc1f9e67c7fa": "\\begin{align} & {} \\qquad DG + DH + DF \\\\ & {} = |DG| + |DH|- |DF| \\\\ & {} = R + r \\end{align} ", "5d7e0a0b8c281e8572ab7e21dc1f38b6": "\\gamma _2 = \\gamma _1 - P(\\lambda _1 ) \\, ", "5d7e58f7206f12574c5e1b58d20e39d2": "b = \\left(586 + 102 \\sqrt{33}\\right)^{1/3}", "5d7e5edc93012bc38596a2421815c0c2": "x = a+r\\,\\cos t; \\,\\!", "5d7e893af570bd3214a887ba86d4f755": "U(t+\\bigtriangleup t) =\\int_0^\\infty U(t+\\bigtriangleup t,w)dw.\\quad (1.9)", "5d7eb989499c3a5fab91ee66475a04be": "\\gamma_{ij}", "5d7edc402bd377a7e02d70db8ebd0d09": "\\scriptstyle \\mathbb{I}", "5d7fbd282525244e9eb91e40fd7a7a00": "f:=\\sum_{U\\in\\mathcal{O}}f_{U}\\,", "5d7fcca1e39d47aa5c2949c3e791d560": "\\alpha = \\beta = 4", "5d80262f6153b1a83d801ccc2d29f04f": "\n\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_y) = \\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & \\cos\\Delta\\theta & 0 & \\sin\\Delta\\theta \\\\\n0 & 0 & 1 & 0 \\\\\n0 & -\\sin\\Delta\\theta & 0 & \\cos\\Delta\\theta \\\\\n\\end{pmatrix} \\,,\n", "5d804347618740dc18fd31175153fff5": "u=0", "5d806aacd045f634152b5cd0498592d5": "\n\\varphi = \\int \\frac{dr}{r^{2} \\sqrt{\\frac{1}{b^{2}} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{1}{a^{2}} + \\frac{1}{r^{2}} \\right)}}.\n", "5d808644c2ecf721772417e25d1701e1": "4\\times 10^2 + 6\\times 10^1 + 5\\times 10^0 = 4\\times 100 + 6\\times 10 + 5\\times 1 = 465", "5d8120f0ede80135979f35b480b5ecc2": "|L| = N^3", "5d813100bc08afcf06c48a8a953182ee": "T_v(p) = p + v,\\,\\!", "5d81386517b73bb1344ae3cf16d56c89": "\\langle a,b \\mid aba=bab, (aba)^4 \\rangle\\,\\!", "5d81481543dcd1e1b75863ff5d993833": "v, v' \\in f", "5d81e2246ea458821915ce2665931d80": "r_{CM}", "5d822428b234fcfd2494b775a9f16189": "Y_{c} = A_{1}r_{1}t + A_{2}r_{2}t", "5d825a52bd50f4f0370c1cf8216f3253": "c_{3,1}(\\widehat{a}, w(S, \\widehat{b}c), \\widehat{d})", "5d8262fb5f3d7e105d38207ac790b583": "\\begin{align}\n&\\delta(x) \\to -n_x\\cdot\\nabla_x\\mathbf{1}_{x\\in D},\\\\\n&\\delta'(x) \\to \\nabla_x^2 \\mathbf{1}_{x\\in D}.\n\\end{align}", "5d82ff54638d8c58d016b57c2b25d732": "N=\\frac{t(t+1)}{2}=s^2", "5d8392d2cb8545b52b9584fe23031ce8": "=\n\\pi\n\\left(\nn\n-\\frac{T_3(n)}{3}\n+\\frac{T_5(n)}{5}\n-\\frac{T_7(n)}{7}\n+\\dots\n\\right)\n.\n", "5d8439207ea01ce154c590561c93409c": "\\begin{bmatrix} 1/\\sqrt{2} & 1/\\sqrt{2} \\\\ -1/\\sqrt{2} & 1/\\sqrt{2} \\end{bmatrix}", "5d84fdc47827bdc67b50b51e3c7194c7": "traces\\left(P\\right) \\subseteq \\Sigma^{\\ast}", "5d853429811edbbadcb5cc4d25f827aa": " a < X < b ", "5d853821147ee12630ef8fc18c482d31": "S'(0)=2", "5d854034fe3c9462ba83c8b04881ee32": "Pacifist(Nixon)", "5d854550278c03dd20fc30189047e8fd": "V_c \\, = \\, 17.5 + \\sum {V_{c,i}}", "5d85618f030d1cb9700d004cdb01212a": " = d_3(m + 2^{-1} {d_2} (m + 2^{-1}{d_1} (m + {d_0} (m)))). ", "5d857079ba6fbdedc4c69b02271b79f2": "\n p(x_1,x_2,x_3) = \\cfrac{b^3}{\\pi^{3/2}}~\\exp[-b^2(x_1^2 + x_2^2 + x_3^2)] ~;~~ b := \\sqrt{\\cfrac{3}{2Nl^2}}\n ", "5d857aa698cdf7abb5a4a49173cd810c": "\\sigma_q = \\frac{d Q}{d S}\\,,\\quad", "5d85af81c5fef1a28abb1feda00e29ca": "\\begin{align}\n\\mathbf{D}(\\omega) & = \\begin{vmatrix}\n\\varepsilon_{1} & -i \\varepsilon_{2} & 0\\\\\ni \\varepsilon_{2} & \\varepsilon_{1} & 0\\\\\n0 & 0 & \\varepsilon_{z}\\\\\n\\end{vmatrix} \\mathbf{E}(\\omega)\\\\\n\\end{align}", "5d85fdeba720b3b22afefba6690ccaa1": "dx^\\mu", "5d862fe2fd39936e4835ede1ff7efc00": "\\Delta \\alpha = \\frac{K}{N}", "5d863c7ead713b0f386b7b468a5e6b81": "\\scriptstyle\\hat{F}_n(t)", "5d86b88e3fbc58fcf4c3b2cbb42b72a6": "\\Delta E_{\\rm c}", "5d86be14bfa21df6d84fb2b2a541d7fe": "\\mathbf{J}=\\mathfrak{m} \\exp\\left(\\frac{\\mu-\\mu_0}{RT}\\right)(-\\nabla \\mu + (\\mbox{external force per gram particle}))\\, , ", "5d86f8e0db03744ee77f714071086523": "z''-y'-z", "5d8762df73a005f822cef0d304dba07f": " Q_1,Q_2,Q_3,Q_4 ", "5d879e735376969c69bd7624894061a0": " \\mathbf{y}_n= \\mathbf{A}\\mathbf{x}_n+ \\mathbf{B}\\mathbf{z}_n +\\mathbf{c}+\\mathbf{e}_n ", "5d87d6d61aeddccd3e8dfaeedcf34d8a": "\\pi/k", "5d87eb59b8c983981e0c11ad55ee037d": "n_B", "5d8810807562cdf75855fc489f327c4a": "h(t)\\,", "5d88298660385d860ceda7ac26658a23": " A_t u [C_i(x) - C_i(x + dx)] + A_t dx \\nu_i r = 0 \\, ", "5d88409b77b98cbadb5e26432cce8bed": "p \\ge 0", "5d884daf48bed882a65517aff0f205b9": "\\textstyle \\mathbb{R}^n", "5d88629697dc04d6bec818d213fa0002": "c_1^\\prime", "5d8897ce31f2d49675b189f0d16f6818": "\\Delta x \\Delta p \\, \\sim \\, h", "5d89cc097141739345d83f7af44cb7cf": "x_i (i=0,1,...,M-1)", "5d89e6ee7408b4496bda8a6f7b7d7147": "E = E_{ex} + E_D + E_{\\lambda} + E_k + E_H\\,", "5d89f30aaca59e84d996123b9e0b0331": "\\Pr(X=x|T(X)=t,\\theta) = \\Pr(X=x|T(X)=t), \\,", "5d8a0ff93f5212cebce88c0e4fc37610": "\\cos\\frac{\\theta}{2}", "5d8a272a99dd27b26757df0a7833e761": "R_{hs}", "5d8a366ec978262a54a6b20258ffba6f": " g(\\tau-t)", "5d8a845080ffb65672caa9d23a7a86cd": "e^{\\lambda_m} = e^{\\lambda_m \\,+\\, q 2 \\pi j}", "5d8aa7a88a64491676cc0775ea502e89": "\\begin{align}\n\\omega & = \\omega_0 + \\alpha t \\\\\n\\theta &= \\theta_0 + \\omega_0t + \\tfrac12\\alpha t^2 \\\\\n\\theta & = \\theta_0 + \\tfrac12(\\omega_0 + \\omega)t \\\\\n\\omega^2 & = \\omega_0^2 + 2\\alpha(\\theta - \\theta_0) \\\\\n\\theta & = \\theta_0 + \\omega t - \\tfrac12\\alpha t^2 \\\\\n\\end{align}\\,\\!", "5d8ab57412a99762a5fc586a4cf8d616": "\nD_1 = \\left (h_1-\\frac{1}{\\gamma} \\right ) \\left (\\frac{1}{h_1} - \\frac{1}{h_2} \\right )\n", "5d8b138bdfbf27a5db134a4bbe240896": "G_i/G_{i+1} \\to U_{L, i}/U_{L, i+1}, i \\ge 0", "5d8b26c0196a2f4f7f8674a03aa00f52": "\\Delta^2 \\psi = 0", "5d8b42955c1b9d04156fa09a4a8585a2": "t\\ne r'", "5d8b490a8af23761f42d4037ab39deb4": "\\boldsymbol{E} = -\\boldsymbol{\\nabla}V", "5d8b53a7a15f6ca4f5823e45047b46ef": "\\inf\\{J(u)|u\\in V\\} = \\lim_{n\\to\\infty} J(u_n) = \\lim_{k\\to \\infty} J(u_{n_k}) \\geq J(u_0) \\geq \\inf\\{J(u)|u\\in V\\}", "5d8b8d50e147a92d3276167a1c4c7c03": "\\lim_{r \\to 0} \\frac1{\\lambda^{n} \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\lambda^{n} (y) = f(x)", "5d8b91b24723dd3f60ecff1a96824d61": "r = f \\sin(\\theta)", "5d8bee3b439abb98e911a4f17c573163": " N=M ", "5d8c6d667cd2b2af904a26d37f963ec6": "V=\\frac{1}{2}(\\dot{e}+\\alpha e)^2", "5d8cc57101f22ed5fc4e7990e484bf62": " \\frac{d\\vec{P}}{dt}=\\mathbf{A}\\vec{P}", "5d8d456e77842ef4aa45591c979004c9": " \\partial p/\\partial n=0", "5d8d5525d66a151a197ec2d8aa51092b": "\\frac{\\partial e}{\\partial p_{1}}=\\frac{\\partial \\mathcal{L}}{\\partial p_{1}}=x_{1}^{h} ", "5d8d9688cecc99f16ef6949369c5eb1d": " \\left(\\frac{d\\phi}{dt}\\right)^2=\\frac{c^2R_s}{2r^3\\sin^2\\theta} ", "5d8e4d9d089f74358080bd2d47e1e062": "\\delta^i_j", "5d8eae58640c5dadc58a071c94bde43f": "\\sqrt{p(1-p)}", "5d8f06effc3f5b5d734b6e6bd71d09cc": "E^{2\\omega}_j", "5d8f3f85160a0c23847e4850a4407e2d": " U \\gg u',v'\\!", "5d8fd6d1d4a2e475cae7f89a301d2b97": "\\mathcal H=\\sum_G\\,-I_{k_\\nu , k_\\mu}\\,\\,S_{k_\\nu}\\cdot S_{k_\\mu}\\,,", "5d9001f4d8a66e4d92a7cd64e2a26f63": "C_{\\hat{X}} = C_{XY} C^{-1}_Y C_{YX}.", "5d9024cb5fe6103a291addc8f04a2097": "x_1+\\cdots+x_n\\ge y_1+\\cdots+y_n.", "5d90421aabd66981c727334bf4242695": "\\displaystyle g_{ij}=\\left(\\frac{\\partial\\mathbf r}{\\partial q^i},\\frac{\\partial\\mathbf r}{\\partial q^j}\\right)\n", "5d9061cefdce04b1cd2983abccc14525": "E_\\text{K} = \\frac{1}{2}m\\mathbf{v}\\cdot\\mathbf{v} = \\frac{1}{2}(mr^2)\\omega^2 = \\frac{1}{2}I\\omega^2.", "5d9075efa0ec4b2e8228505844f11742": "x^*", "5d907ac284ccb9e8c550a4b0f3b277f6": "x_n \\rightarrow \\partial A", "5d908992629e76b0f1246c5c579a33f0": "\\log{\\text{DOR}} \\pm 1.96 \\times \\mathrm{SE}\\left(\\log{\\text{DOR}}\\right)", "5d90bb01e5589b77f1b5ad57d8822310": "\\gamma(A^T\\cdot)", "5d90ca2fc412c3ad15802265b7f334b5": " \\arccsc x = y \\, ", "5d90d54bfe252f5de2e8e1fff0eed20a": "\\Phi=[\\phi_1,...,\\phi_k]=XW", "5d917184df47785e10ac587e6b664851": "m(s)=\\sup_{|z|=e^s} |g(z)|, \\, ", "5d9186603d6f75e2f6c56ad54387a734": "\n\\overline{A} = \\lim_{\\tau \\to \\infty} \\frac{1}{\\tau}\\int_{0}^{\\tau}\\left [ \\sum_{\\alpha, \\beta=1}^{D}c_{\\alpha}^{*}c_{\\beta} A_{\\alpha \\beta} e^{-i \\left (E_{\\beta} - E_{\\alpha} \\right )t/\\hbar} \\right ]~ dt.\n", "5d91b2893b71aa86ca2a39f6d88fa401": " R > C \\,", "5d92106835947ec03ddb7750eb4e31f7": "\\pm_1", "5d9235ea878e6419c088f0c94154596f": "x c_{k+2}(x) = \\frac{1}{k!} - c_k(x),\\text{ for }k = 0, 1, 2, \\ldots\\,.", "5d9252c31d5244e0fe195d50c910228f": "n^{\\Omega(1/\\varepsilon)}", "5d92aa76db1b30b9c4a0cd2ffac4d0a4": " r = k [H_2][NO]^2 \\,", "5d92b2bb8ae3bdaebf6b1ac8934f7369": "L_e(x) = a_ex + b_e > 0", "5d92d054b256c41dadae19f6339b9237": "\\hbar \\omega\\left(n+\\frac{1}{2}\\right)", "5d92e1da75c259dfe43275785b5c85c3": "\\bar{x}_j \\bar{\\mathbf{e}}^j = x_i (\\boldsymbol{\\mathsf{L}}^{-1})_j{}^i \\mathsf{L}_k{}^j \\mathbf{e}^k = x_i \\delta^i{}_k \\mathbf{e}^k = x_i \\mathbf{e}^i", "5d935a6143fff575aa103b7111154520": " \\Re {v(z)\\over z}\\le 0.", "5d93a6f519e1b047daba903da9054b87": "\\overline Y_x", "5d93eb3d576934c9ee923a5bf41e1ffe": "\n\\begin{align}\nD(m) & \\left( = \\sup_{0 \\leq a \\leq b \\leq 1} \\Big| m^{-1} \\# \\{ 1 \\leq j \\leq m \\, | \\, a \\leq s_j \\, \\mathrm{mod} \\, 1 \\leq b \\} - (b-a) \\Big| \\right) \\\\[8pt]\n& \\leq C \\left( \\frac{1}{n} + \\frac{1}{m} \\sum_{k=1}^n \\left| \\sum_{j=1}^m e^{2 \\pi i s_j k} \\right|\\right). \n\\end{align} \\qquad (1)\n", "5d942e1851d5ef1869e00c2dfed0cb31": "(a_0,\\dots, a_{N-1})", "5d9433cdfa8378e6a7838fc16b19e9d4": "F_{air} = -k v^2", "5d94473027e4a6ada744d14b3627f1c6": "K_X(\\mathbf{t}) = A(\\boldsymbol\\theta + \\mathbf{t}) - A(\\boldsymbol\\theta) \\, .", "5d945ed2c0874d54d678b83ec7bd69d9": " \\Phi_{2D}(\\mathbf{x},\\mathbf{x}')=\n\\frac{1}{2\\pi}K_0(k|\\mathbf{x}-\\mathbf{x}'|),\\quad\n\\Phi_{3D}(\\mathbf{x},\\mathbf{x}')=\n\\frac{1}{4\\pi|\\mathbf{x}-\\mathbf{x}'|}\\exp(-k|\\mathbf{x}-\\mathbf{x}'|)\n", "5d949f78023486679077f070d93520af": "\\mathbf{x} = \\sum_{i=1}^K s_i P_i \\mathbf{w}_i", "5d94b357d7633e1cda0e7aeab73a627b": "S_x(f) = \\frac{1}{(2\\pi)^2f^2}h_0", "5d94ebf01eb868b09d6774003cdd431c": "\\hat{x}=\\hat{d}", "5d9508cc0b5671312a4e7b12e12872c3": "{ds_A}^2", "5d952225854444cbc40e6a0f8bf6be6a": " \\operatorname{E}(f(\\text{Aa})) = 2 p q", "5d957fb4a7667bf18d0b5e0793e039b7": "L_\\chi(s)=\\sum\\frac{\\chi(n)a_n}{n^s}", "5d95cf6fea1b56a19c875a32fc40f1e6": "P_\\mathrm{selection}", "5d96140378720a15e7dab720ce2ca987": "\np_{\\rm total} = p_{0} + p_{\\rm osc} \\,\n", "5d962361534261ef9d54bb580e90fea7": "\\theta = constant", "5d9710f0092984cc6d35c0fb45b61d3c": " \\arccot x = y \\, ", "5d972631dabd6123b8eef78d5f1cba65": "\\ p(V-b)=RTe^{-a/RTV}", "5d979f8cb2bf59118e0d80aebb1e2389": "\\|\\mathbf{X}\\|^2 = X^\\mu X_\\mu=(c\\tau)^2 = s^2 \\,,", "5d979fdbbd786ca30d4e3359aef7f3d9": " \\Delta S = S_{\\mathit{final}} - S_{\\mathit{initial}} \\, ", "5d97aafc8bd021cfdfb7c0f5d88d81f7": " f(c) - \\varepsilon < f(x) < f(c) + \\varepsilon.\\,", "5d97f6584375f755d2cba99e716b918f": "\\Z[\\mathrm i]", "5d983b9722f2225540fdc455db41ad60": " \\mathbf{e}_0 = 1", "5d984c1e96eaa34edd83c72f0c93d137": "X \\ \\stackrel{\\mathrm{def}}{=}\\ \\left\\{ x \\mid ( x \\in x ) \\to Y \\right\\}.", "5d9896551b30a050e9b841f9541757ea": "F_\\lambda\\ \\hat{\\lambda}\\,", "5d989ad60bcd230c1aecfde36d8b6635": "a_n = \\mu (n) \\,", "5d98a7cba77ef3c2574c966c70dbc672": "2^{n-1}", "5d98c954040f019c2e0ba8bd38728b6b": " \\vec{r} = \\left(\\vec{q}, \\vec{p} \\right) ", "5d99222195674e5675286702f95f9900": "\\mathit l + \\mathit l^{\\prime} = n", "5d9930083c2c10a63e1c731c940652b7": " i,\\dots,n \\in X ", "5d993ff16581e6d7656170b754ebf2df": " \\psi(x)=\\varphi(Cx). ", "5d99800aa555bb427321c57cfe878d31": " [V,V^{\\dagger}]=|0\\rangle\\langle 0|", "5d9a131531598c625cd7d9bd192e3765": "\\sum_{n=1}^\\infty q^n \\sigma_\\alpha(n) = \\sum_{n=1}^\\infty \\frac{n^\\alpha q^n}{1-q^n}", "5d9a5933f641af105f0f16d71c64fa8f": "\\int_{-\\infty}^\\infty f(r)\\,dx=2\\int_{0}^\\infty f(r)\\,dx, \\;\\; r= \\sqrt{x^2+y^2}.", "5d9a76832d3d686e1dd911d4d2f94931": "\\mathbf{p}_0 + (\\mathbf{p}_1-\\mathbf{p}_0)u + (\\mathbf{p}_2-\\mathbf{p}_0)v, \\quad u,v\\in\\mathbb{R}", "5d9a8daa2702a7a1969251e4677fe50b": "1_{S}", "5d9acb74e88953f1cd4fca437d302d76": " S = \\frac{ \\mu_1 - \\mu_2 }{ 2( \\sigma_1 +\\sigma_2 ) } ", "5d9b1247da0137642fd4da35367bc83c": "R_y = \\frac{R'R''}{\\sum R_\\Delta}", "5d9b2e3c0be3eb7446ca6d53fe1c4403": " A(\\mathit{G}) ", "5d9b387d5f687644be6a147d888e0acf": "\\overline{K}[x]", "5d9c2dc66da064c02ebe64628fec5c37": " = \\frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\\left(x'(s)^2 + y'(s)^2\\right)^{3/2}}\\ , ", "5d9c5e05b30d1a8a0c68e75aeac1200b": "A = A^{\\top}", "5d9c6c8720642f76a188f32d3ae8f50c": "CDecode(l,\\psi) \\stackrel{\\mathrm{def}}{=}(I_{B^{\\ast}}^{W^{\\ast}}(\\Bigl[\\psi\\Bigr]_{0}^{16}),I_{B^{\\ast}}^{Z}(\\Bigl[\\psi\\Bigr]_{16}^{16+l}),I_{B^{\\ast}}^{Z}(\\Bigl[\\psi\\Bigr]_{16+l}^{16+2l}),I_{B^{\\ast}}^{Z}(\\Bigl[\\psi\\Bigr]_{16+2l}^{16+3l}),\\Bigl[\\psi\\Bigr]_{16+3l}^{L(\\psi)}) \\in W^4 \\times Z \\times Z \\times Z \\times B^{\\ast}", "5d9c6f41e4af517ffc051a6c82b6a813": " F: S_1 \\times S_2 \\times \\ldots \\times S_m \\rightarrow \\mathbb{R}. ", "5d9c84591867ff1998ecec39844acb02": "Germany: 30% \\cdot \\euro 1,000,000 \\cdot \\left[ \\frac{1}{3} \\cdot \\frac{\\euro 150,000,000}{\\euro 200,000,000} + \\frac{1}{3} \\cdot \\frac{\\euro 3,000,000}{\\euro 8,000,000} + \\frac{1}{3} \\cdot \\frac{\\euro 135,000,000}{\\euro 200,000,000} \\right] = \\euro 180,000.", "5d9ccd2067f38afaa1c6ac870610ca91": "\\|\\tilde{\\mathcal{M}} f\\|_{L^2(-\\infty,\\infty)}=\\|f\\|_{L^2(0,\\infty)}", "5d9d1cc7b5a92d89ea81b1dccc258256": "\\frac{1}{4} + \\frac{1}{12} = \\frac{1}{3}", "5d9d3ec11a3895964d5366786f107100": "\\textstyle p \\in [0,1] ", "5d9d4ae4482abac5b4bf68d3cf7d7079": "h(t) = t.\\,", "5d9dce9163e98e8dc2363d86d3a0d331": "\\tau_m \\gg \\tau_N", "5d9e3f81a46960a141753a0146ed5215": "\\|\\mu\\|_\\infty<1", "5d9e5db8439b09fe329369e477b2f48a": "R_{(i)}^t ", "5d9e69ce3baa5ac8c305b2a30c9d5b60": "\\,3^3 + 7^3 + 0^3 = 370", "5d9e701fd68b75ef1b50ecf8ec240a3f": "x_1=-\\frac{g(x_0)}{\\omega^2} \\cos(\\omega t)", "5d9e77ecddeedb5ce52d1d9bedca22e6": "\\bold{v}_1 \\bot \\cdot\\bold{v}_2 = \\left | \\bold{v}_1 \\times \\bold{v}_2 \\right | = \\left | \\bold{v}_1 \\right | \\left | \\bold{v}_2 \\right | \\sin\\theta", "5d9e8d99b9db79162495a8a6c80eebc0": " 1- \\left[{1+ {x \\over \\lambda}}\\right]^{-\\alpha}", "5d9fbc4f41772f971a6f745d95be9787": "\\begin{align}\n\\phi_1' &= \\tan^{-1}(\\tan\\phi/B),\\\\\n\\Delta\\phi' &= \\frac{\\Delta \\phi}{B}\\biggl[1 + \\frac{3 e'^2 }{4 B^2}(\\Delta \\phi) \\sin (2 \\phi_1 + \\tfrac23 \\Delta \\phi )\\biggr],\\\\\n\\Delta\\lambda' &= A\\Delta\\lambda,\n\\end{align}\n", "5da00436b140b3a5913d5d8abdd5fe21": " L_{k} = V_{k} \\rightarrow ", "5da03614d1bca10316fff6e9631332d2": "\n\\left[ \\int \\left(\\hat\\theta - \\theta\\right)^2 f \\, dx \\right] \\cdot \\left[ \\int \\left( \\frac{\\partial \\log f}{\\partial\\theta} \\right)^2 f \\, dx \\right] \\geq 1.\n", "5da09e83d855fc5529670b49917147ca": "A(x),\\ [A,B](x),\\ [A,[B,C]](x),\\ [A,[B,[C,D]]](x),\\dotsc\\in T_x(M)", "5da0ad4186d82069fd7389e110100cd0": "\nR = \\frac{1}{t} \\left( \\begin{array}{cc} A + s & B \\\\ C & D + s \\end{array}\\right)\n", "5da0c90358aba6cac39ebfe5f8105787": "\\displaystyle{(Tu,v)=(Tv,u)}", "5da0cb4f399b65e67c0f602a329f5f65": "\nf_\\mathrm{c} = {1 \\over {2\\pi RC}}\n", "5da1010a4873de090cc313e8027df0c6": " |\\epsilon| /|\\lambda +\\epsilon - \\lambda_{\\mathrm{closest~ to~} \\lambda} | ", "5da1014b3190aab2a11751a2277ac188": "S=\\bigcup_{\\alpha < \\lambda} S_{\\alpha}", "5da13418c2810ad87028995776494d5c": "2^{\\aleph_0}.", "5da151835726c4dd71ceb6926a47be52": "T(T(\\mathbf{x})) = \\mathbf{x}", "5da15308c42ebd011485354c92e167c3": "x_1\\geq 0, x_2\\geq 0", "5da18961b04623273683fa5053d30c71": "\\lambda_j-\\alpha\\mu_j \\geq 0.", "5da1edf4cf229adb2f2365a24daec49e": "A \\subseteq \\lambda +1", "5da2415750382e09423f1c5e96c9b4bd": " \\textbf{K}_0 ", "5da2a91bbb060204740ef8a2e111156a": "\\forall g \\in G\\;\\; f_1(g)=1", "5da2b63ee68f205bed66d99c0e2b1872": "a\\in\\mathbb{R}^{n}", "5da302aa5af95581b06e5be6a1045034": "T = \\frac{\\hbar a}{2\\pi c k_\\text{B}},", "5da313f0c69ef255024535e7390a5535": "\\mathrm{^{236}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{237}_{\\ 92}U\\ \\xrightarrow[6.75 \\ d]{\\beta^-} \\ ^{237}_{\\ 93}Np}", "5da322a52d0fe1e5ef1770425fce7008": " \\hat{T}=e^{i\\frac{\\ln\\rho }{2}(\\hat{q}\\hat{p}+\\hat{p}\\hat{q}\n)}e^{-i\\frac{\\dot{\\rho}}{2\\rho}\\hat{q}^{2}}=\ne^{i\\frac{\\ln\\rho}{2}\\frac{d\\hat{q}^2}{dt}}\ne^{-i\\frac{\\hat{q}^{2}}{2}\\frac{d\\ln\\rho}{dt}},", "5da38ee4826f39c928972d531f02c6d4": "\\tan\\frac{\\pi}{15}=\\tan 12^\\circ=\\tfrac{1}{2} \\left[\\sqrt3(3-\\sqrt5)-\\sqrt{2(25-11\\sqrt5)}\\right]\\,", "5da3acb2f076497741aa93ba6979e6d4": "\nCOP_\\text{cooling} = \\frac{\\Delta Q_\\text{cool}}{\\Delta A} \\leq \\frac{T_\\text{cool}}{T_\\text{hot}-T_\\text{cool}},\n", "5da3fa0869178a769280892a0524a135": "x_i \\in \\{0,1\\}", "5da41294f4d283fdfd0dada18a6c4283": "(\\theta \\circ (\\theta,1)) \\circ ((\\theta,1),1)", "5da4385645230c00d46c5c914f6b78fa": "\\tilde{x}_1 = \\mathbb{T}_1 \\, \\tilde{x}", "5da43f839067514bd7e7a65c2e324586": "S(E) = -{dE \\over dx}. ", "5da4438857c4ef7f8f8bcdb921b3ccc3": " \\pi(A) \\psi = 1_A \\psi, \\quad \\int_X^\\oplus H d \\mu(x) \\rightarrow \\int_X^\\oplus H d \\mu(x), ", "5da45240a006c3a5583c46e60f4307c2": "[Z_0:Z_1:Z_2:Z_3]\\ ", "5da4a8ef4fed95b84f222886a425da3a": "|\\mathcal P| < \\infty", "5da513d9ae39736dd1d3d2b65580a816": " x = (R_0 +r \\cos \\theta) \\cos\\zeta \\,", "5da52e7a5418dade891cd6e1f724e7a9": " - {1 \\over z^\\star} = {- z \\over z^\\star z} = {-z \\over \\| z \\|^2} ", "5da56943d0b36222c2530e96031b7e27": "C_{k,n} =\\frac{kn}{2}(n-1)+1.", "5da581bf02ecf90fa8db25fe46bbb9bb": "f(z)= \\sum_{k=0} a_k J_{\\nu+k}(z),", "5da5a363f8b5a9602d2cfb1c8a7e9e1d": "\\boldsymbol{\\mu}_2^{(t+1)} = \\frac{\\sum_{i=1}^n T_{2,i}^{(t)} \\mathbf{x}_i}{\\sum_{i=1}^n T_{2,i}^{(t)}} ", "5da64739390052561681fed2ab74e010": "\\left(\\Gamma^1{}_{11}\\right)_v \\bold{r}_u + \\Gamma^1{}_{11} \\bold{r}_{uv} + \\left(\\Gamma^2{}_{11}\\right)_v \\bold{r}_v + \\Gamma^2{}_{11} \\bold{r}_{vv} + L_v \\bold{n} + L \\bold{n}_v ", "5da6d81d283a118a4aac02a11634b456": "\\rho = \\rho_1 \\simeq \\rho_1 ' \\oplus \\sigma_1 \\quad \\mbox{where} \\quad \\sigma_1 \\simeq \\sigma.", "5da704686468625879ea2df7adfdbbb6": "\\tfrac{\\partial}{\\partial Y}", "5da7184ca6c62b86a9ad3c4a858ea172": "FRA\\,", "5da7c4ac1c2d727c0c3a01d657fac2ea": "p_{Y|X}\\left( y|x\\right) p_{X}\\left( x\\right) ", "5da7ed498ae1b2851bf5f507cf308c4e": "\\beta_f", "5da890551493259695db0a00d10a1a80": "\\frac{R^2}{a^2} + \\frac{Z^2}{b^2} = 1,", "5da8941a49feb978a870a2a2bc25d2f1": "\\Gamma_p", "5da89ecf28eb25c14df305d4462f31b4": "\\,l_x", "5da8a56fbde298a25a77f12ad3871e44": "A_0 A= (A_0 A)^*", "5da8aa9365c99965225fca498528a6a4": "O(N^{-2})", "5da8b4b9449695743e27b8d3ff7451e8": "\\sqrt[3]{m}\\le y\\le\\sqrt{m}", "5da8c449e1f5d67b791e712255ed810b": "\\epsilon_{H^*}", "5da925d6a6f5b8f5529036568412e13e": "b(\\mathsf{i})=i", "5da96d66dd75f9b970dc1972c6170657": "m = - x_2/x_1 = y_2/y_1 \\,\\!", "5da97d01442fbb7855b22e6ab70a0615": "\n\\Gamma(z) = \\lim_{k \\to \\infty} \\frac{k! \\; k^z}{z \\; (z+1)\\cdots(z+k)}, \\qquad\n", "5da9bec93689136279d60744dd68860c": "\\Sigma/\\Delta", "5daa4d48ab7a716f8e11a1e9cc0f0676": "I(X;Y|Z) = I(X;Y,Z) - I(X;Z)", "5daa590540eb94b3d15982a16dbf7ed9": " \\frac{1}{2}v_a^2 = GM \\left( \\frac{2a-r_a}{r_a(2a)} \\right) ", "5daa6ef1bc3600d6bbe70a474ba0c05a": "\n M_{yy}\\Bigr|_{y=-b/2} = -M_{yy}\\Bigr|_{y=b/2}\n", "5daab79abdf2eb2ad281e1bf378c9986": " \\begin{align} y_{k+1} & = \\frac{1-(1-y_k^4)^{1/4}}{1+(1-y_k^4)^{1/4}} \\\\\n a_{k+1} & = a_k(1+y_{k+1})^4 - 2^{2k+3} y_{k+1} (1 + y_{k+1} + y_{k+1}^2)\n \\end{align}\n", "5daabff9ed55b8cff6ff7bc3fccfa002": "\\vert{\\Phi^{*}}\\rangle", "5daba64c45ab9a2e8f90f1ef56c4f0bb": "1 \\otimes 2", "5dabda68008226c2d36dfbee9274617c": "\\mathcal{L}=\\mathcal{L}(\\varphi_i,\\varphi_{i,\\mu})", "5dac24876b860699059a6556e37b0c5d": "s_6(x)=\\frac{225}{8}x^2+\\frac{15}{8}x^4+\\frac{1}{64}x^6;", "5dac3e602675447d9d2136c278d2bb28": "L(p;q)=L(p;1,q).", "5dac54dff4afb5dcd28bd10668712b0b": "-4+2", "5dac6d4a19f2f520fae753a47614ce4f": "\\{1, p, p^2\\}", "5dac87ce4b94c40d0638c02db7f1aeb5": "\\scriptstyle\\;\\rightarrow\\;\\delta'/2= 10,89^\\circ\\;\\rightarrow\\;\\delta'=21,79^\\circ", "5dacd612932a3395668d248c3965ded0": "\\begin{cases}\n \\ -6+ \\frac{\\Gamma \\left(1-\\frac{4}{\\alpha}\\right) -4\\Gamma\\left(1-\\frac{3}{\\alpha}\\right) \\Gamma\\left(1-\\frac{1}{\\alpha}\\right)+3 \\Gamma^2\\left(1-\\frac{2}{\\alpha} \\right)} {\\left[\\Gamma \\left(1-\\frac{2}{\\alpha}\\right) - \\Gamma^2 \\left(1-\\frac{1}{\\alpha}\\right) \\right]^2} & \\text{for } \\alpha>4 \\\\\n \\ \\infty & \\text{otherwise}\n \\end{cases}", "5dacd898a1e44285d79ce00bffddcd16": "z = r \\sqrt{ \\frac{h_c}{k\\, t} }", "5dad4e8a151fd13a4fc69d03a7adb706": " dP = - \\rho g\\,dz\\,", "5dad6d536b86764db4d12e68a0d20905": "\\mathbf{F}\\cdot\\mathbb{J}(\\mathbf{r})", "5dade29adbe5cbef226069a6a39157ac": " n\\geq 5 ", "5dae3f365946755ea4c9f8327798b268": "Z_2 \\times Z_2", "5daea3cb55923fcfaba333ec5da5df23": "\n(\\mbox{div} X) \\; \\mathrm{vol}_n := L_X \\mathrm{vol}_n\n", "5daeb405a228d3b2cfa911ec2b7fc6f5": "\\sqrt{q}", "5daefc23580029877af588d392ce86b1": "p \\mid a", "5daf3554dbfc88a4dea5ed1eb75d6c22": "n_\\ell = 0, 1, \\dots, N_\\ell-1", "5daf8d44a3bd593c76eb7b13f1b5f072": "2\\gamma(x,y)", "5dafabe6c3b14c2b51dcb4595d397c39": " I(\\mathfrak{A}) \\equiv I(\\mathfrak{B})", "5dafd692e45ec3a558891bb3bf620c1a": "f\\colon [A]^{\\omega}\\to Q", "5db00e0e0f2b68db988e13b4134ba9b9": " \\mathbb{A}^s", "5db06671ac2cbea0398f32b3f802e786": "Irish(Fred)", "5db080a66d647d9a4322970f330df13e": "\\int x\\cos ax\\;\\mathrm{d}x = \\frac{\\cos ax}{a^2} + \\frac{x\\sin ax}{a}+C\\,\\!", "5db08fcf4e1067c29d077c31fa6b6cd3": "\n\\mathbf{C}' = \n\\frac{a}{48}(\\mathbf{v}_1 - \\mathbf{v}_2)(\\mathbf{v}_1 - \\mathbf{v}_2)^{\\mathrm{T}}\n+\\frac{a}{16}(\\mathbf{v}_1 + \\mathbf{v}_2 - 2\\mathbf{v}_0)(\\mathbf{v}_1 + \\mathbf{v}_2 - 2\\mathbf{v}_0)^{\\mathrm{T}}\n", "5db09352a7093067d54afc94cebe150f": " c \\equiv m^e \\pmod{n} ", "5db0adb5f446cd253f3b6a5ac8cec833": "p \\log p\\ 2^{O(\\log^* p)}", "5db0e0630976e9d5ea49fef6dd325108": "\\ddot{u}_{\\gamma} = (1 - \\gamma)\\ddot{u}_n + \\gamma \\ddot{u}_{n+1}~~~~0\\leq \\gamma \\leq 1", "5db14be622e745d73b97878b2d943928": "x_1 = m + 2", "5db17edb985fe4aedbb65a44f5c3f800": "\\pi _L", "5db180c4122a91a10275aca21efc5dae": "w(s(\\alpha) - \\alpha) \\ge i+1.", "5db1fe525f8e91f75e8ace4c08396f38": "{p/q \\over r/s} = {p \\over q} \\times {s \\over r} = {ps \\over qr}.", "5db2123449029f0277a031e4b57fd410": " (\\mathbf{a\\times b})_i = \\varepsilon_{ijk} a^j b^k.", "5db21d84c01de55a8693405e275ec8a7": "r_i= y_i - \\sum_{j=1}^{n} X_{ij}\\beta_j", "5db2237c954f3c42f0d3367699a31820": "\n\\lim_{y\\to0} \\left( \\lim_{x\\to0} \\frac{x^2}{x^2+y^2} \\right) = \\lim_{y\\to0} 0 = 0,\n", "5db24a6a93c37d37602cfabfc1629ae5": "\\displaystyle{\\pi(gh) =\\omega(g,h) \\pi(g)\\pi(h)}", "5db2657492435e525144815cf1b3ecfc": " f(x)= \\sum_{n=0}^\\infty \\frac{a_n}{M(n+1)}x^n ", "5db3042c1734d2e62c42fb7b680ff210": "S_1,S_2,S_3\\,", "5db306f96cb07d3f7319da93f2e8e1c5": "\\frac{L_M}{a^2}", "5db3257a4aed0baf890f4d3435305e4c": "\\int\\tan ax\\;\\mathrm{d}x = -\\frac{1}{a}\\ln|\\cos ax|+C = \\frac{1}{a}\\ln|\\sec ax|+C\\,\\!", "5db34b94bf9e0b23e91a2d577d392792": "b=\\sqrt{2}", "5db3589476efafe011b9c3a81600fe4e": "d = \\textrm{MAC}(k_M; c \\| S_2)", "5db3825e21ab218e23760e9ec0d30095": "G '", "5db4037213fd3fee3af2e44c04d08c97": "\\,f( K( \\alpha ) ) = K ( \\alpha ).", "5db4274e5e9eb43ca6c664a9228188b7": "r \\leq \\frac{d}{\\sqrt{3}}.", "5db4ace387139119928634fcaa459760": "y = x^2\\,", "5db52b7ed0b601c20f8ebea778bac204": "GCV = {e^{s_{ln}}\\!\\!-1}", "5db52f276c46db708920a6cdfc211fbf": "\\left|s,m_s=-s\\right\\rangle", "5db53499d0fecb06e05ef83e220693b4": "X_{feature space}=\\left \\{ \\vec x_1, \\vec x_2,...\\right \\}", "5db54180eead7d6a58099feedd5fd321": "\n\\quad\\quad =\\frac{1}{\\tau} \\left( \\int_\\Omega\\int_0^1 \\frac{d}{ds} F(u+s\\tau\\psi) \\,ds\\,dx \\right)\n", "5db559e61e5ac02b642fd8903281046a": "H^d(S)=0", "5db566cb074575457340b7cd89a6acc7": " \\langle X,Y\\rangle_A=\\int\\limits_0^1 {\\rm Tr}[ {\\rm e}^{xA} X^\\dagger{\\rm e}^{(1-x)A}Y]dx", "5db57f6c1f8061efe461a55605fd281a": "\\eta_Y:Y\\to GFY", "5db5cb99b2fa815c58ef00e4a29f461f": "\n\\widehat{\\boldsymbol \\theta}_{JS} = \n\\left( 1 - \\frac{(m-2) \\sigma^2}{\\|{\\mathbf y} - {\\boldsymbol\\nu}\\|^2} \\right) ({\\mathbf y}-{\\boldsymbol\\nu}) + {\\boldsymbol\\nu}.\n", "5db5eceac89ed3ade9642c1eb5bd9cd2": "\\Omega^1_M(\\mathbf V)", "5db6245c9d1782988ecae406eb942a06": "x=c_1a_1 + \\cdots + c_na_n", "5db64a4ad2ad95ee06c01b2277abf27e": "\\pi^n ", "5db666464d92d344ca86828ae9059665": "\\scriptstyle t=D\\,T", "5db677f30bd2b2b511bf8f6b1a149f4d": "\\lambda,\\rho", "5db6f45c41c2e8d2d5fc5c7b8976ea7d": " r = 0^{\\lfloor \\log^2n \\rfloor} ", "5db73a93f1ed3504f88d289da30bee64": "t\\le\\tau\\,", "5db7bc2e618edb798add8b0ba6d666bf": " \\scriptstyle \\phi ", "5db7dc968dff01127edc898e73752846": "q_{i, i+1} = (N-i\\,) \\lambda", "5db8135bd287003dccbf3572074b534e": " D =\\frac{k_1 L_a}{k_2-k_1} (e^{-k_1 t}-e^{-k_2 t}) + D_a e^{-k_2 t} ", "5db847beda7bcdcbe3fff3d16c11b5f1": "\\{ x_n \\}_{n=1}^{\\infty} ", "5db85603ee20ed556262c7a893f13182": "\\vec{v}(t + \\Delta t) = \\vec{v}(t) + \\frac{\\vec{a}(t) + \\vec{a}(t + \\Delta t)}{2} \\Delta t \\,", "5db86779991681d0c2bf1ac52d7f71a3": " : \\hat{f}_1^\\dagger \\, \\hat{f}_2 \\, \\hat{f}_3 : \\,= \\hat{f}_1^\\dagger \\,\\hat{f}_2 \\,\\hat{f}_3 = -\\hat{f}_1^\\dagger \\,\\hat{f}_3 \\,\\hat{f}_2", "5db8ab6872ebec9d8bf64df6ddfaf0e4": " \\frac{1}{4 \\pi R} - \\frac{a}{4 \\pi \\rho R'}, \\,", "5db903db43b049698e179a7456436a6a": "\\psi(x_1, x_2, \\dots) = \\psi(x_2, x_1, \\dots) \\qquad\\quad \\text{for bosons} ", "5db92e9140365518e57d428feaa5fe60": "(3\\times G_2(3)): 2", "5db94e29a6b32e0a4b38af1f14dddde8": "\\big. \\frac{\\partial Q}{\\partial t}\\big.", "5db9dcc88240a121ee2a44d6ccab6be7": " 2\\epsilon", "5db9f2f23d76a12226d4a6e9e0798997": "|T_N|=\\biggl|\\sum_{i=1}^N\\operatorname{E}[X_i]\\biggr| \\le \\sum_{i=1}^N\\operatorname{E}[|X_i|]\\le CN,", "5dba32ab77b4693926b9a4f097ccef97": "(x_n y_n)", "5dba905e9b7bf34d9b21a68229994000": "2\\cdot\\left(\\frac{4!}{2!\\cdot2!}\\right)^2 = 2\\cdot6^2= 2\\cdot36 =72", "5dbaa34dba547f7b901fcee182be5072": "\\operatorname{Ext}^i_R (k, R) = 0", "5dbad057040ec6eb5aa5841786e25d33": "y=x", "5dbaf2a86a80bb2d00583153eee60e9f": " Q(t)^T Q(t) = I ", "5dbbd1f0464f0362830d06a491354b02": "\\mathbf{\\hat{\\boldsymbol{k}}}", "5dbc1b151d9d5a3d5dae947e965d7dde": "\n\\left[ \\bar{X}_{i},\\bar{Z}_{j}\\right] = 0\\ \\ \\ \\ \\ \\forall i\\neq\nj,", "5dbc4597ab9b7a458cbee4a94db26a43": "\\binom nk/n^k=\\frac1{k!}\\times\\frac nn\\times\\frac{n-1}n\\times\\cdots\\times\\frac{n-k+1}n;", "5dbc51af19e5296a02d9ba147cd72070": "(x-7)", "5dbc5ba75d15c404111608667cddbf0b": "(\\lambda z.x)[x := y] = \\lambda z.(x[x := y]) = \\lambda z.y", "5dbc66f5b4a3b4e014e9cc73bd1bd1c1": "\\tilde H_r", "5dbc931d0be9599d1eb9e43a7aabe4c1": " f_{Demand} = \\frac{ \\text{Maximum load in given time period}}{\\text{Maximum possible load}}", "5dbc98dcc983a70728bd082d1a47546e": "S", "5dbcbbfe5017d524e456220d6911c2be": "{\\theta}_{[a,b]}", "5dbcc497d41742c6a1d46cd8a714f1b2": "P(x, \\partial)u(x)=\\sum a_{\\alpha_1, \\alpha_2, \\dots, \\alpha_n}(x) \\partial^{\\alpha_1}\\partial^{\\alpha_2}\\cdots \\partial^{\\alpha_n} u(x) ", "5dbccdb2206ee2438c48681f27a1d806": "n(t)", "5dbd16c4513ea8aee2fdd8a1abc1819b": "P: \\mathcal{H} \\to \\mathcal{H}", "5dbd2bfd7f936527ddd317f33a6d7cd0": " \\mathbf{x}=\\left\\{ x: ax=b, a\\in\\mathbf{a}, b\\in\\mathbf{b} \\right\\} = \\frac{\\mathbf{b}}{\\mathbf{a}}=\\frac{[1,4]}{[1,2]}=[0.5, 4] ", "5dbd6c857aacf98cd03909b79089fdd5": "\\xi(2n) = (-1)^{n+1}\\frac{1}{(2n)!}B_{2n}2^{2n-1}\\pi^{n}(2n^2-n)(n-1)!", "5dbd77f45eeae4ace7fb37192c80c826": "\\mu = g\\vert\\psi(0)\\vert^2/2", "5dbdd8230e4c65a806583b591150d3b3": "Pwf = \\cfrac{2 Pmf}{3}", "5dbdea90cfc7d5fd77d92038b637df92": "-1 \\cdot e", "5dbeadc4a02e25594c3e4c89a4ae0a6d": "\\{ p, q \\} \\vdash (p \\land q)", "5dbef5052da4b5f0df58f51df18e374e": "\\textstyle v^2=\\frac {2gW_S} {\\rho C_L}", "5dbef736ce380660728eb24da9fc461d": " \\mathbf{p}_{+,-} ", "5dbf08bfcb28e71faf553caeabde8918": "B|\\psi_n\\rangle=b_n|\\psi_n\\rangle", "5dbf0cc23cabfbb39926a527d6a70f00": "E(m)", "5dbf3a12e00c2b1ca7b503397b28dab4": "a\\in F", "5dbf3fd0647783d118d8acc44ad69b96": "\nf(n)=\\int_{n-1}^{n} f(n)\\,dx\n\\le\\int_{n-1}^n f(x)\\,dx.\n", "5dc06bc618a258fa1aa6be5cb0f1a040": "\n\\begin{align}\n& & a^2 + b^2 + c^2 & \\geq & & ab+bc+ca \\\\\n\\iff & & 3(a^2 + b^2 + c^2) & \\geq & & (a + b + c)^2 \\\\\n\\iff & & a^2 + b^2 + c^2 & \\geq & & \\sqrt{3 (a+b+c)\\left(\\frac{a+b+c}{3}\\right)^3} \\\\\n\\iff & & a^2 + b^2 + c^2 & \\geq & & \\sqrt{3 (a+b+c)(-a+b+c)(a-b+c)(a+b-c)} \\\\\n\\iff & & a^2 + b^2 + c^2 & \\geq & & 4 \\sqrt3 \\Delta.\n\\end{align}\n", "5dc07d3ddb5e389452a6b6f1942f37f4": "\\delta=1 ", "5dc085466ecf5cb29f978a3bdb692531": "{d^n \\over dx^n} f(g(x))\n=\\sum \\frac{n!}{m_1!\\,m_2!\\,\\cdots\\,m_n!}\\cdot\nf^{(m_1+\\cdots+m_n)}(g(x))\\cdot\n\\prod_{j=1}^n\\left(\\frac{g^{(j)}(x)}{j!}\\right)^{m_j}.", "5dc08578e8d53a832f61844a5f1ed118": "\\rho^{23}", "5dc0a5d6ec26353f573907785799b618": "Commit_k(x,open)=Commit_k(x',open')", "5dc0a920e43a8c13d2008992d7eed692": "\\varphi = \\psi \\circ \\exp_o:S' \\longrightarrow \\mathbb{R}^{3}", "5dc0aeb3a90434cbc8ac7e468f9f7096": " \\omega_{max} ", "5dc0b788590f147f3288380b1ac73629": "{A_{cross}}", "5dc0dae46083239eecdde5a0991bfec0": "\\nabla:\\Gamma(E)\\rightarrow \\Gamma(T^*M\\otimes E)", "5dc0ddc91d086176a60599a3be92be32": "f(\\mathbf{i},\\mathbf{j})=e^{-\\beta V(\\mathbf{i},\\mathbf{j})}-1", "5dc0fc1ca9cdfb68c28fc929fb57ff65": " p,q\\in X", "5dc136839494b7807904fd2e53563d51": "c(x)^2 |\\nabla_x u(x,t)|^2 = |\\partial_t u(x,t)|^2", "5dc17ce1d55b2f99d8c2990f933079b7": "(\\varphi, \\Phi)", "5dc18dd3dc50305055e1faa2a576034e": "\\underline{A}", "5dc1a333301eee02bdbbe837d7110961": " C_{(-)}= i \\sigma_2 = C_{(-)}^* = s_{(2,-)} C_{(-)}^T = s_{(2,-)} C_{(-)}^{-1} ~~~~ s_{(2,-)}=-1 ", "5dc1ed33db0811132d33a667160a29da": "F(x) = .5\\|\\tilde{A}x-b\\|_2^2", "5dc222bdfb8df84b61072f75026457b4": "\n E=\n \\left( { a_1\\, a_2 \\over 2 \\pi L_B}\\right) \\int_0^{\\infty} {k\\;dk \\;} D\\left( k \\right) \\mid_{k_0=k_B=0}\n\\mathcal J_0 \\left ( kr_{B1} \\right) \\mathcal J_0 \\left ( kr_{B2} \\right) \\mathcal J_0 \\left ( kr_{12} \\right)\n", "5dc226cb7b265268495fcf685068107f": "p\\ f = \\lambda x.f\\ (x\\ x) ", "5dc23e48fac74cca7bf943e440e99e95": "x^3-x^2-(3x+1)y^2=0,\\,", "5dc298511bca7e78a4e941f2f18af293": "\\overline{\\mathrm{SL}(2,\\mathbf{R})} \\to \\mathrm{PSL}(2,\\mathbf{R}).", "5dc29acab62ec0c4a84459dae5deab4d": " c_1 = \\frac{2+1}{2} = 1.5 ", "5dc2c3073ee12faa878c3c2b77ea0d94": "p(c_{rational}|f_{p\\mbox{-}int}) = 1\\ ", "5dc32f5d5e8236c2250e9fe26e2ee384": " a^2 +b^2 =c(r+s) \\ . ", "5dc386c6cf915f71771171b8965085cc": "Z_{i1} Z_{i2} = {R_0}^2.", "5dc3a103b72cc55292069f4caad757e6": "\\theta /2", "5dc3ef217d5b7d8e0903c885a1c86489": "2 \\pi R L", "5dc3f46196b491f97e2d9f59a3f7b14f": " k = 2.5 ", "5dc3f80f6c380041911c73ef16fb5aef": "\nq'=\\frac{R\\mathbf{p}\\cdot\\mathbf{M}}{p^3}\n", "5dc459ffee85ef3986056f6e4c4c9d7f": "\\; H_{\\rm{int}}(x)", "5dc51bfb525fc73185cb1b2d0bc25bdf": "\\mathrm{Rep}_{cris}(K)\\subsetneq\\mathrm{Rep}_{st}(K)\\subsetneq \\mathrm{Rep}_{dR}(K)\\subsetneq \\mathrm{Rep}_{HT}(K)\\subsetneq \\mathrm{Rep}_{\\mathbf{Q}_p}(K)", "5dc5276437681e77457253208e57fcdf": "c_i = \\frac{{x_i \\cdot \\rho}}{{M}} = x_i c ", "5dc52f392f2c5fa729d578adb8464662": " \\begin{align}\n\\nu &= \\alpha + \\beta = \\frac{\\mu(1-\\mu)}{\\mathrm{var}}-1, \\text{ where }\\nu =(\\alpha + \\beta) >0,\\text{ therefore: }\\text{ var }< \\mu(1-\\mu)\\\\\n\\alpha&= \\mu \\nu =\\mu \\left(\\frac{\\mu(1-\\mu)}{\\mathrm{var}}-1\\right), \\text{ if }\\text{ var }< \\mu(1-\\mu)\\\\\n\\beta &= (1 - \\mu) \\nu = (1 - \\mu)\\left(\\frac{\\mu(1-\\mu)}{\\mathrm{var}}-1\\right), \\text{ if }\\text{ var }< \\mu(1-\\mu).\n\\end{align}", "5dc56123c9ca5d44ec94a341bb4997da": "E = I Z", "5dc58f0c897160e60ba11e0db08c8f17": "\\forall u \\forall v(uEv \\rightarrow vEu)", "5dc599530d7919f607614de50f9859dc": "\\frac{\\ddot{\\Phi}}{\\Phi} =-n^2", "5dc5eca5570939870a6e3f769adf3555": " \\mathbb{F}^m", "5dc62b79b1ddd07bcff8b197b762369d": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n-{1 \\over \\vec k^2 + m^2}\n,", "5dc63900ee293849a9071cc5ef5b17cc": " \\sigma_{ij} = \\left ( \\frac{\\partial f}{\\partial \\epsilon_{ij}} \\right)_S. ", "5dc6ce4ab0d1851337c334bfe75c66b5": "\\pi_A(\\{ \\langle A=a, B=b \\rangle \\} \\setminus \\{ \\langle A=a, B=b' \\rangle \\}) = \\{ \\langle A=a\\}", "5dc70454f61079b8fb7aa446cb2813a3": " f_{ch}", "5dc714b1e78eeec81e7c3e45f801ebd4": "\\mathcal{S}=\\int \\mathrm{d}^{D-1}x \\, \\mathrm{d}t\n\\mathcal{L} = \\int \\mathrm{d}^{D-1}x \\, \\mathrm{d}t \\left[\\eta^{\\mu\\nu}\\partial_\\mu\\phi^*\\partial_\\nu\\phi\n-V(|\\phi|^2)\\right]", "5dc7fcaa90eecc1cb28c654214405873": " l = \\sqrt{\\frac{2(n+1)}{d}} -1 ", "5dc8193b08f4c2b377b4e3f2df8ef0e5": "\\operatorname{tr}(ABC) \\neq \\operatorname{tr}(ACB)", "5dc8905a679e4314051fe4cb94731aae": "2 \\left (x - 75 \\right )(y+5) + \\left ( z \\cdot 15 \\right )", "5dc8993fc0b3f62450a9f6836d814f59": "2^{1/12} = \\sqrt[12]{2}", "5dc90006e09c82b2ea6bfee3ddac5a86": "(A-\\lambda I)v = 0,", "5dc91cc8ae5a54d75dc5f72246d5853f": "\\left(\\nabla\\times\\mathbf{A}\\right)_{ij} = \\nabla_i A_j - \\nabla_j A_i = 2\\nabla_{[i} A_{j]}", "5dc9467e002b47433b4aeef079230eae": "0<\\alpha_i", "5dc955bb45c28e0c1cdca6d86153e7eb": "|\\hat{f}|", "5dc9b2c43f3c3d7b3e13265f4f8d9f01": "q(0), p(0) \\ ", "5dc9ba8498f0767f3976f414b5628817": "Q \\times \\Gamma^*", "5dc9f4c7dd077465025217dc2763e9ab": "10 \\times 10^{1.5} = 316.23", "5dca1e9ac54dac3903f2bb7d1fb2497b": " q-1 ", "5dca761a931633963c601976312c061e": "n_q", "5dca95864a92e313d08445a810789b4b": "\\vec J_i", "5dcacc5f35de5ca2c703f1a5580f0354": "\\tan\\frac{\\pi}{30}=\\tan 6^\\circ=\\tfrac{1}{2} \\left[\\sqrt{2(5-\\sqrt5)}-\\sqrt3(\\sqrt5-1)\\right]\\,", "5dcbbd12c1ea38d62f989e649bbdc1c4": "\\bar\\Omega", "5dcbf640cdb155217ca0c533e1078ff7": "\\mathcal{L}", "5dcc46d9878ed1b87e0a313060f5092e": "\\begin{alignat}{2}\n6 &&\\; = \\beta_1 (1)^2 \\\\\n5 &&\\; = \\beta_1 (2)^2 \\\\\n7 &&\\; = \\beta_1 (3)^2 \\\\\n10 &&\\; = \\beta_1 (4)^2 \\\\\n\\end{alignat}", "5dcc47434ad8b7f05e8403972e62ebd2": "\\mathcal{Q} = \\left\\{Q \\in \\mathcal{M}_1: E\\left[\\frac{dQ}{dP}|\\mathcal{F}_j\\right] \\leq \\alpha_{j-1} E\\left[\\frac{dQ}{dP}|\\mathcal{F}_{j-1}\\right] \\forall j = 1,...,T\\right\\}", "5dccac2dd138134d3557b83a341cb2f7": "\\varepsilon = \\frac{1}{v h}\\,,\\quad \\mu = \\frac{h}{v}", "5dccc2eeb8a508b4fe3c5167459c2bc2": "- \\frac{dE}{dx} = \\frac{4 \\pi}{m_e c^2} \\cdot \\frac{nz^2}{\\beta^2} \\cdot \\left(\\frac{e^2}{4\\pi\\varepsilon_0}\\right)^2 \\cdot \\left[\\ln \\left(\\frac{2m_e c^2 \\beta^2}{I \\cdot (1-\\beta^2)}\\right) - \\beta^2\\right]", "5dccde159024697af8a8c195dd214c84": "\\propto |x|^n \\cdot \\exp (- x^2)", "5dcdcece8a28dd578b1308aeb2f47a96": "\\rho=\\frac{ab}{a+b}", "5dce6b7f474b59cdba90f6f7cfd3d334": "\\inf\\, \\{ x \\in \\mathbb{Q} : x^3 > 2 \\} = \\sqrt[3]{2}.", "5dce95af9402f11eeda168f2e709e81e": "\\rho(\\vec x)=x_1x_2-x^2_3", "5dcee41a57844b74d2e0ff4c12f9b283": "(R, P) \\approx_\\epsilon (U_\\ell, U_{|P|}) ", "5dcf1890f37ea9302ae361b63e9877c5": "\\oint_{\\partial S} \\bold{B} \\cdot {\\rm d} \\boldsymbol{\\ell} = \\mu_0 ", "5dcf2ad9abacbeba2c6814d64d3a4fcc": " (-i + 1) I = -2\\pi i \\left( \\mathrm{Res}_{z=5} \\frac{f(z)}{5-z} + \\mathrm{Res}_{z=\\infty} \\frac{f(z)}{5-z} \\right).", "5dcf3b4c981f15c9802f03896d2558b0": "t,n \\geq 0", "5dcfda6bd8ed4d6ca4ac1d975f95b1f2": "\\mbox{ }_{U_{I_0}}", "5dd028513fc7ad25ba6c582502d38af7": "D\\frac{1}{(n-2)\\omega_{n}\\|x-y\\|^{n-2}}=G(x-y)", "5dd02d2f1376bc7687f0f606e0af7d85": " n \\geq 2 ", "5dd05c0aaf7ea79d24dbaf51632c3f23": "A= -2i\\Sigma\\partial_x^2+(-i\\Sigma_x-i\\alpha\\Sigma_y\\Sigma+u_yI-\\alpha^3u_x\\Sigma)\\partial_x.\\qquad (3b)", "5dd06a599a31812871b998c12fbbae81": "\\frac{1}{s_n} \\sum_{i=1}^{n} (X_i - \\mu_i) \\ \\xrightarrow{d}\\ \\mathcal{N}(0,\\;1).", "5dd1037ab6998770fd2c0296a44f4e67": "f=q\\circ f\\circ p", "5dd105dddd16de61b52e18e27dface23": "k_T", "5dd11b128898ad7a81abd9ef7d7e912a": "\\{|\\alpha\\rangle\\}", "5dd135d1bcfa7f63c3b7f25425c2a4a1": "Score", "5dd18db2244447566e6df2cacd6777ee": "X \\sim S(\\alpha, \\beta(\\cdot), \\gamma(\\cdot), \\delta(\\cdot))\n", "5dd1de4a59da3272120ff2f3b8d3a664": "\\int_{-\\infty}^\\infty dx\\, \\int_{-\\infty}^\\infty dp\\, P(x,p)P_\\theta(x,p) \\ge 0 ~,", "5dd2199ad68327cc76d583b057aee7d5": "card", "5dd251b13865b545b2f31d1790badab4": "I=f(\\mathbf{x})", "5dd29abb493647d04318ea20094fc2c0": "\nM' = \\begin{pmatrix} a_{1,1} & 0 & 0 & 0 & \\ldots & 0 & a_{k,1} \\\\\na_{1,2} & 1 & 0 & 0 & \\ldots & 0 & a_{k,2} \\\\\na_{1,3} & 0 & 1 & 0 & \\ldots & 0 & a_{k,3} \\\\\na_{1,4} & 0 & 0 & 1 & \\ldots & 0 & a_{k,4} \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\\na_{1,k-1} & 0 & 0 & 0 & \\ldots & 1 & a_{k,k-1} \\\\\na_{1,k} & 0 & 0 & 0 & \\ldots & 0 & a_{k,k}\n\\end{pmatrix}.\n", "5dd2ad65da10f60e09780c535950d743": "b_{ijk}", "5dd2ecd68e92056800ef1252d4349e76": " \\Phi_{C} = ", "5dd2eedbb589fdcb2b075a680d234ce2": "p_1=p_2=\\cdots=p_n=\\frac{1}{n}", "5dd36cb9edcfa879fedfd66008dd6c64": " \\mathrm{d} I = J {\\mathrm{d} A} , \\,\\!", "5dd39207335a0c28b9d3895dce054f4d": "\\scriptstyle{R_{F}(x,y,z)}", "5dd3971e0009554281c3a68ab3a3e3db": "f(y)={\\sum_i {x_i y_i}}", "5dd3b2f7d534bf986acaa38e307a435b": "d = 2753", "5dd3c64c06f66713dec8e6c064a4e74d": " \\delta^{\\alpha\\beta\\gamma\\delta}_{\\mu\\nu\\varrho \\sigma} = \\varepsilon^{\\alpha\\beta\\gamma\\delta} \\varepsilon_{\\mu\\nu\\varrho\\sigma} ", "5dd3e7a0a2c25ccd01664fbe17004da1": "{\\mathit{H}}_n^{(m)}(x)=2^m\\cdot\\frac{n!}{(n-m)!}\\cdot{\\mathit{H}}_{n-m}(x)=2^m \\cdot m!\\cdot{n \\choose m}\\cdot{\\mathit{H}}_{n-m}(x).\\,\\!", "5dd41c465575a22788eed1d67940f69c": "\\Delta k_m k_m", "5dd46fedfa04057c704629b3b4e5f9b9": "[\\mathbf{R}] = \\begin{bmatrix}\n \\mathrm{c}_\\alpha \\, \\mathrm{c}_\\gamma - \\mathrm{s}_\\alpha \\, \\mathrm{c}_\\beta \\, \\mathrm{s}_\\gamma &\n-\\mathrm{c}_\\alpha \\, \\mathrm{s}_\\gamma - \\mathrm{s}_\\alpha \\, \\mathrm{c}_\\beta \\, \\mathrm{c}_\\gamma & \n \\mathrm{s}_\\beta \\, \\mathrm{s}_\\alpha \\\\ \n \\mathrm{s}_\\alpha \\, \\mathrm{c}_\\gamma + \\mathrm{c}_\\alpha \\, \\mathrm{c}_\\beta \\, \\mathrm{s}_\\gamma &\n-\\mathrm{s}_\\alpha \\, \\mathrm{s}_\\gamma + \\mathrm{c}_\\alpha \\, \\mathrm{c}_\\beta \\, \\mathrm{c}_\\gamma & \n-\\mathrm{s}_\\beta \\, \\mathrm{c}_\\alpha \\\\\n \\mathrm{s}_\\beta \\, \\mathrm{s}_\\gamma &\n \\mathrm{s}_\\beta \\, \\mathrm{c}_\\gamma & \n \\mathrm{c}_\\beta\n\\end{bmatrix}\n.", "5dd4c06cbea80f381213972f467c89ae": "s-p", "5dd4e17b90dc0007c99f1a5c590b80ff": "{1-m\\over2},\\ldots,0,\\ldots, {m-1\\over2}", "5dd4f6f4fb405dd77a5823542226458b": "\\vartheta_2", "5dd550c520b67a67e16ceeb459eb1f40": "V(t)|\\psi_1\\rangle+H_0|\\psi_0\\rangle = i\\hbar\\frac{\\partial|\\psi_1\\rangle}{\\partial\\tau}", "5dd58a409fc2147a777473d88cbbebdb": "\\; m", "5dd5cb080a8508ff3c8b05d9f020434c": "Z_{CO} = Z_{CO}^1+Z_{CO}^2", "5dd604195661f3a86f19952be35d98c2": "\\frac{\\neg (P \\or Q)}{\\therefore \\neg P \\and \\neg Q}", "5dd64433c66c2084bf3403ad4589de1f": "{(X_j)}_{j\\in J}", "5dd64574d5e317d08dde91223a08b6df": "-\\frac{\\pi}{a}\\leqq k\\leqq\\frac{\\pi}{a}", "5dd67bd3ad2307e45c60ff596a3db835": "f={\\operatorname{-d}V\\over\\operatorname{d}r}={24\\varepsilon_o\\over r_o}\\left[{2}{r_o\\over r}^{12}-{r_o\\over r}^{6}\\right]", "5dd6c1593cd06c2ca14f13e6bea204a6": "\n\\begin{bmatrix}\n 1 & 0 \\\\\n \\frac{-1}{\\lambda R_B} & 1 \n\\end{bmatrix}.\n", "5dd6d378c534f98bbf7a8b5f13877de9": "H_2", "5dd6d4a1e019603ce2ccc8825747145a": "(U,u)", "5dd6dde829b275932221aa0db29c1483": "\\mathbf{A}_{\\text{Magnetic dipole}}", "5dd7111be425235a713ce4d809483986": "\\bar{r} \\approx \\bar{E} - \\bar{C} + \\gamma \\text{Cov}(E(t),C(t))", "5dd74a88add5425e031971d39e31a49e": " \\mathbf{n} ", "5dd74d63e7a8d9f9b8a023ff81e35bfc": "\\frac{1}{\\lambda} = R_M\\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)", "5dd77c4ea20e9d47c3a897246448cd75": "\\mathbf{r}=(1, \\ldots, 1, r, 1, \\ldots, 1)", "5dd7a091dc73f1ea37ce47e93a0e8dc6": "e_q(t)", "5dd7a7b70d8cde4e0ab8535c9b591a5d": " \\otimes", "5dd7aab72c6e02dc96d418ea17ac5b00": "de_1=0", "5dd7dfb46e65bfa033d50eac04f5a2fb": "\\int\\frac{\\mathrm{d}x}{q \\tan ax + p} = \\frac{1}{p^2 + q^2}(px + \\frac{q}{a}\\ln|q\\sin ax + p\\cos ax|)+C \\qquad\\mbox{(for }p^2 + q^2\\neq 0\\mbox{)}\\,\\!", "5dd7f0354dc7736b312aed16be94c695": "-y-z-y^2*x-x+xyz=0", "5dd89018f35b72f7165d124e636a1927": "A_{\\alpha\\beta\\gamma \\cdots}", "5dd8b498dcd7b44ab362f205ce936f09": " f(x_{1},x_{2}) = c_1 x_1 + c_2 x_2", "5dd98788f16623b5730c3115c25e4703": "\\Pi(g)(A) = \\Pi(g)A\\Pi(g)^{-1}, \\quad A\\in \\mathrm{End}\\,(V),\\ g\\in G.", "5dd98c24e58db10c9bafe305ec7cda0f": "P^{-1}T_0P", "5dd9b74e2f77463d37bc760318f92ace": "a_1x_1 + \\cdots + a_kx_k=b", "5dd9e3e105486edbe676b93b2331a783": "\\operatorname{sgn}(m^{(2)} - \\bar{x}^2)\\sqrt{S}", "5ddab6cbe120d577a3a4c45d2f8e45d9": "\\mathcal{B}(X, Y; Z)", "5ddad16de8209cb023e469ec7343137f": "\\epsilon_{1,2}(p)", "5ddafa925424d5e8165d4ac370255784": " \\mathbf{ \\hat U} (t)|\\psi(0)> ", "5ddb0eee7cf171ac9381fe98e120781c": "U_\\alpha", "5ddb311c48938d13412041f88932d228": "m\\ddot x + c \\dot x + kx = 0", "5ddb6b2194a17a42433f7d839c547eb9": "r_{C}", "5ddb85576b36f22c75aabcfca8beb0df": "y_{n+1}=e^{-r}(y_n+\\epsilon\\cos(2\\pi x_n))\\,", "5ddbbb2e0551267573ff1633bce3d8a3": "p(a,x_1,\\ldots,x_n)=0", "5ddc221717339cbcb30c4f0ea04a9da1": "X \\sim S(\\alpha,\\beta(\\cdot),\\gamma(\\cdot),\\delta(\\cdot))", "5ddc4246169cb38b2a9eb285594f6749": "\\liminf_{y\\to x} f(y)\\ge f(x)\\,", "5ddca0b859bc80caab8b08d1e98218bd": "\\mbox{NP} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NTIME}(n^k)", "5ddca5611553d565ea0d0de5d446bba3": "\\lim_{n\\to\\infty} x_n", "5ddcbe6c99bb24a5eba904c5c7c0a2d2": "H(z) = \\frac{z^{-1}-\\overline{z_0}}{1-z_0z^{-1}} \\ ", "5ddd07ba5f1474c5746e173b9702734f": "x \\mapsto f(x, a_2, \\ldots, a_n),", "5ddd145b1996584ed8c1603259368c17": "e < {N -K+1 \\over 2}", "5ddd197e9b947261bff473736b08ab4d": "\\scriptstyle b^1", "5ddd22e27bb6753a8ccb837b05c40b04": "\\Phi_\\mathrm{v}", "5ddd2f8c258b498795ec2c97fa4fcac9": " a_{N} \\phi_{N} \\,= a_{R} \\phi_{R} + a_{L} \\phi_{L}", "5ddd7d196f6f31ec0827bf3e73e35e05": "G\\, ", "5ddd7d8a2f83314fc0b7ed8941ef778d": "\nu_k \\approx U_k + \\sum_a v_a V^a_k,\n", "5ddda48981a9f7cc5340e4c5f3f8bb33": "A_1,\\ldots,A_5\\in{\\mathbb Q}(x,y)", "5dde1b9aa6059f93992f1f824ffd0a64": "\\limsup_{n\\to\\infty}t_n \\le \\limsup_{n\\to\\infty}s_n = e^x", "5dde7d4f15fe1f723ed86e3be299b214": "0 \\to \\mathbb{Q} \\to \\mathbb{R} \\to \\mathbb{R}/\\mathbb{Q} \\to 0,", "5dde7e1e5442e23995bd213b4b53dadb": "\\chi_k (\\mathbf{r}; \\mathbf{R})", "5ddeb96c8d677aa9e974fdab73106f3d": "e_\\alpha", "5ddf35a84b82fb0b8b032de716edc048": "r=f_1(\\theta)", "5ddf80d66f7944734c7ea4e6d88bf315": " Tr_{12}\\Bigl(\\rho^{123}(-\\log(\\rho^{12})-\\log(\\rho^{23})+\\log(\\rho^2)+\\log(\\rho^{123}))\\Bigr) \\geq 0.", "5ddf8d14ac01a726605746df5de27ee7": "\\chi_\\text{mol} = M\\chi_\\text{mass} = M\\chi_v/\\rho", "5ddffe56ca4741b72c75630414112750": "\n\\frac{ma^{2}}{2} \\left( \\cosh^{2} \\xi - \\cos^{2} \\eta \\right)^{2} \\dot{\\eta}^{2} = -E \\cos^{2} \\eta + \\left( \\frac{\\mu_{1} - \\mu_{2}}{a} \\right) \\cos \\eta + \\gamma\n", "5de03234220d0ad9b89cc29d44363240": "\\log^*(n) =\n\\begin{cases}\n 0, & \\text{if }n \\leq 1 \\\\\n 1 + \\log^*(\\log n), & \\text{if }n>1\n\\end{cases}", "5de06bbbcc466d9b320de9cd169ebd46": "(T^2, g)", "5de0915ff7aa60b3af86ce266b7bef17": "{\\operatorname{d}Q\\over\\operatorname{d}t} = K_f \\times (P_G - P_B - \\Pi_G + \\Pi_B)", "5de0af16b2c58875bd824cc448a4dace": "a_{11}= 9, \\, a_{12}= 8, a_{21}= 1, \\, \\cdots \\, \\, a_{23}= 7 \\, \\cdots ", "5de0c271875b893624eef37706093cad": "g_{ab}", "5de0c966b0c0b6f1f4cdf916c6e791d7": "n^{11/32}", "5de0f4a5e8afe72f3c0847d78f6d3848": "\\Gamma_{e_1} = \\Gamma_{w_2} , \\Gamma_{e_2} = \\Gamma_{w_3} , \\Gamma_{e_3} = \\Gamma_{w_4}", "5de0f90d9784254c56e5d684834fe10e": "\n\\left [ Z-Z_{t} \\right ]\n\\left [ Z+Z^{*} \\right ]\n", "5de15ade4683ed7669090301aa1256d1": " P(Y) = ", "5de1a69d69cedb15f6fcd5c1582a8591": "\\lambda = A_{i i}", "5de1c62b63ae9e3a0a047ab575190504": "[\\overline{P}] - [O]", "5de1e10a635a0f04e81d3a43fed714bf": "G = \\mathrm{clamp}(Y - 0.344 \\times (Cb - 128) - 0.714 \\times (Cr - 128))", "5de1ee944529bb0301018509467af255": "\n\\xi = \\frac{\\exp \\left( \\beta \\mu + \\beta/2 N \\bar{\\Phi}(0)\n\\right) Z'}{\\lambda^{3N} (T)},\n", "5de2399c3e6eae2e580a4c503e969409": "2 N \\log_2 N", "5de2ff7284278059ac0471e54ed3585f": "dr_t = \\theta r_t\\, dt + \\sigma r_t\\, dW_t", "5de316b075c297a8c23c780d0f8ea842": "\n\\begin{align}\n\\mathrm{ad}_x(y) & = d (\\mathrm{Ad}_{x})_{e}(y) \\\\\n& = \\lim_{\\varepsilon \\to 0}\\frac{(I+\\varepsilon x)y(I+\\varepsilon x)^{-1}-y}{\\varepsilon} \\\\\n& = \\lim_{\\varepsilon \\to 0}\\frac{(I+\\varepsilon x)y(I-\\varepsilon x +(\\varepsilon x)^2+O(\\varepsilon^3))-y}{\\varepsilon} \\\\\n& = \\lim_{\\varepsilon \\to 0}\\frac{((I+\\varepsilon x)yI- (I+\\varepsilon x)y\\varepsilon x +(I+\\varepsilon x)y(\\varepsilon x)^2 +O(\\varepsilon^3))-y}{\\varepsilon} \\\\\n& = \\lim_{\\varepsilon \\to 0}\\frac{(I y I+\\varepsilon x y I- I y \\varepsilon x-\\varepsilon x y \\varepsilon x +Iy(\\varepsilon x)^2+\\varepsilon xy(\\varepsilon x)^2 +O(\\varepsilon^3))-y}{\\varepsilon} \\\\\n& = \\lim_{\\varepsilon \\to 0}\\frac{y+ x y \\varepsilon - y x \\varepsilon- x y x \\varepsilon^{2} +y x^{2}\\varepsilon^2 + x y x^{2}\\varepsilon^2 +O(\\varepsilon^3) -y}{\\varepsilon} \\\\\n& = \\lim_{\\varepsilon \\to 0}x y - y x - x y x \\varepsilon +y x^{2}\\varepsilon + x y x^{2}\\varepsilon +O(\\varepsilon^2) \\\\\n& = [x,y]\n\\end{align}\n", "5de34d0440b74ea2fde8c3eb3f62cd01": "\\chi(y_i,x_2,\\dots,x_r)\\chi(y_1,\\dots,y_{i-1},x_1,y_{i+1},\\dots,y_r)\\ge 0", "5de35575cb002e4f9190ec0f560105ea": "\\sum_{l=0}^{p}w_{n}(l)\\left[\\sum_{i=0}^{n}\\lambda^{n-i}\\,x(i-l)x(i-k)\\right]= \\sum_{i=0}^{n}\\lambda^{n-i}d(i)x(i-k)\\qquad k=0,1,\\cdots,p", "5de35b8e59d82b68c32ebc3705b562bd": "g_k ", "5de3c83d66c90d4a5094cb441ed7369e": "h^6", "5de3caa39b65a61c3903cdae628030c2": "\\scriptstyle\\boldsymbol{\\tau}=(\\tau_1,\\tau_2,\\tau_3)", "5de47e93f0c9f4d6feafab2bd60f0344": "1-1/\\sqrt{5}", "5de4b69974b72a0114f5db5563b0f818": "\\frac{\\partial U}{\\partial g}=\\lim_{\\Delta g\\to 0}\\left.\\frac{\\Delta U}{\\Delta g}\\right|_{c.p.}", "5de5241315174bb7cb31db8be44bf995": "Z_{I1}Z_{I2}=R_1R_2=R_o^2", "5de52f0cb64cbe3d5aff8b1a57578c02": "x' = y+1 \\land y'=y \\land z'=z", "5de59acd2bb8a7535a755436c636529d": " v_{i,j} = \\begin{cases}\nn^{- \\frac{1}{2}} & j = 1\\\\\n\\sqrt{\\frac{2}{n}} \\cos(\\frac{\\pi (j - 1)(i - \\frac{1}{2})}{n}) & otherwise\n\\end{cases}", "5de5c0446076ad75c28ce7c37ca2775a": "{\\mathbf A}", "5de5c75fbd5baed7061b7e80fbccfbff": " L^2 \\leq \\frac{\\pi}{4} \\mathrm{area}(\\partial P). ", "5de5ca207e7d3d1fafc34e574d22688d": "\\scriptstyle |\\zeta|>1 ", "5de5d56c2b296e559d7cc7f476c8e7f9": "B_I M = \\oplus_0^\\infty M_n", "5de61ee3ed15fdcee185950d25355f63": "\\scriptstyle{E_{1} = 0}", "5de63d74d81adf49ad470479b3cfbd24": "C_l = C_{l_\\alpha}(\\alpha_\\infty + \\alpha_{geo} - \\alpha_0 - \\alpha_i) \\qquad (3)", "5de6e7a2c4110dc9870679722b0f3a95": " P( | X | > \\epsilon ) \\ge \\frac{ ( 1 - \\epsilon^2 )^2 }{ \\psi - 1 + ( 1 - \\epsilon^2 )^2 } \\ge \\frac{( 1 - \\epsilon^2 )^2 }{ \\psi }", "5de713f254c57d2ade54ee37262c570c": "Y_{3,0} = \\omega_ez_e", "5de73b84220d12ac0c279449ee3fb453": "q = -n", "5de79acbc18da2819f7671d84cf05f86": "\n H^2(X_1,X_2) = 1 \\,-\\, \\sqrt{\\frac{2\\sigma_1\\sigma_2}{\\sigma_1^2+\\sigma_2^2}} \\;\n e^{-\\frac{1}{4}\\frac{(\\mu_1-\\mu_2)^2}{\\sigma_1^2+\\sigma_2^2}}\\ .\n ", "5de82884c37fc73cadd368dafded94c5": "y = a\\phi,", "5de8445986c7e9114b7f1a78ef7923a9": "C_{D}=17.524\\mbox{ mmol/L}", "5de907872f32362d43dc1e520df27db8": "(Comp1)\\quad\\frac{\\displaystyle A \\Rightarrow_{amb}\nA'} {\\displaystyle A\\;\\mid\\;B \\Rightarrow_{amb} A'\\;\\mid\\;B};\\qquad\\qquad\\qquad\\qquad(Comp2)\\quad\\frac{\\displaystyle A \\Rightarrow_{amb} A'\\quad \\displaystyle B \\Rightarrow_{amb} B'} {\\displaystyle A\\;\\mid\\;B \\Rightarrow_{amb} A'\\;\\mid\\;B'}", "5dea3c576915fb609f624a0c0187808b": "\\frac{P_r}{P_t} = G_t(\\theta_t,\\phi_t) G_r(\\theta_r,\\phi_r) \\left( \\frac{\\lambda}{4 \\pi R} \\right)^2 (1-|\\Gamma_t|^2)\n(1-|\\Gamma_r|^2) |\\mathbf{a}_t \\cdot \\mathbf{a}_r^*|^2 e^{-\\alpha R}", "5dea4f7e700ba6972e50acafbc20d314": "g^{\\mathrm{can}}_{ij}=\\langle e_i,e_j\\rangle = \\delta_{ij}.", "5dea94d358801841100741f6852f707c": "\\epsilon_p=-2\\epsilon_m", "5deadccd02d727706289814907c0e562": "\\sqrt{1-x^2}P_\\ell^{m+1}(x) = (\\ell-m+1)P_{\\ell+1}^m(x) - (\\ell+m+1)xP_\\ell^m(x)", "5deafc736c8b221e9c3b7d3225c21cde": "x(U)=\\frac{1}{2\\pi \\sqrt{2m}}\\int_0^U\\frac{T(E)\\,dE}{\\sqrt{U-E}}", "5deb37b9541de12ab4b7bb532e0b0cd3": "\n\\sum^n_{i=1}F^P_id^Q_i = \\sum^n_{i=1}F^Q_id^P_i\n", "5deb39a1e3c969a7850750cd3bcc62d3": " \\mathcal{F}^W_{\\tau_a} ", "5deb7ca3a9352e2a356d004d50ed351e": "R=\\frac{1.22 \\times 400\\,\\mbox{nm}}{1.45\\ +\\ 0.95} = 203\\,\\mbox{nm}", "5deb7ccb701c8d68dc3c9683d024aac9": "K^\\pm\\to\\pi^\\pm(\\pi\\pi)_{atom}\\to\\pi^\\pm\\pi^0\\pi^0", "5debc7770b1c5197960831948adcbb3a": "I = \\sqrt {\\frac{1}{t_1-t_0} \\int_{t_0}^{t_1} i^2(t) dt }", "5debfd5ce2c034be1683173fa10031a8": " { div\\, (\\rho u T )} ={div\\, (k\\, grad\\, T )}+ {S_{T}} \\, ", "5dec02ea32f12e3576ff064e435518be": "u,v \\in D", "5dec089068be4bcc4ff239160f7ca67b": "\\overline{\\overline v} = v,\\quad \\overline{v + w} = \\overline{v} + \\overline{w},\\quad\\text{and}\\quad\n\\overline{\\alpha v} = \\overline\\alpha \\, \\overline{v}.", "5dec10129536da8bf35e75d7cdd43fa2": "p_b=\\frac{2bT}{a^2+b^2-c^2},", "5dec532ab42c0454ee99c9d7d242e0de": "\\langle P,\\varepsilon\\rangle\\le\\langle Q,\\varepsilon^*\\rangle \\iff P\\supseteq Q \\hbox{ and } \\varepsilon\\le\\varepsilon^*", "5dec926a478c8dda64df55f027b8ca6c": "\\mu'_n = \\operatorname{E}(X^n)=\\langle X^n \\rangle \\, ", "5ded506160af9acf1ba5bae3de3922f2": "n < 10^{10}", "5ded752d4b499b87c5db0eb57469505f": "2x-1", "5dee1151370a676d887b7d80ba13b840": "G_1 = \\frac{1}{a1}log(cosh((a1)u))", "5dee7ff7545f03cc755ccc6b5dd6b5e8": "2\\ell + 1", "5def541a09ba724bb9f9defb5ccfaf30": " \n(s_i', t_{si}', t_{ei}')= \n\\begin{cases}\n(s_i', ta_i(s_i'),0) & \\text{if } i = i^*,\\delta_{int}(s_i)=s_i',\\\\\n(s_i', ta_i(s_i'),0) & \\text{if } (\\lambda_{i^*}(s_{i^*}), x_i) \\in C_{yx},\\delta_{ext}(s_i, t_{si}, t_{ei}, x_i)=(s', 1)\\\\\n(s_i', t_{si}, t_{ei}) & \\text{if } (\\lambda_{i^*}(s_{i^*}), x_i) \\in C_{yx},\\delta_{ext}(s_i, t_{si}, t_{ei}, x_i)=(s', 0)\\\\\n(s_i, t_{si}, t_{ei}) & \\text{otherwise}.\n\\end{cases}\n", "5def65ebb4706576155f5206afc68a4c": " {\\rm p} K_c = -\\log_{10} K_c = \\sum_{j=1}^p \\eta_j \\log_{10} \\left [ {\\rm Y}_j \\right ] - \\sum_{i=1}^r \\nu_i \\log_{10} \\left [ {\\rm X}_i \\right ]\n \\,\\!", "5def8a2a72655c52c54d0f01a4aab29f": "h(x) = 0", "5defd0aff15d362a185881612f34b3b6": "\\;\\deg(G)>\\deg(H)", "5df0469e8ea849f647a1e8dd46758f91": " S(\\boldsymbol\\beta+\\boldsymbol\\delta) \\approx \\|\\mathbf{y} - \\mathbf{f}(\\boldsymbol\\beta) - \\mathbf{J}\\boldsymbol\\delta\\|^2", "5df05480edfa84a13e8bf2de2e09f6f0": "\\displaystyle{a_3=a_2^{b_2-b_3}=(a_1^{b_1-b_2})^{b_2-b_3}=a_1^{b_1-b_3}.}", "5df05bfecfeeea789fc18db0f1f225e5": "\\operatorname{Aut}(S_n)", "5df15a2d194932c42597fe39b7547ff5": "\\omega^{\\omega} = \\omega \\times \\omega \\times \\cdots", "5df17255948cf5f65f4122876b8b0ac0": "f_{U}:X\\to[0,1]\\,", "5df214f7e887069635c611c7618f35ca": "\\zeta = \\langle 0|\\hat{U}^\\dagger(t_1,t_0)\\hat{U}(t_1,t_0)|0\\rangle - \\langle 0|\\hat{U}^\\dagger(t_1,t_0)|0\\rangle\\langle 0|\\hat{U}(t_1,t_0)|0\\rangle", "5df240892d46cbc64c6b07b807a5886e": "{}_{\\ 82}^{212}\\mathrm{Pb} \\xrightarrow{\\beta^-\\ } {}_{\\ 83}^{212}\\mathrm{Bi}\\ \\mathrm{(10.64\\ h)}", "5df2768f3c2887d6d44f27a312c56907": "s_2=\\min_{i=1,\\ldots,m} \\{x_i\\}", "5df2d07ad8f35e7d8edf2862dca378c5": "\\bar{q} = -\\boldsymbol{\\psi}(\\mathbf{x})\\cdot\\mathbf{n}", "5df3260d92a5d59223d56265b613e0e2": "A(t)=det(V-tV", "5df365b383126f60158f405604c72292": "x-|M_x| \\le \\sum_{i=k+1}^\\infty |N_{i,x}|< \\sum_{i=k+1}^\\infty {x\\over p_i}", "5df3a0bde61e2a24e532569f0bda5512": "\\Pr \\left (-A < \\frac{\\overline{X}_n - \\mu}{\\frac{S_n}{\\sqrt{n}}} < A \\right)=0.9,", "5df430cfe3f6885aa3a5ad7063cb2852": "\\Delta G = \\sum_{i}^{}{\\Delta G_i}", "5df447f117b7855643625984c2a810b5": "\n \\lim_{r\\to\\infty} f(k; r, p) = \\frac{\\lambda^k}{k!} \\cdot 1 \\cdot \\frac{1}{e^\\lambda},\n ", "5df47590ae817ada4e985fe9c5ab54f5": "J\\;\n\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup\\!\\!\\!|}}\\;I\n\\qquad\\text{and}\\qquad\nJ\\;\n\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoonup}}\\;C", "5df496ddd000a74b5bdae42fda6d7723": " \\left(\\phi\\right) ", "5df535c3e8f4b09e17a4deee29b08b14": "(x+y)\\oplus((x\\oplus a)+(y\\oplus b))=c", "5df55d2b61e08bec3a60b280563d8788": "I(X;Y|Z) = \\mathbb E_Z \\big(I(X;Y)|Z\\big)\n = \\sum_{z\\in Z} p_Z(z) \\sum_{y\\in Y} \\sum_{x\\in X}\n p_{X,Y|Z}(x,y|z) \\log \\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)},", "5df596e76ef8697b154bbb8c85d86175": " f_1(y) = \\begin{cases}\n 3y &\\text{for } 0 \\le y \\le 1/3,\\\\\n 1.5(1-y) &\\text{for } 1/3 \\le y \\le 2/3,\\\\\n 0.5 &\\text{for } 2/3 \\le y \\le 1,\n\\end{cases} ", "5df5a399e80ed75be983b3b2983d241b": " \\nabla^2\\Phi=\\frac{1}{r^2}\\frac{d}{dr}\\left({r^2\\frac{d\\Phi}{dr}}\\right) = 4\\pi G\\rho ", "5df5b9a21608ec653b8c4ac8b150069e": "\\tau_N", "5df5d1952dcaf52073f0fa236f234b93": "A_{c,t}", "5df62ca2a3ade66ceb3414272d586650": "\\hat{\\mathcal{H}}^D_i = \\sum_{i}^{\\rm core} \\epsilon_i E_{ii} + \\sum_r^{\\rm virt} \\epsilon_r E_{rr} ", "5df636fb76faded5b135dd96155f1d22": "B=B_3B_2B_1B_0", "5df689b56b05c65015a9f688d59f040c": "T_{d,ave}", "5df6b15ada84414d3f2f241f5001dbdf": "t\\mapsto \\frac{\\tilde{t}}{\\epsilon}\\,,\\quad r\\mapsto M+\\epsilon\\,\\tilde{r}\\,,\\quad \\phi\\mapsto \\tilde{\\phi}+\\frac{a}{r^2_0\\epsilon}\\tilde{t}\\,,\\quad \\epsilon\\to 0\\,,\\quad \\Big(r^2_0\\,:=\\,M^2+a^2\\Big)", "5df6ccd716d84e9dcee554b72e0b6359": "{\\rm Si}(x) - {\\rm si}(x) = \\int_0^\\infty\\frac{\\sin t}{t}\\,dt = \\frac{\\pi}{2},", "5df6dc06e1afa5d950d3c822f9a1229a": "\\lambda_\\text{max} T = b,", "5df74fdcbf6a039b9c010d721791418e": "\\Delta^2\\varphi=0", "5df762e15bc051f0427e1ffc256eee04": "x_{k}^{\\left(rr'\\right)}", "5df770bb1b7a0098633fbc69728ba1c3": "Q_e = C_eV_e. \\ ", "5df7de1f7a04bceea91c0b92e248efc9": "\\mu_{\\text{eff}}= \\lambda \\langle \\hat{S} \\rangle ", "5df7e7b0f9a17b149f36fec7c1142e01": "F'", "5df7ebd41dc19609a1909c5fe600f2e4": "f(z) = \\sum_{n=0}^\\infty \\left[ a_n r^n \\cos n \\theta - b_n r^n \\sin n \\theta\\right] + i \\sum_{n=1}^\\infty \\left[ a_n r^n \\sin n\\theta + b_n r^n \\cos n \\theta\\right],", "5df7f6b8bb44d15f89e9c28dd5509bfc": "\nL \\psi = E \\psi, \\quad L = - \\Delta + v( x, t ), \\quad \\Delta = \\partial_{x_1}^2 + \\partial_{x_2}^2.\n", "5df823f12812955cdf1b5b71ef0977ec": "\\tfrac{8}{9} n^2", "5df844f3b51e03438ed3490a98d2f15c": "(\\mathrm{Ran}_KT)c=\\int_mTm^{\\mathbf{C}(c,Km)}", "5df85ed45d6d30876280344b5a097b39": "x \\prec y", "5df882bb07ef1f3df1012d87cb8c4c06": " ds^2 = f (-d(x^0)^2 + d(x^1)^2) + g_{ab} \\, dx^a \\, dx^b", "5df8c3bdbd63f34ff53fe06ab9ee4d31": "\\scriptstyle \\frac{d\\mathbf{A}}{dt}", "5df8ed0001f2f6c7f58db90213c0e5b9": "\\begin{cases}\n\\overbrace{ \\begin{bmatrix} \\dot{\\mathbf{x}}_1\\\\ \\dot{z}_2 \\end{bmatrix} }^{\\dot{\\mathbf{x}}_2}\n= \n\\overbrace{ \\begin{bmatrix} f_1(\\mathbf{x}_1) + g_1(\\mathbf{x}_1) z_2 \\\\ 0\\end{bmatrix} }^{f_2(\\mathbf{x}_2)}\n+\n\\overbrace{ \\begin{bmatrix} \\mathbf{0}\\\\ 1\\end{bmatrix} }^{g_2(\\mathbf{x}_2)} z_3 &\\qquad \\text{ ( by Lyapunov function } V_2, \\text{ subsystem stabilized by } u_2(\\textbf{x}_2) \\text{ )}\\\\\n\\dot{z}_3 = u_3\n\\end{cases}", "5df90123039f44d03e83953e4ba67476": "\\sum_j \\beta_j^2", "5df9422fcdbda5252660fef7412a5ac6": " \\mathrm{d}G = \\sum_{i=1}^I \\mu_i \\, \\mathrm{d}N_i + \\sum_{i=1}^I N_i \\,\\mathrm{d}\\mu_i \\,", "5df9f103166522ed0d0cf03f86cbaaea": "0.7078\\sqrt{N}+0.551", "5df9febe652da5b129e1c832929da57e": "x^*(\\theta)=\\arg\\max_{x\\in D}f(x,\\theta)", "5dfa16b072b0f19042e8b6f0f4586d1b": "\\mathrm{Ei}", "5dfa62699149861bb9b4bfc95b9b4bee": "\\frac{d[M_1]}{d[M_2]} = \\frac{[M_1]}{[M_2]} \\left( \\frac{k_{11}\\frac{\\sum[M_1^*]}{\\sum[M_2^*]} + k_{21}} {k_{12}\\frac{\\sum[M_1^*]}{\\sum[M_2^*]} + k_{22}} \\right) \\,", "5dfa7b23aaf82e5bac731f30ef13badb": "\\int_{\\partial M}k_g\\;ds", "5dfabc2766be1601200a09ece712cc78": "d \\approx \\sqrt{ 2\\cdot R \\cdot h}", "5dfac48c27cfb0908bba3b866a2ab94a": "\\tfrac{a}{b} + \\tfrac {c}{d} + \\tfrac{e}{f} = \\tfrac{a(df)+c(bf)+e(bd)}{bdf}", "5dfadafeb50f64e2733c64931590f2b1": "YY'\\,", "5dfb5c07ecc926f6ef9d78d5f81f570f": "f_{sphere}", "5dfb67a660f254ba96300906aad3177a": "p_a(s) = \\begin{cases} \n\\varepsilon & \\text{if } s=\\varepsilon, \\text{ the empty string} \\\\\np_a(t) & \\text{if } s=ta \\\\ \np_a(t)b & \\text{if } s=tb \\text{ and } b\\ne a. \n\\end{cases}", "5dfb71afd0b9563e70888ec4b8b6e5dd": "[E, E+dE]", "5dfb774cdfca891a25c0750fa3cb5b56": "E\\approx 1-e^{-p\\left( s>x \\right)D}", "5dfbc40b3642ec5fb23ff3cdc9b7a98e": "\\mathcal{H\\psi =}\\left( p_{\\perp }^{2}+\\Phi -b^{2}\\right) \\mathcal{\\psi }=0.", "5dfbe57ec9552dc2ce15bab7609635ee": "R_s/R", "5dfbeefe636ca4dc720ddf6a76d6c188": "b_n(t) = r_n(t) \\sin \\left( \\varphi_n(t) \\right) \\ ", "5dfcf0dee5e4800f0a7c4f62bda8950d": "E_6\\cdot\\mathrm{SO}(2)\\,", "5dfd8d32c77c2f5928ab384c0b8c506d": "S_x(s) = S_x^{+}(s) S_x^{-}(s)", "5dfd9f901fdbf8c72467fbf4bbe9911d": "E\\subset M^\\uparrow", "5dfda0ae27cd678d2fe735bed24af6c7": "abbabbabb", "5dfdbbe9ef4744fec7ee0dced68e139b": "2 \\eta^{\\rho \\sigma} \\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\nu \\right) = 2 \\eta^{\\rho \\sigma} (4 \\eta^{\\mu \\nu}) = 8 \\eta^{\\rho \\sigma} \\eta^{\\mu \\nu} .\\,", "5dfe530adf75d86b4a0ca2a244e247fa": "\n\\int_{N_k}^\\infty\\frac{dx}{x\\ln(x)\\cdots\\ln_k(x)}\n=\\ln_{k+1}(x)\\bigr|_{N_k}^\\infty=\\infty.\n", "5dfe6d9ddf8b969432e3cd4bfdb2d96c": " \n\\mu_{ab}^{(c)}(t) = \\left\\{ \\begin{array}{ll}\n \\mu_{ab}(t) &\\mbox{ if } c = c_{ab}^{opt}(t) \\mbox{ and } Q_a^{(c_{ab}^{opt}(t))}(t)-Q_b^{(c_{ab}^{opt}(t))}(t)\\geq 0 \\\\\n 0 & \\mbox{ otherwise} \n \\end{array}\n \\right.\n", "5dfe70497b20d087c7d0c6da536cd6cc": "\\cos 24^\\circ+\\cos 48^\\circ+\\cos 96^\\circ+\\cos 168^\\circ=\\frac{1}{2}.", "5dfe7ef3d20043efdb55136f4e183ce1": "-1.28 < \\lambda \\le -1", "5dfe9666ab14fc60feb5b13198d96acb": "\\dot V_n", "5dfeb133e994ba0320512a79d5fdb70f": "x=\\frac{(m_1 + m_2 + ... + m_{n-1}) - s}{n-2}.", "5dfeba2920a3f5be260e0165c9695bc8": "a < S_i < b", "5dfef54c62f16571229327ee5c95438f": "M \\otimes K", "5dff4c58922e7a4186824c35108b790c": "y=c", "5dff4f393f0be643c24f1ac1a6db2dad": "\\displaystyle{S_\\varepsilon f(\\varphi) =\\int_{\\varepsilon \\le |\\theta|\\le \\pi} f(\\varphi-\\theta)\\theta^{-1}\\,d\\theta}", "5dff99e2e3a2690d002f016325a37e46": "E_{b} = 2 E_{X} - E_{XX}", "5dffbf062cf3ad5cad71d25adbb14b14": "S = \\{ x\\in \\mathbf{Q}|x^2 < 2\\}.", "5dffcc33cbb8a1115d4099bbb2313499": "j^\\nu = t^b j_b^\\nu \\,, \\quad j_b^\\nu = \\bar{\\psi}\\gamma^\\nu t^b \\psi \\,,", "5dffcf7c32b83c599e5d235919b0e864": "E[u_{it}-u_{it-1}|x_{it}-x_{it-1}]=0", "5e00064ddc3550967a0e946badffbb96": "\\mathrm{VO_2\\; max} = Q \\times\\ (\\mathrm{C_aO_2} - \\mathrm{C_vO_2})", "5e0073fab6c6f08ccf845a32e730aaf0": "X=\\mathbb R^3", "5e00e6a9e674ec558c9f6c9c3aa4867f": "Z(\\lambda,\\nu)", "5e019f6856069abde5b4e03bdc2f9a85": " \\widehat{a}\\widehat{U}|\\alpha\\rangle", "5e01c0618cf733badb6f5f5b275757f0": "l_{\\mathrm{QCD}} = \\frac{\\hbar}{m_\\text{p} c}", "5e0211635d5a25f5ed943210ab3fb0c4": "Y\\colon (\\Omega,\\mathcal A) \\mapsto (U,\\Sigma)", "5e02565244ff224016f2f6e73f260f9d": "\\begin{bmatrix} w_1 \\\\ w_2 \\end{bmatrix} :=\n \\begin{bmatrix} w_1 \\\\ w_2 \\end{bmatrix}\n - \\alpha \\begin{bmatrix} 2(w_1 + w_2 x_i - y_i) \\\\ 2x_i(w_1 + w_2 x_i - y_i) \\end{bmatrix}.", "5e025a2e27a2dbcde1afd223ae14d545": "\\displaystyle \\int_Cf(z)e^{\\lambda g(z)}dz", "5e02cefbcd536dd1675d2d6e3380b012": "L(a)=1", "5e03301a9cc5213b10a7927aa308dd9c": "n \\cdot \\frac{k}{d}", "5e037b923f705393249cbb0b6aa8e429": "2|r_1+r_2|", "5e03d39ff8bc51293e35a4406afc9c18": "{}_aD_x^{-\\alpha}f(x) = \\frac{1}{\\Gamma(\\alpha)}\\int_a^x f(t)(x-t)^{\\alpha-1}\\,dt.", "5e03f72d1e0795a55fa236b50a6eab9e": "F\\subseteq \\mathbb N", "5e0445b25f7e330f9e7515e66c1a2d01": "\\mathit{TPR} = \\mathit{TP} / P = \\mathit{TP} / (\\mathit{TP}+\\mathit{FN})", "5e0459eb302ccea79aa0d3b88bc7eb6c": "A_1 \\cos(\\omega_1 t+\\phi_1)+A_2 \\cos(\\omega_2 t+\\phi_2)+A_3 \\cos(\\omega_3 t+\\phi_3)+\\ldots", "5e04993eb6c751a4694ee3b111aacd56": "\\Phi^{(\\infty)}(\\omega)= \\prod_{k=1}^{\\infty} \\frac {1} {\\sqrt 2} H\\left( \\frac {\\omega} {2^k}\\right) \\Phi^{(\\infty)}(0).", "5e04b2efc5e0e824ed22469f065febe0": "\\mathbf{E}_\\nu(z)=\\frac{1}{\\pi} \\int_0^\\pi \\sin (\\nu\\theta-z\\sin\\theta) \\,d\\theta", "5e04c000043a29f917c996b93df89f3f": "\nf(z) = c_0 + c_1z + c_2z^2 + \\cdots = \\sum_{n=0}^\\infty c_nz^n,\n", "5e04c111e8ce8d8f7e82de07f5ec9868": "(1+x)^0 \\ge 1+0x \\, ", "5e04c77f2f885fd5c78c91ab8e52943b": "O(1.246^n)", "5e050c197f994ad23f3af2f986b28aa5": "\\sum_{k=1}^\\infty \\frac{\\zeta(k) - 1}{k} = 0", "5e057f80aceff1894e513a15ddc0d8dc": "F = m \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} = -k x ", "5e059b9e73b393a22edb61d07a01d64e": "\\mathbf B = \\mathbf \\nabla \\times \\mathbf A", "5e05edc9d23d840ec698881f4bd34dfc": " W(t) ", "5e05f137d73913e042ba104a4e801c32": "\\left \\{ a_n \\right \\}_{n=1}^\\infty", "5e061dc11794f39ccaffc44faec3c101": "E_1\\cup E_2\\cup\\ldots=B", "5e06a9ba98927e886b4d9b62d1bcf275": " \\ln x_2 = - \\frac {\\Delta H^\\circ_{fus}}{R} \\left(\\frac{1}{T}- \\frac{1}{T_{fus}}\\right)", "5e07536d8e06d8e106060fb3a80f412b": "\\kappa-2", "5e079a28737d5dd019a3b8f6133ee55e": "O(1)", "5e07df3516c957d0076e572f3589d8ef": "\\begin{bmatrix}\n0 & 1\\\\\n1 & 0\n\\end{bmatrix}.", "5e080f06b9e0393ef720a61b5c9f1354": "\\left(\\mathfrak{B}(V)\\otimes k[\\mathbf{Z}^n]\\otimes\\mathfrak{B}(V^*)\\right)^\\sigma", "5e08103b43a44a8d0d2b6143d79ce13a": "\\operatorname{E}[H(X_i)]", "5e082c62acffc375547537a8cd16313c": " d U = T \\delta S- p\\delta V ", "5e08bb6a7c56b62d137effc4e757a9cc": "||f-p||_\\infty", "5e08c0d2959724ae07069399eacc16e9": "\\mathrm{tf_{ij}}", "5e09978eebff9423b0f976b134f0b403": "P_{\\text{total}} = \\sum_{i=1} ^ n {p_i}", "5e099ec10b3590fa093fd7c7d2b9a8fd": "\\arccos", "5e09b9606c6d9d48d9874259faf124c3": "\\frac{\\bar{v_Y}}{(c-a)^2}", "5e0a1afa5c3e70aaeb7009d028a55b49": "m \\ = \\ \\left({ Q \\over F }\\right)\\left({ M \\over z }\\right) ", "5e0a3bf33446cb0bf66d606e2c142e19": "D_{H,\\mathrm{annulus}} = D_o - D_i", "5e0a54475d243ac00b749dbf9f66b6df": "k(-j) = i\\,", "5e0a5a048a34d5be8a6449608209a4bd": " \\mathbf{\\hat z} ", "5e0b2f5460bd5d7596f324510e25b545": "\n\\overline{\\left (A_{t} - \\overline{A} \\right )^{2}} = \\sum_{\\alpha \\neq \\beta}|c_{\\alpha}|^{2}|c_{\\beta}|^{2}|A_{\\alpha \\beta}|^{2}.\n", "5e0b58467db0c28250102ca71e2c8b9f": "PQ+r=PQ'+r'", "5e0bb2f332e362dc1cfa50a98d0a228b": "\n E_{a^{n}}^{\\dagger}E_{b^{n}}\\notin N\\left(\\mathcal{S}\\right) \\backslash \\mathcal{S},\n", "5e0c019df780e44871990a1eb2f753b3": "g(r) = \\exp \\left [ -\\frac{u(r)}{kT} \\right ] y(r) \\quad \\mathrm{with} \\quad y(r) = 1 + \\sum_{n=1}^{\\infty} \\rho ^n y_n (r)", "5e0c7787c06d47362ed655a05ea16edf": "\\frac{1}{p} + \\frac{1}{q} = 1.", "5e0cf6ba86791668520ed3071458919e": "\\Pi_0(x) = \\sum_2^x \\frac{\\Lambda(n)}{\\ln n} - \\frac12 \\frac{\\Lambda(x)}{\\ln x} = \\sum_{n=1}^\\infty \\frac1n \\pi_0(x^{1/n})", "5e0d0da747780c684231b8c8c7daa5ef": "n(P+N)", "5e0d20d771b46762cb7871770f695aa7": "\n\\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_v + Y^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_v\\Big)\n\n+_*\n\\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_w + Z^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_w\\Big)\n\n=\nX^k\\frac{\\partial}{\\partial x^k}\\Big|_{v+w} + (Y^\\ell+Z^\\ell)\\frac{\\partial}{\\partial v^\\ell}\\Big|_{v+w}\n", "5e0d4d1d722582393a8aec1f8f4c2c60": "p(x)=\\frac{(x-x_1)(x-x_2)\\cdots(x-x_n)}{(x_0-x_1)(x_0-x_2)\\cdots(x_0-x_n)}\\cdot y_0+\\frac{(x-x_0)(x-x_2)\\cdots(x-x_n)}{(x_1-x_0)(x_1-x_2)\\cdots(x_1-x_n)}\\cdot y_1", "5e0de0e93d2bd15d836fdaef58466f42": "|x_n-x|0", "5e2e504aa2243d9faddf0f63d4efbe15": "\\begin{cases}a^2=\\frac{2|p_1|\\Lambda}{\\operatorname{ch}(2|p_1|\\Lambda\\tau)}, \\\\\n b^2=b_0^2e^{2\\Lambda(p_2-|p_1|)\\tau}\\operatorname{ch}(2|p_1|\\Lambda\\tau),\\\\\nc^2=c_0^2e^{2\\Lambda(p_2-|p_1|)\\tau}\\operatorname{ch}(2|p_1|\\Lambda\\tau),\\end{cases}", "5e2e54749d6e8956f26986a3044b9bb6": "\nX_{m+n} = Z_{m-n}((X_m-Z_m)(X_n+Z_n)+(X_m+Z_m)(X_n-Z_n))^2\n", "5e2ed586a7a5a0b826aaffe5da506fb4": "\\mathcal{N}(\\mu, \\sigma^2)", "5e2eeb9d405faa5c9485b99def1dcda3": "\\chi^{(m)} \\!", "5e2f21dc0e07ac935d71cc6e828a1514": "N=2,3", "5e2f881e936ad10eb3b0e9d6696fdcff": "k\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{2m_0E}/\\hbar", "5e2fb740c2fd80fe5e74a0d6242cfc3d": "Eq.2", "5e2fdf7976fa862588d99ce0606eedcb": " X_3 = T_1Y_2X_1Z_2 + Z_1X_2Y_1T_2 = 4 ", "5e305ac803338f56fcc0fd59d0305164": "\\omega^2_{free}", "5e30857cbe4ba0f4b0bb37f1ac41b662": "\\gamma \\to 0", "5e308df17292cb746882b1f2e5299d12": "\\,{}_{t|k}q_x = {}_t p_x \\cdot {}_k q_{x+t} = {l_{x+t} - l_{x+t+k} \\over l_x}", "5e30b0a971cf64a40d91623159e18df0": "\\{0\\} \\times (0,1] ", "5e30d5f182806a75bee001d76a70da01": "\\mathfrak{k}=\\mathfrak{g}_+", "5e311fd9408a8cecf523813d264767a0": "x\\wedge y=x", "5e31bece7cb009fee31aa9cc2e8faf49": "\\approx \\frac{4}{3}m^{3}", "5e31f62a30b050246cb718c4d2acca4b": "\\lambda a + (1 - \\lambda)b = o + \\lambda (a - o) + (1 - \\lambda)(b - o)", "5e31ffc61fb3bcd868e920d287092542": "\nU = \\text{span}\\{u_k,\\ldots,u_n\\}\n", "5e32437bd6f42db2cb4942022e06b5fb": " D(F, G) = 2\\mathbb E\\|X - Y\\| - \\mathbb E\\|X - X'\\| - \\mathbb E\\|Y - Y'\\| \\geq 0,", "5e325655534500f29aa7d840a65162be": "r_{max}", "5e329e912fa3da10f0fa1f5b3212c694": "X\\subset T", "5e32e14e8a0802caea61c97c0bd0cc33": "\\frac{\\omega}{2\\pi}\\text{ Hz,}", "5e32eb8d5c5131b074752208b267d7b2": "\\frac{V_1^2 - V_0^2}{2}", "5e332e69d3cdd40c4e4814805ff32679": "t_{1/2} = \\frac{\\ln (2)}{\\lambda} = \\tau \\ln(2)", "5e3390a9df5b0d1bdec5e889a41ce084": "Ce^{4+} + e^- \\rightleftharpoons Ce^{3+}", "5e33f90401da06ef1096753e37b8af18": "g(y) \\neq 0", "5e33fa3e7637eaeb400c15f06e704fb2": "\n \\int P\\langle x-a \\rangle~dx = P\\cfrac{\\langle x-a \\rangle^2}{2} + C_m\n ", "5e33fe5289dbd842dc9dc5753b88d628": "8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1", "5e34d6d87e5374cfd2429c6a59550049": "y_1(t) = H \\left \\{ x_1(t) \\right \\} ", "5e3541007f2ad5e940eb919c6067ccba": "\n\\begin{matrix}\nA^{\\frac{1}{2}}\\,A^{\\frac{*}{2}}=A;\n\\end{matrix}\n\\qquad\n\\begin{matrix}\nA^{\\frac{1}{2}}\\,A^{-\\frac{1}{2}}=I;\n\\end{matrix}\n\\qquad\n\\begin{matrix}\nA^{-\\frac{*}{2}}\\,A^{\\frac{*}{2}}=I;\n\\end{matrix}\n\\qquad\n\\begin{matrix}\nQ^{\\frac{1}{2}}\\,Q^{\\frac{*}{2}}=Q.\n\\end{matrix}", "5e356e48093d65b9eba9d5d2187e58e2": " E = 2\\pi rb\\gamma_s \\,\\!", "5e3592ba35326baa39f8c188c2acfd7a": "\\textstyle 0\\leq\\alpha\\leq1", "5e362428f7e4c1429427b55b9b489754": "{\\omega t > {\\pi + \\alpha}} : I(\\omega t) = I_{tcr-max} \\sqrt{2} [-cos(\\alpha)-cos(\\omega t)]", "5e364850638ad49b4281be7a558e7f7a": " \\langle \\Phi , \\Phi \\rangle = \\int\\limits_{-\\infty}^{\\infty} d p \\, \\left | \\Phi \\left ( p, t \\right ) \\right |^2 = 1.", "5e36b130448abf6ac91fb8c482c82139": "(\\hbar \\partial_{\\mu} + imc)(\\hbar \\partial^{\\mu} -imc)\\psi = 0", "5e372e0495172d99cbc93bd4922871bf": " \\frac{\\partial^2 F(z,w)}{\\partial z \\partial w} = {f^\\prime(z)f^\\prime(w)\\over(f(z)-f(w))^2}-{1\\over(z-w)^2},", "5e37379ca6aaac49033562c9c69b361d": "H_{ min } = \\frac{ X }{ X + Y } H( X ) + \\frac{ Y }{ X + Y } H( Y )", "5e3794143a9d15825aab25966173afa1": " A ::= s A_0 \\ldots A_{N-1}", "5e37ae2b782cffc7fc025e409afabe42": "\nf_X(x)= \\begin{cases}\n1 & 0\\le x \\le 1 \\\\\n0 & \\text{otherwise}\n\\end{cases}\n", "5e37bc4090dc935f7afa716665b59119": "\\begin{align}\n(x+y)^3 &= (x+y)(x+y)(x+y) \\\\\n&= xxx + xxy + xyx + \\underline{xyy} + yxx + \\underline{yxy} + \\underline{yyx} + yyy \\\\\n&= x^3 + 3x^2y + \\underline{3xy^2} + y^3.\n\\end{align} \\, ", "5e37be7587300cd6b92cc3f70b4e22bc": " f(\\lambda|\\phi=0) = \\frac{1}{2\\pi}.", "5e37e351e9141449b6b71b3fc89a273e": "\\mathcal{L}(\\mathcal{E}) = \\{L(D) | D \\in \\mathcal{E} \\}", "5e3846d52cd869caac3fead2594d0b3d": "b\\sqrt{2} = a", "5e38594f8ac67333eb330b60b274d955": "\\tfrac{(h - y)^2}{h^2}", "5e388ea875f1c760d1e080cb6502ae14": "w(n) = 0.5\\; \\left(1 - \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right) \\right)", "5e38ecac706e8b376f8bb46dc6aa4658": "\\forall x, y, z \\in N", "5e3942fda2cb994aa325e41fcdcb2b2c": "W^n", "5e3949bab2cd48b1af5bd75a767e329b": "-\\frac{}{}I_0 \\, \\cos \\delta", "5e398a8264d7dd2f99e656d58a6bc4c4": "\\operatorname{Tr}(aA+bB)=a\\,\\operatorname{Tr}(A)+b\\,\\operatorname{Tr}(B).", "5e399e6aa707770a6bb03024aaffc80d": " \\int_{-\\infty}^\\infty f(x) \\, \\delta(g(x)) \\, dx = \\sum_{i}\\frac{f(x_i)}{|g'(x_i)|}. ", "5e39ed1763d174a7b8d53c80d5ce33e6": "f(\\mathbf{a}) \\geq f(\\mathbf{b})", "5e3a2c2eb5f4f89b3ffd523134b9b7a3": "\\mathbf{u} \\otimes \\mathbf{v} = \\mathbf{A} = \n\\begin{bmatrix}u_1v_1 & u_1v_2 & \\dots & u_1v_n \\\\ u_2v_1 & u_2v_2 & \\dots & u_2v_n \\\\ \\vdots & \\vdots & \\ddots & \\vdots\\\\ u_mv_1 & u_mv_2 & \\dots & u_mv_n \\end{bmatrix}.", "5e3ab4856a3d395b49bd8baf4bc94b4f": "m_1(t)", "5e3b21b4924a5323a4c8311cd60d7342": "(X,\\mathcal{F},\\mu)", "5e3b85e5075b09ecdb314893b86bf212": "\\mathrm{C} = \\tfrac{1}{n_p-1}\\sum_{i=1}^T \\mathbf{x_i}\\mathbf{x_i^T},", "5e3bb6b500e1c8c25125f5156041930d": "E[Z]_{11} = -\\frac{m}{r^3} \\, \\frac{2-3m/r}{1-2m/r} = -\\frac{2m}{r^3} - \\frac{m^2}{r^4} + O(1/r^5) ", "5e3bcbba22054e412337c35b7f3129e4": "(0,\\pm 1,\\pm 1,0)", "5e3c800690e9bca0f2aa196547a48a58": "\\textstyle p = p_1", "5e3cae553834876d58a77e3ba98a195a": "\\left(\\sum_{n=0}^\\infty a_n\\right) \\cdot \\left(\\sum_{m=0}^\\infty b_m\\right) = \\sum_{j=0}^\\infty c_j,\\qquad\\mathrm{where}\\ c_j=\\sum_{k=0}^j a_k b_{j-k}", "5e3cf75386a64b0c42bc68e7dad724fc": "\\{2, 1, 3\\}", "5e3d1b8843d8b80a0c9465755acbb862": " \\begin{align}\n \\mu(X) & = \\frac{a + 4b + c}{6} \\\\\n \\sigma(X) & = \\frac{c-a}{6}\n\\end{align}", "5e3d20ee3f85e4e942a9d522f5d404a7": "| \\psi_{\\sigma} \\rangle = ( \\sigma^{\\frac{1}{2}} V_1 \\otimes V_2 ) | \\Omega \\rangle ", "5e3d5081313c43d233b6c5f83e661904": " \\lim_{n\\to\\infty} \\frac{|c_{n+1}(z-a)^{n+1}|}{|c_n(z-a)^n|} < 1. ", "5e3da8b5c68db44c71f3459164f4466b": "a_{2}+b_{2}+c_{2}=b_{1}+c_{1}", "5e3dd78045461ee2c68d777f535f8936": "E = m_0 c^2 \\left[1 + \\frac{1}{2} \\left(\\frac{v}{c}\\right)^2 + \\frac{3}{8} \\left(\\frac{v}{c}\\right)^4 + \\frac{5}{16} \\left(\\frac{v}{c}\\right)^6 + \\ldots \\right]. ", "5e3e0241740247c187eb324ecd4a12c1": "2\\gamma", "5e3e09ffb28063c53304ba92e3068494": "\\tfrac{p+q}{2}=50\\tfrac{1}{2}", "5e3e331f96e439afd8bb853797f68771": "\\bar{H}(z)", "5e3e3c3b740352b930d8545d110ee96d": "HV = \\frac{F}{A} \\approx \\frac{0.01819 F}{d^2},", "5e3e6c23f4bf76a768044522a2f0cd3d": "h_{n+1}", "5e3fac061016eb3939b7cd3bae4be8a6": "x^2 + y^2 = L^2", "5e400287763f5303d381816ab7c7c8f7": "x_1,\\ x_2, \\ldots, x_N", "5e40097893b32b52275c0396e4534d0a": "\\sin (\\arcsec x) = \\frac{\\sqrt{x^2-1}}{x}", "5e4053c308f17b168c0985b85c288437": "S=1/2", "5e40682f6d11ca6da8a4795ed65a6bfc": "a + b = b \\,", "5e4086f9aebe12161232b5cbbf477aec": "J^\\mu", "5e40896318473d6b8d7c30912e45a083": "d + m + y + \\left\\lfloor\\frac{y}{4}\\right\\rfloor + c \\mod {7}", "5e40fab9ee573a3c0c5275002ff767de": "\nK = \\xi_+ \\xi_-\n", "5e40fc4ba4f7eddc41380e13de3a6b11": "x_{0} = 0", "5e41323395fda1e290e60fbd54875529": " Y(\\omega) ", "5e414031375386d48642493a70048701": "L := (\\mathbf{P}^1, \\lambda), \\qquad \\lambda := pt \\times \\mathbf{P}^1 \\in A^1(\\mathbf{P}^1 \\times \\mathbf{P}^1)", "5e41a763a2c066b1f0f119d7c0874fa2": "k_{max}", "5e41c2266123574262cf04057b3dc7ec": "\\neg \\Box F \\rightarrow \\neg F", "5e424636610605004dba795145abb4ac": "\\scriptstyle - \\frac{5}{11}", "5e4268bd212b53e5559ffed779649464": "{x}, {s}\\,", "5e430202e926a8c892b56427a88afd96": "M_{\\infty}(K)", "5e4307abcd0114ad5ddc3f2e04dbf5d5": "J_{m_i}^k(\\lambda_i)=\\begin{bmatrix}\n\\lambda_i^k & {k \\choose 1}\\lambda_i^{k-1} & {k \\choose 2}\\lambda_i^{k-2} & \\cdots & {k \\choose m_i-1}\\lambda_i^{k-m_i+1} \\\\\n0 & \\lambda_i^k & {k \\choose 1}\\lambda_i^{k-1} & \\cdots & {k \\choose m_i-2}\\lambda_i^{k-m_i+2} \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\lambda_i^k & {k \\choose 1}\\lambda_i^{k-1} \\\\\n0 & 0 & \\cdots & 0 & \\lambda_i^k\n\\end{bmatrix}", "5e4324d661298efe028c4b54046e832f": "N_2\\subseteq N_1", "5e43efbbe62a3a9062372cdf402d8287": "\\mathfrak u_1 \\oplus \\mathfrak u_1", "5e43f1959bb739e4503afffb6f4c66a0": "\\mathrm{SL}(6,\\mathbb C)", "5e43f5f1b47fb6d4575b36d096658014": "(1-p + pe^t)^n \\!", "5e441593dafbf5e2e7b87f75b96b3689": "\n \\hat{w} = A_1\\cosh(\\beta x) + A_2\\sinh(\\beta x) + A_3\\cos(\\beta x) + A_4\\sin(\\beta x)\n ", "5e44bdf5c4c2306e77f9a7055367cef5": "r \\frac{dr}{dt} = \\frac {(S-1)} { [(L/RT-1) \\cdot L \\rho_l /K T_0 + (\\rho_l R T_0)/ (D \\rho_v) ]}", "5e44d65ffed8cf42eab6d1a77e40c9c0": "T={2 \\pi \\hbar \\over C} L_{cm}", "5e44e1561e0b8ab3c679764ecf0a6de3": "\nx = \\frac{B-\\sqrt{B^2-4A}}{2}\n", "5e44fdf23f2dba5e3ae06d76ddc1a72c": "\n V = L \\frac{\\partial I}{\\partial t}\n", "5e452d9952bda9e0a77d52de356f94eb": "\n\\begin{align}\n(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)\n&= (\\mathbf{y}- \\mathbf{X} \\hat{\\boldsymbol\\beta})^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\hat{\\boldsymbol\\beta}) \\\\\n&+ (\\boldsymbol\\beta - \\hat{\\boldsymbol\\beta})^{\\rm T}(\\mathbf{X}^{\\rm T}\\mathbf{X})(\\boldsymbol\\beta - \\hat{\\boldsymbol\\beta}).\n\\end{align}\n", "5e4584025e84065e5718eff1ffa91ab4": "a = \\left| a \\right| e^{i \\theta_a} \\, \\mbox{ where } \\, \\theta_a = \\mbox{Arg}(a)", "5e45b65e97e36883d4105a9a5523436c": "f(x_1,\\dots,x_j,\\dots,x_i,\\dots,x_n) = -f(x_1,\\dots,x_i,\\dots,x_j,\\dots,x_n).", "5e45c561113ca31523105e77a88fd301": " Z \\le \\frac{ 2 \\lambda^2 } { 2 \\lambda^2 + 1 + \\rho }. ", "5e45caf26a3db7825d4db9f69de16bde": " \\mathbf{S}^1\\hookrightarrow \\mathbf{S}^{2n+1}\\to \\mathbf{CP}^n ", "5e45cb50025d7ad9a5f585bfca9a7023": "(x+1)^4 \\rightarrow x^4+4x^3+6x^2+4x+1", "5e45d3dd189425baabf340e1c3e7a62e": "2\\times \\min(n,m)", "5e45e22b047f8aa899b97b68c1505d43": "\\hat X [0], \\hat X [1], ..., \\hat X [j]", "5e45e7e2b6561ed7ed411ca970a69d23": " E = \\frac{2.21}{r_s^2} - \\frac{0.916}{r_s}", "5e45f1eca449e6a57e216945471d9497": "k\\geq 1 ", "5e45fbfba76495a02951ffaaf3917f6a": "\n \\cfrac{1}{2}\\left(|\\sigma_1|^n + |\\sigma_2|^n\\right) + \\cfrac{1}{2}|\\sigma_1-\\sigma_2|^n = \\sigma_y^n \\,\n", "5e465dbc50358e973c880c71bd1fb616": "\\{Y_i\\} ", "5e46ad80b4f705c55a8d1e6b06ba2e73": "\\textstyle (\\lambda n, \\lambda k)", "5e46c60d7f60348b6697e3bb99300445": "M_*^m(A) = \\liminf_{\\varepsilon \\to 0} \\frac{\\mu(A_\\varepsilon) - \\mu(A)}{\\alpha_{n-m}\\varepsilon^{n-m}}", "5e4707611328cdde130d5adc95c7d1ce": "R(r),", "5e471dfe41efc609ae6d75fe7bbb078f": "-63 \\sqrt {2} / 256", "5e472a17020ba3a7839365776ab444bd": "A=\n\\left[\\!\\!\\!\\begin{array}{*{20}{r}}\n 5 & 4 & 2 & 1 \\\\[2pt]\n 0 & 1 & -1 & -1 \\\\[2pt]\n -1 & -1 & 3 & 0 \\\\[2pt]\n 1 & 1 & -1 & 2\n\\end{array}\\!\\!\\right].", "5e479812c5d2be6660ecf1b5eb7bc979": "a=a_r(q)", "5e47a700f38be642530a24b45342417f": " F(z) = 1 + \\sum_{n\\ge 1} Z(E_n)(f(z), f(z^2), \\ldots, f(z^n)) =\n1 + \\sum_{n\\ge 1} f(z)^n = \\frac{1}{1-f(z)}", "5e47ca5b5530befe3c9724ecf7d34311": "\\scriptstyle Z_{\\mathrm {iT}}", "5e488216549fdbfa9ef151015496cce4": "\\langle m|n \\rangle = \\delta_{mn}", "5e488621b49a4be3124f7046edb28d77": " P = \\zeta + \\zeta^4 + \\zeta^9 + \\cdots ", "5e489f2fe8c149fd56078ec3d757452a": "\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ 2\\varepsilon_{23} \\\\ 2\\varepsilon_{31} \\\\ 2\\varepsilon_{12} \\end{bmatrix} = \n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ \\gamma_{23} \\\\ \\gamma_{31} \\\\ \\gamma_{12} \\end{bmatrix} = \n \\cfrac{1}{E}\n \\begin{bmatrix} 1 & -\\nu & -\\nu & 0 & 0 & 0 \\\\\n -\\nu & 1 & -\\nu & 0 & 0 & 0 \\\\\n -\\nu & -\\nu & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 2(1+\\nu) & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 2(1+\\nu) & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 2(1+\\nu) \\end{bmatrix}\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix}\n ", "5e489fb2b37c1434f87d3bd1a87e0c52": " f(\\mathbf{Y}) = \\operatorname{tr} e^{\\mathbf{H} + \\log(\\mathbf{Y})} ", "5e494805e8aa58d670bb097f2055ed18": "\\lambda_{\\min} (a(x)) > \\alpha \\;\\;\\; \\forall x", "5e495c016950439638f01dc54ffb503a": "\\lambda(x,x)", "5e49ada3f145d3083962e2010a08d4e8": " \\!\\ S_m = a + b ", "5e49c6579e41ae32b7ba5712caa16e03": "\n \\oint_{\\partial\\Omega} \\boldsymbol{A}~ds = \\int_{\\Omega} \\mathbf{n}\\cdot(\\boldsymbol{\\nabla} \\times \\boldsymbol{A})~da\n", "5e49f3534e6c0a54e77800577a6c75c4": "iD) = 37 D^{-2.7}\\ ", "5e4ae14625315fd2450fe4bc8f4fa115": "\\,M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Q_\\textrm{F}, \\sigma_0)", "5e4b2002966d340c7f3251d51e6cbd8b": "O(|V|) = O(b^d)", "5e4bec213f68299d3bdef9fdf309f3ce": "{\\mathcal L}_{xx}^3: L=Lclm(l(\\Phi)). ", "5e4c172305e6b4a772cbe87c21667f7f": "v_{m}", "5e4c192d02d19616f57dcab9fac27a56": " \\langle v \\rangle = \\int_0^{\\infty} v \\, f(v) \\, dv= \\sqrt { \\frac{8kT}{\\pi m}}= \\sqrt { \\frac{8RT}{\\pi M}} = \\frac{2}{\\sqrt{\\pi}} v_p ", "5e4c1bfe69cce1f66c3b8f1b1422ea35": "\\phi_{12}=\\frac{\\pi}{2}-\\theta_{12}\\,\\!", "5e4c68c15104c1b01478321edb4af02e": " E^2_{p,q} = H_p(\\mathbf{S}^{n+1}; H_q(\\Omega \\mathbf{S}^{n+1})). ", "5e4c78d9ac1ba713c4fc467d4c955777": "\\displaystyle{g(a:0)=(a:-\\gamma)= (a^{-\\gamma}:0)= (a(\\gamma a +1 )^{-1}:0)}", "5e4c98a7fb92b5819dc0977c84e19e2d": "\nz=c\\,z_{frac}\\,\\sin(\\beta)\n", "5e4cba0f1ee833fc6c3c42e7fe100147": " U = P_{11} \\cdot U_{11} + P_{21} \\cdot U_{21} + P_{12} \\cdot U_{12} + P_{22} \\cdot U_{22} ", "5e4cca60b687a3ffb0501543cfd6bbf7": "L_4(x)=x^4+4x^2+2 \\,", "5e4d20d519682df5adc5f30c4c70bd34": "c(\\pi, B)", "5e4d6cc7784fa7cfd7c0257bbb8af842": " Q = \\frac{\\sqrt{mn}}{m+1}. ", "5e4de2762fc3837d84a8fe77558da6be": "\\textstyle (\\bar{M}_{\\mathrm f} R^k)", "5e4e139d75eeb7710b75aa1cd2c34e7e": "e^{-(\\mu_1+\\mu_2)+\\mu_1e^{it}+\\mu_2e^{-it}}", "5e4e566d69e12897b052eafa248803b7": "\\Phi (R,D)=-\\frac{\\rho_1\\rho_2\\pi^2\\alpha R}{6 D}", "5e4ef582280033b42c697b37af7f6621": "{m_0}", "5e4f083ab68e1cc8835092eb93c67676": "\\overline{x} \\cdot \\overline{y}", "5e4f35045adcb88c274c6fd64381ba3a": "\n\\begin{align}\n\\mu &\\approx \\frac{E(N+1)-E(N-1)}{2},\\\\\n &=\\frac{-(E(N-1)-E(N))-(E(N)-E(N+1))}{2},\\\\\n &=-\\frac{1}{2}(I+A),\n\\end{align}\n", "5e4f376d8566718c55715781e2cacb83": "M_{2}^{2} =m_{2}^{2}+2m_{w}S+S^{2}.", "5e4f5761b7f3e10086468f69b240b92d": "\\text{Muscle force} = \\text{Total force} \\cdot \\cos \\Phi", "5e4f716b15fb780957c297fb48171dea": " \\mathrm{vol} (A) = \\mathrm{vol} ( \\Phi^t(A) ). \\, ", "5e4f74132191c17b3b540a4157f94dbf": "\\begin{align}\ny(k) & =F[y(k-1),y(k-2),\\ldots ,y(k-n_y),u(k-d),u(k-d-1),\\ldots ,u(k-d-n_u), \\\\ \n& {}\\quad e(k-1),e(k-2),\\ldots ,e(k-n_e)]+e(k)\n\\end{align}", "5e4f9709e341272491d9cd09b92aa96e": " S(z;x)=\\tan( 2^z \\arctan(x))", "5e4ffa8ae2c6e8531b8100cda1126d8d": "\n(\\mbox{equation 2}) \\cdot b_1 - (\\mbox{equation 1}) \\cdot a_2\n", "5e50018db0d19f5d7915478f98e88ff6": " \\operatorname{cl}(\\varnothing) = \\varnothing ", "5e50648998fa77ba9c518dd6babb6c9e": "{dq\\over dt} = C_p \\beta + f(t,T) ", "5e511002c0db476c71a377aa08317125": "S(z\\!+\\!1)=f(S(z)) ~ \\forall z\\in D : z\\!+\\!1 \\in D\\ ", "5e512795322919e57ede890c52ba7b72": "\\pi, \\,", "5e5182c0c6cecf3db426384e76817a21": " f * g = h ", "5e51d8ea98d3eca283185a965c8ba208": "{\\mathbf{x}}_0^r", "5e51f09c967ddf99ec315845691421e8": "\\tilde H_r(f^{-1}(y)) = 0,", "5e51fb0bbf47bcb69ed5d001e3f2f5e5": " \\left\\vert{ \\sum_{n=a}^b e(f(n)) }\\right\\vert \\ll \\left({\\frac{T}{N^\\sigma}}\\right)^k N^l \\ ", "5e52d8f091fb2df2f57d9b5d24e8e45c": "U_{thermal} = N \\cdot f \\cdot \\tfrac{1}{2} kT.", "5e52dd9a43eed0984ccba803b71f5238": "2^{\\aleph_0} = \\aleph_1.", "5e531656cb970c578d87d4544b4ce601": "\\log^52", "5e535490e829ded2cb924948b4720860": "\\scriptstyle X \\;\\sim\\; \\mathrm{Erlang}(k_1,\\, \\lambda)\\,", "5e536b21a6f660f3c7802cdbd8f5a3e1": "a \\vee b \\in F", "5e53a045b175e2dbd2861f28f8e746e7": "(y_{1},y_{2})", "5e53c4b5f1fdf2d5e0a271ce42dd3cdb": "|x| = \\begin{cases} x, & \\mbox{if } x \\ge 0 \\\\ -x, & \\mbox{if } x < 0. \\end{cases} ", "5e53f86beed8252bb042b0fb404a5d5e": "f(x_i+0)\\leq f(x_{i+1}-0)", "5e540e61553451878467c98599ca94cb": "H = \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\dot{\\mathbf{q}} - L. ", "5e544eb3b2b595f39119d2a6299a815d": "\\frac{1}{h_1}+\\frac{1}{h_3}=\\frac{1}{h_2}+\\frac{1}{h_4}.", "5e55544ff053c57e166bccdf7771ae94": " f_k( x ) = 0 \\quad \\text{if} \\quad | x | \\ge \\frac{ 3k }{ 2 }. ", "5e556c354806c78fc1e386ab2bd74523": "I_{12}=\\mathbf{E_{01}\\cdot E_{02}}\\cos\\delta", "5e556f42c4f83dc0fffb140eed3062e9": " | \\frac{\\gamma_\\mathrm{e}}{2\\pi} | = 28\\,024.952\\,66(62) \\mathrm{\\ \\frac{MHz}{ T}}.", "5e55cbe499d4b20cb612cd51ba6aaf29": "\n p_H - p_0 = \\frac{\\Gamma}{V} \\left(\\frac{p_H \\chi V_0}{2} - e_0\\right) \\quad \\text{or} \\quad\n \\frac{\\rho C_0^2 \\chi}{(1 - s\\chi)^2}\\left(1 - \\frac{\\chi}{2}\\,\\frac{\\Gamma}{V}\\,V_0\\right) - p_0 = -\\frac{\\Gamma}{V} e_0 \\,.\n ", "5e566ee8c1bdf16018b8a3f3dd57ea3d": "\\scriptstyle \\int_{-\\infty}^\\infty \\exp ( -\\frac{1}{2} \\phi ( \\boldsymbol{r}_{\\text{rec}},\\, t_{\\text{rec}} ) ) \\, d t_{\\text{rec}}", "5e56a2c9818cb3139c7390d0956be8dd": "\\Lambda^k TM", "5e56d72f011f899870b7f3ea7864fca5": "\\text{Step 1}", "5e56dfbe449bf7adcda89630d1b0350c": " r = \\sqrt{\\frac{xyz}{x+y+z}}", "5e57263b6cb08fb95043f465d923e552": "\\phi: X \\to \\mathbf{P}^n_A = \\operatorname{Proj} A[x_1, \\dots, x_n]", "5e575c8eadcef2fde92c55986de3b5ed": "\\sum_{n=1}^{p} a_{n}^{+} + \\sum_{n=1}^{q} a_{n}^{-} < M \\leq \\sum_{n=1}^{p} a_{n}^{+} + \\sum_{n=1}^{q - 1} a_{n}^{-}.", "5e578eb0eaa0e7e90bcf37b932544989": "\\mathrm{Dir}(\\alpha+\\beta)", "5e5797db41b0a6abb1ee647b8905bd49": " \\Phi^{2}", "5e579ebea6991f51c472ff9cbc3d0579": " \\frac{dp_1}{dt} = m_1 p_1 (1 - p_1) - e p_1 ", "5e57d6cb0b5f00d5b37b97ba544716d7": " \\left[ \\begin{matrix} t \\\\ x \\\\ y \\\\ z \\end{matrix} \\right]\n= \\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\\n 0 & \\cos(\\lambda) & -\\sin(\\lambda) & 0 \\\\ \n 0 & \\sin(\\lambda) & \\cos(\\lambda) & 0 \\\\\n 0 & 0 & 0 & 1 \\end{matrix} \\right]\n\\left[ \\begin{matrix} t_0 \\\\ x_0 \\\\ y_0 \\\\ z_0 \\end{matrix} \\right] ", "5e57f370db55e0bd82e860f464eccd1c": "N(0)=0", "5e5806e56b651ca28091d41a7428b81f": " \\frac{dV}{dI} \\,\\!", "5e5899edc773e2f8fb439302310a0202": "\\mathbf{p} c^2 = E \\mathbf{v} \\,.", "5e58a7547507a2fc3d386b9906e5e892": "\\Phi,\\phi_1,\\Phi_2,\\ldots", "5e58f51cba065501f6d41e0ee2e85ae8": " \\prod_{p} \\Big(1 + \\frac{1}{p(p-1)}\\Big) = \\frac{315}{2\\pi^4}\\zeta(3) = 1.943596... ", "5e590a1a7eea1ace183a6a5d81af0e02": "\\mathbf{N} = m \\left( \\mathbf{x} - t \\mathbf{v} \\right) ", "5e5913ec0dba46e2618f2faf918f06c9": "V_G\\,", "5e592c35abae7c6b39753959f312199a": "C : y^2 = f (x)", "5e594e39765cda802389530e48015c32": "((q^{n+2}-1)/(q-1), (q^{n+1}-1)/(q-1), (q^n-1)/(q-1))", "5e598289d0dc03c12649ce3a69a6f85a": "\\displaystyle{2Q(Q(a)b,b)-L(a)Q(b)L(a)= Q(a)Q(b)+Q(b)Q(a) -Q(ab).}", "5e5a2d72375c1e3269518c68af9ce9e9": " f_n(re^{i\\theta}) = {1\\over 2\\pi}\\int_0^{2\\pi} {1-r^2\\over 1-2r\\cos(\\theta-\\varphi) + r^2 }\\, f_n(\\varphi)\\,d\\phi =\\int_0^{2\\pi}{1-r^2\\over 1-2r\\cos(\\theta-\\varphi) + r^2 } d\\mu_n(\\varphi)", "5e5a38275c67ee501093d858c744eff9": " = R_{XX}(\\tau) - \\mu^2,\\,", "5e5ab7a07533982c828d67076bab29f4": "A \\cap (B \\cup C) = (A \\cap B) \\cup (A \\cap C)\\,\\!", "5e5ac09a41448cd7b45742357dc72e51": " \\left(\\frac{\\partial_\\lambda G_{\\mu\\alpha_2\\ldots\\alpha_{2N}}}{2N}- \\partial_\\mu\nG_{\\lambda\\alpha_2\\ldots\\alpha_{2N}}\\right) q^\\mu_\\tau q^{\\alpha_2}_\\tau\\cdots\nq^{\\alpha_{2N}}_\\tau - (2N-1)G_{\\lambda\\mu\\alpha_3\\ldots\\alpha_{2N}}q^\\mu_{\\tau\\tau} q^{\\alpha_3}_\\tau\\cdots\nq^{\\alpha_{2N}}_\\tau + F_{\\lambda\\mu}q^\\mu_\\tau =0,", "5e5b17b816e1385ecc086b1cd035b7f1": "-c \\cdot u''(c)/u'(c)", "5e5b20a7c4da1c067c6f74a85977e9e9": "\\left |\\xi-\\frac{m}{n}\\right |<\\frac{1}{\\sqrt{5}\\, n^2}.", "5e5bc9af55663c4cd3a73eb7d2e3e7dd": "\\scriptstyle\\frac{x}{y}=\\frac{q}{n_i}+\\frac{r}{yn_i}", "5e5bcf134378d615fb8af6ab1d26d170": "R\\gg\\lambda", "5e5c74530fa31835da912079711cd008": "I_C = \\frac {V_{CC} - V_{out}} {R_C} ", "5e5cb38b68d13c1fe671c40f2de2cf8f": "|x_n-x|\\leq p_n\\downarrow 0", "5e5cca7cabaf7baa647a4d9b4d7ce0d8": "\\tfrac{6}{\\pi^2}", "5e5ccf76fc6182deb2a2ba01df57c1a2": "a_{1}=(1-5/9)*3d", "5e5cd652c2421b40172a0638a3638618": "x_2 = -x_1\\,\\!", "5e5d3e51dbf8cfd5a4ed343100f3dccd": " T'' + k^2 c^2 T=0, \\quad X'' + k^2 X=0,\\,", "5e5d6e0447c85b9f59efe14349e4619a": "\\displaystyle{\\ddot{\\mathbf{v}}=\\kappa(t)\\,\\mathbf{n}(t).}", "5e5dc975cc5db9a448b74ff3854a0fa7": "R_{2,1} = 5 r^2-4 r", "5e5f15f6b28603765cf22987fab28778": "\\displaystyle{M_k(z)={k\\over 2\\pi i^k} {z^k \\over |z|^{k+2}} \\,\\,\\,\\, (k\\ge 1), \\,\\,\\,\\, M_{-k}(z) =\\overline{M_k(z)}.}", "5e5f28dbda84d14c1eb90c3234e035b8": " \\ AR ", "5e5f5e91606769111737c2a51396fdd0": "{K} = \\frac{{I}}{{I_0}}\\,\\frac{{2}}{{\\beta}^3{\\gamma}^3} (1-\\gamma^2f_e)", "5e5fa1b01949bb037dbe2d770f6c7af2": "T_i(2-z)=T_i(z)^{T};\\,", "5e5fbfa206eab30605a84d5ad1413134": "11 + 13 + 17 + 19 + 23 + 29", "5e5fe4c9c03b9156d7c4bf3537787eab": "\\boldsymbol{\\nabla} \\mathbf{w}", "5e5ffb7cf1a74e8c3605eed4fca81b13": "W(x,p;t)=W(x-B^3 t^2, p-B^3 t ;0)={1\\over 2^{1/3} \\pi B} ~ \\mathrm{Ai} \\left(2^{2/3} \\left(Bx + {p^2\\over B^2}- 2Bpt\\right)\\right).", "5e60d7ff62de1a0ae49463fff477fe42": "a(p) = 0", "5e61267b06db58277dcc6f451eb2ebdd": "A \\cdot \\neg B \\cdot C", "5e615dbe7a19c2a3c55120b4b960d37c": "exp(x)", "5e616c5cb261fca1b1031534b2860cc8": "[g_{ij}] = \\begin{pmatrix}\n1 & 0 \\\\\n0 & r^2 \\\\\n\\end{pmatrix}", "5e61a6e13ca1aa182094a8877a06cdc9": " H = -t \\sum_{\\langle i,j \\rangle,\\sigma}( c^{\\dagger}_{i,\\sigma} c^{}_{j,\\sigma}+ h.c.) + U \\sum_{i=1}^{N} n_{i\\uparrow} n_{i\\downarrow}, ", "5e61ecd556c42bfc32a8b8dbebf41dac": "\\{x-a\\}^n = \\begin{cases} 0, & x < a \\\\ (x-a)^n, & x \\ge a. \\end{cases}", "5e6258b473b15eab67a74736aba5015e": "de^{- \\int_t^s V(X_\\tau)\\, d\\tau} =-V(X_s) e^{- \\int_t^s V(X_\\tau)\\, d\\tau} \\,ds,", "5e6267af2edaa355c32e70c3d7ef483e": " R_t ", "5e628be4fbad95dc754fdfe6f9dda87c": " \\frac{d[ES]}{dt} = 0 = k_1[E][S] - k_{-1}[ES] - k_2[ES] ", "5e6304504a613b1aa4a6ed9f5eb3b785": "\\delta(c)", "5e6360716d267c85a7b4edf380b8ffe0": "\\operatorname{Spec} B \\to \\operatorname{Spec} A", "5e637ca1cf1ffc1a2bded39d253786d6": "\\scriptstyle\\frac{3\\pi}{2}", "5e638c005dabdfda2f3e348b3afdd966": "\\phi: \\mathbb{R}^{n+1} \\rightarrow \\{0\\} ", "5e63bd89c45bee2d4b78b4884056255b": "Ix+=-Ix-\\,", "5e63f3ef8b15fe11443a11cd9b3e6260": "\n\\delta[n] = \\begin{cases} 0, & n \\ne 0 \\\\ 1, & n = 0.\\end{cases}", "5e6430b70a0c2be8dd95630811c40805": "\\gamma_\\lambda(\\omega)", "5e6434f870f5dc7c32fb2165936b64bf": "= x(p) = x\\,", "5e650aa4f0a534a76c9f6d1a72b548a1": "\\beta = 1/\\left(-\\sum_i \\pi_i\\mu_{ii}\\right)", "5e651a68b537a474d0b300db7292a83d": "K = \\ker(\\pi_1(M^{-}) \\twoheadrightarrow \\pi_1(W))", "5e6561cfa6699f241bc230417e05100b": "W(x,\\mu ,\\Sigma ) = \\frac{1}{\\sqrt{(2\\pi)^N |\\Sigma|}} exp\\left\\{ -\\frac{1}{2} (x - \\mu)^T \\Sigma^{-1} (x - \\mu) \\right\\}", "5e65b2344eadb34ca77b92548256b0fb": "\n A_i^* = \\{(x,i) : x \\in A_i\\}.\n ", "5e65dc722cd099ed45f0206d959969e1": "c_1 = -42.379, \\,\\!", "5e65e45ef94c35522033af411d98dd40": "dP/dV", "5e65fef6e1b6303f15b9aa9dcea86d8d": "DR_{T}^{V}", "5e666de87dda4c9cc29fb2c9a47ce2c3": "\\prod_{i=1}^\\ell(q-d^*_i-1)= \\sum_{w\\in W}\\det(w)q^{\\dim(V^w)}.", "5e6698d40a001930f5c84bf48507bcf7": "P\\in \\varprojlim_{i \\in I} F(X_i)", "5e679a5b6d81cda443e68dc862f9972a": "\\Phi, \\lambda", "5e67e064e93ff6ca3fe644458688074d": " y' = \\frac{y}{y_c} = \\frac{4.4 ft}{2.3 ft} = 1.9", "5e67f7ecc183be45ef056e2eaa4f1cb8": " \\operatorname{var}(G_1)= \\frac{6n ( n - 1 )}{ ( n - 2 )( n + 1 )( n + 3 ) } .", "5e68849d0c85cae86b8ead1130a6ce42": "\\Phi^{creep} = \\frac {1} {t_c} \\int_0^t exp \\left[ -\\frac {1}{2} \\left( \\frac {(\\dot{\\epsilon_{th}}/ \\dot{\\epsilon_m})-1} {\\dot{\\zeta}^{creep}} \\right)^2 \\right] dt", "5e69b9cbcf3eb15863c2f6ef1004804b": "\\mathrm{I}_\\mathrm{O}\\,", "5e6a025134df9a3346cf31df4e26df57": "= u^{1}_{1}(j^{1}_{p}\\sigma)u^{1}_{2}(j^{1}_{p}\\sigma) - 2x^{2}(j^{1}_{p}\\sigma)u^{1}(j^{1}_{p}\\sigma) \\,", "5e6a6d4fe2b91589877396c515b537a2": "{}^{\\circ}x", "5e6a785ca605717a3ee1bccfed002a7d": "c= {\\rm st}(x_{i_0})", "5e6ad88fda130c015ad0f9e97f04cf0f": "L=L_R+L_M\\;", "5e6adbb1ebcd747a0cbdbfb2e6a709ec": "A\\mathbin{\\Delta}U", "5e6af59b2cee391268c40f23ae61ff61": " \\nabla \\times ( \\mathbf{A} + \\mathbf{B} ) = \\nabla \\times \\mathbf{A} + \\nabla \\times \\mathbf{B} ", "5e6b77e2c4522753da9d08c54e88c11b": "\\scriptstyle i=1,2,3", "5e6b8da8497a9df5e3afc0dfe38f6873": "\\mu=GM \\ ", "5e6bf041c6710190b721232ede9cf0d6": "x_0 \\in R^n", "5e6bf7b72c7525de4d5fbfa2a0bfe0c7": " W = 1.50685 \\pm 0.00015", "5e6c1dc008c3ca1c4d276fa0e8b7106d": "\nr_{c} \\le R(q,u) \\ \\ \\ \\longleftrightarrow \\ \\ \\ \\alpha \\le \\varphi(q,\\alpha,u)\n", "5e6cb836015d55ea21b9fd1f074b52f8": "\\left|f \\left (\\frac{p}{q} \\right) \\right| = \\left| \\sum_{i=0}^n c_i p^i q^{-i} \\right| = \\frac{1}{q^n} \\left| \\sum_{i=0}^n c_i p^i q^{n-i} \\right | \\ge \\frac {1}{q^n} ", "5e6cefbb2180b2ebc567b6ab3a183191": "y \\propto x", "5e6d161b95dfe7695d89ea75af39dbc3": " F_{\\text{viscosity, top}} = - \\eta A \\frac{\\Delta v_x}{\\Delta y}.", "5e6d7235e4da7ca382021ec765431727": " \\hat{\\mu}(k) = \\int \\exp(2 \\pi i k \\theta) \\, d\\mu(\\theta) ", "5e6d7ae0337fc39f212293dc04c3b2de": "k = c \\log n", "5e6d8db7624b23c49d97fbbf85deb4e5": "\\begin{matrix}{5 \\choose 2}{2 \\choose 1}\\end{matrix}", "5e6eb6978be0b77df873d40244a074c6": "\\displaystyle \\frac{1}{|a|}\\hat{f} \\left( \\frac{\\omega}{a} \\right)\\,", "5e6efc169275ecc981365e6876f89979": "\\frac c d = c d^{-1} = b^{\\log_b (c) - \\log_b (d)}. \\,", "5e6f010049849754c918eace49267fad": " w = (d + [(m + 1)2.6] + y +[y/4] + [c/4] - 2c)\\ \\bmod\\ 7", "5e6f2d0dc61076a2098799f6b3452c26": "\\rho_0 := \\langle\\rho\\rangle", "5e6f3b8d9455d8d2fa6b08bade81484d": "=3m_1^2 + m_2^2 + m_3^2 + m_4^2 + 2p_1 \\cdot p_2 - 2p_1 \\cdot p_3 - 2p_1 \\cdot p_4 \\,", "5e6f793870e5bb91673912a4cfa30d80": "\\mathrm{Fr} \\approx 0.5", "5e6f93ca767a83389ec91ca3ada0798f": "w(X)=\\sum_i(-1)^i[A_i]\\in\\widetilde{K}_0(\\mathbb{Z}[\\pi_1(X)])", "5e6fba228da9463572e1fd96dca918d5": "A A^{-1}", "5e6fd202c1f430906b5a474d26d13cd2": " \\sum_{x \\in S_0} x^i = \\sum_{x \\in S_1} x^i", "5e70487e3e2015d053f6fd3bad2e9988": "F(A) = \\sum_{\\sigma \\in S_n} \\left(\\prod_{i = 1}^n a_{\\sigma(i)}^i\\right) F(E^{\\sigma(1)}, \\dots , E^{\\sigma(n)}).", "5e711b962f81e9b0dad340284050d84a": " a_1 a_2 a_3 - 9 a_3^2 - 9 a_1^2 a_4 + 32 a_2 a_4 = 0", "5e712e254f0aec0560ae88a39131924e": "\\frac{\\cos\\theta - 1}{\\theta}", "5e7136f8b35cca6639246a14bf6ea602": "\nF(x) = \\exp\\left[\\sum_{r=3}^\\infty\\kappa_r\\frac{(-D)^r}{r!}\\right]\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]\\,.", "5e7194452ba1d2a5812527e5c882d03f": "\\dot x = y", "5e7198f123c08196004b8964e255ae73": "P = (F_{forward})^{ \\frac{3}{2}} \\times ( \\frac{1}{b})^{\\frac{1}{2}} ", "5e721b39bce14b66be8831d9c7a00c5c": "\n \\boldsymbol{F} = \\boldsymbol{R}\\cdot\\boldsymbol{U}\n", "5e725041770b565a3730ec3545983ea0": "\\exp(A) = 1 + A + \\frac{A^2}{2!} + \\frac{A^3}{3!} + \\cdots ", "5e726171a45a09eeb2d1fbbc61040e87": "e_s(T)", "5e72a3a36355104a8358e02725342e8c": "\\alpha_7 = 255", "5e72d5a697d32aebcd59a9051c0c4f9d": "\\mathrm{height}(u) > k", "5e7326e68136d2145d3800776cef8836": "{\\mathcal{T}}", "5e733875dff87f3435f36a001340f011": "\\hat{\\alpha}\\approx \\tfrac{1}{2} + \\frac{\\hat{G}_{X}}{2(1-\\hat{G}_X-\\hat{G}_{(1-X)})} \\text{ if } \\hat{\\alpha} >1", "5e73482e3b40b91a35edaa7d484ebb6d": "|S_X(t,f)| \\approx |S_x(-f,t)| \\, ", "5e7368d712a5fa19eb138cb5948b9fda": " t \\to \\infty, ", "5e73843d92e437355bd54efd3d06d630": "a \\rightarrow (b \\rightarrow c)", "5e739bcf0d32f17e6b7c1b6a3fba764a": "T:(C,\\otimes,I)\\to(C,\\otimes,I)", "5e73cff26b21e4e2128014e6857e5905": "x-x_i", "5e73e79352e1a10936e6bbddf491afe7": "X=\\mathcal{F}^{n}", "5e7403f6ab051640cbef511efd77e1c2": " \\overline{\\overline{X}} = \\overline{X},", "5e742ff4941a800e254546dab62bb4e7": "\\forall x \\forall y \\forall z [(Rxy \\land Ryz) \\rightarrow Rxz]", "5e743f8d146de605457eccdeb0872bc2": "S_n=\\frac{1}{n!}I_n.", "5e7443ff7782584f98ffc763129ef36a": "\\, J^a", "5e744997af6ce36aeab5733e4458f265": " {\\hat f}^c(\\nu) = {\\hat f}^c(-\\nu) ", "5e746505f47bd4dc85ca983be8cf5d42": "\\Psi^\\prime", "5e74c6ac643339ad174e7f0b0509bf29": "(V,A)", "5e74d1a3cec95183f42762cac3eeaf98": "T^{\\mu \\nu}", "5e74e21d2fb4bbbae3f0c4603c8b58cd": "second / 31 557 600", "5e751572da4f916d0e0d88a1b5b3ed45": "PVNB = (Z_1/(1+R_1)) + (Z_2/(1+R_1)(1+R_2)) + ... + (Z_n/(1+R_1)(1+R_2)...(1+R_n)) ", "5e760659b5d2815c33df8b5cbda51b56": "\\Delta_1+\\Delta_2", "5e76492cee9d2e43138b3da221a1fc21": "\\int e^x \\sin (x) \\,dx = e^x \\sin (x) - \\int e^x \\cos (x) \\,dx. ", "5e76c5d95f939952e981128c3682d500": "\\xi_0 = \\frac{\\tan \\alpha}{\\sqrt{H_0 / L_0}}", "5e777e8084d131ed34f70c15c7d6f122": "\\dot{\\hat x}=f(\\hat x,u)\n +W(\\hat x)L\\Bigl(I(\\hat x,u),E(\\hat x,u,y)\\Bigr)E(\\hat x,u,y)", "5e77be853c5f5f525db58cc1f04bdd2f": "R_{1,0} = -2+3 r", "5e7809e249fe80f96d9c5a4e66edc9a9": " {MX \\over MY} = {XX'' \\over YY''}, ", "5e781b5c74032e25d595ea5670926168": "\\Psi(\\chi) = \\Phi(\\chi)- \\chi \\phi( \\chi ) - \\tfrac{1}{2} ,", "5e7851c170b65f4ebd561f63fb3e5220": "\\Delta S_{ad} = \\frac{\\Delta H_{ad}}{T}", "5e7889ddd70491a0570a7a111017aa02": "(x^2-1){P_{\\ell}^{m}}'(x) = \\sqrt{1-x^2}P_{\\ell}^{m+1}(x) + mxP_{\\ell}^{m}(x)", "5e78b670af767e787820d0cb3bb83dd5": "\\ \\begin{array}{rrcl} & \\dot{u}^* &=& \\dot{Q}u + Q\\dot{u} \\\\\n\\Rightarrow & \\dot{u}^* &=& (w^*Q - Qw)u + Q\\dot{u} \\\\\n\\Rightarrow & \\dot{u}^* &=& w^*u^* - Qwu + Q\\dot{u} \\\\\n\\Rightarrow & (\\dot{u}-wu)^* &=& Q(\\dot{u}-wu) \\\\\n\\Rightarrow & \\bar{u}^* &=& Q\\bar{u}, \\end{array}", "5e78d1c30b14b851430dd64b2d1e0cbe": "\\sec^2", "5e7919eb5f3f319316869df06a525ab6": "P_T", "5e795183d29294f53f3acf1fcdc049e0": "\n\\begin{align}\n\\lfloor x \\rfloor = m &\\;\\;\\mbox{ if and only if } &m &\\le x < m+1,\\\\\n\\lceil x \\rceil = n &\\;\\;\\mbox{ if and only if } &n -1 &< x \\le n,\\\\\n\n\\lfloor x \\rfloor = m &\\;\\;\\mbox{ if and only if } &x-1 &< m \\le x,\\\\\n\\lceil x \\rceil = n &\\;\\;\\mbox{ if and only if } &x &\\le n < x+1.\n\\end{align}\n", "5e7986fdc4be572fecbbd4c742b63d6d": "F\\left(x\\right)\\ge0", "5e7a142f162dcfb00e6133b39a5b08b6": "\\psi_n(\\phi)=e^{i n \\phi}/\\sqrt{2 \\pi}", "5e7a60f071690bfcb998be1ddf8944b5": "X \\to F(X)", "5e7abb7c993e0ff8e257efc7d40d5981": " 0=\\frac{\\dot Q_H}{T_H}+\\dot S_i.", "5e7ac85f2a89bd7c9e7f248b01e3b29e": "\\scriptstyle{L=} \\sqrt{\\scriptstyle{h^2+r^2}}", "5e7aeedebf388972f511d0c8466450af": "\\theta_t\\,", "5e7b122238249d85cfe3bfa8cd203101": "\\frac{1}{\\sqrt{4\\pi t}} e^{-\\frac{x^2}{4t}}", "5e7b9fb97164086caf5c108db5c60065": "d=\\sqrt{(\\Delta x)^2+(\\Delta y)^2}=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\\,", "5e7bc1e91f05e16b4bc94909658beec2": " e_1 = (1,0,0,\\ldots,0) \\, ", "5e7be2a8b6452b22c8c171bd69284691": "\n(x^2-y^2-z^2)^2+(2 x z)^2+(2xy)^2 = (x^2+y^2+z^2)^2", "5e7c1d390fca2ef758d3d364d5ba669c": " rT = s(k) = -\\ln(1-k) \\; ", "5e7c8076106e25a8a02e619443b57c55": "e_1, e_2,\\ldots,e_n", "5e7cdae82244c6cbf7f65009735b7666": "\\mathcal{L}_h = |D_\\mu h|^2 - \\lambda \\left(|h|^2 - \\frac{v^2}{2}\\right)^2", "5e7d61e1d20372c07b21a138ba5a1ab5": " \\| \\mathbf{v} \\times \\mathbf{u} \\| \\leq \\|v\\| \\cdot \\|u\\|.\\, ", "5e7d8814dd00e258c85c7cfa3834c69c": "P\\cdot p=0", "5e7e3fdaeb84e4842526e814b5a0d34f": "x^2 = 1\\,", "5e7e8c134527231ade400072c7079d9a": "a_n=\\sum_{d\\mid n} b_d", "5e7ee935901930f6e3df987c471ee974": "\\widetilde{Ff} \\to Ff", "5e7f441de72bbbf4a034efcdf0e0db7a": "Z_1\\cap Z_2", "5e7f7f0fb84a138ea90a3a4042fa5a7f": "\\Gamma= \\oint_{C} V \\cdot d\\mathbf{s}=\\oint_{C} V\\cos\\theta\\; ds\\,", "5e7f88c3bd23a14796d6d7af8cdd9ce0": "w_1w_2\\cdots w_\\ell", "5e7f8bb11df74727db5ded184fa31d6f": "np = N_cN_v\\text{ exp}\\left(-\\frac{E_g}{kT}\\right) = n_i^2", "5e7fe821ea0fe120873b59e0b3b846ae": "\\mathcal{T}^*(M)", "5e80b46cb092aefef50a208c0c0971df": "\\ell^p", "5e80ecbc0b9877770f4494952d0a7021": "W = \\sum_r W_r = JN \\sum_r \\sigma_r = JN \\sigma", "5e8100a11585e9a6d9c6ac96ecd999b4": "h_1, h_2, h_3, h_4", "5e818ed1b0296984bf1026a59d2dea6a": " 0 \\le x_1^k + x_2^k + \\cdots + x_N^k \\le n,\\,", "5e823a2e978d423e1c8abaa83f61b28b": "i_k.", "5e826a14d4be275298bade9ce95a7ae4": "\\int_{0}^{\\infty} x^{n} e^{-ax^2}\\,\\mathrm{d}x = \n\\begin{cases}\n \\frac{1}{2}\\Gamma \\left(\\frac{n+1}{2}\\right)/a^{\\frac{n+1}{2}} & (n>-1,a>0) \\\\\n \\frac{(2k-1)!!}{2^{k+1}a^k}\\sqrt{\\frac{\\pi}{a}} & (n=2k, k \\;\\text{integer}, a>0) \\\\\n \\frac{k!}{2a^{k+1}} & (n=2k+1,k \\;\\text{integer}, a>0)\n\\end{cases} ", "5e82720a0c6a70de913cf266e69dafdb": "G =\\pi \\Sigma F_{\\Sigma \\rarr S}", "5e82c1d65e47ca9232400cb4947c9294": "X=\\sum_{i=1}^Nx_iM_{i-1}=x_1+m_1(x_2+m_2(\\cdots+m_{N-1}x_{N})\\cdots),", "5e832ee0613817c8de485a680cf1d940": "X\\to\\{1\\}", "5e834ef8ae2827ef68f535e5f4c6d9b8": "\\mathrm{d}f = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x^i}\\, \\mathrm{d}x^i = \\langle \\nabla f,\\cdot \\rangle.", "5e835a66e73358ef3227fe5c4ee6e1e2": "\\omega_2=d\\gamma_2/dt", "5e83777c4a54b882df3ee68c3e625bad": "t = 0\\,", "5e83b721c543704775bc9c4dd2ae2561": "\\frac{n\\alpha}{\\alpha+\\beta}\\!", "5e83e461e81373183cbe4d1d6af66283": "D^k(e^{ax}y)=e^{ax}(D+a)^k y.\\,", "5e83e7ce9b53ae07563fc9c08cc945bb": "\\lambda_1\\lambda_2\\lambda_3=1", "5e84566473f3c8bfafadaeb5725b59b9": " \\mathrm{BIC} = n \\cdot \\ln(\\widehat{\\sigma_e^2}) + k \\cdot \\ln(n) \\ ", "5e8469f6434264af26e741c10e0a967f": "\\lim_{x\\to a}e^{i\\xi x}v_{1}-\\lim_{x\\to b}e^{i\\xi x}v_{1}=\\int_{a}^{b}[e^{i\\xi x}\\,q_{1}\\,v_{2}+e^{i\\xi x}\\,q_{2}\\,v_{3}]\\,dx", "5e84dcd7f66ac1bf8b42084c80b848c0": " \\mathbf{p} = \\gamma m_0 \\mathbf{v}\\,,", "5e8520027417936e6ecf04c034b437f7": "|r-s|^{-2/3}", "5e858adac4bace0b2933102e38d324ed": "2\\cdot S_n^+", "5e85dc17fe72d52c16769ffce77bcd8f": "\\psi '(t) +\\lambda \\alpha \\psi (t)=0 \\ ", "5e85f2318be78cc07f08c52995253fa2": "\\mathrm{Re}(s)\\in [0,n]", "5e860318c47df9dbe8cc219fcb4d5169": "p_{0 \\tfrac{1}{2}} \\leftarrow 2^{\\deg(p)}p(\\tfrac{x}{2})", "5e867a89845d6001d3c3187260e0b0f4": "\\Delta < \\frac{1}{2f_\\max}", "5e874f14b7dc3ceaea4dbaa00a13b1aa": " m_{h^0}^2 \\le m_{Z^0}^2\\cos^2 2\\beta", "5e87a16d5fc53f2e7498f96a0c5d2ef2": "\\epsilon> 0", "5e88482c39c827a8c4a238675218152c": "\\langle v,p\\, |\\, v^2=p^3=1\\rangle", "5e88f343adbf5166e956392d0218cd47": "\\varphi(t) = \\frac{\\exp(-t^2/2)}{\\sqrt{2\\pi}}\\quad{\\rm and}\\quad\\Phi(t) = \\int_{-\\infty}^t \\varphi(s)\\, ds. ", "5e890606537c80ca92207870cbb3d913": "{\\tilde{A}}_{2n-1}", "5e893d08777b7a63072f44a9e6e69ed6": " p\\,", "5e8965a6a43e74dfe6f7bce3e358225c": " A\\geq 0 ", "5e899ee14dfa8e2f6d97efdf5115938f": " g(x) = 0 ", "5e8a817db3d203fd3147be3aa3819e58": " D_{\\mathrm{KL}}\\big(p(\\cdot\\mid y,I) \\mid p(\\cdot\\mid I) \\big) = \\sum_x p(x\\mid y,I) \\log \\frac{p(x\\mid y,I)}{p(x\\mid I)}", "5e8ab64d17aee97b6409bee8b3a6b7d3": "\\operatorname{logit}(\\mathbb{E}[Y_i\\mid \\mathbf{X}_i]) = \\operatorname{logit}(p_i)=\\ln\\left(\\frac{p_i}{1-p_i}\\right) = \\boldsymbol\\beta \\cdot \\mathbf{X}_i", "5e8ad85d3f96df33d31abd8d5ff235fe": "dG=\\lim_{\\delta x \\to 0}(G\\delta x)=Gdx", "5e8b27dad0028defddc2551093562f35": "\\mathbf{O_1} = \\begin{pmatrix}\n 0.9 & 0.0 \\\\\n 0.0 & 0.2\n\\end{pmatrix}\n", "5e8b3ca7f7e8302ccbee1a89f025209d": "\n\\phi _m^{\\mathrm{odd}}(x)=\\frac 1{\\sqrt{a}}\\sin \\frac{m\\pi x}a.\n", "5e8b5642d0fae7254944c31496553a5e": "\\sqrt 2-1", "5e8b5ec19d3cf966461d905130dccd2a": "\\pi: Y\\to X ", "5e8b7d7c4cc13bf409e6235c2bde2551": "\\begin{align}\nA\n &{}= 2 \\pi \\int_0^\\pi \\sin(t) \\sqrt{\\left(\\cos(t)\\right)^2 + \\left(\\sin(t)\\right)^2} \\, dt \\\\\n &{}= 2 \\pi \\int_0^\\pi \\sin(t) \\, dt \\\\\n &{}= 4\\pi.\n\\end{align}", "5e8bb14b88ce7eb1680cd1e01d41fcff": "\\Gamma(z+1)=z\\Gamma(z)", "5e8bb1c768c628e50cdda897a5cd449d": "y-Y=\\frac{dy}{dx}(X) \\cdot (x-X)", "5e8bc9941707961db69ea5151109471c": "\n\\nu^+ \\geq \\mu^+ \\text{ and } \\nu^- \\geq \\mu^- .\n", "5e8be02f18bf0c46ce99bf0c62951660": "\\left ( \\frac{1}{n}, 1 - \\frac{1}{n} \\right )", "5e8c2e965d46ef1ab353c5c63f4147cd": "Q= \\begin{pmatrix} {*} & {\\kappa} & {1} & {1} \\\\ {\\kappa} & {*} & {1} & {1} \\\\ {1} & {1} & {*} & {\\kappa} \\\\ {1} & {1} & {\\kappa} & {*} \\end{pmatrix}", "5e8c3f1200bd6fece710a79b08a06ee7": "(1+\\sigma)^rg/\\Xi", "5e8c3feababb186a8b9284b4c945cea4": "c_k > 0", "5e8ccee3496b42e323f1dbb45017f441": "E[\\pi]=\\theta=E\\left[\\frac{S}{\\sum_{i=1}^{n-1} \\frac{1}{i}}\\right]=2N\\mu", "5e8cd1afcc999d189206758e8da22a47": "\\mathbf{Q}(\\sqrt[3]{2})", "5e8e3421a7d4671e77b05568c572944b": "a_{\\sigma (i)} \\ge a_{\\sigma (i + 1)} ,\\;\\forall \\;i = 1, \\cdots ,n - 1", "5e8e78294b773482e3f5978a5ec59455": " \\left|f(x) - f(p) - \\mathrm{d}f_p(x-p)\\right| < \\varepsilon\\left|x-p\\right| .\\,", "5e8eb012ea1d6ab0fcb5f3aad2f09c75": " 2a \\, ", "5e8edd4a6aaa8399ecb497d146560bc2": "C = wT", "5e8f1ce440b1947548a91deebfe20c3a": "{Dv \\over Dt} = -{1 \\over \\rho}{\\partial P \\over \\partial y} - f \\cdot u", "5e8f75b1bf3d12dc383aa83d7b50f9c3": "\\scriptstyle M_\\text{B}", "5e8fcb212c820ed1fb9cfbec11973393": " x = r ~ \\sin \\theta ~ \\cos \\phi ", "5e900895954402fe9b1699cfad6c0637": "\\frac{1}{-4} = \\frac{3}{0} + \\frac{12}{0}\\,.", "5e906aef8b08c01c9d851f5ea4b19b6e": "\\left(a, p, v\\right)\\succsim \\left(b, q, v\\right)", "5e90af33847cf84986a270641b9c7e82": " D_r = 100% ", "5e91191097782fe2c17f49096aadb97d": "\\left| \\int_a^b f(x)\\,dx - (b - a) f(a) \\right| \\leq {(b - a)^2 \\over 2} \\sup_{a \\leq x \\leq b} \\left| f'(x) \\right|", "5e91a92fc3653815a799ad78f652e757": "\\tilde D", "5e91b4f1f22584886d1be6b9a204434c": "\\mathrm N(\\beta-\\alpha\\gamma)<\\mathrm N(\\alpha).", "5e921071e6f156d13c3641a326cfa2e0": "P(x) = f(a) + f'(a)(x-a) + \\frac{f''(a)}{2!}(x-a)^2 + \\cdots + \\frac{f^{(k)}(a)}{k!}(x-a)^k.", "5e927e4c37018de3b726753a86f182f1": "\\partial_t \\eta + \\sqrt{g\\,h}\\, \\partial_x \\eta + \\tfrac32\\, \\sqrt{\\frac{g}{h}}\\, \\eta\\, \\partial_x \\eta - \\tfrac16\\, h^2\\, \\partial_t\\, \\partial_x^2 \\eta = 0.", "5e932953ccf7575dc3dec3bd06f183f1": "|\\Psi\\rangle_\\nu= |\\Psi_0\\rangle_\\nu \\oplus |\\Psi_1\\rangle_\\nu \\oplus |\\Psi_2\\rangle_\\nu \\oplus \\ldots = a_0 |0\\rangle \\oplus |\\psi_1\\rangle \\oplus \\sum_{ij} a_{ij}|\\psi_{2i}, \\psi_{2j} \\rangle_\\nu \\oplus \\ldots", "5e9390b874f79abfc33d3ddb3502ffa3": "1/\\sqrt{1+\\varepsilon^2}", "5e93c0a9bd885b855a4e7fcc73018a71": "\\mathcal{I}_{\\alpha, a}", "5e94137217dee6fac603933d24a06326": "\\left(\\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "5e941b76da61a1307e67c183a740f4cb": "d\\dot{m} = \\rho c_{x}dy ", "5e9432b01180155328aa6c3e9c2a13bf": "G' = P^\\mathsf{T} G P .", "5e9442689eb5b04b60b46541527eb3cc": "{\\underline P}X \\neq \\emptyset", "5e94d1608504670fe7736d2a05c44287": "d'(i,j) \\ge d(i,j)", "5e94dd794a0acd074ed3818bbbc36497": "= 1.3 \\times 10^{38} \\frac{M_{BH}}{M_{solar}} \\, erg/sec ", "5e94eccc1bcc525ea22550c600745b89": "\\textstyle \\frac{4195835}{3145727} = 1.333820449136241002 ", "5e95666cc9f1fddd20618912431931da": "\n\\frac{\\partial^2( A(x,y) )}{\\partial y^2} = 0\n", "5e9583b3eeafb25ff52c8920a477d80e": " \\and (S_8 \\implies (\\operatorname{equate}[A_8, q] \\and V[F_8] = q)) \\and D[F_8] = D[q] ", "5e95a0ee0b618bc9a890f58ac9886752": "\\varphi = 0", "5e95b37f25c0d9285c711f1e8a47df50": "f(x;\\mu,\\sigma,\\lambda) = \\frac{\\lambda}{2} e^{\\frac{\\lambda}{2} (2 \\mu + \\lambda \\sigma^2 - 2 x)}\n \\operatorname{erfc} (\\frac{\\mu + \\lambda \\sigma^2 - x}{ \\sqrt{2} \\sigma})\n", "5e961bd1bc875fdf9ae25de9097d8930": "\\frac{81}{64}", "5e968764fb93c946455e6b4db9170629": "d(S_1,S_2)= \\min_A \\left \\{ c(A)~{\\rm such~that}~S_2 = A (S_1) \\right \\} ", "5e96b4d4d2b11c7f2619f2b02c2a183a": "\\mathbf{n}_3", "5e96bb55b3e02f056620a24fb5a847ea": " B_{2n}=\\sum_{j=0}^{2n}{\\frac{j!}{j+1}}(-1)^jS(2n,j) \\!", "5e96fb0462d53646ebc51eac65b738e1": "\\sigma^i", "5e9797473f5caa8ccd10ea137a54a544": " \\Delta E_k = W = \\frac{1}{2} m(v^2 - {v_0}^2) ", "5e97cd19df6759cb99fe298e60c89bf2": "\\sqrt{234567}\\approx484\\tfrac{311}{968}", "5e97cf20fe909eb2a7ab210fbf8dee6f": " \\boldsymbol{\\omega}_1 = \\boldsymbol{\\omega}_2 .", "5e97d6509b8573dfdac2105da141e32c": "\\mathbb R.", "5e97dc95820e9b37bd44c201c43cb37c": "\\displaystyle\\delta_i(a_t)=0", "5e9806da425f0bbeb1c8b229fd318a73": "x \\in (0; 1)\\!", "5e980b98391bd983be428711f3d5d169": " \\text{ESR} = \\frac {\\sigma} {\\varepsilon \\omega^2 C} ", "5e988d6478445f6163ae3f564b32c344": " h", "5e98b2c94798d2615a7830ca530e26be": "\\ell_{s}\\sim 10^{-15}", "5e98ba38798cc4dd06e17d6afe8b97eb": "\\sum_{i=1}^m (y_0)_i A_i\n\\prec C", "5e991fec4519ab404367d1ef6b9f4acd": " dS = -k_B\\,\\sum_i dp_i \\ln p_i", "5e992b093a72aa9da555c520701456f1": "\\delta(x)\\!\\,", "5e9935a47e624873452c42fdaec2c522": "~E_{\\rm phys}~", "5e99765b0eced21a8849933b0082ab8c": "\\sin\\sin\\sin\\frac1x", "5e99a246a20814b8878276550e90573b": "f \\in V^*_1 \\otimes V^*_2 \\otimes \\cdots \\otimes V^*_r", "5e99c0846b47071985927d9c58d4260e": "\\scriptstyle I=m r^2 /2 ", "5e9a08a09b2986fe63103dd63a1aa69f": "\nJ=\\left(\\begin{matrix}\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & i & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & i & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 1 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 \\end{matrix}\\right)", "5e9a3a985b84764fe850ec7e9008185e": "q_{\\alpha} = \\langle\\alpha|\\widehat{q}|\\alpha\\rangle", "5e9a8507b368cdf5615d294a422bd628": " \\hat \\Sigma = \\frac{1}{T-kp-1} \\sum_{t=1}^T \\hat \\epsilon_t\\hat \\epsilon_t^'", "5e9a8c7c2a9949d01345b760dd8982d8": "\\lfloor x \\rfloor = m\\;", "5e9ade297dba7ce78117a0e334a3bb43": " \\Delta : C(G) \\to C(G) \\otimes C(G) ", "5e9b31cf67b48b9592073ad31cf5bf27": "\n\\begin{align}\n \\frac{\\pi}{6} \\rho_p d_p^3 \\frac{\\text{d} \\boldsymbol{U}_p}{\\text{d} t}\n &= \\underbrace{3 \\pi \\mu d_p \\left( \\boldsymbol{U}_f - \\boldsymbol{U}_p \\right)}_{\\text{term 1}} \n - \\underbrace{\\frac{\\pi}{6} d_p^3 \\boldsymbol{\\nabla} p}_{\\text{term 2}} \n + \\underbrace{\\frac{\\pi}{12} \\rho_f d_p^3\\, \n \\frac{\\text{d}}{\\text{d} t} \\left( \\boldsymbol{U}_f - \\boldsymbol{U}_p \\right)}_{\\text{term 3}} \n \\\\ &\n + \\underbrace{\\frac{3}{2} d_p^2 \\sqrt{\\pi \\rho_f \\mu} \n \\int_{t_{_0}}^t \\frac{1}{\\sqrt{t-\\tau}}\\, \\frac{\\text{d}}{\\text{d} \\tau} \\left( \\boldsymbol{U}_f - \\boldsymbol{U}_p \\right)\\,\n \\text{d} \\tau}_{\\text{term 4}} \n + \\underbrace{\\sum_k \\boldsymbol{F}_k}_{\\text{term 5}} .\n\\end{align}\n", "5e9b381f1ab92af146a1e8f1d022309f": "K*(P \\wedge Q) \\subseteq (K*P)+Q", "5e9b7a1aff063a10b2cd531fa5babc4a": " \\sqrt{(x - x_0\\cos\\omega t)^2+y^2} - L=0\\,\\!", "5e9b7f3f6bb11b391211beec74f98661": "\\sum_{k=0}^n\\sigma(k)\\sigma(n-k)=\\frac5{12}\\sigma_3(n)-\\frac12n\\sigma(n).", "5e9b97aea19bc99fa884332b2b779053": " \\Im(s) \\to \\pm \\infty ", "5e9bd6d2df629a5660155e5d5d2d09c6": "P_\\ell^{m}(\\cos \\theta)", "5e9bd98ce0f7a250ac1b4df2635be73a": "\\sigma=h^x", "5e9c1258f351f6f5765288c9458ad7ee": "J(v) \\leq J(u)\\text{ for every }u \\in V. ", "5e9c12e637ce30f9bf9eeabf4aa774f8": "g_1^{m_1} g_2^{m_2} ... g_r^{m_r}", "5e9cb83093cab23dcea40e81c7bdabe2": "\\displaystyle \\sum s_n(x)t^n/n! = \\frac{(1+t)^x}{(1+(1+t)^\\lambda)^{-\\mu}}", "5e9cd45203a48b9a32ba0cad013e1bbd": "L=L'/\\gamma", "5e9cfac66c4e8e123a846082dc3e5531": "\\frac{k}{d} \\cdot 2^d", "5e9d1d909eb68f11ff1c5253f727645d": " a_{ij} = a_{n-j+1,n-i+1} ", "5e9d26299835498263087dae828a5626": "C^+(W,p) \\subseteq C(W,p)", "5e9d8edf63d20493f85b4543ac049f56": "\\operatorname{Sl}_{2m+2}(\\theta) = \\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m+2}}", "5e9da65c384b7aa629c254bee3dbe55d": "\\underline{d}(A) = \\liminf_{n \\rightarrow \\infty} \\frac{n}{a_n},", "5e9e2a5001791aa692e4a3b6af18e217": "E - \\sin E", "5e9e4fa16380653f2747c2e2a87e2b99": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 37\\cdot 3.93)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 31.3\\cdot R_{\\bigodot}\n\\end{align}", "5e9e9f17d75254c547300a6fa6ef1a9f": "\\begin{alignat}{7}\n x &&\\; + \\;&& y &&\\; = \\;&& 1 & \\\\\n2x &&\\; + \\;&& y &&\\; = \\;&& 1 & \\\\\n3x &&\\; + \\;&& 2y &&\\; = \\;&& 3 &\n\\end{alignat}", "5e9ec6634f04c39e14c7dd40fc9c105c": "- \\frac{1}{4 \\pi} \\frac{\\partial^2 \\phi}{\\partial x^2} = \\frac{j_i}{ \\sqrt{ v_{i,0}^2 + 2 e (\\phi_{DL}-\\phi)/m_i } } - \\frac{j_e}{ \\sqrt{v_{e,0}^2 + 2 e \\phi / m_e } }.", "5e9ecc4f9faaa67b786cb3fe74ccfaae": "\n\\begin{alignat}{4}\n&\\text{(R1)}&\\qquad \\cos c&=\\cos a\\,\\cos b, \n&\\qquad\\qquad \n&\\text{(R6)}&\\qquad \\tan b&=\\cos A\\,\\tan c,\\\\\n&\\text{(R2)}& \\sin a&=\\sin A\\,\\sin c, \n&&\\text{(R7)}& \\tan a&=\\cos B\\,\\tan c,\\\\\n&\\text{(R3)}& \\sin b&=\\sin B\\,\\sin c, \n&&\\text{(R8)}& \\cos A&=\\sin B\\,\\cos a,\\\\\n&\\text{(R4)}& \\tan a&=\\tan A\\,\\sin b, \n&&\\text{(R9)}& \\cos B&=\\sin A\\,\\cos b,\\\\\n&\\text{(R5)}& \\tan b&=\\tan B\\,\\sin a, \n&&\\text{(R10)}& \\cos c&=\\cot A\\,\\cot B.\n\\end{alignat}\n", "5e9ed6477b0271a549b4af07af968aa1": "x^i({x'}^j), j=0,1,\\dots", "5e9ede4148e58f121befe058cef130a3": " \\mathfrak{gl}(S_{\\pm}) ", "5e9f6ed962acf4f7c39b556a7baf8db5": " p\\circ F ", "5e9f8842d9e116e59fe775b24c4cfe8d": "\\,\\overline{D} = i{\\partial/\\partial t} + {\\partial/\\partial x}\\mathbf{i}+{\\partial/\\partial y}\\mathbf{j}+ {\\partial/\\partial z}\\mathbf{k} \\quad ", "5e9fa75001994d724c2d3ee54655277f": "x = x_L - x_C = \\omega L_M - \\frac{1}{\\omega C_M}", "5ea03d1eed97666536d25ea5b33eeb57": "2 \\pi r (2r + a)", "5ea051565845c5e09771655c3c89d883": "i_B : A \\rightarrow B", "5ea0958696b42931aa4b1dbe6cb9f269": "[2^3,\\,3^2],\\ [5^2,\\,3^3],\\ [2^5,\\,6^2],\\ [11^2,\\,5^3],\\ [3^7,\\,13^3],", "5ea0a5bdd27a8e5ef4df973386b1c4df": "R_{Hn} = -\\frac{1}{nq} = \\frac{V_{Hn}t}{IB}", "5ea11dac256192313284b8a889f063eb": " {\\rm{C}}_\\alpha {\\rm{H}}_\\beta {\\rm{O}}_\\gamma {\\rm{N}}_\\delta + \\left( {a{\\rm{O}}_{\\rm{2}} + b{\\rm{N}}_{\\rm{2}} } \\right) \\to \\nu _1 {\\rm{CO}}_{\\rm{2}} + \\nu _2 {\\rm{H}}_{\\rm{2}} {\\rm{O}} + \\nu _3 {\\rm{N}}_{\\rm{2}} + \\nu _5 {\\rm{CO}} + \\nu _6 {\\rm{H}}_{\\rm{2}} ", "5ea13fd5c94b04e765650a5877532275": "\\text{Percent weight of the liquid phase} = X_l = \\frac{w_s - w_o}{w_s - w_l}", "5ea16ab8134e66ca80f10e766abd18b7": "n_e=", "5ea1a67e1d786d4eeeb1e75308380fd0": "\\dot p = -\\frac{\\partial H}{\\partial q} \\quad\\mbox{and}\\quad \\dot q = \\frac{\\partial H}{\\partial p},", "5ea1a7f6c7cacbb7dd18eb9900fd428c": "\\frac{a}{p}=\\frac{N}{m(b^k-1)}.", "5ea1d4f710643692852eae82a3365ae9": "\\alpha = c \\times \\omega^{\\wedge \\frac{k}{2}}", "5ea21a1f4b1d25bb10ad5180b468a9b2": "e^{X+Y} = e^Xe^Y ~.", "5ea28b4a8e6c9f0bcb5d1de9e69220de": " \\mathrm{Throughput} \\le \\frac {\\mathrm{RWIN}} {\\mathrm{RTT}} \\,\\!", "5ea2dce2b921e387fe940feb893e2f2d": "\\mu_1 + \\mu_2.", "5ea32b48e34fecf8596b535603ab4b99": " |\\cosh(\\sqrt{r}) \\cos(\\sqrt{r})| < 1 ", "5ea394c201933bcd196acb6ea3ce9b42": "g: (Z,\\sigma)\\to (V,\\tau)", "5ea46f1d20504ae5b592d6ae642afc01": "u(x,y), v(x,y)\\,", "5ea47ed78f08224da062fc341e390a0d": "\\mathbf{Q}[x]/(f(x))", "5ea4d93f23ea6550afb64156c364a150": "j+k", "5ea581ad1fabcfe9e0766ad4f4b05b14": " K_n = \\begin{bmatrix}\n 0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n 1 & 1 & 0 & \\cdots & 0 & 0 \\\\\n 1 & 0 & 1 & \\cdots & 0 & 0 \\\\\n & & & \\vdots & & \\\\\n 1 & 0 & 0 & \\cdots & 1 & 0 \\\\\n 1 & 0 & 0 & \\cdots & 0 & 1\n\\end{bmatrix}.\n", "5ea5c0da9b6ce395c0c9c316793be7d0": "M_{BA}^{f}", "5ea5fa8294c812dd3d4237d2d985f2ea": "Pr(X=x|H)=0.", "5ea5ffef9341e8521ce6b55b049c35dc": "\\|W\\|_\\inf\\leq1", "5ea610c590770b67a6983a878fbccf9c": "\\phi(q)=\\sum_{n=-\\infty}^\\infty (-1)^n q^{(3n^2-n)/2}.", "5ea66350ea72a7e07c23a6b3e2593f1c": " S_j \\subseteq \\{X_1,X_2,\\dots,X_n\\}", "5ea70d91992ad79589afa6f727baef9e": "\\Delta \\phi \\;", "5ea7368ee9c3c5b4fcd2b70ae415bc14": "\\displaystyle k_n(x) = \\frac{2}{\\pi}\\int_0^{\\pi/2}\\cos(x\\tan\\theta-n\\theta) \\, d\\theta", "5ea762d2b837fc49628c827d56d7c22e": "\n \\frac{\\partial\\Phi}{\\partial n}\n = \\frac{1}{\\sqrt{ 1 \n + \\left( \\frac{\\partial h}{\\partial x} \\right)^2 \n + \\left( \\frac{\\partial h}{\\partial y} \\right)^2\n }}\\,\n \\left\\{\n \\frac{\\partial \\Phi}{\\partial z} \n + \\frac{\\partial h}{\\partial x}\\, \\frac{\\partial \\Phi}{\\partial x} \n + \\frac{\\partial h}{\\partial y}\\, \\frac{\\partial \\Phi}{\\partial y}\n \\right\\}\n = 0,\n \\qquad \\text{ at } z=-h(x,y),\n", "5ea764195f436638790c87235fcd12bd": "[n\\leq x]", "5ea794e5f2eddc8a31c82effd7e3268f": "c_{ab}^{opt}(t)\\in\\{1, \\ldots, N\\}", "5ea8030143f25e7b4d8f1fe84f320f6a": " L = 0 \\, ", "5ea813995544512a29ff27619e2107e8": "\\ell * \\ell^2 = 0 e^0 + 1 e^L = e^L.", "5ea81bca3d51280b1f5acab5dc447830": "V(x_0) = \\max_{ a_0 } \\{ F(x_0,a_0) + \\beta V(x_1) \\} ", "5ea82fef65f07e4fa5a1b4faaa158688": " L = \n\\langle i \\partial \\bar{\\Psi}^\\dagger \\mathbf{e}_3 \\bar{\\Psi}\n- e A \\bar{\\Psi}^\\dagger \\bar{\\Psi} -m \\Psi \\bar{\\Psi}\n\\rangle_0 ", "5ea855d9c5cd8dfe5b60cd80b706ada5": "\\int_{-r}^r \\pi r^2\\,dx - \\int_{-r}^r \\pi x^2\\,dx = \\pi (r^3 + r^3) - \\frac{\\pi}{3}(r^3 + r^3) = 2\\pi r^3 - \\frac{2\\pi r^3}{3}.", "5ea8ad23cbb541de7ea1c96320fcb3b2": " \\int e^x \\cos x \\,dx = e^x\\sin x + e^x\\cos x - \\int e^x \\cos x \\,dx, ", "5ea8b3e3cef069e9847b570e2793f5d6": "G(x) = g(\\lfloor x\\rfloor)", "5ea8df35078de73a8d583fac0013a944": "6 \\frac{\\sqrt{3}}{4}", "5ea8feb4d86fd14663d1f7fce6228ca9": "h_\\nu = - \\nabla \\cdot \\mathbf{F}_\\nu", "5ea97b32479401ff302a9887a989b339": "\\psi_{\\bold{k}}(\\bold{r}) = \\frac{1}{\\sqrt{\\Omega_r}} \\left[ \\cos(\\bold{k}\\cdot\\bold{r}) + i \\sin(\\bold{k}\\cdot\\bold{r})\\right]", "5ea9d785c3d54dbefcd0a54491c2477b": " \\Tau = \\mathbf{I} \\alpha ", "5eaa19436ffed63356e37f7a6ffdb38e": "x \\cup y = \\bigcup\\{x,y\\}", "5eaa69f6fb720d7f16fd2e83b9e71670": "\\Pr(\\rhos}", "5ed0d757d11aeb3eee6ec5f1dc0ef618": "(M\\times {\\Bbb R}^{>0})\\,", "5ed16def00e6d7060b2506ddd237356a": "T_i^{(n)}\\,\\!", "5ed171ebf28ff5a6c68cb7e24b51244d": "\\widehat{T}(\\Delta\\mathbf{r})\\psi(\\mathbf{r},t) = \\psi(\\mathbf{r} + \\Delta\\mathbf{r},t)", "5ed1dafe84989fa6e88fb57874034a60": "S\\left(\\rho=\\sum_i p_i\\rho_i\\right)= H(p_i) + \\sum_i p_iS(\\rho_i)", "5ed2d4c114d036610b8e20271c5026ef": "1 \\times 1", "5ed2ec358abc7d521a3330e8b84cc7df": "x^{n}", "5ed2f91b2a15fc9b94039f3033ff2148": "\\vec{x}_1(t-r)", "5ed31d8018061e60fb2f52734bf2fe50": "f(x) = 2\\phi(x)\\Phi(\\alpha x). \\,", "5ed40931131a8fcecf6c28da8bcd49c6": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}\\left[-p~\\boldsymbol{\\mathit{1}} + \\cfrac{2}{J^{2/3}}\\left(C_1 + \\bar{I}_1~C_2\\right)\\boldsymbol{B} -\n \\cfrac{2}{J^{4/3}}~C_2~\\boldsymbol{B}\\cdot\\boldsymbol{B} -\\cfrac{2}{3}\\left(C_1\\,\\bar{I}_1 + 2C_2\\,\\bar{I}_2\\right)\\boldsymbol{\\mathit{1}}\\right] \\,.\n ", "5ed447bf9445dab5b2ab46852222a98d": " f(w) ={1\\over \\pi^2} \\int_0^\\infty \\tilde{f}(\\lambda) \\varphi_\\lambda(w) {\\lambda\\pi\\over 2} \\tanh({\\pi\\lambda\\over 2})\\, d\\lambda.", "5ed45c07cd01fdea4cf0b33628631f08": "\\sin(\\theta^1(t) - \\theta^2(t))", "5ed4a442f190807b69d39a2222c04f42": "\n\\begin{align}\n\\text{minimize} \\quad & \\lambda \\\\\n\\text{subject to} \\quad & A^T\\mathbf{u} + \\lambda \\mathbf{d} = \\mathbf{c} \\\\\n& -\\mathbf{b}^T \\mathbf{u} + \\lambda \\beta \\geq \\alpha \\\\\n& \\mathbf{u} \\in \\mathbb{R}_+^n, \\lambda \\in \\mathbb{R},\n\\end{align}\n", "5ed5370d82db193ab9e10bbfb646a01c": "r_f \\equiv ", "5ed5929f483c297f0f49ca8a767f8112": "W_q^{(\\infty)}=\\sup_{k \\geq 0} P^{(k)}", "5ed5935398dc1dbcf4ab096ef816e056": "a_n\\,", "5ed59b05d8aee838d7721c66e848a336": "\\mathfrak{a} \\mathfrak{b} \\mathfrak{c} \\mathfrak{d} \\mathfrak{e} \\mathfrak{f} \\mathfrak{g} \\mathfrak{h} \\mathfrak{i} \\mathfrak{j} \\mathfrak{k} \\mathfrak{l} \\mathfrak{m} \\mathfrak{n} \\mathfrak{o} \\mathfrak{p} \\mathfrak{q} \\mathfrak{r} \\mathfrak{s} \\mathfrak{t} \\mathfrak{u} \\mathfrak{v} \\mathfrak{w} \\mathfrak{x} \\mathfrak{y} \\mathfrak{z} ", "5ed59e891e9ca232c8667f5d83c783dc": "V=\\frac{c\\,B_\\text{int}\\,\\mu_{\\rm N}}{E_\\gamma}(3g_n^e+g_n)", "5ed5a5ed32504eb3df9b6f17dfe80b25": " \\frac {[B_{ad}]}{p_B\\,[S]} = K^B_{eq} ", "5ed639908a220e5792c8d99e848b1ac7": "\\mathcal{M}(P(x)) = |a_0| \\prod_{i=1}^{D} \\max(1,|\\alpha_i|).", "5ed63c723db838889cf312d6c78b6f94": " t_x , t_y \\geq 0.", "5ed65d89279f5bb0299b4ebfa230444d": "\\begin{array}[t]{rcl} wp(x:=x-5;x:=x*2\\ ,\\ x>20) & = & wp(x:=x-5,wp(x:=x*2, x > 20))\\\\ \n & = & wp(x:=x-5,x*2 > 20)\\\\\n & = & (x-5)*2 > 20\\\\\n & = & x > 15\n \\end{array}", "5ed667d9216c641ae6f3daea7ea0ce28": "\\operatorname{Hom}_{\\mathbf{Mod}_R}(\\operatorname{Free}_R(X),M) = \\operatorname{Hom}_{\\mathbf{Set}}(X,\\operatorname{Forget}(M)).", "5ed67d53a560a7f2e4d73c16f3aa3cb2": "Z<1", "5ed6f197d3f610a39523c113a0ccaf73": " 0 0\\,", "5ee02eee60a58990c49580d61799fe43": "E_n(2xa,a^2)= a^{n}U_n(x). \\, ", "5ee0b2d7b2fa7a605f8c7ae954ed96ef": "-\\ln(r+z)\\,", "5ee0d08b7da82d2b38cf814438ad2753": "{\\mathcal Q}_P", "5ee1427847ca0b6f46a72c467c2907f0": "\\dot{\\mathbf{x}} = f_0(\\mathbf{x}) + g_0(\\mathbf{x}) u_x(\\mathbf{x})", "5ee156560e3c68fd895297b3fca569ec": " \\partial_\\mu D_\\mu \\alpha \\,", "5ee1be2f52c2fb7a3d268b56b079f526": "\\Lambda = 1/R_\\mathrm{m}", "5ee1c46248d73e268a8c29d2d5117956": "\\alpha+\\rho+\\tau=1. \\,", "5ee1ddad778cf3f499793712621977a6": " P_Z^\\mathrm{T}P_Z=P_Z P_Z = P_Z", "5ee1ffd9d39056e7e2a9e63770463b77": "y_j=\\beta_0+ \\beta_1 x_{1j}+\\beta_2 x_{2j}+\\cdots+\\varepsilon", "5ee2433b3ba60780a7a18039625a2c07": "\\begin{bmatrix}\n2 & 0 \\\\\n0 & -2\n\\end{bmatrix}", "5ee256361c92e956e1d07dd525527c2b": "A = \\{X \\in L^p(\\mathcal{F}): E[u(X)] \\geq 0\\} = \\{X \\in L^p(\\mathcal{F}): E\\left[e^{-\\theta X}\\right] \\leq 1\\}", "5ee272094ff0bb41bd31c94cc0c2a67b": " \\lambda_d(U_N) \\le \\sum \\{ \\lambda_d(5 \\, C) : C \\in G_n, \\, n > N \\} = 5^d \\sum \\{ \\lambda_d(C) : C \\in G_n, \\, n > N \\} < 5^d \\varepsilon. ", "5ee2b79261fcc5f3156dbfba2eecc82c": "\\omega_{pe} = (4\\pi n_ee^2/m_e)^{1/2} = 5.64 \\times 10^4 n_e^{1/2} \\mbox{rad/s}", "5ee2b8f764e6d46701146406c0f78006": "\\delta\\, = 2\\pi\\,(\\Delta\\,n \\cdot t/\\lambda\\,)", "5ee2e0e0bab98bde6834436e53f87d1d": " M(u_1,\\ldots,u_d) = \\min \\{u_1,\\dots,u_d\\}.", "5ee30b560ca6da1034532b72e5643c71": "\\int \\left[\\frac{d}{dx}(3x^2+1)\\right] = \\int 6x \\,dx = 3x^2+c\\,,", "5ee36e2fdef3ddfeb58e3a1c07bfe476": "\\scriptstyle{E_{\\theta_1}}", "5ee4151268c3abeaaa43f4ef30eebc38": "\\Delta X_t =\\mu+\\Phi D_{t}+\\Pi X_{t-p}+\\Gamma_{p-1}\\Delta X_{t-p+1}+\\cdots+\\Gamma_{1}\\Delta X_{t-1}+\\varepsilon_t,\\quad t=1,\\dots,T", "5ee415f0467ea0c2a0907f14e62bddc1": "dn_{coating}", "5ee4733a52aeaf11d6a82aa7fa426c1d": "M_k=\\sum_{|\\alpha|\\le k}\\sup |\\nabla^\\alpha\\rho|.", "5ee487cdd735b9a26bc263fd93c88e2a": "x = \\alpha n + \\beta", "5ee4b11bb8cd00fe755b2fe05784985a": "[n_x,n_y,n_z]", "5ee4d1f53a550301c06d265d59239945": "w(n,p)=e^{-\\left ( \\frac{n-(N-1)/2}{\\sigma (N-1)/2} \\right)^{p}}", "5ee522ea3f1a5ea438f29ff5d5f7333b": "\n v_{n+1}=\\begin{cases}\n v_n/a &\\mathrm{for}~~ v_n \\in [0,a) \\\\ \\\\\n (1-v_n)/(1-a) &\\mathrm{for}~~ v_n \\in [a,1] \n \\end{cases}", "5ee5362602334ec7b0e892dfc6ac156e": "d_f", "5ee585b49e61ace9a597cbe6e6bd835f": "\\displaystyle{g=\\begin{pmatrix}\\alpha & \\beta \\\\ \\gamma & \\delta\\end{pmatrix},}", "5ee5a261da1f6413b93e399ec392bb6f": "\n \\langle A,B\\rangle_{\\mathbb{S}^n} = {\\rm tr}(A^T B) = \\sum_{i=1,j=1}^n\n A_{ij}B_{ij}.\n", "5ee5c79656b03ce750e6175246e4c0df": "F = \\frac{\\frac{2R}{D}*D_{ob}*\\Phi}{D_{a}} = \\frac{\\frac{2*0.00055}{130}*3474.2*206265}{1878} \\approx 3.22", "5ee5efebe0da0d7a4406ac1f5af890a3": "x\\in\\bigcap_{n=1}^\\infty C_n", "5ee64dfc795fbecfb65db066bb2dcf3f": "v_{a}", "5ee688dc441986165170f3a77d8cbcd1": "11.110001100111...", "5ee6d3bb959f5569d57f55767a5bc66b": "a+bi,", "5ee6e3735356381cdaf3aad25bdee2a1": "\n S_x = \\frac{\\hbar}{2} \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}, \\quad\n S_y = \\frac{\\hbar}{2} \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}, \\quad\n S_z = \\frac{\\hbar}{2} \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}\n", "5ee743e4d36b62739c04c3668e68bca7": " a = {2 \\over \\ln(10)}", "5ee7549cc0a81ac90df5e722e5e479eb": "A \\le_{st} B", "5ee7658da90fbd552511e02177598c3f": "C_{4,4} = 9 + 16", "5ee78efa2676721b69a6d8fcad78bdf2": " \\ U(h)", "5ee7e0e20061e96ccfa1ac47f335c80a": "F(x+1) - F(x) = f(x) \\, .", "5ee8127ef2f89bef2a439988a870a948": "\\frac{1}{N}\\sum_n \\langle n-1|n\\rangle e^{+ika}= S e^{ika}\\frac{1}{N}\\sum_n 1 = S e^{ika} \\ .", "5ee854fc306968db6884a737dbbe363d": "\\begin{align}\\part^\\alpha x^\\beta&= \\frac{\\part^{\\vert\\alpha\\vert}}{\\part x_1^{\\alpha_1} \\cdots \\part x_n^{\\alpha_n}} x_1^{\\beta_1} \\cdots x_n^{\\beta_n}\\\\\n&= \\frac{\\part^{\\alpha_1}}{\\part x_1^{\\alpha_1}} x_1^{\\beta_1} \\cdots\n\\frac{\\part^{\\alpha_n}}{\\part x_n^{\\alpha_n}} x_n^{\\beta_n}.\\end{align}", "5ee8962c7e3b2fa107f884dcc3be616b": "P=\\{0,1\\}^k", "5ee89c7c32378752743d47d36b44d06c": "w_1(M)\\in H^1(M,{\\mathbb Z_2})", "5ee8b763d5b09a20b54d8a058bdb4bed": "\\mathit{{c}_{p}}\\mathit{ln}\\left(\\frac{{V}_{1}}{{V}_{2}}\\right)-\\mathit{R}\\mathit{ln}\\left(\\frac{{p}_{2}}{{p}_{1}}\\right)=0", "5ee8c7cbc3de0b7705f318692b4e9d05": "f^+", "5ee8cac6d43e0ea694f1a7b579cfb9ec": "\\gamma_{\\|} |A|^{2}L", "5ee9052ece1b58e75dbbcf1c2188f191": "A_m(1,5) = 1,5,15,35,70,126,210,330,495,715,\\ldots", "5ee93841d98fdeede4b0efe6c7a68dfe": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} = \n \\int_{-h}^h \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix}\n dx_3 = \\left\\{\n \\int_{-h}^h \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix}\n", "5ee940c5cd57020db37883c68a4ec7ec": " d=d_{\\pm} ", "5ee9a9950392f9fa3277ef8bd1ed6e48": "\\textstyle 2^{l-2}", "5ee9f9fba0e67d96d3b29598d454d1ac": "[\\Omega] \\in H^3(M,R)", "5eea03533801d833671f7db56ed854d3": "d\\mathbf{Y} = \\mathbf{A}\\,dx", "5eea35e45c1d83b9ae6ec79738d20518": "\\lambda = \\frac{1}{(1.74+2 log(\\frac{r}{k}))^2}", "5eea4729c09450568d8a99e560dfa3ad": "\\begin{matrix}\n \\mathbf{y}=\\left[ \\begin{matrix}\n 2 \\\\\n 1 \\\\\n -3 \\\\\n 2 \\\\\n 2 \\\\\n\\end{matrix} \\right], & \\mathbf{X}=\\left[ \\begin{matrix}\n 1 & -1 & 0 & 0 \\\\\n 0 & 0 & 1 & -1 \\\\\n 0 & -1 & 0 & 1 \\\\\n 1 & 0 & 0 & -1 \\\\\n 0 & 1 & -1 & 0 \\\\\n\\end{matrix} \\right], & \\mathbf{r}=\\left[ \\begin{matrix}\n r_{A} \\\\\n r_{B} \\\\\n r_{C} \\\\\n r_{D} \\\\\n\\end{matrix} \\right], & \\mathbf{e}=\\left[ \\begin{matrix}\n \\varepsilon _{1} \\\\\n \\varepsilon _{2} \\\\\n \\varepsilon _{3} \\\\\n \\varepsilon _{4} \\\\\n \\varepsilon _{5} \\\\\n\\end{matrix} \\right] \\\\\n\\end{matrix}", "5eea57f9ca4fcd4715eab936ae7a7c5f": "\\hat{\\mathcal{H}}^D", "5eeb19b97678299a3b3b3e5624f2b85d": "F(b)=\\int_{-\\infty}^b f(b') db'", "5eeb2bcd3eafba4d5b8f1e0980f88c2d": "\\int_a^b\\! e^{M f(z)}\\, dz\\approx \\sqrt{\\frac{2\\pi}{-Mf''(z_0)}}e^{M f(z_0)} \\text{ as } M\\to\\infty. \\,", "5eeb49b11ff5ca8c1df28516dcfe4389": "\nz = x + iy;\\qquad f(z) = w = u + iv\\,\n", "5eeb8fabb1ac8657f8bfb45760e9a347": "\\operatorname{fnchypg}(x;n,m_1,N,\\omega) = \\operatorname{fnchypg}(x;m_1,n,N,\\omega)\\,.", "5eebd680471b7f67408629d724b24eca": "\\begin{align}\n \\sigma_{\\pm} &= \\sqrt{|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2 \\pm \\sqrt{(|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2)^2 - |z_0^2 - z_1^2 - z_2^2 - z_3^2|^2}} \\\\\n &= \\sqrt{|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2 \\pm 2\\sqrt{(\\mathrm{Re}z_0z_1^*)^2 + (\\mathrm{Re}z_0z_2^*)^2 + (\\mathrm{Re}z_0z_3^*)^2 + (\\mathrm{Im}z_1z_2^*)^2 + (\\mathrm{Im}z_2z_3^*)^2 + (\\mathrm{Im}z_3z_1^*)^2}}\n\\end{align}", "5eec167ad71ce798c8f53de2d497f412": "A\\,\\triangle\\,B = B\\,\\triangle\\,A,\\,", "5eec2176574d3fe031c02edc58278a94": "\\pi_1(Ff)", "5eed09393eeb4ef221c3944910e7212a": "\\pi_1(S^2)=\\pi_1(A)*\\pi_1(B)/ker({\\Phi})", "5eed46b0e3cf1e0f9d49db55cd130037": "\n\\frac{\\partial f(x,t)}{\\partial t} = \\frac{1}{2} \\frac{\\partial^2 f(x,t)}{\\partial x^2},\n", "5eed753a9d9c6a3f592e7a8e59f75b48": " \\bar{z}/z = -1 ", "5eedb4a65a6357257e5d6b8eea2fc17e": "\\|{\\boldsymbol{\\beta}}-\\hat{\\boldsymbol{\\beta}}\\|", "5eedd9777e2d581a35c543b52ae74cc2": "r = \\cos\\theta", "5eee0119ea9b1785e900be16795ae955": "b_0z^n + b_1z^{n-1} + \\dots + b_{n-1}z + b_n = 0", "5eee25efaacb6f1cc59f1b4878d60f54": "\n-\\operatorname{Li}_{s}(-z) = {1 \\over \\Gamma(s)}\n\\int_0^\\infty {t^{s-1} \\over e^t/z+1} \\,\\mathrm{d}t \\,.\n", "5eee71095d82905f68e57f544d121fb7": "\\{|+\\rangle, |-\\rangle\\}", "5eee892b5257e4315dcbd67b9ab6fc22": "V = 0.615 \\times R - 0.515 \\times G - 0.100 \\times B", "5eeee50b5fb38fc8ff7b909a87177a08": " \\Sigma_i(L_{p_ip_j}(Dw)w_{x_jx_k})_{x_i} = 0", "5eeeead244c08e60e132ae240c624843": "R^{n^k}", "5eeef97685ac4c5339c6c18d6e081c0a": "001011011010", "5eef5a672d627591ec962c520952034a": "q + q'\\sqrt{-1}", "5eef7263848237961f75b5e0c61f5d2d": "P(s+1,n) - P(s,n) = T_{n-1} = \\frac{n(n-1)}{2}\\, .", "5eefe8e97b5fa3a3650a1396edcfcaa4": "M \\subseteq X", "5eeff84c57117486cd2f61509cbc21cb": " G=\\langle X| r^n\\rangle", "5ef03425e8fa7a0782a9927e3722f0dc": "u^{-1} = \\frac{1- \\mathbf{i} - \\mathbf{j} - \\mathbf{k}}{2}", "5ef0427289a81219f3ba3fecd5ed6af5": " C(u_1,\\ldots,u_n)=F[F_1^{-1}(u_1),\\ldots,F_n^{-1}(u_n)]", "5ef0805d720539671714116f42c050dd": "q''_1(x_0)\\ =2\\ \\frac {3(y_1 - y_0)-(k_1+2k_0)(x_1-x_0)}{{(x_1-x_0)}^2}=0,", "5ef0b5bc555eb09f492bd51e9d29411a": "\\tfrac{2^{1092}-1}{1093}", "5ef1105403a42edc7f66710225cf1ef5": "\\,\\Sigma", "5ef130f72168c0f21ca1b18dcc501977": "h*(g*f)", "5ef1739a4eb6124e61944238ed30f604": "\\Delta^n q^{1-s} = \\sum_{k=0}^n (-1)^{n-k} {n \\choose k} (q+k)^{1-s}", "5ef17fb88fd0486d692a85350d2815dd": "[[x,x],x]=0", "5ef183456776f26627b3dc9950911440": "A_n \\zeta(n) = B_n \\pi^n\\,\\!", "5ef198bbf504ddd8dcc22418812f0802": " d^2 F_x = - \\frac{I I' ds ds'}{2r^2} \\left[\\left[(3-k)cos\\epsilon - 3(1-k) cos(rds) cos(rds')\\right]cos(rx)-(1+k)cos(rds')cos(xds)-(1+k)cos(rds)cos(xds')\\right] ", "5ef1a5b43e86f26d7c682c5befe0d4e0": "\\mu(T^{-1}(E)\\bigtriangleup E)=0", "5ef1d994cd10dca0ec75bb9a9f461ae4": "U(v_i)=T(v_i)", "5ef21879bf8fd3724d3fedf659ea0853": "\\dots, p_{-2} - p_{-1}, p_{-1} - p_0, p_0 - p_1, p_1 - p_2, \\dots", "5ef2695af17634ffa3f0d1ab4e4eee8a": "(A.3)\\quad \\theta_{(\\ell)}=g^{ab}\\nabla_a l_b -\\kappa_{(\\ell)}\\;,", "5ef2d08567bd6ec3d2685a5d6ec0cd6e": "\\left\\| \\mathbf{v} + \\mathbf{w} \\right\\|^2 = \\left\\| \\mathbf{v} \\right\\|^2 + \\left\\| \\mathbf{w} \\right\\|^2 .", "5ef2d31a28ec20a30702317d8dfefee1": " \\int d^2\\theta \\; \\lambda H_{\\bar{5}} \\Sigma H_{5} + \\mu H_{\\bar{5}} H_{5}", "5ef2e8b59513dc1354d892527e4a284d": "\\frac{x}{c} <= r ", "5ef2fb7af01fe4827fcb8ffd6b539498": "\\begin{matrix}\\mathrm{Cabtaxi}(2)&=&91&=&3^3 + 4^3 \\\\&&&=&6^3 - 5^3\\end{matrix}", "5ef34889134228e16086e17276e7bb72": "+I \\otimes S_x", "5ef3b222f9a8c9e5309209e604376a3e": " \\|f_{*}^{\\delta}\\|_{L^n(\\mathbf{S}^{n-1})}\\leqslant C_{\\epsilon}\\delta^{-\\epsilon}\\|f\\|_{L^n(\\mathbf{R}^{n})}. ", "5ef45ae762903ea7868a6eda952dce21": "u^0_{1,12}=u^0_{1,21}", "5ef498fad58cc4fc563b8536d047ea80": "\\sum_{n=1}^\\infty z^n H_{n,m} = \\frac {\\mathrm{Li}_m(z)}{1-z},", "5ef49e02fb1284c79bfa819a977cdb2d": " \\Gamma(m)= \\frac{x^{2m}}{\\sigma^m},\n", "5ef4d40e40f3776c304cd3db12fe3ea1": "C(n)", "5ef4fc5fbabe4fc0623b58a73d08f062": "\nd\\Gamma = \\prod_i dq_i \\, dp_i \\,\n", "5ef524f00a85d6c1231de4e092a77c3e": "a_{11}=\\frac{2}{x_1-x_0}", "5ef55716f9d48f24abd27df34fb31dd5": "\\frac{1}{2}m v^2", "5ef5bb89b4556a7874584a31506a449d": " k_2 ", "5ef5f54ac76882907d71be28baffb32d": "\\alpha=\\frac{\\ x_1 + \\cdots + x_n}n", "5ef64263c3340bddbfd00b59e0474667": "e(\\cdot, \\cdot) : {\\mathbb G}_1 \\times {\\mathbb G}_2 \\rightarrow {\\mathbb G}_T", "5ef66ae3bee8d6d7416ef747de3db9c9": "O(n \\sqrt n)", "5ef6765394a18cb2ac8c8d7b7903f2fc": "\\scriptstyle\\mathbf{a}\\cdot\\mathbf{b} \\,=\\, 0", "5ef77d7cc6617fa4e654b378c1b6b3e6": "P(x) = \\sum^n_{i=0} p_i x^i", "5ef7b6d5d665f2dd0086e983495fa197": "v_b=-C_R u_b", "5ef7dbdd53783364ce9bac0c84dab7d6": "u(\\nu,T) = \\frac{2\\pi h\\nu^2/c^2}{e^\\frac{h\\nu}{kT} - 1} \\approx \\frac{2 \\pi kT\\nu^2}{c^2}", "5ef7ef7ed7ffa219bcff4d915ced4c82": "I = m \\ell ^2", "5ef835e0875377e283aad5b80526b039": "T^{-1/2}", "5ef88b8a07c68f459057a5eadcf98503": " a'=0, \\dots, d-1 ", "5ef8af3db4426642eb8cc007b3787034": "4^5", "5ef8d18382abfe0d9a697cb9dcf7166a": "x/x \\neq 1\\ ", "5ef948f12adfdb88a17a55e9f5078a8c": "F=8L_f \\,", "5ef95fbb2b865f2b20b37c8bbac6a64f": "\\psi_t+u\\psi_x+v\\psi_y+w\\psi_z=0.", "5ef964058b73ebe79226678a06928440": "h(p, u^*)", "5efa05a2b46e6cf1a83770beb5b398ff": "1/2<\\sigma <1", "5efa0ba75b884b1a27ff6f3036be83be": "\\mu^+(E):=\\mu(E\\cap P)\\,", "5efa8f1b2a519bf191cfe20f86c684a6": " \\delta\\ \\mathbf{r}^T \\mathbf{R} -\\delta\\ \\mathbf{r}^T \\sum_{e} (\\mathbf{Q}^{te} + \\mathbf{Q}^{fe}) = \\delta\\ \\mathbf{r}^T \\big( \\sum_{e} \\mathbf{k}^e \\big)\\mathbf{r} + \\delta\\ \\mathbf{r}^T \\sum_{e} \\mathbf{Q}^{oe} ", "5efad05e2e6b92ed5d4be2a7cf14350d": " C_1 ", "5efb31e6ee7e08bd21d9c0151fa61719": "(l^k,r^k)\\in\\mathcal{Z}^k\\times \\mathcal{Z}^k", "5efb69d8b78a3f29c50347d3ef1d656b": "\\mathit l \\hbar", "5efb7cae6a7e6929538bedc89d96eac2": "\\ A^* = (u_1 \\otimes u_2)^* = Qu_1 \\otimes Qu_2 = Q(u_1 \\otimes u_2)Q^T = QAQ^T.", "5efc1e1cb9ecd3df3c1965a8725f9159": "\\Delta H, \\Delta S > 0", "5efc2bd1a35676b9d39f9a71b4b9d34f": "S(t) D(t) = E_{Q_*}[D(T)S(T) | \\mathcal{F}(t)].\\,", "5efc51f926d47770cdc7056edeffcb55": "\n\\vert E_{\\pi/2}(r)\\vert \\, = \\, {1 \\over \\pi\\varepsilon_\\circ c \\, r}\n\\sqrt{{ P_{avg} \\over 2R_a}} \\, = \\, \n{9.91 \\over r} \\sqrt{ P_{avg} } \\quad (L = \\lambda /2) \\, .\n", "5efc758b06a2e5f37abdb26a855910cf": "\\cdot, * \\ast, \\star, \\circ, \\bullet \\!", "5efce06c8503ce11b1a52f7dfbefd583": "\\frac{H}{P}=2\\, ", "5efd4845b5af43b7f5c67d06a73925e2": "g_Y", "5efd4bad26e34c4abbf96ae24a42a845": "A_{n+1}(x)=(2n+1)A_n(x)-xA_n'(x)=(2n+1)A_n(x)-x^2A_{n-1}(x).\\,", "5efd4dda325069eae15410559c69abbf": "\\mathfrak{t}^*\\times \\mathfrak{t}", "5efd4e7d110514e231feb207e6d30038": "\\mathrm{Q} = \\{\\pm 1, \\pm i, \\pm j, \\pm k\\} \\to \\mathrm{GL}(2,3)", "5efd77bb6d43e88bf9add443939be120": "v^{th}", "5efd79fb55aa80e2b4e2e2760dd964a2": " \\begin{align}\n\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{n}}) \n& = I - (-\\Delta\\theta n_k\\varepsilon_{kij} r_j ) \\frac{\\partial}{\\partial r_i}\\\\\n& = I - (\\Delta\\theta n_k\\varepsilon_{kji} r_j ) \\frac{\\partial}{\\partial r_i}\\\\\n& = I - \\Delta \\theta \\hat{\\mathbf{n}} \\cdot (\\mathbf{r} \\times \\nabla ) \\\\\n& = I - \\frac{i\\Delta \\theta }{\\hbar} \\hat{\\mathbf{n}} \\cdot \\widehat{\\mathbf{L}} \\\\\n\\end{align}\n", "5efdf4d9c3512780d0f913060624080f": "\\left(0,0,0,0,0,0,\\pm \\sqrt{2}\\right).", "5efe0c2ef009171b504842531ac8a467": "T_>", "5efe420812a8e74b4e80286faa05dac9": "\\mathrm{Hol}_x(\\nabla) = \\{P_\\gamma \\in \\mathrm{GL}(E_x) \\mid \\gamma \\text{ is a loop based at } x\\}.", "5efe6ac2d6b83f80398e1a2770fc079e": "x'^T C x'", "5efe7173dd3210a4728dcdf3577aad73": "n^{2-o(1)}", "5efec174f0ce3450ac536c00a2e85c4e": "\\begin{align}\n I_\\mathrm{C} &= I_\\mathrm{S} \\left(1 + \\frac{V_\\mathrm{CE}}{V_\\mathrm{A}}\\right) e^{\\frac{V_\\mathrm{BE}}{V_\\mathrm{T}}} \\\\\n \\beta_\\mathrm{F} &= \\beta_\\mathrm{F0} \\left(1 + \\frac{V_\\mathrm{CE}}{V_\\mathrm{A}}\\right)\n\\end{align}", "5efec85ad29e25fb767efd6da75eb1ed": "\\begin{cases}\n \\frac{4}{1-\\alpha^2}\\big(1 - t^{(1+\\alpha)/2}\\big), & \\text{if}\\ \\alpha\\neq\\pm1, \\\\\n t \\ln t, & \\text{if}\\ \\alpha=1, \\\\\n - \\ln t, & \\text{if}\\ \\alpha=-1\n \\end{cases}", "5efeee110b20b2dc76355ac173a44fd9": "\\sin A = \\cos A \\cdot \\tan A \\ ", "5eff39c42afa069ac5523b6f127e6aa4": "\\theta_{i=1 \\dots N}, \\phi_{i=1 \\dots N, j=1 \\dots N}, \\boldsymbol\\phi_{i=1 \\dots N}", "5eff7e357b98472dd8eb83edcfcff938": "\\frac{1}{e^\\frac{hc}{\\lambda kT}-1} \\approx \\frac{1}{\\frac{hc}{\\lambda kT}} = \\frac{\\lambda kT}{hc}.", "5eff89b064b4d29635cb9ea139bf5681": "27^4 +162^3 = 9^7,", "5eff960fd9698e1affbd1f521adc7e7d": "\\begin{pmatrix} 6 & 24 & 1 \\\\ 13 & 16 & 10 \\\\ 20 & 17 & 15 \\end{pmatrix} \\begin{pmatrix} 0 \\\\ 2 \\\\ 19 \\end{pmatrix} = \\begin{pmatrix} 67 \\\\ 222 \\\\ 319 \\end{pmatrix} \\equiv \\begin{pmatrix} 15 \\\\ 14 \\\\ 7 \\end{pmatrix} \\pmod{26}", "5effaadf3b974b9373d4f8a2676d6af0": "I_{n} = \\int x^{-n} e^{ax} dx\\,\\!", "5effca17d8fbf9aa7e379155552df4dc": "u_2 = \\tfrac{(x_1^2+ax_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{10} - 2x_2(x_1 x_9 +bx_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "5f00122ba2b9179fcd06809fc9044dc3": "D^*=\\frac{\\sqrt{Af}}{NEP}", "5f0052355ae1da74691ef95dca81cdc3": "\\|f\\| = \\sup\\{|f(z)|\\mid z\\in \\mathbf{D}\\}=\\max\\{ |f(z)|\\mid z\\in \\overline{\\mathbf{D}}\\},", "5f007134435555d3328cbb34bc4bf43b": " m = 1 ", "5f008192b38eff30b1305773b1434e0e": "s_i^2", "5f008857bd7ce410f4bcebbbcd924b47": "E((h'(X)/h(X) + \\sum \\eta_i T_i'(X))g(X)) = -Eg'(X). ", "5f0094a3f519105e6c983b7b663d7fb5": " n = 1.2 \\sqrt{ \\frac{ 1 }{ \\sum_{ i = 1 }^k { \\frac{ 1 }{ c_i } } } } ", "5f00b47b05eaef26c49ecfb4d9396afa": "y = [x_1, x_2, x_3]^T", "5f010d7a07f9b559f9dfec6d0effc76d": "\nr_{xy}=\\frac{\\sum x_iy_i-n \\bar{x} \\bar{y}}{(n-1) s_x s_y}=\\frac{n\\sum x_iy_i-\\sum x_i\\sum y_i}\n{\\sqrt{(n-1)\\sum x_i^2-(\\sum x_i)^2}~\\sqrt{(n-1)\\sum y_i^2-(\\sum y_i)^2}}.\n", "5f01190ece52b542af4c86d8e4e2b348": "j\\left(\\tau\\right)=\\frac{1728g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}", "5f016edb975be55400ad115e27b32b33": "f(0,0)=0", "5f01ce15d8e8509a598d577da00b1f27": "\\neq 2, 3", "5f01e2e49e2b43e350cc34f6207a23f2": "\\mathbb{Z}=z_1,\\dots,z_N", "5f022541fdb304cc4a1b47d4869ed73f": "\\mathbf{F} = \\frac{\\mathrm{d}\\mathbf{p}}{\\mathrm{d}t} = \\frac{\\mathrm{d}(m\\mathbf v)}{\\mathrm{d}t}.", "5f030fe9a914823daf9ff54809ba6729": "a_n = x_n e^{-\\frac{\\pi i}{N} n^2 }", "5f03363a8566987383b189352e13edde": "\\ln (\\mathcal{L} (\\alpha, \\beta|X) )= (\\alpha - 1)\\sum_{i=1}^N \\ln X_i + (\\beta- 1)\\sum_{i=1}^N \\ln (1-X_i)- N \\ln \\Beta(\\alpha,\\beta) ", "5f0378dbf4f966e254b87a1a28535ac9": "\\ell(x,t) = \\int_0^t \\delta(x-B(s))\\,ds", "5f03b4ffab3ba1d2fcee801e4a693d5a": " \\begin{align} Q\n&= \\frac{\\omega_0}{2 \\zeta \\omega_0}\n= \\frac{\\omega_0}{\\frac{\\omega_0}{Q}}\\\\\n&= \\frac{\\sqrt{\\frac{R_1 + R_\\mathrm{f}}{R_1 R_\\mathrm{f} R_2 C_1 C_2}}}{ \\frac{1}{R_1 C_1} + \\frac{1}{R_2 C_1} + \\frac{1}{R_2 C_2} - \\frac{R_\\mathrm{b}}{R_\\mathrm{a} R_\\mathrm{f} C_1} }\\\\\n&= \\frac{\\sqrt{ (R_1 + R_\\mathrm{f}) R_1 R_\\mathrm{f} R_2 C_1 C_2}}{ R_1 R_\\mathrm{f} (C_1 + C_2) + R_2 C_2 ( R_\\mathrm{f} - \\frac{R_\\mathrm{b}}{R_\\mathrm{a}} R_1 ) }\n\\end{align}", "5f03b69314dda6a0491bb483da548af7": "P_{12} = \\pi_2 \\cdot \\int_{R_1}p(y|H2)\\, dy ", "5f0417ba012e0aebe0037b4303656b04": "\ng^{}_{}(l)=(1-p)^{l-1}p\n", "5f04443a44ad1fa72aacece7e529d7ba": "f \\mathrm{id}_x = f", "5f045f939c7a44511fd6c1864a93fa87": "\n\\wp(2z)=\n\\frac{1}{4}\\left\\{\n\\frac{\\wp''(z)}{\\wp'(z)}\\right\\}^2-2\\wp(z),", "5f05415dae39c170a157869b62fbcadf": "\\Psi(x) = e^{-\\frac{1}{1-x^2}} \\mathbf{1}_{\\{|x|<1\\}}", "5f057fabc0e7956bf8a63e0a21b1f233": " \\begin{align}\n2E_{11}&= \\frac{(dx_1)^2 - (dX_1)^2}{(dX_1)^2} \\\\\nE_{11}&= \\left(\\frac{dx_1-dX_1}{dX_1}\\right)+ \\frac {1}{2} \\left(\\frac{dx_1-dX_1}{dX_1}\\right)^2 \\\\\n&=e_{(\\mathbf I_1)}+\\frac{1}{2}e_{(\\mathbf I_1)}^2\\end{align}\\,\\!", "5f05dbfdd8e9f4cec3521aff411d52d8": "\\begin{align} x_{n+1}^2 = r^2 - y_{n+1}^2 \\end{align}", "5f05e03c87c6998971a33d1339c96e7e": "\\tau(X^*, X^{**})", "5f05f58c73052ac0a40ab49ec2a2d964": "\\mathbf{X}\\,\\!", "5f061f01fb377d795480a66d6ca0b69a": "b=n-1", "5f062aea17584d706d58804df5cdbd01": "u\\!\\in\\!V^+,v,v'\\!,w,w'\\!\\in\\!V^*", "5f064cf68f2c2d2293619dcaa62750c7": "2N \\times 2N", "5f0669050cf5ce07c790e198eafc7627": "\\mathcal{U}(\\alpha, \\tilde{u}) := \\phi^{-1}([0,\\alpha])", "5f06e91b3e2e98e37dee7bb20e2e1147": "\\sigma_E\\,", "5f06f9e894e65af623ebb8c025ef96de": "\\ 0 < \\alpha < \\infin ", "5f071429d495f0c79567aa55af0d1a19": "True \\ Velocity = Ambiguous \\ Velocity + 0.5 \\times N \\left ( \\frac {PRF \\times C} {Transmit \\ Frequency} \\right)", "5f0744fa75a7c82ed1dcdfc4bf1269ae": "Y_{9}^{2}(\\theta,\\varphi)={3\\over 128}\\sqrt{1045\\over \\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(221\\cos^{7}\\theta-273\\cos^{5}\\theta+91\\cos^{3}\\theta-7\\cos\\theta)", "5f07b543a7dcc24799c675b9edce236f": "\\hat{x}_{m+1} = \\hat{x}_m + k_{m+1}(y_{m+1} - a^T_{m+1} \\hat{x}_m)", "5f08376550e55fa69b1ddd2471e5686a": "\\mathfrak r = \\min(\\{|R| : R\\subseteq[\\omega]^\\omega\\land\\forall a\\in[\\omega]^\\omega\\exists b\\in R(|b\\cap a|<\\aleph_0\\lor|b\\setminus a|<\\aleph_0)\\}).", "5f084aa88dd2205ed179296959e85447": "\\kappa=-\\frac{\\alpha}{v}\\log{c}\\,\\!", "5f0854b02efd3befd68dbcdd7968bea1": " w_i = \\sum_{j=2}^{n} a_j(1/j)^i, ", "5f088a2c05cf6388881ea056cefff8fc": "fdr = \\frac{{{p_0}{f_0}\\left( z \\right)}}{{f\\left( z \\right)}} ", "5f088edb34c352a6650cb9a279e9ae96": "g_{ij}=\\sum{p\\partial_i\\ell\\partial_j\\ell}=E(\\partial_i\\ell\\partial_j\\ell)", "5f089d61961d7e9a69993f4ae54591fa": "s=j\\omega", "5f08d18fc697585f4e5e230ef1c70748": "\\boldsymbol{\\Sigma}^1_{2n+2}", "5f08d243717f4bc8bca6dbcb27687181": "\n\\psi(x) = \\sum_i \\psi_i \\phi_i(x).\n\\,", "5f09197ab606afe83f51e41d91365f41": " H(X|Y)=\\sum_{i,j}p(x_{i},y_{j})\\log\\frac{p(y_{j})}{p(x_{i},y_{j})}", "5f099b0b977f181a1163cb16e06d614a": "\\rho\\ \\stackrel{\\mathrm{def}}{=}\\ kr, \\qquad k\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{2m_0E\\over\\hbar^2}", "5f09b3d8b67938f18f27e23cfbe14452": "\\frac{x}{r^3}", "5f09be1de229262c525e50bc008525ad": "(N, \\Sigma, P, S)", "5f0a1c9aa9268c3b11728781b5d5a11a": "\n \\text{(a)}\\quad\n \\tan\\alpha=\\frac{a\\cos\\phi\\,\\delta\\lambda}{a\\,\\delta\\phi}, ", "5f0a466b44ba327fe9a0d3fbbd31c5a9": "E\\left[ x_i x_j x_k x_n\\right] = \\Sigma _{ij}\\Sigma _{kn}+\\Sigma _{ik}\\Sigma _{jn}+\\Sigma _{in}\\Sigma _{jk}.\n", "5f0a4d0e80fea4241388f3ee317fe849": "\nT = \\begin{bmatrix}\n0 & 0 & 0 &1& 0 \\\\\n3 & 0 & 0 &-15& 0 \\\\\n-9 & 0 & 0 &30& 1 \\\\\n9 & 0 & 3 &-1& -2 \\\\\n-3 & 9 & 0 &-45& 0\n\\end{bmatrix} \\quad J = \\begin{bmatrix}\n1 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 2 & 1 & 0 \\\\\n0 & 0 & 0 & 2 & 1 \\\\\n0 & 0 & 0 & 0 & 2\n\\end{bmatrix}\n", "5f0ad8200a5aaa3e6e288b55ca8399d3": "{} = \\sum_{k=0}^{\\infty} e^{-d/100} \\frac{(d/100)^{2\\,k+1}}{(2\\,k+1)!} = e^{-d/100} \\sinh(d/100) = \\frac{1 - e^{-2d/100}}{2}\\,,", "5f0b01ed5bf13eeb50ac4cb4537c646e": "d = m_1 - m_2", "5f0b1e81393ffbe7ca03856a1fa09e9a": "E_y = E_z,~ \\nu_{xy} = \\nu_{xz},~ \\nu_{yx} = \\nu_{zx} ", "5f0b7c137821ae1ed5a6ac1230f5d17b": "B/3", "5f0b9f7bb8eebfb19f06a982ad7bd191": "\\underline{\\underline{\\mathsf{C}}}", "5f0ba7b4fc12a90cf10710f69b56108b": "\\theta_1=\\theta_3\\;", "5f0c5132b6850927a85cd212bf320e24": " \\lang A \\rang = \\sum_j p_j \\lang \\psi_j|A|\\psi_j \\rang = \\sum_j p_j \\operatorname{tr}\\left(|\\psi_j \\rang \\lang \\psi_j|A \\right) = \\sum_j \\operatorname{tr}\\left(p_j |\\psi_j \\rang \\lang \\psi_j|A\\right) = \\operatorname{tr}\\left(\\sum_j p_j |\\psi_j \\rang \\lang \\psi_j|A\\right) = \\operatorname{tr}(\\rho A),", "5f0cc517e979354d57b1aa34e77ff913": "\\mathcal{Z}(R_R)", "5f0d514c34fb66094d1b8c8ea3d72efc": " \\frac {dT}{dx}=\\frac {\\left(T_L-T_o \\right)Ae^\\left(Ax \\right)}{e^\\left(AL \\right)-1}", "5f0db1d136f2cf3aebb43a28a9f49556": "\\operatorname{E}[X^s] = e^{s\\mu + \\tfrac{1}{2}s^2\\sigma^2}.", "5f0db63f23c31422edf0ea7ec6627688": "\\{(\\mathbf{a_i}+(r,0,\\ldots,0),\\mathbf{b_i}+(r k)/q)\\}", "5f0dd750b8b67c1bfd34238836fd6650": "\\hat{\\mathbf{q}} = [q_1\\ q_2\\ q_3\\ q_4]^\\mathrm{T}", "5f0ddfeb06b7b55ca292f4273ef5aefb": "\\iint_D x \\, dx\\, dy = \\iint_T \\rho \\cos \\phi \\rho \\, d\\rho\\, d\\phi.", "5f0def9af36af5128013b1dd3c7fb06a": "\\lambda^*(A) = \\lambda^*(A \\cap E) + \\lambda^*(A \\cap E^c) ", "5f0e0a5e6e3509e99ae88f395b5c0d64": "x_0 + \\gamma\\,\\tan(\\pi\\,(p-\\tfrac{1}{2}))\\!", "5f0e9daaec1597a06cda31d399b9d346": "E_\\text{k} = \\frac{1}{2} \\cdot 80 \\,\\text{kg} \\cdot \\left(18 \\,\\text{m/s}\\right)^2 = 12960 \\,\\text{J} = 12.96 \\,\\text{kJ}", "5f0f267dc1f002563b6df9beec2376b1": "A_0 \\subset R^n", "5f0f2d38aefc37675078b229032980d4": "\\{ \\{ N, N+1, N+2, \\dots \\} : N \\in \\{1,2,3,\\dots\\} \\}", "5f0f3cc85d48068eaed1d867484a9ac0": " l_k=\\sqrt{{\\hbar \\over m\\omega_k}}", "5f0f7b7261cb67b87dfa58553665b8dc": "G(T,P) = \\min_V \\left[ U(V) + E_{ZP}(V) - T S(T,V) + P V \\right]", "5f0f7fc1065aa28170a37b5fbea2e989": "\\eta_{00} \\,=\\, -1,\\; \\eta_{11} \\,=\\, \\eta_{22} \\,=\\, \\eta_{33} \\,=\\, 1", "5f0f8229360f1f2c8a8dc0f42fc1fdf4": "\\left(\\prod_{i=1}^na_i \\right)^{1/n} = \\exp\\left[\\frac1n\\sum_{i=1}^n\\ln a_i\\right]", "5f0f9940af0c6be8ff6434246a9363b7": " \\gamma(E) \\geq \\max \\{0,\\log(\\lambda)\\}. \\, ", "5f0f9bc44ec3f1cf0ad3e41fefe0717f": "{\\partial \\overrightarrow{V_g} \\over \\partial p}", "5f10438cd198d32ec061a5396abeee9e": " \\sqrt{N_j} ", "5f108671e40db92203078dec58333a08": "\nS_\\nu(\\sigma)=e^{i\\nu\\sigma}\\,\\,\\,\\,\\mathrm{and}\\,\\,\\,\\,e^{-i\\nu\\sigma}\n", "5f10cb96f375785fb91847afb83256f6": "O(V^{2/3})", "5f10dcda6fbcbf3a615b994f5be13240": "\\boldsymbol{\\tau}_1", "5f11338764ca6014afc6d2a0973aa9f8": " r(t) ", "5f116450e46e95f029a3f542967a545d": "f=\\left(\\frac{1-D}{4a}\\right)-\\left(\\frac{-D}{4a}\\right)", "5f11b1b4701087157d1a8270f066ce9c": "O(n^{1/3})", "5f11f654b4ad3b88ce72f8353310e281": "m_2(K) = \\int x^2 K(x) \\, dx", "5f1219d98f65a9b6615d2ed2cee49318": "\\int_{\\Omega} \\frac{\\partial L}{\\partial t} \\ dV = - \\int_{\\Omega}\\nabla \\cdot (L\\mathbf{v}) \\ dV - \\int_{\\Omega} Q \\ dV\n\\qquad \\Rightarrow \\qquad\n\\int_{\\Omega} \\left( \\frac{\\partial L}{\\partial t} + \\nabla \\cdot (L\\mathbf{v}) + Q\\ \\right) dV = 0.", "5f127222bf382a394180bfc6f5eccafb": "\\int \\sinh x \\, dx = \\cosh x + C", "5f12b8407ff6b02269f69935fa3bd455": "V_{SW}", "5f12e10ef69d8d737103b17bf5bc2d65": " r_i \\leq 0, \\forall{i}, ", "5f13ee3b6f98dfafb1fea67ac05abae0": "\\land", "5f1407863f72edd5349ca2a2ed1234aa": "\\bar{G_i}=\\bar{H_i}-T\\bar{S_i},", "5f141bf171a6e41d03166676bc167b80": "J^{\\prime\\prime}(u_{0}) = -\\frac{\\beta^2 (1 - \\beta^2)}{u_{0}}", "5f142a3813b4f587074072caae4656f3": " Q = 4 \\pi \\sin ( \\theta ) / \\lambda ", "5f143af24b8168aa0920ab2096dda965": "n = p q", "5f14bdf98a8531f56eeb5911a57f98d7": " \\varepsilon_0 = \\frac{1}{Z_0 c} \\,", "5f151b91f4b1ba60c74d1719c982a819": "\\left(\\frac{a^2}{n}\\right) = 1 \\textrm{\\ or\\ } 0", "5f15a5b04f0fbe4d38ab419d9295f3b0": "n=1,\\ldots,N+2", "5f16035bed14e7f2352409398e282ba9": "|\\psi\\rangle_C = \\alpha |0\\rangle_C + \\beta|1\\rangle_C.", "5f163bea0edd5ddb71c4d519366e962e": "\n \\mathbf{M} = \\mathbf{r} \\times \\mathbf{F}\n ", "5f164e357e76c3dccb0c9fc66a673cd7": "r = \\sqrt{\\frac{(s-a)(s-b)(s-c)}{s}}\\,", "5f168050dfced012fb602b93db8e0752": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi(\\mathbf{r},\\,t) = \\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r})\\right)\\Psi(\\mathbf{r},\\,t)", "5f1682730d00fcba80406448a103d877": "e^{\\pi \\sqrt{d}} \\approx -j((1+\\sqrt{-d})/2) + 744", "5f168f148e49bcd37a6d0bb9f1dbe07a": "S_\\pm = S_x \\pm i S_y. ", "5f1726940f3c57bfdecb4745dfd474af": "\np_\\omega(x) = \\omega(x^{*} x)^{1/2}\n", "5f179c2b92d95063a5f348fc54af3dd6": "\\!V = \\{v_1 \\ldots v_n\\}", "5f17a0ef76add539c2e18f9cb632e110": "{G}_{2}", "5f17f4db14f9bece770d6fe3bdba19c3": "|x_0|< M^{1/n}", "5f17fe54a7cda84670cb97e75d376fcc": "\\sigma_x^2 - m^2", "5f1859de8cf17d8d1dae12c91a679c82": "\\textbf{Q}^0", "5f185e1c298ca8276087da911ee92d0c": " \\vert \\{j; i \\in S_j \\} \\vert \\leq t,", "5f1885cccac539892710c05c9eb107b5": "M^*(A) = \\limsup_{\\varepsilon \\to 0} \\frac{\\mu(A_\\varepsilon) - \\mu(A)}{2\\varepsilon}.", "5f18b08f817ca660b13105cfc91a9bdd": " \\gamma_i^{-} \\, \\sim \\, \\Gamma(\\Delta t_i/\\nu,\\nu \\mu_q), \\quad \\gamma_i^{+} \\, \\sim \\, \\Gamma(\\Delta t_i / \\nu, \\nu \\mu_p),", "5f18ba085b7172597759c4e40feffe65": "\\frac{\\exp((e^t-1)\\mu)}{\\exp(t(1+\\delta)\\mu)} = \\frac{\\exp((1+\\delta - 1)\\mu)}{(1+\\delta)^{(1+\\delta)\\mu}} = \\left[\\frac{\\exp(\\delta)}{(1+\\delta)^{(1+\\delta)}}\\right]^\\mu", "5f18fc8fc59e9372716bb23d54c8272f": " \\psi\\propto e^{im\\phi} e^{-r^2},\\!", "5f191c18d8f11f3112356b812e78dd43": "\ndV = \\frac{a^3 \\sinh \\tau}{\\left( \\cosh \\tau - \\cos\\sigma \\right)^3} \\, d\\sigma \\, d\\tau \\, d\\phi\n", "5f194a95b9d3e19060df3c942df5b632": "\n\\mathbf{ab} = \\mathbf{ab}^\\mathrm{T} = \n\\begin{pmatrix}\n a_1 \\\\\n a_2 \\\\\n \\vdots \\\\\n a_N\n\\end{pmatrix}\\begin{pmatrix}\n b_1 & b_2 & \\cdots & b_N\n\\end{pmatrix}\n= \\begin{pmatrix}\n a_1b_1 & a_1b_2 & \\cdots & a_1b_N \\\\\n a_2b_1 & a_2b_2 & \\cdots & a_2b_N \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_Nb_1 & a_Nb_2 & \\cdots & a_Nb_N\n\\end{pmatrix}.", "5f1950548705fd9e3163d8343d52109e": "F \\equiv \\sum_{even\\ \\ l}(K^{l}_1 + K^{l,l+1}_2) = \\sum_{even\\ \\ l}F^{[l]},", "5f1974cb6510d65d905ca8ada3408f14": "\\delta W = dA", "5f19d2674ceb0432a89238cf9cb2900d": "h= \\frac{\\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}", "5f19de84f75324381e9ffecd6fd3278c": "\ne_U=\\sqrt{e_v^2 - e_h^2} \\sin (2 \\alpha)\n", "5f1a0ef8067b5542c80cad8d4d292384": " \\frac {1}{k} = \\frac{m^*}{m} - 1 ", "5f1af848d26b2623e9059f8e6ec0eaf4": "\\mathrm{_{26}^{52}Fe} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{28}^{56}Ni} + \\gamma", "5f1b1c3d3ab5d423a46dec65f4ab683f": "\\varphi_P(x) = \\sum_{\\tau\\in x} f(\\tau)", "5f1b22021b17e4978233a5339a0aba29": "\\langle m_k|", "5f1b4b6a63e9e46f31ccaef5dd92a4f5": "\\quad\\prod_{i=1}^n\\frac{1}{x_i}=(-1)^{n-1}\\sum_{i=1}^n\\frac{1}{x_i\\Pi_i(x_1,\\ldots,x_n)} ", "5f1b6dfd983a4307995088410bb81398": "\\text{Valve Area (cm}^2\\text{)} =\n\\frac{\\text{Cardiac Output }(\\frac{\\text{ml}}{\\text{min}})}{\\text{Heart rate }(\\frac{\\text{beats}}{\\text{min}})\\cdot \\text{Systolic ejection period (s)}\\cdot 44.3 \\cdot \\sqrt{\\text{mean Gradient (mmHg)}}}", "5f1b9757d9db068b1bb1ccc7ccb77a10": "\\ W=", "5f1b9975e95512d50537dfc1563ca38a": "x_{11}=p_1q_1+D", "5f1be147b119f5495c0b54842433981b": "p_{1}=x_{11}+x_{12}", "5f1c0235f8fbbb0fe35b7cc2e22d465e": "\\boldsymbol{\\tau} = \\frac{d\\mathbf{L}}{dt},", "5f1c0482573005e1edcca9fe91baf2bc": "[\\hat{A}, \\hat{B}] = 0", "5f1c6d62505e5da8cca39ecdd523d7be": "R_{\\beta\\delta;\\varepsilon} \\, - R_{\\beta\\varepsilon;\\delta} \\, + R^\\gamma{}_{\\beta\\delta\\varepsilon;\\gamma} \\, = 0", "5f1c6ff0a1fe01bb85f201294ca07ef0": "\\int_0^\\infty \\frac{x}{e^x-1}\\,dx = \\frac{\\pi^2}{6}", "5f1ca9e014fb587e98a51092130c8b33": "x(t+t_0)=x(t)", "5f1d146d216a751efbd7554d7077620c": "\nL_r = n_e \\sum n_l C_{lk} h \\nu_{lk} + L_{rec} + L_{brems}\n", "5f1d7e09163fdadd1fb7fe14e2113466": "\n\\Phi(\\mathbf{r}) = \\frac{-Q}{2\\pi\\epsilon} \\ln \\rho + \\left( \\frac{1}{2\\pi\\epsilon} \\right) \\sum_{k=1}^{\\infty} \\frac{C_{k} \\cos k\\theta + S_{k} \\sin k\\theta}{\\rho^{k}}\n", "5f1da2d638eb84666ccbc79cfcd5d6db": "((A\\to B)\\to(C\\to D))\\to(E\\to((D\\to A)\\to(C\\to A)))", "5f1e128016ad843a45104b80f999249a": "E= m c^2 + \\frac{p^2}{2m}", "5f1e6ecafcbd5b9e58f2879af1822f02": "M_1 = P_1 | P_2 | \\dots", "5f1e767287899c003654d2b35d3a2a69": "(D_i)", "5f1ebeffcdbeb48c3eff21d42779e381": "G_a", "5f1ec0fc9452e3c393c50d5bb3634d26": "\\ L = \\frac{a}{2}\\cdot t_1^2 ", "5f1f285aac8ec5a5b321b0b5301d10d2": "~h~", "5f1f86dfe1b20585fe675ddbd0d5ac2c": "A^{*} \\rho_{\\infty} (x) = 0, \\quad x \\in \\mathbf{R}^{n}.", "5f1f95a2435dccb0127fcf48bba2c77a": "y_1, \\dots, y_n", "5f20088a6ff639517dea06e0b64fc281": "\\Omega=\\{X_1,X_2,\\ldots,X_n\\}", "5f203401e67c525ab60df5f622908a98": "a = (1/T_{0}) - (1/B) \\ln(R_{0})", "5f2038cfcafe487b8b020cab99e38685": "\n d_{ij} = \\dot \\varepsilon_{ij} = \\frac 1 2 (v_{i,j} + v_{j,i})\n", "5f209e42c42d9deb08a69eeea90ec6f6": "\\mathbf{g}=G_{45} - (G_{poles}-G_{equator})\\cdot \\cos\\left(2\\cdot lat\\cdot \\frac{\\pi}{180}\\right)", "5f20d858f5e206fa0b2fca42732ca91b": "\nU = \n\\begin{pmatrix}\n\\mathbf u_1 & \\mathbf u_2 & \\mathbf u_3\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n12 & -69 & -58/5 \\\\\n6 & 158 & 6/5 \\\\\n-4 & 30 & -33 \n\\end{pmatrix};\n", "5f20ee95c3e0bbaf9425c553bf58a5ed": "\\begin{cases}\n \\ \\frac{\\Gamma\\left(1-\\frac {3}{\\alpha}\\right)-3\\Gamma\\left(1-\\frac {2}{\\alpha}\\right)\\Gamma\\left(1-\\frac {1}{\\alpha}\\right)+2\\Gamma^3\\left(1-\\frac {1}{\\alpha} \\right)}{\\sqrt{ \\left( \\Gamma\\left(1-\\frac{2}{\\alpha}\\right)-\\Gamma^2\\left(1-\\frac{1}{\\alpha}\\right) \\right)^3 }} & \\text{for } \\alpha>3 \\\\\n \\ \\infty & \\text{otherwise}\n \\end{cases}", "5f2142ad80996e6320ddf6f283d20bb9": " = \\begin{matrix} \\frac14 \\end{matrix} \\cdot \\rho \\cdot S \\cdot \\left(v_1 + v_2\\right) \\cdot \\left(v_1^2 - v_2^2\\right) ", "5f21780661bb1eb24f472501cf0929f9": " U_\\mathrm{E}^{\\text{single}}=q\\phi(\\vec r)=\\frac{q }{4\\pi \\varepsilon_0}\\sum_{i=1}^N \\frac{Q_i}{\\left \\|\\mathfrak\\vec r_i \\right \\|} ", "5f21bdd7953e7997ffbc3405f9d7d995": "\\psi(\\alpha)\\leq\\delta", "5f21daa56e8fb7a59977e95782f337c6": "\n p(\\mathbf{x}, t) = \\hat{p}(\\mathbf{x})~e^{-i\\omega t} ~;~~\n \\mathbf{v}(\\mathbf{x}, t) = \\hat{\\mathbf{v}}(\\mathbf{x})~e^{-i\\omega t} ~;~~ i := \\sqrt{-1}\n ", "5f21e23cc804b11603daf4e1b5f35f98": "{\\mathrm {Spin}}^{\\mathbb C}(n) = {\\mathrm {Spin}}(n)\\times_{\\Bbb Z_2} {\\mathrm U}(1)\\, ,", "5f21ec1d523ece468559473e1870a801": "\n\\nabla^2 \\Phi = \\frac{1}{\\sigma^{2} + \\tau^{2}} \n\\left[\n\\frac{1}{\\sigma} \\frac{\\partial}{\\partial \\sigma} \n\\left( \\sigma \\frac{\\partial \\Phi}{\\partial \\sigma} \\right) +\n\\frac{1}{\\tau} \\frac{\\partial}{\\partial \\tau} \n\\left( \\tau \\frac{\\partial \\Phi}{\\partial \\tau} \\right)\\right] +\n\\frac{1}{\\sigma^2\\tau^2}\\frac{\\partial^2 \\Phi}{\\partial \\varphi^2}\n", "5f21ee28b09022407557de84f753217f": "A^T P(t) + P(t) A - P(t) B R^{-1} B^T P(t) + Q = - \\dot{P}(t) \\,", "5f22101d083680e77b0e80c008dc224d": "|x\\cot(x)|", "5f223c241aa2214369022b786b30482a": " F^1 \\subset F^2\\subset \\cdots \\subset F^m \\, ", "5f225e4e14ac9d4da7c3a9a1aa18217e": "\\bold j = \\frac{1}{2m}\\left[\\left(\\Psi^* \\bold{\\hat{p}} \\Psi - \\Psi \\bold{\\hat{p}} \\Psi^*\\right) - \\frac{2q}{c} \\bold{A} |\\Psi|^2 \\right]\\,\\!", "5f2270ae6fab376ce263294e3bba365d": " \\ f_T = \\ {10^a} \\ {Re^{-0.193}} ", "5f22d827dad9a741f2698899f72c34e8": "\\rho_f", "5f22e54db17577bde24108dcab1e8397": " \\sigma^{2}_{P} = \\frac{1}{n} \\sigma^{2}_{i} + \\frac{n-1}{n} \\bar{\\sigma}_{ij} ", "5f22fbef30d25ef42277b96dbaa140d4": "u : n \\mapsto u[n] = \\begin{cases} 1, & n \\ge 0 \\\\ 0, & n < 0 \\end{cases}", "5f2325850d59838a2e47488dea5e26b3": "\\Vert\\tau_a f-f\\Vert_{L^p(\\mathbb{R}^n)} < \\varepsilon \\, \\, \\forall f \\in B, \\forall a", "5f2327428d656a952420063bae588c92": "\\scriptstyle\\epsilon", "5f233d7ba560b7439b3fac781c294d12": "A_i \\in \\mathfrak{A}_i", "5f23b635b3057378f8de453f77aad4ba": "y_i=\\sum_{j=1}^{j=n}X_{ij}\\beta_j + \\epsilon_i\\,", "5f23dbcb46ebed41d2d3fcec30729b75": "\\lambda = \\frac{\\mu_0(T_0+C)}{T_0^{3/2}}\\,", "5f245059bbebbdac69c22a38e04a234d": " f\\stackrel{\\leftarrow}{\\partial}_{i}:=(-1)^{\\left|x^{i}\\right|(|f|+1)}\\partial_{i}f ", "5f24587a779e952714868d7356d8f118": "1 \\land 1 \\le Q", "5f245c23063d965c855abbf37c016d40": "\\sqrt{\\frac{a^2-b^2}{a^2+b^2}}", "5f247f02cf82713f3603253947f8b24d": "[\\Phi_{\\langle\\cdot,\\cdot\\rangle}(v),w] = \\langle v, w\\rangle.", "5f24cab6fbd727b84e15774d4b5cf6e3": "a \\in \\mathfrak{g}", "5f24d8adf5248cb429595f0f361b3a55": " {{D \\over Dt} = {({\\partial \\over \\partial t})_p} + {({\\overrightarrow{V} \\cdot \\nabla})_p} + {\\omega {\\partial \\over \\partial p}}} ", "5f24f0e1680a1b8da2c375f5dcfdbb38": "\\rho = \\frac{M_s + M_w}{V_s + V_w + V_a}= \\frac{M_t}{V_t}", "5f250448ef48268780f46295e888ba04": "\\rho = 1", "5f25088e80408783a4f8805f4212c3d4": " v_i = T_{ij}(\\rho r_j + \\sigma s_j) = \\rho T_{ij} r_j + \\sigma T_{ij} s_j ", "5f251bbaa6211ce91b6ff68c924a780d": "B(S^{-1}S)", "5f259b97b6cdf19a09bdd726ba2ac224": " \\scriptstyle F", "5f25b9942dba99657cc1af82952d2349": "(V, E)", "5f25b9ed51ef7d1ddec32a8e9b800350": " \\theta_i(t)", "5f25c2534de11b9a87cbe2fa136da654": "T_{i_1i_2\\dots i_k}", "5f25e545b82a9b4ce38edf5c0ca8f7ad": "\\scriptstyle P'_D", "5f2654a5f6952d0f611b9ea84a7230c9": " M = Q(I+Y) = QS , \\,\\!", "5f266994ccd4386a4b66ec0d39289702": " \\mathrm{LTE} = y(t_0 + h) - y_1 = \\frac{1}{2} h^2 y''(t_0) + O(h^3). ", "5f26b6ae5e4dd0f71194b32af505486e": "\\rho\\frac{\\partial \\mathbf{u}}{\\partial t}+\\rho\\mathbf{u}\\cdot\\nabla \\mathbf{u} = -\\nabla P+\\mu \\left(\\nabla^2 \\mathbf{u}+\\frac{1}{3}\\nabla\\left(\\nabla\\cdot\\mathbf{u}\\right)\\right)", "5f26beebac55c8d466af342620bb8b7e": " Q = \\frac{\\Delta T}{R} = G \\Delta T ", "5f2717c07bc0ca68295d015a261586e5": "t_e", "5f271c8f656c3715656dfd1f08a7927d": " \\mathrm{P}(-1.96 < X < 1.96) = 0.95. \\,", "5f27717baaefbe93bbc9284fabb4ed4b": "|\\{p\\}\\ \\mathrm{in}\\rangle", "5f27808f8997db0c9d479255a49edfcf": "K_\\mu\\;", "5f2860bf92c0b283d17adba44d2d3e43": "\\omega^{\\omega^{\\omega^{\\varepsilon_0 + 1}}} = \\omega^{{\\varepsilon_0}^\\omega} = \\omega^{{\\varepsilon_0}^{1+\\omega}} = \\omega^{(\\varepsilon_0\\cdot{\\varepsilon_0}^\\omega)} = {(\\omega^{\\varepsilon_0})}^{{\\varepsilon_0}^\\omega} = {\\varepsilon_0}^{{\\varepsilon_0}^\\omega} \\,.", "5f28d32e788ae2eb8a1a76bd1c086567": "m_{D^{*+}} - m_{D^0}", "5f2900486dd4380ff9d84a6d49363b63": "\\displaystyle{T_j=T(z_j)}", "5f2971fe091292065b905755c214c9d5": "F_r = \\frac{GMm}{4r^2 R} \\int \\left( 1 + \\frac{r^2 - R^2}{s^2} \\right) \\, ds.", "5f297956034c6fbdc533645278f445f1": "\\zeta(s) = 1+\\sum_{k=1,2,\\ldots} P_k(s)", "5f29889c67937436c41b99c6ee55c55d": "\\overline{z-w} = \\bar{z} - \\bar{w}, \\,", "5f2a0cbb890ab6ef05a21a9d1e85250e": "I_n=\\int \\frac{dx}{(px+q)^n\\sqrt{ax+b}}\\,\\!", "5f2a2763ad0ab5f41ce542b4de1971bb": "\\tau=\\frac{L^n}{a+bp_\\text{sat}(T)}", "5f2a392834ee00730a7a4741da42e4d3": "a - b = 59", "5f2a5efae83bd8fe8aa7094a2219a391": " x_0 \\, ", "5f2ad324b970cb1f0b22d318383b11f7": " \\prod_{n\\ge 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) \n= \\sum_{n\\in Z}s^{(3n^2+n)/2}(t^{3n}-t^{-3n-1}) ", "5f2b4aea899570c40a2233a402bf0422": "\\begin{bmatrix}1 & 9 & -13 \\\\20 & 5 & -6 \\end{bmatrix}. ", "5f2b72ccde8baac7e7298475ce6361bf": "s_n(x+y)=\\sum_{k=0}^n{n \\choose k}x^ks_{n-k}(y).", "5f2b7589dffc1d274f8363e728e2f4ce": "\\frac{e^{\\epsilon q(d,r)}\\mu(r)}{\\int e^{\\epsilon q(d,r)}\\mu(r)dr}\\,\\!", "5f2bedda447c7861117e05d9025dbf1f": "k_0 \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{3e^2\\rho}{2\\epsilon_0 E_F}} = \\sqrt{\\frac{m e^{2} k_{f}}{\\epsilon _{0} \\pi ^{2} \\hbar ^{2}}}", "5f2c04b83ddf3e851f0fc315ea6192ba": "{d (\\rho m_{ox} ) \\over d t} + div(\\rho m_{ox} u) = div(R_{ox} .grad m_{ox}) + S_{ox} ", "5f2c18f90e523bcaddfe6f2ba0c22db8": "a^n x[n]", "5f2c26aaae0e4938f2328621298172bd": "|\\omega| \\le \\pi\\,", "5f2c64d3f6479204a4444d089872ffe7": "\\gamma_s(1)=Q", "5f2c9ea5757a9fdd0ac24da0db53e5ef": "\\{\\mathcal{F}_t, t\\in T\\}", "5f2cae8af88bf15ddbca9c7e6e5ac675": "\\{\\gamma_0, \\gamma_1, \\gamma_2, \\gamma_3\\}", "5f2cb3ceff14f65e6d4602ae96420d71": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\sqrt{\\frac{3}{2}},\\ 0,\\ \\pm2\\right)", "5f2cc4a4086f8fff69b08ca87c0b8a60": "y = Cz. ", "5f2d191f8bf1038b64c8037bfd5c2b05": "(1/2,0)", "5f2d1d4d68e58c61427590ebaa6efe6d": " \\mathbf {e_5 \\times e_6} =\\mathbf{e_1} , ", "5f2d1e375c33e7bba914b1354449a970": "\\pi_r+\\pi_{ref}=200.0+1.4=201.4", "5f2d222005b11498b5370a5e08e68c16": "t_0=C(p,q),\\qquad t_2=-C(p,-q), \\qquad t_1=-t_0-t_2\\,.", "5f2d270258edbcd69508353e78f6c25c": "\\nabla^2 \\Phi' = 0.", "5f2d351ccf7c5b218e85c1741be5b89d": "\\begin{bmatrix} \\dfrac{-b_{11}}{b_{12}} & \\dfrac{1}{b_{12}} \\\\ \\dfrac{\\Delta \\mathbf{[b]}}{b_{12}} & \\dfrac{-b_{22}}{b_{12}} \\end{bmatrix}", "5f2d695b266f1621b75de4337aa2ce5c": "\\alpha \\in T_n", "5f2d86276bf84df01264036c2d6c9b73": "\\phi(\\mathbf{r},E,t)dr\\,dE", "5f2d9abca6eaaa4c6a0f2f82ab433db3": "9 + 4 =13 \\equiv 1\\pmod{12}", "5f2da22400a5c62d31a4bcf4888c29aa": "U_{sp}(s^\\ast (p),p)<0", "5f2def09caaa297ab9d024f6980b7700": "d_w = 0.944 + (3 * 0.1 (1 - 0.944)) = 0.961", "5f2e1776747820f16a1b61030f1d2ced": "p(\\hat{x_0} + \\sigma)", "5f2e3dc17e1a32a2ec2dc3d0cefb408d": "O(n log n)", "5f2e4b61cf675e8e45bd06adb266f8bc": "K(x,y) = (x^\\top y + c)^d", "5f2e56893dda2640a231e58ff995b90a": "\\mathbf{A}_q^*, ", "5f2ecdafbd4d764da61d414df9e12403": "QT_F = {QT \\over \\sqrt[3]{RR}}", "5f2efce47a32fe63dbc08eceb03079a3": "\\Delta_{ads} G = \\Delta_{ads} H - {T} \\Delta_{ads} S < 0 ", "5f2f3e6a6c38e70f668bb326aba582f0": "{P}^{3}-3PQ\\, ", "5f2f99547722e88655a55e8afa69dce7": "\n M(x) = -EI~ \\cfrac{\\mathrm{d}\\varphi}{\\mathrm{d}x} ~;~~ Q(x) = kAG\\left(\\cfrac{\\mathrm{d}w}{\\mathrm{d}x}-\\varphi\\right) = -EI~\\cfrac{\\mathrm{d}^2\\varphi}{\\mathrm{d}x^2} = \\cfrac{\\mathrm{d}M}{\\mathrm{d}x}\n ", "5f2fb0046bfb3f608bd1e2b77c51557c": "MXM^+", "5f2fdf29528e2cb643be7329af320860": "\\frac{a}{b} = \\phi^{3}, \\, ", "5f304aebd9fd196a8ae65179f2aad39d": " \\frac{dE}{dt} = \\beta \\frac{I}{N} S - (\\mu +a ) E ", "5f30ac843d5c810f563793c4e881f2dd": "10^n \\equiv (-1)^n \\equiv \\begin{cases} 1, & \\mbox{if }n\\mbox{ is even} \\\\ -1, & \\mbox{if }n\\mbox{ is odd} \\end{cases} \\pmod{11}.", "5f30b741e955a73b3d9d65d850e8f453": "\\mathcal{F}\\{s(t+x)\\}\\ = \\hat s(\\nu)\\cdot e^{i 2\\pi \\nu t},", "5f314ae6e1a78d659c59064f23d49e8c": "\\displaystyle b^2=ce,", "5f31e1b8b8833d545eb0e9a971de8ea4": "S\\setminus\\{x\\}", "5f323372bae7cac364d6b3b41b440685": "\\frac{{t_1}}{{R_1}} = -\\frac{{t_2}}{{R_2}}", "5f3283180dbaf3158e18012d78ddc960": "{}^{21}_{11} \\text{Na}_{10} \\rightarrow {}^{21}_{10} \\text{Ne}^*_{11} + \\beta^+ + \\nu_\\text{e}", "5f331741e9319a6f7f72dccd6fb86bd1": "\\sum_{a\\in A} a^i = \\sum_{b\\in B} b^i", "5f331dd3a1fe60ee752bd5dc20a76fb0": "c(t)", "5f33d9b302f0116dd865b14dd6a56b81": "B_1,B_2\\in \\mathbf{H}_n", "5f3426156d0239bffe8b2a5837af7659": "E_{pi} + E_{ki} = E_{pF} + E_{kF}", "5f344a952e29992de54b8cfe645b2d5b": "2^3=8", "5f34ae53a50f7ad4e7bf4081e17a653d": "\\scriptstyle(-1.6\\pm6\\pm1.2)\\times10^{-12}", "5f34ae91b587667c1f5e4d8bb26e4b7b": " \\; \\; \\; \\kappa^2 = \\frac{2 \\beta q^2 c_{\\rm B}}{\\epsilon_0 \\epsilon} ", "5f34bbe6c30ba1702345d2e56155a3e7": "\\vec{P}", "5f34d53fee382d1dd8e75acbeb5930b5": "U_G", "5f3512a2afcfc8c3e774428463abc1c9": "F_L = dE_L/dz\\,", "5f352fd3d02a018613b072a30e99fa0e": "\\lambda(f(L))=0", "5f3541817a016f786f5bb057c64242cd": "V_\\mathit{SWITCH} = I_\\mathit{SWITCH} R_\\mathit{on} = DI_o R_\\mathit{on} ", "5f3596802f95285891f79207881eaae4": "x^3 + u\\,", "5f35ac99130ffddfafcf07324aa668bd": "p(n_0,x_1,\\ldots,x_k)=0\\,", "5f35bee0353674105c07d5b521674167": "ROL", "5f35d03622239aba5847af1c6774cc09": "t_i - \\tfrac{\\delta}{2}", "5f36011b742e54558583155bc9658f2b": "\\mathrm{Hom}(\\mathbb T,\\mathbb T) \\cong \\mathbb Z. \\, ", "5f364cf8e03eb68fa32ad5da47a8b364": "\\mathbf{v}=\\frac{d\\mathbf{r}}{dt}=\\sum_i\\ \\frac{\\partial \\mathbf{r}}{\\partial q_i}\\dot{q}_i+\\frac{\\partial \\mathbf{r}}{\\partial t}\\,\\!.", "5f36706cab908d199c94397268a078fc": "S_{n,m}(u)", "5f36e1b71e335b084697da04486fd04d": "f(y;e_2)/f(y;e_1)", "5f3753b0922cd321fd358e448e63a39f": "\\frac{abRABC}{[\\#]} \\to \\frac{abABC}{[\\#]} \\to \\frac{abaBC}{[\\#f]} \\to \\frac{ababC}{[\\#fg]} \\to \\frac{ababL}{[\\#fg]} \\to \\frac{ababLb}{[\\#f]} \\to \\frac{ababab}{[\\#]}", "5f375c9915350da438317c968dd1e8ab": "01\\,", "5f38a4b871656a2b29a6ee3fad634ae9": " I_m = \\frac { \\sum x ( x - 1 ) } { n m ( m - 1 ) } ", "5f38c13b98075ba8ba6adcc28c6033f6": "\\,2^n \\equiv 2 \\pmod{n}", "5f38fc354bd47cfc60065a4145e437e1": " A = \\bigcup_{r \\in \\mathbb{Q}} \\{(ry,y) \\mid y \\in \\mathbb{R}\\},", "5f390341cf5ad423ab0425c767d89078": "\n \\delta U = \\int_{\\Omega} \\boldsymbol{\\sigma}:\\delta\\boldsymbol{\\epsilon}~{\\rm dV}\n ", "5f39392b7596e9d697717ce36c0d0135": "\n \\psi_R \\rightarrow \n \\begin{pmatrix}\n \\psi_{11} \\\\ \\psi_{21}\n\\end{pmatrix}\n", "5f39403a4ec58d7166f58b9b4c4d6ea1": "\\textstyle \\frac{1}{n}\\log N > R - \\delta", "5f3941afbe2cb409b92b7620727b808f": " T_Z ", "5f3967ac1d50d2a5b0beb100e7912e10": "T = \\frac{1}{2}ab\\sin \\gamma = \\frac{1}{2}bc\\sin \\alpha = \\frac{1}{2}ca\\sin \\beta", "5f39703bd18beec2bf1f28fc835fdcf8": "EIF_i:x\\in\\mathcal{X}\\mapsto n*(T_n(x_1,\\dots,x_{i-1},x,x_{i+1},\\dots,x_n)-T_n(x_1,\\dots,x_{i-1},x_i,x_{i+1},\\dots,x_n))", "5f39d2bc48b392171fb97eca4a8b8493": "\\scriptstyle a \\pmod{b}", "5f3a0555274f8c3661f89798c8c5dfb6": "\\mu_t^{k^{(i)}}", "5f3a6020ce77899641026248d02f6564": " \\mathbf{L} = L \\cos \\alpha_\\mathrm{out} \\sin \\alpha_\\mathrm{in} \\hat{x} + L \\sin \\alpha_\\mathrm{out} \\hat{y} + L \\cos\\alpha_\\mathrm{out} \\cos \\alpha_\\mathrm{in} \\hat{z}", "5f3a68bd63e185327bb6e0fe17cba89f": "L_{g}h(x) = \\frac{\\operatorname{d}h(x)}{\\operatorname{d}x}g(x).", "5f3a81d16551977c339c6e76416dff2c": "D^\\alpha=(-i \\partial_1)^{\\alpha_1} \\dots (-i \\partial_n)^{\\alpha_n}", "5f3a88acf52023ac1deb86697f603c01": "CandS_{k+1} := GenerateCandidateSubspaces(S_k)\\!\\,", "5f3aa61f5fe60cda9cadeddc84e59893": "<\\hat{H} (N) \\Psi , s> = \\lim_{\\Delta \\rightarrow v} \\sum_\\Delta <\\Psi , \\hat{H}_\\Delta (N) s>", "5f3b1aea6314d3893b9c6dfa3acedbdf": "= \\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)", "5f3b487bf6d44335b4fa3fe4d51a8dd1": "\\hat{\\alpha}(q,\\tilde{u}):= \\max\\ \\{\\alpha\\ge 0: r_{c} \\le R(q,u),\\forall u\\in U(\\alpha,\\tilde{u})\\}", "5f3b55eb396f2393cef3acdba4472f27": "\\sigma(S)=S", "5f3b603f8d65b9e8f9c06c793eed1cc7": "\\pi/2\\mbox{ } p ", "5f4935201a1050896922c79e6f43a595": "\nL_{i,j}=k(x_i,x_j) \\,\n", "5f499ff246fb9c199aec6a32e68cfa9b": " S' = R_{space}^{-T} \\, G^{-T} \\, S \\, G^{-1} \\, R_{space}^{-1} = \\begin{bmatrix} \\nu_1 & \\, & \\, \\\\ \\, & \\nu_2 & \\, \\\\ \\, & \\, & \\nu_3 \\end{bmatrix} ", "5f49b2fc04f44f7555ceeb66a41b9837": "X \\overset{g}\\to \\mathbb{A}^n_S \\to \\mathbb{A}^{n-1}_S \\to \\cdots \\to \\mathbb{A}^1_S \\to S", "5f49b39833d8f6dd4c3ff58876712c3e": "\\displaystyle n_i", "5f49fa23dfe4900c168db112fa7de503": "\\int_A\\max\\{f_1,f_2\\}\\,d\\mu = \\int_{A_1} f_1\\,d\\mu+\\int_{A_2} f_2\\,d\\mu \\leq \\nu(A_1)+\\nu(A_2)=\\nu(A),", "5f4a0743633a095cec951231b17f2bf9": "\\ x_i", "5f4bdda44fb1c1a825f4214a992d1d6e": "2\\pi i \\xi_j", "5f4c657810194d4987b9e7185a76df2f": "\\begin{pmatrix} 6 & 24 & 1 \\\\ 13 & 16 & 10 \\\\ 20 & 17 & 15 \\end{pmatrix}", "5f4c6a57c887a59645a06e46fd8f1d45": "\\operatorname{deg}: \\operatorname{Pic}(X) \\to \\mathbf{Z}", "5f4ca928314205fdce835be769fe5ccd": "\\tan(bank)=\\frac{TAS^2/r}{g}", "5f4d565cd2f6887962f2360ca0659481": "V(t)=P(\\alpha=t^{-1},z=t^{1/2}-t^{-1/2}),\\,", "5f4d7121990afced259421187c99ae15": "x_1, x_2, x_3, \\ldots ", "5f4dbe71b781ba4923e3ad18a21c9e7f": "\\text{RMS noise} = \\sqrt{\\frac{\\sum_{i=1}^n (X_i-\\frac{\\sum_{i=1}^n X_i}{n})^2}{n}}", "5f4dee7bb0da9d00f2015090592f23cc": "\\sqrt{\\mathrm{var}[R-R_f]}=\\sqrt{\\mathrm{var}[R]}.", "5f4e575da7eb166fc2414a03729d2615": "P = 3.563 \\, 45 \\times 10^{32} \\left[\\frac{\\mathrm{kg}}{M}\\right]^2 \\mathrm{W} \\;", "5f4e5bdd1e4e5c2f0bfdf57a65668916": " \\int_X^\\oplus A_x d\\mu(x) ", "5f4e881b2d7e1d236dc1cd5862111206": "( a, k)", "5f4e8b60221d9af040d7841edba95664": "\\sqrt{\\det(X_i\\cdot X_j)_{i,j=1\\dots k}}.", "5f4ea608f884d0fa9e00fdfb8264624c": "\n \\begin{bmatrix} M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} := \\int_{-t/2}^{t/2} x_3 \\,\\begin{bmatrix} \\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\\, dx_3 \\,.\n", "5f4ed8ec68d738206302d1d87ccf1d3c": " \\Phi'(0) \\equiv \\left.\\frac{d\\Phi}{d\\varepsilon}\\right|_{\\varepsilon = 0} = \\int_{x_1}^{x_2} \\left.\\frac{dL}{d\\varepsilon}\\right|_{\\varepsilon = 0} dx = 0 \\, . ", "5f4ef77be1d52a389794d7b5b26f5353": "f_y = m \\gamma a_y = m_T a_y, \\,", "5f4f4a2107a33007efb3bf6dc9f5e145": "\n \\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\begin{bmatrix} \\varepsilon_{11} & \\varepsilon_{12} & \\varepsilon_{13} \\\\\n \\varepsilon_{12} & \\varepsilon_{22} & \\varepsilon_{23} \\\\\n \\varepsilon_{13} & \\varepsilon_{23} & \\varepsilon_{33} \\end{bmatrix}\n ", "5f4f9f15a8a1e16f4c018c3bac22456c": "3 \\Rightarrow (2, 1, 1)", "5f4fbe736cb9fb18c1ac145b5f00d877": "\n\\begin{align}\nI_2 I_3 \\ddot{\\omega}_{2}&= (I_3-I_1) (I_1-I_2) \\omega_1\\omega_{2}\\\\\n\\text{i.e.}~~~~ \\ddot{\\omega}_2 &= \\text{(negative quantity)} \\times \\omega_2\n\\end{align}\n", "5f4fe14757d036bc872aeb10ad78ab31": "\\left(1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ -\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "5f4fe71588d3b2e790447d64ffcf27c5": "\n\\ E_{in} = { \\ U_{emf} \\over \\ S }+ {\\sigma * \\ T^4}\n", "5f4ff653acf3e1597ca75a38ea5265c8": " \\frac{\\hbar C}{2 e} \\, \\frac{d^2 \\delta}{dt^2} + \\frac{\\hbar}{2 e R} \\frac{d \\delta}{dt} = I - I_0 \\sin \\delta", "5f50155838d23bd47c814f4c54b5f2af": "\\sum_{k=1}^n k^m z^k = \\left(z \\frac{d}{dz}\\right)^m \\frac{z-z^{n+1}}{1-z}", "5f505c431517c8190c9a0bd027deeeb5": "m = O( n^2 / k^3 + n/k )", "5f505f458f00bd66472369e7128361aa": "\\gamma=\\infty", "5f50f3bbb581bf83b5a3b45613d4ec20": "c(x,y)", "5f511cbbc9ec780a39df93b58bdec062": " E\\left( \\frac{ x }{ y } \\right) = E\\left( x \\frac{ 1 }{ y } \\right) = E( x )E\\left( \\frac{ 1 }{ y } \\right) \\ge E(x)\\frac{ 1 }{ E( y ) } = \\frac{ E( x )}{ E( y ) } ", "5f513c8f31317ab9000a953fcc7b8a73": "\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p", "5f517c1567321fb1fdde7f08e25eda76": "S\\circ R = \\{ (x,z)\\in X\\times Z\\mid \\exists y\\in Y: (x,y)\\in R\\land (y,z)\\in S \\}.", "5f51a6b26779e34fd5948147f9e0122d": "\\nabla^2 = \\nabla \\xi^m \\cdot \\nabla \\xi^n {\\partial^2 \\over \\partial \\xi^m \\partial \\xi^n} + \\nabla^2 \\xi^m {\\partial \\over \\partial \\xi^m }, ", "5f51d64a795a890a13b4f5ee7196f7c9": "\n f(k; r, p) \\equiv \\Pr(X = k) = {k-1 \\choose k-r} (1-p)^r p^{k-r} \\quad\\text{for }k = r, r+1, r+2, \\dots,\n ", "5f51e11ace48d40425ed1eff2271cf70": "\\displaystyle(a;q,p)_n = \\theta(a;p)\\theta(aq;p)...\\theta(aq^{n-1};p)", "5f520a445a3eee9b439ea321806c246c": "\nN(\\mathbf{p}) (Q^\\dagger(\\mathbf{p}))^m|0\\rangle \\;\n= \\left( m - {m(m-1) \\over \\Omega }\n\\right) (Q^\\dagger(\\mathbf{p}))^m|0\\rangle,\n", "5f5294e85a0f61a35eb23b53d235d2f4": "w-a=bu+(a-c)(v-1)", "5f52d3209671ed057e4ab998df05b16e": "+j", "5f532e915a542456f5d3ef54ecd38003": "\\beta > -2 ", "5f5346ff4478eb4c7f7932e4c1555114": "\\tau^n=\\tau \\circ \\tau \\circ \\ldots\\circ\\tau", "5f5358e5c38757097b3d2040715bf408": "\\|\\mathcal F\\|_{q,p} = \\sqrt{p^{1/p}/q^{1/q}}.", "5f536f039b9aef7bc67918adc9e147ab": "W=\\{x\\in R^n\\mid\\forall f\\in F\\,f(x)\\ge0\\},", "5f53da5b76e6129c104098c4d0245782": "B_{i} : \\mathbb{R}^{n} \\to \\mathbb{R}^{n_{i}}.", "5f5424f7b9a971f56d2787177a7d698e": "\\mathrm{O + O_2 \\longrightarrow O_3}", "5f545f56efbec8d7b523e28492db062e": " H \\oplus H \\rightarrow H \\oplus H ", "5f547f91f3af98b9ff645757def7054e": "\\mathrm{Res}(f')=0;\\,", "5f5497d5af58a34b49722bdcdb5b3b65": "\\lambda/\\pi", "5f54e4dbfcc33952a1ccd9e8dd3dae01": "(c)(-c+1+\\alpha +\\beta -1)-\\alpha \\beta )=0.", "5f54eca850f4e29088f7538354524976": "(-1) \\equiv 1 \\pmod 2.", "5f55099abda1c473827bbde5534c46f5": "T_d = \\sqrt{1 - r^2 \\omega^2/c^2}", "5f55217b14d5b722e0e52e727f9a44d9": "\\nabla\\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right) = -\\nabla' \\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right),", "5f554e0c9a14e40944dfd8f369918f36": "| \\psi \\rangle ", "5f5584e43ea1fa5b28187439af408bf6": "Q(I=5/2)=6\\left( 3 - \\frac{48(1+P^2)}{19+26 P^2+3 P^4} \\right)^{-1}", "5f55850cf447f847d126c9347b7b15b0": "\\int_{-\\infty}^{\\infty} e^{-ax^2}\\,\\mathrm{d}x=\\sqrt{\\pi \\over a} \\quad (a>0)", "5f558fa7e9b1567daca23dc3433f5cec": "r\\,", "5f55b598e7d709f632b8e2a745eadfbf": "f(|x|)", "5f55c2a7ab0ecf42dc42488b86aca538": "\\nabla \\times \\mathbf{B} = \\frac{4 \\pi \\mathbf{j}}{c}", "5f55c2f1b9467a7d90b767df0591df12": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = \\boldsymbol{R}\\cdot\\dot{\\boldsymbol{R}^T}\\cdot\\boldsymbol{\\sigma} + \n \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T \n", "5f55d614d1600f5d6fdd6ee72d646ee5": "\\sum_{s=0}^l{l\\choose s}(-1)^s=(1-1)^l=0\\quad\\mbox{for }l>0,", "5f563167b64529af9dada69114226b7d": "\n\\log \\pi=\\sum_{n=2}^{\\infty}\\frac{(2(\\tfrac32)^n-3)(\\zeta(n)-1)}{n}.\n", "5f569369b46d07ce59b79e4d0ccc98a7": "\\hat{n}", "5f577364ae6a3d6ee8e5731ca845c7da": "\n M_{\\alpha\\beta,\\alpha\\beta} - q = 0\n ", "5f57f9ea3b04c517e5af710d5cf42ca7": "\\Delta n", "5f584af2244e19e91cf218577cb9c331": "A^+\\,\\!", "5f585aaf9ab9d2cbf0bddcf7401eba64": " s^G=a ", "5f58880c77c189be54f20794b693fe82": "\\displaystyle{B(a,b)Q(a^b,c) + Q(a)R(b,c) =Q(a^b,c)B(b,a) + R(c,b)Q(a) =Q(a,c)}", "5f58ab02fa3b72ac3f7fb5b614f8ea23": " \\tau_n = o(h) ", "5f58b23951380f7f327ebe83d6c4031f": "Q_{\\rm es} = \\frac{2 \\pi\\cdot F_{\\rm s}\\cdot M_{\\rm ms} \\cdot R_{\\rm e}}{(Bl)^2}", "5f58c6b4f8afeb404a500ae1aa322cf6": "P_n^{(\\alpha, \\beta)} (-z) = (-1)^n P_n^{(\\beta, \\alpha)} (z);", "5f58eace6ec291b294d8f4f972ae686f": "c' \\equiv b^2 \\pmod p", "5f5947e3e03de22e47f3721745aaf4c2": "\n\\begin{align}\n& {} \\qquad \\frac{1}{c} \\int \\frac{ du} {u^2 + A^2} \\\\[9pt]\n& = \\frac{1}{cA} \\int \\frac{du/A}{(u/A)^2 + 1 } \\\\[9pt]\n& = \\frac{1}{cA} \\int \\frac{dw}{w^2 + 1} \\\\[9pt]\n& = \\frac{1}{cA} \\arctan(w) + \\mathrm{constant} \\\\[9pt]\n& = \\frac{1}{cA} \\arctan\\left(\\frac{u}{A}\\right) + \\text{constant} \\\\[9pt]\n& = \\frac{1}{c\\sqrt{\\frac{a}{c} - \\frac{b^2}{4c^2}}} \\arctan\n\\left(\\frac{x + \\frac{b}{2c}}{\\sqrt{\\frac{a}{c} - \\frac{b^2}{4c^2}}}\\right) + \\text{constant} \\\\[9pt]\n& = \\frac{2}{\\sqrt{4ac - b^2\\, }}\n\\arctan\\left(\\frac{2cx + b}{\\sqrt{4ac - b^2}}\\right) + \\text{constant}.\n\\end{align}\n", "5f5956df101d7f74981d54d6330a8fba": "\n \\sigma_{11}-\\sigma_{33} = 2C_1(\\lambda_1^2-\\lambda_3^2) - 2C_2\\left(\\cfrac{1}{\\lambda_1^2}-\\cfrac{1}{\\lambda_3^2}\\right)~;~~\n \\sigma_{22}-\\sigma_{33} = 2C_1(\\lambda_2^2-\\lambda_3^2) - 2C_2\\left(\\cfrac{1}{\\lambda_2^2}-\\cfrac{1}{\\lambda_3^2}\\right)\n ", "5f595f9e365abc5240e636f0c97d4fe0": "\\phi \\land \\lnot \\phi", "5f596397bfb905621141f043b11fbf5b": "\\left| f(z) \\right| = e^R.\\,", "5f597951e303c555de511cd79c15f2d6": "\\dot{W}^{r,p}(\\Omega) = \\{v \\in D'(\\Omega) : |v|_{r,p,\\Omega} < \\infty \\}", "5f5a46992ed4bbf8bc2f7341c06d79c0": "|E(G')| = |E(G)| - d(w) + d(u) > |E(G)|.", "5f5ac6934abbf5a26ebd171d7d392e12": "Z_n=\\frac{\\sum_{i=1}^n (X_i - \\mu)}{\\sigma\\sqrt{n}}\\,", "5f5ad33e8d4e43f7c69ee1aa9c61aeb1": " r_{out} = \\frac{1}{\\lambda I_D}", "5f5b131602c9a549d0ed3eefd9982a42": "\\frac{\\mu}{i\\alpha\\rho} \\left({d^2 \\over d z^2} - \\alpha^2\\right)^2 \\varphi = (U - c)\\left({d^2 \\over d z^2} - \\alpha^2\\right) \\varphi - U'' \\varphi", "5f5b2673f57756b01a7328f58621814b": "\\frac{1}{n+1}\\binom{2n}{n},", "5f5b391ef8af9e3c571b3b02b542243f": "\\langle v_ih_j\\rangle_\\text{data} - \\langle v_ih_j\\rangle_\\text{model}", "5f5b6fe9542e950042d28accda322b26": "O(|E|\\log(|V|))", "5f5b891d1464249332cf31598d9f7632": "VO_2 = \\frac {FR \\cdot (F_{in}O_2 - F_{ex}O_2)} {1 - F_{in}O_2}\n", "5f5bc2be54916f3673360c6ad372923c": "O(|E|\\log |E|)", "5f5bd9cfb19348e9b5d5e393ed760fe9": "\\mathcal{S} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int L \\, dt", "5f5c4bbeb3b93578a6569b91b404f6fe": "\\mathfrak{m}=(x_1,\\ldots,x_n)", "5f5cc0070548b78045007d6755a38dcd": "\\Omega =\\exp \\left[ \\Phi _N\\left( x^\\mu \\right) \\right].", "5f5d88a510fccf83ea87576a8ad1474f": " B = \\begin{bmatrix} A & I - AA^* \\\\ 0 & A^* \\end{bmatrix}.", "5f5da061f0a0a66fcac25365d2148aa9": "\\displaystyle{|\\partial_z H_f(0)|^2 - |\\partial_{\\overline{z}}H_f(0)|^2 = {|\\Phi_z(0,0)|^2 - |\\Phi_{\\overline{z}}(0,0)|^2 \\over |\\Phi_w(0,0)|^2 -|\\Phi_{\\overline{w}}(0,0)|^2} >0.}", "5f5e82ca26ec6568482375b2262cfc7d": "\\scriptstyle T_\\text{C}", "5f5eb28fa163b02f9126167bf45e4e8e": " \\oint_C {1 \\over z}\\,dz = \\int_0^{2\\pi} \\frac{1}{e^{it}} ie^{it}\\,dt = i\\int_0^{2\\pi} 1 \\,dt = [t]_0^{2\\pi} i=(2\\pi-0)i = 2\\pi i.", "5f5ed225e8723bcde038b3154a8ca791": "\n\\begin{align}\n\\operatorname{Var}(X)&=\\operatorname{E}\\left[(X - \\operatorname{E}(X))^2\\right]\\\\\n &=\\operatorname{E}\\left[X^2 - 2X\\operatorname{E}(X) + [\\operatorname{E}(X)]^2\\right]\\\\\n &=\\operatorname{E}(X^2) - \\operatorname{E}[2X\\operatorname{E}(X)] + [\\operatorname{E}(X)]^2\\\\\n &=\\operatorname{E}(X^2) - 2\\operatorname{E}(X)\\operatorname{E}(X) + [\\operatorname{E}(X)]^2\\\\\n &=\\operatorname{E}(X^2) - 2[\\operatorname{E}(X)]^2 + [\\operatorname{E}(X)]^2\\\\\n &=\\operatorname{E}(X^2) - [\\operatorname{E}(X)]^2\n\\end{align}\n", "5f5efc3e03fe65b99db6b4667ad6e26f": " \n0 \\prec A \\preceq H \\Rightarrow f(A) \\preceq f(H)\n", "5f5efc985e36b96563de2b4b1325a96b": "P_{-Q} = \\frac{1}{2}\\left(1 - Q\\right) = \\frac{1}{2}\\left(1 + \\gamma^0\\right)", "5f5f40e1e3f3e713db60d7550277b161": "\\operatorname{AMISE}(h) = \\frac{R(K)}{nh} + \\frac{1}{4} m_2(K)^2 h^4 R(f'')", "5f5f540578a79f2929203fe6a8551d4e": "h(x) = \\sum_{j, k=-\\infty}^\\infty c_{jk} \\psi_{jk}(x)", "5f5f787275ac2d5906d7f537255a46a6": "3\\ f\\ x = f\\ (f\\ (f\\ x)) ", "5f5fbc3f35412b13aed230ddfd8602ad": "n^{\\text{th}}, m^{\\text{th}}", "5f5fdb7c6d9df545ee5d0eec04aec8d1": "{ G_{\\mu \\nu} = 8 \\pi {G \\over c^4} T_{\\mu \\nu} } \\ ", "5f600694ab24b03976fabfab162b15f2": "P\\,=\\,1.854kd^{n-2}", "5f600d73f7b2a65e69e670aa85a29284": "y_{n+1}={1+{1\\over 2}hk \\over 1-{1\\over 2}hk}y_n. ", "5f60199a04399860f5484399b6b7e0db": "n=1,\\ 2,\\ 3,\\ ...", "5f6046fc7e86d26f826e67a83ba9e4e5": "\\mathbf{c} - \\mathbf{A}^T \\mathbf{y}", "5f605a1b7393acf4485a9b1093a221c8": "A'=A(a', \\lambda)", "5f6076f58ba95e12144f81f0759a3de6": "p^1 = p", "5f6110b71f34826c2fda169d822c9178": "z^{k+1}", "5f613e46f65d4132d5eaf2d887a2a700": "\\begin{align} \\\\\n c^2\\Delta t^2 &< \\Delta r^2 \\\\\n s^2 &> 0 \\\\\n\\end{align}", "5f61ea9cfd3c58da381156e1da69f4f9": "V = \\frac{4}{3}\\pi r^3.", "5f622106c3dff2bd4bdae15de3cb3439": "\\frac{dy}{dx}=\\tan \\varphi = \\frac{w}{T_0}x.\\, ", "5f622d306c74246e641030ed6bdf4cf0": "\\int_a^b|f(x)|\\,\\mathrm{d}x", "5f6354209882bf63f85b77ee8a8ccc97": "\\,W\\operatorname{E}[\\,g(Y_t,\\theta)\\,]=0", "5f6372ef0bb3c238971fa4c77cec1a1a": " \\mathcal{L}\\left\\{ { f'(t) } \\right\\} = s \\int_{-\\infty}^\\infty e^{-st} f(t)\\,dt = s \\cdot \\mathcal{L} \\{ f(t) \\}. ", "5f637d309e4c007546f22f486c5750f9": "\\tfrac{1}{p \\ q}", "5f6427a7fd5f4c801eb758a360a2af8e": " \\gamma_{\\rm{particle-matrix}} \\,", "5f644f91e49de1d870aae44e22014e90": "k=1,\\ldots,n-1", "5f65129cab9672382381b89bcea31d5f": "G(V,E)", "5f654fe74bc94045f7a7966e69adcba1": "\\left(\\frac{\\partial T}{\\partial P}\\right)_V \\left(\\frac{\\partial S}{\\partial V}\\right)_P", "5f655794f133ae1954bd1b2b71077a4a": "X_t \\sim I(d-1). \\,", "5f655e01f402fb274d9fb665b3b3415c": "x_{n+1} = x_n - \\frac{f'(x_n)}{f''(x_n)}, \\ n = 0, 1, \\dots", "5f65ec415aa8281f85d9df70bb57b23e": " \\vartheta(K_{n_1,\\dots,n_k}) = \\max_{1 \\leq i \\leq k} n_i ", "5f65fadb9b2d41fd5821cd308729df16": "{\\mathbb C}\\,", "5f6647b7ed6ec966dfdbdbace7d68218": "\\kappa_4=\\mu'_4-4\\mu'_3\\mu'_1-3{\\mu'_2}^2+12\\mu'_2{\\mu'_1}^2-6{\\mu'_1}^4\\,", "5f66a8a0cc13624d1fb7a61f6860377d": "p_c=\\frac{2cT}{a^2-b^2+c^2},", "5f66b3fea89a760df7a7819efca21de9": "Re[\\Delta R M (t)] = \\frac {dR} {dT} \\sum_{m=-M}^M (\\Delta T(m/\\tau + f) + \\Delta T (m/\\tau - f))exp(i2\\pi m t /\\tau)", "5f6700a7758284ad75e75b5f1da39763": " Z_{b}", "5f673a74cdc9f995a9ed8fadc4991f41": "\\forall N\\in\\mathcal{N}_{f(x)}\\exist M\\in\\mathcal{M}_x: M\\subseteq f^{-1}(N)", "5f673d708e2d57cded26f66f42f0e342": " y'(x)^2 = 4y(x) . \\,\\!", "5f678c8771eb8bc02ebbb1b93db356bd": " \\alpha(z), ", "5f67a501e74d1b62c5eb7a6c96d77be4": "Z_\\infty^{p,q} = \\ker(F^p C^{p+q} \\rightarrow C^{p+q+1})", "5f67b79eefacb4f0c542a19b440c4495": "\\hat{B} - \\sum_{i,j} | p_i \\rangle \\langle \\tilde{\\phi}_i | \\hat{B} |\\tilde{\\phi}_j \\rangle \\langle p_j | ", "5f687039303cdd1f5bd3a0e1ff122fbb": " \\nabla \\times \\mathbf{H} = \\frac{1}{c} \\left(4\\pi\\mathbf{J}_\\mathrm{f} + \\frac{\\partial \\mathbf{D}} {\\partial t} \\right)", "5f6874d6ac9d174fc28594d7c433c839": "\\begin{align}\nr_1 &= \\alpha + \\beta\\\\\nr_2 &= \\alpha - \\beta.\n\\end{align}", "5f68d8de753012699d3416eb7b01eca3": "{\\hat c} = cP^{-1}", "5f68ff28a090ba4668878d58b4a455ce": " q _{v \\cap w} \\stackrel{\\mathrm{def}}{=} | A _{S(v) \\cap S(w)}|", "5f690d6ecaca6f93cf2bbe22b93ceea4": "\\exists_f\\colon \\mathcal{P}X\\to \\mathcal{P}Y", "5f698a7fce883a72f3ad6cd2a01913e2": "x^2 + bx + c,\\,\\!", "5f699bf434ef4b6ee0586efb95998190": "(D^k f)(0) = k! a_k, ", "5f69a3d2c541d6a7e4e0acb71b9b24da": "a_{i}^{n + 1}(u)\\,", "5f69a6e1a86937667f5c3e26746c4933": "\\mu(\\cdot, \\omega) : \\mathcal{R}^1 \\to \\mathbb{R}", "5f69bdc82de13e6f9376ab541f8f133f": " n\\in\\mathbb{N}_0", "5f69c7802c227b6c11602f86547ef2fa": "\nX_1 \\approx \\sqrt{R_1 R_2} \\,\n", "5f6a094ae5b7f4a03acc46f0f8868ec6": "\\left . \\begin{array}{rcl} A^2 (1-\\omega^2)^2 & = & \\cos^2\\phi \\\\[6pt] (2 \\zeta \\omega A)^2 & = & \\sin^2\\phi \\end{array} \\right \\} \\Rightarrow A^2[(1-\\omega^2)^2 + (2 \\zeta \\omega)^2] = 1. ", "5f6a292a60d11ac93192fece04c53090": "\\frac{1}{2}mv^2", "5f6a2b54708f721bfc245e8963c6e1af": "\\begin{align}dy_{\\text{1}}\\ =\\ I_{\\text{1}}dt\\ +\\ cdW_{\\text{1}}\\\\\ndy_{\\text{2}}\\ =\\ I_{\\text{2}}dt\\ +\\ cdW_{\\text{2}}\\end{align},\\quad y_{\\text{1}}(0)\\ =\\ y_{\\text{2}}(0) = 0", "5f6a3573c611f963a93fde23e105051e": "A=\\lfloor\\frac{p}{2k}\\rfloor", "5f6a358bf35a0a4c2085ebe06f3f6079": "\n \\sum_{a^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}}}\\Pr\\left\\{E_{a^{n}}\\right\\} \\geq 1-\\epsilon,\n", "5f6a6f9847cd8d6d17dd7d1c9aa8716b": "(3)\\qquad \\langle \\bar T T\\rangle_{ETC} = \\exp{\\left(\\int_{\\Lambda_{TC}}^{M_{ETC}} \\frac{d\\mu}{\\mu}\\gamma_m(\\mu)\\right)}\\,\\langle \\bar T T\\rangle_{TC}\\,,", "5f6a849198ff8efef8c2839b3678e335": "LR ( k )", "5f6a9c573941f81e2956e9d06a1e684f": "f : X\\to Y", "5f6aacca18205e228957e4e1c09bbf1b": "\\binom{r+(m+n-r)}{r}=\\binom{m+n}{r}", "5f6abb0686d469d2614aa05dfead35dd": "\\xi^{(a)}_i", "5f6ad616f37d562acf352eb784240615": "-c^2 \\frac{dM}{dt} = \\frac{K_{\\operatorname{ev}}}{M^2} \\;", "5f6aec8b1b5666ec650f83e2e678db2a": "\\alpha \\in {\\mathcal O}_K", "5f6aecf7369cb39a922c6936ab6b9a61": "X = \\mathbf{E}^{3} \\setminus \\{ (x, y, z) | x > 0, y > 0 \\text{ and } z > 0 \\}", "5f6af3c418723e607142d8578902770c": "R(n_1,\\ldots,n_l,m_1,\\ldots, m_k)\\,", "5f6b3191734c20f1e8a7fc91236b0228": "\\beta=\\gcd(\\alpha_i,\\alpha_{i+1})", "5f6b755b2e797ccf1a85a948bf3b6ed9": " V_{rect}=\\begin{bmatrix} T1 & 0 \\\\ 0 & T2 \\end{bmatrix}", "5f6ba9e29a7f12d3096b885b5b7c0875": "\\underline{\\ell} = 2", "5f6bb37f801640eb5dffc457b49bab0e": "P_{ij}=\\rho\\langle (w_i-V_i) (w_j-V_j) \\rangle", "5f6bf0e5223737187f62d0828b97825a": "I(\\tilde{\\nu}) = 4 \\int_0^\\infty [I(p) - \\tfrac{1}{2}I(p=0)] \\cos (2\\pi\\tilde{\\nu}p) dp. ", "5f6c0c9c925561ef0ac7031c8079edf9": "-\\frac{5}{9}X^4+\\frac{1}{9}X^2-\\frac{1}{3},", "5f6c10a8369c74dd1d4a3ba2e8134471": " \\mu(\\mathbf{x}, \\sigma_{\\mathit{I}}, \\sigma_{\\mathit{D}})", "5f6c25a25db81486c3c5a40ffa1bf110": " \\sim ", "5f6c327a273b7acf7473c18089901871": "C_c^\\infty(K_i)", "5f6ca18fb767f9493717b93806db129f": "\\scriptstyle\\sqrt{\\text{slope of line RH}}", "5f6ca2dd139921864d62df1489bed75a": "T\\varphi = \\frac{\\partial\\varphi}{\\partial x_k}.", "5f6cae8fbeb6a3c2959acf62592ea1b7": "p\\ f\\ x = f (x\\ x) ", "5f6d1884530edbd93d987944863550cd": "p_{\\mathrm{ref}}", "5f6d1c9d0b4345ab80593b33e13ec220": "\\mathfrak{P}^{10}", "5f6d2ebba5917e392958c54afd885832": "H(m)", "5f6da9c025cf60c909918c7fd904e868": "\\eta_c = \\sum_{i=1}^N k_i \\eta_{c_i}", "5f6dee2c4ec4672952c72cea910ad279": "\\sqrt{swap}", "5f6df276d86727175059d62f489b76ce": "\n\\{ (1,[0;0.2]),(2,[0.3;2]),(3,[0.3;2]),(4,[0.1;0.2]),(5,[0.4;2]),(6,[-1;0.1]) \\}\n", "5f6e53e07f6c88858b3fdc36fefe1bb5": "A' =S^\\mathsf{T} A S .", "5f6e7b79eb302b2740c2edccdc7ebd2b": "g(\\exp X)g^{-1} = \\exp(\\mathrm{Ad}_gX)\\,", "5f6ea42f17a0e44a928081d24ceca175": "F(\\nu) = \\sum _ {k : \\nu_k < \\nu} \\frac 12 (a_k^2 + b_k^2).", "5f6eaee6b0b830ab9aeb458052c28463": "(2k-1)!! = \\prod_{i=1}^k (2i-1).", "5f6ec72155d79d6cb1e3584090f3b5fb": "\n (9) \\qquad \n \\cfrac{J}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial t^2} -\\cfrac{mJ}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial t^4} + J~\\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} \n= -\\cfrac{mEI}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} + EI~\\cfrac{\\partial^4 w}{\\partial x^4} + \\cfrac{EI}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial x^2}\n + m~\\frac{\\partial^2 w}{\\partial t^2} - q\n", "5f6ee771aa69c5a03e3de67bdbc0b694": "x_i = \\frac{n_i}{n_{\\rm mix}} ", "5f6efdf6ba553713e827228bd4205439": "L_{\\mathrm{gap}}\\,", "5f6f0a78c3c008fe29af91cafaf0b719": "\n\\sum_{n=1}^{\\infty}\\frac{1}{n^2} = \\frac{\\pi^2}{6}.\n", "5f6f341f7aa22d8ba21c1a2962fa96f5": "\\frac{\\Delta \\alpha}{\\alpha} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\alpha _\\mathrm{prev}-\\alpha _\\mathrm{now}}{\\alpha_\\mathrm{now}} = \\left(-5.7\\pm 1.0 \\right) \\times 10^{-6}.", "5f6fa93f4cfe21f1a45129a2ad918541": "0 \\;\\rightarrow\\; A^{[d]}\\; \\xrightarrow{f}\\; A \\;\\rightarrow\\; A/f\\rightarrow\\; 0\\,,", "5f6fc57145d1c8e9833c9f82145a2d93": "X^{st} = \\begin{bmatrix}A^t & (-1)^{|X|}C^t \\\\ -(-1)^{|X|}B^t & D^t\\end{bmatrix}", "5f6fd22567b26e9c1a1ae9c6eeb3f785": "\\bar{n}", "5f7032e17385deec8294ff079528ceb6": "f \\lambda_{\\epsilon,g}=\\lambda_{\\epsilon,g}+\\mathit{1}", "5f703e849c5b2097ada9041615384fb9": "X = \\sigma_1 = \n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix},\\quad\nY = \\sigma_2 = \n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix},\\quad\nZ = \\sigma_3 =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}", "5f7094c4370b801d33193f2b60894e61": "\\mathbf{P}_{X|z}", "5f71239a0135bf06b8099008003762d8": "PR = \\cos \\alpha \\sin \\beta\\,", "5f712f954d80a9258bca36b46867399c": " \\bar{X}_N ", "5f719a59d4349dab7df6c1c632d79e1f": "\\lang k^{(0)}|", "5f71ab165a87e33a08f5965514138eb1": "S=k_B \\ln \\Omega_{\\rm mic} = k_B (\\ln Z_{\\rm can} + \\beta \\bar E) = k_B (\\ln \\mathcal{Z}_{\\rm gr} + \\beta (\\bar E - \\mu \\bar N)) ", "5f71d3fba4b9bc0fd50874adbe7dbb5b": "\\lfloor n/3\\rfloor", "5f71e3fca565ef0e1ef0b96cf23763d7": "\ne \\times e = 1, \\,\\,\\, m \\times m = 1.\n", "5f71f4e2a4a3136e9cec47905e114160": "\\scriptstyle{\\bar{s}_\\tau}", "5f720f22d6d9ede5719bd57831890385": "-\\frac{R_1}{R_1+R_2}{V_s}", "5f7255e936ee9318e14ea07500b81e23": "\\mathfrak{a}.", "5f72b3bd452df55b07b02be630bc8e52": "\\begin{array}{cccc}\n & & \\begin{array}{ccccc}\n\\sigma_{1}'=+1 & & & & \\sigma_{1}'=-1\\end{array}\\\\\nA & = & \\left[\\begin{array}{ccccccc}\n & & & |\\\\\n & A_{+} & & | & & 0\\\\\n & & & |\\\\\n- & - & - & | & - & - & -\\\\\n & & & |\\\\\n & 0 & & | & & A_{-}\\\\\n & & & |\\end{array}\\right] & \\begin{array}{c}\n\\sigma_{1}=+1\\\\\n\\\\\\\\\\\\\\sigma_{1}=-1\\end{array}\\end{array}", "5f72b48df5a5f2e553e20430c9638d98": "18 / 17 \\approx 99.0 \\text{ cents,}", "5f72dd06dd744775ba0ebe4259878208": " \\mu(\\bigcup_nU_n ) = \\oplus_n\\lambda(U_n)", "5f7343b2ec2afb9b02c096dcd6b984fb": "\\Delta a_T", "5f736199b34d1bc94979b6d44e6c0f2d": "{A_{initial}}", "5f7373b5126595a57b89f42b601d6d5e": "H(W) = -\\sum_{j=1}^n w_j \\ln (w_j).", "5f739be5901bde557ae71c14e4137718": "p_{k,i}^{\\mathcal M}", "5f73b512f983e588219c29468d6158dc": " \\ J0: (())A = A.", "5f73ffab546b49f50131f664f80ffbe7": "\\operatorname{length}(E) = -\\log_2(P(E))", "5f74643bfd384dd8d6062878e1df0bcd": "\n\\begin{array}{rcl}\n R^2 \n &=&\n \\displaystyle \\frac{1}{t^2} \n \\left( \\begin{array}{cc} (A + s)^2 + B C & (A + s)B + B(D + s) \\\\ C(A + s) + (D + s)C & (D + s)^2 + B C \\end{array}\\right)\\\\[3ex]\n {}\n &=&\n \\displaystyle \\frac{1}{A + D + 2 s} \n \\left( \\begin{array}{cc} A(A + D + 2s) & (A + D + 2s)B \\\\ C(A + D + 2 s) & D(A + D + 2 s) \\end{array}\\right) \\;=\\;\n M\n\\end{array}\n", "5f74bb9265455295d533511b035faa00": "D = N - K + 1 ", "5f74bfa9a7b4b78ed6d18fbd49d0a889": "~ \\leftrightarrow ~", "5f74d4f1f15a8ac8b6f74407d9b02adf": "\\frac{f_1}{f_0} = \\cfrac{1}{1 + \\cfrac{k_1 z}{1 + \\cfrac{k_2 z}{1 + \\cfrac{k_3 z}{1 + {}\\ddots}}}}", "5f753fc85828c3a1d95572e0fc00161e": " e^{i \\sigma_z \\omega_r t/2}\\sigma_y e^{-i \\sigma_z \\omega_r t/2} = \\begin{pmatrix}\ne^{i\\omega_r t/2} & 0 \\\\\n0 & e^{-i\\omega_r t/2} \\end{pmatrix}\n\\begin{pmatrix}\n0 & -i \\\\\ni & 0 \\end{pmatrix}\n\\begin{pmatrix}\ne^{-i\\omega_r t/2} & 0 \\\\\n0 & e^{i\\omega_r t/2} \\end{pmatrix}=\n\\begin{pmatrix}\n0 & -i e^{i\\omega_r t} \\\\\ni e^{-i\\omega_r t} & 0 \\end{pmatrix}\n", "5f75d16406681defb1d272b4f95b8204": "\\Psi(t) = - \\log( \\pi ) + Re(\\psi(1/4 + it/2))", "5f76a64cedf54b6c5e7bb8f7b6aedf5c": "\n \\cfrac{\\partial\\mathcal{L}}{\\partial w} = q ~;~~ \\frac{\\partial \\mathcal{L}}{\\partial \\dot{w}} = \\mu\\dot{w} \n ~;~~ \\frac{\\partial \\mathcal{L}}{\\partial w_{xx}} = -EI w_{xx} ~.\n ", "5f77307497464026ce721d18e3ab7b41": "\\rho\\, g\\, a\\, \\frac{\\cosh\\, \\bigl( k\\, (z+h) \\bigr)}{\\cosh\\, (k\\, h)}\\, \\cos\\, \\theta\\,", "5f77702d8b80e342b85a7c6db5c5db18": "\\lg l = \\lg \\lg^2 n = 2 \\lg \\lg n", "5f77a1288672c55a53a1c1676f015f49": "0 < v(x) < 1", "5f77da2ad284f9ae53b6b0076ae8195e": "P(A_i | n_1,\\ldots,n_m, I_m)={n_i + 1 \\over n + m}. ", "5f77f27c740a4d42b1b2ae0501f91732": "(0,1)\\cup(1,2)\\cup\\{3\\}\\cup\\bigl([4,5]\\cap\\Q\\bigr),", "5f77faebb6f780030a6b926bbd0f81e6": "x_1=T(x_0,a_0)", "5f781fa8a5372e7598e4a3f60fcca445": "(8)~~~~~\n \\left(\\frac{\\partial z}{\\partial y}\\right)_x\n = -\n \\left(\\frac{\\partial z}{\\partial x}\\right)_y\n \\left(\\frac{\\partial x}{\\partial y}\\right)_z \n", "5f782b4ced7651ed8bbbbc5b210fd249": "a^s \\leq b^t \\leq a^{s+1}. \\qquad(1)", "5f784874c092bf4be7a7088803ad634b": "{\\partial c_i\\over\\partial t}+\\overline{u}_j{\\partial(c)\\over\\partial x_j} = {\\partial\\over\\partial x_j}\\bigg(K_{jj}{\\partial (c)\\over\\partial x_j}\\bigg)", "5f786c53b29f5dae5be4462e90152d7e": "t_n=0", "5f788d20455a1b8adab1b8cc1630e757": "\\vartheta^{\\parallel}", "5f789a53f5196c9c710c078846857bdc": "L_{E/K}^{\\mathrm{Artin}}(\\sigma, s) = L_{K}^{\\mathrm{Hecke}}(\\chi, s)", "5f78ce69cb0784c8623006b63b0a2930": "[\\phi,\\phi]= \\tfrac12 D\\langle \\phi,\\phi\\rangle", "5f7908af1311995b02bf978d70a91d54": "r=\\frac{h}{3}", "5f7911cc4566a218eee80670c2b028ea": "\\dot{\\vec{u}} = \\vec{f}(\\vec{u})", "5f795bf9506df344c0033a140f89bc39": " u_t=\\frac{3}{2}uu_{x}.\\qquad (4)", "5f7974f6b22ae43b20099b79c8f69000": "\\frac{1}{N}\\sum_{i=1}^N \\ln \\frac{\\hat{c} - Y_i}{\\hat{c}-\\hat{a}} = \\psi(\\hat{\\beta})-\\psi(\\hat{\\alpha} + \\hat{\\beta})= \\ln \\hat{G}_{1-X}", "5f79a731cc492f902c5b3a04d3b1c047": "E\\langle contigs \\rangle = N e^{-R}.", "5f79b938333cadf372237d82797af49d": "\\check{H}", "5f79c92e588d45d4035ea9ccb33cb5a8": " \n\\sum_j {T_{ij} = T_i } ;\n\\sum_i {T_{ij} = T_j } \n", "5f7a169b1c35dd4cd268d0693a82589f": "\\scriptstyle M_{V_{\\ast}}=6.63", "5f7a52280dd8e89f111517b8e12016c7": "q^b", "5f7a66b74b2610b03d6f87474c6c1860": "e^{-yD}f(x)=f(x-y)", "5f7ae208e13d57412dac72100f5b599f": " B_n = \\sum_{N \\ \\text{node of tree-level}\\ n} \\frac{n!}{N!}. ", "5f7aefab6545f6dcdc58cc9a6663e471": " \\theta \\, ", "5f7b0bdca7aa9eb55986c6e499a1f19c": "M_{2} =m_{2} \\cosh \\mathcal{L} +m_{1} \\sinh \\mathcal{L}, ", "5f7b237f462931a7016bd44757d85a4d": "\n \\delta U = - \\int_0^T \\left\\{\\int_{\\Omega^0} \\left[N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta} \n + M_{\\alpha\\beta,\\beta\\alpha}~\\delta w^0\\right]~\\mathrm{d}A\n - \\int_{\\Gamma^0} \\left[n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta} \n + n_\\alpha~M_{\\alpha\\beta,\\beta}~\\delta w^0\n - n_\\beta~M_{\\alpha\\beta}~\\delta w^0_{,\\alpha}\\right]~\\mathrm{d}s \\right\\}\\mathrm{d}t\n", "5f7b32ad1f865121c26339252d327f21": "\\chi_i(t) = \\frac{ \\langle \\mathbf{e}_i'(t), \\mathbf{e}_{i+1}(t) \\rangle}{\\| \\mathbf{\\gamma}'(t) \\|}", "5f7b95678c1085760d61bc93b3390d38": "\\theta(x)\\big|_{x=-j\\infty} = \\angle(x-r_1)\\big|_{x=-j\\infty}+\\angle(x-r_2)\\big|_{x=-j\\infty}+\\cdots+\\angle(x-r_n)\\big|_{x=-j\\infty} = -\\frac{\\pi}{2}N+\\frac{\\pi}{2}P \\quad (14)\\,", "5f7bb6f61e0efd27d2a5c2c761f54199": "c_i = \\langle f_i | \\psi \\rangle", "5f7c44893e8560750a482662f327116d": "\\theta = \\arccos \\left(1 - {2x \\over L}\\right)", "5f7c9db92078e97ada5ba4662fb0848f": "\\left|S_{12}\\right|\\,", "5f7ca7eb132029f2d046e182272193c3": "B(\\cdot)", "5f7d2c41ee1dd76447fa923f3d7984a1": "|{\\psi_{D}}\\rangle = \\sqrt{1-\\epsilon_{n+1}}|{{{\\psi}'}_{D}}\\rangle + \\sqrt{\\epsilon_{n+1}}|{{\\psi'}^{\\bot}_{D}}\\rangle", "5f7d77a43aa5cde6ee3875295ea5d8b4": "9 \\times 6 = 4", "5f7db9864a64190724d73b179bfcc4ff": "\\gamma \\approx \\sqrt {i\\omega CR}", "5f7e0d34f7a0e2bfd80ea8dfc8f56945": "-\\sin x \\, ", "5f7e3ddaf8696b6293ce8df5bb295601": " 3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y = 0 ", "5f7e408a5a6f9d981a20a2a9fb35b50d": "L_{*}(H)", "5f7e93d29b0e776df70bd96f622683a9": "I(t) = I_0 \\cdot e^{- {R_1 \\over L} t} - {1 \\over R_1} V_D", "5f7e97b4c8578e714ad01605abb5bd67": " \\frac{P(\\mu,\\nu)}{P(\\nu,\\mu)}=\\frac{g(\\mu,\\nu)A(\\mu,\\nu)}{g(\\nu,\\mu)A(\\nu,\\mu)}= \\frac{A(\\mu,\\nu)}{A(\\nu,\\mu)}=\\frac{P_\\beta(\\nu)}{P_\\beta(\\mu)}=\\frac{e^{-\\beta(H_\\nu)} /Z}{e^{-\\beta(H_\\mu)} /Z}=e^{-\\beta(H_\\nu-H_\\mu)}.", "5f7eb394522fb618cf132f4b09d01143": "\\bmod v", "5f7eddb78b4580ed3d16190ba491affc": " \\frac{\\partial C_1}{\\partial t}=\\frac{\\partial}{\\partial x} [D_1 \\frac{\\partial C_1}{\\partial x} -C_1 \\nu]", "5f7f0b9144c119d3e726cc15902c8fb3": "\\mathcal{P}(\\Lambda \\times \\pi_2)", "5f7f366945134d8e68935822be95ffb4": "\\hat{\\omega}\n_{i}(t,\\omega)", "5f7f4e397e85e5a0df5fe60d10be318c": " \\|\\vec{n}\\|_1=\\|n_1\\|+\\cdots +\\|n_d\\|. ", "5f7f823f7ed84addea3b9c7ac6c3e190": "\n\\varphi _j^1 = \\left( {j - \\sigma - {1 \\over 2}} \\right)^2 - {1 \\over 4}, \\quad \\varphi _j^0 = \\left( {j - {1 \\over 2}} \\right)^2 - {1 \\over 4}. \\quad \\quad (14)\n", "5f7f890041bf0620470203c2ef678040": "\\sigma_{m} = \\sigma_{D} = \\sigma_{s_2}", "5f7fa05fad61ee9da21f65d9dbbb4727": "\\mathcal{E}_\\lor", "5f7fbf92a59803743ad416973b0359e7": "\\mathrm{Al}^\\times_\\mathrm{Al}", "5f7fd56b8a55d58fc8b7c13eab806485": "A = SDS^{-1} = SDS^T =\n\\begin{bmatrix}\n1/\\sqrt{2}&1/\\sqrt{2}\\\\\n-1/\\sqrt{2}&1/\\sqrt{2}\n\\end{bmatrix}\n\\begin{bmatrix}\n1&0\\\\\n0&9\n\\end{bmatrix}\n\\begin{bmatrix}\n1/\\sqrt{2}&-1/\\sqrt{2}\\\\\n1/\\sqrt{2}&1/\\sqrt{2}\n\\end{bmatrix}.\n", "5f7fe7066967497995739c1fa317107a": "g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}\\,\\!", "5f80088361ef6c5c7435ce8c87687eec": "\\operatorname{Cov}(X,Y) = \\displaystyle\\frac {\\sum_{i=1}^n (x_i - \\bar x)(y_i - \\bar y)}{n}. \\!", "5f8033b979fe7a220e6510d3a6e68ded": "e_2(\\tau) = -\\tfrac{1}{3} \\pi^2(\\vartheta^4(0;\\tau) + \\vartheta_{10}^4(0;\\tau)),", "5f805c865c91779c3ef131601ddbcd27": "red : \\mathbb{R}/R\\mathbb{Z} \\to X, \\; v \\mapsto d^{-1}(v - \\inf\\{ f \\ge 0 \\mid v - f \\in d(X) \\})", "5f8135f56be52619d6ce51bd69df6676": "\\Omega_r", "5f8171a1260236c6ec3affc8f175167f": "\\pi _T < 0 ", "5f818b9922f7b25ca862bb5f90e9b5e0": "F(z) = U_0\\, \\text{e}^{(-1+i)\\, \\sqrt{\\frac{\\Omega}{2\\nu}}\\, z}.", "5f819a3457599dbebf9712856e5240bf": " dv(\\bar{X})", "5f81a8bf18d2d78b17293d3c05908ac6": "M \\approx |na+ma_D| \\,", "5f81edb5a76fdf61a1e977458fc08e66": "\\,\\operatorname{cr}(z_1,z_2,z_3,z_4)=1- \\operatorname{cr}(z_1,z_3,z_2,z_4)", "5f81fabb8087d805e9fdd4795b135bcc": "L_{k}", "5f825c5dc72854d4a171ae9574b2ff75": "\n2^{\\omega(n)}\\le d(n)\\le2^{\\Omega(n)}.\\;\n", "5f825e0b65099ee3b25d092e21abe566": "<\\phi , \\psi>_{Phys} = \\lim_{T \\rightarrow \\infty} <\\phi , \\int_{-T}^T dt e^{i t \\hat{M}_E} \\psi>.", "5f826ac71e189887918abdac7bd3f69b": " f'(5) = \\lim_{h \\to 0} \\left( \\frac{1}{5} - \\frac{h^2}{3 . 5^3} + \\frac{h^4}{5 . 5^5} -\\frac{h^6}{7 . 5^7} + ....... \\right) = \\frac{1}{5}", "5f82dcc04444c7e8ff459dc57d6960e3": "x \\, \\bot \\, y", "5f8302d3c27f5efe57c75c18dcc08f43": " \\frac{1}{S_{t}}\\ ", "5f830a1d0cde63a20ec9393aed9037b4": "\\sigma \\models_K f", "5f833855720a66a7374cf6443d638132": "\\sigma=\\frac{1}{\\rho} = \\frac{J}{E}. \\,\\!", "5f83855c8ad509bc1677d537aea43862": "\\left[ \\begin{matrix} \\exp \\left(\\frac{\\beta}{2}\\right) & 0 \\\\ \n 0 & \\exp \\left(-\\frac{\\beta}{2}\\right) \\end{matrix} \\right] ", "5f83ca31d00cccbd4763141608f1d818": "\\mu' B/\\hbar", "5f841870da96344ae4ae9727b1575e03": "(1-z)^{-u}\n= \\sum_{k=0}^\\infty u^k \\sum_{n=k}^\\infty \\frac {z^n}{n!} \\left[{n\\atop k}\\right] = e^{u\\log(1/(1-z))}", "5f84951df0e091210a87b1491d09d770": "\nU_5=\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 \\\\\n0 & 1 & 2 & 3 & 4 \\\\\n0 & 0 & 1 & 3 & 6 \\\\\n0 & 0 & 0 & 1 & 4 \\\\\n0 & 0 & 0 & 0 & 1\n\\end{pmatrix};\\,\\,\\,", "5f84a0a68b9ab185b8411f83bfeacfe6": "\\sum F_x=0=-F_{AD}\\cos(60)+F_{BD}\\cos(60)+F_{CD}=-\\frac{10}{\\sqrt{3}}\\frac{1}{2}+\\frac{10}{\\sqrt{3} }\\frac{1}{2}+F_{CD} \\Rightarrow F_{CD}=0", "5f84e151c5b0cd64ac8447cb574a6d61": "A\\in K\\backslash\\{-2,2\\}", "5f84e3dc6967c1a91547a57bea1185f5": "\\log_2", "5f851d9e2f46e70da86e1448a0a950d6": "\\binom{b_1}{b_2}=x_1\\binom{a_{11}}{a_{21}}+x_2\\binom{a_{12}}{a_{22}}", "5f852451fe443291e780246345d4fa93": "\\nu_T \n= (C_s \\Delta_g)^2\\sqrt{2\\bar{S}_{ij}\\bar{S}_{ij}} \n= (C_s \\Delta_g)^2 \\left| S \\right|\n", "5f854c83e35bafbabe827fcd51adf494": "x^2\\,", "5f85776ebe3663563724a24f58a9c56d": "\\det A = (\\det L) (\\det U)", "5f858a7de154db1c578f6e5e7ab061d4": "E_{n,m}", "5f8600fb70e114d3e56541a51b74974f": "\\frac{k\\left(x^\\prime -jd\\right)^2}{z}", "5f860d2bc73d0294c45bbc2feb9b30c4": "P(r \\ge k ; n, p)", "5f860eb7c9b9795dc36a9d39b4731ec9": " \\varepsilon(\\alpha_i) = \\left\\langle \\psi(\\alpha_i)|H|\\psi(\\alpha_i) \\right\\rangle. ", "5f8614a42d898b23d6906727d7c59dd0": "x=\\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}", "5f861831116bd5df794413634f4ca602": " E_t ", "5f862eb51df761e902ab16c4ecf4560b": "\\displaystyle \\operatorname{gcd}(\\{\\omega_i\\}) \\displaystyle =\\prod_p p^{\\inf(v_p(\\omega_i))}", "5f8694ee291048b4b26128a3c07fc2c6": "f_{n,-n}(r)=AZ\\alpha\\rho^\\gamma e^{-\\rho/2}", "5f873a0e6209a85b006ddd5160377865": "\n\\begin{align}\n\\delta S[g]&= \\int {1 \\over 2\\kappa} \\left(\\delta f(R) \\sqrt{-g}+f(R) \\delta \\sqrt{-g} \\right)\\, \\mathrm{d}^4x \\\\\n &= \\int {1 \\over 2\\kappa} \\left(F(R) \\delta R \\sqrt{-g}-\\frac{1}{2} \\sqrt{-g} g_{\\mu\\nu} \\delta g^{\\mu\\nu} f(R)\\right) \\, \\mathrm{d}^4x \\\\\n &= \\int {1 \\over 2\\kappa} \\sqrt{-g}\\left(F(R)(R_{\\mu\\nu} \\delta g^{\\mu\\nu}+g_{\\mu\\nu}\\Box \\delta g^{\\mu\\nu}-\\nabla_\\mu \\nabla_\\nu \\delta g^{\\mu\\nu}) -\\frac{1}{2} g_{\\mu\\nu} \\delta g^{\\mu\\nu} f(R) \\right)\\, \\mathrm{d}^4x \n\\end{align}\n", "5f8794ac85d2ba05a1ecf5229d0d1009": "\\text{Alternatively, choosing base volts and base kva values, we have,}", "5f879bd9c0687bb991469c61ffef6f4a": "\\left(q+N^{\\prime}-1\\right)!", "5f880bbc8e8a3c389b7765f8daa97519": "\\iiint", "5f88696d5f6d5706e22a9dea46201767": "\\scriptstyle\\mathbb{C}", "5f88a76871287a6139c26fceef76900e": "\\int_UL(Dw)\\mathrm{d}x", "5f88d2b59215095220a9cc36413c609f": " x_1,\\ldots,x_{2n} ", "5f88db18cfe077944606e76c74a64bdf": "(\\cos(t),\\; \\sin(t))\\quad\\mathrm{for}\\ 0\\leq t < 2\\pi.\\,", "5f89179a5f83ae624a58fa8078fb98ac": "q = \\ell + 2\\ell^5 + 15\\ell^9 + 150\\ell^{13} + 1707\\ell^{17} + 20910\\ell^{21} + 268616\\ell^{25} + \\cdots.", "5f897eb59f97fb4fdaa98eea7f990ab2": "(M_1, V) \\# (M_2, V)", "5f898379773b8cbbe973d009c58ebc9d": "A_d", "5f8988778aaa8e92ffea2e1356b19631": "\\left(Q_2-Q_1\\right)", "5f898ee93c4bce0c0965dc24fa9ab540": "\\max\\{\\, w_{ij} |f(v_i) - f(v_j)| : v_iv_j \\in E \\,\\}", "5f899fc42f2b3f3e44a50639c4493142": "x\\vee y = x", "5f89b78af2e3bc8f17b2a468dea2a2e1": "\\textstyle Z_{N} = \\int \\cdots \\int \\mathrm{e}^{-\\beta U_{N}} \\mathrm{d} \\mathbf{r}_1 \\cdots \\mathrm{d} \\mathbf{r}_N", "5f8a02830031607ec6053e35efb26f45": "E_{in}=f_\\odot (1-\\alpha)\\pi R^2_\\oplus", "5f8a48661511ea8f943df6f4f95fca2d": "g = \\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}\\in SL_2(\\mathbb{R})", "5f8a98d0f09049653f8d3beffe6269ef": "r_{xy}^2", "5f8ade87e95840010570ba3db808d97a": "(\\alpha,\\beta,\\gamma)", "5f8b5b169f94fee686b7b4804478ea8b": "\\mbox{CNR}= F+P-B-K-L ", "5f8b65aaaccfa23fffe4769708dde46b": "\\xi\\mapsto\\varepsilon_\\xi", "5f8b686d0e8117d2cd0ba9e34b883590": "f'' = (f')'\\!", "5f8bac187cbc3a7a2bfec2178b03ee1a": "c_i^{\\dagger},c_i", "5f8bc8f5405f8cf2e5559bc1d6f21b34": "\\tau_{ij} = \\frac{1}{\\Omega} \\sum_{k \\in \\Omega} \\left(-m^{(k)} (u_i^{(k)}- \\bar{u}_i) (u_j^{(k)}- \\bar{u}_j) + \\frac{1}{2} \\sum_{\\ell \\in \\Omega} ( x_i^{(\\ell)} - x_i^{(k)}) f_j^{(k\\ell)}\\right)", "5f8c05dada2a96567383abc837aeb1c5": "S^1 \\times \\mathbb{R}^2", "5f8c14545f100fd3262f82f50cbdf00f": "a=b+c\\,", "5f8c319e7b1d0427935432993cf610de": "\\hat{\\alpha}=\\hat{\\beta}", "5f8c5719ddab9047d23c3a7b2675a861": "X_{C}", "5f8c9b7b6680cf2ad90c69fe46d6b849": "\\theta = k_B T/mc^2 \\,\\!", "5f8ca0ccf982d7455ff319944a1570fb": "f_\\omega(f_0(3)) - 2", "5f8ce5164659bf74740f8fc0ddb00a77": " P' ", "5f8cebea27a3b01b0325ee3816b4a645": "\\hat K(\\xi)", "5f8dc9a8fcb53002a6e8921e4554652f": "\n x(t) \\approx \\cos\\Bigl(\\left(1 + \\tfrac{3}{8}\\, \\varepsilon \\right)\\, t \\Bigr) \n + \\tfrac{1}{32}\\, \\varepsilon\\, \\cos\\Bigl( 3 \\left(1 + \\tfrac{3}{8}\\,\\varepsilon\\, \\right)\\, t \\Bigr). \\,\n", "5f8ddb008f9634077ef5a8a378b16cff": "3+4\\cos\\phi+\\cos2\\phi=2(1+\\cos\\phi)^2\\ge 0,", "5f8e139eb90870dba062e8abcf901b8d": "A_1 = \\begin{bmatrix}0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n1 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 1 & 0 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 0 & 0 \\\\\n0 & 0 & 0 & \\cdots & 1 & 0 \\\\\n\\end{bmatrix}", "5f8e5ecf3a2b0b7cad9a5425e8115bae": "z_{01} = z_{02} = Z_0", "5f8e725ab1f9ae6a3d384d75f2140598": "\n\\bold {A}_2 = \\begin{pmatrix}\n 1 & 0 & 0 \\\\\n 0 & 1 & 0 \\\\\n 0 & 0 & -1 \n\\end{pmatrix}\n", "5f8e9e6c19dbc939526c87c22d4e1391": " I = \\frac{U}{R_x + R_{load}} ", "5f8ea767090e67c716bb42a7f70171d8": " \\begin{bmatrix} 0 & \\mathbf a & c \\\\ 0 & 0_n & \\mathbf b \\\\ 0 & 0 & 0 \\end{bmatrix}, ", "5f8f1e6b2a73a81a32e8beff7a4e2666": " F_r ", "5f8f2ad40bbadaac68d8e0979c6ff012": "\\left(\\frac{\\alpha}{\\pi}\\right)_3 = 1.", "5f8f63474b9159660ac01cee8e039548": " R_i ", "5f8fb29bed1d2af49292d275323379b5": "Q = N (N - 1) / 2", "5f8fe3f80222b10e1712b81b3771195d": "\\theta = (\\beta, \\lambda_0(u))", "5f8ff00ca8b264de29c191896d29f5ad": "\\int x^m\\ln x\\;dx = x^{m+1}\\left(\\frac{\\ln x}{m+1}-\\frac{1}{(m+1)^2}\\right) \\qquad\\mbox{(for }m\\neq -1\\mbox{)}", "5f90ad5085628c9260e771fe5409dc38": "\\prod_{i=0}^{n} ar^i = \\left( \\sqrt{a_1 \\cdot a_{n+1}}\\right)^{n+1}", "5f90be092d48e8a298feaf62d684a8aa": "v=\\dot x", "5f90d44c5201808dea638b712a81507d": "\\mathbf{l}_a + (\\mathbf{l}_b - \\mathbf{l}_a)t", "5f9114e720b86cbb7194f6b95156a6af": "\\begin{align}\n\\pi_{(\\frac{1}{2},0)}(J_i) & = \\frac{1}{2}(\\sigma_i\\otimes 1_{(1)} + 1_{(2)}\\otimes J^{(0)}_i) = \\frac{1}{2}\\sigma_i\\quad\\pi_{(\\frac{1}{2},0)}(K_i) = \\frac{i}{2}(1_{(2)}\\otimes J^{(0)}_i - \\sigma_i \\otimes 1_{(1)}) = -\\frac{i}{2}\\sigma_i,\\\\\n\\pi_{(0,\\frac{1}{2})}(J_i) & = \\frac{1}{2}(J^{(0)}_i\\otimes 1_{(2)} + 1_{(1)}\\otimes \\sigma_i) = \\frac{1}{2}\\sigma_i\\quad\\pi_{(0,\\frac{1}{2})}(K_i) = \\frac{i}{2}(1_{(1)}\\otimes\\sigma_i - J^{(0)}_i \\otimes 1_{(2)}) = +\\frac{i}{2}\\sigma_i.\n\\end{align}", "5f912c9479e2965e1e7a38584c8c09e8": "\\vec{t_1}\\langle s''\\rangle=\\vec{t_1}\\langle s\\rangle", "5f914f8c3c91e7c8a9e8d3fa5ee3788b": "x = a + r \\frac{1-t^2}{1+t^2}\\,", "5f91503881e38a29e88d85ddd624f4f7": "A_{\\alpha \\beta}", "5f9242952e2af5342fa30f50b9b61272": "P = \\int I\\, \\cdot dA", "5f92504421a3e0efbcf151280cd89bec": " f^{''}(x) = \\begin{cases}f(x),\\ x\\in Z\\\\ p \\text{ otherwise.}\\end{cases}", "5f928c1c8fc2dd18d85858cdac1b8ec2": "X + y = \\{ x + y: x \\in X \\} , x + Y = \\{ x + y: y \\in Y \\}", "5f9295ed99c9593ee8f803c3dea0f3f2": "\\bar{z} = x - iy", "5f92ee72c726edf71b39833b8490e843": "\\{a^{r}_{\\textbf{p}},a^{s \\dagger}_{\\textbf{q}}\\} = \\{b^{r}_{\\textbf{p}},b^{s \\dagger}_{\\textbf{q}}\\}=(2 \\pi)^{3} \\delta^{3} (\\textbf{p}-\\textbf{q}) \\delta^{rs},\\,", "5f9376e7f5c9cf99ae37e8bb50014344": "b \\to (b-a)^{-1},\\quad c \\to (c-a)^{-1},\\quad d \\to (d-a)^{-1}.", "5f9380f4525f45a50f0e20e122f313b1": " 0 \\not= v \\in V ", "5f939b7fd3b1e7c168e2d5a2a11e8aa0": "e_i\\ ", "5f93f983524def3dca464469d2cf9f3e": "110", "5f9452aa9a509c22cde534ded53f2b13": "\n\\{p_x, p_y\\}_{DB} = - \\tfrac{q B}{4c}.\n", "5f94656c489cf1820f46b6b5c53f0210": "m_j", "5f94a3c1fdca8221310cf5eee5d7f59f": "W = C_{1} (\\overline{I}_1-3) + C_{2} (\\overline{I}_2-3), \\, ", "5f94b2e8aa3f838158fc4687c1bcdcd1": "\\mathbf{u}_\\mathrm{t} = \\frac {\\mathbf{v}(t)}{v(t)} \\ , ", "5f94fe6f22f4787a096ce7061c61ee37": "\n\\begin{align}\n\n (\\nu x) ( \\; & 0 \\\\\n | \\; 0 \\\\\n | \\; 0 \\; )\n\n\\end{align}\n", "5f9519ede0533433f9f2638fec01fdab": "\\Psi_L\\left(-\\infty\\right)=0,\\qquad \\Psi_G\\left(\\infty\\right)=0.\\,", "5f952d90bd09e5e81099344f1df0d772": " z =\\frac{1}{u} ", "5f95397d2dd117779cc954c1eb0d9e5b": "\n u =\n \\left(\n \\frac{2\\, b^2 \\cosh(\\theta) + 2\\, i\\, b\\, \\sqrt{2-b^2}\\; \\sinh(\\theta)}\n {2\\, \\cosh(\\theta)-\\sqrt{2}\\,\\sqrt{2-b^2} \\cos(a\\, b\\, x)}\n - 1\n \\right)\\;\n a\\; \\exp(i\\, a^2\\, t)\n \\quad\\text{with}\\quad\n \\theta=a^2\\,b\\,\\sqrt{2-b^2}\\;t,\n", "5f9581f7da7901f04252f0848e7a0f29": "C'_w=q^{-\\ell(w)/2}\\sum_{y\\le w} P_{y,w}T_y", "5f958bd1fb7545b8e3381c3f3b74b6f1": "\\beta = \\arctan\\ \\frac{2\\sin b}{\\tan(\\frac{\\gamma}{2}) \\sin (a+b) + \\cot(\\frac{\\gamma}{2})\\sin(a-b) }.", "5f95d62ba9f1ab5dc1f3daf44703ab98": "R=R_0e^{B(\\frac{1}{T} - \\frac{1}{T_0})}", "5f960b8e1187719e53a2d009735b9f04": "\\alpha^j(x)", "5f962418d44a447ceeeacf1687084736": "\\sigma(G)=\\max\\left\\{\\sum_{T\\in\\mathcal T}\\lambda_T\\ :\\ \\forall T\\in {\\mathcal T}\\ \\lambda_T\\geq 0\\mbox{ and }\\forall e\\in E\\ \\sum_{T\\ni e}\\lambda_T\\leq1\\right\\}.", "5f9674b9f96f220e64a5527d3e146089": "x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36", "5f96785e4ba43307dd9b4c92e502a6e7": "s_0\\left( t \\right)", "5f96bfb2d45f1289238fbe71e833df6d": "(2n-1)", "5f971a431b664cb066c782e66984748f": " \\Delta_0 = \\Delta_1 = \\Delta_2 = 1 \\, ", "5f971f1c683571cdd17eb9593fa22afc": "\\int_0^\\infty \\frac {{}e^{-ax}\\sin bx}{x} \\, dx=\\tan^{-1}\\frac{b}{a}", "5f97379f996b989125d8a4a3a80ef424": "\\sqrt{2-a}=2\\cos(\\pi-\\tfrac\\phi2)", "5f979d611fbcebbedd86b849480ab6b8": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 190\\cdot 0.74)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 30\\cdot R_{\\bigodot}\n\\end{align}", "5f97a9a44e7386f6f99aacda39c44a18": "C_G(k)", "5f97b93c15364061bc6cbb0626339bfe": "{\\Psi}(x,t)", "5f97dcf8e31b123e0df42c99d98dc85c": "s= O(n^2)", "5f9816c0f9452829114756dfb5350ea7": "P(x_1) - f(x_1) = + \\varepsilon\\,", "5f982dbdcffcd59e2d93a5424105ee32": "C_k^{(s)}", "5f9847fd2296dbe32db854ec655ba721": "\\sqrt5 = e^{2\\pi i/5} - e^{4\\pi i/5} - e^{6\\pi i/5} + e^{8\\pi i/5}. \\, ", "5f986c720d84c13a3786970611648101": "\n\\nu_{\\rm max} = \\frac{eV}{h}\\,\n", "5f98a8c851a2a9b84fb759d1aa81a5dc": "x_z", "5f99e3be5ecfd2ba46e0916849f17a76": " 0 \\leq a_n + |a_n| \\leq 2|a_n|", "5f99ea665dbe0ee481a8dfdc829b74bb": "\\mathrm{i}\\hat{\\mathbf{L}}=\\sum_{i=1}^{n}\\left[\\frac{\\partial H}{\\partial p_{i}}\\frac{\\partial}{\\partial q^{i}}-\\frac{\\partial H}{\\partial q^{i}}\\frac{\\partial }{\\partial p_{i}}\\right]=\\{\\cdot,H\\}", "5f9a317e5f776cc2ae31898b789c117f": "p_i=\\mathrm{Pr}(X=i)={2 \\choose i}q^i (1-q)^{2-i}", "5f9a7aaca1f5ca55b70d0af224c8f754": "x = 3a(3-t^2)", "5f9aad1bae5572fbbcfbf26a9d24abc5": "a=\\operatorname{id}", "5f9ae72ed1e58f6c08c9701384f3047a": "O_k \\exp(-2\\pi i k/N)", "5f9aee5d0282633aad51428712204bee": "GRLEX", "5f9af97a5832fa20a130ce752f8f9d73": "C_{max}", "5f9b65dff45a91210818a7b2dac38458": "\n\\frac{1}{3}x^5 + \\frac{7}{2} x^2 + 2x + 1 = \\frac{1}{6} ( 2x^5 + 21x^2 + 12x + 6)\n", "5f9b7b4623df997d4831f07478b98da0": " Q(X,Y) = P(X \\mid Y) \\pi(Y) ", "5f9b9bea768db1a3cda4284e17608e25": "2^2 + 2 - 1", "5f9bea3d48c2dd5b51c85471e39e805a": "e_1\\wedge\\ldots\\wedge e_n", "5f9c1646bbe6acb4f3db6ee67f98ebd7": "\\Delta v_2 = \\sqrt{ \\frac{2 \\mu}{r_b} - \\frac{\\mu}{a_2}} - \\sqrt{ \\frac{2 \\mu}{r_b} - \\frac{\\mu}{a_1}} ", "5f9cd0470403447aa2b63a905857c7a5": "\\exists (\\alpha, \\beta)\\in R", "5f9d8764d90be4cf58fb817741a72e36": "\\pi(W,t) = \\frac{\\mu-r}{\\sigma^2\\gamma}", "5f9d87bb434c55b80994e273597f5f00": "Z_\\mathrm{L}", "5f9d88071246fdd622951c151c1ad99e": "\\,\\bar{z}_s=n_s^{-1}\\sum_{i=1}^n z_i I_{H_s}(y_i)", "5f9e3ba82fd4c6f3fd2b6a69851a49eb": "M=I-X(X'X)^{-1}X'", "5f9e4495efaa5fa312924312f774f22d": "\\Phi_n(s) = \\frac{\\sqrt{2 \\alpha_n}} {(s+\\alpha_n)} \\cdot \\frac{(s-\\alpha_1)(s-\\alpha_2) \\cdots (s-\\alpha_{n-1})}\n {(s+\\alpha_1)(s+\\alpha_2) \\cdots (s+\\alpha_{n-1})}", "5f9e6b4c7a7896fa87c37d15787cf0fb": "Redc(T)", "5f9ea427ad79d9fb44ed7a48f8bcaeee": "\\hat{\\mu}_{i,j} = \\frac{x_{i,.}\\times x_{.,j}}{x_{.,.}}", "5f9ec99c248e4a4017e5f98967155ba2": "W \\to X", "5f9eebf71b3dda0303aa29b923903704": "D\\!\\!\\!\\!/\\ \\stackrel{\\mathrm{def}}{=}\\ \\partial\\!\\!\\!/ + i A\\!\\!\\!/", "5f9f64c3dffed109e7398fdcf861df38": "\\rho(\\mathbf r,t_0)(\\nabla u_k(\\mathbf r))^2=0,", "5f9f957b2650e58ab8cf10d7cadbaec1": "Z_{in} = Z^{\\infty}_{in} \\cdot \\frac{1+\\frac{Z^0_{e}}{Z}}{1+\\frac{Z^{\\infty}_{e}}{Z}}", "5f9f9e451c81c32f988dad16fda88c8d": "\\lambda_\\min", "5f9fa7e49038ec723683508cf8746ae9": "E=h \\nu", "5fa02d51b1152580ad912ed70ad6dfc7": " \\sum_{k=0}^n f(k) = \\int_0^n f(x)\\,dx + {f(0) + f(n) \\over 2} + \\int_0^n f'(x) P_1(x)\\,dx\\qquad (1)", "5fa02e3614239d372d4a085e42c9fa3b": " \\pi_2 = ( a_D-a_S ) / ( b_S-b_D ) \\, ", "5fa030e76b5b6a2dfed810fdd783a4aa": "\\vec{E}(\\vec r)= \\frac {1}{4 \\pi \\varepsilon_0} \\iiint \\frac {\\vec r - \\vec r \\,'}{\\left \\| \\vec r - \\vec r \\,' \\right \\|^3} \\rho (\\vec r \\,')\\operatorname{d}^3 r\\,'\n", "5fa0343f444f11acf627c946e288e319": " \\hat{\\Gamma}(G,H) ", "5fa059e1b9f53b3be91baf926371eed3": " a = m^2 - n^2, \\, ", "5fa065805ab27fe1e7c65b78ea1ea405": "q \\sim 10^{-4}", "5fa06f9b95fe35ce4e40e76802b52c7f": "\\operatorname{P}\\left(|T- n H_n| \\geq c\\, n\\right) \\le \\frac{2}{c^2}.", "5fa0c818681c18a3b7636762136a2ac2": "F(x,a)", "5fa120b3d275e10c1cc7a07430314d4d": "\\left({\\frac{k}{2k+2},\\frac{k+l+1}{2k+2}}\\right)", "5fa150ce151fbeefe3687df0e1482e05": "P(Z)=P(X=x)", "5fa1a9fcb4535f2c5e6eed4871dede70": " \\langle \\eta,\\delta \\zeta\\rangle = \\langle \\mathrm{d}\\eta,\\zeta\\rangle. ", "5fa26768f38c3693bcb1a16af8b64a1d": "(1/T)", "5fa26e793b8adcd6153618e3763b83e7": "(x,y,z)=(2,3,4)", "5fa29bf6728f9a6415e7e67487b5a069": "-2q_p(2) \\equiv \\sum_{k=1}^{\\frac{p-1}{2}} \\frac{1}{k} \\pmod{p}.", "5fa2aa2ec7c56ede708c46768bfe3fb6": "g^w=x", "5fa2d4bfb6b87a3f5ec6f4f03c06bf60": "\\Phi : X \\rarr C_b(X)", "5fa2e27fdea9bdaa989fed4e7c5e923c": "e_H = 1 - \\left [ \\frac{\\cos \\theta - ( \\epsilon - \\sin ^2 \\theta )^{ \\frac{1}{2}}}{ \\cos \\theta + ( \\epsilon - \\sin ^2 \\theta )^{ \\frac{1}{2}}} \\right]^2 ", "5fa2fda7b6c74c3532cb2335d27db7fe": "= - \\left (1\\right ) \\left (1\\right ) \\left (1\\right ) \\left (1\\right ) = -1.", "5fa31c5f7e3c3223b2b23de620ac4872": "\n \\mathcal{D} := \\rho~(T~\\dot{\\eta}-\\dot{e}) + \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} \n - \\cfrac{\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T}{T} \\ge 0\n ", "5fa323f6e600b3b930efefba1ff209cc": "n(s)", "5fa37336e9954d89eed1fd6d2e54f372": "u_s = F(u, x(s), t(s)) = 0\\,", "5fa3fd871f6b876bf48074fc8c3d6a46": "|S|= o(N)\\,", "5fa4019dc8fc510cee00730ede03e880": "p_B^b", "5fa4087406958a60a4f5863eddf0a7ec": " a^{\\mu} = f^{\\mu} ", "5fa412eb45620650a7aead00a853bd4f": "CIRP: (1 + i_\\$) = \\frac {F_t} {S_t} (1 + i_c)", "5fa444bf78578c6e8539f4616a288142": "\nG \\equiv \\mathbf{q} \\cdot \\mathbf{p} + G_{3}(\\mathbf{p}, \\mathbf{Q}, t)\n", "5fa45985ab13c36abe20c976f0a2cb4f": "T \\subseteq S", "5fa48cf1377f5f7c023cd9cda6f8b10e": "f^\\rightarrow:\\mathcal{P}(X)\\rightarrow\\mathcal{P}(Y)", "5fa4b621fe981f3c22e4cfa842790c97": "f(c)=(1+Qc)^{-1/Q}", "5fa5112c6b7e3e037bc4026547601b4f": "|h|^2", "5fa5434b0ea2366037c56f9ec4ffed55": "|E_{b}^{v}|", "5fa54eb7f695fe6e94220536578ad3aa": "|-|: s\\mathbf{Set} \\leftrightarrows \\mathbf{Top} : Sing", "5fa55f502ee862042ceb08f7de5aa624": "\n\\begin{align}\nd^nV & = \n\\left|\\det\\frac{\\partial (x_i)}{\\partial(r,\\phi_j)}\\right|\ndr\\,d\\phi_1 \\, d\\phi_2\\cdots d\\phi_{n-1} \\\\[6pt]\n& = r^{n-1}\\sin^{n-2}(\\phi_1)\\sin^{n-3}(\\phi_2)\\cdots \\sin(\\phi_{n-2})\\,\ndr\\,d\\phi_1 \\, d\\phi_2\\cdots d\\phi_{n-1}\n\\end{align}\n", "5fa5a97e5750447cca46ccf8f648d160": "\\frac{\\partial F}{\\partial y}(x, y) = J.", "5fa60f68a72aa6d695042e1cd7110804": "\\varphi=90", "5fa6377fed9d3c31a91ca2b34d88edf7": "K = AA^{\\top}", "5fa65e1824b8c3ddbe72956e8becfe79": "\\sqrt{\\frac{6}{35}}\\!\\,", "5fa680c8dda317133fb26ba15a35e938": "\\tau = \\frac {K \\lambda}{\\beta \\cos \\theta}", "5fa6cd69603d3a6266a84071cb2efd8f": "\n \\bigsqcup_{i\\in I}A_i = \\bigcup_{i\\in I}\\{(x,i) : x \\in A_i\\}.\n ", "5fa6da3680e5281b2201814446df89d8": " e^{i 2\\varphi}=\\cos 2\\varphi +i \\sin 2\\varphi", "5fa705177162f9610c293161ba2d2c38": " Q = \\begin{bmatrix} P & 0 \\\\ 0 & P \\end{bmatrix}.", "5fa836cd140e25ce1647ff692880c3f9": "\\phi_i: U_i \\to P_i", "5fa83d5411b387e5926fbaac1087f183": "\\omega \\in \\Omega_c^m(U)", "5fa862fd27ca8a6b49c6d72b34268f67": "S_\\mathit{wn}(S_w) = \\frac{S_w - S_\\mathit{wi}}{1-S_\\mathit{wi} - S_{orw}}", "5fa8b46fd22986af8918252f988cd06d": "{\\mathbf{x}}_0^r=H_0E({\\mathbf{x}}_0).", "5fa8e82d2cd7954e04a6aa59d5b0d968": "\\sin\\theta_2 = \\frac{n_1}{n_2}\\sin\\theta_1 = \\frac{1.333}{1}\\cdot\\sin\\left(50^\\circ\\right) = 1.333\\cdot 0.766 = 1.021,", "5fa91834e7ceea68a71082ef88eb8a1b": "g\\left( x,t\\right) =h\\left( x\\right) +t", "5fa944460d163df9d2876ec1cadc1727": "+ \\operatorname{Tr}\\left( p\\!\\!\\!/' \\gamma_\\mu m \\gamma_\\nu \\right) + \\operatorname{Tr}\\left(m^2 \\gamma_\\mu \\gamma_\\nu \\right) \\,", "5fa94725ea8cfd20cb455b5b104c882e": "\\frac{1}{L_\\mathrm{total}} = \\frac{L_1+L_2-2M}{L_1L_2-M^2 } ", "5fa96394affe2fec0e965d49e1b6f6b9": "\nV_t \\approx pq\\left(1-\\exp\\left\\{-\\frac{t}{2N_e} \\right\\}\\right)\n", "5fa989799d944debb348aa95a948761c": "\\frac{1+\\sqrt{5}}{2}", "5fa9b5fcb2f9e0dff0152b8efbcbe6e1": "\n\\begin{align}\nd^2 y &= f''(x)\\,(dx)^2 + f'(x)d^2x\\\\\nd^3 y &= f'''(x)\\, (dx)^3 + 3f''(x)dx\\,d^2x + f'(x)d^3x \n\\end{align}", "5fa9b66f19a3700252ad3c2cb4f61b44": "AIC = 2k - 2\\ln(L) = 2k - 2(C-\\chi^2/2) = 2k -2C + \\chi^2 \\,", "5fa9fb3ddbf7522f704a65e188f5d4be": "\\frac{\\hbar}{2} \\left(m \\omega^2 x \\stackrel{\\rightarrow }{\\partial }_{p} - \\frac{p}{m} \\stackrel{\\rightarrow }{\\partial }_{x}\\right) \\cdot W=0", "5faa0c5a1f121d967a92c169ee000cb8": "y=a\\phi", "5faa43c0dd170de8a113499763916313": "L(x, y, t) = (T_t f)(x, y) = g(x, y, t)*f(x, y)", "5faa9042f1c4d164a6aeaf19279c0c0c": "\\nabla \\times (\\nabla \\times \\mathbf{F}) \\equiv -\\nabla^2 \\mathbf{F} + \\nabla \\nabla . \\mathbf{F}", "5faac2e77f389b5a0f34c6f8c4aa8556": "\n\\mathrm{Power}_{ext} = \\mathrm{Power}\\times C_p\n", "5faacf13352ba8d12d72826c60fb013e": "\n\\frac {N^2} {t} = \\frac {L S} {K}\n", "5fab38d29f57ac81158e13d5d2e05c97": "\\left\\{\\begin{array}{lll}\\emptyset & n = 0\\\\\\{0^1\\} & n = 1\\\\ \\{(n - 1)^1, -1^{n - 1}\\} & \\text{otherwise}\\end{array}\\right.", "5fab65556f452184ca3a1d27f1609a00": "\\mathrm{O}^\\times_\\mathrm{O} \\Leftrightarrow \\mathrm{V}^{''}_{Mg}", "5fab7013e309d45fed080d8af8b890ac": "GL_3", "5fab7c7950a211508731ab4c30cbab59": "T^{1/n}", "5faba93b43f9d778592687ec0de97837": "r\\,N^2D^5", "5fabb3d75f74c23c1ba9626251210a2d": "\n\\frac{ \\left(\\mathit{far} - \\mathit{near}\\right) \\cdot z^2 }{ S \\cdot \\mathit{far} \\cdot \\mathit{near} }=\n", "5fabeb7e496b6db744b13c874f612919": "\\lim_{x \\to A_x}y(x) = \\infty", "5fac72ed8b34a71adbbe894c53263414": "d_\\text{match}(\\ell_1, \\ell_2)=\\min_\\sigma\\max_{p\\in C_1}\\delta (p,\\sigma(p))", "5fac85fcddbe42fa8fb77bc6fed5c7ec": "p^* = \\inf_{x \\in X} \\{f(x) + g(Ax)\\}", "5fad35ffe176f6ad7f99da4e952fd6f9": "\\mathbf{P}^n=\\mathbf{P}(K^{n+1})", "5fad4b0b2cf07f15a27650e757549ea8": " \\left(\\frac{\\partial U}{\\partial V}\\right)_{T} = T \\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - p ", "5fade79375b1a8f2ab2c839d4b230cf7": "E[X_0X_t] \\geq 0", "5fadfc25d7856e0cbfca0647b13d45f3": "G_0=\\langle U,D,L,R,F,B\\rangle", "5fae51caeeba87189226be1c940b0e12": " b^2 + 2bs ", "5fae789c78f9102b5cd8cf62b4dbeaba": " f = \\Gamma \\cdot \\nabla V + \\nabla \\times W ", "5fae84979f155adf3317d315c26730f9": "1,2,\\dots,k-1", "5faeaf4b58f62fd513d9d5637b4c474c": "F > \\frac{1}{2}", "5faf4e59727104a7c131e39cf093f5de": "d(z,w)=2 \\tanh^{-1} |z-w|/|1-\\overline{w}z|.", "5faf52c218de3f9238de1f7541d47963": "p_n(0) = 0\\mbox{ for } n \\ge 1. \\,", "5faf8cc9a996d024cc15972784d33f85": " \\cot(z - a_1)\\cdots\\cot(z - a_n) = \\cos\\frac{n\\pi}{2} + \\sum_{k=1}^n A_{n,k} \\cot(z - a_k).", "5faff7ad250f3ba06dab29829d372655": "\\sigma_{h,sand} = \\rho_w g h_w = 1000 \\,\\text{kg/m}^3 \\cdot 9.8 \\,\\text{m/s}^2 \\cdot 10 \\,\\text{m} = 9.8 \\cdot {10^4} {kg/ms^2} = 9.8 \\cdot 10^4 {N/m^2} ", "5fb01067da7e2ab76d6d5ce737799547": "A\\overline{C}", "5fb0405e0eaaf0480e16dd7bf6f0286a": "\\mathrm{FOV} = \\alpha \\frac{D}{d}", "5fb055451c30e3a7647109e5d1380ece": "(x^2-1){P_{\\ell}^{m}}'(x) = {\\ell}xP_{\\ell}^{m}(x) - (\\ell+m)P_{\\ell-1}^{m}(x)", "5fb096307ed3ea6166658de3f9e5ed46": " h : \\mathcal{A} \\to \\mathcal{B} ", "5fb099fc0ca4dcdd3bdc086b2a66ae9f": "\\displaystyle{\\omega(g_1,g_2)=\\mu[\\det (A_3(A_1A_2)^{-1})^{-1/2}].}", "5fb0ae01f5487622468246131c851ac2": "\\scriptstyle [\\boldsymbol\\omega]_{\\times}", "5fb0d49bb0d967ddb5aaa6868b87378f": " \\Diamond_1 P ", "5fb0f4c481e225a8a5534c6354bafd2d": "\\beta \\equiv \\sqrt{1 - M_\\infty^2}", "5fb1236111abb2260e44355fc6b119f3": " \\forall x \\in [0,1], \\qquad \\mu(x)=\\frac{\\rho(x)}{\\frac{\\varphi^2(x)}{4} + \\pi^2\\rho^2(x)}", "5fb139e81ff8d65fc5766a6b7d56b2aa": "\\mu = \\mu_0 \\left( 1+ \\frac{\\omega_0 \\omega_m}{\\omega_0^2 - \\omega^2} \\right) ", "5fb162a980ed1be8dfe123aaea1d8299": "I(\\mathcal{A})", "5fb1a943dbeef61f05c0a4451c9534db": "E_{T}", "5fb2147cb3b5871aaa828114bfbd5c29": "\nim [\\hat{H}, \\hat{x}] = \\hbar \\hat{p}, \\qquad i [\\hat{H}, \\hat{p}] = -\\hbar V'(\\hat{x}).\n", "5fb2449d4c360277fefd7757074a5dba": "|x_n-x_0| \\leq { q^n \\over 1-q } | x_1 - x_0 | = C q^n", "5fb263e9198452dfc5a66348f1ca8484": "\\|x\\|_p = \\left(\\sum_n|x_n|^p\\right)^{1/p}", "5fb282fe6427a259757f291108936344": "\\alpha_p", "5fb2914e39ee5a4f77d642e1e9961dc3": "a(x\\!-\\!\\beta) = (x\\!-\\!\\alpha)t^2", "5fb2d769a292b7f69803b25817a3b087": "\\phi\\in C_0^\\infty", "5fb333c184f32b02dc7c9a5a20fe2580": "{y_1^2 \\over 2} + {qv_1 \\over g} = {y_2^2 \\over 2} + {qv_2 \\over g}", "5fb3630bd8f30f1b1fc45653cd459b0a": "f\\colon S^2\\to \\R^3", "5fb36567c26a76973144c374cf159654": "L_J = \\frac{\\Phi_0}{2\\pi I_J}\\cdot \\frac{1}{\\cos \\phi} \\ ", "5fb3d69153a6782ab9862e1cb9d22d37": " \\Pr(S \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.", "5fb5726c94fed48253e3988e77269bc5": "ES_{\\alpha} = \\frac{1}{\\alpha}\\int_0^{\\alpha} VaR_{1-\\gamma}(X)d\\gamma", "5fb5a0846f791f81788e25e76f4a2d18": "\\{ \\mu_{T} | T \\in \\mathcal{A} (E) \\}.", "5fb5e2af467d57a0b102cdf4bdf4017f": "\\frac{n_x+n_y-p-1}{(n_x+n_y-2)p}t^2 \\sim F(p,n_x+n_y-1-p).", "5fb606d906107467fc731951de9ada3e": "\\lim_{x\\to+\\infty} \\left(1-\\frac{1}{x}\\right)^x=\\frac{1}{e}", "5fb60716526da1bc297f35272b5d802c": "P_{n}^{m}(x)", "5fb653f90570a33e01e4c56beddf5120": "size\\ in\\ bits = sample\\ rate \\times bit\\ depth \\times channels \\times length\\ of\\ time", "5fb667af62fe3609517dab5fe77d037f": "X\\backslash S.", "5fb682beba684f397be6c42ed3ac772d": " u^2 = \\cos^2 \\alpha \\frac{a^2 - b^2}{b^2} \\, ", "5fb695d86281f8b6a36423448cec2989": "\\left(\\frac{\\mathrm{d}s}{\\mathrm{d}t}\\right)^2 \\cong c^2(1 + \\varepsilon\\gamma_{00})", "5fb6e4b0f4fb53e40e8742fc5f9ff711": "\\mathit{SIL}=\\frac{{V_\\mathrm{LL}}^2}{Z_0}", "5fb6fd15d1086edab18e0392908e81e8": "z(r)=\\frac{r^2}{R\\left (1+\\sqrt{1-(1+\\kappa)\\frac{r^2}{R^2}}\\right )}+\\alpha_1 r^2+\\alpha_2 r^4+\\alpha_3 r^6+\\cdots ,", "5fb7131541e85605459724f6f4ebe302": "A \\Rightarrow \\ldots \\Rightarrow A ", "5fb75b30d1f9018bb52caf4c442159fb": " p = 1 - e^{ - m \\log( s^2 / m )( s^2 / m - 1 )^{ -1 } } ", "5fb75f2a76dc62bcbaf93e2f1eae2e7b": "K-e^{-aw}", "5fb790279b89cbb5f233a35c5958ec48": "E_{s} \\ \\ = \\ \\ \\langle x(t), x(t)\\rangle \\ \\ = \\int_{-\\infty}^{\\infty}{|x(t)|^2}dt", "5fb7c0774663cec95c409d4b20e92115": "\\bold{x}_k=\\text{argmin} \\Phi_k(\\bold x) ", "5fb8b9c43044f44c279bd5fff3531fbd": "p_\\text{i=on}", "5fb8dbf16675a738a8aed52aa8a61ddd": " \\mbox{C}(U) = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & x_{00} & x_{01} \\\\ 0 & 0 & x_{10} & x_{11} \\end{bmatrix}", "5fb911e3e2a7c1d7c995b8db62612f47": " A_n = \\{k \\in {^*\\mathbb{N}}: k \\geq n\\} ", "5fb933ca8e81c6d6970b24bb48491018": "color(\\dots,\\dots)", "5fb96616989436fad2353708a4343c21": " \\operatorname{ad}_{I_{V}} = 0 : L(V) \\rightarrow L(V) ", "5fb9bcafcff4b7dab9295a6ff04c83ba": "\\psi(t)=2\\,\\operatorname{sinc}(2t)-\\,\\operatorname{sinc}(t)=\\frac{\\sin(2\\pi t)-\\sin(\\pi t)}{\\pi t}", "5fb9c97c53f4517454d04e84f72a4648": "|\\epsilon v\\rangle", "5fba10f1abb2f0ea433a8bc28b0b30a4": "(\\ln 2)i+(\\ln 3)j+(\\ln 5)k\\le\\ln N,", "5fba11495a188144268a9a0a9d14eb3b": "y_\\lambda", "5fba4b35e3ffc1fcbdeeafd891c4e60f": "= \\frac{y''(s)x'(s)^2-y'(s)x'(s)x''(s)} {\\left(x'(s)^2+y'(s)^2\\right)^{3/2}}\\ , ", "5fba6c1b3978b674cd5c9fefa85993e9": "E_{min}", "5fbb284bd6f2c97f1de62cb05d1e8e5e": "\\gamma (t) \\rightarrow t^{-\\beta}L(t)", "5fbb643309d4f9aca1b02e6e8f84c80e": "\\ \\dot{u}^* = \\dot{Q}u + Q\\dot{u} \\quad \\text{and} \\quad \\dot{A}^* = \\dot{Q}AQ^T + Q\\dot{A}Q^T + QA\\dot{Q}^T.", "5fbbae10c46faf6405b45c7ba3d97916": "\\frac{\\partial u'}{\\partial t} \\, - \\, 2 \\Omega \\sin \\varphi \\, v' \n \\, + \\, \\frac{1}{a \\, \\cos \\varphi} \\, \\frac{\\partial \\Phi'}{\\partial \\lambda} = 0", "5fbbe8eadebbdac0d88bb9d9f6667bc7": "\np(n) = \\prod_{i=1}^T \\frac{1}{n_i!}(M_i)^{n_i}e^{-M_i}\n", "5fbc14b81fa13f89dae8c1750cdd9ed3": "p_{k+1}^* \\leftarrow r_{k+1}^*M^{-1} + \\overline{\\beta_k}\\cdot p_k^*\\,", "5fbc14c3bf8443012155321f6448ad65": "F(0)=1", "5fbc4b6e6276849a7fae952eb88bc34f": "\\begin{matrix} {2 \\choose 1}{3 \\choose 2}{10 \\choose 1}{4 \\choose 2}{36 \\choose 1} \\end{matrix}", "5fbcebbbe8de38bddd6ef8ff11fc0c72": "M(q(t)) = R_\\mathrm{OFF} \\cdot \\left(1-\\frac{\\mu_{v}R_\\mathrm{ON}}{D^2} q(t)\\right)", "5fbd057413211cabb138c3c9ad82657f": " {\\lambda} \\,", "5fbd0c2c1b7c1cb3b60a377177b5b27e": "\\begin{align}\nx&=a\\sinh\\xi\\sin\\eta\\cos\\phi\\\\\ny&=a\\sinh\\xi\\sin\\eta\\sin\\phi\\\\\nz&=a\\cosh\\xi\\cos\\eta\n\\end{align}", "5fbd6872dbd9b9fe63b3bf25ea54454d": "\\displaystyle u_t+45u^2u_x+15u_xu_{xx}+15uu_{xxx}+u_{xxxxx}=0", "5fbd7a76fabcf87ed87d43306acaf3b2": "\\dim(\\mathcal{F}) + 2", "5fbda0124196987e393508c0e4dc5d91": "Y_{10}^{7}(\\theta,\\varphi)={-3\\over 512}\\sqrt{85085\\over \\pi}\\cdot e^{7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot(19\\cos^{3}\\theta-3\\cos\\theta)", "5fbdbb32070e837a1d1bbe1d0aab8020": "f: [0,b] \\rightarrow \\mathbb{R}", "5fbdf9f5511861036443e0c6212178e2": "\\mathrm{bind}: \\mathrm{M} (A + E) \\rarr (A \\rarr \\mathrm{M} (B + E)) \\rarr \\mathrm{M} (B + E) = m \\mapsto f \\mapsto \\mathrm{bind} \\, m \\,\\left( a \\mapsto \\begin{cases} \\mbox{return err } e & \\mbox{if } a = \\mathrm{err} \\, e\\\\ f \\, a' & \\mbox{if } a = \\mathrm{value} \\, a' \\end{cases} \\right)", "5fbe10acdf65af4696fa6a2faf5ee07e": "\nE = q_{1} q_{2} \n\\left[ \\frac{1}{r_{ON}} + \\frac{1}{r_{CH}} - \\frac{1}{r_{OH}} - \\frac{1}{r_{CN}} \\right] \\cdot 332 \\ \\mathrm{kcal/mol}.\n", "5fbe298b799b4bb042c4b0114a377af6": "\\rho \\frac{D \\mathbf{v}}{D t} = -\\nabla p + \\nabla \\cdot\\boldsymbol{\\mathsf{T}} + \\mathbf{f}.", "5fbe4278face375b04978aab450ccef9": "\\frac{W+\\frac{1}{2}T+\\frac{1}{2}OT}{GC}", "5fbe4d082c73f75856c6608b501c1764": " b^*(0)b(0) = \\frac {1} {N}\\ \\cdot \\ \\frac {1}{1 + \\sum_{\\boldsymbol{R_p \\neq 0}} e^{i \\boldsymbol{k \\cdot R_p}} \\alpha (\\boldsymbol{R_p})} \\ , ", "5fbe6407e6445828ae7171c1463ba0c9": "\\tfrac{1}{4}\\tbinom63", "5fbe6593958fd4f6b34632aed5369a0a": "y = q \\sin\\theta\\,", "5fbe7be0524b8ada04a29fee8d3cf17c": "q_{k}", "5fbe7fbda557510a7b89a42d4ee3c9a5": "G \\circ F", "5fbeb424d1d92b5a457cb08536cb633d": "r_1 = 9a_0 + 14a_1 + 11a_2 + 13a_3", "5fbedb9104f3f5c72d93af619619ac91": "\n- \\boldsymbol {\\nabla\\times E} = {\\partial\\boldsymbol B\\over\\partial t}\\quad {\\rm or}\\quad\\boldsymbol {\\nabla\\times (v\\times B)} = {\\partial\\boldsymbol B\\over\\partial t}", "5fbedc8a01387b142e0ba4da67ec5157": "R(M,x) := {x^{*} M x \\over x^{*} x}.", "5fbee351efef4d05aedd16ebe4140ef8": "\n\\arccos(z)\n", "5fbf59677fad55a03e6bb38dbc7f8ebd": " \\vec t_2=v_{21} \\vec r_u + v_{22} \\vec r_v ", "5fbf71ca5602deb635a5768033213cec": "T = \\frac{\\pi}{\\sqrt{I}}", "5fbf8a9b86a81d1bcb9871cd3c112b10": "\\mu_{g} = \\left( \\frac{\\partial A_{g}}{\\partial N_i} \\right)_{T,V, N_{j \\ne i}} \\,\\, = k_{B}T\\ln \\frac{N^{3D}}{Z^{3D}} ", "5fbfb30cd625a486543b60fafa143aa9": "f \\ll s", "5fbfea0d6c913752866e4b28ee8569b3": "\\scriptstyle f = f_\\mathrm{red} = 0.9", "5fc0c59a948c611b2bbaacd9d70d9c95": "n^{\\searrow}\\;\\mid\\;\\overline{n}^{\\searrow}(\\sigma)\\;\\mid\\;\nn^{\\nwarrow}\\;\\mid\\;\\overline{n}^{\\nwarrow}\\;\\mid\\;pino (\\sigma)", "5fc120bae8ee33ce9df4cd0b5d4a8f41": "\\Delta^{(2)}(\\sigma)=(2)_{n=0,\\dots,M-3}", "5fc15ef1d81f3d2408dc24d62a246641": "Z(E^k,T) = \\prod_x \\frac{1}{Z(E^k_x,T)}", "5fc1836a9dbaed94e414aea0e4b51f6a": "\n M_a={\\frac{V}{a}}\n", "5fc1a6a8603ff8155ba9f45d508c1a73": "(D + 1)s + D + k = (D+1)(s+1) + k -1 > N(m - s + 1)", "5fc1aab13b8a3c531ebc07929d15d9dd": "\\frac{(3-\\alpha) \\sqrt 7}{2\\sqrt{\\alpha(6-\\alpha)}}", "5fc1b66a74d2474eab9a776e484cc417": "v_0=c/n", "5fc1babc6a88e204718a86aed7ec6507": "A^n{}_m", "5fc1dd1bcbf25e8a00d8627d7088b99a": "e^{2X} \\sim \\operatorname{F}(n,m) \\, ", "5fc1ebc2dcdd15aae5e1441572a9fab6": "\\mu, s", "5fc2368778eb4cfb56ef7b2bc304e3c3": "\\phi^{}_{}", "5fc274b8955c5510908b90b4c0df2dc0": " p \\times L", "5fc2b9a907767be50a55589d72f98690": "\\Vert \\eta A^{-1}B \\Vert", "5fc2c6677717e6aac811f7d34d0db1f2": "a \\rightarrow b, \\; b \\rightarrow a, \\; a \\rightarrow c,\\; b \\rightarrow d", "5fc2cfbebd69eb2d31b3d274673428d5": "\\lfloor N / 2^k \\rfloor", "5fc3255378cfa40548e20d97f642cfa1": "S(E)", "5fc38563c7ea85a434ad9875ac0c5349": "T = \\sqrt{s\\left(s-a\\right)\\left(s-b\\right)\\left(s-c\\right)}= \\sqrt{8 \\cdot (8-7) \\cdot (8-4) \\cdot (8-5)}", "5fc3ab18ed8ff06cf448b4832e8d64a9": "log BCF=0.76log Kow-0.23", "5fc3af069c11291cf343f8e67b0e779f": "\\Phi_{xy}", "5fc3af50cc0bf134144fbc444100dc13": "R_{\\mu \\nu }-\\frac 12g_{\\mu \\nu }R=\\frac{8\\pi }\\phi \\left[ T_{M\\mu \\nu}+T_{\\phi \\,\\mu \\nu }\\right]", "5fc404f3cea786177c9ef691818a5105": " 2c\\int\\nolimits_{-K}^K \\frac{f(p)}{c^2 +(p-k)^2} dp = 2\\pi f(k)\n-1 \\quad {\\rm and} \\quad \\int\\nolimits_{-K}^K f(p) dp = \\rho \\ , ", "5fc42d09b2af790d3e34f7b3cc3e152b": "\\scriptstyle 10", "5fc9bafb1ff56b3265776e43236c3d92": "E^{(-)}(\\mathbf{r}, t) = E^{(+)}(\\mathbf{r}, t)^\\dagger", "5fc9d0dbcd6032c04415b9657e13b80e": "d\\tilde{V}", "5fc9e8be8720a34f7e16bacbac79b314": "x \\in \\mathbb{R}^d", "5fc9f346b8db98020962d841ec1074fd": " P^{tr}(X,Y)", "5fca00126b3c867664646613e956349a": "p_n(x) = x^n.", "5fcac33616d5cd6ed38fccfa336abc08": "2n\\log k", "5fcb0287d878b7f4c7094997f0af40de": "\\frac{d}{dt}e^{X(t)} = \\int_0^1 e^{\\alpha X(t)} \\frac{dX(t)}{dt} e^{(1-\\alpha) X(t)}\\,d\\alpha ~. ", "5fcb1fe9ac79e1472d1ac98708cf29c3": "x_a = x_o - r_n", "5fcb3678ae17c792d65658ef9c545a82": "\\, \\varepsilon_0", "5fcb6036f96ee9fb93314b6fd1bcd7e2": "j=\\frac{6912c^3}{4c^3+27d^2}", "5fcbfd784570dcbb7c4e5a84fbe1ff84": "\n\\mathbf{P}_\\textrm{EM} \\,|\\, \\mathbf{k},\\mu\\,\\rangle =\n\\mathbf{P}_\\textrm{EM} \\left({a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\,|0\\rangle \\right) = \\hbar\\mathbf{k} \\left( {a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\,|0\\rangle\\right)= \\hbar\\mathbf{k}\\,|\\, \\mathbf{k},\\mu\\,\\rangle .\n", "5fcc524f201002066bde67720485d9f9": " H(s) = \\frac { s C_2 R_1 } {s C_2 R_1 + (1 + s C_1 R_1) (s C_2 R_2 + 1 )} ", "5fcc5977e3e3dbe58610143e29372966": "r_{\\mathrm{g}\\text{ axis}}", "5fcc71d642c19606f60b227451297080": "Q\\left[f(a + \\tfrac{Q}{2}) + f(a + \\tfrac{3Q}{2})+\\cdots+f(b-\\tfrac{Q}{2})\\right].", "5fcd25084de46c01819c2694c679aade": "f = \\frac{pn_s}{120\\ }", "5fcd31f17777bb46c198f720e75f9081": "f_x, f_1, f_2, \\ldots, f_i, \\ldots, f_{k-1}, f_k", "5fcd3689881ce0ff457945d7d0589fec": " { \\frac{\\partial{(\\rho T)}}{\\partial t}} = {div\\, (k\\, grad\\, T )} \\, ", "5fcd6c00cb0732ad419a41f3d7ac9513": "\\sigma_{ij}=C_{ijkl}\\epsilon_{kl}", "5fcd861094f5fb07ef8bcc2c03e3c2a8": " E = T + V_C(R) + \\Delta(\\vec{P})", "5fcd9c71c368d70c494687c77e0edfe5": "\\varepsilon''", "5fcdde95949f6b2481cf286d06899671": " A_{1} \\ldots A_{n} ", "5fce6fb65e297d5b7e9a07717b52fc59": "b\\,", "5fce790df4f1027dc45ebffb655b7c53": "-i|0\\rangle", "5fce9764277ae5e129e02ef732dce3b0": "\\Theta(\\theta)=C\\cos m\\theta + D \\sin m\\theta,\\, ", "5fcea376676fdc186794c73442f5e601": "p(x) = \\sum_{y} p(x,y)\\; , \\; p(y) = \\sum_{x} p(x,y).", "5fcec823cac93cb0808933192b2582b5": "{g_c}", "5fced705d20a0bf02af0b698bcd4e1ee": " \\varepsilon_{n1}, \\; \\varepsilon_{n2} \\sim ", "5fceefd3c84a3eb8fc9c0159ebdd42e1": "e_1, \\ldots, e_s \\in F\\setminus\\{0\\}", "5fcf2bfd35fd0ab1c91e32dd19f0e623": "(j,k)", "5fcf893bee2ce712d39760e6ab18b2a8": " V = k_p(P) \\,T \\,\\!", "5fcfa56c94b5d813b9e7a438218d67f9": "\\boldsymbol{\\sigma\\tau\\upsilon\\phi\\chi\\psi\\omega} \\!", "5fcfb8da793bd29ac27ba9397d6a37d8": "\\lim_{r \\to 0} \\frac1{\\mu \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\mu(y) = f(x)", "5fd00d9b05f7f4a429961ebea5d9aec4": "N_{M_1} V \\otimes_\\mathbb{C} N_{M_2} V.", "5fd07430e8c95425757372079d109109": "\\frac{\\partial u}{\\partial t} + \\nabla \\cdot \\mathbf{J}_u = 0", "5fd09532d1e78e662daa7d74234040f9": "z=x+iy=re^{i\\theta}", "5fd13e511c832e8c3f086adf902c2f57": "\\begin{pmatrix}a&b\\\\0&c\\end{pmatrix}\\begin{pmatrix}x&y\\\\0&z\\end{pmatrix}=\\begin{pmatrix}xa&ya+zb\\\\0&zc\\end{pmatrix}", "5fd16d496acc3bfae3c129e8be53b9a2": "\\Delta V = -i_l r_l \\Delta x\\ ", "5fd1a074937a5e51b36287f81f66ea20": " \n\\int_E f_n\\, d\\mu_n \\geq a \\mu_n(A_n) \\Rightarrow \\liminf_{n\\to \\infty}\\int_E f_n \\, d\\mu_n = \\infty = \\int_E \\phi\\, d\\mu,\n", "5fd1f9826bef17d6354050c15ff35694": "h_A = \\mathrm{Hom}(-, A),", "5fd2591afd988bb3dd307b1b4f2b462d": "\\Delta=\\frac{4A\\omega\\sin\\phi}{\\lambda c} \n", "5fd263a50d1b1ee405d7acfefeb7bb3c": "S,", "5fd268ce76ca0e70a43e80c0280955da": "\\vec{r} \\, ", "5fd27831cbfff473f284a54a0c8c2dd2": "\nu= -\\frac{\\partial\\psi'}{\\partial y},\\qquad\nv= \\frac{\\partial\\psi'}{\\partial x}\n", "5fd27eefba508580e44e3eaa13b269ee": " \\boldsymbol{\\omega} = \\nabla\\times\\mathbf{v} ", "5fd29646d642a895c929c76096d3ebb1": " \\mu = \\frac{-ln({I}_{t}/{I}_{0})}{x} ", "5fd2b014eeb378fb219ccb80e1604ffa": "K_c", "5fd2b5fb3cc5dc632875b19edbd5713c": "\\mu=\\mu^\\ominus +RT \\ln \\frac{f}{p^\\ominus}", "5fd314767bc5a1e70bff8a4adf36ad9f": "\\frac{c}{a}=\\frac{c_1,c_2,c_3,c_4,c_5\\dots}{a_1,a_2,a_3,a_4,a_5\\dots}=b_1,b_2,b_3,b_4,b_5\\dots = b", "5fd317d57921e1b544ad1de5827afade": "\\mbox{Specific Force} = \\frac{\\mathrm{Force_{non-gravitational}}}{{\\mathrm{Mass}}}", "5fd32bd3196be3accb69df1d508a5cb0": "y_{1J}", "5fd38c0c6189dbbe11cdb43862d2d78a": "\\lambda \\in \\hat A_i", "5fd3d2dbae6b5de183724581608e4a9b": " E_{\\text{Compton}} = E_T (\\text{max}) = \\frac{2E^2}{m_{\\text{e}} c^2 + 2E} ", "5fd40eb7c5b10d1970c26ca19a3b4708": "\\langle f,g \\rangle = \\int_0^b x f(x) g(x) \\mathrm{d}x", "5fd447d71cd3362a7c11c98686424333": "Bb \\rightarrow bb ", "5fd4b3ced8c77e01540b6cbf6abc578f": " p^{-1}", "5fd4f46a622317f0554bbf5f111a0f95": "\\int_0^1 \\sum _x f(x) dx=0 ", "5fd61b792ee861dc614575f42e3fe0d8": "\nW = \\prod_i w(n_i,g_i) = \\prod_i \\frac{(n_i+g_i-1)!}{n_i!(g_i-1)!}\n\\approx\\prod_i \\frac{(n_i+g_i)!}{n_i!(g_i-1)!}\n", "5fd651916ddd00318daf393d89e2069b": " Z/pZ ", "5fd69de636bcbbce1abeca215c13c127": "V_{\\mathrm{sen}}", "5fd6c604e8be3c77b16069be6384af86": " g(\\theta_{m+1}|\\theta_m)", "5fd6ea8d3076dd02a2cff1fbd6487eb8": "[(a,b)]\\cdot[(c,d)] := [(ac+bd,ad+bc)].\\,", "5fd6f4c4a13ba397bdfb444c354f56e3": "\nF_r(\\mathbf{r}, \\alpha, \\beta) = - \\frac{3 \\mu_0}{4 \\pi}\\frac{m_2 m_1}{r^4}\\left[2\\cos(\\phi - \\alpha)\\cos(\\phi - \\beta)- \\sin(\\phi - \\alpha)\\sin(\\phi - \\beta)\\right]\n", "5fd74d0b8858c8a4ac1da06316c32297": "x^{n+1}\\,\\bmod\\,\\big(n,x^2-bx-c)", "5fd753b4d5d3cb35ef629ae8ec3dd2fe": "\\textstyle \\Lambda_1,\\Lambda_2,\\dots", "5fd7681d51b49f9dce2ec5573618e684": "\nz = g^{-1} (y') = g^{-1} (y) = g^{-1} (f(x))\n", "5fd76eea234cf8b29a6be6db9f6a57d6": "A_{n}={\\scriptstyle 2\\pi^{n/2}/\\Gamma[\\frac{n}{2}]}", "5fd77f331b1758a7c39724c66fc7c0b6": "\\mathbf{P} = g(\\mathbf{a}) - g(\\mathbf{b})", "5fd7c074e7578ba2ade8e3bce86ace20": "w\\left(t-\\sigma\\right)", "5fd87ac27bdd606f2efae41a474faafb": "f(a)=\\int_0^\\infty e^{-a\\omega} \\frac{\\sin \\omega}{\\omega} d\\omega ;", "5fd8bcf4a881994dc769a6461237c81e": "\\left(\\frac{a}{n}\\right) = -1", "5fd9a0117f74b2cc84318398de69f1e4": "\\mathsf{SAT} \\notin \\mathsf{P/poly}", "5fd9accdfe6c9f8054da195ef034117c": "M = \\left\\lfloor \\frac{-1}{\\log_2(1-p)}\\right\\rfloor", "5fd9b1ac842c9bee660f9cede964ccbe": "I_r^' = \\frac{jX_m}{\\frac{R_r^'}{s} + j(X_r^' + X_m)} I_s", "5fda47bd56a273359db9f924dc6a7a19": "\\forall x \\notin L", "5fda4d2bbed02aaaf6d03a208fdec39a": "PV\\,=\\,{A \\over (i-g)}\\left[ 1- \\left({1+g \\over 1+i}\\right)^n \\right] ", "5fdac75cd7384490e1eacff9f8d87f4e": "U_1=\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}", "5fdadd6d47558c2e67ec4f348d87dbe6": "\\sin (\\alpha - \\beta) = \\sin \\alpha \\cos -\\beta + \\cos \\alpha \\sin -\\beta\\,", "5fdb04e53b0a42e723e05e92a17a495f": " d \\approx 2.44R\\left( \\frac {\\rho_M} {\\rho_m} \\right)^{1/3} ", "5fdb417c054c40a926fd7bd6ad429bb6": "\\langle\\overline\\Psi\\Psi\\rangle", "5fdb5879b05cf4371d9948ba44601fe2": " X^\\#_p = d_e A_p(X) ", "5fdb5cdc7c35fdd6bd89f0d7adb76d24": "\\mathcal{L}\\left\\{\\frac{f(t)}{t}\\right\\}=\\int_{s}^{\\infty}F(p)\\, dp,", "5fdbae498444c348a285d5210174f91e": "M \\otimes_R k(\\mathfrak{p})", "5fdbc1bbcd362093233a26bb1658f8cc": "\\mathcal{O}_X^\\times", "5fdbd1374204cfc08a647d962a6017dc": "\\left(\\frac{\\partial^2U}{\\partial y\\partial x}\\right) = \\left(\\frac{\\partial^2U}{\\partial x\\partial y}\\right)", "5fdbdd854a295adcb81ad057bafeba55": "\n\\begin{align}\n\\boldsymbol\\beta &\\sim \\mathcal{N}(\\mathbf{b}_0, \\mathbf{B}_0) \\\\[3pt]\ny_i^\\ast\\mid\\mathbf{x}_i,\\boldsymbol\\beta &\\sim \\mathcal{N}(\\mathbf{x}'_i\\boldsymbol\\beta, 1) \\\\[3pt]\ny_i &= \\begin{cases} 1 & \\text{if } y_i^\\ast > 0 \\\\ 0 & \\text{otherwise} \\end{cases}\n\\end{align}\n", "5fdc85db3c9a89ef85ccc0166fcab84e": "S - S_0 = k_B \\ln\\Omega = k_B\\ln{1} = 0 ", "5fdc9824eb64d7ed537bccecc42ce0ef": "\n|\\mathbf{v}|^2 = (\\omega r (1 + a'))^2 + (v_{\\infty}(1 - a))^2 = W^2\n", "5fdccb97d0dc348dc17a4993aea06397": "\\omega_r ", "5fdd0a62ff2168602bd0ba57c61c5b39": "(h * h_{inv}) (t) = \\,\\! \\delta (t)", "5fdd460224417226332d3759616f2562": "\\operatorname{Ma}(A,B,C) = (A \\and B) \\oplus (A \\and C) \\oplus (B \\and C)", "5fdd46a0ee9be1992881f7c7893ad0ea": "H = \\frac{1}{2m}\\left(\\sigma\\cdot\\left(p - \\frac{e}{c}A\\right)\\right)^2 + e\\phi.", "5fdd4ac911bcbb1c6d5ac4fa77119a2d": " p((a_1,\\cdots,a_n)) = \\sum_{i=1}^n a_i g_i^{-1} \\otimes_B g_i ", "5fdd50b139df09ecba62e65e0bae1169": "29 \\cdot 782 \\cdot 22678 = 22678^2", "5fdda7a36ddb1cbd06090f9564eda165": "NOT(\\alpha)=s^TNu=1-\\alpha", "5fddbee1fb39cef4c14e2743c4909f68": "alive(0)", "5fddc20b9397d324e11e9c653b7efbf9": "r_n > 1", "5fde2d0266e07cdc2e45f48c072aacd5": "V^* = I^*=P_K : H -> W", "5fde3dde2650f8fe6e7ff3b09fbedd85": "\\Omega=1/N_0", "5fde7bc56a19ba9921585d43be8e8e2b": "\\left[ -3\\, A^2\\, A^\\ast - 2\\, i\\, \\frac{dA}{dt_1} \\right]\\, e^{+it} - A^3\\, e^{+3it} + c.c.", "5fdebdc9edc0b6b956f44686f2bb386c": "\n\\frac{\\ddot{P}}{P}+\\frac{1}{\\rho}\\,\\frac{\\dot{P}}{P}+\\frac{1}{\\rho^2}\\frac{\\ddot{\\Phi}}{\\Phi}+k^2=0\n", "5fdec0af98294310916224006e229ea0": "\\det(\\partial f_i/\\partial x_j)", "5fdf200832f4920886bc8b5c68312332": "\\sigma _1, \\; \\sigma _3", "5fdf39688792e061411e44df75a8e68c": " S=\\frac{n_e}{a_0n_{cr}}", "5fdf7dff690ee217553cbc4cac732d8e": "{A}_{12}^{(2)}", "5fdf815d9af806a3a9bb0c62e0b293ab": "\\Rightarrow (M v_j )' v_i = \\lambda _i v_j' v_i", "5fdf8578d7ddc40067b22c49eff11fd4": "(x-3) (x-1)^9 (x+1)^9 (x+3) (x^2-5)^6", "5fdfd1416b246e3197c8cee92ae68772": "f(x_1, \\cdots, x_n)", "5fe01f15efb0e69cee99ed0ac5704a5e": "\\boldsymbol\\mu_0", "5fe0517427e394e569ca12a5bfca737d": "~= (x \\wedge y \\wedge z)~", "5fe05338720222e7ae7e8978062e11b7": "\\Theta\\in\\text{Hom}(\\wedge^2 TM, TM)", "5fe0ac989bc0107780584baf022a91ec": "\\gamma \\approx 2.1", "5fe0bd4ed6f711718b8a4beacb48e9d6": "F_E = 4\\cdot\\pi\\cdot\\varepsilon_{0}\\cdot\\varepsilon_{r}\\cdot\\text{r}_{H}\\cdot\\zeta\\cdot\\text{E}", "5fe176c1d65909837a1cfa3a2ea8130b": " \\mathbf{j}(\\mathbf{r},t) \\cdot d\\mathbf{S} ", "5fe1ac8ab08a97dc58e379b7f74d6d06": "G_\\mu(s,t)", "5fe1e3640c919f2f0913c13beffad24e": "1 \\le r < 2", "5fe21f8bc5ae77424dec0e1eb5e02608": "\\displaystyle{ S(z)=T(z)\\sigma T(-z),}", "5fe24d1e76d886b9a9de134f9002a244": " 1 - e^2 = (1-f)^2 ", "5fe2507eb5afd1683981b063e5bc1c7f": "\\begin{align}\n\\sigma_\\mathrm{n} &= \\sigma_{ij}n_in_j \\\\\n&=\\sigma_1n_1^2 + \\sigma_2n_2^2 + \\sigma_3n_3^2\\\\\n\\end{align}\n\\,\\!", "5fe261c406df39e54bc841dd9e49ad09": "\\mathrm{p}K", "5fe2cd8f6c4e9ea0cee148db921e3d9e": "p = RT (\\frac{1}{V_m} + \\frac{B_{2}(T)}{V_m^2} + \\frac{B_{3}(T)}{V_m^3} + \\dots)", "5fe34d81a6f883eaec4f7ca2222988a1": "(a\\,,\\,b)", "5fe36ed493192d08c55cba224c4aca50": "E=3.52k_BT_c\\sqrt{1-(T/T_c)}", "5fe37c4aab695accf0aab1ea2b05abdf": "n = 2\\!", "5fe39178118f5972b0cbbc1f4cfc6445": "X=\\frac{m_1x_1+m_2x_2}{m_1+m_2}", "5fe3964c4f8af0c3a78adbef30b3ea34": "0=(k^2+4ik-5)c_3+(k^2-4ik-5)c_4", "5fe3a6cfb3d719396c349e86b865b76c": "P=a^{n+1} r^{\\frac{n(n+1)}{2}}", "5fe3e4c0dd5a981878dd46e02c3e8bc6": "r_{I1}=\\frac{R_1-Z_{I1}}{R_1+Z_{I1}}", "5fe4190aa033468e9f389e174680ab18": "h=2375\\,m", "5fe430796c1b74c1a475a47d71ee6c69": "\\scriptstyle a_n\\geq0,\\ \\sum_n a_n=1", "5fe43fdbf02e09597bcc764c1f95b23e": "m_t = c + \\sum_{i=0}^b \\eta_i d_{t-i}.\\,", "5fe45a7eecc87aeb684031724cb0c598": " \\mathbb{Z}_{m} = \\{0,1,...,m-1\\} ", "5fe51219ea1bd6d0a3b17541722e2ff9": "\\Sigma _{YY} =\\operatorname{Cov}(Y,Y) = \\operatorname{E}[Y Y']", "5fe52a94d7d0e813d5fb8fc56b0be9be": "\\tau = 1/\\lambda", "5fe53629faef39521c78980ffdb675e7": "A_{r} := \\{ x \\in X \\mid d(x, A) \\leq r \\}.", "5fe53d656096c16f4057fb7ad599d1f4": "\\ \\sigma^2(t)", "5fe547a3592be4c781591ea074af61af": " b(r) = \\frac{\\sinh(\\sqrt{2} \\omega \\,r)}{\\sqrt{2} \\omega}, \\; a(r) = \\frac{\\cosh(\\sqrt{2} \\omega r)}{\\omega} + c", "5fe55dabcea9daf7612f29e73f7f8d82": "\\mathbf{H}\\mathbf{x_1}=\\mathbf{H}\\mathbf{x_2}", "5fe5b8524422eeaf271d2241d8b16d00": " f ( a x ) = a f ( x ),\\,", "5fe5da3f8f38f01c16ceacf5b68a90dd": "\\tfrac{1}{2} \\sqrt{(a^2+c^2)(b^2+d^2)}.", "5fe5e3760652cc6006b5ee5af31604be": "O(\\log{N})", "5fe5f08262fec8b29db696c7f4b9d994": "\\sigma_3 = 0", "5fe60d640c9820114965c6818f00f9fa": "\\neg a \\wedge \\neg b", "5fe6ddfdcf480e4a821aed12d657c3cd": "\\operatorname{Ber}(X\\oplus Y) = \\operatorname{Ber}(X)\\mathrm{Ber}(Y)", "5fe749a500a243fa719fe0af2b4f56ab": "\n\\dot C_g(\\theta) = \\frac{n \\theta^{n-1}}{\\lambda_0^n +\\theta^n},~~~\\theta=t-t',~~~1/\\eta_f = q_4 /t\n", "5fe784ff40ce493601263d9b8b8ee770": "\\alpha \\mathbf x = (\\alpha x_1, \\alpha x_2, \\cdots, \\alpha x_n).", "5fe7939b61ae0833805b03035fd15f01": "D = { \\eta_1 \\over 2 } \\times { N_2 \\over \\eta_2 } ", "5fe7a75044d1c637e6254016a699c8b0": "\\tilde{g}_{33}", "5fe7c3e10101c79c918132784ff21602": "\\mathbf N=\\boldsymbol\\lambda+\\mathbf R", "5fe7c7b9f8a35c67d4f2bc6e883bfe49": "\nP_{l}(\\cos \\gamma) = \\frac{4\\pi}{2l + 1} \\sum_{m=-l}^{l} \nY_{lm}(\\theta, \\phi) Y_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "5fe7e32d2ea81fc9d482029ae5178c0e": "(1+i_d) = \\frac {F} S (1+i_f)", "5fe7f0836aaedb1619f130bc16d27b0c": " \\omega = \\omega \\left ( k \\right ) ", "5fe82a35f24d69656b5e8f46bffd90c6": "p + 1", "5fe8945b9dc8a939eee83eb8c9b100e7": "a>1:", "5fe8969113242d10eb884715c773405c": "\\phi\\circ f^{\\circ p}\\circ\\phi^{-1}(\\zeta)=e^{2\\pi i\\theta}\\zeta.", "5fe8bbab5c10552dcd45946977a502a0": "1 \\over 5", "5fe8d76bb372430a49187a0f0eef384c": "\\mathcal C^*(M)", "5fe919889e7c60aacc78c790eac24067": "z^n e^{zx} + A_1 z^{n-1} e^{zx} + \\cdots + A_n e^{zx} = 0.", "5fe91bbee468b8c8e0d2d9e7d977df1c": " \\mathbf {P_{\\mu\\nu}} ", "5fe91cd39fb235602f4cadbda49fa167": " v = \\frac{P}{T} = \\left [ \\frac{T}{P} \\right ] ^{-1}.", "5fe96919cabb72382721241fb2d47edb": "0 \\le v \\le r-1", "5fe9e52136a17567d98dec9321798ba5": "{4 \\choose 1} + {5 \\choose 4}{2 \\choose 1}{3 \\choose 1} = 34", "5fea17036d92d4c2ad457439fdeae47b": "\\sigma = f(r)g(\\theta)", "5fea4e3a75c0208f9a17070538979b45": "1/B(z_1,z_2)", "5fea588ac24f182419f8d13f083375fc": "\\mathbb{C}\\mathbf{P}^1", "5fea5f50e620bb18fdf9c874f42d8ce3": "\\text{Var}\\left(\\hat{\\alpha} + \\hat{\\beta}x_d\\right) = \\text{Var}\\left(\\hat{\\alpha}\\right) + \\left(\\text{Var} \\hat{\\beta}\\right)x_d^2 + 2 x_d\\text{Cov}\\left(\\hat{\\alpha},\\hat{\\beta}\\right) .", "5fea87182a4a976beff667b5a06d93c0": "b^2 - ac", "5feab2dba480810c67b3f94d24326cbe": "p(F_i \\vert C, F_j,F_k,F_l) = p(F_i \\vert C)\\,", "5feab410af9c4cb15e4cbcc80a01d86f": "m \\cdot 2:m\\ ", "5feab572e170085275b468d0bacb27e8": "\n|x|_p = \n\\begin{cases}\n\tp^{-\\nu_p(x)} & \\text{if } x \\neq 0\\\\\n\t0 & \\text{if } x=0\n\\end{cases}\n", "5feade52f51b7f09e56060524e6a7bc0": "M_f", "5feb12dc1f4a23579e272a76268a9d86": "\\{1,i,j,k\\}", "5feb2932f205eb2ddc27ad4e98f47793": "A(x,y) \\rightarrow B(y,z) ", "5feb5d028c0a1a9cc0f0a60dd41ae956": "TT^* f_j \\to TT^* g", "5feb71a1f3b31e14578083ae9989cd84": "size(Y)", "5feb8d704d7321ee07914a88cdf6b14e": "Q_{}", "5feb8e0787771be7a2050a85a6ef3710": "T_2(x) = x (x+1) e^x", "5feba592b2782a5e5d9003e97ee2674e": "|f(y) - f(x)| < \\varepsilon\\,", "5febb603766b648158c99b62906e248a": "\\left(1,\\ 1+\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+3\\sqrt{2},\\ 1+3\\sqrt{2}\\right)", "5fec37c0baecfea7455e78ac46e6234b": "\n\\left[ {\\begin{array}{*{20}{c}}\n 2 & 1 \\\\\n 3 & 5\n\\end{array}} \\right],\\left[ {\\begin{array}{*{20}{c}}\n 1 & 3 \\\\\n 4 & 5\n\\end{array}} \\right],\\left[ {\\begin{array}{*{20}{c}}\n 3 & 1 \\\\\n 4 & 7\n\\end{array}} \\right] \\leftarrow \\text{children}\n", "5fec3acc6939a34ac0766c36cbcd5abb": "\\zeta(-n,x) = - \\frac{B_{n+1}(x)}{n+1} \\ . ", "5fec6a6c880afe01133764837f8994f4": "\\varphi' = \\varphi - \\frac{\\partial \\lambda}{\\partial t}, \\quad \\mathbf A' = \\mathbf A + \\mathbf \\nabla \\lambda", "5fecd3dd1b7e89291cf6cc78919f4f0b": "\\textstyle{\\frac {4 \\log(2)} {\\log(5)}}", "5fed275f7f4582a2f079b3e33f5d91ba": "|\\mathit \\Gamma| \\le 1", "5fed4a0ebee13d3a56b9cd9c39f07387": "P\\left[\\sup_{t \\in [0,T+S]} X_t \\leq c\\right] \\ge P\\left[\\sup_{t \\in [0,T]} X_t \\leq c\\right] P\\left[\\sup_{t \\in [0,S]} X_t \\leq c\\right], \\quad T,S > 0 ", "5fed682c920635e874abe6049d81fd9f": "\nJ = q n \\mu E + q D\\frac{dn}{dx}\n", "5fed7d9577f1a2f064cfa8baa1203b33": " K(n) = a_n K(n-1) + K(n-2) . \\, ", "5fed9089baecdeb0701366b2fa06c80d": "(MN) (MN)^* = MN (NM)^* = MN M^* N^*. \\,", "5fedad21416d28ccf1947873f440ca2f": "\\frac{f(y) - f(x)}{y-x} = \\frac{f(m+h) - f(m-h)}{2\\cdot h} =\n f'(m) + f'''(m)\\cdot\\frac{h^2}{3!} + \\dots ", "5fedb84c78ff35d71244d9199be2f024": "{\\phi}^A \\rightarrow \\alpha^A (\\xi^{\\mu}) = \\phi^A (x^{\\mu}) + \\delta \\phi^A (x^{\\mu})\\,.", "5fee0c2ae9a1c8be36c290dfd0743632": "X = (R - \\mu_1)/\\sqrt{S_1} ", "5fee9cd143732b6572c9a4f52223d5a2": "G\\in K[C]", "5feed59dcca816e0074f551c73a66765": " ~m \\geq 1 ", "5feed624624a398e7583aa5174a981e8": "f = \\overline{f} \\circ p", "5fef24335cda5dccae9cc9a1e4463682": "\\begin{bmatrix}0&0\\\\1&0\\end{bmatrix}", "5fef3f5b86658289da88e35dda774fc0": "f '[B] \\cap g '[C] = (f ' \\circ f)[A] = (g ' \\circ g)[A] \\, ", "5fef70dd53e89f632da83ffbf6699045": "\n\\begin{align}\n x_i &\\sim\\ m_k + \\sigma_k\\times\\mathcal{N}(0,C_k)\n \\\\\n &\\sim\\ m_k + \\sigma_k \\times C_k^{1/2}\\mathcal{N}(0,I) \n\\end{align}\n", "5fefe8b30b1ced5a47dcf6093c88fca2": "\\,r = k(T)[A]^{n'}[B]^{m'}", "5ff0185231fee1a7f0736b464ca48e39": " Q = \\pi_{10} + \\pi_{11} Z\\, ", "5ff01d80bb4fefb733a1c8e0d33519b8": "g_{ji} = \\rho(||x_j-c_i||)", "5ff0608c3a5955975fd19d5da7f67ee7": "\\ p = \\rho g h,", "5ff08ee759b5a5bcd2c7173bfb84f086": "\\,S_c = \\mbox{D}(y, t) S_b \\mbox{D}^{-1}(y, t)", "5ff0a4688578e557cfdd5e19aba744a1": "\\mu^{\\rm eq}_i=\\mu_i(c^{\\rm eq},T)", "5ff0d7a86f31e4a0f7589dd24cf27cc8": " \\det(V) = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n \\alpha_i^{\\sigma(i)-1}, ", "5ff107a99b1f6b359a32d8889b59ab81": "\\ell(\\mathbf{x},\\mathbf{u}) + V(\\mathbf{f}(\\mathbf{x},\\mathbf{u}),i+1)", "5ff108dcfdf17b71d9dc0296e6b64f7e": "C/\\Sigma", "5ff116519e0d5c106fc9eb595e4c494d": " \\text{MTTF} \\approx \\frac{B_{10}}{0.1n_{op}},", "5ff15382e489299e40f3c26ea8e26256": "\\mathbf{E}(\\mathbf{x},t)=\\frac{i Z_0}{k}\\mathbf{\\nabla}\\times\\mathbf{H}(\\mathbf{x},t)", "5ff1ea874b223f87afb3aecb7818c080": "u_s = L/t_r", "5ff1f600f91eb3a8a8e68da4a2e0fc0f": " S = \\frac{\\vec w^T \\Sigma_b \\vec w}{\\vec w^T \\Sigma \\vec w} ", "5ff248f023475c6e023a216e366ebd1b": "l>3d", "5ff2768443d4988941d2e6f0738c6e3e": "\\textstyle f(\\alpha \\mathbf{x}) = g(\\alpha) = \\alpha^k g(1) = \\alpha^k f(\\mathbf{x})", "5ff280676d20b5c4658a1469589fe933": "\\beta < \\alpha", "5ff28d3242cb853938c2af0303e55a19": "P=0", "5ff2f676e9bce7d0a8502ac81b583beb": "C_{ij}(s,t) = \\operatorname{corr}( X_i(s), X_j(t) )", "5ff3822114476ec80b614a2c8211750e": "m^i", "5ff3ae7630be3f69278e6e156b331fc7": "\\cup_{i < 0} \\scriptstyle S_i", "5ff3ffb847976bf2fb7fc5650edbc35a": "\\prod_{i=1}^n (1- x_it) = \\sum_{k=0}^n (-1)^{k} a_k t^k.", "5ff40e0f0b9a00a34f8a070076ac4734": "p_1^{r_1} \\cdots p_k^{r_k}", "5ff431c37e9613e178c3f8a36189fd15": "\\frac {\\delta r}{r} = \\frac{4ik\\tilde n}{1-{\\tilde n}^2}\\frac{d\\tilde n}{d\\eta}\\int_{0}^{\\infty} \\eta(z,t)e^{2i\\tilde nkz}dz+2iku(t)", "5ff441f810b0d078515f02b70e8ad4bb": "n_{ij} = \\frac{k_{ij}}{k_{iH}+k_{iT}}.", "5ff4696cc3495c16c4cfe9cd736e9ba4": "x_B", "5ff480a3aa23e67f002d280c556a9e75": "33^2", "5ff578c9e0b9b1c9e35925ee91485d48": "f= \\sqrt{\\frac{1+\\rho}{1-\\rho}} ,", "5ff579180afbebeb0ceeb440e48a5fad": " a_i\\wedge a_j,\\; i\\neq j,", "5ff59ab7c93039245b970b15608a7253": "\\left\\langle \\sigma_{1}\\right\\rangle =\\frac{1}{Z_{N}}\\sum_{all\\atop spins}\\sigma_{1}\\prod_{all\\atop faces}w\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right).", "5ff5b23b589bec2c847fee516454293c": "(c/n) \\cos(\\psi)", "5ff61efd12211b421822e4a5610b32fb": "P = 2\\pi \\cdot a \\cdot \\sqrt{\\frac{a} {\\mu}}", "5ff63147c5116e9cd9c01709485c5551": "\\sigma_1", "5ff64b55a0c5a8c2b8073e2f77b53488": " \\operatorname{ker} f := \\{g \\in G : f(g) = e_{H}\\}\\mbox{.}", "5ff68c088f8d1098a7eb0ed5e976d613": "\\displaystyle \\overline{f(x)}", "5ff6fb794ba00b47861365961a4f8f64": "u:=s_{i-1};\\quad v:=t_{i-1};", "5ff73aae405d849cfb6fe451fd786631": " H_P = H_R ", "5ff78aa7a0d464b69be53db5213af0b6": "F_P\\cong\\{\\mathbb{Q},+\\}\\;", "5ff78d753911b7ebc61bfd0f51b8a6c7": "Z_0 = \\delta Z + \\frac {Z_0}{1+Z_0 \\delta Y}", "5ff8072e673a1be24c3ce037e0fcd665": "B_\\ell^m", "5ff83735e322cf1fa36ea17100638667": "\\scriptstyle\\frac{e}{p}\\, ,", "5ff84812f0e9a8befe2664f2f4c1a4c8": "\n E_y=- k_x \\sin k_x x \\cos k_y y \\sin k_z z\n ", "5ff8b922acd26d6cebc50959db2dd215": "g\\in{\\rm PSL}(2,\\mathbb{R})", "5ff8f1094af35afa9f7d1d5990a7110f": "\\cos\\,{\\theta_{obs}}= R \\cos\\,{\\theta}", "5ff90f9b85c47a434127783129fadfc5": " \\frac{1}{n} [z^n] \\frac{z^m}{1-z} = \n\\begin{cases} \n\\frac{1}{n}, & \\mbox{if }n\\ge m \\\\ \n0, & \\mbox{otherwise.}\n\\end{cases}\n", "5ff923cec8ad417156cfd995e941b694": "A= (5+\\frac{15}{4}\\sqrt{3}+\\frac{7}{4}\\sqrt{25+10\\sqrt{5}}) a^2\\approx23.5385...a^2", "5ff96c67cb5bbaa484f053fe631e2aa9": " \\frac{\\partial N_x}{\\partial e}\\frac{e}{X} = \\frac{\\partial X}{\\partial e}\\frac{e}{X} - \\frac{\\partial Q}{\\partial e}\\frac{e}{Q} - 1 ", "5ff97857ab9e81abc79aa184397c5cf1": "\\cos \\theta = \\sin \\left ( \\frac{\\pi}{2}-\\theta \\right ) \\simeq 1, \\quad \\theta \\ll 1", "5ffa1fcc1a75cf847457c6d75b4e2ebc": "\\dot{v}_2 = {1 \\over C} {v_3 \\over R} ", "5ffa413231bffe46baa691466a96c170": " \\scriptstyle s_{8,4} ", "5ffa5c647466a2bf86b9aa050c18964f": " Z/pZ \\oplus Z/pZ", "5ffa8c6cd6504d6b10ee1d6dd2e278af": "Q(5,q) :\\ s=q,t=q^2", "5ffab9cd90bce1de7b0c788fb1a13a89": "\\scriptstyle \\epsilon_2 \\, \\approx \\, \\epsilon_{21} \\, \\approx \\, 0", "5ffabad619eb69437338dbf481bfdb77": "\\scriptstyle \\tilde{X}_t ", "5ffac80d889e7f74a30635cfcf0aa3a7": "\n\\begin{align}\ni_{\\alpha}=&\\sqrt3 I\\cos\\theta(t),\\\\\ni_{\\beta}=&\\sqrt3 I\\sin\\theta(t),\\\\\ni_{\\gamma}=&0.\n\\end{align}\n", "5ffadae9db74c15bd39ddbafae4adc76": "SO(10) ", "5ffaf994b7b7a4ffbf433a3b1076c47d": "R=\\pi_{XY}(R)\\bowtie\\pi_{XZ}(R)", "5ffb4a78fb774728d12e45c2103ac7ac": "\\epsilon = 1", "5ffb54d78bc479901a05ce505a7fb4c1": " c_X ", "5ffb69802fa3d8fa02e4f47c30ae0f31": "\\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y} = 1.", "5ffb773f7f93b65d1fa831667a5f79c5": "\\hat{\\beta}(\\tau;Y+X\\gamma,X)=\\hat{\\beta}(\\tau;Y,X)+\\gamma .", "5ffbfeb2193381e262a4a54d56c0071a": "\\left\\langle\\exp\\left(X\\right)\\right\\rangle_{r}\\geq\\exp\\left(\\left\\langle X\\right\\rangle_{r}\\right)\\,", "5ffc00959489a7efd3e95f84f53d9a36": "h_p(X_p,Y_p) = g_p(\\pi_*X,\\pi_*Y).", "5ffc101d62c5cbf15e7531a611ef73c9": "f(z) = \\frac{1+2i}{5} \\sum_{k=0}^\\infty \\left(\\frac{1}{(2i)^{k+1}}-1\\right)z^k.", "5ffc22b94ce018d2326a3f1dd0909178": "\\approx\\sqrt{N} \\,", "5ffcc4899cc1d300c08b044e2096bc17": "(D^+DD^+)_{ij}=D^+_{ij}D_{ij}D^+_{ij}=D^+_{ij} \\Rightarrow D^+DD^+ = D^+", "5ffcee8dfa4b6757ca9b348e72641b70": "\n2T = c^{2} = \n\\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} \\left( \\dot{t} \\right)^{2} - \n\\frac{1}{1 - \\frac{r_{s}}{r}} \\left( \\dot{r} \\right)^{2} - \nr^{2} \\left( \\dot{\\varphi} \\right)^{2}\n", "5ffd47044b8ba03fcaaade30d86801ae": "{P}^{4}-3{P}^{2}Q+{Q}^{2}\\,", "5ffd8265283306db305931d5be9aaa65": " \\binom nk = \\frac{n(n-1)\\ldots(n-k+1)}{k(k-1)\\dots1},", "5ffd861bfa909f94701cc96f5a9b0f64": "\\begin{align}\n\\frac{dH}{dt} & = H(r-cP-q) \\\\\n\\frac{dP}{dt} & = P(acH-m-q) \\\\\n\\end{align}", "5ffda6648ebb3166df8cdaf2709c9e09": "\\{0,1 \\dots p-1\\}", "5ffe15d80713925a7603544ae2391be4": "N = \\begin{bmatrix} n\\\\2 \\end{bmatrix},", "5ffe204100de56fbafa240b9687c9ebc": " x \\mapsto -x. \\, ", "5ffe37f8fcd2595a9586a4de2355a385": "\\omega_\\alpha^\\beta(\\mathbf e\\cdot g) = (g^{-1})_\\gamma^\\beta dg_\\alpha^\\gamma + (g^{-1})_\\gamma^\\beta \\omega_\\delta^\\gamma(\\mathbf e)g_\\alpha^\\delta.", "5ffe4974e514c390540ef434e421e5aa": "h(t) \\cdot \\sum_{k = -\\infty}^{+\\infty} \\delta (t - k T_s) = \\delta (t) ", "5ffe63dd51eb6795ef395eb4382065cc": "O(\\log n)\\,", "5ffe65be40e355aa5ec5848dad5921a3": "I(the : N \\cdot N_0^l) = \\iota(p) \\cdot [p] : E \\cdot E_0^l", "5ffeb2ef131b4248129d48823ad119d7": "\\operatorname{true} \\equiv \\lambda a.\\lambda b.a", "5ffeb97a485876d672fab1a627e5bc7d": " \\rho_w ", "5ffecbebf0d178326e1d98c2a4ef7cbc": "\\frac{\\partial\\,\\textbf J}{\\partial\\,b}\\,", "5ffefd58a56d2a778ee4c8abdc31debf": "(c_1,c_2)=(g^y, m'\\cdot h^y)=(g^y, m'\\cdot(g^x)^y)\\,", "5fff75b1a6b35b71952d406d08a24d72": "\n W = \\tfrac{1}{2}\\lambda~[\\mathrm{tr}(\\boldsymbol{\\varepsilon})]^2 + \\mu~\\mathrm{tr}(\\boldsymbol{\\varepsilon}^2)\n ", "5fffa3b8bdfbf8ee3d876979f30d64ab": "(P+|P|)= 0", "5fffc319763535ee1edb6ef5e4a40a87": "\\scriptstyle Z_0 = \\sqrt{L/C}", "5fffe09f2e2aa70564b896a6d97ad3d6": " \\mu_{y=0},\\mu_{y=1} ", "5fffe888da381625781aaf57a6444c40": "\\begin{align}\n\\hat{\\boldsymbol\\rho} &= \\sin\\theta \\hat{\\mathbf r} + \\cos\\theta \\hat{\\boldsymbol\\theta} \\\\\n\\hat{\\boldsymbol\\phi} &= \\hat{\\boldsymbol\\phi} \\\\\n\\hat{\\mathbf z} &= \\cos\\theta \\hat{\\mathbf r} - \\sin\\theta \\hat{\\boldsymbol\\theta}\n\\end{align}", "5fffea915d09e2380e6512cbaacae40c": " g^R_{ij} = -\\partial_i \\partial_j S(U, N^a) ", "6000b973cd8eb914a43a97394fef11cc": "\\mathbb{S}_+", "6000f679b4af112110bc19ad18dd0872": "2\\uparrow\\uparrow 65536 \\approx 10\\uparrow\\uparrow 65533", "6001172f930f90be866120cb3772bcb5": " IG(A) = H(S) - \\sum_{t \\in T} p(t)H(t) ", "600133aa4beaf036531718ac50e5fc76": "\n \\left|\\psi\\right\\rang = \\frac{1}{\\sqrt{2}} \\bigg (\n \\left|+z\\right\\rang \\otimes \\left|-z\\right\\rang -\n \\left|-z\\right\\rang \\otimes \\left|+z\\right\\rang\n \\bigg)\n", "600149a330016dbd635b602fde7c9792": "\n\\left ( \\frac{d^2}{dx^2} + f(x) \\right ) y(x) = 0\n", "6001aa57a2de82ce1bad527d134980b1": " {Single\\ Loss\\ Expectancy\\ (SLE)} = {Asset\\ Value\\ (AV)\\ } \\times {\\ Exposure\\ Factor\\ (EF)}", "6001d47cb8f8b38602c0f3de728bff45": "\\theta \\in GF(2^n)", "6001f70c594bf3dd6b2c4803a113b32b": " \\pi[w] = e^{i \\lambda w}. \\quad ", "6002013d6018743ededd0d0404283e15": "|a - b| ", "600209a3792e57e1a9716b8309a720f4": " g_\\mathrm{e}(1-\\sigma) ", "600281de6ea2ee98d30ad1c51c5e15dc": " W_e \\subseteq I \\implies \\#(W_e) < f(e)", "600322228a83ad2c2ae85ce4eb183a44": "\\,i^{(2)}/2", "6003610eb2298fdcb4ef9e05353493c9": " \\frac{\\lambda}{2 \\pi} = \\frac{\\hbar}{m c} \\ ", "6003657f52872360adde2b5dc7c17e81": "\\qquad \\qquad A \\equiv_{amb} B", "60037845b9d5c0e1a586bcd1f39d8578": "\\det(M) = \\det(I_n + B A)", "6003ba675c5ffaa1b8512967aeb41e73": "h(q)", "6003ce3e053571e0c69002db113a3c3e": "\\dim C + \\dim C^\\perp = n.", "6003fa48c3b8284c760b2f7b023d9545": "x = 0 \\quad (1')", "60042530404be62bfe227de7d3f7c582": "|0\\rangle\\langle0|, |1\\rangle\\langle1|", "6004874c12958b28623c6d8e0336f750": "V(q) = q - 1 + 2q^{-1} - 2q^{-2} + 2q^{-3} - 2q^{-4} + q^{-5}. \\, ", "600504c3fc483b5caf51b85644ae590b": "D n^a=\\pi m^a+\\bar{\\pi}\\bar{m}^a-(\\varepsilon+\\bar{\\varepsilon})n^a\\,,", "600520ea4ce6e0a22148ea89ed48d8a6": "G_{zz} = {\\partial g_z\\over \\partial z} \\approx {g_z \\left (z + \\tfrac{l}{2} \\right ) - g_z \\left (z - \\tfrac{l}{2} \\right )\\over l}", "60052de181514cfa103b96841ecf9db3": "\\epsilon = v^2/2 - \\mu/r\\,", "600541f865d464e676b50c613959cdaa": "\\frac{\\partial p}{\\partial T}\\ ", "6005922ba6ba273334d65b2e3454896d": "\\mathrm{Ta}<\\mathrm{Ta_c},", "6005e45009bdfe15885e77e2ef1e221a": "R\\left( x \\right) := H\\left( x \\right) * H\\left( x \\right)", "6005f5086c4f1ee767e840446b9852e5": "L_a(x)", "60060e7dba234c6b6097b50a5af85855": " n^2 + m \\le \\lambda_n(D) \\le n^2 + M. ", "600635953c175e48828696199449fe16": "(X,\\sigma(X,Y))", "6006a0fd0610aa3e68856dbb70449f86": "n_i = \\frac {N_i}{V} e^{-\\dfrac{z_i q \\varphi}{k_B T}} = n^{0}_i e^{-\\dfrac{z_i q \\varphi}{k_B T}}", "6006aa893d9a10270d451dc1d78189e7": " \\int_{\\Omega} \\nabla u \\cdot \\nabla v\\, d\\Omega = \\int_{\\Gamma} u\\, \\nabla v\\cdot\\hat{\\nu}\\, d\\Gamma - \\int_\\Omega u\\, \\nabla^2 v\\, d\\Omega,", "6006c997813286548b7094527bc9bc69": "s,", "6007331166227bb82b2c2da4766a6d60": "\\nabla'", "60075a9c374e233d4618d8a93c0de5ec": " \\operatorname{init} = \\operatorname{value}\\ x ", "60075bbf9d3d7282c0d925239517ec72": "\\Big(\\frac{ij}{i+j}\\Big)", "60078c049c41347438613756bdc32f3f": " T_\\infty ", "60078ddb93eedcc0feb5aafed37c25ea": "\\begin{pmatrix}w_1\\\\ w_2\\\\ w_3\\\\ \\end{pmatrix}=\\frac{1}{1+\\frac{v_1u_1+v_2u_2+v_3u_3}{c^2}}\\left\\{\\left[1+\\frac{1}{c^2}\\frac{\\gamma_\\mathbf{v}}{1+\\gamma_\\mathbf{v}}(v_1u_1+v_2u_2+v_3u_3)\\right]\\begin{pmatrix}v_1\\\\ v_2\\\\ v_3\\\\ \\end{pmatrix}+\\frac{1}{\\gamma_\\mathbf{v}}\\begin{pmatrix}u_1\\\\ u_2\\\\ u_3\\\\ \\end{pmatrix}\\right\\}", "6007ca2daae3432a65314ea4d8b0dfba": "m \\times (m+1) \\times (m+2) \\times \\cdots \\times (n-2) \\times (n-1) \\times n \\,\\!", "600854e53c55d80fa53bcf3d8663f453": "\\vec F\\!", "6008647277c4454cecd97d33c069f0ca": "1.5", "600880ecf9ada1507882ea65b0690696": "\\gamma_a=g\\circ\\zeta_a", "60088beeae76504c742737e6a880c6ab": "i=0,1,\\dots,d", "60089d0cca1859b05e0b5faf5a2f0c0b": "x^{2} - n = 0", "6008b4c389730277f6ec8548a8d3a3b8": " a^n + b^n\\!", "60098d1bb40e8df77047739ff4311fd7": "L \\to X", "6009b8a5d4c2ebb93351f4afcbd5e598": " q \\in \\Z, ", "600a3a78f41fff192f99571504c47528": "-1.0316", "600b373c54c61895e8f5bab10ce83dc5": "(V,\\, \\Sigma,\\, ::=,\\, S)", "600b9fcc2080d33bdc0585014fe5f3d9": " E_0 = -(Z-S)^2 = - Z_e^2 ", "600baac2d374749aced9f7179a044646": "MUAA = \\frac{MUAC^2}{4 \\pi}", "600be7f2292243f399632f3d781ac38e": "\\frac{\\delta L}{\\delta r_j} + \\lambda\\frac{\\partial F}{\\partial r_j}=0 ", "600c1c3fd5c3cc5b37a531e011a209c6": "(f) = \\sum_{P \\in C}{\\mathrm{ord}_{P}(f)[P]}", "600c5b2b31a53a1f34b2ea3b52489085": "P_2(x) = f(a) + f'(a)(x-a) + \\frac{f''(a)}{2}(x-a)^2. \\, ", "600c9ada047bdf2b8e99e7ca200474ff": "D(\\alpha,\\beta)=\\sum_{i=1} \\sum_{j=1} |X_0 (i,j)-X_1 (i+\\alpha,j+\\beta)|", "600cf2b2ffa952199c2471a96727ce56": " U_{ni} = \\beta x_{ni} + \\varepsilon_{ni} ", "600d05214fc193e8e285bd11a47ff77c": "\\mathrm{bind} \\colon \\left( \\left( A \\rarr \\mathrm{M} \\, R \\right) \\rarr \\mathrm{M} \\, R \\right) \\rarr \\left( A \\rarr \\left( B \\rarr \\mathrm{M} \\, R \\right) \\rarr \\mathrm{M} \\, R \\right) \\rarr \\left( B \\rarr \\mathrm{M} \\, R \\right) \\rarr \\mathrm{M} \\, R", "600d338a17ce3d4b65c7a9d1bf640447": "L=2^{n-1-b}", "600d4226f506b4695661d7341a50f906": "= \\frac{\\partial \\phi^{\\alpha}}{\\partial x^{i}}\\, dx^{i} + \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}}\\, du^{k} + \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}_{i}}\\, du^{k}_{i} - \\chi^{\\alpha}_{i}dx^{i} - u_{i}^{\\alpha}\\left[ \\frac{\\partial \\rho^{i}}{\\partial x^{m}}\\, dx^{m} + \\frac{\\partial \\rho^{i}}{\\partial u^{k}}\\, du^{k} + \\frac{\\partial \\rho^{i}}{\\partial u^{k}_{m}}\\, du^{k}_{m} \\right ] \\,", "600d668521769a4382d93d538e0a6267": "g_{n+1}=\\mathrm{coth}^2(\\beta/4),", "600dd11eac7179716c42aeb112fccee7": "=2\\sin^2\\left(\\frac{\\alpha}{2}\\right)", "600e2e644ae35a26d28e22544b987cd5": "K_0(R[G])", "600e4e339b0cf7e3751738887b52757d": "v_p = \\omega/k, \\quad v_g = \\frac{\\partial \\omega}{\\partial k}, \\,", "600e6409d35f98acae37988006b8d43e": "u_0^n = u_b(t^n)", "600ee653f975bf85e5730bdef711228f": " q_e = \\int \\lambda_e \\mathrm{d}\\ell ", "600f325140c524e338d91a8dba7bd9ca": "W_{x}(t,\\omega)", "600f36f9782649ab9e7e6e96175abec0": "\\ -\\ln \\left |\\csc x + \\cot x\\right | + C", "600f390f6ba3fa46527e68e8256c0a2a": "\\mathrm{T_n}(\\mathbf{R})_{kp} = \\mathrm{T_n}(\\mathbf{R})_{pk}\n", "600f39507db4a4c0bb5e2e2a6f845088": "\\hat{A}_\\Sigma W_\\gamma [A] = 8 \\pi \\ell_{Planck}^2 \\beta \\sum_I \\sqrt{j_I (j_I + 1)} W_\\gamma [A]", "600f94713e83546a5cf51102bd85123a": "\\hat{f}(\\xi)=F_0(|\\xi|)P(\\xi)", "600fa9abc9850a41d474440b9529f0ad": "i_0 \\amalg i_1 : \\text{pt} \\amalg \\text{pt} \\to I", "600fb93b36baae793cbef142a6fb1349": "\\frac{31}{30}", "600fcc63875f0b17a78ae4ba96053793": " \\quad (5) \\qquad V_t = \\sqrt{\\frac{4 g d}{3 C_d} \\left( \\frac{\\rho_s - \\rho}{\\rho} \\right)} ", "60101c25f3179cba4fab87012bf3a882": " \\mathbb C^n", "60103b235759e5d247727107dffb6446": "V \\sim {\\chi'}^2_k(0)", "601095566d6697426f600ef4e58cf473": "Q_4=f_b(S_b)", "6010b7a1a344fe182cea9247a016c90e": "V=V_0\\oplus V_1.", "60111515ca8acc4d6e3c414024ed0a02": "[A] = \\sqrt[p+q]{K_{\\mathrm{sp}} \\over {(q/p)^q}}", "601177a0ee538026a3a952dd38c42125": "\\lambda_{\\min}", "6011903928e6d3e48bc143cff679c08d": " \\mathbf{A} \\colon \\! \\mathbf{B} = \\mathbf{B} \\colon \\! \\mathbf{A} ", "6011aa9ef1c57929a736a9c86f939acb": " \\phi_1, \\ ... , \\ \\phi_n, \\ \\chi \\vdash \\psi ", "6011d5fdd45fff97a1c753b0ef6ff4f3": "\\left[ \\begin{array}{ccc|c}\n2 & 1 & 0 & 7 \\\\\n0 & 1/2 & 0 & 3/2 \\\\\n0 & 0 & -1 & 1\n\\end{array} \\right] ", "60124975c75bb24aca0f024cdf7d26f2": " c_1 \\frac{dV_1}{dt} + \\frac{V_1}{r_{M1}} = g_{1,2}(V_2 - V_1) + \\frac {I_{electrode}^1}{A_1}", "6012d6c108eb3e00a684db7e880be9ee": "L(\\chi_1, s)= (1-2^{-s})\\zeta(s)\\, ", "6012f929a44bc38550fa596b79391015": "{E}_S[(K-P(S,T))^{+}] = KN(-d_2) - F(t,S,T)N(d_1)\\,", "60133b3de89607c92c3feff63833af71": "A(w)e_q(zw) = \\sum_{n=0}^\\infty p_n(z) w^n", "601341530f30ffce8390632db2aa2e4f": "0^0", "60136729f8a00b689363368bb079ce92": "s{\\in}S", "6013ae39504bb9ce586162b061eb6eb3": "\\psi(x) = \\begin{cases}\n\\psi_{\\mathrm L}(x) = A_{\\mathrm r}e^{ikx} + A_{\\mathrm l}e^{-ikx}, & \\text{ if } x<0; \\\\\n\\psi_{\\mathrm R}(x) = B_{\\mathrm r}e^{ikx} + B_{\\mathrm l}e^{-ikx}, & \\text{ if } x>0,\n\\end{cases}\n", "6013e185af95e14d05f12defebeacd80": "\\hat f: \\hat G \\to \\mathbb{C}", "6013ed9f314f412ac984cea538e31f66": "|a-b|x", "6015f018db7bc94e60e96601d6d16664": "X = X_1 \\cup X_2", "60167ec26ea318128987fa0755b0ac08": "(x,y,z)^T\\in \\R^3\\,\\!", "6016adcd7314e83c18c6d69233b979f6": "A^{\\mu_1,\\mu_2,...,\\mu_n}_{\\;\\nu_1,\\nu_2,...,\\nu_m}", "60172a9ee281f978dbb62491833a8579": " \\ge t ", "60174fc62d9c76e941648d3bae675b6a": "\\alpha = \\frac{2ax + b}{2}", "6017683ab5c7dc8a3279a3dcf10aa6ed": "(b+c)a +0a=ba+ca", "60177a3353e9001be992e763ec880dcc": "N(g)", "6017bd917dcd9b3b72f6bf83e64bb25a": "b_0 + b_1x + ... + b_nx^n", "6017c70ce0674c6f6c4285a4319e62fb": " \\mathcal{P}_2 ", "6018343020a2ad71cd6cede52aedb17e": " n > 2", "60184cfc90edd5d8bd0793cc3a9900e6": "\\xi_j", "60189e96816707e730c877398009b890": "q^{n}+1", "60194dce202c335d16aa45d52c9de2e2": "\\infty m", "60198835eeb147845d42e9bbdf7436d5": "a(i,j)", "6019ee89edbb54c15ee28899b7521b52": "\\mathrm{Mg}^\\times_\\mathrm{Mg}", "601a230a736480a65a33b34fbce0b327": "\ny(x) = y(x_0) + (x-x_0)y'(x_0) + \\frac{(x-x_0)^2}{2!}y''(x_0) + \\frac{(x-x_0)^3}{3!}y'''(x_0) + \\frac{(x-x_0)^4}{4!}y''''(x_0) + \\frac{(x-x_0)^5}{5!}y'''''(x_0) + \\mathcal{O} (h^6)\n", "601a9a01b39175e8f4a1919d2c868f50": "\nL_\\mathrm{W} = L_\\mathrm{p}-10\\, \\log_{10}\\left(\\frac{1}{4\\pi r^2}\\right)\\,\n", "601ad96a19b319298f89c9d8dcef9391": "\\exists \\alpha\\, \\varphi\\,\\!", "601af2f332f5c0c4da8bb00c2f60943b": "P_{rr} = \\frac{\\mathrm{outs}}{\\mathrm{unseen}\\,\\,\\mathrm{cards}} \\times \\frac{\\mathrm{outs} - 1}{\\mathrm{unseen}\\,\\,\\mathrm{cards} - 1}", "601b063074fe4852aff2f4acb1511e70": " EBAC = (0.806 \\cdot 3 \\cdot 1.2)/(0.58 \\cdot 80) - (0.015 \\cdot 2) = 0.032534483 \\approx 0.033 g/dL", "601b3a37119e37e54811909eecdda1fb": "\\mathbf{E}(t)", "601bd4787ffbcd8ec1462fcb3b205d1c": "\\sum_j\\int_0^\\infty \\psi_{ij}(t)=1 .", "601be940a3c8b6fdef899a30f93a22ba": "g=g^+-g^-", "601c0bc8a5f10a99e8cc5352492ff113": " V = q, E = f\\ (q\\ q) ", "601ca947b334bbbfbc7d74451423c43b": "x + y = z.\\ ", "601ce339e2f3b20dd5aac75fe54b7ed5": "\\mathbb C[X]_{n-1}", "601cfc308d6aeb00f422c27ae54cc624": "X = X_1, X_2, ... X_n", "601d32329df459e5fd1f96b724594cf5": "\nQ_R(\\mathbf{p}) = \\sum_\\mathbf{k} F^\\dagger(\\mathbf{k})\n\\left [ c_1(\\mathbf{p}/2-\\mathbf{k})a_1(\\mathbf{p}/2+\\mathbf{k})\n+ c_2(\\mathbf{p}/2+\\mathbf{k})a_2(\\mathbf{p}/2-\\mathbf{k}) \\right ] ", "601d8c13a9799361ececf6cb35813845": "\\scriptstyle{\\Delta x = 1 - x/A}", "601dcc8fefafeb48c4b852c37ee50050": "\\begin{align}\n\\varepsilon_{ij} &= \\frac{1}{2}\\left(u_{i,j}+u_{j,i}\\right) \\\\\n&=\n\\left[\\begin{matrix}\n\\varepsilon_{11} & \\varepsilon_{12} & \\varepsilon_{13} \\\\\n \\varepsilon_{21} & \\varepsilon_{22} & \\varepsilon_{23} \\\\\n \\varepsilon_{31} & \\varepsilon_{32} & \\varepsilon_{33} \\\\\n \\end{matrix}\\right] \\\\\n&=\n\\left[\\begin{matrix}\n \\frac{\\partial u_1}{\\partial x_1} & \\frac{1}{2} \\left(\\frac{\\partial u_1}{\\partial x_2}+\\frac{\\partial u_2}{\\partial x_1}\\right) & \\frac{1}{2} \\left(\\frac{\\partial u_1}{\\partial x_3}+\\frac{\\partial u_3}{\\partial x_1}\\right) \\\\\n \\frac{1}{2} \\left(\\frac{\\partial u_2}{\\partial x_1}+\\frac{\\partial u_1}{\\partial x_2}\\right) & \\frac{\\partial u_2}{\\partial x_2} & \\frac{1}{2} \\left(\\frac{\\partial u_2}{\\partial x_3}+\\frac{\\partial u_3}{\\partial x_2}\\right) \\\\\n \\frac{1}{2} \\left(\\frac{\\partial u_3}{\\partial x_1}+\\frac{\\partial u_1}{\\partial x_3}\\right) & \\frac{1}{2} \\left(\\frac{\\partial u_3}{\\partial x_2}+\\frac{\\partial u_2}{\\partial x_3}\\right) & \\frac{\\partial u_3}{\\partial x_3} \\\\\n \\end{matrix}\\right] \\end{align} ", "601ddd209a9caf0c0489e6a36ed40458": " T \\rightarrow 0", "601e721d021a2a44939399da548b9e94": "\\beta>1", "601f02229c7f0e0f76712989dec83d82": "D_F^y(x,y) = \\|x - y\\|^2", "601f4d7b1dfdfeb231d3895f3a23b28b": "S_k(n,r)", "601f6ccae5d3fec88a222753b0556c31": "\\min(x,y) + \\max(x,y) = x+y,\\;\\;", "601f90a79ff6594962068e22ef7ab07b": "n > 2N", "601fdd0e6033585359232927ef64666b": "S_{xx} = \\overline{ \\int_{-h}^\\eta \\left( p + \\rho \\tilde{u}^2 \\right)\\; \\text{d}z } - \\frac12 \\rho g \\left( h + \\overline{\\eta} \\right)^2,", "601ff8b59c58598fb4449674d7eba553": "F = a\\cdot X_1\\cdot X_2 ", "60204ca632e144d32fbecb4175fcbe86": " V(s) := \\sum_{s'} P_{\\pi(s)} (s,s') \\left( R_{\\pi(s)} (s,s') + \\gamma V(s') \\right) ", "602054ee2908d82d1a4738f1868311cd": " \\langle \\mathbf{u} \\rangle = \\frac{1}{\\rho}\\sum_i \\rho_i \\mathbf{u}_i = \\frac{1}{\\rho}\\sum_i \\mathbf{j}_{{\\rm m}, \\, i} ", "60208407d7a5dcdea8f9ad86e1827f4c": "|\\Psi^{(\\pm)}_\\epsilon \\rangle", "602096312e6f55dfa740b6ba1487c5f3": "c_{1}-a_{1}", "6020b6c3b39a48bc116237031cd65ddb": "a_j \\in A", "6020d07465934a3267b327cea4bc33a7": "C_D(v)= \\text{deg}(v)", "6020e15afda667c8063d98c344959bb1": " \\mathrm{Res}(f,\\infty) = \\mathrm{Res}\\left( {-1\\over z^2}f\\left({1\\over z}\\right), 0 \\right)", "6020eb5e3e44db58bc1a85c5778f73f3": "{p(\\mathbf{\\Sigma})}", "602114dbad7eaf2a61126ebef6404c71": "d \\le 2", "6021245a4bd4461af129bc2d0c9bef90": "4 \\times 4 - 6\\;", "6021278b474e8af293a2337d28c05224": "\\frac{dC_V}{dt}", "6021373695850171f109d5798d32d923": "\n\\delta \\left[ \\rho - \\hat{\\rho} \\right] = \\int D w\ne^{i \\int d \\mathbf{r} w (\\mathbf{r}) \\left[ \\rho (\\mathbf{r}) \n- \\hat{\\rho} (\\mathbf{r}) \\right] }, \\qquad (5)\n", "602173b8f4556414b3947b23c2804f10": "\n F = \\cfrac{1}{2}\\left[\\cfrac{1}{(\\sigma_2^y)^2} + \\cfrac{1}{(\\sigma_3^y)^2} - \\cfrac{1}{(\\sigma_1^y)^2}\\right]\n ", "602190fad2c27ad90866738050b9221a": "\n\\ddot{\\mathbf{r}} = \\ddot{\\mathbf{x}}_{1} - \\ddot{\\mathbf{x}}_{2} = \n\\left( \\frac{\\mathbf{F}_{21}}{m_{1}} - \\frac{\\mathbf{F}_{12}}{m_{2}} \\right) =\n\\left(\\frac{1}{m_{1}} + \\frac{1}{m_{2}} \\right)\\mathbf{F}_{21}\n", "6021adf6e31fb02219756f3f5574627a": "Z(\\omega) = \\{ \\mathcal{H} (x * y) \\} = \\sqrt{2\\pi} \\left( X(\\omega) \\left[ Y(\\omega) + Y(-\\omega) \\right]\n + X(-\\omega) \\left[ Y(\\omega) - Y(-\\omega) \\right] \\right) / 2.", "6021d0f7d7f6bf64144f586f9983ffb2": "V = {C \\over C + M}", "602201f10cc7efca2c53d84736326bfe": "\\sum_{n=1}^\\infty p_n", "60227aedbd6eb9ca02a73069d46a0354": "[ABO]=[ACO] \\,", "6022baed51f380b2f6b17e8af7c54889": " \\, P\\left(x;\\;J_\\nu(x),\\;J_\\nu '(x),\\;J_\\nu ''(x)\\right)\\equiv 0.\\! ", "6022e6112e1254a7e8ea7ecc0ca61709": "\n\\mathbf{p} \\cdot \\mathbf{q} \\equiv \\sum_{k=1}^{N} p_{k} q_{k}.\n", "60230ab3b9d0e9895e469ef812d6da36": " \\langle Y,Z \\rangle ", "60232bd1747f6f7ebf583206ddf7dcca": " \\operatorname{cl}(\\operatorname{cl}(X))=\\operatorname{cl}(X)", "6023429eafa01ab5fc213738ae67f7a3": "\\frac{\\operatorname dV}{\\operatorname dh} = \\pi r^2", "6023a83862eb2d88ef3a2b064aa6397f": "D_k(\\lambda,\\mu,\\nu;z)=s_k(z)", "6023d03fdaf37afac87b837ca220d148": "d_9 = 7 (d_1 + d_4 + d_7) + 3 (d_2 + d_5 + d_8) + 9 (d_3 + d_6) \\mod 10.\\,", "602484fe2d7c0ce7e36d5e38eb60bf73": "0\\;", "6024922a72c7f00da1e1cb15dcb22a00": "g_1, g_2, \\ldots, g_i, \\ldots, g_{k-1}, g_k", "6024d2ca91ee3d1819439fc8d1a0f169": "J(u_0) = \\inf\\{J(u)|u\\in V\\}", "60250179f34d64507aebb84bfc2a36d6": "X=\\sum_i u_i 1_{\\Omega_i}", "60250f0bf2af98f247351bc49d234f2f": " p_k=\\frac{\\partial L}{\\partial \\dot q_k}", "6025466685e4c8f7e706084c15d07372": "\\pi_1 = L^a \\mu^b k^c \\beta^d g^e c_p", "60257fd855d4ef862314974ed209e85c": " \\phi (r) \\,\\! ", "6025f0012646fa16088c93cdc2f4445e": "\\operatorname{Cl}_{2m+2}(\\theta) = \\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m+2}}", "60260fd5301201d24d0339903dc7cf6d": "V(t)", "60265122a798da989713a77292e06714": "z = S(r)=S\\left(\\scriptstyle \\sqrt{x^2+y^2} \\right)", "60266a22db7469d065aeaf457bda2293": " -\\left ( {\\partial p\\over \\partial V} \\right )_T = { 1 \\over {VK_T} } ", "60268e02a624bed3fd6d502d0a027e73": "g_n>0", "6026d6d4be34221ef3ad54e0c69ff219": "AP^2 = v_1 R_1 / r_2 = v_1 Q_3 / r_3", "60271a2a13fa89ae5e835de5d35e5b07": "|X| \\ ", "6027297c268b55c84648377933938117": "\\operatorname{Res}\\limits_{z=i}f(z)={e^{-t}\\over 2i}.", "602729a2e4192aaa037a5daa80d62a2f": "r_1=\\frac{-\\sum_{j=2}^n d_{1j}r_j}{d_{11}}", "60273466a7616d0f503ed1c6a2ebb29a": "|B|=|C|", "6027bf60c60f3a8407f7c60ec7095761": "\\mathrm{height}(t) = 0", "6027cfc43ba4ac89ba22fb1bd188159a": "= (q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). \\,", "602800a683078b95ba4975d0bc7d381c": "f_z\\,", "6028477ca58a508275089bd8303f5b75": " X_t = c + \\sum_{i=1}^p \\varphi_i B^i X_t + \\varepsilon_t ", "6028905a581f06aebca1c6b34887db1e": "t \\,", "6028dba9aba8c04c1a669aa7308f194e": "C_f \\equiv \\frac{\\tau_w}{\\frac{1}{2} \\, \\rho \\, U_\\infty^2},", "6029cd7245ab6d53c19957a6f79af7c9": "x_{t=2 \\dots T}", "602ac17675744ba7ef5eeb27cac1c085": "S_{xx}(-\\omega) = S_{xx}(\\omega)", "602b05401cd939f49882d7c1a8f1b061": " H_{jj} ", "602bcd4705730b08c3609311525b26aa": " {1 \\over {|\\mathcal{A}|\\cdot|\\mathcal{B}|}}\\sum_{x \\in \\mathcal{A}}\\sum_{ y \\in \\mathcal{B}} d(x,y). ", "602bd6cba51b094b522a2d9a89318122": "\\pi \\left(\\frac{{r_2}+{r_1}}{2}\\right)^3 \\left({r_2}-{r_1}\\right)", "602bf929df8af532d61fd54f448ccc92": "-\\beta", "602c9a72c3e81957cf55e0187803e4c2": "0.1\\overline{6}", "602ccba49fa527c828c9dba388a89300": " {\\frac {|AC|} {|DC|} \\sin \\angle\\ DAC = \\sin \\angle ADC}.", "602cd10fc82f85ef3a4238d3c4cd6061": "a - bi - cj - dk", "602cdec3d4a24940eb85720ad00c41da": "\\varphi(P) + \\varphi(Q)+ \\varphi(R) = \\varphi(O) = \\varphi(P+Q+R)", "602ce6af247925d42f7ed39add784f5a": " C_2", "602cf67c414284e2c5f26f07645dadb9": "\\int_{0}^\\infty\\frac{z^{-\\sigma-1}\\phi(z)\\,dz}{{e^{x/z}}+1}=0 ", "602d4d21fa30e1b023f46515904fba4d": "L_\\rho(\\gamma):=\\int_\\gamma \\rho\\,|dz|", "602db2b15830a84927650917233dce77": "v_\\infty^2", "602e7cda25a25ed139da4bf5d1f7c177": "K_{yy}", "602ea94f2adc56df666830c23e6697ae": "(f_{n})", "602eaf20f64a63d1ea8c1d8ef41880a5": "A = (j_1,...., j_{n-k})", "602edaef21388f72122620fcb9cb557f": "\\mu<1", "602f034bcf4bc6932c77645c0b6318bb": "\\begin{bmatrix}0&0\\\\1&0\\end{bmatrix}:\\mathbf b", "602f09ebbf4af8dbc6bf0520a5865ec2": "{\\dot{R}_n}", "602f16106c4e79cb2b31f2c415102694": " \\mathfrak g = \\mathfrak h\\oplus\\mathfrak m", "602fb31b753b99ef938c39eb5ab8809b": "P(s) = \\frac{\\pi}{2}se^{-\\pi s^2/4},", "602feb40ce4c401ba9b1678e68b341e2": "\\scriptstyle s", "602fefbb1a514ee5537732367be1bf88": "\\scriptstyle R(x) \\,-\\, R(x \\,+\\, 1) \\,=\\, f(x)", "60302ba95955af165cbc1a7c75c7e437": "\\frac{x}{\\lambda}=1", "603044414bcd5a0b42a627f174056277": "O(n, F)", "6030986a7961b4991ae4aa0f067336f9": "A_2=A_1\\left(\\frac{\\ell_2}{\\ell_1}\\right)^2", "60309b9033023d6543ad92736e40b897": "M_y=\\int_0^2 -4x^3-8x^2+32x\\,dx", "6031031cd13a960e62a6101b022706e5": "\n\\psi(q) = \\sum_{n\\ge 0} {q^{n^2}\\over (q;q^2)_n}= {1\\over 2 \\prod_{n>0}(1-q^n)}\\sum_{n\\in Z}{(-1)^n(1+q^n)q^{n(3n+1)/2}\\over 1-q^n+q^{2n}}\n", "6031138c4eb94020c56482c5fb15298d": "w = \\frac{F}{6 EI}(3 L x^2 - x^3)\\,~.", "60318b15aa14a266b29ecd3f019d3b01": "x_n, y_n, z_n", "60323e1924cf73c400221f50877c4609": " \\alpha \\in R^+ ", "6032c882a997191f5a152492fbeb8fa6": " y_{n+1} = y_n + hf(t_n,y_n) ", "603350698a32898699ecc6b53e4913fc": "\n\\Sigma_{cr} = \\frac{c^2 D_s}{4\\pi G D_{ds}D_d}\n", "603357d099ad67140d2e77734ac6fcde": "\\alpha_{12}", "603458fb9ee3e6c21b8e67f51c4a6f3e": "\\textstyle\\sum_{n=-\\infty}^{\\infty} x(nT)\\cdot \\delta(t - nT),", "603480e0072f9e0e48525ad26ae7d59e": "\\textstyle \\deg(b(x)) < \\deg(p(x)) = m", "6034ab3e96b1c370dfac50b83dfea4e9": " b_{ij}^{3/4} = \\frac{ b_{ii}^{3/4} + b_{jj}^{3/4} }{2}", "6034d19038d6676222a4693e12f67d92": "\\tfrac{1}{7710}", "6034db17764b35bd0126c0599881ac22": "c_i \\equiv c^{\\phi(n)/p_i} \\mod n", "6034e3e8a232659ce03b2ea10935ba29": "v = \\sqrt{\\frac{p}{\\rho}} = \\sqrt{\\frac{F}{\\mu}} \\,\\!", "6034f94b3764cbcd647fb3b0fc1fd243": "Pmf = \\cfrac{1 - Pmf + \\tfrac{1}{3}Pmf + 2 Pmf}{3}", "603553533a4aedeaa4629afca07d105d": "{\\omega^1}_3 = -\\frac{\\sin\\theta \\, d\\phi}{g}", "603570671f18cbdb7227140cb45b41f8": "F=F_{\\theta_n}", "6035bd6bc12ebaf49f76696fe0e5ac83": " 1 - \\sqrt{1 - \\delta} ", "6035e751148ca86f071dc488e61c9d96": "\\pi\\left(r^2 - y^2\\right)", "6035fdc3f33d9d410fdfe829401578ad": "q = \\sgn(y) \\left\\lfloor \\left| y \\right| + 0.5 \\right\\rfloor = -\\sgn(y) \\left\\lceil -\\left| y \\right| - 0.5 \\right\\rceil \\,", "60365e6e41c3ae270220841b7c718375": "p = \\sum_{C_{i} \\neq C_\\text{max}} \\textstyle \\int\\limits_{x\\in H_{i}}P(x|C_{i})p(C_{i})\\, dx,", "6036c0a0e80c0aa80630ae69a36a2f72": "\\ Z_{\\text{capacitor}} = -j\\frac{1}{\\omega C} = \\frac{1}{j \\omega C}", "6036ea7c8357c26344b90da3c7165620": " H_{2}^{+}", "6037017e1c90f957e50d2ea493b1c7ec": "\\scriptstyle f_\\mathrm{blue}\\,", "60375099dd904249901fb51a27c70110": " G_{\\text{imp}}(\\tau) = - \\langle T c(\\tau) c^{\\dagger}(0)\\rangle ", "60379b33fe667f9aec8975b4e8c05794": "X=\\Phi Y", "6037c00918acaf4761f488a8be58141b": "\\textstyle min_{a^{*}(\\theta_{k}w_{k}=1)}\\ E\\{\\left |W_{k}X(t) \\right |^{2}\\}", "6038012c42c4f4496830e13551e30c94": "e_a^i", "603852952a068ddcfaa5082d0b6164f9": "\\alpha \\to \\beta", "60388f51e8aab9070d79e8dacc391a71": "\n\\begin{align}\nU &= Mgr - mgr \\cos{\\theta}\n\\end{align}\n", "60389bb38068d35594d1dc14f1c32c8c": " \\ln \\left(\\sum_{i=1}^{k} e^{\\eta_i}\\right) = \\ln \\left(1+\\sum_{i=1}^{k-1} e^{\\eta_i}\\right)\n", "60390806286e5ae815be8ee220c6f48b": "{y_u}={y'}{y_c}=0.115(1.459ft)=0.168 ft", "6039640e8ad6820c586f108c95b3dbd9": "\\ F_{propulsive}= lift \\times sin(\\beta) - drag \\times cos(\\beta) ", "6039ea01409cecd93d57139c350cd78d": "\\frac{a\\ b\\ c\\ d}{e\\ f\\ g\\ h} = \\dfrac{d+\\dfrac{c+\\dfrac{b+\\dfrac{a}{e}}{f}}{g}}{h}.", "603a0bc608d86ff1198ccd6ac1596670": "\n\\le (b-a)e^{n (f(x_0) - \\eta)} + e^{n f(x_0)} \\int_{x_0-\\delta}^{x_0 + \\delta} e^{\\frac{n}{2} (f''(x_0)+\\varepsilon)(x-x_0)^2} \\, dx\n\\le (b-a)e^{n (f(x_0) - \\eta)} + e^{n f(x_0)} \\int_{-\\infty}^{+\\infty} e^{\\frac{n}{2} (f''(x_0)+\\varepsilon)(x-x_0)^2} \\, dx\n", "603a4c9e04776ab388551c4221b56c8e": "L(\\sigma) = \\lambda_1, \\,", "603a504afbbfd11c11e327b88e7d603a": " \\frac{v_{k+1} -2v_k + v_{k-1}}{h^2} = \\lambda v_{k}, \\ k = 1,...,n, \\ v'_{0.5} = v'_{n+0.5} = 0.\n\\,\\!", "603a71a0527e19e96ebd6f601a3b1a20": "F(k,\\phi)=\\int_0^\\phi\\frac{d\\theta}{\\sqrt{1-k^2\\sin^2\\theta}}, \\text{ for } \\left|k\\right| \\le 1", "603acf4fb4f19594fbcdf76f59a1aa0f": " \\operatorname{equate}[A, N] \\equiv A = N \\or (\\operatorname{def}[V[N]] \\and A = V[N]) ", "603b2c5ff112780f0b8199a45d92067e": "S(a, b) = \\{ a n + b\\, |\\, n \\in \\mathbb{Z} \\} = a \\mathbb{Z} + b. \\, ", "603b572b62019b578149a839cd885e4a": "H_{\\lambda e} = \\mathrm d(S_{\\lambda})_e (H_{e})", "603b78ce0b62046e4f5e397088c063b7": "a,b>1", "603bc185c1e95940156e64accf7c24f5": "p/2", "603bf622cddd2a05ac6c178a1c750d3f": "n_1,n_2,...\\;", "603bf845683a9ef05fa98fa6f5fb1335": "\\lambda^1 = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}", "603c270e58b9e89d475946c713f7df82": "[g^2]=[L^{D-4}]", "603c502df3521d425e41354d2b14efcc": "T : V \\rightarrow V\\,", "603c6b5f15ba91b924a2e298224507a1": "min(\\lambda_1, \\lambda_2)", "603c9e7feff9b12af41b2527028a9837": "v=V/N", "603caa635d0b7fe894e52c81b9f2e020": "\\phi = q\\cdot ( \\varpi - \\lambda_{\\rm N})", "603cd0f58e39a0504ae807f41c5725d0": "\\begin{bmatrix}x & y & z\\end{bmatrix} . \\begin{bmatrix}A_1 & B_1 & B_2\\\\B_1 & A_2 & B_3\\\\B_2&B_3&A_3\\end{bmatrix} . \\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix} = 0. ", "603cf090806c26fc15da23f1893dd531": "\\lambda\\in \\Bbb C", "603d5a1ace2b48ccca1d8a11c2f91684": "\n\\left [\n\\begin{smallmatrix}\n\\;\\,\\, 2&-3\\\\\n-1&\\;\\,\\, 2\n\\end{smallmatrix}\\right ]\n = \\left [\n\\begin{smallmatrix}\n3&0\\\\\n0&1\n\\end{smallmatrix}\\right ]\n\\left [\n\\begin{smallmatrix}\n2/3&-1\\\\\n-1&\\;2\n\\end{smallmatrix}\\right ].\n", "603d605bebcb774e9dd2a236e52eed94": "q \\geq 2", "603d9f54535c44b62921d64232f0472d": "f''(x) = \\left(\\frac{q}{p} \\right) \\left( \\frac{q}{p}-1 \\right)x^{\\frac{q}{p}-2}", "603df41844a41a6c1181220ced922db1": "\\ t_1 \\ ,\\ t_2\\ ", "603e19130cb4d94da504f8edfa8fb7ba": "\\lim_{n\\to\\infty} \\frac{f(n)}{g(n)} = 1.", "603e2156ca6f70278b9ca8c8f0a2908c": "x = \\sum_{i=1}^k a_i e_i", "603e269fb8efd960da36d8f502dddafe": "(p,l) \\in I", "603ec3f5d1af1d403d01f9e4088092d1": "p_1^2", "603ed4d5f525d8b71df932cfcbbf2c7f": "\\aleph_3", "603ee0a4c9b040246c91494a5deaeae7": "i : x \\mapsto \\overline{\\{x\\}},", "603f0fef59e08685827ad604f41b5784": "\\csc A = \\frac {1}{\\sin A} = \\frac {\\textrm{hypotenuse}} {\\textrm{opposite}} = \\frac {h} {a}. ", "603f8ef9fa1d4bbcb9e8d7792545f060": "\\frac{SS_{Error}}{DF_{Error}}", "603fbddb79f2b0559262aab7f3e2670c": "E\\to M", "603fca3157fde002ba3f81640fa94ebe": "\\nearrow", "603fd5f57c1d3d02e462f64d31ae15dc": "\\dot{x} = -a \\cdot \\sinh E \\cdot \\dot{E}", "6040114c8c5c153a8a78a14b7d1ad0b9": "\\tfrac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}", "604037f8a304c478246b0d0290eb2f22": "f(0)=x", "604058516570717b98c5a478033bb235": " \\mathrm{length}(c_k) \\approx -\\log_2\\left(p_k\\right)", "6040b686ccdf5ebf327cadcf2c14b6dd": "H(X,Y) = H(X) + H(Y|X)", "6040f16e6f9818837311a9f7d1522242": "C^*(\\theta)=x(p,w)", "604107c6f3ffd51df5d994f29693b051": " n \\geq k+1 ", "60411678c8baf51a6cb171d56c835ddd": "\\psi \\circ \\varphi = \\chi", "60416fb01d4788a1af5f028bef00800d": "c_f(u,v) = c(u,v) - f(u,v)", "60419594fb2d31dd798107c2ce0ea55b": " = \\sum_{n=-\\infty}^{\\infty} x(nT)\\cdot \\underbrace{T\\cdot \\mathrm{rect} (Tf) \\cdot e^{-i 2\\pi n T f}}_{\n\\mathcal{F}\\left \\{\n \\mathrm{sinc} \\left( \\frac{t - nT}{T} \\right)\n\\right \\}\n}.", "6041b4db988d70e5079acbb1890c30fe": "P(\\theta)\\propto{}\\exp\\left(-U(\\theta)/kT\\right)", "6041c7457fe328b738a217c78864be0b": "\\begin{bmatrix} H & H\\\\ H & -H\\end{bmatrix}", "6042253c26396745b9ea35db1662c3c4": " \\nabla \\cdot {\\mathbf B} = 0", "60427862d331def645844c27e7272517": "x_t=x_0\\cos\\omega t\\,\\!", "6042d3d93e22a75ae482a05eb22ff2c6": "w_{min}", "6042f011af9dd63a77aedda50ec4d149": "S_n =\\sum_{j=1}^nZ_j.", "6042fa9a8f15854bb8849519c4bc16fa": " J_\\nu^{(3)}(x;q) = \\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty} (x/2)^\\nu {}_1\\phi_1(0;q^{\\nu+1};q,qx^2/4) ", "604306800537f960edd15c5cef9fae98": "\\begin{matrix} {2 \\choose 1}{3 \\choose 2}{40 \\choose 2} \\end{matrix}", "604326f0880a017d9707ae0d13fd418b": "D[\\partial_i||]=\\partial_iD(p||p)=0", "60433f1696a7e10dbfe339a25c55163e": "\\mu_{eff} \\,", "6043b83a8c3319f431a78635850d5ab5": "0 \\equiv f(r + tp^k) \\equiv f(r) + tp^k \\cdot f'(r)\\pmod{p^{k+m}}", "6043f1c6722d95f2e0e886e604414937": "h\\ =\\ \\sqrt{\\frac{2\\gamma_\\mathrm{la}\\left( 1 - \\cos \\theta \\right)} {g\\rho}}.", "604442243078752af0d5d9a8980d2a8e": "VAG(x^3 -7x + 7,(0,1)) ", "60446c3ab00e8ae00bec302bd955498d": "\\mathbb{R}/\\mathbb{Z} \\cong S^1", "6044b73f888dc0a6ca5a60dee3ba7b1c": "(1-t)^{-n-1}", "6044be69bdd80f3acdba30585da4a6a7": "\nq_{yy} = \\frac{\\sum (y-\\bar{y})^2 w(x-\\bar{x},y-\\bar{y}) I(x,y)}{\\sum w(x-\\bar{x},y-\\bar{y}) I(x,y)}\n", "6044f38102238f7e8c0c2c110fe98212": "\\gamma_{\\pm} = \\left({\\gamma_A}^n{\\gamma _B}^m\\right )^{1/(n+m)}", "604531f450b85153555817328183d779": "c_{10}-a_{10}", "60459f855b600ca8362d7b27061a622c": "{13 \\choose 1}{12 \\choose 1}{4 \\choose 1}", "6045d6f00b01ef7d28b38060cf4df362": "B_\\ell^m = Y_\\ell^m (\\theta, \\phi) \\cdot S", "6045d8eb280e424490d94d12c724f26d": "Z = \\sum_{i>\\mathrm{He}} \\frac{m_i}{M} = 1 - X - Y.", "60460eff50998f5a7ab83ff76a1f6abe": "a_0 \\ne 0 ", "60465ce70d3b7baad624802f1b5efff4": " \\exists x_n A(x_1, \\ldots , x_n) ", "6046d5acce886bf199c3608a32607bde": "\\alpha(-1/2)=\\beta(+1/2)=0", "6046ee7d77a915dba00669ece30c5b20": "y = \\mu_{X1}\\, ", "6046fda708b9e9303f6bdc80283b93e4": "x^+ = x^0+x^3", "604737307675c0aca5e55309c0ac5187": "\\deg(P + Q) \\leq \\max(\\deg(P),\\deg(Q))", "6047ff9fdf45432b30efdac4ed4ebee4": "\\begin{bmatrix}0&1\\\\0&-1\\end{bmatrix}:\\mathbf a", "60481097f5726dcc7be18c4b606a07d8": "(n_1, n_2, \\ldots, n_d)", "604820f21468daddf0fb169b78c21152": "(a,a+1)", "60482b35f616cf27b9ab8b67dbb7cd8c": "V_n(R) = V_{n-1}(R) \\cdot R \\cdot B(\\textstyle\\frac{n + 1}{2}, \\textstyle\\frac{1}{2}).", "604849edc21163da8592e70af3082562": "156{\\pi \\over 4}", "604862231f5d6b931b0a44e6bc0628b3": "A = P\\cdot D\\cdot P^{-1}", "6048628c74572c7ce3c49d8a9b76f7ca": "a_0 = 0", "60491ad7a6a5f1f1d3365072c6721d26": "\\cos \\theta - \\cos \\varphi = -2\\sin\\left( {\\theta + \\varphi \\over 2}\\right) \\sin\\left({\\theta - \\varphi \\over 2}\\right)", "6049416f35b42a084638716bfd239c46": "\\gamma \\in \\mathcal{C}", "60497b6427aeb833941a45159af48fb0": "\n\\left[ - \\frac{\\hbar^2}{2m} \\nabla^2 + V\\left(\\mathbf{r}\\right) \\right] \\psi\\left(\\mathbf{r}\\right) = E \\psi \\left(\\mathbf{r}\\right)\n", "60498fe3456a153ee7278eddd8cc8687": "\\rho_T^{\\pm,0}", "604996ab985f15d1aeca4d5c607c5ec3": "\\tau\\colon H_*(B) \\to H_*(E)", "604a1e582c8b0410cfb984705b2ea679": "\\mu=1\\;", "604a57e88a108fe401595787a0315939": " \n Z =\n\\int D\\varphi \\; \\exp \\left \\{ i \\int d^4x\\; \\left [ {1\\over 2} \\left ( \\left ( \\partial \\varphi \\right )^2 - m^2\\varphi^2 \\right ) + J\\varphi \\right ] \\right \\} \n", "604a72385838503ae7811db5f9ffdc6c": "\\omega^n=(-1)^{n/2} x^*_1\\wedge\\ldots \\wedge x^*_n \\wedge y^*_1\\wedge \\ldots \\wedge y^*_n.", "604ab1957beb62a49d5501c191fd5c1e": "H_{00}", "604abb967b88a14a01e78a94da7286d7": "J = \\sum\\limits_{k=0}^{N} \\left( x_k^T Q x_k + u_k^T R u_k \\right)", "604addd7ce17bb5ba850dd9b64378e0e": "r^2-Ar-B=0,", "604b5289c64389d340c5b1a03a0d2b44": "\\ln q_j^{*}(\\mathbf{Z}_j\\mid \\mathbf{X}) = \\operatorname{E}_{i \\neq j} [\\ln p(\\mathbf{Z}, \\mathbf{X})] + \\text{constant}", "604bba36fe43e6ac798be38b52141fba": " \\boldsymbol{\\Pi} = \\mathbf{U} + \\mathbf{V} \\qquad \\mathrm{(1)} ", "604c613f5bda966def5c7b7f0eb12b0a": "X \\sim \\textrm{Landau}(\\mu,c)\\, ", "604cb7da2f48526cafba0ea086b7f636": "Q_{i+1}", "604d7e03da169135c6ca7079cf842e6b": "\\langle f, g\\rangle_w = \\int_a^b f(x)g(x)w(x)\\,dx.", "604ddb30f5d59fd4895a8233b2462692": "P(\\Omega\\setminus E) = 1 - P(E)", "604e0214f79bcb9284df654fe57284b8": "{6\\choose 2}{43\\choose 4}\\over {49\\choose 6}", "604e0e4fcd6ccd5c2b726a1e2dcf390f": " \\mathbf{g} \\left ( \\mathbf{r} \\right ) ", "604e126c524a57d1d7e566a6fef4e862": "\n\\Delta^1_{\\rm LONG}=\n\\frac{\\pi a\\cos\\phi}{180(1 - e^2 \\sin^2 \\phi)^{1/2}}\\,\n", "604e237d211707a2f37c8f7a595bdcdd": "i=51,\\dots,100", "604e3beaaddaed832557aa12333f4e62": "~|\\alpha\\rangle", "604e7aa50613d328dd881e049525b3f7": "\\Vert x \\Vert = \\Vert \\langle x, x \\rangle \\Vert^\\frac{1}{2}.", "604f4e29f3ef0af8ce7e441126b5943f": "\\scriptstyle\\cup", "604f756676cf92f45966895ebb0a9146": "\\frac{\\mathrm{d}p}{\\mathrm{d}x}", "604f8d0aa22253c09e4427dbf8063da4": "\\frac{m+n+3}{2}", "604fbd65b1f16e1c2301be6ff0cb080c": " \\mathcal{M} = \\mathcal{O}_X^\\times", "605013056e15a0d7dd52c5f7055654f3": "\\begin{align} \nW &= \\begin{vmatrix}e^{(2+i)x}&e^{(2-i)x} \\\\ (2+i)e^{(2+i)x}&(2-i)e^{(2-i)x} \\end{vmatrix} = e^{4x}\\begin{vmatrix}1&1\\\\ 2+i&2-i\\end{vmatrix} =-2ie^{4x}\\\\\nu'_1 &=\\frac{1}{W}\\begin{vmatrix}0&e^{(2-i)x}\\\\ \\sin(kx)&(2-i)e^{(2-i)x}\\end{vmatrix} = -\\tfrac{i}{2} \\sin(kx)e^{(-2-i)x}\\\\\nu'_2 &=\\frac{1}{W}\\begin{vmatrix}e^{(2+i)x}&0\\\\ (2+i)e^{(2+i)x}&\\sin(kx)\\end{vmatrix} =\\tfrac{i}{2} \\sin(kx)e^{(-2+i)x}.\n\\end{align}", "605015aef598e65a33231990ca610714": "\\delta Q = T \\mathrm{d}S\\,", "60505b17cddea323fa74a27bf7011535": "T = \\frac{1}{2}ab\\sin (\\alpha+\\beta) = \\frac{1}{2}bc\\sin (\\beta+\\gamma) = \\frac{1}{2}ca\\sin (\\gamma+\\alpha).", "6050749e8cdd0c50cc35f748278a7501": "n_P, n_N", "605132eda0fc57e148f77f8a64e2db6c": "\\mathcal{F}^{-1}\\{H\\}(t)=h(t)=\\frac{1}{2\\pi}\\int\\limits_{-\\infty}^{+\\infty} {e^{-\\frac{\\omega^2}{\\sigma^2}}e^{i\\omega t}} d\\omega=\\frac{\\sigma}{2\\sqrt{\\pi}}e^{-\\frac{1}{4}\\sigma^2t^2}", "60513a88312e620ced2f9cb2d046eefb": "1/\\gamma:1:1\\!", "6051994abd141d91e3083f1224d4e787": "A_x(\\eta,\\tau)=\\int_{-\\infty}^{\\infty}x(t+\\tau /2)x^*(t-\\tau /2)e^{-j2\\pi t\\eta}\\, dt,", "6051ce4caacde801a62e0809d9420ba2": "f : X\\otimes A\\to B", "6051dc1d5c88adcd808560fa67ec015b": "\\rho=a^{-3\\left(1+w\\right)}.", "60521f2949063d9bb3cbb1dad0a99692": "(1 - \\tan\\theta \\tan\\phi) \\tan\\psi + \\tan\\theta + \\tan\\phi = 0\\,", "60525df7caec469eeb4bc7b111343c83": " \\nabla P = \\rho \\mathbf{g}\\,\\!", "6052b6bb7299fa096b89dc0b218fd415": " \\operatorname{E}[X] = \\int_{-\\infty}^\\infty x f(x)\\, \\mathrm{d}x . ", "6052e62a4cf35c534b61436228c6b396": "X_{1/T}(f) = \\mathcal{F}\\left \\{\\sum_{n=-\\infty}^{\\infty} x[n] \\cdot \\delta(t-nT)\\right \\}\n", "6052ffd0e472cde8568ae130b28e483b": "dz^A\\wedge\ndc^i=-dc^i\\wedge dz^A, \\qquad dc^i\\wedge dc^j= dc^j\\wedge\ndc^i", "60530ced5bdb1990a2fcac506b734554": "\\Delta\\varphi=f", "60534d5558bfe084147895078d87d850": "P'\\in z", "605374b5190e48bb71d00f033e6af051": "|\\phi(x)|\\leq p(x)\\qquad\\forall x \\in U", "60538528872516d2e67e5e2c43c71ee4": "\n\\begin{array}{rcl}\n\\frac{\\partial}{\\partial a_i} E\\{e^2[n]\\} &=& 2E\\{ \\big( \\sum_{j=0}^N a_j w[n-j] \\big) w[n-i] \\} - 2E\\{s[n]w[n-i]\\} \\quad i=0,\\, \\ldots,\\, N\\\\\n&=& 2 \\sum_{j=0}^N E\\{w[n-j]w[n-i]\\} a_j - 2E\\{ w[n-i]s[n]\\} .\n\\end{array}\n", "6053b8e68e73f3e1de44802ef8698b61": "\\widehat u|_V", "60544a61a085b7617d36c7f42988e648": "\\tau = \\mathit{\\Omega}_{\\mathrm{P}} L \\sin\\theta,\\!", "6054531d29c98b685f15a8959202cd56": "\\textit{true} \\rightarrow \\textit{open}(1)", "6054866aabd371f281b9d0fe687f3aa8": "S\\subset M", "6054cceb0a097d5442eaa0300b18b285": "x_1\\binom{a_{11}}{a_{21}}+x_2\\binom{a_{12}}{a_{22}}=\\binom{b_1}{b_2}. ", "6054e442472a612dab92250ae26cebd0": "({\\Bbb C}^n\\backslash 0)", "6055241def467fd8902ba503dbc27913": " p_{1^{(n)}} = \\sum_{\\lambda\\vdash n} s_{\\lambda}f^{\\lambda} ", "6055986276ee287d85198233f3dba5e7": "(V,\\psi)\\,", "6055aef04e268c96d6b1edc8908afeea": "=[\\vec{L}\\cdot (\\vec{E}-\\vec{N})]^3=[\\vec{L}\\cdot (\\{\\frac{\\sqrt{3}}{2}-0; \\; \\frac{1}{2}-1; \\; 0-0\\})]^3=[-0.6\\cdot\\frac{\\sqrt{3}}{2}+ 0.8\\cdot (-0.5)+ 0\\cdot 0]^3=(-0.5196-0.4)^3=0.9196^3=0.7777.", "6055c4d0a884b9dc9bb5fe776aa7a753": "w = ke^z", "605628f761753dd0f89040532fcb898f": "r_{\\mathrm{g}\\text{ axis}} = \\sqrt{ \\frac{I_\\text{axis}}{m}},", "605661e75f542a499b2380b9c78544b5": "x \\mapsto f(a_1+c_1 x, a_2+c_2 x, \\ldots, a_n+c_n x),", "6056734da90d6ac81843fc74d2f52f9e": "r=\\sqrt{a^2+b^2}", "605673a735e773244ba0e127793c4d85": "j = 2, 3, \\ldots, k", "60567c2d95443903117a97482170278b": "\n\\nabla \\times \\nabla \\times \\mathbf{E} + \\frac{n^2}{c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{E}\n= -\\frac{1}{\\varepsilon_0 c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{P}^{NL},\n", "6056ed6ae1e9e2fbced4212de5aa2c93": " \\tau_w = \\frac {D \\Delta P} {4 L} ", "60572f5f0da8e7715aac6bf2d8c2d3fe": "A_3 = L U_3", "6057c02ee4513d964403a68a7cc80392": "\\mathbb{T}^1\\;", "6057c45afba5ffa661bd3597844d9d93": "\n\\dot{p}_y = -\\frac{1}{2}\\frac{\\partial V}{\\partial y},\n", "6057c6c6142b6e0522c58bab4796d0d9": "\\xi_i(t)", "6057c9d888f63bfcba31c49a151955bc": " F_q = T_A \\frac{\\partial\\mathbf{\\omega}_A}{\\partial\\dot{q}} - T_B \\frac{\\partial\\mathbf{\\omega}_B}{\\partial\\dot{q}}=0.", "6057d9705c4949b62c2c81d5f3a8b599": " \\mathbf{z}_i \\in \\mathbb{R}^{k} \\; (1 \\leq i \\leq n) ", "6057e7d5cefc634fbb406eecb23592a0": "u(x,0)=u(x,1)=u(0,y)=u(1,y)=0 \\;\\;\\forall (x,y)\\in(0,1)\\times(0,1)", "6057f959c14b61cd70bbe855bf36c0ae": "Fred(\\mathcal H)", "60584d6e7ac41267d682a215f089a5fa": " X(\\omega)", "605885cf2bb14e0dae104c8b81bf61cc": "U^H", "60588c597e9f2432b64327ee77ccfb14": "a=|\\eta|\\,", "6058906b8fe6fc8fb64bd7091d5515ad": "A=\\begin{pmatrix}\n\\pm 1 & 0 & \\cdots & 0 & 0 \\\\\n0 & \\pm 1 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & \\pm 1 & 0 \\\\\n0 & 0 & \\cdots & 0 & \\pm 1 \n\\end{pmatrix}", "6058b2273a981a02856d4895e57308c3": "\\gamma,\\ \\kappa,\\ \\sigma", "6058e2ed80c4931d7196d08e22e1ac5e": "n = 1, C_{01} = C_2, C_{11} = 0, C_{10} = C_1, m=1", "6058eae2847ea123d5078849c2f6c02a": "\\{x_1,x_2, \\ldots, x_n\\}", "6059210432e723a631028d1bb5134e93": "L = {1 \\over 2} m_1 \\mathbf{\\dot{r}}_1^2 + {1 \\over 2} m_2 \\mathbf{\\dot{r}}_2^2 - V(| \\mathbf{r}_1 - \\mathbf{r}_2 | ) \\!\\,", "605987bb2db5b3b208e421220fbd6468": "\\mathrm{d}(\\omega \\wedge \\eta) = \\mathrm{d} \\omega \\wedge \\eta+(-1)^{{\\rm deg\\,}\\omega}(\\omega \\wedge \\mathrm{d} \\eta).", "6059b8822a8dcede91253d8472f9efa3": " N(k,d)\\geq \\sum_{i=0}^{k-1} \\left\\lceil\\frac{d}{2^i}\\right\\rceil", "6059ccfd937925b148989173ea64d782": "(\\tan\\beta_2+\\tan\\alpha_1)\\,", "6059ce58d45ab14127625f29a7befb2b": "\\tfrac{10^n-1}{9}", "6059f0cede0850dae3660f07f920e3c2": "L(\\bar{\\xi}, \\Sigma_L) = R(\\bar{\\eta}, \\Sigma_R)", "605a055546ddb32f3f453a9cecc33aad": "x \\,\\bmod\\, y = \\frac{y}{2} - \\frac{y}{\\pi} \\sum_{k=1}^\\infty\n\\frac{\\sin\\left(\\frac{2 \\pi k x}{y}\\right)} {k}\\qquad\\mbox{for }x\\mbox{ not a multiple of }y.\n", "605a0da76fad6c0cbb6438295f10b57d": "\\sigma_X^2.", "605a425de5afdee44a462b32ba17ac3a": "M(\\vec X,Y) = \\left[ {\\begin{array}{*{20}c}\n {\\begin{array}{*{20}c}\n 0 \\\\\n 0 \\\\\n 0 \\\\\n\\end{array}} & {\\begin{array}{*{20}c}\n {\\mu _2 } \\\\\n 0 \\\\\n {\\Sigma _{22} } \\\\\n\\end{array}} \\\\\n\\end{array}} \\right]\n", "605a44958ecddd35e6ef17b9938fac9e": "(+0) + (+0) = (+0) - (-0) = +0\\,\\!", "605a808f137c4414db5a3a2dcdea05fb": "J_{abc} = R_{ca;b}-R_{cb;a}-\\frac{1}{6}\\left( g_{ca}R_{;b}-g_{cb}R_{;a} \\right)", "605a8e08fe0253ee0f16c489269de008": "f_{\\text{C}}=m_{\\text{e}} c^2/h", "605aa8091508c2c45ed1897ab86ff711": "\\frac{S}{N} = \\sqrt { \\frac{(\\frac{\\Omega}{4\\pi}\\epsilon_n \\mu_A)}{(1+ \\frac{I_{b}}{I_n}(\\mu_{T}+n))} } \\left(\\frac{\\delta\\mu_A}{\\mu_A}I_o^{1/2}\\right),", "605ab1c0a6e9045e49e2498b8e91ddb0": " f''(c) \\ge f''(x_0) - \\varepsilon. ", "605af397e1cc69735741917375f5bfa4": "{\\mathit{He}}_4(x)=x^4-6x^2+3\\,", "605ba4404e91d63999395696dc707d55": " j \\in \\{ |j_1 - j_2|, |j_1 - j_2| - 1 \\cdots j_1 + j_2 - 1, j_1 + j_2 \\} \\,\\!", "605bb6b12fe4c88d4659ff2e03faed8c": "\\zeta=\\eta=0", "605bc93433b5464b4f6aa8d4b14900bd": " \\cos C = -\\cos A \\, \\cos B + \\sin A \\, \\sin B \\, \\cos c .", "605bfc0ee1f0abb54dd7ece6a99dc64c": "\\tau_{j}", "605c3a414f160c2e57013a2c8db8c1c4": "\\mathrm{Ref}_l(v) = 2\\mathrm{Proj}_l(v) - v\\,", "605c3dce87e30cc765b1d8cb2cf35437": "\\left\\{ x\\mid x\\cdot y\\le 1 \\text{ for all } y\\in B \\right\\}.", "605c9523670d7ad74dad8116e9fa2a97": "\\overline{A}A = 1", "605ced173a063706a4bf49f42b81ff06": "O(n^{1+\\rho}kt)", "605d60472d57db165df7fd8aada4b7f0": "N_p(x)=\\frac{1}{(2\\pi i)^n}\\int_{\\mathrm{Log}^{-1}(x)}\\log|p(z)| \\,\\frac{dz_1}{z_1} \\wedge \\frac{d z_2}{z_2}\\wedge\\cdots \\wedge \\frac{d z_n}{z_n},", "605d7d224d31465c6575421e62d3dafd": "\\ \\mathbb{E}\\Big\\{X(f)V^*(f)\\Big\\} = \\mathbb{E}\\Big\\{V(f)X^*(f)\\Big\\} = 0", "605dbc799fcdc81dee82c89831c00d7c": "\\ \\dot{K}=\\frac{I}{K}-\\delta\\ = s \\frac{Y}{K}-\\delta\\ ", "605e1ebc83fe87603ea7a6b890f7ba99": " w_p = \\mathrm{width \\ of \\ plate} ", "605e2f5fae2994bb2121799b9de1ca49": " \\phi_P ", "605e490081451f2c41218f45bcdf63be": "a(t_0) = 1", "605e83571d2a814b70224fc210cc3cac": "\n\\operatorname{Li}_{-n}(z) = \\sum_{k=0}^n \\left( {-z \\over 1-z} \\right)^{k+1} ~\\sum_{j=0}^k (-1)^{j+1} {k \\choose j} (j+1)^n \\qquad (n=0,1,2,\\ldots) \\,.\n", "605eca36d22a3c0ffe044965309d77d5": "\\pi_{A'}", "605ed5fbf505dfdf5f919174f206dc13": " Rec (w',s) ", "605ee9c4da2cbae0710b5a3fd9a7fbf3": "f(n,k)=((f(n-1,k)+k-1) \\bmod n)+1,\\text{ with }f(1,k)=1\\,,", "605ef81b574f8ff82ffca471749d496a": "B(h, v) = \\langle f, v \\rangle \\mbox{ for all } v \\in V.", "605f1aacda090be9b248df5c1041eccd": "\\frac{1}{R_\\mathrm{total}} = \\frac{1}{R} \\times N", "605f3a0042f141632bfeb0eaccd00ace": "A=\\bigcup\\limits_{n=0}^\\infty \\{2^{2n},\\ldots,2^{2n+1}-1\\}", "605f55b195e4d5a830e4d9bc96664d00": "P_r=\\frac{P}{P_c}", "605fa0a40b96174c61781400291cb97c": "\\begin{align}\\frac1{2^{2n-1}}\\int_0^1 x^{4n}(1-x)^{4n}\\,dx\n&=\\frac{1}{2^{2n-1}(8n+1)\\binom{8n}{4n}}\\\\\n&\\sim\\frac{\\sqrt{\\pi n}}{2^{10n-2}(8n+1)},\n\\end{align}", "605fdec423f379cd231e4748f819ff2d": "M_{BC}^{f}", "6060811bd96168ecac28c2045d1e32d8": "0 \\to M' \\to M \\to M'' \\to 0\\ ", "606096914836f44fc7869ffe4cdae485": "\\lim_{h_{n+1} \\to 0} \\frac{ \\| D^nf(x + h_{n+1})(h_1, h_2, \\dots, h_n) - D^nf(x)(h_1, h_2, \\dots, h_n) - A(h_1, h_2, \\dots, h_n, h_{n+1}) \\| }{ \\|h_{n+1}\\| } = 0", "60609c8fb84124cc59aa7522d11d614c": " \\and T_5 = [F_5, S_5, A_5]::[F_4, S_4, A_4]::[F_3, S_3, A_3]::K_2 ", "60611e5ebe6a65c5e47abe1cc86a10a8": "\n\\begin{bmatrix} \n-1 & 0 & +1 \\\\\n-1 & 0 & +1 \\\\\n-1 & 0 & +1\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1\\\\\n1\\\\\n1\n\\end{bmatrix}\n\\begin{bmatrix}\n-1 & 0 & 1\n\\end{bmatrix}\n ", "606178063ebecae9181478925e332c47": "R_1 \\| R_2 \\triangleq \\left(\\frac{1}{R_1} + \\frac{1}{R_2}\\right)^{-1} = \\frac{ R_1 R_2 }{ R_1 + R_2 },\\,", "6061b2e9335913edae628b1bd28caf9a": "H^2(z)= H_0^2 \\left( \\Omega_M a^{-3} + \\Omega_{de}a^{-3\\left(1+w_0 +w_a \\right)}e^{-3w_a(1-a)} \\right).", "6062639bc19ce23285c0b5402fef2b8a": "\\frac{\\partial f}{\\partial x^j} = \\sum_{i=1}^n\\frac{\\partial y^i}{\\partial x^j}\\frac{\\partial f}{\\partial y^i}", "60626a51aaab3658d10e5edad8fb2938": "\nE_S = 4\\pi r_S^2 \\sigma T_S^4\n", "6062700dbd308fef68ebef8dfe5b5a3b": "\\lambda_{2}= -0.0000046 - 5.4280259i ", "60633da4949280fa1021da7108c3e8af": "x_{i+1} = \\frac{1}{3} \\left(\\frac{a}{x_i^2} + 2x_i\\right).", "60635f6ea1dcd91ec7ef4aba3272f958": "\\begin{array}{cc}\n \\begin{array}{c} \\\\ 3 \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & -12 & 0 & -42 \\\\\n & 3 & & \\\\\n \\hline \n 1 & -9 & & \\\\\n \\end{array}\n\\end{array}", "6063ab7e366769ac938d9442966405e8": " F= G\\cdot \\frac{m_1 \\cdot m_2} {d^2}", "6063b185b3606336dfeecb71dc381046": "\nh_{\\mu} = \\frac{1}{2} \\sqrt{\\frac{\\left( \\mu - \\lambda\\right) \\left( \\mu - \\nu\\right)}{S(\\mu)}}\n", "6063bd5712cf04e7c461d897eb9c906c": "\\begin{align}\n(f+g)(x) &= f(x)+g(x) , \\\\\n(f\\cdot g)(x) &= f(x) \\cdot g(x) .\n\\end{align}", "6063f7d81fc3755a79e01c4af8c51e77": "\\mathbf{N} = \\frac{1}{\\mu_0}\\mathbf{E}\\times\\mathbf{B} = \\mathbf{E}\\times\\mathbf{H} \\,\\!", "6064757a5909d7baedc352034bba218d": "(p-c)", "60648a146bc272b0990e87ec29fe1dc0": "x \\in \\mathcal A", "60648d5c99249f71979bc57a4b940c3a": " \\theta = \\omega _1 t - \\frac{1}{2} \\alpha t^2", "60652a06a3f27bcec101a57334976874": "W'_{in}=0 \\,", "6065aa4aada28cf18dca4e8ebbb6d549": "\\int_S {\\mathbf g}\\cdot \\,d{\\mathbf {S}} = -4 \\pi GM", "6065b7da4dbfaebc03948be828401c90": "dv_{ship}", "606622ac99818b38efb2685be9e69570": "E_{x}=\\left [ j\\frac{k_{x\\varepsilon }k_{z}}{\\omega \\varepsilon _{o}\\varepsilon _{r}}(Ae^{-jk_{x\\varepsilon }x}-Be^{jk_{x\\varepsilon }x})+ \\frac{m\\pi }{a}(Ce^{-jk_{x\\varepsilon }x}+De^{jk_{x\\varepsilon }x}) \\right ]sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \t\\ \\ \\ \\ \\ \\ (33) ", "6066b750d320818882e6d274216fffa7": "T_{ab} = T_aT_b + T_{a-1}T_{b-1},\\ ", "6066ddd4dbcb239f0451ff978ea867a3": "\\langle a,\\;b \\mid a^m=e,\\;b^n = e,\\;aba^{-1}=b^{k^{r}}\\;\\rangle", "606709bf187dc66c08adc2eb751ef954": "L^2({\\mathbb R}^n).\\,", "60671dccd48abcace22c92d0fe9639ac": "t' = \\gamma \\left( t - \\frac{vx}{c^2} \\right ) ", "60672cbaf2df0b193d949fc4e62523d5": "\\tfrac{c}{2}", "6067649a8a9885ade13bb74568fbff06": "\\;R(P)=0", "606868804fa58348fa4cfae04acd9885": "g(L,R)\\leq \\lambda g(L_1,R_1)+(1-\\lambda)g(L_2,R_2).", "6068adff9b1e60915a7381f4517f8a48": "R'(W) < 0", "6068da428be49b9815a72157f21012b0": "\\delta(A,Z) = \\begin{cases} -\\delta_{0} & Z,N \\mbox{ even } (A \\mbox{ even}) \\\\ 0 & A \\mbox{ odd} \\\\ +\\delta_{0} & Z,N \\mbox{ odd } (A \\mbox{ even})\\end{cases}", "60693df22db2aa5088d7980bec839a42": "x_{1:t}", "606943e955e55c0250a1fdc0130b91ff": "\nK_{11}x_{1}+K_{12}x_{2}=F_{1}\n", "6069911e8ff39e6228aaa74678f24420": "\\delta _\\nu^2 = P_r \\delta_ \\kappa^2.", "606a566390bc9e33320eb8ef3e634307": "w(a,-) = w(-,b) = w(\\text{mismatch}) = -1", "606a86e8c8cc481cea0ea2d96065160c": "\\partial_\\mu J^\\mu = 0", "606a9f55cb2dcbd155e3a6d166f26456": "1-x+x^2", "606b881a07e897775e160ab5c95ed8ee": "(A - S)", "606bbb16af51957b1f9010788f29aa3a": "\\textstyle g(\\alpha) = f(\\alpha \\mathbf{x})", "606bdbaa70792ab9b6eef71fdc90ea43": "f(P_i)=0", "606bf3bb124417c16781e27d617f7a48": "\\bot_{\\mathrm{Luk}}(a, b) = \\min \\{a+b, 1\\}", "606c11abc678b116937e5a0d2a9156eb": "\np_1 + 2q_1 + r_1 = 0.84325 + 0.15007 + 0.00668 = 1.00000 \\,\n", "606c1abe32296833fec41792c87472ce": "{\\sqrt{n} \\left[ \\log\\left( \\frac{X_n}{n}\\right)-\\log(p)\\right] \\,\\xrightarrow{D}\\,N(0,p (1-p) [1/p]^2)}", "606c68ae5463b4602647b25c3fc043a0": "\\omega(\\mathbf e\\, g) = g^{-1}dg+g^{-1}\\omega(\\mathbf e)g.", "606c9d1b0c612c8bbdae94dbce827893": "[\\cdot]^{-1}", "606cd3e3a2c72c49def0df2802d6aab5": "\\beta_j(p,q) = P(w_{pq}|N^j_{pq}, G)", "606ce1d188bed9d70ac56039d67394c8": "\\left|\\psi\\right\\rang=(\\left|x,y\\right\\rang - \\left|y,x\\right\\rang ) ", "606d0791639c07af9db26f446178f519": "f = \\sum_{l = 1}^\\infty\\sum_{m = -l}^{m = l}f_{lm}Y^l_m.", "606d6029e86e607fbad3fa4a1830bafb": " \\mathbf{R}^n ", "606d691c49b50f53332eb90619db849f": "\\scriptstyle z_{(k)} = (x_{(k)}-\\hat\\mu)/\\hat\\sigma", "606de07fd67c6840f9bc359e37d1b96c": "\\mathbf{F_{12}} = G \\frac{(-m_1) m_2}{r^2}\\mathbf{r_{12}} = G \\frac{m_1 m_2}{r^2}\\mathbf{r_{21}} = \\mathbf{-F_{21}},", "606df22a9cb7c162c9b7d2224fc4f342": "A=A_3A_2A_1A_0", "606e173b0c0ace5da18a7d7e5568eeed": " \\langle x_1 x_2 x_3 ... x_{2n} \\rangle", "606e37afcfccff4dba6b84c106a0ac70": "\\mathcal{O}_x \\subset k(X)", "606e9faf921589a15b864745e4717d08": " {\\rm rank}(\\mathbb Z^n)=n.", "606edfd207d2f54bfcefa8ed7d323d60": "\\Pi(0)", "606f95cbcf3fa38c149446d29a481a32": "\\,\\Delta_1(x-y) = \\Delta_1(y-x).", "606fbdba4d22b51ad762f91e8cfba799": "H_1(M,\\mathbb{R})", "606ff0f46df11a50ff55f75824b956a5": "e_j = - \\frac{X_j^{1-c} \\, \\Omega(X_j^{-1})}{\\Lambda'(X_j^{-1})} \\, ", "6070247d00be973e9a27e0461824006e": "[Y]", "60702b0b5e9ae57ad98756dfffcba74a": " x_t=x_0 e^{-\\theta t} +\\mu (1-e^{-\\theta t})+\n{\\sigma\\over\\sqrt{2\\theta}}e^{-\\theta t}W_{e^{2\\theta t}-1}. ", "60706a5d5f4414128d2ceb27db85905d": " H_t( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ K_t ( \\mathbf{w} ) + \\lambda S_t ( \\mathbf{w} ) ", "6070f49ffbf1199949a6cab3fa717713": "\\mathrm{R_3N + RCl \\longrightarrow R_4N^+ \\ Cl^-}", "60710b66149f5d3c323e860c2976b0ed": "H_1: \\theta=\\theta_1", "6071398d6d9813032afd2713727cec15": "A = A_\\alpha dx^\\alpha", "607164a41d8235bcba18782afdae570f": " p(\\theta| X_t, A_t, O)", "60719614517318b1e403fd5839850b42": "|S(\\rho)-S(\\sigma)| \\le T \\log (d-1) + H[\\{T,1-T\\}] ", "60722028ea16244942e3faacf50382b2": "\\scriptstyle S^-", "60726eba00afb2a0e12fb940d9c8fc56": "\\,a - 1", "6072c8bac637d06929861964b8d7c22e": " \\langle f\\rangle_1 \\ge \\langle f\\rangle_2", "60731a1b4ad0b2d6bae727204ee23d8a": "\\eta(10) = {{73\\pi^{10}} \\over 6842880} \\approx 0.99903951", "6073422b3979035bfe697fdb4417f047": "\\hat{x}(k+1) = A \\hat{x}(k) + L \\left[y(k) - \\hat{y}(k)\\right] + B u(k)", "607350173d9ce745cbe32868fe87b86e": "1_{\\{\\}}", "6073737335b5985a76b1da97c889f49b": "\nT(m,s,x) = G_{m-1,\\,m}^{\\,m,\\,0} \\!\\left( \\left. \\begin{matrix} 0, 0, \\dots, 0 \\\\ s-1, -1, \\dots, -1 \\end{matrix} \\; \\right| \\, x \\right).\n", "60737fa7a569e033e61b853f01b0774b": "V_X-V_Y=0", "6073d071ba09d8c85873f66d0c4330d0": "\\operatorname{E}(\\ln T)=\\psi\\left(\\alpha\\right) - \\ln\\beta", "607427d4436702e7be8065e5069fc661": "\\frac{z}{e^z-1}=\\sum_{n=0}^\\infty \\frac{B_n}{n!} z^n ", "60743b732e59d8fa88fca6a466be9886": "k_4=k_1+k_2\\pm2\\sqrt{k_1k_2}.", "607444deecad172fab58feb0960005d9": "z^K = \\alpha", "60744e149e008feed7e6cd6b5cb465d9": " \\forall p ", "6074a3862d5f462ac5c9dadfc532c930": "\\int x^k d\\mu(x) = m_k", "6074d3f210e4d6fd87e685eff571ffb3": "X(\\omega) = \\begin{cases}\nj & \\omega < 0 \\\\\n0 & \\omega = 0 \\\\\n-j & \\omega > 0\n\\end{cases}", "607544b1224809106369dbbe14ab9ffa": "H=T^{0.5+\\varepsilon}", "6075ebeab0edc6f3922b84131c8c7ed6": "not~p", "6075ee9c77c57c01b16efc87a5aa68b8": " s_\\mu^* s_\\nu = \\delta_{\\mu , \\nu} s_{s ( \\mu )}", "6075f7cdb95e2579592472cc7ebd5b68": " K_p = \\cos\\beta \\frac{\\cos \\beta + \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}{\\cos \\beta - \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}", "6075ff18cd2e5275cf6c3c1a74126e12": "\\mu_{j}", "607600e77c7894c42a73f94c51b20107": "P_{v_i} (v_0,v_1,...,v_n) := (v_0,v_1,...,v_{i-1},\\tilde v_i,v_{i+1},...,v_n)", "60762331aad532190a96ff493f363c42": " 0 \\le i \\le n, \\, ", "607659e7dc542b25f1d168888c7d7c0d": "\\sigma_\\mathrm{bg}", "6076c8766ad6d663bc3208beab03f155": "(1-R)/2", "6076e9f9795b3fe6022b442ddc5667bd": "(-,-,+,\\cdots,+)", "607708482f5a2afc26174c21834baad3": "i\\ ", "607779e5abfd82fde21d512e5c359123": "\\vec{x}_{n+1}=\\vec{x}_n-J_n^{-1}\\vec{F}(\\vec{x}_n).", "6077cfa9a8d2fea7b826bd4241e9cd56": "\\,\\,\\cdot\\!\\!\\!\\!\\!\\bigcup_{i\\in I}A_i", "607852d3ce9964f608580a3b1867b8b0": "A x=b", "607880d0c77c858064ccf3e831fb9632": "\\mathrm{d}S_z= \\rho\\,\\mathrm{d}\\rho\\,\\mathrm{d}\\varphi.", "60792e2b4d701ea72448e054c3e80aed": "(8.a)\\quad \\gamma_{,\\,\\rho\\rho}+\\gamma_{,\\,zz}=-(\\nabla\\psi)^2 ", "60794875e05e63716ddf1f750ea036d4": "r_1 r_2 \\cos\\varphi \\, ", "607971e9d0647ef56471e7d65d8e9f59": "Ly=0", "60797832ab8905bc192ca342dadfe8e6": "s_n = \\sum_{k=0}^n b_k \\, ,", "607aa15b19e37eb66d39fa87f8d08ce1": " \\mathbf{x}^{(0)}", "607aa294004414a908d2e4ff26cd9c36": " \\frac{1}{\\mu} + \\pi_0 \\frac{\\rho (c\\rho)^c}{\\lambda (1-\\rho)^2 c!}.", "607acaa73c762411b20745149a11e90b": "n\\times n", "607adac31c8e0cfcd49ab723c40474ec": "[\\mathbf{x}] = ([x_{11}, x_{12}], \\dots, [x_{n1}, x_{n2}])", "607b008d2994b4edb32bb69278ab7c2a": " MW", "607b03d290a5f458a7c9421d349474f5": "H_{kj}", "607b22aba179f0956858491de77ba81b": "\\mathrm{diam}(M):=\\sup_{p,q\\in M} d(p,q)\\in \\mathbf R_{\\geq 0}\\cup\\{+\\infty\\}.", "607b5feb7428b07f9292d774658c41b3": "\\frac{m}{s}", "607b85a7835f6c738e5e51ebe6628ccc": "\\ominus \\!\\,", "607c1f47a95c7e3a752e3687f9e583bd": " y^2\\ne -b/a ", "607c4944841c6a9e7679217144331a43": "\\hat{n}_i = b^{\\dagger}_i b_i", "607c6270e80a7857c9ab3c38e2cbdfb7": "\\phi(q) = \\sum_{\\delta|q}\\mu\\left(\\frac{q}{\\delta}\\right)\\delta.", "607c73e12e6e36806e12a84263dc2824": "\\mathbf{t} = N \\mathbf{1} =\n \\begin{bmatrix}\n 4 & 4 & 2\\\\\n 2 & 4 & 2\\\\\n 2 & 2 & 2\n \\end{bmatrix}\n \\begin{bmatrix}\n 1\\\\\n 1\\\\\n 1\n \\end{bmatrix}\n =\n \\begin{bmatrix}\n 10\\\\\n 8\\\\\n 6\n \\end{bmatrix}.\n", "607c773073bdfad48823086886d4466f": "E_{\\text{half-cell}} = E^0 - 2.303 \\frac{RT}{nF} \\log_{10} \\{ M^{n+}\\}", "607d398503c57d8ffb5fb4c47917599b": "{2}\\, ", "607d6229f3c01903e396c5de6bac6913": "\\delta_q : C^q(\\mathcal U, \\mathcal F) \\to C^{q+1}(\\mathcal{U},\n \\mathcal{F}) ", "607e044673f8a7b4db06c5d2fc107ac5": " \\begin{align}\n\\varphi_x &= J\\varphi\\Omega+U\\varphi \\\\\n\\varphi_t &= 2J\\varphi\\Omega^2+2U\\varphi\\Omega+(JU^2-JU_x)\\varphi.\n\\end{align} ", "607e0cdf9b5c374bdc22518ca580b518": "\\rho_0(z_0)", "607e29e8b2b604076b33209a17772e88": "Z_0=\\sqrt{\\frac{L}{C}},", "607e7bfbf375d382debbbe0ed445befa": "M_D = -13.657 \\ kN \\cdot m ", "607e8b943b2c22d76978afd3e7bdce68": "\\frac{1}{13} = \\frac{7}{91} = 7 \\times \\frac {1}{91} \\approx 7 \\times \\frac{1}{90}=7 \\times \\frac{40}{3600} = \\frac{280}{3600} = \\frac{4}{60} + \\frac{40}{3600}.", "607e9dedf17bd9e21093fc61d479b44c": "\\vec{g}=-\\frac{Gm_\\oplus}{{R_\\oplus}^2} \\hat{r}", "607eb0fa694cbd4c20141f4bcd3849c0": "\\,E[E[X|Y]]=E[X]", "607ed6f412d0e392984032a7a40822a4": "\\left[\\lambda\\mathbf{R}_{x}(n-1)+\\mathbf{x}(n)\\mathbf{x}^{T}(n)\\right]^{-1}", "607f1abd48f32d5fcd9648c8e9e65407": "dV_\\beta = dV_\\beta^e(1 -V_\\beta /V) \\,\\!", "607f8ee0f9a2b82404f5b122fc5ee794": "m_q = \\sqrt{c} \\, \\bmod \\, q", "607f9b3380f955f5458a73d7e690a957": "d_n:H_n(X_n,X_{n-1}) \\to H_{n-1}(X_{n-1},X_{n-2}) \\,", "607fafc0fe6d50f5f7bf7f132d8b0338": " \\mbox{2. } \\quad \\sum^\\infty_{n=0} a_n = \\infty, \\qquad \\sum^\\infty_{n=0} \\frac{a^2_n}{c^2_n} < \\infty ", "607fb574ae3695b3742e77ac0cb16f5e": "\\scriptstyle M_{V_{\\ast}}=8.23", "607fe99ab5b2d5b243e2897aa5695d83": "X_n\\wedge X_m \\to X_{n+m},", "607ff9133860e61661f6ab8dccfbaa16": "S(A|B)", "607ffa80f053f356e152b2c65f2c739c": "\\operatorname{E}[r] = \\int_0^1 r \\cdot f(r | H=7, T=3) \\, \\mathrm{d}r = \\frac{h+1}{h+t+2} = \\frac{2}{3}\\,.", "608006a2f3ba5ea96a2052e089cf0a30": "\\frac{di_i(U_g)}{dU_g} = -en_i S_F \\frac{eZ_i}{M_i}f\\left(\\sqrt{2eZ_i U_g /M_i}\\right)", "608014242f024f16125fd8ace682feb0": "(M, \\mathcal{B} (M))", "608033a84fa91fc324ecdde4bf4b2a7c": "Rep", "6080417df394047211ecde2ba01a92d8": "{\\mathcal P}:=\\{A,B,C,D,\\infty\\}, \\quad {\\mathcal Z}:= \\{ z \\mid z\\subset{\\mathcal P}, |z|=3\\}", "60806812cdabdc3b795cb75f61831110": "\\frac{1}{12}(i+\\alpha-1)", "6080809ce8711fc376b2406aae53ebca": "p(z)=p_0+p_1 z+p_2 z^2+\\cdots+p_m z^m,\\;q(z)=q_0+q_1 z+q_2 z^2+\\cdots+q_n z^n.", "60812170c085f9c878bc1bc070386795": "B^{-1}C", "608191360d20fa45fb5bae8f833d50a5": "Q(X,p(X)) = 0", "6081c396542461f6ffae96533d47e0fc": "\\displaystyle ~\\theta~", "6081c3e29a1c20fe82aee8fd5db28be0": "\\mathfrak J^{-k}(b)_n=a_n=\\sum_{i=0}^nk^{n-i}\\binom{n}{i}b_i.", "6081c6b7b2d44d2906d417e9b230e841": "X_t = M_t + A_t", "6081debaee5d8ba6145cabda8609518f": "\\frac{1}{2}(y_1+y_2)/R", "6081e74d5013344db5c9db96e72b9b42": "(a_{2,12}) = 1", "608258e7ebce8a6e85d093c3a7995dea": "P_1 \\uparrow G", "60825efb25066ca885941e2cb37b983c": "\\mathcal{D}\\bold{g}", "608267c5612d4da89f8272f75cdbd2f2": "s=\\sigma+i\\omega \\,\\!", "60826c1e50a08157a7d390fcf5ae7fd9": " x_j ", "6082ab8ef4e7e5953974b007c686208b": "s_L=t_L=s- \\frac{P(s)}{\\bar H^{(\\lambda)}(s)}", "6082bd3e7da3524aa27758e9e72c8349": "SD_{Ballistics} = \\frac{M}{d^2} \\approx {p}", "6082bec5cba4cb9c0bd6ac431cb4c3a9": "N_j(\\pi)", "6082eff33b2a0aac3a367e3e53b08de9": " |\\alpha|^2 -|\\beta|^2=1.", "608328dfa2bb00a63c41d69df7a80bf1": "G_1 \\approx 1", "6083438a47bf0e7693e0f012c294685f": "\\bigstar\\bigstar\\bigstar\\bigstar\\bigstar \\mathbf S", "608360c2d3bccdfa965c2339041f3eca": "a_t = \\pi(s_t)", "6083683d42d8a8b48904a1087935c9b3": "x\\,\\bmod\\,1", "6083bf979e453caa307ca492e759a94b": "G = V^* V", "6083e34be23e8b9703191767dfc3a106": "F[r]=\\frac{1}{2}\\int_a^t r^2 dt", "6083ebe2becdfa48f20997f222603bba": "C_r = \\sum_{n=1}^N (C_m)_n", "608430f6762f4fdb9bf662ea88b80762": "I(p_{t_m},p_{t_n},q_{t_m},q_{t_n}) \\cdot I(p_{t_n},p_{t_r},q_{t_n},q_{t_r})=I(p_{t_m},p_{t_r},q_{t_m},q_{t_r})~~\\Leftarrow~~t_m \\le t_n \\le t_r", "60844b0efc7059905121ad2256593457": "m(T)=m^2-\\lambda ^2/2", "6084fd85db44e717d527f1b0ab944145": "\\bar\\eth\\eta = - (\\sin{\\theta})^{-s} \\left\\{ \\frac{\\partial}{\\partial \\theta} - \\frac{i}{\\sin{\\theta}} \\frac{\\partial} {\\partial \\phi} \\right\\} \\left[ (\\sin{\\theta})^{s} \\eta \\right].", "60852d6311b8e740c0cf4d15abbe061c": "r_{A}", "60857013724d29330f10aa1054e9c813": "E\\left({S \\over N} \\mid n,s=0,N=10^k \\right)\\approx {0.434294 \\over nk}", "6085f5fe76a59fac62abd70cb05ae7a9": "\\begin{align}\nx(u,v) &= 2\\cos(v)\\sinh(u) - (2/3)\\cos(3v)\\sinh(3u)\\\\\ny(u,v) &= 2\\sin(v)\\sinh(u) + (2/3)\\sin(3v)\\sinh(3u)\\\\\nz(u,v) &= 2\\cos(2v)\\cosh(2u)\n\\end{align}", "60860858b8669b94a7d747afc65b2e45": "h_\\infty", "608625e28a0293ce5bad63778d3d922a": "s=(p_1+p_2)^2=(p_3+p_4)^2 \\,", "608658cf6882e6a2082bfc5bc986815b": "{\\color{Blue}~6.2}", "6086b1b72695d8b910098efd2c701339": "\\mathcal{L}\\{f(t)\\}\\sum_{i=0}^{n}a_is^i-\\sum_{i=1}^{n}\\sum_{j=1}^{i}a_is^{i-j}f^{(j-1)}(0)=\\mathcal{L}\\{\\phi(t)\\}", "6086b8c24732aedf9ad257425aa7a3d5": "\n\\frac{d\\sigma}{d\\Omega} = |f(\\theta)|^2 \\;.\n", "6086c83727d67c8d436e8792b006f309": "\\phi_n(x)\\to\\phi(x)", "6086cadc9bb407b159483ee892300861": " \\sec \\theta = \\frac{c}{a} \\ . ", "6086d9ae1f8a8f1cff4d5fae1811d8e9": "\\{\\langle SS \\rangle\\}", "60876436a163325c0b25f04f4c0465f2": "\\displaystyle{\\| B \\| \\le {1\\over 2\\pi} \\int_0^{2\\pi} |\\varphi(\\theta)| \\cdot\\|A\\|\\, d\\theta.}", "6087b1a615e15a85f03a88fcb515eb73": "R(\\cdot)", "6087d3f7a9f8c578103310e18abba854": "X(\\omega) = { \\sin[ \\omega (M+1) / 2 ] \\over \\sin( \\omega / 2 ) } \\, e^{ -\\frac{i \\omega M}{2} } \\!", "6087ddc507320df07f5ef13a471339bf": "M_{unit} = {y^2 \\over 2}+{q^2 \\over gy} ", "608819dbd1813b85976b6a9b48158ce6": "u(x,y)^2+v(x,y)^2=1\\qquad\\text{(ellipse)}", "6088334a5121301a79bc8eb5e0545427": "Q_{i-\\Delta}", "608862fa2c1553ce697a941524978e63": "R_{\\mu \\nu} = 0 \\,.", "6088b3652885730b6ee898550674eea4": "(1-x^2)^{-1/2}\\,", "6088c0cc26884bac06151cdf6ed3f493": "f(x) = \\int_{-\\infty}^\\infty \\hat f(\\xi) e^{2 i \\pi x \\xi} \\, d\\xi", "6088ec06bd479d27cf0b483f33d9ac88": " f(x) = \\left(\\sum_{n=0}^\\infty b_n (x-c)^n\\right)\\left(\\sum_{n=0}^\\infty d_n (x-c)^n\\right)", "6088eeaf4290dcc3e3b272727de4037d": " a^ 2 = \\frac{ Q( Q - R ) } { 1 + R( Q - R ) }. ", "608921ee0cc347cd4dc2b7e320523ac1": "\n4\\Phi_n(z) = A_n^2(z) - (-1)^{\\frac{n-1}{2}}nz^2B_n^2(z)\n", "608952936cced06692d143b29459a73d": "\\mathcal{H}_{1}\\approx 0", "6089c80d6e9d28500d495cb370ddbc9c": "\n\\frac{\\exp(-\\coth(2t)(x^2+y^2)/2 - \\text{cosech}(2t)xy)}{\\sqrt{2\\pi\\sinh(2t)}}.\n", "608b12f0b3f3ccdb1ebeffba9922b3cb": " \\frac{u_{i+1}-2u_{i}+u_{i-1}}{h^2}-u_i = 0, \\quad \\forall i={1,2,3,...,n-1}.", "608b1dc4ec0f186103474c4c932ec45c": "\\mathbb{C} = \\mathbb{R}^2", "608b260c59aceab837a8854655857ccf": "(A,+,\\cdot,-,0,1)", "608b7d719a90cfb07e38c4928953bd5f": " T_{\\mu\\nu}:= \\frac{-2}{\\sqrt{-g}}\\frac{\\delta (\\sqrt{-g} \\mathcal{L}_\\mathrm{M})}{\\delta g^{\\mu\\nu}} \n= -2 \\frac{\\delta \\mathcal{L}_\\mathrm{M}}{\\delta g^{\\mu\\nu}} + g_{\\mu\\nu} \\mathcal{L}_\\mathrm{M}.", "608ba9e363a26f695be31219886e1d93": "p-q", "608be6a1fc454dabf666840510d92cf6": "\\frac{g}{l}3}_{\\color{Violet}\\text{predicate}}\\,\\}", "60a005c15c08a2e3e43a5e47c504b499": "\\rho_{\\mathrm{vac}} = \\frac{\\Lambda c^2}{8 \\pi G}", "60a09cdeb2b004f695334001e1f1c4e8": "p_\\alpha \\in C_\\alpha", "60a0ecdfe7d28cc4d28492d921269b45": "\\lambda_n=n(n+1)\\,", "60a11443a27708c46c4f33e0ab1b64b1": "\\mathbf{R}^\\mathbf{R}", "60a11fae59fc199e4f058db3dd0da04d": "\\nabla \\times \\mathbf{E} = \\hat{\\mathbf{k}} \\times \\mathbf{E}_0 f'\\left( \\hat{\\mathbf{k}} \\cdot \\mathbf{x} - c_0 t \\right) = -\\frac{\\partial \\mathbf{B}}{\\partial t}", "60a151d26b732823c54e35a45fc8e548": "v \\in \\mathbb{F}^n", "60a1a1e741f2af1e95b429a7eb6fffa1": "\\hat\\theta(X)", "60a227b9b2aa64aff954573912ef15bf": "m=z^n", "60a24c48cb7626732a929a2a49f4c29c": "\\sin{\\pi H}=0", "60a26af19d78adccebe1bcaf05098146": "g_2= 60\\sum_{(m,n) \\neq (0,0)} (m\\omega_1+n\\omega_2)^{-4} ", "60a2926a60ecc6f8b0714bab35b2e7b8": "\\eta(i) = 2^{-1}\\pi^{-3/4}\\Gamma(1/4)", "60a2fe3874fd727b446720fbbda31d1f": " \\begin{bmatrix} T_{11} & T_{12} & \\ldots & T_{1 j} & \\ldots \\\\\n T_{21} & T_{22} & \\ldots & T_{2 j} & \\ldots \\\\\n \\vdots & \\vdots & & \\vdots \\\\\n T_{k1}& T_{k2} & \\ldots & T_{k j} & \\ldots \\\\\n \\vdots & \\vdots & & \\vdots \n\\end{bmatrix},", "60a31f3436dbe0c6fe2fd0963e9d2715": "\\frac{1}{p^k q^\\ell \\cdots}", "60a31fb4b29d7dc35e0f6d4f636652dc": "O(E f)", "60a350288e29b0392478da9b5e528324": "\\mathbf{W}", "60a383b75b0046ef48b052c0067d9de0": "\\sigma_1^2=\\sigma_2^2=\\sigma_{*}^2", "60a3899b55466f71b9f9532ce5b537ed": "\\Phi_{V}", "60a3f8a813069295dfa17d3eb7145bc7": "\n\\begin{pmatrix}\nF L - k_\\theta & F L \\\\\nk_\\theta & F L - k_\\theta\n\\end{pmatrix}\n\\begin{pmatrix}\n\\theta_1 \\\\\n\\theta_2\n\\end{pmatrix} = \n\\begin{pmatrix}\n0 \\\\\n0\n\\end{pmatrix}\n", "60a45a2ea15c8b018f546e3bf674069a": "|\\mathcal B|", "60a46dd00a2f245797105c95e585547f": "|\\bigstar \\bigstar \\bigstar ||", "60a483ce9c00afd9216fc855817d5de2": "\\mathbf{B} = \\mu \\mathbf{H}", "60a48e0dc622bb8b1f9e271ae03c3544": "\\mathbf{P}(n)=\\lambda^{-1}\\mathbf{P}(n-1)-\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)", "60a526521fa0e070ba74cb3f9cf6dac0": "\\textstyle{F(z) = \\sum_{n=0}^\\infty a_n z^n}", "60a54c0c6c8dfcb9cd67b3eea60889a4": "f=\\text{antilog}_{10} \\frac{\\textstyle\\sum_{i=1}^n F_i}{n}=10^{\\frac{\\textstyle\\sum_{i=1}^n F_i}{n}}", "60a550ae8d843359e5b9519a36d2b085": " \\rho_1 ", "60a58a8a2e4862bd03ee44ecd92da392": " A = 10^{-Loss/20} \\qquad \nZ_{11} = Z_S \\frac {1+A^2} {1-A^2} \\qquad \nZ_{22} = Z_{Load} \\frac {1+A^2} {1-A^2} \\qquad \nZ_{21} = 2 \\frac { A \\sqrt { Z_S Z_{Load}}} {1-A^2} \\, ", "60a5e3f58944d189a674e735065de5a8": "E_{-}=[(E_1+E_2)/2-|W_{12}|]", "60a60fce083241e8d9b3775b7f17b0d8": "0 \\neq \\alpha \\in \\sigma(T\\upharpoonright S)", "60a6505d30652de808a4a5e37d20d936": "\\Delta q", "60a67f795e708f45b6270daccffad065": " A \\subset D \\times D", "60a69f6a06603740e231253ac92b8802": " \\!\\ \\sqrt{2} = 1 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\ddots}}}}. ", "60a6ad36224fc2a272acacfa88165ee7": "\ny_1 = -\\frac{\\ell}{2} \\cos \\theta_1\n", "60a6b6874e530bd42a083e306561a3e6": " \n\\vec{F}_m = - \\vec{F}_f\n", "60a6c404ae84cc98af4a56cb8e1b5a4e": "\\text{diameter} = \\sqrt{\\frac{2 \\cdot \\text{area}}{\\sin A \\sin B \\sin C}}.", "60a6f3d55f69aedd7f1475a09f59c73a": "\\hat{\\lambda}", "60a736d6f8761fc9c46f166c8bcd0e67": "\\displaystyle{2[Q(a,b),L(a)]=2[L(b),Q(a)].}", "60a74f873be2528e076106419e70fa2c": "\n\\rho(r) = 2\\pi\\int\\int f(E,J) v_t dv_t dv_r\n", "60a7da93523110307f4643e345eb5de8": "\\frac{\\partial {\\rm tr}(\\mathbf{X}^n)}{\\partial \\mathbf{X}} =", "60a7e356c28f91b76aa5245d2a351a27": "L_\\alpha\\cup\\{L_\\alpha\\}", "60a818d6cdd19202bf6a9a8f0b6f0e36": "\\varphi_{t+s} = \\varphi_t \\circ \\varphi_s.\\,", "60a81c4c13bf4b15b9b59e8ccad8e5de": "\\psi(x)=\\psi(0,x)\\,", "60a842aa2c6db0e0acdabcbd453fe461": "\\begin{align}\n\\tau_\\mathrm{n} &=\\sqrt{ \\left( T^{(\\mathbf n)} \\right)^2-\\sigma_\\mathrm{n}^2} \\\\\n&= \\sqrt{T_i^{(\\mathbf n)}T_i^{(\\mathbf n)}-\\sigma_\\mathrm{n}^2},\n\\end{align}", "60a84faf28e224388c4b09fbecc150d4": "f_o = \\frac{1}{2\\pi }\\sqrt{\\frac{k}{m_o}}", "60a88e5b46943419fc9458301b583d63": "\\,p^{n+1}", "60a89387afa962a03fa803f8929b408d": "\\langle H\\rangle", "60a8b37684121cdc14c7fd5cf8c41fde": "\\left(X_n\\right)_{n\\in\\{1,\\ldots,N\\}}", "60a8be65c205a8d9ee1a72c686e4b8b8": "\\exists x\\, \\phi(x)", "60a8c457c268ffa9876c3a81831b2620": " \\mathbf{e} \\cdot \\tilde{\\mathbf{y}} = 0 ", "60a9664411f02ec49584ffcadfcca4a3": " \\arccsc z = \\arcsin {(1/z)} \n= z^{-1} + \\left( \\frac {1} {2} \\right) \\frac {z^{-3}} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4 } \\right) \\frac {z^{-5}} {5} +\\cdots\\ \n= \\sum_{n=0}^\\infty \\frac {\\binom{2n} n z^{-(2n+1)}} {4^n (2n+1)}; \\qquad | z | \\ge 1 ", "60a96bbd1dc5d53ace53974a0d5acda3": "\\Delta F=0", "60a99dbc16ab87c804f26229ec3c9201": " \\rho = 0.414682509851111660248109622\\ldots", "60a9a2ea2c84025c46a709cd50ce582d": "\\hat{\\gamma}=\\gamma-1", "60a9cfb652e8bc5be7c99c4d4b49894b": " P(\\operatorname{D}) \\phi = \\sum_\\alpha c_\\alpha D^\\alpha \\phi ", "60a9e115208b886f1248b3513fd50483": "\\frac{\\partial\\bar{x}_i}{\\partial x_k}=\\frac{\\partial}{\\partial x_k}(x_j \\mathsf{L}_{ji})=\\mathsf{L}_{ji}\\frac{\\partial x_j}{\\partial x_k}= \\delta_{kj}\\mathsf{L}_{ji} = \\mathsf{L}_{ki}", "60aa2622c898f8d18922f910e5624d45": "\\gamma_{\\xi} \\approx \\frac{1}{8}\\xi \\cdot 2, \\quad \\gamma \\approx \\frac{1}{8} \\left (\\xi^2-\\xi_0^2 \\right ),", "60aa2ad354250e87f67cd5794aa966bc": "\\hat{H} = \\hat{H}(\\hat{P})", "60aa828a446674a7e6bcb8118a0cc78b": "|n| < m", "60aac42a46e336f78a3ba88774298d6e": "\\left(V_3 - V_1\\right)-\\left(V_4-V_2\\right) = 2L'\\,\\Delta l\\frac{\\partial I}{\\partial t}", "60aba5f0d77650ee736221582a15d87b": "\\Box a", "60ac17c77607d49740922470ef044f97": " c = G = \\hbar = k_\\text{B} = 1 \\ ", "60ac84dc6bf65ce9c7690dd08e366243": "(p_2(t) x_2^\\prime)^\\prime + q_2(t) x_2 = 0, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x_2(a)=1,\\,\\, x_2^\\prime(a)=R_2\\,", "60acb8c5c5eddcaae87e7a23aa1ef0f4": "\\scriptstyle\\sqrt{2}", "60accb07b4b96c596c1c465cec14de5c": "\\mu _A(T_b) = \\mu_A^{\\star}(T_b) + RT\\ln x_A\\ = \\mu_A^{\\star}(g, 1 atm)", "60ad53b259a5f7bd956fc0e204e15274": "(S,R)", "60adac36c05e8c95161867f8faa191f1": "\\int_a^b g(x)\\,dx", "60add7e072e4b4b55f9e92d2abcfac67": "f^{-1}(]-\\infty, c])", "60ae25af669a9b3802e3e34083424063": "~\\alpha~", "60ae317576b238a622aaca7620362166": "\\mathrm{\\phi} = 0.75\\,\\!", "60ae491e4190ed569045437c7e441333": "\\exp(-\\lambda S^4-bS^2)\\,dS", "60aece56114edfe0aa16a83de4694f91": "\\partial_{\\mu} J^{\\mu} \\, = \\, \\partial_{\\mu} \\partial_{\\nu} \\mathcal{D}^{\\mu \\nu} = 0 \\,", "60af3dfb391e361bb76bc78af10008cf": "\\beta^{'}(p,1,a,b) = \\textrm{Dagum}(p,a,b)\\,", "60afa71fcf12056290c0da7ac39ca3f8": " \\mathrm{B}_\\mathrm{1}", "60b021a433cb447735d1936133d1429a": " t \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{T-T_c}{T_c}", "60b029c606e4753f646e22d025bfa2a5": "\\xi:\\Omega\\to \\mathfrak{N}", "60b053e39f0d40a78a6a29cadc7c297a": "\\tfrac{223}{71} < \\pi ", "60b073814b844f213e8bd40313379b33": "E \\rightarrow \\hat{E}^+.", "60b0a4875203b02aefdbcd57910743dc": "|-k\\rangle", "60b118405c6a96ace7f3cb564dfc8c5f": "H_f(\\mathbf{x}_n) + B_n", "60b13da834d44068eccce671e7236343": " \\langle E \\rangle = A + ST.", "60b1aa0207ae6a4f1fe5e6b0f8d950ad": "l_e", "60b1e60aa6d428b4ea2f674e12d395b8": " \\int\n\\exp \\biggl({i \\over 2} \\sum_k k^2 \\phi^*(k) \\phi(k) \\biggr) D\\phi = \\prod_k \\int_{\\phi_k} e^{{i\\over 2} k^2 |\\phi_k|^2 d^dk} \\,", "60b1ed7eeb8cbf5fe7f000c8d17e9d11": "\\pi_* F", "60b23df94d8507e2407ce76752a592d3": "\\frac{z_1}{ z_2} = \\frac{r_1}{ r_2} \\left(\\cos(\\varphi_1 - \\varphi_2) + i \\sin(\\varphi_1 - \\varphi_2)\\right).", "60b2658268b92a71bcb5ad9f40d2c46b": "\\begin{pmatrix} x'\\\\y' \\end{pmatrix}", "60b27d4b6a5d78f4fda0c88eafaa454b": "\\Pr \\left (X_k=\\min\\{X_1,\\dots,X_n\\} \\right )=\\frac{\\lambda_k}{\\lambda_1+\\cdots+\\lambda_n}.", "60b29f2c4b8815c498f5b27060aeb3ff": "r_K=", "60b2afecdebe3a4f642e40c46c6bf29b": "\\mathbf{T} = \\frac{dm}{dt} {v}, \\mathbf{P} = \\frac{1}{2} \\frac{dm}{dt} {v}^2", "60b2e5f2a55fc67b0b4073f8948b0f3f": "\\exp(\\mathrm{ad}(z))\\ ", "60b2f31ff030422761292c9ee237ae52": "\n \\begin{align}\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}(\\mathbf{x},t)~\\text{dV}\\right) & = \n \\int_{\\Omega_0} \\left(\n \\frac{\\partial }{\\partial t}[\\hat{\\mathbf{f}}(\\mathbf{X},t)]~J(\\mathbf{X},t)+\n \\hat{\\mathbf{f}}(\\mathbf{X},t)~J(\\mathbf{X},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\mathbf{x},t)\\right) ~\\text{dV}_0 \\\\\n & = \n \\int_{\\Omega_0} \n \\left(\\frac{\\partial }{\\partial t}[\\hat{\\mathbf{f}}(\\mathbf{X},t)]+\n \\hat{\\mathbf{f}}(\\mathbf{X},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\mathbf{x},t)\\right)~J(\\mathbf{X},t) ~\\text{dV}_0 \\\\\n & = \n \\int_{\\Omega(t)} \n \\left(\\dot{\\mathbf{f}}(\\mathbf{x},t)+\n \\mathbf{f}(\\mathbf{x},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\mathbf{x},t)\\right)~\\text{dV} \n \\end{align}\n", "60b31b7bd0b939a763b64c337e866c5e": "\ng^{(2)}(0) < 1", "60b34a827bd63de5d03305782c8896b6": "//x = x\\ ", "60b35d11c0439c98bbecaf57b15cf74d": " x\\ne 0 ", "60b3752bc8185f50a240e3c4d87d5040": "q\\cdot P(\\Lambda=c \\;|\\; H_0) + P(\\Lambda < c \\; | \\; H_0) = \\alpha ", "60b37897ebb81870c33773c674253712": "\\sup_{y \\isin F} ", "60b3d374ab97d52dab18868ccea3d30f": "( x, f(x) )", "60b3db4e344dfc02b80318342bf559bf": "\\dot \\theta = 0, \\ \\ddot \\theta =0", "60b404cebd635e20d33ad766ccd5977a": "\\theta^4", "60b4418b68ee26de572d0d4a0f7442c2": "|x-x'|", "60b44e0698afc7d189d435ea0cd4ac43": " \\frac{v_0^2}{gh_0} > 1", "60b464b224679f7793e2dc3a79f925ca": " x= {1\\over 2} u \\sqrt{1-4u^2} + {1\\over 4} \\sin^{-1}(2u) \\,", "60b487f829abb59e12be15322a3b25ec": " \\mathrm{var}(\\delta(X_1, X_2, \\ldots, X_n)) \\leq \\mathrm{var}(\\tilde{\\delta}(X_1, X_2, \\ldots, X_n)) ", "60b4baf42c97cb5d1cded8375a049c44": "\\Box \\phi = \\rho \\,", "60b5092fbe4a2c113352a3700764049a": "(x_1, \\ldots, x_n, y_1, \\ldots, y_n).", "60b50c6980070b32404eee4bade37323": "\nf(x,y)=\\frac{1}{2\\pi}(xy)^{-\\frac{1}{2}}e^{-\\frac{x+y}{2}}\n", "60b5197dd98fc9da9911304dcc1b7ef8": "\\lim_{\\varepsilon \\rightarrow 0+} \\left[\\int_{b-\\frac{1}{\\varepsilon}}^{b-\\varepsilon} f(x)\\,\\mathrm{d}x+\\int_{b+\\varepsilon}^{b+\\frac{1}{\\varepsilon}}f(x)\\,\\mathrm{d}x \\right].", "60b5cf79ec88af04f9bc3c975010a806": "P=x_3^3 + x_1 x_2+7,", "60b60d3c400dcf34b2deb3ce8cbbe1ff": "\\hat{S}", "60b61c441d27b9386cd25d951cee53a2": "\\frac{1}{r} \\mathbf{x}", "60b624ca76cea8799b09e84e3b0d5e0e": "\\mathrm{Inv}^1 \\langle X | T\\rangle", "60b63dcf97de0779fb6ceea65a5d7ff2": "e^z = 1 + \\frac{z}{1!} + \\frac{z^2}{2!} + \\frac{z^3}{3!} + \\dots = \\sum_{n=0}^{\\infty} \\frac{z^n}{n!}.", "60b675accbbfca8ef82d22726124bf12": "{\\rho}=\\frac{m_d+m_v}{V}=\\rho_d+\\rho_v \\, ,", "60b6786344374aa6440fd0305a5f1105": "f_0 = { \\omega_0 \\over 2 \\pi} = {1 \\over {2 \\pi \\sqrt{LC}}} ", "60b683360c406e7a3399a8bf647ad3c7": "f(l)", "60b69cdabe2f282ffb6940d5230bb74f": "\\sigma_p^2=\\left(\\frac{\\hbar n\\pi}{L}\\right)^2.", "60b69f4f5c2b5d18487828dc32c3e6a5": "\nM = \\mu(\\mathbf{x}, \\sigma_{\\mathit{I}}, \\sigma_{\\mathit{D}}) =\n\\sigma_D^2 g(\\sigma_I) \\otimes\n\\begin{bmatrix}\nL_{x}^2(\\mathbf{x}, \\sigma_{D}) & L_{x}L_{y}(\\mathbf{x}, \\sigma_{D}) \\\\\nL_{x}L_{y}(\\mathbf{x}, \\sigma_{D}) & L_{y}^2(\\mathbf{x}, \\sigma_{D})\n\\end{bmatrix}\n", "60b6bbbf6bd1a6c01631cbd46eb75700": " p_k(d\\sigma_k | \\eta)=p_k(d\\sigma_k | \\eta_{V_k}) ", "60b6c7d0c5d5a09fb4c380258a463ab9": "dQ = A dx + B dy + C dz", "60b6efc20878e38aead715d06af0028b": "\\displaystyle{L_a(b^{2,a})Q(b) =Q(b)L_b(a^{2,b}).}", "60b703fb6eade867ddaa89b9bfcf70e4": "\\tfrac14\\ + \\tfrac13=\\tfrac{1*3}{4*3}\\ + \\tfrac{1*4}{3*4}=\\tfrac3{12}\\ + \\tfrac4{12}=\\tfrac7{12}", "60b77726e02ae9fd2b0152eab0b8149b": "N\\subseteq_e M", "60b79bcd4dce92c26b49fc426736b1aa": "MTBC=\\frac{1}{\\lambda\\ _{d}} = \\frac{2}{\\lambda\\ \\mu\\ \\tau\\ }", "60b7d3707a1b03baa67a979788967367": "{{P}_{X}}\\left( \\omega \\right)=\\int\\limits_{-\\infty }^{\\infty }{C\\left( \\tau \\right)}{{e}^{-i\\omega \\tau }}d\\tau ", "60b88aae991782fa10df3dbacabeab0d": " p(a/2) > 2p(a) ", "60b90499a5f03cfd58c302a840a12de6": "A{x^\\prime}^2\\cos ^2\\theta\\ -\\ 2Ax^\\prime y^\\prime\\sin \\theta\\cos \\theta\\ +\\ A{y^\\prime}^2\\sin ^2\\theta\\ +\\ B{x^\\prime}^2\\sin \\theta\\cos \\theta\\ +\\ Bx^\\prime y^\\prime\\cos ^2\\theta", "60b9830739f25f53a343fafff3148ad0": "R_{\\mu }\\left( t+1 \\right)=R_{\\mu }\\left( t \\right)", "60ba26bfdb1b8f7d278cc8d43bc75b44": " (hf - hf' + m_e c^2)^2 - m_e^{\\, 2}c^4 = \\left(h f\\right)^2 + \\left(h f'\\right)^2 - 2h^2 ff'\\cos{\\theta}. ,", "60ba67fa45b32117e42f1d30f75c2f10": "m\\{x: \\, \\Omega(f)(x)> \\lambda\\} =m\\{x: \\, \\Omega(f-g)(x)> \\lambda\\} \\le m\\{x: \\, (f-g)^*(x)> \\lambda\\} + m\\{x: \\, |f(x)-g(x)|> \\lambda\\} \\le C\\lambda^{-1}\\|f-g\\|_1.", "60bb377a58debc101e5248b86079efc1": "x = \\sqrt{4} = 2,", "60bb42119177424f137ed63360b0ff22": "\\scriptstyle RC", "60bb54f49a4dc6caa4314c370529702f": " N_i ", "60bb9f3d14c3b8e41000d2fa2af13e66": "p(x) \\ \\stackrel{\\mathrm{def}}{=}\\ \\Pr(T=1 | X=x).", "60bc47242d8992a2c9c3740f67f4d815": "\\begin{align}\n y_1 &= a_0 \\sum_{r = 0}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r (\\gamma)_r} x^r = a_0 \\cdot {{}_2 F_1}(\\alpha, \\beta; \\gamma; x) \\\\ \n y_2 &= a_0 \\sum_{r = 0}^\\infty \\frac{(\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(1 - \\gamma + 1)_r (1 - \\gamma + \\gamma)_r} x^{r + 1 - \\gamma} \\\\\n &= a_0 x^{1 - \\gamma} \\sum_{r = 0}^\\infty \\frac{(\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(1)_r (2 - \\gamma)_r} x^r \\\\ \n &= a_0 x^{1 - \\gamma} {{}_2 F_1}(\\alpha - \\gamma + 1, \\beta - \\gamma + 1; 2 - \\gamma; x)\n\\end{align}", "60bc50479adf58b1e42edd329195926d": "\\frac{d p}{d z} = \\rho g \\quad \\Rightarrow \\quad p = \\rho g z + p_0", "60bc9944697fa7786ed6d2e684203af5": "n_\\eta(\\xi)= \\begin{cases}\n n_B(\\xi), & \\mbox{if } \\eta = +1 \\\\\n n_F(\\xi), & \\mbox{if } \\eta = -1\n\\end{cases}\n", "60bcb01530f7a4284f383a36999df0a0": "A(t-r,\\theta,\\phi)/r \\,", "60bce498e924d3f354cc9a3259cf61e0": "\\mathfrak{su}_2", "60bd1775ba9047e4ea9253c90389f6db": "L(x;\\gamma) = \\frac{\\gamma}{\\pi(x^2+\\gamma^2)}", "60bd5792094627b558d181477f681d6c": " F(w_1,w_2,\\dots,w_m) = \\sum_{n_1=-\\infty}^\\infty \\sum_{n_2=-\\infty}^\\infty \\cdots \\sum_{n_m=-\\infty}^\\infty f(n_1,n_2,\\dots,n_m) e^{-j w_1 n_1 -j w_2 n_2 \\cdots -j w_m n_m}", "60bda3de303a8d0b82368a301e8b5c98": "\\frac{\\partial}{\\partial{c}} P_c^{ n+1}(c) = 2\\cdot{}P_c^n(c)\\cdot\\frac{\\partial}{\\partial{c}} P_c^n(c) + 1", "60be1d3d1873670b072f8f77d899bc1e": "\\sigma_x \\sigma_p \\ge \\frac{\\hbar}{2} \\cdot \\exp\\left(H_x + H_p - \\ln (e \\pi) \\right) \\ge \\frac{\\hbar}{2}~.", "60be40d10d03658056fe5ddac4f58e20": "\n P_{ij} = C_{ijkl}\\frac{\\partial u_k}{\\partial X_l} + \\frac{1}{2}M_{ijklmn}\\frac{\\partial u_k}{\\partial X_l}\\frac{\\partial u_m}{\\partial X_n} \n + \\frac{1}{3}M_{ijklmnpq}\\frac{\\partial u_k}{\\partial X_l}\\frac{\\partial u_m}{\\partial X_n}\\frac{\\partial u_p}{\\partial X_q}+\\cdots,\n ", "60be597eb242dc000999a3260c79ffad": "\\mathit{Expr} \\to \\mathit{Expr} + \\mathit{Term}", "60be6e8886f1557213053fc6dc8c71e9": "n_{\\rm air} 0 & \\text{front-facing} \\\\ = 0 & \\text{parallel} \\\\ < 0 & \\text{back-facing} \\end{cases} ", "60bee0d9bb81ee51105d23de9b1002ae": "\\pm \\epsilon^{ijkl} e_k \\wedge e_l", "60bee0f97f6faeff1fce039f3436669a": "\\frac{h^3}{4}\\sqrt{5(5 + 2\\sqrt{5})}", "60bef5592c206e1ef550c1b83ae27b48": "\\Delta x_k=- \\alpha_k B_k^{-1}\\nabla f(x_k)", "60bef70650b8797d6d65db17b32b2eb8": "\nC^{S_1}_{E_2} = - \\varepsilon^{3}_2 / D\n", "60bf907696e687fef4e41ea9a5660411": "\\mathrm{[A^-] = \\mathit{C_b} + [H^+] - [OH^-]} = C_b + \\Delta", "60bf933a501a073f942932c45efb3b30": "\\omega^\\gamma", "60bffb0b9512bac5126bcd054d17db26": "0\\longrightarrow M\\overset{\\epsilon}{\\longrightarrow}C^0\\overset{d^0}{\\longrightarrow}C^1\\overset{d^1}{\\longrightarrow}C^2\\overset{d^2}{\\longrightarrow}\\cdots\\overset{d^{n-1}}{\\longrightarrow}C^n\\overset{d^n}{\\longrightarrow}\\cdots,", "60c00f4039d7de321d6c6aa1d1c0e9ab": "w\\infty z\\infty = (w z)\\infty", "60c012bb671951dfc335354c45fa8ca1": "\nv = \\frac{1}{\\sqrt{K\\rho}}\n", "60c0a1f484ccc6551a4e6456cb069dde": "\\eta = \\sum_{i=1}^N \\left[ \\ln(1-p_i) -\\ln p_i \\right] ", "60c0f3f05024f6102d77d3b7e6ae4911": "\\mathfrak{M}^{\\mathrm{NC}} \\propto J_{\\mu}^{\\mathrm{(NC)}}(\\nu_{\\mathrm{e}}) \\; J^{\\mathrm{(NC)}\\mu}(\\mathrm{e^{-}})", "60c1109e27e08f7ee4fc594cfb6243b9": " S_x = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \n, S_y = \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}\n, S_z = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix} ", "60c13e05d3ec8c10b8564eae7023d9db": "\\times", "60c14e60522101e5a54d3ee9671f0456": "d \\Phi", "60c16cd5a42f95933bfde502216f31db": "\n\\begin{align}\nX &=\\sum_{i=1}^N \\langle \\phi_i,X\\rangle \\phi_i\\\\\n&=\\sum_{i=1}^N \\phi_i^T X \\phi_i\n\\end{align}\n", "60c1b4c330d27aed7ebde8063fac3720": " S\\ = \\gamma_{SG}-(\\gamma_{SL}+\\gamma_{LG})", "60c1c533173251f7d64566fbba580e2d": "s+5t=3", "60c1f35dcd185137e13d876f4351245d": "\n F(x)= \\begin{cases}\n 0 & \\text{for }x < a \\\\[8pt]\n \\frac{x-a}{b-a} & \\text{for }a \\le x < b \\\\[8pt]\n 1 & \\text{for }x \\ge b\n \\end{cases}\n", "60c2249f59be37909d715a67f66fd67a": "df/f", "60c276633e95582e157373561b42371d": "\\begin{bmatrix}\n1 & 1 & 1^2 & 1^3 & 1^4 \\\\\n0 & 1 & 2\\cdot 1 & 3 \\cdot 1^2 & 4\\cdot 1^3 \\\\\n1 & 2 & 2^2 & 2^3 & 2^4 \\\\\n0 & 1 & 2\\cdot 2 & 3\\cdot 2^2 & 4\\cdot 2^3 \\\\\n0 & 0 & 1 & 3\\cdot 2 & 6\\cdot 2^2\n\\end{bmatrix}\n\\begin{bmatrix} b_{m,0} \\\\ b_{m,1} \\\\ b_{m,2} \\\\ b_{m,3} \\\\ b_{m,4} \\end{bmatrix} =\n\\begin{bmatrix}\n1^{5+m} \\\\\n(5+m)\\cdot 1^{5+m-1} \\\\\n2^{5+m} \\\\\n(5+m)\\cdot 2^{5+m-1} \\\\\n\\tfrac{1}{2}(5+m)(5+m-1)\\cdot 2^{5+m-2}\n\\end{bmatrix}", "60c2b51a81d7630e4bdfa12a655b2644": "x \\in [x_{min},x_{max}]", "60c3627431a571969c645dd6401b6a73": "\\delta E = \\frac{1}{2} m \\left(\\frac{Uky}{r}\\right)^2", "60c3c981b0b3764c2ff2e548de5a639d": "K_a = \\frac{[H] [A]}{[HA]}", "60c415b8780c374f180b6ecd74fbf88c": "R^d{}_{abc} \\,", "60c427c4ac806d2b0f8b36ec3dec6865": "\\tbinom{m}{3}", "60c44ccb9ebcdfbef24bb710d72d5a4b": "A \\leq_m B.", "60c4f177b8ff4ee556f60a35ddb2de2f": "\\beta/\\gamma", "60c507f163ea5f95427fb8afb113721a": "\\frac{2\\cdot\\pi}{3}", "60c508f8051c7a8b4e9aa836947231a4": "\\mathbf{z}^{\\prime}", "60c51b7132d475e3c9f27fd7277b1931": " A = \\begin{bmatrix}a & b \\\\ c & d \\end{bmatrix}", "60c54a1c042106a1f9f18ed751008bf7": " K(w, r\\,; q).", "60c5a9be4e9e42b377ec53764df41026": " (aX+bY)^{'n} = \\sum_{i=0}^n {n\\choose i}a^ib^{n-i} X^{'i} Y^{'{n-i}}", "60c5ad3196c90b8207fcf07df5e015d4": "\\ u ", "60c5b15e36a31ba73f8613087e7c01bb": "\\mathcal{F}_{d}=\\frac{1}{2}K((\\nabla\\cdot\\mathbf{\\hat{n}})^2+(\\nabla\\times\\mathbf{\\hat{n}})^2)=\\frac{1}{2}K\\partial_{\\alpha}n_{\\beta}\\partial_{\\alpha}n_{\\beta}", "60c5daf3faa4d5b36c64b38c42eede24": " \\frac{\\Delta L}{\\Delta Q}", "60c5fb49021957060dfe16257ed736b2": "Hz = H(x+e) =Hx + He = 0 + He = He", "60c606096e92773c2753c35e871fdac4": " P_{pad} \\, ", "60c6d84189e93d5ed9948eb834d50dc5": "\nf_E\\,dE=f_p\\left(\\frac{dp}{dE}\\right)\\,dE =2\\sqrt{\\frac{E}{\\pi}} \\left(\\frac{1}{kT} \\right)^{3/2}\\exp\\left[\\frac{-E}{kT}\\right]\\,dE.\n", "60c6e24be1d91852f7d50559bd25ab8d": "\\sqrt{\\frac{20}{9}} \\frac{\\alpha_2}{\\sqrt{\\alpha_1 \\alpha_3}} = 1", "60c75fe67afffcc7b45c5506d888e7ff": "H\\rightarrow K_k", "60c77cf95ecdf35dcff80e95b005f0f8": "\\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{({\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x})=u_{\\boldsymbol{\\hat a}}(\\boldsymbol{x}_0) \\qquad \\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{(-{\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x})=u_{-\\boldsymbol{\\hat a}}(\\boldsymbol{x}_0)", "60c857ea077ccc3741152549974c8668": "a_2 = V_2^-", "60c8931ae27581b00227d9925d46f9f2": "\\,y' = r \\sin(t + dt) = y + x dt + ...", "60c8992cd79d13bcf1dcd5f399f6ccfc": "a_n \\to a ", "60c8a07b4028d8f9ccb8a317ea653f57": "U_{iy}", "60c8b461c9fd3e0ec0de5b292422c72a": "\\,\\frac{1}{\\sqrt{2 \\pi \\sigma^2}}e^{-\\frac{(x-\\mu-\\sigma^2 h)^2}{2\\sigma ^2}}", "60c8cce59a65b7f95fd617415c19b5a0": "\\alpha_1 + \\alpha_2 = 1 \\,", "60c952cab43c3d1347e15db3d90284bb": "\\int_{0}^{\\infty} xe^{-ax}\\cos bx \\, \\mathrm{d}x = \\frac{a^2-b^2}{(a^2+b^2)^2} \\quad (a>0)", "60c99d294eadf7c071a1394ca2a19a25": "\\left| A_n^\\epsilon \\right| \\leq 2^{n(H(X)+\\epsilon)}", "60c9a970e74ca014324963b2649a43e5": "\\min\\{wx\\ :\\ x \\in \\mathbb{Z}^n,\\ Ax=b,\\ l\\leq x\\leq u\\}\\ .", "60c9cb7a86980f4d73d68d79f2e6dad7": "H \\to G", "60c9e1a2403e7c45757292b36031a503": "{I}", "60c9e5045c6041846fe1afd8869c5125": "\\lambda\\ne0", "60ca027b24b83c21fcd5d7293b39ada8": "[f + g](x) = f(x) + g(x)", "60ca59f6c91e722a4584b07e7decaa21": " ax^2+bx+c", "60ca6cc00332aca8262a102c4f0f8831": "y = R\\, \\sin t", "60ca810f742ab8321c4842fea2c613f7": "B(x:=N)", "60cad9f8ab94d58e27d8bb6042dc2a3d": "w_4 = \\sin(w_1)", "60cb08da89a196cfc223ff560dffc0d9": "Q_l^{(d)} f = \\left(\\sum_{i=1}^l \\left(Q_i^{(1)}-Q_{i-1}^{(1)}\\right)\\otimes Q_{l-i+1}^{(d-1)}\\right)f", "60cb5009c4ba89c7b3c30e52cdf24a61": "H_w", "60cb55aaaf14275dc5a9fdb8912c3316": "[a, b],", "60cb5740d84392679e907d85e4092742": "\\alpha \\star \\beta = \\{(\\alpha',\\beta'): (\\alpha',\\beta') \\mbox{ is well-labelled, } \\rho(\\alpha') = \\alpha, \\rho(\\beta') = \\beta \\}.", "60cba604d5e878a99f01b7e5d3099441": "\\displaystyle d(x,y):=\\|x - y\\|", "60cbb16830f351c327ff2b2aef3576b1": "y_0, y_{\\infty}", "60cbd9b7e30ea0b6ba695eab9ea46e92": "8x+7=4x+35 , \\quad \\frac{4x + 9}{3x + 4} = 2 \\, ,", "60cbfe1116ece0f5a9ab9bf2af15b39f": "z_0, z_1, ..., z_j", "60cc555f0ce16d6ac30ee3e858d26844": "\n\\widehat{f \\ast g}(\\varrho) = \\widehat{f}(\\varrho)\\widehat{g}(\\varrho).\n", "60cc57f602bcaa58734f1722a408a986": "S'=\\{x\\in A: sx=xs\\ \\mbox{for}\\ \\mbox{every}\\ s\\in S\\}.", "60cc6382a52bb019e380be90518a9d61": "[x]_P", "60cc9655a5f88e01bc72f76307e01689": "R_{i}", "60ccb64396787eabacb0730a5c4a907d": "p^{1,000,000}\\neq 0", "60ccdeb0ac144026cc9af5bc9fb57e80": "u_{\\epsilon}(y)", "60cd147d70700124592b15b0bed254c8": "z z^\\ast = (x + y \\epsilon) (x - y \\epsilon) = x^2 + y^2", "60cd3022fd3cf5d18ba373e8c891155e": " \\frac{d^2}{dt^2} y = f(t,y) ", "60cd6cdda79aeba81b5f177dd3f809b0": "\\psi = \\varphi - \\theta", "60cd85a759226b5aab92af1c2ee5599e": "\\Omega_{\\mathrm{planet}} = \\Omega_\\star", "60cda20c585d60889fbac2d8791ff603": " Z= \\cos \\theta_W W_3 - \\sin \\theta_W B", "60cdadf03fd9fa4a31873c456dd80169": "\nX_{3}=[2,7],\n", "60cdcde876dbcb94ca064a5604e79289": "\\left|\\sqrt2 - \\frac{a}{b}\\right| = \\frac{|2b^2-a^2|}{b^2(\\sqrt{2}+a/b)} \\ge \\frac{1}{b^2(\\sqrt2 + a / b)} \\ge \\frac{1}{3b^2},", "60ce04401aa638d5fc2f3f5317ea5427": "\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = \\frac{-1}{H} \\frac{\\partial \\eta}{\\partial t}", "60ce24a3367899eca7e4908d84b437c2": "\\chi_6(n)", "60ce61baf43cbc84e148ffe1ef74b6e9": "\\tfrac{d(d+2)}{2(d+1)}", "60ced935f81a6075adbac887924ba6cd": "\\{u,r\\}", "60cf4dd735f1a5fa89fe5a9cf48cba8d": " \\left| z \\right| = \\sqrt{a^2 + b^2 }", "60cf5a7ca799dedaf5e334bb3ea16613": "(c_\\max)^2\\geq \\frac{m-n}{n(m-1)}", "60cfe0438025981856095561ef7c718b": "\\sum_{i=0}^m|\\Delta F_i|\\!", "60cffcd2c7f7704e9f908e1a068929c6": "T(t)={\\rm e}^{At}:=\\sum_{k=0}^\\infty\\frac{A^k}{k!}t^k", "60d04971500efc3a39c8e6d69ac321f4": "\\bar{\\nabla}_T T = \\nabla_T T + (\\bar{\\nabla}_T T)^\\perp", "60d0a38f9f732f76b21923163d51ffb7": "\\frac{\\partial x}{\\partial \\theta} \\geq 0", "60d0a9bd04b8e72b5e32611b09ab4f5f": "\\scriptstyle\\mathbf{x}", "60d0bc3d091214220476dc8145694333": "H=U+PV.", "60d0f82dd95c0ebadf5e3bbba7aedd07": "\\,\\phi (x_1,x_2, x_3)", "60d10bbf7dda69ffa05b9020b12a0298": "\n\\arctan z = \\cfrac{z} {1+\\cfrac{(1z)^2} {3+\\cfrac{(2z)^2} {5+\\cfrac{(3z)^2} {7+\\cfrac{(4z)^2} {9+\\ddots}}}}},\n", "60d11ae330b9863c91df550280a60655": "\\scriptstyle{i_i}", "60d1359a79a03bc84b892930c0af4b57": "\\rho(x)", "60d1a6779da8592c5b0e5d506c24402c": "\\hat{Y}_i", "60d1bd7821c763944e0be602de1f99de": "\nf_{MB} \\left ( \\mathbf{p}, T_{\\alpha} \\right ) = \\left ( 2 \\pi m k T \\right )^{-3/2}e^{-\\mathbf{p}^{2}/2mkT_{\\alpha}},\n", "60d242868428c76248ff09f9594f8f9c": "\n\\begin{bmatrix}\n1 & 0 & \\dots & 0 & \\lambda_{1, n+1} & \\dots & \\lambda_{1, m}\\\\\n0 & 1 & \\dots & 0 & \\lambda_{2, n+1} & \\dots & \\lambda_{2, m}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\dots & 1 & \\lambda_{n, n+1} & \\dots & \\lambda_{n, m}\n\\end{bmatrix}.\n", "60d24ffe90aaf43f16fb722ea7ecd052": "\\Phi = IA", "60d25297bc0ea8a83f9cf58edec8e72b": "\\Lambda \\,", "60d2a9b091c7906b3bc5dafd956cc37f": " \\ell_2 ", "60d31c9e188c552e230a7dd5c1bae7a1": "g(X_1,X_2,\\dots,X_n) \\in \\Bbb{R} ", "60d33359f430d8a64671fb74eb91ba1f": "\\mathbb{A} (\\textbf{n})", "60d35c97dd32b2cc503587242ae26160": "\\sum_{p\\le n}\\frac1p -\\ln\\ln n-M", "60d38d81d93a48f0282a27590fec0a5c": "\\phi ( \\bold{r} ) = \\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac { \\rho ( \\bold{ r}_0 )} {| \\bold{ r}- \\bold{r}_0 | } d^3 \\bold{ r}_0 \\ + \\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac { \\bold{p} ( \\bold{ r}_0 )\\bold{\\cdot (r - r_0)}} {| \\bold{ r}- \\bold{r}_0 |^3 } d^3 \\bold{ r}_0 , ", "60d39875fbdc40d749497ee15baad1b2": "e^\\pi - \\pi\\approx 19.99909998 ", "60d4d503a4384dee968cd2e45a2264c5": "\\frac{2}{a}-1", "60d4eb29ac00a43097e7dcf77978128d": "f(\\boldsymbol{x}) - f(\\boldsymbol{a})= \\nabla f(\\boldsymbol{a})\\cdot(\\boldsymbol{x}-\\boldsymbol{a}) + \\alpha |\\boldsymbol{x}-\\boldsymbol{a}|", "60d4ee646eb3e303a6f86c769affaa08": "\\displaystyle \\partial_t u + \\partial_x^3 u \\pm 6\\, u^2\\, \\partial_x u = 0", "60d5151a99143f71fbb306b1132a5305": "\\binom{B'}{A} \\subseteq X_i", "60d518597d1c3f6a975cf4a6023ba12a": "A \\cap B = A\\,\\!", "60d52ab6a7537a88a8f9c0fa624c7d9c": "\\textrm{TI} = \\ln(2) \\cdot T_1,\\,", "60d55f68e3984c86f78260217f8800d3": "x_{i+1} = x_i \\left(\\frac{x_i^3 + 2a}{2x_i^3 + a}\\right).", "60d564bacd88faba24183be854234cd7": "\n w(x) = \\frac{P(L-x)}{\\kappa AG} - \\frac{Px}{2EI}\\,\\left(L^2-\\frac{x^2}{3}\\right) + \\frac{PL^3}{3EI} \\,.\n ", "60d5aa23bf6ec2eb3fc7e9de22535629": "10^{12}", "60d5d3052466a542d3aa1ecf695b7203": "\\sigma_0\\propto1/\\alpha", "60d6671e0fe6c395db9275e13d14840e": "i_c(t)", "60d66a7278e1bf88c02377744a16df45": "\\lambda_i \\leftarrow \\lambda_i - \\mu_k c_i(\\bold{x}_k) ", "60d67273c88e6bf4e163b30c78ae3a9b": " a_{i+1,j} = \\pi(a_{i,j}),\\quad\\text{for } 1\\leq i n", "60e58741220d2cee64c9fcc1dec48173": "k = 1,2,\\dots", "60e5894e39d885b92a10e990e2e901ee": "F_1,\\dots,F_m", "60e59321f8be23c8d8c73accde3cad29": "D_a E_b^i = 0", "60e5fbebd0925f11bfd99a82c9171c37": "f_{i-1}\\,", "60e5fe9ffe391a9b5abc5aa5d9da3998": " J_{F^{-1}}(F(p)) = [ J_F(p) ]^{-1}", "60e62d441d9d089802041c0e2d6f56c0": "\\int_S \\sigma_{ji}n_j\\, dS + \\int_V F_i\\, dV = 0\\,\\!", "60e645f7b0a02af98f4db148e90f0917": "\\rho(X) = \\lim_{n \\to \\infty}\\mu_n(X \\cap B_n)", "60e66b900a8e0c3456c14d900aa155ca": "\\Delta U_{system} = U_{final} -U_{initial} =- W", "60e6dac72379c402d0c39a04f36946aa": "\\{0,1\\}^{m+d}", "60e70a812fab167aa6acbd9a60225722": "\\int\\frac{dx}{\\sqrt{a^2-x^2}}", "60e70ed684103a25429b16b58ceb4d65": "( G, \\alpha: E(G) \\rightarrow \\mathbb{Z}_{n} )", "60e73fd7db8cc2cca3b0a637056de9f5": " \\Phi(s) = \\frac{1}{1+\\beta}(1-s)^{1+\\beta}+s", "60e78adbcebc60d66e67c89117d6b6e1": "\\mathcal{T}_\\mbox{seq}", "60e7965f47838f667fc375b977671bf6": " \\nu_E(x) := \\lim_{\\rho \\downarrow 0} \\frac{D\\chi_E(B_\\rho(x))}{|D\\chi_E|(B_\\rho(x))} \\in \\mathbb{R}^n", "60e7af73f880a16d24151eb498243249": "N \\log_2(RM)", "60e7d61b11e415110e91ad0219f46f60": "\\mathbf{F}_{\\mathrm{imp}}", "60e7d69c5b0ce2e47ece0d5cc6b8b653": "\nF_K = \\left[1 + 5.5 \\left( \\frac{y}{\\delta} \\right)^6\n \\right]^{-1}\n", "60e80b65d632f1604b631829f483a5b5": "n \\, ", "60e83fee452dbecbfee305d91acd4b7d": "\n\\times\\left[ G_{p+1,\\,q+1}^{\\,m+1,\\,n} \\!\\left( \\left. \\begin{matrix} 1-\\mathbf{a_p}, h+1 \\\\ 0, 1-\\mathbf{b_q} \\end{matrix} \\; \\right| \\, (-1)^{p-m-n} \\; z \\right) + (-1)^h \\; G_{p+1,\\,q+1}^{\\,m,\\,n+1} \\!\\left( \\left. \\begin{matrix} h+1, 1-\\mathbf{a_p} \\\\ 1-\\mathbf{b_q}, 0 \\end{matrix} \\; \\right| \\, (-1)^{p-m-n} \\; z \\right) \\right] ,\n", "60e87c4b6b93e82fc2c5d0b88580d76c": "(a_1,a_2,\\dots,a_n)\\le (b_1,b_2,\\dots b_n)", "60e8ac6c9b311eb062e6a1ce09d31c1b": "\\frac{A_d}{A_P} = \\frac{T_P}{T_P-T_d} \\times BR", "60e8ad595d12c8c5886a88f0942fc1f9": "q=e^{-\\frac{\\pi K'}{K}}.\\,", "60e925c5b822a443ffd8026f25b32843": "\\prod_{k=1}^n [-k]_q = \\frac{(-1)^n\\,[n]_q!}{q^{n(n+1)/2}}", "60e9443002f934fceb9f6772811b5d07": "\n\\left[b^\\dagger(\\mathbf{k}),b^\\dagger(\\mathbf{l})\\right] = 0, ", "60e94b66fdf9ff3a219ab8a3d4ab686e": " \\boldsymbol{v} \\cdot \\boldsymbol{F} = \\sum_i v_i F_i = 0. ", "60e967768eb1e7d6bf11863117e6a2cf": "x^2+a^2y^2-a^2=0,", "60e9c8a1bb94a3056458948f438abfbb": "\\hat{B}_i", "60e9d8e3a0f22f445e006cc7f83e0315": "!_\\tau", "60ea0cb23ba5efda2c7b31c158850de6": "a_0^*", "60eacc7a58667bb6c7f4149ac8fd6f60": "\\hbox{Cross} \\leftarrow \\hbox{not Train}", "60ead9772a02fc774cba6e3843190196": " C \\,\\!", "60eb0a1e9f503f1077926749418b5456": "P(x)=Q(y)=0", "60eba47fd3cf914cae47af5d8556f2fd": "(q_0, q_1)", "60ebb0f2c48d7f0fc016564632fcc0dd": "a_{E}\\,= D_{E}", "60ebbf359575294a045d9804fc480fa9": "\\frac n{n+1}", "60ebf331d13f9eec59f497a02cf36ee1": " |a_0-b_0| \\leq 1/4 ", "60ebfacc593fc53f8da048942310eeb9": "\n (m',n_m')=(g_{1mm'}(m,u),n_m') \\, \n", "60ec222ee52bf99e7b82a74651103dc3": "R_{\\text{horizontal}} = \\frac{R_{23,41} + R_{41,23} + R_{32,14} + R_{14,32}}{4}", "60ec31a9415b7d4a1380e15efc1a2d16": "\\begin{align}\n f'(t) & = -\\pi^{-1/4}te^{(-t^2/2)} \\\\\n f''(t) & = \\pi^{-1/4}(t^2 - 1)e^{(-t^2/2)}\\\\\nf^{(3)}(t) & = \\pi^{-1/4}(3t - t^3)e^{(-t^2/2)}\n \\end{align}", "60ec508f5ba74c29f00c9d01a4d3b0e1": "\\frac{\\infty}\\infty, \\infty^0, \\ldots", "60ec53e82c03d10dfb350c96b99a8639": "=1(1+q)\\cdots (1+q+\\cdots + q^{n-2}) (1+q+\\cdots + q^{n-1})", "60ecb61293feecefaf69dc1598a90fed": " y = ax^2 + bx + c\\,,", "60ecf42440c68c8dd71654bbfa60fbc6": " xy'(x) +2y(x)= 0 , \\,\\!", "60ed613a2743d971cfe3468fa2482204": "\n\\begin{cases} \n\\{O_{1},O_{2}\\} \\\\ \n\\{O_{3},O_{7},O_{10}\\} \\\\ \n\\{O_{4},O_{6}\\} \\\\\n\\{O_{5},O_{9}\\} \\\\\n\\{O_{8}\\}\\end{cases}\n", "60ed7f36e5a54e1006f3777d02b93ffd": "\\overline{op_1'} = \\overline{T(op_1, op_2)}", "60ed993b2fa542e1c97ae1ff3985d39f": "\\mathbf{p} = \\begin{pmatrix}a\\\\b\\\\c\\end{pmatrix}", "60edc1a5b90887a154121f2914ba9f0d": "m \\ge n\\ge 1", "60edc5612c7fecb87bc7898928264d26": " c_1^2 \\le 3 c_2. \\, ", "60edcce730c1eaac21258707dae7ba4c": "\\frac{\\tau_0}{\\tau} =1+ \\frac{ \\alpha l }{1-R} = 1+\\frac{ \\epsilon l C}{2.303(1-R)}", "60ee56388869ae7d98f901d0c98edab4": "\\sin(\\alpha + 360^\\circ) = \\sin(\\alpha)", "60eea8feaa351c7dc7750c23a82e2de6": "\\mathit{n}(\\mathit{p} - 1)/\\mathit{p}", "60eee259cfd2305c7ac73abd1b0a230e": "f = k \\cdot v", "60eeea2ec48b0015b718388fed3327a6": "c^{q+1}=1", "60eef75e693d0e21552d5ef311f72a52": " S_b = y + \\bar{Y} b", "60ef3846c244ed511b0cd6e93c465382": "Z^A=\\{\\Phi,E^a,I^a\\}", "60ef5f80ff19457aa0faa7ac1136caaf": "f_{v,\\eta}(g) = \\eta(\\rho(g)v)", "60ef9bebd228d1bc505acb00b423edf5": "f(\\chi)=\\sum_{i\\ge 0}\\frac{g_i}{g_0}(\\chi(1)-\\chi(G_i))", "60efb0ac0f9ae0340cb91e33b0645454": "x=x(t),\\quad y=y(t)", "60efdbff589e9f7c8fe4307d653ff1e8": "H_A: \\hat{\\rho}_{XY\\cdot\\mathbf{Z}} \\neq 0", "60effe2ad7fa4e7974e200db945318eb": "\\textstyle{\\operatorname{tr}(AB) = \\sum_{ij} a_{ij}b_{ji}}", "60f042b188fb427e02a436ff43f3b514": "P = 0.", "60f0b7a126af4b47f9ae3c890ff55d0b": "\\,\\Delta_1(x-y) = \\Delta_+ (x-y) + \\Delta_-(x-y).", "60f1584f8b440cf6cfa66f5624579079": " \\Psi(s) = \\lim_{n \\to \\infty} \\frac{\\Delta_n(s)}{(-\\lambda)^n} ", "60f16483fe00ed29427bb66e51b4366c": "M_{3,1}\\,", "60f1f50238aed9ed5517e9f88ae20ff1": "\\int e^{x^2}\\,\\mathrm{d}x = e^{x^2}\\left( \\sum_{j=0}^{n-1}c_{2j}\\,\\frac{1}{x^{2j+1}} \\right )+(2n-1)c_{2n-2} \\int \\frac{e^{x^2}}{x^{2n}}\\;\\mathrm{d}x \\quad \\mbox{valid for } n > 0, ", "60f252f8a247cbcb4d330001da4de616": "V_{OC} \\approx \\frac{nkT}{q} \\ln \\left(\\frac{I_L}{I_0} + 1\\right).", "60f26df58b003b743f4706d7e5feca3c": "\\mathit{StdDev} = \\frac{ \\phi_{ 84 } - \\phi_{ 16 } }{ 4 } + \\frac{ \\phi_{ 95 } - \\phi_{ 5 } }{ 6.6 } ", "60f2916c87ee641e00abb14d7b142c0b": " \\mu B(x,r)\\leq r", "60f29e5f229c996ae3a1e6b7791b99b6": "x_b \\in \\{x_0, x_1\\}", "60f2ab95197e70d5e7fd664415f04d16": "F \\cap E_j = \\emptyset \\and F \\cap F_j \\neq \\emptyset", "60f2d974aac0d17be80b758b8245a121": " k \\, ", "60f2f298f1c5c8855137c23ec2c7917a": "B \\in K", "60f3346e2b7e4ffa4d870ea9c97faebe": "(B \\setminus A) \\cap C = (B \\cap C) \\setminus A = B \\cap (C \\setminus A)\\,\\!", "60f3478112e9dd7b4b066df20b62a9e1": "Y_3 = (T_1Y_1)^2 - (T_1Z_1)^2 + (Z_1Y_1)^2", "60f3516ce2c6d2452257f66690429600": "S \\sim \\sum_{i=1}^n \\sqrt { \\Delta x_i^2 + \\Delta y_i^2 } ", "60f36f1d6a4e65c330d90532acb310a4": "f\\colon (X,\\mathrm{int}) \\to (X' ,\\mathrm{int}') \\, ", "60f39628514338b6ebe97fc8a4c86e6a": "\\|x\\|_p = \\left[ \\sum_{i=1}^{\\infty} |\\xi_i|^p \\right] ^{1/p} \\ x = \\{\\xi_i\\} \\in \\mathit {l}^p \\ ,", "60f412e3c2c51cf487e241fca8231165": "{13 \\choose 1}{4 \\choose 2}{12 \\choose 3}{4 \\choose 1}^3", "60f45f3f52f7f96987d967a978449ddd": "\\theta_{ji}", "60f4b1513e286528f2f64026a1f1d8d9": "\n \\epsilon_j^n = N_j^n - u_j^n\n", "60f4b915bfff0a98454b24e6dcd9a86f": "\\mathrm{d} G = \\mathrm{d}U + p\\,\\mathrm{d}V + V\\mathrm{d}p - T\\mathrm{d}S - S\\mathrm{d}T\\,", "60f4bae5f4bca74e1e5a7b03a02edc35": "D(X) > 0", "60f52d80bc747660327c8847f5b82735": "g(x)=\\sqrt{x}", "60f54fb77a534bb275a1c428696a11cf": " Q_1-Q_2<\\left(1-\\frac{T_2}{T_1}\\right)Q_1", "60f564340440b239b19e93135162b7f7": "\\tan(z) = 2\\sum_{k=0}^{\\infty} \\sum_{n=0}^{\\infty} \\frac{2^{2n+2}}{(2k + 1)^{2n+2}\\pi^{2n+2}} z^{2n + 1},", "60f5d8288570b3e401760cb2061b68a2": " S_k=\\{0,1\\} ", "60f5e367c74f4dff1e4438251147b3dd": " \\alpha = - \\mathrm{e}^{\\mathrm{i} \\arg x_k} \\|\\mathbf{x}\\|", "60f5f9438c12468440ba6489027330a5": "\\mathbf{D_{yy}} ", "60f5f9f8c683f69fede5e4a247013c1d": "B_\\mathrm{domestic}", "60f6ac3a4a42ec96b563fcb2fa1dc836": " f^{(1)}=15.2518 - \\frac{15.6875}{T_r}-13.4721 \\cdot \\ln T_r + 0.43577 \\cdot T_r^6 ", "60f6b6a04c46642487a1502f2da8b223": "\\lambda(y) = \\exp(X(y))", "60f6fa1ff7f44591bfa4c2d76cc7dfa6": "q_{n}({\\mathbf{X}})", "60f7057202d85c52c278ab3017935197": "c=jd", "60f765bc063241d9475eab04170a970d": "X \\mathbf{\\operatorname{o}} Y", "60f7995423c83501e75d7c8b4d1f8a44": "x^{\\alpha} = 0", "60f7be496bc5035f15c7b4262a707323": "t = 2 \\pi \\sqrt { \\frac {L \\cos \\theta} {g} }", "60f87315ef680c1b9b20fdc9da0dd943": "\\int_{\\mathbb{R}} \\delta(tx)\\varphi(x)\\,dx = \\int_{\\mathbb{R}} \\delta(y)\\varphi(y/t)\\,\\frac{dy}{t} = t^{-1}\\varphi(0)", "60f89fc34d80f2218151f803a8734a09": "f : P \\rightarrow Q", "60f9027b0cd6ea71dfdfdcbcdce343cc": "\\left(\\pm1, \\pm1, \\pm(1+\\sqrt{2})\\right).\\ ", "60f9210d1dfeaa9b43bffd247db21c78": "\\bigcup_{k=0}^\\infty V_k = V_\\omega.", "60f93caa3ee724de7cf66c8001f13550": "\\frac1{p} + \\frac1{q} = 1,", "60f9aa2c0eec5f7b682a6ab0367b6bef": "\n\\mathbf{A}(\\mathbf{r}, t) = \\frac{\\mu_0}{4\\pi} \\iint \\frac{\\delta(t' - t_r')}{|\\mathbf{r} - \\mathbf{r}'|} q\\mathbf{v}_s(t') \\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t')) \\, d^3\\mathbf{r}' dt'\n", "60f9e13d8f8609ef785ed9349f3a430d": "p_{n}^{1/n}\\, ", "60f9f5722910fbe60c7dabbe6055bb2c": "\\bold{F}_f = -\\nabla_\\bold{v} F", "60fa09c548c5a187c4acfedd24611353": "\\epsilon \\in \\{ -1,1\\}", "60fa2ec63e32c1227b6e33b1d173a965": " V(f_1,\\ldots, f_k) = \\left\\{(a_1,\\ldots,a_n)\\in k^n \\;|\\;f_1(a_1,\\ldots, a_n)=\\ldots=f_k(a_1,\\ldots, a_n)=0\\right\\}.", "60fa771fd1c501180ccb2c2a0c15c671": "p_H(\\mathbf{X}|\\boldsymbol{\\chi},\\nu) = \\left( \\prod_{i=1}^N h(x_i) \\right) \\dfrac{f(\\boldsymbol{\\chi},\\nu)}{f\\left(\\boldsymbol{\\chi} + \\mathbf{T}(\\mathbf{X}), \\nu+N \\right)}", "60fa7e1d6c36db429a9770b0007d16d1": "\n\\begin{matrix}\nI(Y_{1};Y_{2};Y_{3};Y_{4}) & = & I(Y_{1};Y_{2};Y_{3}|Y_{4})-I(Y_{1};Y_{2};Y_{3}) \\\\\n\\ & = & -2+3 \\\\\n\\ & = & 1\n\\end{matrix}\n", "60fa81f45aef3bc2225f8481c0f18173": "\\gamma_0\\cdot\\bigtriangledown=\\frac{1}{c}\\frac{\\partial}{\\partial t}", "60fab14b5145d881a516f544159e1a90": "\\mathit{FDR} = \\mathit{FP} / (\\mathit{TP} + \\mathit{FP}) = 1 - \\mathit{PPV} ", "60fad8c9274fa9bb9281e863903b1bea": "\\vec \\sigma = \\langle t_{i_1} \\ldots t_{i_n} \\rangle", "60fb1f50b21d7905a3cd91efd1efce8a": "\\sqrt{2.35 \\times \\frac{4}{3}} \\approx 1.7701", "60fb53be1aee6d7ac65fbe52c3b6acde": "a \\uparrow^{n + 1}b", "60fb6cc22b3b5d8321337e2ba831e653": "\\ P = \\rho R_{\\rm specific}T ", "60fb9a78dcc3dad3919de38e0d8ce458": "f^{*}t_{ij} = t_{ij} \\circ f.", "60fbf2d880edf309da70cbd3d6e49ddb": " \\left(\\frac{\\partial}{\\partial x} + i\\frac{\\partial}{\\partial y}\\right)f(x+iy) = 0 ", "60fc185e1bbc082ad62c4c67d279e39d": "(PA)^2 + (PB)^2 +(PC)^2 =(MA)^2 + (MB)^2 + (MC)^2 +3(PM)^2. \\,", "60fc1c3317b99385fcd3b78de16f25a6": "e = x \\backslash x", "60fc2337adb30c78c7ead2793d48a496": "\\begin{align}\nJ_1 &= J^{23} = -J^{32} = i\\biggl(\\begin{smallmatrix}\n0&0&0&0\\\\ 0&0&0&0\\\\ 0&0&0&-1\\\\ 0&0&1&0\\\\\n\\end{smallmatrix}\\biggr),\\\\\nJ_2 &= J^{31} = -J^{13} = i\\biggl(\\begin{smallmatrix}\n0&0&0&0\\\\ 0&0&0&1\\\\ 0&0&0&0\\\\ 0&-1&0&0\\\\\n\\end{smallmatrix}\\biggr),\\\\\nJ_3 &= J^{12} = -J^{21} = i\\biggl(\\begin{smallmatrix}\n0&0&0&0\\\\ 0&0&-1&0\\\\ 0&1&0&0\\\\ 0&0&0&0\\\\\n\\end{smallmatrix}\\biggr),\\\\\nK_1 &= J^{01} = J^{10} = i\\biggl(\\begin{smallmatrix}\n0&1&0&0\\\\ 1&0&0&0\\\\ 0&0&0&0\\\\ 0&0&0&0\\\\\n\\end{smallmatrix}\\biggr),\\\\\nK_2 &= J^{02} = J^{20} = i\\biggl(\\begin{smallmatrix}\n0&0&1&0\\\\ 0&0&0&0\\\\ 1&0&0&0\\\\ 0&0&0&0\\\\\n\\end{smallmatrix}\\biggr),\\\\\nK_3 &= J^{03} = J^{30} = i\\biggl(\\begin{smallmatrix}\n0&0&0&1\\\\ 0&0&0&0\\\\ 0&0&0&0\\\\ 1&0&0&0\\\\\n\\end{smallmatrix}\\biggr).\n\\end{align}", "60fc2e1fe04aa40aea7164a689909cd2": "v_i = \\frac{2 A}{L-1} i - A; \\quad i = 0,1,\\dots, L-1", "60fc538755588f90f89152f8bdc7828b": " x^2 + D = A B^n ", "60fcbeb030a55a3c9e4afbdef73b54b5": "\\gamma_{eff}(A)", "60fccd6e1d8ee9cbab3b6de8ddd7015d": "F_{ts}", "60fcf219a549caae50e948f82886d5f6": "\\cap (D\\setminus \\{c\\})\\setminus c ", "60fdcc4b09b83d2037afe7e4ce9a031d": " P_A O_2 = \\frac{P_E O_2 - P_I O_2 (\\frac{V_D}{V_T})}{1- \\frac{V_D}{V_T}}", "60fe4280740aa5fb63b3ce5f77036e65": "l(1+l^2)", "60fea24a28208d51aead71a28bca381d": "\n T_{11} = \\sigma_{11}/\\lambda = \n \\left(\\lambda - \\cfrac{1}{\\lambda^2}\\right)\\left(\\cfrac{\\mu J_m}{J_m - I_1 + 3}\\right)~.\n ", "60feac1ecd7401f184cd75779df858ff": "\\log D", "60feacc308e3ec8382dd5e5b7a8100ee": "M = \\begin{pmatrix}A & B \\\\ C & D\\end{pmatrix}", "60feb12743cb42d57533ceb3d066f147": "r_p = y^{((p+1)/4)^{L}}~mod~p", "60ff93afd481a2c97dbaa7046aa99d9a": "|a_{11}| \\ge |a_{12}| + |a_{13}|", "61002f74b4fa2bb40d72c9e48057984c": "p\\ge 0", "61004f9bbbe44fd8925cf093f0d8b715": "\nx = b_0 + \\frac{a_1 \\mid}{\\mid b_1} + \\frac{a_2 \\mid}{\\mid b_2} + \\frac{a_3 \\mid}{\\mid b_3}+\\cdots\\,\n", "61007b0aa7e1471ac1ba115b6c30a49a": "\\Phi_{mK}(M)=\\Phi_{mK}(W)", "610090575092a267e005804011b56134": "Z(t)= \\int_{R^n} |f(x_1,\\ldots,x_n)|^s \\, dx ", "6100921ac48d0cb94b6b74851751098a": "b := f(a) \\in Y", "6100973d34baa41cba746704f433253b": " \\begin{align}\n p &= \\frac{3}{2\\pi^3}\\left(1+\\frac{1}{2^3}+\\frac{1}{3^3}+\\text{etc.}\\ \\right) = 0.0581522\\ldots \\\\\n q &= \\frac{15}{2\\pi^{5}}\\left(1+\\frac{1}{2^5}+\\frac{1}{3^5}+\\text{etc.}\\ \\right) = 0.0254132\\ldots.\n\\end{align}", "61009fb83c3dd9c735a44f685e33ea0b": "\\Bbb R^3", "6100b820be03c58abf7616cf5c993131": "(2g)^p", "6100ba51333f8372ad2e981dd045d502": "\\hat h \\leftarrow \\underset{\\omega \\in \\Omega}{\\textrm{argmax}} \\sum_{n=1}^{\\ell} y_n h(\\boldsymbol{x}_n;\\omega) \\lambda_n", "6100f1d8a5b396a43b8e2b70f51b5e11": "mp \\times nq", "61015bfca0f525c49c8ea10df58fbc44": " C_x=d_{xx} F_x n~", "6101a47974671af43866b52985045894": "= \\sum_{n = -\\infty}^{\\infty}{\\left|h[n] z^{- n} \\right|}", "6101a5332190fc64fe3d1d5ae033360a": "\n\\arcsin(z)\n", "6101b14c29f4a710c17032cf515c9449": "\\Delta y_{it}", "6101c11b12b75b5b2b4c234902a3b957": "l_{11} \\cdot u_{12} + 0 \\cdot u_{22} = 3", "6101c8b9cf7004220c0bdd6000b8bf32": " \\mathbf {X} \\,\\!", "6101d5a59f9338f32f0a2ddf855d31a2": "\\mathit{Nu}_{D}={1.86}\\cdot{{{\\left( \\mathit{Re}\\cdot\\mathit{Pr} \\right)}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{3}\\;}}}{{\\left( \\frac{D}{L} \\right)}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{3}\\;}}{{\\left( \\frac{{{\\mu }_{b}}}{{{\\mu }_{w}}} \\right)}^{0.14}}", "610259aa9ac69eb8ae918486b1bec3a9": " \\dot m", "610291e743805688c3dbb52fa654cf99": "\\displaystyle{g\\cdot z=\\alpha z + \\beta \\overline{z}.}", "6102f12d73485f5bf7cf5e3aeac86d47": "\n\\Sigma\n=\n\\begin{bmatrix}\n \\sigma_{CC} & \\sigma_{CS} \\\\\n \\sigma_{SC} & \\sigma_{SS}\n\\end{bmatrix}\n\\quad", "61030ae9afed4baf23761718ca526eee": "P_\\ell^{(m)}(x) = (-1)^m\\,(1-x^2)^{m/2}\\ P_\\ell^{[m]}(x).\\,", "61031976a67cab5fd84dd6440adc33c2": "\n E = 4\\pi^2 m \\nu^2~u^2 = k_B T\n ", "6103744616b91306039e3891371cf71f": "p=2,", "6103e89b446b172c45377308466f1123": "Q_r > K_{eq}~", "61048477ef3f62ca2fef59bbbdb0a153": "f(x'_1,\\ldots,x'_m,x_1,\\ldots x_m)=(h_1(x_1,\\ldots x_m)-x'_1,\\ldots , h_m(x_1,\\ldots, x_m)-x'_m).", "6104ce99134464d1f3b6588e79caab69": "\n \\displaystyle \n S(3,2)\n =\n \\left\\{ \n\t (111),\n\t (112), \n\t (122), \n\t (222) \n \\right\\}\n", "6104dd9ff0f4830de32f6b2ea3490fb3": "\\nabla \\times \\mathbf{B} = \\frac{1}{c}\\mathbf{J} + \\frac{1}{c}\\frac{\\partial \\mathbf{E}} {\\partial t}", "610546dbc8fa42b6555394c301906403": "NIA = Contribution \\times \\frac{Adjusted\\ Closing\\ Balance - Adjusted\\ Opening\\ Balance}{Adjusted\\ Opening\\ Balance}", "610567eb1e0b7688f833b6d3dcb79382": "\\chi_{0,n} = \\chi_{n,n} = 1", "61059f52426fa4be84876a65949baaac": " f(z)", "6105abf061c69e327d9f9db98ff41aa9": "\n k_D^2\n= {4\\pi n e^2 \\over T_e}\n", "610667c36f3e05213368ed1dd850f05a": "\nE\\left\\{S|\\theta,J\\right\\}=\n\\frac{\\theta}{\\theta }+\n\\frac{\\theta}{\\theta+1}+\n\\frac{\\theta}{\\theta+2}+\n\\cdots +\n\\frac{\\theta}{\\theta+J-1}\n", "6106ba82ea3b4d89fc941ecfb5c043fc": "d_{ij}(Z)=0, i\\ne j", "6106c69b73ea28a124e83742f40235a1": " \\langle \\alpha_i,a_j\\rangle = \\delta_{ij}.", "6106c9c56b25fb084465e822f35fd9e2": "\\tilde{\\mathcal{A}}_n(\\mathbf R)=\\mathcal{A}_n (\\mathbf R)+\\nabla_{\\mathbf R\\,}\\beta(\\mathbf R)", "610720ee0d5fd606a9ef32e5840e64f4": " K_H ", "61074768ea49631d9f4a4ff083a73d37": "\\beta = \\frac{v_j}{c}", "610758441457dee3f6d1ec921108791e": "G_k(s)=\\frac{k-1+s}k,", "61078c3eb46bb47f9d2c1e46d182e684": "\\varepsilon _{1} =-\\frac{p_{1}\\cdot P}{\\sqrt{-P^{2}}}=-\\frac{\nP^{2}+p_{1}^{2}-p_{2}^{2}}{2\\sqrt{-P^{2}}}", "6107c9e21839883126344b11fe3b0af5": "l_1 = (l-1)/2", "6107df9515d434535ee9d4b7294174f6": "A_1,A_2,\\ldots,A_n\\in\\mathcal{A}", "61082e1ad477946c00c81a34badbc726": "\\hat{x},\\hat{y}", "610856f5bb3d07d90e71cc36df15017e": "K(x-y,\\tau)", "610867d22b1650037a79b690191cbc13": "\\left|\\frac{d}{dt}\\right|^\\alpha f(t)", "610869bbde380d257720345100b8e238": "\\bar x_i = \\frac {\\sum_{j=1}^n x_{ij}}{n}", "6108cfd1634fa7ae09d9f00e45d02906": "\\mathfrak{g}=\\mathfrak{h}\\oplus\\bigoplus_{\\lambda\\in R}\\mathfrak{g}_\\lambda.", "61090d972ad81aa2b4890f57e7d43c84": " e_1,\\, ...\\, , e_n ", "610925c6dfe87e5cf4676d311d631d30": "\\ell(y) = \\max(0, 1 + \\max_{y \\ne t} \\mathbf{w}_y \\mathbf{x} - \\mathbf{w}_t \\mathbf{x})", "61094f35fe1237b0f846350ba0835d83": "E_7,", "610a41c8952d4d4c59201c55c86a68c6": " I_T = \\int_t^{t+\\Delta t} T_P \\,dt = [ \\theta T_P - (1 - \\theta ) {T_P}^0 ] \\Delta t ", "610a4e3f5de47bd35b82947c21493697": " m_g = m_ae^{-\\tfrac{1}{2}\\sigma^2}.", "610abb8cff315b06178a92ca56d35ffd": "1/s", "610b315d97d25f804eeae203585d596a": "x^*\\in C^1[t_0,t_f]", "610b3360dccca84909e836bc210ca272": "T_{n+1}(x) = 2x\\,T_n(x) - T_{n-1}(x).\\,", "610b4121f442b84d51e7bfbe92d546dd": "P\\in C", "610bb104fde2c2117f633e7cb8fd5d63": "N=\\left\\lfloor0.5-\\log_2\\left(\\frac{\\text{Frequency of this item}}{\\text{Frequency of most common item}}\\right)\\right\\rfloor", "610bedb2abd507d223b735b039b9fb9f": "A_{ji}P_{i}(t\\rightarrow\\infty)=A_{ij}P_{j}(t\\rightarrow\\infty)", "610c0df2ef3ca189ee618373074d69cd": "P(G - uv, k)", "610c5a8445492e4e6291a45449cb4de9": "\\left |\\phi_{n}\\right\\rangle", "610c677119933b894283aa85acdbe4cb": "Z(S) = \\{x \\in \\mathbb A^n \\mid f(x) = 0 \\text{ for all } f\\in S\\}.", "610c9411149920e8e3340a8c58cded44": "u'_1:=0", "610cb9c1568546a641a20ac766f46703": "A \\otimes_k F", "610d481fcd68e8082e4b9b5b9b91b646": "-\\log(p_{i,j})", "610d52fe3fec39d3054f2daf0c2a3858": "\n \\lim_{n\\to\\infty} F_n(x) = F(x),\n ", "610dd5a4dff9d497e60314beaff6c686": "I = \\sqrt{\\pi}", "610e0d3cff6a715bdc2cb65063e6577e": "\\frac {d^2x} {dt^2}", "610e72c53ba0c950f7e296fcac3f592d": "E_\\text{f}", "610eb7885786780756c45e337866b0e4": " \\sigma_{g_1} = \\sqrt { \\frac { 6(n-2) }{ (n+1)(n+3) } } ", "610ef08ed47fb73177ded01d0bd8bee4": "\\mathcal H_{1,\\, 2} =+\\frac{2t_{Mn,\\,O}^2\\,}{U}\\hat S_1\\cdot\\hat S_2\\,,", "610f6d0b70a19be6daf8a7b144ab2598": "g_{22}=\\, r^2", "610f8b82760411678f281640b27f6838": "\\scriptstyle c_A", "611025ecd473742015fee49a197d82da": "end\\,\\!", "611036012d7cd21d84d782b70cb049be": "\\frac{n_3}{n_2} = \\frac{\\tau_{32}}{\\tau_{21}} = \\frac{W_{21}}{W_{32}}", "61107d830f288e534be2e22038bccbe5": " a^2 - b^2 = (a + b)(a - b). ", "61109186825f9c2f8142e6b67a38a0f9": "\\left \\{ \\Gamma^l {}_{ij} \\right \\},", "6110c16170c97947ce780c7a9316ac66": "U=eV", "61110343557a65b42ef7247a33a9d45b": "{T^{ab}}_c", "61113262e69f1ac031232d7795154aa3": "e^{i\\theta}", "61115158e097615298be4e71c6b33ee3": "\\frac {1}{c(w)} =\\frac {1}{c_0} [1- \\frac {(w\\tau_r)^2}{Q_c[1+(w\\tau_r)^2]}] \\quad (4)", "6111521aa3320a95892b5dc5cb0788b1": "n_2 + n_4\\,\\!", "61116c534c877ac02a30225bb661c303": " \\dot =", "61119e41bea274aa30a2acdd3a6fe438": "h^{\\alpha\\beta}\\ \\stackrel{\\mathrm{def}}{=}\\ g^{\\alpha\\beta}-U^\\alpha U^\\beta\\;", "6112404fb4d6c1e81cbdf361926720f3": "\\eta_1 = \\alpha-1,", "6112daea9cd69d6d1b4e1d3b526d7567": " p_k p_\\ell = p_\\ell p_k, \\quad q_k q_\\ell = q_\\ell q_k, \\quad p_k q_\\ell - q_\\ell p_k = \\delta_{k \\ell} z, \\quad z p_k - p_k z =0, \\quad z q_k - q_k z =0.", "611347db34bab19684c77abcab94b8d9": "f = \\tilde{f} \\circ \\pi", "6113c68809304db4e12a8fc861c9bfb3": " b_i ", "6113ca47836dca6c3e6c07bef645e06f": "\\Pr\\Big(n \\text{ coin tosses yield heads at most } k \\text{ times}\\Big)= \\sum_{i=0}^{k} \\binom{n}{i} p^i (1-p)^{n-i}\\,.", "6113e3eb45a44738b0291fb4c9737e34": "e^{tA}=s_0(t)\\,I+s_1(t)\\,A", "6114012cb2e179bb132a87617cb31392": " \\widehat p = \\tfrac i {\\sqrt 2}(\\widehat a^\\dagger - \\widehat a)", "61140f35ab964b8a6581fe111e03da1d": "U(S) = \\frac{Q}{\\phi A} \\frac{\\mathrm{d} f}{\\mathrm{d} S}.", "61142b13bb769503a5e7b3e133092724": "\n\\begin{align}\n\\int\\csc x\\,\\mathrm{d}x&=\\int\\frac{\\mathrm{d}x}{\\sin x}&\\\\\n&=\\int\\frac{\\mathrm{d}t}{t}&t=\\tan\\frac{x}{2}\\\\\n&=\\ln t+C\\\\\n&=\\ln \\tan\\frac{x}{2}+C.\n\\end{align}\n", "61143f91a462402b402f4eb39f74c263": "\\nabla \\mathcal{R} = \\omega \\otimes \\mathcal{R}", "61145f093200daa039cdb98444fcc245": "\\{ \\neg p \\} \\vdash (p \\to r)", "61156337ea33e0dbcdd911e410ab7091": "c \\equiv (a \\cdot (b\\ (\\mbox{mod}\\ m))) \\pmod{m}", "61157caff2544ca7daa45083a43713ef": " a = l + c ", "61158ab48a450b54d0cbd503361226d5": "E = c \\cdot \\sqrt{\\frac{m_e \\cdot \\rho}{\\epsilon_0}}.", "61158bf9d23e9ddd808a0c4b308091bc": "e^\\beta", "6115bafbd46beb29ef5c241203f0cb8f": " \\delta_{ext}(q,x)=(s',0) ", "61160a1c07436d541ed40a67fb749b1f": "\\bigstar \\bigstar ||\\bigstar|", "6116317a20426af170b08fd8fd96d4f1": "\\alpha=1.3", "61165815d6c7feecdd82b30ba56368ed": "=\\frac{\\text{kva base}}{\\text{0/1 X}}", "6116780fa8f18d44490dfa4178b344d3": "\\begin{align}\nF(k;n,p) & = \\Pr(X \\le k) \\\\\n&= I_{1-p}(n-k, k+1) \\\\\n& = (n-k) {n \\choose k} \\int_0^{1-p} t^{n-k-1} (1-t)^k \\, dt.\n\\end{align}", "61167b9fed526e3a82bb2849eb07a6ee": "U(P)= - \\frac{i \\cos \\beta}{\\lambda} \\frac {ae^{ik(r'+s')}}{r's'}\\int_S e^{ikf(x',y')} dx' dy' ", "61168b1eba43b19f075ce48aa19b6bda": " \\sum_n (m_n c^2)^2 - 2\\sum_{n0", "612e8ce0930db9ae04972b18dc382d25": "f_2-f_1", "612ed794341cf0a996b36d70dc6d1140": "X_1^{a_1} \\cdots X_n^{a_n}", "612f5d1d2a28af4143079550c91f16e2": "\n \\boldsymbol{F} = \\cfrac{\\partial \\mathbf{x}}{\\partial \\mathbf{X}} = \\boldsymbol{\\nabla}\\mathbf{x}\n ", "612f8530dcd4faff9f1ae1b42acd97c8": "4\\,\\!", "612fb27dba10b7a95781e34a578f5301": "\n g^{li}~\\frac{\\partial \\varphi}{\\partial q^l} = \n \\left\\{ g^{11}~\\frac{\\partial \\varphi}{\\partial q^1}, g^{22}~\\frac{\\partial \\varphi}{\\partial q^2}, \n g^{33}~\\frac{\\partial \\varphi}{\\partial q^3} \\right\\} = \n \\left\\{ \\cfrac{1}{h_1^2}~\\frac{\\partial \\varphi}{\\partial q^1}, \\cfrac{1}{h_2^2}~\\frac{\\partial \\varphi}{\\partial q^2}, \n \\cfrac{1}{h_3^2}~\\frac{\\partial \\varphi}{\\partial q^3} \\right\\} \n", "613006b51b4c867a99d65233e3a73614": "dU = \\delta W + \\delta Q = -pdV + 0\\,\\!", "613030d36372e3b02004ea4bd9d3e94b": " \\operatorname{var}(Z_i)=\\digamma^{[1]}(\\nu)/(\\mu_i)^2", "613060f4845ba4061d11b645064be649": "\\left|y[n]\\right| = \\left|\\sum_{k=-\\infty}^{\\infty}{h[n-k] x[k]}\\right|", "61308d0faaad2cfa522fb4d768eba2ff": "\n\\begin{pmatrix}\n | & & | \\\\\n \\lambda_1B\\mathbf{v_1} & \\cdots & \\lambda_nB\\mathbf{v_n} \\\\\n | & & | \\\\ \n\\end{pmatrix}\n=\n\n\\begin{pmatrix}\n | & & | \\\\\n B\\mathbf{v_1} & \\cdots & B\\mathbf{v_n} \\\\\n | & & | \\\\ \n\\end{pmatrix}\n\\mathbf{D}\n=\n\\mathbf{B}\\mathbf{P}\\mathbf{D}\n", "6130a338634e0bd38c5f69393732544f": "\\frac{\\lambda^k \\left(1-\\frac{\\lambda}{n}\\right)^{n}}{\\left(1-\\frac{k}{n}\\right)^{n}e^kk!}\\rightarrow\\frac{\\lambda^k e^{-\\lambda}}{e^{-k}e^kk!}=\\frac{\\lambda^k e^{-\\lambda}}{k!}", "6130b8385af84ef1c44b197dd00b5363": "\nE \\in [23V,26V]\n", "6130d460f5b80cfae54fb1c78d277061": "\\sigma(E)=bE^a\\big[1+(q'_o-1)E\\big]^{\\frac{\\beta _o}{q'_o -1}}\\,,", "61316261df77805048589355c1845879": "\\dot x = f(x), x(0) = p", "61319f96303711c3de7cb4af8c44ada3": "... ", "6131a8851d29c6c689cd46adce8a4110": " y = \\hat\\alpha + \\hat\\beta x, \\,", "6131c1a7596099da7858b8a425221fca": "\\theta[\\vec{X}]_{22} \\approx -q \\, \\tan(q u)", "6132147bbdd7b776bc7e67102f90bc0a": " \\Phi(\\mathbf{r}) = \\frac{1}{4\\pi\\varepsilon_0}\\,\\frac{\\mathbf{p}\\cdot\\hat{\\mathbf{r}}}{r^2}", "6132295fcf5570fb8b0a944ef322a598": "Alpha", "613279d3e1c0e55e7dbfe1697cfbea57": " x^{(3)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.8494 \\\\\n -0.6413 \\\\\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8077 \\\\\n -0.6678\n \\end{bmatrix}. ", "613315919eba5723f2bd0e5ced7aa545": "g^b \\bmod p", "61333f6e4e24fbaf38104cebe93608de": "\\Delta y = f'(x)\\,\\Delta x + \\varepsilon\\, \\Delta x\\,", "61338a0b4456f082041751bb742c1a28": "Z b \\bar{b}", "6133f85bdda8c4bf94cc55ef8470e953": "\\ Q(s,a) = \\sum_{s'} P_a(s,s') (R_a(s,s') + \\gamma V(s')).\\ ", "61341fff94eb16f4548614658ddbb58e": "\\mathbf{Q}=\\sum\\limits _{m}q_{m}\\hat{\\mathbf{e}}_{m}", "613420e3086aaae578822ae21728b054": "y|R(\\textbf{x})", "61345f4205ff37aeee7c31e4096ba67b": " E_R = \\sum_{r} n_r \\epsilon_r \\; ", "61347a2eaefecb85177fe2c248140a23": "(2) P={g}(\\overset{+}M)", "61347cd0e8caa196d03555a002b6963e": "a\\,d\\mathbf{X}", "6134b1e64bbdb073d48c5b6147b343c7": "V_{SM} = -\\lambda (H^\\dagger H) + \\mu(H^\\dagger H)^2", "6134d2cc5a11881d73b02280c511d1cc": "\n\\mathbf{C} \\triangleq \\int_{\\Delta} \\rho \\mathbf{x}\\mathbf{x}^{\\mathrm{T}} \\, dA\n", "6134f90848ceab85b8590955fb337d02": "cos^{2}\\vartheta _{m}\\simeq 1-\\left ( \\frac{\\beta }{k_{0}} \\right )^{2}", "613509392ca31169c1c7d46e38f541ac": "\\int x \\phi(x) \\, dx = -\\phi(x) + C", "61357ebfd7af808e7e7c9a20e68c6092": "s = s_k", "6135de4b33c5ba679d666a3985182395": " \\!\\ x^{ni} = \\cos(\\ln x^n) + i \\sin(\\ln x^n ).", "613611ece73911b3593018c9cb137283": "E_{i,j}(\\vec{\\bold{r}}_{n,n'}) = \\langle n,i|H|n',j\\rangle", "613650d0e4a6f3c2cead59617c025803": "\\bar x_i", "613674b4701449d75bb3a5bb9ce64585": "F_4(a, b) = a \\uparrow\\uparrow (b - 1)", "6136a62a200ab5665a8752ed1295c0a8": "\\lambda_J=\\sqrt{\\frac{15k_{B}T}{4\\pi Gm\\rho}}", "6136bc7750b28e1f54011f6d5c63e92e": "M_{\\rm u}", "61371f974f6c5d7612719beb18b28473": "\n- \\frac{\\hbar^2}{2m} \\nabla^2 \\psi(\\mathbf{r}, t) + V(\\mathbf{r}) \\psi(\\mathbf{r}, t) = i \\hbar \\dot{\\psi}(\\mathbf{r}, t)", "61374ce57b7ac292621d725996f34745": "r=\\cos k \\theta", "61376d69d82a2880a320fdcf4629f8dd": "k = 5", "61378eacbfc680ab39d3a48161113507": "(X,D)", "61380159b79d6b462b31facc4e0aac2a": "\\alpha_\\mathrm{QED}(M_Z^2)", "61381c078bebb1a122839fc8b8e1241a": "M \\cup_f H^j", "61387437566c024c911e4099fb69c76a": "(a,b,c)", "6138cbe77dc6836cc918a6108e15f35c": "\n\\begin{array}{c|cc}\n0 & 0 & 0 \\\\\n1 & 1/2 & 1/2\\\\\n\\hline\n & 1/2 & 1/2\\\\\n\\end{array}\n", "6138d20fb641a364f3b52fa39dc20cd9": "\\scriptstyle <6\\times10^{-22}", "6138f6c9e5d3a33f425fd07d8999904d": "f^\\# \\colon \\mathcal{O}_{X, f(y)} \\to \\mathcal{O}_{Y, y}.", "6138f8e98e6db59d6065d3794baac845": "\\frac{P \\to Q,\\; P}{\\therefore Q}", "6139660359ea62e00f42731c127b524d": "I_x(\\alpha,\\beta) = \\tfrac{1}{2} ", "61397be1849c4c0cd388d111b447a438": "z^0=0", "613a181479be041c305ee9464bc349bc": "\\ln(S) ", "613a247a161cc1ae84e2249d63a3602f": " \\vert \\vert \\mathbb{P}_n - P\\vert \\vert_{\\mathcal{F}} \\rightarrow 0 ", "613a2eeff0c21deeacc4031e77b08b77": "F_{MAX}", "613a704a071be22f2d7b9c3fb44c011a": "Ja\\Omega=a^*\\Omega", "613a7509202016d3128d880f2d7566dd": "(x+y)(x-y)=x^2-y^2", "613a751ef235f801586f276cc071e3c0": "\\nabla \\cdot\\boldsymbol{\\mathsf{T}}", "613ab95a10a87655d0deb3186785eda0": "\\frac{d}{dz}w = w", "613b308d741438990c7b480bca6b61b1": "H(x) = 0", "613b381ca31ab897dc4546582036c82d": "p_{0j}\\ge0", "613b982fa9997d2e5096193e8e978736": "\\delta(\\tau) \\, ", "613c67cc5528043e375b6168203ef36e": "y_m = -\\frac{2AD-CE}{4AB-E^2}", "613d246d9f3e6004853d500d9d7d86dc": "\\liminf_{r \\to \\infty} \\sup_{\\alpha < \\theta < \\beta} \\frac{\\log|F(re^{i\\theta})|}{r^\\rho} = 0 \\quad \\text{for some} \\quad 0 \\leq \\rho < \\lambda~,", "613d2e8bb471d6c6615e85ba41f32117": "\\,\\! I_x=I_y^2", "613d7d13aa913eaf29a68374ec36645f": "H_\\mathrm{Heisenberg} = -J \\sum_{} \\left[S_i^z S_j^z + \\cfrac{1}{2}(S_i^+ S_j^- + S_i^- S_j^+)\\right]", "613da37a4d0e78a8cc1a89813666552e": "dG=-SdT+VdP+\\sum_i \\mu_i dn_i,\\,", "613db734e6fda91b0bb4f80b01b5dab6": "\\frac{1}{\\sqrt2} \\begin{pmatrix} 1 \\\\ +i \\end{pmatrix}", "613dcd0c0a2636182aa04d602945cd40": "\\det(M) = \\det(I_p + A B)", "613e6eea7d8f597ddb4ca47f2290bd06": " a_i", "613e8a072021f4f85afec59c3c2fe12c": "\\frac{d[A]}{dt}=-k_f[A]_t+k_b[B]_t", "613ed80bcdc2586cc3e65e8a8935d0d9": "\\scriptstyle L^p(G)", "613edc81abffc42717acfbcaa6d27f7c": "R_2=R,\\,", "613ff7077a571f3af8c8b36215a26490": "U_0(x)=\\frac{\\sin(x)}x;", "61406946d3a5bb3d9ea0856777e2eb37": "\\frac{\\zeta(s-1)}{\\zeta(s)}~\\textrm{for}~s>2", "61407aae9d7b22a57552f42162dac334": "x_1 = x_2 = \\cdots = x_n,", "614134a75648b9863d749870a6511471": "S \\or \\neg S", "614171a4f7104d4cc49cdea8746f81b9": "y = h(x), ", "61419313552820e28f0844d0d93aeb4f": "Q = c_p\\rho u. \\,", "6141a400a8642d048f0add837d71c4f5": "\\rho^3 + 3(460+183\\rho-354\\rho^2-979\\rho^3-575\\rho^4)", "6141b54226207cb06375f57e7b2bc5bc": "X_1\\times X_2 \\to Y_1\\times Y_2", "6141bf6cfb9d93ae59d4d79a9f13265d": "\\mathrm{H}(p, q) = \\mathrm{E}_p[-\\log q] = \\mathrm{H}(p) + D_{\\mathrm{KL}}(p \\| q).\\!", "6141c223b52eaf01caa4d22b1a73cbaa": "1 - \\frac{\\sum_{i=1}^G (t_i^3 - t_i)}{N^3-N}", "6141c943ea75ee2bd39c154d09dc22b9": "\n\\begin{align}\n\\mathbb{E}W & = \\mathbb{E}(W \\mid \\text{fails before } t) \\cdot \\mathbb{P}(\\text{fails before } t) + \\mathbb{E}(W \\mid \\text{does not fail before } t)\\cdot \\mathbb{P}(\\text{does not fail before } t) \\\\\n& = \\frac{t}{2}( 2600 ) + \\frac{2-t}{2} ( 200 ) = 1200t + 200.\n\\end{align}\n", "6141dae0577d9803b17b12e2055f8b60": "\\approx -kL_{e}^{2}\\theta-\\mu B \\theta\\left(\\frac{ H_k}{B+H_k}\\right)^2-\\mu H_k\\theta\\left(\\frac{B}{B+H_k}\\right)^2\\Rightarrow", "6141df481befbc91519c822779105ec0": "\\tfrac{2}{4} = 0.5", "614250ae935c9b7839bde72fc40793be": "U=\\int f(v)E(v)\\,dv", "61430a70a9ecaf4ed9cfe4e586479246": "r_3 = 11a_0 + 13a_1 + 9a_2 + 14a_3", "61431128b5e63397f14c05ba72520d8c": "r = \\frac{1}{1 + e \\cos \\theta},", "614313af38200463b70943d84e641011": "\\textstyle P_{m-1}", "61435b073c8ffef7c02f21e66e132399": "\\frac{\\mathrm{D}^2\\xi^\\alpha}{\\mathrm{d}s^2} = -R^\\alpha{}_{\\beta\\gamma\\delta}\\frac{\\mathrm{d}x^\\alpha}{\\mathrm{d}s}\\xi^\\gamma\\frac{\\mathrm{d}x^\\delta}{\\mathrm{d}s} ", "61435e805c5650f1e79d0c4b6a705fea": "[\\overline{\\nu}]=\\sec ^{-1 }", "6143c55c35650e2d190149c7d35da4ea": "f(x_1, \\ldots, x_n)", "6143cc6f9eab04720642a4a803caa5f7": "\n\\begin{align}\n& P_{0}(x) = 1 \\\\\n& P_{1}(x) = x \\\\\n& P_{2}(x)=\\frac{1}{2} \\left(3x^2-1\\right) \\\\\n& P_{3}(x)=\\frac{1}{2} \\left(5x^3-3x\\right) \\\\\n& P_{4}(x)=\\frac{1}{8} \\left(35x^4-30x^2+3\\right) \\\\\n& P_{5}(x)=\\frac{1}{8} \\left(63x^5-70x^3+15x\\right) \\\\\n\\end{align}\n", "6143f71bd2fd8e660866bebef38c2415": "\\delta , \\eta ", "614433f43db9b435e981c61381642515": "f(x) := \\lim_{n \\rightarrow \\infty} f_n(x)", "614472586d16a81bd98e98506189e8ce": "(x,y)\\neq (p,0) \\mapsto (2(x-p) : (2x+p)(x-p)^2- y^2: y)", "6144995236aa5d510993518932a22791": "a+b\\approx 1", "6144c924899c5ba6da9d5263bd171614": "C\\ell(V,Q) = C\\ell^0(V,Q) \\oplus C\\ell^1(V,Q)", "6144f4092f73102f11147fd30de00cfd": "n_{air} = \\frac{A_{av} N}{V}", "6144f586ac5d4972334d17158536026c": "=\n-\\frac{T_2(n)\\log2}{2}\n-\\frac{T_3(n)\\log3}{3}\n-\\frac{T_4(n)\\log4}{4}\n-\\dots\n,\n", "6144f6364a4dbf9f651ba84b79506ef2": "\\limsup_{n\\to\\infty}x_n := \\lim_{n\\to\\infty}\\Big(\\sup_{m\\geq n}x_m\\Big)", "6145701b117708785ec7363d9d2ece18": " J = \\frac{I}{A} \\,\\!", "6145f66f4fbef0ac726d5939665bdfda": "C[r/s]", "6145fcab5d9df058833de20c4eb7c03b": "x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + x^8 + x^6 + x^5 + x^2 + 1", "61466ab84655cc0c25cbdf896fa05dd9": " D(a,s) = \\sum_{n=1}^\\infty a(n) n^{-s} \\ ", "6146996a853c908535f6b3c595de28f8": " = \\left[\\sum_{i=1}^{N} (x_{i2}-x_{i1})(x_{i2}-x_{i1})' \\right]^{-1} \\sum_{i=1}^{N} (x_{i2}-x_{i1})(y_{i2}-y_{i1}) ={FD}_{T=2}", "6146fb71a4e3caf7aa0ed9e17bac099b": "c_g", "61474abb4840b1f3794b6cf2642de370": "L^{p_0} \\cap L_{p_1}", "6147a30ffae32bd3a8f7af41c90b4e68": "K = k_{col} + k_{rad}", "6147a60b6bb4e4c43c02251224046abb": "\ny^{2} = \\frac{\\left( B - \\lambda \\right) \\left( B - \\mu \\right) \\left( B - \\nu \\right)}{A - B}\n", "6147cf90c0b57ec765fcb4d1786312b5": "\n X = \\frac{\\chi^2_m(\\lambda)}{\\chi^2_m(\\lambda) + \\chi^2_n},\n", "6147e2dafec141c38bd62f4acc5b5bb0": " |x(t)|^2=\\int_{-\\infty}^\\infty W_x(t,f)\\,df \\ \\ ,\\ \\ |X(f)|^2=\\int_{-\\infty}^\\infty W_x(t,f)\\,dt ", "61481b64cfd60ec960b0e09c553c7f74": "\\langle \\vec{x},\\vec{y} \\rangle", "614845e2aa831e760f91f2532cdb1b19": " m_v ", "6148e9990053fb1693141fcb324dcd91": "\\scriptstyle\\overline{B}\\,\\overline{D}", "61491fe756a8400105a972592dfe5117": "A=\\ell^2.", "614934a6bdc5c995c5b724c1f45a3a6b": "Z(s) = \\frac {P(s)}{Q(s)}", "61493aac74078e1ad5cf235b0a5bfb6f": "X = \\bigl(\\begin{smallmatrix}\n\\xi_4 + \\xi_3&\\xi_1 + i\\xi_2\\\\ \\xi_1 - i\\xi_2&\\xi_4 - \\xi_3\\\\\n\\end{smallmatrix}\\bigr) ", "61495160cc06ac56349967fc3f427a68": "\n \\int (A\\,a\\,d\\,f(m+n+p+2)-B (b\\,c\\,e\\,m+a(d\\,e(n+1)+c\\,f(p+1)))+(A\\,b\\,d\\,f(m+n+p+2)+B (a\\,d\\,f\\,m-b(d\\,e(m+n+1)+c\\,f(m+p+1)))) x)(a+b\\,x)^{m-1} (c+d\\,x)^n(e+f\\,x)^p dx\n", "61497101d5ab17bd7d5dfe3aa740d1ac": "\\Sigma _ 1 ^d = \\Sigma _ 3 ^d = I", "61497ef6a1b3ab6324999be13c95e33b": "f^{-1} : \\mathcal{T}(Y) \\to \\mathcal{T}(X).", "6149c3a841533e32116daf9fdee792ef": " \\dfrac{d^a}{dx^a}x^k=\\dfrac{k!}{(k-a)!}x^{k-a}\\;,", "6149c7cdf35e132b83f0fcaeef5072cb": "\\textstyle\\frac{X}{1-Y}", "6149da1077f7fef124cfbe15ea9ba8fe": "A = \\begin{pmatrix}G & -G& 1\\\\-G & G & 0\\\\1 & 0 & 0\\end{pmatrix}", "614a6976ac775dc18c8598911af7d8f5": "n_i=1,2,3,\\ldots", "614a7976f11cf3e91f4b858d15b82f42": "\\scriptstyle Z_n = {\\scriptscriptstyle\\frac{1}{\\sqrt{n}}}\\sum_{i=1}^n X_i", "614aa7228550565dc2923da4c2ef7acb": "(x_{n+1},v_{n+1})", "614ad456e61297eb561e955e3e8fa6e2": "a=\\infty", "614b4c2831f016e769761debef4002b6": "E_{\\text{p}}\\approx 0", "614b56ede77a81bf7dc240798f56d506": "f(V, I)=0", "614b61dcefa1d897088d9f5d5c12536c": " \\Delta S_{SA} \\,", "614b87b4a0a99edc1ecb7c0696c43962": "\\mathfrak{p}_1 \\cap A", "614bb71c8081198bc4027022781c1ea5": "\\lim_{|\\pi|\\to 0} \\sum_{i=0}^{n-1}e^{-s\\tau_i}[g(t_{i+1})-g(t_i)]", "614bca962a6a0ecedca7a7126908ac43": "f=\\left[1-\\left(\\frac{R_\\star}{R}\\right)^{1/2} \\right]^{1/4}", "614bf9c6f2eff357bc09f47fc3cfb0b1": "\\sqrt{\\varphi} \\approx 4/\\pi", "614c0c37b7d009abb3fb1208e82908c0": "p_{t+1}(\\hat{x}|x) = \\frac{p_t(\\hat{x}) \\exp(-\\beta d(x,\\hat{x}))}{\\sum_{\\hat{x}} p_t(\\hat{x}) \\exp(-\\beta d(x,\\hat{x}))}", "614c590788e27f872ed704f92badcfea": "\\langle F(x), y-x \\rangle \\geq 0\\qquad\\forall y \\in K", "614c944dc25e8fab531b99352d172120": "c_P=c_V+\\frac{TV\\alpha^2}{N\\beta_T}", "614ccb5fc7c3a667814f28048f06ede8": "\\begin{matrix} {2 \\choose 2}{46 \\choose 3} \\end{matrix}", "614e715d2523e4a8cbdec2d06bc4cf64": " \n\\overline{Y}_i(t) = \\frac{1}{t}\\sum_{\\tau=0}^{t-1}Y_i(x_1(\\tau), ..., x_N(\\tau)) \n", "614ed0b9665301b4e61bb7613cb6734d": "\\frac{\\partial \\ell(r,p)}{\\partial r} = \\sum_{i=1}^N \\psi(k_i + r) - N\\psi(r) + N\\ln{\\left(\\frac{r}{r + \\sum_{i=1}^N k_i / N}\\right)} =0", "614f1527ed912679fc90f4d0a348eb32": "2^6\\cdot 3^2\\cdot 5\\cdot 7", "614f19f5f03245732edd2b49da30d75c": "\n\\begin{array}{lcl}\n \\mbox{Current Dividend Yield} & = & \\frac{\\mbox{Most Recent Full-Year Dividend}}{\\mbox{Current Share Price}} \\\\\n & = & \\frac{$1}{$20} \\\\\n & = & 0.05 \\\\\n & = & 5% \\\\\n\\end{array}\n", "614f28812f1166a258a66f16c1c139b9": "H_X (\\Beta(\\alpha, \\beta) )=H_{(1-X)}(\\Beta(\\beta, \\alpha) ) \\text{ if } \\alpha, \\beta> 1.", "614f61f11794cfd7f05a81ae2dba5d6c": "\\Pr(S)", "614fc7ad70283e3c88fbb0c3278a7060": " \\Pr \\left \\{ \\lambda_\\max ( Y ) \\geq t \\right \\} \\leq \\inf_{\\theta > 0}\n\\left \\{ e^{-\\theta t } \\cdot \\operatorname{E} \\left [ \\operatorname{tr} e^{\\theta \\mathbf{Y}} \\right ] \\right \\}.\n", "614fd5621b5cceb9cea8bd6a89366db8": "\\mathcal{} \\eta (f) = \\eta (f')", "614fdb476fcdc0390aafd9c7dc5a042b": "\\lambda_s(n) < n\\cdot 2^{\\frac{1}{t!}\\alpha(n)^t\\log\\alpha(n)(1+o(1))}", "61506e1391cc642650d2505648e6e71f": "\\begin{align}\n I_3 - I_x + I_G &= 0 \\\\\n I_1 - I_2 - I_G &= 0\n\\end{align}", "6150a2a60dd2085c9741c444eccdfe36": "k_{r,s}", "6151045479fa81071169e126b82a1722": "\\bigg(\\langle\\phi|A\\bigg) \\; |\\psi\\rangle = \\langle\\phi| \\; \\bigg(A|\\psi\\rangle\\bigg)", "615183ad0804a5a9c8b88a6389a59bb4": "u: \\{ 0, \\ldots, n-1 \\} \\rightarrow A ", "6152154267a719dcd411b727335733b9": "|-\\rang", "61523cd635ff13d0b88bd7e6ef672e59": " A,B", "6152550334901e0f62ea3f7c4a055080": "\\tilde g\\colon T\\to E", "61525a62fae4968fb76b63608402b43a": "\\mathbf{V}", "6152634cee3524b5331de1e8c52aa09d": "V_e\\,", "6152ac6f8b2299ba864154bf1caea95e": "\\begin{Bmatrix} r , q \\\\ s \\ \\ \\end{Bmatrix}", "615303786ca7c6343c24ffc3f8794165": "[q]^n", "615305823a862715ce348cd684bddd26": "f(x) = f(x*)", "615336cce579d1c973e83daf92bbb60e": "\n- \\frac{1}{2m} \\nabla^2 \\psi(\\mathbf{r}, t) + V(\\mathbf{r}) \\psi(\\mathbf{r}, t) = i \\dot{\\psi}(\\mathbf{r}, t)", "61534200176e7f36957266d29b46dbc3": "(L,\\wedge,\\vee,\\ast,\\Rightarrow,0,1)", "615346fe997e74a38d1e2911f7b9dcb9": " \\frac{\\Delta G^\\ominus(T_2)}{T_2} - \\frac{\\Delta G^\\ominus(T_1)}{T_1} = \\Delta H^\\ominus(p)\\left(\\frac{1}{T_2} - \\frac{1}{T_1}\\right) ", "615355bd5804a6077b8a23ba017aec68": " \\begin{align}\n & dx = a_\\text{map} \\left(\\frac{a}{\\sqrt{1-e^2 \\sin^2 \\varphi }}\\right)^{-1} \\sec \\varphi \\, dE ,\\\\\n& dy = a_\\text{map} \\left(\\frac{a(1- e^2)}{\\left(1-e^2 \\sin^2 \\varphi\\right)^{3/2}}\\right)^{-1} \\sec \\varphi \\, dN ,\n \\end{align}\n", "61538f909c00df6d19ec42d07ee52e6b": "\\begin{align}\n C_{ijkl} &= \\lambda \\delta_{ij}\\delta_{kl} + 2\\mu \\delta I_{ijkl}, \\\\\n C_{ijklmn} &= 2C \\delta_{ij}\\delta_{kl}\\delta_{mn} + 2B(\\delta_{ij}I_{klmn} + \\delta_{kl}I_{mnij} + \\delta_{mn}I_{ijkl}) +\\frac{1}{2}A(\\delta_{ik}I_{jkmn} + \\delta_{il}I_{jkmn} + \\delta_{jk}I_{ilmn} + \\delta_{jl}I_{ikmn}),\n\\end{align}\\!\\,", "6153db68896ff5d63a24d8124c65793f": "\\textstyle Z. ", "6153efcd2a1a4e1699e7f0231041740b": "(n-(n-1))/n", "615401bd8f1123368aff50399d8901c8": "\\Vert \\,\\Vert", "61540ea270c8ce2cf31f2dae00aaea80": "\\biggl(\\bigvee_{i\\in I}{x_i}\\biggr)^\\circ =\\bigvee_{i\\in I}(x_i^\\circ).", "61541989cae262850ff92286f7562a26": "S_\\theta = P_{K_\\theta} S|_{K_\\theta}.", "61541efab0e9f8636964f0c00a70487c": " \\mathbb{C}(x,y) ", "61543de65bad8f4d0df7131da6d3f0be": "I_{pl} (\\rho, \\varphi) = I_0 \\rho^l \\left[L_p^l (\\rho)\\right]^2 \\cos^2 (l\\varphi) e^{-\\rho}", "61544a61a71a7de3843222b21cdfee19": "f(q_{1},q_{2},...,q_{n},\\dot{q}_{1},\\dot{q}_{2},...,\\dot{q}_{n},t)=0", "6154cbb9948e123120a040c97bef09a7": "S^0_R", "6154f24c8a8074033371910f24d7c18f": "(id)", "615510a0e9a439b19099ad647e0b2179": "\nV_c = \\frac{1}{1 + i \\omega RC} \\cdot (V_s) = \\frac{1-i\\omega R C}{1+(\\omega R C)^2} \\cdot (V_P e^{i\\theta})\\,\n", "6155480a6b21b2e380f5dad42cd08c9c": "\\bar P", "6155bd205a59f00fba235990d7bb1858": "\\ K \\left(\\frac{\\sigma}{E} \\right)^n = \\alpha \\frac{\\sigma_0}{E} \\left(\\frac{\\sigma}{\\sigma_0} \\right)^n", "6155c0bb435975c73056eab2e688296d": "\\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}\n=\\sum_{j=1}^k (-1)^{k-j} \\frac{j^{n-1}}{(j-1)!(k-j)!}\n=\\frac{1}{k!}\\sum_{j=0}^{k}(-1)^{k-j}{k \\choose j} j^n\n.", "61563fceccfb2101957ed6d92ef0f571": "e^{-2\\overline{X}}\\quad", "61566d26ed66fc1a1be5ce2979fac1ea": "\\dot{d}(t) = d_0 \\dot{a}(t)", "61568aae887458c034e6d1d8f5da6c79": "\\exp\\;[-\\,y^2/\\,(2\\;\\sigma_y^2\\;)\\;]", "61568ccf18445694f15808616755d741": "F_s = \\frac{Q^2}{gA} + zA", "6156afeb84aae46ef3bf1cb324c8f277": "=m_0 c^2 \\left( \\begin{matrix} \\frac{1}{2} \\end{matrix} v^2/c^2 \\right) \\ ", "6156d298187c6601719627d367c00f15": "\\begin{align}\n\\mathbf{J}' & =\\mathbf{J}-\\gamma \\rho \\mathbf{v} +\\left( \\gamma -1 \\right)(\\mathbf{J}\\cdot \\mathbf{\\hat{v}})\\mathbf{\\hat{v}} \\\\\n{\\rho }' & =\\gamma ( \\rho - \\mathbf{J}\\cdot \\mathbf{v}/c^2) \n\\end{align}", "6156ec1e428b7cae59b1d455a936a04d": "r_c = \\frac{\\operatorname{arccosh}(3)}{\\sqrt{2} \\omega}", "615701df97e908cac30c1c4e015efb00": "\n\\begin{array}{l}\n6804/18=378\\\\\n378/18=21\\\\\n21/3=7\n\\end{array}\n", "61571849f156b96907cbee448634a67c": "\\{\\gamma_\\mu\\}", "615763c86ad292bde6fbca8f7d1cf15a": "\\displaystyle R", "61576a493f8a12fcd94aa10a8939eda1": "\\pm\\frac{\\sqrt{1 + \\cot^2 \\theta}}{\\cot \\theta}\\! ", "6157a3815702692843b7598155e6d467": "\\nabla V\\left( t\\right) =\\nabla _{t}f\\left( x,t\\right) ", "6157b6cd771c809b7759bebd04f508d9": "k_{rad}", "6158027633f570993a5a78f9fb12c276": "= \\theta \\circ j^{1}_{p}\\sigma \\, ", "61583c4be3cb11bcc4e51daca77934bd": "\\widehat\\delta:Q \\times \\Sigma^{\\star} \\rightarrow Q", "61583f1f60f3a5c1cd6e8e8d1dd69ca0": " P = {1 \\over 3} \\rho\\ v_{rms}^2", "61585c6f31bd348161582332c917a4fd": " \\frac{2R(x)}{2} = R(x) ", "6158eb4a98f5d2c2186ccb803b6a8a48": "(\\log x)' = x'/x = 1/x,", "6158fb7e8516cc4c02d9055d05f26cbc": "N = [(c+d)/2]^2 - [(c-d)/2]^2.", "6159739f66bfce7999d145d388192fc7": "A^\\top K A = \\begin{bmatrix}1 & -1 & 0 \\\\ -1 & 2 & -1 \\\\ 0 & -1 & 1\\end{bmatrix}", "615a46af313786fc4e349f34118be111": "STOP", "615a58ff02ed9ce563109bacd93b3b8e": "f(n) = 2(n-2^{\\lfloor log_2(n) \\rfloor})+1", "615a905d683475b53ffd5278d0662c7a": "r(R)", "615af7f5e1b05d07b2a39965fe396184": "(a-1-q)A_1-qA_3 =0", "615b39be7d2e1945aa5e3bae99080e53": "f(h(x)) = h(x + 1)\\,\\!", "615b57e5b0732cfe205766e09d5db26f": "\\gamma_\\mathrm{air}\\simeq1.4", "615bdae634779d202bf3883848785ed5": "\\omega=i", "615becb2b5599b04270dfcef8de58a3a": " V_{\\text{out}} = V_{\\text{in}} \\!\\ ", "615bfbf205343bb7340242e21274333f": "0<\\delta_r \\le {\\min_s}{}_{+}\\|x_r-x_s\\| \\quad \\text{and} \\quad 0<\\tau_{r}\\le {\\min_{s}}{}_{+}\\|\\lambda_r-\\lambda_s\\|,\\,", "615c74e530ecffa00f54e7fa44767588": "\\mathcal{G}(3,0)", "615c76e3fd4b8e32a4234d7e4ed50790": "G = (\\{S, X, Y, Z, A\\}, \\{a\\}, S, P)", "615cbdd186a4d159766e3285060c2e8b": "\\hat{c}_V", "615cd56bf575cbc564aff33590a87a96": "A_{n,m}(u)=\\frac{1}{n-m}((2n-1)uA_{n-1,m}(u)-(n+m-1)A_{n-2,m}(u))", "615d51c9228f48948f4743e69d7362e9": "= \\frac{a}{b} \\cdot \\left(\\frac{c}{d} \\cdot \\frac{f}{f} + \\frac{e}{f} \\cdot \\frac{d}{d}\\right) ", "615d5a714b797e018238a77aa8ce3753": "p_{X} (x) = \\mathbb{E} \\mathbf{1}_{X} (x).", "615d77d116450e8a835a93533a653c8f": "N(L) = (2d)^L", "615d7929280845aa2ab4702ebd22eb0a": "\\langle \\dots \\rangle", "615db81026d4af53e2543e9b97098f36": "3(16 - 8\\sqrt{2}) r^2. \\, ", "615de5d5afd303d7c2dc726459e33004": "E(t)=[1-\\int_{t-r}^{t}E(t')dt']S(x)dt", "615e4671d3cdf7bd4987e42b61290959": "\\psi(\\Omega^{\\psi(\\psi(\\psi(0)))})", "615e6214030146cfe52332b58085142b": "\\|Ax^{(n)} - b\\|", "615e7847036f58981a51a0f9f09b3887": "[A]=[L^\\frac{2-D}{2}]", "615e88843ee9faecdb301e1b0b69f226": "\\textstyle\\{ (1,2),-(1,3),-(3,2)\\}", "615e91aa7ebe2d4ad419ceff051732c5": "\\ C_3^3 (3)=\\frac{16}{49}", "615ebd9eb1e2cf30efa4b48a72f7ea67": "\\displaystyle{(f_1,f_2)= {1\\over \\pi} \\iint_{\\Bbb C} f_1(z)\\overline{f_2(z)} e^{-|z|^2} \\, dxdy.}", "615f0472fb751d67e1259ec0ee9dfc93": "a_V", "615fb3faac54e52a1759fd190af812e6": "V_{ij} = V_{sel} - V_{on|off}", "615fba706b25f1d8da6746ebe749de08": "n(2+2\\log 2+o(1)) \\leq 3.4n + o(n)", "615fd28ef71cdf208c97bb66625094cf": "x_1^{k_1}\\cdots x_n^{k_n}", "615fd28f7eb58d91a7e262342191e19c": "\\boldsymbol{\\nabla}_\\mathbf{X} f = \\partial f/\\partial \\mathbf{X}", "616006aaa45621a1ad521ef1896e40e8": "H(s) = \\frac{ sRC - 1 }{ sRC + 1 }, \\,", "6160264952f304f534f989b3c032f554": "L^{0} = \\{0\\}", "61602834cd07cd9988a531ec5d8b7526": " {E(b,N)} \\approx {b {\\log_b (N)}} = {b {\\ln(N) \\over \\ln(b)}} ", "6160871fcf2702898c9433d767cf1e59": "T_t = \\sum T_i", "6160a685bdfd51ac8f134faed0b6421d": " TM_{10} ", "6160c4ad5dcf4acbbe9364ba7f0f733d": "{\\mathbf{w}}", "61611dcf5e86a460dc491071226f9bfe": "n_0 = \\left (\\frac{V_{\\rm g}}{V_{\\rm l}}\\right )^2 \\frac{3}{256\\pi\\ell^3}", "61611def371a143df9e6302967b1c2a8": " \\Delta VC = {w \\Delta L}", "61620618a76e6b7fe9191d43aeb4b5c7": "\\mathbf{r}_2 = (a/2)(\\hat{y} + \\hat{z})", "61623e09689a6b3fcd33a1a4b2bfed35": "\\frac{q}{A} = h_o(T_o - T_s)", "61625ffd2aacc1bd33fb1293701bb4ef": " K_\\text{g} = {4b \\sigma_\\text{l} \\sigma_\\text{f} T_m^0 \\over k \\Delta h}", "61629359603e2a8d8f47b6fdba0e3e8f": "f_{dp}=f_2-f_1", "6162c2c5370f70a73a2d873fe0b4cb25": "[B]^\\Phi=\\{K \\in A/\\Phi: K \\cap B \\neq\\emptyset\\}", "61630d2794a56415b51a0938c5d75d94": "\\begin{cases} \\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = f_1(\\mathbf{x},z_1) + g_1(\\mathbf{x},z_1) z_2\\\\\n\\dot{z}_2 = f_2(\\mathbf{x},z_1,z_2) + g_2(\\mathbf{x},z_1,z_2) z_3\\\\\n\\vdots\\\\\n\\dot{z}_i = f_i(\\mathbf{x},z_1, z_2, \\ldots, z_{i-1}, z_i) + g_i(\\mathbf{x},z_1, z_2, \\ldots, z_{i-1}, z_i) z_{i+1} \\quad \\text{ for } 1 \\leq i < k-1\\\\\n\\vdots\\\\\n\\dot{z}_{k-1} = f_{k-1}(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}) + g_{k-1}(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}) z_k\\\\\n\\dot{z}_k = f_k(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}, z_k) + g_k(\\mathbf{x},z_1, z_2, \\dots, z_{k-1}, z_k) u\\end{cases}", "61631dcde6a69416dcb70109d1e4ffb9": "a_3\\times (2\\rho^2-1)", "61636a17b310de16cff5eb6e9d9d44b4": " \\xi \\in (\\min\\{x_0,\\dots,x_n\\},\\max\\{x_0,\\dots,x_n\\}) \\,", "6163f5df31f381c2c7dd7c715a28984f": " \\sigma_\\theta = \\dfrac{Pr}{2t} \\ ", "6164814f1c11f8926ff4a1fa83f360f4": "P(R_{n-1})", "616488afa8a4efef228d7c39ad7ed080": "i=0,\\dots,n-1", "616495c58774d199325ff3265f37d14a": "\\gamma \\delta \\neq 0", "6165275c26b997ea3a37782898b24f98": "{\\zeta_X(s)}", "616528cf2817c831d40c1d881e8651a4": "g_\\gamma=g_{ij}\\dot\\gamma^i\\dot\\gamma^j", "616531c8af8f631632a5f8655a73a66b": "v_a = \\sqrt{\\frac{\\mu(1-e)}{a(1+e)}}\\,", "61657720a50630e40ccdbcf393a77721": "\\,NPV\\,", "6165ba49754d1ef1c0b4b6f6d84dbf93": "\\theta^{(1)}", "6165d3e367ccbc4cc33ff0c421ac97c1": "x_{1}(t) = x_{2}(t), \\quad \\forall \\ t \\le t_{0},", "6165e5c884f1f75c959d9a612e6f4841": "\\frac{dy}{dx}=\\frac{1}{5y^{4}-1},", "61665046df957fb69a6654a1136232fe": "\\alpha = \\arctan\\ \\frac{2\\sin a}{\\tan(\\frac{\\gamma}{2}) \\sin (b+a) + \\cot(\\frac{\\gamma}{2})\\sin(b-a)}", "616680f2782e73cc8e2714bb099823c1": "x_1,\\ldots,x_N", "6166bdcc3f958ab2959465ca984302c9": "(-1, 1)^T", "6167bbc6f04d6a83cd45c49d8566c33f": "\\phi_1=\\phi_3=\\phi_6=1", "6167fd6d35bc99c0369810ae07bad763": "F(d, k) = F(d', k)", "61686265e6061f081062ec4d083ea216": "q_K(x,y) = N_{K/\\mathbf Q}(\\omega_1 x + \\omega_2 y)", "616868c9e3ef516a9807db9f488425fc": "E = \\left\\langle \\frac{\\partial}{\\partial u}, \\frac{\\partial}{\\partial u} \\right\\rangle = 1 \\qquad F = \\left\\langle \\frac{\\partial}{\\partial u}, \\frac{\\partial}{\\partial v} \\right\\rangle = \\left\\langle \\frac{\\partial}{\\partial v}, \\frac{\\partial}{\\partial u} \\right\\rangle = 0 \\qquad G = \\left\\langle \\frac{\\partial}{\\partial v}, \\frac{\\partial}{\\partial v} \\right\\rangle = e^{u} ", "61686c111f5e3cc04dc18c82d13a15a2": "{a\\pi\\over 4}\\ {b\\pi\\over 3}\\ {c\\pi\\over 2}", "6168e389b97125ceac804d2233cd3cb7": "U_{ix}\\subset\\mathbb{R}^{n-p}", "6168f7877ab6ea222bfdc7ca973e2a45": "Dx - xD = 1.\\,", "6169331388a7b837d96d27f67b307686": "\\bot \\to \\phi ", "61693fc571c2cd2a0f49efee884b361c": "\\scriptstyle \\eta \\in \\{0,1\\}^{Z^d} ", "6169448e85e6cc09773971899abdd179": "\\sigma_m", "6169855db61c6f0b7ef4b675291d7d6f": "n > n_0 ", "6169a1e5998355348e675333791be450": "\\, O((N / B)^{1-1/d} + T / B) I/Os", "616a0357eb39a236815788795483184f": " \\zeta(n) = \\frac{\\left(-1\\right)^{\\frac{n}{2}-1}B_n\\left(2\\pi\\right)^n}{2(n!)}", "616a54d677e034592fad4437c0a324e7": "q \\approx 2^{256}", "616a61d175736a156261d0d149f6433f": "\\mathit{USp}(4) \\supset O(4)", "616ae49a7a86dda422bfa90b01987f1a": "O(n^{\\frac{11}{2}+\\varepsilon})", "616b232df54d0566c4bf62c815dced44": "2 ^{\\mathfrak c}", "616b5690c381606ba89ed3659e79a8f3": "N/V", "616b5a5a8f95c141c28298134f9eff0b": " \\bold{x} \\equiv x \\equiv x(t), \\quad r \\equiv r(t), \\quad s \\equiv s(t) \\cdots \\,\\!", "616b712576ba0073c91131216be49b6c": "\n f\\;a\\;:\\;B\\times (C\\times 1) \\to D\n", "616b75eac2fd92660a02698720bdaccf": "S_{r_1, r_2}'(r_2)", "616b82d5bf764a6234cc40454e639898": "1/2 = 0.1_! = 0.0\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10..._!", "616ba0099f0d9d5695f65a2992d5a2d8": " d\\tau = \\sqrt{ \n \\left( 1 - \\frac{2m}{r} \\right) dt^2\n - \\frac{1}{c^2} \\left( 1 - \\frac{2m}{r} \\right)^{-1} dr^2 \n - \\frac{r^2}{c^2} d\\phi^2\n - \\frac{r^2}{c^2} \\sin^2(\\phi ) \\, d\\theta^2\n }, ", "616bcd42a0612d2d3a90cb2986a36975": "\\text{Sing}(X)", "616bf54a8a459fd7cfddb4cfd901de64": "\\begin{align}\n\\nabla^2 B_x &= {\\partial^2 B_x \\over \\partial x^2} + {\\partial^2 B_x \\over \\partial y^2} + {\\partial^2 B_x \\over \\partial z^2}\\\\ \n&= {\\partial \\over \\partial x} {\\partial B_x \\over \\partial x} + {\\partial \\over \\partial y} {\\partial B_x \\over \\partial y} + {\\partial \\over \\partial z} {\\partial B_x \\over \\partial z}\n\\end{align}", "616c2434a7180e0fb181a72f60072025": "P_{j}=\\frac{\\exp\\left(-\\frac{E_{j}}{k_{\\mathrm B} T}\\right)}{Z}", "616c5742d2d5eed9cfe4340564b04b09": " I_{q(\\theta)}^{-1} = \\dot{q}(\\theta)I^{-1}(\\theta)\\dot{q}(\\theta)'", "616c73c39659ea9ad3afc71829022144": "\nf_{3dB} = \\frac {1}{2 \\pi \\tau}. ", "616cc1c6c2a76c666162655d13de6636": "\\int_0^1 x J_\\alpha(x u_{\\alpha,m}) J_\\alpha(x u_{\\alpha,n}) \\,dx = \\frac{\\delta_{m,n}}{2} [J_{\\alpha+1}(u_{\\alpha,m})]^2 = \\frac{\\delta_{m,n}}{2} [J_{\\alpha}'(u_{\\alpha,m})]^2,\\!", "616cec89c88b61ca911510d74489818b": "\\mathbf{u}\\oplus\\mathbf{v}", "616e167c1b5c75697b5607b7b38c208f": "\\operatorname{sqsum}", "616e17599523231de861240470fed358": "n_e << m_e\\omega^2\\,/\\,4\\pi e^2", "616e39b481526b05138e20c3e11e1e16": " G \\approx 4.302 \\times 10^{-3} {\\rm \\ pc}\\, M_\\odot^{-1} \\, {\\rm (km/s)}^2. \\, ", "616e5c048b46f3977bf8947a19822db5": "\n\\int_0^{\\infty} e^{- \\omega x} \\; x^{- \\alpha} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\eta x \\right) dx =\n\\omega^{\\alpha - 1} \\; G_{p + 1,\\,q}^{\\,m,\\,n+1} \\!\\left( \\left. \\begin{matrix} \\alpha, \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\frac{\\eta}{\\omega} \\right) ,\n", "616e5da17ec242df541d1aa447043b21": "(1,0,\\ldots, 0)^T", "616e8c2a1598d07bf60db91b48fceb3c": "0.6 \\le \\mathrm{Pr} \\le 160", "616f13c6b020183b545b4bdd3e9bd343": "p^{4}", "616f18fd7ed694a73af54a6b05bb6892": "x = 3/2", "616f32628a13c33d64b3ce8c8375bea6": "g'=p_s", "616f7eba475d37204ba06d94f389ece5": "\\Delta^{n-1} \\twoheadrightarrow P.", "616f86990f60e1c5ac78a07cfbbe7998": "(P_{\\mu \\mu}=P_{\\tau \\tau} \\simeq 7/18)", "616f8d204de80223511ecf6b67b205e8": "\n F_{ij}=\\frac{\\partial u_i}{\\partial X_j} + \\delta_{ij},\n ", "616fb717ed0ab1dbf5ded834a72ae83e": "\\ f", "616fba2f664b9d05905308d0d3750776": "\\mathcal{X} \\subseteq \\mathcal{P}(\\mathbb{N})", "616fc61ddfc41e7407fa87ee8608bfc5": "w(r)=-C/r^6", "61701196c2d7311ef58e3a5cb98bd0d2": "(Kf)(x) = \\int_{X} k(x,y)f(y)\\,dy", "6170581266439e97809c15d611db94bc": "\\frac{(s_{n+1})(s_{n-1})}{(s_n)^2}=\\left(\\frac{n+1}{n}\\right)^n, ~ n\\geq 1.", "6170813e96d857b6890b141fffccdf90": "y = \\sqrt x", "617088f9a9d0c522eebcf07844eb1263": " B_n < \\left( \\frac{0.792 n}{\\ln( n+1)} \\right)^n ~;", "6170a2abe9e9b72ae1f3826f2eec7ced": "1/\\rho^2", "617154513a86a05abed0691106fc324e": " b_{k+1} = \\frac{(A - \\mu I)^{-1}b_k}{C_k}, ", "61717d2aeb4d80840d6f893dfd52c583": " \\delta / 2 ", "6171d6dc3ff326fb26dcc933e104ee15": "\\, x^2", "6171db7e4b765b9a1505937438823387": "\\begin{align}\n \\mu_{X \\cup Y} &= \\frac{1}{N_{X \\cup Y}}\\left(N_X\\mu_X + N_Y\\mu_Y - N_{X \\cap Y}\\mu_{X \\cap Y}\\right)\\\\\n \\sigma_{X \\cup Y} &= \\sqrt{\\frac{1}{N_{X \\cup Y}}\\left(N_X[\\sigma_X^2 + \\mu _X^2] + N_Y[\\sigma_Y^2 + \\mu _Y^2] - N_{X \\cap Y}[\\sigma_{X \\cap Y}^2 + \\mu _{X \\cap Y}^2]\\right) - \\mu_{X\\cup Y}^2}\n\\end{align}", "6171ea472d34874118e288be08cf6028": "\\frac{20! \\times 3^{19} \\times 30! \\times 2^{15}}{4} \\approx 6.14 \\times 10^{63}", "61727f4d8bddc91fd53fbccbe42639dd": "\\lambda\\neq\\lambda_j", "617286a1aa6828142e32d55417a38849": "\n\\overline{E} = \\frac{1}{2} \\overline{ u_i u_i }\n", "6172a687ea44c244cf54a731583cfa71": "n_\\text{upper}", "6172aa5482884f05aa5052a795825076": "N=N_\\mathbf{P}", "6172db56e844c4b80b89b03e764beda4": "\\mathbb{Z}/(2) \\times S_3", "6172e6f06686f597d4aacf4f888599dc": "=\\underset{q\\in R}{\\mbox{arg min}} \\left[ (\\tau-1)\\sum_{y_{i}} J_{ij} \\sigma_i \\sigma_j -\\mu \\sum_{j} h_j\\sigma_j", "6174dfa6b8a8f3d244843cb932247481": "\\sigma = \\sum_j J_i\\frac{\\partial F_j}{\\partial x_j} ", "617515f80c9f1835b3761c44439b24de": "\\textstyle (\\sigma_0, \\sigma_1, \\ldots)", "617532d0a398246d7c51beff2feafd5e": "{\\mathbf W}.\\,", "617575c390961deca568e2912b964c09": "\\rho = \\sum_i p_i \\rho_i^A \\otimes \\rho_i^B, ", "6175d2cb059132861821a8dc82fbdda3": "\\Pi (t,f) = g(t)\\, W_h(t,f) ", "6175ec96d862072b16f2cbd29f4b2a26": "C_k^n", "6176118b10b048155f380d0236969ef4": "x = t-\\frac{b}{3a}", "61765e96edbe240f84b337423216497d": "\n\\lim_{t\\rightarrow\\infty} \\frac{E[Q_i(t)]}{t} = 0 \n", "6176780a36c8e5fca06fd6fff88cdf39": "F = \\rho g V ", "6176cd1d435722b6d21b816cd1f0c3f6": "t_1, t_2, \\ldots", "6176f0877e5a549a56701c4b82984d71": "\\Lambda (\\omega_1, \\omega_2)=\\{ m\\omega_1 +n\\omega_2 : m,n\\in \\mathbf{Z} \\}", "61772e85f5409d9dab60df96e5427b3a": "\\Delta I^2 \\ \\stackrel{\\mathrm{def}}{=}\\ \\langle\\left(I-\\langle I\\rangle\n\\right)^2\\rangle \\propto I. ", "617746a779e9cb29eab60884a824d977": "\\int \\left| (ax + b)^n \\right|\\,dx = \\sgn(ax + b) {(ax + b)^{n+1} \\over a(n+1)} + C \\quad [\\,n\\text{ is odd, and } n \\neq -1\\,] \\,.", "6177498345ed4005f6255a57ec9ea41e": "p_1(x) \\equiv p_2(x)", "6177a77ee8de8761111f7b1e0732e976": "x-h", "6177ddaa4cfaaf7e1a97e2aa8af2c82c": "d \\in N ", "6177ef4e7f735f126045a414010b6fc2": " D^j_{m'm}(\\alpha,\\beta,\\gamma) \\equiv\n\\langle jm' | \\mathcal{R}(\\alpha,\\beta,\\gamma)| jm \\rangle =\n e^{-im'\\alpha } d^j_{m'm}(\\beta)e^{-i m\\gamma}.\n", "617887f54856ae9a8ae286d75bd6226a": "(2v/m + s)n", "61788b8047992ea9d615d4658a0e3c5e": "x \\sqrt{N}- 1", "617bb76e62b0ce7c0ecd625959326dc4": "Q = - \\hbar^2 \\,\\partial^2_x {\\sqrt{\\rho}} / (2m {\\sqrt{\\rho}})", "617bc3ddb9e84930d3b503b6da03c413": "C_*(M, (f_1, g_1))", "617c08c6eab7c14df972b8666d67c2e6": "B_z =\\frac{\\mu_0 I}{4\\pi} \\frac{1}{L} \\frac{1}{ \\sqrt{a \\rho}} \\left[ \\zeta k \\left(K(k^2) + \\frac{a-\\rho}{a+\\rho} \\Pi(h^2,k^2)\\right)\\right]_{\\zeta_-}^{\\zeta_+}.", "617c2b4949cf05928065b3ffb7bb1995": "p:X\\to Y", "617c39750a2a08cf6c808401c3346f6c": "E[\\widehat{\\boldsymbol\\Sigma}] = \\frac{n-1}{n} \\boldsymbol\\Sigma.", "617cfb04d73cb39907316216954829b1": "(r, \\mu, \\nu)", "617d3fce54684e88b5734c3fda915fce": "H = G \\oplus G^{\\perp},", "617d7c7426411e9e06ae443938d8f415": "J,t\\,", "617e0b5f69e0b549df8b3bea2196d00e": "t_i=s_j", "617e190bd035ce2a53a2115b00c1d814": "\\operatorname{dist}(T(\\mathcal{M}), \\mathcal{S}) = \\sum_{m \\in T(\\mathcal{M})} \\sum_{s \\in \\mathcal{S}} (m - s)^2", "617e6617f534e39d042b400a7706dbdb": "f(f(xy)z)=f(xf(yz))\\; ,\\; f(f(xy)z)=f(xf(yz))=f(xyz) ", "617e6cca778588e76f8ebf66a0cfed6e": "\\hat e_3", "617e8c75814aabdb3ad872496e091e34": "\\sum\\limits_{i=1}^{N_{\\lambda}}{n_i} =N.", "617ea30aac687a51a7d5328c78bee723": "t_a=t_0 k ) \\le \\max\\left( \\left[ \\frac{ r }{( r + 1 ) k } \\right]^r E| X^r |, \\frac{ s }{( s - 1 ) k^r } E| X^r | - \\frac{ 1 }{ s - 1 } \\right) ", "617f415a930907e8153d5f741314f45a": " j^{1}_{p}\\sigma = \\{ \\psi : \\psi \\in \\Gamma_{p}(\\pi); \\bar{\\psi}(p) = \\bar{\\sigma}(p); d\\bar{\\psi}_{p} = d\\bar{\\sigma}_{p} \\}. \\,", "617f4427c78f5f0e0ec9007657fb40a9": " \\ \\ \\ \\ \\ \\ \\ \\overleftarrow{Y} \\ \\ \\ and \\ \\ \\ \\overrightarrow{Y} \\ \\ ", "617f4f912ae7b12bd22b1f6a27bd7c7c": "E\\mathbf{x} = 0", "617f985ec699fb48a5487c66ab180c2a": "\\boldsymbol \\mu", "617fa4d51397b168b3f3f2b61f379d52": " \\beta = VBd \\!", "617fb3091d4504d500ae1f2d830102b3": " \\gamma = \\cosh\\phi = { e^{\\phi} + e^{-\\phi} \\over 2 }, ", "617fb7adebd83ac29f151d3eed3729d7": " \\vec S_i, \\vec S_j ", "617fc62123afce0b83c3b88b6b62cee1": "{p}={p_d}+{e} \\, .", "617fcc9d56693fa0103bacc1470a97ec": "q^i = \\mathbf{v}\\cdot \\mathbf{e}^i = (q_j\\mathbf{e}^j)\\cdot \\mathbf{e}^i = q_je^{ji}. \\, ", "617fdfd9705de48499add574c7cf012c": "\\Gamma(X, TX)", "618003a7e749e8b060ab1c0c0062ec75": "\\eta \\rightarrow e^{i s \\psi}\\eta", "6180420df254f1060cb6eac4b97356d1": "\\mathrm{arcsinh} (x) = \\sum^{\\infty}_{n=0} \\frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\\quad\\text{ for }|x| \\le 1\\!", "6180f8fc214f7824943782adbb472232": "\\delta W = \\sum_{j=1}^m \\sum_{i=1}^n ( \\mathbf {F}_{i} - m_i \\mathbf{a}_i )\\cdot \\frac {\\partial \\mathbf {r}_i} {\\partial q_j} \\delta q_j= 0.", "61810e7ae7e660b24303b7a9f8970a8a": "F(a, c, x) = \\frac{\\Gamma(c)}{\\Gamma(a)} \\sum_{n = 0}^\\infty \\frac{\\Gamma(a + n) x^n}{\\Gamma(c + n) n!}", "6181387d9882d3b9c67e6f6750fc97a0": "z_2(x,y)=G(y)e^{-x}", "61814ce8672b0ba6ae60001af18fea17": "4e^2/h", "6182299993d0aae0c1cf7a1bec8526b9": "w(-\\infty)=0", "6182439bc4a162217bcd21d9cacab690": "c_{k'}^{(1)}", "618250452d9e552bc6c0a5329013cbb1": "p=0.5", "61826af78cdac7eabe5621684a829847": "\\frac{\\ddot a}{a} = - \\frac{4\\pi G}{3}\\left(\\rho + \\frac{3p}{c^{2}}\\right) + \\frac{\\Lambda c^{2}}{3}", "6182855ea33468d63285a155a4df4bec": " a < b ", "6182b8e7f74dbd4a3f19a7e191538f42": "F^{-1}_{\\theta | \\sigma_k} = \\left[\\begin{array}{cc}\\sigma_k^2 C_k&0\\\\ 0&2 C_k\\otimes C_k\\end{array}\\right]", "6182e0661bc2aa04f945337c9a12c551": "\\alpha, \\beta, \\delta >0", "6183399aa39fc92f4acecd764018a338": "x \\not\\in A \\cap B", "6183404b8333d2b268f4a72a017fc3e2": "I_\\mathrm{e}", "61835756a9bbea914aa81076e52f2406": "i \\in [t] ", "6183c5dfda33c379f4d36fac2b8e7e60": "L= L_{1} [\\dot{+}] L_{2}", "6183fa89dd86c2a39614e520460430c7": "\\mu=\\cos\\theta", "618405a62965153cf05bf298e89ab95d": "r_p^{(7/3)} = r_{d_1}^{(7/3)} + r_{d_2}^{(7/3)} + r_{d_3}^{(7/3)} + ... + r_{d_n}^{(7/3)} ", "61844ddbd26e27998ddc28000f276941": " M_1,\\dots,M_m ", "618473d36513b8d78fba846a7372341e": "\\scriptstyle AD-BC=1", "618474991c0bcb69a5eff6e5c4b2d633": "\\sum_{k=n}^{n+13} F_k = 29 F_{n+8}", "61847a7cfa2fddbeea36ff57f4221db0": "0^{(1)}", "6184c7b1951c8c611e2cad266c387b56": "f_t\\colon A \\rightarrow Y", "6184d52cd75e0bc77998e870760466f4": "\\mathrm{Res}_{z=5} \\frac{f(z)}{5-z} = - 5^{\\frac{3}{4}} \\exp \\left (\\tfrac{\\log(-2)}{4} \\right).", "6184f12d18b81312ad3f4956c4d422e6": "\\frac{1}{15} + \\frac{1}{30} = \\frac{1}{10}", "6184f4454dd343b95dab9b19eee3c795": "\\mathbf{a}_2 = \\mathbf{a} - \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {\\mathbf{b} \\cdot \\mathbf{b}}{\\mathbf{b}}.", "618541af8d5f7e824de12bb8d03da86f": "f(\\alpha)=f(\\beta)=\\gamma\\,", "61856511e214379f0a44d72b7a21ade4": " f(\\chi) - (\\chi(1)-\\chi(G_0)), ", "61856a8bb093e7e53289c4efd2ebdc62": "V = \\left(1 - \\frac{r}{2GM}\\right)^{1/2}e^{r/4GM}\\cosh\\left(\\frac{t}{4GM}\\right)", "61859abb6fea11675d6f951d10f67a2f": " Q_n(x)=(-1)^n2^nn!\\binom{2n}{n}^{-1}F_n(2x+1)", "6185e7e76f5f49cc561f32f3437c3da4": " C = \\{\\mathbf{c}_1,\\ldots,\\mathbf{c}_n\\} ", "618601a7f27829db43d1b37f38f48b7a": "\\mu : M\\to \\mathfrak{g}^*", "61860fcb9fd0353a983ec8e59fea94ed": "z^{2} + c", "618634eae20f619226ef0c47115adcd3": "y = \\frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_1) + f(x_1)", "618696ebc92ed71b983105484562da2c": "y = \\frac{1}{3}e^{t} + \\frac{2}{3}e^{-5t}. \\,\\!", "6186a23946988e1f9b0c64091d25391a": "\\gamma^n \\subset \\Gamma^n", "6186da229a0198d20c081afb789e1053": " h^*", "6187409509ac80801cb8a2b62c611366": "\n[C4-5] \\quad \\frac{[sc] \\vdash C_1 \\quad [sc] \\vdash C_2}{[sc] \\vdash C_1\\; ;\\; C_2} \\qquad \\frac{\\vdash exp \\;:\\; sc \\quad [sc] \\vdash C}{[sc] \\vdash \\textbf{while}\\ exp\\ \\textbf{do}\\ C}\n", "6187646bc03b8ef89acf77b2afc6ce6f": " u = \\mathrm{exp}(x) ", "6187de6e06d02239845951f290639300": "G(z,x)=0", "6187def73554a108555ddb898c9833f5": "G_\\epsilon(V,E)", "618839d87dcd089d1c1b80c634b953ca": "jmd", "618895ca555494b94f6c71acc7b28e86": "\\beta = \\frac{\\mu_1 - \\mu_2}{\\sqrt{\\sigma_1^2 + \\sigma_2^2 - 2\\sigma_{12} }}.", "6188ccf13c3ad8a467eae75a87094b25": " L(1, \\chi) =\n\\begin{cases}\n-\\dfrac{\\pi}{q^{3/2}}\\sum_{m=1}^{q-1} m \\left( \\dfrac{m}{q} \\right), & q \\equiv 3 \\mod 4; \\\\\n-\\dfrac{1}{q^{1/2}}\\sum_{m=1}^{q-1} \\left( \\dfrac{m}{q} \\right) \\ln 2\\sin \\dfrac{m\\pi}{q} , & q \\equiv 1 \\mod 4.\n\\end{cases}", "6188e1acc52edea4dd2f88d528b7fc94": "(N + 1)(p - 1)/p = \\mathit{(p - 1)}\\mathit{p}^{k-1}", "6188f42ac8ef04ad0a00bef522a02fc0": "P_m", "6189e23a97ffce534aa58ea8c71a59c2": "{A^k}_i = {R^k}_i + {S^k}_i", "618a8286e1d0df17bf1b1cbe0784cb0f": "M_{\\mathrm{left}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\left \\{ - \\frac{q dx \\, x (L-x)^2 }{L^2} \\right \\}= - \\frac{q L^2}{12} ", "618addf8513933aff5e0a3ffcfb12879": " \\langle\\alpha|\\widehat{q}|\\alpha\\rangle = 2^{-1/2}(\\alpha^{*} + \\alpha) = q_{\\alpha}", "618b868797be7dda8490c859563b8519": "C_{16} = G_{12} + P_{12} \\cdot C_{12}", "618bbcc669b8d23e4712b9c858b9ea21": "\n\\exists i \\in \\mathrm{N} \\; s.t. \\; \\nu\\ _i (a) < \\nu\\ _i (b) \n", "618bdbc7cacbfa5c464a32f014555078": "h_{\\mathbf{a},b} (\\boldsymbol{\\upsilon}) : \n\\mathcal{R}^d\n\\to \\mathcal{N} ", "618c1d34b37eec0aa44e76d32ef913c0": "S\\subseteq\\kappa", "618c33d6bcfed0dd9e358f24b253d6fa": "D(E(m_1, r_1)\\cdot g^{m_2} \\mod n^2) = m_1 + m_2 \\mod n. \\, ", "618c361ddf05cfb6a88befee58f4e50f": "t_{\\mathrm{min}}", "618c4a98083ba6d4f9966aec29557cae": "\\vec{v}_{i}", "618c7ab1829406bbf1b272e02ebed86a": "S_N(f)=f * D_N\\,", "618c834a8f1e95c97dddf6144ddc9403": " B_k + \\frac {y_k y_k^T}{y_k^{T} \\Delta x_k} - \\frac {B_k \\Delta x_k (B_k \\Delta x_k)^T} {\\Delta x_k^{T} B_k \\, \\Delta x_k}", "618cb53604a67151088cab8362403588": "\\scriptstyle bc + d = a", "618cb8e93c7c8db1af96a542fc676758": "\\psi_{2m}\\in 2y\\mathbb{Z}[x,A,B]", "618d65257a909c9bc855d7dc1d2cd626": "\\frac{d}{dx}\\left(u(x)v(x)\\right) = v(x) \\frac{d}{dx}\\left(u(x)\\right) + u(x) \\frac{d}{dx}\\left(v(x)\\right).\\!", "618d78914fd55ee5cfd27785bcbbfc5e": " \\Gamma^k_j \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathcal{M}^k_i {d \\bar{\\mathcal{M}}^i_j \\over ds} ", "618dc97b6924bf4d3285df88a85b89a6": "2 = {1 + 3 \\over 2}.", "618df4fc9a40a981f2eeea243e5ce48f": " \\lambda^g ", "618e97faa888f50b8eef08e058a23b65": "\\Delta W = \\frac{\\mu_a}{\\mu_t} W", "618f04179845c7092e7415e1df6723a5": "^*\\mathbb{R}", "618f215ee5c52227d0aae2ea15e860e7": "\\Pr(C=c)>0", "618f265519dc306841d599be9c90c493": "\n\\frac{d\\eta}{d\\tau} = \\frac{d}{dt} \\left( \\frac{1}{y} \\right) \\frac{dt}{d\\tau} = - \\frac{\\dot{y}}{y^{2}} y^{2} = -\\dot{y}\n", "618f2893614fc767d57024e804033344": "- \\sqrt {2} /2", "618f4c7b176d64ef0212e65e368f08b2": "ax^2 + bx + c = 0", "618ffd7adcf41ebb71548f4ebf07ad00": "X=\\left \\{X_1=5, X_2=1, X_3=5\\right \\}", "6190016c5a62392c07deebb445e4da83": "f^\\dagger\\colon B\\to A", "61904baabd131f4d689ea66301a33c06": "K_X", "61905b133aac9bdd43dd1005303e18f0": "z=e^{i(\\theta-\\mu)}", "619066113a9ff31ee03e4d646fee3430": " \n I[f_1,f_2,\\dots,f_m] = \\int_{\\Omega} \\mathcal{L}(x_1, \\dots , x_n, f_1, \\dots, f_m, f_{1,1}, \\dots , f_{1,n}, \\dots, f_{m,1}, \\dots, f_{m,n}) \\, \\mathrm{d}\\mathbf{x}\\,\\! ~;~~\n f_{j,i} := \\cfrac{\\partial f_j}{\\partial x_i}\n", "6190e3c8c913a81db18bb31b35de3f81": "LU = I - \\operatorname{diag}(1,0,\\ldots, 0).", "619100895695114a2083e0207033ab9a": "A \\to B \\iff \\neg A \\or B", "619104a8cdb03c9dae1b83586cf4f000": "\\mathbb{N}_{0}", "61911c2a720207e78f588213c56f7cd4": "P^{\\prime }=(w,\\vec{0}),", "61914f3cf9331a3a1935800ece4a5b87": "m \\mapsto T_m", "6191a9719f6d2385e80f03be6b8677eb": " 1 - F_{IT} = (1 - F_{IS})\\,(1 - F_{ST}). \\!", "619208e50451c6fbb059f98d5a262c9a": "\\sqrt{ax^2\\!+\\!bx\\!+\\!c}", "6192423deeded63e1c5c96a5f014cbba": "\\{1,\\ldots,N\\}", "619245af71c560211c9a69e9b83a3cde": " A_{\\{1} A_2 A_{3\\}} + B_{\\{1} B_2 B_{3\\}} + C_{\\{1} C_2 C_{3\\}} + A_\\mu \\Gamma^{\\alpha \\alpha'}_\\mu \\psi\\psi \n + B_\\mu \\Gamma^{\\beta \\beta'}_\\mu \\psi\\psi + C_\\mu \\Gamma^{\\gamma \\gamma'}_\\mu \\psi\\psi", "6192a30c4c8b695319998afef6339ba6": "(1,\\varepsilon,A,1,\\alpha)", "6192a3c328d9430366e3aa460f0c60c2": "C_1C_2R_1R_2=1\\,", "6192b3deddce24bb785663f2acd5a4fb": " \\lim_{x \\to c}f(x) = L \\, ", "6192dd0af591eebd788d18c62b632717": "Slip \\ Ratio \\ % = \\frac {Vehicle \\ Speed - Wheel \\ Speed}{Vehicle \\ Speed} \\times 100", "61936ccc7e8cec37c632014e460ac00f": "\n\\Pi(n,k) = \\int_0^{\\pi/2} \\frac{\\mathrm{d} \\theta} {(1 - n \\sin^2 \\theta) \n\\sqrt{1 - k^2 \\sin^2 \\theta}} = \\frac {\\pi} {2} \\,F_1(\\tfrac 1 2, 1, \\tfrac 1 2, 1; \nn,k^2) ~.\n", "61937f9a87292ed0fd4a6fb872511a4f": "\\int\\frac{x^2\\;dx}{s^7}\n= \\frac{1}{a^4}\\left[\\frac{1}{3}\\frac{x^3}{s^3}-\\frac{1}{5}\\frac{x^5}{s^5}\\right]", "6193f48db4599d0e048374ed326608a5": "(R+h)^2 = R^2 + d^2 \\,\\!", "619420954a75a7debc1f09314c6e0e99": "K^{ab}_{mn}=c_1g^{ab}g_{mn}+\nc_2\\delta^a_m\\delta^b_n\n+c_3\\delta^a_n\\delta^b_m+c_4u^au^bg_{mn}.\n", "6195269473d6a885bfdeb7ad3c51d525": "\\varepsilon_{\\color{Orange}{2}\\color{Violet}{3}\\color{BrickRed}{1}} = -\\varepsilon_{\\color{BrickRed}{1}\\color{Violet}{3}\\color{Orange}{2}} = -(-\\varepsilon_{\\color{BrickRed}{1}\\color{Orange}{2}\\color{Violet}{3}}) = 1", "61954599c987faf35967c6d14fcbf423": "f^{1,2}(\\theta)", "6195f12817920c48dcb2b486bc1d243b": "p \\to [a]q\\,\\!", "61967551dac80f3fbdefb26b7bc792c1": "B = \\int_0^1 L(X) dX. ", "619697f065488cb3ecd343e049a1fadb": "h'_k = 2.25 h'_{k-1} + 1", "6196ebb9ef3d825a877e2a247daa5aaf": "n = qm", "6196f492ca03b977a408351328e03696": "x_{0,k} + h", "619734c6a354bd510f84f7f2efb2df72": "F = 93", "6197b3fc31983897fa1fda1e689e5feb": "P(x,y) = P_n(x,y) + P_{n-1}(x,y) + \\cdots + P_1(x,y) + P_0", "6197b4d5e08225ca83d871b83950531b": " \\bar{\\phi} (-p) ", "6197b9738fca4218dedd4213c9588289": " \\mathbf{f_B} = - \\nabla p. \\, ", "6197c2080fd2edfdf14e621e48826a25": " \\vec{b} ", "6197ca5e43517dc60742ee80fc523a20": "\\displaystyle m_3", "61981304a0e2a2916aa91fdb4129a2d5": "(T_1,\\dots,T_p),", "6198260ebd10a855e587e86981028a5e": "\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}", "619849989338410bf4d5d9774d845e5f": "\\xi_{\\alpha}", "6198687be715b0a20202b94bfdd8b2f6": "0 \\le x \\,\\bmod\\, y aF(K,L), ", "61b6d7ad70bb63d70dea4ddbd31fc39f": "p_{K, m}=\\sup_{x\\in K}\\left|f^{(m)}(x)\\right|", "61b6f72077e1a010ae65ca2c55a7e034": "[1,0,0,10000r^2], [0,1,0,10000r],", "61b6f8adb694288539583a3c143682b2": "\n\\begin{bmatrix}\n10 & 17 & 13 & 28 & 23 \\\\\n17 & 22 & 16 & 29 & 23 \\\\\n24 & 28 & 22 & 34 & 24 \\\\\n11 & 13 & 6 & 17 & 7 \\\\\n45 & 44 & 32 & 37 & 23 \\\\\n36 & 33 & 19 & 21 & 6 \\\\\n75 & 66 & 51 & 53 & 34 \\end{bmatrix}", "61b70e911fcd3c32a87d997b4d286ced": "Em", "61b7135def83f08a5387e13f85237808": "3^9", "61b8110a0558e183a8dfae360dcf861f": "e^{i\\theta} {(a\\oplus_M {z})} ", "61b839daa6ff5656834a11f74e2204f8": " p\\ge-|m| ", "61b84c071f14e74a31dabfb74d27caff": "\n(\\ddot{q}_d-\\ddot{q}+\\alpha \\dot{e}) = -\\frac{\\kappa}{2}(\\dot{e}+\\alpha e)\n", "61b8683ec06b16c921a4dae4f961b1bb": "\\Delta^\\text{w}_\\text{o}\\phi = \\frac{\\Delta^\\text{w}_\\text{o}\\phi^\\ominus_\\text{C+}+\\Delta^\\text{w}_\\text{o}\\phi^\\ominus_\\text{A-}}{2} + \\frac{RT}{2F}\\ln{\\left(\\frac{\\gamma^\\text{o}_\\text{C+}\\gamma^\\text{w}_\\text{A-}}{\\gamma^\\text{w}_\\text{C+}\\gamma^\\text{o}_\\text{A-}}\\right)}", "61b88eddde54a2cd767e3e14c7de1b26": "\\dot v=F(x),\\;\\dot x \\equiv v", "61b914e594ffd52644b589155436b0f9": "M=(Q, \\Sigma, \\iota, \\sqcup, A, \\delta)", "61b92f7c60a48609e9cd9a33f4dd3b6e": "\n\\begin{align}\n \\min&f_l(\\mathbf{x})\\\\\n \\text{s.t. }&f_j(\\mathbf{x})\\leq\\mathbf{y}^*_j,\\;j=1,\\dotsc,l-1,\\\\\n &\\mathbf{x}\\in X,\n\\end{align}\n", "61b945ae308bd67293489ae77099cce5": "K^{m \\times n}.", "61b9562a976b91c9279c846f46713915": "\\mathrm{R{-}O{-}O{-}R\\longrightarrow\\ 2\\ R{-}O\\cdot}", "61b9820b75ed0c5637e5f0599b161ce2": "2\\le \\lim_{n\\to\\infty} D(n) \\Big/ \\binom{n}{\\lfloor n/2\\rfloor}\\le 2\\frac{3}{11}", "61b98671b75b8738fa7634384a3160ec": "N=\\sum_{i=1}^{N_G}n_i", "61b98bb724d72e0b009c3f523c50cd82": "1/x", "61b99ce7aa66c5fb0c5ec0f15f6343d4": "\\gamma_i = \\beta_i / \\alpha_i", "61b9cfe9a5a062e57da9c72e8fd4edb4": "(19)\\quad\\quad e_1 + \\frac{1}{2} u_1^2 + p_1/\\rho_1 = e_2 + \\frac{1}{2} u_2^2 + p_2/\\rho_2,", "61ba13f07ba13a358fe38ba2df9ef25c": " h\\otimes x\\mapsto \\pm x", "61ba20ec5e1117a8a1f34a9b6ca5324f": "A = LU", "61ba536257afe894cd49ffe767e64ecc": "m_i(x,y)", "61baecd4b7a9ef93b306b0eca3f0c76d": "r_0(\\theta) = x+\\left(\\frac{1}{2}-x\\right)\\cos2\\theta", "61bb153af86b1a4b6d6d6a4a096f9e5e": "\\mathcal{L}_X [Y,Z] = [\\mathcal{L}_X Y,Z] + [Y,\\mathcal{L}_X Z]", "61bb3423bd8ab9c7e9b2ed1a90fb838a": "\nD = \\begin{vmatrix} A_{xx} & A_{xy}\\\\A_{xy} & A_{yy} \\end{vmatrix} < 0\\,\n", "61bb9bfa764bb65075eb46f87e5ed029": "\\mathcal K(\\mathbb{P}^n_k),\\,", "61bbc88211b50ad7b4fd586654e275d3": " V = x, E = x\\ q = f\\ (q\\ q), L = f\\ (x\\ x) ", "61bbf099a6408df2e7bf6de4bcc02aaa": "\n\\begin{align}\nP(x) &= 2 - 8(x+1) + 28(x+1) ^2 - 21 (x+1)^3 + 15x(x+1)^3 - 10x^2(x+1)^3 \\\\\n&\\quad{} + 4x^3(x+1)^3 -1x^3(x+1)^3(x-1)+x^3(x+1)^3(x-1)^2 \\\\\n&=2 - 8 + 28 - 21 - 8x + 56x - 63x + 15x + 28x^2 - 63x^2 + 45x^2 - 10x^2 - 21x^3 \\\\\n&\\quad {}+ 45x^3 - 30x^3 + 4x^3 + x^3 + x^3 + 15x^4 - 30x^4 + 12x^4 + 2x^4 + x^4 \\\\\n&\\quad {}- 10x^5 + 12x^5 - 2x^5 + 4x^5 - 2x^5 - 2x^5 - x^6 + x^6 - x^7 + x^7 + x^8 \\\\\n&= x^8 + 1.\n\\end{align}\n", "61bbf72c2e36b1b877cf9a7ea8de89eb": "f_n(x)\\to f(x)", "61bbf8add0b80b1227db1ac90ef10948": "f^{(2)}>0", "61bc4933bcb6c83725a2f5e9f15bdf63": "h_t^j", "61bc4b3184327571dba3dd2f1474c551": "p=2^{32}+15", "61bc5857c4a6c17ad759689753baf8c3": "\\eta:I\\rightarrow GF", "61bcd82c408964f422bfa11415675d38": "\\rho\\propto a^{-3}=V^{-1}", "61bd06491a5cafd1c5c305454221c97f": " \\theta = \\sqrt{2} \\omega \\, \\frac{C^\\prime \\left( \\frac{q^2}{\\omega^2},\\frac{q^2}{2 \\omega^2},\\omega u \\right)}{C \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u \\right)}", "61bd35d7fdd3eb42a6a41eedb5b7a48d": "\\langle x, y\\rangle = \\sum_{i\\in I} \\langle x_i, y_i\\rangle_{H_i}.", "61bd6ce87cfb58e409c4c38a4955abf2": "\\int_{0 mm}^{0.01 mm} (2.69 \\ast 10^{25} molecules \\cdot m^{-3}) dx = 2.69 \\ast 10^{25} molecules \\cdot m^{-3} \\cdot 0.01 mm - 2.69 \\ast 10^{25} molecules \\cdot m^{-3} \\cdot 0 mm ", "61bd797e2c71d5010a0b17a99e23d568": "\nQ = \\frac{1}{2m} \\sum_{ij} \\left[ A_{ij} - \\frac{k_i*k_j}{2m} \\right] \\frac{s_{i} s_{j}+1}{2} (3)\n", "61bda155ae4bfd92763df293f39c924e": " \n\\begin{align}\nd(uv) & {} = (u + du)(v + dv) -uv \\\\\n & {} = uv + u\\cdot dv + v\\cdot du + du\\cdot dv - uv \\\\\n & {} = u\\cdot dv + v\\cdot du + du\\cdot dv \\\\\n & {} = u\\cdot dv + v\\cdot du\\,\\!\n\\end{align}\n", "61bda2e1190a01160d44115843b56345": "V_\\textrm{eff}(r)=V(r)-\\frac{\\ell(\\ell+1)\\hbar^2}{2mr^2}", "61bdcd2739648fb7666c881f3599e685": "f(x)=\\sum_{k=0}^\\infty\\frac{\\Delta^k [f](a)}{k!} ~(x-a)_k\n= \\sum_{k=0}^\\infty {x-a \\choose k}~ \\Delta^k [f](a) ~,\n", "61bdd067049f22225f43df143dc767f7": "\\omega_{jk} \\omega_{lm} = \\omega_{lm} \\omega_{jk} \\,", "61bdf559ff5674dac3651de84ae28214": "\n[B] \\ \\ \\ \\ \\tilde{S}(\\omega) \\mapsto \\tilde{S}_1(\\omega) + \\tilde{S}_2(\\omega) \\text{ implies } G \\mapsto G_1 + G_2\n", "61be1d06240e3ead8eb5f65a088987a3": "\\mu = -{1\\over 4}({g^{(s)}}_p + {g^{(s)}}_n) + {3\\over 4} = 0.310", "61be23d88e986fae3a8e7854e219f4cf": "\\Delta = D_x^2 + D_y^2", "61be3c54370a5870466d807a851891da": "Y_s = \\mathbf{E}_{\\mathbf{P}} ( Y_t | \\Sigma_s ),", "61be792f6845da2bd641cbcb931b0625": "\\scriptstyle [-\\frac{1}{2},\\, 0]", "61be8da43bbbcebbd7e6cd432c5a9112": " f'\\rightarrow 1", "61be9247c917161697b79a5349366125": "\\frac{\\mbox{Votes}}{\\mbox{Seats}+1}+1", "61bea20a796a858a797d97dc758be2bf": "dev(D_1)", "61bf30cf4894df36f23c9658b671ee01": "Cone_\\omega(\\mathbb Z^2, d)", "61bf4c3a994f2872b9da993a139cf066": "\\chi_M^I(n-1) = \\chi_{M'}^I(n-1) + \\chi_{M''}^I(n-1) - \\ell((I^n M \\cap M')/I^n M')", "61bf5c3f3ce57c7d4bd23ba77ed9a78f": "E\\neq V_0", "61bfa13c6ad1c75a4543deb44f2da8eb": "1(2^1/1!!)\\pi^0 ", "61bfaf932bfd158b0cefa1543f706389": "f=k^{O(1)}", "61bfc5e4928b273874b2a788c420826c": "\\mathrm{Score}(i,j)", "61bffbee668c9185b24f68acb502f1e5": "T_i=t_i", "61c026a31ff60e2aaaf35362d7b1275c": " W = \\int_C \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{r} \\,\\!", "61c026cc80b93cea327f50083202d2ae": "f:(X,p)\\rightarrow (Y,q)", "61c0c0474abfe0f01c54015390db9e98": "\\operatorname{Aut}(A_6),", "61c1170ce3fdf21fbddaeb4acb5b3535": "\\eta_\\epsilon(L)", "61c1384eb050cabbe132e059bd4f92d9": "\\,v_d = \\mu E", "61c156dd35fad21fbc8bd913ee018c6f": "\\nabla_T T", "61c19e1f5f727e7baa1d35f6764a7a3c": "T_2,T_3,... ,T_n", "61c1cb6488cb7fcebd1714dde29a872e": " \\int_0^\\infty x^{s-1} ({\\lambda(0)-x\\lambda(1)+x^{2}\\lambda(2)-\\cdots}) \\, dx = \\frac{\\pi}{\\sin(\\pi s)}\\lambda(-s) ", "61c27434578412d25dfe9948816e90a2": "\\exp(j \\omega t)\\,", "61c2bf03908e37d6b236a721ea1e0c29": "\\{l_a\\,,n_a\\,,m_a\\,,\\bar{m}_a\\}", "61c3049192cefabd7f7d3ff66a523c08": "{\\partial u \\over \\partial t}=u-v+ \\int_{R^2}\\omega(x-x',y-y')f(u-\\theta)\\,dxdy + \\zeta(x,y,t),", "61c3480bb572b958f15605f2965b89b7": " \\operatorname{sink}[(\\lambda p.\\operatorname{sink}[(\\lambda q.q)\\ (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f))]\\ p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x))] ", "61c36b635516181d19b6589d8988f45e": "\\dot{q} = \\mathrm{d} q / \\mathrm{d} t", "61c3c5a026aba9e84b1d359713a20a93": " \\mathbb{R,C,H}", "61c40a20e8afda91b07db09d3b43e037": "J_{zz} = J_{xx}+J_{yy} = \\frac{\\pi r^4}{4} + \\frac{\\pi r^4}{4} = \\frac{\\pi r^4}{2}", "61c40a52c63c4dee355cc04945c92427": "\\Phi(x):R \\rightarrow [0,1]", "61c432f56606db41480f9408d4fa9c0f": "G_q", "61c435ff0bce4bb2057fca6fd6b2c077": "\\Pi_i", "61c46fb20a87088fece8b4f3aa33445a": "E = \\mu_c/e", "61c4c43ae159a0a494c995c70e02d687": "[f\\wedge g](x) = f(x)\\wedge g(x).", "61c4e9b685a935c12ca374448d36bd66": "f_i(x,c,t)", "61c5108f32ef54feada7b13acf156c22": "K(m)=\\int_0^{\\frac{\\pi}{2}} \\frac{d\\theta}{\\sqrt {1-m \\sin^2 \\theta}}", "61c52d0e7c5a27b782416546f52070b4": "\\dfrac{dN}{dt} \\dfrac{1}{N} = r", "61c56a52041f850e43d558b5fed3fd96": " \\phi \\; = \\; \\left (1 + \\varepsilon {c_1}^2 \\right ) \\left [ \\theta_5 - \\frac { \\varepsilon }{24}x^4 \\tan \\phi '(9 - 10 {c_1} ^2) \\right ] - \\varepsilon {c_1}^2 \\phi '", "61c574a7483b1e75950a907e432f0e57": " f - g = d_B h + h d_A.", "61c579fed1c4c9163648a1f1c4274871": "\\dot{Q}_L", "61c5a18c82a46b9e0719b79fb52c6e32": "L^{(v)} = \\frac{\\beta_v}{\\epsilon_v}\\begin{pmatrix}\n0 & 0 & 0\\\\\n0 & \\frac{\\gamma^2 {\\mathcal H}_v}{\\beta_v} & -\\gamma\\phi_v\\\\\n0 & -\\gamma\\phi_v & 1\\end{pmatrix}", "61c5cc2e70f74b501af0ecd61ba110d2": "d_Y (f(b), f(c)) \\leq K \\cdot d_X (b, c)", "61c60746717ef006a6a4dbbf83a81c3c": "c_\\max^2", "61c6ab7cac01e145d15f3261d8e53c33": "\n F(\\boldsymbol{\\sigma}, \\dot{\\boldsymbol{\\sigma}}, \\boldsymbol{\\varepsilon}, \\dot{\\boldsymbol{\\varepsilon}}, \\mathbf{x}, t) = 0 \\,.\n ", "61c6de4dacb42b75847839affe63dd33": "k_0=-0.6875\\ ,\\ k_1=-0.1250\\ ,\\ k_2=1.5625", "61c7b1efc055ca468e751e097cbce209": "(0.85 \\cdot 1 \\cdot 1.77 + 0.05 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.1030) = 1 \\cdot 1.77", "61c7ceb584175b2892588ff408826318": "x(xy) = (xx)y", "61c7f26a9e44d0d553292acc92987f3d": "\nE_{n} = - \\frac{m k^{2}}{2\\hbar^{2} n^{2}} ~ ,\n", "61c802d479acc7529da1ef98eb45f317": "x \\in (0, \\infty)", "61c8a4eb6ee1619c0c3fd1f6d4841376": "k\\cdot 2^n + 1\\text{ divides }2^{2^m} + 1. \\, ", "61c8a51f89f1c0780af0e81ca2887301": "f(y)f\\left(y+\\frac{1}{2}\\right)=\\frac{\\sqrt{\\pi}}{2^{2y-1}}f(2y)", "61c8ed6405f008772852bbddbb95df59": "\nr=r_p=7^{1/6}r_0\\approx 1.38 r_0\n", "61c8f4626fa627d8a7edc32c77740164": "\\hat J_j(i)", "61c8f942160d484870a8785d29ec4859": "P(\\mathbf{X}) = \\frac{1}{Z} \\prod_{f_j} f_j(x_j)", "61c928ba9410b63064952240c6ca013f": " {d \\bar u^i \\over ds} + \\Gamma^i_j \\bar u^j + \\bar R^i_j \\bar h^j = 0 ", "61c979066ce4ffecad32e59e4fd0c03c": "\\operatorname{Tr }\\Lambda^2(K) = \\frac{1}{2!} \\iint K(x,x)K(y,y)-K(x,y) K(y,x)\\,dxdy", "61c9a024f3162faf5ee56bfaf9f5f941": "g(z)=\\frac{z^2}{z^2+2z+2}", "61c9a40cfd609e81ed618b3caf080225": " \\lambda_j < (p\\sigma^2)/\\boldsymbol{\\beta}^T\\boldsymbol{\\beta} ", "61c9d58cc2e672e5ebd63e90f53036ea": " \\frac {a + b} {c + d}. ", "61c9eda5f54dd9d76007e5b55a74df92": "\\sin(16\\tfrac78 ^\\circ) = \\frac12\\sqrt{2-\\sqrt{2+\\sqrt{2-\\sqrt{2}}}};", "61ca12281d911bccaac65cfa0ef752da": "{\\mathbf a} = a_1{\\mathbf e}_1 + a_2{\\mathbf e}_2 + a_3{\\mathbf e}_3.", "61ca74be4abf9cfa61e3f45f76d6b75c": " \\delta V = \\frac 2 3 \\pi a^3 ", "61caebd284f4c72e8937513a3c8988db": "\\varphi_{X_k}(t)=1-p+pe^{it} \\qquad \\varphi_Z(t)=\\left(1-p+pe^{it}\\right)^2", "61cb3bfc136777b9c32cabe827f20769": " (h+k)^2 - hk", "61cb92c70b0084f6070ceb6caffd9fd8": "\\lambda(M) = \\lambda(p\\cdot q) = \\operatorname{lcm}(p-1, q-1)", "61cb9be21efe13d55096edb04403545c": " T(x, y, z) = (y z, z x, x y) = (U,V,W),\\, ", "61cba8c14181899f0b0ac1c8f82ef2b8": " \\Sigma _{21} (\\Sigma _{11} )^{ - 1} \\Sigma _{12} \n", "61cc0f6ccc4c9ac99391dddb79f7070a": "p_k: A_1 \\oplus \\cdots \\oplus A_n \\to A_k", "61cc3bf84515d4b81ff2f878123fe035": "\\sum_{i=1}^M F_i = I.", "61cc3f10309df76612c4fcfc15ebcd7e": "R_{uv}-\\frac{1}{2}g_{uv}R+g_{uv} \\Lambda =\\frac{8 \\pi G}{c^4} T_{uv}", "61cc51f3e026ba6ae1e957fd7257021c": "\n\\forall a^{n} \\in T_{\\delta}^{\\mathbf{p}^{n}}:\\Pr\\left\\{ E_{a^{n}\n}\\right\\} \\leq2^{-n\\left[ H\\left( \\mathbf{p}\\right) +\\delta\\right] },", "61cca3f98bc6c14c192ed4ad500acad2": "{\\mathbf M}({\\rm Riesz}(z)) = \\int_0^\\infty {\\rm Riesz(z)} z^s \\frac{dz}{z}", "61ccd556453bc0304847d1708ed1ced1": "\\chi_G(k) = \\chi_{G-a}(k) + \\chi_{G/a}(k) - \\chi_{G_a}(k)", "61cd1f6afa32ed5e3fbb255b81f00b5f": "\\Delta_{\\mathrm{LB}}=\\frac{1}{\\sqrt{\\det(g)}} \\sum_{i=1}^m \\frac{\\partial}{\\partial x_i} \\left(\\sqrt{\\det(g)} \\sum_{j=1}^m g^{ij} \\frac{\\partial}{\\partial x_j} \\right),", "61cd32257cb95a1fd57008642d764159": "A=\\cup_{i=0}^{\\infty}C_{i}", "61cda76c0d76870ec9b628c79e63c05e": "L(x,y^*) = \\begin{cases} \n f(x) + y^*(g(x)) & \\text{if } y^* \\in \\mathbb{R}^d_+\\\\\n -\\infty & \\text{else}\n\\end{cases}", "61cdcca4f3546533c55230be3d724836": "\\Gamma_{jk}", "61cdd04c7e404d8e13b5f55d35240683": "\n (\\or R_1)\n ", "61cdfd05ddc9c3471e229ea498b5e66f": "e = (v,w)", "61ce025a87d5f89bc1c1a8498fa59317": "Q^{-1}\\left|y\\right\\rangle \\left|f(x_0)\\right\\rangle", "61ce271a422ddc2b55ee90e96364e172": "V=D^+ + D^- .", "61ce30d22c262a73ef3fd0918baba102": "(X, \\Sigma,\\mu)", "61ce486b13d0c868eafc8512007e005b": "2^{2^2} + 2^2 - 1", "61ceb522eae467ce34403ee1dea512a4": "{\\it {M \\ll N}}", "61cef8b82c7e224a0c17b52ec970021c": "\\scriptstyle m\\theta", "61cf048fa5cfd1371e42b2739263b2a3": "|L|=\\sqrt{L^2}=\\hbar \\sqrt{6}", "61cf08dae1ae5d76bd1db43d0c81fddb": "\\left(2v^B-1\\right)", "61cf0ebd3e588e6a9674f8f91477443b": "M(V) = (V/V_d)^{-2/3}", "61cf623821de6dd846f1ea528e783483": "\\mathrm{Da}_{\\mathrm{II}} = \\frac{k C_0^{n-1}}{k_g a}", "61d01abbab3aa32f9e90c7b10e48544b": "f_1 \\circ h \\simeq f_0", "61d0c0bed013f53c8c8f7de5bc900b49": "g^{x_3}, g^{x_4}, B = g^{(x_1 + x_2 + x_3) x_4 s}", "61d142aad9e962188b5482c904451142": "(s_t,a_t,s_{t+1})", "61d17790550bd8b033eee0d336c86570": " \\int_0^t Z_s \\, ds ", "61d1876450275737c6ef17f5a2be4643": "P[s(t)|\\{t_i\\}]", "61d1efc3eae784a973090e35a74bae99": "{5 + 3\\varphi}", "61d20065f49019ef5810ca360a2e057f": " A\\,", "61d21e45c7e8a4edd86fb69523f68f08": "\n\\Phi : C^*(S) \\rightarrow C^*(V) \\quad \\text{by} \\quad \\Phi(T_f + K) = \\oplus_{\\alpha \\in A} (T_f + K) \\oplus f(U).\n", "61d235736e622497a52726f602397caf": "\\partial f", "61d2cf248838cd603ec3b205df17e7fb": "n=n_kp^k+n_{k-1}p^{k-1}+\\cdots +n_1p+n_0", "61d32108ffcf09e52cc0163c6d02a2ef": "B_4C", "61d347a6b8c79a667530c3deac84e98d": "\\scriptstyle y(2k)=ay(2k-9)+x(2k)", "61d362462499a5e62836c7a536d3786b": "[y_0:\\cdots:y_n]", "61d3d82f9605c7fbf6670f05a61016e5": "\\varphi(x) \\to \\varphi'(x) = T(\\omega)\\varphi(x).", "61d40d73a1fd51d28ce78c248aeeaa70": "(M,\\varphi:M\\to \\mathbb{R})\\ ", "61d43e62b7c5e7cc164b1a169f7d3719": "\n\\begin{alignat}{2}\nQ_\\mathbf{x} &= \\ell_\\mathbf{x}+ \\mathbf{f}_\\mathbf{x}^\\mathsf{T} V'_\\mathbf{x} \\\\\nQ_\\mathbf{u} &= \\ell_\\mathbf{u}+ \\mathbf{f}_\\mathbf{u}^\\mathsf{T} V'_\\mathbf{x} \\\\\nQ_{\\mathbf{x}\\mathbf{x}} &= \\ell_{\\mathbf{x}\\mathbf{x}} + \\mathbf{f}_\\mathbf{x}^\\mathsf{T} V'_{\\mathbf{x}\\mathbf{x}}\\mathbf{f}_\\mathbf{x}+V_\\mathbf{x}'\\cdot\\mathbf{f}_{\\mathbf{x}\\mathbf{x}}\\\\\nQ_{\\mathbf{u}\\mathbf{u}} &= \\ell_{\\mathbf{u}\\mathbf{u}} + \\mathbf{f}_\\mathbf{u}^\\mathsf{T} V'_{\\mathbf{x}\\mathbf{x}}\\mathbf{f}_\\mathbf{u}+{V'_\\mathbf{x}} \\cdot\\mathbf{f}_{\\mathbf{u} \\mathbf{u}}\\\\\nQ_{\\mathbf{u}\\mathbf{x}} &= \\ell_{\\mathbf{u}\\mathbf{x}} + \\mathbf{f}_\\mathbf{u}^\\mathsf{T} V'_{\\mathbf{x}\\mathbf{x}}\\mathbf{f}_\\mathbf{x} + {V'_\\mathbf{x}} \\cdot \\mathbf{f}_{\\mathbf{u} \\mathbf{x}}.\n\\end{alignat}\n", "61d44a0125ea835d9f50d16b7f954acf": "k_c,\\bar{k}", "61d45264e31240aa71567107eb1963d1": "-\\infty c \\big] \\approx - \\frac{1}{2 T},", "61f01b3d303cc5b63fe6754945cd8b39": "E(\\pi_$)", "61f03435eeb9d85dfe86e2b5397f7af5": "\n(\\mathbf{a \\times b})_i = \\sum_{j=1}^3 \\sum_{k=1}^3 \\varepsilon_{ijk} a^j b^k.\n", "61f08ab498ebf9a99a8c6fb2ccd3ea8e": "\\,z_i", "61f0b6cd2494473dd82ca3a377283e7a": "N=\\lfloor u^2 \\rfloor", "61f0ecf3f173582187f6b00eb3b0bd2c": "\n W(j_1j_2Jj_3;J_{12}J_{23}) = (-1)^{j_1+j_2+j_3+J}\n\\begin{Bmatrix}\n j_1 & j_2 & J_{12}\\\\\n j_3 & J & J_{23}\n\\end{Bmatrix}.\n", "61f13be66facf5b102ddbcd19b450efc": "S=Nk\\ln(2)", "61f1a4d221c38bf7e04f060aef01f7f9": "\\frac{\\left(\\frac{\\partial P}{\\partial T}\\right)_{S}}{\\left(\\frac{\\partial V}{\\partial T}\\right)_{S}} = \\left(\\frac{\\partial P}{\\partial T}\\right)_{S}\\left(\\frac{\\partial T}{\\partial V}\\right)_{S}=\\left(\\frac{\\partial P}{\\partial V}\\right)_{S}\\,", "61f20520f89c6b1e91aef8ba533b180d": " \\vec F = -k \\vec x \\, ", "61f2101b87fc64d422d3a209841eca7a": "\\operatorname{Aut}(G)", "61f2207ead511206f558d34273fa6800": "\\#S = n.", "61f22e11fe448de318dd762bc3460629": "H/\\Phi(H)", "61f238c38d9f8ad3d6df00cc1725fc4b": "p=0.3/2=0.15", "61f2a3998056d4b92cd1a450b917990b": " \\lambda_0 v_0 + \\lambda_1 v_1 + \\cdots +\\lambda_n v_n + \\lambda_{n+1} v_{n+1} = 0. ", "61f2dd93a0e9cc20815f5fdefa9f5dbb": "\\min(3,4,1,4,7) = 1", "61f352f01348c680d6b1c0ab0e77e675": "\\, a'=\\lambda_1 a+\\lambda_2 b ", "61f36d3c570d448ddc0c78dd95492165": "\\gamma\\in\\Delta", "61f37f2ca60ff53268f7414d3b983656": "E_i^c", "61f4892f625df472bf408290fc1bc039": "\\mathcal{P}[f]\\colon t \\mapsto f(-t),", "61f59ce01b6bcb278b0c3ec9ffedc74b": "\\tfrac{dS}{dT} = \\delta MS - \\beta SI - \\mu S ", "61f5bcb718e8aceed50ab8566369a6b4": "u_i:\\mathbf{C}\\to \\R", "61f60bf4ef5913239f7eba5b0ff4b13e": "\nF = \\langle \\partial \\bar{A} \\rangle_V \\,.\n", "61f62a34418c605e6aa6e1f9515311ab": "\\hat{\\tau}", "61f632011c97147fd6d540c29635f57d": "r_{I2}\\,\\!", "61f633589052f9c9b4946cb93226b1c6": "\\sigma^\\mu", "61f65778874a154ec1ec89c87260be12": "z^K = -\\alpha", "61f6739ce6b19a11b7260d2932fda5c2": "\\bar{h}^{\\alpha \\beta} \\equiv \\eta^{\\alpha \\beta} - \\sqrt{|\\det g|} g^{\\alpha \\beta} \\,", "61f6b7c4be4108125238d48845c65997": "\\rho_{\\mathrm{bound}} = -\\nabla\\cdot \\mathbf{P}", "61f6d3504579879ffab14a0f8a919905": "\\mathcal{E}\\subset \\mathcal{T}", "61f6d8f4a44b22a41920441f79d6ab53": "i \\,", "61f719a86931e88b097259d509b54fc2": "\\alpha\\rightarrow\\alpha", "61f750dc144df1d561d5640d4de2218b": "H_{eg}=\\mathrm d(R_g)_e (H_{e})", "61f7b194fbc106608e5e24f6e2db92da": "f(x,y) = f_x(y) = x^2 + xy + y^2.\\,", "61f801ea19802650c2d73c20e7de03de": "\n \\mu =\n {\\omega_p^2 r_B\\over \\omega_H c} \n= \\left( {2e^2r_B\\over L_B \\hbar c }\\right) {\\nu \\over \\sqrt{1+{\\omega_p^2\\over \\omega_c^2}}}\n= 2 \\alpha \\left( { r_B\\over L_B }\\right) \\left({1 \\over \\sqrt{1+{\\omega_p^2\\over \\omega_c^2}}}\\right) \\nu\n", "61f81ff846bc435a12a3c66e3bf62fcd": "h_{ij} = h_{ji} \\qquad \\qquad k_{ij} = k_{ji} ", "61f885f41c0bae61c7a25f7c665bd5c7": "\\frac{\\mathrm{d}n_i}{\\mathrm{d}t} = \\sum\\limits_{j=1}^N\\frac{n_j}{\\tau_{ji}}-n_i\\sum\\limits_{j=1}^N\\frac{1}{\\tau_{ij}}+I(\\delta_{iN}-\\delta_{i1})", "61f8e6b6208b31795f9236428feb604e": "u'(t)=Au(t),~~~u(0)=x,", "61f8ec04ef88cec879064840d4f26555": " P = ", "61f8f987305f322d7a2e56d987ec18d9": "\\Pi=\\{P_1, P_2, \\dots,P_l\\}", "61f940f1fc3921ab50ee42a4c9b9267e": "\\sin \\lambda =\\ r_n\\ =\\ \\sin i \\ \\sin u\\,", "61f95e24e18e07133eb9a3a1d6f0a7e5": "y \\in V.", "61f9667c36de23f9d17d2a6c8aff2888": "Aw=\\lambda Bw", "61f9a5b77186ee46e0961f478e1846c5": " Q = T_H \\sum_{p}^{}{\\Delta S_i} ", "61f9a9fcf22190762bc9fb523dd07b4e": "\\xi \\mapsto H_\\xi", "61f9d765eeec5dfd531b90842ad04e2f": "f(X)=X^n+c_{n-1}X^{n-1}+\\cdots+c_0", "61f9e34501bfea3a9791948bacbd59c9": " B + X \\rightarrow Y + D", "61fa66ec333a25cf05ca90c8c1ec560d": "\\mathbf{P}(t) = \\mathbf{P}_0 + \\left(\\frac{\\mathbf{V}+ \\mathbf{V}_0}{2}\\right) t .", "61fb1e7d93a3ea55b0891c82e4a2e159": " \\left\\{|\\psi_i\\rangle \\right\\}, \\left\\{ |\\phi_j \\rangle \\right\\} ", "61fb36611c25095d34ea025b5ea2e8e3": "\\hat\\mu \\pm i\\hat\\gamma = i\\frac{1+\\hat\\zeta}{1-\\hat\\zeta}.", "61fb4d8924c283b64934f7614d1c1695": "u_k=-\\mathcal{F}\\boxtimes_{n=1}^N\\mathbf{w}_{k,n}(p_n(t))\\mathbf{x}(t).", "61fc08630fa58d925fc81459ea1f6a13": " \\Lambda = {64\\over {\\it \\mathrm{Re}}} \\; , \\quad\\quad \\mathrm{Re} = {2\\rho v r\\over \\eta} \\; , ", "61fc1cef0c0d2e0f8b39f7dfa7039d13": "x,y\\in\\{0,1\\}^k", "61fc4d79a6d8b596a0295b02351be882": "u^2+v^2=w^2.\\,", "61fc5c6d1de487c83f427b20bae3af92": "\\Rightarrow |\\overline{AF}|=|\\overline{FD}| ", "61fc7ff5447e1b59446652537bd9f936": "\\psi(\\alpha) \\leq \\delta", "61fcfd6c61ed734a254192fbcbd4b8e3": "\\scriptstyle i^2 \\;=\\; -1", "61fd1ae2d57578ee0e0f8a4ea69c814e": "h = h(x\\cdot\\xi)", "61fd5920a98d17f4c52f2e6b2de7b5ce": " T:X \\rightarrow X", "61fd5b530bf756840de218b7e8a9830c": "f_n\\colon R^n \\to R", "61fd6517e7c94f106165cdfdf7e59e2b": "(\\mathrm{d}_t f) (+1) \\in \\mathrm{T}_{\\alpha (t)} M.", "61fd8670e31b2bf4fbc6e636c1c004df": "\n\\Psi(\\mathbf{r},\\mathbf{R}) = \\chi_1(\\mathbf{r};\\mathbf{R})\\Phi_1(\\mathbf{R})+\n\\chi_2(\\mathbf{r};\\mathbf{R})\\Phi_2(\\mathbf{R}),\n", "61fdb085efc24e77bb16484c66e1f067": "m - 1\\,", "61fe31306761b144a783b3b2bfbf271c": "\\psi_0", "61fe990299c5c9f2ab5baf98100733a5": "\\alpha^2*A_L=1000000", "61feb4081c78ac87749b6df6cf49d722": "C_v = dU/dT", "61fecd32fe0916d910250bea4bcdc327": "\\frac{1}{N}\\left(\\sum_{n=1}^N\\sum_{i=1}^n\\sum_{k=0}^i a_k b_{n-k}\\right)\\to AB.", "61fee6f5158a7a85af97f5a7ea07f2f3": "e_{(\\mathbf N)}\\,\\!", "61ff37f0999fffd9aa2797bad93fb5f3": " A_{{\\lang i_1,\\dots,i_m} \\rang} \\Lambda = \\bold{0} . \\,\\!", "61ff4af6861686bb6474f878d0e61e05": "\\tilde{L}", "61ff5e49d1da46195be9f2dc7ded5907": "\\sum_{0\\leq{k}\\leq{n}}\\binom nk = 2^n", "61ff8d92fc28224a4d5059201c2f1261": "t=x-\\mu", "62002234e0458ec4ade2d18c4bf1693c": "dL/dT", "620027baf7659bfd51268a5728f4a546": "t(\\theta) = V(x(\\theta),\\theta) - U(\\theta)", "62003ae7ba74346e59a91d4089d8f7ee": "P_c\\,", "62005e0ee012f4317994af5eb934217f": "H_{\\beta} := \\{X \\in 2^\\omega : X\\ \\mathrm{has\\ effective\\ Hausdorff\\ dimension\\ } \\beta \\}", "62008ff2a23e9d9bf662b6474fc028f7": "GL_2^+(\\mathcal O_F)", "6200e8d8059e2b09f4a8ab689fcb073d": "v_\\mathrm{p} = \\frac{\\omega}{k}.", "6201005a0cbc2853ea32dc78a1ed3990": "\\bar{Y}_i", "62010e4eca231d853affac2586545033": "M=\\mathcal{T}(B)", "620179672b1e1cf0e378d3764301dbb9": "R=c/H", "6201ad72f5b97daf928f7be20e035160": "\\tilde{F}^{\\mu\\nu}=\\frac{1}{2}\\varepsilon^{\\mu\\nu\\rho\\sigma}F_{\\rho\\sigma}", "62021614bf2fa608f4459231684d061d": "nQ\\,", "62023ac8857daf5c3e8c09ecc0688e4f": "v = v(x,y),\\,", "6202665c9b7a3066f2fe68f23d4e1bff": "\n\\frac{1}{2}\\rho v_{\\infty}^2 + P_{\\infty} = \\frac{1}{2}\\rho \\left(v_{\\infty}(1 - a)\\right)^2 + P_{D+}\n", "6202f543005a110a4ffcce4a0d82ac56": "\\tbinom\\alpha k", "620326bcbddd72fc6d24edeb14967c0f": "z_5 = x_5 y_1 - x_6 y_2 - x_7 y_3 - x_8 y_4 + x_1 y_5 - x_2 y_6 - x_3 y_7 - x_4 y_8", "620337cae3166e420363f30b633ed651": "\\rho = \\lambda / \\mu", "62033a940b5495b1708687ca3b8c0e97": " \\sin^2 2 \\theta_{13} = 0.113 \\pm 0.013({\\rm stat.}) \\pm 0.019({\\rm syst.}) ", "620376cd04372a98d07f19739c826f81": "z(r)=\\frac{r^2}{R\\left (1+\\sqrt{1-(1+K)\\frac{r^2}{R^2}}\\right )}+\\alpha_1 r^2+\\alpha_2 r^4+\\alpha_3 r^6+\\cdots ,", "6203b85cbea2081b36c7fd13516bb0d3": " (p|q)_8 = (q|p)_8 = (aB-bA|q)_4 (cD-dC|q)_2 \\ .", "6203e990073063aa7d9b23bfa6a9bfd9": "\\sum^{k}_{i=1}1/\\lambda_{i}\\,", "6203faa8d98e5054c6696c16c5a81bdb": "\\begin{align}\nF_{\\alpha\\beta} & = \\eta_{\\alpha\\gamma} \\eta_{\\beta\\delta} F^{\\gamma\\delta} \\\\\n& = \\eta_{\\alpha 0} \\eta_{\\beta 0} F^{0 0} + \\eta_{\\alpha i} \\eta_{\\beta 0} F^{i 0}\n+ \\eta_{\\alpha 0} \\eta_{\\beta i} F^{0 i} + \\eta_{\\alpha i} \\eta_{\\beta j} F^{i j}\n\\end{align}\n\\,", "62041226de36b44f7ef4eda5b71ebe34": "\\Delta(a) = \\sum_i b_i \\otimes c_i", "62043ce8444816d6b9e0457b5ca94de9": "\\iint\\bold{E}\\cdot {\\rm d}\\bold{S} = \\frac{q_e}{\\epsilon_0} \\Rightarrow \\bold{\\nabla}\\cdot\\bold{E}=\\frac{\\rho_e}{\\epsilon_0}", "6204dc9182a3f4d546e232bb752b383d": " \\gamma= \\frac {{a}}{{x}} ", "620501a80e8450c08c344f760caa75bc": " \\sum_{J \\subseteq [n]} (-1)^{|J|} |A_J|.", "62052494ba769dd3aa96ef9b742ce194": " t_1 = \\frac {2 \\pi R }{c - R \\omega}. ", "620578bb70a9cde90a26462bdbf07219": "\\mu_k \\propto", "620596225439f2a6ffd3f40c16d01a55": "(in)^k", "6205b5173c711ba122e0b10ad7034f07": "\\left(\\frac{a\\psi+b}{c\\psi+d}\\right)", "6206155f034cf741a4935d13f4c77f2d": "2\\sqrt{a} \\int \\frac{du}{(u^2-b)^{\\frac{3}{2}}}", "62062f29ab49ad34e53e45705248347f": "(\\partial H)_P=C_P", "62068da31d9e857b077d12e7b3bd6a55": "\\bar{a}", "6206a9ee7e590a21ea5ed19d420e5c2a": "\nE=\\sum_{n=1}^{\\infty}\\frac{\\sigma_0(n)}{2^n}\n", "6206c4a667e1146059844468ee0c188f": "\\mathcal{L}_K = [d,i_K] =d\\,{\\circ}\\, i_K-(-1)^{k-1}i_K{\\circ}\\, d", "62073170b71ecd7642d07fb485efcc88": " d\\colon \\pi_{1}(F) \\rightarrow \\pi_{1}(E) \\! ", "62073f18d4247c4822203082d4530619": "I_{D0}", "62076bfcc63c06ce6d02869eb642b2fb": "s_d =", "62077808e9c445bf7081bdb925c81d83": "\nf(z) = b_0 + \\cfrac{a_1z}{1 - \\cfrac{a_2z}{1 - \\cfrac{a_3z}{1 - \\cfrac{a_4z}{1 - \\ddots}}}}.\n", "62077a05ba9bb1f0a3edd172b838730e": "\\sigma=\\frac{q}{A}", "62079bbd2ff617c64b3fe67b28bf8efd": "\\pi^2\\approx10;", "6207a80403dcccc1aa3b5b7303315c4b": "H_1", "6207a89f95913bce5c15fad4d74773f5": "\\sqrt{6} - \\sqrt{2}", "6208291902fa6765ca5d3436e0ea1813": "x=3.\\;", "620871f8e626adaed7db19a497a23ef8": "\\begin{pmatrix}\n 1 & x_{12} & x_{13} \\\\\n x_{12} & 1 & x_{23} \\\\\n x_{13} & x_{23} & 1\n\\end{pmatrix} \\succeq 0", "6208a6db53d9941ebb3b72b19362276e": "z_{n+1} = {z_n}^2 + c", "6208c0b53045b472b1fbcbcb5fb3d49a": " \\operatorname{get-lambda}[F, G = \\lambda V.E] ", "6208f7023e360d42ca93cccd5de7b0b9": "\\delta \\approx \\theta_0 - \\alpha + \\Big( n \\, \\Big[ \\Big(\\alpha - \\frac{1}{n} \\, \\theta_0 \\Big) \\Big] \\Big) = \\theta_0 - \\alpha + n \\alpha - \\theta_0 = (n - 1) \\alpha \\ .", "6209373261a91d85797a10f3a5dce7b2": "D = \\varepsilon E+ (\\chi - i \\kappa) \\sqrt{\\varepsilon \\mu} H", "62099e142e663a8f686d8498d935d2be": "1 \\leq j \\leq n_B", "620a0d654179802c6f9a5eeeb1b96200": "{\\frac{wet\\; sample\\; weight - dry \\; sample \\; weight}{dry \\; sample \\; weight}} \\;*100 \\; = WME\\;", "620a162fd8f23b9e712bea05ce5fb092": "a=0.4275\\frac{R^2T_c^{2.5}}{P_c}", "620a745dd89cfdcb45b14329f0ba6383": "Pr(\\mathbb{Z}\\mid\\boldsymbol{\\alpha})=\\operatorname{DirMult}(\\mathbb{Z}\\mid\\boldsymbol{\\alpha})=\\frac{\\Gamma\\left(\\sum_k \\alpha_k\\right)}\n{\\Gamma\\left(\\sum_k n_k+\\alpha_k\\right)}\\prod_{k=1}^K\\frac{\\Gamma(n_{k}+\\alpha_{k})}{\\Gamma(\\alpha_{k})}", "620a8fca2c335d3f07c3200a744a9a15": "\n Q_\\alpha = \\mathcal{M}_{,\\alpha}\n ", "620aa8b66baefd3dfdbdaaeaadd43f09": "x \\notin x", "620b528c12137c68b463120836baf4ce": "\\exists x ( x^2 = 1 \\land 0 = x)", "620bba764a67108610f99b224d484fa6": "\\pi_{t}", "620bd720bc77ecbf9d67c56330a32bde": "\\forall c \\in a (\\forall d \\in c (d \\in a \\and \\forall e \\in d (e \\in c))).", "620bde729dcacf3416fa9bda59b21c7c": "M(x)=\\frac{1}{x^2}.", "620c128e3e38acb9fb12b769ca94de82": "F \\rightarrow \\Delta", "620c233532dfe0409d8d5e5ff275f852": "a = b = \\pm \\frac{1}{\\sqrt{2}}.", "620c4df7e160d733353cbe75c8b7944d": "m \\neq M", "620c7002e9daa468364adf48859c33ec": "\\left(\\mathbb{Z}/n\\mathbb{Z}\\right)^*", "620cb75b586498104bfc45ae983c783b": "4 \\times 2^2 - 6", "620de3b218f830e4699a77d5e3d507ad": "(p,q)_\\infty(p,q)_2(p,q)_p(p,q)_q=1", "620e033a21d3b206a59681b80cb76a1f": "\\overline{s} \\in \\alpha, t \\notin \\gamma", "620e4f4a664c68d65333413811e7a044": "\\sigma^2|\\alpha, \\beta \\sim \\Gamma^{-1}(\\alpha,\\beta) \\!", "620e62973515b6531456503a35a58cd0": "\\psi(\\mathbf{r}) = \\mathrm{e}^{\\mathrm{i} \\mathbf{k}\\cdot\\mathbf{r}} u(\\mathbf{r})", "620e76e8b31a3b33e7e310a7a27e92f3": "R_{0 0} = K \\left(T_{0 0} - {1 \\over 2} T g_{0 0}\\right)", "620ecf577c9c833ec46f9e3d93eab295": "\n\\vec{X_2}=\\vec{X_{\\beta}}-\\vec{a_2}.(\\vec{D_{\\beta}})\n", "620f4e5b49433448e1b4e91e6e0a0c33": "f(x) = 0.6 \\, \\frac {1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} + 0.4 \\, \\delta(x-3.5).", "620f92ecd4925cb03e88b2a6037611fb": "F(x,y,t)=t^2 + t(y-x-k) + kx = 0\\,", "620fbc6808e429dea8b4255ab8d7cdd1": "\\mathbf{v} = cv \\wedge \\gamma_0 / (v \\cdot \\gamma_0).", "620fd7f915a8b7c0ca5419b74517f0d6": "\\mathcal{E}(f)=\\{\\alpha\\in\\overline{\\mathbf{Q}}\\,:\\,f(\\alpha)\\in\\overline{\\mathbf{Q}}\\}.", "620ff603df36f48ca6acb0e6254aee8c": "f * g", "620ff8bbb0c4b447bf30e120f662906f": "g(x,y) = U(x,y,z)\\big|_{z=d} ", "621008d67a114de7219c6bb458b80332": " X \\sim \\textrm{Frechet}(\\alpha,s,m)\\,", "62100d639bd172be77fe05ccd02d4460": "h:G\\rightarrow H^{*}", "6210a33b45e85c6792a3687323051ae2": "\\mbox{LOP}=340", "6210d4bfc7e59253a7c6ea426e6ab7bb": "\\beta_+ = \\beta_1 + \\beta_2 = -\\beta_3, \\quad \\beta_- = \\frac{\\beta_1 - \\beta_2}{\\sqrt{3}}", "621177d63718aaed8a7ddd042ba7d938": "\\|Ax\\|\\le \\|x\\|.", "6211b00c67407e1498b7bc359ec7837e": "M(T) \\propto \\left(T-T_c\\right)^\\beta,", "62124055f954c6dcc7f15341a35fd99c": "\\phi \\leftrightarrow \\chi ", "621284a18df3ac3651bf9e4ebd1376a6": "52,608\\,", "6212d2469928b5c44113d63fa51e9b37": "\\displaystyle{\\rho(a,T,b)=(-a^*,-T^*,-b^*).}", "6212debfc9d77c137afc00139f17758e": "\\phi_{ij}=0", "621458e0c0a891396d271edf08f25380": "rho_p", "6214f773bb80afffa3f1714abc9aa831": "y_2~", "62151cdd2231c3f55534dc2b8266183e": "l(u,v)", "62153e888d242012ca3dbebefef721a6": "\\overrightarrow{Y}=Y_{o} ", "621549acbdb0fb102742a41ab9dc9bc4": "\\frac{\\partial f}{\\partial r} = 0 \\quad \\Rightarrow \\quad f = f(\\theta)", "62154a7d485c59ab6121329a3fcca0e7": "\\frac{1}{2} - \\frac{1}{4} + \\frac{1}{6} - \\frac{1}{8} + \\frac{1}{10} + \\cdots + \\frac{1}{2(2k - 1)} - \\frac{1}{2(2k)} + \\cdots", "62155f6ec6baa4274df9a3d8ce9b7f24": "G = ", "62157bb44d98af0eb7c12243cc422958": "V_0\\ =\\ \\sqrt{\\frac{\\mu}{r}}", "6215d107d1cb4e5c4502d1c3ba05b2e0": "\\partial_i(1)=0 \\quad \\partial_i(v_j)=\\delta_{ij} ", "6216496d8b087690b992cc4bbb2e9c7c": " G= \\frac{\\left\\langle\\hat a\\right\\rangle _{\\rm final}}{\\left\\langle\\hat a\\right\\rangle _{\\rm initial}} ", "62167659ee6ffb21c81d130f9444aaeb": "\\eqcirc \\circeq \\triangleq \\bumpeq \\Bumpeq \\doteqdot \\risingdotseq \\fallingdotseq \\!", "6216855aa1bb0a10416fdffb1feea0d8": "\n\\begin{align}\n\\log \\frac{1+z}{1-z} & = 2z \\left[1 + \\frac{z^2}{3} + \\frac{z^4}{5} + \\cdots\\right] \\\\[8pt]\n& = 2z \\left[1 + \\frac{z^2}{3} + \\left(\\frac{z^2}{3}\\right)\\frac{z^2}{5/3} + \n\\left(\\frac{z^2}{3}\\right)\\left(\\frac{z^2}{5/3}\\right)\\frac{z^2}{7/5} + \\cdots\\right]\n\\end{align}\n", "621685934c0419a14669639732416803": "h_0 = h + \\frac{V^2}{2}\\,", "6216890e27e1d36a1f12aa60169bb40c": "x^2a^{-1} = 1\\ ", "62169cf9e4ec5356560d9b08f595b487": " k \\mapsto |f_k \\rangle ", "6216b1122659845f923a043580f3ea94": "EL(\\Gamma)=\\frac{\\log(r_2/r_1)}{2\\pi}.", "6217179095745ee6865ced1ddd4287d9": "\\varepsilon_{jmn} \\varepsilon^{imn} = (\\varepsilon^{imn})^2 = 1", "6217207001da87024aa9c593fe6d62f4": "\\dot{\\boldsymbol{x}}(t) = \\boldsymbol{F}(\\boldsymbol{x}(t)), \\qquad \\boldsymbol{x}(0)=\\boldsymbol{x}_0.", "6217a85d79208f24db49697d27839cfd": "f\\in\\Lambda^n,", "6217c53ce7d133fec6517e12ed4e5f93": "K(\\sqrt{1-k^2})", "6217c86f78616ec8231ececd3b6ab73f": "M_X(t) = {-3\\left(e^{at}(4+(a^2+2a(-2+b)+b^2)t)- e^{bt} (4 + (-4b + (a+b)^2)t)\\right) \\over (a-b)^3 t^2 }", "62180106dcba27ddf61de342df68d7df": "\\Omega\\approx-\\frac{\\textrm{Li}_{\\alpha+1}(z)}{\\left(\\beta E_c\\right)^\\alpha}.", "62182987b2daf2ea234f6a0088b7e86e": "\nO_n\\left\\{\\sum_{k=-\\infty}^{\\infty} c_k\\cdot x_k[m];\\ m\\right\\} = \n\\sum_{k=-\\infty}^{\\infty} c_k\\cdot O_n\\{x_k\\}.\n\\,", "62186ea538fcd31d081ca7aa8e64d374": "\\,\\,\\sigma_{ij} = 2\\mu\\varepsilon_{ij} + \\lambda\\varepsilon_{kk}\\delta_{ij}", "6218b78f71ba6211806fc62af67bc7e6": "a_{n,0}", "6218c6f33c70828d738949d8fda439ed": "J_n", "62191840c853d60f0d18f0f72196c236": "\\pm\\left(\\pm\\sqrt{10},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{5}{\\sqrt{3}},\\ \\pm1\\right)", "62194b875336cee3d25dc0df1122e4e7": " h=x_i-x_{i-1}", "62199d4d19a0d23fcd038699b103bdd5": " T = \\frac{1}{2}M\\mathbf{V}\\cdot\\mathbf{V} + \\frac{1}{2}\\vec{\\omega}\\cdot [I_R]\\vec{\\omega},", "6219fce9b5c938c46d0df57ad1938e48": " \\alpha = {\\sigma_D}/{\\sigma_M} ", "621a0160e870cc2aff50f500cf8a77d0": " F=F_1\\ast F_2", "621a2abced7ede6d87fa599cb59b7c64": "P(A \\cap B) = P(A|B)P(B)= P(B \\cap A) = P(B|A)P(A)", "621a6ab78ab71dc413b122463c3ac8d0": "|\\xi-\\alpha| 0", "6228f7994a21ee53499c6684fac51774": " E ", "62292e0177684f9697f4261e30b51f49": "u(y) \\approx 2 u_0 \\frac{y}{h} = \\theta y ", "62294640553a2e5fa78e51617ff1ae57": "t \\in \\mathbf{R} \\ \\longrightarrow \\ f^*(|t|) / 2.", "6229dbadee0b3c395c05c270e50ebfe6": "(5+3)x\\,\\!", "6229e1e3823305c4103e187fdbf2ae8a": "\n\\vec \\eta_2 = \\begin{pmatrix} x^2_1(t) \\\\ x^2_2(t) \\end{pmatrix} = c_2 \\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix} \\cos{(\\omega_2 t + \\varphi_2)}\n", "622a095dc9edee90d21fc439afb1ab02": "a=2.", "622a4ac483348c233f7c922b809d6731": "S_C", "622a5cd880df50eeccc7cf9611fd84b0": "\\{x_0, x_1, \\dots, x_{m-1}\\}", "622a65828cbb7dff39d58e9c6b2e5bdf": "I_k \\otimes \\Phi : \\mathbb{C} ^{k \\times k} \\otimes \\mathbb{C} ^{n \\times n} \\rightarrow \\mathbb{C} ^{k \\times k} \\otimes \\mathbb{C} ^{m \\times m}", "622b163b4b037c91c06acf90379cf4ef": "\\left|\\tau_{cap}(\\omega)\\right|^2", "622b18927a01202995a6ef9a789ec8f1": "\\Psi_0 = \\{\\omega, \\theta, \\xi, \\rho\\} \\,", "622b78f6707a3a8baf5843887d080d94": "x_1 = x_2 = 0", "622bc4b0bac8c91f9affc30e4773391d": "\\zeta(z;\\Lambda)=\\frac{1}{z}-\\sum_{k=1}^{\\infty}\\mathcal{G}_{2k+2}(\\Lambda)z^{2k+1}", "622bf508cb929db828242a91d6183527": "m_3=\\left.\\Delta(1+6\\mu+\\Delta^2)\\right.\\,", "622c3ac4f76b0c2ff5f5758025945ec6": "p_1 = n_\\text{mean}+\\delta\\,,\\quad n_1 = n_\\text{mean}-\\delta\\,,", "622c8f56ba60996524cb225c834119bc": "\\mathrm{DD} = \\begin{cases}\n\\mathrm{D} &= \\operatorname{trunc}(\\mathrm{DD}) \\\\\n\\mathrm{M} &= \\operatorname{trunc}(|\\mathrm{DD}| * 60) \\bmod 60 \\\\\n\\mathrm{S} &= \\left(|\\mathrm{DD}| * 3600 \\right) \\bmod 60\n\\end{cases}", "622c9dfc9ba07db168127f9e1b171d1a": " = \\frac{R_{wr}}{R_w}", "622ca36e5d9b3763376283244d914baf": "n\\Delta=b-a", "622d44b403526febb3eca79f280f99fe": "k=\\frac{\\sqrt{2m(E-V)}}{\\hbar}.", "622d563b68ec09b0a5d24d49a9ee3233": "x_1M_1 = b", "622d5b2f0484d7a890b71bcb3e4e48df": "\\int_0^{\\pi} \\Bigl| \\frac{f(x_0 + t) + f(x_0 - t)}2 - \\ell \\Bigr| \\, \\frac{\\mathrm{d}t }{t} < \\infty,", "622d72ecb45b7df392f4f2d77a13732b": "p(\\sigma) = \\frac{1}{\\sigma_{av}} e^{-\\frac{\\sigma}{\\sigma_{av}}}", "622dc14abad8e9deade10eb48409a73c": "S_n=\\sum_{i=1}^n \\log \\Lambda(x_i)=\\frac{\\theta_1-\\theta_0}{\\theta_0 \\theta_1} \\sum_{i=1}^n x_i - n \\log \\frac{\\theta_1}{\\theta_0}", "622dcd7cff85e8f92441e407f11462be": "\\begin{align}\n\\mathbf{U} &= \\begin{bmatrix}\n 0 & 0 & 1 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 \\\\\n 1 & 0 & 0 & 0\n \\end{bmatrix} \\\\\n\n\\boldsymbol{\\Sigma} &= \\begin{bmatrix}\n 4 & 0 & 0 & 0 & 0 \\\\\n 0 & 3 & 0 & 0 & 0 \\\\\n 0 & 0 & \\sqrt{5} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0\n \\end{bmatrix} \\\\\n\n\\mathbf{V}^* &= \\begin{bmatrix}\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n \\sqrt{0.2} & 0 & 0 & 0 & \\sqrt{0.8} \\\\\n 0 & 0 & 0 & 1 & 0 \\\\\n -\\sqrt{0.8} & 0 & 0 & 0 & \\sqrt{0.2}\n \\end{bmatrix}\n\\end{align}", "622de910083e82ace0d173d72c9cc316": "\\scriptstyle r \\omega /c \\gg 1", "622ed0a1a80f2a9684ed38a0b972824f": " \\lambda_n(t) = {n \\choose 2} \\frac{2 \\beta S(t)}{I(t)}", "622f1eacb2bd36db46231e852bf9be3c": "\nds^{2} = dr^{2} + r^{2} d\\theta^{2} + r^{2} \\sin^{2} \\theta d\\varphi^{2} \\,\\!\n", "622f678a260eff06ce8189d1987c00a3": "\\rho_{tch}(x)=\\frac{8}{\\pi}\\sqrt{x(1-x)}", "622f744c22c0024978395fe315a58a74": "\\textstyle K = \\mathbb{Q}(\\sqrt{2})", "6230199e62572e07c5cd426f8558d31d": "b = a \\sqrt{1-e^2},\\,", "623020187ecf74a8ec6ea44a4d9c8c07": "D < 0", "6230532bec2f57a453aca28dc09509f8": "P := \\mathbb{P}(N_M V \\oplus \\mathbb{C}).", "6230644f6cad1d273c017d12971992e3": "\\overline{\\delta}_D(-1,i) = 0\\,\\!", "623088490933d7a7ad1ba5ba8b876769": "X_1, X_2, \\dots\\,", "6230c71b28ef9093de4c7df208f1fae6": "\\sum_{i=1}^n x_i (1 - k_i) = 0", "6230f915b3eaadb97939232ccfd68014": "\\frac1{128}\\int_0^1 x^{16}(1-x)^{16}\\,dx=\\frac1{2\\,538\\,963\\,567\\,360},", "623109c178225ccead203dedbf3f9217": "NP \\cap coNP", "62310d6cbe7a64ac87a86172d0b22a34": "VAS(x^3+x^2-2x-1,\\frac{2x+3}{x+2}) ", "62312b251af68ff96676ef6de8495804": "\\phi_i = \\sum_{j = 1}^n\\frac{Q_j}{4\\pi\\epsilon_0S_j}\\int_{S_j}\\frac{f_j da_j}{R_{ji}}", "62312fcae7f64b26e3c39e41511213dc": "E = E_\\text{exch} + E_\\text{anis} + E_\\text{Z} + E_\\text{demag}", "62314c7493f1f3802fdf56b2829d7720": "\nH = \\sum_i \\hbar\\omega_i \\Big(a^\\dagger(i) a(i) +\\tfrac{1}{2} \\Big).\n", "623151b6e196f47c9ba708fa97b1b282": "f^{(k)}(x_0) \\neq 0", "62315c719fa935f014043d0bac2ce36d": "\\bar{\\Delta} \\cong \\sigma_-\\Delta^*.", "6231603dece403bafef38445049e7f81": "{O}(n \\log^{k-1}n)", "62318b1d5e86c9a604170031e8559a9c": "\\int_{0}^{p} V_{m}(p^\\prime,0)\\mathrm{d}p^\\prime = V_{m0}p", "6231d784e77de2777299119b45490e44": " \\mathcal L = i \\psi_L^\\dagger \\bar\\sigma^\\mu \\partial_\\mu \\psi_L ", "6231e18865aef7c0257085a2837336d8": "(0\\le \\lambda\\le 1)", "6231e30c4ac198f5e137ce2870c89fc9": "\n\\begin{matrix}\n&&&&&1\\\\\n&&&&1&&1\\\\\n&&&1&&2&&1\\\\\n&&1&&3&&3&&1\\\\\n&1&&4&&6&&4&&1\n\\end{matrix}\n", "62321be00dc425851716c81cfed45f74": "g_{\\rm indirect}(r)=\\exp\\{-\\beta[u(r)-w(r)]\\}", "6232549e06568877cb055a10d4acc20d": " \\forall h \\cong 0, \\ |f(x+h) - f(x)| \\cong 0 ", "6232f640dcf38d6a4005ff8878b96595": "\\overset{{{A}'}}{\\mathop{\\left[ \\begin{matrix}\n 2 & 1 & -1 \\\\\n -2 & 2 & 2 \\\\\n -2 & 1 & 3\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_1 \\\\\n b_1 \\\\\n c_1\n\\end{matrix} \\right],\\quad \\text{ }\\overset{{{B}'}}{\\mathop{\\left[ \\begin{matrix}\n 2 & 1 & 1 \\\\\n 2 & -2 & 2 \\\\\n 2 & -1 & 3\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c \\\\\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_2 \\\\\n b_2 \\\\\n c_2\n\\end{matrix} \\right],\\quad \\text{ }\\overset{{{C}'}}{\\mathop{\\left[ \\begin{matrix}\n 2 & -1 & 1 \\\\\n 2 & 2 & 2 \\\\\n 2 & 1 & 3 \\\\\n\\end{matrix} \\right]}} \\left[ \\begin{matrix}\n a \\\\\n b \\\\\n c \\\\\n\\end{matrix} \\right]=\\left[ \\begin{matrix}\n a_3 \\\\\n b_3 \\\\\n c_3\n\\end{matrix} \\right]", "62332b90585db5420b50adf4cce0a0af": "Y_{O,0}", "6233706e41607f529c06771528747b93": "J^*(x) = \\int_c^\\infty dy\\ f(x, \\ y)", "62339d06797c8e9d58d408f98298d3e9": " I = \\hat{U}^{\\dagger} \\hat{U} \\approx \\left ( I - i\\hat{H}^{\\dagger} \\right ) \\left ( I + i\\hat{H} \\right ) \\approx I - i\\hat{H}^{\\dagger} + i\\hat{H}. ", "6233eb7838217c8eca210d2a647f6d32": "\\rho=\\rho_c", "62341a2e241062446e9088efe4e1e223": "\\boldsymbol{u}_g=\\frac{1}{f}\\hat{\\boldsymbol{z}}\\times\\nabla\\phi,", "62341e65911e66bec9bf93c2a47d2732": " r\\theta~\\sin\\theta \\,", "62344c3c6a4cbe8940c397c46b437219": "M_{bol_{\\rm star}} - M_{bol_{\\rm Sun}} = -2.5 \\log_{10} {\\frac{L_{\\rm star}}{L_{\\odot}}}", "62346a52214226667a432238f8baa8ac": " I(X_1;\\ldots;X_n) = -\\sum_{ T \\subseteq \\{1,\\ldots,n\\} }(-1)^{|T|}H(T) ", "6234af8a81ab3107825b285ceb56a307": "u[1] := 2*atan(\\sqrt((a0+b1)/(a0-b1))*tan((1/2)*\\sqrt(a0^2-b1^2)*\\eta))+(1/2)*\\pi", "6234e02d0e9eac03e710e9d06809589c": "\\Pr[p_i = 0] \\leq \\frac{1}{2}\\cdot \\Pr[y = 0] + \\frac{1}{2}\\cdot \\Pr[y \\neq 0]", "6234e3d1324d1f81c2ec620833d13f99": "C = \\frac{\\dot{m}}{K} + \\left(C_{o}-\\frac{\\dot{m}}{K}\\right) e^{-\\frac{K \\cdot t}{V}} \\qquad(2)", "6234ed275b8f8a91c6ff71936b7d228c": "s_k = \\operatorname{tr} ( A^k )\\,.", "6236076309d03fc2040724635987451b": "\\begin{align}\nf(0_A) &= 0_B \\\\\nf(S_A (n)) &= S_B (f (n))\n\\end{align}", "623646423b3c2d881a7740450436580f": "3 + 0", "62365e63ef1c0529908583cce95f2323": "\\frac{\\neg \\neg P}{P}", "62370fd089d1262230bb43616f6afab9": "\\forall u,v \\in \\{1,2,\\dots,n\\}:~a_{iu},a_{iv} \\neq 0, ~~~~ \\sum_{i=1}^n a_{iu}^{-1}\\,a_{iv} = \n \\begin{cases}\n n, & u = v\\\\\n 0, & u \\neq v\n \n \\end{cases}\n", "623718514a7ec69b41fd16cae23fc408": "RevPAR = Rooms Revenue /Rooms Available \\,", "6237abe1522797baf41fdc718b417dec": "\\frac{d H}{d t} = \\nabla H \\cdot \\frac{d \\mathbf r}{d t} = \\nabla H \\cdot \\mathbf f(\\mathbf r, t)", "6237fb211334db845459e9936736e552": "W = \\left(\\frac{\\omega^{jk}}{\\sqrt{N}}\\right)_{j,k=0,\\ldots,N-1} ", "62380416612b55e03e7924b0437dd86c": "k \\geq 0, \\,\\delta_{n}+ \\delta_{n+1}+ \\cdots +\\delta_{n+k} \\in [ \\alpha, \\beta] \\}", "6238e500ab8775ef7a92b78f96ca2186": "\\Delta\\alpha/\\alpha", "62395ca434190d166fe2965dec9ddd9d": "\\langle \\chi_N,\\mu^{(g)}\\rangle = \\langle \\chi_N^{(g)},\\mu^{(g)}\\rangle = \\langle \\chi_N,\\mu \\rangle ", "62396d164be91c402f6b212ae5f22e6b": "\\omega_c = eH/m^*c", "62398d26581867511067cdad33a3a2f9": "\\theta_W \\,\\!", "62399f773a40997349af7ef78cd0426e": "\\mathbb{E}^g", "6239c3bd7ae66d5d4f7e9f8b421b2fcf": "\\lambda_K", "6239c6e275a6b86a158bfb3ff736ded7": " \\mathrm{E}[ ] ", "6239e05d611b78da98475f26d3b20017": "y_T=\\frac{1}{z_T}\\,", "623a531f26230e8ed1abc2c590c2e2a3": "t=\\frac{1}{\\lambda} \\ln (J \\times R+1)", "623b6ea8028e8e3bbf60c211864f63ef": "S_0 \\rightarrow S", "623b9b2f324cdf758ed5a7a4a0fd8e94": "\\langle X \\rangle\\,", "623becdeba978c7387cef05d4c534a32": " \\frac{\\partial \\bar{u_i}}{\\partial t} + \\frac{\\partial \\bar{u_i}\\bar{u_j}}{\\partial x_j}\n= - \\frac{1}{\\rho} \\frac{\\partial \\bar{p}}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 \\bar{u_i}}{\\partial x_j \\partial x_j}\n- \\left(\\overline{ \\frac{\\partial u_iu_j}{\\partial x_j}} - \\frac{\\partial \\bar{u_i}\\bar{u_j}}{\\partial x_j}\\right).\n", "623bf478b0cac3fec66ee3706aa98374": "\\mu = 5.855", "623bf7e75de1e9665a1baf17f3644f59": "Z = \\rho v\\,\\!", "623c00a0d45d443875ef6b2ddb74f292": "dt^2", "623c8c595cab4400224bf4743adf6841": "\\Delta\\theta^j = (\\theta - \\theta_0)^j", "623cb06508b7bc302d26cf33a275d003": " D \\approx D_{cl}t^* / t_c \\quad \n[\\text{assuming }t_c \\gg t^*] ", "623ce4ee9b146516003af19773c3cb41": "dQ", "623d39a5a41e35e20f4a207d6f1cfd79": "\np_{xy}^k =\n\\frac\n{ (\\tau_{xy}^{\\alpha}) (\\eta_{xy}^{\\beta}) }\n{ \\sum_{y\\in \\mathrm{allowed}_y} (\\tau_{xy}^{\\alpha}) (\\eta_{xy}^{\\beta}) }\n", "623d404a19d2d913945b40a5f32e5869": "C_2(G)", "623d66089eebb7dc2ef9ec22408a5bd6": "f^n(\\bot) \\leq f^{n+1}(\\bot)", "623daa666e0a5bed20c4a16fbf44e6a5": "\\{Y_i\\}", "623e1ef41361cc2571e11c906239faa6": " \\Delta r_i\\mathbf{e}_i = \\mathbf{r}_i-\\mathbf{R}, \\quad \\mathbf{v}_i = \\omega \\Delta r_i\\mathbf{t}_i + \\mathbf{V},\\quad i=1,\\dots, n.", "623e3c1a96427439c1d6de8b48fcd484": "N(1-R) \\delta_4\\,", "623e473fb3bed7ab372729d78c699f2d": "\n\\begin{align}\n|f_n(t)-f_m(t)| & {} \\le |f_n(t) - f_n(x_k)| + |f_n(x_k) - f_m(x_k)| + |f_m(x_k) - f_m(t)| \\\\\n& {} < \\varepsilon/3 + \\varepsilon/3 +\\varepsilon/3\n\\end{align}\n", "623e9cc6168ca12db7ec212b9b6adfee": "\\gamma_n(x) = \\frac{1}{n!} x^n", "623eae2ed3afb4d0f8805145c306f750": "\\sqrt{2\\pi}ac", "623edc1e37c5bfc44b0471df7e2fb28f": "\\left\\lfloor \\frac{m}{n} \\right \\rfloor + \\left\\lfloor \\frac{2m}{n} \\right \\rfloor + \\dots + \\left\\lfloor \\frac{(n-1)m}{n} \\right \\rfloor =\n\\left\\lfloor \\frac{n}{m} \\right \\rfloor + \\left\\lfloor \\frac{2n}{m} \\right \\rfloor + \\dots + \\left\\lfloor \\frac{(m-1)n}{m} \\right \\rfloor.\n", "623f0a7dd22a9a463abf70f18c803611": "(3.5) \\,", "623f57e56872bb722dcbf7ac66faba46": "\\|T\\|_{1,\\infty} = \\sup_N\\frac{\\sum_{i=1}^N \\mu_i(T)}{\\log(N)}", "623f92cd8044e8b73ee37686f68a94f7": "\\phi \\, \\Box \\phi = -4 \\pi \\, T_{\\rm matter}", "623fbe36cfcc07cebe62faf4ff45d435": "\nH(\\mathbf{s}) = - \\sum_{i\\neq j} J_{ij}\\; \\mathbf{s}_i\\cdot\\mathbf{s}_j -\\sum_{j} \\mathbf{h}_j\\cdot \\mathbf{s}_j\n=- \\sum_{i\\neq j} J_{ij}\\; \\cos(\\theta_i-\\theta_j) -\\sum_{j} \nh_j\\cos\\theta_j\n", "62400812b03e8fda1548585a6b1ef1a0": "2^{93} + 1 = 3 \\times 3 \\times 529510939 \\times 715827883 \\times 2903110321", "62400909bef6108005c490f89e796c9c": " \\tau_m = \\frac {G} {2 \\pi\\ }. \\, ", "62401aa9559df2dd09b93a33670d9e75": "a = \\frac{2-x}3, ", "6240248900517c203ce72b2abf57b771": "B \\leq_T A ", "624027919b1cca6f7552be724688e282": "f(x) = x^2 + 1", "624041fa0c0db52ea379c4ee713a89f8": "Ax+b\\geq 0", "6240eab9e6dc47fda05bc259ede6b327": "Q(A \\times B) = \\sum_{x \\in A, y \\in B} Q(x,y). ", "6240f0dd9702783141a8ab2f63bdce08": "J_{ij}^{s} = \\cfrac{i}{2}(S_i^+ S_j^- - S_i^- S_j^+) ", "62410f210f0421aa2d7691666d8c32af": "E[Z]_{33} = \\frac{m}{r^3} ", "6241621dd58a419adbd4c2b47f78c2bb": "\\sum_i c_i \\; D_i", "6241df9269755dada666c8c67aa33f8f": "u_h", "6242a8cb185954d5589c1c5703282a0d": "1,g_2,g_3, g_2g_3", "62432af10f5574d71f67af3312eabdbd": "\\frac{24 + 10m}{24 - 7m}", "62433159967656783a8de1f55490e955": "V_\\max / 2", "6243aea0721bebeae1892196a0b0e776": "F={\\mathbb C}", "6243be2ffb234372b967408092ecf923": "(\\mathcal{V}^{\\mathcal{A}})^{op}", "624409364d4b57a15086eeea7a49a3b6": "\\hat{c}", "62441aae76df07e1770c95d56e042e9f": "\\color{Black}\\tfrac{360^\\circ}{n}", "6244265118ae779466973895294eba7b": " c_R^{\\text{Gauss}}(u)\n= \\frac{1}{\\sqrt{\\det{R}}}\\exp\\left(-\\frac{1}{2}\n\\begin{pmatrix}\\Phi^{-1}(u_1)\\\\ \\vdots \\\\ \\Phi^{-1}(u_d)\\end{pmatrix}^T \\cdot\n\\left(R^{-1}-\\mathbf{I}\\right) \\cdot\n\\begin{pmatrix}\\Phi^{-1}(u_1)\\\\ \\vdots \\\\ \\Phi^{-1}(u_d)\\end{pmatrix}\n\\right), ", "624484ef9c1be03c5e13f40b2cb0e3aa": " 0 + \\tfrac{4000}{100} = 40 ", "6244a7b39f898cee4b00021cd198fd35": "(-1)^{\\text{sign}} \\times 2^{\\text{exponent} - \\text{exponent bias}} \\times 1.\\text{mantissa}", "6244ba12b2f45fd56cbe55cf8503f24f": "G/U", "6244dee23d997bc57901a7571f03d5f7": "P(E)=0", "624554a12c97331b474f38f6400e7e5f": " L_2(X) ", "62457a24a8b54dc10119a6f72b7281d3": "\\lambda \\left ( M + \\frac{1}{n+1}\\cdot K\\right ) - \\lambda \\left ( M + \\frac{1}{n}\\cdot K\\right ) = \\alpha (K), ", "624585f8770fa4294966ea173511c7e4": "\\textstyle \\mu_w := 1 / \\sum_{i=1}^\\mu w_i^2 \\approx \\lambda/4", "6245ccd088c5395845d706667a138ad6": "\\text{Cl}_2(-\\theta) = -\\text{Cl}_2(\\theta) ", "624609b263090463ab74ae0f6dfe0962": "u V(x) \\geq 0", "6246d129acd894269dc490c0ac09a712": "\\lambda_{\\epsilon}", "6247192e492bed27e707cf99765cd2ad": "(3-2\\ln2)/6\\approx 0.27", "624779b6cf0cb0204b4c80e930af3950": " =e^{j(k_x x + k_y y)} e^{j k_z z}", "6247f081e0ed4930a75d00a4c51c90dd": "\\begin{matrix}4\\end{matrix}", "6247f5ba64ab620a042fda00e42b80a9": "\\textrm{pK}_{b} = - \\log_{10} (K_{b}) = - \\log_{10} \\left ( \\frac{[\\mbox{O}\\mbox{H}^-][\\mbox{HA}]}{[\\mbox{A}^-]} \\right )", "62480f9e5a8d132f182d3b4fe85a6521": "z_n = \\sum_{i=0}^{M-1} c_i x_{n-i}", "624834b125cc436fab1978a6281a704d": "\\lim_{n \\to \\infty} X_n f = f,\\text{ for every }f \\in C([a,b]). \\, ", "6248575f10a2be1fb741f904eab445e3": " x/\\operatorname{ln}(x)\\!", "624862710bf3835ba028cc428358a6ae": "\\sin[\\arctan(x)]=\\frac{x}{\\sqrt{1+x^2}}", "6248cf36c3c1307e9bb4e8e5be819df2": "\\hat{y} = \\hat{f}(\\mathbf{x}) = \\hat{b} + \\sum_i x_i", "6249425032c04e45be0de48f545685b3": "~ C_{p \\text{ or } V}= \\frac{T}{N}\\left ( {\\partial S\\over \\partial T} \\right )_{p \\text{ or } V} ~", "624966966e2e543170aa496bc84a31e5": "X(t, f)", "6249a0ba0f149ebcd83e7f7843378ecf": "x^*\\in X", "6249e7d9e53ec0d4354d23c58d6fa1c0": "d\\, {\\textbf{F}} = \\textbf{0}", "624a089c89d28329c208f918a5ebbf2c": "\\xi_b = \\frac{\\tan \\alpha}{\\sqrt{H_b / L_0}},", "624a462fb8e4bef49106fde11afe6c32": " s \\to 0 ", "624a7d2991da61ad76058e3247106df6": "k_3(s) = l_0 + u_1 s^1 + u_2 s^2 + l_3 s^3 + l_4 s^4 + u_5 s^5 + \\cdots \\,", "624ab4865ac583cc139b9744eac9e260": "\\frac{7 \\cdot \\pi}{12}", "624ad9aa9b27345fc5aae9110e9112ae": "V_1 = 2.", "624adb5bb37ac611d95415849fc5222f": "I_y\\,\\!(a,b)", "624b0276b7b5e6d6eac75cdff7918caf": "~~~\\leftrightarrow~~~", "624b0e2c57955ae2c16cc3b37997590c": "w=(w_1,w_2,w_3)", "624b27e9bfc18f6ee02871c5a21434ef": "\\begin{align}\nf'(g(f(y))) g'(f(y)) &= 1 \\\\\nf'(y) g'(f(y)) &= 1 \\\\\nf'(y) = \\frac{1}{g'(f(y))}.\n\\end{align}", "624b422065abdda558ae07df85370f38": "\\nabla_\\sigma g^{\\mu\\nu} = 0 ", "624b50f6aef8e37d7792368f6ad3d1a8": "(T,t_0) = (S^1,x_0) \\times (S^1,y_0)", "624b6dc21c971901a620592d330aab42": " \\sigma(X)", "624ba97c27b92e0838bdbd3a55f6b1ee": " \\mathrm{S}_L ", "624bde41a22f86be69333a251929afcf": "\\nabla f(a) = \\left(\\frac{\\partial f}{\\partial x_1}(a), \\ldots, \\frac{\\partial f}{\\partial x_n}(a)\\right).", "624c54021cda44b56c92aa798ebadd9e": "11100", "624ca5d2e7c2d0f08d14cc713c45244f": "k_{z}", "624d0304ee4e7f78a6f93e21c5e6b2be": "\ne'_i = \\begin{cases}\ne_i & \\text{if }\\; B(e_i,e_i)=0 \\\\\ne_i/\\sqrt{B(e_i,e_i)} & \\text{if }\\; B(e_i,e_i) \\neq 0\\\\\n\\end{cases}\n", "624d09fdb4d779cc2d0b8f761f97dfba": "Z_0 = \\mu_0 c_0 = 376.730313 ", "624d1076d0564f20de0c1c9c4d4678ab": "00} (\\lambda + \\rho,\\alpha)\\over \\prod_{\\alpha>0} (\\rho,\\alpha)}.", "6269cf35caf7214817a5c0964861b3dd": " B=\\frac{V_t}{V_d}\\ \\ln(A_x)+\\ln\\left(y'(0)+\\sqrt{{y'(0)}^2+1}\\right) ", "6269e3732812b1244350f756caa731a9": "\\nabla_\\theta \\log\\pi(x|\\theta)", "626a168340a22db42775219ba57a848c": "\\bar{\\delta} = 0", "626a1c6756a882872abe603a9f31eff4": "H_{\\mathrm{dR}}^{1}", "626a5fe809001716c8960f713c0af803": "\\forall x \\in X \\, \\forall y \\in X \\, ( x < y \\, \\lor \\, y < x \\, \\lor \\, x = y ) \\,.", "626a76c7318b0abb6fe9cf14ef8b7378": "L_{2} = \\left(1 - \\alpha - \\beta\\right) \\left(1 - \\alpha - \\beta - \\gamma\\right)", "626aad09e4841f69523de939c3573995": "\\Delta S_{\\text{bath}} +\\Delta S= -\\frac{\\Delta U -T\\Delta S+ W}{T} \\,", "626ad58cfa5f5a55de3b301137d3a39e": "S=\\int d^4x \\left[aX^2-bX\\right]", "626ad9d1e70121fed937d0a560fab2f1": "T(s)=\\frac{3.48s}{13.2s^2+1.32s+0.33}", "626af7d1df00942d64f85bad20eddc71": " \\alpha(x,t)= e^{it(\\omega'_0 k_0-\\omega_0)}\\int_{-\\infty}^\\infty dk \\, A(k) e^{ik(x-\\omega'_0 t)}.", "626b3c02c08236ffece4e7e95054e12d": "\\langle x,y\\rangle=\\sum_jx_jy_j", "626b66e32506042d9d21dc26610b2dcf": " \\chi(\\lambda) = \\left( \\lambda - 8 \\pi \\mu \\right) \\, \\lambda^3 ", "626bd7d06cbff1bdaf55f9ac506e3985": "\\scriptstyle C'_i \\;=\\; P'_i \\,\\oplus\\, 2^i M", "626be28ceaf31fe3f04d58fdf1549009": " E_p = - W_{\\infty r} \\,\\!", "626c22a207c5c96345eebf6e68556d17": "T_{D,0}", "626ca6c89c39a1752aca26a9d12fcdda": "t_2 - t_1", "626cae003ef60291bfeb4765b0a7f4c0": "\n{\\boldsymbol{p}^{t+1}(\\boldsymbol{x})} = \\sum_{ \\boldsymbol{y\\in \\mathcal{N}(x)} }\nw_{\\boldsymbol{xy}} {\\boldsymbol{p}^t (\\boldsymbol{y})}\n", "626cea2ce1962abc444962970b23a9c7": "{dQ_k \\over dt} = F_k (C_{art} - {{Q_k} \\over {P_k V_k}})", "626cf8a261535eac96f60b4c825077ca": "\\bar R= \\frac{1}{2} m(n+1).", "626d0b4c50f3b69de20298a84198cc3c": "I_{o_{\\text{lim}}}=\\frac{V_i\\, D\\, T}{2L}\\left(1-D\\right)", "626d222f24484c6a4bc0ccb301b3cad8": "E(R)", "626d28681715f1bde5fc48a14d3e61fe": "q^*=a-bi-cj-dk", "626d6acb9469719580c0ee934a3890e5": "\\frac{J_{X_t+1}}{X_t} = \\frac{J_{X_t+1}}{X_t+1}\\frac{X_t+1}{X_t} = \\frac{J_{n+1}}{n+1}\\frac{n+1}{n} \\to \\mathbb{E}S_1\\cdot 1 ", "626db741b433c565f75d66fb483556a9": "\\ell_P", "626e21821601a91213cd329709d1c954": "U^{*}", "626e249023c05762121b99436afec133": "|z|<1/(1-q)", "626e2be782e852c57ec4d03899293d26": "\nz = \\frac{r}{c} \\sqrt{\\frac{\\left( \\mu^{2} - c^{2} \\right) \\left( \\nu^{2} - c^{2} \\right)}{\\left( c^{2} - b^{2} \\right)} }\n", "626e63280e3a4230044ba5ca70bcdbd6": "f_K[x(t)|0\\frac{|\\mathcal{N}|-1}{2} ", "62763a06f5ca76db4fc5a938276f39ee": "\\mathbf{F} = \\frac{1}{4\\pi}\\frac{q_1 q_2}{r^2} \\mathbf{\\hat r}", "62767800acee05127d9f75b9894a4a13": " 2\\ln 3 = 3\\ln 2-\\sum_{k\\ge 1}\\frac{(-1)^k}{k8^k}.", "62769e316ea455bbff17007bc0cf5b23": "\\sec x = 1 + \\frac{1}{2} x^2 + \\frac{5}{24} x^4 + \\frac{61}{720} x^6 + \\frac{277}{8064} x^8 + \\cdots", "6276a69751cab82550ead80013d2ea41": " \\mu = \\frac {\\pi } {3} \\frac {w}{L} \n \\left( {\\frac {d}{L}} \\right)^{3} n \\left( n-1 \\right) \n \\left( n-2 \\right) ", "6276f11ff03e21838f7d897acfd36e30": "P(T) \\approx 10^{-10.4}~T^{-2} ~~~~~~(10^{5.4} < T < 10^{5.75} K) ", "627704222a58db93aa7496069d738352": "P(q^c)=c", "62773807dd31a41a4d12e63708e7ebae": "\\mathrm{d}\\, {*\\mathrm{d}\\mathbf{A}} = \\mathbf{J} .", "627750e43cd398a5965a8196d7aeecda": "S'\\leq S-1 ", "62778e9427aec9c304828da088a859fa": "\\tfrac{4n}{2n-1}", "6277ba31346a2bba9c8c87e31fd75ab7": "H^i(V,\\mathbb{Z}_\\ell) = \\varprojlim H^i(V, \\mathbb{Z}/\\ell^k\\mathbb{Z})", "62782e5606267dccd5634594b5fc103e": "E_{vib} = \\left(n+\\frac{1}{2} \\right)\\hbar \\omega \\ \\ \\ \\ \\ n=0,1,2,.... \\,", "62784531697307af2c31b6cc18659b38": "R(\\bar{\\beta}) - M\\beta^2 \\left( \\frac{\\partial^2 f}{\\partial c^2} + 2 K \\beta^2 \\right) ", "6278793d4196028a5dda1ada39189d2a": "-\\left(\\tfrac{1}{\\rho_0 c^2}\\right)^2 \\tfrac{P^2}{V}", "627879967ccabd8f1198471c710997ea": "R = \\rho \\ell /A ", "6278e6f5e2f82b8147c07dcbb46aaf20": "\\begin{align}\n\\nabla^2_{norm} L(x, y; t) &\\approx \\frac{t}{\\Delta t} \\left( L(x, y; t+\\Delta t) - L(x, y; t-\\Delta t) \\right) \n\\end{align}", "6279057fc09a0eb9cfb20e4ac3608b8b": " k \\to \\infty ", "627913b0d290c2eb8d528bfd6e5fac5a": "\\int \\frac{dx}{x^{2^n} + 1} = \\sum_{k=1}^{2^{n-1}} \\left \\{ \\frac{1}{2^{n-1}} \\left [ \\sin \\left(\\frac{(2k -1) \\pi}{2^n}\\right) \\arctan\\left[\\left(x - \\cos \\left(\\frac{(2k -1) \\pi}{2^n} \\right) \\right ) \\csc \\left(\\frac{(2k -1) \\pi}{2^n} \\right) \\right] \\right] - \\frac{1}{2^n} \\left [ \\cos \\left(\\frac{(2k -1) \\pi}{2^n} \\right) \\ln \\left | x^2 - 2 x \\cos \\left(\\frac{(2k -1) \\pi}{2^n} \\right) + 1 \\right | \\right ] \\right \\} + C ", "627918151706b1749efe4013cc7c0399": "\\text{M} + \\text{IE}_\\text{M} \\to \\text{M}^+ + \\text{e}^-", "627937202b0dd33afdd0bf82c95b4365": " C_J = \\begin{matrix}\\frac {dQ_J}{dV_Q}\\end{matrix}", "6279f7ada415de895e666cc6299ab6f6": " \\hat{\\beta} = \\underset{\\beta}{\\operatorname{argmin}} (\\| y-X \\beta \\|^2 + \\lambda_2 \\|\\beta\\|^2 + \\lambda_1 \\|\\beta\\|_1) .", "627a05af14eccc2275d649db516ce467": "\n\\ln \\gamma_2 = \\left( \\ln \\gamma_2^\\infty + 2 \\left( \\ln \\gamma_1^\\infty - \\ln \\gamma_2^\\infty \\right) \\Phi_2 \\right) \\Phi_1^2 \n", "627a0f40601ca1d3c5a869b6eb2bcbca": "\nm\\ddot{x} = - \\frac{\\partial V}{\\partial x} + \\frac{q B}{c}\\dot{y}\n", "627a1f9ceb13116e9c3b9fcd1609e42d": "c_0\\,", "627a584f0f123ad26ca93a731c220448": "\\lim_{n \\to \\infty} a_n = A", "627ab23e435f67f399eaaccdfb615cc7": "\\mathbf \\theta (x)= \\int \\frac{M(x)}{EI}= -\\frac{5}{3} x^3 + \\frac{75}{2} x^2 -1406.25(\\frac{m}{m})", "627ab282c22257e7fc844c39ddb155e8": "\\Delta P = \\rho g \\Delta H \\,", "627ace48f5d95a810f9c8149f84d8b5b": " \\hat{\\Phi}(0) = 0", "627b750665b887a94c916c88a135bd66": "\\varphi (x) + \\psi (y) \\leq c(x, y).", "627bc16fb14b34269ebe67291be29393": "\\{ C (\\vec{N}) , C (\\vec{M}) \\} = C (\\mathcal{L}_\\vec{N} \\vec{M})", "627bcb128e8983b2ec05d71a4b9e61c1": "f(a+)", "627bd11f37c6452c9f88c2b25cbaabcb": "n_A \\approx n_B", "627be670769bdb9d5e97531ef15564f3": "so(1,3)_\\mathbb{C} = so (1,3)_\\mathbb{C}^+ + so (1,3)_\\mathbb{C}^- ", "627be93339a25efc521760e01275bd0e": "\\log L(\\theta^{t};\\mathbf{x},\\mathbf{Z})", "627bf9a688d6f00f7a24f9e4595776e9": " e(m,n) = 0 ", "627c40de07e0008e2b0dd7c1c3139594": "\\sum_{i\\in\\Z}a_ib_{k-i},", "627cc0d13954e2de0aa49d3e55ec3b9d": "\\vec{r}_2", "627cea5d2faf3900463f7ec4fed9f909": "(a_i)_{i \\in I}", "627cf4c8a08c0d07443de556e91b27bc": "u_1^q \\ne 1 rem P", "627cf532ad574dabfe39a0a5570c0e9f": "M_{\\pi \\ominus \\sigma} = \\begin{bmatrix} 0 & M_\\pi \\\\ M_\\sigma & 0 \\end{bmatrix}", "627d052810d1d818675f4131132401bd": "T:R^{p|q}\\to R^{r|s}\\,", "627d221d67535ee84ebfd15d9ad28be3": " \\frac{v_0^2}{gh_0} < 1", "627d44dc2ccdbec477e63e87acf4afbd": "f(x) = \\frac{1}{x}", "627d87d1903d8a1aadffb7e56a1999eb": "x\\in L\\iff\\exists y_1\\,\\forall y_2\\dots Qy_k\\,R(x,y_1,\\dots,y_k),", "627d981a4b082e7396585b995ef51925": "i=j", "627dbbacf09851ef0f3dc94cdcbd1c01": "t \\in \\{0,1,...,T-1\\}: \\rho_{t+1}(X) = \\rho_{t+1}(Y) \\Rightarrow \\rho_{t}(X) = \\rho_{t}(Y)", "627deba627f2659603d72afea5f31e5a": "\\textrm{erfc}(z)", "627dfe3da8c871a94e9f209e28cc7121": "M\\times M", "627ee950639dcc2786d18fec77d12de1": "\nL \\propto \\sigma^{15.0} \n", "627f80671fb069c2b24094021a2795f6": " k = \\sqrt{k_x^2 + k_y^2 + k_z^2} ", "627fcdb6cc9a5e16d657ca6cdef0a6bb": "st", "627fd8fed7a4ab0ebf5f07414ce734f2": "\\Phi(k)=\\Phi_0-(k-k_0)", "62806ea1e5e300b9d24ec3f94110af63": "\\operatorname{nassoc}(S, \\overline{S})", "62807ca7af143a7f5760bd628f09a67c": "\\tfrac{1}{2} \\left( 1 - {\\varepsilon \\over 3}\\right)", "6280b8138be7d08dcfbb20b4082c37ae": " \\mathrm{N}(\\alpha\\beta)=\\alpha\\beta\\overline{(\\alpha\\beta)}=\\alpha\\beta\\overline{\\beta}\\overline{\\alpha}=\\alpha \\mathrm{N}(\\beta)\\bar\\alpha=\\alpha\\bar\\alpha \\mathrm{N}(\\beta)= \\mathrm{N}(\\alpha) \\mathrm{N}(\\beta).", "6280bf20ae6d6192b20eee2335257460": "\\forall x \\exist y Lyx", "628100e629726571c2c031d56d782a26": "\\frac{1}{A^n}=\\frac{1}{(n-1)!}\\int^\\infty_0 du \\, u^{n-1}e^{-uA},", "62811d3eba811958c151d34b8d9b10d0": "P = ( lift \\times ( sin(\\beta) -{(L/D)_{\\alpha}} ^{-1} \\times cos(\\beta)) )^{ \\frac{3}{2}} \\times ( \\frac{1}{b})^{\\frac{1}{2}} ", "628121f35a2d4e8d366d92053b07ad4a": "a+10b", "62813e30b388f07a90231699280bf5d8": " g(\\sup(X)) = f(\\sup(X), x_0^{'}))", "62814a82632a4761059e974e1c480e4b": "\\sum_{n=s}^j f(n) + \\sum_{n=j+1}^t f(n) = \\sum_{n=s}^t f(n)", "6281a3b2582a5cfd4a609ee3b4c2d04c": "\\frac{\\overline{x^2}}{2t}=D=\\mu k_BT=\\frac{\\mu RT}{N}=\\frac{RT}{6\\pi\\eta rN}.", "6281bf297934c64a3eb2fd7bb07bd939": " \\Box (p \\rightarrow q) \\rightarrow (\\Box p \\rightarrow \\Box q)", "6281fb27fc1990f2305c0ec15b8a9707": "G_{k, \\sigma} (y)= 1-e^{-y/\\sigma} ", "62822f7d5d833e62049d7e9c0ed3f051": "\\frac{a_1b_1+\\cdots+a_nb_n}{n} \\geq \\frac{a_1+\\cdots+a_n}{n} \\; \\frac{b_1+\\cdots+b_n}{n}.", "6282406f1184a36a6877ed7fc08062b8": "A, B, C...", "62829466580de80d630c4e727b0abccf": "\\omega_2^2 = \\Omega^2(k_2).\\,", "6282a109884dd531e3f50dd95f2cd122": "J(f)\\, = \\partial K(f) =\\partial A_{f}(\\infty)", "6282a4440e035f73656b1c4b3f36daf2": "\\ln(1 + \\sqrt2) + \\sqrt2 = 2.29558714939\\dots", "6282fc1f6045734dfd38ea9e354ab006": "\\mathbf{\\omega'}_1 = \\mathbf{\\omega}_1 - j_r \\mathbf{I}_1^{-1} ( \\mathbf{r}_1 \\times \\mathbf{\\hat{n}} )", "62830ba9b334d6d431ccee44d8649c00": "\\cos\\frac{\\pi}{20}=\\cos 9^\\circ=\\tfrac{1}{8} \\left[\\sqrt2(\\sqrt5+1)+2\\sqrt{5-\\sqrt5}\\right]\\,", "62834f39f1da8f59377fb50efc38cc7e": "k_\\perp=\\pi/a+\\kappa", "628355254ce4fac2cfe68dc507b44340": "\\hat{p}(x)", "628428e0b46334ec948d8703d631d357": "s_{c'}(t) = \\left\\{ \\begin{array}{ll} A e^{2 i \\pi \\left (f_0 \\,+\\, \\frac{\\Delta f}{2T}t\\right) t} &\\mbox{if}\\; -\\frac{T}{2} \\leq t < \\frac{T}{2} \\\\ 0 &\\mbox{otherwise}\\end{array}\\right.", "6284864fa5b77a53cec5338a45937e4d": "QT_B = {QT \\over \\sqrt{RR}}", "628492e962c03bfe202ab3be36b4e9ee": " p_{\\theta} = \\mathbb{P}( q(X) > q^* | \\theta) ", "6284f7aaaba0910b55b66404a9fec98a": "d\\Gamma", "62851316b7df0770dc950a4c562337e1": "P \\lor \\neg P", "62858b37de0c2873445b323f94a79afd": "\\displaystyle \\Phi(\\bullet)=\\pi^{D/2}(q_\\mu^2)^{-zc/2}{\\Gamma(1+cz)\\over cz}.", "6285b1733b983b78ff053aa1df865eb7": "\\mathcal{N} (X) \\cong [X,G/O]", "62863ebfc2312452fd6b07406ec79be0": " \\frac{f(t)}{t} \\ ", "628682f9bbdd99266846372fa154bd2b": "A:H\\to H", "6286a0d9f39c345cf4d008cc9ddffc7e": "\\frac{V}{n}=k", "6286a6023afeda41c070832e0bfcb694": "\\alpha[\\mathbf{f}]\\longrightarrow \\alpha[\\mathbf{f'}]", "6286c3731d8856b0bdffd6cdadaaa726": " \\langle\\mathbf{\\hat X}\\rangle \\rightarrow \\langle\\mathbf{\\hat X}\\rangle + \\varepsilon ", "6286c95e44e94aa0c672c4e5cd2071e8": "\\ln Z_{ij} = \\lim_{n\\to 0}\\dfrac{Z^{n}-1}{n}", "62874b07f30d502231ce71d7a5220c14": "\\tau (\\omega) := \\inf \\{ t \\geq 0 | Y_{t} (\\omega) = 0 \\}", "628753da2016c1fc53479d8f4166c78c": "A, \\neg A\\lor B\\vdash B.", "628793715f36acf20d6cc4adce443bce": " P(X \\ge 1) \\le \\frac{ \\sigma^2 }{ 1 + \\sigma^2 } ", "6287ca286a452084b2212b8cb3d8602e": " s \\leftarrow s+1 ", "6287e920f043f859fff5d4154c221ce3": "\\theta=\\arctan(x)", "6288499ed73769db76f508732a1a8adb": " -e_1", "62885bedcaf8a9684fe1c2d0dcb313ae": "L(x, y; 0) = f(x, y)", "6288c64e8b11d1597cff519aa3d674c5": " R_{wr}, G_{wr}, ", "6288d352882211045a376e5074eed7c7": " \\tau(w) = (3,3,2,1) ", "6288fefda408be430720d41003e89843": "\\scriptstyle 1-2/(9k)", "6289363938d650a438b2c450e2868c57": "{\\mathrm{h}} \\ = \\frac{k 0.14 \\mathrm{Ra}_L^{1/3}} {L} \\, \\quad 2\\times 10^7 \\le \\mathrm{Ra}_L \\le 3\\times 10^{10}", "62895e7fd931b95e8a38b911d7c997ae": "S_{TD} = (1.6 \\times 10^{-6}\\;\\mathrm{m})^2 - \\ ", "62898edfae82770ef57be87e245227f1": "\\delta_0 (U) = 0", "6289a66c159ede3dc586829765cc301a": "F(y)=0\\,\\!", "6289aa5fa16774560a05d835a36f85b9": "f \\in \\mathcal H", "6289b9151f430ea1eb616ec41b7baedd": "(a+b+c+d)(x+y+z)=ax+ay+az+bx+by+bz+cx+cy+cz+dx+dy+dz", "6289cef6fad530f4ac96bad72cffe34c": "T = \\sum_{n=1}^N \\lambda_n \\langle f_n, \\cdot \\rangle g_n\\,.", "628a01cc01c878b518ed8fbb8fb5ed83": "\\mathcal{I}_{\\alpha, a} =\\frac{\\operatorname{E} \\left[\\frac{1-X}{X} \\right ]}{c-a}= \\frac{\\beta}{(\\alpha-1)(c-a)} \\text{ if }\\alpha > 1", "628a46c69368e0ec62d4249e59e78749": "Cone_\\omega(X)\\,", "628a46eccb7bd336eaa3f6cf173ac781": "\\sum_{i\\in I} r_i=\\sup\\left\\lbrace\\sum_{i\\in J} r_i : |J|<\\aleph_0, J\\subseteq I\\right\\rbrace.", "628aa9d0fe768a12352ea4fa9c05a151": "\\hat\\phi \\colon C\\to\\hat D", "628aaa5b24b6a479c0d38417f0b498f9": "(...A(t)...A(t')...)", "628afdbdc61c55117bcc35f08f1843d0": "p_{\\rm ML}(x_{n+1} \\mid x_1, \\ldots, x_n) = \\left( \\frac1{\\overline{x}} \\right) \\exp \\left( - \\frac{x_{n+1}}{\\overline{x}} \\right)", "628b2d0a9c258dfa6e9c26bf2f50050c": " x_j^\\star=\\frac{w \\alpha_j}{p_j},\\quad \\forall j", "628b3e1a67ebdb94c519e5b82e24d92d": "f(x)=\\lim_{n\\rightarrow\\infty} \\cos (\\pi x)^{2n}", "628b5a73e9d22358c33b12437ab750af": "\\begin{matrix} {r \\choose 1}{4 \\choose 2}{r - 1 \\choose 2}{4 \\choose 1}^2{52 - 4r \\choose 1} \\end{matrix}", "628b6a2349c6e1213b64b82ae96dd536": "1-\\sum_{j=5}^{30} f(j) = 1 - I_{0.4}(5, 30-5+1) \\approx 1 - 0.99849 = 0.00151. ", "628b9081942a350378141bf47acc0a4e": "\nC_{3,1/2}=\n\\begin{bmatrix}\nc_1 & c_2 & c_3\\\\\n-c_2 &c_1& -c_4\\\\\n-c_3&c_4&c_1\\\\\n-c_4&-c_3&c_2\\\\\nc_1^* & c_2^*&c_3^*\\\\\n-c_2^* &c_1^*& -c_4^*\\\\\n-c_3^*&c_4^*&c_1^*\\\\\n-c_4^*&-c_3^*&c_2^*\n\\end{bmatrix}\n\\quad\\text{and}\\quad\nC_{3,3/4}=\n\\begin{bmatrix}\nc_1&c_2&\\frac{c_3}{\\sqrt 2}\\\\\n-c_2^*&c_1^*&\\frac{c_3}{\\sqrt 2}\\\\\n\\frac{c_3^*}{\\sqrt 2}&\\frac{c_3^*}{\\sqrt 2}&\\frac{\\left(-c_1-c_1^*+c_2-c_2*\\right)}{2}\\\\\n\\frac{c_3^*}{\\sqrt 2}&-\\frac{c_3^*}{\\sqrt 2}&\\frac{\\left(c_2+c_2^*+c_1-c_1^*\\right)}{2}.\n\\end{bmatrix}\n", "628ba05db8b7d238a18ea155210b6676": "(y_i)_{i\\geq 1}", "628ba6b100eb91cde6bb768c19ddc99b": "\\textbf{P}_{k\\mid k} = \\textrm{cov}(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k})", "628bc7b913f5b303aa7828b08be44936": "\\mathbf{P}: \\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix} \\mapsto \\begin{pmatrix}-x\\\\-y\\\\-z\\end{pmatrix}.", "628bd3287ca15047178762a1c364b8ca": "g=g.e=\\phi(g).e", "628c06c34c203cea445f1c7af0428e02": "\\mathbf{v}=v^a\\mathbf{e}_a=v'^aR\\mathbf{e}_a.", "628c2787a24773e87a1907f83f40c125": "20%u($10,000)+80%u($0) < u($1000) \\, ", "628c681409b1e8843f28c8256522fe0e": "I_{S}", "628cacf5e5b1d1878fc4a8a5db428260": "X \\in \\mathfrak{h}", "628cdfd0f2f71ba0d06d92a704601ff9": "\\mathbb Z_p", "628d08fc59b6c988f0d292cd7164de6b": "E^{(2)} - E^{(1)} = \\left(\\phi^{(2)} - \\phi^{(S)} - \\frac {\\mu_j^{(2)}} {z_j F}\\right) - \\left(\\phi^{(1)} - \\phi^{(S)} - \\frac {\\mu_j^{(1)}} {z_j F}\\right) = \n\\left(\\phi^{(2)} - \\phi^{(1)}\\right) - \\left(\\frac {\\mu_j^{(2)} - \\mu_j^{(1)}} {z_j F}\\right)\n", "628d75a9ec8fa448dbfd309e324ea4e8": "\\begin{matrix} {1 \\choose 1}{3 \\choose 2} \\end{matrix}", "628d8efadd1f6637742218c30eb3ed5d": "\n2.~~~~d\\sigma^{2}=\\gamma_{ij} dx^{i} dx^{j}\n", "628ddab8df020d9ff9c7f6236c02d01f": "L = \\Delta x", "628e5921a0018c8324aaf6ae0faca6f5": "\\sum_{n=1}^\\infty \\frac{1}{(n+2)^a}\\sum_{n=1}^\\infty \\frac{\\bar{H}_n^{(c)}}{(n+1)^b}=\\zeta(a,b,\\bar{c}) ", "628e65abd44ac9e99f654f2f92721c10": "2^\\sqrt 2", "628ebcef39508cf679824c0caaf263af": "\\psi\\left(x,z,t\\right)=e^{i\\alpha\\left(x-ct\\right)}\\Psi\\left(z\\right),\\,", "628ec44e68fe6fff0347f654fd9c3979": "\\{r_1x_1+\\dots+r_nx_n \\mid n\\in\\mathbb{N}, r_i\\in R, x_i\\in X\\}.\\,", "628ec6892ad002f1ad24a7083849f2bb": "f(t)=at+b", "628ed3b9798d081fe9509e9092c3197d": "\\int |\\psi|^2~da \\approx A +2~\\mathrm{Re}\\left(\\frac{f(0)}{z}\\int_{-\\infty}^{\\infty} e^{ikx^2/2z}dx\\int_{-\\infty}^{\\infty} e^{iky^2/2z}dy\\right)", "628f3747e2df349db7b34f9edb1db13b": "\\Sigma^2 = (\\mathbf{A \\cdot A })(\\mathbf{B \\cdot B })-(\\mathbf{A \\cdot B })(\\mathbf{B \\cdot A })=\\Gamma(\\mathbf A,\\ \\mathbf B ) \\ , ", "628fb1a37373bcd22b210df64d735e3d": " \\varepsilon_m = \\begin{cases} 1 & m\\equiv 1\\mod 4 \\\\ i & m\\equiv 3\\mod 4 \\end{cases}", "628fc4f691390ae54ab5989eaddbdbbe": " \n\\boldsymbol{\\sigma} = \\mathsf{C} \\boldsymbol{\\varepsilon} \\ ,\n", "628fc5aba803e5d86cc4966aad4ee4a9": "s^2-v^2=a^2(-2+2\\cosh \\tfrac{x_2-x_1}{a})=4a^2\\sinh^2 \\tfrac{h}{2a},\\,", "628ff1ac7499c3e0c103ec52529c1c5c": "\\Pr(\\mu-2\\sigma \\le x \\le \\mu+2\\sigma)\n = \\Phi(2) - \\Phi(-2) \n \\approx 0.9772 - (1 - 0.9772) \n \\approx 0.9545\n", "629086f1b22011f883b59529998a566a": "\\sin (2\\pi(xy+\\sigma))", "6290d3198c9c9dc707e5bf9860014fa4": "\\begin{align}\n\\tau_\\mathrm{n}^2 &= \\left( T^{(n)} \\right)^2-\\sigma_\\mathrm{n}^2 \\\\\n&=\\sigma_1^2n_1^2+\\sigma_2^2n_2^2+\\sigma_3^2n_3^2-\\left(\\sigma_1n_1^2+\\sigma_2n_2^2+\\sigma_3n_3^2\\right)^2 \\\\\n&=(\\sigma_1^2-\\sigma_2^2)n_1^2+(\\sigma_2^2-\\sigma_3^2)n_2^2+\\sigma_3^2-\\left[\\left(\\sigma_1-\\sigma_3\\right)n_1^2+\\left(\\sigma_2-\\sigma_2\\right)n_2^2+\\sigma_3\\right]^2 \\\\\n&= (\\sigma_1-\\sigma_2)^2n_1^2n_2^2+(\\sigma_2-\\sigma_3)^2n_2^2n_3^2+(\\sigma_1-\\sigma_3)^2n_1^2n_3^2 \\\\\n\\end{align}\n\\,\\!", "6290ec192bda820a64a95a5c744c6a3d": " \\mathrm{Du} = \\frac{\\kappa^{\\sigma}}{{\\Kappa_m} a}", "6290f39a154a09255fdd3220b047d506": "x_j\\,\\!", "62913487793d95cfbb874eddf96e7137": " \\left\\langle\\mathbf{e}_+ , \\mathbf{e}_{+} \\right\\rangle = \\left\\langle\\mathbf{e}_{-} , \\mathbf{e}_{-} \\right\\rangle = \\left\\langle\\mathbf{e}_0 , \\mathbf{e}_0 \\right\\rangle = 1 ", "629134db902b266368eb79f28d76678e": "\\delta N", "62917ffc981d59ba238bcd743cd98f73": "\\Omega(k^{1/2}/\\log^3 k)", "629192821aab3b8c9ac4fa3276320318": "R T^n = P_H S U^n \\vert_H \\; \\forall n \\geq 0,", "62919e0d69d891a8c2dad6c670c8c398": " H = \\begin{bmatrix} H_{11} & H_{12} \\\\ H_{12}^\\ast & H_{22} \\end{bmatrix} ", "6291d4afe9d99b6d99fff7e8c2f3d802": "(t-c_1)^n_1(t-c_2)^n_2\\cdots(t-c_k)^n_k,", "6291e0756e88a70169f0b832b9309d14": "\\{ w=(w_1, w_2, \\dots, w_n) \\in {\\mathbf{C}}^n \\mid \\vert z_k - w_k \\vert < r_k, \\mbox{ for all } k = 1,\\dots,n \\}.", "6291e4da429d9d412f33c08a9e63033f": "\\ n\\,! \\sim C\\, \\sqrt{n}\\left(\\frac{n}{\\mathrm{e}}\\right)^n", "6291e5aae9fed08a5b10c8336c2160ba": "-\\frac1n\\log a(n,k,X)", "6291f9fb3afd707d378e618f27b6491e": "\\begin{bmatrix} \\ln \\dfrac{p_1}{p_k} \\\\[10pt] \\vdots \\\\[5pt] \\ln \\dfrac{p_{k-1}}{p_k} \\\\[15pt] 0 \\end{bmatrix} =", "62922133cffa09d143e95184f5fdab27": " \\begin{align}\nk_1 &= f(t_n, y_n) \\\\\nk_2 &= f(t_n + \\tfrac12 h_n, y_n + \\tfrac12 h k_1) \\\\\nk_3 &= f(t_n + \\tfrac34 h_n, y_n + \\tfrac34 h k_2) \\\\\ny_{n+1} &= y_n + \\tfrac29 h k_1 + \\tfrac13 h k_2 + \\tfrac49 h k_3 \\\\\nk_4 &= f(t_n + h_n, y_{n+1}) \\\\\nz_{n+1} &= y_n + \\tfrac7{24} h k_1 + \\tfrac14 h k_2 + \\tfrac13 h k_3 + \\tfrac18 h k_4.\n\\end{align} ", "6292957d1dd5383c35ac81ca868ba0a8": "P_0 = 0", "6292a73c84809fc41f87c14207bc4b76": "\\tfrac13\\, \\left( \\tilde{\\zeta}' \\right)^2 \\approx -\\zeta^3 + 2\\, \\tilde{R}\\, \\tilde{\\zeta}^2 - 2\\, \\tilde{S}\\, \\tilde{\\zeta} + 1,", "6292c2c690956c1405b435d2abfc9c1b": "[\\ A]", "6292dd45e87a011c4aa10d1fbb40db9a": "y_u=a \\, \\frac {vf} {v-f} +b", "6293be0f287f3a36f8375f27c359485d": "\\begin{bmatrix} 0&1\\\\ 0&0\\end{bmatrix}", "6293de7a8db8a8e61207a0287c81d82b": "P = \\frac{\\Delta P Q}{\\eta}", "629400a974903336aa9ec91c17778dfb": " \\eta = \\frac{4}{3} \\left( 4\\sqrt{2} - 5 \\right) ", "629438171c422fe54f37657451846fcb": " \\alpha_x = \\alpha_y \\ \\stackrel{\\mathrm{def}}{=}\\ \\alpha ", "6294a11ff4c5ce6d2d9b98336b42baca": "(\\!|f|\\!)", "6294f60f74a0d2222d0f48c7b7a3fa0d": "r_D=\\frac {n \\cdot V_T}{I_Q}", "629554856fdf5a856f52151111d93cc6": "- [F , G]^{IJ} = [*F,*G]^{IJ} .", "629555933249f07d9ef3ec1daa74c369": "\\nabla \\cdot \\mathbf{E} = \\frac{\\rho}{\\mathcal{E}_0}.", "62957d0aa9614e0c63092dab3db3fb3e": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ \\pm4\\right)", "6295df3f53eb1d5edcbd5e7f70d2d45e": "\\phi:H_1\\times H_2\\to C", "6295e6a6a149697b27678fa63f697aa6": "\\rho=5, \\ \\theta=20^{\\circ}, \\ \\phi=45^{\\circ}", "62967ba54e56dbe7f876bee83c66d902": "\\omega\\in\\R", "62967e0409aad0267e753238111d22a7": "\nx(t) = (A+Bt)\\,e^{-\\omega_0 t} \\,\n", "6296d48808ee05568c1d11d8989c00bd": "\\lambda(n) = (-1)^{\\Omega(n)}", "6296d8280d471ac6f1aec60f27897f0e": " \\delta_h[f](x) = f(x+\\tfrac12h)-f(x-\\tfrac12h). \\ ", "629702a9f2a0a0ef67d1c42c3d703799": "\\text{hex } 3b2", "6297220b980f244a5b9bf77223caf85b": " | e_{\\phi^*} - e_\\phi | \\le \\delta_\\phi . ", "62975a7d90f99bf7aef6b7103bb591c4": "f(x) \\neq f(y).", "629761150eecd0cc48ef2688cca0c574": "(K_1,\\ldots,K_c) \\in \\mathbb{N}^c", "6297ac9ab73a5fae2fddb4b56a118492": "\\sup_{Q\\subseteq\\mathbf{R}^{n}}\\frac{1}{|Q|}\\int_{Q}e^{\\frac{|f-f_{Q}|}{A}}\\mathrm{d}x<\\infty.", "6297d194582834864a513fb3bbb63ac9": "t = 3 + 4s", "629806cbe7a7b1518dae1246649f08ea": "p_2/p_1", "6298179364d84764ef1a6aa4f64e3714": "I_x(q_i),I_y(q_i),I_t(q_i)", "629825b0604f606dc77ed8e609209ca6": "AB \\longrightarrow A + B", "6298508c548df4693dfea0c4bae7aa60": " f(z) = \\frac1{\\sin(1/z)} ", "6298587542a9352bf5399c3353c2730d": "V\\ ", "62988603b0d13342d515c82e41ade1c1": " U_{i+1}(v) U_i(u+v) U_{i+1}(u) = U_i(u) U_{i+1}(u+v) U_i(v)\\,\\!", "6298aba357d588ae748e78748023082a": "G(\\bold{v})", "6298dbc87e1aceb95bf2282cf645156a": "y' \\in \\mathbb{R}^m", "62992ad646d0a220b309845c94c677a9": "\\Delta_2(k_\\lambda) = k_\\lambda \\otimes k_\\lambda", "62997adc6ba3dcc2c80bec4f9bed2d4f": " n(n+1)~r^{-n-2}~\\cos(n\\theta) \\,", "6299c4961b149a7f356788d7ec9bdb76": "p^u", "6299cbab29e6541e35c6a70bc799d67b": "\\frac{p_{n}(x)}{x-x_{i}} = a_{n}x^{n-1} + s(x)", "6299e98183fa9592ed95fc3d75bc8e70": "\\lambda = fS,", "629a173cfd24f44b13bd5892caa38d59": "\n\\omega_{12} = \\lambda_{12} + f\\sin\\alpha_0 I(\\sigma_1, \\sigma_2; \\alpha_0).\n", "629a224c00b5ce561ed155f35511502f": "\\gamma = \\frac{D}{H}", "629aaf0a8ddd1092ca371c4fe9c87b7a": "\\omega_{rad}\\,", "629ad9b1abe116e5ee6f77714e2b069f": " B_{0}\\neq B ", "629b1a18b3ce09cf6279b6ef190efb59": "C_M(O_2(M)) \\le O_2(M)", "629b553175aacf4b17d16c861d579a7d": "z = z + s_i (\\alpha_i - \\beta_i)\\,\\!", "629ba19ea783c72707c3111f94f68594": " L_{n-\\ell-1}^{2\\ell+1}(\\rho) ", "629bbc2d32d484d1504cfc621e43621e": "\\zeta=\\psi+i\\lambda", "629c2267804e4b3513cf3b7782dbe328": "\\frac{1}{1 - z + {\\scriptstyle\\frac{1}{2}}z^2 - {\\scriptstyle\\frac{1}{6}}z^3}", "629c67235ce960bbda345adf6145e4ad": "C_n(\\varphi)=-\\frac{4\\sqrt{2n+1}}{n(n+1)}", "629cea1b8b44e17d0626a05e991caff1": "(x+1)", "629d00b8469fdc594d77d8d7c2736e33": "f=\\sum_{i=1}^n\\alpha_i 1_{A_i}", "629d172a7e0bce097bf36bb7f969c16d": " z = f(x,y), \\quad \\vec r(x,y) = (x, y, f(x,y)).", "629d18b55337098260cb6192de60fd6c": "I_1,\\dots, I_{N-1} ", "629d2ee26eb76c142dc5b210fcbdbca2": "n=0,3,6,12,24,48...", "629d64d3360372f670bc2a3502475750": "\\mathbf{I}(t)", "629d8ca42a1e660bd0f55965b4e6eb0f": "a_1 = 0, \\; 27 \\, a_3^2 + 8 a_2^3 = 0, \\; 12 \\, a_4 + a_2^2 = 0", "629da5a1f3bb1138d1621242f56def33": "\\ln(\\zeta_M(s+it,0))\\;", "629daca194efc19651981d497fa273d6": "+ p_1 ( 1 - p_2 ) [ N(1-R) \\delta_2 - \\frac{Nc}{4} \\delta_1 ]", "629db11eb51254cdf1bcfd0ca425a1bb": "E[(y_i - g_i)^2] = E[(y_i - f_i + f_i - g_i)^2]", "629df2cdbecf81e739fb83b6b858b12d": "\\lambda = \\nu S P_s", "629e00189fd9286e09014ff8e9606c31": " \\mathcal{P} ", "629e42b53041e1bf5002337225dcbdae": "P(t)=\\frac{1}{B(\\alpha ,\\beta )}t^{\\alpha -1}\\cdot (1-t)^{\\beta -1},\\quad 1\\leq \\alpha ,\\beta \\leq +\\infty ", "629ea95e4b57bbd6ef2900a57f79c008": "P\\left(x\\right)", "629ebecd1a36b5d998dcbfac7ea5fce0": "x^4=1\\ ", "629f038b99df766c805d90d885807d9c": "H_{\\frac{1}{a}, 3} = \\frac{1}{2a}\\left(2\\cdot3\\zeta(4)-\\frac{3\\cdot4}{a}\\zeta(5)+\\frac{4\\cdot5}{a^2}\\zeta(6)-\\frac{5\\cdot6}{a^3}\\zeta(7)+\\cdots\\right).", "629f170f68af36d8946f0379fd3ea8ea": "1 \\times \\sqrt{2} \\times \\sqrt{2}", "629f289561d24cf122590f1e90d7b150": "-\\tfrac 3 2\\pi<\\arg z\\le\\tfrac 1 2\\pi", "629f31085339a4f4d233ef2e2a626ecc": "(\\mu_{(1)}(t), \\cdots, \\mu_{(n)}(t))", "629f4d007123bfcfa47bc8568b23950a": " 2 < x \\le 3 ", "629f5f8f6dae33230b1d8a0a225f19f5": "w\\Vdash(A\\lor B)[e]", "629f9869b7662c1e17fda51fe65ab39d": "I^2 = J^2 = K^2 = IJK = -1.\\,", "629fd7521f24c203e7400825035b8621": "L = - \\int d^4 x \\sqrt{- det (g)} (- g^{\\mu \\rho} g^{\\nu \\sigma} F_{\\mu \\nu} F_{\\rho \\sigma})", "62a014f2081978b371e0bab273abfc1b": "\\mathbb{Z}/7\\mathbb{Z}", "62a072242e8a7e511a16150611841c96": " E_P\\left [\\exp\\left (\\frac{1}{2}\\int_0^T Y_s^2\\, ds\\right )\\right ] < \\infty. ", "62a099e57c14de9e897dafd0ab660c2e": "n \\mapsto \\omega^n", "62a0e5c8c6c1fedcdec74cd16534f8bc": "-(x-1)^2 x^2 (x+2)^3 (x^2-3) (x^2-4 x-9)", "62a138eb13f0ef7cde6f02357eb2e1d6": "p_{f,g} = \\frac{\\sum_{i=1}^N a_{f,i}^2}{|S_1| |S_2|}", "62a167a3e2fd0d2e2ab3dbb506d98374": "E_i = \\sum_{n=0}^\\infty l_{in}", "62a227571d248e4ad9bf8ea0595d9dda": "(A(1), \\ldots, A(n))", "62a26e19e90c81161092430eac1c147a": "\\lambda_{max} \\cdot T\\ =\\ 0.290\\ \\mathrm{cm \\cdot K}", "62a297de97bf311031d4556018fd0f29": " G(z+1)=\\Gamma(z)\\, G(z) ", "62a29d76c7645ee75127aa9f54d5e0ff": "\\rho (\\vec r)", "62a2c00d2b39cfbb52faaaf328e4349e": "\\textstyle \\delta_\\eta = 0", "62a2fd6b6e254d0aeea42cfccb91e7e5": "\\|f\\|=\\sum_{n=-\\infty}^\\infty |\\hat{f}(n)|,\\,", "62a330c46d7d513d644ee20f3cdfa9da": "1^m+2^3=3^2\\;", "62a3332181ff66ad879999180b2b76f0": "\\dot{V}_1\n= \\dot{V}_x(\\mathbf{x}) + \\frac{1}{2}\\left( 2 e_1 \\dot{e}_1 \\right)\n= \\dot{V}_x(\\mathbf{x}) + e_1 \\dot{e}_1\n= \\dot{V}_x(\\mathbf{x}) + e_1 \\overbrace{v_1}^{\\dot{e}_1}\n= \\overbrace{\\frac{\\partial V_x}{\\partial \\mathbf{x}} \\underbrace{\\dot{\\mathbf{x}}}_{\\text{(i.e., }\\frac{\\operatorname{d}\\mathbf{x}}{\\operatorname{d}t}\\text{)}}}^{\\dot{V}_x\\text{ (i.e.,} \\frac{\\operatorname{d}V_x}{\\operatorname{d}t}\\text{)}} + e_1 v_1\n= \\overbrace{\\frac{\\partial V_x}{\\partial \\mathbf{x}} \\underbrace{\\left( (f_x(\\mathbf{x}) + g_x(\\mathbf{x})u_x(\\mathbf{x})) + g_x(\\mathbf{x}) e_1 \\right)}_{\\dot{\\mathbf{x}}}}^{\\dot{V}_x} + e_1 v_1", "62a3514fb21e9e30fabe08f39dcc71aa": " \\sum_{e} \\int_{S^e} \\delta\\ \\mathbf{u}^T \\mathbf{T}^e \\, dS^e + \\sum_{e} \\int_{V^e} \\delta\\ \\mathbf{u}^T \\mathbf{f}^e \\, dV^e ", "62a36a0f31014829dc475fdc7834d6f2": "ds^{2} = \\left(1 - \\frac{2GM}{rc^{2}}\\right)c^{2}dt^{2} - \\left(1 - \\frac{2GM}{rc^{2}}\\right)^{-1}dr^{2} - r^{2}(\\textrm{sin}^{2}\\theta d\\phi^{2} + d\\theta^{2})", "62a38ccbb38818233418b51a3174a3cc": " \\frac{ K }{ K - 1 } ", "62a3d1f6b4eaafb82603893222ee8a94": "[J_a,J_b]=i\\epsilon_{abc}J_c", "62a3f4476cdeb61fe8ef512baa152552": "= \\arctan \\frac {10}{24}", "62a41c0fd48057376b5982825ca358d3": " \\boldsymbol{ v}_G(t) = \\boldsymbol{\\Omega} \\times \\boldsymbol{ r}_{G/O}.", "62a51281b713103062ba82c69d4c2942": "T^t(b_2),", "62a56ec72948b3116cb0f33220a817bf": "\\displaystyle (s-a)(s-b)=s(s-c)", "62a5759227d1fadb02a76deddecbbb39": "\\mathrm{return}: A \\rarr \\mathrm{M} (A + E) = a \\mapsto \\mathrm{return} (\\mathrm{value}\\,a)", "62a59f9a3d4a54de5c6b98a8eb06ca35": "r_1 = 3", "62a5c0e27625599a0f89002501b13906": "S_{yt}", "62a5cd219e74954fc08b965776c637a6": "K_o(x,y)= \\sum_{\\gamma\\in o}f(x^{-1}\\gamma y) = \\sum_{\\delta\\in \\Gamma_\\gamma\\backslash \\Gamma}f(x^{-1}\\delta^{-1}\\gamma\\delta y)", "62a5de419fb5d04c65305f08cb2f3082": "I_c\\,", "62a658c2e041a1f05838b8d3aca0c9eb": "\n \\begin{align}\n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} & = 0\n \\end{align}\n", "62a6b48e8d50e269a31d3abc7a78ded7": "\\{K_{AB}, A\\}_{K_{BS}}", "62a6fbe35ba04ca314c628264aee43e3": "h(F) := (F \\wedge F) (I \\wedge \\Delta_X) - (I \\wedge \\Delta_Y) (I \\wedge F).", "62a7abfd3cdf19f5423f96a9d3e0c604": "\\sigma=[v_0,v_1,v_2,...,v_n]", "62a7c748fa318a08a346a83b4a3298f0": "\\frac{2\\left(-\\beta t\\right)^{\\!\\!\\frac{\\alpha}{2}}}{\\Gamma(\\alpha)}K_{\\alpha}\\left(\\sqrt{-4\\beta t}\\right)", "62a8714b9b95950baba166cd8f762a0a": " L_3 = g^{-1}(P_1 \\times P_2). ", "62a888bbfbe3d6bb568a20547fd58ab2": "f\\left({az+b\\over cz+d}\\right) = (cz+d)^k f(z)\\,\\!", "62a8b6c4f04ee3184cdade0476448c2b": " LWE_{q, \\Psi \\leq \\alpha} ", "62a98c0aaeb0277c15cf423873c11ccf": "S:=\\{s_1,\\dots,s_m\\}", "62a9d12799b84223fcbb7883313da00b": "\\scriptstyle \\zeta (\\rho_n)=0 ", "62a9e7523e1a12b0256b8b09c0837000": "g \\in \\mathbb{Z}^*_{n^{s+1}}", "62aa4242dad724cf522beeb6c045b507": "\\sigma\\leftrightarrow\\tau", "62aa58a161c2647c0fb608f64d845779": "\\frac{P}{Q}=K\\left(1 - \\frac{Q}{URR}\\right) \\qquad \\mbox{(2)} \\!", "62aaced6e784a6a5b344b43850f98398": "B_{1}", "62ab0f352fa62815f657e057a341cfb1": "\\mu \\alpha . T", "62aba1c80310e0bff57f385a02168fc5": "\\epsilon = {\\mu \\over{a}} {\\left [ {(1-e^2) \\over{{2(1-e)}}^2} - {1 \\over{(1-e)}} \\right ] } = {\\mu \\over{a}} {\\left [ {{(1-e)(1+e)} \\over{{2(1-e)}}^2} - {1 \\over{(1-e)}} \\right ] } = {\\mu \\over{a}} {\\left [ {(1+e) \\over{{2(1-e)}}} - {2 \\over{2(1-e)}} \\right ] } = {\\mu \\over{a}} {\\left [ {{e-1} \\over{2(1-e)}} \\right ] }\\,\\!", "62abdf933f2f9942b05ec0fbafb1cfbc": " \\hat{T}_n = \\frac{\\bold{p}_n\\cdot\\bold{p}_n}{2m_n} ", "62ace64d7a51e66a5fe7202a1264f880": "\\frac{1}{12} + \\frac{1}{24} = \\frac{1}{8}", "62ad6b3f391e96554bb24f53dc62838d": "H(\\mathcal{S}) = -\\sum_i p_i \\sum_j p_i(j) \\sum_k p_{i,j}(k)\\ \\log_2 \\ p_{i,j}(k). \\,\\!", "62ad71a29f93ab0a8b5acce1e0b39427": "E_n={2\\hbar^2 v_n^2\\over m L^2}", "62ade385b33b68d83db6bb33101aa053": "\\langle 110 \\rangle", "62ae26cf9a37de03cb48aec45c92d826": "Q_{i=1 \\cdots 4}", "62ae730d85bebc92d5cb9d85f14447c1": "~\\Phi_2(x) = x+1", "62ae8237f5e62545dde2b61d6cd8e41b": "v_1 \\wedge v_2 \\wedge \\dots \\wedge v_n \\mapsto A v_1 \\wedge A v_2 \\wedge \\dots \\wedge A v_n.", "62aed7286096bc41b4515c137f29d8bb": "\n \\begin{bmatrix} x_1(X_1,X_2,X_3,t) \\\\ x_2(X_1,X_2,X_3,t) \\\\ x_3(X_1,X_2,X_3,t) \\end{bmatrix}\n = \\begin{bmatrix}\n F_{11}(t) & F_{12}(t) & F_{13}(t) \\\\ F_{21}(t) & F_{22}(t) & F_{23}(t) \\\\ F_{31}(t) & F_{32}(t) & F_{33}(t)\n \\end{bmatrix} \\begin{bmatrix} X_1 \\\\ X_2 \\\\ X_3 \\end{bmatrix} +\n \\begin{bmatrix} c_1(t) \\\\ c_2(t) \\\\ c_3(t) \\end{bmatrix}\n ", "62aeee3976241a9164bcce588c62ec3d": "K\\le \\tfrac{1}{4}(a+c)(b+d)", "62af11d8ab18d1a5f2a9d965560c4270": "\\scriptstyle{ 2\n\\begin{bmatrix}\n\\scriptstyle{ Q_{xx}-M_{xx} + Q_{xx} Y_{xx} + Q_{xy} Y_{xy} } & \\scriptstyle{ Q_{xy}-M_{xy} + Q_{xx} Y_{xy} + Q_{xy} Y_{yy} } \\\\\n\\scriptstyle{ Q_{yx}-M_{yx} + Q_{yx} Y_{xx} + Q_{yy} Y_{xy} } & \\scriptstyle{ Q_{yy}-M_{yy} + Q_{yx} Y_{xy} + Q_{yy} Y_{yy} }\n\\end{bmatrix}}", "62affe45a6f6ddbac8b3de5e92cc5f35": "( x(\\tau) e^{-j2 \\pi f \\tau} )", "62b0022e1de5bf7c282ea588aae5614d": "Gr(r,n) = O(n)/(O(r) \\times O(n - r))", "62b0b410d6e456b3c4ad6afb9ee79b38": "\\cos(\\theta_T)", "62b0dad57c5181c5b882f6cb6df377d7": " T_f ", "62b10c5b6ddfc236ad7c8c8f79bcc608": "ST_x", "62b11f3c494c23542c30d66aec11eb39": " r_\\pi + (\\beta_0 + 1) R_\\mathrm{E}\\ ", "62b13020b83ad404573f185471feff58": "{\\mathcal K}(L^2(G))", "62b1314ca614e6b9a150f3632c46fa66": "\\displaystyle\n\\sum\\limits_{m}^{K}i_{m}V_{m}dt=\\sum\\limits_{m,n=1}^{K}i_{m}L_{m,n}di_{n}\n\\overset{!}{=}\\sum\\limits_{n=1}^{K}\\frac{\\partial W\\left( i\\right) }{\\partial i_{n}}di_{n}.", "62b1a4c15a3a31c5ff951e520d33e492": " \\alpha = \\cos\\left(\\frac{\\theta}{2}\\right) ", "62b20c5ca30f67b137d9713c9fc570eb": "(S-I)+(T-G)=(NX)", "62b20fd1bbbdfe52e947f30f5deeadad": "\\gamma t/\\lambda^2.", "62b232ba3878e5163efc84d5814bab2a": "\\lambda^{(1)} = (1),~\\lambda^{(2)},~\\lambda^{(3)},~\\ldots,", "62b2afc9151f5fbc4463dfc2d2de12b5": "\\|x\\|_p = \\left( \\sum_{i=1}^n |x_i|^p \\right) ^{1/p} \\ , ", "62b2d6eba954166dddf4248e89111476": "\\tau_b*", "62b2da1105b66f97fbd8a72beb920ab5": " xy=yx \\leftrightarrow x^{-1}y=yx^{-1}", "62b3005b19c6812c524a3373476a3eb9": "\\frac{3\\cdot\\pi}{2}", "62b330acd0b9665a79b70ab7ca546505": "x = \\ln y", "62b364d6226ba354a1bea949ff62c0fc": "\\le H(W|Y^n) + I(X^n(W);Y^{n})", "62b450e7f46e64e23eac1f75dca6e863": "F(EG,X)^G", "62b452aa831e5ff126f147387522ef5d": "\\scriptstyle H'", "62b463d910020cab2b7735503a17ceb7": " \\langle x(t) \\rangle - \\langle x \\rangle_0 ", "62b489bedcd081667c08aec7ed74a19f": "\\begin{array}{cc} P_{j}(d_{j})=\\left\\{\n \\begin{array}{lll}\n 0 & \\text{if} & |d_{j}| \\leq q_{j} \\\\\n \\\\\n \\frac{|d_{j}|-q_{j}}{p_{j}-q_{j}} & \\text{if} & q_{j}<|d_{j}| \\leq p_{j} \\\\\n\\\\\n 1 & \\text{if} & |d_{j}| > p_{j}\\\\\n \\end{array}\n \\right.\n\\end{array}", "62b4ccc6a07a7e44fda8173e44bc56db": "a_{n}=r a_{n-1}", "62b4e76061e49c016961d07c9bc9351f": "O(n*2^{n/2})", "62b59b848c8ab467c6a984176dac74f6": "\\mathbb{R}^N", "62b5dbdb7dc529ed5542fb20800d4885": "\\mathfrak{so}_{12}(\\mathbf K)", "62b68f4cb59f2cd1f1c71e5fa83c126e": "\nn_i = \\frac{g_i}{e^{\\varepsilon_i/kT}/z-1},\n", "62b69332ea9bad7542c115be36ae5ad5": "P[M=m] = \\binom{N+m}{m}\\left(\\frac{1}{2}\\right)^{N+1+m}", "62b6bd4c74e0fd0a38ae59f4b5a75d48": "\\left\\{\\begin{matrix} n \\\\ n-1 \\end{matrix}\\right\\} =\n{n \\choose 2}. ", "62b6c31d18e58d22287173a75f6a6c63": " \\int_{-\\infty}^b f(x) \\, dx ", "62b6c5fd75d16e8ee3960ab71f61d85b": "\\mu(M)", "62b6d10eb704e70af9c26fa7c050fe28": "a \\oplus b = b \\oplus a ", "62b6d523110c8e916fc8e9448191510b": "AOP_m(x) = x^m + x^{m-1} + \\cdots + x + 1", "62b6ecbcc366b5a13f61304194889553": "\\eta_c = 1 - \\left | \\frac{Q_L}{Q_H} \\right | = 1-\\frac{T_L}{T_H}\\,\\!", "62b7651a898303a2b185a0ccf43ae04b": "\\mathbf{F}_\\mathrm{Cfgl} = - m \\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{x}_B ) =m\\omega_R^2 R \\mathbf{u}_R\\ ,", "62b792deed3dcab8c99e601922299505": "S\\to SS ~|~ (S) ~|~ \\varepsilon", "62b7c031638809f6f625abe9642b65a9": "W^s(f,p) = \\{q\\in X: d(f^n(q),f^n(p))\\to 0 \\mbox { for } n\\to \\infty \\}", "62b8343b3dc033c79528c04df2a34943": " (\\gamma, \\delta)", "62b850c026da32ab71be5ac5ae7b38eb": " z = a_0 + ad + a_2d^2 + \\cdots + a_7 d^7 ", "62b8555eec1093b555329dd1588fecfc": "w_{r}", "62b85c0a2baf218e83f02be8016e875f": "n=0,1,2,\\ldots", "62b8af140ad36fb1aa3e3f4d413e9c98": " a_n>0,a_{n-1}>0,a_{n-3} >0, \\ldots,\\, \\Delta_4>0,\\Delta_2>0", "62b936f75d3a2279c242abb4d54dc9b1": "M(2,1,5)", "62b965a448632a9aa051018076ebbdaa": "\n EI\\dfrac{d^2w}{dx^2} = \\dfrac{Pbx}{L} - P\\langle x-a \\rangle\n", "62b9bd596fdb131e00a7ee347b2130d8": "\\frac{7!27!}{34!}", "62b9d43dcc2f64f02bd746fcfff95775": "45^2", "62b9f9467b95a63fb43baa85469e8c5d": "(V_g)", "62ba715f3d6694ce125847edf3ecc7b5": "C(\\mathcal{N})", "62baede7346c0f669bafc4efbd4f9b5c": "A \\cup \\emptyset = A", "62bc264d68cb18b6d6aef2d922b3481d": " R_u \\le \\phi * R_n ", "62bc3e196ab6a0e8c46c35f2e3a65ec6": "z = \\sin^2 x + \\cos^2 x\\,", "62bc7893ae9e7fe2ab21ac46af063a41": "t \\to t_c", "62bcbd2a76f560be905ebd42b061234a": " \\beta = 1.5 ", "62bce6d0b81dc0a2440faaa333a80fab": "|\\psi\\rangle ^{\\otimes m} = \\sum_{x_{1},x_{2},...,x_{m}}\\sqrt{p(x_{1})p(x_{2})...p(x_{m})}|x_{1A}x_{2A}...x_{mA}\\rangle | x_{1B}x_{2B}...x_{mB}\\rangle\n", "62bd22f2c7b055d76596d1bef66cf893": "\n F(z k;\\,k) \\leq (z e^{1-z})^{k/2}.\n ", "62bd81cd950ab78ba10a1f1d07b8f1c4": "\\mathfrak{r}(\\mathfrak{g}) = \\mathfrak{z}(\\mathfrak{g}).", "62bd9ddeffac8147a2fe43b3a271e374": "r^{-\\ell_2}", "62bda00c58154d3cf1b699b4c4f03b45": "\\omega_{pi}/\\omega_{ci} = 0.137\\,\\mu^{1/2}n_i^{1/2}B^{-1}", "62bdabaff0e3f8a96c477a5e578a6316": "\\frac{-\\nu}{\\sigma^2} \\left[ d_1 d_2 (1 - d_1 d_2) + d_1^2 + d_2^2 \\right]", "62bdbd7af3afa2a9f3fe571b4ca845b3": "H_{so}=(e^2/(4m^2c^2r^3))[\\vec{J}^2-\\vec{L}^2-\\vec{S}^2]", "62be2f78bee7e590f3f72d72e18ddf8f": "K_{sp} = \\frac{{(-1)}^x {(x)}^x {(N_{AxBy(\\Delta)})}^x {(-1)}^y {(y)}^y {(N_{AxBy(\\Delta)})}^y}{V^{(x+y)}}\\,", "62beb576310617ffebbc4ef3f2cc8bf8": "v = 3n-3-n_r", "62bec94f3281650ecd7aef2173a1bc96": " \\pi _T", "62bee98aa7284a9bfc512b32c7e83057": "\\partial U_n/ \\partial r = 0", "62bfae47cc422610538fbb2038b88b8f": "B^{\\lambda N}", "62bffb18cef78e8876ab31ce329496f0": "(x_1, x_2, \\dots, x_N) = (-1, 1, \\dots, 1)", "62c01722512fc9fb10448f7b6c32fc8d": " \\frac kq ", "62c02c1c3f96819f2698e7b77859cf7b": "\\sum |a_k|", "62c04a3c5ff328db9dbf7a4e7ebc9611": "\\bar{\\mathbf{x}} = \\left(\\Sigma_1^{-1} + \\Sigma_2^{-1}\\right)^{-1} \\left(\\Sigma_1^{-1} \\mathbf{x}_1 + \\Sigma_2^{-1} \\mathbf{x}_2\\right)", "62c0516ced2ebc9f095e5127d50ae44a": " Q \\equiv ", "62c0617bb7436128a4cc637804f72654": "| \\gamma' | (t)", "62c0ee7b1ac6a53dd732678fe3c42560": "u_i (s)", "62c1042169675d9d929e45f2505d8fde": "V_i=\\begin{pmatrix} \n1 & 0 & \\cdots & 0 & 0 & 0 & \\cdots & 0 \\\\\n0 & 1 & \\cdots & 0 & 0 & 0 & \\cdots & 0 \\\\\n\\vdots &\\vdots & \\ddots & \\vdots & \\vdots &\\ddots & \\vdots & 0 \\\\\n0 & 0 & \\cdots & 1 & 0 & 0 & \\cdots & 0 \\\\\n0 & 0 & \\cdots & 0 & X^i & X^{i-1}& \\cdots & 1 \n\\end{pmatrix}.", "62c10ba38f9bbf6572bf170aa878a86d": "f(z)={\\sin{z} \\over z(z-1)}", "62c1306e74778af6e8cee6c122452c98": "\\scriptstyle \\mathbb{R}^1", "62c1464046f220dee53f03722ba7fd0e": "\\{e^{-2\\pi \\mathit{i} \\mathit{l}}/\\mathit{m}", "62c1a7e3dadd65c215f5c95c8c27c293": "\\omega _c -\\omega _+", "62c1f2aa68d9106065a4b08c89bfb1e8": "\\ x,", "62c232bad75cead42fe5b3d1c416a3e1": "\\sum_{j=1}^N \\epsilon_{ij} [K_0] \\mathbf{x}_{0j} + [\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} [M_0] \\sum_{j=1}^N \\epsilon_{ij} \\mathbf{x}_{0j} + \\lambda_{0i} [\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i [M_0] \\mathbf{x}_{0i}.", "62c23732e25723cea3a16dc177da05bb": "m_1\\,\\!", "62c251286a60701d4b6f4fc5854e97d6": "x^5 + 2 x^3 y^2 + 9 x y^4", "62c2555e59b1cb257f5eb6104c6fdf65": "\\delta S= \\delta\\int_{\\mathbf{A}}^{\\mathbf{B}} n \\, ds =0 ", "62c283bab5bc4b4b1c834b898b85c178": "\\begin{cases} p_a & \\text{if } u=a; \\\\ p_b & \\text{if } u=b \\\\\n\\frac{1-p_a-p_b}{b-a-1} & \\text{if } a0", "62cd5718700d0ee27de4f13ef42f223f": "E_k\\!", "62ce058d8f79e9549ea46d7f178e5fc1": " \\varphi(A) = \\varphi(A \\cap E) + \\varphi(A \\cap E^c). ", "62ce1cc0d413cba3fff291cdd1bd88ab": "-\\sin(\\theta-\\alpha)=\\cos(\\theta)\\sin(\\alpha)-\\sin(\\theta)\\cos(\\alpha)", "62ce248a65cd559f4a5f13432b04eadf": "\\,\\tfrac{\\partial\\omega_k}{\\partial k}", "62ce5e1892c0cc3e7b306e6dc4b1026d": "S^k\\to X.", "62ce94e1a01448d1e51c030a1a7d8e06": "X(z)=c_0 + c_1(1-z_0z^{-1}) + c_2(1-z_0z^{-1})(1-z_1z^{-1}) + ... + C_{N-1}\\prod_{k=0}^{N-2}(1-z_kz^{-1}),", "62ce9ddfaf8e660143933879bd067f91": "\\xi^{\\mu}", "62cf106ff047701d1cfec6cc1f056d95": "\\sqrt{1000}~\\mathrm{V} \\approx 31.62~\\mathrm{V}", "62cf262c304ccd66e139c7cff0848718": "V_\\mathrm{+} = \\frac{R_2}{R_1+R_2} \\cdot V_\\mathrm{in} + \\frac{R_1}{R_1+R_2} \\cdot V_\\mathrm{s}", "62cf2892cb84816ac9a62b64a72bfced": "s_{i}=1", "62cf6cce12ae63855f9f57a2033bc2f5": "\n\\begin{align}\n\\left\\| y - \\bar{y} \\mathbf{1} \\right\\|^2 &= \\left\\| \\hat{y} - \\bar{y} \\mathbf{1} \\right\\|^2 + \\left\\| \\hat{\\varepsilon} \\right\\|^2, \\quad \\mathbf{1} = (1, 1, \\ldots, 1)^T ,\\\\\n\\sum_{i = 1}^n (y_i - \\bar{y})^2 &= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n (y_i - \\hat{y}_i)^2 ,\\\\\n\\sum_{i = 1}^n (y_i - \\bar{y})^2 &= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n \\hat{\\varepsilon}_i^2 .\\\\\n\\end{align}\n", "62cf8ca297a55d775a60c39c31e53107": " LR+ = \\frac{\\text{sensitivity}}{1 - \\text{specificity}} ", "62cf8f8f58fe52c09a4bfaab4eb72e02": " \\tilde{X}_N ", "62cfbb65c1f1dec37bcd2496f1a620a7": "p^0=E/c", "62cfdf661412c9aa24a1aa5b09e6c212": "AA^g\\bold{b}=\\bold{b}.", "62d003687b2d3b1fe737ebac15463c48": "Y\\ f", "62d082217c9e9fc83c4daf78afab5efd": "\\sum^{k\\ell}_{n=0}p(n,k,\\ell)x^n = {k+\\ell \\choose \\ell}_x. ", "62d0b1f78ed9e88355f2a2a99b1cbd29": "W(1) = 0.567143\\,", "62d0ba9d0579da95b98e56e4587ebd6e": "y = \\sum_{1 \\leq i\\leq k} (\\alpha_iv_i)", "62d0cfc81ef3956d4769615e47b45e79": "\n1\\,", "62d0da517d4f767bbe353a5c39b6e094": "\\Phi_p(x)=\\sum_{i=0}^{p-1}x^i", "62d125020a7d444d0d6860b3c69449d0": "\\displaystyle{R(Q(a)b,b)=R(a,Q(b)a).}", "62d13cc23133553f7ac7d840cfc9ded8": "Q_A = -\\tfrac{5qL}{8}", "62d19e8634f1af7a77f8ca2f625b3e0e": "\\ cdf_y(i) = iK", "62d2165c3199ddd47c0bab6bf4c2f09c": "\n\\delta \\boldsymbol B = \\boldsymbol {\\nabla\\times (\\xi \\times B)}", "62d22656076f6c61f022f970aa9f377d": "g_i(\\sigma)(a) = \\sigma_i(a) + \\text{Gain}_i(\\sigma,a)\\ ", "62d22d9d150e3b8453ac5c9e843b2a8d": "r^{-3}", "62d23cf10db625f89f80ba95b033b00c": " J = \\{ A \\in B(H) : \\mu(A) \\in j \\} . ", "62d26e16afceb1d6029812632360c7aa": "D = \\{ (x,y,z,w) | x^2+y^2\\leq r_1^2,\\ z^2+w^2\\leq r_2^2 \\}", "62d27ba9d177da28938fe40daacdeac2": "CP^1", "62d29d0002941da5a32e8dc8adf6e362": "E[f(\\vec{X})]=B_0.", "62d2aa2e47892024130868e1028de49c": "\\text{if } a \\equiv a' \\pmod 4 \\text{ and } b \\equiv b' \\pmod 4,\\;\n\\bigg(\\frac{a}{b}\\bigg) \\left(\\frac{b}{a}\\right)\n=\\left(\\frac{a'}{b'}\\right) \\left(\\frac{b'}{a'}\\right).\n", "62d2fb25942f2ccafdd073597239681f": "R = \\{ (a_1,\\ldots,a_n ) \\in M^n : \\mathcal{M} \\vDash \\phi(a_1,\\ldots,a_n)\\}.", "62d3367c5267b5d875d1f0444952f160": "A = 2 \\frac{V^2_\\mathrm{S}}{V^2_\\mathrm{P}} \\left ( \\frac{\\Delta \\rho}{\\rho} + 2 \\frac{\\Delta V_\\mathrm{S}}{V_\\mathrm{S}} \\right ) ", "62d355c1d5f09c09056533c9b3e4cbec": "\\log \\pi =\\sum_{n=2}^\\infty \\frac{2(3/2)^n-3}{n}\\left[\\zeta(n)-1\\right]", "62d3aae27afbb310591a4f644cb27923": "\n \\tilde{OI}=\\phi\\left[|m|\\right]\\left[\\left(2v^B-1\\right)\\left(1-\\frac{|m|}{V}\\right)+\\frac{m}{V}\\right]+\\left(1-\\phi\\left[|m|\\right]\\right)\\left(2v^B-1\\right) \\;.\n ", "62d422f5d7728f1128fa87c6b3fbaef5": "f:j\\to k", "62d4253bf2add6343e1c782e9f478a2c": "L_A(\\Omega)", "62d4575487d3faf38aa992014fffd58f": "\n\\sigma_s(n)=\nn^s\n\\zeta(s+1)\n\\left(\n\\frac{c_1(n)}{1^{s+1}}+\n\\frac{c_2(n)}{2^{s+1}}+\n\\frac{c_3(n)}{3^{s+1}}+\n\\dots\n\\right)\n", "62d480ec4c6def7184b3bab2e11d1813": "\\displaystyle z", "62d4c46e87f5b2dceaa04709f13bdfb6": "\\displaystyle{N_\\pm=\\mathbf{N}_\\pm\\cap G_{\\mathbf{C}},\\,\\,\\, T_{\\mathbf{C}} = \\mathbf{T}_{\\mathbf{C}}\\cap G_{\\mathbf{C}}.}", "62d513527d63f4ad47a0ace7ced9f96c": "\\Delta(x)", "62d66a4cf03e5b9e16ad2b0ef7731e9f": "\\ r(x,y) = \\delta(x + \\Delta x, y + \\Delta y)", "62d67840f113a21d068f0b9341e334ac": "d_0\\colon F\\rightarrow G_0(F)", "62d68922872327ca82528713a8e44f59": "\\mathbf{NL \\subseteq SPACE}(\\log^2 n) \\ \\ \\ \\ \\text{equivalently, } \\mathbf{NL \\subseteq L}^2.", "62d6b207a148ab10a05c786b1e993086": "\\operatorname{Id}(B)", "62d6fda83b4b2d34616358acd1e6e467": "(\\bar{3},1,3)", "62d7189c569a9c55a22be0756dcb20ba": "\\begin{align}\\operatorname{arsinh} \\;u + \\operatorname{arcosh} \\;v & = \\operatorname{arsinh} \\left(u v + \\sqrt{(1 + u^2) (v^2 - 1)}\\right) \\\\\n & = \\operatorname{arcosh} \\left(v \\sqrt{1 + u^2} + u \\sqrt{v^2 - 1}\\right) \\end{align}", "62d7197f3b52ebfd100e7514a710eddd": "f(-x) = f(x),\\,\\!", "62d7980f82c6583ded2fb9d29e406b49": "P u", "62d7f219c161d95e50cca08260265b57": "w=f(z)", "62d8bad80cd4bbf8fdba760ee56a5786": "\n \\begin{align}\n \\theta_s & = \\frac{L'_s}{2R'_c} \\\\\n & = \\frac{\\tfrac{1}{\\sqrt{6}}} {2 \\times \\tfrac{3}{\\sqrt{6}}} \\\\\n & = 0.1667 \\ \\mbox{radian} \\\\\n \\end{align}\n", "62d8f46412dc7ac85d74a001a423c92d": "(Q,\\Sigma,\\delta,q_0,F)", "62d8fb87cb0f7d1c5eb5ea17004ad56a": "\\displaystyle \\frac{2\\,\\operatorname{rect}\\left(\\displaystyle\\frac{\\nu}{2} \\right)}{\\sqrt{1 - \\nu^2}}", "62d91952078396bc7013e8f3de9ea2cf": "\\frac{1}{\\sqrt{f}} = -2 \\log_{10} \\left(\\frac{\\varepsilon}{12R_\\mathrm{h}} + \\frac{2.51}{\\mathrm{Re}\\sqrt{f}}\\right).", "62d9edbd078954b3f50b273472cca007": "N=\\{1, 2, \\ldots\\}", "62dace6932807e380b1bc5be142a95d7": "f(x + y) = f(x) + f(y)\\,\\!", "62db60ba2796d15f9cec60a9ad66863c": "s_r \\rightarrow s^{*}_{r}", "62db975aabfd8121ce652e9b9264b6d9": "V(1/t)", "62dbbdb0f4c2e7a205dd396057c92f2d": "\\frac {s - D_{\\mathrm N}} {D_{\\mathrm F} - s}\n\\approx \\frac {f^2 - Ncs} {f^2 + Ncs} = \\frac {H - s} {H + s}\\,.", "62dbcf70862797d7fbfce4d666297eaa": "S^1 \\times W_n", "62dbf2085ceba7a366b6ba06d65e7ef2": "B_i (0,1)", "62dc07fa3f9dec3fa4c65b91b726aaa4": "\na =\\frac{b}{x_3} \n", "62dc201fde028a7e45ca74e357133dd7": "\n\\begin{align}\nQ_i[\\mathcal{L}] & = \\sum_\\alpha m_\\alpha \\dot{x}_\\alpha^i-\\sum_{\\alpha<\\beta}\\partial_i V_{\\alpha\\beta}(\\vec{x}_\\beta-\\vec{x}_\\alpha)(t-t) \\\\\n& = \\sum_\\alpha m_\\alpha \\dot{x}_\\alpha^i.\n\\end{align}\n", "62dc340e9739cd9f5b5966bb6254e76b": "\ns=(\\phi_B-\\phi_A)\\sec\\alpha.\n", "62dc3b43531b21c832b30e72234c0732": "y^2 = x^3 - x", "62dc7e21c691ee48d631607cdeea82bd": "\\forall x \\in y \\; \\psi(x)", "62dca39abdc80817128adcf1b951d6b2": "s_3 = c_2 = \\frac{1}{16}", "62dcba9a1d2852eb2cf0ff59b3d160b7": "|f_{t}(x,t)|\\leq b(t)", "62dccea0ac0301e80df1b818cce9cd9c": "R = A[G]", "62dcd9c051dd1135d49ec702f497771a": "\\text{McCutcheon index}= {\\text{Displacement}\\over \\text{Distance}}.", "62dcef8b4e8288ae656d95db247a2599": "\\scriptstyle{3/2\\sqrt{2}}", "62dd7f2e727990e95e88526b791b9b48": "E\\{Z(x)\\}=\\sum_{k=0}^p \\beta_k f_k(x)", "62dd9d25b072bcfcec9ae0da3fd0d56d": "\\begin{align} Log_e (upper~limit) &= \\mu_{log} + t_{0.975,n-1} \\times\\sqrt{\\frac{n+1}{n}} \\times \\sigma_{log}\\\\ \n&= 1.67 + 2.20\\times\\sqrt{\\frac{13}{12}} \\times 0.079 = 1.85 \\end{align}", "62ddaf89fc23588f1b89e7566a0ba903": "G=(V,E,A)", "62ddf62fd150a6ff8fe20fb0da517acf": "N_r = 10 - 1 = 9, W = |1.5+1.5-3-4-5-6+7+8+9| = 9.", "62de02bd46dfbbe8bad1f84d309dd414": "Pr(X \\geq x |H)", "62de5f2b6faf6aeecf31424115e2b38c": "\n \\sigma^* = \\cfrac{\\sigma}{\\sigma_{\\rm HEL}} ~;~\n p^* = \\cfrac{p}{p_{\\rm HEL}} ~;~~ T^* = \\cfrac{T}{\\sigma_h}\n ", "62dee8a6d76fe51a2c7411900bccf18b": " = \\frac{3}{2}", "62deef77d992e61eebc3943d0b15c495": "\\chi = \\frac{{(\\delta_1-\\delta_2})^2 \\times \\bar{V}}{RT}. \\ ", "62df680b037cbf85c9fadd58e7a1d8ed": " x=v \\cos u,\\quad y=v \\sin u,\\quad z= \\sin nu. ", "62df74337749dce57dba775eaf59fce6": " R(n) =\\operatorname{max}\\left (Z_1, Z_2, \\dots, Z_n \\right )-\n \\operatorname{min}\\left (Z_1, Z_2, \\dots, Z_n \\right ). ", "62dfa5537637a7a989da96d3920e3c73": "\\binom {n-1}{k} - \\binom{n-1}{k-1} = \\frac{n-2k}{n} \\binom{n}{k}.", "62dfc1d87f0f4946a5c557ce60a7d4fe": "||x||", "62dfcf969f6f2a6a62cea5d68f67a242": "c_\\Lambda=0", "62dfd63f68482a41bf15cfe17fc92e7c": "\\,\\mathrm{st}(x)=x_0.", "62e02ed8595e352a2bbb90b48da0f139": "\\mathbf{F}=\\mathbf{AL\\Pi f}", "62e06ef53bbde83f39be78b7530a5d4b": "K(m) = \\int_0^{\\pi/2} \\frac{d \\theta}{\\sqrt{1 - m \\sin^2 \\theta}} ", "62e0f4533d94f602915346951d942a90": "KST = MOV(ROC1,AVG1)*W1 + MOV(ROC2,AVG2)*W2 + MOV(ROC3,AVG3)*W3 + MOV(ROC4,AVG4)*W4", "62e106f48bb44a3a430045ab23fbb0f0": "f'^{-1}(s)", "62e10eac2184f2eea906316df3d2d944": "C_0(G) \\rtimes_{\\rho} G ", "62e12681bb052fc81205766372aed16b": "r_{\\rm vir}", "62e18d0b22da522f703160525f3ac4a1": "e_a", "62e1ac03c9c3bc7216572a00d682310a": "x < y \\ ", "62e1e126c6e86d9d818097be985c18e9": "\\displaystyle{D(f,g)=(\\Delta f,g)}", "62e1ecd46b320d6f0cee3e398e7236fb": "\\hat{p}=\\frac{r-1}{r+k-1}.", "62e235141a945e87b029255394aa2b48": "e^{-ikz}", "62e2357ca0cffde0f9dc15b078b707b9": "D^1\\,.", "62e245d1359555a75086dddb1e756aa1": "\n\\min\\limits_{x\\in X}\\{ g(x) = f(x)+E[Q(x,\\xi)] \\}\n", "62e281035f350a660c00be0add1f73f3": "\\gamma =\\,", "62e2f726bb75d287b7184ea8d2edbc73": "\\frac{\\lambda}{\\lambda-1}", "62e3170efea514c77abbe2f99255956f": "\\alpha(G,\\varepsilon) = \\max \\left\\{ 1-\\frac{\\left|A_{(\\varepsilon D)}\\right|}{|G|}\\,:\\, A\\subseteq G, |A|>|G|/2 \\right\\}. ", "62e322f60bfc2e662f0d33dc5722b7a4": "\\mathrm{Area} = \\frac{\\pi d^2}{4} \\approx 0{.}7854d^2,", "62e3f5c99d60cab8bd887605503e143e": "\\,z=1", "62e4b969cb996d3965653002f6a8405b": "A \\subseteq A'", "62e5d353d8d52a2782af15b018633ec2": "\\exp(\\lambda(z - 1))", "62e6353009d766f80eb37d61f598f2f7": "C = \\frac{U \\times \\dot{V}}{P}", "62e6391218268155d6dbcaab4442e8b3": "\\begin{matrix}a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n&=&b_1\\\\a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n&=&b_2\\\\\\vdots&\\vdots&\\vdots\\\\a_{n1}x_1+a_{n2}x_2+\\cdots+a_{nn}x_n&=&b_n.\\end{matrix}", "62e64f5b125f03aad5f687d71f66db74": "\\scriptstyle \\theta \\;=\\; \\arg (Z)", "62e66d17bd17639fa741af5dcdeafa76": "\\mbox{power} = \\frac{\\mbox{force} \\times \\mbox{linear distance}}{\\mbox{time}}=\\frac{\\left(\\frac{\\mbox{torque}}{\\displaystyle{r}}\\right) \\times (r \\times \\mbox{angular speed} \\times t)} {t} = \\mbox{torque} \\times \\mbox{angular speed}.", "62e684548efd6284ec8f126b9770834b": "G(x) = \\int d\\tau {1\\over (\\sqrt{2\\pi\\tau})^d} e^{- {x^2\\over 4\\tau} -t \\tau}", "62e6b3fa09264b866c4bb580e8bde2ad": " \\oint_C \\frac{\\mathrm{D} \\boldsymbol{u}}{\\mathrm{D}t} \\cdot \\boldsymbol{ds} = \\int_A \\boldsymbol{\\nabla} \\times \\left( -\\frac{1}{\\rho} \\boldsymbol{\\nabla} p + \\boldsymbol{\\nabla} \\Phi \\right) \\cdot \\boldsymbol{n} \\, \\mathrm{d}S = \\int_A \\frac{1}{\\rho^2} \\left( \\boldsymbol{\\nabla} \\rho \\times \\boldsymbol{\\nabla} p \\right) \\cdot \\boldsymbol{n} \\, \\mathrm{d}S = 0. ", "62e6c29722d2ed3598ba12f388e96b15": "\nG(2n,2n,2n) = \\bigl[x_1^{2n}x_2^{2n}x_3^{2n}\\bigl] (x_2 - x_3)^{2n} (x_3 - x_1)^{2n} (x_1 - x_2)^{2n} \\, = \\, \\sum_{k=0}^{2n} (-1)^k \\binom{2n}{k}^3,\n", "62e6c6b1f98e408907d8996af3e1a716": "\\tau = { 2 Q \\over \\omega_0 } = {1 \\over \\zeta \\omega_0} = {1 \\over \\alpha} ", "62e6d476905987c8a63ad7cd9e7a62e7": "\\Lambda_{(\\mathbf N)}^2=C_{KL}N_KN_L\\,\\!", "62e6fbd27aae55240b849446d38c6bef": "bdcaeb", "62e6feac06f63bf89f9270f908ea9499": "\\mathbf{R}\\times(-W\\vec{k})= \\mathbf{r}_1\\times\\mathbf{F}_1+\\mathbf{r}_2\\times\\mathbf{F}_2+\\mathbf{r}_3\\times\\mathbf{F}_3. ", "62e7a0116d80f6ff3c87959517ca81bc": "\\rho a^2", "62e81791b02c8cf96c8c8b70666d371b": "\\geq \\frac{1}{2}", "62e8232e7efdf0b3d9334335a11cc6c8": " 0\\le s_0,t_0 \\le 1 ", "62e8515f1acdf68d7a58ebfccfb519b5": "(Z)", "62e8753a562438ce124da5122bc27622": "H(x,\\xi(x)) = 0", "62e88d1c7e88c13a8ae03ae87de15b78": "A\\mapsto P^{-1} A P", "62e93d4b3c640fb78e857c2ea0677fa4": "S(n)= \\sqrt{\\frac{1}{n} \\sum_{i=1}^{n}\\left ( X_{i} - m \\right )^{2}}. ", "62e95887e2cacb263eadfb56b312dfa1": "a_{n+1} - a_n \\,", "62e98b550ff1cb4fba336f9d6cbcfaef": " \\theta = \\frac{ 1 }{ n } - \\frac{ 1 }{ N } ", "62ea81a0a1c3f0b3b5c8dabd78845022": "\\mathcal{E}(\\mathcal{C}) = \\{ e(x) : x \\in \\mathcal{C}\\}", "62eaac9a6a9a00e4b3e24be35f7a2062": "\\int_0^\\frac{1}{2} -2\\log(2)\\,dx=-\\log(2)\\,", "62eaee554a10f93bcd707e1235ac95ce": "\\varphi (x) = k_0 n L(x)", "62eb0384c72272f38104fbfa19db94e6": "H(\\mathbf{q}, \\mathbf{p})", "62eb2db7e26db25e979ec153ae78a300": "\\int_1^0 \\frac{dt}{t} = -\\infty", "62eb61148e258d9ae980bdaee3c376cf": "> 90^\\circ", "62eba08965c497e54f9bf98245c22a1d": "I,", "62ebdf1f36f4448fe892a41622d57b39": "|\\beta\\rangle", "62ebeb0d9c515de3b2fa71f89a146813": "\\displaystyle{\\mathfrak{g}=\\mathfrak{k} \\oplus \\mathfrak{p}}", "62ec2104a8ab75b2ad60a2a31323e2a5": "L \\in G", "62ec2b2c776be1c1ad9ead2b1383f86c": "\\prod_{i \\in I} X_i = \\{ g : I \\to \\bigcup_{i \\in I} X_i\\ |\\ \\forall i\\ g(i) \\in X_i \\}.", "62ec330d0c8c2d2f0f8cdd29efd90044": "R_{r} = \\frac{2 \\pi}{3} Z_{0} \\left( \\frac{\\ell}{\\lambda}\\right)^{2}", "62ec947ecdf8ff9b8c17f54a7e5d75c9": "p(\\theta|\\alpha)", "62ece6255cf51cec1b567549edb8189e": "V({x})", "62edd95cf548ad979b746a641ec6a18d": "2^n(3 \\cdot 2^{n - 1} - 1)(3 \\cdot 2^n - 1)", "62edfc453dc4ccccb3605364ed0a6c2a": "\\Lambda \\rightarrow \\infty", "62ee0f80eee282e427883fd8d521e976": "D_{2N}=A_{2N}-A_N.", "62ee1ac32fb67e651efdfbe1fb3f0cdf": "a_{n+1} - f_n a_n = g_n", "62ee546dcce97b6cc07db82ede2615ac": "x^4+y^4+z^4=1", "62ee66a26754124fc3ba49efb3be282c": "\\text{torque} = \\text{radius} \\times \\text{force}", "62ee727629a5a05de6bda09caa0bd722": "\\tilde\\phi", "62eea0e957a5f35bea745aa1eae057cf": "S=k\\log W \\,", "62eea9adc412c1547d98336b3da60193": "\\alpha \\approx \\sqrt \\frac{\\omega CR}{2}", "62ef4b049b3e9c12397fba68a22f9d2e": "_2^0\\text{S} + \\text{E} \\underset{\\text{k}_{2(1)}}{\\overset{\\text{k}_{1(1)}}{\\rightleftarrows}}\n\\text{C}_1 \\overset{\\text{k}_{3(1)}}{\\rightarrow} {_2^0} \\text{P} + \\text{E}, ", "62ef706bad82a8b558eee0b7be395a30": "p^*(z)=z^n\\bar p(z^{-1})=z^n\\overline p(\\bar z^{-1})", "62ef8af8f37513b9ab7a26d7391d1072": "\\text{If }S_{i+1} = \\forall, \\quad f_i(a_1,\\dots,a_i) = f_{i+1}(a_1,\\dots,a_i,0) \\cdot f_{i+1}(a_1,\\dots,a_i,1) ", "62ef91184bb32ebb99f350600a85f4ff": "D = 2Eh^3/[3(1-\\nu^2)]", "62efb1af22281b8205e0dd6fe2a4e9bb": "\\text{If }\\psi + \\theta + \\phi = \\tfrac{\\pi}{2} = \\text{quarter circle,}\\, ", "62efe8a00abeb55d4926735bb23d24f0": "\\mathrm {DOF} = 6 N c", "62f0f24266e6d028ff18700cdf7b9009": " k", "62f191ba52ff2730f1dfccaa11edc00a": "\\mathrm{d}E=\\delta Q+\\delta W", "62f1f322cc53024e855b52dc6c67f77b": "\\{x\\in F : w(x)\\geq 0\\}", "62f2115c04c732a382009ca2150b78eb": "N \\ne M", "62f2f0999724811d2eee10f9cb5091ac": "\\scriptstyle{\\int_0^\\infty f(x)\\,dx}", "62f2f7a9b506ade2c94c3d68b8edd27c": "\\|\\theta\\|\\,\\!", "62f345867aece06472a6b5a08c012658": "|\\tau(p)|\\le p^{11/2}.", "62f37bab32a9b19fc66f3be46b589127": "\\phi_2 = N_2 (1s_A - 1s_B),\\,", "62f3aae25d66c62870e5c7bff9e775a1": "p^{k+m}", "62f436fc435c58255553f5d31f71976c": "\\chi_\\nu", "62f4b995708c4d5858a6e5f751360e48": "\\varphi_{h(x)} = \\varphi_{\\varphi_x(x)}", "62f4f658ac86f8373d32aeb0d4f7c7de": "L(G) = \\{ w \\in T^{*} : S \\Rightarrow^{*} w \\}", "62f512934e3d08aebbebf50723fcf909": "E[M]=\\ \\sum_tF_t", "62f54e25729a7cc235813c0b4c7a75f1": "(Z,\\sigma)", "62f5948dc21f41111de0a66eb8855e8e": "t[x := r]", "62f5b333a097ebaf0f1709d2c1b09a11": "\\displaystyle{(D(X),X^2)=0.}", "62f5d4b5d06222cf1d60b925f6c8f019": "\nL^2=\\frac{\\gamma Dt}{4\\eta}", "62f6dc3f643eadadc2aab12548f7c3e4": "A \\to \\alpha", "62f714e87dd07a3a9d1a841f7ce3e63b": "w_X = (u_X \\otimes 1)(1 \\otimes v_X)", "62f760c82b2e9d64c56b191d893d2c76": "\\arctan \\frac{a_1 b_2 + a_2 b_1}{b_1 b_2 - a_1 a_2}", "62f7764f165dceafe53b7d6d3d9304df": "\\mathbf{G} =\\mathrm{d}\\boldsymbol{\\mathcal{A}} \\mp g_s\\,\\boldsymbol{\\mathcal{A}}\\wedge \\boldsymbol{\\mathcal{A}}\\,,", "62f79cea6decd1357c9dde7d2c02e50f": "\\{(\\Gamma, \\phi)\\}\\,", "62f7ce1542b40a92a13ed793cd4011db": "p\\mathbf{Z}[i]\\mbox{ is a prime ideal if }p\\equiv 3 \\,(\\operatorname{mod}\\, 4)", "62f819f6b8841e486f8fe488051e35b4": " y(t+h) \\approx y(t) + hf(t,y(t)) \\qquad\\qquad (2)", "62f85872921eb53c91163e404891f493": "\\mathrm{erf}(x)=\n\\frac{2x}{\\sqrt{\\pi}}\\,_1F_1\\left(\\tfrac12,\\tfrac32,-x^2\\right).", "62f87de35579eb9e9ed22d82fe3735b2": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}e^{2\\pi i (\\ell \\cdot x_{j})}=0.", "62f899608cd47ec2e9bbefb2f1c2c940": "T^m_n(V) \\cong\nL(V^* \\otimes \\dots \\otimes V^* \\otimes V \\otimes \\dots \\otimes V ; \\mathbb{R})\n\\cong L^{m+n}(V^*,\\dots,V^*,V,\\dots,V;\\mathbb{R}).", "62f8c77e4242f0de98ceb291bebfb183": "\\begin{align}\n\\left(a^2 + nb^2\\right)\\left(c^2 + nd^2\\right) & {}= \\left(ac-nbd\\right)^2 + n\\left(ad+bc\\right)^2 & & & (1) \\\\\n & {}= \\left(ac+nbd\\right)^2 + n\\left(ad-bc\\right)^2, & & & (2)\n\\end{align}", "62f8d7db659dbad4337253d50cf31ab7": "\\theta(t)\\mathrm e^{-\\gamma t}\\frac{1}{\\omega}\\sin(\\omega t)", "62f8f28fc8a4548ce6084847dbd82cc5": " A =\n \\begin{bmatrix} 10 & -1 & 0 & 1\\\\\n 0.2 & 8 & 0.2 & 0.2\\\\\n 1 & 1 & 2 & 1\\\\\n -1 & -1 & -1 & -11\\\\\n \\end{bmatrix}.", "62f9470ebecfc132ae226e36a74bfb5c": "r^\\mu - p^\\mu", "62f9be300eb90d7ed28d238287031285": "\nr=\\dot{e}+\\alpha e\n", "62f9d9ab57dc22c514e4b9286188be24": "K(r)=\\sum_{s=1}^m K(r-s)F(s), \\, \\, \\, \\, r=1, 2, \\dots\\,. ", "62fa05f831b14976a742ab329f356af4": "A_F", "62fa14a75e2f8da8f427bdf730095f1c": "m = u - u_{xx} + \\kappa, \\,", "62fa61f2bdef7773554b23ceb8049ea1": "R_{ab}", "62fade5052f4c8b67baf64a7a563869b": "\ny = g(x) = \\sqrt{x}\\ = x^{1/2}\\,\n", "62faea33a5a570ac91aa8c2b3441c43c": "u_f=U\\cos(\\theta-\\alpha)", "62faed513c35ed9c02af66e98a28b941": "E(2\\omega,z=l)=-\\frac{i\\omega d_{\\text{eff}}}{n_{2\\omega}c}E^2(\\omega)\\int_0^l{e^{i\\Delta k z}dz}=-\\frac{i\\omega d_{\\text{eff}}}{n_{2\\omega}c}E^2(\\omega)l\\, \\frac{\\sin{(\\Delta k l/2)}}{\\Delta k l/2}e^{i\\Delta k l/2}", "62fb1c9a1d67ea3f3e7112f0c96101fb": "\\mathbf A\\cdot\\mathbf B = \\|\\mathbf A\\|\\,\\|\\mathbf B\\|\\cos\\theta,", "62fb2c05e7efd4522e7e57b89545cd3d": "Literature \\geq good", "62fb9f430c1fa8baff7c46d36b860f0d": "H_U=-\\int_\\Gamma \\frac{1}{2\\pi}\\ln\\left(\\frac{1}{2\\pi}\\right)\\,d\\theta = \\ln(2\\pi)", "62fbd8f18eec7cdf6dcf1deca7e3969b": "m\\frac{d^2x}{dt^2} + c \\frac{dx}{dt} + kx=0,", "62fd110e77af83751ea6dedc9252b8b0": "\\left(\\tfrac{ab}n\\right)=\\left(\\tfrac an\\right)\\left(\\tfrac bn\\right)", "62fd5605a5c1a4f8729b6d6726fe3a31": "\\cos 36^\\circ=\\frac{\\sqrt5+1}{4}\\,", "62fd6ae321067bb490def6a806568a9f": " \\sum_{1 \\leq i < j \\leq N} \\log \\|x_i - x_j \\|^{-1} ", "62fdeacf4147e496acca72152e8cd055": "1- (1/2)^{k}", "62fdf92cad39ac06f8ff380c6cc3e21e": "R_n^{(2)}={2^n-1\\over{2-1}}={2^n-1}\\qquad\\mbox{for }n\\ge1.", "62fdfeda2c6be6ee1a89252d140cfc7e": "T^{1/p}\\otimes1-1\\otimes T^{1/p}", "62fe514d98546fa5dcc1ecf821256118": "d^2z = r\\,dx^2 + 2s\\,dx\\,dy + t\\,dy^2\\,?", "62fe5fb321ea703d0431a53bdcded2dc": "\\{X_{5},X_{6},X_{7}\\}", "62feb30f9b2b7b2f7f34d111b8786fbf": " f(x) \\equiv \\frac{1}{1 + e^x} ", "62ff011fffe0d91a0cad32c1821376ac": "X_{\\epsilon}", "62ff3cbf347580075abb5017c138d439": "w\\Vdash A[e]", "62ff99ccfea62dfb0a447951bd3479cc": "|f_{2j}\\rangle", "62ffd20d51af4ceae1d7e19df379adf0": "c^2 \\Delta t^2", "62fffffa4e93897f28c8922adb2107e8": "g(t) = \\sum_{n=0}^{N} \\frac{g^{(n)}(0)}{n!}\\,t^n + \\frac{g^{(N+1)}(t^*)}{(N+1)!}\\,t^{N+1}", "63001ac4e0e936fa0649a654b19aa34e": "\\mathfrak{so}(6)\\cong \\mathfrak{su}(4)", "630065b5572147aef903da61b5179ae8": "H\\to G\\to G/H.\\,", "63006aee046c02906fe4c4e3a8d505f4": "\\vec J = \\vec J_c + \\vec J_T + \\vec J_\\sigma + \\vec J_N", "63007720ad5de080ab982ad5739b3b9c": " \\delta", "6300f0748129031a22c39c6468c85da5": "GDP = C + I + G + \\left (X - M \\right ) ", "63013141b58787ceaf38b27679c05856": "( \\pi^\\alpha )", "6301395aead7f17b9ade2195091c9a2f": "\n\n\\dot{\\hat{x}} = \\left[ \\frac{\\partial H(\\hat{x})}{\\partial x}\n\\right]^{-1} M(\\hat{x}) \\operatorname{sgn}(V(t) - H(\\hat{x}))+B(\\hat{x})u.\n", "6301c6b8cfce31a294e93913e203f17f": "\\bold a = \\boldsymbol\\alpha \\times \\bold r ", "6301f0d821a0dc65231892843359179c": "\\frac{\\dot c} {c} = \\frac{r - \\rho} {\\theta} \\,", "63022293118db3a522db7c413b305582": "\\|Ax -b\\|_2 \\ge \\|Az -b\\|_2", "630236f5e1cf935bdf179d9f8d6dc672": "\\frac{P}{K}= \\frac{g_n}{s_c}", "63028e1eeb9644fd5ceaa43977fa0d2a": "M = M^{T} > 0", "6302acab36f8bc570dfc7de51982c95c": "\\rho^*:=\\inf_{r>0}\\{r:IF(x;T;F)=0, |x|>r\\}", "6302b5e10a6e052c987a2a384b7461ed": "(\\exists x \\in \\mathbb{N})\\phi(x)", "6302d6699c779282231eecb73fb755a2": "G_a^\\mu", "63035b9bf9dc5e1f70b72ef3d92c5fc5": "\\operatorname{arccis} \\, x = \\frac{\\ln x}{i} = -i \\ln x = \\operatorname{arg} \\, x \\,", "6303795e131a2b6399844c953baa1914": "\\alpha=\\frac{\\gamma V_o}{2\\pi^2 \\beta^{3/2}}\\,,", "6303a843999c337a7b0bf3932f48a395": " f_r(z)=f(rz).", "6303efc76084176ebe7d601576fdb6df": "\nJ_{\\pm 1} = \\mp \\frac{1}{\\sqrt{2}} \\left( A_{1} \\pm i A_{2} \\right) ~.\n", "6304c778191931685d56ddfa27358068": "\\textstyle {N}(B)", "63052a0b7014b1822b010a771e8ec737": "365^N", "63053b9ea01ed4845bb9c7449696b9c4": "\\omega_g=qB/m", "630550c1321191757789081c8284cb31": "\\varphi_V^{-1}\\circ\\varphi_U (x,v) = \\left (x,g_{UV}(x)v \\right)", "63055baeaedbde62d3941b2c7e994764": "F_{0\\%} = \\left({\\frac{1 + \\sqrt{5}}{2}}\\right)^{-\\infty} = 0 \\,", "63059c003796c8871ac580da8c6355c0": "P(A^{-1}B | T=t) = P^A(B) \\quad \\text{for all }t\\,", "6305c22bdcab48063ce0cebc4f53d6bc": "\n\\epsilon = \\{\\left|\\epsilon\\right|\\cos 2\\phi, \\left|\\epsilon\\right| \\sin 2\\phi\\}\n", "6305dc217177b56f40533e10d0e0972b": "\\scriptstyle \\{x(u-\\tau);\\ u\\}", "63060898b4131d5f4a05b470faca983f": "S^{\\prime} \\subseteq (S^{\\prime})^{\\prime \\prime} = S^{\\prime \\prime \\prime}", "630612b4d8494eaf7a5fd872595bb7aa": "\\mu_*(T)=\\sup\\{\\mu(S):S\\in\\Sigma\\text{ and }S\\subseteq T\\}.", "63065299d4ab151138cc135014f54036": "\\frac{ \\pi^{2} }{ 3 },\\,\\,\\lambda\\,=\\,0", "63067ddfe42393b4c952c259a3e912cc": "F^+\\cap \\overline{F^+} = \\sum_i \\mathbb{Z}c_i", "63068a48fedffc4d30fdef9ba5de2958": "\\forall x. F", "6306ac4b60d94fac57c003fe9882d2a7": "\\begin{align}\n&\\phi(x + y, z + w) = \\phi(x, z) + \\phi(x, w) + \\phi(y, z) + \\phi(y, w)\\\\\n&\\phi(a x, b y) = \\bar a b\\,\\phi(x,y)\\end{align}", "6306b847c79092a1a17f2063ff35a01e": "2 + 5 + 8 + 11 + 14 = \\frac{5(2 + 14)}{2} = \\frac{5 \\times 16}{2} = 40.", "63070baa7956a58ec08158d53a26b79d": "1+z = \\frac{\\lambda_{\\mathrm{obsv}}}{\\lambda_{\\mathrm{emit}}}", "63072ce75b75263e4c743b22c689b517": " [e^3(e + 2)(a + 1)^2 + 1 - o^2]^2 - ", "630757ad13e9a3d8481219d627881e6e": "{\\tilde{F}}_4", "630788a4247532c419a5f5169522219c": "u\\left(Sara\\right) > u\\left(Roger\\right) > u\\left(abstain\\right)", "6307fe81ebd09a9aaa3be51ebbe0e12f": "k_xk_yE_x + \\left(-k_x^2-k_z^2+\\frac{\\omega^2n_y^2}{c^2}\\right)E_y + k_yk_zE_z =0", "63082eabf4af2b15f7a568f151e8f283": "z \\neq 0,\\,", "630830a3751d69a2673fd452d1c5cc7f": "f(g)= M^*F(g)= \\int_{\\mathfrak{a}_+^*} \\tilde{F}(2\\lambda) M^*\\Phi_{2\\lambda}(g) 2^{{\\rm dim}\\, A} |d(2\\lambda)|^2 \\, d\\lambda\n= \\int_{\\mathfrak{a}_+^*} \\tilde{f}(\\lambda) \\varphi_\\lambda(g) \\,\\,b(\\lambda) 2^{{\\rm dim}\\, A} |d(2\\lambda)|^2 \\, d\\lambda.\n", "6308692280589f29d9e4fdbd8fc7fe96": "\\sum_{d | n } J_k(d) = n^k. \\, ", "63088a9ba72f1909c5ff4c72d2850459": " k(n,m) ", "63091f7a69f25dafa83df298427781ec": "p^2/2 -\n\\cos q", "6309277fda508d909b639873aff8879b": "n^2-3n+3", "630957f84ecbcc99a37c5d766556493b": "P(N \\mid n) = \\frac{ P(n \\mid N)P(N) }{P(n)} = \\frac{ 1}{c} \\frac{1}{N \\ln(\\Omega) } \\frac { \\ln(\\Omega) c }{\\ln(\\frac{\\Omega}{ n} ) } = \\frac{1}{N \\ln(\\frac{\\Omega}{ n} ) } ", "63097d5a48f216fce45d6fc5ef0aa452": "\\partial h/\\partial z", "63098453d0720beec7f6b53b07e0fb2d": " \\theta \\approx 0.664 \\sqrt{{\\nu x}\\over u_o}", "63098db735f9596d49568d6b69cf480b": "\\sum_{k=0}^n (-1)^{k} e_k(x_1,\\ldots,x_n) t^k=\\prod_{i=1}^n (1- x_it).", "6309c66f2afeda2813a55fe0cd34677f": " P = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v \\cdot (v_1^2 - v_2^2) = \\rho \\cdot S \\cdot v^2 \\cdot (v_1-v_2) ", "6309caeb7e2df2b3dd49f2857d45cb0e": "\\, Nx^2 + k = y^2\\implies Nm^2x^2-N^2x^2+k(m^2-N) = m^2y^2-Ny^2", "6309d568b62297320fdd7ec75c4b99f6": "x = R\\lambda", "6309f8fbc292ce49eaccc6d943aca723": "\\{g_\\alpha\\} \\equiv \\{g_\\alpha(k)\\}", "630a3682a944da82e6f49e5a74c9d737": "\\psi(\\Omega^2 3 + \\Omega)", "630a4203c3283092bf02c99dd483c271": " X_t = c + \\varphi_1 X_{t-1} + \\varepsilon_t", "630a511a9ea56f9ff866501144433206": "\ny_t=a_2+b_2x_{1t} + c_2x_{2t} + \\varepsilon. \\,\n", "630ae0c6c80691a9d9a434b6773d79ae": "\\sin(2U)", "630ae3259d980fd93c71940e5eb23a48": "\\textit{list}(X)", "630b0ab9b4366dd48f7e19151a3a2a6c": " A = \\begin{bmatrix}\nr_1 & r_2 & r_3 & \\cdots & r_n \\\\\nr_2 & r_3 & r_4 & \\cdots & r_{n+1} \\\\\nr_3 & r_4 & r_5 & \\cdots & r_{n+2} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nr_n & r_{n+1} & r_{n+2} & \\cdots & r_{2n-1}\n\\end{bmatrix}.\n", "630b821a212e3e61fe2513021fa78d01": "F_{grav}=\\frac{GM_{BH}m_p}{r^2}", "630bc25418468439730237b605f3d060": "\\Pi_{i=1}^nX_i", "630c225f449df13c6bb2e8a36e3c77bb": " \\frac{dI}{dt} = \\left( \\beta N - \\gamma\\right) I - \\beta I^2 ", "630c27e36d7e0f6ddf0453839fa6b70a": " |\\nu_{i}(t)\\rangle = e^{ -i ( E_{i} t - \\vec{p}_{i} \\cdot \\vec{x}) }|\\nu_{i}(0)\\rangle,", "630c71a265c80d3dd498825888727575": " f(\\epsilon_{ij}) = \\lambda \\left ( \\sum_{i=1}^{3} \\epsilon_{ii}\\right)^2+2\\mu \\sum_{i=1}^{3} \\sum_{j=1}^{3} \\epsilon_{ij}^2", "630cb2f8d511bee6767c2970fb4afc28": "\\mathbf{v}^{(t)}", "630cbefca24e0726f6f613522b42175b": "H(\\mathbf{X}) = \\mathrm{I}_{\\{\\mathbf{x}\\in A\\}}", "630ce707c95956fb0f1704061ad42274": " K_{W}(t,s) = \\operatorname{Cov}(W_t,W_s) = \\min (s,t). ", "630cf6d10d1f9a5b3e4b768049dfd8b6": "\\mathbf{y}_p = e^{tA}\\begin{bmatrix}\n-{1 \\over 24}e^{3t}(3e^t(4t-1)-16) \\\\ \\\\\n{1 \\over 24}e^{3t}(3e^t(4t+4)-16) \\\\ \\\\\n{1 \\over 24}e^{3t}(3e^t(4t-1)-16)\\end{bmatrix}+\n\\begin{bmatrix}\n 2e^t - 2te^{2t} & -2te^{2t} & 0 \\\\ \\\\\n-2e^t + 2(t+1)e^{2t} & 2(t+1)e^{2t} & 0 \\\\ \\\\\n 2te^{2t} & 2te^{2t} & 2e^t\\end{bmatrix}\\begin{bmatrix}c_1 \\\\c_2 \\\\c_3\\end{bmatrix} ~,", "630d2f143dbce53ac85250991bdd9c4e": "P'(X_{n+1}=1 \\mid X_i=x_i\\text{ for }i=1,\\dots,n)={s \\over n}.", "630d3273deb04f68acf8fffaa0343c83": "\\chi_{\\mathrm{plus}}(x,y) = t = \\frac{y}{1-x}{}", "630d40ddfe160189eeba20d4fe604e74": "B^*,", "630d46613f24215fa632c1e584c9c0ef": "\\left( \\sum_{r=0}^n a_r x^r \\right)' =\n\\sum_{r=0}^n \\left(a_r x^r\\right)' =\n\\sum_{r=0}^n a_r \\left(x^r\\right)' =\n\\sum_{r=0}^n ra_rx^{r-1}.", "630d512d044b18ad618f0246de68ae48": " w^2+x(x^2+y^3)=0 ", "630d6ab45dbb9756ff44beccbc28469e": "s = |AB|+|A'B|=|AB|+|AB'|=|AC|+|A'C|", "630d81627ae6daf4c8ead0df5cfa65ca": "T = \\rho V^2 D^2 [ f_1(\\frac {ND}{V_a}), f_2(\\frac {v}{V_a D}), f_3(\\frac {gD}{V_a^2}) ]", "630d93bcac99ab670f527223f94b5346": " A \\exp \\left(- \\frac {(x-\\mu)^2}{2 \\sigma_2^2}\\right) \\quad \\text{otherwise,}", "630dc530c3d14f93c407e29a45ec2308": "\\displaystyle{\\mu_{g\\circ f^{-1}}\\circ f={f_z\\over \\overline{f_z}} {\\mu_g-\\mu_f\\over 1 -\\overline{\\mu_f}\\mu_g}.}", "630dd1a517b2ba3c69578dfbc18585ad": "\nW^{m,p}(\\Omega)\\hookrightarrow L_A(\\Omega)\n", "630e0ab33de425e65035c208c98dbde7": "x_1,\\dots,x_{l+u}", "630e1a9ac79dc54f6b967eeac3a0b834": "\\nabla^2 \\varphi =-\\frac{\\rho }{\\varepsilon }=\\frac{q}{\\varepsilon }\\left( \\underbrace{{{n}_{0}}-{{p}_{0}}}_{\\begin{smallmatrix}\n \\text{equilibrium concentration} \\\\\n \\text{difference of free charges (}\\approx \\text{0)}\n\\end{smallmatrix}}+\\underbrace{{{N}_{A}}-{{N}_{D}}}_{\\begin{smallmatrix}\n \\text{concentration difference} \\\\\n \\text{of acceptor and donor atoms}\n\\end{smallmatrix}} \\right)", "630e2b181438e36576de8df260528f8e": "Eq.9 / Eq.10", "630e2dcb1ecf0b0b9eac121d8cdcabcd": "L^2=L_x^2+L_y^2+L_z^2.", "630e39bddf737242c64400a0a8244b31": " \\ln r ", "630ecd91d524ecd6cc70b8567320d2f9": "\\int\\frac{x}{(ax + b)^n} \\, dx= \\frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} + C \\qquad\\text{(for } n\\not\\in \\{1, 2\\}\\mbox{)}", "630f161466f923562fd0ea6fbc794052": "b = \\frac {fm_\\mathrm s} {N} \\frac { D - s } { D }\\,.", "630f19d374360a38fa2dc4feef9dc989": "2^{2 + 1}", "630f358791d22118ba7c31db27b07bfb": "\\scriptstyle E=\\tfrac12\\, \\dot{x}^2 + \\tfrac12\\, x^2 + \\tfrac14\\, \\varepsilon\\, x^4", "630f3db72368c02551625d7132932be0": "\\left|\\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x \\right |\\leq V(u,\\Omega)\\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\n\\qquad \\forall \\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n)\n", "630f4eecfe10d62896ad6ee0d598cf8e": "a^{15} = a^3 \\times ([a^3]^2)^2 \\!", "631008fcbb456adeb9833550caee219b": "\\pi+20\\,", "63103d42f400fef98762df4bf0c2c15b": " u_o= \\mu_o E ", "6310c3cb677375073f82ef598e12dbad": "(\\ker T)^\\perp \\to W", "6310ec03e3a7ecfde36c5943847968b0": "\\Pr(A_n^\\epsilon)> 1-\\delta ", "6310f20c0dc2ed6a7c7841efc6d29c9b": " \\frac{\\delta J}{\\delta\\rho(\\boldsymbol{r})} = \\int \\frac {\\rho(\\boldsymbol{r}') }{\\vert \\boldsymbol{r}-\\boldsymbol{r}' \\vert} d\\boldsymbol{r}' \\, . ", "63110f4c43c7c9284d8580fbb1aa5ac1": "(a, b) = 1,\\;", "631151d7bea46c13d3e7a31685cb7080": "1\\leq j \\leq k", "631162cb98d95ee7f013645fcc51b942": "[\\ ,\\ ,\\ ] \\!\\,", "631224d7d92b4ec99ebea12ee1ba097c": "x_j\\notin A_1", "63125a8dcab38d89ffef13e5d4ecadcd": " u(t) \\le \\alpha(t) + \\int_a^t\\alpha(s)\\beta(s)\\exp\\biggl(\\int_s^t\\beta(r)\\,\\mathrm{d}r\\biggr)\\mathrm{d}s,\\qquad t\\in I.", "631260ded9b7fd865ff1c45e27dbce2a": "\\sideset{}{_{i=0}^\\infty}\\bigcup L^i", "63126407d3637960fab13a8b72e7966d": " X \\sim \\mathrm{Suzuki}(\\mu, \\sigma)\\,", "631294137cc28ef64fe6db7febb76b71": "(\\mathcal{H}, -F, -\\Gamma).", "6312ce9bedc1b80ede20855be40a5a1b": "\\lambda=\\textstyle{\\frac{1+i}{2}}", "631382ff769120ceb4434d949126218e": "\\Theta \\mapsto \\left|{\\mathbf I} - 2i\\,{\\mathbf\\Theta}{\\mathbf V}\\right|^{-n/2}", "6313b9c5ea15618e5cb73f6b318de4ae": "\\sigma_{s}=\\sigma(R=R_{s})", "6313c06c105a07be69b11a99245726bf": "\\;_1F_1 (a;b;z)=M(a;b;z)", "6313e8ff20ed0bdca968194271d488f8": "\\hat t(s)", "631415dce462fb019cccebdb08a59d73": "10^{2466}", "631437f51afbd59612d9643bb5f1f715": "x=(1,1)", "6314853f4e4b0481df2312771edb2043": "\\rho \\rightarrow U \\rho \\;U^*,", "63149da9ce1ca6918e7566909744e932": " \\mathbb{F}=\\mathbb{C}\\mbox{ or }\\mathbb{R}", "6314b2f7933c38ffd3b9ed83acf582de": "\\textstyle f_{f_{(p)}}(x) ", "6314e4ef93bd5c99e52e0eb3487c063b": "\\sum_{k=q}^n \\tbinom n k \\tbinom k q = 2^{n-q}\\tbinom n q", "63150374a65acfddde7390f9f763cc1f": "\\sum_{m=0}^n P(m)=P(n+5)-2.", "631563ad64298eacef9779144d8ad401": "\\gamma(1),...,\\gamma(n)", "631595b5fe7ba05dff8ca5311f2b6bb7": "\n \\mathcal{L}_\\varphi[\\boldsymbol{\\tau}] = \\boldsymbol{F}\\cdot\n \\left[\\cfrac{d}{dt}\\left(\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\tau}\\cdot\\boldsymbol{F}^{-T}\\right)\\right]\n \\cdot\\boldsymbol{F}^T ~.\n", "6315d022f1f8618586f9b42eb1124005": " \\mu_X^\\pi = \\mathbb{E}_{Y} [\\mathcal{C}_{X \\mid Y} \\phi(Y) ] = \\mathcal{C}_{X\\mid Y} \\mathbb{E}_{Y} [\\phi(Y)] = \\mathcal{C}_{X\\mid Y} \\mu_Y^\\pi ", "631610c9d4f90c9d10d12979a41e40bc": "d = \\sqrt{x^2 +y^2}\\,", "63169d980311f7f5928105611c17d5f4": "p_1/p_2<\\pi_1/\\pi_2", "6316eedfde78f9cd4dfced0b518aaa9a": "\\sigma[n]", "6316f35d204cdb059bf150aeffc358b1": "P[X=1] = \\frac{G^1 e^{-G}}{1!} = Ge^{-G}", "63175a3f143451ba2d9b974deacaa42e": "a=\\sqrt{k_e/2D_e}", "6317a1414bd6ab45db59c4fdc89d84c9": "\\eta \\in \\Lambda^k (V)", "6317b0337b8f8d762c7bf21932904a09": "a^m+b^n=c^k,", "6317f7f79de7cde48107c1d195d31dca": "L^\\infty([0,1] \\cup \\{1,2, \\ldots, n\\}), \\quad n \\geq 1 ", "631803d37d016e42616c56f0197a0916": "P(s,n) = \\frac{n(s-2)(n-1)}{2}+n", "6318434a5158cf2f75d477db091d6654": "|U_O + U_R|^2=U_O U_R^*+|U_R|^2+|U_O|^2+ U_O^*U_R", "631849c5efce5823fe36c562ad0f5d93": " \\sigma_y>0 ", "63186e53703978191593c845509bfc81": " \nq \\overset{\\alpha}{\\rightarrow} q'\n ", "6318f97a3fd3c689e2d9914b37b68b57": " f_\\xi(x) = \\langle \\xi \\mid \\pi(x) \\xi \\rangle ", "631959d9ecebc09e69a0efb7045e669c": " \\lambda_j = -\\frac{(2j - 1)^2 \\pi^2}{4 L^2} ", "63198f139e0619c1e34d30c458d12b69": "\\mathbf{L}f^*L^{Y/Z}_\\bullet \\to L^{X/Z}_\\bullet \\to L^{X/Y}_\\bullet \\to \\mathbf{L}f^*L^{Y/Z}_\\bullet[1].", "6319a965810321e2c378501a77eb9b1b": "P \\circ Q\n\\;\\mid\\;\\sigma(~)\\;\\mid\\;\\sigma(P)", "6319aa4ea1bb039db15cc8259c17aa29": "x_S \\in \\{0,1\\}", "6319b21eea4598a949e5aee7761b7181": "\\,\\!\\gamma", "6319ebcd263edb1b9594fe5b5e0a5146": "\\kappa_p(V)=\\|V\\|_p\\|V^{-1}\\|_p", "6319f1603f1a5e0e345319564447834a": "\\dim_{\\mathbb C}V^{\\mathbb C} = \\dim_{\\mathbb R}V.", "6319fb109c716654512b10577548c3b6": "\\Box A^{\\sigma} = \\mu_{0} \\, J^{\\sigma}\\,", "631a4e0d5ebf69d5bb0e557ada639cfd": "\\boldsymbol{q}", "631aa9baa73f9c36d433a2e85ec06e98": "G\\subset F", "631ab3f0050c4c535546c1802a47430e": "\\mathbf{u}(t)=0", "631abadfecbdd849726a6200154b623a": " y^e(\\mathbf{x})=y^m(\\mathbf{x},\\boldsymbol{\\theta}^*)+\\delta(\\mathbf{x})+\\varepsilon ", "631acd260f3a78b476614ec0cb84eee3": "f^{(4)}.", "631ad48768a449a34b862ae52200a14b": "\\scriptstyle\\operatorname{var} \\,=\\, \\sigma^2", "631ad96d88886346f863d1a4e9402841": "z_0\\in\\mathit{\\Omega}\\backslash\\mathrm{spec}A", "631b67754fc36315fbc7229fde1af3b8": "\\ell_x", "631b73371cdbf446479bc2ab8de825d5": "\\sqrt[12]{2}=2^{\\frac{1}{12}}\\approx 1.059463094359295264561825294946341700779204317494185628559208431458761646", "631c023b7c70a74e364ef068f03d9895": "E_{k,l}=\\operatorname{diag}(z^{k_1},z^{k_2},z^{k_3})\\backslash SU(3)/\\operatorname{diag}(z^{l_1},z^{l_2},z^{l_3})^{-1}.", "631c185e945f8024d4c1c0970d78e382": "{\\upsilon}_{out} = A_v \\ {\\upsilon}_{in} \\begin{matrix} \\frac {R_{L}}{R_L + R_{out}} \\approx A_v \\ {\\upsilon}_{in} \\frac {R_{L}}{R_{out}} = \\frac {A_v }{R_{out}}\\ {\\upsilon}_{in} R_L \\approx -g_{m2} R_L {\\upsilon}_{in}\\end{matrix}", "631c1efec82518f28e65512859226efa": "\n \\begin{bmatrix} x_1(X_1,X_2,X_3,t) \\\\ x_2(X_1,X_2,X_3,t) \\\\ x_3(X_1,X_2,X_3,t) \\end{bmatrix}\n = \\begin{bmatrix}\n Q_{11}(t) & Q_{12}(t) & Q_{13}(t) \\\\ Q_{21}(t) & Q_{22}(t) & Q_{23}(t) \\\\ Q_{31}(t) & Q_{32}(t) & Q_{33}(t)\n \\end{bmatrix} \\begin{bmatrix} X_1 \\\\ X_2 \\\\ X_3 \\end{bmatrix} +\n \\begin{bmatrix} c_1(t) \\\\ c_2(t) \\\\ c_3(t) \\end{bmatrix}\n ", "631d3b2c06ee84468a66b3c5547c9cac": "B_{2n-1}(x)=\\frac{2(-1)^n(2n-1)!}{(2\\pi)^{2n-1}} \\, \\sum_{k=1}^{\\infty}\\frac{\\sin 2\\pi kx}{k^{2n-1}}", "631e2da67cec7f1b1f5b5b2f679af3f0": " \\frac{1}{\\epsilon} y''(\\tau ) + \\left( {1 + \\epsilon } \\right)\\frac{1}{\\epsilon }y'(\\tau ) + y(\\tau ) = 0,\\,", "631e4cf898acbd6014b8ae79cddfda86": " \\rho \\rightarrow \\sum_n P_n \\rho P_n.", "631e592e1151b025416fe4cf76e304b4": "\nS=\\sum_{i}m_{i}\\int_{C_{i}}ds_{i}+\\frac{1}{2}\\sum_{i,j}\\int\\int_{C_{i},C_{j}}q_{i}q_{j}\\delta\\left(\\left\\Vert P_{i}P_{j}\\right\\Vert \\right)d\\mathbf{s}_{i}d\\mathbf{s}_{j}\n", "631e6a4e43ca1ad68b30933172e0dd88": "X+\\!_f \\,Y", "631e7d8e7175a871dd05267ab4821877": "\\left(\\frac{\\partial V}{\\partial P}\\right)_T = -\\beta_T V", "631e825f9f10728c9b92ec193aef84a5": "\\mathbf{u_2G}", "631f0fa8445b22e6c33738d8790f7369": "A_i \\rightarrow A^D.", "631f29b841e723485a16585679a60284": "\\zeta_i = \\frac{m_i}{m_{\\rm mix}- m_i} ", "631f5c91cb2fbdcd5a5aba7b57c666d9": "\\mathbf{S}_{jk}=\\left \\langle b_j|b_k \\right \\rangle=\\int \\Psi_j^* \\Psi_k \\, d\\tau", "631f8422f129799a4a1592f045698dfa": "\n C_{ij} = C'_{ij} \\quad \\implies \\quad C_{ij}~(\\epsilon_i~\\epsilon_j - \\epsilon'_i~\\epsilon'_j) = 0 ~.\n ", "631f9fe401bab128521ad7e54b17a528": "\\frac{R^2 + Z_o^2/\\Omega^2}{(1+R)^2+Z_o^2/\\Omega^2}", "631feac79cc9bb6705a14bad2cc7b05f": "\\mu t\\left (\\frac{\\partial U_{i}}{\\partial x_{j}}+\\frac{\\partial U_{i}}{\\partial x_{i}}\\right )-\\frac{2}{3}\\rho k\\delta_{ij}", "63202cc20d1d4a4a68e792c8057a4956": " w ", "6320a65ad657fc0ac61bbc47e054fd86": "\\nabla \\cdot \\left(\\alpha \\mathbf u \\right) = \\beta", "6320bc27ba8560d1601d7939e85d0154": "[t_m,t_{m+1} \\; m = 0 \\ldots M-1 ", "6320c499e1c971adb3529cf953ac4e39": "s^2 = (d-l)^2 + d^2 + l^2 = 2(d-\\frac 1 2 \\,l)^2 + \\frac 3 2 \\,l^2", "6320ea2cbb973c1803ba148ece1155c4": "g \\circ f \\sim Id_A", "632112e7a6a9194f8288b54b360a9589": "\\mathbf{n} ", "63212027aa193b0fdafa49fe96bb5a81": "\\mathcal{M}\\models\\varphi[a_1,\\ldots,a_m,b_1,\\ldots,b_n]", "63214fa0783b8ba6f326f0cd755591aa": "\\bigcap_{i=1}^{\\infty} A_i.", "63217ec2f8e238bda7d12ff4a11bb890": "G(s)=K_p + \\frac{K_i}{s} + K_d{s}=\\frac{K_d{s^2} + K_p{s} + K_i}{s}", "632188480c876063b3c94435e49daf9c": "\\phi_{mm}, \\ ", "6321c7833cd05341f3e9d633e5842d56": "L(P||Q)=\\frac{(|P|+|Q|)!}{|P|!|Q|!} L(P)L(Q),", "6321cef986f186dc006e29f7a1f80989": "\\mathrm{Pr}\\ll 1", "632209e035dae3e3981e3f504109b001": "J_{n}(t)\\,\\!", "632272243d6e65b5a7a24dbded410860": "\\hat{\\mathbf{r}} (\\mathbf{E}_1 \\cdot \\mathbf{E}_2) / Z", "6322831c52d97b712d1bd49bec2c5a1e": "\\Omega(\\det(t_{ij})^s) = s(s+1)\\cdots(s+n-1)\\det(t_{ij})^{s-1}", "63229536523f407f37a9f052c6b5af8f": "f: \\mathbb{R} \\to \\mathbb{R}", "6322cc3a0acd516bd7f1e402f3539e63": "\\mathfrak{p}'_i", "6322e5bae1cbf13442684fdf21e8d334": "t_2=x_1/c", "63231375c32401a0a420524a2841e41f": "1971 = [1, 0, 35]_{44}", "6323822e7d614edff7e28ab20f7e3f87": "O_{p',p}(G)/O_{p'}(G) = O_p(G/O_{p'}(G))", "63239d617cbace6d2a4db5cdb75a7afc": "P_{r} = A \\frac{G}{4 \\pi r^{2}} P_{t}", "6323a757b8dcfdd546122792d28657e8": "\\alpha \\approx \\frac{p}{D}", "6323a9f22649c4d66c80e07988007b71": "\\{1/n | n \\in \\mathbb N\\}", "6323dbd151c7c5d936422c97ab9318ec": "q_1,q_2,q_3\\,\\!", "6323ddea3fee25b3692262cddd1c0022": "\\psi_{i}", "63247d19088991b5e451350fe4356db0": "{g_0}", "63248c2c37cf0313abf5c661b2e4a827": "x<\\left(\\frac{n}{n-1}\\right)b", "63249b9fd79c478fa7947d87360361e4": "B^2 - 4AC < 0.", "6324c216f1087b5975d5a21cc5111fc7": "\\Delta t \\approx \\frac {4 \\pi R^2 \\omega} {c^2} = \\frac {4 A \\omega} { c^2}, ", "6324c7ea754ee6c6179e689ab157edb5": "FP(\\mu, \\sigma, \\gamma, 1, 1) = P(III)(\\mu, \\sigma, \\gamma)", "6325798162138e4fd93c73348a057c9b": "\\lambda_{m}", "63259a7b112615a49b439ab420f798d2": "I_0 = I_c \\sin\\phi_0\\,", "6325a65b73ad3051d411effdc6a97e85": "\\pi \\!\\,", "6325a8fe8569893829b7dab655223d08": " \\Phi = S - U/T \\,\\!", "6325baf542d35a0cb2c4f38c99168810": " \\vec v = \\frac {d}{dt} \\vec r(t) = R\\frac {d \\hat u_R } {dt} = R \\frac {d \\theta } {dt} \\hat u_\\theta \\ = R \\omega \\hat u_\\theta \\ . ", "6325e10084959a2e991cfb2c980a53ab": "C_\\text {UL}(\\alpha,n,N) = \\left [ 1+ \\frac{N-1}{F_\\text {c}(\\alpha/N,(n-1),(N-1)(n-1))} \\right ]^{-1} .", "6325f732464f1d221e6ba7328370dc6e": "H_{\\frac{1}{a}, 2} = \\frac{1}{a}\\left(2\\zeta(3)-\\frac{3}{a}\\zeta(4)+\\frac{4}{a^2}\\zeta(5)-\\frac{5}{a^3}\\zeta(6)+\\cdots\\right)", "6326046d52a86c67f7e8b7adda9bdf78": "S_n= {\\alpha\\,\\!x^n \\over n}", "63261c63a7627f8a719d1be3865e1b24": "\n( {\\nabla}^2 u )_{ij} = \\frac{1}{\\Delta x^2} (u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4 u_{ij}) = g_{ij}\n", "632624554d4b3e6a7f41d8ca7ca0c350": "\\scriptstyle 0.05^{1/5} \\;\\approx\\; 0.55,", "63265920a5f71e9461c02540b7843cf9": "\\begin{align}\n {n_o}^2 & = 0 \\qquad & n_\\text{o} \\cdot n_\\infty & = -1 \\qquad & n_\\text{o} \\cdot \\mathbf{x} & = 0 \\\\\n {n_\\infty}^2 & = 0 \\qquad & n_\\text{o} \\wedge n_\\infty & = e_{-}e_{+} \\qquad & n_\\infty \\cdot \\mathbf{x} & = 0\n\\end{align}", "6326621dd9ee628bfa825a602377c252": "{E[\\vec{X}]^a}_a = R_{mn} \\, X^m \\, X^n", "6326cc385dbb5d3854eef862841f7c1e": "\\exp(-|x|)", "63271377e181bda6a4f6644de822f6a1": "\\epsilon>0.", "6327268b508aa9da04d2d0b69cda61ce": "T_m(T_n(x)) = T_{mn}(x).\\,", "63276215bd461c8244d6a26f3f76f800": "\n \\langle j_1, m_1; j_1, {-m_1} | 2j_1, 0\\rangle = \\frac{(2j_1)!^2}{(j_1 - m_1)! (j_1 + m_1)! \\sqrt{(4 j_1)!}}.\n", "63277cbadd35f9e7487d62a2fe287fdb": "8128=2^6 \\cdot (2^7-1)", "632783c0ff8e5e3db04dd7a29ccb5206": "h = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e}) c_0 \\alpha^2}{R_{\\infty}} \\frac{\\sqrt{2}d^3_{220}}{V_{\\rm m}({\\rm Si})}.", "6328260fc7164263e2e9bbc834b2a143": " c = {f A \\over S_2} = {f^2 \\over N S_2} \\,.", "6328f1fa1ae384ed6453f5916d04656d": "I_{20.2, 14.5} = \\tfrac{21 - 20.2}{21 - 20} \\cdot 150.5 + \\tfrac{20.2 - 20}{21 - 20} \\cdot 128.5 = 146.1", "63290a6cf6302cb7b5f590d152472f12": "\\partial^j\\theta^i=\\partial^i\\theta^j", "632910fd14f5fff74afbf6e63b0e84b0": "u^{\\prime}", "63295493e069d39bd3033ecf7af90b18": " 1.4 M_{\\odot} ", "632a778b1c69455c0a4c8d2aa2139202": "\n\\hat{P} \\mathcal{A}\\Psi(1,\\ldots, N) \\equiv \\hat{P} \\Psi'(1,\\ldots, N)=(-1)^\\pi \\Psi'(1,\\ldots, N),\n", "632b04ed843d3d0e8b503916b95c17ff": " b\\neq 0 ", "632b280ab27fb1a1bdef1bb88a318fe3": "[n]_q!", "632b5365cfdedb0ee32d607081302e25": "s_3 = E(h_3, K_3)", "632b546d05510facb33b0caae2b0ce4d": "=\\frac{1}{2}\\frac{m}{L^3}v^2\\int_0^L y^2\\,dy", "632b6039c7aa360483a662ffaea77fc4": "P_{ME}=\\frac{\\sum [p_{c,t_n}\\cdot \\frac{1}{2}\\cdot(q_{c,t_0}+q_{c,t_n})]}{\\sum [p_{c,t_0}\\cdot \\frac{1}{2}\\cdot(q_{c,t_0}+q_{c,t_n})]}=\\frac{\\sum [p_{c,t_n}\\cdot (q_{c,t_0}+q_{c,t_n})]}{\\sum [p_{c,t_0}\\cdot (q_{c,t_0}+q_{c,t_n})]}", "632b666a8271c26e7ad642068167818c": " \n\\phi_{cm}(r)=\\left\\{ \\begin{array}{ll}\n\\frac{r\\left(3r+1\\right)}{\\left(r+1\\right)^{2}}, \\quad r>0, \\quad\\lim_{r\\rightarrow\\infty}\\phi_{cm}(r)=3 \\\\\n0 \\quad \\quad\\, , \\quad r\\le 0\n\\end{array}\\right.\n", "632b9a19784f26556ea96d413f5d37d4": "p_{xy} = \\begin{cases}\n\\frac{1}{d}\\frac{g(y)}{\\sum_{z \\in \\Theta: z \\sim_j x} g(z) } & x \\sim_j y \\\\\n0 & \\text{otherwise}\n\\end{cases}\n ", "632b9f1ffd3aa9cdfa60dadcd6731ab9": "\\sqrt{273.15}", "632bab115c09ea1f9c816e50f0a65b27": "c\\to s + W^+", "632bb766aa431c76ea2c91f81053a046": "\\overline{n}\\,=\\,1+Q(M)", "632c3f9f8f5c8ec37fff5f1b01470746": " \\|Tf\\|_{q_\\theta} = \\sup_{\\|g\\|_{p_\\theta} \\leq 1} \\left| \\int (Tf)g \\, d\\mu_2\\right|", "632c4abe735f9712b0d8cf5b910d720e": "C(\\cdot)", "632c67b9a031c48064442fd73d842cd8": "\\begin{align}\n & \\frac{\\partial\\mathcal{L}}{\\partial\\beta'} = -\\frac{1}{2\\sigma^2}\\Big(-2X'y + 2X'X\\beta\\Big)=0 \\quad\\Rightarrow\\quad \\hat\\beta = (X'X)^{-1}X'y \\\\\n & \\frac{\\partial\\mathcal{L}}{\\partial\\sigma^2} = -\\frac{n}{2}\\frac{1}{\\sigma^2} + \\frac{1}{2\\sigma^4}(y-X\\beta)'(y-X\\beta)=0 \\quad\\Rightarrow\\quad \\hat\\sigma^2 = \\frac{1}{n}(y-X\\hat\\beta)'(y-X\\hat\\beta) = \\frac{1}{n}S(\\hat\\beta)\n \\end{align}", "632c8c4831af63ee5afe339bb4f6cf3a": "{{\\Delta \\hat g} \\over {\\hat g}}\\,\\,\\,\\, \\approx \\,\\,\\,\\,{1 \\over {\\hat g}}\\,\\,{{\\partial \\hat g} \\over {\\partial L}}\\Delta L\\,\\,\\, + \\,\\,\\,\\,{1 \\over {\\hat g}}\\,\\,{{\\partial \\hat g} \\over {\\partial T}}\\Delta T\\,\\,\\, + \\,\\,\\,\\,{1 \\over {\\hat g}}\\,\\,{{\\partial \\hat g} \\over {\\partial \\theta }}\\Delta \\theta", "632c937d489f5d60fe127323cbe86b13": "f^*(x')\\approx f^*(x)", "632cebe123a2183988ea8ea2216bbd0c": "3x + y = 9", "632cec7788a22365340f93be7d3bdf6e": "\\left(A + B K \\left(I - D K\\right)^{-1} C \\right)", "632d3a4d18bc75a7d80f25cdb76e2c4f": "y = kx\\,", "632d633f743071a036af556232c4d885": "F\\colon \\mathcal C \\rightarrow \\mathcal D", "632d7ed7748f3dfc9c9e287f544335f2": "\\sum_{n=0}^{\\infty} f_n\\mid_K", "632d80e773a88ddc8671692218d269fa": " \\det \\mathbf{E} = 0\n", "632dcc3500d8d884f197786d2625e785": "\\begin{matrix} \\frac{312}{1326}=\\frac{12}{51} \\approx 0.2353 \\end{matrix}", "632dff0b7e1163be1d35ac7d2ccfa946": "\\delta^M:\\mathcal{X} \\rightarrow \\Theta \\,\\!", "632e1b7619d4033b90e22c2b77c0ddcf": "L_\\mathrm {L2}=\\sqrt{Z_\\mathrm {i T} Y_\\mathrm {i \\Pi}} \\ e^{\\gamma_\\mathrm L}", "632e2d5824735010ca174147cd3ec405": "\\hat{\\mathbf{Z}}=\\lim_{\\longleftarrow}\\mathbf{Z}/n \\mathbf{Z}.\\,", "632e4db9d1ce4a9d16a47cfaac00a0c4": "g_{mi}=\\frac{C_i}{C_T}g_m\\quad\\mbox{for}\\quad i=[1,N]", "632e74054b3fbfef425ce613dba5a568": "f^\\leftarrow(B) = \\{ a \\in X \\;|\\; f(a) \\in B\\}", "632e8a5896b02c515f8e43d2b18dfa68": "I, F, R \\subset Q", "632e9d3c390ae15f3afd73a396e01b93": "[\\cdot,\\cdot] : A\\otimes A\\to A", "632f0297f3f3f77bb4d31292ec9bd8c2": "\\lambda_1 \\langle e_1, e_1 \\rangle = \\langle A (e_1), e_1 \\rangle = \\langle e_1, A(e_1) \\rangle = \\bar\\lambda_1 \\langle e_1, e_1 \\rangle ", "632f3d36e759d7c586be403be408377c": "\\Delta\\varphi=-\\Delta a\\,f(\\xi_1,\\alpha+\\Delta\\alpha)+\\int_a^b[f(x,\\alpha+\\Delta\\alpha)-f(x,\\alpha)]\\;\\mathrm{d}x+\\Delta b\\,f(\\xi_2,\\alpha+\\Delta\\alpha).\\,", "632f85f256a1253f998766e6ed39bb74": "\\boldsymbol{a}=\\frac{\\text{d}\\boldsymbol{v}}{\\text{d}t}=\\frac{\\text{d}^2\\boldsymbol{s}}{\\text{d}t^2}", "633018776c144a01831196d4288f4fe4": "y_1, \\ldots, y_K", "6330630b8761ea652443221ca69fd982": "u_{\\Omega} = \\frac{1}{|\\Omega|} \\int_{\\Omega} u(y) \\, \\mathrm{d} y", "63306b02e45aaac86b6b79a9c236ba7e": " \\lim_{n\\rightarrow +\\infty} \\sqrt n\\;W_n=\\sqrt{\\pi /2}", "6330d651fe9504af49e852648ab48845": "{d \\over dt}\\left\\{ B \\right\\} =-\\beta k_+ \\left\\{ A \\right\\}^\\alpha \\left\\{B \\right\\}^\\beta +\\beta k_{-} \\left\\{S \\right\\}^\\sigma\\left\\{T \\right\\}^\\tau \\,", "6331252c55007505d34d62068bd10117": "\\begin{pmatrix}x&y\\\\0&z\\end{pmatrix}\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}=\\begin{pmatrix}0&x\\\\0&0\\end{pmatrix}", "6331304f0ffbbe3b7cebe443b56d6332": "\\forall z[z p[Z,X]", "6342e86675a25f8730aab206f1615ab4": " h_{ji}^{\\nu} = \\sum_{k=1,k\\neq j}^{n} w_{ik}^{\\nu-1}\\epsilon_{k}^{\\nu} ", "63432955e4f828ffd79b398ee9fd4947": "U(\\{c_t\\}_{t=t_1}^{t_2})=\\sum_{t=t_1}^{t_2}\\delta^{t-t_1}(u(c_t)),", "63432c15d31e117f4f567ed9551a5538": "U(x) = \\sum_{i} T^{i}_{i}(x)", "6343826feef02fe023442848e376b0dd": "\\tan(bank)=\\frac{TAS(kt)}{364}", "634386399fe695792977d53cf8ace8fa": "k\\;", "6343a26c300dc295ccd0bc5280697e28": "\\mathrm{A_{2}}", "6343edbaab8a1e305907a1672c60c5ff": "h(f)=\\lim_{\\epsilon\\to 0} \\left(\\limsup_{n\\to \\infty} \\frac{1}{n}\\log N(n,\\epsilon)\\right).", "6343ef358b41383b05356222e0986eb5": "\n\\qquad\nT_1=\\left[\n\\begin{array}{cccc}\n1&0&0&\\vdots\\\\0&0&0&\\vdots\\\\0&0&0&\\vdots\\\\\\cdots&\\cdots&\\cdots&\\ddots\\end{array}\n\\right],\n\\qquad\nT_2=\\left[\n\\begin{array}{cccc}\n0&0&0&\\vdots\\\\0&1&0&\\vdots\\\\0&0&0&\\vdots\\\\\\cdots&\\cdots&\\cdots&\\ddots\\end{array}\n\\right],\n\\qquad\nT_3=\\left[\n\\begin{array}{cccc}\n0&0&0&\\vdots\\\\0&0&0&\\vdots\\\\0&0&1&\\vdots\\\\\\cdots&\\cdots&\\cdots&\\ddots\\end{array}\n\\right],\n\\qquad\n\\dots.\n", "6343f6497b34f1d1aed84185cada87a4": "[0,\\sigma]", "634443b00bb2696b951be84451c3c5bc": "\\Big[\\mbox{un-}\\Big] \\Big[ \\big[\\mbox{easi}\\big] \\big[\\mbox{-er}\\big] \\Big]", "63448d15bd3ab916345f47ea38cedfc7": "\\int_{-1}^1 e^{i k x^2} \\, dx = \\sqrt{\\frac{\\pi}{k}} e^{i \\pi / 4} + \\mathcal O \\mathopen{}\\left(\\frac{1}{k}\\right)\\mathclose{}", "634491d0bffb61d0aaeebd12e76dd950": "x_m=y_m=z_m=a\\,\\frac{\\sqrt{2}}{2}", "6344ad20208ad4f67ed92af510be3f5a": "\\tan \\theta _1 = - \\cot \\theta _2", "6344ad6ec29d9d93bb66017950c3bc89": " T_E = \\frac{ 2\\log( N ) } { Var( r ) } ( \\log( K ) - \\frac{ \\log( N ) } { 2 }) ", "6344f1bb97311c8fafb25ebb3ad306ce": "bcbdddcced", "634501b3200048851c517efa578ac0a0": "(r,~\\theta,~z)", "63451db17f08121e76fb9291015d6b7c": "v_K", "63452fe629a9042f6c8533591afcf8d4": " [X,Y]_t = \\lim_{\\Vert P\\Vert \\to 0}\\sum_{k=1}^{n}\\left(X_{t_k}-X_{t_{k-1}}\\right)\\left(Y_{t_k}-Y_{t_{k-1}}\\right).", "634557a188331af7a271534c8a67db9c": "\n\\int_0^1 \\Bigl(\\,\\int_x^1 (1-y)^{n-1} \\frac{dy}{y}\\Bigr)q(tx)\\,dx\n\\leq t^{\\alpha-1}\\text{ for all }t\\in[0,+\\infty),", "6345bea4822ff7ec50611aaf668ebc9b": "i^{th} \\,", "634630a44b5c83d619e35eb116e04390": "l(x)=\\min \\{j\\mid f_j(x)\\le x_j>0\\}", "634642ceec1b2ab172a30c919e7d0ece": "\\delta W = P \\pi {a^2} \\,\\!", "6346b9d2e9ae0fbd5b8b4f8e6998c1f6": "\n\\begin{align}\n\\operatorname{E}[H(X)] & = \\int_{-\\infty}^\\infty H(x) \\sum_{i = 1}^n w_i p_i(x) \\, dx \\\\\n& = \\sum_{i = 1}^n w_i \\int_{-\\infty}^\\infty p_i(x) H(x) \\, dx = \\sum_{i = 1}^n w_i \\operatorname{E}[H(X_i)].\n\\end{align}\n", "6346b9fa5d66dd9a337939cb91c67cf5": "L_t = [L,A] \\equiv LA - AL \\,", "6346d094e7d4caa5eef68ebc5b700ec5": "(P_i P_j-P_{i-1}P_{j-1})", "6347323e1ffa8eadfc0f71b9bb334e25": "H = \\{x \\in X: x^*(x) = x^*\\left(x_0\\right)\\}", "6347458c5572cad6ece23a7e686c97e4": "1 - G(z)", "6347c0ed1e219a72f9f440a6ef8607b7": "\nA=\\left( \\begin{array}{ccccc}\n1& -1 & 0 & 0 & \\dots \\\\\n0 & 1 & -1 & 0 & \\dots \\\\\n& & \\ddots & \\ddots & \\\\\n0& \\dots & 0 & 1 & -1 \\\\\n0 &\\dots & 0 & 0& 0 \\\\\n\\end{array} \\right) , \\quad B=\\left( \\begin{array}{cccc}\n0& 0 & \\dots & 0 \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n0& 0 & \\dots & 0 \\\\\n1 &1 & \\dots & 1 \\\\\n\\end{array} \\right).\n", "63482f0213d926514a1de91f1893ee69": "v = \\omega r", "6348fc2e40f237cbfe3933b5e184d400": "a^2=b^3", "63493cc0a36c4f9e0afd21bb1f517975": "I = I_0\\cos^2\\theta\\,\\!", "634946fae50f86bf1756d2b33895f559": " \\tau = {L \\over R} ", "6349796dbe2a63138de1ac91c408befb": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "634983008da14f846b7cb01b161832b0": "f(n) = n^2 + n + 41", "6349b778c71fa8b8e1582600326d1277": " S_\\varphi = S \\cot \\varphi . \\,", "6349e25aa69d83e27a76cba37df5c6ae": "Z = \\int_0^{2\\pi} d\\phi \\int_0^{\\pi}d\\theta \\sin\\theta \\exp( \\mu B\\beta \\cos\\theta).", "6349f9b7983d619a1073373f7e1948c6": "f_d = {\\mu}_d f_r", "634a2df3a62e675ccc72419c006071bc": "\\displaystyle{ B(z_1,z_2)=x_1\\cdot y_2-y_1\\cdot x_2=\\Im\\, z_1\\cdot\\overline {z_2}.}", "634a375984801fa49c7ff7ab455720e7": "C\\subseteq [q]^n ", "634a5446342b1bbfc379e0cf050e3e40": "\\mathcal{R}(M_2)", "634a89b180e901b71beeb4e42da49d49": "1090.89^{+0.68}_{-0.69}", "634adeced89e01de75f35d9b67f825ac": "\\operatorname{nec}(U)", "634b8676d56ff2d0c5611dda29384651": "L^{(\\alpha)}_k(\\gamma r^2)", "634ba5aeaceb2c0aa841e616eccf684a": "\\cap, \\Cap, \\sqcap, \\bigcap \\!", "634bb0d5ffe1a58d6ae61e7d176f16e1": "-\\ln(r^2-z^2)\\,", "634c080bac6c9e13114a5c4422d09eea": "Pm\\overline{3}n", "634c25b287abeed0b73df6b5d7a07363": "\\begin{array}{cl}\n \\underset{\\boldsymbol{w},\\boldsymbol{\\xi}}{\\min} & \\|\\boldsymbol{w}\\|^2 + C \\sum_{n=1}^{\\ell} \\xi_n\\\\\n \\textrm{s.t.} & \\boldsymbol{w}' \\Psi(\\boldsymbol{x}_n,y_n) - \\boldsymbol{w}' \\Psi(\\boldsymbol{x}_n,y) + \\xi_n \\geq \\Delta(y_n,y),\\qquad n=1,\\dots,\\ell,\\quad \\forall y \\in \\mathcal{Y}\n \\end{array}", "634c4ad45a4b42214b0f28b9f97e129c": "\\xi\\, =\\, \\sqrt{3}\\, \\frac{x}{h}", "634c5ee978169f4921b7d7b0972228e8": "\\int\\limits_C \\mathbf{F}(\\mathbf{r})\\cdot\\,d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t))\\cdot\\mathbf{r}'(t)\\,dt.", "634cb4ece0555b3f9df29631af40031b": "\n\\mathfrak{g} = \\bigoplus \\mathfrak{g} (i).\n", "634ccdbd9fa946b2ae0f98a1cb5135f2": " ({P}_{\\alpha})_{\\alpha \\in F} ", "634d2ec500544b7560c205bb4b33cf94": "\\displaystyle{a^m a^n = a^{m+n},}", "634d631032e53166296164f491ce3276": "AD\\ volume\\ line = Advanced\\ Volume - Declined\\ Volume + yesterday's\\ AD\\ volume\\ line", "634e5b931abcb479c2dc0d14721e8b8e": "R_{xx}(j)", "634e61c791a9eaa08a1107017323458f": "\n\\begin{align}\nI(x,y) &= I(a_1, g_1) = I(a_2, g_2) = \\cdots\\\\\n &= I\\bigl(M(x,y),M(x,y)\\bigr) = \\pi/\\bigr(2M(x,y)\\bigl).\n\\end{align}\n", "634eb0deb0a9e00c939f469691147295": "a_r(\\mathbf{p})", "634eb476d9a4a03786de4fb01ba35011": "\\tan h = {\\sin A \\over \\cos A \\sin\\phi_o + \\tan a \\cos\\phi_o} \\qquad\\qquad \\begin{cases}\n \\cos\\delta \\sin h = \\cos a \\sin A \\\\\n \\cos\\delta \\cos h = \\sin a \\cos\\phi_o + \\cos a \\cos A \\sin\\phi_o\n\\end{cases}", "634edebfd4cb5474dc84ece4bf6b53f5": " \\varphi_1\\circ (\\varphi_2\\circ f) = (\\varphi_1\\star \\varphi_2)\\circ f.", "634f18676ed3870d5506817adc1975c4": "\\Delta I_{L_{On}} + \\Delta I_{L_{Off}}=0", "634f21e5b94fea85af905dc4943338f7": "\\frac{\\partial F}{\\partial x} \\frac{dx}{dx} + \\frac{\\partial F}{\\partial y} \\frac{dy}{dx} = 0,", "634f43be9a76ad0006353401e7860fca": "\\scriptstyle V \\to V^*,", "634f861bf9d88ea631f9e6da5917a72f": "{\\hat \\Psi} ^{\\otimes m}", "634fd54bf0d363851579915a0cc2c17c": "\\sqrt{u_0^2-v^2} = \\begin{cases} v \\tan v, & \\mbox{(symmetric case) } \\\\ -v \\cot v, & \\mbox{(antisymmetric case) } \\end{cases}", "634ffd5fd009421aadd43c202b4b4a1f": "\\int\\frac{r\\;dx}{x} = r-a\\ln\\left|\\frac{a+r}{x}\\right| = r - a\\, \\operatorname{arsinh}\\frac{a}{x}", "6350320bfd3e0348acbbe63fb46146db": "0=h'(c)=f'(c)-r\\, g'(c) \\Rightarrow(g(b)-g(a))\\,f'(c)=(g(b)-g(a))\\,r\\,g'(c)=(f(b)-f(a))\\,g'(c)", "6350ab05ebe0615d06baf693651ab310": "\\displaystyle \\gamma_\\mu = \\eta_{\\mu \\nu} \\gamma^\\nu = \\left\\{-\\gamma^0, +\\gamma^1, +\\gamma^2, +\\gamma^3 \\right\\}", "6350eb6fb3863c970097cbeba1699c7f": "\\operatorname{inc}^n \\operatorname{con} = \\operatorname{val} (f^{n-1} x) ", "6351188ea09191b6813864420ef91f48": "\\ s_n=x_1+x_2+ \\cdots + x_n.", "63511a826e7432e9a4d4f923be74ffa6": "G=\\langle X|S\\rangle ", "63514a6e48f973c158499d81ccae2b0e": "(E,\\,\\nu)", "6351521c0b10995f11acd268b3f53e3f": "X_1,X_2,\\dots,\\,", "635162074712e5b71630f791dc972432": "O(n^{5})", "63516d884e98ef310b5fde2e7386538c": " {d^2 x^\\lambda \\over dT^2} =- {d x^\\nu \\over dT} {d x^\\alpha \\over dT} \\left[{\\partial^2 X^\\mu \\over \\partial x^\\nu\\partial x^\\alpha} {\\partial x^\\lambda \\over \\partial X^\\mu}\\right]", "6352a5a7747a614a6b7897f4a1f6fb42": " 10MCalories=corresponds to=1 pesticide/transportational.impact.vegetarian.city ", "6352c9b4653b9aa9c32e2bef800fd027": "F = \\{r\\}", "63538ce17623ad28b1d43d49407b56a3": "(\\operatorname{Spec}\\ \\mathbb{F}_p)", "6353a29fedfc10c6b87405102120b0c4": "H(z)=\\frac{Ir^2}{2(r^2+z^2)^{3/2} }", "6353e2ebdad61e95744895db3d3e22a5": "p_m=\\frac{p_b \\cdot p_r}{p_b - p_r}.", "635413210d937ad5e42c365aced835d9": "({\\mathbf r}_2 - {\\mathbf r}_0) \\wedge ({\\mathbf r}_1 - {\\mathbf r}_0) \\wedge ({\\mathbf r} - {\\mathbf r}_0) = 0.", "6354368d860bb41b992b9417f7bbf887": "\\begin{matrix}\\left({66 + \\left\\lfloor{\\frac{66}{4}}\\right\\rfloor}\\right) \\bmod 7+\\rm{Wednesday} & = & \\left(66+16\\right) \\bmod 7+\\rm{Wednesday} \\\\\n\\ & = & \\rm{Monday}\\end{matrix}", "635458c58b321a84bb909579b11620d3": "\n\\sigma^*_i(a)(u_i(a_i, \\sigma^*_{-i}) - u_i(\\sigma^*_i, \\sigma^*_{-i}))\n=\n\\sigma^*_i(a)\\text{Gain}_i(\\sigma^*, a)\n", "6354c01b4bf6bed5ee529f91f3de6714": "\\left[\\frac {|a+\\omega b|^q + |a-\\omega b|^q} 2 \\right]^{1/q}\n \\le \\left[\\frac {|a+b|^p + |a-b|^p} 2 \\right]^{1/p}", "6354ccea36d61da58e3872a67a796853": "R \\gg \\frac{1}{2}(x_1 + x_2)", "6355a90da58c5e522521fd8afb461d6d": "\\begin{align}\n f\\colon& [0,1] \\to \\mathbb{RP}^2 , &\\qquad&\\text{(projective plane path)} \\\\\n g\\colon& S^2 \\to \\mathbb{RP}^2 , &\\qquad&\\text{(covering map)} \\\\\n h\\colon& [0,1] \\to S^2 . &\\qquad&\\text{(sphere path)} \n\\end{align}", "6355b75b739edacaf9254b62231a2797": "L_n[\\alpha,c] = L_n[0, c] = e^{(c + o(1)) \\ln\\ln n} = (\\ln n)^{c + o(1)}\\,", "6355c2ee894f8f9681cc9646b28eab64": "dy/dt(0)=y'_0", "6355f51010942d0870ff498e67fae9b6": "\\stackrel{\\mathbf{\\nabla \\times E = - \\part_t B}}{}", "63560e1997389c1c6b74008f226be4e0": "\\langle\\Omega|\\mathcal{T}\\{{\\phi}(x_1)\\cdots {\\phi}(x_n)\\}|\\Omega\\rangle=\\frac{\\int \\mathcal{D}\\phi \\phi(x_1)\\cdots \\phi(x_n) e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{\\lambda\\over 4!}\\phi^4\\right)}}{\\int \\mathcal{D}\\phi e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{\\lambda\\over 4!}\\phi^4\\right)}}.", "63561b78f09aa67db9ff8705c9201020": "\\Psi_{1\\ldots N}(j^N\\alpha JM) ", "6356577d957065330449d7c1931c4f3f": "f_y = 0", "6356764866e63f74861d0b5b12680cca": "e_i f_j ^T", "635679e7f8b9b91178ef40e4426c8b9c": "f \\mapsto \\int_{x_0}^{x_1} \\sqrt{ 1+|f'(x)|^2 } \\; \\mathrm{d}x", "6356b908587fdae25e93a4d63994f12f": "G_k(X)", "635729275f58d8d3d6003589d105f73c": "n < p < 2n", "635759e23aaf6e02541e3b72d65268d0": "\\triangle ABC", "6357bcafcecbf5fa736737775ca2d1b1": "\n V = \\pi r^2 l = \\text{constant} \\qquad \\implies \\qquad \\delta V = 2\\pi r l \\delta r + \\pi r^2 \\delta l = 0 \\implies \\delta r = -\\cfrac{r}{2l}\\delta l ~.\n ", "6357c60bf7e309b3222285639817d04b": "\\bar y_i= \\leftarrow y_i - \\gamma \\Bigg\\{ w_{internal} \\bigg[ \\alpha \\frac{\\partial ^2 y}{\\partial s^2} (\\bar v_i)+\\beta \\frac{\\partial ^4 y}{\\partial s^4} (\\bar v_i) \\bigg]", "635823616f6d3767c522bd41419a812d": "x \\Rightarrow_G y \\mbox{ iff } \\exists u, v, p, q \\in (\\Sigma \\cup N)^*: (x = upv) \\wedge (p \\rightarrow q \\in P) \\wedge (y = uqv)", "63584be660ea62c75dd225c1b75f124b": "\\arg\\max_{k}{(T_1[k,T])} ", "6358641e113fca304844a79cba7edf7a": "e^\\mathbf{B} = e^{\\beta\\frac{\\mathbf{B}}{\\beta}} = \\cos{\\beta} + \\frac{\\mathbf{B}}{\\beta}\\sin{\\beta}.", "635886859e3f0aa0e3464f7644ff8aa1": "\\begin{align}\na^2d^2 = aadd&: 0+0+3+3 &&& abcd&: 0+1+2+3 &&& ac^3 = accc&: 0+2+2+2 \\\\\nb^3d = bbbd&: 1+1+1+3 &&& b^2c^2=bbcc&: 1+1+2+2.\n\\end{align}", "6358c1bbaba9cfaea2907303aa3084f8": " \nR^m_{\\ell}(\\mathbf{r}) \\equiv \\sqrt{\\frac{4\\pi}{2\\ell+1}}\\; r^\\ell Y^m_{\\ell}(\\theta,\\varphi)\n", "63592391560aeaf05a973bb1db7421e2": "\\hbox{The quadratic formula is } \\textstyle{-b \\pm \\sqrt{b^2 - 4ac} \\over 2a}", "63592662d618c0e3c3d04b51e16382a6": "\\Gamma_{\\alpha}(\\theta)", "635975bd9861be3cefef98e40a657cec": "A = 2 \\pi \\frac{(a^2 + h^2)}{2h} h = \\pi (a^2 + h^2).", "6359813d297669889fc1c7bf653cead7": " \\ell_1 ", "63599542bbd9ec7b9790b876d5ae9050": "\\frac{|v\\rangle - i |w\\rangle}{\\sqrt{2}}", "6359af0ecf8cc70b618df458491ef355": "A=N\\pi l d_o", "6359bda75c641c14e5b922f8f6c94331": "xy^2z", "635a1a6f725ca23660f4fcddf11ecabd": "q_j\\,\\!", "635a23e9ee1dd244401ddcc5af5443ab": "{(a)}_{{n}}", "635a2b8f05e66a34287acf4582ac856c": "H\\leftarrow s(H) \\oplus s^{k}(h( c_1 )) \\oplus h(c_{k+1})", "635a697fc8a0da13dc145f8a0dc94dc4": "\\left (s^{3}-s^{2} \\right )y''+ \\left ((2-\\gamma )s^2 +(\\alpha +\\beta -1)s\\right )y'-\\alpha \\beta y = 0.", "635a6fa83b2cb9128827ad048fc2a762": " \\mathfrak{P} ", "635aa2ab5f90a483d97cdd44f8ad2b5a": "w_{t+1}", "635b01129291f121a0a2051576697576": "V(x - v^b(1),1) = 1 - \\frac{1}{3} \\exp(-1.10 x/10) \\exp(1.10 v^b(1)/10)\\left[1 + 2 \\exp(-1)\\right]", "635b5575fc39c03e1e00de729acb51af": "q=\\frac{Q}{b}", "635b8a8307a0dea10ca7af8419a8355e": "H_{1,S}", "635b967af4952c175037345b89c5b0a7": "\\mathop{c.h.}\\mathop{\\rm supp}\\,\\phi\\ast \\psi=\\mathop{c.h.}\\mathop{\\rm supp}\\,\\phi+\\mathop{c.h.}\\mathop{\\rm supp}\\,\\psi.", "635bcab1dbabc25621a87784132056b1": "A= A^*\\,", "635bdf00ecd82bbf4e7febefc512d4f9": "\\Delta z\\,\\, \\approx \\,\\,a\\,\\,\\frac{{\\Delta x}}{\\mu }", "635bebb2d590777041493b76b9de8a7d": "\\alpha_1, \\alpha_2, \\alpha_3", "635bfbe220bf90e8a18d80fbe293007f": " k_1 = \\frac { \\sqrt {(1 + u^2)} - 1}{ \\sqrt {(1 + u^2)} + 1}", "635c008aad72c4fd2582bf0afb5eaf15": "F^\\nabla \\in \\Omega^2(\\mathrm{End}\\,E) = \\Gamma(\\mathrm{End}\\,E\\otimes\\Lambda^2T^*M).", "635c652f402a9324166c4586a90a91b9": "\\scriptstyle\\bar{\\partial}_b", "635c91279330545e6434b1a9c3598dc6": "l(D) \\leq \\frac{deg D}2+1.", "635cba97cabca7b2c1aa2cda02ed0b83": " \\mbox{IP}(\\cdot,\\cdot), (\\cdot,\\cdot),\\langle \\cdot,\\cdot \\rangle", "635cd3d50699005eff82f8da32beb485": " \\langle x, y + z \\rangle = \\overline{\\langle y + z, x \\rangle} = \\overline{\\langle y, x \\rangle} + \\overline{\\langle z, x \\rangle} = \\langle x, y \\rangle + \\langle x, z \\rangle,", "635cda7fdd365f1d4fa69eb8ad6f2af7": "\\neg locked(door,result(opens,s))", "635d04b2e53017dc90a381ab1389a14e": "A \\cdot B = \\sum_i A_iB_i = \\sum_i ( A_i \\land B_i)", "635d0be9b27ed245d8477f9c0e51bd12": " \\phi_{0} ", "635d541faff18a3e7e772ece046b281a": "\\scriptstyle\\nu!", "635d8c12f3ea7f2b3d86eb5753323fcd": "E_{r}=\\text{ energy of eigenstate }r\\,", "635dd88c39e81112df90428420e64bbf": "\\mathit{D}_G \\varphi", "635e0bf2249b8310445806c87255142a": "\\begin{matrix} {4 \\choose 3}{3 \\choose 1}^3{36 \\choose 2} \\end{matrix}", "635e24c281a51ea9b4b566c6bde0760a": "2, \\sqrt{6}, \\sqrt{6}", "635e7f052fb27f85cecdfcf3849c80d0": "\\mathcal{L}_X R^a{}_{bcd} = \\delta ^a{}_d \\psi_{b;c} - \\delta ^a{}_c \\psi_{b;d}", "635e8326c45c057c8810d87fee642fad": "ds^2 = Edu^2+2Fdudv+Gdv^2 \\,", "635ee334fea8aa97c9346025051ba764": "\\frac{G\\cdot I}{c^{2}\\cdot R^{3}_{E}}", "635f69a306c37d25e4ca1c80b912247b": "x^* \\,", "635f786ec744ec30e4c0be09907c9b14": "2\\times 3 = 6", "635f93a791cf333a8848a690b74cd52f": "\\mathbb{E}\\bigl[|XY|\\big|\\,\\mathcal{G}\\bigr]=0\\qquad\\text{a.s. on }\\{U=0\\}\\cup\\{V=0\\}", "635fda96efe88f9b0f68d523985ef08f": " ds^2 = \\lambda| \\, dz +\\mu \\, d\\overline{z}|^2,", "63601e0164ebf662d1caa10438c5b477": "2. \\quad f\\star g-g\\star f = \\mathrm i\\hbar\\{f,g\\} + \\mathcal O(\\hbar^2) \\equiv \\mathrm i\\hbar \\{\\{f,g\\}\\}", "6360c46bd3fc2b35954f16375da805f4": " \\dot{\\varphi}_i = \\omega_i(I_1, \\dots, I_n) ", "6361dc3b459fc3252283a9bc0edf36af": "\\begin{bmatrix}\nX&amount&of&money\\end{bmatrix}", "6361e231ac9013657644a2435a5610b9": "s_1, s_2, \\dots", "63621fda6d508dd36c62a12b50f02361": "\\Delta\\varphi^*=-\\frac{2\\pi}{h}2m_e\\int_\\delta \\vec{v}_s\\cdot\\vec{\\mathrm{d}s}.", "63622f1197ba20e300124449fb5435d5": "\\zeta(s)=\\sum_{n=1}^{\\infty} \\frac{1}{n^s},", "636232f204fb068afc7701735624a1b9": "Z(s)=R\\,\\!", "636268abea656caaa589fb270314647c": "\\overline{U'_r V'}", "6362bbbe9fdefe2185a39441763f770b": "\\Pi_3", "6362bd8029d7a6bdd0d48844aef84af0": "-j\\,", "6362f925cc541b86a3e2e5c6c1a3e03a": "\\displaystyle{R(a,b)=R(a^b,b-Q(b)a)=R(a-Q(a)b,b^a)}", "63630852e162bf1ff38dbb88c1f81683": "\\lim_{z \\to z_0} g(z).", "6363270099491b8cd17315f7b40a5c5d": "\\int_{{-c}}^{{c}}\\sin {x}\\;\\mathrm{d}x = 0 \\!", "6363c2d8efce549c3804e5a7b95d0548": " F_T = GMu\\frac{2dr-r^2}{d^4-2d^3r+r^2d^2}", "6363dab104367c67a3add07c01f53a63": "J(x):=\\left\\{x'\\in X':\\|x'\\|_{X'}^2=\\|x\\|_{X}^2=\\langle x',x\\rangle \\right\\}.", "63649ae0e15b786d5916e8541a779581": "p_{\\text{A}} \\cdot p_{\\text{B}} = \\frac {1}{r \\cdot (f_{\\text{A}} - 1) (f_{\\text{B}} - 1)}", "6364c41dba05363ad05ba135794a9caf": "\\text{E}[e^{-a \\epsilon}]=e^{-a \\mu + \\frac{a^2}{2}\\sigma^2}.", "63650d3f40381cab1c33c9df8f7e7cb4": "p = a, q = -b, r = -c, s = -d", "6365731c02c7aeacb0713431c7bbc62e": "(n-R_2)", "6365735e89f21e2d34fbbf2fd0ab6a2f": "(\\hat{x} - x_0 \\hat{I}) \\cdot \\hat{p} \\, | \\psi \\rangle = (\\hat{x} - x_0 \\hat{I}) \\cdot p_0 \\, | \\psi \\rangle = (x_0 \\hat{I} - x_0 \\hat{I}) \\cdot p_0 \\, | \\psi \\rangle=0.", "63657d6f7990e965921e9c4729e899bf": "g(x) \\,", "6365df228d734f3483f408a11af9791c": " ||X||_1 = Tr|X| = Tr \\sqrt{X^\\dagger X} ", "6365ffaa01acc13741c7677935b80784": " \\delta\\psi = e^{-i\\mu t}(u(\\boldsymbol{r})e^{-i\\omega t} - v^*(\\boldsymbol{r})e^{i\\omega t})", "636619543157bb3995f5f2da91434c7b": " \\partial D ", "63662d24a7c08c5e9daddd350b56a1fc": "U(x)= {1\\over2}kx^2", "636637bf28b766f0db85c39acb4ed131": "f(x) = 1/x", "636649411c9d3f1d6f3479353db0d925": " \\textbf{V}_P = \\frac{d}{dt}(R(t)\\textbf{e}_r + Z(t)\\vec{k}) = \\dot{R}\\textbf{e}_r + R\\dot{\\theta}\\textbf{e}_t + \\dot{Z}\\vec{k},", "6366e6afda0d348f0d092d589a37ae60": "x_k= x_k^*", "6366ee0dfe101c89c20e5d050f52ad66": "C_{xx}=\\{\\}", "63671e2617e3b277045b213be3e80a66": "\\mathbf{\\mu}", "63672a9f7b0a5898f1a673cb1fd2f0c7": "u(x,t)=\\int_{0}^{t}\\int_{0}^{\\infty} \\frac{1}{\\sqrt{4\\pi k(t-s)}} \\left(\\exp\\left(-\\frac{(x-y)^2}{4k(t-s)}\\right)+\\exp\\left(-\\frac{(x+y)^2}{4k(t-s)}\\right)\\right)\nf(y,s)\\,dy\\,ds ", "636744d6f2085661ceb8a90b6a27e185": "n=|N|", "636758c24817ea6eb229a5018fe81882": "P^{(0)}=0", "6367d38595164c782b5aa9e657fab59b": " Reactant \\xrightarrow[enzyme]{} Product ", "6367d8f071a68bed84275ac049605bd6": "A=\\arcsin \\left({p \\over 2r}\\right)", "6367e36e2b32576a99160143d33bc780": "z\\in\\mathbb{C}", "63681f28b751647595884d2f89f10d51": "Y_{9}^{7}(\\theta,\\varphi)={-3\\over 512}\\sqrt{13585\\over \\pi}\\cdot e^{7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot(17\\cos^{2}\\theta-1)", "63682037cf9b0bb803a41d1a7dd585b8": "\\delta_v H\\,", "636868b3a885b79ed199e6e12bedbf13": "G/K", "636903ed98c06a438353578522d04b93": " f(x)=\\frac{1}{x} ", "63695def300fbc26f55be0759425d9d5": "\\hat{m} = (\\hat\\mu(x_1),\\ldots,\\hat\\mu(x_n))^T", "6369650cb3eb6b9da8657c86b09c8724": "s^2 = \\sum \\frac{(x_i - \\bar{x})^2}{n-1}", "6369707843d6a8bc9ea6271f02eebd23": "l_\\text{n}", "636970941f2f8c28b39960cfe24781bc": " \\sigma_t^2=\\alpha_0 + \\alpha_1 \\epsilon_{t-1}^2 + \\cdots + \\alpha_q \\epsilon_{t-q}^2 + \\beta_1 \\sigma_{t-1}^2 + \\cdots + \\beta_p\\sigma_{t-p}^2 = \\alpha_0 + \\sum_{i=1}^q \\alpha_i \\epsilon_{t-i}^2 + \\sum_{i=1}^p \\beta_i \\sigma_{t-i}^2 ", "6369856a00e483bee41ed60eb067c3f4": " P = F_8 + F_4 + F_2 + F_1 + F_0 ", "63699f5b97f25276a6f64649f2f13aee": "c'_1 = \\frac{c_1}{b_1}\\,", "6369f37bb5ea43a235cd48059a2dc22b": "5 + 2 = 7\\;", "636a186172c27d808e56314e6918dff3": "\\begin{pmatrix}\n 1 & a & c\\\\\n 0 & 1 & b\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n 1 & a' & c'\\\\\n 0 & 1 & b'\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}=\n\\begin{pmatrix}\n 1 & a+a' & c+c'+ab'\\\\\n 0 & 1 & b+b'\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}\\, .", "636a480f5bcfd111bc4b1a85be07d778": "c = m B'+e=(-104, -79).\\,", "636a4fda7bcf0fa6c491450aa3367ee0": "\\textstyle\\left(\\!\\!{n\\choose k}\\!\\!\\right)", "636a96d86db865d3b29547eba8bd3b33": "1\\le n \\le 10^3", "636a99e97bfd27ed4f71db94e69c11a1": "X \\subset Y", "636ad03ca2161087c4fafe3f64e0ead9": " \n -\\left ( \\partial^2 + m^2\\right ) D\\left ( x-y \\right ) = \\delta^4\\left ( x-y \\right ) \n", "636b0597e42fa9ea8e0b59dad48a30e2": "((P \\and Q) \\and R) \\leftrightarrow (P \\and (Q \\and R))", "636b1a89e99cd5b3818ae4c3ab4a5501": "\\mathbf{y}_1, \\dots, \\mathbf{y}_N", "636b3e6a3e0b11df64f50cb94f971f04": "f(x)=\\sin x = \\sum^{\\infin}_{n=0} \\frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\cdots\\text{ for all } x", "636b4e98f66809c3dd67fc9a2be6aa93": "f=A_0 \\left(1+{-1\\over 2}x^2+{-1 \\over 8}x^4+{-7 \\over 240}x^6+\\cdots\\right) + A_1\\left(x+{1\\over 6}x^3+{1 \\over 24}x^5+{1 \\over 112}x^7+\\cdots\\right)", "636c43b95fc387dd90aff76e1bf4a6cd": " |\\mu|(f) = \\sup_{0\\le |g|\\le f} |\\mu(g)|.", "636ce3acd9194af0ce67d4639032b05a": "\\chi_{12} = z\\Delta w/kT \\,", "636d006919ac2d0150041b1dfcfe46a4": "\\ g_a", "636d95826e1c5d70764eb2012eb94cf4": " x^2 + y^2 = 1 + d x^2 y^2 \\, ", "636dbc8e0d9a89c88f5010ea76a2948c": "\\int x^4 r\\;dx= \\frac{x^3r^3}{6}-\\frac{a^2xr^3}{8}+\\frac{a^4xr}{16}+\\frac{a^6}{16}\\ln\\left(x+r\\right)", "636df6112c7e0c56573f4f03e2045a84": "ax^2+y^2= 1+dx^2y^2", "636e370dd3de273ba89f8f6b2919fa9d": "\\begin{align}\nf^\\star(p)=\\int_0^p F^{-1}(q) \\, dq & = (p-1)F^{-1}(p)+\\operatorname{E}\\left[\\min(F^{-1}(p),X)\\right] \\\\\n& = p F^{-1}(p)-\\operatorname{E}\\left[\\max(0,F^{-1}(p)-X)\\right].\\end{align}", "636e8c6ee61c640bac46876e164b82f7": "y^\\alpha e^{-y} K_n^{(\\alpha)}(\\cdot, y) \\rightarrow \\delta(y- \\, \\cdot),", "636ebba80504869d8445cddc600b48c9": "I < 0", "636efa88d291a29cf3a0bc181ff5741c": "T_{\\delta}^{Y^{n}|x^{n}}", "636f08a20031eb0280dc669718bca33e": "\\Delta l^a=(\\gamma+\\bar{\\gamma})l^a-\\bar{\\tau}m^a-\\tau\\bar{m}^a\\,,", "636f19b8ef9a19da846adf0ae182a0bb": " \\overline{u}{\\partial \\overline{u} \\over \\partial x}+\\overline{v}{\\partial \\overline{u} \\over \\partial y}=-{1\\over \\rho} {\\partial \\overline{p} \\over \\partial x}+ \\nu \\left({\\partial^2 \\overline{u}\\over \\partial x^2}+{\\partial^2 \\overline{u}\\over \\partial y^2}\\right)-\\frac{\\partial}{\\partial y}(\\overline{u'v'})-\\frac{\\partial}{\\partial x}(\\overline{u'^2}) ", "636f4dac33a6ac13dd315ed279aca1ae": "p = \\frac{(\\tfrac{f}{N} + c) f}{c} = \\frac{f^2}{N c} + f", "636f5392b613a9219fc7b4e4b2ce73e0": " \\underline{Z} = R + j X = Z cos(\\phi) + j Zsin(\\phi)", "636f7be6072aa5837c7a2b9bbd505859": "T[] \\to \\epsilon", "636f9be63b75804c37a6ef47191ef734": "M^{-1}(x)=\\frac{dx-b}{-cx+a}", "636faa7a9f609c7b0915fb1e73ae65fe": "\\, \\! V_-=0", "636ff259993fee9a1fa78b372afcde0d": "\n \\sigma_y = \\left(\\tfrac{1}{2}|\\sigma_2-\\sigma_3|^n + \\tfrac{1}{2}|\\sigma_3-\\sigma_1|^n + \\tfrac{1}{2}|\\sigma_1-\\sigma_2|^n\\right)^{1/n} \\,.\n ", "637011d8fbe77118fdbbacad05927413": "\nv_{w, z} = (1 - 2a)v_{\\infty}\n", "637062c4ada8ee0fd08e02153c2f4409": "\\mathit{Assoc}2(R) = \\{(d,(a,(b,c))) \\mid (d,((a,b),c)) \\in R\\}", "6370a79da09c6cb6b86632fd2b21a1e3": "a=\\sqrt{2}", "6370d87bfa748767f7a6eece0a59f746": "F = \\frac{q_1q_2}{\\epsilon r^2},", "6370dae92aac1dd525844c3126b28f5e": "\\mathbf{w} = \\sum_{i=1}^n{\\alpha_i y_i\\mathbf{x_i}}", "6370ef8f57650fbaa6d8e99c1136436f": "(-,+,+,+):", "6371191c6baccb8a8a30666416f08873": "S = \\frac{-i}{\\sqrt{2}}\n\\begin{pmatrix}\n 0 & 1 & 0 & -1 \\\\\n 1 & 0 & 1 & 0 \\\\\n 0 & 1 & 0 & 1 \\\\\n-1 & 0 & 1 & 0\n\\end{pmatrix}\n", "637121019077720f1d285a8d4382ade3": " \n {A'}_{i j} = \\frac{\\partial x^l}{\\partial {x'}^i}\n \\frac{\\partial x^m}{\\partial {x'}^j} A_{l m}\n", "63715b71f2623ecb5683d579cf700ef3": "AP^{-1}y=b", "6371c3334c2f381e5fe266e8d78241e8": "A:S\\times R \\to \\mathbb{R}", "6371c44a0120b903def49a83afe46a91": " \\Phi_I", "6371c7cb923325b3777279fff5b0e0f7": "S\\cap T \\in W", "637226d05e7854996ad9bd2c0be7a123": "\n\\begin{align}\n\\lim_{x \\to 0} \\operatorname{sinc}(x)\n& = \\lim_{x \\to 0} \\frac{\\sin\\pi x}{\\pi x} \\\\\n& = \\lim_{y \\to 0} \\frac{\\sin y}{y} \\\\\n& = \\lim_{y \\to 0} \\frac{\\cos y}{1} \\\\\n& = 1.\n\\end{align}\n", "63722d3a65782d5dc344833e779972db": "x I-A", "6372371395906b5a9e36ffbbf9545e2e": "Y=\\{\\} ", "6372508c5f5988f2cf884af63377f6b5": "q=\\left( {1\\over{{1 \\over h}+{t \\over k}}} \\right) \\cdot A \\cdot \\Delta T", "637259c906c3a6910d28c1d311936048": "\\mathbf{i} = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}", "6372dc30f971a18cd52a2a818c35ad06": "93^2", "637310031094516a49c212b9892ead6b": "\n\\langle \\ln\\tau_D \\rangle = \\ln\\tau + { \\Psi(\\beta) + {\\rm Eu} \\over \\alpha }\n", "637351dc4bcc5e70b92005a1945f445e": "\n \\tau_m = \\tfrac{1}{2}(\\sigma_1-\\sigma_3) ~;~~ \\sigma_m = \\tfrac{1}{2}(\\sigma_1+\\sigma_3) ~.\n ", "6373accf16c083723e8abae2f5401af2": "x\\,\\!", "6374084270370a92e9a263ef5c610600": " A = G_1 \\times ... \\times G_n = \\prod_{k=1}^n G_k ", "6374227f5923aefaf289e690b9b2c81d": "\\nu_i = 4.80 \\times 10^{-8} Z^4 \\mu^{-1/2} n_i\\,\\ln\\Lambda\\,T_i^{-3/2} \\mbox{s}^{-1}", "637446532aacf4411a76aae72188541d": "Z=\\sum_{n=0}^{\\infty } \\left (\\frac{\\sqrt{5}-1}{2} \\right )^{8n} \\frac{(42n\\sqrt{5} +30n + 5\\sqrt{5}-1) \\left ( \\frac{1}{2} \\right )^3_n} {{64^n}(n!)^3}\\!", "63749adbdce708fbd2c79ead183e015f": "4\\pi\\varepsilon_0 \\alpha' = \\alpha", "6374b4c39e3f7b90bfa07feefdecc931": "\\sin{\\pi n}=0", "63750509afd02d03180d04f16226926a": "x^2 \\left( 1 - {c^2 \\over a^2} \\right) + y^2 = a^2 - c^2", "6375305c0128f52deb23a0feaca85927": "E_{pot}^g =s_g(x_d - x)", "6375a2d924cec1920fd79024533fbe8c": " z = \\frac {R(t_0)}{R(t_e)} - 1 \\approx \\frac {R(t_0)} {R(t_0)\\left(1+(t_e-t_0)H(t_0)\\right)}-1 \\approx (t_0-t_e)H(t_0) \\ , ", "6375c3ee1e23ebbaac29a46643627b7e": "(x \\top y) \\top (u \\top z) = (x \\top u) \\top (y \\top z)", "6375e5fafe5494648644267dde7741c4": "\\mathsf{WKL}_0", "63762080c837a94ee45e1df67ac608da": "x_{n+1} = x_n - m\\frac{f(x_n)}{f'(x_n)}. \\,\\!", "637649b943cd757db43e5caa6014ea0d": "B = \\frac{2 \\sqrt{2\\pi} \\sqrt{u^2+v^2}}{[(u+1)^2+v^2]^{1/4}}", "6376a59be50c7a90bb4be5a166babdfc": "\\mathbf{u} = {1\\over{2 \\mu}} \\left[ \\nabla ( \\mathbf{x} \\cdot \\mathbf{\\Phi} + \\chi) - 2 \\mathbf{\\Phi} \\right]", "6376c8569fabd8be68f981ab43847f81": " H=-\\sum_{\\langle i,j \\rangle} \\hbar J_{ij} \\left({\\sigma}_x^{i}{\\sigma}_x^{j} +{\\sigma}_y^{i}{\\sigma}_y^{j}+\\gamma {\\sigma}_z^{i}{\\sigma}_z^{j}\\right)-\\sum_{i=1}^{N} \\hbar B_i \\sigma_z^{i} ", "637763947ada214e7b9a4e87dec1a96d": "\\left|\\int_{\\mathbb{R}^n} f(x) \\overline{g(x)}\\,dx\\right|^2\\leq\\int_{\\mathbb{R}^n} \\left|f(x)\\right|^2\\,dx \\cdot \\int_{\\mathbb{R}^n}\\left|g(x)\\right|^2\\,dx.", "63776e5489fa10aa6eb34fde63110eb2": " \\mathsf{S}\\times \\mathsf{T} = |\\mathsf{S}||\\mathsf{T}| \\sin\\hat{z} \\mathsf{N}. ", "637795b79b2a91b6378676bf4129a055": " x = x \\to y ", "637802ee4dcba118e0896f2aab0cc33e": "\\text{Tr}\\left\\{ \\Pi_{\\rho_{x^{n}},\\delta}\\right\\} \\leq2^{n\\left[\nH\\left( Y|X\\right) +\\delta\\right] },", "63789b1f628836f7327a314fa1b57129": "P_b = \\frac{1}{2}e^{-E_b/N_0},", "6378a68ec0be5fedd2718871f8d6024a": "n = 1000 \\log_2 \\left( \\frac{a}{b} \\right) \\approx 3322 \\log_{10} \\left( \\frac{a}{b} \\right)", "6378ca68af34af8f1744d172520890fa": "\\omega_{N_\\ell} = \\exp(-2\\pi i/N_\\ell)", "6379060464fec74c5bbd6f6de378dc5f": "Q = z e", "63793bddc44d88016f02d12919204ba5": " T_2 \\cos\\theta - T_1\\sin\\theta \\, ", "63798943524dcee7c2971d8c38973687": " O(n^\\epsilon \\log n)", "6379db01a83f8c380cea1321f5e398cb": "T^*\\mathbf{P}^1(\\mathbf{C})", "637a00d058ab6ea287dee7ebe9c8e82a": "w(y)", "637a166fa51235d63d3c3e0e6a530400": " A \\to \\alpha \\in P ", "637a8b435f8a6e51ec2d18f0e58c296b": "x^k_1", "637ac69e560c69ea806e0ff6309da57d": " u(x) = \\frac{1}{2} e^{-x} \\sin(2x) ", "637ada355897ac580f389e9513cac7d3": "\\ \\gamma \\ ", "637b4380ed50dd25cbc3c494a26149c7": "e^+e^-\\to2\\gamma", "637b4ca109ad78ea8a3ee6219aa4e19d": "{\\Pr(A)=\\Pr(A|B_1)}\\cdot{\\Pr(B_1)}+{\\Pr(A|B_2)}\\cdot{\\Pr(B_2)}={99 \\over 100}\\cdot{6 \\over 10}+{95 \\over 100}\\cdot{4 \\over 10}={{594 + 380} \\over 1000}={974 \\over 1000}", "637b6b1d02f5906a019a29240a219b56": "A \\cup A\\,\\!", "637b7036a53a8603be45c7898b2c9792": " (H \\phi)(j) = E_j \\phi(j) + \\sum_{k \\neq j} V(|k-j|) \\phi(k)~, ", "637b834668e627630b1397145004d008": "\\vec \\omega = \\dot\\alpha \\bold u_1\n\n +\\dot\\beta \\bold u_2\n +\\dot\\gamma \\bold u_3", "637b936989b8dc0db26efe5ac7dcf7e1": "\\hat{c}^{\\dagger 2}", "637bad1441c9b635393642036b0ee253": "\\begin{pmatrix} 1 & 0 & 1 & 0 & 1 \\\\ 1 & 0 & 1 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 0 & 1 \\\\ 1 & 0 & 1 & 0 & 1 \\end{pmatrix} ", "637bd402823decb931ac239a07b9cd27": "\\{ (x,y) ~|~ 0 < y < f(x) \\}", "637c5ce8d76218c13b0ce962b991bcaf": "\\Pi_{Z \\gamma}", "637cd89544088e588e1ee9364f55d8ed": "\\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \nf( (x+b_ka) \\mod 1 ) = \\int_0^1 f(y)\\,dy ", "637d0dc0a81e51c20b8812bf6565a176": "\\mathbf{\\gamma_t}(i)", "637d199fa99056c805e3042a3edecffc": "x_0,\\ x_1,\\ x_2,\\dots,\\ x_n,\\dots", "637d1a60cb53a560cea8fd42a4f0c4ce": "\\frac{\\partial u}{\\partial t} = F\\left(u, x, t, \\frac{\\partial^2 u}{\\partial x^2}\\right)", "637dba127d5dacf0182562af78218cb3": "\\text{Berry phase} = \\frac{1}{\\hbar} \\oint \\vec{A} \\cdot d\\vec{X}", "637dcfd2d5e5a2937dd03342821bdb1f": " y_H (A_0, A_1) (x) = \\left(A_0 \\sinh \\left (\\sqrt{\\omega^2 - \\omega_0^2} x \\right ) + A_1 \\cosh \\left ( \\sqrt{\\omega^2 - \\omega_0^2} x \\right ) \\right) e^{-\\omega x}. ", "637dd68021bf8ff575f356e1ef2d8b31": "\nP^{-1} = \n\\begin{bmatrix}\n\\frac{right-left}{2} & 0 & 0 & \\frac{left+right}{2} \\\\\n0 & \\frac{top-bottom}{2} & 0 & \\frac{top+bottom}{2} \\\\\n0 & 0 & \\frac{far-near}{-2} & \\frac{far+near}{-2} \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix}\n", "637e2b26d5b9756f93632f0c1b21491a": "\\text{S}_j", "637e3c5e33c10a250d2ed3e2e5d0cf08": "\\nu=0.5", "637e62807b201f584a840bd31400f956": "\\beta_R", "637e7d5a4fcebf0a9e2850df6f213572": "\\sqrt{k}(l+\\frac{5}{2})=E_{l}.", "637eadf2f2190538b70cc6396601e9d4": "d(x,y) 0", "639d8780819c992786361384a3241707": "\\sigma(g) \\in H", "639d9f9322aed3b94250a7cde2d67621": "M \\gg m", "639da43e4dfae72e078edaca0386d8be": " \\lim_{n \\to \\infty} \\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=\\ell.\\ ", "639dd820337291494d5b6aa5a9fe1404": "\\frac{N_J}{N_0}=e^{-\\frac{E_J}{kT}} =e^{-\\frac {BhcJ(J+1)}{kT}}", "639e122f7656c6da0b767e7ca30b6c15": "K = 4C(2C-1)", "639e98fe99bd41a873d871a74e547fe7": "v' = RvR^{-1}.\\,", "639e9cd9cf31b2d0386c179fd3049856": "I = m R^2", "639f816d257efa21bd03c5c52d3ad164": " \\|a-b \\|_2 = \\sqrt{\\sum_i (a_i-b_i)^2} ", "639f93990828d1cd1074b462bf7dfa76": "\\sum_{j=0}^{n} \\binom{n}{j} (-1)^j (n-j)^k.", "639f9f7bcd1b0a904fd72716c11e029d": "\\mu = \\mu_+ - \\mu_-, \\;", "639fe3f7f727254f04adf4b5d5def72d": " T_m(0) = t_{m,0}=t_{m,m}=0 ", "63a01b8d6cd743525e825d1152a11f04": "\\mathbf{A} = A_\\rho\\boldsymbol{\\hat \\rho} + A_\\phi\\boldsymbol{\\hat \\phi} + A_z\\boldsymbol{\\hat z}", "63a01bbf56bcdc1f52bb72939d16232f": "\\zeta = \\frac{\\eta \\lambda E_{s}}{\\varepsilon_{r}\\varepsilon _{0}(\\rho -\\rho _{0})g }", "63a07c55c812d2f6982a1881869f82bd": "(g_j)_{j \\in I}", "63a1400a5cad69bc48bb3843e475ece5": "h_i(k) = \\#\\{q \\ne p_i : (q - p_i) \\in \\mbox{bin}(k)\\}", "63a1a3b185b1df7174d8b986f76b655b": "U^* U = UU^* = I \\,", "63a1b25a3e2867923ca3f011fa6cba8c": "b:1{\\to}B", "63a1e1a5a5a28b0cf5e7687836075240": "2^{3}", "63a1e9eaf1a00af2c986a017325dd8a1": "z = \\frac{v_{Hubble}}{c}", "63a2b3d26b6154f1c8995bc927fe1303": "\\sigma^j", "63a2eb6f60b1db7f68eb78f52fa9863f": "V(a,z)V(b,w) = V(b,w) V(a,z) = V(V(a,z-w)b,w).\\,", "63a323d6f4fbcc46395c1395f103bf50": "\\lbrace w_t \\rbrace", "63a3407b982c3d8c456cd68b7f0d2cf1": "\\sqrt{g^{ac}\\dot{r}_c g_{ab}\\dot{r}^b}", "63a35af4d67266ef2733bed42fa6ba08": "A_\\mu(x)", "63a367fbcb3e19ca5c715b9082b11851": "A_1=(x_1,y_1)", "63a38f0173c9b3c35ce2caba310a92fe": "A^s=(\\lambda I+A_{\\lambda,1})^s=\\sum_{r=0}^s\\binom{s}{r}\\lambda^{s-r}A_{\\lambda,r}", "63a3b5ef8f44df57f98202bab411164e": "e^{i x} = \\cos x + i \\sin x,", "63a3dca70ae44b4b03066d4622ba1989": "\\alpha=-\\mu/kT", "63a42b3d9b9fc9ea43b3659d815c9fb0": "y(t)-r=0", "63a54aafd4b442d5b0a20bff9722de7e": "4a^2x^2 + 4abx = -4ac", "63a56568f5fed102b3b6ac8a5d724ab2": "\n \\nabla\\cdot\\boldsymbol{s} = (2\\mu + \\lambda)~\\nabla(\\nabla\\cdot\\mathbf{v}) -\n \\mu~\\nabla\\times\\nabla\\times\\mathbf{v}~.\n ", "63a6017ef19741fd7dc8ea7782b15869": "\\frac{(2 d_1 + d_2 - 2) \\sqrt{8 (d_2-4)}}{(d_2-6) \\sqrt{d_1 (d_1 + d_2 -2)}}\\!", "63a6157e34c39c210c89c9dd1be03575": "f = V_S / V_O\\;", "63a63f2f60f21d5211ce1194efb0ad60": "w_{n,i_n}(p_n(t))\\in [0,1]", "63a661e2f2142f6229d7a9ee3ba1c3e5": "S(x)=S_0(x)+C(x)\\,", "63a666a02cb76155f8f658d1f5be57d0": "s := \\langle s_{\\alpha}| \\alpha < \\gamma\\rangle", "63a6bbb4faa2bdfa1ff56af5e093e4d0": "v_n(m_1)", "63a6d275b01cb47e4a7838623ed5f6e6": " ( \\lambda \\mathbf{A} )^k = \\lambda^k\\mathbf{A}^k", "63a6f689e184506539bb2973020fe413": "\\displaystyle k=0", "63a6ff9626a22e917c8a6e2bf20d6a10": "b_0+b_1 (x-\\lambda)\\!", "63a71b7722cae714c1fe5127a760b0b7": "2^2\\cdot 3^2", "63a78ba67abd76ed5f9cc4d4df183dae": " x_1 \\ldots x_n ", "63a7a5d3640f98d9df1b901e099db616": "PR(u) = \\sum_{v \\in B_u} \\frac{PR(v)}{L(v)}", "63a7d6e64ae96a33d5cdd5aada6bd25a": "C_p/C_v\\,\\!", "63a7fb18e93a613bcc58cf8ba6a16cd9": " Z^+ \\subseteq (X^+ \\cup Y^+)\\setminus\\{e\\} \\mbox{ and }", "63a86b2ed813c105b2ac4b35a071038b": "N_A=N_1+2N_2\\,", "63a8bcc42b5bf1cd1805b33762128c09": "ds^2 = (f_1(x) + f_2(y))(dx^2+dy^2).\\,", "63a8c13997fce97016fc39e18028b06d": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 59.1\\cdot 0.95)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 12.0\\cdot R_{\\bigodot}\n\\end{align}", "63a9178d7b2336fc3b32a51c4e709058": "G(\\mathbf M) = (\\mathbf M \\otimes \\mathbf 1') * (\\mathbf 1' \\otimes \\mathbf M)", "63a92c43ee41b4a27685505a6d967bf5": "\\mathcal{Z}\\left( \\mathcal{S}\n\\right) ", "63a95c98c2c4f1c22f12774f749956d9": "\\kappa(X_1,\\dots,X_n)=\\sum_\\pi \\kappa(\\kappa(X_{\\pi_1}\\mid Y),\\dots,\\kappa(X_{\\pi_b}\\mid Y))", "63a9eeedd8f57a2774b804314ed0a7f2": "W_4 \\subset W_3", "63a9f0a2d5786ad738d39467135bafa8": "\\operatorname{Id}_k(n)=n^k", "63aa1624af8f677f56924bced334e571": "f(X; \\theta) = \\theta_1 g_1(X) + \\theta_2 g_2(X) + (1-\\theta_1-\\theta_2) g_3(X)\\,,", "63aa3a9a7f42a47c22bb9d39b1804a59": "g(x, y) = c. \\,", "63aa437ce51a24ed5e26e24f9233896c": " f \\sim f_0 + t (f_1-f_0)\\,", "63aac5767908d779dc3b29f37e0aba4c": " n = {\\frac{m}{M}} ", "63abad05c04d6fb531325911ff965267": "\\alpha_1,\\ldots,\\alpha_n", "63abcabc4e167c774b1bb9d292b85ede": "E(e)", "63ac20036f0e370f216f83531a7adc13": "\\mathfrak{u}", "63ac22d26e6420127b88efd79df9860e": "S_w(p) = \\int w(r) S_0(p-r)\\,d r", "63ac833f70ff0fc132533c200699c626": "\\vdash^*", "63ad7b4d2e9525afccbaf6f7d71a7a50": "\\mu(X)=c", "63ad8c9de4d3e52f94707bf9198b07b6": "\\{ (c_1,m_1),(m_1,c_2)\\}", "63adc14d26559dfbf220bafbf5c7af4b": "V_n(i) = \\{ v \\in V_n : v \\text{ has weight } i \\}", "63add0d8cf9bd27be95e086c00143e2a": "M_K", "63ade2164ebc9814a03923c123cfb346": "\ny_i^{[n]} = \\sum_{j=1}^s b_{ij} h F_j + \\sum_{j=1}^r v_{ij} y_j^{[n-1]}, \\qquad i=1, 2, \\dots, r.\n", "63ae5ba92c4eef209df64a387b946702": "\n\\frac{x^2}{a^2+\\lambda}+\\frac{y^2}{b^2+\\lambda}+\\frac{z^2}{c^2+\\lambda}=1.\n", "63aef544cf49d88f9f3661cd516c759e": " \\bigcup_k B_k = A, ", "63af623f3a71ddae5a592d90389b87c0": "\\scriptstyle{\\varkappa_{\\alpha}^{\\beta}=\\frac{\\partial \\gamma_{\\alpha}^{\\beta}}{\\partial t}}", "63af8e19cadc17353fc0b23606f2e967": "l=L_B(P)\\,", "63b03883845ec64aa5c0c15f73e8ffdf": "C = \\frac{\\partial}{\\partial T}\\int E\\, f(E)\\, g(E) \\,dE", "63b06826766930a1a7113b55ad1a7d8c": "D = .0003(np^2)^{1 \\over 3}", "63b08cc45ec2589545911a46086fe88b": " -n(n-1)~r^{-n}~\\sin(n\\theta)\\, ", "63b09ac9d621fb20544b9f7468a91c85": "= Y + U \\sin (\\omega t) + V \\cos (\\omega t) +", "63b0c9797888c86c04e4bd9019056e0d": " P( | X - \\operatorname{ E } ( X ) | \\ge k ) \\le \\frac{ \\operatorname{ E }(| X - \\operatorname{ E }( X ) |^n ) } { k^n } ", "63b0f6b6e1e28a26ba045adc0ad8af25": " \\max \\left [ F_{r} , \\left ( D_{r} + \\frac{ F_{r}}{2} \\right ), 0 \\right ] ", "63b1ffbcd6631ab49327451f7cab2ed4": "D=[K(Q):K]", "63b217ef1393d133c609963298198b68": "a=0\\quad", "63b2361312245858561cc22c6063d437": "A \\subset \\{1,\\dots, N\\}", "63b2e0b6e73f21035ab96273870270e8": "D = \\{(x,y)|a\\le x\\le b, g_1(x) \\le y \\le g_2(x)\\}", "63b30539a315d80df4af48dd2a4c7f49": "-\\lambda ", "63b3428c90cbf31476f1e80403df1261": " g_{ij} = \\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\\cdot\\cfrac{\\partial\\mathbf{x}}{\\partial q^j}\n= \\left( h_{ki}\\mathbf{e}_k\\right)\\cdot\\left( h_{mj}\\mathbf{e}_m\\right)\n= h_{ki}h_{kj} ", "63b394ea90a939d9236639907af73cc3": "\\theta=\\exp(-[\\beta_1X_1 + \\cdots + \\beta_pX_p])", "63b3db91e46e7965353f1866f5b56d82": "z_8=\\chi_{\\psi_{8,8}}(z_8,\\rho_{\\psi_{4,8}}(z_4))=\\chi_0(z_8,\\rho_{1}(z_4))=\\sin x_1+x_1", "63b3dcaf9256d30fc8e8b6c3800092b8": "f * g = h \\, ", "63b3e60ffa582772b2b05fcd5be19818": "B(x, r)", "63b3edeb03ac1fba397d89d5f1bc98c9": "2 \\int_0^\\infty e^{-x^2} dx=2\\int_0^\\infty \\frac{1}{2}\\ e^{-t} \\ t^{-\\frac{1}{2}} dt = \\Gamma\\left(\\frac{1}{2}\\right) = \\sqrt{\\pi}", "63b4169030c03c91ba58a575a2375670": "\\sin C = \\frac{\\sinh c}{\\sinh b} \\,", "63b41ccd44924bb76743d5ba6ca7403a": "t_0 = a, t_1, t_2, \\ldots, t_n, t_{n + 1} = b", "63b46c9e6e0a335b59304aae9856779f": "\\scriptstyle z = \\cos\\frac{1}{2}\\alpha + \\sin\\frac{1}{2}\\alpha \\hat{\\mathbf{v}}", "63b48723fadace43c7de8b3da6a02051": "\\forall x (F(x) \\leftrightarrow x=n)", "63b490345ceaf9174a8a7f6ee505e8ce": "\\lim_{n\\to\\infty}\\frac{F_{n+\\alpha}}{F_n}=\\varphi^\\alpha", "63b5202ce3e1456f8d7c85f27559d2f6": "x=u", "63b58c2c401451b5f5e72f8868bf5490": "\\theta'_i", "63b607b325fb8efa1b11dc5c97569f63": "\\xi =\\frac{\\rho \\omega}{3c\\gamma^3}\\left ( 1+\\gamma^2 \\theta^2 \\right )^{3/2}", "63b67a43dce25f096f402802c309ef98": "\\left(E + m \\right) \\chi = \\left(\\vec{\\sigma}\\vec{p} \\right) \\phi \\,", "63b69293ec52ce6c81e2aba4842ebd61": "f n,", "63b8ce854193e1e2b9f64db8b7313179": "a_0.\\,", "63b8d55955cf2409cb46f945812724e0": "m\\colon U \\rightarrow [0,1].", "63b8e8d8478b132018fb9ca875404daa": "g(k) = \\frac{2 e^{-k}}{(1+e^{-k})^2} \\mbox{ for } k\\geq 0. \\!", "63b92425205dbfce6d37e5e32b473e11": "k_\\theta", "63b9d36e0aff2b0f7567a583304b8eb9": " \\vec{X} = X^j \\, \\partial_{x^j}", "63b9e8b9d48bc24aa95810d2d1b54451": "S_3 = \\_", "63ba0e3b678a5067d9a877bed5eb17cd": "f(\\mathbf{x},\\sigma)", "63ba629b0a85717df5cde5b27955dc21": "\\rho\\circ\\tau=\\rho\\circ\\sigma", "63ba826c38c0ad525aaf198768ca51c4": " \\sigma(E) \\propto \\pi (R+\\lambda(E))^2 ", "63ba9b10ea9786a26e53d705e2c71e19": "h:\\Gamma^*\\to \\Sigma^*", "63baa5179591ecbc88dd96763421ee27": "\\theta\\in\\Theta_i", "63bace9ac8d62073a5b4f115e7936928": "V\\ =\\ \\sqrt{\\frac{\\mu}{r}},", "63bae38477e83a189d7347171ed83cb0": "\\mathbf{l}_{k}", "63bb3bad605f0b650d291965efede0ab": "h / n", "63bb8d64ab27aba8bd8653f53dd56345": " \\begin{align} a_{n+1} & = \\frac{a_n + b_n}{2}, \\\\\n b_{n+1} & = \\sqrt{a_n b_n}, \\\\\n t_{n+1} & = t_n - p_n((a_{n+1})^2 - (b_{n+1})^2), \\\\\n p_{n+1} & = 2p_n.\n \\end{align}\n", "63bb971dae66fffcd9236215b0620c4a": "\\int\\frac{\\mathrm{d}x}{\\cos ax} = \\frac{1}{a}\\ln\\left|\\tan\\left(\\frac{ax}{2}+\\frac{\\pi}{4}\\right)\\right|+C", "63bbe3b1a6233afe82aa9aee5cbaa310": "\n\\frac{[\\mathrm{Ox}]}{[\\mathrm{Red}]}\n= \\frac{\\exp \\left(-[\\mbox{barrier for losing an electron}]/kT\\right)}\n{\\exp \\left(-[\\mbox{barrier for gaining an electron}]/kT\\right)}\n= \\exp \\left(\\mu_c / kT \\right).\n", "63bc23156008da5f4f2fc1f0fe2a4a9f": "\\top", "63bc4a49649a592b2e82393751273896": "\\psi_\\mu", "63bc88fe497737781b7d2de0f7bccf96": "I_{sp} = c \\ \\sqrt{2 \\eta - \\eta^2}", "63bcaaada600cf71e36361111c2de574": " \\frac{b-a}{90} (7 f_0 + 32 f_1 + 12 f_2 + 32 f_3 + 7 f_4) ", "63bcabf86a9a991864777c631c5b7617": "delta", "63bcccca3e81a56a34e3f90ff614edbe": "|f| = \\sum_{\\,\\,v\\in V} f_{sv} ", "63bcd23182f50c590dd575e7ccb51750": "\\frac{1}{2}(|0\\rangle + |1\\rangle)(|0\\rangle - |1\\rangle).", "63bd6e2aa265cc11da540c570820d37f": "b+c+\\dotsb > 1", "63bd8831cf30f45d43bbddb803aaf887": "\\bar{c} = \\sqrt{\\frac{8 k_BT}{\\pi m}}", "63bdfaba97505af660ce93284a87fcfe": "I(rain;dark|cloud) \\leq I(rain;dark)", "63be0b805d1e0f33f2579f130b3ba3fd": "T(n) = 73n^3 + 22n^2 + 58", "63be1641c7292fa21ac9649549331b9b": " f_1 = \\frac{\\nu}{\\lambda} = \\sqrt{\\frac{F}{\\mu}}.", "63be1cbfbaaaf7defd274974d925c8fb": "\\frac{bpm(sp)}{60}+1=B", "63be61a6199c6a67d39fb6f7d0e19cf7": "A_{i} \\in \\mathbb{P} \\{expr_{1}, \\dots, expr_{n}\\} \\setminus \\{\\{expr_{1}, \\dots, expr_{n}\\}\\}", "63be687f5487e1db6a76c88d5d2d5c3e": " \\nabla \\times \\left( \\mathbf{\\nabla \\times F} \\right) = \\mathbf{\\nabla} (\\mathbf{\\nabla \\cdot F}) - \\nabla^2 \\mathbf{F} \\ , ", "63be848adda82f04ccf165e829b15326": "\\mu_{12}", "63be84aae0ca1995af2d779096090fdc": "[x]=[y]", "63bef89194f78f1c76d0f9cd4e00d3c0": "\\Delta(\\tau) = (2 \\pi)^{12} \\eta(\\tau)^{24}\\,", "63bf09998be3ed2ddd91ba88c45dd9bf": " K = \\frac {\\dot{m}}{C_{\\infty}} \\qquad (10b)", "63bf217b3183727aec6628d8d51b886d": "N = log_{B} M", "63bf2f0ff20c76c3582ae3e19d9480ff": " f(z) = \\alpha x + \\beta y + \\eta(z)z \\,", "63bf37488f23acc1503b938826f05962": "F_D = \\frac {3 \\pi \\eta V d}{C_c}", "63bfe9386fca6b444c418ab4c038ccf1": "dw^2 + dr^2 + r^2 d\\varphi^2 = -c^2 d\\tau^2 = \\frac{dr^2}{1 - \\frac{r_s}{r}} + r^2 d\\varphi^2", "63c0051740e25b1ee7f2d6905169ba6f": "b^2=pa", "63c0325d9525bcb190cca8148c62902c": "\nE_{k}(r_{k}^{A}) - E_{k}(r_{k}^{B}) + \\sum_{l=1}^{N} \\min_{X} \\left(E_{kl}(r_{k}^{A}, r_{l}^{X}) - E_{kl}(r_{k}^{B}, r_{l}^{X})\\right) > 0\n", "63c06eb4722c0bcb70e8ede8aee23c8b": "f(x)=\n \\left\\{\\begin{matrix}\n a, & \\mbox{if }x=1 \\\\ d, & \\mbox{if }x=2 \\\\ c, & \\mbox{if }x=3. \n \\end{matrix}\\right.\n ", "63c0786ef710cd318d2c2e41f653ce72": "dA_1/dt = \\kappa_1 A_1 + (Q_{11} A_1^2 + Q_{12} A_2^2) A_1", "63c0c0f2e9d73be3531d081b1f54d981": "\\delta(-x) = \\delta(x)", "63c0e8adf87fe71f52154e5c857ecb14": "v_g = \\frac{\\Delta x}{\\Delta t} = \\lambda_{mod}\\Delta f =\\lambda^2 \\frac{\\Delta f}{\\Delta \\lambda} \\ . ", "63c0edc4713a9ec99e9e8625e7479804": " |\\mathbf{R}_{AB}| > |\\mathbf{r}_{Bj}-\\mathbf{r}_{Ai}| \\quad\\hbox{for all}\\quad i,j.\n", "63c10f885c7eff546f4bb47b26e75dd3": "x = x(q_1,\\ q_2,\\ q_3)\\ ;", "63c1975f5459921e94f242fa80cca4b4": "KIE = {k_1 \\over k_2} = \\frac{\\ln (1-F_1)}{\\ln (1-F_2) }", "63c265ce4ec4c2a1eedb1216e18e3ea1": "S=\\{a_1+\\cdots+a_n:\\ a_1\\in A_1,\\ldots,a_n\\in A_n \\ \\mathrm{and}\\ P(a_1,\\ldots,a_n)\\not=0\\},", "63c2a1ff76d8da9bc94074bf5456838c": "k_eQ_1Q_2/r", "63c2cb7ce8ea56baf745b364ad412051": "L(\\mathbf{x})", "63c31a4ed9783c9ae83bb74a710eb897": "\n(7.f)\\quad e^{2\\psi}=\\Phi^2-2C\\Phi+1\\,.\n", "63c33d3e0e375c7ba430ad351816f330": "\\frac{d^2 T_n}{d x^2} = n \\frac{n T_n - x U_{n - 1}}{x^2 - 1} = n \\frac{(n + 1)T_n - U_n}{x^2 - 1}.\\,", "63c39e63dd49b36b58e308a3c733a777": "x\\in\\mathcal{C}", "63c40222a649bc3f45f6e883b1206341": " \\sum_{i=0}^{\\infty} b^t \\left[u_0+u_1c_1-\\frac{u_2} {2} c_t ^2 \\right],", "63c42c5c7a3de9b4d24d49d2cc096c02": "\n J_1 = \\int_A \\left[\\cfrac{\\partial W}{\\partial x_1} - \n \\sigma_{jk}~\\cfrac{\\partial\\epsilon_{jk}}{\\partial x_1}\\right]~dA\n ", "63c459245ea9aa13b579ea6cf4665ae2": " q = \\left\\lfloor\\frac{n+b}{d}\\right\\rfloor d - b", "63c466d94804559a0cb2e5a4bb7b53c5": "\\chi^2=\\frac{(ad-bc)^2 N}{ABCD}\\;(=336,\\text{ for data in Table 3; }P<0.001)", "63c479ce5a1efc8d4ea325470ef601ee": " \\mathbf{v}_\\mathrm{p} = \\lambda f \\mathbf{\\hat{e}}_{\\parallel} = \\left ( \\omega/k \\right ) \\mathbf{\\hat{e}}_{\\parallel} \\,\\!", "63c4fe140ed46ad48f64711e0a8b896c": "cos\\phi\\,", "63c56daa797f1183f7a78ff26b361c05": "|Coin_i\\rangle =\\frac{1}{\\sqrt{2}}|0,0,\\ldots,0\\rangle + \\frac{1}{\\sqrt{2}}|1,1,\\ldots,1\\rangle", "63c56f51417d99359e45a6f1eeb5fbfd": "\n\\begin{align}\n g'(x)^4 \n& & \\leftrightarrow & & 1+1+1+1 \n& & \\leftrightarrow & & f''''(g(x)) \n& & \\leftrightarrow & & 1 \n\\\\[12pt]\n g''(x)g'(x)^2 \n& & \\leftrightarrow & & 2+1+1 \n& & \\leftrightarrow & & f'''(g(x)) \n& & \\leftrightarrow & & 6 \n\\\\[12pt]\ng''(x)^2 \n& & \\leftrightarrow & & 2+2 \n& & \\leftrightarrow & & f''(g(x)) \n& & \\leftrightarrow & & 3 \n\\\\[12pt]\ng'''(x)g'(x) \n& & \\leftrightarrow & & 3+1 \n& & \\leftrightarrow & & f''(g(x)) \n& & \\leftrightarrow & & 4 \n\\\\[12pt]\ng''''(x) \n& & \\leftrightarrow & & 4 \n& & \\leftrightarrow & & f'(g(x)) \n& & \\leftrightarrow & & 1.\n\\end{align}\n", "63c575e70ff01dd8756830a417918ae9": "(2-\\epsilon)k", "63c58de86a9608f6a5b380454b12a733": "\\frac{2} {3} \\times ALI + \\frac{1} {3} \\times GEI", "63c5957f61a61c9ddeb002bdb1b05f74": "\\|-x\\| = \\|x\\|.", "63c5d99ffd0943799b20f31944fa09a7": "O(n+k)", "63c5de39ed351e6fbd2c82319332f6df": "\\nu(Y')=1", "63c671e5c39dd5c235b6de993988a362": "MU^2/2", "63c6bf7acd6099fcfd4f4a6b2e8c27cc": "\\frac{\\partial g(u)}{\\partial u} \\frac{\\partial u}{\\partial \\mathbf{X}} ", "63c6e8c1c979a43e6efdba8d17f4760a": "V_H", "63c70c5df4cb882cd125891947037c31": "X = Y \\cup Z", "63c73877d4bb2418330232fabbbe2934": "x_1=x_2=-\\frac{b}{2a}", "63c73a266f684283ebd6689e806e5897": "\\lambda = \\ell(\\ell+1)", "63c741bcc6731a4410e322679ee87000": "w>x>y>z", "63c763b0c9ae22d061528e1bc510f3f5": "T\\Omega\\leqq2\\pi \\,", "63c7746cd6d1fe6701d12c6be2320bec": "=\\,\\dot{m}\\,\\bigg[v_e + \\bigg(\\frac{p_e - p_o}{\\dot{m}}\\bigg)A_e\\bigg]", "63c78bbf4bcffc7199725dc23c6a12f2": "\\mathrm{k_{leg}}=\\frac{\\mathrm{peak\\ force}}{\\mathrm{peak\\ displacement}}", "63c813fd31abe4c89bc7c27c82e80613": " \\lambda = 10.66a ", "63c823c8c11684305b273ba83450b4b2": "|X|^{|Y|} = \\left|X^Y\\right|", "63c825d178d6df16217d7510fe35ae54": "a_i^\\dagger (k)", "63c85b78148359757e2a03dd0933cb72": "x\\in X^{m}\\,\\!", "63c885a08c30438884928785e3993057": "\\left(b, z\\right)> \\left(c, y\\right)", "63c8f14e02cb3cd962d99590c16da7ce": "\\Rightarrow V_\\mathrm{BE} = V_\\mathrm T \\ln \\frac{I_\\mathrm C}{I_\\mathrm{SO}}", "63c93dd51d8b6b915aa1440102e48318": " {d \\over d\\tau} {\\partial \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} \\over \\partial \\dot x^\\lambda} = {\\partial \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu} \\over \\partial x^\\lambda} ", "63c953fdadc267ebb405a910ebbc72fd": "\\mathrm{d}s^2=\\left ({\\ln(W)\\over \\pi k}\\right )^2\\, \\mathrm{d}\\varphi^2+e^{2\\ln(W)\\varphi\\over \\pi}\\eta_{\\mu\\nu}\\,\\mathrm{d}x^\\mu\\, \\mathrm{d}x^\\nu.", "63c9712bd94a7b9c20394c7078340e62": "MPS=\\frac{dS}{dY}", "63c9e57adf519d70aab73ffc86a20f06": "\\mathbf{u}, \\mathbf{v} \\in V\\,\\!", "63ca187391bfa365392d53ece0abf80a": "\\langle\\chi|\\psi\\rangle = \\int\\chi^*\\psi \\mathrm{d}x^1\\wedge\\mathrm{d}x^2\\wedge\\mathrm{d}x^3 ", "63ca45631d98b0c8a5cd486963e6e4f7": "y^3 + y^2 = \\frac {ca^2}{b^3}", "63ca88288940ed3193e9306c553f8e06": "I(x) = \\sup_{\\theta > 0} [\\theta x - \\lambda(\\theta)],", "63ca9f39e1e806dc314614bc55a26e3c": "V_c = \\sqrt { \\pi } V_r \\mathrm{e}^{ {V_r}^2 } \\mathrm{erf} ( {V_r} ) ", "63cac60c2ca957d20d4784bab8d8d945": "\\Pi_{i \\in I}A_i", "63cace2657ddd32f7cc6b770a34ec37e": "c_{n+1} \\circ \\gamma_n + \\gamma_{n-1} \\circ c_n = \\operatorname{id}_{C_n}", "63cb092cba7e1031d7c8729f4bace63e": "2\\boldsymbol{\\alpha}{S}^{-2}\\mathbf{1}-(\\boldsymbol{\\alpha}{S}^{-1}\\mathbf{1})^{2}", "63cb382cfa69486fa94a0a964ec23966": " \\mathbb{Z}/2^n \\mathbb{Z} ", "63cb4a9a9aa2a9c685da60882253ff46": "V_0=1\\qquad V_{n+1}=S_n/(n+1)", "63cb64b1d811624ca33b91719a7cbf9b": "\\widehat{G}", "63cb8d95b623e14fce1983593f7c3ad0": "\\nabla \\cdot \\textbf{H} = 0\\,", "63cbcef3ce78e0003b571c581527c48e": "\\textstyle 5.\\ Capon's\\ Beamformer", "63cc037a761e4efb8afc05813b3a053e": " w = x + y \\!", "63cc16c7fec027cd92eecfb5c57136cc": "\\nabla_S u = \\nabla u - \\mathbf{\\hat n} (\\mathbf{\\hat n} \\cdot \\nabla u)", "63cc49b0113f53fa7e9e45ea70f47e3e": "\n f(\\theta x + (1 - \\theta) y) \\geq f(x)^{\\theta} f(y)^{1 - \\theta}\n ", "63cc5baddcfcd257838701d376627fd2": "\\frac{d^2\\mathbf{r}}{dt^2} = \\left( \\ddot r - r \\left( \\dot\\theta ' +\\Omega\\right) ^2 \\right) \\hat{\\mathbf{r}} + \\left( r\\ddot\\theta ' + 2\\dot r \\left(\\dot\\theta ' + \\Omega \\right) \\right)\\hat{\\boldsymbol\\theta}", "63cc84bdd1c27a99858aa491c6d00501": "D f : U \\to L(V, W) \\,", "63cce6b836de9495cc85781300051fbe": "I=GV", "63cd13a569c897702b57a27631018266": "\\neg K_i \\varphi \\implies K_i \\neg K_i \\varphi", "63cd61f3104245f6b10d94e1a03f8b11": "\n{e}^{tA}\\!=\\!\\begin{pmatrix}{e}^{t} & t{e}^{t} & \\left( 8t-48\\right) {e}^{t}\\!+\\left( 4t+48\\right){e}^{3t/4} & \\left( 16-2\\,t\\right){e}^{t}\\!+\\left( -2t-16\\right){e}^{3t/4}\\\\ 0 & {e}^{t} & 8{e}^{t}\\!+\\left( -t-8\\right) {e}^{3t/4} & -\\frac{4{e}^{t}+\\left(-t-4\\right){e}^{3t/4}}{2}\\\\ 0 & 0 & \\frac{\\left( t+4\\right) {e}^{3t/4}}{4} & -\\frac{t {e}^{3t/4}}{8}\\\\ 0 & 0 & \\frac{t{e}^{3t/4}}{2} & -\\frac{\\left( t-4\\right) {e}^{3t/4}}{4}\\end{pmatrix}\n", "63cdaf8d87b72c6318f539e4ed096eb4": "B \\neq \\emptyset", "63ce22813e8b654f788c3c4964ac1a5d": "F_\\oplus(x,y) = \\min\\{1, x + y \\}.", "63ce276f4926b9aac3aab8d267ca0020": "\\Xi = \\frac{P V}{T} + \\sum_{i=1}^s (- \\frac{\\mu_i N}{T}) -\\frac{P V}{T}", "63ce4c1c0ea8bb6b37d77684205f54dc": "S_{mn}(\\sigma) = A_3\\,y_1(n^2/2m,\\sigma\\sqrt{2m}) + A_4\\,y_2(n^2/2m,\\sigma\\sqrt{2m})", "63ce5dc5f9706732c513c48743712f32": " 5 c_1(X)^2 - c_2(X) + 30 \\ge 0 \\quad (c_1^2(X)\\text{ odd}). ", "63ce74cb37d9c3034de31f813e42fe61": "(\\phi^{\\Rightarrow x} \\otimes \\psi)^{\\Rightarrow x} = \\phi^{\\Rightarrow x} \\otimes \\psi^{\\Rightarrow x}\\,", "63ceb76422da83e86c6b68ee806b5b98": "\\Pi'(n,k) = \\int_0^{\\pi/2}\\frac{d\\theta}{(1+n\\sin^2\\theta)\\sqrt {1-k^2 \\sin^2\\theta}}.", "63cecbc114cf407281d23ec3ba8c5f8e": "{n-1\\choose k-1}", "63cf3d05728b50146a8427b6ffc23fb9": "{dQ_g \\over dt} = F_g (C_{art} - {{Q_g} \\over {P_g V_g}}) + K_a Q_{ing}", "63cf4b20c974eb3f9caf6f3831e75e43": "\\left(\\frac{d t}{d x}\\right)^{n - 2} = (2 - n) \\int f(x) dx + C_1", "63cf6480136fe7e2399f99bab070b1f6": "\\alpha = \\arctan{\\frac{\\Delta h}{d}}", "63cfe1106fbc42b82605bffa7bdf1833": "Vd = \\frac {Ab}{Cp}\\,", "63d01961bc36e21dadf8a88e71085d3f": "\\ \\begin{align}\n\\sum F_{x'} &= \\sigma_\\mathrm{n} dA - \\sigma_x dA \\cos ^2 \\theta - \\sigma_y dA \\sin ^2 \\theta - \\tau_{xy} dA \\cos \\theta \\sin \\theta - \\tau_{xy} dA \\sin \\theta \\cos \\theta = 0 \\\\\n\\sigma_\\mathrm{n} &= \\sigma_x \\cos ^2 \\theta + \\sigma_y \\sin ^2 \\theta + 2\\tau_{xy} \\sin \\theta \\cos \\theta \\\\\n\\end{align}", "63d093cca6fe8eeccb53fbe5ec978ef6": "\\left(\\frac{1}{z}\\frac{d}{dz}\\right)^m\\left(z^{n+1}f_n(z)\\right)=z^{(n-m)+1}f_{(n-m)}(z).", "63d0de849125b812a695a5b6e8a93974": "S = \\int d^4x \\sqrt{-g} \\left[ \\frac{1}{2\\kappa}\\left(\\Phi R - V(\\Phi)\\right) + \\mathcal{L}_{\\text{m}}\\right].", "63d142a9d6d0a1235d19a2a906dbc11b": "\n{A_\\mathrm{v}} \\approx \\begin{matrix} \\frac {g_m R_\\mathrm{L}} {1+g_mR_S} \\end{matrix}", "63d1500e0709c14a464f859c52ae63d2": "\\mathbb{F}_{p^n}", "63d175aad8c8713a7199908f464d15a0": " W_\\mu \\otimes W_\\nu = \\bigoplus_\\lambda c_{\\mu \\nu}^{\\lambda} W_\\lambda. ", "63d191b2da85695fe7ec1e35ccfd0bbb": "\\bigcup_{\\sigma\\in S_n}A_{n,\\sigma}(s,t)\\subset I_{s,t}^n.", "63d1e0add360a49e0fbd4eca72e4b978": "2:\\pi ", "63d274612755ea45ab13105decb4ef3c": "s_1=h_1(x_1,\\ldots,x_m),", "63d2c3abbbe86eac500dbdbf09362ab4": "1-2x+x^2-yz^2=0", "63d2ea5103f3999cecab8c4729697d9f": " \\lambda(\\tau) ", "63d34efe3647bdd7c997142e08080cf2": "^\\dagger:X\\rightarrow X^\\dagger", "63d3a220ab36645f069ddd47700f9063": "\\operatorname{Bun}_G (X) = \\operatorname{Map}(X, BG)", "63d3dfc84b7425e579e597521584c346": "T(x_o,y_o) = \\iint T(k_x,k_y) ~ e^{j(k_x x_o + k_y y_o)} ~ dk_x \\, dk_y. ", "63d3fb532eddb9e6e2b2f9b6a35e006e": "A_{arbelos}=\\frac{\\pi r}{4}-\\frac{\\pi r^2}{4}=A_{circle}", "63d3fd4eca9710b68714a7d5b3d0610c": "\\tfrac23\\times\\tfrac55=\\tfrac{10}{15}", "63d4146d4fb839616725d1f9d35d8583": " \\pi_k(n) \\sim \\left( \\frac{n}{\\log n} \\right) \\frac{(\\log\\log n)^{k-1}}{(k - 1)!},", "63d4629e3e1fc04ad1bf602d71695913": "F = \\int d^dx A H^2.", "63d4bfb16c86be52e65ed3505ad1eba2": " n = x^2 + x. \\,", "63d5056a829453e19effa16078990845": "N(S)\\,", "63d56859bb5ab96ec81fe5a99a8c6c30": " \\overline{\\lambda}, \\underline{\\lambda}\n\\in \\{ \\lambda_t, t = 1, 2, \\ldots , n \\} ", "63d56c5320ef77012443e9e9726439fb": "Q(x'; x^t)=\\mathcal{N}(x^t;\\mathbf{\\Sigma})", "63d570cbee90917ce008978b7bc3bb96": "\\begin{array}{rc}\n& \\scriptstyle{\\alpha\\,\\, \\longrightarrow \\,\\, \\beta} \\\\\n& \\nwarrow_\\gamma \\swarrow \n\\end{array}\n", "63d57999fce93adc36e3526d4139f250": "\\nabla\\times(\\mathbf{J}/\\varepsilon)", "63d5adb8a6f84d997241caffe10caa3f": "C(X) > X^{0.332}", "63d5bab7d686b61054e01a4c0a897eb2": " e^- + A \\rarr A^* + e^- ", "63d5e2a45b7fb1baecd8c361c3053214": "\\bigcup_{c \\in C} X_{c}", "63d62aab93d08e51a457e7e5e5caa503": "\n\\begin{align}\n \\Psi(z) &=& Ae^{ik z}+ Be^{i[k -(2\\pi/a) ]z}.\n\\end{align}\n", "63d6dbb39f463133928dca47dbdb77f1": "x_{crit}=v t_{crit}", "63d6faad78d046eee50c0ec6a5c64f67": "H_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\mu}(C \\ e^{jk_{x\\varepsilon }w}+D \\ e^{-jk_{x\\varepsilon }w})e^{jk_{xo}(x+w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ (50) ", "63d6fc25f282df015fc1cd86bf636d30": " \\left[\\begin{array}{c}A\\\\\\hline I\\end{array}\\right],", "63d73162b36860483ef8e9b72213ddb2": "A\\in\\mathbb{M}_n \\left(\\mathrm{C}^0(\\mathbb{C}^d)\\right)", "63d75cb69dcc1e4a932c28f9031ceef9": "\\, b=N_\\mathrm{A} b'", "63d76fbad533d20ed8c3c9eb9ef08380": "\\vec{m}_0", "63d7cf3d5561d0d436089251932864a7": "\\Pi_0(x) = \\frac12 \\bigg(\\sum_{p^n < x} \\frac1n\\ + \\sum_{p^n \\le x} \\frac1n\\bigg)", "63d7df974c1ff4c9fb7c91d64e549206": "|p_z,i;\\uparrow\\rangle", "63d7e3dfe14b36708e11c13ea2c02778": " \\widehat{\\Omega}\\left|\\mathbf{r}(t)\\right\\rangle = \\left|\\mathbf{r}'(t')\\right\\rangle ", "63d856686afb4d2157b431240ac96b5d": "\\psi(x) = \\sum_{n\\le x} \\Lambda(n).", "63d8803f9e7f8e5ddc19cbd144822c9e": "\n s_n(T) = \\inf\\big\\{\\, \\|T-L\\| : L\\text{ is an operator of finite rank }0}\\prod_{\\alpha\\in \\Delta^+}\\frac{1}{(\\alpha^\\or, m_1\\lambda_1+\\cdots+m_r\\lambda_r)^{s_\\alpha}},", "63dd22a49d69507e52a4a0d787f79c36": "= \\frac{2L_{L} \\gamma (v)}{c}", "63dd4e7362e4bf395b74956df3e25aed": "\\left(\\frac{q}{p}\\right) = (-1)^{\\left\\lfloor\\frac{q}{p}\\right\\rfloor +\\left\\lfloor\\frac{2q}{p}\\right\\rfloor +\\dots +\\left\\lfloor\\frac{mq}{p}\\right\\rfloor }\n", "63dd9635d123422676b27d94fbed9a41": "[\\mathbf{x}] \\in [\\mathbb{R}]^{m}", "63ddd3e92a56e5e044abf5256a74879a": "G(t_n;\\frac{1}{2})=\\sum_{n=1}^{\\infty} \\frac{t_n}{2^n}", "63dde4f236ef672d6254db791ac703c2": "A(t)-\\alpha t {}\\approx \\hat{A}(t)", "63ddeff701c20735b23199862e5b69a0": "[0, c)", "63de26d32454170aca8dd69865d5d38e": "R(t)= \\left| \\frac{ \\left(1+4\\cdot t^2 \\right)^{\\frac{3}{2}}}{2} \\right| ", "63de4688a3fc7a54fe52d8010cd761cb": "S(\\boldsymbol \\beta^s)", "63de4764b9a526f93a65b2ab2a7481d0": " \\and (S_4 \\implies (\\operatorname{equate}[A_4, p] \\and V[F_4] = p)) \\and D[F_4] = D[p] ", "63de628745b8365de1aa28b218212c63": "u_{jk}(x,y)=\\sin(\\pi jx)\\sin(\\pi ky)", "63dea22b564ef28f7301888c33798e4f": "1, 8, 23, 77, 281, \\ldots", "63deb686c0c084598ecacbc2e3f42b7c": "\n\\begin{align}\nq & = { 0.15771 \\over 2} = 0.07886 \\\\ \\\\\np_1 & = (p + q)^2 = 0.84325 \\\\\n2q_1 & = 2(p + q)(q + r) = 0.15007 \\\\\nr_1 & = (q + r)^2 = 0.00668\n\\end{align}\n", "63df11768064755075cc94a31a855fdc": "\\Delta\\psi = \\frac{\\partial^2\\psi}{\\partial x^2} + \\frac{\\partial^2\\psi}{\\partial y^2} = 0.", "63df26b2c000d2a3476e1f6efe7bd8c8": "\n\\begin{align}\n\\epsilon&=-\\sum_{ij}(J_h\\mu_{ij}+J_v\\alpha_{ij}+J\\alpha_{ij}\\mu_{ij}+J'\\alpha_{i+1,j}\\mu_{ij}+J''\\alpha_{ij}\\alpha_{i+1,j})\n\\end{align}\n", "63df3d1dff8341e7661c012c1338250e": "A=\\frac{e^2B}{2.303 \\times 8\\pi\\epsilon_0\\epsilon_r kT}", "63df901248cc4d9c55267971fd0586f9": "\\gamma[0]", "63df9cf120715b082d986a5fb5254039": "\\frac{ad}{bc} = \\frac{a}{b} \\times \\frac{d}{c}.", "63dfcadb1a105c8940110da96a8f9809": "\\Phi(t_2,\\Phi(t_1,x)) = \\Phi(t_1 + t_2, x),\\, ", "63dfee90ecd2baf30869d89cf550076d": "\\mu (A)", "63e0002b410f9773616deb27573916d0": " \\frac{\\zeta ' (s)}{\\zeta(s)}= -\\sum_{n=1}^{\\infty} \\Lambda (n) e^{-slnn} ", "63e02264126dc1a1edf25e2443be79ef": "A(G)=\\{ \\rho : \\rho \\mbox{ defined as above }\\}", "63e0268a7206b4bdb867cdb4f9824f9a": " E = - \\frac{Z^2}{2n^2}E_\\textrm{h},\\qquad n=1,2,\\ldots . \n", "63e026a7948b0e67c9815b12c4acd0d0": " \\mathcal{H}", "63e0501636219bdb39fccc8969a59492": "\\begin{align} 2\\cdot R_*\n & = \\frac{(48\\cdot 2.09\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 22\\cdot R_{\\bigodot}\n\\end{align}", "63e0ad048170bcdcbb08cf610b9007dc": " \\lambda_n(t) = {n \\choose 2} \\frac{2 f(t)}{I(t)^2}", "63e0c90a3341612ee5a9ef2a9ee791f3": "Q_1(X)E_1(X)", "63e159ec07dc6268eea321fd8af1be17": " \\lim_{h\\rightarrow 0} \\frac {D_{KL}(X_{\\theta+h}\\|X_\\theta)} {h^2}\n \\ge \\lim_{h\\rightarrow 0} \\frac {\\Psi^*_\\theta (\\mu_{\\theta+h})}{h^2},\n", "63e15c96c34f9bb9316cf81bf734598b": "\\frac{1}{2}MV^2=\\frac{1}{2}M\\left(\\frac{mv}{M}\\right)^2=\\frac{m}{M}\\frac{1}{2}mv^2", "63e1e8e2823ede98f93216235dc9605b": " \\frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \\frac{(-x)^n}{n!} L_i^{(n-i)}(x);", "63e1f312e5e38b83e03131e4a781a20b": "S(z;u)=\\mathcal{X}^{-1}(z+\\mathcal{X}(u)),", "63e1fd88239155e07f0bd345e56863bf": "ax^2 + by^2 + cxy", "63e274def0f164c3fafc73b233e86b9b": "x_i-\\mu_i=l_{i1} F_1 + \\cdots + l_{ik} F_k + \\varepsilon_i.\\, ", "63e2edd7e8267896b64cf72c745b83c0": "\\,\\!\\zeta", "63e309e377a2fc1aab863807e2ff0fcb": "\nEY_{ij} = \\mu + \\alpha_i + \\beta_j + \\lambda\\alpha_i\\beta_j\n", "63e30ab562eca80290370e40e21ae00a": "\\left | \\alpha - \\frac{p}{q} \\right | = \\left|\\frac{f(\\tfrac{p}{q})}{f'(x_0)}\\right| \\ge \\frac{1}{Mq^n} > \\frac{A}{q^n} \\ge \\left| \\alpha - \\frac{p}{q} \\right|", "63e31baeae65d7074f5b011cff89728e": "P \\;", "63e36943a72ae0096fd356112417bd5c": "Scenario \\ II: \\qquad d_B = {\\left ( {\\frac {149,597,871 km}{696,000 km}} \\right )} {\\left ( 3.3 AU \\right )} = \\quad 710 R_{\\odot}", "63e37078c9a3a274ea495ac9706d389d": "\\tau \\in \\mathbb{H}_n", "63e4012f323214e11ccd89b5d29f6e98": "\n\\tilde{G}_\\epsilon(p) = e^{-\\epsilon {p^2/2} }\n\\,", "63e43ca2d6abd67c62c6874bd66bdc01": "\\alpha_1^{\\beta_1}\\alpha_2^{\\beta_2}\\cdots\\alpha_n^{\\beta_n}", "63e463b1205f654ac337aac9bf497698": "\\mathbb P \\Gamma (X, \\mathcal L (D))", "63e54bd97afb3f5ea43fb9f0b2490d3f": "(H^{s_0}_{p_0}, H^{s_1}_{p_1})_\\theta = H^{s_\\theta}_{p_\\theta}, \\quad s_0 \\ne s_1, \\ 1 < p_0, p_1 < \\infty,", "63e564b65d8f6f3d71aca59427dfb1e0": "a^2 - 61b^2 = k", "63e5873ac2fac09e82269c26184621df": "\n{\\mathcal L}_v=-\\frac{K}{32\\pi G}\\left[g^{\\alpha\\beta}g^{\\mu\\nu}(B_{\\alpha\\mu}B_{\\beta\\nu})+2\\frac{\\lambda}{K}(g^{\\mu\\nu}u_\\mu u_\\nu-1)\\right]\\sqrt{-g}\n", "63e59595e2fa5416385c6f3ee9132f8f": "(1-a)S \\pi r^2 = 4 \\pi r^2 \\epsilon \\sigma T^4", "63e5ea8efc20aa753dfc24915323a6c4": "\nT_{ij} = A_i B_j T_i T_j e^{ - \\beta C_{ij} } \n", "63e6402016495289e5fd89352af99208": " eRPF = RPF \\times extraction ratio", "63e64a758323f1351f1a7f3b72305753": "\\phi^*(\\mathcal{O}(1))", "63e673ed2da060bc80a4191cf213ead6": "Q_A = d_A \\times G", "63e6b28db300daa12a165e0a8ba56d3b": "\\beta=\\xi", "63e7591e19d81e34b711d1bf09fdf4a0": "\\begin{align}\n I_{o_{lim}} &= \\frac{V_i}{2L}D\\left(1-D\\right)T\\\\\n &= \\frac{I_o}{2\\left|I_o\\right|} D\\left(1-D\\right)\n\\end{align}", "63e7931d84a91fc5070c1c5aed277aa9": "\\scriptstyle [m,\\, 1.16m]", "63e7a2547d6e5088192e325b6772fbc8": "2pq", "63e7b98958ff451379ca29d04cffce57": "\\frac{256}{255}", "63e7c0326969ee637ebb96aac9e20ff3": "( x - x_1 )( y_2 - y_1 ) - ( y - y_1 )( x_2 - x_1 )=0.", "63e7d294770a5869e76ae9721d4d5a6f": "\\xi_{\\pm}", "63e83feabd3828ff115febf42300e335": " \\hbar ", "63e8ac14db05515beb4ae8296fced2f3": "C_2 = x^2+y^2+z^2+2X\\overline{X}+2Y\\overline{Y}+2Z\\overline{Z} ", "63e8d64e151b2f2a4cd6d942dd90e9d3": " x = \\sqrt[3]{6+x}. ", "63e90025f8fc65ede52b81b754369ea0": "\\int_{\\min_x \\in X}^{x_1} f_{\\theta_1}(x_1) f_{\\theta_0}(x_0) \\, dx_0 ", "63e91b4469b2575b6f18b36ab8807615": " R = \\frac{ a + x }{ b + y } ", "63ea818bed8e02373b6e97a64e6f3ca5": "I_{\\mathrm{ASE}}", "63eaf3a21c722c1d9495424d8ec5b75b": "J_z |\\psi\\rangle", "63eb1d94460708dfa1f88b1dc7301700": "\\mathfrak{so}(n, F)", "63eb32398cca9c6a02641c5e5e20ce19": "x_4=-0.5878", "63eb4bc49d020ad190c1bef47b986b6f": "4{\\frac{}{}}", "63eb558216b26a9e0ce91ab95c3d0ff6": "R_{CF}=R_{P} \\left( \\frac{B_{surf}^2}{\\mu_{0} \\rho V_{SW}^2} \\right) ^{\\frac{1}{6}}", "63eb79cc07920abeb0293a14da98064b": "a(u,v-u) \\geq f(v-u)\\qquad\\forall v \\in K.\\,", "63ebe0ea6f3d97a97834ccca237b452e": "\\left( f_{n(k)} \\right) \\subseteq (f_{n}) \\subset \\mathrm{Reg}([0, T]; X)", "63ec0a7a30a50562e97917a7fa9e1a3a": "I_i = \\left( \\frac{\\left( \\frac{1}{Z_i} \\right)}{\t\\left( \\frac{1}{Z_1} \\right) + \\left( \\frac{1}{Z_2} \\right) + \\,\\cdots\\, + \\left( \\frac{1}{Z_n} \\right)} \\right)I", "63ec24c1463acb7f60365327b8efbc6d": "(-\\pi<\\theta\\le\\pi)", "63ec4779a1f91c0bbc1fa8291469ddf2": "D_3 \\cong A_3, E_4 \\cong A_4, E_5 \\cong D_5,", "63ec5baafad80efadaaab519853faa76": "P_1 V_1 = P_2 V_2. \\,", "63ec6078f9f129b4500dee601f3718eb": "A(\\mathbf{x}) = \\sum_{p,q} w(p,q)\n\\begin{bmatrix}\nI_{x}^2(\\mathbf{x}) & I_{x}I_{y}(\\mathbf{x}) \\\\\nI_{x}I_{y}(\\mathbf{x}) & I_{y}^2(\\mathbf{x})\\\\ \n\\end{bmatrix}\n", "63ec96bbad0bd267940f0031644deec5": "E_{T,L}^{x_c}=0", "63eca690061cd93d9dbcb3efdb95eaf0": "\\mathbf{y}=(y_1,y_2,y_3)", "63ecc1d4da0a7c92073cc228bbba5209": " \\Leftrightarrow V^{'}_{A} + i^{\\bullet}_{A} \\Leftrightarrow V^{'}_{X} + i^{\\bullet}_{X}", "63ece1da08d7089781754d60a3632a4f": "x,y \\in R^d", "63ecf058e4c5990c9cb3c6fbfcb2fed1": "= (f_1' f_2 f_3 \\cdots f_n) + (f_1 f_2' f_3 \\cdots f_n) + (f_1 f_2 f_3' \\cdots f_n) + \\cdots +(f_1 f_2 \\cdots f_{n-1} f_n').", "63ed01c2b2494eb3879001448ca5cb26": "N_i/V", "63edb55616f17ea02f7770ece0ec63f9": "1 + i_t = 1 + r_{t+1} + \\pi_{t+1} + r_{t+1} \\pi_{t+1}", "63edc95dbc2fba2412e1684c093d7f74": "H=\\{Q_1,Q_1\\}=\\{Q_2,Q_2\\}=\\frac{(p+\\Im\\{W\\})^2}{2}+\\frac{{\\Re\\{W\\}}^2}{2}+\\frac{\\Re\\{W\\}'}{2}(bb^\\dagger-b^\\dagger b)", "63ee0c336f0afcce24a214cdd3ab84bd": " Q =q", "63ee1a95443748cf833eec148dc1f843": "\\sigma_{\\hat{X}}^2 = C_{XY}C_Y^{-1}C_{YX} = \\Big(\\frac{\\sigma_X^2}{\\sigma_X^2 + \\sigma_Z^2/N}\\Big) \\sigma_X^2.", "63ee53203bab37adcd024a08be9f66b4": "b_1Z_1", "63ee7ce7ebe4b2532ed52a15e2e43cdb": "\nv \\sim v_{ph} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\omega}{k},\n", "63ef1ef402b7e3fe41df96938fcb22da": "k(f)|_{f=f_z}=0.", "63ef2bc0ba1190af7b4d258a24ced7dc": "\\sin^4\\theta \\cos^4\\theta = \\frac{3-4\\cos 4\\theta + \\cos 8\\theta}{128}\\!", "63ef308998ab06feca51d4903e8743b8": "\nV(x,y) = \\frac{1}{2}(x^2+y^2+2 x^2y - \\frac{2}{3}y^3)\n", "63ef3b0dfba08501727ceee8136e10f3": "8t x+16t^2y=-1 \\,", "63ef55c9bdeccd37dbb2d6862bc9f925": "\\frac{d^2E}{d\\sigma^2} = \\pm\\left.\\frac{d\\mathbf{N}}{ds}\\right/\\frac{d\\sigma}{ds} = -\\frac{1}{RR'}\\frac{dE}{d\\sigma}.", "63f00003704da48c1c19272d74e65f38": "\\lnot (x \\leq y)", "63f00a89f6c4370fdbfc9942a5dae464": " \\forall g \\in G: \\ \\ \\sum_{i=1}^n a_i g(b_i) = \\delta_{g,1_G}1_A ", "63f03d39a920471350be7f47a850378a": "~\\sigma_{\\rm ap} ~", "63f0680df66871d09a2f795746dd06e2": "2^{nR}", "63f084792d39535322378abcb76a8c54": "\n\\frac{5}{2}", "63f0919827c942c3b497d0cb87eb8726": "{F}", "63f0ce97fabb45faa2f676e91ea9dce6": "\n\\mathcal {D} = \\int_0^L\\int_0^T \\left \\{ \\sqrt{\\left({\\partial \\vec{r}(s,t) \\over \\partial t}\\right)^2} + \\lambda \\left[\\sqrt{\\left({\\partial \\vec{r}(s,t) \\over \\partial s}\\right)^2} - 1\\right] \\right\\} \\, ds \\, dt\n", "63f0e85894a01fc8079cfbdc5463219b": "\nw_{k^*j}^{U(new)} = w_{k^*j}^{U(old)} - \\eta (x_{ij}^p-z_{k^*j}^{old})\n", "63f0e97a6eb917e84a634e068c195c99": "\\hat{y}_i = y_i", "63f11dbb03def2b44686756a69d21540": "\\Pr [X \\in A ] = \\int_{X^{-1}A} \\, d P = \\int_A f \\, d \\mu", "63f13a257d970b225cd3eab1fcd74698": " M_\\epsilon + K_\\epsilon = \\hat{G}.", "63f142229023d024d8039aa743d95db0": "y|u", "63f1633bab00590152835572e6970016": "c \\Rightarrow a;", "63f16c4ef98d75111b459ef841b93671": "\n\\begin{pmatrix}\nA'^0 \\\\ A'^1 \\\\ A'^2 \\\\ A'^3\n\\end{pmatrix}\n=\\begin{pmatrix}\n\\cosh\\phi &-\\sinh\\phi & 0 & 0 \\\\\n-\\sinh\\phi & \\cosh\\phi & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nA^0 \\\\ A^1 \\\\ A^2 \\\\ A^3\n\\end{pmatrix} \n", "63f190eb3d1328c5ea4a069c476c922e": "R\\subset F_m", "63f19f2b7a0236396f45486ba7e2e25e": "\\bar{y}_{i\\bullet}-\\bar{y}_{j\\bullet} \\pm \\frac{q_{\\alpha;r;N-r}}{\\sqrt{2}}\\widehat{\\sigma}_\\varepsilon \\sqrt{\\frac{1}{n}_{i} + \\frac{1}{n}_{j}} \\qquad ", "63f2013549b15e9481defc758642a641": " r_{1} + r_{2} = r", "63f282ddd333b576a45b061d559ca6fd": "\\frac 1\\alpha +\\frac 1\\beta=2.", "63f289794768b2e1b74aa02da18b2acb": "P_{TOT}=\\frac{3V_P I_P}{2}\\cos\\varphi", "63f38bded30acaadaba07d9b6c8b936f": "p\\in \\mathbb{P}", "63f3a1e2af7b8a4ba6821ab5e134b6f9": "\n\\underset{\\boldsymbol \\beta}{ \\operatorname{arg\\,min} }\n \\big\\| \\mathbf y - X \\boldsymbol \\beta \\|_p\n =\n\\underset{\\boldsymbol \\beta}{ \\operatorname{arg\\,min} }\n \\sum_{i=1}^n \\left| y_i - X_i \\boldsymbol\\beta \\right|^p ,\n", "63f3c2ecde59a2bf1c8d127c6b4b5ed9": "\\tau \\ \\stackrel{\\mathrm{def}}{=}\\ t/t_{0}", "63f3fc41951437e56c57248a98fa4dbd": "Oxy \\leftrightarrow \\exists z[Pzx \\and Pzy ].", "63f42029a38357e3c4082739de57f4de": "\n \\operatorname{E}[A_t - B_t] = \\frac{\\mu \\alpha_t(1-\\delta_t)}{\\epsilon + \\mu \\alpha_t(1-\\delta_t)} (S_G - \\operatorname{E}[S_t]) + \\frac{\\mu \\alpha_t\\delta_t}{\\epsilon + \\mu \\alpha_t\\delta_t} (\\operatorname{E}[S_t] - S_B) \\;.\n ", "63f53e9d8752551802633e57eddb63d8": "dTf(\\omega)=f(\\Omega^k)", "63f540fd0ba9ed9709c259a061139071": "\\bar{x}=\\frac{\\sum{f\\,x}}{\\sum{f}} = \\frac{405}{20} = 20.25", "63f5603ca479b4512d810b12f60c7dc7": "\\{ \\varphi_i: U_i \\to X \\}_{i\\in I}", "63f5a8d656ec5384790b2740daccc6ea": "\\scriptstyle R \\;\\rightarrow\\; R/I_1 \\,\\times\\, \\cdots \\,\\times\\, R/I_k", "63f5c346c803ada017d3c338923cf067": "\\frac{10}{\\sqrt{a}} = \\frac{10}{\\sqrt{a}} \\cdot \\frac{\\sqrt{a}}{\\sqrt{a}} = \\frac{{10\\sqrt{a}}}{\\sqrt{a}^2}", "63f5ce3729da658267c1eeb4708a70a9": "\\max_{|\\alpha|\\le k}\\ \\sup_{y\\in B'_\\delta(x_{2k})}|\\nabla^\\alpha (\\rho_{2k}s_{2k})|\\le 2^{-k}.", "63f62f09e14fb2797378749c237d1af7": "\\hat{A}\\hat{A}^* = (\\hat{a}_0+\\mathsf{A})(\\hat{a}_0 - \\mathsf{A}) = \\hat{a}_0^2 + \\mathsf{A}\\cdot\\mathsf{A}.\\!", "63f67edcacb10ce65ecaaa6e1b98d3fd": "s=s'", "63f683d0119942b3e83fdd7162bad0cf": "\\nu \\approx 0.18", "63f71e283caf3eb102e02780a024f093": " \\boldsymbol{u}(\\boldsymbol{x}) = \\int_0^\\ell \\frac{\\boldsymbol{f}(s)}{8\\pi\\mu} \\cdot \\left( \\frac{\\mathbf{I}}{|\\boldsymbol{x} - \\boldsymbol{X}|} + \\frac{(\\boldsymbol{x} - \\boldsymbol{X})(\\boldsymbol{x} - \\boldsymbol{X})}{|\\boldsymbol{x} - \\boldsymbol{X}|^3} \\right) \\, \\mathrm{d}s ", "63f7b1d300dff03f2362fc8cbe9c8c72": "a \\in (\\dots,-2,-1,0,1,2,\\dots)\\,", "63f820a996b5ac8e2e74dd61c8ff9c0f": "= -3.5", "63f8aef748983c97acef09dcd0a1f340": "\\mathbf{Adj}(C,T)", "63f8e7bd299f16d25086f1bd810ac9c3": "\\vec{v}_\\mathrm{B|A}", "63f9570e97ea65b625762a306c360e5c": "\\rho \\frac{Du_i}{Dt} = -\\frac{\\partial p}{\\partial x_i} + \\mu \\left(\n\\frac{\\partial^2 u_i}{\\partial x_j \\partial x_j} \\right),", "63f986fa7940ea8a0724eb66d8eba360": "\\phi ~\\mathcal{R}~ \\psi", "63f99b110f11176decd5c5bc9c3cb430": " \\frac{\\partial V}{\\partial t} + \\nabla \\left(\\frac{V^2}{2} + h \\right) = V \\times \\mathbf{\\omega} + T \\nabla s + \\mathbf{f},", "63f9c6eb8db38d6ce462d84db76d7466": "y = x(c-a) + a, \\text{ therefore }x = \\frac{y-a}{c-a}.", "63f9cfc40f5d32cffc0eef69f1937010": "|V_0|\\ll E_f", "63fa8a277af418e4a0dffab2f7be0600": "u'=\\frac{1}{EI}\\left(\\frac{3}{2}N\\langle x-0\\rangle^2\\ -\\ 1Nm^{-1}\\langle x-2m\\rangle^3\\ +\\ \\frac{9}{2}N\\langle x-4m\\rangle^2\\ +\\ c\\right)\\,", "63faa25949a30bf50e4134bd7e91fec4": "\\boldsymbol{\\sigma} = (\\sigma_1, \\sigma_2, \\ldots, \\sigma_n)", "63fae940b04ef636ab50e296240174cf": "pN\\equiv\\lambda", "63fb48248aeeed8c5f992db32f45e242": "\\left( \\begin{array}{c} a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n \\end{array} \\right) || \\left( \\begin{array}{c} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n \\end{array} \\right) = \\left( \\begin{array}{c} a_1 || b_1 \\\\ a_2 || b_2 \\\\ \\vdots \\\\ a_n || b_n \\end{array} \\right)", "63fb7a55bd39a8bc213fd1159ce3863a": "\\int_c^\\infty dy \\ \\left(\\int_a^\\infty dx\\ f(x,\\ y) \\right )", "63fb89b1471eb84a0129947cec520c89": "X_1=0, ~~~ X_k= (1 k)+ (2 k)+\\cdots+(k-1\\ k), ~~~ k=2,\\dots,n. ", "63fbc2af9fd3780d6cb5fe62036dae20": "\n\\begin{align}\n\\beta_1 & = \\frac{a \\rho_R - \\rho_0 u_R}{2a\\rho_0} \\\\[8pt]\n\\beta_2 & = \\frac{a \\rho_R + \\rho_0 u_R}{2a\\rho_0}\n\\end{align}\n", "63fbdbd5b32d9a47cf398d356b31b243": "F(y)=\\int_{-\\infty}^\\infty f\\!\\left(\\sqrt{x^2 + y^2}\\right)\\,dx", "63fc26067ed309307b1d92fd66495fe8": " -\\frac{d}{dx}\\left[p(x)\\frac{dy}{ dx}\\right]+q(x)y=\\lambda w(x)y,", "63fcab474fadb14cbd0bb0167d9aff7d": " \\hat u(\\xi) = \\frac{1}{P(\\xi)} \\hat f(\\xi) ", "63fcf2e27c6ed39cf907acea2cc39ef2": "P\\cdot \\tfrac 12r^2 \\frac{d\\theta}{dt}=\\pi a b", "63fd1fdad145d96653444996c1604a97": "\\textit{SMA} = { p_M + p_{M-1} + \\cdots + p_{M-(n-1)} \\over n }", "63fd3f560da7e6740def0aa2d8c43439": "\\scriptstyle \\left\\langle\\rho^2\\right\\rangle", "63fd5198cdad8a8d9eb53d09deb5c1d8": "\\tfrac{\\pi^{12}}{12!}\\approx 0.001930", "63fe0e3e3b6034afee58bbeebd403488": "\\tbinom{2n}{n+1}=\\tfrac{n}{n+1}\\tbinom{2n}n", "63fee8feaab455fe424851596b9f194d": " \\lambda r=(\\sigma r)\\lambda+\\delta r. ", "63fefbd184280e6745d3055c67051db6": "\\kappa = 0.37464 + 1.54226\\,\\omega - 0.26992\\,\\omega^2", "63ff1f43737944b869775476e26e3e16": " 2L_w = h ", "63ffc2cd65abc27b76cf79b97fd87468": "\\ M_x = \\frac{u_x}{a_x} = \\frac{W}{a_1},", "63ffd3844726d5232aab5543cff565f5": "\n\\begin{align} \n& V_{\\text{obs, r}}=V_{\\text{star, r}}-V_{\\text{sun, r}}=V\\cos\\left(\\alpha\\right)-V_{0}\\sin\\left(l\\right) \\\\\n& V_{\\text{obs, t}}=V_{\\text{star, t}}-V_{\\text{sun, t}}=V\\sin\\left(\\alpha\\right)-V_{0}\\cos\\left(l\\right) \\\\\n\\end{align}\n", "640001e098ff21c9ec698f44f7cfb2d1": " \\frac{\\partial \\bar{u_i}}{\\partial t} \n+ \\bar{u_j}\\frac{\\partial \\bar{u_i} }{\\partial x_j}\n= \\bar{f_i}\n- \\frac{1}{\\rho}\\frac{\\partial \\bar{p}}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 \\bar{u_i}}{\\partial x_j \\partial x_j}\n- \\frac{\\partial \\overline{u_i^\\prime u_j^\\prime }}{\\partial x_j}.\n", "64000d0015f77481804660e933598419": "\\langle p | p'\\rangle = \\delta(p-p')", "6400128d08a385f0a4b9fc452cc8f25b": "a = (x_2 - x_1)^2 + (y_2 - y_1)^2\\,", "640015ebaf805da45498f16b57ab08b3": "a=m^2-mn+n^2 \\, ", "64002dea80b9f6a09362c5c0713a55af": "\\mathrm{SNR} = \\frac{ |h^\\mathrm{H}s|^2 }{ h^\\mathrm{H}R_vh }.", "6400718cc1cbc88bfeb85c97ae7db5e3": "V_i ", "64007d69c4160dc34d63ca6d3acd3b4d": " \\mathrm{tr}(\\mathbf{A} \\otimes \\mathbf{B}) = \\mathrm{tr} \\mathbf{A} \\, \\mathrm{tr} \\mathbf{B} \\quad\\mbox{and}\\quad \\det(\\mathbf{A} \\otimes \\mathbf{B}) = (\\det \\mathbf{A})^m (\\det \\mathbf{B})^n. ", "64009c009cad256aa427e05017fd1bb3": " (g_1,\\epsilon_1)\\cdot (g_2,\\epsilon_2)=(g_1 g_2, \\epsilon),", "6400a9600fb7df208d8b582f64608a02": " k ", "64010963099c36aa35bdaae31ac715d8": "1.\\overline{428571}", "6401300c2e18be634cf948704f0e86d7": "\\epsilon_{\\mathrm{paint}}", "64014eb5af2f389cc9367ca40311acbc": "\\omega = [(\\Omega_i\\Omega_e)^{-1}+\\omega_{pi}^{-2}]^{-1/2}", "6402305dbefcd81977c8dc93429d2638": "V_r=0", "640265596e82da8eba52ef816a115cef": "\ne_{ij} = \\left ( \\frac{\\partial D_i}{\\partial S_j} \\right )^E\n = -\\left ( \\frac{\\partial T_j}{\\partial E_i} \\right )^S\n", "6402aa22b808837ffd1d3e5c6a108421": " H^{'}_{11}=H^{'}_{22}=E ", "6402b72a99eb0486ccbbd87c5f77e049": "\\mathrm{PS} (A) = \\det (I - A), \\, ", "6402e21213f86e281e5fbd2b74ffbece": "\\mathbb{F}_5", "6403096ee2d4aa4d00e370ead654f92e": "E = \\hbar^2 k^2/2m", "64034dccdea655856b15e8b632c0ef26": "\\left(\\sqrt[3]{u} + \\sqrt[3]{v}\\right)^3 = 3 \\sqrt[3]{uv} \\left(\\sqrt[3]{u} + \\sqrt[3]{v}\\right) + u + v", "64035d06a5421197107ea6a0e6c1342e": " \\langle A_\\mu(k) A_\\nu(k') \\rangle = \\delta(k+k') {g_{\\mu\\nu} \\over k^2 }.", "64036c0cc9511731855d26adeb678f15": "\\lim_{|x|\\rightarrow \\infty}|\\Phi|(x)=1.", "6403b9e4a65435bc8ef1d0a2dc9014bf": "\\left(A \\cap B \\right)^{c}=A^{c} \\cup B^{c} .", "6403df14fbf26c62d09c820e2466baf0": "\\int_{0}^{Dmax}", "64048309548c21182933fccecb536b51": "T^2(\\alpha)", "64049ea16b141b5d5df763f4245d45f5": "\\mathit{L \\to s | dFd}", "6404b575c8460ec90d974b5f1d771550": "\nC_A = \n\\begin{bmatrix}\n\\Gamma_1 \\\\\n\\Gamma_3 D_{\\Gamma_1}\n\\end{bmatrix}\n: Ran(A+1) \\rightarrow\n\\begin{matrix}\nRan(A+1) \\\\\n\\oplus \\\\\nRan(A+1)^{\\perp}\n\\end{matrix}.\n", "6404fbebf30f72d6841fd5971c81dceb": "\\sigma_{x,y,z}", "64054195df9c60972760ec8319f18059": "\\operatorname{Var}\\Big(\\sum_{i=1}^n X_i\\Big) = \\sum_{i=1}^n \\operatorname{Var}(X_i).", "64055f5ed0eef7844d9869bec43ff6e8": "\\{1,1\\}", "64057664781f5e251291e35f872d4db2": "\\int\\frac{1}{(ax^2+bx+c)^n} \\, dx= \\frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+\\frac{(2n-3)2a}{(n-1)(4ac-b^2)}\\int\\frac{1}{(ax^2+bx+c)^{n-1}} \\, dx + C", "6405ceb057aee12988e6504144268fbc": " ( -{g^{00}} ) ^{-1/2}", "6405d868f7b69c909d525f5ee9e34b4f": "q = (266^{10} * 2100^4", "64061bf3938f5f72de4f351e350c8013": " \\Psi(\\bold{r},t) = \\, A e^{ i (\\bold{k}\\cdot{\\bold{r}} - \\omega t)} ", "64062f3f3218dda2b068723a561f7607": "y'= \\frac{y+b}{(x-a)^{g+1}} ", "6406cdb3ff9096b6ca152192e36a4436": "T = \\frac{K}{S}", "6406edec96c5e26434a10a55bdade63f": "\\ \\displaystyle (q,\\alpha) \\ ", "640703146c86aa66a735314b6d8caccf": "{1 \\over 2}({m^2\\over n} + n)", "640706fecc98892a5cafa2ca00c6376d": "\n\\begin{align}\nc^k(\\ell,m,\\ell',m') &= c^k(\\ell,-m,\\ell',-m')\\\\\n&=(-1)^{m-m'}c^k(\\ell',m',\\ell,m)\\\\\n&=(-1)^{m-m'}\\sqrt{\\frac{2\\ell+1}{2k+1}}c^\\ell(\\ell',m',k,m'-m)\\\\\n& = (-1)^{m'}\\sqrt{\\frac{2\\ell'+1}{2k+1}}c^{\\ell'}(k,m-m',\\ell,m).\\\\\n\\sum_{m=-\\ell}^{\\ell} c^k(\\ell,m,\\ell,m) &= (2\\ell+1)\\delta_{k,0}.\\\\\n\\sum_{m=-\\ell}^\\ell \\sum_{m'=-\\ell'}^{\\ell'} c^k(\\ell,m,\\ell',m')^2 &= \\sqrt{(2\\ell+1)(2\\ell'+1)}\\cdot c^k(\\ell,0,\\ell',0).\\\\\n\\sum_{m=-\\ell}^\\ell c^k(\\ell,m,\\ell',m')^2 & = \\sqrt{\\frac{2\\ell+1}{2\\ell'+1}}\\cdot c^k(\\ell,0,\\ell',0).\\\\\n\\sum_{m=-\\ell}^\\ell c^k(\\ell,m,\\ell',m')c^k(\\ell,m,\\tilde\\ell,m') &= \\delta_{\\ell',\\tilde\\ell}\\cdot\\sqrt{\\frac{2\\ell+1}{2\\ell'+1}}\\cdot c^k(\\ell,0,\\ell',0).\\\\\n\\sum_m c^k(\\ell,m+r,\\ell',m) c^k(\\ell,m+r,\\tilde\\ell,m) &= \\delta_{\\ell,\\tilde\\ell} \\cdot \\frac{\\sqrt{(2\\ell+1)(2\\ell'+1)}}{2k+1}\\cdot c^k(\\ell,0,\\ell',0).\\\\\n\\sum_m c^k(\\ell,m+r,\\ell',m)c^q(\\ell,m+r,\\ell',m) &= \\delta_{k,q}\\cdot\\frac{\\sqrt{(2\\ell+1)(2\\ell'+1)}}{2k+1}\\cdot c^k(\\ell,0,\\ell',0).\n\\end{align}\n", "640712e64e186a7f3d5982e1a8887835": "\\det (\\mathrm{ad}_L x - t)", "6407e15c1d69122d86e4e019ef7e0db8": "\n \\begin{align}\n w_2 = \\frac{1}{24EI}\\Bigl[ & -3125 (-1645 + 4 M_c + 64 R_a) + \\\\\n & 50 (-4025 + 6 M_c + 120 R_a) x + 120 (5 + R_a) x^2 - 4 R_a x^3 + x^4\\Bigr] \\,.\n \\end{align}\n ", "6407eab3070be6a204ffad89fea5a34a": "S(X)=\\bigcup_{\\rho >0}\\bigcap_{\\mu >0}\\left[(X\\ominus \\rho B)-(X\\ominus \\rho B)\\circ \\mu \\overline B\\right]", "640826832471bee817b637995e62e6b8": "4\\pi R^2", "6408df592b6d2481b4cb8fa05f5fc489": "\\mu\\ge n", "6408e079aefee9702aa00f77228dd941": "2/4", "64091afb9d78de1462edac19be0e5166": "(\\mathbb{F}_p) = p+1", "640954d9e89922751eef07a50bba0929": "\\mu_\\mathrm{sat}= \\mu_\\mathrm{dry}", "64095da5ad4f01774266c7157506dafc": "\\displaystyle{M\\rightarrow\\mathfrak{t}^*}", "6409b18d2f54370c398e0e016eecce4d": "g(x):=\\det\\Phi(x) \\exp\\left(-\\int_{x_0}^x \\mathrm{tr}\\,A(\\xi) \\,\\textrm{d}\\xi\\right), \\qquad x\\in I,", "6409c34a3545b038c03fb4d9b5b27c37": "\\rho_E=2\\cdot \\pi\\cdot a\\cdot R_W\\,", "6409ed4a635a38cf08feaf8b69fa3904": "4K \\,", "640a325aa404adef8f32c67fc72fb843": " g( \\epsilon ) \\ ", "640a3f545e13243d19408f61306797e4": "\\begin{align}\nF_{3n+1} &= F_{n+1}^3 + 3 F_{n+1}F_n^2 - F_n^3 \\\\\nF_{3n+2} &= F_{n+1}^3 + 3 F_{n+1}^2F_n + F_n^3 \\\\\nF_{4n} &= 4F_nF_{n+1} \\left (F_{n+1}^2 + 2F_n^2 \\right ) - 3F_n^2 \\left (F_n^2 + 2F_{n+1}^2 \\right )\n\\end{align}", "640a3f66a0a6f3325b7547f68e360979": "F_{\\text{damping}}=-Dv", "640a90f3c98032df2616d26bf4727a94": "a = \\frac {p}{e^2-1}", "640b00786fdf3b949c6a5645b576a1c4": "D_{\\delta\\delta}(X, X) = |\\mu_x-\\mu_x| = 0.", "640b0994e5394f103305095367eb9325": " W_t^2 - t ", "640b97d83ee2ef7bb585e0416faefeb2": "\\phi(\\alpha, \\beta)", "640b9880d0c458b3e437425743e688c7": "x_1^3", "640c1a7a3ae6d39c8980b7f0d009fe35": "\\nu > 0", "640c32ff74d8bb395d9fc25063218300": "\\subset \\C", "640c6728a60b3c6e1394010c5d382264": "G_{\\nu}(\\pi)=1", "640c9783a9f3fe92b2b1eed635e71893": "\\gamma^*(s,x) = \\sum_{k=0}^\\infty \\frac{(-x)^k}{k!\\,\\Gamma(s)(s+k)}.", "640cd1f69e1954403293a08cb0523ee7": "(x^\\mu,\\dot x^\\mu), \\qquad \\dot x'^\\mu=\\frac{\\partial x'^\\mu}{\\partial x^\\nu}\\dot x^\\nu, \\qquad\\qquad (1)", "640d0e9ece2aab72bb898fc67c7d4942": "H_3 = 0\\,", "640d446296a472011b7168156a9fadd9": "\\sum_{i=0}^{N-1} x_i |i\\rangle", "640d785f3813b991e3c77c559b2c2729": "n > m", "640d8a7465cd7cb7bf42cd80021b9a9f": "r=\\alpha\\cdot\\sqrt{n}.", "640e0113519d709798e75c0d56009a7d": "\n f(\\alpha v) = \\overline{\\alpha}f(v).\n ", "640e35203724ee1f7b15b48e4ac8ab22": "f(x) = \\lim_{r\\to 0^+} M^rf(x). \\, ", "640e6519a8ea15d88c2540c5c56a98d6": "2 (2^{13-1}-1)/13 = 630 ", "640e90295c0f81639a7711357317e4c6": "\\left(\\frac{1}{\\sqrt{10}},\\ \\frac{5}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "640f1e3bbc83bb1b80bbb1302add10d1": "s=C \\theta.", "640f8bbd7c54d5dd1efd8aa0a6215420": "b+1=c", "640fb5ccd02cc688d5b58dd8b422a394": "576 m/s^2", "6410202d2ec1f7329f1ff50ff426d291": "Z=\\sum_{n=0}^{\\infty } \\frac{(8n+1)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^3{9}^{n}}\\!", "64105bcc4e5b87e14d6ed0e44569013e": "\\mu = 0", "64107148061f71263cdb9d6b5883964e": "\\mu_\\text{B}", "64109d83d7594248cc00d0a825790305": "\\scriptstyle\\Gamma(5/2) = \\frac {3 \\sqrt{\\pi}} {4} ", "6410b4e361a8ebef9c91f9370b27bb76": "ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2. \\,", "641169c460cf9963a846723c1c742854": "\\phi \\otimes \\psi = \\phi \\vee \\psi\\,", "6411a7662b12a4f8625b13b5116d9e4e": "P = 100 \\tan d\\!", "6411bf0feb9830c8807a3671d7029a86": "B_v(K_v)/f(A_v(K_v))\\rightarrow H^1(G_{K_v},A_v[f])", "641215ae947d8fd2ad02007e3631b2d2": "\\scriptstyle T_{input}", "64124c93b0d17f47119cac32d9bc2070": "\n W = -\\cfrac{\\mu J_m}{2} \\ln\\left(1 - \\cfrac{I_1-3}{J_m}\\right)\n", "64125f52266cdca13ecb7cbacd384241": "\\Delta {{V}_{BE3}}=\\frac{\\Delta {{I}_{IN}}}{{{g}_{m3}}}", "64127e82180913db0a2373a7b77e22f5": "xy = (-1)^{|x||y|} yx", "6412a0edb9feeb4970fffda82c4e4038": "C = \\begin{bmatrix} 1 & 3 & 4 \\\\ 2 & 7 & 9 \\\\ 1 & 5 & 1 \\\\ 1 & 2 & 8 \\end{bmatrix}\\text{,}\\qquad\nF = \\begin{bmatrix} 1 & 0 & -2 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix}\\text{.}", "64134b561af29db2752c05a4cae184d0": "\\textstyle q", "64138640288b6567ad19e71eaec25051": " f(\\alpha {\\mathbf v}) = \\alpha f({\\mathbf v})", "64138dd486101c60d898aee819d75733": "\\frac{\\partial^2 f(X; \\theta)}{\\partial\\theta_i \\, \\partial\\theta_j}", "6413d825f3fd5d7303bfe7e571cd6617": "P_n = {{n + 2} \\choose 3} + {{n + 1} \\choose 3}.", "6413e463dbf677d3eeced2d8c97f94ba": " m \\leftarrow \\tfrac{3}{2} ", "64140bf8b2c3730328862f6de8025b89": "T_B^2", "64143f926f5ba07da93cc89bfaccc5a7": "r > \\frac{p(p - q - pq - q^2 + pq^2)}{p^2(1 - q) + q^2(1 - p)^2}", "641448333ffb0238a1c762d8b4280523": "p\\circ i", "64144f40de88373bc8931b129f6f7cf8": "\\Delta_\\mu = \\det (x_i^{p_j}y_i^{q_j})_{1\\le i,j,\\le n}", "641452848ff9c24ecf2fbfb56917149e": "c_1 : \\mathrm {Pic}(V)\\to H^2(V, \\mathbb Z),", "64149d163facbcf19468587f92cd44c9": "f(r) = zp^k", "6414abba2ca24277e2af9548b90870ca": "-1.645 < \\lambda \\le -1.28", "6414b8b34de578dc63348a77b8539dd1": "V \\; (\\alpha - dy/dx) = w(x) = \\frac{1} {(2 \\pi)} \\int_{0}^{c} \\frac {\\gamma (x')}{(x-x')} dx'", "6415682413815b848a8215be6d1390f2": " \\mu(t) = \\text{E}(X(t)) ", "6415769084bd2131bdc1f00926b03b94": " xyxyx^3 ", "64158f0e87ea6f02206443ee39f9ee45": " \\displaystyle{\\sum_{n\\ge 0} |b_n|^2 \\le \\exp \\sum_{n\\ge 1} n |a_n|^2. }", "64159c1e5fe7074aab8bf678b0d4f255": "P\\left(n\\right) - a {p^n}", "6415bd0135e733f39967bac00bf7e9bf": "\\int_0^x x'J_0(x')dx'=xJ_1(x)", "6415fdfac4d45c10d4862d378a2e11c5": "\\sup_\\theta R(\\theta,\\tilde{\\delta}) = \\inf_\\delta \\sup_\\theta R(\\theta,\\delta).", "64160aa561c5b03b7ce50737f33839c6": "[S_gX_p,Y_p] = [S_gY_p,X_p] \\,", "64162a29bc1ffa6424ce73bd1bfda9fd": "A=\\begin{bmatrix}1&2&1\\\\-2&-3&1\\\\3&5&0\\end{bmatrix}", "64163c05e03f0d0bd2e487a84ab9042f": "K d + d K = j_1^* - j_0 ^*.", "64166affa31d22ffbcaaf18e10a0dc28": " \\mathbf{\\hat{A}} = \\sum_{j=1}^n \\mathbf{e}_j \\hat{A}_j ", "6416f8be315da213d4f481f9d1591017": "d \\leqslant n", "64175d56ca1ca1a7c7455901c132be51": "MPK=\\frac{\\partial F}{\\partial K}", "64184e0f5bb3b97427fc45b8ada2fc58": " (F,\\preceq) ", "6418dc73b70aabe10c2baf4035962b0e": "\\displaystyle f(-\\xi)\\,", "6418dce34b1a2a231dce9f856157a620": "\\langle g \\;| g \\;\\rangle_{\\{p\\}}\\;", "64191e59c55efd1253d6cf395b7551cf": "\\operatorname{cov}(\\mathbf{AX} + \\mathbf{a}, \\mathbf{B}^{\\rm T}\\mathbf{Y} + \\mathbf{b}) = \\mathbf{A}\\, \\operatorname{cov}(\\mathbf{X}, \\mathbf{Y}) \\,\\mathbf{B}", "6419a0fcec12419fed8114ce28ba775a": "B + Z \\rightleftharpoons (BZ)", "641a310eb70669afd0dcd6922fdbd66d": "\\mathcal{N}=1", "641a5929bae7d7207357b81d1576cf0d": "y=x_1", "641b4ee45c55d60480d271ca98c59467": "\\frac{V_M}{\\sqrt{2E}} = \\left(\\frac{M}{C}+\\frac{1}{3}\\right)^{-1/2}", "641bb6ed0050c16e226d2ffe01ea48a0": " D_{l} + \\max( F_{l} , 0) ", "641bc3cf420f10324d1658c6342fa348": " \\sum_m |u_m|^2 < \\infty \\, ", "641bd781c056e3f8367a12554820fc98": "P(cancer|do(smoking))", "641bdbfe74ad0ce3389cda23e33c16dd": "k < t", "641bfbbf493fe966610b9e7ae44bcd84": "Z_2^3", "641c08acd36d76772e70726c6abab40c": "AS\\cdot BC=BS\\cdot AC=CS\\cdot AB", "641c3b0d723d561f288a03223fbb265a": "\\rho = {1\\over 2} \\begin{pmatrix} \n1 & 1 \\\\ \n1 & 1 \\end{pmatrix} ", "641c90d83590aad7049892c33d001e9a": "\\mathbb{K} = \\mathbb{Q}", "641c931733cf21f71513d0849740322f": "X_t=\\cos (t+Y) \\quad \\text{ for } t \\in \\mathbb{R}. ", "641c9d25ddd783b170f9a470e0874ec3": " \\begin{Bmatrix} X\\end{Bmatrix}e^{i\\omega t}", "641cb8d56b1f2aeea2f08bd1826b3bc8": "\\delta_{i}(z) = \\delta_{i,1}z^{1} + \\delta_{i,2}z^{2} + ... + \\delta_{i,k}z^{k}", "641ceb8e989be064a4deb000768ca434": "\\langle n, k \\rangle", "641d19750a61179980f22eff66c93e2d": "\\mathbf A = A_{1} \\times \\cdots \\times A_{1} ", "641d5ecb675ddf17dac4242ada1ee5bc": "\\{\\mathcal{H}_1,\\mathcal{H}_1\\}=0", "641db10b7347f300e9f80a0dc938639a": "X_i,Y_i ", "641e2b0537947968f7ff3bc9dd6f7f70": "\\kappa = \\aleph_{\\kappa}", "641e411f5c344c4f739fdb5f71b9edb7": "\\, e^{it\\mu -\\theta|t|}", "641ee5257bf7564ed8eca23ef4a19234": "y\\ll 1", "641eef58244e92b350a7708f085db9ab": "\\lim_{x\\rightarrow -\\infty} F(x)=0\\,;", "641f3586c00db6084ebf0f5c1926bbb7": "{n_\\mathrm{D}}^{real}", "641f4959f59f4622651922aaddf69fde": "\\circ_0", "641fafdd9eea57eb560854a30f064891": " 0 = \\frac{k_3[I]K_m[ES]}{[S]} - k_{-3}[EI] ", "641fdf3d2af7e31b1b0ee86916e7bb1c": "\\frac{\\langle E(s) \\rangle}{A} = \n-\\frac {\\hbar c^{1-s} \\pi^{2-s}}{2a^{3-s}} \\frac{1}{3-s}\n\\sum_n \\vert n\\vert ^{3-s}.", "641ff6e77c06a798b00f3f9471a3d32d": "k (x, y) = (1 + x^2 + y^2)^{1/2} . (x, y)", "64205b9d97ea77b5ea9110ab6ff37c95": "0\\le k < 2n", "642077e7fe5b7ee0215851720520864d": "u(x,t) = F(x+ct)+G(x-ct)\\,", "64208002ee49ed39063a2adc136bf9b0": "\\frac{\\Gamma \\vdash \\Sigma_1, A, \\Sigma_2, B, \\Sigma_3}{\\Gamma \\vdash \\Sigma_1, B, \\Sigma_2, A, \\Sigma_3}", "64209a76680575e68f433a89b6189fd2": "ID_{<\\omega}", "6420ce1f5a2f137aeba69e5fc86d2a37": "\\sigma=\\frac{(\\alpha_E-\\alpha_{Cu})\\Delta T A_E E_E E_{Cu} }{A_E E_E+A_{Cu}E_{Cu} },\\quad \\text{for }\\sigma\\le S_Y", "642104252a5ee43d6311552025c2ad55": "\\{\\pm 1, 2^{k-1} \\pm 1\\}\\cong \\mathrm{C}_2 \\times \\mathrm{C}_2,", "642146e30b387d044d05c7a9e7914455": " \\tau(\\omega)=i^{p(p-1)+l}\\star \\omega\\quad,\\quad\\omega \\in \\Omega^p(M) ", "642155d0d58b2eee08e5543a9b2aa5c5": "\\delta \\int \\psi d\\tau = 0 \\,", "6421a7f944e0d927c655e16b2072f2d9": "p = \\gamma m_0v ", "6421d261a0b01a1b9b3ba3a16d958297": "\\operatorname{wnchypg}(x;n,m_1,m_2,\\omega) = \\operatorname{wnchypg}(n-x;n,m_2,m_1,1/\\omega)\\,.", "64220be8fae60b7b0459ceb28634f48d": "\\frac{MS_{Treatment}}{MS_{Error}}", "64225ad25ccd4c3c4e898aa96fb7da6b": "dN/dt = 0", "642275cb033115ded74d4df1c51716fb": "R(D, C^*x)", "6422d20ce9d66935be0bc3e10b79c429": "\\textstyle (\\dots,1,\\dots)", "642302f3db19b7834bd31939dcd73fe5": "m(n) = \\theta(2^n \\cdot n)", "64236a80686244843ae40e5a9b89043f": "\\operatorname {cn}\\; u = \\cos \\phi", "642375568b3f16a275d09f7ed473348a": "\\mathbf{g}_v(X,Y) := \\frac{1}{2}\\left.\\frac{\\partial^2}{\\partial s\\partial t}\\left[F(v + sX + tY)^2\\right]\\right|_{s=t=0},", "642379aa2fc5a977d4c55100748923ac": " V(\\mathbf{x}) = - \\frac{GM}{|\\mathbf{x}|} - \\frac{G}{|\\mathbf{x}|} \\int \\left(\\frac{r}{|\\mathbf{x}|}\\right)^2 \\frac {3 \\cos^2 \\theta - 1}{2} dm(\\mathbf{r}) + \\cdots", "6423d51971beeaf9e09689ce2e37b317": " \\mathcal{L} = y_t\\, m_{\\tilde{t}}\\, a\\; h_u \\tilde{q}_3 \\tilde{u}^c_3", "6423db26e5106d4bed254e973a45153d": "|\\mathbf{A}-\\mathbf{u}|^2 = r^2", "6423eb9b81f6f6cac7161a6234c6fde1": "\\displaystyle{(0)\\rightarrow H^2_0(\\Omega) \\rightarrow H^2(\\Omega) \\rightarrow H^{3/2}(\\partial\\Omega) \\oplus H^{1/2}(\\partial\\Omega) \\rightarrow (0),}", "6423ec792346ced168643bc6d9fe6e2b": "(M_f)", "642411964f8d648264beb88cfb7df970": " \\mathop{\\mathrm{im}}(h) := h(G) :=\\left\\{h(u)\\colon u\\in G\\right\\}\\mbox{.} \\! ", "64243e0bad1ef9c86eff2d45e13900b5": "\\scriptstyle p \\;=\\; 2", "6424af78b265a37de4a08fb31d409fa9": "A_1,\\dots,A_n/B", "642558146ad6def158bb6e5a27f0ccce": "\\sigma^2 = X^TVX", "6425dea043f5672610c7f61e3c6e8582": "(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd.", "6425e727486741db1029b1b0b95ae069": "m = \\frac{x^3-y^2}{x} \\pmod{N}", "6425ea979d5e7f81c3492a5f53e4aa5f": "\\{x^2+2xy+y^2 = 0\\} = \\{(x+y)^2=0\\}=\\{x+y=0\\} \\cup \\{x+y=0\\} = \\{x+y=0\\}. \\,", "64263cfa42af37568b7bd7998168dd55": " x(t) = \\left( \\frac{9}{2} \\mu t^2 \\right)^{ \\frac{1}{3} } ", "64265ba6b03ce86ac567a80aad97951a": " v_\\mathrm{ap} = \\sqrt{ \\tfrac{(1-e)\\mu}{(1+e)a} } \\,", "64269c9f88cb35e6ae4d506711264be5": "f\\colon I \\rightarrow \\mathbf R.", "6426abb7ac815ac57cee40f85a55b143": "g=\\begin{bmatrix}\n \\alpha & \\beta \\\\\n \\gamma & \\delta \\\\\n \\end{bmatrix}", "6426b4d2115e697500db6bb4556f9269": "d \\approx 1.32 \\sqrt{h} \\,.", "6426c3b1ded8aeb0fa04e88619f83857": "\\sigma_{ij,kk\\ell\\ell} = 0", "6426d9a8d1b505c2ce395398b1847d58": "\\psi_n^{(k)}(0)=\\begin{cases}n!&\\text{if }k=n,\\\\0&\\text{otherwise,}\\end{cases}\\quad k,n\\in\\mathbb{N}_0,", "642717e4e00c70aee8efb9b82ac51a97": "0 = \\sum_{j=1}^m u_j ", "64272c4158e8ea8d02af4e80757a5802": "x=g(x)=x-\\frac{f(x)}{f'(x)}", "642748009714c04958f26d4d73e946cf": "{\\color{Blue}~2.21}", "64278217d289cf52da3b7ecfb00b0019": " \n\\binom{2m + 1}{m} = \n\\frac{1}{2} \\left[\\binom{2m + 1}{m} + \\binom{2m + 1}{m + 1}\\right] \n< \\frac{1}{2}\\sum_{k = 0}^{2m+1} \\binom{2m + 1}{k}\n= \\frac{1}{2}(1 + 1)^{2m + 1} \n= 4^m.\n", "6427c0ab0d9d16e9c34a90de37bf8ff5": " r \\ge R(t) ", "6427cb8ba4857c3ac009a8a9c585ea84": "[P_i,C]=-iK_i,[K_i,C]=0,\\,\\!", "6427db6c7a25418f88d6cf57a1808d9c": "\\begin{align}\n y &= G \\left \\{ \\frac{(1 - x)^{\\Delta}}{(\\Delta+1)_{-\\Delta - 1}} \\ \\sum_{r = -\\Delta}^\\infty \\frac{(\\Delta + \\alpha )_r (\\Delta + \\beta)_r}{(1)_r (1)_{r + \\Delta}} (1 - x)^r \\right \\} + \\\\ \n&\\quad + H \\left \\{ \\sum_{r = 0}^\\infty \\frac{(\\Delta)(\\Delta + \\alpha)_r (\\Delta + \\beta)_r}{(\\Delta + 1)_r (1)_r}\\left (\\ln(1 - x) - \\frac{1}{\\Delta} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{\\alpha + k} + \\frac{1}{\\beta + k} - \\frac{1}{-\\Delta + 1 + k} - \\frac{1}{1 + k} \\right)\\right ) (1 - x)^r \\right \\} \n\\end{align}", "6427e350c69c580a8090f0e3cb3fc4e6": " \\frac{d\\mu}{d\\nu}=\\left(\\frac{d\\nu}{d\\mu}\\right)^{-1}\\quad\\nu\\text{-almost everywhere}.", "6427edc02b831aeba937bd031e2022e0": "\\begin{align} H= \\begin{pmatrix}E_{1}&0\\\\0&E_{2}\\end{pmatrix} \\end{align}\\,\\!", "6428672435fc93e2e57c2c7d43b85b7e": " (7)\\,", "6428b92c46a16e1ce4f880a8147085a8": "d(x,y) >\\delta", "6428c886217bbd8851ccc61ea515a6d0": "a + y + b + z > b + x + c + y", "6428e423c22635a6744fc6d417f3382c": "f(x, y) = g(x)", "642922b32c71844e4b6330a50c90bc29": "\\left\\{ X,Z\\right\\} ", "64295780ce3b447ada4364ff34aeba17": "\\theta_n:=\\mathbb I_{\\left[-N,N\\right]}\\frac{\\nabla f}{\\left|\\nabla f\\right|}", "642982f487713885dc815ba5ec74e7d1": " |p_1| > |z_1| > |z_2| > |p_2| ", "642a0d24cbaff4ee5537321531879ef8": "\\hbar^{-1}", "642a4159ef6865d37a3711dde2c574c4": "S_{21} = \\frac{b_2}{a_1} = \\frac{V_2^-}{V_1^+}\\,", "642a8d2ab18efff3f029ed51d7179999": "E_{XC}^{\\rm GGA}[n_\\uparrow,n_\\downarrow]=\\int\\epsilon_{XC}(n_\\uparrow,n_\\downarrow,\\vec{\\nabla}n_\\uparrow,\\vec{\\nabla}n_\\downarrow)\nn (\\vec{r}) {\\rm d}^3r.", "642a931cd768f043700b5178d0f2da0f": "\\textstyle \n \\begin{bmatrix} 0 & 1 \\\\ 1 & -w \\end{bmatrix} \n \\cdot \n \\begin{bmatrix} A & B & x \\\\ C & D & y \\end{bmatrix} \n = \\begin{bmatrix} C & D &y \\\\ A - wC & B - wD & x-wy \\end{bmatrix}\n ", "642a9ae864e2899b03bddc693c1bec36": "x\\Vert A\\Rightarrow f(x)\\Vert f(A)", "642b24877753c38e70e5a600a4a7a3cf": " i = 0.5 S/ \\sqrt{t}\\ + A_1", "642b48b53dfb7aa6a9d3834db9844d26": "H_n(\\underline q, \\underline x)", "642b78cc46ed407555baa9d4f704938b": " \\operatorname{Var}(\\bar x_w) = \\sum_{h=1}^H W_h^2 \\,\\operatorname{Var}(\\bar x_h). ", "642b7cdf2581ae937e9668a07c46f6c5": "\\{X,Y\\} \\in T_p \\mathbf{CP}^n", "642b97f3d56202f4ad19bc73344df864": "X_n\\ \\xrightarrow{p}\\ X", "642bba0d27d9b5eb1f1555cf9d58ed78": " P(E) = \\sum_i P(E|S_i) P(S_i)\\, . ", "642bc7dc13cf0a54488c7b1d8df3b88e": "(a;q)_n = \\frac{(a;q)_\\infty} {(aq^n;q)_\\infty}, ", "642c62a78de14dec104a9d489e2ea242": "\\mathrm{ft}^3", "642c84b2a890c6f09cd181134f3876f1": "\\{ ww^{+} : w \\in \\{a,b\\}^{*} \\}", "642ca36ce9352b01fec7c4348ecbc053": "{\\upsilon}_{in} = {\\upsilon}_s \\begin{matrix} \\frac {R_{in}}{R_S + R_{in}} \\end{matrix}", "642d08e9d261944c573bf234ddfa492e": "B(z)=\\prod_n B(a_n,z)", "642d676254317841d4c1bd2a28c5bcb9": " \\frac{1}{2}[(\\kappa+1) \\theta~\\cos\\theta - \\{1 + (\\kappa-1) \\ln r\\} ~\\sin\\theta] \\,", "642d7eed13a0f2a9ab628efcfa649fa5": "\\psi(\\mathbf{r},t) = \\begin{bmatrix} \\psi_{\\sigma=s}(\\mathbf{r},t) \\\\ \\psi_{\\sigma=s - 1}(\\mathbf{r},t) \\\\ \\vdots \\\\ \\psi_{\\sigma=-s + 1}(\\mathbf{r},t) \\\\ \\psi_{\\sigma=-s}(\\mathbf{r},t) \\end{bmatrix}\\quad\\rightleftharpoons\\quad {\\psi(\\mathbf{r},t)}^\\dagger = \\begin{bmatrix} {\\psi_{\\sigma=s}(\\mathbf{r},t)}^\\star & {\\psi_{\\sigma=s - 1}(\\mathbf{r},t)}^\\star & \\cdots & {\\psi_{\\sigma=-s + 1}(\\mathbf{r},t)}^\\star & {\\psi_{\\sigma=-s}(\\mathbf{r},t)}^\\star \\end{bmatrix}", "642dee9ce3e46a2c90c67fffe02131e8": "\\text{excess kurtosis} (\\Beta(\\alpha, \\beta) )= \\text{excess kurtosis} (\\Beta(\\beta, \\alpha) )", "642e552240e5988fa09a7c57a67de54e": "Z = \\int_x h(x) e^{\\boldsymbol\\eta \\cdot \\mathbf{T}(x)} dx.", "642ecefc60715e49bcd4faa752d7472a": "\\langle\\psi_\\Lambda(t)|E_1^{(-)}(t)E_2^{(+)}(t)|\\psi_\\Lambda(t)\\rangle=\\kappa'\\langle 1_{\\nu_1}0_{\\nu_2}|a_1^\\dagger a_2|0_{\\nu_1}1_{\\nu_2}\\rangle exp\\lbrack i(\\nu_1-\\nu_2)t\\rbrack\\langle b|c\\rangle=\\kappa' exp\\lbrack i(\\nu_1-\\nu_2)t\\rbrack\\langle b|c\\rangle", "642ee59175f426c4ea05d1aecb26edba": "\\ell^{(0)}=\\frac{2}{1-\\alpha}p^{\\frac{1-\\alpha}{2}}=2\\sqrt{p}", "642f4b968703498d500ed824700181d4": "\\eta(s)=\\sum_{\\lambda\\ne 0} \\frac{\\operatorname{sign}(\\lambda)}{|\\lambda|^s}", "642f5aeefbb375a065df65fd7e409e85": "A\\cdot v=w", "642f602750e596d2089607e91a211c14": "O\\left(m^{g(k)}n^k\\right)", "642fb905c4643eb10148efcb0f5af9dc": "1 \\equiv a^{N-1}\\pmod{p}", "642fcd006d23ad07986bcdc61a789533": "T_{n+1}(x)=x \\left(1+T(x) \\right)^n.", "642ff95bb5a5ee3504db4fc921f0d18a": "\\frac{\\mathrm{A_{1}}(\\theta,\\Phi)}{\\mathrm{G_{1}}(\\theta,\\Phi)} = \\mathrm{constant}", "6430105967a1cd535ee6b4416736f279": "\\|f_r - f\\|_1 \\to 0", "6430176e3a8549c053837b8e21b9607f": "{a+b \\over 2}", "643026275fe3d4c4179aa9bebdd34c16": "X = \\{X_t : t\\in T\\}", "643050d1d0d6f235acd67a5c8050c7bc": "~\\psi^{(\\alpha)}(x,t)=\\left(\\frac{m\\omega}{\\pi\\hbar}\\right)^{1/4}e^{-\\frac{m\\omega}{2\\hbar}\\left(x-\\sqrt{\\frac{2\\hbar}{m\\omega}}\\Re[\\alpha(t)]\\right)^2+i\\sqrt{\\frac{2m\\omega}{\\hbar}}\\Im[\\alpha(t)]x+i\\delta(t)}\\;,", "6430673f878015eec24d38eebb9cab17": "\\frac{e^{-k|z|}}{k}=\\sqrt{\\frac{2|z|}{\\pi k}}K_{-1/2}(k|z|)\\,", "64308d6959c132efa7aaf60f0d981646": "\\Delta U = 0, \\quad \\Delta W = \\Delta Q \\,\\!", "6430db6b5c72c7418ea94fad09ae19c0": " \\sigma(x)=\\|X\\|", "643160d50b0dd9761a84e05d59b40d13": "G_{45}", "64317f1d6a88a21ebf1b986e51657344": "\nMDV = \\frac{\\lambda}{2}\n\\left(\n\\frac{4 v_p}{B} \\sqrt{(\\sin(AZ)\\sin(EL))^2 + (\\cos(AZ)\\cos(EL))^2}\n\\right)\n", "6431a105c272c40cfce8a117ba4e5d3c": "\n2 \\langle T \\rangle = n \\langle V_\\text{TOT} \\rangle.\n", "6431dd1ed3c828d8015ebf5f9596e193": "\\beta U(r) = Z^2 \\lambda_B \\, \\left(\\frac{\\exp(\\kappa a)}{1 + \\kappa a}\\right)^2 \\,\n\\frac{\\exp(-\\kappa r)}{r},\n", "64326e11dc91de16c2341dd6a9234b1e": " \\operatorname{tr}_R \\tilde{\\chi} = \\tilde{\\rho} ", "6432c6dbb84ae5e89fb61a17b7899810": "\\frac{k_i(k_i-1)}{2}", "643307a9835ff906e4585a3713118012": " \\mu(\\{x\\in X:|f(x)|\\geq \\varepsilon \\}) \\leq {1\\over \\varepsilon}\\int_X |f|\\,d\\mu.", "643358eebe576bc926d981c8cb6c7204": "\\frac{dE}{dy} = \\frac {d}{dy} \\frac{1}{2}(t - y)^2 ", "6433b8dc8c592d3f6f86ce57d1a5a7e9": "(a_n x+a_{n-1})x", "643409ef7eb1ac09b76a30ab330ca1bf": "\n\\frac{\\partial R}{\\partial t} = -\\frac{1}{2m} \\left[ R \\nabla^2 S + 2 \\nabla R \\cdot \\nabla S \\right] \\; ,\n", "64345888de843e3c0b7c1a62beded292": "\\lim_{x\\to c}\\frac{f(x)}{g(x)}= \\lim_{x\\to c}\\frac{f(x)-0}{g(x)-0} = \\lim_{x\\to c}\\frac{f(x)-f(c)}{g(x)-g(c)} = \\lim_{x\\to c}\\frac{\\left(\\frac{f(x)-f(c)}{x-c}\\right)}{\\left(\\frac{g(x)-g(c)}{x-c}\\right)} =\\frac{\\lim\\limits_{x\\to c}\\left(\\frac{f(x)-f(c)}{x-c}\\right)}{\\lim\\limits_{x\\to c}\\left(\\frac{g(x)-g(c)}{x-c}\\right)}= \\frac{f'(c)}{g'(c)} = \\lim_{x\\to c}\\frac{f'(x)}{g'(x)}.", "6434da8e5064e8fc4c36e4dddb9dfa07": "\\ln(\\Gamma(k/2))+\\,", "6434df915f36db341f9e08a860765380": "r_k(t)\\,", "64350f9918649b6e890f1bb3348f9fe8": "\\ PMAD = \\frac{\\sum_{t=1}^{N} |E_t|}{\\sum_{t=1}^{N} |Y_t|} ", "643557606e4cb35177fd3e340e382759": "\\begin{pmatrix} 3 & 3 \\\\ 2 & 5 \\end{pmatrix} \\begin{pmatrix} 11 \\\\ 15 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 19 \\end{pmatrix}", "643564cc412531b4472e76f87bd2fbbf": "p \\in Q", "64356524d6b3e1bce6c6bd31c29e1c28": "\\operatorname{Der}_K(A,M)\\subset \\operatorname{Der}_k(A,M),\\,", "643568907197d06a14c450693f905170": "Y = \\frac {1} {1 - b + bt - t} (I) ", "64359a9a733e58f897a8149bbcc1b8cd": "\n\\sigma _z^2 \\approx \\,\\,\\,\\left( {a\\,r\\,\\mu ^{r - 1} } \\right)^2 {{\\sigma ^2 } \\over n}\\,\\,\\, = \\,\\,\\,\\,{{\\left( {a\\,r\\,\\mu ^r } \\right)^2 } \\over n}\\left( {{\\sigma \\over \\mu }} \\right)^2\n", "6435d75397acd8054bbb6490f9eb4553": "I_0 = \\left(\\frac{\\pi}{16}-\\frac{4}{9\\pi}\\right)r^4", "6435edd36db05d4b0adc5781c0c2672c": "\\lim_{h \\to 0} \\frac{|f(a + h) - f(a) - f'(a)h|}{|h|} = 0", "6435fec3db0dab5b9f92c64120a08d3d": "\\sqrt{-2\\ln(u_1)};", "64365962cdd28aa32d05bcbb975bdf81": "\\varepsilon(U)=U", "64366e0f3f5dbda8635d142f3c4091f6": " f : \\mathrm{Bd}^n K \\to L ", "64367884d9f3a572f9b8c090fca374a2": "\\operatorname{E}[T_N]=\\operatorname{E}\\!\\biggl[\\sum_{n=1}^N \\operatorname{E}[X_n]\\biggr]=\\operatorname{E}[X_1]\\operatorname{E}\\!\\biggl[\\underbrace{\\sum_{n=1}^N 1}_{=\\,N}\\biggr]=\\operatorname{E}[N]\\operatorname{E}[X_1].", "6436b2bde4fcc1096e8bb4bbb978738b": "\\{\\Delta \\ X_1\\}", "6436b42dcc8137f74ce1f83e6c8f260d": " ({\\mathcal H}, \\langle \\cdot, \\cdot \\rangle) ", "6436c0cbe9303ed3fa6cbe09b50acf11": "|f_j(z) - f_k(z)| < \\epsilon / 3 \\,", "6436e85b84fb4ffae37cbc95d1fc74e4": "Cl_t^{\\leq}", "6436ff993997504d4e191af50555029f": "{\\mathcal C}\\subset\\{C: C \\mbox{ is measurable subset of }\\mathcal{S}\\}", "643722671d71735df9886aad08220304": " g:\\mathbb{R} \\rightarrow \\mathbb{R} ", "643738a6fa38ae178884ec810aca2888": "\\sin[\\arccos(x)]=\\sqrt{1-x^2} \\,", "6437a4ed15edfe4a319039d6de784e4a": " dA_{\\bold{x}}\\,dA_{\\bold{y}} = dA_{\\bold{y}}\\,dA_{\\bold{x}} , \\,\\!", "6437ae661e048d35b1d2bc5f1f3070c6": "\\mathbf{S}_A", "6438370a48e3365bb01ca63d6e3e5172": "\\mathbf{L}\\cdot\\mathbf{V}(\\mathbf{x})=i\\mathbf{\\nabla}\\cdot(\\mathbf{x}\\times\\mathbf{V}(\\mathbf{x}))", "64388e5a833504b1b6bb68fd71017dc1": "\\scriptstyle \\sqrt{2}", "6438c70a2bd3ee033acb7dd58c5dc6c7": "{\\nabla_{1a}}^2", "6438e120755f023bb1a4ce047b4caa1a": "\\int_a^b f(x)\\, dx = G(b) = F(b) - F(a).", "6439543baad709c298a08168287330cb": "\\mathrm{adj}(\\mathrm{adj}(\\mathbf{A})) = \\det(\\mathbf{A})^{n - 2}\\mathbf{A} ", "64398d394bf1b47c220694e27616e770": "\\{x,u,u_{1}, y_{2}\\}\\,", "64399fee713d085eb7389d042941d316": "\\mathrm E(|X|:A)\\leqslant\\epsilon", "6439e1efc6d8ce8dc284a34fa3d0a34a": "\n\\hat{\\mathbf{x}}(t_0)=E\\bigl[\\mathbf{x}(t_0)\\bigr] \\text{, } \\mathbf{P}(t_0)=Var\\bigl[\\mathbf{x}(t_0)\\bigr]\n", "643a59d4d9b00262495bd493acf110db": "[0, 1)^2", "643a9abda3f33f313710175df30df665": " \\frac{r(1-p)}{p^2} \\,", "643ab73049a873e3abc1b9c0be346a73": "Y_{10}^{-9}(\\theta,\\varphi)={1\\over 512}\\sqrt{4849845\\over \\pi}\\cdot e^{-9i\\varphi}\\cdot\\sin^{9}\\theta\\cdot\\cos\\theta", "643ad4a341644266f66a566cd8b026f1": "D_\\mu =C[\\partial x,\\partial y]\\,\\Delta_\\mu", "643b3d733a0b396710045ab459e6f274": "y_{1i}-y_{0i}=\\beta_{i}", "643b56a2f5e605457085cbeaf0ba0501": "d_{2,2}^{2} = \\frac{1}{4}\\left(1 +\\cos \\theta\\right)^2", "643c023cb2ad7bbbbdc0a6609bb71443": "x \\in Cl_s \\cup Cl_{s+1} \\cup Cl_t", "643c412f3caa554317097175ca5cb0fd": "f(x,a)=0 \\,", "643ca579c9bf6e1c791fb510c89389a3": "p_i = \\frac{\\partial u}{\\partial x_i}.", "643cbaa96f9a8fc9361154627a1e9564": "H_d(z) = H_a \\left( \\frac{2}{T} \\frac{z-1}{z+1}\\right) \\ ", "643d20801451654cd0a0a2b64eabc441": "\\int\\frac{\\cos ax\\;\\mathrm{d}x}{\\cos ax + \\sin ax} = \\frac{x}{2} + \\frac{1}{2a}\\ln\\left|\\sin ax + \\cos ax\\right|+C", "643d5bcf7a6ba893e5a25c0ba896f4ad": "\\mathbf{P}( X \\ge a) \\le e^{\\frac{-a^2}{2n}}, \\qquad a > 0 ", "643d7b5e5a54a3d18e2942c8e747b5b0": "\\omega_i=\\omega(\\mathbf{k}_i),", "643d82ec5eeb6ccf8b2c50536cf27a5a": "(\\varphi,f)", "643d8453a20e4ebcd0af31756c8a8e7e": "\n\\frac{n_\\text{H}}{n_\\text{p} n_\\text{e}} = \\left(\\frac{m_\\text{e} k_\\text{B} T}{2 \\pi \\hbar^2}\\right)^{-3/2} \\exp\\left(\\frac{Q}{k_\\text{B} T}\\right),\n", "643d8512f6116d67c7c59a02721f3447": "E_\\mathrm{cell}^\\ominus , \\Delta E^\\ominus", "643dcb3bc852027f19a464f543621930": "F(x)|\\nabla T(x)|=1.", "643e395c3992f9c3a3585c476571b478": ">\\Pi", "643e4804f3da2d95a13785712c32b782": "F=(\\omega+dd'\\phi)^m/\\omega^m", "643e61126e126e143c709ec887ce3b9d": "\\beta = 2: \\quad \\operatorname{E}\\left [- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial c^2} \\right ] = {\\mathcal{I}}_{c, c}", "643e640ad1bb703ddf9374aba1c715fb": "\\begin{align} 2\\cdot R_*\n & = \\frac{(24.4\\cdot 0.74\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 3.88\\cdot R_{\\bigodot}\n\\end{align}", "643e73c5a23cf6a6cb8e08091ffc7536": "C_n = \\sum_{i=0}^n a_{n-i}B_i", "643ea20f07009e3d474b695fb856cfaf": "\\rho_v", "643eac5f68d75c3e929ce91a55326671": "f(x) = (x+1)/2", "643ebc7f4eaf759f5fbe3be3662c50d9": "\\Omega^2U\\simeq U .\\,", "643ed2c697fd025c606e048bf0c95252": " F(T;H) \\ge T^{- c_1},\\quad G(s_0; \\Delta) \\ge T^{-c_2}, ", "643fadd7312f15e4b2abc3f219e94fc1": "-\\boldsymbol{\\mu}_L = \\dfrac{e\\hbar}{2m}\\mathbf{L} = \\mu_B\\mathbf{L}", "643fccce657fa08715a871c68fe9bde1": "a_{x}=\\frac{f_{x}}{m\\gamma^{3}},\\ a_{y}=\\frac{f_{y}}{m\\gamma}", "64400a8ef236c947a51c46b1ba42f1a6": "\\frac{K \\cdot t}{V} = -ln ((1-URR) - 0.008 \\cdot t) + (4-3.5 (1-URR)) \\cdot \\frac {0.55 \\cdot UF}{V}", "64403c341774bfcf302618a9f4622b1e": "S_0 = 0\\,\\!", "6440b919bd8b358371e23082cd2ce6e0": " supp(\\mathbf{b}) = \\{i : \\mathbf{b}_i \\neq 0\\} ", "6440cd6a9025dc6c361d9443655c2eaa": " \\frac{\\partial}{\\partial t} Q_i( t ) = -\\frac{1}{\\Delta x} \\left( f( q( t, x_{i+1/2} ) ) - f( q( t, x_{i-1/2} ) ) \\right), ", "644167a8b38669a0c71880703aaaa995": "I_a+I_b=I_c", "64418f40231ac91465463c89e9fecb1e": "\\det(V) = \\prod_{i,j=0, i \\alpha \\right \\}\\Bigr| \\leq \\frac{1}{\\alpha} \\, \\|f - g\\|_{L^1} < \\frac{1}{\\alpha} \\, \\varepsilon", "6461925ba218cd40117184c830108af6": "n=j", "6461c3ac143f0b8d240daa035721437b": "2005.2\\times 10^{-6}", "6461c48ea003df76e9c146d1bee5419f": "\\gamma^{(m)}(t)\\to \\frac{1}{2}[(t+1)^{2H}-2t^{2H}+(t-1)^{2H}]", "6461c950dcf8210186ce088922708184": "Pr[e=H\\mid q]=\\frac{f(q\\mid H)}{f(q\\mid H)+f(q\\mid L)}", "6461d855a116f0ab99c2939a91a261f9": " d+2 ", "6461dedcf5d12ce9eaafdd102bf405e2": "mN \\equiv TkN \\pmod{R}", "6462377a08068546c75173bd5d84b8c4": "p_i: P \\to P_i", "64623ec808fd31a8d5c1c472b1bfed3b": "s(h,k)=\\sum_{n=1}^{k-1} \\frac{n}{k} \n\\left( \\frac{hn}{k} - \\left\\lfloor \\frac{hn}{k} \\right\\rfloor -\\frac{1}{2} \\right).", "6462d50e2bc1309e7f7ec633c785b717": "cos\\Theta_{obs} = R*cos\\Theta ", "6462dd52cc0684f3848ca550ddd52da6": "\n\\langle \\Psi \\rangle_S \\rightarrow\n \\frac{ \\psi_{11} + \\psi_{22} }{2}\\begin{pmatrix}\n 1 & 0 \\\\ 0 & 1\n \\end{pmatrix} = \\frac{Tr[\\psi]}{2} \\mathbf{1}_{2\\times 2}\n", "6462f7481ac098d89ccf42844b71e8d3": "\n\\frac{1 - X}{X} = n_\\text{p} \\left(\\frac{m_\\text{e} k_\\text{B} T}{2 \\pi \\hbar^2}\\right)^{-3/2} \\exp\\left(\\frac{Q}{k_\\text{B} T}\\right).\n", "6463319019f3e8517abd0226eb8109ec": "\\mathcal{G}B(\\phi)=\\neg\\mathcal{F}\\neg B(\\phi)", "6463435e813fe3cc8bef703f6896a18a": "x \\to x-a\\,", "6463885a2ccba9f7aa32b92480a21dc9": "\\operatorname{sinh}(z)", "64639171e367b30e2eaaf51866d1c70d": " G(k_x,k_y) ~ = ~ H(k_x,k_y) ~ F(k_x,k_y)", "6463ad7b539ff13e364e757ec978431b": "\\text{E}\\left(e^{-t x}\\right)=\\eta e^{\\eta}\\text{E}_{t/b}\\left(\\eta\\right)", "6463fedef39c364ee1194019e8c93e49": " h_{\\alpha A}(x)=\\alpha h_A(x), \\qquad \\alpha \\ge 0, x\\in \\mathbb{R}^n", "6464864d778af8da668d2e2637101ffe": "v_1,\\dots, v_n", "64650b073c76444a19ab2cd2547dca60": "P_{rad}=\\frac{\\langle S\\rangle}{c},", "646525b1bd81e88375c63e6f49e63998": "\\boldsymbol\\mu_n", "64653ab20486a5ea296de58748110002": "\\bar{X}_1, \\bar{X}_2", "6465412fd8c759f3f57ff2fa0566c429": "\\Delta v = n {Ve} \\cdot ln (Mratio) ", "6465657b3eb9a46e8777a3a6b855b0f5": "\\langle\\widehat{\\mathbf{a}}_j^m\\widehat{\\mathbf{a}}_k^{\\dagger n}\\rangle=\\int Q(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)\\alpha_j^m\\alpha_k^{*n} \\, d^{2N}\\mathbf{\\alpha}", "6465a7b7218a6669ce749fe2bbae3ae0": "\\frac{RC}{27}", "6465aecfc2664a55160e61a2369a2b62": " f_-(z) = 1 - wz", "6465b4e1e33f8e2fa795cac4a051dfb9": "\\displaystyle{\\mathbf{SL}(2,k) = \\mathbf{B} \\cup \\mathbf{B}\\cdot J\\cdot \\mathbf{B},}", "6465bf4d8bbaf923861fcf594ff3230d": "\\textstyle \\mathcal{M}", "646623952e1707f57305d48ecd02ce38": "P_0(c)= 0 \\,", "6466d6f85c56d0ced48e189b3a2d5183": "J = \\begin{pmatrix}p & q \\\\ r & -p \\end{pmatrix}, \\quad p^2 + qr + 1 = 0", "6466f6a193bac42f93c87739ea503be8": "\\int\\cosh ax\\,dx = \\frac{1}{a}\\sinh ax+C\\,", "6467ff932f2b31ec04998345a4459fb9": "g : B \\rightarrow A", "64682aaa1ee386e173d0cce201606a9c": "\\mathbb{Q}(\\sqrt 2)", "646866739a5a01f9b630fdb7be31c2f4": "d_f = \\frac{\\lambda}{2 n \\theta}", "64688b9a2f7539a38a66e9fbb003291d": "\n \\operatorname{div}~\\mathbf{v} = \\boldsymbol{\\nabla}\\cdot\\mathbf{v} = \\text{tr}(\\boldsymbol{\\nabla}\\mathbf{v})\n ", "64688c255091b2490e0c0fc3ed106a0f": "1-\\tfrac1{24}(wh)^3+\\mathcal O(h^5)", "6468917a235bfcd86b674f9e1a464b43": "\\frac{16}{\\lambda(2\\tau)} - 8", "64691a34e7c8981f8330e937bc927469": "\\forall m \\forall n [m+Sn = S(m+n)].", "64694d71640004d29f8cb99a483a288a": "\\mapsto", "64697a4688536c5ab8c0c0fc138d69e8": "\\displaystyle{(Bf,g) ={1\\over 2\\pi} \\int_0^{2\\pi} \\varphi(\\theta) (U_\\theta A^{(1)} U_\\theta^*f,g) \\, d\\theta}", "6469b266c27b11e24e16973960da73be": "\\sigma_i\\in\\{-1,1\\}", "6469f2d12bd400775e7dbc34aba171df": " (c \\oplus d)", "646a09a8631be48ee5896b98b6a5f64c": "w_n^2 + w_{n + N}^2 = 1", "646a4c6cad9ece9daa0c6a8760ad154d": "\\bigcup M", "646aab232c0bd6def0b735d9f3194300": "\\,\\phi(v_i)=\\tanh\\left(\\beta_1 + \\beta_0 \\sum_j v_{i,j} x_j \\right)", "646aafc45f99e5745e66c1f15d5b37b6": "\\displaystyle \\Delta v", "646b0c3a3169ebe9fc4d036e61287e54": "dF_r = \\frac{Gm \\;dM}{s^2} \\cos\\phi.", "646b306fb226138674efe975c1d67ba1": "B \\cdot N_0", "646b616c18e3ca2e18a23b389ed8efb7": "\n \\sigma_{11} = 2C_1\\left(\\lambda^2 - \\cfrac{1}{\\lambda^4}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right] = \\sigma_{22} ~.\n ", "646b7382394f363f9d6d4337e2f36b9c": "m\\ge0", "646c0c729526fade24dc7c7e4713042b": " J \\;= \\lambda_Z a \\phi^{-1} F^2 P_{\\mathrm{F}} \\mathrm{exp}[- \\nu_{\\mathrm{F}} b \\phi^{3/2} / F ], ..........(29) ", "646c24d71f7691a121e3e13e5c06b7df": "\\mathbf{Y_0}", "646c293a894a302cd5a9655a4b6a80c9": "R(t) = 1 - F(t).", "646c6d191eff2fda8388ec9fe0c0cd15": "\\omega>1", "646c8405027a091d39493c8502a9feb8": " \\mathrm{Hg} = -\\frac{1}{\\rho}\\frac{\\mathrm{d} p}{\\mathrm{d} x}\\frac{L^3}{\\nu^2} ", "646cbd925a1e37f549b91cf4091eaaed": "\\frac{x}{1+x}=[0;a_1+1, a_2, a_3,\\cdots].", "646ce06fb4e2c0aedf80f822b2fbc971": "C(\\mathbf{h})", "646cf2de7e0994f8411a88d88a7bb29d": "\\sum_{p=0}^{q-1}\\zeta(s,a+p/q)=q^s\\,\\zeta(s,qa).", "646cf779932201ed869aae8898a1d67b": "\\begin{align}\nf(x)&=\\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{ipx}\\left(\\int_{-\\infty}^\\infty e^{-ip\\alpha }f(\\alpha)\\ d \\alpha \\right) \\ dp \\\\\n&=\\frac{1}{2\\pi} \\int_{-\\infty}^\\infty \\left(\\int_{-\\infty}^\\infty e^{ipx} e^{-ip\\alpha } \\ dp \\right)f(\\alpha)\\ d \\alpha =\\int_{-\\infty}^\\infty \\delta (x-\\alpha) f(\\alpha) \\ d \\alpha,\n\\end{align}", "646d318046cd806bb1f7b72a89a12539": "|S| = \\left( \\coprod_{n=0}^{\\infty} S_n \\times \\Delta^n \\right)/_{\\sim}", "646d33dff934c32559ed1a6eb8e60aca": "A = \\frac{{\\mathrm d}Q'}{{\\mathrm d}\\xi}. \\, ", "646d3404583997c0811238d8631d8f9e": "x(t)=0.", "646d3c5420e5299a9e21479409e79dd4": "\\alpha=1 \\dots 8", "646d6131d8e78c7f2eb9cd250f3e39c1": "E = 0.772 m_1(\\Delta v)^2", "646d681317206e644d927fb2f3531927": "C\\subseteq P_1\\cup\\ldots\\cup P_m\\subset \\R^n,", "646d86a5c780599dd3289dd461284bbd": "\\hat{L_z}", "646de8cbb8a23d9eb2c76f7554fd57f8": "v_p(\\Delta_\\lambda)v_p(\\Psi)", "646e6e8b1ae81d1e2ea0a9f83cf84a46": "\\langle s,t,\\phi(s,t) \\rangle", "646e8e7bcf38c528cfb1edce1324a6ee": "\\phi(\\eta)=\\theta^i\\eta_i-\\psi(\\theta)", "646ef25a9535caffa1245002d1de1cb0": "A\\subset\\{1,2,\\dots,n\\}", "646efb28757f6ff83f699b27e242cc05": "ln K_{eq} = a+ \\frac {b}{T} + \\frac {c}{T^2} ,", "646f3e15ae798fc664ec7279ea51045d": "\\frac{y_n}{x_n}\\frac{X}{Y}=\\frac{x/x_n}{y/y_n}", "646f8d4abdb4073b27bf86bd45a62f1e": "B_2 = {\\gamma_0, \\gamma_1, \\ldots, \\gamma_{m-1}}", "647028a59807d16ccd40567457225fcb": "m=\\frac{y_v}{y_u} \\,;", "64705f826162cbff8d5e376ea452d525": "T_fT_g - T_{fg} \\,", "6470d563c42d53efd1cfcd580d7b9923": "\nP^{MAP}_{j}=\\max_{i,q}\\frac{P_j(i,q)g(j-i)}{1-G(j-i-1)}, \\forall j \\sqrt{5}\\;", "647f934e0f9933c1a40616e97dd72554": "\\exp(2\\epsilon\\Delta q)\\,\\!", "647f991ca1b7f4b0a493f6958309b98e": "\\begin{align} x_{n+1}^2 = x_n^2 - 2y_n - 1 \\end{align}", "647fc464206f0197c12e072f97b296a9": "\\{f_j^{(1/p)}\\}", "647feb2726f339635e59060e1e6f8a38": "f(z)=az+b\\overline{z}", "64800529ffae6e4fdd56027fc8ceba55": " ~\\sigma^2 ", "6480117150ea4c7cad44cb410a461a5f": "\\mathbf{E} = -\\nabla\\phi", "6480182b26628508a525e8aab1f9e9d4": " y^G ", "6480cf0b73912d49e530a1ba8329525c": "\\dot{x_1}(t) = x_2(t)", "6480db890ec38d3f73d28e5431aed491": "u_i=1/n", "6481068f3959dc685f6903dee1ba73a2": "gcd(m,n) = m", "6481077b932f9ae0a17276a600958288": "\\exists n_1 \\forall n_2 \\exists n_3 \\forall n_4 \\cdots Q n_m \\rho(n_1,\\ldots,n_m,x_1,\\ldots,x_k),", "64815e79177b832f4b63f2445a6851db": "\n\\Lambda_{m} \\geq0\\ \\ \\ \\ \\forall m", "64816fc368e485447022b4ada94540d6": "\\mathbf{a}_\\mathrm{A} = \\mathbf{a}_\\mathrm{B} +\\ 2\\sum_{j=1}^3 v_j \\boldsymbol{\\Omega} \\times \\mathbf{u}_j (t) + \\sum_{j=1}^3 x_j \\frac{d\\boldsymbol{\\Omega}}{dt} \\times \\mathbf{u}_j \\ + \\sum_{j=1}^3 x_j \\boldsymbol{\\Omega} \\times \\left[ \\boldsymbol{\\Omega} \\times \\mathbf{u}_j (t) \\right]", "6481a0878a62cb4f985284fcf6abd556": "f=\\sum_{\\{x\\mid f(x)\\ne 0\\}} f(x) e_x \\mapsto \\sum_{\\{x\\mid f(x)\\ne 0\\}} f(x) x,", "6481a79833d7ddc1720ef37942e8667b": "\\alpha_1=\\alpha_2=\\alpha_3=\\alpha", "6481af37dc7c0400a46f1cbd6631349c": "E_{\\lambda}=E[\\psi_{\\lambda},\\lambda],", "6481e0ce1912c1e5e2b65e993ceb0f54": "\\scriptstyle q_i", "6481e931773f85782007d7ad3e266fd6": "b \\ge 2", "6481f36f125f79c10e198c921624b05b": "\\text{see} : (S\\backslash N)/N", "6481f5231e257e960321df03627c1f81": "((p \\lor q) \\land \\neg p) \\vdash q", "648254917b615e6196c94f20b7d5ef63": "\\{w_1,\\ldots,w_G\\}", "648271eee7fc0613f729b5a4807cc20f": "\n\\begin{pmatrix}\\alpha^{-1}+\\alpha^{6}x&1\\\\ 1&0\\end{pmatrix}\n\\begin{pmatrix}x^6\\\\\n\\alpha^{4}+\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3+\\alpha^{1}x^4+\\alpha^{-1}x^5+(\\alpha^{-1}+\\alpha^{-1})x^6+(\\alpha^{6}+\\alpha^{6})x^7\n\\end{pmatrix}=\n", "64829d12b2bb1b03cd02ce092ba893d5": "(-1)^{n-k}x_1^{b_1}\\cdots x_n^{b_n} x^k,", "6482a917ed52c9988809c71048c6c810": " 2^\\kappa\\le 2^\\lambda\\,", "6482b151e9b0acd2770650e62bca660e": "\\sin(\\theta - \\phi) = n \\sin(\\psi - \\phi)", "6482ffb7334692394c0c10624f8b1383": "t \\pmod l_i", "64835756c6b9744950239ad5bfb4bb59": " \\left ( \\frac{\\operatorname{d}i}{\\operatorname{d}t} \\right ) _c ", "6483d1cb50d8f4cf1c02f202033a6e74": "p_r = \\frac{(r+1) S(N_{r+1})}{N S(N_r)}.", "6483e190e3bd384bcecfe30f2222f907": "\\chi(x)", "6483e4926d40d640b9e1f5e15102e501": "\n M = N_{xx}^{\\mathrm{topface}}~(2h+f) + 2~M^{\\mathrm{topface}}_{xx}\n ", "6483fac31d553a2de021a68f674c0196": "F(x;\\mu,\\sigma, a,b) = \\frac{\\Phi(\\xi) - \\Phi(\\alpha)}{Z}", "6484311d1f5b2330a2e688e7462b93dd": "S=\\mathbb Q", "64848da6dfbff6b2140946021764fce9": "xAA + xBB \\rightarrow AA-(BB-AA)_{x-1}-BB", "6484a2c965c894ce5e99b96f2ce2f3eb": "v\\in \\mathbb{F}_{q^n}", "6484c514c03cd08ed6a79baf9ab52779": "\\mathcal{L}=\\overline{\\psi} \\left(i\\partial\\!\\!\\!/-m \\right) \\psi + \\frac{g}{2}\\left(\\overline{\\psi} \\psi\\right)^2,", "6484efd1be9ff1c12d6d2947d95eb753": "\\! C< x \\mbox{ and } sy < y\n\\end{cases}", "648d92a116a5487265496a409b2a9cb0": "x^3 + 7x^2 + 8x + 2 = (1)^3 + 7(1)^2 + 8(1) + 2", "648daeb7ee9cc2fb3f3e42fe1a413706": " S_{i,j}^t=2 ", "648dbc9187b24cbfbe6b10c5a3b570ed": "\\cos(\\theta_2)=1/\\sqrt{1+b^2}", "648ddc9e774ac25163a3bb4974066528": "\nU =: \\begin{bmatrix} U_1 & U_2\\end{bmatrix}, \\quad \n\\Sigma =: \\begin{bmatrix} \\Sigma_1 & 0 \\\\ 0 & \\Sigma_2 \\end{bmatrix}, \\quad\\text{and}\\quad \nV =: \\begin{bmatrix} V_1 & V_2 \\end{bmatrix},\n", "648ddefb20fd03880018724ac94e1920": "\\alpha^n\\ge x_1 x_2 \\cdots x_n\\,", "648dffb97cf583a7503ac444c2816a4e": "C_{2} = 2 \\pi \\,.", "648e0ba295a0fdf2ad41ea79604a9335": "\\textstyle x^{j}b(x)", "648e22ed8fe6fe056b247884bc5bae6f": " {n \\choose k} = \\frac{n!}{k! \\, (n-k)!} , ", "648e3238f049399d50d003f83036ec77": "\\mathbf{L}=I\\boldsymbol{\\omega},", "648f2f0b6546aabaef69502b6819bfc4": "\nW = k^D (2 \\pi)^{D/2}\n", "648f5a4054789035a6be8152d09e5f6e": "L_t=L(w_t,L_{t-1})\\,\\!", "648f740a06038f474a3157c99421b5c9": "z\\!\\ge\\! 0", "648fabe5da4f3fbbd86b2e503f350dc2": "\\hat{H}_{S} \\big(\\hat{H}_{B}\\big)", "6490084f130e61c3ec13ebdaa963ea69": "ds^2=0", "649034e4b73d779dc43952cce9c7dfde": "E = \\sum_{n=-\\infty}^{\\infty} {x[n] \\cdot y^*[n]} \\!", "64903b328cb524463140613869eeb3b8": "P_c^* = 0.127 + 0.0023\\mu^{*2}", "6490586659b90b7d76945a02f9c8cd94": "-2/3", "6490594ecbf428efb43998a476e5686b": "v=v(x_1,x_2,t)", "6490caaee226b27129f3b1f388f8440d": "\\sqrt{\\frac{Z_1Z_2}{E_0d}}", "6490f1aee72a5031c5b8229627319eeb": "b^{-n} = {1}/{b^n} .", "6490fb32abfdcb013788bcd0696150a8": "\\mathcal R", "6491224e90b0cb16ab83148b7c02921f": "{\\mathbf A_{ij}}", "6491300cfbdfed0fc0a3b487b806cfee": "\\sqrt{1836}", "649153ae2a612227c39426d46cecbb8c": "\\displaystyle P(w|d) = \\sum_{z=1}^Z P(w,x|z)P(z|d)", "6491d8b93575729f33994c18fdccd78d": "j = i - 1", "6491e67baaa4516c5594b0e88ee44d1c": "v_2=\\sqrt{\\frac{2 \\times G \\times M}{R}}", "649222326958e787200a075e29538b2b": "\\langle u_x \\rangle = \\frac12\\, \\lambda\\frac{\\mathrm{d}u_x}{\\mathrm{d}y}", "649230c861af6f698b8d1407f15ed100": "\n\\hat{H}_\\mathrm{nuc} \\approx \\frac{1}{2} \\sum_{t=1}^{3N-6} \\left[-\\hbar^2 \\frac{\\partial^2}{\\partial q_{t}^2} + f_t q_t^2 \\right] .\n", "64925e843581f3a940f3e0a84099a123": "\\left.\\frac{\\partial \\phi(\\boldsymbol{x},y)}{\\partial y}\\right|_{(\\boldsymbol{x},y) = (\\boldsymbol{a},b)} \\neq 0 ", "6492afcbb4b4035800b9874a24e39ed3": "h_i=\\frac{1}{2} \\sqrt{\\frac{(q_j-q_i)(q_k-q_i)}{(a^2-q_i)(b^2-q_i)(c^2-q_i)}}", "649407c347130e2afe9acbf38e7da594": "\\nabla_i\\equiv \\nabla_{\\mathrm{e}_i}", "64946157b1113f2f5e89e519638c202e": "x_0=-\\tfrac{3}{2}", "64946a880cafdd8e4999dd317af38539": " \\boldsymbol{\\omega} \\times \\mathbf{B} = \\mathbf{B'}, ", "64947f44dfc22807e9f84f8e1b8e8a19": "\nG(\\boldsymbol{x} - \\boldsymbol{r}) = \\left( \\frac{ 6 }{ \\pi \\Delta^{2} } \\right)^{\\frac{1}{2}} \\exp{ \\left( - \\frac{6 (\\boldsymbol{x-r})^2}{\\Delta^2} \\right) }.\n", "6494885733239a1a89a2dac97b312014": "\\mathfrak{f}", "6494d83fce97efb882e466b3661ac0c1": "\\boldsymbol{r}_0", "6494f8b31977372d9d0c74ccd9d0e3c5": " \\lambda_c = \\frac {1}{P_{total}} \\int p(\\lambda) \\lambda\\, d\\lambda", "64950e334e2255745a41f3d6421eb30e": " \\det(\\lambda I+A(\\zeta,z))=0, ", "64952b4b18105da0e26da393dd533ee0": "\n\\mathbf{e}_i \\mathbf{e}_j = - \\mathbf{e}_j \\mathbf{e}_i \\,\\,; i \\neq j \n", "649532104e50cce985223742acc18ece": "g = \\begin{bmatrix}\n1&0&\\cdots\\ 0\\\\\n0&&\\\\\n\\vdots &&g_{\\phi\\phi}(r,\\phi)\\\\\n0&&\n\\end{bmatrix}.", "649549e0a00c8d036f422407f034f186": " \\{\\varphi: U \\rightarrow {\\mathbf R}^2\\}", "64957ae70cb8c1ff15b171dba6508a77": "\\lambda_3 = -2", "64958c057fdb68806e265e4f810535f6": "1.7990", "64959ff530e7d9c005deddc10df98911": "r = Z \\rho [A][B] \\exp \\left( \\frac{-E_{a}}{RT} \\right)", "6495a770f0137cd10720de5d38b16824": "s=0.114 \\mathrm{nm}^2", "6495b1a7368c452d2a5b36b1d667dc31": "\\tfrac{157}{50}", "6495f46fc395c552e46b49cf0e54ed49": " \\sum_{k=1}^{N}\\frac{1}{k}", "6495fa9596f821e70c91359df92971f0": "s[n]", "64961759816ec6502a3e4db2df2999f9": "\\alpha. \\, \\!", "64965193cdcbad51d3cab092bd6d574f": "\\begin{align}\n & \\hat\\beta = \\frac{ \\sum_{i=1}^{n} (x_{i}-\\bar{x})(y_{i}-\\bar{y}) }{ \\sum_{i=1}^{n} (x_{i}-\\bar{x})^2 }\n = \\frac{ \\overline{xy} - \\bar{x}\\bar{y} }{ \\overline{x^2} - \\bar{x}^2 }\n = \\frac{ \\operatorname{Cov}[x,y] }{ \\operatorname{Var}[x] }\n = r_{xy} \\frac{s_y}{s_x}, \\\\\n & \\hat\\alpha = \\bar{y} - \\hat\\beta\\,\\bar{x},\n \\end{align}", "64965510bdc34122ba0c3da047b31d0c": " (\\varphi^*\\alpha)_x(X) = \\alpha_{\\varphi(x)}(\\mathrm d\\varphi_x(X))", "64968b10db82448889b1da305d7108ac": "\\left(\\frac{n^2+1}{2}\\right) n", "6496b08106f404b6a442f0869a498e5e": "\\lambda_{\\rm min} = {2L \\over \\sqrt[3]{N}}\\,,", "64970007d570d2ab5fa252d649c70561": "\\Delta x_{i+1} = \\frac{f^{\\prime\\prime} (\\alpha)}{2 f^\\prime (\\alpha)} (\\Delta x_{i})^2 + O[\\Delta x_{i}]^3 \\,,", "64971e837a6f4b1d59577818f3280822": "\\vert\\psi_{n}\\rangle", "649783e922b02210a6230acc58950ab1": "(\\hat{\\alpha})", "6497a0d87a39f25c31d2856f0b07a9a4": "\\lambda>0\\,", "64985602e9794e182cdaf22493d52090": "\n\\frac{\\partial}{\\partial\\theta} \\int \\left[ \\hat\\theta(x) - \\theta \\right] \\cdot f(x ;\\theta) \\, dx\n= \\int \\left(\\hat\\theta-\\theta\\right) \\frac{\\partial f}{\\partial\\theta} \\, dx - \\int f \\, dx = 0.\n", "6498ccffc389c96e99ccd78db750f255": "\\,x \\prec y", "6498e46433e2517e3b44a761f5f5ef3a": "f(x) = a_0 + a_1\\binom{x}{1} + \\cdots + a_d\\binom{x}{d}", "6499255e7346fef53284ee776280cd17": "\n\\begin{align}\n\\lim_{x\\to 0}{\\frac{2\\sin x-\\sin 2x}{x-\\sin x}}\n& =\\lim_{x\\to 0}{\\frac{2\\cos x -2\\cos 2x}{1-\\cos x}} \\\\\n& = \\lim_{x\\to 0}{\\frac{-2\\sin x +4\\sin 2x}{\\sin x}} \\\\\n& = \\lim_{x\\to 0}{\\frac{-2\\cos x +8\\cos 2x}{\\cos x}} \\\\\n& ={\\frac{-2 +8}{1}} \\\\\n& =6.\n\\end{align}\n", "64997d406f7b38dfa6d52ea8cd8acad7": "y_iE_2(\\alpha_i) = Q_2(\\alpha_i", "6499a29dff9e6eb28c69a37b1fb9f868": "\\beta_j\\,", "6499ae0492419a0685dc835ca206180b": "4^3 = 64", "6499cb6f05f99a0c04162b9634d87dbf": "(S,\\mathfrak{m}_S)", "6499ef638a56b989ab528dbeb2dd4567": "\\frac{d^2\\beta}{dt^2}+(\\frac{2k}{MV}+\\frac{2ka^2}{VI})\\frac{d\\beta}{dt}-(\\frac{2ka}{I})\\beta=0", "6499efcf9e1e4b814d4ee7a6aa795fa8": "Q_{\\lambda}^{\\mu}(z) = \\frac{\\sqrt{\\pi}\\ \\Gamma(\\lambda+\\mu+1)}{2^{\\lambda+1}\\Gamma(\\lambda+3/2)}\\frac{e^{i\\mu\\pi}(z^2-1)^{\\mu/2}}{z^{\\lambda+\\mu+1}} \\,_2F_1 \\left(\\frac{\\lambda+\\mu+1}{2}, \\frac{\\lambda+\\mu+2}{2}; \\lambda+\\frac{3}{2}; \\frac{1}{z^2}\\right),\\qquad \\text{for}\\ \\ |z|>1.", "649aa7048d42f693d59dd8e1a87d23b4": "(X_0, X_1)_{\\theta, q} = (X_1, X_0)_{1 - \\theta, q}, \\quad 0 < \\theta < 1, \\ 1 \\le q \\le \\infty.\\,", "649ac3a0e8907c56fd20360b2cf7490e": "\n \\sigma = \\mathbf{t}\\cdot\\mathbf{n} = n_i~\\sigma_{ij}~n_j ~~\\text{(repeated indices indicate summation)}\n ", "649ae700a008148c06bd510a1833b377": "\\ln\\ell-\\ln n = \\ln (\\ell /n)", "649aec1b90b93e044d7bed745a0ee104": "y = a \\sec \\varphi = a \\cosh \\tfrac{x}{a},\\,", "649b533ff5974834a703d55ea4db4cd7": "\\theta=2\\pi", "649b91ea365dcad9736822a7f73357c2": "\nh_{\\sigma} = h_{\\tau} = \\frac{a}{\\cosh \\tau - \\cos\\sigma}\n", "649bc2463b533e034ecac919590bb720": "\\scriptstyle - \\frac{1}{2}", "649bc7bef14af4c3280bd1c668f6b7b0": "\\tan 50^\\circ\\cdot\\tan 60^\\circ\\cdot\\tan 70^\\circ=\\tan 80^\\circ.", "649bdaa0b32bc7a3d1c4fc536b717ee7": "f^-(\\bold{x})=\\min(\\sum_S \\alpha_S f(S):\\sum_S \\alpha_S 1_S=\\bold{x},\\sum_S \\alpha_S=1,\\alpha_S\\geq 0)", "649c2bf4fb65e15fde366890535354ac": "m^2-n^2,", "649c520f70df7d95fe081a903fc3b0ac": " \\|T\\| _{p} := \\bigg( \\sum _{n\\ge 1} s^p_n(T)\\bigg)^{1/p} ", "649c576015f48558c002d5f41bd9ddcd": "8 \\sum_{n=1}^\\infty \\left(\\frac{9}{10}\\right)^{n-1} = 80.", "649c59203995093bd6edeb93505f7936": "\\ m_\\mathrm{indif}=m_\\mathrm{budget} ", "649c82a1499dc305e39e02a7530d8a56": " n D(n) = 3(2n-1)D(n-1) - (n-1)D(n-2) , ", "649cc7ee434fc7d8ca73b70d26f88e26": "\\hat{C}_{pk}", "649ce969f414fe222db402d7cfef907e": "r_E = r_0 + \\frac{D}{E}(r_0 - r_D)(1-T_C)", "649d3176dfc64673afdaf504f6de7660": "\\widehat K\\to \\widehat K/\\widehat H", "649d4867e80aa02a5a8e9522c3267d04": "\\phi(t)=1", "649d832a8e8f5019c397d91fbdf0e7b2": "C_k = (b - 2)b^{k - 1} + C_{k - 1} + (b - 2)\\,", "649e0be6ba5230e2ac038ac6604aee82": "Q = \\prod_{\\text{primes}~q \\in (B_1, B_2]} (H^q - 1)", "649e4701312b8d6be967325581eac4f2": "\\displaystyle \\sqrt{\\frac{\\pi}{\\alpha}}\\cdot e^{-\\frac{\\nu^2}{4 \\alpha}}", "649e56a383e5dd8ab312fa005479f92c": " x_i\\in \\Phi", "649e59815fec445e84dcdebc2d8dd960": "M_x = \\frac{1}{a} \\frac{\\partial \\Phi}{\\partial x},", "649e793887ab29fb7c0c50a1bf3f7c06": "\\operatorname{dim} X = 0", "649eb48d487f9da07d829da6cc55a154": "M_{unit,1}= {{{y_1}^2}\\over 2}+{{q^2}\\over gy_1} = {{{16.3}^2}\\over 2} +{{{\\left({100\\over10}\\right)}^2}\\over{(32.2)(16.3)}}", "649edbc0f859ee2cb941ca8110408c15": "\\phi_{\\lambda}^{\\mathrm{L}}(\\mathbf{k})", "649fb6f351446a3879ffa62563c9e669": "\\mathbf F_i", "64a05381c273fe7f4454bfd1e3d11ab9": "\\sqrt{(t - p)(t - q)(t - r)(t - s)}.", "64a06ecd9c66f49ace2caf7aa4a4a3fd": "\\mathbb{E}\\Phi(||\\mathbb{P}_n - P||_{\\mathcal{F}}) \\leq \\mathbb{E}_{\\varepsilon} \\mathbb{E} \\Phi \\left( \\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n \\varepsilon_i [f(X_i) - f(Y_i)] \\right|\\right|_{\\mathcal{F}} \\right)", "64a09a16750a1dc9794c12b6ba4e90ca": "-n a^n u[-n-1]", "64a09ef43094a4a1b59db2ae5a9a7efe": "\n \\langle 2\\rangle = \\langle 2\\rangle_\\mathrm{S} + \\Delta \\langle 2\\rangle\n", "64a0ba100ac76251e52c46072b92c2d4": "\\begin{align}\ng^{\\mu\\nu}\\frac{\\partial}{\\partial{x^{\\mu}}}\\frac{\\partial}{\\partial{x^{\\nu}}} = \n& \\frac{1}{c^{2}\\Delta}\\left(r^{2} + \\alpha^{2} + \\frac{r_{s}r\\alpha^{2}}{\\rho^{2}}\\sin^{2}\\theta\\right)\\left(\\frac{\\partial}{\\partial{t}}\\right)^{2} + \\frac{2r_{s}r\\alpha}{c\\rho^{2}\\Delta}\\frac{\\partial}{\\partial{\\phi}}\\frac{\\partial}{\\partial{t}} \\\\\n& - \\frac{1}{\\Delta\\sin^{2}\\theta}\\left(1 - \\frac{r_{s}r}{\\rho^{2}}\\right)\\left(\\frac{\\partial}{\\partial{\\phi}}\\right)^{2} - \\frac{\\Delta}{\\rho^{2}}\\left(\\frac{\\partial}{\\partial{r}}\\right)^{2} - \\frac{1}{\\rho^{2}}\\left(\\frac{\\partial}{\\partial{\\theta}}\\right)^{2} \\end{align}", "64a10853e4ca0b0ea7a859b1d9cdcd3f": "\\ F(aK,aL)=aF(K,L) ", "64a12bf0a90fbc6769cf83afe6759725": "O(2^nn)", "64a17a4b82f650cc87c5fe149b084439": " \\vec M_m(\\vec S_m, n), m = 1..M.", "64a1db4efc72eba98aa31f007cf2ac19": "O(\\tfrac{d}{\\epsilon}\\log\\tfrac{1}{\\epsilon})", "64a21200ae0aded94cdcb39d011de3fd": "f(t,H)", "64a219723f4c9a06cdb280f404e32ffa": "k\\leq p", "64a26772bbc0eaef73abdd3575e07d40": "H_{i+1}", "64a2718756d275a3d5c41e04dde09a28": "j_\\mu^{em} = \\frac23\\overline U_i\\gamma_\\mu U_i -\\frac13\\overline D_i\\gamma_\\mu D_i - \\overline l_i\\gamma_\\mu l_i.", "64a28232d635e4fe307b3b177264b654": "u'' + \\frac{2}{x}\\,u' + \\left[\\frac{\\lambda}{x} - \\frac{1}{4} - \\frac{\\alpha^2-1}{4x^2}\\right]\\,u = 0\\text{ with } \\lambda = n+\\frac{\\alpha+1}{2}.\\,", "64a2b2dccd49552eb34c6f408843a2c0": " p_i^{(0)}(n) ", "64a34932da993998680198d8c2d51b58": "A e^{i\\theta}\\,", "64a383636fe3f78868743ecd8c43fd11": "\\scriptstyle TV_a", "64a3eea6ad7b44044577b78cadac8b62": "B_i (0,0)", "64a40ab1a22635ff35e8419fed76a703": "\\textstyle I_0", "64a437bfb48f0a2b171fb5b7ae72a049": "\n \\mathbf{F}~\\mathbf{N}_i = \\lambda_i~(\\mathbf{R}~\\mathbf{N}_i) = \\lambda_i~\\mathbf{n}_i\n\\,\\!", "64a44b190b285a91bdf46901222a20c9": "G = (V,E)", "64a473dbb4923118872329d185720d1e": "\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}", "64a47af7327be4f4a442b06ad2f06cb6": "f(x;\\beta) = \\begin{cases}\n\\frac{1}{\\beta} e^{-\\frac{x}{\\beta}} & x \\ge 0, \\\\\n0 & x < 0.\n\\end{cases}", "64a4cccb084f7ec390527b78b70ebb50": "\\Delta < 0", "64a520f0b88c9472808098640c8a448f": "\\mathbf{G}^i", "64a55bc3906bce49ca74fe0a301c4a28": "{u_{ij}^{n+1}-u_{ij}^n\\over \\Delta t} = \n{1 \\over 2}\\left(\\delta_x^2+\\delta_y^2\\right)\n\\left(u_{ij}^{n+1}+u_{ij}^n\\right)", "64a57c6be2fa57aa716353c04dfc036d": "L(t_1,\\ldots,t_n)=L(t_1',\\ldots,t_n')", "64a5a9449a96168e0e698684e1bbdd67": "\n\\mathbf{r} = (x, \\ y ) = r (\\cos \\varphi ,\\ \\sin \\varphi)\n", "64a5b9be64f62d1df70029c2ca18d9cb": "f,g : (\\mathbb{C}^n,0) \\to (\\mathbb{C},0)", "64a6705867c857aee51225ace82a03b8": " x = \\sqrt[3]{6+\\sqrt[3]{6+\\sqrt[3]{6+\\sqrt[3]{6+\\cdots}}}} ", "64a6b3c2b5705fc94f644cec9835733c": "p_0 \\leq q_0", "64a6d7ff7f3ce5073d603d9e1ee57f1e": "\\exp(a_1(e^{ti}-1)+a_2(e^{2ti}-1))\\,", "64a6de9ad267e2e8da715ff3f64429b0": " l_{31} = a_{01} - \\mathcal{L}(p_8)+p_3p_8+p_2p_9,", "64a6e2524afb80ffbf10fd57f37b1b29": " \\mathbf n = (\\cos \\theta,\\sin \\theta)", "64a7de1d07c60a454c6786df00c86c82": "10^{10^{10^{10^{2.08}}}}", "64a8188fcb7a83890f8cae2c812c02a7": "\\mbox{eGFR} = \\mbox{32788}\\ \\times \\ \\mbox{Serum Creatinine}^{-1.154} \\ \\times \\ \\mbox{Age}^{-0.203} \\ \\times \\ {[1.210\\ if\\ Black]} \\ \\times \\ {[0.742\\ if\\ Female]}", "64a8dbc6adf934c9d433ab578f5a992e": "\\sum_{i_1+\\cdots+i_k=n, i_1>1}\\zeta(i_1,\\cdots,i_k)=\\zeta(n)", "64a8e86f00d0f35498e51cebd7ebea6a": "f(0)= 0, f(1) = 1, f(N) = K", "64a8fe2b059a27a17c4834007ef892b1": "c \\in _{R}\\boldsymbol{\\Zeta}_{q} ", "64a9697fff3b99fb1e7388e662fb9a1d": "I(...)", "64a96d8dd9db1c2228febc65dcb20290": "V_\\min = V_f - V_r = V_f - \\rho V_f = V_f ( 1 - \\rho).\\,", "64a98232de310bcbd62fb0c65b442e91": "R_a = \\frac{1}{n} \\sum_{i=1}^{n} \\left | y_i \\right |", "64a983ce58ba6f995dcf6c354acda4a7": "\\forall m [m+0=m].", "64a99186f978952d4f7bcccb25ac3d02": "[S_g^{-1}\\alpha,\\beta] = [S_g^{-1}\\beta,\\alpha]", "64a9b060013721f6181bd6eac975f240": "(p_\\sigma,", "64aa2abd83f5fa96e666872cc0750087": "x \\not\\approx x", "64aa8db04acdedd5f4a3f6ae35aa7c88": "5/6", "64aaee874d0fab4c7c7f31e1779f94f2": "a_m, \\, \\exists b_k \\in \\Gamma(a_{m_k}), \\, b_k \\rarr b ", "64aafdadfe9c16da4cfd57c257e0ed36": "(\\mathbf x_k)_k", "64ab0be297b8fa212b8052b1f510210d": "E=\\Big( \\frac{\\phi_0^2\\omega^2}{2}+U(\\frac{\\phi_0}{\\sqrt{2}})\\Big) \\frac{4}{3}\\pi R^3.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)", "64ab2f3722a6f10f0b32f90b860e93e5": " i= ", "64ab3d80fdc461b43278f69d931edc32": "(\\sum_i a_ix^i)'=\\sum_i (a_ix^i)'=\\sum_i ((a_i)'x^i+a_i(x^i)')=\\sum_i(0x^i+a_i(\\sum_{j=1}^ix^{j-1}(x')x^{i-j}))=\\sum_i\\sum_{j=1}^i a_ix^{i-1}.", "64ab5acc58393a4c6d4c08a50417287b": "X \\to p \\to 2 = f^p(1)", "64ab6c623b36e364fdc4fbf1d922cd1e": "\\begin{align}\n\\int_\\delta^\\infty t^{\\lambda + n} e^{-xt}\\,dt &= \\int_0^\\infty (s+\\delta)^{\\lambda + n} e^{-x(s+\\delta)}\\,ds \\\\\n&= e^{-\\delta x} \\int_0^\\infty (s+\\delta)^{\\lambda + n} e^{-xs}\\,ds \\\\\n&\\leq e^{-\\delta x} \\int_0^\\infty (s+\\delta)^{\\lambda + n} e^{-s}\\,ds\n\\end{align}", "64ab9b38cad54c42bd3a7cc970ba520d": "\\scriptstyle (\\rho)", "64abe517a8c97c684e9763685138c654": "Q_{P}(h)\\,\\!", "64ac24e531effd9c5e2a1a21fadfe458": "x \\# y \\;\\leftrightarrow\\; x < y \\;\\vee\\; y < x", "64ac50fdfe74467a93a451ca7ff67164": " \\mu_t = \\rho I_t + (1 - \\rho) \\mu_ {t-1} ", "64acf1f27add05f7d2d930094da7e069": "\\displaystyle{Q(a)Q(b,c)a=Q(Q(a)b,a)c.}", "64ad0a37b10c708a868ad3af460f24a9": " \\frac 1 {p_\\theta} = \\frac{1 - \\theta}{p_0} + \\frac{\\theta}{p_1}, \\ \\frac 1 {q_\\theta} = \\frac{1 - \\theta}{q_0} + \\frac{\\theta}{q_1}, \\ \\ s_\\theta = (1 - \\theta) s_0 + \\theta s_1.", "64ad8c36309144aa5730de1e8f89798f": "\n\\begin{align}\ns' & = m''^d\\pmod n \\\\\n & = ((mr)^e\\pmod n)^d\\pmod n \\\\\n & = (mr)^{ed} \\pmod n \\\\\n & = m \\cdot r \\pmod n \\mbox{, since } ed \\equiv 1 \\pmod{\\phi(n)}\\\\\n\\end{align}\n", "64ad99e08d2201ced0490948d58ad059": "\\zeta(x+2iy)", "64adcade2df4a413c3f61c8d6e59205c": "{ {\\underbrace{a \\cdot a \\cdot a \\cdot a \\cdot \\ldots \\cdot a}} \\atop{b} }", "64adcb5df7924630df4dbc8d0acb39d1": "G = \\frac{\\pi \\sigma^2 a}{E}\\,", "64ade629ffe12b5a44f38004f75a2b89": "\\omega_q", "64adf40211b86365c5d5c5b90de74806": "Y_{p}=\\frac{n_{p}(t)-n_{p}(t=0)}{n_{k}(t=0)+\\int_0^t\\dot{n}_{k,\\text{in}}(\\tau)d\\tau}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "64ae290fc88b3a53ee32c2696eef7c04": " \\dot{\\tilde{\\chi}}=-i[\\tilde{H}_{BS},\\tilde{\\chi}] \\, ", "64ae5679f280da98d8f1cc5ed6f1b7c2": "\nS_{HSI} =\n \\begin{cases}\n 0, &\\mbox{if } C=0 \\\\\n 1 - \\frac{m}{I}, &\\mbox{otherwise}\n \\end{cases}\n", "64aeda1ff25abe30518b3549bdeb8281": " z = t + y \\!", "64af230d051ff25c125668a4d8cbe613": " \\ldots \\longrightarrow H_n(L) \\stackrel{H_n(f)}{\\longrightarrow} H_n(M) \\stackrel{H_n(g)}{\\longrightarrow} H_n(N) \\stackrel{\\delta_n}{\\longrightarrow} H_{n-1}(L) \\stackrel{H_{n-1}(f)}{\\longrightarrow} H_{n-1}(M) \\longrightarrow \\ldots, ", "64af2702cad487fa39cc3497bad50f52": " p(x)= (x-t_1)\\cdots(x-t_r) (x-z_1)(x - \\overline{z_1}) \\cdots (x-z_s)(x - \\overline{z_s})", "64af3d8ef671708bd45dcdc96f94b1cd": "dV", "64af4db2f5e662e69578f119fb6cf6ed": " {\\lambda \\over {\\alpha -1}} \\text{ for } \\alpha > 1", "64af857212bde4d74218f78928d5183c": "q=\\frac{n(\\text{Quadrant I})+n(\\text{Quadrant III})-n(\\text{Quadrant II})-n(\\text{Quadrant IV})}{N},", "64af8e8ed4502c3e3d8ec07bfb5d219a": "\\frac{g(n+1)}{g(n)}\\,", "64afa53563b01a2cd639c032add576ad": "\\lim_{n\\rightarrow\\infty}f_n(x)=f(x).", "64afdd458d0c8a913110f56d9e26b7ca": "\\text{Margin of Safety}=\\frac{\\text{Failure Load}}{\\text{Design Load}}-1", "64b01da677f0ca766df7daec6871b0b1": " 3x^2 \\, ", "64b02eba52194f0020f4aab46617331a": " \\approx \\frac{1}{N}+ (1-\\frac{1}{N})f_{n-1}. ", "64b077ff23c26ac2d10c7a84609fa09d": "2^1", "64b084d834ba276514b6918cc6c89b75": "(a \\rightarrow b) \\vee (b \\rightarrow a)", "64b0dd2ed387a82654bb950a708ab2b2": "\n\\partial_{t}u = \\nabla^{2}u+\\frac{1}{2}\\partial_{t}h -\n{\\mathbf \\nabla}\\cdot {\\mathbf q_{u}}({\\mathbf r},t) \n", "64b0f9d0147a5b252a05d7b79d62262b": "A_{i,j} = A_{i-1,j+1}.\\ ", "64b10c2cb51fe5354c51a8fbdf52471c": "t(d,n) \\leq (d \\log n)^2", "64b115d2ad16fdc3b1dbc25530e8cba7": "Q(e^{2\\pi i/q})", "64b13d3ed2e583cf3a4e035d04d61af9": "\n\\begin{align}\n\\gamma=0 \\ \\ &{for}\\ \\ S_0\\to B \\\\\n\\gamma=1 \\ \\ &{for}\\ \\ |S_0-B|\\gg 0 \n\\end{align}\n", "64b14168a8f68d9d14d51e12b5c5a515": "\\varphi_P", "64b14ab02e6e41301126d124d616a77b": "i_{{RMS}}=\\sqrt{2\\,e\\,I\\,\\Delta f}", "64b16010f9e64b552d2c6182ea3de759": "\\mu=x+ct, \\eta=x-ct\\,", "64b1c8a7e42e2e785c8e539c1d92f79c": " d(T^n(x_0), x^*) \\le \\frac {q^n} {1-q} d(T(x_0) ,x_0) ", "64b1dfe1f10c6624cd53b57c41a78bcf": "m^{\\star}", "64b1ea1db133bcfc451ad893cbca25e7": "n^2 + 4\\sum_{k = 1}^{n - 1} k = 3n^2-2n", "64b1ebaa55db6924008eae10b95bb321": "a_{b,i} = \\gamma_{b,i}\\, \\frac{b_i}{b^{\\ominus}}", "64b20c13d9e8fbc097544cc0ed41e32b": "\\Delta G = \\Delta H - T \\Delta S \\,", "64b21a1a34ddc565eded5385abf15c6e": "N\\subseteq A\\subseteq G", "64b246ca3aa161d69643a840af0a9214": " \\frac{dx}{dt} = -xy \\text{ and } \\frac{dy}{dt} = -y+x^2 - 2y^2", "64b273d0095f7598bca0700deaf74558": "U\\setminus \\{a\\}", "64b34f1c3b65d16ded43bb25559af524": "f(c)=g(c)=0\\,", "64b38378ba7bc5ffc83694e8dcf414d3": "Q = {V\\Delta H_0[H.G]}", "64b387104b936e0a2533ea303900d755": "\\xi \\gg 1", "64b42b3b5d837f523df966e8c940c75c": " P(\\partial_t)u(t) = F(t) \\,", "64b467aacf2a8408a6af93929c0cc9fc": " (X, \\mathcal O_X) ", "64b58b0d193df213ac3a8e647f61617f": " S:R\\to R ", "64b5a47431c45e83ec2837b3da92506f": "\\tfrac{G(3\\lambda + 2G)}{\\lambda + G}", "64b5f417561f4867cf5fab23ea20c852": "N_1^* A = A N_2^*", "64b64ba3c8f3387990cdeda0efa2a1aa": "Mf(x) = \\sup_{r > 0} \\frac1{\\lambda^{n} \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} | f(y) | \\, \\mathrm{d} \\lambda^{n} (y),", "64b665294f911d50f39fe6b8ce4479d5": "\\Psi_0:= C_{abcd} l^a m^b l^c m^d\\,\\hat{=}\\,0", "64b68a5f46f675aeba08af069128c978": " X_1", "64b69df423be5107aecb0ae2d0917fb0": "X \\to c_{m,n}(\\alpha, \\beta, ...)", "64b6a916d77b33cc3a0397eac0bc4991": "\\ 2", "64b6ddb4f01029079233b72ac7793c86": "(\\dot \\xi ^2+\\dot \\eta ^2+\\dot \\zeta^2) =v^2=\\mu\\left({{2 \\over{r}} - {1 \\over{a}}}\\right)", "64b7566e597dd099fe6d7aee517e008c": "cr^{1+\\varepsilon}\\leq L(r)\\leq Cr^{5/3}\\,", "64b7781ce0550cb2dc5f19f973476a11": "P_A~dA = \\frac{dN}{N} = \\frac{dg}{N\\Phi}", "64b7c39bcbe3601596fac0f35d0c15e0": "W_T", "64b7da83d0b3ecfdac2fc4abe04f5c4e": "2^3P[S_3=k]", "64b7f65358f783aea30955b66686b64f": "V_2=\\sum_{i=1}^n {w_i^2},", "64b81cb3ea2c21f32e7051b064e6ca11": "d^k\\,", "64b8326896b6dfc0059fe1666c23bdb1": "\\gamma <1", "64b88aa21c9b42987ee340d0f5e95a12": "\\mathrm{2 \\ H_3N{-}Hg{-}H \\ \\xrightarrow{\\Delta T} \\ 2 \\ NH_3 + H_2 + 2 \\ Hg}", "64b89f54aee40b2fe243064133f685dc": " \\scriptstyle \\mathbf{\\hat{e}}_r \\,\\!", "64b8b9dae94792d19bd233947cb220cc": "\\mathrm{Lie}(A)", "64b91286ad4026a1b17a94a465dad28c": " 0 \\mathbf{u}_1 + 0 \\mathbf{u}_2 + \\cdots + 0 \\mathbf{u}_n = \\mathbf{0}, ", "64b95ebf21790e780387c352d8e0828b": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} b(k) \\frac{z^k}{k} =\n\\sum_{n\\ge 1} \\left( \\sum_{\\sigma\\in S_n} b(\\sigma) \\right) \\frac{z^n}{n!}", "64b96c317de051351f65758eeba8ea0e": "\\sigma_{mk} \\in D_R", "64b98c47f203ab31b9167f641a7797f7": " \\int_{\\Bbb Z_p} \\log_p(x+u) \\, {\\rm d}u = \\psi_p(x) ", "64b9c8a83e3abc01fdce4c52037fbbe6": "\\frac{7\\cdot\\pi}{3(\\sqrt{6}+\\sqrt{2})}", "64b9e59a7dbe61ab0eb27a16aa45dfea": "\n\\begin{array}{lll}\n& L_7=\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & . & . & . & . & . & . \\\\\n1 & . & . & . & . & . & . \\\\\n. & 2 & . & . & . & . & . \\\\\n. & . & 3 & . & . & . & . \\\\\n. & . & . & 4 & . & . & . \\\\\n. & . & . & . & 5 & . & . \\\\\n. & . & . & . & . & 6 & .\n\n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n1 & . & . & . & . & . & . \\\\\n1 & 1 & . & . & . & . & . \\\\\n1 & 2 & 1 & . & . & . & . \\\\\n1 & 3 & 3 & 1 & . & . & . \\\\\n1 & 4 & 6 & 4 & 1 & . & . \\\\\n1 & 5 & 10 & 10 & 5 & 1 & . \\\\\n1 & 6 & 15 & 20 & 15 & 6 & 1 \n\\end{smallmatrix}\n\\right ]\n;\\quad\n\\\\\n\\\\\n& U_7=\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & 1 & . & . & . & . & . \\\\\n. & . & 2 & . & . & . & . \\\\\n. & . & . & 3 & . & . & . \\\\\n. & . & . & . & 4 & . & . \\\\\n. & . & . & . & . & 5 & . \\\\\n. & . & . & . & . & . & 6 \\\\\n. & . & . & . & . & . & . \n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n. & 1 & 2 & 3 & 4 & 5 & 6 \\\\\n. & . & 1 & 3 & 6 & 10 & 15 \\\\\n. & . & . & 1 & 4 & 10 & 20 \\\\\n. & . & . & . & 1 & 5 & 15 \\\\\n. & . & . & . & . & 1 & 6 \\\\\n. & . & . & . & . & . & 1 \n\\end{smallmatrix}\n\\right ]\n;\n\\\\\n\\\\\n\n\\therefore & S_7\n=\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & . & . & . & . & . & . \\\\\n1 & . & . & . & . & . & . \\\\\n. & 2 & . & . & . & . & . \\\\\n. & . & 3 & . & . & . & . \\\\\n. & . & . & 4 & . & . & . \\\\\n. & . & . & . & 5 & . & . \\\\\n. & . & . & . & . & 6 & .\n\n\\end{smallmatrix}\n\\right ]\n\\right )\n\\exp\n\\left (\n\\left [\n\\begin{smallmatrix}\n. & 1 & . & . & . & . & . \\\\\n. & . & 2 & . & . & . & . \\\\\n. & . & . & 3 & . & . & . \\\\\n. & . & . & . & 4 & . & . \\\\\n. & . & . & . & . & 5 & . \\\\\n. & . & . & . & . & . & 6 \\\\\n. & . & . & . & . & . & . \n\\end{smallmatrix}\n\\right ]\n\\right )\n=\n\\left [\n\\begin{smallmatrix}\n1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n1 & 3 & 6 & 10 & 15 & 21 & 28 \\\\\n1 & 4 & 10 & 20 & 35 & 56 & 84 \\\\\n1 & 5 & 15 & 35 & 70 & 126 & 210 \\\\\n1 & 6 & 21 & 56 & 126 & 252 & 462 \\\\\n1 & 7 & 28 & 84 & 210 & 462 & 924\n\\end{smallmatrix}\n\\right ].\n\\end{array}\n", "64ba42825f77377bd7660b2c0a5a3246": "\\mu\\, =\\, 0.63\\, +\\, 0.37\\, \\left( \\frac{A_2}{A_1} \\right)^3.", "64ba42eac4f4964de2131447b3b0a852": "\\langle x, ay_1+by_2\\rangle = \\bar{a}\\langle x, y_1\\rangle + \\bar{b}\\langle x, y_2\\rangle.", "64ba67361c4fa48753096c67f58be385": "h(z)=p_0+p_1z+p_2z^2+\\cdots. \\, ", "64ba7510c723deb566053596b83058c6": "a(p^k)", "64bab76a85950679a1e8e7aa40de36b6": "\\mathsf{NatInd} : \\begin{matrix}\n \\forall P : \\mathsf{Nat} \\rightarrow \\star \\Rightarrow P\\ \\mathsf{zero} \\rightarrow \\\\\n (\\forall n : \\mathsf{Nat} \\Rightarrow P\\ n \\rightarrow P\\ (\\mathsf{suc}\\ n)) \\rightarrow\\\\\n \\forall n : \\mathsf{Nat} \\Rightarrow P\\ n\n \\end{matrix}", "64bad97a5d7146a39b68fc68878c7743": "(c_{1}-b_{1})+(c_{1}-a_{1})", "64bb7566c39f099a8a02fb1a826930f4": "\n \\begin{align}\n k &= \\tfrac{5 + 5\\nu}{6 + 5\\nu} \\quad \\text{rectangular cross-section}\\\\\n &= \\tfrac{6 + 12\\nu + 6\\nu^2}{7 + 12\\nu + 4\\nu^2} \\quad \\text{circular cross-section}\n \\end{align}\n ", "64bb9f4918d135fc774523b0f4a4145e": "\\tau_{I2}\\,\\!", "64bbc7b5efb97fe4f42bd3e91d88c605": "\\Theta = D\\theta.\\, ", "64bc1dccd8d74f48801d1cea3c21b94e": "X\\rightarrow Y", "64bc293b7be0dbb7ee903bdcd0aab44c": "P= {\\left \\{ (0,1), (1,2), (2,3), \\dots \\right \\} } ", "64bc63eb5d6565cd37ca2171a9ac5bf3": "m_n = \\int_{-\\infty}^\\infty x^n\\,d \\mu(x)\\ ?", "64bca77a66fb7d6eb21ec2436f828b94": "\\Theta \\geq 1/\\alpha", "64bcb60f7356a757a529c7d9353e97a0": "\\Phi(f)", "64bd22f14e94758184a788da3eac9daf": "b^{n} = {b^{n+1}}/{b}, \\quad n \\ge 1 .", "64bd259744e866215309a24e883ec2ea": "\n\\sqrt[3]2 = \\cfrac{5}{4}+\\cfrac{0.5} {50+\\cfrac{2} {5+\\cfrac{4} {150+\\cfrac{5} {5+\\cfrac{7} {250+\\cfrac{8} {5+\\cfrac{10} {350+\\cfrac{11} {5+\\ddots}}}}}}}} = \\cfrac{5}{4}+\\cfrac{2.5 \\cdot 1} {253-1-\\cfrac{2 \\cdot 4} {759-\\cfrac{5 \\cdot 7} {1265-\\cfrac{8 \\cdot 10} {1771-\\ddots}}}}.\n", "64bd29798bf6e790e0cf999b4cff5f6d": "I_H = \\frac{h}{eL_{QA}} = \\frac{e\\omega_B}{4\\pi} \\ ", "64bd7e7d5e8fe76d53d16b1650122a97": "I(\\mathbf{r}) \\propto U(\\mathbf{r}) \\overline{U} (\\mathbf{r})", "64bd85cd1a26e6f1daa739dd82adf794": "2\\sqrt{2}/3 = 0.94....", "64bd88ed0c034943ce32ce961de94120": "\\rho = x^2 + y^2", "64be32bd1fc7717dd66b62ea9a371164": "Lu(x)=\\int LG(x,s) f(s) \\, ds.", "64be35ea08f6b65d1d0afedc705d63c5": "(a + c, b + d)", "64beb4cb02a5c39a93f680ecb4aeb908": " D \\cap S = \\partial\\!D.", "64bef5d0f52a08dc372060c3d67035a3": "S_1, S_2,\\ldots,S_r", "64bf0145d5c024bcba1b7fbfc60d50f1": " L^x(t) =\\lim_{\\varepsilon\\downarrow 0} \\frac{1}{2\\varepsilon} \\int_0^t 1_{\\{ x- \\varepsilon < b(s) < x+\\varepsilon \\}} \\, ds,", "64bf2573815d1415bf92b995d71f8597": "\\delta( q, \\sigma, d(t))", "64bf842b59aa91511c4a5b5a002b703c": "\n{d\\sigma \\over d\\Omega} = (2\\pi)^4 m_i m_f {p_f \\over p_i} |T_{fi}|^2,\n", "64bf9b560204e8bce8b8a92912aad6b8": " A=m\\ell+b", "64bf9e4e6a51a9b8b683f373046402ed": "SL(3,\\mathbb{R})\\times SL(3,\\mathbb{R})", "64bfc3796e8daa535dae4b1d3d5fe97c": " \\mbox{C} = (\\mbox{DL} + \\mbox{DW}) \\bmod 7 ", "64bfe7548f539b0258faa75aa1b1f2cf": " \\sum_{y \\in C_2} {{|}} x + y \\rangle", "64c000831a2f13d2d9b56582302ce4ef": "A_{n-1}(r) = r^{n-1} A_{n-1}(1).", "64c0165a912ba625138d64174f12bbb8": "X\\subset\\{0,1\\}^{n}", "64c01ba0cd227d2152b9bc41cfe3caf8": "\\operatorname{MCG}(\\mathbf{T}^n) = \\operatorname{Aut}(\\pi_1(X)) = \\operatorname{Aut}(\\mathbf{Z}^n) = \\operatorname{GL}(n,\\mathbf{Z}).", "64c052b30718ec11bce4fa8c5cad5396": "\\left(\\mathcal{F},K\\right)", "64c09791c9acd9516157f0d74c9dbdcc": "|s(t)| + |\\hat{s}(t)| \\le C (1+|t|)^{-1-\\delta}", "64c0bdf6b4f8ca9024ffa23cab8d24d6": " (I_n \\otimes A + B^T \\otimes I_n) \\operatorname{vec}X = \\operatorname{vec}C,", "64c199b1d247dc83655529123d898f58": "1 \\le j \\le n", "64c219b4fdf35b1238ba84605f338574": "y_{op} = \\frac{F_c}{k_{op}} = \\frac{2m\\Omega X_{ip} \\omega_r \\cos(\\omega_r t)}{k_{op}}", "64c24648cb404779849df8796d43e6ea": "I \\subseteq \\{1,\\dots,n\\}", "64c28022dd04688d8d970b0b971917cf": "\\scriptstyle \\gamma(K_n)\\, =\\, 2n - 1", "64c2a065639590d1db8fd6a81b042bb1": "m_6", "64c2f3f3ab76a588cec29b9176043413": "S_{mk}^{}=\\alpha_{mk}/R-p_{m+1,k}/R+p_{m+1,k+1}", "64c31ff062268803e0d11126b6c6da4d": "\\scriptstyle w'\\left(\\eta\\right)\\,", "64c3431b0da1b13ed719fa128d5c16fd": "I_\\nu(z)=\\frac z {2 \\nu} (I_{\\nu-1}(z)-I_{\\nu+1}(z));", "64c35006471499dce2e37de6061088c5": "W_0(x)/x", "64c38520dce3cada618b54801b554db3": "C=\\sum_{i=1}^N(C_i \\cdot f_i)", "64c3b2f27718ea11e1cd18ac975ef786": "G(\\tau=0^-)", "64c3c03ddee527b1236072e425b60cf8": "R(\\beta) = - \\frac{M}{N} \\beta^2 f''", "64c3d4a0a8264249cb391af3204d897d": " x_\\mathrm{d} = x_\\mathrm{u} + x_\\mathrm{u}(1 + K_1r^2 + K_2r^4 + \\cdots) + \n(P_1(r^2 + 2x_\\mathrm{u}^2) + 2P_2 x_\\mathrm{u}y_\\mathrm{u})(1 + P_3r^2 + P_4r^4 \\cdots)\n", "64c3dd609d21aa4d236df60bc580cffc": "A\\in\\Sigma", "64c403ca2d465e6a62342a9dfd54aed1": " s_2 = { 1 \\over V_0}{u_2 \\over (1 + v_1 u_1) } = \\sqrt{1-v^2}\\, { u_2 \\over 1 + v_1 u_1 } ", "64c4bf16e8b0b304268c70ff68e450bb": "K_i(R) = \\pi_i(K_R)", "64c4c21a4710e3da32610a35b19fd288": "(f\\circ f\\circ f)(x) = f(f(f(x))) = f^3(x)", "64c502922eed8c458e5c5ffe90243803": "\\tau_p", "64c5418fbc73d1c94eebb24752d982b8": " A' = \\sigma \\ell N = \\alpha' \\ell \\,", "64c57f915f3069eba9eb936997dc9dee": " \\hat{\\sigma}_2^2 = \\frac{-L(\\mu)}{N} \\left[\\sum_{x_i: x_i>\\mu} (x_i-\\mu)^2 \\right]^{2/3},", "64c58449d93c71b7e157448ea9328ab9": "D[\\partial_i||]=-D[||\\partial_i]", "64c5d50e790fdba7e21a7b824e6fbd30": "2^{14} \\times 3^{9} \\times 5^6 \\times \\cdots \\times 1451", "64c5faefaa7267e03627457fde545f01": "S^2(n)=\\frac{1}{n}\\sum_{i=1}^n X^2_{(i)}-(\\frac{1}{n})^2Y^2_n", "64c61c521df1e4f8a9c2e06357809e19": "\\vec k=(k_x,k_y,k_z),", "64c659ed44332e0d57180aa6ccfabd0b": "\\left(-\\frac{\\hbar^2}{2m} \\frac{d^2}{d x^2} + \\frac{1}{2}m \\omega^2 x^2\\right) \\psi(x) = E \\psi(x).", "64c6794850ac1fc26303d4a4cde563fe": " i>j ", "64c68bb083ba4acc8335a1e4fa26380f": "\\scriptstyle\\tfrac{h - y}{h}", "64c77cd303e754381fbde5370ee98665": "I = J = 0", "64c78f4948c2c921e4396d739e68e40a": "\\frac{20! \\times 3^{19}}{120} \\approx 2.36 \\times 10^{25}", "64c79e96e0c42c0b0a72ab011e5647cc": "e = \\textrm{HASH}(m)", "64c7e47d973392e889356562501943c5": "\\text{failed}(j)", "64c804c5988021f7a263f14f9b709fc5": " dh = v \\, dp. ", "64c933301c294c0aeb534cb1ae4b536f": "(x_n)_{n\\in\\mathbf{N}} + (y_n)_{n\\in\\mathbf{N}} \\stackrel{\\rm{def}}{=} (x_n + y_n)_{n\\in\\mathbf{N}}", "64c93ad61697be2c1bab1d84cf60709a": "H_{n-1}(X_{n-1},X_{n-2})\\,", "64c96fdf751808d949b9991b28cd2013": "\\zeta(s,x)", "64c9757105ce720ab1c213dc899fbda4": "(D_1 \\circ D_2)(f) = D_1(D_2(f)).\\,", "64c9da6937ddc460dc415d8460ce723f": "e^x \\ge 1+x.\\,", "64ca14e7b0e667f28731ef94633d9586": "26^{n^2}(1-1/2)(1-1/2^2)\\cdots(1-1/2^n)(1-1/13)(1-1/13^2)\\cdots(1-1/13^n).", "64ca6c9f2e8d6004c36112d5e507f1d7": "\\lambda_n=\\rho(x_n)", "64ca75e16c59ebc5937dda423cb177ef": "\\frac{\\partial r_i}{\\partial \\beta_j}=-J_{ij}", "64ca90b741d0b0215cea07f4b0749d3c": "\\tfrac{(1+\\epsilon) \\ln n}{n}", "64ca9351fc6556ac9ce5fc3c54265fd9": "\\operatorname{LEQ} = \\lambda m.\\lambda n.\\operatorname{IsZero}\\ (\\operatorname{minus}\\ m\\ n)", "64ca98f5bcab8f22865795b7021d7a05": "\\scriptstyle\\frac{-\\pi}{2}", "64caa3cabd23ec9ae2396b83cc486871": "c>\\operatorname{rad}(abc)^{1+\\varepsilon}", "64caef0ad39b96ad6256176b687fbb82": "u_{i}^{n + 1} = u_{i}^{n} + r \\left(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n}\n\\right)", "64cb033bb1595586b505aca87fe9d4e9": "\\lbrace p_1,p_2,p_3,p_4 \\rbrace = \\lbrace 1,1,1,0 \\rbrace", "64cb2b393e79a02dc6497c866534b6a1": "N_1\\# N_2 ", "64cb4b41bd890e657fe948e9f377b0ed": "R\\to S", "64cb53657773ea419cd0bc76be05efdc": "\\ N (M) \\propto e^{0.307 M} (1 - e^{3(M^{*} - M)} ) \\,.", "64cb644c5df9993f89ebdbe920c67253": "\\mathbf{E} = -\\nabla\\phi-\\frac{1}{c}\\frac{\\partial \\mathbf{A}}{\\partial t}", "64cb725a46f76fff0a3e72b41c1b29c9": "\n \\left.\\sigma_{rr}\\right|_{z=h,r=a} = \\frac{3qa^2}{16h^2} = \\frac{3qa^2}{4H^2}\n", "64cb790012ee67e161dbe4f68d395bff": "N(0,\\sigma^2)", "64cbab92ff06d8fdd05af8909315e4df": "\n\\int_E \\operatorname{div} \\boldsymbol{\\phi}(x) \\, \\mathrm{d}x = - \\int_{\\partial^* E} \\boldsymbol{\\phi}(x) \\cdot \\nu_E(x) \\, \\mathrm{d}\\mathcal{H}^{n-1}(x) \\qquad \\boldsymbol{\\phi} \\in C^1_c(\\Omega, \\mathbb{R}^n)\n", "64cbb1ce0807deaae9473904d9e442d7": " d(x)=\\lim_{\\varepsilon\\to 0} d_{\\varepsilon}(x)", "64cbf1adc3c8b06a33dc98cc735bdc83": "\\text{Base ohms }=\\frac{\\text{base volts}}{\\text{base amperes}}", "64cc02e5710dcbe2e14d11b35f60fc63": "4\\pi\\varepsilon_0 = 1", "64ccf05702ce05adfdb114bbd279b4b7": "\\bigwedge^j V", "64cd24d7150770cd957176d739722172": "\\ MU_x/MU_y=P_x/P_y ", "64cd4b4ae271116bf39d5fa9670cdc93": "\\frac{e^{i k \\|\\mathbf{x}-\\mathbf{x'}\\|_2}}{\\|\\mathbf{x}-\\mathbf{x'}\\|_2} \\rightarrow \\frac{e^{i k r}}{r}(-i k)(\\mathbf{n}\\cdot\\mathbf{x'})", "64cd4e5b2a11f3cf974979db02eea4bb": " Cr \\lbrace A \\rbrace = 1 ", "64cd7f3d10f42ae9d15ef09978b6b8ce": " E_\\mathrm{tot} = \\sum_k E_k N_k", "64cddd48b1474bdd667d7b33b5e731f7": "\\alpha_G=\\frac{m_e^2}{4\\pi}", "64cdf6710fc4753e207d9f37bf86af76": "DCF = \\frac{CF_1}{(1+r)^1} + \\frac{CF_2}{(1+r)^2} + \\dotsb +\n\\frac{CF_n}{(1+r)^n}", "64ce043e998cc76d9d3bd5ad92a8031c": "\\deg(G)=\\deg(\\overline{G})", "64ce591b5e37d96813f0d8bf45393a33": " c_{i,j} = \\dfrac{c_{i,j}}{a_{i + 1, j + 1}}.\\,\\!", "64ceb5091e821f6241530a071a46173b": "S = \\{x \\in \\mathbb{R} : 1 < x < 3 \\}", "64cf0adfe5e4b41d8b0a917d36f27af5": "F(x)=\\int_{a(x)}^{b(x)}f(x,t)\\,dt,", "64cf7ee33a9b5aebfb97311d733e873c": "g_{j} : \\mathbb{R}^{d - 1} \\to [0, + \\infty),", "64cf8ad730229a31f3be7442785f7729": " g(y) = \\frac{ 1 }{ y^2 } f\\left( \\frac{ 1 }{ y } \\right) . ", "64cf8f07e81cc498f292dde5aebed90f": " y^T =\n \\begin{bmatrix}\n h_1 & h_2 & h_3 & \\ldots & h_{m-1} & h_m\n \\end{bmatrix}\n \\begin{bmatrix}\n x_1 & x_2 & x_3 & \\ldots & x_n & 0 & 0 & 0& \\ldots & 0 \\\\\n 0 & x_1 & x_2 & x_3 & \\ldots & x_n & 0 & 0 & \\ldots & 0 \\\\\n 0 & 0 & x_1 & x_2 & x_3 & \\ldots & x_n & 0 & \\ldots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ldots & \\vdots & \\vdots & \\ldots & 0 \\\\\n 0 & \\ldots & 0 & 0 & x_1 & \\ldots & x_{n-2} & x_{n-1} & x_n & \\vdots \\\\\n 0 & \\ldots & 0 & 0 & 0 & x_1 & \\ldots & x_{n-2} & x_{n-1} & x_n\n \\end{bmatrix}.\n", "64cfce9fb8ecdcbe3e739f77b3ef1854": "(n+1)! + n + 1", "64d05d47e81067ffed1b7599f4c9873b": "\\Delta_\\xi", "64d05dbdc853ae315287d56bcf5802cd": "|x_m - x_n| < \\varepsilon, ", "64d08412b8f3f3c9997c13bb50b5d249": "(\\nu\\; x)P", "64d0c55a8b273966a086643636169a02": "|S|_0 \\subset |S|_1 \\subset \\cdots \\subset |S|, ", "64d0cebbf758d77f768b2ff5e244ba9c": "\\oint_{C} \\frac{f'(z)}{f(z)}\\, dz", "64d124460b9dfde2970411f69be62d54": "\\int \\frac{1}{3} \\phi^3 - \\left( \\partial_x \\phi \\right)^2\\, \\text{d}x.", "64d169c87ea017f27464c28d24e3b6ed": "d = 2 - 4 = -2", "64d16d8bf6e9692b7b13e45142501357": "{\\mathfrak m}\\colon {\\mathrm {Mp}}(n,{\\mathbb R})\\to {\\mathrm U}(L^2({\\mathbb R}^n)),\\,", "64d181c0cf95d36cd0ae6442ca206949": "U+PV-TS", "64d1ca96968adf1ff1fb5016b8f13de8": "\\begin{alignat}{5}\nln\\left[ \\frac {n\\left( T\\right)} {T}\\right] = -\\alpha ln\\left( T\\right) + b ; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ..... Eq:1\n\\end{alignat}", "64d1d7850be1ef2f0fe2d054d7c4b197": " \\frac{e}{\\sqrt{2-e^2}} ", "64d1e7c134bf9eb52286523c43a6b675": "n_W=\\frac{2^W}{\\mbox{GCD}(\\Delta F,2^W)}-1", "64d1e810c0f4ad46a6a96d87d5063649": "\\textstyle\\vec{A}", "64d1f09fe811198cfa7f11414ec5f29a": "\\neg (P \\and Q)", "64d20df73e70845501acebdabbe6f795": "\\prod_{j=1}^r\\partial_{x_j}^{n_j}\\quad f(x)^{s+1}\n=\\prod_{j=1}^r\\prod_{i=1}^{n_j}(n_js+i)\\quad f(x)^s", "64d22f71759ec2cf476270d8f020c0f3": "d_+", "64d23568e4d24286b4ce884a92e564ff": "|a|^2", "64d2ca63c00adadd3b387a4f0802f775": "\\rm MPa^{-1}", "64d33ca4c68cfea02b5f9a8c578133cd": "\\gamma \\equiv \\gamma'", "64d3929559b16c40d75d4a01516b1574": "\\int_0^{\\theta}\\log(1-\\cos x)\\,dx=-2\\text{Cl}_2(\\theta)-\\theta\\log 2", "64d3a59bd2cefec2fd0513565826f8c3": " \\mathbb{Z}[x]/f(x)", "64d3aa1bb42d4f312b538e4b4ae2f1af": "r \\cdot r = r", "64d3b946be32fdc42c5a4fa951318541": "\\Psi(x,t) = \\psi(r)\\exp\\left(-i\\dfrac{E}{\\hbar}t\\right)", "64d42895ff7746d00cc5091982067ffa": "\\text{return}\\colon A \\to A^{*} = a \\mapsto \\text{cons} \\, a \\, \\text{nil}", "64d42d26d73a5561c222c50e498413ec": "I(x) = \\frac{1}{Z_0}(V^+ e^{-\\gamma x} - V^- e^{\\gamma x}). \\,", "64d489861b1a153a280f40539ad680ba": "\\mathcal{M}\\in \\mathcal{P}", "64d489e4e5eebc3d36817d6f21b25ce3": " \\operatorname{Cov}(Y) = \\operatorname{E}[Y Y^\\top] = M^{-1/2} \\operatorname{E}[X X^\\top] (M^{-1/2})^\\top ", "64d4d36a74499e491be9a0368765dbf5": " D_x(r)=1-e^{-\\lambda |b(x,r)|}, ", "64d5012edcb34d71a3f97338baec6b2b": " J = \\int f_{\\rm D}(\\lambda) \\, \\epsilon_{\\rm A}(\\lambda) \\, \\lambda^4 \\, d\\lambda ", "64d50900bae4023bef3e325653530ad9": "\\tau^{\\pm}\\equiv {1 \\over 2}(\\tau^1{\\pm}i\\tau^2)=\n\\begin{pmatrix}\n 0 & 1 \\\\\n 0 & 0 \n\\end{pmatrix}\n\\; .", "64d5933875ebc571be56b47cb6e095ff": "Q_{F1} = e\\sqrt{\\frac{S_{F0}}{S_B}} = 2\\cdot 10^6 e \\ ", "64d5adbc02ceda49d9aa06cf942ada86": "s(s-a)+(s-b)(s-c)=\\frac{1}{4}(a+b+c)(-a+b+c) + \\frac{1}{4}(a-b+c)(a+b-c) = \\frac{1}{4}[(b+c)^2-a^2] + \\frac{1}{4}[a^2-(b-c)^2] = \\frac{1}{4}[(b+c)^2 - (b-c)^2] = cb", "64d5c935a194fceb418f28eed88a3d51": "\n\ty^2 = x^3 -n^2x\n\t\\,\\!\n", "64d5d222deb05377bdcfdadf47d1b4af": "f=Ax \\, ", "64d68406ee4df8faf7192b71f2cd3fb2": "\\,\\! -\\omega^2 A \\mathrm{e}^{\\mathrm{i} (\\omega \\tau + \\phi)} + 2 \\zeta \\mathrm{i} \\omega A \\mathrm{e}^{\\mathrm{i}(\\omega \\tau + \\phi)} + A \\mathrm{e}^{\\mathrm{i}(\\omega \\tau + \\phi)} = (-\\omega^2 A \\, + \\, 2 \\zeta \\mathrm{i} \\omega A \\, + \\, A) \\mathrm{e}^{\\mathrm{i} (\\omega \\tau + \\phi)} = \\mathrm{e}^{\\mathrm{i} \\omega \\tau} .", "64d6bfb01483d722422854c89b04ebac": "\\scriptstyle{|2\\rangle}", "64d6e2599a9550371ded283acfa6ae8b": "p_{A}", "64d76da9558577547d65d8e96d101a42": "V-=0\\;V\\,", "64d77dd946a94d21516273fde5557afa": "\\exists x \\, \\exists y \\, P(x,y) \\Leftrightarrow \\exists y \\, \\exists x \\, P(x,y)", "64d77fb2a337ce9f19af993bebfbffcf": " F_C ", "64d845bdbc0d429350b991906390ec8e": "\\varphi = c", "64d84d8b65cb2bf1e215d1ae163b79f4": "f, g : (M,x) \\to (N,y)", "64d8841752e604ab81b545bded2cdff0": ":\\Leftrightarrow \\!\\,", "64d88df500bea4bcf807104a2625284d": "F^\\%", "64d8b337a22df8fa0d2a0d268d89650f": "\\textstyle c(\\Gamma, \\Lambda)", "64d8c0adedb4bcf1453a906c2b22deed": "\n\\overbrace{\\rho \\Big(\n\\underbrace{\\frac{\\partial \\mathbf{v}}{\\partial t}}_{\n\\begin{smallmatrix}\n \\text{Eulerian}\\\\\n \\text{acceleration}\n\\end{smallmatrix}} +\n\\underbrace{\\mathbf{v} \\cdot \\nabla \\mathbf{v}}_{\n\\begin{smallmatrix}\n \\text{Advection}\n\\end{smallmatrix}}\\Big)}^{\\text{Inertia (per volume)}} =\n\\overbrace{\\underbrace{-\\nabla p}_{\n\\begin{smallmatrix}\n \\text{Pressure} \\\\\n \\text{gradient}\n\\end{smallmatrix}} +\n\\underbrace{\\mu \\nabla^2 \\mathbf{v}}_{\\text{Viscosity}}}^{\\text{Divergence of stress}} +\n\\underbrace{\\mathbf{f}}_{\n\\begin{smallmatrix}\n \\text{Other} \\\\\n \\text{body} \\\\\n \\text{forces}\n\\end{smallmatrix}}.\n", "64d8ce463b2ec29613382904a1508370": " \\partial_{\\beta}\\left[\\frac{\\partial \\mathcal{L}}{\\partial (\\partial_{\\beta}A_{\\alpha})}\\right] - \\frac{\\partial \\mathcal{L}}{\\partial A_{\\alpha}}=0 \\,.", "64d8d9c2277fa93442ed512b4d7bc0e1": "y_i = h_i(x), i \\in p", "64d909a8e24e28cc4431eb3461d6f6f6": " \\omega = \\arctan2({e_y}, {e_x})", "64d90dd7f6449e1276b65fe4c782a9a5": "x\\le_1 y", "64d9830357444080ac1a09382704a157": "\\mbox{Free}(\\exists v \\phi) = \\mbox{Free}(\\phi) \\backslash \\{v\\}", "64da591acb15ac9271a4ce63e1ef7c24": "\\left(f_1 \\lor \\dots \\lor f_n \\Leftrightarrow f\\right) \\land\n\\bigwedge_{i \\epsilon > 0", "64f7fbb299dd9f3510328f24c75760e9": " x_0, ..., x_J ", "64f830cc9da13bf106675513aaf50c22": " \\mathbf{J}_\\mathrm{D} = \\varepsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t} + \\frac{\\partial \\mathbf{P}}{\\partial t}.", "64f87b62ec08263520bd6de1a2464d55": "(1)\\quad ds^2=-e^{2\\psi(\\rho,z)}dt^2+e^{2\\gamma(\\rho,z)-2\\psi(\\rho,z)}(d\\rho^2+dz^2)+e^{-2\\psi(\\rho,z)}\\rho^2 d\\phi^2\\,,", "64f888100bfb9586546727da8bf7a62c": "\\{a \\in \\mathbb{R}^m | \\text{for some }x\\in X, g_i(x) \\leq a_i, i \\in \\{1,\\ldots,m\\}.", "64f89951f3b261ec484114df96a52f93": "b(1)e^{\\gamma(1)}+\\cdots+ b(n)e^{\\gamma(n)}\\ne 0.", "64f912f3eca87e883d635310384ee534": "g : B \\rightarrow B'", "64f91856074ae709e2b2883979348f6b": "S \\cdot T", "64f939973de2bb1ae7e55c90f678853f": " L(Q) = \\sum_{H} Q(H) \\ln \\frac{P(H,V)}{Q(H)} ", "64f93ec2e1909a1ef32043e7000c3ad8": "a^2 = \\frac{3 k_B T}{\\gamma}(\\Gamma^{-1})_{ii}", "64ff6e7adf1d321c99cc041e3217b06b": "(3_3, 1_2)", "64ffd22b4747befbe976620eab96b9a8": "\\delta \\left(q_i,[x_1,x_2]\\right)=\\delta ' \\left(q_i,[x_1,x_2]\\right)", "64ffd7e963a2eef8dade61c5bc6e0e3f": "(1+r)((1+r)((1+r)P-c)-c)-c = (1+r)^3P - (1+(1+r)+(1+r)^2)c", "64ffe4d4f34d1339f3207a549765cf93": "-0.5 \\le \\lambda < -0.25", "6500f1f395449065d1a624e9a9a52c87": "\\omega = 2 + \\sqrt{3}", "650111b50259020fdcc3cdb499072a90": " I(x,y) = i(x,y) + I(x-1,y) + I(x,y-1) - I(x-1,y-1)\\,", "650120280d9d50702d1debea9fe46de9": "\\frac{1}{4} C\\pi\\,\\!", "650161b834989ccee6ba4dc3b4f11cac": "\\mathcal{T}", "65016385f62d960f796bda143fd881ee": "E^{*}", "650213ba140ae34c0e3372d45b107548": "x_{1} \\dots x_{n}", "6502162f9449ed65d3efa61b1b397e3e": "y_i = \\frac{x_i}{1 + t}", "65026d958326eb42cd97f0c6d947f516": " \\frac{1}{MPL}", "6502bdf9ec7cf9b00c0453876a5f4642": "\\lim_{d \\to \\infty} \\frac{\\operatorname{dist}_\\max - \\operatorname{dist}_\\min}{\\operatorname{dist}_\\min} \\to 0", "6502dbbb27902982836d545abc5b6734": "s_2\\,\\!", "650371f6891f3e9aa2383a862374cfae": "S_{GS} = \\int B_{2}\\wedge X_8", "6503a39a43a16ffdf748035b8f40045c": "\\scriptstyle\\int", "6503afb547e87c6a30ccfa4791a81cd7": "h=rv", "6503d99303659a056a554a1f00ec53cd": "R=K[X]/\\langle f\\rangle, ", "650423bd9449f19c6d48f9f32138ec2d": "y(t) = e^{-\\frac{t}{5}}", "650432d072ed3e38e911be8d54191062": "{\\tilde{A}}_{1}", "650448c07caf5a37c252bbe808e69079": "I_R\n\\le 2RM_R \\int_0^{\\pi/2} e^{-2aR\\theta/\\pi}\\,d\\theta\n=\\frac{\\pi}{a} (1-e^{-a R}) M_R\\le\\frac\\pi{a}M_R\\,.", "65045136c5f9ddf461774341b0cda910": "n=r=s=1", "65045aa62225bb6bd2b8946045d06eba": "\\hat{V}=eB(L_z+2S_z)/2m", "6504963693167117e59f24502efb2383": "dV = R\\,dI", "6504a5629d650f2253332530e59ff42e": "{{B}_{x}}^{\\prime }(v)\\ge \\left(\\frac{v-{{B}_{x}}(v)}{v-B(v)}\\right)B(v)", "6504f17d7bd7828878476df6da126554": "E(|Z^2_i|) < \\infty", "6504f3cd64f68c729b9c6ab2e9a19c91": "\\phi^{X}(z)=Y(z)", "65051c047573bf5d5d00735cbbfb71de": "{T_v} \\approx T+\\frac{w}{6}\\, .", "650575bac89146bea6a373e651f3815a": "\\displaystyle{(x,y)_S= (Sx,y)}.", "65057e88b8f11eccb60ff870481096de": "P_n \\sim 1/n^a", "65058fea05cf25ed4a8c47551e7a43c1": "\n\\frac{ih}{2\\pi} \\{X,P\\}_\\mathrm{PB} \\qquad \\qquad \\longmapsto \\qquad \\qquad [ X , P ] \\equiv XP - PX = \\frac{ih}{2\\pi}\\,\n", "65059b04c125f861f92ddbb7b11bae35": "\\textbf{f}_{dyn} = M \\frac{d\\textbf{v}_M}{dt} = -\\frac{4\\pi \\mbox{Ln}(\\Lambda) G^2 M^2 \\rho}{v_M^3}\\left[\\mbox{erf}(X)-\\frac{2X}{\\sqrt{\\pi}}e^{-X^2}\\right]\\textbf{v}_M", "6505a22300582cbea98a4cac6275e79c": "\\!x", "6505b9bde39fa71d770f510801632cf2": " \\frac { \\tau_1} { \\tau_2} = \\alpha \\beta A_0, ", "6505de669ffc6ab1532637b354578c45": "T^2 \\propto a^3 \\,", "6506116505d0661db736c38279422f88": "a_i(x)", "650638512cefc4a767379e792d478946": "G = \\{(1,2),(1,3),(2,7)\\}", "65064170401e286c799d8268d5a70649": "c_5 = 9.41695 \\times 10^{-3},\\,\\!", "6506562086fdabe6079b6e7ecaab1d95": " f(z) = \\sin(Az) + \\sin(Bz) ", "6506849715fa9132832782897f95d99b": "K =c \\sigma \\sqrt{a}", "6506aa967bc3ba41396ce5c89ca1fcab": "\\frac{1}{f} = \\frac{1}{f_1} + \\frac{1}{f_2}", "6506c182450da8fbc8d7675e72ac89da": "S(A_1)", "65075fdb4108e7a30701fbebe27c36ca": "{}^{2S+1}\\Lambda_{\\Omega}", "650763b271b47dfde9ffe78d1ca8e2a1": "\\nabla_{\\bold{v}} (cf) = c\\nabla_{\\bold{v}} f", "6507a5505e29be91874147bcc905d0f6": "E(\\mathbf k) = E_0 + \\frac{\\hbar^2}{2 m_x^*}(k_x - k_{0,x})^2 + \\frac{\\hbar^2}{2 m_y^*}(k_y - k_{0,y})^2 + \\frac{\\hbar^2}{2 m_z^*}(k_z - k_{0,z})^2", "6507ec2d8f62dc95b403d7a7994d8bbf": "\\mathbf{s_n}", "6507f6b80be89bd0242676edfdd539c4": "[FLu](\\omega,m,n) = \\frac{ \\omega \\, [Fp](m,n) + [Fq](m,n)}{\\omega^2 + m^2 + n^2}", "65081a23b96ce743c3d98863054e6227": "\n \\lambda_1\\lambda_2 = 1\n ", "65083e71f5539f475c2b2b557483861d": "x\\rightarrow -x", "65087f226623fdcf33f5a5d60866f29f": " \\gamma([0,1])\\subseteq Z_2", "65089796c2db46afe0bf014f30bca5c8": "\\Delta p = -\\eta_nZ\\dot V_n.", "6508aa0a51e39e01394d49f989187d01": "\\oplus : B \\times C \\rightarrow C", "650921833b608fe298c0641acb0f98e1": "\\psi(\\Omega)^{\\psi(\\Omega)} = \\phi_2(0)^{\\phi_2(0)}", "65094208fcb4016dbc43346c73de5723": "\\ln(2) = \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} = 1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots.", "650960d47308607d66206dbd00662ff7": "R_2 = -\\frac{1}{8} \\, {S^a}_b \\, {S^b}_c \\, {S^c}_a", "65097f798a2f88cb29353b0f9f8285e3": "\\sqrt{3}/2", "6509aba51273a7ebd3c5f54a92df3cac": "\n\\lambda(n) = \\operatorname{lcm}[\\lambda(p_1^{a_1}),\\;\\lambda(p_2^{a_2}),\\dots,\\lambda(p_{\\omega(n)}^{a_{\\omega(n)}}) ].\n", "6509ed91af4e79c7f7c749ec74211587": "\\{U_{j}\\}", "650a6ebd9c7cfb7ba14d5b17e03005d6": "|X_k - X_{k-1}| < c_k, \\, ", "650ad4459c88bad800cd000ce94f6092": "\\overline{M L}", "650b373ba7b6d51ceb8615fd18ccd360": "\\mathrm{GF}(q^p)", "650b38b31a8a9c11c97e3f70fd1b0826": "\\boldsymbol{ \\Omega} = \\omega \\begin{pmatrix} 0 \\\\ \\cos \\varphi \\\\ \\sin \\varphi \\end{pmatrix}\\ ,", "650b404a8f7e51f3967b3a9331e9a585": "\n\\varphi \\mathbf{(r)} = \\frac{1}{4 \\pi \\varepsilon_0}\n\\int \\frac{\\rho(\\mathbf{r'})}{|\\mathbf{r}-\\mathbf{r'}|}\\, \\mathrm{d^3}\\mathbf{r'}\n", "650b42a2cef3c66887fee3207b6b8da1": "A+\\overline{D}", "650b502b498dd5a9ce04f2251806f6cf": "a \\lor b", "650b51c44dd591460ee3956a2f592eb6": " a_i^T ", "650bbcb065e7ad470ccde9c32eb3516d": "\\mathbb E(W_q) \\approx \\left( \\frac{\\rho}{1-\\rho} \\right) \\left( \\frac{c_a^2+c_s^2}{2}\\right) \\tau", "650bc6a6218e60948495cf89786bbfbf": "\\gamma = \\frac{1}{\\sqrt{1-v^2/c^2\\,}}.", "650bef12cfe1d7eca8a597e2454ebc94": "M_n=X_0+\\sum_{k=1}^{n}\\bigl(X_k-\\mathbb{E}[X_k]\\bigr),\\quad n\\in\\mathbb{N}_0.", "650bf51a37663c3e4be369eabb68979b": "\\chi = k_0 - k_1 + k_2 - k_3 + \\cdots,\\ ", "650c40c5e701aced7ebd485472f6a5da": "y = 0.30 - j0.54\\,", "650c4a6ab62435827c866d8cb9e571a8": " \\frac{V_A}{V_B} = \\frac{\\dot{S}}{-\\dot{R}} = 2,", "650c68c98b418ace338a47787e130a82": "\\frac{{\\partial}u'}{{\\partial}x'}+\\frac{{\\partial}v'}{{\\partial}y'}+\\frac{{\\partial}w'}{{\\partial}z'}=0 \\,\\!", "650c8b0d91fd5b7324a19de9d589f85d": "\\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{k=1}^n a_k=a.", "650cc07793ac34fe429a0c31060f6621": "\n\\begin{align}\n\\mathbf{a} \\cdot \\mathbf{b} &= a_1 b_1 \\vec{r}_u\\cdot\\vec{r}_u + a_1b_2 \\vec{r}_u\\cdot\\vec{r}_v + b_1a_2 \\vec{r}_v\\cdot\\vec{r}_u + a_2 b_2 \\vec{r}_v\\cdot\\vec{r}_v\\\\\n&= a_1 b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G \\\\\n&=\\begin{bmatrix}\na_1 & a_2\n\\end{bmatrix}\n\\begin{bmatrix}\nE&F\\\\F&G\n\\end{bmatrix}\n\\begin{bmatrix}\nb_1 \\\\ b_2\n\\end{bmatrix}\n\\end{align}.\n", "650cdc7e32e0d62e980e2c2c99499994": "G_{\\infty} = G\\ \\Big |_{T \\rightarrow \\infty}\\ , ", "650cfaf9edddab8f4289674ee48001c3": "<\\mathbf u(0) - \\mathbf u_0(0),\\mathbf F(\\mathbf u_0(0),\\lambda_0)> = 0", "650d8706d326873c33f333e023b0a9a0": "C_d A = \\dfrac{\\dot{m}}{\\sqrt{{2}{g_c}{\\rho}{\\Delta} {P}}}", "650db3b0a4e52a5c771939806b561067": "\n\\xi_{\\times +}(\\Delta\\theta)=\\xi_{+ \\times}(\\Delta\\theta) = \\langle \\gamma_+(\\vec{\\theta}) \\gamma_\\times(\\vec{\\theta}+\\vec{\\Delta\\theta}) \\rangle\n", "650dda49fcd143572c382117e36512b1": "x^2-a^2y^2=a^2,", "650dea7c23ed77bde1f16c4fba0ee97b": "\\varepsilon_l=\\frac{eB}{m^*}\\left(\\ell +\\frac{1}{2}\\right)", "650e050bb7eb172aac20beb7e708d434": "Cp = \\frac {W\\Delta H}{M\\Delta T}", "650e5786dd5298125aad206899bd8dff": "\\frac{\\partial {\\rm tr}(a\\mathbf{U})}{\\partial \\mathbf{X}} =", "650e76a7a02db7bd2b2ce8858f9f7eab": "3\\cdot8 + 1 = 2\\cdot10 + 5 = 25", "650eb608b65b1642c86efba50ccf65d9": "C = \\overline{A \\cdot B}", "650ecf6edceb4f9be19b9db605579e5c": "A=\\begin{pmatrix}\n0 & 0 & \\dots & 0 & -P_0 \\\\\n1 & 0 & \\dots & 0 & -P_1 \\\\\n0 & 1 & \\dots & 0 & -P_2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\dots & 1 & -P_{n-1}\n\\end{pmatrix}\\,.", "650f05723cc7dd5459c49cf87e5f574d": "\\bar{h}^{\\alpha \\beta} \\,", "650f06d0c081b7c886afba6575ae200b": " U_E(r) - U_E(r_{\\rm ref}) = -W_{r_{\\rm ref} \\rightarrow r } = -\\int_{{r}_{\\rm ref}}^r q\\mathbf{E} \\cdot \\mathrm{d} \\mathbf{s} ", "650f88d92d1a78f0cfe4f3e8d9ecb677": "\\sigma(R)=\\sigma \\left(1-\\frac{2\\delta}{R}+\\ldots \\right)", "650fcf9ed28c83fed9ab22401da5708e": " P = \\frac{PVx}{K} + \\frac{D}{K} ", "65101fcb22d33c11213c89f653e434e1": "\\ f_{x,y}", "65103e1d5922f55793770e684b8cea8f": "ac^*dc^*a,", "6510789420c5a0c960ab1c9bf68066c1": " \\varphi^{\\mathop{\\rm QMC}}(f)=\\frac 1n \\sum_{i=1}^nf(x_i),", "6510c5b51673a21d8f96d6d2376c1930": "\\scriptstyle \\leq-1.5\\times10^{-15}", "6510d908ed79834b50f2794fec831d0c": "\\begin{Bmatrix} r, q , p \\end{Bmatrix}", "65117cbbd6bde3ef7abaf0c47184f849": "\\mathbf{E}^{x} [f(X_{\\tau})] = f(x) + \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau} A f (X_{s}) \\, \\mathrm{d} s \\right].\\ ", "6511a9fa99c6c755374b34f15993aaa8": "x=\\frac{2v^2\\cos^2\\theta}{g}\\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)", "6511da886286d78f34f492db84f04560": " D_{1}, D_{2}, D_{3}, D_{4}, D_{5}, D_{6} ", "6511ffcd6c9751bb0d76f9be8586b097": "u_t(x,0)=h(x)\\,", "65122edff8104dc64daeffe00e0d5160": "\\operatorname{lambda-lift}[(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)), \\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))] \\equiv \\operatorname{let} x\\ f\\ y = f \\ (y\\ y) \\operatorname{in} \\lambda f.(x\\ f)\\ (x\\ f) ", "6512a5aa050ee4cad6a02d99b5782ff7": "\nT_1 > \\chi_{\\alpha,k-1}^2\n", "6512b78590dedb6b75ae05057e0ea47e": "\\sqrt{k}", "6512bd43d9caa6e02c990b0a82652dca": "11", "6512c585c2d47cd47d33ef45af0caa1c": "T_I \\varpropto \\frac {M}{R}", "6512d9feb33a91d92f089447f863ef28": "\\mathsf{s}", "651322dea31b5410c8272407e5938f2c": "0 = E \\{ \\hat{x} - x \\}", "651334d57c643591d56cabf0fd7dfbf6": "\\varphi_\\alpha(0) = \\alpha", "651403e33d51fcfec3f2cc160dc334a6": "U'_a:=\\bigcap_{b\\in B_0(a)} U_{a,b}", "6514475d6011c890d3fa741aab0697c1": " l = K \\times q", "65145eebe5af326ccac9b4a0a34c302f": "Q=\\{(s,t_s, t_e)| s \\in S, t_s\\in \\mathbb{T}^\\infty, t_e \\in (\\mathbb{T} \\cap [0, t_s])\\}", "65147d085ca9ac3518c8b88f1d97bb49": " A \\times B ", "6514873e76a582f6fb7f69820c9ce475": "\\scriptstyle\\pi\\,", "6514b60330eb2db4109507c5610bddb4": "\\zeta=2z/h_{z}", "65153d1787f6afd5b5d4861518e6933e": "A = 4t^2 \\cot \\frac{\\pi}{16} = 4t^2 (\\sqrt{2}+1)(\\sqrt{4-2\\sqrt{2}}+1)", "65155dd594ac1694dfe64f31219fe439": "h_{00}(t)", "6515666ff947165e2465ef22f3bbede0": "f_i(r_1,\\dots,r_{i-1},z)", "651576abbf949a1c1676bae7e2b9dc82": "d=2:", "6515880c5d42d911e1952c69fb1fd6a6": "\\mathbf{\\nabla}\\phi", "6515d116a45dfd6ccda13a7a1dd17961": "\n\\eta = \\cosh^{-1} \\frac{ml + (z - z_S)(z - z_R)}{r_S r_R}\n", "6515db16d890b2bbef9741ba291dbf53": "k \\leq i < n", "65169052f0eca940b83c4d9f62568585": "\\ \\mathbf v", "6516d48a4fee8b1c8c5541be8493f5bf": " -a", "6516f9cd73722f446789890e6152ed1a": "f^{-1}(a)\\,", "6518202e8327ec177335491b3a8783ac": "h([\\![e]\\!]_{s}) = [\\![e']\\!]_{s}", "651879302a9e23cc126b0cceb544605f": "\\Phi_N (f)", "65187c3c7252bcab1df70d9efe5f6f38": " \\mathit l_B = \\sqrt{\\hbar c\\over e B} ", "651910919f12f55be1f70b5fcc3e843e": "ij=k\\sqrt{-1}", "6519c185de6549dfbfd64a5abb998a84": "(h',g)", "6519cf060f57a07edb7b1f29a31c082b": "\n\\text{Effective processor performance} = \\text{MIPS} = \\frac{\\text{clock frequency}}{\\text{CPI} \\times 1000000} = \\frac{400 \\times 1000000}{1.55 \\times 1000000} = 258 \\, \\text{MIPS}\n", "651a44957a54d0723a3f059c9017ab0a": "t_1 = X^c g_{1}^{s_1}g_{2}^{s_2}", "651a456ffb22f10041d317cf41252ee2": "y.\\ Q \\rightarrow x:=y\\ ,\\ R)\\ =\\ P \\wedge \\forall z, Q[y \\leftarrow z] \\Rightarrow R[x \\leftarrow z]", "651a70bdbc7a29cbb210238418a517f4": "\\{0, 1, 2, ..., p-1\\}", "651a794fccf6145eab6ed8a0c2ebcebb": "\\scriptstyle P_b", "651a9bb99e56c7f532d80949f0d9ba2d": "g:\\mathbb{R}^n\\to \\mathbb{R}", "651ab8454e84d1102fad6883fd42c265": "d_n", "651ac4c10f0dd334a7ebf0ea1c87d865": "[c] = LT^{-1} \\ ", "651b341cf23f26f7d5c119c15b874026": "(p',q)", "651b5897794a214fe468ff1a7c644666": " \\widehat{Tf}(\\xi) := m(\\xi) \\hat{f}(\\xi).", "651b5d07e8003c821983579a655485ae": " \\mathcal{F} ", "651b959b09cc0232ad615f9b5a6fbec4": "\\frac{5n-2m}{m-2n}", "651b9e4d4e6cedced911224f49bbf73a": "\\partial_x f|_{(0,y)}=-y", "651bd4086162a87e21f0b0642f64957a": "(2) \\ n(z,t)=n+\\Delta n \\cos (\\omega t - kz), \\,", "651c558c3b98b84a38cf429cbf815b1f": " \\nu = \\eta / \\rho ", "651c6b47cce572574e312358639636fa": "d_{z^2} = N_2^c \\frac{3z^2 - r^2}{2r^2\\sqrt{3}} = Y_2^0", "651c703aa551af59682d236c94ab44be": " u_e = \\frac{1}{2} \\varepsilon_0 \\left|{\\mathbf{E}}\\right|^2.", "651c8ad089a018bdebc8941a88cfddc7": "T_n(\\cos(\\vartheta))=\\cos(n\\vartheta) \\,\\!", "651cfe212d57c58156cca6cafcadb7ea": "\\alpha = \\frac{e^2}{4\\pi\\varepsilon_0\\hbar c}", "651d2d2b7b1b8956443a7091c44db2ac": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n -\\frac{(m+n+2 n\\,p+1) x^{m+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{2 a\\,n^2 (p+1) (2p+1)}\\,-\\,\n \\frac{x^{m+1} \\left(2 a+b\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{2 a\\,n(2p+1)}\\,+\\,\n \\frac{(m+n(2 p+1)+1)(m+2 n (p+1)+1)}{2 a\\,n^2 (p+1) (2p+1)} \\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}dx\n", "651d9dd3e34265561c73eea2ad52f3ba": " ~p^v_0 = \\exp-{\\beta [ vP(z,T) + a\\sigma (z,T) ]}, ", "651dc734f77a28fd93f2ae3a267658cf": "p(X_1, \\ldots, X_n)", "651de41201112cff53f0b1dffe961be6": "f \\overleftrightarrow{\\partial_x} g = f \\partial_x g - g \\partial_x f.", "651e110127ae9d8f468eddc89f5aa063": "j = 1, \\ldots, n", "651e32ba954f7d1f033f80d058ca8d72": "\n \\cfrac{\\partial^2 M_{11}}{\\partial x_1^2} + 2\\cfrac{\\partial^2 M_{12}}{\\partial x_1 \\partial x_2} +\n \\cfrac{\\partial^2 M_{22}}{\\partial x_2^2} = q\n", "651e34111523c3e8096a8aa0930d79e8": "\\mathbf{S}(\\mathbf{p}(t))=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_{n}(p_n(t)),", "651ea0238d67e11748add0c62768adf9": "A \\lor B, \\neg A \\vdash B", "651ef3bf72352ae48ffd56a3dda1a679": "\na_i x_{i - 1} + b_i x_i + c_i x_{i + 1} = d_i , \\,\\!", "651f17f6ad5b9b600cc1b3a632a03cc1": "rpm_{fan} = rpm_{motor}\\,\\bigg(\\frac{\\,D_{motor}}{D_{fan}}\\bigg)", "651f54fec7daccd0ce54d04422398b8e": "\\sim \\exp\\left(S/k\\right)", "651fc2886d9a3ad8d3085cf4207770f3": "\\scriptstyle \\alpha^0", "651feb49045bbb30bbb0d7a6333aacd1": "\\operatorname{RMSD}=\\sqrt{\\frac{\\sum_{t=1}^n (y_t - \\hat y_t)^2}{n}}.", "651fffb87903b1a99eb265d3c1f5b18e": "A=\\frac{1}{n}\\sum_{i=1}^{n} a_i", "65200397ade131023e6519bb089e6cc6": "\\vec{\\nabla} \\cdot \\vec{B} = 0", "6520b0f3a71e982f0eac42b7668d5f1d": "\\underline{\\mathrm{data}} \\; \\left(\\frac{}{\\mathsf{Nat} : \\star}\\right) \\; \\underline{\\mathrm{where}} \\;\n \\left(\\frac{}{\\mathsf{zero} : \\mathsf{Nat}}\\right) \\; ; \\;\n \\left(\\frac{n : \\mathsf{Nat}}{\\mathsf{suc}\\ n : \\mathsf{Nat}}\\right)", "6520eae9adc0ad2b62f405bde7d88297": "u(b,t) = u_b(t)\\,", "652138780913a1d4ff517dc712678eae": "[x]_{(\\mathrm{RED}-\\{a\\})} \\neq [x]_P", "65214e7ec98d43e0309361059a7af8bb": " \\lambda(v-1) = r(k-1). \\,", "65216ce1f18957a72e9916eb72067d7d": "S^{\\prime \\prime}", "65217a1df56730fc77b281ab1a099ba8": "\n\\mathbf{F}_{\\mathrm{Coriolis}} = \n-2m \\boldsymbol\\Omega \\times \\mathbf{v}_{\\mathrm{r}}\n", "65217bba4a27408e3334dee6151f41f0": "K_a = \\frac{[H^+][A^-]}{[HA]}", "652227cef3ab0cd46c7dcda7fd6ae902": " \\sum_{k=0}^s a_k y_{n+k} = h \\beta f(t_{n+s}, y_{n+s}), ", "65222ea1a0162298a34422381ebd80ec": "d_1(0)=D", "65226dd0001f1252715dd41abb8e52c1": "\\left\\{\n\\begin{array}{lc}\n\\frac{\\partial u}{\\partial x}=f\\left(x,y,u,\\frac{\\partial u}{\\partial y}\\right) & (x,y)\\in\\mathbb{R}^+\\times[a,b]\\\\\nu(0,y)=U(y) & y\\in[a,b]\\Subset\\mathbb{R}\n\\end{array}\\right.,\n", "6522a4c19a06039435eedd51e4139589": "\\lnot Q \\Rightarrow \\lnot P", "6522de4de0cafa4ff7850883acd0b90d": "T(n) = 2 T\\left(\\frac{n}{2}\\right) + n^2", "6522de72f599288be187323fa4a1ee22": "\\displaystyle \\frac{\\displaystyle\\delta\\left(\\xi-\\frac{a}{2\\pi}\\right)-\\delta\\left(\\xi+\\frac{a}{2\\pi}\\right)}{2i}", "65231f7c53e2259036d6ccc9f055c299": " \\rho = \\sum_j \\eta_j |j\\rang \\lang j | ~, ", "652378fe77784b721945d1a067be7e79": " \\cdots \\rightarrow H_n(A) \\rightarrow H_n(B) \\rightarrow H_n(C) \\rightarrow H_{n-1}(A) \\rightarrow H_{n-1}(B) \\rightarrow H_{n-1}(C) \\rightarrow H_{n-2}(A) \\rightarrow \\cdots. \\,", "652382b6de0d0cfbc700a23a48e099d2": "\\scriptstyle\\triangle\\theta=(\\delta'-90^\\circ)-(\\delta-90^\\circ)=\\triangle\\delta", "652382c27c6289f255665f7c940e3af1": " \\tfrac{1}{q}+\\tfrac{1}{p}=1.", "6523b9791b4128ee9a29fd0a879ab9cc": "\\! G(n) < 2 n\\log n + 2 n\\log\\log n + 12 n.", "6523c8f44780045831e14454714b3ed7": " \\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{b} \\cdot \\mathbf{a}.", "652447a3e6c026557b318f19aac4c1c6": "\\int_0^t A(\\tau) \\, d\\tau = -\\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{Q} \\int_0^t C_p(\\tau) \\, d\\tau + \\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{A}", "652522a0c5f6a31bb821eaf1d533efc5": " \\forall x\\ \\forall y\\ xRy\\Rightarrow (\\exists z\\ xRz \\land zRy).", "65259218483ab3bc856ce801053b548a": "U = - m \\left(G \\frac{ M_1}{r_1}+ G \\frac{ M_2}{r_2}\\right) ", "6525b40da1ed3251eb8b9b3a53aeb104": "P_j[i]=p_{ij}", "6525d310482917fb09b8ee8c07317a56": "{U(R)}^\\dagger \\widehat{T}_{q'}^{(k)} U(R) = \\sum_{q} D^{(k)}_{qq'} \\widehat{T}_{q}^{(k)} ", "65267e47e705a356e68750d9a36573bd": "\\mathit{g}(\\mathit{x}) = \\mathit{c}_0 + \\mathit{c}_1\\mathit{x}+\\mathit{c}_2\\mathit{x}^2+...+\\mathit{c}_{N-k }\\mathit{x}^{N-k}", "6526f6a6e429fd47d84f75ccf799c873": "\\mathrm{Bowling~average} = \\frac{\\mathrm{Runs~conceded}}{\\mathrm{Wickets~taken}}", "6527628062e81464caf9ea12aef42b18": "\\gamma=\\|\\vec{\\gamma}\\|=\\|\\nabla U\\|", "6527fa120b10db463597869691cb46f9": "x_{k+1} = x_k - \\frac{f(x_k)}{f'(x_k)}", "65281607d033a51e89990fe2f62c539d": "1\\to \\mathrm{Pic}^0(V)\\to\\mathrm{Pic}(V)\\to \\mathrm{NS}(V)\\to 0", "65282a93fc589d017c969086df0c7b41": "r=R_f+\\beta_3(K_m-R_f)+b_s\\cdot\\mathit{SMB}+b_v\\cdot\\mathit{HML}+\\alpha", "652869ff5de9f226ad3041f87e8b877a": "c = \\Psi_M (w) = w + \\sum_{m=0}^{\\infty} b_m w^{-m} = w -\\frac{1}{2} + \\frac{1}{8w} - \\frac{1}{4w^2} + \\frac{15}{128w^3} + ...\\,", "6528b2b828472f27bb6859e376b4fa26": "\\left(m=m_1\\ldots m_n,\\; n\\in\\N\\right.", "6528cb4ecf08a19101aa61f2261bc20b": " k[\\Delta] = \\bigoplus_{\\sigma\\in\\Delta}k[\\Delta]_\\sigma,", "65291ff5bec21540c3ec63fc8db4a3a4": " \n\\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & 1 \\\\ \n0 & 0 & -1 & 0\n\\end{pmatrix}\n\\quad\n", "6529a68052696a3be32b6e2f1c4ab2cd": "p(V) = \\frac{1}{\\sqrt{2\\pi}} e^{-V^2/2}", "6529e1f71bf750b8a55115300117da30": "S\\subseteq MN/M \\cong N/(M \\cap N)", "652a3513933e6066ccc55e22774d73d3": "\\ |y[n]| \\leq B \\quad \\forall n \\in \\mathbb{Z}", "652a36b81c65052e5529e7217583f77c": "\\int_0^\\infty \\frac {x}{e^{x}+1}\\ dx=\\frac{1}{1^2}-\\frac{1}{2^2}+\\frac{1}{3^2}-\\frac{1}{4^2}+\\dots=\\frac{\\pi^2}{12}", "652a88965e996eed11541d207881c3de": "A=C^\\infty (M),", "652a9b81974894c313c646ddd894f2d1": " 2^{128} ", "652ac46736d04ae7f68139cc0c1e51fc": "h[f] = \\operatorname{E}[-\\ln (f(x))] = -\\int_\\mathbb X f(x) \\ln (f(x))\\, dx.", "652b336606f88fe4ea58a08585dd1758": "\n \\cfrac{\\Gamma \\vdash A, A, \\Delta}{\\Gamma \\vdash A, \\Delta} \\quad (\\mathit{CR})\n ", "652b4938135aee28b4166ffe17e279eb": "\\delta\\theta=\\theta-\\alpha \\,", "652b560116b07528fec1d2978559d462": "f'_+\\,", "652bf92413345fa17d6f6dfb3d89cc68": "x(y) = f(g^{-1}(y))", "652cb15d3ba445a530454636fbb74953": "B[i] = 1", "652cbd0a325fa17fd95785504e04f7cd": "p^* - d^*", "652d9f6a1b503cd9f1955fda2849ee7b": "\\sum_{i=1}^n x_i (1 - k_i) (1 - k_j)^{-1} = 0", "652dc50a042dae33fbf5169c4707c668": "H_{\\mathrm{b}}(p) = - p \\log_2 p - (1-p)\\log_2 (1-p).\\,", "652e044095e4733246b2833955accdb7": "\\ 2s = l+m+n ", "652e06e1bd4c40f76ce973501d532031": "I_{2}(\\sigma_{xx}\\sigma_{yy} - \\sigma^2_{xy}) - I_{3}(\\sigma_{xx}+\\sigma_{yy})", "652e477a72db192306953f371d75b800": "\\gcd{(q , p - 1)} = 1", "652e4a907c7fb9b6e1c1a3b0e25aae28": "H[\\xi]=\\int_0^1\\Phi^{-1}(\\alpha)\\mbox{ln}\\frac{\\alpha}{1-\\alpha}d\\alpha", "652ed0c430730a42bddb447ef7762eff": "\\operatorname{arcsch}(z)", "652ed1e97300480365fa979b41a90dba": "y_{Q20} = 0.65 - j1.20\\,", "652ee5721018cd5e6ae158afd15895b3": "\\partial V\\subset V", "652ee587ccc4ef1151765fe96a91ad43": "\n{\\mathfrak{T}}^\\alpha_\\beta =\n\\sgn\\left( \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right)\n\\left( \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right)^{W} \\, \\frac{\\partial {x}^{\\alpha}}{\\partial \\bar{x}^{\\delta}} \\, \\frac{\\partial \\bar{x}^{\\epsilon}}{\\partial {x}^{\\beta}} \\, \\bar{\\mathfrak{T}}^{\\delta}_{\\epsilon}\n\\,,", "652ee62ef7e2edca0c44cff6cfa1f6c0": "G_{ij} = B(v_i,v_j) \\, ", "652f428861527378c0c5807a8adcd844": "\\lambda x_1 \\ldots x_{A_i} . \\lambda c_1 \\ldots c_N . c_i (x_1 c_1 \\ldots c_N) \\ldots (x_{A_i} c_1 \\ldots c_N)", "652f659bbce2df4f92e201c8162e38e4": "\\mathrm{Re}=\\frac{u'L}{\\nu}.", "652f8058d09a86ced315f61b885080eb": "H=F(X)*(*_{i\\in I} g_i A_ig_i^{-1})* (*_{j\\in J} f_jB_jf_j^{-1}).", "652f8b58afc1f43ab5fbe1cf3ef07c32": "[X,Y] := J_Y X - J_X Y", "652fce556960eb088f2e3b9d8bcddbd8": "=\\sum_{i=0}^{n}\\lambda^{n-i}d(i)\\mathbf{x}(i)", "653042174562c7c0e533b9fe21e50595": "B = (Y_1,Y_2),~ D = (X_1,X_2).", "6530c4b886d0dd4303cfafea7b076855": "\\forall c\\in C", "6530ff04a5d85d26027761f29d16f908": " \\mu \\le \\nu \\le \\theta ", "6530ff3ecd8f36c4335fc28e67761855": "M=\\begin{bmatrix}A_1 & B_1 & B_2\\\\B_1 & A_2 & B_3\\\\B_2&B_3&A_3\\end{bmatrix}", "653120d801fb81f9f0c7b97e56a9d603": "y_{21}", "653125b801a2ee5b149f0be78d3c4232": "\\lambda_1 = 1", "65315416e0e3f5645fe1a0e78c916737": "x^m", "653163677990ae91904212ba4a82b819": "\\,f_i(u,v)", "6531660c71060ccd2806e89fe19d75fc": "C = 2 \\pi r \\,\\!", "65316bca3f3dc69af6592498dc1172a3": " \\dot{\\tilde{\\rho}}= - \\int^t_0 dt' \\operatorname{tr}_R\\{[\\tilde{H}_{BS}(t),[\\tilde{H}_{BS}(t'),\\tilde{\\rho}(t)R_0]]\\} ", "65317aa26ad6725b6041c7c4ca821c2a": " \\begin{pmatrix} n \\\\ m \\end{pmatrix} ", "6531aa1952a26cbacfdfa16a2a6c32c8": "\\operatorname{cov}(X,Y)=\\operatorname{E}[(X-\\mu_X)(Y-\\mu_Y)'], ", "6531add96a69b2b23f50ed0cc135dcbd": "\\forall i,j \\in \\{0,\\ldots,n-1\\}: L_i \\in GF(2^m) \\and L_i \\neq L_j \\and g(L_i) \\neq 0", "6531bc23f53e66a306782ed73d688f5e": "N = 1.0", "6531c8f3a9dc6c2f4d85426aea2b33b0": "m_{i} = [p_{i},...,p_{i_{k(i)}}]", "6531d7f33352be0806e55d89ae7bd29e": " I(x,y) = \\sum_{\\begin{smallmatrix} x' \\le x \\\\ y' \\le y\\end{smallmatrix}} i(x',y')", "65326afdcaf0d7e3343953cc2471dc86": "T^{\\mathrm{Y}}_p", "653284422b1b9965d93d633acd3b08c1": "\\lambda^4 := \\beta\\omega^2", "6532bbe5782b72558d75fe73c8d69e3f": "T : S \\times F \\times I \\rightarrow S \\times U \\times O", "65336bd022fb8a936a7fa8243012d53c": "\\widehat u(t).\\,", "653389fbe0cc3469bee08ac37a125b84": "~v(\\omega)~", "65338a65ff83cabbe94e4876ee6be763": "p_0(x)=1", "6533d0e1b2baca86aaa18c5a117667b1": "\\mathcal{J}(\\theta^*) \n = - \\left. \n \\nabla \\nabla^{\\top} \n \\ell(\\theta)\n \\right|_{\\theta=\\theta^*} \n", "6533d7e8aa0bcd4170ddfa48cf9f5a52": "g(x)=\\int_0^1 f(x,y)\\,dy.", "6533f3f59f74b2038e32dc32874c3e56": " \\theta>0 ", "65340763b65eb5457d9b002f5b2bfe77": " A_{i+1} = 2A_i - A_i A A_i, \\, ", "65341b86454fcd74e0210bf2cb60f132": "\\ y' = 0.60", "653427b1955f423a189afa6298903cbd": "X_n = (X_0, \\mathcal{O}_X/\\mathcal{J}^{n+1})", "65342c74e6ba96c4ecb70c65ead5c90c": "\\sum_{m=3}^{n/2} \\frac{1}{\\ln m} {1 \\over \\ln (n-m)} \\approx \\frac{n}{2 \\ln^2 n}.", "6534761787bf06bd4d378ce0b089fd9b": "h_{ij}=2(\\phi g_{ij}+S_{ij})", "6534cea23a8b9420c9abfe1dfbb9556d": "\n\\sum_{\\delta|n}d^{\\;3}(\\delta) = \\left(\\sum_{\\delta|n}d(\\delta)\\right)^2.\\;\n", "65351975510b64caca664dd2784fedba": "A\\in \\mathbb{R}^{m,n}", "6535205b6e257fa3f48008d06361f94b": "\\Sigma^{-1}", "65357371986dbb6bd1f0dcc148f3474f": "L_\\eta = \\{s\\in\\Sigma^* \\vert vP_s Q_\\text{accept} > \\eta\\}", "6535a355dc1dae12b8a8ae751447c216": " \\Phi_B = \\iint\\limits_{\\Sigma(t)} \\mathbf{B}(\\mathbf{r}, t) \\cdot d \\mathbf{A}\\ , ", "6535e13ec82a2b9d8a8364c0d1b8cd85": "\\begin{align}\nW(u) = -0.577216 - \\ln(u) + u - \\frac{u^2}{2 \\times 2!} + \\frac{u^3}{3 \\times 3!} - \\frac{u^4}{4 \\times 4!} + \\cdots\n\\end{align}", "6535ea33e496d54975b1f15fddecfa30": "{\\rho_0}", "6535fbbd5ca45863a936fd7cf5a347dc": " G = \\ G_{ \\infty } \\frac {T} {1+T} + G_0 \\frac {1} { 1+T} \\ \\ , ", "653603f6f4cea8db545bb87820f984db": "\\mu(x):\\mathbb{N}{\\rightarrow}\\mathbb{R}", "653608394b69fdd0b9097d3ef90c2d5c": "\\beta_j \\in \\bigg[\\ \n \\hat\\beta_j \\pm q^{\\mathcal{N}(0,1)}_{1-\\alpha/2}\\!\\sqrt{\\tfrac{1}{n}\\hat\\sigma^2\\big[Q_{xx}^{-1}\\big]_{jj}}\n \\ \\bigg]", "653628e6b999451191d76a79228355e7": "\\qquad \\qquad b_{\\kappa,\\alpha} = \\frac{1}{N^{1/2}}\\sum_{\\kappa_p,\\alpha} e^{-i(\\boldsymbol{\\kappa}_p\\cdot\\mathbf{x})}\\mathbf{s}_\\alpha(\\boldsymbol{\\kappa}_p)\\cdot[(\\frac{m\\omega_{p,\\alpha}}{2\\hbar})^{1/2}\\mathbf{d}(\\mathbf{x})+i(\\frac{1}{2\\hbar m\\omega_{p,\\alpha}})^{1/2}\\mathbf{p}(\\mathbf{x})],", "65367b18358bafb6ef47623f20a11e5f": "\\xi^d_{f_{min}}(k,i)\\,\\!", "65372762784e9710380b8fcbf1356252": "x_2=4(1/3)(2/3)=8/9", "6537569f0355e834f5fb284a642afced": " K_f = \\frac{\\textrm{GFR}}{\\textrm{Net\\ Filt.\\ Pressure}}=\\frac{\\textrm{GFR}}{(P_G - P_B - \\Pi_G + \\Pi_B)}", "65377ae40c2bb578c55c9f036c90941d": "-\\boldsymbol{\\nabla}c", "65377cab0ef2f08e6b54c98b9af2413b": " \\mathcal{K}^{-2} = \\frac{1}{\\mbox{ energy converted per input energy}} ", "65377e5844ed7485902d5d2d5a1c6051": "a = U_\\infty/W_0", "6537a6e0c1d7dee6fae82deace439bc0": "=2^2\\cdot5\\cdot193\\cdot401", "6537b81c96869e65cc8be6c4f174b9c7": "c_{\\sigma}=\\left(1+e^{-\\sigma^{2}}-2e^{-\\frac{3}{4}\\sigma^{2}}\\right)^{-\\frac{1}{2}}", "6537d0a7b8d06c25883bbd21c6bc8e2d": "V_x", "6538a76220a2414da9f82e270addd174": "T_{vw} = T_w \\circ T_v.", "6538a767b6c4cc996585cca1764fc9b2": " Y_i = \\alpha + \\beta x_i + \\varepsilon_i, ", "6538a7ff8f02a825ec27f168b493f8bb": "R = 0.18", "6538de488b27312120b569ff601653ab": "\\partial^{\\,2}V / \\partial c^{\\,2}<0", "6538e767bd81b82c60a7dbec4472d868": "\\oplus_{i=1}^mV_i", "653928af8c802a6ec14cc0284f2046fb": "t = k_\\mathrm{DM} \\times \\left(\\frac{\\mathrm{DM}}{\\nu^2}\\right)", "6539c71e592ddb37ffe1c3f7870905c3": " c_n(t) = c_n(0) + \\frac{-i}{\\hbar} \\sum_k \\int_0^t dt' \\;\\lang n|V(t')|k\\rang \\,c_k(t')\\, e^{-i(E_k - E_n)t'/\\hbar} ", "6539dd56b21065b690cbd4a9aaf130bb": "score(X,Y) := d(X, Y) - d(Y, X)", "653a28555fec85f8f35becad28bde417": "\\mathbf{Q}_k", "653a40792261e39e68b32ace636fc4ae": "\\sqrt {\\frac {d_1} {d_2}} ", "653aea47314ec781acd55ba6ed28a440": " A \\mapsto (I+A)(I-A)^{-1} , \\,\\!", "653b5afddfcc37ffa4138d4d7bee4591": "y^2 = x^3 + 3(x+1)^2", "653b6a6e8c4de0ff01f9602c79cc48e9": "\\displaystyle{(\\pi(g,\\gamma)1,1)=\\gamma^{-1}.}", "653b851e23bbe73d5cf69917dd75591e": "\\textstyle \\Omega_1 \\setminus A_1 ", "653b9542a897dd48ba9cc94d444c3244": "v_1 \\leftarrow \\, ", "653bbb778862b79fa8c61337988e372c": " E_2 = \\tfrac {1}{2} m_y v_2^2 \\,\\! ", "653bfceca5db566df128dd69349f8371": "\nE_{spin} = \\frac{1}{2} I \\Omega_H^2\n", "653c064b3be83d89ba16cf3147fe0abc": "z = r\\cdot(\\cos\\varphi + i\\sin\\varphi)", "653ca4dbc47520d577370290007575f7": "1 < k < 6 ", "653ca6e0652c329d4b9cd7c600b5e782": "\\frac{\\Delta t}{m}", "653cd972b2bf4fc7614c4ea32265f370": "g_{\\mu 2}=\\, 0", "653cdf7511a038c8ac34b28ee64ce360": "G(S) = \\frac{1}{n-1}\\left (n+1 - 2 \\left ( \\frac{\\Sigma_{i=1}^n \\; (n+1-i)y_i}{\\Sigma_{i=1}^n y_i}\\right ) \\right )", "653cfcd3cfc1e212a336d82aed3a472e": " I = I_0 \\cos^2 \\theta_i \\quad ,", "653d062bda8939434e2b54c060a12d4e": "X\\subseteq\\mathbb{C}", "653d2667c130511e74b1218e5a92ceaa": "r = \\pm\\theta^{1/2}\\,", "653d9b15c466b9357258198b24c05b42": "\\mathrm{P}(u,v)", "653dd9622590fc6de207ee5c1e25f669": "\\delta=\\delta_S=1+\\sqrt2", "653e08bd84f3611b8ec536516df3fb5f": "(A,\\lambda)=(2.1,1).", "653e4fba60ccbe006b3ea03aecdfce8f": "\\forall x \\forall z . P(x,f(x),z)", "653e52acc14ddb5f8bc15962595c32b0": "{}^4_3", "653e702e5ece661c310a82af2f07e081": "= \\frac{\\lambda}{1+\\delta} + \n\\sum_\\rho \\frac {\\Gamma(1+\\delta)\\Gamma(\\rho)}{\\Gamma(1+\\delta+\\rho)}\n+\\sum_n c_n \\lambda^{-n}.\n", "653eafddfaa1f18ea9aab5e0eb33a6a3": "\n \\begin{cases}\n 0 & \\mathrm{for\\ } x < a, \\\\[2pt]\n \\frac{(x-a)^2}{(b-a)(c-a)} & \\mathrm{for\\ } a \\le x \\leq c, \\\\[4pt]\n 1-\\frac{(b-x)^2}{(b-a)(b-c)} & \\mathrm{for\\ } c < x \\le b, \\\\[4pt]\n 1 & \\mathrm{for\\ } b < x.\n \\end{cases}\n ", "653efac2d799b5bcafd6f14fe5077b9a": " F( \\nu - x ) + F( \\nu + x ) \\ge 1 \\text{ for all } x, ", "653f011f34f1ed630a0fca4080c1ed70": "\\mathrm{RCA}_0", "653f0c36b285c4c9e3c32344e10c546a": "\\sigma = -S \\hbar , -(S-1) \\hbar , \\dots, 0, \\dots ,+(S-1) \\hbar ,+S \\hbar \\,.", "653f0e424f492a651cee5489b31d845b": "\\sqrt[n]{x} \\,=\\, x^{1/n}", "653f114d697131c3a41cf3cb5e814ca7": "\\mathbf{1}_{A} (x) = \\frac{1}{1 + \\chi_{A} (x)}", "653f334c66f6276d866661d9dd107777": " K = \\frac{m}{2}\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}} =\\frac{1}{2}m{\\mathbf{v^{2}}}", "653f863e7465e145332069e9dfdeab30": "\\mathbb{F}_2^4", "653fab5c7c767a056cbf9d04f0be86c0": " P: C^\\infty(E) \\to C^\\infty(F) ", "653fe4db0325559fa5b496a5d42d98cc": "f : \\mathbb{R}^{d} \\to \\mathbb{R}^{d}", "65403667968955f25a91bd1b69a321d4": "G \\approx 2\\gamma = 2 \\,\\, J/m^2", "65407c820556db31613ac8f3bf59d3db": " \\epsilon_0 ", "6540aa8adc36ed35701f1c173109130e": "t\\in [0,\\infty)", "654106c78210d392a87412759c7522dd": "\n\\dot{e}+\\alpha e= (\\dot{e}(0)+\\alpha e(0))e^{-\\frac{\\kappa}{2} t} \n", "65415340d8c2540be2a31da8a089a9d0": "1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\ddots}}}}", "654170baa3f76185c958f68a36022ee7": "\\mathbf{y} = \\mathbf{s}", "654180c7e2291ffefb5a8ee9ec480b20": "\\sqrt{M}", "6541998eec22179e480de28cec1ca3fe": "\\hbox{not } C_{1},\\dots,\\hbox{not } C_{n}", "6541da1195ed464deb96f1b95634da0f": "T_{hold}", "6541dece88ecc7b91ce05c769e440bb3": "\\mathbf{p_c} = -e \\sum_n \\int\\ d^3 r \\,\\, \\mathbf{r} |W_n(\\mathbf{r})|^2 \\ , ", "6541ee214b472678c2de2259692b89e6": "(x_1,...,x_n)", "6542072522c497dcedffcd47abfd41c4": " A_{i'} \\, ", "6542482c99199bddbc965bae1c42111a": "a \\mbox{ } b", "65428c96da2f2a4b203c1c78b2b70d98": "\\mathbf{C}=\\mathbf{C'}", "654298a90aae2f20e44a30986776dfd4": "\\Vert C \\Vert^2=\\inf\\{\\mu :\\,AA^*\\leq\\mu BB^*\\}.", "6542b78a465c454f159c3fe51e45f1c8": "[J_m,K_n] = i \\epsilon_{mnk} K_k ~,", "6542b79297cc01b4ab59ec88cae0214d": "\\gamma<1", "6542df48a29920c4f75512b30bae1a29": "\\Sigma_1^*", "6542f7931cc58538cfe71c2edeb5ebed": "\\mathrm{vol} ( s \\, \\varphi K + t \\, \\psi L )^{1/n} \\leq C \\left( s\\, \\mathrm{vol} ( \\varphi K )^{1/n} + t\\, \\mathrm{vol} ( \\psi L )^{1/n} \\right)~.", "65430496be80498ddc31529054cbb3cb": "T(P) \\wedge \\forall p \\neg (T(p) \\wedge p 2^{k-1}.\n", "655e53e0afeedfcc4e471b28302d54f7": "\\mathrm{S}_4 \\twoheadrightarrow \\mathrm{S}_3,", "655e6e2dafb5ac839c51dd6c7f7493b5": "G\\,\\!", "655ec1e563435a1b0bd47e91cacbcf17": " k=\\mathrm e^{\\frac{ a s_1-\\sum\\log u_i}{m a}}", "655f025b39d920c1065a04923a941db6": "I_{n,2^{2n}+1}=[2^n,\\infty)", "655f3ead6798aa3f000e0e1fc4333a5b": "D^2 f : U \\to L^2(V\\times V, W) \\,", "655f6d884daee47574563044f5301314": " \\textbf{I}(\\alpha) = C - \\frac{\\alpha^2}{2}", "655fd6dbbb63c13a21f64fff0fa58398": "1 - {1 \\over 2} + {1 \\over 3} - {1 \\over 4} + {1 \\over 5} - \\cdots =\\sum_{n=1}^\\infty (-1)^{n+1} {1 \\over n}=\\ln(2).", "655fef892c818f4cd1a4f02ac968590f": "\\,\\gamma\\,", "655ffb1f252b671cae9b3a4469bef8b3": "\\Pr(|X-\\textrm{E}(X)| \\geq a) \\leq \\frac{\\textrm{Var}(X)}{a^2},", "65604f412e100aa195736d17ef1c9dbf": "S^{-1}R", "65607b42001683ea501fd7f5ed1ba749": " \\{ a^n b^n c^n d^n : n \\ge 1 \\} ", "65607d5b17c5987fdcc9554a09555693": "\\phi: B \\to A", "6560ccb006d2759923fc5dfbdd7b2b34": "\\delta.", "6560f1c2afc30bf0ed5daeadd88d6722": "\\tilde{f}(x) = f(x) + I_{\\mathbb{R}^d_+}(-g(x))", "65610cbda6c1da79deebcb02a263339c": "H[n]=\\begin{cases} 0, & n < 0, \\\\ 1, & n \\ge 0, \\end{cases} ", "6561bc64e47d0b36d53a156abe4f10a0": "\\begin{align}\n f(j\\omega) & = (-1)^{n/2}\\big[a_0\\omega^n+a_1\\omega^{n-2}+a_2\\omega^{n-4}+\\cdots \\big] & {} \\quad (22)\\\\\n & + j(-1)^{(n/2)-1}\\big[b_0\\omega^{n-1}+b_1\\omega^{n-3}+b_2\\omega^{n-5}+\\cdots \\big] & {} \\\\\n\\end{align}", "65622a880ce817388f32cb6c17253664": "\\mathcal{R}(\\mathbf{r},t) = \\mathcal{L}(\\mathbf{r},t)-\\pi(\\mathbf{r},t)\\varphi(\\mathbf{r},t)\\,.", "65626d77fa280588a75e16e6ec06520e": "O(n^{-1})", "6562d83ccd7e1ab6eb1b3767f5c4bb3a": "2k-1", "65631ae427aa0e927cc2fe1fe1a2de61": "\\psi(p)= \\langle p|\\psi\\rangle", "65634b036ee09086307a6b4f7ac89edb": "\\alpha\\leq\\omega_\\alpha.", "65638954e71e25a478d007d9ef0d45a4": "\\int_{a}^{\\infty} f(x)\\,dx = \\lim_{b \\to \\infty} \\int_{a}^{b} f(x)\\,dx", "65639194432a41a5259afcc8b8eed0b3": "\\psi(b',e) = (b', \\mbox{proj}_2(\\varphi(e))).\\,", "6563bc0d6b94ac43769ce0f43fdde869": " b= \\alpha + i \\beta. \\,", "6563c080b15ebcdc8e3515d8f51404d1": "\\Phi(t, x) = x \\,", "6563f7b6cbc1e61f7cd5df112bfeaf89": "\\mathbf{C}[G] \\hookrightarrow C^*_{\\text{max}}(G).", "656447188d60ee8c91440234bd229c30": "\\mathrm{area}-\\frac{\\sqrt{3}}{2}(\\mathrm{sys})^2\\geq \\mathrm{var}(f),", "656463430baeecc3a2cbc376070bb79e": "\\pi_2(u)\\!:\\!\\tau_2", "6564878b88265871ac155e2bc039867d": " R_{i,j} \\geqslant 0 ", "6565162d4156b513d251d05fcb3df26f": "\\vartriangle^0_n", "65652b19eee239674cb2c3104bfba754": "1+0.01x", "6565a95bf8df529e0a6731b8d41ff87c": "5Z < Z", "6565bf9c041748ac14a0ce6489c0a180": "\\scriptstyle kn \\, + \\, 1", "6566141048c0ad8e1e7c96b708af8169": "{{\\sigma }_{abs}}=4\\pi k{{R}^{3}}Im\\left| \\frac{{{\\varepsilon }_{particle}}-{{\\varepsilon }_{medium}}}{{{\\varepsilon }_{particle}}+2{{\\varepsilon }_{medium}}} \\right|", "65662327e32ab5757f51cd8d03dc86b2": "N_I(f)", "65666f0988d979cf7e589068660fa938": " F_{ST} = \\frac{\\operatorname{var}(\\mathbf{p})}{p\\,(1 - p)} \\!", "65668904600b3fc07f7003f21a910f57": "\\left[\\begin{array}{c} X \\\\ Y \\\\ Z \\end{array}\\right]=\\left[\\begin{array}{ccc}X_w/X'_w & 0 & 0 \\\\ 0 & Y_w/Y'_w & 0 \\\\ 0 & 0 & Z_w/Z'_w\\end{array}\\right]\\left[\\begin{array}{c}X' \\\\ Y' \\\\ Z' \\end{array}\\right]", "6566a83dfaa0840b6e3db5b7bcca1d8a": "\\displaystyle{g(z e\\oplus x_{1/2} \\oplus x_0)={\\alpha z +\\beta \\over \\gamma z +\\delta}\\cdot e \\oplus (\\gamma z +\\delta)^{-1} x_{1/2} \\oplus x_0 - (\\gamma z + \\delta)^{-1}P_0(x_{1/2}^2).}", "6566e5bd217e953f57c59f6191960547": "\\Delta N=\\frac{\\Delta L_{A}-\\Delta L_{B}}{\\lambda}", "6566e634c136cd48bdfc6af1931bed6e": "t' = \\gamma (t - \\frac{vx}{c_0^2}), \\quad t = \\gamma(t' + \\frac{vx'}{c_0^2}).", "6566f64f00116db43670795f66820a89": "e = \\begin{bmatrix}1 & e^{j \\omega} & e^{j 2 \\omega} & \\cdots & e^{j (M-1) \\omega}\\end{bmatrix}^T.", "6567348deee5be0c96ab82133a0ec7ba": "\nA \\rightarrow A + \\partial \\chi \\,,\n", "656756d91f9012784070fa7c7fd5f2ca": "t_2\\ ", "656761f16102be2e29cf1398fb2065e2": " t_0 \\ ", "6567771634caff534f23c2e93052c77a": "\\text{turns ratio} = \\sqrt{\\frac{\\text{source resistance}}{\\text{load resistance}}}", "65682806072dcf74dadc3c350c0cb157": "\\sigma_\\alpha", "6568287665c5d4f470fe90ad51f9049f": " R = ab ", "656834b57cf12b84aff4f4162fd05f17": "1 \\times \\sqrt{10}", "65685e8323813416d17cc6012d1b589a": "\n\\mathbf{f} = \\mathbf{r} + A\\mathbf{x},\n", "656862c3075e276e847749612221daf6": "\\frac{D}{Dt}\\frac{\\beta y + \\zeta}{H} = 0", "65686afa34d098035ded7ad12ad1c9b9": "\\ \\displaystyle R(q,u)\\ ", "65689e904378a91b70f6f11e5e8096d1": "\\omega_{ab} = {h^m}_a \\, {h^n}_b X_{[m;n]}", "6568b0f6561365863b125047093352a9": "b_i(\\sigma_{-i})\\, ", "6568e58672ba9efbb982ef49e9f7d41e": " f_\\text{max} = f(\\sigma;\\sigma) = \\frac{1}{\\sigma} e^{-\\frac{1}{2}} \\approx \\frac{1}{\\sigma} 0.606", "6568f8c18e7d53567bb7477c9abe5f03": "\\scriptstyle [x,y]=z, [x,z]=[y,z]=0.", "6568ffda9ddcc076c9e45de58b516985": "i=pd", "6569580b0a4268d2fd447fba85c0f453": "u, u'", "656960e0871e9538fa067e29dc80e261": "\\epsilon_a = \\dfrac {\\Delta a}{a\\Delta c}", "65698b3aa0aee20204f37a19aba97c3e": " H_*^G(E_F(G))\\rightarrow H_*^G(\\{\\cdot\\}) ", "6569b0fb2af44f6f16a183be279546b3": " M = M_0 \\exp \\left (- \\frac{Q}{RT} \\right ) \\,\\! ", "656a3cab618370c206f0ba078343ffc9": "P(n)\\sim A\\frac{1}{\\sqrt{a}}\\frac{\\sqrt{n}}{\\log n}", "656a4aefb06c8133f6b89ebb606a2942": "\n F(t) = f(t) + f'(t)(x-t) + \\frac{f''(t)}{2!}(x-t)^2 + \\cdots + \\frac{f^{(k)}(t)}{k!}(x-t)^k.\n", "656a5da83fd4b67b1a98e972cdf482aa": "E_c = \\frac{3}{2}y_c", "656a7db9ffec69c7a060e46280f5684a": "K^\\ominus = \\frac{\\left\\{\\mbox{Ca} ^{2+}(aq)\\right\\}\\left\\{\\mbox{SO}_4^{2-}(aq)\\right\\}}{ \\left\\{\\mbox{CaSO}_4(s)\\right\\}}\n=\\left\\{\\mbox{Ca} ^{2+}(aq)\\right\\}\\left\\{\\mbox{SO}_4^{2-}(aq)\\right\\}\n", "656a90748b4845786d2c0e6e347c405a": "\\displaystyle (ab)=a_1b_2-a_2b_1.", "656ad4eb6889e21bae2db3b300ca0593": " d_H(X,Y) = \\epsilon ", "656ad6c885d4d391b93abeee0d27724f": "(\\lambda x.\\Box A(x))(t)", "656b0b396254d1c78f0bc1a931bd3be3": "\\begin{align}\nc &= \\arctan\\frac\n{\\sqrt{(\\sin a\\cos b - \\cos a \\sin b \\cos \\gamma)^2 + (\\sin b\\sin\\gamma)^2}}\n{\\cos a \\cos b + \\sin a\\sin b\\cos\\gamma},\\\\\n\\alpha &= \\arctan\\frac\n{\\sin a\\sin\\gamma}\n{\\sin b\\cos a - \\cos b\\sin a\\cos\\gamma},\\\\\n\\beta &= \\arctan\\frac\n{\\sin b\\sin\\gamma}\n{\\sin a\\cos b - \\cos a\\sin b\\cos\\gamma},\n\\end{align}", "656b45234c1280de1ac291cdbe3e73e5": "x^2-x_k^2=(x-x_k)(x+x_k)", "656babb3d194f21dcd9ae147c262e848": " \\langle F\\mid G\\rangle = \\pi^{-n} \\int_{C^n} \\overline{F(z)}G(z)\\exp(-|z|^2)\\,dz. ", "656bfbaec222f587fa698407d3f8a0fa": " \\sup_{k \\in \\mathbf{N}} f_k, \\quad \\liminf_{k \\in \\mathbf{N}} f_k, \\quad \\limsup_{k \\in \\mathbf{N}} f_k ", "656ca508787d4c5078fa03a292b74b14": "\\mathbf{P}(X_t\\mid X_{t-1})", "656ca74898e281c783fe8b4ff8f76c20": "- \\infty", "656ccc608b6a60b30afdcff6b2e2e6c3": "m^q / 2", "656cfb3ca4c3f547b45db39e95bb8937": "\\mathrm{CodeValue} = \\sum_{i=1}^N s_i \\cdot 2^{i-1}", "656d65c4fe23e71e373a3a76280b0058": "M \\in \\mathbb{Z_N} ", "656d74e3b168c5c07dd2b6f1b2511a19": "U=\\frac{1}{2}K_2\\left(\\frac{d\\theta}{dz}\\right)^2-\\frac{1}{2}\\epsilon_0\\Delta\\chi_eE^2\\sin^2{\\theta}", "656e09a71c83071c9f39c3459d6027fc": "S^+V = \\bigcup_{x \\in V}S^+_xV \\subset T^*V.", "656e33d940dd51c0c0378ee0f629926a": "\\omega \\in \\Omega_{Z,[t_l, t_u]}", "656e3b6abc3548d98f5922000c12b5ad": " \\sum_{i=1}^n a_i = 0. ", "656e5ea84e6bb123597538b9b9b81f76": "\\gamma_1 = 2", "656ec876e1452387fdb42c85c8b2def6": "f^{*}=p/a-q/b .", "656eeb30f403648af1adf9d6444c84b0": "X\\oplus Y", "656f57c7eaf2d6a20adecb8788ee74cf": "\\boldsymbol{\\mathsf{I}}", "656fe20ff3d7cd68288114ea5190f60c": "S_{11}, S_{22}, S_{33}=0", "6570dddeb2d85678adecae436b4d7767": "\\frac{1}{2}r^2\\theta", "65718bfb46ee5e99afae12cebe3138bf": "G \\cong \\mathbb{Z}_n, +_n", "6571d4319a5e7e724c8f83c6b3d436da": "\\chi=\\sum \\text{index}_D = \\text{constant}\\,", "65721f4a0cfe89b5614279605a633367": "F_S(T,T)=S(T)", "65723b3b986d1a3090b9f1d09abc048d": " \\Lambda(B)=M^1(B)=E[{N}(B)]. ", "65725766598eafef3700c217527c9be4": " (S,P,T) ", "65734a12c659a0c7a62bfe6258d4184a": "f(x, y): B \\to \\mathbb{R} ", "65736f3daef8be19f1b5768bf9bf00b1": "\\delta t=0.6\\pm0.4\\ (\\mathrm{stat.}) \\pm3.0\\ (\\mathrm{sys.})", "65737800dc1a5f18369c286c48225ed0": "\nI = \\frac{e^2}{\\pi\\hbar} V \\sum_n T_n \\ ,\n", "6573924dfed9e0db4dec50f14c08f6f0": "g(z) = c_2 + c_3 (z - a) + \\cdots \\, .", "6573b94b02030b92866585422e25c8bc": "\\Delta \\beta_j\\,", "6573c6047833d0212bba98b3fbe33b0f": " \\forall x \\, Q(x) ", "6573ca709a20315cbaee42f5a511d4da": "\\mathcal{M}_1,\\dots,\\mathcal{M}_n", "6573fa6c85f6093bd9342c2851dfe836": "\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }o(t^\\varepsilon).", "657408e22de1ef7de630067162e4790c": "P_\\mathrm{avg} = {(6~\\mathrm{V})^2 \\over 2(8~\\Omega)}\\,\\ = 2.25~\\mathrm{W}", "65740ad0f1ebeb1c0a34b34dfcee083b": " \\langle f_i | e_j \\rangle = \\delta_{ij} ", "657421da61fc7c9e5fb624a3c899ae47": "x_0 = x\\,\\!", "657445ad5fac2b06136e10974a99f25d": "[ u ]_{H^{1/2} (\\mathbf{T}^{2})}^{2} = \\sum_{k \\in \\mathbf{Z}^{2}} | k | \\big| \\hat{u} (k) \\big|^{2} < + \\infty:", "65747dad63e8326ae46d04c073e2aca9": "f_m", "6574eba98e86f1b52d0a082121926097": "\\Phi(x,y,z,t) = f(z;x,y)\\, \\varphi(x,y,t)", "6574eecff464d1d67c63b6bda4158b79": "\\mathcal{E}(g)", "657509ce2095ec29efe434a26456faf0": " x + iy = e^{\\rho+i\\theta} \\, ", "657526ec1b74b43cb3e7b31377840a87": "\\forall \\alpha\\in\\mathbb{R} \\ : \\ \\Pr\\left[\\frac{j(n,X)}{a(n,X)}\\geq \\alpha \\right]\\leq \\frac{1}{\\alpha},", "657573e4f0de3261acc7a69e18450edc": "\n\\begin{align}\n0 &= \\sum F_{\\mathrm{Contact \\ line}} \\\\\n &= \\gamma_{LG} \\cos(\\theta) + \\gamma_{SL} - \\gamma_{SG}\n\\end{align}\n", "657576c84d661670021db73e49afacec": "\n\\sigma ^{2}=1.0299 \\left ( \\frac{d}{r_{0}} \\right )^{5/3}\n", "657596088c9c5954b797ee3536ac763c": "\n \\varepsilon_x = \\frac{\\text{extension}}{\\text{original length}} = \\frac{\\mathrm{length}(ab)-\\mathrm{length}(AB)}{\\mathrm{length}(AB)}\n = \\frac{\\partial u_x}{\\partial x}\n ", "6575b896b01bfb394b8a1daa77a4d141": "\\mathcal{B} ", "6575b8a3a0b413bb49c7da9f31b124d2": " L(\\lambda)|_{\\mathfrak{g}_1} = \\bigoplus_{\\sigma} L(\\sigma(1)),", "6576b48952cbff1699bfbfcb99447a9f": "L_m>0", "6576c76bf6b5f2a6de9a0dd26f1ddabd": "V = G \\cdot\\ \\theta\\ ", "6576eb2391eb9e39e7f898a276975283": " \\frac{C(X_1, \\ldots, X_n)}{n-1} \\leq D(X_1, \\ldots, X_n) \\leq (n-1) \\; C(X_1, \\ldots, X_n) .", "65775d21c3738127908ddd67be220579": " x^5- 20 x^3 +320 x^2 +540 x + 6368 ", "6577a0bf4df07f182dc8372b8e8662e7": "k=1,\\,n", "65780dd61bf5a2a9a6a71647fa2ecdb1": "\\frac{\\Gamma \\vdash \\Sigma}{\\Gamma, A \\vdash \\Sigma}", "6578292c54e87d3d26926b7e78f294a6": "v_e = \\sqrt{\\frac{2GM}{r}}", "65784150664b4b4b11079d08e57fdd2d": "T_s = -\\frac{\\ln(0.02)}{\\zeta \\omega_n}\\approx\\frac{3.9}{\\zeta \\omega_n}", "657857e733c98f38d38e014da0007ead": "\\scriptstyle P_{\\text{CO}_2}", "6578a85478f4edc001cc4d9d1fd51e8c": "\\begin{bmatrix}0&0\\\\0&1\\end{bmatrix}:\\mathbf b", "6578cf272986b11d75f1f5ee1e2a51f7": "\\mathbb{Z}/p\\mathbb{Z}", "6578f94ea3e90e1b34cac29bf8ea998d": " H_i = (P_{i,1}, P_{i,2},\\ldots,P_{i,n_i}) ", "65793b5be3701b7b1e3832fbcfa7e4a4": "slip=", "65794b6fa40e3e9a8099ccd87e9251b7": "\\gamma(r)", "65794e258ef3e5b1ae9a4ef1970a8c7d": "\\frac{dN}{dT} = aN\\left(1-\\frac{N}{K}\\right),", "65796c6ce678f1d865e2c02bc03cc472": "H+\\vec{u}+\\vec{v}+\\vec{w}", "6579792b8d8073a7b7a32e3047960f25": "\\textstyle{n^3}", "6579d5ab8a733e80f2569b10b7de69f5": "H_{p-1}", "6579ee77b701a6a894b1718eb8ac0dce": "L_{-n}=(-1)^nL_n.\\!", "657a335368e7364f1435ed54f9983d19": " Pr[(m, n)] \\approx 1-e^{-m^2 / 2n}.\\ ", "657a517816a219a7de0e58431ae6540e": "\\Gamma'=(V',E',s',t')", "657a66f24964a213f90cfef34ca050df": "\\displaystyle \\partial_t u + \\partial_x^3 u - 6\\, u\\, \\partial_x u + u/t = 0", "657adef67c6249a16f11d2299232f0d1": "p_{y|x}(y_k|\\hat{x})", "657ae622bd37c95546ab5a5677e1b087": "\\sqrt 3", "657b340603957ba378249de826f37fb1": "H = -K\\sum_{i=1}^k p(i) \\log p(i),", "657b5d863ab57f6370c00bc4d6cbf6f7": "\\,F(-\\omega,x)", "657b63cc4b318fcc168c53f912426edf": "N - i", "657b9e68bd85e6c0fb16c85c01e626f4": "~\\Phi_8(x) = x^4 + 1", "657bc373215e5bc387fb14df1fa0ad31": "\\langle\\alpha,\\beta\\rangle=\\sum_i \\langle \\alpha_i,\\beta_i \\rangle.", "657bc46327db0d87b0f2845513198f0d": "B(\\alpha,\\beta)=\\varepsilon(\\alpha,\\beta)\\varepsilon(\\beta,\\alpha).", "657be89b174f43fc973a026589d419b9": "d\\tau", "657bfdea04818b48aab761b5a02fc391": "\\operatorname{var}(X_i)=np_i(1-p_i).\\,", "657c95fc04b7681d7bb99d33815755fb": "Q_0(f)", "657ca5e3d9da23f79d20fad03f174b3f": "x=A\\sin(at+\\delta),\\quad y=B\\sin(bt),", "657cc6781364d13001fbf31ef2c16f3a": "\\pi_{11}", "657ce372da0ad8fc6127c7240c53f1c1": "MW=24-[5 \\times (5+1)/2]=9", "657d23429fb2043eb31e67a10d0bc40b": "\n\\left[{\\begin{matrix}\n\\sigma _{xx} - \\rho_{x} f_{y} & \\sigma _{xy} & \\sigma _{xz} \\\\\n\\sigma _{xy} & \\sigma _{yy} - \\rho_{y} f_{y} & \\sigma _{yz} \\\\\n\\sigma _{xz} & \\sigma _{yz} & \\sigma _{zz} - \\rho_{z} f_{y} \\\\\n\\end{matrix}}\\right]\n", "657d2e17053824572cd14f93f49ea56e": "f_i:\\mathbb{R}^3\\to \\mathbb{R}^3.", "657d44b902eaad8dc5f11a21c6906039": "se_{\\beta}", "657d51bc105abcc7dca2234efdd3572a": "M_{\\max}=\\min(\\dim({{H_A}}),\\dim({{H_B}}))", "657dd721ca3a03e5d696437ccada1a21": "~x_i~", "657e4adad7075ebe664309beeef1bf79": "J_{i}", "657f2effed068e5a39fb393655c57612": " \\nu =0 ", "657f7e7dbea6aac87fbd4dca0c47cc3f": " y=f(a)+f'(a)(x-a).\\,", "657f83839e68e333f4bfb429b7255201": "f(x;n)=1/n I_{\\{1,2,\\ldots,n\\}}(x)", "657feb19ce8202d1910a12acd4df688d": "\\mathbf{CP}^n = \\left\\{ \\mathbf{Z} = [Z_0,Z_1,\\ldots,Z_n] \\in {\\mathbf C}^{n+1}\\setminus\\{0\\}\\, \\right\\} / \\{ \\mathbf{Z} \\sim c\\mathbf{Z}, c \\in \\mathbf{C}^* \\}.", "658053a65eeb907feb7dda27dac8eb90": "\\log_2 N", "6580b0563b78e5e3822bcf307cfacdfa": "\\mathrm{coeq}(f, g) = \\mathrm{coker}(g - f)", "6580e94f1fc6677e6460f3d81b4532aa": "\\frac{1}{\\frac{1}{N}\\sum_{i=1}^N \\frac{\\hat{c} - \\hat{a}}{Y_i - \\hat{a}}} = \\frac{\\hat{\\alpha} - 1}{\\hat{\\alpha}+\\hat{\\beta} - 1}= \\hat{H}_X", "6580f482bd5f36612d00fb8888832bd3": " \\Phi'=-\\frac{1}{\\pi}\\ln\\left(\\frac{t}{|c|}\\right)", "65813bbdc87129ca9d9ab276054fe4f0": "\\frac{g(1)}{g(0)}=\\frac{1}{2}\\,", "65813dc301661b6f02d8f69c8e8d518a": "\\kappa(A)", "658145ddf78744bc928f02945babfd9f": "C_P \\phi _P= \\left(\\dot m_w - \\frac {(\\dot m_s)^2} {\\dot m_w} \\right) \\phi_W + \\left(\\dot m_s + \\frac {(\\dot m_s) ^2} {\\dot m_w} \\right)\\phi_{SW} + 0.\\phi_S\\text{ for } 0<\\theta\\le 45", "65815cb9ae3431248525350c0c766dc3": "f(x,y) = x^2 + y^2 - 1", "6581681f11cb65b445c49703b5bb9abe": "\\boldsymbol{H}=\\begin{pmatrix} 1\\ 0\\ 0\\ 1\\ 0\\ 1\\ 1 \\\\ 0\\ 1\\ 0\\ 1\\ 1\\ 1\\ 0 \\\\ 0\\ 0\\ 1\\ 0\\ 1\\ 1\\ 1 \\end{pmatrix}", "658176535b5c4f561bfd59473b7dbca5": "\\Delta_Z", "6581fc9d0b5ba382c807edddb52fb03a": " |n_1, n_2; ?\\rang = \\mbox{constant} \\times \\bigg( |n_1\\rang |n_2\\rang + i |n_2\\rang |n_1\\rang \\bigg) ", "65821ca27bf98350051a6dce3798cf06": "\\sin(\\tfrac{2\\pi nx}{P}+\\phi_n) \\equiv \\sin(\\phi_n) \\cos(\\tfrac{2\\pi nx}{P}) + \\cos(\\phi_n) \\sin(\\tfrac{2\\pi nx}{P})", "658229005a54189de0966633cc004a47": "\\delta_\\epsilon S=\\int_{M^d} \\Omega^{(d)}(\\epsilon)", "65827cc165ec19ca5319bbb6d8b43dbb": "\\int_{-\\infty}^\\infty {e^{itx} \\over x^2+1}\\,dx", "6582d62d917ee5ecdd2bea1883482471": "\\mathcal{L}_\\mathrm{Yukawa}(\\phi,\\psi) = -g\\bar\\psi \\phi \\psi", "6582ffd981e78e22c8c56b0b1eedf8d0": "p/3", "65830f2510bd095a11f27c635db82769": "AE = C+I+G+Xn", "65836eaaff2c06d91d2ffc34b378aa1a": "\\log x", "65837e79d357c38e6aaf909556ca42d7": "|M(x)| \\le \\sqrt x.", "6583971a4135762cce30eca8253f5146": "S(p/q) = [p+q; p+2q, p+3q, p+4q, \\dots],", "65839e0aef20d9a19fef30bfda00eb28": " \\frac{\\lambda}{\\mu} + \\pi_0 \\frac{\\rho (c\\rho)^c}{(1-\\rho)^2 c!}", "65840e8ae2c23a0257d4ca331972fa6e": " \\mu^x_s (\\sigma)=\\mu^x_{comb}+\\mu_{disp}+\\int p^x (\\sigma) \\mu_sd\\sigma", "6584131dd9152ccb71317118849e1269": "= {4 \\times 80 \\times 20 \\over 80 + 20}", "65841c777f99e25d6c1e53468335a572": "=\\int_{-\\infty}^\\infty\n\\left[\\int_{-\\infty}^\\infty f(x,y)\\,dy\\right]\\,e^{-2\\pi ixk_x} dx\n", "65844eeb83d06b5cc354d5bc73847d80": "\\ \\mathbf U(\\mathbf x,t) = \\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad \\text{or}\\qquad U_J = \\delta_{Ji}x_i - X_J ", "658464da3ae8c3b7baf044eeb0bad40f": "\\scriptstyle\\tilde\\beta", "65847959e636619e604b72779ab43466": "\\left \\langle \\sum_{g \\in G} a_g g, \\sum_{g \\in G} b_g g \\right \\rangle = \\frac{1}{|G|} \\sum_{g \\in G} \\bar{a}_g b_g.", "658495ea0002393faa774ca09bf2633e": "\\hat x_d(\\omega)", "6584c1cb0f65a4e3120e2309419414b7": "u = \\frac{4x}{-2x+12y+3}, \\quad v = \\frac{6y}{-2x+12y+3}", "6584c515fed3de60e5394e2847e00e67": "\\int_\\gamma \\langle V(x), \\mathrm{d}x \\rangle = \\int_a^b \\langle V(\\gamma(t)), \\gamma'(t)\\;\\mathrm{d}t \\rangle.", "6584dab87dc2f1d9d0681fb372ca6e7e": "\\frac{2}{2k-1}\\,", "65852cea2a438f3e34972e949642c22e": "\\mathcal{SR}", "658575c7182542918b3f0acfe4ddba8b": "V_{\\bar p,\\hat\\omega}", "6585786eee774400c2f240fac452b6b2": "e = \\sum_{k=1}^\\infty \\frac{k^7}{877(k!)}", "6585920376531a612edd378fffcecae0": "r_i+1", "6585969d8c9084f8d06f7d6a5fd317b3": "\\nabla ( \\phi_1 - \\phi_0 )", "6585c3cb01176ce7f122a8dbf9b9d56c": "S_{ij}=2(r_i,r_j)", "65862ef6ee0e99100870bcb3110c032e": "i=\\langle C_{i}, R_{i}, I_{i}, A_{i}\\rangle", "65865decdbfc13aac4bf993605efb54d": "p^*\\colon A\\to P(1).", "65867d0c73dff8211c7fa429a8bc8289": "\n\\log\\left(\\frac{Z_\\eta(t)}{Z_\\mu(t)}\\right) \n= \\log\\left(\\frac{\\mu_{(m+1)}(T) + \\cdots + \\mu_{(n)}(T)}\n{\\mu_{(m+1)}(0) + \\cdots + \\mu_{(n)}(0)} \\right) \n+ \\frac{1}{2}\\int_0^T \\frac{\\mu_{(m+1)}(t)}{\\mu_{(m+1)}(t) \n+ \\cdots + \\mu_{(n)}(t)}.\n", "6586e78d7b82bff46ce4823b1a31ed40": "1 = C_{n} \\cdot \\Delta_1 \\cdot df_{1} + C_{n} \\cdot \\Delta_2 \\cdot df_{2} + C_{n} \\cdot \\Delta_3 \\cdot df_{3} + \\cdots + (1+ C_{n} \\cdot \\Delta_n ) \\cdot df_n ", "658706504a291d40da5ca84120c0a53d": "(Df)_x E^s_x = E^s_{f(x)}", "658715340533d8c7a5ce7780a87d067a": "\\omega_i = \\min(\\Delta(C_\\text{in}(y_i'), y_i), {d \\over 2}) \\le \\Delta(C_\\text{in}(y_i'), y_i) = \\Delta(c_i, y_i) = e_i", "65872ba8a42a5fb871d11253900543ad": "\\text{ER}_{g,\\beta}(X)=\\inf_{t>0,\\mu\\in\\R}\\left\\lbrace\nt\\left[\n\\mu+\\text{E}_P\\left(\ng^*\\left(\n\\frac{X}{t}-\\mu+\\beta\n\\right)\n\\right)\n\\right]\n\\right\\rbrace\\,", "658763141e31c7d192c6100aff1d19df": "-+", "658784733e802c334713b97d71d95678": "P_{m,\\mu}", "6587e73ed8a7cb721256ca82351eaa1a": "\n \\begin{align} X & \\sim \\operatorname{Bin}(n,p) \\\\\n\n \\text{then } P(X=k|p,n) & = L( k | p ) = {n\\choose k}p^k(1-p)^{n-k}\n \\end{align}\n", "6587f61ee0695ff7f10af1db4d1aaccc": "\\delta^{\\beta_k}(\\gamma_k-1) + \\delta^{\\beta_k-1}\\delta", "6588c4082483741f48d56a8d30374a62": " \\bar{\\boldsymbol\\omega} = {1\\over 2\\pi c} \\sqrt{k \\over \\mu }", "6588c95074f2609674f5fe10ab63f88f": "\\sim", "65897e0b738d8279f814159da9378b71": "a\\cap b", "65899a442e9669b9415134a570f56f43": "{C}_3", "658a0e37ccc8af805816ef3f86f87d64": "x>\\mu+s", "658b3afc123df83b73da3cdeebc56ad6": "\\det \\left(\\frac {it\\Omega}{2\\pi} +I\\right) = \\sum_k c_k(V) t^k", "658b4cdc87e95c9a6241a22f4a993b1c": " p_2=-\\omega, ", "658ba7ee15241042a0ed67c993b58f18": "\nh_{\\sigma} = h_{\\tau} = \\sqrt{\\sigma^{2} + \\tau^{2}}\n", "658bb28e7070fcab549dd76c42b4e579": " \\mathbf{x}=\\mathbf{0}\\,", "658bb5591990fa49f1b2df7e43238745": "\n \\begin{bmatrix}\n 1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 \\\\ \n 0 & 1 & 0 & -\\tfrac{25}{7} & 0 & \\tfrac{2}{7} & -\\tfrac{10}{7} & -\\tfrac{130}{7} \\\\ \n 0 & 0 & 1 & \\tfrac{1}{7} & 0 & \\tfrac{3}{7} & -\\tfrac{1}{7} & \\tfrac{15}{7} \\\\\n 0 & 0 & 0 & \\tfrac{11}{7} & 1 & -\\tfrac{2}{7} & \\tfrac{3}{7} & \\tfrac{25}{7}\n \\end{bmatrix}\n", "658c1453e5f60eebfd9ca1d1c7932cf1": "\\eta=\\frac{9\\mu}{(2\\mu+1)(\\mu+2)-2\\left(\\frac{a}{b}\\right)^{3}(\\mu-1)^2}", "658cb6153c1e3f143236df900b14836a": " \\quad (5) \\qquad \\qquad \\bar{\\rho}_{i}\\left( t_{2}\\right) =\\frac{1}{\\Delta x_i}\\int_{x_{i-\\frac{1}{2}}}^{x_{i+\\frac{1}{2}}}\\left\\{ \\rho\\left( x,t_{1}\\right) - \\int_{t_{1}}^{t_2} f_{x} \\left( x,t \\right) dt \\right\\} dx.", "658ce3c735704c458c74b3308d63551a": "R[T_h]", "658d199f2ad7d14a57bc948c92093f03": "\\{c_{n} \\}", "658da433c14606d013fd452541afe8da": "^{18}\\text{O}", "658dd3b2f64bef5b2d0a7c238330602e": "\\mathcal{N}(y)", "658de495f1e84b90bdca4d3e2f188146": " A(S_D) = \\iint_D\\left |\\vec{r}_u\\times\\vec{r}_v\\right | \\, du \\, dv. ", "658e4f1e16bf0a91320bad63a69b8b38": "a_1 \\otimes \\cdots \\otimes a_n", "658f317b8e71212de9eb768088c4b978": "\n\\sum_{n=0}^{\\infin} \\frac{f^{(n)}(0)}{n!}\\ x^{n}\n", "658f3f363e9eca830ef2d5ca2ef40feb": "u_i\\,\\!", "658f6105cf22a957355fb408cf60e5a5": "GH^\\infty", "658fc160c37901808d4aafa059beed50": "T^\\prime_\\varepsilon(D)f(w) = -{1\\over \\pi} \\iint_{D\\backslash V_\\varepsilon} \\frac{f(z)}{(z-w)^2} dxdy,", "658ffaf828031d207ad32c48b2971b0c": "\n[b] =[a]\\cdot [x_3] \n", "65902732dddadfb6dc7097098aed5ee7": "f(x;\\lambda) = \\mathrm \\lambda e^{-\\lambda x} H(x)", "659076d8f3d0aa0bc73ee8365bb9f506": "|3\\rangle", "659089701398ac4248118cfb90492f32": "S_z |s,m\\rangle = \\hbar m |s,m\\rangle.", "65909f07cf79f4c61446c0f2be5110c6": "\\Omega(n) := \\sum a_i \\qquad\\mbox{if}\\qquad n = \\prod p_i^{a_i}.", "6590cf02bcfd453be9ab4bdac9e2bd4c": "n\\,\\log_{10}\\varphi\\approx0.2090\\,n", "6590e45b0fb67a429f76754eec8258ae": "\\Omega(Sp/U) \\simeq U/O", "6590e6d1fe3b76f4ea2dffed31c42a3d": "\\Delta y = h/n", "6590fcdd91e95d593f60727115162467": "\\hat H = \\hbar \\omega \\left( a^\\dagger \\, a + \\frac{1}{2}\\right).", "65915bab400e92a9a5ddbff82f0b5b9c": "\\sqrt{45\\over 224}", "6591a5b3a714df11bc50d41f56d30f82": "c_i = \\left [ {\\rm X}_i \\right ] = \\frac{\\mathrm{d} n_i}{\\mathrm{d} V} ", "6591fd218734b9ff79fcbc0baa15bade": " 180^\\circ ", "65923bdde28d8ef307b42941e34fb0ae": "\\sigma=\\frac{\\sqrt{6}}{6}", "65924dd185b600aaed0ee8b313a9883a": "y = aP + u.", "65927757ff8edba55d93a5081c745be9": "L_q\\left[1/3,c\\right]", "65929d14f296d59af470885d546c6743": "dz(t)=\\varepsilon_t \\sqrt{dt}", "6592e012e36a21930ae070538dcb508f": " |\\mathbf{F}| = \\sqrt{(B_x-A_x)^2+(B_y-A_y)^2+(B_z-A_z)^2}. ", "6592e16f120c848fa51288f8a00a4986": "| \\triangle SCB|=| \\triangle SDA|", "6592ef7aa594186b8ab5f33f38bd7ada": " \\left( f_1 \\Box \\cdots \\Box f_m \\right)^\\star = f_1^\\star + \\cdots + f_m^\\star. ", "6592fd0549eb34d05a10bfcf6f8e30f0": "p_{c,t}\\,", "6593b9861598a160f9182ba6da425e10": "\\operatorname{P}(X=0)^2=e^{-2\\lambda}.\\quad", "6593cbf2fd699168703106417928d721": " A + H_2O \\rightarrow \\ B ", "6593eac1646cf69c9599df5733f6235d": "\n\\hat{A}\\hat{C} = [\\hat{C}^-]\\hat{A} = \\begin{bmatrix} C^- & 0 \\\\ D^- & C^- \\end{bmatrix}\\begin{Bmatrix} A \\\\ B\\end{Bmatrix}.\n", "6593f09ebc020af18235935eee55a69e": "U^a = \\frac{dX^a}{d\\tau} = \\gamma \\left[c, \\frac{dx}{dt}, \\frac{dy}{dt}, \\frac{dz}{dt}\\right]", "65948aab930872be27672fff8d54ee62": "\\ y", "6595115e36931f418d32c0ae88bfac12": "c_w = \\nu \\lambda ", "659524d0d329d5a6ce365ab2fdcbe9da": " \\mbox{SL}(n,\\mathbb{R}) \\ltimes \\mathbb{R}^n.", "6595a638e0820d6e517324521a2cf8e0": "\\phi=\\rho_A-\\rho_B", "6595cb3aa03f068f8884c1e7ce3fb898": "B^\\prime .", "6595d679e306a127a3fe53268bcaddb2": "n^2", "65962ff1e40fd82be7657cf2d936f701": "V=\\frac{1}{3}(5+4\\sqrt{5})a^3\\approx4.64809...a^3", "65963b56bf43602a0389b6d1ad1f078d": "\np(x)= \\alpha k^{\\alpha} x^{- \\alpha -1},\\ \\alpha ,k>0,\\ x \\ge k\n", "6596fb998b6dd0d69d9ca24d7bfa90cd": " \\sigma_{min} \\,\\!", "65970831ae9c947b1f6429dc32f80ba1": " Q = \\frac{P}{\\eta} \\cdot \\frac{1}{LHV_{gas}} ", "65971b022bb42872ba7f20e8892f89d6": "(I - A_i)S_i", "6597420a4fafed71741087c8ee1eedc7": "A_1;\\dots;A_k \\leftarrow B_{1},\\dots,B_{m},\\hbox{not } C_{1},\\dots,\\hbox{not } C_{n}", "65974921ce363b46a22f0dd3e1cb3568": "2^\\Omega", "6597675521321c773af0ebf8c7dadb83": "\\mathbf{x}_{ij}", "6597d279ace31039abf0c0cae327c74a": "\\frac{T_\\nu}{T_\\gamma} = \\left(\\frac{4}{11}\\right)^{1/3}", "659866a526697fe0b6ce186eb7f9e6b0": "|x_m - x_n|", "6599514e61725fbf478d4d89fceeb42c": "\\displaystyle \\delta_s\\delta_t=\\delta_{st}", "6599645726ddb1133fc08df352ce7bbe": " b_2 - b_1 \\ge 1 ", "6599769e667f3a952c75a78ad3632599": "= \\frac{1}{2}\\left(1 - \\frac{1}{2} + \\frac{1}{3} + \\cdots\\right) = \\frac{1}{2} \\ln(2)", "65998170db033969dc71c38dc6b37df2": "\\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \\psi_{l}).", "65999972a0f75d548ab1590ae3207b15": "A_{mn}", "6599b1e79a465cb8bf80bb58fa6c1bca": "\nf(a) = \\frac{1}{2\\pi i} \\oint_C \\frac{f(z)}{z-a} dz,\n", "6599ecb968c79b795b1d9bc590e1c1a1": "a_k=b_k", "6599ffd0fe7ab34a8c10d14b6d27d563": "\\sigma_L^P", "659a4837c2939770ff8617a9adaeae26": "\\int a^{cx}\\;\\mathrm{d}x = \\frac{1}{c\\cdot \\ln a} a^{cx}", "659a5f7cd985e5478aec24b6df20ebbc": "<\\phi , \\psi>_1", "659a9f4bf001eb24b7f99841ac59807f": " a = y - \\hat{y} ", "659acfd21bbc650408362aa927952e96": "\\sum_x \\tan ax = i x-\\frac1a \\psi _{e^{2 i a}}\\left(x-\\frac{\\pi }{2 a}\\right) + C \\,,\\,\\,a\\ne \\frac{n\\pi}2", "659b06b05fc82cd756485c10ad620419": " {d \\mathbf p_1 \\over dt} = -\\nabla_1 H ", "659b1d12101edd8ead8a8c3a7da6e30a": "(\\infty,a)", "659b28d352b60e54d1f2a5e48859eeb3": "CIRC", "659b80ffaff9bdea39637d42cbde08b0": " \\frac{1}{c} | \\vec x - \\vec{x_0}\\, | \\quad \\hbox{is stationary for} \\quad \\vec{x_0} \\in S. \\,", "659b8cc13d093fbc43cee8efd1b923bc": "P= \\rho v^2", "659bbc74c82bcfa7185faa43d08c20eb": " h A {(T_{\\infty} - T_{0} )}+\\epsilon \\sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 ", "659bbe9d23f47e445c6aed7f5b2c0e78": "p^{ij} \\,\\! ", "659bdb088b1468f066ca572b8cd90ec6": "m=-\\ell,-\\ell+1,\\dots,\\ell", "659bf28c23fbed5f737724938ce6fab1": "V(q) = \\frac{q^3 + q - q^{3-n} + q^{-n}}{q+1}.", "659bf561da3e96f297e3bd10d0f1003b": "\\exists x\\, F(x)", "659c3493e62bb03656ca900104e39e17": "u = \\Phi_A(\\mathrm{id}_A).\\,", "659c998d6b9097f58297f4c09578f6d8": "v \\to (-1)^{l(v)}", "659ceb8dd6e5633169123567613aee94": "\\scriptstyle -\\hat v \\hat x \\hat v", "659d213dfb9c3b1f86b7df28b36264d1": "MF = \\frac {1}{\\left [ 1- \\left ( \\frac{f_d}{f_n} \\right ) ^2 \\right ]^2 + \\left [ 2 \\zeta \\left ( \\frac {f_d}{f_n} \\right )^2 \\right ]^{\\frac {1}{2}} }", "659d23f0ed16cdb87b1d41c7b58b52f4": "\\Delta ", "659d5d51eee37905268c9da792378566": "\\mathrm{Nu} =\\frac{hd}{k}", "659d8420ebcd2f432bf7205c41aa390a": "e_1,e_2,e_3", "659d86c9871d8ab7f94ac16c67ab2270": "C_2 = \\begin{bmatrix}0 & 0 & 0 & \\cdots & 0 & 1 \\\\\n\\end{bmatrix}", "659da5da0ffd8b3cc08f2ad667668ac8": "\\lambda_a\\,", "659dbe9d318decbd8fca1a2fd7c97e2e": "u_1 \\in \\phi^{-1}([0,1]),", "659dc11b13139a92506aaea554787911": "p_{n,t}", "659e660fd9b9729d3b19a56eb41de76d": "\\mathbf{H}^n \\cong \\mathbf{R}^{4n}", "659e8bf057c793a896b75e2e39296a8b": "W_f", "659ea2f159c904d7f0f53a0732ff5345": "\\lambda_0(t)", "659ec2e8349b07d3887be887d068577f": "\\mathbb P(V) ", "659ec5fe078e46486739bc56eab6f4b3": "k = \\frac{1}{d}\\frac{\\partial}{\\partial T}\\int E f(E)\\, g(E)\\, \\nu(E)\\, \\Lambda(E)\\,dE", "659ed556f8ecdb5edf46fb62dc05d0fd": "\\tilde{g}_{22}", "659f2e8be535f67be9ef4fb70e5c2368": "I_{t} = I_{0} + I(r) + b (C_{t} - C_{t-1})", "659f5c827ee99bf2cf8e8de5496f764c": " g_0\\ll 1 ", "659f740f24ea26a4bf03291be722e110": "D_B={1\\over 8}(\\boldsymbol\\mu_1-\\boldsymbol\\mu_2)^T \\boldsymbol\\Sigma^{-1}(\\boldsymbol\\mu_1-\\boldsymbol\\mu_2)+{1\\over 2}\\ln \\,\\left({\\det \\boldsymbol\\Sigma \\over \\sqrt{\\det \\boldsymbol\\Sigma_1 \\, \\det \\boldsymbol\\Sigma_2} }\\right)", "659fde0cd780516bb5cd3214da48f39e": " {1 \\over f} = {1 \\over f_1} + {1 \\over S_1} \\,,", "65a005dea9dfa6f52954c4a88390a367": "\\widehat{\\beta}_{RE}-\\widehat{\\beta}_{FE}", "65a00e8887ae58c4936938cea58c7f14": "\\scriptstyle \\zeta", "65a014c9f4d44886453754ca571d64b2": "\\langle f, g \\rangle := \\int_\\Omega f(x) g(x)\\ w(x)\\ dx.", "65a07732917bc176a0bfb44f0cd6acd4": "\\det: \\mathrm{GL}_n \\rightarrow \\mathbb G_m.\\,", "65a07f497062d3ea30a833e3e13b62aa": "\\overline{d}_{\\dot{\\alpha}}\\Lambda=0", "65a0aa5fd6ef523944d7869667728c24": "\\Phi_n=\\cos(n\\varphi)\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,\\sin(n\\varphi)", "65a1060cb8eb8d3919dde2de1d0cea38": "\\rho(G)(L) = L.", "65a10ad1232d4cdd91f128b655f2ba57": "\\begin{align}\n\\text{Volume} &= \\iiint_D f(x,y,z) \\, dx\\, dy\\, dz \\\\\n&= \\iiint_D 1 \\, dV \\\\\n&= \\iiint_S \\rho^2 \\sin \\phi \\, d\\rho\\, d\\theta\\, d\\phi \\\\\n&= \\int_0^{2 \\pi }\\, d \\theta \\int_0^{ \\pi } \\sin \\phi\\, d \\phi \\int_0^R \\rho^2\\, d \\rho \\\\\n&= 2 \\pi \\int_0^{ \\pi } \\sin \\phi\\, d \\phi \\int_0^R \\rho^2\\, d \\rho \\\\\n&= 2 \\pi \\int_0^{ \\pi } \\sin \\phi \\frac{R^3}{3 }\\, d \\phi \\\\\n&= \\frac{2}{3 } \\pi R^3 [- \\cos \\phi]_0^{ \\pi } = \\frac{4}{3 } \\pi R^3.\n\\end{align}", "65a139fa6365c83c5796a08709a5dce4": " |\\langle\\mathbf{u},\\mathbf{v}\\rangle| = \\cos(\\theta)\\ \\|\\mathbf{u}\\|\\ \\|\\mathbf{v}\\|.", "65a19cc1d383371ee9a6eb47b0261b97": " \\mathbb{P} ", "65a19d650423ab0779b8cddb0d3336f2": "\\begin{bmatrix}\n0&0&0&0&0&0&0&1 \\\\\n1&0&0&0&0&0&0&0 \\\\\n0&1&0&0&0&0&0&0 \\\\\n0&0&1&0&0&0&0&0 \\\\\n0&0&0&1&0&0&0&0 \\\\\n0&0&0&0&1&0&0&0 \\\\\n0&0&0&0&0&1&0&0 \\\\\n0&0&0&0&0&0&1&0\\end{bmatrix}", "65a1a8534f268e890568e4f565913ba6": "1, 2, 3, \\ldots, p-1. \\quad\\quad\\quad (B) ", "65a1b30325f249b32f32992e77fd4e8b": "p_i = P_{\\partial /\\partial q^i}", "65a1b5d5ff88dc8afd967b883d0ae10e": "\\boldsymbol{\\omega} = \\nabla \\times \\boldsymbol{v}.", "65a1e0eff9c44d44870675dee996be71": " \\lambda = L + 1.915^\\circ \\sin g + 0.020^\\circ \\sin 2g", "65a286b5807a8484a2f7710153218bd3": "E(XYZ)=\\kappa(X,Y,Z)+\\kappa(X,Y)\\kappa(Z)+\\kappa(X,Z)\\kappa(Y)\n+\\kappa(Y,Z)\\kappa(X)+\\kappa(X)\\kappa(Y)\\kappa(Z).\\,", "65a2a3a26e703ddeb85fa7f5dd68365f": "\\displaystyle a\\cdot \\hat{f}(\\omega) + b\\cdot \\hat{g}(\\omega)\\,", "65a2ce704b4fd7e4360e26ae03e15126": "(p_{01}:p_{02}:p_{03}:p_{23}:p_{31}:p_{12}) \\,\\!", "65a304fd0af2bafe708d0b00e7ec3926": "\\forall p: \\mathcal{B}p \\to \\neg\\mathcal{B}\\neg p", "65a34f1c9ae79f359db439895b0a086d": "~h", "65a3a4f8275a0df5223ab6e2b3b17176": "y' = y.\\,", "65a3ebbf53bcd0709478d2fc1d1098c9": "\n\\lim_{y\\to0} \\left( \\lim_{x\\to0} \\frac{xy}{x^2+y^2} \\right) = \\lim_{y\\to0} 0 = 0,\n", "65a4213ef1ed18797a622629a52e6835": "\n \\mathbf{e}_i = \\mathbf{b}_j~\\cfrac{\\partial q^j}{\\partial x_i}\n ", "65a44e4bac3b94cdb0afb4bd4d370bd9": "b_1=b_2=p \\in [c,v].", "65a45a75008dbd38d78adeb37460fa9d": "\\operatorname{std}(f) = 2\\pi^{-n(n+1)/4}\\left(\\prod_{j=1}^n\\Gamma(j/2)\\right) \\zeta(2)\\zeta(4)\\cdots \\zeta(n-2)\\zeta_D(n/2)", "65a4b55f490a5ebe2fd5c7ba26bb49bf": "E[n]", "65a5183862a58b4190eb5967c26c7985": "\\lnot \\exists x \\, P(x) \\Leftrightarrow \\forall x \\, \\lnot P(x)", "65a532828c157a7743927672dcda515b": "\n(6.2)\\quad\nd_W(\\mathcal{L}(W),N(0,1)) \\leq 2\\sum_{i=1}^n \\left(\n \\frac{1}{2}E|X_i X_{A_i}^2|\n+ E|X_i X_{A_i}X_{B_i \\setminus A_i}|\n+ E|X_i X_{A_i}| E|X_{B_i}|\n\\right).\n", "65a5411f46b18306d57045afd81ff428": "\\forall x,y\\in A\\qquad \\|xy\\|\\le\\|x\\|\\|y\\|", "65a601e0d6943f2ee09042ac70807823": "(k-2)", "65a69cd35d16700d9e2c8cdb4d3fb9c9": "\\Delta \\alpha=\\frac{1}{k}\\frac{\\partial \\phi(x)}{\\partial x}", "65a7bf78a912cd620ac81234792795f5": "\\tfrac{dS}{dT} = \\delta MS - \\beta SI - \\mu S", "65a7e6a4b506d4d25dd85010e094c64d": "\\frac{e^{ut}}{2\\pi i}", "65a7faea4366396bb09cf6bc6d86b79f": " Proj \\oplus_n f_*\\omega_Y^{\\otimes n} \\to X", "65a850fa6d04cea8c3fb639a3a189120": "S_1=S_2", "65a867ae67a4aedf412b2b4e910d6e51": "b_i = 0", "65a8e3e803a4deb21e3633b5ea729266": "b_i,", "65a8f9446dd0e3c5986d238968bc46bc": "\\,\\varphi", "65a9023e705200dab172df817318b138": "O(\\log .\\max \\{K(x\\mid y), K(y\\mid x)\\} )", "65a929ac99220a087dde0eb4c84da51d": "\\{\\neg, \\land, \\lor, \\to, \\leftrightarrow\\}", "65a93f41b75b1df05753cf44ff0e3e8e": "R = \\frac {\\overline{x} S} {s \\sigma} - \\frac{\\overline{X}- \\mu} {\\sigma}\n = \\frac {\\overline{x}} {s} \\frac {\\sqrt{U}} {\\sqrt{n}} ~-~ \\frac {Z} {\\sqrt{n}} ,", "65a942a083a6841c3908c5823f01d53d": " \\vec{\\psi}_{P}=\\vec{a}_{P}-\\vec{\\omega}\\times\\vec{v}_{P} ", "65a9c20cdd4a70d8da09c7aa14d23cc0": "A_{\\alpha}{}^{\\beta}{}_{\\gamma}{}^{\\delta\\cdots}", "65aa43704cbb21cd2b72f3e2558961fc": "\\lambda([0,1]\\times [0, 1]\\times \\cdots \\times [0, 1])=1.", "65aaddeaeb0ca01e2c17d248ae04a7f1": "\\bigl[ \\begin{smallmatrix} 1 \\\\ 1 \\end{smallmatrix} \\bigr]", "65aaf2ffd18bb01089a0c0b3f8762370": "E=\\frac{mc^{2}}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "65ab22ae36392e0a8c1f370e745b4f5d": "\\alpha_k \\le \\frac{7k-34}{7k}\\quad(k\\ge 58)\\ .", "65ab26f6149ff87c93ecb8b6444e420f": "\\beta_{h,v}", "65ab38a93d03ebe81ee4fc9e171a4912": "~V(x)~", "65ab4cb19f33d893525ebcfbdcaf1d0b": "k -1", "65ab4da84b0cee44ab2ad4a2c96e722d": " f(x_1 x_2)=f(x_1) f(x_2) ", "65ab9501357ae8fab349da8ea086f566": "1\\over {\\sqrt 2}", "65abcb8f65126800901e79b2dc367fd3": "[t_l, t_u] \\subset \\mathbb{T} ", "65abff736425bf1fba594e2950cd233d": "t \\geq T", "65ac39f0e3c543e97b4e69d39788803e": "y_{n+s-1}, \\ldots, y_n", "65ac7a1b4449a2a67b37260ffab212af": "\\mathcal{O}(p^2)", "65ac84e537f56fa264d63244191805de": "\\mathbf{X_0} = \\frac {\\mathbf{V_1}} {\\mathbf{I_m}} ", "65ac8e662ba88fd1c75bbd8691089f8d": "K=CH^{\\mathrm{T}}\\left( HCH^{\\mathrm{T}}+R\\right) ^{-1}", "65acb931b011d8d2aaba2d3b0f879aa9": "a_n=\\textstyle\\sum \\varepsilon_i \\varepsilon_{i+1}", "65ace44084b30941148f92b6747932d6": "\\approx 147.273", "65acff6f8bea25d2fd70bca32e885d6b": "\\alpha_j \\leq \\beta_j \\leq \\alpha_{n-m+j}.", "65ad415f0c49571155f82c2937288355": " \\nabla \\cdot \\mathbf{E} = \\frac{\\rho}{\\varepsilon_0} ", "65ad43e437e8cb649b04bef821077158": "\\operatorname{diag}(\\xi)", "65ad6ceda437106dfffcde417c9e3b9c": "XOR(\\alpha,\\alpha')=s^TX(u\\otimes v)=\\alpha+\\alpha'-2\\alpha\\alpha'", "65ad91467a98e9368bf69c204ea1fe3f": " n = max(child_1, child_2), if child_1 \\ne child_2 ", "65adb142d1cca8be371beea2ccd138ec": "(x+1)^2-(x-1)^2", "65adc20b0dc0257d2d9f11aebedaf1e7": "\\mathbf S: \\begin{cases}\nX^2-TZ=0\\\\\nY^2-aXZ-bZ^2-TX=0\n\\end{cases}", "65adc4b897aa60664021c6bada94a071": "p>1", "65adf01008c49e3ecb5944ef266d2cd7": "\\Pi_{Age,Weight}(Person)", "65ae0b5445dec5967f12b38f92cda381": "\\scriptstyle{\\sqrt{122+71\\sqrt{2}}}", "65ae220d4bb23712254e05e9755f3579": "\\sigma_\\varphi(R) = \\{ \\ t : t \\in R, \\ \\varphi(t) \\ \\}", "65ae2926bcd2e7c6231e391a3be377aa": " t_i \\cdot \\pi \\cdot \\frac{K_{i}}{K}", "65ae37f9aa3b634af2a7a49c0be69167": "\n\\begin{align}\nV_{\\text{r}}' &= V_{\\text{r}}\\cos(\\omega_\\mathrm{m} t+\\varphi) \\propto P_{\\text{r}}\\cos(\\omega_\\mathrm{m} t + \\varphi). \\\\\n\\end{align}\n", "65ae5a480e20e60118de8ef59a14cba9": "\n\\langle E (\\cup _i B_i) x, y \\rangle = \\sum_i \\langle E (B_i) x, y \\rangle \n", "65ae78cb355a6d844dbbf1489e8bdb97": " e_y\\,", "65ae8ec31c17a7eb781f510de2e3ad17": "(A,B)\\mapsto \\frac{1}{p-1}({\\rm Tr}(B^{1-p}A^p)-{\\rm Tr}\\, A)", "65aecdf4a20432fcac1c0f69ec476b2c": "\n P_q = \\int_0^a\\left[\\left(\\int_{-b/2}^{b/2}q(x,y)\\,\\text{d}y\\right)w_x + \\left(\\int_{-b/2}^{b/2}yq(x,y)\\,\\text{d}y\\right)\\theta_x\\right]\\,dx \\,,\n", "65af4ddf59582423860ac110232e1335": "f(\\vec{x}) = [a_1 \\dots a_n] \\begin{bmatrix}x_1\\\\ \\vdots\\\\ x_n\\end{bmatrix}.", "65af64f73ef5c697b618d845f5448d74": "f\\circ s \\, = \\, g\\circ t", "65afb5e720b96cfd5646e3781c3f31c5": "p = p_0 + \\frac{m.g}{A} ", "65afe385f84332a95296a291ca53d03f": "\\psi:\\mathcal{X}\\times\\Theta\\rightarrow\\mathbb{R}^r", "65b03b3851926822f03ebead8650e0bf": "\\omega_p^{-1}", "65b0498db11b1284dbcba998eccf3a26": "\\eta_{max} = \\frac{1 - \\sin{\\phi}}{1 + \\sin{\\phi}} ", "65b0a9b64036fb4c3991025fdf9c1d8a": "i': \\widehat{X} \\to X, i: \\widehat{S} \\to S", "65b0f89ca7b9a4e63a392148ed0991ae": "Q=\\mathbb R\\times M", "65b10d7de5785e067786bbed47d36165": "\\deg(\\gcd(p,q)) = m+n-\\mathrm{rank}~S_{p,q}", "65b118a3b9a82c0914caeb780d662893": "Y = \\left(\\frac{X}{\\lambda}\\right)^k", "65b11fdce1a0ce0436c4942a73066efd": "x=0,~~ \\frac{\\lambda}{4},~~\\frac{\\lambda}{2},~~ \\frac{3\\lambda}{4},~ \\dots", "65b142365df4902ff3d3b5388a8fa233": "\\| s \\|_{\\infty} = \\sup _n |s_n| .", "65b155acbdffcaba7fd89f2f8c463b79": "\\tfrac{1}{q}", "65b166c26ac0ffef8ef7927013490551": " \\begin{align}\n\\mathbf{v}_1 & = [ (v_1)^1, (v_1)^2, \\cdots (v_1)^n ] \\\\\n\\mathbf{v}_2 & = [ (v_2)^1, (v_2)^2, \\cdots (v_2)^n ] \\\\\n\\vdots \\\\\n\\mathbf{v}_n & = [ (v_n)^1, (v_n)^2, \\cdots (v_n)^n ] \\\\\n\\end{align}", "65b1869476818e000a796bddc1cb88ad": "e=0\\,\\!", "65b1b5bbfb6ebb856c8b898af8fc277a": "\\delta > 0", "65b1bd30e5abf122d37a67723a148531": "\\boldsymbol{F}=m \\boldsymbol{a},", "65b1be59be07ad372610bfc8957dcbc2": "R=\\left\\{+1, -1, -2\\right\\}.\\,\\!", "65b2601da86dcc3cbf1026b1ef92779f": "l_{\\rm P}", "65b26fcfad1043b276eaba2253a52bfa": "\\displaystyle{(0)\\rightarrow H^k_0(\\Omega) \\rightarrow H^k(\\Omega) \\rightarrow \\bigoplus_{j=1}^k H^{j-/2}(\\partial\\Omega) \\rightarrow (0).}", "65b2b9f90fb4bf08f716024e7dd4b094": "h(t)= (\\pi(t), H_\\alpha)", "65b2c5b4ff18cc9a8920bb23b56cbed4": "p(\\ell) = \\frac{P(\\ell)}{\\sum_{\\ell=l_\\min}^N P(\\ell)}", "65b32a0244abc7106ece1b3d03946e81": "r = \\tfrac{R}{R+G+B}", "65b3a8a64e332107bd4c3248f0af1e80": "\\rho_o - \\rho = \\beta \\rho(T - T_o)", "65b3d8f62eced8ef742b77b6601c61ae": "{\\rm ci}(x) = -\\int_x^\\infty\\frac{\\cos t}{t}\\,dt", "65b3e7317671321fee87fb3ac50d9516": "{}^2\\!A_n", "65b405f8154c5e93c92b071b823b763d": "F_{P}=\\frac{3}{4\\pi^2}\\left(\\frac{\\lambda_{c}}{n}\\right)^3\\left(\\frac{Q}{V}\\right)\\,,", "65b41affda1d713e8518312582554193": "\\frac{9}{25}+\\frac{16}{25}=1", "65b4806170e1216b06cb900a546d84e3": "R E R.", "65b4859168fd47b0e0e2d0225565d68a": " \\begin{align}\\mathcal Z & = \\exp(0(\\mu - 0)/k_B T) + \\exp(1(\\mu - \\epsilon)/k_B T) \\\\ & = 1 + \\exp((\\mu - \\epsilon)/k_B T)\\end{align}", "65b4a3e475d7c218a7ac9431b108989c": "\\bold{\\dot{r}}(\\theta(t),\\dot{\\theta}(t))=( \\ell\\, \\dot{\\theta}\\cos\\theta, \\ell\\,\\dot{\\theta}\\sin \\theta)", "65b4b6e2ed053f94c15e62e8c102e5aa": "E\\supseteq L", "65b50c31a219975a1c362e78abc76e74": " EV = e(p_0,u_1) -w ", "65b540ad71e2ef86641f8672cd429947": "\\cong \\mathfrak{m}_{f^{-1}P}/(\\mathfrak{m}_{f^{-1}P}^2+I)", "65b5678f1fa73cddbfe6982c79048c0a": "1\\ang 90", "65b581a525a04c5128df9a82a3a1e5a4": "U=\\sum_i\\sum_j r_i W_{ij} r_j\\,", "65b5bbcf173514df876d083d70161e83": "A^{?}", "65b5de50969ae17ee2361c97d985b560": " Z(\\beta)={\\rm Tr} V^L= \\lambda_1^L + \\lambda_2^L= \\lambda_1^L\\left[1+ \\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^L\\right]", "65b66ea3ddb6976ae778ac3487a87a8c": "v_i = l_W l_i", "65b6b953f722100674d490722a59910c": "-\\pi/2", "65b6fac9e6a7913b08ebc8818c53ec3a": "\\int_{0}^{\\infty }\\frac{\\cos ax}{\\cosh bx}\\ dx=\\frac {\\pi}{2b}\\frac{1}{\\cosh \\frac{a \\pi}{2b}}", "65b7190ebf53f26ba05e88d024a0c8ba": "x(j)", "65b734ef5559e5c32879271b5e474bb8": "(C_\\alpha,C_\\beta)", "65b74cf6446fb7ad4aa11ec6f2ab90a6": "\\vec B(\\mathbf r)", "65b7766cb99d438753be52adcb6d6480": " { (MX)^2 \\over (MY)^2} = {(PM)^2 - (MX)^2 \\over (PM)^2 - (MY)^2}. ", "65b7c346227ab489ac0bc80e7f304565": " \\sum_{i=1}^n \\left(x_i \\sum_{j=1}^{n_i} \\widehat\\varepsilon_{ij} \\right) = 0. \\,", "65b7f847f703ceef47a6a5deee5f8559": "\\Pi(z) = \\Gamma(z+1) = z \\Gamma(z) = \\int_0^\\infty e^{-t} t^z\\,{\\rm d}t,", "65b8361c48755b873383fc9cc2a68ad8": "\nP(Q) \\, dQ = \\frac{e^{-Q/2}}{(2\\pi)^{k/2}} \\int_\\mathcal{V} dx_1\\,dx_2\\cdots dx_k\n", "65b8a53c2d6a3e5ce9616a6ba35a7a2c": "\\left. \\frac{d}{dt} \\left( \\psi\\circ f \\right) (t) \\right|_{t=0}= \\sum_{i=1}^n\\left.\\frac{d}{dt}(x^i\\circ f)(t)\\right|_{t=0}\\ (D_i\\psi)\\circ f(0)", "65b8b598a3fb1fdf8a8c0d727bdbd66b": "\\textstyle (\\Omega, \\mathcal F, \\mathbb P)", "65b8d03be691d542c932f61c5785a762": "(1, k)", "65b94fab1924320c4285e3ad23c1ca17": " :v(z)^2:= \\sum_{n<0} v_n z^{-n-1} v(z) + v(z)\\sum_{n\\ge 0} v_n z^{-n-1}.", "65b96f6cda082e2061777a6bcf5ca9f6": "EAS ={a_0} M \\sqrt{P\\over P_0}", "65b9b4f96a75764727df9c35d1f899bf": "\\begin{bmatrix}A & B/2\\\\B/2 & C\\end{bmatrix} ", "65b9bbdaeec375ca1a96b9150979ad15": "\\alpha : [0, a) \\rightarrow [0, \\infty)", "65b9f78223a6b2fa4556f967911ee39a": " \\underset{\\boldsymbol{\\beta}_{*} \\in \\mathbb{R}^{p}}{\\text{min}} \\; \\|\\mathbf{Y} - \\mathbf{X}\\boldsymbol{\\beta}_*\\|^2 ", "65ba00f311c85bb4424de3dbb8fa15f0": "\\hat{x}_{k+1}", "65ba15d9408d52f445fa01ea495de877": "\\delta:{\\mathcal{X}}\\rightarrow {\\mathcal{A}}", "65ba293a25fe6df34e9863f428e9c897": "r = \\cos(3\\varphi)\\,", "65ba374e15dd8143f9051598a81282f0": "\\mathcal{H}=\\mathcal{H}_0\\oplus\\mathcal{H}_1", "65ba3d60d28353bb9d59f06de68d936b": " \\log \\frac{1}{1-z} = \\sum_{k\\ge 1} \\frac{z^k}{k}", "65ba4b57aa8aa6def49c4bd3a657b64c": " c \\geq 24 ]", "65bab95c44e14a05f5bd98eceb46a166": "3.141024 < \\pi < 3.142074", "65badc6ae384e819ef83f56b9bc9ac84": "\\frac{\\mbox{Net Sales}}{\\mbox{Total Assets}}", "65bae11bfed14ad3eb5ed6ed4b4a8213": "\\mathrm{res}(X,Y) = \\det{\\begin{pmatrix} a_{00} & a_{01} \\\\ a_{10} & a_{11} \\end{pmatrix}}^{\\deg P} \\cdot \\mathrm{res}(P,Q)", "65bb4ab6e611a3ca435a301e503a0b1a": "3 + {5 \\over 2} + {7 \\over 4} + {9 \\over 8} + {11 \\over 16} + \\cdots=\\sum_{n=0}^\\infty{(3+2n) \\over 2^n}.", "65bb69ae174f449f58b5b60de94d137e": "C=\\pi d.", "65bbcfa2035e20e3775be7de9592420b": "f(i) = 555 + 74(i-1)", "65bbe2c1f8110dc2ec28aa14d22eac1f": "\\scriptstyle V_{\\rm b} - V_{\\rm c} = E(T_{\\rm h}) - E(T_{\\rm c})", "65bbed33fdf300ece5252e48a02de2a4": " \\bar{N} ", "65bc23e778424a5f38d814d8f4f73670": "\\lambda_i + \\lambda_{p+q - i + 1} = a+b, \\quad {i = 1, \\dots, q}.", "65bcd21a963a0c630215c4429816a25a": "\\mu^*=\\frac{\\mu_n\\mu_p(n-p)}{p\\mu_p+n\\mu_n}", "65bce46c955a02b743766c0762c32a76": " \\frac{d^2}{d t^2}N_1 = c_2 c_1 N_1 ", "65bd28b77ea00fa020e99f82671bc2a5": " \\operatorname{Var}(X) =\\operatorname{E}(X^2) - (\\operatorname{E}(X))^2. ", "65bd2a17ba305a8e4bbf2c2354f3ac82": "H_{3x}=\\frac{1}{3}\\left(H_{x}+H_{x-\\frac{1}{3}}+H_{x-\\frac{2}{3}}\\right)+\\ln{3},", "65bd9c10c87dc38fc33807c65fa773d7": "\\frac{CGER - 0} {100 - 0}", "65bdf7db7e9c4a973f3ed98438d83aa1": "S_{22}\\,", "65be086577c8af0f7c75c82fe242bab6": "\\scriptstyle{^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B}}", "65be54808667c3b5f66bdf5d3d85bee7": "\nH(j\\omega) = {1 \\over 1+j{\\omega \\over {{\\omega_\\mathrm{c}}}}}.\n", "65be65608a476bcc6bcc136c3629d878": "S(R_j) = - R_j\\,", "65beb1d1e49d8c09623457481bad3a25": "\\cos x = \\frac{t^2 - 1}{t^2 + 1},\\quad \\sin x = \\frac{2 t}{t^2 + 1}.", "65beb89c159fa7013fe1ac4374e8438b": "\\int_{0}^{T} \\tfrac{2A}{r^2}dt = \\int_{0}^{2\\pi} d\\theta = \\mathrm{constant}.", "65bf5b3504986352289a0707c14d0b57": "f(x \\to y) = f(x) \\to f(y),", "65bf5c2296f176a14401afbe1601854e": "D = \\sum_{P \\in C}{c_P [P]}", "65bf6f33310d7ab58212bdeaac4d780f": "a^{2^0}", "65bf711357fe523c3a367c3f6c3f8f7d": " E\\Psi = -\\frac{\\hbar^2}{2m}\\frac{d^2}{d x^2}\\Psi + V\\Psi ", "65bf83d38011471fdc432a4304827f18": "\\theta_{min}", "65bfd47d9c5b815a782faaaec44900f4": "\\frac{2\\pi}{\\alpha} \\,", "65bfeeab174c6358c4da45671b63fe44": " \\|\\varphi\\| = \\sup_{f} \\|\\pi_f(\\varphi)\\| ", "65bff8c6d44c576e63de9baf14251d91": "Nu = \\frac{(f/8)(Re - 1000)Pr}{1+12.7(f/8)^{0.5}(Pr^{\\frac{2}{3}}-1)}", "65c0262ef255187c6e22b34674f9accb": "\\Delta S = \\int \\frac{{\\rm d}Q}{T}.", "65c04441a47525d80af1d41e8c8050dd": " \\mathbb{C}^n \\otimes \\mathbb{C}^m ", "65c08101968ca6158de15f9c50828cd3": "\\textstyle \\pi(H_0) = \\pi(H_1) = 0.5", "65c0972d9b5499acd054d395c2867138": "K_1M_1^{1+a_1}=K_2M_2^{1+a_2}", "65c0fb3a61314e2bd3e6537216eee38a": "E(y | x_d)=\\hat{y}_d\\!", "65c10a18abd03c5ded2e062b354310e2": "-I \\in SO(2k)", "65c1798a57058e490501d97a7a297de5": "\nf^{*} = \\operatorname{arg min}_{f \\in H_k} \\left\\lbrace E\\left( (x_1, y_1, f(x_1)), ..., (x_n, y_n, f(x_n)) \\right) + R(f) \\right \\rbrace \\quad (\\ddagger)\n", "65c180243bdfedc833b5711588131dfc": "p \\ge u", "65c190d563e72042d540be9c5d1940f9": "n \\choose 2", "65c19391ad01c532137b528b355e02b0": "Ff(\\mathbf{x}\\wedge\\mathbf{y}) = \\phi(\\mathbf{x},\\mathbf{y}).", "65c1b629c220c69b754b3731e1fb8f46": " y_2 = y_c ", "65c224bd12583f97ebc978a6ec3cda27": "\\vec{C}=(1,1)", "65c24a91c853c509fa6e072a9351478e": "\\frac{\\partial^2f}{\\partial x_i\\, \\partial x_j} = \\frac{\\partial^2f} {\\partial x_j\\, \\partial x_i}.", "65c25ccb4c7df1954aa83112f5d0e62f": "n^2 \\cdot k", "65c26e00dc3acd674df07f9039366217": "^{\\;}\\xi", "65c27d3c86e4fff9bfdcca3d1c85562c": " D=-2ln\\lambda(y_i)=-2ln\\frac{likelihood\\ under\\ fitted\\ model\\ if\\ null\\ hypothesis\\ is\\ true}{likelihood\\ under\\ saturated\\ model\\ }", "65c2a0eb933da10867bd682672069165": "\\mathcal{L} = \\left\\{\\sum_{i=1}^{n} a_i \\mathbf{v}_i \\quad | \\quad a_i \\in R, \\mathbf{v}_i \\in B \\right\\}.", "65c30a5f9a9555d3091d599bfd20cad4": "x_{n+1}=M_{t_n\\to t_{n+1}}(x_n)", "65c392fd9945c4b100cddcc409e99962": "\\displaystyle \\frac{(-i)^n}{a}", "65c39e2164693c5009e5d7bda93a408c": "\n \\begin{align}\n \\cfrac{\\partial W}{\\partial\\boldsymbol{C}} & = \n \\cfrac{\\partial W}{\\partial \\lambda_1}~\\cfrac{\\partial \\lambda_1}{\\partial\\boldsymbol{C}} +\n \\cfrac{\\partial W}{\\partial \\lambda_2}~\\cfrac{\\partial \\lambda_2}{\\partial\\boldsymbol{C}} +\n \\cfrac{\\partial W}{\\partial \\lambda_3}~\\cfrac{\\partial \\lambda_3}{\\partial\\boldsymbol{C}} \\\\\n & = \\boldsymbol{R}^T\\cdot\\left[\\cfrac{1}{2\\lambda_1}~\\cfrac{\\partial W}{\\partial \\lambda_1}~\\mathbf{n}_1\\otimes\\mathbf{n}_1 +\n \\cfrac{1}{2\\lambda_2}~\\cfrac{\\partial W}{\\partial \\lambda_2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2 +\n \\cfrac{1}{2\\lambda_3}~\\cfrac{\\partial W}{\\partial \\lambda_3}~\\mathbf{n}_3\\otimes\\mathbf{n}_3\\right]\\cdot\\boldsymbol{R} \n \\end{align}\n ", "65c3af540d6ad2f491b695b89a36c685": "\n h^D(c) = \\left(\\cfrac{4\\sigma_0 a}{\\pi E^*}\\right)\\left[\\sqrt{m^2-1}\\cos^{-1}\\left(\\cfrac{1}{m}\\right) + 1-m\\right]\n ", "65c3c8d57c5d8379d02e3eb1ba2072f3": "\\{ \\phi_i(x,y) \\}", "65c3f9aa2113a537702f9ef072523634": "t\\times n", "65c492cfdaa41522f69e9b89ebd99b9b": "\\text{then }\\cot(x) + \\cot(y) + \\cot(z) = \\cot(x)\\cot(y)\\cot(z).\\,", "65c4e38ba9ea4d27b80f67dc79b1e1b8": "Position = Position + Velocity", "65c4e7db65a7958b4408eecce19ef516": "f^{(k)}(x_0) (x-x_0)^k,", "65c4fe5b4976b03992a45d357676624c": "t\\in \\mathbb{R}", "65c596cabbfb7c04baf8f06b10264b55": "\\text{EMA}_{\\text{today}} = { p_1 + (1-\\alpha) p_2 + (1-\\alpha)^2 p_3 + (1-\\alpha)^3 p_4 + \\cdots \\over 1 + (1-\\alpha) + (1-\\alpha)^2 + (1-\\alpha)^3 + \\cdots }", "65c5b690ad9ad257b2e349de43d6d5bf": "\\mathit{alg}(A,B)", "65c5c49b58905a5c7bd47e1461d913eb": "\\left | d(u) - d(w) \\right | = \\left | (r+1)c(u) - (r+1)c(w) \\right |", "65c5ca759e3405d5e5362535c9f0b83e": "M1 M2", "65c5e52b0c1834390610aef61f3cd8b2": " \\psi^{(0)}(\\vec{r}_1, \\vec{r}_2) = \\psi_{n_1,l_1,m_1}(\\vec{r}_1) \\psi_{n_2,l_2,m_2}(\\vec{r}_2) ", "65c5e6facb37c8f8706f71a9b1aef187": "B = \\operatorname{core}B", "65c6401ff5507aeee65023db0cab1970": "\n~\\mu_j=\n\\left(\\frac{\\partial \\phi}{\\partial N_j}\\right)_{X,Y,\\{N_{i\\ne j}\\}}\n", "65c6b6cc3c62b6d3eeadb952dddcafcd": "~e_q(z) ~ e_{1/q} (-z) =1", "65c6f28253d26856a5d59b0725bc7043": "x+y\\omega", "65c71a1f502f44428038543d48832027": " \\operatorname{prox}_{\\varphi}(u) = \\operatorname{arg}\\min_{x\\in\\mathcal{H}} \\varphi(x)+\\frac{1}{2}\\|u-x\\|_2^2,", "65c73d2977015e789e42ef74c826571c": " F(z)=f(z)/\\lambda z,", "65c74ff017a56bc13f4c4de40ec5f505": "\\delta^2 E<0\\,", "65c75ee26b23deb3da3355326505350b": "\\textstyle \\R^2 ", "65c7a34accda27a027709b5aa2dfb3bd": "F_{\\alpha \\beta} = \\partial_{\\alpha} A_{\\beta} - \\partial_{\\beta} A_{\\alpha} \\ +\\partial^{\\mu}(\\epsilon_{\\alpha\\beta\\mu\\nu}P^{\\nu}),", "65c81126049ede7f6b0d15c97aa3818a": "H=H_0 e^{j \\omega t} \\qquad B=B_0 e^{j\\left(\\omega t - \\delta \\right)}", "65c86980e6f3fed18b742fb97cfb0f6c": "\n \\boldsymbol{F} = F_{11}\\mathbf{e}_1\\otimes\\mathbf{e}_1 + F_{12}\\mathbf{e}_1\\otimes\\mathbf{e}_2 + F_{21}\\mathbf{e}_2\\otimes\\mathbf{e}_1 + F_{22}\\mathbf{e}_2\\otimes\\mathbf{e}_2 + \\mathbf{e}_3\\otimes\\mathbf{e}_3\n ", "65c87e38f4ad5243b16c4d30cad7484d": " P(1) = \\lim_{x\\to 1} \\frac{\\prod_{k=1}^n (1-x^k)}{\\prod_{(i,j)\\in \\lambda} (1-x^{h_{(i,j)}})}. ", "65c884f742c8591808a121a828bc09f8": "a+b", "65c8ccdf03dbc09a76a8b426a66977d2": "\n \\| \\hat{m}(\\theta) \\|^2_{W} = \\hat{m}(\\theta)'\\,W\\hat{m}(\\theta),\n ", "65c8f4a22f56c89b2807a4e72686307d": "\\tau_x \\varphi(y) = \\varphi(y-x)", "65c97ac593f6546566415b3259d020e3": "n = 3\\;", "65ca315ac85783e22d6ff73faf6dc784": "d(sw)=ds\\cdot w + s\\cdot dw,", "65ca39517a68f83e8b9407ac5e40f1aa": "\\text{Holant}(H, \\text{NAND}_2, \\text{EQUAL}_3),", "65ca95edb1cbf6a21096e05ca01b2422": "\n \\dot{u}_i = \\frac{\\partial u_i}{\\partial t} ~;~~ \\ddot{u}_i = \\frac{\\partial^2 u_i}{\\partial t^2} ~;~~\n u_{i,\\alpha} = \\frac{\\partial u_i}{\\partial x_\\alpha} ~;~~ u_{i,\\alpha\\beta} = \\frac{\\partial^2 u_i}{\\partial x_\\alpha \\partial x_\\beta} \n", "65cab178ccaa5d60d87ab70af0f5724e": "\n\tE_1 = \\begin{pmatrix}0.0291933231647860588\\\\ -0.328712055763188997\\\\ 0.791411145833126331\\\\ -0.514552749997152907\\end{pmatrix}\n", "65cb4687bed0d3190f6bed29c34a048e": "f: \\{0,1\\} \\rightarrow \\{0,1\\}", "65cba941d1e21eba2a77b983a245119d": "= 2 e^4 \\frac{t^2 +u^2}{s^2} \\,", "65cbfc6f3b8ea387f10b3943ebf6d25f": "A=M_N", "65cc1d06309f44641e9a35dc3559eb2b": "c_q(n)c_r(n)=c_{qr}(n).\\;\n", "65cc367e1402d9f0e47d647a55ce3439": "0<\\gamma<1", "65cc41b418b69747977819633eaa7fdf": "T=\\frac{1}{2} \\dot q^\\mathrm{T} M \\dot q", "65cc5a275fa8b686cf105188736ad84f": "\\frac{\\partial \\log \\mathcal{L}(\\alpha,\\beta \\,|\\, x)}{\\partial \\beta} = \\frac{\\alpha}{\\beta} - x", "65cc7c427ce37d2f79cf05ed1a363194": "\\frac{1}{2}\\left(-2v(x(s),\\tau)+2 \\frac{\\partial\\tau}{\\partial t} \\frac{\\partial v}{\\partial \\tau} +S\\left(\\left(2 \\frac{\\partial x}{\\partial S} + S\\frac{\\partial^2 x}{\\partial S^2}\\right) \n\\frac{\\partial v}{\\partial x} + \nS \\left(\\frac{\\partial x}{\\partial S}\\right)^2 \\frac{\\partial^2 v}{\\partial x^2}\\right)\\right)=0.\n", "65ce3b42f8e2d3728a1fb7e749a4eb6c": "\\mathrm{C}_n", "65ce647f714018e37b11fffee5bb3630": " d_1 \\, ", "65ced70f43c2aee9aa879ef1bb920c26": " \\mathbf{x} \\times \\mathbf{y} = -\\mathbf{y} \\times \\mathbf{x} ", "65cf025bd56eb12a440ff707111a4fe0": "\\Gamma^l_{ij} = \\tfrac{1}{2} \\left ( \\partial_i g_{jk}- \\partial_k g_{ij} + \\partial_j g_{ik} \\right ) g^{kl}.", "65cf17e20ab9ae7d69c5cbd6e7c020f7": "P = \\sigma \\cdot A \\cdot T^4", "65cf392709d3a0167fcf592d85be42ac": " \\Gamma'=\\Gamma", "65cf5b32fac691be7eab9282b3e669ef": "\\delta_{ij} = q^{ab} E^i_a E^j_b", "65cf673411a4c0be7cdc2a17b235688f": "x_{n-b}", "65cfd3974938e0910458593e0af18b2b": "C^*_2 = \\sqrt{ {a^*_2}^2 + {b^*_2}^2 }", "65cffbfed89a085078b743de58a27580": "1 + \\epsilon", "65d02cb79a4314df36df762e6156e2f4": "p(\\bar{c})\\textstyle \\sum_{i=1}^n p(f_i|\\bar{c})\\log p(f_i|\\bar{c})", "65d0372625be255954035b2212683e74": "(x,y) \\overset{\\mathrm{def}}{=} \\{\\{\\{x\\},\\emptyset\\},\\{\\{y\\}\\}\\}", "65d046ae9387923e71238ee178aba44e": "\\Delta T=\\sqrt{\\left(\\Delta t_1\\right)^2+\\left(\\Delta t_2\\right)^2}\\geq 0", "65d0597eb307169c04cabce091d3d221": " -\\frac{d \\phi_a (\\omega)}{d \\omega} =\n\n\\frac{ \\left| a \\right| - \\cos( \\omega - \\theta_a )\n\n}{ \n\n\\left| a \\right| + \\left| a \\right|^{-1} - 2 \\cos( \\omega - \\theta_a )\n\n }", "65d06fbc3719ee99ce807bcb8771e5dd": "i = 0, 1, 2", "65d076286f62eb364664d8e336980969": "\\theta_1(u;q) = \\sum_{n=-\\infty}^{n=\\infty} (-1)^{n-1/2} q^{(n+1/2)^2} e^{(2n+1)i u} ", "65d09d076cf7b5ca13ca5e17a23aab5d": " f(x) = f_0 (x) + \\varepsilon f_1 (x)", "65d09d34ad56f50f8440c011143cfdfb": "k^{v} = A_{t}^{v}k^{t}", "65d15db0352b8df189854c8e02216648": "f_{\\mathrm{max}}", "65d207862fbccf4e1ccfd4239189fef1": "\nd = \\frac {L^3 F}{48 I E } ", "65d210ff7545fe76877c179643f608ee": " A_t = \\pi r_o^2 ", "65d21bd50693e277723396bd00791bad": "s \\approx_x s'", "65d248f5e73077bb98db30df3ea04fec": "T^{\\mathrm{H}}_0", "65d25e991a8893d1bdd508bc6789ef6d": "a_i(x) \\in \\{0,-1,1\\}\\text{ for }i=1,2,\\ldots, s,", "65d281067d363e907977b293071f3d28": "d-2.", "65d2d7f66a39728b28e1bb775d8c1466": "X \\subseteq R", "65d31e9bf4b41a0abba060438528e8ac": "H_A \\otimes H_B.", "65d36b56bd11e515ef234c111f6cca83": "S^{-1}", "65d3a8c6c0adfc7209899b58005ec6e4": "k_x", "65d3e33889a2e8a9597175ad958e3829": " \\partial V ", "65d3e5ec5de873fc1cfc4239b6faef4e": "[M_i,p_0]=0", "65d4b99505aca7aec36e975a25c09b90": "z = c_1", "65d4ec8de58519665a48a6d223a3b697": "\\sigma_{11}^{\\mathrm{eng}}= 2C_1 \\left(\\lambda - \\cfrac{1}{\\lambda^2}\\right)", "65d53d1825e72ff74505286b0e961a97": "\\mathrm{NA}=\\sqrt{n_\\mathrm{core}^2-n_\\mathrm{clad}^2}", "65d569582ff107c880076be15e40768e": "\\lambda=2\\pi c/\\omega", "65d57010df47726d4f9d76d9ba867a79": " \\rho^L_{i - \\frac{1}{2}} = \\rho_{i-1} + 0.5 \\phi \\left( r_{i-1} \\right) \\left( \\rho_{i} - \\rho_{i-1} \\right),\n \\rho^R_{i - \\frac{1}{2}} = \\rho_{i} - 0.5 \\phi \\left( r_{i} \\right) \\left( \\rho_{i+1} - \\rho_{i} \\right).", "65d58fccb131b768c6d97ac6810b02a6": "y=\\frac{w}{2T_0}x^2 + \\beta\\,", "65d5b372ace6d18a68d37c9ca1ae9215": "\\tfrac{\\text{kg}}{\\text{m}^3} \\!", "65d5e80e923b0a059afcf328e3c5ec36": "t = {1 \\over 2\\alpha g} \\ln \\frac{1 + \\alpha v}{1 - \\alpha v}", "65d5ffbe431c64c6c78aac011585746f": "0.99 = 1 - \\lambda\\sum_{j=1}^{20}f(j)m(j)", "65d68f1974ee7b0ea6ea6aeba01b0e97": "A = \\mathbf{F}_{p^2}", "65d6b8129e6f0d147d58bbc3dfa27055": " {_a^CD_t^\\alpha} f(t)=\\frac{1}{\\Gamma(n-\\alpha)}\\int_a^t \\frac{f^{(n)}(\\tau)d\\tau}{(t-\\tau)^{\\alpha+1-n}}", "65d6d0e94b1fdd714a9730dab0c70123": "A^\\mathrm{T}=A^{-1}, \\,", "65d72296d86ea05b3a09fb73b5a09931": " |0\\rangle ", "65d76e5afa53e31d5ba94e2a891fe0cd": "\nF(\\rho_1,\\rho_2) = \\mbox{tr}( \\sqrt{ \\sqrt{\\rho_1}\\rho_2\\sqrt{\\rho_1}}) ", "65d7cc4d514cbfdb96c7c5f104666b0e": "\\frac{\\widehat{P}_{\\eta}(z^{T})}{\\widehat{P}_{\\eta}(z^{0})}.", "65d7ddab0483b42935ebd49660ef313e": "\\{3,4\\}", "65d7fdf048c77ce65d22b84efc408da4": "\\phi = \\partial_x \\psi \\,", "65d881b011abdbbed76f1ec6d9c154fd": "G = \\frac{{\\rm d}}{{\\rm d}z}\\ln(P)=\\frac{ {\\rm d}P /{\\rm d} z}{P}", "65d890738d753c9263b2b9483eaeca4d": "\nC_2^+(\\beta) = \\frac{\\alpha}{2} \\log \\left( 1 + c_{31}^2 P_1^{(1)} \\right)\n + \\frac{1-\\alpha}{2} \\log \\left( 1 + c_{31}^2 P_1^{(2)} + c_{32}^2 P_2 + 2 \\sqrt{ \\beta C_{31}^2 P_1^{(2)} C_{32}^2 P_2} \\right)\n", "65d890f437a8231722f7a83934b2a80c": "q_1^*=\\frac{a + \\frac{\\partial C_2 (q_2)}{\\partial q_2}- 2 \\cdot \\frac{\\partial C_1 (q_1)}{\\partial q_1}}{2b}.", "65d89dff7817425f17b4d744fca4c282": "(\\mathbb{R}, dF_{X})", "65d8a7a479560f4cef7438864cbfd20e": "F_i", "65d8aa7858f07904d5d8f69ad022001e": "\\gamma^{\\alpha \\beta} = -g^{\\alpha \\beta}.\\,", "65d8c6d044183841951ba5370717e3d5": "a \\;", "65d9c359948dbef82c8f62146620b2c2": "bcdabe", "65d9d2e36747594bcebd440c7a42d2d7": "\n\\begin{alignat}{2}\n\\|Ax -b\\|_2^2 &= \\|Az -b\\|_2^2 + (A(x-z))^*(Az-b) + \\text{c.c.} + \\|A(x - z)\\|_2^2 \\\\\n&= \\|Az -b\\|_2^2 + (x-z)^*A^*(Az-b) + \\text{c.c.} + \\|A(x - z)\\|_2^2 \\\\\n&= \\|Az -b\\|_2^2 + \\|A(x - z)\\|_2^2\\\\\n& \\ge \\|Az -b\\|_2^2\n\\end{alignat}\n", "65d9e2036afd1ab1efe18f52dd412500": "\\frac{1}{12\\sqrt{3}},", "65d9f58689042686575fd8a69425e97f": "\\hat{x}_{\\mathrm{MMSE}}(y) = E \\left\\{x | y \\right\\}.", "65da0136fc8d89172ac7a228c9650807": "M_{35142} = \\begin{bmatrix} &&1&& \\\\ &&&&1 \\\\ 1&&&& \\\\ &&&1& \\\\ &1&&&\\end{bmatrix}", "65dab44747cf7d72ff5b85ff6fa5134f": " \\vdash \\ \\ A \\rightarrow \\lnot\\lnot A ", "65dac05239370a3ee9e1c4bfa8d60513": "\\Delta,A\\vdash B", "65dadb6727a88330ff8d5c00d3069d10": "\n \\operatorname{E}[V^B - V^S] = \\alpha (1 - \\delta) (\\epsilon - (\\mu + \\epsilon)) + \\alpha \\delta (\\mu + \\epsilon - \\epsilon) + (1 - \\alpha) (\\epsilon - \\epsilon) = \\alpha \\mu (1 - 2 \\delta) \\;.\n ", "65dae34b1fb6681b4238e1dd7f3542ec": "0\\leq t \\leq t'\\leq c", "65db614cc65a62379a4308010e1fc898": " \\frac{1}{2}K^2n \\leq \\left \\langle {(\\Delta p)}^{2} \\right \\rangle \\leq \\frac{1}{2}K^2n^2 ", "65db90e72d2b5e8cf81eb828c6eedb41": "\\leq_{f}\\,", "65dbcbe296bada6e71c4de56bcaf5fe5": "\\frac{1}{\\pi}=\\frac{10,00,00\\dots}{31,41,59\\dots}=b_1,b_2,b_3\\dots = b", "65dc3e15ccafad019e3f193bb2677f91": "f_{n} (x) := x^{n}", "65dc7c2464c268f2d457a5819f5dc400": "\\psi_{i+1}", "65dc80630c273f7030432f8370289884": "\n= \\delta(\\mathbf{p}^{\\prime}-\\mathbf{p})\n\\overline {\\Delta} (\\mathbf{p},\\mathbf{p})\n", "65dcdcfa7041e8b29a0af239613b4555": "I(v) = \\Omega (|v|)", "65dd597637bf24648a98f5a234805f12": "t, t' \\in T", "65dd5cb60aee3c1185cf93ffbadc6844": "t_{k+1} = t_k + \\Delta t", "65ddd45d97266c967eadb88f40f8ad3b": "E(z) = 2\\int_0^ze^{-\\pi u^2}\\,du", "65dddfa2f41f9d6dc0615a1ef447d937": " \\left(\\frac{m}{n}\\right) \\left(\\frac{n}{m}\\right) = (-1)^{(m-1)(n-1)/4}.", "65ddf3172b846f49501fc33013804159": "\\text{time required for refresh} = (\\text{length of refresh cycle})(\\text{rows}) = (30\\, \\text{ns})(8192) = 0.246\\,\\text{ms} \\,", "65ddfab1d7c5e07e6cf1af2113e48f04": "ip={pARk\\over 60}", "65de024a372aa1ec2fcc7f040ec4a9ba": " 2 \\sqrt{3}\\, \\sum_{n=1}^\\infty \\frac{1}{n\n{2n \\choose n}} \\sum_{k=n}^{2n-1} \\frac{1}{k} = \n6 \\int \\limits_{0}^{\\pi / 3} \n\\log \\left( \\frac{1}{2 \\sin t} \\right) \\, dt = ", "65de0eb1a4e4a38143e3d6350085a50d": "{\\mathcal A}={\\mathcal H}(\\mathbb D)\\cap {\\mathcal C}(\\bar{\\mathbb D})", "65de90a5dd88e729d2bb72825352dde7": "_{s.4.right\\,}\\!", "65de9999b9d545ec9b96b839928adf45": "\\operatorname{cov}(X, X) + \\lambda I_X", "65de9d250f2e90e23abebd9db6998f75": "\\beta^x=-v_s(t)f\\left(r_s(t)\\right)", "65deb208978b2f2a1353598efb4e5848": " \\eta > 0 ", "65dedd12ec48f3988dd37a1dc6e7cd58": "\\!-\\!\\left(1\\!+\\!\\frac{\\nu}{2}\\right)\\psi\\left(\\frac{\\nu}{2}\\right)", "65dee32b298828eabd2e60651cc016f1": "17+12\\sqrt{2}=33.97056\\ldots", "65df55e1e0dcc1f59465838772ca0ba5": "B(aab, b)", "65df68d3413fdf51219a14ac37bcb905": " \\sin \\hat{z} = \\sin\\phi + d \\cos\\phi, \\,\\,\\, \\cos\\hat{z} = \\cos\\phi - d \\sin\\phi.\\!", "65dfae788ab2b3e782c9eaf27b06c6de": "{\\rm E}_n(x) = \\int_1^\\infty \\frac{e^{-xt}}{t^n}\\, dt,", "65dfe67dd67652351220e610ce9d54de": "\\displaystyle{VT_{K,0}V^*=T_\\Omega.}", "65dffa02a755c291fe2ad6c379b5d2c8": "f(t_i)(x_i - x_{i+1}) = f(t_i)(x_i - y_{j+1}) + f(t_i)(y_{j+1} - x_{i+1}).", "65e01f080ca937e366070dd4a41bab34": "(-P_x/P_y)", "65e0559a12f45b1eb0ae422408f00137": "\\, SO(3)", "65e092120b9416492c966826d1238756": "1/\\phi^2", "65e11d887d16b426534d12e0ceb4c023": "\\mathbf{w}_j", "65e1c68d7ded7252ee1b479e3b976a11": "y=a\\,", "65e1e531f6c74c901155d76812f8c5f2": " f(\\mathbf{x}) ", "65e1ed8968e5f79da6108d0ed521f0ee": "\nt_{ij;\\sigma \\sigma'}^{x,y}=\\langle p_z,i;\\sigma | H | p_{x,y},j ;\\sigma'\\rangle\n", "65e1f6f514e567cdf6b91e7963f353a8": "p_0(x), p_1(x), p_2(x), \\ldots , p_n(x)", "65e202386c60acae6b4d75eb6fa87b60": "\\begin{align}\\theta(\\omega) &= \\angle[D_\\text{cal}(\\omega)D^\\ast(\\omega,\\Omega)]\\\\\n&= \\phi(\\omega-\\Omega) - \\phi(\\omega)\\end{align}", "65e24b8d07d7086e82aaeede7d845ecf": "s_\\alpha", "65e27564e8440c8eca3cd15eb4733124": " D[g] = [x, \\operatorname{false}, \\_]::[o, \\_, p]::[y, \\_, n]::\\_ ", "65e2b529689dc1844b21c3938c00e3e9": "(b,a) \\in R, \\exists c \\in E", "65e2d4ffe4d98015acbfcc03d766938f": "c \\in C", "65e333331009cf9dc8b973c90c65b55d": "\\frac{1}{2}\\sqrt{r-r^2}.", "65e33b6d36f4ab4162d2b5c12d08f256": "pdet[f]>0", "65e3825c0cac842c9df0eed61d809d8b": "\n\\rho(r) = \\frac{\\sigma_{V}^{2}}{2\\pi G r^{2}}\n", "65e3841e21e8fc0d96b6ea1d5c0f09a3": "\\Sigma^f_{ij}= \\sum_k^n \\sum_\\ell^n A_{ik} \\Sigma^x_{k\\ell} A_{j\\ell}: \\Sigma^f=\\mathbf{A} \\Sigma^x \\mathbf{A}^\\top", "65e384a109ad26e4549feed96befd943": "S_{k} = \\frac{\\lambda}{k} \\left( \\rho^{\\prime} \\right)^{k} \\sin k\\theta^{\\prime} . ", "65e3a766bf309125a04e75bc8453010c": "t\\,\\, \\epsilon \\text{ }\\mathbb{R}", "65e3aa269b4b4650d7583447e720f376": "\\mathfrak{a}_0", "65e3bea9f5b90f76db6d4cc461b9fa20": "|x|^\\alpha = x_+^\\alpha + x_-^\\alpha", "65e3d9b783f0f118cd93aab95ae9934f": "0.\\overline{54}", "65e430ba06ddce1bec5790f24ed86bdd": "a \\times a_i", "65e451a425f25296369baac5e3285d1f": "o(...)", "65e4e53d6f515cdf4d08f6ea668d95fd": "1 \\;\\xrightarrow{}\\; \\operatorname{Inn}(\\mathfrak{g}) \\;\\xrightarrow{}\\; \\operatorname{Aut}(\\mathfrak{g}) \\;\\xrightarrow{}\\; \\operatorname{Out}(\\mathfrak{g}) \\;\\xrightarrow{}\\; 1", "65e527ddc534da2731422b7de368213a": "\\mathbf{l}", "65e532d5d77f1af10a18942afd601b41": "L = \\{ a^n b^n c^n : n \\ge 1 \\} ", "65e55a7c81227b5542c716c4cc3bf369": "I\\!I(X,X) = k_1(X\\cdot u_1)^2 + k_2(X\\cdot u_2)^2.", "65e5a8d31233f9890ab7af2dcb001d7c": " y_{n+1} = y_n + hf(t_n,y_n),\\, ", "65e61b7ce6389c051101564f1320e001": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 9.106679\\log_e(T+273.15) - \\frac {7556.611} {T+273.15} + 76.86698 + 5.403634 \\times 10^{-6} (T+273.15)^2", "65e61f6426495da1e4d14e2836ff71fb": "\n\\Phi(\\rho, \\theta) = \\frac{-\\lambda}{2\\pi\\epsilon} \\ln R \n= \\frac{-\\lambda}{4\\pi\\epsilon} \\ln \\left| \\rho^{2} + \n\\left( \\rho^{\\prime} \\right)^{2} - 2\\rho\\rho^{\\prime}\\cos (\\theta-\\theta^{\\prime} ) \\right|\n", "65e628df11905f118cd76a2746c8e477": "\\varepsilon = -1,\\quad a = -\\left( m^2 - n^2 \\right),\\quad b = -2mn;", "65e64b0975a8ac45035b1e6d4b503b2b": " b < r < c \\, ", "65e65571c6922a9d2ced3db7939841ce": "c(\\lambda)=c_0\\prod_{\\alpha\\in\\Sigma_0^+}{2^{-i(\\lambda,\\alpha_0)}\\Gamma(i(\\lambda,\\alpha_0))\\over\\Gamma({1\\over 2} ({1\\over 2}m_\\alpha + 1+ i(\\lambda,\\alpha_0)) \\Gamma({1\\over 2} ({1\\over 2}m_\\alpha + m_{2\\alpha} + i(\\lambda,\\alpha_0))},", "65e6c4d10d9211fceebb2c5a50c89249": "f(r,\\theta,\\varphi)", "65e6f81c818fd6d38ac98e3a52243399": "D_w - D_m = \\frac{q A_t}{2\\pi K (D_b - D_m) N F_w} \\ln \\left( \\frac{R_i}{R_w} \\right)", "65e71e1d88e5a56295681e55c097e166": "\\mathcal{K}(\\boldsymbol{A},\\boldsymbol{r}_0)=\\{\\boldsymbol{r}_0,\\boldsymbol{Ar}_0,\\boldsymbol{A}^2\\boldsymbol{r}_0,\\ldots\\}", "65e73b4544fc667aa6e933b3b9f91261": "\\frac{\\eta}{(\\xi_1-\\xi_2)^2}\\left(\\frac{2(n_\\eta(\\xi_1)-n_\\eta(\\xi_2))}{\\xi_1-\\xi_2}-(n_\\eta^\\prime(\\xi_1)+n_\\eta^\\prime(\\xi_2))\\right)", "65e754245293235f9a4c5a769b2e89a1": "\\mathrm{div}(f)+G-P_{i_1} - \\dots - P_{i_{n-d}}> 0", "65e7a0203efb14df374029665abd4829": "L_D=x_d-x_m", "65e7d59d1aaecba38240fe7603b8ee42": "\\langle C(u,v)w,z \\rangle=-\\langle C(u,v)z,w \\rangle^{}_{}", "65e85bcb496a618476ff936a695b1e5a": "(T,\\eta,\\mu)", "65e8ada1b104c10e922054ebc005ad6a": "\\lambda^a{}_{;b}\\equiv \\partial_b \\lambda^a+\\Gamma^a{}_{bc}\\lambda^c", "65e8b5f12eb7c63f01eafeffeb54de6f": "\\mathbf{A}\\mathbf{B} =\\begin{pmatrix}\n \\left(\\mathbf{AB}\\right)_{11} & \\left(\\mathbf{AB}\\right)_{12} & \\cdots & \\left(\\mathbf{AB}\\right)_{1p} \\\\\n \\left(\\mathbf{AB}\\right)_{21} & \\left(\\mathbf{AB}\\right)_{22} & \\cdots & \\left(\\mathbf{AB}\\right)_{2p} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\left(\\mathbf{AB}\\right)_{n1} & \\left(\\mathbf{AB}\\right)_{n2} & \\cdots & \\left(\\mathbf{AB}\\right)_{np} \\\\\n\\end{pmatrix}", "65e8cadfc7d5f2364d9a74a932f53b45": " I=mr^2.", "65e8f93906568bf486ac075da3136b42": "\\mathbf{b}_i", "65e8feeff3b7cb1f24f605c87d8741e5": "2\\mathbf{X}^{+}", "65e9d67248dc16a9cf504dab9c2b0db6": "\\textstyle P- \\sum_{j=1}^{r}Q_jA_j", "65ea0e23329953a1b3df73c55fe5b96c": "\\left| \\Psi \\right\\rangle = \\frac{1}{\\sqrt{2}}\\left(|00\\rangle + |11\\rangle\\right),", "65ea0f02ec5d0789052b191d1e77686e": "g\\circ(f_1,\\dots, f_n): X\\to \\mathbb{R}", "65ea301486a931535f6f3d5e193ba09d": "\\int_{-\\infty}^\\infty \\delta(x-y)f(x) dx=f(y),", "65ea93a0512011fbadf68ea005d65c1a": "S \\to \\epsilon ~|~ aSa ~|~ bSb", "65eaa25abf3c3804197a25db02ed0880": " r_\\mathrm{ corr } = \\frac{ r }{ 1 + \\theta ( c_x^2 - c_{ xy } ) } ", "65eaafb5133befb904153487039833ae": " {\\hat y}^T {\\hat y} = \ny^T X (X^T X)^{-1} X^T X (X^T X)^{-1} X^T y = y^T X (X^T X)^{-1} X^T y = y^T {\\hat y}, ", "65eac6162b297c2f8ac1f276152924e1": "S_i^{(t)} = \\big \\{ x_p : \\big \\| x_p - m^{(t)}_i \\big \\|^2 \\le \\big \\| x_p - m^{(t)}_j \\big \\|^2 \\ \\forall j, 1 \\le j \\le k \\big\\},", "65eb073892b6afaddd44468fd952191a": "k_{av} \\ge k_a \\ge k_f", "65eb2fcaacaadc314d337e2dca22e79f": "N_\\epsilon", "65eb58ee6fdf107144508bdb35c1aa9b": "\n\\begin{align}\nQ &= \\text{Flow in cubic meter per second } & \\left[ \\frac{m^3}{s} \\right] \\\\\nn &= \\text{revolution per second} & \\left[ \\frac{rev}{s} \\right] \\\\\nV_{stroke} &= \\text{Swept volume in cubic meters} & \\left[ m^3 \\right] \\\\\n\\eta_{vol} &= \\text{Volumetric efficiency} & \\left[\\right]\n\\end{align}\n", "65eb773a175aa3bf92db0665c4bb7fd6": " L = \\frac{\\partial f}{\\partial y}(y^*); \\qquad \\mathcal{N} = f(y) - L y.", "65ebccdb0cedb33ccee05ea4b49a195a": "\\Delta \\lambda / \\lambda", "65ebcde780a72986cf328861215cfa08": "K_a^i = K_{ab} e^{bi}", "65ebd7ee0780081cbb708708d67e4fed": "r \\!\\, ", "65ebe73c520528b6825b8ff4002086d7": "\\sqrt{6}", "65ec20dc7d654ab09be95f4e3aa604fd": "\\{\\psi_{m,n}:m,n\\in\\Z\\}", "65ec42a039ec9de60f04d10de7302936": "W(n,k-x)", "65ec691961463a10cc20952130b6efa2": "\np_{ik}\n= \\Sigma n^\\sigma m^\\sigma \\langle v_iv_k\\rangle^\\sigma\n- V_iV_k\\Sigma m^\\sigma n^\\sigma,\n", "65ecb15f084075fcae0a30201a31b8cc": "(\\sigma, \\tau, \\phi)", "65ecb98c82790afdecfccd5b905083f8": "P_P = \\frac{\\sum (p_{n}\\cdot q_{n})}{\\sum (p_{0}\\cdot q_{n})}", "65ecd9df06da9b45096710a99ba2a1b6": "\\frac{p_n}{q_n}", "65ed77a04c82501550ee68d3c6c2692a": "b = mn \\, ", "65ed8b84462ab91c86a3be2c8a64966e": "\\sqrt{-g}", "65eda8da07b4deaf408c9ed1b3bf31c4": "p\\in M\\,", "65edf0a9ac5c41ca234ecca445ec62c0": " Q_0\\cdot C_{\\mathrm{A},0} + r_A\\cdot V = Q\\cdot C_{\\mathrm{A}} + \\frac{dn_{\\mathrm{A}}}{dt} ", "65ee3b0cb00b2b3b1700217d6d8f861c": "\\varepsilon_{ij}\n= \\frac{1}{9K}\\delta_{ij}\\sigma_{kk} + \\frac{1}{2\\mu}\\left(\\sigma_{ij}-\\textstyle{\\frac{1}{3}}\\delta_{ij}\\sigma_{kk}\\right)\n\\,\\!", "65ee6b82ff6103f41848d881331c8a9d": "AH + CH_3^+ \\to CH_4 + A^+", "65ee6e91f0251421c5377e877e80fa73": "\n\\frac{ \\partial \\bar{u_i} }{ \\partial x_i } = 0\n", "65eeba52dce33c6fea8fc3a5bfbc3441": "\\left.T\\left(\\frac{\\partial}{\\partial x},\\frac{\\partial}{\\partial t}\\right)\\right|_{x=0} = \\left.\\nabla_{\\frac{\\partial}{\\partial x}}\\frac{\\partial}{\\partial t}\\right|_{x=0}.", "65eeeea42e9eaff497378e41b1abe84d": "dH", "65ef61a59bfa0ce4c256ad76f20e8b20": "X \\sim \\mbox{Scale-inv-}\\chi^2(\\nu, 1/\\nu) \\,", "65ef81ea17106f7a8439dbeb54ce3d83": "{T_a}^b{}_{; b} = {T_a}^b{}_{,b} + {\\Gamma^b}_{cb} \\, {T_a}^c - {\\Gamma^c}_{ab} \\, {T_c}^b = 0", "65ef828e8b1cb3c00f276c54022a56d0": "V(A,0)\\,", "65efb0f38d61fc25fe7d56d8e138d3cf": "\\int_{-\\infty}^\\infty e^{-z^2}\\,dz = \\sqrt{\\pi},", "65efec8392c9d69e0dbcd89fc3827c97": "\\mathbf{e}_3 \\times \\mathbf{e}_4 = \\mathbf{e}_6, \\quad \\mathbf{e}_4 \\times \\mathbf{e}_6 = \\mathbf{e}_3, \\quad \\mathbf{e}_6 \\times \\mathbf{e}_3 = \\mathbf{e}_4,", "65eff85bc1f4890086c339fcef3dc775": "X_2 - X_1", "65f012f601e3316fc819c4a6f3f6f535": "x^2 - y^2 + x - y = (x + y)(x - y) + x - y = (x - y)(x + y + 1)", "65f06e8a424fdbacb3f9ab9ab14e4075": "\\textstyle C", "65f083da1282ae57c928975dfbb1b1c7": "X = \\{ (x,y) \\in \\mathbb{R}^2 \\, : \\, x^2 + y^2 \\leq 8 \\, , \\, x \\neq 0 \\, , \\, y \\neq 0 \\}", "65f0b3a5a959fe5b2957c6538e790327": " \\mathcal{P}(S) = \\{ \\{ \\} \\} ", "65f0d18a1e73d86531bc342418072960": "\\therefore \\tan\\frac{\\delta}{2} = \\frac{\\sin\\delta}{1+\\cos\\delta}\n=\\frac{2}{3+\\surd 5} \\approx 0.3819660", "65f0ee159ebaa8379aa5f08d2737803a": "D_{st}", "65f1004a06cd5e5a60999869fbc41153": "\\oint_{\\partial S} \\boldsymbol{B} \\cdot d\\boldsymbol{\\ell} = \\mu_0 \\int_S (\\boldsymbol{J} + \\epsilon_0 \\frac {\\partial \\boldsymbol{E}}{\\partial t}) \\cdot d\\boldsymbol{S} \\, ", "65f1020247b7a52cc89eeddfec1d8d11": "(1)' = 0 \\!", "65f133ea6f6f936d4857d73c6f4443dc": "(x \\ge 3) \\to [x := x+1](x \\ge 4)\\,\\!", "65f1479be0254193960dafd7799bfb4a": "q+pe^{it}\\,", "65f14b38d653995af578fcbb0e437a87": "\\,mg = \\pi d \\gamma", "65f1c831b053f659f2ed59b7c0662c1c": "\\|x\\|_\\infty\\le\\|x\\|_2\\le\\sqrt{n}\\|x\\|_\\infty", "65f2005614732869aed23a394c6f2977": "0 \\le \\beta \\varepsilon s_n\\}} (X_k - \\mu_k)^2 \\,\\mathrm{d}\\mathbb{P} = 0, \\quad \\text{ for all } \\varepsilon > 0,", "6607b782c7c957c4dfd512452e5f8c0d": "(b_{8}-a_{8})+(b_{2}-a_{2})", "6607be6a11079fc33f26cd7139730355": "(\\sigma < 0)", "6608625e889e1856de5335c84467ec24": "\\langle l, r \\rangle_w = w_1\\ldots w_{l-1} \\bullet w_l\\ldots w_{r-1} \\bullet w_r\\ldots w_n", "66086e18ce9e3cedfdb912fe6ce45213": "V_g=\\sqrt{V_a^2+V_w^2-2V_aV_w\\cos(d - w+\\Delta a)}", "6608de233c318cc0a3ecf7c79653e3c1": "\\log_\\varphi(\\sqrt 5 (n+2)) - 2 = { \\log_2(\\sqrt 5 (n+2)) \\over \\log_2(\\varphi) } - 2 = \\log_\\varphi(2) \\cdot \\log_2(\\sqrt 5 (n+2)) - 2 \\approx 1.44\\log_2(n+2) - 0.328", "66092513467265a0cce36da8aaac59b4": "\\vec{F} = - \\frac{mv^2 \\hat{r}}{r}", "660996aeab34e0803bc78e5556ae9f92": "\\hbox{ch}(V)=\\hbox{tr}\\left(\\exp\\left(\\frac{i\\Omega}{2\\pi}\\right)\\right)", "6609c082ca817a6609aa2411a71fb426": "g(0,s)=c_1 \\cdot 1+c_2 \\cdot 0=0, \\quad c_1 = 0", "660a06d4cd6f7d827194fb8247968086": "X_{(i)}", "660a2becaa0047e3c4c0ac041f755647": "\\rho = \\frac{M_{\\rm air}}{V_{\\rm m}} ", "660a3ffd0ecee104b7aebf6bec9fd077": "\\Omega_{d} = \\frac{2\\pi^\\frac{d}{2}}{\\Gamma\\left(\\frac{d}{2}\\right)}", "660a6c35d3ddee22b0b78a2de9ac860c": "\\begin{alignat}{7}\na_{11} x_1 &&\\; + \\;&& a_{12} x_2 &&\\; + \\cdots + \\;&& a_{1n} x_n &&\\; = \\;&&& b_1 \\\\\na_{21} x_1 &&\\; + \\;&& a_{22} x_2 &&\\; + \\cdots + \\;&& a_{2n} x_n &&\\; = \\;&&& b_2 \\\\\n\\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && &&& \\;\\vdots \\\\\na_{m1} x_1 &&\\; + \\;&& a_{m2} x_2 &&\\; + \\cdots + \\;&& a_{mn} x_n &&\\; = \\;&&& b_m. \\\\\n\\end{alignat}", "660a87c67c7826258dc3df5f5f75762c": "\\mathrm{Pe}_L = \\frac{L U}{D} = \\mathrm{Re}_L \\, \\mathrm{Sc}", "660ad11ec883c2e7e35bc161e9b84a12": "\\frac{1}{m}\\sum_{j=1}^m 1-\\mbox{erf}(r_i(x_j)/\\sqrt{c}) = 1-\\mbox{erf}(\\sqrt{c})", "660ae0e10f199e7712fd26a821a018b0": " p_j ", "660b1f54148a9c2c8dc70d15d0618780": "C_m\\,", "660b940eb6fe0ba4ce262ef4abb1f6e2": "\\hat{\\mathbf{L}}_\\mathrm{NR} = \\frac{\\partial}{\\partial t} + \\frac{\\mathbf{p}}{m} \\cdot \\nabla + \\mathbf{F}\\cdot\\frac{\\partial}{\\partial \\mathbf{p}}\\,.", "660bf59f78ca05eb1b1874b6e273ca85": "g_sg_t=g_tg_s=g_{s+t} \\,", "660c52a6d65b3b753bf4b472bc518021": "|V| < 6", "660cc43222995849dbd41e27b19737d8": " \\xi \\ge \\xi_\\mathrm{cutoff} ", "660ce20d3c5a93feb41a05219b8a48a8": "P_j = C_j \\oplus O_j", "660d6a20c8c4ebe19494cb2d93f6ffac": "\\left[{13 \\choose 5} - 10\\right]\\left[{4 \\choose 1}^5 - 4\\right]", "660d7169437c354c60d0628987d7d369": "\\Pr(A=1| B=1) = \\cdots = \\Pr(A=1| B=k).", "660d79f271c266548167fba94f875de3": "\\langle A\\rangle", "660e0d296a665e04ce60fddd5af7d284": "5x-6=10x+2", "660e0d476a33814e336630f31baed9b0": " r_2 ,", "660e9d484bfc5de2cf74067b8bca60a1": "y^T A = 0", "660ebcb8cb45fb310dda58b9300b6ef2": "p(-1)=1", "660f338c11113302ae2c2edfabc9f4f8": "B^{\\alpha }_{\\alpha }", "660f9203ec6e8958e63ed5639f040081": "\\frac{\\pi}{12} \\ (15^\\circ)", "660ff7c7dc41fa812b2e6b9ffd7e0deb": "p_a \\in (0,1)", "661003ce3d1c39d8eaabca88d58e4a41": " \\mathbf{A}^{-1}=\\frac{1}{\\det (\\mathbf{A})}\\left[ \\frac{1}{6}\\left( (\\mathrm{tr}\\mathbf{A})^{3}-3\\mathrm{tr}\\mathbf{A}\\mathrm{tr}\\mathbf{A}^{2}+2\\mathrm{tr}\\mathbf{A}^{3}\\right) -\\frac{1}{2}\\mathbf{A}\\left( (\\mathrm{tr}\\mathbf{A})^{2}-\\mathrm{tr}\\mathbf{A}^{2}\\right) +\\mathbf{A}^{2}\\mathrm{tr}\\mathbf{A}-\\mathbf{A}^{3}\\right]. ", "66104cd3e21f11e794bc973fec9e9cd5": "\n\\begin{align}\n\\log P(\\mathbf{X}) & = D_{\\mathrm{KL}}(Q||P) - \\sum_\\mathbf{Z} Q(\\mathbf{Z}) \\log \\frac{Q(\\mathbf{Z})}{P(\\mathbf{Z},\\mathbf{X})} \\\\\n& = D_{\\mathrm{KL}}(Q||P) + \\mathcal{L}(Q).\n\\end{align}\n", "6610bfe9a6a34d43f7f92b9e3e60e528": "L((y+x)^n)", "6610e90a866df920aef970e64d297339": "y^i", "6610e90f41127affd0e0e9526b169ea3": "\n\\Pr \\{X_{ji}=1\\} =\\frac{e^{{\\delta_j} - {\\delta_i}}}{1 + e^{{\\delta_j} - {\\delta_i}}}\n= \\operatorname{logit}^{-1} (\\delta_j - \\delta_i),\n", "6610fba90da3ab4e16de0c9c72ba8ed7": "A\\equiv A", "661132338c7e4318aee0af52049b50f8": " \\int e^{\\phi^* M \\phi + h^* \\phi + \\phi^* h } D\\phi^* D\\phi = {e^{h^* M^{-1} h} \\over \\mathrm{Det}(M)}", "66113f2c16d07d87020ca2d580ff72af": "\\!\\cosh", "661166122bb69dc1a3db6bf649f57f45": "\\wedge : \\Omega^p(M,E_1) \\times \\Omega^q(M,E_2) \\to \\Omega^{p+q}(M,E_1\\otimes E_2).", "6611a2aa3113fa84f9bd2cda47ba1d3a": "\n\\alpha(t) = \\text{control action on slot t (chosen after observing } \\omega(t) \\text{)} \n", "66122b5c2ecb7fb5a5dfa36f262d6f6c": "\\sum_{n=1}^N n^{-s}", "66124032b3efc4c6a78a016d850db21b": " \\det e^{tB} = e^{\\mathrm{tr} \\left(tB\\right)}, ", "66128092b1d0349728b51d7d12f9c889": "d = |r_1 - r_2|", "6612ae21fa37f83939f8fb3c5c717ff3": "\\textstyle\\deg\\left( P- \\sum_{j=1}^{r}Q_jA_j \\right) < \\deg(Q)", "6612c8466407a76c68d3f34a465de645": "(1-1/n)", "6612e4bf4991907cec52e51ac84dd7e0": "(-\\infty,b]=\\{x\\,|\\,x\\leq b\\}", "661319c19e8c245dc704a7975d6bfd85": "\ns^{(k)}(z):=(\\downarrow 2)(a^*(z)\\cdot s^{(k+1)}(z))\n", "6613863160b534018b3281a858523f13": "[\\dot{T}(t)] = [S][T(t)],", "661387a20d15230fc23311e27c7ce287": "\\bar{ \\mathbb{F}}_p", "6613e0cca4b1eda45b7d5c4f2a74f356": "\\sum _x f(Tx)=x f(Tx) + C\\,", "6613e9266b78d082d4577f37211160aa": "\\mathrm i^2=-1", "6613ec3cc637e965e284be019f6edb2c": "\\gamma < 0", "66141947bc57603b1ce719b13cd72dc7": "\\eta \\in P_1", "66143e2fac1ece6c34d762c9883a2061": "x^2 - 3,\\ ", "6614e11ad4c1acdeabc9a560ed696700": "\\left|\\frac{x}{\\sqrt{1-x^2}\\arcsin(x)}\\right|", "66153037913a1738c0692076b4e397fc": "x^2 - x - 1", "66154834a92daf85f94c4f8ed4bc5e55": " R = R(z, w) = \\left(1 - 2zw + w^2\\right)^{\\frac{1}{2}}~, ", "66154f49658cd942a22aeff74d4ccd0e": " X_0 = \\sum_{n=0}^{N-1} x_n,", "6615502ff3ab59f74aa1f3aaa416ecc1": "(F,\\Phi)", "6615564942ae949825879a22665eb918": "g^{ab}\\omega_{ab}=0", "6615d3f5cafbaad9d8c629baeb800189": " \\sum_{p \\le n} \\frac{\\ln p}{p} - \\ln n", "6616141a9281e19f3baccb006c6f6223": "\\scriptstyle{R_s}", "66163d207670b2289dd63c4885fef056": "\n\\begin{array}{lrclr}\n\\max\\limits_{x_{0}} & E[Q_{1}(W_{1},\\xi_{[1]})] & \\\\\n\\text{subject to} & W_{1} &=& \\sum_{i=1}^{n}\\xi_{i,1}x_{i0} \\\\\n &\\sum_{i=1}^{n}x_{i0}&=&W_{0}\\\\\n\t\t & x_{0} &\\geq& 0\n\\end{array}\n", "66165f0d16ff9f01824495ed17cffb0d": "\\bigg[\\ F \\frac{\\quad}{\\quad} Xe^+ \\ {}^-\\!F \\quad \\longleftrightarrow \\quad F^- \\ {}^+\\!Xe \\frac{\\quad}{\\quad} F\\ \\bigg]", "6616d099f81704ed23e7f395971f51c5": "y^{(n)}(x) + \\sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = 0.\\quad\\quad {\\rm (ii)}", "66175ce775c20c4ef958d228946cfca9": " z(s,t) = {-3 - (s^2 + s t + t^2) (s + t) \\over t (s^2 + s t + t^2) - 3}. ", "661776bb3f238f50f4a74d217b3efaf3": "\\frac{dy(t)}{dt}=(c-a)*x(t)-x(t)*f+c*y(t), ", "66178a0f99039c4c5da1b7ec503d8135": "\nF^{+} \\subseteq F^{*}_{A}\n", "66179af5e896d9187651bb1ac8a733a4": "\\arcsin z = \\arctan \\frac{z}{\\sqrt{1 - z^2}} \\quad z \\neq -1, +1 \\,", "6617ac33d003b8352233b830dcf88879": "\\ \\delta I_G", "661876bde94f4fa48421a032066c6465": "p_{n}(x)", "661879a03b8563f0e7b1c15a3630434a": "\\text{(1)} \\qquad \n \\sigma_y(\\varepsilon_{\\rm{p}},\\dot{\\varepsilon_{\\rm{p}}},T) = \n \\left[A + B (\\varepsilon_{\\rm{p}})^n\\right]\\left[1 + C \\ln(\\dot{\\varepsilon_{\\rm{p}}}^{*})\\right]\n \\left[1 - (T^*)^m\\right]\n", "66188f83e0905e24091a2cd5eebbef1e": "s = \\frac{2 | X \\cap Y |}{| X | + | Y |} ", "6618bf5f98bb67c08f41d398058e0949": "y(t)=h(x(t))", "6618c2e0bf0d6ee4fdafe3b5e87b4fff": "f : R^n \\times R^m \\to R^n", "6618d2a891be7bb1eb3627443f90d373": "({\\Bbb C}\\times{\\Bbb R})", "6619037442392cd2bb426613bc056f6f": "\\log 1 , \\log 2 , \\log 3 , \\log 2 , \\log 5 , \\log 1 , \\log 7 , \\log 2 , \\log 3,...", "66194240ee807ca5f3b15718f8f838a7": "\\mathbf{E}^{0} [\\tau_{R}] = R^{2}.", "661959207ea987cb5e8cde7285d07378": "m_J (E,a_E,a) = m_{lJ}(E,a_E,a) \\sum_j (E)L(E)Q_{trial-J}(L(E))x_{L-J}", "6619a3e154304250c256b32b37f21992": "p_c=0", "661abb661fe37f5a491fb27963eecd8f": " \\mathcal N", "661adc6e0004b6c9fbb3a0853dd29cfa": "\\hat u(k) = \\lambda_k \\hat f(k) ,\\;\\;k=1,2,\\dots", "661ae73477d86d3dbd800aa7b5abd8f8": "b_{-1} = a - \\sqrt{a^2 - b^2} \\, ", "661b45962681dfc52324d3f711ed46a7": "p_I", "661bb1807a0701886f214db9b5fe0e98": "\\eta(\\vec x,t)", "661bb1e83cb7a069f954c441e38ba633": " e^{i\\alpha}", "661c04e90402f3e8542200d54f1fd481": "P^{-}(v)", "661c335eae7083714c770b461767fd15": "ZFW + FOB = TOW", "661c65e1b948e8afe760cd2fab448a1c": "\\{\\epsilon_{i,1},\\ldots,\\epsilon_{i,m}\\}", "661c9586496d45f774acea9221874ed7": "a_{n-5}", "661cfb8756c9182fde38264d2577d13f": "\\mathfrak{sl}(5,\\mathbb C)", "661cfd54f0f4fca9b64ba6bed5c12f8a": "\\text{if }\\Sigma \\models \\phi\\text{ then }\\Sigma \\vdash \\phi", "661d3cebfa2e72b232b2af0a69ea357d": "{\\pi\\over 5}\\ {2\\pi\\over 3}\\ {\\pi\\over 3}", "661d46dce11bfd103e569b01f8785b29": "D^{\\le 0}\\cap D^{\\ge 0}", "661d9dcc64476af31b8c293f789afbd4": "\\forall x \\in A^c \\cup B^c, x \\in (A\\cap B)^c", "661da3f3838a9c152f002719a6cdb7ec": "Q_r = K_{eq}~", "661dd7ca3c0ccbd55d94f566778f5497": "\\begin{align}\nL_{4k}(\\mathbf{Z}) &= \\mathbf{Z} && \\text{signature}/8\\\\\nL_{4k+1}(\\mathbf{Z}) &= 0\\\\\nL_{4k+2}(\\mathbf{Z}) &= \\mathbf{Z}/2 && \\text{Arf invariant}\\\\\nL_{4k+3}(\\mathbf{Z}) &= 0.\n\\end{align}", "661de8484bf15765e8531461ea65888a": "\\frac{kg}{m^2}", "661e11204be70ef47af104a4173f2ec4": " \\eta = \\frac{\\nu \\sigma_f^F}{\\sigma_a^F} ", "661e1a414d31a9911c7dc37886b33afb": "\\lim_{\\lambda\\to\\infty}\\int_0^\\lambda\\left(1-\\frac{x}{\\lambda}\\right)^\\alpha f(x)\\, dx ", "661e3b1db7ff2b036f0f7f13d8b86a5c": "\\phi(z,w)=\\frac{1}{2i}\\log\\frac{(z-w)(z-\\bar{w})}{(\\bar{z}-w)(\\bar{z}-\\bar{w})}.", "661e87b56c43e0e809b1a00669be42c3": "\\lim_{x\\rightarrow a}h(x) = 0", "661e8c815de1b266bd5aa1991c8df653": "\\alpha_{mk} \\in (-1,0,1)", "661ea4433f06e527ecb29039852a7ca9": " F(x + m) = F(x) +m ", "661eebbd3ddeb0a49ee50042a751f711": "\\begin{align} Log_e (lower~limit) &= \\mu_{log} - t_{0.975,n-1} \\times\\sqrt{\\frac{n+1}{n}} \\times \\sigma_{log}\\\\ \n&= 1.67 - 2.20\\times\\sqrt{\\frac{13}{12}} \\times 0.079 = 1.49, \\end{align}", "661f02c40878cd3454299ad3339c8261": "\n\\sum_\\stackrel{d\\mid n}{\\gcd(d,k)=1} d\\;\\frac{\\mu(\\tfrac{n}{d})}{\\phi(d)} =\n\\frac{\\mu(n) c_n(k)}{\\phi(n)},\n", "661fc209124685f663a8c42f805c55c8": "(Pu)_\\nu = \\sum_\\mu P_{\\nu\\mu}u_\\mu", "66205b936941a49cd5bf3c38efbf8d99": "f|_{x_i = g} = f (x_1, \\ldots, x_{i-1}, g(x_1, x_2, \\ldots, x_n), x_{i+1}, \\ldots, x_n).", "6620cb828c801f37dbca0e7a16cb337a": " \\langle U_\\text{B} - U_\\text{A} \\rangle_\\text{B} \\le \\Delta F \\le \\langle U_\\text{B} - U_\\text{A} \\rangle_\\text{A} ", "6620dda8c224adff079ff9ddb53068b9": "N_\\uparrow\n+N_\\downarrow", "662110431ec6edacc06bec19b58179c9": " X^{(n)}(t) = \\begin{cases} - \\int_a^x X^{(n-1)}(t) p(t)^{-1} y_0(t)^{-2}\\,\\mathrm{d}t & n \\text{ odd}, \\\\ \n\\int_a^x X^{(n-1)}(t)y_0(t)^{2} w(t) \\,\\mathrm{d}t & n \\text{ even} \\end{cases}", "66214b56426b80e3463c1a6d75778b52": "[\\!(f)\\!]", "6621ddd18b344105f2ccbcb741bef9e8": "\\scriptstyle\\sqrt{4.9}", "6622132c5f86ef2be9b462c31d3d282f": "\\mu_2 = k_2 = a_1 +4a_2", "66226f257f6884a689e916e8d8eabc8d": "X_0 (i,j)", "6622778241f8044bd2386c9ad77a5761": "0\\cdot 0 = 0\\ ", "662352bcca7577eba94e4c905baf49e8": "\\ \\forall c\\in C : \\exists p_c\\in \\text{Sym}(n) : \\forall e\\in C : p_c(e+c) \\in C.", "6623a9127ba8a089b43da39fc57a1f28": "\\left(\\frac{f}{\\sigma_f} + \\frac{1-f}{\\sigma_m}\\right)^{-1} \\leq \\sigma_c \\leq f\\sigma_f + \\left(1-f\\right)\\sigma_m ", "6623ae99838ea3d267abf8c09f1fb00d": " f(z(x,y))=u(x,y)+iv(x,y) ", "6623fbc6c83579c8cb5d5e1febd6ee58": "\\gamma=\\frac{1}{\\sqrt{1-\\beta^2}}=\\frac{1}{\\sqrt{1-\\left(\\frac{v}{c}\\right)^2}}", "66244bc86d6de697ded36faec24168df": "G\\cup H", "6624d2d34944049ffb6f282a5dbd8932": " p \\, ", "6624d3196eff54dcb5bc1b68cf4a4b79": "p_s = \\min(\\lambda x^*_s, 1)", "6625062a765640e469b9b523b2255aaf": "\n\\begin{align}\n& P(\\boldsymbol{Z}, \\boldsymbol{W};\\alpha,\\beta) = \\int_{\\boldsymbol{\\theta}} \\int_{\\boldsymbol{\\varphi}} P(\\boldsymbol{W}, \\boldsymbol{Z}, \\boldsymbol{\\theta}, \\boldsymbol{\\varphi};\\alpha,\\beta) \\, d\\boldsymbol{\\varphi} \\, d\\boldsymbol{\\theta} \\\\\n = & \\int_{\\boldsymbol{\\varphi}} \\prod_{i=1}^K P(\\varphi_i;\\beta) \\prod_{j=1}^M \\prod_{t=1}^N P(W_{j,t}|\\varphi_{Z_{j,t}}) \\, d\\boldsymbol{\\varphi} \\int_{\\boldsymbol{\\theta}} \\prod_{j=1}^M P(\\theta_j;\\alpha) \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) \\, d\\boldsymbol{\\theta}.\n\\end{align}\n", "66251141832b0a54d4816d7a9c1967f7": "\\ R = \\frac{1}{k+1} ", "6625563cde820f411801fbf13dc69f8d": "\\frac{P_{t+1}}{P_t} = 1-\\frac{1}{2N_e^{(F)}}. ", "66258b34bd46b56cfd8fa1534b008939": "r=\\sqrt{x^2 + y^2 + z^2}", "6625adfede05f5e583db9e1dd9136574": "isopt(\\theta)=\\left\\{X\\mid \\left( \\overrightarrow{XC}, \\overrightarrow{XD} \\right)=\\theta +2k\\pi\\right\\}.", "6625d8fc9f07bb1362bfb1c5616602ee": "r= \\frac {K_1 k_2 C_A C_S}{K_1 C_A+1}", "6625e8c5c56a4b7a0a51dd3bfd072a11": "(9-8i)\\cdot(29+4i) = 293-196i", "662668ef23159d0a9ee3f2cd5ed63a08": "(23)\\quad \\mathcal{L}_{\\ell}\\theta_{(\\ell)}=-\\frac{1}{2}\\theta_{(\\ell)}^2+\\tilde{\\kappa}_{(\\ell)}\\theta_{(\\ell)}-\\sigma_{ab}\\sigma^{ab}+\\tilde{\\omega}_{ab}\\tilde{\\omega}^{ab}-R_{ab}l^a l^b\\,,", "66268a04b238347e8edaa11b92baa734": "MAI=Y(t)/t", "66268e9bf7ec584aec19b858ab025e25": "\\hat{v_i}\\equiv i[\\hat{H}_0,x_i]", "6626bd14c86b2a71b594f03ba23cbcf6": " \\frac{\\det \\left(-\\frac{d^2}{dx^2} + A\\right)}{\\det \\left(-\\frac{d^2}{dx^2}\\right)} = \\prod_{n=1}^{+\\infty} \\frac{\\frac{n^2\\pi^2}{L^2} + A}{\\frac{n^2\\pi^2}{L^2}} = \\prod_{n=1}^{+\\infty} \\left(1 + \\frac{L^2A}{n^2\\pi^2}\\right). ", "662703a6427624bb954f8457d78b606d": "a_1, a_2, \\ldots, a_n, b_1, b_2, \\ldots, b_n \\in A", "6627177c56944b7b43e6b6e1d88c2267": " b=0 ", "66271d76a269e72d2b63003f7f32c34a": " L(f) = \\frac {l_0 + l_{\\infty}(\\frac{f}{f_m})^b }{1 + (\\frac{f}{f_m})^b} \\, ", "6627415e39f7fdeb39fdd9c4fef4e2b8": "\n \\cfrac{\\mathrm{d}}{\\mathrm{d}t}\\int_{\\Omega} f~\\text{dV} = \n \\int_{\\Omega} \\frac{\\partial f}{\\partial t}~\\text{dV} ~,\n", "662741e655938d1df00cb523385d16ba": " DE-2CB=2AD-CE =0 \\,", "66277030a5efe9c7f8a87e673a5200fa": "\\frac{d^2f}{dj^2} + \n\\frac{1-1968j + 2654208j^2}{4j^2 (1-1728j)^2} f=0.\\,", "66285c222b1b71bb7e94c83c8ec4d82e": "\\ x[n] = 1.5^n u[n]", "66286bb9a201e6180e5fd40d2de36963": "P_a=0.9877^a", "66286c7944bb5133768f163f3f00ba55": "\\mathrm{S\\scriptstyle ET}(\\mathrm{S\\scriptstyle ET}_{\\ge 1}(\\mathcal{Z})).", "6628a94096b773fc0615bc88af8eabf9": "Publicly Held Currency = \\gamma \\cdot Deposits = \\frac{\\gamma}{\\alpha + \\beta + \\gamma}", "66296473314809d3363dca2187de0e6e": "\\scriptstyle -\\sqrt{5}", "662970f07e92de00f14a33dee5f296bd": "\\ f^{\\prime\\prime}(x) < 0", "6629bd1ae1f4b87d3f7dc28e9040d2fe": "\\begin{cases}\n\\lambda \\left(\\frac{k-1}{k} \\right)^{\\frac{1}{k}}\\, &k>1\\\\\n0 &k=1\\end{cases}", "6629f7b2c2579cb5e6b05be5e0a2415c": "C_M = \\mu_r \\mu_0\\frac{S}{l}", "662a8911303f9c96d0280b3e3d0d1b9d": "T_{AMB}", "662adcbbb2d3c007a44ca057029a0256": "y_O", "662ae041b5754a7948193d220224478d": "\\mathbb D=\\{z \\in \\mathbb C:|z|<1\\}", "662afb925882b0cf8285ef78ffa6170f": "\\eta^\\mathrm{v}", "662b08124c062a7e626fd25b0cb79267": "2\\uparrow\\uparrow\\uparrow n", "662b38b7acb752d2ec554a3536399183": "ds_{VIII}^2 = \\tfrac{1}{4}c^2 \\left ( d\\xi^2 - dz^2 \\right ) - ab \\left \\{ \\sin^2{y} \\left ( \\operatorname{ch} z - \\chi \\right ) dx^2 + \\left ( \\operatorname{ch} z + \\chi \\right ) + 2 \\operatorname{sh} z \\sin{y}\\ dx\\ dy \\right \\}.", "662b4ab4774a5ef625244097be84dc80": "{\\mathbf{v} \\over c}", "662b559fc068cfaf714f3d4265a00768": "\\,M", "662b66319bffa0a4fbdca491f4d10444": "1 \\le i \\le k", "662b683195d0f9ebda8e06ab2b6d9f87": "(F^T, G^T, \\eta, \\mu^T):C\\to C^T", "662b8c10e63eb8e7a2f71c18acd59dea": "b_0 = \\frac{1}{16\\pi^2}\\frac{1}{3}(11N_c-2N_f) \\;\\;\\;\\; \\text{ and }\\;\\;\\;\\; b_1 = -\\frac{1}{(16\\pi^2)^2}\\left(\\frac{34}{3}N_c^2 - \\frac{1}{2}N_f\\left(2 \\frac{N_c^2 -1}{N_c} + \\frac{20}{3}N_c \\right) \\right)", "662ba217c0973487538b3145d1c96699": " c '", "662bf4df1b15886df547892ebbac1790": "\nK_{a} = \\frac{\\left[ \\text{AbAg} \\right]}{\\left[ \\text{Ab} \\right] \\left[ \\text{Ag} \\right]} = \\frac{1}{K_{d}} \n", "662c3763f88ca2442ccbf6cf0459137a": "\\langle u, v\\rangle_{\\Phi\\times\\Phi^*} = (u, v)_H", "662c6172b371943e3ef82525e25df976": "H\\left[w_1(y_1,\\dots,y_n), \\dots, w_n(y_1, \\dots, y_n))\\right]", "662c6b96932b1b7bcea24bbd9d1b6043": " \\Delta G_{micelle}=\\Delta G_{HP} +\\Delta G_{EL} +\\Delta G_{IF} ", "662c78995ad60c88637fe2ed511e206d": "\nF_{\\beta} = \\frac {(\\beta^2 + 1)\\cdot P \\cdot R } {\\beta^2 \\cdot P + R}\n", "662c983cf2b8fd741d7cef07008f5249": "\\,^{z_5 = x_5 y_1 - x_6 y_2 - x_7 y_3 - x_8 y_4 + x_1 y_5 + x_2 y_6 + x_3 y_7 + x_4 y_8 + u_5 y_9 - u_6 y_{10} - u_7 y_{11} - u_8 y_{12} + u_1 y_{13} + u_2 y_{14} + u_3 y_{15} + u_4 y_{16}}", "662caf18daee51377ddfeb5a2e89c22f": "\\psi _{GS}\\left( z\\right) =\\arctan \\left( z/z_\\mathrm{R}\\right)", "662cb0ddeee9df59a955647b458e675e": "\\frac{\\left \\langle E \\right \\rangle}{N} = \\frac{3}{2} kT + \\frac{\\left \\langle U_{N} \\right \\rangle}{N} = \\frac{3}{2} kT + \\frac{\\rho}{2}\\int_V \\mathrm{d} \\mathbf{r} \\, u(r)g(r, \\rho, T) ", "662d1912798c0a6765f2e6ac9078b124": "\\alpha\\in\\tilde{P}_i", "662d1e2188c4033271cc003e1ec92413": "\\scriptstyle w :\\; S \\,\\to\\, [0,\\, + \\infty)", "662d9bb99e2fc837e420340dfc45ebe7": "F_\\mathrm{app} = q E_\\mathrm{app}", "662dbc90d407364b25bdae9415f1b706": "\nN=\\frac{g_0\nz}{1-z}+\\left(\\frac{Vf}{\\Lambda^3}\\right)\\textrm{Li}_{3/2}(z)\n", "662dc7715a5e4d60513d83b39523da73": "\\tilde{n}=n-i\\kappa", "662e286a19cc793723c12997d6f6ac39": "T=D=\\frac{C_D}{C_L}W", "662e29724f5837644aa071cbb193be07": "2 \\alpha \\cos(\\omega K)", "662eedc986023f7d4e37df2ef2fa7ac5": "\\scriptstyle|1\\rangle", "662f2f63485160658057cdacac083dc8": "n=8", "662f631c1b028949689a651ba5491d92": "126\\,\\text{GeV} < m_\\text{H} < 174\\,\\text{GeV}", "662fa46a13f922ff66ab17c3bc6de693": "\\omega_c = \\frac{1}{C R_0}", "662fa9d0b238cd292d9db478ce2460d5": "Z=\\frac{P}{\\rho k_B T}=1-\\ln\\mathbf{B}+\\frac{1}{\\rho}\\int\\limits_{0}^{\\rho}\\ln\\mathbf{B}\\,d\\rho'", "662ff8431d8381e07d5ddc8f03b48a55": "\\varphi = \\operatorname{arctan}\\left(\\frac{y}{x}\\right)", "663048787f1e44083dc8e21d4d4d1e01": "\\varepsilon_{a...c} \\epsilon^{a...c} = n!", "663070cac5780bea41ef26fb88da4372": " f^{(2)}(r) = \\frac{\\partial f(r)}{\\partial N} \\cong |\\phi_{\\text{LUMO}}(r)|^2-|\\phi_{\\text{HOMO}}(r)|^2", "66309e519008f5e3a6ddb5b6fab3c7ad": "\\tau\\, =\\, \\sqrt{3}\\, t\\, \\sqrt{\\frac{g}{h}}", "6630c0bf4ff8ff40d0fcce4e9300ee41": "c = \\frac{-d}{D} \\begin{vmatrix}\nx_1 & y_1 & 1 \\\\\nx_2 & y_2 & 1 \\\\\nx_3 & y_3 & 1\n\\end{vmatrix}.", "6630daee7f60e1e6a567ed53d189d0d8": "(x_1, \\ldots, x_k)", "663113e08e3481dcc0baa63c0d914c1a": "x_n = y\\,\\!", "6631167fa87d287b3812ea41af16af1e": "H^* \\cong G^*", "66311f9d796ec272d6cb81fde8038f73": "\\mathrm{d}H = C_p\\mathrm{d}T", "66314435c3709e9285dcec1c8f96d1e1": "H_{\\bar{q}}=-eV_{\\bar{q}} =\\frac{-e^{2}}{\\bar{q}^{2}\\varepsilon \\varepsilon _{r} V}=\\frac{-e^{2}}{(\\bar{q}^{2} + q_{s}^{2})\\varepsilon \\varepsilon _{r} V} \\;\\; (9)", "66322ae86eaa306fe226bf7853ad821b": "\\mathcal J_{9-p}", "66322d43e4db25b8ab1b71ed2490b6c7": "\n\\bar{x} = \\frac{4300}{50} = 86.\n", "66325be95ff15253ac1ed9faff463529": "\\scriptstyle M_C \\;=\\; E_K(M_P)", "6632d6b4f16d5a583565cf3b5db3b36b": "\\succ", "6632ef3014cc4f3a4c0de9b312e99aae": "i\\eta_{s+1}", "663306b040d8040d35a33326f4db75cb": "\n\\cdots \\to \nX^{-1} \\xrightarrow{d^{-1}}\nX^0 \\xrightarrow{d^0}\nX^1 \\xrightarrow{d^1}\nX^2 \\to \\cdots", "6633423f8ffad7667d5bf2a9aa77ecad": " \\sum_j W_j = 1 \\,\\!", "6633faf904396777b378e614f2a50efb": "D_{\\mu} = \\partial_{\\mu}+\\frac{1}{2}\\Omega_{\\mu}", "663468b80ed572ba613b9cc062c10c66": "S_1 = {v}(\\alpha)", "6634f76859b04512749f50f4484df632": " {\\rm co} \\exists^{\\rm P} \\mathcal{C} = \\forall^{\\rm P} {\\rm co} \\mathcal{C} ", "6634f98ec0eb82d8e1703eaa042c1736": "G = \\beta(2)", "66352c8334c4d7c84c99ae2def70360e": "\\scriptstyle{DTFT}\\displaystyle \\{x_N\\}", "663533375a0f90e95d4428462dea0555": "I_{FROG}(\\omega,\\tau) = \\left | E_{sig}(\\omega,\\tau) \\right | ^2 = \\left | FT[ E_{sig}(t,\\tau)] \\right | ^ 2 = \\left | \\int_{- \\infty}^{\\infty} E_{sig}(t,\\tau) e^{-i \\omega t} dt \\right | ^2", "66353c57a180c82d91b90b390db3ff59": "H^0( K^{\\otimes k} ),", "66353e0cf2d0e5da7250877e0d39e9da": " \\mathbb C^{64} ", "66354d95aae2ea6abf4b0f4f72ccfd9c": "\\scriptstyle x_2 + B < A + B", "66354ea18a54a97d7937971ead64a88a": " \\ \\textbf{f}^{-1} = 1\\pmod p ", "663585fb75935e0e88afecc356efdeed": "\\frac{A}{P} = \\frac{i}{1 - (1+i)^{-n}}", "6635a4988c0255f11e0a11c41f2c44c2": "I = L \\frac{10^{10}}{4\\pi}", "6635c0698c5e4b980ebce73337ab536b": "\n \\boldsymbol{\\sigma} = \\begin{bmatrix} -p^* +2(C_1-C_2)+2C_1\\gamma^2 & 2(C_1+C_2)\\gamma & 0 \\\\ 2(C_1+C_2)\\gamma & -p^* + 2(C_1 -C_2) - 2C_2\\gamma^2 & 0 \\\\ 0 & 0 & -p^* + 2(C_1 - C_2)\n \\end{bmatrix}\n ", "6635cefd4901765f24ddd0ff30a64b96": " K_\\nu (x) = \\frac{1}{2} \\; G_{0,2}^{\\,2,0} \\!\\left( \\left. \\begin{matrix} - \\\\ \\frac{\\nu}{2}, \\frac{-\\nu}{2} \\end{matrix} \\; \\right| \\, \\frac{x^2}{4} \\right), \\qquad \\frac{-\\pi}{2} < \\arg x \\leq \\frac{\\pi}{2} ", "6635fc121604e5a662b6496276971baa": "r_0,\\ldots, r_{k+1}", "66364b86f1fe9e263133afa0a7ae559a": "\n\\xi^{Hill}_{(k(n),n)} = \\frac{1}{k(n)} \\sum_{i=n-k(n)+1}^{n} \\ln(X_{(i,n)}) - \\ln (X_{(n-k(n)+1,n)})\n", "663665927c29ccc5f5cfad6abea570b3": " F_\\text{n} =\\frac{l_\\text{m} + h_\\text{cg}(\\frac{a_\\text{x}}{g})}{l_\\text{m} + l_\\text{n}} W.", "663667d7af172fb62b2553fea9c870e3": "\\operatorname{nec}(U \\cap V) = \\min ( \\operatorname{nec}(U), \\operatorname{nec}(V))", "6636a44f6c1829acea55f77fdcdc3bd6": "1+2\\cos\\theta", "6636d239f777b0e6baee47e1206e38c4": "\\phi_a (\\mathbf{r})=\\mbox{Re}[\\mbox{FT}[(R(\\mathbf{k})\\otimes I(\\mathbf{k}))K(\\mathbf{k})]]", "6636d2406c4057a5a0e77386e12efdb9": "\\frac{\\partial \\theta}{\\partial t}= \\frac{\\partial}{\\partial z} \\left[ K \\frac{\\partial h}{\\partial z}\\right] ", "6636d9afb059d3130c5cab1bef088760": "D_{E}/(D_{E}+H_{E})\\approx D_{E}/H_{E},", "6636ddd33b094a2434bf7837e803fe89": "H_\\varepsilon f(x)=\\frac{1}{\\pi}\\int_{|y-x|\\ge \\varepsilon} \\frac{f(y)-f(x)}{y-x}\\,dy=\\frac{1}{\\pi} \\int_{|y-x|\\ge \\varepsilon} \\int_0^1 f^\\prime(x+t(y-x))\\,dt\\, dy", "663769854da9e71eadcc20414d1aec8b": "1RM = \\frac{100 \\cdot w}{52.2 + 41.9 \\cdot e^{-0.055 \\cdot r} }", "6637fd4ad21bb5a07e42fdec3cc97eb1": "\\frac{\\hbar}{m_\\text{P} c} = \\frac{G m_\\text{P}}{c^2}", "663812976c79003033f479c8623e60fc": "c_{XX}(\\tau) = \\frac{C_{XX}(\\tau)}{\\sigma^2}.\\,", "6638e0d63e27fb1771eda35db93a3de0": "\\ e^x - (1 + x + x^2/2)", "663933ddc644fd105c318b0d5bff864a": "\n\\exp_r(\\kappa)^+\\longrightarrow(\\kappa^+)^{r+1}_\\kappa\n", "66394979b1d173694e8a1ed528ce4bb1": "\\left[\\Delta(x)p_{1^{(n)}}\\right]_{l_1,\\cdots,l_k}", "66395934d7db2918404c1ff5a6006b4d": "\\sum_{i=1}^n-\\log f(x_i)", "663959cd96b8941dced6ee2aec6ae588": " \\begin{align} a_0 & = \\sqrt{2} \\\\\n b_0 & = 0 \\\\\n p_0 & = 2 + \\sqrt{2}\n \\end{align}\n", "6639a5b64f8401ccb93e087ac7e14c32": "\\nabla^2 \\lambda - \\mu_0 \\varepsilon_0 \\frac{\\partial^2 \\lambda }{\\partial t^2}= - \\mathbf \\nabla \\cdot \\mathbf A - \\mu_0 \\varepsilon_0 \\frac{\\partial \\varphi}{\\partial t}", "6639b08ff4a8a693650d2023e470f1b5": " h(x,t) ", "6639e0206b54694312dcf03ff5198098": "|\\psi\\rangle_A \\otimes |e\\rangle_B \\, ", "663ac5e94cdc756e88b62ed3c48eef33": "f(Q)-f(P) = \\varphi(Q-P)", "663ad2a6b76610d8784ec0d1cd44f4bf": "\\alpha = 2\\arccot \\left\\{\\tan\\left(\\frac12(\\beta-\\gamma)\\right) \\frac{\\sin \\left(\\frac12(b+c)\\right)}{\\sin \\left(\\frac12(b-c)\\right)} \\right\\}.", "663ae768931906ad41545d5b7e4779a7": "\\frac{d\\,\\ln au}{dx} = \\frac{1}{au}\\frac{d(au)}{dx} = \\frac{1}{au}a\\frac{du}{dx} = \\frac{1}{u}\\frac{du}{dx} = \\frac{d\\,\\ln u}{dx}.", "663b9d9b3d4780c1adc6e1b0fa7739a0": "1000009 = 293 \\cdot 3413", "663bb095fcab7e0ceffd7a48cf24c3ed": "\\mathfrak c^{\\aleph_0} = \\left(2^{\\aleph_0}\\right)^{\\aleph_0} = 2^{{\\aleph_0}\\times{\\aleph_0}} = 2^{\\aleph_0} = \\mathfrak{c},", "663bb42c26dc086fa9733ef6d21f01f6": "Fib(3+4i) \\approx -5248.5 - 14195.9 i", "663beb2fbdf0d4a2f2fb95370c221cdc": "Z=R\\,\\!", "663c2319352d386f02ba1ee7c10d4e1f": "\n A = 2\\pi r^2 + 2\\pi r l \\qquad \\implies \\qquad \\delta A = 4\\pi r\\delta r + 2\\pi l\\delta r + 2\\pi r\\delta l\n ", "663c2af0f93f958518968b4474b06e01": "\\ln(e^x) = x.\\,\\! ", "663c3e794a99cf1b034cabfc2aeccd2c": "k^n - 0 \\to \\mathbb{P}^n", "663c6fa47b379eed3e1e6f1e239ccff6": " \\|f\\|_\\mathcal{B} = |f(0)| + \\sup_{z \\in \\mathbf{D}} (1-|z|^2) |f'(z)|. ", "663c8e84f4e909d5e17d36e34bac71ba": "\\rho\\equiv 2Cr", "663c98f83fe8424fbc8724e0691cd1ff": "\\begin{align}\n \\operatorname{arsech} x &= \\operatorname{arcosh} \\frac{1}{x} \\\\\n \\operatorname{arcsch} x &= \\operatorname{arsinh} \\frac{1}{x} \\\\\n \\operatorname{arcoth} x &= \\operatorname{artanh} \\frac{1}{x}\n\\end{align}", "663cbbaea241e0d8ba942071c8ca7919": "(x_1 + s(x_2-x_1) - x_c)^2 + (y_1 + s(y_2 - y_1) - y_c)^2 = r^2.\\,", "663cc337989b2eb80e636fd87a275202": "\\lVert \\phi \\rVert = \\lVert \\psi \\rVert = 1", "663cf06d3524eac145bfc4586281c2f5": "P = \\frac {{(2S-575)^2}-625} {5000} ; \\quad S = \\frac {575 + \\sqrt {625(8P+1)}} {2}", "663d5612f3ec23ca938ddb9c53bd8786": "V={\\scriptstyle\\frac{1}{4}}\\pi{D^2}", "663d679c52950c5e995961bdff17d29a": "3600 = seconds\\ per\\ hour", "663dd18f8057be597518f6b4063a7960": "Z_6", "663df35350ad073d51a7422fb9381db2": "S_N=\\sum_{k=1}^N Z_k e_k(t)", "663e10c259a91ab8ed27215e3b003e2a": "\\frac{X - \\overline{X}}{s}", "663e4ea17901a1c416ef9772ecb096b2": "Y_{m_n} = m_1Y_1 + m_2Y_2 + m_3Y_3 + \\dots + m_NY_N", "663e52f66de1ceac0358c456eb0e3809": "C_1 \\cup C_2", "663e5d9064da30808d5190b3c14004bc": "\\Delta S_{reaction}^\\ominus = \\sum S_{(products)}^{\\ominus} - \\sum S_{(reactants)}^{\\ominus}.", "663f738b9899ab989abb62d5bce87c8e": "{4}\\sin^{4}\\!\\left(\\frac{\\pi}{n}\\right)\\!{\\color{Blue}R}^{4}-{4}\\sin^{3}\\!\\left(\\frac{\\pi}{n}\\right)\\!{\\color{Blue}R}^{3}+({4}\\cos\\!\\left(\\frac{\\pi}{n}\\right)-7)\\sin^{2}\\!\\left(\\frac{\\pi}{n}\\right)\\!{\\color{Blue}R}^{2}+{2}({1}+\\cos\\!\\left(\\frac{\\pi}{n}\\right))\\sin\\!\\left(\\frac{\\pi}{n}\\right)\\!{\\color{Blue}R}-{3}\\cos^2\\!\\left(\\frac{\\pi}{n}\\right)+2=0", "663ff627b3093b4f8baa5f56e1b4ed6f": "Corr^r(k)(X, Y) := \\bigoplus_i A^{d_i+r}(X_i \\times Y)", "66400983f4ecf0304a992d5dca48fe07": "\\Re\\{\\cdot\\}", "66405036883fefe5f4fc8c49090afba2": "\\hat\\theta \\xrightarrow{p} \\theta_0\\ \\text{as}\\ T\\to\\infty", "6640a9b6ab340c3258f3edef5374236f": "{4 \\choose 3}{4 \\choose 2} = 24", "6640d8e386d391f6ad1df5833185e2c4": "\\mathbf{x}_{t+1}", "66412472df2ec7c9dfa16f89cdf72b8f": "\\begin{align}\n\\mathbb{E}[\\ln x] &= \\frac{ \\partial A(\\eta_1,\\eta_2) }{ \\partial \\eta_1 } = \\frac{ \\partial }{ \\partial \\eta_1 } \\left(\\ln\\Gamma(\\eta_1+1) - (\\eta_1+1) \\ln(-\\eta_2)\\right) \\\\\n&= \\psi(\\eta_1+1) - \\ln(-\\eta_2) \\\\\n&= \\psi(\\alpha) - \\ln \\beta,\n\\end{align}", "664130843e1dfcd8309a14e518b0d58c": "\\Lambda'(\\alpha^{-i_k})=\n\\lambda_0\\alpha^{i_k}\\prod_{\\ell\\in\\{1,\\dots,v\\}\\setminus\\{k\\}}(\\alpha^{i_\\ell}\\alpha^{-i_k}-1)\n.", "664184fa036906f0f17d5b9c5d57b03f": " \\Theta(\\alpha ^2) ", "664203e00420d31b8586c7d21491439b": "{\\mathbf E} = -\\nabla\\varphi - \\frac{\\partial{\\mathbf A}}{\\partial t}\\,, \\quad {\\mathbf B} = \\nabla\\times{\\mathbf A}.", "66421e056c40917ff0ce6c4988caec8c": " B =\\frac{(1-r-g)G}{g}", "664239e2d07965f2601f9721793bc226": " z^{}_{}=xyx^{-1}y^{-1},\\ xz=zx,\\ yz=zy ", "66427ebf83a9597efabf21dceaa96fe3": "\\mathfrak{su}_3", "664295b94a361d20e0aba7f3834149a4": "r_d = \\frac {n_d - s_d} {2} ", "6642ba652cfb10994f4bf09b246a46aa": "m=i", "664304d2a90eb91166afe549e1647e1d": "F,V : \\mathcal{K} \\to \\mathcal{C}(R)", "66435930f7f4fb9a1346577cd5930790": "\nC = \\max_{f(x) \\text{ s.t. }E \\left( X^2 \\right) \\leq P} I(X;Y)\n\\,\\!", "66436528f4811e8b92dc55d0bf337dc6": " SH = {m_v \\over (m_v + m_a)}. ", "66444091aa7464a7194093d901d2d6c8": "\\Delta H = \\Delta G + T\\Delta S ", "664461206e919c99ecfdade36d23e611": "Ra", "664486878d8d3583fa13b9ec865baea3": "x (\\tau)", "66449ed33286d0c1c71e3ebe8346ab6a": "_{\\sim x}\\!", "6644c96ed5c681bb005b521e31b53202": "\\mathbf{F}\\left(\\mathbf{r}\\right)", "6646535694066da9bb48618978444163": "1\\ang \\theta ", "66466dd0e55b9de37a522364c3fcfb2f": "u:A", "6646730ef67974d6bfacc2614e153eda": "\n(x \\triangleleft y) \\triangleleft z - x \\triangleleft (y \\triangleleft z) = (x \\triangleleft z) \\triangleleft y - x \\triangleleft (z \\triangleleft y).\n", "6646aa84eebf433c1c10267bab599a45": "\\scriptstyle\\delta t_{\\text{clock},i} (t)", "6646f301368a16997ee01fc055d81fba": "P= {n \\over{n \\over K+R}-d}-{n \\over{n \\over K}-d}", "6646f88d427cbf02f42be01746b449ab": "\\bold{x}=A^{-1}\\bold{b}", "6647274b24b8b0669d74d3210efc65db": "S_{\\rm prolate}=2\\pi a^2\\left(1+\\frac{b}{ae}\\sin^{-1}e\\right)\\qquad\\mbox{where}\\qquad e^2=1-\\frac{a^2}{b^2}", "6647b92b955cd2006e4f17c612a532f0": "|A_1-B_1| + |A_2-B_2| + |A_3-B_3| = |2-6| + |3-4| + |5-1| = 4+1+4 = 9, ", "6647c101ddbc86b2064798c33784d013": "\\hat{R}_x^\\alpha(\\tau) = \\lim_{T \\rightarrow \\infty} \\frac{1}{T} \n\\int_{-T/2}^{T/2} x(u+\\tau/2) x^*(u-\\tau/2) e^{-i 2\\pi \\alpha u}\\, du.", "6647df88d695e6ae6a1b1055221a6bb6": "C_1, C_2, C_3", "6647fbb0c7b217961da3bf42969fef7d": "\\psi(\\Omega^\\Omega 3)", "66487c0277e15a669d68adea00265d7e": "[\\underline{P}(A), \\overline{P}(A)]", "66487df85f08f0ff44a65a04de624d09": "\\ddot x=-\\nabla V(x)", "664885f408112b54a9531b8b6564f05e": "\nax^2 + bx + c = 0\\,\n", "6648fe956a53ed3af539937e25234188": "2+\\zeta(\\tfrac12)", "664902b7773424ec9607c4f344088614": "\\frac{2 \\pi}{\\alpha} ", "664918b1eefaa52b3e4e51cb746d0658": "{{\\phi }^{'}}(u)=\\frac{\\int\\limits_{-\\infty }^{\\infty }{\\xi {{P}_{V}}{{f}_{a}}(u,\\xi )d\\xi }}{\\int\\limits_{-\\infty }^{\\infty }{{{P}_{V}}{{f}_{a}}(u,\\xi )d\\xi }}", "66494012c295a266293963b7505e17be": "P(x_1, ..., x_n)", "6649b72898c835b4536f59a4cde6284c": "0.22120334\\ldots", "6649d6ef1c5e79584c4da0a1843ccea7": " \\log {g(z)-g(\\zeta)\\over z-\\zeta} = - \\sum_{n\\ge 1} a_n(\\zeta^{-1})z^{-n}.", "6649de41b6ae0e70a566a921e1f8e559": " a_1,b_1,a_2,b_2,\\ldots ", "6649f5988135ebbc6238e657c659544b": "\\boldsymbol{\\hat{k}} = (0, 0, 1)", "664a01263f7fb3ea2edd4f593eeed1a1": "\n \\begin{align}\n \\varepsilon_{\\alpha\\beta} & = \\frac{1}{2}\\left(\\frac{\\partial u_\\alpha}{\\partial x_\\beta} + \n \\frac{\\partial u_\\beta}{\\partial x_\\alpha}\\right) \\equiv \\frac{1}{2}(u_{\\alpha,\\beta}+u_{\\beta,\\alpha})\\\\\n \\varepsilon_{\\alpha 3} & = \\frac{1}{2}\\left(\\frac{\\partial u_\\alpha}{\\partial x_3} + \n \\frac{\\partial u_3}{\\partial x_\\alpha}\\right) \\equiv \\frac{1}{2}(u_{\\alpha,3}+u_{3,\\alpha})\\\\\n \\varepsilon_{33} & = \\frac{\\partial u_3}{\\partial x_3} \\equiv u_{3,3}\n \\end{align}\n", "664a07202c4ee2185be05a1e15f8e913": "\\boldsymbol\\Theta", "664a245cbb75d3b8c3d402e249bfbcfa": "\\frac{\\mathrm{d}^2}{\\mathrm{d}x^2} \\Psi(x) = U_1 \\cdot (x - x_1) \\cdot \\Psi(x).", "664a3e9a99d37a49d33a9bb506475809": "{}^{212}_{82}\\text{Pb}\\to{}^{212}_{83}\\text{Bi}", "664a6d451fc3a062b28e9ca95743e66c": "\\left(-4\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "664ac2105896ae1c2292e395339baed6": "\\qquad \\qquad M::= u \\;\\mid\\; [\\;M\\;]_u\\;\\mid\\;M \\| M", "664b293fb6bf3e6b837ce8942765ecc1": "(t,q^i,p_i)", "664b2b699ed29bcf2c35dc4154f61b07": "i\\hbar|\\dot\\psi\\rangle=H|\\psi\\rangle", "664b57251624a3abdcc5cf213ad221f5": "M_0 = 2.282 \\, 71 \\times 10^5 \\left[\\frac{t_\\mathrm{ev}}{\\mathrm{s}}\\right]^{1/3} \\mathrm{kg}\n\\ \\ \\approx\\ 7.2 \\times 10^7 \\left[\\frac{t_\\mathrm{ev}}{\\mathrm{yr}}\\right]^{1/3} \\mathrm{kg} \\;", "664b6c9283190adcdfa7ced62407c095": "\\phi: {\\mathbb R}^n \\times Z \\rightarrow {\\mathbb R}", "664bc1e4b2721a27e30a5e191a8843db": "14 - 13 = 1", "664bc5819bc131a98c5f8f1d78e282eb": "[t/2,\\ 1-t,\\ t/2]", "664be1572029397f70624808fb85bc57": "x,y\\in\\mathbb{R}^n", "664bec813107ea68663c288d1b67c844": "150\\,\\text{lb}_m \\cdot 8\\,\\tfrac{\\text{ft}}{\\text{s}^2} = 1200\\,\\text{pdl}", "664c06d23b9f8e7743fad6d839645c68": "\\textrm{Kendrick~mass} = \\textrm{IUPAC~mass} \\times \\frac{14.00000}{14.01565} ", "664c0d6743f35461a17b57d1b74aca61": "(P)", "664c21c976ec18b91e2b4d904b747971": "tA+(1-t)C \\succeq t B+(1-t)C", "664c576f42760dbca3e68acda3bd187e": " (b_1,b_2,\\dots b_n)", "664c8c39546ac0826a5bddca5157f72d": "d-a_{15}", "664cee48f62a48ab24fae35cd02b9b83": "\nx = \\underset{1}{\\overset{\\infty}{\\mathrm K}} \\frac{a}{b},\\,\n", "664df37e1b0fac89250e3a1f730eca1b": "B \\and A", "664df9f0d5d7a75fa4fc2f00cef31c99": "Q = 1/2", "664e0359363e75399c9fc7ba0a87941d": " \\Pr_{x,y}[C(x, y) = f(x)] = E_y[p_y] \\, ", "664e03db4eb4537170678f78111f9b00": " \\lambda = \\lim_{t \\to \\infty} \\lim_{\\delta \\mathbf{Z}_0 \\to 0} \n\\frac{1}{t} \\ln\\frac{| \\delta\\mathbf{Z}(t)|}{|\\delta \\mathbf{Z}_0|}.", "664e047f60d0227ddc651b02a45e1e1c": "e^{ikr}/r", "664e08186b9c69004a8c71edec31db14": "U_1(q) = \\sum_{n\\ge 0} {q^{(n+1)^2} (-q;q^2)_n \\over (-q^2;q^4)_{n+1}}", "664e40248cb496bb72899fcbcefa3628": "\\tfrac{1}{100}", "664e7c90156887bc819b8bae10d39dba": "9(2^5/9!!) \\pi^4 = (32/105)\\pi^4 ", "664fd38f237431420dc024098bf29753": "\\Phi_t", "664fe2a4dafa40abbdc8bc32ee17041d": "P = \\frac{m_0 v}{\\sqrt{1-\\frac{v^2}{c^2}}}. ", "66506113b68a4a541e04ca99ae2f7b4f": "x-1", "6650fef508c72ccd19f9046b5cd1f65b": "C_{k+1} := C_{k+1} \\cup C^{cand}", "66510022995f87ac4a55b481eaa58f08": "M(p)=\\lim_{\\tau \\rightarrow 0} \\|p\\|_{\\tau} = \\exp\\left( \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\ln(|p(e^{i\\theta})|)\\, d\\theta \\right).", "66512b246461090416f308a84b08c768": "2^{n/2}", "665139250463c06af00a3879e6a33a22": "\\ddot{y} = \\frac{d^2y}{dt^2} \\,,", "665155d52f2e319f5bc886f6f3c90466": "(1/R^3)(dR^4\\Omega^2/dR)\\ ,", "665160566e3fcec1849e15209934a708": "{\\dfrac{l}{nd+1}}", "6651c01c673d9a263db17a6c859ad3fd": "\\eta_{th} \\equiv \\frac{W_{out}}{Q_{in}} = 1 - \\frac{Q_{out}}{Q_{in}}", "6652053e5585672b581b67eb68a3463f": "\\sum_{j=1}^n x_j", "66521f5c8787735787367510b1b78f6c": "V = P \\frac{\\ln(1+kD)}{k}\\,", "665224530503f53e9d614967358f7e8c": "G = (N, A=A_1 \\times \\dotsb \\times A_N, u\\colon A \\to R^N)", "66525feab52566d2af9e76ed99013236": "\\mathcal{E}(m) = (g^r,m\\cdot h^r)", "66528de02779ea199e6274ffaa3a8c02": "t = T", "6652aa6f2c8dc4af25c0e961a8cd7b08": "a \\otimes b", "6652f1ad897d31a3a60f80ea4b794a55": "H_o", "6652f454422725f736108dd74944a041": " (t,x,y,z) \\mapsto t^2-x^2-y^2-z^2", "665366998b67915603e10bbcbaea8bb6": " \\lambda = (t)^{-1/p}", "66536df0096bfe632c91a9dc2c43284d": " m = \\frac { w_1 } { 1 - w_2 } ", "6653a70e565c9b4011af267417fbf372": " P(cancer~WHOIFPI) \\approx \\frac{1}{500} * 1 = \\frac{1}{500} = 0.002 ", "6653b258340ebab429c37ae3f97d1512": "S_{\\omega^2}", "6653ffde725d3bb4022888372fd8d918": "Y \\cup \\{B\\}", "66540249c49aa4c7ada27ad2a767fd14": "x'(s)x''(s) + y'(s)y''(s) = 0 \\ . ", "665419d1ffad14bc0ebd3ee5bdf4a0b0": "F_2 = \\frac{M_0^\\mathrm{act} M_2^\\mathrm{pass}}{r^2}", "66543d540d85b65046958240f202ac15": "\\Omega=\\beta A_m^{\\rm fp}", "66549b9625c2bbfb6d6d8dfdd18af348": "\\nabla_{\\mathbf{x}_0}f", "6654e0d774d1977c3b0b7ad6f1dff3be": "\\langle\\cdot\\rangle_c", "6654fa9ff24153542a14c7722c2d9d72": "\\sqrt{T}=\\frac{2\\sqrt{k_1k_2}}{k_1+k_2}", "66554bf750b3c87c65c885a857fd7705": "\\begin{align}\ne^{\\pi \\sqrt{163}} &= \\left( \\frac{e^{\\pi i/24} \\eta(\\tau)}{\\eta(2\\tau)} \\right)^{24}-24.00000000000000105\\dots\\\\\ne^{\\pi \\sqrt{163}} &= \\left( \\frac{e^{\\pi i/12} \\eta(\\tau)}{\\eta(3\\tau)} \\right)^{12}-12.00000000000000021\\dots\\\\\ne^{\\pi \\sqrt{163}} &= \\left( \\frac{e^{\\pi i/6} \\eta(\\tau)}{\\eta(5\\tau)} \\right)^{6}-6.000000000000000034\\dots\n\\end{align}\n", "66555f48682d18a8157f77cef33b172e": "\\begin{bmatrix}0&0\\\\1&-1\\end{bmatrix}", "665570987075bc028beed1994b94b2f7": "\\frac{\\partial E(2\\omega)}{\\partial z}=-\\frac{i\\omega}{n_{2\\omega}c}d_{\\text{eff}}E^2(\\omega)e^{i\\Delta k z}", "66558d93d241d8f2a5a7721cf6b6271b": "\n \\mathcal{S} = ka\\, \\frac{3 - \\tanh^2\\, kh}{4\\, \\tanh^3\\, kh}.\n", "6655c2e3b26205cd95868f8d96b86c3e": "w+n_w", "6655fdb21c1a954f14df5b904ac72a5e": "\\hat{y}_0=x'_0\\hat\\beta", "665644ee3a13a5d1103791425cd64189": "V_n(r) = \\frac{\\pi^{n/2}}{\\Gamma(\\frac{n}{2}+1)}r^n", "66567a0c648d11aec88af6da2e0a5361": "e:=(u,v)", "665691f1022f06c081a3131fa444e1c6": "\\{|s_i\\rangle\\}", "6656a81cca46b0d37bb797233ab04392": "\n \\begin{align}\n & \\frac{\\mathrm{d}^2}{\\mathrm{d} x^2}\\left(EI\\frac{\\mathrm{d} \\varphi}{\\mathrm{d} x}\\right) = q(x,t) \\\\\n & \\frac{\\mathrm{d} w}{\\mathrm{d} x} = \\varphi - \\frac{1}{\\kappa AG} \\frac{\\mathrm{d}}{\\mathrm{d} x}\\left(EI\\frac{\\mathrm{d} \\varphi}{\\mathrm{d} x}\\right).\n \\end{align}\n", "6656ab29646eff7f6c59843ccbabce31": "\\begin{align}\n\\text{minimize } & c(\\mathbf y - \\mathbf x_0) + \\mathbb E_{\\mathbf d} [ G(\\mathbf y, \\mathbf d) ] \\\\\n\\text{subject to } & \\mathbf y \\geq \\mathbf x_0,\\end{align}", "6656baf640e26b98e9d0102328e1dcbd": "\\sigma(A \\oplus B) \\ge \\sigma A + \\sigma B - \\sigma A \\cdot \\sigma B.", "6656e94b4f04c66f08eeebd58986441c": "\\ \\mathcal{L}_{\\mathrm{QED}} = \\bar\\psi(i\\hbar c \\, \\gamma^\\mu D_\\mu - m c^2 )\\psi - \\frac{1}{4 \\mu_0}F_{\\mu\\nu}F^{\\mu\\nu}", "66579cb11ffdd12cc18a9004f349d6d4": "\n T = \\frac{1}{2} \\left[ \\frac{L_x^2}{I_1} + \\frac{L_y^2}{I_2}+ \\frac{L_z^2}{I_3}\\right].\n", "6657d98be694135e9d019901379cd9b5": "M' \\to N", "665803a95a830c59f423b5dfac0846e6": "m(x,y,z)=(x\\vee y)\\wedge(x\\vee z)\\wedge(y\\vee z)=(x\\wedge y)\\vee(x\\wedge z)\\vee(y\\wedge z)", "6658240e79f20249642d73c074f7e718": " m< 0.08,", "66585e41dcca39696e672d75924f4f3d": "Pcm = \\%C + \\frac{\\%Si}{30} + \\frac{\\%Mn+%Cu+%Cr}{20} + \\frac{\\%Ni}{60} + \\frac{\\%Mo}{15}+\\frac{\\%V}{10}+5B", "6658600e037385b530d4cb7266fb7953": "\\|T\\|_\\infty\\leq 1", "66587e4515cc8b727586efd4e5e66826": "H(X) := \\sum_{i=0}^{N-1} \\left[p_i \\cdot \\lg\\left(\\frac{1}{p_i}\\right)\\right]", "66588467db1097c8290dfc9685bd320e": "\\mu V/L = \\mu /T_{conv} ", "6658b870a4201713e9e4309a2a4b7995": "\\theta \\mapsto n\\theta", "6658dc9d6ea9e39573f27f1407fd14fe": "K_T", "665916cd50832d3cd51e6da4afb03393": "\\tilde{R}_n", "66595785fee9b4a205aba5d4cf7b29c8": "\\|g\\|_1 \\to 0", "6659b89cccc6aaa1180006f85e4cc9d5": "e=2", "665af5cdc24c8dc11d7d4fe78e180363": "A\\in \\mathbb{M}_{K\\times M}", "665b47cbac3702ec94ab4ef161c95ed7": "|S(j \\omega)|", "665b57e0745d60e87dcd6e0bce7fdecc": "\\omega_n\\,", "665b788a608dbe399031f013fd133f41": "[(X+E) \\; (Y+F)] \\begin{bmatrix} B\\\\ -I_k\\end{bmatrix} = 0", "665bb0397e0e986f4516164552151f5e": "A \\succeq B", "665bbef3a9b993f25b2eee0443bf94e4": "C(\\alpha,\\beta)", "665bea66b0eaca13caebaeb866d12ace": "\\epsilon u u j", "665c02262e7fe82263a64993f6042641": " \\beta(p_{1}, ...\\,, p_{n}) = \\prod_{i=1}^M \\alpha(p_{S_{i}})", "665c5706c5375691e95b1490eb6eee67": "m=\\iiint_V m_0\\, n(x,y,z)\\;dV ", "665d0076433016cde991724bec3351aa": "A \\triangleq \\frac{ R_{\\text{f}} }{ R_1 }", "665d5796d4bf5cf845ae1f38c507231b": "\\int_S \\mathbf g \\cdot \\,d{\\mathbf {S}} = g(r) \\int_S \\,d{S} = g(r) 4\\pi r^2 ", "665d749ef1767f3002d0e292b9cc6c78": "X:=(Y,Z)", "665d9e05965b28ca3c3d6559ddf24e56": "f*g= \\mathcal{F}^{-1}\\big\\{\\mathcal{F}\\{f\\}\\cdot\\mathcal{F}\\{g\\}\\big\\}", "665dd30fd32270d75e7871a23e9ae2b3": "x_I", "665e0f58eaa5431e349ad8f68d521cd6": "f(x;a,b,p)= \\frac{a p}{x} \\left( \\frac{(\\tfrac{x}{b})^{a p}}{\\left((\\tfrac{x}{b})^a + 1 \\right)^{p+1}} \\right) .", "665e1da0e326112f64cba12ca0ca7db1": " C^i{}_{jki} = 0 ", "665e2c666752cc0bbeddf324ce4807c7": "(1+N_i/g_i)^{g_i}\\approx e^{N_i}", "665e54d92b8966df26ee2d6535f67315": "\\mathrm{Im}(\\partial_1) = \\{(b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3] | b_1,b_2,b_3\\in \\mathbb{Z}\\}", "665e98cc759cb748fd791e9ec68f352f": " \\frac{\\partial \\psi}{\\partial t} = i\\left(\\omega_1 \\begin{pmatrix}\n0 & e^{i\\omega_r t}\\left(\\cos{\\omega_r t} - i\\sin{\\omega_r t}\\right) \\\\\ne^{-i\\omega_r t}\\left(\\cos{\\omega_r t} + i\\sin{\\omega_r t}\\right) & 0 \\end{pmatrix}+\\left(w_0+\\frac{\\omega_r}{2}\\right)\\sigma_z\\right)\\psi ", "665eaf52a4bc7bf3fdc12174d27e1af4": "g(\\varepsilon)\\,d\\varepsilon=2\\frac{1}{8}4\\pi n^{2}\\,dn=\\frac{8\\pi L^{3}}{h^{3}c^{3}}\\varepsilon^{2}\\,d\\varepsilon.", "665ed561b4c5061f312330eb41970757": " F = p, G = p, V = f, E = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) ", "665f3446f7538743b42353a965e84c5f": "\\scriptstyle v^*", "665fa9959c2dcc5dd8e6ee3e78e98d8f": " \\succeq", "6660782225469b414b72c176875eb01f": "n = 2\\,", "66609f73298420d6a1b307dac4296ee0": "\\frac{p_2}{p_1} =\n 1+\\frac{2\\gamma}{\\gamma+1}(M_1^2\\sin^2\\beta-1)", "6660e889639c7f58c078bdb29ec38a61": "\\langle v\\rangle ", "66619193acf17e2ca9c507abd1fee2d3": " {\\omega^1}_2 = p_y dx - p_x dy.", "6661a621386ed4fd388c1d3d4a9d0872": "\\frac{E_{c,t_0}}{p_{c,t_0}} = q_{c,t_0}", "6661e1a48a81dd932920471324445804": " V_{\\mathbb R} \\otimes_{\\mathbb{R}} {\\mathbb C} \\to V ", "6661f55d31f8ab6cf64b7588a5f41dc6": "R=A \\cdot e^{\\frac{B}{T}}", "66621307eae2857e90ee416b86de787b": "\\begin{matrix} \\frac{9}{5} \\end{matrix}", "66621333aab80f476f9fe77049747f76": "{\\epsilon^2}_n \\, ", "66621f0541e97179c760b51b340c508e": "\\left( \\frac{\\partial \\ln K}{\\partial \\frac{1}{T}} \\right)_\\theta=-\\frac{\\Delta H}{R}.", "66622883084abc484ba5adbec6a475af": "=(\\lambda _{1}+\\lambda _{2})[-b^{2}(-P^{2};m_{1}^{2},m_{2}^{2})+p^{2}+\\Phi\n(x_{\\perp })]\\Psi =0\\,,", "66624eee7f2cfa4f6492b1e8cd475373": "\\varphi+\\theta_0 = \\angle PQ''P' = \\psi = \\theta_1-\\theta = (q-1)\\theta+\\theta_0", "6662a1896e15643f3f78481ffed87e8a": "ES_{0.20}", "666338bf2a35725c3faafc6c8d573d76": "\\mathbf{\\nabla}\\phi=0", "66634b3a4b3d9c563ad8c51c87d28b9e": "w=1/4", "666370f27f8eaebae79e2299cbffcae8": "\\textstyle \\omega^2 := \\frac{k_i}{m}", "6663b3731800fb00e0367f52c5a5d3ba": "\\rho _\\infty", "66640bf1fdfc1470b0cf706a48db06ba": "\\mathbf{\\hat{e}}_1, \\mathbf{\\hat{e}}_2, \\mathbf{\\hat{e}}_3", "66640e7e72879b2ffd7fd2e7bb2345e0": "\n\\mu (\\mathbf{x}, \\Sigma_I, \\Sigma_D) = \\det(\\Sigma_D) g(\\Sigma_I) * ( \\nabla L(\\mathbf{x}, \\Sigma_D)\\nabla L(\\mathbf{x}, \\Sigma_D)^T)\n", "66645f5a566808260deb3b58eaa1772d": "f = f_0[1-0.04 \\ \\mbox{ppm}(T-T_0)^2]", "66647f939ff05705fa8836d460083e99": "p = {m_0 v \\over {\\sqrt{1 - \\frac{v^2}{c^2}}}} \\!", "6664b042bcd4b7a677a4df7f47f088c7": " S=LWy ", "6664b4c1486d0fb9b0bc4c3635086fa5": "1/(2\\ell+1)!!", "6664bc1eb7598dcd683dbbb4175b0c8d": "\\alpha : H \\rightarrow L(H)", "6664c62c32a60ce1510de69d7bdde1c4": "f\\colon R^k \\to R", "6664df717f323b0ad4f0ebe6b4309b92": "|q(T_w)(\\Pi_j(x))|<\\frac{1}{2}", "6664e53c8914417246ee51e0ff75042a": "\\Delta\\Theta \\approx 4 \\times 10^{-5}\\mbox{ rad} = 0.002^\\circ", "66652049b3a9f93a94dc93da48805f22": "T_{surr} \\Delta S = mc_p \\delta T - Vdp", "66654893bc39049a21a418acefb13679": " \\begin{bmatrix} V_1 \\\\ V_2 \\\\V_3 \\end{bmatrix} = \\begin{bmatrix} Z_{11} & Z_{12} & Z_{13} \\\\ Z_{21} & Z_{22} &Z_{23} \\\\ Z_{31} & Z_{32} & Z_{33} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ I_2 \\\\I_3 \\end{bmatrix} ", "66655513271eb2abc656056d120ddccc": " \\lambda_i = \\frac{\\mu'_{20} + \\mu'_{02}}{2} \\pm \\frac{\\sqrt{4{\\mu'}_{11}^2 + ({\\mu'}_{20}-{\\mu'}_{02})^2 }}{2}, ", "66657f62029d3c3a223823858d56a9d2": "(1+x)^n", "66659e8197ff3255ead3717a06c14eb2": "V(p_{1},p_{2},Y)", "6665adc5a53210bc53f2d00e70a02e84": "\\textstyle (A_0, A_1, \\ldots)", "66662d2be1b174d9f00b0e8d513baeb1": " \\varepsilon_0 \\ ", "66665e9f1cc27d676959ea35da3ee62a": "\\int_{B_x} |f|dy > t|B_x|.", "6666716ddb469e1a10b4dedf9a566dcb": "\\pi r^2", "66668e71e00ecea937a45efe8e413514": "\\scriptstyle{4.6%}", "666695e69da3f338b37d22e349ad953f": "\\geqq \\!\\,", "6666971a916445dc856087d036c7e76c": "v_{i}=\\left( z_{i}^{\\prime\n}|x_{i}^{\\prime}\\right) ", "6666cd76f96956469e7be39d750cc7d9": "/", "6666f69a90353142ba91fb508a547a4f": " Q(y) \\ ", "666738ed3914d30ac87a8995658e96a7": "s^2 = \\frac{\\sum_{i=1}^N w_i}{{(\\sum_{i=1}^N w_i})^2 - {\\sum_{i=1}^N w_i^2} } \\ . \\ {\\sum_{i=1}^N w_i (x_i - \\overline{x}^{\\,*})^2}", "66674ad2721a048a37cdb2b2258e230a": "\\boldsymbol\\mu", "66677068723335d7be107521182e62eb": "w^3+x^3+y^3+z^3=0.", "666790313f6bb58a2f62e2b17077f7c5": "\\mathcal{E}(e^{e^x})=\\emptyset.", "6667b4c41a280ba66174fbc87ced0b5d": "\n{\\rm E}[\\hat g]\\,\\,\\, = \\,\\,\\,{k \\over {\\mu _T^2}}\\,\\,\\, + \\,\\,\\,{1 \\over 2}\\left( {{{6\\,k} \\over {\\mu _T^4 }}} \\right)\\sigma _T^2{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(16)}}", "6667c88968f9cc97de1319b3ba1b380a": "\\varphi : O(2) \\to \\mathrm{Aut}(\\mathbb{R}^2)", "6667c8ac01151eeb66ce2a92718c18fb": "\\forall x \\forall y [\\forall z [z \\in x \\leftrightarrow z \\in y] \\rightarrow x = y].", "6667d711dc91effdc4a5d8a9d1c5fb2b": "\\eta^{a b}", "6667e91435a349ca9a9095f039d32f46": "\\mathbf{b_{1}}", "666859f1d66979b75d6c110e9876ab14": "\\pi = \\frac{C}{d} \\approx 3.14159", "66688cea0a1a1fdb47f0d018915723e4": "N_{Y}\\left(E+\\delta E\\right)", "666898c9a19bd31dc674d72f2c157332": "d(X) > 0", "66689b3fbe0dd5a8404acc7da0589803": "\\Delta G_m = \\Delta H_m - T\\Delta S_m \\,", "6668cae114cc8d4dc991ac6152f99da5": "v(x)", "66698fb025b35665217b25df186b0e93": "\\nu_e", "6669b6528a696adef120ead62fb32ed7": " F \\ge 0, Q_i \\ge 0, R_i > 0. \\, ", "666a1d08824b9f0dd92b7191410802dc": " (d_3 2^3 + d_2 2^2 + d_1 2^1 + d_0 2^0)m = d_3 2^3 m + d_2 2^2 m + d_1 2^1 m + d_0 2^0 m ", "666a321d404ee9c091bb7058bab1a3bd": "\\int\\cos a_1x\\cos a_2x\\;\\mathrm{d}x = \\frac{\\sin(a_2-a_1)x}{2(a_2-a_1)}+\\frac{\\sin(a_2+a_1)x}{2(a_2+a_1)}+C \\qquad\\mbox{(for }|a_1|\\neq|a_2|\\mbox{)}\\,\\!", "666a6483a41761be370a95c6665ab57f": "\\phi_1=0^\\circ", "666a6b0962eaff37824ecdf5b7ffe57c": "a^{\\frac{p-1}{2}}\\equiv -1\\ \\pmod{p}", "666a9a17f2d28dfaf73342d5df3e0550": "B_{k+1}=\n(I-\\gamma_k y_k s_k^T) B_k (I-\\gamma_k s_k y_k^T)+\\gamma_k y_k y_k^T,", "666a9e33e6d55f398bc22176a434523c": "\\{E_{mb}T_{na}g\\}_{m,n\\in Z}", "666ac70d6a2b38f2bc71e5750f054899": "\\displaystyle{\\beta(g)^{2}=b(g).}", "666afb242b04464e76ad25baf445aa5e": "\\frac{[\\Gamma(\\tfrac14)]^4}{128\\pi^3} = \\frac{1}{\\sqrt{u}} \\sum_{k = 0}^{\\infty} \\frac{(6k)!(2w)^k}{(k!)^{3}(3k)! 6486^{3k}}", "666b1dba4092b19b6fc7a3d78ba72822": "\\sigma_i=\\sigma(i)", "666b5a336dcb08e14747a11ed19afe09": "\\!\\, \\ge M ", "666b9913af4b374e53931018c8dfa008": " \\dfrac{\\partial}{\\partial x^\\alpha} = \\left(\\frac{1}{c}\\frac{\\partial}{\\partial t}, \\nabla\\right) = \\partial_\\alpha = {}_{,\\alpha}", "666bb823a25d5acaa64826021464925f": "\\lim_{n \\rightarrow \\infty} D_{\\mathrm{KL}}(P_n\\|Q) = 0", "666bbd48d271be6224b8318ea274a5c5": "\\bigg(\\frac{p_0}{1 - p'e^{it}}\\bigg)^{\\!k_0}", "666bc50c46d5d5549688a9e01545f84d": "E = \\frac{3}{5} \\left( \\frac{1}{4 \\pi \\epsilon_{0}} \\right) \\frac{Q^{2}}{R}", "666c05600b56fcc63c87ecad4902d574": "\nm(\\varphi)=a(1-n)^2(1+n)\\left[D_0\\varphi-D_2\\sin 2\\varphi+D_4\\sin4\\varphi-D_6\\sin6\\varphi+\\cdots\\right], \\, \n", "666c0ecf538c79d2e2acad36fd1c8221": "f(x)=y", "666c4f3a25bbfd43c2352d76088aa4f2": "\\displaystyle q^1,\\,\\ldots,\\,q^n,\\,w^1,\\,\\ldots,\\,w^n\n", "666c61ee40ab2655f1b6788cd863af91": "p = \\frac{I\\cdot (1+i)+\\sum_{t=0}^T \\frac{I \\cdot b}{(1+i)^t}}{\\sum_{t=0}^T\\frac{E\\cdot (1-v)^t}{(1+i)^t}}", "666c6dd1afc7ddc4d821e9037e912104": "\\tfrac{13}{52} + \\tfrac{12}{52} - \\tfrac{3}{52} = \\tfrac{11}{26}", "666cbc966992a765f097b1dc10978c05": "\n\\left\\{\\begin{align}\n\\frac{\\partial}{\\partial\\bar{z}_1} &= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x_1}+ i \\frac{\\partial}{\\partial y_1} \\right) \\\\\n&\\qquad\\qquad\\vdots \\\\\n\\frac{\\partial}{\\partial\\bar{z}_n} &= \\frac{1}{2} \\left( \\frac{\\partial}{\\partial x_n}+ i \\frac{\\partial}{\\partial y_n} \\right) \\\\\n\\end{align}\\right..", "666d10bb70921fdbf97258403ab7d153": "\\,\\!\\alpha_p", "666d348b4b438926364755ba70e43567": " \\mathcal{C}_{XY} = \\mathbb{E}_{XY} [\\mathbf{e}_X \\otimes e_Y] = \\bigg( P(X=s, Y=t) \\bigg)_{s,t \\in \\{1,\\dots,K\\}} ", "666d53e85dbd182853c1e957acc8ca17": " Q = \\Delta U + W_\\text{boundary} + W_\\text{isochoric}.", "666d736d14e32cb6bcf41f77c26bc865": "n=d_1+10d_2+\\dotsb+10^{m-1}d_m", "666d973be2940f3ed0b1b83e89998a1c": "\\|\\mathbf{a}_1\\| \\;\\mathrm{e}_1 = (14, 0, 0)^T.", "666e1a83ccb2d5fa1b88ec346716bc1a": "\\textstyle 1\\leq p<+\\infty", "666e22c07d1b79d9bb236299436e1759": "\\|A^{-1}\\|\\leq - {\\frac {1}{\\mu(A)}}.", "666e613dfe44154f2acf44ed1aa7debd": " \n\\mathbf{f}_{k' k}\\equiv\\langle\\,\\chi_{k'}(\\mathbf{r};\\mathbf{R})\\,|\\, \\hat{\\nabla}_\\mathbf{R}\\chi_k(\\mathbf{r};\\mathbf{R})\\rangle_{(\\mathbf{r})} \n", "666e78617d4793f6bdb9bb59e1d7915e": "T = A + B \\cdot \\left( \\left( \\delta {}^{18} \\text{O} \\right) \\text{calcite} - \\left( \\delta {}^{18} \\text{O} \\right) \\text{water} \\right)", "666e87f1bb549bac99030eb88dd007d2": "\\ \\Phi \\mapsto \\Phi' = G \\Phi ", "666ea600977df62e7e7301242bcaae24": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\dfrac{\\partial T_{i_1i_2\\cdots i_q\\cdots i_n}}{\\partial x_{i_q}}dV=", "666eb9d04e5efc87f919d0d7f995b674": "z^{2n} - 1 \\equiv 0 \\pmod{p}", "666ec504992fa310051d6e7f8a88eed0": "p = w\\rho", "666ec5163ef010e8841acf33f0cb322c": "5_+^{1+2}: 4S_4", "666f0af9f053fb71121dccf1261dfa7c": "\nS_vp =\n\\begin{bmatrix}\nv_x & 0 & 0 & 0 \\\\\n0 & v_y & 0 & 0 \\\\\n0 & 0 & v_z & 0 \\\\\n0 & 0 & 0 & 1 \n\\end{bmatrix}\n\\begin{bmatrix}\np_x \\\\ p_y \\\\ p_z \\\\ 1 \n\\end{bmatrix}\n=\n\\begin{bmatrix}\nv_xp_x \\\\ v_yp_y \\\\ v_zp_z \\\\ 1 \n\\end{bmatrix}.\n", "666f3979d495bbdd33e7858e12efd561": " F = h \\circ f \\circ g ", "666f8b1445cc31e21fac1a34cba517af": "\\ \\| u \\|_{L^{p} (S, w \\, \\mathrm{d} \\mu)} \\equiv \\left( \\int_{S} w(x) | u(x) |^{p} \\, \\mathrm{d} \\mu (x) \\right)^{\\frac{1}{p}}", "666f9742f45edfeb8fd91f1b434efdbc": "\\eta=W/Q_h", "666fd3b321cdb390829dca0f7ec78906": "\\ u^2x^2+v^2y^2+w^2z^2-2vwyz+2wuzx+2uvxy=0", "6670cf56fbec6ee7402676715c283e87": "M^{(2\\eta + 2)} = M^{(2\\eta+1)}\\text{diag}(s^{(\\eta+1)})", "6670d57a881cad795ac18e808cd634d3": "S = 1 - 1 + 1 - 1 + 1 - 1 + \\cdots", "6670e63339f77c9df1ae0b8886721c94": "\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta} = \\cot \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\cot \\theta} ", "6671c1e7e4a0cc3bb6a679a1c3e48834": "F=\\operatorname{succ}(I,[a_1,a_2,\\ldots,a_n])", "6671e658ada3ccfef2ca2dc9199b6d8d": "\\textstyle \\lambda_{\\diamond N}", "6671fed5416fbdf8dfb6dc1628835789": "4\\,\\pi^2\\,\\ell^2\\le A(\\rho)\\cdot\\,2\\,\\pi\\,\\log(r_2/r_1).", "66723723f47d9d5297bc5f1e2dfbeae2": "p_{ij} = \\begin{cases} q_{ij}/\\gamma &\\text{ if } i \\neq j \\\\ 1 - \\sum_{j \\neq i} q_{ij}/\\gamma &\\text{ if } i=j \\end{cases}", "6672a0c3ffef031cf16739e2a928586c": "w_1,\\ldots ,w_k", "667312d250e3533ac2ca914b2e55983e": "\\mathbb{R}^7", "667321f7cf1259a916b9fbcce267056f": "\\Lambda(\\alpha^i) = 0", "667323a6ac6d93dfd9c9d0cebefe90e3": "Y_1^{n}", "6673244c52f1508cffb6caed3a0c2fdd": "f_n(z) = \\frac{g(z)^n}{|G|}.", "6673c7664515563ae83dff59a332ff05": "\\theta(\\Omega^2)", "667410168426dc9501bb98d4a5c3d4b6": "MD = - \\frac{1}{P}\\frac{{\\partial P}}{{\\partial y}}", "66741cd4016c245cc43a7bc9a62b56b7": "\\gcd(N_i,N_j)", "667462eb07a774f698f20b4fb83273fc": "U_n(p)", "6674a60f8b06153b0a1d29f4d5e2460b": "RMS_{Total} =\n\\sqrt {{{RMS_{DC}}^2 + {RMS_{AC}}^2} }\n", "6674dafdc08d9677f7b6f52e90ff7cdf": "\\sum F_x=-F_{AB}-F_{BD}\\cos(60)=\\frac{5}{\\sqrt{3}}-\\frac{10}{\\sqrt{3}}\\frac{1}{2}=0 \\Rightarrow verified", "667529788dc6106c7f77a172dc4554b4": "0 \\bmod p", "667574c1a3072a41f63078b4df1893d9": "u=-v-1", "667574e20a6513ee5fabac3f64e286fc": "C_{QY} = \\frac{\\epsilon_0}{\\lambda_0} = 3.6492417 F/m^2 \\ ", "6675aef2b22563f2e711857aba502a65": " N > 3 ", "6675ef80515f35381fbe4a1460c2ac81": " \\scriptstyle \\Delta \\omega", "667651141c762810b57f0742ed1bb432": "0 \\div 0 = c", "66767bc634c059ec57aa92fc8927d0da": "\\mathcal{G}(A)", "6676cc316fb77ee745b2a82ec6298478": "U^{-1} = \\begin{pmatrix}\n 5 & -3 \\\\ -3 &2\\\\\n \\end{pmatrix}", "6676e60fab2d8562d9cf1016ba7e8bb6": "x < b", "6676fc7068cf798f3cf7f43d6df960c2": " G = \\frac{1}{2} \\int_0^1 \\mathrm{K}(t)\\,dt \\!", "66772a2c34d52439c5d20c0a0fc11b2d": "\\mathrm{I\\!I}(v,w) = -\\langle d\\nu(v),w\\rangle\\nu", "667733cb87372b2a457e507cd4620494": "Q = (3,0)", "667744bcf6a6b8fd9b35f14a88f9182a": "\\forall A, \\exists B, \\forall C, \\forall D, P(C,D) \\rightarrow (C \\in A \\rightarrow D \\in B).", "66775c7b619cadbec356483fab96f7b9": "S_\\mathbf{BC}=S_\\mathbf{AD}", "66776c0329b42d595e2ed8478bc180e4": " l = 0.07L, ", "66777729f1c620eb3c8d7d4b4f563202": "r_1^2=x^2+y^2+z^2 \\, ", "6677a95e317ae2e43609a53d5cebfe8f": "x = X/Z^2", "6677d8970fd5621f4f4f8e729d8fc2da": "2x \\equiv 2 \\pmod {6}", "6677f578953af4312c17cc69bb63500a": "(p \\leftrightarrow q) \\vdash (q \\to p)", "66780fc0b179e4382ea01caee5ab73c2": "\\scriptstyle{(Rc)}", "66782d654d3129cc0abf92757df4be52": "v_j, j = 1, 2, . . ., n", "66785731ef0dd9a54916b95c055da700": "P_W = {{\\frac{A^N}{N!} \\frac{N}{N - A}} \\over \\sum_{i=0}^{N-1} \\frac{A^i}{i!} + \\frac{A^N}{N!} \\frac{N}{N - A}} \\,", "6679397deb4da08e1fc1530a4fae1c3f": "\n\\left(\\frac{du}{d\\varphi}\\right)^{2} = C - u^2 - G\\left(u\\right)\n", "66797ff24f48704810175f8b2d2f4b44": "T=\\sqrt{n}\\frac{\\bar{D}}{\\hat{\\sigma}_D}.", "6679977e1925462dd7477e3c184ae464": "\\frac{1}{1+e^{-\\eta}} = \\frac{e^\\eta}{1+e^{\\eta}}", "6679a5c222cab29d14c45a15416b1b1b": "{\\hat \\lambda}_i = \\left(1 - \\frac{c}{b + \\sum_{i=1}^p X_i}\\right) X_i, \\qquad i=1,\\dots,p.", "667a5b9586e5f361659b1d965933019e": "\nF - S = S \\left( \\frac{1+r_d T}{1+r_f T} -1 \\right) = \\frac{S (r_d - r_f) T}{1+r_f T} \\approx S \\left( r_d - r_f \\right) T ,\n", "667b04b909ed22c3e2878a83622364a0": "S^{(t)}", "667b417d65156212dcfe27bd869c778a": "I^*", "667bae4221ecc37363af67c33f45d345": "(A \\rightarrow (A \\rightarrow B)) \\rightarrow (A \\rightarrow B)", "667c03261998c94489c974b5176c1c0d": "\\hat{K}", "667c0a19339d6dbf2c454dc84ea31bbb": "\\frac{AB}{P_2 B}=\\frac{\\sin \\alpha_2}{\\sin \\phi}", "667c1389ea19f0b396195cf3588fab80": "\\frac{d F}{dt} = -\\int d^n x \\left|\\nabla\\mu\\right|^2,", "667c4e269474200eaddb432a12d3ebd4": " \\mathbb{Z}_p ", "667c750334667fd85fac7ccfcbb8c926": " f_L ", "667cf0f206a6c3fb69467674a81a7510": "\\cos \\varphi = \\zeta. \\, ", "667d1a67de5848ee115975d4c4017f8e": "N>4\\sqrt{q}", "667d56fef56202fbe9e1d2716a2c68b3": " \\rho_s (z,E) = \\frac{1}{\\epsilon} \\sum_{E- \\epsilon}^{E}|\\psi_n (z)|^2 ", "667d905e6da6f7c7b7b61c8cb41d9082": " M(0,H) \\simeq |H|^{1/ \\delta} \\mathrm{sign}(H)", "667da6d5a8c2b68fe5a5e2671e06e3c2": " \\Sigma(3,4,5)", "667e3a8ebde68147ca4a67874c008f43": "\\ G^{*}(f)=G^{'}+jG^{''} ", "667e7fcc0eda1a71a97fedbc37282e3e": " \\frac{1}{1-a z^{-1}}", "667e8b242782a6c281f4ecce25fb63ab": "\\langle 100 \\rangle", "667e9eff1caf57134418378ba5424316": " \\pi_2(x) < 4.5 \\frac {x}{(\\log x)^2}", "667ebd08e8a42cfe217a5bc3f8aabcbe": "\\scriptstyle C(f^*) \\leq C(f)", "667f1a1393d297c4a48bd7ceda03e68a": "\n \\operatorname{Var}[\\,T\\,]\\ \\geq\\ \\mathcal{I}_\\theta^{-1},\n ", "667fd32408dbc7377974e146007903cc": "p_{2T}", "6680254750ddf128e72ec156e5e34c47": "Df:U\\to B(V,W) ; x \\mapsto Df(x)", "668027dd5140772d1162776e88489d55": "H \\otimes N", "66807739f16f8aa6daa24ccf3cbad883": "F(x, f(x)) = c.\\,", "6680c47e12aaa2525a1611973546303f": "\nAF = \\frac{9.73}{\\lambda \\sqrt{G} }\n", "668132102d9c1c7faa5dbe6ee0a7f858": "{} 3.141024 < \\pi < 3.142704.", "66820444aaa06903e7f9774f1a1738bd": "x \\cup \\{x\\} \\in y", "668251ced559091f9834f844c6d25ed9": "\\phi : X \\rightarrow Y", "668287e5c2a91ef935bc71bd6a8c08a5": "S^1 a = \\{sa \\mid s \\in S^1\\}", "6682b412b92926f9cb8b14a8e427f387": "\n= 1\\,\\mathrm{k}\\Omega + \\left({1 \\over ( 1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega )} + {1\\over (2\\,\\mathrm{k}\\Omega ) }\\right)^{-1} = 2\\,\\mathrm{k}\\Omega.\n", "6682dfd04db5d52191b69be352b30669": "| \\psi(t) \\rangle = e^{-iEt / \\hbar} | \\psi(0) \\rangle.", "6682e03e1b8a67beccc123ce4490004a": "A \\hookrightarrow X", "668326c81e8557feb8f0921f2ba63326": "-G_g^fa I_g^f W_g^r + (R_g^a + R_m^a) I^a + (L_g^a + L_m^a) I^a + G_m^fa I_m^f W_m^r = 0", "66835d25a4b1f987875c9fb69a739adc": "\\nabla h=0", "6683646f4b321bcd08ec86441406720b": " \\bold{\\hat{p}} = -i \\hbar \\nabla \\,\\!", "668368b5a6bced26e233f2e512d833c1": "\\scriptstyle \\mathbf{x}=(x_1,x_2)", "668388ee8e1d390648de6f25d49fb985": "\\kappa=k/\\rho c_p", "6683c151ccc77efb0f7062920d20251c": "\\{W_{U}:U\\in\\mathcal{O},\\text{ meets }N\\}\\,", "6683d41bcfc14dfa1eba63e839c76ed8": "D(fg) = f \\cdot (Dg) + (Df) \\cdot g,", "6683d7c7f73dd6718b2c98911c055d23": "1 \\leq i \\leq 2^n", "668409c401748a016472ee95dbfff91b": "\n \\boldsymbol{N}^T\\cdot\\mathbf{n}_0 = \\boldsymbol{F}\\cdot\\boldsymbol{S}^T\\cdot\\mathbf{n}_0\n", "668464ed7597bca86527aca63757ee96": "\\mu^'_1=\\sqrt{\\frac{\\pi}{2}}L_{1/2}^{(k/2-1)}\\left(\\frac{-\\lambda^2}{2}\\right)", "668474a998f9c10c7a68defcc2cca43c": "f_\\mathrm{E}=Nr\\frac{d{\\omega}}{dt}.", "66847b7785fa37b7b8668e578a8b4872": "\\Delta pH = pH_A - pH_B", "668495cbced9ae630b6da7f4e5235cd1": "a_{14}+b_{14}", "66850441b61b8ce0f2b52fb597896b14": " \\delta W = \\left(\\sum_{i=1}^n \\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}_1}\\right)\\delta q_1 + \\ldots + (\\sum_{1=1}^n \\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}_m})\\delta q_m. ", "66854a20cdacbdb9f0d45976e60278d6": "EXP \\subseteq NEXP", "66854de345331a16d6402efce5dc478d": "\n\\begin{align}\n\\widehat\\alpha, \\widehat\\beta & = \\text{least-squares estimators}, \\\\\nSE_{\\widehat\\alpha}, SE_{\\widehat\\beta} & = \\text{the standard errors of least-squares estimators}.\n\\end{align}\n", "668563348f4921e763ca8023c24c3ce0": "({x^3}1+{x}10) - ({x^2}2+{x^0}1) = {x^0}5", "66857ca28b8fe84f91df166d28049c4d": "\\langle \\hat{\\mu}\\xi,\\eta\\rangle_{H_\\sigma} = \\int_G \\langle \\overline{U}^{(\\sigma)}_g\\xi,\\eta\\rangle\\,d\\mu(g)", "66858fc6ae96c10ea65df7d2c0c90266": "{\\delta}_{[k,j,c]}({t}_{1},\\cdots,{t}_{k}):=\n({t}_{1},\\cdots,{t}_{j-1},c,{t}_{j+1},\\cdots ,{t}_{k})\n", "668592fb8695ebc80f242028a7f07811": "\n \\quad (5) \\qquad \\epsilon_m(x,t) = e^{at} e^{ik_m x}\n", "66859db4e184ca4c6819ec1eae564ed8": "H(X_{1},X_{2},\\ldots ,X_{n})", "6685c272ef1ea24d138a7b9dc053ef58": "-1.0726", "6685e51449ddb0807122570a727a0961": "\\Gamma_q(x) = (1-q)^{1-x}\\prod_{n=0}^\\infty \n\\frac{1-q^{n+1}}{1-q^{n+x}}=(1-q)^{1-x}\\,\\frac{(q;q)_\\infty}{(q^x;q)_\\infty}\n", "6685e5884cae8db41de38d5228bec597": "E\\backslash \\bigcup_{i=0}^\\infty f_i\\left(\\mathbb{R}^m\\right)", "6685eb7a5c332d39e1d84e50d245cab8": "\n\\begin{align}\n \\frac{\\partial \\Phi_3}{\\partial t} + g\\, \\eta_3 = & \n - \\eta_1\\, \\frac{\\partial^2 \\Phi_2}{\\partial t\\, \\partial z} \n - \\eta_2\\, \\frac{\\partial^2 \\Phi_1}{\\partial t\\, \\partial z} \n - \\mathbf{u}_1 \\cdot \\mathbf{u}_2\n \\\\ &\n - \\tfrac12\\, \\eta_1^2\\, \\frac{\\partial^3 \\Phi_1}{\\partial t\\, \\partial z^2}\n - \\eta_1\\, \\frac{\\partial}{\\partial z} \\left( \\tfrac12\\, \\left| \\mathbf{u}_1 \\right|^2 \\right).\n\\end{align} \n", "6685fc155538bb21b7b8fb5745f368ff": "r = \\frac{\\sum ^n _{i=1}(X_i - \\bar{X})(Y_i - \\bar{Y})}{\\sqrt{\\sum ^n _{i=1}(X_i - \\bar{X})^2} \\sqrt{\\sum ^n _{i=1}(Y_i - \\bar{Y})^2}}", "668612fa3936c788a62a1f80e97ae3c9": " E = E_{00} + \\frac{R\\;T}{F} \\left( \\ln \\left( a[\\mathrm{H_3 O^+}] \\right) - \\frac{1}{2}\\ln\\left( \\;p[\\mathrm{H_2}] \\right) \\right)", "66865b101216f9a4ac6efb39aa248172": "\\begin{align} v & = a\\left( r - r_0 \\right)\\left( \\frac{2}{v+v_0} \\right )+v_0 \\\\\nv\\left( v+v_0 \\right ) & = 2a\\left( r - r_0 \\right)+v_0\\left( v+v_0 \\right ) \\\\\nv^2+vv_0 & = 2a\\left( r - r_0 \\right)+v_0v+v_0^2 \\\\\nv^2 & = v_0^2 + 2a\\left( r - r_0 \\right)\\quad [4] \\\\\n\\end{align}", "66866f87be70af503b0f88ebb01ba4c6": "\\mu_s^0(l,p)=\\mu_s(l,x_s,p+\\Pi)", "668688881721a7f64076ab23f6821424": "{\\mathcal L}_{xx}^2: L=Lclm(l_2,l_1);", "66869a2f1e40e14c20cf99851780fa1f": "\\ a_{i}", "6686e0d0fda5627c015ebc472384baed": "k\\in\\mathbb{Z}", "66871eec2e790b80f90dadd57dcb7ab5": "\\frac{1}{2k - 1} - \\frac{1}{2(2k - 1)} = \\frac{1}{2(2k - 1)},", "66872789236ef9e1f149fe6867dbf77c": "\\left(\\begin{array}{c}\nX_{3}\\\\\nY_{3}\\\\\nZ_{3}\n\\end{array}\\right)=\\left(\\begin{array}{ccc}\n\\cos\\Phi & \\sin\\Phi & 0\\\\\n-\\sin\\Phi & \\cos\\Phi & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{c}\nX_{2}\\\\\nY_{2}\\\\\nZ_{2}\n\\end{array}\\right).", "6687415a7b27f20e50324338b9290039": "\\displaystyle f(x) = \\int_{\\mathbf{R}^n} \\hat{f}_1(\\xi) e^{2 \\pi i x\\cdot \\xi}\\, d\\xi \\ ", "6687a70eca9c46560e12535479ca1184": "(w_1, w_2, \\dots, w_k)", "6687bc584e999ebd371ddfabaef13029": "\\begin{align}\ny &= h(x)\\\\\n\\dot{y} &= L_{f}h(x)\\\\\n\\ddot{y} &= L_{f}^{2}h(x)\\\\\n&\\vdots\\\\\ny^{(n-1)} &= L_{f}^{n-1}h(x)\\\\\ny^{(n)} &= L_{f}^{n}h(x) + L_{g}L_{f}^{n-1}h(x)u\n\\end{align}", "6687c1d775a254e6a7f1968cdf21283f": "X=\\frac{y'}{yx'-xy'}", "6687dd5007ca9163a97ed1db976cb11a": "\\hat{\\beta} = \\frac{\\bar{X}_P - \\bar{X}_N}{\\sqrt{s_P^2+s_N^2}}.", "6688179088578bf8c426e3ca5bd1df0e": "U(p):=\\int_{-\\infty}^0 v(x)\\frac{d}{dx}(w(F(x)))\\,dx+\\int_0^{+\\infty} v(x)\\frac{d}{dx}(-w(1-F(x)))\\,dx,", "668837d3264d17847da2005a7f319a75": "V(S, t)", "668843d50f4eefb34a49d1e62302eadd": "{1 \\over 3}\\times \\left( {2 \\over 3} + {1 \\over 3} + {1 \\over 2} \\right) = {1 \\over 2}", "6688741c7d60de5204298e0b2481eb16": "c = \\sqrt{E_0/\\rho}", "66889fb8b893e48090a6ef784a355f9c": "\\, \\eta_ts\\, = \\, \\frac{\\psi\\,u_2^2}{[C_p\\,T_{01}(1-(\\frac{p_3}{p_{01}})^{\\frac{(\\gamma - 1)}{\\gamma}}]}", "6688a994080b6dcb5112d1e8ce1db06b": " n = (1,\\cdots,k)", "6688bf070a3030ab87dcc34603f63c72": "\\bar{\\Phi}(s;L)", "668a28b1bfba178a13cb2d561630833f": "\n \\begin{align}\n EI\\dfrac{dw}{dx}(a_{+}) &= \\dfrac{Pba^2}{2L} + D_1 & &\\quad\\mathrm{(vii)}\\\\\n EI w(a_{+}) &= \\dfrac{Pba^3}{6L} + D_1 a + D_2 & &\\quad\\mathrm{(viii)}\n \\end{align}\n ", "668a5209f9adc30bfa5074a535387c05": "{}_{1}F_{1}(\\cdot;\\cdot,\\cdot)", "668a5460af8e4ea2b541127147b4424b": " C_D = C_D ( C_L , M , Re ) \\equiv ", "668a618251444980ca29fdb157449792": "F(x+iy) = \\int_{\\mathbb{R}^n}e^{2\\pi z\\cdot t}f(t)\\,dt.", "668a723e4021a1a126a0a22b3d9cdb64": "(M-1)q_1 q_2...q_n", "668a763346ec654724b762de44b5397f": "a=1:", "668a7c6116c34b31af75551bb73f3ae5": "\\lim_{\\rho \\to 1} \\frac{x^{1-\\rho}-1}{1-\\rho} = \\lim_{\\rho \\to 1} \\frac{-x^{1-\\rho}\\ln x}{-1} = \\ln x.", "668abed38fd3130015b9b1c699076e5d": "1/0=\\infty", "668b12695425c31e6bcc7a0e00d94bec": "\\lambda_{\\hat{x}}^\\top \\mathbf{P}_{X|z}", "668b368266d86888ee9154b94a93f3ee": "\\delta \\boldsymbol\\phi", "668b5f7825c9ecb0975b5a35954d886d": "\\chi(\\omega)", "668b7bb6596f44886fd8b89bc6fdd58b": "s\\sqsubseteq t ", "668b8586484e3737be4e8c9892f2589e": "W=U\\cap V=\\{(a^n,b^nc^n)\\mid n\\in\\mathbb N\\}", "668bac40b650182111d8b93e1b127c98": "x:1{\\to}\\tau", "668c24b5b73957437751fe0a536cf074": "\\textstyle b_i = 0", "668c574d747d80eefeda9aa662381e6d": "\\tan \\delta (T)", "668c67d05c13d5bd7d4620bab057d724": "\nK_{d} = \\frac{\\left[ \\mathrm{P} \\right] \\left[ \\mathrm{L} \\right]}{\\left[ \\mathrm{C} \\right]}\n", "668c77c1217824938432e6d8f3ab95d3": "n_{a}", "668c7b55a37300c330dcd565d9e076da": "(0,1]", "668c7ec1714eb37c6cadf8b658cf3be2": "1/2^{O(T)}", "668d10ed3ce57bbebe8a3b05d5ec719f": "2x + 4 = 12", "668d9a540e454d99de49895b261e13d4": "\\mu_Y^\\pi = \\sum_{i=1}^{\\widetilde{n}} \\alpha_i \\phi(\\widetilde{y}_i)", "668db72f9aff1e46c2c3891f9599122f": "\\mathcal{C}_{m}=\\mathcal{C}_{1}\\mathcal{C}_{m-1}-\\mathcal{S}_{1}\\mathcal{S}_{m-1}, \\mathcal{S}_{m}=\\mathcal{S}_{1}\\mathcal{C}_{m-1}+\\mathcal{C}_{1}\\mathcal{S}_{m-1}, \\mathcal{S}_{0}=0, \\mathcal{S}_{1}=\\mathbf{R}\\cdot\\mathbf{\\hat{e}}_2, \\mathcal{C}_{0}=1, \\mathcal{C}_{1}=\\mathbf{R}\\cdot\\mathbf{\\hat{e}}_1", "668db8ba595fbbb68b8845611ae1cea5": "\\displaystyle g_{ij}(x,\\xi)", "668ddc04febef33cff91fe515ee4ec3a": "p(\\tfrac{x}{2})", "668e4982a9e0dc7559a1db0f260affe1": "\\Gamma_{12}(u, v, 0) = \\iint I(l, m) e^{-2 \\pi i (ul + vm)} \\, dl \\, dm", "668e680de0a96c5639634b1679da7f61": "(R, m)", "668e70725bb38f0eeead643f1ad1fe69": "\\Theta_{ii}", "668e764c1445231d57012ba41eb15aba": "\\Delta=\\lambda_1\\lambda_2", "668ec86dfe5be1ed6e29ff4743264698": "G_{1}", "668ee6aa7cafb5e4b1c27edf6b3f7f27": "\\nabla = \\boldsymbol{\\hat \\rho}\\frac{\\partial}{\\partial \\rho} + \\boldsymbol{\\hat \\varphi}\\frac{1}{\\rho}\\frac{\\partial}{\\partial \\varphi} + \\mathbf{\\hat z}\\frac{\\partial}{\\partial z},", "668f270975371dc9e3395cfb18fbdec9": "f(m)", "668fa6e4ae46110c23ec8e76a75de2ea": "\\left(\\frac{V_{CE}}{BV_{CBO}}\\right)^{\\!n}= 1-\\alpha \\iff V_{CE}=BV_{CEO} = \\sqrt[n]{(1-\\alpha)}BV_{CBO}=\\frac{BV_{CBO}}{\\sqrt[n]{\\beta+1}}", "668fba1f15a6a85cd816b7a8f0b62947": " \\Delta P = \\frac{2 \\gamma}{R} , ", "668fd6d2697c709432be55b3aca5ed67": " G^*=\\alpha S^* + (1-\\alpha)e e^{T}/N", "668ff2a777c2565bcc1f77d8dd0ca57e": "\\Phi~", "668ff71db33be59665571c19ecd81e21": "d(X,Y) := \\frac{\\left| E(X,Y) \\right|}{|X||Y|}", "6690652eb79b054147539685d74b36ec": "x^{\\left [ 0 \\right ]} = \\log x", "669079cb0eaa06c092a800f2f967536d": "A:x \\mapsto 2\\sqrt{x+\\tfrac{3}{8}} \\, ", "6690ab019bdb942d299029f21ea5fcb5": "w_0(n)\\,", "6690ba82b5e7ad7835f4d6acc64d4a27": "F=dA", "66911070bee73ca216ba503ab7667746": "B_{k}", "669122e03508f9f4ea73fdbb73e5deff": "f(x+\\Delta x) = f(x) + {D_x}^\\alpha f(x+)\\frac{(\\Delta x)^\\alpha}{\\Gamma(\\alpha+1)} + {D_x}^\\alpha {D_x}^\\alpha f(x+)\\frac{(\\Delta x)^{2\\alpha}}{\\Gamma(2\\alpha+1)} + \\cdots.", "66913963194417e9b8533d2be5d9b874": "E=E^0 + \\frac{RT}{nF} \\ln \\frac{[\\text{oxidized species}]}{[\\text{reduced species}]}", "66914cea3ee1f27af36e09148e7f6eef": "\n x(t) = A s(t) = \\begin{bmatrix} A^\\mathfrak{s} & A^\\mathfrak{n} \\end{bmatrix} \\begin{bmatrix} s^\\mathfrak{s}(t) \\\\ s^\\mathfrak{n}(t) \\\\ \\end{bmatrix},\n", "669165b2d87746f4241c75db37fb7953": "\\lambda^{1/2}_k", "669175c8834ea7fe41d422efb03d6dbe": "G(k, \\chi)", "66917dad7c728ec651c8760e2838f044": "\\sec(M_i)", "66917e0aa43baa38e4f1e34ec6481022": "N_{th}=N_{tr} + \\frac{1}{\\alpha\\tau_p\\Gamma}", "66924f45b787b28fbec58c5ee1514b23": "\\begin{align}\n {[}\\,\\underbrace{2053 - 2050}_{\\begin{smallmatrix} \\text{Deviation from} \\\\ \\text{the population} \\\\ \\text{mean} \\end{smallmatrix}}\\,]^2 & = [\\,\\overbrace{(\\,\\underbrace{2053 - 2052}_{\\begin{smallmatrix} \\text{Deviation from} \\\\ \\text{the sample mean} \\end{smallmatrix}}\\,)}^{\\text{This is }a.} + \\overbrace{(2052 - 2050)}^{\\text{This is }b.}\\,]^2 \\\\\n & = \\overbrace{(2053 - 2052)^2}^{\\text{This is }a^2.} + \\overbrace{2(2053 - 2052)(2052 - 2050)}^{\\text{This is }2ab.} + \\overbrace{(2052 - 2050)^2}^{\\text{This is }b^2.}\n\\end{align}", "669254775ef0db366c017dccf53374dd": "M_{ij}", "6692ad2e7e7ac8c411f0dd26a75cfd6d": "Y \\subseteq X \\subseteq H~\\Rightarrow~X \\rightarrow Y \\in S^+", "6692c28b3cfe6c43572d4dd5c0ac8251": "\\lim_{a\\rightarrow\\infty}\\int_{-2a}^a\\frac{2x\\,\\mathrm{d}x}{x^2+1}=-\\ln 4.", "66930e973050b4798da65d304c0e78fc": "f^{\\star} \\left( x^{*} \\right) := - \\inf \\left \\{ \\left. f \\left( x \\right) - \\left\\langle x^{*} , x \\right\\rangle \\right| x \\in X \\right\\}.", "66937a02aabecbe52bcdbae81a259e8c": "s_w \\geq \\frac{I}{Y} \\to s_w \\geq g_nk", "6693be49ee15e3d455aa0feeed5d1a52": "\\varepsilon=0.01", "669436b283fae575cb4c6e17b6d3ed4d": "EP_f", "6694b11db72292c83cb29d5f6d8c0279": "y(x)=\\begin{cases}\n 1&\\mbox{ if }x\\geq\\beta\\\\\n 0&\\mbox{ if }x\\leq\\alpha \\\\\n k&\\mbox{ if }\\alpha 1", "66960d54855bf80d64b70a0258f1dd33": "|x| = x \\sgn(x),", "66967dd2ca7bd84bc6f2fb40f7e3c094": "y_0 = Y_0", "66967ef13d82d30dbc2706328faf6c54": "(\\lambda,\\,\\nu)", "6696a8a6706903c3448094fa7c25b034": "W(\\alpha,\\alpha^*) = (-1)^n\\frac{2}{\\pi} e^{-2|\\alpha|^2} L_n\\left(4|\\alpha|^2\\right) ~,", "6696d14d1f941d27562fe5f327d8d68d": " \\det A_{{\\lang i_1,\\dots,i_m} \\rang} = 0 \\,\\!", "6696e8482afe35fa47228b42a799fe50": " x \\preceq y \\preceq x \\implies x = y", "66974acdedc4b8f714a9f9d838520b75": "\\mathcal{L}_\\mathrm{H} = \\varphi^\\dagger\n\\left({\\partial^\\mu}-\n{i\\over2} \\left( g'Y_\\mathrm{W}B^\\mu + g\\vec\\tau\\vec W^\\mu \\right)\\right)\n\\left(\\partial_\\mu + {i\\over2} \\left( g'Y_\\mathrm{W}B_\\mu\n+g\\vec\\tau\\vec W_\\mu \\right)\\right)\\varphi \\ - \\ {\\lambda^2\\over4}\\left(\\varphi^\\dagger\\varphi-v^2\\right)^2\\;,", "6697d8b44f59f03a62ad318118adefbc": "v = v(u)", "6698095eb77507643d863a9b702987cb": "4n-2", "66982da809eb00444b167bd38b0fbf1b": " \\varphi = \\tan^{-1} \\left( \\frac{a t + b}{c} \\right) + d ", "669848b783ded7d0a0c4e139e75a3126": "\\begin{align}\nd(x_{(k + 1) + 1}, x_{k + 1}) & = d(x_{k + 2}, x_{k + 1}) \\\\\n& = d(T(x_{k + 1}), T(x_k)) \\\\\n& \\leq q d(x_{k + 1}, x_k) \\\\\n& \\leq q q^kd(x_1, x_0) && \\text{Induction Hypothesis}\\\\\n& = q^{k + 1}d(x_1, x_0).\n\\end{align}", "6698644366af506ab7ce0b74556de237": "\\scriptstyle \\ell", "66988293b1043aec92fde0de465c0ac1": "\\mathcal{L}u(x,t)=0", "6698eb64b369383cec5fec797975771f": "\\delta\\sigma=\\frac{2k}{R_{0}}", "66990031507b88fe6f66a6809bc8f586": "\\mathbb {M}(D)", "66990a57c57755f7ce6ad4bef4c93af1": " \\scriptstyle{Z_\\circ=\\sqrt{{\\mu_\\circ \\over \\varepsilon_\\circ}}= 376.730313461\\ \\Omega}", "66992cc6310b5e8922f808a3f95f97ca": "\nE_{\\mathrm{tot}} = \\frac{1}{2} m \\dot{r}^{2} + \\frac{1}{2} m r^{2} \\dot{\\varphi}^{2} + U(r) = \\frac{1}{2} m \\dot{r}^{2} + \\frac{m h^{2}}{2 r^{2}} + U(r)\n", "6699364c936be9501f2a8073be55f796": "A \\oplus B = \\{z \\in E| (B^{s})_{z} \\cap A\\neq \\varnothing\\}", "66993800040db708a34362282c5f36a8": " \\frac{\\sin\\alpha}{\\sin\\beta} < \\frac{\\alpha}{\\beta} < \\frac{\\tan\\alpha}{\\tan\\beta}. ", "66995f570cdf1ce2ebb0107b14fea707": " r_t ", "6699765d55395121aaf936e9189b5212": "\\ln(1 + u) = \\sum_{k = 1}^{\\infty} \\frac{(-1)^{k-1}u^{k}}{k} ", "669a9413754c67ad6f6edb37f147631c": "(a,b)_2 = (-1)^{\\epsilon(u)\\epsilon(v) + \\alpha\\omega(v) + \\beta\\omega(u)}", "669ab1435b5b7e8983eaa70a50d96fb3": "V_{\\mathbb R} \\otimes_{\\mathbb{R}} \\mathbb{C}= V_{\\mathbb{R}} \\oplus iV_{\\mathbb{R}}\\,", "669acc1302717779cad82f9ae4c55771": "(\\mathbb{Z}_n, +_n)", "669b216138217b88438db411cb0c73c2": " T^p_nI(x) = -x+n ", "669b32c7221884f8721622366a3cd4ed": "\\min_{\\lambda\\in\\sigma(A)}|\\lambda-\\mu|\\leq\\ \\kappa_p(V)\\|\\delta A\\|_p.\\,", "669b48e8b9ea14ce5573adc8ae098765": "a=2(k^2-m^2), \\, ", "669b4f2459b09ea77bddd604c30a30b2": " \\mu(\\lambda A + (1-\\lambda) B) \\geq \\mu(A)^\\lambda \\mu(B)^{1-\\lambda}, ", "669b60b7fae7fdce0c788bcbe67f964d": "\\theta_\\text{min}", "669b6bdd37676868a05983734bc062da": "S^1 \\times D^2", "669b7a64ab6d987645894df1d0623f7b": " E_{11}E_{22}-E_{21}E_{12}", "669b81f4251eedfaa5cde1bc81d9e0b5": "f(t, \\bar{u}) = f(t, x, y, z)", "669c1ba7ba6aab1770f182d14aa9aa1f": "\\text{ARR} = \\frac{\\text{Average return during period }}{\\text{Average investment}}", "669c29b7c8bc996ded520250da09111c": " \\left ( \\frac{\\operatorname{d}v}{\\operatorname{d}t} \\right ) _c ", "669ca3787cafcdc4011129d786b20184": "\\int_X (a * b)_A \\smile c = GW_{0, 3}^{X, A}(a, b, c).", "669cc91d521fc677abfc84cbfe155cff": "\\{-1,1\\} ", "669d38c4ea175be9a23e81db18733b8d": "a_1,\\ldots,a_m\\in R", "669d6733109d9df401da0a1e69adcc44": "\\bigcap A_\\alpha^C", "669d706a07d958c2adce9ba3b1e62565": "\n\nR_{stack} \\left( \\eta \\right)\\,\\,\\, = \\,\\,\\,{{F_{stack} \\left[ {\\dot C\\left( \\eta \\right)\\,\\,\\, + \\,\\,\\,\\lambda \\,\\int_0^\\eta {\\dot C\\left( \\tau \\right)\\,\\,d\\tau } } \\right]} \\over {\\varepsilon \\,k\\,F_m \\,\\phi }}", "669d79de669abdd90a3fe5497002dcfc": "Y = \\frac {1} {Z} = \\frac {1} {R + j X} = \\left( \\frac {R} {R^2+X^2} \\right) + j \\left( \\frac{-X} {R^2+X^2} \\right) \\,", "669dbce353a049ff18739c92fc4b6734": " C=c_0I+c_{1}P+c_{2}P^2+\\ldots+c_{n-1}P^{n-1}=f(P).", "669dc35b2c19ffdb62bea23791cfb030": " \\dot{M} = \\frac{4 \\pi \\rho G^2 M^2 }{c_s^3} ", "669de994f9046fcb5d69858c53b49c66": "\nA \\times (B \\vec{r}) - C\\vec{r} =\n\\begin{bmatrix}\n 2 & 3 \\\\\n 3 & 4\n\\end{bmatrix}\n\\left(\n\\begin{bmatrix}\n 1 & 0 \\\\\n 1 & 2\n\\end{bmatrix}\n\\begin{bmatrix}1 \\\\ 0\\end{bmatrix}\n\\right)\n-\n\\begin{bmatrix}\n 6 & 5 \\\\\n 8 & 7\n\\end{bmatrix}\n\\begin{bmatrix}1 \\\\ 0\\end{bmatrix}\n=\n\\begin{bmatrix}-1 \\\\ -1\\end{bmatrix}.\n", "669e5b838dfe152b49ac05e638bb6bb3": "N(t)=M(t)+A(t)", "669ebb4957bc4ef27c990f8623a09000": "P = g m V_g (K_1+s) + K_2 V_g^3", "669ec1a75c42e3d59eb74de6019449f2": "{\\lambda_i}^{+}", "669eec66f6ac35e04fb4d8c10c23eee7": "\\rightarrow_{c2}", "669f4dab88f070e0573f1b782a2fc790": "\\begin{align} P(q=x|s,f) & = {{{s+f \\choose s} x^{s+\\alpha-1}(1-x)^{f+\\beta-1} / \\Beta(\\alpha,\\beta)} \\over \\int_{y=0}^1 \\left({s+f \\choose s} y^{s+\\alpha-1}(1-y)^{f+\\beta-1} / \\Beta(\\alpha,\\beta)\\right) dy} \\\\\n& = {x^{s+\\alpha-1}(1-x)^{f+\\beta-1} \\over \\Beta(s+\\alpha,f+\\beta)}, \\\\ \n\\end{align}", "669fd51a9dca11e66ecf731c785ab885": "a \\not= b", "66a01006249200431a52740f3e0ceace": " \\phi: X \\setminus N \\rightarrow Y \\setminus M, \\quad ", "66a077796cedb382e0f8a968de7336f6": "(m,n)\\,", "66a0812b5622607b18cd55c2e2fc2c86": "G_s = \\frac{\\rho_s} {\\rho_w}", "66a081b830b46e44866d1c2cdf336f5e": "0<\\theta\\le\\pi", "66a088a08eab72ddadec66e5483d7137": "c x_1^{j_1}\\ldots x_n^{j_n}\\,\\!", "66a095b1b447f1db26cd692c4b1a1ef9": "0\\leq\\alpha", "66a0b8a4385839048b44e032fd8522c9": "1/a(v) = b(v) = 1/\\sqrt{1 - v^2/c^2} \\,,", "66a0bb494c967b4bafd807e6bfe29651": "\\scriptstyle D_F(6\\rightarrow 4)= 4(1)-2+0-2=0", "66a0bd633c16ae76630fccf1059bca76": " TV^{\\gamma - 1} = \\operatorname{constant} ", "66a123e5bbbdfcbf891f615a1ae836e7": "\\boldsymbol{p}(x) = h_{00}(t)\\boldsymbol{p}_{k} + h_{10}(t)(x_{k+1}-x_k)\\boldsymbol{m}_{k} + h_{01}(t)\\boldsymbol{p}_{k+1} + h_{11}(t)(x_{k+1}-x_k)\\boldsymbol{m}_{k+1}.", "66a12b1e8201b1605dd27a271e968526": "X_0=0 \\,", "66a174f53846dd7e3e6e660d660cefd4": "d(q-1)", "66a20e3d8d7c88b6a3da02114e891cdd": " |z| < 1", "66a2495256d3ffc0fdf8e8284c516f96": " \\begin{bmatrix} 1 & \\epsilon^n_b \\\\ \\epsilon^n_f & 1 \\end{bmatrix} \\begin{bmatrix} \\alpha^n_f & \\alpha^n_b \\\\ \\beta^n_f & \\beta^n_b \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}.", "66a26bc5983706f81f58bb84ea01f4e3": " \\sigma_{ij}", "66a31e22c7a9954809453c932efafa32": "\n{\\rm d}S = \\frac{{\\rm d}M}{T}.\n", "66a322b2b564a1e479e80b721d6d5d59": "G_n(\\mathbf{C}^\\infty)={\\lim_\\to}\\;_{k\\to\\infty}G_n(\\mathbf{C}^k)", "66a3495b0ad016dbb636893487732a29": "0 \\leq i", "66a3793e4235b7247f05ab5085fef61c": "u, u=u_1, u=u_2 ", "66a3cd4807d9a8b0c003aefdba8eafc0": " [f * g](t) = \\int_G f(s) g \\left (s^{-1} t \\right )\\, d \\mu(s).", "66a3d0b2d337d008b5cf1605605e7624": "c_{i+1} = a_i - a_{i+1}\\!", "66a3ecf1b62a7af6df713a34e6d9ea49": "\\delta^M \\,\\!", "66a4adac1180e238633d74e285cbe768": "H_{n}\\left(X\\right)", "66a4bce59fb42d78286c2b14e33aedc8": "b = B_x - A_x\\,", "66a4fc47ef3cee4aca500bf146987777": "D(x)=\\sum_{k=1}^x \\left\\lfloor\\frac{x}{k}\\right\\rfloor = 2 \\sum_{k=1}^u \\left\\lfloor\\frac{x}{k}\\right\\rfloor - u^2", "66a5116fdf9c8bf5425799579690e3e0": "y^2 = ex^4 + 2ax^2 + 1", "66a58adc3e15be015fb0fe7fedde53d7": " \nE[L(t)] - E[L(0)] + V\\sum_{\\tau=0}^{t-1}E[p(\\tau)] \\leq (B+Vp^*)t - \\epsilon\\sum_{\\tau=0}^{t-1}\\sum_{i=1}^NE[Q_i(\\tau)] \n", "66a5a220357d278500d9bbf0f7b78c32": "\\varphi_{\\beta+1}(0) [0] = 0 \\,", "66a5a5ab1308e6f3f680df6a7b168e20": "\ns^{2} = \n\\left( x_{2} - x_{1} \\right)^{2} + \\left( y_{2} - y_{1} \\right)^{2} + \n\\left( z_{2} - z_{1} \\right)^{2} - c_0^{2} \\left(t_{2} - t_{1}\\right)^{2}\n", "66a5b73df7546eecc1e27c0793ce781d": "a>a_t ", "66a5ff515572034b4478244fcbd0ef94": "V = \\bigoplus_{i \\in I} V_i.", "66a61ed595ce053df0dd332c4d3068c8": "\n\\begin{align}\n\\int_0^{2\\pi} F'(p) \\, dp & =4\\int_0^{\\frac{\\pi}{2}} F'(p)\\, dp=F(2\\pi)-F(0)=4(F(\\begin{matrix}\\frac{\\pi}{2}\\end{matrix})-F(0)), \\\\[10pt]\n& =2\\pi F[0,2\\pi]=2\\pi F'(0 < P < 2\\pi), \\\\[10pt]\n& =2\\pi F[0,\\begin{matrix}\\frac{\\pi}{2}\\end{matrix}] =2\\pi F'(0 < P < \\begin{matrix}\\frac{\\pi}{2}\\end{matrix}).\n\\end{align}\n", "66a650421c98856a151786e4b02b2d32": "\\textstyle -\\mathbf{b}_2", "66a65dcb38a8d4b6aaeb184586883baf": "\\sin A = \\frac{2\\sqrt{(s-a)(s-b)(s-c)(s-d)}}{(ad+bc)},", "66a6701f15bdc3680d9e65dfb343263d": " \\mathrm{Li}(x) \\sim \\frac{x}{\\ln x} \\sum_{k=0}^\\infty \\frac{k!}{(\\ln x)^k}\n= \\frac{x}{\\ln x} + \\frac{x}{(\\ln x)^2} + \\frac{2x}{(\\ln x)^3} + \\cdots. ", "66a69fa45cefd10091421d70363fc21e": "\\lambda_x(\\alpha)", "66a6ca988d64495597e916ddeb5d7fb6": "A(z)P(z)", "66a6e985cfd5c5a873fe55369396b20a": "i_p = 0.4463 \\ nFAC \\left(\\frac{nFvD}{RT}\\right)^{\\frac{1}{2}}", "66a7c962b8bdb307d23e4a826bc3d38d": "\\theta_{i=1 \\dots N}, \\phi_{i=1 \\dots N}, \\boldsymbol\\phi", "66a7fc0532dff8cb2fb6ce7bfb1e56d1": "T_c=\\left(\\frac{n}{\\zeta(3/2)}\\right)^{2/3}\\frac{h^2}{2\\pi m k_B}", "66a8012f365363fb763d8ceec9beae93": "P_f-U=0", "66a8584ed720d983c4f4d5552535f265": "5F^2_{\\frac{p \\pm 1}{2}} \\equiv \\begin{cases}\n\\tfrac{1}{2} \\left (5\\left(\\frac{p}{5}\\right)\\pm 5 \\right ) \\pmod p & \\textrm{if}\\;p \\equiv 1 \\pmod 4\\\\\n\\tfrac{1}{2} \\left (5\\left(\\frac{p}{5}\\right)\\mp 3 \\right ) \\pmod p & \\textrm{if}\\;p \\equiv 3 \\pmod 4.\n\\end{cases}", "66a8828f94aabb7b92227846a2435490": "C = 0.6", "66a89a6eafdaf834ccd12f4ce9690424": "h_j(x^*) = 0, \\mbox{ for all } j = 1, \\ldots, l \\,\\!", "66a89f21778e27682a7f3b0a086b753a": "x_c = \\frac{F_0}{k}.", "66a8e0543112450ca0499b5c72fad110": "\\delta \\mathbf r_i", "66a90ac5ad56557cd29ed019a3eb5da1": " p(\\psi) = \\frac{1}{2 \\pi}", "66a93a0a60eff65dd4c12ce270c23ce4": " c(\\lambda)=c_{s_0}(\\lambda).", "66a9f9a889f254332a984334428897f6": "\\ y[n] = \\frac{1}{a_{0}} \\left(\\sum_{i=0}^P b_{i}x[n-i] - \\sum_{j=1}^Q a_{j} y[n-j]\\right)", "66aa8f083676b58a600672456808f691": " C_je^{\\alpha_j x} = C_j e^{\\chi_j x}\\cos(\\gamma_j x + \\phi_j)\\,\\!", "66aaa1e6b98627d184cb2c0961777e41": "E^{(t)}_1", "66ab193f9d1d2959360712b6d0132ea3": "r_0:=a;\\quad r_1:=b;", "66ab374621adfab56f681a9bc7ac359c": " W(t) = \\prod_{i=0}^n (t-x_i) ", "66ab886f1a6ed288835e588fbc79b7b3": " R_{\\rm specific} = \\frac{k_{\\rm B}}{m} ", "66ac06a084d0ad2f45199ad3c4015397": "\nG > \\frac{N-1}{\\sqrt{N}} \\sqrt{\\frac{t_{\\alpha/(2N),N-2}^2}{N - 2 + t_{\\alpha/(2N),N-2}^2}}\n", "66ac31d90fb5d652b836efc33a88aa0a": "\n\\psi _{\\mu }(\\tau )\\simeq 1/\\nu \\tau ^{\\mu +1},\\qquad \\tau \\rightarrow\n\\infty ,\n", "66ac3b4ba37dac46fa3841acb52610ab": "E(x^{2n})=\\frac{1}{2}\\int_{-1}^1 [1+\\cos(x\\pi)]x^{2n}\\,dx ", "66ac7aff5d4e9199cb09ff6d7fa67bdd": "f'(x) = \\begin{cases}-\\mathord{\\cos(\\tfrac{1}{x})} + 2x\\sin(\\tfrac{1}{x}) & \\mbox{if }x \\neq 0, \\\\ 0 &\\mbox{if }x = 0.\\end{cases}", "66ac86a3b60337d7549af36b78fc1369": "\n\n\\dot C_{RW} (t)\\,\\,\\, = \\,\\,\\,{{\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 } \\over {\\lambda ^2 }}{v \\over L}\\left[ {\\lambda \\,t\\,\\, - \\,\\,1\\,\\, + \\,\\,\\exp ( - \\lambda \\,t)} \\right]\\,\\,\\, + \\,\\,\\,{{\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 } \\over \\lambda }\\left( {1 - {{v\\,t} \\over L}} \\right)\\left[ {1\\,\\, - \\,\\,\\exp ( - \\lambda \\,t)} \\right]", "66acee2ddb5414b81da151c0705f25a9": "\\mathbf x[k+1] = e^{\\mathbf AT} \\left[ e^{\\mathbf AkT}\\mathbf x(0) + \\int_0^{kT} e^{\\mathbf A(kT-\\tau)} \\mathbf B\\mathbf u(\\tau) d \\tau \\right]+ \\int_{kT}^{(k+1)T} e^{\\mathbf A(kT+T-\\tau)} \\mathbf B\\mathbf u(\\tau) d \\tau", "66ad2ab0175330803439504eca21bf4d": "n\\tau_0", "66ad6d795a428a60ff7dc0891841846b": "\\sum_{k=0}^\\infty \\frac {\\zeta(k+n+2)-1}{2^k} \n{{n+k+1} \\choose {n+1}}=\\left(2^{n+2}-1\\right)\\zeta(n+2)-1", "66adcefcab7b03d060b03c9239e8994b": " f \\mapsto \\frac{1}{\\sqrt{2 \\pi}} \\left\\{\\int_{-\\pi}^\\pi f(t) e^{-i k t} \\, dt \\right\\}_{k \\in \\mathbb{Z}} ", "66ae76738843aded5c1e1236a8ab71ac": "\\Delta=p^2-4q \\,.", "66aebf6458dd9235d054c9d4d8611cbc": "\\prime", "66aefed2dce402e2a059457723b3602d": "\\lambda^A(x) \\otimes \\lambda^B(y) = f^{A,B,C}(x) \\lambda_C (y)", "66af0c867649b7dc1c0c2d3ac8edeb0f": "\\|(x_1, x_2, \\dots , x_n) \\| = \\sqrt{\\sum_{i=1}^{n} x_i^2}", "66af1fc97570a318833f295a60193216": "2H", "66af2e0b5543dc0022e6e11f8aaa4fdf": "x = \\sum_{n=0}^\\infty a_n q^{-n}", "66af38df4dc0f94d82fa9220049326cd": "Z-n+1", "66afcea5373350363fc70c17c32eeb75": "\\mathrm{ld}(x)=\\log_2(x)", "66afe6085a56212671abe5101224a8ca": "A_{0,m}=\\left\\{x\\in A|d(f_n(x),f(x)) \\le 1\\ \\forall n\\geq m\\right\\}", "66b0759368beb47ab3427f5243e6e1c3": "T\\colon \\mathbb{R}^2 \\to \\mathbb{R}^2,", "66b093278b8972c92ad734191901dd81": "\\begin{align}\nJ_\\alpha(iz) &=e^{\\frac{\\alpha i\\pi}{2}} I_\\alpha(z)\\\\\nY_\\alpha(iz) &=e^{\\frac{(\\alpha+1)i\\pi}{2}}I_\\alpha(z)-\\frac{2}{\\pi}e^{-\\frac{\\alpha i\\pi}{2}}K_\\alpha(z).\n\\end{align}", "66b0d69d7d795879bcc9daf3a80467dc": "f^*(x_{i_0})\\geq f^*(x_i)", "66b0e754940259da50a7937ef0bd1a42": "x^4+4x^3+10x^2+15x+5xy+15y+10y^2+4y^3+y^4.", "66b1130708a4518d1bb47659d0a104c5": "\nT(z) = \\frac{1-\\vert z \\vert^2}{2} + 2 \\log{ \\left| \\frac{1-z+\\sqrt{(1-z e^{-i \\varepsilon})(1-z e^{i \\varepsilon})}}{2\\sin{\\frac{\\varepsilon}{2}}}\\right| }\n", "66b129bc25090be160005101c55c8095": "\\mathbf{L} ", "66b134107df6d224ea0c94152895ee55": "y=2a\\{ft\\}-a\\,", "66b1356df85c7a2f48e35895132338bc": "(1-\\sqrt2)^n \\approx (-0.41421)^n", "66b15f3391fae370c8a9f3ee6b729dec": "\\sum H(X_{i})", "66b168bbc07ee3e0b9abe112fa012aa6": "F_{3n} = 2F_n^3 + 3F_n F_{n+1} F_{n-1} = 5F_n^3 + 3 (-1)^n F_n", "66b184bfe84ac1285ff4fca72228145c": " A(q,\\dot{q},t)\\ddot{q} = b(q,\\dot{q},t), ", "66b1cfcb9291bedae195eb982ab7f075": "\\frac{\\partial \\omega}{\\partial k}", "66b1fa59f12cdfef01011b65e809872b": " T \\frac{ds}{dn} = \\frac{dh_0}{dn} + ||V||\\omega ", "66b1ff30a638397833c92644f6aadc0b": "rank(a'^{(g+1)})", "66b2083016bb2aa14f2775fdbb08c1c2": "(U_1,U_2,\\dots,U_d)", "66b2131291035cad1293d0c8ea4a0b51": "L \\to aW~~~~~~~~~~\\text{probabilities of generating 4 possible single bases on the left}", "66b242812601e3a24cc12ef5262aa75c": "|z-\\rangle_B", "66b284af2af4c8adb22eb1ad5e59d825": "\\mathcal{B}_s", "66b2870a8930bb6e6ab2e75e2a03f7ee": "x^*(q)= \\arg \\max p(x;q) ", "66b28d5efcd1139796ba1ae53ba93cb8": "\\scriptstyle\\ P_\\text{I}\\,", "66b2ba0702c7bb38310dcd500a2ddfcb": "5x^2+4", "66b2e550a17fbdb3f10453617320c814": "p(\\cdot,\\cdot)", "66b2fe14b76ca54d02e5837300c47a41": "\\ y_1 ", "66b313681cf85a363f92a5d7eff56326": "p_0=p, p_1, \\dots, p_m", "66b31906bc26bd3e3e0cd79860cdc231": "\\frac{3 \\times (-2.5^2\\pi) + 5 \\times 10^2 + 13.33 \\times \\frac{10^2}{2}}{ -2.5^2\\pi + 10^2 + \\frac{10^2}{2}} \\approx 8.5 ", "66b3595485185356a281f506729c4903": " \\begin{align} \nz = \\frac{[Z]}{[Z]_0 } = G(v_1, v_2, J_1, J_2) &= \\frac{ 2 v_1 J_2}{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}.\\\\\n\\end{align}", "66b387a76c7e12d0b748ef619bec89ae": "\\mu_n(X\\cap B_n)", "66b3a11f5e71a2c5357b732907f2d9a4": "B=\\frac{{\\eta}I}{{\\epsilon_x} {\\epsilon_y}}", "66b48084fcd1e835c3e8f0357d1906d3": "I\\in {}^H_H\\mathcal{YD}", "66b4c5e0261c5b85b62ffc7f4d986c38": "L(0) = \\{K_{5}, K_{3,3}\\}", "66b4f5f54063edc97d0283a1fb3e2b5d": "\\begin{align}\\hat{S}_x = {\\hbar \\over 2} \\sigma_x\\\\\n\\hat{S}_y = {\\hbar \\over 2} \\sigma_y\\\\\n\\hat{S}_z = {\\hbar \\over 2} \\sigma_z \n\\end{align}", "66b502532c606700fdffa5518d96a0d6": "t(t-1)(t-2)...(t-(n-1))", "66b513de776f5bb36919ba536b87c18d": "\\overline{A}\\in\\mathcal{N}_x", "66b525865417699bae8e52f0504b34d1": "Q(x + \\alpha,y + \\beta) = \\sum_{i,j} a_{i,j} (x + \\alpha)^i (y + \\beta)^j", "66b52dd8d055e2ea9d4984dc09794241": "\\mathcal{A} (E) := \\{ T \\in \\mathrm{Lin} (E; F_{T}) | T \\mbox{ surjective and } \\dim_{\\mathbb{R}} F_{T} < + \\infty \\}.", "66b578cc6ebff34a902b1567f494832f": "\\ddot{x_0}+\\ddot{x_1}=\\left[g(x_0)+x_1 g'(x_0)\\right]\\cos(\\omega t)", "66b594a4d8fd1ccdaa3617a68643307c": " \\Omega = \\{ X=x_1 \\} \\uplus \\{ X=x_2 \\} \\uplus \\dots ", "66b59bb629285df4bb51446fbc1cf496": "\\omega _\\pm = \\frac{\\omega _c}{2} \\pm \\sqrt{\\left({\\frac{\\omega _c}{2}}\\right)^2-{\\frac{\\omega _t^2}{2}}}", "66b5d5020c9de6dcd1ab8c13f046ae8b": " \\mathcal{B}^2 = 4 \\mathbb{I} - [A_0,A_1][B_0,B_1] ", "66b60a56c50b11b5d8f89985bacf4ff6": " \\mathrm{adj}(\\mathbf{A}) = \\mathbf{C}^\\mathsf{T} ", "66b6682ff9ae498b1c423ec32339b1e6": "t_{i_1}\\ge t_{i_2}\\ge\\dots\\ge t_{i_n}\\,.", "66b66df233bacf82de6b884a178c34cc": "\\mathbf{X}=\\exp (\\mathfrak{m}_+)\\cdot K_{\\mathbb{C}} \\cdot \\exp(\\mathfrak{m}_-)=\\exp (\\mathfrak{m}_+)\\cdot P", "66b6b3497d05b48842511daca4ccd880": "\\mathit{G}", "66b6c04b2ef796b322b39da5978d4c97": " \\mu(A)=\\lim_{k\\to\\infty} k \\cdot \\lambda\\left(A \\cap \\left(0,\\frac{1}{k}\\right)\\right),", "66b6ec50555207a2f864f7bccab52746": " a^2c ", "66b6eff1be220665e77872b1ec27c85c": "f^n(e)", "66b716bb6eb3813b9ac1041712ae5092": "f_2(d) = f_1(k/d).", "66b77596a8d90b1d6b47e798fc872b9c": "D_M X \\smile D_M Y", "66b7b3b09d2ad63b405d2411c98df55d": " \\lim_{k\\to\\infty} \\frac{|x_k-\\xi|}{|x_{k-1}-\\xi|^\\mu} = \\left| \\frac{f'''(\\xi)}{6f'(\\xi)} \\right|^{(\\mu-1)/2}, ", "66b7d24d45118e36d0b178104d708496": "I=\\ln \\dfrac{a}{b}", "66b7d6c703e047c314e52d647c7d2a29": "\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = 0.", "66b7d9f0d4ab1acebc91961e1d183bad": "f(T) = \\frac{1}{2 \\pi i} \\int_{\\Gamma} f(z)(z - T)^{-1} dz.", "66b7e0ab630026c7d3e9ffa0d9901158": "\n\\mathcal{L} = \\frac{1}{2} \\left(\n \\varphi_t^2 - a^2(\\varphi) \\varphi_x^2\n + \\sin^2 \\varphi \\left[ \\psi_t^2 - b^2(\\varphi) \\psi_x^2 \\right]\n\\right),\n\\qquad\na(\\varphi) := \\sqrt{\\alpha \\sin^2 \\varphi + \\gamma \\cos^2 \\varphi},\n\\quad\nb(\\varphi) := \\sqrt{\\beta \\sin^2 \\varphi + \\gamma \\cos^2 \\varphi}.\n", "66b7f032907f7100f446bb4d2404609b": "\\frac{d\\beta}{dt}=\\frac{Y_\\beta}{mU}\\beta-r+\\frac{Y_\\zeta}{mU}\\zeta", "66b87f7a9400ae1e26bef4ed90ec046d": "\n\\theta = \\arctan \\left( { \\sin (\\pi\\alpha) \\over ( \\tau_D / \\tau )^{\\alpha} + \\cos (\\pi\\alpha) } \\right) + \\pi\n", "66b8947603ced7be7bef7d20008a12fa": " = \\frac{y''(s)}{x'(s)} = -\\frac{x''(s)}{y'(s)} \\ ,", "66b897b82244ab2008b0690d6ca05157": "\\scriptstyle\\mathbb{S}", "66b8e007ab87b2bdf93f060981989c13": " V_\\parallel(\\beta) = e^{i \\phi} \\int E_s(s) \\exp\\left(i k_\\beta s\\right)\\,\\mathrm d s ", "66b9434cedb3aef60961c0078e4f1d85": "L_{\\bold{v}} f(\\bold{p})", "66b9596a5c8d10fa41a47806233d53e3": "\\mathrm{d}V=r^2\\sin\\theta\\,\\mathrm{d}r\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\varphi", "66b9787bf37612cdaed8eb7b34447996": "a\\in \\partial{A}\\cap \\partial{B}", "66b9795e5e0d53139d41389c332b2f60": "\\alpha_1=-4(\\gamma+1)\\,", "66b9c345baaced0b8852807918bc8184": "1/\\sqrt{8}", "66b9db5e45e6b06097139e8e615887cb": "l_w ", "66ba26d94e78276b461b38ad4399e204": "\\lambda (x \\oplus y) = \\lambda x \\oplus \\lambda y", "66baaefb71cd771bd347eb4b48b3036d": "38+\t22+\t11+\t54+\t43+\t27+\t6+\t59\t=\t\t260", "66bb140b8d756e3ccf443644315d2613": "T_p M = T_p N \\oplus N_p N := (T_p N)^\\perp", "66bb27a76e74b335a9e802ba9b4a9666": "\\tilde{\\textbf{y}}_k", "66bb950b78deb25bba2888a3288a7fcd": "\\mathcal{H}^1\\otimes \\mathcal{H}^2", "66bb9511ce2fcdd59ad389b6073b5074": "N = \\left\\{\\left. (x_0,\\ldots,x_{n+1}) \\right| -2x_0x_{n+1} + x_1^2 +\\cdots+x_n^2 = 0 \\right\\}.", "66bbaab99cfc0f468a6e2bc064b980b3": "\n \\begin{bmatrix}\n \\epsilon_{{\\rm xx}} \\\\ \\epsilon_{\\rm yy} \\\\ \\epsilon_{\\rm zz} \\\\ 2\\epsilon_{\\rm yz} \\\\ 2\\epsilon_{\\rm zx} \\\\ 2\\epsilon_{\\rm xy}\n \\end{bmatrix}\n = \\begin{bmatrix}\n \\tfrac{1}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & \\tfrac{1}{E_{\\rm y}} & - \\tfrac{\\nu_{\\rm yz}}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yz}}{E_{\\rm y}} & \\tfrac{1}{E_{\\rm y}} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\tfrac{2(1+\\nu_{\\rm yz})}{E_{\\rm y}} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm xy}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm xy}} \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\sigma_{\\rm xx} \\\\ \\sigma_{\\rm yy} \\\\ \\sigma_{\\rm zz} \\\\ \\sigma_{\\rm yz} \\\\ \\sigma_{\\rm zx} \\\\ \\sigma_{\\rm xy}\n \\end{bmatrix}\n ", "66bbad30b7b065522a79c11bac9082ab": "\n\\left \\{\n\\begin{array}{l}\nf_1(Z)=(1-cP)+bQ + 1, \\\\\nf_2(Z)=aP - lQ +1. \n\\end{array}\n\\right .\n", "66bbb61da735066fa89d3599f0515706": "H_{n,0}= n", "66bc9e135d806a105d9cbe0daa318f7d": "\\left\\{ {W^i} \\right\\}_{i = 1}^n ", "66bcf7b50e985e279ca486e81efb6e57": "g\\in C^\\infty(\\R)", "66bcff8f099e133a53f76cb01ab8a35e": "\\neg (p \\lor q) \\vdash (\\neg p \\land \\neg q)", "66bd12178de37955fa5c3775894a04c8": "I \\subset\n\\{1,2,\\dots,n\\}", "66bd6ef33d0fd40d8251529427a4b5ab": "\n \\psi^\\alpha(x,t) = \\langle x | \\alpha(t)\\rangle \n", "66bd83c3f434f6a8957edade458b231f": "\\frac{1}{\\rho} = \\frac{d\\theta}{ds}\\ . ", "66bdbe908c5dcda2ae828d5a70e610dd": "\\int_0^\\infty e^{-ax}\\cos bx \\, dx=\\frac{a}{a^2+b^2}", "66be194337d5fcebb8ef73823c69ffb1": "\\mathrm{Cu}^{++} + \\mathrm{Arsenomolybdic\\ acid}\\xrightarrow{\\mathrm{Oxidation}} \\mathrm{Arsenomolybdenum\\ oxide}", "66be5fa947ded1b525c1c6f949d87b40": "\\underline{b} \\ ", "66be8f5fe0b1dc44a1f087e8d0c60f11": "\\scriptstyle \\overline{B}(x, r)", "66bed02b5ebe2be18a94bf883aa4c346": "Z= \\mathrm{tr}(T^N).", "66bfa69c0d5be2337d75557c3bbf4472": "\\vartheta(x)<1.01624x", "66bfb36f5200559894203cee7b8fecbb": "\\sigma_{t}", "66bfdcffb8cfdc926bc0953ef744a219": "i=1,\\ldots, m", "66c006ff65d33c0de462c1075f8dcbd5": "T = \\frac{c_2}{A \\ln \\left(\\frac{C}{S} + 1\\right)} - \\frac{B}{A}", "66c007c15aeb02e04e6264a83bdfc8f1": "1_A(x) = \\begin{cases}1 &\\text{if }x \\in A\\\\ 0 &\\text{else}\\end{cases}", "66c008e2d1668504bd9e4c3b0b742a0a": "y \\le \\; ?(x) < y + 2d", "66c00f97359cc792a36f0b9b6ae0c331": "\\scriptstyle \\C[x]", "66c01652645b45bc1e24beb598ef02eb": "\n\\mathbf{Q} = \n\\frac{\\partial G_{2}}{\\partial \\mathbf{P}} = \\mathbf{g}(\\mathbf{q}; t)\n", "66c022baddf4f6fb08c5127aacb974ad": "\\ M_{pitch}= D_{pitch} \\times lift ", "66c0343ed73773bbd617b78d78ef2ef8": "-1/m", "66c04fed0d464310c5a7a9fd65f35660": "z(x,y) \\rightarrow z(x,y) - 4", "66c09ce6a3885a9e08d8682719104398": " f: X \\rightarrow \\mathbb{R} ", "66c0c87b1852d4cd8c9ac44cdd558edd": "\\| \\cdot \\|' : H^{k} (\\Omega) \\to \\mathbf{R}: u \\mapsto \\| u \\|' := \\sqrt{\\| u \\|_{L^{1} (\\Omega)}^{2} + \\sum_{| \\alpha | = k} \\| \\mathrm{D}^{\\alpha} u \\|_{L^{2} (\\Omega)}^{2}}.", "66c0ce18f277ae9b48c9105720da41ae": "H^{-i}(j_x^* IC_p) ", "66c12b7746f428f909e7703431c100ac": "\\begin{align}\nf_{X_1^n}(x_1^n)\n &= \\prod_{i=1}^n {1 \\over \\theta} \\, e^{ {-1 \\over \\theta}x_i }\n = {1 \\over \\theta^n}\\, e^{ {-1 \\over \\theta} \\sum_{i=1}^nx_i }.\n\\end{align}", "66c14d1f5bcfbfb6441fa4fef00229fc": "P \\and Q \\and R", "66c1be48a7696cf49ea5a4f1672d1033": "\\frac{1}{\\sqrt{n}}(\\dots, 0, 1, \\lambda^{-1}, \\lambda^{-2}, \\dots, \\lambda^{1 - n}, 0, \\dots)", "66c1e9c9357288ba6a1a52d275a97896": "i\\,", "66c1fe7985d706c74a62f2bfa156bbb0": " \ndf_{n+1}\\left(\\vartheta\\right)/d\\vartheta=f_n\\left(\\vartheta\\right)\n", "66c2264b4805e2b8e3ff916bc6a8154a": "\\begin{align}\n & \\hat \\varphi_{\\eta_j}(v) = \\frac{\\hat\\varphi_{x_j}(v,0)}{\\hat\\varphi_{x^*_j}(v)}, \\quad \\text{where }\n \\hat\\varphi_{x_j}(v_1,v_2) = \\frac{1}{T}\\sum_{t=1}^T e^{iv_1x_{1tj}+iv_2x_{2tj}}, \\ \\ \n \\hat\\varphi_{x^*_j}(v) = \\exp \\int_0^v \\frac{\\partial\\hat\\varphi_{x_j}(0,v_2)/\\partial v_1}{\\hat\\varphi_{x_j}(0,v_2)}dv_2, \\\\\n & \\hat \\varphi_x(u) = \\frac{1}{2T}\\sum_{t=1}^T \\Big( e^{iu'x_{1t}} + e^{iu'x_{2t}} \\Big), \\quad\n \\hat \\varphi_{x^*}(u) = \\frac{\\hat\\varphi_x(u)}{\\prod_{j=1}^k \\hat\\varphi_{\\eta_j}(u_j)}.\n \\end{align}", "66c234e35d6631897ea32edb0e490407": "\\frac{\\ddot{a}}{a} = -\\frac{4 \\pi G}{3}\\left(\\rho+\\frac{3p}{c^2}\\right) + \\frac{\\Lambda c^2}{3}", "66c24a196de169a5a366387891c135f6": "\\mathbb{C}^n", "66c290cc1262b28c293d9a2da75b987c": "\\lambda v ", "66c2c046537c84d605d004b77183f338": "(\\forall x\\forall y(f(x) = f(y) \\rightarrow x=y) \\land \\exists y\\neg\\exists x(y = f(x))),", "66c32b60bed38a4b54b1aabfa67426d9": "|x|1.\\,", "66c4d9bd69eb11d709a659ce6530a57b": "z_0\\in D", "66c529e5081b0c22898785410059d9ec": "V_S - 0\\,\\mathrm{V} = V_S", "66c5ac7222b60b2563584015073be8be": "x_\\mathrm{m} \\sqrt[\\alpha]{2}", "66c5e520b37d75faf97c440b779e883c": "abababbca", "66c6d2535054a9324f2d33de767d919c": "\\Sigma _i x_i^* = \\omega + \\Sigma _j y_j^*", "66c72bc9ea897abd506ea3fc372b5729": "\\left(\\frac{a}{p}\\right) = \\varepsilon(\\pi_a)", "66c73a06d746134f4dd57b470138a477": "\\mathcal{M} ", "66c7507ff9d114f0cd1e309599d7bde6": "F_y = -T \\frac{y}{L}-M\\;g", "66c761c8291cb4a1ecb1d5d14ad63a89": " R_i = {1 \\over \\sqrt{1 + \\tan^2 \\gamma_i}} \\begin{bmatrix} 1 & -\\tan \\gamma_i \\\\ \\tan \\gamma_i & 1 \\end{bmatrix} ", "66c7d0731a5b3cd4c3b6bb34afd93549": "y=r(s)", "66c80034655832bb5d7ba3676243ca80": "p_i(r) = \\sum_{i=1}^N c_i\\phi_i(r), ", "66c804354f8dca01847a1a368e763eb2": "\\scriptstyle\\sin(\\delta-90^{\\circ})=-\\sin(90^\\circ-\\delta)=-\\cos\\delta\\;,\\;\\cos(\\delta-90^\\circ)=\\cos(90^\\circ-\\delta)=\\sin\\delta", "66c866ffa506d266734bd6c795311744": "U=U(S,V).", "66c8b07882fd6eb044b61270ff8cb11d": "Q=\\frac{z^2[1-(a-b)^2] +z[2c(a+b-1)-4ab] +c(2-c)}{4z^2(1-z)^2}", "66c93470fe5ccef4c71b32aa629d3b39": " \\frac{d^2}{ds^2}\\,x(s) + k(s)\\,x(s) = \\frac{1}{R} \\, \\frac{\\Delta p}{p} ", "66c952e4f8b6c8a3365fd9fdca356045": "p=\\left( x_1,x_2,\\ldots ,x_N \\right)", "66c9a57b9fadd4484d2b6611efa1a89e": "E_{\\rm CISDTQ} \\approx E_{\\rm CISD} + \\Delta E_Q, \\ ", "66c9bb252cc053a5bbd93dad6f7580e6": " L = 280.460^\\circ + 0.9856474^\\circ n ", "66c9d1e1cefbbfd9c7e451cd6e7cbd3d": " \\varepsilon(p_1,\\ldots,p_r;X,L) = {d \\over \\sqrt{r}}.", "66c9d27a203f123b9532c8079c4c6db4": "\\omega = f_I\\mathrm{d}x^I=f_{i_1,i_2\\cdots i_k}\\mathrm{d}x^{i_1}\\wedge \\mathrm{d}x^{i_2}\\wedge\\cdots\\wedge \\mathrm{d}x^{i_k}.", "66ca2a77913aa1f20d91122c0e9a3a60": "\\frac{d^2 f}{dx^2} + \\frac{1} {x} \\frac{df}{dx} + \\left (1 - \\frac {\\alpha^2} {x^2} \\right )f = 0", "66ca6ab3364acd3b6174b39754a8ad64": "\\operatorname{dim}R \\le \\operatorname{dim}_k \\mathfrak{m}/\\mathfrak{m}^2", "66caaedf440dacdefd036ae0c14ba4b0": "(mask_e,genState) \\leftarrow GenWords(r,genState)", "66cae63d5841d357ffc7df26e21cc3f3": "d(x,y)\\leq\\max\\left\\{d(x,z),d(z,y)\\right\\}", "66cb38b54a11b38481c514d0a4e8471d": "\\left(\\frac{\\alpha}{\\beta}\\right)_n", "66cb574ddff4e6226cfd01030e751d42": "n!\\sim [{\\rm constant}]\\cdot n^{n+1/2} e^{-n}.", "66cb64301ce1edf081d4b0334dccf50d": "f(x)\\geq x", "66cb676b5dc25300d176511cbc9c5434": "\\{poly,interval\\}", "66cb6f31719a6c368ff63471b01856a0": "X_n\\ \\xrightarrow{a.s.}\\ X", "66cb91bb0786b98553245e67596f1592": " \\frac{d\\vec{P}}{dt}= \\int^t_0 \\mathbf{A}(t- \\tau )\\vec{P}( \\tau )d \\tau ", "66cba41eb365c4449a75bf50bc4b9fb5": "|d^{+} \\rangle | d^{-} \\rangle", "66cbec553109e9ab6f3f07a931cd1c58": "R_{\\mu\\nu} - {1\\over 2}R g_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = T_{\\mu\\nu}", "66cbfd2a07bbef35ecc750e299c4c835": "x = a - x,", "66cc236d89b0046db61722e959f63f1f": "\\vec a_g = - \\hat r ~ G ~ \\frac{M}{R^2} ~ \\frac{1}{(1 \\pm \\Delta r / R)^2}", "66cc43609badf5353e2e804ab261e77d": "I/I^2[1]", "66cc9f8f271b6045566c3c767bbcd334": "J=-{\\delta\\over\\delta \\phi}\\Gamma[\\phi]", "66cca6b4f7c74e7f99bfb075bafc8465": "\\widehat{E} = i \\hbar \\frac{\\partial }{\\partial t}", "66ccb33dd6cb6c05f7f30d8aa00a07ee": " {\\rm 2p}\\sigma_{\\rm u}^{}", "66ccc5c720269bc7e302c7a48151f5b5": "\\frac{\\partial}{\\partial x} \\to \\frac{1}{1 + \\frac{i\\sigma(x)}{\\omega}} \\frac{\\partial}{\\partial x}", "66cce31b4c0caec8c98c58b80174ae09": " \\frac2\\pi = \\frac{\\sqrt2}2 \\cdot \\frac{\\sqrt{2+\\sqrt2}}2 \\cdot \\frac{\\sqrt{2+\\sqrt{2+\\sqrt2}}}2 \\cdots", "66cd4f5283c8d2aca1c4eb8fab2499c3": "\\theta_1:=\\min \\left\\{ \\arccos \\left( \\left. \\frac{ |\\langle u,w\\rangle| }{\\|u\\| \\|w\\|}\\right) \\right| u\\in \\mathcal{U}, w\\in \\mathcal{W}\\right\\}=\\angle(u_1,w_1),", "66cd8ca9d36c0a4d60094cabf9f57c68": "HL= A\\int_{T_L}^{T_H} k(T) \\mathrm{d}T", "66cdaaece9407d6d4221d8004c0adcac": "\n\\frac{1}{2}\\rho(v_{\\infty}^2 - v_w^2)A_D\n", "66cdbbeb5762a3adf595981bde2263cb": "x \\equiv \\pm 8\\pmod {17}", "66ce05fc5df2390c6d94373914c06676": " I=(1\\le i_1<\\cdots 16", "66d1584294f8ca878c6403f733c31f82": "\\exists n < t \\, \\phi", "66d1a02831fc0f5ac19b75f07816ef4c": "m^{\\varphi(n)/p_i}\\mod n \\qquad\\mbox{ for } i=1,\\ldots,k", "66d1e8661af33997b351f6bb3aedc1c2": "\\mathbf{a} \\wedge \\mathbf{b} \\wedge \\mathbf{c}", "66d2154d6cd9d6d9de1ecd25083beb13": "\\delta_m", "66d2437e2fef32c36ca5e2a676bc0dd0": "\\mathbf{a}_1 = a_1 \\mathbf{\\hat b} = (|\\mathbf{a}| \\cos \\theta) \\mathbf{\\hat b}", "66d25cd3f774d0d4857ed95ed7a73789": "R_+M = M", "66d28146289245c68722caffe78b7878": "f(x) = x(x-1)(x-\\lambda) ", "66d29731a2e8e77529f8a5a4e61b79f2": "d^s_{m'm}(\\beta)= \\langle sm' |e^{-i\\beta s_y} | sm \\rangle", "66d2b87be847f345c74238474c9ba2e2": "\\mathcal{H}[\\rho] = [\\hat{H},\\rho] \\equiv \\hat{H}\\rho - \\rho\\hat{H}", "66d30dd03833488ec7cf9e9750105198": " \\operatorname{de-let}[M\\ N] ", "66d36a5ddd243d4d108902b83cfa4d42": "\\begin{smallmatrix} \\mu = \\sqrt{ {\\mu_\\delta}^2 + {\\mu_\\alpha}^2 \\cdot \\cos^2 \\delta } = 77.63\\, \\end{smallmatrix}", "66d39b48cd65caf6521f077c3c4f58a9": "1/\\beta = kT", "66d3a58b2f85500ac59e12cc6549c062": "\n|1000\\dots0\\rangle\\equiv|f_1\\rangle=|u_1,t_1,s_1,r_1\\rangle\\equiv|100,100,100,100\\rangle\n", "66d4550a9c4e8e38be94b93e601d1967": "P_{br}=\\frac{P_a}{\\eta_j}", "66d45c8d229ef648a05c6932ed942de2": "\\frac{\\langle t^n \\rangle}{\\langle t \\rangle^n} = n!", "66d4e7a518afe1d68f990397e474a9ef": "|x|<\\left |\\frac{b}{d}\\right |", "66d4fd75968d5e93a2206999d26f33dc": "\\frac {F_{t,T}} {S_t} - 1 = \\frac {(i_$ - i_c)} {(1 + i_c)} = E(e)", "66d5222d01850927e152106d9364b83e": "V _{L}(t) = L \\frac{di_{L}}{dt}\\,", "66d53380cdcde420fb7c14a649c06c52": "E \\in \\Sigma", "66d534ef004b022468c19a0d2c93d2a0": "c(i,j)", "66d558bd2256919667ddbd1fb8bd6c88": "v_\\infty\\,\\!", "66d58197b2dc72c6ec6cff5833c24275": "M={s\\over{r}}={1\\over{2n-1}}{9n^4-18n^3+18n^2-9n+2\\over{2}}={9n^4-18n^3+18n^2-9n+2\\over{4n-2}}", "66d5937d6b1c74e20565447bb3bcf10f": "\\displaystyle{{1\\over \\pi^n} \\int_{{\\mathbf C}^n} |f(z)|^2 e^{-|z|^2} \\, dx\\cdot dy}", "66d62465393366a249562b337fbbca94": "\\operatorname{E}[S_N]=\\operatorname{E}[T_N].", "66d641b8c2fe74dc6d39db717d332007": "E_{b}", "66d6ac42d872d857874a6fb318267561": "\\cos(\\bold{k}\\cdot\\bold{r})", "66d6df3b78d1efd2bb930c8b2f24861e": "d_1=\\begin{pmatrix}\n1&0&0 \\\\ \n\\alpha & 1-\\alpha&\\zeta \\\\ \n\\beta&-\\beta&\\eta\n\\end{pmatrix}.", "66d730d0dd70f709068e91fc5e7663a4": "c_1(M)\\in H^2(M,{\\mathbb Z})", "66d79ac937f65f6f40178a7f2772a6f9": "\\Gamma\\!\\left(\\frac{\\nu}{2},\\frac{1}{2x}\\right)\n\\bigg/\\, \\Gamma\\!\\left(\\frac{\\nu}{2}\\right)\\!", "66d7e71cd0f627de93e8a718abf797dd": "g_{t t} = - \\left( c^2 + 2 \\Phi \\right) \\,", "66d803aa2ae78ad989e798dd598e0490": "\\scriptstyle \\lambda=1 ", "66d82d73cb286c3c1839e312ace359af": "\\frac{m - 1}{\\tilde N - 1} = \\frac{1}{2}", "66d8370d413e619a3e9c05155c06613f": " t_0+t\\ ", "66d882f32947c08916f2a2261776fa32": "\n(a)_{b,c} = \\prod_{i=0}^{b-1}(a+ic) = \n\\begin{cases}\na^b & \\text{if }c = 0, \\\\ \\\\\n\\dfrac{c^b\\,\\Gamma(a/c + b)}{\\Gamma(a/c)} & \\text{otherwise}. \n\\end{cases}\n", "66d88ffd8b0db89a9239ef08275a2a80": " \\ddot {\\bar r } = -\\mu \\cdot \\frac {\\hat r } {r^2}\\ \\ ", "66d89afad9cb8112a3d26eb7271e82c5": "\\sin^2{\\frac{\\alpha}{2}}+\\sin^2{\\frac{\\beta}{2}}+\\sin^2{\\frac{\\gamma}{2}}+2\\sin{\\frac{\\alpha}{2}}\\sin{\\frac{\\beta}{2}}\\sin{\\frac{\\gamma}{2}}=1.", "66d9477997c74c5efda92d7ef900c2fa": "t_r\\cong\\frac{0.34}{BW}\\quad\\Longleftrightarrow\\quad BW\\cdot t_r\\cong 0.34", "66d96901747fb9e2a627b6992e0977a9": "t(M)=\\mathcal{Z}(M)", "66d96b58e32a4f4d19cbc2ee221592d0": "\\mathbf{r}_{AB}", "66d9826d670cf971d93ec5093fe73a78": "(A_0 \\and A_1 \\and \\cdots)", "66d9b09db7fc83b2f1b5d3985be27c19": "\\mathcal{P} = -{I}^{kl}\n\\overleftarrow{\n\\frac{\\partial} {\\partial \\xi^{k}}\n}\n\\overrightarrow{\n\\frac{\\partial} {\\partial \\xi^{l}}}", "66d9b40cf4bcfde75374ffb5b2f5611d": "\\tfrac{50GeV}{c^{2}}", "66d9e8ef0e1e5fc73a2d852bad5e2351": "p_{vanna}", "66d9f1682f6c2ffdd6d9474fc4153706": " S^3", "66d9f81f886314c3b519717b25f24cb7": "x\\leq y \\text{ if and only if } f(x)\\leq f(y).", "66da01b3d10a44d9738829f9edfce724": "y=\\begin{cases}a & \\{ft\\} < D \\\\ 0 & \\{ft\\} > D \\end{cases}", "66da2fa24c409e818b8c2a723ca6b74e": "k \\times m", "66da497c7135c6788da98c5910be1640": "w(n)=1 - \\left(\\frac{n-\\frac{N-1}{2}}{\\frac{N+1}{2}}\\right)^2", "66da5d478dbe63c9b033c670ffa5a1d9": "1 - 2S_n(s) = T_n(1 - 2s).\\,", "66da66b0484abde2c5c1be4c52bd45b0": "1/\\lambda^2", "66dafb34822880973159f1c97be22fc4": "h\\, =\\, z\\, +\\, \\frac{p}{\\rho g}", "66db31faf874b528d2ede5454f1357af": "\\mu_{2,1}= \\mu_{2,1} - 4\\mu_{1,1}= \\frac{13}{3}- 4= \\frac{1}{3}(< \\frac{1}{2})", "66db3c85b604124b08b8985b7282d2fd": "|\\Psi \\rangle = a|\\psi\\rangle + b|\\phi\\rangle", "66db934d39aab7a6e1d49de6c5fa4a10": "dw = \\gamma dA", "66dbafe67ebf1b7127d9f091c8121ba4": "\\dot{V}_{CO} =\\frac {\\Delta{[CO]} * V_A} {\\Delta{t}} ", "66dbd0a4ec6d2679bf14f0d5bb564e5b": "\\nabla \\cdot (f \\vec v) = f (\\nabla \\cdot \\vec v) + \\vec v \\cdot (\\nabla f", "66dc03679e67743929ae5f374f48ceaf": "J(t)=y^k(t)e_k(t)", "66dc7b017500aa977ace8f29e3f38445": "f(x) = \\frac{\\pi}{2\\sinh\\pi} \\sinh x.", "66dccd5a9d4b19bb0dc225f0c1de4512": "\\omega_{0} = \\gamma B_0\\,", "66dce52ae2dc205c232142189e5bf7f3": " H_B(r)=P({N}(rB)=0). ", "66dcea1181c41c8e47346bb55f476c99": "p(\\lambda) = \\det(\\lambda I_n - A)", "66dd1f5858a09cc5b2836ef81978c58b": "Y=C+S+T", "66dd2e1ca106dd0f68d0a4393cbfeff6": " R_0 = ", "66dd6fecdf13b6439cf8406b726f395e": "\\frac{1}{\\sqrt[3]{a}+\\sqrt[3]{b}} = \\frac{\\sqrt[3]{a^2}-\\sqrt[3]{ab}+\\sqrt[3]{b^2}}{(\\sqrt[3]{a}+\\sqrt[3]{b})(\\sqrt[3]{a^2}-\\sqrt[3]{ab}+\\sqrt[3]{b^2})} = \\frac{\\sqrt[3]{a^2}-\\sqrt[3]{ab}+\\sqrt[3]{b^2}}{a+b} \\,.", "66dd8a3f8fb480a973284e77b52cf834": "{1 \\over \\pi} \\log \\left | {t + {1 \\over 2} \\over t - {1 \\over 2}} \\right |", "66dd9dcb3984673d9682797dd5d66cc8": "\\! a(t) = \\frac{1}{1 + z}", "66ddb76221739dffb6f22c48f6114fab": " \\gamma_L \\cos \\theta_\\mathrm{C} \\,=\\gamma_\\mathrm{SV} - \\gamma_\\mathrm{SL} - \\pi_\\mathrm{e}", "66ddc7e7b2bf72e77b63ed6e07948871": "\\frac{2\\pi^{d/2}}{\\Gamma\\left(\\frac{d}{2}\\right)}", "66ddfdd8cac86cf9e7430ec3d328b942": "dl = c\\ dt", "66de2122743b2cfb833e20c5921ce832": "P_{drag} \\,\\!", "66df44adc5d9a7920ffa8ae091d5e791": " m_{ut}(\\cdot) = \\sum_{i=1}^n \\beta_{ut}^i \\phi(x_t^i) ", "66df467ce480831d078d9519e4121e08": "f_1(x),f_2(x), \\dots, f_{m-1}(x)\\,", "66df86d7d224fcd81e7542966ce9ba02": "5^{ab}", "66dfb961bffa06da867aaf90637494e4": "Y=\\{a,b,c\\}", "66e000273fcfe0f471ff13908e0168b0": "H_{\\mathrm{kinetic}}=-\\frac{p^{4}}{8m^{3}c^{2}}.", "66e01e29b6e238e8229cb9b955cd654b": "\n\\begin{array}{c|c}\n1/2 & 1/2 \\\\\n\\hline\n & 1\n\\end{array}\n", "66e0592243d27f0a70b2b462a02d0425": "\n\\begin{alignat}{2}\nA^*(Az - b) & = A^*(A A^+ b - b)\\\\\n& = A^*(P b - b) \\\\\n& = A^*P^* b - A^*b \\\\\n& = (PA)^* b - A^*b \\\\\n& = 0\n\\end{alignat}\n", "66e065ea60da4127d4670c91641b6af9": "[0.5,1.0]", "66e0a3861502ba6e4282f25c1470b4e5": "UCL=\\overline{x}+2.66\\overline{MR}", "66e0b6d6a1ab1afd893964f9220e836c": " j^{th}", "66e0d4f5a589a6b4653aa9e679e34a1a": "\\mathbf{N}_i\\,\\!", "66e0d7dae015c2b05076502012b6bde3": "v = \\frac{ds}{dt}.", "66e0f4334744ebc0f7e870e478aab692": "F(\\Omega)=|\\Omega|f_\\infty+|\\partial\\Omega|f_x+...", "66e14906e6dac6118c45487a5d0d4440": "\\phi_n (a) = a \\otimes I_r,", "66e1527dcd5872c8206971cfc77cc39d": "\nU =\n\\begin{pmatrix}\n u_{11} & u_{12}\\\\\n u_{21} & u_{22}\\\\\n\\end{pmatrix},\n\\quad\nV =\n\\begin{pmatrix} \n v_{11} & v_{12}\\\\\n v_{21} & v_{22}\\\\\n\\end{pmatrix},\n", "66e16a7222fa6966da2df3bac764cdd3": "r_+^2=M^2+Q^2", "66e178a6e4d8ffdc94692bd73c12e7ac": "\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1", "66e17d8857461ff389ed1b41d7ba1701": "M(t)=\\frac{\\Gamma(\\alpha-t) \\Gamma(\\alpha+t) }{ (\\Gamma(\\alpha))^2 }, \\quad -\\alpha0", "66e3398cb550464cd580e3398307d4bd": " Q(x,\\xi)", "66e33ad7a04c44c569050f81d3783bdc": "\\Pi = (\\pi_1 , \\pi_2 , \\pi_3 , \\pi_4)", "66e38136d2a41ffc439e095e3f7a5d2f": "(xy)^\\circ = y^\\circ x^\\circ", "66e38a243ff8b4c84c10c65641beec5e": "\\rho_\\parallel \\ ", "66e395e2c81b71fbbe0aa725fd4e8c82": "1/m", "66e3bdff5dfed974075a100bedeeb152": "{\\tilde{B}}_6", "66e3dfa8db9e44f799e39d23f91a2f00": " \\textstyle \\prod_{i=1}^r (2s-2 + s(p_i)^{-1/2})+ s_{m}^{-1} ", "66e3eaf1bff719b3cbed9730efe40d24": "{\\mathbf P}(t)", "66e43b276ac7d1b604f2cbbb011260ec": "f^{(n)} = \\underbrace{f \\circ f \\circ \\dots \\circ f}_{n \\mathrm{\\, times}} : X \\to X.", "66e4686da8b253eb7754fde2904cee58": "\\alpha = \\frac{(0.744\\ln(Re)) - 1.41}{(1+ 1.32\\sqrt{\\frac{\\varepsilon}{D}})}", "66e4b05a60b3e627c1d72f8f191f817f": "Var(Z_t) = \\sigma_Z^2", "66e4bc5b1b666a4acef9ae69125f02b6": "k = { 2 \\pi \\over \\lambda } \\,", "66e4eb97715b6e33271dfed1e7ab8d31": "i \\hbar \\sum_n (\\dot{c_n}\\psi_n + c_n\\dot{\\psi_n} + i c_n\\psi_n\\dot{\\theta_n})e^{i\\theta_n} = \\sum_n c_n\\hat H\\psi_n e^{i\\theta_n}", "66e50cf4c3e3f1d57f5811887e4d23ae": " \\sigma(X) \\le \\frac{1}{3} e(X), ", "66e5414b33d4c6c97f79b891dc28b226": "F(T;H)", "66e58545938222564d2541223d8ac5c6": "\n{\\partial^2h\\over\\partial t^2} + \\sigma_u^2 {\\partial^2h\\over\\partial x^2} - F_z(x,t) =0,\\,\n", "66e5c76a64532814b5daf4f212fbff2f": "\n\\Omega' \\ \\stackrel{\\mathrm{def}}{=}\\ \\{1,\\dots,1,0,\\dots,0\\}\n", "66e60b61523aae460334961fe87b1261": "n^{0}_i", "66e63e9df310cfabb6270dbb65db2f73": "\\tfrac{4}{13}=\\tfrac{1}{4}+\\tfrac{1}{18}+\\tfrac{1}{468}", "66e6da6dc10097574037558f243c5830": "\n[C6-7] \\quad \\frac{\\vdash exp \\;:\\; sc \\quad [sc] \\vdash C_1 \\quad [sc] \\vdash C_2}{[sc] \\vdash \\textbf{if}\\ exp\\ \\textbf{then}\\ C_1\\ \\textbf{else}\\ C_2} \\qquad \\frac{[high] \\vdash C}{[low] \\vdash C}\n", "66e7197b5466876447cbb73a5726ded4": "\\dot{\\hat{\\mathbf{x}}} = A \\hat{\\mathbf{x}} + B \\mathbf{u} + L v(\\hat{x}_1 - x_1)", "66e813830f03f8f417344476020785b9": "H_4 = +12\\,", "66e8588337e5b734ce26dcaa1033b540": "\n\\pm z_{1-\\alpha/2}\\sqrt{\\frac{1}{N}\\left(1+2\\sum_{i=1}^{k} y_i^2\\right)}\n", "66e877ed3c2c3f1e4d1d3e86a37df606": "Y = a (X_1 + X_2 + \\cdots + X_{N_p})", "66e8d426b00b590b125549207564aba7": "Z: \\mathbb{R}^n\\rightarrow\\mathbb{R}", "66e92ec26dd510e635ee418998cf2847": "\\{1,2,3,\\dots,2m\\}\\,\\!", "66e9353d6b0cb98faebbd0ef28275554": " \\cos(\\alpha) ", "66e93f3eb2bf87374e45cd4e43447ece": "n \\times m\\,", "66e98dcba0bfc7f7c683f442e244bf16": "\\chi_G(k) = \\chi_{G-e}(k) - \\chi_{G/e}(k)", "66e9b030e414a2adb5d5390e72ef8885": "\\scriptstyle\\varphi(r)", "66e9e2abae9adf4c9726aa2977276b09": "\\lambda = \\varpi ", "66ea0d7f51567743708a93efd006378c": " u_{2n}^2 = u_n U_{2n} . \\,\\!", "66ea62fd7ed2759f100d38e98bdb37ef": "e_{ji}", "66ea6a3e6010b1a5d7f2e2d8782d0f59": "8 a_3^2.", "66ea7320bba0561d666c2f9c5e99704d": "\\mathbb P(N(t)=k) = \\frac{m(t)^k}{k!}e^{-m(t)}.", "66ea8742da0fe006660df6c2c3099082": "D=\\frac{1.858 \\cdot 10^{-3}T^{3/2}\\sqrt{1/M_1+1/M_2}}{p\\sigma_{12}^2\\Omega}", "66ea8a576dd4c0cbbfe5568d9e478967": "P(t+s)=P(t)P(s)\\,", "66eab8a23b6b6f80032f16777cb3ccde": "dx\\wedge dy,", "66eac8a2c28e1ed08f3cd063efcc5987": "Q_i( )", "66eb12e0f1aaa7ee1c52f44482fe4706": " i\\epsilon ", "66eb389475d6e6f6b595a5030fc8b974": " X \\sim U(0,1) \\,", "66eb521937727189cb21339fae469b9d": "\\mathfrak C \\mapsto ({\\mathfrak H}^{-1})^\\dagger {\\mathfrak C} ({\\mathfrak H}^{-1})^*.", "66eb7e008f67acec950126edbfdc7490": "O(n^{c+\\varepsilon })", "66ebb18e6482c1ed281e55e02157ba23": "f_1(x)\\,", "66ebc32081e019cbc9b636ae42ab8e99": "r\\sin\\theta=mr\\cos\\theta+b.\\,", "66ec0c14f94f69d5937f0cdc90393869": "\\mathrm{LSq} = \\sqrt{ \\frac{1}{2} \\sum_{i=1}^n ( V_i-S_i ) ^2}", "66ec4ffedcf66525041a5445f76b8d33": "\\mathrm{Li}_s(z)", "66ec65702d7d999a5d9f6244b2f6d7ec": "\\mathcal{M}_\\epsilon", "66eca3332bb1d16c053c16e2e76487c2": "\\frac{d\\bold{x}}{dt} = \\bold{F}(\\bold{x_0},t) + D\\bold{F}(\\bold{x_0},t) \\cdot (\\bold{x} - \\bold{x_0})", "66eca3b3418df881d8f76ee558438070": "\n\\mathbf{F} = \\frac{k}{r^{2}} \\mathbf{\\hat{r}}\n", "66ecefdb9145c61d8ad11d8d3ae28f05": "\n \\forall A,B\\in {\\mathcal A}\\qquad \\exists C\\in {\\mathcal A}\\qquad A\\cup B\\subseteq C,\n ", "66ed2ee5455009c32b19ba6740b5c2ec": "\n\\begin{align}\n0 & = L [\\mathbf{q} [t_2], \\dot{\\mathbf{q}} [t_2], t_2] T - L [\\mathbf{q} [t_1], \\dot{\\mathbf{q}} [t_1], t_1] T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_2] T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} [t_1] T \\\\[6pt]\n& {} + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\epsilon} [t_2] - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\epsilon} [t_1].\n\\end{align}\n", "66eda923f042b0e379289f4e70ad4791": "\\mathbf a\\mathbf b = \\sum_{i=1}^8 f_i(\\mathbf a)g_i(\\mathbf b)w_i", "66edf8eac6779757e2ca795b61d28493": "\\mathsf{I}", "66ee23528c59996bb2df044e4a3282ea": "f(u)=u^{\\alpha}", "66ee23c31a3d09adb474a90fab11ec86": "\\cos c = \\sin\\varphi_0 \\sin\\varphi + \\cos\\varphi_0 \\cos\\varphi \\cos\\left(\\lambda - \\lambda_0\\right)\\,", "66ee9e7fff72f6555b7c429c0a7597a4": "x = x\\,", "66ef1c762efd75abfd71a5f102090c76": "u_{ij} = \\frac{1}{2}(\\partial_i u_j + \\partial_j u_i )", "66ef771516c364937b031aef30864527": "\\det\\,A = 0", "66ef88c11a00968e9b929731d9c3996e": "\\mathit{N} + 1 = \\mathit{p^n}", "66ef9fdc4eeaff42f2e928e21c62fe70": "v_1 '", "66efbb32acd4c9f9123f577321648313": "~\\exp(x)~", "66eff23dfa3b0c7905a45c3553b4ac95": " |n_1, n_2; A\\rang \\equiv \\mbox{constant} \\times \\bigg( |n_1\\rang |n_2\\rang - |n_2\\rang |n_1\\rang \\bigg) ", "66f008cd7700ffcb0a05dcfe309c6434": "\n \\quad (9) \\qquad \\left\\vert 1 - \\frac{4\\alpha \\Delta t}{\\Delta x^2} \\sin^2 (k_m \\Delta x/2) \\right\\vert \\leq 1\n", "66f02dd267030396e8f0ed2d4cc866ab": " \\nabla _\\mu T_{M\\;\\nu }^{.\\;\\mu }=\\frac 1{8\\pi} \\frac {\\nabla _\\nu \\phi }{\\phi }\\Box \\phi =\\frac 12 \\frac {\\nabla _\\nu \\phi }{\\phi } T_{M\\;}^{\\;}", "66f07f8e7958b43afc15bdb0a121cde5": "x \\wedge \\bigvee S\n= \\bigvee \\left \\{ x \\wedge s \\mid s \\in S \\right \\}", "66f0854568892a4aa8e33ee6821e51ab": "\\pi_0 = \\left[\\sum_{k=0}^c \\frac{\\lambda^k}{\\mu^k k!} + \\frac{\\lambda^c}{\\mu^c c!}\\sum_{k=c+1}^K \\frac{\\lambda^k}{\\mu^k c^k}\\right]^{-1}", "66f0d53c2543225eb9fc5e0a03a33d4f": "\\langle \\ |A|\\ \\rangle", "66f0f976aba1675284df7dd809fffbd2": "2^m,0,\\infty", "66f0fd6916325b961a72e30baf076125": "\\mathbf{p}_i \\,", "66f12a652b74dd3893b20c19e3bb384a": "= (I_\\mathrm{RMS})^2R\\,\\!", "66f190cf0ab68b9a6424f33e5909a41d": "1728=12^3=8 \\cdot 12 \\cdot 18", "66f1b15e18ff7f7149d5d749729ec4ba": " \\Delta g_{ij} ", "66f1fea775a5b4d9dbb4e86271e23de6": "\nk_X \\equiv {\\omega_p^2 \\over \\omega_H} \n", "66f23ddde4ad0bf4f83d40c58000062c": "EM_{9}(endo,exo,fendo,fexo)=RE", "66f24c5ae88feb45f0f3b7daa06b35bd": "x_0, x_1, \\ldots", "66f29b656ac15504def6bffeb26cebbe": "f(x;\\mu,\\sigma, a,b) = \\frac{1}{\\sigma Z}\\phi(\\xi)", "66f29f812d1a4b54a0a63d9efcc99c64": "R\\ge \\sqrt{2}r", "66f2bcdea590c7d6a0c173e00ef3ead1": " \\int \\limits_{0}^{\\infty} exp \\left(\\! -2 \\int \\limits_{0}^{x} \\frac {1-e^{-y}}{y} dy\\right)\\! dx = {e^{-2 \\gamma}} \\int \\limits_{0}^{\\infty} \\frac{e^{-2 \\Gamma(0,n)}}{n^2} ", "66f2f7ac9f1444f140f05d5fef06de53": "A_{ij}[\\mathbf{k}]=\\alpha^2 k_ik_j+\\beta^2(k_mk_m\\delta_{ij}-k_ik_j)\\,\\!", "66f31cd9538756c95815a567844a8927": "{\\nabla}^2 \\phi = 0,", "66f3341399d5f5110479e1dabd3049fe": "P = E*G*100", "66f336f8e0cd6b64798eaa55baf7ce35": "\\mu = \\mu_\\mathrm{k}\\,", "66f348622c9a4ada0581e9f8116777dd": "\\phi(x)\\phi(y)=\\phi(y)\\phi(x)\\,", "66f36ffc5552341448265a333f444f66": "d_k = \\lfloor x/\\beta^k\\rfloor", "66f3940597089f244304cecf1caad01c": "Z_4\\,", "66f4213883e49d6a98b71d4f4edcaf23": " \\alpha \\sum_{i=1}^{M} N_i P_i ", "66f45c13987f067c74cf9ae5c5f3634a": " L(F*g)=(LF)*g=\\delta(x)*g(x)=\\int_{-\\infty}^{\\infty} \\delta (x-y) g(y) dy=g(x) ", "66f47d044700cbf8bf6235e8bae11ac6": "F(u) = u \\cdot d - |d| |u| \\cos \\theta", "66f491bef00f534030e400cb3a17bd67": "N=[N_x, N_y, N_z]", "66f4d734b5b9c748e16a3caa636fd96f": "\nT_i \\downarrow - \\tau_i (1-R_{i-1}) T_{i-1} \\downarrow - \\tau_i R_{i-1} T_i \\uparrow \n = (1 - \\tau_i) T_i\n", "66f505b61613caf9a8fa3826abf4bc42": "\\mathbf{n} / \\mathbf{N}", "66f52a60d338920c59761340a67830cc": " \\mathrm{pH} = pK_a + \\log( \\frac{n_{OH^- added}}{n_{HA initial}-n_{OH^- added}} )", "66f550283f5614717b76f47ab1a9e278": " i, j > 1 ", "66f56d0eda5bf243a92cb8e25be845d6": "C:F^k \\to F^n", "66f59f25bbe43bd3769c8251284d57fb": "U_\\mathrm{max} \\frac{m}{2} \\left ( \\omega A \\right )^2 \\,\\!", "66f63045f4a1228b71104a749e68c0d0": "\\sigma^t = \\rho g H \\,", "66f65f1872835afd7bb05b5270f1270e": " s_{p} - s_{q} = \\sum_{n=q+1}^{p} x_{n}.", "66f6c06ec546ff954d8512f7977b9399": " h_{j,k-1} \\leftarrow q_j^* q_k \\, ", "66f73234ec9937b25dc60199e7775206": "x'=\\gamma\\phi(x-vt),\\ y'=\\phi y,\\ z'=\\phi z,\\ t'=\\gamma\\phi\\left(t-\\frac{vx}{c^{2}}\\right)", "66f7462bbb7a84a3e3e52d2fd351f692": "S_\\ast(B)\\xrightarrow{\\triangle} S_\\ast(B\\times B)\\xrightarrow{\\simeq}S_\\ast(B)\\otimes S_\\ast(B).", "66f748be411ab10db52e8ab3da23a847": "I_{\\text{ext}}", "66f75e3452b72745d559ceeb95131d45": "\\exp(X) \\sim \\operatorname{Log-\\mathcal{N}}(\\mu, \\sigma^2).", "66f7a2f0c14511b6c65d3bf625640198": " \\mathrm{idf}(\\mathsf{this}, D) = \\log \\frac{2}{2} = 0", "66f7dfcb9be3ce1b7398d384f143bc60": "(1+x)(1-x)=1", "66f832baeae47e5fa3934eee70e19f27": "\n\\begin{align}\n\\mu'_x & = \\frac{\\sin\\theta(\\mu_x \\mu_z \\cos\\varphi - \\mu_y \\sin\\varphi)}{\\sqrt{1-\\mu_z^2}}+ \\mu_x \\cos\\theta \\\\\n\\mu'_y & = \\frac{\\sin\\theta(\\mu_y \\mu_z \\cos\\varphi + \\mu_x \\sin\\varphi)}{\\sqrt{1-\\mu_z^2}}+ \\mu_y \\cos\\theta \\\\\n\\mu'_z & = -\\sqrt{1-\\mu_z^2}\\sin\\theta\\cos\\varphi + \\mu_z\\cos\\theta \\\\\n\\end{align}\n", "66f8418c3dc27b16bbc7ce4d57826231": "\\lambda=(\\lambda_1,\\ldots,\\lambda_m)", "66f84c0ad5600eacf2ae6a170c3417e5": " e^{-\\varphi} \\partial_x \\partial_y e^{\\varphi}= e^{-\\varphi} \\partial_x e^{\\varphi}\ne^{-\\varphi} \\partial_y e^{\\varphi}=(\\partial_x+\\varphi_x) \\circ (\\partial_y+\\varphi_y)", "66f878f5a9df98d7a28f18675e3cd1f6": "h_{-l}=h_{|l|-1}", "66f8f33c9ced80eaa3cfbf6bf92102d1": "I_{L3}", "66f8fb0b1d20ff8c16d74438de5f3a43": "g(\\mu) ", "66f8fda557aa80dd6e3ac5216ee67f01": "\n\\lim_{x \\to \\infty} e^{\\lambda x}\\Pr[X>x] = \\infty \\quad \\mbox{for all } \\lambda>0.\\,\n", "66f901524094fb7b96701cddefa1672c": "\n\\begin{align}\n\\begin{bmatrix} A & U \\\\ V & C \\end{bmatrix}^{-1}\n& = \\begin{bmatrix} I & 0 \\\\ C^{-1}V & I\\end{bmatrix}^{-1} \\begin{bmatrix} A-UC^{-1}V & 0 \\\\ 0 & C \\end{bmatrix}^{-1} \\begin{bmatrix} I & UC^{-1} \\\\ 0 & I \\end{bmatrix}^{-1} \\\\[8pt]\n& = \\begin{bmatrix} I & 0 \\\\ -C^{-1}V & I\\end{bmatrix} \\begin{bmatrix} (A-UC^{-1}V)^{-1} & 0 \\\\ 0 & C^{-1} \\end{bmatrix} \\begin{bmatrix} I & -UC^{-1} \\\\ 0 & I \\end{bmatrix} \\\\[8pt]\n& = \\begin{bmatrix} (A-UC^{-1}V)^{-1} & -(A-UC^{-1}V)^{-1}UC^{-1} \\\\ -C^{-1}V(A-UC^{-1}V)^{-1} & C^{-1}V(A-UC^{-1}V)^{-1}UC^{-1}+C^{-1} \\end{bmatrix} \\qquad\\mathrm{(2)}\n\\end{align}\n", "66f97dd88316d246c546496149d91283": "\\displaystyle{f_m^\\prime = Tf_m=\\lambda_m f_m.}", "66f9cceba2ca6857774d72b5ed291fb2": "x^2+1,", "66fa68d9005175a83c5788c489a926ad": " \\frac{A(\\mu,\\nu)}{A(\\nu,\\mu)}=e^{-\\beta(H_\\nu-H_\\mu)}.", "66fa867838adc02174e667b0a602c3af": " q \\in Q_A ", "66fa8871345805f7970cf076d8c32517": "m \\cdot 6:m\\ ", "66fb0cc28336f4f737936ec6fe6e086f": "T_0 = T + \\frac{V^2}{2C_p}\\,", "66fb3c4cc09ce7108637ebd0c3cb8386": "\n\\frac{1}{\\tau_l}(s) = \\frac{r_0^2 c N}{8\\pi\\gamma^2\\sigma_x\\sigma_y\\sigma_z\\delta_\\mathrm{acc}^3}F(\\varepsilon_m).\n", "66fb8be20f781be618e15f8d3ff90f0b": "y(x_0)", "66fba72dc455f49f9ca3c438132ad507": "\\ P_{i} =P_{total}m_i ", "66fbecf6e3593c287522c8dd6acd82b2": " x_1 = \\frac{b_1}{l_{1,1}}, ", "66fc5e2d7b568b58536d6ab30c5422f7": "\\{\\rho_n\\}", "66fc7d6a0f31b74428fdc4aad22f7377": " 3\\left(\\log{s(R)} - \\log{s(v)}\\right) ", "66fcf82e80b2eb47114911d2bc4440ef": "[y_0,\\ldots,y_j]", "66fd6f2377b806241d155b2c8c051b56": " \n\\begin{align}\n\\frac{d}{dt}\\langle p\\rangle =& \\int \\Phi^* V(x,t)\\nabla\\Phi~dx^3 - \\int \\Phi^* (\\nabla V(x,t))\\Phi ~dx^3 - \\int \\Phi^* V(x,t)\\nabla\\Phi~dx^3 \\\\\n=& - \\int \\Phi^* (\\nabla V(x,t))\\Phi ~dx^3 \\\\\n=& \\langle -\\nabla V(x,t)\\rangle = \\langle F \\rangle,\n\\end{align}\n", "66fd7b3cac28d684b3382ce131063d04": "PHM_{3} = \\gamma \\left(1 - \\alpha - \\beta - \\gamma\\right)^2", "66fd86d0fa1d47f10824723edcb201a9": "(e,e,e)", "66fd9563d2fdfab69f27dcae83549378": " \\cfrac{\\Gamma, A \\vdash \\Delta} {\\Gamma, A \\and B \\vdash \\Delta} \\quad ({\\and}L_1)\n ", "66fdf2d5362ead05439634632152ae67": "\nf(\\mathbf{x}) = \\langle f,k(\\mathbf{x},\\cdot) \\rangle_k, \\quad \\forall \\ f \\in \\mathcal{H}_k,\n", "66fe1feb6f26c657dfa21ba45d5fe5e3": "V + v", "66fe5f9f63325461075bad939a39e26d": "\nT_s= T_e 2^{1/4} = 1.189 T_e \\qquad T_a=T_e\n", "66fe9c4633030fd0c7290ef3143deec2": "\nG = \\frac{\\textit{precision}}{r}\n", "66fed786a66bade034eeb35eda45cc9a": "X_n - E(X_n) = O_p(\\sqrt{\\operatorname{var}(X_n)}) \\,", "66ff009cf39d024d669a03a4cbf024d4": "\\mathrm{Hom}(X\\wedge A,Y) \\cong \\mathrm{Hom}(X,\\mathrm{Hom}(A,Y))", "66ff0ba743ee58db96643c4012eb158b": "\\sum_{n=0}^{\\max\\{j,k\\}} s(n,j) S(k,n) = \\delta_{jk}", "66ff3c9d574f1d98509d026052b73d42": "2t-1", "66ffa14b6a40e6f97e9d49329837109a": "db=2az\\;", "66ffaecabbda6f5f13f3a5337d2facd9": "f(x) = \\sum_{n=0}^\\infty a_n e^{-nx} = \\sum_{n=0}^\\infty a_n z^n,", "66ffb1041ab310d216bb9b84d64fc8ca": "\\phi_{23} : A \\otimes A \\to A \\otimes A \\otimes A", "66ffcbd493a0cce29ab1b07d8952c8ba": "P = k_1d_1^{n_1} = k_2d_2^{n_2} = k_3d_3^{n_3} = ...", "670046dea315e407f2c0b2e787cfde3b": "K \\colon \\mathcal{A} \\to \\mathcal{S}", "6700744abc1e5108b0bd7c857db2b131": "x(yz) = e", "6700a8d8fcda3f9bfc2caa13ae2bce45": "f_\\mathrm{eq}", "6700b8e850327480a5f0397ac56d8d0a": "p(x-y):= p_0(x^0-y^0) - \\vec{p} \\cdot (\\vec{x}-\\vec{y})", "6700e027a273679b151ca34b51fcc114": " X_k = \\sum_{n=0}^{N-1} x_n e^{-{i 2\\pi k \\frac{n}{N}}}\n\\qquad\nk = 0,\\dots,N-1. ", "67017143e31bfd1a27023aedd3a3d980": "ax-bx=d-c", "6701c929774a40aa7360b172ea940cca": "\n \\mathbf{e}^k\\cdot\\mathbf{b}_i = \\frac{\\partial x_k}{\\partial q^i} \\quad \\Rightarrow \\quad\n \\frac{\\partial x_k}{\\partial q^i}~\\mathbf{b}^i = \\mathbf{e}^k\\cdot(\\mathbf{b}_i\\otimes\\mathbf{b}^i) = \\mathbf{e}^k\n", "67020f0f42df1c56f5ba93c5a216df0a": " \\mathrm{Tr}\\left( \\Pi_i \\Pi_j \\right) = \\left| \\langle \\psi_i | \\psi_j \\rangle \\right|^2 = \\frac{1}{d+1} \\quad i \\ne j", "670233ee957e3962df6f5b0addbf4ca7": "\\gamma \\in \\Gamma ", "6702d07828c1b73104cb865a32a6d2c3": "\\sum_{k=0}^\\infty (-1)^k {k+\\nu+1 \\choose k+1} \\left[\\zeta(k+\\nu+2)-1\\right] \n= 2^{-(\\nu+1)} ", "6702ed047a4e3f9943fd8ea170f44de9": "\\epsilon = 1 + \\frac{1}{B^2}c^2 \\mu_0 \\rho", "6702f6b4b98b700c3280976ed5d7e7bc": "\\Delta m_l = 0, \\pm 1", "67033a96fc8025ba231f8779db2eb203": "\\{ z \\in {\\mathbb{C}} \\mid a + b z \\in G \\}\\subset\\mathbb{C}", "67037af81ed75d39b33b66fee3cd0434": " \\scriptstyle D,", "67044e1943743b84e7bf81d79f82f7d1": "= 100.02", "6704546f5f5d76607eb43b509b2ae7ec": "I_y=E_yL+M_y=R-\\frac{1}{2}(E_xL+M_x)=R-\\frac{1}{2}I_x", "670495248d3401971fbc3472b403a15a": "a_j^{\\dagger} a_j - \\frac{1}{2} = f^{\\dagger}_j f_j - \\frac{1}{2}.", "6704e1ed5713d79fe7d242800f9ec087": "\\vec r(u,v)=(au+bv,cu+dv, 0)", "6704fe271ec8b548f0624a543f47bb16": "\\frac{$200 - $100}{$200} * 100% = 50%", "670515129a88a330ae1eb229fffd4c5b": "\\sin(\\arcsin x) = x\\!", "6705264cf18897b2993d6033cef2c5ba": "I_{L}", "67054dbbfd3195b873027cec1e106c22": "e \\sin(\\varpi)", "670559bde9679acc634c9d2a299d6d5e": "\\{|\\phi_i\\rang\\}", "67055bdda3719a1e1be52d665f3caf6f": "\\mathcal{H}(x,y,z,p_x,p_y,p_z)=\\frac{1}{2}\\left( p_x^2 + p_y^2 \\right).", "6705777b712ee811e76fb07162081d63": "TOP", "6705cbdc24ae97ca8b4d2b36cf655039": "s=(p_1+p_2)^2=p_1^2 + p_2^2 + 2p_1 \\cdot p_2 \\,", "6705e636d8da317c951cb8bc769dcaaf": " \\Delta S = \\frac {Q}{T}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)", "670643346999691a280874b79eec7f75": " S_i", "67067a675ff4472fac5475e7886e244d": " \\sigma_{ij}=\\sigma_{ji}\\,\\!", "67067dff6cd65086c155314c9d8d785c": " v(\\emptyset)=0", "670687f8110a0d12ce92a3be84940fa1": "x=\\frac{a/b}{c/d}=\\frac{ad}{bc}\\;(=39.7,\\text{ in Table 3 })", "67068e96a5b62b7f11d8b079645d3376": "E \\left \\{ \\left| h_{ij} \\right |^2 \\right \\} =\ng_{ij} \\quad \\forall i,j", "6707307e7fe229aa9b9b1643d8efacd8": "x^{\\deg(p)}p\\left (\\frac{1}{x} \\right )", "67076950d74dd747844d68991e4aadf2": "f(x)=ax^2+bx+c,\\quad a \\ne 0.", "6707d823e404dde5787a8ee09e5b6ba5": "P(M, t) = \\sum \\ell(M_n) t^n", "6707f1a4fdb8954582e4e9c431fd12b7": " \\tfrac29 - \\tfrac1{15} \\sqrt{15} ", "67085ae6f883e8d679d26475b8e80480": "P[\\{t_i\\}|s(t)]", "6708726c9afa57907ae4b66d42135319": " a'_{kk} = c^2 a_{kk} + s^2 a_{\\ell\\ell} - 2 s c a_{k\\ell} \\,\\! ", "670878a3980a0932b3bd8d88e79d0afd": "k_v(k_v - 1)/2", "6709646677c0f6771045f6b53d132bb7": "\\exp \\left(c + \\sum_{i} w_i f_i(X) \\right)\\,,", "67096b0dc54c6df5765cf3898c6c8c0d": "t_a\\;", "67096d2fb4aceb2ed6b005f1e792eed8": "((p \\vee q) \\wedge \\neg p) \\rightarrow q", "670a0751eef9b7782754616a02221eda": "\\sum_{n=0}^{\\infty}a_n = a_0 + a_1 + a_2 + \\cdots ", "670a3ff1db69ca1974f06990a1d65694": "SDec\\,", "670a675775af3490bd5e7f88081bf7f8": "R_{\\text{in}}", "670b831e8cbafd7fdc995a26d75cc06d": "v = (v_x, v_y)^T", "670ba4ac59ff057abf9a9714ea1523d7": "3\\,", "670baa1052b4848e0b0382b55ab15d73": "\n\\bold{V}_d := \\begin{pmatrix}\n{v_{1,1,1}}&{v_{2,1,1}}&{\\dots}&{v_{d,1,1}}\\\\ \n{v_{1,2,1}}&{v_{2,2,1}}&{\\dots}&{v_{d,2,1}}\\\\ \n{\\vdots}&{\\vdots}&{\\ddots}&{\\vdots}\\\\ \n{v_{1,d,1}}&{v_{2,d,1}}&{\\dots}&{v_{d,d,1}}\n\\end{pmatrix}\n", "670c0564ae94d2a62b1d0b84d0d9b669": "\\mu(-\\infty, \\xi_j] \\leq \\rho_{m-1}(\\xi_1) + \\cdots + \\rho_{m-1}(\\xi_j) \\leq \\mu(-\\infty,\\xi_{j+1}).", "670c087a4780ba4238a34aa42e43cafc": " \\lim_{n \\to \\infty} \\frac{a_n}{b_n} = c", "670c394ea7851a773e3bef66f3b49301": " h_A(\\alpha x)=\\alpha h_A(x), \\qquad \\alpha \\ge 0, x\\in \\mathbb{R}^n,", "670c7d29d617d3f6822d11cc16c33598": "\n\\begin{align}\n&F_\\theta = -\\frac{1}{r} \\frac{\\partial u }{\\partial \\theta} = -J_3 \\frac{1}{r^5} \\frac{3}{2} \\cos\\theta \\left(5 \\sin^2\\theta -1\\right) \\\\\n&F_r = -\\frac{\\partial u }{\\partial r} = J_3 \\frac{1}{r^5} 2 \\sin\\theta \\left(5\\sin^2\\theta - 3\\right)\n\\end{align}\n", "670c8f3ce50d76de19ac45aa09fef70d": "\\frac{r_0 e^{i\\varphi_0}}{r_1 e^{i\\varphi_1}}=\\frac{r_0}{r_1}e^{i(\\varphi_0 - \\varphi_1)} \\,", "670d3f61ae4847f0c5e2b8d0b14a1bab": "r\\in \\mathbb{R}", "670d6af47cb9730f7c22634059f6d9cb": " \\ddot{\\mathbf{r}} = \\frac{G m_1}{r^2} \\mathbf{\\hat{r}}", "670d913f748ce60ee2886ad33ec8696b": "-\\sigma_0", "670dacde93476fc39c7e01efae203bf7": "\\tau_1\\Big.", "670daf0a0e7ce178c5a5b93d64a45090": "1 - 1 + 1 - 1 + \\cdots ", "670e000edb9faf3ec289f4790d58fedf": "P(S)", "670e1b3517d3fc39a38c5f0ce815221e": " d = e^{-\\sigma\\sqrt {2\\Delta t}} = \\frac{1}{u} \\,", "670e3e773ba91d4646ce750a739e008c": "(\\Omega, \\mathcal{F}, \\{\\mathcal{F}_t\\}_{t\\in I}, \\mathbb{P})", "670e67d57c193774747cf9bddf75847c": "\n2 p_0 K(p) = {i \\over p_0 - \\sqrt{\\vec{p}^2 + m^2}} + {i \\over p_0 + \\sqrt{\\vec{p}^2 + m^2}}\n", "670f1785706fa5fe126ad4b45f77eb21": " \\varepsilon (x),", "670f4b070b441a6f985cea23f8cb1227": "n F_{max} = \\frac{2 \\lambda}{D_{crit}}\\,\\!", "670f93c5bf80654ad8c96f65333a1704": "\n\\sqrt{11} = 3 + \\cfrac{2}{6 + \\cfrac{2}{6 + \\cfrac{2}{6 + \\cfrac{2}{6 + \\cfrac{2}{6 + \\ddots}}}}} = 3 + \\cfrac{6\\cdot 1}{20-1 - \\cfrac{1}{20 - \\cfrac{1}{20 - \\cfrac{1}{20 - \\ddots}}}}.\n", "670faa23bed51f0be9f362f5aa830088": "\\mathrm{ozone}", "670fc2220e8a8f81808953582a2092b2": " \\and T_2 = [F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_ ", "670fe662c91a22e44de0774f326aff2f": "\\mathcal G(A)", "670fef0949a1b01869a7c4fcc2c5cc4c": "-B := \\{a - b: a < 0 \\and b \\in ( \\textbf{Q} \\setminus B ) \\}", "67103f2ac5632c9d7c428bf4c3f321d0": "f(z)=h(z)/(z-a)^2 = c_2 + c_3 (z - a) + \\cdots \\, .", "6710625c727469cc3182ce49915fa30e": "F(x,2) = \\frac{\\alpha}{\\alpha+\\beta}-\\frac{\\beta\\left(\\lambda-\\mu\\right)}{\\alpha+\\beta}e^{\\left(\\frac{\\beta}{\\mu}-\\frac{\\alpha}{\\lambda-\\mu}\\right) x}", "67107c20e4c3b08c5b5ff0cb714ee71a": " e(n) = d(n)-\\hat{d}(n)", "6710a82fedfc903e3fbe0d54366b4477": "\\phi_{i=1 \\dots K}, \\boldsymbol\\phi:", "6710e493a711d0a2df926659fa20ee2b": "\\sigma = \\sigma_0\\cdot e^\\frac{-E}{R\\cdot T}", "671107ac3af549113e74e6a8ba387eaf": "u\\in\\Phi\\subset H", "6711bb9b7e5fc73d0ee0710db59bcf62": " \\nu \\rightarrow \\infty ", "6712941095a86defac038977ea46a037": "N_i = \\{v_j : e_{ij} \\in E \\and e_{ji} \\in E\\}.", "6712c4d6d53bb0bcadaf4b4c6df0a388": "\\ell_R=\\ell_r", "6712ff0ab87ed6e352771c20abaa1a7d": "r' = r - \\tfrac{1}{3}", "671313929a6dedf9d57caa0487b87d06": "n_p=0", "67134260660913678ef311a26984b5ab": "s(t) = \\left\\{ \\begin{array}{ll} A e^{2 i \\pi f_0 t} &\\text{if} \\; 0 \\leq t < T \\\\ 0 &\\text{otherwise} \\end{array}\\right.", "6713bbbd03a2573855e6558d5c7cddfe": "L_v^2 L_{vv} = L_x^2 \\, L_{xx} + 2 \\, L_x \\, L_y \\, L_{xy} + L_y^2 \\, L_{yy} = 0,", "6713c3ca24e3bc3731a3a21367da96b7": " \\frac{\\partial f_t(z)}{\\partial t} = -z f^\\prime_t(z)\\frac{\\zeta(t)+z}{\\zeta(t)-z}", "67145e7ab15ef9077c0a85615698d17c": "\\tilde{G_0}", "67147bc62cabea3ed46d0b8dcfe78612": " H = -\\frac{\\hbar^2}{2m} (\\nabla_1^2 + \\nabla_2^2) + V(\\vec{r_1}, \\, \\vec{r_2}) + \\int \\frac{e^2 \\rho(\\vec{r'})} {|\\vec{r} - \\vec{r'}|} d^3r' ", "671487cc12d800ef73b67cfa73ac7642": " \\bar{W}_{ij}(s;L)= \\bar{W}_{1L}(s;L)/ \\bar{W}_{1\\tilde{L}}(s;\\tilde{L})", "671498e0b8f4a631e1b279d315486057": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^{m+1} \\left(A \\left(b\\,c\\,d-b^2 e+2 a\\,c\\,e\\right)-a\\,B (2 c\\,d-b\\,e)+c (A (2 c\\,d-b\\,e)-B (b\\,d-2 a\\,e)) x\\right)\\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{(p+1)\\left(b^2-4 a\\,c\\right) \\left(c\\,d^2-b\\,d\\,e+a\\,e^2\\right)}\\,+\\,\n", "67158f0f97590a9acbba478308e800ab": "y(t_{n+1})", "671631c2a51b4fe6ae339a7456396459": "F(2) + F(7)", "67167dd0ede76bbddccbe017d147eec5": "\n\\xi_1 = 0.68 \\left( POL - 1 \\right) +\n\\left[3.4 - 2.4\n\\exp \\left( -0.002687 \\left( \\alpha_1 \\beta_1 \\right)^{1.5} \\right)\n\\right]^{\\left( 293 K/T \\right)^2}\n", "6716b1e4041657438fb65a1c87ac3aef": "\\displaystyle{|\\alpha|^2 - |\\beta|^2=1,}", "6716bc89ea2b50e556903be8773ff055": "dist(u,\\mathcal{N})= 0,\\forall u\\in \\mathcal{N}", "6716d94996790dbe25eeca3b81f2632d": "\\left(X, \\Sigma\\right)", "67172373c85ad6afa6f03fc08691c871": "v = \\frac{6y}{12y - 2x + 3}", "6717607a0c5e002ab20c4163be6284f2": "\\phi(G) := \\min_{S \\subseteq V; 0\\leq a(S)\\leq a(V)/2}\\frac{\\sum_{i \\in S, j \\in \\bar S}a_{ij}}{a(S)}.\\,", "671774be9f792f48e38b82e518c161a0": " \\langle\\psi|\\mathbf{[\\hat P, \\hat H]}|\\psi\\rangle = 0 ", "671799c240707c7a8be90a3cf15c304e": "R_F\\left(x,y,z\\right)=R_F\\left(\\mu,\\mu,\\mu\\right)=\\mu^{-1/2}.", "6717f1cbf2b2259297ed1b6c404ef435": "V=\\sum _{i=1}^n m_i \\bar{v_i},", "671828e7354cb18f8c1404c7a41fb06b": "H_{out}\\ =\\ f(H_{in}, m)", "671836c90779b4efb6e1a2954e148660": "\\mathbb{P}(y \\mbox{ sent} \\mid x \\mbox{ received})", "671855bbc40ddf3b1d341438ae434371": "\\displaystyle x>R+r, ", "67191935b77d03c725e65eb3d093d21a": "\\frac{\\mu}{\\lambda}\\sum_{i>c} \\frac{c!}{i!}\\left( \\frac{\\lambda}{\\mu} \\right)^{i-c}", "6719256c8f049e356bd6d24b5bee43c0": " \\bar{P}_\\mathbf{\\mathbf{k}} ", "6719b012179f777c5f343427674cb981": "\\scriptstyle \\overline{a \\,+\\, bX} \\;\\rightarrow\\; a \\,+\\, ib", "6719b4c0956e140c6ad9a231108d719a": "c - \\sqrt{c^2 - N}", "6719c3430cf4e8b7aa09d387dc314f22": "\\theta_{t,d}", "6719d7a930aaebe879a15ab4fafe9bd3": "\\psi(a,b)", "6719e01fa2379c1e70b6162504503c21": "\\Omega=\\{1,2,3,4\\}", "6719f676dd12ac5456f9cb83ef9aa6c6": "\\Omega(x) = S(x)\\,\\Lambda(x) \\pmod{x^{2t}} \\, ", "6719f86e7d4702ecc570c4fd05f54fcc": "{F_y}' = \\gamma F_y.", "671a1767504ff105af2fea81ddc6475d": "E = \\{ xy - yx \\mid x, y \\in X \\}", "671a84fc97d3cfa54450625dcd1b81fb": "y=hs+\\rho n \\,", "671a884b16c14338901e96de1055e495": "t\\!", "671b0e6f6bf9d6e3f13cb3f9c58450e3": " P(Condition~WHOIFPI) \\approx RR_{condition} * P(Condition~in~population)", "671b1bcccea34d47207e34dab6cefb33": "\\mathbb{F}_{q^m} ", "671b3d0cb6ca7a95c2a0dce474f37c31": "(A \\oplus B)", "671b46928b05b894aff25ca53729676e": "\nA_{\\alpha \\beta} \\equiv \\langle E_{\\alpha} | \\hat{A} | E_{\\beta} \\rangle .\n", "671b7b2a62c0f896c0ae1c1c794ab0fb": "\\mu > 0.", "671bb25e68ab5061b3d0b9ffdcfff4e9": "\\varepsilon^J = \\varepsilon^{-1}", "671bc158861dfb99da2fe3e79ebbfdb5": "\na_{i,0}= \\left[ \\frac{n+[L/2]-1}{[L/2]} \\right]. \n", "671bdbf7855207d7a8efa7b0c6c05f4b": "|a, b, c\\rangle", "671c0ee27bfcb06b660b5742445527c4": "B^{ij} \\wedge B^{kl}", "671c78017854a850828d9a52a41fa701": "w \\mapsto \\phi(w,z)", "671d1cf8d65863d21c47318c166a02b3": "\nG = Y - \\left\\lfloor \\frac{C_b + C_r}{4} \\right\\rfloor ;\nR = C_R + G ;\nB = C_B + G.\n", "671d45ce685e027226a85eeb1153a75a": "PV\\,", "671d7f2887d355ca699c26f5854ac2cf": "\\frac {d^{n}y(x)} {dx^{n}} + A_{1}\\frac {d^{n-1}y(x)} {dx^{n-1}} + \\cdots + A_{n}y(x) = f(x).", "671d8c12a57b08e6858bebb89855eb12": "M(\\omega)={1\\over{2-\\omega}} \\left ( {1\\over\\omega} D + L \\right ) \\left ( {1\\over\\omega} D \\right)^{-1} \\left ( {1\\over\\omega D} + L\\right)", "671deabb84777802a688fe6332c32131": "= \\frac{1}{T}. \\ ", "671e1d323e50d2d9260afcd6b8a865b0": "p(t)=P^t p(0) \\,", "671e23a2f9ecc65c57682bad3837b713": "\\alpha^{-1}(n)=\\mu(n)", "671e285142b68e611597a622025c4e7c": " K_D = \\frac{C_E \\cdot V_E}{C_B \\cdot t} \\qquad(6d)", "671e6496a2f256b5d3f8f3ac7a9a702e": " Z=\\lim_{\\nu_2\\to\\infty}\\nu_1 F ", "671e7be9bca39125f087ea1ce58e7622": "f(x_1,\\ldots,x_n) = \\sum_{i=1}^n \\operatorname{softmax}(i,x_1,\\ldots,x_n) x_i \\approx \\max(x_1,\\ldots,x_n)", "671e9afb1e2221e8fe78c9d460bb5921": "\\mathrm{S} = (\\frac{\\mathrm{PV}}{\\mathrm{TP}})\\times{TP_s}", "671ea8fb8afe796276773172e871faf9": "a_3 a_2 a_1 a_0", "671f3801328139ba1685d535023f155a": "\\langle njm|x|n'j'm'\\rangle = \\langle njm|\\frac{T_{-1}^{1}-T^1_1}{\\sqrt{2}}|n'j'm'\\rangle = \\frac{1}{\\sqrt{2}}\\langle nj||T^1||n'j'\\rangle (C^{jm}_{1(-1)j'm'}-C^{jm}_{11j'm'})", "671f7fe2f53d57f97dd20ee06279b0b9": "Z[j]=\\int [dA]\\exp\\left[- \\frac{i}{2} \\int d^4x\\operatorname{Tr}(F^{\\mu \\nu} F_{\\mu \\nu})+i\\int d^4x \\, j^a_\\mu(x)A^{a\\mu}(x)\\right] ,", "671f7ff4cb32ad26150c5d065466db34": "\\frac{\\partial \\sigma}{\\partial K}", "671f840d6f00f064dccab8991e0f4966": "M(n)/n < 0.38201", "671f8aac1136d02eda77fb7890fe3741": "\\sum_{x=M}^{M+N}\\exp(2\\pi if(x))=O\\left(N^{1+\\varepsilon}\\left({t\\over q}+{1\\over N}+{t\\over N^{k-1}}+{q\\over N^k}\\right)^{2^{1-k}}\\right)\\text{ as }N\\to\\infty.", "671fa0630e0b52a71639562be994fcac": "\\operatorname{E}((X - \\mu)(X - \\mu)^{\\dagger})", "671fa1fbbbdea70da05de22e7ab89405": "\n g_1\n = 3 \\uparrow\\uparrow (3 \\uparrow\\uparrow (3 \\uparrow\\uparrow \\ \\dots \\ (3 \\uparrow\\uparrow 3) \\dots ))\n \\quad \\text{where the number of 3s is}\n \\quad 3 \\uparrow \\uparrow (3 \\uparrow \\uparrow 3)\n", "671fdabfdac3a938ab913ce61a8eed19": " \\sigma (f) = \\{ f^{-1}(S) \\, | \\, S\\in B \\}. ", "672005cd0455343dd4a7c2b6095a5993": "\\phi(0)=X", "67200d5c4b744ed0129f67cd52304aa8": "6 a_2^3.", "672043c4b1bfdb130b2e4a094c49ebc1": "\\dot{J}_{ab} = -J_{am} \\, {J^m}_b - {E[\\vec{X}]}_{ab} + W_{a;b}", "6720822d02191a1d1366b2058c4d254a": " \\frac {a}{b}= \\frac {\\left(\\frac {a}{c}\\right)} {\\left(\\frac {b}{c}\\right) }.", "6720ad9987111f41acb9d6f81f19a962": "\\{", "6721330c9e55eafb0c35d83921f879d2": "\\displaystyle{G/T=G_{\\mathbf{C}}/B.}", "672167a470956d73f98fb83f255b218c": "M_{DC} = 0.2 \\times 5.785 + 12.5 = 13.66 ", "67216f3ec29e09732b2c112fa8ae228b": "SF\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\n1\n\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;I\n", "6721c4ef9d303bec60e083f632e35ebb": "\\gamma_0", "6721e5aa6a12615344931f88973d0cb4": "p=1/2^{45}", "6721fe6e2e16fad82c1e391fa64b26eb": "\\varphi_{\\frac 1 n X}(t)= \\varphi_X(\\tfrac t n) \\quad \\text{and} \\quad\n \\varphi_{X+Y}(t)=\\varphi_X(t) \\varphi_Y(t) \\quad ", "6722032b92d4e92461a5266aaf5a936f": "\\text{length}(\\gamma)=\\int_a^b | \\gamma '(t) | \\, dt. ", "67223586542dd371631ec893cf26b3b1": "\\sigma (a_i) = a_{i+1},", "67223f58a228110b09fbac4d10c5da57": "\\lambda_i=-\\mu_j", "672271012852169a0ab2b2d0336502f7": " \\mathbf{P}(X_t=x_i,X_{t+1}=x_j) \\neq \\mathbf{P}(X_t=x_i) \\mathbf{P}(X_{t+1}=x_j) ", "672290562d0123bcebcfb50edac57887": "\\textstyle M_{\\mathrm{V}} = m_{\\mathrm{V}} - 5\\log_{10} \\left(\\frac{100}{\\mathrm{parallax\\ in\\ milliarcseconds}}\\right)", "6722c218a6f30869ef6886dc4b050a37": " x ", "672300a72c05a1f4b9f705b4fcdbeee4": "= \\left( \\frac{d\\boldsymbol{f}}{dt}\\right)_r+\\boldsymbol{\\Omega \\times f}(t)\\ ,", "672346c526a076308dad52654f6ae5c2": "T_{[abc]} = \\frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}) .", "672349796278e3aab44ea3652c542871": "\\Phi_S=", "6723aba320e6a6816fa03d3353a8ea3f": "\\forall A \\forall B \\, ( [ \\exists E \\, ( A \\in E ) \\and \\forall C \\, ( C \\in B \\implies C \\in A ) ] \\implies \\exists E \\, [ B \\in E ] ) \\,,", "67240e96375abda910ce7537e7b9d06b": "(w/p)_{-}(x) = \\max \\{ -w(x)/p(x),0\\})", "672427d9c224f604424a374a8ea1aa5e": "\\ell, \\ell_1, \\ell_2, \\ldots, \\ell_n ", "6724295a2b7eee288914766e580aaee8": "t^3+pt+q=0", "672449bcd44f7ea5e24b00636b051b86": " m_{\\rm o} = \\sqrt{m_{\\rm p}^2 + m_{\\rm F}^2} ", "6725b74ddc5bd4fc7646e21acf8a4f2a": "\\bar {x_i}=c", "6726816518b82562438e0c814dc88ec8": " \\langle \\mathcal{B} \\rangle \\le 2\\sqrt{2} ", "6726a418c5a763b25926060665d1139d": "\\theta_o=90^{\\circ}\\,", "6726b2e5c5f5b86f0253d647676e638d": "U(S,V)\\,", "6726dcb6cc4058d4de07f6c8c5e02806": "W_X=V_X-V_Y", "6726e56c069df8311a3a371946be68e2": "\\scriptstyle \\hat p", "6726f1c0bd86fd472a2f4b9e0a7dec36": "\\alpha : F(A) \\longrightarrow A", "67274c7d335d87f21c9c38ab9e61eb3f": " \\{ |0\\rangle, |1\\rangle \\} ,", "672751fefaa1fb49bc329e9edb449778": "\\frac{\\partial(f_{i_1}, \\ldots, f_{i_k})}{\\partial(x_{j_1}, \\ldots, x_{j_k})}", "672784e4d955f5103a727d635e3e571e": "m_t(\\theta) = A(z_t) \\frac{1}{S}\\sum_{s=1}^S H(x_t,y_t,z_t,v_{ts};\\theta) \\sum_{j=1}^J\\!\\gamma_j v_{ts}^j,", "6727a4cd2d6ec2f21694679a9e84f114": "x \\rightarrow \\pm \\infty", "6727b0940b39129acc37897cc706bc53": "\\mathbf{g}=f_1\\mathbf{d}t \\otimes \\mathbf{d} t +f_2\\boldsymbol{\\eta}\\,", "6727f2e703cdda015e64626bd7f42a48": " x y = U.\\,", "672828f60af79cf9310cb22b687bfede": "\\textbf{D}", "67284986a58ea976bf5984dd3c75344c": "E_r^{p,q} \\Rightarrow_p E_\\infty^{p,q}", "67285a5794139bfbf458e0f7bf9d0248": "|S| \\le \\binom{n}{\\lfloor n/2\\rfloor}.", "67287ea2df417145b8c1c09b4ba71d05": "\\varphi*1=\\operatorname{Id}", "67289cfe49d8cfb8016594fe695efabd": "\\omega_0^2\\, =\\, g\\, k\\, \\tanh\\, (k h).", "67289d87d33506ecd8ea7fbd3e02da8a": " S = \\gamma_{SA} + (\\gamma_{CA} - \\gamma_{SC}) ", "67289ebc59ab3a68fc113208929c4bd3": " p_{Y|X}(y|x)", "6728a162e79963eabfae143914f67cc3": " \\prod_{p} \\Big(1 + \\frac{1}{p^2+p-1}\\Big) = 1.419562... ", "6728a4ff195fc6ae282a63fbd5c7835c": "p_{00}", "6728c22bb4ee30881adb39e3bdfbe607": "MV\\frac{d\\psi}{dt}=Y ", "6728eba779a6f39a9d12cbb243f96655": "\\log_2 n", "6728fa599a6d73bd21b7ba6ebf16e562": "\\left| f_Y(y)\\, dy\\right| = \\left| f_X(x)\\, dx\\right|,", "6729284677d5bbb1f885555b94a1b118": "E_{\\text{Core State}}", "672a17242380cc378b28008da76c4c0c": "n - \\text{number of disjoint cycles in the decomposition of }\\sigma", "672a46b5b34a86710da1abb0e94d56b5": "\\sqrt{E}", "672ad8c87752aee162e7407738909cf9": "\\boldsymbol{B}=-\\frac{4}{5}\\frac\n{\\boldsymbol{\\omega} m R^2}{r^3}\\cos\\theta.", "672b059bae4164271d0217843ac6811b": "E_{12}", "672b0ce6faf65678b272138c1d46972a": "(0,-\\sqrt{3})", "672b2686a3fe066177e8962bba99d186": "(T_{fus})", "672b579944432ca37e3941fb8cc429b4": "l^a=(0,1,0,0)\\,,\\quad n^a=(1,-\\frac{F}{2},0,0)\\,,\\quad m^a=\\frac{1}{\\sqrt{2}\\,r}(0,0,1,i\\,\\csc\\theta)\\,,", "672b76a6d0b47e87a049243b9a991cbc": " \\text{AGE}z \\times 2 + \\text{BAT}z + \\text{HR}/\\text{FB}z + (\\text{SBA}z + \\text{SBR}z + \\text{XBT}z)\\times .33 + \\text{BL}z + \\text{UZR}z + \\text{PAY}z + \\text{LUCK}", "672bb0b01fd240c650e4db79c7cfb211": "O((k+\\lambda)^{-4})", "672bed2d0d4b36923bc8e980650b198c": "x_0,c_0", "672bfc7542c6036865df9a1023c2a0c9": " x = \\left[\\frac{3}{2} \\left( \\frac{\\pi}{2}- t \\sqrt{ \\frac{2\\mu}{ {y_0}^3 } } \\right) \\right]^{2/3} ", "672c2866cb0c6abfe8f974c4960e9d50": "X\\subset A", "672c449974cc854bb1ec6451ee5e0dee": "{\\nabla} \\times \\mathbf{F} =0\\boldsymbol{\\hat{x}}+0\\boldsymbol{\\hat{y}}+ {\\frac{\\partial}{\\partial x}}(-x^2) \\boldsymbol{\\hat{z}}=-2x\\boldsymbol{\\hat{z}}.\n", "672ce1ad0dc97454b51494e25780219c": "\\det S = \\exp\\left(-\\zeta_S'(0)\\right).", "672ce62e2f1201c235cc0eac72eff24c": "\n\\begin{align}\nf_k(x) & = 1 - \\frac{x^2}k+\\frac{x^4}{2! k(k+1)}-\\frac{x^6}{3! k(k+1)(k+2)} + \\cdots \\\\\n& {} \\quad (k\\notin\\{0,-1,-2,\\ldots\\}).\n\\end{align}\n", "672d1242f899a6c7e9ef8b0432b4232f": "i(t) = \\frac{\\operatorname{d} Q_c}{\\operatorname{d} t}", "672d23caf6ea2c292740844efeb23739": " \\tilde{q}\\tilde{\\bar{q}} \\rightarrow q \\tilde{N}^0_1 \\bar{q} \\tilde{N}^0_1 \\rightarrow ", "672d2bfa2451f91d6ee5d9a7be34251e": "\\begin{align}\n P(k \\mid H_0) & = {n\\choose k}(0.5)^k(1-0.5)^{n-k} \\approx 1.95 \\times 10^{-4} \\\\\n P(k \\mid H_1) & = \\int_0^1 {n\\choose k}u^k (1-u)^{n-k} du = {n\\choose k} \\mathrm{\\Beta}(k + 1, n - k + 1) \\approx 1.02 \\times 10^{-5}\n\\end{align}", "672d8c1d5280deb83fda12c07ebfde1b": "W''_i, W''_{1-i}", "672da17724b53d124d5470d531151c32": " \\frac{1}{p!} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} \\delta^{\\nu_1 \\dots \\nu_p}_{\\rho_1 \\dots \\rho_p} \n= \\delta^{\\mu_1 \\dots \\mu_p}_{\\rho_1 \\dots \\rho_p} ,", "672db80b5d9fac6167b8bc399d98765a": "10^2+96=196=14^2", "672dfca4af43396d24c07dfe5c10e4b9": "\\ell < 2^k", "672e0e90865bfa3f7ec6662087ae72af": "\\mathrm{idx}_J \\lambda\\,", "672e16acf12fbc2fcbc8e53f8c36b4d4": "(\\cos\\alpha,\\sin\\alpha)", "672e613a6ad384f4381c9a4d05c1dace": "\n\\mathbf{y} = \\frac{1}{x_{3}} \\, \\tilde{\\mathbf{x}}\n", "672e8da14e807aadd021b6cb562c2084": "S(x) = \\sum_{n=1}^\\infty n\\,\\delta\\left(x-\\frac{1}{n}\\right)", "672ebf090e490e99608dabeb24d64c3e": "\\langle\\mid r\\mid \\rangle ", "672ef1fd5a56c43f3e30fbbb6d2c9fec": " \\Gamma_{kl}=\\sum_{j=1}^J (\\lambda_j \\wedge \\mu_j)[p_{jk}(\\delta_{kl}-p_{jl})+c_j^2(p_{jk}-\\delta_{jk})(p_{jl}-\\delta_{jl})]+\\alpha_k c_{0,k}^2 \\delta_{kl} ", "672ef290b1e5c7da47994e6b507609cf": "(x = 0) \\lor (x = 1) \\lor (x = 2) \\lor \\cdots.", "672f804992ee78a7baeecb460fccd672": "G_{m,n} (x,y) = C_{m,n} |x-y|^{2m-n} \\int_1^{\\frac{\\left||x|y - \\frac{x}{|x|}\\right|}{|x-y|}} (v^2-1)^{m-1} v^{1-n} dv", "672f85bbacbcb3c035a7f36e831639fe": " f(x_1,\\dots,x_n) = g(x_1,\\dots,x_{i-1},\\,h(x_i,\\dots,x_j),\\,x_{j+1},\\dots,x_n), ", "672fb386fcee90a27af0abb35688984b": "q=q_0+iq_1+jq_2+kq_3", "672fb8840aab805777ce9b0e55f0c8a3": "\\mathbf e_k", "672fdfef67f441e74ddcc7450714199c": "i \\in \\{ 1,2,\\ldots ,I \\}", "673005b08ece9425a01662917ca7bc2b": " A y = B x \\, ", "67302d198f962c634c8b77c3d199c939": "p \\log p \\,", "67308446843c9738fadb1640400114ab": "\\chi_\\parallel", "6730a6113b94f6f618c86cf749ce055a": "N_\\mathit{w}", "6730c92e65a577f837fa9f980d7865d1": "|\\Psi _{BETA}(\\omega \\alpha ,\\beta )|", "6730d2baa94355e022bce537a2e27720": "FDCR = E\\left( {{c_0}{V_0} + \\frac{{\\sum\\limits_{}^{} {{c_i}{V_i}} }}{{{c_0}{R_0} + \\sum\\limits_{}^{} {{c_i}{R_i}} }}} \\right)", "6731265b0d22c865db65e94ff42d4c8f": "-\\sum_t\\chi(t)\\phi\\left ( \\frac{t}{\\pi} \\right )^{(p-1)/m}", "673173325e44fd1911a18c90e831aa24": "\n\\int{ e^{i \\int{ f(x) g(x) dx}} }[Df] = \\delta[g] = \\prod_x\\delta( g(x) )\n", "673177b2e372a4f7f5305ed79bc3165c": "\n\\frac{(n(P+N))^\\frac{n}{2}}{(nN)^\\frac{n}{2}} = 2^{\\frac{n}{2}\\log(1+P/N)}\n\\,\\!", "6731c562ccf16a13afd60ee46066d1f1": "\\frac{1}{\\lambda}\\ = \\frac{4}{h} \\left( \\frac{1}{n_1^2} - \\frac{1}{n_2^2} \\right)= R_H \\left( \\frac{1}{n_1^2} - \\frac{1}{n_2^2} \\right)", "673226916e1383a5a04b8de5e975d030": "\\frac{\\delta P}{\\delta t} =\\rho a \\frac{\\delta v}{\\delta t} ", "6732440bfd9a11bbf7c5feaa3a960934": "\\sup_K u(t-\\tau,\\cdot)\\le C\\inf_K u(t,\\cdot).\\,", "67330bc95c409d1cd2e07d5750ccb837": "a \\cdot (b+c)=a \\cdot b+a \\cdot c \\quad\\forall a,b,c \\in Q", "673334628837bc539297e7e2c3b9128e": "0 = \\int^{c_L}_{c_R} (x-X_M) \\mathrm{d}c", "67333c7443ac970ded49749b7c5f84ba": " \\mathbf{[V]}= \\mathbf{[Z][I]} ", "67336104860310cc51fb18bb6e49e0cd": "a_{i0}=\\left (\\sum_{j=1}^{m}a_{ij}-1 \\right )\\mu", "673362ec7fd593e7117fa6d71fe86a6e": "H^k(M) \\cong H_{n-k}(M).", "673364a9fd34f9dce216b6fe06000ea2": "y_{n + 1} = y_n + h f(t_n + h, y_{n + 1}). \\,", "673367bdc125a98b7e44b9466b8e6e8b": "\\exists a \\,\\forall x\\, \\forall y\\,[(\\phi(x) \\and \\psi(y)) \\rightarrow Baxy] \\rightarrow \\exists b\\, \\forall x\\, \\forall y\\,[(\\phi(x) \\and \\psi(y)) \\rightarrow Bxby].", "673412b8e74feeff44c95710a629500b": "\\tfrac{1}{2}\\left[X(z)+X^*(z^*) \\right]", "673419aa2ae7a6718c25bb30bc35e785": " 3.333e8 W/m_{2} ", "67342959000deb8465cec36a63f7781b": "\\Lambda(n) = \\begin{cases} \\log p & \\text{if }n=p^k \\text{ for some prime } p \\text{ and integer } k \\ge 1, \\\\ 0 & \\text{otherwise.} \\end{cases}", "6734448c034247e12983b84b930b937c": "X_T=C+\\mu RX \\, ", "673459fbb33e0ee18a4f7a75f61f0fca": "E\\left\\{ e|Ctx \\right\\}", "6734622ce34ad6cb220210ae01cd0d6a": "\\text{extend}: ((C \\times A) \\rarr B) \\rarr C \\times A \\rarr C \\times B = f \\mapsto (c, a) \\mapsto (c, f \\, (c, a))", "673462f9fadfde68c3c4e8a321fd9cab": "t(z)=\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}:z = \\frac{1\\cdot z+1}{0 \\cdot z + 1} = z+1", "673470727341b500d9a2ea7371a55bdd": "\\mathrm{^{241}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{242}_{\\ 95}Am\\ \\left(\\ \\xrightarrow [16.02 \\ h]{\\beta^-} \\ ^{242}_{\\ 96}Cm \\right)}", "673470e816b097887d4a830c3731214a": "J_i(\\mathbf{x},\\mathbf{U}_i)=\\sum_{j=i}^{N-1}\\ell(\\mathbf{x}_j,\\mathbf{u}_j) + \\ell_f(\\mathbf{x}_N).", "67347add246d896d05c4bd70b19a2b03": "f(x_i) \\Delta = \\int_{i\\Delta}^{(i+1)\\Delta} f(x)\\, dx", "6735132c7587d607ed58ea8fdac11345": " \\nabla s = C_p \\ln(\\overline{T}) +R \\ln(\\overline{p}).", "67354478f02a56581e2902a1d390ab3d": "\\sum\\limits_{i=1}^{N_{\\lambda}}{m_i} =N_{\\mathbf{v}}.", "673596d4bc48779ea3c57f34468eea2f": "\\begin{array}{c|ccccccc}\n& 0&1&2&3&4&5&6 \\\\\n\\hline\n0&1&0&0&0&0&0&0 \\\\\n1&0&1&1&1&0&0&0 \\\\\n2&0&1&2&1&1&0&0 \\\\\n3&0&1&1&2&1&1&0 \\\\\n4&0&0&1&1&2&1&0 \\\\\n5&0&0&0&1&1&1&0 \\\\\n6&0&0&0&0&0&0&1\n\\end{array}\n", "673598c55863d384f636d1e8e9d0bbd3": "E (\\mathbf{r}, t)", "6735bd00833d6db009e591a20e0c714c": "\\lambda=\\nu+\\lambda_p", "6735e706d851db706bbb23ded82f64ed": "N_p/p = 1 + O(1/\\sqrt{p})\\ ", "67362f3cc5c47b71bc6658af9ac23828": "\\sigma(f(z_1),f(z_2)) \\leq \\rho(z_1,z_2)", "67364e2b169f80430054d57fa1c8ff86": "N \\ge 1", "673664f8cc808d347b22916dc52620bd": " \\left(\\mathbf{ab}\\right):\\left(\\mathbf{cd}\\right) = \\mathbf{c}\\cdot\\left(\\mathbf{ab}\\right)\\cdot\\mathbf{d} = \\left(\\mathbf{a}\\cdot\\mathbf{c}\\right)\\left(\\mathbf{b}\\cdot\\mathbf{d}\\right) ", "6736d94def29817bb906fdd855f46e62": "=2s\\ \\omega \\ \\mathbf{u}_{\\theta}-\\omega^2 R(t)\\ \\mathbf{u}_R \\ . ", "6737140c44f4e62842d2f85bb7fc5b4f": "\\ell = w\\,", "67374d978203681f0227aefda65b06d8": "F_{\\mathrm M}", "67377381bcad888e3d20f732f855f9ff": "\n g_2 = 60\\sum_{(m,n) \\neq (0,0)} (m + n\\tau)^{-4},\\qquad\n g_3 = 140\\sum_{(m,n) \\neq (0,0)} (m + n\\tau)^{-6}\n", "6737bd4f9a0322979124e7a881b878ea": "| f(z) | \\leq M \\text{ for all } z \\in \\Omega. \\,", "6737fc09041e6750ad1078c6c998bb2c": "g = n g_D ", "673802fdecafc7b1d38ad9fcdf8dba64": " D=\\delta V=0|_{(s_0,w_0)}", "67389a192351ccbf80b01ae174d703b0": "2^nn!", "6738afd4a543ed1b09fe3aa440d3c414": "k\\frac{d}{dx}\\left(A_c\\frac{dT}{dx}\\right) + Ph\\left (T-T_\\infty\\right) = 0", "6738c6dc534fa4cc6f2907fe793d425d": "\nR_\\mathrm{eq} = R_1 \\| R_2 = {R_1 R_2 \\over R_1 + R_2}.\n", "673931bb5733bdfd9a77245199545659": "\\{a\\vee b, a\\wedge b\\} = \\{a, b\\}", "6739388fddfabf9ca79b644c2fe6cc7e": "=\\mathbf{ \\Omega \\ \\times } \\left( \\mathbf{ \\Omega \\times X}_{AB}\\right) + \\mathbf{a}_B + 2\\ \\boldsymbol{\\Omega} \\times\\mathbf{v}_B\\ ", "6739a9e29ec39b90d65688178eabe741": "X^*_{c(X^*, X)}", "6739b962cdf2b78034c13b5a49e19862": "\\mathrm{Ein}", "6739be0fbd187a9ccfc227cfd7b87dad": "\\begin{align}\\operatorname{arsech}\\, x = \\operatorname{arcosh} \\frac1x & = \\ln \\frac{2}{x} - \\left( \\left( \\frac {1} {2} \\right) \\frac {x^{2}} {2} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {x^{4}} {4} + \\left( \\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} \\right) \\frac {x^{6}} {6} +\\cdots \\right) \\\\\n & = \\ln \\frac{2}{x} - \\sum_{n=1}^\\infty \\left( \\frac {(2n)!} {2^{2n}(n!)^2} \\right) \\frac {x^{2n}} {2n} , \\qquad 0 < x \\le 1 \\end{align} ", "6739c716219bb34db391840648692d73": "s := \\#\\{r \\in \\{0,1\\}^T\\mid f(x, r)=1\\}", "6739fca92dfbdd7683b1177aa4573a38": "\\left( t_{TOF}\\right)\\,", "673a0a8e8e4b3825cf8c4bb907fc2e03": "\\; \\sum_k p_k = 1.", "673a55fb4e219bd4191d6002684243e9": "\\rho=1 ", "673a695badc73ffc822f1733182fede6": " \\sigma_x, \\sigma_z ", "673aed5a1add10b0976b6e9dbf2eee05": "\\langle\\hat{n}_{\\mathbf{k},s}\\rangle=\\frac{1}{e^{\\hbar \\omega / k_B T}-1}.", "673b1bd7272db4c9f0d8740a932aecc4": "\\delta g_B", "673b900d39eeadb365cb3ceda790124c": "[HA]_{0^{ }}", "673b9a4ee8e8088244c2a28a1fd78495": "\\ \\displaystyle \\varepsilon > 0\\ .", "673ba34b63a04c2a72d11b2a90449c25": "h(x_k)", "673be60e97f3f4938b42de7214a83ff9": "X_1,X_2,\\dots,X_k", "673c0a2666b8185ffb0d96e19300d87b": "(f_1 f_2 f_3 \\cdots f_n)'/(f_1 f_2 f_3 \\cdots f_n) \\!", "673c152c3f2606876446d3e76247e859": "\\left(1/6,\\ -\\sqrt{7/4},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "673c2728fe5dd9c39b1e0c6ae5c7396a": "x = p+td", "673d4fa391e1136b2867be580ca1861d": "\n{\\left( {{Area_{octane}/M_{octane}} \\over {Area_{nonane}/M_{nonane}}} \\right)}_1\n= F =\n{\\left( {{Area_{octane}/M_{octane}} \\over {Area_{nonane}/M_{nonane}}} \\right)}_2\n", "673d67a01fc0db35badd0e054bacc7eb": "P_0e^{rT}=\\int\\limits_{0}^{T} M_ae^{r(T-t)}\\, dt=\\frac{M_a(e^{rT}-1)}{r}.", "673d76206710e2629417900ee83fb56c": "x^{-2}", "673d81b02fc055303117e3ed37616081": "T_m=\\frac{T_b \\cdot T_r}{\\sqrt{T_b^2+T_r^2-2 \\cdot T_b \\cdot T_r \\cdot \\cos(\\alpha_r-\\alpha_b)}}", "673dca29fbb2e6268ed94e7d70b1059a": "jh_y\\leq y \\leq (j+1)h_y\\,", "673de06ef62c0379ee325c98ffe1b15f": "3 \\times 10^0", "673e27b20b2b21e281e253d280131f3d": "\\mathbf \\phi_2 = \\left (\\frac{2\\alpha_2}{\\pi} \\right ) ^{3/4}e^{-\\alpha_2 r^2}", "673e4946fd674a9d6373cd1c3a9c74d3": "\\mathcal{L}(\\psi) = \n\\bar{\\psi}(i\\partial\\!\\!\\!/-m)\\psi - g\\bar\\psi_L \\phi \\psi_R", "673e8993859476cb32af1256f6b82bcc": "\\Delta\\,L{{=}}\\alpha\\,\\!_{CET}L\\Delta\\,T", "673ed25d994ba41fbdc9ecae3dd02238": "a\\in\\text{cl}(A)\\,", "674031ab6f42b9c670cbe5aefa1c3781": "f(t) = \\sum_{k=-\\infty}^\\infty f\\left(\\frac{k\\pi}{\\sigma}\\right)\\frac{\\sin(\\sigma t-k\\pi)}{\\sigma t-k\\pi} \\qquad (t\\in \\mathbb{R})", "674089a92b0cba3cc035420634c19655": "\\mu = {{m_{\\mathrm{N}} m_{\\mathrm{e}}}\\over{m_{\\mathrm{N}}+m_{\\mathrm{e}}}}", "67414162d6b1b54b817446f24fc9ea49": "k = \\pi / (2 a)", "674141c3bf39fb98011f41d06917aa7e": "\nP=\\begin{bmatrix}\n0 & 0.5 & 0.5\\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0\\end{bmatrix}, \n\\quad \n\\mu=\\begin{bmatrix}\n\\mu_1(x_1)\\\\\n\\mu_2(x_2)\\\\\n\\mu_3(x_3)\\end{bmatrix}\n=\\begin{bmatrix}\n15\\\\\n12\\\\\n10\\end{bmatrix} \n\\text{ for all }x_i>0", "674161ef52a872889eb6f481e0d9601a": "f_N=1/(2\\Delta t)", "6741740a785189473b058a67e482f007": "{w} = {Mz} + {q} \\,", "6741b89f2885e1fa3ea4767ff13b4b37": "\\sum^\\infty_{n=1} {\\frac{1}{n(n+k)}} = \\frac{H_k}{k} ", "6741daf8ae2a45a5756f52522753d08f": "\\Delta = I_{-\\infty}^{+\\infty}\\frac{f_2(x)}{f_1(x)}= V(-\\infty) - V(+\\infty) \\quad (27)\\,", "6741fc40315cbbd47947f2c823594024": "\\omega_{\\eta}^*", "674241e05c3f95da1c257ed809e32dea": " \\Delta \\nu \\ge \\frac{1}{\\Delta t}. ", "674247b569b89e74a14fffedcc37ff9b": "\\rho:H\\rightarrow\\mathrm{Aut}(\\mathbb{V})", "67424b6f27cbd14646406411b3cab95e": "\\sigma_{s} = E \\varepsilon", "6742831ce149933bf8e421012b3f37bf": "\\left [\\begin{smallmatrix}\n-1 & 0 & 2 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{smallmatrix}\\right ]\n", "6742913222be27552e33aebabb016269": "\\rho^{op}: G^{op} \\to \\mathrm{Aut}(X)", "6742d62dc9e77198fac9861c242e78c2": " \\left(\\mathbf{ab}\\right)\n\\!\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\cdot\n\\end{array}\\!\\!\\!\n\\left(\\mathbf{cd}\\right)=\\left(\\mathbf{a}\\times\\mathbf{c}\\right)\\left(\\mathbf{b}\\cdot\\mathbf{d}\\right)", "67434e101a2cb6f4e972098eacda1581": "\\partial P_\\delta", "6743b9fcc57cf1b169fb4ef6c7a955d2": "A ", "6743e3e7bc2229d204db43ba412921be": "H_0=p^2/2m", "67443609c89279f01bae291ab7f470e4": " \\langle A \\rangle_\\psi = \\sum_j a_j |\\langle \\psi | \\phi_j \\rangle|^2 ", "6744ca179e970a52e523fefb4818d9fa": "p\\left(\\vec\\theta\\right) \\propto \\sqrt{\\det \\mathcal{I}\\left(\\vec\\theta\\right)}.\\,", "67453862e222477ab0177e21d1f8f6a3": "\\sum_{k=0}^\\infty\\frac{k!}{(2k+1)!!}=\\sum_{k=0}^\\infty\\frac{2^k k!^2}{(2k+1)!}=\\frac{\\pi}{2}\\!", "6745dc63f4ee0671a4a0e21adeede66f": "1/n!.", "674619449be50673ae48a33c79e05147": "c_2I=c_1E-S_e^{-1}\\left (\\frac{E}{k_e-r_eE} \\right )+P", "6746892bad2763fcdbd4a75787e21c00": "E''= \\frac{1}{y''} + \\frac{y''^2}{2}", "674692e91cae6d87a8a52206469ee9d8": "\\displaystyle \\frac{1}{\\left\\|\\boldsymbol \\sigma\\right\\|\\left(2\\pi\\right)^{n/2}} e^{-\\frac{1}{2} \\mathbf x^{\\mathrm T} \\boldsymbol \\sigma^{-\\mathrm T} \\boldsymbol \\sigma^{-1} \\mathbf x}", "6746b8f1ea968371bc6bac35fe3bc573": "K(0) = \\tfrac {\\pi} {2} ", "6746d71e779d764b38f43f3f3b74aff9": "T:\\mathcal{H} \\to \\mathcal{H}", "6746e7af4646913c2ffd6b50f5488b23": "\\rho^{2} = r^{2} + \\alpha^{2} \\cos^{2} \\theta", "6746ee58a97e2bcf0e871d007aa7619e": "z_{2i}=z_{2i+1}=x_i.", "6746f50cb1d3a693734293473f9857a7": " \\left[\\text{d}_t^2 - (v \\Delta t/\\Delta x)^2 \\text{d}_x^2 \\right] \\Psi(x,t) = 0. ", "6746f620c7129d70f91f8ecff2be55b0": "R_\\text{H}", "6747143cf3583363068c93d9a1badce2": "M_X=(g_\\theta,Z)", "674769e3326f8cf937af4282f2815c02": "SP", "67477424db29786ccab00955a1fcb48a": " |\\operatorname{det}(M)| \\leq n^{n/2}. ", "67477e1f789ad805f4e7777be7ce7ed8": " \\mathbf{ \\hat U} (t) \\mathbf{ \\hat T} (t)|\\psi(0)> ", "6747b096190df83b0eebfdac53455af7": "\n\\overline{n}_\\mu =\\sum\\limits_{n=0}^\\infty nP_\\mu (n,t)=\\frac{\\nu t^\\mu }{\n\\Gamma (\\mu +1)}. \n", "6747fd3e3e28c3e66022225fa06c44b3": "\\lambda_p \\leq \\lambda_s \\leq \\lambda_i", "67480dffabfee67f614d2e169336c577": "\\Omega\\simeq 500\\rm Hz", "6748271fa3a89f39527eaa800bcd80aa": "G_1(\\gamma)=\\sum_{k\\neq0}\\sum_{j\\in Z} |\\hat{\\psi}(a^j\\gamma)\\hat{\\psi}(a^j\\gamma+\\frac{k}{b})|", "674835ad8b15db5781549d566ea624b7": "\\,\\!\\omega", "67485fdef4b9c3e539fdf43db583470d": "[X,fY]=\\rho(X)f\\cdot Y + f[X,Y]", "6748752818256f55d23fb773b645a051": " \\boldsymbol{\\sigma}^* ", "6748f5709d659875fb8ede9e99c59c96": "\\frac{2^{m+1}\\Gamma(m/2+1)}{k^{m+2}\\Gamma(-m/2)}\\,", "67498d78dbaf3c3d08f514cb07b96207": "J(\\Omega)", "6749b9d0651f2abcd4fdb43e66464bf2": "1-d", "6749bb5ee65b23099404c81fc6042f21": " \\Phi=1", "6749be05e187739c24349476c702fe6f": "\\tilde{\\Phi} = \\sqrt{3} \\ln{\\Phi}", "6749d8f4ed4ac8d0dc1728b43a181354": "\\Lambda = \\hat{C} \\hat{\\Phi}", "6749e7258ba3bcf2c14426ef18569122": "x^*=x", "674a65af826cd08f119e910af9ef39aa": "=N b^*(0)b(0)\\sum_{\\boldsymbol{R_p}} e^{i \\boldsymbol{k \\cdot R_p}}\\ \\int d^3 r \\ \\varphi^* (\\boldsymbol{r}) \\varphi (\\boldsymbol{r-R_p})\\ ,", "674b1a587564849d5c8f548934593b95": "\\chi_1,\\chi_2, \\ldots , \\chi_n ", "674b28f3a8f1677e9c95bc54fd8906f1": " \\langle Tx \\mid x \\rangle ", "674b408e90955d0f2f4cb5a301a47f10": "\\sigma_\\varepsilon^2", "674b4e820c534bb44a249a05318089c6": "h(c) = h^H(c) + h^D(c) = h_0", "674bc71cb2626276ddea5328f312965e": "\\{p_1,p_2, p_3, p_4\\}", "674bd6fd09e4aeb6ff686e7cba7f84dc": "(x-3)(x+3)(x^4-5x^2+3)^6. \\, ", "674bdecd6ab5aa271e1e87e7c8d8f906": "\\mathcal{P}x\\in U", "674bec5b30c6402dd38e1d95ff27a3e8": "\\mathbb{W}^{k}", "674c06961f8ea5c9d4578553c38619e8": " H_0(\\mathbb{Z}S) = \\frac{\\ker \\partial_0}{\\mathrm{im} \\partial_1} = \\mathbb{Z} \\langle v \\rangle \\cong \\Z, ", "674c5ebd97ddb4ef75ecbcde7e3347bd": " R_{ij} ", "674cd90f8c9550e6e353aab1789ecba0": "\\int_{\\mathcal{X}}\\psi(x,\\theta) \\, dF(x)=0", "674cdae4dbc65d149759090f2ffbd642": "\\Delta v/c=(-1.3\\pm2.7)\\times10^{-6}", "674cfea98858d356bf25d8b41918a5e5": " |f(x) - L| < \\varepsilon,", "674d0858d4dc1ba38ffb5b0055a9af1d": "\n \\tilde{x}_j \\equiv -\\frac{\\hat{\\phi}_j^{\\mathrm{refl}}}{2 k_p}\n = \\hat{x}(t_j) + \\hat{x}_{\\mathrm{fl}}(t_j) \\,,\n", "674d0a357f276666d924a2548c0e51ba": "\\int f(x)d_qx = (1-q)\\sum_{j=0}^\\infty xq^jf(xq^j)", "674d0a6ee3ada4b25ee52adb46b5c32d": "k = \\frac{1}{\\tau} = \\frac{\\ln 2}{T} = \\frac{\\ln \\left( 1 + \\frac{r}{100} \\right)}{p}\\,", "674de2990d396548089677f235ebe9fd": "= fl((x_1*y_1)(1+\\delta_1)+(x_2*y_2)(1+\\delta_2))", "674ea2ee74a61b3ce945c28ab6f8428b": "\\hat{z} = \\pm \\hat{p} \\,", "674eb881d118b33a6a87da8c322df467": "\\frac{dC_W}{dt}", "674ec1a8b9792757638e0497ed8c2b87": "\\frac{1}{1} + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{8} + \\cdots = \\psi.", "674edffa1d964a5df1d70deec2c6991a": " \\frac{1}{2T}\\int_{-T}^{T}|F(a+it)|^{2} dt= \\sum_{n=1}^{\\infty} [f(n)]^{2}n^{-2a} \\text{ as } T \\sim \\infty. ", "674f33f0ebdb2bb9e1e635a36bb71394": "\\mathbb{S}_{n\\geq 6}", "674f3e246961cadaf4236ff578d37fae": "C=h \\, ", "674f47a6e2f8431926074304b65d7f3b": "\\Omega (|V|^3)", "674f4af40c8758fa089458b39867875d": " X^\\star", "674f8186ebbd31a6d6c3533050f91c39": "Re(a) = Re(b)", "674fc0578297da1d314048695f44bd93": "\n\\begin{align}\n\\{Q_\\alpha^A , \\bar{Q}_{\\dot{\\beta} B} \\} & = 2 \\sigma_{\\alpha \\dot{\\beta}}^m P_m \\delta^A_B\\\\\n\\{Q_\\alpha^A , Q_\\beta^B \\} & = 2 \\epsilon_{\\alpha \\beta} \\epsilon^{A B} \\bar{Z}\\\\\n\\{ \\bar{Q}_{\\dot{\\alpha} A} , \\bar{Q}_{\\dot{\\beta} B} \\} & = -2 \\epsilon_{\\dot{\\alpha} \\dot{\\beta}} \\epsilon_{AB} Z\\\\\n\\end{align}\n", "6750153c68e3cb7383fd0af7c2353425": "\\sum_{i=1}^n k\\mathbf{F}\\times \\mathbf{F}_i = k\\mathbf{F}\\times(\\sum_{i=1}^n \\mathbf{F}_i )=0,", "67501c1fe0a7feba025d4d3cb13e1bc4": "\\begin{align}\np(\\mu|\\sigma^2; \\mu_0, n_0) &\\sim \\mathcal{N}(\\mu_0,\\sigma^2/n_0) = \\frac{1}{\\sqrt{2\\pi\\frac{\\sigma^2}{n_0}}} \\exp\\left(-\\frac{n_0}{2\\sigma^2}(\\mu-\\mu_0)^2\\right) \\\\\n&\\propto (\\sigma^2)^{-1/2} \\exp\\left(-\\frac{n_0}{2\\sigma^2}(\\mu-\\mu_0)^2\\right) \\\\\np(\\sigma^2; \\nu_0,\\sigma_0^2) &\\sim I\\chi^2(\\nu_0,\\sigma_0^2) = IG(\\nu_0/2, \\nu_0\\sigma_0^2/2) \\\\\n&= \\frac{(\\sigma_0^2\\nu_0/2)^{\\nu_0/2}}{\\Gamma(\\nu_0/2)}~\\frac{\\exp\\left[ \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right]}{(\\sigma^2)^{1+\\nu_0/2}} \\\\\n&\\propto {(\\sigma^2)^{-(1+\\nu_0/2)}} \\exp\\left[ \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right]\n\\end{align}", "67501eecf53fe0e5b9c945b992fcadaa": "s^2 > 0", "6750565f0bbd54db7d5035c93bf73682": "V_s", "67508c6569a1524ecb05c7ec5a6c093f": " \\log_b(b) = 1 \\!\\, ", "67509eaa7bcfe857d149606178c75197": "\n\\epsilon_\\mu^j(p) \\cdot \\epsilon_\\mu^{k*}(p) = 0 \\;\\; \\text{for} \\;\\; k \\ne j.\n", "6750ea55f25cf944e9201345a0b6700b": "\\mu_{i,j}(p_{S_{i}\\cap S_{j}})", "67510c897cafa081695b6cdc716b44e6": "\\omega(t) = \\phi^\\prime(t) = {d \\over dt} \\phi(t),\\,", "67510d44601785776471e2a944173828": "x(i)", "67512204570af726a5d4e1318460a8b2": "\\int_0^{2\\pi} \\int_0^{\\theta} \\sin\\theta' \\, d \\theta' \\, d \\phi = 2\\pi\\int_0^{\\theta} \\sin\\theta' \\, d \\theta' = 2\\pi\\left[ -\\cos\\theta' \\right]_0^{\\theta} = 2\\pi\\left(1 - \\cos\\theta \\right)", "675142b6ce2577e15eb8c58059e22e2b": "T = I + \\frac{1}{2}B - 1", "675144980f32614c21e18d222afb419c": "\\textstyle \\left ( \\frac{2}{1} \\right )^{1/2} \\left (\\frac{2^2}{1 \\cdot 3} \\right )^{1/3} \\left (\\frac{2^3 \\cdot 4}{1 \\cdot 3^3} \\right )^{1/4}\n\\left (\\frac{2^4 \\cdot 4^4}{1 \\cdot 3^6 \\cdot 5} \\right )^{1/5}\\cdots ", "6751725897a0cd6c278a5f42ca4dc9ae": "qr^*", "6751af89c70a44b8f1d1ad2972f1f666": "\\| B \\|_{\\infty} > c \\iff \\sqrt{\\varepsilon} B \\in A := \\big\\{ \\omega \\in C_{0} \\big| | \\omega(t) | > 1 \\mbox{ for some } t \\in [0, T] \\big\\}.", "6751b9a02d4ae5a09d393ea8756b97f6": "I_\\lambda=\\sum_k \\epsilon_{k,\\lambda} c_k", "67520142374978d1a0d40a2c783c5d05": "\\lambda_{n-k}<0", "6752210ad5ecd8c5fe9964765ef395f9": "k=2,\\ldots,n-1", "675238c2dd76e6a1c337808b51f0af28": "A^2 b", "67524fd850939bd06f292e0d3ec84357": "h_{n+1} = \\frac{2}{\\frac{1}{g_n} + \\frac{1}{h_n}}", "6752ce57fed55a83d72f76897c8a5dbb": "\\alpha\\beta^2", "6752e1c436aa274ab4ac677bbe5eb50e": "\\mathbf{e} = \\hat{\\mathbf{\\theta}} - \\mathbf{\\theta}", "6752f2530a48accd3c84b2d6e2f3f5a6": "-616\\pm 4.1%", "6753001e27cca8e3f1fe387691aa7891": "\\vec{R}_{1}", "67531572c33652bf08282f3d8dc1d2cd": "\\frac1{A(z)} = \\frac1{ 1 + \\sum_{i=1}^P a_i z^{-i} }", "67534c479fb100eff9511fa2575dfc62": "\\operatorname{P}\\left[E_1\\mid E_1\\cup E_2\\right]=\\frac{\\operatorname{P}[E_1]}{\\operatorname{P}[E_1]+\\operatorname{P}[E_2]}", "6753bbf7a2a2661600f2c4270418eb1a": "J \\left (C^{-1},y \\right ) f \\left (C^{-1}(y) \\right )", "675479da6bfabcfa66d252d33083fe1f": "V_{IOS}", "67550451e8f0ab7c8c6dd801bf77a3ae": "L_{\\omega_1,\\omega}", "67552f0eb60ab78310ebe3184ff00e90": "R[x] / \\mathfrak{p} R[x] = (R/\\mathfrak{p}) [x]", "675531f55e8e7ac86a1618cd310ff300": "\\alpha=1\\,\\mathrm{cm}/\\mathrm{s}\\,\\!", "6755408fd9a62a1f8586a3e0e17b3664": "n \\geq 8", "675550c1ae4159b73bbec492d9ce0414": " \\frac{1}{k^2 \\sigma_\\alpha^2} \\int_\\Omega \\|X - \\mu\\|_\\alpha^2 \\, \\mathrm d \\mathbb P = \\frac{\\mathbb E\\|X - \\mu\\|_\\alpha^2}{k^2 \\sigma_\\alpha^2} = \\frac{\\sigma_\\alpha^2}{k^2 \\sigma_\\alpha^2} = \\frac{1}{k^2}. ", "6755c8d284c1f669abaf849b7969cbb9": "\\mathbf{P} ( X > (1+\\delta)\\mu) < \\left(\\frac{e^\\delta}{(1+\\delta)^{(1+\\delta)}}\\right)^\\mu.", "6756047bb5f2f9034a8e3cd233a1bbc9": "\\Delta V=\\alpha_V L^3\\Delta T", "675614ca061aaccef71231ae5ae75c7c": "\\Lambda_n[v_n]/\\langle v_n^2-\\Delta\\rangle", "67567fdbc4ed9fd479dea4cc63b10234": "F(EG, (X)_p)^G", "6756c77c23bf61f5b3077a8bec517971": "u(a)=u(b)=0.\\,", "6756d12f2698be264a9a6fc319de9ad1": " k^1=\\frac{t_r-t_M}{t_M(1-(t_r/t_c))}", "67570b537bded14926ddfb03c4a7ec4a": "A = \\frac{1}{2}VC_{ijkl}\\epsilon_{ij}\\epsilon_{kl} - ST + \\sum_i \\mu_i N_i\\,", "67572ea4fa3dfeb6d1f85e5da0fa4dd4": " \\sum_{i=0}^{n} (p_{i+1} - 1) \\cdot p_i\\# = p_{n+1}\\# - 1 ", "675829ce2e39af84ed909644d2c89d73": "L(\\vec{r},\\hat{s},t)=\\frac{1}{4\\pi}\\Phi(\\vec{r},t)+\\frac{3}{4\\pi}\\vec{J}(\\vec{r},t)\\cdot \\hat{s}", "675853b623dcbb0c0c81eda21df8295c": "<_x", "6758b3481757b0ad5648009f22980a07": "s\\left( t \\right)=\\sum_p B_p h_p(t)", "6758b5d9a4cd798dea1c2bfdc31228b1": " y\\left( m \\right) = \\beta _0 + \\beta _1\\frac{{\\left[ {1 - \\exp \\left( { - m/\\tau} \\right)} \\right]}}{m/\\tau} + \\beta _2 {\\left(\\frac{{\\left[ {1 - \\exp \\left( { - m/\\tau} \\right)} \\right]}}{m/\\tau} - \\exp \\left( { - m/\\tau}\\right)\\right)} ", "6758d559cc7a416c903d1f916de9b663": " |B_{2 n}| \\sim 4 \\sqrt{\\pi n} \\left(\\frac{n}{ \\pi e} \\right)^{2n}. ", "6758ea31be918249fbc64f155fe4d929": "L_C M", "675909d1d7ed39f07f5e61776eda65ee": "\\Rightarrow e^{\\ln x} \\cdot \\ln x = \\ln z\\,", "67590b4a8c787342ac55cfb402a05261": "\n\\left(\\frac{\\partial \\log L(\\theta | x)}{\\partial \\theta}\\right)_{\\theta=\\theta_0} \\geq C\n", "675a3584a7f4eb469fb3aeb6caee8b86": " \n\\int_E \\varphi \\, d\\mu = \\int_A \\varphi \\, d\\mu = \\int_{A_k} \\varphi \\, d\\mu + \\int_{A-A_k} \\varphi \\, d\\mu.\n", "675a45b27e9044c495e0478a818cfb19": "Pi = k(1 -k)^i-1", "675ad23c39adaf7ac3346abc9cb2123f": "\\hat{\\eta}", "675b0a5a7b2df17885bc7e103c1cc4c3": " (\\mathbf{a} \\wedge \\mathbf{b})^2 = (\\mathbf{a} \\cdot \\mathbf{b})^2 - \\mathbf{a}^2\\mathbf{b}^2 = \\left|\\mathbf{a}\\right|^2\\left|\\mathbf{b}\\right|^2( \\cos^2 \\theta - 1) = -\\left|\\mathbf{a}\\right|^2\\left|\\mathbf{b}\\right|^2\\sin^2 \\theta", "675b4190ce93ec096c5bfbbc0a1365f0": " \\partial_\\sigma X^\\mu (\\tau, 0) = 0, \\partial_\\sigma X^\\mu (\\tau, \\pi) = 0 ", "675be45e699b145bb50c6c7b13377e9e": "\\,\\ \\csc x", "675c45b8a984cdc28f418b02dbeda7b3": "G_n=\\sigma\\bigg(\\bigcup_{k=n}^\\infty F_k\\bigg)", "675ce4ffbfa5b2c4ccbc066c737718c7": "1/\\sigma", "675d5d5134afc6ef1fad16054bbb9865": " \\boldsymbol{m}_k = (1-c)\\frac{\\boldsymbol{p}_{k+1}-\\boldsymbol{p}_{k-1}}{t_{k+1}-t_{k-1}}", "675d99a36d899e468a9873008c25ef84": "L_i := \\min(", "675da6ae468f08c49f9ed7f8dfc5c0f1": "\\operatorname{rank}(A^T A) = \\operatorname{rank}(A A^T) = \\operatorname{rank}(A) = \\operatorname{rank}(A^T)", "675deb0002b460d2d3ef157f2373c5a1": "\\frac{V_1}{V_2}", "675df4b0aa61682cd8dc6b5cc791bf22": "\nS(\\lambda_k,\\theta) = \\sum_{k=1}^\\infty a_k \\cos(\\lambda_k\\theta) \\qquad \nS(\\lambda_k,\\theta,\\omega) = \\sum_{k=1}^\\infty a_k \\cos(\\lambda_k\\theta + \\omega) \\,\n", "675e0ac6c4147ffa0467c3c98fc863e8": "\\frac{\\theta \\models \\phi}{\\theta \\vdash \\phi}", "675e0e318938b04f16178f5615651d40": "A X = B\\,", "675e758ffce613a202cdb93c7ec3451c": "Q^T=Q^{-1}", "675e98f82fd0347f4286c0a6f2606158": "\nG^{\\star}_n (x) \\sim \\begin{cases}\n e^{-|\\alpha_n| x} & \\text{:} \\, \\alpha_n^2 < 0, \\, \\text{ evanescent or trapped} \\\\\n e^{i \\alpha_n x} & \\text{:} \\, \\alpha_n^2 > 0, \\, \\text{ propagating}\\\\\n e^{\\left( \\kappa - \\frac{1}{2} \\right) x} & \\text{:} \\, h_n = H / (1- \\kappa), F_n(x)=0 \\, \\forall x, \\, \\text{ Lamb waves (free solutions)}\n\\end{cases}\n", "675eaca57216c629963c1b1617f34102": "\\displaystyle \\beta=\\max\\{0,\\,\\beta^{PR}\\}", "675fc6e7bd351ed6cf2bc5c6e8bacf51": "\\alpha > 0 ", "675fd8b6839088b7dc317fae035c4ea1": "\\nabla^2\\mathbf{F}=\\nabla(\\nabla\\cdot\\mathbf{F})-\\nabla\\times(\\nabla\\times\\mathbf{F})", "676010c28bf3542c153bf4950fe0e3bb": "R_{TE}+X_{TE}", "67603225c38653edba8aa7e7bf3413ee": " \\mu_{log} = \\ln(m) - \\frac12 \\ln\\!\\left(1 + \\!\\left(\\frac{s.d.}{m}\\right)^2 \\right) ", "676069e314d5f21021b26a56c5c14019": "E_\\theta= i\\frac{Z \\,I_0 \\delta \\ell\\, k}{4\\pi r} e^{i(\\omega t-k\\,r)}\\,\\sin(\\theta)", "6761520bc95d37cdd304106cac199a41": "t=1,\\dots,T-1", "6761ed0627cb4b6174a53993b055c960": "\\vec f_{op}", "67620619840b96a2f19fa411d8b6faba": "\\textbf{x}_r = \\textbf{x}_o + \\alpha (\\textbf{x}_o - \\textbf{x}_{n+1})", "676217ced2e35e92088d8c85bf43bfcc": " f_u(z) = \\sum_n j(u(n)) z^n \\ . ", "6762184d21d63d23feb01ac0eab808b5": "V= \\sqrt{\\cosh\\tau-\\cos\\sigma}\\,\\,S_\\nu(\\sigma)T_{\\mu\\nu}(\\tau)\\Phi_\\mu(\\phi)\\,", "67627aa6a85e7cd88c6e69bcaa093ff8": "\\sqrt{(r_1-r_2)^2 + (g_1-g_2)^2 + (b_1-b_2)^2}.", "6762ac9715728f396e69068c2f9d83aa": "\\frac{n^2}{(n+1)^2}", "6762b74ce9a4f6213e4dc0214eee5151": "\\mathbf{C}^{n+1}\\setminus\\{0\\} \\stackrel{(a)}\\longrightarrow S^{2n+1} \\stackrel{(b)}\\longrightarrow \\mathbf{CP}^n", "6762fdb636b2028d5f00de363e96e275": "y\\sec\\varphi=y\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2},", "6762fede4382123115d7c56b55f0cad2": "\\dot{v}_2 = {1 \\over C} i_3 ", "676324b5c9aeb2cb243e15f6c58dea84": "J_{\\gamma,n}", "67632ee46043acb9f5571f699f38f5ec": "l=|r_2-r_1|", "67639851afd58d06505c16729efdefe6": " I_{b,t+1} = I_{b,t} \\times {(1+R_{S,t+1})\\over (1+R_{M,t+1})^b}", "6763be4fa81d0855d0536993877d6c07": " \\langle\\mathbf{v},\\mathbf{u}\\rangle ", "6763f23efbf5aa733dd9e3a5130a4bc9": "O\\left(\\log n\\right)", "676410df3fb29cc67e1434bc150ef018": "\\forall \\bar{x} [\\psi(\\bar{x})]", "676430d670b431c1c8391a40ead76a8f": "f(x_1, \\ldots, x_k)", "6764322ec76b8042adb1cbb556be7d0f": "m_0=m_1 e^{\\Delta V\\ / v_e}", "67648bb3c8040daba82d6ecdf27e7a4f": "\\operatorname{Hom}(G,\\mathbf{Z}/p),", "6764974cd7c882c3213892b5e919b770": "\\sigma A + \\sigma B - \\sigma A \\cdot \\sigma B = 1 - (1 - \\sigma A)(1 - \\sigma B)", "6764add8ef177a0b19bff7c6f41daed1": "\\begin{bmatrix}\\cos(\\pi / 6^{R}) & -\\sin(\\pi / 6^{R})\\\\ \\sin(\\pi / 6^{R}) & \\cos(\\pi / 6^{R})\\end{bmatrix}", "6764ff9d114f13e0d5eed32e3f1d34b2": "I_{\\parallel}", "67653892000c4279276628e31298d09a": "R(r)=Bh^{(1)}_l\\left(i\\sqrt{-2m_0E\\over\\hbar^2}r\\right),\\qquad r>r_0", "6765447bf87c4dcc6036c1f603b79053": "R=\\frac{2^kQ-(Q+1)}{(Q+1)-2^k}", "676553b54854a9a45bcd9152c1005b9d": "Cx + Dy = b", "67658120fb98f04dc85ae8467215de7c": "\n\\Beta \\left( \\frac{n+1}{2},\\frac{1}{2} \\right) =\n2\\int_0^{\\pi/2}(\\sin\\theta)^{n}(\\cos\\theta)^{0}\\,d\\theta\n= 2\\int_0^{\\pi/2}(\\sin\\theta)^{n}\\,d\\theta\n= 2 W_n\n", "67658b01a5b0eb73241d5f4766eedecf": "b^2 = 78^2-5959", "67658f3382de31bdfbac2a3555779f6c": "\\mathbf{X}(t) = {\\lbrack X(1,t), X(2,t), \\dots , X(p,t)\n\\rbrack}^T", "676590f88598dd9c34432b18f7449a4f": "\\exists z", "6765dd9615bfdc8d1dcdcdceb7e67daf": "\\| u \\|'_{W^{k, p}(\\Omega)} := \\begin{cases} \\sum_{| \\alpha | \\leq k} \\| D^{\\alpha}u \\|_{L^{p}(\\Omega)}, & 1 \\leq p < + \\infty; \\\\ \\sum_{| \\alpha | \\leq k} \\| D^{\\alpha}u \\|_{L^{\\infty}(\\Omega)}, & p = + \\infty. \\end{cases}", "6766253ddd9de315047366f0c1adcf4c": " G = G_1 \\supseteq G_2 \\supseteq \\ldots. ", "676658f2797f4875db109be72a65721c": "\n\\left( \\frac{\\mathrm{d}S_{\\sigma}}{\\mathrm{d}\\sigma} \\right)^{2} + \\left( \\frac{\\mathrm{d}S_{\\tau}}{\\mathrm{d}\\tau} \\right)^{2} + 2m U_{\\sigma}(\\sigma) + 2m U_{\\tau}(\\tau) = 2m \\left( \\sigma^{2} + \\tau^{2} \\right) \\left( E - \\Gamma_{z} \\right)\n", "6766c02a6bf5242cdfee786cb677dd0b": "\\{m \\setminus L \\,\\vert\\; m\\in L\\}", "6767323c24d0986763fe22d19bc296aa": "\\frac{x^{2^{n-1}-1}}{(1-x^2)(1-x^3)\\cdots(1-x^n)}", "67674c46d15e47bc9a631883242e5852": " |\\mathbf{V}|^2= |\\mathbf{V}_0|^2 + 2 |\\mathbf{A}|(|\\mathbf{P}-\\mathbf{P}_0|).", "6767633b5a16caefbbf132eb68dfdbc0": "m = [12.3, 7.6]^T, \\quad M = \\begin{bmatrix}1.44 & 0 \\\\0 & 2.89\\end{bmatrix}", "67676607952d3eaae4c8277fc80b6708": "V_0, V_1, \\ldots ,V_k", "676791c429fbd222fd384dbd1323e001": "\\rm \\ 4Co_3O_4 + 18Na_2O + 7O_2 \\rightarrow 12Na_3CoO_4", "6767e4859a48a6224048525fcfb8d589": "u_0\\in V", "67680dc23480988db41ed248d01926a4": "\\tau = \\mu \\frac{\\mathrm{d}u_x}{\\mathrm{d}y}", "676840b41a7bd9e1500d20bc9cac838b": "v_{th}=\\sqrt{\\frac{8k_B T}{m\\pi}}", "67685870a664639bcbf41b90ac894112": " x^2 - 2y^2 = -1 ", "67686f604a949d865b1edec78a9840ea": "\\frac{Y-Y_n}{a} = -b(U-U_n) ", "6768e13eef87040f50db3a2ea020e9bf": "t=\\left\\lfloor\\frac{d-1}{2}\\right\\rfloor.", "6768f32641c4d1cf757515b0f2aaa2ab": "\\frac{1}{c^2}\\frac{\\partial^2\\varphi}{\\partial t^2} - \\nabla^2{\\varphi} = \\frac{\\rho}{\\varepsilon_0}", "6769214604565155ceca6d226858e510": " J_v = \\frac{1}{A_M}*\\frac{dV}{dt} = \\frac{\\Delta P}{\\mu *(R_u + R_c)}", "6769443a6cade99a1d321cc18568b3a8": "\\begin{cases}\n \\infty & \\text{for }\\alpha\\in(1,2] \\\\\n \\frac{x_\\mathrm{m}^2\\alpha}{(\\alpha-1)^2(\\alpha-2)} & \\text{for }\\alpha>2\n \\end{cases}", "67696009d1fcbbd7dda4c0eb0bd04341": "\\sigma=5/8", "676963e6d108ad1f87180310b92da812": "((s-1)\\bmod n)+1", "676978ad408fa44e87359637ca865557": "p,q\\in S^2", "6769a5553be7f51e206f3d28d4c39af3": "|A|-r(A)", "6769ba8e6534b18b0068e2b2bac648db": "\\begin{bmatrix} ^\\diagdown m_{r\\diagdown} \\end{bmatrix} \\begin{Bmatrix}\\ddot{q}\\end{Bmatrix}+\\begin{bmatrix}^\\diagdown k_{r\\diagdown}\\end{bmatrix} \\begin{Bmatrix} q\\end{Bmatrix}=0.", "6769ff9572afce21d16f7c77e507c51a": "2.7257", "676a55c74c4b8a98ceb532bfe5a4c1e9": "\\sigma_c=\\sigma_a=\\limsup_{n\\to\\infty}\\frac{\\log |a_n|}{\\lambda_n}.", "676a705afd70cb613edf2b15bc39b969": "C_{fb}", "676af827e6f2995c22fdc5d77fa8931e": "f(z+1) = f(z)\\,", "676af9246f0f0f7cb5692322e163b5fc": "\\langle G,\\rho\\rangle", "676b4059945a0c13d0ecfd44c66d5315": "\\mathcal{F} = \\mathcal{O}_{\\mathbf{P}^r}(n),", "676c67c510428e6f4030f42fba6ebc21": "\\overline \\Gamma", "676d3e3a7195eea73df6d579ca079fb8": "\\mathrm{su}(2)_R \\,", "676d4e641324a51693c5abebed3358ef": "c_n=1+c_{n/2}", "676d5b5d17ef4364a29f38e9fe1b1ace": "\\frac{m^*v_{emit}^2}{2} \\approx \\hbar \\omega_{phonon (opt.)}", "676d6935fb94f82533e3e6997dacb264": "Y_0 = \\frac{1}{Z_0}\\,", "676d7d825c24a7f836c7823970a840d3": " R = \\{ (t,y) \\in \\mathbf{R}\\times\\mathbf{R}^n \\,:\\, |t-t_0| \\le a, |y-y_0| \\le b \\}. ", "676e3d63fedc67f16c5d83eb6f3beef0": "c_{p} = \\frac {c_{p0}} {\\sqrt {1-{M}^2}}.", "676e7788ca29a38e9277a41da096409f": "\\mathrm{APF} = \\frac{N_\\mathrm{atoms}\\cdot \nV_\\mathrm{atom}}{V_\\mathrm{crystal}} = \\frac{6\\cdot (4/3)\\pi \nr^3}{[(3\\sqrt{3})/2](a^2)(c)}", "676eb80060867ddb1118771e52eb1d3c": "\\frac{1}{12}~-~\\log A~+~\\frac{z}{2}\\log 2\\pi~+~\\left(\\frac{z^2}{2} -\\frac{1}{12}\\right)\\log z~-~\\frac{3z^2}{4}~+~\n\\sum_{k=1}^{N}\\frac{B_{2k + 2}}{4k\\left(k + 1\\right)z^{2k}}~+~O\\left(\\frac{1}{z^{2N + 2}}\\right).", "676ecc23c29b67df378c22c94ea4e7f8": "\\forall t \\, (T^*\\ E\\ t)", "676eed7f1ca729176cfffa086580dd97": "| F(x, y, p) | \\leq a(x, | y |, | p |)", "676f3ee227bc3d49b1fd45d820818eb1": "S(\\bullet{}abbabbabb\\bullet{}) \\Rightarrow A(\\bullet{}abb\\bullet{}abbabb, abb\\bullet{}abb\\bullet{}abb, abbabb\\bullet{}abb\\bullet{}) \\Rightarrow A(a\\bullet{}bb\\bullet{}abbabb, abba\\bullet{}bb\\bullet{}abb, abbabba\\bullet{}bb\\bullet{})", "676f4128c2d619d5d30edc7e3da1ddec": "\\mathcal{F}_k = \\mathcal{F} \\otimes_{\\mathcal{O}_S} (\\mathcal{O}_S/{\\mathcal{I}}^{k+1})", "676f931bf9c89d9614f0c35e77190bd3": " C_{i i}= 0 ", "676fdc0325fe42e8976190d3681b2608": "\\mathrm{D}", "67703ad0a43345c5951b4c606a1fe955": " g(\\mu) ", "67707fc911afef6cd753f3c961910157": "\\left \\{ a_n \\right \\} \\,", "677088a0d00e75a1f5473a5bd886fbab": " E_{pot} = m g h ", "6770a3b5ee06da843a64cb187247d56a": "\\,\\!F_0", "6771351261db5d17c38f866156d7a682": "P^+", "67715a38765234e072141b0cefe35e29": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi}t_g\\ \\frac{V_r}{V_t}\\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin i \\cos^2 u\\ \\sin u\\ du\\ =\n-3\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}(e_g\\ \\sin u\\ -\\ e_h\\ \\cos u)\\ \\frac{p}{r}\\ \\cos u\\ \\sin^2 u\\ du \\ = \\\\ \n&-3\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}(e_g\\ \\sin u\\ -\\ e_h\\ \\cos u)\\ (1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u)\\ \\cos u\\ \\sin^2 u\\ du\\ = \\\\\n&3\\ \\sin^2 i \\ e_h\\int\\limits_{0}^{2\\pi}\\ \\ \\cos^2 u\\ \\sin^2 u\\ du\\ =\\ 2\\pi \\frac{3}{8} \\sin^2 i \\ e_h\n\\end{align}\n", "6771a7faebc99d51df8c529462b2b8e0": " \\beta = 0", "6771f396cc3958988a6b08459a5add1c": "(1-z)\\sum_{n=1}^m \\frac{z^n}{n}=z -\\sum_{n=2}^m \\frac{z^n}{n(n-1)} - \\frac{z^{m+1}}{m},", "677227b01fd74f5174071b922c1c25f7": "\\frac{f(a+h)-f(a)}{h}.", "67723114f78742a1def74d66ca1bd5c8": "V(\\rho,\\varphi,z)=\\sum_{n=0}^\\infty \\sum_{r=0}^\\infty\\, A_{nr} J_n(k_{nr}\\rho)\\cos(n(\\varphi-\\varphi_0))\\sinh(k_{nr}(L+z))\\,\\,\\,\\,\\,z\\le z_0", "67728aba43780fab1ce79e1702f3c9c1": "\\frac{\\mbox{Expected Return}}{\\mbox{Economic Capital}}", "6772c881969123dd179e01ec57d3ac3d": "L/D_{max}=\\frac{4(M+3)}{M}", "677323556f6306d7018be9c34a1cad62": " \\sum_{n>0} n|c_n|^2 r^{-2n} \\le \\sum_{n>0} n |c_{-n}|^2 r^{2n}.", "67733c40aa108f417f1941bcb3815f1c": "L\\mathbf{v}_i = \\lambda_i \\mathbf{v}_i", "67734d8eb4d2c6ee9e97855b4edf2421": "\\displaystyle{{\\partial_{n+}u\\over \\lambda + {1\\over 2}}={\\partial_{n_-}u\\over\\lambda - {1\\over 2}}=\\varphi.}", "677370510d34567d8a8a4a09dac608e7": "W_N", "6773728e9c3ea8aebc19c838a68fda76": "\\text{Ext}: \\{0,1\\}^n \\times \\{0,1\\}^d \\to \\{0,1\\}^m", "6773820e13545b9ab6b8f5a78ce72692": " \\mathbf{x}_{k} = \\begin{pmatrix} x_{1k} \\\\ x_{2k} \\end{pmatrix} ", "6773bd0958cc4b8da2dc1167ecf39efc": "(10)~~ ~~ A=\\frac{\\alpha}{1+F/F_0} ", "6774007d80c1338af47d597c097baa3a": "x = L \\cot^2(\\theta/2)", "67743173470c6b87f7ef386dfbf40a24": " \\textstyle{2 -\\frac{\\log(\\sqrt{2})}{\\log(2)}=\\frac{3}{2}}", "67749e9e64e3bd0cfd0172c37eb102c8": " J_x = J_1 = i\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n0 & 0 & 1 & 0 \\\\\n\\end{pmatrix} \\,, ", "6774d317584229d1a2413bb55a6a168e": "-\\mathrm{d}\\gamma\\ = \\Gamma_1\\mathrm{d}\\mu_1\\, + \\Gamma_2\\mathrm{d}\\mu_2\\,.", "67755b8b6c4104fc6dec5d4647f4511c": " \\nabla_X(fY) = (Xf) Y + f \\nabla_X Y", "67755d2855051e731d485ad83e82d715": "\n\\mathrm{EVM (%)} = \\sqrt{ {P_\\mathrm{error} \\over P_\\mathrm{reference}} } * 100%\n", "67758167c42674c6cdf125a6e1f8e648": "\\alpha+\\beta=180^\\circ-\\gamma", "67759ebb43d9117243fef050a33c896d": "s_n(x+y)=\\sum_{k=0}^n{n \\choose k}p_k(x)s_{n-k}(y).", "6775c651d40fa3091cf59c1276c5421a": "\\scriptstyle z_{1} \\;=\\; -\\frac{1}{2} \\,+\\, j\\frac{\\sqrt{3}}{2}", "67765223780b7adffe61b028701fa46b": " { \\frac{\\partial{(\\rho \\phi)}}{\\partial t}} + { div\\, (\\rho u \\phi )} ={div\\, (k\\, grad\\, \\phi )} + {S_{\\phi}} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,(1) ", "67765bc2964f7a61fdcfcc7c316a0b31": "\\frac{\\partial F}{\\partial n_3} = n_3\\sigma_3^2-2n_3\\sigma_3\\left(\\sigma_1 n_1^2+\\sigma_2 n_2^2+\\sigma_3 n_3^2\\right)+\\lambda n_3 = 0\\,\\!", "6776a0d8913e28ffe10351d61f041015": "y\\in T", "6776e2e74a269c9abb2cf9e638c4137e": "\n2 d\\sin\\theta = n\\lambda,\\!\n", "6776f0f1d49f66e4543a7bf756cc5415": "\\operatorname{}^{n - 1}T_n\n = \n\\left[\n\\begin{array}{ccc|c}\n \\cos\\theta_n & -\\sin\\theta_n \\cos\\alpha_n & \\sin\\theta_n \\sin\\alpha_n & r_n \\cos\\theta_n \\\\\n \\sin\\theta_n & \\cos\\theta_n \\cos\\alpha_n & -\\cos\\theta_n \\sin\\alpha_n & r_n \\sin\\theta_n \\\\\n 0 & \\sin\\alpha_n & \\cos\\alpha_n & d_n \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n\n=\n\n\\left[\n\\begin{array}{ccc|c}\n & & & \\\\\n & R & & T \\\\\n & & & \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n", "6776f855d83690092777225ae01fdec9": "\\left(x - a \\right)^2 + \\left( y - b \\right)^2=r^2.", "677776be867050a681a2ee8c30711d5b": "u(x,y) = \\int_{R^n}P_y(t)f(x-t)dt", "6777bb38587d6ace80d2a60418dd48a5": "\\mathbf{b}_1 = \\dfrac{\\mathbf{h}_1}{h_1}; \\;\n\\mathbf{b}_2 = \\dfrac{\\mathbf{h}_2}{h_2}; \\;\n\\mathbf{b}_3 = \\dfrac{\\mathbf{h}_3}{h_3}.", "67783032a8b6225704760a4f8bdb2625": "\\frac{1}{2 \\pi i} \\int_{\\Gamma} \\frac{\\zeta ^k}{\\zeta - T} d \\zeta = T^k", "677851e8797e5a33c77b4424dc584b65": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log (\\frac{\\varepsilon}{3.715D} + \\frac{15}{Re})\n", "67787658a3efc937f111e253a1ef44eb": "\\left(A_1 \\oplus A_2 \\oplus A_3 \\oplus\\ldots\\right)^k=A^k_1 \\oplus A_2^k \\oplus A_3^k \\oplus\\ldots", "67787f289876b012e1c40c59e3246bc3": "L_{c}(X, Y)", "677883f27e5fd7241b44cf2d12b8581a": "P \\;\\vert\\vert\\vert\\; Q", "6778b2f67f62acc85c73f267c9eede19": "\\sqrt{p_{n+1}} - \\sqrt{p_n} < 1 ", "6778b7175d768738b03c2717e9c4a7ca": "I_4 := 0", "6778b7d718b0c8ed1459af8de3cda683": "\\mathbf{v}(x,t)", "6778bfd7cddbc5c8d05d2ba015ee1e15": "\n\\mathcal{A} \\Psi(1,2, \\ldots, N) \n= \\frac{1}{N!} \n\\begin{vmatrix}\n\\psi_{n_1}(1) & \\psi_{n_1}(2) & \\cdots & \\psi_{n_1}(N) \\\\\n\\psi_{n_2}(1) & \\psi_{n_2}(2) & \\cdots & \\psi_{n_2}(N) \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n\\psi_{n_N}(1) & \\psi_{n_N}(2) & \\cdots & \\psi_{n_N}(N) \\\\\n\\end{vmatrix}\n", "6778c8212e2e236847cbf27e869c89a1": " R \\otimes I_W ", "6778ff0bab3c102eb321efcab71a2fba": "\\Re(s)\\in (0,n)", "67790400c15140ee95a8d8051cf4f14b": "M=\\bigcup_{n_m=1}^I \\R^{d_m}", "67791e9bc9280633b1d52901c33bfee2": "g:\\mathbb{R}^m \\times \\mathbb{R}^p \\to \\mathbb{R} ", "677950c8ab4a7db87280eb5157b1accb": " {{}^3 R}_{1212} = {{}^3 R}_{1313} = q^2 \\, \\sin(\\omega u)^2 ", "67798d88ae7f73a27a6ca56b73915fc2": "h(n,s)=\\begin{cases}n&\\text{if }n<2\\\\ s[n-2]+s[n-1]&\\text{if }n\\geq2\\end{cases}", "6779996bc763b9ec720d40ef5af6f268": " H_\\bullet(F)", "6779a13036ec5154c6f8ed621c050096": " \\frac{1}{\\sqrt {f}} = -2\\log_{10} \\left({\\varepsilon/D\\over 3.71} + {2.18 S \\over \\mbox{Re}}\\right) ", "6779e51b07b1a75c13b940bf67ccd3ed": "GH:\\mathcal{A}\\to R\\operatorname{-Mod}", "677a01fa328d01b2d1e99872282726e9": "Pmf = \\cfrac{1 Pwo + 3 Pwf}{3}", "677a053592b8d524e1bfe1651c36bab8": " \\Psi = \\Psi(\\mathbf{r},t) ", "677a3054f965c7e81630e8359d4b9d2a": "(m_e/m_p)^{1/2} = 2.33\\times10^{-2} = 1/42.9 \\,", "677a3d8ba1bec29d96e98197bc549c1f": "\\nabla_v w ", "677a40ae8e6a200a9fe379e8a7e99bb5": "f(s) = \\frac{1}{a_1^s} + \\frac{1}{a_2^s} + \\frac{1}{a_3^s}+ \\cdots ", "677a500a12fb75d301afcc686679a571": "L_{qq}", "677ad6fe27c8086329708073ece3bd72": " \\rightarrow (\\lambda x . z) ((\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w))", "677b39ff39edc7d1c572057e103091d9": "dU=TdS-pdV.\\,", "677b72196234ddb2847c91f0c3897f9a": "R^{-1} \\in \\mathbb Z_m ^*", "677bb561bfcd307247226ab9b3912e3c": " S \\subseteq \\kappa \\,,", "677c44065579a185ca0c23694aa4eb70": " v(\\vec r)", "677c53b8b83473b3f7973a7cb4bfd698": " G_1 = -y_{21} R_L \\, ", "677c5b07d9db3713830e57c35cc91884": "\\equiv_{amb}", "677c8cefcd66a464fd25786cefce2025": "\\Psi_{i+1}= \\Psi_i-\\alpha( H- \\varepsilon \\mathbf{1} )^2 \\Psi_i", "677cd66e889a6e3179a71add707c9294": "{f}", "677d079c24c43c02bd77df664b64e529": "y_0 = A", "677da4fc49d517341c3fe50fff8db5f9": "g(v,w) = \\mathbf{v}[\\mathbf{f}]^\\mathrm{T} G[\\mathbf{f}] \\mathbf{w}[\\mathbf{f}] = \\mathbf{w}[\\mathbf{f}]^\\mathrm{T} G[\\mathbf{f}]\\mathbf{v}[\\mathbf{f}]", "677dc89db7caad21fe536449aec37984": "\\lceil n/m \\rceil", "677dcecfa0d6d829f48b96849c8daf31": "\\alpha+n,\\, \\beta+\\sum_{i=1}^n x_i\\!", "677dd994bec83e7e2e4f536b49c6c247": "\\delta Q\\leq T\\mathrm{d}S\\, ", "677e5d281a8bae0b25632d1778d86531": "\\int_X K(x,z)\\; K(z, y) \\; d\\nu (z) = K(x,y)\\; .", "677e6710f926acac3b0ddf26899173e2": "\\nabla\\cdot\\vec A", "677e69ba91b06388db4232424a7b2913": "F(x) = \\Psi(x) + \\Phi(x)", "677ea1faa6448be1a0ba71358d3b62f0": "(n_1,n_2, \\ldots, n_r)", "677f25530f6ebaf6fb548f55f6230806": "\n\\lim_{\\mathrm{Re}(s) \\rightarrow \\infty} \\operatorname{Li}_s(z) = z\n", "677f45ce4e0ed4a37ec78064a00c0418": "(d-a)^{2}+4bc", "677f7a70d9ea677c19345573efbaa544": "\\frac {4 \\cdot C_{13}^5 - 40} {C_{52}^5} = \\frac {4 \\cdot 1{,}287 - 40} {2{,}598{,}960} = \\frac {5{,}108} {2{,}598{,}960} \\approx 0.196\\% ", "677ff8eded308ee33fe017e736dc877d": " \\cos(x) + \\cos(y) + \\cos(z) = 0 \\ ", "6780151f91a6c0f41a6f119356d660ea": "M \\sqcup M'", "6780272a3c3eb78fe8d9878019dea651": " \\vdash (r,1,Z)", "678041ca7ab0fb2dc3cb249d61bca369": "f_2(q, \\dot{q}, t)", "67804fa09ec1cae89b724ff449305a92": "p(t)=\\frac{1}{V}e^{(-t/V)}", "678060ceece520052ecdc7ba70640ded": "\\textstyle\\vec{G}", "67810ab9cf358c0107b5e8a8d48cc606": "2^2\\cdot 3^2\\cdot 5", "6781228e0a0a6072d89c076e9a5ab4db": "A_{ij}", "67814348b8a85d0f60e4bdb540653560": "y''''-2y'''+2y''-2y'+y=0", "67816eb7cf6eda91e491098d88bd5300": "dU=TdS-PdV+\\sum_i\\mu_i\\,dN_i", "678176f7b8c938e570694b3a18cef0d5": "\\mathbf{b} = b_1 \\mathbf{i} + b_2 \\mathbf{j} + b_3 \\mathbf{k}", "6781c2ddcc83015ba99c7a53337f1d4e": "\nm(f,z_0)=\\lambda = \n\\begin{cases} \n f_c'(z_0), &\\mbox{if }z_0\\ne \\infty \\\\\n \\frac{1}{f_c'(z_0)}, & \\mbox{if }z_0 = \\infty \n\\end{cases}\n", "6781d43d1db6cb661759c03bc4b2826f": "df_i=\\sum_j\\ c_{ij} dq_j+c_i dt=0,\\,", "67827d9540a54dcb20da40a60a42ff1e": "\\phi(q)\\,", "6782ca637c208707f315979437dc4351": "p_{ij} = \\frac{\\mathrm{tf}_{ij}}{\\mathrm{gf}_i}", "6782e618f20e6554a83a24757c8754b8": "E_\\sigma = \\frac{3}{2} \\lambda \\sigma (\\alpha_1 \\gamma_1 +\\alpha_2 \\gamma_2 + \\alpha_3 \\gamma_3)^2", "6783056bd2754c9ed0cc1e91299c39c1": "\\sigma_{33}\\,\\!", "678315ffd9b613f7361d8186ebd0d520": "\\displaystyle{\\sum_{n\\ge 0} s^n H_n(x)H_n(y) = (F_{\\sigma,x},F_{\\sigma,y})_{\\mathcal F},}", "6783298e435bf9e42715fa389f759574": "F(x,t)=\\left((1-t)+{t\\over \\|x\\|}\\right) x.", "6783344d6c88dfee55f2b05b54ea7078": "R^{\\frac{2v_e}{c}} = \\exp \\left[ \\frac{2v_e}{c} \\ln R \\right]", "678338cc1ae747af17dd6961584719c6": "\\scriptstyle x-\\tfrac1x=c", "6784166831809c6a808c871068189e9a": "W = \\int_C \\mathbf{F}(\\mathbf{r}) \\cdot \\mathrm{d}\\mathbf{r} \\, .", "6784209cbfa2d51f35cd5d9a095136cb": "s_1^3+s_2^3", "67849467b02faecb6f92cd163f291d36": "= \\pm\\frac{\\sqrt{\\sec^2 \\theta - 1}}{\\sec \\theta} ", "6784c05a2acbacfe01e923d0a7c85b19": " p(\\mathbf{x})\\propto\\exp\\left( -\\frac{1}{2}(\\mathbf{x}-\\mathbf{\\mu })^{\\mathrm{T}}Q^{-1}(\\mathbf{x}-\\mathbf{\\mu})\\right) . ", "6784f240b4d142d8700eca5e4a3480e4": "a_n(x-b)^n=0", "67852ee198f2ec42eee2a92cf438cb29": "\\scriptstyle G \\ = \\ H \\ - \\ TS", "67853bcb5b94ba78d37a3fae44e10cc5": "\\sigma(t) = \\inf\\{s \\in \\mathbb{T} : s>t\\}", "67855427722de8f6c4331b15bdfc250e": "\\mathbf{Ax} = \\mathbf{b}", "678593e3b7bc8df19e8de4cace9e1f6b": "V = (M + 1) M^N", "6785a9554d07f36cfd60ee7ba27ee25d": " -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N) + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N)\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N) = E\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N) ", "6785dd99c32c3545fd12987ddeebb9ca": "\\textstyle \\mathbb{F}_q", "678617e543116d3d3a2cea050d78e565": "\\left\\langle \\mathbf{e}_+ , \\mathbf{e}_{-} \\right\\rangle = \\left\\langle \\mathbf{e}_{-} , \\mathbf{e}_0 \\right\\rangle = \\left\\langle \\mathbf{e}_0 , \\mathbf{e}_+ \\right\\rangle = 0 ", "678680f39103a7a2e872737e66c019ec": " g ", "6786c409c82fed566cd29d0a2d5c8794": "M_w", "67873a63da394975e0fba1be44dca8b3": "v=\\partial/\\partial x^j", "67876121164a84781a3c3980b1b4ee5e": "\\ \\displaystyle q\\in \\mathcal{Q}\\ ", "67876a6ad426a821f035373255f9d3d0": " i_{OUT} = \\frac {v_{IN}} {R_{R}} -i_B \\ , ", "67877a27b0a8a31b0c5c2dad453ce1a4": "\\operatorname{tr}(AB) = \\operatorname{tr}(BA)", "678791f6dc77be26af3788e82b6786aa": "1<\\alpha \\leq 2 ", "6787d8c5d75494d7e6e7de9f2fa6393a": "\\Psi = \\begin{bmatrix}\n 0 & 0 & 0 & 1 & 1 & 0 & 0 & 12 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}", "67883a26a6866e844a30ecb980db566d": "\\tbinom{5}{1}=1\\times\\tfrac{5}{1}=5", "6788835ea2bcb425baee36a6bfddb16e": "\n\\begin{align}\np & = {2 \\times \\mathrm{obs}(\\text{AA}) + \\mathrm{obs}(\\text{Aa}) \\over 2 \\times (\\mathrm{obs}(\\text{AA}) + \\mathrm{obs}(\\text{Aa}) + \\mathrm{obs}(\\text{aa}))} \\\\ \\\\\n& = {1469 \\times 2 + 138 \\over 2 \\times (1469+138+5)} \\\\ \\\\\n& = { 3076 \\over 3224} \\\\ \\\\\n& = 0.954\n\\end{align}\n", "6788d8188d7b93d62861b8a38f8a2a2c": "Q = \\sum_{i=1}^{N} Q_i", "678941a3a1a014214477ce1b5a1522e9": "\\,\\Gamma", "6789b4736a80a0c0a151f6e246c7fcc4": "\\sum_{k=0}^\\infty \\frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\\operatorname{csch} z, |z|<\\pi\\,\\!", "6789df4fc30b0464fbec5f5214d72b68": "\\mathbf N\\,\\!", "6789ef48ab957991bb30e4714bb33d49": "{\\rm Pr}_{y_1,\\dots,y_m}\\Bigl( \\forall z \\bigvee_i A(x,y_i \\oplus z)\\Bigr)=1 - {\\rm Pr}_{y_1,...,y_m}(\\exists z A(x,y_1 \\oplus z)=\\dots=A(x,y_m \\oplus z)=0).", "678a0450fdb7f949051cc8ceb262896a": "A v", "678a22049c51728b8457e145e1c654f4": "\\!R", "678a30ef82dddc109432874a5b55c194": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & 0\n \\end{Bmatrix}\n = \n \\frac{\\delta_{j_3,j_6} \\delta_{j_7,j_8}}{\\sqrt{(2j_3+1)(2j_7+1)}}\n (-1)^{j_2+j_3+j_4+j_7}\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_5 & j_4 & j_7\n \\end{Bmatrix}.\n", "678a3c6c98e09f978b45883e2f1a6f14": "w\\in\\mathcal{B}", "678b09d643ac57bc21697579f070a4eb": "p=12", "678b25ffa0091875ae67122ec827cae0": "\\mathbf{B}(p\\times p)", "678b4c1f3c052b17d25e3bed57e41d71": "(n^2-n)/2", "678b4e6424b646986826f34d70ad8ad4": "\\int x^{2k+1} \\phi(x) \\, dx = -\\phi(x) \\sum_{j=0}^k \\frac{(2k)!!}{(2j)!!}x^{2j} + C", "678b66a189a7fde46152c47eaccd392b": "\\ell_X(g)=\\min\\{ d(x,gx) | x\\in VX\\}.", "678b76ba5000cec01c99de0e609ff793": "A_{i} \\,", "678bc2b585d16777e02210c48b1b0fbd": "\\mathbf{B} \\approx \\frac{e^{i (kr-\\omega t)}}{kr} \\sum_{l,m} (-i)^{l+1} \\left[ a_E(l,m) \\mathbf{\\Phi}_{l,m} + a_M(l,m) \\mathbf{\\hat{r}} \\times \\mathbf{\\Phi}_{l,m} \\right]", "678bccda1972833bf98add295aa95dec": "T_{1/2} = \\frac{\\ln 2}{\\lambda _c} = \\frac{\\ln 2}{\\lambda_1 + \\lambda_2 + \\lambda_3} = \\frac{t_1 t_2 t_3}{(t_1 t_2) + (t_1 t_3) + (t_2 t_3)}.", "678be308bc4bd801092cc9d2902bf3cb": "v > v_{ph}", "678c01a68d411ddb83ff4f849719ff07": "\\begin{matrix} x & y \\\\ z & v\n\\end{matrix}", "678c0cee788802c3fb9558475cc6220c": "k^* = \\left( \\frac{s}{n + g + \\delta} \\right)^{\\frac{1}{1-\\alpha}} \\,", "678c0ea37019cdd2811dc201e9ece77f": "F_r(t)\\,", "678c344a242f44b2753a5bfa09ff81ff": "\\mu \\{ F \\geq M \\} \\geq 1/2, \\, \\mu \\{ F \\leq M \\} \\geq 1/2.", "678c5dfd897990a85c8e9d32c836aa80": "G(k) = 2 \\pi \\int_0^{\\infty} g(r) J_0 (2 \\pi k r) r \\,dr = \\frac {1} {\\Lambda (4 \\pi^2 k^2 + q^2)^{1/2}}", "678c71fc653d64a300d06579a77252f0": "P_{11}(x)=x^5+6x^2.\\,", "678cb1ca07ce2fe3939e37f4c994a2fe": "Q = A^+ A", "678cc5fcdb19359bbfabeed420f3e970": " \\varepsilon > 0 ", "678cea920214df86fcb54bf85ea93ff4": "\n\\sum\\left(\\frac{X_i-\\mu}{\\sigma}\\right)^2=\n\\sum\\left(\\frac{X_i-\\overline{X}}{\\sigma}\\right)^2\n+n\\left(\\frac{\\overline{X}-\\mu}{\\sigma}\\right)^2\n=Q_1+Q_2.\n", "678cf6974ca0cfdc230b2ec18f629419": "b_n=\\sum_{k=1}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\} a_k,", "678d5a3bc8790a14d7f5119f13d94e42": " \\operatorname{Hom}(P,-)\\colon\\mathcal{C}\\to\\mathbf{Set}", "678d9b33ba0c85016fa50ea0c5262a89": "f : M \\to \\mathbb R.\\,", "678d9b44d266b14f26dadb767105cf7a": "\n\\mu=\\frac{\\pi M}{2 m_p}.\n", "678da9d6319f2644b509216587fa85d8": "\\mathbb{R}^{6}", "678dbf52857e5ca2d1cc5a759f302780": " \\frac{r_\\pi}{r_\\pi + 2R_E} ", "678dd0c93dabdae4c3553066c110bfec": "\n \\cfrac{\\partial^2 \\varphi}{\\partial t^2} - c_0^2~\\nabla^2 \\varphi = 0\n ", "678de1f4a8b031a57939e65b30bf97c2": "\n\\lim_{x\\to0} \\left( \\lim_{y\\to0} \\frac{x^2}{x^2+y^2} \\right) = \\lim_{x\\to0} 1 = 1.\n", "678dedff55c2a80805c4c315df8b9f87": "n \\approx \\sqrt { 2 \\times 2^{32} \\times 2^{-20}} = \\sqrt { 2^{1+32-20} } = \\sqrt { 2^{13} } = 2^{6.5} \\approx 90.5 ", "678e28d22b520d6f0782c2759d9c34a9": "a^2+b^2=3 \\cdot (s^2+t^2)", "678e5655c3da4f3faecfd0e2077ecf34": "\\mathbf{x}_{0i}^\\top", "678efa86dca8ee7308aa4962297c907e": "\\mu= G(m_1+m_2)\\,", "678f28b130fa2e0050917f54c3a266a6": "\\tilde{\\rho}(rs)\\cdot v = \\tilde{\\rho}(r)\\cdot \\tilde{\\rho}(s)\\cdot v. ", "678f4ce8a7b76bc5e750f54503383ebe": "2/7", "678fa2fbab2c6f4fe7c7523c5f2d0e6e": "={2 * 320 * 600 \\over 320 +600 +\\sqrt(320^2+600^2)}=240", "679064262cf0aad1feaf5535914dc9ee": " x/y ", "67906e84d8d79e8cd45058ba4f895a38": "\\{1, 3, 9, 11\\}", "679089e4fdd7d32dad1521e3587ea2de": " X_1,\\ldots,X_n ", "6790ec89867774cf15fabb1c605eaffe": " M^{\\mathrm{core}} \\,", "679103d483282ee82bbfe93e8ed94fd4": "\\langle\\ln\\tau\\rangle = \\left( 1 - { 1 \\over \\beta } \\right) {\\rm Eu} + \\ln \\tau_K ", "67912e41ad9fc1218dc3b045adc42445": "V_{0 \\dots n}=\\{q_0, q_1, q_2, \\dots, q_n \\}", "67916cf26f74e6d900019349ff38792e": "F\\;'=F^2", "67920379187eae9b9f5091b90dac8884": "I_{3322}", "679209bf95bec22fa3aef85fb86f28b1": "\\,\\gamma_0", "679235327ba14941805de6a78641c3b3": "\\hat{r}(x)", "6792c045d10e9cce6161328f03dfb55f": " MA = \\frac{F_B}{F_A} = \\frac{a}{b},", "6792d120d1cce1a9b36666e93d28c2c3": "P \\sim N^{-\\alpha}", "67932b6a3467ef4c117d568227c9d96a": "\\mathrm{P}(u,v)=0, ", "67933c4b6b083dfbb42ab6976708e25a": "C_1; C_2", "67934cfe40418f25b050c7ea0587efd2": "\\text{and in three-phase systems:}", "6793a935c91421729e5ae993acd64d02": "\\ell=4,\\quad m = -4,-3,-2,-1,0,+1,+2,+3,+4", "6793b2d102ffe71f129d4ffba6291509": "B = \\frac{u^2}{1024} \\left\\{ 256 + u^2 \\left[ -128 + u^2 (74-47 u^2) \\right] \\right\\} ", "6793d42680437164cbbd009f0178374d": "\\mu:\\mathcal{B}\\rightarrow[0,1]", "6793f81fd3bccfabb09bfe5e68a30344": "\\gamma^0 \\,", "6793fae47540fe4ffed327d805103b2d": "0.087\\pm0.014", "67940f052d2a8cd5eb73dbc918a90785": "\\frac{1}{\\tau} = \\lambda = \\lambda_B + \\lambda_C = \\frac{1}{\\tau_B} + \\frac{1}{\\tau_C}\\,", "679465842450b10512740be99eb33362": "g'(z) = g(z) + t(z^2)\\cdot h(z)", "67946745f65bb312101ad34860d21cea": " \\mu\\left(\\bigcap_{i=1}^\\infty E_i\\right) = \\lim_{i\\to\\infty} \\mu(E_i).", "67949455061a7af042427b8408aaae22": "L_2(X^*, \\mu^*)", "67953b8bd6a2723d4a189ea4939ef85d": "\\theta(g_{n})", "6795a6901b76acccfd9d4268d380115c": "PCER", "6795b490aa9058d16dc50cfca2aa016c": "Pu=0\\,", "6796819ed952f19e0f33e3c9de69eb0c": "f'(b)", "67968a5b02bcd736b60d9d28ceab64e9": "f \\ll p \\ll T_\\mathrm{smax}\\,", "67968fb290ed04565443ab3703d4b1fe": "\\mathrm{Verify}(\\sigma,y,\\pi)=\\mathrm{accept}", "6796bfc7511158cf0de2a715be25f403": "4\\pi ", "6796fe4dc874fbfc1dd0da5c28afc8ff": " R:X\\rightarrow Y ", "679715a75782f1a9203d604e5f3cc3b3": "y_1 \\ne 0", "6797487a2d047207de1f13192a875b13": "T = (\\gamma - 1) mc^2\\,", "67974e105cfd1ba7d28956db893837a4": "\\kappa = (\\aleph_{\\alpha})^+ \\,,", "67974f688f3c6a881e98bc1a90c9f01b": "\\hat H |\\Psi\\rangle=E_{\\Psi} |\\Psi\\rangle", "6797af224aff8b199ebbf15606ed782c": "\\chi_{\\rho^*} = \\overline {\\chi_\\rho}", "6797b9a52fc2538f72ece40cdd7e11d7": "\\beta>0", "679841ca88f27d2026525c89be483c23": "\\lim_{(x,y) \\to (0, 0)} \\frac{x^2 y}{x^2+y^2}", "6798512765a7e05fb7c6719b84b92927": "\\nabla f(\\mathbf{x}_k)", "67992771127562e28a00f075c42908d1": "U = \\int E\\, f(E)\\, g(E)\\,dE", "67997a24b6f31c1b5c82c149f9990c12": "\\displaystyle{H_n(x)=\\|F_n\\|^{-1}F_n(x) =p_n(x) e^{-x^2/2}.}", "67998bd995d57435851f7ba101896164": " V = (v_1, v_2, \\ldots, v_n) ", "6799972015129903443e94f3fb23a247": " \\left((1-x^2)P_n'(x)\\right)'+n(n+1)P_n(x)=0", "6799ac722dae996649ca7f635db54258": " a \\longrightarrow b \\longrightarrow c \\qquad \\text{and} \\qquad a \\longleftarrow b \\longleftarrow c ", "6799f18bc6e3025d6c3434cd6936735d": "\\mathcal{H}", "679a055f4252ec59fb126aeedbb18b9a": "f(x) = \\frac{n_1^{\\frac{n_1}{2}} n_2^{\\frac{n_2}{2}}}{B(\\frac{n_1}{2},\\frac{n_2}{2})} \\frac{x^{\\frac{n_1}{2} - 1}}{(n_2 + n_1 x)^{\\frac{n_1 + n2}{2}}}", "679a5355a9629a187094c34f3c140dd3": "(\\omega_{pm})", "679a89d3b0a1306d5ba01d7ee693a29d": "\\hat{\\zeta}_0(\\omega)", "679a9e958b586657beb870c8c327a9ec": "* [F , G]^{IJ} = [*F , G]^{IJ}", "679ade11b0c7d47de239d1cefc01cf7e": "\\,P_s = 1 - \\left(1 - P_{sc}\\right)^2", "679af7203b6e7f1cdc3f5cf7faf6ce59": "f(x,y) = \\sin^{2}\\left(3\\pi x\\right)+\\left(x-1\\right)^{2}\\left(1+\\sin^{2}\\left(3\\pi y\\right)\\right)", "679b3144f71ef556d85688ff956279b9": "\\operatorname{VIF}(\\hat \\beta_i)", "679b5ce007346b10075a393aa9a6d02e": "4*\\pi *10^3 *F*\\frac{d}{300}", "679b61c48d3654775a7619285671f8f5": "\\phi, \\psi", "679b985cb284579c6a945d8a3168cffe": "p_{10}/(p_{10}+p_{00})", "679ba7f87826bfe2cafe05054571a98d": "\\begin{bmatrix} x& \\\\ y& \\\\ z& \\end{bmatrix} \\;=\\; t_1 \\!\\begin{bmatrix} 2& \\\\ 5& \\\\ -1& \\end{bmatrix} + t_2 \\!\\begin{bmatrix} 3& \\\\ -4& \\\\ 2& \\end{bmatrix}.", "679bb1d633f0616b7c4dcd6ff85fbfcd": "\\sum_{j=1}^np_{ij}=1", "679bc6176ddbd3e16e37e81468c62155": "(1 + T_{sys} / T_{ant})", "679c18a7bb0f547b609d5ca13438eecd": "{\\rm Var}(a^{\\dagger}a)=0", "679c4a2037013e1481635cf57a7a81f3": "\ni = (8-1)\\ (8-3)\\ (8-5)\\ (8-7)\n = 7 \\ .\\ 5\\ .\\ 3\\ .\\ 1\n = 105\n", "679c4c927f816045befe573024ddd21b": "S^1", "679c770d7c58abb799a9a316308e3548": "B=\\dfrac{dN}{dt} = k_n(c-c^*)^n", "679c81686819e6581eb4fb57cee6ef04": "x_i = x(iT) \\, ", "679d16b2887296d34266cf9a154dddae": " \\lim_{x \\to c} f(x)^{g(x)} = \\exp \\lim_{x \\to c} \\frac{\\ln f(x)}{1/g(x)} \\! ", "679d1b46486b0eeb79ea3e0ec19122e6": "\\blacksquare \\!\\,", "679d58870d423a1421795b65c196cc1d": "\nqs=ps+rs\\Rightarrow q=p+r.\n", "679d5df491288ab604ef98291d954ba4": "\\displaystyle \\mu(B) = P \\big( f^{-1}(B) \\big) ", "679d6b00b5210e3821e1b33ab7a3686f": "S=2", "679d9dcf3ea20718853ee22a8684cdd5": "VC(y)", "679dd00de940060b0d64da28957f3e6e": "(\\tau_R)^c", "679e33a0b52ca4e0bac6901a4567af66": "(\\mathrm{Funct}(C,D))^{op} \\cong \\mathrm{Funct}(C^{op},D^{op})", "679e61fb799ca1db5418b8db1305859c": "\\mathbf{0}^{\\rm T}", "679e708342c6d96fb530576c4009d920": "\n M=mat(\\textbf{x})=\\begin{bmatrix}a_{11}a_1+a_{12}b_1+a_{13}s+a_{14}t&a_{31}a_1+a_{32}b_1+a_{33}s+a_{34}t\\\\a_{21}a_1+a_{22}b_1+a_{23}s+a_{24}t&a_{41}a_1+a_{42}b_1+a_{43}s+a_{44}t\\end{bmatrix}\n", "679e8cafe6ab2e06e4c47dc9214b81f3": "\\|\\mathbf{A}+\\mathbf{B}\\| \\leq \\|\\mathbf{A}\\|+\\|\\mathbf{B}\\|", "679ed4b2fe61e453f7bb9258a9243526": "\\frac{\\mbox{Net Earnings}}{\\mbox{Number of Shares}}", "679f0a4b9035083cfec41258fe9c8db5": "= \\frac{-1}{i \\hbar} \\langle \\psi|\\left[H,Q\\right]|\\psi \\rangle + \\langle \\psi | \\frac{dQ}{dt} | \\psi \\rangle \\,", "679f49b42f7bd52adc3d14f47cd88d8c": "\\{e_{i_1}e_{i_2}\\cdots e_{i_k} \\mid 1\\le i_1 < i_2 < \\cdots < i_k \\le n\\mbox{ and } 0\\le k\\le n\\}", "679f82dd6286814059cd65b602c2745a": "\\inf_{x \\in X} f(x). \\, ", "679f900bbba47016dd77d3ad1f3702dc": "Fe_2O_3", "67a0191484a4093b893f97b2ae4927d8": " \\alpha \\geq \\omega^2 ", "67a03714c2ce014b4a2a7c2fe4d06f4b": "-e^{-2\\alpha}", "67a0bdd0b8c6e120b2bb7ffae0e03869": "\\textstyle \\Gamma_{\\mathrm e}(V)", "67a0cea2d47e7102450aeb669e30caf1": "3.14159265359", "67a16d4a2a63074b034f14b51ddb9fdd": "\\,a_{\\overline{n|}i} = v + v^2 + \\cdots + v^n = \\frac{1-v^n}{i}", "67a1944cd76febf6ab9a37d9799f659c": "\\vartheta (z,\\tau)", "67a1f021cdd9f1b35d9c2fba5d03b948": "\\begin{matrix} {4 \\choose 3} = 4 \\end{matrix}", "67a278c3124887b7dd39191d32fe7348": "E[d_j S(t)]=h(S(t^-)) \\, dt \\int_z z \\eta(S(t^-),z) \\, dz.", "67a301df972ae509ba992f9c1b087afb": "\\rho_{\\alpha}^t(X) = \\operatorname*{ess\\sup}_{Q \\in \\tilde{\\mathcal{Q}}_{\\alpha}^t} E^Q[-X\\mid\\mathcal{F}_t]", "67a34bf79e1a837f9fdcf53a5c51cddb": "V_{\\text{bd}}= E_{\\text{ds}} d", "67a3d20057a71cb641a396f57f0982b1": "\\mathrm{OC}^2 = \\mathrm{OA} \\times \\mathrm{OB} \\,.", "67a429e605d212b41bfa42b649d300bc": "\\mathrm{Lie}(G)", "67a4c374b8b7f894c3415a785ee6a773": " L(n,k) = {n-1 \\choose k-1} \\frac{n!}{k!} = {n \\choose k} \\frac{(n-1)!}{(k-1)!}", "67a4d3348d655f163beed330ed3ed561": "(w_1, w_2, w_3 )", "67a4fd54298749a18761a4e0eb9b85c4": "\\rho > 1 ", "67a527af819fda7abebed20be2a2a11f": "(\\xi + \\eta)_{sup}(\\alpha)=\\xi_{sup}(\\alpha)+\\eta_{sup}{\\alpha}", "67a5399170a907c7b6d4dae760f6af65": "\\textstyle X^-=\\{v_i:\\lambda_i<0\\}", "67a556d15d2946b7965f0627670c9e44": "\\bold{\\hat{n}} = \\bold{\\hat{e}}_r\\times\\bold{\\hat{e}}_\\theta \\,\\!", "67a56260c2b585097636b41bfb7b0eb3": "\\textstyle y=\\frac{1}{\\sqrt{x}}", "67a59930179b1b6a88d834cf08a2ff7e": "\\tan(y) = x \\ \\Leftrightarrow\\ y = \\arctan(x) + k\\pi", "67a66fe1295266a1f40aaff2d8d54aab": "\\theta U^2 = \\mu W \\frac{rd^2}{4Ck}", "67a6e473579bf9f29814d2515862e101": "\\frac{\\eta_{sp}}{c} = [\\eta] + k [\\eta]^2 c + \\cdots,\\,", "67a70bcd0b0988157c57ed95394ac8cc": " \\mathbf{X} = (X_1, X_2, ... X_k)", "67a721b74ee2f23fa7fe13e53eca62c6": "O(n \\log n + z)", "67a7e4deca6003bf94f6d39288d3bb63": "\\frac{\\{I \\land C\\}\\;S\\;\\{I\\}} {\\{I\\}\\;\\mathbf{while}\\;C\\; \\mathbf{do}\\; S \\;\\{I\\land\\lnot C\\}},", "67a7e74d411a626d907cdce6b1ff87a9": "{BE}_{7}", "67a80874b619df4de41241e1891e36b5": "\\theta_{12}=\\theta_{23}=\\pi/4", "67a828e34b018a235f46edbf9a4a53de": "\\langle \\varphi,f \\rangle := \\sum_{k=1}^\\infty \\xi_k c_k", "67a87c77c62bd92a6b6e5d265b581d59": " (x,y) \\to (x,y,x^2,y^2,xy) ", "67a8f1f018400ab5d0940e0c761e38a5": "[X,Y] := \\sum_{i=1}^n\\left(X(Y^i) - Y(X^i)\\right) \\partial_i = \\sum_{i=1}^n \\sum_{j=1}^n \\left(X^j \\partial_j Y^i - Y^j \\partial_j X^i \\right) \\partial_i ", "67a9301782eb1b2cabc39f460142637b": " \\frac{\\partial\\varepsilon}{\\partial c_k^*} = \\frac{\\displaystyle\\sum_{j=1}^Nc_j(H_{kj}-\\varepsilon S_{kj})}{B} = 0, ", "67a93ac3b9306c753efb4c2fa2c26120": "\\overline{\\psi}\\to \\overline{\\psi}e^{i\\gamma_{d+1}\\alpha(x)}", "67a966ee109af7e05db3b75f6cf86bd6": "E=(t-y)^2 \\,", "67a9ba3229dd60e2d0e8f94a3ebd0f89": "p_{n+1}(x) = (x - (\\log T)')p_n(x).\\,", "67a9e0519082ee9a43230ed27c67ab49": " | A(t,y_1,h,f) - A(t,y_2,h,f) | \\le L |y_1-y_2|. ", "67aa25a92ce80014c6d7e663d62b1844": "v \\in G", "67aabd2294f2b230938f1b1a58f8a076": "- 5 \\sqrt {2}/16", "67aaca6cbb03c26d59fd6a7aa885069c": " \\liminf_{n \\to \\infty} \\frac{\\psi(2n)}{\\psi(n)} = 1 . ", "67ab6dc871ed9ab07df27e7d7c8e14c3": "p^{x-1} \\,", "67ac5979c493452570f71f5dc719139d": "x \\stackrel{*}{\\leftarrow} w \\stackrel{*}{\\rightarrow} y", "67ac645aeff27be6157838ecfc816795": "f_H\\,", "67ac6ee799a921f7d325b36e4e41e7ea": "x_1 = \\sqrt{\\alpha^2-r^2}\\cosh(t/\\alpha)", "67acc79fa1e7dbf2e3f291b8fd815ed4": " \\tilde{\\mathcal{A}} ", "67ad914d9900227be1a036350c1c62fa": "-a < r < a", "67adc8ad967f886509d72208177f2920": "p_{xy}^k", "67ade65f1bc30c59a341c1eb61690ecd": "\\varepsilon_{i,j}", "67ae048c46bb394b424ed367e4e9a2dd": "\\psi_{1} = 1", "67ae214d913e60676c8eea4c20b1f221": "\n\\bar{f}(\\bar{x}^j) = J^w f(x^i)\n", "67ae3889e5b67f1c26bf6ba3ac31e1a3": " f(c,\\lambda)=(f,\\xi_1(\\lambda)), \\quad f_x(c,\\lambda)=(f,\\xi_2(\\lambda)),", "67ae425ede1932cf7da98879db5351c5": "Q(u)", "67ae8f11260be8d55a915b3e0ec4ef1d": "N(\\lambda,h)", "67aee0b9596c636c46b430c0ffb2a2ed": "\n \\bar{C}^\\prime = \\frac{C_1^\\prime + C_2^\\prime}{2} \\mbox{ and }\n \\Delta{C'}=C'_2-C'_1 \\quad\n \\mbox{where }\n C_1^\\prime = \\sqrt{a_1^{'^2} + b_1^{*^2}} \\quad\n C_2^\\prime = \\sqrt{a_2^{'^2} + b_2^{*^2}} \\quad\n", "67af25748c16b0a16a795cea006643f5": "F\\,'_{\\rm D}", "67af57b308dff14ffd3035c300acc14a": "f(t) = \\frac{1}{2}(\\delta(t+1)+\\delta(t-1)).", "67afd1884ae6d215d41a41d50681e5d2": "\n\\left[ x\\frac{d^2}{dx^2} + (2l+2-x) \\frac{d}{dx} +(\\nu -l-1)\\right] f_l(x) = 0 \\quad\\hbox{with}\\quad \\nu = (-2W)^{-\\frac{1}{2}}.\n", "67afe5e0063d3e9550cad2963550c022": "\\mathbf{F}^\\beta\\ ", "67b00e634e46068b739053c5feef746e": "\\mathrm{rad}(\\infty,D) := \\frac{1}{\\mathrm{rad}(\\infty,E)} := \\lim_{z\\to\\infty} \\frac{f(z)}{z},", "67b036d35ce8d1b0cdd87eda50f0a96e": "w = 10", "67b09d9f85cf9861a6335a1ca1fdf965": "\\mathbf u(\\mathbf X,t)=u_i\\mathbf e_i=u_i(\\alpha_{iJ}\\mathbf E_J)=U_J\\mathbf E_J=\\mathbf U(\\mathbf x,t)\\,\\!", "67b0d16c12558c07961cb818c834e410": "I_n[w]", "67b0d2b5e7c7687357cbc1456a932647": "E \\left [X_n \\right ]\\ge c_2", "67b0fa682cb3046fa188e016cbc5546c": "\\mathcal{L}\\left\\{a f(t)\\right\\} = a \\mathcal{L}\\left\\{ f(t)\\right\\}", "67b1204d4b30d67d29cbfde5065f1db9": "h^*_n \\mapsto h_n", "67b17ad63309a9d5d76469569bf7e38a": "f^{(n)}(x) = \\frac{d^n f}{dx^n}.", "67b2190d4475566448e9d92ae3987a9a": "I(q) \\approx I'_0 \\exp \\left( \\frac{- q^2}{2\\sigma^2} \\right) \\ ,", "67b247c7f4d7064614e93185eaca58fd": "y=m\\frac{(x_0+my_0-mk)}{m^2+1}+k.", "67b2a185e5faf0e293c02f3cc88f817f": "\\left\\langle \\vec{x},0 \\big| U^* \\operatorname{E}_{F(x)} U\n\\big|\\vec{x},0 \\right\\rangle = \\left\\langle \\operatorname{E}_{F(x)} U( |\\vec{x},0\\rangle) \\big| U( |\\vec{x},0\\rangle) \\right\\rangle \\geq 1 - \\epsilon.", "67b2accab1fcf890fc457fdcee1fb9ed": "\\tfrac{a}{b}", "67b2f26e6b775cc9dbf503903bedf0e9": " F_a(s)F_b(s) = \\left( \\sum_{m=1}^{\\infty}\\frac{a(m)}{m^s} \\right)\\left( \\sum_{n=1}^{\\infty}\\frac{b(n)}{n^s} \\right) . ", "67b2f95b4e4b604a92a98dbe9c95a32b": "N_{i+1}/N_i \\mbox{ is simple for }i=0,\\dots,n-1", "67b3013ed7dfe02d2bbd9629ddc50db7": "\\nabla A", "67b30682b08328f9f678fcd355a38b4c": "N_{\\mathrm{heads}}", "67b319c863e0fd4ed8844a20e043f6db": "\n\\begin{align}\nT &= \\frac{1}{2} M \\dot{x}^2 + \\frac{1}{2} m \\left( \\dot{x}_\\mathrm{pend}^2 + \\dot{y}_\\mathrm{pend}^2 \\right) \\\\\n&= \\frac{1}{2} M \\dot{x}^2 + \\frac{1}{2} m \\left[ \\left( \\dot x + \\ell \\dot\\theta \\cos \\theta \\right)^2 + \\left( \\ell \\dot\\theta \\sin \\theta \\right)^2 \\right], \n\\end{align}", "67b31d49c0358463d6dbefd1c0c6c18c": "\\epsilon_{i}", "67b347d189c69e57ba64952ced1cde5a": "n>2", "67b39c946e8e6f2401518df005120be6": "P(\\omega_r\\mid\\xi)", "67b3d8ce5813a4fe9324329b94b4959c": "D_{12}=\\frac{3}{8nA_1({\\nu})\\Gamma(3-\\frac{2}{\\nu-1})}\\left[\\frac{kT(m_1+m_2)}{2\\pi m_1m_2}\\right]^{1/2} \\left(\\frac{2kT}{\\kappa_{12}}\\right)^{\\frac{2}{\\nu-1}}", "67b3e4c5b6391604ab40a388088ef966": "E+dE", "67b400e5ec254bb910ff980e776947ca": "\\sigma_1(n) = 2n", "67b45e73185a226ed0fb2287cc626705": "U_{ij}=\\varphi_{i}\\left(W_{ij}\\right),\\,", "67b49a61bbdee2808edaf97a990e697c": "M, N, \\ldots", "67b4b99474a6571a0a98c26d9d29f4d6": "\\hat{m}_x(t)_{T} = \\frac{1}{2T} \\int_{-T}^{T} x(t) \\, dt.", "67b4cd4994e85ab9021626aa8f6eb60a": "P, Q \\vdash P \\and Q", "67b4f82fab50b14e85ae1a18ca4532db": "\\displaystyle{U(x) f(t)= f(t-x),\\,\\,\\, V(y)f(t)=(y,t) f(t)}", "67b5054e2660ae35b3b9772c18944507": "F(t)=\\frac{a_N}{t^N} + \\frac{a_{N-1}}{t^{N-1}} + \\cdots", "67b5131ff53537928027fcabe5cd4ef2": "p[A,B]", "67b5171bedd51fa46679ea3ab47e1ba4": "p^a", "67b52b6e2ba10c0400f3bdf222e5341b": "dD", "67b5a0b94b0d98c0d0f609649944d8a8": "n!_k", "67b5e691e087684a9ae4738d7e1acfef": "\\scriptstyle \\sin(2\\pi ft)\\cos(\\phi)", "67b608f25496800c1b931fa76f4937f1": "s^\\prime\\in S_i", "67b686a50bf3ea915e85f440d0e98045": "t_1=0", "67b68721103b5a16194f4b3e3ec222db": "x_{n}", "67b6adff8bc99d6cc2602a6dbe53ba44": "I_c=c/ \\pi \\Delta \\nu", "67b6fd785fbe21b320e90922f512c842": "\n \\begin{bmatrix}Q_1 \\\\ Q_2 \\end{bmatrix} = \\cfrac{\\kappa}{2}\\left\\{\n \\int_{-h}^h \\begin{bmatrix} C_{55} & 0 \\\\ 0 & C_{44} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} w^0_{,1} - \\varphi_1 \\\\ w^0_{,2} - \\varphi_2 \\end{bmatrix}\n", "67b70264ea0d3d32facae26ab169e9a8": "dq \\ \\stackrel{\\mathrm{def}}{=}\\ \\sigma dS", "67b755017be66a0ccfee13011e8d4127": "\\tfrac{Time of slower stroke}{Time of quicker stroke}", "67b75781b18ab757d39edfb13cf90113": "hms", "67b75f91afe882d446dcf6cec91bb2c3": "\\hat{g}(f) = \\operatorname{rect} \\left(\\frac{f}{\\Delta f} \\right) =\\chi_{[-\\Delta f/2,\\Delta f/2]}(f)\n := \\begin{cases}1 & |f|\\le\\Delta f/2 \\\\ 0 & \\text{else} \\end{cases} ", "67b78c8dd348ce6955eabfb9142e3576": "(s_n)_{n \\ge 0}", "67b79b4d9b14f655df2e674ace0089aa": " R_N = 2B \\, ", "67b7d90faab7e190e44f1440c7cc436b": "y_3 = (2x_1+x_1+A)l-Bl^3-y_1 = \\frac{(2x_1+x_1+A)(3{x_1}^2+2Ax_1+1)}{2By_1}-\\frac{B(3{x_1}^2+2Ax_1+1)^3}{(2By_1)^3}-y_1", "67b8178425c26b1d3626c295a16cb170": "P(H_1)", "67b833a5d613a71db2672991f97d0db3": "x_0, x_1, x_2, \\ldots,x_{n-1} ", "67b886c77635aecb75a17305ce73220c": "P=w\\rho c^2", "67b8b3be8f573dd74947cbf473718acd": "\nV(x)=\\infty ,\\qquad \\ x>a\\qquad \\qquad \\ (\\mathrm{iii})\n", "67b8c6ea5b075405d5205eee386458d1": "x[n] * y[n] \\!", "67b8e0da39efc0292cf18acd83ddb027": "\\mbox{S}^{0}", "67b8e7ebdc05d7043eea107ee649a78f": "W^{*}_{t}", "67b8edc1167ac94c604efc54fd7e49b9": "\\sigma(A+B)\\geq \\alpha^{\\frac{1}{1-k}}\\ . ", "67b997d84b53ee27cb5198061d2af07f": "(1-x^2)\\,y'' -2xy' + \\left[\\lambda - \\frac{m^2}{1-x^2}\\right]\\,y = 0\\qquad \\mathrm{with}\\qquad\\lambda = \\ell(\\ell+1).\\,", "67b9a7261ff5ee5e8548444f1dda5f73": "\\mu^{\\prime}_{\\rm p} = \\frac{\\mu^{\\prime}_{\\rm p}}{\\mu_{\\rm e}} \\frac{g_{\\rm e} \\mu_{\\rm B}}{2}", "67b9cf06ba10665a1b9e2e58482d8bb5": "i_G", "67b9e7c8e195a6ff220de8692c631fc4": "I_xV_x+I_yV_y=-I_t", "67b9f88b2747ca27b0560354579d83e8": "\\frac{Du}{Dt} = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x} + f v + \\frac{1}{\\rho} \\frac{\\partial \\tau_x}{\\partial z}", "67ba2d11218685e9b23146964e349dc4": "L=n\\sqrt{1+\\dot{x}_1^2+\\dot{x}_2^2}", "67ba2fe438784dcd0cc8397bbee29f21": " \\sigma^3 = r \\sin\\theta \\, d\\phi", "67ba40f6312840cd6f5cbb43cab039dc": "\\mathbf{n}\\cdot\\mathbf{x'}", "67ba8afa26d028e7376762a33ac33319": "O(1.6262^n)", "67baa1a8bff3f9061a1a62fa2989281a": "N - 1 = 2\\cdot 5^2\\cdot 227", "67bab0c0b18f9e06e43c3948f2d9cec0": "\\sec\\,z - 1", "67bafcc0ad91c5711cdf9493c83d2630": "B_\\nu \\propto \\sqrt{\\frac{1+v}{1-v}\\ }", "67bb8186bbc2e4a242c8d9f730435fbd": "e^+ + e^- \\rightarrow \\gamma + \\gamma", "67bb9bf77e2d0b4c316303fe0809749c": "d(x,y) \\le \\frac{d(T(x), x) + d(T(y),y)}{1-q},", "67bb9d66b951f4ec1bcee6f9a5c0506a": "S^{*} = \\{ (o_{i}, o_{j}) | o_{i} \\in X_{k_{1}}, o_{j} \\in X_{k_{2}}, o_{i}, o_{j} \\in Y_{l}\\}", "67bbc413b90037113c1e8f8531c67401": "\\mathbf R^{\\mathbf N}", "67bbea977f6c3d0e21882ef9422ccd3a": "\\mathbf{n} = (n_1, n_2, \\dots, n_d)", "67bbfd721d0294e15eafe202664dc994": " p_i(x), p_i(x^q), p_i \\left (x^{q^2} \\right ), p_i \\left (x^{q^{d-1}} \\right ).", "67bc2fe412f510083965c1d127514962": "L = L\\left [ \\mathbf{q}(t), \\mathbf{\\dot{q}}(t), t \\right ] ", "67bc44f2fad9a58652d2ce075d12be95": "F_i:A_{-i}\\longrightarrow A_i", "67bc4f81de536800b28895fdf51abc78": "e = \\left [ \\sum_{k=0}^\\infty \\frac{4k+3}{2^{2k+1}\\,(2k+1)!} \\right ]^2", "67bc5af78e751269f4448739feeb21de": "On(\\text{coin},\\text{black area})", "67bd8474039d50beb9e314f3223700aa": "\n\\lim_{n\\rightarrow\\infty}\n\\frac{\\text{number of copies of }\\beta\\text{ in a }\\beta\\text{-optimal permutation of length }n}{\\displaystyle{n\\choose k}}.\n", "67bd906f418f0d14eee49592783e203c": "i,\\,", "67bda0a5b755f87473e9cf5c38543d85": "\\displaystyle \\alpha_i^\\vee=2(\\alpha_i,\\alpha_i)^{-1}\\alpha_i,", "67bda22ebd8b99b7fee6081db8f5b7f0": "H(U)", "67bdaf95f549f34f6464b52b78876653": "\n \\mathcal{L} := \\tfrac{1}{2} \\mu \\left( \\frac{\\partial w}{\\partial t} \\right)^2 - \\tfrac{1}{2} EI \\left( \\frac{ \\partial^2 w}{\\partial x^2} \\right)^2 + q(x) w(x,t) = \\tfrac{\\mu}{2}\\dot{w}^2 - \\tfrac{EI}{2}w_{xx}^2 + qw \\equiv \\mathcal{L}(x, t, w, \\dot{w}, w_{xx})\n ", "67bde50e9744b54a7fb491ddab5e59a2": "\n\\Bigg(\\frac{q}{p}\\Bigg)_4 \\Bigg(\\frac{p}{q}\\Bigg)_4 =\\Bigg(\\frac{a+bj}{q}\\Bigg)=\\Bigg(\\frac{c+di}{p}\\Bigg).\n", "67bdedf5dd38f246ed229a831fb3ef9e": "v_r=-\\frac12 \\alpha r,", "67be02543ac77a7e83157645076e4b24": "SA = 4\\pi r^2\\!", "67be773ce4d151383da256f22a73a41f": " \\nabla_{\\partial_a}\\partial_b=\\Gamma_{ab}^{c}\\partial_c", "67bec51aaf031e1b217246ce45a35e6e": " \\Delta p \\equiv ", "67bf06e66d0170dacb708eef493f4be9": "[A,Q]=-\\frac{1}{2}Q", "67bfd972c154e31e29795d29f78dde8d": "Em = Pmf Pwf Emmfwf + Pmf Pwo Emmfwo + Pmo Pwo Emmowo + Pmo Pwf Emmowf", "67c0087bf71317dd6e06d5fbaf0afe87": "\n E= \n - {E_0\\over 2} \\left[ 1- {1\\over 8}\\mu^2\\right]\n", "67c09f60982fed3f9f9420b4d53024c5": "{\\mathbf h}=(h_1,\\dots ,h_k)\\in Z^k", "67c0b34b39f4d86224d4cf9733f58995": "\\psi(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)dr\\,dE\\,d\\Omega", "67c0c6f071f4db87fc8e0e102fa25ae4": "\\mathrm{rad}(504) = 2 \\cdot 3 \\cdot 7 = 42", "67c0e910c573d4b8814dedaad240ec66": "A\\in\\mathcal{V}", "67c11ef49203b1fe69e77399f7d8fe1a": "\\vec{h}_1 = \\sqrt{1-2m/r} \\, \\partial_r ", "67c15502f6dcdff94e61c54026f52e52": "\\hat k=\\frac{\\sum_i F_i y_i}{\\sum_i {F_i}^2}.", "67c16d41728d88312746c435033c1105": "A_i(x_1, \\dots, x_{r-1}) \\leq B_j(x_1, \\dots, x_{r-1})", "67c196ea60cd3f8aad7f64b9c3c1dc2d": "d_q\\equiv d \\bmod (q-1)", "67c1eaffa94d8227bad5d337a268eac2": "\\phi_j ", "67c208ac5575708664c420c0f70c2ce9": "-\\left(\\frac{\\partial T}{\\partial V}\\right)_P \\left(\\frac{\\partial S}{\\partial P}\\right)_V", "67c221e01ef8b997041a77fdf4f89a21": " \\tfrac{1}{2}(a+b)", "67c25032df217668fa16df3fb6d7dc68": "\\sin_k(\\angle\\zeta^i)=(1/j)(2^{-1}\\bmod{p})\\cdot(\\zeta^{ik} - \\zeta^{-ik}),", "67c28d1b86cac9010b2e3679603a5ecc": "J_{n-1}", "67c28d4167d9af832784f175449f7557": "(x-\\lfloor x \\rfloor)/2", "67c392179f7215f70ac32caa68bd853c": "f(z)=\\frac{z}{1-|z|^2}", "67c3af44a78023318d24f5b4d3700587": "\n T_{11} = \\sigma_{11}/\\lambda = \n 2~\\left(\\lambda - \\cfrac{1}{\\lambda^2}\\right)~\\cfrac{\\partial W}{\\partial I_1}~.\n ", "67c3c01210c724ebbd3983c6a57ea9b9": "s = \\frac {\\mu} {H^2} \\cdot \\left ( 1 + e \\cdot \\cos (\\theta-\\theta_0)\\right )", "67c3c20c4188d69dcc28bdb345d23128": "x^{1/2}", "67c3cbca2f34d01f8aa904df89705e5b": " \\mathbf{W} = \\begin{bmatrix} \\mathbf{w_1} \\\\ \\vdots \\\\ \\mathbf{w_C} \\end{bmatrix}", "67c44fd5f7965dfb570acec0827c4412": "\\phi_i : \\pi^{-1}(U_i) \\to U_i \\times F\\,", "67c4725ef5d7afd22133bd359ba6cc8b": "S_{\\sqrt{3}}", "67c4b70d17f50c0a8ef5dd1fed54fcd8": "\\frac{\\partial n}{\\partial t}=D \\Delta n^m \\, ,", "67c543889e3f54b6a19f10c26ad1e2eb": "\n \\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^2 + \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda + \n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}} = \n \\det(\\lambda~\\boldsymbol{\\mathit{1}} + \\boldsymbol{A})~[(\\lambda~\\boldsymbol{\\mathit{1}}+\\boldsymbol{A})^{-1}]^T \n", "67c5449e16b0df97295790629d8bda6a": "E_i=\\frac{N}{n}\\, ,", "67c5747e13c2038865eb2d0bfcac45b6": "\\chi'(G)=\\chi(L(G)). \\, ", "67c57ea371d2fc03214092d632dfe9e7": "x^n=\\left( x_1, \\ldots, x_n \\right)\\in \\mathcal{X}^n", "67c5defaa3a3c20a0d00669f29ca56b6": "Q(\\beta, V)= \\operatorname{Tr} (\\hat{\\rho} (\\beta)) = \\operatorname{Tr} \\lbrace \\exp \\left[ -\\beta \\hat{H} \\right] \\rbrace", "67c60a1de20a59dfe7492cb9a4bccbc8": "\\kappa^{2} \\equiv \\frac{2 \\Omega}{R}\\frac{d}{dR}(R^2 \\Omega)", "67c6408f2ea0e19f5369ca1eae33e61a": "p(x+1)=(x+1)^3 -7(x+1) + 7\\Rightarrow p(x+1)=x^3+3x-4x+1, v_2=2", "67c66b41c1acdc05c49f0cd7653c9403": "h_f = \\frac{Nu k_{air}}{D_h}", "67c6e456bd1e48f002d5dbaf25596d34": "F(K, AL)", "67c7358487ed8f63288336a1627a3403": "SU(N_f)_L^2 U(1)_B", "67c77cc00e83a60647d826334509d2b3": "y^*", "67c7bb36cc9dadee2e76f50e7995b563": "\\overrightarrow{F}_y = (-q)\\overrightarrow{\\xi}_y + (-q)[\\overrightarrow{v}_n \\times \\overrightarrow{B}_z] = 0", "67c7ea03700a901baf266d24a0a6d44e": "R \\not= r", "67c8efa7e1bd8f8802a8b9b00c5b9c23": "(1, \\xi) = (1 / \\xi, 1) = (\\zeta, 1)", "67c8efcb8104a08a30cd52cc43339f9b": "p = (x^3 - y^3) / (x - y), x = y + 1", "67c91c4e96ac44bb74de20d18d751723": "y\\, ", "67c96d1b0dc840521467d15e6e32d4fc": "(B', \\pi_{B'})", "67c9b0e48a24ce8069bf634f186cafc0": "\\begin{align}\n \\mathbf{M}^* \\mathbf{M} &= \\mathbf{V} \\boldsymbol{\\Sigma^*} \\mathbf{U}^*\\, \\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^* = \\mathbf{V} (\\boldsymbol{\\Sigma}^* \\boldsymbol{\\Sigma}) \\mathbf{V}^* \\\\\n \\mathbf{M} \\mathbf{M}^* &= \\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^*\\, \\mathbf{V} \\boldsymbol{\\Sigma}^* \\mathbf{U}^* = \\mathbf{U} (\\boldsymbol{\\Sigma} \\boldsymbol{\\Sigma}^*) \\mathbf{U}^*\n\\end{align}", "67c9f8676f15fe621d323682caf1c258": " {\\Gamma ^{\\mu}}_{\\alpha \\beta} ", "67c9fba94edecb151dcdeb3976fa561b": "g_2\\in\\langle y, z\\rangle, g_3\\in\\langle z\\rangle", "67c9fd16d1086f9d9cbce8496159660b": "-\\gamma", "67ca1f86ab50d1871a2ba7ccd75a3e56": "\\begin{align}\nA_0 &= 0\\\\\nA_k &= A_{k-1}+\\frac{x_k-A_{k-1}}{k}\n\\end{align}", "67caa68ecf955f35e806461e1365b732": "\n W^\\perp = \\{ \\varphi \\in V' : W \\subset \\ker \\varphi\\}. \\,\n ", "67cafec0f8efe3b701037b5448ff7327": "\\overline{O_L P}", "67cb0df27615ec7e23d18d5c99a4d35b": "u A = \\lambda u", "67cb1fd512ff488afffbe932114e09a1": "\\lVert \\phi \\land \\psi \\rVert = 1", "67cb3592fa323b14889f1f49295b6b1c": "\\tfrac{-8}{5}", "67cb41238a66bc3216b9c217fca51dae": "{\\partial \\mathcal{H} \\over \\partial \\lambda} = 0", "67cb61967e8c2c3b882e39bba359ecf1": "*m'", "67cbff1a0d20f30e17f0be5880fef7f8": " \\sigma \\in \\Sigma", "67cc045f9d9b30ca79f5cb7e30fcf07b": "\\scriptstyle (N \\,+\\, 1)", "67cc79ca3b8943ac980041a6e442cf66": "y = \\Phi(t) ", "67cc82895e5220b15607262acd08e1b3": "\nZstep=\\sqrt{HbTws},\n", "67ccbe9d7cf3eef267f2aef8bacddea4": "\\begin{align}\n\\nu' & = \\gamma \\nu \\left ( 1 - \\beta \\right )\\\\\n& = \\nu \\frac{1}{\\sqrt{1 - \\beta^2}} \\left ( 1 - \\beta \\right ) \\\\\n& = \\nu \\frac{1}{\\sqrt{\\left ( 1 - \\beta \\right ) \\left ( 1 + \\beta \\right ) }} \\left ( 1 - \\beta \\right ) \\\\\n& = \\nu \\frac{\\sqrt{1 - \\beta}}{\\sqrt{1 + \\beta}}\n\\end{align}", "67ccc8b29b6729c8241eb91010bb9030": "i=0,1\\,", "67ccf054921e965ade0d188a16be446c": " t \\,", "67ccf53640bae6b92d28577e05b391b8": "|(j_1j_2)JJ\\rangle", "67cd2930c7b2076f28fc1100d1dfb475": " \\text{d}_x ", "67cda031f1dc3c23fb4b2bf3e09bafb4": "\\frac{\\delta K_{2}(\\delta \\gamma)}{\\gamma K_1(\\delta\\gamma)} + \\frac{\\beta^2\\delta^2}{\\gamma^2}\\left(\\frac{K_{3}(\\delta\\gamma)}{K_{1}(\\delta\\gamma)} -\\frac{K_{2}^2(\\delta\\gamma)}{K_{1}^2(\\delta\\gamma)} \\right)", "67ce1d698be0abccd24a576f016ba9e8": "\n \\sqrt{n}(\\hat\\theta_n - \\theta_0)\\ \\xrightarrow{d}\\ \\mathcal{N}(0, V).\n ", "67ce52b6701de453d63f9f2f18bbb337": "\\binom{n}{2}^{-1}", "67ce8985f527ed5662b5a01a22715646": "\\vec{T}", "67cead227bd6fe422f3083174c22899c": "f([0.1, 0.8], [0.06, 0.08])", "67cecd377758a188db4ceb5ab92b422a": "x^2 + c^2 + y^2 = a^2 + {c^2 \\over a^2}x^2", "67ced5d1ce090ddf4d56e85fdc780416": " b=\n \\begin{bmatrix}\n 11 \\\\\n 13 \\\\\n \\end{bmatrix}.\n", "67cf255f5acadcca025273edb3813b31": "\\Rightarrow-\\dfrac{1}{3}=\\dots 1313_5", "67cf2cee3e7a56eba3c0368ae66d54c6": "\\mathbf{E} = - \\mathbf{\\nabla} V_\\mathbf{E}. \\, ", "67cfcf0e8260c561cdb5476d3998885d": "\\int\\frac{\\mathrm{d}x}{1 - \\cot ax} = \\int\\frac{\\tan ax\\;\\mathrm{d}x}{\\tan ax-1}\\,\\!", "67cfeda0c24d5368867a6ef69309e71f": " (e\\rho\\sigma)^{1/\\rho} = \\limsup_{n\\to\\infty} n^{1/\\rho} |a_n|^{1/n}.", "67d012c86ce173b7766abf719d98dc04": " X^{\\prime}(t) = F\\left( X(t) \\right) X(t), F^{\\prime}(X) \\le 0 ", "67d020203dcdd24db6bbaedb75133e9b": "\\rho_t(1_A X) = 1_A \\rho_t(X)", "67d037bd4ad9bdec84e4dd60f2146695": "m \\Psi_0^2 = \\rho_s", "67d046bd9debf4927aa79f0bf025b1b6": "\\sum_i w_i = 1.", "67d097ac2b90926ae4bce671dbf12640": "\n{\\rm \\hat{var}}(\\hat{\\beta}_j) = \\frac{s^2}{(n-1)\\widehat{\\rm var}(X_j)}\\cdot \\frac{1}{1-R_j^2},\n", "67d0d256b8fda07faf7cda234b340bfa": "\\begin{align}\n & H_1(x_t,y_t,z_t,v_{ts};\\theta) = y_t - g(\\hat\\pi'z_t + \\sigma v_{ts}, \\beta), \\\\\n & H_2(x_t,y_t,z_t,v_{ts};\\theta) = z_t y_t - (\\hat\\pi'z_t + \\sigma v_{ts}) g(\\hat\\pi'z_t + \\sigma v_{ts}, \\beta)\n \\end{align}", "67d0dbb3c96a894c7ee0f79fec309e78": "(a,b)\\in E", "67d13178d25bf9754f49b93333237b51": "W(D) = -\\frac{2 \\pi C \\rho _1 \\rho _2}{12} \\int_{z=0}^{z=2R}\\frac {(2R-z)zdz}{(D+z)^3} \\approx -\\frac{ \\pi ^2 C \\rho _1 \\rho _2 R}{6D}", "67d1667a1b234f16bf151235a3f41ed8": "s_d(V)\\ ", "67d1a9146df6b510d3f9a4e6fe0dc848": "x=20\\frac{1298205}{2362256}", "67d2139123248a409e15ccf760040291": "w\\in A_p", "67d2527db2b46219053d784c1fca7dbd": "v \\tan v", "67d25a48b1450fb4f887f3329a504ab8": "V=0", "67d2768cac23aa4ebdc11371d4003647": "\\text{300 km range} = \\frac{C}{2 \\times 500}", "67d3394e973a8bec0ad9cb3e454ec7b5": "\\beta = \\frac{1}{2(\\pi_A + \\pi_G)(\\pi_C + \\pi_T) + 2\\kappa[(\\pi_A\\pi_G) + (\\pi_C\\pi_T)]} ", "67d365bea75abc54469d49d3fa2b1d2d": "\\mathbf{J}_i", "67d3a0c22b2422863a7b1d0fa6d734d8": " \\bigl( \\begin{smallmatrix}\\\\ 1&0\\\\ 0&\\pm 1\\end{smallmatrix} \\bigr),", "67d3cba3726e639a506fe7067cd5c367": "\\varepsilon_{\\cdots i_p \\cdots i_q \\cdots }=-\\varepsilon_{\\cdots i_q \\cdots i_p \\cdots } .", "67d474f8b10467f44715eca2c9ac5770": "10^{50}", "67d47a99df625d8206c5cca4f0491bfb": "\\Phi_{22} = \\Phi_{33} = m/r^3", "67d49c4c1d1245b90f17b2efbf8cfab1": "\\beta=\\epsilon", "67d4e34d7b2d9e8adcd3fbd2ed7f7709": "S_2= E(\\sin(2\\theta)) = E(2\\cos\\theta\\sin\\theta)\\,", "67d51cc5e72a23241ad6a96df3429398": "0 \\leq i \\leq L", "67d54161ab6c3494f7e751cdff438f47": "\nv_{i;j}=\\nabla_j v_i=\\frac{\\partial v_i}{\\partial x^j}-\\Gamma^k{}_{ij} v_k\n", "67d599273d5d3d6c77a3742ef99f916a": "{D_{n,k} \\over n!}.", "67d5ee426c1482fb7c7536356d7f9eea": "X(\\omega_k)", "67d60f3942b1eebda3573c805d30219a": "\\frac{3b}{8}", "67d688390fa7c8a47ca27b4d06a76d39": "\\mu_{2,1}= \\frac{\\langle b_{2}, b_{1}^{*} \\rangle}{B_{1}}=\n\\frac{\\begin{bmatrix}-1\\\\0\\\\2\\end{bmatrix} \\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}}{3}=\\frac{1}{3}(< \\frac{1}{2})", "67d68a18322eb5d61d1e5dce21105f5e": "\\mathrm{Ha} = B L\\sqrt{\\frac{\\sigma} {\\mu}}", "67d6c3c4ed35f8d9451ade162f7c99ed": "x = v t \\cos \\theta , t = \\frac{x}{v \\cos \\theta}", "67d7399e097ab67fb3fd285c2ac32b17": "\\phi (\\mathbf{r}, t) = \\frac{1}{4 \\pi \\epsilon_0} \\int \\mathrm{d}^3 x^\\prime \\frac{\\rho( \\mathbf{r}^\\prime, t_r)}{ \\left| \\mathbf{r} - \\mathbf{r}^\\prime \\right|}", "67d783e19652d66faa03d4d01207663a": " 0=A^n+c_{n-1}A^{n-1}+\\cdots+c_1A+c_0I_n= p(A).", "67d78dbbe9fd8807166b3639a95e05e3": "V = x^3+y^4+a x y^2 +bxy+cx+dy+ey^2", "67d792c1318781d165a99a1509c16b06": "\\Omega(n \\, {\\log n})", "67d79be2203b996750cd2112c44f2ca7": "\n\\sgn(x) = \\begin{cases}\n1 & \\text{ if }x > 0\\\\\n0 & \\text{ if }x = 0\\\\\n-1 & \\text{ if }x < 0\n\\end{cases}\n", "67d7f942d27ffa837ffd4a9185f6d48e": "\\underset{n\\to \\infty }{\\mathop{\\lim }}\\,{{G}_{n}}(\\zeta )=\\alpha ", "67d7ff9f1c6a34bfe218f3b9c7b7ea4f": "\\widehat\\sigma^2 ", "67d8635982e9ca986e5820ae3680a562": " 0 = f(a)\\,", "67d875f9bca25ed6f6124c82f9d2d841": " \\cos \\left(\\frac{c}{R}\\right)=\\cos \\left(\\frac{a}{R}\\right)\\cos \\left(\\frac{b}{R}\\right).", "67d880871b9270d9a27b6b2dc5a018ad": "c(n)={n\\choose (n-1)/2}(2^{-(n+1)/2} + n^2 2^{-n-4}).", "67d8ada0849257ef03c88e4c5ce6a29f": " \\operatorname{Perf}(f,r_{g(n,1/\\epsilon)}) \\geq 1-\\epsilon, ", "67d8bd969cb2cf1b378f5e534a19d936": "x^y", "67d8de3deecf9d07d40e29b6a3d00203": "\n\\hat{\\mu} =\n\\begin{cases} \n\\mu & \\text{, if the mean is known.} \\\\\n\\bar{X} = \\frac{1}{n} \\sum_{i = 1}^n X_i & \\text{, otherwise.}\n\\end{cases}\n", "67d8fca2479f2c237f09d73c58d60c14": "[-(b+1),+(b+1)]", "67d8fd46c48b4d89142e437ddc83727d": "\\rho b_i = p_i\\,\\!", "67d908be0d665511fd8ddd03cc057e9a": "[x_1,x_n]\\,", "67d90fe9eeae0773f768f9a8c176fccd": "a_j ", "67d9d1bd79040558991c9eae8c931760": "h_{L}", "67d9dfc543365cc29937e78364f20fd1": "E(G)", "67d9ee11f17c448d2e28e7ec573fd918": "\\sigma= \\frac{4 \\pi}{k^2} \\sin^2 \\delta_s =4 \\pi a_s^2 ", "67d9f4b510f255971898141f550bcd8c": "n_e=n_p", "67da0dd79546da91e482e99b745f0b71": "H_{ij} = \\int \\phi_iH\\phi_j\\mathrm{d}v\\,", "67dac527351c312b74efa24e618629bb": "m = p\\times d", "67db40f372705f7817c5047b7fbe81b8": "OA\\equiv\\Box(\\lnot A\\to s)", "67db6855bf4e572bb95ed9510a5034b3": "\\displaystyle{g=\\begin{pmatrix} -DB^{-1} & DAB^{-1}-B \\\\ B^{-1} & -AB^{-1}\\end{pmatrix}}", "67db687e9d24829e070dd2f3cba2084e": "+ - +", "67db8c40e7f7fba1a6055c5eddbb3d14": "\\sum_{m=-\\infty}^{\\infty} h[n-m] \\, z^m", "67dbdd89636435b2526995fd675becfa": "-x^2.", "67dc63f365e1d285a284775c66024486": " \\mathbf J = q_2 \\mathbf v_2 \\delta \\left( \\mathbf r - \\mathbf r_2 \\right),", "67dd1c0d4e68c367b980f8723c781cb1": "-m", "67dd8df6fd2f59f0b94c2e747e5762a6": "b = \\sigma_1 \\sigma_2", "67dd94c462ffab6e35616e353b16cd7a": "H^p(X \\times_S \\operatorname{Spec} -, \\mathcal{F} \\otimes_A -) \\to H^p(K^\\bullet \\otimes_A -), p \\ge 0", "67dd965194ec6bcd4bbe447da0db04fa": "M = {\\Bbb Q}(q) \\otimes_A L;", "67ddad02efa6ab1ba5086e21916aa3a5": " \\alpha >2 ", "67ddf034d728b528cf41fe998c82c608": "\\frac{dy}{dx}\\neq 0", "67de03eb19c0b3ca2724f73d02e0e482": "\\psi = \\psi(\\mathbf r)", "67de03eeab0e1c0e066011b39378c0f1": "W^{I}(z,x)=\\beta ^{I}(z),", "67de2045c5a785921df3994b022ddbf0": "\\begin{matrix}B\\left(\\cos ^2\\theta\\ -\\ \\sin ^2\\theta\\right) &=& 0 \\\\ \\\\\nB\\cos 2\\theta &=& 0 \\\\ \\\\\n2\\theta &=& \\frac{\\pi}{2} \\\\ \\\\\n\\theta &=& \\frac{\\pi}{4}\\end{matrix}", "67de348d30735325e4716dd034b17078": "A = \\begin{bmatrix} 4 & 1\\\\6 & 3 \\end{bmatrix}", "67de76e8ee022cdcf4ad5ff09b5602fc": "p\\ne 0", "67de7ea19a73525b34b77714fb12009c": "{f_x}(m),(x \\ge 1)", "67df5b00b65c8432fcf7fd4d3fab6249": "\ndg=\\frac{\\pi}{2}~f n^2\\,dn = \\frac{4\\pi fV}{h^3}~ p^2\\,dp\n", "67df63f173d9db68f1071188803a341f": "\\left\\langle 2, Z_2 \\right\\rangle", "67df6725d84b5f9c27ccc40553685b8a": "r(n) = \\hat{y}(n)-y(n)", "67e01d4fe24ceacc55c7638662a675af": "m = 2410 \\log_{10}(1.6\\times10^{-3} f + 1)", "67e088abe937323a9b54126ac63ba687": "\\alpha\\pm 1.5 \\cdot 10^{-9}", "67e0909da7ffae7465efff80c5f8905f": "\\delta_i^{-2}", "67e0c442dfc41ae1aa1b76c800f16082": " u=\\frac{2gr^2}{9\\eta_2}(\\rho_2-\\rho_1)\\!", "67e0ed041b3c1cd3107c356b559fb1c2": "\n\\begin{align}\nz^n & = |z|^n \\left(\\cos \\left(n\\arccos \\frac a{|z|}\\right) + i \\sin \\left(n\\arccos \\frac a{|z|}\\right)\\right) \\\\\n& = |z|^n T_n\\left(\\frac a{|z|}\\right) + ib\\ |z|^{n - 1}\\ U_{n-1}\\left(\\frac a{|z|}\\right).\n\\end{align}\n", "67e0edbaf798ed33e344fa8b2f4e3708": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {\\alpha A} \\right) =\\alpha\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A} \\right).", "67e0f67583b6afca5b5fffc4e9a9e1b2": " \\frac{ b - a }{ \\log_e( b ) - \\log_e( a ) }", "67e19561f9ea959dcd277453cb1023ef": "\\bold{AC}^0", "67e1acc048b208e008282037a2aa5bd4": " \\delta(H) ", "67e21f2a6c8b91c622b608f7658676b8": "\\pm\\frac12\\sqrt{x}", "67e2be5f1a62ffeb7990a361c39c63d0": "r^2=2st", "67e2f6532b7297def8802db14ef12235": " \\tau = {\\rm{i}}\\frac{{}_2F_1 \\left (\\frac{1}{2},\\frac{1}{2};1;1-z \\right )}{{}_2F_1 \\left (\\frac{1}{2},\\frac{1}{2};1;z \\right )}", "67e35f7a692a68e93d6f2848efe68378": "e^{A+B} = \\lim_{N \\rightarrow \\infty} (e^{A/N}e^{B/N})^N, ", "67e3a5abe6caecebb6b4d006dadad6c9": "\nH = H_{e} + H_{ph} + H_{e-ph} \n", "67e3c8e389309b855c252ecef4f36f57": " L_5 = \\frac {-K_P K_V K_C K_M } {s^3 L_M M} \\, ", "67e4261548d9e3cfc90442089a0da700": " \\mathcal{G}(n) = H_{R_2^\\omega(n+1)}(1) - 1, ", "67e4909883919130f94ce85e532d447b": " \\text{OMA} \\approx 2P_{av} ", "67e495341e99fc21292be0a888702564": "g_1,g_2,\\dots , g_n", "67e49d027d84e5ed5e3e528355e530b8": "\\langle \\bar{T}T\\rangle_{TC} \\cong 4 \\pi F_{EW}^3", "67e5645c0609ff69cb134a3202df6f54": "P_0(x)=false", "67e611bdd83741e1584296fff82b7fae": "\\ (x,y)", "67e631d2dc3dfa47d3da75b4c2c08ce1": " \\tilde{h} ", "67e63d43d191788c12c22fc88055972a": "L_\\beta N_r > N_\\beta L_r", "67e675ad310de212cb66831b7cf1ce27": "\\sum_{u,v}", "67e67de573fd92f78f7be3d487cb1b2e": "A^\\dagger", "67e683e0c987f4694a9b6506ad2e9148": "L_{FS}", "67e6c96ffd7d1b8d4d6a8f52269aea59": " 1 + k_i \\psi_i + k_i^2 \\psi_i^2 + \\cdots ", "67e6ea5e85682d66a41d32f0c3040ed1": "f'(x_0) = K", "67e799dc92f6f4e7f53ffeb9ee40ab93": " \\tilde{\\mathbf{x}}' ", "67e7c4748177aabf1a43cac9f9c05c83": " \\ln \\Gamma\\left(\\alpha\\right)-\\alpha\\ln\\beta-\\frac12\\ln\\lambda", "67e846390979079a69a41b924809e0e8": "H(f) = \\begin{cases}\n T,\n & |f| \\leq \\frac{1 - \\beta}{2T} \\\\\n \\frac{T}{2}\\left[1 + \\cos\\left(\\frac{\\pi T}{\\beta}\\left[|f| - \\frac{1 - \\beta}{2T}\\right]\\right)\\right],\n & \\frac{1 - \\beta}{2T} < |f| \\leq \\frac{1 + \\beta}{2T} \\\\\n 0,\n & \\mbox{otherwise}\n\\end{cases}", "67e860c90e57a87a27b0e1ca30dff495": " V_{i+k,j} = \\begin{cases} 0, & \\text{if } j \\le k; \\\\ \\frac{(j-1)!}{(j-k-1)!} \\alpha_i^{j-k-1}, & \\text{if } j > k. \\end{cases} ", "67e862d9dcc7af1a98932675a3012782": " \\kappa=1/\\sqrt 2", "67e86df8acee81f3e1c4c640dac58ddd": "\\mathcal{L}_X R^a{}_{bcd}=0", "67e8716ee52db5f4dfc395e280ca4cf7": "C = 2Rsin(\\frac {\\Delta}{2})", "67e896f785bfe8b995a8ecea929a5c55": "j: V \\to M", "67e89e1b30d1371fbb6abb1103108da4": "\\exp\\colon(\\mathbb R, +) \\to \\mathrm{SO}^{+}(1,1)", "67e8d40c537234712f6ba914d5b27699": "e^{-i a n}", "67e9023d44deac09ce0dae0a8d3682ef": "\\cos\\theta\\approx 1", "67e93c076033cf210dfc1252912d97b8": " \\hat{\\bold{r}}=\\cos(\\theta)\\hat{\\bold{x}} + \\sin(\\theta)\\hat{\\bold{y}} ", "67e961b7f07e0bbb61d57ea02104c123": "v_e\\to c", "67e9677d769ad01d084e90a19e31e44e": "\n\\partial_t \\boldsymbol{q} = \\underline{\\underline{\\boldsymbol{D}}} \\,\\nabla^2 \\boldsymbol{q}\n+ \\boldsymbol{R}(\\boldsymbol{q}), ", "67e96b821a7ffd64e0a8d4a8ff89e297": "W\\left( \\mathbf{p,z}\\right) =\\sum_n \\left( \\int\\limits_0^{z_n(p_n) }p_n( z) dz\\right) -C\\left( \\mathbf{z}\\right). ", "67e97b08adb5b7717b5bdf67c79780b6": "\\frac{\\zeta(s)\\zeta(s-2) - \\zeta(s-1)^2}{\\zeta(s)^2}~\\textrm{for}~s>3", "67e981e0c1540498bf061404042ea3c7": " 1 - e^{- 2 \\gamma^2 k}, ", "67e9da00f58fedf24e8f89393212f8a4": "\\text{Spec }L", "67e9da9c479101f21fe6c2354569d7a3": "-1=1", "67e9e3363da3c5c506b66562e461efb9": "\\left(F_{\\mu\\nu}F^{\\mu\\nu}\\right)_B = Z_3\\, F_{\\mu\\nu}F^{\\mu\\nu}.", "67e9ea8e7db5a8d9f1f3e1e30d50e228": "\\Delta T_{LM}", "67ea1ee2f1543ec4398c6c39bab54f43": "y(4)", "67ea334bf119d1f173695db948937f44": "E=B\\oplus B^d", "67eaabb4b79bf2111f9ce8f0303904ea": "\\begin{align}\n\\left[f*_{2\\pi}g\\right](x) \\ &\\stackrel{\\mathrm{def}}{=} \\int_{-\\pi}^{\\pi} f(u)\\cdot g[\\text{pv}(x-u)] du, && \n\\big(\\text{and }\\underbrace{\\text{pv}(x) \\ \\stackrel{\\mathrm{def}}{=} \\text{Arg}\\left(e^{ix}\\right)\n}_{\\text{principal value}}\\big)\\\\\n&= \\int_{-\\pi}^{\\pi} f(u)\\cdot g(x-u)\\, du, &&\\scriptstyle \\text{when g(x) is 2}\\pi\\text{-periodic.}\\\\\n&= \\int_{2\\pi} f(u)\\cdot g(x-u)\\, du, &&\\scriptstyle \\text{when both functions are 2}\\pi\\text{-periodic, and the integral is over any 2}\\pi\\text{ interval.}\n\\end{align}", "67eaae63b53f2d694d517d851191dee1": "\\int\\limits_{-1}^1\\frac{dx}{\\sqrt{1-x^2}} = \\pi\\!", "67eab4128a835a803fc1910cae4ca084": "\\mathit{W}_{1-2}+\\mathit{Q}_{2-3}+\\mathit{W}_{3-4}+\\mathit{Q}_{4-1} = \\left(\\mathit{u}_1-\\mathit{u}_2\\right)+\\left(\\mathit{u}_2-\\mathit{u}_3\\right)+\\left(\\mathit{u}_3-\\mathit{u}_4\\right)+\\left(\\mathit{u}_4-\\mathit{u}_1\\right)=0", "67eb3aabe3799c78bc1b966f107d5f46": "\\begin{matrix}\na_i &=& \\sqrt{m\\omega \\over 2\\hbar} \\left(x_i + {i \\over m \\omega} p_i \\right) \\\\\na^{\\dagger}_i &=& \\sqrt{m \\omega \\over 2\\hbar} \\left( x_i - {i \\over m \\omega} p_i \\right)\n\\end{matrix}", "67eb644415be9d6c48bf9a740b54b248": "\\sigma= \\sigma_\\lambda^i(x^\\nu) dx^\\lambda\\otimes\\partial_i ", "67eb6fba80f4d254d291558863958f6c": "(3^3\\times 3_+^{1+2})\\cdot 3_+^{1+2}: 2S_4", "67eb984ea2e46b2208d019e3f001995d": "F_{p - \\left(\\frac{{p}}{{5}}\\right)}", "67ebc0638bd4707a276a7909f85219c8": "\\scriptstyle |\\psi_n\\rang. ", "67ebcee919fa5358229085972ec5430d": "16x^10-64x^9+160x^8-384x^7+512x^6-544x^5+456x^4+126x^3+3x^2-4x-177162=0", "67ebfa81725325cdecf5ff7725c1e9bb": "\\begin{align}\n s(x) &= \\sum_{k=0}^{\\infty} F_k x^k \\\\\n &= F_0 + F_1x + \\sum_{k=2}^{\\infty} \\left( F_{k-1} + F_{k-2} \\right) x^k \\\\\n &= x + \\sum_{k=2}^{\\infty} F_{k-1} x^k + \\sum_{k=2}^{\\infty} F_{k-2} x^k \\\\\n &= x + x\\sum_{k=0}^{\\infty} F_k x^k + x^2\\sum_{k=0}^{\\infty} F_k x^k \\\\\n &= x + x s(x) + x^2 s(x).\n \\end{align}", "67ec5814956d2e71aa4f475a45239493": "LWE_{q,\\Psi_\\alpha}", "67ec6e49b4ee04a1367d99f7040f0527": "\\{\\{64x^3+192x^2-256x+64,(1,2)\\},\\{64x^3+192x^2+80x+8,(2,4)\\}\\}", "67ec7e0e5cd36d1be64d70e530425e45": "\\begin{bmatrix}0 \\\\ 1\\end{bmatrix}", "67eccacff66f9a60e7cc580636aa4a29": "x \\leq y\\Leftrightarrow x \\subseteq y", "67eccb988f11a2a48560e6ac010a7ddd": "p_{j} ", "67eceffe2b25234566fc15fbfa08ed02": " A_{[\\mu \\nu]} = \\frac{1}{2!} [ A_{\\mu \\nu} - A_{\\nu \\mu} ] ", "67ed0587eb4ecc8a6eaabcf79feae2dc": "\\mathrm{F\\ m^{-1}=A^{2}kg^{-1}m^{-3}s^{4}}", "67ed7039894e5147e582ce99dcf0def6": " u : M \\rightarrow \\mathbb{R} ", "67ed83eb81204bf5e77c65dc6ea06673": "\\scriptstyle \\mathbf J", "67ede5558c6b215f7ade51b72e77b8f3": "\nC =\n \\begin{align}\n (1 - \\left\\vert 2 L - 1 \\right\\vert) \\times S_{HSL}\n \\end{align}\n", "67ee217ebd03828093b3dcf1613d28cc": "\\hat{X}_i", "67ee33f3d1239cad0e242989edbef246": "v_l, v_r", "67ee8644e5ec6a6bb53fed019c12cb76": "\\frac{|\\overline{PR}|}{|\\overline{PS}|} = \\frac{|\\overline{TV}|}{|\\overline{TU}|}.", "67ee889554f7daa56e8b05cd7b94e2ae": "x^2 - 67y^2 = 1", "67eebd1886d78daef450d973d349b8ac": "A_{ij}(t)", "67eeeeb7b50b035ee3a2f2643e585e06": "(p,a,A,q,\\alpha)", "67ef21846cec20c9f69074dfcf7bdc32": " f(x_0^+) - f(x_0^-) = a \\neq 0.", "67efaca07d12ac053f23e214e5c1d642": "x-10, k0\\,", "67fb9792bd763c00de733d5da30b9b39": "\n\\frac{{e^{ik R} }}\n{R} = \\int\\limits_0^\\infty I_0(\\lambda r) e^{ - \\mu \\left| z \\right| } \\frac{{\\lambda d \\lambda}}{{\\mu}}\n", "67fb9a4678156da9f6ab90af4630371e": "ROC3 = (1-Price/Price(X3))*100;", "67fbb213ca0f66c81a0a7c5e0c12a552": "\\sqrt{n}(\\hat\\beta - \\beta)\\ \\xrightarrow{d}\\ \\mathcal{N}(0,\\,\\Omega^{-1}),", "67fbbde3f857dd0abe2ea0c7ffc078b8": " L = n{h \\over 2\\pi} = n\\hbar", "67fc08a895fceacef7bf55ab833e35cb": "P_n f", "67fc2ad01d72fd6259e441b4e99fe321": "\\tau_b*=f\\left(\\mathrm{Re}_p*\\right)", "67fc8b4da26397cf2f961fa0a6abca67": "m=\\lim_{x\\rightarrow+\\infty}f(x)/x=\\lim_{x\\rightarrow+\\infty}\\frac{2x^2+3x+1}{x^2}=2", "67fccc7025e32f4365367ba86ab900bd": "z\\mapsto z^k", "67fcd29f57b234da3e34a6b2898a3db3": "f_2(t)\\,", "67fd02dfedb24a5b8e817b26d6bcc90f": " f(x, \\lambda x) =\\frac{\\lambda x}{x^2+\\lambda^2}", "67fdc2944bbad63c9accc6fc1b84af75": "~~~\nD(\\omega)=\n\\frac{\\frac{a(\\omega)}{\\sigma_{\\rm e}(\\omega) v(\\omega)}}\n{\\frac{n_1}{n_2} \\frac{\\sigma_{\\rm a}(\\omega)}{\\sigma_{\\rm e}(\\omega)}-1}\n~~~~~~~~~~~~~~{\\rm (D1)} ", "67fdd6651261a0b920e76776f8352d6d": "Obs1\\,", "67fdf0ea0048ff0eb4e79c86934def5f": "\\textstyle E_{\\mathrm p} = m_0 c^2,", "67fdf737525e2115cd13e9fcb0f3c4e0": "(x_n)_n, (x'_n)_n", "67fe94ccf1ef92477a66a87edfa9c1bf": "t_{m,1-\\alpha}", "67fea295ad73936bb5b92fcd2453d88f": "\\chi \\to \\phi \\lor \\chi ", "67fee5fe92583988ea855527ee590d73": "2a-R", "67ff2683098a7f5b81c5ab29b9639651": "T_{y^{-1}}^{-1} = \\sum_xD(R_{x,y})q^{-\\ell(x)}T_x.", "67ff5c7f640ba208afc8e3b5bf0be2be": "\\lim_{x\\to\\infty}N^x=\\begin{cases} \\infty, & N > 1 \\\\ 1, & N = 1 \\\\ 0, & 0 < N < 1 \\end{cases}", "67ff6c1c8a4b8f4c160b1a78de2f1b7c": "G_{p,q}^{m,n}", "67ff6e9f1518f7b8c8732d3a8db28f05": "Z_i = \\frac {\\hat{p}_i - \\bar p}{\\sqrt {\\frac {\\bar p (1 - \\bar p)}{n_i}}}", "67fff58bdbd422326bfdd7e2f020c7ff": " I = (|e_{x}|^2 + |e_{y}|^2) \\, \\frac{1}{2 \\eta} ", "67fffc960715cfbe1602477c2729334e": "\\frac{n}{K+1}(1-o(1))", "680016fea1178d185a25cd816ab3cfc2": "f_5", "680048d090eb0779b4432f4b6c1c86fc": "\\mathcal{B}_B", "680056a35a42089c66edc2e870564e03": "\\mbox{If }X \\vdash a \\mbox{ and } X \\subseteq_f x \\mbox{ then } a \\in x.", "6800ce69e5c311d2dffc0f7c35265229": "\\mathit{p}", "6800fddabf13a59153e4081213570f5d": "(A, +)\\;", "680117d66dc82ac117252d06192a8332": "t_1 \\equiv t_{p-1} t_1 + D u_{p-1} u_1 \\quad \\mbox{and} \\quad u_1 \\equiv t_{p-1} u_1 + t_1 u_{p-1}", "68014ed12af035086a7b31d33540f338": "\\alpha\\in\\mathbb R_+", "68018ad26f1cf94db91d2a88e4d1392b": "\\rho(u) = 1-(1-\\log(u-1))\\log(u) + \\operatorname{Li}_2(1 - u) + \\frac{\\pi^2}{12}", "6801f0aae5fa295783d76d81c524b455": "E(\\Gamma)", "68020484089b8267804c41a72887338f": "MU_*(X)\\otimes_{MU_*}M", "680216b079c29a6e9ec03b985f360a1b": "A[t/x]", "6802338bc849266552c0469d0d8b126d": " \\varphi(\\theta) = \\sum_{n \\in \\mathbb{Z}} f(n) e^{-in\\theta} \\, ", "68027d223f687bccf80538b7c2005d69": "g(2^n,2^k) = 2^k", "6802802c7d7426538af6f7e25df45691": "M_w \\geq 8.0", "6802b2c9042f949e0a66d1344ef07769": " k_n = \\left( \\frac{C_1 v_1 C_2 v_2}{C_1 v_1 + C_2 v_2} \\right) \\left( \\frac{d_1 + d_2}{2} \\right) ", "6802c919bfd5f6f8531893cbf48602d3": "Z_t\\!", "6803331dccd70aef2eca6902cc576cdc": "\\operatorname{Aim}(X)", "6803fd660a615a5df646c8779cde24aa": "C_n = [\\sqrt{2}, \\sqrt{2}+1/n]", "68043e677cb3a2c572cf39d0c5dedb82": "\\bar{0}_2", "68047fffccca335dcf86a93f207cc2ec": "\\tfrac{pq}{2}", "6804b86f41f199c48c547267963949d2": "\\{p,r\\}", "6804c4998ca6b9db613b6395b7d902a1": "q^{n-k}", "6804f4e949c046539857b496e8496530": "\\delta(B) \\ge 1", "68055767893c717bd5854db354a541b5": "\\mathbf{T}(s) = \\gamma'(s)", "68055ce0cb02bb57ed38e803dfdc6ad8": "\\displaystyle \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{\\infty} f(x) e^{-i \\omega x}\\, dx", "680585faa345d1f5a193ae8d66dcd77a": "\\tfrac{1}{2} + \\tfrac{1}{6} + \\tfrac{1}{21}.", "6805d50cabbc452e6d05d82c5e82e527": "E\\{\\hat{x}_{\\mathrm{MMSE}}(y)\\} = E \\left\\{ E\\{x|y\\} \\right\\} = E\\{x\\}.", "6805fe08783db75d523540a8fd4d83fb": "\\mathrm{GF}(q^p).", "68064e7eb2226741b412551c1429218b": "\\scriptstyle{\\omega}", "6807011f9a139a51916f67a456ad324a": "H^2(\\mathbb{C}\\mathbf{P}^1, \\mathbb{Z})", "68075baa0645aa49141e8eb2247bf81e": "\\begin{pmatrix}x' \\\\ t' \\end{pmatrix} = \\begin{pmatrix}1 & v \\\\ 0 & 1 \\end{pmatrix}\\begin{pmatrix}x \\\\ t \\end{pmatrix}.", "680771f000ba69a65af4e0ae63fb5fe2": "R(r) = J_0(\\lambda r).", "68077a8e260324f17767f2d08565738a": "c_1y_1+c_2y_2", "6807c423cfd9e34ec15ec98f041a9620": "\\#S", "6808170e79d5f5e072a9140f1ce53504": "=\\frac{1}{4a}", "68088c04b7bfafbaf80b3c2ba5e1526f": "\\limsup_{x\\to a} f(x) = \\inf_{\\varepsilon > 0} (\\sup \\{ f(x) : x \\in E \\cap B(a;\\varepsilon) - \\{a\\} \\}) ", "68089a6bdfe2e87fafe7bbdb284c5c65": "\n\\left(\\frac{\\partial ^2}{\\partial x^2} + \\frac{\\partial ^2}{\\partial y^2} \\right) \\phi (x,y) = 4\\pi b e^{2b \\phi (x,y)} \n", "68091306721854a576ed08e5d80a0568": " \\phi_> = \\left(Br + \\frac {C}{r^2} \\right ) \\cos \\theta \\ . ", "6809379f20eff7fdecb8b5a829838fad": "\\operatorname{Cov}(X, Y) = p_B - p_X p_Y,", "68098b91b8348f8dd5c82cec634b356a": "x(x^3-x-1)^{1/n} = \\pm (x^3+x^2-1)^{1/n} ", "6809bfba07c2f7efb0ab3df1aa1a7bbf": "\\widehat{\\sigma_e^2}", "6809c59370e21b3e6e8fd117442fd377": "O(n^{3})", "6809f56a73e5b3b4a74fd32a2f1c791d": "\\text{i}", "680a9c20a50f71573b4f59a28a0bed6b": " d \\sigma_t = (\\beta_t-\\sigma_t)\\,dt + \\sqrt{\\sigma_t}\\,\\eta_t\\, dW_t", "680ad2ebf9c5e390d8306e8ecf95b599": "c_2 = -\\frac{\\cos ks}{k} \\quad;\\quad c_3 = -\\frac{\\sin ks}{k}", "680ae11c09b2a3b92445635a54e73f6d": "s\\,=\\,q\\,\\sigma", "680aeef603dfdb60c6241506bfdc3f50": "\\textbf{x}_{n+1}", "680b0473f3635e0cda039a1e75b05c21": "\\frac{1}{\\prod_{\\alpha>0}(1-e^{-\\alpha})}", "680b8badd9f70687faf6d66fe178a519": "\n \\begin{align}\n\\dot X&=f(X,U)\\\\\n Y &=h(X,U) ,\n\\end{align}\n", "680b9533b7dccd6b1be931fc810eb6f1": " a_D = \\frac{dF}{da} \\,", "680ba47123b0eb2b27bf66d42c85af86": "W^{k,p}(M)\\subset W^{l,q}(M)", "680ba73c2dc2320c972764e9c16cf6ba": "c_{3,1}(\\widehat{a}, T, \\widehat{d})", "680bc80a5d75522f41acda91717f6465": "\\sum_{n=0}^\\infty {a_n}\\cdot 2^n", "680c2fbd5dbf4a642774f6a33bdf9ae8": "A^\\top", "680cce0d7e250fc263761d8197a2d277": "x_1^{},\\dots, x_n", "680cd3db05c5dc5bf0c142018f8948df": "F_2 = (z^{N/2}+1)", "680cdbf55e72af7daaed288e756db14d": "\\frac{d\\mu}{dt}=\\frac{Y_\\beta}{mU}\\beta + \\frac {Y_r}{mU}r + \\frac{Y_p}{mU}p + \\frac{g}{U}\\phi", "680cfae7f3b63028a37219c627c58328": " \\operatorname{const} ", "680d0d4601961d467020cd3b47ac5da9": "\\Gamma=\\frac{m_\\text{initial}}{m_\\text{payload}}", "680d1094bf4782cef1be350496ece683": "g^{\\beta\\delta}(R_{\\beta\\delta;\\varepsilon} \\, - R_{\\beta\\varepsilon;\\delta} \\, + R^\\gamma{}_{\\beta\\delta\\varepsilon;\\gamma}) \\, = 0", "680d3cf3ca694934186f99cb6bad3eac": "\\scriptstyle \\frac{\\partial f}{\\partial t} (\\gamma,\\, t) \\;>\\; 0", "680dac822606427525d3cf92b6cab43b": " h_2 (X_1, X_2) = \\frac{2!1!}{2!}X_1^2 +\\frac{1!1!}{2!}X_1X_2 +\\frac{1!1!}{2!}X_2X_1 + \\frac{1!2!}{2!}X_2^2 = X_1^2+X_1X_2+X_2^2.", "680e0dc638d8e9da9d37b9bfa5f49401": "\\gamma_k = \\gamma_1b^{k-1}.", "680ef7c83bb46965da9f41305b5833dc": " \\varphi = \\operatorname{atan2} \\left( b, a \\right).", "680f3f0f3cb2d7f5d53e93a791c9772c": "y\\le K", "680f46151f98f49509fc99150592b7e7": "\\Omega(k)\\, =\\, \\sqrt{g\\, k}", "680f60fad4410bc1ec384a099a3c5056": "{\\mathbf{}}+S(t)B(t)R^{-1}(t)B'(t)S(t).", "680fc105f99203ddc6ceecf14b79c746": " \\frac{dx}{dt}=\\frac{K}{(t_c-t)^2}=\\frac{x^2}{K}", "680fe34c0da4cbd87132fb48b63bb048": "f^*(x^*) = \\delta(x^*|C^o) = \\delta^*(x^*|C) = \\sup_{x \\in C} \\langle x^*,x \\rangle", "680fea088a227cbc3135ca60f64e4729": "\\frac{1}{T_{p}} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{\\sigma_{p}} \\frac{d\\sigma_{p}}{dt}", "6810d65be9853c40038cdd869456b73b": "- \\mathbf{E} = \\nabla V + \\frac{\\partial \\mathbf{A}}{\\partial t}\\,\\!", "6810e73462aa18e1100aa47524cb9c19": "z_{k-2}", "68111e23a178d1a9e59f2ba66091be8e": "L_1=L_2", "681121adac0a5c5157d794e7e1365280": "\\Phi_f(q)", "6811de369d3f2094ad277dfd6cd90932": "E_{kin,i/f}", "6812395516b21fe68450d8fa677b8529": "ds^2 = E \\,du^2 + 2F \\,du\\, dv + G\\, dv^2\\,", "681254f3a0a19782c20f1b4d865982f3": "\\scriptstyle\\varphi_1", "6812842efb57a263e6c81c9bc68b4baf": " P(IV)(\\mu, \\sigma, 1, \\alpha) = P(II)(\\mu, \\sigma, \\alpha),", "6812968c551ee9b097130cc3210c090a": "T^*M|_{U}\\simeq U \\times \\mathbb{R}^n", "68129f0075345eab8c31a2b4dfaa813d": "\\mathit{x^N - 1} = \\mathit{g(x)h(x)}", "6813033ef72d7df1c237b4750e7c3a71": "\\phi_{e5}", "6813280f28d62dec06159daf20fa14cf": "\\tan\\beta=\\sec\\phi \\tan\\alpha.\\,", "68136f57ec9360508f5e175e4e56d877": "\nB = \\left| \\mathbf{L} \\right|^2 + a^2 \\left| \\mathbf{p} \\right|^2 \n-2a \\left[\\mu_{1} \\cos \\theta_1 + \\mu_2 \\cos \\theta_2 \\right]\n", "681384ead7b1b4f77732632c04868db5": "\\Delta\\varepsilon = \\varepsilon_{s}-\\varepsilon_{\\infty}", "6813ad429e4b3c2405c53bd6dfcd9925": " m(r_0), m(r_1), m(r_2), \\ldots, m(r_n) ", "681430ad7a22efdc9d272731fd104443": "I_{1/2}(z)=\\sqrt{\\frac{2}{\\pi z}}\\sinh(z) \\sim \\frac{e^z}{\\sqrt{2\\pi z}}\\text{ for }|\\arg z|<\\pi/2 ,", "68147c80a8c56aed946e709f056867d8": "e^{-x} \\; {}_2F_2(a,1+d;c,d;x)= {}_2F_2(c-a-1,f+1;c,f;-x)", "681485982dd1c1bb86f21bed1d7ecd50": "\\lambda L + \\mu L^\\prime = 0.\\ ", "68148dec30e2321ed539235f28694fb2": "J_{K,N}(q)", "6814e1ccf7a8e2e30d635d48ae26ba8f": "\\text{winding number} = \\frac{1}{2\\pi} \\oint_C \\,\\frac{x}{r^2}\\,dy - \\frac{y}{r^2}\n\\,dx.", "6814e7f9f1bc710cc8d6d2a6866458fb": "\\mathrm{_{28}^{56}Ni} + \\mathrm{_2^4He} + \\gamma \\rightarrow \\mathrm{_{30}^{60}Zn}", "681500395fa71ed717e167b9a4471dad": "\\chi_i", "68150cb5ebf30d3e0c4d434706d92400": "e_1^2 = -1", "68156a879df9c5350830defda6ffd266": "\\vec{\\xi }\\in \\mathbb{R}^{k}", "68156bd21ef91aaf051954c36d8ff41d": "\nE^{(2)} = \\sum_{k>0} \\frac{\\langle \\psi^0_0 | V_\\mathrm{int} | \\psi^0_k \\rangle \\langle \\psi^0_k | V_\\mathrm{int} | \\psi^0_0 \\rangle}{E^{(0)}_0 - E^{(0)}_k}\n=- \\frac{1}{2} \\sum_{i,j=1}^3 F_i \\alpha_{ij} F_j\n", "6815b91ead0b2d57c91efe8e0f24aca6": "1+\\varphi", "6815b97769753c83dcaa1e78a9fd4e2b": "\\int_{UTM} f\\,d\\mu = \\int_M dV(p) \\int_{UT_pM} \\left.f\\right|_{UT_pM}\\,d\\mu_p", "6815e5a4cd0834d48e4af94fec80db9b": "\\psi(\\alpha) =\\frac {\\partial\\ln \\Gamma(\\alpha)}{\\partial \\alpha}", "6815f278f3d8c311c9d4591ba915c78f": "\\rho(x) (v \\otimes w) = (\\sigma(x) v)\\otimes w + v \\otimes (\\tau(x) w) ", "68160331e66ebd9acaf6437ebdd37499": "\\langle (U^*)^k [ e_ {n+1} ], [ e_ {n+1} ] \\rangle = \\int_{\\mathbf{T}} z^{k} dm .", "681686ff275832f293f728c9831732c8": "z_M = \\sqrt{r_{M}^2 + x_{M}^2}", "6816a7fdf86af63bf25c611fc59b871e": "\\sqrt{24x+1} \\equiv 5 \\mod 6", "6816c1221a94050d888e16d1c95b9cd5": "\\displaystyle{J_{f\\circ g}(a)= J_f(0)\\cdot |g_z(a)|^2.}", "681719f24c273837dd7f0b2a93a655d4": "\nf_Y^{\\text{GIG}}(y|r_1,\\dots,r_p;\\lambda_1,\\dots,\\lambda_p)\\,=\\,K\\sum^p_{j=1}P_j(y)\\,e^{-\\lambda_j\\,y}\\,,~~~~(y>0)\n", "68175869c7aea0fba0211a8e9820a122": "\\displaystyle A+B+C=\\pi+ \\frac{4\\pi \\times \\text{Area of triangle}}{\\text{Area of the sphere}}.", "68177878749f258cbbbab7a314c5b193": "L=mrv_T=mr^2\\omega\\,", "68178b74eb6aa4d65788442492f61e27": "P+Q\\sqrt{a},", "6817aaedd91997d82c92a5e9fd69c6c3": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_1+\\theta_2)\\sin(\\theta_3+\\theta_4) \\, ", "6817e5b4abd0193b64c58773aecfc97b": " D_n(z)=d_n \\prod_{k=1}^n (z-z_k) ", "6817ede1fd205d4793d4f88dcce6630a": "C_{\\alpha \\beta}=\\langle \\psi_\\alpha |\\psi_\\beta\\rangle", "68184205d9d98504122b9a7c06a9c549": "S(a,q,x)", "681875ff7eeeb9f097882f023eb103d2": "0 < \\theta \\le 90 ", "68189f50be4628fe1c3371286095e338": "\\mathbf{F}^*_{i \\pm \\frac{1}{2} } ", "6818b66686dba66a46c14efd6dcfb3c0": "X \\succeq 0 \\Leftrightarrow S = C - B^T A^{-1} B \\succeq 0", "68195d0b2d8e74b90537b8b85edf7620": "\\bar{\\alpha} = \\operatorname{atan2}\\left(\\frac{1}{n}\\cdot\\sum_{j=1}^n \\sin\\alpha_j, \\frac{1}{n}\\cdot\\sum_{j=1}^n \\cos\\alpha_j\\right) ", "6819d71b8aa9d3e42105caedad6c0cb4": "\\mathbf{X}\\boldsymbol{\\beta}=-\\mu^{-1}\\,\\!", "6819ff5508cdd86b53cc26712ee49131": "\n\\frac{1}{2} W^2Nc(c_l\\sin\\phi - c_d\\cos\\phi) = 4\\pi U_{\\infty}(1 - a)\\times\\Omega a'r^2\n", "681a15e2a4a9cff83868b6988c43c27c": "\n g = \\det([g_{ij}]) = \\sum_i g_{ij}~A^{ij} \\quad \\Rightarrow \\quad \n \\frac{\\partial g}{\\partial g_{ij}} = A^{ij}\n", "681a38496b7e48f21dfc974a02dd20b2": "(n-1)(2n+1) ", "681a44120d191057d6b295422a56a305": "\\mathrm{2\\ BaO\\ +\\ air\\ \\xrightarrow {500 ^\\circ C}\\ \\ 2\\ BaO_2\\ \\xrightarrow {700 ^\\circ C}\\ 2\\ BaO\\ +\\ O_2\\ (pure) }", "681a55136eb19c849abc2fabe8a7670d": "\\omega(v_1,\\ldots,v_n)= o(v_1,\\ldots,v_n)\\mu(v_1,\\ldots,v_n).", "681a9df8edcf4e0f3475984a038e6118": "D_{MLD} : \\sum^n \\rightarrow C", "681ac5d6d9d6173a3d01b6443f5ffcd7": "\\mathbf{p_r} = \\begin{pmatrix}\n 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} ", "681ad8aa94765baf8e89b3d2f2ea15e3": "x(S_j) \\leq 2t-1, \\forall j \\text{ where }x(S_j) := \\sum_{i \\in S_j} x_i.", "681aee3eb166411af0300c24050a0f0c": "f_2 = r f_1 \\,", "681b33a5fdfd88485387259e0da02859": "C= -\\frac{1}{2} + \\frac{6 \\ln 2}{\\pi^{2}}(4\\gamma -24\\pi^{2}\\zeta'(2) + 3\\ln 2 - 2) \\approx 1.467", "681b861315f9ffd00a6e633693770f3f": "(x,u) \\ \\stackrel{\\mathrm{def}}{=}\\ (x^{i},u^{\\alpha})\\,", "681ba55a5df0af2da2de613a268537bb": "\\tau H_i M \\to \\mathrm{Hom}_{\\Bbb Z}(\\tau H_{n-i-1} M, \\Bbb Q/\\Bbb Z)", "681bacc982435551617095f48bf6358b": "LK(x,y) = \\delta(x-y)", "681bea21dbc11b02ddefa642f41b7218": "T_b(f)=f\\bullet b-f", "681c1bb410c6304d80ee58bd338ebdc2": "k,\\, \\theta\\!", "681c628ffd0cea7bdf0cf2e8ca4fef5f": "k_{t+1}", "681ca163e6b4c4642938843cfe901b40": "\\tfrac{N}{3}\\le n\\le 2N", "681cc88f33af921ec0e2218e4fbf03b4": "\\tau = \\mu \\dot\\epsilon", "681cd607dabef3ce7462129d352f81ec": " E_a = E_k + E_t ", "681ce02c9c96308b3cf2e8b6818ef606": "\\left(y_1''\\right)^3 = \\left(\\frac{y_c}{y_1}\\right)^3 = \\frac{q^2}{gy_1^3}", "681d36f4f2b1f35f27e2181b82d13689": "x\\in", "681d3d0251ddbd2df7f2d0aede13bb9f": "(1-z)^{-b} = \\cfrac{1}{1 + \\cfrac{-b z}{1 + \\cfrac{(b-1) z}{2 + \\cfrac{-(b+1) z}{3 + \\cfrac{2(b-2) z}{4 + {}\\ddots}}}}}", "681d3f00f696c1b51fa481217e792515": "H=G", "681d6cbf173de332ee2c8f22eaf62253": "p_n=\\frac{100}{N}\\left(n-\\frac{1}{2}\\right).", "681dabbb5746235cb6baa9d9363fd078": "\\boldsymbol\\Sigma_0^{-1}", "681dd9c37ed485d07ef98de52db5c4cc": "f(n)=n+(n-1)+(n-2)+\\cdots+1+f(0).\\,", "681df82ee95b42c115b8f6e1aa15adbf": "C_n\\not\\in U,\\forall n<\\omega", "681e02ea48254b7b2c184bb3f920ac3d": "|A| \\leq |B_n|", "681e1604a512166c9e5312453906cbed": "g\\colon \\mathbb{R} \\rightarrow \\mathbb{R}", "681e49cbcbe4f74a60534989d152bfcb": " \\forall S_1 \\neq S_2 \\subseteq [n]", "681e7dcd052e6502f8a3b4f534a8f27a": "\nx_{ij}^r \n", "681e820f5ca24d1e2428780a50278736": " \\psi_{\\pm}(\\theta) = \\frac{1}{\\sqrt{2 \\pi}}\\, e^{\\pm i \\frac{r}{\\hbar} \\sqrt{2 m E} \\theta } ", "681e83822d971a775664357b3bb217d7": "\\lfloor N / 2 \\rfloor", "681f2b9edde86b2d443ee8adb8052b7a": "\\sum_{k=1}^\\infty \\frac{(-1)^{k+1}}{\\sum_{j=1}^k {F_{j}}^2} = \\frac{\\sqrt{5}-1}{2}.", "681f3d66a828ba2f1df43cb0b9b6f94c": "\\log_{2}4^{n} = 2n", "681f3eda32bac403efb5ac0e25175d83": " x_{i+1} = x_{i} - \\frac {6f {f^{\\prime}}^2 - 3f^2 f^{\\prime\\prime} }\n {6{f^{\\prime}}^3 -6 f f^{\\prime}f^{\\prime\\prime} + f^2f^{\\prime\\prime\\prime}} ", "681f4d5bb203cf1a9df6105e5ab94c7e": "[\\omega_1,\\omega_2,\\omega_3]", "681f5d7d8b76a4510ed955436b22022a": "\\rho_{AB}= |\\Psi\\rangle\\langle\\Psi|", "681f9a0a04b41d496a05efaa0dd572b1": " k_\\mathrm{e} = \\frac{1}{4 \\pi \\epsilon_0} = 1 ", "681fa0c477bb2274b014caabb3025234": "\\scriptstyle x\\,", "681fa93288f983cd8bf8c3f2ecac94c9": "x * y\\ =\\ \\scriptstyle \\text{DTFT}^{-1} \\displaystyle \\left[\\scriptstyle \\text{DTFT} \\displaystyle \\{x\\}\\cdot \\scriptstyle \\text{DTFT} \\displaystyle \\{y\\}\\right].", "6820124d7988c319a7206f32183f825d": "\\nu_{ik} = \\frac{\\partial N_i}{\\partial \\xi_k} \\,", "6820ea2ae817b1aaf438d5585280527d": "P(\\boldsymbol{Z}|\\boldsymbol{W};\\alpha,\\beta)", "6820f4d7fa35f1769b8266fbddcf7d4c": "f = f \\circ \\mbox {id} \\le f \\circ (f^r \\circ f)\n= (f \\circ f^r) \\circ f \\le \\mbox {id} \\circ f = f.", "682172dfd8d25f804fa413a6b170ca84": "\\boldsymbol{\\hat{u}}", "68218831c007cf0e100ea2650182e531": "e \\in B^{\\ast}", "682211704b32beb8b73bab7c93fcddd6": "m+m'", "6823807ece65ca32fc868190c1de8f98": "\\langle B^2\\rangle", "6823865c5cd52e439f523dfde6c3ad99": "\\theta(u,v)", "682395ac2dd8c3575f32a3549862918a": "gate1", "68239bbd3d4cb7568f853d7d10d99c43": "\\Diamond p", "68246c696384981f3c1b31c3188c3f40": "G^{p^k} = \\langle g_1^{p^k},\\ldots,g_d^{p^k}\\rangle", "6824ad211e4b028e35f99691c3a66dfc": "\\left(1 - \\frac{it}{\\lambda}\\right)^{-1}", "682528484b31053f63a0e8653e9d31f0": " \\Rightarrow |\\psi \\rangle = \\frac{1}{2}\\bigg[|0 \\rangle \\bigg(|f_k \\rangle |f_k' \\rangle + |f_k' \\rangle |f_k \\rangle \\bigg) + |1 \\rangle \\bigg(|f_k \\rangle |f_k' \\rangle - |f_k' \\rangle |f_k \\rangle \\bigg)\\bigg]", "68254b3f596943ec976cd4fafe9f7bf8": " \\qquad \\qquad \\mathrm{H}_e \\psi_{e,\\mathbf{x}}(\\mathbf{x}) = E_e(\\boldsymbol{\\kappa}_e) \\psi_{e,\\mathbf{x}}(\\mathbf{x}),", "6825994293a3b63eb7464155885b7be3": "\n\\frac {\\mbox{GTPase}*\\mbox{GTP}} {\\mbox{GTPase}*\\mbox{GDP}} =\n\\frac {k_\\mbox{diss.GDP}} {k_\\mbox{cat.GTP}}\n", "6825a8e8fed565b9a366e11d8b57fe82": " p_{BA}(t) = t^{n-m} p_{AB}(t).\\,", "682606998e51361d345fa07e68c1d1de": " \\int \\frac{dx}{x^{2} R}=- \\frac{ R}{cx}-\\frac{b}{2c} \\int \\frac{dx}{x R}", "682646de4a776538aaba84c16d411bff": "\\left\\{ y~\\backepsilon~(y\\succcurlyeq x)\\land\\lnot(x\\succcurlyeq y)\\right\\}", "68268620bd656b2c6b7f17783f24a0f0": " \\Pi_A", "6826c8cc13c66ef083cc340db6805edc": "p_{\\tfrac{1}{2}1} \\leftarrow 64x^3+576x^2-64x-64", "6826e1322c64bc62c9a137ef63699878": "L=\\bigoplus L_i:\\mathcal E^\\bullet\\rightarrow\\mathcal E^\\bullet", "6826f181b31492d9d81260408f6f353e": " \\left ((r+c-1)(r+c-2)+(2-\\gamma )(r+c-1) \\right ) a_{r-1} +\\left ( -(r+c)(r+c-1)+(\\alpha +\\beta -1)(r+c)-\\alpha \\beta \\right ) a_r=0", "68274cc0f46589370fd54bcc1b4c4023": "E=\\frac {1}{2} R N^2 I^2 f \\left( \\xi,\\delta\\right)", "68278140484adb3fc62c4685c505dafd": "\\!\\,p=x^2+2y^2\\text{ if and only if } p=2 \\text{ or } p\\equiv 1, 3 \\pmod8,", "6827b30c254a3497bae29e7242d17db3": "s=4", "6827d89885abb0fda0c3713800f76b7c": "(0,\\pm 1,0,\\pm 1)", "6827f130d860c44ab8d8085047ee2cae": "\\scriptstyle V_\\mathrm{peak}", "6828046cb0fc01564c74aded5a90824d": "\n\\Delta \\varphi = 2\\pi \\frac{m}{n}\n", "6828497e0d719c2b712044af3731d1fe": "\\Gamma=", "68289fed4035fe97297b5ce12d21511f": "\\alpha(k) = 2^k", "6828dfe38eb032014b6351ad13fcb3fd": "\\tilde{\\nu}_\\mu", "6828f4b0e73521bdfe9936b2bc6f7017": "2t^3-3t^2+1", "6828feb224035a25980fcbdb76126b02": "N^{2}", "68291209d15b2b9dfccd6bf23e360c2c": "\\displaystyle{\\|P^\\perp(Y_{n+1})\\|^2\\le \\left({m-1\\over m}\\right)\\|P^\\perp(Y_n)\\|^2.}", "68291cc80834999af6754d8a55dd4133": "G_{\\alpha\\beta} =\\partial_{\\alpha}\\mathcal{A}_\\beta-\\partial_\\beta\\mathcal{A}_\\alpha \\pm ig_s[\\mathcal{A}_{\\alpha}, \\mathcal{A}_{\\beta}]", "68292d2017e9ed978191f05c7133bf4c": "\n\\frac {\\mathrm d} {\\mathrm dt} (\\vec{x}+\\delta \\vec{x}) \\approx\n\t\\vec{v} + \\nabla \\vec{v} \\cdot \\delta \\vec{x}\n", "68292e860d84dbd90de5b06c5a44e88c": " \\sqrt{N} ", "6829ecd80494128adb7f7d15d445bd4e": " \\left( \\frac{\\partial \\mu}{\\partial T} \\right)_{p,n} =- \\left( \\frac{\\partial S}{\\partial n} \\right)_{T,p}", "682a63da4d1659caa395d17dceb94209": "E_0 = mc^2 \\,\\!", "682a8b815b24d979231c4c9d1311e50a": "m_i - \\left(i+1\\right)\\cdot m = 1", "682ab2900ecd3f319f111df673c3f9e0": "x\\in A_i.", "682ab4a3ca13a7de800fe3ad3e830dd9": "C = \\{ i | a_i(x) = 1 \\}, ", "682ac301cef8301d20efb3f62aed3452": "\n\\left.\n \\begin{matrix}\n G &=&3\\underbrace{\\uparrow \\uparrow \\cdots\\cdots\\cdots\\cdots\\cdots \\uparrow}3 \\\\\n & &3\\underbrace{\\uparrow \\uparrow \\cdots\\cdots\\cdots\\cdots \\uparrow}3 \\\\\n & &\\underbrace{\\qquad\\;\\; \\vdots \\qquad\\;\\;} \\\\\n & &3\\underbrace{\\uparrow \\uparrow \\cdots\\cdot\\cdot \\uparrow}3 \\\\\n & &3\\uparrow \\uparrow \\uparrow \\uparrow3\n \\end{matrix}\n\\right \\} \\text{64 layers}\n", "682aef6011d687df5fab1accbb84af32": "\\textstyle p = 0.5. ", "682af88e91089d69344d8e6ea0101c3d": "\\sum_{l}\\hat{A}^{\\dagger}_{l}\\hat{A_{l}} = \\hat{I}_{S}.", "682af975079c759e2381759b8093e971": "c_p\\mathbb{E}([M]_t^{p/2})\\le \\mathbb{E}((M^*_t)^p)\\le C_p\\mathbb{E}([M]_t^{p/2}).", "682afa28512a76afd0f7debbdba9b32c": "\\mathbf{D} = \\varepsilon\\mathbf{E}\\,\\quad \\mathbf{H} = \\mathbf{B}/\\mu", "682b0b7c95143b4210a33a9faa666957": "\n\\sin[2\\pi ft + \\phi(t)]\\ =\\ \\underbrace{\n\\sin(2\\pi ft)\\cdot \\cos[\\phi(t)]}_{\\text {in-phase}\n}\\ +\\ \\underbrace{\n\\overbrace{\n\\sin\\left(2\\pi ft + \\tfrac{\\pi}{2} \\right)}^{\\cos(2\\pi ft)\n}\\cdot \\sin[\\phi(t)]\n}_{\\text {quadrature}}.\n", "682b2bb3e8ed17ac432705e6033c23e4": "\\min_{\\gamma \\in \\Gamma(\\mu, \\nu)} \\int_{\\mathbf{R}^2} c(x, y) \\, \\mathrm{d} \\gamma (x, y) = \\int_0^1 c \\left( F_{\\mu}^{-1} (s), F_{\\nu}^{-1} (s) \\right) \\, \\mathrm{d} s.", "682b41c11cfb8d2db9db3d49e0fc80ac": "\\,\\alpha_1", "682baaf8ebee13fab1dba820aeb71fe3": " [I|\\Delta\\mathbf{r}|^2-\\Delta\\mathbf{r}\\Delta\\mathbf{r}^T].", "682bdf41c64199293e560bd903a9a5e0": "f(c^-) = f(c^+)", "682c4920690e7fe9afab1f4071301b8d": "\\{p_{2}\\}", "682c61d9214ade2ba4a86e5b21c3ed63": "f(x,q)\\,\\!", "682c6669904bc4bd1502d756572bbdab": "\\frac{\\eta}{s}\\approx\\frac{\\hbar}{4\\pi k}", "682cb2df61bfab3cfe69c9e0e404affb": "C_n(x; \\mu)= {}_2F_0(-n,-x,-1/\\mu)=(-1)^n n! L_n^{(-1-x)}\\left(-\\frac 1 \\mu \\right),\\,", "682cbd8ce2d4d187ceb4507f714adce7": " \\zeta^a ", "682cd2fc60d78c68c95eefddd9cb47bb": "f_{WC}(\\theta;\\mu,\\gamma) =\\frac{1}{2\\pi}\\left(1+2\\sum_{n=1}^\\infty\\phi_n\\cos(n\\theta)\\right)", "682cdc1b3310a36af3f98cb31cd0b7b2": "\\hat{y}=E\\{y|x\\}=\\beta x.", "682d2d6c6265678c975188fe26e60d8c": "(x_{1},x_{2})", "682d36a39cc50094b461aa48b003ccee": "n(p;H)\\approx \\sqrt{2H\\ln\\frac{1}{1-p}},", "682d3fdc64b02d6989961c336a4038a0": "\\dot m = \\iint_A \\rho \\bold{v} \\cdot {\\rm d}\\bold{A} = \\iint_A \\bold{j}_{\\rm m} \\cdot {\\rm d}\\bold{A} ", "682d541366bca0fc02bfda9f7e597c0a": "x^2-w^2=1", "682d5fa74231595c3e109c1992173159": "\\int x^p (1-x)^q \\, dx = B (x; 1+p, 1+q),", "682d93f8631b21e32c94abd79645df2f": "\\operatorname{FPC} = \\sqrt{\\frac{N-n}{N-1}}.", "682d9675806b48c4a4fbd9b6109fdaa2": "\\frac{g(T,P)-g^\\mathrm{gas}(T,p^u)}{RT}=\\ln\\frac{f}{p^u}", "682db65f263faa91477ac2bcb25c26bc": "\\Theta(\\log |V|)", "682e84a6dc05dfa762e677bca884a2e3": "U{}^3_5", "682eb135cff8b1597ecca74143c105c5": "(\\lambda)", "682eb635a04fbabbe8b3011e96e742dd": "F=\\frac{\\sum [\\frac{c'+((W/b)-u)\\tan\\phi'}{\\psi}]}{\\sum[(W/b)\\sin\\alpha]}", "682eb9fa0268cb566898ad0628dc29d8": "\\Delta = b^2 - 4 a c \\, . ", "682ef1be9f8c59755b6a466f0deb65cc": "\\prod_a^b (1+f(x)\\,dx) =\\exp\\left(\\int_a^b f(x) \\, dx\\right),", "682efd48cb60a8fc66d6dc29afb250b9": "\\Gamma \\, : \\, (x,y) \\to (2x+y,x+y) \\bmod 1.", "682f1e29e6bdbab1b95ec27972fac7da": " \\mathbf{V}^{*} \\mathbf{V} = \\mathbf{L} \\mathbf{L}^{*}, ", "682f4b8e6b4642dee275190b6651c203": "\\mathrm{S^{\\gamma}}", "682f5cc8d2a225b6f212181bdb877d5d": "f(f(...(f(e))...))", "682f6df1691327076f7a47dded1abd4a": "\\Psi_S=L_Si_S-Mi_P", "682fc1d5035a8ed5f4928b8d37bd9196": "\\swarrow", "682fd60db5451317d3501c5dc033a5d3": "\n\\Delta U = 4 \\pi G \\rho \\,\n", "682fded2fdc4e1993b1a79e941d54bc4": "\\psi_1 = C \\sin(k' x) + D \\cos(k' x)\\quad", "682fea87335a31df180c386f07d2bef5": "X\\sim GD(\\alpha_1,\\ldots,\\alpha_k;\\beta_1,\\ldots,\\beta_k)", "682ff9b05a34b6047303ee39d32a559f": "\\int x^2 dx = \\tfrac{1}{3} x^3.", "683016f88698889379c1100d2d0d0798": "D(s)", "683020c56e1cc1f41b522ea3eeaa0f08": " = -\\frac{4 \\pi r_p^3}{3} \\frac{V_t^2}{r}\\rho_f .", "683049481dfc40b038b3613b984b7783": "\n E_x= k_y \\cos k_x x \\sin k_y y \\sin k_z z\n ", "68304e71c32c6e4fda6162321b51a469": "\\mathbf{L}_i=\\mathbf{S}_{1i}+\\mathbf{S}_{2i}", "683073f162a30d37484f90d3c71f11d6": " a_1 \\mathbf{e}_1 + a_2 \\mathbf{e}_2 + \\cdots + a_n \\mathbf{e}_n = (a_1 ,a_2 ,\\ldots, a_n) , \\,\\!", "68311cea046cdf94ee649cef2ec961ec": "\n\\begin{align}\ny & = mx + b \\\\\ny & = \\frac{(\\Delta y)}{(\\Delta x)} x + b \\\\\n(\\Delta x) y & = (\\Delta y) x + (\\Delta x) b \\\\\n0 & = (\\Delta y) x - (\\Delta x) y + (\\Delta x) b\n\\end{align}\n", "68314af75f47e796395d731ee285fc86": "\\frac{|SA|}{|SB|}=\\frac{|AC|}{|BD|} ", "683156242e1d152a5e5fc401c3c43cd2": "(r \\times r)", "683168aa1dc061d9713008459f69b020": "\ni\\frac{\\partial}{\\partial t}u=-\\frac{\\partial^2}{\\partial x\\,^2}u+g(|u|^2)u,\n\\qquad\nu(x,t)\\in\\C,\\quad x\\in\\R,\\quad t\\in\\R,\n", "68319d59ae96bbc86cca930f07ca9379": "{]-\\infty,+\\infty]}", "6831bdb17d2bd0780af70686955e6315": "\\mathbb{A}_4\\times\\mathbb{Z}_2", "6832174a3013a71cef777132c2c218d9": " e^\\pi = (e^{i\\pi})^{-i} = (-1)^{-i},", "68322f1509fc1a010ee84df37201ee6a": "\\mathrm{not}~p \\equiv \\mathrm{not}~q \\or r", "6832301800968d9a27cf0e8362ed01af": "\\sqrt{abcd}/s", "68325b7703d79fc61b7526bb7b211ff4": "p\\Delta V\\;", "68327f179b16f6314777b781bc586661": "|a_{n+1}+a_{n+2}+\\cdots+a_{n+p}|<\\varepsilon", "6832b4b1748ac06e7068cc54bf9d625a": "(x-3) (x-2)^6 (x-1)^3 x^4 (x+1)^3 (x+2)^6 (x+3),\\ ", "6832da3f173382915fccf1cb375e9f23": " d^2(O,P) = \\| \\bold{p} \\|^2 = \\sum_{r=1}^n p_r^2 ", "6832df8d4d9cc978737672b5eecc0ec6": "f=\\frac{m}{2} \\sum_i\\dot{x}_i^2-V(x).", "683311871ae221a1a1f234e0dde6c6f1": "z>0 ", "68331d01c1ca8ec5904351eae34e30c0": "\\Sigma_0, \\Sigma_1", "683337b235f5d7f3df5510ac122c73b9": "_{\\Rightarrow\\,}\\!", "683363779c05af0264ff8730abd29eae": "K=1, -3,", "6833b39e0b8da8875ee98942e4c08b9b": " \\begin{bmatrix} \\ln x \\\\ \\frac{1}{x} \\end{bmatrix} ", "6833c9ff9154b33e134788005f150d0b": "\\lambda_i \\,\\!", "6833cdc256de9cabff1c3668528c0b30": "H(i,j)", "6833f4eaccfb60d5c13fdf6b6cc30aef": "\\sqrt[3]{x}", "68342c08791759605e1b55feab783792": "S_o\\ ", "6834882bd8890b828a8aa7bb13952dea": "\\displaystyle{[J_m,G_r^\\pm]= \\pm G_{m+r}^\\pm}", "68348bd46de4f5f2d3a134d68d39ba7b": "1_F = \\varepsilon F\\circ F\\eta", "6834d37e6d80dd96e139c4e43bdebf88": "_{s.5.right\\,}\\!", "6835031ea459c71955e4b260529c1205": "\\operatorname{Li}_5(\\tfrac12) = -\\zeta(\\bar1, \\bar1, 1,1,1) \\,.", "6835742e2663f44242d081b7682ecb59": " \\mathbf{E} ( \\mathbf{r}, t ) = \\mathbf{E}_0 \\cos( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} + \\phi_0 ) ", "683590a0c6a1c3a77515ee29d578dbb9": "\\nabla^2", "6835d284b7e817818c68593137d78e1a": "\\Vert g_n \\Vert \\le 1", "683605b5e3204e2b0f350bb707361306": "E=\\frac{1}{2}\\,\\left(I_1\\omega_1^2+I_2\\omega_2^2+I_3\\omega_3^2\\right)", "683651ee604fa3412ab9fe6de28049ba": "0 < i 1. \\\\\n \\end{cases}\n ", "683be4a50ed82b8ad8804fa18cdc3202": "\\frac{1}{8\\pi^2}\\int d^4x \\sqrt{-g}\\, G = \\chi(M)", "683c1312220507b3ba280ee9a328cc2e": "A\\mid B\\Leftrightarrow\\forall a\\in A\\hbox{ and }b\\in B\\colon ab\\not\\in E.", "683c9de648ab67210c53dd16c1784fb3": "\\sigma \\propto d^2\\,", "683d1a0aa9f396d69caec811c453bd7b": "(E)", "683d222dfae0befb784c7aaeb4714d09": "2ax + b = \\pm\\sqrt{b^2 - 4ac}", "683d2f6bcfb9f0410353169f34c98614": "J_4~", "683d5c4e8f31d62029a16f74100c9eef": " {1 \\over R} \\triangleq {i \\over v} = \\frac {1-A}{R_1} + {1 \\over R_{in}} ", "683da6825018e46e0c4d3468f5a67221": "\\mathcal{G}(\\mathbf{x} ,\\tau|\\mathbf{0},0)", "683df6631fa97dbdecb891080c5a3f36": "g(x) = \\frac{\\langle Tx, x \\rangle}{\\|x\\|^2}, \\qquad 0 \\ne x \\in \\mathbf{C}^n.", "683e052cfc54eebf3eb6d5317ab0ee2e": "\\mathcal{F}\\{f\\star g\\}=(\\mathcal{F}\\{f\\})^* \\cdot \\mathcal{F}\\{g\\},", "683e286faf2728de470183a8744bf2da": "f(p,q)\\rightarrow pq/(pq + (1-p)(1-q))", "683e45d003eea8fa199d2a183ac030ad": "X_s(f)\\ \\stackrel{\\mathrm{def}}{=} \\sum_{k=-\\infty}^{\\infty} X\\left(f - k f_s\\right) = \\sum_{n=-\\infty}^{\\infty} \\underbrace{T\\cdot x(nT)}_{x[n]}\\ e^{-i 2\\pi n T f},", "683e6644de1e4b063e9788db58d489b7": "\\beta_0 = \\sqrt{\\frac{2}{K^2}\\left( K\\sum_{k=0}^{K-1}\\sin ^2\\left(\\omega t_k-\\theta_0\\right) -\\left[\\sum_{l=0}^{K-1}\\sin\\left(\\omega t_k-\\theta_0\\right)\\right]^2\\right)}.", "683e6bd8bf009d1ddb98a76a3bf0b10f": "\\textstyle \\textbf{R}^d", "683e860d15f7769064b37f5f208f4c6e": "\\varphi''(x)", "683e98e1c7a59585ca81401440cea503": "\\varphi_1(\\beta) = \\varepsilon_\\beta", "683ee8a3f7aa9fdfa1606bf3e4f4db90": "M^{0.11}", "683f07db35dee4cb5a7405bac09fc8bd": " \\forall x (\\neg(\\phi \\lor \\psi) \\lor \\forall z \\rho)", "683f4f479bab9cb4f1111b674ed2db07": "\\displaystyle{\\overline{H^\\varepsilon f} = - u^{-1} H^\\varepsilon( u \\overline{f}).}", "683f80232b19c64f20f895b97978fd3c": " (\\lambda x.f\\ (x\\ x))\\ \\operatorname{get-lambda}[x, x\\ q = f\\ (q\\ q)] ", "683fa57ed8862bc0bc828f1fb68089a0": "1.\\overline{571428}", "68401b92157ce3740de96b0366e9ed60": "u_{-}", "684040d8d243e96256167104809dd351": "b=\\sqrt{\\frac{\\sum_{i=1}^t c_i^2}{\\sum_{i=1}^t c_i^2/n_i}}. ", "684079f6ed569516e338da165147d62e": "f\\circ h=h\\circ g", "68407d798026a8bb34d5c3e8582f15ff": "R_2(\\xi,x)=\\frac{(t+1)x^2-1}{(t-1)x^2+1}", "6841408057a90f147c6b2621d25c8e3f": "g_{\\mu\\nu} \\,", "68418a18716121d0c874663f9fe18bc8": "s(t_2)-s(t_1) = \\int_{t_1}^{t_2}{v}\\, dt, ", "68419a17efc862ab08fc6f97795b3a48": "\\ \\Phi (m) = A e^{(m-m_0)^2/2{\\sigma}^2} \\,.", "6841bb76261e1ebe7b08c4591f7ce0c7": "\\displaystyle n=3\\,N-K", "6841d2635a50a0626f0726d652015c29": "4.\\overline{6}", "684281d894ae7ae63dd6b753b5f0b7e1": "\\dot \\epsilon_\\infty = \\frac 1 {2 \\lambda}", "684335e74ebd3522454538824f8f29f2": "\\bigcup_{e,f\\in E}", "684343b0c33143c8f6da9b442e711310": " p_L(\\lambda) = \\lambda^n + a_1\\lambda^{n-1} + \\dotsb + a_{n-1}\\lambda + a_n\\,", "68434e9ad269b0f88b4bab037208105d": " R(\\pi)\\phi(x) \\phi(-x) \\, ", "684374de45633f19ab4a432080052860": "q\\in S", "6843afd8d533ee6c044a46b4ac6a8d3f": "M_0 \\times \\{0\\}", "6843cc53d509508538756e7f2db8f249": "\\mathrm{Q}", "684401eccd67892a7ccd889ef7a49151": "I_{1}(\\sigma_{xx}\\sigma_{yy} - \\sigma^2_{xy}) - I_{3}", "684457e03a22a04f621aa0b351e094b1": " \\nabla^2 \\psi = k^2\\psi ", "6844a047fc10ff924f530cf9e53dcb55": " \\frac{\\sec^2\\theta\\,\\Delta\\theta}{2} \\le \\frac{\\tan(\\theta + \\Delta\\theta) - \\tan(\\theta)}{2} \\le \\frac{\\sec^2(\\theta + \\Delta\\theta)\\,\\Delta\\theta}{2} ", "6844a3ad56c80b87f7881b1f0a811af9": "I_{\\mathcal Q}(-)\\colon Q\\to Q", "68452a5d040f6eca7f1a88a0051867a5": "M_{earth}\\,", "6845a8fd91329f0716f3d072e7c8d9cb": "\\sigma_{\\mathrm e}", "6845bd3dd0f65f90849b4c1538854ede": "\\cos \\theta_W = \\frac{M_W}{M_Z} = \\frac{|g|}{\\sqrt{g^2+{g'}^2}}", "684640b22fc73b547c2c25c9f6f57b5d": "C = 64r_{444}+ 48r_{443} + 32r_{442} + 36r_{433}.", "6846922cbfeaf178dc1763cb89e24670": "V_P\\,", "6846ec03b51aeed6832e65091743aa84": "Q = \\begin{pmatrix} {*} & {\\mu\\over 4} & {\\mu\\over 4} & {\\mu\\over 4} \\\\ {\\mu\\over 4} & {*} & {\\mu\\over 4}& {\\mu\\over 4}\\\\ {\\mu\\over 4}& {\\mu\\over 4}& {*} & {\\mu\\over 4}\\\\ {\\mu\\over 4}& {\\mu\\over 4}& {\\mu\\over 4}& {*} \\end{pmatrix}", "68471f6c1fae82cede22b14185483f9e": "\\liminf_{x \\to \\infty} \\frac{A(x)\\sqrt{\\log x}}{\\sqrt{x}}\\leq 1", "68473bd4a73fe27a0664f757fab22e2a": "\\diamondsuit, \\heartsuit, \\clubsuit, \\spadesuit, \\Game, \\flat, \\natural, \\sharp \\!", "68476db83cdeac4dda12c104e630986b": "\n\\langle m |\\psi_\\alpha|n \\rangle\\langle n |\\psi_\\beta^\\dagger|m \\rangle =\n\\delta_{\\xi_\\alpha,\\xi_\\beta}\\delta_{E_n,E_m+\\xi_\\alpha}\\langle m |\\psi_\\alpha|n \\rangle\\langle n |\\psi_\\beta^\\dagger|m \\rangle.\n", "68480051fdb56c837e8111f570f05be2": "\\prod _x x^a = C\\, \\Gamma (x)^a \\,", "68480457c91263653688feb4d7a2cbd5": "x = (0,x')", "68482ad07f1b5fbc7283e31d8ac3a910": " \\int_a^b \\phi _i^* (x)\\phi _j (x) \\, dx =\\delta_{ij},", "68485beb73218ae956f201c1f399c63c": "\\frac{100}{x + 1} %", "68487b14388972a3515e253ed2a2d02d": " \\ \\alpha_{geo} ", "6848865e503020ed4e8a66268c968964": "f=g(x)\\text{ for }x\\leq 0, \\quad f_{y}=0\\text{ when }x>0", "6848c71cda235e6c315615a9ef71bb5a": " \\delta_i ", "6848ff622ad8f829da8bf5ed5cd80048": "P_{I, M} = P_{I, M'}", "68491c85ba1b09686ae5b0fa3c0a90b7": "\\scriptstyle P(\\alpha) \\,", "68496003794a0a025e8e68d56adacc31": "\\textstyle j-i = g(2l-1)+r", "6849613b8775b8cda79ce265f9affdd0": "\\displaystyle \\alpha_n=\\arg \\min_{\\alpha} f(x_n+\\alpha s_n)", "684a023f31743a65a1f692dcb95cdcbe": "c_j = \\frac{\\displaystyle \\prod_{i=1}^{hype} \\cos^2 \\alpha_i}{\\displaystyle \\sum_{j=1}^{config} \\prod_{i=1}^{hype} \\cos^2 \\alpha_i}", "684a364e0098c277b95ae0326b2e0f10": "\\operatorname{Ext}_R^1(A,B).", "684a55fe554b0806fb8ea766c959bb94": "P_r = 3R_r^{'}I_r^{'2}", "684a7db9d6a279bc504ee97ea56e16f0": "V_{L1}=V_i", "684a82125480261f0a1ba2c1115d2a83": "{B}_{\\,4}", "684abf24217c37e795f12c1bae0a6095": "\\rho = \\alpha \\cosh(nt + \\epsilon) + \\beta \\sinh(nt + \\epsilon)", "684ad7c113675f1ffbc3c92a2a06d359": "\\beta_0,\\beta_1,\\beta_2,\\dots \\,.", "684af2872090fb660df04e208600e6ec": "\\forall p \\forall x \\, (x \\notin p)", "684af6c4fe22c60379664787bb4bc9d7": "\\mathcal{F}^2(f)=f(-t)", "684b191d8dec246584d38e7d40f7e0da": "1 \\,+\\, 3\\left(\\frac{1}{9}\\right) \\,+\\, 12\\left(\\frac{1}{9}\\right)^2 \\,+\\, 48\\left(\\frac{1}{9}\\right)^3 \\,+\\, \\cdots.", "684b628f388ccfc8a698e155ca7373df": "\\operatorname{Ber}(X^T) = \\operatorname{Ber}(X)", "684bcdda058c9a3da6aeaaf1503320e5": " \\Delta_0 = 1 ", "684bd77987a3b78d2e1a8377d29dac27": "y(t) = \\sin\\left( \\theta(t) \\right) = \\sin(\\omega t) = \\sin(2 \\pi f t).\\,", "684c19fc12eb08aae3717fc5485f61b4": "D_n(u + \\alpha/u,\\alpha) = u^n + (\\alpha/u)^n \\, ; ", "684c7b46d5b6771a4054cf06c9442d9b": "Y_i=\\frac{X_i-\\hat{\\mu}}{\\hat{\\sigma}}.", "684cf5b9b370ba100d237ce29727348e": "\n\\mathbf{A} \\ast \\mathbf{B} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{A}_{11} \\otimes \\mathbf{B}_{11} & \\mathbf{A}_{12} \\otimes \\mathbf{B}_{12} \\\\\n\\hline\n\\mathbf{A}_{21} \\otimes \\mathbf{B}_{21} & \\mathbf{A}_{22} \\otimes \\mathbf{B}_{22} \n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array} {c c | c c}\n1 & 2 & 12 & 21 \\\\\n4 & 5 & 24 & 42 \\\\\n\\hline\n14 & 16 & 45 & 72 \\\\\n21 & 24 & 54 & 81\n\\end{array}\n\\right].\n", "684d1b587d7df661fc22babd86e4f216": "\\frac{d A^*}{d t} = J(t)a^* - k_{A^*}A^*", "684d2458b7154226f1a2b3910eea38a6": "\n \\frac{\\partial f}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\frac{\\partial f_1}{\\partial f_2}~\\frac{\\partial f_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} \n ", "684d392db8ab0f40ff6ec3392303e007": "S^7\\hookrightarrow S^{15}\\to S^8.", "684d52a5c550ea9dbf48e116c2f13894": "\\hat{\\mathbf{E}}= E(r)\\mathbf{\\Phi}_{lm}", "684d53ddc1ff0744340ab68467467c31": "\n \\underline{\\underline{\\mathsf{C}}}^{-1} = \\frac{1}{\\Delta}\n \\begin{bmatrix}\n C_{11} C_{33} - C_{13}^2 & C_{13}^2 - C_{12} C_{33} & (C_{12} - C_{11}) C_{13} & 0 & 0 & 0 \\\\ \n C_{13}^2 - C_{12} C_{33} & C_{11} C_{33} - C_{13}^2 & (C_{12} - C_{11}) C_{13} & 0 & 0 & 0 \\\\\n (C_{12} - C_{11}) C_{13} & (C_{12} - C_{11}) C_{13} & C_{11}^2 - C_{12}^2 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\frac{\\Delta}{C_{44}} & 0 & 0 \\\\\n 0& 0 & 0 & 0 & \\frac{\\Delta}{C_{44}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\frac{2 \\Delta}{(C_{11}-C_{12})}\n \\end{bmatrix}\n ", "684d60b73502bda7771d59c45f5fa5e1": " H(s) = \\frac { s C_2 R_1 } {C_1 C_2 R_1 R_2 s^2 + (C_2 R_1 + C_2 R_2 + C_1 R_1) s + 1 } ", "684d7f4ef9d4c044e0be2c3092ab88e4": "\\operatorname{ker}(T) = \\operatorname{ran}(T^*)^\\bot.", "684e63ffe7069028d1dadf0f0efc7853": "E\\left[ x_i^4\\right] = 3\\Sigma _{ii}^2", "684e826d50b20e89447e91ec69faff34": "\\liminf X := \\inf \\{ x \\in Y : x \\text{ is a limit point of } X \\}\\,", "684f1061f31b74e48deba9130e78aecb": "\\ell(M_n)", "684f4fbdb759952ff9938bc763188d3f": " \\rho(x,u,u_{1}) = 1 + u_{1}u_{1} \\,", "684f5ee478ce99acf0b247e91a83ff25": "Q z = A^+ A A^+ b = A^+ b = z", "684f87b2fb62575b2d6bd4dd8da76b3b": " -t = a x + \\frac{a d}{c} ", "684fd0bf1470c572b871fd002c7e342e": "\n F_c = -2\\Delta\\gamma\\pi R\\,\n ", "68500b17a956dd205cfbd221cba8cdd8": "\\phi_{i+2}", "6850105bcdaa0443940207192672bc44": "c_{j,k} = \\operatorname{Res}_{z=0} \\left(\\frac{a_{-m}}{z^{m+j+1}} + \\frac{a_{-m+1}}{z^{m+j}} + \\cdots + \\frac{a_j}{z} + \\cdots\\right) = a_j,", "685013d02734d51398f165843196b741": "n=1,2,3,\\ldots", "685022c5c9fa397812535d9303c35b55": "\\mathbf{z}(t)", "68509c1a553949ac1904d358c3ae98d5": "-\\mathfrak{a}_+", "6850e5f646d105fccbdfcf6455bc065c": "\n\\begin{align}\nI_1\\dot{\\omega}_{1}&=(I_2-I_3)\\omega_2\\omega_3~~~~~~~~~~~~~~~~~~~~\\text{(1)}\\\\\nI_2\\dot{\\omega}_{2}&=(I_3-I_1)\\omega_3\\omega_1~~~~~~~~~~~~~~~~~~~~\\text{(2)}\\\\\nI_3\\dot{\\omega}_{3}&=(I_1-I_2)\\omega_1\\omega_2~~~~~~~~~~~~~~~~~~~~\\text{(3)}\n\\end{align}\n", "6851413829ad82c0dc823cde24804c26": "\nGT = K \\cdot V\n", "68515f534d5b5452a3eace630020457d": "n!!", "68516897c7419de15c938759c8abace2": " A= sI + (A-sI)~,", "685168cb1a5ab453acf8bdcdae7df936": "P_n = h^0(V, K^n) = \\operatorname{dim}\\ H^0(V, K^n)", "68518101d8a03cfce2784bb69080e15c": " \\tfrac{Z}{\\beta} ", "6851a07d03e93997931d58b03c102917": "q_i=\\frac{\\partial \\bold{U}}{\\partial Q_i}.", "6851cd4e9a07d6cc7fe770e54c1d59f2": "P = \\frac{E}{K} = \\frac{(X+D)}{K} = \\frac{X}{K}+\\frac{D}{K} ", "6851d84e41a9aa5b7ac3cff0674e4f9e": "g(n) = \\Omega(n^c)", "6851e69e19ffdcfe3a7b209fd010137c": "\\text{PI}=1000\\times\\frac{\\sqrt[3]{mass}}{height}", "68520d69990e1a31446818d14e659795": "\nv_a \\geq 0,\\;x_{ij}^r \\geq 0\n", "68521c77521906a7d23ed7b503b64fd3": "\\operatorname{E}[\\ln X] = \\psi(\\alpha) - \\psi(\\alpha + \\beta)\\!", "68527b70c9fce52c3c36d36e7fd8421e": "0^{(\\lambda)}", "6852885385c6c242c022282cd1a47592": " \\hat{B} ", "68535a2929e6badc51b8511bb99d9440": "C\\operatorname{-}\\min", "6853b8b8017aae6fb64201e7108027e3": "\\{ V_i \\}", "6853d5e8c62b3101270af7138abd2a52": "V_0,V_1,\\ldots,V_b", "6854053cb9b0e180aa0ac524b72ad86c": " (f\\in A) ", "685413f5b4c21573ed3038e7838ca41c": " V \\times \\omega =v \\nabla p_0 ", "68542e363501dab9b55a05905e83a4a4": "Y(t+\\Delta t)", "6854750c7276126aa3388a3b7344d7d2": "\\{\\begin{smallmatrix} k\\\\ ij \\end{smallmatrix}\\}", "685477a59d54388c044d6c198f779228": "\\begin{array}{l}\n \\left( {\\frac{\\partial \\mu _i / T}{\\partial T}} \\right)_P = \\frac{\\partial\n}{\\partial T}\\left( {R\\ln x_i } \\right) \\Rightarrow R\\ln x_i = -\n\\frac{H_i\n^\\circ }{T} + K \\\\\n \\\\\n \\end{array}\n", "68548904b82d49dad0275b8e6ecd4462": "\\{\\sqrt{2}\\sin(2\\pi n x) \\; | \\; n\\in\\mathbb{N} \\} \\cup \\{\\sqrt{2} \\cos(2\\pi n x) \\; | \\; n\\in\\mathbb{N} \\} \\cup\\{1\\}", "6854b056ed61cc6d5a199d1f093227ca": "t_r =", "6854e7a9d81ffedd5566c93bbded522f": "d\\geq 3 ", "685553812b7f8dae48572e09ce96729d": "\\beta_{0}", "6855a414b5070e1cfd7b47b177fb4aec": " \\frac{\\part L}{\\part f} -\\frac{d}{dx} \\frac{\\part L}{\\part f'}=0 ", "6855bac244f74b268636b2377853ec05": "\\log f(x_i,\\boldsymbol \\beta)=\\log \\alpha + \\beta x_i", "6855f19d2f53818ef77fc4bd2108b6bb": "M(i,j) = \\frac{\\lambda m(j)A(i,j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)}", "68562ba86ed08c5ee215ac6bc8f326e1": "\n\\begin{align}\n& \\left(\\frac{2D^{\\mathrm{face}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} - \\left(1+\\frac{2D^{\\mathrm{face}}}{D^{\\mathrm{beam}}}\\right)\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} + \\left(\\cfrac{1}{S^{\\mathrm{core}}}\\right)~\\cfrac{\\mathrm{d} Q}{\\mathrm{d} x} = \\frac{M}{D^{\\mathrm{beam}}} \\\\\n& \\left(\\frac{D^{\\mathrm{beam}}}{S^{\\mathrm{core}}}\\right)\\cfrac{\\mathrm{d}^3 w_s}{\\mathrm{d} x^3} - \\left(1+\\frac{D^{\\mathrm{beam}}}{2D^{\\mathrm{face}}}\\right)\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x} - \\cfrac{1}{S^{\\mathrm{core}}}~\\cfrac{\\mathrm{d} M}{\\mathrm{d} x} = -\\left(1+\\cfrac{D^{\\mathrm{beam}}}{2D^{\\mathrm{face}}}\\right)\\frac{Q}{S^{\\mathrm{core}}}\\,\n\\end{align}\n", "685697ffd6beb4b974c49fa5cc56491a": "\\lambda(p) = p-1", "6856a57ec2f9529337d6110dc808b760": " |e_i \\rangle ", "6856e42a905357cf142f864dbad0e911": "\\displaystyle{L_y(a)=[L(a),L(y)] + L(ay).}", "6856f8bb099cab23011fb5c788aaf971": "V(T(p)) = T(V(p)) \\qquad (T \\in \\mathrm{O}(n, \\mathbf{R}))", "6857c369ca7c231f2f00c2f8dc471558": "X=\\lbrace x_1,x_2,\\dots \\rbrace", "6857d71951ca137bcda4fe14dccae251": "n_g", "6858a3f5e7fd9ce8a20e21a91c8ceaa6": "\\operatorname{bel}(A) = \\sum_{B \\mid B \\subseteq A} m(B). \\, ", "6858beecedf479241eb9787fa0b758a6": "t_{j}", "6858bf3c117bf25eb0ec178fba163780": "\\hat{\\mathbf{x}}=\\mathbf{P} \\, \\hat{\\mathbf{X}}", "6858cb6deb5c47e2decf6dcfa2f7b75d": "\\frac{d[E]}{dt} = \\frac{d[ES]}{dt} = \\frac{d[EI]}{dt} = 0. ", "6859510ac33db2d6d75ed65e280335e2": "\\frac{ \\partial^2 f}{ \\partial x^2} = f_{xx} = \\partial_{xx} f.", "68599e4e7f2e02df7c966eab72606424": "dx;", "6859b56eb6795761b1af9d099df86ced": "\n\\left(\\frac{1}{n}, \\ldots, \\frac{1}{n}\\right)\\prec_w \\left(\\frac{1}{n-1}, \\ldots, \\frac{1}{n-1},1\\right).\n", "6859d54013007531536400056d9e5700": "\\{\\pi(x)\\xi:x\\in A\\}", "6859f29c8f4944a0ae8f5816701d2dc2": " a^\\dagger", "685a9840655ed737392302ae4fa4848c": "V_{\\mathbb R}\\,", "685a985b73824e7f8e53fc6ddaccc74a": "\nE + S \\, \\overset{k_f}\\underset{k_r} \\rightleftharpoons \\, ES \\, \\overset{k_{cat}} {\\longrightarrow} \\, E + P\n", "685a9e68d7d82ff929fa2df0d811e29d": "c_{3,2}\\equiv\\frac{\\partial e_3}{\\partial x}-\\frac{\\partial^2e_2}{\\partial y^2}", "685aa4873bcfee0c45eb1f5b7aa0a593": "\\displaystyle{Q(c,y)L(x) +L(x)Q(c,y) = Q(yx,c) + Q(cx,y).}", "685abd196f805e4721a13c590cd3c1fd": "\\mathbf{w_p} \\leftarrow \\mathbf{w_p} - \\sum_{j = 1}^{p-1} \\mathbf{w_p}^T\\mathbf{w_j}\\mathbf{w_j}", "685b27ee7f5a43d6bc60a17bc0e39c87": "\\psi^{(-3)}(2)=\\ln(2\\pi)+2\\ln A-\\frac34", "685b2fc31078b9197149ef252ffc9842": "\\sigma_v\\,", "685b465129ce8e908d9c3a3175016d61": "= \\frac{GM}{R^2}\\int_R^\\infty \\rho dr", "685b92779ed06a3ce55e417947c51611": "T(K)", "685c1c03ff244e44923ea41f4e26ecab": "\n\\tau_{ij}^r - \\frac{1}{3} \\tau_{ij} \\delta_{ij} = -2 \\nu_{T} \\bar{S}_{ij}\n", "685c25bd46f765867714ebc993fce33f": "R^*T/M g", "685c2ae5f0a8540e720c53b0c10c7a62": "F \\to \\Gamma(F) , \\,\\!", "685c383d0cd4ebae734d8b6d0eed1f33": "I = \\{4,5\\}", "685c5173ba38220102bc53addc62793a": " \\Delta t = t_2 - t_1 \\,\\!", "685c7f68b512a44bd557e116049cb726": "V=\\bigoplus_{\\lambda\\in\\mathfrak{g}^*} V_\\lambda", "685d26dfb50cb547a29bbd01c813deb2": "\\rho_{AB}=\\rho_{A}\\otimes\\rho_{B}", "685d3058591361b757263db2079aa6cf": " \\psi^' = 1- J(\\phi^')\\, ", "685d3f1f056a4c365bfbd95cd0fc4974": "d\\psi", "685d4c711a7552f0ecf9c9d35ef26563": "\n\\operatorname{Li}_s(e^{\\pm i \\theta}) = Ci_s(\\theta) \\pm i \\,Si_s(\\theta) \\,.\n", "685d5c16762b4ab4b73e46f433c829f2": "\\Phi\\,\\!", "685d7b8a7abc14916d7d7fb977867683": "\\tbinom{c_{k-1}}{k-1}\\leq N - \\tbinom{c_k}k", "685d9b3124d3d6e4e6260e32506211d7": "A|B", "685dc50f0138b26f0a4201902a92ff29": "\\vec a \\times (\\vec b \\times \\vec c) \\neq (\\vec a \\times \\vec b ) \\times \\vec c \\qquad \\mbox{ for some } \\vec a,\\vec b,\\vec c \\in \\mathbb{R}^3", "685dccac767d7ad06e8b4e61be78566a": "x_{16}", "685dd398b057403f14bfee73dc7e2282": " \\Pr\\left[ T_+ \\le x \\right] = \\frac{2}{\\pi}\\arcsin\\left(\\sqrt{x}\\right), \\qquad \\forall x \\in [0,1].", "685de6e7de1f879003927a28355a60b7": "|T| \\leq d", "685e14e650f58e81ca67e7726b7b3d36": "\\tilde{h}(E)", "685e42a9b0da6dc10c19e5dd98478193": "\\langle x, A^* y \\rangle = \\langle Ax, y \\rangle", "685f014efbae109d9622d06686081933": "\\neg,", "685f53c5d14de57a5fb2c8228370107d": "\\displaystyle R(\\sum_{n} a_n z^n )= \\sum_{n< 0} a_n z^n.", "685f7f8f4b5d16fd6bc4404c305aead5": "\n\\begin{pmatrix}\nA^0 \\\\ A^1 \\\\ A^2 \\\\ A^3 \n\\end{pmatrix} = \n\\begin{pmatrix}\n\\phi / c \\\\ A_x \\\\ A_y \\\\ A_z \n\\end{pmatrix}\n", "685f9eba71e8294446289c79146e7cb5": " = \\lim_i \\frac{f'(y_i)}{g'(y_i)} + \\lim_i \\epsilon_i = \\lim_i \\frac{f'(y_i)}{g'(y_i)} ", "68600ba524dc5f0a7577504d7f70b317": "\\mathbf{T}^{-1} = \\left(T^a_{\\ b}\\right)^{-1}", "686060ff4e9dcb80db0369240380a1fc": "(x^2+y^2)^2=cx^2+dy^2 \\, ", "68606385b33897bee638c9bdaf83ada5": "\\mathrm{div} \\left( \\frac{\\nabla u_{n} (x, y)}{\\sqrt{1 + | \\nabla u_{n} (x, y) |^{2}}} \\right) \\equiv 0", "68606942e3670aed95f94fc7578401a0": "\\eta = \\eta_\\mathrm{Receiver} \\cdot \\eta_\\mathrm{Carnot} ", "68607e8f7f2cb8321e2fcd38ddf7afd6": "\\eta_0", "68609622d97abf7af02fba747835856f": "L(0,0)=1", "6860b8e5d56a812cb9552d552f224954": " \\psi_0 = \\psi_1 + \\psi_2. \\qquad \\qquad (1) ", "6860b99966134426ac8ba1e546c0642f": "H(s)=\\frac{G_0}{\\prod_{k=1}^n (s-s_k)/\\omega_c}.", "6860ef0c10f3eca1f07a8d59e9544640": "\\sin(2\\pi t/33)", "6860f1192f992e38b2e2cfea683afabe": "D_x \\in \\Gamma (x, M^*_X/\\mathcal O^*_X)", "686105f5e617fb1636de8681ab1e547c": "\\big\\{|0\\rangle_{1}\\otimes|0\\rangle_{2}, |0\\rangle_{1}\\otimes|1\\rangle_{2}, |1\\rangle_{1}\\otimes|0\\rangle_{2}, |1\\rangle_{1}\\otimes|1\\rangle_{2}\\big\\}", "68611a70a09169f330c71ad37f42d018": "LR=1", "686120e9603436f68f29521e02c9087e": "10 \\uparrow \\uparrow \\uparrow \\uparrow \\uparrow 2 = 10\\uparrow\\uparrow\\uparrow\\uparrow 10=(10 \\uparrow \\uparrow\\uparrow)^{10} 1", "686121c3a1a26cb99e4eca16a3e97d5f": "\\ C=\\frac{P_tG^2\\lambda^2}{(4\\pi)^3R^4}\\pi R^2(\\theta/2)^2\\sigma^o", "68617042b29e3205914f999cbb277050": "f(x) = f^{\\star\\star}(x) ~.", "68619124a9e58f9bb8cb500c6d7b3691": "f_{\\mathbf{X}}", "6861915f22df04ab6ae9e5a2f9540889": "\\displaystyle{d(f(x),f(y))\\le (1+\\delta)^{-1} d(x,y).}", "6861cf4dc69ec8aa7ed6574303a047ac": "\\mathcal{Z}(z,\\beta,V) = \\prod_i \\left(1-ze^{-\\beta\\epsilon_i}\\right)^{-g_i}", "6861d9d1ab535534b76e5d32648b8ba7": "\\Pr(X_{n+1}=j|X_0, X_1, \\ldots, X_n=i)=\\Pr(X_{n+1}=j|X_n=i)\\, \\forall n \\ge1, i,j \\in \\mathrm{S} ", "6861f6b2a86fdbd7c81dca834472996e": " x=y,\\quad x=-y. ", "68620f2738524006b737d92e65ff298c": "\nx^2 + bx + c = 0\\,\n", "68628064adb14f2c57d2c61e7a452c9f": " \\sigma Q_D (l_A a_B + l_B) l_D ", "68634d3031eaa343b35720d2b1885da4": "(M, g)", "68635e02a82f99684c5d4113925f6cd4": "f_v.", "68637b80077101bf6529a8a7ecc40fc2": "\\begin{align}\n \\Beta_a e^{\\lambda_a t} + \\Beta_b e^{\\lambda_b t}\n &= \\frac{1}{2} A_i e^{\\phi_i j} e^{(\\sigma_i + j \\omega_i) t} +\n \\frac{1}{2}A_i e^{-\\phi_i j} e^{(\\sigma_i - j \\omega_i) t} \\\\\n &= A_i e^{\\sigma_i t} \\cos(\\omega_i t + \\phi_i).\n\\end{align}", "6863922fe1650a961a9a67e26ae211af": "PV=\\frac{\\zeta(4)}{\\zeta(3)}NkT \\approx .9NkT ", "6863b0a08b9733149ee8ef3e8f61f5bc": "k = Cn", "686454cfbf01f385f8880865ebe1d6a3": "R_k(a, b)", "6864fed6a6ca719efd78f6d90c506b62": "\\mathcal{L}_X \\phi^A = \\frac{\\partial \\phi^A}{\\partial x^{\\mu}} X^{\\mu}\\,.", "686535135afefd14e8627e4c73f54585": "L^2_0(G(K)\\backslash G(\\mathbf{A}),\\omega)=\\hat{\\bigoplus}_{(\\pi,V_\\pi)}m_\\pi V_\\pi", "68658cdb9d93e330ad7b027037a4f9e9": " \\Phi(a'_{i},a_{-i})-\\Phi(a''_{i},a_{-i}) = u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})", "686596c03cdb0417a5c8e5839a207539": "\\omega = \\sigma", "6865aeb3a9ed28f9a79ec454b259e5d0": "cd", "68664548dca62f4355e7784827c16371": "\\; I_{A_1}", "68668de33662d7cc191bf942d6e83dad": "\\neg \\neg A", "6866def88c92106cee0ea9eebc61d79f": "Budget Deficit = Savings + Trade Deficit - Investment.", "6866e1edd8f6ab988c3da14031d09aa5": "[2;1,2,1,1,4,1,1,6,1,1,8,1,\\ldots\\;].", "6866f6c189fd8a2c58b87bad6cf1f7ed": "E, E'", "68670c45a2d6fa599d47365c548c5315": " \\frac{dS_{t}}{S_{t}}\\ - d(\\log S_{t}) = \\frac{\\sigma^2}{2}\\ dt ", "686715b2dc349692c5f57a80e8f3aaae": "\\mu_{10} = 0,\\,\\!", "68676d2a9c0cd84fa5083bfb8821725e": "\\tbinom{m - 1}{k - 1}\\big/\\tbinom Nk", "6867815b174305958320806eb5d77b4e": "q = 7 \\times 10^{-3}", "68678a032003b3d8ae478372d972a708": "\\varepsilon_1 \\leq \\varepsilon_2 \\leq\n\\ldots \\leq \\varepsilon_n \\leq \\ldots", "6867b47bc23742f6a5716839f52dc3bd": "\\alpha+\\beta+\\theta=2\\pi", "68680a6b26fee4430d03edcdbc9cd335": "\\psi(q,t)\\in\\mathbb{C}", "68685ec1bbd864490cfc8af31aaedff7": "d_{ib}", "686887f277b59e16a96bce3124b95940": "f^{-1}(\\text{Spec }B)", "6868ded08e29bebcce21d820589f4eac": "\\frac{\\partial\\phi(\\mathbf{r},t)}{\\partial t} = \\nabla \\cdot \\big[ D(\\phi,\\mathbf{r}) \\ \\nabla\\phi(\\mathbf{r},t) \\big], ", "6868f373981b67c785a961b7775ef094": "\nA = QR = Q \\begin{bmatrix} R_1 \\\\ 0 \\end{bmatrix} \n = \\begin{bmatrix} Q_1, Q_2 \\end{bmatrix} \\begin{bmatrix} R_1 \\\\ 0 \\end{bmatrix}\n = Q_1 R_1,\n", "68693e3251aa25ac0e9813368e2c5d90": "\\mathrm{_{14}^{28}Si} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{16}^{32}S} + \\gamma + Q", "6869eae828515b5fc9d0113b738908e0": " f^+= \\frac{|f| + f}{2}\\,", "686a8e809328960f4a9d40ae3a638f48": " \\mbox{const} \\cdot \\frac {K'}{V} = \\mbox{const} \\cdot \\frac {\\dot{m}}{C_o \\cdot V} \\qquad(6)", "686b16afc8dcc624bafe879fd13c234f": "|1/2,\\ m_1\\rangle|1/2,\\ m_2\\rangle", "686b2b74b43e0cf40a711b290345dad3": " \\widehat{D}^{\\dagger}(\\delta\\alpha)\\widehat{a}\\widehat{D}(\\delta\\alpha)=\\widehat{a} + \\delta\\alpha", "686b8d2e70998e910e767e5bcec077af": "\\mu_1 = 1", "686b9e13d543674ac0f3b6d02d629782": "\\{ u_1, \\ldots, u_n \\} \\subset H_1", "686ba00a46511db943d3988a1524266a": "M \\in \\mathcal{M}_k \\cup \\mathcal{M}_{k+1}", "686ba47ee32fa6fa8b074aeada8b6328": "f_x := \\inf\\{ f' > 0 \\mid d(x) + f' \\in d(X) \\}", "686c1e6ed8ffe8a3ce90bdad5dba9712": "1 - H(p)", "686c67177398af25f418f7df482e58c7": "2^\\kappa=2^{<\\kappa}\\times\\gimel(\\kappa)", "686c7454b5a10c110d67d32c327707b5": "(11)\\quad T_{ab}k^ak^b\\geq 0\\;,", "686ca2522d5d1ba0736151150e90c08d": " E_x", "686cd0a7e0a0d37c5d47bff3834e8178": "\\hat{s}=(h^*h)^{-1}h^*y.", "686cd8c1c9a2e7886df725ebe7842b12": "\\hat{\\boldsymbol\\theta}", "686d0d9cf26087f5805cecd386ecd534": "\\Delta d=x+x'-l=\\sqrt{(h_t+h_r )^2 +d^2}-\\sqrt{(h_t- h_r) ^2 +d^2}", "686d2b8bf5c241e09a17d276922f205b": " -\\tilde J = - J_a = D \\nabla_c ", "686d3942fdd79a73de3a38dd44f99cbc": "\\mathcal{M}=\\bigcup_{C\\in \\mathcal{C}} C.", "686db38d3c2db67dcd62722c68f43b0e": "x_{2,i}", "686dc00a8f30573f6494d7a71a2236d7": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & -1 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 0 & 0 & 2\n\\end{smallmatrix}\\right ]", "686e558411bf8715595ccb682141f677": "x \\,\\bmod\\, y", "686e6c30618d5e24db7cd594d5c16a1d": "\\scriptstyle X \\;>\\; 0", "686e6e974f3b6b0e9ce37e334dff6582": "S^{||}{}_{ij}=(\\nabla_i\\nabla_j-\\frac{1}{3}g_{ij}\\nabla^2)B", "686ead9f32045a2012eb334defbb2a15": " \\sum_{n\\le x}\\log n=x\\log x-x+O(\\log x),", "686ebe532b0c1ccb46fb6ee6a6661544": "\\frac{B_4}{{v_0}^3}", "686ed33125e6dc52e0b2bd55adbef94c": "mr \\omega^{2}", "686ee05471be4eff75593de2da9ae7f7": "\\dot p = - \\frac{ \\partial H }{ \\partial x } ", "686f8a9b9a8a7d36dbd27f30855319e7": "V(x(0), 0) = \\min_u \\left\\{ \\int_0^T C[x(t),u(t)]\\,dt + D[x(T)] \\right\\}", "686f9384f88ab672e4555a1c1d09a1a5": "\\kappa = \\delta^{-1}\\beta\\delta", "686fc5d790d485defe6a8f31942c3351": "\n N_{Ij} = J~F_{Ik}^{-1}~\\sigma_{kj} \\qquad \\text{and} \\qquad\n P_{iJ} = J~\\sigma_{ki}~F^{-1}_{Jk}\n", "68700cf988c5d159a392c42063ea3502": "\nf(x) = \\sum_{n=0}^\\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots.\n", "68706ffe053e200160db0d4a6135d6a6": " \\textstyle f_n(t) = \\sin 2\\pi n t", "68708a5008b3a2e52aacbe620927ce6e": "d\\ge 0", "68708ba49e16ae72b38c8e95756faca8": "\\alpha + \\sum_{i=1}^n x_i,\\, \\beta + rn\\!", "68709c437ede95a1796677e98627c875": " U2 =U1= U", "6870b7c3539e393f2a000c4f097efb31": "\nA z \\bar z + B z + C \\bar z + D = 0\n", "6870ea74e9490b7cf7417d6a287a7bb4": "\\tfrac{D}{D_\\max}", "687142a8a5f5b3418a29262767bef52f": "\\mathrm{BMI}=\\mathrm{pounds}\\frac{703}{\\mathrm{inches}^2}", "68715cb2154a1a972427dc25cfc34b72": "(i,j)^{th}", "687162509fba47fa4ba634feb8a4f24c": "x(t)=\\phi_t(x_0)", "6871c10afd691e6bcec116093e2dbf50": "\\exp(x+y) = \\exp(x) \\cdot \\exp(y)", "6871cf6f1ee973486cb4d2cbeba548bc": " \\frac{\\partial}{\\partial y}[E(u(X)|X>y)] = \\frac{f(y)}{1-F(y)}[E(u(X)|X>y) - u(y)] ", "68720a487e28f43371c69ab0879a683c": "\\Rightarrow \\!\\,", "687258a207b9ccd7e0cb9c43d3862f6a": "\\scriptstyle 1/(\\omega C)", "68728feb51a61d4c1a0aafcc771611d9": "\nQ_{lm} \\ \\stackrel{\\mathrm{def}}{=}\\ \nq \\left( r^{\\prime} \\right)^{l} \n\\sqrt{\\frac{4\\pi}{2l+1}}\nY_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})", "6872bf6c24b8886c2b6defc03a73ff15": "d_{1}+d_{2}", "68730265d90eb35761f4045e8f39f8d8": " \\pm f V ", "6873326d67e043d7d8b01e240c923b3e": "\\delta=\\ln(1+r)\\,", "68734786de092533e91395781d31d0b6": "\\mathcal{O}(nt)", "6873a466a793870fead3b4ae5614d0ae": "\\textstyle{u}\\,", "6873e53762c3197b69bab0158caafaf0": "\\frac{12}{7}", "68740af7f8a3459941a6f6319984c510": "k[t_1, \\ldots, t_n]", "68741189294ee1e8bf2fc4f2dfbffc4c": " \\mathbf{O}_{2} ", "68744552d490b338c0dd9c783c239fe4": " X = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}", "6874569c0a4888b1b20d6d64d7127628": "\\Delta\\sigma=\\sqrt{\\Delta x^{2}-c^{2}\\Delta t^{2}}", "68748cff134f16f120cbbcd25854a0c7": " \\binom{N}{N_R} = \\frac{N!}{N_R!(N-N_R)!}", "6875532ac2e6addb7c77e6834c78f885": "p(\\epsilon)", "6875e4e6415a08694acfab0d91e07ade": "F_{\\mathrm{net}} = M a_{\\mathrm{cm}}\\;\\!", "6875f11abc0d38d163ff94f77d2b232c": "\\varrho_i\\,", "68760d16a4ed18c999a38bf89559f3ab": "\\frac{1}{R-x}+\\frac{1}{R+x}=\\frac{1}{r}", "68767ba99ea99e3c98fcba62c99486c7": "\\Gamma (X, \\mathcal L (D))", "6876943221ad4e1fc3be0d0fc516116b": "\\sqrt{n}/\\hat{\\sigma}_D > 1.64- z_{0.10}=1.64+1.28\\approx 3", "6876a6a7876540b2cef2a2527df21e6f": "\\mathcal{O}_U/(f_1, \\ldots, f_k)", "68772f632a553df9c26852c32480c564": "{|\\psi_k\\rangle}", "6877c1f83d35552a71b1ff433f10091c": " \\displaystyle{W_{\\mathcal F}(a)E_0=e^{-|a|^2/2} E_a.}", "687817e3860b662ea4a5f8fe20189087": "b=a=1", "687825bd96bc4cb1976f5589b802d8e1": "(n, k)", "68784e5be93eecf99a8d89d6d0f06837": " r =\\sqrt{x_{\\perp }^{2}}\\,,", "687877bcc482e86de31cc761f4883458": " |\\psi\\rangle = \\psi_R |R\\rangle + \\psi_L |L\\rangle ", "6878b223d81122c2eff792cd7577e6bd": "A = 2 \\pi r \\Delta x", "6878d7f710d46a124dc657736943cc6b": "f = a e^{\\pm bA}\\,", "6878ff734c1514aedeb724b36f41fb24": "0 \\to E \\to F \\to X \\times \\mathbb C \\to 0", "687908b8c9ffd50f122058ef3da05a72": "\\displaystyle \\int_{-\\infty}^{\\infty}f(x) e^{-i \\nu x}\\, dx ", "687949812cb3b24bb5a924fb9db0bbf2": "\\rho_{\\rm out} \\in \\mathcal{S}(\\mathcal{H}_{\\rm out})", "68794b84ba512988d38e8dc99fb8ba89": "\n|V_{\\infty}(t)| = A\\frac{1}{\\left(\\omega^2 +(1/\\tau)^2\\right)^{1/2}} = A\\tau \\frac{1}{\\sqrt{1+(\\omega \\tau)^2 }}.", "68797561033e5334475e8c7ae67ac3cd": "\\theta(t) = t,\\,", "68797b97fd531a143dd6eeb1ef9ec6b3": " x_2 \\in X_2 ", "68799ee3af73ee2d654152e86b29dbbc": "D^\\mathrm{JW} = D^{(j,0)}\\oplus D^{(0,j)}\\,.", "6879b03a0ca1a76eb30b89acc4695e79": "((\\Sigma\\cup\\{\\# \\})^n)^*", "6879c7fb68fcc4df0dbe740c0cac04c1": "E_k = m_0 ( \\gamma -1 ) c^2 = \\frac{m_0 c^2}{\\sqrt{1-\\frac{v^2}{c^2}}} - m_0 c^2,", "6879d6b7ffd78808bcdb5dbd7b573fdb": "\\Box p ", "687a225386b8b27be8de2dbb36409af7": " \\mathbf{u}(s) = \\mathbf{u}(\\gamma(s)), ", "687afe112e894201e52aaed86afdbfda": "y^\\star_j \\in \\{-1, 1\\}.\\,", "687b22f5506c44a97002b82a43da148a": "P_{\\mathrm{h}}=q P_{\\mathrm{s}}.", "687b37a62c684575d9e4fa6a743b1634": "\\textrm{versin} (\\theta) := 2\\sin^2\\!\\left(\\frac{\\theta}{2}\\right) = 1 - \\cos (\\theta) \\,", "687b669a7f01500f179ac248277d6237": "\\mathrm{FWHM} = 2 \\; \\operatorname{arsech} \\left( \\frac{1}{2} \\right) X = 2 \\ln (2 + \\sqrt{3}) \\; X \\approx 2.634 \\; X ", "687b799f3f2f4ba1e236f48e62262658": " C", "687bd151d0c385b7fdc6424ffa337394": "\\vec{q} = [a,b]\\!", "687c64ccffac8834178146074871d3f7": "\\kappa=C/s", "687c6b575040215d35d6aa12a029b085": "E(\\bold{k}) = \\frac {\\hbar^2 k^2}{2 m} ", "687d7001731c8bad6731b5a40eb8e712": "\\mathrm{St}", "687df030f5f3541c2fac32860c6e20a7": "\n\\omega \\ll \\frac{1}{RC}\n", "687e25085bb67a44a54184caca2bd4d1": "\n\\lim_{\\Delta k \\Lambda\\to\\pi}\\frac{1-(-1)^N \\cos(\\Delta k \\Lambda N)}{1+\\cos(\\Delta k \\Lambda)}=N^2\n", "687ea7f07ca3ab21b90bc39a71a7457f": " P \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix}\nx + 0.5 \\cdot z \\cdot \\cos \\alpha \\\\ y + 0.5 \\cdot z \\cdot \\sin \\alpha \\\\ 0 \\end{pmatrix}", "687eb99e0083780e60f406daaca4f68f": "(S, \\pi, \\mathcal{K}_1, ..., \\mathcal{K}_n)", "687ee4bdcbe3307db3a12b172d17d8ab": "z_i\\mapsto r_iz_{\\sigma(i)} (r_i>0)", "687f9112bae68e2c759481c616059b75": "y_i \\in \\{1, -b \\}", "687fbe0a268a2bbb6b83f5002a8999b0": "260708.", "6880071b2f0a0b165dc739731233f3ea": "S_m(n) = \\int_0^n (\\mathbf{B}+x)^m\\,dx ", "6880480909098535c4980f87a7c225f9": "F_{\\hat\\theta_n}", "68814b7565d0ea4a491915d602373e76": "A = \\frac{E[\\mathrm{Uptime}]}{E[\\mathrm{Uptime}]+E[\\mathrm{Downtime}]}", "6881a5c36a489c38c3d10437733778db": "F_n\\,", "6881c6ec6464ec235c6fa1c774d43b4e": "\\theta_x : T_xM\\rightarrow V_{o(x)} E", "6881c87772fb5c6be9bd0e1cad3d8837": "R_c + 2nd", "688243277e96dcd7b58a67ec9e55fbcc": "\\liminf_{n \\to \\infty} \\frac{\\phi(n)}{n} = 1", "6882534dea04144a49090da68c177833": "D_{**}(X_i, Y_i)\\,", "688258915a7cde039747a532aab33c6a": "\\propto \\frac{a^{\\alpha-1} \\beta^{\\alpha c}}{\\Gamma(\\alpha)^b}", "6882595450f397b92bd8a330d4205e9f": "S = -k_B \\sum p_i \\ln p_i \\,", "68826998ed537b0a38ad4513df5a23b5": "g^{-1}(t)", "6882b97f99ae605d5b88d7302c39e8a6": "c=5", "6882eb1db8368aec0b30c91f29262108": "2^{\\frac 1 {72}} \\approx 1.0097", "6883104dc00db9b4dbc315ba11a035f6": "\\{c\\tilde{s}[n+d],\\tilde{h}[n-d]/c\\}", "68840c45d2f08e4128da1e9fd4d1dd44": "\\{c_r,z_r\\}", "688476fb24d2cd8b96553efc99a41b37": "x_1^{n}", "6884a858fbb78556d7006a4945844c2d": "\\epsilon(t) = \\frac{L(t) - L_0}{L_0}", "6884dcb4de0a71c5ec2e6ce94e4d0e80": "t_f = \\frac {2 A_m\\sqrt {h_m}} {A_g \\sqrt {2g}} ", "688532bba82a14124383dffa3c1ed1d1": "T_o(p_{i,j})=p_{i,j}+I_o", "688534ca3f9328362e29b7ccd6592e32": "\n f(x) = \\frac{1}{\\sqrt{2\\pi}}\\; e^{-x^2/2}.\n ", "68855e131e7ca88d846a3ef4468b835c": "x_{i} \\in X,\\, y_{i} \\in Y = \\{-1, +1\\}", "6885718fddcdcc338ebe80767d874c76": "\\operatorname{Pref} (L) = L", "6885b378c2899cc75de886ea99c98833": "\\log n + O(k)", "6885d763e8308b8a95176e64a2b31752": "\\|Wu\\|_{(k+1)} \\le C(\\|VWu\\|_{(k)} + \\|W^2u\\|_{(k)})\n\\le C\\|(\\Delta - V^2 -A)u\\|_{(k)} + C \\|(WV+B)u\\|_{(k)} \\le C_1 \\|\\Delta_1 u\\|_{(k)} + C_1 \\|u\\|_{(k+1)}.", "68865ad099dd587af50c1064aa4d3e98": "H \\propto \\varepsilon^{0.308}", "6886887a4d1fe8d448632f109531f92c": "[e^i,e^j]=\\sum_k{c^{ij}}_k e^k.", "68870cee2ba7f546241c1f458c41bf32": "(a\\succcurlyeq b)~\\Leftarrow~[f(a)\\le f(b)]", "68872fc777e69a97fcef92e36826f0e3": "\\rho_{SB}(t) = \\hat{U}(t)[\\rho_{S}(0)\\otimes\\rho_{B}(0)]\\hat{U^{\\dagger}}(t).", "6887b263836683d5e29c94f58b9ad8f9": "~ \\left ( {\\partial T\\over \\partial V} \\right )_{S,N} \n= -\\left ( {\\partial p\\over \\partial S} \\right )_{V,N} ~", "6887c0dd043fa81c365d26fc2709318e": "(k_1z_1+k_2z_2+k_3z_3+k_4z_4)^2=2\\,(k_1^2z_1^2+k_2^2z_2^2+k_3^2z_3^2+k_4^2z_4^2).", "6887c167270697a61981c597491fac59": "h(f_s(z)) =f_s^\\prime(0) h(z).", "6887cc20307964f927708ea7293b1a94": "\\phi(y)", "6887ef1912397d564bb83df11ad55407": "h_{\\ast}\\colon \\pi_k(X;A,B) \\to H_k(X;A,B) \\,\\!", "6887f858131cabe5f9ccabd78a77ef1a": " a^3 - b^3 = (a - b)(a^2 + ab + b^2).\\,\\!", "688809ffac05994915fa26ed782c5f63": "d((x_n),(y_n)) = 1/k", "68882df103d1aa0d8734ea3d2977f1e9": "\\iint_D (x+y) \\, dx \\, dy.", "6888cec00a584019a6390510d6c3ffc6": "\\text{V} = \\text{A} \\cdot \\Omega= \\dfrac{\\text{W}}{\\text{A}} = \\dfrac{\\text{J}}{\\text{C}}.", "6888df84bc5367697761bf0885561ea3": "B_{\\kappa\\mu}^\\nu(i,j)", "6888fe2a883153de4a628756a8c4a95b": "\\phi(S)=\\{\\phi(x)\\mid x\\in S\\}\\in\\mathrm{RAT}(M) ", "68891f72949cfa71a4726eb1c142bc2c": "\\mathrm{spectrogram}(t,\\omega)=\\left|\\mathrm{STFT}(t,\\omega)\\right|^2", "68894a30e6b971ab8f17318f684fec3e": "e^j", "6889ffe640c78923e84845ed0eb199f6": "f^{\\mathcal A}=I(f)", "688a20a0d622dfbf87d70f54584f7664": "\\hat{A}^\\dagger", "688a45d54e230dbc62b264afb3654287": "PSO(2n)", "688a70f32175fba6ec3be2fe8b4abb53": "\\Delta\\phi = 2\\pi n", "688a947e6903cb4ebad4852b468573d9": "Vp=Vn=0\\,", "688aab22fa191c05df8705dc97e88ba6": "V_{90}", "688b033df298d275639a9333bf26138d": "(2)\\quad \\psi_{SS}=\\frac{1}{2}\\ln\\frac{L-M}{L+M}\\,,\\quad \\gamma_{SS}=\\frac{1}{2}\\ln\\frac{L^2-M^2}{l_+ l_-}\\,,", "688b34cb71cf0f1a036e58c8fbc47489": "\\lambda + V_p", "688ba9eff96d61516cf7a4d987b3ebe6": "z_0\\,", "688bb7676b4e19049604afb7e6e952b8": "(1 + r_1)(1 + r_2) \\cdots (1 + r_n) - 1", "688be7cef9914bb41d33356623730f17": "d \\Xi = \\frac {U} {T^2} d T + \\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i - \\frac{P}{T} d V - \\frac {V}{T} d P + \\frac {P V}{T^2} d T", "688c26b0351320825aed112b1ab7d81c": "x_0=", "688c4a34dcf7096cdb4fef9957853d8c": "e:\\tau_1{\\to}\\tau_2", "688c7a18ba213f7ae23df400d7788f4f": " H=k[\\mathbb{Z}/2\\mathbb{Z}] ", "688c7e0ebcc5821f6ddefdeacc85080c": "\n\\hat{H} = \\frac{\\hat{p}^2}{2m} + U(\\hat{x}).\n", "688cc569fcb7bf5a5dbb4b200c022bc3": "k_{\\infty}", "688cf4f061a17f698c42567ce0de0449": "\\varphi:V\\to{\\mathbb F}", "688d36609a0e4d4f954b2b1f6ecef1d3": " F = \\frac{\\operatorname{E}{(f(\\text{Aa}))} - \\operatorname{O}(f(\\text{Aa}))} {\\operatorname{E}(f(\\text{Aa}))} = 1 - \\frac{\\operatorname{O}(f(\\text{Aa}))} {\\operatorname{E}(f(\\text{Aa}))}", "688d58fbeb9ed923fff84bbe3f6d50dc": "w_{ii}=0\\qquad \\forall i", "688d5a319d74574e765ac1d29da8907c": "\\displaystyle{\\Re\\, p_s(z) > 0}", "688d87f66f5d2f69137666d6021e2002": "\n-\\biggl\\langle\\sum_{k=1}^{N} \\mathbf{q}_{k} \\cdot \\mathbf{F}_{k}\\biggr\\rangle = P \\oint_{\\mathrm{surface}} \\mathbf{q} \\cdot d\\mathbf{S},\n", "688d90e4119102bc1e7f3e1d86254e99": "\\left \\lceil \\log_2 \\frac{P + 1}{\\log_2 17} \\right \\rceil \\,", "688dbbfc794144a3dc64bd19cb660830": "X=m\\frac{dU}{dt}\\cos(\\beta)-mU\\frac{d(\\beta+\\psi)}{dt}\\sin(\\beta)", "688dcfa593ccac21560eb664c3c0cb52": "\\; C_\\Phi=(I_n\\otimes\\Phi)(\\sum_{ij}E_{ij}\\otimes E_{ij}) = \\sum_{ij}E_{ij}\\otimes\\Phi(E_{ij}) \\in \\mathbb{C} ^{nm \\times nm}", "688e765854aa927570feaf8b1169b600": "\\tfrac{10}{1-\\tfrac{2n}{3}}", "688f4712d507aefa62ad2d93afb00fb7": "\\tau^M", "688fbb4372adb85774fe2b059c865578": "\\begin{align}\n y_i &= y^*_i + \\varepsilon_i, \\\\\n x_i &= x^*_i + \\eta_i,\n \\end{align}", "689019bb6132bc248344fc6edf63e5b6": "t+h/2.", "689035bbffaadb374a5367d4200614a0": "(27)\\quad ds^2\\,=-e^{2\\psi(r,\\theta)}dt^2+e^{2\\gamma(r,\\theta)-2\\psi(r,\\theta)}(dr^2+r^2d\\theta^2)+e^{-2\\psi(r,\\theta)}\\rho^2 d\\phi^2\\,,", "68906169cf6fb094ad30c86109160422": "x_1, \\dots, x_n,\\,", "689087e8a6823cf2f971d6e021e95294": "(b,a) \\in R, L(b) = \\mathit{out}", "6890a0337e0612aa2f75ae399402c329": "\\partial_2", "6890bb52f730f04884dbb032e1719018": "K^\\ominus = \\left\\{\\mathrm{{C}_{12}{H}_{22}{O}_{11}}(aq)\\right\\}", "689162f89597071d2411f3a774b1d3c8": "\n\\bar{A}_j = \\frac{1}{n_j}\\sum_{i=1}^{n_j}A_{ij}\\quad j=1,2,\\ldots, k\n", "689203cba6e1a3b09ccf54e4557c8d1e": " {\\Delta x} \\, ", "6892546f31b87bf7b3f70d8fc83adcd9": "\\gamma = \\lim_{n \\rightarrow \\infty } \\left( \\sum_{k=1}^n \\frac{1}{k} - \\ln(n) \\right)=\\lim_{b \\rightarrow \\infty } \\int_1^b\\left({1\\over\\lfloor x\\rfloor}-{1\\over x}\\right)\\,dx.", "689291e4b1d685e3a829d70b23d82c7f": "\\frac{1}{\\sqrt{D}} = \\sqrt{\\frac{T}{\\tau}}", "6892bc333b45f3cde8277dcd69333215": "\\pm\\Delta\\upsilon \\,", "6892d88725349b2fc84a51741ccb96c8": "x^2 - x + 1 = 0", "6892e0977ed77b1a051b88bee6ee6495": "\\textstyle a(x) + x^bb(x)", "68934a3e9455fa72420237eb05902327": "false", "68935e73ac553426386c6d5b5748d8f9": "\n\\sigma_{r\\theta} = - \\frac{\\sigma}{2}\\left(1 - 3\\frac{a^4}{r^4} + 2\\frac{a^2}{r^2}\\right)\\sin 2\\theta\n", "689375ea117f6500d1fe3b0f9cdda25e": "\\! w=1/3", "689415bfaae344af16078be33e77a901": "{d \\over dt} x(t) = {i \\over \\hbar } [ H , x(t) ]=\\frac {p}{m}", "6894714c5fb2659d9d5eb94d04fc4717": "v=(v_1,\\dots,v_n)", "6894b8163fa9de70dd02819e3f1442af": "P_{\\rm net} = 100 \\ \\mathrm{W}.", "6894bf1750e26ad46ba871ef43556a8a": "\\mathrm{Li^+} + \\mathrm{e^-} + \\mathrm{LiCoO_2} \\rightarrow \\mathrm{Li_2O} + \\mathrm{CoO}", "6894c11ad3446346113fce9039e795db": "\\Phi_{i,j}^{k,m}: \\bigoplus_{\\ell} H_{i,j}^\\ell \\otimes H_{\\ell,k}^m \\to \\bigoplus_n H_{i,n}^m \\otimes H_{j,k}^n", "68953fe60b3c08582d8f82333033fadb": "\\varepsilon_{tot} = \\varepsilon_{m} = \\varepsilon_{s_1}", "68954cfd94feebdc91f3a355c3d9c5bb": " \n\\begin{cases}\n(q, \\omega, (s', ta(s'),0)) \\in \\Delta& \\textrm{if } ~t_e = t_s, y = \\lambda(s), \\delta_{int}(s)=s',\\\\\n(q, \\omega, \\bar{s} )\\in \\Delta& \\textrm{otherwise}.\n\\end{cases}\n", "68957bacf4ff90dbed7957c9413372ef": "\nG(t) = G_\\infty + \\Sigma_{i=1}^{N} G_i \\exp(-t/\\tau_i)\n", "68958c2532cbd7e62cfae0766da7bbb5": "\\gamma^0 \\gamma^1 \\gamma^2", "68958f3eabfb81fac68d4186309e61e2": " \\mathcal{V}_1(S_i) ", "6895cd81b784b732886fe67c8d9ee702": "(2x + 1)^2", "6896348917f6517eef39419f73f3a416": " \\mathcal{P}(k)", "689651eb584a0659bef9f8ff39d60046": "\\sum_k a_k 1_{S_k}", "689678092ff065f9c0d868965266ba82": "\\frac{K}{P}=1+Ae^{-kt}", "6896e459b4842ac3464d100a32783143": "S^N", "6896f6e5e12a49ae2ff6bb610a64be6b": "\\mathbf{\\Sigma a}_i", "68970db3c1bed15cb26fcf9b49901fae": "c_{i,j}", "68971d492c48d8ba89e94d28f9a15859": "\\mathbf{y}_i\\in \\mathbb{Z}_{\\geq0}^{3}", "6897255a3b631da30108e3040dc324a7": "f(x; \\theta) > 0", "68973844cc5419538cc606d2dc283e04": "\\prod_{n=1}^{\\infty}\\left(1-x^{n}\\right)=\\sum_{k=-\\infty}^{\\infty}\\left(-1\\right)^{k}x^{\\frac{k}{2}\\left(3k-1\\right)}=1+\\sum_{k=1}^\\infty(-1)^k\\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\\right).", "68978caa54ccd655c543ea76f5cdf0a9": "\n\\begin{align}\nU &= \\begin{bmatrix}\nU_{e 1} & U_{e 2} & U_{e 3} \\\\\nU_{\\mu 1} & U_{\\mu 2} & U_{\\mu 3} \\\\\nU_{\\tau 1} & U_{\\tau 2} & U_{\\tau 3}\n\\end{bmatrix} \\\\\n&= \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & c_{23} & s_{23} \\\\\n0 & -s_{23} & c_{23}\n\\end{bmatrix}\n\\begin{bmatrix}\nc_{13} & 0 & s_{13} e^{-i\\delta} \\\\\n0 & 1 & 0 \\\\\n-s_{13} e^{i\\delta} & 0 & c_{13}\n\\end{bmatrix}\n\\begin{bmatrix}\nc_{12} & s_{12} & 0 \\\\\n-s_{12} & c_{12} & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\ne^{i\\alpha_1 / 2} & 0 & 0 \\\\\n0 & e^{i\\alpha_2 / 2} & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix} \\\\\n&= \\begin{bmatrix}\nc_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\\delta} \\\\\n- s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \\delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \\delta} & s_{23} c_{13}\\\\\ns_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \\delta} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \\delta} & c_{23} c_{13}\n\\end{bmatrix}\n\\begin{bmatrix}\ne^{i\\alpha_1 / 2} & 0 & 0 \\\\\n0 & e^{i\\alpha_2 / 2} & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix} \\\\\n\\end{align}\n", "68978d8c5a042c460432e79c57a8dcd0": "\\tilde H_i(G;\\mathbf{Z}) = 0.", "6898015fe74b3b15f1eebc86596ed91d": "t\\gg\\frac{(x_c-x_{0})^2}{4D}", "689802a07506384a335e7842c390b015": "\\tilde{u}(\\vec e_j) = \\sum_{i=1}^n (u_i \\, \\tilde{\\omega}^i) \\vec e_j = \\sum_i u_i (\\tilde{\\omega}^i (\\vec e_j)) ", "68980e9feaa5e479b15d8034b145f2db": " \\frac{e^{3+2\\gamma}}{2\\, \\pi} = \\prod_{n=1}^\\infty e^{-2+2/n}\\,\\left (1+\\frac{2}{n} \\right )^n. ", "68987d77600ff32d79d847a3d4a19bbe": "Z_F(x_1, x_2, \\dots) = \\sum_{n \\ge 0} \\frac{1}{n!} \\left( \\sum_{\\sigma \\in S_n} |\\mathrm{Fix}(F[\\sigma])| x_1^{\\sigma_1} x_2^{\\sigma_2} \\cdots \\right).", "689882ca5d2f81b8771b1c7ca4cad7dd": "X = (X_t)_{t \\in [0,T]}", "6898f2ab20069590dc9a9315cb36064e": "p(3)=4", "689910a6dc68b63c4671508c0d0d2eab": "\n{\\textbf C}_X = \\frac{1}{N'} {\\textbf D}^{\\rm t} {\\textbf D}. \n", "68994e71d4b28e9e7345ff9bea026aa8": "\\begin{matrix} {4 \\choose 1}{3 \\choose 3}{45 \\choose 1} \\end{matrix}", "68997c81bc9435a791f25216f3ecce50": "~A \\cup B \\cup C", "689984053edcbdaf32494ff7f53b20a4": "\n \\Delta_k := \\det(\\underline{\\underline{\\mathsf{S}_k}}) > 0\n ", "6899e1115219167d0dbfd12f93026d77": "\\nabla f = \\sum_i \\mathbf e^i \\frac{\\partial f}{\\partial q^i} = \\frac{\\partial f}{\\partial x} \\mathbf e^1 + \\frac{\\partial f}{\\partial y} \\mathbf e^2 + \\frac{\\partial f}{\\partial z} \\mathbf e^3", "6899e7842ea3de92e01811efa4cec38a": "\n\\left ( M + m \\right ) \\ddot x - m \\ell \\ddot \\theta \\cos \\theta + m \\ell \\dot \\theta^2 \\sin \\theta = F\n", "689a085197aac0f1ffe8e7adce266aee": "V_{\\text{out}} = A_{OL} (V_{\\text{in}} - \\beta \\cdot V_{\\text{out}})", "689aa692afce6fc439b4b5992fe2914f": "{\\omega^\\hat{m}}_\\hat{n}", "689ac5d48dac3a21846e81fa035d4ed4": "\\sigma =\n\\begin{cases}\n0.07 & \\text{if }\\omega \\le 5.24 / T_1, \\\\\n0.09 & \\text{if }\\omega > 5.24 / T_1.\n\\end{cases}\n", "689b04adf53f829d4d6010d6f1430b5e": "u, v, w", "689b7a8c8cb423c7aab56dd4060bfc33": "\\frac{f(1)}{f(0)}=\\frac{1}{2}\\,", "689b92d80becd306336abdb1915c2ef3": "{q}\\times{\\alpha}\\times\\frac{1}{q}", "689bb62a23d0bc22bdd0621807218f24": "L(x) = \\coth(x) - \\frac{1}{x}", "689bd731d2644b3a747263877cece6ad": "L(P,f,g) = \\sum_{i=1}^n \\inf_{x\\in [x_i,x_{i+1}]} f(x)\\,\\,(g(x_{i+1})-g(x_i)).", "689c225adf7fba342ceb65409b53eb22": "|z|0,", "68d2d46eabb725908a2a2e1d0d978686": " \\left|\\Psi\\right\\rang ", "68d3640c5cddce16c9d1cead2e00b15d": " \\check H^q(\\mathcal U,\\mathcal F)= H^q(X,\\mathcal F), ", "68d3dcbb18ece9f6454d4313cc88c1a3": " C_{P,m} - C_{V,m} = \\frac{C_{P} - C_{V}}{n} = \\frac{n R}{n} = R", "68d40a4b1fd58587fdf3f5d647207efd": "n^{-\\frac{1}{2}} (\\nu - m)", "68d43680560c63759aa513246f8ad463": "\\scriptstyle \\sigma_{ij}", "68d43d74e3b383cebf057804e69024fb": "\\epsilon t", "68d47a6259fdde98835be46e0c729e0b": " \\frac{T-T_{f}}{T_{i} - T_{f}} = \\operatorname{exp} \\left [ \\frac{-hAt}{\\rho C_p V} \\right ] ", "68d56982c924b234db2ec52d1c9d4b88": "\\sqrt[4]{2} \\approx 1.19", "68d582a097568cc3776ddd393aa1c24d": "\\zeta(3) = \\frac{5}{2} \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^3\\binom{2n}{n}}.", "68d593eaf1eb417d21c4bcf89098bb1f": "M_f(x_1,x_2) = f^{-1}\\left( \\frac{f(x_1)+f(x_2)}2 \\right).", "68d594d75eac531367728d4a41a8a4cc": "y_k=\\nabla f(x_k+\\Delta x_k)-\\nabla f(x_k)", "68d5b05adfd4fbb329603ede1bb17ea6": "x = u/2R", "68d5ddce68fe4f83b2df76dd9bc8fb52": "\\varphi \\Rightarrow \\psi", "68d60302440fcd8726cc0792b5acb1a9": "\\triangle\\delta = -\\frac{v}{c}\\cdot\\sin(\\delta-\\alpha)\\cdot\\frac{180^\\circ}{\\pi}", "68d6910ef246a67596496bb75dc3ba3f": "q \\, \\!", "68d6f5be3f90fa649e01e1dd523da657": "\\hat \\theta_n", "68d6fe08dccb1503cfe1f4ad63e27fd3": "\\mathit{Q} = {\\mathit{a_i} - \\mathit{b_i} \\mod \\mathit{p} : i = 1, 2,..., N}", "68d708d5fa9fa6a1214527720cce71ec": " 45 \\rightarrow 24_0 \\oplus 10_{-4} \\oplus \\overline{10}_4 \\oplus 1_0", "68d708e55a7fd08884d0fdd4b367fe3a": " \\begin{align} P(Hypercalcemia~WHOIFPI~by~no~disease) = \\\\\n P(no~disease~WHOIFPI) * r_{no~disease \\rightarrow hypercalcemia} = \\\\\n 0.997 * 0.0014 \\approx 0.0014 \\end{align}", "68d70b6c590d542c809c8d143df86f26": "\n\\max \\{ R_A(x) \\mid x \\in U \\} \\geq \\lambda_k\n", "68d70feaf86cf225418bec4c88015bb0": "u_{\\alpha\\beta}", "68d72d2aaf72549a0c21fd80ee87eb78": " E_K = h\\nu - I\\,", "68d72e511eeafd09d2f830b0b7865bf1": " P_0 - \\left(\\sum_{t=1}^T\\frac{C}{(1+r)^t}\\right) - \\frac{F}{(1+r)^T}.", "68d76131e8f058fad05c9cc810cc9bc5": "Y_{10}^{3}(\\theta,\\varphi)={-3\\over 256}\\sqrt{5005\\over \\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(323\\cos^{7}\\theta-357\\cos^{5}\\theta+105\\cos^{3}\\theta-7\\cos\\theta)", "68d763bd42949ce21a12bb5a6bb84f3c": "\\nabla\\cdot\\left(A^*\\nabla u\\right) = f", "68d7ac0e2d1475a6ab3cef80cf3c6a37": "T = \\sum_{\\lambda\\in\\sigma(T)} \\lambda E_\\lambda,", "68d7b6cacd839dc9019cc94fb1dd981e": "\\eta_c = 100%", "68d81844fb51d9e0cfbdac59ecb3916f": "\\omega = \\Omega(k).\\,", "68d87b2e99577921a3f5d2d419bc7ab7": " \\hat{B}_{m_j} ", "68d8ce2a97f17e68d261811b14dc5039": "t' = \\gamma t + \\frac{ \\left( 1 - { \\gamma^2} \\right)x}{ \\gamma v}", "68d9113d19bf6c118b06b24428f72b46": " \\mathrm P(A, B)=\\mathrm P(B \\mid A) \\mathrm P(A) = 2/3 \\times 1/2 = 1/3", "68d91d22b2ff8b7412b501f9b8e72237": "p(x,y,z) = -A^2B \\left(a^2+x^2+y^2+z^2\\right)^{-3},", "68d951dee6d77bfb938b86348f394f2a": "1/x \\in o(1)", "68d9827d67b9139a7584b71f5c2cada9": "i^\\text{th}", "68d9c6dce2f0a1778127a17669bb8c12": "f^{\\,vap}(T_s,P_s)", "68da07b1ac344006c67faa23ab78860a": "f_{\\mathrm{FD}}(E) = \\frac{1}{\\exp\\left(\\frac{E-\\mu}{k_\\mathrm{B} T}\\right)+1}", "68da86f651458a1c0cff9cba073d3c07": "A = R[x] / f(x)", "68da970df9f2310c089fa86a4e345faf": "\\begin{align}\n x_{n+1} &= y_n+1-a x_n^2,\\\\\n y_{n+1} &= b x_n.\n\\end{align}", "68dad66750b0a500e3505c875d30c5d8": "\\frac{x^2}{a^2}\\pm\\frac{y^2}{b^2}=1.", "68db034ff55740367945e8a6470b33cd": "\\scriptstyle \\leq3\\times10^{-23}", "68dbdc73bf3bb5d66bc74592f4c591a9": "\\,Z=v^T=(1+i)^{-T} = e^{-\\delta T} ", "68dc102e197ca6c60c2418c7524b2d14": "J^k_{x_0}f\\cdot J^k_{x_0}g=J^k_{x_0}(f\\cdot g)", "68dc10e74d3940d32c60e79766b0a9bb": " \\det(\\mathrm{II}-\\kappa\\mathrm{I})=0, \\quad\n\\det\\left|\\begin{matrix}L-\\kappa E & M-\\kappa F \\\\ M-\\kappa F & N-\\kappa G \\end{matrix}\\right| = 0. ", "68dc2895d2d9ebf80dbf7ffab55d1581": "e^{i\\,\\theta} = e^{2\\,i \\pi s} = (-1)^{2\\,s}", "68dc46967ef892884b8336ad7b58d308": "0.8\\overline{3}", "68dc944057a4df0344df36911ca5d2c1": "p\\rightarrow 1", "68dc99bce78f7fcab8eff126e8160c9c": "\n\\bar{h}^{i j} (t,\\vec{x}) \\approx\n-\\frac{4}{r}\\, \\frac{\\mathrm{d}^2}{\\mathrm{d}t^2}\\, \\int\\, x'^i x'^j \\tau^{0 0} (t-r,\\vec{x}')\\, \\mathrm{d}^3x'\n", "68dc9f1cbc716213b7f029e85b8e11f3": " \\text{national savings} = \\text{domestic investment} + \\text{net foreign investment}\\,\\!", "68dcb4d401ffa39b6b45e05d22d9e52b": "lastblock \\leftarrow 0", "68dcba296e1be813a23bdfa299d79fc6": "-ke^{-2\\alpha}", "68dd09c5239969657ec10d94aae7266b": "q_2:=\\sqrt{{q_1}^2-4m_1m_2}\\,\\!", "68dd0edf8ae66d4e70779835c2773f7f": "\\cdots \\to \\Omega^2 B \\to \\Omega F \\to \\Omega E \\to \\Omega B \\to F \\to E \\to B \\, ", "68dd0fc28f2f0c7f05953e2925e56873": "g(\\mu,X)", "68dd17c4783604d733e627e7463ad301": " G(\\tau)=G(0)\\frac{1}{(1+(\\tau/\\tau_D)^\\alpha)(1+a^{-2}(\\tau/\\tau_D)^\\alpha)^{1/2}} +G(\\infty),", "68ddab024709c43db7fb430dab2c724b": "\nM(\\theta + \\Delta \\theta) = M + \\Delta M = F L (\\sin \\theta + \\Delta \\theta \\cos \\theta ) - k_\\theta (\\theta + \\Delta \\theta) \n", "68ddf43ddcdc12aff5de942579fef84f": "\\mathit{N} = \\mathit{p}^\\mathit{k} - 1", "68de5b973fdbafe6c7136f07f189c232": "|E_n| \\leq \\frac{\\varepsilon \\log_2 n}{1 - \\varepsilon \\log_2 n} \\sum_{i=1}^n |x_i| ", "68de8a0fc46e392a66741d2caa6037d8": "\\frac{R_3}{R_1} = \\frac{R_a}{R_c}.", "68dea5396b69598dbd62394e54153816": "y\\,f\\left({f(z) \\over y}\\right)=z\\,f\\left({f(y) \\over z}\\right)", "68df48c5f04a63638f9435c52b5d2efa": "\nd(n) = y(n) + \\nu(n)\n", "68df4fedda3a6ee42485d71ce568ba2e": "\\pi_j", "68df629e2e4920882b5659f622065bd7": "\\displaystyle u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x ", "68df847a5518d0804eb48635ac7eb2b4": "U_{(\\lceil np \\rceil)} \\sim AN\\left(p,\\frac{p(1-p)}{n}\\right).", "68df9d44e7a70bd555ba404609336669": "\\cos(\\theta) = (m_2-m_3)/(l_2+l_3+1)", "68dfc5bdba5355e360ab65fe2ca0d838": "x = \\pm \\sqrt{-r}", "68dfdc927f024a3e963592b940b2b643": "f(z)=\\sum_{n=-m}^\\infty a_n q^n.", "68dfea137096355fda9ed68060c18bb8": "\\scriptstyle E=\\frac{1}{2}\\rho g a^2,", "68dff3751cd4477d575b60648150ce6e": "\\; P(0,y) ", "68e029d119be896acf05055d625cbac6": "z\\approx Z-\\dfrac17Y^2-\\dfrac13Y^2Z", "68e0588cd05494f93c95d9637bad7399": " X(\\omega) = \\int_{-\\infty}^{\\infty} \\left[ \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, d\\tau \\right] \\, e^{-j \\omega t} \\, dt ", "68e0aafe37878345b56618d1d7bc8b5d": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n1 & 0 & 1 & 1\\\\\n1 & 0 & 1 & 1\n\\end{array}\n\\right] .\n", "68e0fba291b48e59de5bba6aa3cef935": "S_\\nu(x)", "68e127b744f2ca515e7a4ef7aaa84ab2": "\\hat e_1\\ ,\\ \\hat e_2,\\ \\hat e_3", "68e12a181c7416339ad6f7215c3c4f9c": "N_{wave}", "68e1334766c4744566674e148fae4a31": " \\int_0^1 g_2 (f_2(y)) \\, \\mathrm{d}y = \\tfrac13. ", "68e185345528f3beb5b663e555e1f0e1": "B_n, \\mathfrak{so}_{2n+1}(\\mathbf{C}): \\mathfrak{so}_{n,n+1}(\\mathbf{R})", "68e18cdcb9fb90a0b12603ba772d405f": "\\omega\\colon \\pi \\to \\left\\{\\pm 1\\right\\}", "68e1eda932fb8e1716851829f9528c73": "k=[G:H]<\\infty", "68e289f4267728b3ddd2f4f5131a03d9": "\\mathrm{GL}(\\mathfrak{g})", "68e2eac013b36043ff20f655b8fde836": " \\operatorname{G}(A^*) = (\\operatorname{J}\\operatorname{G}(A))^\\perp\n= \\{\n(x,y) \\in H \\oplus H : \\langle (x,y)|(-A\\xi,\\xi) \\rangle = 0\n\\;\\;\\forall \\xi \\in \\operatorname{dom}(A)\\}\n", "68e3159e23f04dedd681e91bc0fe40a2": "G=(V,w)", "68e3314e465e67bfef72409cd0fddcba": "\n\\mathbf{F}_{\\mathrm{imp}} = m \\mathbf{a}_{\\mathrm{i}}\n", "68e3c0ffc8b9c234afe54a76614d5a0a": "q_i, p_i, q_e, p_e", "68e3c745bafe198808f866782b267b9e": "|A_i| = |B_j| = 2", "68e3d9b1b1320cc1d46d33f30882c4fe": " V = - N \\frac{\\mathrm{d}\\Phi_B}{\\mathrm{d} t} , \\,\\!", "68e406247a1c089946766016a83afca7": "\\scriptstyle P^{\\prime}", "68e41d656b1445b783253a23136f5d40": "\\{U_\\alpha\\}", "68e4255e4baad71f8530d4a5bf09a8f2": "\n \\mu(T) = 2.414*10^{-5} \\times 10^{247.8/(T - 140)} \n", "68e4415df336c79a50db94218e7042ae": "M(\\phi)", "68e558bea236d6c5e628289a13c3024c": "f(\\boldsymbol{x}) := \\textrm{sign} \\left(\\sum_{j=1}^J \\alpha_j h_j (\\boldsymbol{x})\\right)", "68e565dcbc92e0c7a39d395c109fb8eb": "e^{-\\alpha t} \\sin(\\omega t) \\cdot u(t) \\ ", "68e58b16c7c95fa974fbc755aab4ec6e": "\\frac{2}{\\sigma^2}(1-\\mu\\sigma\\gamma_1-\\sigma^2)", "68e59111cfa3857334d53c9c8dbd467a": "r = \\sqrt{r_0^2 + t^2}", "68e5993707f86001892aeb4d9236f905": "\\left ( \\frac{r_e}{r_m} \\right )\\approx 1.09051", "68e59a27104bac1a5df6d5a62c735fa1": "\\frac{dx'}{dt'}=\\frac{ \\gamma \\left ( dx - v dt \\right ) }{ \\gamma \\left ( dt - \\frac{v dx}{c^2} \\right ) }", "68e5e3c947a56b45df4e9485f842e225": " \\sum \\|u_n\\|_p^p < \\infty", "68e633c799f45ba8e9fcfc5c6e8a2697": "\\psi: \\mathbb{R}^{9}\\times \\mathbb{R} \\to \\mathbb{C}^{2}\\otimes \\mathbb{C}^{2} \\otimes \\mathbb{C}^{3}", "68e66d202e2957a688e459c07b5d67b2": "(\\lambda^3-27) \\frac{\\partial^2 \\omega_\\lambda}{\\partial \\lambda^2} \n+3\\lambda^2 \\frac{\\partial \\omega_\\lambda}{\\partial \\lambda} + \\lambda \\omega_\\lambda =0. ", "68e696bcf9bb19bae8786c9e4469d187": " l_{\\rm{interparticle}} \\,", "68e6b65f879821c56a2806a6acd78ef0": "p \\quad", "68e75496bcfb60b549b585132013f576": " \\mu(E) = \\sup \\{\\mu(K): K \\subseteq E, K \\mbox{ compact}\\} ", "68e795e9fb20e7fda2e68445ee42d0ea": " \\frac{\\partial \\Phi(\\vec{r},t)}{c\\partial t} + \\mu_a\\Phi(\\vec{r},t) + \\nabla \\cdot \\vec{J}(\\vec{r},t) = S(\\vec{r},t)", "68e7a818b658ce2dfb85a2d9cd57f7cb": " L_1(\\beta)=L_2(\\beta) = 1 ", "68e82059ac40a85ff8b291b1c25fae01": " K_e= \\{ x \\in S\\;|\\; \\exists n>0:\\; x^n \\in G_e \\} ", "68e83a36293278d5e61c0bd1ed60297c": "G_U=G_\\infty\\;", "68e8542f410a9dea66c3d3889b75effd": "I_{S}(s):=\\begin{cases} \\begin{array}{ccc} 1 &,& s\\in S\\\\\n0 &,& s\\notin S\n\\end{array}\n\\end{cases} \\ , \\ s\\in X", "68e8cdc2c8ef02b20c9f820c667cce92": "\nS_p = \\sum_{n=0}^p a_nz^n.\\,\n", "68e8d88d346e3608b4ddcbf2b31b3e05": "\\psi(x)-x > K\\sqrt{x}.", "68e909af9f483587b8bfc6d75dc3366c": "D_k", "68e921112b896c25cd06d20f1871c577": "f_{z^3} = N_3^c \\frac{z (2 z^2 - 3 x^2 - 3 y^2)}{2 r^3 \\sqrt{15}} = Y_3^0", "68e92bd7d9878c99406d6f534f99f10a": "\\sqrt{\\gamma}", "68e985849080db0128e47caf9b230057": "\\psi(\\alpha) ", "68e9b65f463cb637c19055c100ab7368": "\\Pr(|\\chi(E)|>\\lambda)<2 \\exp(-\\lambda^2/(2n))", "68e9dc0c41a209bf4865f684fd0a3647": "\\{(x, y, \\sin{x^2}\\cdot \\cos{y^2}) : x,y \\in \\mathbb{R}\\}", "68e9eb8f9dda003f1d4ac435869041d0": " V_p = 1150 \\ S_\\mathrm{mils} \\ d_\\mathrm{in} \\ \\log_{10}(D/d)", "68e9ef9cb29aca2b5452bd9336a601e4": "\\ \\displaystyle \\alpha\\ge 0 \\ ", "68ea198041545061aeb8410e3709214d": "1+GH", "68ea2e9d86b2690800658d9f7403b088": " b^{*} = \\frac{K}{1-A-B}.", "68ea599d079aacf50d09624048b3087d": "\\operatorname{succ}(C,[a_1,a_2,\\ldots,a_n])=\\operatorname{succ}(\\operatorname{succ}(C,a_1),[a_2,\\ldots,a_n])", "68ea9509dd24e33a0dab9f48e9f1d271": " \\mathbf{m}_i = \\mathbf{r}_i m_i \\,\\!", "68eaea8a7859128ca6095a8ac4403bc5": "K(u) = \\frac{\\pi}{4}\\cos\\left(\\frac{\\pi}{2}u\\right) \\mathbf{1}_{\\{|u|\\leq1\\}}", "68eba04a4b57ee1327db7b0d516dd173": "z \\in \\{1, 2, ..., n\\}", "68eba3c704ba18f3ee4f6bdc3795e695": "(\\exists) \\frac{\\exists x . \\delta(x)}{\\delta(c)}", "68ebb928836794ba14e4eea093bfecb9": "u^* \\left( x,y \\right)=\\sum\\limits_{i=1}^N \\alpha_i\\phi \\left( r_i \\right)", "68ebf1c047bc22af8a7044485f0450fd": "\n\\begin{matrix}\n{\\bold 1}=\\left(\\begin{matrix}1&0\\\\0&1\\end{matrix}\\right),&\n{\\bold 1}'=\\left(\\begin{matrix}1&0\\\\0&-1\\end{matrix}\\right),\\\\\nP=\\left(\\begin{matrix}0&1\\\\1&0\\end{matrix}\\right),&\nQ=\\left(\\begin{matrix}0&i\\\\-i&0\\end{matrix}\\right)\n\\end{matrix}\n", "68ec93baed5905eac95a6337b5fb8e83": " f_{st} = \\frac{ m_{ox, 0}}{sm_{fu, 1} + m_{ox ,0}} ", "68ecc278db517852258ac8475c6feb77": "\\{\\hat{\\pi}_i\\}", "68ed5795973eb3f0be85207905082c43": " \\beta(M) = (-1)^{r(M)-1} p_M'(1) \\ ", "68ed66c0d1c1fa53f220690f4e44780d": "\\left \\vert L \\right \\vert", "68ed906bda1010f9cdaea437dce40526": "\n\\begin{align}\n \\mathrm{ozone} = &\\ 5.2 \\\\\n& + 0.93 \\max(0, \\mathrm{temp} - 58) \\\\\n& - 0.64 \\max(0, \\mathrm{temp} - 68) \\\\\n& - 0.046 \\max(0, 234 - \\mathrm{ibt}) \\\\\n& - 0.016 \\max(0, \\mathrm{wind} - 7) \\max(0, 200 - \\mathrm{vis})\\\\\n\\end{align}\n", "68edb012cadeeb4e2f4facfb2587c9d8": "X=\\bigcup_{k\\in\\mathbb{N}}kU", "68edebc4d78df957b01a15b7f3471f54": "\n\\Pr \\left\\{ \\lambda_{\\text{max}} \\left( \\mathbf{H}(\\mathbf{z}) - \\mathbb{E}\\,\\mathbf{H}(\\mathbf{z}) \\right) \\geq t \\right\\} \\leq d \\cdot e^{-t^2/8\\sigma^2},\n", "68edf4df22b0800e5b55618e308aecca": "\\ [x,y,z] \\mapsto \\left[\\frac{x}{1-z},\\frac{y}{1-z}\\right].", "68ee0400a8e041cf64e941118c802249": "v_1\\cdots v_k \\mapsto \\frac{1}{k!} \\sum_{\\sigma \\in S_k} v_{\\sigma(1)}\\otimes \\cdots \\otimes v_{\\sigma(k)}.", "68ee1c83c18f2bfa5b210549c8b17715": "v_i\\in U'", "68ee274468b6229b396c155f1b805c02": "A(\\mathbb{T})\\subset C(\\mathbb{T})", "68ee5003f9280dc602cf5d20fa4db106": "\nZ=\n\\begin{bmatrix}\n1 & 1 & \\cdots & 1 \\\\\ny_{p-1} & y_{p} & \\cdots & y_{T-1}\\\\\ny_{p-2} & y_{p-1} & \\cdots & y_{T-2}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_{0} & y_{1} & \\cdots & y_{T-p}\n\\end{bmatrix} =\n\\begin{bmatrix}\n1 & 1 & \\cdots & 1 \\\\\ny_{1,p-1} & y_{1,p} & \\cdots & y_{1,T-1} \\\\\ny_{2,p-1} & y_{2,p} & \\cdots & y_{2,T-1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_{k,p-1} & y_{k,p} & \\cdots & y_{k,T-1} \\\\\ny_{1,p-2} & y_{1,p-1} & \\cdots & y_{1,T-2} \\\\\ny_{2,p-2} & y_{2,p-1} & \\cdots & y_{2,T-2} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_{k,p-2} & y_{k,p-1} & \\cdots & y_{k,T-2} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_{1,0} & y_{1,1} & \\cdots & y_{1,T-p} \\\\\ny_{2,0} & y_{2,1} & \\cdots & y_{2,T-p} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_{k,0} & y_{k,1} & \\cdots & y_{k,T-p}\n\\end{bmatrix}\n", "68ee565343e1fa95e54202ecb915ecde": "a^2 = ac\\cos\\beta + ab\\cos\\gamma,\\,", "68ee66b870ee33254c638dcf04b86d43": "\\text{SSG} = \\frac{\\rho_o - \\rho_f}{\\rho_f}", "68eed943ff4c9f69d648b2342c676317": "M(x)=\\frac{ax+b}{cx+d}, \\qquad a,b,c,d \\in \\mathbb{Z}_{>0}", "68ef88eda4f3fc090552390d08620158": " {1 \\over 2} {h_1 \\over h_0}\\left({h_1 \\over h_0} + 1\\right) - Fr^2 = 0, ", "68efed545d5790bbdab11ce3d413982d": "\\langle T_m\\varphi,\\psi\\rangle = \\int_U m(x)\\varphi(x)\\psi(x)\\,dx = \\langle\\varphi, T_m\\psi\\rangle", "68f09f9bdc2eaa3fd492ca85c093ca9f": "\\mathbf{w}^{\\textrm{W-MMSE}}_k = \\sqrt{p_k} \\frac{( \\mathbf{I} + \\sum_{i \\neq k} q_i \\mathbf{h}_i \\mathbf{h}_i^H )^{-1} \\mathbf{h}_k}{\\|( \\mathbf{I} + \\sum_{i \\neq k} q_i \\mathbf{h}_i \\mathbf{h}_i^H )^{-1} \\mathbf{h}_k\\|} ", "68f0b34fea8c9f5f0a00c847302298bd": "\nf(z) = 1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\cfrac{z}{\\ddots}}}}.\\,\n", "68f0f2f22a84a005ab7a1bffa2d5f6f8": "X_{o^{N}}", "68f0ffa628cbbb127b77047e9be5c710": " \\mathbf{H} = \\left( \\begin{array}{cccccccc}\n\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nP_z^{\\dagger} & P+\\Delta & \\sqrt{2}Q^{\\dagger} & -S^{\\dagger}/\\sqrt{2} & -\\sqrt{2}P_{+}^{\\dagger} & 0 & -\\sqrt{3/2}S & -\\sqrt{2}R \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\nE_{el} & P_z & \\sqrt{2}P_z & -\\sqrt{3}P_{+} & 0 & \\sqrt{2}P_{-} & P_{-} & 0 \\\\\n\n\\end{array} \\right) \n", "68f1358647e81ea94e86b803c3807dfe": "(\\mathbb Z_2)^n \\rtimes S_n", "68f13814637bb32e522dac146fecd590": "\\Z^k", "68f19adea0bef7a4f017e773e09e627e": "d_{1/2,-1/2}^{1/2} = -\\sin (\\theta/2)", "68f1a324772d24086fd10d63c4b574ad": "H(u,v)\\,", "68f223565b2551ebe591d13701b3e8c5": "\\tbinom nb", "68f27e30cac27a9dc4bbf5592b07962e": "(1+i) ", "68f289b9a5fea658578201bf61a5dbd4": "\\frac{\\mathrm{d}F}{\\mathrm{d}x_0}=\\frac{\\partial F}{\\partial x_0}+\\sum_{i=1}^{n}\\frac{\\partial F}{\\partial x_i}\\frac{\\mathrm{d}x_i}{\\mathrm{d}x_0}.", "68f2d039b7df1c794ea797b705797445": "\\frac{d}{dt}\\langle A\\rangle = \\frac{1}{i\\hbar}\\int \\Phi^* (AH-HA) \\Phi~dx^3 + \\left\\langle \\frac{\\partial A}{\\partial t}\\right\\rangle = \\frac{1}{i\\hbar}\\langle [A,H]\\rangle + \\left\\langle \\frac{\\partial A}{\\partial t}\\right\\rangle.", "68f2d180f01b86e24f7c4f839c34f911": "\\langle d(x,\\hat{x}) \\rangle", "68f399064dd22164672ce5b678524c85": "s(t,\\vec r)", "68f3bc45006b6e7cc7063da029201694": "\\tau*=\\frac{\\tau_b}{(\\rho_s-\\rho)(g)(D)}", "68f411175825720dd792fd157ed9996d": "(E,r)", "68f42c788e15de6b6a06028213025fea": "F = \\kappa(X)", "68f4e32405f454badf33ac0edf81ca32": "(E(1/X))", "68f4eca93c5033275d4b27a6c5fe319e": "F_\\rightarrow", "68f4faadebbee5c4aacfb8c6154bfc4d": "\\breve{ }", "68f50f8758676a9f6d91647b2b310468": " \\left( - \\rho \\overline{u_i^\\prime u_j^\\prime} \\right)", "68f5150c714ffc64d6fb761899077057": "\np_{11} = \\frac{1 + (p_{1\\cdot}+p_{\\cdot 1})(R-1) - S}{2(R-1)}\n", "68f55c996a5168b88c8ba018e3c1af39": "\\mathrm{\\frac{Q}{M T}}", "68f56120c2c71aa01562c07c1a9d7a45": "\\scriptstyle P_\\mathrm i", "68f582fe885181c6d7d568d8976962c5": "e_i = 1", "68f5f2ce1f90dc1fb1990f3ebbfd6925": "\\bar{x}=\\frac{M_{10}}{M_{00}}", "68f5fccd1a258040d36ef4d503e89af6": " A,B \\in \\mathcal{A}", "68f6029163b57ca4fda3fa107c35d07c": " r^3-r^2-r-1<0. \\,", "68f6259394b592f8c7eeed7de5e661b0": "\\frac{1-p}{p}\\!", "68f63afd652a989ff777475971f58108": "\\Gamma\\left(\\frac{\\nu}{2},\\frac{\\tau^2\\nu}{2x}\\right)\n\\left/\\Gamma\\left(\\frac{\\nu}{2}\\right)\\right.", "68f69bcdb77512c30031cd84c25a6ce2": "\n\\begin{align}\n\\int_r^\\infty gm \\left(\\frac{r}{s}\\right)^2 \\, ds\n& = gmr^2 \\int_r^\\infty s^{-2}\\,ds\n= gmr^2 \\left[-s^{-1}\\right]_{s=r}^{s=\\infty} \\\\\n& = gmr^2\\left(0-(-r^{-1})\\right)=gmr.\n\\end{align}\n", "68f6aae6184d908d8809726b0249adaf": " a_j = \\partial /\\partial z_j ", "68f7320d65eed481d48f9a7105274629": "\\displaystyle{2L(ab)L(a) + L(a^2)L(b)=2L(a)L(b)L(a) + L(a^2b).}", "68f7992007fc6ea95250c7c2d2153038": "\\nabla \\cdot (a(x) \\nabla u(x)) + b(x)^T \\nabla u(x) + cu(x) = f(x)", "68f7f41a2c606459bef7463f5e46bcda": "E_y", "68f85c3512b2714518a0d748c4c7d2f0": "v(\\vec{p}, 1) = \\sqrt{E+m} \\begin{bmatrix}\n\\frac{p_1 - i p_2}{E+m} \\\\\n\\frac{-p_3}{E+m} \\\\\n0\\\\\n1\n\\end{bmatrix} \\quad \\mathrm{and} \\quad\nv(\\vec{p}, 2) = \\sqrt{E+m} \\begin{bmatrix}\n\\frac{p_3}{E+m} \\\\\n\\frac{p_1 + i p_2}{E+m} \\\\\n1\\\\\n0\\\\\n\\end{bmatrix} ", "68f874b5063e96fa5e2fc98549b33d9e": "|\\alpha| = q^{d/2}.", "68f891a79d7246dc372d8ff1ce9c1d81": "x = a^2 + b_i^2", "68f8a0e68b1e09a2d9827ba3d37b654b": "g(v)= \\sum_{s \\neq v \\neq t}\\frac{\\sigma_{st}(v)}{\\sigma_{st}}", "68f8a1725b942e9a06a930cccb2492f7": "D_R = \\{-r_1,-r_1+1,\\dots, -1,0,1,\\dots,r_2-1,r_2 \\}", "68f8a553979707e322e2a9c1cf98e7da": "{d^2 \\theta^{ij} \\over dt^2} = \\left( {q^{ij} \\over J^i} + {q^{ji} \\over J^j} \\right) \\tau^{ij} + \\sum_{k\\neq j} S(ij,ik) {q^{ik} \\over J^i} \\tau^{ik} + \\sum_{l\\neq j} S(ij,jl) {q^{jl} \\over J^j} \\tau^{jl}", "68f92f0707cab39be91e26ec891bfdc9": "\\scriptstyle t < 0", "68f988b9ca184fc7bfdb94c648f41336": "(s,t)\\in E", "68f9c50683a347b17cbd8d69392f3aad": "\\lambda = 2\\pi \\sqrt{\\frac{rd}{2k}}", "68fa56c4be7d15c8aaa321bae58c6c6d": " PET = K_c * RET", "68fa8105b2a60ff4e4d4e355bd19d685": "m \\times p", "68fab0cbae20d306feda9db7a53184b0": "\nE\\tau = \\frac{(R_2^2-R_1^2)}D\\left[\\log \\frac{1}{\\varepsilon} +\n\\log 2 + 2\\beta^2 \\right] +\\frac{1}{2}\\frac{R_2^2}{1-\\beta^2}\\log\\frac{1}{\\beta}- \\frac{1}{4}R_2^2 +\n O(\\varepsilon,\\beta^4)R_2^2.\n", "68fac1497ee3e110b9c58fb3d86e82db": "q = p/(p-1)", "68fadf3b70f0743095a0edc3c0ef8cf0": "K=\\pi \\left(2 \\ln 2+3 \\ln \\pi + 2 \\gamma - 4 \\ln \\Gamma \\left(\\frac{1}{4}\\right)\\right)\\approx 2.58498 17595 79253 21706 58935 87383\\dots", "68fb1b1c12107ddb6ee46461bfb917aa": "1.97 \\pm 0.04 M_\\odot", "68fb80c3b62c64a1e63b7eafe2540c39": "c_{start}", "68fba06fbe686b4e3c29734ea80370bb": " ( \\nabla f(x) - \\nabla f(y) )^T (x-y) \\ge m \\|x-y\\|_2^2 ", "68fbcc0cb9e69b5630204a9ee21eae6e": " u = \\frac{ c( x, y ) }{ n^2 - n } ", "68fc086cef28df57e60969012b713031": "z \\in Y + \\{\\omega\\}", "68fc8eb07168fedd9796cc996428b731": "\\operatorname{O}(M) \\times_{\\sigma_+} \\{-1,+1\\}", "68fc94471fbfbcd5d502b324e2e2335f": "P \\to (P \\and Q)", "68fca1c2604b89461da49f8c31b9abfe": "-\\tfrac{a_1}{n}", "68fcc607b76135310b9656d968a41f4f": "\\mathrm {DoF} = \\frac\n{2 N c \\left ( m + 1 \\right )}\n{m^2 - \\left ( \\frac {N c} {f} \\right )^2} \\,,\n", "68fcf0d1e505b89ff4f79330f85a1a63": "\\mathrm{N}_{\\mathfrak{L}}(S)=\\{ x \\in \\mathfrak{L} \\mid [x,s]\\in S \\text{ for all } s\\in S \\}", "68fd0b76843498bd7aa38030d5aa8453": "\\begin{align}\n1\\,\\mathrm{lbf} &= 1\\,\\mathrm{lbm} \\times g_{\\rm n} \\\\\n&= 1\\,\\mathrm{lbm} \\times 32.174049\\,\\mathrm{\\tfrac{ft}{s^2}}\\\\\n&= 32.174049\\,\\mathrm{\\tfrac{ft {\\cdot} lbm}{s^2}}\\end{align}", "68fd2c13945ee1c9eed24a4b5876c0bd": "{e}_1 \\wedge {e}_2", "68fd47fb53b6fc4e0297ad1580a598a0": "\\sin{(\\chi_{nk}/2)} = 0", "68fdf072e36aea0bc1c8415d3d83e2dc": "\\mathcal{O}_n", "68fe72b9655ca24be6578a30254559bc": "A=\\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{bmatrix}, \\qquad \\mathbf{x} = \\begin{bmatrix} x_{1} \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix} , \\qquad \\mathbf{b} = \\begin{bmatrix} b_{1} \\\\ b_2 \\\\ \\vdots \\\\ b_n \\end{bmatrix}.", "68ff08435556296d0df19d450d2611b1": "(i-1,j-1,,*)", "68ff29f352794a6786d299b93c196003": "C_{pL}", "68ff3fd48bbd7cb5f47df4b1ed722dea": "{\\scriptstyle \\Sigma}", "68ff4da217a82a24e11ddaac676cb062": "i\\neq j, r_j-{2(r_i,r_j)\\over (r_i,r_i)}r_i", "68ff7405f8981822e653193e91865da0": "y=\\arctan(x).", "68ffb15ec9c21f35144d6922bf0a4bd2": "c (e) = i", "68ffcf5da3fd65f04a89c80ca15ab3d5": "n \\tan{\\tfrac{\\pi}{n}}", "690006c3c0f61662e189912910290a0d": "\\omega_G", "690025aad991d788e27ad2de432511a7": "\\mathcal{E}(u,u) \\geq 0", "69002f15903a8a001d8ea043a3e64563": "\nP_i= \\cfrac{\\rho\\ g\\ H\\ Q}{\\eta}\n", "690054a6fe53f6b00fc3cbb9c8d2dc5f": "\\tfrac{p+1-q}{p+1}", "6900c59e02e821ee8d70529806ce2811": "\\mu_1(n-1)/\\sqrt{n-1}.", "690124d62b6710273d34f98af897baa0": "d+1.", "69017f4daaf4a8fd50f94546a8c8cea9": "gevol", "690188b60bfda1c760b517f47a57b4e3": "\\zeta(n)", "690206e28110111d2ae6a779b4eacab3": "\\inf_{n\\to\\infty}P(X_n>0)>0", "69025c27ca36fe9f1d3f70be1e995747": " \\varepsilon \\ \\stackrel{\\mathrm{def}}{=}\\ f[x(t)] - \\varphi [x(t)] = x(t+1)- c[x(t),t] - \\varphi [x(t)] = x(t+1) - d(t+1) ", "69028bed061de491c1fe567e448e5d50": " e(k) = \\hat{x}_U(k) - x(k) ", "69029c75f7c2dc11aeee9d794ed8d6ed": "(\\xi_{1},\\xi_{2},...,\\xi_{N})", "6902a726295c73ef2612715d4094f4f9": " \\mu_s^{(j+1)} = \\frac{\\sum_{t =1}^N h_s^{(j)}(t) x^{(t)}}{\\sum_{t =1}^N h_s^{(j)}(t)} ", "6902b83426ddf055df7b09288f295b08": "N(M)=L(G)", "6902d08abda48a40baf7dae477476050": "r = \\sqrt {x^2 - y^2} .", "6902eacd71c91d4d11f32d14127a3fdb": "\\boldsymbol Y=\\exp(\\boldsymbol X)", "6903041bc01e99e6451a9c2e1df727a5": " \\mathbf{P}(t) = (p_A(t),\\ p_G(t),\\ p_C(t),\\ p_T(t))^T", "69030b15ea1d69d9494b74a9f1c6fe40": "dy/dx=2ax+b", "690342493c15bce830c0338756798d43": " u^2 - dv^2 = \\pm 2 \\, ", "690368e11a62197b017b727010044f26": "\\{x \\mid \\neg\\phi\\} = V-A", "690382ddccb8abc7367a136262e1978f": "di", "6903aff4d2aaa2d391b17c46ffcbb362": "\\mathbf{y}(t) \\in \\mathbb{R}^q", "6903dee1ff89a5648d7dc7d939829420": "E_0(x)=1\\,", "69042490ad18967c2164b9c4b9e417f7": " T = \\langle u \\bar{v} w \\rangle_I", "6904ba03b266ca596605fccefdd5fc13": "\\text{CBC-MAC-ELB}(m, (k_1, k_2)) = E(k_2, \\text{CBC-MAC}(k_1, m))", "6904e712bb8038ef8c8df7097d8bb94e": " 0 \\to \\mathfrak{nil}(\\mathfrak g)\\to \\mathfrak g\\to \\mathfrak{g}^{\\mathrm{red}}\\to 0", "6905042fd540dd0093e2ee549cbd353f": "S \\subseteq N \\in \\Sigma \\mbox{ and } \\mu(N) = 0\\ \\Rightarrow\\ S \\in \\Sigma.", "69051bc7f8bca0b32dac20b2d29d120b": "\\left( R = \\frac{a^2}{b} \\right) ", "690541814b51e1d6a1fe5e342fa55acd": "\\frac{|2 - 3| + |2 - 3| + |3 - 3| + |4 - 3| + |14 - 3|}{5} = 2.8", "69057ef46388cc18c02798218f3e913c": "n=\\prod_p p^{\\nu_p(n)}.", "690599953bfcd8bc4009a53a10eaf61d": "\\log T", "6905e19b9920a3712152f0cb2de57ee4": "n \\geq 4", "690673aba96d7ca34a22c88eab091ee8": "r=\\frac{K}{|a-c|}=\\frac{K}{|b-d|}", "6906850d01c1904867d55a21c408654b": "L = \\sqrt{m r}", "6906871add96f780c07a795f1244cbed": "\\mathcal{X} \\times \\mathbb{R}", "69068cb2968aad300225c9718b8ad190": "2\\sqrt{|V|}", "6906fed25064b240b2cd929bd36817dd": "\\mathbf{k^\\prime}", "69073612d1eca08713080b94ace7418e": "J_-|j\\,-j\\rangle = 0. \\,", "6907504bba3956f9059f262b8fd40b7c": "h(y_2, \\dots, y_n | y_1; \\theta)", "6907a34f345b6ef75619f4e646b34d02": "K_\\mathrm{sat}^{(1)}", "6908473fd79a083cbf2abce0a59c9cef": "V = W \\oplus W'.", "690862e011389162085d2c6e86d7a575": "\\exp(j\\theta) = \\cosh(\\theta) + j\\sinh(\\theta).\\,", "6908a9c97b094e1cf11d35d2f5d33865": "\\,\\sum_{i < j} V(r_i - r_j)", "690917683c8bff707eeb1f4cbca974f5": "U_1 - U_2 = v_1 - \\left(V^{M}_{N \\setminus \\{b_1\\}} - V^{M \\setminus \\{t_1\\}}_{N \\setminus \\{b_1\\}}\\right) - v_j + \\left(V^{M}_{N \\setminus \\{b_1\\}}-V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_1\\}}\\right) = \\left[v_1 + V^{M \\setminus \\{t_1\\}}_{N \\setminus \\{b_1\\}}\\right] - \\left[v_j + V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_1\\}}\\right] ", "69091e0d32ad567e881646083ad54479": "\\mathfrak{a} \\cap (\\mathfrak{b} + \\mathfrak{c}) = (\\mathfrak{a} \\cap \\mathfrak{b}) + (\\mathfrak{a} \\cap \\mathfrak{c})", "690975189a7f06bf8e4ecae2daeb0d49": "\\eta_{s}", "690a0009352ef753ee713b31b8a37df6": "\\mathcal{SN}(\\mu,\\, \\sigma^2,\\xi)", "690a030fa7f8ad4c686ad3c18b63e9fe": " \\frac{(uv)'}{uv} = \\frac{u'v + uv'}{uv} = \\frac{u'}{u} + \\frac{v'}{v} .\\! ", "690ab982aa57a9e0b935a2788192c208": "i^{\\ast}_{Q}(TE) \\to Q\\,", "690abc63ebca65a7ae8484722b263d36": "(\\vec{\\xi}^{\\prime}, z)", "690bbd8a7458f8890d9befc06d47dd84": " h \\in R^m ", "690c26cf9e7795bfeeba38a30de04b1a": "Fall: W = 0.000015241L^{2.94}\\!\\,", "690c2f46bbf0d37f99ecb519d0d7b5e8": " {dL \\over dx} - {\\part L \\over \\part u'}u'' - {\\part L \\over \\part x} -u'\\frac{d}{dx} \\frac{\\part L}{\\part u'} = 0 \\, . ", "690c3fb62b00e8fafd7477e6c702d428": "Q(a)", "690c7077db62450baae7f6eb916dbdc4": "\\bar{v}(U) = \\sup_{i\\in I}v_i(U) \\quad \\forall U\\in \\mathcal{T}. \\,", "690c94027ae3cf3363102ed1e6203be6": "L_0 \\subset S^n", "690cbda6ad460bc135fec1cae57d95a7": "p\\cdot q = q\\cdot p", "690d43ab8caa92818e0d291dad1528a9": "\\phi = \\beta - \\alpha", "690dad3de9950895e449309b6c82ae1d": "\\mathbb{R}^{n-1,1}", "690dc1ae84f2ff13f0fae38dd4f6eeab": "T_g = P M_g \\vert_{H^2},", "690dd0514c67d6da6fae0ffd885d3ac4": "Y' = 0.299 R' + 0.587 G' + 0.114 B'", "690e1b5634c64d2474f85e907bf50789": "\\frac{D\\zeta}{D t}=\\zeta\\frac{\\partial v_z}{\\partial z}+\\nu\\nabla^2\\zeta,", "690ea223281a941f3f868a075164e539": " \n\\langle\\theta^2\\rangle = 2 D_r t \\!\n", "690eab28b1335577fd9c46a572549127": "( C'_y(B^{'2}_x + B^{'2}_y) - B'_y(C^{'2}_x + C^{'2}_y) )/ D', \\,", "690ee1e91608f9c8c0ebd8f407bf1dc1": "\n\\text{If } X \\sim \\mbox{qExp}(q,\\lambda) \\text{ and } Y \\sim \\left[\\text{Pareto} \n\\left(\nx_m = {1 \\over {\\lambda (q-1)}}, \\alpha = { {2-q} \\over {q-1}} \n\\right) -x_m\n\\right],\n\\text{ then } X \\sim Y \\, \n", "690f3c9bcfb267d4f5d4eb005a0a75a4": " p_z=\\hbar k_z .\\, ", "69103fffb7da39dd5de6e85916604812": "f : C \\to X\\,", "69104612199b4eb6347b24bc1023e1a4": "h_{\\alpha \\beta} \\eta^{\\beta \\gamma} \\,", "691049c13b63c45067573149b1beeb99": "\\!V = \\frac{4}{3}\\pi r^3.", "691075217eb13799c1fc2c5ae2f3b215": "\\frac{1}{k}", "6910a9f76bb6d927d2321012c45e146c": " \\arccsc x ", "6910ad536c6da9e02f355f6e25fa38db": "\\begin{smallmatrix} \\lambda_b = (2.898 \\times 10^6 \\operatorname{nm\\ K})/(35,500\\ \\operatorname{K}) \\approx 82\\, \\end{smallmatrix}", "6910da7102b9b9edbb9a7d42d6e19ef8": "\\left\\{I,X,Y,Z\\right\\}", "6911018d946240169abb3e411af7aa0c": "\n\\begin{align}\n\\langle\\Psi|\\hat{G}|\\Psi\\rangle &= \\frac{1}{2}\\sum_{i=1}^{N}\\sum_{j=1\\atop{j\\neq i}}^{N}\\ \\bigg(\\langle\\phi_{i}\\phi_{j}|\\hat{g}|\\phi_{i}\\phi_{j}\\rangle - \\langle\\phi_{i}\\phi_{j}|\\hat{g}|\\phi_{j}\\phi_{i}\\rangle\\bigg), \\\\\n\\langle\\Psi|\\hat{G}|\\Psi_{m}^{p}\\rangle &= \\sum_{i=1}^{N}\\ \\bigg(\\langle\\phi_{m}\\phi_{i}|\\hat{g}|\\phi_{p}\\phi_{i}\\rangle - \\langle\\phi_{m}\\phi_{i}|\\hat{g}|\\phi_{i}\\phi_{p}\\rangle\\bigg), \\\\\n\\langle\\Psi|\\hat{G}|\\Psi_{mn}^{pq}\\rangle &= \\langle\\phi_{m}\\phi_{n}|\\hat{g}|\\phi_{p}\\phi_{q}\\rangle - \\langle\\phi_{m}\\phi_{n}|\\hat{g}|\\phi_{q}\\phi_{p}\\rangle,\n\\end{align}\n", "6911430df049ab794c5d5c02d447c50e": "\\left\\{ A_{i}\\right\\} _{i\\in\\mathbb{Z}^{+}}\n", "69117c5414dc6a0e6f5ce56bd3bfb3e9": "100/60 = 1.66", "691208c959a9b0c86c7f5e0dc8f84784": "{R_2}", "69124f458ecc89c200a9a22edf2f9d54": "\n\\begin{matrix}\nL & \\equiv & |L|e^{i2\\theta} \\\\\n & \\equiv & Q +iU. \\\\\n\\end{matrix}\n", "6912684a12693a9da1d0f36d397c8b43": "283 \\times 204 = 57,732\\,", "691270e252825e8de3f8c1b2909d0618": "\\pi_i(B\\Sigma_\\infty^+)\\cong \\pi_i(\\Omega^\\infty S^\\infty)\\cong \\lim_{n\\rightarrow \\infty} \\pi_{n+i}(S^n)=\\pi_i^s", "6912b6ac1e5dc61e79cf328d457a0cff": "E(x,t) = \\,", "6912d1245d0375cd1d63c1abbc81a93e": " S = \\Delta \\omega_r/\\Delta_d", "6914758f99a27731c87c90e923a04f79": " s_1 \\in S_1, s_2 \\in S_2, \\ldots, s_m \\in S_m ", "691478451b6077de6679f45c104f3bd1": "\\textstyle \\sum_{i \\in I} x_i", "691492d4922904ed102508d864cdf20e": "\n g(z) =\n \\sum_{n=1}^{\\infty}\n \\left( \\lim_{w \\to 0} \n \\left( \\frac {\\mathrm{d}^{n-1}}{\\mathrm{d}w^{n-1}}\n \\left( \\frac{w}{w/\\phi(w)} \\right)^n\n \\right)\n \\frac{z^n}{n!}\n \\right)\n", "6914c3e569b51021cf86de659b002691": "\\textrm{pOH} = \\textrm{pK}_{b}+ \\log_{10} \\left ( \\frac{[\\textrm{BH}^+]}{[\\textrm{B}]} \\right )", "69151c55e6b2dbe131eada3ddde38ad2": "\\text{PCSA} = {\\text{muscle volume} \\over \\text{fiber length}} = \n {\\text{muscle mass} \\over {\\rho \\cdot \\text{fiber length}}},", "69153baa605689ccea02a33c4e708ec2": " \\frac{1}{\\sqrt{2}} \\prod_{p = 3\\,\\text{mod}\\,4} \\Big(1 - \\frac{1}{p^2}\\Big)^{-1/2} = 0.764223... ", "6915a62b71f13ae14cb96ebe2bb3c4df": "r_i =\\left\\| \\left( x,y \\right)-\\left( s x_i,s y_i \\right) \\right\\|", "6915ba5270361dcf506808b41485c21b": "{\\omega \\over {\\omega_\\mathrm{c}}}", "6915e6a56f06f5eeda9b158f7f0bca86": "\\lim_{n\\rightarrow \\infty}\\frac{1}{n^2} \\sum_{k=1}^n (n\\;\\bmod\\;k) = 1-\\frac{\\pi^2}{12}\\!", "6916152e61b669967ba7f0accc00778b": "\\mathrm{U}(1) \\times \\mathrm{SU}(2) \\times \\mathrm{SU}(3)", "6916218fdd10b9b36d3f5ad08a57866c": "g^{efghcdba} = g^{abcdefgh}", "69162d65821a2ff36f88cd87256c64a2": " M(\\ldots, x_i, \\ldots, x_j, \\ldots ) = M(\\ldots, x_j, \\ldots, x_i, \\ldots) ", "69163be101a6bc213d542b2dab6a6d1e": "\\kappa(\\gamma) = - \\mu^{-1} \\gamma", "6916415ce55e7a2c0c3e285918cf1b3e": "\n \\begin{align}\n w & = w^K + \\frac{\\mathcal{M}^K}{\\kappa G h}\\left(1 - \\frac{\\mathcal{B} c^2}{2}\\right)\n - \\Phi + \\Psi \\\\\n \\varphi_1 & = - \\frac{\\partial w^K}{\\partial x_1}\n - \\frac{1}{\\kappa G h}\\left(1 - \\frac{1}{\\mathcal{A}} - \\frac{\\mathcal{B} c^2}{2}\\right)Q_1^K\n + \\frac{\\partial }{\\partial x_1}\\left(\\frac{D}{\\kappa G h \\mathcal{A}}\\nabla^2 \\Phi + \\Phi - \\Psi\\right)\n + \\frac{1}{c^2}\\frac{\\partial \\Omega}{\\partial x_2} \\\\\n \\varphi_2 & = - \\frac{\\partial w^K}{\\partial x_2}\n - \\frac{1}{\\kappa G h}\\left(1 - \\frac{1}{\\mathcal{A}} - \\frac{\\mathcal{B} c^2}{2}\\right)Q_2^K\n + \\frac{\\partial }{\\partial x_2}\\left(\\frac{D}{\\kappa G h \\mathcal{A}}\\nabla^2 \\Phi + \\Phi - \\Psi\\right)\n + \\frac{1}{c^2}\\frac{\\partial \\Omega}{\\partial x_1}\n \\end{align}\n", "69170e3ba7995bace49c10263f8f9a93": "{P}^{3}-2PQ\\, ", "69176e506aec36265d496e8fb30d60af": " s^2 = ( c n^d ) ( m^b ) ", "691794d4e6a3988f36bf020a55645b36": " F(q_1 , q_2 ,\\ldots,q_n) ", "6917adee329815a7e89984db97702c8b": "\\neg D(f(d))", "6917c9232806350b8b38a23962eade5b": "\n X_n\\ \\xrightarrow{d}\\ c \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{p}\\ c,\n ", "6917d3adbbb8f16a67ecc8af9e4fb2e6": "\\mathrm{_{20}^{40}Ca} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{22}^{44}Ti} + \\gamma", "6918cfd0c7678f849483b0b50088f045": "A_T= A \\frac{2r^2s}{abc}", "69190bf5e73df99eb121a8628fd2e0bb": "\\Delta G = \\sum_{p}^{}{(\\Delta G^0_i -(T_i-T^0)\\Delta S^0_i)} + \\sum_{n}^{}{(\\Delta G^0_i -(T_i-T^0)\\Delta S^0_i)} ", "6919802e19e5fb2067383c3edd352333": " \\lambda_X^*: \\mathcal F\\rightarrow (\\lambda_X)_* \\mathcal F^\\mathrm{an} ", "6919f1ed6ea013459d7d1e7757333070": "\n\\nabla^2 \\mathbf{E} - \\frac{n^2}{c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{E}\n= \\frac{1}{\\varepsilon_0 c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{P}^{NL}.", "691a08aa3025d1fec2a869bec1bcb270": "\\|x+y\\|_p \\le \\|x\\|_p + \\|y\\|_p \\ .", "691a79866f3c3702d53becacbcb93d64": "y \\in \\mathbb{F}_q", "691ab6500912ab0fecd20f2a30bcc627": "P\\to Q, \\neg Q \\vdash \\neg P", "691b1bd90b91dab43c165e08f6a764d9": "\n\\sigma^2 =\\frac{1}{5}\\frac{GM}{R}\n", "691b1f92bea3183c6c967405ce2a41ee": "\np = 69 + 12\\times\\log_2 { \\left(\\frac {f}{440\\; \\mbox{Hz}} \\right) }\n", "691b2133e4add0543c2cef4c78071845": "M=(F-B)/D", "691b4b15041befb6f2af412d667a6633": "\\displaystyle{D(f,g)=(u_x,v_x) + (u_y,v_y).}", "691b5ed463962900bf7f07e833263ee3": " \\lim_{n \\to \\infty}P\\left(\\frac{M_n-b_n}{a_n}\\leq x\\right) = F(x)", "691b7c73ab235a3aa235bc888fad127d": "\\hat{\\mathbf{j}}", "691b9b6cd2958163487b171e5e440ce5": "H_{\\mathbf{k}} = \\frac{p^2}{2m} + \\frac{\\hbar \\mathbf{k}\\cdot\\mathbf{p}}{m} + \\frac{\\hbar^2 k^2}{2m} + V ", "691c0fa7458d18c44954c15dd30951e7": "\\; E_{jk}+E_{kj}", "691c12dcbf200d442e8d3884e7bd4d93": "disc(\\mathcal{H}) \\leq \\lambda. \\ \\Box", "691c5c4b29420e50b4553333d175f0f3": "g : \\mathbb{C} \\to \\mathbb{H}. \\, ", "691c911fd96c2f38623aef91334d8880": "\\scriptstyle X_{ip} \\omega_r \\cos(\\omega_r t)", "691ca0f34786cc10cfa45521b71f21ec": "\\mathfrak{P}^{47}", "691ca23aa2b806ffc4c9cd322d6c46a4": "\n \\varphi_\\alpha = w^0_{,\\alpha}\n ", "691ca76be720e8056843484903c0430a": "\\zeta_0 \\leq \\alpha \\leq \\Omega", "691cbc79bca63daea2627c60b5071ce6": "F^{\\mu \\nu} \\, \\stackrel{\\mathrm{def}}{=} \\, \\eta^{\\mu \\alpha} \\, F_{\\alpha \\beta} \\, \\eta^{\\beta \\nu} = \\left( \\begin{matrix}\n0 & -E_x/c & -E_y/c & -E_z/c \\\\\nE_x/c & 0 & -B_z & B_y \\\\\nE_y/c & B_z & 0 & -B_x \\\\\nE_z/c & -B_y & B_x & 0\n\\end{matrix} \\right)\\,.", "691cd6fd688c88e391ee41b7f3535bd7": "\n \\operatorname{Pr}\\Big( \\omega\\in\\Omega:\\, d\\big(X_n(\\omega),X(\\omega)\\big)\\,\\underset{n\\to\\infty}{\\longrightarrow}\\,0 \\Big) = 1\n ", "691d13d3ca2872b923e11f2e4d341925": "t 1", "69290c5bf85fbc75dfa53a9ceb086427": " v_\\mathrm{rms} = \\sqrt{\\langle v^2 \\rangle} = \\sqrt{\\frac{3k_B T}{m}} \\,\\!", "6929de4a0af9904a1df60709e8fb76a2": " -\\sum_{k,l}\\frac{\\partial}{\\partial x_k}\\left(a^{kl}\\frac{\\partial u}{\\partial x_l}\\right) = f", "692a1304afa01b15ec78593d563bbe8b": "\\|f\\|_{H^{s,p}(\\Omega)} := \\inf \\left \\{\\|g\\|_{H^{s,p}(\\mathbb{R}^n)} : g \\in H^{s,p}(\\mathbb{R}^n), g|_{\\Omega} = f \\right \\} ", "692a28478be9b03adfea66cf9aea531e": " \\phi_1\\ \\land\\ \\phi_2\\ \\land\\ \\dots\\ \\land \\ \\phi_n\\ \\ \\le\\ \\ \\psi", "692a48f8dbc8c6dfb3e61a7daee5e154": "(d\\Psi)_{x} : T_{x}G \\to T_{\\Psi(x)}\\mathrm{Aut}(G) ", "692a7dbe9e76bd1911e304507cfffd2d": "\\|\\vec{s}\\|", "692a83e6f7253832eb58788d2a0048cd": "\\begin{align}\n\\frac{\\partial A_i}{\\partial \\lambda_j} &= \\frac{[A_i,A_j]}{\\lambda_i-\\lambda_j} \\qquad \\qquad j\\neq i \\\\\n\\frac{\\partial A_i}{\\partial \\lambda_i} &= -\\sum_{j\\neq i}\\frac{[A_i,A_j]}{\\lambda_i-\\lambda_j}.\n\\end{align}", "692adfd934e838e3f144bd1948f0a3d7": "{\\Bbb P}(V).", "692b036036379f453ed2b6cc77c53193": "\\alpha>1/2", "692b1c570b1b18e01080896a6432ece2": "W_{\\mathcal F}", "692b575f0fbab02c6b92a0926c54c32a": "\\lim_{n\\rightarrow\\infty}f_n=f\\ \\mbox{uniformly}", "692b73d48a71d5b943afb3656d1069d7": "x = \\sqrt{c/a} \\tan\\theta ", "692be7256b90b06e2f67d7d41d695609": "P_{top} - P_{bottom} = - \\rho \\cdot g \\cdot h.", "692c4c02d38a54fbef623527cf26a094": "\\ll \\!\\,", "692c674a747c885bdd468bd90ac7f154": "\\mathsf{R}(a \\wedge b)=-\\mathcal{P}_B (\\mathsf{S}(a) \\times \\mathsf{S}(b)).", "692cb95b56c37d8d58c5177df5ac434a": "\\rho = \\frac{1}{h^n C} e^{\\frac{A - E}{k T}},", "692cff9e0a99ea0d3803dc9b13eeaa52": "v(ab) = v(a) v(b) ", "692d1c7bf90438fe8e7ce5683e743eeb": "S(f) \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{n=-\\infty}^\\infty S[n]\\cdot \\delta \\left(f-\\frac{n}{P}\\right),", "692d4bffca8e84ffb45cf9d5facf31d6": "a\\ b", "692dc27f71496f6390fade05411f9688": "\\mu_t(U_{t+1})=[E_t U_{t+1}^\\alpha]^{1/\\alpha}", "692e5b829a6a0621e69fc43cfcc9a6a0": "\\alpha^{-2}", "692ec34d9441d55f65c5307efa81a238": "(p-1)!\\ \\equiv\\ -1 \\pmod p", "692ecc52baf861acdedd006bcf52ac0d": " [0,2 / \\pi] ", "692f0c97d9a0b2bcc3975b4d0181ba7e": "\\frac{d x}{d t} = f(x) \\quad \\Rightarrow \\quad \\frac{d t}{d x} = \\frac{1}{f(x)} \\quad \\Rightarrow \\quad t + C = \\int \\frac{dx}{f(x)}", "692f37d23568f0d5e3228bdc7476dbf9": "E_2:\\textrm{ a\\ 7\\ is\\ rolled\\ (a\\ loss) }", "692f644f3de967d36ea993c903808bfa": "x \\in \\mathcal{X},", "692f6c4f30b2f303e335bacafef68088": "(10\\cdot x+y)^{10}-1000\\cdot x^{10}=y^{10}+1000\\cdot x^{\\mathrm{1T}}\\cdot y+ 100\\cdot x\\cdot y^{\\mathrm{1T}}=\n\\begin{cases}\n\\mathrm{T}+\\mathrm{T000}\\cdot x^{\\mathrm{1T}}+100\\cdot x, & y=\\mathrm{T}\\\\\n0, & y=0\\\\\n1+1000\\cdot x^{\\mathrm{1T}}+100\\cdot x, & y=1\n\\end{cases}\n\n", "692f97970321cdadbe106ec15e623117": "r = \\frac{a+b-c}{2} = \\frac{ab}{a+b+c}.", "692f9f3271c9e0c87d72627f6c234ac3": "\\mathrm{Glucose} + \\mathrm{O}_{2}\\xrightarrow[\\mathrm{Oxidation}] {\\mathrm{glucose\\ oxidase}}\\textrm{D-glucono-1,5-lactone} + \\mathrm{H_{2}O_{2}} ", "692fb4d49289a197b5a32dc5550194cd": " [u,v,[w,x,y]] = [[u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y]].", "6930df778f37c295d44960b581c5ebbe": "\\frac{|v-c|}{c}<10^{-18}", "69317d9b65cc71aa45b34d86af904d77": "h(r)\\,", "69321a67be4088c00c142e1fe039a6e5": "E_{z,x^2-y^2} = \\frac{\\sqrt{3}}{2} n(l^2 - m^2) V_{pd\\sigma} - n(l^2 - m^2) V_{pd\\pi}", "69322874459d8bc38e78ea01045f5ecc": "x_1 = 0,\\; y_1 = 0", "693252df811653ce7c703c882ffddc4c": "y(t-k)=T( x(t-k), t-k )", "6932c4c255e761a7eb966e7db75a798d": " [I_R] = (-\\sum_{i=1}^n m_i [r_i - C]^2) + (\\sum_{i=1}^n m_i[r_i - C])[d] + [d](\\sum_{i=1}^n m_i[r_i - C]) + (-\\sum_{i=1}^n m_i)[d][d].", "6932eb22ea102e265c40cbd0e7671cfd": "{\\gamma}_{23}", "6932f87ad3ba67b638e86b23dd6258d7": "\\begin{align}\nf_Z(z) &= \\int_{-\\infty}^\\infty \\frac{1}{\\sqrt{2\\pi}\\sigma_Y} e^{-{(z-x-\\mu_Y)^2 \\over 2\\sigma_Y^2}} \\frac{1}{\\sqrt{2\\pi}\\sigma_X} e^{-{(x-\\mu_X)^2 \\over 2\\sigma_X^2}} dx \\\\\n&= \\int_{-\\infty}^\\infty \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma_X^2+\\sigma_Y^2}} \\exp \\left[ - { (z-(\\mu_X+\\mu_Y))^2 \\over 2(\\sigma_X^2+\\sigma_Y^2) } \\right] \\frac{1}{\\sqrt{2\\pi}\\frac{\\sigma_X\\sigma_Y}{\\sqrt{\\sigma_X^2+\\sigma_Y^2}}} \\exp \\left[ - \\frac{\\left(x-\\frac{\\sigma_X^2(z-\\mu_Y)+\\sigma_Y^2\\mu_X}{\\sigma_X^2+\\sigma_Y^2}\\right)^2}{2\\left(\\frac{\\sigma_X\\sigma_Y}{\\sqrt{\\sigma_X^2+\\sigma_Y^2}}\\right)^2} \\right] dx \\\\\n&= \\frac{1}{\\sqrt{2\\pi(\\sigma_X^2+\\sigma_Y^2)}} \\exp \\left[ - { (z-(\\mu_X+\\mu_Y))^2 \\over 2(\\sigma_X^2+\\sigma_Y^2) } \\right] \\int_{-\\infty}^\\infty \\frac{1}{\\sqrt{2\\pi}\\frac{\\sigma_X\\sigma_Y}{\\sqrt{\\sigma_X^2+\\sigma_Y^2}}} \\exp \\left[ - \\frac{\\left(x-\\frac{\\sigma_X^2(z-\\mu_Y)+\\sigma_Y^2\\mu_X}{\\sigma_X^2+\\sigma_Y^2}\\right)^2}{2\\left(\\frac{\\sigma_X\\sigma_Y}{\\sqrt{\\sigma_X^2+\\sigma_Y^2}}\\right)^2} \\right] dx\n\\end{align}", "69330b80bb8ca62b997c413ad875a78b": " \\mathrm{Re} = {{\\rho {\\mathbf v} D_H} \\over {\\mu}} = {{{\\mathbf v} D_H} \\over {\\nu}} = {{{\\mathbf Q} D_H} \\over {\\nu}A} ", "693338e9d1f3691c468883f4aec6cc1e": "X_n \\to \\Omega X_{n+1}", "69337add0fdd781e3d6024b919727dda": "\\langle f|g\\rangle = \\langle(\\hat{A}-\\langle \\hat{A} \\rangle)\\Psi|(\\hat{B}-\\langle \\hat{B} \\rangle)\\Psi\\rangle,", "6933e8042a08493194552d0105e0be68": "u[8] := 2*atan(\\sqrt((a0+b1)/(b1-a0))*cot(\\sqrt(b1^2-a0^2)*\\eta+sec(h*\\sqrt(b1^2-a0^2)*\\eta)))+(1/2)*pi", "69342d9321b655f3b93d6fbaa5123eb5": "A = \\prod_p\\left({1 - \\frac{p-1}{p^2(p+1)} }\\right) \\ . ", "693439806376a50a86b6dffb0e35eb6c": "d(a, b) \\ge 0 ", "693454559192ef0f254bc9bbf9a35716": "|Y_{i}-Y_{j}|^2 \\leq N \\tau \\,\\!", "6934aee5801e650a78597bd93db1cbf3": "\\sqrt[5]{\\pi^3+1}\\approx 2,", "6934c6a0b279c80b9a5b2c8a22fc9a8b": "\\beta_n = \\frac{\\pi n}{M},", "69352906ffcf0a6cb66c7b272009e194": " z_{cr} = 0 \\,", "69353450d2f92ada8eb9054d3d36949e": "X=\\sigma\\sqrt{-2 \\ln(U)}\\,", "6935415b0de3e278ef00ce5a65df9269": "x(na + mb) + yc = (xn)a + (xm)b + yc = \\gcd(a,b,c).\\,", "6935526e92b02b7a335114c24a22d599": "\n\\begin{matrix}\nA \\wedge \\left ( B \\wedge C \\right ) \\ true \\\\\n\\vdots \\\\\nB \\ true\n\\end{matrix}\n", "69361f5cfb0682e1796e69b9ad8d6c72": "v^2 = v_r^2 + v_{\\theta}^2 + v_z^2", "69363d6ab3ab1d67450118597dc1d99b": "{\\rm coATIME}(C,j)=\\Pi_j {\\rm TIME}(C)", "69366a1ec4453cc2b3dbc8fb932a302e": "\\sum_j W_i(x_j) h_i(x_j) y_j > 0", "69366fed5234c254a0a21e714fe40129": "I_0 = I_\\mathrm{rms} \\sqrt{2}\\,\\!", "6936d7b379908cda4c45639650bca081": "TIR= I\\times RR ", "693710bc761280d82dd7ca64dc35237e": "\\ \\sqrt{2\\pi}", "69373157980be6249e1229254152b7c6": "G(v) = \\mathfrak{D}-X_n(C_n)[v_{\\mathfrak{D}} - v]^{\\frac{2n}{n-2}}.", "69375c918842b089df0233edb6bf7b72": "f^* = \\frac{1} {L{*}}", "69375f21da6078657c9d4db0f7a39398": "\\|(y+t a)-x\\|^2-\\|y-x\\|^2=2t\\langle y-x,a\\rangle+t^2 \\|a\\|^2=2t\\langle y-x,a\\rangle+O(t^2)", "693795226b619ac051d6d0fd162c9f56": "\\mathbf{F}_{13}", "69379a3f13765926a805e9771b0c9b3c": "(\\forall n\\in\\mathbb{N}\\setminus\\{1\\}):x^2I_n(x)=2n(2n-1)I_{n-1}(x)-4n(n-1)I_{n-2}(x)", "6937f376cb9e777384ab8309d89988bb": "~x'=\\rho' \\cos \\omega'; y'=\\rho' \\sin \\omega'", "693809668896c5269fdac8569393c83c": "\\varepsilon \\nabla^2 \\phi + \\rho = 0.", "69381aaf692412b559e7f1abdecac0c7": "\\mathrm{EOT} = t_\\text{high-k} \\left( \\frac{k_{\\text{SiO}_2}}{k_\\text{high-k}}\\right)", "693821090942b624059dd1ac5d197a52": "L-u'\\frac{\\part L}{\\part u'}=C \\, ,", "69384345b8807967c47e5544c68c7afc": "x_4\\,\\!", "69384e3f26a46f67fb7106f41f8fac6e": "f,g\\in \\mathcal{F}(X)", "6938c1fa1fcd0b360473c72eb321a4d2": "O\\left(\\frac{N (\\log \\log N)^5}{\\log N}\\right)", "6938d9d43bb3763f4e744d9d9abeb86a": " k_r[ES]", "69397a07d696ffc71e9c788eaec5172d": "\\delta(g(x)) = \\tilde{g}(\\delta(x)).", "6939a40bf5e178a28894862de09e9025": "\\partial^2 / \\partial t^2(\\boldsymbol{u}^{(0)}) = \\partial^2 / \\partial t^2(\\boldsymbol{X}) = 0 ", "6939fbe0229857e4ab2cc68640698ca5": "T_{old} - T_{new}", "6939ff508b8745ab0749d94583f5e403": " \\delta W = \\sum_{j=1}^m Q_j\\delta q_j,", "693a106766f807b3e507595a02d7a77e": "\n \\sup_{x\\in\\mathbb R} |F_n(x) - F(x)|\\stackrel{d}{=} \\sup_{x \\in \\mathbb R} | G_n (F(x)) - F(x) | \\le \\sup_{0 \\le t \\le 1} | G_n (t) -t | ,\n ", "693a315e08a9b1bd7ff34452ce9a6cc7": "\\left ( \\frac{12345}{331}\\right )=\\left ( \\frac{98}{331}\\right )=\\left ( \\frac{2 \\cdot 7^2}{331}\\right )=\\left ( \\frac{2}{331}\\right )=(-1)^\\tfrac{331^2-1}{8}=-1.", "693a3b974c23e87e8c941211cd45cfb8": "A_i", "693b3a8f7bcfc067f6203f4ad5c87a2a": "X\\left(t\\right)=\\textrm{log} P\\left(t,T\\right).", "693b4567d10c687ca1b9e818ff6a040f": "\\operatorname{head} \\equiv \\lambda l.l\\ (\\lambda h.\\lambda t.h)\\ \\operatorname{false}", "693bb63d3f3141b007d1355383643e5a": "(k,h)", "693bce814c51ba351f57429b36787af2": "\\alpha\\Vdash p_j", "693cf691f84cba3ba32457dbdc51f33b": "\n1, 2, 11, 170, \\ldots = \\prod_{j=0}^{n-1} \\left( 3j + 1\\right)\\frac{ (2j)!(6j)!}{(4j)!(4j + 1)!}\n", "693d08500e823f0efce072d8eec2f8c1": "\n\\begin{align}\n& \\int_{\\boldsymbol{\\varphi}} \\prod_{i=1}^K P(\\varphi_i;\\beta) \\prod_{j=1}^M \\prod_{t=1}^N P(W_{j,t}|\\varphi_{Z_{j,t}}) \\, d\\boldsymbol{\\varphi} \\\\\n= & \\prod_{i=1}^K \\int_{\\varphi_i} P(\\varphi_i;\\beta) \\prod_{j=1}^M \\prod_{t=1}^N P(W_{j,t}|\\varphi_{Z_{j,t}}) \\, d\\varphi_i\\\\\n= & \\prod_{i=1}^K \\int_{\\varphi_i} \\frac{\\Gamma\\bigl(\\sum_{r=1}^V \\beta_r \\bigr)}{\\prod_{r=1}^V \\Gamma(\\beta_r)} \\prod_{r=1}^V \\varphi_{i,r}^{\\beta_r - 1} \\prod_{r=1}^V \\varphi_{i,r}^{n_{(\\cdot),r}^i} \\, d\\varphi_i \\\\\n= & \\prod_{i=1}^K \\int_{\\varphi_i} \\frac{\\Gamma\\bigl(\\sum_{r=1}^V \\beta_r \\bigr)}{\\prod_{r=1}^V \\Gamma(\\beta_r)} \\prod_{r=1}^V \\varphi_{i,r}^{n_{(\\cdot),r}^i+\\beta_r - 1} \\, d\\varphi_i \\\\\n= & \\prod_{i=1}^K \\frac{\\Gamma\\bigl(\\sum_{r=1}^V \\beta_r\n\\bigr)}{\\prod_{r=1}^V \\Gamma(\\beta_r)}\\frac{\\prod_{r=1}^V\n\\Gamma(n_{(\\cdot),r}^i+\\beta_r)}{\\Gamma\\bigl(\\sum_{r=1}^V\nn_{(\\cdot),r}^i+\\beta_r \\bigr)} .\n\\end{align}\n", "693d57d85b93a386aed9fdb75ec85c28": "(1 + x) \\sum_{k=0}^n \\; {\\alpha \\choose k} \\; x^k =\\sum_{k=0}^n \\; {\\alpha+1\\choose k} \\; x^k + {\\alpha \\choose n} \\;x^{n+1}, ", "693df994ad0664958e81a1c849d2be9d": "S[p_1-1] \\neq S[p_2-1]", "693e13de7523a77a73240eefb3111ad9": "x_1 -x_2 \\le 1", "693e37a646399b8581861319f50bc7be": " \\ln(Y) = \\ln(A) + \\frac{\\ln[\\alpha K^\\gamma + (1-\\alpha) L^\\gamma]}{\\gamma}", "693e682af8a6287c8b6ebd6811f8fa4b": "\nP(x=v|c)=\\tfrac{1}{\\sqrt{2\\pi\\sigma^2_c}}\\,e^{ -\\frac{(v-\\mu_c)^2}{2\\sigma^2_c} }\n", "693e6ccc4f588567348f12d82a96a772": "\\tilde{6}\\cdot m", "693fd4b43ebe9484558c8a1e97801455": "\\int_0^{t_{cr}}F_{cr}(t)\\,dt=-m_f v_f=m_p v_p", "693fdc63fe22ba62f04e3941c96fffa3": " \nN_{nl} = \\left[\\frac{ 2^{n+l+2} \\,\\gamma^{l+\\frac{3}{2}} } {\\pi^{\\frac{1}{2}}}\n \\right]^{\\frac{1}{2}}\n \\left[\\frac{ [\\frac{1}{2}(n-l)]!\\;[\\frac{1}{2}(n+l)]!}{(n+l+1)!}\n \\right]^{\\frac{1}{2}} .\n", "694047faefad1ec0f853e16cdc7364a8": "\\frac{Rw_i}{(w_1+w_2+...+w_N)}", "694062068a6a4f5131a8d841176dee48": " \n\\phi(x)\\sim-\\frac{1}{2\\pi}\\ln{m|x|} \\qquad\n{\\rm for}\\quad x\\sim 0\n", "69411b1640f25270ccfa1e8ea83ec55d": "\\epsilon = | \\epsilon_a - \\beta \\epsilon_G | ", "69413ab64fe0f6f45ff4fd98d8dbb7a5": "r\\colon L\\rightarrow R", "69416a72a7c6a779b5b562c9a5a5df62": " SU(n) \\supset SO(n),", "69420624b726d3dddd4611adbe9ff24b": "Q_{n + 1} = 2 M (Q_n^{-1} M + M^\\mathrm{T} Q_n)^{-1}", "6942c014e92f4b83ccc8c55a6975e44a": "(C \\sqcap D)^{\\mathcal{I}} = C^{\\mathcal{I}} \\cap D^{\\mathcal{I}}", "6942cf05cb0188b1e8e3129445991760": "\\,f", "6942d3aa4faf680bc83f8a9a08c07bf1": "\n \\begin{bmatrix}\\sigma_{11}\\\\ \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix} = \n \\begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\\\\n c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\\\\nc_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\\\\nc_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\\\\nc_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\\\\nc_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212} \n \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11}\\\\ \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ 2\\varepsilon_{23} \\\\ 2\\varepsilon_{31} \\\\ 2\\varepsilon_{12} \\end{bmatrix}\n ", "6942e4b02efe951b6b2053543c1610fa": "\n\\int\\left[\\phi^2+\\left(\\nabla\\phi\\right)^2\\right]\\,dV.\n", "6942f97c51562162565e762c44d89451": "\\dot{u} \\equiv \\text{d}u/\\text{d}t", "6943205ecfba95b8145c1421379fbd05": " c_9 = 0 \\,\\!", "69435d5078b3fdb35cba67db55ee3d48": "m_{\\rm e}", "694373150b0ac013a1f22014a5bf2344": "r_\\infty", "6943756c2d28cc3a4c813dfaa368067b": "\n\\begin{align}\n z &= x + iy =\\zeta+\\frac{1}{\\zeta}\n\\\\\n &= \\chi + i \\eta + \\frac{1}{\\chi + i \\eta}\n\\\\\n &= \\chi + i \\eta + \\frac{(\\chi - i \\eta)}{\\chi^2 + \\eta^2}\n\\\\\n &= \\frac{\\chi (\\chi^2 + \\eta^2 + 1)}{\\chi^2 + \\eta^2} + i\\frac{\\eta (\\chi^2 + \\eta^2 - 1)}{\\chi^2 + \\eta^2}.\n\\end{align}\n", "6943b00593d4335f50e8d4b4c00efc3c": "S \\otimes_R S", "6943c7db5cb0d7ab5fef248f84d89e31": "a_{vf}", "6943c9b2189e2fafc0cb6661bcec05d9": "|\\psi\\rangle = \\alpha_0|0\\rangle + \\alpha_1|1\\rangle", "6943fe1ef0b252e033ccc9f9a08677d9": "y_{t=1 \\dots T}", "694406bfa7a779e074029609028674bc": "2P_{3/2} \\to 1S_{1/2}.", "69442a943053c66099258bcb79e81634": "\\mathrm{NSPACE}(g(n))\\subseteq\\bigcup_{c>0}{\\rm ATIME}(c\\times g(n)^2),", "69442c700196a079f83cc059968e722d": "p(\\tilde{x}|\\alpha) = \\int_{\\theta} p(\\tilde{x}|\\theta) p(\\theta|\\alpha) \\operatorname{d}\\!\\theta", "69444b0ee409e3646866fd34a8aea5e3": "\\Delta f = \\sum_{i=1}^n \\frac {\\partial^2 f}{\\partial x^2_i}", "69448a3a3ca20e57a6fe1e86ed3619ef": "\\theta = \\arctan \\left( \\frac{y}{x} \\right)", "69449f5143112c1857ce618e1f30f7fb": "\\ln x \\approx \\frac{\\pi}{2 M(1,4/s)} - m \\ln 2,", "6944dbd273e1cc29f7a43731ef0f9b02": "_{p \\nleftarrow 0=0}\\!", "694507c1e894c547f997acdb23add672": "f_x(0,0) = p_x(0,0) = a_{10}", "69458df6239725ca5df74809d71177da": " \\mathbf{F}_\\mathrm{ext}", "6945a3380bf20c0bd181081d0dc876da": "E_{AB}^{\\rm disp}", "6945f329a8320b2388b5862fe9fecf55": "m \\left.\\frac{\\partial C_L}{\\partial x}\\right|_{x=0} > \\frac{\\partial T}{\\partial x}", "6945fe7d56248b88df0b734ff07e6711": " M^* = \\sup_{n \\ge 0} \\, |M_n|.", "69462e0d2573e1bcf3f118ef72868be0": "p \\Rightarrow r", "69467b281edca763a8bd912bc4797219": "c_1 y_1 + c_2 y_2 = c_1e^{2x} (\\cos x + i \\sin x) + c_2 e^{2x} (\\cos x - i \\sin x) = (c_1 + c_2) e^{2x} \\cos x + i(c_1 - c_2) e^{2x} \\sin x", "6946c21a45f8fd8b30ae1c446c7d89c7": "F^\\chi(s)=\\sum_{n=1}^\\infty\\frac{\\chi(n)a_n}{n^s}", "6946fc5461b81677080ff6d490495d49": "\n w(r,t) = \\sum_{n=1}^\\infty C_n\\left[J_0(\\lambda_n r) - \\frac{J_0(\\lambda_n a)}{I_0(\\lambda_n a)}I_0(\\lambda_n r)\\right]\n [A_n e^{i\\omega_n t} + B_n e^{-i\\omega_n t}] \\,.\n", "69471f2f3526a28e3485b0955bf55d90": "\\dot{\\theta}", "69472b46df7423e94d3111119e2f798c": "s_C", "69473f48c19a7a970824b423def05d20": "\\pi_1(G)", "6947c70867d2fc66c7b504de7def3f8f": "\nH_{\\text{kin}} = \\frac{1}{2}mv^2 = \\frac{p^2}{2m},\n", "694873715955037832eeddb9cacc8546": "f_i(n+1)", "69488ed85c5d0f55251c7c62374ed7ac": "\\Delta(e_i) = e_i \\otimes e_i", "6948905ae746a3fdb3c2788e39e8f5d8": "\\frac{v^{2}}{2 g}", "6948e6e130b7bade8b8bf147e1fa40a0": "A_\\varepsilon(y_1,\\ldots,y_N)=A_\\varepsilon(y_{\\sigma(1)},\\ldots,y_{\\sigma(N)})", "6948f574102efdd34a31a10c660e910d": "f(2,3) = (2) - 2(3) + 2 = 2 - 6 + 2 = -2", "694997266d002d2cf63fabcf6d529343": "\\textbf{L}_{k} = \n I - \\textbf{C}_{k} ", "69499b8066f298cf9ca2f341261e77ad": "\\int_{0}^{1}\\int_{0}^{1}\\frac{-\\log(xy)}{1-xy}\\tilde{P_{n}}(x)\\tilde{P_{n}}(y)dxdy=\\frac{A_{n}+B_{n}\\zeta(3)}{\\operatorname{lcm}\\left[1,\\ldots,n\\right]^{3}}", "694a3f5250913dbdb060f431f4cdf87a": "\\aleph_\\omega^{\\aleph_0}", "694aab3155c9f9c792fa1748afcaf170": "\\mathrm{adiabatic},\\,{A\\to O}\\,", "694ab0364110605ed84febd7cd0059e6": "\n \\overset{\\triangle}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{\\sigma}\\cdot\\boldsymbol{w}\n - \\boldsymbol{w}\\cdot\\boldsymbol{\\sigma} \n", "694ad7f242d234152dc0d6ea33ffbab3": "\\{ \\mathbf{x} \\in \\mathbb{R}^n : \\sigma^{\\text{T}}(\\mathbf{x})\\dot{\\sigma}(\\mathbf{x}) < 0 \\}", "694afc4ca61d07ab9610b98b9524bb10": "\\mathfrak P_3(K)", "694b606eacd8946a6f319214f7dd625e": " u(\\mathbf{x}) = \\int_{\\mathbf{x}'} d\\mathbf{x}'G(\\mathbf{x},\\mathbf{x'})f(\\mathbf{x'})", "694b63b2df852b9d561a43fa6e46317a": "\\log \\left|\\frac{g(0)}{h(0)}\\right| = \\log \\left |r^{m-n} \\frac{a_1\\ldots a_n}{b_1\\ldots b_m}\\right| + \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log|f(re^{i\\theta})| \\, d\\theta.", "694bcbf178e66e5f2398017945cd481c": "\\deg f(x) = 2g+2", "694bf2c5924932717bebea477396e22c": "\\mathbf{i} = \\begin{pmatrix}\n 1 \\\\\n 0 \\\\\n 0\n\\end{pmatrix}, \\mathbf{j} = \\begin{pmatrix}\n 0 \\\\\n 1 \\\\\n 0\n\\end{pmatrix}, \\mathbf{k} = \\begin{pmatrix}\n 0 \\\\\n 0 \\\\\n 1\n\\end{pmatrix}\n", "694bfe90837c9f7cffbc94a609e953f1": "\\mathbb{C} \\!\\,", "694c05fe02d474b3332ec3214d8077cf": " t \\approx \\frac{70 + (r - 2)/3}{r} ", "694cbda8b04985b1c0f609345f00fed4": "b\\in \\hat\\Sigma", "694cffca52ce5c29fca9d4b063d34b6b": "S\\mu^T\\cdot\\mu^STT\\cdot SlT", "694d420236e02e4d42c22208715e6e7a": "F_k(a, b) = a^b", "694d6a52ef0913358e4ca1f250afab1e": "H^*(G_1\\times G_2;k)\\cong H^*(G_1;k)\\otimes H^*(G_2;k).\\,", "694da293d9a640435489755d80fd89e6": "Y_{4}^{3}(\\theta,\\varphi)={-3\\over 8}\\sqrt{35\\over \\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot\\cos\\theta\n= \\frac{- 3}{8} \\sqrt{\\frac{35}{\\pi}} \\cdot \\frac{(x + i y)^3 z}{r^4}", "694dddfd33b6d691d380b527dba90042": "\\scriptstyle T_\\text{UB}", "694e2938c556fc90499f6ac46c849e5e": "\nL = p_{\\phi} = m \\, r^{2} \\frac{d\\varphi}{d\\tau}\n\\,,", "694e3bed576b5e152bdbfa6df177f83d": "a_{i-1}^3", "694e72d0858191d2348c0a3ac40c2264": " a_n ", "694ebc1c3348737a1c6752e10893d386": "P^S_{solid}", "694ec67c42acde23a9a381990c4136f0": "\\Delta g\\,", "694ed80e5007bc9da34bdcdc11e153b8": "\\scriptstyle{\\cos\\psi}", "694f3e988821faacac843d6f952f876c": "\\lambda_5 = \\begin{pmatrix} 0 & 0 & -i \\\\ 0 & 0 & 0 \\\\ i & 0 & 0 \\end{pmatrix}", "694f47b8770df00e3a511d846e925d93": "H(E_i)", "694f488990f945b2816976ee0d1f8e3c": " \\mathrm{O}(n) \\supset \\mathrm{O}(n-1) ", "694f6e46a3f8def895cb0e020994b31a": "\\int_\\Omega f(x)\\ dx", "694f7ba8589937689509c7380a5c8a33": "(x+2)(x+2)=0(x+2)", "694f95b840dd412eda980370ef192277": "\\textit{odd} \\subseteq \\textit{int}", "694fcf72b18e8d65a9144e63c6bc62f1": "F(x+h)", "6950135e61f0f81e8be9e1077c83c79d": " \\hat O ", "69502ccf989f5363750b819839e91a70": " X_n := \\left(0,\\frac{1}{n}\\right) \\cup (n,n+1) = \\left\\{ x \\in {\\bold R}^+ : 0 < x < \\frac{1}{n} \\ \\text{ or } \\ n < x < n+1 \\right\\}. ", "69503f17ebdd030584925b36751ac25e": "= e_\\gamma^I R_{\\alpha \\beta I}^{\\;\\;\\;\\;\\;\\; J} e_J^\\delta V_\\delta", "695073dd03b03a54c9ab04d4c46d6368": "\\left|\\Psi_m^{(0)}\\right\\rangle = \\left|\\Phi_c \\Psi_m^v\\right\\rangle ", "69508e98cfdc0ce6c08c49f7ce209251": "\\sigma_r\\gg\\sigma_t", "6950db756cb1afdc71929fee2aa24375": "\\mu(X)=\\#(X(k))", "695144a458cd65216df5a238e8e6812f": " \\mathbf{f} = \\nabla\\cdot\\boldsymbol{\\sigma} - \\dfrac{1}{c^2} \\dfrac{\\partial \\mathbf{S}}{\\partial t} \\,\\!", "6951818f911351ede3f1ffd93fd56165": "\\log(x) + \\log(y) = \\log(xy)", "6951e4fc125306804acf01a826ba5d0c": "= 1 - \\frac{x}{zx+x+1} - \\frac{y}{xy+y+1} - \\frac{z}{yz+z+1} ", "695214cc14751c01dbea162d75ad5a57": "j \\in C_i", "695284a18b1ad54ef027caac8dc7e590": "E_A = \\frac 1 {1 + 10^{(R_B - R_A)/400}}.", "69529809f2bfb3885c1f8756616587fd": " f^-(x) = -\\min(f(x),0) = \\begin{cases} -f(x) & \\mbox{ if } f(x) < 0 \\\\ 0 & \\mbox{ otherwise.} \\end{cases} ", "6952d774eb1ebbda69f7fb3b3978d762": "\\sigma_\\infty", "6952efba0eaa9706c98f1d4a3a4819f2": "(-2)\\cdot((-1)\\cdot(-1)-2\\cdot3)+1\\cdot((-2)\\cdot(-1)-2\\cdot(-3))", "69531e65f50b49ae15591d8df6260ace": "(p_n\\circ q)(x)=\\sum_{k=0}^n a_{n,k}q_k(x)=\\sum_{0\\le k \\le \\ell \\le n} a_{n,k}b_{k,\\ell}x^\\ell", "69532e0dad08b96d7bc2e77a6c545e96": " i \\in p ", "695377ea91bd0edf1a10402dae7b5380": " \\mathrm{ ICS } = a m^{ b - 1 } - 1 ", "6953d4becc964d29cb30f3a363d480bb": "M_{\\rm limit} \\approx N^2 \\left(\\frac{\\hbar c}{G}\\right)^{3/2}.", "69540f87836180c95cfdcefde7e2dfc8": "X,Y\\in M", "69548a594929897e23a11219831e0b27": "\\,g=e^{iaQ+ibP+ic},", "6954f88658d42213f37585e1c9ce0835": "[I_k, I_k] \\subseteq I_{k+1}", "6954faf6df2c21cc6290546729562690": "s_p", "69551a2a5afdc25ba6ed536092ebd79f": " 0 < a < b, a, b \\in \\R", "69556a7bf51d1ea624becfc0fa1195de": "\n\\sqrt{n}(\\hat\\beta_n - \\beta) \\xrightarrow{d} N(0,\\Omega),\n", "6955bc00f11df19efc33a68a01c440b2": " \\frac{v_\\text{object}}{v_\\text{sound}} ", "695617aa4d2b4e1c93f4ffa59c529d83": "A_z\\,\\!", "695626df2402feca7bcad2689c1c02be": "N\\setminus S \\notin W", "6956673d628b66acfdc44579054af545": " v^2 = u^3 - 27D(D^3 + 8)u + 54(D^6 - 20 D^3 - 8), \\, ", "6956891ab0704d9adc206a5c62358428": " \\mathrm{div\\,} \\mathbf{A} = {\\partial A_x \\over \\partial x} + {\\partial A_y \\over \\partial y} + {\\partial A_z \\over \\partial z}", "695696f77982f93a3ee83122533327c1": "P = 0", "695707b15e05178cf921ce452770c1e8": "\\scriptstyle |f| = \\sum_{v:(s,v) \\in E} f_{sv}", "6957156d2e3327aac765587f4634f084": "\\begin{bmatrix} h_{11} & h_{12} \\\\ h_{21} & h_{22} \\end{bmatrix}", "69573a170b1ff983d504f8dad5cca959": " K \\, ", "69575ad56b64046d3bf577954452da15": "a(h_R)\\;=\\;(1.1\\log f\\;-\\;0.7)h_R\\;-\\;(1.56\\log f\\;-\\;0.8)", "69577cad5848705a266d3f20fe0164bb": "\\Pr(P_i|X)", "695798a034782dc09992eb6b0a03002b": " E = \\sqrt{ \\frac{2 U(\\phi_{0})}{\\phi_{0}^{2}}}~ Q .", "6957dbcec6d2618bd49b994dd5ace1b9": "\ne= 2+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{2}{3+\\cfrac{3}{4+\\cfrac{4}{5+\\ddots}}}}} = 2+\\cfrac{2}{2+\\cfrac{3}{3+\\cfrac{4}{4+\\cfrac{5}{5+\\cfrac{6}{6+\\ddots\\,}}}}}\n", "6957fa114f868074726775374bc992e2": "P'Q+P-MC=0", "695852000f3edf221ea53d8ff96b1c2b": "k=1,2,\\dots", "695862dadf1b99271e9b37f6ebc5f90e": "\n(Tp)(z)=\\bar a_0p(z)-a_n p^*(z).\n", "695866264ac3dee7fd30eec022678535": " \\frac{dV}{dt} = - \\mu V, V(n T^+) = V(n T^-) + p S(n T^-) n=0,1,2,\\dots", "6958d13d4ae2a72ff62fbaf5f6115602": "V_0,V_1,...,V_b", "6958d2ee65b0ad26e16f9cbc195917f8": "\\{n: \\varphi_n \\in A\\}", "6958d64d02bd5c0581769a43f9f39214": "|f(x)| \\le \\; M |g(x)|", "6958ef725e3dd9e14074970d1f9dfcec": " \\!\\ S_m^2 = ma + mb + 1 ", "6958ef798babcb1a297e55a9826a8339": "2^\\sqrt{2},", "69590888c0eabc6eda94dc7ff86a5a60": "\\bold{r}\\rightarrow \\bold{r} + \\bold{a}", "695947217cbe4caf0661d8b1f5ba4e5f": "\\hat{\\mathbf{x}}_{k\\mid k} = \\hat{\\mathbf{x}}_{k\\mid k-1} + \\mathbf{K}_k(\\mathbf{z}_k-\\mathbf{H}_k\\hat{\\mathbf{x}}_{k\\mid k-1})", "695953a13213af3ff218bbe96180e9b2": "M_J \\sim \\rho^{\\frac{3}{2}\\left(\\gamma-\\frac{4}{3}\\right)}.", "6959b96ff79542a989b191acaf631fe0": " x_{i+\\frac{1}{2}} \\ ", "695a3c044801ec66b58771183f875223": "i_{limiting} = \\frac {nFD} {\\delta} C^*", "695a4afdd72a64f2593357183b1da102": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{q,p}^{\\,n,m} \\!\\left( \\left. \\begin{matrix} 1-\\mathbf{b_q} \\\\ 1-\\mathbf{a_p} \\end{matrix} \\; \\right| \\, z^{-1} \\right),\n", "695a51deea636fb19cd79069246cf907": "\\sum_{s'\\in S} \\mathcal{T}(s,s')=1", "695a7c721a047ddb2f379404e822058f": "{n}", "695a96a2a19cc62b38e71a64acaf7297": "\\nabla_\\alpha", "695aabe593f2ac6de18b7709813c5cf0": "a,b,c\\in S", "695aca03c8a1c1bdfe234315170d621c": "n+o(n)", "695b27210cc33c8effda43ecd9fc20b0": "T+1", "695b61c1e6f83980cf01f93970073e2b": "w = (a + b)z(1 - z)", "695b6f1d9a834509b7d5e9d33cd87bc3": "E\\times iC", "695b835da2df832721761e74b46b4b9c": "\\frac{d}{dx}{(\\mu{y})} = \\mu{q(x)}", "695b9f2bb362ce19683a58d35572318b": "(q \\to p)", "695ba69e05f25cef7e10a9e37acf32f9": " k = R/n ", "695c1d876f408cbe2148443c51964e64": "\\, 2^3=8", "695c24cf7e57a8fd384dc4c6fa36aa10": "\n\\begin{array}{lcl}\n\\Pr(w_{dn}=v\\mid\\mathbb{W}^{(-dn)},\\mathbb{Z},\\boldsymbol\\beta)\\ &\\propto\\ & \\#\\mathbb{W}_v^{k,(-dn)} + \\beta_v \\\\\n\\Pr(z_{dn}=k\\mid\\mathbb{Z}^{(-dn)},w_{dn}=v,\\mathbb{W}^{(-dn)},\\boldsymbol\\alpha)\\ &\\propto\\ &(\\#\\mathbb{Z}_k^{d,(-dn)} + \\alpha_k) \\Pr(w_{dn}=v\\mid\\mathbb{W}^{(-dn)},\\mathbb{Z},\\boldsymbol\\beta) \\\\\n\\end{array}\n", "695c38a6fdb95d111d2d600db7e238c7": "\\ C_{rr} = 0.0048 (18/D)^{\\frac{1}{2}}(100/W)^{\\frac{1}{4}} ", "695c668c0a6aa61c99d74534e44e07fc": "\\mathrm{d}x", "695ce209341d5cd62c1c86d070a21634": "\\text{M}", "695d32fd1ab17389c8a12f92ee7c16da": " L_2 = -a_2 Z^{-2} \\, ", "695d8274acef11031b3a3bcadf99fb0a": " 1/5", "695dd73cde38f068109c88ba341d1250": "\\Gamma \\varphi(t) = y_0 + \\int_{t_0}^{t} f(s,\\varphi(s)) \\, ds.", "695ddfa27543b13b92c2505bd332b8e1": "\n\\begin{align}\nD & = \\left[ A(x_0+1) + B (y_0+1/2) + C \\right] - \\left[ A x_0 + B y_0 + C \\right] \\\\\n & = \\left[ Ax_0 + B y_0+ C + A + \\frac {1}{2} B\\right] - \\left[ A x_0 + B y_0 + C \\right]\\\\\n & = A + \\frac{1}{2} B\n\\end{align}\n", "695e1e73d8b9b5be137c2849debe8177": " \\gamma_{ttx} ", "695e2a4d4b550b77f38620d236048b34": "AA^+A = U\\Sigma V^*V\\Sigma^+U^*U\\Sigma V^* = U\\Sigma\\Sigma^+\\Sigma V^* = U\\Sigma V^* = A", "695e74b16237b4811c658cbf431180f7": "R_1 \\cap R_2 \\rightarrow R_1 - R_2", "695e8a7f845316b56ff03fa229b3ea95": " a \\mid b ", "695f0e6642f244bffc26ae9b68e63fb4": "\\omega^{n_1} c_1 + \\omega^{n_2} c_2 + \\cdots + \\omega^{n_k} c_k", "695fd3740341a0ca949a9c8cdc60ab60": "\\text{Min} =\n\\begin{cases}\n f\\left(1.34941, -1.34941\\right) & = -2.06261 \\\\\n f\\left(1.34941, 1.34941\\right) & = -2.06261 \\\\\n f\\left(-1.34941, 1.34941\\right) & = -2.06261 \\\\\n f\\left(-1.34941,-1.34941\\right) & = -2.06261 \\\\\n\\end{cases}\n", "69601e3d6e0626830e7712475065445a": "\\displaystyle{\\|\\int_X T(x)v\\, d\\mu(x)\\| \\le \\sqrt{AB} \\cdot \\|v\\|.}", "69604af04ae779274c7e655651db3bc0": "4^4 + 3^3 + 8^8 + 5^5 + 7^7 + 9^9 + 0^0 + 8^8 + 8^8", "69605a3495f028d91c275c8f6efd4977": " (L \\partial) = \n \\mathbf{e}_0 \\partial_0 L + (\\partial_1 L) \\mathbf{e}_1 +\n (\\partial_2 L)\\mathbf{e}_2 + (\\partial_3 L) \\mathbf{e}_3 \n ", "69607bd4b145de451ebc2978cbfca056": "\\theta = \\big( \\boldsymbol{\\tau},\\boldsymbol{\\mu}_1,\\boldsymbol{\\mu}_2,\\sigma_1,\\sigma_2 \\big)", "6960ae9ac801a36f8e848425ec2f5009": "I(n) = \\int_0^\\pi \\sin^nxdx", "6960cf8af38172908224cf5175c029bf": " \\frac {1} {s L_M} \\, ", "6960ebea899d313cef91de89cc1cb28a": "\\{x\\}\\,", "6960eee62ffe573b10851e5b8f4b5c5b": "B_2 (r, \\mu ) = \\frac{ \\left \\langle\\sigma_y^2(2, T, \\tau ) \\right \\rangle}{ \\left \\langle\\sigma_y^2(2, \\tau, \\tau ) \\right\\rangle}", "6960fe7572bdbb2007ac49d8d945a1fb": "U\\subseteq F", "69611cb9d44a3365ac7cc6cd0ca4e94a": "\\rho_0\\,", "6961275176c229284b6f71a58af3adf5": "\\langle \\hat{X} \\rangle", "6961433d9fbd9b05109323b78b6b687f": "\\Omega_n", "6961e5398a1ae3a8ea8c36d63bceac9e": "\\displaystyle{(x,y)_1=\\int_{H\\backslash G} (gx,gy) \\, dg.}", "6961e78be39c8a325c23ee3b4b5bf3ca": "\\Phi^{-1}", "6962349e059b9c5e07650c1bd8bdc54a": "D_\\infty= \\frac{d}{2}(\\infty_1+\\infty_2)", "6962924699fd6a74c84c02915f61b7e9": "\\mathrm{HA1} = \\mathrm{MD5}\\Big(\\mathrm{A1}\\Big) = \\mathrm{MD5}\\Big(\\mathrm{MD5}\\Big( \\mathrm{username} : \\mathrm{realm} : \\mathrm{password} \\Big) : \\mathrm{nonce} : \\mathrm{cnonce} \\Big)", "6962a44ea39a1470ee9c6ae014cf9938": "H_B=H_U=0", "6962c4008c657ca6654ce8c89490ebe4": "P \\phi_l=\\frac{1}{2}(P+|P|)[f_l^+\\phi_P+(1-f_l^+)\\phi_{LL}]+\\frac{1}{2}(P-|P|)[f_l^-\\phi_L+(1-f_l^-)\\phi_R]", "6962e6382060679319452db725e1ed27": "(x)^+ =\\{^{x\\ \\ x\\geq0}_{0\\ \\ x<0}", "6962ed3aff508306f4f41c9ddb9b8288": " (a) = (b) ", "696301548765009d82cf4c07756445d3": "D_{\\mathrm N} = \\frac H 2 \\,.", "69636a84978e486b98787d9179b972ad": " \\left( rs^2 \\right)^{1/3} ", "6963928bc664e8e862bf6c974df51843": "e^{b\\theta_\\mathrm{right}}\\, = \\varphi", "69639ce1a797219f52bb4b98b8b083d1": " \\frac{\\pi}{4} = \\frac{3}{4} \\times \\frac{5}{4} \\times \\frac{7}{8} \\times \\frac{11}{12} \\times \\frac{13}{12} \\times \\frac{17}{16} \\times \\frac{19}{20} \\times \\frac{23}{24} \\times \\frac{29}{28} \\times \\frac{31}{32} \\times \\cdots \\! ", "6963e39b2ff8436cb098cdb26fbe4aea": "\\log \\frac{k}{k_0} = \\sigma\\rho. ", "69645ae103ea4dcf4f7418d35aef8962": "C_1: (x_1(t),y_1(t)), \\ C_2: (x_2(s),y_2(s)).", "69647f42c96b06a85aaa072436bab494": "T_0^2=T_1^2+T_2^2+T_3^2.", "6964ad62b418fe6b767e51075b9628fc": "\\textit{changeopen}(t)", "6964d8a1d385b5c138997ad7fcaf97c3": " X \\in L^2(\\mathcal{T}) ", "6964e1aaf8d7d5a8840403b640d10a81": "h_0=h^\\mu_0 \\partial_\\mu", "696507b8410079a04ee882bd6b370b8c": " |\\varphi_i\\rangle ", "69654577027dbd944b9dd2a59ad23fd0": "n + 1 = (\\sigma(n) + \\varphi(n))/2, \\, ", "69655b3c7735692905a683efd2762cbd": "v\\, ", "6965947d5cc63d6697fd7a1382337cb9": "x_0 \\hookrightarrow X", "6965c84485c97d29ea851a2bb9a42514": "\\phi_i =\\phi_j \\circ f_{ij}", "6966307caedcfd7717cb825e60c2be15": "(m, \\sigma)", "6966a13dac3b8af7591b4a2aab0d1afd": "\\|\\Psi_tx\\|\\geq k_t\\|{\\rm e}^{At}x\\|", "6966a19f87ad04d7049d8f8606f36ca8": "\\epsilon_2,\\epsilon_3,\\epsilon_4", "6967331a14c1a51cff73951f61ea77e1": "\\{x,z\\} \\in R_i", "69673e60f668b7bef4700371526a2000": "w_i'", "69676ceacaccdba7ef59367d0dc18cd5": "F_3=2 \\text{ and } F_4=3.", "6967c3d3195de5d2eb1ebfc096d94a69": " X, Y, Z, W", "696889c8abe18743b3f69930ede7877e": "H = -\\sum_{i=1}^N p_i \\log p_i .", "6968e0c4fc98c76caa1492e4d068ec21": "m^2+x^2+y^2=3mxy\\,", "6968f0749e4314d5152d6d122740fc19": "H_B", "6968f97539e0380d84782f73ad495a40": "Rec(w',s)=w' \\triangle v=w", "6968fa9230a36b8f12306e96fedd5191": "2.4477", "69691c7bdcc3ce6d5d8a1361f22d04ac": "M", "6969302dcf6c7defb89084eb1987ab26": " \\textstyle \\prod_{i=1}^r ((p_{i} - 3)/(p_{i} - 1))^{1/2} \\geqslant 2^{-r/2} ", "696937026472b96277f96229ea706c78": "P = \\left(\\frac{84 - 6r}{1225}\\right) \\times n - P_{ma}= \\frac{n(84-6r)-1,225P_{ma}}{1,225}.", "696943c8f268bd78501f6017e728aa07": "\\mathbf{Y}_{00}= \\sqrt{\\frac{1}{4\\pi}}\\hat{\\mathbf{r}}", "6969ad761679c232554292bf3276cb67": "\\mathcal{A}_q^n\\text{,}", "6969ef1a4c542b216c2ea965129d34c2": "\\underline{u}", "696a02d3c67df377438449772be936b5": "\\mathbf{M}_{*}", "696a152af28eb29d436cf3f98564953e": " f(r)= {u^2 - \\epsilon^2\\over R^2}\\,", "696a32c74565f44acca637bc789fbecf": "j_{\\mu}^{Y}=2(j_{\\mu}^{em}-j_{\\mu}^3)", "696a58c061097f513ea031315cea715e": "s-c=(m+n)k^{2} \\, ", "696a5c6f1ff986839af7d6da0c638566": "R(u, s) = \\mathcal{N}\\{f(t)\\} = \\int_0^\\infty f(ut)e^{-st}\\,dt.\\qquad(1)", "696a7a1c1dad8ce68172daac42e2c7bc": "N^2 \\int_0^\\infty \\left(r^{n-1}e^{-\\zeta r}\\right)^2 r^2 dr =1 \\Longrightarrow N = (2\\zeta)^n \\sqrt{\\frac{2\\zeta}{(2n)!}}. ", "696aacbcf375281ab6c39b0caa6ce2f6": "|s_k| < \\epsilon\\!", "696ad89c7ab6ccdd61c5166aade67108": "\\left[X,Y\\right]=X \\triangleleft Y-Y \\triangleleft X", "696ae8d0d1e23ebd86d9c6a342d98e10": "\n0\\leq [D_{0}]_{i,j}<\\infty\\;\\;\\;\\; i\\neq j\n", "696b123eba2b155036d95da47a10a46d": "y' = y \\,", "696b4d38646aea3afad76a11d151e4d6": "= \\mbox{T}_x(-y dt) \\mbox{T}_y(x dt)|x, y, z\\rangle", "696b774c48daaff2aac0ce5585a43704": "(\\varphi\\land\\alpha)\\rightarrow\\psi", "696c17513bb447fe345681e57356e747": "V_{TS}", "696c223ba7734bd1ab662036774bc06d": "\nQ > \\chi_{1-\\alpha,h}^2\n", "696c33ee55e757688f5cbb01ca252373": "{n \\choose i} = \\frac{n!}{i!(n - i)!}", "696c419baef217488f9c52e4fd64d681": "\\sum_i m_i n_i C_i(x,t)", "696c54d5dad01098de68502dec5d1494": "dT_\\text{man}/dt", "696c5be463aa3015c8f9b94c7750c640": "\\bar{10}_{-1}", "696c92c85a1b2d9f8528fcad3ddad501": "\\mu_i \\ge 0, \\mbox{ for all } i = 1, \\ldots, m", "696cb380fbfbc6284a045f9bd8b3d07d": "R_{n+1}(x)=2\\,\\frac{x-1}{x+1}R_n(x)-R_{n-1}(x)\\quad\\mathrm{for\\,n\\ge 1}", "696d4b269e7ce19dc86607e4f76e86ed": "R_S^\\prime=a^2R_S", "696d9b9f38a461859c1520524ae646f6": "\\pi_{E}\\colon E\\to M\\,", "696db6e8131c09e6fa8372db5d3f4915": " P(t) = pQ, \\!", "696db88db9d9066b4097170dc5dd3742": "\\hat{h}_E(Q) \\ge \\frac{C(E/K)}{D}", "696de7240ea53e1220ef352d18e8a2cd": "E_1", "696e2cfb7e1a9237134e99e43f98efb1": "sen(\\sigma):sen(\\Sigma)\\to sen(\\Sigma')", "696e541444d1f628fe893903b0174210": "\\forall x,y \\in X : x\\not = y ", "696e9b2cfc3c26fffb6b4ae4d89f316b": "\\begin{align}\n\\left\\langle x, \\frac{1}{2\\pi i } \\oint_C \\frac{\\varphi}{\\lambda I - L} d \\lambda\\right\\rangle &= \\frac{1}{2\\pi i }\\oint_C d \\lambda \\left \\langle x, \\frac{\\varphi}{\\lambda I - L} \\right \\rangle\\\\\n&= \\frac{1}{2\\pi i } \\oint_C d \\lambda \\int dy \\left \\langle x, \\frac{y}{\\lambda I - L} \\right \\rangle \\langle y, \\varphi \\rangle\n\\end{align}", "696eaabc22c408cde583399b61e031bb": "H^k(X, \\mathbf{C}) = \\bigoplus_{p+q=k} H^{p,q}(X),\\,", "696eed16864632ea3e10790641e7fd61": "(k-1)/\\lambda", "696f0cbb6e6311087ab3adcb1d4dcd25": "\n \\mathbf{b}^1 = \\cfrac{1}{J}(\\mathbf{b}_2\\times\\mathbf{b}_3) ~;~~\n \\mathbf{b}^2 = \\cfrac{1}{J}(\\mathbf{b}_3\\times\\mathbf{b}_1) ~;~~\n \\mathbf{b}^3 = \\cfrac{1}{J}(\\mathbf{b}_1\\times\\mathbf{b}_2)\n", "696f0d542828809d3d9dae40625c37d1": "FT(\\theta)= \\sum_{j=(1-m)/2}^{j=(m-1)/2} C_j \\cos(j\\theta)", "696f31032a9b78f53b42d109965422fc": "H \\in \\mathcal{R}^1", "696f3129e75f935049aaeed993f32058": "r\\ r", "696f4f997e6ffe04a9c18321fccdb234": "\\mathfrak{g}_\\alpha\\ne (0)", "696f865958420e3a05ef422aca298db8": "\n\\begin{align}\nx_1^T \\omega x_1 &= \\left ( h_1 + j h_2 \\right )^T \\omega \\left ( h_1 + j h_2 \\right ) \\\\\n &= \\left ( h_1^T + j h_2^T \\right ) \\omega \\left ( h_1 + j h_2 \\right ) \\\\\n &= h_1^T \\omega h_1 + j \\left ( h_2^T \\omega h_2 \\right ) \\\\\n &= 0\n\\end{align}\n", "696f93a1b94a52dbb9ab3d0cb8ae94d0": "\\left ( N_t \\right )_{0 \\le t < \\infty}", "696feb6fc4523bb3ce2fe8e964e3782b": "99 - (98 - 0.98x) = x", "6970374b8658a9342e1acfebbaaf6a8f": "\\int f\\left( \\theta\\right) \\mathrm{d}\\theta=\\int f\\left( \\theta\\left( \\xi\\right) \\right) \\left( \\det D\\right) ^{-1}\\mathrm{d}\\xi.", "69704db1be0a64988c0508514f617f7d": "\\langle a | \\hat{O} | b\\rangle", "6970af6a2c0d3f86a0cfd41fcb0ad147": " \\mathbb{K}\\langle x_1,\\ldots,x_n\\rangle, ", "6970bfcd96ba601b5bab5b38a64a5b60": "{2\\pi}/k", "6970e005c43b1837722a06413650ea56": "\\phi_{*}\\colon\\mathfrak g \\to \\mathfrak h", "6971595d320023dcd3051111fb411589": "F_{\\alpha\\beta} = \\partial_{[\\alpha} A_{\\beta]}", "69715a1770633c25b8d2c77134e14ae3": "z_1, z_2, \\ldots, z_i, \\ldots, z_{k-1}, z_k", "697164ec504ee05632b7adfbaf5033b3": "\n\n{\\partial F\\over\\partial t} = {1\\over\\sin\\theta} {\\partial\\over\\partial\\theta} \\left(\\sin\\theta{\\langle\\Delta\\xi^2\\rangle\\over 4} {\\partial F\\over\\partial\\theta}\\right). \n\n", "697168fed18775ab1f44a6d006b20372": "x_2 (t)", "6971abf1fbc60bf769998b675e321ced": "\n\\mathcal{I}(\\theta)\n=\n\\mathbb{E}\n\\left\\{\\left.\n \\left[\n \\frac{\\partial}{\\partial\\theta} \\log L(\\theta;X)\n \\right]^2\n\\right|\\theta\\right\\}.\n", "6971c987b0e7412c505d879f67512489": "Z = \\frac{1}{s C}", "6972b394080b855b34e403cf18173dfc": "\n\\frac{d}{dt} \\mathbf{A} = \\frac{d}{dt} \\left( \\mathbf{p} \\times \\mathbf{L} \\right) - \\frac{d}{dt} \\left( mk\\mathbf{\\hat{r}} \\right) = \\mathbf{0}\n", "6972f52765ffc7529e1eb8ad78a5dfc9": "\\sigma_1 = \\frac{E}{1+\\nu}\\varepsilon_1 + \\frac{E\\nu}{(1+\\nu)(1-2\\nu)}(\\varepsilon_1 + \\varepsilon_2 +\\varepsilon_3)", "6972fdf25a6bd5bb71e274b2390b36dc": "m^2+n^2", "697335de4aa975feb91c36f3d2ae1821": "\\alpha_\\tau \\beta_\\tau +\\alpha_\\tau \\gamma_\\tau+\\beta_\\tau \\gamma_\\tau = \\frac{1}{4}\\left(\\lambda^2a^4+\\mu^2b^4+\\nu^2c^4-2\\lambda \\mu a^2b^2-2\\lambda \\nu a^2c^2-2\\mu \\nu b^2c^2 \\right).", "6973ad366e7032babdf4c46c44051db8": "\nB_m(x,y) \\equiv\n\\frac{1}{2i} \\left[ (x+iy)^m - (x-iy)^m \\right]= \\sum_{p=0}^m \\binom{m}{p} x^p y^{m-p} \\sin (m-p) \\frac{\\pi}{2}.\n", "6973cc06a4a8a7dbc30d600a23ed6491": "M_{2} =m_{2}+S_{2},", "69745a1e7931bf2b68c21a70a280bd0f": "\n\\begin{align}\n G(\\phi,\\alpha^2,k) &= \\int_0^\\phi\n \\frac{\\sqrt{1 - k^2\\sin^2\\theta}}{1 - \\alpha^2\\sin^2\\theta}\\,d\\theta\\\\\n &=\\frac{k^2}{\\alpha^2}F(\\phi, k)\n +\\biggl(1-\\frac{k^2}{\\alpha^2}\\biggr)\\Pi(\\phi, \\alpha^2, k),\\\\\n H(\\phi, \\alpha^2, k)\n &= \\int_0^\\phi\n \\frac{\\cos^2\\theta}{(1-\\alpha^2\\sin^2\\theta)\\sqrt{1-k^2\\sin^2\\theta}}\n \\,d\\theta \\\\\n &=\n \\frac1{\\alpha^2} F(\\phi, k) +\n \\biggl(1 - \\frac1{\\alpha^2}\\biggr) \\Pi(\\phi, \\alpha^2, k),\n \\end{align}\n", "697473006c20e81f5b6ed99431cca32b": "\\scriptstyle \\varphi \\,-\\, \\frac{\\pi}{2}", "6974b6401e041a2a74f4bd22fdab05e7": "c^{2} (\\mu) = \\iiint_{\\mathbb{R}^{2}} c(x, y, z)^{2} \\, \\mathrm{d} \\mu (x) \\mathrm{d} \\mu (y) \\mathrm{d} \\mu (z).", "6974c76441791fad741e111d02407321": "\\delta S = S - S_0 = \\frac{\\delta Q}{T} ", "697503310c88de3800ea4de3c837c4d9": " {\\dot{m}_S} ", "697541f03e73b17a522aa5632c0fa5d0": " 2^k \\cdot 509203", "69762da72e7badc20e00703dc85bc8e3": "\\scriptstyle (x,y)\\,", "69767b65d083cc3b9f039a10f4e96135": " \\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} y_{11} & y_{12} \\\\ y_{21} & y_{22} \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} ", "6977037ac3cb82930ceff4ce8d1c4b2f": "\\mathbb{Q} ", "6977378e4cfc27017c9eb5e2e20d4e24": "\\scriptstyle\\operatorname{sgn}(x) \\ ", "6977bcda55c580cc13d19a67018f14c2": "D(T_w)=T_{w^{-1}}^{-1}", "6977c33130c410aa627e27ac41b3419a": "\n (mk)^2= A^2+ p^2 L^{2} + 2 \\mathbf{L} \\cdot (\\mathbf{p} \\times \\mathbf{A}) ~.\n", "6977cb303efb2550fa7e1dd27d35223d": "\\sqrt{2^{\\sqrt{2}}}=\\sqrt{2}^{\\sqrt{2}}=1.6325269\\ldots", "697816528915e39ce2c3c07a94812ed5": "\\mathcal M(\\mathrm F_{SO}(M))\\to \\mathrm F_{SO}(M)", "6978ca63ea9f9ebde3bd5a29569f6f34": "\n\\mathcal{S}_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int \\mathbf{p} \\cdot d\\mathbf{q} = \n \\int ds \\sqrt{2}\\sqrt{E_{tot} - V(\\mathbf{q})}\n", "69790f8d18285f49867d93298420f097": "\\vec{E} = -\\vec{\\nabla}\\phi.", "69795b1a890b038a8f6d414450798df0": " \\frac{dR}{dt} = \\nu I - \\mu R. ", "69798d68394babf52aab09d465109198": "x^2=ny^2", "6979fd8dab00390ca0a9452ce4a65bd8": "V_{cr2}", "697a669f62ecc0af2d72f7c41f92d6ec": "\\mathbf{a} \\times \\mathbf{b} = \\left\\| \\mathbf{a} \\right\\| \\left\\| \\mathbf{b} \\right\\| \\sin \\theta \\ \\mathbf{n}", "697a92da0b2027786f11f7d9a2dcdbcc": "{\\tilde{A}}_{9}", "697a9675767af0f54e8a2c1baf18bdb2": "\\textstyle\\frac {2}{2-1}=6", "697aaf22834dcf4cd5bfb7f23cdee1ce": "\n\\overline{H}\\left( x^{n}\\right) \\equiv-\\frac{1}{n}\\log\\left( p_{X^{n}\n}\\left( x^{n}\\right) \\right) ,", "697ad03a31dcd442e4cf7e22f377edd8": "S_B(1) = 14.2\\%", "697af39875f482db89e3f2be1628c64a": "\\begin{align}\naVR &= -\\frac{I + II}{2}\\\\\naVL &= I - \\frac{II}{2}\\\\\naVF &= II - \\frac{I}{2}\n\\end{align}", "697b35dffa905a667ea70089023f9a00": "{n_N} = 0", "697b8a862c2fd5443d4cb0cbd312072b": "\\mathbb{E}\\left[ \\ln\\left((1 + r) + \\sum\\limits_{k=1}^n u_k(r_k -r) \\right) \\right]", "697bcdc0ebdd929f47bd06124935b96b": " V[t] = m \\zeta [\\vec{x} [t],t] .", "697be2dea18a831d946deb1de32beabd": "\\ \\phi(r_i, s_j, t_k) ", "697be97655589fd7b6644e7735d2e69f": "{\\tilde{D}}_{4+}", "697bec9912484ab7721ccc1b7d881c98": "\\varepsilon_{i_1 \\dots i_k~i_{k+1}\\dots i_n} \\varepsilon^{i_1 \\dots i_k~j_{k+1}\\dots j_n}= k!(n-k)!~\\delta_{[ i_{k+1}}{}^{j_{k+1}} \\dots \\delta_{i_n ]}{}^{j_n} = k!~\\delta^{j_{k+1} \\dots j_n}_{i_{k+1} \\dots i_n} ", "697beef48d45d6fadac81808f4442686": "T^{(i)}_j", "697c3cb47c98cf8b466abb232ae91f5d": "O(s^{-\\frac{\\alpha}{1-\\alpha}}),", "697c4eee255f06da87310b4408f3c6b3": "N=\\left \\{\\begin{pmatrix}I_{n_1} & * & \\cdots & * \\\\ 0 & I_{n_2} & \\cdots & * \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & I_{n_r}\\end{pmatrix}\\right\\},", "697c929385b6e2465c78d3a732503442": "\\lim_{n\\rightarrow +\\infty}a_n=0", "697c97919f7e72c3a3c7d858f074c2ce": "\\cot[\\arcsin (x)]=\\frac{\\sqrt{1 - x^2}}{x}", "697c987fd7ecb6427169bfe19b46849c": "\\int_{-\\infty}^\\infty |F(x + iy)|^2\\,dx < K", "697c9982e098c9bba6207785323c1fe4": "AUC_{k,l}", "697ce749673d6892a30a5e35733483d0": "X\\subset\\mathbb{R}^{m}", "697d3a532527bf0cded7cb1cbb099686": "\\frac{\\partial\\Phi'}{\\partial z} = \\frac{\\partial\\eta}{\\partial t}", "697d3b15b6b41b37526cccd537be1635": "GF(p^2) \\cong \\{y_1 \\alpha + y_2 \\alpha^2 : \\alpha^2+\\alpha+1=0, y_1, y_2 \\in GF(p)\\}.", "697d3c6b83bef03f86dfea30a4c76961": "\\Pr(\\mathbb{W}\\mid\\boldsymbol\\alpha,\\mathbb{Z}) = \\prod_{k=1}^K \\operatorname{DirMult}(\\mathbb{W}_k\\mid\\mathbb{Z},\\boldsymbol\\alpha) = \\prod_{k=1}^K \\left[\\frac{\\Gamma\\left(\\sum_v \\alpha_v\\right)}\n{\\Gamma\\left(\\sum_v n_v^{k}+\\alpha_v\\right)}\\prod_{v=1}^V\\frac{\\Gamma(n_v^{k}+\\alpha_{v})}{\\Gamma(\\alpha_{v})} \\right]", "697d484a80cf0526ea81a9b4861105dd": "\\sin x = 2\\sin\\frac{x}{2}\\cos\\frac{x}{2},", "697d6180396cdb38699c17050c03f493": "M\\mathbf{\\ddot{q}}+C\\mathbf{\\dot{q}}+K\\mathbf q=\\mathbf f", "697d93eb08dd00ab19a0817137ace442": "v_A = \\frac{B}{\\sqrt{4 \\pi n_i m_i}}~~", "697da43677d43cabfe222af55116083e": "\\mu(A_1 ) \\le \\mu(A_2) \\mbox{ whenever } A_1\\subset A_2", "697df9e1cd8a3c5433fba879e1884694": "{K_1}/{k_2}", "697e0ff9ed40a1ba89f4335e7c81e486": "E_q", "697e2873ada6eb153c885f33a4ded6e9": "\nC_{QP}=C_{P_1P}=q_0\\left(\\frac{x_M-x_U}{v_f}\\right)-k_0(x_M-x_U)=0 \\qquad (7)\n", "697e686d6f18ccfa9cd8784c922c5b10": "p(D\\vert C)={p(D\\cap C)\\over p(C)}", "697e7285eb5380a1ee1ca85b9cce0abf": "H_{x}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\\frac{m\\pi }{a}\\frac{1}{\\varepsilon _{r}}(A \\ e^{jk_{x\\varepsilon }w}+B \\ e^{-jk_{x\\varepsilon }w})-\\frac{jk_{xo}k_{z}}{\\omega \\mu }(C \\ e^{jk_{x\\varepsilon }w}+D \\ e^{-jk_{x\\varepsilon }w})]e^{jk_{xo}(x+w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (48) ", "697ecfdd4b7acf899b01ba4cf46604ff": "\\Sigma_{XX}, \\,\\, \\Sigma_{YY}", "697f020bb4f0f7402430d11e8cc9ff35": "A\\cap B=A\\setminus(A\\setminus B),", "697f961a7a4c67214ab0ff3b349603eb": " \\int_\\Omega\\vert f\\vert dH^n+ \\int_\\Omega\\vert g\\vert dH^{n-1}<+\\infty.", "69800bf5a204142183f04bcb9daa8b96": "\\frac{dU\\left(x_0,y_0\\right)}{dx} = 0\\Leftrightarrow\\frac{dy}{dx}=-\\frac{U_1(x_0,y_0)}{U_2(x_0,y_0)}", "698011d37d98b300daa1d7e112fdbee1": "\\,y(t)=(C+m(t))\\left(\\tfrac{1}{2} + \\tfrac{1}{2}\\cos(2\\omega t)\\right).", "69801ce2cc306153d4be7c83f8303860": "= du_{1} - u_{2}dx \\,", "69804cfb95cc8aba121c0e0df0b74651": "H^k(U_{i_1} \\cap \\cdots \\cap U_{i_n}, \\mathcal{F}) = 0", "6980565ced8dc6085db887439917aeaf": "L^2(\\mathbb{R})=L^2(-\\infty,0) \\oplus L^2(0,\\infty)", "698087945ed58d7d6167413d35beb75e": "\\left(i,f\\left(i\\right)\\right)\\,\\!", "69808cb552d7896309da3ead383060f6": "V = a(T_h - T_c)\\,\\!", "698118c2dd638f59a0908a008a826075": "F_{i+1}", "6981289162e0818ba1fb88b3f66c299a": "\\mathfrak{H}^3=\\{x+y i + t j|t>0\\}", "698142f1ed8e023dc8edd36014ce54f2": "R = Q \\frac{|V_\\parallel|^2}{\\omega W},", "698174afa9f55541a8dad06625b4ca84": "\\frac{}{}R", "6982466e5a882edb0ff86dcced502e2f": "f_i(X)=f_{i-1}(X)^d \\mod E(X)", "6982542574c45ef4df267caeff2c5455": "\\tau_Y=\\left\\{ U \\subseteq Y : q^{-1}(U) \\in \\tau_X \\right\\}.", "69825f96b4fe2cfbee361160f040a462": "\\frac{\\alpha(m+1)}{2m}", "69826ee3c09c352a9727eac90161d615": "\\deg f = \\lim_{x\\rarr\\infty}\\frac{\\log |f(x)|}{\\log x}.", "698295f739a7333e4e6cbd08ec3a545c": "p_n = {{p_{n - 1} + p_{n + 1}} \\over 2}.", "698310dbf01ff8999dfe33a8ae68d2a9": "\\frac{F(y)-F(x)}{y-x}", "698349a6f790639f51bd23475e688f97": "a_2=3=2+1", "69835ddc94ea5744ce323a6366bf0b43": " \\frac{d\\Phi}{dr}= - \\frac{1}{\\rho}\\frac{dP}{dr} ", "6983f070dc0e278b4ac6545d168f4cae": "\\nu(d)\\leq\\frac{(d+1)(d+2)}{2}", "6984549d7be7b4c4021c370c9411cef3": "W_x", "6984f5eaa304af61c9c627ca718d61e9": "\\overrightarrow{A}", "6985185b6162b9d201519af248619f42": "y = \\dot x ", "69852976333a7c9f64c935e594040a9f": "|\\Psi^-\\rangle_{AB} = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |1\\rangle_{B} - |1\\rangle_A \\otimes |0\\rangle_{B})", "69853a3f21e0425789bc4c2e6e5f929f": "(C,~ 0\\! \\rightarrow\\! 1)", "698559334d002b9b147e1c60437e9552": "Vs_\\mathrm{actual} ", "698560ebcb787de7454688ebbb4efe2c": "(a_k)_{k=0}^\\infty = ( a_0, a_1, a_2,... ).", "698564d9c5a7b9b4ca14287c97b6103f": "T_3", "698582a47fe6a98f0cfc08f4b2b88eb5": "2mV_0a^2/\\hbar^2", "6985ad0bc7b2c08a17f051b210e50144": "(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2, \\, ", "6985cc1eb0f67041f9a7e5b4617ae1ab": "f^n(A) \\cap B \\neq \\varnothing", "6985f22583395607e22782ef8c346212": " e_i=Y_i-\\hat{Y}_i ", "6985f6f13d01b727d58dc87dca206737": " \n\\int_{E}\\phi \\,d\\mu\\leq \\liminf_{n\\rightarrow \\infty} \\int_{E}f_n\\,d\\mu_n\n", "698627260d19ebd046744c73b3b90a60": "T = \\frac{1}{2} \\dot q^\\mathrm{T} M \\dot q", "69863db36b71018fb95ab67c19879686": " a = 3b ", "698671018fbf2bdddaa2776b56a71139": "\nf(x) = g(a \\cdot x)\n\\Rightarrow\nf^\\star(p) = g^\\star\\left(\\frac{p}{a}\\right).\n", "69869db1e4fc53829deff9d43da1e3b5": "L=\\mathcal{L}(x^\\lambda,y^i,y^i_\\lambda) \\, d^nx,\\qquad\n\\delta_i L =\\partial_i\\mathcal{L} - d_\\lambda\n\\partial_i^\\lambda\\mathcal{L}.", "6986a897dc1394bb2f4a2ed51106d77f": " (dF)_{i_1... i_k i_{k+1}} = F_{(i_1... i_k,i_{k+1})}", "6986cf9ed9be2967458708402ebca96a": "D_\\infty=\\infty_1+\\infty_2 ", "6986dfca452fd0e8eb6c2f5460779205": "T_{ib}", "69875f197c2fa049b72c312206df341b": "O(k^2n)", "6987b5da0e5402371c8d21774e76934b": "p_n = \\sum_{j=1}^n (-1)^{j-1} e_j p_{n-j} \\,.", "6987db1f1593b0e27d70d899a70c7071": "\\overline d(A)=\\lim_{m \\rightarrow \\infty} \\frac{1+2^2+\\cdots +2^{2m}}{2^{2m+1}-1}\n= \\lim_{m \\rightarrow \\infty} \\frac{2^{2m+2}-1}{3(2^{2m+1}-1)}\n= \\frac 23\\, ,", "6987f5f9dadaea452e20093fb7921ef3": "\\Sigma\\,\\!", "69881116f6120e78524a37d46409aeb9": " r(\\nu) = \\frac{p}{1+e\\cos(\\nu)} ", "698812956581ece7b6a5628a8534dff5": "f(x|X>y) = \\frac{g(x)}{1-F(y)}", "69881d934d4b6c0ac7515f2195404345": "\\bar{Y}_i = \\frac{1}{n_i} \\sum_{j=1}^{n_i} Y_{ij}", "6988438afc625eb18538972673279035": "\\therefore I_3\\ = \\tfrac{1}{3} \\cos^2 x \\sin x + \\tfrac{2}{3}\\sin x + C_2, \\quad C_2\\ = \\tfrac{2}{3} C_1,\\,", "6989034da7fb332426bd46d6b17c5a8d": "C^n_k", "69891d9f3acaf91fdb88ff62dfebb662": "\\Omega^k_{\\mathrm c}(X)", "69892b3eaae23ee08a29cd2b0ce7f292": "(z_i)", "69894969475040c439e6c3672d5acb5c": "\\Delta{v_i}= {2v\\, \\sin \\left(\\frac{\\Delta{i}}{2} \\right)}", "6989e2d18b18242c257a42a811057ec0": "N^1", "698a0908944fb467326708398505376e": "|V(x,y,t)-B(x,y)|> \\mathrm{Th} \\, ", "698a1cfbd82558fc79e128847ec337c8": "\nk_{AB} = \\Phi_{1,0} \\prod_{i=1}^{n-1} P_A (i+1|i)\n", "698a5e688c04cf3241d2d849a0dc5634": " Q(t_3) ", "698a6f63a337c02261a93e3d04e5b908": "\\text{Cl}_{2m}\\left( \\frac{q\\pi}{p}\\right)= \\sum_{k=1}^{\\infty}\\frac{\\sin (kq\\pi/p)}{k^{2m}} ", "698a83383e6178667d47be974006dabb": "f_{x}", "698aec33eca0d4d6252b5b1cbf9ce166": "\\frac{v^2}{2}+g z+\\frac{p}{\\rho}=C", "698af5c550bd5aae134a4229fb79f40d": "f = ax+b \\in \\mathbb{Z}_N[x]", "698b57e05ac6f85e4fac6369b9c25672": "\\psi (x)", "698b99325a724bd750afe66ab9e2c001": "S(a, b)", "698bd20b767372a24cebea5d969e76db": " NPSH_A = \\left( \\frac{p_i}{\\rho g} + \\frac{V_i^2}{2 g} \\right) - \\frac{p_{v}}{\\rho g}", "698be555a9510fade5b3396efa912f1e": " \\Gamma_{a_1 a_2} ~,~ \\Gamma_{a_1 a_2 a_3}", "698c6994ec258fc8461c26b6b89f49df": "\\Delta y = \\Delta I * \\frac{1}{(1 - b_C)(1 - b_T) + b_M}", "698c6d2ff4ad1afbf66a2905aae6ebff": "BS(1,2)", "698ca1206d3ceab7224847bef07a469f": "\\underline{\\varphi \\vdash \\psi}\\,\\!", "698ca667cd4e89aac50bebc89007c7b8": "\\mathbf{v} = \\frac{1}{m} \\left( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right) ", "698cc023ca7db5c5310642c1eddab279": "\\left| \\sum \\mathbf{r}_{i} \\right| ", "698ce5b586741479d22ff9a0a5f7366b": "N(x) = 1", "698cfbca3dc657626a5d0dffa82b3207": "\\sum_{\\beta}h_{\\beta}^{k-1}", "698d030c348c8292731cdd6fe878e36e": "\\sqrt{\\sqrt{1} \\cdot \\sqrt{10}} = \\sqrt[4]{10} \\approx 2 \\,", "698d51a19d8a121ce581499d7b701668": "111", "698d72c5abc725a3d63672966aa8f4d3": "g(x,y,\\epsilon)", "698e2f0526f676551f84e7c06d0272bc": "X_i-\\overline{X}_n\\,", "698e90bfd1ce338e1677ca268d73b52e": "R = k\\frac{\\gamma l}{g \\omega} ", "698ecbcccc431527564779ac6fde48ca": "u(n)", "698f07eee9d1bafa4b96f5d9c5258d9b": "\n\\zeta(s) = \\frac{\\eta(s)}{1-\\frac{2}{2^s}}\n", "698f5967563bc585d83607eb340c4be7": "F_{thrust-gate} = \\gamma b(M_{unit,1} - M_{unit,2}) ", "698f6bf4c4a7866d882a0aad9903956f": "\\beta(x) = \\int_x^\\infty u^{-1/2}e^{-\\pi u} du", "699002621026160567b1f42097c5b710": "dt=\\gamma\\left(dt' + \\frac{vdx'}{c^2}\\right)=\\gamma\\left( 1+ \\frac{vu'{x}}{c^2}\\right)dt' ", "6990500bc5ef431f11337daee3ddd2ae": "h_a(\\bar{x}) = h_\\mathrm{int} \\left( \\big(\\sum_{i=0}^\\ell x_i\\cdot a^i \\big) \\bmod ~p \\right)", "69908053924df497a2295c4291df6d21": "\\int_{R^n} m(\\xi) e^{2\\pi i x \\cdot \\xi}\\ d\\xi", "6990c1340552a75212f742c82ada9706": "D=\\frac{1}{3} \\ell v_T = \\frac{2}{3}\\sqrt{\\frac{k_{\\rm B}^3}{\\pi^3 m}}\\frac{T^{3/2}}{Pd^2}\\, ,", "6990ccd0675fac22e3badf2c847dfc2d": "\n\\boldsymbol{\\gamma}^T \\equiv\n\\begin{pmatrix}\n {\\partial z \\over \\partial x_1} & {\\partial z \\over \\partial x_2} & {\\partial z \\over \\partial x_3} & \\cdots & {\\partial z \\over \\partial x_p }\n\\end{pmatrix}", "699113af6afff42bc2ecdda44f79cafb": "\\left\\langle \\mathbf x, \\boldsymbol \\xi \\right\\rangle", "699162662bd18750fc700c73f47ddbea": " n=(a\\pm r)^2=a^2\\pm 2ar+r^2\\ ", "6991833fa5cfbe7ce0ad279ea44793ee": "\\begin{align}\ny_p &= u_1(x) y_1(x) + u_2(x) y_2(x) = \\frac{i}{2(3+4i+k^2)}\\left((2+i)\\sin(kx)+k\\cos(kx)\\right) +\\frac{i}{2(3-4i+k^2)}\\left((i-2)\\sin(kx)-k\\cos(kx)\\right) \\\\\n&=\\frac{(5-k^2)\\sin(kx)+4k\\cos(kx)}{(3+k^2)^2+16}.\n\\end{align}", "699183e277aae4234a391b359f101f35": "\nM(\\vec X,Y) = \\left[ {\\begin{array}{*{20}c}\n {\\begin{array}{*{20}c}\n {\\bar \\mu _1 } \\\\\n {\\bar \\Sigma _{11} } \\\\\n {\\bar \\Sigma _{21} } \\\\\n\\end{array}} & {\\begin{array}{*{20}c}\n {\\bar \\mu _2 } \\\\\n {\\bar \\Sigma _{12} } \\\\\n {\\bar \\Sigma _{22} } \\\\\n\\end{array}} \\\\\n\\end{array}} \\right]\n", "6991aaff96b9b83f7b3e42e492d8796d": "\\mathbf{J}\\approx\\frac{\\hat{z}\\epsilon_0}{2\\omega}\\int \\left(|{E}_L|^2-|{E}_R|^2\\right)d^{3}\\mathbf{r} +\\frac{\\hat{z}\\epsilon_0}{2i\\omega}\\int \\sum_{i=x,y,z}\\left({E^i}^\\ast \\frac{\\partial}{\\partial \\phi}E^{i}\\right)d^{3}\\mathbf{r} .", "69920884fd97ea7f342920aa29d631e0": "f \\pitchfork Z", "69921d57bfc6a6bc02aaad2c287f1347": "q(q^*)=a^2-b^2-c^2-d^2", "699228aa9419e6159bee8fc55bfce433": " x^{\\alpha-1} (1-x) \\leq 1 ", "69924cd4f98c2c5b1d059a5eacd28b4d": "J^k\\! f\\,", "6992528977b7515afef01363a0c480e8": "\\mathrm{C\\ m^{-2}=A\\ m^{-2}s}", "69926d65c5e0e100b794b7c47162a72d": "BCF=\\frac{Concentration_{Biota}}{Concentration_{Water}}", "69929ca69568753d5da3eb7c1c887eaa": "\\hat{\\mathbf{y}}", "6992c9478372772d774115aa67b73ed8": " \\hat{f}_1^{(i)} = [I - \\sum_{\\alpha = 0}^{i-1}(S_1 S_2)^\\alpha(I-S_1)]Y ", "69931ecc3d3da8ef6876382648b38b62": "c \\colon E \\to \\mathbb{R}^+", "69931f5b701b99d8abf52c9862cb620a": "I(Y_{1};Y_{2};Y_{3})", "699327b1945be6e74127ee406cc394db": "a \\triangleright b", "699365af55c725a22c8d91eb3bdcfe69": "\\textstyle\\text{rate(chain transfer)} = k_{tr}[\\text{M}^+] [\\text{M}]", "6993b3fd4583b64c4d7e65e779c42f89": "y'(x) + f(x) y(x) = g(x),", "699403fcb8196e36757a2494ea6ee905": "y(t)=e^{-15t}\\,", "6994072682a178abf78d57e7284b4a20": "|\\mathcal{P}(S)| = 2^n", "699437d105fe6a94d483bda8d0859111": " \\lambda a, b, c.c\\ (\\lambda x.\\lambda a, b, c.b\\ \\operatorname{mse}[f]\\ \\operatorname{mse}[x\\ x])", "6994693a4c3cd09e2815d2a245e3d396": "|\\nabla u|^2 = 1,", "6994693ad2afd913276cf0356470f7c7": "\\psi' = \\exp{\\left(\\frac{1}{8} \\omega_{\\mu\\nu} [\\gamma_{\\mu}, \\gamma_{\\nu}]\\right)} \\psi", "6995627f123743c70905ffeaae312497": "E_g=\\left\\{x\\in X: g(x)>s\\right\\} \\, ", "699583eef14cec61ce33bb982caa6696": "\\eta(z)=1-\\frac{1}{2^z}+\\frac{1}{3^z}-\\frac{1}{4^z}+\\cdots=\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n^z},", "699642eef6271965a11da74bc093f71f": "B(a,b)", "69965d7ef5187a59f868daa16c7a077b": " b_1,b_2 \\in [0,1].", "69969b7ef5dbb2d19995ab4b2d6c4904": "\\theta=1", "6996a72676527bbcfa977b9e9bd440f5": "\\phi_\\lambda (\\mathbf{k})", "6996ea5f3a4e90779cef3a3fbcdcb807": "\\begin{matrix} {12 \\choose 1}{4 \\choose 3} \\end{matrix}", "69975efb05c86c96fe5fcea123752e38": "\n\\begin{align}\n& p=\\operatorname{prox}_f(x) \\Leftrightarrow x-p \\in \\partial f(p) & (\\forall(x,p) \\in \\mathbb{R}^N \\times \\mathbb{R}^N) \n\\end{align}\n", "6997a28bc8de9c079aea1730e4a78089": " F_x= -\\oint_C p \\sin\\phi\\, ds \\quad, \\qquad F_y= \\oint_C p \\cos\\phi\\, ds. ", "69981045aff9b06c229170592226927d": " d_k ", "6998173b2b7c339bc638d91724d928b4": "c=L-1+L(1-a)", "69982324229c68f1958b6890b7554592": "\\delta = \\delta_\\text{c} \\cup \\delta_\\text{r} \\cup \\delta_\\text{int}", "69988be828bcae9b2eb94e4dae9c45f0": " \\chi_1(\\omega) = {1 \\over \\pi} \\mathcal{P}\\!\\!\\! \\int \\limits_{-\\infty}^\\infty {\\omega' \\chi_2(\\omega') \\over \\omega'^2 - \\omega^2}\\, d\\omega' + {\\omega \\over \\pi} \\mathcal{P}\\!\\!\\! \\int \\limits_{-\\infty}^\\infty {\\chi_2(\\omega') \\over \\omega'^2 - \\omega^2}\\,d\\omega'. ", "6998f2f3f512cf00211bcc52511bd65e": "H_{\\omega^\\omega}(1) - 1", "699915366c70fa33fa33a7a30fe23830": "B\\ge { {|C|Vol(0,e)} \\over {q^n}} ", "69991784719922b7311ed81e5dc0a3a3": " \\Psi(x,y,z,0) \\,\\!", "6999204e457d39825292c558589b3687": "x^{\\lambda(n)/k}", "69994ffcaf7b89326cfb3b9469f77acf": "\\mathbf{z} = \\mathbf{Hr} = \\begin{pmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\ \\end{pmatrix} \n\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 1 \\\\ 1 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 4 \\\\ 3 \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix} ", "69999e1dfc00d1b00dcf92534ab52b4b": "\\left(1- \\frac{\\lambda}{n}\\right)^{n-k}=\\left(1- \\frac{\\lambda}{n}\\right)^{n}\\left(1- \\frac{\\lambda}{n}\\right)^{-k}", "6999bb7c0a2dcbef2b610490155e9595": "\\int\\frac{\\tan^n ax\\;\\mathrm{d}x}{\\cos^2 ax} = \\frac{1}{a(n+1)}\\tan^{n+1} ax +C\\qquad\\mbox{(for }n\\neq -1\\mbox{)}\\,\\!", "6999c3d69cf36775d4303277b2abf92e": " -n~r^{-n-1}~\\cos(n\\theta) \\,", "6999dce00c87dd5a86cb5990a2543236": "\\Phi(f(y_1,\\ldots,y_n),y_1,\\ldots,y_n)", "6999e59e7d5bded272f26cb248f42b5b": "\\frac{1}{s_{pm}^\\pm}=\n\\pm \\sinh\\left(\\frac{1}{n}\\mathrm{arsinh}\\left(\\frac{1}{\\varepsilon}\\right)\\right)\\sin(\\theta_m)", "6999f8dc165ddd073e195d9db10135f1": "E^+", "6999fb1d72fb29891b910f440c3d6d9e": "(x-1)^2-x^2=0 ", "699af64d15fb64afc050d74cc4714400": "\\begin{align}&E(a+b)=E(a)\\cdot E(b),\\\\\n&E(0_F)=1_F \\end{align}", "699b8f2d826aa8c270df1496b542d371": "\\alpha_2=c+d+e;\\quad \\beta_2=a+d+e+f;", "699ba5b0bfce9ebfefd6771496c64572": "\\Psi = \\Psi_0 + \\Psi_\\pi + \\Psi_p + \\Psi_s + \\Psi_v + \\Psi_m ", "699c5ec266e5810d436aee16db6204b2": "\ndx_u:=\\prod_{j\\in u} dx_j\n", "699c684741cd43af2ee77fdb1a05aa75": " r = 2a(1 + \\cos \\theta)\\,", "699c768cd20eb3999e02e317d1feab54": "\ndr(t)=\\mu(t,r(t)) \\, dt + \\sigma(t,r(t)) \\, dW(t)\n", "699c9605005f1836059c779c8114042c": "\\scriptstyle 2n, 2n-2, 2n-4, \\dots, 4, 2, 1, 3, \\dots, 2n-3, 2n-1", "699d0dd40b50695fa4d1c4e5204baea2": " {}_RQ_P = 0 ", "699d2a923c41583e5e13a3a28be6a5a4": "\\tan ( \\phi/2) = \\frac{1-\\cos(\\phi)}{\\sin (\\phi)} \\ ,", "699d42fc524506dcf4e3a5e5af3ac909": "t^a(d,n) \\geq d\\log\\frac{n}{d}", "699d5134a0980b7eb453c2164477e614": "\\begin{align}\nV(\\mathbf{x}) &= - \\int_{\\mathbb{R}^3} \\frac{G}{ \\sqrt{|\\mathbf{x}|^2 -2 \\mathbf{x} \\cdot \\mathbf{r} + |\\mathbf{r}|^2}}\\,dm(\\mathbf{r})\\\\\n{}&=- \\frac{1}{|\\mathbf{x}|}\\int_{\\mathbb{R}^3} G \\, \\left/ \\, \\sqrt{1 -2 \\frac{r}{|\\mathbf{x}|} \\cos \\theta + \\left( \\frac{r}{|\\mathbf{x}|} \\right)^2}\\right.\\,dm(\\mathbf{r})\n\\end{align}", "699d6bb10ba25e4f035bccb25d7625e3": "\\langle (x_1,\\ldots, x_n),(y_1,\\ldots, y_n)\\rangle := x^\\mathsf{T} y = \\sum_{i=1}^{n} x_i y_i = x_1 y_1 + \\cdots + x_n y_n,", "699daaa84360e94aabbb51340e16ba0b": "\nK=\\ln(1+\\delta_{ij^*}z^{i}\\bar{z}^{j^*})\n", "699dd53f54db20145b6bef96b58f1720": "\n\\begin{align}\nO_n\\{\\delta[m-k];\\ m\\}\\ &\\stackrel{\\quad}{=}\\ O_{n-k}\\{\\delta[m];\\ m\\}\\\\\n&\\stackrel{\\text{def}}{=}\\ h[n-k].\\,\n\\end{align}\n", "699e1dfc6a7ea35d4fee8e058d816f6a": "V(XY)= E(X)^2 V(Y) + E(Y)^2 V(X) + E((X-E(X))^2 (Y-E(Y))^2)^2", "699e902f5598ca623370f833cffb1a57": " C ", "699ed85ddac8abd657ef7edf6dbe329b": "G'(t) = N_1 +N_2t^1 + N_3t^2 +\\cdots \\,", "699efcc765ae8484d76e8cc5f3ca779a": "\\int_\\Omega g(x) |\\nabla u(x)|\\, dx = \\int_{-\\infty}^\\infty \\left(\\int_{u^{-1}(t)}g(x)\\,dH_{n-1}(x)\\right)\\,dt", "699f0b74c377b8d9515c00af35aff9c4": "\\mathcal{C}_a", "699f5dadba57c4e925b2dfcf5f697590": "\\begin{align}\n\\dim_{\\mathrm{H}} (K) &{} = \\liminf_{n\\to\\infty} \\frac{n \\log 2}{- \\log a_n} \\, , \\\\\n\\dim_{\\mathrm{P}} (K) &{} = \\limsup_{n\\to\\infty} \\frac{n \\log 2}{- \\log a_n} \\, .\n\\end{align}", "699f749d9a478540e6b866100898fb16": "x^i~", "699fc6648d8a27fb385d46f9a8eead6c": "\ny' = g(z) = \\int\\limits_{0}^{z} p_z (u) du\n", "699ff35b4f873564493ff1f9f969a339": "\n E_H = \\tfrac{1}{2} \\frac{p_H \\chi}{\\rho_0} = \\tfrac{1}{2} \\frac{p_H \\chi V_0}{\\rho V} \\quad \\text{or} \\quad\n e_H = \\tfrac{1}{2} p_H \\chi V_0\n ", "69a00dfada185cb5a9e1c0942f3bfdb9": "\\beta(5)\\;=\\;\\frac{5\\pi^5}{1536},", "69a0280c54e97213ed505198e02719ab": "\\mathcal{F}^{-1}(\\mathbf{x}) = \\textrm{swap}(\\mathcal{F}(\\textrm{swap}(\\mathbf{x}))) / N", "69a0833a93e552c9d841bcf797e78894": "z^mF(a+m,b+m;1+m;z).", "69a0839dc038b4348d89b94923aa6ffc": "E_k = C (N_p^{5/3} + N_n^{5/3})", "69a0a1273491057003f007aba5895f4c": "\\boldsymbol{M}_{k}", "69a0bfc17c5c04ffb8b66419aee1afed": "y=x^ke^{zx}", "69a0e7829371fc49dc7a37737441b766": "(A) \\lor (C \\land B) \\lor (F)", "69a117e67bbcff2279bdb4c22eb6ab2a": "K = A^{-1} B^{-1} A B \\in G", "69a13660f6ec6ac97898bbbfb8a29ad5": "K_n(R)", "69a162e260d0b07ab7b75e10501952ff": "P_W = \\frac {\\sum\\left(p_{t} \\cdot \\sqrt{q_{0}\\cdot q_{t}}\\right)}{\\sum\\left(p_{0} \\cdot \\sqrt{q_{0}\\cdot q_{t}}\\right)}", "69a1f050306c66be704e2bc5f2c620d1": "\n\\left(\\frac{\\alpha}{\\beta}\\right)_n = \\left(\\frac{\\alpha}{(\\beta) }\\right)_n,\n", "69a202e22c75bfe82c3599f4d395f9ca": "z_{mn} = z_{00} + m \\Delta x + in \\Delta y", "69a28f826963b1277a813d28392add9f": "\\alpha\\approx A\\sin(\\tilde\\omega t+B)", "69a2a9cc575f047909dc08223ad64ad5": "\\|Tu\\|_{L^{p}(\\partial \\Omega)}\\le C \\|u\\|_{W^{1, p}(\\Omega)}", "69a2e2be07af25735ef3c3beab2c8543": "m_\\text{P} = c^{1/2}G^{-1/2}\\hbar^{1/2} = \\sqrt{\\frac{c\\hbar}{G}}. ", "69a37a3873afdef94c21d5b063947d70": "\\Sigma_{a \\in T^*} \\Sigma_{j \\in S(a)} p_j \\leq \\Sigma_{a \\in T^*} W(a)", "69a3adddd8d335ec568875257d2b982e": "d\\omega = (d\\omega^\\alpha)e_\\alpha.\\,", "69a40040adc8ee686f0f080a8d5d5460": "S[\\bold{A},\\sigma]=\\int d^dx \\frac{1}{4g^2}\\eta((\\bold{g}^{-1}\\otimes \\bold{g}^{-1})(\\bold{F},\\bold{F}))+\\frac{1}{2}\\alpha(\\bold{g}^{-1}(D\\sigma,D\\sigma))", "69a44b32573dd038d7b56bd8cd5ad62f": "\n\\theta_2(t) = k \\theta_1(t)\\,\\!\n", "69a452b268dcb4a272938faebfeda625": "\\scriptstyle \\sqrt{2mE}", "69a478e736052fdb1d1c8ffd48cb7494": "i_0 \\leftarrow i_0 + 1", "69a4a2c0528d551f9ac71d1cc889d437": "\\sum_{i,j=1}^n a_{ij} u_i u_j > \\alpha \\sum_{i=1}^n u_i^2 \\;\\;\\; \\forall u \\in \\mathbb{R}^n", "69a4d9d82052be653e729c46c020b879": "K=3", "69a502c5a68cfdfc3f59f09774fdfc40": "T(V)", "69a524a4d0d4f6a557d752ef9a30d72e": "\nu_{1} + u_{2} = \\frac{-2\\alpha}{mh^{2}}\n", "69a532ad3e58c9a3ae3693abb97588ed": "\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\end{bmatrix} \\implies \\begin{bmatrix} Y \\\\ I \\\\ Q \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\end{bmatrix}", "69a55b20e8c8fbc3ce61ac8caec68c97": "\\Phi_{S}\\ne0", "69a5700807e5a9889213cce0314a5324": "\n \\frac{\\partial^2 p}{\\partial r^2} + \\frac{1}{r}\\frac{\\partial p}{\\partial r} +\n \\frac{1}{r^2}~\\frac{\\partial^2 p}{\\partial\\theta^2} + \\frac{\\omega^2\\rho_0}{\\kappa}~p = 0\n ", "69a5c21fa684c32bba849dec1d65ce6b": " 1:\\sqrt{\\varphi}:\\varphi .\\, ", "69a5cfe9a71489b6f40a3e4a7cc0275b": "\n\\frac{\\partial f_{i}}{\\partial x_j}=P_{i,j}(\\boldsymbol{x},f_{1}(\\boldsymbol{x}),\\ldots,f_{i}(\\boldsymbol{x}))\n", "69a670546df5d44720f30d8750f289e8": "\\Gamma_0(N) g \\Gamma_0(N)", "69a732cb7e9b93e23dc4e50b30327c61": "\\omega = \\exp( \\pi i /3).", "69a79b58eb4b49c112b3c7aed2e690a9": "y = \\frac{1}{M}\\sum_{i=1}^{M}[w_i*x(t-\\Delta t_i)]......(3) ", "69a7a01354913c275e68a653674f840a": "\\sigma_{kl} \\equiv \\frac{i}{2}[\\gamma_k,\\gamma_l]", "69a7ada8486f9e732519574728d1f7f7": "u = (u_i)_{i \\in \\mathcal{I}}: T \\rightarrow \\Re^\\mathcal{I}", "69a7ce8d77e0c7afc71b23796e1825c4": "\\phi_i = \\frac {V_i}{V}", "69a7dd8ba437617f33dc83db351adf14": "\\frac{\\pi x^2}{6}-\\frac{3 \\sqrt{x^2-(\\frac{x}{2})^2}}{2}+\\frac{\\sqrt{x^2-(\\frac{x}{2})^2}}{2}", "69a7e53cbb352805e0aff7758cfb88cd": "\\log_d", "69a7f1acba99da1cd6cea4d66341fe1a": "E^1", "69a835487d94462b5bcd2b95f9e7e645": "\\int\\frac{\\cos^n ax\\;\\mathrm{d}x}{\\sin^m ax} = \\frac{\\cos^{n-1} ax}{a(n-m)\\sin^{m-1} ax} + \\frac{n-1}{n-m}\\int\\frac{\\cos^{n-2} ax\\;\\mathrm{d}x}{\\sin^m ax} \\qquad\\mbox{(for }m\\neq n\\mbox{)}\\,\\!", "69a838a928cad450d67471964ff29659": "Q(\\lambda)", "69a844baf3dd92846dcb3b9b0fda56c7": "f_{BP}", "69a8dfd369c0363810ef6830e2504487": "bX \\sim \\operatorname{Logistic}(0,b).", "69a8f72c2a806aab9d25249233b58778": " f_X(\\mathbf{x}|\\boldsymbol \\theta) = h(\\mathbf{x}) \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(\\mathbf{x}) - A({\\boldsymbol \\theta})\\Big)", "69a91212e3e7df48a4fd9c08fd9e493a": "\\mathit{n}_\\mathit{b}", "69a99348fa863f97f55bc3e178a258f8": "b_n\\in \\mathbb R ", "69a9c0353705a8046635f4baea1e2c31": "\\frac {(0.01 \\times 300,000,000\\ \\mathrm{m/s})} {(10.23 \\times 10^6 / \\mathrm{s})}", "69aa1565a5ccfefb812ac05b762c6358": "\\tau(p)\\equiv 0\\ \\bmod\\ 23\\text{ if }\\left(\\frac{p}{23}\\right)=-1", "69aa2cf3dfe9d41cb1db567d1b0ad275": "\\nabla f(\\mathbf{x})", "69aa2e0512dbb1b7ac60e75b5232bc9c": " V = L ", "69aa8ea35eef69ec3137fb837bef876e": "S(h_{(1)})h_{(2)}=\\varepsilon(h)=h_{(1)}S(h_{(2)})", "69aae92df2c7af8b99790acbe46a4c79": " \\phi(n)=\\sum_{i=1}^{n}[\\gcd(i,n)=1],\\qquad\\text{for }n\\in\\mathbb N^+.", "69aaec872909a6652572aa1f210b3735": "\n J_{\\Gamma_1} = -J_{\\Gamma_2} \\,.\n ", "69aaecea3e601f3b9d7c3b10ea2e2c59": " BS=CAL + REF", "69ab103a3817d41462b875d81408a463": "\n\\hat{S} = \\frac{\\vec{R}_1}{|| \\vec{R}_1||} \n", "69ab19d14c8ac9b72593b32b7e7a2984": " x > 0 ", "69ab2a6077161cec901efeb5db1e6153": "\\ \\cos (2j \\pi x) \\cos (2k \\pi y)", "69ab425f734e101ce42440bc9e90505b": "A: X \\to Y", "69abc87bc61ffde579d8db902485ca25": "\\sigma_+", "69ac448e5438d4606217a74f307f5279": "v I", "69ac49315fb75559bc7125a373ed5735": "t_1", "69ac563f2c40414b7d8eda5032830e1f": " \\omega_{1} = -\\gamma B_{1} ", "69ac595661e9a93314275896625d917a": "J(x,y) = \\begin{bmatrix} \n\\alpha - \\beta y & -\\beta x \\\\\n\\delta y & \\delta x - \\gamma \\\\\n\\end{bmatrix}.", "69ac8d9214b617f9952df0518fb430b8": "[\\mathcal{L}_k f](x) = \\frac{1}{k}\\sum_{n=0}^{k-1}f\\left(\\frac{x+n}{k}\\right)", "69ac918aa1c84bdf2a1739810eecd24f": " h \\cdot f(y_0) = 1 \\cdot 1 = 1. \\qquad \\qquad", "69ac93dc9dd067fe7eba8e137bb0ed54": "\\phi(\\mathbf{r})=0", "69ac9b7b34967eab2b702787e37e3d07": "\\mathbf{\\mu} = \\frac{ge}{2m_\\mu}\\mathbf{S}", "69acf82a92780ccf127f374588d9564d": "\\frac{1}{\\gamma} \\frac{{\\rm d}\\mathbf{m}}{{\\rm d}t} = \\mathbf{m \\times H_\\text{eff}} - \\frac{\\lambda}{\\gamma m}\\mathbf{m} \\times \\frac{{\\rm d}\\mathbf{m}}{{\\rm d}t}", "69ad419c1d4a7c51ad22a205ca721f20": "J^{\\star} = 2451545.0009 + \\dfrac{l_w}{360^\\circ} + n", "69adc26f0440864cc3214949a7255d2c": "(0,2a)", "69ae1d171c17f6c98b0908e58446011d": "f = \\phi P\\,", "69ae4831e5e82cfc3dd163b98bccb063": "(\\mathbf{A}\\otimes \\mathbf{B})^T = \\mathbf{A}^T \\otimes \\mathbf{B}^T", "69aef69712461f576488eb46dfcd0687": "\\mathfrak{D}", "69af23572c2f2e52d0d7aac913cffa01": " \\mu = \\frac {\\pi n \\left( n-1 \\right) \\left( n-2 \\right) \n \\cdots \\left( n - \\left( k-1 \\right) \\right) } \n {k \\left( k-2 \\right) !} \\left( \\frac {w}{L} \\right)^{k-2} \n \\left( {\\frac {d}{L}} \\right)^{k} ", "69af4729530d58005596c6f3cd6778f0": "J := J_1, ..., J_n", "69afb6f4bc3db5e19a5bd2bc1cfd075a": "\\eta_d", "69b0544c015cbe9f1b8127a006f0651c": " {\\mathbf{}}L_i = (B^\\mathrm T_iS_{i+1}B_i + R_i)^{-1}B^\\mathrm T_iS_{i+1}A_i. ", "69b057f37b605d95bf336066371811ec": "z_4 = \\frac{1}{24} (X^2Y^2 - 2XYXY - Y^2X^2 + 2YXYX).", "69b123e865f0fcc704f3fd0146578461": "\\textstyle \\mathbf{w}\\cdot\\varphi(\\mathbf{x}) = \\sum_i \\alpha_i y_i k(\\mathbf{x}_i, \\mathbf{x})", "69b13d4c0d139cb19c0f1b2184244ff7": "\\eta^{\\mu\\nu}\\partial_\\mu\\partial_\\nu\\phi+m^2\\phi=\\partial^2_t\\phi-\\nabla^2\\phi+m^2\\phi=0", "69b15249406ed21ca2106fb2853cb082": "n \\stackrel{.}{-} m = \\max\\{n-m, 0\\}", "69b1a5da5215813d2a8463cd6f621a65": "p > N", "69b1a9a5da532e0900fe7db02bd10a5b": "\np(k)= \\frac{1}{k(k-1)} \\qquad (k=2,3,\\dots,N). \\,\n", "69b1c49dccc9c5d8e8ce4aee366861bc": "G(z,x) = -\\frac{1}{2\\pi} \\log \\vert z-x\\vert + \\gamma(z,x)", "69b2701fbbcc7761e923b4456cbde224": "\\mu : X \\to \\mathbb{R}.", "69b28fd5117196299fb9e9c45bbd3e15": "\\beta = \\frac{x^2}{2}", "69b2a229ef76169054f7e7d2e875c3f2": "\\frac{\\rho}{1-\\rho} \\text{ C}(c,\\lambda/\\mu) + c \\rho.", "69b2b3988bfb4dc51280dbb9f91d2fa4": "S(\\varrho)", "69b33207584d4dcda1f437260f4369a7": " A(z) = \\sum_{k=0}^\\infty z^{2^k}, ", "69b3b7fce19048a6c3732877f98f711f": "\n\\begin{bmatrix}x \\\\ y \\\\ z\\end{bmatrix} = \\frac{1}{ a \\wedge b \\wedge c}\n\\begin{bmatrix}\n d \\wedge b \\wedge c \\\\\n a \\wedge d \\wedge c \\\\\n a \\wedge b \\wedge d\n\\end{bmatrix}", "69b42d543d1c5c91757fafa647ea12c5": " \\overline G(X)=G(X)/Z", "69b4404f18244d18b65760f9df07db25": "\\pi_S", "69b49214681c4d33feb91e2e42374c75": "a_1\\times \\rho \\cos(\\theta)", "69b49458b26d68aa5731cb01b48a9220": "K \\otimes_\\Q \\mathbb{R} \\cong \\mathbb{R}^2 ", "69b4cdcddb5a3190d8d4c80a518e237e": "\n\\frac{dI}{dx} = -I n \\sigma \\ \\stackrel{\\mathrm{def}}{=}\\ -\\frac{I}{\\ell}\n", "69b546a323011887fb4c0a97111d6a54": " u(t,x) = \\frac{1}{(2 \\pi)^n} \\int \\frac{e^{i (\\langle x,\\xi \\rangle + c t | \\xi |)}}{2 i |\\xi |} \\hat f (\\xi) \\, d \\xi - \\frac{1}{(2 \\pi)^n} \\int \\frac{e^{i (\\langle x,\\xi \\rangle - c t | \\xi |)}}{2 i |\\xi |} \\hat f (\\xi) \\, d \\xi .", "69b5549b9be010d39b4be899cd9136fd": "u_{xy}=u_{yx}", "69b58a868c1cbe6d6042546589b8fa77": "\nR_{x_1,x_2} (\\tau) = \\sum_{n=-\\infty}^{\\infty} x_1(n)\\ x_2(n+\\tau)\n", "69b5c3b481dc9baefdc9b841d4574fee": "q''(x_2)=2\\frac {a-2b}{{(x_2-x_1)}^2}", "69b601d5ac0dd0dbc0cd462a5ba9f1e9": "\\overset{d}{=}", "69b6519c54461c2f7cc0d90673bb9def": "(\\neg 2) \\frac{\\neg \\neg A}{A}", "69b65473b96417e91547b02ce03a4173": "x\\in[0,\\infty)", "69b69c9eb3d029e510dc4cc10f8eaf6a": "F = C_c \\times E = Max Q \\times S_c \\times C_c", "69b6fb3548d1fd05856055349e59e784": "283 \\times {4 \\choose 1} + 222 \\times {4 \\choose 1}{10 \\choose 1}{3 \\choose 1} = 27,772", "69b74749c4b38fb87aaf6725e70a6887": "z_k(s)", "69b82ec432cbaf58eaf3356828306331": "\\scriptstyle+\\infty", "69b86e7ed5d75923a04b2923f23b68f4": "x^- \\to e^{-\\beta}x^-", "69b886448947b43753f6f36f9e0cba5c": "\\frac{1 + {\\scriptstyle\\frac{2}{3}}z + {\\scriptstyle\\frac{1}{5}}z^2 + {\\scriptstyle\\frac{1}{30}}z^3+ {\\scriptstyle\\frac{1}{360}}z^4}\n{1 - {\\scriptstyle\\frac{1}{3}}z + {\\scriptstyle\\frac{1}{30}}z^2}", "69b88c64b89391a0415b0dee3bbe171e": "\\bullet \\bullet \\bullet \\bullet \\bullet \\bullet \\mid \\bullet \\bullet \\mid \\bullet \\bullet \\bullet \\mid \\bullet \\bullet \\bullet \\bullet \\bullet \\bullet \\bullet ", "69b895a236077cf55cb360716da25c3f": "S \\rightarrow aBSc", "69b8a5dacad10105ff5ec5b7dec7a6b2": "=\\int_{X^{2m}}Pr[\\sigma(x)\\in R]dP^{2m}(x)", "69b8b75b6e2fb32c93c0ca066dfb48cf": "\\textstyle \\frac{1}{2}", "69b8e5f6777323b545f6969a00aa5c63": "SP-DP", "69b925053fad51c310d4ae9a664f46fa": "\\scriptstyle f ,", "69b936c1d8af1ee042eb128c361ca246": "r=\\,\\mathbf{K}q", "69b937053a9db5ad271b4b219b9e596e": "V_t \\ = \\ a_0 \\cdot \\sqrt{5\\left[\\left(\\frac{q_c}{P}+1\\right)^\\frac{2}{7}-1\\right] \\cdot \\frac{T}{T_0}}", "69b998a59cb021612767187decfa4763": " \\dot{x}(t) = A(t) x(t) ", "69b9ad5384aa46feec966b280b41e6c6": "\\mathrm{SR} \\ge 2\\pi f \\times V_{\\mathrm{pk}}, ", "69b9bf6c492ca8ea1f468121f9d0b2a1": " \nv_{k} = 2 \\beta v_{k-0.5} - v_{k-1}\n\\,\\!", "69b9c2fa90a7a0ab0efe6a46d3f9dcdf": "{\\rm Tr}_{q^{n+2}/q}:{\\rm GF}(q^{n+2})\\rightarrow{\\rm GF}(q)", "69b9ff6ff69c3135a6acd6dff7ab1dc5": "\\rho = RA/l \\,\\!", "69ba1a4408d38b6810b165a53b968d4e": " \\; x_1^2 x_2^2, \\; x_1^2 x_3^2 \\; \\ldots \\; ", "69ba34777f7d235d6ef47148f9fd16ee": "\\lim_{n \\to \\infty} a_n = 0.", "69ba9c05e4885e8db338843b6c041d5f": " K=\\{x\\in E|f_n(x)\\rightarrow f(x)\\} ", "69bab188314bce0a0139857de2757fcf": "\\Delta v_3 = \\sqrt{ \\frac{2 \\mu}{r_f} - \\frac{\\mu}{a_2}} - \\sqrt{\\frac{\\mu}{r_f}} ", "69bab5e63adf6cf13a87cea370318bf6": "\\frac{dP}{dz} = -g\\rho", "69babb6335e711173750e8bc6c75d124": " t' = \\frac{2h}c \\frac{1}{\\sqrt{1 - \\frac{v^2}{c^2}}} = \\frac{t}{\\sqrt{1 - \\frac{v^2}{c^2}}}", "69babe99d17012dab87b9c05b76dde85": "\\mbox{distance ratio} \\equiv \\frac {d_{in}} {d_{out}}= \\frac {2 \\pi r}{l} \\,", "69bada09f4b04cb9760e95fb53b56eff": "E^{(t)}_0", "69bb1f667d3ef43df92c3db66c722fe0": "I_1, J_2, J_3", "69bb4f1dcc2abc6711471d18a8652f6b": "X_m = \\lbrace e_i | x_m \\in e_i \\rbrace. ", "69bc49f3d23cc1cc42e164fa9906454a": "F(z,w)= \\log \\left ( \\frac{f(z)-f(w)}{z-w} \\right ),", "69bc4ab3972fffe3821c2cad3fad50c3": "\n\\mathbf{p} \\cdot \\dot{\\mathbf{q}} - H(\\mathbf{q}, \\mathbf{p}, t) = \n-\\mathbf{Q} \\cdot \\dot{\\mathbf{P}} - K(\\mathbf{Q}, \\mathbf{P}, t) + \\frac{\\partial G_{2}}{\\partial t} + \\frac{\\partial G_{2}}{\\partial \\mathbf{q}} \\cdot \\dot{\\mathbf{q}} + \\frac{\\partial G_{2}}{\\partial \\mathbf{P}} \\cdot \\dot{\\mathbf{P}} \n", "69bc7dd67618fb03d9729ef550c268f7": "x^2+(b_{2}+c_{15})x-b_{2}c_{15}=0", "69bc944a5af3a43bcebf0db454f6a2a7": "\\frac{\\overline{IA} \\cdot \\overline{IA}}{\\overline{CA} \\cdot \\overline{AB}} + \\frac{\\overline{IB} \\cdot \\overline{IB}}{\\overline{AB} \\cdot \\overline{BC}} + \\frac{\\overline{IC} \\cdot \\overline{IC}}{\\overline{BC} \\cdot \\overline{CA}} = 1.", "69bc961f30d73b6c174686eab1e2456e": " R^m_\\ell(\\mathbf{r}+\\mathbf{a}) = \\sum_{\\lambda=0}^\\ell\\binom{2\\ell}{2\\lambda}^{1/2} \\sum_{\\mu=-\\lambda}^\\lambda R^\\mu_{\\lambda}(\\mathbf{r}) R^{m-\\mu}_{\\ell-\\lambda}(\\mathbf{a})\\;\n\\langle \\lambda, \\mu; \\ell-\\lambda, m-\\mu| \\ell m \\rangle,\n", "69bcd958c7fd4213c1403be58e61fe9f": "S_1, S_2, \\ldots, S_n", "69bd04c1cd41a19eef39d1719492a4aa": " I = E du^2 + 2F du dv + G dv^2 ", "69bd4f8e2cab508322bf785d73b2eed9": "\\mathbf{B} = \\log{(\\mathbf{mn})}.", "69bd7351c5c93fbde5f16d52e2e3729e": "T_{w,2}=1000(x_{n,2}-x_2)-650(y_{n,2}-y_2)", "69bd842f15253c4fc5a1533b6d387895": "\\Box \\phi = 4 \\pi \\rho", "69bda58ea097e2e6fcd087c726e82635": "\\alpha\\ =\\ \\frac{e^2}{\\hbar c_0 \\ 4 \\pi \\epsilon_0}\\ =\\ \\frac{e^2 c_0 \\mu_0}{2 h}", "69bdb2bdf1b5fece752a2da381825a10": "\\begin{align}\n\n\\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i=1}^n A_{i,\\sigma_i}\n\n&=\\sgn([1,2,3]) \\prod_{i=1}^n A_{i,[1,2,3]_i} + \\sgn([1,3,2]) \\prod_{i=1}^n A_{i,[1,3,2]_i} + \\sgn([2,1,3]) \\prod_{i=1}^n A_{i,[2,1,3]_i} \\\\ &+ \\sgn([2,3,1]) \\prod_{i=1}^n A_{i,[2,3,1]_i} + \\sgn([3,1,2]) \\prod_{i=1}^n A_{i,[3,1,2]_i} + \\sgn([3,2,1]) \\prod_{i=1}^n A_{i,[3,2,1]_i}\n\n\\\\\n\n&=\\prod_{i=1}^n A_{i,[1,2,3]_i} - \\prod_{i=1}^n A_{i,[1,3,2]_i} - \\prod_{i=1}^n A_{i,[2,1,3]_i} + \\prod_{i=1}^n A_{i,[2,3,1]_i} + \\prod_{i=1}^n A_{i,[3,1,2]_i} - \\prod_{i=1}^n A_{i,[3,2,1]_i}\n\n\\\\\n\n&=A_{1,1}A_{2,2}A_{3,3}-A_{1,1}A_{2,3}A_{3,2}-A_{1,2}A_{2,1}A_{3,3}+A_{1,2}A_{2,3}A_{3,1} \\\\\n& \\qquad +A_{1,3}A_{2,1}A_{3,2}-A_{1,3}A_{2,2}A_{3,1}.\n\n\\end{align}", "69bdbb47d47922e00ad7e095c90fb887": "\\int_V f(x) \\, d\\mu(x)=1.", "69bdca4bec8062051816ed614a0d598b": "\\begin{smallmatrix} \\tau_{\\rm MS}\\ \\approx \\ 10^{10} \\text{years} \\cdot \\left[ \\frac{M}{M_{\\bigodot}} \\right] \\cdot \\left[ \\frac{L_{\\bigodot}}{L} \\right]\\ =\\ 10^{10} \\text{years} \\cdot \\left[ \\frac{M}{M_{\\bigodot}} \\right]^{-2.5} \\end{smallmatrix}", "69bdd2eb76eef1415b6456b8f35882fd": "O(m^2 nl\\tbinom ld 3^d)", "69bde3aa661278921eb31cb5b50e5b22": "~D=\\frac{c}{b} \\sin\\beta", "69be12a6ec1f661b01daa135d39eec07": "G \\in \\mathcal G", "69be39ec6f9f32dd44a4c3759c1e8061": "\nP(\\sup_x|F(x)-F_n(x)|>\\varepsilon)\\le2e^{-2n\\varepsilon^2}.\n", "69be58a2d58cf375d4f86d1deb2ab157": "D>0", "69beb65f2df7cee23ee3a90ee9a551f8": "\nQ = R \\sqrt{\\frac{C}{L}} = \\frac{R}{\\omega_0 L}\n", "69beca35c1f0c4af80604a9aa0e61707": "j^2=-1.\\,", "69bf071c5a6c220722f428b8498deded": "\n\\beta_0 = \\frac{\\mathbf{r}_1^\\mathrm{T} \\mathbf{r}_1}{\\mathbf{r}_0^\\mathrm{T} \\mathbf{r}_0} =\n\\frac{\\begin{bmatrix} -0.2810 & 0.7492 \\end{bmatrix} \\begin{bmatrix} -0.2810 \\\\ 0.7492 \\end{bmatrix}}{\\begin{bmatrix} -8 & -3 \\end{bmatrix} \\begin{bmatrix} -8 \\\\ -3 \\end{bmatrix}} = 0.0088.\n", "69bf2afe5c38032a0fa1e1862aa6de28": "{{\\theta}}", "69bf6745bbdd4980db3d749fbdfd69f0": "\\Gamma \\left ( \\mu^- \\rarr e^- + \\bar{\\nu_e} +\\nu_\\mu \\right ) = \\Gamma \\left ( \\tau^- \\rarr \\mu^- + \\bar{\\nu_\\mu} +\\nu_\\tau \\right ).", "69bf9d452d0b6b5560a0e65c75d49638": "\\mathbf a = \\left( \\mathbf {J^T W} \\mathbf{J} \\right)^{-1} \\mathbf{J^T W} \\mathbf{y}\\qquad W_{i,i} \\ne 1", "69bfa74829418a04ff4e80a3e6bca6b3": "\\textbf{d}_j^T \\textbf{d}_q = \\textbf{d}_q^T \\textbf{d}_j", "69bfde9023f1dfa9998c9c2bf42e08b7": " z^{n-1} det^{col}(d/dz - E/z) = det^{col}(zd/dz - E - diag(n-1,n-2,...,1,0) ) ", "69bffffd432bd9e746c7f511e7957291": "I_P", "69c07842461b447c5d6e424051799c1c": "P(r|s) = \\prod_{l} \\left [ \\prod_{i,j} v_i ( t_{ijl} | s)dt \\right ] exp \\left [ -\\sum_{i} \\int_{0}^{T} dtv_i(t | s) \\right ] ", "69c08fd50b84a775dadef0f64117b16f": " O(N^{1.25} \\, \\log N). ", "69c0c581ece9facecf37379569030c55": "y_n = f(\\cos[n\\pi/N]) + f(-\\cos[n\\pi/N]) . \\!", "69c0c97287a1405a4db0947a76fae64c": "\\displaystyle{f \\in C^k(\\overline{\\Omega})}", "69c1104148d089ff1b679d354b0c4848": "T_{i i}", "69c125a18f06240904d5497ad0bfe515": "\\lambda_z = \\mu \\, \\hbar", "69c12eeab0fac9f1cebd2f99de0cfac3": "\\displaystyle{([b^2,a,b],c)=(b,[a,b^2,c]).}", "69c187d2722016662eda74baf6a0019c": "\\hat{N_i}|\\Psi\\rangle_\\nu=N_i|\\Psi\\rangle_\\nu", "69c1a3cb0b731270bf59c2beca3d1b37": "X^\\mathrm{opt} = \\alpha W + \\beta \\mu", "69c22fd1ede7a14a54fc9f23fed7fcde": "S_{12} = S_{21} = 0 \\,", "69c244ca302be6728b311df37da42e6c": "\\delta = 2\\pi - q\\pi\\left(1-{2\\over p}\\right).", "69c256f0e03a71c32e50843d98b98921": "=>F/8 + C/4=12 \\,", "69c25aaa3893a9c2c5882dfdea9bfc7d": "\\text{transient inertial forces + convective inertial forces}=\\text{gravitational force + Pressure force + viscous forces}\\,\\!", "69c27b50f42372b282e4279015045f0b": "-\\Delta u_i^{(k)} = f_i, \\qquad u_i^{(k)}|_{\\partial\\Omega} = 0, \\quad u^{(k)}_i|_\\Gamma = \\lambda^{(k)}", "69c27bbd855748506738f599669a6187": "FOV_C= \\frac{FOV_P}{mag}", "69c2c487b704f4f5a8ad2d50d70091cb": " \\dot{V} = \\dot{V}_A + \\dot{V}_D ", "69c2c7e22b94ba23db702238a8e421c9": "\\textstyle L_i,R_i ", "69c2e3600730e5663a9014f7aef7b221": "F_n=\\frac{2}{c_n\\pi}\\int_{0}^\\infty f(x)R_n(x)\\omega(x)\\,dx.", "69c37432df99bf140c10ff5fad9e2f08": "\\mathcal L_{\\Psi}+\\mathcal L_{\\Psi\\Phi}=\\Psi^+(i\\partial_0+i\\vec{\\partial}\\vec{\\sigma})\\Psi-ig\\Phi\\Psi^+\\sigma_2\\Psi^*+\nig\\Phi^*\\Psi^T\\sigma_2\\Psi", "69c3c17b513ace3c2424ff3f078171c2": "[x_0:\\cdots: x_n] \\mapsto \\left (\\frac{x_0}{x_i}, \\dots, \\widehat{\\frac{x_i}{x_i}}, \\dots, \\frac{x_n}{x_i} \\right )", "69c435943fff70e1b13fd5766c50c279": "\\frac{1}{2}(T-t)", "69c47484034c4bb06e621829098f2baa": "\\left\\{ \\overline{D}_{\\hat{\\dot{\\alpha}}}, \\overline{D}_{\\hat{\\dot{\\beta}}} \\right\\} = c_{\\hat{\\dot{\\alpha}}\\hat{\\dot{\\beta}}}^{\\hat{\\dot{\\gamma}}} \\overline{D}_{\\hat{\\dot{\\gamma}}}", "69c47ac1d70cae522e483f90f211ac2a": "\\boldsymbol{\\Phi}_r, \\ \\boldsymbol{\\Phi}_s\\,\\!", "69c48e3db3d39fcab0ace23a16a6c773": "\\frac {4\\cdot 10}{2{,}598{,}960} \\approx 0.0015\\%", "69c4af8e9be14ccaf6390ac2d530923a": "\\Delta = \\{(x, -x) \\mid x \\in \\mathbb{R}\\}", "69c4be3786610ae67bef1b42986dcf40": "|P(V)|", "69c5225e19f9ca99ad2febe7069bac11": "\\Delta S_{ext} = - {Q \\over T},", "69c56dfb42aee03bf0d905e12a56d6e2": "\\overline{\\mathbb{Q}} = \\mathbb{R}", "69c56f4bde90c96728d61e215786ca74": "p^2 < a^2 + b^2", "69c5aca1796adcf513469e981c06571e": "\n B_y + i B_x = C_n \\cdot ( x + iy )^{n-1}\n", "69c5c96d20164ab87a18c9e1e0f5d9fc": " \\Gamma \\sim \\Lambda^{-2/3}\\ ", "69c62c99c53eb327f6ccab80eb08147d": "\\textstyle [\\mathbf{v}_1, \\ldots, \\mathbf{v}_m] = \\sum_{i_1 < \\cdots < i_n} \\vert \\det(\\mathbf{v}_{i_1} \\cdots \\mathbf{v}_{i_n}) \\vert", "69c653bcbbb6133e016cbd1abe25627e": "v_e = I_{sp} \\cdot g_0", "69c659dc39e141d748035d9b02e342b0": "b_2 = -a_2 \\,", "69c6623f663981659580e4ef09ee3e50": "m_e v_e \\frac{\\partial v_e}{\\partial x} = - e E.", "69c669d76d618d54a2e181826e43d39f": "E\\rightarrow M", "69c66b3f1c4471027a00aaf826343e22": "x_0, x_1, ... x_{n+1}", "69c68127659a013953bca3b29fbf4a12": "\n\\sum_{n=2}^{n_\\text{max}}(2n+1) = n_\\text{max}(n_\\text{max}+1) + n_\\text{max} - 3 = 130317", "69c6a00ae1922d79e6fd8e38b5c55350": "p\\le n", "69c6bf6c6f8a10bbdddcf0f8caf1ea8c": "P_{\\mathrm{in}} = F V,\\!", "69c6d4c89d6cd2c9af492656ebe001c8": " \\infty", "69c6dbfbd47cee6a752b35c135a21a77": "\\sin_k(i)\\equiv -\\sin_k(t). \\, ", "69c6f3ed88e6417c3068aa1914b32d88": "K_\\mathit{rw}^o", "69c6f5f3c13cfc14594ee42e8c0fb68b": "\\forall a_0,\\dots, a_{n-1}\\;\\exists s\\;\\forall i < n \\; \\beta(s,i) = a_i", "69c6ffbe1bf2fd2f2af0257baf0de656": "\\hat{h}_i^+ = \\hat{h}_i - \\mu\\frac{\\partial E}{\\partial \\hat{h}_i}", "69c788a50267142124aa1f67c603fc40": "n=2^en'", "69c7d546266ce5690ac5e4c76760885b": "\\scriptstyle (b,\\, x_2)", "69c7dfc8dea21fefff1f993d473700cc": "(p,00111,Z) \\vdash (p,0111,AZ) \\vdash (q,0111,AZ)", "69c81f747f9ed9bed8bcdada74b6c11c": " (1)\\,", "69c8210748a5eccda2ef0895cfa9d6c0": "\\overline{\\lambda}", "69c84ed567804bed7066f7660325ee07": " \\bigcup_{i\\in\\mathcal{I}}C_i ", "69c8aaaf1469ae61e51e79c38299b4d6": " r = e^\\rho ", "69c8e548e799c8c94176b7a51ae2a539": "q^m-1", "69c91d877552dbd5f07e5c48394383fa": "C \\in C(F)", "69ca767e1f49cbc99d4becc708cefd66": "\\bar{L}^{(k)}_{n+k} = (-1)^k (n+k)! L^{(k)}_n", "69ca76d08da3b72b60ce184803a25c16": "{2a \\times b \\over a+b}=d", "69caae98d3ccb0a487ff6824693d6fdf": "l_0 = - \\infty", "69caafb2595bdf341b249207517463f0": "\\mathrm{azeq}_x", "69cabecfa2e0aa98d2cca4802efadff3": " \\hat{\\mathbf{e}}^3", "69cae349aad419f769b2b72e7b5da568": "\\bar y = \\frac{1}{n} \\sum_i y(t)_i=x(t)+ \\frac{1}{n} \\sum_i w(t)_i", "69cb40c80daf43cb60cce6d9005cdf1e": "\\Omega = N!/N_1!N_2!\\,", "69cb5f146629127ae157048a72703633": "X\\cdot a = a\\cdot X +\\delta(a).", "69cb86d97707d1c83eb8d775d98a8ba0": " 1 - \\alpha", "69cb8f5c50d169bb147bc61e64350509": "\\chi(H)", "69cbaea04e0247a6d36e4c68a2656309": " F_{e} \\phi_{E}-F_{w} \\phi_{P}\\,= D_{e}(\\phi_{E}-\\phi_{P})-D_{w}(\\phi_{P}-\\phi_{W})", "69cc155f34d1662b83d59590f920c8f2": "{\\mathit{He}}_n^{[-\\alpha]}(x)\\,\\!", "69cc246a897196109350da832145e44f": " \\operatorname{build-param-lists}[q, D, V, K_7] ", "69cc36dbd0062a8c9355fffd6ced3ca8": "n = 0\\;", "69cc3ffc1948f6ec8255b2cd6c7149b7": "w\\to \\infty", "69cc851a82f9655fc6866bd9fb569db9": "\\mathbb N^k", "69cc96c9661d46637ec3aee75b10feed": "\\vec F_g = - \\hat r ~ G ~ \\frac{M m}{R^2}", "69cc9dbd37c451ab6f21b798548c7821": "\\scriptstyle \\nu(mg) = |m| \\nu(g)", "69cd892f3c5c268a1f926c920ff7be07": "l^an_a=-1", "69cd94441c83561cb16234b84036dd66": "\\Delta {{V}_{BE2}}\\approx \\frac{\\Delta {{I}_{IN}}}{{{g}_{m2}}}", "69cda15ae0bb11dec9951bd7407b776c": " \\frac{d}{dt} y(t) = Ay(t), \\quad y(0) = y_0, ", "69ce2ab67d3b0060556859825e56a592": " = -\\frac{1}{4\\pi}\\iiint_{\\vec{r}'} \\frac{\\vec{E}(\\vec{r}') \\bullet (\\vec{r} - \\vec{r}')}{\\|\\vec{r} - \\vec{r}'\\|^3}d\\tau' ", "69ce57d22236f8e89c4aa915c0f68ae9": "e= \\lim_{n \\to \\infty}n^{\\pi(n)/n} ", "69ce8d777bea84d927fb19acc64e07f8": "\\int x^2\\,\\operatorname{arcosh}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{arcosh}(a\\,x)}{3}-\\frac{\\left(a^2\\,x^2+2\\right)\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}}{9\\,a^3}+C", "69ced1301149a4f7d469e9a362a50228": "Y=C+I+G+NX \\, ", "69ceed6570eaf9478cc7b2a4dd9fe460": "{\\oplus}\\hbox{P}", "69cf10ed8faa7ddd3d6328d97ad799ec": " E = It^p \\ ", "69cf4895c2ed823f96098109ad114bba": "\n(1-x^2) {d^2 y \\over d x^2} - x {d y \\over d x} + p^2 y = 0 \n", "69cf88b43a0308371490516a2dd0b19c": " h \\over \\lambda ", "69cfa596a18dd6adc7eb6067967d26b9": "L_2(9) \\cong A_6", "69d03f26e0951d054d628d9208b37299": "p = \\frac{{\\rm d}P}{{\\rm d}\\mu}", "69d04ad93c7a9514e189967f7a45e81d": "m = \\frac{1}{2\\,b_2} \\!", "69d128311955ce4f5882da1b5e21692b": "| 1 - \\omega \\lambda_j |", "69d14c45a7582e20218de7eb5041b019": "\nF_{eq} \\ \\stackrel{\\mathrm{def}}{=}\\ \\left( \\frac{4}{3} \\right) \\frac{(1/p)^{2} - p^{2}}{2 - S \\left[ 2 - (1/p)^{2} \\right]}\n", "69d16b2c4ce4b208fd16b401a1abd061": "a^2 = \\frac{2\\mu_{2}\\beta_{2}}{3-\\beta_{2}}", "69d180f6a294eb21275cdadbee4aa60f": "i\\in\\mathbf{Z}", "69d1a17f46c79a38201ac2a0862681fe": "\\ g_{\\phi}= \\left(9.7803267714 ~ \\frac {1 + 0.00193185138639\\sin^2\\phi}{\\sqrt{1 - 0.00669437999013\\sin^2\\phi}} \\right)\\,\\frac{\\mathrm{m}}{\\mathrm{s}^2}", "69d1b6be51b5c3ebeb613a3aee8b4cee": "\\wp(z;\\Lambda)=\\wp(z;\\omega_1,\\omega_2)", "69d1ee39578b80a2d41ec21b9c719706": "\\alpha = R \\left( \\frac{\\omega}{\\nu} \\right)^{1/2} \\ = R \\left( \\frac{\\omega \\rho}{\\mu} \\right)^{1/2} \\, ", "69d2347b7c5add9b92f9de169aef8881": "M := b_1(G,t) := \\operatorname{rank}H_1(G,t),", "69d245f5dddcefd583a3033220e4f794": "2^{2^{1}} + 1= 2^{2} +1 = 5,", "69d24ed333e0d75c70777e7f47c70e3e": " R_N^k(n) = \\sum_{i=0}^n r_N^k(i)", "69d2bb2584cdabb9ab7e5c92ea0af60d": " \\exists x (K(x) \\land \\forall y (K(y) \\rightarrow x=y) \\land B(x)) ", "69d39a7becea657dcc297f01d084ed1d": "h'_{L}", "69d408ca4f75393c2ae1928a93ef5868": "R_{\\text{B}}", "69d4ada8d91374ef4290d42e3b6e77a9": "\nT_{\\delta}^{Y^{n}|x^{n}}\\equiv\\left\\{ y^{n}:\\left\\vert \\overline{H}\\left(\ny^{n}|x^{n}\\right) -H\\left( Y|X\\right) \\right\\vert \\leq\\delta\\right\\} .\n", "69d56f7830813ea631d57512c5d6f101": "\nv_j(x) = \n\\left\\{\n\\begin{array}{lr}\nL^{- \\frac{1}{2}} & j = 1\\\\\n\\sqrt{\\frac{2}{L}} \\cos (\\frac{(j - 1) \\pi x}{L} ) & otherwise\n\\end{array}\n\\right.\n", "69d577d74f05d4303c995fcebdf8b6c3": "\\cos (\\beta) = Z_3.", "69d6132f1992dbc592dc09d6b0d8a099": " 2\n^{O(\\sqrt{\\log n})} ", "69d6225b9f07b2942581212bbe194eda": "T_E", "69d65d9f0bf8ddc676f80c3f54147450": "A^{- 2}", "69d6822b6f2cd456c5708dc6fc6447bb": "O(\\log V)", "69d6921fbe644cb7fa6a49fbbcb6a76e": "Q = \\oint\\limits_{\\text{cycle}} I \\, dt = -\\oint (G^{31} \\, dX_1 + G^{32} \\, dX_2)", "69d6d63cf318821824a72c6f293a2aae": "(x_1, x_2, y_1, y_2, z)", "69d6dd69c0bfe32268a9df14b2b18e42": "n_i=n_{i+1}=0", "69d72f2290a9f27af4758043863f121c": "F \\to E \\to B,", "69d79e10ecc821cf5ca0ae8b27378b52": "g_{ab} = \\partial_a X^\\mu \\partial_b X^\\nu g_{\\mu\\nu} (X^\\alpha) \\ ", "69d7e53ae9249ab88003aa5ad46f46a2": "z\\in\\mathbb{H}", "69d80006d83e2e481b34961ea974c1f9": "t \\uparrow 0)", "69d81b7557ad1cdc3ea6bfa5effcb40f": "\\mathbf{E}^{(M)}(\\mathbf{x})=\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell a_{\\ell m}^{(M)} h_\\ell^{(1)}(kr) \\mathbf{X}_{\\ell m}(\\theta, \\phi)", "69d84f084748742e5fe1cb0ef403d1fe": "\\Psi = \\frac{E^*h'}{\\sigma _0}> \\tfrac{2}{3}~.", "69d88173074acf55593d710370bed1fe": "\\max_{\\mathrm{a}}P[A = a]", "69d899d81d834ef2f894297f0868fe5f": "z=f(t)", "69d8c865f0ccce249eff9ec226511fb6": "V_{71}", "69d8d76bf5aed5a167b4487597d816ac": "C_a m^{p_a} \\lambda_a^m", "69d95c7deec7d3f9683d6528a0e6f0ba": "\\beta_n^{HS} = -\\frac{\\Delta x_n^\\top (\\Delta x_n-\\Delta x_{n-1})}\n{s_{n-1}^\\top (\\Delta x_n-\\Delta x_{n-1})}\n", "69da147b53b2c8008b98871c0f96c554": "\\mathbf{a} = [ a_1\\ a_2\\ a_3 ].", "69da2e7d2833d127b0263d10cd871ff0": "P_\\pi \n= \n\\begin{bmatrix}\n\\mathbf{e}_{\\pi(1)} \\\\\n\\mathbf{e}_{\\pi(2)} \\\\\n\\mathbf{e}_{\\pi(3)} \\\\\n\\mathbf{e}_{\\pi(4)} \\\\\n\\mathbf{e}_{\\pi(5)} \n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\mathbf{e}_{1} \\\\\n\\mathbf{e}_{4} \\\\\n\\mathbf{e}_{2} \\\\\n\\mathbf{e}_{5} \\\\\n\\mathbf{e}_{3} \n\\end{bmatrix}\n=\n\\begin{bmatrix} \n1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 & 0 \n\\end{bmatrix}.\n", "69da2fae354e01181f6e04ce2fdfe503": "\\bold{r} = \\sum_{i=1}^n x_i \\bold{e}_i = x_1 \\bold{e}_1 + x_2 \\bold{e}_2 + \\cdots x_n \\bold{e}_n ", "69da9a64c3bb58c3a2de6ffa2665336c": "k=1,\\ldots,n", "69daf2ce1092adf45f50dd44b9056c4a": "\n EI~\\cfrac{\\mathrm{d}^4 \\hat{w}}{\\mathrm{d} x^4} + m\\omega^2\\left(\\cfrac{J}{m} + \\cfrac{E I}{k A G}\\right)\\cfrac{\\mathrm{d}^2 \\hat{w}}{\\mathrm{d} x^2} + m\\omega^2\\left(\\cfrac{\\omega^2 J}{k A G}-1\\right)~\\hat{w} = 0\n ", "69db2eb6df0ed89a758ddb95e31da400": "-1 > y > x + 1", "69db47cfdb953244c3f9f56df0def1ac": "\\frac{d y}{d x}=\\frac{d y}{d u}\\, \\frac{du}{d x}", "69db54528c7c50a5ca8ec3b494582cb7": "x_i\\ne 0", "69dba7c729fb00b64479d88272b4ab87": "\\alpha T = 1", "69dc14ff62d990478b5781885950895f": "x_1\\rightarrow x_s(t)", "69dcabda4737ec62c910ea43dc75f2e2": "\\lim_{n\\to\\infty} L_{n} = \\frac{1}{e}", "69dcc08efd1f913df282f9610f2a043a": " S_{6} := \n\\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\\\ \n 1 & 1 & z & z & z^2 & z^2 \\\\\n 1 & z & 1 & z^2&z^2 & z \\\\\n 1 & z & z^2& 1& z & z^2 \\\\\n 1 & z^2& z^2& z& 1 & z \\\\\n 1 & z^2& z & z^2& z & 1 \\\\\n \\end{bmatrix}\n\\in H(3,6)\n", "69dd0710365a3fb1eb57318aa6aee3b9": "\\tilde A = \\left(\\begin{array}{c}{\\rm Id}_n\\\\ \\hline{\\rm A}\\end{array}\\right)", "69dd4774f90b54256181a65037800eed": "\\mathcal F(f(x + x_0)) = \\mathcal F(f(x)) e^{2\\pi i \\xi x_0}", "69dd924ffe38689a611022f5bf9271ea": "\\hat{p} \\psi (q) = -i \\hbar d \\psi (q) / dq", "69dd95b227095e431a9e388f3d96d277": "\\!\\,r=\\cos(k\\theta)", "69dd97e84f6c983125d1f16d8b3153f0": " | \\psi \\rangle", "69ddf37e4ff5a97516d98b8458bca4f3": "\\mathbf a \\cdot \\mathbf b = \\sum_i a^i b_i = a^1 b^1 + a^2 b^2 + a^3 b^3 + \\sin(\\phi) (a^1 b^3 + a^3 b^1).", "69de1cfeafbd4cf04d3f6f12401439d3": "f_k\\,", "69de324cb1b60e08c34a9a92e244e4d4": "e^{2/n} = \\left[1; \\frac{n-1}{2}, 6n, \\frac{5n-1}{2}, 1, 1, \\frac{7n-1}{2}, 18n, \\frac{11n-1}{2}, 1, 1, \\frac{13n-1}{2}, 30n, \\frac{17n-1}{2}, 1, 1, \\dots \\right] \\,\\!,", "69de5cebaaa4d2304dc435300f7087d9": "E(\\left|X-c\\right|)\\,", "69debfce1e156d6649c24577a4f868de": "\\tilde{\\mu}_n(D):=\\inf_{p\\in\\mathcal Q} \\sup_{z\\in D} |p(z)|", "69ded76a759be31b54f9ad3463745ca2": " \\epsilon _{i,j,k} ", "69ded91eebb8b8b18a19f05b53fcce63": "S(T) = \\frac{c_1}{\\lambda^5\\left[\\exp\\left(\\frac{c_2}{\\lambda T}\\right)-1\\right]}", "69def4f836c18efe0d2bdc373ac5076d": "(A-\\lambda_\\star I)x=0", "69defa66f67cbb1a20b0d31179d5882f": "f_\\ell(k)", "69df248a7391c5c1e58f88d5ad1776ab": "k = -\\frac{1}{i}\\ln(iP + F)", "69df619198038d92bb67896128c0e99f": "\\mathbb{E}\\left[g(X_1,\\dots,X_d)\\right]", "69df7379ebc7a95b5708fe9e88d80b71": "H+\\vec{w}", "69dfd1bca22af8d6ebca681239d64030": "F(b)=c", "69dfdf4e6a7c8489262f9d8b9958c9b3": "(a)", "69e066e0f473a7066daa2ddeccb1458e": "N(\\overline{X},1+(1/n)),", "69e06efc0fa66d8fb63da15281a3ddaa": "G(x) = \\int_0^x f(t) \\, dt\\,", "69e0780af7eaec510a7303246c1dcff9": "\\pi_2(BGL(R)) = 0 \\to \\pi_2(B(S^{-1}S)^0) \\to \\pi_1 (Ff) \\to \\pi_1(BGL(R)) = GL(R) \\to K_1(R)", "69e0791fa4ca883614bf61d93cee5971": "V(x - v^b(1),1)", "69e0c48ce42004519be1e576190e060d": "Y[x,y]=\\frac{x'}{xy'-yx'}", "69e0e4255bf4bc1888fd029c9627db38": "\\Sigma^{-1} C", "69e1062c65580db27a1e97b289d2a126": "\\sqrt{n}(z-\\zeta) \\Rightarrow N(0,1)", "69e1116a4f540d27d7e3fe26ce3cd75f": "\\sum\\limits _{m}\\left[p_{m}\\nabla^{2}q_{m}-q_{m}\\nabla^{2}p_{m}\\right]=\\mathbf{P}\\cdot\\nabla^{2}\\mathbf{Q}-\\mathbf{Q}\\cdot\\nabla^{2}\\mathbf{P}.", "69e12274e073a1eb3144c7d2e0492848": " F = \\frac{lk_BT}{ \\langle \\delta x^2 \\rangle} ", "69e130a830ddbf5cf1ca65ac79828bbe": "P^{m}\\{|Q_{P}(h)-\\widehat{Q_{s}}(h)|\\leq\\frac{\\epsilon}{2}\\}\\geq\\frac{1}{2}\\,\\!", "69e1bfe12929db6d6973a504581c2990": "\\lambda(C_1 ) \\le \\lambda(C_2) \\mbox{ whenever } C_1\\subset C_2", "69e204457f55e39d7c85126d12951f6f": "\\deg(r_0(x)) < \\deg(b(x))", "69e23e8c1536f1fab7582863a44aea76": "X_i = \\sum_{j=1}^i F_j", "69e2463b67d54ed6d75f7040c00ffbe6": "\\,\\! L(x_1,x_2,\\lambda):= u(x_1,x_2)+\\lambda(m-p_1x_1-p_2x_2)", "69e26b78cf6a02723c56daf416588eb3": "r( \\text{out}, \\text{out})", "69e28c16c8dd95d7679579f0b186f7bb": "a(t) = \\frac{1}{1 + z}", "69e346830894461c8235c24be3ad7003": "(x_1^2+x_2^2+x_3^2+\\cdots+x_n^2)(y_1^2+y_2^2+y_3^2+\\cdots+y_n^2) = z_1^2+z_2^2+z_3^2+\\cdots+z_n^2\\,", "69e3609ec2a8c06044a5eb84bd9fe5cc": "dF = -S dT - P dV\\,", "69e3966668f4dabe833bedf0903ccb0c": "\\vec{B}", "69e3c721bacc030bfc58ed139825d72b": "L = \\mathbf{Z}b_1\\oplus \\ldots \\oplus \\mathbf{Z}b_k", "69e4364d425821a20b2c42aeeaced4a6": "\\mathbf{s_1}=\\mathbf{H_1}\\mathbf{x}", "69e4387ac96f3073516c6938bd424a52": "x_c=\\begin{cases}\n-0.2661239 \\frac{10^9}{T^3} - 0.2343580 \\frac{10^6}{T^2} + 0.8776956 \\frac{10^3}{T} + 0.179910 & 1667\\text{K} \\leq T \\leq 4000\\text{K} \\\\\n-3.0258469 \\frac{10^9}{T^3}+2.1070379 \\frac{10^6}{T^2} + 0.2226347 \\frac{10^3}{T} + 0.240390 & 4000\\text{K} \\leq T \\leq 25000\\text{K}\n\\end{cases}", "69e487825d1006dcebc210f8dd25b73b": "k=0,1,2,3", "69e4a5c0e66f43df00bb966cc09a994f": "\\scriptstyle \\sqrt 5", "69e4e28a87f55552ee80e19a37094af6": "\\boldsymbol{l}", "69e52f5b75679450b8c9afd0b938593a": " \\lambda_i e_i(t)= [T_K e_i](t) = \\int_a^b K(t,s) e_i(s)\\, ds. ", "69e5312f3db60c99d3f38089c6464e7f": "S = R \\cup \\left\\{ (x, y) : (y, x) \\in R \\right\\}. \\, ", "69e5520722bfb4dbdd74b512f02cacad": "d_p(x,y) = \\sum_{i=1}^n |x_i-y_i|^p", "69e55d92c886ca7a21ae897f6d2542ee": " dM = Qdx ", "69e6139e6279853c1f31625327a6cf08": "S_{55}\\,", "69e642e81fb946617091f4a39b90e189": "\\Big( \\pi \\models \\phi_1 \\land \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big(\\pi \\models \\phi_1 \\big) \\land \\big(\\pi \\models \\phi_2 \\big) \\Big)", "69e70a4acd06e1cdd7e6441a225f5fdc": " \\partial\\Phi_n +d_\\nabla(\\Phi_{n-1})+[\\omega_2,\\Phi_{n-2}]+\\cdots+[\\omega_n,\\Phi_0] = 0.", "69e725b78773eb805a66589fa7047521": " \\Phi^{2}(a,b) := [[\\Delta,L_{a}],L_{b}]1 =: (-1)^{\\left|a\\right|}(a,b) ", "69e77a55afd73a630a5c750aa2d517ea": " P(x')=y' ", "69e78f0f09622c539cd070d7879829c8": "M_{xy}'(t) = e^{+i \\Omega t} M_{xy}(t)\\,", "69e7fd6595bbbdae564388de1ca051ce": "\\pi_j = \\frac{C}{M_j}\\,,", "69e8103c8029eab5ba897a896a1ff6d3": "\nR^2 = s^2 - r^2 = p \\,\n", "69e81b5ad207b093e06764b7b31101d3": "\\operatorname{Pr}(K\\leq x)=1-2\\sum_{k=1}^\\infty (-1)^{k-1} e^{-2k^2 x^2}=\\frac{\\sqrt{2\\pi}}{x}\\sum_{k=1}^\\infty e^{-(2k-1)^2\\pi^2/(8x^2)}.", "69e8596810a6a739a70a7cf2558992ec": "\\oint_C {1 \\over z}\\,dz = 2\\pi i", "69e87600b6b1f05873da0725df312ec7": "\\!\\,x=\\cos(kt)\\sin(t)", "69e8cb73da28e0650b8910d6423b0064": "S^{2k} \\times S^{2k}", "69e8da41d1612ae666486fdb9f2b7fa3": "K(\\cdot \\mid \\cdot)", "69e9008ed1cd6a1d7b3aef08a9fe3a2e": "\\delta\\lambda_i = \\frac{\\mathbf{x}^\\top_{0i}([\\delta K] - \\lambda_{0i}[\\delta M] )\\mathbf{x}_{0i}}{\\mathbf{x}_{0i}^\\top[M_0] \\mathbf{x}_{0i}}", "69e90d3a0f21f68ef4127f66b88e73e6": "[a_k,a_k^\\dagger] = 1", "69e913c707e319da2b344b9ff16f17c9": "w'''", "69e9506efa942791b81ce0bbe5504699": "\\begin{align}\n\\int_{0}^{1}f(x) \\, dx &= \\lim_{n \\to \\infty} U_{f,P_n} =\\lim_{n \\to \\infty} L_{f,P_n} = \\frac{1}{2}\n\\end{align}", "69e95c167280a2b4e18161768e4ea54b": "B_p", "69e9c5c47f0e672c58c0d2c399eaaa74": "\\nu = p/q,\\ ", "69e9efb79c96187fb02108f297c29ef8": "\\gamma()", "69ea0c9e9b8db57e7dd2314f9e6a7694": "\\textstyle\\frac{6}{\\nu-4}", "69ea2c9df8ec19583f4d68b0274d874a": "j_e = -n_e e v_e", "69ea5c292df13683acd640129af3fa70": " s = \\frac{a+b+c}{2 } ", "69ea8e3a41f0945b6efd9779c90fc0a9": "p_k=\\frac{m+1}{m(k_0+1)+1}\\frac{\\mathbf{B}(k+k_0,2+1/m)}{\\mathbf{B}(k_0,2+1/m)},", "69ea9635052be3fbd8fc2b1a48560b7e": "O \\left(\\frac{m^{1/3}}{(\\log m)^{2/3}} \\right)", "69eac6c97204f5b3ea7d903b99e0fb79": "\n\\mathit{LCSubstr}(S, T) = \\max_{1 \\leq i \\leq m, 1 \\leq j \\leq n} \\mathit{LCSuff}(S_{1..i}, T_{1..j}) \\;\n", "69ead8eafb6ee3dd1dd7933b79336046": " \\mathbf{F}_G=-\\frac{GmM\\hat{\\mathbf{r}}}{r^2}, ", "69eb2b20a538c696d32803f187b7b5a6": " \\langle S \\rangle = {\\rm Tr} (ST) = \\sum_{n=0}^\\infty \\langle S e_n , e_n \\rangle \\lambda(n,T) = v_T(\\{ \\langle S e_n , e_n \\rangle \\}_{n=0}^\\infty ) ", "69eb2dcb7b7546ee8508cec8178df7f1": "{\\vec\\partial}^2\\sigma - (dm(\\sigma)/d\\sigma)\\langle\\overline\\Psi\\Psi\\rangle - dU(\\sigma)/d\\sigma=0", "69eb3025218859c5cdddb82869b9edee": "Recall", "69eb46f46d297de51d0f3abba3e759c1": "\\psi(t,\\cdot)", "69eb4bb5f723e448875d0fd7dbd805be": "a = \\frac{C}{C_u} - 1 ", "69ec03798f16914f33cbcbfdaa98d735": "H_\\alpha(X)", "69ec51d4e77a217b690022543bc6d95b": "\\operatorname{Var}(X)=-\\frac{2 \\operatorname{Li}_3(1-p)}{\\beta^2\\ln p}-\\left(\\frac{ \\operatorname{Li}_2(1-p)}{\\beta\\ln p}\\right)^2.", "69ec5d3a868349fe7c8b15f6b8174c23": " \\big. \\frac{\\Delta Q}{\\Delta t} = -k A \\frac{\\Delta T}{\\Delta x} ", "69ec75aeec0eee7a7d40b8088a423379": "\\ell^1", "69ec7c25b30aae4fcebfd248c500e423": " 3/4 = 0.75", "69ece0aebeb5596b8d0ef4a93cde850e": "R = \\mu - 3 \\times \\sigma", "69ece313cfc723cfea1c6fa6717069c4": "\\rho \\rightarrow \\rho - \\frac{\\Lambda c^2}{8 \\pi G}", "69ecf0249b55c12867e7dc732f372d14": "f_B = 0.00737 E^{-0.9} \\;", "69ed0524b8d77a974ca0dd6fccea7026": "\\textstyle x^bb(x)", "69ed0f9d90e2d8426f37a69956a5cfd2": "r_{Ace}", "69ed83dc4d4e7456ddceed29501909c4": " \\mathbf{H} = \\sum_{\\alpha}\\hbar\\omega_{\\alpha}a_{\\alpha}^{\\dagger}a_{\\alpha}", "69ed8f3834ac817cee85b4ede606ed04": "\n\\begin{align}\na_r & = a_u - a_s \\\\\n & = (\\Omega + \\omega_r)^2 R - \\Omega^2 R \\\\\n & = \\Omega^2 R + 2 \\Omega \\omega_r R + \\omega_r^2 R - \\Omega^2 R \\\\\n & = 2 \\Omega \\omega_r R + \\omega_r^2 R \\\\\n & = 2 \\Omega u + u^2 / R \\\\\n\\end{align}\n", "69ed956896f9e0ac08f9eff6ce5b807d": "\\displaystyle AB^2+CD^2 = AC^2+BD^2 = AD^2+BC^2.", "69ee45adb0beacfdfab60a430a68a04f": "|B_k (a_k - a_{k+1})| \\leq M(a_k - a_{k+1})", "69ee7b637d1e5ddcff03c17c2afa232f": "\n\\hat{H} = \\frac{\\hat{p}^2}{2m} + V(\\hat{x}).\n", "69eea0567e55e9fd6cab0d54897431a0": "R_{\\text{f}}", "69ef16976b4cd8a01cf598c70ad83e05": "f(\\rho \\sin \\phi \\cos \\theta, \\rho \\sin \\phi \\sin \\theta, \\rho \\cos \\phi) = \\rho^2,", "69ef17318cb6fbe5d5eecab1f4d6e7a7": "p_1/p_2=\\pi_1/\\pi_2", "69ef188454eb4431bb03d04d0ded3f66": " P(T) = \\text{max}\\left( k A(0,T) - S(T), 0 \\right).", "69ef469de30f29f808f4879582ebe2ce": "X_2,\\ldots,X_{K-1}", "69ef5fe9ff3fade64ccbfe45f592274d": "(X,Y) = (x,y) + R \\mathbf{N} = (x-R\\sin\\varphi,y+R\\cos\\varphi)", "69ef90d0ccdfb87c44594046d56b21fa": " ~\\epsilon_t~ ", "69efaf1ae25993d8111cd8f2c2b4d6b5": "t^{-2}m^0 \\ell^1", "69efb608737015b48ca757449489fab6": "H_{\\min}(A|B)_{\\rho} \\equiv -\\inf_{\\sigma_B}D_{\\max}(\\rho_{AB}||I_A \\otimes \\sigma_B)", "69efc1d041e04a1d040387b75bdad6f2": "\\phi \\rightarrow \\phi + \\delta\\phi", "69f0136dedb37f62797b3c1c71bad608": " P_d = \\frac{R_s}{2}\\int{|\\overrightarrow{H}|^2 dS} ", "69f05a9781a6f511f23000c1d37358b3": "2,-2", "69f07a3dfe3de5566e7f94f7c28bb620": " \\frac{\\partial F(x)}{\\partial y}", "69f07be04a20e376750441efac058c88": " \\operatorname{build-param-lists}[f\\ (p\\ p\\ f), D, \\_] \\and \\operatorname{build-list}[\\lambda q.\\lambda x.x\\ (q\\ q\\ x), D, D[p]] ", "69f0c529fe672c2aa691aa28eb290a7c": "w = 2^r > 1", "69f1069bba820910f1640df1073a292c": "\n\\begin{pmatrix} a \\\\ b \\end{pmatrix} =\n\\mathbf{M} \\begin{pmatrix} r_{N-1} \\\\ 0 \\end{pmatrix} =\n\\mathbf{M} \\begin{pmatrix} g \\\\ 0 \\end{pmatrix}\n", "69f164dd4f0ddad3708d076c2c56a1c7": "\\Phi(M,x)", "69f1a6f7d8176d94ed6be9bf73daf1c8": "\\langle\\hat{T}\\rangle = \\bigg\\langle\\psi \\bigg\\vert \\sum_{i=1}^N \\frac{-\\hbar^2}{2 m_\\text{e}} \\nabla^2_i \\bigg\\vert \\psi \\bigg\\rangle = -\\frac{\\hbar^2}{2 m_\\text{e}} \\sum_{i=1}^N \\bigg\\langle\\psi \\bigg\\vert \\nabla^2_i \\bigg\\vert \\psi \\bigg\\rangle", "69f1e9bff685189b0f4d514276d51256": "V_{\\text{out}} = A {V_{\\text{in}}}^{\\gamma}", "69f22f1c204e3dda746e03e95c74bcae": "W_t\\,", "69f2a59b88ba98eb4d1a2d737142a381": "\\mathrm{ HA_{(aq)} \\, \\leftrightarrow \\, H^+\\,_{(aq)} +\\, A^-\\,_{(aq)} }", "69f2b1e9a9970c402a2938215f3b05aa": "I(c_v)=\\log_2 |C_v|", "69f2b4f7422fc6b867bd722d373cabac": "\\Delta(y, {P(\\alpha_i)_i}) \\le e", "69f2bd1ae7785b4f328990f77158e262": "v(t, x) = \\mathbf{E}^{x} \\left[ \\exp \\left( - \\int_{0}^{t} q(X_{s}) \\, \\mathrm{d} s \\right) f(X_{t}) \\right].", "69f2fb04afafa186062f1d32d8e1f0e8": "H_{4,3} = \\frac {H_{1,2}}{1 \\cdot 2} + \\frac {H_{2,2}}{2 \\cdot 3} + \\frac {H_{3,2}}{3 \\cdot 4} + \\frac {H_{4,2}}{4} ", "69f336cbb465ef01dc03ccc08c0f1093": "\n \\sqrt{0.0062} = 0.079", "69f33e6539c6f3bc92a10819d4c36981": "\\tilde {u}", "69f35a261b72d9aa842d84a44a2e8dcc": " {A} {x} - {b} = {A} {Q}^{-1}({A}^{T} {\\lambda} - {c}) -{b} \\,", "69f39a3b83276f030b2828c909d509ba": "[A_i,B_j]=0", "69f3a4b52756f643a98ceb45aeb2dae2": " dG = Vdp-SdT+\\sum_{i=1}^k \\mu_i dN_i ", "69f3bfb67285462575796721757d1538": "\\scriptstyle f(x_1,x_2,x_3)=0", "69f3e81cd918330f549c3e7997c164a8": "\\Rightarrow_{r_5} a A A A \\Rightarrow_{r_5} a a A A \\Rightarrow_{r_5} a a a A \\Rightarrow_{r_5} a a a a", "69f3f556b0820c3863de9855ac7e61be": "f:\\{0,1\\}^n\\to\\{0,1\\}", "69f42a057a670c96d904e4179c41aa82": "\\delta_{i,j}=\\left\\{\\begin{matrix}1 & \\mathrm{if}\\ i=j \\\\ 0 & \\mathrm{if}\\ i\\neq j\\end{matrix}\\right.", "69f4733b047c6f7e11beea6454711eb7": "\\mathrm{RHC=CH_2\\ +\\ CO\\ +\\ H_2\\ \\xrightarrow {Rh_4(CO)_{10} / PPh_3}\\ \\ RCH_2CH_2CHO}", "69f4afbf197967cb0c33133dab647666": " f : X \\to Y", "69f4d154e91517fd08b153c17d96694d": "(x,y)=(z,w) \\equiv x=z \\wedge y=w", "69f5372922f36b74055f350da91f01c3": "\\cos \\beta = \\sqrt{\\frac{1}{2} \\left( 1 + \\frac{L_{xx}-L_{yy}}{\\sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2}} \\right)}", "69f5390f3afe7f9e368e15ec5435696f": "\n\\begin{align}\n& \\mathbf{x}_0 := \\text{Some initial guess} \\\\\n& \\mathbf{r}_0 := \\mathbf{b} - \\mathbf{A x}_0 \\\\\n& \\mathbf{p}_0 := \\mathbf{r}_0 \\\\\n& \\text{Iterate, with } k \\text{ starting at } 0:\\\\\n& \\qquad \\alpha_k := \\frac{\\mathbf{r}_k^\\mathrm{T} \\mathbf{A r}_k}{(\\mathbf{A p}_k)^\\mathrm{T} \\mathbf{A p}_k} \\\\\n& \\qquad \\mathbf{x}_{k+1} := \\mathbf{x}_k + \\alpha_k \\mathbf{p}_k \\\\\n& \\qquad \\mathbf{r}_{k+1} := \\mathbf{r}_k - \\alpha_k \\mathbf{A p}_k \\\\\n& \\qquad \\beta_k := \\frac{\\mathbf{r}_{k+1}^\\mathrm{T} \\mathbf{A r}_{k+1}}{\\mathbf{r}_k^\\mathrm{T} \\mathbf{A r}_k} \\\\\n& \\qquad \\mathbf{p}_{k+1} := \\mathbf{r}_{k+1} + \\beta_k \\mathbf{p}_k \\\\\n& \\qquad \\mathbf{A p}_{k + 1} := \\mathbf{A r}_{k+1} + \\beta_k \\mathbf{A p}_k \\\\\n& \\qquad k := k + 1 \n\\end{align}\n", "69f5f7c9f84efc9a3f721d3214cc7b17": "R_\\mathrm{fb}= 2(R_1+R_2+R_3) + \\frac{2R_1R_3}{R_2} + \\frac{C_2R_2+C_2R_3+C_3R_3}{C_1}", "69f6189b8451306a0a071a4398d75223": " \\tanh \\varphi = \\beta \\,\\!", "69f6235b49d7c8420a268aa1dd267c6c": " \\frac{G}{G_w} = \\frac{G_{c}}{G_{wr}}", "69f667d5f06156fd50e6e57b17a4da70": "\\frac b{\\sin \\vartheta}\\; \\left| \\frac{db}{d\\vartheta}\\right|\\ ,", "69f6c39b8c572136d80921aa21f056b0": "(\\overline{gate2}\\vee gate1)\\wedge (\\overline{gate2}\\vee x2)\\wedge (\\overline{x2}\\vee gate2\\vee \\overline{gate1})", "69f6d835352c7aed975377cf2cb273e4": "\\psi _{beta}(t|\\alpha ,\\beta )\\leftrightarrow \\Psi _{BETA}(\\omega |\\alpha ,\\beta )", "69f6ea43abb817262f8d1dd02f167f75": "\\hat{t}(t,\\omega),\n\\hat{\\omega}(t,\\omega)", "69f7023cc082a9e95607491f286d4d5b": "\n\\begin{align}\nx &= 0.999\\ldots \\\\\n10 x &= 9.999\\ldots \\\\\n10 x - x &= 9.999\\ldots - 0.999\\ldots \\\\\n9 x &= 9 \\\\\nx &= 1\n\\end{align}\n", "69f7335e5711d41df1f8a905d2c143e0": "\\scriptstyle\\nabla\\times\\textbf{u}'=0.\\,", "69f74741d18d2314aa906d0b2f005aeb": "m_0=m", "69f7784f16143fd6e3fb5d91d777787d": "V = V_1 \\, \\cup \\, V_2", "69f7c4f0e4faef4b0fae2001e219f6b3": "p_n(x) = b_0(x)-x b_1(x)-a_0=\\frac{1}{2}\\left[b_0(x) - b_2(x)\\right].", "69f7ddd501b1b05f3389a9873b686ad3": "P(n) = \\frac{(\\rho V)^n e^{-\\rho V}}{n!}", "69f84b49111150d06717fab79e099cf3": "\\partial_{\\text{in}}(S)", "69f871226779a821cf44a6ed329be049": "f^*(p)=\\sup_{x\\in I}(px-f(x))", "69f88e28d5b3836756eb837c0a663f7d": "{\\Bbb R}^{>0}", "69f88e89c6805f3b6916164e2852af78": "O_0^{(\\alpha)}(t)=\\frac 1 t,", "69f8ee44a26b9e46a8f59de70dfa2794": "\\begin{align}\n f(t) + (d-r(t)){f'(t)\\over r'(t)} \n & = ae^{it} + (d-ce^{i(a/c)t}){aie^{it}\\over aie^{i(a/c)t}} \\\\\n & = (a-c)e^{it} + de^{i(1-a/c)t}.\n\\end{align}", "69f8fccc13a03eb18f2562e88c6a71c8": "0=u'_1y_1+u'_2y_2+\\cdots+u'_ny_n", "69f9b8ac2f87e27712b3ec86b1eab2b6": "\\mu_{\\delta}^{s} (E) = \\inf \\left\\{ \\left. \\sum_{i = 1}^{\\infty} \\mathrm{diam} (C_{i})^{s} \\right| \\mathrm{diam} (C_{i}) \\leq \\delta, \\bigcup_{i = 1}^{\\infty} C_{i} \\supseteq E \\right\\}.", "69f9ba92d3c79dc5b9e7f562581eb977": "a \\equiv b \\pmod N", "69f9c89f0d5097e951224d21ceb0049b": "y = a_0 x^c \\sum_{r = 0}^\\infty \\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r^2} x^r = a_0 x^c \\sum_{r = 0}^\\infty M_r x^r.", "69f9d14fdf94a42f1dd560789cb1e791": "\\scriptstyle t_{\\text{rec}}", "69f9dde129db9d617022a2bbdabc4f03": "n_m\\,", "69f9e7630b2b6b3a263e20030173f3f4": "\\rho = \\sum_k p_k \\rho_1 ^k \\otimes \\cdots \\otimes \\rho_n ^k.", "69fa2406d586e8ac3ddd3c20679ce330": "\\mu \\ne \\overline{X}", "69fa3b6c78fc098601b3dab87c0dba9d": " 1.11 \\pm 0.01", "69fa5bd418eef29a539165e7fd9f382c": " 5.69 ", "69fa6c9b13caa7dc4a0ea407c2aefd0e": " \\theta^{*}_{(1-\\alpha)} ", "69fb10e53f8e27e9e6159f29dcaf2aff": "\\binom{n}{\\tfrac{n+m}2}-\\binom{n}{\\tfrac{n+m}2+1} = \\frac{m+1}{\\tfrac{n+m}2+1}\\binom{n}{\\tfrac{n+m}2}.", "69fb1eb3927145aa886ff40daa0bba12": "\\left \\langle \\nabla_{ \\partial_i }\\partial_j, \\partial_k \\right \\rangle = \\tfrac{1}{2} \\left ( \\partial_i g_{jk}- \\partial_k g_{ij} + \\partial_j g_{ik} \\right ).", "69fb35b37004c367e166fd5ad5830609": "\nE[K(K-1)...(K-n+1)]=n!\\boldsymbol{\\tau}(I-{T})^{-n}{T}^{n-1}\\mathbf{1},\n", "69fb62bb265bb25f3e55ac6a7f8f322e": "\\operatorname{skewness}(X) = \\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} ", "69fb7f5c2614e628ac2576caadc77fe3": "X_{isth}, \\; i = 1, \\dots, N, \\; s = 1, \\dots, S, \\; t = 1, \\dots, T, \\; h = 1, \\dots, H, \\, ", "69fb87b0ebcac708ad9fc88abcb7e2ae": "10\\uparrow\\uparrow\\uparrow\\uparrow 3=(10 \\uparrow \\uparrow\\uparrow)^3 1", "69fb9370ff9c5846253ec750c3effa66": "\\int_{-1}^1 x^2\\,dx = \\left[\\frac{x^3}{3}\\right]_{-1}^1= \\frac{1^3}{3} - \\frac{(-1)^3}{3}=\\frac{2}{3}", "69fbc3a839d0c3a77478b61eb4f0087a": "\n\\hat{A} = \\frac{1}{N} \\sum_{n=0}^{N-1}x[n]\n", "69fbf3e19ca1695459bee025d222df9e": "D^{0}f\\left( x \\right) = f\\left( x \\right)", "69fbfcd9197c59c0a56824ae65409e31": "\nf_n(x)=\\begin{cases}-\\frac1n&\\text{for }x\\in [n,2n],\\\\\n0&\\text{otherwise.}\n\\end{cases}", "69fc00a1ff269e909cdd4031cbfaaf70": "S \\Rightarrow_{r_1} SSS \\Rightarrow_{r_2} aSbScS \\Rightarrow_{r_2} aaSbbSccS \\Rightarrow_{r_2} aaaSbbbScccS \\Rightarrow_{r_3} aaabbbccc", "69fc13b627eda2b4de24420ad9313ced": "n = \\sum_{i=1}^n a_i = \\prod_{i=1}^n a_i", "69fc360cf6a0fdd5836659f2a0b00afb": " T = \\frac{T_{N}A_{N} + T_{E}A_{E}}{A_{N} + A_{E}}", "69fc5ae3b97a853b755071cd1b0f10a8": "\\scriptstyle\\eta", "69fc9bf8fccd1dbdf59611b22b8a50e3": "5-8", "69fcbc056064f510f76220b160e249a5": "\\forall x \\in \\mathbb{R}-\\mathbb{Q}", "69fd0b12905b22fef66e241b186129e1": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n P^{-1}(A\\mathbf{x}_n-\\mathbf{b}),\\ n \\ge 0.", "69fd0d3bcc2241e36fff3da7461fc3fe": "\\eta_{\\alpha\\beta} = \\begin{pmatrix}\n-1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\n\\end{pmatrix}", "69fd5ac68ea2904b3b12115ff8b7c281": "\nJ_{tot} = \\sum_{A} J_{A} = 0 \n", "69fd76b13ec5de7722df34342500dc03": "\\left \\lfloor x \\right \\rfloor", "69fe2228d8592da0c98db5a74ccedafa": "\n P_z = \\int_S p(x,y)~ \\mathrm{d}A ~;~~ Q_x = \\int_S q_x(x,y)~ \\mathrm{d}A ~;~~ Q_y = \\int_S q_y(x,y)~ \\mathrm{d}A\n ", "69fe52d96142317407bf7bcdb7f48036": " \\Delta S = R ln \\sigma ", "69fe56cd97347341e1f848e2b8c42e26": "{X-Y=\\sigma(l_2-l_1)}\\,", "69fe75971f1fab64189e0f6b21578feb": "\\sum_{i=1}^n F_i = \\operatorname{I}_H.", "69fed06831c2bd810607ab1d8d93f4e1": "[AFO]=[FCO]=[DBO]=[ADO]=[BEO]=[CEO] \\,", "69feda49ee6073aee13b318426e91bbb": "0 = \\partial_{\\gamma} F_{\\alpha\\beta} + \\partial_{\\beta} F_{\\gamma\\alpha} + \\partial_{\\alpha} F_{\\beta\\gamma} = \\nabla_{\\gamma} F_{\\alpha\\beta} + \\nabla_{\\beta} F_{\\gamma\\alpha} + \\nabla_{\\alpha} F_{\\beta\\gamma}.\\,", "69fedcd0debf4ab923ec2d24fbc7ea59": " H", "69fede4be6ff4bab8df3e83c856701aa": "\\left|2^X\\right|", "69ff1b6c877b3806f31904fc8354a325": "C^*_\\text{internal} := \\left \\{y\\in X: \\langle y , x \\rangle \\geq 0 \\quad \\forall x\\in C \\right \\}.", "69ff245352b8ecc7122d7e8c00a979bf": "\\frac{t}{t_0}=e^{-A^2\\left(\\xi_0-\\xi\\right)}.", "69ff7c84494dc1900410718c5080a480": "\n \\frac{\\partial}{\\partial t}n+\\nabla\\cdot (n\\mathbf{u})=0\n", "69ff7d881b818cde520160309b28a153": "\\scriptstyle \\eta_D", "6a0049c0c4e2c065d9492efe7bc63680": "0 < \\cdots < T < M.", "6a0061d6d96e1991796cafdf7b6bee92": "\\mathbb{E}: \\mathcal{H} \\to \\mathbb{R}", "6a008c447f06740f55c2091f7a2e800a": "f(\\gamma x ) = \\gamma f\\left( x\\right) ", "6a00ab0ec01cea6ed34c03094f7bb491": "[0, \\mathit{g(x)}]", "6a017adf8ee754141eb175e325d142db": "\\phi \\, (t) = P(t)e^{tB}", "6a019f446ecd9946c7f3be10ec16d29e": "\n \\Pr\\Bigl(\\sup_{x\\in\\mathbb R} |F_n(x) - F(x)| > \\varepsilon \\Bigr) \\le 2e^{-2n\\varepsilon^2}\\qquad \\text{for every }\\varepsilon>0.\n ", "6a01a3d4015fa7d2911703bcd7f61888": " \\lambda + \\sigma + \\gamma =0 \\,", "6a0278760da9bd7b5bde936de44d25be": "P_{F} : L^{2} (\\Omega, \\Sigma, \\mathbf{P}; \\mathbf{R}^{n}) \\to L^{2} (\\Omega, F, \\mathbf{P}; \\mathbf{R}^{n})", "6a028d146da364fecf7fcaad364f6dfd": " -\\frac{dx}{dt} = {(k_f + k_b)x} - {k_b [A]_0}\\,", "6a02a87402eec73f3b42c742130fa035": "{N_n}^{2/3}", "6a02b94ab4e7d01f8c32fdcce68a9239": "\\begin{align}\\operatorname{E}[\\,\\hat\\beta] &= \\operatorname{E}\\Big[(X'X)^{-1}X'(X\\beta+\\varepsilon)\\Big] \\\\\n&= \\beta + \\operatorname{E}\\Big[(X'X)^{-1}X'\\varepsilon\\Big] \\\\\n&= \\beta + \\operatorname{E}\\Big[\\operatorname{E}\\Big[(X'X)^{-1}X'\\varepsilon|X \\Big]\\Big] \\\\\n&= \\beta + \\operatorname{E}\\Big[(X'X)^{-1}X'\\operatorname{E}[\\varepsilon|X]\\Big]\n&= \\beta,\\\\\n\\end{align}", "6a02ccd1495fb69ae931cc476bb75ee6": "\\,\\!\\beta_n", "6a02f067fe4e2590ac3c5fd34b3d8b28": " \\textstyle \\hbar ", "6a030d2e05a9050de6b89b54d7188b83": " P(\\mathbf{x})=\\prod_{i=1}^n\\frac{\\lambda^x e^{-\\lambda}}{x!}=\\frac{1}{\\prod_{i=1}^n x_i!} \\times \\lambda^{\\sum_{i=1}^n x_i}e^{-n\\lambda} ", "6a0339d54403e41588f646cc16f8f3ec": "d(x)", "6a035a5942944e0ef358fff85a33c4df": "\\eta(x,t) = \\tfrac12\\, H\\, \\cos\\, \\theta,", "6a03908b751b516410e1ad744c8d349c": "\\tfrac{4}{17}", "6a041ee80c88e17a1d32de0daa11fc51": "\\Gamma(x)\\,", "6a043440488fe099fc08db8be4f73833": "\\scriptstyle x(t)\\,", "6a045c0439c6966024745ffeb15e122b": "\\beta\\left(\\pi\\left(x_0,m\\right),i\\right) = a_i", "6a04b34bcf2b3028343363a28d307d2e": "C_{2k+1} < \\operatorname{Spin}(n)", "6a04e2d61f3bb26e51bee817fb3d71a4": "\n\\begin{align}\nf'(x) &= 4 x^{(4-1)}+ \\frac{d\\left(x^2\\right)}{dx}\\cos (x^2) - \\frac{d\\left(\\ln {x}\\right)}{dx} e^x - \\ln{x} \\frac{d\\left(e^x\\right)}{dx} + 0 \\\\\n &= 4x^3 + 2x\\cos (x^2) - \\frac{1}{x} e^x - \\ln(x) e^x.\n\\end{align}\n", "6a0515ab1d37ab8e640e5d21eec23c5a": "\\rho_{*}", "6a05176c4fded15e4dfd83382cfb32a4": "\\,w= Q_s R_s ", "6a055ce1807ef6cc5464cced68c91589": "\\ N(\\theta) = \\frac{I_R(\\theta) - I_S(\\theta)} {I_R(90) - I_S(90)}", "6a058d102910f33a7d4cf9ea23067b8c": "T_2", "6a05c8126bb4a208fcd93531d420be41": "2r_o -1", "6a05ff96c7336f9a0349f61ac6a2dbab": "a>0\\,\\!", "6a0669b0f19c3238067e0dc3f0a5a897": "\\text{Semiperimeter}=s=(a+b+c)/2=mn(m+n) \\, ", "6a073b00e927535ef92242d1a199afa5": "x \\in X_{1}", "6a07917583113c798af354930ead66e8": "\\frac{\\Gamma^2(\\tfrac{1}{4})}{4\\sqrt{\\pi}}", "6a07c01556899098b89a08e98f8159ec": "MS_W = S_W/f_W = 68/15 \\approx 4.5", "6a08533650c500484f79f4b354499b41": "B = A[t_1, \\dots, t_n]/(P_1, \\dots, P_m)", "6a0895999bf2468ee69d7b4b0088177a": "\n \\begin{align}\n \\sigma & = n_1^2 \\sigma_{1} + n_2^2 \\sigma_{2} + n_3^2 \\sigma_{3} \\\\\n \\tau & = \\sqrt{(n_1\\sigma_{1})^2 + (n_2\\sigma_{2})^2 + (n_3\\sigma_{3})^2 - \\sigma^2} \\\\\n & = \\sqrt{n_1^2 n_2^2 (\\sigma_1-\\sigma_2)^2 + n_2^2 n_3^2 (\\sigma_2-\\sigma_3)^2 + \n n_3^2 n_1^2 (\\sigma_3 - \\sigma_1)^2}. \n \\end{align}\n ", "6a08cafcc31efff4693af10f671b7336": "\\ F_{forward} = lift ", "6a08e4659b98d55fbcbb072489034a6f": "g(x_n) = \\frac{f(x_n + f(x_n)) - f(x_n)}{f(x_n)}.", "6a0958831c5a9251890e8f9ba6c9e811": "J = J^\\mu \\gamma_\\mu = c \\rho \\gamma_0 + J^k \\gamma_k = \\gamma_0(c \\rho - J^k \\sigma_k).", "6a09905eb5cf64518722ffcf56d2fa22": "\\ A=\\pi R^2\\tan^2\\theta/2", "6a09d8026a8eb3b09e809adf6768dca8": "A_k=\\sin\\frac{ (2k-1)\\pi }{ 2n },\\qquad k = 1,2,3,\\dots, n ", "6a0a22d0eece92a46ca299ccea519e42": "\\frac{t:\\!\\!-~~ \\alpha \\times \\beta ~\\vdash~ \\gamma}{\\lambda t:\\!\\!-~~ \\alpha ~\\vdash~ \\beta \\rightarrow \\gamma}", "6a0a263338cb38de8c825cec4cc2094a": " U:f\\mapsto \\tilde{f},\\,\\, L^2(K\\backslash G/K) \\rightarrow L^2({\\Bbb R}, \\lambda^2\\,d\\lambda)", "6a0a30f761669621a4bab63b246570b3": " \\prod_{T>0}(1+B_T^x)<\\infty", "6a0a3ef58380ed214aa7233ee8d2a3f2": "\\!e", "6a0a59cf7b5ccb4acf57d95831a7fa54": "\\left\\{\\left(x, x\\right) \\mid x \\in A\\right\\}", "6a0a69c3158294f86d94f566b4f972cd": " \\begin{bmatrix} x_a - x_0 \\\\ y_a - y_0 \\\\ z_a - z_0 \\end{bmatrix} = \\begin{bmatrix} x_a - x_b & x_1 - x_0 & x_2 - x_0 \\\\ y_a - y_b & y_1 - y_0 & y_2 - y_0 \\\\ z_a - z_b & z_1 - z_0 & z_2 - z_0 \\end{bmatrix} \\begin{bmatrix} t \\\\ u \\\\ v \\end{bmatrix} ", "6a0a6ffe80d54b2edf4fd0d806af5cc5": "{\\sin\\theta}^2 > 0", "6a0aae5a095ed5b2429a5e3dd49a7332": "v = \\sqrt{\\frac{T}{A} \\cdot \\frac{1}{2 \\rho}}.", "6a0ac943102428526627e3d0b6622a1c": " \\gamma\\!_{polymer} =\\sum_{i=1}^n f_i \\gamma\\!_i ", "6a0ae96bca240860dbe02b511632fb31": "\nc^{2} d\\tau^{2} =\n\\left( 1 - \\frac{r_{s} r}{\\rho^{2}} \\right) c^{2} dt^{2}\n- \\frac{\\rho^{2}}{\\Lambda^{2}} dr^{2}\n- \\rho^{2} d\\theta^{2}\n", "6a0b2d3da520275a14b7ef9897acf2f4": " \nT = \\begin{bmatrix} T_{11} & T_{12} \\\\ T_{21} & T_{22} \\end{bmatrix} : \\begin{matrix}W \\\\ \\oplus \\\\ W' \\end{matrix} \\rightarrow \\begin{matrix}W \\\\ \\oplus \\\\ W' \\end{matrix},\n", "6a0b32a31bbb305be12f67eda22e4c99": "\\log_b 1,000,000 = 3", "6a0b3c7a69c0a5c6ef3826d9cb0648c0": " J = -M \\left( \\frac{d}{dx} \\right) \\frac{df}{dc}", "6a0b71f920d88650aff8449248c72655": " P=\\frac{r}{c} \\quad \\text{and} \\quad H=\\frac{m}{ac}", "6a0bb2b6d70f9092b89d30a11fa475a5": "\\Delta\\text{H}_{\\text{f}}", "6a0c272dc6d811f43b1c3f87de105682": "R(t')=|\\mathbf R|,", "6a0cbc20c1ae77a138b264faa2b6c440": "\\bar p(n) \\approx \\left(\\frac{364}{365}\\right)^{C(n,2)}.", "6a0d390c9b9951abc66f57cc4c47c744": "|x\\tan(x)|", "6a0dd2b48df0fca7c20370a91b0a0c67": " \\pi (x) \\approx \\sum_{n=1}^{\\infty} \\frac{\\log^{n}(x)}{n\\cdot n!\\zeta (n+1)} ", "6a0debf9a1756a280a10891dae9dbd16": "\\mathbb{G}_m(\\mathbf{F}_q) = \\mathbf{F}_q^\\times", "6a0e125a769e5a1d57f2b67fd1338337": "z_1,z_2 \\in U.", "6a0e2abe3c263e4838632fc73349baef": "\\wedge^1_n = \\vartriangle^0_n", "6a0e4ad965995127fea8942ed1858414": " \\boldsymbol{\\beta}^m ", "6a0ebaec42e5abfbb1a2a77841d0f48f": "(\\hbar m)", "6a0ebea125e27a4dddcaa28f6e9bcbf5": "U_{cm} = \\frac{U_1 + U_2}{2}", "6a0ee7227bfcff27f1cf32edeabda847": "F_z\\ \\hat{z}", "6a0f6b89833ef233329bc5b3b8c4943a": " x\\ f\\ y = f\\ (y\\ y) \\and q\\ x = \\lambda f.f\\ ((x\\ f)\\ (x\\ f)) ", "6a0f9fdf60ace9dfd2de1816eecf10b8": "X_2 = X_1 - [20 a_1 + a_2] a_2\\cdot10^2 \\geq 0.", "6a0fbb36e9643d11934f0d93b574c018": "\\psi(\\Omega) = \\zeta_0", "6a10b33c1f976ca934b3e8e9615dbfd6": "eV_{if}", "6a1131eda57a4ce94d3e7573a07f4677": "(\\mathbb{Z}/n\\mathbb{Z})^\\times.", "6a11363389e3bd14bb2591fb4bfb3d2c": "\\omega=2\\pi{f}", "6a118da2a01da27d5a80c3bcdd3a0178": "q = g_k\\,\\!", "6a1269b33f10e6904f137b9d33e5fdb5": "p_4=q_5\\ .", "6a12a7d2d0a636481647ba3d45e6e466": "A', B', C'", "6a12c81cf0820195068cf4cb3907c400": "\\sqrt{ax^2\\!+\\!bx\\!+\\!c} \\;=\\; (x\\!-\\!\\alpha)t", "6a132e41f03b70ac9cd2f7b57ae53a9a": "\\frac {1}{\\varphi(d)}.\\ ", "6a1348058daa8abf604bb65bf944d66d": "\\int_0^{2\\pi} \\frac{dx}{a+b\\cos x}=\\frac{2\\pi}{\\sqrt{a^2-b^2}}", "6a137b2c2c7e8f2e41a568bb234b1bb6": "E = \\frac{1}{2}C[V_{DC} + V_{AC}\\sin(\\omega_0 t)]^2 = \\frac{1}{2}C[2V_{DC}V_{AC}\\sin(\\omega_0 t) - \\frac{1}{2}V_{AC}^2 \\cos(2\\omega_0 t)]", "6a13dd84eb9dfa665dd7cfb3640c3049": "{\\tau _{fc}^\\star= \\frac {\\rho u \\,\\Delta x \\,\\Delta y \\sin 2\\theta}{4(\\Delta y \\sin ^3 \\theta+\\Delta x \\cos ^3 \\theta)}}", "6a14040019db1d03f0fec2130f74d67a": "\\{2, 2, 4, 5, 5, 5\\}", "6a14c444d3edb8a24074a231abb56caf": "\\neg p \\to (p \\to q)", "6a15386112b222419745a8dc454c8027": "i=p", "6a15461f8540cbf574e96aa1f9494d57": " \\bigl[ \\begin{smallmatrix}\n a&c\\\\ c&b\n\\end{smallmatrix} \\bigr] \n\\bigl[ \\begin{smallmatrix}\n u\\\\ v\n\\end{smallmatrix} \\bigr] =\n\\lambda \\bigl[ \\begin{smallmatrix}\n u\\\\ v\n\\end{smallmatrix} \\bigr].\n ", "6a1579b3f752a695a0b57de5e979d20d": "F(x_1,\\dots,x_d)=\\sum_{(a_1,\\dots,a_d)\\in {\\rm cone}} x_1^{a_1}\\cdots x_d^{a_d}.", "6a15d04f72b1600c5cca2a0e4ed53c1a": " \n\\eta(4i,j)=\\eta(4i+1,j)=1,\\quad \\eta(4i+2,j)=\\eta(4i+3,j)=0\n ", "6a15e80b5f929739718cedafda94c0f2": "\nf(z) = \\phi(z) + \\frac{B_1}{z-a} + \\frac{B_2}{(z-a)^2} + \\cdots + \\frac{B_n}{(z-a)^n},\\quad\nB_i, z,a \\in \\mathbb{C},\n", "6a16898a75986397142ef2ee9bcd1831": "\\overline{k} = k_s", "6a1689db204fe93847ee74fb5921a279": "\\aleph_{G(\\alpha)}", "6a16b061bec19d4cb27b6a578d452987": " pV \\equiv nRT \\equiv k_B T \\equiv TS \\,\\!", "6a17165cc8bd2870b628562e395773c2": "R(\\theta,\\delta)", "6a17464afb09b0184f5f4d5ef8a01fb5": "\\sqrt{x+y}\\leq \\sqrt{x}+\\sqrt{y}.", "6a1807a7f1ca60d49e6522787e239daa": " {C_1 \\over C_2} = {R_4 \\over R_3} - {R_2 \\over R_1}", "6a182744b7a90836b187cd8d25097133": "\\text{True}", "6a188995d17bf0b3204d5b4cd1f922c4": "\\neg B\\or\\neg D", "6a1915ae5ec7300b0af09085b2f4753d": "|\\phi^{(k)}(x)|\\ge 1", "6a193a8adb8f46a140f98a4036505e90": "\\displaystyle{R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b)}", "6a1969d76dc9e757af22bd2e7d7d1af8": "\\{a^n b^n c^n d^n | 1 \\le n \\}", "6a198260bb5ffffc903616f18b3ae04e": "R=\\frac {\\left|A'\\right|^2} {\\left|A\\right|^2}, \\quad T=\\frac {\\left|B\\right|^2} {\\left|A\\right|^2} \\,", "6a19acfc9b1d377b0023178ef0a9f618": "c_p = \\frac{{\\gamma R}}\n{{\\gamma - 1}}", "6a19c553e6647620274421c87d046198": "Q\\left(\\frac{p}{q}\\right)", "6a19d4bdab891f3aa952d73a92841d8b": "a d - b c \\neq 0", "6a1a14fd633f83df8161db1a2c7aaf8f": "\\displaystyle \\hat{f}(\\omega_x,\\omega_y)=", "6a1a61fbd76c7d462a15a408ad952bdd": "i \\hbar \\frac{\\partial}{\\partial t}|\\psi (t)\\rangle = \\hat H |\\psi (t)\\rangle", "6a1a8bbb803bd4736cf8b61da610c5fb": "A\\textbf{x}=\\textbf{0}", "6a1ab0af1952410b299e7f241b967c72": " \\Delta t < \\frac{h^2}{2a}", "6a1b00f7d58d6c097a3696a791d39b0b": " X = \\frac{ 1 }{ 2 } [ ( \\frac{ T }{ \\beta } )^{ 0.5 } - ( \\frac{ T }{ \\beta } )^{ -0.5 } ]", "6a1b0f69e177cda8a6c7faba2131fb0a": "\\Psi(x)(y) = d(x,y) \\quad\\mbox{for all}\\quad x,y\\in X", "6a1b3aa6435acd2d22937e6bde112e57": " count[j,h_j(a)] \\leftarrow count[j,h_j(a)]%d + 1 ", "6a1b74393d54ef0630b3a4f62ff88e6d": "\\frac{AW}{WB}=\\frac{DY}{YC}", "6a1baff6f6eca909a1d6c5491066a3ca": "\\mathbf{\\boldsymbol\\Delta\\beta=V \\boldsymbol\\Sigma^{-1}\\left( U^T\\ \\boldsymbol\\Delta y \\right)}_n. \\, ", "6a1c00fe55292d25ec590ef102019dca": "r_0 \\leftarrow b-A\\, x_0\\,", "6a1c2bbbeafc2436fd8a15514cfb95d2": "\\alpha = 1 - \\frac{1}{(1-p)(1+2^{\\mathsf{H}(p)/(1-p)})},", "6a1c5c3b9f484ca112494aeeea8506e9": "1\\le i, j, p,q\\le k", "6a1cb7560d6e6f9ac3b13adef69459a5": " \\ F(a_1, \\ldots, a_n) = \\min(a_1, \\ldots, a_n) ", "6a1ce5bc2db96806a6a5b90d45f98cf1": "\\left[K_a ,K_b\\right] = -i\\varepsilon_{abc}J_c", "6a1d1c4b3d78ce95df4a5c2c619d099b": "\\mathbf{P} \\subseteq \\mathbf{BPP} \\subseteq \\mathbf{BQP}\\subseteq \\mathbf{AWPP} \\subseteq \\mathbf{PP} \\subseteq \\mathbf{PSPACE}", "6a1d2667400bb21240efe5e2e790ae63": " \\exists j \\in T_2 \\setminus T_1 ", "6a1d2af74cac401d6a59fcc79a45a4ec": "\\operatorname{Br}(k) \\to \\operatorname{Br}(F)", "6a1d6c4fa101b339c3b7ba10c428a4b4": "\nh=\\exp \\left [ \\frac{T_h(37GHz)}{\\alpha T_v(37GHz)} \\right ] - \\gamma\n", "6a1db6d675056f662bc31c3bd477e8c3": "\\! F_0(x)", "6a1de0e814785555919abca8eca65ed2": "\\langle \\epsilon\\rangle", "6a1e21eb96bd977da5b0e9b20a4e21ad": "\n{n \\choose 6} {6 \\choose 2, 2, 2} \\frac{1}{6} +\n{n \\choose 5} {5 \\choose 3} \\times 2 +\n{n \\choose 4} \\times 6 =\n{n \\choose 2} {n \\choose 4}.", "6a1eb2da7328bb07fa3b3ac8257094a9": "\\tau = \\omega t,\\,", "6a1ed34029024b47ebbfdf242ce3f6c5": "n>3\\,", "6a1efb290f3ce6c77c849404eef9af26": " \\bold\\lambda=(\\lambda_1, \\lambda_2, \\dots, \\lambda_m),\\,", "6a1f7a9b39aa10e85110fb4d594869d8": "\\lambda=6573 \\AA", "6a1f971cce727a82bc81b365cc5624d5": "H(x^*(t),u^*(t),\\lambda^*(t),t) \\leq H(x^*(t),u,\\lambda^*(t),t), \\quad \\forall u \\in \\mathcal{U}, \\quad t \\in [t_0, t_f]", "6a1fb5170e19935d694b76298f79a3df": "P(X_i=c \\mid X_{i-2}=a \\cap X_{i-1}=b)", "6a1fc80df5e8189e306f706dd5fdc2e4": "\\frac{(Wh-H\\Omega)k}{AJ}", "6a1fd3fbac2ae23654062eeff4e0764a": "M=\\lambda_B \\,", "6a1fd8a69d38ca02e9bf2d18dca9d114": "\n Y_m = A_m \\cosh\\frac{m\\pi y}{a} + B_m\\frac{m\\pi y}{a} \\cosh\\frac{m\\pi y}{a} + \n C_m \\sinh\\frac{m\\pi y}{a} + D_m\\frac{m\\pi y}{a} \\sinh\\frac{m\\pi y}{a} \n", "6a1ff7c40d5b01010c8da1ba2b37a9c5": "K=\\sum_{i=1}^{s}\\frac{p_i^2}{2m}", "6a2017339a52f9b8f610bb7e1b2cbbb0": " \\operatorname{pred}(0) = 0 ", "6a2025f181273844ec5f867779eab737": "\\mathbf{w} = \\frac{\n\\left|\\mathbf{u}\\right| \\mathbf{v} +\n\\left|\\mathbf{v}\\right| \\mathbf{u}}\n{\\left|\\mathbf{u}\\right| +\n\\left|\\mathbf{v}\\right|}.", "6a2026c1023edc2f52c60247e6a66bc6": "\\rho_{f}", "6a207a17582230623ada4df34ef227da": "\\chi_{\\text{1}}(\\omega)=\\frac{Nq^2}{m\\varepsilon_0} \\frac{1}{\\omega_\\mathrm{0}^2-\\omega^2-\\tfrac{i\\omega}{\\tau}} ", "6a20b38d00eb92283d3115612128e672": "E_{n_1,n_2,n_3}\\left(r\\right)=\\left(r+\\frac{1}{2}\\right)\\frac{hc}{2L}\\sqrt{n_1^2 + n_2^2 + n_3^2}. \\qquad \\text{(1)}", "6a20c28f63c41476245a7ed8849c286e": "\\overline X", "6a211709d5a3fdda2283de1c37c66dee": "\\overline{x}={1 \\over n}\\sum_{i=1}^n x_i", "6a21e3b7abef11bc17fe00deb86cc7ec": "\\tan{\\theta} = \\mu_s\\,", "6a21e92009f1783cdfeee853c49c834b": "\\mathbf{e}^i \\cdot \\mathbf{e}_j = \\delta^i_j,", "6a2255d4c5f15e11b5302ad251a9834e": "[VIS_{Op} | P]", "6a2282c875f24da2929ae8d3f877efd7": "S - X", "6a22a7935fdf03dd76ef19816fe314d4": " 11 ", "6a23324b1b28d4a3e46716fcf0a2a3ed": " \\leq\\sum_{a^{n}\\in T_{\\delta}^{A^{n}}}\\Pr\\left\\{ E_{a^{n}}\\right\\}\n\\Pr_{\\mathcal{S}}\\left\\{ \\exists E_{b^{n}}:b^{n}\\in T_{\\delta}^{\\mathbf{p}\n^{n}},\\ b^{n}\\neq a^{n},\\ E_{a^{n}}^{\\dagger}E_{b^{n}}\\in N\\left( \\mathcal{S}\n\\right) \\right\\} ", "6a23605f292c49b550ce5d2e782d7acd": "\\eth\\left({}_sY_{\\ell m}\\right) = +\\sqrt{(\\ell-s)(\\ell+s+1)}\\ {}_{s+1}Y_{\\ell m};", "6a239fb852634855e9a6cbf0e41d653e": " f, ", "6a248a65651a1bedca53b82649e1ffd5": "\\ 0 = F_{forward} + F_{hullcourse}", "6a24d7dd5a3990e860f2d1621aa682f3": " M^n = \\begin{bmatrix}a^n & b^n-a^n \\\\ 0 &b^n \\end{bmatrix}, ", "6a250fbb83b4b63584a304c74ccadd8a": "\\sum_ \\mathrm{sym} x^2 y^0 \\ge \\sum_\\mathrm{sym} x^1 y^1.\\ ", "6a258a2e756bf9e28b4ec6309203f595": "\\alpha(x_n)_{n\\in\\mathbf{N}} := (\\alpha x_n)_{n\\in\\mathbf{N}}.", "6a25a9c89ab7dc5d6608a1e3ad5e4370": "C_d ", "6a25bd09892b8c46c7fe2e672c27135f": "1 = r_0 + r_1 x + \\cdots + r_n x^n, \\quad r_i \\in \\mathfrak{p}R", "6a25d307a4345b98e5a98602dd552b63": "\n \\sigma_{11} = -p + \\cfrac{\\lambda^2\\mu J_m}{J_m - I_1 + 3} ~;~~\n \\sigma_{22} = -p + \\cfrac{\\mu J_m}{\\lambda(J_m - I_1 + 3)} = \\sigma_{33} ~.\n ", "6a25ec158ae21a208c49e82ac37820c8": " \\begin{align} \\hat{T}_x & = \\frac{1}{2m}\\left(-i \\hbar \\frac{\\partial }{\\partial x } - q A_x \\right)^2 \\\\\n\\hat{T}_y & = \\frac{1}{2m}\\left(-i \\hbar \\frac{\\partial }{\\partial y} - q A_y \\right)^2 \\\\\n\\hat{T}_z & = \\frac{1}{2m}\\left(-i \\hbar \\frac{\\partial }{\\partial z} - q A_z \\right)^2 \n\\end{align}\\,\\!", "6a26226f94e82ecea3d90bab9c8172a3": "2^\\kappa\\not\\rightarrow(\\kappa^+)^2", "6a2639aebe1b88572e48f0f7a1c74ab9": "\\bar{a}_p\\bar{b}_q= a_i \\mathsf{L}_i{}_p b_j \\mathsf{L}_j{}_q = \\mathsf{L}_i{}_p\\mathsf{L}_j{}_q a_i b_j ", "6a267007228f9f654a0d28dec6932c31": "y=1", "6a2678dc1ed2c4283bc793e9ad96e536": "A_{j+1}", "6a2689c843966b3465bbd70e522a32a0": "{\\rm Re}(x)>0", "6a26ad40135d339f9962e41407df2390": "\\operatorname{Aff}(G)", "6a26d3ace9ad8d8cfd6b3dc29a1c2389": "V=[0 0 0 1]", "6a26d85dc9dd7b138b8e99d31f0d9478": "0\\rightarrow \\mathfrak{a}\\rightarrow\\mathfrak{e}\\rightarrow\\mathfrak{g}\\rightarrow 0", "6a26e0e618f3e6b927b16cf70d57e712": "\\{\\{x^3+x^2-2x-1,\\frac{2x+3}{x+2}\\},\\{x^3+2x^2-x-1,\\frac{x+3}{x+2}\\},\\{x^3+6x^2+5x+1,x+2\\}\\}", "6a2720b7811c779e7b5689e21ebbdcab": "s^2\\left(\\frac{1}{3}-\\frac{2}{\\pi^2}\\right)\\,", "6a2748cfb40db35d81758a834280efd5": "R_{t}", "6a275f4bb2face484608efaacd87cfd1": "\\langle 2, e\\rangle", "6a280cc6689a2d33848a35a4308601b8": "\\bigcup_{i=1}^n A_i=A_1\\cup A_2\\cup\\ldots\\cup A_n", "6a281a5ab4f182b0fe01c0fc261aba0e": "\\mathbb{E}^n", "6a2878b52c145b3dcd4bb54b2b6c439d": " K_D", "6a28c9c67fe0f4115c2c5f853d5ca49d": "\n\\det\n\\begin{bmatrix}\n \\cos\\theta & -\\sin\\theta & 0\\\\\n \\sin\\theta & \\cos\\theta & 0\\\\\n 0 & 0 & 1\n\\end{bmatrix}=1\n", "6a28cc10be504d71adef3acdcf25ee86": "\\ \\displaystyle \\tilde{u}\\ ", "6a291053927c17318a4c71ea51defddd": " \\binom{n}{k} = \\frac{n}{k} \\binom{n-1}{k-1}", "6a292e36eec4fe39fa739053d1265fc9": "T= \\mathbb{R}^2 / \\mathbb{Z}^2", "6a29410cce4cfe5868f372dd361e8c98": "P(a) = \\left\\{ \\begin{matrix}\nMN - 1 & \\mbox{if } a = MN - 1, \\\\\nNa \\mod MN - 1 & \\mbox{otherwise},\n\\end{matrix} \\right.\n", "6a29482d390f8017d28408f68c9f23f9": "\n\\mu = \\frac{\\sum_i^N x_i}{||\\sum_i^N x_i||} ,\n", "6a29e38a3a67d2d7d3fe14fe84f64fa0": "n\\log n.", "6a2a0ccd0dec83e9f0d5041389959f96": "T(a,b,x)=c \\,", "6a2a133eb5a234ec5b24b9007e5caabc": "y_i = V_m u_i^{(m)}", "6a2a19caff8dcc3b43cdc7d302ea1bb7": "Pro(C)=Ind(C^{op})^{op}", "6a2a54f74e96aa7fe99d5384c3341280": "A\\subset X", "6a2a9d8290a3362beede7b0213f82e88": " f(u-\\theta)=H(u-\\theta) ", "6a2aa2ecb335b4ccb201abe2a1f7b75c": "(\\ ) \\!\\,", "6a2b519c7cad3bd29d9c68012d2f0e02": "= \\operatorname{tr} \\left( \\gamma^5 \\gamma^5 \\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\right) \\,", "6a2b55fbb453199cbf76ac4e0db6ecbe": "g=f-e", "6a2b5bc9caaa741c3f29b3dd6933b2dc": "\\log x(t) = \\log x_0 + t \\cdot \\log (1+r).", "6a2bc73397d9fcc91382b845493e7432": "= 0.032416667", "6a2c2829d476ecf3eee5c0fe388a1d7d": "{1}\\, ", "6a2c2bc9aa43538efdb73a6299cecefa": "\\mbox{curl}\\,(\\mbox{grad}\\,f ) = \\nabla \\times (\\nabla f) = 0", "6a2c479469770c01476e9c33c3342a90": "N_G^j(k) = \\frac{S_G^j(k)}{T_G^j}", "6a2c4b75c8bc2f3f2ed45e44df362349": "\\mathbb{C}-\\mathbb{R}_{\\le 0}", "6a2d024426c0c93cbf6641b8380e2dd6": "xy-1", "6a2d22648a30f3a88d0c61dc1bec2048": "y_i \\in F(p)", "6a2d2ff68b18b19244cf9ce4fa9127dd": "\\boldsymbol{\\mathsf{F}} =\\frac{\\mathrm{d}\\boldsymbol{\\mathsf{P}} }{\\mathrm{d} \\tau} = \\gamma m \\left( c\\frac{\\mathrm{d}\\gamma}{\\mathrm{d}t}, \\frac{\\mathrm{d}\\gamma}{\\mathrm{d}t} \\mathbf{u} + \\gamma \\mathbf{a} \\right)", "6a2d588b947d863da3cefe5d769efbad": "-S", "6a2d92ea5492c8788680e26ab7248613": "\\lambda x\\!:\\!\\texttt{int}.~x", "6a2dcf29d9bc62b1fd2cd5ce594236ee": "\\gamma_V", "6a2df23df4b8cf730e11d34637fbdfbe": "Y_{-n}(x) = (-1)^n Y_n(x).\\,", "6a2e8754a516f1540f62bb7a798e0840": "\\frac{dq(t)}{dt} =-", "6a2eb5f1c282e4f62a7d63628d88a5cf": " \\vec{v}_\\text{new} = q \\vec{v} q^{-1}", "6a2eba137bea3df2c19f01ff528da548": "|\\langle x, y\\rangle| \\ge \\sqrt{q(x)q(y)}\\,,", "6a2f1affd5b1c95899a9c0234e38d629": "\\Omega^i_{\\ j}", "6a2f7205d1505b58b89a16ce986d324b": "\\omega^2\\, =\\, g\\, k\\, \\tanh\\, ( k h ),", "6a2f889782253228bc82a6a62225ecd6": "\\frac{R''(r)}{R(r)}+\\frac{R'(r)}{rR(r)} + \\frac{\\Theta''(\\theta)}{r^2\\Theta(\\theta)}=-\\lambda^2", "6a2fa2eb3e77fafbfec423a11dc3583c": "x < x_c", "6a2fe841f10fbc8deec05182749818b2": "A=\\sup_{y\\ne 0} \\int_{|x|\\ge 2|y|}|K(x-y) - K(x)|\\, dx <\\infty,", "6a2fffc1a628c42b5dcb39c9c3ad845e": "A=\\lim_{n\\rightarrow\\infty} \\frac{(2\\pi)^{n/2} n^{n^2/2-1/12} e^{-3n^2/4+1/12}}{G(n+1)}", "6a3014c3ff90031618a4e8a43c3bd0f4": "n=\\sqrt{2 + 2 k}", "6a302a0fa10d42cf543c6b3c225102c3": "\\nu' = \\gamma \\nu", "6a30446ae06e244a487ee12fc4ed5b84": " \\Omega,", "6a30509bf873cf5671b60d6b97fdd2b9": "\nv = \\sum_{i=1}^k \\alpha_i u_i\n", "6a30752559fba361fc4a5f658b3c10cd": " \\int\\limits_{-\\infty}^0 dx \\cdot 0 + \\int\\limits_0^L dx \\, A^2 + \\int\\limits_L^\\infty dx \\cdot 0 = 1 , ", "6a30825090684f994b878b1ea2955bb6": "Pr(X|H),", "6a3098614f6752d8a9f8e72be7a05ebd": "\\{p \\in \\mathbb{P}| \\langle\\sigma, p\\rangle \\in \\eta\\}", "6a30cf677d15820f1387aa8743eb9fbe": " \\gamma_0= \\sigma_1 ~,~ \\gamma_1= -i \\sigma_2 ", "6a3142c9e9a14dcc696dc1bf04813f33": "\\left(1-\\left[1-\\frac{1}{m}\\right]^{kn}\\right)^k \\approx \\left( 1-e^{-kn/m} \\right)^k.", "6a314ae875b7ebb2d69eb15da1ed6103": "TQ = E^+ \\oplus E^0 \\oplus E^-", "6a318b9323f4ab4a44319b293dc4256b": " {1\\over D_0 D_1 ... D_n} = \\int_0^\\infty ...\\int_0^\\infty e^{-u_0 D_0 ... -u_n D_n} du_0 ... du_n \\,.", "6a3221107121ff93564637b7f6fcca49": "\\Delta W = 0, \\quad \\Delta Q = \\Delta U\\,\\!", "6a3283a49c74ed168c464a840c017859": "\\Sigma r (Q_0 + \\Delta Q)^{n} = 0", "6a3298d4ccd9f6b91c28ea697645b0b5": "\\rho_{a/b}(R) = \\{ \\ t[a/b] : t \\in R \\ \\}", "6a32a5efaf68e72d2628d0d8ff00386a": "u_2 = (r * w)", "6a3328fbc42c6188d2f13e4d6ad7771e": " P( | X | \\ge k) \\le \\left( 1 - \\left[ \\frac{ k^r }{ ( r + 1 ) \\operatorname{ E }( | X | )^r } \\right]^{ 1 / r } \\right) \\quad \\text{if} \\quad k^r \\le \\frac{ r^r } { ( r + 1 )^{ r + 1 } } \\operatorname{ E }( | X |^r ). ", "6a333509dc267f8008b434d52459aaf5": " {} = -2 \\cdot (-6) + 5 \\cdot (-12) - 8 \\cdot (-6) = 0. ", "6a333deb177a1820a47647cbafc9707a": "(yx)^m", "6a3370cf1995077dd01e0704b452908a": "p_i \\in [0,1]", "6a337242d92bf245d72bb8c17564233e": "\\pi\\over 2", "6a33aa9b0b715467ec01529727fbb5dc": "\\mathrm{E}[D]=0", "6a33b28fa6ba4bdec12547f7b3db9ff1": "\\frac{k}{2m}=\\frac{k}{m} - \\frac{k}{m} + \\frac{k}{m} - \\cdots.", "6a33b55f8fba4b29208658c4ef0a62cd": "\\sqrt[n]{x} = \\exp \\left(\\frac{1}{n}\\int^x_1\\frac{dt}{t}\\right).", "6a33dd97c4b957db9ba8012c0e466abb": "f : FY \\to X ", "6a33e3f3691b887940a326129167351c": "\\rho = \\operatorname{corr}(a' X, b' Y)", "6a344eefe20f0f34768e07d347b67439": " \\displaystyle \\frac 1 2 mv_\\mathrm{rms}^2 = \\frac 3 2 k T.", "6a3491a5464337c1346eaf12bfc0bca8": "S(T) = C T^A", "6a34b9ef141950d045313f95f35ef754": " W_x(t,f)=\\int_{-\\infty}^{\\infty} C_x(t+\\tau/2,t-\\tau/2) \\, e^{-2\\pi i\\tau f} \\, d\\tau \\, .", "6a34e1b82f5e3aa4e954943fdfcec7dc": " E_n = \\frac{4\\pi^2 n^2}{W^2(ne^{-1})} ", "6a34e553946f13983c922a27084cb675": " x\\rightarrow x+b_1x^2+b_2x^3+\\cdots", "6a3533ff18b1397b2a4c9c079d484a89": "a' + b' + c'", "6a3565fef462c4d7b4993f28cd4fc38a": "\\delta V=2\\pi y^{3}\\delta p / Ec", "6a3595bc59c7de8c891f08f1d3a5b408": "\nf(E) = E - \\varepsilon \\sin(E) - M(t)\n", "6a35a8b1eb3471c3ce53197a058b5999": "\\mathbf{M}_{6} := (\\mathbf{A}_{2,1} - \\mathbf{A}_{1,1}) (\\mathbf{B}_{1,1} + \\mathbf{B}_{1,2})", "6a35d65a8c289ba2c3bd5d1a72ebfa1a": "C(n) = \\sum_{j=1}^\\infty \\beta (1 - \\beta)^{j-1} (1-p)^{j-1}", "6a363368abd0231cdad6a3300c5e78bc": "\\left\\vert \\langle \\psi_{x-} \\vert \\psi \\rangle \\right\\vert ^2", "6a365791292525c9bc79cecfca71fabc": "g_q(p) = X \\to p \\to q", "6a3662017404106ce7da148508c2e440": " \\frac{f(1-F)}{F} = r_2 - r_1\\left(\\frac{f^2}{F}\\right) \\,", "6a368012fd824606194b1a6056cb91e4": " \\frac{1}{2m} \\left( \\nabla S \\right)^{2} + U + \\frac{\\partial S}{\\partial t} = \\frac{i\\hbar}{2m} \\nabla^{2} S. ", "6a36a499cbd6dd81de0596ef6a8a8271": " \\leq\\,", "6a36a8ab082195e46f4393c20c7ee353": "Gain-Bandwidth = \\frac{\\mu_{\\rm e}V}{\\pi L^2}", "6a36ad607054f01c18d881c3eb8afc34": "\nL \\approx L_0 \\exp(\\bar \\lambda_1 t)\n", "6a370e5888698de93d68466e6a3d897c": " U(\\mathbf{B}_{\\theta,\\phi})\n=\\begin{bmatrix}\n\\cos \\theta & -e^{i\\phi}\\sin \\theta \\\\\ne^{-i\\phi} \\sin \\theta & \\cos \\theta \\end{bmatrix}\\,,", "6a372a79f7cef97532bf485522e225ac": "\\omega_0=\\omega(k_0)", "6a373dd9a3c65fa4102f8097ea662613": "\\frac{(d-2)(d-3)^2(d-4)}{12}-\\frac{g(d^2-7d+13-g)}{2}.", "6a37660ca7039933528c340affec85cd": "Q'_{lid} = Q'_{cond} + Q'_{conv} + Q'_{rad}", "6a3787b9ab2820bc4bfe2613a7b39206": "\\Phi(\\tau+48)\\,", "6a37f4e7664711769956033e6b9ba961": "{\\overline P}X = \\{O_{1}, O_{2}\\} \\cup \\{O_{4}\\} \\cup \\{O_{3}, O_{7}, O_{10}\\}", "6a38249d2967cd4d8f842c15c04ab122": "e^{(1)}_{i+1} = a_{i+1}", "6a386402110bcd3992502c239e379f8a": "\\operatorname{tr}(a\\!\\!\\!/b\\!\\!\\!/) = 4 (a \\cdot b)", "6a38b63c6f08ba07e62e4ff7f1b88e62": " \\begin{align}\nS_{n} (\\alpha, \\beta, \\gamma) & =\n\\int_0^1 \\cdots \\int_0^1 \\prod_{i=1}^n t_i^{\\alpha-1}(1-t_i)^{\\beta-1}\n\\prod_{1 \\le i < j \\le n} |t_i - t_j |^{2 \\gamma}\\,dt_1 \\cdots dt_n \\\\\n\n & = \\prod_{j = 0}^{n-1} \n\\frac {\\Gamma(\\alpha + j \\gamma) \\Gamma(\\beta + j \\gamma) \\Gamma (1 + (j+1)\\gamma)} \n{\\Gamma(\\alpha + \\beta + (n+j-1)\\gamma) \\Gamma(1+\\gamma)}\n\\end{align}\n", "6a38d25beaefbcb26f080be13fef789f": "\\{X_1(p), \\dots,X_n(p)\\}", "6a38e3d9c7cfc4c910ae1d13a0a81ae0": "n_\\mathrm{D} =N_\\mathrm{D}/N ", "6a394f48721972442846ed5b00aae279": "\\textstyle\\frac{1}{10}", "6a39cee6e78082ea396fc0a0f7cfe20d": "R^{p-1} f_* \\mathcal{F} \\otimes_{\\mathcal{O}_S} k(s) \\to H^{p-1}(X_s, \\mathcal{F}_s)", "6a3a2ec34aea7546f17e715d0d22a00e": "X^*_{b(X, X^*)}", "6a3a802e9ce9bbaf52e9bc81d9e97e13": "X_{hG}\\sim X\\times BG.", "6a3a813bd16c56b76065e4efc392c567": "= nRT", "6a3aa8480cfd164d4f51cde11acc1f18": "(k_{f_1},k_{b_1},k_{f_2},k_{b_2}, ... ,k_{f_{n+1}},k_{b_{n+1}})", "6a3ad1cf76fcdb52b858c82f92871f04": "\\chi_V", "6a3aee007e657d94998d5fa5c88f1aef": "0 \\rightarrow \\ker h \\rightarrow H^n(C; G) \\stackrel{h}{\\rightarrow} \\text{Hom}(H_n(C),G) \\rightarrow 0.", "6a3b3f5f9af209b2261f7e49dd8fcc22": "\\arg(L( j\\omega)) = \\frac{\\alpha \\pi}{2}", "6a3b3f9f95a408d8a5ad2617aa12aae7": "\\ \\Delta S = \\frac{\\Delta s}{c_p} = \\ln\\left[M^\\frac{\\gamma - 1}{\\gamma}\\left(\\left[\\frac{2}{\\gamma + 1}\\right]\\left[1 + \\frac{\\gamma - 1}{2}M^2\\right]\\right)^\\frac{-(\\gamma + 1)}{2\\gamma}\\right] ", "6a3b68c544aff9a6fb69de06e1d55c1f": " m\\in \\mathcal{H}_L, l\\in\\mathcal{H}_H", "6a3bea688251344171b26fc3dc495ebc": "\n\\begin{align}\nS_p(x,y)\n& = \\lim_{(\\xi,\\eta)\\to(x,y)}\n\\left({\\frac{\\xi^p-\\eta^p}{p (\\xi-\\eta)}}\\right)^{1/(p-1)} \\\\[10pt]\n& = \\begin{cases}\nx & \\text{if }x=y \\\\\n\\left({\\frac{x^p-y^p}{p (x-y)}}\\right)^{1/(p-1)} & \\text{else}\n\\end{cases}\n\\end{align}\n", "6a3c5f7f8b3eee4ec1338cca862f2919": "\n \\begin{align}\n & \\rho_1\\,u_s = \\rho_2(u_s - u_2) &\\qquad \\text{Conservation of mass}\\\\\n & p_2 - p_1 = \\rho_2\\,u_2\\,(u_s - u_2) = \\rho_1\\,u_s\\,u_2 &\\qquad \\text{Conservation of momentum}\\\\\n & p_2\\,u_2 = \\rho_1\\,u_s\\,\\left(\\tfrac{1}{2}\\,u_2^2 + E_2 - E_1\\right) &\\qquad \\text{Conservation of energy}\n \\end{align}\n ", "6a3c6d1fc886dc4a88ee228e8da535f0": "Beta_{\\text{x,t}}=\\overline{R_{\\text{t}}}\\cdot\\left[\\frac{P}{E}\\right]_{\\text{x,t}}\\cdot[1-T] ", "6a3c95c3d538287eb05eca3aaa2118a5": " \\Pr(T = T_i|\\theta) = f(T_i|\\theta) .", "6a3cba3f16392d162446a99c07b47079": "\\mathtt{int}", "6a3cbb91ad176344aced59fa2ccf0c50": " k=0,1,... ", "6a3cd893c1b9116c599bccaaa551c34d": "\\hat h(P)\\ge c(E/K)/[K(P):K]^{1+\\epsilon}", "6a3eb00a4df5c9aa6e8c549ff3ff61b6": "N\\ll M", "6a3eb348808b2c5160700cbd4f92457b": "\\xi_{-1}(\\hat{z}) = \\begin{pmatrix}\n0\\\\\n1\n\\end{pmatrix} \\,", "6a3f100c1adcfd18a7e2ea97d2dd8b04": "[A]_{\\text{seq}}= \\{x\\in X : \\{a_n\\}\\to x, a_n\\in A \\}", "6a3f28ac7236f98793efdfe0dadee47c": "\\mathbf{p}_{\\mathrm{2}} = (m + \\mathrm{d}m)(\\mathbf{v} + \\mathrm{d}\\mathbf{v}) = m\\mathbf{v} + m\\mathrm{d}\\mathbf{v} + \\mathbf{v}\\mathrm{d}m + \\mathrm{d}m\\mathrm{d}\\mathbf{v}", "6a3f73bcba4b7072f0ab0c268dee07ee": " s^2 + \\omega_0^2 = 0 ", "6a3f8fa8e34b1a15c3096f69f0fd842c": "f \\star g", "6a3fa4c43eb52b1203eb4cfc29131f84": "B(q)", "6a3fdfcacadbc88192765deef61ee8fd": "V(t,S(t))", "6a402e3e7b6831e84aec89f181912a07": "\\left \\lfloor A^{3^{n}}\\right \\rfloor \\text{ and } \\left \\lfloor 2^{\\dots^{2^{2^\\mu}}} \\right \\rfloor", "6a4039b5055bdab767356ef5177ee795": "\\left( \\begin{array}{ccc}\n1&a&c\\\\\n0&1&b\\\\\n0&0&1\n\\end{array}\\right)= e^{by} e^{cz} e^{ax}~.\n", "6a406abb37c3ef7c2f3bb59c9e4209b3": " y = y_a + \\left( y_b-y_a \\right) \\frac{x-x_a}{x_b-x_a} \\text{ at the point } \\left( x,y \\right) ", "6a4080b760086ac855df6ecea5a181e0": "\\forall t \\in p ((a \\in t \\and \\forall x \\in t (x = a)) \\or (a \\in t \\and b \\in t \\and \\forall x \\in t (x = a \\or x = b))).", "6a40b9ce577fc83cf5f6867dfa414081": "(k-m)", "6a4146d438f40cbf215596857e9b3db6": "\\textstyle r_\\text{cm} ", "6a41854458725519834fa6823ba1c620": "H\\left(\\frac{b+c}{2},\\frac{e+f}{2}\\right)", "6a418c3c4c4b7293f0ab9714db7f5fea": " J_{n+1} = 2^n - J_n. \\,", "6a418d988179c74473a9b7c79d020dd9": "\\frac{\\partial f(x_i, \\boldsymbol \\beta)}{\\partial \\beta_j} \\approx \\frac{\\delta f(x_i, \\boldsymbol \\beta)}{\\delta \\beta_j}", "6a41b9d03cbe6f1d6dc566ca9df27015": "\\lambda^{\\beta}\\delta \\mathbf{u}(r)", "6a41bc22dc776b6f75aa2723a54aae17": " \\frac{\\partial u_i}{\\partial p_j} \\ge 0 ", "6a42192d887e7a78cdbac9847db08d34": "v'_1=v_1\\frac{\\sqrt{m_1^2+m_2^2+2m_1m_2\\cos \\theta}}{m_1+m_2},\\qquad\nv'_2=v_1\\frac{2m_1}{m_1+m_2}\\sin \\frac{\\theta}{2}.", "6a4279b6eeb65be688ba9e4378a7b079": "\\wp'(z)^2 = 4\\wp(z)^3 -g_2\\wp(z) - g_3", "6a42b9c161a2c13818f23889a519d0ea": "f_{U_{(i)},U_{(j)}}(u,v)\\,du\\,dv= n!{u^{i-1}\\over (i-1)!}{(v-u)^{j-i-1}\\over(j-i-1)!}{(1-v)^{n-j}\\over (n-j)!}\\,du\\,dv", "6a4332ba0e1d0572acf8d45052cd675e": "a = \\frac{r_m + \\sqrt {d^2 + y^2}}{2} \\quad (15')", "6a433c02dd2cf7ecae0175813c6395ba": "0 \\mathbin{:} \\mathbb{N} ", "6a436708f145829e175bbae81db862a9": "\\tfrac{\\Delta y}{\\Delta x}", "6a43d11e234212bcec67f4ce6e282b51": "m - M = 5 log (d/10 pc)", "6a43f0f77033a3b91ba2a8169d7fd359": " \\ln n + \\ln\\ln n - 1 < \\frac{p_n}{n} < \\ln n + \\ln \\ln n \\quad\\text{for } n \\ge 6. ", "6a447e3ef077d70365ce80dc2a9b74f9": "x' \\rightarrow 0", "6a44b69cf2b8142ed03862f8d0ec881b": "\\boldsymbol\\theta=(\\theta_1,\\ldots,\\theta_k)", "6a450e2b46fc78d1b2f4d5ff0ee8adaa": "\\Sigma^{0,\\emptyset^{(n)}}_1", "6a45163a5aa07da49def4c4af28ed7a1": "\\cos^2 \\varphi - \\sin^2 \\varphi\\ = \\cos 2\\varphi", "6a45799b55dd92e94538da5988673941": "= V^{1} + \\phi(x,u,u_{1},u_{2})\\frac{\\partial}{\\partial u_{2}} \\,", "6a457ffac0fa3cb7bffab71412cdc529": "\\textrm{pH}_{new} = \\textrm{pK}_{a}+ \\log \\left ( \\frac{[\\textrm{A}^-]_{original}}{[\\textrm{HA}]_{original}} \\right )+ \\log \\left ( 4 \\right )", "6a45d6521f2fed5722a83a336ded0269": "\n\\left( \\begin{array}{c}\\rho_L\\\\P_L\\\\v_L\\end{array}\\right)\n=\n\\left( \\begin{array}{c}1.0\\\\1.0\\\\0.0\\end{array} \\right)\n", "6a46457b9a44a97da4bfc5a3c414e330": "T_1 \\vdash \\varphi", "6a46771ece215ee6dfbd8162606f43c3": "(d,s)", "6a4695f483ff422bbc6f5182f9a81216": " \\hat{E} = i \\hbar \\partial / \\partial t \\,\\!", "6a46b57ab16433504708430d58d0c31b": "1,2,\\dots,N", "6a46b8aced648b2b3190d5e384b34e6a": "5252.113122 \\approx \\frac {33,000} {2 \\pi}. \\,", "6a46c35223684c4e3cdbd9f363ae78a4": "x_k[n] \\ \\stackrel{\\mathrm{def}}{=}\n\\begin{cases}\nx[n+kL] & 1 \\le n \\le L+M-1\\\\\n0 & \\textrm{otherwise}.\n\\end{cases}\n", "6a46cc7018e51e94837594bce090143a": "x_{1},...,x_{m}\\,\\!", "6a46d3973671c1f7e0687749646e5fd6": " g:X\\rightarrow S^n ", "6a46d6e142ee5d73984185074859a67b": "H_{r\\phi\\phi}=\\frac{-GM\\sin^2 \\theta}{3(1-2GM/r)}", "6a4708c776ca48ec192884d8ea454879": "\\mathbf{F} \\cdot d \\mathbf{x} = \\mathbf{F} \\cdot \\mathbf{v} d t = \\frac{d \\mathbf{p}}{d t} \\cdot \\mathbf{v} d t = \\mathbf{v} \\cdot d \\mathbf{p} = \\mathbf{v} \\cdot d (m \\mathbf{v})\\,,", "6a474d4efb0a90757cbcb48b66de9ced": "(q^1,\\ldots,q^n,\\dot q^1,\\ldots,\\dot q^n)", "6a47a9a80cde9a483db353da3b0c9c6e": " Z_6 =\\{0, 3\\} \\oplus \\{0, 2, 4\\}", "6a48ce69edfddbd13a4b99d1794099ef": " R\\equiv ", "6a491cc4b426639040c3f70a187476e1": "d=\\deg(D_1)", "6a49274d139be54a29829bcbf7efed26": "j'=j", "6a49dd589d2091f7d1fe20753ad80df5": "\n\\operatorname{Li}_s(e^\\mu) = \\Gamma(1 \\!-\\! s) \\sum_{k=-\\infty}^\\infty (2k \\pi i - \\mu)^{s-1} \\,.\n", "6a4a142151dd55fe9d815ae9c131f138": "\\check{H}^q(\\mathcal{U}, \\mathcal{F}) := H^q((C^{\\textbf{.}}(\\mathcal U, \\mathcal F), \\delta)) = Z^q(\\mathcal{U}, \\mathcal{F}) / B^q(\\mathcal{U}, \\mathcal{F})", "6a4a458985ea95c60cd60c38fd6d0618": "\\begin{matrix} \\Delta J = 0, \\pm 1 \\\\ (J = 0 \\not \\leftrightarrow 0)\\end{matrix}", "6a4a57314988f3f8ceddfafbbbf6820d": "(19)\\quad ds^2=-\\frac{L^2-(M^2-Q^2)}{(L+M)^2}dt^2+\\frac{(L+M)^2}{l_+ l_-}(d\\rho^2+dz^2)+\\frac{(L+M)^2}{L^2-(M^2-Q^2)}\\rho^2 d\\phi^2\\,,", "6a4a7b8505170b58b12fc0ff0685f4e5": "DU = 1 + \\frac{1}{2} \\sum n_i(v_i-2)", "6a4aaf81d05e13480ffe899730a7ed73": "A = f^{-1}(B).", "6a4ae11f809536684e23272f5f8fb501": "\\hat{\\alpha} > 3 \\pi/2", "6a4b1c62e2788357bc8796d9f0649325": "Q_C = CV_C \\ ", "6a4b20acd1f687cfe65e8354d7fbca51": "\n\\left(\\mathbf{A}\\times\\mathbf{B}\\right)\\times\\left(\\mathbf{C}\\times\\mathbf{D}\\right)\n=\\left(\\mathbf{A}\\cdot\\left(\\mathbf{B}\\times\\mathbf{D}\\right)\\right)\\mathbf{C}-\\left(\\mathbf{A}\\cdot\\left(\\mathbf{B}\\times\\mathbf{C}\\right)\\right)\\mathbf{D}", "6a4b3e85a4a753e141a42c571d2c68c9": "rate = \\frac{SNPs}{(2 T_{CHLCA}16553)}", "6a4b4bf1b98345fe52fce9c46113b4fa": "\\,\n{\\mathbf{u}}_{||} ={\\mathbf{v} \\cdot \\mathbf{u} \\over |\\mathbf{v} |^2 } \\mathbf{v} \\ , \\quad {\\mathbf{u}}_{\\perp} = \\mathbf{u} - {\\mathbf{u}}_{||}\n", "6a4bb30c49798b28771d51d956dc5c55": "V=\\frac{\\sqrt{2}}{6}a^3.", "6a4bb4ee8e90ef233c0d5c07a7ff6904": "\nP(x)=x \\int_{x}^{1}\\frac{1}{t}\\,dt = -x \\log(x).\n", "6a4bc3b861d83664a0d6b18aca82216a": " \\widehat{\\Omega}\\psi(\\mathbf{r},t) = \\psi(\\mathbf{r}',t') ", "6a4bc9249a64c3c4a07c41c5220a4af2": " R^\\times ", "6a4c1236131dc49f7465797aba89c0bd": "\\bar w", "6a4c2fc612cec93902f666b906aabdcc": "x = y\\,", "6a4c9200bd0811bd7a39f92cebe2ee2b": " \\vartheta(C_n) =\n\\begin{cases}\n \\frac{n \\cos(\\pi/n)}{1 + \\cos(\\pi/n)} & \\text{for odd } n, \\\\\n \\frac{n}{2} & \\text{for even } n\n\\end{cases}\n", "6a4cee2c73972ef029b5a6e55a5d39c5": " \\Delta \\mathbf{L} = \\int_{t_1}^{t_2} \\boldsymbol{\\tau}\\mathrm{d} t \\,\\!", "6a4d07487c3ee37317b5c35d1813024d": " P_n(x) = (A_n x + B_n) P_{n-1}(x) + C_n P_{n-2}(x)~.", "6a4d09c502438c9643637a202b2f768c": "\n\\sum_{i 0} : x \\in \\lambda K \\} ", "6a4efa55940754ce6369f90dbc75593e": "x(t) = p - \\frac{1}{5} p^2 - \\frac{3}{175}p^3\n - \\frac{23}{7875}p^4 - \\frac{1894}{3931875}p^5 - \\frac{3293}{21896875}p^6 - \\frac{2418092}{62077640625}p^7 - \\ \\cdots \\ \n \\bigg| { p = \\left( \\tfrac{3}{2} t \\right)^{2/3} } ", "6a4efae42f256389cd2a27d86a735b7b": " B_{0}^{-1} c_0 = c,\\quad B_{0}^{-1}B_i = A_{i}\\text{ for }i = 1, \\dots, p\\text{ and }B_{0}^{-1}\\epsilon_t = e_t", "6a4f49d9c7ccd88b3874f114f0ea47b7": "c_{i+j}", "6a4f57f8490b2c18e52f4026a4701439": "a_0 = \\frac{1}{2}.", "6a4f6423dcfabaa037f2dc67df93b014": "\n\\mathcal{Q} = \\{ Q_{Y^n|X^n}(Y^n|X^n,X_0): E[d(X^n,Y^n)] \\leq D \\}\n", "6a4f7ae405f8e9b19b48f183669865e2": "\\det(I-\\lambda K) = \\left[\n\\sum_{n=0}^\\infty (-\\lambda)^n \\operatorname{Tr } K^n \\right]=\n \\exp{(\\sum_{n=1}^\\infty(-1)^{n+1}\\frac{\\operatorname{Tr} K^n}{n}\\lambda^n})", "6a4fa47e4c8bfc3816051ada31ea3b06": "\\phi=-\\begin{matrix}\\frac{\\pi}{2}\\end{matrix}", "6a4faca61feab5e4fa40d37d5453147c": "f'(x) = f(x)\\times \\Bigg\\{\\frac{g'(x)}{g(x)}+\\frac{h'(x)}{h(x)}\\Bigg\\}=\ng(x)h(x)\\times \\Bigg\\{\\frac{g'(x)}{g(x)}+\\frac{h'(x)}{h(x)}\\Bigg\\}", "6a50261954395fceadb338ed37e98932": "v_{i+1/2\\,}", "6a5062be3f652dae531334a021f3de64": "(U \\otimes V) \\otimes W \\cong U \\otimes (V \\otimes W)", "6a50b8b759395e637faf9d532b827125": "P(n) = \\sum_{d\\mid n} d \\varphi(n/d)", "6a50ef61ca681fc82005155fd8a3c5f8": "E(\\alpha_i) \\ne 0", "6a511bdaf6fecb96e54c0f283adcfcb8": "c>0\\ ", "6a51459275b9d83d12817fb53aa91f7d": " i(t) = B_3 e^{-\\alpha t} \\sin (\\omega_d t + \\varphi) \\,", "6a5171cccebf3ce11ddf9e141db79856": "{BSA}= \\sqrt{W \\times Ht} / {6} ", "6a51bd4eb3cbbd306814c8057317e7f3": " S_x ", "6a51e35aed2d336001f9b247cd8b3ded": "\\pi_1=\\dot{m}/\\eta r", "6a51e75b899db6e5911a3865a3eec8d7": "\\Rightarrow \\left|1,45\\right\\rang \\left|2,H\\right\\rang ", "6a51f650b1024d7383a0e7417ed729f0": "\\hat{D}=D", "6a5206550e2336bebe1bc0c3a45ed86d": "\\displaystyle m_i", "6a5213d96ec3f466ace8ff5fabee4ff2": "S_a + S_b = C_S,\\, ", "6a521cb0c77827d78869bac64369ea55": " F(h) = 2 \\pi R_{\\rm eff} W(h),", "6a523b1c7e952f39e1d94b1a4aafdc3b": " z_{ij} = \\mu_j + \\epsilon_{ij} ", "6a526bb209a14c5fe8cda1f8250dd6d4": " \\rho_n=\\frac{\\Gamma(2\\nu+n+1)}{\\Gamma(n+1)}(\\sigma_n-\\sigma_{n+1}) ", "6a526ef34e2febea22aadecb82af8583": "1/x, 1-x, x/(x-1)", "6a5275c96519e481f2b0f26ef59a24ec": "\\gamma > 0", "6a52cf99ca1e62c2970428a8b73782c0": " X_j ", "6a52e4b1c77d4af65103be17b63cb5e1": "\\lim_{r \\rightarrow \\infty} h_{ab} = O(1/r)", "6a531385831217dcbeb8c16bffe6d0ef": "\\nu^\\prime", "6a5340c1958ab10c5e6bd8ab120354da": "\\left\\{{3\\atop5/2}\\right\\}", "6a5384912c69019ea053b3e761564eb5": "\\prod_{m,n} \\left( 1 - \\frac{u}{mw_1 + nw_2} \\right)", "6a539677b854171889aff9ff431e8baf": "\\tfrac{dE}{dT} = \\beta SI - (\\varepsilon + \\mu)E ", "6a53abf563c8e8aaef4011c08fa0683b": "N_{BER}", "6a53bdc32eec1120ec1a5daef0a2a42b": "\\dot{\\vec{\\beta}}", "6a53d5319540609f7a8720a5b9283ee1": " P''(\\rho) - \\nu^2 P(\\rho) = 0 ", "6a541730b6f25de4c7083adc4a484433": "R \\to S=R[x_1,\\ldots,x_n]_g/(f_1,\\ldots, f_n)", "6a5433a7044dd09195555d60a10881f6": "{n\\choose k}\\frac{\\mathrm{B}(k+\\alpha,n-k+\\beta)} {\\mathrm{B}(\\alpha,\\beta)}\\!", "6a54573334635a9c018172c0a071d3f6": "(u-iv)t^2-2wt+(u+iv)=0.\\,", "6a55027ff1e537c017429e27e4341001": "\\gamma:{\\mathbb R}\\rightarrow M", "6a55341c8a44f386941e6f9a341fd5d4": "\\mu= \\mu_0", "6a5534a5cbd6fb4bbffce9bce2978f8f": "T^{1/2}", "6a556713216edbe9f3110e600d716673": "W/\\mathbb{F}_p", "6a55d74278a67c76354ff8a13b3fcb19": "C:\\{0,1\\}^k\\to{0,1}^n", "6a55dad4999b4c29b2963542681d5ef1": "\nx^{(L)}_k \\sim \\pi(x_k|x^{(L)}_{0:k-1},y_{0:k})\n", "6a55edbc4b8d80c4a2ae42bd633b7285": "Z_m", "6a5604d933c94ef747dff56a88a30bf0": " ( a_1 , a_2 , a_3) = ( a_1 ,0,0) + (0, a_2 ,0) + (0,0, a_3) \\,", "6a563411789aa6f5d7bd2885c9e6716a": " \\psi = {-A y \\over r^2} = {-A y \\over x^2 + y^2}, ", "6a565aba8a71688e45342a7ef56412bb": "\n D_{\\alpha,\\beta}(p \\parallel q) = \\frac{2}{(1-\\alpha)(1-\\beta)} \\int \n \\Big(1 - \\Big(\\tfrac{q(x)}{p(x)}\\Big)^{\\!\\!\\frac{1-\\alpha}{2}} \\Big) \n \\Big(1 - \\Big(\\tfrac{q(x)}{p(x)}\\Big)^{\\!\\!\\frac{1-\\beta}{2}} \\Big) \n p(x) dx\n ", "6a5675efba676a38cc55b847fa6566fb": " \\left[\\begin{array}{c}B\\\\\\hline C\\end{array}\\right].", "6a567fb06cbfd49bec62ad48ec23f43a": " \\nabla_{\\vec{p}_0} \\, \\vec{p}_0 = \\frac{-\\omega^2 \\,r}{1 - \\omega^2 \\, r^2} \\, \\vec{p}_2", "6a5690dd21cea1163e1a20e6a076e6aa": "\\tbinom{m+1}{k}", "6a572bc02125a8eeeae6be2e96000720": " \\mathbf{} v_t = \\frac{(\\rho-\\rho_0)Vg}{b}", "6a5796d1ea63793e4e8879a330fe9ae7": "2l+1", "6a57b25e28c9179f6690841befd1d9d5": "n = \\dim \\mathcal{H}_A", "6a57cd3266f405a3fe1e161a80d8c25e": "\\scriptstyle c', (v \\ll c') ", "6a580bd06cc4a9c969b96c6731ac2cf8": "U_{\\bold{G}}", "6a581092d183b83ff168037a83558d8a": "\\exp(a r) = \\cosh(a) + r \\ \\sinh(a) ", "6a582764e3bf7c5bbe1a533179d0bef3": "\\mathcal{F}(\\mathbf{x})=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(x_n) ", "6a583d026dca60a9c4457540791fd9ce": "\\mathbf{g}=-\\nabla\\phi.", "6a5861a7f650d2b1b5d85ae2b61242a1": "),\\Delta J = \\pm 1 ", "6a586fc1cf16cb1198865b867943d998": "T_{ref} = 10\\log_{10}(1-\\left|\\Gamma_{21}\\right|^2) dB", "6a5888b6cd44c729f0ca0ff471a4c80e": "\\sigma^{\\mu\\nu} = -\\frac{i}{4}[\\gamma^{\\mu}\\gamma^{\\nu} - \\gamma^{\\nu}\\gamma^{\\mu}].", "6a5899ab37525f709545ddad0291e70a": "E(m) = E(a\\cdot m) + E((1-a)\\cdot m) + c\\cdot\\sqrt{m} \\text{ if } m>b", "6a58afb8c24f447ec1c7ce723b57cf12": " \\boldsymbol{\\Lambda} = \\left(\\mathbf{G} + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\widetilde{\\mathbf{G}} \\text{diag}(\\boldsymbol{\\alpha}), \\mathbf{D} = \\text{diag}\\left(\\left(\\mathbf{G} + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\widetilde{\\mathbf{G}} \\boldsymbol{\\alpha} \\right) ", "6a58cfb018d4a59c6708b8a6479af566": "J_\\text{row} = 1\\ v\\ w\\ v^2\\ vw\\ w^2\\ v^3\\ v^2w\\ vw^2\\ w^3", "6a593d010d86e923351b8fab294dfe25": "T_2 - T_1", "6a59a3eb095083bb93aaff59370d8f42": "\\text{Recall}=\\frac{tp}{tp+fn} \\, ", "6a59c26dbd3df3096436672d2fdd7b62": "\\widehat{\\mathfrak{g}}=\\mathfrak{g}\\otimes\\mathbb{C}[t,t^{-1}]\\oplus\\mathbb{C}c,", "6a59dfab289797c38bebe62a0333f9ee": "\\scriptstyle{E_2(t)}", "6a5a639f4b1d2395bfae924b018a511a": "f(x)= \\left\\vert x \\right\\vert", "6a5a968c0698bb272afcbf9ad0bf7877": "E[Y] = E[E[Y|Z]] = E[Z].", "6a5ac5a373cf73906257cef8296b1786": "\\,t_{12} \\cdot t_{34}+t_{41} \\cdot t_{23}=t_{13}\\cdot t_{24}.", "6a5b36b5a747479caa4ff937361de453": "X(z) = \\sum_{n=-\\infty}^{\\infty} x[n] \\,z^{-n},", "6a5b448e62b4fd77d73793a46be1641c": "\\Delta H_m = k T N_1\\phi_2\\chi_{12} \\,", "6a5b46b1bb22feee16ce16b40cad5de5": "\\lfloor n\\phi^2\\rfloor", "6a5b5775d16017a995717380726dd898": "\\cos \\widehat C = \\frac{a^2+b^2-c^2}{2ab}", "6a5b66240930efba49a6230d9e8512a5": "R_{xx}(j) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=0}^{N-1}x_n\\,\\overline{x}_{n-j}.", "6a5b7cb7b971bf724a4276c121c37494": "\\mu'_s=\\mu_s (1-g) \\,", "6a5bc287b7de096207fc6351c8e3ca9e": "\\widehat{H} = \\hbar\\omega (\\widehat{a}^{\\dagger}\\widehat{a} + 1/2)", "6a5bcc02d80c61c76f97ab9c27969fdb": "S_k(r) = r \\; \\mathrm{sinc} \\, (r \\sqrt{k})", "6a5be6279cd68dd5493170ba4a2484a3": " \\,Q_0 \\subseteq Q", "6a5beab304e8a7994213adce2f58b4ce": "Z = \\frac {\\sum_{j=1}^J (O_{1j} - E_{1j})} {\\sqrt {\\sum_{j=1}^J V_j}}.", "6a5bf130a89b1edfbf95366bb815c8bc": "\\eta = \\sinh^{-1}\\left(\\frac{w}{c}\\right) = \\tanh^{-1}\\left(\\frac{v}{c}\\right) = \\pm \\cosh^{-1}\\left(\\gamma\\right) ", "6a5c4f6b9eb5f21b1d7ce109379a809f": " \\frac{V^2}{R} = \\frac{1}{\\rho}\\left|\\frac{\\partial p}{\\partial n}\\right| - \\left| f \\right| V", "6a5c7c7680891397848a0ded395b1e3c": "N\\bar\\psi(x)\\gamma^\\mu\\psi(x)\\bar\\psi(x')\\gamma^\\nu\\psi(x')\\underline{A_\\mu(x)A_\\nu(x')}\\;,", "6a5cc036320c6050f1821ecfa40360c7": "ds^2 = -f(r)^2 \\, dt^2 + g(r)^2 \\, dr^2 + r^2 \\left( d\\theta^2 + \\sin^2\\theta \\, d\\phi^2 \\right), ", "6a5d383ce94c9855371f6078404079bb": "X\\mapsto (C_\\bullet(X),\\partial_\\bullet)", "6a5db7062fb2ad8375b51532b4a9df6e": " \\mathbf{r} - \\mathbf{r}_0 ", "6a5dd4cba56ba6fd29cc6d9dd56e3ab3": "\\Pr(6\\text{ heads}) = f(6) = \\Pr(X = 6) = {6\\choose 6}0.3^6 (1-0.3)^{6-6} \\approx 0.0007", "6a5df91f465fb100bff7ad6d5c18025e": " y_{n+s} = - \\sum_{k=0}^{s-1} a_{n+k} y_{n+k} + h \\sum_{k=0}^s b_k f(t_{n+k}, y_{n+k}). ", "6a5e121c28719bdd59c0846b54ef97dc": " \\frac{1}{|Q|}\\int_{Q}|u(y)-u_Q|\\,\\mathrm{d}y", "6a5e1fd0b5a7f23256806201f98bcffe": "\\mathcal{I} \\models", "6a5e20047f1fb259be834b31181d4a59": "\n \\beta_t = \\int \\sup_{0\\leq\\phi\\leq1} \\Big| \\mathcal{E}_t\\phi(x) - \\int \\phi dQ \\Big| dQ.\n ", "6a5e366a046236f0f4824432f94f0d52": "brilliance=\\frac{photons}{second \\cdot mrad^2 \\cdot mm^2 \\cdot 0.1% BW}", "6a5e6b95266fe560b40279ded9be7108": "a =\n \\begin{cases}\n a + \\epsilon, & \\text{if }a = 0 \\\\\n a, & \\text{if }a > 0\n \\end{cases}\n ", "6a5ecd2895e0ff73d367b42cbe1348a4": "P_a - P_G = R_a \\times Q_a", "6a5f6ab41b3ba04945b781546aa1befa": "\\begin{align}\n\\lim_{x \\to a} \\frac{(x - a) P_1(x)}{P_2(x)} &=\\lim_{x \\to 0} \\frac{(x - 0)(\\gamma - (1 + \\alpha + \\beta)x)}{x(1 - x)}=\\lim_{x \\to 0} \\frac{x(\\gamma - (1 + \\alpha + \\beta)x)}{x(1 - x)}= \\gamma \\\\ \n \\lim_{x \\to a} \\frac{(x - a)^2 P_0(x)}{P_2(x)} &= \\lim_{x \\to 0} \\frac{(x - 0)^2(-\\alpha \\beta)}{x(1 - x)} = \\lim_{x \\to 0} \\frac{x^2 (-\\alpha \\beta)}{x(1 - x)} = 0\n\\end{align}", "6a5f8e9fb97f2d795c98ffb2d8e5c6fc": "d(p, q) = \\sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+...+(p_i - q_i)^2+...+(p_n - q_n)^2}.", "6a5faeb54089ec85913637d77717d77b": "\\mathbf M (T) := \\sup\\{ T(\\omega)\\colon \\sup_x |\\vert\\omega(x)|\\vert\\le 1\\}. ", "6a6035314aafef910b16719d14a1db5a": "(1-|f(x)|)_+", "6a603ed7603de99fd468bbafc431f1d4": "\n \\sigma_h = \\mathcal{H}(\\rho, \\mu) = p_{\\rm HEL}(\\rho) + \\cfrac{2}{3}~\\sigma_{\\rm HEL}(\\rho, \\mu)\n ", "6a6054948814aa4d03b91d300c0850f1": "\\sigma_1 = M c/I = 3 P L / 2 b D^2", "6a6094f85ec47451e7e96a894febb7a8": "\\frac {\\sin \\theta_1}{\\cos \\theta_1 }=\n\\frac {F_1}{mg}\\Rightarrow F_1= mg \\tan \\theta_1 ", "6a60b3ade2b712c631c529e04ae42614": "V=Spec(B)\\to U=Spec(A)", "6a61699ace6953cb9cb6eb4e3a2ab4bd": "S^n\\ ", "6a6176cb87785bbfdec2570d86d9ead4": "\\textstyle{\\frac {\\log(4)} {\\log(3)}}", "6a61f5b7d6dd24534b8e1ae605d81aef": " B \\, := J A J^T ", "6a62138b6cc7285f1751aea4ddf7c3e3": "\\sum_{k=0}^n\\frac{a_k}{{n \\choose k}} \\le 1.", "6a623bacdeb2ebe825b467edd2238d9f": "r_m = (10%)(100% - 38%) = 6.2% \\, ", "6a6262ca8177e6821fe0d45c2d419e5f": "w = x = 0", "6a62a29a817f1f39905cfa358eb06fdc": "f_n^*(x)", "6a62e1e19323cb84eb9c7fc34dded7fe": "\\sum_{0\\le k< 100} f(k)", "6a62ebd3eece0a855c027aad515d05fd": "\\Lambda \\alpha . \\lambda x^\\alpha . \\lambda f^{\\alpha\\to\\alpha} . f (f (f x))", "6a6307d16154f15997a051d92e6d97b7": "(y,z) \\in \\mathbb{R}^m \\times \\mathbb{R}^{m \\times d}", "6a631b71762d6783dc0dcdc9427ac07d": " V_p = | \\vec a_1 \\cdot ( \\vec a_2 \\times \\vec a_3 ) |.", "6a631ce25c0d9ee4aee6db085a623f48": " = f_0 \\sqrt{\\frac{\\mathrm{SNR} + 1 + \\frac{B^2}{12f_0^2}}{\\mathrm{SNR} + 1}}", "6a636dbdc3a6daec3373385a6a3ffceb": " {\\partial \\overline{p} \\over \\partial y}=0 ", "6a6370655dd400270e69d3814c1fb3a1": "\\mathbf E_J \\cdot \\mathbf e_i = \\alpha_{Ji}=\\alpha_{iJ}\\,\\!", "6a637fef21ee557c6defb3615a010ca1": "G_\\Psi=G_H", "6a63a887b6186a769c9658a4795d7a98": "\\sqrt{\\Lambda}\\rho_{x}\\sqrt{\\Lambda}", "6a643cfe658d27dc15c80bb027216cc8": "C_0 \\# C^r_1 \\# C_2 \\# C^r_3 \\# ... \\# C_n", "6a64501a4f90697379302474dea89d7f": "P(x_1, \\ldots, x_l) = \\sum\\limits_{i_1+\\ldots+i_l\\le d}c_{i_1,\\ldots,i_l}x_1^{i_1}x_2^{i_2}\\cdots x_l^{i_l}", "6a64f5186280850d47e4c6138da037b0": "\\phi_{k=1 \\dots K,w=1 \\dots V}", "6a660c3cc95bb50277e344bc5da8b35e": "\\alpha^\\kappa = \\limsup_{\\lambda < \\kappa} \\, \\alpha^\\lambda", "6a66320e00289a36beab03c92b14452c": "a \\leq b, c \\leq d \\Longrightarrow ac \\leq bd", "6a66b420eed547f449a1827c34f2d587": "x_2 = -y_1^{-1}x_1\\left(y_1\\left(n^2-a\\right)-x_1^2y_1^{-1}\\right)^{-1}", "6a66bfe58155182bcb9dbf667d1bbf95": "\\int_\\Theta R(\\theta,\\delta)\\,d\\Pi(\\theta).", "6a675051dad38032a511d0c889aa7eab": "Z_{\\text{in}} \\approx \\infin", "6a67519171c359afb67ee326c79da591": "t_{n+1,n+1} - \\beta", "6a67b0409ff44844f36dca8737960c2a": "\\tau = \\frac{t_{1/2}}{\\ln 2} \\approx 1.44 \\cdot t_{1/2}.", "6a67b16eb03cb1ca22fcf7578128d049": "(x+c)^2 + y^2 = \\left ( 2a - \\sqrt{(x-c)^2+y^2} \\right )^2", "6a67dad3017fe3ecd426acbe05319cb0": "\n\\begin{align}\ny&=x^2 \\\\\n\\frac{dy}{dx}&=2x.\n\\end{align}\n", "6a6853bd0beaec0fb75cab830f86aa1b": " R = \\left \\{ x \\in X \\ : \\ \\forall n \\in \\mathbf{N} : \\sup\\nolimits_{T \\in F_n} \\|Tx\\|_Y = \\infty \\right \\}", "6a68776026ef3e3c9c506aed16def6fe": "pM + qL ... \\leftrightharpoons M_p L_q...", "6a68c96e79ad44ca04595c70c8abbc4e": "Y_p", "6a68d2bf3f590668b99d2749c6eb24aa": "\n\\Pr[X \\leq E(X)-\\lambda]\\leq e^{-\\frac{\\lambda^2}{2(Var(X)+\\sum_{i=1}^n a_i^2+M\\lambda/3)}}\n ", "6a68d37348c01394ed79eb8c268d8bb2": "a=\\frac{1}{4f}.", "6a68d8b8a708caa39fcc01b5874a2b2d": "x_1,\\ldots,x_n", "6a68f9d19f273eaaa4f54696d69951a1": "\\scriptstyle Pn\\bar{3}m", "6a6927f281ae3ea7e80241b4083a91ef": " H_k(\\omega_0,\\gamma) = \\frac{Y(k\\omega_0,\\gamma)}{U^k(\\omega_0,\\gamma)} ", "6a69a687c8b5a6d011895be8fadda1a4": "\\alpha<0", "6a69bf09fd4f551142c91eaa0831c34d": "\\xi^a = K u^a\\,", "6a6a6639cbe2cc6e6d1d3d95b64c2078": "\\scriptstyle \\alpha_j\\,", "6a6a70261c7541c409b8c6b44a499cab": "\\kappa(\\alpha) = \\alpha^*", "6a6a8bf12f77442afe83ffa0351618ab": "\\theta = ", "6a6acf55cd60e0ec24389903f189453c": " T \\rightarrow \\infty ", "6a6aef79bfa38898627166a2061e7b15": "R = R(t) = \\sqrt{R_0^2 + 4 \\nu t}", "6a6be0d5c1e32ccdd5b043aaa218433e": "\n{} -1600qr^3s+144pq^2r^3-900p^3rs^2+2000pr^2s^2-3750pqs^3+825p^2q^2s^2\n", "6a6c084a996236d9b5e5892f3f7e33a3": " w = \\overline{w} + w' ", "6a6c2bc8d8fa7335af632245b8301fc2": "\\theta \\leftarrow \\theta + \\eta \\nabla_{\\theta} J(\\theta) ", "6a6c4ae98240f00db4db878480476854": "\\forall S \\subseteq X\\ (S \\neq \\varnothing \\to \\exists m \\in S\\;\\; \\forall s \\in S\\;\\, ( s, m) \\notin R)", "6a6c4ca079c42f376faf78cd4726f228": "c_\\eta(a,b)\\equiv-\\frac{n_\\eta(a+b)-n_\\eta(a-b)}{2b}", "6a6c4df55ab8a13c108f22c7094fc41c": "*\\,\\!", "6a6c79baec3a84393032aa0d4d1310ea": "R_t^m=\\alpha+\\sum\\limits_{i=1}^I\\beta^iR_t^i+\\epsilon_t", "6a6cd1f2ab0c7f8b0888f2e00eacad05": "\\scriptstyle |\\zeta|\\leq R_p ", "6a6d017fcb39dc4abd7a26375bbd7fab": "\\omega _n=\\sqrt{\\frac{k}{m}}", "6a6dec5853281b1270ca6244c2a745b3": " \\varepsilon^v_S = \\frac{\\partial v}{\\partial S} \\frac{S}{v} = \\frac{\\partial \\ln v}{\\partial \\ln S} ", "6a6e223cb1448b0636c6603cfbf18e09": "x\\cdot 1 = x", "6a6e2728a9e27b7444b5d650c0ecca89": "\\varphi_2", "6a6e3ca1087ed8a950230f2a5a7bb0eb": "\\varphi(y)=a\\cos\\frac{\\pi y}{2}+a'\\cos 3\\frac{\\pi y}{2}+a''\\cos5\\frac{\\pi y}{2}+\\cdots.", "6a6e444501631d1bccc3709572ffb24c": "\\tau_\\nu(s_1,s_2)", "6a6e6984795b840c3c2cec75d33d9718": " {A_\\mathrm{v}}= g_{21} = \\begin{matrix} {v_\\mathrm{out} \\over v_\\mathrm{in} }\\end{matrix} \\Big|_{i_{out}=0} ", "6a6e931b3e3d6d78e3c72b6f98a6162d": " X_p = \\left.\\frac{d}{dt}\\right|_{t=0} A(t,p). ", "6a6ed53c3047a6770927dff1a7ad2e40": "(x,y) \\mapsto \\alpha(x,y) + \\alpha(y,x)", "6a6f18e39fbfdeef2477425aa39ea783": "\n\\begin{align}\n& P(\\boldsymbol{Z}, \\boldsymbol{W};\\alpha,\\beta) \\\\\n= & \\prod_{j=1}^M \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i\n\\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)}\\frac{\\prod_{i=1}^K\n\\Gamma(n_{j,(\\cdot)}^i+\\alpha_i)}{\\Gamma\\bigl(\\sum_{i=1}^K\nn_{j,(\\cdot)}^i+\\alpha_i \\bigr)} \\times \\prod_{i=1}^K\n\\frac{\\Gamma\\bigl(\\sum_{r=1}^V \\beta_r \\bigr)}{\\prod_{r=1}^V\n\\Gamma(\\beta_r)}\\frac{\\prod_{r=1}^V\n\\Gamma(n_{(\\cdot),r}^i+\\beta_r)}{\\Gamma\\bigl(\\sum_{r=1}^V\nn_{(\\cdot),r}^i+\\beta_r \\bigr)} . \n\\end{align}\n", "6a6f8838f3f3ac73eb0f7912170f4944": "f(x) = \\frac{x^{\\alpha-1} (1+x)^{-\\alpha -\\beta}}{B(\\alpha,\\beta)}\\!", "6a6f9127b289f1515f683afead11a274": "\\left|\\frac{AF}{FB} \\cdot \\frac{BD}{DC} \\cdot \\frac{CE}{EA} \\right|= 1,", "6a6fa4fc06ac963620a56605360cc876": " T_c", "6a6ff587142ce2b8e8c72c5cfda37ce6": "f_{U_{(1)},U_{(2)},\\ldots,U_{(n)}}(u_{1},u_{2},\\ldots,u_{n})\\,du_1\\cdots du_n = n! \\, du_1\\cdots du_n.", "6a70072aa2eced936a040145910f986d": "\\mathbf{A}=\\begin{pmatrix}1 & 0\\\\ 0 & -1\\end{pmatrix}", "6a709bc7b69727c4a7b319ba58012f4f": "\\ \\psi ", "6a70b4a5f49df2dea74ef112e70de775": "\\partial p = \\rho g \\left(\\partial h \\right)", "6a70c8767b38c8f04d3a23d197a80a31": "A_{jk}(t)", "6a70e736ab62fb9ab18b44edb428b078": "D_k(x)=\\sum_{s=-k}^k {\\rm e}^{isx}", "6a714d922006ba2bb97266903fb6afe5": "\nP_{nxi} =\\frac{\\exp({\\beta_n} - {\\tau_{ki}})}{1 + \\exp({\\beta_n} - {\\tau_{ki}})},\\ k=x,\\,\n", "6a71bf270071241c5402295cbb1c8ee4": "f(x), x\\in[0,L]", "6a724871fb56900a99dafe65d4845004": "\\sum_{x \\in \\mathbb{A}}{x^2}", "6a72929e57dcba811a0785e7249a4981": "k_i(\\epsilon_i,t, t_i)", "6a72d508f3c15b56c51109ea750fce4d": " \\theta_{a} = \\lambda_{p}(\\theta_{a,0}+w)+(1-\\lambda_{p})\\theta_{air} ", "6a72d535819924b42a129bc33355f56c": "Ho(L_C M)", "6a72e8487ab9524ba6b5c2a89f3efd60": "(\\mathcal{L}_X Y)_x := \\lim_{t \\to 0}\\frac{(\\mathrm{d}\\Phi^X_{-t}) Y_{\\Phi^X_t(x)} - Y_x}t = \\left.\\frac{\\mathrm{d}}{\\mathrm{d} t}\\right|_{t=0} (\\mathrm{d}\\Phi^X_{-t}) Y_{\\Phi^X_t(x)}", "6a733e79930966ac96699e9bb0488cee": "s_{xy} = \\frac{1}{N} \\sum_{n=1}^N (x_n - \\bar{x})(y_n - \\bar{y}) .", "6a7404691135a789e1ee50a101bc566b": "((p \\to q) \\land p) \\vdash q", "6a7441043cd46be488dc87bd0d679702": "|0_S\\rangle=\\frac{1}{2\\sqrt{2}}(|000\\rangle + |111\\rangle) \\otimes (|000\\rangle + |111\\rangle) \\otimes (|000\\rangle + |111\\rangle)", "6a7455cb23c108ed9af011ec4f3ad241": "I_q", "6a745f6f1e661f264806555dd61765a4": "\n\\left[(1-u^2)\\psi'(u)\\right]'+\\lambda(\\lambda+1)\\psi(u)+\\frac{2E}{1-u^2}\\psi(u)=0\n", "6a74b9fd35f2c05a39f1d240dcbe105c": "\\textstyle\\sum_{i=1}^k e^{\\eta_i}=1", "6a75296b18d038094067201dd975ee4e": "\\psi : (\\mathbb{C},0) \\to (\\mathbb{C},0)", "6a75864c8a397f7bf4d45e6169232253": "\n g\\left( mr\\right) \n= 1+{2\\over mr}-{2 \\over \\left( mr \\right)^2 } \\left( e^{mr} -1 \\right)\n\n.", "6a75988cd743cc5444b3f6f9ca24cd13": " |X\\,\\triangle\\,Y\\,| ", "6a75bb5ad923b0022da8bbebb24fc07d": "\\,L \\prec M\\prec N\\,", "6a76164b76127a21ce95de089fc07841": "t\\in \n\\mathbb{R}^{L}", "6a763c70edda0b153f734db19eae5640": " \n{\\rho}", "6a7643be14182a76bd81c413f7504225": "\\nabla \\cdot E = 0", "6a764aa7b1fc9b8bfa431c564640b33c": "H_p \\le \\frac{1}{2} \\ln ( 2e\\pi \\sigma_p^2 \\ell^2 / \\hbar^2 )~,", "6a7669c01e3f4b484c56913304118eee": "\n\\left[ {\\begin{array}{*{20}{c}}\n 1 & 1 \\\\\n 2 & 3\n\\end{array}} \\right] \\leftarrow \\text{parent}\n", "6a7688e8fe6fc6f3e53f7300337b6e61": " \\quad (9) \\qquad \\qquad \\int _{v_{i}} {{\\partial {\\mathbf u}} \\over {\\partial t}}\\, dv \n+ \\int _{v_{i}} \\nabla \\cdot {\\mathbf f}\\left( {\\mathbf u } \\right)\\, dv = {\\mathbf 0} .", "6a7711774bbef45fe14f103f5618adbc": "Y_{i}", "6a771d9a113e620fb4344e825ca7f446": "\\frac{1}{r^4} P^3_3(\\sin\\theta) \\cos 3\\varphi = \\frac{1}{r^4} 15 \\cos^3 \\theta \\cos 3\\varphi", "6a7738e75dda2b20c33b08fa39b32cbc": "\\bold{V} = \\frac{\\bold{p}_1 + \\bold{p}_2}{m_1+m_2} = \\frac{m_1\\bold{u}_1 + m_2\\bold{u}_2}{m_1+m_2}\\,\\! ", "6a774eee889436b012fd5b3d4599ce55": " \\lceil b/2 \\rceil \\le m \\le b", "6a78152e51eff97f521c8d90d65590fc": " \\pi(Y) ", "6a78221ded2c73accf4e01f031d688b9": "\\mathbf{S}' = \\mathbf{S} + \\nabla\\times\\mathbf F\\,\\Rightarrow\\,\\nabla\\cdot\\mathbf{S}' = \\nabla\\cdot\\mathbf{S}\\,,", "6a784a4f6de2b47da93d17096c97f7bd": "Velocity > \\left (\\frac {C}{2 \\times Transmit Frequency \\times Pulse Width} \\right)", "6a7969027952ecf915cb68951d5370b3": "\\cdot^{\\mathcal{I}}", "6a798d9d8db0cca4199f5f8fc1d79fd5": "\\frac{1}{2}\\hbar\\omega_{q}", "6a79c22edc0ad3d30a161f6d501c63fc": "U(n,k) = \\frac{1}{2}{n \\choose k} k^{n-k}.", "6a79c897110edccb258e16f2eabe65d6": "\n\\begin{align}\n\\left[K_0\\right]\\mathbf{x}_{0i} & + [\\delta K]\\mathbf{x}_{0i} + [K_0]\\delta \\mathbf{x}_i + [\\delta K]\\delta \\mathbf{x}_i \\\\[6pt]\n& = \\lambda_{0i}[M_0]\\mathbf{x}_{0i}+\n \\lambda_{0i}[M_0]\\delta\\mathbf{x}_i + \n \\lambda_{0i}[\\delta M]\\mathbf{x}_{0i} +\n \\delta\\lambda_i[M_0]\\mathbf{x}_{0i} \\\\[6pt]\n& {} + \\lambda_{0i}[\\delta M]\\delta\\mathbf{x}_i + \n \\delta\\lambda_i[\\delta M]\\mathbf{x}_{0i} +\n \\delta\\lambda_i[M_0]\\delta\\mathbf{x}_i + \n \\delta\\lambda_i[\\delta M]\\delta\\mathbf{x}_i.\n\\end{align}\n", "6a79fe3cf32b49d14b6b0a23e93cf3f2": "F_1, F_2", "6a7a01aebf8a439cc2271b1a0a9e168f": "\\Delta I_{L_{On}}", "6a7a835732f41f378dadb2f8acc13e45": " f_A(y)= [F \\ast a_1 | . . . | F \\ast a_{m/n}]y \\bmod \\ q ", "6a7a89d95b853a12cee522c6726001aa": "\n{{\\sigma _z^2 } \\over {z^2 }}\\,\\, \\approx \\,\\,\\,\\left( {{{2\\sigma _x } \\over x}} \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {{{\\sigma _y } \\over y}} \\right)^2 \\, + \\,\\,\\,4\\left( {{{\\sigma _{x,y} } \\over {x\\,y}}} \\right)\n", "6a7aaaef6622b8d489c93d8960ee35e8": "\\frac{N'(d_1)}{S\\sigma\\sqrt{T - t}}\\,", "6a7acb9d9147fbcd91bf4a54e2a816dc": "\\tau_c = \\frac{\\tau_1 \\tau_2}{\\tau_1 + \\tau_2}. ", "6a7ad0a609af2a7099a74311ed73a0fd": "\n-F^P_1 + \\sum^n_{i=2}F^P_id^Q_i = F^Q\\times 0 \\iff F^P_1 = \\sum^n_{i=2}F^P_id^Q_i\n", "6a7ae92761e6acc709d0e56c44972833": "\n\tE^u(k) \\approx E^b(k) \\approx A (\\Pi V_A)^{1/2} k^{-3/2}.\n", "6a7b80c5cbb1c29e1090edc7c888c57d": "x\\stackrel{*}{\\leftrightarrow}y", "6a7c01277af9a1ae167eae081d32e2b7": "M = (\\mu,\\alpha,\\dot{\\alpha})", "6a7c49c5d7936b7168e71df36286feb2": " {u_z}_\\mathrm{max} ", "6a7c4a68ec3f9298958e91f0ba6952df": "\n\\int_0^T \\sum_k P_{mk}(t) {dX_{kn} \\over dt} dt \\,\\, \\stackrel{\\scriptstyle?}{\\approx} \\,\\, J_{mn} ~.\n", "6a7cb233df17694fe7ab162972b0ea7a": "y(\\pi/2)=2", "6a7cb638fccf192c024fc153f58c0bd5": "\\Sigma=M_1 \\cup_F M_2", "6a7cfd566ee7c707e00893b31f7b3f37": " \\geqslant \\!", "6a7d67338c2d8927ca27fe3b27adbe51": "\\frac{dM}{dy} = y_c - \\frac{q^2}{gy_c^2} ", "6a7d97554862a60b5679bed3916c72a7": "\\log_2(n!)", "6a7da446b72a6c6c1655ac525da2c960": "\\Gamma = \\frac{\\gamma - 1}{\\gamma + 1}", "6a7dbe49019e0b8d8c9c69a5e86a0181": "C\\ |p,\\sigma,n\\rangle \\ \\propto \\ |p,\\sigma,n^c\\rangle ,", "6a7de63531b1a4098c8da3d4b444c6b8": "(x,y)\\mapsto (1/x, y/x^{g+1})", "6a7deb48e90b66d5b96d328314f53e73": "\\int_{V} \\boldsymbol{\\epsilon}^{*T} \\boldsymbol{\\sigma} \\, dV ", "6a7e3046704d05b82e3f92c57c14ece1": "\\, {\\hat{B}} = {T^{-1}}B", "6a7e6c10d581264fd75bb615959f1d2b": "\\frac{R_o}{R_E} = 1+e\\cos(\\theta-\\varpi) = 1+e\\cos(\\tfrac{\\pi}{2}-\\varpi) = 1 + e \\sin(\\varpi)", "6a7f2939a1d8c9baf3936c99a8d22930": "\\hat{Y}_i = \\beta_0 + \\beta_1 \\phi_1(X_{i1}) + \\cdots + \\beta_p \\phi_p(X_{ip}) \\qquad (i = 1, \\ldots, n), ", "6a7f69c39f647b204bfc4722e6c1a421": "\\sum_{k=0}^{N-1}[\\cos_k(i)\\sin_k(t)]\\equiv 0. ", "6a7facd7d641cec7c9fca320fa41ddd7": " O(n) ", "6a8023cf80c325cf11856164509307c9": " L = \\{ \\mathbf{u}+t\\mathbf{v} \\mid t\\in(0,1)\\}", "6a804a5bfab2b5c3b56d0e910a9973fd": "C^\\infty(X)", "6a807877407caf8abc3b5378b49e8e23": " \\vec w \\cdot \\vec x > c ", "6a807b332186ff6da9b28d881aa2b66c": " C_{nV} = \\partial^2 Q/\\partial n \\partial T \\,\\!", "6a80a405fcd9da62452d017d7e8f521c": " e^A_M,\\psi_M,A_{MNP}", "6a810335d25e75229146d95997a73132": "\\mathbf{k}_3 = \\mathbf{k}_1 + \\mathbf{k}_2.", "6a8124a4ef16261e9d9d53a5727f457b": "\ndS_t = \\mu S_t\\,dt + \\sqrt{\\nu_t} S_t\\,dW^S_t \\,\n", "6a813a3dad044fd770d2f2d7721df9a7": "\\mathrm{d}^2=0", "6a814e01cde2365b0a13604f67e129f3": "q_i\\,\\!", "6a815340b171976e8f809f9bbe98d448": "\\ge 200", "6a8175813a1318431cc530919263e1c3": "\\textstyle a ", "6a818a72c832d46b43d02693e4a43314": "m \\arctan\\frac{1}{x} + n \\arctan\\frac{1}{y} = k \\frac{\\pi}{4}", "6a82410f1ba90cfbd4bfe4d47e018f87": "\n\\lambda_k = -\\frac{4}{h^2}(\\sin^2(\\frac{k \\pi}{n})), \\ k = 1,...,n-1.\n", "6a825440d9aba7e757d54346d26f12d5": "\n\\Bigg[\\frac{\\mu}{\\nu}\\Bigg]_2 \\left[\\frac{\\nu}{\\mu}\\right]_2 = \n(-1)^{\\frac{m-1}{2}\\frac{n-1}{2}}\n\\chi(\\mu)^{m\\frac{n-1}{2}}\n\\chi(\\nu)^{-n\\frac{m-1}{2}}.\n", "6a828e66175ca131ca9739ada64ff2aa": "\\exists x_1\\ldots\\exists x_m\\,\\varphi(x_1,\\ldots,x_m)", "6a82b1ba53a93b75cd1d46f1afd58a0d": "(4) \\qquad \\frac{\\partial\\Phi}{\\partial t}\\, +\\, g\\, \\eta\\, =\\, 0 \\quad \\text{ at } z\\, =\\, \\eta(x,t).", "6a8346039c50440605ab8ca2b130df09": "R_{std}", "6a836b170bee54eb7d991f14fb5d7383": "f(z_1) \\neq f(z_2)", "6a8377cd608bec98724fa40f013a18a4": " p=A \\cdot \\left( 1 - \\frac{\\omega}{R_1 \\cdot V} \\right) \\cdot \\exp (-R_1 \\cdot V) + B \\cdot \\left( 1 - \\frac{\\omega}{R_2 \\cdot V} \\right) \\cdot \\exp (-R_2 \\cdot V) + \\frac{\\omega \\cdot e_0}{V} ", "6a83abae6cdd24c3c14a13b19142f62c": "4^n - 2^{n + 1} - 1", "6a83b401fd83ab7042022b0b9fa97117": "s^2=\\frac{1}{n-1}\\sum_{i=1}^n(X_i-\\overline{X}\\,)^2,", "6a83c985490ad8c2ef0424403f3bd90a": "E_2-E_1 = \\Delta E = h\\nu_{12},", "6a83d11ac84dfbe5379d543699981d6c": "p(z)=z^d+p_1z^{d-1}+\\cdots+p_{d-1}z+p_d:=(z-\\zeta_1)\\cdot\\cdots\\cdot(z-\\zeta_d)", "6a83fcdf1e160b39495541d0a5d17bad": "- m c^2 \\frac{d \\tau[t]}{d t} = - m c \\sqrt {- g_{\\alpha\\beta}[x[t]] \\frac{d x^{\\alpha}[t]}{d t} \\frac{d x^{\\beta}[t]}{d t}}.", "6a840e99fe1a51bfd9b0c2efac011f6d": "\\,z_1, \\ldots, z_4", "6a8426c27422ca2e1b53ce6f1e4ae912": "\n\\begin{align}\n\\varepsilon_0 & \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{c_0^2\\mu_0} = \\frac{1}{35950207149.4727056\\pi}\\ \\frac{\\text{F}}{\\text{m}} \\approx 8.8541878176\\ldots\\times 10^{-12}\\ \\text{F/m}\n\\end{align}\n", "6a8460b64d394ec84a5d71727b9439c9": "Q = \\left(\\cos\\left(\\frac{\\theta}{2}\\right), \\omega \\sin\\left(\\frac{\\theta}{2}\\right)\\right)", "6a8482c780a6dda638c6d20f1b84081a": "\\text{rank}(\\boldsymbol\\Sigma)", "6a84bb41011461818a673de9c0d28768": "-\\mathbf{E} = \\nabla\\varphi +\\frac{\\partial\\mathbf{A}}{\\partial t}\\,,\\quad \\mathbf{B}=\\nabla\\times\\mathbf A\\,.", "6a84fea162d828192012a3052c28fba7": " a=[1,4] ", "6a85123ec65f0d39374e8f5af3eee81e": "10^{10^{10^{10^{10^{1.1}}}}} \\mbox{ years}", "6a8525737d67549541badeb41abb116d": "\\left.x \\cup y\\right. \\overset{\\mathrm{def.}}{=} \\left\\{z : z \\in x \\vee z \\in y\\right\\}", "6a8543178c3aa0f6a1433fcb351a98f5": "b^\\dagger_r(\\mathbf{p})", "6a854ab273d2e7551e4b75337ab3b0d3": "\n\\int \\chi^*_{nlm}(r)(-\\frac{\\nabla^2}{2})\\chi_{n'l'm'}(r)d^3r\n=\n\\frac{1}{2}\\delta_{ll'}\\delta_{mm'}\n\\int_0^\\infty dr e^{-(\\zeta+\\zeta')r}\n\\left[\n[l'(l'+1)-n'(n'-1)]r^{n+n'-2}+2\\zeta'n'r^{n+n'-1}-\\zeta'^2r^{n+n'}\n\\right],\n", "6a85727427c365b7f4bcfb67200110fa": "\\mathbf{u} \\cdot \\mathbf{v}", "6a85978dd8669479920ca23ccfa653df": "\\displaystyle g_{\\mu\\nu} = \\eta_{\\mu\\nu} + \\varepsilon\\gamma_{\\mu\\nu}", "6a85de05cd3de88f842882c7fc591458": " Osm_a = \\frac{n_a}{TBW_b - V_{lost}} ", "6a865e69a81a0c14159428d61775ebad": " \\mathbf{T} = {d\\mathbf{r} \\over ds}. \\qquad \\qquad (1) ", "6a86c5aaa05b78391ed3ff2d3cdfc192": "\\{d_0,d_1,\\ldots\\}", "6a86cffa275e840a3f77701e9eac165c": "M=\\begin{bmatrix}m&0&R d \\cos\\alpha\\\\0 & m & R d \\sin\\alpha \\\\ R d \\cos\\alpha & R d \\sin\\alpha & R^2 m\\end{bmatrix}", "6a86ea374b87f303e7f46544278a4086": "\\mathbb{C}^\\times = \\mathbb{C} - \\{0\\}", "6a8709076b1ea6d98ffeb8d1985d545a": "\\psi(x) = \\int_0^{\\infty}\\left(\\frac{e^{-t}}{t} - \\frac{e^{-xt}}{1 - e^{-t}}\\right)\\,dt", "6a87b3b54e51bd49be24b0c744daaf26": "S_g=0", "6a87b4ab9088df3f8ed3665ce354467d": "t_{nn} - \\beta", "6a8830afc12311436131da1b972f4ad1": "T_{h}\\cdot x = 0", "6a889ad7386466669c7a3190e499a58c": "\\int_{-\\infty}^\\infty |f(x)|^2 \\,dx=1.", "6a88ce158773dc1f4d460d354b7cf614": "q, t", "6a88e89d83324abc2f323451cd6fa847": "g=h\\circ\\sigma", "6a89624feb3ba4abb3c94a828e7b2932": "(\\Delta E \\sim 1/r^6)", "6a8973b7ff88db67396f64ea4106ccf0": "M_{BC}=3.867 \\mathrm{\\,kN \\,m}", "6a898f1f6df97f6674f77373fa8de04d": "(\\Delta \\otimes 1)(R) = R_{13} \\ R_{23}", "6a89bdeb2629596c7fe68fdcb483a668": "x \\in P", "6a89d1820dde547be08f92f3f5fe082a": " \\sigma(\\textbf{q}, i) = \\frac{\\exp(q_i)}{\\sum_{j=1}^n\\exp(q_j)} \\text{,} ", "6a89f053e22a4d83b4a1ef60eafe4f54": " \\tfrac43\\pi r^3", "6a8a3a192918c8dcc718a60a796f3b75": "\\Delta \\alpha (\\omega)", "6a8ae903670c9b02b7d46adb1f17235a": "p \\equiv 1 \\pmod 4,", "6a8ae9793759b4c47d7f1bc205cb7757": "\\Chi \\, \\chi \\,", "6a8b13ebd9e569322726c37d81fe6944": "I_\\mathrm{max} \\propto ||C_A|+|C_B||^2 ", "6a8b1d1968e18ddf39aeb8674893fb75": "E_n - \\hbar \\omega", "6a8b3a8eff087d1b9b3915c9eecd6e8b": "GL_{n+1}(R)", "6a8b6d3abc56f2779e409222c3bb08d9": "\n\\frac{\\partial \\rho}{\\partial t}\n+u_j\\nabla_j\\rho\n=\n\\nabla_i\\kappa\\nabla_i\\rho\n", "6a8b7a3dba301492a93d7370a3ab9dbf": "E_\\mathrm{LO} ", "6a8bb29e21caaf113cfc5a65a4ee6609": "H_2 \\overline{C_2 P_n},", "6a8c4e6da998a3246d90c6b957ebf5dc": "\\tilde{M}\\to E^* = H_1(M,\\mathbb{R})", "6a8c5dd8eb685e41923151bafb2bfeff": "\n\\int_0^{+\\infty}\\frac{h(t)}{t}\\,\\frac{dt}{1+t^{2\\alpha}}\\leq\n\\frac{\\pi}{2} \\prod_{k=1}^{n-1} \\Bigl(1+\\frac{\\alpha}{k}\\Bigr)=\n\\frac{\\pi}{2\\alpha} \\cdot \\frac{1}{\\mathrm B (\\alpha, n)}.\\,\n", "6a8c8544f8ac1bf0623a2160cff872c4": "\\mathbf{*2\\cdot11}. \\ \\ \\vdash . \\ p \\ \\vee \\thicksim p", "6a8cf0652a5ca81341b8601a55e80e96": "F_{MH}\\;=\\;\\begin{cases}\\;\\;\\frac{h_M} {3} \\;\\;\\;\\;\\mbox{ if, } h_M > 3 \\\\ \\Big( \\frac {h_M}{3} \\Big)^2 \\mbox{ if, }h_M \\le 3 \\end{cases}", "6a8d7fe2cb95b0f5af842c17da488715": "I_yI_x^-=I_uI_z^-", "6a8dabe136e3550e1d79d0ab1e6f8d72": "\\frac{1}{|A_n|}=\\frac{2}{n!}", "6a8df6bd9d9a24957739591acfb25799": "f(s) = \\sum_{n=0}^\\infty (-1)^n {s\\choose n} a_n = \\sum_{n=0}^\\infty \\frac{(-s)_n}{n!} a_n", "6a8dfb5933c03e9fdbf7231868ee2553": "f = \\frac{f_0}{\\gamma}", "6a8e14377c7145e80b1aa3f7b13d2541": "l,m,l',m'", "6a8e42e8ca81d5adbd8ecdc725cd5da3": "\nA_r^2 + A_i^2 + B_r^2 + B_i^2 + C_r^2 + C_i^2 = 1\n\\,", "6a8e48524e41f5266fbf441ad5c62770": "c + d", "6a8e9e819d1dee4e2fbf67a6f203587b": "\\begin{bmatrix}\n3/2 & 0 \\\\\n0 & 3/2 \\end{bmatrix}", "6a8ee4724ecee6659d143b3b7363fe98": "A^T=A^{-1}", "6a8f0c04fb5077e0eb45e0bb06f298e4": "p-H", "6a8f5aedcf59f7ee05ce164a2e479635": "\\mu (A) = \\sup \\{ \\mu (F) | F \\subseteq A, F \\mbox{ compact and measurable} \\}", "6a8f5d4ca6c43bd892dceea1b74dd73c": "su(2)\\oplus u(1)_A\\oplus u(1)_B", "6a90016d9f08f06b1697450af6732634": " f_1 \\propto \\sqrt{F}.", "6a9008a82f0064b2fd7999de60bc9bf0": "W_X(t,f) = W_x(-f,t) \\, ", "6a9008ec3627ae59c725b77434320d08": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{k}) = \\sigma^2 \\sideset{}{}\\sum_{j = k+1}^{p}\\frac{\\mathbf{v}_j\\mathbf{v}_j^{T}}{\\lambda_j} ", "6a901807a5bb8402a5643c76e82a3c3c": "\\nu_6\\,\\!", "6a902442b55439011724e7dfbc1b82f3": "\\mu_\\text{N}", "6a906429cd6e028b18d14ca049a6a848": " k_x^2+k_y^2 > k^2 ", "6a9065ac6e2aa132d35bc22fa9179eec": "Z_\\mathrm{iT}=iL\\sqrt{\\omega^2-\\omega_c^2}", "6a907fe269b3d1c39672735d240e1b4c": "\\vec{p}=\\gamma m \\dot{\\vec{x}}[t] \\,,", "6a9097aaabb3dc46ddb288c39ccdc6dd": " \\frac{\\dot{W}_{turbine}}{\\dot{m}} = h_3-h_4 \\approx (h_3-h_4) \\eta_{turbine} ", "6a90a177bca5ee33f4ec94b6a079cc3f": "\\begin{matrix} {2 \\choose 2}{2 \\choose 1}{2 \\choose 1}{44 \\choose 1} \\end{matrix}", "6a90b87be777464f4377675ca0cec4db": "\n\\langle \\lambda, \\mu; \\ell-\\lambda, m-\\mu| \\ell m \\rangle\n= \\binom{\\ell+m}{\\lambda+\\mu}^{1/2} \\binom{\\ell-m}{\\lambda-\\mu}^{1/2} \\binom{2\\ell}{2\\lambda}^{-1/2}.\n", "6a90d1c805b7ab9e5a0ec347df96440c": "\\frac{\\partial\\beta_T}{\\partial\\theta} = \\frac{\\partial}{\\partial\\theta} \\left[\\frac{\\beta\\sin\\theta}{1-\\beta\\cos\\theta}\\right] = \\frac{\\beta\\cos\\theta}{1-\\beta\\cos\\theta} - \\frac{(\\beta\\sin\\theta)^2}{(1-\\beta\\cos\\theta)^2} = 0", "6a90d7a7f4bb83ec245542fd8c3f3d69": "0\\longrightarrow M\\overset{\\epsilon}{\\longrightarrow}C^\\bullet.", "6a90da410f421e4e5d6a630c969df572": "E_q(z) = \\;_{1}\\phi_0 (0;q,z) = \\prod_{n=0}^\\infty \n\\frac {1}{1-q^n z} ~. ", "6a90e27107ca5ba81f61d22ffdf3b5c5": "\\neg a ", "6a9148e784d16ae4e210cb2f192f860b": "f(f(\\dots f(z^*))) = z^*", "6a91495790838a1a66f61a058155151e": "(-,-)", "6a919cb1034bece2255dfcaac77d0a43": "\n\\begin{align}\n u_{0} &= a_{0}(y) + b_{0}(y) x + L_{x}^{-1} \\rho(x, y) \\\\\n u_{1} &= a_{1}(y) + b_{1}(y) x - L_{x}^{-1} L_{y} u_{0} + b \\int dx A_{0} \\\\ \n &\\cdots \\\\\n u_{n} &= a_{n}(y) + b_{n}(y) x - L_{x}^{-1} L_{y} u_{n-1} + b \\int dx A_{n-1} \\quad 0 < n < \\infty \n\\end{align}\n", "6a91bb2c5526636272ea3d911cb50782": "\\displaystyle \\sqrt{\\frac{2}{\\pi}} \\frac{ (-i)^n T_n (\\omega) \\operatorname{rect} \\left( \\displaystyle\\frac{\\omega}{2} \\right)}{\\sqrt{1 - \\omega^2}} ", "6a9275b7f966e45ffb33492e358c8dff": "a_0", "6a927c6b62547c84efcd58c52a1b4361": " \\nu=\\rho \\phi_0/B.", "6a92b21d1cceace5e5e42ab16245fa18": "\\tbinom{n}{n/2}", "6a932140f20e620cc1937e6b4afbbf93": "P_1(y)=((-b/c)y)P_2(y),", "6a932de571c35734f5a24054f966ee7e": "V(q) = q^2 - q + 1 - q^{-1} + q^{-2}.\\ ", "6a939e1950f22910ecfbfebf35b53e2b": " [a , b] \\subset \\mathbb{R}", "6a93f78a4f07fc6f2f251df4ac3a1ea3": "g\\in \\mathbb Z^{*}_{n^{2}}", "6a940c73e78ab0df0b26728abf3b382d": "\\gamma_{i}=\\beta_i-\\alpha_i", "6a9433b98ad7edcf21356099b83836ab": "n\\hbar\\omega/2", "6a94aac415735f368d9bc7e649103ace": "C \\underline{\\quad} D", "6a94b794b485b529b25de737bede4b4b": "[x_2^k x_3^{2n-k}](x_2 - x_3)^{2n}, \\ \\ [x_3^k x_1^{2n-k}](x_3 - x_1)^{2n}, \\ \\ [x_1^k x_2^{2n-k}](x_1 - x_2)^{2n},", "6a94e6758075a64e5479272fdf6bea42": "\\Lambda>0.8U_M", "6a952d979f4ce8e9db543efce0028673": "V(Q)", "6a957ae36ebe795ec6fb9010ebffa70c": "\\{rs(\\overline{rs})^{-1}|r\\in R, s\\in S\\}", "6a95a895398d739e726272a3594ee505": " R = g^{\\mu\\nu} R_{\\mu\\nu}.\\!", "6a95d6f6a5a1c87622237d14d9ad911e": "W < W_{\\alpha = 0.05, 9} = 39\n \\therefore \\text{fail to reject } H_0.", "6a96031ff463518a2bb0cc57377a531b": "\\dot y = \\frac{1}{\\mu} x.", "6a9608870aa2e0894b01885310f7542c": "{4 \\choose 1} = 4", "6a9616f54ff21dfddbd54e48e831975a": "\\;_{1}\\phi_0 (a;q,z) =\\frac{(az;q)_\\infty}{(z;q)_\\infty}= \\prod_{n=0}^\\infty \n\\frac {1-aq^n z}{1-q^n z}", "6a962563e235e1789e663e356ac8d9e4": "tm", "6a96458d9d6abf9edc28b5e72e35d538": "V(\\mathbf{x},i)= \\min_{\\mathbf{u}}[\\ell(\\mathbf{x},\\mathbf{u}) + V(\\mathbf{f}(\\mathbf{x},\\mathbf{u}),i+1)].", "6a968b9fe4433a7280078de5ce112509": "\\scriptstyle f_i:\\, M \\,\\to\\, k", "6a96ad848011a49b1737845e6c42e8ba": "g(x,y)=0", "6a96cc54b3955d67fa171facc5952988": "R_{0 0} \\approx \\Gamma^i_{0 0 , i} \\,.", "6a971cca6d9879909fcad192bdaa7e1e": "\\scriptstyle \\{1,...,6\\}", "6a974089c50769c70fc4250cfd493db6": "q=a_{ijkl}E_{ij}E_{kl} \\qquad Q=b_{ijkl}E_{ij}E_{kl}", "6a977618d671c954623ec2d826c224c0": "\\Q(\\alpha)", "6a97c88b431f712f2d941235043d609d": "\n\\Zeta(x) =\n \\begin{cases}\n \\displaystyle \\sum_{k = 1}^\\infty k^{-x}, & \\text{for } x > 1 \\\\[10pt]\n \\displaystyle 2^x\\pi^{x-1}\\sin\\left(\\frac{x\\pi}{2}\\right)\\Gamma(1-x)\\zeta(1-x), & \\text{for } x < 1 \\\\\n \\end{cases}\n", "6a97d30fd2fd64cf5d63691cc7d91ab5": "iu_t+pu_{xx} +q|u|^2u=i\\gamma u", "6a986512a9d9a0c84a903bef76bbacd4": "\\scriptstyle \\mathbf{H}", "6a989ec44b50222f45825c3b4e15cc84": "f(\\mathbf{r}) = \\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} F(\\mathbf{q}) \\mathrm{e}^{\\mathrm{i}\\mathbf{q}\\cdot\\mathbf{r}} =\n\\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{q}) \\right|\\mathrm{e}^{\\mathrm{i}\\phi(\\mathbf{q})} \\mathrm{e}^{\\mathrm{i}\\mathbf{q}\\cdot\\mathbf{r}} ", "6a98c03363e52571a13a98c987342d24": "\\lambda_2 = -1/2 + \\mathbf{i}\\sqrt{3}/2\\quad\\quad", "6a99073a199615c424082560f14d81fc": "\\gamma _{i\\perp }^{\\mu } =(\\eta ^{\\mu \\nu }+\\hat{P}^{\\mu }\\hat{P}^{\\nu\n})\\gamma _{\\nu i}, ", "6a9916958927fc6481cc514da3549dbb": "\\boldsymbol{X}(s,t)", "6a99238da58454c119de49a151a99bd8": "\\mathrm{_{18}^{36}Ar} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{20}^{40}Ca} + \\gamma", "6a9947d98c4b55e1d80a74641a9268a7": "q_{out}=\\frac{A_sq_{in}+B_s}{C_sq_{in}+D_s}", "6a9950a0e7f3d54e853f080ccc808c87": "(\\Psi_nx)_k=CA^kx", "6a995d338ba2731a11e321ff7b32e043": "\\forall x (P \\lor Q(x))", "6a9973c19ce034d8fd77bc6aaec63b61": "\\delta \\psi_{in} = u \\delta y + v \\delta x.\\,", "6a9996a6e8d176fc7cc2f21d4647b2a9": "RAB_{t-1}", "6a9a154490f9a4fd421cbd63f2f60b31": "|X| + |Y| = | X \\cup Y|.", "6a9a5aebb38a1a9080954ba0eec7b14f": "(M,{\\mathcal A})", "6a9a5d8f9da578deea4a0af9b4a255f1": "|\\psi_\\Lambda(t)\\rangle=\\sum_{i=a,b, c}c_i'|i,0\\rangle+c_1'|b,1_{\\nu_1}\\rangle+c_2'|c,1_{\\nu_2}\\rangle", "6a9abd103c9688f81df5319d64eda28f": "\\sum_{a\\in A}C(a,f(a))", "6a9ad2adbb2e75baca075ea3fad9caff": "a\\in S", "6a9b028fb680a123cf1bf1ddfa814972": "\\mathrm{K}", "6a9b6a65e1445f82fc3c4284d57662b0": "\n\\vartheta_{00}(z; \\tau) = \\sum_{n=-\\infty}^\\infty \\exp (\\pi i n^2 \\tau + 2 \\pi i n z)\n", "6a9b8a686bff2987a727b053b00df712": "X' \\to X \\times_S P", "6a9bc7a7c7eea9c27846e1c6b0d64eff": "\\frac{19}{6} ", "6a9be3d0c26d3dcf28c76f88e0cdb658": "\\beta(\\infty) = (4\\pi)\\sqrt{2Mr} \\;", "6a9be75529ba863d394084ffef9e2b80": "(\\bar{3},1)_{-\\frac{2}{3}}", "6a9c07a2362d28d3506d7a17ec9d2ffb": "P = e^{\\frac{\\Omega + \\mu N - E}{k T}},", "6a9c0b937fe56f989d0f3ee526eed998": "\\{\\land, \\lor, \\rightarrow\\}", "6a9c1ce69d8e2cf71a5ddac2e8b56402": "\\hat{e} = \\arg \\max_e E \\left[ u(w(y(e))) - c(e) \\right] \\geq \\bar{u}", "6a9c3aaf0581850bc25e3aa7d2c8ac7d": "\\hat{C}^{(1)}", "6a9c5b753a7bef301bb6a9266ff0637c": "w \\log(n/w) > r", "6a9c9f0a92846765718a83cb9c4979b6": "\n\\begin{align}\nE_{10} & = 10! \\left( - \\frac{1}{10!} + \\frac{2}{2!8!} + \\frac{2}{4!6!}\n\t- \\frac{3}{2!^2 6!}- \\frac{3}{2!4!^2} +\\frac{4}{2!^3 4!} - \\frac{1}{2!^5}\\right) \\\\\n& = 9! \\left( - \\frac{1}{9!} + \\frac{3}{1!^27!} + \\frac{6}{1!3!5!}\n\t+\\frac{1}{3!^3}- \\frac{5}{1!^45!} -\\frac{10}{1!^33!^2} + \\frac{7}{1!^6 3!} - \\frac{1}{1!^9}\\right) \\\\\n& = -50,521.\n\\end{align}\n", "6a9ca3f943d84ad69e546813c5413207": " {^{0}a} = 1 ", "6a9cd3c353f73110c1d98c86ec51edcd": "\\frac{\\partial \\mathbf{f(g)}}{\\partial \\mathbf{g}} \\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{u}}{\\partial x}", "6a9cdd5428b5de73d351ac791008cf1b": "f_1(w)=\\int_K f(gkw) \\, dk,", "6a9d2039c919a02dcdcd84c893fd49ba": "\\left(\\frac{\\partial H}{\\partial p}\\right)_T = V - T\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "6a9d4f07a4ae7c7c4529ec7c6f33af88": "x_n\\,\\!", "6a9d58873378c41d7c147f67651f2dad": "\\scriptstyle \\vec{v}", "6a9d953539ddab8cc02ca03f5ca09cd4": "R_1 \\gg R_2", "6a9da43cf613a644515bb7ea9737a809": "\\forall j~\\sum_{i=1}^M \\mu_{ij} = 1", "6a9dc930d9c343aae9422fe3d646d65b": "\\Omega = \\{H,T\\}", "6a9f11e86d5e6a86c1baf08a7305a73e": "\\mbox{mex}(\\left \\{0, 1, 2, 3, \\ldots, \\omega\\right \\}) = \\omega+1", "6a9f34b8ce62537adc2d5dda3f3848ee": "= R_{\\alpha \\beta I}^{\\;\\;\\;\\;\\; J} V_J + (\\nabla_\\alpha C_{\\beta I}^{\\;\\;\\; J} - \\nabla_\\beta C_{\\alpha I}^{\\;\\;\\; J} + C_{\\alpha I}^{\\;\\;\\; K} C_{\\beta K}^{\\;\\;\\; J} - C_{\\beta_I}^{\\;\\;\\; K} C_{\\alpha K}^{\\;\\;\\; J}) V_J ", "6a9f399332a797a2d4f5555c10df2c65": "=\\left[ G(\\omega) + j \\omega C(\\omega)\\right] V(\\omega) = \\frac {V(\\omega)}{Z(\\omega)} \\ , ", "6a9f48a1435445f753966a26f1e6f5b1": "\\frac{\\partial v'}{\\partial t} \\, + \\, 2 \\Omega \\sin \\varphi \\, u' \\, +\n \\, \\frac{1}{a} \\, \\frac{\\partial \\Phi'}{\\partial \\varphi} = 0", "6a9f574ca6070e5724c136051e605649": "\\; D_{\\mathrm{REE}} (\\rho) = 0", "6a9fef9c03f6ced6b339be26958d739a": "D(M)_S(X) := \\mathrm{Hom}(M, \\mathbb{G}_{m,S}(X)).", "6aa006c4375e9bfb8e49d168087d4d3b": "\\scriptstyle{B}", "6aa0102ec095242232f2b8c8141d1eec": "\n\\mathbf x\n= \n\\begin{bmatrix}\n\\mathbf A, & \\mathbf B\n\\end{bmatrix}\n^{+}\\,\n\\mathbf d\n= \n\\begin{bmatrix}\n(\\mathbf P_B^{\\perp} \\mathbf A)^{+}\\\\\n(\\mathbf P_A^{\\perp} \\mathbf B)^{+} \n\\end{bmatrix}\n\\mathbf d\n.\n", "6aa024ea2066258b76b469b29970087a": " a\\equiv m_{i}^{2} a_{i}\\pmod {p_{i}} \\;\\text{and }\\; b\\equiv m_{i} b_{i}\\pmod {p_{i}} \\text{. }", "6aa02fcb2d1367849b0fc7bff8271acc": "R = NL/G", "6aa05b6e83242942cb8a3f9609431303": "\n\\frac{1}{2} Y^{2} \\dot{\\varphi}_{r}^{2} = E \\chi_{r} - \\omega_{r} + \\gamma_{r},\n", "6aa05eca8efd0af08a2f05fb18b472a6": "V=S(\\sigma)\\,T(\\tau)\\,Z(z)", "6aa08271bc79be92a7c2114fe2f1cb7c": " R=(Y_1\\cdot X_2\\cdot Z_2-Y_2\\cdot X_1\\cdot Z_1 : X_1\\cdot Y_2\\cdot Z_2-X_2\\cdot Y_1\\cdot Z_1 : Z_1\\cdot X_2\\cdot Y_2-Z_2\\cdot X_1\\cdot Y_1) ", "6aa08e7902a5cc66786324371daaf4bf": "c(a_i) \\in {\\mathbf R}^+_0", "6aa0b3d8e977553f119e4f109ec66fd0": "l_0/2", "6aa0cea833d56961b2b14e97ced23a29": "|z|^{2}=zz^{*},", "6aa114237631d844af04acb2fa944207": "\\sum_{p\\le x}\\frac1p=\\log\\log x+M+o(1/\\log x).", "6aa1d524fe3fa5a4578c49ae907cac5a": "|\\{v \\in R | N(v) \\cap V(y) \\,", "6aa21f1593703bdec064c011f3663b25": "A\\oplus B \\subseteq C\\oplus B", "6aa29376839d21b1d5f01153d544a78b": "SO(3) ", "6aa2c8de260a72b7a1a52993ad948fd1": "\\{A \\mid \\forall F\\,(\\emptyset \\in F \\wedge \\forall x,y\\,(x \\in F \\rightarrow x \\cup \\{y\\} \\in F) \\rightarrow A \\in F)\\}", "6aa399863d6fdceac30b30e87627ddc2": " \\tau \\mapsto -1/\\tau \\ :\\ \\lambda \\mapsto 1 - \\lambda \\ . ", "6aa4623df0b1b48ce309004822a71754": "O\\left(n (\\log n)^{(O(c \\sqrt{d}))^{d-1}}\\right),", "6aa4a88dd6272f941298a8aed8b67df1": "f_\\text{mod} = f_\\text{b} + \\frac{v_\\text{p}}{\\Delta s} = f_\\text{b} + \\frac{2v}{\\lambda} \\sin \\varphi", "6aa4bb25d8459f43688882481c73348a": "GL(n,{\\Bbb C}) \\times GL(m, {\\Bbb C})", "6aa4be42f058103a2e2181e7b7be072d": "\\,{}^{n}a", "6aa4c2d8249413289fb9a8365f355ac1": "\\scriptstyle{\\psi}", "6aa4deddf72249faca9245a871655944": "\\frac{1}{x_1} + \\frac{1}{x_2} = \\frac{1}{f}= \\frac{2}{r}\\,\\!", "6aa50ffca0d2920300d4fc58a4c68a68": "\\rho(\\vec x)=x^2_1+a_0x_1x_2+b_0x^2_2-x_3x_4", "6aa55764e8b736f68d1f16b3fa9a8ecc": " HH_n(A,M) = \\text{Tor}_n^{A^e}(A, M)", "6aa5c516339e712e562f250f2a6328fa": "A \\le_T^P B", "6aa5d16b8b6fea42cd5c76d1a31b3eb0": " x \\in C", "6aa5ffefab6945366f899af62dbe2d57": "= [\\, a + b\\sigma'(x) + c\\sigma''(x) + e\\sigma'''(x)\\,]dx\\,", "6aa63ff7b06e88ff4e53e7b2ced92f15": "n=max(deg(f),deg(g))", "6aa68d5f92319906403b2185b270d274": "\\begin{matrix} {2 \\choose 1}{3 \\choose 3}{45 \\choose 1} \\end{matrix}", "6aa7d28baa91347d8baf0e5d325409a1": "c=(-1)^{m} \\sum w_i ", "6aa7f68b7aeeada26d5e49aa85759baf": " V_{\\rm{eff}}(r,e,l)=-\\frac{M}{r}+\\frac{l^2-a^2(e^2-1)}{2r^2}-\\frac{M(l-ae^2)}{r^3},~~~\na\\equiv \\frac{J}{M} ", "6aa88cf12ad79ef7cdf20ecc311b3874": "v = (m,0)", "6aa8be17f7b8e21af46bcc4cf03fa44c": "M := \\{ r, p, s \\}", "6aa8efc5341b3a8eceb4273f89bc302b": "\\{z \\in \\mathbb{C} \\quad : \\quad \\mbox{Re}(z) > 0\\}", "6aa963808c1e0dd514832b4095f35358": "a(i) \\ll b(i)", "6aa99f27bc03baba4a5f388484e184de": "~J_n", "6aa9bea1ba6a4e895bbfcefad89783ee": "m \\in \\mathbb{N}", "6aaa300722da8f16a75542c39938648f": "\\rho_{sl}", "6aaa3f4376ded03e34a695e67563cd8c": "d\\star d u =0", "6aaa4cf8efb2c057e88f31a2f31ae1eb": "\n\\mathbf{f} = \\epsilon_0 \\left(\\boldsymbol{\\nabla}\\cdot \\mathbf{E} \\right)\\mathbf{E} + \\frac{1}{\\mu_0} \\left(\\boldsymbol{\\nabla}\\times \\mathbf{B} \\right) \\times \\mathbf{B} - \\epsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t} \\times \\mathbf{B}\\,\n", "6aaaca94b45a43730c44f7470c370785": "\\left[ A, B \\right]_- \\equiv AB - BA", "6aab2bd447640922fa2a6706cf8b8794": "\\scriptstyle \\sqrt{2E/\\omega^2m}", "6aab3a594ebc9963677a0efd35671d9c": "K(p) = \\lim_{r\\to 0^+} 3\\frac{2\\pi r-C(r)}{\\pi r^3} = \\lim_{r\\to 0^+}12\\frac{\\pi r^2-A(r)}{\\pi r^4}.", "6aab452c23e7171cd5a14b8f3cd8eb5f": "\\sigma_T = \\frac{8\\pi}{3}\\alpha^2\\bar{\\lambda}_e^2 \\simeq 66.5~\\textrm{fm}^2 ", "6aab6d600aec80395431f30b7ab1a336": " D_2: d_{ ij } = | r_i - r_j | ", "6aab9cd87e2429a8dc7b79cc3b96e1ef": "B(x) = \\frac{1}{\\hbar} \\sum_{k=0}^\\infty \\hbar^k B_k(x)", "6aabb0e8220896d3059fd0613ba0eedd": "Z = \\sqrt{R_L^2\\ +\\ X_C^2} = 1020\\, \\Omega", "6aac29ebc45445913a07ca5f3906a547": "\\Phi_{00}=\\Phi_{10}=\\Phi_{20}=\\Phi_{11}=\\Phi_{12}=\\Lambda=0\\,,\\quad \\Phi_{22}=-\\frac{M(u)_{\\,,\\,u}}{r^2}\\,,", "6aac9c0e9179112c03dad39ea47b4dc3": "[U] [\\Sigma] [V]*", "6aac9cbb7d3c90fb09e62af7aa5dece4": "\\delta = \\frac {p+q-1- \\frac {g}{k}}{pq} ", "6aaca6618796d88911e06625ecf5dce4": "\\scriptstyle \\leq-5.8\\times10^{-12}", "6aacbb6bcb93a87e8744fdfb324ab62c": "\n \\boldsymbol{\\epsilon} = \\frac{1}{2} [\\boldsymbol{\\nabla}\\mathbf{u} + (\\boldsymbol{\\nabla}\\mathbf{u})^T]\n", "6aad064f4b23dd35570b0befe416b30c": "p_1 < p_2 < \\cdots < p_k", "6aad0a9d8accd5e0979266ec3d08b097": "\\scriptstyle r_2^2 = (\\alpha - \\beta)^2", "6aad126368bed67b6de260c126977c8c": "\n\\int_a^b f(\\phi(t))\\phi'(t)\\,dt\n", "6aad89c2f8edf052250efde4d0d56213": " \\Psi = \\Psi \\left (\\mathbf{r}, \\mathbf{s_z}, t \\right ) ", "6aad9a3ddb3f2d14d624482df13fdb6b": "C=\\left(\\frac{1}{2}\\sqrt{11+4\\sqrt{5}}\\right)a\\approx2.23295...a", "6aadfaf5e5694410589b96dcb958d74b": "V_{r2}\\,", "6aae307e51d970c5e64b52b85e53ef99": "\\phi^{\\# }", "6aae6267a7b29be117283f8a70b119b0": " \\psi(x) - \\psi(x_1) = \\psi(x-x_1) = \\psi(z_1 + z/a) = \\phi(z_1 + z/a) \\geq 0~, ", "6aaed8496e8fa52dc5d107c84fa4ba21": "\nE_{\\mathrm{tot}} = U(r) + \\frac{L^{2}}{2 m r^{2}}\n", "6aaee42ac640c5f346e3d7b04132afd0": "C_s(h)=C(0,h)=C(x,x+h)\\,", "6aaf0ad3b555c447d36dbe2e42786548": " y\\ =r \\sin \\theta", "6aaf3ac93c841f10d4f46b93e7cd90e4": "\\pm\\frac{1}{\\sqrt{\\sec^2 \\theta - 1}}\\! ", "6aaf68d39bd6b362ce9853cfcf5c50c9": "\\delta_S", "6aaf6a0a5962d0baf382a0b373348480": "\\Sigma^{\\ast}", "6aaf70c1166a7410f8e0eeda55727ced": "-0.5 \\le \\beta < -0.25", "6aafaa7cb802314652dc5f4141cd7049": " \\left({e_w}\\right) ", "6ab03eae82ebae654073c932e01673fa": "k \\equiv m \\pmod p. \\,\\!", "6ab09618374de5af91a5a235725205c0": " y(t) = e^{At} y_0. \\, ", "6ab0abed1be560322a2129bc3ad189cb": "K_p = \\frac{(\\mathrm{p_T})(4\\alpha^2)}{(1+\\alpha)(1-\\alpha)} = \\frac{(\\mathrm{p_T})(4\\alpha^2)}{(1-\\alpha^2)}", "6ab0ace6675cc10f4957b0cdd713b7af": " \\int_a^b f(x) \\, dx \\leq \\int_a^b g(x) \\, dx. ", "6ab0f3be9ae84f203c64e7e4b15655f3": " p = \\frac{RT}{V_\\mathrm{m}-b}-\\frac{a}{V_\\mathrm{m}^2}", "6ab113c9c0248fb17eda1c4e6e8077c5": "p=q", "6ab11c58573b9ef41e40be176a1f7def": "\\mathbb{F}_l", "6ab126b1dc2e66a7030ca429d1b78da1": "\nr_{pb} \\sqrt{ \\frac{n_1+n_0-2}{1-r_{pb}^2}}\n", "6ab1769b13037d89d3eeaeab11bcee75": "\n \\big(\\mathbb{H}_\\mathrm{n}(\\mathbf{R})\\big)_{k'k} = \\langle\\chi_{k'}(\\mathbf{r};\\mathbf{R}) |\nT_\\mathrm{n}|\\chi_k(\\mathbf{r};\\mathbf{R})\\rangle_{(\\mathbf{r})}.\n", "6ab1b57e9ce6071ae283f548aecf6259": "{64 \\times 64}", "6ab1beed73dfa2cbd3fec4a528fe16b5": " \\log_{b_1}a_1 \\,\\cdots\\, \\log_{b_n}a_n\n= \\log_{b_{\\pi(1)}}a_1\\, \\cdots\\, \\log_{b_{\\pi(n)}}a_n, \\, ", "6ab1c32ea9c4b8f9b842f528e6ecb633": "D_{\\mathrm F} - s = \\frac {Ncs(s - f)} {f^2 - Nc(s - f)}\\,.", "6ab21986c255b3d09f5fdd99c28ef93d": "\\mathrm{Ref}_\\mathbf{p}(\\mathbf{a}) = 2\\mathbf{p} - \\mathbf{a}.", "6ab2333560318a935d463db230502f3c": "\\sigma(t) =t", "6ab2f6ebb803cda225f7b0744d827026": "\\sin \\theta = \\frac{\\mathrm{opposite}}{\\mathrm{hypotenuse}}= \\frac{b}{c}", "6ab3385d66ab1593cc69f944226db859": "E\\left( \\tfrac{\\sqrt{2}}{2} \\right) = \\pi^{\\frac{3}{2}} \\Gamma\\left( \\tfrac{1}{4} \\right)^{-2} + \\tfrac{1}{8 \\sqrt{\\pi}} \\Gamma\\left( \\tfrac{1}{4} \\right)^2 ", "6ab376d2730ea4675daee1911b79682b": "v^2/(2c^2)", "6ab378e824966cb017450d0e72633808": "(\\mathcal{O}_k / \\mathfrak{p})^\\times.", "6ab3c3f3b8c98530ad856c1a939fe028": "\\mathcal{H}_A = \\{ (a_i) | \\sum a_i^{*}a_i\\text{ converges in }A \\}.", "6ab3f11870fbdebd926083d2a20c059d": "F = \\frac{a^2}{L\\lambda} \\ge 1", "6ab4216f6b45e9118bdf7a40ba6f3237": "\\lambda_{\\epsilon}= \\frac{\\sum_{i=1}^{B} {m_{i,\\epsilon}^2 p_{i,\\epsilon}-\\mu_{\\epsilon}^2}}{\\mu_{\\epsilon}^2} = \\frac{\\sigma_{\\epsilon}^2}{\\mu_{\\epsilon}^2}", "6ab422248291bd549ec71a267bbb26ef": "F(x, y, z, t)=G(x, y, z^2-t^2, 2zt),\\,\\!", "6ab4308e08dbb5f15daab8076b23690d": " var( \\frac { 1 } { x } ) = \\frac { m [ E( 1 / x - 1) ] } { n m^2 } ", "6ab46abf27f3a6eaf579fd2278af39f2": " g ", "6ab4e1f36e5c202cf44080fcdeddd85e": "X=X^{(1)}+X^{(2)}+X^{(3)}", "6ab58bfd50dd60fe7f35967c20e22a91": "\n -N\\frac{\\partial^2w(x,t)}{\\partial x^2}+\\rho\nA\\frac{\\partial^2w(x,t)}{\\partial t^2}=\\delta(x-vt)P\\ .\n ", "6ab5975b8cfe55b6aeb496c7d3a5b6da": "\np(t) = A D^{-1} p(t-1).\n", "6ab59bf2f5662c4b3bfbe4dd5aaf565e": "\\left(H^{(\\lambda)}(z)\\right)_{\\lambda=0,1,2,\\dots}", "6ab5ac6f973ff355d2d4e2278fdf4c16": "\\hat{\\boldsymbol{r}} = \\frac{ \\partial \\mathbf{r}}{ \\partial r} = (\\cos\\theta,\\ \\sin\\theta)", "6ab5b1719c45c4ea3f970679ce7005ca": " \\log V_x = \\log V_1 + b\\log x \\, ", "6ab602b4a763a660dfc551a987d652f3": "|x_3-x_2|=|f(x_2)-f(x_1)|\\leq L|x_2-x_1|", "6ab653bd280fbb7d3cba91ca6ce32236": "\\ \\mathrm{Lie}_q\\,\\mathcal{F}=\\{g(q)\\mid g\\in \\mathrm{Lie}\\,\\mathcal{F}\\}", "6ab65ed26d2af984aed30457654744bb": " \\gamma = {C_{P} \\over C_{V}} = \\frac{f + 2}{f}, ", "6ab676f58ca1f5ebb9fa94b8301ce1f5": "U_{(uv)(lm)}(k)= \\delta_{vl} \\sigma_{(uv)(vm)}(k) \\textrm{e}^{i kL_{(uv)}}.\\,", "6ab6773e8860f034055e6ed18721b398": "O(n\\log k)", "6ab6a0124e250f8634471e66937019d6": "W_{2}^{II}(x,x)\\geq W_{2}^{I}(x,x)", "6ab6d1edf2719dc6ab9d67bfbf99cd8e": "\\frac{\\partial \\mathbf{M}}{\\partial t} = \\frac{\\mu}{\\rho} \\nabla^2 \\mathbf{M} -\\mathbf{v} \\cdot \\nabla \\mathbf{M} + (\\mathbf{f}-\\nabla \\text{P})", "6ab7a066ab47645597049379dfe0c37d": "\\boldsymbol{\\zeta} = \\frac{{\\rm d} \\boldsymbol{\\alpha}}{{\\rm d} t} = \\bold{\\hat{n}}\\frac{{\\rm d}^2 \\omega}{{\\rm d} t^2} = \\bold{\\hat{n}}\\frac{{\\rm d}^3 \\theta}{{\\rm d} t^3} \\,\\!", "6ab7bc7488583bddbfd4b61b45d28040": "x_n \\to x", "6ab7e25ad955d565fa66872dced5195c": "-{\\mu\\over{r}}=2\\epsilon", "6ab7f83114997ac01e0516f7201ffb3e": " P( X > 0 ) \\ge \\frac{ 2 }{ 3 + \\psi + \\sqrt{ ( 1 + \\psi )^2 - 4 } } ", "6ab80cd9850786e92f1748f476cc9502": "\\lim_{n\\rightarrow\\infty}P\\left(\\left|X_n-X\\right|\\geq\\varepsilon\\right)=0", "6ab86ac8cd67ee76a7e26537494d6224": " E_c = fE_f + \\left(1-f\\right)E_m.", "6ab872fd4b92f33d23c8eb300691d7eb": "P(\\textrm{spike}) \\propto f(\\mathbf{k_1}\\!\\cdot\\!\\mathbf{x},\\; \\mathbf{k_2}\\!\\cdot\\!\\mathbf{x},\\; \\ldots,\\; \\mathbf{k_n}\\!\\cdot\\!\\mathbf{x})", "6ab8768ad0508511c6608252b9d60dc2": " {n \\choose x} ", "6ab8eec9ca9b1e506005369be189a9e2": "K=K(\\mathbf{J})", "6ab9b39267e896afbc2b4986f18671db": "\n\\log L(\\theta_0+h|x) \\approx \\log L(\\theta_0|x) + h\\times\n\\left(\\frac{\\partial \\log L(\\theta | x)}{\\partial \\theta}\\right)_{\\theta=\\theta_0}\n", "6aba063ca1bc0dc7e34ddf6b0a9b39a7": "Q*", "6aba354b3ff03acd5bab98c04c2dde5f": "(m_p/m_e)^4 \\approx 10^{13}", "6aba850d389447e06f967b5756764478": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "6aba88b5169e6266d6d64817110c0dff": "q_i(\\theta) = \\sum_{j=0}^{[x_i/2]} \\frac{\\theta^j}{(x_i-2j)!j!}", "6abaab2a0519a39a8473a73bdd6050e3": "=x\\cdot\\lim_{h\\to 0}\\frac{\\ln\\left(1+h\\right)}{h} \\quad \\mbox{ where }h=\\frac{x}{n}", "6abac91b0873fd682fb1d50d6284983a": "F(r) = \\frac{GMm}{4r^2 R} \\int_{R-r}^{R+r} \\left( \\frac{1}{s^{p-2}} + \\frac{r^2 - R^2}{s^p} \\right) \\, ds", "6abbd44e055d5c9778a025e392ef71e4": " \\scriptstyle \\theta ", "6abc02caa6c0e50de9c510212570e63a": "\n\\omega_R = \\frac{\\hbar k_z^2}{2 m}.\n", "6abc59a5759c3d9e19e39c1ac535c777": " x\n= ( w \\wedge u \\wedge v)\\frac{1}{ u \\wedge v}\n= \\frac{1}{ u \\wedge v}( u \\wedge v \\wedge w).\n", "6abc8a664a607709088633d2d5b2df8d": "\\{P_X,P_Y\\}=-P_{[X,Y]}.\\,", "6abccc9391cca6ecf884f8e00580444e": "\\scriptstyle x_n", "6abcdf8a50830f3c4d63233631784110": "\\sum_{i\\in I}a_i = \\sup \\Bigl\\{ \\sum_{i\\in A}a_i\\,\\big| A \\text{ finite, } A \\subset I\\Bigr\\} \\in [0, +\\infty].", "6abce1aa4aabee49c4c86dad11603e2f": "\\alpha_\\lambda=n! / \\text{dim } V_\\lambda", "6abcf5df7c8974ab3a0bfad3504dae9e": "g_ig_j=g_k\\ ", "6abd0a3afc274c8129e81a36a858ffda": " e= 2 \\left ( \\frac{2}{1} \\right )^{1/2} \\left ( \\frac{2}{3}\\; \\frac{4}{3} \\right )^{1/4} \\left ( \\frac{4}{5}\\; \\frac{6}{5}\\; \\frac{6}{7}\\; \\frac{8}{7} \\right )^{1/8} \\cdots ", "6abdcc5fe350509a2a66db07373224c9": "\\chi(\\mathcal{O}(D)) = \\operatorname{deg} D + 1 - g", "6abdf47e53e0e81cd634a2faa7105c9b": "CP^2", "6abe0282b964fd5031d87cf7792dcbb6": "r=\\mu^t a", "6abe6c6452cd99d35861993325872c55": "Enc(x)", "6abe890cd92483bfc4b8777bc1164863": "P\\left(A\\mid X\\right)", "6abed0755f308ef6ca1cfefc78e6c051": " a'/c' ", "6abefdea996d4a3817223163cdcfea68": "D_{ik}", "6abf6e00583d408557e6c31f894f9619": "\\scriptstyle \\int x^m (\\ln x)^n\\; dx", "6abf96a985109ebfb50c2f1984d94683": "\n\\Delta \\Omega\\ =\\ -2\\pi\\ \\frac{J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos i\n", "6ac063809fd688ef025c91b54399725d": "U(\\mathbf{r},t)= U_oe^{i(\\mathbf{k}\\cdot\\mathbf{r} - \\omega t )}", "6ac07c3b5806b651bc93e1e26a647af9": "C \\times T", "6ac0b8b9d7a00f57222c825ac9ef1897": "p_i \\geq 0,\\quad 0 < a \\leq x_i \\leq b\\text{ for }i=1, \\dots ,n.", "6ac0f1962fa1493ad0fb9715cc1b37ae": " \\Gamma \\backslash \\mathbb{H}^2 ", "6ac1652273275b3031500604a7471a73": "c_{t}+b_{t+1}\\leq y_{t}+(1+r)b_{t}", "6ac1744d92b063a747b0a31fc823bc9e": "\\operatorname{MSE}(\\widehat{\\theta}) = \\operatorname{var}(\\widehat\\theta) + (B(\\widehat{\\theta}))^2,", "6ac19fb8dadf8e76002e3ea4fc6d3d5c": "\\lbrace p_1,p_2,p_3,p_4 \\rbrace = \\lbrace 1,0.5,0.5,0.25 \\rbrace", "6ac24386ca887ccc7401e3a09b14fbc1": " y_j^{te} ", "6ac24695e6c4c07bd7737a5071e7a8c1": "\\ \\mathbf{x}=x_i\\mathbf e_i", "6ac2bd03a6d62f1ebc210aa04e47235e": "\\ u_x = W.", "6ac306461dd73f954a5031db446f923d": "(q) \\subsetneq \\beta,", "6ac306a0101baf84ff1152e0d84580b0": "\\Pr(w \\mid z)", "6ac375ddcd4a8efdc63246f8a7e70cb4": "s\\begin{Bmatrix} p \\\\ 2 \\end{Bmatrix}", "6ac3c3e6906f0e44b0687aac5e2ec5f0": "\\mathrm{Ptr}(k,t)", "6ac3dc44856fb58bab1cc62df84af4bb": "Z^n(G,M) = \\operatorname{ker}(d^n) ", "6ac4bc6913551539a85b10f9c5c88c38": "\\forall{x} A", "6ac518c0bd24d797255f7828dea30e74": "kh_k=\\sum_{i=1}^kh_{k-i}p_i,", "6ac518f1be4a9f4b3ddf96e587122428": "I=\\displaystyle s_wY+(s_c-s_w)P", "6ac52dedab9e5898ae527aada3a8856e": "\\rho=\\frac{1}{2}\\sum_{\\alpha\\in \\Delta^+} \\alpha", "6ac54074bf8ad6ebc83bb7b3d3e287c5": " u_3 =0.37428 ", "6ac5409d75b2db47f330bab3c729c8bf": "\\scriptstyle N \\times N", "6ac5ac3a9cb714df31f1e98f429ce882": "\\boldsymbol {\\nabla \\times E} = -\\frac{\\partial }{\\partial t} \\boldsymbol B \\ , ", "6ac5cd79b881cd9565f295a1be29976a": "\\sum_{v \\in X} a(v) \\ge T.", "6ac5d344f3acd61445822d0acfdbfa88": " |u|=r ", "6ac60e7429ae663509d2392aa274b9a9": "f=u+iv", "6ac6323607e650edd53419203cc8a4ce": "\\scriptstyle v_1,\\dots,v_n", "6ac644a4c4926af42ceea65f8f7a09c3": "[\\mathbf{a}]_{\\times} = \\mathbf{d}\\mathbf{c}^\\mathrm T - \\mathbf{c}\\mathbf{d}^\\mathrm T.", "6ac671a4e7306c9809b840e1795b34d3": " \\frac{d(q-1)q^d}{2}", "6ac682dc8bb7b76b72b95aa1a2fa276d": "D(E)=\\sum_i \\delta(E-\\epsilon_i)", "6ac699d84828a944da47db7f508b6cd7": "g_{\\mu 4}=\\, 0", "6ac6beb186f4e8f569d72d1476893534": "\\widehat E_i", "6ac6d627f81e417f4eb6ea15390326c4": "f:K \\rightarrow \\mathbb{R}", "6ac6e2320729c73b00c4e9e6acd7133f": "A_i \\nabla_j - A_j \\nabla_i = 2 A_{[i} \\nabla_{j]} ", "6ac7119f212a35ed735b27a9c4c1f422": "r_s > 1", "6ac778b9837f33e3e5ea9c98b5e4ef3a": "\\zeta^2+\\zeta=-1", "6ac79f11d78e68989b35284a5f5b17c4": "K=\\tfrac{1}{2}\\sqrt{p^2q^2-(m^2-n^2)^2}.", "6ac7dd452d65c8ac837bb1c958ffc77c": " \\lim_{y \\to b}\\frac{\\partial}{\\partial y}[E(u(X)|X>y)] = \\frac{1}{2}u'(b) ", "6ac8470965b721692fb546905e304535": "[\\overline 11\\overline2]", "6ac87ecef1ab0fafa918fa3195de9a9a": "E\\to\nX", "6ac91ffc73cb5fc5b1e7ee911d6a7717": " \\frac{V_1}{T_1}=\\frac{V_2}{T_2} \\,", "6ac92089e8c4d7be43437888ec8bf52f": "[\\sigma_\\mathrm{avg}, 0]\\,\\!", "6ac93c0066186345d3b51cc905f24ed6": " E_0(x,\\alpha) = 1 \\,", "6ac9bfb9dac2b40397237db8f7444054": "\\frac{d}{dx}\\, \\operatorname{arcsch}\\,x =-\\frac{1}{\\left| x \\right|\\sqrt{1+x^{2}}}", "6ac9d46f68bee1df1f458e7967cd789c": "ZZ_3 = Z_3^2", "6ac9e14ba92bd2634dc5c67dbf195988": " g(r) = e^{-\\beta w^{(2)}(r)} ", "6ac9faf12f1ef04a9cba76b00f4b01b2": "\n\\mathcal{E}_{\\mathbf{z}}(f) = \\frac{1}{m} \\sum_{i=1}^{m} \\left(f(x_{i}) - y_{i}\\right)^{2}.\n", "6acaa1394fd7fdfd0cbee9d558d1be30": "\n\\gamma^{\\prime} = \\lambda \\gamma\n", "6acaf5896be6d0b018462641b0b9074d": "D_{t+1} \\,", "6acb028b6823fa4fac988e6a10507967": " n = P_{recon}{SG_{recon} \\over SG_{beer}}", "6acb3796df278d74f8c0bc1170bef07f": "O\\left[m + n + \\sum_{j=1}^m \\log(|i_{j + 1} - i_j| + 1)\\right]", "6acbbb7b67aa2605d272a0af21113718": "H_\\text{R}", "6acc3cb651f90b4b3fb78610926b7967": "\\tfrac{7}{3}", "6acc3eede1c9915f6c669f134319dc54": "y = Y/Z^2", "6accc04dae96d833eacfe23affb6745c": "(1+\\sqrt2)^n=H_n+P_n\\sqrt2 ", "6acce1dee368fa1c5a67a7fd4f32c329": "Y = \\lim_{\\nu_2 \\to \\infty} X\\,", "6accf2b120992f3d99956131ced1ae65": "[L_z, L_x] = i \\hbar L_y", "6acd97427e7d91f0ad247a53517137f2": " C_{\\mu} = 0.09 ", "6acdc7c05b11680d6a3e325c9269d621": "(l)=(1-\\zeta_l)^{l-1},", "6acdeb8958b2724dc3519fa7bb17de28": "r v^2", "6ace068afd8e9008709132e1b437c009": "\\phi(x,1)", "6acec122d1ec97f40c3745a6904eaedf": "\na \\le u(t) \\le b \\quad t \\in [0,T] \n", "6acedd1417f694444c63173a950d6af3": "\\alpha^{\\frac{\\mathrm{N} \\pi - 1}{3}}\\equiv \\omega^k \\pmod{\\pi}", "6acf7bd02475b8666403657049cb6520": "\n\\Delta \\vartheta(\\vec r) = -\\tan \\vartheta_B \\frac{\\Delta d}{d}(\\vec r) \\pm \\Delta \\varphi(\\vec r)\n", "6acf82ecd887881f65909d9eed3f063b": "x^3\\frac{\\mathrm{d}f}{\\mathrm{d}x}+2f=0.", "6acfaf032acd36699f9c577e26369ff5": "\\Gamma^a_{bc}=\\Gamma^a_{cb}", "6acfc83fd831e5d29d0e00c732bc37ba": "\n\\hat{l}^2 Y_{lm}(\\theta,\\phi)\\equiv \\left\\{ -\\frac{1}{\\sin^2\\theta} \\left[\n\\sin\\theta\\frac{\\partial}{\\partial\\theta} \\Big(\\sin\\theta\\frac{\\partial}{\\partial\\theta}\\Big)\n+\\frac{\\partial^2}{\\partial \\phi^2}\\right]\\right\\} Y_{lm}(\\theta,\\phi) \n= l(l+1)Y_{lm}(\\theta,\\phi).", "6ad0749f79e75e4b2da39b2bd6cdfa15": "\\operatorname{curl}\\,\\mathbf{F} = \\nabla \\times \\mathbf{F} = \\left(\\frac{\\partial F_3}{\\partial y}- \\frac{\\partial F_2}{\\partial z}\\right)\\mathbf{e}_1 - \\left(\\frac{\\partial F_3}{\\partial x}- \\frac{\\partial F_1}{\\partial z}\\right)\\mathbf{e}_2 + \\left(\\frac{\\partial F_2}{\\partial x}- \\frac{\\partial F_1}{\\partial y}\\right)\\mathbf{e}_3.", "6ad0ac40b8db864e6314f382799c8299": "\\mathcal{GW}^{-1}(\\mathbf{\\Psi},\\nu,\\mathbf{S})", "6ad10610ed10fa4a93d48b383d1a753c": "\n\\Delta_{\\psi} \\hat{A} \\, \\Delta_{\\psi} \\hat{B} \\ge \\frac{1}{2} \\left|\\left\\langle\\left[{\\hat{A}},{\\hat{B}}\\right]\\right\\rangle_\\psi\\right|\n", "6ad115ac5315312700f48bdecaa8f5d2": "r' = x' - y'^n = B^n x + \\alpha - (B y + \\beta)^n", "6ad1a8771c3d01be55d1f4d46aa9f4e9": " \\tan \\theta =\\frac{b}{a} \\ , ", "6ad1cb0f7e09115bbe822ea396ca4267": "(0,1,a)", "6ad1d4e1bd357493dc965ec60dab5171": " k = {y \\over x}.", "6ad2ff4e5b2fd3e05cca2417322ed661": "T_\\text{A} = \\frac{m_\\text{e} e^4}{\\hbar^2 (4 \\pi \\epsilon_0)^2 k_\\text{B}}", "6ad34d83eafb5cae524587c4689b7a3b": "V_{\\text{BE}}", "6ad39ddb9687380f87fb3251af29442a": "\\lim_{n \\rarr \\infty} p_{ij}^{(n)} = \\frac{C}{M_j}.", "6ad4e946a52724ffecdca383e692484d": "\\|\\mathbf{v}\\| = 0 \\iff \\mathbf{v} = 0", "6ad504efb1b06f41159c63cc51b2d909": "t^{\\prime} = \\gamma \\left(t - \\frac{v x}{c_0^{2}}\\right)", "6ad537323ff7c9814cef80e13bae392e": "~\\rho = \\sum_j \\eta_j |j\\rangle\\langle j|", "6ad59557a6a058b187563b3097f509e6": "\\eta_c = \\pi a b N / L^2", "6ad5c869270583e74702774c699150a0": "A(S)", "6ad647d784b2eb0b2066ef9414e8a314": "\\frac{\\partial \\mathcal{A}}{\\partial t}\\, +\\, \\nabla\\cdot\\left[ \\left(\\boldsymbol{U}+\\boldsymbol{c}_g\\right)\\, \\mathcal{A}\\right]\\, =\\, 0,", "6ad68d1e9e2b69c18e634efe8a7808b1": "S_n(r)", "6ad6a865978438b16273923ae7723946": "\\sum_n \\dot{c_n}\\psi_n e^{i\\theta_n} = - \\sum_n c_n\\dot{\\psi_n}e^{i\\theta_n}", "6ad6ad19ed6897fd82692eb9ffdc920d": "M_R=\\max_{\\theta\\in[0,\\pi]}\\frac1{|1+R^2e^{2i\\theta}|}=\\frac1{R^2-1}\\,,", "6ad6b21aa528a750e8b4db33a0e6a154": " \\mathbf{r}_i = 0 ", "6ad6cc9e275b47761c71d3024f7822ff": "L (ESL)", "6ad6daf4db2f9f3de3decd10947cd9de": "E(k) = \\frac{\\pi}{2} \\sum_{n=0}^{\\infty} \\left[\\frac{(2n)!}{2^{2 n} (n!)^2}\\right]^2 \\frac{k^{2n}}{1-2 n},", "6ad7362fc4d52bf842e604ce1131d0f8": "\\mathrm{A}_n \\to \\mathrm{C}_3,", "6ad77dac562ee6afbc8badc3fc6bb60f": " L(a) = \\log(\\cosh(a)) ", "6ad7a117939e685f0f351a7d5d8efe6d": "{\\widehat{AH}}_3", "6ad7a8a3846ad49a130524807c0bef13": "\\omega_{0} \\in \\Omega", "6ad81554661fac459cb2a7480c3440f0": " \\mathsf{R}=(\\mathbf{F}-\\mathbf{F}, \\mathbf{A}\\times\\mathbf{F} - \\mathbf{B}\\times\\mathbf{F}) = (0, (\\mathbf{A}-\\mathbf{B})\\times\\mathbf{F}).", "6ad81671f4705aae5a743c7f88f92ed1": "\\scriptstyle {Z_0}^2=L/C", "6ad866ebb15291aeaf4dcf7863461354": "\\langle \\vec R \\rangle", "6ad87015d0726a0a03d77f2c6466a89d": "W(\\mathbf{Q}) \\cong \\mathbf{Z} \\oplus \\mathbf{Z}/2 \\oplus \\prod_{p\\ne2} W(\\mathbf{F}_p) \\ ", "6ad87263ab31975296189af26a20f84a": "\\rho_\\text{sample}\\,", "6ad877d4894242261b2a3393ef6e1cb8": "p=x^2 - 4xy + 7y^2", "6ad878be1a0f7653b8bada7611a14577": "p = -\\frac{\\partial U}{\\partial V},", "6ad8afa634fbd19c4a7e49bf82369415": "\\dim_\\mathbb{C}M=a(M)=\\operatorname{tr.deg.}_\\mathbb{C}\\mathbb{C}(M).", "6ad8b2e676525b2c8029f4f76bf12262": " T(h,0) = 0 ", "6ad8feccf51c135c102ba0bfe4142a73": " (2ax + b)^2 = b^2 - 4ac ", "6ad90b4dcf8d960c5dabfe4aecb19100": "\\int_{}^{}", "6ad970f70e906c85c04b089f607abe76": "\\frac{\\partial}{\\partial t} \\psi(x, p) = \\left[- \\frac{\\partial H(x, p)}{\\partial p} \\frac{\\partial}{\\partial x} + \\frac{\\partial H(x, p)}{\\partial x} \\frac{\\partial}{\\partial p} \\right] \\psi(x, p),", "6ad9960b1c77224f2a2cd76a41a2999d": "\\textstyle p(x,y) = p(t_y, b_{-1}, b_{0}, b_{1}, b_{2})", "6ad99d6a128a69531882702dfa314f88": "G = k \\cdot g", "6ad9a48daaee312cd75a6a564f36d4d8": " i\\ ", "6ada575ba55d1e07c4a4cc1b73cce6a8": "{\\rm ad} (x+y)={\\rm ad} (x)+{\\rm ad} (y)", "6adabb0d38fff718475c18a746603685": "a \\,a^\\dagger\\,,", "6adb0ec2e51b2130b35c4d850ae6efdc": "\\scriptstyle P_{sc}", "6adb2d4c5388a9d8f09b208c021c0126": "1_A(h)", "6adb7981a6868e92e0f76494aaafd035": " \\frac{F_p}{F_s} = \\frac{M_p \\cdot d_s^2}{M_s \\cdot d_p^2} ", "6adc10c6b69fd049c5ea9ccb98c4046f": "X^p - X + \\alpha,\\,", "6adc1514b336ab8ffba38b4f29213f65": "n = \\frac{\\sqrt{8x+1}+1}{4}.", "6adc261115ea2f2bce98382d371334c3": "\n\\sigma_x =\n\\begin{pmatrix}\n0 & 1\\\\\n1 & 0\n\\end{pmatrix}\n,\\quad\n\\sigma_y =\n\\begin{pmatrix}\n0 & -i\\\\\ni & 0\n\\end{pmatrix}\n,\\quad\n\\sigma_z =\n\\begin{pmatrix}\n1 & 0\\\\\n0 & -1\n\\end{pmatrix}.\n", "6adc6e5ee8a23d36a3d9254ff0aece8b": " \n(s_i', t_{ei}')= \n\\begin{cases}\n(\\delta_{int}(s_i),0) & \\text{if } i = i^*\\\\\n(\\delta_{ext}(s_i, t_{ei}, x_i),0) & \\text{if } (\\lambda_{i^*}(s_{i^*}), x_i) \\in C_{yx}\\\\\n(s_i, t_{ei}) & \\text{otherwise}.\n\\end{cases}\n", "6adc95c4dd84f47a2a516466000c436d": "e \\,\\! ", "6adcc56a7049fa406c037b66c1ff1057": "E_\\text{P} = {m_\\text{P}} {c^2},", "6adcc77708cae62ac45f2c5d976eadbe": "ST_{i}", "6addb4231b6b95ba2827813909131f1d": "2x+5=3", "6addcc5cc8598c59c39ececbe90d17f7": "Q(x) = x^{12} - x^7 - x^6 - x^5 + 1,", "6adddd603e6243977602f94b35552745": " \\sum_{v \\in V(G)} \\|f(v)^2\\| < \\infty ", "6addf9901f1672752851df961d17baa6": "\nf(\\mathbf{x}) = \\sum_i k(\\mathbf{x}_i,\\mathbf{x})c_i\n", "6ade0542adbc066617ab2a173c896ef8": "\\left\\langle\\nabla^2\\left(\\frac{-e^2}{4\\pi\\epsilon_0r}\\right)\\right\\rangle_{at}=\\frac{e^2}{\\epsilon_0}|\\psi_{2S}(0)|^2=\\frac{e^2}{8\\pi\\epsilon_0a_0^3}", "6ade1aba1c50e998578952a5411522e9": "\\textstyle r>-1", "6ade59a87494de848d46573e3680f337": "\\alpha_t(x_t) = p(y_t|x_t)\\sum_{x_{t-1}}p(x_t|x_{t-1})\\alpha_{t-1}(x_{t-1})", "6aded55368ed9de699ed7a52915aafd4": "f(x) = - \\int_{D} A f (y) \\, G(x, \\mathrm{d} y).", "6adef84e8e8bcce4695e893b3a6b5fcc": "x = \\sqrt{z^2 - r^2}", "6adefd17b19f73695dca21ac8466759b": "\\mathbf{r} = \\mathbf{r}_0\\ f(s) + \\mathbf{v}_0\\ g(s)", "6adf159b822d947c891ba56c011a9cf9": "\\exp\\left\\{-\\exp\\left[\\left(-\\frac{x-\\mu}{\\sigma}\\right)\\right]\\right\\}", "6adf708c8e2aaaf24e50c3315e36e258": "\n\\mathbf{F}(r)= -k \\mathbf{r} ,\n", "6adfdd6a98dcc8af443fb229741b4799": "V(\\lambda)\\,", "6adfdff47cf01a19ffb706730289307b": " L u = -\\frac{\\mathrm{d}^2u}{\\mathrm{d}x^2} = \\lambda u", "6ae03b813fe88375dd7aa0266d454a44": "A_{21}=\\Gamma_{21}=\\frac{1}{\\tau_{21}}.", "6ae04aad481ea23a8025db0e0d66965c": "\\tilde{Fr_1} = \\frac{1}{\\sqrt{y''^3}}", "6ae07946b76111aea57411bc92a55005": "P_n=\\begin{cases}0&\\mbox{if }n=0;\\\\H_{n-1}+P_{n-1}&\\mbox{otherwise.}\\end{cases}", "6ae1706da784b8488b1772dc28d2c422": " \\textbf{W} \\in \\mathbb{Z}^{n \\times n}_{q}", "6ae1b77d49933e4a41b32877adedb4f4": "x_0 \\in (a, b)", "6ae223298baff71dd6f0bbb31685c7a1": "\\mathcal{P}\\rho \\equiv {{\\rho }_\\mathrm{rel}},", "6ae2282e42f99a0a24e3b80484391933": "A_t=\\inf\\{s>0:W^{(\\gamma)}(s)=\\delta t\\}.", "6ae23c8281eb4365e7fb1998ddc3d663": "p(x|\\nu,\\mu,\\sigma^2) = \\frac{\\Gamma(\\frac{\\nu + 1}{2})}{\\Gamma(\\frac{\\nu}{2})\\sqrt{\\pi\\nu\\sigma^2}} \\left(1+\\frac{1}{\\nu}\\frac{(x-\\mu)^2}{\\sigma^2}\\right)^{-\\frac{\\nu+1}{2}} ", "6ae260100a3ead6746c178240f45de2f": "\\displaystyle \\cap^{\\infty}_{n=1} U_n", "6ae284d5e6067082b94a351a33d94b43": "\\left(\\rho_s=2650 \\frac{kg}{m^3} \\right)", "6ae2cd7a61fe50bb7e51b47591970186": " K=-{1\\over 2H} \\left[\\partial_x(G_x/H) +\\partial_y(E_y/H)\\right].", "6ae2dfb066867668e72b2cc0e38e479b": "\\qquad \\dot{Y}=g(\\bar{x}(Y),Y)=:G(Y).", "6ae2f8f5e6ad77cacfadde8a2d351008": "\\displaystyle R=\\frac{1}{4}\\sqrt{\\frac{(ab+cd)(ac+bd)(ad+bc)}{abcd}}.", "6ae2fbed3df5a9585245347c21da0a44": "f^2 x = f f\\, x = f(f(x))", "6ae2ffb57770e5ce69560b314c1c6a25": "\\sigma(L^1,L^\\infty)", "6ae34df70fece53040fa431f8a6ef43a": "D_b^{D_a}", "6ae381115a1f917c2738b8de2e18f276": "\n \\mu \\Delta\\Delta w + \\hat{q} + \\rho w_{tt}= 0\\,.\n ", "6ae3d8b44baea0e3f5598c8e6ee04992": " \\Delta \\Phi = \\gamma v_x \\Delta m_1", "6ae4068d689bad7d90d165d09ccea49a": "k \\neq 0", "6ae4a8dc20ef36c0fefb6da53d3caff5": "F_n(x)", "6ae4afa4606957adaffa8746e40a1eb6": "\\exists p: \\mathcal{B}p \\wedge \\mathcal{B\\neg B}p", "6ae530c11f67a8249a312e91ff8fa19d": "\\Delta H^*", "6ae531b0bfbb996cdb262a89554992b7": "q = 10.\\frac{ft^2}{s}", "6ae5436fd5e00309f40dd7ec3f75e179": "h = \\theta_L - \\alpha ", "6ae558362de6b60e3f5a033f16555341": " \\, \\propto ", "6ae584d58474fa2b4b2ed92098d4363d": "T = \\frac{2\\pi}{M(1, \\cos(\\theta_0/2))} \\sqrt\\frac{\\ell}{g}.", "6ae5857636ee9459180bd1f137e93faa": "8_8", "6ae589fe4468015da15b3406f9e112ee": "\\Sigma = G_0^{-1} - G^{-1}.", "6ae59e6a3b6fda3a95a62b7b754d5b34": "\\mathrm{CINT}_x(u_{-1}, u_0, u_1, u_2) = \\mathbf{v}_x \\cdot \\left( u_{-1}, u_0, u_1, u_2 \\right)", "6ae5dea3ac5c9f9210a6cbce84e367f6": "\\forall a \\in A \\, ( f(a) \\in \\varphi(a) ) \\,.", "6ae64c6c3c3837080278c5a5c7dd97fe": "\\log P(x, \\hat{\\pi}|\\theta)", "6ae65468d7f798c495ade58b40ac1b42": "\\mathbf F = \\int_V \\mathbf a\\,dm = \\int_V \\mathbf a\\rho\\,dV = \\int_S \\mathbf{t} dS + \\int_V \\mathbf b\\rho\\,dV", "6ae68dfc8d0723f1730eb2d53604aeac": "f(x) \\sim \\sum_{n=0}^\\infty c_n\\varphi_n(x),", "6ae6a21e0bd9deedf34b881392efcd68": " p = \\frac{RT}{V_\\mathrm{m}-b}.", "6ae6cc954391ff96673d8e08ca8b53be": "y\\in A_i\\cap C_j", "6ae71afab14dd84f4f48d714e03ddc72": " p_u = \\begin{cases}\n\\dfrac {f_u} {\\sum_{i=1}^{20}f_i \\, + \\, w\\sum_{k=1}^{\\lambda} \\tau_k}, & (1 \\le u \\le 20)\n\\\\[10pt]\n\\dfrac {w \\tau_{u-20}} {\\sum_{i=1}^{20} f_i \\, + \\, w\\sum_{k=1}^{\\lambda} \\tau_k}, & (20+1 \\le u \\le 20+\\lambda)\n\\end{cases}\n\\qquad \\text{(4)}\n", "6ae754b5b871ff18a18de95f08542964": "\\int_{\\theta=0}^{\\frac{\\pi}{2}} \\int_{x=0}^{m(\\theta)} \\frac{4}{t\\pi}\\,dx\\,d\\theta ,", "6ae787bde8f16f207b980a80e99115e6": "x_1, x_2, x_3", "6ae7aa7d4a7520d29d3f94ec9cf0f8ef": "KR_x", "6ae7c1666f4a2c27c5cf9d437c8c54ea": "|\\boldsymbol{F}|=k_e{|q_1q_2|\\over r^2}", "6ae7e6aaa642dbe986d52e8898204b32": "\\sqrt{2}k", "6ae802575d250515f54e2ae4586a9b02": "R_{(1)}^t \\leq R_{(2)}^t \\leq \\dots \\leq R_{(i)}^t \\leq \\dots \\leq R_{(n)}^t. ", "6ae8266b6546571e608dac91f87864fb": "A \\Rightarrow^{ac}_{p_1} w_1 \\Rightarrow^{ac}_{p_2} w_2 \\Rightarrow^{ac}_{p_3} ... \\Rightarrow^{ac}_{p_n} w", "6ae83b7185f2a85cf1403367012ea2f3": "\\beta_n^{ }(b) = c", "6ae87bcf630a9f789d3e5f8af9a1ddc9": "\\left(-2\\sqrt{\\frac{2}{5}},\\ 0,\\ \\sqrt{3},\\ \\pm3\\right)", "6ae8f5fc3244ccbe3212af48fb7e40a3": "\\cos \\varphi = \\textrm{cn} \\; u, \\qquad \\textrm{and} \\qquad \\sqrt{1 - m \\sin^2 \\varphi} = \\textrm{dn}\\; u.", "6ae8fd5dbc418e768498abab19ee447e": "\\theta _R=\\frac{hc \\bar B}{k_{B}}=\\frac{\\hbar ^{2}}{2k_{B}I}", "6ae92d9dfdeb6e90a069430b2aae270b": "(S, R, V)", "6ae9412e3a3a252a0b7c66e971e9af43": "\nF(r) = \\frac{A}{r^{2}} + B r\n", "6ae9575d563a5bb6721259b689ea2180": "\\sum_{n=1}^{\\infty}\\frac{s_{n}}{n},\\!", "6ae9992e14b22d4a47c1fa37240476a5": "2 \\pi/(\\Delta \\omega_{l} - \\Delta \\omega_{j})", "6ae9b301812e3f20101aef258e715158": "n_1=n_3=0\\,\\!", "6ae9f055a7e045240964312dab411481": "(X_1 \\or Y_1) \\and (X_2 \\or Y_2) \\and \\dots \\and (X_n \\or Y_n)", "6aea2f4056c888899c8cb9c804455b1a": "\\displaystyle\\Box A = m^2_{} A + |u|^2 ", "6aea4c3e9ceb8a49bbd63c0facb18dc0": "\\Phi_w =(\\Phi_W + \\Phi_P)/2", "6aea7a9e2798c74061a8045654e7cd52": "\\mathcal{B} = M(G)", "6aeaca7de6896bd59103e8f8b5ce9eb4": "P \\propto \\cos\\left[\\left(\\Delta + \\frac{2\\pi\\Omega d}{\\lambda}+\\phi\\right)\\frac{2d}{v}\\right]", "6aeaddaace97bd88e065916c36f3b66d": "\nf(r; \\alpha,\\beta)=2r \\frac{\\beta-1}{\\alpha^2} \\left[1+\\left(\\frac{r^2}{\\alpha^2}\\right)\\right]^{-\\beta} . \\,\n", "6aeaf08d211b10d50beaaf11de42481c": "\nH(x,\\lambda,u,t)=\\lambda^T(t)f(x,u,t)+L(x,u,t) \\,\n", "6aeb36f8936f9cd744b77b6089e083b2": " \\mathcal{S} ", "6aeb47ce57a9ac28da4f1753c310ff8a": " \\frac{n+1}{n}T(X). ", "6aeb5f4250c0eb6d10022900717b087e": " Z(S_n) = \\frac{B_n(0!\\,a_1, 1!\\,a_2, \\dots, (n-1)!\\,a_n)}{n!}.", "6aeb8bfec5d8ec7e3653b27cca42a153": "20^\\circ", "6aeb98a2d1f5d5292e8ab85ed16321b6": "\\phi(t)=c_1 e^{\\lambda_1 t} + c_2 e^{\\lambda_2 t}", "6aeba896cbf60ebf5f19b12a94392e12": "\\sum_{\\stackrel{p_{n+1}-p_n>x^{1/2}}{x\\le p_n\\le 2x}}p_{n+1}-p_n\\ll x^{2/3}.", "6aec1ee96aaf60227fab1feb070503b2": "\\overline{B}(x_n, r_n) \\subset B(x_{n - 1}, r_{n - 1}) \\cap U_n", "6aec23b7dfc6291e9219340ea9db4ea2": "\\epsilon_{\\mbox{creep}} /\n\\epsilon_{\\mbox{initial}}", "6aec4231030d289b56bf839069f3930d": "X(M) \\to X(N),", "6aec4da2793b46b91df58c3d006c636f": "n\\equiv m", "6aec54de94a62bd61376c3451594898a": "\\mathrm{d}\\omega_j^i = -\\omega^i_k\\wedge\\omega^k_j.", "6aec6482c5d7177a25eb180cb94b6532": "\\nu = 1/n", "6aec8ff9cd9c51ff8f46fc01bf8b11af": "S=\\mathbf{r^T Q Q^Tr = r^Tr}", "6aec9b02554a60bd0cb191a0b909c4c4": "\\eta_2 = \\frac{H}{m}\\, \\left( 1 - m - \\frac{E(m)}{K(m)} \\right).", "6aecc7955c1d0dcbdcf4d3dad235ab83": "\\mathit{p}_i \\neq \\mathit{q}_i", "6aece3501598717744a0e7702fa12b46": " \\hat{g}_c (s) = \\frac{(2/c)^{1/3}}{\\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \\ \\ \\ s \\in \\mathbf{R} ", "6aecf205179ac9b596002cc8cf0bebee": "\\int_{-\\infty}^a", "6aecf9fe4b77f19af0caa894250b2a04": " \\lim_n \\int g_n \\, d \\mu = \\int f \\, d \\mu. ", "6aed00855831922f1c6e0a62bc4a1365": "c_9 = 1.02102 \\times 10^{-5},\\,\\!", "6aed4dae738d5a8842031bed78258faa": "\\left\\{F\\!\\left(x\\right) : x \\in A\\right\\}", "6aed6ae2f8d82fff69061a2de99dd2ad": "(\\Omega, \\mathcal F, \\mathfrak P)", "6aed8442c87b91af41503f5fb956b01f": "p_\\alpha = ( E/c, - \\bold{p}) = mu_{\\alpha} \\,", "6aedbb7707786c6416af01943e706b5a": "\\left(\\frac{\\partial F}{\\partial V}\\right)_{T,\\{N_i\\}}=-p", "6aeddd4a95d9eb5d3c0c4e9d042b756f": "\\oint\ndx\\sqrt{2m\\left( E-\\varphi \\left( x,V\\right) \\right) }", "6aede2f38d83524aac203e260445e431": "N_k \\,", "6aee76374ccf5a7741692442168be6a1": "\n\\frac{d \\theta_i}{d t} = \\omega_{i}+\\zeta_{i}+\\dfrac{K}{N}\\sum_{j=1}^N\\sin(\\theta_{j}-\\theta_{i})\n", "6aee8b79a8d967d15c1ec83bbf6eacfc": " b_0=\\frac{4 \\beta_2-3 \\beta_1}{10 \\beta_2 -12\\beta_1 -18} \\mu_2 ,", "6aeea33026423e76890b44e5a446c71e": "[\\operatorname{Spec}L/G]", "6aeed7fc552494db5c6cac6dd2f4a224": "\\sigma(T) = \\sigma_{\\mathrm{ac}}(T) \\cup \\sigma_{\\mathrm{sc}}(T) \\cup {\\bar \\sigma_{\\mathrm{pp}}(T)}.", "6aef1b1a4f9e0031ac9bce16aa85bd44": "{7140 \\choose 1} = {120 \\choose 2} = {36 \\choose 3}", "6aefae9b2d10418f7dd9d913f7663689": "\\sum\\limits_{i=1}^\\infty \\|a_i\\|<\\infty", "6aefc3b5a72cbf0d321e33a846194fc0": " \\tau(w) ", "6af00848b4699bdec21aac5b65d4fe08": "\\boldsymbol{\\Upsilon}_s = \\left(\\phi(x_s^1),\\dots, \\phi(x_s^n)\\right)", "6af0bf1879e3eb345dc94c45643f6756": "\\begin{pmatrix}\n1 & 1 \\\\\n0 & 1 \\\\\n\\end{pmatrix}", "6af0c125f3ddb6d19ada109d1801c2dd": "\\mathbf{Ax}\\leq\\mathbf{b}, \\mathbf{x}\\geq 0", "6af10378b02e7d8f7bc40bfdb3062a85": "\\bar{y} = E\\{y\\},", "6af10c847ff2fc8c121ec7dce6dd5539": "-\\frac{d}{d x}\\left(\\ln\\left(\\frac{d t}{d x}\\right)\\right) = f(x)", "6af14c449345f1660f887beb4ab7a1c2": "\\det(M(\\mu,\\sigma))", "6af1be4c0f601582526d9a7b48e35f82": "\\varrho_{A, B}= trace_\\Lambda(\\varrho_{A, B, \\Lambda})", "6af276bc15e6578fe10df861dcbbc049": "\\gamma\\left( \\omega \\right) ", "6af276ca6275a79c759281fdd379d2e7": "u =\\,", "6af28b77ebdf5f096be6cc253bffd337": "R^{**}_{S_{b_j,\\beta_j}}(t)= \\frac{{^{b_j}_{a_j}}S^{\\beta_j}_j (t)}{^0_{a_j}S_j(t)} ", "6af2a2526f99c54cad283b9162d71bf1": " W=\\oint \\mathbf{F}\\cdot d\\mathbf{r}=0. ", "6af2d5f71fbe63e3f51bfe383e2d1645": "S_{b/\\$}", "6af3351d4c2c30c242cf51fd95befe63": "n{K\\over N}{(N-K)\\over N}{N-n\\over N-1}", "6af35ab12d0fb0ace30f5a0a4c70f157": "m_{od}", "6af360b2a808cf39dbca3e2aa3f4861d": " \\Vert \\mathbf{s} \\Vert = \\sqrt{s \\, (s+1)} \\, \\hbar", "6af374df6ecb8854ad7283087d509e60": "\\sigma_1,\\sigma_2\\in T_\\mu S", "6af3afad0b1b5714bc4faa10c0a18312": "\\neg,\\or,\\and,\\rightarrow", "6af3d5c6c82d1b3af1b99d415d15f9cb": "\n\\ x(t) = \\sum_{n=-\\infty}^{+\\infty} \\sqcap(t - nT) = \\sum_{n=-\\infty}^{+\\infty} \\left( u \\left[t - nT + {1 \\over 2} \\right] - u \\left[t - nT - {1 \\over 2} \\right] \\right) \n", "6af408ab39a99c5073957a9610c4be09": "R = \\sqrt{\\frac{a^2b^2c^2}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}.", "6af4628ff1af2d08a879e20e71bbb6c9": "\\aleph_0 < \\mathfrak c.", "6af4780331993f2a7d4f18de3cb8179d": "10^{10^{76}}", "6af492786786b98878e24c525ceeb280": "G \\triangleright SL(a,q)\\text{ and }p^d=q^a;", "6af4b1dd2a14e511577a4f23e474f4be": "{\\rm tr}(\\mathbf{A}\\mathbf{g}'(x\\mathbf{A}))", "6af52181a42cdbd68a46588b5c74edd9": "m_e \\approx \\frac{1}{3} m_d", "6af52fda27fe621b239fcb83750acff0": "g = \\begin{pmatrix} A & B \\\\ C & D\\end{pmatrix}", "6af5c8b31daaf768a44b534437b2bdf4": " -\\frac{\\mbox{total runs conceded in match 1 + total runs conceded in match 2 + ...}}{\\mbox{total overs bowled in match 1 + total overs bowled in match 2 + ...}} ", "6af60a46ece3dc87ef524c8cfc0769d1": "\nA(x,y)=\\langle \\psi|x,y\\rangle = \\langle \\psi | ( |x\\rangle \\otimes |y\\rangle )\n", "6af651f451eafdff42f7fc17ecddb252": " T_{00} \\rightarrow u^{\\alpha} \\left ( A T_{\\alpha \\beta} + B T \\eta_{\\alpha \\beta} \\right ) u^{\\beta} ", "6af6a06270c8fe92fc046cff9f84df9b": " \\lambda_X^{-1} \\mathcal F \\otimes_{\\lambda_X^{-1} \\mathcal O_X} \\mathcal O_X^\\mathrm{an} ", "6af6a7155b2b734a595f3c99cbd36a70": "\\exists x\\, \\lnot q(x,G(P))", "6af707477aff6767e0d0172d6b4362bf": "q'_i(x_i) = q'_{i+1}(x_i)", "6af750225984288f8215575b98f553e1": "21\\cdot 2^k-24k-24", "6af75b7d0fd9abdbf6055bf18924f1a1": "\\Phi \\left(\\eta,\\tau \\right) = \\exp \\left(-i 2\\pi \\frac{\\eta \\tau}{2} \\right), \\, ", "6af7688b8a3dbcec096c5514a53969a5": "\\int_0^\\infty e^{-a x^2}\\,dx = \\frac{1}{2} \\sqrt \\frac {\\pi} {a} ", "6af7fdba4c06b8c0f466585d3234150f": "L = 2r", "6af821e0cd1eff3afbc54e98daf8dd59": "\\arccos\\mathrm{sech}\\,x = \\vert\\mathrm{gd}\\,x\\vert = \\arcsec(\\cosh x)", "6af82dfa58ebee9ec63b1d2d064eda28": "-282\\pm 30", "6af858b46558902a6e952a288e65036b": "\nr_{xx} (\\tau) = \\int_{-\\infty}^\\infty S_{xx}(f) e^{2\\pi i\\tau f} df.\n", "6af8a0624df121821f8635b028d29443": "x(t)=\\frac{K}{t_c-t}", "6af8bcb7d778a0a1eb3032e2080af3bc": "\\, L^k X_{t} = X_{t-k}.\\,", "6af8beb8cad506e259bf32b8a7d9c383": "\\| \\mathbf y - X\\hat{\\boldsymbol{\\beta}} \\|", "6af8d6432d1c15fd907e171fb52542c2": "P(X=a) = 1\\Leftrightarrow \\operatorname{Var}(X)= 0.", "6af978bf56f84e2e8eab1291eecaafa1": "I_z = m r^2\\!", "6af99813afdae7cbd20e3edf3f10ce6c": "t=f(x_1,x_2)", "6af9c372f82064a747914fd2c133863e": "g\\left(\\lambda(x)\\right)\\lambda'(x)", "6afa571ec2de11ec97b305bd23b989b6": "L = \\sum_{n=0}^{\\infty} a_n \\Leftrightarrow L = \\lim_{k \\rightarrow \\infty} S_k.", "6afa7324ee3b6fa6420b1e930bd908eb": "L=N-M+1", "6afa824d291d89e047e584c77674bcd4": "q = 12 * 8", "6afac1788b3c6b1648846e838471a521": "H_0 = 0.", "6afae006e6625c83e9361c13c1d2193d": "a = b = c = d, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "6afb872feae02d9d9c06aa174df73b45": "\n \\begin{bmatrix}\n x_{ecliptic} \\\\\n y_{ecliptic} \\\\\n z_{ecliptic} \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos \\epsilon & \\sin \\epsilon \\\\\n 0 & -\\sin \\epsilon & \\cos \\epsilon \\\\\n \\end{bmatrix} \\! \\cdot \\! \n \\begin{bmatrix}\n x_{equatorial} \\\\\n y_{equatorial} \\\\\n z_{equatorial} \\\\\n \\end{bmatrix}\n", "6afb8cbcebadddb05af122aa8a72e80d": " p_{0,3}(x) \\, ", "6afbda56f33cfb218c7f267b50ef52e8": "e_{i_1}e_{i_2}\\cdots e_{i_k} \\mapsto e_{i_1}\\wedge e_{i_2}\\wedge \\cdots \\wedge e_{i_k}.", "6afbe29fef14b6f8e6c17b8e08bc3b1e": "\\mathbf{v}(t) \\sim N(0,\\mathbf R)", "6afbe83a47ca8682aa961a1784eec1f2": "\\Re j", "6afbf7c91775753eb114f9222ba92c0c": " i = 1, \\ldots, n", "6afc736f1a220152f144806de3aba4a5": "I(z)", "6afc95a6b15ac43fde9c8f38ddace033": "G_{ab}", "6afc9938521bed4e806b944a5602aab9": "{\\tau}= arg \\max\\{csp_{ij}(k)\\}\n", "6afca11940a341774ac40d692fa9bda5": "\\alpha^{{\\rm N}(\\mathfrak p)} \\equiv \\alpha \\bmod {\\mathfrak p}", "6afcb76d2d6809dc5fd168a82b23ec02": " S = \\int_1^2 \\mathcal{L}\\, d\\Omega ", "6afced8082428bee150e61c8fad76110": "^\\breve{\\ }", "6afcf9dfe4e911d604c4abe8cc800d6e": "\\sqrt{x_1^2 + x_2^2 + \\cdots + x_n^2} = x_1 \\oplus x_2 \\oplus \\cdots \\oplus x_n", "6afd16d444e49b0bd83444dcdaf2cb81": "\\tfrac{2G(1-\\nu)}{1-2\\nu} ", "6afd29080171fa1ec952259d389e9c20": "b_{ij}=-\\frac{w_{ij}\\delta_{ij}}{d_{ij}(Z)}", "6afd92c583aacca8b412557492474a5b": "r_0(\\theta)", "6afe0bb1625ff3cd8fa66123ee84e50b": "\\begin{align}\n g\\colon \\mathbb{Z} &\\to \\mathbb{Z} \\\\\n x &\\mapsto 4-x.\n\\end{align}", "6afe51c6574ec6474558099622d33a58": "w = \\frac{\\mathrm{d}W}{\\mathrm{d}V},", "6afe910d4948196b88dc88692f124535": "\\operatorname{pd}_R M \\ge \\operatorname{pd}_{R_1} (M \\otimes R_1)", "6afe912e65e11922fdfd989a83e36de7": "\\sum_i m_i \\mathbf{V}_i=0\\,", "6afe914c253fc42b6ab9d84acf2c873b": "\\begin{align}\n\\Pr\\left(\\bigcap_{n=N}^{\\infty}E_n^{c}\\right) \n&= \\prod^{\\infty}_{n=N}\\Pr\\left(E_n^{c}\\right) \\\\\n&= \\prod^{\\infty}_{n=N}\\left(1-\\Pr\\left(E_n\\right)\\right) \\\\\n&\\leq \\prod^{\\infty}_{n=N}\\left(1-\\Pr(E_n)+\\frac{(\\Pr(E_n))^{2}}{2!}-\\frac{(\\Pr(E_n))^{3}}{3!}+\\cdots\\right) \\\\\n& = \\prod^{\\infty}_{n=N}\\left(\\sum^{\\infty}_{m=0}\\frac{(-\\Pr(E_n))^{m}}{m!}\\right) \\\\\n&=\\prod^{\\infty}_{n=N}\\exp\\left(-\\Pr\\left(E_n\\right)\\right)\\\\\n&=\\exp\\left(-\\sum^{\\infty}_{n=N}\\Pr(E_n)\\right)\\\\\n&= 0.\n\\end{align}\n", "6aff2873c3ed304c033163d9e0cdb82f": "\\displaystyle{g(t)= f(e^{Y/2} e^{tX} e^{Y/2})= \\sum e^{\\mu_i t} \\|Ad(e^{Y/2})f_i\\|^2_\\sigma.}", "6aff2ab4faf1d65142ba874dabd12ef7": "(a_1b_4 + a_2b_3 - a_3b_2 + a_4b_1 + a_5b_8 - a_6b_7 + a_7b_6 - a_8b_5)^2+\\,", "6aff368d999190125294dcb04de35cb1": "q = \\frac{\\text{Market value of installed capital}}{\\text{Replacement cost of capital}}", "6aff4c5b0f9cd322ed8f9b108f729836": "\\mbox {id} \\le f^r \\circ f\\qquad\\mbox{(right unit)}", "6aff631601f2821b9b249819add09b6a": "\\chi'=K_{\\chi}\\chi.\\,", "6aff74626947457933bad306d735ce8a": "\\beta (g) = 0", "6afff2b3cc77253749690aa8dc8e9c73": " \\ z : F_p(z,f) = 0", "6b000b6e08a957dbea67a96296c02fff": "E^{p,q}_2\n\\cong \\text{gr}_p H^{p+q}(B^\\bull)\n= \\begin{cases}\n0 & \\text{if } p < 0 \\text{ or } p > 1 \\\\\nH^q(B^\\bull)/H^q(A^\\bull) & \\text{if } p = 0 \\\\\n\\text{im } H^{q+1}f^\\bull : H^{q+1}(A^\\bull) \\rightarrow H^{q+1}(B^\\bull) &\\text{if } p = 1 \\end{cases}", "6b0016d6cc3a345615da4bc46101b580": " P = \\alpha C V^2 f ", "6b0034e242ba9e2a24ffbdc68f44c0fa": "V(x,t) \\ = \\ { f_1(\\omega t - kx) + f_2(\\omega t + kx)} \\ ", "6b00b65c6f6b472aad9f848db2103e3b": "\\mu \\mathbf{H} = \\nabla \\times \\mathbf{A}", "6b00c23cd4935573e0b838066b001e0f": "n=1\\,", "6b00e0ee5bd09ec633b77065479ea319": "20 \\cdot a + 2 \\cdot b,", "6b012ec7bdf6a04440864248a6bf555c": "\\ln(L) = -\\frac{1}{2}\\ln (|\\boldsymbol\\Sigma|\\,) -\\frac{1}{2}(\\mathbf{z}-\\boldsymbol\\mu)^\\dagger\\boldsymbol\\Sigma^{-1}(\\mathbf{z}-\\boldsymbol\\mu) - \\frac{k}{2}\\ln(2\\pi)", "6b01353973befcb29794c66b8bdc9f6a": "\\frac{\\sqrt{5}}{2}(3\\sin^2(\\phi)-1)", "6b0140d6db466db053ba8f2581cf7156": "n_{\\rm oil}>n_{\\rm water}", "6b0153044b1d55e2ade741fdd5ad142e": " \\Rightarrow L(y) \\equiv \\frac{p(y|H2)}{p(y|H1)} \\ge \\frac{\\pi_1 \\cdot (U_{11} - U_{21})}{\\pi_2 \\cdot (U_{22} - U_{12})} \\equiv \\tau_B ", "6b017035c97c9e68f3da10366d8c207e": "i \\geq 2", "6b01ac55d6181931253978e23ddaf68b": " \\lim_k f_k(x) \\geq g(x) ", "6b01d4ba33ad0ba7fbe985459750b1de": "\nNS_i = e_i^t \\left( G \\right)\n", "6b021c48d38f263e73435de40cde6cdd": "d \\sin \\theta \\approx d \\theta", "6b02479770149f678f6ffaee1dad2078": "\nV_R = e_1 - e_2\n", "6b02c1f66bba3191665535ea87f0c98f": "B_n(x+y)=\\sum_{k=0}^n {n \\choose k} B_k(x) y^{n-k}", "6b02dfa816b66d98183e198046302c06": "{\\overline{b}}=(A^{-1}b_1A,\\ldots,A^{-1}b_nA)", "6b03085630d82d0920076d4d00cadc07": "{\\it{l}}-1", "6b04159a88951151a792e7a08951f61a": "\\operatorname{Aut}(S_6)=S_6 \\rtimes C_2", "6b044c88d7dc8ac3d1c52c58aeb6177a": "\\left \\| A \\right \\| \\ge |\\lambda| , ", "6b04635ae4b16796c5a22cb6dd2d3e7b": " h \\in \\mathcal{H}_{R,m} ", "6b04635f2e8d4e4b611b32d4a4fa6637": " n!=\\prod_{k=1}^n k \\!", "6b0470e6f7f368461731183b7e7e24b8": "P(C_1)", "6b04754828703b1f146418fd5d24a390": "\n\\sin\\phi = \\frac{U_{\\infty}}{W}(1 - a) \\rightarrow \\sin^2\\phi = \\left(\\frac{U_{\\infty}}{W}(1-a)\\right)^2\n", "6b047737cd0e474c3708d4c11ff6cf96": "H_0 | \\phi \\rangle = E | \\phi \\rangle. \\,", "6b04ffd8a28bd89c15a7e491123e784e": "f^{(n)}(a) = {n! \\over 2\\pi i} \\int_C {f(w) \\over (w-a)^{n+1}}\\, dw.", "6b059c7f4c36b975bb4d3bb01ea46ec0": "\\frac{1}{2}L\\cdot I^2", "6b05e14f9892a751300b964affbc7850": "\\|\\varphi-\\psi\\circ \\bar h\\|_\\infty=\\inf_h \\|\\varphi-\\psi\\circ h\\|_\\infty\\ ", "6b06098f74084e6871e15d785460e493": "\nC^J_{E_1} = \\varepsilon^{2}_1 \\varepsilon^{3}_2 / D\n", "6b065b1837c58cb73ba631d7e8e77f82": "\\theta\\!", "6b06642a8a0b0ca4a0f7377a8aba0ccb": "x=\\tan y,\\, y=\\tan^{-1}x,\\, dy=\\frac{dx}{1+x^2}\\,", "6b0692392f827023e7befd4c9f6ea813": "\\left[ 1 + R(t) \\right]^ {(\\frac{1}{252})} ", "6b06cf35c46a5389a6f3dbec28486e77": "w =\n\\begin{cases}\n2, & d < -4; \\\\\n4, & d = -4; \\\\\n6, & d = -3.\n\\end{cases}\n", "6b06d23408eb6f1a34432ea137156935": "= I_{C1} + \\frac{I_{C1}}{\\beta_1} + \\frac{I_{C2}}{\\beta_2}", "6b06d76919cd3f27b249de157a84ce16": "O(VE \\log V)", "6b06e725967e329036846e01c7a3f82f": " f=O_NE_N ", "6b08406b60a062247a7ba1c6b1a4cb97": "\\Gamma^k_{mj}", "6b0848fbaca9ce7c7f493670c5cfc744": "\\begin{align}\n d_1 &= \\frac{1}{\\sigma\\sqrt{\\tau}} \\left[\\left(x + \\frac{1}{2} \\sigma^{2}\\tau\\right) + \\frac{1}{2} \\sigma^2 \\tau\\right] \\\\\n d_2 &= \\frac{1}{\\sigma\\sqrt{\\tau}} \\left[\\left(x + \\frac{1}{2} \\sigma^{2}\\tau\\right) - \\frac{1}{2} \\sigma^2 \\tau\\right]\n\\end{align}", "6b084b4afefb99658c5af6ca6b5e214c": "\\left\\{\\begin{matrix} n \\\\ k \\end{matrix} \\right\\}", "6b087d95bafcc8d2bc8d0ce51e3affd8": "f: X \\to \\mathbb{R} \\cup \\{\\pm \\infty\\}", "6b08cc363439e04057b83f7986b12849": "\\scriptstyle R\\left( z \\right) = R_0 + A_k \\cos \\left( kz \\right)", "6b090db9ad7ff3b6cbb5f4b85160c3fe": "\nP_0 = \\max [p(t)]\n", "6b0923b8bc7a9f6f3c1821684a8bd7c6": "T_{abcd}=C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} - \\frac{3}{2} g_{a[b} C_{jk]cf} C^{jk}{}_{d}{}^{f}", "6b093859434788e37e0e381394ba2e61": "\\pi_n(f)\\colon \\pi_n(X) \\twoheadrightarrow \\pi_n(Y)", "6b0954871e663a216e913b471df47a0c": "\\mathit{DIC} = D(\\bar{\\theta})+2 p_D.", "6b0993f0c3a47fccddc22ab237ba0f24": " \\exp(i(ax^2+bx)) \\,", "6b099fdecdb2400280bd539f60679b4f": "Bn_P(Cl_2^{\\leq}) = Bn_P(Cl_3^{\\geq}) = \\emptyset", "6b0a40e036849ed314f11c482fd2202a": "\\mathcal{F},", "6b0a6117780f7098028920bdd058d731": "(\\tau = (1 +\\surd 5 )/2 )", "6b0a6366b51f98549b1a92b2fcb436fc": "\\frac{\\partial \\mathbf\\phi(\\mathbf X)} {\\partial \\mathbf{X}}=\n\\begin{bmatrix}\n \\frac{\\partial \\mathbf\\phi}{\\partial x_{1,1}} & \\cdots & \\frac{\\partial \\mathbf\\phi}{\\partial x_{1,q}}\\\\\n \\vdots & \\ddots & \\vdots\\\\\n \\frac{\\partial \\mathbf\\phi}{\\partial x_{n,1}} & \\cdots & \\frac{\\partial \\mathbf\\phi}{\\partial x_{n,q}}\\\\\n\\end{bmatrix}\n", "6b0aa05ce76947e66ea840a868fb383d": " -\\omega ", "6b0ae3576f3b8fd4a558453203d44aa8": "V_3 = V_{LN}\\angle +120^\\circ", "6b0afe762f27d8a5955b64d3eae0477f": "\\frac{d\\psi}{dt} = \\frac{T_x}{C\\omega\\sin\\epsilon}", "6b0b7c81ee37baefa1c0bcdf34af3eb0": "G_{cd} = \\frac{1}{2}\\epsilon_{abcd}F^{ab} = \\begin{bmatrix} 0 & B_x & B_y & B_z \\\\ -B_x & 0 & E_z/c & -E_y/c \\\\ -B_y & -E_z/c & 0 & E_x/c \\\\ -B_z & E_y/c & -E_x/c & 0 \\end{bmatrix}", "6b0bbf1d37f27b98b11e3f65f7af33ad": "d_{n_k}", "6b0c5d6ab132fb744320ceeb01b969bb": " m\\frac{d^2x}{dt^2} + c\\frac{dx}{dt} + kx = 0 ", "6b0c8ce2c184d7e8b05893a31bd084be": "Q(x; k)\\,", "6b0cb06cf05cae7965a1a69ea1ea491a": " T = S \\setminus \\{ e \\} \\!", "6b0d20bae1ba026299d4f64062443822": "\nP_s = < P_i > \\; ; \\; i < j \\implies |R(P_i)| < |R(P_j)| \n", "6b0d3279a3eea2aa5673790f7b01a041": "\\scriptstyle\\cos \\theta = u/R = u/\\sqrt{s}", "6b0d67030a244de23ebf439e27e97429": " \\omega = \\frac{1}{2}|\\bold{k}|^2. ", "6b0d6753b15a4b78acf09d8d53488a51": "I:f^\\infty", "6b0d69d44871fa7e0c77d63a8a399019": "\\mathbb{E}[\\tau^{2}] \\leq 4 \\mathbb{E}[X^{4}].", "6b0d6d527932507dcd3b1e09ffeaa5d3": "\\ R = +\\frac{ee}{2} + 50% ", "6b0d8647e2ea48e79fcb94bab499ee86": " p_i^2 ", "6b0df862dcf31a0975a0ff7c1873b139": " t = 4 ", "6b0dfffb00660ade180aee1ff680f8aa": "\\begin{pmatrix}\n P | I\n\\end{pmatrix}", "6b0edaf64e272021a233c8d2985a797d": "A:X\\to X^*", "6b0efa441c6192105f0c5a264d6099a9": "{\\hat c} = cP^{-1} = m{\\hat G}P^{-1} + zP^{-1} = mSG + zP^{-1}", "6b0f15134fa59cc5e0a86ce3eeafb308": "(\\tfrac{p}{q})=1.", "6b0f6833a704e4d3becc6ceefe4c54af": " \\phi_y - \\phi_x ", "6b0f7c163ab5c6dbfc2e00e41d71dc72": "k = \\omega \\sqrt{\\mu \\epsilon'} = \\frac {2 \\pi} {\\lambda}", "6b1011c9c2dc51e590b2564569604ecd": " (\\mathrm{III}_T f)(t) = \\sum_{k=-\\infty}^{\\infty} f(t-kT)\\delta(t-kT)", "6b10f25894770ce470fe38a81556975b": "{l} = \\frac{L}{d}", "6b113e264b373bf837d13f3521fe1d65": "A_\\mu A_\\mu+B_\\mu B_\\mu+C_\\mu C_\\mu +\\psi^{\\alpha \\beta \\gamma}\\psi^{\\alpha' \\beta' \\gamma'}\\varepsilon_{\\alpha \\alpha'}\\varepsilon_{\\beta \\beta'}\\varepsilon_{\\gamma \\gamma'}", "6b11743d7e47bcbb9b135646fc71e599": "(a_1b_3 - a_2b_4 + a_3b_1 + a_4b_2)^2+\\,", "6b119fe606d726977998b43e33be2a03": "Qc= \\tfrac{i}{2}[c,c]_L", "6b11b89c695ce4e1ee8277a1758dcb62": "\\left(\\frac{1}{z^3}\\left (3-\\frac{1}{z} \\right )\\right)^{\\frac{1}{4}} = \\frac{1}{z} (3z-1)^{\\frac{1}{4}} = \\frac{1}{z}\\exp(\\tfrac{\\pi i}{4}) (1-3z)^{\\frac{1}{4}}, ", "6b11c35396f3db148dbb2077ba333c39": " \\mathbf{J}(\\mathbf{r}) = \\sigma(\\mathbf{r}) \\mathbf{E}(\\mathbf{r}) \\,\\, \\rightleftharpoons \\,\\, \\mathbf{E}(\\mathbf{r}) = \\rho(\\mathbf{r}) \\mathbf{J}(\\mathbf{r}), \\,\\!", "6b125b39f6804f959d99e10340742f6a": "\\frac{3}{4} \\pi", "6b126e2c437194033e6e54c7d6afe7ed": "B \\rightarrow S: B, N_B, \\{A, N_A, T_B\\}_{K_{BS}}", "6b1293407b924f000e5c9109e7a9b3c2": " \\left (\\ddot{\\theta} \\right )", "6b12dbfdb665fa1c97cbf7f5636d29f4": " \\Delta F = \\ln(\\cosh(4J)).", "6b12ddbba96f154c3640f38a7a835f71": "I(q) = P(q)S(q) ,", "6b12eb5de4bc3bc7f13f99c9c31aec13": "\\underbrace{\\frac{I_0\\left(\\pi \\alpha \\sqrt{1 - \\left(\\frac{2t}{(N-1)T}\\right)^2}\\right)} {I_0(\\pi \\alpha)}}_{w_0(t)}\n \\quad \\stackrel{\\mathcal{F}}{\\Longleftrightarrow}\\quad\n\\underbrace{\\frac{(N-1)T\\cdot\\sinh\\left(\\pi \\sqrt{\\alpha^2-\\left((N-1)T\\cdot f\\right)^2}\\right)}{I_0(\\pi \\alpha)\\cdot\\pi \\sqrt{\\alpha^2-\\left((N-1)T\\cdot f\\right)^2}}}_{W_0(f)}.\n", "6b12f8da641a2781d1d8f2fc684430b0": "K_{ij} = 0, |\\mathbf{R}_i - \\mathbf{R}_j| > R_c", "6b13725ec77a8bb976b658e176e20721": "Ce^{\\lambda t}", "6b1384f6329fa8122a1920cd5ec45183": "\\mu_\\pi=\\mu_m \\,\\!,", "6b13aabe2c960a9d05e71206afd29142": "|\\phi|", "6b13cacac9832f244dfca2876f3d6073": "x^2 + y^2 = (-r)^2,", "6b13d31bcc21bae5185a5985651f5ce1": "c_2=0.9", "6b1474d8220df2b45a03e6cdf3eb0c84": " \\log \\sum\\limits_{i = 1}^n {p_i^2 } \\ge \\log \\sup _i p_i^2 = 2\\log \\sup_i p_i ", "6b1488bbe2a60612ec079480878195c8": "b^{-\\alpha}", "6b14d8857e5f90f94e581ce57487278d": "T_B = \\frac{-a+b+c}{a} : 0 : \\frac{a+b-c}{c}", "6b14fe770e1ad72f822f0fc27c06fb8d": " r = -\\frac{d[A]}{dt}=k", "6b15032d5a0e1f45927ca1f5320a31dc": "\\vec{S}(1)", "6b1539fa5f455983a9d3cf148a4ac486": "\\Delta T_{sat} = 22.5 \\cdot {q}^{0.5} \\exp (-P/8.7)", "6b154cbd6859585005c66d34a34d6bf6": "(q,xa,yb,s) \\in \\delta^*", "6b154d909f7ad4b8b3c77f12f19c31af": "\\Delta f=\\max_{D_1,D_2} \\lVert f(D_1)-f(D_2) \\rVert_{1}\\,\\!", "6b156725683f251a86ef8df9c49342de": "d(uv) = \\sum_{\\delta\\;|\\gcd(u,v)}\\mu(\\delta)d\\left(\\frac{u}{\\delta}\\right)d\\left(\\frac{v}{\\delta}\\right).\\;\n", "6b1570424873d809141f8b9ac5b45d0a": "\\dot {N}_m", "6b15c3ea77969e6d51a684ee217b2e36": "V_f, V_g", "6b15f1c51e19a364ebbbcd784991ff53": "E_n(2)", "6b15f1eb523780210e37596a6d2b67bb": "i:=m+1", "6b164e58f6d4a3d1e87b0bc8077f1e8d": "\\text{sign}(\\sigma)", "6b164fd28edf493646fd13d81b75f2f5": "\\epsilon_{ijkl}", "6b16c1223df1d82a685e2ebf62f98ab4": "(W_1 , \\dots, W_n).", "6b16f2be9a45a154b2250e660c9311d6": " \\begin{array}{l}\n\\int\\nolimits_{a}^{\\infty }x^{m-s} dx =\\frac{m-s}{2} \\int\\nolimits_{a}^{\\infty }x^{m-1-s} dx +\\zeta (s-m)-\\sum\\limits_{i=1}^{a}i^{m-s} +a^{m-s} \\\\\n-\\sum\\limits_{r=1}^{\\infty }\\frac{B_{2r} \\Gamma (m-s+1)}{(2r)!\\Gamma (m-2r+2-s)} (m-2r+1-s)\\int\\nolimits_{a}^{\\infty }x^{m-2r-s} dx \\end{array} ", "6b1724d30b559a4e232ddf450310f4d5": "\\begin{pmatrix} 1 & 2 & \\cdots & m \\\\ \\pi(1) & \\pi(2) & \\cdots & \\pi(m) \\end{pmatrix},", "6b1759eef3ff6faed355cdbc526aace6": "d/\\lambda=0.6", "6b176d54af288a0d3b8d5dcc4de14aa0": "\\epsilon = (1-\\beta)n0", "6b1777f1c9eec5c9b7b26f0fcf599173": "\\rho^{\\otimes n}", "6b178c68d8980c6973467bdd7a3618e6": "F_1(q) = \\sum_{n\\ge 0} {q^{2n^2+2n}\\over (q;q^2)_{n+1}}", "6b17f65639ab52225797fc69af220c21": "[\\![\\underline{~\\,}]\\!]_i : \\phi \\rightarrow 2^S ", "6b18347969402bf6e61a8c63ce0d0632": "\\Omega^k(M)=\\Gamma(\\Lambda^k(T^*M))", "6b184fb3a7cb0dab1bbb4b2b45d1c637": " \\and T_8 = [F_8, S_8, A_8]::[F_7, S_7, A_7]::[F_6, S_6, A_6]::K_1 ", "6b1871279b3c5ac6127ea8ef9e159f38": "\\mbox{fractional yield} = \\frac{\\mbox{actual yield}}{\\mbox{theoretical yield}}", "6b187b271782a6082fc3e377c375c6aa": " \\left(E_n^{(0)} - H_0 \\right) |n^{(1)}\\rang = \\sum_{k \\not\\in D} \\left(\\langle k^{(0)}|V|n^{(0)} \\rangle \\right) |k^{(0)}\\rang. ", "6b195f3d9c0777f61057e81ceb4cf5e3": "V=\\sqrt{\\frac{2\\gamma}{\\rho h}}", "6b19966cc4d01b555803c53274342ca2": "\\epsilon < 0,", "6b19d9b53fc7574dff5c209756ecd7df": "KP(x|y_1,\\ldots,y_{k}) = \\min \\{ \\ell(p)\\ |\\ U\\ (p:\\ z\\ \\ )\\ y_1\\ \\ldots\\ y_{k} = \\langle x,z \\rangle \\}", "6b19f6824e0eac9f8f72f48ef566aea6": "\\,J = i\\rho+ J_1\\mathbf{i}+ J_2\\mathbf{j}+ J_3\\mathbf{k}\\quad ", "6b1a0ae31e722f579c3df78b70dff5fe": "\\{U\\in\\mathcal{O}:U\\text{ meets }N\\}\\,", "6b1a1adb32cc5bb62df6cd31ebc7d488": "\\mathit{alg}(A_1,B_1)", "6b1a5d61add5758ae5357abba57d63ec": "K=0.5", "6b1a8f660cd525c4c5161c410f977759": "\\zeta = \\pm 1", "6b1a93cd277e47d16e563d1b8f375cf7": "\\int_{C_2} f(z)\\,dz = \\int_{-R}^{R} f(x)\\,dx\\,.", "6b1bb83f323150f6a588fb51c481eea1": "E\\left(r_j\\right) = r_f + b_{j1}RP_1 + b_{j2}RP_2 + \\cdots + b_{jn}RP_n", "6b1bbeb547ca99a807d8e96efa57fda4": "\n\\begin{align}\nL(\\beta_1,\\beta_2) =\\Big( \\prod & P(Y_1=1,Y_2=1\\mid\\beta_1,\\beta_2)^{Y_1Y_2} P(Y_1=0,Y_2=1\\mid\\beta_1,\\beta_2)^{(1-Y_1)Y_2} \\\\[8pt]\n& {}\\qquad P(Y_1=1,Y_2=0\\mid\\beta_1,\\beta_2)^{Y_1(1-Y_2)}\nP(Y_1=0,Y_2=0\\mid\\beta_1,\\beta_2)^{(1-Y_1)(1-Y_2)} \\Big)\n\\end{align}\n", "6b1c0a68055a14294463fc372d97c46b": "X_{c} = \\{ (x,y) \\in L(c) : y \\in \\mathbb{Q} \\}", "6b1c107d8f0dee083af13c93b401d95f": "F_i=\\hat{\\pi}_A \\hat{\\pi}_i \\hat{\\pi}_A", "6b1ca5393f0a236c5786ac9dd2d4ed09": "M_{\\mathfrak{p}}(\\lambda)", "6b1ced0092315864ae988d4af18b5b34": "\\ \\Delta^s(\\alpha_{i,j,k}) = \\alpha_{i,j+1,k} - \\alpha_{i,j,k} ", "6b1d4b1b9148548ceccea327f8b4bf24": "H_{x,2} = \\sum_{n=1}^{\\infin}(-1)^{n+1}(n+1)x^n\\zeta(n+2)", "6b1d5f7bbcd07c16d9d00e319e2214ac": " \\Gamma(y) ", "6b1d83bb033b3704e695be4192976a5a": "\\mathcal{F}^{-n} = (\\mathcal{F}^{-1})^n", "6b1d94340129c1ee2903078e688e1bc7": "\\left\\{[x : 1] \\in \\mathbf P^1(K) \\mid x \\in K\\right\\}.", "6b1d9a53a34dd74461d69e2bc8365175": " \\operatorname{E}\\left(X\\right) = G'(1^-).", "6b1e400789a4145366c4852d0d4ab7b8": "\\gamma^{-1}= \\frac{\\gamma^{*}}{\\gamma(0)\\gamma(1)}. ", "6b1e488820a1083f21069bf56e900365": " \\mathcal{F}\\left\\{ R(x) \\right\\}(f) ", "6b1e4956b32bb4ecc4ed26ab259e65a3": "\n2H^+ (aq) + 2 e^- \\rightarrow H_{2} (g)\n", "6b1e5c9d8eb71818c0521f0c05733354": "\\mathbf{J_r}", "6b1e806d58d88768ada99a8f2dd02a35": "f : G \\rightarrow \\mathbb{C}\\,", "6b1e8a2ff884245438996bdcdeea7b81": "\\,\\Delta w(t) ~ = ~ \\eta(t) \\left(\\mathbf{y}(t) \\mathbf{x}(t)^{\\mathrm{T}} - \\mathrm{LT}[\\mathbf{y}(t)\\mathbf{y}(t)^{\\mathrm{T}}] w(t)\\right)", "6b1eaba20c43eba999fce464d6fd6777": "\\Delta\\theta = \\frac{\\Delta\\chi}{d_A(z)}\\!", "6b1ee1aa189e51dbfb660c2a81c71a56": "\\begin{matrix}{4 \\choose 1}^2{52 - 4r \\choose 2}\\end{matrix}", "6b1f1a3d5351c3c52745d5e88a0c068d": "\\mathfrak{g} \\to \\Gamma(TM)", "6b1f230b388318eaa672f6de2da4b27b": "\\boldsymbol{\\tau_{d}}=-\\frac{\\alpha}{\\gamma m} \\left( \\mathbf{m} \\times \\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}\\right)", "6b1f37eb1b1eb09a86fbc9c60b075bc1": "\n\\mathrm{SNR_{dB}} \\approx 20 \\log_{10} (2^n \\sqrt {3/2}) \\approx 6.02 \\cdot n + 1.761\n", "6b1f65d5dc3b40b58315c92f04afd082": "a = (1,2,\\dots,n), b = (1,2)", "6b1f6718d26a706f7b50a81701cbf484": "\\Omega\\subset\\mathbb{R}^3", "6b1fb7adb17cdff11a858704177172f0": " J=\\sqrt{2P(\\Delta_c^2 + \\kappa^2)/(\\hbar\\omega_c \\kappa)} ", "6b1fd6a5ab4e088ebd27e472b5c10a1c": "S = \\sum_{N=1}^M T_N \\ ", "6b1fe98b32848a5c38efcfc582ab62e4": "\\rho(\\mathbf{r})", "6b200ed709987633da4bbb1f02006f2d": "\\langle \\cdot , \\cdot\\rangle", "6b2025106a11b19fa274fa21a0938ad8": " \\psi=N~[\\tan~\\zeta~\\ln(r/r_0)~ - ~\\theta]", "6b206a28e60f665e235f89f460448467": "x\\,", "6b207291f99b84e0eaeb5e14bcd998be": "(-1/b)\\ln(z^\\star)\\text{, for } \\eta > 0.5", "6b207a0b325c8fafd5b91523bf3fdb93": "C_m = \\frac{1}{(P_{11} + P_{22})-(P_{12} + P_{21})}", "6b208761169c62aa4387d9b6210d4443": " \\|x\\|_{\\theta, q; K} \\simeq \\Bigl( \\sum_{n \\in \\mathbf{Z}} \\bigl( 2^{-\\theta n} K(x, 2^n; X_0, X_1) \\bigr)^q \\Bigr)^{1/q}.", "6b20e361d6dbcfb4a21b4332bd0d5eb9": " \\Omega_{n+1} \\tilde{H}_{n+1} = \\begin{bmatrix} R_n & r_{n+1} \\\\ 0 & r_{n+1,n+1} \\\\ 0 & 0 \\end{bmatrix} \\quad\\text{with}\\quad r_{n+1,n+1} = \\sqrt{\\rho^2+\\sigma^2} ", "6b20ecf14db584cced2372f58184b381": "x(t) = \\frac{1}{2}a_0(t) \\ + \\ \\sum_{n=1}^\\infty \\left[a_n(t)\\cos \\left(2 \\pi n \\int_{0}^{t} f_0(\\tau)\\, d\\tau \\right) - b_n(t)\\sin \\left( 2 \\pi n \\int_0^t f_0(\\tau)\\, d\\tau \\right) \\right]", "6b21013f0209f462df96e33b3d05a2a9": "F_\\text{seq}(0)=\\big\\{7,-7,0,6\\big\\},v_0=2", "6b212f81e015334f9d169913dd7d9e6e": "\nE(V) = E_0 + \\frac{9V_0B_0}{16}\n\\left\\{\n\\left[\\left(\\frac{V_0}{V}\\right)^\\frac{2}{3}-1\\right]^3B_0^\\prime + \n\\left[\\left(\\frac{V_0}{V}\\right)^\\frac{2}{3}-1\\right]^2\n\\left[6-4\\left(\\frac{V_0}{V}\\right)^\\frac{2}{3}\\right]\\right\\}.\n", "6b21480196391b2666e360e49e9697e3": " P_{R}(t)=\\sum_{i=0}^{d}\\frac{f_{i-1}t^i}{(1-t)^{i}}=\n\\frac{h_0+h_1t+\\cdots+h_d t^d}{(1-t)^d}. ", "6b21b06b2476c9a91789714916fff798": "\\ Y", "6b222358878e9a50b8901c6e76e902d6": "x^{y}", "6b2267dda6c4dde1b79152ef8269f59a": "\\Sigma_i", "6b22b929ffa0dd00cf2dbf8c0ceda3cd": "\\sigma A = \\inf_n \\frac{A(n)}{n},", "6b22c428b1c5d3b2e16e8c1354ec57b9": "a \\lor \\left(b \\lor c\\right) = \\left(a \\lor b\\right) \\lor c", "6b22e6913bb3bb8354f88232f45b5e07": " { \\partial x \\over x_m } A = { \\partial V \\over V_m } = { -1 \\over \\ \\gamma } {\\partial p \\over p_m } ", "6b231d51b3347554c03b63bdf079dcfb": "\\hat{h}(\\xi) = \\overline{\\hat{f}(\\xi)}\\,\\hat{f}(\\xi) = |\\hat{f}(\\xi)|^2.", "6b235e581bb1386e90cdf775339e1f4b": "\\vec k\\perp\\vec B_0,\\ \\vec E_1\\|\\vec B_0", "6b236519c311711264c8d31d7ecee0bd": "\\mathbf{v'}_i", "6b238785ba63c981115fe01a65c0b38b": "Z_\\mathrm{iT}=L\\sqrt{\\omega_c^2-\\omega^2}", "6b2394f323c5881408f748a246c77e8e": " - {d[A] \\over dT} = k [A^n] ", "6b239f44fcee1ff2e4ac66f7fe51ac34": "\\mathbf{J}_{n,\\text{drift}}/(-q) = - n \\mu_n \\mathbf{E}, \\qquad \\mathbf{J}_{p,\\text{drift}}/q = p \\mu_p \\mathbf{E} ", "6b23b9e7b68239409e4472a70222300f": "\\begin{smallmatrix}L=4 \\pi R^2 \\sigma T_{\\rm eff}^4 \\end{smallmatrix}", "6b23bbc3589157c91e963054d3ef5f10": "\\scriptstyle x_{i}|x_{1}\\cdots x_{i-1}x_{i+1}\\cdots x_{n}+1", "6b25171b4309effd300ce2e9f09b3263": "BO(n) \\to BO", "6b259200812d6e98d54b842725479bf2": "\\operatorname{erf}(x)\\approx 1-(a_1t+a_2t^2+\\cdots+a_5t^5)e^{-x^2},\\quad t=\\frac{1}{1+px}", "6b25b369bcb9775cea8ff42bd6eaebeb": "w''=w\\,\\!", "6b25dfd0706d358cd2d4665a9bf319fe": "z = r e^{i \\theta}", "6b2669979649e99014d1f04242ba30fa": "A_1+A_2 \\rightleftharpoons 2A_3", "6b26761e352355ef743e445f6108bf82": "\\gamma = \\sqrt{2} + \\sqrt{3} ", "6b2682f1b3d43f52f9ffe28b8f0ab35b": "n \\geq 1.", "6b26b522beb6eda1ba50ca58e25b5591": " \\mathbf{L} = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}) + \\left(\\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\right)\\times\\mathbf{v}.", "6b26c5bf19b2559f839f6330ff0d84dd": "\\{t_n\\}", "6b26ec828c180dc445e9a706d3187fb0": "\\int x^{k_1}\\cdots x^{k_{2N}} \\, \\exp\\left( -\\frac{1}{2} \\sum_{i,j=1}^{n}A_{ij} x_i x_j \\right) \\, d^nx =\\sqrt{\\frac{(2\\pi)^n}{\\det A}} \\, \\frac{1}{2^N N!} \\, \\sum_{\\sigma \\in S_{2N}}(A^{-1})^{k_{\\sigma(1)}k_{\\sigma(2)}} \\cdots (A^{-1})^{k_{\\sigma(2N-1)}k_{\\sigma(2N)}}", "6b26f2ba43e27b02dacfced4922cbdd4": " \\lambda_D = \\sqrt{\\frac{\\varepsilon_0 k_B T_e}{n_e q_e^2}}", "6b275230aa685c4b0f7b1532d1eecfb6": "X(z)=\\sum_{k=0}^{N-1}\\frac{L_k(z)}{L_k(z_k)}\\hat X [k],", "6b279383f42b2636f59053141b0fab57": "T h = \\alpha \\langle h, v\\rangle u \\quad \\mbox{for all} \\quad h \\in H ,", "6b2793a71bccd6aac05809bd486869bd": "\\max_{x\\in X}\\ \\{f(x): g(x,u)\\le b, \\forall u\\in U\\}", "6b27b172fca1db864be24075312ecd35": "c_i = n_i/V", "6b27d59ca46cde79a0253e0cba1f5583": "\\text{Cl}_{2m}\\left( \\frac{q\\pi}{p}\\right)= \\frac{1}{(2p)^{2m}(2m-1)!} \\, \\sum_{j=1}^{p} \\sin\\left(\\tfrac{qj\\pi}{p}\\right)\\, \\left[\\psi_{2m-1}\\left(\\tfrac{j}{2p}\\right)+(-1)^q\\psi_{2m-1}\\left(\\tfrac{j+p}{2p}\\right)\\right] ", "6b282d61b91cb991d076f7a88b94f73a": "F[y]=\\frac{1}{3}\\frac{y''''}{(y'')^{5/3}}-\\frac{5}{9}\\frac{y'''^2}{(y'')^{8/3}}", "6b28b34071a3f2cedba9e1db89a78d5b": "F(x)=\\begin{cases}-\\frac{1}{x}+C_1\\quad x<0\\\\-\\frac{1}{x}+C_2\\quad x>0\\end{cases}", "6b28b551b9d40828ae9b97e66239ed4b": " n/3", "6b28c17e6c51c1200f7cd40eb7262bcf": " i = 1, \\cdots, q ", "6b28f4c3cd410c609f9f2f3f840421c1": "(\\phi_x,\\phi_y)=(1.5,0.2)", "6b2922bc169a1f526f915cc8e124b1e4": "\\omega_0 = \\frac{1}{\\sqrt{LC}} = \\frac{1}{\\sqrt{L'C'}} ", "6b292cc228363bc05b73fad2cf49aaee": "km^{-1}", "6b296032a4383f4d4bdba21f64bbc019": "K_{\\alpha}(\\cdot)", "6b29610e0e3555caafa406a6b1e52794": "\\nabla^{2}\\Phi_{M}=0", "6b29f9f883450d474982546a0b862946": "1+z = \\frac{a_\\mathrm{now}}{a_\\mathrm{then}}", "6b2a0eda24b8a1a3e2a04613586885d5": "d_r", "6b2a106638ed4c240621f8b932711535": "\\mathfrak{p}_1", "6b2a7355175c2db6d9356acc03781c7a": " \\phi(x) \\leq N(x), \\quad x \\in U.", "6b2a990397ac606dec60dda3c7e0bba2": "\\dim \\mathbf{H}_\\ell = \\binom{n+\\ell-1}{n-1}-\\binom{n+\\ell-3}{n- 1}.", "6b2b45be494665f6ef2bf8e6257a294f": "Diagnostic \\ Cost = Maintenance \\ Down \\ Time \\times Labor \\ Rate \\times \\ Team \\ Size", "6b2beea29f5ae838d1e2ba2bc40b7a0e": "\\dot \\theta = \\frac {\\ell}{\\mu r^2}, \\, ", "6b2c17b26f3d83eef46f4495e0697ee2": "\\mathbf{l} = \\{0.707107, -0.707107\\}, ~ \\mathbf{n} = \\{0,1\\}, ~ r = \\frac{n_1}{n_2} = 0.9", "6b2c3772e956bd7d191e4e9417c49700": "a,b,c,d", "6b2c4c684ec2c9668bcec7b8ec2b23bf": "\\mathbf{y}- X \\boldsymbol \\beta", "6b2cb90b35867a3616b4b351f038d1f1": "\\Complex/\\Lambda", "6b2cbd2f97f9199a23ecddfb4dcc3ddd": "R = \\frac {\\textrm''{Isentropic \\,\\, heat \\,\\, drop\\,\\, in \\,\\, rotor}''}{\\textrm''{Isentropic \\,\\, heat \\,\\, drop \\,\\,in \\,\\,stage}''} ", "6b2ce934a5de14267f669c40512fcdab": "W_{e}", "6b2d39e5db88df61b67baa8eaf9843db": "\nE_2=\\frac{2P^2m_2}{(m_1+m_2)^2}\n", "6b2d45aee1dbabf27494342d0302c6d8": "k=l", "6b2d75cc5eb0ea8c7fdf5686d758947a": "1 / 1", "6b2dabc9245bdcf3e3a65e8e731bc992": "3A+2B+C = 0,\\,", "6b2e46dd874bc09deb9ce7af41256e22": "n\\rightarrow\\inf", "6b2e6c672aae11e3f2bff27f14062f40": " \\langle q| \\mathbf{ \\hat T}(-\\lambda) = \\langle q + \\lambda| ", "6b2ea48df396bb815b016e86deec918f": "-\\otimes_R-", "6b2eaf304e86fb16eef95efd3a210d1e": "V = V_\\mathrm{Eq}-Z_\\mathrm{Eq}I \\!", "6b2eedb9494a6aebbdc6380e0e3d8b9b": "\\Delta K^y", "6b2ef0a32e9c6fb294c8f296d57453a2": "y''+(2+\\frac{1}{x})y'-\\frac{4}{x^{2}}y=\nLclm\\Big(D+\\frac{2}{x}-\\frac{2x-2}{x^{2}-2x+ {\\frac{3}{2}}}, \nD+2+\\frac{2}{x}-\\frac{1}{x+{\\frac{3}{2}}}\\Big)y=0. ", "6b2f0a436b177bcab215cdf59a98f585": "\\omega\\,", "6b2f10dc8342a33f743f0b94a1dbe4c2": "\\operatorname{div}\\, \\mathbf F \n= \\nabla\\cdot\\mathbf F \n= \\frac1r \\frac{\\partial}{\\partial r} (rF_r) + \\frac1r \\frac{\\partial F_\\theta}{\\partial\\theta} + \\frac{\\partial F_z}{\\partial z}\\, .\n", "6b2f27c61233f7ba58c19dc5bf91839d": " \\mathbb{R} \\times S^{2}", "6b2f2b86783f2e55cc0cfba5cf16cf14": "\\begin{align} \\mathrm{Size} & = 2 \\cdot \\min \\left( 100\\,\\mathrm{GB}, 350\\,\\mathrm{GB} \\right) \\\\\n& = 2 \\cdot 100\\,\\mathrm{GB} \\\\\n& = 200\\,\\mathrm{GB} \\end{align}", "6b2f4dc3854c4dfd5926f44ce2258668": "z^{|\\omega|}/|\\omega|!", "6b2f4e165089c10f4b316442ed3ad64f": "\\scriptstyle -\\hat x \\hat x \\hat y \\hat y \\;=\\; -1", "6b2f9d9bc341d9ec3d039da8cbb14894": "K=\\frac{k}{k_{-1}}=\\frac{\\theta}{(1-\\theta)P}", "6b2fc989ac88c990fe2549d280d7779a": "|x| = \\begin{cases}0 & x\\in V_0\\\\1 & x\\in V_1\\end{cases}", "6b304a7cd4b027b47600e3103f0db866": "|p_{i}-2^{-m}| < 2^{-m} \\epsilon(n)", "6b30592c940c4ddfc4bf711213fdacd5": "f^{-1}(U) = \\psi(U)", "6b3071dec74b3c8e23847ec361756c0c": "(143/11)P = 13P \\text{ and }(143/13)P = 11P.", "6b3086e418feb3fab71bfe89ed921e34": "\n A=\\frac{\\kappa}{E^*h'}F\n ", "6b30f50c65f8e95c215b213a14faaed9": "\nM(\\theta) = F L \\sin \\theta - k_\\theta \\theta\n", "6b3191e4ee482e3876fa9125061f17de": "I_{\\text{E}p}", "6b31db9b35b192e8334444f71844f638": "\\frac{0.083\\ \\mathrm{N}}{(3.5\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=0.0024", "6b31e0cbdf94abcf8ef161879f331fab": "\\Delta G^\\circ > 0", "6b31ea674f29ba632b485b52d5e559d8": "\\textstyle \\bigcup_{i=1}^l [x_{i1},x_{i2}]", "6b3280c0c935f27eba689a6c94c523bd": "\\sigma_{t,T}=\\sqrt{\\frac{(T-t_0)\\sigma_{t_0,T}^2-(t-t_0)\\sigma_{t_0,t}^2}{T-t}}", "6b328b04717afd778381cc85fc94eed9": " 4\\pi \\varepsilon a ", "6b32997ff1150b51d8adb961697b273e": "\\,=A(I-B)(I-B)'A' = \\Big(A(I-B)\\Big)\\Big(A(I-B)\\Big)' \\geq 0", "6b329c67ef015a34e6976bf02e077fbc": "y = -x", "6b33147a430db92e814b07b5483071b8": "\n[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0\n\\,", "6b33bd7981200f74ec96142117d67ba1": "(a,b):\\Sigma_{(x:A)} B(x)", "6b33c9700ce72992fbd3d19e05b7b871": " \\vDash ", "6b33e997ffd8a2202f680f62760a4e8f": "\\mathcal{F}^\\vee", "6b3417e1751448cf7ad9860d06950826": "\\sum_{n=0}^\\infty\\frac{B_n}{n!}z^n=e^{e^z-1}.", "6b3469aaff2957118d499ffc2d3287a2": "21(y-8)", "6b347c4be8afd43aad92004a6588292c": " 0 \\le \\alpha_i,\\beta_i\\le 1", "6b348eb63aef0d28c634b23d591cff9d": "=\\left(\\ell_A\\right)^{n_A}\\left(\\ell_B\\right)^{n_B}\n\\left(\\underbrace{\\frac{N!}{n_A!n_B!}}_{combination}\\right)\n\\left(\\underbrace{\\frac{n_A\\pi^{n_A/2}}{(n_A/2)!} (2E_A)^{\\frac{n_A-1}{2}}}_{n_A-sphere}\\right)\n\\left(\\underbrace{\\frac{n_B\\pi^{n_B/2}}{(n_B/2)!} (2E_B)^{\\frac{n_B-1}{2}}}_{n_B-sphere}\\right)\n", "6b348f1cbb3fefd68761b52b1f3a7eec": "f_\\theta(x,t) = f_\\theta(x)", "6b3522b2f5cff2bac9307be13e38f734": "\\not\\in\\mathbb R", "6b354fe57163eac745b0e9f2a7b9e292": "\\alpha_k = \\frac{1}{2 \\pi}\\int_0 ^{2 \\pi} e^{-ikt}\\,d \\mu(t).", "6b3565dd315856d16fff1614bd9c92df": "NAND(\\alpha,\\alpha')=1-AND(\\alpha,\\alpha')", "6b35795e7309704b35ba3d1ffd0cc796": "E^\\nu (q, \\mathbf{k})", "6b358eb265b06efb9c06510afa1666d0": "K = \\frac{ -R + \\frac{BA}{\\pi A / 180}}{T}", "6b3594bfe4e856702d52b467825d3656": "\\det(V) = \\prod_{1\\le i\\aleph_1", "6b46e648e8ae2cd97367c09486fb5921": "s \\in \\mathbb{N} \\cup \\{ 0 \\}", "6b47043dad09169ddc9a2639eb57f3ba": "\\upsilon(T)", "6b47128caa1c4de9af30e5cdf18ab8f9": "expr \\in \\{expr_{1}, \\dots, expr_{n}\\}", "6b4743bc5104610a4f0576bd9953f6a0": "F\\colon M \\times N \\to P", "6b47601667ccd718e63cfb8ba4f3b7ea": "\\frac{\\partial^2 p}{\\partial t^2} = c^2 \\nabla^2 p\\,,", "6b476ce62e1e42c6c25849ef4d80c532": "\\Psi = \\Psi _0 \\exp(i\\varphi)", "6b47728d0741874816b5584049fae4e3": "p\\left( S\\ge x \\right)=1-\\exp \\left( -e^{-\\lambda \\left( x-\\mu \\right)} \\right)", "6b4787d8260e4e91dc04ca489416bf3f": "\\varphi\\approx17^\\circ", "6b478f5ea9c6a14df536c41134493c6f": " \\frac{V_1}{i_1} = \\frac {r_M(r+r_M)}{r+2r_M} ", "6b479a69c92829eacbbacc2d7cc20aee": "|W|^2=0", "6b482b263d8dc8540ac37b927f00975d": "N_c N_p = N", "6b4865872c6719f83b0a4d770477d4eb": "(i \\hbar \\beta^{a} \\partial_a - m c) \\psi = 0", "6b4884b527668b377578e709fc23aa83": "{\\color{Blue}~2.29}", "6b4898cd30a921c4bca389493ab092d4": "x(10-x) = 25-(x-5)^2\\le 25", "6b4988abcec16eee4e302de74ba72816": "x\\in \\lbrace 0, ..., N-1 \\rbrace", "6b49b3e15040b545164aeecec64c3eef": "T_N", "6b49c8dd78792e8c2a6e693469376651": " c = e^{-r T} [FN(d_1) - KN(d_2)]", "6b49f2351b0b1259f77bd9fdb476cbc1": "[\\![\\phi \\vee \\psi]\\!]_i = [\\![\\phi]\\!]_i \\cup [\\![\\psi]\\!]_i", "6b4a14e19009f63d0599f8a184e0018b": "\\widetilde{\\rho} = \\rho^*", "6b4a29aed9fa1f2069fea5228af579fd": "\\mathfrak{so}_8", "6b4ab8c8ee7f520811816863d79ad4a1": " f(K^*,L^*) = f(K,L) + \\frac{1}{2} kT \\log(\\sinh 2K \\sinh 2L) ", "6b4af10c8dc0a6c53cbf9258b72edfe8": " Y(s) \\rightarrow \\infty", "6b4b0af74db7d17533f1929373148735": "f_n(x)=\\frac{\\alpha_n}{n!\\,\\lambda_n^n}\\psi_n(\\lambda_n x),\\qquad n\\in\\mathbb{N}_0,\\;x\\in\\mathbb{R}.", "6b4b1b600993088c69bbd94a285541f9": "\\begin{align}\n\\lim_{x\\to a} \\frac{f(x) - P(x)}{(x-a)^k} &= \\lim_{x\\to a} \\frac{\\frac{d}{dx}(f(x) - P(x))}{\\frac{d}{dx}(x-a)^k} = \\cdots = \\lim_{x\\to a} \\frac{\\frac{d^{k-1}}{dx^{k-1}}(f(x) - P(x))}{\\frac{d^{k-1}}{dx^{k-1}}(x-a)^k}\\\\\n&=\\frac{1}{k!}\\lim_{x\\to a} \\frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\\\\n&=\\frac{1}{k!}(f^{(k)}(a) - f^{(k)}(a)) = 0\n\\end{align}", "6b4b8df6f1a7805fc23211d9b86a1b75": "d^2G := n dS \\cos{\\theta} d\\theta \\ ", "6b4be04d42c501df430426ebc6debb4b": "{8 \\choose 1} + {9 \\choose 1}{7 \\choose 1} = 71", "6b4c10c9c9fec1c516dcadf3ae0cf3e7": " PK = g^{SK}", "6b4c4dceb390c1f882b6d81e12b371da": "BP^2 = v2 R_2 / r_1 = v_2 Q_3 / r_3", "6b4d2497f51d6f1b0697c0c0bc287205": "\\chi_E(x)=1", "6b4d3b000b6980ab2bb3b0bf0bca2049": " c_3 ", "6b4d56f4adcdeb012cd1721afc07bbee": "\\mathbb{Z}^b ", "6b4dc70f65d3fb075d9748808950d4e1": "S>2F", "6b4e20ee930b5aa354fc337d5e1dc472": " \\scriptstyle\\mathcal{F} ", "6b4e3107f3c57d45e6ecd0d70cfc2b6b": "p(D|\\theta)", "6b4e664623c955a72f7117bd5053d064": "\\zeta,", "6b4e6efffa12f32d74b6f25d5110b6ba": "\\mathbf{F} = q(\\mathbf{E} + \\mathbf{v}\\times\\mathbf{B})", "6b4ec10bfc6c41ba7da66f3887445ced": " dr_t = (\\theta_t-r_t)\\,dt + \\sqrt{r_t}\\,\\sigma_t\\, dW_t,", "6b4ef6642d62dffafc71fd868ada0f71": "\\begin{matrix} 10 \\times {4 \\choose 2} + {4 \\choose 3} = 64 \\end{matrix}", "6b4f48d6159440db0c3f5f1ae99cf61a": "A(x)=\\sum_{n\\le x} \\chi(n)", "6b4f601bf3c9b160d224a502a62ea8cb": "\\mathbf{\\sigma}^o ", "6b4f7f2d43f86c8ce9b92395368031ba": "I(v)=\\log_2 n", "6b4fb609cda7dc71453e86b6cbac8b42": "O\\left({N^{1/2}}\\right)", "6b5019b7d8917d05eccc3ee65b364f6f": "\n{\\left(\\frac{2}{n}\\right) \n= (-1)^{(n^2-1)/8} \n= \\left\\{\\begin{array}{cl} 1 & \\textrm{if}\\;n \\equiv 1\\;\\textrm{ or }\\;7 \\pmod 8\\\\ -1 &\\textrm{if}\\;n \\equiv 3\\;\\textrm{ or }\\;5\\pmod 8\\end{array}\\right.}\n", "6b50407a7a3466b2c3d4e86608f0ca70": "\\displaystyle{(f,g)=\\sum_{x\\in A} f(x)\\overline{g(x)}.}", "6b505f15011857f9fd71467461881c04": "\\exp \\circ\\, \\psi", "6b5112c1bcf53b31d97f9e9665e6997a": "R =\n \\begin{bmatrix} 7.8102 & 4.4813 & 2.5607 \\\\\n 0 & 4.6817 & 0.9664 \\\\\n 0 & 0 & -4.1843 \\\\\n \\end{bmatrix}", "6b5134199e8471ed2dcc6b12caedf692": "dU\\, ", "6b51487bfcd983f9ce639403f25b118c": "V = \\sum_{i=1}^{n}PV_i = \\sum_{i=1}^{n}CF_i \\cdot e^{-y \\cdot t_i} ", "6b51543e5a09f5d1bf9f7191ff97d855": "\\Gamma/2\\pi\\,", "6b5167b144311add0d00d108e0b7c29b": "1.9619", "6b519c433060e7d2ccef9cffb606b63b": "10\\uparrow\\uparrow 65,534", "6b51c9e6e8134198fa771accb8cd1c7f": "x_i = 0", "6b51e70fd79c0bdefdb16ede2cc529cf": "\\tilde{U}\\,", "6b523e3ff04690a5fcfa860d0896aa4e": "i\\ne k", "6b52661e7ef143761eaa903363232831": "x_1, ..., x_d", "6b52cb3ef6f8a3eca7b9a2bd183da889": "\\left(\\frac{19}{45}\\right) = 1\\quad\\textrm{ and }\\quad19^{(45-1)/2} \\equiv 1\\pmod{45}", "6b52d8e5e968884c3fbb7c951f9d6f3d": "x_1, \\dots, x_s", "6b5321a79ca72923412bfce8f7f8d15f": "u, v\\in\\sum^n", "6b53475eb108bee4b04054eb3853c483": "(5+i)^4\\cdot(239-i)=-114244-114244i.", "6b53917c198e37da3cf9baf4d273b0ef": "\\operatorname{Tr}(x) = \\sum_{i=0}^{n-1} x^{q^i}", "6b53b53127b29478dd2804042c47bf39": "d_{1,0}^{2} = -\\sqrt{\\frac{3}{8}} \\sin 2 \\theta", "6b53b5acfeac9a52271a3860a83acfad": "\\tfrac{11}{4} =2\\tfrac{3}{4}", "6b53fa5501159a699f5bd9cc23de77a8": "\\textstyle \\phi(\\mathbf{q}) = f(\\mathbf{q}) \\sum_{j=1}^{N} \\exp (-i \\mathbf{q} \\mathbf{R}_{j})", "6b5454bd9fd5ce85d140c1289c1f2066": "L_2\\!", "6b5494be66beeb4b63e90be8f8123806": "\\Delta f(x)=f(x+1)-f(x)\\,", "6b54ac559c8614593ed3fe4123920773": "0\\le x\\le 1", "6b54cf8cf8dc34722df883b20b0be18b": "\\mathbf{G_0}", "6b557f2f96c9871b83cfa0902485984e": "5 + 3 + 1 = 9", "6b55876c527c799143df0e98d8a41d4b": "1 \\over 100", "6b55ae5307a0ff35352564e3922f242c": "\\{1,2,3,4\\} ", "6b55ce83c7bb7caae0121cb4e562e362": "\\mathrm{d}U = T\\mathrm{d}S - p\\mathrm{d}V+\\sum_i \\mu_i\\,\\mathrm{d}N_i\\,", "6b55dfd717aecea35d888d745d445581": "\\pm t", "6b56aea242e46caa1614cf5a284ec77a": "\\le w(e)", "6b56da94e98c25e064568f4b360d5715": "F = 1 + \\frac{T_e}{T_0}", "6b56dede647b5a669d84214430ba4cf5": "(\\nu x)P \\rightarrow (\\nu x)Q", "6b5724c1006bcfcf0366d10c6f1afa02": "f(x,y,z)=\\sum_{i=0}^3 \\sum_{j=0}^3 \\sum_{k=0}^3 a_{ijk} x^i y^j z^k.", "6b57670c7cf291baf2eaa9eea1015592": "\\mathrm{d}{\\bold{J}}=0.", "6b57d27a06bb4ae1fdeeb0744b4f301b": "\\tfrac {1}{2} \\,", "6b583a3fe94d0c050a27eb23d7c22ad1": "\\exp(x) = \\lim_{n \\rightarrow \\infty} \\left(1+\\frac x n \\right)^n ", "6b583b99e83dab2ba9cf6bb8946c4dbf": "A \\Delta x = b", "6b585aaedc976a7d22d164ce84643a5b": "R(\\theta_1 + \\theta_2 , \\mathbf{e}_i) = R(\\theta_1 \\mathbf{e}_i)R(\\theta_2 \\mathbf{e}_i)\\,,\\quad [R(\\theta_1 \\mathbf{e}_i),R(\\theta_2 \\mathbf{e}_i)]=0\\,.", "6b5863745411db217e0879af5631d04c": "p_i(\\mathbf{x})", "6b586c4c7fc5e29305c5efd3d356aa7f": " R = \\int_m^n f^{(2p+1)}(x) {P_{(2p + 1)}(x) \\over (2p + 1)!}\\,dx ", "6b5880f48d4f468c9a16ac6070b5de28": "\\Delta p = F \\Delta t\\,.", "6b588cea119dd23a867702f855e8fc9d": "P(x_1,r_2,\\ldots,r_n)", "6b58b0c3818ac41197682312bbe2086c": "(x \\triangleleft y) \\triangleleft z - x \\triangleleft (y \\triangleleft z) = (x' y)'z - x'y'z = x'y'z x''yz - z'y'z = x''yz", "6b58b6d0eaf727f880431a83b85c8851": "w^r", "6b58def4ac8ac76599de7042ce4bef40": "\\sum_{i=1}^{n}x_{i0}=W_0", "6b59603e0914a625fdbf5fd6925fc758": " \\langle \\psi_1 | H | \\psi_2 \\rangle = 0", "6b59730651c1344d678dee427c18b498": "N0", "6b598e400a46300a5bcdae78e5074b5b": "{\\vdash}={\\,|\\!\\!\\!\\sim}", "6b59bc315c28725e04e9331e71527914": "[0,\\pi/2]", "6b5a3bd3ccba039520f633705222b7e3": "\\arcsin(x)", "6b5a9c1ce2140095c866908a41099d45": "\\Box^i p", "6b5aa1f4f5bb2f7a60fd85c9ef13143a": "{d\\vec{x}_S\\over ds} \\times \\vec{u}(\\vec{x}_S) = 0,", "6b5abbc61ab1d8bbf2b647b7be70a3fa": " \\begin{align} & {} \\quad \\frac{\\partial \\Pi(\\mathbf{\\xi}, t)}{\\partial t} - \\Omega^{1/2} \\sum_{i = 1}^N \\frac{\\partial \\phi_i}{\\partial t} \\frac{\\partial \\Pi(\\mathbf{\\xi}, t)}{\\partial \\xi_i} \\\\\n& = \\Omega \\sum_{j = 1}^R \\left( -\\Omega^{-1/2} \\sum_i S_{ij} \\frac{\\partial}{\\partial \\xi_i} + \\frac{\\Omega^{-1}}{2} \\sum_i \\sum_k S_{ij} S_{kj} \\frac{\\partial^2}{\\partial \\xi_i \\, \\partial \\xi_k} + O(\\Omega^{-3/2}) \\right) \\\\\n& {} \\qquad \\times \\left( f_j(\\mathbf{\\phi}) + \\Omega^{-1/2} \\sum_i \\frac{\\partial f'_j(\\mathbf{\\phi})}{\\partial \\phi_i} \\xi_i + O(\\Omega^{-1}) \\right) \\Pi(\\mathbf{\\xi}, t). \\end{align}", "6b5ac2ca746123779a3ac84d6c1b27ce": "\\alpha<\\zeta_0", "6b5b11a441e1fb84cf2ac882e6e4b878": "-2V_{nn}\\left(\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_3}V_{k_3n}}{E_{nk_3}^2E_{nk_4}E_{nk_5}}-\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_2}V_{k_2n}}{E_{nk_2}E_{nk_4}^2E_{nk_5}}+\\frac{|V_{nk_5}|^2}{E_{nk_5}^2}\\frac{|V_{nk_3}|^2}{E_{nk_3}^2}+2E_n^{(2)}\\frac{|V_{nk_5}|^2}{E_{nk_5}^3}\\right)", "6b5b187501838e1bbae8690119cf0794": "\\delta f_1 = f_1 -f_2", "6b5b372ea7da88b5e16d32cdbfcadc02": " \\Gamma_L = 0 \\, ", "6b5b51f632177b925e883a82740688ca": " K =k_1 + \\cdots + k_n", "6b5b541356163b9b5f7e060ef1761b7a": "\\tilde\\psi(\\alpha)", "6b5b817cdad197f22e2a0352770951ec": " k_1 ", "6b5b890b0b25cd795292be0d48fed912": "\n \\int_{G_\\text{Aff}}| b, a\\rangle\\langle b,a|\\;\\frac {db\\;da}{a^2} = I\n", "6b5bd0d8c9d4602dbac0ea27c5039154": "-{{1}\\over{2\\pi i}} \\oint_{\\Gamma_s} {D'(s) \\over D(s)}\\, ds=N=Z-P ", "6b5bd4de0fe15ec82701bf5cdb1e6733": "\\left( \\cos\\frac{E}{2}\\ ,\\ \\sin\\frac{E}{2} \\right)", "6b5c12b96b682011ff7c8a18b562c38e": "\\left\\{ \n|0\\rangle =\\! \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \n|1\\rangle =\\! \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \n\\right\\}", "6b5c2e16325faba3f81458353feb3c60": " \\left[(\\mathbf{AB})^\\star\\right]_{ij} = \\left[\\sum_k \\left(\\mathbf{A}\\right)_{ik}\\left(\\mathbf{B}\\right)_{kj}\\right]^\\star = \\sum_k \\left(\\mathbf{A}\\right)^\\star_{ik}\\left(\\mathbf{B}\\right)^\\star_{kj} = \\sum_k \\left(\\mathbf{A}^\\star\\right)_{ik}\\left(\\mathbf{B}^\\star\\right)_{kj} = \\left(\\mathbf{A}^\\star \\mathbf{B}^\\star\\right)_{ij} ", "6b5c835230bc26e163f9d5d6a776e808": "D = N - Z", "6b5ca5c8021a41019b69b10dfa6e8570": "D_1=6P_1+ 4P_2", "6b5d02137759c3b5a41c140a8f0e1170": "(X:Z)", "6b5d052dc4ded3367278e36056eeff5b": "\\eta_p = \\frac{2}{1 + \\frac{v_e}{v}}", "6b5d355b466bc83eeb6d2ea817035d12": "|f_n(x)| \\le M", "6b5d4e141ae385b0ac5c5d41829e53c9": "f^{-1}(U )\\cap Y", "6b5d6a53c31d2a26b57e5ea617cbd6d4": "t\\to\\infty", "6b5d74f175c6555931c310d099b9ab20": "F_{24}", "6b5dea2422345ee8af6d27832b2d414b": "0 \\le x \\le x_2", "6b5df891b1daf108bf53e57da74107a2": "f^{m} \\circ f^{n} = f^{n} \\circ f^{m} = f^{m+n}~.\\,", "6b5e9a5ffa1ba5c47d939ee355390f5f": "\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}\\,", "6b5ea4a80a00267c5989ade6990e5279": "b_i(x)^m = \\pm 1", "6b5eaa31568b521152a5aacec91548c2": "s_{x}", "6b5f2cce772cde8dcd21d37fbadf0d24": "\\rho (\\tfrac{v}{c})", "6b5f48d2324fc941288fb08209d2760d": "\\Phi_B:=B_1{\\rm d}x_2\\wedge {\\rm d}x_3 +\\cdots ,", "6b5ffa7e5e70076e7656cfc9146a45d8": "\\psi_I(x) = A_I e^{i \\rho y} + A_{II} e^{-i\\rho y}\\quad y<-1 ", "6b5ffdbcdc8577dd9bb984fca1700ca0": "M+2L \\rightleftharpoons ML_2: [ML_2]=\\beta_{12}[M][L]^2", "6b600fc6266c9e1568f95521038eac6f": "\n\\begin{align}\na &= \\left\\lceil \\mu - \\frac{1}{2} \\left\\lceil \\sqrt{1+12c^2\\mu^2} \\right\\rceil \\right\\rceil \\\\[8pt]\nb &= \\left\\lfloor \\mu + \\frac{1}{2} \\left\\lceil \\sqrt{1+12c^2\\mu^2} \\right\\rceil \\right\\rfloor \\\\[8pt]\np_b &= \\frac{(c^2+1)\\mu^2-A-(a^2-A)(2\\mu-a-b)/(a-b)}{a^2+b^2-2A} \\\\[8pt]\np_a &= \\frac{2\\mu-a-b}{a-b}+p_b \\\\[12pt]\n\\text{where } A & = \\frac{2a^2+a+2ab-b+2b^2}{6}.\n\\end{align}\n", "6b6043a27a579b245f94bcc58a7b9415": "\\sin(x)-\\pi", "6b6094211b4ec43790c3f137da1f1dfe": " \\pi_iQ_{ij} = \\pi_jQ_{ji}", "6b60f29344111cc4bc52cfad3baf77ba": "\n\\begin{align}\nA^2 e^{A t} =& (3/4)^2 B_{1_1} e^{3/4 t} + \\left( (3/4)^2 t + ( 3/4 + 1 \\cdot 3/4) \\right) B_{1_2} e^{3/4 t} + B_{2_1} e^{1 t}\\\\ +& \\left(1^2 t + (1 + 1 \\cdot 1 )\\right) B_{2_2} e^{1 t} \\\\ =& (3/4)^2 B_{1_1} e^{3/4 t} + \\left( (3/4)^2 t + 3/2 \\right) B_{1_2} e^{3/4 t} + B_{2_1} e^{t} + \\left(t + 2\\right) B_{2_2} e^{t} ~,\n\\end{align}\n", "6b611d473b2d860d1b59736257cc3c1c": "J=-D \\nabla \\phi \\ , \\mbox{ it is the product of a tensor and a vector: } \\;\\; J_i=-\\sum_{j=1}^3D_{ij} \\frac{\\partial \\phi}{\\partial x_j} \\ .", "6b612a17ec832776b82457d06d1892e3": "\\langle S, \\land \\rangle", "6b6139e466bb6d3c9a1bb31516ede0ac": "x^5+3x^2+2x-1 ", "6b618a15653494846816382a5fd1099f": "\\omega_{\\text{i}} \\cdot \\mathbf n", "6b61f50c7a6546616a147a612deb245b": "S=cA^z", "6b623bcf78099c519f69e9dbba46fbf2": "L_i", "6b629debf8d4908c0481b5f68f111aff": "\\mathcal{N}_k(x)", "6b637ffaa076adabfbcdcd31baf14b9f": "Y_i \\ \\sim \\operatorname{Bin}(n_i,p_i),\\text{ for }i = 1, \\dots , n", "6b63bf56fc0f977338ea71236f755087": "\n\\begin{align}\n\\Phi_3(-z) \n&=z^2-z+1 \\\\\n&= (z+1)^2 - 3z\n\\end{align}\n", "6b63db7c29478d49642cfe705c68a538": "M \\setminus \\{t_j\\}", "6b6446587a3a4b0afc4952801f7ec220": " \\lim_{p \\to 0} p \\log p \\;=\\; 0", "6b645f7106ae8aaaa45a9b7e49d92e68": "x\\ast y=0 \\and y\\ast x=0\\implies x=y", "6b64bd98064b4e557ffef006d853f168": "\\leq_T", "6b654820832dc7e75cafb5f7e26972a5": "H_1 \\simeq H_2", "6b657b6dc7d345231d276b6f46af7375": "\\mathbb{K}[x][D_x]", "6b65d4f05d6b72b203c272eb5c963abf": " U(\\mu, \\nu, z) = \\frac{U_{\\mu}(\\mu) + U_{\\nu}(\\nu)}{\\sinh^{2} \\mu + \\sin^{2} \\nu} + U_{z}(z) ", "6b65f0a205de9dd6c8fe8e47d70f0f30": "\\mathbb{N}\\times\\mathbb{N}", "6b6604f508e49742aba3ca3be8e8338c": "(\\hat{c}-\\hat{a}), \\hat{\\alpha}, \\hat{\\nu} = \\hat{\\alpha}+\\hat{\\beta}", "6b6614d47975b36dc106fc1663735568": " y(w)", "6b66906cd6c7658f828b5f993d32342c": "e^{\\pm i H_{0,S} t/\\hbar}", "6b66a5095d6955802f333a77c5d8104b": "\\frac{-j}{k}=i", "6b66b49dbaf5ae1fc4e4801bf8a1d21d": "O_{p',p}(G)", "6b66b5c24afdb6c393d0c2bc4d435fa1": "\\bar{Z_i}(\\lambda n_1,\\lambda n_2,\\cdots )=\\bar{Z_i}(n_1,n_2,\\cdots ).", "6b66edfdcc169b1662d713c4fc28f790": " q' = {1 \\over {2-q}}", "6b670dadea2f6236c88a676abe835f12": "p(\\mathbf{X}|\\boldsymbol\\eta) =\\left(\\prod_{i=1}^n h(x_i) \\right) g(\\boldsymbol\\eta)^n \\exp\\left(\\boldsymbol\\eta^{\\rm T}\\sum_{i=1}^n \\mathbf{T}(x_i) \\right)", "6b673c85a83a022507577fa245ef74a9": "h^{\\prime} \\in \\left\\{1,...,N-1\\right\\}", "6b67dd4bc6c45d6a3a5121d977125aa5": "\\Delta U=U(S_2)-U(S_1)\\,", "6b6822c500ef69b67922d18107773f29": "u=xaby", "6b6842d11898f51a2df76d4c1c0ec2ba": "\\scriptstyle \\mathbf{R}_T", "6b686984b6c2b3646d230a0864d97e37": "X'=X+\\varepsilon U", "6b68a07cfe63caa276d9bc7fea866bf9": "p(X^o,x^m,h)\\,", "6b68be9c2262f68eff68583793f76a46": "n_F(a+b)-n_F(a-b)=-\\mathrm{tanh}\\frac{\\beta b}{2}+n_F^{\\prime\\prime}(b)a^2+\\cdots", "6b68cd764cee114cf396e0402c09a170": "M_{-1}(x_1,\\dots,x_n) = \\frac{n}{\\frac{1}{x_1}+\\dots+\\frac{1}{x_n}}", "6b68dc4c062915d0ae219db518c8b6e9": "n/n^4", "6b68e0abe196d60b0b2fa2809915d463": " d\\nu (x) \\equiv \\frac{1}{\\pi} d^2\\alpha.", "6b6916dc719f0070115683ee61bb7cf9": "\\|", "6b697d4e26a01f9a7a82a2dca96637ad": " \\Delta S \\approx - \\Delta t \\alpha IS/N", "6b69853ec4288927203cb59da10a283b": " u^{3}=-{q\\over 2} + \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}", "6b69c594c7d1fe0fb479213810d5e543": "x\\in\\R^n,\\,t\\in\\R", "6b69d2011923fde7eb90737d49f4363c": "v: \\R \\mapsto \\R", "6b69f0cae7e7fc0891898db2c3355fba": "S = X_1 + \\cdots + X_n", "6b69f33e693f2371b8cff168167a7a78": " \\Psi(r_j) = \\Psi(r_{x,j}) \\Psi(r_{j,s})", "6b69fa0d9c5f7aab7434c3f7741cd297": " k = n-m \\, ", "6b69febd50221607b125466c094e81e9": "(A\\to(B\\to C))\\to(B\\to(A\\to C))", "6b6a0c070c9bad2a734a18ce645211e7": "X=\\sum_{i=1}^n X^i \\partial_i", "6b6a4364d2c68ee53548202030e465fa": "\\sigma_{xx}\\sigma_{yy} - \\sigma^2_{xy}", "6b6aaec17eb08ef976b5e8387d24aed8": "L_\\odot", "6b6ab328c856a5610e4ce359f594f423": "\\Lambda(f) : \\Lambda(V)\\rightarrow \\Lambda(W)", "6b6b7d56772ac5c167bbb445490d4423": "\\mathfrak f\\times V", "6b6b8dceb54edac5a3b6716a72faee95": "E(\\mathbf{r}) = A(\\mathbf{r}) e^{ikz} ", "6b6bbbd7105bde5f42fb5c192fc03d18": "R_s x_j", "6b6bc739e890cac0ec56bb747e56db1e": "y=\\epsilon^{1/4}+y^5", "6b6c0a4c7ea1e4a9e32983fe1b4608ce": "\\int_C {\\sqrt{z} \\over z^2+6z+8}\\,dz=2\\int_0^\\infty {\\sqrt{x} \\over x^2+6x+8}\\,dx.", "6b6c231088db003651577a696df24dff": "M_{\\alpha\\beta,\\alpha\\beta} = -q", "6b6c290388d447a6db24405a19d0c9e4": "\\neg P \\or \\neg R", "6b6c54665ead8195b213ea25ece9462c": "k = \\omega\\sqrt{\\mu \\epsilon}", "6b6caa7ec3417d5e8375e9ca38b85c41": "A_\\mathrm{d}", "6b6cdc7211ae736038447db354a9a9d6": "(a, b)^* (a, b)\n = (a a + b b, a b - b a) = (a^2 + b^2, 0),\\,", "6b6d6e9f0fdc7132eaa4301ae41f9088": "h_k = \\lceil h'_k \\rceil", "6b6df1426c990e7f6652f1dd594f4321": "t_0 > 0", "6b6ec59a5663906dd02258f411a4823a": " \\{0, 1, \\ldots, N_1-1\\} \\times \\cdots \\times \\{0, 1, \\ldots, N_d-1\\} \\to \\mathbb{C}. ", "6b6edbc7f8234cc20320874ceafe89a8": "R_2 =0", "6b6ee32ebbb80565ffa98ac08bf0d28a": "\\gamma^\\mu \\gamma^\\nu \\gamma_\\mu \\,", "6b6f07a4ce501b74158184d202405293": "\\exp(q) = \\sum_{n=0}^\\infty \\frac{q^n}{n!}=e^{a} \\left(\\cos \\|\\mathbf{v}\\| + \\frac{\\mathbf{v}}{\\|\\mathbf{v}\\|} \\sin \\|\\mathbf{v}\\|\\right) ", "6b6f2a355ade3b00e011b1814ad34015": "\\textstyle q > 3", "6b6f7524cc8c15edfff3b7db44c756d7": " \\lambda = 0", "6b6f7e1780b11e559e733ab521e23104": "y_{n-k} = y_{n-k},", "6b6fe468f078bde2388267d60418b404": "[SU(5)\\times U(1)_\\chi]/\\mathbb{Z}_5", "6b7011208b9fae687eee69f1519827b4": "\\scriptstyle M_n(\\theta)", "6b7026478713d8b65d2cd9fd7bad7e45": " \\frac{50 -0}{12.0 -0}(12.0-0) +0=50", "6b702fc1f6ad882906a58dca403142d9": "\\Delta \\theta_d", "6b704e3e2e83baf487b833711057d1a9": "T_3=0", "6b707e124cb7f19556a60c956c365928": "\\scriptstyle\\hat\\theta^H_n \\;=\\; \\bar{x}\\cdot\\mathbf{1}\\{|\\bar x|\\,\\geq\\,n^{-1/4}\\}", "6b70980bcdba1df76ac115d16d085fec": "\n \\displaystyle\n u = \\log_b (nm) = \\log_b (n) + \\log_b (m).\n", "6b70ac1297906b202be9b9961d934573": "x \\div 12", "6b70aea3e0b5547d986c42de2389f606": "I_{sp}(vac) =\\,\\frac{v_e}{g_o} + \\frac{p_e\\,A_e}{\\dot{m}\\,g_o}", "6b712fca9f984d59f954ce3e94aa02a2": "\\Box A^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ \\partial_{\\beta} \\partial^{\\beta} A^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ {A^{\\mu , \\beta}}_{\\beta} = - \\frac{4 \\pi}{c} J^{\\mu}", "6b715ba589e9b66860d624850f769367": " \\left ( {\\partial T\\over \\partial S} \\right )_V = { T \\over C_V } ", "6b71b2d727fbf2041a1674f7c218111d": "|F_n| = \\frac{1}{2}(n+3)n-\\sum_{d=2}^n|F_{\\lfloor n/d\\rfloor}|,", "6b71cf6899fedefa3328c9e9587791db": "{[x_1, x_2]}^n = [0, \\max \\{x_1^n, x_2^n \\} ]", "6b71d7b28649663245134de6c51ae60d": "x * h", "6b71f1679979d374bd87f5751d96eceb": "\\scriptstyle S_{CHSH} \\;=\\; 2\\sqrt{2}", "6b721d50f898861be397eb4cade8f69b": "U(a,x)=D_{-a-\\tfrac12}(x),", "6b723ed87ffe17bba95a43a299ce5cfd": "F = m r \\omega^2. \\,", "6b724df5e094fa7a67f72a4651808959": "{2{-}(1{+}\\sqrt{2})\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{\\sqrt{n}}} = \\gamma + \\sum_{n=1}^\\infty \\frac{(-1)^{2n} \\; \\gamma_n}{2^n n!} ", "6b7258d9354b07e65d6ce049c5c958af": "\\textstyle \\zeta ( \\alpha - d + 1)", "6b725a3ca8be31fb57b4cb911bd992ff": " \\tilde{O}(m) ", "6b7267776b7b4b8a32894333c3876872": "\\scriptstyle E=\\rho g m_0\\,", "6b72781c22ddf4e906a2c278f5132a62": "\n\\phi(\\mathbf{x}) = \\frac{1}{4\\pi\\sigma_{\\mathrm{outside}}} \\oint_{\\mathrm{membrane}}\n\\frac{\\partial}{\\partial n} \\frac{1}{\\left| \\mathbf{x} - \\boldsymbol\\xi \\right|}\n\\left[ \\sigma_{\\mathrm{outside}} \\phi_{\\mathrm{outside}}(\\boldsymbol\\xi) - \\sigma_{\\mathrm{inside}}\\phi_{\\mathrm{inside}}(\\boldsymbol\\xi) \\right] dS\n", "6b728a91ef1d9479957493be669172a4": "{ dI \\over I(z)} \\left[ 1 + \\bar{g}(\\nu) { I(z) \\over I_S } \\right] = \\gamma_0(\\nu)\\cdot dz ", "6b72b46b87e92dd1e698bf3a1d28757a": "\\text{pHad}(x)= x \\cdot G'", "6b72b6fccfddc3691aa44c9bae373454": "13\\,X^2+25\\,X-49,", "6b72da5bba870f8ae2044455e739993e": " f(x) = \\sum_{k=0}^\\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \\cdots, \\qquad |x-a|2n", "6b781f32370de39818b477d033bfe2a0": "succ : N \\longrightarrow N", "6b782bc67a994b921c0293a68ea288fa": "\n\\bigg(\\frac{\\alpha}{\\mathfrak{a} }\\bigg)_n \n\\left(\\frac{\\beta}{\\mathfrak{a} }\\right)_n \n=\n\\left(\\frac{\\alpha\\beta}{\\mathfrak{a} }\\right)_n. \n", "6b786b43e4c2d2fa81cb3de4bc2d291c": "\\log_2( 6^5 )", "6b786f6d68b0e3547acd27bcc4a23935": " \\lambda = (-t)^{-1/p} ", "6b787279895e93e93db2a1d0156b15f4": "E_F = \\frac{\\hbar^2}{2m} \\left( \\frac{3 \\pi^2 N}{V} \\right)^{2/3} \\,", "6b788e026c166e9718d725342ce7de51": "k^\\text{th}", "6b78ec7561d022e82f7b760c59e4e82f": "{\\underline P}Q_i", "6b7922b1220562162594e95dc131c7dd": "Y\\Vdash B", "6b793a9666a9d9d1e6ac6f98e4e09799": " 12 a_4 + a_2^2 - 3 a_1 a_3 = 0 ", "6b798694094380a1513ede2df7e1fb77": "\n \\begin{align}\n Y_{1,-1} & = p_y = i \\sqrt{\\frac{1}{2}} \\left( Y_1^{- 1} + Y_1^1 \\right) = \\sqrt{\\frac{3}{4 \\pi}} \\cdot \\frac{y}{r} \\\\\n Y_{10} & = p_z = Y_1^0 = \\sqrt{\\frac{3}{4 \\pi}} \\cdot \\frac{z}{r} \\\\\n Y_{11} & = p_x = \\sqrt{\\frac{1}{2}} \\left( Y_1^{- 1} - Y_1^1 \\right) = \\sqrt{\\frac{3}{4 \\pi}} \\cdot \\frac{x}{r}\n \\end{align}\n", "6b79899759918cc7f97ceac6b2926166": "\\tilde{n}=n+i\\kappa.", "6b79b5a883fe0330182c22b2537e800a": "\\mathrm{If} \\; X \\leq Y \\; \\mathrm{then} \\; \\rho_t(X) \\geq \\rho_t(Y)", "6b79bd8a6f5b543534b793f826e2a925": "\\frac{M_H}{M_{\\oplus}} = 7.539 \\times 10^{-3} = 0.754 \\; \\% \\;", "6b7a2063a61a6c324b11a90deb86685b": "\\scriptstyle g(X_n)\\xrightarrow{p}g(X)", "6b7a320283dd0ee1b30956434fd4524e": "e^{-\\pi R_{12,34}/R_s}+e^{-\\pi R_{23,41}/R_s}=1", "6b7a452ef1fb3c95848bb664a334444d": "\\bar{x}(t) = \\frac{m_{1} \\cdot x_{1}(t)+m_{2} \\cdot x_{2}(t)}{m_{1}+m_{2}}", "6b7a5b32a4b39771d148d2fbdea01c22": "n = k + d", "6b7aae87d819bda0a19215fa230c682b": "k^2-(k-1)^2=2k-1", "6b7b3d2ed530b117d23dd62a2e98b5e6": " \\delta(k_1 + k_2) \\over k_1^2 ", "6b7b4783a1e38fb75856a1e56e15db21": "f = \\frac{1}{2 \\pi R C}", "6b7bffa72ece210d1e78badaaca302eb": "y_i = f(x_i ;\\beta ) \\cdot TE_i", "6b7c0a301fcd7731d269165bc2fa806c": "\\langle (\\Delta N)^2 \\rangle = \\langle N\\rangle ", "6b7c48523b5d40d693471391c72ca031": "\\mathbf{n_{i+ 1}} = \\mathbf{Ln_i} = \\lambda \\mathbf{n_i}", "6b7c856df6cb59ca209123eb5e407443": "\\boldsymbol\\Delta \\mathbf y", "6b7d636d318592300e8822afd3da1bfb": " \\left(\\sum_{i=1}^n \\alpha_i e_i \\right) \\cdot \\left(\\sum_{i=1}^n \\beta_i e_i \\right) = \\sum_{i=1}^n \\alpha_i\\,\\beta_i ", "6b7d638fcf22a421efa9adbced6c5b35": "\\sigma S = \\frac{k_{\\rm B}}{-e} \\int \\frac{E - \\mu}{k_{\\rm B}T} c(E) \\Bigg( -\\frac{df(E)}{dE} \\Bigg) \\, dE", "6b7d6bf13b3d9693fbe935f0bd63e57b": "L_n^{(\\alpha)}(-x) = \\frac{(n+1)^{\\frac{\\alpha}{2}-\\frac{1}{4}}}{2\\sqrt{\\pi}} \\frac{e^{-\\frac{x}{2}}}{x^{\\frac{\\alpha}{2}+\\frac{1}{4}}} e^{2 \\sqrt{x(n+1)}} \\cdot\\left(1+O\\left(\\frac{1}{\\sqrt{n+1}}\\right)\\right),", "6b7d91279e6c6b661fb4f1f545eed51c": "p_{ij} =\n\\begin{cases}\n \\frac{e^{-||Ax_i - Ax_j||^2}}{\\sum_k e^{-||Ax_i - Ax_k||^2}}, & \\mbox{if} j \\ne i \\\\\n 0, & \\mbox{if} j = i\n\\end{cases}\n", "6b7dc9eefadf29f69269a59fcf65d107": "{Z}_{{{\\mathbf{k}}}}(R)", "6b7e6b0633bb1741b8d6484a09ec6b9e": " \\tilde{f} \\colon X\\times I \\to Y ", "6b7e8d27433774bf9dbfcdb9b6d94fd9": "\\frac{d^2f}{dx^2}=w^4\\frac{d^2f}{dw^2}+2w^3\\frac{df}{dw}", "6b7f3858a63e47dc0a16241fe8464294": "\\mod l^n", "6b7fa2565d30aaea57802c02176fd69b": " 1, 2, 3, 6, 9, 18, 27, 54. \\, ", "6b7faf60aaeba0ad0941b7a0098fe908": "\\ (\\lambda, \\phi)", "6b7fee658c67cd12b63e75af3c4d60d8": "M=J-1", "6b802002027797c8ba330df9d588d85a": "X(s)-Y(s)H(s) = Z(s) = \\dfrac{Y(s)}{G(s)} \\Rightarrow X(s) = Y(s) \\left[{1+G(s)H(s)} \\right]/G(s)", "6b806e361654426bde105e3b7b797183": "\\underline{d}(A)", "6b80cabe74a74adffadf94ae14ca2d9a": "2 \\leq k \\leq K", "6b80f91cf8ceb3c0d18b791c86ae14be": "[A]_{\\text{seq}}", "6b814b9d7f2b14423b2ca0659d549162": "N^2 \\le Q < 2N^2", "6b8168334e09719aac8b8765f9a8eb2b": "10^{13} ", "6b8181f87c14e8342074a4a95483ea3a": "(3)-(6)=-3", "6b81a65356bfa5e9072bdb28664d066d": "(\\Omega,F,P)", "6b81eaab1a430c134134849eea54897c": "{\\partial \\over \\partial t} \\iiint_V \\rho \\, dV = - \\, {} ", "6b8233e5b184828c4ece7c8d0853a351": "\\omega^{(\\perp)}~=~\\omega^{(\\|)}~+~\\omega^{(\\|)}~-~\\omega^{(\\|)}", "6b825dae86466d8edeabf3c04af7e732": " S( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{t=1}^\\infty S_t( \\mathbf{w} ) ", "6b82e989453c18d2b4ef00b671f73d0d": "\\frac{D}{ds}m u^\\lambda = -\\frac{1}{2}u^\\pi S^{\\rho\\sigma} R^\\lambda{}_{\\pi\\rho\\sigma}", "6b83607a58e5fdd24ed270a53695199d": "[\\phi^{(i)}(x),\\phi^{(i)}(y)]=0", "6b8390a213fd6a3526d23551a69b3a73": "\\Omega\\ .", "6b839f9d94c78e57aa48aa61c6cd6db2": " : \\hat{b}_2 \\, \\hat{b}_1^\\dagger \\, \\hat{b}_3 : \\,= \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\, \\hat{b}_3 ", "6b83ae6fd70f3c6d2a49df3613a66555": " Q = 2024.86 ", "6b83c494621f84870577949aeb7c9049": " c_8 = 0.000274954, \\,\\!", "6b842e75e4ae127791ac01dab2587271": "[i] := T^{q^i} - T, \\, ", "6b842fa83e66a6bf2af91a57ed8e03ae": "d(\\tau)", "6b84351c656731b42505d60c3a1b05ec": " p(\\alpha_i) = y_i ", "6b844534e2693973cf042e4b42fb15ba": "\\displaystyle{\\int_{\\partial\\Omega} K(z,w) |dw| ={1\\over 2\\pi}\\int_{s-\\pi L}^{s+\\pi L} \\partial_t \\arg (z(s) - z(t))\\, dt =1/2.}", "6b8457b2cd065c0c84d496397247a750": " \\int_{-\\infty}^\\infty e^{-x^2} \\, dx = \\sqrt\\pi.", "6b847bb1386c519be152b2a1502c45cc": "H=-\\log_2 w", "6b848743224eb293ca4f458c7c0277a3": "\\{\\psi_n(x)\\}", "6b84a73d789ae9b0e01585b27d9189e8": "\\underline{\\lambda} = -\\mathbf J_\\sigma^{-1} \\underline{\\sigma}", "6b84ad65d196846d71213275719a1890": "2^{681} 5^{378} = 1010 = 2^{301} 5^{416} \\pmod{1019}", "6b8512729f3fd92a58018d55e23033ab": "\\Phi(s,t)=-\\ln(-t)\\text{ for }s=1,\\,", "6b852525c625a77d28c8ce618c045bf6": " q(z) = z + z_0 i\\ ,", "6b8552dc027c22faa1dbdeb209aa7936": "\\textstyle D = (1000011,1)", "6b862906fac6c52bc172428d40309cd5": "\n\\frac{1}{4}S_0(1-\\alpha_p)=\\left( 1-\\frac{\\epsilon}{2} \\right) \\sigma T_s^4\n", "6b866d88d0ccbc92fd0a0a93efa91280": " \\Leftrightarrow V_\\mathrm{Ba}'' + V_\\mathrm{Ti}'''' + 3V_\\mathrm{O}^{\\bullet \\bullet}", "6b867b02aa96387ba99550d8bf24f89d": "\n\\begin{align}\np(\\varphi) & = p(\\theta) \\left|\\frac{d\\theta}{d\\varphi}\\right|\n\\propto \\sqrt{I(\\theta) \\left(\\frac{d\\theta}{d\\varphi}\\right)^2}\n= \\sqrt{\\operatorname{E}\\!\\left[\\left(\\frac{d \\ln L}{d\\theta}\\right)^2\\right] \\left(\\frac{d\\theta}{d\\varphi}\\right)^2} \\\\\n& = \\sqrt{\\operatorname{E}\\!\\left[\\left(\\frac{d \\ln L}{d\\theta} \\frac{d\\theta}{d\\varphi}\\right)^2\\right]}\n= \\sqrt{\\operatorname{E}\\!\\left[\\left(\\frac{d \\ln L}{d\\varphi}\\right)^2\\right]}\n= \\sqrt{I(\\varphi)}.\n\\end{align}\n", "6b86e9f9a550e82404873e09dc219675": "2x_1 -x_2", "6b87381f26725d27ceab0ab2a364629d": "A=(3+\\frac{5\\sqrt{3}}{2})a^2\\approx7.33013...a^2", "6b878691c7a571678307faa4a47e56ab": "\\text{winding number} = \\frac{1}{2\\pi i} \\oint_C \\frac{dz}{z}.", "6b87a122b7f0dbeccd5bd4c1f9163ffd": "\\eta=1,\\,\\!", "6b87dd8033f2c78b0f720542db8e5528": "h(Z,w)\\,", "6b87e88d9324cc43e72920b4e7fc8b3f": "(\\operatorname{coth}\\,x )' =\n\n -\\,\\operatorname{csch}^2\\,x", "6b880842a5d80f7802b63a7e783ea33b": "[H^{+_{ }}]_i", "6b8878ac05a20adc75a8853f80181763": "\\frac{(6.02\\ast 10^{23} \\frac {molecules}{mol}) \\cdot (101325 Pa)}{8.314 \\frac{Pa \\cdot m^{3}}{mol \\cdot K}\\cdot 273 K} = 2.69 \\ast 10^{25} molecules \\cdot m^{-3}", "6b88e8779f90b531fc15790b14e886a0": " m = \\lfloor 4^k/n \\rfloor", "6b88fe6512f412b48c8215aa1d87733a": "y \\cdot dy - x \\cdot dx = 0.\\,", "6b89382acfe6a8306feb7d7213a51b14": " \\nu=\\frac{\\nu^*}{2p\\nu^*\\pm 1}.", "6b894c4f0b71fffc1e1b09d871171e49": "n_e=xn", "6b89ce70a7aa9179adacbf56b47469f2": "a_1(t) = \\frac{q(t)}{q(0)} \\ ", "6b89dead944911a169364d3de75e85d5": "\n d\\boldsymbol{\\sigma}:d\\boldsymbol{\\varepsilon} \\ge 0 \\,.\n ", "6b89f17b7b02faab7fb3b8054d8ee824": " m = s. ", "6b8a07179e881ddb838040a1ddeabe1d": "U = \\sum_i E_i \\!", "6b8a834e23c2d588d90e7ad40d9708e5": "p(\\psi)", "6b8a95bfec6b144b78e399e3fa499dd8": "c = \\sqrt{\\frac{2}{3}}(4r).", "6b8ab7abff6cc4b9f56e40d77947b2db": "BD = 2 \\left(R + T \\right) \\tan{ \\frac{A}{2}} - BA", "6b8ac09697aeb7e1d681eba381d8d7b1": " 4^{4^4} + 3 ", "6b8b5b843927fd19808288b3397cf210": "\\frac{U(1,b,z)}{\\Gamma(b-1)}+\\frac{M(1,b,z)}{\\Gamma(b)}=z^{1-b}\\exp(z)", "6b8b5f8088be784ec6c0b0f1c2fb4ce6": "X = \\operatorname{Spec} A", "6b8b7c7d1c594799bea27daab0602ad2": "\\alpha(x;\\theta) ", "6b8b985697b71dc5c3c9799ecadb64e2": "0(9)\\, ", "6b8bc1867bc2c38c434b80e4f2d7a25d": " FC = \\frac{SM \\times AM}{SM-AM} = 411.78443 d", "6b8bd5ae94b9ea6924a2e57026f3932b": "T(s)=\\frac{kG(s)}{1+kG(s)}", "6b8be99dbca3b8e1ea63b7f6f4a976bb": "c=D \\sin \\left(\\frac{\\theta }{2}\\right)", "6b8cb2ea7798d887d42c9cce64f051f4": " |k' \\rangle", "6b8cdfe6588639bb05d2f7bbcb5e4ca4": "\\frac{d^3y}{dx^3},\\quad f'''(x),\\quad\\text{or }\\frac{d^3}{dx^3}[f(x)].", "6b8d04cb49e170f5ffefdd51f799e471": "K=-1", "6b8d6c4f4446a04d7dbd7be2c6813e1b": "V_v = - \\frac{C_v}{(Z-Z_0)^3}+O\\left(\\frac{1}{Z^5}\\right).", "6b8da5f1fd4bf32941fdc2851732172b": "(v,x)", "6b8dcf2786ee91cc81a5472c8b06c7d5": "P_{k}(\\zeta)", "6b8e2ff3a6fc85ac4a14e777dd17f622": " H = \\sum_{k=-\\infty}^{\\infty} \\hbar\\omega_k a_k^\\dagger a_k = \\sum_{k=-\\infty}^{\\infty} \\hbar\\omega_k N_k", "6b8e303567e90cfd5c7f7842ad29daf8": "\n\\left[ Q_R(\\mathbf{p}^{\\prime}),\nQ_L(\\mathbf{p}) \\right] = 0, \\;\n\\left[ Q_R(\\mathbf{p}^{\\prime}),\nQ_L^\\dagger(\\mathbf{p}) \\right] = 0, \\quad\\quad\\quad\\quad (10)\n", "6b8e741ba741e34ee9d4c364783f3231": "[x_{\\sigma(1)}, x_{\\sigma(2)}, x_{\\sigma(3)}] = \\sgn(\\sigma)[x_1,x_2,x_3]", "6b8e9732a03142d4852ec498ae2e1bb4": "F_{X,Y}(x,y)", "6b8e9bcd266a13bf346be8e4cbd10c29": "\\theta=t", "6b8eba05082a145dccb600312492ba2c": "3 + 3 + 3 + 3 = 12", "6b8ebe398fb69a1bb98471eec024e156": "SIVP_{t}", "6b8ebf2a81e65033e376f6682e08c205": "\\Omega_*^{\\text{SO}}\\otimes \\mathbf{Q}=\\mathbf{Q}[y_{4i}|i \\geq 1],", "6b8ef013e9dca79ce5ff82c5d399e343": "Q\\,y'' + L\\,y' + \\lambda y = 0\\,", "6b8f0029ce30f9b4d5fe0def33875511": "FC", "6b8f38d1075aa4172b41e8317b74fc63": "\\mathrm{sinh}(t)", "6b8f86627987a52272d0078de64e8682": "\\int_{-\\infty}^\\infty \\psi_n(x)\\psi_m(x)\\, \\mathrm{d}x = \\delta_{n\\,m}\\,", "6b8fa21c46de1eb13980f8f9084eda66": "\\, c_2 - c_1", "6b8fb5f0aee8bba95cd54f64bb80e7f6": "(\\alpha, \\beta, f)\\mapsto \\beta", "6b903836c007f13336eefd065c54e746": "\\mathrm{^{235}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{236}_{\\ 92}U_m\\ \\xrightarrow[120 \\ ns]{} \\ ^{236}_{\\ 92}U\\ +\\ \\gamma}", "6b904b65fdbdee8c9cb3a2d3e515952a": "\\int_S T_i^{(n)}dS + \\int_V F_i dV = 0\\,\\!", "6b908dcdedcd80cf27a29a90b78ff51c": "\\sin 2\\pi x + \\ln e\\,\\!", "6b9142c6a7e1e484d664b05dcc871d5f": "\\rho = \\text{constant}.", "6b914a510d7445e2068407277d96d352": "\\Pr(A)=\\sum_n \\Pr(A\\cap B_n)\\,", "6b9164a36bd9618189d34f12417eca16": " \\scriptstyle \\bold{\\hat{e}}_r \\,\\!", "6b9183d9801e3445005d555ffd74c3bf": " \\frac{mu(1).output(1)+mu(2).output(2)+mu(3).output(3)+mu(4).output(4)}{mu(1)+mu(2)+mu(3)+mu(4)} ", "6b9244d0a084c7a6d3d812c437bc9cdf": "G = (V_N, V_T, X_0, M).", "6b925522ba100cf7372fe02b1df3b7c4": " \\frac{\\partial x}{\\partial \\lambda} = -y, \\; \\frac{\\partial y}{\\partial \\lambda} = x, \\; x(0) = x_0, \\; y(0) = y_0. ", "6b9286102fb808bd7998885cbc35a629": "\\aleph_{\\lambda} = \\bigcup_{\\beta < \\lambda} \\aleph_\\beta", "6b929faf9f52b6d307f63ac93f99dbb2": "RC = \\frac{(H+BB) \\times TB}{AB+BB}", "6b92d76f285eef92fac1345487a72dac": "\n C_7 = -\\frac{1250}{3EI}(-625 + M_c + 30 R_a) \\quad \\text{and} \\quad\n C_8 = \\frac{125}{EI}(-125 + 6 R_a) \\,.\n ", "6b92f8f74b0f2765504447d42d797283": "T=+(n_\\mathrm{t}-n_\\mathrm{\\bar{t}}),", "6b9330b9bcb1282b2ae26cde18a87ed0": "x_1^s \\, = \\, \\frac{K[x_1^l/(1-x_1^l)]}{1+K[x_1^l/(1-x_1^l)]}", "6b93397f29d732eb5b140c47d8bc769e": "\n\\begin{array}{cc}\n\\text{value} & \\text{frequency} \\\\\n\\hline\n0 & 1 \\\\\n1 & 2 \\\\\n2 & 4 \\\\\n3 & 5 \\\\\n4 & 3 \\\\\n5 & 3 \\\\\n6 & 1 \\\\\n7 & 0 \\\\\n8 & 1 \\\\\n>8 & 0\n\\end{array}\n", "6b942fe23424246258356ce216dd5746": "z \\mapsto \\frac{az+b}{cz+d}", "6b946feb4ff30ded6a2d9504541e2f27": "\\mathbf{w} = \\mathrm{sign}(P)\\mathbf{i} + \\mathrm{sign}(Q)\\mathbf{j}", "6b948b3d5b2d93e5886e4018d274f0d3": "\\frac{K_m}{V_\\max}", "6b948bcba1e0c31f728e9316c0a5531d": " K_1(A,I) \\rightarrow K_1(A) \\rightarrow K_1(A/I) \\rightarrow K_0(A,I) \\rightarrow K_0(A) \\rightarrow K_0(A/I) \\ . ", "6b949159251cae81e882c2261cd38bdf": "c_n = \\sum_{m=0}^{d-1}a_m b_{n-m\\ \\mathrm{mod}\\ d} \\qquad\\qquad\\qquad n=0,1\\dots,d-1", "6b94af866947a3da07fdea82f0fe946f": "\\mathbf{u}_1 = \\begin{bmatrix}1/\\sqrt{2}\\\\-1/\\sqrt{2}\\end{bmatrix},\\quad \\mathbf{u}_2=\\begin{bmatrix}1/\\sqrt{2}\\\\1/\\sqrt{2}\\end{bmatrix}.", "6b94b5dace978eeb07aad6f3b42aa43f": " \\langle q|\\mathbf{\\hat T}(-\\varepsilon)|\\psi\\rangle = \\psi(q+\\varepsilon) ", "6b94c1a786647ecce0a29577d33226b5": "a_{\\mathrm{norm}}= N^b a_b \\,", "6b950dfbd0cfc4ec7df12a0bf97ae032": "|3| \\ge |-2| + |1|", "6b95229092b1cf7e551994ed92e712ab": "u(x_0) = 0", "6b95ca74db8636fbccec57a9b659cb48": " \\mathrm{Pr}[ |T_j| < \\frac{t}{2d}] \\leq e^{\\frac{-t}{12d}} = e^{\\frac{-cd\\log n}{12}} \\leq n^{-2d} [", "6b95f86625cebd9cd8493a36ac927504": "\\dot{v}_2 = {1 \\over C} { { v_1 - v_2 } \\over R} ", "6b96320cc9839f097711cf20d384a755": "eval \\circ (\\lambda t \\circ \\pi_1, \\pi_2) = t, \\lambda eval = id", "6b96529e3efec5f9d24889b8e07b6d00": "\\mathbf{S}=\\mathbf{S}(\\mathbf{E},\\dot{\\mathbf{E}}) \\quad\\rightarrow\\quad \\mathbf{S}=\\mathbf{S}(\\mathbf{E})", "6b9661ac125627f5008cb2d30a61d0af": "\\frac{\\mathrm{d} (T_{h})_{*} (\\gamma^{n})}{\\mathrm{d} \\gamma^{n}} (x) = \\exp \\left( \\left \\langle h, x \\right \\rangle_{\\mathbf{R}^{n}} - \\tfrac{1}{2} \\| h \\|_{\\mathbf{R}^{n}}^{2} \\right),", "6b97cde50bf31cafe8d530558223b067": " d(x,y)= | \\log(x) - \\log(y) |", "6b97e306b1c4e76b7aed8a373bfd7836": "kth", "6b9809c1ad35c6e96966d0920c8f99ff": "A_{t}=\\kappa RA_{d}R^{T},", "6b9812bbdab3fd16be338c627476200a": " x\\ =\\ -ae+r \\cos \\theta", "6b9826e64e05e5ffa2db52f1a6cf3c18": " \\mathcal{B}p \\to \\mathcal{B}q ", "6b989851cd3857fe4f4b661a2ce98198": " x^\\mu=(t,\\vec{x}) ", "6b99240acdf19e0882608f6ea8efbc33": "W = 2 *\\operatorname{atan}[ \\tan(\\pi/4 - \\phi) \\tan(\\frac{1}{2}(\\alpha+\\beta+C))].", "6b9943b09d61da89afccd7067263b3ef": "ALL_{PDA}", "6b998366ea8df9f0f2cd00e1e17b74b6": " -n ", "6b99900e4e41f8b8374bb45ae6788cdc": "B \\cap A \\neq \\varnothing", "6b99b18439228e2ef16159b4bee03246": "\\mathfrak{m}_A \\cap B = \\mathfrak{m}_B", "6b99e50e0ec6a737f9670480541a9631": " E \\ = 1 - \\exp[-NTU] ", "6b99ea8aa22939577edd5dbeec1750c7": "E_1(l, m, t) = A \\left( l, m, t - \\frac{R_1}{c} \\right) \\frac{e^{-i \\omega \\left( t - \\frac{R_1}{c} \\right) }}{R_1}", "6b9a860d3590380f798d319e3760285c": "z_{dn}", "6b9a994aa9e09f29709530600ff0c0af": " \\overline{X} = \\frac{1}{n}\\sum_{i=1}^n X_i\\text{ and }\\overline{Y} = \\frac{1}{m}\\sum_{i=1}^m Y_i", "6b9abf7126ef0af19e68fae358757a43": " \n \\|x\\|_T = \\sqrt{ \\|x\\|^2 + \\|Tx\\|^2 }\\ . \n ", "6b9ae7e3ca4e13e690e525f51c8b8cea": "\\gamma A \\zeta \\delta", "6b9afca68d95493a7c23640382f8b154": "c = c_0, c_1, \\ldots", "6b9b0200ec71d987f83ff9ef61647f50": "p \\times n", "6b9b20ebc451138b4b15f95f946c2683": "m^{\\star} = m\\exp(\\lambda(U_1-0.5))\\ ", "6b9b49a26ac5c61bca001a09a3ff3aac": "Tr = \\frac{E_m}{E_x}", "6b9b4bf589e0b256a212f37a5cfa8dd2": "\\mathrm{Ber~J}=\\det\\left( A-BD^{-1}C\\right) \\det D^{-1}", "6b9b58d8058ecd838982d36286fea579": "F(z) = \\int_\\gamma f(\\zeta)\\,d\\zeta.\\,", "6b9ba3674283240c3dbbed752f6f008a": "M_y ", "6b9bf2e437dbffed37edf65178cd17e8": "1, \\sqrt{2}, \\sqrt{3}", "6b9c0305704c903e350d60c347b1d05c": "\n\\frac{\\left\\| \\nabla\\Phi \\right\\|}{a_0} \\nabla\\Phi - \\nabla\\Phi_N = \\nabla \\times \\mathbf{h}.\n", "6b9c06d94079e0180e6ca22532d67d39": "\\tbinom \\cdot\\cdot", "6b9c214eb8fa87375c9888885a49308c": " \\min(x_1,\\ldots,x_n) \\leq M(x_1,\\ldots,x_n) \\leq \\max(x_1,\\ldots,x_n)", "6b9c7bd06ed47a6702f375eeae016662": "\\sum_{k=0}^\\infty k^n \\frac{z^k}{k!} = z \\frac{d}{dz} \\sum_{k=0}^\\infty k^{n-1} \\frac{z^k}{k!}\\,\\! = e^z T_{n}(z) ", "6b9c904f1c3982c7292e35c03d6340cc": "\\operatorname{char}K=2", "6b9cbf55d5996193ba144ccc8ea3edfe": "\\phi\\mapsto\\langle v,\\phi\\rangle", "6b9cf6b8c1d3b8c91215bcddbaa9f327": "(10^8)^{(10^8)}=10^{8\\cdot 10^8}", "6b9d2db30f7d9666031c14d23a6bc234": "Z=\\sum_{n=0}^{\\infty } \\frac{(6n+1)\\left ( \\frac{1}{2} \\right )^3_n} {{4^n}(n!)^3}\\!", "6b9d3812c0cd9414c2dc0226ed880431": " = \\frac{12!\\cdot 8!}{2} \\cdot \\frac{2^{12}}{2} \\cdot \\frac{3^8}{3} \\sim 10^{20}", "6b9d5bc27c23f58cda8195c02d6f37f8": "z' = \\frac{a}{a + b}", "6b9d62d6c5af07e5ac65c8c4d44383c9": "\\psi(\\Omega2) = \\tilde\\psi(\\Omega+1)", "6b9d6b844da0c53566097172e50624b8": "\n\\begin{align}\n&\\eta_i=\\sum F_i\\ell^{(-\\alpha)}\\\\\n&\\partial_j\\eta_i=g_{ij}=\\sum{\\partial_i\\ell^{(\\alpha)}\\partial_j\\ell^{(-\\alpha)}}=\\sum F_i\\partial_j\\ell^{(-\\alpha)}\\\\\n&\\Psi^{(\\alpha\\neq -1)}(\\theta)=\\frac{2}{1+\\alpha}\\sum p\\\\\n&\\Psi^{(\\alpha=-1)}(\\theta)=\\sum p(\\log p-1)\\\\\n&\\psi(\\theta)=\\Psi^{(\\alpha)}\\\\\n&\\phi(\\theta)=\\Psi^{(-\\alpha)}-\\sum C(x)\\ell^{(-\\alpha)}\\\\\n&D^{\\alpha}(p||q)=\\Psi^{(\\alpha)}+\\Psi^{(-\\alpha)}-\\sum\\ell_p^{(\\alpha)}\\ell_q^{(-\\alpha)}\\\\\n&D^{\\alpha\\neq\\pm 1}(p||q)=\\frac{4}{1-\\alpha^2}\\sum\\{\\frac{1-\\alpha}{2}p+\\frac{1+\\alpha}{2}q-p^{\\frac{1-\\alpha}{2}}q^{\\frac{1+\\alpha}{2}}\\}\\\\\n&D^{\\alpha=\\pm 1}(p||q)=\\sum \\{p-q+p\\log\\frac{p}{q}\\}\\\\\n&\\theta^i\\eta'_i=\\sum\\{\\ell^{(\\alpha)}(v;\\theta)-C(v)\\}\\ell^{(-\\alpha)}(v;\\theta')\\\\\n&D(\\theta||\\theta')=\\psi(\\theta)+\\phi(\\theta)-\\theta^i\\eta'_i\n\\end{align}\n", "6b9d956446d49b04c32ec487ec5e93ba": "X \\sim \\operatorname{Beta}(d_1/2,d_2/2)", "6b9da47bb2e1e45fa6514ae40f51d65d": "\\mbox{lrd}(A):=1/\\left(\\frac{\\sum_{B\\in N_k(A)}\\mbox{reachability-distance}_k(A, B)}{|N_k(A)|}\\right)", "6b9dbdefdd0b4b76cb4f167dfd6d8a7a": "(y_{r}, \\ldots, y_{n})", "6b9e68070b0fb6a866d93fb07a547bf2": "2^{n+1} ", "6b9e69cd27aeb61d91101d27a8f61cac": " \\boldsymbol{ \\nabla \\times B} = \\mu_0 \\boldsymbol J_f \\ , ", "6b9e899fee0f51141f10b8e25e5d49d5": "D_{192} \\approx \\tfrac{1}{4} D_{96}", "6b9f53e616cd5bce4ddf7d2403230610": "+ 7 \\cdot 8^3 + 7 \\cdot 8^2 + 7 \\cdot 8 + 7", "6b9f71c88842207650c93a5ab9ff540d": "\\chi = \\frac{y - y_f}{y_g - y_f}", "6b9fab20827972879546f16f75c12c61": "\\mathbf{E}_{scat}= \\sum_{n=1}^\\infty \\sum_{m=-n}^n f_{mn} \\mathbf{M}^3_{mn}+ g_{mn} \\mathbf{N}^3_{mn}.", "6b9fc65c93e82ed58810769242b14930": "h_1 + h_2 = L\\,", "6ba05e5afcee0ee566d3d6c10d69a95f": "T^2", "6ba07fa19946c919b7a13eaff876b3c0": "q \\in [0;\\infty)", "6ba0c349ca9a9bdd1a1ccd89714df575": "|z| < 9", "6ba13d2d9426f3f6bb2c03073b8de69c": "V_{max} \\ = \\ r_{m}I", "6ba156769337b146780e8210ac4be629": "T0.5", "6baacb591204bb720da2aac8de453cb0": "\\scriptstyle\\pi_0(x)", "6bab3d9dc2286fc4f0b3da72f939dbdd": "F_n = F_{n-1} + F_{n-2}.", "6bab4d3e96e2783d5a05e647a4e7ecaa": "\\Psi_0 \\,", "6bab7acc6209d8d36e9dba68307e59b4": "\\operatorname{sgn}f = \\begin{cases} +1, & \\text{if }f\\mbox { is even} \\\\ -1, & \\text{if }f \\text{ is odd}. \\end{cases}", "6babafa07a18a18d0b2197646c48a3a9": " x^2+y^2+z^2 =R^2,", "6babd05ba0ae7e68eebcdce3f6a9e4d7": "\n\\| u \\|_{L^{4}} \\leq \\sqrt[4]{2} \\, \\| u \\|_{L^{2}}^{1/2} \\, \\| \\nabla u \\|_{L^{2}}^{1/2}.\n", "6bac52c61e7a413495425f7bbfae252a": "\\gamma_{\\pm} = \\left(\\gamma_\\mathrm{Na^+}\\gamma _\\mathrm{Cl^-}\\right )^{1/2}", "6bac8e8f77d7c4c0cbd452540917840c": "C_{abcd} \\, C^{abcd}", "6bad23d0aa165b8cef61d4c71a3e0b38": "y''+4y'+4y=0.\\;\\!", "6bad702492e4c6be3040288ee0c94b31": "\\overline{B}", "6bad9d407bce14e8ba2dfbc38023d310": "pos_i", "6badb8014505449c027df5149dfb99b3": "m = 2595 \\log_{10}\\left(1 + \\frac{f}{700}\\right) = 1127 \\log_e\\left(1 + \\frac{f}{700}\\right) \\ ", "6badf1ed8de6f4f403830cc32466b725": "I(\\beth_n(T), \\lambda)= \\min(\\beth_n(I(T,\\lambda)), 2^\\lambda)", "6bae7134358a250be76ad5f8149446f4": "\n\\lambda_0 + \\lambda_1 X_{1i} + \\lambda_2 X_{2i} + \\cdots + \\lambda_k X_{ki} = 0\n", "6bae91179279d652d139c976883783b5": "\\varsigma = \\frac{c}{2\\sqrt{k m}}", "6baf326d8b1a76caf91d5062149f6349": "\\,s^3", "6baf57f45ebbab44c62d0c7b347c783e": "\\frac {1}{z_+ z_-}\\log \\gamma_{\\pm}", "6baf6f967fb59fa5cae927b1f96efca4": "f(x)=\\begin{cases}\nmx+y_1-mx_1 & \\text{if } 0 \\leq x < x_1 \\\\ \n0 & \\text{if } x \\geq x_1 \n\\end{cases}", "6bafb5b6eb4d27fb0d14fab997312683": " \\operatorname{tr}_{H_A}(P(\\sigma)) = \\operatorname{tr}_{H_A} \\left(\\sum_k (V_k \\otimes I_{H_B})^* \\sigma (V_k \\otimes I_{H_B} )\\right) ", "6bafbe1494abf1cfdeef5eb8d903bf18": "\\scriptstyle x^2\\ -\\ y^2\\ =\\ 1", "6bafd81182ac9530bcd123a4a1a261e8": "1^n", "6baff0a02691064a4c36986f245776b6": "a=\\frac{x}{10^{\\left( \\frac{\\sigma_m / \\sigma_w - 1}{-0.411} -1\\right)}}", "6bb002d9cbd27eeffcf61d7c3e5b928b": "w\\sqrt{T}/{P}= w\\sqrt{\\theta}/{\\delta} * \\sqrt{288.15}/{101.325}", "6bb09c5afb2bf7304c263ae99f0e5976": " B = Ke \\oplus U_e \\oplus Z_e ", "6bb0cbea7eec048135ec2fc6d9f9d5af": "(\\forall x \\phi ) \\rightarrow \\psi", "6bb1223f4aded3c5b118e4fb60f8e276": "N_{\\alpha}", "6bb14d89f1fa5ea2437c2f2749896555": "\\frac{\\sigma_{1-s}(m)}{\\zeta(s)}=\\sum_{n=1}^\\infty\\frac{c_n(m)}{n^s}.", "6bb17db02d87cb39b59ed7cd4efd595e": "w(\\sigma) \\leq \\frac{\\sum_i{w_i}}{\\max_j{s_j}}.", "6bb1981c87fea6dddf06398b833e7dc3": "\\arctan(2)", "6bb1a5c269ec556003d0ec19475cfa1a": "+ {13 \\choose 1}{4 \\choose 2}{12 \\choose 3}{4 \\choose 1}^3 - 97,536", "6bb1a9cf4743a43aafe447380c9c6992": "V =\\hat{\\mathbf{k}} \\cdot \\frac{\\nabla\\times\\mathbf{\\tau}}{\\beta}", "6bb1b81bb9ed977236d23e36ef20168e": "M = \\{t_1,\\ldots,t_m\\}", "6bb1b8c8516d29d06f4f8eae8790985e": " \\tilde{\\chi}_1^0, \\ldots, \\tilde{\\chi}_4^0", "6bb1e313ef511439990030cdeb84bc1c": "\\alpha_{t+1}", "6bb1eebadb4cedf511478c482c5e27f2": "Z_0 = 50 \\ \\Omega\\,", "6bb2600ee8f5612c850517d1f825c42b": "e\\# f", "6bb27376b75df6d38bed556f51fa59e5": "O(1/\\epsilon)", "6bb28b16d71bb2a6c5d2ca1b7a06425d": "\\textstyle\\binom{n - k}{k}", "6bb2c16cdf85300ca8a969befcc246c3": "\\Gamma(t)", "6bb2ce8c6c69341b43f38f908bdd5ca2": "k = \\frac{1}{4 \\pi \\epsilon_0} \\approx 8.988 \\times 10^9 Nm^2{C^-}^2", "6bb2f92f5975b27ccd3ae427a3898f9f": "\\operatorname{Perf}(f,r) = \\operatorname{Prob}(f(x)=r(x)) - \\operatorname{Prob}(f(x) \\neq r(x)).", "6bb31a9324095deda461c0dbd25d6c10": " \\bigwedge_{j\\in J}\\bigvee_{k\\in K(j)} x_{j,k} = \n \\bigvee_{f\\in F}\\bigwedge_{j\\in J} x_{j,f(j)}\n ", "6bb332ab91a13302c0113ceac53e2767": " S_j=\\frac{V_j}{V} ", "6bb3807265ecf60386db16dcfde6407f": "=E[\\epsilon^2] + E[(f_i-g_i)^2] + 2\\left ( E[f_i y_i] - E[f_i^2] -E[g_i y_i] + E[g_i f_i] \\right ) ", "6bb3c90b1e5ab29647bf0304e40d7c04": "f: A \\rightarrow A'", "6bb4094c4d9e7644439ffdb2267508e8": "\\forall \\ \\exists \\ \\nabla \\ \\wedge \\ \\infty.", "6bb41639eb166d4e0a34625effa493d4": "\\{c, \\neg c \\vee d, a\\}", "6bb44508205acf6d7c4886629d5890b1": "\\mathbf{p}=(p_1,p_2,p_3)", "6bb47e05f16534365e334ca2ad88b16c": " L(x) = \\exp \\left( \\eta(x) + \\int_B^x \\frac{\\varepsilon(t)}{t} \\,dt \\right) ", "6bb48f60c583a61a368578c1a298f7a1": "\\rho(f) = \\frac{1}{V}\\int_\\Omega \\delta(f - F(\\boldsymbol{r})) \\,d\\boldsymbol{r}", "6bb4955d4a6b93fd5216f4f2c3fda1c7": "n < N", "6bb4ef297d971dfada1dd0aad3f2d751": "\\begin{align}\n\\mathbf{L}^2 &= -r^2\\nabla^2 + \\left(r\\frac{\\partial}{\\partial r}+1\\right)r\\frac{\\partial}{\\partial r}\\\\\n&= -{1 \\over \\sin\\theta}{\\partial \\over \\partial \\theta}\\sin\\theta {\\partial \\over \\partial \\theta} - {1 \\over \\sin^2\\theta}{\\partial^2 \\over \\partial \\varphi^2}.\n\\end{align}", "6bb52049062e33ed64ac40abc81fde6a": "[\\forall x_1 \\exists y_1 \\forall x_2 \\exists y_2\\, \\phi (x_1, x_2, y_1, y_2)] \\wedge [\\forall x_2 \\exists y_2 \\forall x_1 \\exists y_1\\, \\phi (x_1, x_2, y_1, y_2)]", "6bb59841e02e2edeb188423baea94603": "\\Delta t' = \\gamma \\left(\\Delta t - \\frac{v \\,\\Delta x}{c^{2}} \\right)", "6bb59f0e2c99b208d6fdca145923300f": "h_{\\overrightarrow{X}}(I)=H(X_I)=H(X_{i_1},X_{i_2},\\dots,X_{i_k})", "6bb5a5de0a2cfccac223d9e0a981b2db": "\\langle j_1j_2;m_1m_2|j_1j_2;jm\\rangle=(-1)^{j-j_1-j_2} \\langle j_2j_1;m_2m_1|j_2j_1;jm\\rangle", "6bb5e7f82b512c05bb769d601e487d8a": "\\frac {ct}{r_e} \\approx 10^{40},", "6bb5ede3b15a43b1ca48a6cb509fcadc": "\\mu_n(X+c)=\\mu_n(X).\\,", "6bb606b9a24f7da1bb7091c39dc91663": "\\alpha=2\\eta\\omega^2/(3\\rho c^3)", "6bb61e3b7bce0931da574d19d1d82c88": "-1", "6bb6937641e3e2943f12182bd66796f8": "F(\\nu) = \\mathcal{F}\\{f\\} = \\int_{\\mathbb{R}^n} f(x) e^{-2 \\pi i x\\cdot\\nu} \\, \\mathrm{d}x ", "6bb75e2a1d5722be6e06de1069216502": "\\Delta t(M)=M+\\lambda_p - \\arctan_{M+\\lambda p}\\left(\\cos\\varepsilon\\tan\\lambda\\right). ", "6bb792e42d6ab3fc4587e5bf41809acb": "\\frac{d^2y}{dj^2} + \\frac{1}{j} \\frac{dy}{dj} + \n\\frac{31j -4}{144j^2(1-j)^2} y=0.\\,", "6bb7d737d1f85448fd96da2da2314e7c": "x', y',\\ldots", "6bb7e8c8635d81cd8d9d7528c5c0129d": "\\Delta f = \\delta \\, df. \\,", "6bb88d5dbdc61bae1a1ef6230dce7c7e": "\\mathbf{a} = \\lim_{\\Delta t \\to 0} {\\Delta \\mathbf{v} \\over \\Delta t} ", "6bb9556d3371d1db1944fcbf9f9bcc47": " \\Delta \\tau = \\int_{t_1}^{t_2} f(r) dt ", "6bb966e6d45985d461767f4a5156756b": "\n\\begin{align}\n\\frac{s}{p_1 p_2}\n&=p_3 \\cdots p_m \\\\\n&=q_3 \\cdots q_n.\n\\end{align}\n", "6bb96da23e9dffaf22f2fb0dd7cdacd3": "\\nabla_{\\nu}", "6bb97373b8bf1159939b2f2ff850c1ec": "a^2 + b^2 = 5c^2.", "6bb9da7f589ca0fadbb8dc64e3d7774b": " \\scriptstyle {\\color{white}...\\color{black}C_1\\;\\;\\;\\; C\\#\\;\\;\\;\\, D\\;\\;\\; D\\#\\;\\;\\, E\\;\\;\\;\\;\\, F\\;\\;\\;\\, F\\#\\;\\;\\; G\\;\\;\\;\\; G\\#\\;\\;\\; A\\;\\;\\;\\, A\\#\\;\\;\\;\\, B\\;\\;\\; C_2} ", "6bba0bfb25d9f14528e0189ea6725059": " \\begin{bmatrix} 1 & \\mathbf a & c \\\\ 0 & I_n & \\mathbf b \\\\ 0 & 0 & 1 \\end{bmatrix} \\cdot \\begin{bmatrix}1 & \\mathbf a' & c' \\\\ 0 & I_n & \\mathbf b' \\\\ 0 & 0 & 1 \\end{bmatrix} = \\begin{bmatrix} 1 & \\mathbf a+ \\mathbf a' & c+c' +\\mathbf a \\cdot \\mathbf b' \\\\ 0 & I_n & \\mathbf b+\\mathbf b' \\\\ 0 & 0 & 1 \\end{bmatrix} ", "6bba1052f183c050e01d9b8fd5bc332f": "\\{\\mathcal{F}^*g\\}(s) = \\{\\mathcal{L}^*g\\}(\\mathrm{i}s), \\quad s \\in \\mathbb{R}.", "6bba3f84e3f2f20609adac81137aef07": "{\\partial^n \\over \\partial x_1 \\cdots \\partial x_n}f(y)\n= \\sum_{\\pi\\in\\Pi} f^{(\\left|\\pi\\right|)}(y)\\cdot\\prod_{B\\in\\pi}\n{\\partial^{\\left|B\\right|}y \\over \\prod_{j\\in B} \\partial x_j}", "6bba4d01ae87d23e9c559fc10358cf47": "\\mathcal{N}^{DIFF}_\\partial (S^n \\times I) = \\Omega^{alm}_{n+1}", "6bba8f34e20d4d1c0ce47f84c439271e": "\\sigma_1^{ }", "6bba99f7c7d390634cb180548d90b7bf": "f:I\\rightarrow\\mathbb{R}^+", "6bbb333c6a026a842c6085a07e9b6e40": "r\\equiv\\alpha\\sinh \\rho", "6bbb81f9c84306d7261b8d7f24cbc948": " \\tau_{dynamical} \\simeq \\frac{R}{v} = \\sqrt{\\frac{R^3}{2GM}} \\sim 1/\\sqrt{G\\rho} ", "6bbb880619e97178e4117bac9e542af5": "\\ddot{M}_{i,j} = H_{i,j(i)} M_{i,j} ", "6bbb9f296d967df4cde88ef2f0053d9d": "\\begin{bmatrix}0 \\\\ 0\\end{bmatrix}", "6bbbeae9922c8e5dd31d464be6b0a562": "\\frac{y}{x^2} = C\\,", "6bbc8d191bf3903f5e9c9ad777083727": "\n[M_{ij}, Y_{kl}]= -\\delta_{jk} Q_{il} ~~~~~\n", "6bbca52d234b18378198d49925b02ddb": "f(\\phi) = 0", "6bbcae9dbfd813a864f26c5d82546663": "o \\odot c = a \\odot (b \\odot c)", "6bbcc80372650af9a2bbdcbb6380fcf1": "\\mathcal{B}A(t) \\equiv \\sum_{k=0}^\\infty \\frac{1}{k!}t^k = e^t,", "6bbd00ef44eace4b0543a6370faa823d": "R = {R^a}_a, \\; \\; G = {G^a}_a = -R", "6bbd0baf55c028a2276408143d33cd44": "(y+1)", "6bbd300b2741da9b38de3496edc4bf25": "m!=\\Gamma(m+1)", "6bbd641e34613e4f0e4c5b35fea14291": " X^{ab} \\rightarrow \\frac{1}{2} \\, {C^{ab}}_{mn} X^{mn} ", "6bbd675ade73965b9ca9913d1782ee48": "B^s_{2,2}(\\R)", "6bbd8794f719671730e55941a45b3bb7": "I_M(\\tau) = \\int_{-\\infty}^{+\\infty}|(E(t)+E(t-\\tau))^2|^2dt", "6bbde4a55f40dbc3b5aa02c6cdfcd265": "f(p) = X \\to p", "6bbdf8847a4f0835005469b2e9705f28": "n + o(n)", "6bbdfc90ff89ed639eb935c388a3987b": "2^{5\\,.}\\mathrm{GL}_{5}(\\mathbb{F}_{2})", "6bbe062e2414d1c37f83605665b1f9af": " \\operatorname{E} (X | Y=y) \\operatorname{P}(Y=y) = \\sum_{x \\in \\mathcal{X}} x \\ \\operatorname{P}(X=x,Y=y), ", "6bbe08083b7a9f13b0c1199162cc0450": "P(j,t,q)", "6bbeb7a11ff7498297494e49ee8f1187": " \\nabla \\psi \\cdot \\nabla \\psi = n^2, \\,", "6bbec1f8f64b180bd8ce865eb3fe63f4": "\\forall x, y, z\\,\\left(\\left(x, y\\right) \\in F \\wedge \\left(x, z\\right) \\in F \\to y = z\\right)", "6bbed6ad08a86b35e4494be441af74ba": "\\delta(x)=\\frac{\\int_{-\\infty}^{\\infty}{\\theta f(x_1-\\theta,\\dots,x_n-\\theta)d\\theta}}{\\int_{-\\infty}^{\\infty}{f(x_1-\\theta,\\dots,x_n-\\theta)d\\theta}}.", "6bbee0dbe3eeaff465e3fb574e623c4b": "\n \\Pr\\!\\big[\\,\\operatorname{rank}(X) = p\\,\\big] = 1\n ", "6bbf15cbe8692960591566892b1c99ad": "\\scriptstyle \\left(\\partial/ \\partial n\\right)", "6bbf5c2c1c9872faf067a560b00e2855": " E\\left[e^{-tX}\\right] = \\frac1{ \\lambda^k\\, t^k} \\, \\frac{ p^k \\, \\sqrt{q/p}} {(\\sqrt{2 \\pi})^{q+p-2}} \\, G_{p,q}^{\\,q,p} \\!\\left( \\left. \\begin{matrix} \\frac{1-k}{p}, \\frac{2-k}{p}, \\dots, \\frac{p-k}{p} \\\\ \\frac{0}{q}, \\frac{1}{q}, \\dots, \\frac{q-1}{q} \\end{matrix} \\; \\right| \\, \\frac {p^p} {\\left( q \\, \\lambda^k \\, t^k \\right)^q} \\right) ", "6bbf6a0b4badb095886da44504a88a64": "\\frac{7}{6}", "6bbff560380251237fa7f52f837b1eb3": "\\det(XD+(n-i)\\delta_{ij}) = \\det(X) \\det(D). \\, ", "6bbffdda7878d8727b8139b2f38fb496": "\\textit{dau}(x_{me},x_{ht})", "6bc05e7d5effcb21d93520e2b649cf19": "m(a\\otimes b) = ab", "6bc0a105f548ad0d49b11dd692f2c9b2": "S_{v\\times v}", "6bc0c2ddc318414c494c5451cebea3b9": "\\zeta(s)=\\sum_{n=1}^\\infty \\frac{1}{n^s}", "6bc0f6e22042e2dba53e7ccd536279c5": "\\mathrm{lift}: \\mathrm{M} \\, A \\rarr \\mathrm{M} (A + E) = m \\mapsto \\mathrm{bind} \\, m \\, (a \\mapsto \\mathrm{return} (\\mathrm{value} \\, a))", "6bc11e49bbdb4d287d7cbb6a358d3245": "\n\\begin{bmatrix}\n0 & j^T\\\\\nj & Q\\end{bmatrix}\n", "6bc11ff03106e8cd353c94708198e8cf": "p'=h^rg^x", "6bc17f2246bb8c53833366a4d79a90b5": "\\alpha=-7/2\\,", "6bc1c7fc54a06edc92d1c813c576b0cc": "\\alpha d = 1", "6bc1fc02ec6b8dc225c59b1df4625fdb": " \\rho = r \\cos \\theta \\,", "6bc21551734f5cd98e04f360506db9fc": "B_{n,m}^{(\\alpha,\\beta)}(x)=\\frac{a_{n,m}^{(\\alpha,\\beta)}}{x^{\\alpha+m} e^{\\frac{(-\\beta)}{x}}} \\left(\\frac{d}{dx}\\right)^{n-m} (x^{\\alpha+2n} e^{\\frac{(-\\beta)}{x}})", "6bc26e54096880381c8b67a8a0d475b5": "=\\frac{(q;q)_n}{(1-q)^n}.", "6bc29ea1a0c94d13b63575578fb357ca": "E_j = \\sum_i E_j^i=E_j^{trans}+E_j^{rot}+E_j^{vib}+E_j^{e}", "6bc2d2f1bc9fe679fef7019baf6ab472": "|\\mu(\\varepsilon)|^2", "6bc33805d0c66333cda031d517221360": "\\sum_i \\gamma_i c_{V,i} = \\sum_i \\left[- \\frac{V}{\\omega_i} \\frac{\\partial \\omega_i}{ \\partial V} \\right] \\left[ k_B \\left(\\frac{\\hbar \\omega_i}{k_B T}\\right)^2 \\frac{\\exp\\left( \\frac{\\hbar \\omega_i}{k_B T} \\right)}{\\left[\\exp\\left(\\frac{\\hbar \\omega_i}{k_B T}\\right) - 1\\right]^2} \\right]", "6bc34fa1a1427bc28b3419bde366e613": " F_\\text{P} = \\frac{m_\\text{P} l_\\text{P}}{t_\\text{P}^2} = \\frac{1}{4 \\pi \\varepsilon_0} \\frac{q_\\text{P}^2}{l_\\text{P}^2} ", "6bc3c871f952f45f8e1c900f3ef213a8": " \\mu = Gm \\,\\!", "6bc418c88a959ebccd31db84014953a1": "E \\in \\operatorname{FV}[G] \\and E \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] = (\\lambda E.G\\ H)\\ Y ", "6bc45184fd51eec719589431bf49525c": " U_p ", "6bc4c333dcbcce9459dc2fbcde5fec1b": "\\textstyle \\zeta_G(\\alpha)=2\\zeta(\\alpha)", "6bc4cf7c3397a03746d50bbb64aab04b": "L = \\begin{bmatrix}\n l_1 e^{-\\rho z} & l_2 e^{-\\rho z} \\\\\n m_1 e^{\\rho z} & m_2 e^{\\rho z} \n\\end{bmatrix},", "6bc4de285c5fb08d2069848ce8dd6ddf": "\\rm \\dot h =", "6bc4e539dc3f4aedfbec13c3d0288bf6": " f(x) \\sim \\sum_{n=0}^\\infty a_n \\varphi_n(x) \\ (x \\rightarrow L).", "6bc4fc4f749b4975414689ede9bbc88d": "N = \\sum_{d=1}^{M} N_d", "6bc51077d06984cc114d07885ad0cea7": "\\mathbf{X} \\in \\mathbb{R}^{n \\times p}", "6bc51732702312366017a7aadb013b3d": "-\\pi < \\phi \\le \\pi \\,", "6bc51a15d4bfce4b5e7428c6cd601235": "\\left\\{{3\\atop5}\\right\\}", "6bc538175aedc7dae21056b0d3069c45": " M[f] ", "6bc5533001ac598cdd45f03c5c5e9da5": "E_n(x) = (x + n) \\mod {26}.", "6bc5c0850a989d967e85c37544241718": "\nA_1 = 10^\\frac{L_\\mathrm{dB}}{20} A_0 \\,\n", "6bc6327c2b955ecca143966066c9631b": "(s-b_q)", "6bc67798a38cb4dc85eb48267ff7a826": "\\mathcal{F}(f) := \\hat{f},", "6bc696dd378ba1ab50333f8e0d8f3ec5": "\\sin\\alpha =\\frac {O}{H},\\,\\cos\\alpha =\\frac {A}{H},\\,\\tan\\alpha =\\frac {O}{A},\\,\\sec\\alpha =\\frac {H}{A},\\,\\cot\\alpha =\\frac {A}{O},\\,\\csc\\alpha =\\frac {H}{O}.", "6bc69c349a5b28f22218f641b1bfbbd4": "\\tilde{H}_n(SY)\\cong \\tilde{H}_{n-1}(Y)", "6bc6d30a1dceafc6185991f96bd0f7e5": "Z\\left(\\beta\\right)=\\sum_{r=0}^{\\infty} e^{-\\beta E\\left(r\\right)}=\\frac{e^{-\\beta\\varepsilon/2}}{1-e^{-\\beta\\varepsilon}}.", "6bc6d5a076620cf969477d56faf03887": "\\ x \\ne c", "6bc6e2ececfad17ecee62811d0568109": "E[x_1x_2] = E[x_1]E[x_2]", "6bc701b3ecf6da63b18b8d88408e0cef": "Vect_1(X)", "6bc763821f09911baf1dbdaf44d21304": " v^2= v_0^2 + 2a(r-r_0).", "6bc78515e5d194a5fb8a661c44353232": "2am+b=0", "6bc7b7306bff35387c5728fc859d486c": "\\scriptstyle{\\hat{U}(t_1,t_0)}", "6bc7e010fd052fb71ce43eb7a4f471b9": "2N-1", "6bc7ed1524b8fa7e94ebd40e3f92aca2": "\\sqrt {\\rho \\ \\exp (a m)} = \\sqrt {\\rho} \\ \\exp (a m /2)", "6bc829c82e2759e16c194653d8c37756": "=\\kappa_4(W)\\kappa_1(Y)+4\\kappa_3(W)\\kappa_1(W)\\kappa_2(Y)\n+3\\kappa_2(W)^2 \\kappa_2(Y)\\,", "6bc837aec783cbe95361f2b944d44f79": "\\theta_{n+1} = \\theta_n + \\Omega - K \\sin(\\theta_n)", "6bc87ef48d2059eccdeef7c531a1a516": " D(k) = \\sum_{m=1}^{M} \\sum_{n=1}^{N}{ E (m,n,k+1) - E (m,n,k) }\\,\\!", "6bc89234fe0aa90e01d542c3d1a1908f": "\\Psi \\propto (-\\tau)^{\\beta}", "6bc8c259a9f91471167ba6ea1e9cec4b": "\\boldsymbol{\\sigma}(\\mathbf{x},t)", "6bc8cc869db1b2ced5e59c7ee2e95fd6": "\\bigcup_n \\sigma(E_1, \\ldots, E_n)", "6bc8e2f70ff22f2d3662f7252606b856": "A_m(2,2) = 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, \\ldots = A_{m+1}(2,1)", "6bc8ff6d30934e37c8594d3f2273d9e6": "\\Phi'\\colon X\\to X^*", "6bc9007e3d9279dcbeadd575450f4a6a": "\\displaystyle \\mathrm{d}s^2 = (\\mathrm{d}x^0)^2 - (\\mathrm{d}x^1)^2 - (\\mathrm{d}x^2)^2 - (\\mathrm{d}x^3)^2", "6bc93f3aa45bb1c379e2b4c1b6518e65": "\\left\\{ {{x}_{n}} \\right\\}", "6bc944d8e18c50a73ec3f8ef5ef92f5a": "\\theta=4N\\mu\\,", "6bc9d5eb52c54d237b8af32e1a701763": "\\varphi^2 = \\varphi + 1,\\,", "6bca1a9d76b04a76161e4ffd54db4d62": "\\left( {1 - \\varepsilon } \\right)N", "6bca4969547c3a77d4fb04653eebc44b": "-\\log_2 p(X=x)", "6bca576a372962df6c451f08929f9df4": " \\mathbf{F} = \\sum_{i=1}^N \\mathbf{F}_i,\\quad \\mathbf{T} = \\sum_{i=1}^N (\\mathbf{R}_i-\\mathbf{R})\\times \\mathbf{F}_i, ", "6bcab63945b00493eded1984f98b01e0": "C(1)", "6bcad7baf9be1c2c442cea20be66a9de": "\n\\text{Performance rating} = \\frac{\\text{Total of opponents' ratings } + 400 \\times (\\text{Wins} - \\text{Losses})}{\\text{Games}}", "6bcb3457b39bcdb6012a3116408c352a": "S(D)", "6bcb5aa3be8aaab499bb108591f73d65": "\n (I)\n ", "6bcbe6e7ad99ff900f56509cce48c022": "\\zeta_\\text{max}", "6bcc11479d49bc6498dccf36e7231a19": "\\tbinom n0=1=\\tbinom nn", "6bcc2436762d46d651b9fc9921d77ef6": "1/r\\,", "6bcc3b4c88f6654f6a01b5a61b5b26fd": "4^3", "6bcc91dcbd7fbcc5ad1437f3eb7967a2": " s_{(d,-)} ", "6bcc96ee51d0ad8e2bfb17896387b997": "\\angle H(i\\omega)= -\\arctan {\\left (\\frac{2 \\zeta r}{1-r^2} \\right)}. ", "6bccaa8f393e07d366370c49dce4604b": "\n\\begin{align}x \\gamma_0 &= x^0 + x^k \\gamma_k \\gamma_0 \\\\ \\gamma_0 x &= x^0 - x^k \\gamma_k \\gamma_0 \\end{align} \n", "6bccbfd0aff91156ba346a6105c8184a": "\\pi\\tau", "6bccc000dcd6cdd54eeac7d8bb825cd4": "\\Gamma, \\Delta, \\Sigma", "6bcd193107c58dc09ae5e7232b82fdbd": "\\varphi(e_i) = \\sum_{j = 1}^n A_{j,i} e_j \\quad\\text{for }i=1,\\ldots,n.", "6bce3f0b250aca2ba2f40bcb31779178": "\\alpha(p_{1}, p_{4})", "6bce49cdb54502761abef3625856250e": "f(z)=f(z+\\omega)", "6bce8dea34dca6de99f898552ac31489": "q \\in Q", "6bce973652c11f8d5404b54c6556b63c": "m_p \\le p+1+2\\sqrt{p} = (\\sqrt{p} + 1)^2 \\le (N^{1/4} + 1)^2 < q", "6bcea55566ae289372a3ff09be7c3c07": "p\\in(0,1)", "6bceb596946ecd82d98e85aa1ae75af9": "n_i = nx_i\\,", "6bcf0883f89b9518cb7bd0aa9fa7a84c": "\\ln_q(x) = {{x^{1-q} - 1} \\over {1-q}} ", "6bcf2afc6addf801736d0ff07b9f929c": "\\displaystyle s= O\\left(\\frac{n}{\\epsilon \\log (n/\\epsilon)}\\right)^2", "6bcf49137ca9b27cc7985487ee45131b": "\\Gamma_\\gamma(s)=\\{\\dot\\gamma^i\\dot\\gamma^j\\Gamma_{ij,k}+\\ddot\\gamma^jg_{ij}\\}\\dot\\gamma^k/g_\\gamma", "6bcf652349a0eee5f74c745b35a7bfd1": "\\Pr[y_i^{\\prime\\prime} = ?] = {2\\omega_i \\over d}.", "6bcf7e50c4c001c19377210ab204d6b8": "\\beta<\\gamma", "6bcfdb27ecd9ec8d171cfba220aaec65": " \\lim_{x \\to p}f(x) = L, \\, ", "6bd039b12fc0eecdc5cc0592f1441b25": "{\\mathbf{u}}(t)", "6bd043a97e6b00bd20254c036d29c210": "d\\quad", "6bd047d896b100951a4fd2db7075e2c0": "D = R1(A)", "6bd095a839d168313c0e0ee93b8ea58d": "a_1 = \\frac{-b_1 - \\sqrt{b_1^2 - 4 b_2 b_0}}{2 b_2}, \\!", "6bd09a47d507e8246226735a73907149": "n+\\frac{1}{2} = \\frac{\\lambda}{\\Delta x}\\,\\!", "6bd0f76d4552a6b237d9795aca2aee2c": "u(x)=\\sum_{k=1}^n u_k v_k(x)", "6bd11ddc45de5eb67349ec8df9c56faa": "\\Omega(\\mathbf e) = d\\omega(\\mathbf e) + \\omega(\\mathbf e)\\wedge\\omega(\\mathbf e).", "6bd16ebc4082dd2a680e49123f03acfd": "r, p", "6bd1d3b6a20088ae0c799497e1a30f7b": "\\displaystyle{\\int_{\\partial\\Omega} u \\, \\partial_nv - \\partial_nu \\, v=\\iint_\\Omega \\, \\Delta u \\, v - u \\, \\Delta v.}", "6bd23146029e919c7abdd78242adab29": "t_0=0", "6bd24c34cd49f6f667a2a6e1ba88f94c": " C_i = C_j g ", "6bd2564e8b19c5f635b5a94459659a1b": "2^{1-n}\\Lambda_n(-z^2) = \\Lambda_n(z)+\\Lambda_n(-z)\\ ", "6bd2752284a9f4b8489c0c22a0fb07f7": "\\mathbb{P}(X_1>0) = \\mathbb{P}(X_1>0, X_2 \\in \\mathbb{R})", "6bd2b153d9df088f454c8856b6d622ec": "m_1, m_2, ..., m_k", "6bd2c8b76cece6262136c7379e122ad3": "|\\{v \\in R | N(v) \\cap V(y)\\,", "6bd2f31bd7f65e5a993013cb1ef8fd8c": "V_d\\ ", "6bd2f7973049727cc5d6b63a092e42bb": "dx/dt = v_{A|O}", "6bd3140eed3649372e6222eb1c81b51a": "\nf(x,y) = \\begin{bmatrix} \\ 0.00 & \\ 0.00 \\ \\\\ 0.00 & \\ 0.16 \\end{bmatrix} \\begin{bmatrix} \\ x \\\\ y \\end{bmatrix}\n", "6bd3287bbde7eef27e12b786da68e550": "_{(p\\tilde{\\leftarrow}q)'\\,}\\!", "6bd3412f8b7f05a3530debe117d3fe11": "\\Delta \\,", "6bd34a0945f05d8732c0db4297b3520d": " \\sin(x)^3+\\cos(3x)=0 \\, ", "6bd3e35fdb1f9bda2c5a85d06090e282": "\\frac{\\partial \\mathbf{\\hat{r}}} {\\partial \\theta} =\\cos \\theta \\cos \\varphi\\mathbf{\\hat{x}} + \\cos \\theta \\sin \\varphi\\mathbf{\\hat{y}} - \\sin \\theta\\mathbf{\\hat{z}}= \\boldsymbol{\\hat \\theta}", "6bd41df23809f5ea29926c76a82834b7": "T(x) = \\sum_{n=-N}^N c_n \\mathrm{e}^{\\mathrm{i}nx} \\qquad (x \\in \\mathbf{R}).", "6bd4833bb2d0d63104c363d4b83746e3": " R_H={h\\over \\nu e^2}, ", "6bd494bfb21a082bf7900150a9ac8764": "gp_1(N)=p_1(h)p_1(N)=p_1(hN)=p_1(Nh)=p_1(N)g", "6bd4ba2283976337e1a478ee9bd16043": "\nf=\\frac{d}{4\\rho_ic}\\cdot\\frac{\\delta^2V^e}{\\delta x^2}\n", "6bd565831b064fa766cf041868a0dd2d": "M=\\langle Q, \\Gamma, s, b, F, \\delta \\rangle", "6bd588984a54f3327b65dfba73c54390": "1{-}\\mathsf{H}(p)", "6bd59ff4906c58bb3cd35fa5c458f8d2": "\\rho'", "6bd5b90fdf9c9d7b5838d604bd52fc4b": "p^2 = p", "6bd5c2f58eb6f32f7050f19db71de0b6": " K, K z = \\overline{z}, ", "6bd6052eb59e38e5ce8d3e0f8fb89418": "\\frac{a^2 M}2=\\frac{a^2 E}2-\\frac {a\\varepsilon\\cdot a\\sin E}2", "6bd613e89559ee94ef0c4800428c5cd6": "T_f := \\bigsqcup_{n\\ge 0}f^{-n}(t),", "6bd63cca2fcdc2a81cf40151b7597caf": "\\begin{matrix} \\frac{7}{3} \\end{matrix}", "6bd63f3e9f184d8b2671ae90c1f99797": "O(r)", "6bd6acaa7775828ce228f2f92dfe5869": "\\bar{X}=(X_1+\\cdots+X_n)/n\\,,", "6bd6bca34d1311f82b93933c2199a93e": "\\widehat{\\mathbf{P}} = \\left(\\frac{\\widehat{E}}{c},-\\widehat{\\mathbf{p}}\\right) = i\\hbar\\left(\\frac{1}{c}\\frac{\\partial}{\\partial t},\\nabla\\right) \\,, ", "6bd6d8f048f2389d3cb82500fd9721d9": "S \\to X", "6bd70f2bf173cde481b0a0ce4230acff": "3^k-2^k\\left\\lfloor \\left(\\tfrac32\\right)^k \\right\\rfloor > 2^k-\\left\\lfloor \\left(\\tfrac32\\right)^k \\right\\rfloor -2\\;\\;?", "6bd72c9da0c47486b9a2075920fab3ba": "\\begin{matrix}\np \\rightarrow q & \\quad & \\quad & p \\Rightarrow q \\\\\n\\mbox{not}\\ p \\ \\mbox{or}\\ q & \\quad & \\quad & p \\ \\mbox{implies}\\ q\n\\end{matrix}", "6bd7310f2e5c97f8fd3bce09c93d5bcf": " \\oplus_1 ^n X \\in L( {\\tilde H} ) ", "6bd77228b12f4061b5cc49af7cf03b62": "\\gamma_D", "6bd7a8ef3618ba7478e9f4eedd661237": "{z_1}^{z_2} = e^{z_2 \\log(z_1)}", "6bd7fd3f25b6def881c798770723ac9c": "\\scriptstyle (\\pi R^2 D),", "6bd8118415adbe7a26445c50514e649d": "A_\\alpha^{\\;\\; IJ}", "6bd8132544a921ac20fb0c0be4694dc7": " y(x) = ue^{r_{1}x} \\, ", "6bd813d7fa790517a7b81437ca1aab0f": "\\sigma^2 = \\sum_{i=1}^n (x_i - \\bar{x} )^2 / (n-1)", "6bd81e6452f93726a958e9462be4a805": "\\bigsqcup_{i \\in \\mathbb N} F^i(\\{\\}). ", "6bd84950160b00f3b6f87211a17ff650": "\\hat{\\mu}^0_{ij}", "6bd890b331839fa7c4850678e7384111": "\\scriptstyle b", "6bd8ab38587eb74c3132e48c70f57a49": "\n\\begin{bmatrix}\n 0 & B_{11} & B_{12}\\\\\n B_{21} & B_{22} & B_{23} \\\\\n B_{32} & B_{33} & B_{34} \\\\\n B_{43} & B_{44} & B_{45} \\\\\n B_{54} & B_{55} & B_{56} \\\\\n B_{65} & B_{66} & 0\n\\end{bmatrix}.\n", "6bd8bbd64749ddb274cc93f77e72dca9": "\\mathrm{CAPE} = \\int_{z_\\mathrm{f}}^{z_\\mathrm{n}} g \\left(\\frac{T_\\mathrm{v,parcel} - T_\\mathrm{v,env}}{T_\\mathrm{v,env}}\\right) \\, dz", "6bd8bce62aea1d8a4cf1d51e37eece61": "[F , G]^{IJ} := F^{IK} G_K^{\\;\\; J} - G^{IK} F_K^{\\;\\; J},", "6bd8f37dd7d1805bb1688650bac21adb": "A_{\\mathrm{Tail}}=\\left(\\frac {2}{3}-\\frac {\\pi}{4\\sqrt {2}}\\right)a^2", "6bd8f65ef988537b9d959aa86786d441": "F_{1 \\rarr (2,3)}=F_{1 \\rarr 2}+F_{1\\rarr 3}", "6bd8f8eff85901580703ab1c417270a6": " \\mathbf{T}(s) = \\gamma'(s), ", "6bd956763d98a9b116d915a8683b769a": "\\sum_{i=1}^{m}{x_{i,m+j}}=a_i", "6bd957131224de5dd97f7e59fc5dc4ed": "P^{-1}\\left({d \\over dx}\\right)\\,", "6bd96b2159f23cfa7d25b721a0c17620": "ac(z)=z \\pi^{-\\operatorname{ord}(z)}", "6bd97488dad1aedd13b423d4a3cc04a0": "\n\\begin{align}\n\\textbf{a}_s^* &= G_{11}^*\\textbf{a}^* + G_{12}^*\\textbf{b}^*,\\\\\n\\textbf{b}_s^* &= G_{21}^*\\textbf{a}^* + G_{22}^*\\textbf{b}^*.\n\\end{align}\n", "6bd9abb7b4b333669a7cce108b56a8bc": "\\frac{1}{2} \\sqrt{2}", "6bda118005111b050c5b701bccb18066": "B_1 (N, r, \\mu ) = \\frac{ \\left \\langle\\sigma_y^2(N, T, \\tau ) \\right \\rangle}{ \\left \\langle\\sigma_y^2(2, T, \\tau ) \\right\\rangle}", "6bda47f784160a7429ccfbe8a1116201": "wp(\\textbf{abort},R) \\ =\\ \\textbf{false}", "6bda8af54c40bc23ed858e9e9f5c11d2": "f'", "6bdaa9f28e4d4292fc55798391ecfa68": "C_{out} \\circ (C_{in}^1 ,...,C_{in}^N )", "6bdb0b18d115198ba7c1fd2e21a36c64": " \\Delta\\ E \\approx\\;\\rho\\;Gb^2 or \\approx\\;3\\gamma\\;_s/d_s \\,\\! ", "6bdb4f1f1a522703c726b65548dfeea8": "\\sigma^{-1}(m - m'),", "6bdbecaffe6f71e8c5cbbf335aa5ecdb": "\\left(x-\\frac{2za}{c}, y-\\frac{2zb}{c}, -z\\right).", "6bdc2ec47b62432281ce647354154744": "y(t) = \\sum_{k=1}^{K} r_k(t) \\cos\\left(2 \\pi \\int_0^t f_k(u)\\ du + \\phi_k \\right)", "6bdc6444aab0b9e18996b66c35f6977d": "f(x) = \\int \\, \\delta(x-\\xi) f (\\xi)\\, d\\xi, ", "6bdc7b8e6abd35a3793c8fc9a4425be7": "SL(2,5)", "6bdc9daad79d4dde92d091c23cd965ed": "u\\in A^2", "6bdccaec0d1e62f1fc3b92d7143a106b": "e_V = 1 - \\left [ \\frac{\\epsilon \\cos \\theta - ( \\epsilon - \\sin ^2 \\theta )^{ \\frac{1}{2}}}{ \\epsilon \\cos \\theta + ( \\epsilon - \\sin ^2 \\theta )^{ \\frac{1}{2}}} \\right]^2 ", "6bdccd093a5b855ccd96abfa4e691eff": "a\\ne b \\ne c \\ne d, \\alpha \\ne 90 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "6bdcdff138508f635b3c1716098df0bd": "\nu = \\sum_{i = 1}^n w_{i} x_{i}\n", "6bdceabcfbe4c4c1d0504fa8ef01431a": " \nu_s\\left(\\omega\\right)=A~w\\left(\\omega\\right)exp\\left[-i\\omega T_0\\right]\n\\Gamma\\left(\\phi,\\delta t\\right)\\cdot\\hat p\n", "6bdd18f0995b8e1956bbd415a40cfa68": "1-t_1f_1, \\ldots, 1-t_kf_k,", "6bdd364fc31fd22d4e23fcd9fd0140de": "Y=\\{1,4,6\\}", "6bdd44f281fd819244f083ae1319a1ba": "y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + \\cdots + A_p y_{t-p} + e_t", "6bdd4578b5dff8a283a3749d94540c23": "R = (A/\\mathfrak{p}) [f^{-1}]", "6bdd8d2e7d4f05e6f9f0b18a46267f55": "\\begin{bmatrix} \\,\\\\ \\,\\end{bmatrix}", "6bddf89af43ad57bd48c64dd6f0ca3fe": "l_{ab}", "6bde4e18b95ea9cf423712f37a587953": " 8.4% = \\frac {10% + 7.5% + 7.7%}{3} ", "6bde9af3c24fdd54db980e25d8a8e80f": "{}\\,\\rho\\mathbf{u}\\cdot d\\mathbf{S}", "6bdeead2f3c20b17462a285b528683d0": "h=0.3", "6bdf121b3352267db92df83311aa17ae": "\\langle\\mu|\\xi\\rangle = \\bar{\\mu}_1\\xi_2-\\bar{\\mu}_2\\xi_1", "6bdf4503b2b5441fd9307833ba6675d1": " G = \\frac{ n }{( n - 1 )( n - 2 ) } \\sum_{ i = 1 }^n \\left( \\frac{ x_i - \\overline{ x } }{ s } \\right)^3 ", "6bdf4d3013aff383b726f024099c089f": "\\partial/\\partial r", "6bdfb86f971fda3f861f2e55280cc49d": "\ng \\cos \\theta \\,d\\theta = - k^2 \\,ds \\,\n", "6bdfd5bb3f63ebbd2147083fb2d84248": "y_i = a+bx_i+\\varepsilon_i\\,", "6bdfe7e42e6fecf61931ffeb96fefab9": "z = r \\ang \\varphi . \\,", "6be021dcd7d8676b033baf672548066e": " \\mathbf{X} ", "6be03f076ad8c5bbb948813e43d19a4f": "x=\\frac{\\ln z}{W(\\ln z)},", "6be050dba41cdee73a69a7478dd45588": " \\qquad \\qquad \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\epsilon_{e,c,\\omega} = \\frac{4\\pi^2}{\\omega^2V}\\sum_{i\\isin \\mathrm{VB},j\\isin \\mathrm{CB}}\\sum_{\\kappa}w_\\kappa |p_{ij}|^2\\delta(E_{\\kappa,j}-E_{\\kappa,i}-\\hbar\\omega),\\ \\ \\ \\qquad \\ \\ ", "6be0cfd47c03fa4ee0b7974725fb2010": "S=\\{0\\}\\cup [1, 2]", "6be0eb60ae4bd59326c83fbec44f810a": "\\det(B)^{n/2} \\exp\\left(-{1 \\over 2} \\operatorname{tr} (B) \\right)", "6be158019cafaaa3bac5b828defabbe3": "\\vec{e}_3 = \\frac{1}{r} \\, \\partial_\\phi + O \\left( \\frac{1}{r^2} \\right)", "6be19f19d489da4e3d0b89df6d657961": "(a(x)g(x))\\mod g(x)", "6be1cef64ac1b96847ab63e675c6894f": "ncp_t=2c_t r_0 e^{-\\theta t}", "6be201349d55b907f35460f5838eb7fd": "E_s(u) = \\exp\\left[-\\left(\\frac{\\pi\\alpha}{\\lambda}\\right)^2 \\left(\\frac{\\delta\\Chi(u)}{\\delta u}\\right)^2\\right] = \\exp\\left[-\\left(\\frac{\\pi\\alpha}{\\lambda}\\right)^2(C_s\\lambda^3u^3+\\Delta f\\lambda u)^2\\right],", "6be2083c6f1b5d16e7a4f0e31b253c61": "S^2_{n-1}", "6be20c5795213d90a16eef2ad8eaec1f": "\\frac{N(N+1)(2N+1)}{6}.", "6be2135cc584da4aabaf961a02816189": " (\\ldots 0 \\overbrace{1 \\ldots 1}^{n} 0 \\ldots)_{2} \\equiv (\\ldots 1 \\overbrace{0 \\ldots 0}^{n} 0 \\ldots)_{2} - (\\ldots 0 \\overbrace{0 \\ldots 1}^{n} 0 \\ldots)_2. ", "6be24c838b86f09528741ae390ce326c": "E_\\text{K} = \\frac{1}{2}I_C \\omega^2 + \\frac{1}{2}M\\mathbf{V}\\cdot\\mathbf{V}.", "6be24ff2f3b7e12a66bf68ecca98d3b3": "\\frac{\\partial}{\\partial y}(y-b) = \\frac{\\partial}{\\partial y}\\sqrt{m^2-x^2} = 0,", "6be25fd94262843991b6184d3c384aee": "E= E^0 + \\frac{RT}{nF} \\ln a_{H^{+}}", "6be2b06f5f191f7db053a5868aba4989": " \\langle T_\\pm,\\phi\\rangle =\\lim_{\\epsilon\\downarrow 0} \\int f(x\\pm i\\epsilon) \\phi(x) \\, dx.", "6be33178005c09ae4418d125df49001e": "SU(2)_L \\times SU(2)_R \\times U(1)_V \\times U(1)_A ~.", "6be3861f8aba5c7077fddffdada56913": "\\sigma(\\mathfrak{p}''_n) = \\mathfrak{p}'_n", "6be3a5c0031d89ab4fa433f3c96c51bc": "b^1 = b", "6be3ee2930b398d26378cb1b33e48b9a": "Q=A\\,\\bar{u}", "6be44b8802e41f8c352480f8e14aac97": "\n\\pi_{n,a}(x) \\sim \\frac{1}{\\phi(n)}\\mathrm{Li}(x),\n", "6be45452573bd574e5d18bf6138c9b3e": "\\,x_i\\,", "6be468f4df4d1ef6ebd3cee4672d7a21": "u(t, x) = \\int_{\\mathbf{R}^{d}} p(t, x, y) f(y) \\, \\mathrm{d} y", "6be47851b752df0ee0a040ad30d70c78": "\nD = D_s + D_u - N = 2 D_s - N = N - 2 \\gamma\n", "6be48ba6ea892010d6dd8e54993ae68a": "\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(c)}\\,_2F_1(a,b;c;z),", "6be4b0b01ac22b26bd3ad773dcfe6b1d": "\\phi_\\lambda (\\mathbf{r})", "6be4bbfcc469d9fc067d98b6893e2ef6": "\\Lambda = 2 \\lambda^2", "6be5608a6c323cd3031260628d1aaaf8": "\\operatorname{pf}(A) = \\sum_{M \\in PM(n)} \\operatorname{sgn}(M) \\prod_{(i,j) \\in M} A_{i j},", "6be56680bd734c038c60ae51312da748": "\\lambda^\\to", "6be5e0bc800b552a4c7539853b757e94": "u_{l}= x_{l} - la", "6be60149aef38db4fe72e13e2dcbdcf2": "\\rho(1_{\\{\\tau_X\\le T\\}},1_{\\{\\tau_Y\\le T\\}})=\\frac{P(XY)-P(X)P(Y)}{\\sqrt{P(X)-(P(X))^2}\\sqrt{P(Y)-(P(Y))^2}}", "6be63491ea3801822ef2731c357995bb": "\\mathrm{CML} : E(r) = r_f + \\sigma \\frac{E(r_M) - r_f}{\\sigma_M}.", "6be70d8b7d1fc2afc2f91cb8267fbe5c": "A \\to A_P", "6be79078001e05d279e5622018db383a": " n=\\bar k \\sum_{k=1}^n l_i = \\bar k L ", "6be81ec86e0cf52e25ccf060d726db24": "\\langle \\mathbf{r}|m\\rangle\\propto e^{i m \\phi} .", "6be82f029db8c2460602d182fe065d69": " \\mathcal{H} = \\sum_{j=1}^{L-1} (1 - e_j) ", "6be8344b2b1ca77b617c6aac8c67cdb2": "R = (h/2) + (a^2/6h)", "6be85126ec9c9bad614da1508e75b2ed": " \\varepsilon ^0 = \\begin{bmatrix} \\varepsilon^0_x & \\varepsilon^0_y & \\tau^0_{xy} \\end{bmatrix}^T ", "6be8c5ca15c0f9cdc4bc8ddf543b5799": "\\mathbf{C}=\\begin{bmatrix} 1 & \\gamma & 0 \\\\\n\\gamma & 1+\\gamma^2 & 0 \\\\ \n0 & 0 & 1 \\end{bmatrix}\\,\\!", "6be8ec4bf695294c1070b1360b4bd51d": "H=\\sum \\frac{1}{2m_i}\\hat{p}_i^2 + V(\\hat{q})", "6be9ababf42276b08b3f2d568eba1a61": "\\Chi^2 ", "6bea04532f7993560814db0f1e4f02d3": "\\omega^2 = (-1)^{n(n-1)/2}(-1)^q = (-1)^{(p-q)(p-q-1)/2} =\n\\begin{cases}+1 & p-q \\equiv 0,1 \\mod{4}\\\\ -1 & p-q \\equiv 2,3 \\mod{4}.\\end{cases}", "6bea6a3e7e95d7d5e0f0346742d9edb2": "MRP = MPP \\times \\text{AR}\\,\\!", "6bea89983136cd77cea74d9b859e62de": "E = T + V =H \\,\\!", "6bea9034bb729f932536f92f887d22f2": "q_i=p_i^{r_i}", "6beac9cdcebed7292f6cdb15f0ed848b": " \\displaystyle{\\varphi(z)=g^{-1}(f(z)).} ", "6beacb0ffae9862fdc3fcc174f891fc1": "\\scriptstyle \\bar{x} \\pm 2\\sigma", "6beadf12d5e95049ad6be8a56cc2ecdd": "\\rho_{\\text{realized }}", "6beb0739b4ed41f0c17850b9de27c119": "\\textstyle t_{k}", "6beb4344dcb5f6a5255d61bf9fe32992": "\\scriptstyle \\sqrt{-5}", "6beb6115f446653d637960acfeaef9ad": " \\pi_2 =\\frac{l^3 g \\rho^2}{\\mu^2}", "6bebc3e3939a3c3fdf6cc4f41a902617": " U \\subseteq \\mathbb{R}^m", "6bebcd26c176fc31d5873240bb474a98": "L \\gg \\frac{a^2}{D} \\bar w = \\frac{\\mathit{Pe}_d\\, d}{4}", "6bec2f4b18db6820a14d89eec3be1abf": " E: M \\rightarrow \\mathbb{R} ", "6bec3682c750fa8e6b9eb6775eb4d3a6": " \\operatorname{de-let}[V] \\equiv V ", "6becfd571e9805c0c1696c4437d94be9": "\\Delta_0 u = - K e^{2u}.", "6bed03a26c84deafcaae35a6e245f4e2": "\\sum_{k=0}^{\\infin}\\frac {1}{2^k}=1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}+\\cdots=2.", "6bed0e53c0781f618a5cd9855afb38e7": "f,g \\in \\mathcal{H}", "6bed0eb25add714c48ff8e0cf23d2a7d": "\n\\frac{\\partial(r, \\theta, h)}{\\partial(\\rho, \\theta, \\phi)} =\n\\begin{pmatrix}\n\\sin\\phi & 0 & \\rho\\cos\\phi \\\\\n0 & 1 & 0 \\\\\n\\cos\\phi & 0 & -\\rho\\sin\\phi\n\\end{pmatrix}\n", "6bedb45eac271428d440cb00f6be92f9": "\\Delta^1_n", "6bedb7099782209abc18fb0febb41f23": "g_1(x)", "6bedbd344d35a1f4b5902bad2f2430ca": " V_i^G", "6bede46fe9e54e1bdc77dbcae66226a8": "16 - 6 = 10\\;", "6bef13a8f77ded7333fe4040c5ed6e0e": "v_D", "6bef20a9b0a1beedd07fc84ef7be5f9a": "P V = N k_\\mathrm{B} T \\,", "6bef5ca51be2518b9754d63c98cfdec8": "\\textstyle [2040,1784,33]_2", "6befb4eb19959a429a19c367b4c34f2c": "r > 6m", "6befb9baedf6b0264453203d5c9a8b7b": "B^{-1}= \\begin{pmatrix}\n \\frac{1}{7} & 0 \\\\ 0 & \\frac{1}{3} \\\\ \n \\end{pmatrix}", "6bf0173d4383ed68bb467bd0174e4e09": "\\frac{(P-MC)}{P}=-\\frac{1}{PED}.", "6bf019ae49459c88b50eb5b9a13ebe9f": "\\Rightarrow \\frac{P-x}{x} = \\frac{r_1}{r_2} \\,\\!", "6bf03c05c7b43cd817d981fb90f5200b": " \\left( 6\\,rs^5 \\right)^{1/6} ", "6bf071bef8fa6dd2f6bee57624bb9379": "f(r) = \\frac{\\partial \\rho(r)}{\\partial N}", "6bf0946b0da3fe3f99c53aee78c7f0d6": "\\mathbf{u}(t) = K \\mathbf{y}(t)", "6bf0a5fb0fef182c5ca28da23a270991": "\\displaystyle e^{i \\pi} + 1 = 0\\, .", "6bf0b58375e30523d99291a7b44b4525": "2^\\binom{n}{2}", "6bf0eae666c81afe460ce58933d88dee": "n_e (\\phi_2) = n_e(\\phi_1) e^{- (\\phi_2-\\phi_1)/k_B T_e}", "6bf0ffa32e1f88e6a117cc99fb249415": " \\tau_c ", "6bf127bc96b887e603406dd5b923cff8": "\nS_{mn} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{2N^{2}}\\sum_{i=1}^{N}\\sum_{j=1}^{N} (r_{m}^{(i)} - r_{m}^{(j)}) (r_{n}^{(i)} - r_{n}^{(j)})\n", "6bf1771953527a02849b0fd62c669a81": "f - g", "6bf1d41035c15656ec4c62385fad9990": "\\left\\lfloor\\frac{\\binom{n}{2}-6n/13}{3}\\right\\rfloor=\\left\\lfloor\\frac{n^2}{6}-\\frac{25n}{78}\\right\\rfloor.", "6bf1e44c6e8e54d0b1685ee82ea48de4": "r\\in \\boldsymbol{\\Zeta}_{q^*} ", "6bf1ebe83288b4ae16777841388def05": "\\frac{\\$ 100/(1+I)}{1 - 1/(1+I)} \\;=\\; \\frac{\\$ 100}{I}.", "6bf21782161689d664253bd8b90370da": "\\Psi_{mn} = \\mathrm{exp}\\frac{-a_{mn}}{T}", "6bf22c6be280ff0b8cb663a829f0b424": "O(kL+kd\\cdot d^d)", "6bf33675bb9194d555b213cce45a9d58": "\n\\frac{1}{r} = \\frac{mk}{L^{2}} \\left( 1 + \\frac{A}{mk} \\cos\\theta \\right)\n", "6bf393b32918534879170cc3c3ca5645": "e^{-q \\tau} \\frac{\\phi(d_1)\\left(d_1 d_2 - 1\\right)}{S\\sigma^2\\sqrt{\\tau}} = \\Gamma\\cdot\\left(\\frac{d_1 d_2 -1}{\\sigma}\\right) \\, ", "6bf39edec59211fa0dae89f0fddaaf51": " {\\it f_{input}} \\over {\\it f_{output}} ", "6bf3a9eefef1c36fcb94cf936b8266ce": " -\\frac{1}{2}[(\\kappa-1) \\theta~\\sin\\theta + \\{1 + (\\kappa+1) \\ln r\\} ~\\cos\\theta]\\, ", "6bf3af626464d47159e46dfab01e7f6a": " M(p) = \\langle a,b,c : a^p = b^p = 1, c^p = 1, ba=abc, ca=ac, cb=bc \\rangle ", "6bf3de27474b93b6a278444e333d1bf5": "(\\forall f)(\\exists e)(\\forall n)(\\exists u)[ \\mathbf{T}(e,n,f(n),u) ]", "6bf40403a33cb3f0ce1e37637a471e6d": "\\scriptstyle{\\dot{\\hat H}}", "6bf428d0a126bbde56b8559864727ed4": "\\omega(z) = \\sum_{n=0}^{+\\infty} \\frac{q_n(\\omega_a)}{(1+\\omega_a)^{2n-1}}\\frac{(z-a)^n}{n!}", "6bf429f77678d39c8074e838457cfa80": "T_1 \\times \\ldots \\times T_n = T^n", "6bf4cddc14af5d1392bd6f076dec9410": "V=\\begin{bmatrix} e^{\\beta(h+J)}&e^{-\\beta J}\\\\ e^{-\\beta J}&e^{-\\beta(h-J)} \\end{bmatrix}.", "6bf50771d0bdf57f5b631289965d97bb": "\\top(z, x) \\le y", "6bf5172aa356b933cdd581356c277ee8": " \\Delta t' = {\\Delta t\\over 1-u/c }.", "6bf54946bffb3ff2d5d5e6e7b1e5f66c": "T_\\rho\\circ S_\\rho \\left( f\\right)=\\frac{\\rho}{\\mu}\\times (f).", "6bf597944312d67237e74116198faab9": "\\textbf{E} = -j \\omega \\mu \\int_V d \\textbf{r}^{\\prime} \\textbf{G}(\\textbf{r}, \\textbf{r}^{\\prime}) \\cdot \\textbf{J}(\\textbf{r}^{\\prime}) \\,", "6bf5c8b4d1ab03bd0c66e2f8384dd4da": "T_\\text{max} =\n\\begin{cases}\nb/(M -r) & \\text{ if } r < M \\\\\n\\infty & \\text{ otherwise }\n\\end{cases}\n", "6bf5d745d336cedc36b6d2e3d388fa42": "j=1,2,\\dots\\,.", "6bf61258a8f7a62b30c346fcc7f4abc8": "df = \\begin{bmatrix} du/dx & du/dy \\\\ dv/dx & dv/dy \\end{bmatrix},", "6bf6656a8aeb552bcddf988207ec1fe4": "\n\\mathbf{U'}^\\dagger \\mathbf{R} \\mathbf{U'} = \\begin{pmatrix}\n\\cos\\phi & -\\sin\\phi & 0 \\\\\n\\sin\\phi & \\cos\\phi & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}\n\\quad\\text{ with }\\quad \\mathbf{U'}\n= \\mathbf{U}\n\\begin{pmatrix}\n\\frac{1}{\\sqrt{2}} & \\frac{i}{\\sqrt{2}} & 0 \\\\\n\\frac{1}{\\sqrt{2}} & \\frac{-i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix} .\n", "6bf6abdd64400267d303b84ea9a4e02f": " \\bigcap_i Y_i ", "6bf6e8069081003506c64a2654e6adc9": "M,w\\models\\triangle\\phi \\iff \\forall u, M,u\\models\\phi\\Rightarrow Rwu", "6bf761373b43eba5ea2a4d14bb6b3231": "f:(-2,+\\infty)\\rightarrow \\mathbb{R}", "6bf774235bb052a366c90d42996b3987": "U_B = -\\boldsymbol{\\mu}\\cdot\\mathbf{B} = \\mu_B B (M_L + g_S M_S)", "6bf785a53b0969abab540d4a62ef288b": "dg = \\frac{1}{\\Gamma(\\alpha)}\\,\\frac{E^{\\,\\alpha-1}}{ E_c^{\\alpha}} ~dE", "6bf7ce34fcd5b4b249fd8292f889b99a": "{p_i, p_0,\\eta}", "6bf8802c9e8e4fd4acf351c7c150cf13": "\\mathfrak{h}_1", "6bf950aa4d480757d0e70b209b585bce": "\n(\\mathbf{\\gamma_5})^T = \\alpha\\begin{pmatrix}0.8673 \\\\ 0.1327 \\end{pmatrix}\\circ \\begin{pmatrix}1.0000 \\\\ 1.0000 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.8673 \\\\ 0.1327 \\end{pmatrix}=\\begin{pmatrix}0.8673 \\\\ 0.1327 \\end{pmatrix}\n", "6bf95a8d8b0147d485457bcb89393106": "\\rho: A \\to B", "6bf973879fc8b2f9037d6b69095085a1": "\\pi \\approx \\frac{256}{81} \\approx 3.1605", "6bf99c392781425ed0f289cd78f2dc5c": "\\ \\Phi ^* = \\Phi. ", "6bf9efaeee22e15570c401be7795cf5c": "X_M = \\frac{1}{c_L-c_R} \\int^{c_L}_{c_R} x \\mathrm{d}c ", "6bfa18583b8b4643f418e08621dfb151": "dU = T dS - P dV", "6bfaac2fc17220bd6280e198622c28d6": "p(c_{even}|f_{p\\mbox{-}int}) = 0.5\\ ", "6bfac9ff66c6eed0d2a05b23a09e1ce7": "V_0 = v_\\text{resistor}(t) + v_\\text{capacitor}(t) = i(t)R + \\frac{1}{C}\\int_{t_0}^t i(\\tau) \\mathrm{d}\\tau", "6bfb38e8fa9b39603cf2e22b2389669d": "\\mathbf x_\\mathfrak p, \\mathbf y_\\mathfrak p", "6bfb3f2a5b6abc2859baa358340221f4": "\\text{C} = \\text{P} - \\text{V}", "6bfb727b656d02ac43818be9f0eb9951": "n \\times m", "6bfb786e62894eb2f4a23d9b3d675639": "_{k+1}V^i_1(x,y)=_kV^r_3(x,y-1)", "6bfc0e5bca50c81b9024c8bd0357200b": "\\dot{V}(x) \\leq 0 \\quad \\forall x \\in \\mathcal{B}\\setminus\\{0\\}", "6bfc769d273d00a7886386aa2e4df543": "c_\\text{shallow} = \\lim_{\\lambda\\rightarrow\\infty} c = \\sqrt{gd}.", "6bfc9d6c6688651e0d7cc1d73fadb10f": "p^7", "6bfccb9e70be200adb8cd0731ca2a524": " P1 \\, ", "6bfcedee3bbc91b2742a42bc429b5594": "\\langle f,\\, g \\rangle \\;\\stackrel{\\mathrm{def}}{=} \\; \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} f(x)\\overline{g(x)}\\,dx.", "6bfd08a4b79b6b0746022b825dbb1e05": "d_{AB} \\le d_{AC} + d_{CB}", "6bfd12e412fa0a9f681ec16f711806e5": "K_1=nK, K_2=\\frac{n-1}{2}K, \\ldots K_n=\\frac{1}{n}K", "6bfd59d8cf45cfe0d18e26637e45e25b": " E_d= - 1 ", "6bfd97d4a7959ac621f2653eb53443c5": "\\textstyle f : \\Omega \\to \\mathbb{R} ", "6bfe3dabd62bf3bf0330654c740960cb": " S_x=\\{\\gamma | f(\\gamma) \\in \\beta \\times \\{x\\}\\}", "6bfe61716d9d92c36bbe25ff7ee80ee4": "\\pi_i(x_\\alpha)\\to \\pi_i(x)", "6bfe92eb133be3d5ddc2a7f4507021ac": "H = \\{ h : U \\to [m] \\}", "6bfec203b07efddd3aaca1679e2d2bf5": "2^2 + 2^0", "6bff43d4c71b1a279e73283a01fc9bfd": " \\textbf{P}(t) = R\\textbf{e}_r + Z_0\\vec{k},", "6bff53fcbd4e2670905b54390a6435a1": " E(x',y',0) ", "6bff6ac09841488adea59e5324854398": "C_R \\,", "6bff90669cde40f6d8930285577a49b7": "I = {V \\over R} \\, ,", "6bffd4ee206020eb8d7546d7648ab90e": "f(z)=\\frac{az+b}{cz+d}", "6bffe32c29d85d7010dafe4f487208c1": "N=-\\frac{dp}{dt}\\int xz dm +\\frac{dr}{dt}\\int x^2 + y^2 dm =-E\\frac{dp}{dt}+C\\frac{dr}{dt}", "6c003ff6a7d34fdc5df7327ada3b5438": "\\mathit(V)\\!", "6c00975dd13444ade47550977ee223a3": "i \\in \\{1, \\dots, n\\}", "6c00d2b948ca874154d177737f78ad9d": "x \\equiv y^2 v_1^{-a_1}v_2^{-a_2} \\cdots v_k^{-a_k}\\pmod{n}", "6c00dc76e22925fb5d27af347994560a": "rN_t", "6c01287f3ac23ead48fdffa67adf0f37": "\\mathcal T\\Pi_{k=1}^m\\phi(x_k)=\\mathopen{:}\\Pi\\phi_i(x_k)\\mathclose{:}+\\sum_{\\alpha,\\beta}\\overline{\\phi(x_\\alpha)\\phi(x_\\beta)}\\mathopen{:}\\Pi_{k\\not=\\alpha,\\beta}\\phi_i(x_k)\\mathclose{:}+\n", "6c01350fce69cc9e556abc08548906e1": "S\\sim W_p(V,n).", "6c019747e5eb4a721fa1c6be88119fff": "Fu", "6c01e77ce7b27213ef12ccbb2bfe8d0c": " \\lambda_i = \\lambda \\left.\\binom{v-i}{t-i} \\right/ \\binom{k-i}{t-i} \\text{ for } i = 0,1,\\ldots,t, ", "6c01efbc41835d6bd0fba349130644a5": "2 _1^0\\text{S} + \\text{E} \\overset{\\xrightarrow{\\text{k}_{1(1)}}}{\\xleftarrow[\\text{k}_{2(1)}]{}}\n\\text{C}_1 \\xrightarrow{\\text{k}_{3(1)}} {_2^0} \\text{P} + \\text{E}, ", "6c02b6b828654810ebe87f714d385c63": "\\nu=(D_1-D_2)\\frac{\\partial N_1}{\\partial x}=(D_2-D_1)\\frac{\\partial N_2}{\\partial x}", "6c0339d4038b833e75a07216da1f7a47": "q' \\in Q", "6c034d7c35eee2cc69fb0bb7675bba93": "r=a\\cdot\\sqrt[3]{2}", "6c03680da7a00138f3ec151f95f30793": "\\sin 0=0\\,", "6c03a9a19ee96fe69d86ac1a7be45091": "\\frac{1}{\\lambda \\pm \\Delta \\lambda}=\\frac {1}{\\lambda}\\ \\frac{1}{1\\pm\\Delta \\lambda / \\lambda }\\approx \\frac{1}{\\lambda}\\mp \\frac {\\Delta \\lambda}{\\lambda^2} .", "6c03b78835193728e709d3a613f4c3d0": "\\alpha_{j+1}=\\tau\\alpha_j,\\,", "6c04070f9d8c6793160eb002a81e071c": "P(T_i)", "6c040a0ae02d7fdba1b3ff8bc650c74a": "\\phi : K \\to L", "6c0420c44f1b12c07eac8dcb8ae68511": "N_{\\mathrm{H}}", "6c042e4c90b0d803a639a9b7217c12e9": "\\gamma_j=\\beta_j-\\alpha_{j+1}-\\beta_{j+1}", "6c04325db019fb0d1622a09d5409137b": "\\hat{\\rho} = r \\left[1 - \\frac{1 - r^2}{2\\left(n - 1\\right)}\\right]", "6c0436dddd7d20c48731878303c41038": "e^{-r(T-t)}F(t)", "6c043cb1cae4008c958b0cc3a9c829a5": "\\scriptstyle\\bar s", "6c04eb96b869f272eaa1e3f225017d68": " \\Gamma^a_{bc} = \\frac{1}{2} g^{ad} \\left( g_{cd,b} + g_{bd,c} - g_{bc,d} \\right) ", "6c04fbd928df142ebf7ec76d270a481c": "((a), [c])\\notin I", "6c0527698299fb4cc3c92daca64c7641": "r=kC_S C_B\\frac{K_1C_A}{K_1C_A+1}", "6c052d80d92f8fad86378e01cc733b37": "d_A", "6c0540b206b7c225aeed941f3576d35b": "G(x-y):=\\frac{1}{\\omega_{n}}\\frac{x-y}{\\|x-y\\|^n}", "6c05c2ac81359197b79e15bd47350f95": "\\widehat \\Sigma = {1 \\over n} S,", "6c05dc26810646388d97a955bb009058": "[A,B]", "6c05f3f811c0a5138366c2a11b094d03": "F_i(x):=\\mu\\bigl(\\{(y_1,\\ldots,y_n)\\in{\\mathbb R}^n\\mid y_i\\le x\\}\\bigr),\\qquad x\\in{\\mathbb R}", "6c06458ce65dbb226170234024075462": "a_n = 1 \\,", "6c0676928ead07ef1c49e6c3deba75ac": "\\mu(S)", "6c0688bd18b9d219120fc81d056ca722": "\\operatorname{Func}(M)^0=\\operatorname{Morse}(M)", "6c0688c731ef7ee0c92db0dbd52d08dc": " \\theta,\\; \\alpha_x,\\; \\mbox{and} \\; \\alpha_y ", "6c06b68a697cf688b9dfae6163400125": "{{R}_{Y}}\\left[ k \\right]=\\frac{1}{N}f\\Theta \\bar{f}\\left[ k \\right]", "6c06cb26910c720c36f3f834b1cc208a": " W_N(x)\\, ", "6c06e2553d6051322e8c31f458efcdf2": "\\frac{2x^2+7}{3x^2+x+12}, y=\\frac{2}{3}", "6c0721f12911c769edff00320184b5fa": "1 \\over 60", "6c072cc41de3c77177a7c6677d2c87eb": "G_X(t,f) = G_x(-t,-f) \\, ", "6c072d4dbbed7caf8038433a4911c1d2": "\\displaystyle \\sigma^2", "6c07f293c23e9dfc83bd6889b7d7db27": "v = \\sum_{i=1}^n v^i[\\mathbf{f}] X_i = \\mathbf{f}\\ \\mathbf{v}[\\mathbf{f}].", "6c08034eb30cc81d873b0f2b192912f8": " \\int_{\\Sigma} \\nabla \\times \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{\\Sigma} = \\oint_{\\partial\\Sigma} \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{r}, ", "6c08425c0348ea8c6bcec86d662a4450": "\\Psi(\\bar{x})", "6c08ec2275a922086ec990f72a50781d": "p(\\theta) \\propto \\sqrt{I(\\theta)}\\,", "6c092bf86fee1e2ca5be60e4748b4909": "H = \\begin{bmatrix} \n1 & \\frac{1}{2} & \\frac{1}{3} & \\frac{1}{4} & \\frac{1}{5} \\\\[4pt]\n\\frac{1}{2} & \\frac{1}{3} & \\frac{1}{4} & \\frac{1}{5} & \\frac{1}{6} \\\\[4pt]\n\\frac{1}{3} & \\frac{1}{4} & \\frac{1}{5} & \\frac{1}{6} & \\frac{1}{7} \\\\[4pt]\n\\frac{1}{4} & \\frac{1}{5} & \\frac{1}{6} & \\frac{1}{7} & \\frac{1}{8} \\\\[4pt]\n\\frac{1}{5} & \\frac{1}{6} & \\frac{1}{7} & \\frac{1}{8} & \\frac{1}{9} \\end{bmatrix}.", "6c093fa2c33f592a79277b8fc67d3052": "D_3\\left(E\\right)", "6c096064e0f5cbb5b26354d42b942c2c": "x(t) = \\frac{\\nu}{{t^2+\\nu}}.", "6c0a0cbc70cc70a51709a50ab44d9d64": " \\delta(X) = \\frac{T^2}{2} ", "6c0a2917f760aed3c3a3b4ace6c0a02a": "\nX \\perp\\!\\!\\!\\perp A,B\n\\quad \\Rightarrow \\quad\nX \\perp\\!\\!\\!\\perp A \\mid B\n", "6c0a4bbda3d2326d436f458e71148dec": "T = \\frac{1}{4} \\sqrt{(a+b-c) (a-b+c) (-a+b+c) (a+b+c)}.", "6c0a8586446e93bc25d10dc5190a2fa0": " \\acute{\\mathbf{x} } ", "6c0a87ba2e7bedb613315dbefc1505c3": "conc(\\langle x \\rangle, \\langle y \\rangle, \\langle z \\rangle) = \\langle xyz \\rangle", "6c0ab424955a877d4f533824dd7f617d": "\\begin{align}\n & X = \\sqrt{- 2 \\ln U} \\, \\cos(2 \\pi V) , \\\\\n & Y = \\sqrt{- 2 \\ln U} \\, \\sin(2 \\pi V) .\n \\end{align}", "6c0ac66fd0c2e0be2db06c682d86a282": "\n\\sum_{i=1}^n (y_{i}-\\bar{y})^2=\\sum_{i=1}^n (y_i - \\hat{y}_{i})^2+\\sum_{i=1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i=1}^n 2(\\hat{y}_{i}-\\bar{y})(y_i - \\hat{y}_i).\n", "6c0b04c256d8a67b6f258e2889ab94ea": "d = 1 - \\frac{2 | X \\cap Y |}{| X | + | Y |} ", "6c0b082448f9c63f63618ade6a62b716": "q^m - 1", "6c0b1369e5530270c4be965696518265": " \\mathbf X V \\Sigma^+ U^{\\rm T} = U P U^{\\rm T} ", "6c0b3e691d5696345d459b9d0e31d026": "V\\cos\\theta\\,", "6c0b524a611340b48d2cc05d4c1a2321": "w(+\\infty)-w(-\\infty)=2", "6c0b536ed79fe254a5d7ef89c0563bda": "A^{n,K}_{cv}(k_e,k_h)", "6c0b5df4d91eb7cf97950ded0018fc1e": "h,m\\!\n\\in\\! \\mathcal{N}, u\\in V^+,v,v',w,w' \\in V^*", "6c0b771ba86c456ea2db8e5d41421aa2": "TM\\otimes TM.", "6c0b972032a864c6b31d679757a8ffe3": "0\\text{ for }q<2", "6c0c5e81e7e5b81da4a66ab04e079f1d": "X_n Y_n\\xrightarrow{a.s.}\\ XY", "6c0c835412a5d60df47897e845176299": " S(E_i) \\leftarrow S(E_i) + f", "6c0c8b745c165b02b821e9dd6d913bb8": "\\mathbb{Z} (K) = \\mathbb{Z} [K \\setminus \\{0, 1\\}]", "6c0cb6fec28f715d77eac47dd984bfac": "\\left\\{{n+1\\atop k}\\right\\} = k \\left\\{{ n \\atop k }\\right\\} + \\left\\{{n\\atop k-1}\\right\\}\n", "6c0cc57a8284458f7fdaa7bafa85a697": "\\ln(2)/\\lambda\\,", "6c0d05249147c85036e79fdb7509b31c": "(\\mathbf{Z}/p^n)^\\times", "6c0d0dc51ee6904422ba920464d04610": "V^{+}=\\{t\\in V | \\sigma t = t\\}", "6c0d7ab07282c22fd107f09255dfb6c5": "X=\\ell^p", "6c0dbad9f0e6bfffef168814eae154b3": "\\Q", "6c0e19691697eac056c50a3d261d44a7": "\\text{OR}_2^{\\otimes |U|} \\text{EQUAL}_3^{\\otimes |V|}", "6c0e44c352760771b7baef0099204b86": "n+\\nu ,\\, \\left(\\mathbf{V}^{-1} + \\sum_{i=1}^n (\\mathbf{x_i} - \\boldsymbol\\mu) (\\mathbf{x_i} - \\boldsymbol\\mu)^T\\right)^{-1} ", "6c0e8018381471830877a81e9cdd2816": "\\gamma(\\lambda_0) = x'", "6c0ecfb10f01e13e465ed5e79fda8bc9": "g(r)_I = 4\\pi r^2\\rho dr", "6c0ed44fc015f47d4b0cb58f1de17d90": "n\\operatorname{sinc} \\left(\\frac{nt}T\\right) - (n-1)\\operatorname{sinc} \\left( \\frac{(n-1)t}T \\right) ", "6c0f07533789b0f79850d2a53db5bd4d": "\\mathbf{r} = \\mathbf{x} +\\mathbf{e}_i", "6c0f8a516da5cf10aef6d840b8761cda": "1/p_L(ik) .", "6c0fbf33754e8e6f73e0b1e5c4897d65": "f_c", "6c100bb7fadb5b326c435158c17a6662": "G=\\pi_3|_M\\colon M\\to{\\mathbb{R}}, (\\theta,\\phi) \\mapsto \\sin \\theta \\,", "6c1042eb5e8b83e18233cfe801169067": "ds^2 = \\sum_{i=1}^p dx_i^2 - \\sum_{j=1}^{q+1} dt_j^2", "6c109c09b8a52cc64ddcdc8db0d97c3f": "\n \\tau_{xz}(x,z) = \\int \\frac{\\partial \\sigma_{xx}}{\\partial x}~\\mathrm{d}z + C(x)\n = \\int \\cfrac{z~E(z)}{D}~\\frac{\\mathrm{d} M_{x}}{\\mathrm{d} x}~\\mathrm{d}z + C(x)\n", "6c109f80e26cec4841a6f680805e6548": "\\Gamma(t) \\; 0 \\leq t \\leq T ", "6c10d12c09a8e9f0cd0b50aef4ccfc3d": "\\sum_{n=0}^{\\infty} 2^{n}f(2^{n})", "6c10d691725a367e8179d5988eb72a6a": "B \\leq_T \\emptyset^{(n)}", "6c112c10c07aabc8433a381f8613e29f": "D^{'}", "6c11ebb252d0b0849a14ee867bac2b90": "\\sqrt{x^2+y^2}", "6c1270fd05e39f9e443ce5cab98cb08b": "{z}=\\rho \\, \\cos\\theta\\quad ", "6c12ae7e3b774dfb6596da583836a818": "\\forall R.C", "6c1314e4eb65cafe8d4c2722bfd2129a": "6\\ ", "6c13489e22867d41cb681c7e53af1c53": "\\Phi_e = L_eI_e. \\ ", "6c137f8747841a4e5780d83e314429ef": "i = 1, 2, 3, \\cdots, n", "6c138f349c8f90584741e07ab2c39a54": "p_i = \\frac{\\partial L}{\\partial \\dot{q}_i}", "6c13aefd2d5c9fcc21022f2d22623b75": "[\\mathtt{Abs}]", "6c13faeabeaed1bf03b319acd6a3d56c": " s = \\left( 2/5, 2/5, 1/5 \\right) ", "6c14007ae1f73140cda0f549768bc38e": " y_i = \\alpha + \\beta x_i + \\gamma z_i + u_i", "6c140b6600456dc60b47a7c814936ea0": "E(\\pi_$) = \\frac {(i_$ - \\rho_$)} {(1 + \\rho_$)} \\approx i_$ - \\rho_$", "6c1427724b43c6cd590963a51cb9ceae": "\\omega = \\sqrt{|\\det g|}\\, dx^1\\wedge\\cdots\\wedge dx^n", "6c147412fd7524e0a92cf005d3c3a195": "\\mu(x) = e^{\\int -\\frac{x}{x^3}\\,\\mathrm{d}x}=e^{\\int -\\frac{1}{x^2}\\, \\mathrm{d}x}=e^{\\frac{1}{x}},", "6c14fc2853e77fae333ced2174da0cea": "\\begin{align}\n & AI=\\sum\\limits_{i=160}^{8000}{SN{{R}_{i}}*W{{1}_{i}}} \\\\ \n & SN{{R}_{i}}=V{{S}_{i}}-T{{L}_{i}}-M{{S}_{i}} \\\\ \n & T{{L}_{i}}=S{{S}_{i}}-R{{S}_{i}} \\\\ \n & PI=100*(1-AI) \\\\ \n\\end{align}", "6c153720a9814dc381d71b0db5de771b": "\\mu'_{11} = \\mu_{11} / \\mu_{00} = M_{11}/M_{00} - \\bar{x}\\bar{y}", "6c156adc495ffccc74e241d41272a14f": "\n\\begin{align}\n \\lambda(x,y)\n&=\n\\frac{x}{c_1(k_0\\nu_1)}\n-\\frac{x^{3}V_3}{3!c_1(k_0\\nu_1)^3}\n-\\frac{x^{5}V_5}{5!c_1(k_0\\nu_1)^5}\n-\\frac{x^{7}V_7}{7!c_1(k_0\\nu_1)^7},\n\\\\\n\\phi(x,y)&=\\phi_1\n-\\frac{x^2 \\beta_1t_1 }{2(k_0\\nu_1)^2}\n-\\frac{x^4 \\beta_1t_1 U_4}{4!(k_0\\nu_1)^4}\n-\\frac{x^6 \\beta_1t_1 U_6}{6!(k_0\\nu_1)^6}\n-\\frac{x^8 \\beta_1t_1 U_8}{8!(k_0\\nu_1)^8}.\n\\end{align}\n", "6c157b9ed67d8a2c4d43c1c45e759a87": "\\int_A g\\,d\\mu=\\lim_{n\\to\\infty} \\int_A f_n\\,d\\mu \\leq \\nu(A)", "6c15ca4c325a854e35007401df57949d": "\\langle\\hat{d}| \\hat{a}\\rangle +\\langle\\hat{e}| \\hat{b}\\rangle+\\langle\\hat{f}| \\hat{c}\\rangle", "6c1610cb7593be9fff4b8b9a09aa9613": "dv = e^x \\,dx \\Rightarrow v = \\int e^x \\,dx = e^x.", "6c169d9817abe005a83cd236a197a9c6": "(\\mu+1)\\ddot{r} - r\\dot{\\theta}^2 + g(\\mu - \\cos{\\theta}) = 0", "6c16aff00dc23fe22d6236e0a28f24c6": "\\mathcal{U}\\,", "6c16b5459537fee14f7e0654aa3edb73": "\\frac{\\partial f}{\\partial t} + \\frac{\\mathbf{p}}{m}\\cdot\\nabla f + \\mathbf{F}\\cdot\\frac{\\partial f}{\\partial \\mathbf{p}} = \\left(\\frac{\\partial f}{\\partial t} \\right)_\\mathrm{coll}", "6c16c3df6e04dc19eeed9cfcf9b61f4c": "\\mbox{P} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{DTIME}(n^k)", "6c16da75622828fa57ebfd7f206cf9f0": "\\forall n, m>M\\colon |g(n,m)| \\le C(n+m),", "6c1702bdaae94f5c6213917cecb8658f": "V=(\\frac{1}{12}(2\\sqrt{2}+3\\sqrt{3}))a^3\\approx0.668715...a^3", "6c17e18540b07bbcc2116e4af2b30403": "w_1, w_2, \\ldots, w_k", "6c185331b7ff1d5e74a137d59e7d991e": "s_1 = e_1 \\alpha^{(c + 1)\\,i_1} + e_2 \\alpha^{(c + 1)\\,i_2} + \\cdots \\, ", "6c185a2367e3e92ec695c87b4be2f4ea": "\\frac{\\partial E}{\\partial t}\\, +\\, \\nabla\\cdot\\left[\\left( \\boldsymbol{U}+\\boldsymbol{c}_g\\right)\\, E \\right]\\, +\\, \\mathbb{S}:\\left(\\nabla\\boldsymbol{U}\\right)\\, =\\, 0,", "6c189bf036f1bfb762b2cc1d77bb4b22": "T\\mathrm{d}S\\,", "6c18aad9d43ac2d4b94fca0358172270": "\\overline{x}=\\frac{1}{n}\\sum_{i=1}^n x_i", "6c1911d869de62bf442a929cfd501c25": "|\\beta|=1", "6c19c4b41d8067951486517f2f2046d2": "(i-r)", "6c19d8ddffeae299de7d1abfae725722": "\\Delta \\tau = 2 T_c / \\sqrt{ 1 + (a \\ T_a/c)^2 } + 4 c / a \\ \\text{arsinh}( a \\ T_a/c )", "6c19e27fb2c01499febba74e2d4ab9ab": "\\phi(x)=y", "6c19e2b50de8a2a44bc54cd646418784": "u(x): x \\rightarrow \\mathbb{R}, \\quad x \\in \\mathbf{D} \\sub \\mathbb{R}^n", "6c19f9d1d34ca06a1628eaa09f1f1643": "2^{10/12} = \\sqrt[6]{32}", "6c1a427f5a564d4a20a0c894de4eaec5": "\\forall x,y\\in M_1: \\frac{1}{A}\\; d_1(x,y)-B\\leq d_2(f(x),f(y))\\leq A\\; d_1(x,y)+B.", "6c1a4b2f9bf7f068b2b1919de70ea7b5": " FE_{Na} {{=}} 100 \\times \\frac {\\rm sodium_{urinary} \\times creatinine_{plasma}}{\\rm sodium_{plasma} \\times creatinine_{urinary}} ", "6c1a5e99f50101282473bac2c29f9c3d": "\\bar{s}", "6c1a663ecb4008cb511fe92fe526f918": "c+2v\\,\\!", "6c1a8d01bdbc34883abf55d1cb67b202": "(V_{if}-V)^{\\frac{1}{2}}", "6c1aa487a526a7bba508acd6a1c7db74": "S = {s_1, s_2, ...}", "6c1b0d223be40b190f9c815fe846c266": "( r_S, \\theta_S, z_S )", "6c1b1cd43460fc69a6f1705cdc2d8ac6": "\\tfrac{1}{2} mv^2", "6c1b7ba8adf9f96604dcef246c09dfaa": "\\nu > 0\\,", "6c1bb0436a193e6a5fd155b1b37d233e": "\\tfrac{2^{n-1}-1}{n}\\not\\equiv 0 \\pmod{p}", "6c1bd8d098aac668920104badd70d6bc": "\\sigma_\\mathrm n \\,\\!", "6c1ca40e210f8d0959d1f3a0418555e1": " Q^{(i)} ", "6c1cde6ec5874264583346d7ef71bcd1": "\\tilde{\\mathbf{v}}", "6c1ce661127d73d2175d59b9a7c9f103": "|m|\\lambda", "6c1cf35ce378452ff874e37b35aaca5d": "\\arcsec z = \\arccos \\frac{1}{z} \\quad z \\neq -1, 0, +1 \\,", "6c1d02f66c026ec972d2e694cf088531": " 0 < t < 1\\, ", "6c1da7b8766b28a126a994c66de2e985": "nF\\Delta E = nF\\Delta E^\\circ - R T \\ln Q \\, \\,", "6c1dc2aacb8919c8762727bf936da28a": "X^{(2)}", "6c1dccc920919b3de0d28fcc6dff9468": "\\begin{align} \n\\Delta\\ =\\ &256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 \\\\ \n&+ 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e \\\\\n&- 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2\n\\end{align} ", "6c1de59ff9ba87a40a3078b35d08d279": "\\alpha_y", "6c1e06dba1bc349ad9ef0374855c249f": "G M^2 < J c ", "6c1ebf6a6f73be0435b433388487ad22": "g:(X\\times Y)\\to Z", "6c1f1d6354f208f4cf6af15a4c8da758": "\\displaystyle{{1\\over \\pi} \\iint_{\\Bbb C} |f(z)|^2 e^{-|z|^2} \\, dxdy}", "6c1f63fc222e4e972592911266236e37": "\\{\\cdot,\\cdot\\}", "6c1f6693bbd675592e6c15add479430a": "\\textstyle (P,L)", "6c1f83eebb3d5df3850adb1715eb9b8a": "f(\\mu) f(x | \\mu)=\\pi(\\mu) L(\\mu) = \\frac{1}{\\sqrt{2 \\pi} \\sigma_m} \\exp\\left(-\\frac{1}{2} \\left(\\frac{\\mu-\\mu_0}{\\sigma_m}\\right)^2\\right) \\prod_{j=1}^n \\frac{1}{\\sqrt{2 \\pi} \\sigma_v} \\exp\\left(-\\frac{1}{2} \\left(\\frac{x_j - \\mu}{\\sigma_v}\\right)^2\\right),", "6c1faf8c3b0835c93d59d98d435672fe": "\\log_{10} 2 = .3010... \\approx .3,", "6c1fde718bad9c3c295e00ab1e5afa5f": "\\alpha_i, \\beta_j \\geq 0 ", "6c200b7213131c3db41963a4b489b6b7": "\\rho = \\Omega^2 3 + \\psi(\\Omega^2 3 + \\Omega)", "6c203dc81aa491795cf134a9b7e7e153": " ( X_{n_1}+\\cdots+X_{n_k} ) / \\sqrt k ", "6c205409d68cd792a5710256621ce333": "\n (1.1)\\quad \n d(P,Q) \n = \\sup_{h\\in\\mathcal{H}}\\left|\\int h dP - \\int h dQ \\right|\n = \\sup_{h\\in\\mathcal{H}}\\left|E h(W) - E h(Y) \\right|\n\n", "6c2056f92f704230b7f9e3a80b8587e5": "\\mathbb{F}_{q} ", "6c20690293f8d363f4e70bb4a322b13e": "I_s\\,", "6c2082ee73db8a7559e15e61c4915f36": " \\Delta A =2 A_{zz}-(A_{xx}+A_{yy}). \\, ", "6c212424c73b21231c8ff2a40623c7d6": "y=\\pm b/2", "6c213e75d42f4610d22991e3c53e6570": "K_i, \\ i \\geq 0", "6c215a999607388684154dda639b2efc": "f(x+h) = \\sum_{|\\alpha| \\le n}{\\frac{\\partial^{\\alpha}f(x)}{\\alpha !}h^\\alpha}+R_{n}(x,h),", "6c215f7de79657ee6c153537597be3ed": "\\rho = 1 - \\frac {6 \\sum D^2}{N(N^2-1)}", "6c2183aec3d575cb92197883019c69ed": "T_{\\pi} : \\Sigma \\to \\Sigma", "6c2188187ef37ef0ee8baa965f8b8b98": "\\lambda (\\partial A) := \\liminf_{\\delta \\to 0} \\frac{\\mu \\left( A + \\overline{B_{\\delta}} \\right) - \\mu (A)}{\\delta},", "6c2193d9cac11784e696a0ec0adc470d": "\\int_{-\\infty}^\\infty \\left| f(x) \\right|^2\\,dx = \\int_{-\\infty}^\\infty \\left| \\hat{f}(\\xi) \\right|^2\\,d\\xi. ", "6c21b4ff66df40e3745d52244d14eb55": "\\Pr(N > 20n) = \\frac{1}{\\sqrt{20}} \\simeq 22.3\\%. ", "6c21ea312b03258db2fd1ef91d09e379": " Z(o,s') ", "6c21ed53bc0943fccaf6763e8c296254": "(R_1\\bowtie R_2)\\bowtie R_3=R_1\\bowtie(R_2\\bowtie R_3)\\,", "6c220ee9974688993bcc9eaff4fe0c4b": "\\{H,\\Pi\\}", "6c227e3cddc89dbae7e413ad2aac0be5": "r_{n,n+1} = \\frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}\\exp(-2k_{n}k_{n+1}{\\sigma_{n,n+1}}^2) ", "6c2299bbe5dc6299e02ccc3d54ec8aac": " \\begin{bmatrix} \\frac{du}{dt} \\\\ \\frac{dv}{dt} \\end{bmatrix} = A \\begin{bmatrix} u \\\\ v \\end{bmatrix}, ", "6c22ea2afed9255a52344ea6946c6cd5": "\\Delta L_{A}=\\frac{2\\left(L_{L}-L_{T}\\right)}{\\sqrt{1-v_{A}^{2}/c^{2}}},\\qquad\\Delta L_{B}=\\frac{2\\left(L_{L}-L_{T}\\right)}{\\sqrt{1-v_{B}^{2}/c^{2}}}", "6c22f44b1d5a096634a83a32deb68edd": "\\mathrm{CuFeS_2 + 4 \\ Fe^{\\,3+} \\longrightarrow Cu^{\\,2+} + 5 \\ Fe^{\\,2+} + 2 \\ S_0}", "6c231a565ee8eba573933c06d788f186": " \\text{Inv-Gamma}( a_0,b_0)", "6c2353032b2e8ec3a4c5dfe4d5b3b98d": "\\mathbb{R}/\\mathbb{N}", "6c23a5be868e3bf5c451b352f0d946a5": " f'(x) = \\sgn(x)", "6c23cb45e834d65ff63689f757d799e9": "n \\,", "6c23d2caa6bcb762e0a04a08d512ad35": "A(r)=\\left(1+\\frac{1}{Sr}\\right)^{-1}", "6c24515f7bc481a9762bc271360dcfca": "E'(\\theta)=0\\,", "6c2462f3c0f4982546b43a5d0e70a1b2": "(M,N_m)", "6c247ea57f5c5e516d9f5d87b41ad664": "g_{ij} = |\\mathbf{p}_i - \\mathbf{p}_j| = |\\mathbf{p'}_i - \\mathbf{p'}_j|", "6c24e9cd1683c300d3a83bce6b083baa": " \\lim_{k\\to\\infty} \\left\\| T\\psi_k - \\lambda\\psi_k \\right\\| = 0. ", "6c24f4e7e0a9cd2d04a064dd373c4ee7": "HTi = \\frac {1}{60} \\int\\limits_{0}^t [90-SpO_2(t)] dt", "6c25251cf54c5283e222d5b7294f34b8": " \\frac{\\sin^2 \\theta}{\\sin^2 \\theta} + \\frac{\\cos^2 \\theta}{\\sin^2 \\theta} = \\frac{1}{\\sin^2 \\theta}\\!", "6c2525e73be48f15082c84848eab9584": "\\mathit \\Gamma = -1\\,\\!", "6c2550c6310134a92c85ab9fb44c9e81": " W_mH^k(X; \\mathbf{C}) ", "6c2589d5fd74944ae0f07879befe52c1": "\\mathbf{V}_{B/A} = \\mathbf{V}_{B} -\\mathbf{V}_{A} \\,\\! .", "6c259c35ab18deca851a9f4b06d7de8f": "B = \\frac{4}{a-d}", "6c25a4d29c5fae67be2eed0432607949": "\\mathbf{F} = \\frac{q_1 q_2}{r^2} \\mathbf{\\hat r}", "6c25b56b121235589c4287ce31450469": "2nb \\sin\\left(\\frac{\\pi}{n}\\right)", "6c25c4dd66c50622b4ba89af3ceadf5a": " \\mu(gS) = \\log((g\\cdot b)/(g\\cdot a)) = \\log(b/a) = \\mu(S) ", "6c262a285c95eb4443739ff9266aa352": " \\sigma_{yy} = \\frac {\\mu b} {2 \\pi (1-\\nu)} \\frac {y(x^2 -y^2)} {(x^2 +y^2)^2}", "6c26482a5edf8c1129935b9df9b7fcee": "\\Gamma_{ij,k}^{(-1)}=\\sum\\partial_i\\partial_j(\\xi^ip_i)\\partial_k\\ell^{(1)}=0", "6c266448b77f7b76eb793fa199d1fecf": "\\rho = \\sum_i p_i \\rho_i", "6c26663134fdca8647acc7b7397abde9": "\\int_{H^3} (Mf)\\cdot F \\, dV =\\int_{H^2} f\\cdot (M*F)\\, dV.", "6c266ebe06ac14a588ab202dd65c1aa1": "V_i^L || V_i^R = V_i", "6c268c625ecd099165b0884c171f13a9": "\\{x,p\\} = 1 \\, .", "6c26a3b1a761007f8540711414d4cbd6": "\\cdots\\rightarrow 2\\rightarrow 1\\rightarrow 0", "6c26d6696d81e63fa8a77fa00d78a838": "v^{(k)}(x_n)", "6c276952d6f53d47801f6370f95c4535": "\\begin{align}\n\\arg \\colon \\mathbb{R}^+\\times\\mathbb{R} &\\to \\mathbb{R} \\\\\n(r,\\ \\phi) &\\mapsto \\phi\n\\end{align}", "6c276a0911b686561b66a6e752811f3a": "\\rho_{s}(\\mathbf{r},t)=\\rho(\\mathbf{r},t).", "6c28080a914c088290224922b70a80b1": "E = T + U \\,\\!", "6c2813416cc11d40f4031fb827732d68": "1\\to\\mbox{SU}(n)\\to\\mbox{U}(n)\\to\\mbox{U}(1)\\to 1.", "6c283481904a6ab30a216ba63f7afb1e": "\\!\\,y=\\cos(kt)\\cos(t)", "6c287a653768df438c87951331979513": "\\binom{n}{3}_F = \\binom{n}{n-3}_F = \\frac{F_n F_{n-1} F_{n-2}}{F_3 F_2 F_1} = F_n F_{n-1} F_{n-2} /2,", "6c28919a03cdfaf6987165f98977e976": "{q\\choose d} + {q\\choose d+c} + {q\\choose d+2c} + \\cdots = \\frac{1}{c}\\cdot \\sum_{k=0}^{c-1} \\left(2 \\cos\\frac{\\pi k}{c}\\right )^q\\cdot \\cos \\frac{\\pi(q-2d)k}{c}.", "6c291665ae86530ef5baa39d5ddfa704": "k(2\\pi/n)", "6c298723202539942199927e9f37c966": "\\delta \\colon M \\to R", "6c29a2547e2fb9a4707455d17aebee7c": "\\scriptstyle\\{x_1,..x_k\\}", "6c29daacc6317820fb25dd0a4a94dff1": "\\textstyle N_\\mathrm{S}/N_\\mathrm{P}", "6c2a1e4656221a8263d1c69379071eca": "h_n = \\det (b_{i+j-2})_{1 \\le i,j \\le n+1}.", "6c2a57cea8ac14c5fe79cf7a58fccf87": "\n\\begin{pmatrix}\\rm Li &\\rm Be \\\\\\rm Na &\\rm Mg \\end{pmatrix}\n\\otimes\n\\begin{pmatrix}\\rm Li &\\rm Be \\\\\\rm Na &\\rm Mg \\end{pmatrix}\n=\n\\begin{pmatrix}\n\\rm Li_2 &\\rm LiBe &\\rm BeLi &\\rm Be_2 \\\\\n\\rm LiNa &\\rm LiMg &\\rm BeNa &\\rm BeMg \\\\\n\\rm NaLi &\\rm NaBe &\\rm MgLi &\\rm MgBe \\\\\n\\rm Na_2 &\\rm NaMg &\\rm MgNa &\\rm Mg_2 \\\\\n\\end{pmatrix}\n", "6c2a5f21307599f4b0d953b781045c77": " \\delta W = (\\sum_{i=1}^n \\mathbf{F}_i)\\cdot\\mathbf{d}\\times \\vec{\\omega}\\delta t+ (\\sum_{i=1}^n \\mathbf{F}_i)\\cdot\\mathbf{v}\\delta t + (\\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i)\\cdot\\vec{\\omega}\\delta t = (\\sum_{i=1}^n \\mathbf{F}_i)\\cdot(\\mathbf{v}+\\mathbf{d}\\times \\vec{\\omega}) \\delta t + (\\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i)\\cdot\\vec{\\omega}\\delta t .", "6c2ab028533ca44d10f18854074fa9ac": "\\dot{x_2} = \\ddot{x_1}", "6c2abb783bb7efa7ceede2a959cd777b": "\\boldsymbol{{x}}(t)=\\Phi(t,\\boldsymbol{{x}}_0)", "6c2abdfd2c69a68b21f45f8272f96001": "|x_n|_\\infty = \\sup (2^{-n}, 0) = 2^{-n} \\rightarrow 0", "6c2ac4e8856813657519a207f862ee78": "I_n =\\int f(x,n) \\,dx, ", "6c2ac791be73bd4d2822dc3c0dced5fa": "\\dim X = \\dim Y ", "6c2ad153b6588dce1b84352daae5040b": "h_{\\mathrm{ZOH}}(t)\\,= \\frac{1}{T} \\mathrm{rect} \\left(\\frac{t}{T}-\\frac{1}{2} \\right)\n = \\begin{cases}\n\\frac{1}{T} & \\mbox{if } 0 \\le t < T \\\\\n0 & \\mbox{otherwise}\n\\end{cases} \\ ", "6c2aedcd1d441cb5be0dd98825b9d14b": "A\\subset\\mathcal{X}", "6c2b07b877a2afd7c69f1b85b2fa0e90": "\\nabla_{e_i}e_j = \\sum_{k=1}^n\\Gamma_{ij}^k(\\mathbf e)e_k.", "6c2bf831d977cc2c78cad91f3dd259d8": "r P_A", "6c2c5773e602bef1ca875cc55a8bc93d": "\\mu^D", "6c2ced3bea0641c19acc969cdd6bb827": " [I_a,I_b]= c^d_{ab}I_d, \\qquad [F_\\alpha,F_\\beta]= c^d_{\\alpha\\beta}I_d,\n\\qquad [F_\\alpha,I_b]= c^\\beta_{\\alpha b}F_\\beta. ", "6c2cf6334faf34a7be2835f8152de5cc": "c.c.", "6c2d098e0d56bf498b2b80e5d7c1910e": "\n f:= \\cfrac{1+2R}{1+R}(|\\sigma_1|^m + |\\sigma_2|^m) - \\cfrac{R}{1+R} |\\sigma_1 + \\sigma_2|^m - \\sigma_y^m \\le 0\n", "6c2ddfc49e0e070fbcd40c77a49360cd": "\\frac{\\delta M}{\\delta t}= D_0 \\nabla^2 M - \\frac{M}{T_{2b}}", "6c2e021c27bd336ee96485cb6f7342ff": "{}^6_5", "6c2e17ef64d66b8a33f141608149bd3c": "\\forall x (P \\land Q(x))", "6c2e290ab33db4de555592e75c3d62ff": "\\mathbb{R}^*", "6c2e3d9c9415d71f3442e718b1410a9b": " x_1:A_1, x_2:A_2, \\ldots \\vdash t:B", "6c2e3e2e98abd1fd9a66519db9da8d90": "\\frac{1}{6}", "6c2e4a2e1a96cef4e2615f9648330ff4": "\\{p_1, p_2, \\ldots, p_n\\}", "6c2e81a3b5cb77c6cd18f1fbfb0a326b": "\\frac {\\sum_{i=1}^m n_i}{m}", "6c2e8d745802633da1c37b90cdf0d943": "\\varphi : \\mathbb{R} \\times \\Omega \\times X \\to X", "6c2f01ffeefb70a98edd4345dc2544c1": "\\mathcal{H}_H", "6c2f3a464ccd09615ef8b95a4ca0cab5": "(\\lambda 1 1) (\\lambda \\lambda \\lambda 1 (\\lambda \\lambda \\lambda \\lambda 3 (\\lambda 5 (3 (\\lambda 2 (3 (\\lambda \\lambda 3 (\\lambda 1 2 3))) (4 (\\lambda 4 (\\lambda 3 1 (2 1)))))) (1 (2 (\\lambda 1 2)) (\\lambda 4 (\\lambda 4 (\\lambda 2 (1 4))) 5)))) (3 3) 2) (\\lambda 1 ((\\lambda 1 1) (\\lambda 1 1)))", "6c2f922640f32977db38c9d46724f0d5": "E = \\hbar \\omega \\;", "6c2fda78e6d71a41db2f378133f858c5": "d > 2w", "6c2ffa555c9f4d47fc6276787bf18425": "\\mathbf T^n \\begin{bmatrix} \\vec f^{n-1} \\\\ 0 \\\\ \\end{bmatrix} =\n \\begin{bmatrix}\n \\ & \\ & \\ & t_{-n+1} \\\\\n \\ & \\mathbf T^{n-1} & \\ & t_{-n+2} \\\\\n \\ & \\ & \\ & \\vdots \\\\\n t_{n-1} & t_{n-2} & \\dots & t_0 \\\\\n \\end{bmatrix} \n \\begin{bmatrix} \\ \\\\\n \\vec f^{n-1} \\\\\n \\ \\\\\n 0 \\\\\n \\ \\\\\n \\end{bmatrix} = \n \\begin{bmatrix} 1 \\\\ \n 0 \\\\\n \\vdots \\\\\n 0 \\\\\n \\epsilon_f^n \\\\\n \\end{bmatrix}. ", "6c302e70515c1aa6a48fcab808f73558": " (\\hbar\\omega)^2 = (c \\hbar k)^2 + (m_0c^2)^2 \\,,", "6c30352ea1f3742bd9882bad44c8dcdb": "2r", "6c305bb891de7f451dbd3fb8082b64ac": "\n E=\n \\left( { a_1\\, a_2 \\over 2 \\pi L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_{Ds}^2 }\n \\mathcal J_0 \\left ( kr_{12} \\right)\n= \\left( { a_1\\, a_2 \\over 2 \\pi L_B}\\right) K_0 \\left( k_{Ds} r_{12} \\right) \n", "6c30b42101939c7bdf95f4c1052d615c": "A,B", "6c30dc3d05f4c5d73cda3936e0ea962a": "M=\\pi R^2 \\rho ", "6c31777188843b75b66fe34ad58475a1": "\\left[\\tau_{ind} + \\tau_{cap}\\right] = 1", "6c318d9e4efbbddad5f7261aab660830": "S_n=E(\\sin (n\\theta))\\,", "6c31a148eb1f973d163e8dce9cb4eb5f": "R^{(3)}_{ij}", "6c31cdf34805b9c337b8e63e44d088fe": "V_L=L\\frac{dI_L}{dt}", "6c320ee9e9e4033fe5b6ac07b4388195": "\\lim_{k \\to \\infty}\\|A^k\\|^{1/k} = \\rho(A).\\,\\,\\square", "6c322064e529c811ed8e569f08688ba8": "c_i+\\alpha_i", "6c32397d589a727cab119434b4bd42e5": "\\displaystyle{\\alpha_m={1\\over N} \\sum_{n=1}^N\\lambda_n z_n^m.}", "6c32be0184bd24bdeee2d6cf4255444e": "\\langle A \\rangle _r", "6c32dc57ef37350da00ba53985a59451": "i>b_n", "6c32ef0df34abc20dcce087fa3f40aed": "n_i n_i = n_1^2+n_2^2+n_3^2 = 1", "6c334a60d38cf2e2f97085e39ac8309e": "F^\\prime", "6c33b02943dea6a75e0d90b1ae41dc44": "x^q \\equiv 1\\;(\\text{mod }p)", "6c33b18d5223183ab30a81aa2c4002fd": " \\sum_{n=0}^{\\infty} \\frac{1}{n+a}= - \\psi (a) ", "6c33b35bb880253f01b173828fb85831": "1.645 > \\lambda \\ge 1.28", "6c33c579b3ce5915ffd3e7f5ea0a1e60": "\\lambda=1/\\epsilon\\,\\!", "6c33e6060a8aec5c79dd69093a67f69b": "y_0,y_1,...,y_j", "6c33ebc1422e51ef400fd6041a82997f": "b=p^2-q^2k^2,", "6c3443b00c1e4dcab1e29b267894d64d": "\\kappa(vw) = \\kappa(w) \\kappa(v)", "6c34527773da9f06105eef47a2505022": "T^\\alpha \\nabla_\\alpha", "6c345a5786387214e7f88d8063b8bb02": " \nu(\\rho,\\phi,0) \\propto \\rho^{p+|m|}e^{-\\rho^2+im\\phi}.\n", "6c34f84f6649bdbf2515c2ad07d22e18": "r = R - \\delta = \\alpha k^{\\alpha-1} - \\delta \\,", "6c35186076a5a16ebc3c102813d59776": "G(a,b,c)=\\sum_{n=0}^{c-1} e\\left(\\frac{a n^2+bn}{c}\\right),", "6c35f1debc0401de74e17c339e54007c": "\\tilde{\\theta}_i = \\hat{B}_i\\,\\hat{\\mu} + (1-\\hat{B}_i)\\hat{\\theta}_i", "6c3631283c59fa8cd26165a630f010bd": "\\mathrm{A}(M) = \\frac{9,850 + 0}{9,850 + 150 + 0 + 0} = 98.5%", "6c366d565f408e4c706aafd0a24cdab9": "f(x)=x^5+x\\,", "6c3680d60adb19d648360df20b40f7b8": " (\\partial V)_P=-(\\partial P)_V=\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "6c36c7beda53069d09bfc7d0258f6a5c": "z \\,{\\partial \\operatorname{Li}_s(z) \\over \\partial z} = \\operatorname{Li}_{s-1}(z)", "6c36fcc4abaee84bacada51b96df30b5": "\\sum_{i=1}^n i", "6c3700c46d8e52ffaadbc23f02f584b1": " b_{2k+1}=\\frac{(1-x_{2k-2}x_{2k-1})}{(1-x_{2k+2}x_{2k+3})} x_{2k+0}", "6c37180a4c2ff028527d39ef67139360": "(X^*_{\\sigma})^*", "6c374cbbca78c61cf3681753500beb47": "\\sqrt{S}\\sigma_P\n=\\frac{\\sigma}{\\alpha}(1-\\exp(-\\alpha(T-S)))\\sqrt{\\frac{1-\\exp(-2\\alpha S)}{2\\alpha}}\\,", "6c37825a0647b440ea3e74376b3b965f": "\\dot V", "6c379e838e4c70d09709c0ebaa2b1e2c": "\\varphi = 1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\;\\;\\ddots\\,}}}", "6c37ff9fa569f7753bb1f1ea2aaae4f0": "C_\\mathrm{statistical-CSI} = \\max_{\\mathbf{Q}} E\\left[\\log_2 \\det\\left(\\mathbf{I} + \\rho \\mathbf{H}\\mathbf{Q}\\mathbf{H}^{H}\\right)\\right].", "6c38107b022bdbe93b335da0e175f953": "\\mathrm{MA}={25\\ \\mathrm{cm}\\over f}+1\\quad", "6c382609dac1b68597aa9c9c45fc96c0": " \\sim 10^{8126}\\,", "6c38880e886c852451827932896c31ac": " \\scriptstyle *:A^n \\to \\mathfrak{G} ", "6c39088eb8729d9408058d3f28c0b212": "\\widehat{U}(\\theta)|\\alpha\\rangle = |\\alpha e^{-i\\theta}\\rangle", "6c395d58b614bf9453f743863d251678": "2\\alpha + \\beta - 3 \\le 0", "6c395ff1d5d07bb810f6de1d39cddbd4": "\\eta=\\frac{W}{Q_H}=1-\\frac{T_C}{T_H}\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad(1)", "6c3998b8061450a8564d955346403d36": "\n\\ R = 8.314510 \\cdot \\mathrm{J \\cdot mol^{-1}} \\cdot K^{-1}\\,\n", "6c39fdee44f1ae891931176889988639": "(\\cdot|\\cdot) ", "6c3a3571086109ad1c5a0cf342a20877": "D \\to \\infty", "6c3afe60195af589c58c25ce1ef9fb42": "\\Omega_k = 1-\\Omega_m-\\Omega_\\Lambda", "6c3b518ff156bdf7d332fa32d680bbba": "x^5 - x + 1 ", "6c3bad0a0acf2713ec1e3905022dff7b": "g=1/|J|", "6c3bf2657e73cd5f4160bbebad8d4f29": "c_k = \\frac{1}{2\\pi}\\int_{0}^{T} x(t) \\, e^{-i 2 \\pi k t / T}\\, dt", "6c3c2967e3c4d94d4a59b9f05be10b6e": " \\frac{\\rho_2}{\\rho_1} = \\frac{\\left(\\gamma + 1\\right)M_1^2}{\\left(\\gamma - 1\\right)M_1^2 + 2}", "6c3c5c91d8f526548f6e3ed53ab2e900": " \\psi(x, t)=e^{it(\\hat D+\\hat N)}\\psi(x, 0)", "6c3c5cf076489a11e82662d8771e2cdd": "\\mathrm d \\colon S \\to \\Omega_{S/R}", "6c3c712988116b561e5aecab86dcc001": "\\hat{\\rho}(x,\\hat{u}):= \\max_{\\rho\\ge 0}\\ \\{\\rho: u\\in S(x), \\forall u\\in B(\\rho,\\hat{u})\\}", "6c3c845b7998658500d628faa02ed06b": "\\dot {\\bar{v}} = -\\mu \\cdot \\frac {\\hat{r}} {r^2}", "6c3d15370241af59d52d91121e522917": "V_{\\rm m} = {M\\over\\rho}", "6c3d478a1aa6d89e545d2122c4ba49fa": "\\Delta T = i\\cdot K_b \\cdot m", "6c3d4fa744870cbfd596c25d4bc439fa": " S_h = \\sum_{n=0} {{h^n} \\over {n!} }D^n ", "6c3d4ff91bd8579c4b08b9420a319278": "\\color{ProcessBlue}\\text{ProcessBlue}", "6c3d6b199c1179928603fc81626bde03": "\\cos(c) = \\mathbf{v} \\cdot \\mathbf{w}", "6c3d86686c6092479505d5caff1ef732": "g_p : T_pM\\times T_pM \\to \\mathbf{R}.", "6c3db9ac212a7ff88522ddd0c0274a1b": "u={i}_0{i}_1\\cdots {i}_k", "6c3e23081b28bc9cae7516e2ab1d5b85": "1/6 + 1/3 + 1/2+ 1/1 = 2", "6c3e45036ca7b875bd45c2bca5370f3e": " N_{i}^{e}=\\frac{1}{2}\\xi\\left(1\\mp\\zeta\\right),\\qquad i=1,4 ", "6c3e82e53a824bf9eb03f902cb6b1abf": "I = 100 \\times \\left [ n + ( N - n ) \\frac{t_{r (unknown)} - t_{r (n)} }{t_{r (N)} - t_{r (n)} } \\right ] ", "6c3e85ec728a7aeccc2c89ccbcf09d6c": "\\varphi(a)\\mid\\varphi(b).", "6c3e8c3067d3e8ee1a01c720b7dbe937": "{S^k}_i", "6c3f145ce394f5768c03201c766b0d0a": "\n \\overset{\\triangle}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{\\sigma}\\cdot\\boldsymbol{w} - \\boldsymbol{w}\\cdot\\boldsymbol{\\sigma} \n", "6c3f3c8f893fbdb3956a9327cebfb488": "(r_1,\\ldots,r_n) \\in \\mathbb{R}^n", "6c3f72b7768f7c9f43490cc315e3da2d": "(3/2)^{53}", "6c3f7557e89899003eb08206e9974522": "\\int_0^\\infty \\frac {\\sin mx}{e^{2\\pi x}-1}\\ dx=\\frac{1}{4} \\coth\\frac{m}{2}- \\frac{1}{2m}", "6c3f9d2aae31d87f054c6de110a46097": " z = {2\\over a^2} x y ", "6c3fa2d5ce83402ff2bf0b3e058f6f6a": "\\varepsilon(P_j) = 0\\,", "6c3fd6ccaec84f489e04800b1d0f2511": "T(y_0)\\,", "6c3fef37caca1c335ae66fc85370bcd2": " \\nu_B (x,y) := ( \\lfloor x\\rfloor, \\lfloor y\\rfloor ), ", "6c400eb43391b4295a0c6fe231be8729": "Q_L = H_m.\\,", "6c407732c9b3c4ef43e6d686dc10f350": " \\frac {1}{c^2} \\frac{\\partial^2}{\\partial t^2} \\psi - \\nabla^2 \\psi + \\frac {m^2 c^2}{\\hbar^2} \\psi = 0. ", "6c40cf3f0d349639e730b8599b9d9236": "\\ \\mathbf v = \\dot{\\mathbf x} =\\frac{d\\mathbf x}{dt}=\\frac{\\partial \\chi(\\mathbf X,t)}{\\partial t} ", "6c40eb38389f6c0a4ac77749cc8911a6": " \\chi(\\omega') /( \\omega'-\\omega)", "6c40edc23b334387c28e5cc734ffce4d": " T(\\epsilon) = 1 - \\epsilon {d \\over dx} = 1 - {i \\over \\hbar} \\epsilon \\left ( - i \\hbar{ d \\over dx} \\right )", "6c4108cea39595ecd123e9bfb40020c0": " R^5_5(\\rho) = \\rho^5 \\,", "6c410ff6c8acf705c792627e17b71830": "G_L = G_R = \\frac{1}{\\sqrt{2}}", "6c41526a30f891615eb245a494b87fc3": "\n\\ \\nu(t)=\\sum_{k=0}^{N-1}X_k e^{j2\\pi kt/T}, \\quad 0\\le t \\|T\\|.", "6c42e1f7bca1b36297efa6485a340f27": " A_\\alpha =\\operatorname{sgn}(\\pi)a_{i_1,j_1}a_{i_2,j_2}\\cdots a_{i_n,j_n}.", "6c42e24c9c2de797b1875eae4003a9dc": "\\tilde{\\Lambda}=\\left(\\Lambda+\\Lambda^T\\right)/2", "6c436a904d8305dad5d8d8de88da9442": "\\mathcal{D}(\\mathcal{I}(D)) \\cong D", "6c43756dac4dba6dbf5de9a1df889afe": "\\mathbb{Q}_p^*/\\left(\\mathbb{Q}_p^*\\right)^b", "6c43a5fb7e6d8248654b3ff1f79fdc9c": "\\mathrm H_2\\mathrm O\\underset{hv}{\\longrightarrow}\\mathrm H^++\\mathrm{OH}^\\bullet", "6c43c1deccc70fb56bdcc3814fe1bc53": "\\frac{2,000,000 \\mbox{ tons} \\times 63 \\mbox{ MW}}{1000 \\mbox{ MW}} = 126,000 \\mbox{ tons}", "6c4431c47c33943d5ca154ffb947c30d": "r_i = R_F +\\beta_i(R_M-R_F)+\\alpha_i+\\epsilon_i", "6c44370a4eae6f98f92a1d31e56df1a1": "n^2 = (n-1)^2 + (n-1) + n = (n-1)^2 + (2n-1)", "6c44a2691f1d2d25a5d750c748196d99": " \\hat\\psi_n(\\xi) = (-i)^n {\\psi}_n(\\xi) ", "6c44fe140bd3a2ce640b7da66125a813": "Aa~Gradient=\\left(F_iO_2(P_{atm}-P_{H_2O})-\\frac{P_aCO_2}{0.8}\\right)-P_aO_2", "6c45583e2c854599b903ef4102d9cd4b": "B(x) = \\frac{1}{\\hbar} \\sum_{n=0}^\\infty \\hbar^n B_n(x).", "6c456523461433fcefc80a231536cb51": "A + P + U\\, ", "6c459f5b10712b971144cd818059860d": "P(d|c)", "6c45eb1a810f46c1380002257c1569ce": "\\scriptstyle \\{ \\mathbf{v}_i \\}_{i\\in I}", "6c4625e43e1afc63bee0a0ae9f682fb2": "\\frac{V}{V}", "6c4628a5ad4c677477b5e213fa97d7ed": "\n\\begin{align}\nx & = & 1100&.1\\overline{01110}\\ldots \\\\\nx\\times 2^6 & = & 1100101110&.\\overline{01110}\\ldots \\\\\nx\\times 2 & = & 11001&.\\overline{01110}\\ldots \\\\\nx\\times(2^6-2) & = & 1100010101 \\\\\nx & = & 1100010101/111110 \\\\\nx & = & (789/62)_{10}\n\\end{align}\n", "6c462b08a2ac9ffbd61f54894265a068": "\\mathbb{E}\\left[\\mbox{ Arnold }|\\mbox{ Charles calls }\\right] = \\frac{42-9-4}{42} \\cdot (P+3) = \\frac{29}{42} \\cdot (P+3)", "6c464c50ecf2402dfacf3446e12b3a90": "\\mathbf {L} \\boldsymbol {\\beta} = \\mathbf {d}", "6c46521861322a931b85c2171a82315b": "P(x)P(x\\rightarrow x') = P(x')P(x'\\rightarrow x)", "6c46b14a780c9bd6122fade3f89d97c0": "Fr\\,", "6c46fcdd2f1d8c3ce91664e4adf02ced": "G\\rtimes_{\\phi}A", "6c472c85a58c0d88dd9b23343e4baef1": "{2,1,3}\\,\\!", "6c476072652f8d08b75d4f25087427e5": "c_0,\\dots,c_k", "6c4777f86f226e8d7333a89a0581e57f": "\nDB = \\frac {1} {n} \\sum_{i=1}^{n} \\max_{i\\neq j}\\left(\\frac{\\sigma_i + \\sigma_j} {d(c_i,c_j)}\\right)\n", "6c47ba3092a7523d771c4c105099fdf3": "\\begin{cases}\n \\sigma & \\mbox{if } \\sigma=n^\\searrow or \\sigma=n^\\nwarrow or \\sigma=\\overline{n}^\\nwarrow \\\\\n \\overline{n}^\\searrow\\;\\|\\; S(\\rho) & \\mbox{if } \\sigma=\\overline{n}^\\searrow(\\rho)\\\\\n pino\\;\\|\\; S(\\rho) & \\mbox{if } \\sigma=pino(\\rho)\\\\\n \\mathcal{S}(a)\\;\\|\\;\\mathcal{S}(\\rho) & \\mbox{if } \\sigma=a.\\rho\\\\\n \\mathcal{S}(\\tau)\\;\\|\\;\\mathcal{S}(\\rho) & \\mbox{if } \\sigma = \\tau\\,\\mid\\,\\rho\\\\\n \\lambda & \\mbox{if } \\sigma=0\n\\end{cases}", "6c47dbffa09c84b4f6fe4597561acc12": " (t_1,\\dots,t_k)\\in \\mathbb{R}^k ", "6c47ee99483d4c60ded7dbc88762ddd3": "\\lambda_1, \\lambda_2", "6c484c01354de1672dc9296063a71c4b": "(\\mathbf{x},z_1) = (\\mathbf{0},0)", "6c48db323dbd71466605b61e3fd2058b": "\\beta \\,\\, = \\,\\,\\,0", "6c495cdb47aa4b464971b50752cf2062": "E\\setminus S", "6c497b5bb4010e3ef31c0bb234088e1d": "d=\\sum_j \\partial_j \\, dx_j", "6c49e5aa25e4023b9ab59640370372f7": "\\left(\\frac{\\partial U}{\\partial V}\\right)_T = T\\left(\\frac{\\partial P}{\\partial T}\\right)_V - P", "6c4a053d88f91cb5d4de21ba43130cc4": "\\exp\\left(\\pi\\sqrt\\frac23 \\frac{\\sqrt n}{k} \\right) , ", "6c4a8a399acef64616df5ac59ba73fb3": "f(x)=\\sum_{k=1}^n a_k {\\mathbf 1}_{A_k}(x),", "6c4a8bcfc062cb5943e51b78f8b8eeba": "\\tau_{\\phi}(\\omega) = -\\begin{matrix}\\frac{\\phi(\\omega)}{\\omega}\\end{matrix}", "6c4ab7bb37324a3a917501ed8727030e": "\\langle d \\rangle^{-1}=41 R^{-0.21}", "6c4ad1857efa69f75bff0b2ac1ede31d": "(\\mathbf{c}_i, y_i)", "6c4b0155ff2ade733a5fd7f8cb647eb1": "~\\hat{\\rho}(0)~", "6c4b7e43ada8e9de098c0bfea442d07f": "\\nabla_XT", "6c4bdb85e14cc9704b1fc37e86113d1e": "f(\\textbf{v}_1,\\ldots,\\textbf{v}_n) = \\sum_{j_1=1}^{d_1} \\cdots \\sum_{j_n=1}^{d_n} \\sum_{k=1}^{d} A_{j_1\\cdots j_n}^k v_{1j_1}\\cdots v_{nj_n} \\textbf{b}_k.", "6c4be144a1df2becec02383425e8d097": "Y(z)=E(z) + \\left[ X(z)-Y(z)z^{-1} \\right] \\left( \\frac{1}{1-z^{-1}} \\right). ", "6c4bf3c754629451dce767439cc114ec": "\\lambda_f(t) = \\omega\\left\\{x\\in X\\mid |f(x)| > t\\right\\}.", "6c4bf9e4825fdb2c176366183aecca7a": "(\\mathcal{F}g)(\\xi) = (\\mathcal{F}f)(\\xi - \\eta)", "6c4c1b0b65387fb4401c88e45045af43": "T_{L} - T_{T} = \\frac{2 (L_{L} - L_{T}) }{c} ", "6c4c749e8d415c408edb012b6aba419d": "\\{1,2,3,...,2m\\}\\,\\!", "6c4cab7b8f9201371695bf57671af654": "\\text{Margin of Safety}=\\frac{\\text{Failure Load}}{\\text{Design Load*Design Safety Factor}}-1", "6c4d8ac6db1239392691150b5e19f777": "R_{exp} = \\frac{3hc10^{3} ln(10)}{32\\pi^{3}N_A} \\int \\frac{\\Delta\\epsilon}{\\nu} d{\\nu}", "6c4dc351cbcbbad29c6334cadb5c1b45": "|\\phi^+\\rangle", "6c4e09fd0d23df4edcc239c48743c15d": "\n\\mathbf{F} = F(r) \\hat{\\mathbf{r}} = m\\mathbf{a} = m \\ddot{\\mathbf{r}}\n", "6c4e7860548aa6bac4d2b3f5f528be23": "\\displaystyle{\\varphi_{0,t}(z)=e^{-t}(z+a_2(t)z^2 + a_3(t) z^3 +\\cdots)}", "6c4ec6f0b33a2588d96a99d0166e916f": "\\mathop{\\rm Char}P=\\{(x,\\xi)\\subset T^*\\R^n\\backslash 0:\\sigma_p(P)(x,\\xi)=0\\},\\text{ with }\\sigma_p(x,\\xi)=\\sum_{|\\alpha|=m}i^{|\\alpha|}A_\\alpha(x)\\xi^\\alpha\\,", "6c4edd215bfb46deac06326e175735e1": "\n\\begin{align}\np(\\tilde{x}=i\\mid\\mathbb{X},\\boldsymbol{\\alpha}) &= \\int_{\\mathbf{p}}p(\\tilde{x}=i\\mid\\mathbf{p})\\,p(\\mathbf{p}\\mid\\mathbb{X},\\boldsymbol{\\alpha})\\,\\textrm{d}\\mathbf{p} \\\\\n&=\\, \\frac{c_i + \\alpha_i}{N+\\sum_k \\alpha_k} \\\\\n&=\\, \\mathbb{E}[p_i \\mid \\mathbb{X},\\boldsymbol\\alpha] \\\\\n&\\propto\\, c_i + \\alpha_i. \\\\\n\\end{align}\n", "6c4ee66cc536772f42827d776646dabd": "t+\\delta t", "6c4f3474e7ef90e972ed0751b9dee635": "b^n = \\underbrace{b \\times b \\times \\cdots \\times b}_{n \\text{ factors}}.", "6c4f5cc9bf70de7c3dc4fc30f7a438d6": " \\tfrac12 + \\tfrac1{10} \\sqrt{15} ", "6c4fd968e86080996c82bdec9d359bad": "\\phi \\;=\\; He^{\\frac{1}{H}i\\theta}", "6c4fe9527b8487b84373be7945287517": "\\neg AX\\phi \\equiv EX\\neg\\phi", "6c4ff69dbcc329835a33b80fe3a145c7": "h_t", "6c50bb8ec12938684b83b21eee95fcd5": "S = A \\oplus B \\oplus C_{in}", "6c50c0379c1be7c1b58d4622832e85e2": "\\omega_\\mathrm{res} = \\sqrt{\\omega^2 - \\left ( \\frac{\\kappa}{4m} \\right )^2 } \\,\\!", "6c50c97476e9ae9af34de2bdb481ede1": "\\dot{\\rho} = -3 H \\left(\\rho + \\frac{p}{c^2}\\right),", "6c50d974d1a5d10ee1af215c53cdfbfc": "T_{\\text{H}} = \\frac{\\kappa}{2\\pi}.", "6c50ec6729c75d3d4b168e784564b849": "H_2O + H_2O \\rightleftharpoons H_3O^+ + OH^-", "6c50ffb84a36916604d54f3ce3ee4900": "g_{UU}(x) = I, \\quad g_{UV}(x)g_{VW}(x)g_{WU}(x) = I", "6c511a6dcb3948418f584551ca360946": "u_4", "6c5175891e1aacf6db15b7f9990c990a": "\n\\mathcal{S} = \\int_{t_1}^{t_2}\\; L(x,\\dot{x},t)\\,dt\n", "6c517d62dd7821952891516a5206d345": "d d = 0.", "6c51a86f3bd7d0dec36be9047635d8e6": "a=x_0 0", "6c63ac7fa259c224effd3e5f3b9bfdfd": "[-L,0]", "6c63bb20c57d5925776346bf3302dee1": "R_n(\\xi,1)=1", "6c63c2c4e5d5f695b3be9ae4bcda1daa": "I_h", "6c63df29780290d3c3aab151b6009589": "a_k", "6c63f734f0edc8844522d2a0e52feaab": "\\mathbf{I} = \\mu \\mathbf{M} \\,\\!", "6c63f83735d4e90c40c4f19f03e13923": "\\mathbb{E}[Y_n] = F_e(\\boldsymbol\\beta \\cdot \\mathbf{s_n})", "6c6404adc033dfed51422fdaf7fa0494": "\\mathbf{A}", "6c6451acf73fca94ef5b5690f6544461": "[\\lambda \\bold{x_\\alpha} \\bold{B}_\\beta] \\bold{A}_\\alpha \n= \\bold{S}^{\\bold x_\\alpha}_{A_\\alpha}\\bold{B}_\\beta", "6c6475b4967bf0722a796c4adba34d9c": " g(x)=\\min S_x", "6c64774ff8ef35ec285f2f107cc91a2f": "P_{\\mu}=mc\\,U_{\\mu}", "6c64a5291a133d4bb0617d08dd2ee2c2": "O(n^m)", "6c6511c6ad78f438dc10b207534b4c14": "L_1, \\bar{L}_1", "6c6524ebd012d37e91255f98d3a64b6a": "C_{\\alpha I}^{\\;\\;\\; J}", "6c6526273c612767dbca7ac35fc98daa": "[0:1]", "6c656980b16b6e2b777c32f22f16e93e": "V(h)=\\left\\{ \\begin{matrix}0 & \\mbox{if}\\quad h\\geq \\Big( \\frac{D+d}{2} \\Big)\\\\ \\infty & \\mbox{if}\\quad h< \\Big( \\frac{D+d}{2} \\Big) \\end{matrix} \\right. ", "6c65715630a5921d92ab38d8a1959a96": "<\\phi , \\psi>_{Kin}", "6c65cb9fa3af99826552952da62df793": "\\scriptstyle E(w),", "6c66028ac27dfcfaaa639a6aee537a21": " \\langle x, y \\rangle", "6c661ea68e87db2d8d92de7d4ecba1bf": " \\dot{g} = A g \\, ", "6c664619af1c95cdbe4f2c8307320c41": "\\hat h(P)\\ge c(E/K)/[K(P):K]^{3+\\epsilon}", "6c66901228069751c4198c1b59c1f3e9": "2^{42643801}-1", "6c66dae541ab399e38f713c3630160f0": "M/(N_1\\cap N_2)", "6c67bfcb97ad4ab5d167ebf5c6c2567c": " x = r \\, \\cos \\theta \\, \\cos \\phi, ", "6c67d983aa42529176e7d7e6a066628f": "D_i = 1", "6c67dab7285706a5df5e07bbeee9719e": "f(x_1,\\ldots,x_k)\\!", "6c67dda897c3d39d5c9605603980b590": "\\|y - Dx\\|_p \\le \\epsilon", "6c68563070ff102212203c4688c4f926": "A(x,t)\\,", "6c68636d55bd53016d2b0f7500b23550": "(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_\\theta = B^{s_\\theta}_{p_\\theta, q_\\theta}, \\quad s_0 \\ne s_1, \\ 1 \\le p_0, p_1, q_0, q_1 \\le \\infty.", "6c68a358d997321edf2e58cb0b240764": "\n\\begin{align}\ng(v) & = \\int \\int \\exp(ik[y f(z) - z + x]) g(z) (1-y f'(z)) \\, \\frac{dk}{2\\pi} \\, dz \\\\[10pt]\n& =\\sum_{n=0}^\\infty \\int \\int \\frac{(ik y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)}\\, \\frac{dk}{2\\pi} \\, dz \\\\[10pt]\n& =\\sum_{n=0}^\\infty \\left(\\frac{\\partial}{\\partial x}\\right)^n\\int \\int \\frac{(y f(z))^n}{n!} g(z) (1-y f'(z)) e^{ik(x-z)} \\, \\frac{dk}{2\\pi} \\, dz\n\\end{align}\n", "6c68bc640ded0abc6d46abc30dff3b1c": "(Q_1, P_1)", "6c68cd2bfd95f2b0aba0526378af4924": " G < 3\\rightarrow 3\\rightarrow 65\\rightarrow 2 <(10 \\to 10 \\to 65\\to 2)=f^{65}(1)", "6c69301ddf255690a508d0eaf5d84fbc": "|xy|=|x| |y|", "6c69470432434ce2cfcdbe5bb10cd3af": " \\omega = \\dot{\\theta},", "6c696abff1b5525fa342f80b3199320f": "\n\\sum_{A=1}^N \\vec{q}^{\\,A}_r = \\vec{0} \\quad\\mathrm{and}\\quad \\sum_{A=1}^N \\vec{R}^0_A\\times\n\\vec{q}^A_r = \\vec{0}.\n", "6c696b1d359480d9884e10686e95a1fa": "w_{ij} \\in \\left\\{0,1 \\right\\}", "6c69ce24f45e2f765c7e48a7da76c719": " \\rho(r) ", "6c69eced43863ae6cae8f94433db30e4": "f:M\\longrightarrow BG", "6c69fda452e0c010b07527af034a7959": " 4000 \\ \\mbox{g}\\,H_2 O \\cdot \\frac{1 \\ \\mbox{mol}\\,H_2 O}{18 \\ \\mbox{g}\\,H_2 O} \\cdot \\frac{10 \\ \\mbox{mol}\\,e^{-}}{1 \\ \\mbox{mol}\\,H_2 O} \\cdot \\frac{96,000 \\ \\mbox{C}\\,}{1 \\ \\mbox{mol}\\,e^{-}} = 2.1 \\times 10^{8} C \\ \\, \\ ", "6c6a22550977ba00e1ea7423f8289875": "\\textstyle d=2", "6c6a2fc4b37474f047c589847ce8b363": "r \\to \\infty", "6c6a72a3e1a069f12773e86ddf96d739": "p = {s \\over c+v}", "6c6a81c17bbe5c6eda250ab047a073a4": "E = VQ = VIt = Pt = \\frac{V^2 t}{R} = {I^2}Rt \\,\\!", "6c6a83ef3c05834740a1974a2a79e71a": "\\eta_{opt} = \\frac {U^2} {(1 + \\sqrt{1 + U^2}) ^ 2}", "6c6a9f580105866100bc2dbd04dbe7a0": "F_5(x)=\\frac{407}{960}x-\\frac{49}{96}x^3-\\frac{21}{320}x^5", "6c6b139c179a9fb694855dc464563d1c": "E = {Z_3}^2 = 4", "6c6b7d1eeef89ddb360d0008dadda126": "A \\mathbf{v} = \\lambda \\mathbf{v},\\,", "6c6bbe79422eb3352740f42b55fecab5": "\\sqrt{\\frac{(m - 1)(k - 1)(m + 1 - k)}{(k - 2)^2(k - 3)}}", "6c6bd0b339f565b3da293a1c13f8da17": "|1\\rangle = \\begin{pmatrix} 0 \\\\ 1\\end{pmatrix}", "6c6c0f057b2b0766dbdefefbccbbc09f": "b\\in\\mathbb{Z}^k", "6c6c50e2c6e14feabb24c24fa3841316": " \\delta W = \\mathbf{T}\\cdot\\vec{\\omega}\\delta t,", "6c6c6d29a5927493a3d2664d1eea5ff0": "L^*[v] = \\sum_{i,\\ j =1}^n \\frac {\\partial ^2 (a_{i,j} v) }{\\partial x_i \\partial x_j} - \\sum_{i=1}^n \\frac {\\partial (b_i v)}{\\partial x_i} + cv. \\, ", "6c6c7d7dd20e58ba084d2b35c5dfd92b": "\\theta_e \\approx \\theta_L\\exp\\left[\\frac{ r_s(T_L) L_v(T_L) }{ c_{pd} T_L }\\right]", "6c6cbe6603cb9d2c7d0f820212bfd679": "t_n = \\frac{(t_1 + u_1 \\sqrt{D})^n + (t_1 - u_1 \\sqrt{D})^n}{2}, \\qquad u_n = \\frac{(t_1 + u_1 \\sqrt{D})^n - (t_1 - u_1 \\sqrt{D})^n}{2 \\sqrt{D}}", "6c6d3e8961877ea1f571c902b0d82225": "v_{\\ell}", "6c6d9340919714f21e395e28d7e53269": "\\mathrm{d} s = \\left(\\frac{\\partial s}{\\partial v}\\right)_T \\mathrm{d} v + \\left(\\frac{\\partial s}{\\partial T}\\right)_v \\mathrm{d} T.", "6c6dae5904821f6718b592b5a1d50b78": "1 = \\sqrt{1} = \\sqrt{(-1)(-1)} = \\sqrt{-1}\\sqrt{-1}=i \\cdot i = -1.", "6c6dd7720ebe9e3c2ac2cfddf009b5bf": "SaS \\cup aS \\cup Sa \\cup \\{a\\}", "6c6defe66e0cbdc2a0444f7e7c5f35b3": " u(x)=u^*(x) \\text{ for } x \\in \\partial\\Omega ", "6c6e8edb236612bace4dd6b3d666f127": "[A,M]=[A,D]=[A,P]=[A,K]=0", "6c6ea02da2bd31199f6ef17c368125ac": " A \\rightarrow \\; B \\rightarrow \\; C ", "6c6efbd2ea6fe06a26191b4ca5aff835": "\\scriptstyle g", "6c6f087b6af6a2b8f59f4ccbaf6cd149": "w_n = F_{f_1} ( F_{f_2} ( \\cdots F_{f_n}(\\varepsilon) \\cdots ) ) \\ .", "6c6f1883d3b5078b3471bac25b00baa9": "x_1^n ", "6c6f7f7facee5e38fb6307d54bb99a9a": "f_\\epsilon\\circ h(x)=h\\circ f(x), \\quad \\forall x\\in \\Omega(f).", "6c7012fe4b95ff6c1039cd0c095cb747": "R_{in} = -30 \\ \\Omega", "6c7025b1a6e9cc9b05fb02748a89117e": "\nD(\\theta'||\\theta) = \\int f(x; \\theta)\\log\\frac{f(x;\\theta)}{f(x; \\theta')} dx\n", "6c703d5921bb43735290ef15d5472105": "X \\backslash \\{x\\}", "6c70ac343ef7b11acb109cc234d53037": "v_{Ar}=v_{Al}", "6c70ea7d694d89ad3251fd4ac3599ea7": "U(x_1,x_2) = x_1^{a}x_2^{1-a}", "6c70f0fb5fa9292dcf5ce9ace3234cd5": "\\Delta Q(V_a,V_b;T^+)\\,=\\,\\,\\,\\,\\,\\,\\,\\,\\int_{V_a}^{V_b} C^{(V)}_T(V,T^+)\\, dV\\ ", "6c71043e2d7aa9e53a6f5007127a45b7": "\\textrm{Gal}(K_{\\infty}/K)", "6c7117e4b6031bb6830b0eba1b47f22c": " c = 1 - 6 {(p-q)^2 \\over pq}", "6c71333defee27d25251546d8e102d59": "r_{i+1}:=\\text{rem}(\\gamma^{d_i +1}r_{i-1},r_{i})/\\beta_i;", "6c7142ab79d8d7a113d173d33b8d7bf3": " \\forall a \\in \\mathrm{A}, \\; \\forall \\sigma\\ _{-n} \\in \\Sigma\\ ^{-n} \\; \\exist \\sigma\\ _n \\in \\Sigma\\ ^n \\; s.t. \\; \\Gamma\\ (\\sigma\\ _{-n},\\sigma\\ _n) = a ", "6c714e1ebfec56ac805424d0f3c4bc30": "\\mathrm{Q} = \\{\\pm 1, \\pm i, \\pm j, \\pm k\\} \\to \\mathrm{GL}_{2}(\\mathbf{C})", "6c7166a77f46408c6c29562a8c6cf088": "Q=40P-2P_{rg}", "6c71749c5a903e8b9caa9290b45c6b04": "O(n^2 \\log n)", "6c71a06c4e6d84822625bd344157b8d1": "\\scriptstyle d_n", "6c71b7f795904df4e067b1ffcc8cfcbb": "\\int_M \\omega^{{dim_{\\Bbb C} M}}>0.", "6c71ce08c91109e755066d9fadb2e886": "|{a_i}|", "6c7273cfe7765964de0c2f18e1032896": "\\Diamond A \\equiv \\lnot\\Box\\lnot A", "6c728854e4dfac1ee2b18b9fa6079c8e": "\n\\pi(x_k|x_{0:k-1},y_{0:k}) = p(x_k|x_{k-1},y_{k}). \\,\n", "6c72a1b85d33fe94e9230489fcee02d0": "\\Gamma^i =0", "6c72b24f95910e313ce2b6ce64eb763b": "(\\dagger)", "6c730ba3235d220b07b29d9a285746b9": "\\scriptstyle{p \\le q}", "6c733ccd0cab3342003fc77a81476444": "- \\pi < \\omega < \\pi", "6c7382e15f5d96f0a0eb28861e892faf": "M>|Q|", "6c73ef0edc943ef14da38ca36503aad4": "(x^\\lambda, \\sigma^m, y^i, \\sigma^m_\\lambda ,y^i_\\lambda). ", "6c7483f212fad8c78603130a4bfb3aa1": "S(q) = \\frac{1}{N} \\left | \\frac{1 - \\mathrm{e}^{-i N q a}}{1 - \\mathrm{e}^{-i q a}} \\right | ^2 = \n\\frac{1}{N} \\left [ \\frac{\\sin(N q a/2)}{\\sin(q a/2)} \\right ] ^2 ", "6c74ba5233291f8e9b56aa28099c5e93": "f[y]\\,", "6c74bb7c7ead57a8cb52a9c373ef67bd": " \\langle A,k \\rangle \\in \\mathit{CM}", "6c75658f2a5e908e65050a44c4546d38": "\\int\\operatorname{arcosh}(a\\,x)^n\\,dx=\n -\\frac{x\\,\\operatorname{arcosh}(a\\,x)^{n+2}}{(n+1)\\,(n+2)}\\,+\\,\n \\frac{\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}\\,\\operatorname{arcosh}(a\\,x)^{n+1}}{a\\,(n+1)}\\,+\\,\n \\frac{1}{(n+1)\\,(n+2)}\\int\\operatorname{arcosh}(a\\,x)^{n+2}\\,dx\\quad(n\\ne-1,-2)", "6c75752d4ee9ec643b0224534e18b16d": "\\left[n_{ij}\\right]", "6c7594fa6ae23c525f4d6ef5160743bf": "\\left\\{4,{4\\atop4}\\right\\}", "6c75dc5cbb39ee50860e9be527db73d3": "f \\left({n \\over {2W}} \\right) = {1 \\over 2\\pi} \\int_{-2\\pi W}^{2\\pi W} F(\\omega) e^{i\\omega {n \\over {2W}}}\\;{\\rm d}\\omega.", "6c761ec583c1a27a0228bbc5a5d8f7a8": "\\, E", "6c76ac8b88aedb298db378e441589816": "V\\subseteq W^s(f,p)\\cup W^u(f,q)", "6c76ae7d54e3a665455c1693a6d4c727": "Y=1-\\exp(-X)", "6c76ea6412041b8a8c9890fed3a8d9df": "\\alpha_\\xi = \\alpha_i(x,\\xi) dx^i|_{(x,\\xi)}\\in T^*_\\xi TM", "6c76f15c104caae409cec934a996b61e": "\\mathcal{O}(m^2)", "6c7705d644f98593e0cc6d5c6d17a358": "\\scriptstyle \\,\\Pi", "6c770eb44b6acd2d08cc1da5cfecf294": "a\\ne0", "6c77189143ceed30ab7005bc40468399": "(C_e)_{m+1} = (I - k_{m+1}a^T_{m+1})(C_e)_m .", "6c7739bea04b8359b153575865a18ed0": " \\psi_L \\ \\stackrel{\\mathrm{def}}{=}\\ \\left ( {\\cos\\theta -i\\sin\\theta \\exp \\left ( i \\delta \\right ) \\over \\sqrt{2} } \\right ) \\exp \\left ( i \\alpha_x \\right ) ", "6c773b2b7798e5713845e475d0c4b4c7": "n_1", "6c774838b0671b3e3a190db7128814ee": " \\mathbf{v} \\cdot \\mathbf{a}", "6c77525b0360f299fcb35be7959a9569": "\\hat{x}=\\sum_i c_{i}p_{i}", "6c77e4d6b2218ba3bfeef43a4ef7f954": "\n\\begin{cases}\nt^3-t=0\\\\\nx=\\frac{t^2+2t-1}{3t^2-1}\\\\\ny=\\frac{t^2-2t-1}{3t^2-1}\\\\\n\\end{cases}\n", "6c77fe317a48bbbc8fe540463190b1ca": "\\mathcal{F}^2(f)(x) = f(-x),", "6c7890aa109c56738ddfbe3c3e8f8b7d": "\\Delta^0_\\alpha", "6c78961d107b12ef5b8e5448f3489d43": "\\nu = \\frac{1}{2 \\pi c} \\sqrt{\\frac{k}{\\mu}}", "6c78a303f4005293aa336d4edef04926": "z=\\frac{t}{\\sqrt{n-1}}", "6c78acf82b8374c50134451c514dd6fb": "\\!\\lambda", "6c78c61373079f676222921d8dccffbe": "\\mathrm{comp}", "6c78c8894088005faa769cda23e65eaa": "\\begin{align}\n\\mathbf{a}\\cdot(\\mathbf{a}+\\mathbf{b}) &= \\|\\mathbf{a}\\|^2 &\\quad\\mathrm{since}\\quad \\mathbf{a}\\cdot\\mathbf{a}+\\mathbf{a}\\cdot\\mathbf{b} &= \\|a\\|^2+0 \\\\\n\\mathbf{a}\\cdot(-\\mathbf{a}+\\mathbf{b}) &= -\\|\\mathbf{a}\\|^2 &\\quad\\mathrm{since}\\quad -\\mathbf{a}\\cdot\\mathbf{a}+\\mathbf{a}\\cdot\\mathbf{b} &= -\\|a\\|^2+0 \\\\\n\\mathbf{b}\\cdot(\\mathbf{a}+\\mathbf{b}) &= \\|\\mathbf{b}\\|^2 &\\quad\\mathrm{since}\\quad \\mathbf{b}\\cdot\\mathbf{a}+\\mathbf{b}\\cdot\\mathbf{b} &= 0+\\|b\\|^2 \\\\\n\\mathbf{b}\\cdot(\\mathbf{a}-\\mathbf{b}) &= -\\|\\mathbf{b}\\|^2 &\\quad\\mathrm{since}\\quad \\mathbf{b}\\cdot\\mathbf{a}-\\mathbf{b}\\cdot\\mathbf{b} &= 0-\\|b\\|^2\n\\end{align}", "6c790d6fc055da125d90f8ab6dbbd644": "\n\\phi(\\omega+\\Omega)+\\phi(\\omega-\\Omega)=2\\phi0+\\phi''(\\omega)\\Omega^2+...+\\frac{2}{(2n)!}\\phi^{2n'}(\\omega)\\Omega^{2n}\n", "6c7930308b9e34357f0e3ee996123218": "\nf = 0.25\\frac{v}{h}\n", "6c799a468b8ce4e37c6f7aabae10af92": " \\left[ \\widehat{S}_a , \\widehat{J}_b \\right] = 0 ", "6c7a16e70800b88407c670f4a56a1b32": "\nT \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{2} \\sum_{k=1}^{N} \\left| \\frac{d\\mathbf{r}_{k}}{dt}\\right|^{2}\n", "6c7a3dca563a7b99399c495e1008d377": "T_\\epsilon f(x)=f(x-\\epsilon)\\approx f(x) - \\epsilon f'(x).", "6c7a5d752afd5ab22357ee2ef1d7adee": "Y_{t}=mI_{t}=\\frac{1}{1-c}I_{t}", "6c7a914584e986ad4b21f5d04658d895": "K=\\left(IA\\cdot IC+IB\\cdot ID\\right)\\sin\\frac{A+C}{2}", "6c7aa0978bb68160349fe26bbc390d2e": "\nd_1 = \\frac{4MG}{3Rc^2}\n", "6c7acc1dbe157343d492c6e8a7131fdc": "\\chi = \\frac{M}{H} =\\frac{M \\mu_0}{B} =\\frac{C}{T} .", "6c7adc79410363f477c512b902fb4195": "\\frac{a_{rel}}{a_0}=\\sqrt{1-\\left(\\frac{Z}{nc}\\right)^2} ", "6c7ae56cf9a6e1b910ecf20803be90f5": "\nu^{-1}_{-1}(\\mathbf{-p}) = u^{+1}_{+1}(\\mathbf{p}),\n", "6c7b4e8b178127515b748bf4b96f0afb": " f_s", "6c7b5415687fdcebef0277e8e6f0412a": "p_i=n\\cdot{I}\\cdot{b(T)}\\frac{N^2\\alpha^2}{V^{n+1}}\\,", "6c7b5ad53194895d0534bd216db048bb": "g^M_p \\colon T_pM\\times T_pM\\longrightarrow \\mathbf R,", "6c7b6a8f64ed3dec2f4b9e65f6bb3123": "\n\\mathbf{G_x} = \\begin{bmatrix} \n-1 & 0 & +1 \\\\\n-1 & 0 & +1 \\\\\n-1 & 0 & +1 \n\\end{bmatrix} * \\mathbf{A}\n\\quad \\mbox{and} \\quad \n\\mathbf{G_y} = \\begin{bmatrix} \n+1 & +1 & +1 \\\\\n0 & 0 & 0 \\\\\n-1 & -1 & -1\n\\end{bmatrix} * \\mathbf{A}\n", "6c7b6e18d90a4c85dbf874173a482763": "\\neg,\\land,\\lor,\\rightarrow", "6c7b8c140adbebed5afa3ba426af81c4": "p_2 > 0, \\ldots , p_n > 0", "6c7bb7d6bc384adb7e8d50a1646a692a": "[G]_{eq}", "6c7bcc373e33ff025ee6aac2baea0eda": " \\dot{x}_k = h_k + g_{kl} \\xi_l - \\frac{1}{2} \\frac{\\partial g_{kl}}{\\partial {x_m}} g_{ml}.", "6c7c5ace67df55e86d3c1fd1a5e12cb2": "\\vec{t_2}", "6c7c5d9fad03f595beef2ce7683420e2": "PM_{NS}=A \\times Size^E \\times \\prod^n_{i=1} EM_i", "6c7c72b33619a3bea8a452e63a9acf3d": "f^2(3 \\times 10^5)", "6c7c845b4d5d2fc53945b0f16a088ae2": "\\ \\tau ", "6c7cce826cc19f58689dd6b2b95c227a": " U^\\alpha {}_\\beta V^\\gamma = (U \\otimes V)^\\alpha {}_\\beta {}^\\gamma ", "6c7d79878147a69b2f031417285e1caf": "\\varphi_i : X_i \\to X\\,", "6c7df187fe9eccacd06c536a8b2dc938": "(x^\\lambda,y^i,y^i_\\Lambda)", "6c7e258d01c9a7d6b8fc2d1a857c5273": "r_\\alpha", "6c7e365c9f4832dc879d82a1530170c6": " {\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1}", "6c7ea2bd48e8fa344ee999171f36b486": "x = L", "6c7f3f01be7c3f097b2f5f4eb6e91d04": " U_{r}= \\frac{\\sigma_{y}^2}{2E}", "6c7f7ba404c7e1bef82d606ee2e3c218": "gx+h", "6c7f9ead077f7a631bdf3de40a9e548a": "0\\le\\omega\\le\\omega_{\\rm D}", "6c7fbebdfe4a82a2bd7b6800d6f54349": "\\int\\arccos(a\\,x)^n\\,dx=\n \\frac{x\\arccos(a\\,x)^{n+2}}{(n+1)\\,(n+2)}\\,-\\,\n \\frac{\\sqrt{1-a^2\\,x^2}\\arccos(a\\,x)^{n+1}}{a\\,(n+1)}\\,-\\,\n \\frac{1}{(n+1)\\,(n+2)}\\int\\arccos(a\\,x)^{n+2}\\,dx\\quad(n\\ne-1,-2)", "6c7fef6f4dce2ea56e11ec78bc6e0599": "\\rho(A,B) = \\mu(A \\triangle B)", "6c7ffc8121ec892f37258400c23485da": " m \\in \\mathbb{R}", "6c8084341debd18893407a06fb6d1a61": "\\lambda 0 \\\\\n0 &:\\ x \\le 0.\n\\end{cases}\n", "6c8660e9bde02ac1bfc6c6fde145891a": " = N \\big( 3/2 log(\\scriptstyle{\\frac 2 3} \\displaystyle E/N)+log(V/N)\\big )", "6c867e17349cf82c47d2e11de0ebfae3": "d=2uv(v^2-u^2), \\,", "6c86880d0e498442b8570acb747d8804": " R = \\frac{ts}{Ot} ", "6c87495036a03fed48dc6ad34e0db154": "=> x = \\frac{1}{\\sqrt{3}}", "6c87561b9c2ec241224aa5c385e016b9": "\\left\\{ \\Lambda_{m}\\right\\} _{m}", "6c876e87cf4c9bfed7ff82200e4fcf8a": "\n K_{\\rm I} = \\sigma\\sqrt{\\pi a}\\left[\\frac{1 + 3\\frac{a}{b}}{2\\sqrt{\\pi\\frac{a}{b}}\\left(1-\\frac{a}{b}\\right)^{3/2}}\\right] \\,.\n", "6c87ab577a2e4bb191270d37f8258309": "0\\leq\\alpha<1, ", "6c87f00d4c85046b56e5ea29afda7c7c": "O(2^kn)", "6c87f5ad2083cddbea242870f7f78fb3": "\\forall x P(x) ", "6c8804cb1f8c45bdca9773bdbda647a8": "\\omega_1,\\ldots,\\omega_n", "6c881cb56d3363e98422e6bfa7c4a5ec": "\\beta = 180^\\circ - \\alpha - \\gamma.", "6c881de6cb7f06295d8a0cfa1b4cb7fe": "\\hat{O}^{-1}", "6c886a1ffab586698d2121c0ef79c374": " \\delta = \\frac{ N^2 }{ 2 } d_{ max } ", "6c889cb9c6756d3bbffe9697fb8510db": "F(\\mathbf u,\\lambda) = 0", "6c88c43eda9233bfd40affbb01d6ac88": "\\Gamma=4\\pi V_\\infty R \\sin \\left(\\ \\alpha + \\sin^{-1} \\left( \\frac{\\mu_y}{R} \\right)\\right). ", "6c88cf1649a6c2f7e9be276aa21fd4cd": "q=kv \\, ", "6c8925c7fe6437b1cc87e63cc3b5ea60": "f(x1, x2, x3) = -x1 * -x2 * -x3 + x1 * x2 + x2 * x3", "6c89552d8c5bedfd0016fc86878bdd62": "t^*_{n-2}", "6c899fa687fe07c4671897c08e21a77d": "\\ell = (\\sigma n)^{-1},", "6c89d97f0118f27d618c07e334d1c08c": "\\mathrm{\\phi}\\,\\!", "6c89e7a5d3350ad0d5694ff959ee2c61": " g(i) \\rightarrow g(i) + f ", "6c89fd7d39d18b6d22631a39a2ed3ba2": "\\scriptstyle -(t_i - t_{\\text{rec}}) \\;=\\; \\tilde{r}_i/c \\,+\\, \\delta t_{\\text{clock},i} \\,-\\, \\delta t_{\\text{clock,rec}}", "6c8b5a17ddc45c8b35c666857fb39f5d": " w_i =\\frac{\\partial R}{\\partial a_i} ", "6c8ba22ebaa1e425175a39b1819892f1": "g(X,Y) = X^i Y_i = g_{ik}X^i Y^k = g^{ik}X_i Y_k.\\ ", "6c8bb5cd6e42b527604c933b5972ec75": "4\\cdot18-2=70", "6c8bce3b58ac3bdd22275decfb311b5c": "\n f(k; r, p) = \\frac{\\Gamma(k+r)}{k!\\cdot\\Gamma(r)}(1-p)^kp^r = \\frac{\\lambda^k}{k!} \\cdot \\frac{\\Gamma(r+k)}{\\Gamma(r)\\;(r+\\lambda)^k} \\cdot \\frac{1}{\\left(1+\\frac{\\lambda}{r}\\right)^{r}}\n ", "6c8c3e48bbcc4c217848350bf65ba0d1": "\\langle A \\rangle = \\frac{\\langle A / w \\rangle_\\pi}{\\langle 1 / w \\rangle_\\pi},", "6c8c47ffec337d5d449b498bd60f8e35": "(\\operatorname{trace}_{V}(T))_{k_1 \\dots k_N }^{\\ell_1 \\dots \\ell_N} (\\operatorname{trace}_{V}(T))_{j_1 \\dots j_N}^{k_1 \\dots k_N} ", "6c8cbdee7e04016271b5f9de94a75ded": "S_0'^2=S_0'\\,", "6c8cce6cedf579200d2cbf89b8326a70": "\\text{AMR}=\\bar{x} = \\frac{1}{n}\\sum_{i=1}^n x_i = \\frac{1}{n} (x_1+\\cdots+x_n) \n \n = \\frac{ 5% + 5% + 30.%}{3} = 13.33%.", "6c8cdb6a38f286edb78f8803e4dbc955": "|{\\Psi}\\rangle=\\frac{1}{\\sqrt{\\sum\\limits_{{\\alpha_l}=1}^{{\\chi}_c}{|\\lambda^{[l]}_{{\\alpha}_l}|}^2}}\\cdot\\sum\\limits_{{{\\alpha}_l}=1}^{{\\chi}_c}\\lambda^{[l]}_{{\\alpha}_l}|{\\Phi^{[1..l]}_{\\alpha_l}}\\rangle|{ \\Phi^{[l+1..N]}_{\\alpha_l}}\\rangle,", "6c8cf78548e5802f9ce052d1bc930ba0": "\\det(E+(s-i)\\delta_{ij})_{KL} = \\sum_{I=(1\\le i_10} A_\\alpha,", "6c8f1ee7da31592176b7de5d43faae4e": "\\ p =a-b(Nq)= \\frac{a + Nc} {N+1},", "6c8f311640ba50a0a422712ad4b5a323": "\\begin{align} F_L &= F_Z\\\\\ne\\cdot v\\cdot B &= m\\cdot\\frac{v^2}{r}\n\\end{align}", "6c8f7eaa019729b7e35ce2648555293c": "\\mathcal{N}(\\rho) = \\lambda \\rho + (1-\\lambda) \\mathsf{id}/2", "6c8fc295bb2cc54f7cc8206ce4787c9a": "y(j)=P \\left\\{ \\begin{matrix} \n0 & 1 & \\infty & \\; \\\\ \n{1/6} & {1/4} & 0 & j \\\\\n{-1/6\\;} & {3/4} & 0 & \\;\n\\end{matrix} \\right\\}\\,", "6c8fe3574822917bc360f93d2966d7b7": "g : TM\\times_M TM\\to \\mathbf{R}", "6c8fe68bccef816b7f740e67e7dfc02f": "\\Rightarrow A=Y_1 - \\left(B+L_1^2 C\\right) L_1", "6c900bc5b45f56ad0c5fa86210ed8ba9": "\\scriptstyle{t_0}", "6c9048775ccdfbd18533276125faa987": "x_{abc} = K^{-1}x_{dqo} = \\sqrt{\\frac{2}{3}}\\begin{bmatrix}\\cos(\\theta)& - \\sin(\\theta)&\\frac{\\sqrt{2}}{2}\\\\\n\\cos(\\theta - \\frac{2\\pi}{3})& - \\sin(\\theta - \\frac{2\\pi}{3})&\\frac{\\sqrt{2}}{2}\\\\\n\\cos(\\theta + \\frac{2\\pi}{3})& - \\sin(\\theta + \\frac{2\\pi}{3})&\\frac{\\sqrt{2}}{2}\\end{bmatrix}\n\\begin{bmatrix}x_d\\\\x_q\\\\x_o\\end{bmatrix}", "6c9111f6fa9fd52dd3324a30728fbdc5": " {}^{239}U \\rightarrow \\; {}^{239}Np \\rightarrow \\; {}^{239}Pu\\! ", "6c9162dd29fcd590207f524663989030": "f_{1...\\lambda}", "6c916e4e4463cc0d28b780701aca44ec": "\\frac {p_0} {q_0} > 0\\,", "6c9170484f8f40c00dba04b3e38dc956": "\\begin{pmatrix} p_m & p_{m-1} & \\cdots & p_1 & p_0 & 0 & \\cdots & 0 \\end{pmatrix}.", "6c918c268db8fd3126baff905db7efdd": "a_i = \\pm 1", "6c91d39e6a2f9b1002405e75a64701d2": "M, v \\models \\Diamond \\varphi", "6c924ecbed17a3dbb64622f4e4f80066": "\\omega_C", "6c927703f437b8d2433f02698ec3e579": "Q(\\alpha,\\alpha^*)=\\frac{1}{\\pi}\\langle \\alpha|\\hat{\\rho}|\\alpha\\rangle =\\frac{1}{\\pi}|\\langle n|\\alpha\\rangle|^2 =\\frac{1}{\\pi n!}|\\langle 0|\\hat{a}^n|\\alpha\\rangle|^2 = \\frac{|\\alpha|^{2n}}{\\pi n!}e^{-|\\alpha|^2}", "6c92b3dbe4510eb82dfb5a84038a84b0": "T3 = \\frac{\\sum_{h=5}^{H}{a_h}}{\\sum_{h=1}^{H}{a_h}}", "6c92ea3e1c76158137e53b6b3ab5c118": "\\mathbb{R}^2 \\cong \\mathbb{C}", "6c93226fbaf82d33db22cf10cc0dd790": "\\scriptstyle \\mathbf{b}_1", "6c935c5e18ef7013b5f7820cac0da872": "\n\\Phi(z,s,a)=-\\frac{\\Gamma(1-s)}{2\\pi i}\\int_0^{(+\\infty)}\n\\frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\\,dt", "6c93b458edf5731ef5bd6f75f5058d28": "\\scriptstyle S_1,S_2", "6c93ea7e88cf7ff1aa1263576ed39461": "\\log_2 9", "6c941acfb4f553a887b6fc0877bfc2a6": "\\operatorname{succ}", "6c94652fa3c6aeee09c7a9e03f0f1bc9": "\\left [\\begin{smallmatrix}\n2&-1&0&0&0&0\\\\\n-1&2&-1&0&0&0\\\\\n0&-1&2&-1&0&-1\\\\\n0&0&-1&2&-1&0\\\\\n0&0&0&-1&2&0\\\\\n0&0&-1&0&0&2\n\\end{smallmatrix}\\right ]", "6c94872d97b103d0adf74a9a3f69d18f": " n - 1 ", "6c949d9c0fe6ca97d4ec59658bbac707": "P=\\frac{R^2}{Q\\beta^3}", "6c94ac790e50c0ff6006926062138c98": "\\sigma_1 - \\sigma_2 > 0", "6c94c25941e3da4e00464c8fe8f74474": " \\boldsymbol{\\nabla}\\times(\\mathbf{c}\\cdot\\boldsymbol{S}) = e_{ijk}~c_m~S_{mj,i}~\\mathbf{e}_k = (e_{ijk}~S_{mj,i}~\\mathbf{e}_k\\otimes\\mathbf{e}_m)\\cdot\\mathbf{c} = (\\boldsymbol{\\nabla}\\times\\boldsymbol{S})\\cdot\\mathbf{c}\n ", "6c94e36c8a6f73713591cb933e6152f1": "\\operatorname{log}(R)=\\log\\left( \\frac{{p_1}/(1-p_1)}{{p_2}/(1-p_2)} \\right) =\\log\\left( \\frac{p_1}{1-p_1} \\right) - \\log\\left(\\frac{p_2}{1-p_2}\\right)=\\operatorname{logit}(p_1)-\\operatorname{logit}(p_2). \\!\\,", "6c9511e1e8461919468e04ed86df9df5": "P_{3}^{-1}(x)=-\\begin{matrix}\\frac{1}{12}\\end{matrix}P_{3}^{1}(x)", "6c9520c1d29d9e21e9a60fcd50015a96": "\\mathcal{T}: \\mathcal{P} \\rightarrow \\mathcal{M}", "6c955a6b6aec0c4410604f5061afcde6": "ST_x(p) \\equiv P(x)", "6c9589d5773f7a7f4aed7644c00b2237": "e^{i\\theta_k}", "6c958ed2058511235e6583fdffc75073": "| \\psi \\rangle \\langle \\psi |", "6c9599f105509d0ba21bc1c2aaee393d": "\\frac {dm} {dt} = {k_r} (C_i-C_e)^{n1}", "6c96316e9ec547af026b9cb4b3a909da": " ( p-A)^2 + (m+S)^2 =0. \\,", "6c96bb53e80ae5384b65d65c62b36a86": "\\inf\\{J(u)|u\\in V\\} > -\\infty.\\,", "6c96f2679c8a3357623ddd322a6c07cf": "P_\\pi\\mathbf{g} = \\mathbf{g}'", "6c96fc5a069d56eb48f47ff5348ec5bb": "w = b^{-1} a", "6c97334452551df9885195f8b8f71117": "\\psi_1(x_1)", "6c9799dce95c82ab2dbc5029eca19578": "\\mathcal{S}:\\mathcal{C}\\rightarrow \\mathbb{R}", "6c97abc21c195ada0e9786d624a8491b": "n_i=c_i^{\\dagger}c_i", "6c9804b19dcfbd4cd5716b7b20e453c6": "\\hat{v_i}'=\\hat{v_i}", "6c9818f551d7b42d2c66e516dbbd8296": "m\\times 1\\!", "6c981deb9dcb02d48bdc368f618464c1": "R_{\\ell ijk}=g_{\\ell s}R^s{}_{ijk}", "6c982fd0bd83cb7f6e01fa150b42fcd0": "(\\sqrt{-g})_{;c} = (\\sqrt{-g})_{,c} - \\sqrt{-g}\\,\\Gamma^{d}{}_{d c}", "6c98316927b2533eee5daf0c4c3fe87b": "\\hat{f}_i^\\dagger", "6c9954d6c26e63f2300f90aa99a2aa19": "P(t) = exp(kt)", "6c996a01f304833bad679dca0ecabc20": "X^*(s)=\\frac{1}{T}\\sum_{m=-\\infty}^\\infty X(s+jm\\omega_s)+\\frac{x(0)}{2}", "6c99e3e04f601389849133130d698bc6": "= \\left( \\frac{1 - e^{-i 2\\pi fT}}{i 2 \\pi fT} \\right)^2 \\ ", "6c9a19fc24cfb96fdfd9099433b98b45": "\\tilde{\\mathbf{E}}^+ = \\varprojlim_{x\\mapsto x^p} \\mathcal{O}_{\\mathbf{C}_p}/(p)", "6c9a572ea809b703a030e569936f5f82": " \\mathbf{L} = -i\\hbar\\, (\\mathbf{r} \\times \\mathbf{\\nabla}) .\n", "6c9abec4d5c60ca3e26e2358f02dfb52": "75+100=175", "6c9acfe37ad81460e755a44db63353b0": "N(t) = N_0 e^{kt/T} \\,", "6c9ad56d822c3a7ac8565584dae11cb5": "P_1, P_2", "6c9afc393081e6abf33932add5cab38a": "GL(m)", "6c9b0abde68445773bafc0900453f7e4": " \\Im=\\{Q\\ll P:D_{KL}(Q||P)\\leq-\\ln\\alpha\\} ", "6c9b52fbb0d1ea41a8df30d7884e9cb4": "\\tfrac{1}{2}(2 - \\tfrac{6}{4}) = \\tfrac{1}{4}", "6c9ba029ea39c01e0c27bc43b7fce21f": "\n\\int_0^\\infty \\cos(2x)\\prod_{n=1}^\\infty \\cos\\left(\\frac{x}{n}\\right)dx \\approx \\frac{\\pi}{8}\n", "6c9bdc735349dc350b8b21e30c15e0f8": " |\\alpha| \\le q^{d/2+1/2}", "6c9bddb7373bc4806d89a6a18e3031ed": "\\left(-4\\sqrt{\\frac{2}{5}},\\ 0,\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "6c9c245a450a00e5a5ba97b70052a71f": "\\scriptstyle f(t).", "6c9ca7d6d46bb8836b78f986da4f6651": "\\delta : X \\rightarrow P(X) \\,", "6c9cac5b1af44b0f3cfe3a2540fce40a": "\n d\\mathbf{A} = dA~\\mathbf{N} ~;~~ d\\mathbf{a} = da~\\mathbf{n}\n\\,\\!", "6c9ce4b466952143d74f03dc391816a9": "\nI(\\omega,\\tau)=\\left|\\int{E(t)g(t-\\tau)e^{i\\omega t}\\mathrm{d}t}\\right|^2\n", "6c9d36f500a0f5af4e1f82f9415e7b79": " \\overrightarrow{Dj} ", "6c9d3d6bf6b5bdca831b17b032a8a0a7": "d_i = d + \\epsilon_i, \\; \\epsilon_i \\sim N(0, \\sigma_d^2).", "6c9d61d285cd0f0ce59f8b06c86cb445": "\\theta = \\arctan\\left(\\frac{y}{x}\\right) - \\omega t.", "6c9d79e9fc1cf4ffa2d80f1720eb8e33": "\\mathrm{E}_1(z) = \\int_1^\\infty \\frac{e^{-tz}}{t}\\, dt = \\int_0^1 \\frac{e^{-z/u}}{u}\\, du ,\\qquad \\Re(z) \\ge 0.", "6c9da6d437348164a313fa0cdcf7a128": "\\frac{\\Gamma \\vdash \\alpha \\rightarrow \\beta \\qquad \\Gamma \\vdash \\alpha}{\\Gamma \\vdash \\beta} \\rightarrow E", "6c9dbf302e594163f9bfbaa915d6915d": " T_v(p) = \\begin{bmatrix} p_x + v_x \\\\ p_y + v_y \\end{bmatrix}. ", "6c9de7079dbd7abe7bf5c24ece3acfb3": "+6\\frac{3}{5}", "6c9f010fc9e70f1b46dd23cb9e4d880d": "m(\\varphi)=a(1-e^2)\\Pi(e^2 ; \\varphi \\,|\\,e^2).", "6c9f2ae6d6678e5824849eb92b3c5d47": "V_\\text{TWN}", "6c9f2bbbbc74fc09d0cff7342deeba7e": "E(b,N) = b \\lfloor \\log_b (N) +1 \\rfloor \\, . ", "6c9f32358bbcbcb658e93b8364969d88": "\\varphi(k) = \\frac{\\sin(k/2)}{k/2},\\,", "6c9f330f6001916c88436764e1b574b5": "b \\in L^1(\\Omega)", "6c9f728510296acefe79c4d00af83974": " \\int_0^2 \\! \\int_{0}^\\sqrt{2^2-x^2} \\! \\int_0^2 \\! f(x,y,z) \\, dz \\, dy \\, dx = 16 + 10 \\pi", "6c9f834dc0e00754af46b1c1cedfee00": "q = 53 * 8", "6c9fc8ddb4476ca164a9775c0193b753": "k'=k_e", "6ca07ae71b804c9a6c1707cb7e56cd66": "c_{12}-d", "6ca0882ecad3fc8cbc898d8b7e0cee66": "T_{N}", "6ca0a1ab6648ca7f19afeae604ed033f": "\\mathcal{F}_X", "6ca0e3ee3fd50f4b355a3a5f28881243": " \\Delta\\tau = \\cfrac{Gb}{l_{\\rm{interparticle}}-2r_{\\rm{particle}}} ", "6ca0fe5845289a55a0ee8e0c083a6540": " r_{iA} \\equiv |\\mathbf{r}_i - \\mathbf{R}_A|", "6ca1656ecc58f2ed3ba82a0c001e3594": "I = \\int_{\\Omega}f(\\overline{\\mathbf{x}}) \\, d\\overline{\\mathbf{x}}", "6ca185ea56b3cd7b93cd4dc5065180bc": "\\beta = \\frac {1 + R_\\text{f} C_\\text{f}s} {1 + R_\\text{f} {(C_\\text{i} + C_\\text{f})} s}", "6ca1aed4f123e60be6ef74d0d753e6aa": "A_1=\\int_{y_1}^{y_2}x(y)dy", "6ca1beb23f774272e1c979d3a667784d": "\\Gamma_{rad}", "6ca1dd5da23a89a67d6acf34ae07de61": "\\,N", "6ca218db42ceb524969bc935797d5b4d": "\n\\bar{\\Gamma}_{ij}(s)=\\prod_{c=0}^{i\\mp1}\\bar{\\psi}_{c\\pm 1c}(s),\n", "6ca29a77343d1e708a8546098c6b66f4": "(D_j, Y_{D_j})", "6ca29a7de885766808df5645685c67fd": "(+++)\\,", "6ca31ff3a41391cb66f6fd6e15f31bc6": "x = {a \\over 2}\\sqrt{2}- {a \\over 2}", "6ca331725afd265ae74a3a55e90f1b25": " D_{\\lambda}(P\\|Q) = \\lambda D_{\\mathrm{KL}}(P\\|\\lambda P + (1-\\lambda)Q) + (1-\\lambda) D_{\\mathrm{KL}}(Q\\|\\lambda P + (1-\\lambda)Q),\\, \\!", "6ca335815ac86a0042ab4ba349f6b0ac": "dU = \\delta Q - \\delta W.\\;", "6ca3996f6b453581d6ce6749298d52c9": "\\dagger\\colon \\mathbb{C}^{op}\\rightarrow\\mathbb{C}", "6ca3a5776f4b9dcec145ca150d0194be": "D_{96} \\approx \\tfrac{1}{4} D_{48}", "6ca3e26d6e34724b8273060a67d69484": "x/y \\cdot y = x = y/y \\cdot x", "6ca464b6adcf6e19cf2f37bc43810bab": "f(x)=\\sum_i |p_ix|^2", "6ca4953910856ab13a7acd0140df9ba3": "\\pi(\\cos \\phi_0)^2\\,", "6ca4a5a9031dce03e197985c703f7828": "\\rho (t),", "6ca4a5e4a9fa86cf58fcd5a62c7e3172": " \\mathbb C^3\\otimes\\mathbb C^3 ", "6ca4bc0f1d3758ef906d063ed6086bc1": " \\max\\{\\,x_1,\\ldots,x_n,x_\\mathrm{m}\\},\\, k+n\\!", "6ca56888a34c4b37e01fd47f5eebc066": "\\scriptstyle \\mathrm{Ker} F_j", "6ca58b2efff7dcc9429dfec78ff42c9e": "4m_N^2", "6ca5986ea8cee966b1bf4873edf1be36": " -C = - A^0 B^0 + A^1 B^1 + A^2 B^2 + A^3 B^3 = - {A'}^0 {B'}^0 + {A'}^1 {B'}^1 + {A'}^2 {B'}^2 + {A'}^3{B'}^3", "6ca5cb839e5a57734924ad5040196b77": "z = \\frac {x-x_c}{w} - \\frac {w}{t_0}", "6ca5f6366c3ebc570bf0941d9e057857": "\\lim_{x \\to -\\infty}{x^{-1}} = -0", "6ca6021f270534d6fa248be9c6a26684": "1-4p", "6ca6317dd2a458af42244417c133698f": "30^\\circ", "6ca66c4f9e7c752eb32a9cd86c4c793e": "Y = \\{ x \\in X | x\\,R\\,x\\} \\subseteq X", "6ca685f0f43cc8f28e39564acfd2ad86": "\\ BL_s = EBV \\ln \\frac{H_i}{H_m} ", "6ca6937a57014978508545810b5fab55": "\\mathbb E(W) = \\frac{\\lambda}{\\alpha \\mu + \\beta(\\lambda-\\mu)}.", "6ca6c1280f6eb33fa6aff45e199ea123": "f(x) = \\sum_i \\left( \\frac{a_{i1}}{x - x_i} + \\frac{a_{i2}}{( x - x_i)^2} + \\cdots + \\frac{a_{i k_i}}{(x - x_i)^{k_i}} \\right). ", "6ca6c17d97972a255266caf2eb0dabf7": "p=p_c\\,\\!", "6ca6ebaf4b734bddfe43305b2ca8d816": "\nE_\\text{out} = K_1 E_\\text{in} + K_2 E_\\text{in}^2 + K_3 E_\\text{in}^3 + K_4 E_\\text{in}^4 + K_5 E_\\text{in}^5 + ... \n", "6ca7000010152beb3b0af634511b7d1e": " \\mathbf{A} = \\left( \\phi /c, \\mathbf{a} \\right) ", "6ca70b42c6afaa9f765ebf7f90ef182e": "{C}_{9}^{(1)}", "6ca741d86a4a9b3d88c5e67f68e063a5": "\\begin{align} A_x &= E[Z] = E[v^T] \\\\\n &= \\sum_{t=1}^\\infty v^{t} Pr[T = t] = \\sum_{t=0}^\\infty v^{t+1} Pr[T(G, x) = t+1] \\\\\n &= \\sum_{t=0}^\\infty v^{t+1} Pr[t < G - x \\leq t+1 \\mid G > x] \\\\\n &= \\sum_{t=0}^\\infty v^{t+1} \\left(\\frac{Pr[G>x+t]}{Pr[G>x]}\\right)\\left(\\frac{Pr[x+tx+t]}\\right) \\\\ \n &= \\sum_{t=0}^\\infty v^{t+1} {}_t p_x \\cdot q_{x+t} \\end{align} \\ ", "6ca79363f79991245b74e36aceaeb079": "\\Psi(v)", "6ca8a0216a550c8d563b9bdb261cae42": "f(\\dots)\\,", "6ca8a124a0fe96f23426aacc6c5841be": "{ x \\in \\left[ x_{i-\\frac{1}{2}} , x_{i+\\frac{1}{2}} \\right] }\\ ", "6ca8c824c79dbb80005f071431350618": "\\frac{2}{3}", "6ca8c8b0dd1fa0760863527102db5a8d": "\\mathbf{x}^{\\prime}", "6ca933360bb6bd352b5648b7ac13f386": "l=1,2", "6ca94cd68edb93f66ba0789b46c6240b": "{{O}}(M^2\\cdot\\chi^2)", "6ca9553de9960d820b78804ccfc16c00": "\\mu_0 ", "6ca9c2a995628de2b8876ff119c65024": "\np_k=\\frac{\\partial L}{\\partial \\dot x_k}\n", "6caa019f45507d45cb46768487de3f6e": "\\scriptstyle\\gamma=1/3", "6caa3d93965ae68cf1457082d9ae8f0b": " v(t) = \\frac{\\mathrm{d} x}{\\mathrm{d} t} = - A\\omega \\sin(\\omega t+\\varphi),", "6caa5ab14d58842f6f79c32421150d07": " H = 0.\\ ", "6caa992d99621dba0e10b8ef44862e3a": "(\\kappa a)^2", "6caa9e43374b216f2f6ed8166a4c8909": "\\begin{align}\n D((\\partial_i)_p \\parallel q) \\ \\ &\\stackrel\\mathrm{def}=\\ \\ \\tfrac{\\partial}{\\partial\\theta^i_p} D(p \\parallel q), \\\\\n D((\\partial_i\\partial_j)_p \\parallel (\\partial_k)_q) \\ \\ &\\stackrel\\mathrm{def}=\\ \\ \\tfrac{\\partial}{\\partial\\theta^i_p} \\tfrac{\\partial}{\\partial\\theta^j_p}\\tfrac{\\partial}{\\partial\\theta^k_q}D(p \\parallel q), \\ \\ \\mathrm{etc.}\n \\end{align}", "6caadc12f6534a8aca4b8f392fce9502": "|\\mathbf{E}| \\cos\\theta", "6caaf0b588dbfa1ba2a408ca5382ad33": " \\frac { \\partial (C_{M'}) }{ \\partial (C_L)} = 0 ", "6cab11e1454298d832d1ee928208d731": "\\scriptstyle{1/\\sqrt{\\mu_0\\varepsilon_0}}", "6cab6dba8661eea55b8163da35613bd0": "{n \\choose k} = {n-1 \\choose k-1} + {n-1 \\choose k}", "6cab9f65132495f90af3260ed4ff55e9": "w(x) = 1", "6cabe4d66428d46fc6ae2f0c8e7d32e4": "\\Delta{H_\\mathrm{neutralization}} = C_\\mathrm{cal} \\cdot \\Delta{T}", "6cac2f1d0ad6a27ffc40eb9ce3a03fab": " \\left\\lfloor { {n} \\,\\varphi} \\right\\rfloor - \\left\\lfloor {\\left( {n - 1} \\right)\\,\\varphi } \\right\\rfloor -1 ", "6cacc3857585ba5cc5f2bb11a9b36532": "\n\\partial_\\mu J^\\mu |0\\rangle = k^\\mu k_\\mu |\\pi\\rangle = m_\\pi^2|\\pi\\rangle = 0\n\\,.", "6cacc836d52f6f4a625defcdb8d140bd": "Z(E(K), T) = \\frac{1 - aT + qT^2}{(1 - qT)(1 - T)}", "6cad0ac428bb8461634807ec89ddc4e1": "1-(1 - p)^k\\!", "6cad2567f999ea61c055ff1a86026ed7": "\n\\begin{align}\nP_s(T)& = \\frac{100}{R\\!H}P_\\text{a}(T) = a\\exp\\left(\\frac{bT}{c+T}\\right);\\\\[8pt]\nP_\\text{a}(T) & = \\frac{R\\!H}{100}P_s(T)=a\\exp(\\gamma(T,R\\!H)),\\\\\n&\\approx P_s(T_\\text{w}) - B\\!P_\\text{mb} 0.00066 \\left[1 + (0.00115T_\\text{w} \\right)]\\left(T-T_\\text{w}\\right);\\\\[5pt]\nT_\\text{dp} & = \\frac{c\\ln(P_\\text{a}(T)/a)}{b-\\ln(P_\\text{a}(T)/a)};\\end{align}", "6cadb2f91ba83253bac6b4ed8959b3af": " \\left(\\frac{L}{1}\\right)", "6cadfe7f0106eca9441a0a9050acc376": "M_{11}", "6cae0a4ec23d341fff26be1be8a66fb0": " 2[\\gamma(\\epsilon_{ij})A(\\epsilon_{ij})-\\gamma_0 A_0]=2\\int(Af_{ij} d\\epsilon_{ij}) ", "6cae99eaa80b469ff7d42a4f53584516": "\\hat{a}_i \\,\\hat{a}_j^\\dagger \\, \\hat{a}_k= (\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij})\\hat{a}_k = \\hat{a}_j^\\dagger \\,\\hat{a}_i\\, \\hat{a}_k + \\delta_{ij}\\hat{a}_k = \\hat{a}_j^\\dagger \\,\\hat{a}_i\\,\\hat{a}_k + \\hat{a}_i^\\bullet \\,\\hat{a}_j^{\\dagger\\bullet} \\hat{a}_k =\\,\\mathopen{:}\\,\\hat{a}_i\\, \\hat{a}_j^\\dagger \\hat{a}_k \\,\\mathclose{:} + \\mathopen{:}\\,\\hat{a}_i^\\bullet \\,\\hat{a}_j^{\\dagger\\bullet} \\,\\hat{a}_k \\mathclose{:} ", "6caeccadee9247bf304083eeb4203e98": " Y", "6caf4e25c037d5cf29ed95c48c26e330": "X=x_i \\rightarrow Y=y_j ", "6caf7da982c1c5a206d7ff16085fee2b": "\nn \\frac{\\partial \\langle{v_j}\\rangle}{\\partial t}\n+ \\sum_i n \\langle{v_i}\\rangle \\frac{\\partial{\\langle{v_j}\\rangle}}{\\partial x_i}\n= -n \\frac{\\partial \\Phi}{\\partial x_j} - \\sum_i \\frac{\\partial (n \\sigma_{ij}^2)}{\\partial x_i} \\qquad (j=1, 2, 3.)\n", "6cafa0412d81730147bfe56b9971c76d": "n(r)=n_2 \\mathrm{\\ for\\ } r \\ge \\alpha", "6cafe36e464a094a2273b9a9ac27e77c": "\\frac{P_t}{P_{t-1}} = \\prod_{i=1}^{n}\\left(\\frac{p_{it}}{p_{i,t-1}}\\right)^{\\frac{1}{2} \\left[\\frac{p_{i,t-1}q_{i,t-1}}{\\sum_{j=1}^{n}\\left(p_{j,t-1}q_{j,t-1}\\right)}+ \\frac{p_{i,t}q_{i,t}}{\\sum_{j=1}^{n}\\left(p_{j,t}q_{j,t}\\right)}\\right]}", "6cb0d41d0b2c950a49d12ba7b6dec81f": "t \\in [0, T].", "6cb10db011e41065f4fd2bf967cc881f": "y \\in \\operatorname{recc}(A) \\backslash \\{0\\}", "6cb115f0aa99849c294fb4feb03f42a0": "M\\ge 2", "6cb12960b0e3262031cb61aa8d7130ea": "X(e^{i \\omega}) \\cdot Y(e^{i \\omega}) \\!", "6cb131ed6bf4638eea26df8f7e97c1b2": "{}-h(v_1,v_4)k(v_2,v_3)-h(v_2,v_3)k(v_1,v_4)\\,", "6cb157c41932eb7d50e5ecc77b5a0afc": "\n\\Psi_{1\\ldots N}(j^N\\alpha JM)= \\sum_{\\alpha_1 J_1}\n(j^{N-1}\\alpha_1 J_1;j|\\}j^N \\alpha J)\n\\left[\\Psi_{1\\ldots N-1}(j^{N-1}\\alpha_1 J_1)\\otimes\n\\psi_N(j)\\right]^J_M\n ", "6cb1985bdf4dc1abfb62a4d466fe7459": "1 < n(\\lambda_{\\rm red}) < n(\\lambda_{\\rm yellow}) < n(\\lambda_{\\rm blue})\\ ,", "6cb1c4e33f721d3ef625a2679480e772": "y = \\frac{\\sin(\\phi)}{\\mathrm{sinc}(\\alpha)}\\,", "6cb1c69561e16ba9c001601ece0f2eab": " \\int_0^1 e(t) \\, dt \\ \\stackrel{d}{=} \\ \n\\int_0^1 W_0 (t) \\, dt - \\inf_{0 \\le t \\le 1} W_0 (t) .\n", "6cb1dc2dceb94e23e52b476e7805c567": " x : \\mathcal{A} \\to (0,1) ", "6cb23fb49ac165a03a563dcb47540587": "\\mathbf{w}={\\arg \\max}_\\mathbf{w} \\frac{||\\mathbf{wX}_1||^2}{||\\mathbf{wX}_2||^2}", "6cb3009bb1cf140162f898bada390ecf": "\\beta_0 + \\beta_1 z + \\beta_2 z^2 + \\dots = \\sum_{n \\geqslant 0} \\beta_n z^n", "6cb3017f3cdbd14f4aaa42a53423012e": " \\beta_i(X) ", "6cb371c05abd4ec5adec0ba4530fdd68": "\\scriptstyle H_0 \\;\\approx\\; 4.41 \\,\\times\\, 10^{13}", "6cb3915bf880b4eab2388e2f29ac7cd8": " \\frac{d^{1/2}}{dx^{1/2}} V^{-1}(x)=2 \\sqrt \\pi \\frac{dN(x)}{dx} ", "6cb39328354397c8a463a9005c1b7032": "f(x) = 1", "6cb3bb7963fbafe774eef70fc71a39f3": "[L : K] = [l : k]", "6cb3c546c0855d0afdca207640a49c75": "\\displaystyle\\Delta=-4y^{2}(\\partial_x^2 +\\partial_y^2).", "6cb3e78d7801ed1d99fccbc72a143f6f": " \\mathbb{E} \\left [ (H\\cdot B_t)^2\\right ] = \\mathbb{E} \\left [\\int_0^tH_s^2\\,ds\\right ].", "6cb458c4f5e963b5efa1d3d9935f646f": "-1\\leq r < 0", "6cb45dd30fd728ff9d784676ce0ebaf1": "\\Epsilon_v =0", "6cb46c838f2c9ac74b88fba5290d4412": "\n\\begin{align}\n(\\Delta e_g,\\Delta e_h)\\ =\\ & -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ (-e_h ,e_g)\\ + \\ 2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos^2 i (-e_h ,e_g )\\ = \\\\\n& -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ 3 \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)\\ (-e_h ,e_g )\n\\end{align}\n", "6cb46fd0d4cc537b0c56e120c7e0ba46": "\\log^c p", "6cb5574d030170d5c73d8178e607b889": "B\\mathbf{v} = \\lambda \\mathbf{v}", "6cb57db3db0a1d6b0f769ccfbaea3a16": " T= 2 \\pi{\\sqrt{\\frac{m}{k}}}", "6cb58c8d5736917cc71192391d17b806": "kT/E_{F}", "6cb5ad464deb7f6445892eb2bdae6e99": "\\alpha=2.35", "6cb5afd940fd5f540f531948f101375a": " \\ m_1(v_1-u_1)=m_2(u_2-v_2)", "6cb6490b88cfb4f3587ea27396ce658e": "Z=f(X_0,\\ldots,X_{n-1})", "6cb66c2b44f9bd264b735f22c43aaadb": "\n M_{xz} = -\\left[\\int_z\\int_{-h/2}^{h/2} y\\,(-y\\sigma_0)\\,dy\\,dz\\right] = \\sigma_0\\,I\n ", "6cb677d7f93fc0b0dbc4b65d17f29b98": "-T \\Delta S_{mix}\\,", "6cb6ce23a38b9d103e202d61b863d478": "\n\\begin{array}{l}\n \\left\\{ {{\\begin{array}{*{20}c}\n {\\forall i < n \\Rightarrow \\ln x_i + \\frac{H_i ^\\circ }{RT} -\n\\frac{H_i^\\circ }{RT_i^\\circ } = 0} \\\\\n {\\ln \\left( {1 - \\sum\\limits_{i = 1}^{n - 1} {x_i } } \\right) +\n \\frac{H_n\n^\\circ }{RT} - \\frac{H_n^\\circ }{RT_n^\\circ } = 0} \\\\\n\\end{array} }} \\right. \\\\\n \\\\\n \\end{array}\n", "6cb7305fdd47d99253e8171a8082a45b": "{ \\partial^2 u \\over \\partial t^2 } = c^2 \\nabla^2 u ", "6cb7461aabf989e01e19630dfcd7ec16": "\n1/\\cos(\\alpha(z)) \\approx 1 + {\\alpha(z)^2 \\over 2}\n", "6cb7632c2834075eccdeba14dd4f53ce": "c_0,\\ldots,c_K", "6cb76631b1a1f5a89a9f8183afb45d43": "\\!v_2", "6cb7c8f5357972c0e21c12dc5540d498": "u=-\\sqrt[3]{q} \\text{ and } v = 0", "6cb88312fda866c6c4f90664df4d6cf8": "\\;\\lambda=0", "6cb8a05439ebdcde685b69b3755f5679": "40 x 4 x 26 cm^{3}", "6cb8fa7707dd780ce99ab0864e6659bf": "h_f", "6cb906ced26525601b140dabd5d4ced1": "a=2\\cos(\\phi)", "6cb91f4160279c33a237583160148ecc": "E_{\\overline{K}}(\\overline{P})=\\overline{C},", "6cb99cc09d663ac9eddd6d4fd025c39e": "\\textbf{end for}", "6cb9d0dea2ee9ba8e21db93160b11401": "R = f(N_1)-N_1 \\cdot f'(N_1)\\,", "6cb9e05f293fdb3a4311e6576447c6e0": "\nX^n Y = C\n", "6cba02ea4430f8b787570b2bc02684b9": "P_{\\text{Magnetic dipole}}=\\frac{Z_0}{12 \\pi}k^4\\|\\mathbf{m}\\|_2^2", "6cba28470ad70473f58638a45c4212bc": " \\left( \\frac{\\mathrm{d}S_{\\nu}}{\\mathrm{d}\\nu} \\right)^{2} + 2m a^{2} U_{\\nu}(\\nu) + 2ma^{2} \\left(\\Gamma_{z} - E \\right) \\sin^{2} \\nu = \\Gamma_{\\nu} ", "6cba2a388598339d0244fda68dd67969": "S_l \\simeq \\delta / \\tau_b \\simeq \\sqrt {\\kappa / \\tau_b} ", "6cba3051d5ba722b088fcf1cc6f44def": "\\operatorname{cl}(A) = \\Bigl\\{x\\in E\\mid r(A)=r\\bigl(A\\cup\\{x\\}\\bigr)\\Bigr\\}", "6cba6e250cf2c644bf43ffc79bb7de5a": "A(\\mathbf{u}-\\mathbf{v}) = A\\mathbf{u} - A\\mathbf{v} = \\mathbf{b} - \\mathbf{b} = \\mathbf{0}\\,", "6cbb0464ad97f9b9fdaddc1e7b641b03": "n^{O(\\frac{\\log \\log k}{k \\log k})}", "6cbb4c92fba51093f2c754de78ff52fc": "k_1=\\sqrt{2m (E-V_0)/\\hbar^{2}}\\quad 00 ", "6cca1af54a8771a1cdc2fd3919e4706e": " \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty G_x(t,f)G_y^*(t,f)\\,df\\,dt = \\int_{-\\infty}^\\infty x(\\tau)y^*(\\tau)\\, d\\tau ", "6cca38a9b358c0a9e0422a08639b24d7": "e^{(1-r/c)}", "6cca566f6cf98a5511da1be68fb0efbb": " \\partial_k g_{ij} = \\left \\langle \\nabla_{\\partial_k}\\partial_i, \\partial_j \\rangle + \\langle \\partial_i, \\nabla_{\\partial_k} \\partial_j \\right \\rangle.", "6cca69d7af241ce1fb46986dedd475ee": "(p\\dot{x}-\\mathcal{L})", "6cca93353085fcb16e8609f918b3e06d": "\\part^\\alpha x^\\beta", "6cca9d3acbfc60b8616dfec8b496cba9": "u_t + F_x\\left(u \\right) = Q_x \\left( u , u_x \\right), ", "6ccad365211f525caa50e83fa1b4a426": "F^-=B_\\nu J(J+1) -2B_\\nu\\zeta_r (J+1)", "6ccaed15ffdec0e788e2cd52104471a7": " 3 > \\beta \\ge 2", "6ccaf3790ceca055317952e1775449b4": "g(x) = \\int_0^x f(x')\\, dx'.", "6ccb2c4414e9d2f049938d97ee02b912": "\\preccurlyeq, \\curlyeqprec \\,", "6ccbbb218aa1a404d0845501a8128c90": "D_1,D_2\\in\\mathcal{D}^n\\,\\!", "6ccbc317b874088ba68ab02ef66c18cf": " a L_x + b L_y + c L_z + d L_w + \\text{plane}_D = \\text{indicator} ", "6ccc071329ef3f32e7599ee67a045c88": "\n\\begin{align}\nP(r)\n&= \\sum_{i=1}^{n}\nP\\left(\\text{applicant } i \\text{ is selected} | \\text{applicant } i \\text{ is the best}\\right) \\times\nP\\left(\\text{applicant } i \\text{ is the best}\\right)\n\\\\\n&= \\left[ \\sum_{i=1}^{r-1} 0 \\times \\frac{1}{n} \\right]\n+ \\left[ \\sum_{i=r}^{n} P\\left( \\left.\n\\begin{array}{l}\n\\text{the second best of the first } i \\text{ applicants} \\\\\n\\text{is in the first } r-1 \\text{ applicants}\n\\end{array} \\right| \\text{applicant } i \\text{ is the best}\n\\right) \\times \\frac{1}{n} \\right]\n\\\\\n&= \\sum_{i=r}^{n} \\frac{r-1}{i-1} \\times \\frac{1}{n}\n\\quad=\\quad \\frac{r-1}{n} \\sum_{i=r}^{n} \\frac{1}{i-1}.\n\\end{align}\n", "6ccc39c816fa71ff2039f05d996c6f79": "n\\to\\infty.", "6ccc3d02e0384739508764d57040650c": "\\text{Coefficient of Restitution } (C_R) = \\frac{\\text{Relative Speed After Collision}}{\\text{Relative Speed Before Collision}}", "6cccaefe630ca2fc0570d7eb22e1ec1b": "\\left (2^x=e^1\\Rightarrow x=\\tfrac{1}{\\ln 2}\\right )", "6cccbcd9cd1904cccab6124274d19590": "E(X_1, X_2, ... , X_N) = \\sum_{i C", "6ce0eec7a97ee6f0d025ba56d8426db8": "\\mathbf{r} (t) = (x(t), y(t))", "6ce172c8c7f2e6519168260b68e9e450": "u_t = Lu", "6ce198c0f774c8e4aca45b9937ece455": "(g,e')N", "6ce1c2531d08671881ab023cca81dc21": "K_{\\mathrm a} = \\mathrm{\\frac{[A^-] [H^+]}{[HA]}}", "6ce2149cf1278bb2dec69eef0dbd1b02": "-\\pi<\\lambda<+\\pi\\,\\!", "6ce291f843cf856ceee00569208295c5": "N_0||X- \\widehat{X}||^2/2", "6ce2c2764e26bd67cf8910689e24722d": "A_0[x_0, \\dots, x_{n-1}]", "6ce2efe6ebfb747c54214f866ce54b21": "A \\begin{bmatrix} 1\\\\0\\\\0 \\end{bmatrix} = \\begin{bmatrix} 1\\\\0\\\\0 \\end{bmatrix} = 1 \\cdot \\begin{bmatrix} 1\\\\0\\\\0 \\end{bmatrix},\\quad\\quad", "6ce2fefe0a018265ab06b130edd5afad": "k = (1-H_q(\\delta)-\\varepsilon)n", "6ce2ff802d04a2ce8b387d8669d8b9a9": "y_0 = \\sqrt {{r_m}^2 - d^2} \\quad (12')", "6ce355df4df9aacf338f76524f82ec00": "\\beta_d = \\sqrt{\\sum_{i \\in \\Theta_d^*} \\frac{\\sigma^2_{dl_d^i}}{\\sigma^2_{il_d^i}}\\beta_i^2}", "6ce37ac3b3150e2309e8993584fcee18": "\\int_0^x{dt\\over \\sqrt{1-t^4}} + \\int_0^y{dt\\over \\sqrt{1-t^4}} = \\int_0^{F(x,y)}{dt\\over \\sqrt{1-t^4}}.", "6ce39943592b72da2120aedbeeded124": "\n J_1 := \\int_{-h}^h \\rho~dx_3 = 2~\\rho~h ~;~~ \n J_2 := \\int_{-h}^h x_3~\\rho~dx_3 = 0 ~;~~ J_3 := \\int_{-h}^h x_3^2~\\rho~dx_3 = \\frac{2}{3}~\\rho~h^3\n", "6ce47115cf26ee03bc464283a5fa37a1": "\\displaystyle{\\|g\\|_p^p \\le (2\\alpha)^{p-1}\\|f\\|_1.}", "6ce4b71af7dfde7d75c807ae9c8ee5d0": "\\Vert\\cdot\\Vert", "6ce4e3adcc56dd62e8fd483e34297f43": "MPC=\\frac{dC}{dY}", "6ce4ec861b18ec19e53c5cf37d650ff2": "X \\times |N(I / i)|", "6ce53f54a4abf821bbbcbde049b26111": "c_i(q_i)", "6ce5a60ac4198d0a4f952c830f2f3734": "\\frac{Sieve_{Largest}}{Aggregate_{max-size}}", "6ce5f9052437664842c4b7bc0cd8c08a": "5\\times(5\\times3)=(5\\times5)\\times3=75 \\,", "6ce655e3033cc394676aef313009c852": "\\mathrm{vec}(A) = \\begin{bmatrix} a \\\\ c \\\\ b \\\\ d \\end{bmatrix}", "6ce6f674028cf40c9a35f7be48409ea6": "\\gcd(m,n) = 1", "6ce70cf6d9b749efd00a6f3fa5e1332a": "F=m\\frac{d^2x}{dt^2}", "6ce7608836ab3c7f52db48bbf260b5ba": "\\mathbb{Q}[j(\\tau), \\tau]/\\mathbb{Q}(\\tau)", "6ce76e088ddf9070a5c5aec40810b6be": " p_1'", "6ce79b8b4d294a225a6e50c82af8d1b0": "\n\\begin{align}\n\\left[\\frac{\\alpha}{\\mathfrak{p} }\\right]_2 & \\equiv \\alpha^\\frac{\\mathrm{N} \\mathfrak{p} - 1}{2}\\pmod{\\mathfrak{p} } \\\\&=\n\n\\begin{cases}\n+1 \\text{ if }\\alpha\\not\\in \\mathfrak{p} \\text{ and there is an }\\eta \\in \\mathcal{O}_k \\text{ such that } \\alpha - \\eta^2 \\in \\mathfrak{p} \\\\\n-1 \\text{ if } \\alpha\\not\\in \\mathfrak{p} \\text{ and there is no such }\\eta \\\\\n\\;\\;\\;0 \\text{ if } \\alpha\\in \\mathfrak{p}, \n\\end{cases}\n\\end{align}\n", "6ce83d792bc22c9b268a48140ba527ba": "a(\\phi)\\,", "6ce872a0cf85a6d69697cc0ea2cfd2a2": "]\\!]x[\\![", "6ce8b60d813e2c2eb87efed3f2b2f602": "\\tau_L = L/R\\,\\!", "6ce8c036c09a2fe04b0eff61c2b717c2": " 3^3 + 6^3 = 3^5 ", "6ce8ea27d2df4810f86f5921a45dc789": "\\ F = pA", "6ce91cf855c249e18fe797525efbc5da": "\\left|G_{n-1}(z)\\cdot g_n(G_{n-1}(z))\\right|\\le C \\beta_n.", "6ce94c2dac2e97367ea3e9242c47bf51": "{\\partial x_j \\over \\partial y_1} = f_j(\\psi(y)) = f_j(x)", "6ce951f44cb1f88e4f747a0045fa7666": "b \\approx \\frac {fm_\\mathrm s} N \\,.", "6ce990f2ef4ce0aea2ae2e4088feb021": "\n\\mathcal{L}_\\mathrm{QCD} \n= \\bar{\\psi}_i i \\left( (\\gamma^\\mu D_\\mu)_{ij} - m\\, \\delta_{ij}\\right) \\psi_j - \\frac{1}{4}G^a_{\\mu \\nu} G^{\\mu \\nu}_a \n", "6ce9cffb886d0c59852e99bf6cb01383": "\\nabla \\times \\mathbf{E}, \\quad \\nabla \\times \\mathbf{B}\\,.", "6ce9f01132ff43a9b7c90370c2b88351": "\\frac{2+3}{2}=2.5\\ ", "6cea0404798fd28c89cf00d00c804bcf": "\\sin(U)", "6cea0eef2c8cdd00819ed77aa72153bf": "\\rho_G<\\rho_L\\,", "6cea1ff4a52e5360e50d64f710a5cab2": "\\gamma^{\\gamma}", "6cea3a9846465ca804458111950fecf7": "\na^p_i=\\frac{1}{\\pi}\\int\\limits_{-\\pi}^{\\pi} f^p(x)\\sin(ix)dx,\n", "6cea3aab9b7c4704c68560fd1a500150": "\\scriptstyle{0.6\\,\\lambda}\\,\\!", "6cea9b7935f6abc996164464e8f841f0": "\\mu:= \\bar{m}^a\\delta n_a=\\bar{m}^a m^b\\nabla_b n_a\\,, \\quad \\lambda:= \\bar{m}^a\\bar{\\delta} n_a=\\bar{m}^a \\bar{m}^b \\nabla_b n_a\\,;", "6ceaa4498eaca0328b61c1a23774e482": "TE = - G \\frac{M m}{2r}\\ ", "6ceab40362c00629c440090db0a95bda": "v^\\min_\\text{free} = \\frac{q_\\max}{k_\\text{crit}}", "6ceab6c9765c033d8206aa926ea128bd": "q = \\frac16 \\sqrt{2 \\pi} \\Delta P \\frac{d^3}{ l \\sqrt{\\rho_1}},", "6ceacc693d288f94a9a035b9343634ab": "\\{A,B\\} \\longmapsto \\tfrac{1}{i \\hbar} [\\hat{A},\\hat{B}] ~.", "6ceb03a1d4c864520a3a0436320252d5": "\nf(x) = a \\cdot g(x)\n\\Rightarrow\nf^\\star(p) = a \\cdot g^\\star\\left(\\frac{p}{a}\\right)\n", "6ceb3c2fb944d8aaa3f42d6ec1c30456": "s_2=\\alpha^{1},", "6ceb62acfca41c341af4c6b208cf94de": "h = \\begin{pmatrix}1 & 1 \\\\ 0 & 1 \\end{pmatrix}", "6cebd3b5d5b7ab789c824d197a48ab8b": " fold (\\uparrow^{n - 1}, [2, 2]) ", "6cec13b173777ba11c1eda72af8f7c83": " \\frac{\\pi}{2 \\ln(1+\\sqrt{2})}", "6cec28ff34922f9ef090229c6d8e7905": "C_c = 2 \\sqrt{\\kappa I}\\,", "6cec3449b5e2e22cef1d48bef4ae4bc3": "\\hat{F}", "6cec5039984a470c3852014803e472fd": "\\mu=0, \\sigma=1", "6cec773592de4168a9698e3e5c02d595": "F/4 + C/8=12 \\,", "6cece206721501e7fce47ba3383ceeb2": "\\frac{132332}{(1+0.10)^{11}}", "6ced10b19b0553198119c4a4b7d0ab3e": "\\scriptstyle \\overline{B(0,1)}\\setminus B(1,\\delta)", "6ced1473a29e774690948ddd5e528c27": "\\sigma_1 = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0\\end{bmatrix}", "6ced1a01b068c0a3748dd7de19007847": "-\\sqrt{\\frac{3}{7}}\\!\\,", "6ced3f41a667e5564c8e3e5d3a156c10": " Df(x) = \\lim_{h \\to 0} \\tfrac{1}{h}\\big( f(x+h) - f(x) \\big) ", "6ced430ea6532a2cd649115718fd137b": "S^{\\rho\\sigma} = \\varepsilon^{\\mu\\nu \\rho\\sigma} U_\\mu S_\\nu ", "6cedaafa96b82f3a96a68903d11e9bfa": "c \\in I", "6cedc0ce66a074b2722bd79ea0041b84": "(A \\oplus B) \\oplus C = A \\oplus (B \\oplus C)", "6cee01a3870c1ee64d382d78c739d739": "\n\\sigma _z^2 \\,\\, \\approx \\,\\,\\boldsymbol{\\gamma}^T \\,\\mathbf{C}\\,\\,\\boldsymbol{\\gamma} {\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(15)}}", "6cee11ace44a780f82d3ac64ec201054": "\n\\frac{1}{\\ddot{a}_{\\overline{n}|i}} - \\frac{1}{\\ddot{s}_{\\overline{n}|i}} = d\n", "6cee60161afb4aba0f83eac88cab387c": "\n \\cfrac{1}{\\sqrt{3}}~\\sigma_t - A - B\\sigma_t - C\\sigma_t^2 = 0 ~.\n ", "6cee759b6f414f21891e5a45b488fc42": "\\bot_{\\mathrm{max}}(a, b) = \\max \\{a, b\\}", "6cee7a98dc553ba82a2818719c8269b9": "4n - 1", "6cee7fcf24a4248ce5057b9756bc5b58": "\\epsilon = 24 \\pi^3 \\frac{\\alpha^2}{T^2 c^2 (1-e^2)}", "6cee9c63214807d18f8785fc2bbec662": " (\\partial H)_T=-(\\partial T)_H=-V+T\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "6ceef60e61dc045b968d06cbad4d2ef5": " \\tau_n/h ", "6cef16c7007db075be9033987dce2fae": "=1\\,", "6cef4d7a40bba690470853e244eeb42a": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} = \\Delta_rG_{T,p}= \\Delta_rG^{\\ominus} + RT \\ln Q_r ", "6cef4eaef2d8397f86adc556eb70616f": "B_k=\\frac{T_k}{\\sqrt{P_kQ_k}}", "6cef708e7cfbbb02bcebff315b4b4884": " L = c_1 \\bar X_1 + c_2 \\bar X_2 + ... + c_k \\bar X_k = \\sum c_j \\bar X_j ", "6cefc0b87834f65d6e7dbd67e933b9aa": "{(\\mathcal{I}(\\theta))}_{i, j}=\\operatorname{E} \\left [\\left (\\frac{\\partial}{\\partial\\theta_i} \\ln \\mathcal{L} \\right) \\left(\\frac{\\partial}{\\partial\\theta_j} \\ln \\mathcal{L} \\right) \\right ].", "6cefc6dc4a4ab58a27dfec07244ced9d": "N_{j}\\in P^{p}", "6cf02607f972488c26f0a9012bde0655": " = \\int_{-\\infty}^\\infty s\\left(e^{\\log t - u}\\right)r(e^u) \\, du.", "6cf0892448322ccba5a024538209fbca": "n=\\Pi_{i\\sum_{j=1}^r d_j", "6d11982eb4c553f35c4683786976c519": "\\frac{\\partial L(x, y, y')}{\\partial y'} = \\frac{y'}{\\sqrt{1 + y'^2}} \\quad \\text{and} \\quad\n\\frac{\\partial L(x, y, y')}{\\partial y} = 0.", "6d11a3fee8ac752726729c6088b1183d": "\\tbinom{n}{n/2}^n", "6d11c47069fcbb86947cefb783ef87c5": "\\vec{v} = \\mathbf{i} + \\mathbf{j} + \\mathbf{k}", "6d11e623d3442a0d8537cca15f60f9b3": "\\Delta E\\approx0.1\\,eV", "6d12126b92a6ebf6990f919f44211ad8": "\\displaystyle \\det(\\partial_{i\\bar j}\\phi) = ", "6d1275c6343c1b5a2307d61c8bee8c34": "\n \\begin{align}\n \\left[\\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^3 \\right.&\n \\left.+ \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda^2 + \n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}}~\\lambda + \\frac{\\partial I_4}{\\partial \\boldsymbol{A}}\\right]\\boldsymbol{\\mathit{1}} +\n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_0}{\\partial \\boldsymbol{A}}~\\lambda^3 + \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^2 + \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda + \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_3}{\\partial \\boldsymbol{A}} \\\\ \n &= \n \\left[I_0~\\lambda^3 + I_1~\\lambda^2 + I_2~\\lambda + I_3\\right]\n \\boldsymbol{\\mathit{1}} ~.\n \\end{align}\n", "6d127840bc33c07e5f3ebaef119c679b": "E=\\Theta(V^2)", "6d128a3781d2442836fefd703f46b039": "L_{m+n}(x)=L_m(x)L_n(x)-(-1)^nL_{m-n}(x)\\,", "6d12f2317ac6a666a971db773e7a2b80": "\\varphi _j^n \\le \\varphi _{j + 1}^n \\le \\cdots \\le \\varphi _{j + m}^n ", "6d13dcc500b173992663302bd3ba8734": "x_n^\\ast=x_n(t)+\\left[v_n(t)+v_n(t+\\tau)\\right]\\tau/2-v_n(t+\\tau)^2/2b_n ", "6d1406f009ecfd4238c745ea9ee022d3": "\\ \\displaystyle \\hat{\\alpha}(q, r_{c}) \\ ", "6d144d2e4850aedfdccf8c724dd0d0a6": "E_*", "6d145a39b1816994c2d83bf6f626cfd6": "=(1-g)", "6d1484f6af4b8b7c1ece0869db0b81cc": " \\alpha_m=\\arctan\\left(\\frac{T_b\\cdot\\sin\\alpha_r-T_r\\cdot\\sin\\alpha_b}{T_b \\cdot\\cos\\alpha_r-T_r\\cdot\\cos\\alpha_b}\\right)", "6d1486ff9d02ba830b5bbf4fd10af304": "1\\le i < k\\le n", "6d14e55bc88631cea5d2cc893c1fe8f9": "\\alpha = -0.5", "6d1503666d7b8fc9c81fa4331a307625": "U=U(x)", "6d153e790fdd0144b3ca1c8a486fcc04": "g_{\\rm indirect}(r)", "6d1549b1d0541bcc6582875c95da3f41": " \\begin{bmatrix}0 & 1 \\\\ 0 & 0 \\end{bmatrix} ", "6d158491362caab50c6e7cdfca60f5bc": " c(h,k)c(g,hk)= c(g,h) c(gh,k)", "6d168cd7fe413f9129556f3c8cae8065": "2^{3\\,.}\\mathrm{GL}_{3}(\\mathbb{F}_{2})", "6d175cf9266920cbc379fc7c1b7b430a": "\\tfrac{(c+a)}{b}", "6d177eefa175cefd51d5c6f678a25e2e": " m \\in \\{1,.., p\\}", "6d179e368c326740119039261b4b015c": "\\ell_S=\\sqrt{\\frac{e^2}{4 \\pi \\varepsilon_0 m c^2}\\cdot\\frac{G m}{c^2}}", "6d179e615b041a9ffc6ef9db0f5319f7": " P(M_N > x) < \\exp(-NI(x)) ", "6d17b06cb019e62eec70bb918487cc81": " E\\left[\\sum_{x\\in {N}}f(x)\\right]=\\int_{\\textbf{R}^d} f(x)\\Lambda (dx), ", "6d17d108397d4b6318e26008564f4338": "f(x|\\sigma) = \\frac{e^{-(x - \\mu)^2 / 2 \\sigma^2}}{\\sqrt{2 \\pi \\sigma^2}},", "6d17d6d9c69d0f6a426edc37ed1b84ec": "x=6", "6d17dbfd9ff212d44ffa5d85ed52c9cf": "\n\\left|\\phi\\right\\rang = \\frac{1}{\\sqrt{2}} \\bigg(\\left|+\\right\\rang_a \\otimes \\left|+\\right\\rang_b + \\left|-\\right\\rang_a \\otimes \\left|-\\right\\rang_b \\bigg) \\otimes \\frac{1}{\\sqrt{2}} \\bigg(\\left|+\\right\\rang_c \\otimes \\left|+\\right\\rang_d + \\left|-\\right\\rang_c \\otimes \\left|-\\right\\rang_d \\bigg) ", "6d18047ba11c8d2d93899497f6ab733e": "\\zeta_q=e^{\\frac{2\\pi i}{q}}.", "6d18249f4ac59baf28d7142c2d223de8": "\\scriptstyle \\sin(2\\pi ft+\\phi),", "6d182de885b916c16243b15ffa91e5e1": "\\mathcal{F}_\\alpha(f)(\\omega) = \n\\sqrt{\\frac{1-i\\cot(\\alpha)}{2\\pi}} \ne^{i \\cot(\\alpha) \\omega^2/2} \n\\int_{-\\infty}^\\infty \ne^{-i\\csc(\\alpha) \\omega t + i \\cot(\\alpha) t^2/2}\nf(t)\\, dt. \n", "6d18b581dd04ec911d65b55cecc92832": "f \\overrightarrow{\\partial_x} g = f \\partial_x g", "6d193595251a8e2293ae0bd4a3817723": "\\langle j_1j_2;m_1m_2|j_1j_2;jm\\rangle=", "6d1957171cd1d1bba425afafbc84101d": "g^*(R^if_*\\mathcal{F}) \\to R^if'_*(g'^*\\mathcal{F})", "6d197626cb25070e4fb2d8264f91fbea": "(\\mathbf{W}^i -\\mathbf{G}^i)\\cdot(\\mathbf{W}^i -\\mathbf{G}^i)=R^2,\\quad i=1,\\ldots, 5.", "6d19e1c1f76f3e4931fb7c0dbea53dcb": " H^n(X;G) \\cong \\langle X,K(G,n) \\rangle ", "6d19f2adadb21fc89ee968cf0434b252": "\\mathcal{C}^2", "6d1a242b2de70cc09d29ab9271c74ed6": "\\ell(\\gamma)=\\int_a^b {2|\\gamma^\\prime(t)|\\, dt\\over 1 -|\\gamma(t)|^2}.", "6d1a4f379fe85b3ab493b5108aee4cfd": "I=\\frac{1}{rHC}", "6d1ab3c8e952c606474a585f9279687c": "K \\subset G", "6d1abaa684ceb52f1939fb19787505f0": "\\det(\\varepsilon)", "6d1ae6a3fb8ea126f038244ab1097f0b": "\\frac{M_1}{(R+r)^2}+\\frac{M_2}{r^2}=\\left(\\frac{M_1}{M_1+M_2}R+r\\right)\\frac{M_1+M_2}{R^3}", "6d1b451a248c13a623a8df658d79196c": "P=MAN", "6d1b87381cf30a68b186e3603da87a50": "\\sum_{k=1}^\\infty \\frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\\tan z, |z|<\\frac{\\pi}{2}\\,\\!", "6d1b896e323d356ee0389219ca28e667": "1:\\sqrt{\\varphi}:\\varphi", "6d1bb2435983cea8c44d7a2a6dd1a99f": "\\psi_{n_x,n_y} = \\sqrt{\\frac{4}{L_x L_y}} \\sin \\left( k_{n_x} x \\right) \\sin \\left( k_{n_y} y\\right)", "6d1c3351b040c167310c0a83cd5d5ed0": "x\\sqrt[x]{a}-\\operatorname{Ei}\\left(\\frac{\\ln a}{x}\\right)\\ln a\\,", "6d1c49eb8f823a5b65b9a47f539568ff": "X \\cup Y", "6d1c4c3cc1cc7651051a07db3a558231": "E_v", "6d1c538960890fda112da82a7a4dfc83": "(x+y) z = xz + yz", "6d1c5a430334de17aa2b4facb7c17f81": "c_i(f^*(E)) = f^*(c_i(E))", "6d1c97eee4f4ef9e0cf6cea004bbfb53": "w''(x-)", "6d1d6744202a0094dfd9ae1389bff84f": "B=\\partial x/\\partial \\xi,\\ C=\\partial\\theta/\\partial y", "6d1e26ecfe6ebe7d0bccb1637c8e075d": "\\mathbf{\\mathit{e}}^{i\\phi}", "6d1e3e0d6a93cc0641ca64242bae1ed9": "\\varphi = \\frac{\\sqrt{5} + 1}{2}", "6d1e884f279fee5209475a05e14e65b4": "h\\sim{2 \\times 10^{-17}/\\sqrt{\\mathit{Hz}}} ", "6d1e8b8d469f91739f27310b8e267a1a": "\\textstyle \\sum_{e \\in P} d_e(x_e) > \\sum_{e \\in Q} d_e(x_e)", "6d1eae8b82ff5e8260e186a4b2e4c0e0": "F(C) \\cdot F(k) \\cdot Y(k)", "6d1f50b5eaf4f375f610d641e4656f35": "F(x) = \\sum^{\\infty}_{n=0}\\frac{f_n}{n!}x^n", "6d1f6adadc278a89a9ef3ca9d27be196": "N_f = mg\\left(\\frac{b}{L} + \\mu \\frac{h}{L}\\right)", "6d1f7e670f98d35c939ca1ebe7ab254a": "K_{1,3}", "6d1fca1b8172f7526d6191eb90bd9ce9": "d-w(v)+w(u)\\geq d", "6d1fcd2aeaf3ce592b6191d86063f596": "H \\ge T^{27/82+\\varepsilon}", "6d1fd729f8852de1381a8ed8606c7da5": "\\eta_\\varepsilon(x)= \\begin{cases}\n\\frac{2}{\\pi \\varepsilon^2}\\sqrt{\\varepsilon^2 - x^2}, & -\\varepsilon < x < \\varepsilon \\\\\n0, & \\text{otherwise}\n\\end{cases}", "6d1ffd2f32d52db1ea420a57bf3e6333": "_aD_t^{-\\alpha} f(t)={_aI_t^\\alpha}f(t)=\\frac{1}{\\Gamma(\\alpha)}\\int_a^t (t-\\tau)^{\\alpha-1}f(\\tau)d\\tau ", "6d20a1bcec95ff62910f28b487b06595": "T_s=\\phi=\\frac{\\tau_b}{\\tau_c}", "6d20b8e5d59c70f275c47a8913438771": "y_{R_{1}}(x), \\ldots , y_{R_{h}}(x) ", "6d211562301d6a78c9f82e4e046471e9": "\\operatorname{GL}(A) = \\operatorname{colim} \\operatorname{GL}(n, A)", "6d217162de2c6bf6ce86d200a3b17a63": "\\sqrt[n]{|c_n|}\\geq t+\\epsilon", "6d2176d300cd4847502ad3f9b73579aa": "\\beta_1 + \\beta_2 = -1", "6d217a2d7de4301d1ffbd9537aba4dca": "\\qquad R = N_{\\rm A} k_{\\rm B},\\,", "6d21f8e985e25a32ad7ec47e97b45eb2": "t^3_i = 24", "6d2203600a48b7f5cc524a02a2c01256": "\\begin{matrix} {9 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "6d2239333dc043d0d9ca04b5e2c1322a": "x^{ 6 }+x^{ 5 }+1", "6d226b92edd399722bf7a4699b2c62ad": "\nf_i^{*}\n", "6d22f3a374f50a389b9046ee70996956": "\\Phi_2", "6d231582e00642057cbcc9239d764c06": "S_{11} = \\frac{b_1}{a_1} = \\frac{V_1^-}{V_1^+}\\,", "6d235c5aa42af9d6d16583042c78ae10": "\\ \\mathrm{Lie}\\,\\mathcal{F}", "6d23ab3381d8dcf0e6ef1cb47fce5242": "y\\sim f(x)=3.67\\,", "6d2414be51179f03ba2cae9669332616": "W = \\int_{t_1}^{t_2}\\mathbf{F} \\cdot \\mathbf{v}dt = \\int_{t_1}^{t_2}\\mathbf{F} \\cdot {\\tfrac{d\\mathbf{x}}{dt}}dt =\\int_C \\mathbf{F} \\cdot d\\mathbf{x},", "6d241e348e918478425c5031a01a7e4e": "aX+bY \\leq z", "6d2438324b7601221416fda94af4cfc1": "a(t) = \\left(1 + \\frac {r} {n}\\right) ^ {nt} ", "6d245212486d07a1702baafad851de8a": "E^\\mathrm{free} ", "6d246ded100ac51946beb116e69a9a85": "\\sigma_p^2 = \\int_{-\\infty}^\\infty p^2 \\cdot |\\phi(p)|^2 \\, dp - \\left( \\int_{-\\infty}^\\infty p \\cdot |\\phi(p)|^2 \\, dp \\right)^2.", "6d2488e0d30e0872a7c407bb5337bc25": "\\langle x,y\\mid xyx^{-1}=y^{-1}, x^{2^k}=y^n\\rangle", "6d24b341cf27e31cf98a8c468f3fca7d": "\\mathrm{FeS_2 + 6 \\ Fe^{\\,3+} + 3 \\ H_2O \\longrightarrow 7 \\ Fe^{\\,2+} + S_2O_3^{\\,2-} + 6 \\ H^+}", "6d24e2bc97c5e4283dd8e34674afe7ea": "n=1", "6d24f654f9c63363eb3de67d79fa77ad": "\n\\left[\\mathbf{Q}^{\\ell_A} \\otimes \\mathbf{Q}^{\\ell_B} \\right]^{\\ell_A+\\ell_B}_M \\equiv\n\\sum_{m_A=-\\ell_A}^{\\ell_A} \\sum_{m_B=-\\ell_B}^{\\ell_B}\\;\n Q_{m_A}^{\\ell_A} Q_{m_B}^{\\ell_B}\\;\\langle \\ell_A, m_A; \\ell_B, m_B| \\ell_A+\\ell_B, M \\rangle. \n", "6d25034c0254a35f919c659a08f10bd6": "B_\\mathrm{out,par} = \\frac{B_\\mathrm{in,dau1} (d_\\mathrm{dau1}/d_\\mathrm{par})^{3/2}} {\\sqrt{R_\\mathrm{m,dau1}/R_\\mathrm{m,par}}} + \\frac{B_\\mathrm{in,dau2} (d_\\mathrm{dau2}/d_\\mathrm{par})^{3/2}} {\\sqrt{R_\\mathrm{m,dau2}/R_\\mathrm{m,par}}} + \\ldots", "6d258a5d43b66e27d97f5c57e23279e9": "\n b \\in [-\\infty ,\\infty ] ,\n", "6d2598a11ab8bf164d46357d40c2ebd8": "\\nabla\\cdot \\mathbf{u}^{n+1}", "6d25ceeb6938846d9e9334a24d73ce27": "y \\not = x", "6d2637fe2d1c790e6e2d8ea0ee70525d": "t_{0.975,n-1}", "6d2638f3b017bed72452bddbc28cbd6a": "\\scriptstyle k", "6d268b8d6016df603cb361ebcd50d51a": " t_j", "6d269d77e1f2888d1ba810cf45e9d28b": " r_1 = r_2 ", "6d26b14a8440e42f07e46d19720c68da": "P = (x_0, \\ldots, x_n) \\,\\!", "6d2720f446458051af9db13c1365eda4": "b\\mapsto b^e \\mod n,", "6d2760e7007b3274662ea64f7efcd9d6": " H(x_1, \\ldots , x_n)= \\frac{(G(x_1, \\ldots , x_n))^n}{A(x_2x_3 \\cdots x_n, x_1x_3 \\cdots x_n, \\ldots , x_1x_2 \\cdots x_{n-1})} = \\frac{(G(x_1, \\ldots , x_n))^n}{A\\left( \\frac{\\prod_{i=1}^n x_i}{x_1}, \\frac{\\prod_{i=1}^n x_i}{x_2}, \\ldots , \\frac{\\prod_{i=1}^n x_i}{x_n} \\right)} .", "6d27dad607e443422d9cc57060a2375c": "R_\\mathrm{opposite}", "6d2809701a2c0d3460e7f7ed7bbbc842": "\n \\left|B\\right|_{ij} = \\left|A\\right|_{ij} .\n", "6d2825a9f044e220e03610f80c2dcc9c": "E=(n+\\alpha)\\hbar\\omega_c, \\alpha=1/2", "6d283c6d13c384cd964e60810a2fed18": "-\\frac{\\partial S}{\\partial t} = H\\,,", "6d2851e7c7714bdcb7a4bf4482c85b0d": " H = -t \\sum_{ \\left\\langle i, j \\right\\rangle } b^{\\dagger}_i b_j + \\frac{U}{2} \\sum_{i} \\hat{n}_i \\left( \\hat{n}_i - 1 \\right) - \\mu \\sum_i \\hat{n}_i ", "6d28ace970185dfc99b3ec3283736f07": "i=\\frac{e\\upsilon}{d}", "6d28f4cb1da16fb47277bbfb4ed28394": "\\frac{m v_{\\perp}^2}{r_g} = qv_{\\perp}B", "6d2930c657554af418f6dc3a2813dc5d": "\\mu\\{x\\}\\,=\\,0", "6d2941df36b370b07dbd3ebffdd94cd0": "i \\omega", "6d29937d32524d0c31998f8b5ecbf1a5": "a = m^2 + mn + n^2, \\, ", "6d29b947fc8b9a97280244d6a23ddd70": "\\operatorname{lcm}(a,\\gcd(b,c)) = \\gcd(\\operatorname{lcm}(a,b),\\operatorname{lcm}(a,c)),\\;", "6d29d0ca11a6585c849e19c9959bb519": "\n\\mbox{If }\nm \\equiv n \\pmod q\n\\mbox{ then }\nc_q(m) = \nc_q(n)\n.\n", "6d29e0bbe7019626e302ab096e85586c": "[0,1]\\cup[2,3]", "6d29e351d360c1c1c1584f913c5e6609": "=-2\\gamma^\\sigma\\gamma^\\rho\\gamma^\\nu \\,", "6d29f9d1384374dec29c34ec04246c07": "[ E_{ij}, E_{kl}] = \\delta_{jk}E_{il}- \\delta_{il}E_{kj}.~~~~~~~~~", "6d2a2586d4eb1febd0c27b9ed663c63e": "\\scriptscriptstyle\n\\begin{pmatrix}\n t_i \\\\\n E_i \\\\\n\\end{pmatrix}\n\\leftarrow \n\\begin{pmatrix}\n t_i \\\\\n E_i \\\\\n\\end{pmatrix}\n- \n\\begin{pmatrix}\n 1 && 0 \\\\\n \\frac{\\dot{M}_i (t_i)}{1 - e_i \\cos E_i} && -\\frac{1}{1 - e_i \\cos E_i} \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n \\Delta t_i \\\\\n \\Delta M_i \\\\\n\\end{pmatrix}\n", "6d2a63c4130f117d0338a4ee274d903b": "=\\quad -{k_1 \\over n}(\\log {k_1 \\over n}) - {k_2 \\over n}(\\log {k_2 \\over n})", "6d2b371910bb4399e9bef12bccbabfb2": "\n\\alpha = {K \\over K-1 } \\left(1 - {\\sum_{i=1}^K P_{i}Q_{i}\\over \\sigma^2_X}\\right)\n", "6d2b4c3c1671bd7647831bd61e2b9f68": "\\mathbb{F}_q", "6d2b9dc8a88acd9d5aae976c0bc42c9a": "x(k)=0, k=-p,\\dots,-1", "6d2bba59c21c5208b65fa6ecd1e0550d": "A_\\phi = \\frac{\\mu_0 I}{2\\pi } \\frac{1}{L} \\sqrt{\\frac{a}{\\rho}} \\left[ \\zeta k \\left( \\frac{k^2+h^2-h^2k^2}{h^2k^2}K(k^2)-\\frac{1}{k^2}E(k^2) +\\frac{h^2-1}{h^2} \\Pi(h^2,k^2) \\right) \\right]_{\\zeta_-}^{\\zeta_+},", "6d2c15b219e3586a9b30cde997f25832": " V_{\\beta+1} := \\mathcal{P} (V_\\beta) .", "6d2c2caa7b9054399a9b2c78254b6485": "\\rho{}", "6d2c7c083eded0c20c6eb8645170abe0": "e(m_1+m_2,n)=e(m_1,n)+e(m_2,n)", "6d2c9bdb79a9752e686272eb0fd426e8": "a_N=\\sum_{i 1\n\\end{align}", "6d305b9e99cbfcf408a236e3f8a65012": "a_k x^k + \\dotsb + a_1 x^1 + a_0", "6d30cddad506b1bbf7348675f508e37c": "[A,B]=C,\\,\\,[B,C]=A,\\,\\, [C,A]=B.", "6d30f7877207f5cd6598ed11137e784f": "\\frac{1}{in} 1_{n \\neq 0}", "6d315abd46a919359efda501829da1a3": "Pr(s,a,s')", "6d31a507948e95d20121a23e1ef18dcc": "P_b \\approx \\frac{1}{k}P_s", "6d31f0d93e40616f3bef1137de4685dc": " r, g, b ", "6d31f3c5756345b24ee0c91352e92531": "\\mathbf x_{k\\alpha} + \\lambda_k \\frac{\\partial \\sigma_k(t)}{\\partial \\mathbf x_{k\\alpha}}", "6d32204423e12ef8c4bdab27dbfb099a": "Z_{in0} = \\frac{V_1}{I_{in0}} = Z.", "6d32420df194fb45a65c40ee7c438751": "a_1=1", "6d331bbff972c37aa581f8f412173cb4": " MC = \\frac{\\Delta VC}{\\Delta Q}", "6d332e344fb0fd8a457b3d01fe9afdc2": "C_{ijkl} = \\lambda\\delta_{ij}\\delta_{kl} + \\mu(\\delta_{ik}\\delta_{jl}+\\delta_{il}\\delta_{jk})", "6d3389729bf4f5d551b800e7bb0233ca": "k={K-1\\over K+1}.", "6d33f64ea9383101109c7837cea9ab53": "|a|_{\\ast}>1", "6d345b89d0caad072333fc146e8d56d1": "G = \\beta(2) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)^2} = \\frac{1}{1^2} - \\frac{1}{3^2} + \\frac{1}{5^2} - \\frac{1}{7^2} + \\cdots \\!", "6d3470b5a80b5ad10f51c37f26ba92c1": " r^2=a ", "6d348775c8857d88105ad4c1c320efde": "\\scriptstyle x_0,\\ldots,x_n", "6d34bde4650d6d5eb73c46d2e1bad46b": "\\eta_{pump},\\eta_{turb}", "6d34bec727ff2952d2ae76de1661176b": "k=QAe^{-E/RT}", "6d3512b10901e122ccf931969c979f05": "\\textbf{R}", "6d35a50a4fc62df5d468f98014393982": "L = L_{1} + L_{2}", "6d35c7f5613d1a5cdd68ec2da092fa56": "(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2)(b_1^2+b_2^2+b_3^2+b_4^2+b_5^2+b_6^2+b_7^2+b_8^2)=\\,", "6d35cca231c2456e3693a9b6f2c0f2e1": "(1-x)^\\alpha (1+x)^\\beta,\\quad \\alpha, \\beta > -1\\,", "6d36003db37a7b633beeaa55cb8ba359": "\\mathbf{b}^i=\\dfrac{\\nabla q_i}{\\left|\\nabla q_i\\right|} = h_i\\nabla q_i", "6d362e44321f246ca14bdda116fc87a9": " \\omega_n = \\sqrt{\\frac{mgr}{I_P}},", "6d36493aaa0789b8b9802de777f73009": "L=\\frac{-1}{E_d}", "6d364dc0f8e92c7df0030a059a2b426b": " \\mathbf{E}(\\mathbf{r},t) = \\mathrm{Re} \\left \\{ E(\\mathbf{r}) e^{ i \\omega t } \\right \\} \\mathbf{\\hat{z}} ", "6d367ab7487b2638c4f7678ed59f6fce": "S^{p-1}", "6d36a7ccd6c1e8241a32411ccdd3d519": "D_n(z)\\,", "6d36d8524473bc5b2307a658c0f74e3e": "ax+c=bx+d", "6d36e30935cc4544262e89f162969ad8": "\\frac{\\Delta PE}{\\Delta t} = m g u S", "6d36f2b64267324323436a70259d97c7": "\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\mathbf{J}_\\rho = 0", "6d373b11e93199920ecd52a6333c688f": "{\\tilde{B}}_{6}", "6d37449dc74a648fb1827196060353ac": "\\gamma^0 = \\begin{pmatrix} I & 0 \\\\ 0 & -I \\end{pmatrix},\\quad \\gamma^i = \\begin{pmatrix} 0 & \\sigma^i \\\\ -\\sigma^i & 0 \\end{pmatrix} \\,", "6d38231bd75ff001f7df4ee8493c5b84": "J_\\mu(x_1,\\ldots,x_{n-1})=0", "6d38329e158f41b4c2789d9e3bd07550": " \\cos t = {1 \\over 2} \\left(e^{it}+e^{-it}\\right) = {1 \\over 2} \\left(z+{1 \\over z}\\right)", "6d389077aa1149c648ab884560c387eb": "p_1 = \\pm \\frac{\\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{cE}", "6d38ae009d08030df888ad19d7859487": "\\exp \\left[-i\\left(k(\\omega)x-\\omega t \\right)\\right]", "6d38f915da93b920730128584dde0e93": "A_\\lambda = -\\ln(I_1/I_0)\\,", "6d393bf5fd57a35aebd8d9de0a5bb3e3": "\\mathit l=1", "6d39727ba8ba48d7910e62ea3d44b57e": " \\ PV \\ = \\ {Ae^{-g}(1-e^{-(r-g)t}) \\over e^{(r-g)} - 1} ", "6d3a1be477328b8492532b9fcd731d00": "R_{abcd}+R_{acdb}+R_{adbc}^{}=0", "6d3a8a52d7e7e774ed3e5bc1e591b244": "p_N(x)=1+x+x^2+ \\cdots +x^{N-1} = \\frac{x^N-1}{x-1}.", "6d3ab8dc9c91ff722ea08acc9df57886": "(P,\\le) ", "6d3ac7d55adceb3991d8063efaaa3c8b": "1254542875143750\\, X-1654608338437500,", "6d3ae1a5425e1a65c98fa49258ec2eeb": " A.A=A", "6d3aeb0d2f2c314c1927a88c91ea264a": "\\mathbf{s}_j=(\\cos \\theta_j, \\sin\\theta_j)", "6d3b1e19013593a04e4b70c666b263b9": " V^{\\dagger}=a^{\\dagger}\\frac{1}{\\sqrt{aa^{\\dagger}}}", "6d3b2c09aba6ab8dfd6a203ffbf28827": " y = \\alpha + \\beta^2 x", "6d3b77674c82f4f823cbc5ce49a0e0ab": "\\bigg[G:\\bigcap_{i=1}^kG_i\\bigg]\\ \\bigg|\\ \\prod_{i=1}^k[G:G_i]", "6d3bfaedf7b37aeb0b13746902211402": "W_\\alpha = Z_\\alpha + dZ_\\alpha", "6d3c19692553a998f3e47951fc4305a9": "F_{I}", "6d3c52c17ee28d7d1a74135968b30716": "a^2 = a_2,\\,", "6d3ca3a8fac918b574a303000d1252d9": "-\\sqrt{\\frac{1}{20}}\\!\\,", "6d3ca5680cca2cb282a0eda887b4f791": "\n \\hat f_{x^*|x}(x^*|x) = \\frac{\\hat f_{x^*}(x^*)}{\\hat f_{x}(x)} \\prod_{j=1}^k \\hat f_{\\eta_{j}}\\big( x_{j} - x^*_{j} \\big),\n ", "6d3ce2954e255d26fb6ee30f63ccc539": "(x_1,\\ldots,x_I)", "6d3ceecba72c1b5ea1f5821a525c9d7e": "a = d + e,", "6d3d0e2f9707710cdb714a8ffb7131f5": "s_x = \\frac{m}{k}v_{xo}(1-e^{-\\frac{k}{m}t})", "6d3d9a8fb976990262f1bb62f5cf16c4": "g'(x) \\ne 0~ mod~ p", "6d3db7fe8290762347210d5410cf015e": "Z = 2\\pi \\int_{-1}^ 1 d y \\exp( \\mu B\\beta y) =\n2\\pi{\\exp( \\mu B\\beta )-\\exp(-\\mu B\\beta ) \\over \\mu B\\beta }=\n{4\\pi\\sinh( \\mu B\\beta ) \\over \\mu B\\beta .}\n", "6d3dbb1dda54934eaaec30c8247fe274": "H(u*v) = H(u)*v = u*H(v)", "6d3de3101bb478ea1f1684e3e146c518": "\\displaystyle u_t+u_x=v^2-u^2=v_x-v_t", "6d3e5ae25bfcca52f7c4431b1d3acca2": "1 - ( 1 - P_1^k ) ^ L", "6d3e8470663dd6c812a574caa909a33f": "\\theta_\\mathrm r", "6d3e855fbdef18b03007fe081d0127d7": "P(B|A) = \\frac{P(A|B) P(B)}{P(A)}.", "6d3ea633df9eef11c1ca320f396d281b": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi \\epsilon_0} \\left(\\frac{q}{|\\mathbf{r}-\\mathbf{r}_s| (1 - \\boldsymbol{\\beta}_s \\cdot (\\mathbf{r}-\\mathbf{r}_s)/|\\mathbf{r}-\\mathbf{r}_s|)}\\right)_{t_r} = \\frac{1}{4\\pi \\epsilon_0} \\left(\\frac{q}{(1-\\mathbf{n}\\cdot \\boldsymbol{\\beta}_s)|\\mathbf{r}-\\mathbf{r}_s|}\\right)_{t_r}\n", "6d3eb568cb7de84843b691118a125f8d": " T = T_1 \\circ \\pi, \\ \\ \\ T : X \\ \\overset{\\pi}{\\longrightarrow}\\ X / \\operatorname{Ker}(T) \\ \\overset{T_1}{\\longrightarrow} \\ Y ", "6d3ee4437d747def7036460a26d59d99": "h(p)=h(q)", "6d3f0923592efd34b087d4ee1609f733": "\n \\boldsymbol{\\sigma} = \\cfrac{2C_1}{J}\\left[\\bar{\\boldsymbol{B}} - \\tfrac{1}{3}\\bar{I}_1\\boldsymbol{\\mathit{1}}\\right] + 2D_1(J-1)\\boldsymbol{\\mathit{1}} = \\cfrac{2C_1}{J}\\mathrm{dev}(\\bar{\\boldsymbol{B}}) + 2D_1(J-1)\\boldsymbol{\\mathit{1}}\n ", "6d3f27c5144e4edbf56d500cc7ad2002": "\\mu_S\\approx 2\\frac{e}{2m_e}\\frac{\\hbar}{2}=\\mu_B.", "6d3f5f789f196696ba5cc3fb048473b3": "f(d,s)=d^{2}-s^{2}", "6d3f7b96c4e115aca6bc13d62cac9c99": "{\\tilde{I}}_1", "6d3f8d308d4f98a53880605371a1996c": "\\epsilon(u)=\\epsilon(v)", "6d3f9a920166784f78cbdd6811bcb253": "CLI = 0.0588{L} - 0.296{S} - 15.8\\,\\!", "6d3fb1243abbe7c6cb01ba1a09e68ae8": "N\\!", "6d3fba3df69bfab3401600dd6d7376b7": "(u_i)_{i \\in I}", "6d409217e99aed0322deab002997953a": "1-\\gamma", "6d40c421e6008e1cb294080952e78cf6": "D^n_-", "6d40e07ce503b57f7587ab853761daf7": "X = \\mathbb R", "6d412ec76391cb3789a10eb6ca8aa5b6": " Q_1 = \\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\\n 0 & \\cos(\\theta) & -\\sin(\\theta) & 0 \\\\\n 0 & \\sin(\\theta) & \\cos(\\theta) & 0 \\\\\n 0 & 0 & 0 & 1 \\end{matrix} \\right]=\\exp \\left ( \\theta\n\\left[ \\begin{matrix} 0 & 0 & 0 & 0 \\\\\n 0 & 0 & -1 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 \\end{matrix} \\right]\n\\right ) ~. ", "6d416f7786dbe588eaa6ab48666588d1": "En", "6d4194286816a2b4451f8e2136d9dcb1": " | \\delta\\mathbf{Z}(t) | \\approx e^{\\lambda t} | \\delta \\mathbf{Z}_0 |\\ ", "6d4209f14789b4755fa49dec15c1f548": "f(a) = f(b)", "6d4285922a4a97b3ffdbca15875c4790": "\\tau_{x'y'}=40\\textrm{ MPa}", "6d432e493b5ff8701d5ad91ed7d4c69b": " F = -cv \\, ,", "6d43953bfe7f365ee58c55cea7d4bcb0": "Z/p_t^{k_t}Z", "6d4398bde46e1e3f9bd98fddf8f8bee9": " q_t(V) = \\int_V Q(x,t)\\,d x \\quad ", "6d43dfa7ab023a5c4ee105ccc2b4872f": "\\|\\boldsymbol{x}\\|_2", "6d44274f194f7d8347631d2f90d4aaf4": "\\big \\langle E_1(l, m, t) E_2^*(l, m, t) \\big \\rangle = \\Bigg \\langle A \\left( l, m, t - \\frac{R_1}{c} \\right) A^* \\left( l, m, t - \\frac{R_2}{c} \\right) \\Bigg \\rangle \\times \\frac{e^{-i \\omega \\left( t - \\frac{R_1}{c} \\right) }}{R_1} \\times \\frac{e^{i \\omega \\left(t - \\frac{R_2}{c} \\right)}}{R_2}", "6d44aedfe43c28f46d7f69edf74f95b0": "\\frac{IM_p}{IM_q}=\\frac{IA\\cdot IC}{IB\\cdot ID}=\\frac{e+g}{f+h}", "6d4539a67fa3baa78cc2f03e666e3cfe": "Z_{nm} = {V_n \\over I_m } \\bigg|_{I_k = 0 \\text{ for } k \\ne m}", "6d456059e92091937cf60566eafa81c7": "f(k) = 2^{k2^{k-1}+3k}", "6d45636cd6589a955a003744177e3c9d": "ds^2 = dt^2-t^2(d \\chi ^2+\\sinh^2{\\chi} d\\Omega^2)\\ ", "6d45add2033044382ddde7cba3da2485": "\nJ(\\mathbf{W}) = \\frac{\\left|\\mathbf{W}^{\\text{T}}\\mathbf{S}_B^{\\phi}\\mathbf{W}\\right|}{\\left|\\mathbf{W}^{\\text{T}}\\mathbf{S}_W^{\\phi}\\mathbf{W}\\right|}.\n", "6d45bb9197479f61990497f3479c63ba": "\\frac{\\partial \\theta_f \\bold{u}_f}{\\partial t} + \\nabla \\cdot ( \\theta_f \\bold{u}_f \\bold{u}_f ) = - \\frac{\\nabla p}{\\rho_f} - \\frac{\\bold{F}}{\\rho_f}+\\theta_f \\bold{g}", "6d45f9c4ebf15cdbd4656f6aa06d9a25": "\\hat O", "6d46328fd73cee1e6237a8b2ed93c60d": "\\overline{Y}_i - \\overline{Y}", "6d4654e793a5d3b38dd2bb22804328a8": "\\forall x f\\ f\\ x = f\\ x", "6d468c2e92848011bba808531eb33e2d": "J_{jk}=\\frac{\\partial y_j^{calc}}{\\partial p_k}", "6d46fb26fe0fc9c7166cf0793ae5a803": "I_{D1} = I_{L1} - I_{L2} ", "6d470d572a9724edb27317e05d6d6adb": "\\mathrm{920ECF10}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathbf{kiebgt}", "6d47431bcfd22e6c8e3490792a525a7b": "\\mathbb{F}_{q^n}[x]", "6d476cc0e6d13ea5d5b78145bdd407b6": "Var[r_t|r_0]=r_0 \\frac{\\sigma^2}{\\theta} (e^{-\\theta t}-e^{-2\\theta t}) + \\frac{\\mu\\sigma^2}{2\\theta}(1-e^{-\\theta t})^2", "6d47d3b2ec3103a80f99549e313e2852": "d=\\sqrt{2 R^\\prime h} \\,.", "6d47fd6ce2578aacee4b8d135c14f646": "T^2 = \\frac{T_1^2 + T_2^2}{2} + \\frac{T_1^2 - T_2^2}{2} \\left ( \\frac {h_1 + h_2}{h_1-h_2} \\right ) \\, \\qquad \\qquad \\qquad (2)", "6d4836c85460df0b957fac6bc208c9a9": "\n\\sum_{l=1}^{n}lk_{l} = n.\n", "6d48a0bd2ce96a538a4fa1f6ffd57a96": "R = \\rho \\frac{\\ell}{A},", "6d48a3ea3b1aadd4d12aae797bc509ed": "W_{in} = W_{out} \\,", "6d4918c1d973aaf80f3e1bcd6090db60": "\\textbf{c}=(a,b)", "6d497cff635980a5c909467a6c7c0e35": "{\\textbf{x}}_3", "6d498f784dfbdd2e1ab7592b2a18feb3": "\\begin{array}{rl}\nx_{n+1} =& x_n + 2\\,\\frac {\\left(1/f\\right)'(x_n)} {\\left(1/f\\right)''(x_n)}\\\\[1em]\n=& x_n + \\frac{-2f(x_n)\\,f'(x_n)}{-f(x_n)f''(x_n)+2f'(x_n)^2}\\\\[1em]\n=& x_n - \\frac{f(x_n)f'(x_n)}{f'(x_n)^2-\\tfrac12f(x_n)f''(x_n)}\\\\[1em]\n=& x_n + h_n\\;\\frac{1}{1+\\frac12(f''/f')(x_n)\\,h_n}.\n\\end{array}\n", "6d49cc754139e3cb378cee90a137804e": "\\scriptstyle a(k)", "6d49f25b7d32ebd504aee545596f5bd5": "\n\\zeta = \\frac{P+\\sqrt{D}}{Q}\n", "6d4aaaef4381489760c616e8ff6b1140": " \\sgn(x) := \\begin{cases}\n-1 & \\text{if } x < 0, \\\\\n0 & \\text{if } x = 0, \\\\\n1 & \\text{if } x > 0. \\end{cases}", "6d4addb95547e9b379f2875dfd31e1e2": "Q(x,y) = \\sum_{i = 0, j = 0} ^{i = m, j = p} a_{i,j} x^i y^j", "6d4ae6d3015696ef375f9229925474f2": "\nf_X(x)= \\begin{cases}\n\\frac{1}{4}(x+2)^3 & -2\\le x \\le -1\\\\\n\\frac{1}{4}\\left(3|x|^3 - 6x^2 +4 \\right)& -1\\le x \\le 1\\\\\n\\frac{1}{4}(2-x)^3 & 1\\le x \\le 2\n\\end{cases}\n", "6d4b3381968ee82db9e0bd6a980af09d": " E=\\hbar \\omega ", "6d4b45718b0cae66bee8695061b6c553": "k_3 = B_\\max k_\\mathrm{on}", "6d4b52f4b949e3d53ce986c6af15c368": "3.\\overline{6}", "6d4b740a4f99017ae217bb03f273e32d": "D = \\frac{1}{v_g^2} \\frac{dv_g}{d\\lambda}", "6d4b84f9e1564230250eeb61981a8f82": "\\mathit{N} = pt", "6d4bd8fba3b65cf8c38b11afc697ec49": " \\theta = \\log(\\lambda) ", "6d4bdf9ba201e94828f54412f1f015d9": "m_{eff} \\sim \\rho^\\alpha", "6d4bfa1f6f43d4d79d8263cdf9b9f858": "H=H_\\mathrm{sys}(\\mathbf{Z})+\\frac{1}{2}\\sum_n\\left((p_n-\\kappa_nX)^2+\\omega_n^2q_n^2\\right)\\,,", "6d4c091598658cc4b098151c93647350": "m^2 \\equiv c\\pmod{r}", "6d4c9424f6be32d863568bd1ae502a4a": " k = { \\omega \\over v } = { 2 \\pi f \\over v } = { 2 \\pi \\over \\lambda }", "6d4c94a74fdb34c07e27bcf5683b4122": "\\begin{align}\nR_{m}^2 & = (v \\tau_{m} + R_{0} )^2 \\\\\nR_{m}^2 & = (v \\tau_{m})^2 + 2 v \\tau_{m} R_{0} + R_{0}^2 \\\\\n0 & = (v \\tau_{m})^2 + 2 v \\tau_{m} R_{0} + R_{0}^2 - R_{m}^2 \\\\\n0 & = (v \\tau_{m}) + 2 R_{0} + \\frac {(R_{0}^2 - R_{m}^2)} {v \\tau_{m}}. \\\\\n\\end{align} ", "6d4ca0fbd737e636f5961531ff160227": "\\mathrm{Cov}[X_i,X_j] = \\frac{- \\alpha_i \\alpha_j}{\\alpha_0^2 (\\alpha_0+1)}.", "6d4cec6c9ca9429537d4726a1b352c54": " c_+ ", "6d4cf3e820441b7a5b0fe87c0dd476be": "\\alpha_n := \\tan^{-1}\\left(\\frac{\\Omega \\sqrt{n+1}}{\\delta}\\right)", "6d4cf4a395065a8e3fa6077febaae3b9": "\n\\nabla \\cdot \\mathbf{H}(x) = 0.\n", "6d4d0f77dd34e9fa61c70bad518b7013": "f_1 (x_1, y_1, x_2, y_2) = \\mathbf{r}_1\\cdot \\mathbf{r}_1 - L_1^2 = 0, \\quad f_2 (x_1, y_1, x_2, y_2) = (\\mathbf{r}_2-\\mathbf{r}_1) \\cdot (\\mathbf{r}_2-\\mathbf{r}_1) - L_2^2 = 0.", "6d4d855e5cbca52f3c32bd1c350ee5c2": "\\scriptstyle 2^x{n\\choose x}", "6d4df7ad92ee4d74f1eee9ae9e5e00d8": "g_{k,n}(z)\\in S", "6d4dfdb9e57db38036fdd873f296f2fa": "\\widetilde{\\gamma} : [0,T] \\to \\widetilde{M}", "6d4e0075d8a11fbd9038f9005d6ba244": "\\Lambda=\\bigcap_{i\\ge 0}T_i", "6d4e3a9f1843f0fea1e6664d8454d731": " \\phi = \\frac{m_{\\rm C_2H_6}/m_{\\rm O_2}}{(m_{\\rm C_2H_6}/m_{\\rm O_2})_{st}} = \\tfrac{0.938}{0.268} = 3.5 ", "6d4e53fdd333a6e95f68d278f8cb5147": " H^k(X;\\mathbf{C}) = \\mathbb{H}^k(Y, \\Omega^{\\bullet}_Y(\\log D))", "6d4e6691c3fd40e42b1cb1e7d228e2df": " \n L_n :=\n \\begin{cases}\n 2 & \\text{if } n = 0; \\\\\n 1 & \\text{if } n = 1; \\\\\n L_{n-1}+L_{n-2} & \\text{if } n > 1. \\\\\n \\end{cases}\n ", "6d4e6e158807c40b11a847621ab78c4f": "C(\\varepsilon,R)", "6d4f9e24de9a13c38ccc705cd72b81e0": "(4n-1, 2n-1, n-1)", "6d4fb54fd2eee0be7445ce694151f182": "M_{\\text{o}}", "6d4fe3e9335798f81b153b982e3c3b80": " \\rightarrow (\\mathbf{\\lambda} x . x x x) (\\lambda x . x x x) (\\lambda x . x x x) (\\lambda x . x x x)", "6d4feb259633e832b1b8545d07632e12": "\\bar r(\\lambda), \\bar g(\\lambda), \\bar b(\\lambda)", "6d50390e1fe97035176956350d4438ff": "D^+_{\\alpha}q=0", "6d5060b9db522b7ddf8df133712d7240": "-\\nabla^2\\mathbf{B} =\\alpha^2 \\mathbf{B} ", "6d506e2d3512a55a27b377803aa5b470": "\\omega(X,D)(E)=1", "6d50b9651cd889446a22033112d456df": "[\\operatorname A,\\operatorname B] = \\operatorname A\\operatorname B-\\operatorname B\\operatorname A", "6d513f121c3056bc1efbea38ef150a3f": "\n10.000 \\mbox{ metres} = \\frac{L + B + 1/3G +3d + 1/3\\sqrt{S} - F}{2}\n", "6d51492cdf70815c4e7a5a3532a37b5d": " \\mid \\psi_L \\mid^2 ", "6d5150ac60aae2289d1c850092503ae9": "\\ddot{a} > 0", "6d516d0896424d854a7e0c69c9930668": "1 \\le i\\le r", "6d51819f090f59bf13c6932db27ec2f8": "\\left\\{\\begin{matrix} n \\\\ 2 \\end{matrix}\\right\\} = 2^{n-1}-1.", "6d519471dcd27b5e41a351378a67deb7": "W = -\\Delta PE.", "6d51db7238883161d4282f04a161f4e9": "\\scriptstyle \\|z\\| \\;\\le\\; 1,", "6d51df6c3d59b0e227aa8cd5e6e13faa": "X \\vdash d\\mbox{ iff }d \\sqsubseteq \\bigsqcup X.", "6d51f83f47620b7ea84162d702981845": "\\gamma |S||T|", "6d51f8c18528ad2c78a9473246374cb8": "C_n^{(\\alpha+1)}(x) = \\frac{1}{2\\alpha}\\! \\ \\frac{d}{dx}C_{n+1}^{(\\alpha)}(x)", "6d523d2156a1f903c9cd55ab12627d5f": "t_{0}", "6d52b2c9399f786db5f6067a48a5e86d": "\\mathcal{I}_{\\beta, \\beta}", "6d52ef85dd59e4bf6cbb2fe66f50603c": "\\Pi V", "6d531aaa9261e1b0a11725070f290586": "E(s') - E(s)", "6d5388affd3e9938631bad607ab82545": "\n\\mathbf{a}_{k} = \\frac{d\\mathbf{v}_{k}}{dt} = \n\\boldsymbol\\alpha \\times \\mathbf{r}_{k} + \\boldsymbol\\omega \\times \\mathbf{v}_{k}\n", "6d540e57af3f5df09f5b767835f40a92": " \\tau_g(\\omega) = -\\frac{d\\phi}{d\\omega}", "6d54536ca3238905904ed7df761c10c8": "\\frac{n-p}{p(n-1)}t^2 \\sim F_{p,n-p} ,", "6d54a262e172b63a2ab14f5e4c7f19e8": "c/2", "6d54b893a4c544b8063952825d11a787": " \\sum_{i=1}^{m} n_i \\lambda_i = r(k-1) ", "6d54cabb2557fbf4f53c58e035f97455": " \n\\begin{align}\nP_{ni} & = Prob(\\beta z_{ni}+\\varepsilon_{ni} > \\beta z_{nj} + \\varepsilon_{nj}, \\; \\forall j \\; \\ne \\; i) \\\\\n & = \\int I(\\beta z_{ni}+\\varepsilon_{ni} > \\beta z_{nj} + \\varepsilon_{nj}, \\; \\forall j \\; \\ne \\; i) \\; \\phi(\\varepsilon_n | \\Omega) \\;d \\varepsilon_n,\n\\end{align} \n", "6d54e47e197496a61bc8a8a92dded529": "\\chi(\\Sigma)=\\frac{1}{2\\pi} \\int_{\\Sigma} K(u)\\,dA,", "6d55051f95f866291b4a21ad8b74a2af": "\n S_u = (0.8,-0.6,0.0)\\cdot \\vec{\\sigma} = \\begin{bmatrix}\n 0.0 & 0.8+0.6i \\\\\n 0.8-0.6i & 0.0\n \\end{bmatrix}\n", "6d55079ca3f6563716a2abd06a2e431b": "\\overline{f}\\colon (M/\\sim,d')\\longrightarrow (X,\\delta).", "6d5507c51c8e8da7efeeab7ee24fbcc5": " B^{\\prime}=\\frac{dB}{dr}, B=1-\\frac{R_s}{r} ", "6d554ca4707bbae79cf11243c0e2dc3f": "HME_0(X)=\\pi_0({\\rm Homeo}(X))=MCG^*(X)", "6d5553d5a8f704395eb0f0bc01b95ff8": "P_k(x) \\begin{cases} 0, & \\text{if } x\\le q_k \\\\ \\frac{x-q_k}{p_k-q_k}, & \\text{if } q_kp_k \\end{cases}", "6d558db8c5b36837587bf5167115d1f3": "\\tilde{F}(\\omega)", "6d55b445152a8d5d583a757658ddb6a2": "e^{i(k\\cdot x - \\omega t)}", "6d55dcdfa08e4f3234c037110659ad60": "{2p-1 \\choose p-1} \\equiv 1 \\pmod{p^6},", "6d55f7ceecbce456a2be9975cee418ef": " F_{ST} \\approx 1-\\frac{T_0}{T}", "6d561a004826a5a5ee1d0c4fdcff34c1": " \\hat\\mu_j ", "6d5634e0daebc94cb03a9c6328c4ac8a": "x \\in \\left[\\pm {1 \\over \\sqrt{\\beta(1-q)}}\\right] ", "6d5638cea3d241406245af1c5fd3928a": "\\tfrac{2\\pi}{5}", "6d568d41dc309a5e9e8126803b5bc26e": "d=\\infty", "6d56a48d128149a8d1ed4933fa830a08": "e^{-t(x)},\\,", "6d56c151928625fa615ba77f0c554930": "\\sum_{k=0}^n {n \\choose k} = 2^n", "6d570b09b2991489b37f1f4b15ceb8d7": "3^\\frac{4}{13}", "6d573adf4698ac56571a11ada21d672d": "\\mathbb{HP}^{\\infty}", "6d579904a51ed91fe40a0f24cebdb991": "P_0=\\dfrac{x[1-(1+i)^{-m}]}{i}", "6d58066fd24d3f6d74093e6fa48a5526": "\\forall i < n \\; \\left(\\dots\\right)", "6d5832988e4a0aa03a93eba456404608": "\n\\mathbf{u}=\\frac{F_z}{4\\pi\\mu r}\\left[\\frac{1}{4(1-\\nu)}\\,\\frac{\\rho z}{r^2}\\hat{\\mathbf{\\rho}} + \\left(1-\\frac{1}{4(1-\\nu)}\\,\\frac{\\rho^2}{r^2}\\right)\\hat{\\mathbf{z}}\\right]\n\\,\\!", "6d5847f0abecde2339aa48623b175346": "\\gamma=\\sqrt{2}", "6d586ade2c3a9540a8dff03bbf9e8110": "c=\\pm 5/\\sqrt{6}", "6d5874be52711a05267745f66de826e0": "(2D)^2 = 2 p^2 \\cdot \\frac{1+\\cos \\alpha}{\\sin^2 \\alpha}", "6d5948d55cdbc6b7fd7b50afb8e342cc": "E_{s,a}", "6d5997e48fde3ff908e9670807fea250": " = -\\frac{T_0}{c} \\int \\mathrm{d}^2 \\Sigma \\sqrt{(\\dot{X} \\cdot X')^2 - (\\dot{X})^2 (X')^2} \\ ", "6d5a0eede86f5b7a82fac9397f485b05": "\\textstyle P(A\\mid[x]) \\geq \\alpha", "6d5b0682d07791d04e4d1bc094fdb47f": "\\left ( {\\partial p\\over\\partial T} \\right )_V = \\left ( {\\partial S\\over\\partial V} \\right )_T", "6d5b30d144eba2c28061360332322c32": "\\dot\\gamma", "6d5b6048f7cad645d3fdef0ba1a77d25": "f_4(Pr)= \\left[1+ \\left ( \\frac {0.5}{Pr} \\right )^\\frac{9}{16} \\right]^\\frac{-16}{9}", "6d5b89145ccf7904416d196a5ad7a75a": "\\scriptstyle{\\mathbb{R}}^3", "6d5baf7eaeef1ace69cbf62aefac86ac": "X^{\\beta_{\\gamma+1}}_n\\setminus\\beta_\\gamma", "6d5bdb050b4a5eab5d544049700c33a5": "\\text{Subject to } \\begin{cases}\n y_i \\left[ {w^T \\phi (x_i ) + b} \\right] \\ge 1 - \\xi _i , & i = 1, \\ldots ,N , \\\\\n \\xi _i \\ge 0, & i = 1, \\ldots ,N ,\n\\end{cases}", "6d5bf61b10fd65680e4b2c38403154a5": "f_u^{\\otimes |U|}.", "6d5bf623a08d3e5c577322c1e54fb3c3": "\\mathit{Var}_g(f; N) = \\mathit{Var}(f/g; N)", "6d5c5ff1c9202d6a3bad431cca4fa553": "\\mathbf{X}_I", "6d5c94418d8309cec946bb3e9f76d735": "f(z)=\\sum_{k=0} a_k^\\nu J_{\\nu+2k}(z)\\!", "6d5cdd3c70728f7c49ab3a8bfa89357a": " x^3 \\, ", "6d5d11478eaf1006057ec60b5ceda7e6": "X^{\\beta}", "6d5d3479ced44b9c69ead344c8f2dd1a": "\\mathbb{CFM}_I(R)\\,", "6d5d442605eee6d8ea223cc7926e1d65": "9 \\times 3", "6d5d448b20e9cdcbe7d6539906533625": " c(x,y,t) ", "6d5d776bf8c0b9a38d0badcd47d9ffba": "f(x) \\sim \\sum_{n=0}^\\infty b_n J_\\alpha(\\gamma_n x/b)", "6d5d86bdb13512fefa714cc56b742f1e": "r \\ge 3", "6d5db3aef2fafd39daca1545f38caf3e": "k + m < j < k + m + 1 \\,", "6d5e1369b22e8816644e8cd854bc756a": "P \\subseteq \\mathbb{A}", "6d5e1f5928a8c2337fc6dfa3ce17146a": "n = 2^m-1", "6d5e66f3b23bb1f54c888f339ea2eb31": "\\scriptstyle{AB}", "6d5ecef716f823931d95d919dfec16ce": "W_{out} = F_w \\cdot \\text{Rise} \\,", "6d5efc44451c764b60c6e3f5fbe2550c": "\\scriptstyle R_w[m]", "6d5f29ada41291e71b2a9400dd386e1c": "\\beta^2, e^{\\beta x}", "6d5fb52b1be5cc55da5edbff039a0a0a": " E_E = ", "6d5ff94d25eba78069d9a90077e17e04": "\\frac{f_1(x)}{f_2(x)}=2x^3-2x^2-x+1-\\frac{4}{2x-1}.", "6d602cd90c97c603bce1e8378f8dedaa": " \\mathcal{N}(\\rho) = \\sum_i \\frac{|\\lambda_{i}|-\\lambda_{i}}{2}", "6d6040f298aef497e03a694b3e588316": "\\theta(t) = -\\frac{t}{2} \\log \\pi + \\sum_{k=0}^{\\infty} \\frac{(-1)^k \\psi^{(2k)}\\left(\\frac{1}{4}\\right) }{(2k+1)!} \\left(\\frac{t}{2}\\right)^{2k+1}", "6d6047ac1de7e493b1a32284d815ea1a": " \\mathbf{x}_{k} \\in \\mathbb{R}^{p} ", "6d604a411cb41956b6a968c2e49e065e": "\\scriptstyle I,\\, J", "6d6099c9d70886a668d8c9700706679a": " \\frac{\\partial \\varepsilon}{\\partial E} = \\frac{1}{L} \\frac{\\partial u}{\\partial E} = \\frac{-P}{E^2A} < 0 ", "6d60cc3d04953772a845a8a4cedde1da": " \\mathbf{A} \\otimes \\mathbf{B} = \\begin{pmatrix} \nA_{11}\\mathbf{B} & A_{12}\\mathbf{B} & \\cdots & A_{1n}\\mathbf{B} \\\\\n A_{21}\\mathbf{B} & A_{22}\\mathbf{B} & \\cdots & A_{2n}\\mathbf{B} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{m1}\\mathbf{B} & A_{m2}\\mathbf{B} & \\cdots & A_{mn}\\mathbf{B} \\\\\n\\end{pmatrix}.", "6d61110a29a6d804064dab013722fa2d": "H(vu) = v+u", "6d61aee85dcd95b121472365214e0638": "\\frac1{\\tfrac{n}2+1}\\binom{n}{\\tfrac{n}2}", "6d624ab6cabc3f7fb4b538c6ece63516": "a_i \\ ", "6d629c4b594fbaec0ae5eab538a9bd63": "g_c", "6d62c24e0f0cb409ce833e1bee1b2f9f": "G = \\frac{4 \\pi A}{\\lambda^2}e_A = \\frac{\\pi^2d^2}{\\lambda^2}e_A", "6d62c6b14d602f8cf096f0d5412007a8": "(\\gamma_n)_{n=1}^\\infty", "6d62c9ca26cb4fcb655d5471f23fef4f": " H_0 |n^{(0)}\\rang = E_n^{(0)} |n^{(0)}\\rang \\quad,\\quad n = 1, 2, 3, \\cdots ", "6d62d65c59ca9f95bbf3efc4d52eec0d": " \\lambda = \\frac{2\\pi}{k}", "6d62e126dff4e41690f9823fc0268ccb": " \\exists{n}{\\in}\\mathbb{N}\\, P(n,n,25) ", "6d62f76c5c360fee4a94a6a117390920": "\nm = \\frac{\\ln\\left(F/K\\right)}{\\sigma\\sqrt{\\tau}}.\n", "6d630cb955fe416c8f60ca3c2e06bca7": "{q}\\times{\\beta} = \\alpha.", "6d6332d0043e9a4a120d592d110f1028": " D_\\text{eff} = 1 + (m - 1) \\rho .", "6d639c9fd20ff28a23f4bcd5cd0856b4": "\n\\gamma_{P}(Q) = \\frac{\\left | \\sum_{i=1}^N {\\underline P}Q_i \\right |} {\\left | \\mathbb{U} \\right |} \\leq 1\n", "6d640a7ba407aec2da9cd9317fc6963e": "B_{n - 1} = I_n, \\qquad B_{i - 1} - A\\cdot B_i = c_i I_n\\quad \\text{for }1 < i < n-1, \\qquad -A B_0 = c_0 I_n.", "6d642c8d93425a96fae47df9c35a7a8d": " \\langle q| \\mathbf{ \\hat T}(\\lambda) = \\langle q - \\lambda| ", "6d64912fc3db54983fd78031cfd14673": "X = -|\\alpha|", "6d64ac8bf51a8a5a80750ceebc8a3f8f": "\\begin{align}\nf_*(x) &:=\\sup_{y\\in X}\\left\\{f(y)-\\omega(|x-y|)\\right\\}, \\\\\nf^*(x) &:=\\inf_{y\\in X}\\left\\{f(y)+\\omega(|x-y|)\\right\\}.\n\\end{align}", "6d64bd1cfa08ced7c0b289d22ad0df2f": "\nK = \\left( 1 + \\frac{1}{c_v^2} \\right) \\ \\frac{(n - 1) \\ \\hat{c}_v^2}{1 + (n - 1) \\ \\hat{c}_v^2/n} \n", "6d6502eb887967a9761d8a3c01b8efcf": "1\\leq j\\leq t\\,\\!", "6d65d3fadae30a75fdcac38bc7c810e4": "\\text{RR}_k = \\frac{1}{N-k} \\sum_{j-i=k}^{N-k} \\mathbf{R}(i,j),", "6d66142acd8d848fa6d13b561038b05f": "\nF_s(\\mu) = -\\operatorname{Li}_{s+1}(-e^\\mu) \\,.\n", "6d6623d4c6a4036a5371ac3cd552abd1": "\\scriptstyle k_2 = 0.03", "6d666d0154a6bccae24de1d5cde8f007": "n \\in \\mathrm{T}_p M", "6d66e3f3172a6e709bfd5d7a4c7a3a7d": "\\mathbf{S}_W", "6d672554356db4deab1d0321e7b79735": "\\lambda^* \\in BV[t_0,t_f]", "6d672db142986d8c09f7b8ce3a5cb937": "\\langle x-a\\rangle ^n", "6d674689869c01161838642c88184b9e": "\\lim_{n \\to \\infty}\\frac{L_{n+1}}{L_{n}} = \\lambda", "6d675d79433b6322cdaf3e9d35a96fd6": "\nI \n= I_{ion}^{sat} \\left( -1 + \\,e^{e(V_2-V_{fl})/k_BT_e} \\right)\n= -I_{ion}^{sat} \\left( -1 + \\,e^{e(V_1-V_{fl})/k_BT_e} \\right)\n", "6d6773ae112deb1cb537835214c357e0": "\\mathfrak{D}_{k,m}^{(l)}(\\alpha,\\beta,\\gamma)", "6d6779428485f2114ef12842112b8864": "R = rt", "6d6825021918c0e505581488e675a1fb": "k=j'-j", "6d6837ee5c1d78f1a3d3a793ca7f51e4": "\\frac{d\\phi_x}{dt} = \\Omega_\\alpha|_{\\phi_x(t)}.", "6d68678ce706a5ff26823454fef0c0ea": "\\, w", "6d6893ae6f802b7967a2f4b793a07437": "{v^2\\over{2}}=-\\epsilon", "6d68b77e4606f572ad49ddba80d4dec8": "\\frac{c_{i1}}{p_i}=\\sum_{\\alpha_j:p_i(\\alpha_j)=0}\\frac{c_{i1}(\\alpha_j)}{p'_i(\\alpha_j)}\\frac{1}{x-\\alpha_j}.", "6d68c3d220a75cd4c7bc6e2c5fc3a9f1": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}{10 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "6d68f0777f89c70699da282d6e5753d4": "\\sigma(\\mathbf{x}) \\neq 0", "6d68fa6f37091dc5d5a18453d4181e3f": "\\gamma_2 = \\varphi_1 \\gamma_1 + \\varphi_2 \\gamma_0", "6d6928242b702afb1f3d98fb1ceaa806": "\\sum (a_n + |a_n|)", "6d6947763a23abe8ebadcdc61ccfd142": "M^{\\mathbf r}=\\left(I-H \\right) M.", "6d69cc8c373b80a7015fad3538e0710b": "(a_j)_x = \\sum_{i \\neq j}^n G \\frac{M_i}{( (x_i - x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2 )^{3/2}} (x_i - x_j)", "6d69de32f838ac5bbd0ad624fc7dff75": "a\\in\\mathbb{R}.", "6d69fe98b7a0f98c1d28ab046421f724": " f(a) ", "6d6a18cdfc240729a57802e9c93af1b6": "\\max(x^2, y^2) = 1", "6d6a25fd2f183d562f77afcad3282388": "G \\to EG \\to BG", "6d6a3d988efb500a1ca775df02b2e80f": "\\mathbb{H} \\!\\,", "6d6a5eec69587a760359bc551f7dc98d": "y_1=Ay_0,", "6d6a65cb8286ce96e858e55948bc0da8": " \\displaystyle{X} ", "6d6a880a322630454cfc9bb6ec82dbce": "(v+dv,1)", "6d6a89d4169a2a351fcf73ebf466c428": "93/3 = 31", "6d6ab91b40e666a2c767104a9306f576": "Y \\xrightarrow{v\\,} Z \\xrightarrow{l} X' \\xrightarrow {i} ", "6d6b27d205cb6c7a3d329f98704792b3": "P(A|E)=P(A)", "6d6b878f83faead3082cc84cee335d19": "|n_{k_i}\\rangle \\rightarrow|n_{k_i}+1\\rangle", "6d6b96be948a3e07486e95523e544264": " \\mathrm{M} = \\frac{{v}}{{v_\\mathrm{sound}}}", "6d6bb60599ae51049eedaa69d59f6d68": "\\det A = \\det U\\,\\det P = re^{i\\theta}", "6d6c0a7bd51f222efab7f8a1f9eba050": "M/L", "6d6c4ff3fed968db1f944d6791f3271c": "\\; \\Phi(1) = 1", "6d6c5649e4c29981ad4dbe866cb3408b": "\n \\theta = \\tfrac{1}{3}\\cos^{-1}\\left(\\cfrac{3\\sqrt{3}}{2}~\\cfrac{J_3}{J_2^{3/2}}\\right) ~.\n ", "6d6c5c0d545fdf923b56d8a77dafce84": "\\gamma = c_p / c_v", "6d6cb5d87db3924d3d70b7b14744153c": "\\psi(\\Omega)^2", "6d6ccefb65c8f72bcbfeb86784cbe47e": "Q(r)\\simeq rV_\\xi\\eta n_Q", "6d6d17f91ce1bc764b52c9c81ed1a6f9": "\\scriptstyle \\frac12 r(r-1)", "6d6d551154beffbc292a6312f7a00309": "x\\,T(3,1,x) = {\\rm E}_1(x)", "6d6d576ba9c5c7d0fe8e2760c57e01ec": "\\displaystyle q^{\\prime}=\\frac{\\partial H}{\\partial p} = p-q^2-t/2", "6d6d877769a58c3aab26fa9810aa221d": "\\sigma_{yz}\n=-\\frac{\\partial^2 A}{\\partial x^2}\n +\\frac{\\partial^2 B}{\\partial y \\partial x}\n +\\frac{\\partial^2 C}{\\partial z \\partial x}", "6d6d8cd1b555a212111822096d818712": "T = \\begin{bmatrix} T_{1} & 0 \\\\ 0 & T_{2}\\end{bmatrix}.", "6d6da86ae7b0af7a27da0410b15da954": " y_{n+6} - \\tfrac{360}{147} y_{n+5} + \\tfrac{450}{147} y_{n+4} - \\tfrac{400}{147} y_{n+3} + \\tfrac{225}{147} y_{n+2} - \\tfrac{72}{147} y_{n+1} + \\tfrac{10}{147} y_n = \\tfrac{60}{147} h f(t_{n+6}, y_{n+6}). ", "6d6dbd427fc65375eb52922ad98c1de2": "e_v = e", "6d6dc0422542171d1c24a5d89fe3e86a": "x_1,x_2,\\ldots,x_n", "6d6dd5f252f51014c189db766da9d186": "{y_1, y_2, \\ldots, y_n}", "6d6df5df11fe5ab56cea337f31147fd1": "c_1, c_2", "6d6e24ddcd88eb72d2c5c93f4cb37d58": " y_{t}=\\mathbf{X_{t}\\gamma}+\\sigma\\epsilon_{t}.\\,", "6d6e31b4f6f511594adc51cef4776758": "x_t = \\Phi^{t}(x)\\,", "6d6e3c495b1fd0a71083f46e9ebf107f": "\n[-\\gamma^\\mu (i\\hbar \\partial_\\mu - eA_\\mu)+mc]\\Psi = 0\n", "6d6e5482ca2f74590d54d53eb1ded6bb": "F(\\{a_i\\},\\{A_i\\})=\\sum_j A_j \\left(\\frac{\\partial F}{\\partial A_j}\\right),", "6d6ea8ecd3d08d5dac624529b4e5d782": "l\\leq j", "6d6eaf3f3246286047e051c4f8152aaa": "f(t+1, \\bar{u}) = h(t,\\bar{u},f(t,\\bar{u}))", "6d6ec513ebc1c3b162b66d4bec309ef8": "\\langle x,\\tau \\rangle", "6d6ef8e391dc1d119107e311894c74bf": " S = -[V_{1}(W-1)+V_{2}(L-1)] ", "6d6f0659f1b6b0b1f06433f962ef9a01": "\\{v\\colon va\\}", "6d70c1bfd23a9f9bb9828bc2995404f5": "N_{E}/N_{NE},", "6d70ebc9ac42ce53072cbcb3878030d0": "n\\cdot(n-1)\\cdot(n-2)\\cdots(n-k+1)", "6d7113567ec516cfe4e2536f3a09c3c7": "E_\\theta = \\frac{-i Z I_0}{2\\pi r} \\frac{\\cos\\left(\\frac{\\pi}{2}\\cos\\theta\\right)}{\\sin\\theta} e^{i(\\omega t - kr)},", "6d714f21e89d7bcbde7520b446eae59a": "\\Delta E_i = -k_B\\,T\\ln(p_\\text{i=off}) - (-k_B\\,T\\ln(p_\\text{i=on}))", "6d7160df075211114d9ff13c8a4ea059": "\\oplus_{i \\in I} V_i", "6d716b1b6f8438db93b6e2a9f58827ad": "~(x \\and y)~", "6d71acc3107825a7155fd0c385eb25f9": "S_{\\delta f}", "6d71c0ac438eb329c70cb03ba938a5ea": "(x_\\alpha)_{\\alpha \\in A}", "6d71fcb88e65b281fdd1d8a9b57572aa": " \\rho \\left(\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} + v \\frac{\\partial u}{\\partial y} + w \\frac{\\partial u}{\\partial z}\\right) = -\\frac{\\partial p}{\\partial x} + \n\\frac{\\partial}{\\partial x}\\left(2 \\mu \\frac{\\partial u}{\\partial x} - \\frac{2\\mu}{3} \\nabla \\cdot \\mathbf{v}\\right) + \n\\frac{\\partial}{\\partial y}\\left(\\mu\\left(\\frac{\\partial u}{\\partial y} + \\frac{\\partial v}{\\partial x}\\right)\\right) + \n\\frac{\\partial}{\\partial z}\\left(\\mu\\left(\\frac{\\partial u}{\\partial z} + \\frac{\\partial w}{\\partial x}\\right)\\right) + \n\\rho g_x", "6d721fe30df112ee5fd2af5a955254c1": " B_n ", "6d7254028d7d44409de71c077a20d423": "x_i ", "6d72753e8e047430d27b0e3689d8a6c6": "\\left.p(x_1, \\dots, x_n)\\right.", "6d72ca4f41e31e6c5e706b08ec9bc942": "\\ C_b = \\frac{12.5M_\\max}{2.5M_\\max + 3M_A + 4M_B + 3M_C} ", "6d72d03fbbb5c1f6617aaebe62c6df53": " \\mathrm{Pr}(X=1) = p,", "6d72dc36d4576ee20da9d25c8a3fb76e": " \\mathbb{E} \\left[ \\int_0^t H^2 ds \\right ] < \\infty,", "6d72fbed0cc29ee941533d71cc983b79": " M^{\\mathrm{face}} \\,", "6d72fc7fc7de67d63c2e10a473386e89": "4\\pi\\,10^{-7}\\,\\textrm{Wb/A.m}", "6d73b6da9b700d8fc89d99a7ea63bd46": " \\sigma_1 = \\arctan \\left ( \\frac{ \\tan U_1}{ \\cos \\alpha_1} \\right ) \\, ", "6d73c7b146bd62b75a3cc30b954b04b0": "x_{1,2}=\\frac{-b \\pm \\sqrt {\\Delta}}{2a}=\\frac{-b \\pm \\sqrt {b^2-4ac}}{2a}", "6d7400eb161e83d990529b79a8204f96": "S^{2m+1}", "6d74043e190226d26ab79ff8413e5879": "\\sup_X |f_j - f_k| < \\epsilon", "6d74175f95b3a8f84997d6e208dda95c": "\nh_{\\phi} = a\\sqrt{(1+\\zeta^2)(1 - \\xi^2)}\n", "6d74439abfff88c7a07b51dd3e8cd6c2": "\\mathbf{y}^2", "6d7444a35de8cb29b730fb00a46c7ffe": "\\quad u(t,x)\\ge0", "6d7488b37e6b204097144b09830eecd5": "n = \\prod_{i=1}^{\\omega(n)} p_i^{\\alpha_i}", "6d74950de7441ce8488b45c65f0437c9": "r=\\frac{k_{C}e^2}{m_0 c^2}=\\frac{\\alpha\\hbar}{m_0 c}", "6d749f48c92076919daf9ae8bf4bea8e": "\\scriptstyle f:\\mathbb{R}^p\\rightarrow\\mathbb{R}", "6d74d8f1c6d1196d2e75893f266ae552": "y = mx + b", "6d750e62e3ca09decddd2d320ec9ada4": "z \\approx \\frac{v}{c}", "6d75755c7d3581fc164c475948a2fbd7": "b_1 > b_2", "6d7580da5fb2058e69851d464a120b24": "O(log_2(n))", "6d75c2c1f9b9655d4adc6a108d60782c": "i = j", "6d7624053913697a0d57bd8c132c93ef": "-2x = -1", "6d76312ed700e79cafcc723151e0416c": "-x_1 +x_2 \\le -2", "6d7637d1267c540f6af1a5523a4e26e5": " W:= (M\\times I)\\; \\cup_{S^p\\times D^q\\times \\{1\\}} (D^{p+1}\\!\\times\\!D^q)", "6d76ff5e26a13366a7fc8a15abafd2a7": "T\\mathbf{x} = \\sum_{k=1}^s \\mathbf{a}_k (\\mathbf{a}_k\\cdot\\mathbf{x}),", "6d773a6407d9f1a28e2653eee6eb756d": "\\delta_{uw}", "6d77a7980f839c17adc9eaf28b8c33d0": " \\left(0,\\ 0,\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "6d77c263d159ee8faf267cae7151a9f5": "\\langle R'(X,Y)Z, W\\rangle = \\langle R(X,Y)Z, W\\rangle + \\langle \\alpha(X,Z), \\alpha(Y,W)\\rangle -\\langle \\alpha(Y,Z), \\alpha(X,W)\\rangle ", "6d7808a66c939798638afea2b9d230af": "\n (1) \\qquad p - p_0 = \\frac{\\Gamma}{V} (e - e_0)\n ", "6d7850a60d4ee24ae2cdc1911c42f8fc": "00", "6d8d9fa92836afeb1b6591ff7781b8c0": "R_{k + 1}(a, b) = \n \\frac{C}{\\left|I(a)\\right| \\left|I(b)\\right|}\n \\sum_{i=1}^{\\left|I(a)\\right|}\\sum_{j=1}^{\\left|I(b)\\right|}\n R_k(I_i(a), I_j(b))", "6d8e102a73e11d38b981a834334d6374": "\\int_t \\tfrac{1}{2} (r(t))^2 d(\\theta(t)) = \\int_t \\tfrac{1}{2} (r(t))^2\\, \\dot \\theta(t)\\,dt.", "6d8e5ffae5742785003a71ddd6c3e68b": " E = N - B - T,", "6d8fcb3b30710e3fb464d5a570192d1d": "ds^2 = -\\frac{(r^2 - r_+^2)(r^2 - r_-^2)}{l^2 r^2}dt^2 + \\frac{l^2 r^2 dr^2}{(r^2 - r_+^2)(r^2 - r_-^2)} + r^2 \\left(d\\phi - \\frac{r_+ r_-}{l r^2} dt \\right)^2 ", "6d8fdeb44070491c1e8087d73906fb30": "a R b", "6d901e52454aaf0c245e281898b8f56b": "\\sum_{k=0}^r{m \\choose k}{n \\choose r-k}.", "6d903a15b2b9b2ea710d6d603e9cf9a7": "a=r_{0}", "6d9078ea9d6b256ca5aeef9ed54f8166": "xy'' + (n+1)y' = y \\qquad", "6d90cda332d955057b3608d5f866a510": "\\mathbf{g}(\\mathbf{r}) = -\\mathbf{e_r}\\frac{\\partial \\phi}{\\partial r}.", "6d90d95bc26bfe549928c8a714f2103d": "E_n = E_n^{(0)} + \\lambda\\langle n^{(0)} | V | n^{(0)} \\rangle + \\lambda^2\\sum_{k \\ne n} \\frac{|\\langle k^{(0)}|V|n^{(0)} \\rangle|^2} {E_n^{(0)} - E_k^{(0)}} + O(\\lambda^3)", "6d91296c994630fb98024c2224945f84": " q=q_0", "6d9242d87ef17b7104dc3db29cf1ffe5": "\\|x\\|\\to \\infty.", "6d924de501217b47bf7182005fb0a14b": "{\\varpi} ", "6d9271f920282b23e810b88086b822d0": "\\textstyle (\\Omega,\\mathcal{F},P). ", "6d928f0dd975a368d532b20592172a91": "\\phi_s(n)=n^s(1-p_1^{-s})(1-p_2^{-s})(1-p_3^{-s})\\dots,\n", "6d92e764babd8d714bdabb1dfd4d34b3": "\\mathbf{L} = I\\boldsymbol{\\omega},", "6d93a014eb7ac9d2039aae36fd80fc11": "f(x) = 1-x^2.\\!", "6d93ab46eb20d93d34f4255bb7f285aa": " 0 > {\\lambda_1}^{-} > {\\lambda_2}^{-} > {\\lambda_3}^{-} > \\cdots > {\\lambda_n}^{-} > \\cdots \\to - \\infty; \\, ", "6d93dcda9ebbabd2aaa89d33e2d32570": "\\mathbb N ", "6d93ede9be72877274db6baf87fd32bd": "f(1)=b,", "6d94098745c0c6f3022bfc470f925b83": "P_B-P_\\infty = \\frac{2S}{R}", "6d945bb41b3f80d49d99bb02f723a793": "A^2(b^2+c^2-a^2)+B^2(a^2+c^2-b^2)+C^2(a^2+b^2-c^2)\\geq 16Ff,\\,", "6d954fe9de1c3b0b92593d96fac7b616": "d:T^k\\to\\mathbb{N}^k", "6d9555c900ecb3e4665a99b8431902c4": "\nR^{m}_{\\ell}(-\\mathbf{r}) = (-1)^{\\ell} R^{m}_{\\ell}(\\mathbf{r}) .\n", "6d956c2d8ca982330c9dfa5c1a35a202": "P \\vee Q", "6d95890a0edbb9f2e4e9887ad91e7903": "(c, c)", "6d959981d2e7890efab92d3851a1c024": "s_3=\\alpha^{1},", "6d96685aee73de0adbf6030dbeeb681b": " Q^{n}_i = \\frac{1}{\\Delta x} \\int_{x_{i-1/2}} ^ { x_{i+1/2} } q(t^n, x)\\, dx ", "6d97338a1651b1e3596df5d5e3ec077a": "\\displaystyle u=xu_x+yu_y+f(u_x,u_y).", "6d978331320f2e7a3387e148f5972a63": "\\approx V_{cc} \\frac{R_2}{(R_1+R_2)}", "6d97a39abf3c690ffbef832d4b39d541": "s^2=c^2t^2-r^2", "6d97a950bc3894e49ef35ed70c1f3f9f": "\\Box \\!\\,", "6d97efba171c113b80a8d43e78249b46": "(h,k)", "6d982155911a2296b1672c68aaae91b8": "OPD = 2n_2d\\left(\\frac{1-\\sin^2(\\theta_2)}{\\cos(\\theta_2)}\\right)", "6d98e9deff124f831b3f1c821169649d": "(K,\\,G)", "6d991e657f90d57a5360739a9c59f355": "\\det(A)= \\Bigl( (\\mathrm{tr}A)^4 - 6 \\mathrm{tr}(A^2)(\\mbox{tr}A)^2+3(\\mbox{tr}(A^2))^2 +8\\mbox{tr}(A^3)~\\mbox{tr}A -6\\mbox{tr}(A^4)\\Bigr)/24~.", "6d999bbc8a6ecbb820c04121088529a1": "k \\times n", "6d999c011402c25829dd01eba94fd766": "x_{2} = x(t_{2})", "6d99ad183eb37593234a4c2a5ec74aea": "\\varepsilon\\left(v\\right)=v", "6d9a255d578c2e7d4e42cfcb829005c1": "1/r^6", "6d9a4b494c0961c27b17e4a0b7fb01a8": "\\tfrac{\\sqrt{4}}{2}", "6d9a8edebe00df982841d83625107b1f": "= {52,900 \\pi \\over 211,600}", "6d9aac10b073637b7b2b4ab984f2a5a7": "\\mathbb{Q}(\\sqrt{5})", "6d9addc8cfa2f7bd8209ef91f75268b3": "(n_1,\\ldots,n_k)", "6d9b59856dde33cd5db7899a66989573": "D_A(B)-D_B(A)", "6d9c201bb271d06716b0159db9acbe44": "-e_1", "6d9c227739ae8fb64d816745bba11883": "\\mathbb{CFM}_I(D)", "6d9c2e3b016b840236811363180c9060": " \\mathrm{var}(\\log\\left(1 + e^{-X} \\right)) = \\frac{ \\partial^2 A(\\eta) }{ \\partial \\eta^2 } = \\frac{ \\partial }{ \\partial \\eta } \\left[\\frac{1}{-\\eta}\\right] = \\frac{1}{(-\\eta)^2} = \\frac{1}{\\theta^2}.", "6d9c6e2a5e4da69cd683965e5530fca0": "|a_n| \\le |b_n|", "6d9c737860512afee0ee857d54b0d8c9": " l_i\\frac{du_i }{dt }=0", "6d9c7e7171a7e9a90b001b9f2e244cf4": "\\neg A,A\\vdash B", "6d9c9a97b58a186da6b3b671ed730405": "n:=n+1\\,\\!", "6d9d0ff8c50639aef90acc67c28bc4f5": "1/\\sinh{(\\chi_{nk}/2)}", "6d9d78b441977ce8ef6adc330d652da5": "\\partial_r^2", "6d9d7d0c698ade6192304df7c416f0a9": "GM/r^2", "6d9d94d902ebb505b03a7b3d715ad8d4": "i=0, 1, \\dots, n,", "6d9d9a1e4e7566265994e772a9b40d84": "P[t, T, r(t)] = E_t^{\\ast}[e^{-R(t,T)}]", "6d9da13006434e469257e8b092c10a7c": "\\varphi(x) = 0.1818e^{-3.2x} +0.5099e^{-0.9423x} +0.2802e^{-0.4029x} +0.02817e^{-0.2016x} ", "6d9df7ed7a01b7464ef67014dd9f5ee7": "I_{o_{\\text{lim}}}=\\frac{V_i\\, D\\, T}{2L}\\frac{V_i}{V_o}\\left(-D\\right)", "6d9dfa1c847a4e802adae34cf89b4299": "I(z) \\propto |e^{-\\alpha_{abs} z/2}\\mathbf{E}_0 e^{i(k z - \\omega t)}|^2 = |\\mathbf{E}_0|^2 e^{-\\alpha_{abs} z}", "6d9e02d4330bc39a0981bc0c0643d28f": "T(a)", "6d9ec05d3fd3f0ac223fd1aeb05e957d": "K \\,", "6d9f28173490440308d5413ee7e0e6cb": "\\mathcal{ALUE}", "6d9f5bfc09546d9718c3b58d4861548e": " R_1 = R_0 \\, \\sec (\\Phi_1-\\Phi_0 ), \\; \\; R_2 = R_0 \\, \\sec (\\Phi_2-\\Phi_0 )", "6d9f92b26fc2366d6d32199119d00938": "\\ s = (25 + E + P) * M + H + S", "6d9fa5dcc5e677fe8a64a8acc1d16bd0": "\\mathrm{for\\ bases\\ with\\ } \\big(pK_a - pH\\big) > 1,\\ log\\ D_{bases} \\cong log\\ P - pK_a + pH", "6d9fa81e660c74c9ae9979d05d5eb9f7": "\\mathbf{B} = \\mathbf{P}^{-1} \\mathbf{A} \\mathbf{P}", "6d9fe520c8874392f045299c218adbac": "G^n", "6d9fe7050a920b431f140a03d48dc2ec": " \\frac{f(x)}{g(x)} ", "6d9fee46f80b5c64133706a70e29bd4e": " m\\ddot{u}(t) + c\\dot{u}(t) + F(t) = f(t) ", "6da0bec30ec13d5161394c8e7fe575c7": "x = \\frac{x(t)}{w(t)}", "6da0c925590c7def2cc9a664dbb88f3d": "\\frac{-0}{ \\left| x \\right| } = -0\\,\\!", "6da0da8d3a8855658ffcf0709476539c": " \\overbrace{ EE \\cdots E }^{e\\text{ times}}\\underbrace{ NN \\cdots N }_{n \\text{ times}}", "6da0e51f5df0f11d8701d0013d5bdc12": "q^{-1}", "6da1794149fce407a2c1f59d269ab205": "0<|\\alpha|\\leq |\\beta| <1", "6da1816b9b4303003b50fff358519b16": "x/{\\sqrt{\\ln(x)}}.", "6da1ac726b5769d679bb7d5d6b4cc56b": "<\\phi , \\psi>_1 =

_{Kin}", "6da1c5b83f5e5cdd5c0ed101b287ceb2": "\nD = \\frac{k_B T}{4 \\pi \\eta_m h} \\left[\\ln(2/\\epsilon) - \\gamma + 4\\epsilon/\\pi - (\\epsilon^2/2)\\ln(2/\\epsilon)\\right] \\left[1 - (\\epsilon^3/\\pi) \\ln(2/\\epsilon) + c_1 \\epsilon^{b_1} / (1 + c_2 \\epsilon^{b_2}) \\right]^{-1}\n", "6da21397deea14df9a930039d425a42f": "U\\left(n,q^2\\right)", "6da213b7924f9eeffc005fe91144c980": "\\nu = -\\frac{d\\varepsilon_\\mathrm{trans}}{d\\varepsilon_\\mathrm{axial}} = -\\frac{d\\varepsilon_\\mathrm{y}}{d\\varepsilon_\\mathrm{x}}= -\\frac{d\\varepsilon_\\mathrm{z}}{d\\varepsilon_\\mathrm{x}} ", "6da23aba241d273be7b2cabbda61f227": "(k~~k+1),", "6da263bd4b4379b656fc2466bf39ee04": "\\phi_i = a_i \\phi_{i+1} - b_i c_i \\phi_{i+2} \\quad \\text{ for } i=n-1,\\ldots,1", "6da28842c3099f9e1294d442c1e7e7bb": " \\angle P_1 P_2 P_3.", "6da29cf073a1c12adf326eea07ce22ad": " R_n \\rightarrow T_n. ", "6da2cabacbe65e4035152a0d1fe4750a": "\\exists a \\in T(s)", "6da2f4582a757ab446c767ee416d6355": "\\scriptstyle x'=\\gamma\\cdot(x-\\beta\\cdot ct)", "6da3aae2fe05b43d6c346876c7027753": "f_s / 2 ", "6da3e375094d434633f85f5f9e9f1195": "\n\\ddot{y} = \\frac{1}{m} F_{y} = \\frac{1}{m} F(r) \\, \\frac{y}{r}\n", "6da40a3508785ed9039676b25d99f9d6": "xy = yx", "6da4149c00d7e9cab8e729a5b612dbe7": "\\zeta,\\gamma", "6da46e126e0a3f0dbbad7d700f27dcd4": "\\mathrm{Power~factor} = \\sigma S^2.", "6da485aab71d0ebf5981ff579c658519": " f: U \\to \\mathbb{R} ", "6da49cbc77d24364bfa366dce1621703": "T=\\frac {V_\\mathrm t}{V_\\mathrm i} = \\frac {2Z_{02}}{Z_{02}+Z_{01}}", "6da4df6c915405d6b10c64e71dcc19e0": "\\phi_X(t) = {3i\\left(e^{iate^{ibt}} (4i - (-4b + (a+b)^2)t)\\right) \\over (a-b)^3 t^2 }", "6da4eeaa6f3fca6d14341dc81d70a360": "a^{+}", "6da4ff1942fa0034211fda7191f16eed": " \\frac{\\det \\left(-\\frac{d^2}{dx^2} + A\\right)}{\\det \\left(-\\frac{d^2}{dx^2}\\right)} = \\prod_{n=1}^{+\\infty} \\left(1 + \\frac{L^2A}{n^2\\pi^2}\\right) = \\frac{\\sinh L\\sqrt A}{L\\sqrt A}. ", "6da5083cdf71229f6a293a22091b627c": "\\mathrm{E_1}", "6da5147c9b443a0e9f87e370ac7a3b7e": "10.101010100010...", "6da5616f9f85616fdb4ff3677e9cd8ee": " \\langle F ( \\hat{A} ) \\rangle = \\int_R \\psi(\\mathbf{r})^{*} \\left [ F ( \\hat{A} ) \\psi(\\mathbf{r}) \\right ] \\mathrm{d}^3 \\mathbf{r} = \\langle \\psi | F ( \\hat{A} ) | \\psi \\rangle , ", "6da574064b2bdd4b86fa48378acc390c": "Y \\left ( \\frac{a}{W} \\right ) = 1.12 - \\frac{0.41}{\\sqrt \\pi} \\frac{a}{W} + \\frac{18.7}{\\sqrt \\pi} \\left ( \\frac{a}{W} \\right )^2 - \\cdots\\,", "6da5a30d5780784d8b506ffc2064764c": "\\frac{d q(t)}{d t} + ", "6da5fb894d1d91e83328ffbb24f218e5": " \\gamma_2 ", "6da6a8a727f5ec59cbc568434eb37bd8": "\\tbinom{[n]}m", "6da6d672739ef54b4bedb9fecbff84e1": "\\hat{p}_{01}", "6da6dd42805b0322d9a32dce4e80364f": "F = \\frac1{2} \\sum_{i = 1}^{n} A_{i}^{2} + A_{0}.", "6da7035e712c9be1c3ebfab707ef0970": "\n \\begin{align}\n K(m) &= \\frac{\\pi}{2}\\, \\left[ 1 + \\left( \\frac12 \\right)^2\\, m + \\left( \\frac{1\\,\\cdot\\,3}{2\\,\\cdot\\,4} \\right)^2\\, m^2 + \\left( \\frac{1\\,\\cdot\\,3\\,\\cdot\\,5}{2\\,\\cdot\\,4\\,\\cdot\\,6} \\right)^2\\, m^3 + \\cdots \\right],\n \\\\\n E(m) &= \\frac{\\pi}{2}\\, \\left[ 1 - \\left( \\frac12 \\right)^2\\, \\frac{m}{1} - \\left( \\frac{1\\,\\cdot\\,3}{2\\,\\cdot\\,4} \\right)^2\\, \\frac{m^2}{3} - \\left( \\frac{1\\,\\cdot\\,3\\,\\cdot\\,5}{2\\,\\cdot\\,4\\,\\cdot\\,6} \\right)^2\\, \\frac{m^3}{5} - \\cdots \\right].\n \\end{align}\n", "6da7273f89fe7e4634e1b499c60b718f": "\\ x = c(1 - \\cos (\\theta ))/2", "6da79e61a5439b2b58798fa130db5e52": "\\int \\frac{dC}{C} = \\int -\\frac{K}{V}\\, dt. \\qquad(2a)", "6da7a289486895e2ba7ba62898a4bbd6": "\\frac{\\part^2 \\Phi(R) }{\\part R^2} + \\frac{2}{R} \\frac{\\part \\Phi(R) }{\\part R} = (\\kappa a)^2 \\Phi(R).", "6da7a5ef498a7ef390901dd8d5932c53": "\\int_V \\operatorname{div} \\nabla u\\, dV = \\int_{\\partial V} \\nabla u \\cdot \\mathbf{n}\\, dS = 0.", "6da80b0506fb6ce6b4a53f376deb8764": "\\mathbb{R}^n.\\,", "6da81b51ddcc77f5ba6af32ebda9c766": "\\lim_{s \\to t} \\mathbf{E} \\left[ \\frac{\\big| X_{s} - X_{t} \\big|}{1 + \\big| X_{s} - X_{t} \\big|} \\right] = 0.", "6da8557b425a509b958631abe975ea39": "\\operatorname{RHom}_Y(Rf_! M, N) \\to Rf_*\\operatorname{RHom}_X(M, f^!N),", "6da8774e9a4942e96344f27ad5a92481": "TCE_{\\alpha}(X) = E[-X\\mid X \\leq -VaR_{\\alpha}(X)]", "6da888a3f566e692b0756c64dcf12ee7": "1 \\to \\mathbf{Z}_2 \\to \\operatorname{Spin}(n) \\to \\operatorname{SO}(n) \\to 1.", "6da8c52d13e3beda7f9735bdc50f1551": "H^2(P,Q) = \\frac{1}{2}\\displaystyle \\int \\left(\\sqrt{\\frac{{\\rm d}P}{{\\rm d}\\lambda}} - \\sqrt{\\frac{{\\rm d}Q}{{\\rm d}\\lambda}}\\right)^2 {\\rm d}\\lambda. ", "6da8e16bed260b5ccf9ab617314cdc0f": " I(X;Y) = \\int_Y \\int_X f(x,y) \\log \\frac{f(x,y)}{f(x)f(y)} \\,dx \\,dy ", "6da8ef88e80ea9b04a4cd9a5cfaef329": "\\Theta(N^{3/2})", "6da95180948c2ea3f24e31261dafed87": " F'(x) = f(x,b(x))\\,b'(x) - f(x,a(x))\\,a'(x) + \\int_{a(x)}^{b(x)} \\frac{\\partial}{\\partial x}\\, f(x,t)\\; dt\\,. ", "6da985833b026440d031f41976970653": "(i\\omega-\\xi)^{-2}", "6da9aa23fa599414fda4a33a31890679": "\\kappa _T(T,p)\\ = -\\frac{\\left.\\cfrac{\\partial V}{\\partial p}\\right|_{(T,p)}}{V(T,p)} ", "6daa38fd425eafb4f9e550ab5b1f223c": "0 < |a| < 1 ", "6daa96981cc3e4e404b2661c1551b9bb": "T=(e^{100,000}-1)\\,\\mathrm{s}\\,\\!\\approx2.8\\times10^{43,429}\\,\\mathrm{s}\\,\\!", "6daaccf6835f0e6bb09f7044c6905215": "i\\,(a + bi) = ai + bi^2 = -b + ai.", "6dab3aa9e64d2a22ab7ef8e86cb8c945": "q=-kA\\frac{dT}{dx}", "6dab4fd46be4ba64cf90cb623e0f004d": "d\\Omega = \\sin\\theta\\,d\\theta\\,d\\varphi", "6dac3df177e89ff70e45a4ce690868da": "3, 6, 8, 8, \\text{and }10", "6dacb7d2d48bc71af96f9730e770d0bf": "\\mu = {\\langle F_X \\rangle\\over M_{pl}}", "6daccb755d5622c4fc8e3291da97577e": " p_n ", "6dad12cf854f2d4377001df6f0cd3a31": " \\int \\sigma \\big ( \\left \\vert y - e_i \\right \\vert \\big ) \\, dy =1", "6dad740fb40fcc08d5ceaf622e2bcb68": "b(e)=e", "6dae21dee40f5761873d8b0efebf1b5b": "\\theta_2 \\; d_{\\rm S} = - \\; \\theta_{\\rm S}\\; d_{\\rm S} + \\alpha_2 \\; d_{\\rm LS}", "6dae4621489f7615eaf5d75c795e6751": "R_{xx}=(...,14,11,11,14,11,11,...)", "6dae7a47d33be8700598874ea7a0a853": "W=\\{Y\\}", "6dae806a5487513797a41f7e87120af8": "\\sum_{k=1}^\\infty c_k D^k", "6daefbe0428efd37faed840230bb5fda": "A_j", "6daf21a39ba3a31c9f19d71e66628404": "x_{n_j}", "6daf25fb4c8d0f1cd398800f2a494128": "T_o=1/\\beta _o", "6daf97e5acc128d8f080f590f52fa161": "\\int_0^\\infty J_\\alpha(z) J_\\beta(z) \\frac {dz} z= \\frac 2 \\pi \\frac{\\sin\\left(\\frac \\pi 2 (\\alpha-\\beta) \\right)}{\\alpha^2 -\\beta^2}.", "6dafc72e41ad207f40bd317bcb8396b2": " \\gamma = 1/(1+r) ", "6dafd0aa93619ff9dbbbf28f2a60d6ca": "\\exp ( \\epsilon )", "6dafda4b8901c06ac04a34d3131ed041": "Happens(a,t) \\equiv\n(a=open \\wedge t=0) \\vee (a=exit \\wedge t=1) \\vee \\cdots", "6dafe17741185a513e2496146eac9d5b": "0\\leq v(t)\\quad\\perp\\quad i(t)\\geq 0", "6dafe9620dec1ea6d3308f9024410bef": " [2; -4, -4, -4, ...] = 2 - \\cfrac{1}{4 - \\cfrac{1}{4 - \\cfrac{1}{4 - \\ddots}}}", "6db085a7b41fc5692b987e972899477c": "x^2+yz=0,", "6db08d9f516bf086b27de2ba4017f72f": " P_1 ", "6db0a2e73d6fd9366632c0fae996df0f": "\\int \\Phi(x)\\, dx = x\\Phi(x) + \\phi(x)", "6db0a9793f1033a79aabd9b6d176c754": "\\leq p+q", "6db11e2edbc5241a160f8d079bf5481c": "\n \\cfrac{1}{\\sqrt{3}}~\\sigma_c - A + B\\sigma_c - C\\sigma_c^2 = 0 ~.\n ", "6db1c4abb04f2e8e0e2d6e13e9d8bebd": "u \\in H^{s+1}(\\Omega)", "6db1f4bb8ab9ff047e627524ab0adbc3": " \\left| \\sum_{k=m}^n a_k \\right| < \\varepsilon, ", "6db27ebeb6acf89586422ac71211734f": "(f * g )(t)\\ \\ \\,", "6db284c8bd6669b7f2180b1f1056c9de": "\\text{d}P = - g_0 \\rho \\text{d}h \\,", "6db2a878c3143750bf0ba00e507fa486": "\\textstyle f ", "6db2ae4a765c688aa743ecb68141bcf7": "1. \\; \\; \\mathrm{CFCS} \\; \\xrightarrow{h \\nu} \\; {\\mathrm{Cl} \\cdot}", "6db2f2e12b209b9ec3b1fa0e41e8da80": "i=1,2,\\ldots,s-1", "6db30763a893818b49786b294e486193": " \\widetilde{\\mathbf{G}} ", "6db327d0af1589c0923ba1cb1d270d44": "\n \\begin{align}\n\\frac{\\partial^2 X^\\mu}{\\partial x^i \\partial x^j} & = \\frac{\\partial X^\\mu}{\\partial x^m}\\,_{(x)}\\Gamma^m_{ij} - \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} \\\\\n\\frac{\\partial^2 x^m}{\\partial X^\\alpha \\partial X^\\beta} & = \\frac{\\partial x^m}{\\partial X^\\mu}\\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} - \\frac{\\partial x^i}{\\partial X^\\alpha}~\\frac{\\partial x^j}{\\partial X^\\beta} \\,_{(x)}\\Gamma^m_{ij}\n \\end{align}\n", "6db3875e41c4985fd5b8b408992edd4a": "\n{\\rm E}[z] \\approx \\,\\,\\,z\\left( {\\mu _1 \\,\\,\\mu _2 } \\right)\\,\\,\\, + \\,\\,\\,{1 \\over 2}\\left\\{ {{{\\partial ^2 z} \\over {\\partial x_1^2 }}\\,\\,\\sigma _1^2 \\,\\, + \\,\\,\\,{{\\partial ^2 z} \\over {\\partial x_2^2 }}\\,\\,\\sigma _2^2 } \\right\\}\\,\\,\\, + \\,\\,\\,{{\\partial ^2 z} \\over {\\partial x_1 \\partial x_2 }}\\,\\,\\sigma _{1,2}", "6db3bed4ba991b69adc9c3c1e966a756": "\\hat{r}=y-Hy", "6db461dcff1e8ec4569170da2da0ed84": "\\ \\displaystyle d\\ ", "6db476efbcf9fce5b1e597c7f665cdd7": "(x^2)^2 = w(x)^2 x^2", "6db4908d080d340241a76edc85703863": "\\bar{u} = \\sqrt{\\langle u^2 \\rangle}", "6db4c7e37fe48f44bdfb59498fbadd00": " \\frac{\\partial I}{\\partial t} = - \\nabla E_I = \\mathrm{div}(g'\\left( \\| \\nabla I(x)\\|^2 \\right) \\nabla I) ", "6db4f4fcf133561bab2db82728299c5f": " R = 0 ", "6db509a3fb84c3e91dab13b5df2769ff": "\\ell_\\text{P}", "6db5cd0411d20c7e9b6e97c90537ea76": "\\mathbf\\tau_{12} (=\\mathbf\\tau_{12}(\\mathbf{q}))", "6db60f3126d32d40eac1af348dc5cdad": "H_i = E_{g(H_{i-1})}(m_i)\\oplus H_{i-1}\\oplus m_i.", "6db616bebec3d7da8709bd6163a89ae5": " \n\\rho = q (c_+ - c_-)\n", "6db63cfb9e72421c6c1fce2e60182223": "J_{\\phi} = D_{\\phi} \\frac{ \\partial \\phi }{ \\partial x_i }", "6db6999957817f8074d44e0ef7ca342f": " p_{0,0}(x) = y_0 \\, ", "6db70b63e7f5ba8fcd3e19c82f919dda": "\\mbox{dist}(T,\\mathcal{A})\\le K \\beta(T,\\mathcal{A})", "6db7678839510d6b0cd735686a2b80e5": " (x * y ) * x = x * (y * x) = \\langle x|x \\rangle y \\ . ", "6db7bc4d2277e228e2d78e252bf904ed": " (\\Omega,\\mathcal{F},\\mathbb{P})", "6db7c1293327298c2ba43d465d5f76c8": "g_{uc}", "6db7de9eccaff7637ec1a25382567c2d": "{(\\hat k_F)}^{\\kappa\\lambda\\mu\\nu}", "6db8151cc7e2684b282359763fcc2bb9": "g h(r) \\ ", "6db83857608c5b7bbb69da8067122b7f": "(2k-1)!! = (2k)!/(2^k k!)", "6db845b4692117ffff3f4484d7dca5df": "\\mu ={G}(m_1 + m_2)", "6db8b806bc562f6cf830c9acb81d3c63": "\\widehat{M}", "6db90b0f971c8c4a243d2843c89deab5": "F_X(x)", "6db91ca5542f048e4230f82cc605a3af": "\n B = A_{i+1} \\times A_{i+2} \\times \\cdots \\times A_n.\n", "6db968927fab89a3e4b9ba33537460cd": "\n\\begin{align}\n\\textbf{a}_s &= G_{11}\\textbf{a} + G_{12}\\textbf{b},\\\\\n\\textbf{b}_s &= G_{21}\\textbf{a} + G_{22}\\textbf{b}.\n\\end{align}\n", "6db96f798f1fd8bec0c87de2c7f431e2": "\\tbinom{2n}{n}:=\\frac{(2n)!}{(n!)^2}", "6db9b670aa775b9d784cd86bb5d2597a": "\\begin{matrix} \\frac{1}{6} \\end{matrix}", "6db9eec42ae091af7ebe53db94275c7d": "\\scriptstyle{f(\\pi r^2)}", "6dba3f3dd3e49e357b03381707c8270b": "P(H|I)=P(T|I)", "6dba50c88fbda22fb047ae53add9e662": " S(j,k) = s(v_j, v_k) \\quad j,k \\in (1,\\ldots,n)", "6dba8f832646e8249e0ff7bec983532c": "\\begin{align}\n\\min\\{x_1,x_2,\\ldots,x_{n}\\} \n& = \\sum_{i=1}^n x_i - \\sum_{i C\\sqrt{|x|}.", "6dbd2c3dd32a72d376ac287b7f60824a": "m[w]= \\max_{w_i \\le w}(v_i+m[w-w_i])", "6dbd62a3c58db43feb9162149c8a41cf": " \\frac{1}{|\\mathbf{x} - \\mathbf{x'}|} = \n\\frac{2}{\\pi} \\int_0^\\infty K_0 \\biggl( k\\sqrt{R^2+{R^\\prime}^2-2RR^\\prime\\cos(\\varphi-\\varphi^\\prime)}\\biggr)\n\\cos{k(z-z^\\prime)}\\,dk. ", "6dbda8599dddf0555af60acffe568643": "\\frac{1}{1-x^2}=1+x^2+(x^2)^2+(x^2)^3+\\dotsb.", "6dbe1aab0b65939c7c478e36f1c657d2": "\\mu _{sur} (micelle) = \\mu ^0 _{sur} + RTln[S]", "6dbe39ccb422cd0da8226a6773c4f61d": "{{v}_{2}}", "6dbe71697fbc78bca20f0a963a31271d": "10^{2n(n+1)}", "6dbe7433345c98f0f976032ee61edeb5": "Z_L^\\prime=\\frac{V_P}{I_P}=\\frac{aV_S}{I_S/a}=a^2\\times\\frac{V_S}{I_S}=a^2\\times{Z_L}", "6dbecc5a50a4fe4acd8f0efa7ba57580": "\\ u_k=t", "6dbf34f9ed26ba071db675ddbdd1bb9f": "(y_1,y_2,y_3,s)", "6dbf469a8bb38c1322988d2ffd7592e1": "n=1,2,\\cdots,N", "6dbf56b32eebada3b98dd6fa7ab6160e": "x_2(t)=x_1(t-t_0)", "6dbfb36840f6cef134651dbce7569486": "h_{rf}", "6dbfd314f9d673dc9930e326be91ad42": " \\begin{align}\n\\hat{p}_x & = -i \\hbar \\frac{\\partial }{\\partial x} \\\\\n\\hat{p}_y & = -i \\hbar \\frac{\\partial }{\\partial y} \\\\\n\\hat{p}_z & = -i \\hbar \\frac{\\partial }{\\partial z} \n\\end{align}", "6dc06d757f0b3c873fb87b7a4bd878db": "\\mathcal{L}\\{f'\\}\n = s \\mathcal{L}\\{f\\} - f(0)", "6dc0892d1130a85bd06ff3b2e9605bfb": "I(z) = I_0 e^{-z/\\delta_{pen}}", "6dc09a8b8daa150ffa26c527f3c21216": "S = \\int_{t_0}^{t_1} L(t, \\mathbf{x}(t), \\mathbf{\\dot{x}}(t))\\,\\mathrm{d}t", "6dc09fd85f975a853d049ed33348a634": "F(x, y, u, v) = 0\\,", "6dc0a0ca3348f420a79a30c2bc443a32": "Kf", "6dc107047b385dbae7b125db63679967": "\\textstyle{\\frac {\\log(2)}{\\log(\\tfrac {8}{3})}}", "6dc13743c75ed7e07a561d2567c13f2c": "H_{(1-X)} = \\frac{\\beta-1}{2\\beta-1},", "6dc1491a9a36a1c1bad404f63f13f60a": "u0,\\ a \\ne r_0),", "6dd607808f5a49399eb41f3908f1699f": " m = \\int_V \\sqrt{g_{tt}} \\left( 2 T_{ab} - T g_{ab} \\right) u^a u^b dV", "6dd620aa14b43cccaafa3f5afc3ea5e9": "\\Gamma(s,x)= (s-1)\\Gamma(s-1,x) + x^{s-1} e^{-x}", "6dd62359d19efe02603e69f7e771d2c8": "I_{m,n} = \\int \\frac{\\cos^m{ax}}{\\sin^n{ax}}dx\\,\\!", "6dd69b2141ea8de1322560efe1916dff": "\\bar{v_i}", "6dd69ee97b69ff98e5f1f04a775a3729": "\n\\left(\n 1 - \\sum_{i=1}^p \\phi_i B^i\n\\right)\n\\left(\n 1-B\n\\right)^d\nX_t\n=\n\\left(\n 1 + \\sum_{i=1}^q \\theta_i B^i\n\\right) \\varepsilon_t \\, .\n", "6dd6e040acd5970e29cd0b5336d6a573": "1+\\epsilon^2\\mathrm{cd}^2\\left(\\frac{nwK_n}{K},\\frac{1}{L_n}\\right)=0\\,", "6dd757cbdd852a16f222a7d1a07eab3e": "Mb", "6dd79aea36b7784766fe02226d703629": "\n\\begin{align}\nc_q(n)\n&=\\frac{\\mu\\left(\\frac{q}{\\gcd(q, n)}\\right)}{\\phi\\left(\\frac{q}{\\gcd(q, n)}\\right)}\\phi(q)\\\\\n&=\\sum_{\\delta|\\gcd(q,n)}\\mu\\left(\\frac{q}{\\delta}\\right)\\delta.\n\\end{align}\n", "6dd7b3a3198248ee5f77a0f72e2d769c": " U = \\frac{\\mu_0}{2}\\int{|\\overrightarrow{H}|^2 dV}", "6dd7e8c3ff22e73a69771fc3f0b68409": "O|q_j\\rangle=O\\sum_k c_k(0) |e_k\\rangle = q_j |q_j\\rangle", "6dd7f3cbd4d6dcdb0f6f2297e52f2587": " C_d ", "6dd8348ffa91d22584b9c38e9b2d65fc": "\\textstyle -D_1\\frac{\\partial C_1}{\\partial y}", "6dd83adb8e30de279b70be8285486574": "\\delta \\vec {x}_{0i}", "6dd854c100674622bf8f8037e4b39e43": "\\mathbb{C}^n = \\bigoplus_{i = 1}^k X_i", "6dd8783e0160e603c3a627b8e4ccd398": "T = \\rho V_a^2 D^2 \\times f_r (\\frac {ND}{V_a})", "6dd8a2f7a411de0b9d76d1d2c04e773b": "\nY_\\ell^m Y_{\\ell'}^{m'} = \\sum_{\\ell''}(-1)^{m'} c^{\\ell''}(\\ell,-m,\\ell',m',) Y_{\\ell''}^{m+m'},\n", "6dd8a9a5c71e6e37f5eae6b2f97ea38d": "G_e(f)=f\\oplus b-f", "6dd9392f51410e9aa894cb416bc2b4bd": "q^{\\mathrm I}", "6dd999e94b7cb26e1030a4dd648ff16d": "\\begin{bmatrix}\n1 & \\lambda_1 & \\lambda_1^2 & \\cdots & \\lambda_1^{n-1} \\\\\n0 & \\lambda_2-\\lambda_1 & \\lambda_2^2-\\lambda_1^2 & \\cdots & \\lambda_2^{n-1}-\\lambda_1^{n-1} \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n1 & \\lambda_n & \\lambda_n^2 & \\cdots & \\lambda_n^{n-1}\n\\end{bmatrix}=\n\\begin{bmatrix}\n\\lambda_1^{n+m} \\\\\n\\lambda_2^{n+m}-\\lambda_1^{n+m} \\\\\n\\vdots \\\\\n\\lambda_n^{n+m}\n\\end{bmatrix}", "6dda35625bdbd136a18b7ea4f31d1c1c": " \\pi = \\sum_{i=0}^\\infty \\frac{1}{16^i} \\left( \\frac{4}{8i + 1} - \\frac{2}{8i + 4} - \\frac{1}{8i + 5} - \\frac{1}{8i + 6}\\right)", "6dda8a3dcd004bc4334d9bdafe02a699": " (-1)^{(\\left|x^{i}\\right|+1)(\\left|x^{k}\\right|+1)}\\pi^{i\\ell}\\partial_{\\ell}\\pi^{jk} + {\\rm cyclic}(i,j,k) = 0 ", "6ddb0d9f4ed5eee99bcc0f3b70f106d6": "\\lim_{x\\to a} f'(x) = {+\\infty}\\text{,}", "6ddb0ddae03684dbc544329f2d2984c2": " Z:=\\sum_{\\Gamma}w(\\Gamma)\\left[ \\sum_{j_f,i_e}\\prod_f A_f(j_f) \\prod_e A_e(j_f,i_e)\\prod_v A_v(j_f,i_e) \\right]", "6ddb33429517039558c532af4eb658f8": "BM =\\frac{I}{V} \\ ", "6ddbb034822549d8b019770e676a762b": "p = \\frac{x^3 - y^3}{x - y},\\ x = y + 1,\\ y>0", "6ddbf17ce040e5219c4803fb0916020f": "\\left[ \\frac{ \\text{length}^2 }{ \\text{time} } \\right]", "6ddc0632b43c1884be3e1fa09dbe7a67": " 6{q-1 \\over 7-5q} \\text{ for } q < {7 \\over 5} ", "6ddc26e279fa3336a5267c8d4a2dbcd4": "\nx(i) = (x(i-S)-x(i-R)-cy(i-1))\\ \\bmod\\ M\n", "6ddc365ac9a27cb6031b41856d59a550": "O(k \\log k 2^{O(\\log^*k)})", "6ddc45f3e82b29e1b269d7cdb827bf31": "\\eta_V\\colon V \\to V^*.", "6ddc64f0911f3c80afbf24234f6d821f": "mac \\leftarrow AXUHash(k_{AXU},lastBlock,enc) \\in W^4", "6ddcf85e51d747dad561e443c7bf24b9": "\ng_{jk}(\\theta)\n=\n\\int_X\n \\frac{\\partial \\log p(x,\\theta)}{\\partial \\theta_j}\n \\frac{\\partial \\log p(x,\\theta)}{\\partial \\theta_k}\n p(x,\\theta) \\, dx.\n", "6ddd09364afb3d5342a750f2ef2560c2": "\n\\ell = \\ln L^* = \\sum_{i = 1}^n \\left[ Y_i \\ln P_i + \\left( 1 - Y_i \\right)\\ln \\left( 1 - P_i \\right) \\right]\n", "6ddd4ce08f1f9941dabe8a805b74a08c": "\\begin{align}\n \\overbrace{(2051 - 2052)^2}^{\\text{This is }a^2.}\\ +\\ \\overbrace{2(2051 - 2052)(2052 - 2050)}^{\\text{This is }2ab.}\\ +\\ \\overbrace{(2052 - 2050)^2}^{\\text{This is }b^2.} \\\\\n (2053 - 2052)^2\\ +\\ 2(2053 - 2052)(2052 - 2050)\\ +\\ (2052 - 2050)^2 \\\\\n (2055 - 2052)^2\\ +\\ 2(2055 - 2052)(2052 - 2050)\\ +\\ (2052 - 2050)^2 \\\\\n (2050 - 2052)^2\\ +\\ 2(2050 - 2052)(2052 - 2050)\\ +\\ (2052 - 2050)^2 \\\\\n (2051 - 2052)^2\\ +\\ \\underbrace{2(2051 - 2052)(2052 - 2050)}_{\\begin{smallmatrix} \\text{The sum of the entries in this} \\\\ \\text{middle column must be 0.} \\end{smallmatrix}}\\ +\\ (2052 - 2050)^2\n\\end{align}", "6ddd90f74adb0be515f0b267f4f2310d": "L \\in \\mathsf{S}_2^P - \\mathsf{SIZE}(n^k)", "6ddda292d5a3569abee799fb516e9e9c": "GL(n,\\mathbb R) ", "6dde591255047c402d73c1d1cfc0a27a": "H_{\\mathbf{k}} = \\frac{p^2}{2m} + \\frac{\\hbar \\mathbf{k}\\cdot\\mathbf{p}}{m} + \\frac{\\hbar^2 k^2}{2m} + V + \\frac{1}{4 m^2 c^2} (\\vec \\sigma \\times \\nabla V)\\cdot (\\hbar\\mathbf{k}+\\mathbf{p})", "6ddf9be7cf675f3a872687406b1beebb": "\\mathbf{(A \\times B) \\cdot (A \\times B) = |A \\times B|^2 = (A \\cdot A) (B \\cdot B)-(A \\cdot B)^2 } ", "6ddfab0f1448dcbe2c49a391cee5743d": "Z_{IN} = Z_0 \\frac{Z_L + j Z_0 \\tan (\\beta l)}{Z_0 + j Z_L \\tan (\\beta l)}\\,", "6ddfd46d77f653c43948122eb1645973": "\n \\begin{align}\n \\mathcal{M} & = \\mathcal{M}^K + \\frac{\\mathcal{B}}{1+\\nu}\\,q + D \\nabla^2 \\Phi ~;~~ \\mathcal{M}^K := -D\\nabla^2 w^K \\\\\n Q_1^K & = -D\\frac{\\partial }{\\partial x_1}\\left(\\nabla^2 w^K\\right) ~,~~\n Q_2^K = -D\\frac{\\partial }{\\partial x_2}\\left(\\nabla^2 w^K\\right) \\\\\n \\Omega & = \\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1} ~,~~ \\nabla^2 \\Omega = c^2\\Omega \\,.\n \\end{align}\n", "6ddfd8b088810e33365d195988ad000c": "\\{x\\in F;\\,a\\in x\\},", "6de022c013e3bbec2275c72ff2f3cdbd": " C_\\text{new} = (1-\\lambda) C+\\lambda I\\,", "6de07fcec074f5d44b3673a6b08fc320": "r = \\frac{m^b(x)k_R}{m^b(x)(k_R+k_G+k_B)} = \\frac{k_R}{k_R+k_G+k_B}", "6de09fcbf5ebd669f64b2401e89bfddf": " D = \\frac{\\mu_q \\, k_B T}{q} ", "6de0d3cc4295589ef839abea6552b7fd": "R \\cap L", "6de19833c9cf9fd45338c8425bb920aa": " max(K-S,0) ", "6de1ac75bfd59b247c775adf52b0010d": "A_g(z) = c_0 + \\sum_{k=1}^{N} \\frac{c_k}{z+k}", "6de215d4cd7f8c9ca8c4c2e373479601": "x_U \\in \\mathcal F(U)", "6de2962b26165683c9303bc5f5c2f183": "\\mathbf{x}(1)", "6de2e71e654a8e34d4d6996e2cb45430": "T(\\bot)=\\bot,", "6de304e4637e1fcafe1ac6910628f108": "\\| f \\|_{L^{2} (E, \\gamma; \\mathbb{R})} = \\| j(f) \\|_{H}", "6de34407bf61efbff82c0ebac52234d9": "\\mu_{3,1}= \\mu_{3,1} - r= \\frac{14}{3}-5= \\frac{-1}{3}", "6de349e1add309a187f17020dac6c015": "i>3", "6de35a105d39014b7953af9059afc259": "\\max\\{|z_i|\\} - \\min\\{|z_i|\\}", "6de35bfc0ab00d5e29a9d356739fb0c1": " -2\\sqrt{\\eta_1\\eta_2} -\\frac12\\ln(-2\\eta_2)", "6de3d56aa950f953123013b386e63b45": "{\\delta}/{\\delta} t", "6de3ebe8e20e064d80a769dd013c685e": "\\alpha_{2}", "6de40c24b8f2354857786741de950969": "\\mathbf{x}_{t+1}=f(\\mathbf{x}_t, \\mathbf{a}_t)", "6de4170055b7fe35fef455ed7825e31c": "D_\\alpha(P(A)P(X)\\|Q(A)Q(X)) = D_\\alpha(P(A)\\|Q(A)) + D_\\alpha(P(X)\\|Q(X)).", "6de449ce800fde679da3c656f31ae7ba": " \\frac{1-\\hat p}{\\hat p \\, n}+\\frac{1-\\hat q}{\\hat q \\, m} ", "6de463aa15d401a003be6165ed24780b": " \\det \\begin{bmatrix} \n 0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & 1 \\\\\n d(AB)^2 & 0 & d(BC)^2 & d(BD)^2 & 1 \\\\\n d(AC)^2 & d(BC)^2 & 0 & d(CD)^2 & 1 \\\\\n d(AD)^2 & d(BD)^2 & d(CD)^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 1 & 0\n\\end{bmatrix} = 0, ", "6de4a2b8357a2690b2cf02f0c981ced9": "\\operatorname{Gal}(\\overline{\\mathbf{F}}_q / \\mathbf{F}_q)", "6de54246965aa5858b0af8ae8fb30c61": "|B_q(y,e) \\cap C| \\ge Vol_q(y,e) {{|C|} \\over {q^n}} = {{|C|Vol_q(0,e)} \\over {q^n}}", "6de59736175e1a908a126e44d83b5085": "O(n \\log n)", "6de5a52b4099f728475fb6411638ede7": " \\hat{f}: \\hat{R}\\to\\hat{S}. ", "6de5c91898829de67368f8b7a1be78d0": "\n -N\\frac{\\partial^2w(x,t)}{\\partial x^2}=\\delta(x-vt)P-\\delta(x-vt)\\,m\\frac{\\mbox{d}^2w(vt,t)}{\\mbox{d}t^2}\\ .\n", "6de675ff5dd2ee9d579ac62248b52702": " W' + E_2 \\overset{a_2}\\underset{d_2}\\rightleftharpoons W'E_2 \\overset{k_2}\\rightarrow W + E_2", "6de68e415ef7acd2507053711a531dca": "c_\\mathrm d = \\dfrac{2 F_\\mathrm d}{\\rho v^2 A}\\ = c_\\mathrm p + c_\\mathrm f = \\underbrace{ \\dfrac{1}{\\rho v^2 A}\\ \\textstyle \\int\\limits_{S} (p-p_o). \\hat n. \\hat i dA }_{ c_\\mathrm p }+ \\underbrace{ \\dfrac{1}{\\rho v^2 A}\\ \\textstyle \\int\\limits_{S} T_w . \\hat t. \\hat i dA }_{ c_\\mathrm f} ", "6de6c49b61fc80c431d0f21495ca76e4": "|\\mathbb{E}(\\theta-\\hat{\\theta})|<\\delta", "6de6f09e7d799cf71f6d3dcc22477f1c": "+, -, \\pm, \\mp, \\dotplus \\!", "6de79d1d96df5eb0ef464fc402bffa97": "\\boldsymbol\\tau = (-\\sum_{i=1}^n m_i[\\Delta r_i]^2)\\boldsymbol\\alpha + \\boldsymbol\\omega\\times(-\\sum_{i=1}^n m_i [\\Delta r_i]^2)\\boldsymbol\\omega", "6de7c9233d38ed01a6b74f2e3e1e6bec": "(Z,W) = \\operatorname{NW}(X,Y)", "6de7ec79b4822282b6c84204597af78a": " \\bar{H}_n^{(b)}=-1+\\frac{1}{2^b}-\\frac{1}{3^b}+\\cdots", "6de816c91e34ef4bb9d0d9344809c9aa": "\\mathcal{M}\\vDash \\forall x ( x = m \\leftrightarrow \\phi(x)).", "6de878d5b8fce4e8ff788fb4093b0c5e": " F(x) = \\mathrm{constant}\\cdot x^m. ", "6de887eea1c14c47188ae8faa45b04fb": "\n\\prod_{i=1}^{n}\\left(1-a_{i}\\bar{a}_{i}\\right)\\prod_{i=1}^{n}\\left(1-b_{i}\\bar{b}_{i}\\right)=\\left(1-\\sum_{i=1}^{n}a_{i}\\bar{a}_{i}+\\sum_{i0\\} =\\frac{\\exp{{\\sum_{k=1}^x (\\beta_n} - {\\tau_{ki}})}}{1+ \\sum_{x=1}^m \\exp{{\\sum_{k=1}^x (\\beta_n} - {\\tau_{ki}})}} ;\n", "6de9da0f9c8cfff3b26a09d08620b41e": " f \\in \\mathcal{H}", "6de9f1ddee5053edbf206254c60074ef": "\\mathbb Z \\rightarrow \\mathbb Z [x, y, z] / \\langle y^3-x^4+z \\rangle ", "6dea5a1fe715ada453c552e7ffae30a8": "f(V_i) \\sub W_{i}", "6dea9b91a7942d22cb20b6b0a84ae8dd": "R = {V\\over I}, \\qquad G = {I\\over V}, \\qquad G = \\frac{1}{R}", "6deab11fcfff7622b3933b7a0ad99083": "y=ax^2+bx+c ", "6deacc4039bbb54fe339613010f5fe00": "f'(v)=f'(0)", "6deaed16243f60dec46f4104c3d0c9d7": "X_{\\alpha}", "6deafc40c0b838b5e35d0a0fbc95b2fe": "\\frac{\\partial \\xi}{\\partial t} = -\\frac{x}{4 t^{3/2}} = -\\frac{\\xi}{2t}", "6deb119b06155cdf72c71c1a80fdb586": "A f(x) = \\theta(\\mu - x) f'(x) + \\frac{\\sigma^{2}}{2} f''(x).", "6deb76422bb8f94e5c3c2f0994a2dd09": " \\mathbf{A}'\\cdot\\mathbf{B}' = {A'}^0 {B'}^0 - {A'}^1 {B'}^1 - {A'}^2 {B'}^2 - {A'}^3 {B'}^3 = C' ", "6deb8203b8b3377eb8d521ca3354f6d3": " H(s) = \\frac{d}{s} + c(sI-A)^{-1}b \\quad \\quad", "6debf65e1b715c7fc56bf0dbe282d25d": "L \\to \\frac{R}{R'} \\,L", "6dec1e5fca9803c157841fc53e730cca": "\\ltimes{\\Bbb R}", "6dec39eb9f690d419d2ff07b283ff527": " \\tan \\delta_m = \\frac{\\mu''} {\\mu'} ", "6dec9313efe4e483d5b25a757a32a2e0": " p \\notin \\{p_1,\\ldots, p_r\\}.", "6decbfd5160aeb86871d52755d2775fa": "\\mathbf{f}\\mapsto \\mathbf{f}\\cdot R = \\left( R_1^i \\mathbf{e}_i, \\dots, R_N^i\\mathbf{e}_i\\right)", "6ded1512cce372d757fdad1aac93722b": "f(t_n,y_n)", "6ded78dffe764ee08418549b6b733ae9": "\\forall x \\forall y[x=y \\rightarrow \\forall P(Px \\leftrightarrow Py)]", "6ded9927e9863f2d063f54943df17997": "V(\\mathbf{x}) = - \\int_{\\mathbb{R}^3} \\frac{G}{|\\mathbf{x}-\\mathbf{r}|}\\ dm(\\mathbf{r}).", "6dedaa31b442070770cb644a7797087e": "1-p_B", "6dee65b4d613ff745d86cbb894289929": " \\left( B_\\text{n} \\right) ", "6dee65f750eb63c79a41a6829f9fb510": "H_{n,t}", "6deee7a6e00f4a2f7103e2176e7aa414": "\\underline{u}_1 \\neq \\underline{u}_2", "6def006149ae82d84fce6d32f82858e3": "O(\\sqrt{p}).", "6def03a35829c0066852e9f496ac576a": "U_1 - U_2 \\geq 0", "6def24947be247f32d113cfa7d3edd53": " PAQ = LU, \\, ", "6def973132e6b27f5ee251eec68e15b7": " F + 101 = \\begin{bmatrix}\n 1 & 0 & 1 \\\\\n 0 & 1 & 1 \\\\\n 0 & 0 & 0\n\\end{bmatrix} = F^{(1,3)_R(1,3)_C}=F^{(1,3,2)_R(1,3,2)_C}.", "6defb8bdb52006351725d46eea4cc063": " (\\lambda p.(\\lambda q.q\\ p)\\ \\operatorname{get-lambda}[q, q\\ p\\ f = (p\\ f)\\ (p\\ f)])\\ \\lambda f.\\lambda x.f\\ (x\\ x) ", "6defbe48e273a9febbaa41e8e753ce45": " -k^a \\nabla^b k_a = \\kappa k^b", "6df051b7709a980ac26b25ee3a7c225c": "\\cos(\\theta_i-\\theta_j)=\\cos(\\theta_j-\\theta_i)\\,", "6df0cf799a2929266c2aa55b7d1696c9": "0.a_1a_2a_3\\ldots", "6df11251998d2cb3414ce30cea5e2dc2": " \\frac{\\mathrm{d}P}{\\mathrm{d}t} ", "6df11c4945a47dd7f0cd362b7e98dd2b": "\\ (\\Delta x, \\Delta y) = \\arg \\max_{(x, y)}\\{r\\}", "6df11d38db92daa4fa6c07b62cf3b68c": "p_A(x,y)", "6df1332062e9dc821faf3ae00b9e2dad": "2.1345", "6df163f333a5d98bcaa7830f3f69ac54": "\n\\left\\lbrace \\begin{array}{cl}\nU(r) &= a + \\displaystyle{\\frac{1}{1+ 4t/Re} e^{-r^2/(1+ 4t/Re)}}, \\\\\nV(r) &= 0, \\\\\nW(r) &= q \\displaystyle{\\frac{1-e^{-r^2/(1+ 4t/Re)}}{r}},\n\\end{array}\\right.\n", "6df1722a3010f2ffa80d90aef48e76a0": " \\vartheta(x) = \\sum_{p=2}^{x} \\ln (p) ", "6df1c2c84160ea0f6bff0dbb055c325c": "\\scriptstyle{\\varkappa_a^a}", "6df1d13a0543a12a8199fed41c59c906": "E - e\\phi \\approx mc^2\\,,\\quad \\mathbf{p} \\approx m \\mathbf{v}\\,,", "6df20afd5be8c55bd23f404fc0b85555": "{{B}_{x}}^{\\prime }(v)=\\frac{f(v|x)}{F(v|x)}(v-{{B}_{x}}(v))", "6df21e63503fe3e32aa148a3728e0ee6": "A_o (k_1) = N\\ e^{-\\sigma^2 (k_1-k)^2 / 2} \\ , ", "6df2326cd96d1d3dd16e5e4851d2dd55": "P(x)Q(x) = 0", "6df2726a98eef5b3b4786aa789f22e59": "mod \\mathit{p}", "6df2ab831dccb6590640e4ef8a62df85": " \\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a}^{\\mathrm{T}} \\mathbf{b},", "6df2df3456a69c1bd0b65d74af2d2a6a": "v^{i'} = \\sigma {{\\partial x^{i'}} \\over {\\partial x^{i}}} v^i,", "6df393b2dc997352d9eac8068453d538": " c_X + c_Y = c \\,", "6df39c7effa40a773ff08fdb70aabb64": "\ne_i = y_i - f(x_i;\\hat{\\beta}),\n", "6df41695778e8fa94d230bd883665551": "\n{\\Vert \\mathbf{u} \\Vert}^2 {\\Vert \\mathbf{v} \\Vert}^2\n= ({\\mathbf{u} \\cdot \\mathbf{v}})^2 - (\\mathbf{u} \\wedge \\mathbf{v})^2\n", "6df4205fb3ffe5e3969bcee8d161bf3c": " \\mathbf{A}+\\mathbf{B}=\\mathbf{B}+\\mathbf{A} ", "6df435860c6b27b8fe6bfa6010b36855": "=O(n^2).", "6df43dce8aece1cfffad9ffb0b52b3ce": "\\Pr_{h \\in H}[h(f)=d|h(a)=c]\\,", "6df4406f70f7b64795420e25c5eaa839": "\\omega_s^{nl}\\equiv(-\\frac{1}{4\\zeta^2})^s\\frac{(n-s)!}{s!(n-l-2s)!}", "6df45777be34d7feecdd6418fc4c8396": " j: \\Sigma \\to M ", "6df47ed7535fdc313441f0d78e184a58": "L(P,t) = \\#(tP \\cap \\mathbb{Z}^d)", "6df4b51fad75569f5c5ca88d02c6334e": "E_1 \\geq E_2 \\geq E_3", "6df4bdae1cff72004feaac6ac4233b62": "\\mathrm{P}(u,v)=0, \\forall u,v : \\sqrt{u^2+v^2}>R,", "6df4d2610659a98f06717c83a16f40cf": "q_0 w_0 w_1 ...", "6df5032cd8b3a16839236b709d6e4e4a": "Z\\sim\\mathcal{N}\\left(\\mathbf{b}\\cdot\\boldsymbol\\mu, \\mathbf{b}^{\\rm T}\\boldsymbol\\Sigma \\mathbf{b}\\right)", "6df5050273ef9fbf1f15e1aaacbc1154": "t_g", "6df5168987376d93a30f715611aec341": "\\ln \\phi = \\int_0^P {\\frac{Z - 1}{p} dP} = Z - 1 - \\ln {(Z-B\\, P)} - \\frac{A^2}{B}\\, \\ln {(1+\\frac{B\\, P}{Z})}", "6df5aa4cce222eae3673d753516bcadf": " k = \\left(\\frac{k_\\mathrm{B}T}{h}\\right) \\mathrm{exp}\\left(\\frac{\\Delta S^\\ddagger}{R}\\right) \\mathrm{exp}\\left(-\\frac{\\Delta H^\\ddagger}{RT}\\right)", "6df5d9e799071842d4c937213ef98a49": "\\scriptstyle \\pi", "6df62918ea4e70190084b1d0bb118b33": "f'' = f + f' + (d(x) + d(x') - d(gs(x, x'))) \\ge 0", "6df66dac472b721813e6c0388d3bc00f": "~E_2 = m_2 \\frac{v_2^2}{2} + \\frac{m_1} {m_1+m_2} U_{21} = Const_2(t) ", "6df6b1896b065d3129a0478d4d8c46d4": "\n\\begin{align}\n\\left[\\frac{2}{p}\\right]_3 =1 &\\mbox{ if and only if } 3|b\\\\\n\\left[\\frac{3}{p}\\right]_3 =1 &\\mbox{ if and only if } 9|b; \\mbox{ or }9|(a\\pm b)\\\\\n\\left[\\frac{5}{p}\\right]_3 =1 &\\mbox{ if and only if } 15|b; \\mbox{ or }3|b \\mbox{ and }5|a; \\mbox{ or } 15|(a\\pm b); \\mbox{ or } 15|(2a\\pm b)\\\\\n\\left[\\frac{6}{p}\\right]_3 =1 &\\mbox{ if and only if } 9|b; \\mbox{ or }9|(a\\pm 2b)\\\\\n\\end{align}\n", "6df6d1da287b37d06d1e5674ea40782e": "|m|=1", "6df7213ad3324629253c2e8635a6fb1f": "T:c_0\\to c_0", "6df72b5c42ebac24c1756f4274deb026": "\\epsilon(x_0)=0", "6df76ea69b9b7a21f07e862771c1badc": "|p| \\leq \\pi.", "6df7fd975b6aaa93e6604a3db13d2018": "\\sigma=\\int {d \\sigma \\over d \\Omega} \\, d\\Omega.", "6df80872c557c90b7181e3a0d2766ad2": "d_2 = d_1 - \\sigma\\sqrt{t}", "6df8103084bc53dd4b30f2162414c60d": "\\|u-u_n\\| \\le \\frac{C}{c} \\inf_{v_n\\in V_n} \\|u-v_n\\|.", "6df8143c2c6953c744ca0a6e349676d0": " x = (x_1,x_2,\\dots,x_5) \\in R^5 ", "6df819416c883b65dbaf3417e53fc418": "Ax^n - By^m = C", "6df8babde0640628faff9a24227d7a35": "\n \\ln \\frac{p_{n,\\theta+h/\\sqrt{n}}}{p_{n,\\theta}} = \n h' \\Bigg(\\frac{1}{\\sqrt{n}} \\sum_{i=1}^n\\frac{\\partial \\ln f(x_i,\\theta)}{\\partial\\theta}\\Bigg) \\;-\\; \n \\frac12 h' \\Bigg( \\frac1n \\sum_{i=1}^n - \\frac{\\partial^2 \\ln f(x_i,\\theta)}{\\partial\\theta\\,\\partial\\theta'} \\Bigg) h \\;+\\;\n o_p(1).\n ", "6df8ed9feaf51807d8696bf6d3aecc20": "\\scriptstyle{a}", "6df8f00996de4b94d5a8eef4441328fe": "E = E_0 \\oplus E_1", "6df8f759aeecb42061935928063bedbf": "D(L) \\subseteq X", "6df91e0056d42598c001cfa81e01f958": "d(R)=x\\,", "6df95e9e2cc0e651add7583bd2553658": "\\left(m_a+m_X - m_b - m_Y\\right) c^2 = K_b + K_Y - K_a - K_X ", "6df96cd8770ecd05524965380d07186e": "\\gamma=2.55268-0.959456i", "6df98d92c9b68249566c24eae68f9dae": "\n\\hat{Y}_{HT} = \\sum_{i=1}^n \\pi_i ^{-1} Y_i,\n", "6df99e5c2fd9e5dcdb64a86d5e29816f": "C_{qs}^+", "6df9adfb507b456822ef7a38da38ea9f": "u_k x \\equiv \\pm t_k", "6df9bdb6b45e1741b16cedc980792004": "\\Gamma = \\nu e^{-E_\\mathrm{diff} /k_BT} \\qquad \\text{(eq. 1)}", "6df9e452de451f480b4b788588ad97eb": "N\\leq Min(\\alpha ,\\beta )-1", "6df9fdc006242e9cef148045fed18b17": "\\gamma _2", "6dfa3f4f0a867c097d8946e9c1f02985": "h_\\gamma [A] = \\mathcal{P} \\exp \\Big\\{ - \\int_{s_0}^{s_1} ds \\dot{\\gamma}^a A_a^i (\\gamma (s)) T_i \\Big\\} \\approx I - (s_k^a) A_a^i T_i", "6dfa4ee07ee7155f406b01271319790d": "\\left (\\left (\\textit{coin} \\rightarrow \\textit{choc} \\rightarrow \\textit{STO}P\\right ) \\Box \\left (\\textit{card} \\rightarrow \\textit{STOP}\\right )\\right ) \\setminus \\left\\{\\textit{coin}, card\\right\\}", "6dfa81a9ee264b1c870d531ed02c7b16": "M(x,y)", "6dfa8241043b972bb64d45e2118877e1": "\n D_x w^0_{,1111} + 2 D_{xy} w^0_{,1122} + D_y w^0_{,2222} = -q\n ", "6dfa8737a5280dec15992b8b50a828f3": "\\nabla^2 H + t H = - {\\lambda \\over 6} H^3.", "6dfab4737d6b9541d0797dd9ef3aad5c": "x < 1", "6dfab4fe7d91c2a1b020fd0b32119e0e": "(x, y) \\vee (z , w ) = (z , y)", "6dfba5a300a50bbec8b3596e735a0e3f": "\\mathbf{A} = (A_x,A_y)", "6dfbebc6cd78220d1773400446110001": "\n\\varepsilon = \\sqrt{\\left(I_n^f\\right)^2 - \\left(I_0\\right)^2}.\n", "6dfc272cfc6478faf174e5046fdb6410": "M^{ij} = x^i p^j - x^j p^i = L^{ij} ", "6dfc3572b064847a5a28ae350519eff6": "[a(\\vec{k}_1),a^\\dagger(\\vec{k}_2)]=(2\\pi)^3 2E \\delta(\\vec{k}_1-\\vec{k}_2).", "6dfc3fde3fb2549af8fb23aa3b798054": "D^{-1}", "6dfc549ac9f8fe431c7db41b78093dda": "\\frac{d}{dt}\\langle A(t)\\rangle = \\left\\langle\\frac{\\partial A(t)}{\\partial t}\\right\\rangle + \\frac{1}{i \\hbar}\\langle[A(t),H)]\\rangle", "6dfc5fe985ea5a17f81b064eaea1b789": "b^n=1", "6dfc63ca257c9259e2317040b4933fc4": "\\alpha \\equiv e^{\\frac{2}{3}\\pi i}", "6dfcbbfda44a9461c842bc00e841a8af": "(\\omega_1,\\omega)", "6dfccb1197c4fddd0189c1aca07f3688": "\\mathbb{Z}_{3}", "6dfcf3f2ce02b2f73aabe31f493e49cb": "\\text{Contribution Margin Ratio} = (\\text{Sales - Variable Costs}) / \\text{Sales}", "6dfd9a856eca317fb1b0e52ceb1e6d23": "\\mathit{V}_o", "6dfdc1337651f14be72eea6ced266acc": "F_d + F_c + F_b = 0 ", "6dfe1e76e2c20b957c913c3c01a4c540": "z \\le (x \\Rightarrow y)", "6dfe8545ae7fe5c7f65a47f76d2dc336": "S^1\\times S^3,", "6dfe9a9b979c2b526907babac7a7ca0c": " \\operatorname{perm}(A)=\\sum_{k=0}^{m-1} (-1)^{k}\\binom{n-m+k}{k}\\sigma_{n-m+k}.", "6dff00022c51fe08349188e9677b788c": "A^n = U \\operatorname{diag}(\\sigma_i^n) U'", "6dff0a0d0a77936d3bf67ff7fd90f87f": "\\left.X\\right.", "6dff33f6f2d826f52f496632088fdafe": "F(x)=\\sum_{n=1}^\\infty\\frac{1}{2^n}(x-x_n)^{1/3}.", "6dff530458b8859dbb58a58481c3e548": "e = \\sqrt{4 \\pi \\alpha \\hbar c}", "6dffa0e8c53de1e84f9f62310c527682": ".d_1d_2\\ldots d_n10", "6e005b60fd836f00e720dda0bb5cfaa4": "\\operatorname{Li}_2\\left(-\\frac{1}{3}\\right)-\\frac{1}{3}\\operatorname{Li}_2\\left(\\frac{1}{9}\\right)=-\\frac{{\\pi}^2}{18}+\\frac{1}{6}\\ln^23", "6e007345a78f718fb1db2256d00ee83b": "-0.3>{\\alpha}>-1.6", "6e0074cabe0c06751d6ca1d120609756": "|\\boldsymbol{F}|=k_\\text{e}{|q_1q_2|\\over r^2}", "6e00b925e1462015c93ab8047d1df75a": "M_{unit,1}= 132 \\quad ft^2", "6e00c6c0bd4bc7f36621ef3f9215c91c": " S= \\int {1\\over 2}(\\partial^\\mu A^\\nu \\partial_\\mu A_\\nu - \\lambda (\\partial_\\mu A^\\mu)^2)", "6e00eee3d35a6eee7acff4f7a4726b18": "E\\times_M E", "6e010ba4eb60e088fb8a3ddadc820497": "D_1, D_2 \\in \\mathrm{Div}^0(C)", "6e01249f9ee4c004af49a014a2e7025f": "dt\\,", "6e01261d96ee5d3bdb5c0eab89dc439f": " k^G=(a/(n+d))^{1/(1-a)} ", "6e012b5915f0354ae804e7f0ddc14191": "B\\vee a\\vee B", "6e015a1a5a3facda634ea37cfccf878d": "\\mathbf{p}=\\varepsilon_0\\alpha \\mathbf{E_{\\text{local}}}", "6e01e9566eb64a41ea35526b0ce87f93": "\\forall x \\forall y\\forall z\\;x \\vee (y \\wedge (x \\vee z)) = (x \\vee y) \\wedge (x \\vee z)", "6e0247aef9796f5916532981f0b22117": "C[P]", "6e026c11a5cd9541760edf2ff1025a42": "i > k", "6e02deb75d1fd0f16ffe41580580f5e9": "R(x_t,u_t)", "6e032463365cb735203c5631484855f4": "\\{q-w|q\\in C\\}", "6e03f3fbecf5ad18b39326fc3b6a8623": "(q_{0},w,Z)", "6e044b638ebe1ea318862162b0086893": "\n\\hat{\\boldsymbol\\varphi} \\cdot \\hat{\\boldsymbol\\varphi} = (-\\sin \\varphi )^{2} + (\\cos \\varphi)^{2} = 1\n", "6e047542ae1fb7e739917f4b5c021427": "Spin^c", "6e05676501924cda0f889561a484c7e5": "2\\frac{(b\\!-\\!c)e^{at}\\!-\\!(b\\!-\\!a)e^{ct}\\!+\\!(c\\!-\\!a)e^{bt}}\n{(b-a)(c-a)(b-c)t^2}", "6e05ab66c522505ca7347b2a71f4512f": "(A \\to \\lnot B) \\to (C \\to (A \\to \\lnot B))", "6e05b55a397ab196379445c9b3b0040f": " I = \\frac {I_{high} -I_{low}}{ C_{high} -C_{low} }(C-C_{low}) +I_{low} ", "6e05eb580fc6c1fe7e04359bb74d6a0b": "2\\sqrt{m + 3/8} - 1/(4\\sqrt{m})", "6e05ff171fb4cb5a9d16f6fc84e5ce8f": " = \\sum^\\infty_{k = 0} ({2\\pi}{\\overline{a_k} r^{2k}})(\\frac{1}{2{\\pi}i}\\int^{2\\pi}_0 \\frac{{f(re^{i\\vartheta})}}{(r e^{i\\vartheta})^{k+1}} {rie^{i\\vartheta}}) \\mathrm{d}\\vartheta", "6e066fb3c2393f53ac2fd8b64030f4ef": "\\tilde{a}_k = a_k^*", "6e0675513cac9f311b07c65c4d9dd303": " \\mu^{-1} ", "6e0715f798a63cb7df8584ae0f962986": " b_2 ", "6e0726fc6865726f53444d739b8b9166": "\\hat{M},\\hat{\\alpha}", "6e07339c8796a30aee78a31e1ba36cfd": "b\\times c=a", "6e0745018b52b81ff8fdc0f5742b029c": "\\frac{d[3]}{dt}=k_2[1][SO_2]^2", "6e0760a6ee3bd7a3d2b6a919262ec9a9": "\n \\begin{bmatrix}\n H & 0 \\\\\n 0 & H\n \\end{bmatrix}\n", "6e077fae0f7a51b42957d7b22a770a7b": " \\left([-1,1] + \\frac{1}{2}\\right)^2 -\\frac{1}{4} =\n \\left[-\\frac{1}{2}, \\frac{3}{2}\\right]^2 -\\frac{1}{4} = \\left[0, \\frac{9}{4}\\right] -\\frac{1}{4} = \\left[-\\frac{1}{4},2\\right]", "6e07ae6ca3c3a09b79696a6197ba6985": "[a < b] + [a = b] + [a > b] = 1. \\, ", "6e07e0e771bdb0c19274e89b677162e3": "L_1, L_2, ...", "6e0826ae52c017c224669a5dcd33ae3f": "(A \\cup B)^C = A^C \\cap B^C\\,\\!", "6e08306a7eb7ecdba535722ce1ab19ca": "w^z = e^{z \\log w} = e^{z(\\log r + i\\theta)}", "6e0874b4a7d9ce850e63c7d0d05e104d": "\\Pi_{2} = \\prod_{p>2}\\left(1-\\frac{1}{(p-1)^{2}}\\right) = 0.6601618...", "6e0888397872966c3597146ad41b94a6": "\\operatorname{distance}(ax+by+c=0, (x_0, y_0)) = \\frac{|ax_0+by_0+c|}{\\sqrt{a^2+b^2}}. ", "6e08cae2179b5def99702c29ad83a479": "-80", "6e08db6e08df464d9a28a3b30d78db37": "\\begin{align}H(X\\mid Y) & = \\log N - D_{\\mathrm{KL}}(P(X,Y) \\| P_U(X) P(Y) ) \\\\\n& = \\mathrm{(i)} \\,\\, \\log N - D_{\\mathrm{KL}}(P(X,Y) \\| P(X) P(Y) ) - D_{\\mathrm{KL}}(P(X) \\| P_U(X)) \\\\\n& = H(X) - I(X;Y) \\\\\n& = \\mathrm{(ii)} \\, \\log N - \\mathbb{E}_Y \\{ D_{\\mathrm{KL}}(P(X|Y) \\| P_U(X)) \\}\\end{align}", "6e08e2ca6c74ac7f5bca7b79689ca82d": "q = \\cos \\frac{\\alpha}{2} + \\vec{u} \\sin \\frac{\\alpha}{2}", "6e08e713d87587ba190887a6a590fdc5": "m_\\text{star}", "6e09496db98d09202291a99e51845e8d": "G_Q=2e^2/h", "6e096c67d8ffb52d909332ef936894e5": "\\frac{\\partial p}{\\partial z} = 0", "6e0981a1e5d4a75a509e0eefaadf46d4": " \\liminf_{n \\to \\infty} \\frac{\\phi(n) \\ln \\ln n}{n} = e^{-\\gamma}, ", "6e09ac663f334ce4f11f90dc71bf3096": " (I \\downarrow X \\setminus A)", "6e09e5ca2814fef209bfa90fb81f5a13": "e^{e^x} \\,", "6e0a77a710bcbb3df2d86f5ce02c70e5": "r_g \\ ", "6e0a9d6f12e7c8f1a38f4c22eb3702cc": "\\det(A) = \\Bigl((\\mathrm{tr}A)^3 - 3 \\mathrm{tr}A ~ \\mathrm{tr}(A^2) + 2 \\mathrm{tr}(A^3)\\Bigr)/6, \\,", "6e0aad1d905333f8b4af9f05d0a4f9ea": "f^{2}", "6e0aba03d973afb74c68b43d730ee006": " 4AB-E^2 <0 \\,", "6e0acb24fca188688561c98871419dbb": "\\scriptstyle 1 \\,+\\, 2\\cos(\\theta)", "6e0ad7556bdd70a4ff001fa8b3c3e6bf": "\\mathbf{E_T}=\\mathbf{E_0}e^{-\\kappa z}e^{i(kx-\\omega t)}", "6e0ae4b0ab564b30c9e18f80a07430b0": "q(x+y) = q(x) + q(y) + 2\\langle x,y \\rangle.", "6e0b4caada70dc39504a227754a6459c": "\nD = 2 \\sigma_0 / E \\epsilon'_n\n", "6e0c05647d37f43d17e963d0959e4275": "F_u(y) = P(X-u \\leq y | X>u) = \\frac{F(u+y)-F(u)}{1-F(u)} \\, ", "6e0c56ee8a8ec667430d5d78a297564a": "\nH_{nm} = H_{mn}\n\\,", "6e0c72f8a29029c0386571672d56b695": "w(A, B) = \\sum \\limits_{i \\in A, j \\in B} w_{ij}", "6e0c7f296575b0c611fc3171fb768ff7": "Df(a)", "6e0c867e849998b867f43a6349887e5a": " H_d(z) \\ ", "6e0cb076a38a6a52f00276db8bc8137c": "V_t(x,t) = A \\sin (\\omega t - kx) + \\rho A \\sin (\\omega t + kx).\\,", "6e0cbb1a980168a47e51ed07c88ec981": "Cr = 0.499 \\times R - 0.418 \\times G - 0.0813 \\times B + 128", "6e0cbf7a5114058b5c22247a97f3d447": "i\\leq n \\,", "6e0d5b49059b4ffb5ee12c5358305c47": "\\scriptstyle d=2,T=2 ", "6e0db27424394356865d44ae4e3e86d2": "E(z)", "6e0de5d43c8db222d43775a42d9b6b9c": "\\Delta G_{mix}\\,", "6e0e3b28216903b736776d46ec5b1f73": "\\mathbf{x}^{\\rm T}(\\mathbf{A} + \\mathbf{A}^{\\rm T})", "6e0e9d7d564cf711b347f117969cdd7a": "x^2 + x - 1 = 0\\,.", "6e0ea06bf38074a4bf97f4e1778e8817": "\\mathbf U(\\mathbf x,t) = \\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad \\text{or}\\qquad U_J = \\delta_{Ji}x_i - X_J \\,\\!", "6e0ece4e91f35f5094e1122eafc55148": "w(c,d),\\; c, d\\in\\Sigma\\cup\\{'-'\\}", "6e0ee4b07e0a236e30580f486d098d12": "c_1=\\sqrt[3]{3\\sqrt{33}+17}", "6e0f16b4163f67b9e5b447a6c0c107d7": "\\mathbf{c} \\cdot \\mathbf{x}", "6e0f37d0903779fa5f2dc9b295f4fba9": "\\frac1q = \\frac{a_1}{q(q+a_1)} + \\frac{a_2}{(q+a_1)(q+a_1+a_2)} + \\dots + \\frac{a_{n-1}}{q+a_1+\\dots+a_{n-2})(q+a_1+\\dots+a_{n-1})} + \\frac{a_n}{a_n(q+a_1+\\dots+a_{n-1})}", "6e0f40129c717cb8f016b6c7d9c9294e": "\\begin{align}\n & \\operatorname{E}[X] = e^{\\mu + \\tfrac{1}{2}\\sigma^2}, \\\\\n & \\operatorname{Var}[X] = (e^{\\sigma^2} - 1) e^{2\\mu + \\sigma^2} = (e^{\\sigma^2} - 1)(\\operatorname{E}[X])^2\\\\\n & \\operatorname{s.d.}[X] = \\sqrt{\\operatorname{Var}[X]} = e^{\\mu + \\tfrac{1}{2}\\sigma^2}\\sqrt{e^{\\sigma^2} - 1}.\n \\end{align}", "6e0f62c4cb783d3d9305b5da91add20a": " r_{ij} ", "6e0f6f4fdfef174a38af9ff919042b91": " F\\colon D^n \\to D^n \\mbox{ with } F(rx) = rf(x) \\mbox{ for all } r \\in [0,1] \\mbox{ and } x \\in S^{n-1}", "6e0fbb800f96cd0bb3699a1d41151231": "(\\sigma,\\theta)", "6e0fd8f769de7bded882f66ee4c8ad07": "\\mathrm{d}S_\\theta=r\\,\\sin\\theta\\,\\mathrm{d}\\varphi\\,\\mathrm{d}r.", "6e10782fbe30d6203ca0c2abeb797a7d": "D(i,j) \\ne D(j,i)", "6e108e59493ab65e031ad7feaea8762e": "g~", "6e109c87e7497bcc0b76c73628f8b4c5": "\\hat{A_{l}}, \\hat{A}^{\\dagger}_{l}", "6e109cf744649ecbe61565845b13dce5": "\n\\lim _ {N\\rightarrow \\infty} \\frac 1N \\sum_{n=0}^{N-1} x_n^2.\n", "6e10a7a4a8d48b2e3fe1c3e4d19c90b7": " \\ \\mbox{Adopters} = \\int_{0} ^{t} \\mbox{New adopters }\\,dt ", "6e10bc2c4ea880bd16e4f5a07f246db5": "|(\\phi f)^\\wedge(\\xi)| \\le C_N(1+|\\xi|)^{-N}\\quad\\mbox{for all }\\ \\xi\\in\\Gamma", "6e10cae9abc250445890a3fa0c99f904": "|\\alpha\\rangle=e^{\\alpha \\hat a^\\dagger - \\alpha^*\\hat a}|0\\rangle = D(\\alpha)|0\\rangle", "6e112ae12d304205202d82d2830f6ff3": "G = g_0, \\cdots, g_{2^n-1}", "6e11377d3585a78750de6c64f8d9c1bd": "\\frac{k}{i} = k\\frac{1}{i} = ki^{-1} = k(-i) = -(ki) = -(j) = -j", "6e114134c7a4fb7c0bcac81a14847a20": "W_{CO}", "6e114b8bfc2decdae8132789720d8b1a": "N_2 > N_1 ", "6e120389c5a55fa519bd5b854111cb27": "\\displaystyle{Mp(2,\\mathbf R)=\\{(g,G): \\,G(Z)^2=m(g,Z)\\}}", "6e124c11a86d851a4a43b38791b0849e": " \\vec{k}", "6e12621deaaa7d984118789f44b2f050": "\n\\overline{E}_\\mathrm{dip-dip} = -\\frac{2 kT}{3R_{AB}^6 }|\\mu^A|^2|\\mu^B|^2.\n", "6e12cb8cf16a396442893ab8c8c43373": "\\overline{x}^{\\,*} = \\frac{\\sum_{i=1}^N w_i x_i}{\\sum_{i=1}^N w_i}", "6e12dd5b7b0630ef349004df6e3f05cf": " N=2 ", "6e1342eb2b0c507d084fecf0ffa14b98": " \\pi = \\frac{4}{1} - \\frac{4}{3} + \\frac{4}{5} - \\frac{4}{7} + \\frac{4}{9} - \\frac{4}{11} + \\frac{4}{13} - \\cdots", "6e1368d7bf597a99ed20685bb201fe7b": "\\bar{\\boldsymbol{F}}:=J^{-1/3}\\boldsymbol{F}", "6e13a21498058c32181dab400a751f75": "\\textstyle c = \\left(u, v\\right) \\in \\mathcal{C}", "6e1435b12b1c571e9078467e33681bc7": "\\scriptstyle \\eta_t ", "6e14387aba6096f330d65a9da1485fa8": "\\,x\\in\\Sigma^*", "6e143b1e005c0e8d6d9487200630955e": "\\cfrac{\\cfrac{stC \\qquad \\overline{s} tD}{tCD} \\, \\operatorname{var}(s) \\qquad \\overline{t} E}{CDE} \\, \\operatorname{var}(t) \\Rightarrow \n\\cfrac{\\cfrac{stC \\qquad \\overline{t} E}{sCE}\\, \\operatorname{var}(t) \\qquad \\cfrac{\\overline{t} E \\qquad \\overline{s} tD}{\\overline{s}DE}\\, \\operatorname{var}(t)}{CDE} \\, \\operatorname{var}(s)", "6e146bcf204ef86a3a5f47ad9a98b7d6": "(\\exist{Y}{\\in}{p}:{x}{\\in}{Y})\\and(\\forall{Y_{1},Y_{2}}{\\in}{p}:Y_{1}\\ne Y_{2}\\rarr ({x}{\\notin}{Y_{1}}\\or{x}{\\notin}{Y_{2}})).", "6e14743f5885f1b6d9dba8b0d1a8e07b": "\\frac{\\Delta E_i}{T} = \\ln(p_\\text{i=on}) - \\ln(1 - p_\\text{i=on})", "6e14a565deeae9abffc19001632b9606": "\\Delta p=\\frac{2\\sigma_{s}}{R_{s}}", "6e15dbd26c76d88ad8ef8ade9b2b4cc9": "\n\\overline{Z}(x) = a\\left[\\left(b-1\\right)\\overline{x}^3-b\\overline{x}^2+\\overline{x}\\right]\n", "6e15df026a8f31c998871b9c8140d3aa": "C=\\{e_1,\\ldots,e_n\\}", "6e162e14188fb4caf1085c0f2387e891": "V \\approx U.", "6e1689b283b52674841c166cf0c9cfd4": "\\psi(x)=\\sum_{i=1}^p \\psi^{(i)}(x)", "6e168bd22c158a8b79e79ea706ffc0a2": "I_A \\otimes \\sigma_B \\geq \\rho_{AB}~.", "6e16912845b294891b50afec79a1c7d2": "{\\rm Tr}\\, e^{A+B}\\leq {\\rm Tr}\\, e^A e^B.", "6e171443c3782eab54be07cbebf8988a": "\\frac{\\Delta P}{L} = \\frac{150\\mu (1-s)^2 u_0}{s^3 d_p^2} + \\frac{1.75 (1-s) \\rho u_0^2}{s^3 d_p}", "6e17288d836fb2e178d3b4988b86d89a": "\\mathrm{If}\\ %B =2 \\frac{f_H-f_L}{f_H+f_L} = p%,\\ B = \\frac{200+p}{200-p} ", "6e172a9fd264a188856ed959e7171dde": "A_j \\rightarrow \\delta_1 | \\ldots | \\delta_k", "6e173450cbdea0a35e0138f510b4802c": "\\|I^\\alpha f\\|_1 \\le \\frac{|b-a|^{\\operatorname{re}(\\alpha)}}{\\operatorname{re}(\\alpha)|\\Gamma(\\alpha)|}\\|f\\|_1.", "6e175eac85a59e9a97ff5765204aa4e6": "\\omega_g=L_{g^{-1}}^*\\omega_e", "6e1772350675054b65efd2740900bda3": "3\\;2/m", "6e179dfee8ec7a0ac9532fcc33000ec0": "2.441^{+0.088}_{-0.092}\\times10^{-9}", "6e17a0b4fdfe2c0135751f32bd80f9d8": "\\int_1^2 \\sum _x f(x) dx=0 ", "6e17a6af931fe626ae87567c479da53d": " \\| \\mathbf{e}^{T} \\, \\mathbf{Y} \\| ", "6e17e024d2d439c1395980f5f95a2339": " R^* \\times r = 1 \\times r ", "6e181fbf2655c42ab860f429ce90c0ef": "\\tau_n(N)", "6e1822f4389eb315d0d1997e1256d770": "F_T", "6e1836a3f5a183c968aefcdf4c6dcff9": " x \\mapsto \\{\\chi \\mapsto \\chi(x) \\}\\mbox{ i.e. } x(\\chi):=\\chi(x).", "6e183b2e91e051897cb273cc72107cc4": "\\lim_{t\\to\\infty}{1\\over t} \\int_0^t f(x(s))\\,ds = {1\\over \\mu(X)} \\int_X f(x)\\,\\mu(dx)", "6e187ed5a70f95eb60751fb662512018": "\\frac{\\partial{H_z}}{\\partial{y}} = \\varepsilon\\frac{\\partial{E_x}}{\\partial{t}}", "6e18b5fcc24047656b5548c4c4b82534": "v\\not=0", "6e18e12cdbffe41602afe15961c07cd7": "I_{sp} g_\\mathrm{n} = v_{e}", "6e18faa83f2d19478e9f66c060fdf984": "\\left|G_n(z)\\cdot \\frac{G_n(z)^2}{n^3} \\right|<(0.02)\\frac{1}{n^3}=C\\beta_n", "6e191dd0ee6cb8d6e90e85def31dcc58": "\\forall x Tx", "6e195ccefef246fe219bdaca2a467cc0": "\\tilde H", "6e1968e68b7adfa4a08a38822722b472": "\n Y=\\cfrac{1.99-a/W\\,(1-a/W)(2.15-3.93a/W+2.7(a/W)^{2})}{(1+2a/W)(1-a/W)^{3/2}} \\,.\n", "6e197bac125aca8eb523b6b2e3c4a38f": "\n\\lambda_i(\\vec{x},t) \\equiv \\frac {1} {2 t} \\ln {h_i(\\vec{x},t)}\n", "6e1a866675d9c9775be2fc62413ee8a6": "L_n(x)\\,", "6e1ada549710778ac18fe8561d414866": "\\Gamma^{a}_{bc}", "6e1ae5c5ba79cdda5c459d59f958cdf3": "\\Leftarrow\\,", "6e1b0615b5c5897fbf5e09529a430e32": "x-\\frac{1}{3}*x^3+\\frac{2}{15}*x^5-\\frac{17}{315}*x^7", "6e1b06bb6d96b860170ce491698a6108": "\\mathcal{S}_{y|x}", "6e1b3e64bd1f599a8d3f4144ee847209": "f(\\mathbf{x},t)", "6e1b4c8da9f190eb278b1bf063355277": " u(x) = \\bar{u} ", "6e1b9cc62ec5651ad3306fd61aaeb1d5": " y_c = e^{ -b\\frac{x}{2}} \\left [ C_1 \\sin{\\left ( \\sqrt{\\left | b^2-4c \\right |}\\frac{x}{2} \\right )} + C_2\\cos{\\left ( \\sqrt{\\left | b^2-4c \\right |}\\frac{x}{2} \\right )} \\right ] \\,\\!", "6e1bca332766611d8b4bfe4281c3ec3f": " 0 \\text{ for } q < {3 \\over 2} ", "6e1beac7044b719b55d657acd5d7527e": "s(x) = \\frac{x}{\\pi}, \\quad \\mathrm{for } -\\pi < x < \\pi,", "6e1c050547d926ac4fa5b0e79fdc0e39": "\\begin{matrix} {3 \\choose 3}{45 \\choose 2} \\end{matrix}", "6e1c13b5ec0f7e9fd819ab52e1054352": "1,0 \\Leftrightarrow 1", "6e1c3316c80e6c89f4f2613476dcc435": " \\mathbf{w}(t+1) = \\mathbf{w}(t) - \\nu \\frac {d} {d\\mathbf{w}} H_t(\\mathbf{w}) ", "6e1d2e467d84c09edfe447d4ee3be7ec": "B_x", "6e1d51ff109625e080e5d48f9f1bf074": " \\mathbf{v} = v^1\\mathbf{b}_1 + v^2\\mathbf{b}_2 + v^3\\mathbf{b}_3 = v_1\\mathbf{b}^1 + v_2\\mathbf{b}^2 + v_3\\mathbf{b}^3 ", "6e1d53fb8dbefd0cd95b90cb1ee2ec30": "f(X\\times \\mathbb{Z})=\\mathbb{Z}", "6e1db950a24991eda354158e19566621": "\n y_i = x'_i\\beta + \\varepsilon_i, \\,\n ", "6e1dcacd28c7971d2e43d800504acfd8": "\\mu(U)\\in\\ [0, \\infty] ", "6e1dea99fd4771d009812463b19fab42": "[-(l-1)/2,(l-1)/2]", "6e1e44d3a0793f37787e41cda885e413": "f^{*}\\mathrm dg = \\mathrm d(g \\circ f).", "6e1eb54bcc53e49443504d65ea8e4b58": " d_p ", "6e1f171b878f75b270c8496f8c59ebf3": "D\\subseteq\\Sigma^*", "6e1f5e99ea38112254fc3f55a9cd4862": "\\gamma_1=10", "6e1fad47891f4dedec97990ec014d34d": "\\delta_x = \\frac {q x^2} {24 E I}(6L^2 - 4L x + x^2)", "6e1fae78100c6f6ec7e73c09d5b09398": "(x, y) \\in \\Omega \\times \\mathbb{R}^m", "6e1fc13debea1df02af4ee35cc800245": "\\left(\\frac{\\sum_{k=1}^n X_k}{n}\\right)_{n\\in \\mathbb N}", "6e2009a650d09220098234df67d5e25b": "F=\\left\\{(x,\\ y):a \\le x \\le b, p(x) \\le y \\le q(x) \\right\\} \\,", "6e2067e3064f871e98b069b3156ed23a": "\nS \\ \\stackrel{\\mathrm{def}}{=}\\ 2 \\frac{\\mathrm{atan} \\ \\xi}{\\xi}\n", "6e2075ff3de701442ba4e369bc7cc4cc": "\n[b] =[b]\\cap [1,2]\n", "6e208034f9ad43b632cf8a36b1fbb66e": "A_1,A_2,\\dots,A_n,\\dots ", "6e212474383571a6b741521d2f03502f": "\n\\begin{array}{lcl}\nminimize: V(\\vec w, \\vec \\xi) = {1 \\over 2} \\vec w \\cdot \\vec w + C_{ontant} \\sum{\\xi_{i,j,k}} \\\\\ns.t. \\\\ \\begin{array}{lcl}\n \\forall \\xi_{i,j,k} \\geqq 0\\\\\n \\forall (c_i, c_j)\\in r_k^'\\\\\n \\vec w (\\Phi(q_1,c_i)-\\Phi(q_1,c_j)) \\geqq 1- \\xi_{i,j,1};\\\\\n ...\\\\\n \\vec w (\\Phi(q_n,c_i)-\\Phi(q_n,c_j)) \\geqq 1- \\xi_{i,j,n};\\\\\nwhere\\ k \\in \\left \\{ 1,2,...n \\right \\},\\ i,j \\in \\left \\{ 1,2,... \\right \\}.\\\\\n \\end{array}\n\\end{array}\n", "6e218eb0164882c23fc499e7b4895e14": "vLu = v \\left[\\frac {d}{dx} \\left( p(x) \\frac {du}{dx} \\right) + q(x) u \\right]. ", "6e2197f8a4723ef5a9e385178ab9bcd6": "8 a_3^2.\\,", "6e21d1bb7d24d3f2985328bab0b0eb21": "{\\it{p^{th}}}", "6e21db0fe346f983c9c83bec3c905189": "n_1 = 2", "6e21fc6878240fe42850bc6a56322909": "d(\\det(A)) = \\mathrm{tr}(\\mathrm{adj}(A) \\,dA).\\ \\square", "6e222ae2468257cebf8ebefb5e76272c": "\nV_b=\\frac{S \\rho_i}{S_b \\rho_b - S \\rho_b + S \\rho_i}\n", "6e22474bba2c110e29de4fce47df8bc2": "\n\\begin{align}\n\\mathbf{P}(\\tau, \\mu | \\mathbf{X}) & \\propto \\tau^{\\frac{n}{2} + \\alpha_0 - \\frac{1}{2}} \\exp \\left[-\\tau \\left( \\frac{1}{2} n s + \\beta_0 \\right) \\right] \\exp \\left[- \\frac{\\tau}{2} \\left( \\left(\\lambda_0 + n \\right) \\left(\\mu- \\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n} \\right)^2 + \\frac{\\lambda_0 n (\\bar{x} - \\mu_0 )^2}{\\lambda_0 +n} \\right) \\right]\\\\\n& \\propto \\tau^{\\frac{n}{2} + \\alpha_0 - \\frac{1}{2}} \\exp \\left[-\\tau \\left( \\frac{1}{2} n s + \\beta_0 + \\frac{\\lambda_0 n (x - \\mu_0 )^2}{2(\\lambda_0 +n)} \\right) \\right] \\exp \\left[- \\frac{\\tau}{2} \\left(\\lambda_0 + n \\right) \\left(\\mu- \\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n} \\right)^2 \\right]\n\\end{align}\n", "6e224db4e4dc9dffd12c2c25577a9c27": "C_n(x)=-x y_n(x)=-\\sqrt{\\frac{\\pi x}{2}} \\, Y_{n+\\frac{1}{2}}(x)", "6e22931baf51157ef748669f21d2ed6e": "\\{y_0,y_1,\\ldots\\}", "6e22b1135faa92f9fc0ce1ae69c7896c": "\\mathbb{X}=(x_1,\\ldots,x_{N})", "6e23381a07ece15efb72e40ebe1617e6": "r(\\varphi) = {r_0}\\sec(\\varphi-\\gamma). \\,", "6e2341eb7e08430ca0e5fdafc67c0d88": "f(x,y) = \\frac{x^2y}{x^4+y^2}", "6e23904c88237553ec177c76c0afdc7b": "r_k^2+S_k^2+W_k^2=1", "6e23913f0a27bb839ebc09fbf2f6ae9b": "\\sin(\\theta_3-\\theta_1)\\sin(\\theta_4-\\theta_2) = \\sin(\\theta_2-\\theta_1)\\sin(\\theta_4-\\theta_3) + \\sin(\\theta_4-\\theta_1)\\sin(\\theta_3-\\theta_2)\\,", "6e239238d9aea3cb9b77a763e817a96f": "E(R) = \\frac{1}{ (1 - \\epsilon ^2) }\n \\left( 1 - J_0^2(x) - J_1^2(x) + \\epsilon ^2 \\left[ 1 - J_0^2 (\\epsilon x) - J_1^2(\\epsilon x) \\right] - 4 \\epsilon \\int_0^x \\frac {J_1(t) J_1(\\epsilon t)}{t}\\,dt \\right)\n", "6e23a78c418c19cb1924d6c76b1d4ff6": " d_2 = \\frac{\\ln(S/K) + (r - q - \\sigma^2/2)\\tau}{\\sigma\\sqrt{\\tau}} = d_1 - \\sigma\\sqrt{\\tau} ", "6e23d19ad30b594bb7f2e6ca2393c13f": "S_{i,j}=\\log \\frac {p_i \\cdot M_{i,j}} {p_i \\cdot p_j}=\\log \\frac {M_{i,j}}{p_j}=\\log \\frac {observed\\;frequency} {expected\\;frequency}", "6e23e6cc956e77902f51cddc15c8ddc4": "C_p = 4a(1 - a)^2", "6e241243ecbb97c7c453499d29d98fd6": "E\\left(X\\right)=p", "6e2417afb29560607b23c48e6126d01e": " \\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n}}\\equiv{-1}\\pmod p \\!", "6e2433484e197b5136d992b24e613ef2": "S : P \\rightarrow V", "6e246910e7c7d1a2eff38aa3613a1834": "\\int_{\\mathbb{R}^{n}} \\prod_{i = 1}^{m} f_{i} (x) \\, \\mathrm{d} x \\leq \\prod_{i = 1}^{m} \\| f_{i} \\|_{p_{i}}.", "6e25333aafb0314b682805fdfe034acf": " \\phi_{l}\\,= \\left [\\left (1 + \\frac{2}{ Pe_{l} }\\right) \\frac{\\phi_{L}}{2} + \\left (1 - \\frac{2}{ Pe_{l} }\\right) \\frac{\\phi_{R}}{2}\\right ] \\,", "6e25620f53547804b13da2eda52fd542": "\\nabla_a T^b = \\nabla_a (T_c g^{bc}) = g^{bc} \\nabla_a T_c", "6e25a18bfd06fac66bd267ce630e2504": "\n = \\left(\\sum_{a \\in A_i} \\sigma^*_i(a)u_i(a_i, \\sigma^*_{-i})\\right) - u_i(\\sigma^*_i, \\sigma^*_{-i})\n", "6e25bf4abf4cb2ecd5ee300a50032ad2": "\\scriptstyle f: U \\rightarrow \\mathbb{V}", "6e25f2c1ec1c80b2abb3ca8ca3ad0d5b": "\\psi = 3x^2y-y^3", "6e2608f682c112a53afa312c60fb0447": " P_0 (T) ", "6e26229c0d20303368756e519243474b": "a^{a^{\\cdot^{\\cdot^{a^x}}}}", "6e26ad118a173670d9a3fd14b62bfba0": "\\delta = \\sqrt{\\frac{\\Delta H_v -RT}{V_m}}", "6e2725a46e7b2bc2cc36e46679154329": "\\frac{44,100 \\times 16 \\times 2 \\times 4,800}{8} = 846,720,000\\ \\mathrm{bytes} \\approx 847\\ \\mathrm{MB}", "6e272a9b76bb1afd4bc908d77974f09b": "b=+\\infty", "6e274ee107089742f6d02a71aac38307": "\\nabla f = \\frac{\\partial f}{\\partial x} \\mathbf{i} +\n\\frac{\\partial f}{\\partial y} \\mathbf{j} +\n\\frac{\\partial f}{\\partial z} \\mathbf{k}", "6e276540d36ee2b8cb5a9c503fca6e2b": "Z^M", "6e27876918aa3494be4c5a79602f48f2": "\\mathtt{let}\\ x = e_1\\ \\mathtt{in}\\ e_2", "6e27f2fe620c35c2cc9384eda9363660": "h_\\ell^{(1)}(kr)=(-i)^{\\ell+1}\\frac{e^{i k r}}{k r}+O(1/r^2)", "6e281da26f78ba0e8bbf63e1f5bb3609": "d\\,\\sin\\theta_\\text{min} = \\lambda", "6e282042e1843995c312486118f43ad6": "c^{2} d\\tau^{2} = c^{2} dt^{2} - \\frac{\\rho^{2}}{r^{2} + \\alpha^{2}} dr^{2} - \\rho^{2} d\\theta^{2}- \\left( r^{2} + \\alpha^{2} \\right) \\sin^{2}\\theta d\\phi^{2} ", "6e2839f7213ffd2670f1ce2fbbc5c04a": "(\\cosh x )'= \\sinh x = \\frac{e^x - e^{-x}}{2}", "6e285750457ec3b98bb6dd4af8c21f64": " \\mathbf{a} =\\left ( \\frac{\\mathrm{d}^2 r}{\\mathrm{d}t^2} - r\\omega^2\\right )\\bold{\\hat{e}}_r + \\left ( r \\alpha + 2 \\omega \\frac{\\mathrm{d}r}{{\\rm d}t} \\right )\\bold{\\hat{e}}_\\theta ", "6e28c73cdcdcf3f658116486e467863d": "A-n=\\{k\\in\\mathbf{N}\\mid k+n\\in A\\}.", "6e28cfdd116e21ea9217cdcedc2a25e3": "\n\\int_0^1 (1-r)^{\\alpha-1} r^{\\beta-1} \\; dr = B(\\alpha, \\beta) = \\dfrac{\\Gamma(\\alpha)\\,\\Gamma(\\beta)}{\\Gamma(\\alpha+\\beta)}\n", "6e28fa997e76d1fa3d488750c4f092ae": "\\sum_{j=k}^n\\binom nj(-1)^{n-j}\n=-\\sum_{j=0}^{k-1}\\binom nj(-1)^{n-j}\n=(-1)^{n-k}\\binom{n-1}{k-1}.", "6e293a340fdf01f84b68185379727146": "\\sigma (p)=E (p) \\epsilon (p)", "6e29cf7cfa8ae422eb4e9814c79286c4": "p\\!\\!\\!/ = \\gamma^\\mu p_\\mu \\,", "6e29d179b51d3bd551670b8aafca7d6e": "\n\\begin{align}\nU_n(x) & = \\frac{(x+\\sqrt{x^2-1})^{n+1} - (x-\\sqrt{x^2-1})^{n+1}}{2\\sqrt{x^2-1}} \\\\\n& = \\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{n+1}{2k+1} (x^2-1)^k x^{n-2k} \\\\\n& = x^n \\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{n+1}{2k+1} (1 - x^{-2})^k \\\\\n& =\\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{2k-(n+1)}{k}~(2x)^{n-2k} \\quad (n>0)\\\\\n& =\\sum_{k=0}^{\\lfloor n/2\\rfloor}(-1)^k \\binom{n-k}{k}~(2x)^{n-2k} \\quad (n>0)\\\\\n& = \\sum_{k=0}^{n}(-2)^{k} \\frac{(n+k+1)!} {(n-k)!(2k+1)!}(1 - x)^k \\quad (n>0)\\\\\n& = (n+1) \\, _2F_1\\left(-n,n+2; \\tfrac{3}{2}; \\tfrac{1}{2}\\left[1-x\\right] \\right)\n\\end{align}\n", "6e2a23b9e4b61d0a4e53b3cd0addba12": "x_n=\\sum_{k=1}^r a_k x_{n-k}+ e_n", "6e2a3a199f91be3135485cd6da276d38": "\\bar {u} \\frac{\\partial \\bar{T}}{\\partial x} + \\bar {v} \\frac{\\partial \\bar{T}}{\\partial y} = \\frac{\\partial}{\\partial y} \\left [(\\alpha + \\epsilon_H) \\frac{\\partial \\bar{T}}{\\partial y}\\right].", "6e2ac695b32456b032ea3df18d214722": "\\int \\cos (\\ln x)\\;dx = \\frac{x}{2}(\\sin (\\ln x) + \\cos (\\ln x))", "6e2acbef7be5cfb917b0931dee54a68e": "A\\models\\exists x\\phi(x) \\text{ iff there is an } a\\in A\\text{ such that }A\\models\\phi[a]", "6e2af87ed79de1b0ed0e2d650eb08c64": "s - \\frac{(s)_3}{3!} + \\frac{(s)_5}{5!} - \\frac{(s)_7}{7!} + \\cdots\\,", "6e2b3e935350835fae78e21285853f56": "d = \\sqrt{\\left(\\frac{b_1m-b_2m}{m^2+1}\\right)^2 + \\left(\\frac{b_2-b_1}{m^2+1}\\right)^2}\\,,", "6e2b705795bb2bf92c16c3aa5dc8e93e": " \\mathbf{F} = \\frac {d \\mathbf{P}} {d \\tau} = \\gamma(\\mathbf{u})\\left(\\frac{1}{c}\\frac{dE}{dt},\\frac{d\\mathbf{p}}{dt}\\right) = \\gamma(\\mathbf{u})(P/c,\\mathbf{f}) ", "6e2b8d684a28795d34510d842080bd0c": "\\frac{3\\alpha a}{2}", "6e2c290a62ce0401935e9a7f8c7cb316": "\\lambda_n = \\frac{\\lambda_\\text{u}}{2\\gamma^2n}\\left ( 1+\\frac{K^2}{2}+\\gamma^2\\theta^2 \\right )\\qquad (13)", "6e2c4d6f09639b974a60080d549b80c3": "x^i(t)=t\\frac{dx^i}{dt}(0)+\\frac{t^2}{2}\\frac{d^2x^i}{dt^2}.", "6e2c61e3891e5e4499b9f04c50009a25": "-v \\cot v", "6e2c8b46891e5d0f82c09c567247b5e8": "\nh_{\\omega}(t) = h(t) e^{j \\omega t} \n", "6e2cbd623aa7bdcc1cf40f3da9d0b8b1": "\\ X", "6e2cd761b586c4b02418763e6f0a7095": "\n\\hat{\\mathbf{x}}_{0|0}=E\\bigl[\\mathbf{x}(t_0)\\bigr], \\mathbf{P}_{0|0}=Var\\bigl[\\mathbf{x}(t_0)\\bigr]\n", "6e2cea9df92a26d165cf80d0d96f6110": "p \\cdot (\\omega + y_j + \\Sigma_h y_h^*) \\leq r = p \\cdot (\\omega + y_j^* + \\Sigma_h y_h^*)", "6e2d22151d1a8f44300184121d186949": "\\begin{align}\nL \\colon [a, b] \\times TX & \\to \\mathbb{R} \\\\\n (t, x, v) & \\mapsto L(t, x, v).\n\\end{align}", "6e2d2b5985611c97f002fe2ae25d3a48": "|\\alpha|=1", "6e2d2ca441ecf5abacb1390c05c042b5": "\\scriptstyle (x,y)", "6e2d3db79b1344c4f78ef58b87ca47ef": "M X_M = D X_C + W X_C = X_C (D + W)", "6e2d44b0665b4772e0d6df0930c37a4c": "\\textstyle n(t + T) =\nR_0 \\, n(t)", "6e2d9f73d7248fd71b40b24ffbc82f69": "A/{\\mathfrak{p}_i}", "6e2e01ff3980f9adf65dadcf3f50d9b6": "\np(x|z) = {{p(z|x) p(x)}\\over{p(z)}}\n", "6e2e604151b95d8aeeee84fa66f4cc42": "P_{a \\parallel v} = \\frac{q^2 a^2 \\gamma^6}{6 \\pi \\varepsilon_0 c^3},", "6e2e69f8a09f5900819f323c66bee545": " D_f(P\\parallel Q) = \\int_{\\Omega} f\\left(\\frac{p(x)}{q(x)}\\right)q(x)\\,d\\mu(x).", "6e2e76078284eceebe87a745c21c6485": "\\phi(q)=(q;q)_\\infty", "6e2e7c02f8b62667edb73a80861e0b33": "\n\\phi=-\\sum_{k=1}^\\infty\\frac{\\varphi(k)}{k}\\log\\left(1-\\frac{1}{\\phi^k}\\right)\n", "6e2ef44a5fdcccb24c7249cb4a897916": "\\text{Digestible energy} = {{GE}_p(D_p)} + {{GE}_f(D_f)} + {{GE}_{cho}(D_{cho})}\\,", "6e2f4230951cee5f0803885393222567": "A[\\rho]=A[\\Psi[\\rho]],\\,", "6e2f89b7398be6486dc816cd6395b1cd": " \\rho v_0^2h_0 + {1 \\over 2} \\rho gh_0^2 = \\rho v_1^2h_1 + {1 \\over 2} \\rho gh_1^2.", "6e2fcb373c3dc3d1d15e9e998d2a028e": "\\textstyle D^\\alpha u", "6e2fce1d5dc8d90514f796a3398725e5": "(v,v+dv)", "6e3050bf38260c7d54df3092b90750db": "(a,b) + (c,d) = (ad+bc,bd) \\,", "6e305d9055d6429a2f585fe21aab3ee0": "\\vec a_P = \\frac{{\\rm d} \\vec v_P}{{\\rm d} t}", "6e30baf83c3d8b7c2c798ba0efe0850b": "F_{ijk} =", "6e30e92d08c61a285ca901c77d8e5bb1": "\\frac{d}{dx} \\ln(x) = \\frac{1}{x}.\\,", "6e30f35dc9f8b3d30cd1ab7614e8c152": "\\pi(x;q,a)", "6e310dac289bab5b3d501f4ed4f9213c": " \\nabla\\times\\left(\\mathbf{A}\\times\\mathbf{B}\\right)=\\mathbf{A}\\left(\\nabla\\cdot\\mathbf{B}\\right)-\\mathbf{B}\\left(\\nabla\\cdot\\mathbf{A}\\right)+\\left(\\mathbf{B}\\cdot\\nabla\\right)\\mathbf{A}-\\left(\\mathbf{A}\\cdot\\nabla\\right)\\mathbf{B} ", "6e3136d1135f474ebec6fe02c53f2097": "\\mathbf{A}+\\mathbf{B} = (A^0, A^1, A^2,A^3) + (B^0, B^1, B^2,B^3) = (A^0 + B^0, A^1 + B^1, A^2 + B^2, A^3 + B^3) ", "6e3197bc2a8f3dd2b324cbc2a22ca316": "F = \\frac{\\left(X_1^2+X_2^2+\\cdots+X_n^2\\right)/n}{\\left(Y_1^2+Y_2^2+\\cdots+Y_m^2\\right)/m}\\ \\sim\\ F_{n,\\,m}.", "6e31facb65e3d27c6a36509bc052e435": "\n\\langle 1;x_1|1;y_1\\rangle = \\delta(x_1-y_1)\n\\,", "6e31fff11ed2604ca5da9e0fd444e1d1": "f[x_i, x_i, x_i]=\\frac{f''(x_i)}{2}", "6e3238190282b398acf109c98de853db": "p(a,b)", "6e3261082be216ce21cef40620a81d1d": "\\operatorname{tr}_B", "6e3268fae436dc38a578acc7ca96927f": "\\tilde A_5 = A_5^{(1)} = A_5^{+}", "6e326d63189c40bbf48f6b62d226bdc0": "\\mathbf{x}_1,\\ldots,\\mathbf{x}_{k-1}", "6e33378555863080c4ab9fe6f34ebc9c": " u(x_j,t_n) = u_{j}^n ", "6e33902f928e6220ab9c16116514b3af": " F(\\xi) = \\frac{1}{\\sqrt{2\\pi}}, \\,", "6e33e7002360bad968b71209b6ad5175": "q_c(A|B) = 2^{-H_{\\min}(A|B)}", "6e33e8c76991c9a8fe569f36d23bfbce": "\\iiint_T f(a,b,c) \\rho^2 \\sin \\phi \\, d\\rho\\, d\\theta\\, d\\phi.", "6e3477bcc23d193f7aa8489590a3df89": "\nP_{\\nu_\\mu\\rightarrow\\nu_\\tau}^{(1)}=-Re(\\delta h_{\\mu\\tau})L\\,\\sin{(\\Delta m^2_{32}L/2E)}.\n", "6e347f9a66c20735d65249fc26ad32fe": "|\\Phi^{<}|", "6e34872eb1e4e86b36ec007a83d4fbb1": "V^{\\textrm{loc}}_{\\textit{ps}}", "6e35714e6616c41fa18fa4a4672c9645": "I\\subset\\R", "6e3579a8acc2e2ae32c12e59a87711ad": "\\begin{align}\n2\\theta_{p1}&=180-53.13^\\circ=126.87^\\circ \\\\\n\\theta_{p1}&=63.435^\\circ \\\\\n\\end{align}", "6e358008c5e439a8fa8d2305ab35fb83": "l_G", "6e35b65e296195736226ef48275e777d": " \\mathrm{SNR} = \\frac{P Q_e t}{\\sqrt{P Q_e t + D t + N_r^2}} ", "6e35bd42ec9ae1a5e318c0b00016db7c": " \\frac{\\Delta y}{\\Delta x} = \\frac{(x + \\Delta x)^2 - x^2}{\\Delta x} = 2 x + \\Delta x \\approx 2 x", "6e36506ff22e861e6dc063bcfdbdb7dc": " k = xy\\,", "6e365e61edd6f98b209a727bfb9d0e17": "\\textbf{1} = (1,\\dots,1) \\in R^d", "6e370739e14c7cd8cf23ae44d48423d6": "[-{\\hbar^2 \\over 2\\mu} \\nabla^2 - eE \\cdot r ] \\psi(r) = \\epsilon \\psi(r)", "6e3727e068b8f55c94242f94cc6eceb1": " R = f_p \\log_2(M), \\, ", "6e376daeebdd7942297645ef928b2b68": "~180^\\circ - \\gamma", "6e379e4aac69e1b763e7b072594bb2e3": "\\lambda^2 + 5\\lambda - \\lambda - 5 = 0 \\,\\!", "6e37efed332e57f8b152601f87f6ee21": "\\mu \\left( A + \\overline{B_{\\delta}} \\right) = \\mu (A) + \\lambda (\\partial A) \\delta + \\sum_{i = 2}^{n - 1} \\lambda_{i} (A) \\delta^{i} + \\omega_{n} \\delta^{n},", "6e38286a0f7aebf216dc640f1b5ba1e0": "\\lambda_t", "6e383e950ce93238df6b84ed0619d8e2": "m_\\text{Atom} = g_J \\mu_B \\sqrt{J(J+1)}", "6e385feca4bceb05adb35933741dc84b": "\\chi \\rho", "6e3868f7de5e8253f65e626a7e209bad": "\n\\left(\\frac{\\partial \\mu}{\\partial P}\\right)_{S, N} =\n\\left(\\frac{\\partial V}{\\partial N}\\right)_{S, P}\\qquad=\n\\frac{\\partial^2 H }{\\partial P \\partial N}\n", "6e388723455f26c62a55ca10139d5b70": "\n\\rho(x_1,x_2)=\\sup_{y\\in\\mathfrak{Y}}\\left| H(x_1,y)-H(x_2,y)\\right|.\n", "6e390b0c3ec604e5c2ca7e08379195c6": " S = \\sum_{i=1}^n |y_i - f(x_i)|. ", "6e391fb81f5e59977aab60128efe4921": "R_{abcd;e}^{}+R_{abde;c}^{}+R_{abec;d}^{}=0", "6e3935bb1907e05aef6649527518e040": "y=x^2", "6e395efde5d8ebb0a935c998313dee81": "\\operatorname{Out}(S_n) = 1", "6e39778329daca77e4cf78f942622aff": "\\hat a_{1}= 2^{-1/2}(\\hat\\alpha - \\alpha_{LO})", "6e39d611e63f7b379ee6eadc33eae6e3": " \n\\hat{E}\\left\\{\\mathbf{x}(n) \\, e^{*}(n)\\right\\}=\\mathbf{x}(n) \\, e^{*}(n)\n", "6e3a3b69b2f8c2e826f16fc4a5a6312c": "\\mathbf{D(r)}=\\varepsilon\\mathbf{E(r)}", "6e3ae759512c3d16f3683cd2afb667b0": " C_1\\triangleleft C_2\\triangleleft C_6 \\triangleleft C_{12}", "6e3b77e7887e34d088e9d8ef5df3d605": "f=\\text{uxp}", "6e3bbf316b10627928963451723dcf8e": "\\bold{S} = \\frac{1}{\\mu_{0}} \\bold{E} \\times \\bold{B} ", "6e3bdcadb6e34d56ea739e7b208ccc0c": "Y^{\\prime}[\\sigma]", "6e3be5632094795cbd0303c44f6726ee": " S_n + k ", "6e3c2701f9aa8b25fa35275d8ffb76aa": "u_{k}(\\mathbf{q})", "6e3c30b35a99bfe153732ffaf03732c3": "D_{t}", "6e3c4fc1d41b5ce561c3cf00e5b5dfb2": " L_o = n\\, l^{5/3}\\, \\Gamma^{1/3}", "6e3c947ceb8bcc17e08864f767822f2b": "p_i = c_i + v_i (1 + \\sigma) = l_A a_i + l_W l_i (1 + \\sigma)", "6e3cde96b7022e00e88e2faad416200c": "\\scriptstyle{\\pi}", "6e3cf524688bb529c6f569a41bb8993d": "Y_{t} = X_{t} \\mbox{ for } t < \\zeta,", "6e3d17e825f03e4768ba834adc099d8a": "1 \\cdot a = a \\cdot 1 = a", "6e3d835be88aad3084c2c8e087aae531": "2cos(a)cos(b) = cos(a+b)+cos(a-b)", "6e3d8a42da5560e59b394e97d80e230c": "s_{i+1} = T(s_i)", "6e3e38b4e9c426554adb2c5e480e0396": "a_i = e^{\\left ( \\mu_i - \\mu^{\\ominus}_i \\right )/RT}", "6e3eb3daaf8f7cf01390a64785801315": "s^n s^m = s^{n + m}", "6e3ec140f2c2dc5a7602672a8715662f": "\\underline{\\mathrm{W}}(\\mu,E)=\\inf\\left\\{\\mu(A)|A\\in\\Sigma\\text{ and }A\\subset E \\right\\}\\qquad\\forall E\\in\\Sigma", "6e3ed397975b5bbbd003fb4f4b46bf91": "\n\\begin{align}\n\\sin\\alpha_0 &= \\sin\\alpha \\cos\\beta = \\tan\\omega \\cot\\sigma,\\\\\n\\cos\\sigma &= \\cos\\beta \\cos\\omega = \\tan\\alpha_0 \\cot\\alpha,\\\\\n\\cos\\alpha &= \\cos\\omega \\cos\\alpha_0 = \\cot\\sigma \\tan\\beta,\\\\\n\\sin\\beta &= \\cos\\alpha_0 \\sin\\sigma = \\cot\\alpha \\tan\\omega,\\\\\n\\sin\\omega &= \\sin\\sigma \\sin\\alpha = \\tan\\beta \\tan\\alpha_0.\n\\end{align}\n", "6e3f0abf6562e1f81a75299ff6778e88": "\\boldsymbol s_{\\boldsymbol\\Theta}", "6e3f4bdb9fb183ba9afce6f55fd72dad": "{\\tilde{C}}_4", "6e3f704231841152c931a6d823df4e76": "\\epsilon_{sh\\infty} \\approx \\epsilon_{s\\infty} E(607) / (E(t_0 + \\tau_{sh})", "6e3f92186980f20d4cd2f45cb0c54d40": "y \\wedge x\\wedge y = y", "6e3fb3cebec8d37059c0a94072cd3ff0": "S \\to a", "6e3ff2f1b97b3048803040213be25cd8": "\\max_i |\\sigma_{i,v}| \\leq \\delta", "6e402ac09fddb3ff55bd3cddc6d254e8": "\\{n \\in \\mathbf N: n^2 \\le 4\\} = \\left\\{0, 1, 2\\right\\} \\,\\!", "6e4031a2553373d8e6685da91c373eba": "\\frac{\\partial^2 |\\mathbf{U}|}{\\partial x^2} =", "6e40338822f0f1d84e921f2ec8e3a110": "V_L =V_U + T_C D\\,", "6e4047804716a7c867057d7371ecdebc": "K_\\mathrm{max} = hf - \\Phi\\,\\!", "6e40acdce34f7981a6a1bf32466092c2": "S^2_{n-1} = \\frac{1}{n-1}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2\n=\\frac{1}{n-1}\\left(\\sum_{i=1}^n X_i^2-n\\overline{X}^2\\right).", "6e40b09facf8e543061a50be9ec8d6fc": "(\\pm 1,0,\\pm\\varphi)", "6e4110ed2cd91a13e79b120fb9723907": "x_1 \\oplus x_2", "6e41232a8f2dfb735e582a287e0f3018": "x\\wedge y=y", "6e4245aa1fbf8b125fc2469a7dfbb853": "\\; m_1g-T=m_1a", "6e4261be65c2a56a51d38b2c3a411c49": "\\mathbf{H}_\\alpha(x)", "6e428a96a4bbdb0b7681b21e3b373307": "A = 20\\sqrt{3} + 3\\sqrt{25+10\\sqrt{5}} \\approx 55.28674495844515", "6e42b6c46e4afeba3407cfb77f6dda8e": "R_p=\\{g\\in G, \\exists n\\in \\mathbb{N}, g^{p^n}=I\\}\\;", "6e4315b8b985bb1cebf8fb992dd0e4dd": "\\lambda_c = \\frac{h}{mc}", "6e431d46cecdca6a5bef2d36790d7f9a": "\\int x^4 r^3\\;dx= \\frac{x^3r^5}{8}-\\frac{a^2xr^5}{16}+\\frac{a^4xr^3}{64}+\\frac{3a^6xr}{128}+\\frac{3a^8}{128}\\ln\\left(x+r\\right)", "6e432a25dbfdc249c59731dfc7c41d1a": "\\varphi(t - k)", "6e435152e766773c60408981b818914f": "\\ \\alpha^m\\beta^n;", "6e435a1f64d909f00fea51137acc5237": "V_n = f^{-n}(H_n)", "6e4383d86f2cb2dd2a67a01761542453": "\\gamma_5 = \\begin{pmatrix} 0 & I_{2} \\\\ I_{2} & 0 \\end{pmatrix}.", "6e43ae74ff399e7eddcf9d2e02ebe568": "x \\in H_1(M;\\mathbb{Z}_2)", "6e43d6252f5d8d31e9d56cb3cf2e16e9": "X \\cup \\{A\\}", "6e43ff36ed8c3ccce1d4fc54cb2db5f1": "\\begin{cases} a^2 + b^2 = d^2\\\\ a^2 + c^2 = e^2\\\\ b^2 + c^2 = f^2\\end{cases}", "6e4453863304d54479405c3b8a55b6a5": "\\mathcal{L}={1\\over 2}(\\varphi_t^2 - \\varphi_x^2) - \\cosh\\varphi.\\,", "6e446dde540d0f30700bf6996604635c": "X_0 \\prec X \\prec X_1", "6e44a26c963b1b08ba229ec2fffe298c": "\\int_T\\!f(\\mathbf{x})\\,d\\mathbf{x}.", "6e44c1770ff588d99fe9c468158ca5cb": "\nR_{ik}=\\frac{\\partial\\Gamma^\\ell{}_{ik}}{\\partial x^\\ell} - \\Gamma^m{}_{i\\ell}\\Gamma^\\ell{}_{km} - \\nabla_k\\left(\\frac{\\partial}{\\partial x^i}\\left(\\log\\sqrt{|g|}\\right)\\right).\\ \n", "6e44f87d8fc3fbc10e57fc1f0ffa7094": " \\mu(A) = \\int_A g \\, \\mathrm{d} \\lambda", "6e45645df8ceb87b9b9fb0ab8058d7e3": "\\displaystyle{(Tf,f)=(u,u)_{(1)}\\ge 0,}", "6e45828aa6b1d23e9accac78b9ab86f3": "A(z) = \\sum_{k = 0}^\\infty z^k,", "6e459e7bd0dc6cbad9f7ded35ed92324": "\\mathbf{R}_{x}(n)", "6e45b63d70e3e8796b69da8a039fbb2f": "y \\in A", "6e4614527200512c8b1b2afbd3ef07d5": "S = \\cfrac{BH^2}{6}-\\cfrac{bh^3}{6H}", "6e465715355b99250f42988c7837765a": "\\varphi(S_{k}) = \\lfloor (k-1)/2\\rfloor+1", "6e469320146d938c326ec88f76a8dbcc": "\\phi(0)\\ne 0.", "6e47289d73e0b977a5859da1c7167953": " x^{(k+1)}_i = (1-\\omega)x^{(k)}_i + \\frac{\\omega}{a_{ii}} \\left(b_i - \\sum_{ji} a_{ij}x^{(k)}_j \\right),\\quad i=1,2,\\ldots,n. ", "6e475ab430b843665b4d592db45cb730": "[A^+ ] = [B^- ] = \\alpha c_0 = \\sqrt{K_d c_0 } ", "6e476cfae8505c3412619b1cbc885c07": "\\hat\\mu = \\bar{x} = \\sum^n_{i=1}x_i/n. ", "6e47b9646721a344e44acd47989b6275": "k \\sim 2(500) = 1000 \\mathrm{s}^{-1}", "6e47cb495120950740c5141225743a09": "\\displaystyle a^n = e\\mbox{.}", "6e47d34fb3cf440e7798de76e860ec45": "\\hat{\\mathcal O}_{Y,y}", "6e47db4d9fefe2bb76a0e68ab386d8f6": "\\begin{matrix}\n x_0 & f(x_0) & & \\\\\n & & {f(x_1)-f(x_0)\\over x_1 - x_0} & \\\\\n x_1 & f(x_1) & & {{f(x_2)-f(x_1)\\over x_2 - x_1}-{f(x_1)-f(x_0)\\over x_1 - x_0} \\over x_2 - x_0} \\\\\n & & {f(x_2)-f(x_1)\\over x_2 - x_1} & \\\\\n x_2 & f(x_2) & & \\vdots \\\\\n & & \\vdots & \\\\\n\\vdots & & & \\vdots \\\\\n & & \\vdots & \\\\\n x_n & f(x_n) & & \\\\\n\\end{matrix}", "6e4800c1ea252c58e26cc0b7b0a75be4": "\\frac {X_b}{X_b^0 - X_b} = \\frac {X_c}{1-X_c} exp \\left [ \\frac{-\\Delta\\,G - Z_1\\omega\\,\\frac{X_b}{X_b^0}}{RT} \\right ]", "6e4815e58c13cde118f05c3273cb65cc": "\\displaystyle{S=\\psi(a)}", "6e482e5656076509a5e49033e413ce4f": " \\mathcal{R}(\\alpha,\\beta,\\gamma) ", "6e4853e170e5a7835e4f77ee4b5aa1ef": "ax^2+y^2=1+dx^2y^2.", "6e486228d0bde0b1a337f697497ca05f": " v(r) = \\frac{1}{4 \\eta}r^2\\frac{\\Delta P}{\\Delta x} + A\\ln(r) + B. ", "6e486504d33a2e31649ab0789bbae5ed": " E \\ = \\frac {1 - \\exp[-NTU(1 - C_{r})]}{1 - C_{r}\\exp[-NTU(1 - C_{r})]} ", "6e48b8e43e9dc15af6315d0f6c3e0345": "(1+u_x^2)u_{yy} - 2u_xu_yu_{xy} + (1+u_y^2)u_{xx}=0", "6e48d295f8d4a6c520a16ad567d5f4b8": "{\\int_K|f|\\,\\mathrm{d}x}={\\int_\\Omega|f\\chi_K|\\,\\mathrm{d}x}\\leq\\left|{\\int_K|f|^p \\,\\mathrm{d}x}\\right|^{1/p}\\left|{\\int_K \\mathrm{d}x}\\right|^{1/q}=\\|f\\|_p|\\mu(K)|^{1/q}<+\\infty,", "6e49106b1cc051b54ceecdae4e6a6a9a": "0 \\to TN \\to TM\\vert_{i(N)} \\to T_{M/N} := TM\\vert_{i(N)} / TN \\to 0", "6e495cfbe6ecbcb4c521ec7ce5fe5f84": "\\frac{\\Gamma \\vdash a : X \\circ Y^{*} \\qquad \\Delta, b : X, c : Y^{*}, \\Delta' \\vdash d : Z}{\\Delta, \\Gamma, \\Delta' \\vdash d[b := \\epsilon, c := a] : Z}[\\circ E_{strong}]", "6e499793f52b225f3883bb74b4e46de6": "P_i=(\\,\\cos\\alpha_i,\\sin\\alpha_i \\,)\\text{ where }\\alpha_i \\in \\,[\\,0,2\\pi).\\,", "6e49a72b9c58062d218338cf39578bd3": "\\lambda_\\min = \\frac{h c}{e V} \\approx \\frac{1239.8 \\text{ pm}}{V\\text{ in kV}} \\,", "6e49ba540eb15c8577ddb1368ea1815a": " \\mbox{cosh}(s_1-s_2)=\\mbox{cosh}(s_3-s_4) ", "6e49ca5d8b2d68ecde3c9fda8a5c2f67": "\n\\cdots \\rightarrow\n(0, 0, 128, 128) \\rightarrow\n(0, 128, 0, 128) \\rightarrow\n(128, 128, 128, 128) \\rightarrow\n(0, 0, 0, 0)\n", "6e4a0e22ea09b13960450e90c2b2d53f": "U=", "6e4a7b7cd0e5077909fa362982f34a45": "m_{0j} = \\sum_{k=1}^j \\ m_k \\ . ", "6e4aa5ea49ad8e9400f2ccbba8eaa7d3": "\\begin{align}\nx^k(1-x)^l&=(1-2x+x^2)x^k(1-x)^{l-2}\\\\\n&=(1+x^2)\\,x^k(1-x)^{l-2}-2x^{k+1}(1-x)^{l-2}.\n\\end{align}", "6e4ab2a28fb6bd1bc0b168139e316aa1": "\\; \\Phi_E(f) = V \\pi (f) V^*.", "6e4ac1faa785b1e10ca4c09297db143f": "D/Dt", "6e4b930791ee4964b35cfcf36fecc694": "FOV_C= \\frac{FOV_P}{(\\frac{f_T}{f_E})}", "6e4b9c0b565d72cbc5816c7fa2b9534d": "(-1/b)e^{\\eta}\\text{Ei}\\left(-\\eta\\right)", "6e4ba56bfe6cbcee52a67c50b4b4610a": "\\ddot{\\varphi}+\\xi^{-1}\\dot{\\varphi}-\\varphi^{\\prime\\prime}+4\\rho^2 \\varphi = 0.", "6e4c16d6c48871352b9472a85af5e78e": "\n\\sigma_3 = d \\psi + \\cos \\theta \\, d \\phi.\n", "6e4c376ada10bb393016852aa8c860d7": "t'=T_1", "6e4cad75baac14e5eebd3be8aa23fcf6": "dim(a) c", "6e544a0a9e6108a6142de6e7f640d9bf": "\\operatorname{E}(c) = \\frac{2^c - 1}{2}", "6e546f997b520bbf03ac3791888f05a9": " r < 0 ", "6e54a061f27fbeb83f66f438cb7f86d2": "\n\\qquad\n\\frac{A\\hbox{ prop}}{\\neg A\\hbox{ prop}}\\ \\neg_F\n", "6e550337b79bc62c04165b93e29586fa": "\\Delta t' = \\gamma \\Delta t. \\,", "6e552c7571be506dd0c7e1b971815dac": " \\begin{bmatrix}\n x_1 \\\\\n x_2^* \\end{bmatrix} = \n\\begin{bmatrix}\n h_1 & -h_2 \\\\\n h_2^* & h_1^*\n\\end{bmatrix}\n\\begin{bmatrix}\n S_1 \\\\\n S_2^* \\end{bmatrix} +\n\\begin{bmatrix}\n n_1 \\\\\n n_2^* \\end{bmatrix}\n", "6e55a546c8c077b06ae87dab6477dbd8": "g = (g_1,\\dotsc,g_N)", "6e55fab13f65880ae22db6cba2407873": "\\,\nS = \\left | \\frac{V \\pm U} {1 \\pm V U} \\right |\n", "6e562258e55d14a78b8fabd1b8b9807e": "\\log_{10} \\mbox{ second}", "6e563d1fe25f4dcbbf2375f1bec989be": "\\frac{\\pi}{2} = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\frac{8}{7} \\cdot \\frac{8}{9} \\cdots = \\prod_{n=1}^{\\infty} \\left( \\frac{ 4 \\cdot n^2 }{ 4 \\cdot n^2 - 1 } \\right). ", "6e5642144822b26971691a61ef4eee79": "C\\approx\\pi\\left(a+b\\right)\\left(1+\\frac{3h}{10+\\sqrt{4-3h}}\\right).\\!\\,", "6e5679dd517998afc29db47e645ab274": "F(x,\\lambda)=0", "6e5711c6ee29a8923abda14601465c9d": " E: \\mathbb{R}^n \\rightarrow T_{p}M ", "6e57359483643fc885713984760812c2": "\\chi_{01}", "6e5769e0c0da17652cde5639d11908b4": "\\scriptstyle k_{mnl}", "6e576dc1cc74bd27f43854abeb8e6967": "\\left(D^2-\\alpha^2\\right)\\Psi_j=0,\\,\\,\\,\\ D=\\frac{d}{dz},\\,\\,\\,\\ j=L,G.\\,", "6e578799ec3673c7de09f500b7b41721": "K_{\\rm a}", "6e57987f48bf723d5f9fa8584e530a22": "x'= \\frac{2 \\pi \\rho}{\\lambda z} \\rho'", "6e57aa98eb4e6dedf4c671b3083baac7": "\\{ \\mathbf{q}(t) \\in \\mathbb{R}^N \\,:\\,t\\ge 0,t\\in \\mathbb{R}\\}\\subseteq\\mathcal{C}\\,,", "6e57be449067da70a9daf7f86a86d87a": "\\tau_{del, jam}^{(a)}", "6e5804e005a915dfb236a8b1855d9362": "\\delta = \\epsilon = 1, f(z) = \\tanh(z)", "6e5831e9e8e6c97fa0d4f71a189183ea": "D(a) = \\{ p \\in \\operatorname{Proj}\\, S \\mid a \\;\\not\\subseteq\\; p \\}.", "6e586e32b0d8f7685f6f4bd9680c32a4": "n=2k+1", "6e58a47776dd2596aecda9293966e15a": " \\mathbf {Q_{A}} = \\mathbf {Z_{A}} - \\mathbf {GAP_{A}}", "6e58bd85197f58e3dfe63d29ecaf2db4": " \\mu \\left ( \\left \\{x\\in Q: d(x, X\\backslash Q)\\leq t\\delta^k \\right \\} \\right ) \\leq C_3 t^\\eta \\mu(Q).", "6e58c44bed99dd65bcf119a1ccd8cb9c": "\n \\frac{\\partial }{\\partial t}(\\rho~\\eta) \\ge\n -\\boldsymbol{\\nabla} \\cdot (\\rho~\\eta~\\mathbf{v}) - \n \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) + \n \\cfrac{\\rho~s}{T}.\n ", "6e58efb00d046b692778f268a4247149": "\\hat H^n(G,A) = H^n(G,A)", "6e592ccc9575ac1840a4e40d8329d14b": "\\kappa \\equiv 0 (\\kappa_0 \\equiv 1)", "6e59c214ce20e77c49f64e2321509bd6": "C=\\frac{Z\\alpha}n\\frac{\\mu c^2}{\\hbar c}.", "6e5adb75ab8002c425b78f7aa96c1bf3": "\\partial_t=\\frac{\\partial}{\\partial t}", "6e5b3baaa9a444f4029182068b794e25": " \\stackrel{\\mathfrak{I}}{}\\,", "6e5b8673ac8723ff41257c55b102c284": "\\scriptstyle{\\mathbf{r}}", "6e5ba114a00ceb9dccd7e07a821f254b": "x \\cdot S(y) = xy + x\\ ", "6e5ba364aeebd24b2a93d542101fb8fb": "B=\\omega_{0}/Q", "6e5bed5b5adab55c25fd620395008e85": " n\\# = 1 < 4 = 4^1.", "6e5bf9c192969ba5b8096fd1cf66a85b": " M(r) ", "6e5c357ee2469fae772adc30cc8a2b70": "\n\\min_{y}\\{ q(y,\\xi)| T(\\xi)x+W(\\xi) y = h(\\xi)\\}\n", "6e5c3e16ed322a00decaa1a4f6f32772": "\\operatorname{samenum} = \\lambda n.\\lambda f.\\lambda x.\\operatorname{extract}\\ (n \\operatorname{inc} \\operatorname{init}) = \\lambda n.\\lambda f.\\lambda x.\\operatorname{extract}\\ (\\operatorname{value}\\ (n\\ f\\ x)) = \\lambda n.\\lambda f.\\lambda x.n\\ f\\ x = \\lambda n.n", "6e5c5ae909f67682eacf375e267e414e": " Df(a)(h) = \\sum_{i=1}^{n} h_i \\frac{\\partial f}{\\partial x_i}(a). ", "6e5c87a1d51cf9df4a7cfc18feddc349": "12\\tfrac{3}{11} \\div 13\\tfrac{1}{2} = \\tfrac{10}{11}", "6e5cb863f6155d170ebffc3abd9686e3": " 2 x^3 + 2 x y^2 + 2 x^3 = 2 a^2 y \\frac{dy}{dx} - 2 x^2 y \\frac{dy}{dx} ", "6e5cdace1dc2a892114bbda2601caccd": "Z = A^+B", "6e5ce3f4af5a3d197abe9d7c410a4891": "\\left [\\begin{smallmatrix}2&-2\\\\-4&2\\end{smallmatrix}\\right ]", "6e5d13905f51be16a6a663a0b9c90b4c": "S_{600}", "6e5d3aeb70430c1271630b22d785b8d1": "Q=a+b X_1+c X_2+d X_3+\\dotsb", "6e5d6f966609ec55a9ee8c2bece67fe6": "\\,F_{\\gamma} = \\pi d \\gamma", "6e5da3493e80b268ffd4c27df3fd89ef": "V_n(i) \\cup V_n(i+1)", "6e5de61cf8a50b74b60b548d146e2833": "(X(t)|X(t-1) = i)", "6e5e35bdec4e52512e92fe8a9480550b": " p\\rightarrow p ", "6e5e814cc8fd1b1d837f981ae16abd7e": "\\mathcal A=(A, \\sigma, I)", "6e5e84839173bb61b56803d63ba8198c": "\\frac{\\partial}{\\partial x_{ij}}", "6e5e8d4f5278b6a73e60a4d69797285b": "p = \\frac{F}{A}\\ \\mbox{or}\\ p = \\frac{dF_n}{dA}", "6e5e95ef064960aad74e7b7b80d6a0d0": "\n\\frac{d p}{d \\tau} = e \\langle F u \\rangle_{R}\\,.\n", "6e5ee18ad65ba64b8cc41b8cc69e6d69": " x[n] = -A (-1)^{a[n]} \\ ", "6e5f2663b7dae9166c0684c928694141": "\\mathcal{F}| _U", "6e5f814bdf259cb2b458d7dcdb6b4cee": "x^{\\prime}=\\gamma lx^{*},\\quad y^{\\prime}=ly,\\quad z^{\\prime}=lz,\\quad t^{\\prime}=\\frac{l}{\\gamma}t-\\gamma lx^{*}\\frac{v}{c^{2}}", "6e5fd6160859b24a0c6ecf47370a9776": " u(t,x) = \\frac{1}{2\\sqrt{\\pi \\alpha t}} \\exp\\left(-\\frac{x^2}{4 \\alpha t} \\right). \\,", "6e601da083d437899637d76d99db1c7c": "R_{abcd}^{}=R_{cdab}", "6e605eefa102c5de3b0df3e2d695ce8f": "f_{4}", "6e608e16822eb744b4c26acc54fc7d0a": " \\frac{\\sum[M_1^*]}{\\sum[M_2^*]} = \\frac{k_{21}[M_1]}{k_{12}[M_2]}\\,", "6e60c1c7f6b2316a90370cfbe983cf15": "k_F+k_V=k_{SHG}", "6e61053c90ec1cdf9e29de15194e5343": "\\bar{Z}_1^{p,q}", "6e615c42d961b74a11fcd07626b8a363": "y = 2 \\sqrt{\\frac{\\pi}{4 + \\pi}} R \\sin \\theta \\approx 1.3265004\\, R \\sin \\theta", "6e616796169d85a908090f010fca111d": " n =P(P^{-1}(m) + 1 - \\frac {\\rho_\\text{EtOH}(A_w)} { \\rho_\\text{water}})", "6e61fcc3ee6553109e8f0571a9ce2ff5": "\\left.\\varnothing\\right. \\overset{\\mathrm{def.}}{=} \\left\\{x : x \\neq x\\right\\}", "6e622d72b90c249c3e04aebf0eb12ca4": "(N)", "6e6246a7782d94e972e7a2f366e9c8af": "\\left|e^{i\\theta}\\right|=\\sqrt{\\cos^2{\\theta}+\\sin^2{\\theta}}=1", "6e624ae25c8333d4c5b1c1f52f1bfc02": "\\left(\\frac nm\\right)=\\left(\\frac{m^*}n\\right)=(-1)^{\\frac{n'-1}2\\frac{m'-1}2}\\left(\\frac mn\\right).", "6e62c4f14c764c307ccf200755a97316": "\\displaystyle{c=D_c(Q(a)a^{-1})=2[(L(a)L(c)+L(c)L(a)-L(ac))a^{-1}] +Q(a)D_c(a^{-1})=2c +Q(a)D_c(a^{-1}).}", "6e62fc163504153ed69137c52c8e761c": "|\\mathbf{S}| = \\hbar\\sqrt{s(s+1)}\\,\\!", "6e6339bddc77f993e82a9d6aa159ef59": "\\Delta W_\\mathrm{SLV}", "6e634ff2588f9f94720ceb4ff489976c": "\\lbrace C_\\mathfrak{s} | \\mathfrak{s}\\in M(\\lambda)\\rbrace", "6e638be0e22a138dce32482d24ca39fd": "\n\\begin{align}\nH = & \\sum_{ i, j \\text{ neighbors}} J\\left(\\tau(\\sigma(i)), \\tau(\\sigma( j))\\right) \\left(1 - \\delta(\\sigma( i), \\sigma( j))\\right) \\\\\n+ & \\sum_{ i} \\lambda_\\text{volume}[V(\\sigma( i)) - V_\\text{target}(\\sigma( i))]^2\\\\\n+ & \\sum_{ i} \\lambda_\\text{surface}[S(\\sigma(i)) - S_\\text{target}(\\sigma( i))]^2 .\\\\ \n\\end{align}\n", "6e638f96fe1fe551122a063867b630e8": "\n\\begin{align}\nds^2 & = dx^2 + dy^2 \\\\\n \\ & = \\left ( 1 + \\left ( \\frac{dy}{dx} \\right ) ^2 \\right ) \\,dx^2 \\\\\n \\ & = ( 1 + \\tan^2 \\theta )\\, dx^2 \\\\\n \\ & = \\sec^2 \\theta \\,dx^2 \\\\\n ds & = \\sec \\theta \\,dx\n\\end{align}\n", "6e63ad4213bc5e780f47566ddf751088": "=\\frac{1}{3}\\cdot\\left(\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{4}\\right)", "6e63e07b0928a1f96995e4b111ade22a": "F/L", "6e640136181dff678981708c8ec95105": "0.0385=1/26", "6e6414aa5da2615050a3c089d09ce3e1": "R_{k+1}(a, b)", "6e641ee5bdf44dbd127e4b5ec1830a7d": "\\dot{x} = Ax + Bu", "6e643ef9d5ed52970f589c63b3e42540": "[Z,X]", "6e6462a1b348efb607d581640c1a880c": " \\psi = \\sqrt{\\frac{2^n}{\\prod_{i} L_i}} \\prod_{i}\\sin(k_i i)", "6e6491252b2d2e0c0df8db87891df7d6": " f(r,\\theta) = \\frac{1}{\\sqrt{2\\pi}}\\sum_{m=-\\infty}^\\infty f_m(r) e^{im\\theta},\n", "6e649334c78ae05e67c3849ce51edb48": "1-1/x", "6e64c6413192faa658fc3d451c8fb6d3": "C_X A^T(AC_XA^T + C_Z)^{-1} = (A^TC_Z^{-1}A + C_X^{-1})^{-1} A^T C_Z^{-1},", "6e652252c3f0b960123db7bdd5802d89": "\\frac{d \\arcsin x}{dx} = \\frac{d \\theta}{d \\sin \\theta} = \\frac{d \\theta}{\\cos \\theta d \\theta} = \\frac{1} {\\cos \\theta} = \\frac{1} {\\sqrt{1-\\sin^2 \\theta}} = \\frac{1}{\\sqrt{1-x^2}}", "6e65323ed70266bb05b00c7547066896": "J = \\frac{4}{9} \\left(\\frac{2e}{m_i}\\right)^{1/2} \\frac{|\\varphi_w|^{3/2}}{4\\pi d^2}", "6e6578cfdda58d2a07cd34bf40247e7a": "= 10^{\\left ( \\frac{1}{2}\\times\\log_{10}\\left ( {2\\pi n} \\right ) + n\\times\\log_{10}\\left ( \\frac{n}{e} \\right ) \\right )} ", "6e65d72c9f3d06379332e4ba03147a6c": "\\begin{align}\n\\int_{\\gamma} \\nabla\\varphi(\\mathbf{u}) \\cdot d\\mathbf{u} &=\\int_a^b \\nabla\\varphi(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t)dt \\\\\n&=\\int_a^b \\frac{d}{dt}\\varphi(\\mathbf{r}(t))dt =\\varphi(\\mathbf{r}(b))-\\varphi(\\mathbf{r}(a))=\\varphi\\left(\\mathbf{q}\\right)-\\varphi\\left(\\mathbf{p}\\right)\n\\end{align} ", "6e65ec0a3b3198442523a5acc3b204b5": " 1\\leq j \\leq n", "6e65ed386a62e3647a26e5e90874273f": "{1\\over 3}N_f < N_c < {2\\over 3}N_f", "6e6615e9f5b1561ee6851c95090f38ed": "\\phi(z) = \\phi_0 e^{-kz}\\,", "6e66299c09bf6164825959be6d8688e0": "m = \\tan (\\theta)\\!", "6e6693bc1543bea862637d74dc677c46": "Q_\\mathrm{absorbed}", "6e66fe4c3b03e5098d6ce4c5ccb882cf": "y_i \\in \\{0,1\\}", "6e670ad199f55689902e82204e78fb69": " P_f=P_m\\theta^{\\frac{1}{1-a}}", "6e673219d9cc4219879bc59c7573536e": "\\sum_{m=2}^{\\infty} \\sum_{k=2}^{\\infty}\\frac{1}{m^k}=1.", "6e67573c1d22d74c534f488a77b4c9fc": "\\phi,", "6e67b1ec2fdcf80ce38c66c83ac4aee6": "\nu^{+1}_{+1}(\\mathbf{-p}) = u^{-1}_{-1}(\\mathbf{p}),\n", "6e67e80080dd0d67931bba76850c645c": "\\varepsilon_1 = \\sup\\{\\varepsilon_0 + 1, \\omega^{\\varepsilon_0 + 1}, \\omega^{\\omega^{\\varepsilon_0 + 1}}, \\omega^{\\omega^{\\omega^{\\varepsilon_0 + 1}}}, \\dots\\},", "6e67f94a9404388b8784306d4bd083a0": "\\cfrac{\\cfrac{stC \\qquad \\overline{s} D}{tCD} \\, \\operatorname{var}(s) \\qquad \\overline{t} E}{CDE} \\, \\operatorname{var}(t) \\Rightarrow \n\\cfrac{\\cfrac{stC \\qquad \\overline{t} E}{sCE} \\, \\operatorname{var}(t) \\qquad \\overline{s} D}{CDE} \\, \\operatorname{var}(s)", "6e6820e504660ce0cfe1569c32a8da5c": "Arf(K) = \\sum\\limits^g_{i=1}lk(a_i, a_i^{+})lk(b_i,b_i^{+}) \\pmod 2.", "6e6885010409a0fded6158e0ad9dfff4": "P_L = \\frac{\\sum (p_{t}\\cdot q_{0})}{\\sum (p_{0}\\cdot q_{0})}", "6e68a50d704a10a6e061a7a1acfb4c9d": "\\hat{\\alpha}, \\hat{\\beta} > 1", "6e68a82431a62794a901e67b2529203c": "H[A^{-1}[i]+1,A^{-1}[j]]", "6e68ab9b1c979bf190c7d4fa348687bd": "T = \\left( \\tau A \\right) \\left( r \\right) = \\left( \\frac {2 \\pi r \\mu N}{c} \\right) \\left( 2 \\pi r l \\right) \\left( r \\right) = \\frac {4 \\pi^2 r^3 l \\mu N}{c}", "6e68cb5d2d9deb4bc6d2869434364cb0": "x-\\mu=LF+\\varepsilon. \\,", "6e68d79b56a59bb45282248403b4e1ae": "\n D = \\cfrac{2Eh^3}{3(1-\\nu^2)} \\,.\n", "6e696a5c519c45d9067b147714c3d56f": "\\alpha (s)", "6e69dbcd94181af437fb4f9a26027007": " P_{\\mbox{lacunary}}", "6e6a1b13f21391e38d625aef2ccda5da": "(C)\\int \\lambda f \\,d\\nu = \\lambda (C)\\int f\\, d\\nu,", "6e6a4e78a06761ce18ff3262a4c43a0b": "\\tau = \\hbar/\\Gamma", "6e6a5157f0373f1c89616164c64739e4": "f(x) = a(x - h)^2 + k \\,\\!", "6e6a61c35c9ee279ccdd17eb0ad14dd3": "\\Delta_c = 200", "6e6a651761005b89057d2be34cea046d": "r\\phi (P) = \\phi (rP)", "6e6ac42771b5046a5700ce37d108f121": "\\int_{-\\infty}^\\infty e^{x}\\,\\mathrm{d}x", "6e6b0cc4871830f2a300ec6836e942f2": "Q_t \\cdot Ca_{O_2}", "6e6b2cd1737c3bbfaf9dd43f48f78542": " E_{kin} = \\frac{1}{2} m \\omega^2 r^2 ", "6e6b9e0f7b7dc7f417af9ea6b0ab460d": "N_i\\colon I \\to S^l", "6e6bbd96686d006d4f9a0addef6f3a33": "r = \\sqrt { x^2 + y^2 } \\, ", "6e6bd687380767e5cf05eda2be4fb247": "\\bigl(G_n(c)\\bigr)_{c\\in\\mathcal{C}}", "6e6c30b00683c3ddca8857cbdc96d57f": "C_{ijkl}=C_{IJ}", "6e6c506c673ff1f055a732edf0ac8035": "J = \\lambda_1\\lambda_2\\lambda_3", "6e6c8abe9b73a8c2b3a6135a5bf4dce7": "\\mathbf{r}_A (t) = vt\\ \\left( \\cos (\\theta ), \\ \\sin (\\theta )\\right) \\ . ", "6e6ca7977b1edc976d717a0b1532d03d": "\\Longleftrightarrow R > B \\; ", "6e6cc43aad94f78e93f16bba1bb154b4": "f_\\text{a} < f_\\text{b}", "6e6cef0abfe629dcf367e419172e4ab4": "\\mbox{If } \\frac{x+y}{a-b} = \\frac{2}{3} \\mbox{ , then } \\frac{9x+9y}{10a-10b} = ? ", "6e6cf99b11c27cbd2d72a13b2d2c4518": "A_k(\\omega)=\\int_{[a,b]} X_t(\\omega) f_k(t)\\,dt", "6e6d0a8519fb7a9405299198940466a5": "\\mathbb{P} (E) = \\mathbb{P} (\\vartheta_{s}^{-1} (E))", "6e6d0ca7a82dc50540ac84ed51bd838c": "\\sigma^{u_1}\\tau^{v_1}\\sigma^{u_2}\\tau^{v_2} \\cdots \\sigma^{u_n}\\tau^{v_n}", "6e6d40fe587470a43e6e094bf980609e": "\n2\\int_0^L p \\, dq = nh\n", "6e6d5f69f582fa777dec6f533dd08465": " \\mathcal O_X ", "6e6d603f51b167eede531d0bc92f5852": "|\\Psi\\rangle = \\mathcal{A}(\\phi_{1}(\\mathbf{r}_{1}\\sigma_{1})\\phi_{2}(\\mathbf{r}_{2}\\sigma_{2})\\cdots\\phi_{m}(\\mathbf{r}_{m}\\sigma_{m})\\phi_{n}(\\mathbf{r}_{n}\\sigma_{n})\\cdots\\phi_{N}(\\mathbf{r}_{N}\\sigma_{N})).", "6e6d87fcd1c54aa22cf3bce37740fe51": "A_1=S(a)M \\cup M \\cup B", "6e6d9dd1ff9e05e8bb786ad3eb3faeed": "{P_0}=1,\\quad \\quad P_{k+1}=P_k\\cdot \\left( \\xi -x_k \\right) ~,", "6e6dce84822774d62701f2a7cc161f41": "R[f^{-1}].", "6e6dd5a8541a85d639bd8c38ceaad6dd": "(0.878,0.478)", "6e6dfef9f1fb025c5f717959e8e8ed13": "O(\\log{\\log{n}})", "6e6e0e9e7460d1d3310d1418414fb08b": "\\lambda 's", "6e6e77e7b466e780ca737f9abb4bec98": "\\langle\\tau\\rangle \\equiv \\int_0^\\infty dt\\, e^{ - \\left( {t /\\tau_K } \\right)^\\beta } = {\\tau_K \\over \\beta }\\Gamma ({1 \\over \\beta })", "6e6eafd5f8fb9c2d26be042cfdea65a4": "\\nabla:TM\\times TM\\rightarrow TM", "6e6ebd280a1e481093f8b2179abe06b5": "(\\R^n, \\langle\\cdot;\\cdot\\rangle)", "6e6edb818573e1e1ac1220509903f23b": " X \\to E \\to Y \\to \\Omega^{-1}(X).\\, ", "6e6efa542f43148f34e28e9cfe3c7b06": "q_1=e^{-\\frac{\\pi K}{K'}}. \\, ", "6e6efefc0bb50a975d85d768ebc4481b": "\\|Ax\\|.", "6e6f3ea0450504f04a89609281f50917": " V_d/V_t = \\frac{F_a - F_e}{F_a}", "6e6f4e610249a923e3bbe742e0d167d4": "P = \\frac{\\mu_o q^2 a^2 \\gamma^6}{6 \\pi c},", "6e70250b46e48fb5417ffb0f5085bf76": "\\Delta I = I_f - I_i = \\begin{cases} 0 & I_i = I_f = 0 \\\\ 1 & I_i = 0 \\text{ and } I_f = 1 \\end{cases}", "6e7063382d0e013022a5fece37b45a8f": "\\mathrm{D} F : C_{0} \\to \\mathrm{Lin} (C_{0}; \\mathbb{R})", "6e709d85cd5e95a48425d220998a2ac6": "f(q) = u q u^{-1}", "6e70d967d4a8b4f798482aa76ed5ec0a": "\\widetilde\\tau", "6e70eaf991f152e6835005fa99b3470a": " d_1 \\leq d_2 ", "6e70f02971e7c75b87c4e7fdd5b25b71": "\\sinh x = x + \\frac {x^3} {3!} + \\frac {x^5} {5!} + \\frac {x^7} {7!} +\\cdots = \\sum_{n=0}^\\infty \\frac{x^{2n+1}}{(2n+1)!}", "6e711c3a81b46e262dc7c49e07f725ae": "x_1 + x_2 + \\ldots + x_n = k.", "6e716b3ed2981e1118e0849f4621cd14": "a^p \\equiv a \\pmod p.", "6e716f470ba94b6ae2490f731bdb86e6": "\\sin^{\\frac{1}{2}}(x),", "6e717585ecdc8e04c570433b0f7d8bfe": "\\nabla \\cdot\\mathbf{U} = 0", "6e71970ee9eef8bd830c73d0c0168962": " (o_1, ..., o_t) ", "6e71a678c6a0c45ee1c70d1c269aa78d": "u = \\frac{J_2\\ P^0_2(\\sin\\theta)}{r^3} = J_2 \\frac{1}{r^3} \\frac{1}{2} (3\\sin^2\\theta -1) = J_2 \\frac{1}{r^5} \\frac{1}{2} (3 z^2 -r^2)", "6e71af6737127176c34b0932346463a7": "t_2-t_1 \\ = \\ \\delta t", "6e723090b0489e12382dacedb6bc746f": "\\begin{cases}4|a|,&a\\equiv2\\pmod 4,\\\\|a|,&\\text{otherwise.}\\end{cases}", "6e726202e8479646ae8c68cbbcdc30fe": "\\mathrm{div}(f)> D", "6e729d452be44108bf47051bfa34fb6e": "L^2 (\\mathbb R, dx )", "6e72a3b21252f6b2365df8aae3512210": "VATX", "6e72a40844f9a3474890d59f2ef59618": "\\gamma(q) = \\sum_{n\\ge 0} {q^{n^2}(q;q)_n\\over (q^3;q^3)_{n}}", "6e736e178cfef28f9718fe24683f4fac": "f^{-1} : H \\rightarrow G", "6e73991ce1f447e14ccb89e3cdae19bf": "T_E \\ne T_D\\,,", "6e739c59a78c6ee936fcda6b63b4a814": " w^2+y(x^2+y^{n-2}) = 0 ", "6e73cb84a5516d3876f271a0f2275aba": "E(u)=\\frac{1}{2}\\int_{\\R}\\left(|\\frac{\\partial u}{\\partial x}|^2+G(|u|^2)\\right)\\,dx", "6e73ce48f8ab465c1978fe7e9b279c56": "s=\\frac{C \\theta}{400}.", "6e742c36d6ea60d3330f5543b5c51719": "\\varepsilon_{ij} = \\begin{bmatrix}\n\\varepsilon_{11} & \\varepsilon_{12} & 0 \\\\\n\\varepsilon_{21} & \\varepsilon_{22} & 0 \\\\\n0 & 0 & \\varepsilon_{33}\\end{bmatrix}\\,\\!", "6e7465f4041d51f1a75324d91f585128": "v(t) = \\frac{1}{2\\pi} \\frac{d\\Phi(t)}{dt}", "6e748544e55cb51c43b00855b1bcb0ec": " \\to ", "6e74c5dfc3a2591136cda23d1570bb85": "\\scriptstyle \\sum_{n>0} C n p^n=p/(1-p) ", "6e754826d01dc9feec2a3c6b8a3a473e": "\\frac{\\mathrm{d}v}{\\mathrm{d}t}=g-\\frac{kv^2}{m}", "6e7558c9644b20ca0243229ef4636e92": "H|\\uparrow\\rangle = \\frac{1}{\\sqrt{2}}|\\uparrow\\rangle + \\frac{1}{\\sqrt{2}}|\\downarrow\\rangle", "6e759166cf2e587725da6282a8d0f0b3": "\\sigma_{yy} - \\frac{\\sigma_{yz}\\sigma_{xy}}{\\sigma_{xz}}", "6e75c424aac44ed5b7aecab42d6230a3": "\\int d^2 \\theta\\; \\lambda_2\\; Q D^c L ", "6e767c4e040672c8eb0c309d75840eab": "3.7 \\times \\triangledown^{0.1667}", "6e76a93c86536966a2d2f078e850d1d8": " E(R_i) ", "6e76b6b805d7e882ac2e65829cd842a8": "g(v,w) = \\phi_*(v)\\cdot \\phi_*(w).", "6e76e26f51d39c0ae142f3590f7e400e": "\\nabla \\times V=0", "6e7700c407568d6a7f67e535971c85cd": "\\begin{align}\n \\operatorname {arsinh} (x) &= \\ln \\left(x + \\sqrt{x^{2} + 1} \\right) \\\\\n\n \\operatorname {arcosh} (x) &= \\ln \\left(x + \\sqrt{x^{2} - 1} \\right); x \\ge 1 \\\\\n\n \\operatorname {artanh} (x) &= \\frac{1}{2}\\ln \\left( \\frac{1 + x}{1 - x} \\right); \\left| x \\right| < 1 \\\\\n\n \\operatorname {arcoth} (x) &= \\frac{1}{2}\\ln \\left( \\frac{x + 1}{x - 1} \\right); \\left| x \\right| > 1 \\\\\n\n \\operatorname {arsech} (x) &= \\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1 - x^{2}}}{x} \\right); 0 < x \\le 1 \\\\\n\n \\operatorname {arcsch} (x) &= \\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}}}{\\left| x \\right|} \\right); x \\ne 0\n\\end{align}", "6e772d2b7685f0153ebca621a471624c": "(f_B)^p \\leq \\frac{c}{\\omega(B)} \\int_B f(x)^p \\, \\omega(x)\\,dx", "6e775f9c8da5c436f9a89c7fed67f238": "\\mathbf{S}_{1}\\psi \\equiv (\\mathcal{S}_{10}\\cosh (\\Delta )+\\mathcal{S}\n_{20}\\sinh (\\Delta )~)\\psi =0,", "6e77aec83a7a59c33c71a250f01d26ea": "\\operatorname{Ass}_R(M)\\,", "6e77d96fdfad753f198f1bb22ff46241": "\\phi(k)\\,", "6e77e2571c27ac506cc3b2e60b170c52": "C[C'^n[u]] \\in L", "6e78208f78003725e4e476dbd4dc5070": " \\hat{v_k}. ", "6e782b9ada370a25018044618b1219d5": " \\mu = \\operatorname{E}(Y) = A'(\\theta). \\,\\!", "6e7845551930f9e578fb1bbf171f3795": "R= \\frac{I(X;Y)}{H(X)+H(Y)}", "6e7848dbc92b28745ac7098d92796eb5": "\\sum_{t=1}^\\infty\\left [R_t(L_t)-w_t(L_t,L_{t-1})L_t\\right]\\left(\\frac{1}{1+r}\\right)^{t-1}\\,\\!", "6e786176dfe0682308f307623c21b5aa": "(Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 (1-s_3).\\,", "6e788f1c912420816bb44a1dccfa69b8": "(x,y)\\to(x-\\bar{x},y-\\bar{y})", "6e78c730152d968aff30b9dabcb27eb2": "\\sum_{i=1}^n x_{ij} = 1", "6e78cc9aadb2bfd1954aab866d9a7820": "\n= \n {1 \\over 4 \\pi r } \\exp \\left ( - m r \\right ) \\left\\{ 1+ {2\\over mr } \n- {2\\over \\left(mr\\right)^2 } \\left( e^{mr} -1 \\right) \\right \\}\n\\left\\{\\mathbf 1 - {1\\over 2} \\left[\\mathbf 1 - \\mathbf{\\hat r} \\mathbf{\\hat r}\\right] \\right\\}\n+\n\\int_0^{\\infty} {k^2 dk \\over \\left ( 2 \\pi \\right )^2 } \\int_{-1}^{1} du {\\exp\\left( ikru \\right) \\over k^2 + m^2} \n{1\\over 2} \\left[ \\mathbf 1 - \\mathbf{\\hat r} \\mathbf{\\hat r} \\right]\n", "6e7924c724a3424fe7805a9c84eda9f7": " k_B ", "6e792b3f66839454d532024a781f374f": " V_{anticyclone} = \\frac{ V_{inertial} }{2} - \\sqrt{ \\frac{ V_{inertial}^2 }{4} - V_{cyclostrophic}^2 } ", "6e796f28dff4427b9910798374218ee4": " L = \\lambda - (1-C) f \\sin \\alpha \\left\\{ \\sigma + C \\sin \\sigma \\left[\\cos (2 \\sigma_m) + C \\cos \\sigma (-1 + 2 \\cos^2 (2 \\sigma_m)) \\right]\\right\\} \\, ", "6e79b2f5cec4a5c90580af56ebe66571": "\\mathbf{w}_{n}", "6e79d33737d84686cd743002f493f295": " \\int\\limits_{y = g(x_1, \\cdots, x_n)} \\frac{f(x_1,\\cdots, x_n)}{\\sqrt{\\sum_{j=1}^n \\frac{\\partial g}{\\partial x_j}(x_1, \\cdots, x_n)^2}} \\; dV", "6e7a4035855b898f369b9804fad110ac": "a_{15}=(1-(4/5))a_{11}", "6e7a6a9d1fb274c9459bd41fc5fb5ae9": "\\mbox{ASUI} = \\frac{\\sum{U_i N_i}}{\\sum{N_i} \\times 8760} = 1- \\mbox{ASAI}", "6e7a7e157d78768daf3d93ae7f7e99c0": "\\rho=\\frac{M}{\\frac{4}{3}\\pi R^3}", "6e7ac20beffe4dd1937e10ab1da95728": "(v/c)^2", "6e7ade4bf86ddef9f0077f9019a9a112": "P=RT\\frac{\\exp{(\\frac{-a}{V_mRT})}}{V_m-b}", "6e7b986df762e4c9e3e09774285af8b7": "x_b=(x^1_b,x^2_b,x^3_b) ", "6e7c9d32592b901e39d38a5ada5b88e2": "p_n(x)=x(x-an)^{n-1} \\,", "6e7cc61cb8e5897850698d1b6840422b": "\\scriptstyle \\frac{1}{\\lambda}=\\frac{f}{c}", "6e7ccac47440f9f64c35eb75b39a99bf": "d(t) = \\epsilon (M(t) + O(\\epsilon))", "6e7d0c320fe49de7653d240750e51f44": " 0<\\varepsilon <1", "6e7d4824178f8630c7d113b05150e745": " \\propto (\\sigma^{2})^{-n/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)\\right)", "6e7d84595535d43c24efd6a49f058cdc": "x\\in f(y)\\,", "6e7dce09097de2e838837123de743483": "M_{0}=q", "6e7e304f6ad4637d1a6ec3ad82bb31c4": "Y_{\\ell, m}(\\theta, \\phi) = \\sqrt{\\frac{(2\\ell+1)(\\ell-m)!}{4\\pi(\\ell+m)!}}\\ P_\\ell^{m}(\\cos \\theta)\\ e^{im\\phi}\\qquad -\\ell \\le m \\le \\ell.\n", "6e7e49085e1eeda0ff23802da2a1fbc8": "m = \\frac{x+y}{2}, h=\\frac{y-x}{2}", "6e7e6199d23e05d9ae29d1ef6eba789c": "\\phi(\\mathbf{r})", "6e7ed14b1b5740d136274eab233296ea": "\\begin{bmatrix} \\eta_1+\\frac12 \\\\ -\\eta_2 + \\dfrac{\\eta_3^2}{4\\eta_4} \\\\ -\\dfrac{\\eta_3}{2\\eta_4} \\\\ -2\\eta_4 \\end{bmatrix} ", "6e7f3d8085796f0fe74747fbde317630": "M_2 = M_3", "6e7f4bc7871eb3966a8c02dd49150487": "\\mathbf{x}, \\mathbf{y} \\in [\\mathbf{x}]", "6e7f8e3c15ea004a250599ca9e2a22d1": "\\forall u\\,(w\\; R\\; u \\to u\\Vdash A)", "6e7fffd4ad4ff39ab28c79249111e47f": "R_{\\max }=\\frac{\\min (H(X),H(Y))}{H(X)+H(Y)} ", "6e80b375b0acb35d1de42330ae358313": "I_\\mathrm{SO}\\,", "6e80c0c55496c388fd86b3bf316446c2": "I_{R1} = I_{C1} + I_{B1} + I_{B2} \\ ", "6e80c3b99f10fe83ad6b0f4d55148464": "a(v, w) =a(w, v)\\, ", "6e80eb0f2ed8e72d12d3f3467d22baed": "\\mathrm{lcm}", "6e80f11e7d4fa4186c822b1e325ebee6": "\n \\nabla ^2 U(x,y,z) = \\frac{\\partial^2 U}{\\partial x^2} + \\frac{\\partial^2 U}{\\partial y^2} + \\frac{\\partial^2 U}{\\partial z^2} = 0,\n", "6e8157da2e47ed23716729e05b3ed178": "\\zeta\\left(\\frac12 + it\\right) \\mbox{ is }\\mathcal{O}(t^\\varepsilon),", "6e81740ec6f2d7a64dece8bf913393bd": "g_{ab} = \\eta_{ab} + h_{ab}", "6e819457db3c93b7d215e5b76cbbc2a5": "P_G(k)", "6e81f4aa8777163b10dbadfe5a5d6f51": "\\lang n^{(0)} | n^{(1)} \\rang + \\lang n^{(1)} | n^{(0)} \\rang = 0", "6e82749dae024737b3919f32257bbd74": "\\alpha(d) > d+1", "6e82cf1ecc0332e3b124ea9ba2ecb893": " c_h, c_M\\mbox{ and }\\chi_T \\geq 0 ", "6e82effe0f2034cafc2863741a545398": "L=120000", "6e832c143e94f508ac67ddc0d93c99e6": "Hom(A,B)", "6e836ee762ab8e4a167d9d6521d73b42": "\n\\begin{align}\n& {} \\qquad \\frac{n!}{k!(n-k)!} + \\frac{n!}{(k-1)!(n-k+1)!} \\\\\\\\\n& = \\frac{(n-k+1)n!}{(n-k+1)k!(n-k)!}+\\frac{kn!}{k(k-1)!(n-k+1)!} \\\\\\\\\n& = \\frac{(n-k+1)n!+kn!}{k!(n-k+1)!} \\\\\\\\\n& = \\frac{(n+1)n!}{k!((n+1)-k)!} \\\\\\\\\n& = \\frac{(n+1)!}{k!((n+1)-k)!} \\\\\\\\\n& = { n+1 \\choose k }.\n\\end{align}\n", "6e83d0dc0ecb1b88216a3c4246346408": "= x(t) \\ T \\sum_{n=-\\infty}^{\\infty} \\delta(t - nT) \\ ", "6e841668ed6f696bd16f9f7bdf2f9a1c": "\n\\begin{align}\n\\phi = \\sum_i c_i \\mathbf{v}_i.\n\\end{align}\n", "6e844a6c91f30f658c3229d85567d408": " J_1 ", "6e84b0b6fe4f22ba5b79b95d90d30282": "K=\\int d^3r\\vec{A}\\cdot\\vec{B}", "6e85289a3e3df758f5e468bb85c4fef7": "{a\\choose b}", "6e854882b4a120efbc8a98d445fdeb05": "\n\\beta_{L} = \\frac{cov(\\frac{EBIT(1-T)}{E_{L}},r_{M})}{\\sigma^{2}(r_{M})}\n", "6e85c72537b46bcf69adbf37850c4e7d": "12.61 \\le \\beta \\le 17", "6e85ed16b150a2e8c2dd03d09406ff4e": "O(n)\\,", "6e864cb7b00122144938fa6a23fb9e7f": "\\sum_{k=2}^\\infty (\\zeta(k) -1) = 1", "6e868c81904ec10c810ca2614e60bca6": " \\sigma\\ = \\frac{b-a}{6} ", "6e86bb551b6ff8aa9ef63e1afc5b8413": "\n \\begin{cases}\n 0 & \\text{for }k<0 \\\\ q & \\text{for }0\\leq k<1 \\\\ 1 & \\text{for }k\\geq 1\n \\end{cases}\n ", "6e86c46b8f3fba55d0f3fd56f45c25f4": "d_{1,2}", "6e86cb4db88787448641d4193a669939": " \\frac{I}{2} ", "6e871d1cc5eb878abd1a479377eb3ea2": "C_{\\beta I}^{\\;\\;\\; K} e^{[\\alpha}_K e^{\\beta]}_J + C_{\\beta J}^{\\;\\;\\; K} e^{[\\alpha}_I e^{\\beta]}_K = 0 .", "6e8735e4707c0e1f4fc8a413091cc275": "\\bar p(n) = \\prod_{k=1}^{n-1}\\left(1-{k \\over 365}\\right) < \\prod_{k=1}^{n-1}\\left(e^{-k/365}\\right) = e^{-(n(n-1))/(2\\times 365)} .", "6e8753bf44d1444621a910953f8b719b": " \\; \\epsilon_{\\mathrm{n}} = \\epsilon - K_{\\mathrm{p}}...........(16) ", "6e879aff489a761c89b6b5f033107628": " ( )\\,", "6e8838ac99d26c76dde5dfb26e65ab67": "F(x,1) = \\frac{\\beta}{\\alpha+\\beta}\\left(1-e^{\\left(\\frac{\\beta}{\\mu}-\\frac{\\alpha}{\\lambda-\\mu}\\right) x}\\right)", "6e887311c40c4d8331f08833691ce027": "=\\frac{gt(2v-gt)}{2h(2v-gt)}", "6e88c91a42fd3beacd02250476d47af5": "s^2 \\,", "6e88fbbd152ab0a25a7bdf0d886ee089": "\n \\sigma_{11} - \\sigma_{33} = \\lambda_1~\\cfrac{\\partial{W}}{\\partial \\lambda_1} - \\lambda_3~\\cfrac{\\partial{W}}{\\partial \\lambda_3} ~;~~\n \\sigma_{22} - \\sigma_{33} = \\lambda_2~\\cfrac{\\partial{W}}{\\partial \\lambda_2} - \\lambda_3~\\cfrac{\\partial{W}}{\\partial \\lambda_3}\n ", "6e88fda817872745579e0187978e3d2a": "x_i'", "6e893ea1b7e53b406b124c17ff1bcad2": "\\mathrm{crd}^2 \\theta + \\mathrm{crd}^2 (180^\\circ - \\theta) = 4 \\, ", "6e8965abeb77b7a28ffb527d81b062aa": "\n\\frac{\\Delta L}{L} = \\alpha_L\\Delta T\n", "6e89c78ef5699f99c974074e17b13cf9": " \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} a x^2}\\,dx = \\sqrt{2\\pi \\over a} ", "6e8a467dbee08e119340e653326b2103": "J(K)", "6e8aa828650eac980d5d6c350281f442": " \\epsilon \\rightarrow \\infty ", "6e8ad7ec5b4e0fb0be2aadc17d93ee09": "{2}( \\alpha + \\beta ) =180^\\circ ", "6e8b492c981d4af6cc13c6ff4a59e3a9": "d_1,d_2,\\ldots,d_k", "6e8b5ad7b289d8de318b467529fcec35": "g:2^X\\to[0,1]", "6e8b5cb39bf7208f9a8ee74b262da556": "r=\\mu^{'}_1/\\mu^{1/2}_2", "6e8c505f801f6eccccf69ec3d2f1f81b": " df(k)/dk ", "6e8c756549b1454ab7d7fc4406062cb7": "\\beta_i\\beta_k\\sigma^2", "6e8c761044fd431af6d3a40da4186aca": "r = f \\cdot \\theta", "6e8d13b36803bc84dc7464b119d31e6d": "p \\cdot (\\Sigma _i x_i) \\geq r", "6e8d604e6ccb048d3b6c844158ccce3a": " \\forall \\varepsilon > 0\\ \\exists \\ \\delta 1 > 0 : \\forall x\\ (0 < |x - a| < \\delta 1 \\ \\Rightarrow \\ - \\varepsilon < g(x) - L < \\varepsilon).", "6e8d6ff574248a3b54b42d17e8593cd3": "\\frac{(P \\leftrightarrow Q)}{\\therefore (P \\to Q)}", "6e8dae2ad64d5130e984691d0b16a143": "U_\\tau(\\lambda,a,b)\\;f(z)=\\lambda (S_a \\circ T_b f)(z) = \n\\lambda \\exp (i\\pi b^2 \\tau +2\\pi ibz) f(z+a+b\\tau)", "6e8dd841e51476e7c7096eb5d1ba1918": "(x_1, \\dots, x_i)", "6e8de35022734ea8ce658c2d8dff02bd": "\\beta_3", "6e8e106b827fd92dedc534bb99557601": "\\epsilon \\circ \\eta = \\eta_0 : K \\to K", "6e8e15d795fc65c5432fa9904f8fc511": "\\mathrm{d}\\,S = \\frac{\\delta Q_\\text{rev}}{T}", "6e8e46121257e9cc84eb3597c1557a0c": "\n\\begin{align}\n\\int \\frac{\\delta V}{\\delta \\rho(\\boldsymbol{r})} \\ \\phi(\\boldsymbol{r}) \\ d\\boldsymbol{r} \n& {} = \\left [ \\frac{d}{d\\varepsilon} \\int \\frac{\\rho(\\boldsymbol{r}) + \\varepsilon \\phi(\\boldsymbol{r})}{|\\boldsymbol{r}|} \\ d\\boldsymbol{r} \\right ]_{\\varepsilon=0} \\\\\n& {} = \\int \\frac {1} {|\\boldsymbol{r}|} \\, \\phi(\\boldsymbol{r}) \\ d\\boldsymbol{r} \\, .\n\\end{align}\n", "6e8e48773f21b9ab72e86c508dd91b47": "P(x_1, ..., x_n) \\log_2[P(x_1, ..., x_n)]", "6e8e85a8f629051aabd8ed01987a8887": "P_n\\sqrt2", "6e8f5058f578bb91e676c8e345d13cf8": "\\mathbf{F} = \\Lambda\\alpha{.}\\lambda x^{\\alpha} \\lambda y^{\\alpha}{.}y", "6e8f681439d81eb5b088aaf275ef4d6a": "g_{ij} = 1", "6e8f725a1b2001564d97ff24ee57da58": "\\alpha\\colon S^1 \\to X", "6e8fd65cabf7cca7d8404151b1052295": "K \\subseteq \\mathbb{R}^d", "6e901ec6b465da454a191f9e77d533c5": "(a_0, a_1, a_2, a_3, \\ldots) = \\sum_{i=0}^\\infty a_i X^i, \\qquad (1)", "6e90a757647942e4ee2b03528a0c4785": " 10^6 ", "6e90feb672c2c68072f5fba8847065f1": " p(x)=x^n+p_{n-1}x^{n-1}+\\cdots+p_0 ", "6e910dd46738ebf411da92947827f81b": "\\mathrm{tr}[R]", "6e912f285c0fbc839f50ea987d0846c7": " g(E_f) ", "6e91ec88558f1cf5357c65ab7ee6469b": "\\tau(f).", "6e91fc37942c41c7a30318ecc91bc720": "\\frac{d^2\\Gamma}{x^2dxd\\cos\\theta} \\sim (3-3x) + \\frac{2}{3}\\rho (4x-3) + P_{\\mu}\\xi\\cos\\theta\n[(1-x)+\\frac{2}{3}\\delta(4x-3)],", "6e9214b3e8269a940ece937666eb6cdd": "\n\\alpha_{H_2 A}={{[H^+]^2} \\over {[H^+]^2 + [H^+]K_1 + K_1 K_2}}= {{[H_2 A]} \\over {[H_2 A]+[HA^-]+[A^{2-} ]}}\n", "6e924e04b5c9d4c5be131609a038b821": "\\textstyle 1", "6e92e58fc27669559781d045290675e4": "\\text{E}(u(c))=\\text{E}[1-e^{-a (c(x)+ \\epsilon)}] = \\text{E}[1-e^{-a c(x)}e^{-a \\epsilon}] = 1 - e^{-ac(x)}\\text{E}[e^{-a \\epsilon}] = 1 - e^{-ac(x)}e^{-a \\mu + \\frac{a^2}{2}\\sigma^2}.", "6e9319dbcc08a996a9b986e97a61d890": " d_{\\mathrm H}(A,B) = \\| h_A-h_B\\|_\\infty", "6e9332bbf8cf3fbdd170fe6a6edf68c3": "ax^2+2bx+c=0", "6e936371f5b706f27386bf72dd4240f7": "d2=dimeter - 1.082532 * P", "6e93733a9654c6ff45a8f7faa3f4732f": " \\mathbf{F} = \\frac{\\mathrm{d}\\mathbf{p}}{\\mathrm{\\mathrm{d}}t} = \\frac{\\mathrm{d} (m\\mathbf{v})}{\\mathrm{\\mathrm{d}}t}.", "6e939b49fd18d343fb579eb06765bc46": " L = \\sum_{i=0}^{N-1} p_i x_i ", "6e939e4f0de42bcfc4917ed6c7af81d1": "|F_n(S)|=1", "6e93b54092a96360a80966b91a211d65": "\\left|\\tfrac{Q}{k!}\\right|<1", "6e944b7a70df72bdc4c3989fcf464642": " x^{(3)} = \\cdots. \\, ", "6e9471489dc4ac91a60e79bbc6ae1669": "D_{n+1} \\to B_n", "6e94c59d3031f96e0a84127f609a8442": "{\\Bbb E}(\\operatorname{cr}_H) \\geq {\\Bbb E(e_H)} - 3 {\\Bbb E}(n_H).", "6e952fb26581d5cfb531d8fb6013e37a": "\\psi(x_i)", "6e9542faa6e377862cb83aa9095e63d3": "a_{i\\, j} = ({a_i\\, a_j})^{1/2}", "6e9569526d86be8f8f6ecd57a0c5f556": "a^2 + b^2 = c^2\\, ", "6e958f93ce5c274d283e028a5ba7fb30": "\n\\frac{{\\rm d^{2}}x}{{\\rm d}z^{2}}+\nD(x)\\frac{{\\rm d}x}{{\\rm d}z}+\n\\Phi'(x) =0,\n", "6e95925b90ccdc837ed624a5f2053436": "\n\\sigma _z^2 \\approx \\,\\,\\,\\frac{a^2 }{n}\\,\\,\\left[ {\\left( \\alpha \\,\\mu _1^{\\alpha - 1} \\mu _2^\\beta \\right)^2 \\sigma _1^2 \\,\\,\\, + \\,\\,\\,\\left( \\beta \\,\\mu _1^\\alpha \\mu _2^{\\beta - 1} \\right)^2 \\sigma _2^2 \\,\\,\\, + \\,\\,\\,\\left( 2\\alpha \\,\\beta \\,\\mu _1^{2\\alpha - 1} \\mu _2^{2\\beta - 1} \\right)\\sigma _{1,2} } \\right]", "6e95acf129c477dcf341c0eb951139c3": "\n a_{mn} = \\frac{4q_0}{ab}\n \\int_0^a \\int_0^b \\sin\\frac{m\\pi x}{a}\\sin\\frac{n\\pi y}{b}\\,\\text{d}x\\text{d}y \\,.\n", "6e95ce08b4a84dd52395ee02e8790eb6": "\\alpha_1 \\,", "6e95d5641ba3de4d0672b8053e08746e": "\\frac{3x + 5}{(1-2x)^2} = \\frac{13/2}{(1-2x)^2} + \\frac{-3/2}{(1-2x)}.", "6e95e102ecb67723b5eb4385aca91834": "\\neg(A \\or B)", "6e95ff2805b9a5144fc1500439c64c87": "S^6=G_2/SU(3), Sp(2)/SU(2)\\times U(1), SU(3)/U(1)\\times U(1), S^3\\times S^3", "6e967854a28dfad133b97022a0fc2378": "\n\\mathbf{g}_c = \\omega^2 \\mathbf{p},\n", "6e969e5f988c128843b97f0e030ff42c": "g^*(R^r f_* \\mathcal{F}) \\to R^r f^'_*(g'^*\\mathcal{F})", "6e96b4824efcc017d938818063debdec": "\\mathrm FM", "6e96f1fa5865f826e8b89f84e8c05f7d": "h_{B_1}(x)=|x|", "6e9714f93d82d9ba0a568ce05cecde99": "\\tfrac{3M+E-S}{8}", "6e9721113f08e9fcb796ca7241a4996c": "\\log (q+1)", "6e974f0da6839290924bde8320b30b01": " \\left ( \\frac{d}{dx} - r_{1} \\right )^{k} ue^{r_{1}x} = \\frac{d^{k}}{dx^{k}}(u)e^{r_{1}x} = 0", "6e97ca9406ba49dfc162ba6c313d2f85": "A = B^2=B^*B.\\,", "6e9821a99a8744c9b94aa92df67d4787": " \\sqrt{(-i\\hbar\\mathbf{\\nabla})^2 c^2 + m^2 c^4} \\psi = i \\hbar \\frac{\\partial}{\\partial t}\\psi. ", "6e983e85c6e735a02684a18f1937b7d4": "\\dot{\\eta}", "6e985a69d43946fb778bec1b3e546ba0": "1.6 \\times 10^{-12} \\frac{\\text{Sv}/\\text{h}}{\\text{Bq}/\\text{m}^2}", "6e989a013e7ddb92e9323c3d3d9234d6": "(-1)^k=\\binom{-1}{k}=\\left(\\!\\!\\binom{-k}{k}\\!\\!\\right) \\,.", "6e989b38dabbd252f524306685166ae4": "\nz_{min} \\le x + y \\le z_{max} \n", "6e98afcd2a43412c82c241587a4e91e1": " F: \\R^{(2n+m+1)} \\to \\R^{(n+m)}. ", "6e98b531bbe0b7765969aeeeb4ed132f": "f(S)=I(S;\\Omega-S)", "6e994d81313e5a8b0e12c80e4e450c4d": "s_0,\\ldots,s_{m-1}", "6e99888e74b6b2f6c58d90005570f1f1": "P(Z_n \\le z) \\approx F\\left(\\frac{z-a_n}{b_n}\\right) .", "6e9998ee14e84faf3c8e8d7ca40fa324": "|{\\downarrow}\\rangle", "6e99f4d0b5d148ec8da72bac56634b13": "\\dots \\rightarrow K_0 \\times K_0 \\rightarrow K_0.", "6e9a07cd3aeb2db7bd61a94aff6ffe5c": "y^2=-\\sqrt{q}", "6e9a3c574b81ad5ddbf113c5c4e04b31": "\\phi'_X \\, ", "6e9aa4d8d0905e4c0e579455f9a67333": "\\hat{N}_\\mathit{eff} < N_{thr}", "6e9ae46f327e69a5fc214b1d9712097f": "\n\\lim_{\\varepsilon\\rightarrow 0^+} \\int_a^b \\frac{f(x)}{x\\pm i \\varepsilon}\\,dx = \\mp i \\pi \\lim_{\\varepsilon\\rightarrow 0^+} \\int_a^b \\frac{\\varepsilon}{\\pi(x^2+\\varepsilon^2)}f(x)\\,dx + \\lim_{\\varepsilon\\rightarrow 0^+} \\int_a^b \\frac{x^2}{x^2+\\varepsilon^2} \\, \\frac{f(x)}{x}\\, dx.", "6e9ae91a96b2c8c73ab98590ad98d3f8": "p_k(E)=e(E)\\smile e(E),", "6e9b45b055b35a5e791e06a90ebb935a": "\n\\begin{align}\nR_{dq} &= \\sqrt{ \\frac{1}{N} \\sum_{i=1}^{N} \\Delta_i^2}\n\\end{align}\n", "6e9b5459cec6bc0973d5d27b22a02f06": "p = 10/1000 = 0.01", "6e9b964dcad288591d7dcebd9ce85fd8": "x_1^ix_2^j\\cdots x_D^k", "6e9c60db917045fed233c0c61bc63b09": "p_n\\, ", "6e9cbeffde87ed4c3b9c28638e964a73": "(X, \\omega)", "6e9cf2c6d9d50c4b91b23f829498eaf1": "\\mathbb{P}_{i_{1} \\dots i_{k}}^{X} (S) := \\mathbb{P} \\left\\{ \\omega \\in \\Omega \\left| \\left( X_{i_{1}} (\\omega), \\dots, X_{i_{k}} (\\omega) \\right) \\in S \\right. \\right\\}.", "6e9d96db84b99fdb16c5c9606727695d": "r_a \\approx \\alpha + \\beta r_b", "6e9dae4ee9488b53667f8eb59c96f712": "\\omega^2=[(\\Omega_c\\omega_c)^{-1}+\\omega_i^{-2}]^{-1}", "6e9de1d4502d3794fd0d62a11df59300": "\\frac{\\lambda ^2}{(-i t+\\lambda ) (i t+\\lambda )} = \\frac{\\lambda ^2}{t^2+\\lambda ^2}", "6e9df88bb0478788f5702dce1978e12c": " \\mathbf{f}^e ", "6e9e0a035104dc727dbe1e5759e98e54": "w=\\sum_{m>0} a_mx^m", "6e9e22075cd2eeece89d03d3f00c9fea": "|S_2\\rangle", "6e9e2c76e03fde56979b52855def6524": "Z_e = \\int_{0}^{Dmax}", "6e9eac448662ae17921e95311ca0d18a": "{{i}_{C}}={{I}_{S}}\\exp \\left( \\frac{{{v}_{BE}}}{{{V}_{T}}} \\right)", "6e9eae8bf79769365274eef6cf74e466": "\nf_X(x;n) = \\frac{1}{\\left(n-1\\right)!}\\sum_{j=0}^{n-1} a_j(k,n) x^j\n", "6e9eb2cec5674853d9cef46fda240e9a": "\\Delta{i}\\,", "6e9ef2cc0c749b80914383ab3d4387d8": " b_{s-j-1} = \\frac{(-1)^j}{j!(s-j-1)!} \\int_0^1 \\prod_{i=0 \\atop i\\ne j}^{s-1} (u+i) \\,du, \\qquad \\text{for } j=0,\\ldots,s-1. ", "6e9f0ad041bde9546bcab8369635bf70": "\\mathbb{A}^{n + 1}", "6e9f1cd5a0c95759c4774b695015f21a": "R_{\\text{INIC}} \\gg R_s", "6e9f5b37a7c89b69d43d63b373234f3b": "\\sqsupset", "6e9fa12b920eca6b8ae1b041fcf98e5a": "ax \\equiv 1 \\pmod{m},", "6e9fa928d250baf2d07c069eeb3cd056": " U_sU_\\omega = \\begin{pmatrix} -1 & 0 \\\\ 2/\\sqrt{N} & 1 \\end{pmatrix}\n\\begin{pmatrix}\n-1 & -2/\\sqrt{N} \\\\\n0 & 1 \\end{pmatrix}\n = \n\\begin{pmatrix}\n1 & 2/\\sqrt{N} \\\\\n-2/\\sqrt{N} & 1-4/N \\end{pmatrix}.", "6e9fa9809f5695d013419c81492707ff": "R = I\\cos\\theta + \\sin\\theta[\\mathbf u]_{\\times} + (1-\\cos\\theta)\\mathbf{u}\\otimes\\mathbf{u},", "6ea01c98dedc424dc844d68d8663c208": "\\boldsymbol\\theta = \\mathbf{b}(\\boldsymbol\\theta')", "6ea01d7af3ceaf751b78826d06fdb1bb": "r \\ ", "6ea029f091edf703a07b95b188dba843": "e^{-sT}", "6ea06b270ed39119b786d42537f05dd3": "r_{i+1}:=\\text{rem}(r_{i-1},r_{i});\\quad i:=i+1;", "6ea07f4de3f6f0aa6269d757127c44d1": " A_j \\supseteq A_{j+1}", "6ea0809ee74184fed05646cc5631d23e": " p \\rightarrow p'", "6ea0a95f41192a6ad436a6874796e0cb": "c\\in [0,1]", "6ea0f59bd79d2a40386c8175f0afac98": "\\sim \\in \\{ <, \\leq, \\geq, > \\}", "6ea10357eab4a025b6b048f92b890a31": " \\max E \\left[ \\int_0^T e^{-\\rho s}u(c_s) \\, ds + e^{-\\rho T}u(W_T) \\right] ", "6ea1380a1f8d63b81eddfecf79ef8d55": "\\theta=\\frac{t-\\pi}{2}", "6ea1b938e9d61b6e778e72f3d00b185f": "s^\\prime", "6ea1d9e1ea9c7898bf5558a338abc15d": " X_k ", "6ea20392c5f6dc32417c012dc709ae7c": "I=-G^{31}\\dot{X}_1", "6ea215c1b29ea59d546f194e79a92e47": "\nD_{ }^{ } = O^{-1} A O = O^T A O\n", "6ea283b920169bb348ef0938ee614fa7": " c'", "6ea2a9fb5fa0930d3c41ab1a903b0b80": "-\\mathbf{e}_{23}", "6ea313de80283b54db6d2273ea9b55f2": " d\\nu_t = \\theta(\\omega - \\nu_t)dt + \\xi \\sqrt{\\nu_t}\\,dB_t \\,", "6ea34e1eb3ca576c3600c828bbff5d6c": "I_2\\setminus I_1", "6ea36687b62a275f3279f544c5bf97c4": "\\langle \\phi(k_1) ... \\phi(k_{2n})\\rangle = \\sum \\prod_{i,j} {\\delta(k_i - k_j) \\over k_i^2 } ", "6ea36a11b8855c711a0aa9c38613d4e0": "\n\\begin{align}\nU(x,z)\n&= \\hat {f} \\left [a \\left[\\mathrm {rect} \\left (\\frac{x-S/2}{W} \\right) + \\mathrm {rect} \\left (\\frac{x+S/2}{W} \\right) \\right ] \\right ]\\\\\n&= 2W \\left[ e^{- i \\pi Sx/\\lambda z}+e^{ i \\pi Sx/\\lambda z} \\right] \\frac {\\sin { \\frac {\\pi Wx} {\\lambda z}}}{ \\frac {\\pi Wx} {\\lambda z}}\\\\\n&= 2a \\cos {\\frac { \\pi S x }{\\lambda z}} W ~\\mathrm{sinc} \\frac { \\pi Wx}{\\lambda z}\n\\end{align}\n", "6ea3834c81691edf874f74399933fbbf": " \\beta = \\beta(x)", "6ea3babc7aefe9b25a8af08fd567b3ea": "\\omega_B = eB/m- \\ ", "6ea416b527e628ecc48f0744181d7370": "\\mathbf{\\Sigma}^0_2", "6ea4f980252940bb01b35eebbf8d0657": "\\mathbf{v} \\cdot \\nabla \\rho + \\rho \\nabla \\cdot \\mathbf{v} = \\nabla \\cdot (\\rho \\mathbf{v})", "6ea5053184269d83e6858cb3b3beef08": "\\frac {d \\phi}{dx}", "6ea5775bd58544e9eb936d86d768d5ae": "M_{\\mathrm{max}}", "6ea583cc8c6e11f0b541f40154a4c083": " \\sigma(\\bar x)= \\sqrt {\\sum_{i=1}^n {w_i^2 \\sigma^2_i}}.", "6ea5ea758a46222235db13babe087adb": "\\begin{align}\n\\mathcal{F}\\left \\{\\sum_{n=-\\infty}^{\\infty} T\\cdot x(nT) \\cdot \\delta(t-nT)\\right \\} &=\\mathcal{F}\\left \\{x(t)\\cdot T \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\\right \\}\\\\\n&= X(f) * \\mathcal{F}\\left \\{T \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)\\right \\} \\\\\n&= X(f) * \\sum_{k=-\\infty}^{\\infty} \\delta \\left(f - \\frac{k}{T}\\right) \\\\\n&= \\sum_{k=-\\infty}^{\\infty} X\\left(f - \\frac{k}{T}\\right).\n\\end{align}", "6ea60fb071c61d2fb49a4e8654353d82": "\\mathbf{A} =\\frac{d\\mathbf{U}}{d\\tau}=\\left(\\gamma_u\\dot\\gamma_u c,\\gamma_u^2\\mathbf a+\\gamma_u\\dot\\gamma_u\\mathbf u\\right)\n=\\left(\\gamma_u^4\\frac{\\mathbf{a}\\cdot\\mathbf{u}}{c},\\gamma_u^2\\mathbf{a}+\\gamma_u^4\\frac{\\left(\\mathbf{a}\\cdot\\mathbf{u}\\right)}{c^2}\\mathbf{u}\\right)", "6ea6a95bb895cfe3a0f185f967ab4065": " S=\\frac{-1}{6} \\int d^{D}x (F_{\\alpha\\beta\\gamma\\mu}F^{\\alpha\\beta\\gamma\\mu}-3F_{\\alpha\\beta}F^{\\alpha\\beta}).", "6ea6be6bda5ce80d7d6449d04cdec5aa": "\\alpha=-0.5", "6ea6bf7b2354f8e0e9402ed4ede4bcd9": "\\begin{bmatrix}\nX_1^1 & X_2^1 & \\cdots & X_\\nu^1 \\\\\nX_1^2 & X_2^2 & \\cdots & X_\\nu^2 \\\\\n\\vdots & \\vdots && \\vdots \\\\\nX_1^{n-k} & X_2^{n-k} & \\cdots & X_\\nu^{n-k} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nY_1 \\\\ Y_2 \\\\ \\vdots \\\\ Y_\\nu\n\\end{bmatrix}\n= \n\\begin{bmatrix}\nS_1 \\\\ S_2 \\\\ \\vdots \\\\ S_{n-k}\n\\end{bmatrix}\n", "6ea6f3f1d6a5d00200ddde3e8aa01443": "y_2=y_1+h(\\tfrac14k_1 + \\tfrac34k_2)=\\underline{1.141332181}", "6ea71278a21cf7e9e8e4435c55870641": "P(k^{}_j=i)=w(n,i)\\big/{\\sum}_\\ell w(n,\\ell)", "6ea73441977231ef55e5dab0457d4fef": "\\scriptstyle{(r_{1},\\theta_{1})}", "6ea7b009bc9b37d1238538212a0e078a": "v(n)", "6ea7f840190b22b0a9a2f87dc2ed429a": " L_{total} = \\frac{L+M}{2}", "6ea84129e4cb5d7ada4781cd7735f65b": "\\mathrm{Sel}^{(f)}(A/K)=\\bigcap_v\\mathrm{ker}(H^1(G_K,\\mathrm{ker}(f))\\rightarrow H^1(G_{K_v},A_v[f])/\\mathrm{im}(\\kappa_v))", "6ea85fa12ef03cb0401914a7bf9512a9": "\\frac{1}{V} = \\frac{\\phi}{V_f} + \\frac{1 - {\\phi}}{V_{mat}}", "6ea8826d29b650bb2d8190fb81da8874": "x\\in{\\rm cl}_X(Z)", "6ea8882d1dd784cc2d343cc05b08cc29": "P_{3}^{3}(x)=-15(1-x^2)^{3/2}", "6ea8aaf56f03c02816c13cea241125d1": "\\mathbf{B} = 0 = \\nabla \\times \\mathbf{A}", "6ea8c93a7d2142799353e6437255d165": "N_l \\leq \\frac{1}{2l+1} \\frac{2m}{\\hbar^2} \\int_0^\\infty r |V(r)|_{V<0}\\, dr", "6ea8ce228929b9cf1a3de3c2f428ea00": "\\langle i\\rangle", "6ea8d09fcc72183bcd25a102effdc305": " FT(f) = F_1(w) ", "6ea9146fcced3e372745f9b547c4eb18": " \\phi \\ ", "6ea9788639fe8ffc3537e39ed74db6af": "V_\\mathrm {dc}=V_\\mathrm {av}=\\frac{2V_\\mathrm {peak}}{\\pi}", "6ea97b34acb40945427539a76a2c8af0": "\\operatorname{logit}^{-1}", "6ea97c8bbe7189c0ed35f316cfbb2f92": " e^{-q \\tau} \\frac{\\phi(d_1)}{S\\sigma\\sqrt{\\tau}} \\, ", "6ea9ab1baa0efb9e19094440c317e21b": "29", "6ea9af8a6cf452cd6238c77e9d24df46": "L=V", "6ea9b54e20289c3c32b98d267d924c29": "\\mathrm{soc}(R_R)=\\{0\\}", "6ea9cfae257c570a5c466e05b43e0c90": "\\sum_{i=1}^{n}\\left(y_{i}-\\bar{y}\\right)^2", "6ea9e58e2348c751ea31db3470c27904": "g_s=2", "6eaa2f53b54abf22448feca78fe7be10": "\\mathbf{g}(\\mathbf{r}) = -Gm\\frac{\\mathbf{e_r}}{r^2}", "6eaa38e4f3ce426aaba996fb32d04fc8": "\\mathbf{u}_2", "6eaa427f381f00698669991f7bc004b5": "(r \\cdot p) \\cdot s = p \\cdot s = s.", "6eaa4e2e1640af857ee021d49afefc25": "y^{(3)}(t)", "6eaa54a6296de261443ad99740b3fe93": "\\left(a + \\omega \\right)", "6eaa5685791b5cc8a5548745ae0eecc9": "a_{k} = {d_{1}^{k}}", "6eaa790602fc52a9dad6dda22e392911": " \\varnothing ", "6eaaea65cda6d77da1e5e3eacf35238c": "\n\\frac{dT}{d\\boldsymbol\\omega} = \\mathbf{I} \\cdot \\boldsymbol\\omega = \\mathbf{L}.\n", "6eaaf67c80f8c01ba2ecac6496b725c7": "~~~~~\\frac {1}{T},V,\\{N_i\\}\\,", "6eab0b68ab9c9d821809178e58cd5c72": "x^n = \\underbrace{ b^\\frac{1}{n} \\times b^\\frac{1}{n} \\times \\cdots \\times b^\\frac{1}{n} }_n = b^{\\left( \\frac{1}{n} + \\frac{1}{n} + \\cdots + \\frac{1}{n} \\right)} = b^\\frac{n}{n} = b^1 = b", "6eab9ef8ae01b367b73c90229da3f728": "\\frac{dN}{dE}", "6eabb3e30b5f046f76e2bfb4c7f6713c": "{}_s\\!\\dagger\\!(X)_N", "6eac04646425d0817ca26e873a7e4de1": "\\epsilon^{\\prime}", "6eac110a9c4830c391e7b2cd6786e949": "H^i(X, F)", "6eac15e28ad4754eac13bce85c304a5e": "f \\ast \\Phi_t(x)", "6eac51753c30e3043e269ed2328b9830": "\n\\lim_{x\\to0} \\left( \\lim_{y\\to0} \\frac{xy}{x^2+y^2} \\right) = \\lim_{x\\to0} 0 = 0\n", "6eac76ebde69029cb67f7061f802a3c3": " A(0, R, \\infty) ", "6eac879ea94abddd9430c1a718d28e9a": "(t_i)_{i\\in I}", "6eaca9df7d2323117920dc8b53429fed": "base \\equiv_{(base - 1)} 1", "6eacde1db0b5749f68f834dc86e4dcbe": "|F_n| \\sim \\frac {3n^2}{\\pi^2}.", "6ead23967c764a64208595161e8712d3": " \\dot x(0) = 0.\\,", "6ead5985efe4f984e4981b74284b1094": "2^k - 1", "6ead6ca9e8a3959fdd246847848f6d80": "\\alpha \\! \\left( \\nu \\right) = \\frac{\\partial \\log S \\! \\left( \\nu \\right)}{\\partial \\log \\nu}.", "6eadd93217f3cfe088641860ded77ab5": "T_{11}^{\\mathrm{eng}} = T_{11}\\alpha_2\\alpha_3 = \\cfrac{T_{11}}{\\alpha} ", "6eadf6e2b6992099f3310b169bfe03d2": "{\\tilde{A}}_{10}", "6eae1b1f7063017a69e133f117267e6b": "\\textstyle z^p - \\sum_{i=1}^p \\varphi_i z^{p-i}", "6eae2c3f0e72d0d077d75bfa229e2064": " \\epsilon = \\epsilon_k+\\epsilon_p \\!", "6eae908ea31c325fb70f9b7ec4ae7405": "X(t)=\\sum_{j=1}^{n}a_j(t)\\exp\\left(i\\int\\omega_j(t)dt\\right).\\,", "6eaecd41bd117d8c9ffeb40ab4315ad3": " \\mathit{x}_\\mathit{e} ", "6eaee22b4fe5a3256f13dfc41277d786": "e \\in BSC_{p}", "6eaf07bacf3347c099f40d28b53f0767": " \\mathbf{B} = \\mu_0 \\mathbf{H}, \\,\\!", "6eaf1337edd369fb541ef610aece6690": "WXYZRSTUVPQ", "6eaf1b69a19e53d58725ee114e484154": "P(x)=(x+1)(x-2)(2x^2-x+4).\\,\\!", "6eaf2d97a359e4713cfcbed2ce01378a": "x := e\\,\\!", "6eaf2f852382b534056a9cc9334ef92c": "\\mathbf{R}=\\frac{1}{M}\\sum_i m_i \\mathbf{r}_i", "6eafca7d3bbdb6fa36c3c07f10586c09": " f_{ii}^{(n)} = \\Pr(T_i = n)", "6eafe433af408c43ee062c28a804bd33": "b_1 = \\tilde{e}_i b_2", "6eaff967c9b841630f326ca807862cc2": "2^{61}-1", "6eb01dad68d5bbca91c82ebd2eacc1ad": "\n (10)(x) - R_a(x-10) - R_b(x-25) + (1)(15)(x-17.5) + M_3 = 0 \\,.\n ", "6eb02da07022427924c60f268b929443": " \\frac{1}{p!} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} a_{\\mu_1 \\dots \\mu_p} = a_{\\lbrack \\nu_1 \\dots \\nu_p \\rbrack} .", "6eb03e7a3959e7ac5f46d847cc4450dd": "\\begin{align}\nR &=& Y &&& + 1.402 & \\cdot (C_R-128) \\\\\nG &=& Y & - 0.34414 & \\cdot (C_B-128)& - 0.71414 & \\cdot (C_R-128) \\\\\nB &=& Y & + 1.772 & \\cdot (C_B-128)&\n\\end{align}", "6eb046afeb3adedc74a4644bc657161c": "E^2", "6eb0b1b4f7cbb0549693a8881595ec2b": "\\ ln\\left [ \\frac{C_{ending}}{C_{initial}}\\right ] \\quad={-}\\frac{Q}{V} \\cdot (t_{ending}-t_{initial})\\quad ", "6eb0b56fc1e8118a8bb11a28ae8b7102": "a_1 - a_2 \\equiv b_1 - b_2 \\pmod n\\,", "6eb0f5dc3d08b1bf0e18cf27d224e1c0": "\\neg (P \\and Q) \\to (\\neg P \\or \\neg Q)", "6eb1c9af90185fbfea3a53db8aafc235": "\n \\cfrac{\\Gamma, A[y/x] \\vdash \\Delta}{\\Gamma, \\exist x A \\vdash \\Delta} \\quad ({\\exist}L)\n ", "6eb20d594d5ce841cf1b6989696a4cad": "\nE(x,y) = \\arg\\max_{a^*} \\; E_{Pr[a|x,y]}(acc(a^*,a))\n", "6eb216ca40cdc9228fcbee176be74e0b": "\\begin{align}\n\\Vert\\vec a\\Vert^2 &= \\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - 2 \\Vert \\vec b\\Vert\\Vert\\vec c\\Vert\\cos\\theta \\\\\n\\Vert\\vec b - \\vec c \\Vert^2 &= \\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - 2 \\Vert \\vec b\\Vert\\Vert\\vec c \\Vert\\cos\\theta \\\\\n2 \\Vert \\vec b\\Vert\\Vert\\vec c \\Vert\\cos\\theta &= \\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - \\Vert\\vec b - \\vec c \\Vert^2 \\\\\n\\Vert \\vec b\\Vert\\Vert\\vec c \\Vert\\cos\\theta &= \\frac{\\Vert\\vec b \\Vert^2 + \\Vert\\vec c \\Vert^2 - (\\Vert\\vec b \\Vert^2 - 2 \\vec b \\cdot \\vec c + \\Vert \\vec c \\Vert^2)}{2} \\\\\n\\Vert \\vec b\\Vert\\Vert\\vec c \\Vert\\cos\\theta &= \\vec b \\cdot \\vec c \\\\\n\\end{align}", "6eb21ec81fd2f6fad9e359b4b125314f": "\\cos\\theta_1 = -\\mathbf{n}\\cdot \\mathbf{l}", "6eb22e2309d907105757e7bb811c4034": "(\\text{Prim},\\, \\Sigma,\\, \\triangleleft,\\, S)", "6eb2346043ced1ff1a8aa153a45a301d": "a_x = b_y c_z - b_z c_y. \\, ", "6eb25a45a6afacb6080afdfe8fc4d910": "\\, n", "6eb2a905314fcb6c40045735d7e238ff": "\\delta_C = \\frac{5 q L^4} {384 E I}", "6eb2b4a8aadc6c0a6bf6b72f38367340": "\\Delta f\\geq 0", "6eb30488598f3af08ba013c76dc947b1": "\\tfrac{(\\alpha+\\beta+2n)(\\beta-\\alpha)}{(\\alpha+\\beta+2)}\\sqrt{\\tfrac{1+\\alpha+\\beta}{n\\alpha\\beta(n+\\alpha+\\beta)}}\\!", "6eb306caa59f0d1d95d403e3ef507bad": "O_k(P)=E_k(P)", "6eb35a2b26fbb8e851a4d26d46e22622": "x_{1}^{h}", "6eb3cf6be703ea04c6d973e548d85162": "LC_{50} (mixture) \\le 1000 \\tfrac{mL}{m^3}", "6eb469c2e482707337c1352da3545a21": "\\mu y R(y,x_1,\\ldots,x_k)", "6eb487916326330f4b877e2cbfe79c07": "P_{out} = \\frac{h}{c} \\cdot \\nu \\cdot \\Pi \\cdot R_{sp}\\frac{\\exp[(g-\\alpha)L]-1}{g-\\alpha}", "6eb4a96caa34628508486dce3bfee603": "\\,S_p = p", "6eb4ae69ebedb0a5b3bcb8a7fa161807": "m_q = c^{\\frac{1}{4}(q+1)} \\, \\bmod \\, q", "6eb4dd0af61210e1ce5ec50f6f781cfa": " y_n = y_{n-1} + h A(t_{n-1}, y_{n-1}, h, f) ", "6eb514521f51599d8498980c52558d3a": "\\{t,\\rho,z,\\phi\\}", "6eb521805dcc72c9fb7b911de01b4a4f": "\\Delta d = -d \\cdot \\left( 1 - {\\left( 1 + {{\\Delta L} \\over L} \\right)}^{-\\nu} \\right)", "6eb53616f8d312aa7852c4ab9f74fa96": " \\Gamma(x)=\\frac{e^{-\\gamma x}}{x}\\prod_{n=1}^\\infty \\left(1+\\frac{x}{n}\\right)^{-1} e^{x/n} \\!", "6eb611265b90b5a8910a5d400b621a28": "a_i = \\exp\\left (\\frac{\\mu_i - \\mu^{\\ominus}_i}{RT}\\right )", "6eb645509d93c2a4a35722758a2f1167": "i, p, q \\in N", "6ec30de45cbcff42f7b46c50be2dddec": "\\boldsymbol{g}", "6ec336eaf821d8a6937f59710d935681": "\\sum_{x \\epsilon {0,1}}|\\alpha_{x}|^{2} = 1", "6ec3f82e6dcbb2cfb49bbad6c0a4dd92": "a_0=1, a_1=-\\mu, a_2=1", "6ec4179858ee31b2b883b2f435563826": " tq_x", "6ec4624301b0a8d0ce0cb6cc9934ee7e": "\\, G_{ret}(x-y) = -\\Delta(x-y) \\Theta(x_0-y_0) ", "6ec5795a3ffb71aaf2cfe6145220dcdb": "C\\ell_{3,0}(\\mathbb{R})", "6ec5d9b7c4380b28d217f0531a949067": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1=A_{12} \\left( \\frac{A_{21} X_2}{A_{12} X_1 +A_{21} X_2} \\right)^2\n\\\\ \\ln\\ \\gamma_2=A_{21} \\left (\\frac{A_{12} X_1} { A_{12} X_1 +A_{21} X_2} \\right )^2\n\\end{matrix}\\right.", "6ec6082e094a2ff3e2fe64fc43a43699": "\\sum_n (1-|a_n|) <\\infty.", "6ec6230ad50a8eaab0d5534bfcf584e1": "S^{n-1}", "6ec649c050e489ff07e62f4a14ebe8d3": "2^{-56.6}", "6ec65ef6cda21969383521097b072a2f": "\n\\phi=\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{c}\n\\phi^1 + i\\phi^2 \\\\ \\phi^0+i\\phi^3\n\\end{array}\n\\right)\\;,\n", "6ec66e124fb93c190e9207b0b82f542a": "na", "6ec6852ffef017acbcd3dc2231779ae3": " \\mathbf{r}^o_X = \\mathbf{B}_X^T \\Big[ \\mathbf{f}\n\\Big( \\mathbf{B}_R \\mathbf{R} + \\mathbf{Q}_v \\Big) + \\mathbf{q}^{o} \\Big] ", "6ec686ac2e68a43f843d0d108963bbc5": "U_i=\\beta(\\kappa_i)", "6ec6aa052f884f28d571173683cf290e": "u_i \\neq 0", "6ec6fa32ad90af2d81f2ce712d2c47da": "f = \\nu = R_\\mathrm{v} \\left( \\frac{3}{4}\\right) (Z-1)^2 = (2.46 \\times 10^{15} \\operatorname{Hz})(Z-1)^2.", "6ec72955b6264c794d27869df023536e": "P = \\frac{2\\cdot P_{max}} {3}", "6ec76aeeeab6b823ec93317ea850d398": "S(n) \\subset T(f(n)).", "6ec7af56ce32d513add49e2fe82a1960": "G_{ab} = R_{ab} - 1/2 \\, g_{ab} R", "6ec961852747843a540c95dda66cfcb7": "\\rho \\frac{D \\mathbf{v}}{D t} = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\rho \\mathbf{f}. ", "6ec9c5a736832be65b6fa2257f610cb7": "p/p_0", "6eca059e7eeb4c26070fc89c49bcc353": "\\textbf{t}_i^T \\textbf{t}_p", "6eca1af69c2723fd660cf073ff6e3f9f": "d=4\\ln\\varphi\\approx 1.925,", "6eca292adb8e4e5fe18036ea94737c0f": "p\\in U", "6eca5c858b5a294ee307954cfb2439f8": "e^{i \\pi\\ }", "6eca703cac82a6b04460f90c4e159f6f": "k \\in \\{a,a+1,\\dots,b-1,b\\}\\,", "6eca87838327f27b1267d00067a5bf96": "\\{x_n,n=0,1,2...\\}", "6ecaedd0796010cc1be582eabd4b1064": "\\bar E", "6ecaf35c0d812a85b1052e6dc58a81ac": "g = (b,a)", "6ecb5e93bbfe766552ad20d8e851d162": "\n \\sqrt{n}(\\hat\\theta_n - \\theta)\\ \\xrightarrow{d}\\ L_\\theta\\ .\n ", "6ecc08432936f65c3b8464ad99335d16": "\\lambda\\, ", "6ecc4a3f2b48b12cb03c72690941a2c3": "j = \\lim \\limits_{A \\rightarrow 0}\\frac{I}{A}=\\frac{dI}{dA}", "6ecc4d3678a1fd20394dc16c50a97802": "\\int \\mathcal{D}\\phi F[\\phi(x)]F[\\phi(\\bar{x})]^* e^{-S[\\phi]}=\\int \\mathcal{D}\\phi_0 \\int_{\\phi_+(\\tau=0)=\\phi_0} \\mathcal{D}\\phi_+ F[\\phi_+]e^{-S_+[\\phi_+]}\\int_{\\phi_-(\\tau=0)=\\phi_0} \\mathcal{D}\\phi_- F[\\bar{\\phi}_-]^* e^{-S_-[\\phi_-]}.", "6eccc92dec5218862795bb512ca921d6": "x \\sim \\mathcal{D}", "6ecce108e86c8596935f3666d023b12c": "dp_2 \\ ", "6eccec4fa3be34f5804ef308124616c6": "\n\\vartheta_2(z) = \\sum_{n=-\\infty}^\\infty q^{(n+1/2)^2} \\exp ((2 n + 1) i z)", "6eccfb35da99edc50a5c9a28ba880fd0": "V_D \\approx 0.65 V", "6ecd46a9780cd29a1be20cb109005a17": "j = {jI \\over I} ", "6ecd60ea472a6d87f332cde4abae35d4": " \\alpha \\in \\mathbb{C}", "6ecd6421435086faed532a7a6924f61e": "\\Phi_{22}", "6ecd6c800b963d83547ee868bae639d7": "C_{ym}=0.5(c_{y2}-c_{y1})", "6ecd74514d539f21417e8e95a927ba77": " \\hat{r}", "6ecd7a78fcf6d5099f1b57f922b9a6d5": "k \\leq n-log{n \\choose{t}}", "6ecde7a0ff012f7e21376292eebf3eaa": "\\nabla\\cdot\\mathbf{H}_\\text{d} = -\\nabla\\cdot\\mathbf{M}", "6ecdefe545aff8dddf1bb5515a96d478": "\\frac{d\\phi}{dt} = \\frac{c}{r\\textrm{sin}\\theta}\\sqrt{1 - \\frac{R_s}{r}}", "6ece31523d0875dde9daed5119866cdf": "\\sigma_{xx}", "6ece3570e5e25a52759d3d7d7d990d39": "\nV_{2,n} = \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n \\frac{1}{2}(x_i - x_j)^2 = \\frac{1}{n} \\sum_{i=1}^n (x_i - \\bar x)^2,\n", "6ece45e3f78470bcf0e7db1d3c539a09": "h(t)", "6ece6bb9e36507571a5bc15449aed8ce": " \\qquad \\qquad \\mathrm{rotational}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ E_{f,r,j} = \\frac{\\hbar^2}{2I_f} \\ \\ \\mathrm{and} \\ \\ Z_{f,r}\\sum_{j = 0}^\\infty (2j+1)\\mathrm{exp}[-\\frac{-\\hbar^2j(j+1)}{2I_fk_\\mathrm{B}T}] \\approx \\frac{T}{T_{f,r}}(1+\\frac{T_{f,r}}{3T}+\\frac{T_{f,r}^2}{15T^2}+...),", "6ececa1a179a26ebb56abb386309fdac": "[\\quad]_p:\\mathcal{P}(X) \\to \\mathcal{P}(X)", "6eced1df39fd2219e6d9d24bb2eb47b4": "\n\\begin{align}\n& {} \\qquad (1+x)(1+x)^k \\ge (1+x)(1+kx)\\quad\\text{(by hypothesis, since }(1+x)\\ge 0) \\\\\n& \\iff (1+x)^{k+1} \\ge 1+kx+x+kx^2, \\\\\n& \\iff (1+x)^{k+1} \\ge 1+(k+1)x+kx^2.\n\\end{align}\n", "6ecfac215d1e0f17df8e05a158c7846a": " 1-q = \\frac {1} {R_0}, ", "6ed0221b0b2df5cf5da4f9cdf6f026f3": "J^+(p)", "6ed06e921585051324c9494dc6f95cac": "\\exists\\kappa \\exists\\lambda. x \\Leftrightarrow \\exists \\lambda \\exists\\kappa. x", "6ed0a2579fe917cc0b5c50b3ee8fe341": "e^{\\hat{M}}_N (x,\\overline{\\theta},\\theta)^* = e^{\\hat{M}^*}_{N^*}(x,\\theta,\\overline{\\theta})", "6ed17c909b43e62f93cd472709df1dc6": "\\hat{\\psi}(\\gamma)", "6ed2302d48491ece0e88183bc255a8f8": "S=\\times_{i\\in I}S^i", "6ed2453a981f623ec6c6d9a183542b23": "\\operatorname{std}_p(f)= {1\\over 2(1-p^{-2})(1-p^{-4})\\cdots(1-p^{1-n}) } ", "6ed25ef3cedf3971b2d1bad89be94542": "\\left[\\,\\right]", "6ed2aebd63fd8705d52c9bf2e489443b": "O(A \\mid B)", "6ed2db669833b32bf7d22c2ab64ce530": "\\sigma(X) = \\sup \\left\\{ \\left. \\sqrt{\\operatorname{E} [\\langle \\ell, X \\rangle^{2}]} \\,\\right|\\, \\ell \\in B^{\\ast}, \\| \\ell \\| \\leq 1 \\right\\}.", "6ed2f98f4f4eb79d9c77f2b22837cc83": "f(x_3)=0.931596", "6ed3c805832a5ea4b593dee06ae4bdf0": " y^{(1)}, \\dots , y^{(m/n)} \\in \\lbrace 0, \\dots , d-1 \\rbrace ^n ", "6ed411d11b57013f04268bcb1e6c6675": "\\cos(a) = 1 - \\frac{a^2}{2} + O(a^4),\\, \\sin(a) = a + O(a^3)", "6ed430efe645c7f0cc686aab7118f3a7": "O( n^2 / 2^{3j} + n / 2^j )", "6ed4a6ec2418df5cd4ceb9feb5450815": "r_{jt}", "6ed4acb8b7ce22fbc2b43ee5477b2860": "\\scriptstyle\\partial A\\setminus\\Sigma", "6ed4e1b6c751499d7d092ad693603193": "\\mathbf{\\mathit{\\rho}}_S(t)", "6ed5c1ee433392912eaa60ca9ba58d93": "m_n=\\langle z^n\\rangle=\\int_\\Gamma z^n\\,f(x|\\mu,\\kappa)\\,dx", "6ed5e88f497e285837d4de50c952d90b": " {\\partial D \\over\\partial b_k} = 0 \\Rightarrow b_k = {y_k + y_{k+1} \\over 2} ", "6ed5ed82ec77b3e08a6ffdafcffe6885": "GL_2^+ (\\mathcal O_F)", "6ed634248d41d80a8bd93d4c93de0038": "A(\\boldsymbol\\eta) = - \\ln g(\\boldsymbol\\eta) = \\ln Z.", "6ed64a5608f1119cb0554954b75cefc8": "c_\\mathrm{a}^2(0, T) = \\frac{\\gamma_0 R T}{A_\\mathrm{r}(\\mathrm{Ar}) M_\\mathrm{u}},", "6ed664c11edfc4510be3f8c43ba3c3cb": "V = C_{n} \\, r^{n} \\left( \\frac{1}{2}\\, - \\,\\frac{r-h}{r} \\,\\frac{\\Gamma[1+\\frac{n}{2}]}{\\sqrt{\\pi}\\,\\Gamma[\\frac{n+1}{2}]}\n{\\,\\,}_{2}F_{1}\\left(\\tfrac{1}{2},\\tfrac{1-n}{2};\\tfrac{3}{2};\\left(\\tfrac{r-h}{r}\\right)^{2}\\right)\\right)\n=\\frac{1}{2}C_{n} \\, r^n I_{(2rh-h^2)/r^2} \\left(\\frac{n+1}{2}, \\frac{1}{2} \\right)", "6ed6a38d77612a424e9d268e08b57c6f": "S_n=2S_{n-1}+S_{n-2}\\qquad\\text{for all }n\\geq 2.", "6ed6afd05a0643c24291f761c6ae6c81": "\\{F,G^A\\}", "6ed6c7788ef839c6e2d25e3f1d2b0911": "\\rho = (\\sigma\\otimes \\tau) \\circ \\Delta.", "6ed6d2d20b6fa63845d884ea860f534d": "| \\alpha |^2", "6ed6eca4964a0b35c0948229c762fe53": "a_{1,2}x_1x_2 \\oplus \\cdots \\oplus a_{n-1,n}x_{n-1}x_n \\oplus \\!", "6ed70d462c0e543b779e69157ec60633": "\\omega \\to \\infty", "6ed72a54567c91190c0c74f4b253f355": "\\frac{(n-j)k}{2}", "6ed78503571d9e613882034903e11674": "u_j=Av_j", "6ed8796a5b5910cd5574573bc9a4f64c": "\n\\lim_{n\\,\\rightarrow\\,\\infty}\\,PV(i,n,R)\\,=\\,\\frac{R}{i}\n", "6ed8b3d6a8eb4dd6d3c21b2545e64ea1": "D_A:\\Gamma (W^+)\\to \\Gamma (W^-),", "6ed9642349e211b7edb127f1828b29c6": " - n u[-n-1] \\,", "6ed9833a69ca3e78ac25eec4e47eb7f1": "\\,^{z_{15} = x_{15} y_1 + x_{16} y_2 + x_{13} y_3 - x_{14} y_4 - x_{11} y_5 + x_{12} y_6 + x_9 y_7 - x_{10} y_8 + x_7 y_9 + x_8 y_{10} + x_5 y_{11} - x_6 y_{12} - x_3 y_{13} + x_4 y_{14} + x_1 y_{15} - x_2 y_{16}}", "6ed9e572a11be4821ad7a4ff901e03a8": "\\beta=\\frac{1}{k_{B}T}", "6ed9ebf31617725d0e51563ca62b804c": "1-\\tau(p)u+p^{11}u^2", "6ed9ec3db60e4837329fded03af2ed8c": "p \\cdot x(p',m') \\leq m ~ \\wedge ~ x(p',m') \\neq x(p,m) ~\\Rightarrow ~ p' \\cdot x(p,m) > m'~", "6eda47afae6f075ae1ea44bdcfe43f3f": "\\phi_n\\left(x\\right) = \\int K_n\\left(x,z\\right)f\\left(z\\right)dz", "6eda9b0c7f5ff0679e0ce4e896a8328f": "S = k \\cdot \\log W \\,", "6edaebf40c911cf70ec8cbca5686f582": "\\displaystyle (2\\pi i\\xi)^n \\hat{f}(\\xi)\\,", "6edb0aafc09db59d32821cb470676f43": "H^0(M,\\mathbf{K}) \\xrightarrow{\\varphi} H^0(M,\\mathbf{K}/\\mathbf{O}).", "6edbe5c8522bd39472459d8929a523fc": "p_\\text{P} = \\frac{F_\\text{P}}{l_\\text{P}^2} = \\frac{\\hbar}{l_\\text{P}^3 t_\\text{P}} =\\frac{c^7}{\\hbar G^2} ", "6edc042e8c5fd2aca00389b19a7d6974": "V_{\\text{out}} = -\\frac{R_{\\text{f}}}{R_{\\text{in}}} V_{\\text{in}}\\!\\,", "6edc3e31c3dc06c36ad75ace4eaa6c8a": "\\text{Specificity} = \\frac{TN}{TN+FP}", "6edc4aaf2c0a11c03040e21ceb8d7758": "\\mu=Gm\\,\\!", "6edc6241b183c8624ef9de8ea36af1b5": "\\frac{\\mathrm{d}}{\\mathrm{d}x} f^{(-1)}(x) = \\frac{\\mathrm{d}}{\\mathrm{d}x}\\int_a^x f(t)\\,\\mathrm{d}t = f(x)", "6edc6f6db2cc4e9de831933a3487106f": "0.999999-\\delta", "6edc97e1a0738719cf02eed3b561bcd0": " 0\\leq t\\,", "6edcbaa9104a6de7c834f2170a6356e2": "\\frac{d\\bold{x}}{dt} = \\bold{F}(\\bold{x},t)", "6edcd3b9a2e5f58c0c7d1bf0467701ac": "f_1(X, Y)\\, = Y^2 - g(X)", "6edcf8941daf690904e066871fe1cdac": "D_{1} = \\left(1 - \\alpha - \\beta - \\gamma\\right)", "6edcfd17bc921dbbd865eb4958fa80c6": " \\mathcal{M}^k_i {d h^i \\over ds} = \\mathcal{M}^k_i {d \\bar{\\mathcal{M}}^i_j \\bar h^j \\over ds} ", "6edd1c169092256e9ba6e6802140ea58": "\\scriptstyle\\mathcal{N}(\\mu,\\sigma^2)", "6edd2e6b7e3dc9c45b4af48383b532f4": "F(r) \\boldsymbol{\\hat r}", "6edd33d6ddb0ac452a58aa3fc166dbec": "2x^3 + 3x^2 - 4x + 5 = 0", "6eddbd247e9d34a001b481391ff49d5f": "\\langle\\phi_n\\rangle", "6ede0a4513c2948011360611add4e8b7": "\\chi_f(G)\\ge n(G)/\\alpha(G)", "6ede33dad843593dba151d0b58599730": "> 1", "6ede4c69ae8ddc3d44a42ad4fd7c9946": "E(\\lambda)=\\chi_{[0,\\lambda]}(T),", "6ede6129817cc616ca88f76bebc49eb2": " \\sum_{i=1}^n a_{i} = 1 ", "6ede831e4b081f3deb3d8506d614d2cd": " \\int_\\mathcal{V} \\partial_\\gamma \\mathcal{J}^{\\alpha\\beta\\gamma} cdt dxdydz = \\oint_{\\partial \\mathcal{V}} \\mathcal{J}^{\\alpha\\beta\\gamma} d^3 \\Sigma_\\gamma = 0 ", "6edeb5352d3f224c881b5fae62263844": "\\!\\mu_4(v_1)", "6edeb6a4c1d3d17daf138491c56d2f13": "\\frac{\\partial f}{\\partial y}(X,Y) = 0", "6edef18d57f078ffc4d90cac9068ebfa": "~\\delta(t)", "6edef4afde0a321634bfb6fa484494a2": "\\rho \\boldsymbol{u}(\\boldsymbol{x},t)", "6edf763703b89d98480ec50f62ab6b47": "10^{-4}", "6ee005265c88344767fcaf1a1e6d03a2": "x^3+(A-Bl^2)x^2+(1-2Blm)x-Bm^2=0", "6ee01944f6480d30d5b7801c027cdd0c": "\n \\operatorname{det}(\\boldsymbol{Q}(\\boldsymbol{N}) - \\rho_0 c^2 \\boldsymbol{I}) = 0,\n ", "6ee04f46df57679f558d157677d96f8f": " ta(s)= \\min\\{ ta_i(si) - t_{ei}| i \\in D\\}. \n", "6ee069b2ed53468dbf5cd490b495e6f0": " I(z) = I_0 e^{i\\omega t} \\cos kz,", "6ee09ebb6925981dcb6b1815701ad0d0": " a_1+a_2+\\cdots, \\, ", "6ee0fcdf09d2c41cbbef763cc63d98cf": "\n \\sigma_i = 2C_1\\left(\\cfrac{1}{\\lambda^3} - \\cfrac{1}{\\lambda}\\right) + 2D_1(\\lambda^3-1)\n ", "6ee10a562bcff806f01445b3480ade67": "\\left\\{\\begin{array}{ll}\nz: (a,b) \\rightarrow (0, \\infty)\\ ,&\\textrm{if}\\ \\alpha\\in \\mathbb{R}\\setminus\\{1,2\\},\\\\\nz: (a,b) \\rightarrow \\mathbb{R}\\setminus\\{0\\}\\ ,&\\textrm{if}\\ \\alpha = 2,\\\\\\end{array}\\right.", "6ee11dae5f60f38214989e6eb64e8ded": " \\, t_e = t - t_l ", "6ee14121af85474e944f2441219980df": "H-\\mu N", "6ee150b948f7ee2f82dcdac99a2212ac": "n_{11}", "6ee1a34a8ae97c24cd87a35b9724cba0": "r=\\tfrac{1}{2}\\sqrt{ab}.", "6ee24fad0bf9c351eda6d974f47c4b17": "c_{n-1}/c_{n}", "6ee2770d149459bbf760147e13536ebb": "f(x+y) = f(x) f(y)", "6ee35f9c93c60009477692d674d5a4d1": " \\Theta=\\mathrm d\\theta+\\omega\\wedge\\theta. ", "6ee39df685981cce5b3645e6977cc6c7": "\\chi(S,\\mathcal{O}_S):=\\sum_i (-1)^i h^i(S,\\mathcal{O}_S)=2.", "6ee3d70cd8a9ac2134276f9b9ca864ab": "w^{(0)},w^{(1)},...,w^{(l-1)}", "6ee406bd7f10550acc5c67e713e8762e": "\\widehat{\\beta}_0, \\widehat{\\beta}_1", "6ee41a04281a94e36be03f154c473030": " wp(x := E, R)\\ =\\ R[x \\leftarrow E] ", "6ee42678c190288422cbf95e43bb772a": "\n \\det\\left|\\frac{\\partial x^i}{\\partial X^\\alpha}\\right| \\ne 0\n", "6ee4314d8e1e31bc9138d462fa34fded": "-\\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{{\\partial \\alpha}{\\partial \\beta}} = \\operatorname{cov}[\\ln {X,(1-X)}] = -\\psi_1(\\alpha+\\beta) ={\\mathcal{I}}_{\\alpha, \\beta}= \\operatorname{E} \\left [- \\frac{1}{N}\\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha \\partial \\beta} \\right ] = \\ln(\\operatorname{cov_{G{X,(1-X)}}})", "6ee457b0c0d6dc88053e74614ff8c0bc": "\\left|{\\alpha \\choose k} \\right|^2\\leq \\exp\\left(- 2(1+\\mathrm{Re}\\,\\alpha )(1+\\log k) +2|1+\\alpha|^2 \\right) = \\frac{M^2}{k^{2(1+\\mathrm{Re}\\,\\alpha )} }", "6ee496cdccdf9ae5102d14c54491868c": "L\\left(\\sigma^2;\\mathbf{x}\\right)\\propto \\left(\\sigma^2\\right)^{-n/2} \\exp\\left\\{-\\frac{\\sum_{i=1}^n \\left(x_i-\\mu\\right)^2}{2\\sigma^2}\\right\\}.", "6ee4ab537da00a9a7b4e42caf1d2bf0f": "W^+", "6ee4ad8cbc30464a9e23cfafcfcdbef5": "S\\bar\\psi_e\\psi_\\tau", "6ee4b5cefcc8112f2b2a780d286f194c": "\\boldsymbol{F} +m r\\dot\\theta^2\\hat{\\mathbf{r}} - m 2\\dot r \\dot\\theta\\hat{\\boldsymbol\\theta} = m \\tilde{\\boldsymbol{a}}= m\\ddot r \\hat{\\boldsymbol{r}} +m r\\ddot\\theta\\hat{\\boldsymbol\\theta} \\ , ", "6ee51bc441a5772511b4f96f8af442a5": "\\ G", "6ee51e3277db0f140723c59c3e6bb617": "\\mathcal{I}\\neq\\emptyset", "6ee52e6366b514bf93b5b1a307b9f5fe": "(n_1, n_2, n_3)= (\\mathbf{\\hat{z}}\\cdot \\hat {\\mathbf{e}}^1,\\mathbf{\\hat{z}}\\cdot \\hat {\\mathbf{e}}^2,\\mathbf{\\hat{z}}\\cdot \\hat {\\mathbf{e}}^3) ", "6ee53373f0f2afc927d06aadeb9d0ebf": "\nT^n = P_H S^n \\vert_H = P_H (Q_{H'} U \\vert_{H'})^n \\vert_H = P_H U^n \\vert_H.\n", "6ee535def0a7839b150657a243e518aa": "\\Pi_{ZZ}(q^2) = \\Pi_{ZZ}(0) + q^2 \\Pi_{ZZ}^{\\prime}(0) + ...", "6ee54bde508dd543d2887b3a30d39bb4": "|f| = |f|^{1-\\theta}|f|^{\\theta}", "6ee54e9eae369e7f40a4f4fd3faeef61": "a^{\\frac{1}{2}}", "6ee55795e43ffda63824d7b406d5d70e": " f(x) = \\frac{1}{\\sqrt{\\pi}} \\exp \\left\\{ - \\ln^2 x \\right\\};", "6ee5e38a118b40c4a8d92031fdcf1caf": "\\alpha! = \\Pi_{i=1}^N (\\alpha_i!)", "6ee60077946b0f8b23cf9485d6c1a874": "S_k \\equiv 0\\pmod{p}", "6ee6e846248e70878e275c269542ad46": "A \\to a", "6ee7525e0101bf40340f1aaf263836c4": "\\mathbf{x}(\\tau)", "6ee75fbbfb21f501be313e69b33dc905": "s= \\,", "6ee7fb8d198508cf4251cb73dde429b2": "\\begin{align}\nP(v; f_1,\\dots, f_n) & = v\\\\\ne_i(v; f_1,\\dots, f_n) & = f_i, \\qquad i=1,2,\\dots,n.\n\\end{align}\n", "6ee881603793b46a9be7f0b687b13d3f": " K(P,Q,t) = \\sum_{n=1}^{\\infty} f_n(P)f_n(Q) e^{- \\lambda_{n}t} ", "6ee8b7be6b4662acc26d72920f1899fb": "|D|\\geq O\\left(\\frac{k\\cdot VCDIM(H)\\log(1/\\alpha)}{\\alpha^{3}\\epsilon}+\\frac{\\log(1/\\delta)}{\\alpha\\epsilon}\\right)\\,\\!", "6ee9b82e8151fb0d1f5f024db07cbad2": "pg(q-d,q-\\frac{q}{d},q-\\frac{q}{d}-d+1)", "6ee9bbd8b0ca6af55121eabddfa77acb": "\\scriptstyle\\left[m,\\, m/0.05^{1/k}\\right] \\;=\\; \\left[m,\\, m \\cdot 20^{1/k}\\right]", "6ee9ee4ac5f1334e41e9413a3920e2db": "\n\\begin{array}{l}\nF(u,\\lambda)=0\\\\\n\\dot u^*_0(u-u_0)+\\dot \\lambda_0 (\\lambda-\\lambda_0) = \\Delta s\\\\\n\\end{array}\\,\n", "6eea0a80a34781528c798bcb92b5c985": " h(X_1, X_2) ", "6eea1c5d7349f43ba4112c1115da6935": "e_j = - \\frac{\\Omega(X_j^{-1})}{\\Lambda'(X_j^{-1})}", "6eea90da6893a8b16cb1d4c845ef011a": " K \\,", "6eeaa1b797ce1e5152b3b05ddd7478b5": "p(z)= \\frac{1}{\\pi} \\frac{1}{1 + z^2} ", "6eeab21416bf1065a56749e6b9ff492f": " l_{(+)} = {l\\over\\sqrt{1-v^2/c^2}} ", "6eeaf0b348a409853a7b226b7e1c539a": "-\\omega_c", "6eeb24157adc0237ec04e8b7a5b8a8f9": "\\sigma = \\lim_{r \\to \\infty} 4 \\pi r^{2} \\frac{S_{s}}{S_{i}}", "6eeb81cbaff54e719aa207c865369dea": "\\begin{align}\nJ_a(g) &= \\begin{pmatrix} g'(a) \\end{pmatrix}, \\\\\nJ_{g(a)}(f) &= \\begin{pmatrix} f'(g(a)) \\end{pmatrix}.\n\\end{align}", "6eebb5494e1c773cd62b78d49f60642e": "X(\\omega) = j \\omega", "6eebbbda2ef9ca65b925f7452d45cde6": " \\frac{1}{2}\\frac{d}{dt}(\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}}) = \\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}, ", "6eec283d70ebae59e3dab2ff666fad38": "C(0)=\\frac{3(K-2)}{4(K-1)}", "6eec5fce38d800fcdd0398199f1fe302": "\\pi r^2\\ \\text{or}\\ \\frac{\\pi d^2}{4} \\,\\!", "6eecb50ad74529967b93d19d0fddac83": "x_{20} = (1+5\\epsilon,\\ 6\\epsilon,\\ 1+4\\epsilon,\\ 5\\epsilon,\\ 1+3\\epsilon,\\ 4\\epsilon,\\ 1+2\\epsilon,\\ 3\\epsilon,\\ 1+\\epsilon,\\ 2\\epsilon,\\ 1+2\\epsilon,\\ \\epsilon,\\ 1+3\\epsilon,\\ 2\\epsilon,\\ 1+4\\epsilon,\\ 3\\epsilon,\\ 1+5\\epsilon,\\ 4\\epsilon,\\ 1+6\\epsilon,\\ 5\\epsilon)", "6eece4cd54a5105de750bc277345c4b8": "f(x)=\\sqrt{x^2-5x+6}", "6eece5a53c148bb0318bfc6d47c3265d": "\nY_{\\ell}^{m} =\n\\begin{cases}\n\\displaystyle {1 \\over \\sqrt{2}} \\left(Y_{\\ell |m|} - i Y_{\\ell,-|m|}\\right) & \\text{if}\\ m<0 \\\\\n\\displaystyle Y_{\\ell 0} &\\text{if}\\ m=0\\\\\n\\displaystyle {(-1)^m \\over \\sqrt{2}} \\left(Y_{\\ell |m|} + i Y_{\\ell,-|m|}\\right) & \\text{if}\\ m>0.\n\\end{cases}\n", "6eed5523c1db973b899633b2c7387ae0": "\\left| \\mathbf{p} \\right|", "6eed7cd11738935022940b94b03b9c82": "\\scriptstyle b-x", "6eedaba37cabc36269dfe616afeb9000": "\\frac{\\delta \\mathcal{S}}{\\delta \\varphi_i} = 0,\\,", "6eedb1117ee4730522b12b75a54c0065": "V \\propto n\\,", "6eedb405118f0546214ef5297e6ea162": " \\scriptstyle Du\\in\\mathcal M(\\Omega,\\mathbb{R}^n)", "6eedcc59ed663e709e2c64ace3847b19": " d_3G_3=Sq^3G_3+H\\cup G_3=G_3\\cup G_3+H\\cup G_3=0\n", "6eedd77ebdd3504e9b062b3ecbb7d813": "\\mathrm{Beta}(\\alpha,\\beta)\\,\\!", "6eee0ffe0a07a22d4ea87155296f04a1": "\\text{A} \\not\\in \\text{PTAS} \\implies \\text{B} \\not\\in \\text{PTAS}", "6eee4550525265d85b76b0e454aa02d6": " \\equiv S ", "6eee7e6a9e8c54d152ced25796f176fa": "A=\\exp \\begin{pmatrix}0 & a \\\\ a & 0 \\end{pmatrix} = \n\\begin{pmatrix}\\cosh a & \\sinh a \\\\ \\sinh a & \\cosh a \\end{pmatrix} =\n\\begin{pmatrix}1.25 & .75\\\\ .75 & 1.25 \\end{pmatrix}", "6eee87265d2db1fb461161768ff3a9ef": "\\Box = \\partial_{\\beta} \\partial^{\\beta} = \\nabla^2 - {1 \\over c^2} \\frac{ \\partial^2} { \\partial t^2}", "6eeea4dc0e11cfa6868c91858c6740c8": "Q^\\mathsf{T} Q = Q Q^\\mathsf{T} = I,", "6eeeb0551b56cef258c118c5b443bafb": "|L_x(y)| \\le \\|x\\|_q\\,\\|y\\|_p", "6eeeb6488fa1ddcd9ff2ad21e4c538fb": "f(p)\\cap\\{x,y\\}^2=f(p')\\cap \\{x,y\\}^2", "6eeed4fd287e47737ac81a0f78b29123": "\\sqrt{1 - x^2}", "6eef482e97f7853e11f0c72e8b29e402": "I \\rightarrow B: \\{N_B\\}_{K_{PB}}", "6eef600f8caac9d01c42155c85e976d1": "c_c= 2 \\sqrt{k m}.", "6ef06830f4845c5a93861fcaefb51087": "A^{(a-1)/2}\\equiv -1 \\pmod a\\;", "6ef141eb29502c6ed99273bfa7b69336": "\\operatorname{Spin}(n) \\to \\operatorname{SO}(n),", "6ef142ac7fa2f32a7af2479a6036eec6": "f : M \\to X", "6ef20f43af4eef383a47e75862d5fa14": "V=\\frac{abc}{6}.", "6ef21471bea23df5e33ac0ad09589b5f": "H = \\frac{1}{3} \\left(A + \\sqrt{A B} +B \\right).", "6ef2ac71d8b267e9866edc898f773aee": "\\sigma_c, \\sigma_b, \\sigma_t", "6ef314ca067027201c78aee53247e296": "\nr^m \\sin^m\\theta \\cos m\\varphi = \\frac{1}{2} \\left[ (r \\sin\\theta e^{i\\varphi})^m \n+ (r \\sin\\theta e^{-i\\varphi})^m \\right] =\n\\frac{1}{2} \\left[ (x+iy)^m + (x-iy)^m \\right]\n", "6ef3308ebc70b1d28b0855ea1fdfb404": " A = A\\!\\left(\\frac{h}{t}\\right) + a_0\\left(\\frac{h}{t}\\right)^{k_0} + O(h^{k_1}) .", "6ef3884379a491060d0e8a1428c7b5cf": "H(q) =\\sum_{n=0}^\\infty \\frac {q^{n^2+n}} {(q;q)_n} = \n\\frac {1}{(q^2;q^5)_\\infty (q^3; q^5)_\\infty}\n=1+q^2 +q^3 +q^4+q^5 +2q^6+\\cdots \\,\n", "6ef419f0729a23c625cb47a2b85d4b50": " \\overline{\\Gamma^*_n} \\subset \\Gamma_n,", "6ef4858891b314299696d63f799b6a17": "\n\\begin{align}\n \\cos(z \\cos \\theta) &= J_0(z)+2 \\sum_{n=1}^{\\infty}(-1)^n J_{2n}(z) \\cos(2n \\theta),\n \\\\\n \\sin(z \\cos \\theta) &= -2 \\sum_{n=1}^{\\infty}(-1)^n J_{2n-1}(z) \\cos\\left[\\left(2n-1\\right) \\theta\\right],\n \\\\\n \\cos(z \\sin \\theta) &= J_0(z)+2 \\sum_{n=1}^{\\infty} J_{2n}(z) \\cos(2n \\theta),\n \\\\\n \\sin(z \\sin \\theta) &= 2 \\sum_{n=1}^{\\infty} J_{2n-1}(z) \\sin\\left[\\left(2n-1\\right) \\theta\\right].\n\\end{align}\n", "6ef4c8b87802be9a0ad99575797ac3cb": " \n\\int_E f_n\\, d\\mu \\geq a \\mu(A_n) \\Rightarrow \\liminf_{n\\to \\infty} \\int_E f_n \\, d\\mu = \\infty = \\int_E \\varphi\\, d\\mu,\n", "6ef4ed1a75164b79e055eb8f731f12ee": "\\scriptstyle G", "6ef4fae10da0f7cd435b225d763f1709": "k x) = \\frac{\\Gamma_e(x)}{\\Gamma_e(0)} = \\exp{\\left(-\\frac{x}{\\lambda_e}\\right)}\\qquad\\qquad(11)", "6f02cb440b242a6251798d8a7feca9d7": "P(t,x) = c_n\\frac{t}{(t^2+|x|^2)^{(n+1)/2}}", "6f02cd11075297b8a8a70c3a8ff47136": " k^i \\sin \\theta^i = k \\frac{\\sin \\theta}{M}. ", "6f02fd70b0e78a18aa6a29a2dd8ee52b": "g^{(n)}(\\infty)= 1 ", "6f0313c5604ece6ff45bbeb3728d44a8": " \\tan \\psi = \\frac {-v_y(t_d)} {v_x(t_d)} = \\frac {\\sqrt { v^2 \\sin^2 \\theta + 2 g y_0 }} { v \\cos \\theta}", "6f0328484eb0348f2853e1603148b128": "B_I R", "6f03966aac87aa8c68eff7bfb9cde892": "\\rho_{01}(p)", "6f03b69a7cf5a050e8fa337edf66a7be": "\\mathbb{R}^{\\mathbb{N}}", "6f03f6d5f6072e1274bb02837686cd34": "Q_3=f_b(S_a)", "6f04309d268ccb6897af51b7b056fda8": "\\dim\\partial W-1=n-1\\geq 2(k+1)", "6f04d360562d1a43ce48cc5e429fe9ae": "v_{\\infty}=\\sqrt{\\frac{2mg}{\\rho C_D A}} \\, .", "6f04ebfe5518967b8886e016f501ea3e": "\na_2 = \\frac{Q(x_t \\mid x')}{Q(x'\\mid x_t)}\n", "6f054fee2077675eea752fc72558229f": "\\kappa=\\lambda^{-1}", "6f056c5c424582a25a3eb9f1c756c6bd": " f = \\tfrac{q}{d} ", "6f05a94f5f23a52ef960263e42cdf629": "K_{b}", "6f05d4c99a26eb9cd141d33e25be9041": "\\boxplus", "6f0602c0f0c0c39d1677d08bb0b5ff90": "|x|\\gg 1", "6f06437ec05f4bc09d985be601aa367d": "\\bold{u}, \\bold{v}", "6f0665baf2839d3fe6dd4a264dd757a6": "\n\\Bigl\\langle p_{k} \\frac{\\partial H}{\\partial p_{k}} \\Bigr\\rangle = \\Bigl\\langle q_{k} \\frac{\\partial H}{\\partial q_{k}} \\Bigr\\rangle = k_{\\rm B} T.\n", "6f06b21b1c2db6394f02ffe78af8ef34": "O = (0, 0)", "6f06d75977e99b2bc3891bd346c4a16c": "\\vec f = -\\vec f_{op}", "6f06da7ec280fe98ad827c20524dc322": "\\mathbf c^{\\rm T}\\Sigma = \\operatorname{cov}(\\mathbf c^{\\rm T}\\mathbf X,\\mathbf X)", "6f0702ec07d2f72d84cb2e0af2baf25e": "f'(x+):=\\lim_{h \\to 0^+}\\frac{f(x+h)-f(x)}{h}", "6f0715c178ae59cf99d22300565b69e9": "1918 = [31, 58]_{60}", "6f0722fd9a9395aacc53295f78e9d725": "\\nabla_{\\dot{\\gamma}(0)} e_\\alpha = \\sum_\\beta e_\\beta \\omega_\\alpha^\\beta(\\dot{\\gamma}(0))", "6f073acfb2e7d73fccc1fb098ba8a38e": "\\scriptstyle\\widehat{u_i} ", "6f0742941894eb318ac076c8ffb880b4": "a_{22}", "6f078175654d37a05dbbdbb089317a0e": "\\langle x, y \\rangle = \\sum_{b \\in B} x(b)\\overline{y(b)}", "6f079ebfb95f37599f05607071167c6c": "\\hat{\\Psi}_{\\sigma}(\\omega) = c_\\sigma \\pi^{-\\frac{1}{4}} \\left( e^{-\\frac{1}{2}(\\sigma-\\omega)^2} - \\kappa_\\sigma e^{-\\frac{1}{2}\\omega^{2}} \\right)", "6f07cee4a4c60cfd86546065717ed085": "NaCl_{(aq)} + AgNO_{3(aq)} \\longrightarrow NaNO_{3(aq)} + AgCl_{(s)}", "6f080c817321cc2d2769ee2c0648038a": "\\varphi(t) = \\mu((-\\infty,t])~.", "6f08542d2ad82075eb9eaad25cd89071": "P(X_k\\ |\\ o_{1:t})", "6f08972d74d6a5caaa9195c7eb2e11b1": "\\sigma = 10", "6f0912f8716a9ab0133e2a2e1e5c0a20": "M\\left ( \\mathrm{d} I_1/\\mathrm{d} t \\right )=-NV_2\\,\\!", "6f092f765ca168635f111b0a8af46c2a": "u^*=\\frac{u}{V}", "6f096afdfbb9ea896c01752271a1b0e7": "\\rho(A) = |\\det A|^{-s},\\quad A\\in \\operatorname{GL}(n)", "6f097c1201029b3fcd3421cd6a907fb3": "\n\\begin{align}\n\\nabla \\times \\mathbf F & =\n\\frac{\\hat{ \\mathbf e}_1}{h_2 h_3}\n\\left[\n\\frac{\\partial}{\\partial q^2} \\left( h_3 F_3 \\right) -\n\\frac{\\partial}{\\partial q^3} \\left( h_2 F_2 \\right)\n\\right] +\n\\frac{\\hat{ \\mathbf e}_2}{h_3 h_1}\n\\left[\n\\frac{\\partial}{\\partial q^3} \\left( h_1 F_1 \\right) -\n\\frac{\\partial}{\\partial q^1} \\left( h_3 F_3 \\right)\n\\right] \\\\[10pt]\n& + \\frac{\\hat{ \\mathbf e}_3}{h_1 h_2}\n\\left[\n\\frac{\\partial}{\\partial q^1} \\left( h_2 F_2 \\right) -\n\\frac{\\partial}{\\partial q^2} \\left( h_1 F_1 \\right)\n\\right] \n=\\frac{1}{h_1 h_2 h_3}\n\\begin{vmatrix}\nh_1\\hat{\\mathbf{e}}_1 & h_2\\hat{\\mathbf{e}}_2 & h_3\\hat{\\mathbf{e}}_3 \\\\\n\\dfrac{\\partial}{\\partial q^1} & \\dfrac{\\partial}{\\partial q^2} & \\dfrac{\\partial}{\\partial q^3} \\\\\nh_1 F_1 & h_2 F_2 & h_3 F_3\n\\end{vmatrix}\n\\end{align}\n", "6f098289d25732d51ab75f35ff1515f9": "\\{z=0\\}", "6f0a2ec51e5018513b066e790ee7b0f5": "a_{n+1} = \\frac{a_n + \\frac{2}{a_n}}{2}=\\frac{a_n}{2}+\\frac{1}{a_n}. ", "6f0a3af982eb0aec9093845294ff2fcb": "\\pi r^2 h = \\pi r^2 (2r) = (\\tfrac{2}{3} \\pi r^3) \\times 3.", "6f0aae9cdf89b316c69d8e0c8d2d4767": "J = \\int_D |\\nabla u|^2 \\mathrm{d}x", "6f0b8f221e04d473b0b126d6f42ed3ff": "|w| \\leq p(|x|) ", "6f0b9862cddf39f5a9d3a6e71bba0b17": "\n \\begin{align}\n m_1 & = \\int_{-b/2}^{b/2}m_x(y)\\,\\text{d}y ~,~~ m_2 = \\int_{-b/2}^{b/2}y\\,m_x(y)\\,\\text{d}y ~,~~\n q_{x1} = \\int_{-b/2}^{b/2}q_x(y)\\,\\text{d}y \\\\\n t & = q_{x2} + m_3 = \\int_{-b/2}^{b/2}y\\,q_x(y)\\,\\text{d}y + \\int_{-b/2}^{b/2}m_{xy}(y)\\,\\text{d}y \\,.\n \\end{align}\n", "6f0c38ef7e2491c3012916d00444961e": "\\rho ' = \\rho\\ - \\rho _w", "6f0cde671c674c9b2e7eaded690fec04": "\\{\\ X\\} = \\{\\ X^/\\} + \\{\\Delta \\ X\\}", "6f0ce37752b7fea4c46194972b21da71": " s \\neq 0 ", "6f0d1a0c56187e6f264ea94d4f7d8370": " Var( \\theta ) = \\frac{ ( N - 1 )^2 } { N^3 } - \\frac { 2N - 3 } { nN^2 }", "6f0d73a1fa13bd3a8f566e4f74b6ee03": "(\\mathbf{p} - \\mathbf{a})\\cdot \\frac{d\\mathbf{r}(t)}{dt} = 0", "6f0d9258f1756a3c0e7cb352dc092633": "J_+|\\psi\\rangle", "6f0da409509bd3e214340897252fd33a": "\n \\left(2D_1 - \\cfrac{C_1}{\\lambda^4}\\right)J^2 + \\cfrac{3C_1}{\\lambda^4}J^{4/3} - 3D_1J - 2C_1\\lambda^2 = 0\n", "6f0e287281e52b4f2aa0fab5feae169f": "c,\\phi", "6f0e3624a941032817e1773dce10bc61": "\\Delta P_p", "6f0ed41ab973bcfa268883871e836f88": "B_n = \\sum_{k=1}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}\n\\mbox{ and } B_0 = 1,", "6f0eed6879138982370ada86f3b2c833": " \\delta W = \\mathbf{F}_1\\cdot \\left(\\sum_{j=1}^m \\frac{\\partial \\mathbf{V}_1}{\\partial \\dot{q}_j}\\delta q_j\\right) + \\ldots + \\mathbf{F}_n\\cdot(\\sum_{j=1}^m \\frac{\\partial \\mathbf{V}_n}{\\partial \\dot{q}_j}\\delta q_j)", "6f0f4d7841eb2b6fbacd8b62a5c00256": "\\widehat{a}_j\\,", "6f0fdc35e491e4863b20d32b2110b830": "V=|V| e^{i\\omega t} ", "6f0fe1a88f68329b8e6d6740e9cfbf79": "\\begin{matrix} \\frac{m}{s^2} \\end{matrix}s = \\begin{matrix} \\frac{m}{s} \\end{matrix}", "6f101e73503dc6c7c7d0b19a2b6be84a": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^{m+1} (A\\,c\\,e (m+2 p+2)-B (c\\,d+2 c\\,d\\,p-b\\,e\\,p)+B\\,c\\,e(m+2 p+1) x)\\left(a+b\\,x+c\\,x^2\\right)^p}{c\\,e^2(m+2 p+1) (m+2 p+2)}\\,-\\,\n \\frac{p}{c\\,e^2(m+2 p+1) (m+2 p+2)}\\,\\cdot\n", "6f1096746ac00d500eab6d4851e55321": "\\mathit{a, b, c}", "6f10c6ce7c11e88631a4406eda069a94": "f_{\\theta_0}(\\theta) := D_{\\mathrm{KL}}(P \\| Q) = \\sum_{jk}\\Delta\\theta^j\\Delta\\theta^k g_{jk}(\\theta_0)", "6f10fa39c345daf042e7b87a935b081a": "s^\\prime = -\\sin{\\theta_\\mathrm{c}} d + \\cos{\\theta_\\mathrm{c}} s. ", "6f1105b6b51c6fcdc5a3efb60b5c0346": " \n\\begin{alignat}{2}\n\\log(S_t) &=\\log\\left(S_0\\exp\\left(\\left(\\mu - \\frac{\\sigma^2}{2} \\right)t + \\sigma W_t\\right)\\right)\\\\&\n=\\log(S_0)+\\left(\\mu - \\frac{\\sigma^2}{2} \\right)t + \\sigma W_t.\n\\end{alignat}\n", "6f11192f4cee20687e72d7af19339349": "\\langle V_M \\rangle = \\frac{\\mu_B}{\\hbar} \\vec J(g_L\\frac{\\vec L \\cdot \\vec J}{J^2} + g_S\\frac{\\vec S \\cdot \\vec J}{J^2}) \\cdot \\vec B.", "6f114df668ebb56175fef0bf9b8b5ace": "E = Emergency = f(S,T,D)", "6f1237082821cf3bb292bf91c03d718c": " e^{\\pi\\sqrt{163}}", "6f123bfc357d4dcbb01fd2c6bc02d10f": "C=(z + r d_A) s^{-1} \\times G", "6f1289c7057cc035473181813d713d24": "19601-13860\\sqrt{2}=0.00002\\ldots", "6f1338f6fa05a04c50e5c64c1a06d648": "T \\to S", "6f135b3bc2fdd11828a486401f0f6674": " \\mu_{3,1}= \\mu_{3,l} - r\\mu_{1,1} = \\frac{-1}{3}(< \\frac{1}{2}) ", "6f13963aa5c06b8aaf6c21f39e196edc": "B\\subseteq A", "6f13a228146d52094cb2017c94d9ef43": "\\Sigma(12) > 3 \\uparrow\\uparrow\\uparrow\\uparrow 3 = g_1, ", "6f13a69529c7fbd293f2fdaf5dd75514": "\\|Tf\\|_q \\le C_p \\|f\\|_q.", "6f13ce15bfa5acd33fb1388d83fbb4ed": " I_S = I_R + Md^2, \\, ", "6f13d78b529b641838b1a05b0e446ae2": "-1.00253", "6f149402164d0693c11544324cfdf105": "\\Gamma_q(x+1)=[x]_q\\Gamma_q(x)\\,", "6f149b4130e51b5232251a88da0d73fb": "dP = \\left(\\frac{\\partial P}{\\partial S}\\right)_{T}dS+\\left(\\frac{\\partial P}{\\partial T}\\right)_{S}dT\\,", "6f14eed5731128938c74c8542e7588e6": "\nz_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{D}{sg}\n", "6f15019814a2844b569f254acfabac34": "\\Delta G^\\ominus = -RT ln K.", "6f152a3004abf47558fab72daa060ac2": "\\frac{1} {L^*} = \\frac{L^*} {T^{*2}}", "6f154411609d4e7503aea04c26125127": "[\\![\\phi \\wedge \\psi]\\!]_i = [\\![\\phi]\\!]_i \\cap [\\![\\psi]\\!]_i", "6f15a42ec6360f187ff7567f88f88d70": "\\displaystyle S_\\mu S_n=\\sum_\\lambda S_\\lambda", "6f15a963d2e2d47270acb9da02296c04": "nx_r\\,", "6f15e6516c66c423401f346887b0baab": "B_{2n} =(-1)^{n-1}\\frac{2n}{4^{2n}-2^{2n}} A_{2n-1}", "6f15f7c10f48f31581fc7d31b3bfe734": "\nx = -a\\ \\cosh\\ \\mu\n", "6f163520f133ce52380ce918e312e71c": "M_T = {32\\over 3\\pi G} \n\\langle R \\left(2V_R^2 + V_T^2\\right)\\rangle.", "6f165583a7df940439fc387ca2b4fad8": "\n\\begin{align}\n\\lambda_0(\\mu-\\mu_0)^2 + n(\\bar{x} -\\mu)^2&=\\lambda_0 \\mu^2 - 2 \\lambda_0 \\mu \\mu_0 + \\lambda_0 \\mu_0^2 + n \\mu^2 - 2 n \\bar{x} \\mu + n \\bar{x}^2 \\\\\n&= (\\lambda_0 + n) \\mu^2 - 2(\\lambda_0 \\mu_0 + n \\bar{x}) \\mu + \\lambda_0 \\mu_0^2 +n \\bar{x}^2 \\\\\n&= (\\lambda_0 + n)( \\mu^2 - 2 \\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n} \\mu ) + \\lambda_0 \\mu_0^2 +n \\bar{x}^2 \\\\\n&= (\\lambda_0 + n)\\left(\\mu - \\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n} \\right) ^2 + \\lambda_0 \\mu_0^2 +n \\bar{x}^2 - \\frac{\\left(\\lambda_0 \\mu_0 +n \\bar{x}\\right)^2} {\\lambda_0 + n} \\\\\n&= (\\lambda_0 + n)\\left(\\mu - \\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n} \\right) ^2 + \\frac{\\lambda_0 n (\\bar{x} - \\mu_0 )^2}{\\lambda_0 +n}\n\\end{align}\n", "6f1656dffdd7e26a47c17b9ea3652b25": " \\rho_N \\circ f = (f \\otimes 1) \\circ \\rho_M ", "6f1669182023a0d8ae13db23b17a93b2": "\\mu_{eff} = \\sqrt{\\vec{L}(\\vec{L}+1)+ 4\\vec{S}(\\vec{S}+1)} \\mu_B", "6f169b161675f23b38fa44ece531aeab": " Z=\\{0,1\\}", "6f16ad05d8cd9a5865878186cf1793a2": "B_\\lambda(T) =\\frac{C\\lambda^{-5}}{e^{\\frac{c}{\\lambda T}} - 1},", "6f16be476ff3e4d07f2f7f7e20da7a97": "\n\\begin{align}\nT = Nt & = \\frac{t}{\\log_2 1/r} \\log_2 \\frac{2D}{W} \\\\\n& = \\frac{t}{\\log_2 1/r} + \\frac{t}{\\log_2 1/r} \\log_2 \\frac{D}{W}.\n\\end{align}\n", "6f1788f885728318811477b56bb6c2b2": "y = -2x + 4 \\,", "6f179937097e91bdc169b6105a8c623e": " 2 \\dot{r} \\ddot{r} = \\frac{d}{dr} \\left( (E^2-V) \\, (1+m/r)^4 \\right) \\; \\dot{r} ", "6f17a7b01c0d8ec4e4772b9454570ad8": "G/\\{V\\} ", "6f17c28e68cad552a27fe7715670aa64": "W=T_o \\Delta I", "6f17c7d878f7b3f34cfe415039996a88": "F = \\mathbb{R}", "6f17f69ef60091c060a8dfd6a43af33c": "|\\mathbf{D}| = \\frac{Q}{A}", "6f186da9cf3689720464d5b0a9e972d1": "(\\mathbb{Z}, \\le)", "6f18cc95f4097886794ac7e530d63980": "y_q", "6f18fb5b3e8371aa6e234963dee89eca": "X(\\omega) = \\frac{1}{1-e^{-i \\omega}} + \\pi \\cdot \\delta (\\omega)\\!", "6f190807b2921431340d5b564d45d82f": "\\delta^3", "6f192703237fd1d3cf3be0b425aee1f6": "\\delta u^0_\\alpha", "6f1977122c0c280329b722370e6621a5": "\n\\arcsin z = z + \\left( \\frac {1} {2} \\right) \\frac {z^3} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {z^5} {5} + \\left( \\frac{1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6 } \\right) \\frac{z^7} {7} + \\cdots\\,.", "6f1a05a8c1cd65cb79439a7bfc34af39": "t^2 - d u^2 = 4", "6f1a33044432a26701b873b6f06ad557": " e = \\frac{c}{a} ", "6f1a353023b665b6921932ff828f4016": " 1\\equiv \\sum_x | x \\rangle\\langle x|", "6f1a47065366dca302e3deac4b5b0596": "e^{\\lambda(r)} = \\left(1 - \\frac{r_s}{r} \\right)^{-1} \\;", "6f1a5f21af01ae1d8c4de505061f3146": "c=(m^2 - n^2)^2 - m^2 n^2, \\, ", "6f1a791df07934d871f37229273003e8": "\n\\phi = \\arctan \\left( { (\\omega\\tau)^\\alpha \\sin(\\pi\\alpha/2) \\over\n1 + (\\omega\\tau)^\\alpha \\cos(\\pi\\alpha/2) } \\right)\n", "6f1a80e0ab71d746d0e0f5d9a71633bd": "\\hat{H}_{\\lambda}|\\psi(\\lambda)\\rangle = E_{\\lambda}|\\psi(\\lambda)\\rangle,", "6f1a8699625aa571c5e4f086d4bff734": "\nE_\\mu(x) = i \\sum_\\mathbf{p} {\\sqrt{p_0} \\over \\sqrt{2 V }}\\left\\{\n\\left[Q_R(\\mathbf{p}) \\epsilon_\\mu^1(\\mathbf{p})\n+ Q_L(\\mathbf{p}) \\epsilon_\\mu^2(\\mathbf{p}) \\right]e^{i p x}\n\\right.\n", "6f1b5493bedf6b5b62f7105207408b00": "Q(A) \\times Q(B)", "6f1b7f4d41c849ce40ea8310d4ddfdf3": "n_e = 1", "6f1bb75af40d58d0bd019b5dff3dbb8c": " \\mathrm{N} \\mathfrak{p} ", "6f1bc9af34e8519130445a6bd10136f3": "p_{*}(\\pi_1(\\tilde{X},\\tilde{x}))", "6f1bdd8f1338765aebfd948ad12f6b0d": "\\beta_j 'x", "6f1bfb6d1c1d00ea01703d73aefa07f5": " \\leq ", "6f1c020ff030987cfce3dd8bba0f4b4c": "\\vec{V}\\cdot\\widehat{n}=0,", "6f1c292298ed9f979f6392acd277b37b": "\\mathfrak{so}.", "6f1c441907e7013e3be0fdb02a860145": "a^r\\;", "6f1cfc305f92f76823574a4f2a9a8501": "{\\rm ad} (x){\\rm ad} (a+b)(y) = [x,[a+b,y]\\,].", "6f1d1730a501b20cef031197bbde9405": " e_f = (k_\\mathrm{B}T^2/m)(\\partial \\mathrm{ln}Z_f/\\partial T)|_{N,V}.", "6f1d47b9d61c7480e0dce218ec0ee1cb": "\\scriptstyle Z(s) ", "6f1d519b5160ad489a338a6f4a8a5704": "x=\\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}", "6f1d7d45ce591cb233383b55fd00fc2b": "[x]b", "6f1d8b32bc7f10eead61bc26fff023ee": "F(ax_1 + bx_2) = aF(x_1) + bF(x_2)\\,", "6f1dfea45500ba24d176186892f4592c": "e^{(a_1i+a_2j+a_3k)\\pi} + 1 = 0. \\,", "6f1e3e2f071c5c906f75e11ca654ace3": "\\begin{align}p(\\mu|D, I) \\propto & \\; A^{-\\frac{\\nu + 1}{2}} \\\\\n\\propto & \\left( 1 + \\frac{n(\\mu - \\bar{x})^2}{\\nu s^2} \\right)^{-\\frac{\\nu + 1}{2}} \\end{align}", "6f1e49c19c27db0d29ef2d502be30d90": "\\sum\\xi^i=1", "6f1e49e63dca7927734101b43d7678b8": "\\mathrm{Res}_{L/k}X(S) = X(S \\times_k L)", "6f1ed865fda743f0aef33b3257a8e901": " ds^2 = dx^2 + dy^2 + dz^2, \\; -\\infty < x,y,z < \\infty", "6f1f1d67cb8fefbe1eaf754609f4e171": "\\sigma=\\sigma(\\dot{\\varepsilon})", "6f1f634924d38ca7865131b9f70418cb": "\\mathbb{D}(S^*,\\hat{S}_N)\\rightarrow 0 ", "6f1f6f3994e6c422c4b8cc6056954990": " \\sqrt[3]{n+\\sqrt[3]{n+\\sqrt[3]{n+\\sqrt[3]{n+\\cdots}}}}", "6f1f9c73279ad1872b1fa74df9858c5c": "b\\,\\!", "6f202c23b1f9880a45e706e10c227b7d": "k(M)=\\max_S \\left\\lceil\\frac{|S|}{r(S)}\\right\\rceil,", "6f205494e831315a8eadf785eeadb428": "X_1 \\sim \\mathrm{Herm}(a_1,a_2)", "6f2059ac253836e27640b61c95bc79bb": "X = \\{piano, car, apartment\\}", "6f208efa8933f3ebcfca6664d6cd11d9": " d(v,w)=\\log\\frac{M(v/w)}{m(v/w)}. ", "6f20d140306cd1514bb9821276781f64": "\n\\Bigg(\\frac{\\omega}{\\alpha}\\Bigg)_3 = \\omega^\\frac{1-a-b}{3}= \\omega^{-m-n},\\;\\;\\;\n\\Bigg(\\frac{1-\\omega}{\\alpha}\\Bigg)_3 = \\omega^\\frac{a-1}{3}= \\omega^m,\\;\\;\\;\n\\Bigg(\\frac{3}{\\alpha}\\Bigg)_3 = \\omega^\\frac{b}{3}= \\omega^n.\n", "6f210fbd35f5c6dcc5f066574e04b605": " \\textbf{P}_{k\\mid k} = \\textbf{P}_{k\\mid k-1} - \\textbf{K}_k \\textbf{H}_k \\textbf{P}_{k\\mid k-1} = (I - \\textbf{K}_{k} \\textbf{H}_{k}) \\textbf{P}_{k\\mid k-1}.", "6f211eb31fb5bc6bb4acdde57f81bfbc": "\\scriptstyle[\\mathrm{Ca}^{2+}] \\simeq \\frac{[\\mathrm{A}^-]}{2}", "6f217013c7cbba8021e880c8209ceb8a": "(c, \\tau)", "6f21a8529bef1019b3bf1e4893005c6e": "q \\sim 5 \\times 10^{-3}", "6f21ecba796d292ba17f88fc59db3e48": "\\Delta x = 1.5 \\text{m}", "6f2206ad131d28f624a3d9bdc08aebf0": " \\operatorname{de-let}[V] ", "6f2239d16a38395b134c58d373de836c": "\n\\psi (\\mathbf{r},t)=\\frac 1{(2\\pi \\hbar )^3}\\int d^3pe^{i \\mathbf{p}\\cdot\\mathbf{r}/\\hbar}\\varphi (\\mathbf{p},t),\\qquad \\varphi (\\mathbf{p},t)=\\int d^3re^{-i\n\\mathbf{p}\\cdot\\mathbf{r}/\\hbar }\\psi (\\mathbf{r},t).\n", "6f2252ba81c29f3f42258377e8cd5488": "\\varphi i,j", "6f22a499ffdaaa0b53d4c80e637043d2": "\\mathbf{x}[k+1] = \\mathbf A_d \\mathbf{x}[k] + \\mathbf B_d \\mathbf{u}[k] + \\mathbf{w}[k]", "6f22bb37c1146d427da4382cbec8b3b4": "L=\\sup\\left\\{t \\in [0,1] \\, \\colon \\, W_t = 0 \\right\\} ", "6f23196b49914ebd0e26d4c8511609fa": "\\Rrightarrow 1 \\text{ rad} = \\frac{360^\\circ}{2\\pi}", "6f2334e97a81ce45d23842f98ffd9b5b": " \\vec{e}_3", "6f235644d2f0adda87915f0114951bc8": "\\sum_{k=j}^p(k+1)_j a_{p,k}=(p)_j", "6f235baebbf82c5df96bd395fa0128f6": "\\overline \\nu_1", "6f2371cca227700ef8e01c4a095fc753": "g_{ij}=-D[\\partial_i||\\partial_j]=-\\partial'_j\\partial_i\\sum{p(\\log;p-\\log;q)}=\\sum\\frac{\\partial_ip\\partial_jq}{q}=\\sum\\partial_ip\\partial_j\\log q=[p=q]\n=\\sum{p\\partial_i\\ell\\partial_j\\ell}", "6f237e373950282db04385e667e6fd3c": "g(\\vec{p})", "6f238c746c327b91c3219d8bb5daa1e1": "\\textstyle Z_{4}", "6f2406e1d31ed7a24460633d360f9ce9": "(n+1)! + k", "6f24222b2dd9002642e55600898055e9": " d\\colon H \\longrightarrow G, \\! ", "6f244b84d23fbd47617a735ea1dd6746": "P=P_{\\mathbf{r} \\in V}(t) = \\int_V \\Psi^{*} \\Psi d V = \\int_V |\\Psi|^2 d V \\,", "6f248c8ca16cc1aa995d43eed6111f5a": "y = 2^{-((15-n)/2)}", "6f249a151487ce07c8e0f8ac35a23787": " C (\\mathrm {dB}) = 20 \\log | \\kappa | \\ ", "6f24fa5affe2ef25a035f057de7f4cde": "[\\mathbf{A}] \\in [\\mathbb{R}]^{n\\times m}", "6f2502311e90b40f2269435636db202f": " x^{-1} = \\frac 1 x ", "6f2539ce4555d33c4a8ac1c2586bd46c": "\\dot{\\tilde{\\mu}}=D\\tilde{\\mu}-\\partial_{\\tilde{\\mu}} F(\\tilde{s},\\tilde{\\mu})", "6f258ae4f948779db085427c021bc8ad": "\\displaystyle PV = N k_B T ,", "6f259302c9b150e951829991ffc9b544": "R+\\tau_{(123)}R+\\tau_{(132)}R = 0.", "6f25b7e94e0fd85337cb019c5eb300ee": "Q_i\\;", "6f25e43ce43538accb9a25df18e76568": "\n\\left\\{ L_{i}, L_{j}\\right\\} = \\sum_{s=1}^{3} \\epsilon_{ijs} L_{s} ~,\n", "6f25e6a6b19b362e9642a94af015a53e": "\\rho(X + a) = \\rho(X) - a", "6f262c29aa930dd1b65f60bea1203f3f": "E = e+\\frac{1}{2}u^{2},", "6f26358cc328d5d192c5086e46723a56": " \\csc^2(x) + \\sec^2(x) - \\cot^2(x) = 2\\ + \\tan^2(x) ", "6f26a8c59e18f5c0da95bf2847225ea3": "\\neg A, \\sim A", "6f26f1c73dc6e430d323298428736b8b": "\\exp \\colon M_n(\\mathbb C) \\to \\mathrm{GL}(n,\\mathbb C)", "6f27007654263fcbdd61d779394518ac": "\\left|\\log {zf^\\prime(z)\\over f(z)}\\right|\\le \\log {1+|z|\\over 1-|z|}.", "6f27864b9a7fe5584359466214f7fefb": "AdS_7\\times S^4", "6f27bed5ff7ebde292c9a3f6d8585fbd": "x^* = \\frac{R^2}{|x|^2} x.", "6f27f79a984a3ade9c9a375ccce6d7c4": "D \\rightarrow D^{'} ", "6f2806db835e9fa2449ca5f06f24e186": "\\int_{\\Lambda^{m\\mid n}}f\\left( x,\\theta\\right) \\mathrm{d}\\theta \\mathrm{d}x=\\int_{\\mathbb{R}^{m}}\\mathrm{d}x\\int_{\\Lambda^{n}}f\\left( x,\\theta\\right) \\mathrm{d}\\theta.", "6f2808ee9a86215ec6c05c529021794a": "f_{xy}(0,0) = p_{xy}(0,0) = a_{11}", "6f282877fd88473ce1daf5f17376491d": "\\ \\Delta P = 2H\\gamma", "6f283085118a6b00fd4614b1aa8158a2": "v^\\gamma A_{\\alpha ;\\gamma} \\,.", "6f283a858a8ccd3d68c6b97cadae94ae": "w=f(z),", "6f2882ee980593a9478871960af1977c": "Y^2 = X^3", "6f2883f63ced1039b48878ab789260f9": "U,\\mu", "6f28ecc1d172249b0ed8496c04c37030": "\\operatorname{lift-choice}[\\lambda f.f\\ ((x\\ f)\\ (x\\ f))] ", "6f2915841bbadb776a43e70d39aafdc6": "\\phi^2-t\\phi+ q = 0", "6f292709ace97fd405dde77533686fc7": "S(A_n),", "6f29afecb29ac4392840cb27fd3ff62e": "\\mathrm{Im} \\phi < G", "6f29c10d55082851e7c2e55cd46255e3": "\\text {Work}=F_{av}\\times d =\\text{2Wd}\\,\\!", "6f2a612a4065fc5d1a50e479b934e352": "\\sigma = \\sqrt{\\frac{1}{N}\\left[(x_1-\\mu)^2 + (x_2-\\mu)^2 + \\cdots + (x_N - \\mu)^2\\right]}, {\\rm \\ \\ where\\ \\ } \\mu = \\frac{1}{N} (x_1 + \\cdots + x_N),", "6f2ac003277e769619d66dc640239f42": "R_{f}", "6f2ac125f87ba1c140097354c01deaae": "T = \\frac{T_{a}}{D}", "6f2b637e38e1c391fe810e6c3182485b": "{\\mathbf{}}P_{i+1} = A_i \\left( P_i - P_i C'_i \\left( C_i P_i C'_i+W_i \\right)^{-1} C_i P_i \\right) A'_i+V_i+\\tau_{\\perp i+1}\\Psi^1_i \\tau'_{\\perp i+1}, P_0=E \\left( {\\mathbf{x}}_0{\\mathbf{x'}}_0 \\right)", "6f2b82f8a49667a7d0b2ec0428f35f4b": "\\text{Step 6}", "6f2b896ab9d8126432813a694895b7a0": "G = \n\\begin{pmatrix}\n\\uparrow & \\uparrow & & \\uparrow\\\\ \ng_0 & g_1 & \\dots & g_{2^n-1} \\\\ \n\\downarrow & \\downarrow & & \\downarrow\n\\end{pmatrix}", "6f2b8a421009ad8e29fb164c860be112": "g^*:=\\prod_{i\\in I}(X-\\alpha_i^*),\\ h^*:=\\prod_{j\\in J}(X-\\alpha_j^*)", "6f2bc61c058480e39a806be24c25e75f": "\\lambda(z), \\mu(z), \\rho(z)", "6f2be84e2dd8541ca501c95c29f120de": "\\int\\frac{dx}{\\cosh^n ax} = \\frac{\\sinh ax}{a(n-1)\\cosh^{n-1} ax}+\\frac{n-2}{n-1}\\int\\frac{dx}{\\cosh^{n-2} ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "6f2c4e49fc3723882e9f541dd6ce18c2": "T(n) = \\Theta\\left( n^{\\log_b a} \\right) = \\Theta\\left( n^{3} \\right)", "6f2c99caac84ee04d4c7167df353d32b": "\\mathrm{NPV}_0<0", "6f2c9bd5e308247d91011256a5d06193": "d\\tau = \\sqrt{dT^2 - (r \\omega /c)^2 \\sin^2(\\omega T)\\; dT^2 - (r \\omega /c)^2 \\cos^2(\\omega T) \\; dT^2} = dT\\sqrt{1 - \\left ( \\frac{r\\omega}{c} \\right )^2}.", "6f2cb57c39fd9f22f24808dd7bd8353a": "S= C - B^T A^{-1} B . \\, ", "6f2cb5bcf99224782405ab8d1b09e86f": "{t_{near}}", "6f2da52c8333fc6738fd8d9bf1d212f9": "v * (1,-1)^{n+1}", "6f2de05c9465892abd951d295ac1e2b7": "\\sigma_{GB}", "6f2e1f1755bf503e0ae35097703a5625": "\\phi(x\\cdot a) = \\phi(x)\\cdot a\\qquad \\phi(a\\cdot x) = (-1)^{|a||\\phi|}a\\cdot\\phi(x).", "6f2e32eae9b3ee0f6b42908651d064ae": "{y} = \\sqrt {r^2 + a^2} \\sin\\theta\\sin\\phi", "6f2e7f4fcdc1e6167b5da8513b983e34": "(a + bi)1 + (c + di)j.\\ ", "6f2eb01a599c191bc4c2e7d742036612": " \n\\begin{align}\n \\vec{\\nabla} \\times \\vec{B} = \\alpha\\vec{B} \\\\\n\\vec{v} = \\pm\\beta\\vec{B}\n\\end{align}\n", "6f2ee62b19c500e50d0790aefb08de53": "\\Gamma_k^\\text{gh}", "6f2eef0181888383899cbc3a64c5c28c": "\\mathbf{\\beta}", "6f2ef07912d4ee5649f7f38204e2f793": " \\nabla^2 \\mathbf{v} = \\nabla(\\nabla \\cdot \\mathbf{v}) - \\nabla \\times (\\nabla \\times \\mathbf{v}) ", "6f2ef0e6245964c200235cacc6b2695d": "A = 4 \\pi^2 R r = \\left( 2\\pi r \\right) \\left(2 \\pi R \\right) ", "6f2f0bd84a8ba3b5ad829c3f886ceb2a": "\\sigma \\geq 0", "6f2f1cd117019e09cec80575504c3091": "T_{v}", "6f2f56e8221a42da156eefac502d6dc8": "\\begin{align}\n&{} D(X, \\xi) = \\int\\limits_{0}^L |x-\\xi||\\psi_m(x)|^2dx = \\\\\n&{} = \\frac{\\xi^2}{L} - \\xi +L\\left(\\frac{1}{2}-\\frac{\\sin^2(\\frac{m\\pi\\xi}{L})}{m^2\\pi^2}\\right).\n\\end{align}\n", "6f2f7043580253da4fd802f1ab022ef0": "m_{i,i+1} = 3", "6f2fed62e101b00eb35c2bf909c408e6": "f_y=\\sin^2\\theta", "6f2ff7b8b81f630bc1cefafb212e4339": "K_N f", "6f301277ceadc2374467699a69182284": "\\{x\\in U\\mid m(x)=1\\}", "6f3022a8b2ca7f205c2c6825da5b3cb2": "f: X \\rightarrow X", "6f3038430191987da0d70651e5e45b85": " \\widehat \\mbox{Cov} (\\mbox{Vec}(\\hat B)) =({ZZ'})^{-1} \\otimes\\hat \\Sigma.\\, ", "6f30735a33804c6c111c4d5b4969812d": "\\phi :A\\rightarrow S(W)", "6f3075f5ec1b218c4bd3ea44d7f2d7f3": "\\Gamma_1(a)=\\Gamma(a)", "6f30827df86e30a983faa9e5fabed4be": "EMV = BMV \\times (1+R)+ \\sum_{i=1}^n F_i \\times (1+R)^ \\frac{T - t_i}{T})", "6f30be699d7c3bf62095c89cf805ed9a": "(t',x') = (t,x)\\begin{pmatrix}1 & v \\\\0 & 1 \\end{pmatrix}", "6f30ebbcafd46cebb71397c7c414da18": "(\\alpha_k, \\beta_k)", "6f30ee1009a3754e5fc9bf470e43d562": "y = b/c", "6f311b1f534589caca1e097b8c00603e": "\\|I\\|_{q,p} = \\mu(S)^{\\frac{1}{p} - \\frac{1}{q}}", "6f31212c189f12d174be8a76a6fee688": "\n\\sigma_{-s}(n)=\n\\zeta(s+1)\n\\left(\n\\frac{c_1(n)}{1^{s+1}}+\n\\frac{c_2(n)}{2^{s+1}}+\n\\frac{c_3(n)}{3^{s+1}}+\n\\dots\n\\right).\n", "6f31c2aa2eccbb29380780adde55854b": " \\displaystyle{H_\\varepsilon f \\rightarrow -i f}", "6f31d55b1e8a97bd4496729fedbbcc32": "(\\mathbf{a} - \\mathbf{p}) \\cdot \\mathbf{n}", "6f3225e7b771997a691bdd7bec817c9a": "\\frac AE = \\gamma D_0 f (E)", "6f324dfd5cbb2b8d21f9e369a6e02ccf": " \\left\\{ H_n(i, j)~, \\, 1 \\leq i < j \\leq n \\right\\} ", "6f32a1dd56b1d1cf720804265bfc6f63": "r:K\\to X", "6f32a546ca859e5beb48ee1528f01688": "w\\in D", "6f32d246cf2144de480d63c82ee2d3dd": "\\forall\\alpha.\\alpha\\rightarrow\\alpha", "6f3322c3451a373289fdcb0447c39a59": "y_2(t) = y(t + \\delta) = 10 \\,x(t + \\delta)", "6f333ffc72cae462654295315bbb42d8": " G_{\\infty} = \\lim_{T \\to \\infty }G = G_0 - \\frac {1} {\\beta} \\ .", "6f3357ae1d6de5c7df30cf8503177f87": "T'", "6f33cff6ac8ea37d5f6f976f1f1df411": "Q_j\\,", "6f33e200c35025fcf2cc525843cb2259": "- \\frac{\\partial u}{\\partial t} = \\nabla\\cdot\\mathbf{S} + \\mathbf{J}\\cdot\\mathbf{E},", "6f3416cd6f585c4f077331608a2d7718": "Rf(\\alpha,s) = \\int_{\\mathbf{x}\\cdot\\alpha = s} f(\\mathbf{x})\\, d\\sigma(\\mathbf{x}).", "6f34623ccb8bbdddf1931f7c0b0f2566": "a, b, c, d = -2634, 955, 1770, 5400", "6f34aa8b2a3969bb2ddd8fa8231d00b5": "\n\\frac{d}{dt} \\left( \\mathbf{r} \\cdot \\mathbf{v} \\right) = v^{2} + \\mathbf{r} \\cdot \\frac{d\\mathbf{v}}{dt},\n", "6f353d11424de3355b1fea17a4295c3d": " B_1(x) = \\frac{\\mu_0 n I R^2}{2(R^2+x^2)^{3/2}}", "6f357fa44462d46d8d229b2034d8c291": "\\operatorname{E}_Y (X\\cdot Y) = \\sum_y X \\cdot y \\cdot P(Y=y),", "6f358f01419482742cc48fd3a77c0e52": "x, y \\in B", "6f35bb7c96aefc536b8dc4db99a24ec2": " \\in (0,\\infty)", "6f35c7bd6263a88609ec623f068a4304": "Qu'(c_t) = Q\\beta Ru'(c_{t+1})", "6f35daba95e92123f86e0351b06d7f14": "s = \\frac{v_t}{r\\omega^2} = \\frac{m}{6\\pi \\eta r_0} ", "6f35f8879eefd53ea15c1145766c37e1": "L^2, J^2 , L_z , J_z", "6f364d2ba8c274f6bc3ca7e3f5dae92b": " \\neg (\\operatorname{ask}[S] \\and FV[A] \\subset V) \\to \\operatorname{drop-formal}[[F, S, A]::Z, \\lambda F.Y, V] \\equiv \\lambda F.\\operatorname{drop-formal}[[F, S, A]::Z, Y, V] ", "6f3658e846d78e5829d462a33755d20b": "\\mathbb{Z}/8\\mathbb{Z}", "6f36a392bd9cf3e46ecd3678d9820bb1": "d_h", "6f370f94d00189248b48b1988ec94aaa": " x_2=l^2+a_1\\cdot l-2\\cdot x_1 ", "6f3733ee34fd9a01ecc48423b238126d": " ((1+1/3)d)^2 ((2/3) h)", "6f374060aebf4d4f9a846337dd989c5a": "curry", "6f3773615fcba1b81eaa8c1dc6b44217": "2 \\pi \\ln a \\rightarrow \\infty ", "6f37aa32b773eeecc43508c6b73eedea": "X+\\delta", "6f37b4743d28f7914ff17685bedd52ee": " \\, k = 1, \\ldots, N ", "6f38083d934b32f52fe41460b8260105": "T_r=\\frac{T}{T_c}", "6f38112f55b386b3543a196e62b42351": " \\int_{x_1}^{x_5} f(x)\\,dx ", "6f382bc81c00d94253aa07f42ed536dd": "[M] = [M_0]+[\\delta M] \\, ", "6f383928d21e276f8ae241dcc751c510": "w=\\Pr \\{x0\\}", "6f409321a00b0f18fefa5059bd20db04": "\\sum_{2 \\leq i \\leq r} (b_ir_i) = 0", "6f409906c2a29da07030a8cb6b0b76e0": "i - 1", "6f40b28b49f3da111d4e72fdaa961215": " A_n^\\epsilon =\\; \\{x_1^n : \\left|-\\frac{1}{n} \\log p(X_1, X_2, ..., X_n) - \\bar{H_n}(X)\\right|<\\epsilon \\}.", "6f40e4cc57a6764c5da5e67b3e8bb39c": "\\varphi^n = \\varphi^{n-1} + \\varphi^{n-2}\\, ", "6f41208f6b446f338950638ec2d1b44e": "1-p(n) = \\bar p(n) = \\prod_{k=1}^{n-1}\\left(1-{k \\over 365}\\right) .", "6f4186738c24148c03963fe2ad8b4db5": "\\mathcal{F}_\\infty", "6f41eee67c3c4f63171e4db304aab904": "\\left(-4\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "6f42033419437e0f821e772e08b493b5": "a, b\\in A; p, q \\in P", "6f421193ee738f482cd5efff80976346": "(\\forall x \\exists y. p(x,y)) \\rightarrow (\\exists y \\forall x. p(x,y))", "6f421bcaa37de80cc88d9cf1e918a19a": "f(x)=e^{-1/x^2}", "6f428a025e008c7c56c4de664c52000c": " \\hbar = \\frac{1}{\\alpha} \\ ", "6f43602d57bc26fdcacef967d01b5484": "{v_1,v_2,\\ldots,v_n}", "6f43a2952e363d7e8c06abb79aaa1a37": "-\\frac{\\zeta(3)}{3}-\\frac{\\pi^2\\gamma}{12}-\\frac{\\gamma^3}{6}", "6f43c2a3587b92e8f3e0e83f78cd775e": "J^{\\mu}_{\\text{free}} \\, = \\, \\partial_{\\nu} \\mathcal{D}^{\\mu \\nu} \\,.", "6f43c43c04f907f33f6fa958040b976d": "r(h,q)= \\begin{cases}\nq \\sin (\\pi|h|/q)&\\text{for }h \\in C_{1}(q)\\\\\n1 &\\text{for }h = 0\n\n\\end{cases}", "6f43d8aabb48811c16451aea389dbf93": "\\sum_x \\log i_x", "6f43fce9178d5cc846915958b90b7d5b": "S_0=\\emptyset", "6f441f33adf3e5160dcb5225090063b9": "\\cos (2 \\theta) = 2 \\cos^2 \\theta - 1\\,", "6f44978543da5dc4e4139cac72ba7cc7": "R_{m,n}", "6f452df059c651dd74e546e4458a8d3d": "P\\times EG", "6f453bada4c221af9fd51834434055f1": "(u_1,u_2)= f(v_1,v_2)", "6f455cb651e92c947fe8ff0e2ba620a7": "~C", "6f457195d14ad3c5d84d4fff1ddf8edc": "P(e)", "6f45cd70aa49a027aca4a7fa2ac1fd84": "E(Q)=4\\sigma_0(\\pi\\lambda)^{1/4}/3\\left((4\\pi)^{1/3}(3/4)^{5/3}Q^{4/3}+\\beta e^2Q^2\\right)^{3/4}.", "6f45d5b690d93e79c129349ae3fb7467": " \\operatorname{E}(\\theta_i|y_i) \\approx (y_i + 1) { {\\#\\{Y_j = y_i + 1\\}} \\over {\\#\\{ Y_j = y_i\\}} },", "6f45f3b8b34c6e6451f67a8ea1544c70": "\\,q = ia + b\\mathbf{i}+ c\\mathbf{j}+ d\\mathbf{k}", "6f465426ab72dfdc10dcc491c7c27ce7": "\\scriptstyle z \\,=\\, f(a_0,a_1,\\dots,a_n)", "6f4674ffa45962ec75af5a0c6145673c": "\n\\omega_{n}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\omega_{0}^{2} \\left( 1 - \\frac{b^{2}}{4} \\right)\n", "6f46b725dafc9c0edb2c278127bf69c3": "n = 20", "6f46f450181b405a70f4cb4fdc2c6e4c": "M_z(t) = M_{z,\\mathrm{eq}}\\left( 1 - e^{-t/T_1} \\right)", "6f471f8f3ce9ad94c6a203b88b7ab8c0": "\\frac{1}{2T}", "6f4737b9975d4f6cc12cd7ad99e56b4f": "\\psi(x)-x+\\log(\\pi)", "6f475668aad3b3017e4f271a72b6d4c9": "u(y) = \\theta(x) y ", "6f4776ef8d2b662bdd29193e904ab4b2": "g(x_1, x_2, \\ldots) = g_1(x_1) g_2(x_2) \\cdots", "6f4792fee47bd9433038a187fa9bb80f": "\\displaystyle{\\overline{H}=G\\cdot \\exp(\\overline{C}) =\\exp(\\overline{C})\\cdot G.}", "6f4793ee1955723fa8b82f97e634f84b": "C_l = C_{l_{\\alpha}}\\left(\\alpha_{\\infty} + \\alpha_{geo} - \\alpha_0 - \\alpha_i + \\frac{p y}{s}\\right) \\qquad (3)", "6f47bd61a060fdce482f0fb3ebd31b3f": "Z \\ ", "6f47c30fc23fde4b4bd379032c0bf32f": "\\vec{\\nabla}_{\\mathbf{k}} \\, \\omega", "6f47c3d628501dc3203cf2f51a617a70": "\n{\\rm E}_1( {\\rm i}\\!~ x) = i\\left(-\\frac{\\pi}{2} + {\\rm Si}(x)\\right)-{\\rm Ci}(x) = i~{\\rm si}(x) - {\\rm ci}(x) \\qquad(x>0)\n", "6f47d8e07a1691f6bccf4ac5a1a7e8ac": "P_\\mu(x_1,\\ldots,x_n;t) =\\sum_\\lambda K_{\\lambda\\mu}(t)s_\\lambda(x_1,\\ldots,x_n).\\ ", "6f47ecef419a4392b905e556ddd2d25b": "\\scriptstyle{R_J(x,y,z,p)}", "6f484651074e75a96ce1ffc2e64e3739": "||p_\\sigma||)", "6f486fe26007cd924be12c88d72488b2": "f_{j} (x) = g_{j} (x)^{d - 1},", "6f48719e59a31970a903698d8316edeb": "\\scriptstyle \\det(J) = -\\sqrt{k}.", "6f4872dbc18d0f1d5e4f48f6c6fc51b4": "Y=\\alpha + \\beta V+\\sigma \\sqrt{V}X,", "6f48766e60995063b7135c52861c9b80": "10^{10^{100}}", "6f487f9dccba1e9eed81cf04c2dac025": "\\Delta r = 2h \\cos i \\;", "6f48829027f87504c7c0c654981a9f39": "e=\\rho (c^2+e^C)", "6f4922f45568161a8cdf4ad2299f6d23": "18", "6f493bfa603642a6668bf2b2634425f9": "P_{reflect} = \\frac{2E_f}{c} \\cos^2 \\alpha ", "6f4978f834fb052e851bf10922342307": "F_4(a, b) = e^{e^{\\ln(\\ln(a))\\ln(\\ln(b))}}", "6f4987b01bb75f1331442acbc7a32da6": "f_{me}", "6f498d7d5b1bba2f2980a5d37989ef31": "{\\mathbf Q}_{j}", "6f499f8236475df8cbc8f22adb5754ce": "M_A", "6f4a3eca9fd821d5ee8c3415f0d221b4": " E= \\frac{\\hbar ^2}{2I} ", "6f4a46560f35d1b89105f7b6b7446a09": "\n Y = X - (z_1 + z_2) \\frac{X}{s} + z_1 z_2 \\frac{X}{s^2}+ (p_1+p_2)\\frac{Y}{s} - p_1 p_2 \\frac{Y}{s^2}.\n", "6f4a60ce7cdbf3708d833e4856ab92a1": "H(p_1, p_2, \\ldots , p_k) = -\\sum_{i=1}^k p_i \\log p_i,", "6f4af212b7d4b1f5236163ffbc3cb24b": "\\frac{G \\hbar}{c^3}", "6f4af4c6806643497def80d6f218e6a7": "\\begin{align}\n \\Rightarrow I(n) &= \\int_0^\\pi \\sin^nxdx=\\int_0^\\pi u dv = uv |_{x=0}^{x=\\pi}-\\int_0^\\pi v du \\\\\n {} &= -\\sin^{n-1}x\\cos x |_{x=0}^{x=\\pi} - \\int_0^\\pi - \\cos x(n-1) \\sin^{n-2}x \\cos x dx \\\\\n {} &= 0 - (n-1) \\int_0^\\pi -\\cos^2x \\sin^{n-2}x dx, n > 1 \\\\\n {} &= (n - 1) \\int_0^\\pi (1-\\sin^2 x) \\sin^{n-2}x dx \\\\\n {} &= (n - 1) \\int_0^\\pi \\sin^{n-2}x dx - (n - 1) \\int_0^\\pi \\sin^{n}x dx \\\\\n {} &= (n - 1) I(n-2)-(n-1) I(n) \\\\\n {} &= \\frac{n-1}{n} I(n-2) \\\\\n \\Rightarrow \\frac{I(n)}{I(n-2)}\n &= \\frac{n-1}{n} \\\\\n \\Rightarrow \\frac{I(2n-1)}{I(2n+1)}\n &=\\frac{2n+1}{2n}\n\\end{align}", "6f4b2250a32507fc7a1e1bd99b9360b8": "\n\\int_0^T p_r \\,dq_r = n h\n\\,", "6f4b53a1a225a9d3ba27a49b1698bfd4": "p_AO_2 \\approx F_IO_2(P_{ATM}-pH_2O)-\\frac{p_aCO_2}{RQ}", "6f4b98c229b033a03ab880cdcfa6122c": "j_o", "6f4bb3d52cfefbb64371021eec6fb3f8": "\\begin{alignat}{5}\n-4y &&\\; + \\;&& 12z &&\\; = \\;&& -8 & \\\\\n-2y &&\\; + \\;&& 7z &&\\; = \\;&& -2 &\n\\end{alignat}", "6f4bbeb044bbe28d44a227ffe4540389": "M=1", "6f4bec9abeeeb3e7ba87e0047c07d588": "\n\\frac{du}{dt}=A(u),\n\\qquad\nu(t)\\in X,\n\\quad t\\in\\R,\n", "6f4c180deeb887e88466892628f067b8": "x_1M_1 + x_2M_2", "6f4c441b145ac4428d1a6df2e8a4ed9e": "\\! X", "6f4c66779f986108df67eb633651d803": "\\Phi_{A}", "6f4c7117238ae46d0fc104fa974c0d82": "\nF(x) = \\sgn(x) \\frac{\\ln(1+ \\mu |x|)}{\\ln(1+\\mu)}~~~~-1 \\leq x \\leq 1\n", "6f4c8b6b74a95db47e66d7d7e3d5b337": "a(x) \\leftarrow \\sum_{y \\rightarrow x} h(y)", "6f4cdd9335072a2534e43357ed01763a": "|S(\\rho)-S(\\sigma)| \\le 2T \\log (d) + 1 / (e \\log 2) ", "6f4cfd2855a770e971c8be553fa7a35b": "\ny^*_{n-1/2} = y_{n-2} + h \\left( \\frac9 8 f( y_{n-1} ) + \\frac3 8 f( y_{n-2} ) \\right).\n", "6f4d02de6dc0403cc954c4e87658a57f": "\\frac{\\partial H}{\\partial u} = p - \\lambda(t) - 2\\frac{u(t)}{x(t)} = 0", "6f4d343df83f2c436e1ae778d6e7ec25": "R\\left( t,s \\right)", "6f4da61ef323c875674ec83473bc1f06": "\\operatorname{pd}_R M < \\infty", "6f4e851a8ab5f6b1ba8b58d3ca025455": "X=\\frac{1}{2}Y = \\frac{1}{2}IT", "6f4eee6087c7cb81989cc88a1978f1e9": "\\displaystyle{G=KAN.}", "6f4f0ed74238d025cf4fed833b2f6e25": "\\lambda = \\mp 1/\\sqrt{2}", "6f4f6771823b1c727cdd02fb8619f649": "\\alpha_A", "6f4f974733eefe1d4b125e2d9def0722": "dS = {dQ\\over T} = 8\\pi M dQ.", "6f4fc39108c7782915245e9c9039c9f5": "2^{-s} F(s;q) = F\\left(s,\\frac{q}{2}\\right)\n+ F\\left(s,\\frac{q+1}{2}\\right).", "6f4fdf8203476166e377b5611909a3fc": "\nd_{\\pm} (\\lambda )~=~{\\textstyle\\frac{1}{2}}q (\\lambda+1) \n\\pm {\\textstyle\\frac{1}{2}}\n\\left\\{ q^2 (1+\\lambda )^{2}-4\\,\\lambda q^2 \\lbrack 1-e^{-2d_{\\pm }(\\lambda\n)R}]\\right\\} ^{1/2} \n", "6f4fe75921a6ba1c8418318466374a23": "\\dot{\\omega}_3", "6f501643ba850860d75580f20c2bee04": "C[a, b] \\,", "6f503ed677ce9860ce6ac28fb78a9d93": "b(x) = x^{l(2t-1)}b'(x) ", "6f506e616de40848a86137a4f6c1db79": "\\scriptstyle g_i", "6f507c647d5fdd424c6da93e03bc6043": "CMUAMA = \\frac{\\left ( MUAC - \\left ( \\pi \\times \\frac{TSF}{10} \\right ) \\right )^2 - 10}{4 \\pi}", "6f50c5d61e67f948867488fb91aa27e9": "\\begin{smallmatrix}R \\end{smallmatrix}", "6f511b5d32dda346dd3acd8c8566cbc1": "G(z) = \\frac{1 - \\sqrt{1 - 4z}}{2}.", "6f5129fc3cf40b1405e256352e4ed069": "\\frac{|x^*|^q}{q} ", "6f512f894070aa14160661b423617d23": "\\,f_{23}=-B_z", "6f5155cd999952f7c02ab3cd32440b50": "C_{m}=Q/V", "6f51a6f29518266663ac0e81f0c86b60": "\\psi_n(x,t)", "6f51c20fbe6732227b115a43016132d6": "u_{1}=-v_{1}\\,\\!", "6f5207a98fcc1d0a19f8e79f5250a41a": "L_{sd} = 1 \\mu m", "6f529eb06c8b0c580140fc772956022e": "A(\\uparrow) + B(\\downarrow) \\rightarrow A(\\downarrow) + B(\\uparrow)", "6f52a931c0334cb82884ab8852cd4acd": "f(x)+a[\\frac{f(x)-f(x-b)}{b}]\\le f(x+a)+K", "6f52c73313bf3ecdba42ac99e6229ea6": "\\sum_{i=1}^n c_i x_i", "6f531da41598b3a6feb22d99e2ad29e3": "\\lim_{n\\rightarrow\\infty} x^n=0\\ \\mbox{pointwise}\\ \\mbox{on}\\ \\mbox{the}\\ \\mbox{interval}\\ [0,1),\\ \\mbox{but}\\ \\mbox{not}\\ \\mbox{uniformly}\\ \\mbox{on}\\ \\mbox{the}\\ \\mbox{interval}\\ [0,1).", "6f53222d0945aeb43c3050b7dc9db99f": "b=n(m^{2}-n^{2}), \\, ", "6f539eee529d3ed0de0ff141420cfb61": "\\prod_{i=0}^{k-1}(x-i)^n.", "6f53ae4cac5b17a39ab4f38ea0706b1d": "p=4n+1", "6f53d84feac7f3dcb0187fe653d61134": " X_t-dX_{t-1}+\\frac{d(d-1)}{2!}X_{t-2} -\\cdots = \\varepsilon_t . ", "6f53fbc47a2d2194e8574073b267ece3": "{v} = \\frac {\\Delta f}{f} \\frac {c}{2} \\,", "6f542862fc279f056e4019be5518c575": "\\sqrt{R}\\,", "6f5466a198d438d844c1cdbad2d80828": "4\\pi r^2", "6f546cbfa57acca1d136d4dc5dee3bda": " \\langle f,g\\rangle = \\int f^*(x) g(x)\\,dx ", "6f547ce590d711d600b6525faa35c006": "Ae\\cong Ai(e)", "6f54ec393eccf4f3873fb6318d72e6c1": "I_D= \\mu C_i \\frac{W}{L} \\left( (V_{GS}-V_{th})V_{DS}-\\frac{V_{DS}^2}{2} \\right)", "6f54f3b28bf1893d985ee6e5b167febe": "\\left( \\frac{\\partial V}{\\partial T} \\right)_P = \\frac{\\partial}{\\partial T} \\left(\\frac{\\partial G}{\\partial P}\\right)_T = \\frac{\\partial}{\\partial P} \\left(\\frac{\\partial G}{\\partial T}\\right)_P = - \\left( \\frac{\\partial S}{\\partial P} \\right)_T", "6f550d81e93964e6b191b4c8b8599092": "\\operatorname{GL}(n, \\mathbf{C})", "6f5519f66c774ee7b507b81111585b92": "\\omega_1^{CK}", "6f552f0c989e3e483be41e7a9c5e9c53": "[-1, 0, 1]\\text{ and }[-1, 0, 1]^T.\\,", "6f554e17f499536ca7fd80974d523d26": "\\tau_{hom} \\,\\!", "6f55837d6a9200dbd641b3aa51497c76": "\\varphi_\\lambda(g) = \\int_K \\lambda^\\prime(gk)^{-1}\\, dk. ", "6f55b355a49f4ab3b80a491e848eae4c": "x=\\frac{-b \\pm \\sqrt {b^2-4ac}}{2a} \\quad \\Rightarrow \\quad x = {y_2 \\over y_1} = \\frac{-{1 \\over 2} \\pm \\sqrt {{1 \\over 2}^2-4\\left({1 \\over 2}\\right)\\left(-F_{r_1}^2\\right)}}{2\\left({1 \\over 2}\\right)} = \\frac{-{1 \\over 2} \\pm \\sqrt {{1 \\over 4}+2{F_{r_1}^2}}}{1}", "6f562d7477c74dcc0f28f43990243f87": "\\mathbf{A} = \\begin{bmatrix}\n\n4&1&-2&2 \\\\\n1 & 2 &0&1 \\\\\n-2 & 0 &3& -2 \\\\\n2 & 1 & -2&-1 \\end{bmatrix},", "6f564f67774e2201a102fc2b92a5bfe3": "\\omega _0 =\\frac{1}{\\sqrt{LC}}", "6f56748a475cbb48f3802329286fb16a": "{}_pF_q(a_1,\\dots,a_p;b_1,\\dots,b_q;z)", "6f56a2e19b8c761a782409db4e231219": "|x| + |y| = 2", "6f56b0e7f95874a86f9dd077707ac2ba": "|\\psi_n\\rangle", "6f56c19a7ff0452a5599ff9e58b9051b": "\\tau_{i_0,v}", "6f571f159844cb23a0341386566ffe14": "C \\rightarrow (B)^A_k", "6f575d88d679aa65f242c83ae32c411a": "\\alpha = K ({\\sigma_0}/{E})^{n-1}\\,", "6f5774e0b9b12866346babe443a7797d": "\\displaystyle{({1\\over 2} I-T_K)\\varphi= -S\\psi.}", "6f57828a43917d488657b2ffb9315c8a": "\\begin{align}\n\\mathrm d\\theta^j - \\sum_i\\omega^j_i\\wedge\\theta^i &=0\\\\\n\\mathrm d\\omega^j_i - \\sum_k \\omega^j_k\\wedge\\omega^k_i &=0.\n\\end{align}", "6f57ef4c4668d86fdb8b62a7e099c1dd": "\\nu_m", "6f582cf63543a01a7858fb9526014e92": "\\kappa \\equiv 0,", "6f5854b526f558fb2704b31f84df44ae": "\\phi\\colon L \\to \\mathfrak{gl}(V)", "6f585b35918f8092c6cd0a6b4a9dafe6": "\\left[Y=\\sum_{i=1}^N R_i^2\\right] \\sim \\Gamma(N,\\frac{1}{2\\sigma^2}) .", "6f585df5b3729ad672d28b2bd7c5b25d": "[P]", "6f58730f154756d9dc7efb13fc938933": "c_n", "6f58c6c65d96bd9dc9dee0a19a2cf2ae": "P(t) = \\begin{cases}\n 0 & tt_o+t_p \\\\ \n\\end{cases}\n", "6f5917e7c5ea56d39ddadcee29b2a921": "-\\frac{dN(t)}{dt} = N\\lambda _1 + N\\lambda _2 = (\\lambda _1 + \\lambda _2)N.", "6f59186bb5f7a76acd5b5ee5597fc161": "\n \\bar{\\sigma}_N \\; = \\; C_s (l_0 / D)^{n/m}, \\; C_s \\; = \\; [\\Gamma(1+m^{-1})\\Psi^{-1/m}s_0\n", "6f592de43b6584a6fac7f28b387b2bbb": "r^{\\frac{3}{4}} \\exp(3\\tfrac{0 \\pi i}{4}) (3-r)^{\\frac{1}{4}} \\exp(\\tfrac{0 \\pi i}{4}) = r^{\\frac{3}{4}} (3-r)^{\\frac{1}{4}}.", "6f593df4a70a5a0413296dfb0308fdd7": "E=\\frac{1}{2}L\\, I_{\\text{L}}^2", "6f59d848fad060ea0a466b892b559dc4": "\\dot{V}_1(\\mathbf{x}, z_1) \\leq -W(\\mathbf{x}) < 0", "6f5acba127c81fdf9e8439be71ef9794": "\\rho = N_{j+1} / N_j < 1", "6f5afd5e37b3034981bf88b92fd6b245": "X_5 = \\,\\!", "6f5b27720c44318960914aabfe13fbdf": "f \\circ T^n", "6f5b83932ab082d90b4bb2bb19f1ef4f": "-0.9074790760", "6f5bd6d9e30ce90b5d86380a57137ed4": "(a^2-b^2)^2+(2a b)^2 = (a^2+b^2)^2", "6f5c15a3d30789fe16fd8fd6186aa54b": "A_{ij}=\\frac{\\partial \\beta_i}{\\partial \\theta_j}=\\delta_{ij} - \\frac{\\partial \\alpha_i}{\\partial \\theta_j}\n= \\delta_{ij} - \\frac{\\partial^2 \\psi}{\\partial \\theta_i \\partial \\theta_j}", "6f5c35ae281f3fcd56e95ac6436d6e20": "(A, r, \\xi)", "6f5c615a4d715fe6b36711d0c514aba7": "\n\\int_a^b f(x)\\,dx = \\frac{b-a}{2} \\int_{-1}^1 f\\left(\\frac{b-a}{2}z \n+ \\frac{a+b}{2}\\right)\\,dz. \n", "6f5c705aeafc93bb42cde16d345e8755": "\\alpha + \\beta x_d", "6f5c8539e2f3bc022d50004aeb5c77f8": "r_3 = (S \\to \\epsilon,S \\to \\epsilon,S \\to \\epsilon)", "6f5c9a78906245def363533d57d7c9e2": "a_0=S\\,\\!", "6f5ca7f1a26f5d44026dc238a0c29286": "N:=\\max(N_1,N_2)", "6f5cc1304d5e977858ac0cefe4284804": "PA \\subsetneq REG", "6f5ce6c85010ef83700d89d0fb9366d1": "id_X", "6f5d1434932730c96a74db6c217a5e86": "[0,\\frac{1}{2}]", "6f5d2c75207640edbb91a1d646194055": " \\exists j ", "6f5d5ca45333d7af814224a6535b6774": "[A,B]^{\\mathsf{T}} = B^{\\mathsf{T}}A^{\\mathsf{T}}-A^{\\mathsf{T}}B^{\\mathsf{T}} = BA-AB = -[A,B] \\, .", "6f5d896f5a5a7d60aa6d86c824f9e575": "\\textstyle \\mathbb{F}_2", "6f5e1e3f3009ea0740edcede4fb0b313": "K(y, x) = \\overline{K(x, y)}, \\quad x, y \\in [0, 1].", "6f5e3a8e1c8466c3503af36bfbff2002": "ab>0", "6f5e9d2923f0e861a1d328670de3c001": "k_1 = 1.7745", "6f5ec70cc6ce3df64ee4fe4f49bbfc06": "\\Phi(m,n)=\\Phi(m,n-1)-\\Phi\\left(\\frac{m}{p_n},n-1\\right)", "6f5ef944a2d6b5db7b0f5eb7664fbe8d": "B_2", "6f5f551304fad46831bbc3f558b96327": " G^{ab} + \\Lambda \\, g^{ab} = 8 \\pi \\, T^{ab}", "6f5f7e93b93f5e1ef05d850229552604": "\\rho_i = \\rho_0\\,b_i M_i,\\ b_i=\\frac{\\rho_i}{\\rho_0 M_i},", "6f5f7fb2c85bcf44d5b6db7c1ca569c5": "x = \\log^*(\\exp^*x) = \\exp^*(\\log^*x)", "6f5f8dbe57794350fa6e3c81ac6977e9": "h(tx) = \\alpha(t)h(x)", "6f5ff0d45e67bf457292eeade4474a27": "N^2\\log^{-3}N\\ll\\left(\\hbox{number of ways N can be written as a sum of three primes}\\right).", "6f603231002b03c6c9a0666e94bb4393": "g \\in F", "6f6078995c3b4fb2fed4b912f002f4c6": "\\Delta \\sum_x f(x) = f(x) \\, .", "6f608dd32612eaa497eedb99ad8ab4f0": "P \\Leftarrow Q", "6f609808a6daa37769984ff8370bbb5a": "z=0^+", "6f609be43cff6ab4312adf173c7c6d89": "L_\\rho(\\Gamma)^2/A(\\rho)", "6f60f3431a7fa2005924ff0e5be55674": "= 100 (a\\cdot c) + 10 (b\\cdot c) + 10 (a\\cdot d)+ b\\cdot d", "6f619619172a2add8de0b59584cd9119": "m(n) = \\Omega(2^n \\cdot \\sqrt{n / \\log n})", "6f61a7ff03fe19a5c706a5a097549020": "1/{k_m}", "6f61c1d475224f669ae2e906a1ebd66c": "Q = \\frac{1}{2\\zeta}.", "6f621c5f2c3fb35b187236d7fe7e5ae3": "\\mathrm{C_B = [B] + \\Sigma q \\beta_{pqr}[A]^p[B]^q[H]^r}", "6f6226bba7ec4398e678d26bc93103ec": " \\vec{\\mu} = g \\frac {\\mu_B} {\\hbar} \\vec {J} = \\gamma \\vec {J} ", "6f62439bae84c672a3c453dee06d7985": " G = - W_0(-v e^{-v}), v=n/k ", "6f6333e2b24b3c5dd9a5c0e1899c099f": "{{F}_{2}}=\\left\\{ \\{1,2\\},\\{1,3\\},\\{2,3\\} \\right\\}", "6f633885f20d709ef6e83da285a21c4c": "x,y\\in E_i", "6f6384a279e33b82cba81a9e0952a71e": " \\Theta(s)=s\\frac{d}{ds}", "6f63a79ff3d7a80d3a24d4429fa8f893": "\\langle f,g\\rangle = \\int_{[0,2\\pi]^n}f(e^{i\\theta_1},\\cdots,e^{i\\theta_n})\\overline{g(e^{i\\theta_1},\\cdots,e^{i\\theta_n})}\\prod_{1\\le j0\\ (a>4)", "6fb9cc2458dfe3fb85ea2db460f7e59e": "\\scriptstyle \\Delta p / p", "6fb9ef169d39fee41b151978ad6db750": "F(x,y) = \\operatorname{P}(X\\leq x,Y\\leq y),", "6fba0d5f59838f237676241e7d5851f7": "t^{2 / d_w}", "6fba19fa1129f4dd6e462dfb8b030e72": "L=\\int_{S_o}^{S}n(x,y,z)\\sqrt{x'^{2}+y'^{2}+z'^{2}}\\, ds", "6fba82bcb457115aae690414c707f141": " \\lambda T ", "6fba9ef5c2f8d85627d511350f064a81": "Q:=\\bigcup\\nolimits_{\\sigma\\in\\sigma(T_0)} Q_{\\sigma}", "6fbaac8130635ed599dc970915cac887": "\\langle H(x_1) H(x_2)\\cdots H(x_n) \\rangle = { \\int DH P(H) H(x_1) H(x_2) \\cdots H(x_n) \\over \\int DH P(H) }.", "6fbaee8bace8ff93f5d3888c713d45eb": "(\\lambda,\\,M)", "6fbbc4ae7a41841af4dc65ec5628c112": "\\mathbf{j} = \\rho \\mathbf{v} .", "6fbc090b420ae726e373649f1c55bd6b": "a^r \\equiv 1\\ \\mbox{mod}\\ N.\\,", "6fbc13563cc416d59eea0991e6981103": "\\kappa_m", "6fbc3f3a280f70ac4f59f1c0c1688c2f": "x_2 \\in X", "6fbc7b7dfc984a9b12d70c2b44c49c7d": "( f(\\alpha_1), f(\\alpha_2),\\ldots,f(\\alpha_n))", "6fbc7ee218637295b8e1adb0436c5e86": " S^{2n-1}(p)\\rightarrow \\Omega \\widehat S^{2n}(p)\\rightarrow \\Omega S^{2pn-1}(p)", "6fbd168857890c1b94d7d96243106b24": "T_{k,n}(z)=g_{k,n}\\left(T_{k-1,n}(z) \\right)", "6fbd62b16c55a29c513676bdbc7e7662": "\\theta=0.5", "6fbda170e6e139435f91ea40fc633db8": "l,s, j", "6fbdbfc78c6e4231903bdbb287597cc7": "\\begin{matrix}{4 \\choose 2}{4 \\choose 1}^2\\end{matrix}", "6fbdc272d6e306385acfe671fafa95d0": "\n\\tan \\beta = \\sin \\eta \\cot \\lambda. \\,\n ", "6fbdf291cda891b99cf211417ad1df18": "\\bar{x}", "6fbe88f46f1d10c5b88ec551aecd0780": "=> \\frac{\\sqrt{3}}{x} = 3", "6fbf08fc26f314978246b62a9615e2c6": "Ef := \\begin{cases} f & \\textrm{on} \\ \\Omega, \\\\ 0 & \\textrm{otherwise} \\end{cases}", "6fbf1f344dc09d5c54b9976790cb0f15": "\\binom{t}{k} =\\frac{(t)_k}{k!}=\\frac{(t)_k}{(k)_k}= \\frac{t(t-1)(t-2)\\cdots(t-k+1)}{k(k-1)(k-2)\\cdots2 \\cdot 1};\\,\\!", "6fbf4efc1c6e57ed676a9c0a015de484": "B|\\psi_{nr}\\rangle=\\sum_{s=1}^g c_{rs}|\\psi_{ns}\\rangle", "6fbfa4a02f5673bcd16f0210c1753fb4": " \\mathbf{s}(x) = \\sum_{i=\\ell-n}^{\\ell} \\mathbf{d}_i N_i^n(x) ", "6fc00f3dd896e305327e5a953ebfb4e1": " \\tau^{ab}{}_{;cd}-\\tau^{ab}{}_{;dc}=-R^a{}_{ecd}\\tau^{eb}-R^b{}_{ecd}\\tau^{ae}", "6fc01dcc961a9228e3f22c34d2e63ac1": "g(s)", "6fc0255b71c910811fac175e10455789": "1 - \\sqrt{R}", "6fc06e4ff62705e8993f62272cc72b57": "2^3 = 8 < 26 < 27 = 3^3", "6fc08fc50f8fdeac7e3bad956a921354": "\\mathbf{F}(\\mathbf{r}) = -\\frac{G M m}{r^2}\\hat{\\mathbf{r}},", "6fc0bd65c8cd7833e6bbb673112edc41": "b_1 = 2.74819", "6fc1c64266b92fe36d235dc4fdee3a3f": "S1\\ \\delta^i\\ S2", "6fc1d29e26f20446d5ba39be313b1df7": "f(z) = \\sum_{n = 0}^{\\infty} c_{n} (z-a)^{n}", "6fc219cdeba331e561d143441aa359eb": "\\lim", "6fc240c96086793e0f9750145c0a3269": " n = 3 ", "6fc249f7387374d049b3045c9f2a5824": " \\left (\\ddot{r} \\right )", "6fc25b4ded8b50c852a143a8292e4c49": " \\sigma_0 ", "6fc27541de63d6421cdb5dbbfed22aa3": "\\{ x \\in X :A_x \\text{ is meager (resp. comeager) in Y} \\}", "6fc2bc099084f11d5dbfb02a75940784": "N_{\\mathbf{v}}=N", "6fc2c5113331cdf38dc126dfb8b54ac6": " \\mathcal{B}(0,1) ", "6fc373a384c187a09168b8d8aaabe45a": "\nk_\\mu \\langle N(p) |\\pi(k) N(p') \\rangle = \\langle N(p) | J_\\mu |N(p')\\rangle \n\\,.", "6fc433f102c265cad11cfb813f2c524b": "S_{pure}=Ge^{-2G}", "6fc4f28ab5c577c3ff3642fbf45fc047": "\\mathbf{A}=\\begin{pmatrix}0 & -1\\\\ 1 & 0\\end{pmatrix}", "6fc52f0f9159d23839d3ff6f912dde94": "\\partial_u, \\partial_v", "6fc539a326ca02423a740da0f607e4e2": "\\sum_{j=1}^{n}e_{j}(x)\\tilde{\\Gamma}_{e_{j}(x)}", "6fc5437907abbced43dc2379a59bbc6e": "T_C = 10 + \\left ( \\frac{N_{60}-40}{7} \\right ).", "6fc58701691d3b801268f5a5980ca865": "G_{ret}(x,y) = \\lim_{\\epsilon \\to 0} \\frac{1}{(2 \\pi)^4} \\int d^4p \\, \\frac{e^{-ip(x-y)}}{(p_0+i\\epsilon)^2 - \\vec{p}^2 - m^2} = \\left\\{ \\begin{matrix} \n\\frac{1}{2\\pi} \\delta(\\tau_{xy}^2) - \\frac{m J_1(m \\tau_{xy})}{4 \\pi \\tau_{xy}} & \\textrm{ if }\\, y \\prec x \\\\\n0 & \\textrm{otherwise} \n\\end{matrix} \\right.", "6fc5c07bfdd9ce8a42d5c8e0f47fbbe9": "10\\zeta(2)\\zeta(5)+\\zeta(3)\\zeta(4)-18\\zeta(7)", "6fc61739ea889908578c5568eab84c3a": "\\mathbf{k}^m ", "6fc6380fdf07dc502f60d675a891691f": "G\\rightarrow H", "6fc65d6dc086ac04f316af6d0bedc91d": "P(M_0,0) + P(M_1,1) + P(M_2,2)", "6fc6ea570ae8221f8985b4b5325d4da4": "s_n(F_2)\\sim n\\cdot n!.", "6fc6f7f789072bb0b3efb9ee27894e3a": "c_a=\\frac{X/X_n}{Y/Y_n}-1=\\frac{X/X_n-Y/Y_n}{Y/Y_n}", "6fc6fc88fb8516e759a07f0b3d226415": "i \\hbar \\frac{d\\Psi}{dt} = - \\frac{\\hbar^2}{2 m l^2} \\frac {\\mathrm{d}^2 \\Psi} {\\mathrm{d} \\phi^2}+m g l (1-\\cos(\\phi)) \\Psi ", "6fc741547ab2a3e0e08850b64d8d4546": "K_{w^{ }} = [H^+]_i[OH^-]_i", "6fc80887f82d9bccefa6667ada583a8f": "[k]^k", "6fc86ca65940223b28b3065ae96cc5e7": "\\begin{align}\n0 \\leq I_\\varepsilon(x)& \\leq \\mathbf{1}_{{x}\\in D}\\quad \\forall \\varepsilon >0\\\\\n\\underset{\\varepsilon \\searrow 0}\\lim\\; I_\\varepsilon(x)&=\\mathbf{1}_{x\\in D}\n\\end{align}", "6fc87c987364cf588e93431452430145": " \\tfrac{1}{t}\\bigl( \\begin{smallmatrix}\\\\ \\pm s&\\mp r\\\\ \\mp r&\\mp s\\end{smallmatrix} \\bigr),", "6fc89b6bbfac73adea805d51a2deca87": "F_{\\hat\\theta_n}(x)", "6fc9086dd9ed8e07615c02743be63d34": "\\begin{bmatrix} d+1 \\\\ n_1\\ n_2\\ \\dots\\ n_p \\end{bmatrix}", "6fc9d0b791115ddb126e8630fc7b2d69": "\n\\left[\n\\begin{array}{rrrrrrrr}\n-416 & -33 & -60 & 32 & 48 & -40 & 0 & 0 \\\\\n0 & -24 & -56 & 19 & 26 & 0 & 0 & 0 \\\\\n-42 & 13 & 80 & -24 & -40 & 0 & 0 & 0 \\\\\n-42 & 17 & 44 & -29 & 0 & 0 & 0 & 0 \\\\\n18 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{array}\n\\right]\n", "6fc9d28e2919b9898aa0ac428cbcc7e0": "f_{uc}(\\{\\langle \\text{Straße} \\rangle, \\langle u2 \\rangle, \\langle Go! \\rangle\\}) = \\{ \\langle STRASSE \\rangle \\} \\cup \\{\\langle U \\rangle\\} \\cup \\{\\} = \\{ \\langle STRASSE \\rangle, \\langle U \\rangle\\}", "6fc9f498286fd8a4eb6997dd4d101550": "4\\sqrt{1\\ \\mathrm{m}\\over g}K\\left( {\\sin {10^\\circ\\over 2}} \\right) \\approx 2.0102\\ \\mathrm{s}", "6fca8221b5c2a5ca27763adaca3adbce": " \\Phi_{00}, \\Phi_{11}, \\Phi_{22}, \\Lambda ", "6fca8259bbd3639d33200708b16edb62": "Q_{t+n}^{I}", "6fcad87f72d1ffce51da4f3273039d6c": "R_f=R[t]/(1-ft),", "6fcb17c76122b90b7bb9693d59192d8e": "f(f(...f(z^*))) = z^*", "6fcb552a5aac0f63c4082f118862f346": "\nR = \\det(A) - \\alpha \\operatorname{trace}^2(A) = \\lambda_1 \\lambda_2 - \\alpha (\\lambda_1 + \\lambda_2)^2\n", "6fcb6af380fa8bebbb07b500e73e6ffe": "\\rho(x_1,x_2)", "6fcbc181f2b4fb686cd8383a7b79a813": "\\mathrm{DM} = \\int_0^D n_e(s) ds,", "6fcbff456dbffe89aeff2043052cfb6e": "\\theta_1 \\geq \\theta_0", "6fcc3df12a6485e3b84983a894e8ac7a": "A = \\begin{bmatrix}1&-3\\\\1&2\\end{bmatrix} . \\,\\!", "6fcc6f5052f3ab9094208f3f73c9779f": "\n \\mathbf{e}^i(\\mathbf{e}_j) = \\delta_{ij}\n ", "6fcca76369f6bd19b434bd63a1634a69": "\\Delta\\tau\\rightarrow 0", "6fccb9693e518a2cef3ecdadc3b9703f": "\\textstyle A_i", "6fccfa80d6a846b625ab15a7b1819cea": "\\{\\tilde{s}[n],\\tilde{h}[n]\\}", "6fcd15c6f5c6cd4b6415cf98980f63cc": "n = 2 \\frac{1}{(2\\pi)^3} \\frac{4}{3} \\pi k_F^3 \\quad , \\quad \\mu = \\frac{\\hbar^2 k_F^2}{2m}\\quad , \\quad\nn \\propto \\mu^{3/2}.", "6fcd3e941e1ccadff26bf7ce050c2d18": "\\left.\\frac{\\partial^p}{\\partial x_1^{p_1}\\partial x_2^{p_2}\\ldots\\partial x_n^{p_n}} f(\\boldsymbol{x})\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}", "6fcd428cfd56357699bc8692aa5dea80": "2^{2^{2^{n^k}}}", "6fcd4ef741a268d5346310b7e73b34dc": "a_{3}+b_{3}+c_{3}=a_{1}+c_{1}", "6fcd7283c754146ad8c0a7b4c43fe8ac": "(A_x\\ ,\\ 1.21688\\ A_x)", "6fcda20b82fe8647d09229e0da096c71": "S=(z_1, z_2, \\dots, z_m) \\in Z^m", "6fcdba9bc595c59331ee85cc563f464e": "Y_{\\mathrm{f}}", "6fcdfab03edf93b472a19c440c718473": "\\begin{bmatrix}\\alpha & \\beta \\\\ \\gamma & \\delta\\end{bmatrix}", "6fce26a4291f880d71c692c970ef2538": "W=(64\\pi k_B TR\\rho_{\\infty} \\gamma ^2 /\\kappa ^2 )e^{-\\kappa D} \n", "6fce4efae47cffe95d79feee9c3e098a": "\\mathrm{d}\\omega \\cong (D \\wedge A)^{\\dagger} \\cdot \\mathrm{d}^{k+1}X = (D \\wedge A) \\cdot \\left(\\mathrm{d}^{k+1}X \\right)^{\\dagger},", "6fce543c6280a62e27a1a29985c89838": "\\sum_i p_i x_i ", "6fce7eee67f9e51a646eb9bcd1a27f8a": "1/\\rho", "6fcea58e0b450cb5a925ab793e63a04f": "{{\\tau }_{1,2}}=\\frac{1\\pm \\sqrt{1-4u(u+v)}}{2\\omega u}", "6fceadacabc85f7fc0a889ff7039f3a6": "\\rho_{AB} = x_{12}, \\ \\rho_{AC} = x_{13}, \\ \\rho_{BC} = x_{23} ", "6fceb2ff8bb5a47fafbbc338c0c17ba7": " R_{\\rm eff} = R/2.", "6fceca88febf0e7eed98bed241347320": "\n\\begin{align}\n\\Theta_+&=\\sphericalangle(\\mathbf{p}_+,\\mathbf{k}),\\\\\n\\Theta_-&=\\sphericalangle(\\mathbf{p}_-,\\mathbf{k}),\\\\\n\\Phi&=\\text{Angle between the planes } (\\mathbf{p}_+,\\mathbf{k}) \\text{ and } (\\mathbf{p}_-,\\mathbf{k}),\n\\end{align}\n", "6fced8387d3cbad4c3bef7892b09729f": "p(t) := {{2m+1} \\choose 1}t^m - {{2m+1} \\choose 3}t^{m-1} \\pm \\cdots + (-1)^m{{2m+1} \\choose {2m+1}}.", "6fcf1982ee205ffa1868c40bcee24e85": "m^2-n^2", "6fcf893639a4b56e0480131e4e19e070": " T_2 = {{m_2 g (2 m_1 + {{I} \\over {r^2}} + {{\\tau_{friction}} \\over {r g}})} \\over {m_1 + m_2 + {{I} \\over {r^2}}}}", "6fcfa6a489f15ed62c76f078b7ee8c64": "\\mathrm{Div}^0(C) = \\{D \\in \\mathrm{Div}(C) | \\deg(D) = 0\\}", "6fcfacd4507d836d456173df8b58c38b": " \\chi(0) = \\rho(0) R_0 ", "6fcfb6516c58dacf441af9075d90443b": "g^{(2)}(\\tau) \\le g^{(2)}(0)", "6fcfceb1cec03d20496afc3dbadfeb0b": "\ni_n = \\sqrt {{4 k_B T \\Delta f } \\over R}.\n", "6fd021841545329bdf01a4ce02dc28ee": "\\scriptstyle\\boldsymbol{u}(\\boldsymbol{x})=\\left(u_1(\\boldsymbol{x}),u_2(\\boldsymbol{x}),u_3(\\boldsymbol{x})\\right)", "6fd03a7f723e2c0cec23d875825c6d61": "A(a/x)", "6fd084c165d2b6f5419caa5676d60481": "x ^ 9\\,", "6fd0d1b9594a00f016857751d8dcfa19": " 217 \\rightarrow 352 \\rightarrow 160 \\rightarrow 217 \\rightarrow ... ", "6fd16bb68d9c3b3821dc8d2982c69e56": " \\omega_0 = \\frac {1}{\\sqrt {LC}} ", "6fd18b030fba20b42b027bb899cf81b3": "\\mathcal{L} = \\mathcal{L}_f(\\mathcal{D})", "6fd1b1a973e35a9f15075190b21fc9c1": "ds^2=-dt^2+h_{ab}dx^adx^b", "6fd23ba2b2b0ad072691c8c0ec407a7b": "\\begin{matrix}1&4&4\\\\3&5\\\\4&7\\\\6\\end{matrix}", "6fd27e1a8907d818d5fb22e9a9933f22": "\\Phi = \\Phi_0 + \\operatorname{arctan} \\left( \\frac{ 2 \\, \\sqrt{(E^2-P^2)\\, R_0^2-L^2} + 2 \\, (E^2-P^2) \\,s}{2 \\, L} \\right) ", "6fd2b49a7a30fbad9ec42c8f94c55687": "i, j, v", "6fd2d9634885fa69e787483217389cca": "\\mathrm{Bo} = \\mathrm{Eo} = 2\\, \\mathrm{Go}^2 = 2\\, \\mathrm{De}^2\\,", "6fd2f0a1b947b34a1c159df2f852176f": "I(\\mathcal{B}).", "6fd30f2111010637f08de188eb1c0bfc": "41-29\\sqrt{2}=-0.01219\\ldots", "6fd30f3998d80a8a4e5b9a8a6998cf4b": "S = - k_B \\, \\sum_{n=1}^W P_n \\ln P_n, ", "6fd31183e4a868366c50ffc7f7d2fc63": "\n\\frac{dY}{dt}= k_3 X - k_4 Y\n", "6fd33517d1998ab14fa1bfa2854353ec": " [ a_0 , \\left( \\begin{array}{c} \\cosh \\tau \\\\ \\sinh \\tau \\end{array} \\right) ] = i \\lambda ~ \\sinh \\tau \\left( \\begin{array}{c} \\sinh \\tau \\\\ \\cosh \\tau \\end{array} \\right) \\, ", "6fd33ee4ebe00eaf5be8226d94766d1a": "2.7668", "6fd37f40deffd8415d17f560fa7eebfa": "\\overline{g}(x)=g(x)", "6fd3e11055c7d5a64acea246b4aca409": "x=X(\\eta),t=T(\\eta)", "6fd4002085d9b61de4e0a54bfbf38aa2": "\\Pi^{n}", "6fd4ab60e4f9d592d79d9e3c5d6f9216": " F_w = g(m_b - \\rho_a{m_b\\over \\rho_b} + V\\rho_w - V\\rho_a). ", "6fd4c2122442a4084a2ecadde43e4e1c": "n^{1/2}n^n/e^n", "6fd4cf17e2be84d14f87374a1751375c": "C_L", "6fd5110211f6e5b2031c6fd54e8e676c": "\\mathbf{M} = \\begin{bmatrix}\n 1 & 0 & 0 & 0 & 2 \\\\\n 0 & 0 & 3 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 4 & 0 & 0 & 0\n \\end{bmatrix}\n", "6fd51631ff94c73ec27546649daf39d0": "N \\log_2 N", "6fd5366d84f507fbc752c27fb7fd9a1e": " \\frac{1}{\\sqrt{f}}= 1.7384\\ldots -2 \\log_{10} \\left( \\frac { 2 \\varepsilon}\n{D_\\mathrm{h}} + \\frac {18.574} {\\mathrm{Re} \\sqrt{f}} \\right)", "6fd54a73b2259828d3554db52b8b6d89": "\\Big[(-1)^{i+j}(r-M(1-\\cos\\theta))+r-M(1+\\cos\\theta) \\Big]", "6fd564dca8b2e6c5725d8ab802842eef": " \\omega^a=dx^a. \\, ", "6fd566f8fb8804cb139dbf95f94fa56e": "\\sqrt{ \\frac{t_1}T }", "6fd584f931de173aabb6c9f6695945a0": "R_1 R_2\\exp(2g_\\text{threshold}\\,l) \\exp(-2\\alpha l) = 1", "6fd5a8f1e91ebcfa2bbe1d7a9b67ca44": "x_{\\lambda}=\\frac{\\dim(|\\lambda|)}{2\\dim(g)}(\\lambda, \\lambda +2\\rho)", "6fd5d894f11d807af317dea11369cc2d": "CH_2O", "6fd5e59f02b8549350d2f212c67a2c83": "f(P, V, T) = PV - nRT = 0 ", "6fd60c3e3e28278a718bea2213feb2b8": "A_a^i", "6fd629fbe0a4f879d891298de5f3315e": "h(x)=x_1^4-x_1^2x_2^2+x_2^4", "6fd64a8eafc5224488e3523dd225bb7b": "e_{ij}", "6fd6f6844e69b1cd2909af80e7fef85d": "|\\Delta^n| = \\{(x_0, \\dots, x_n) \\in \\mathbb{R}^{n+1}: 0\\leq x_i \\leq 1, \\sum x_i = 1 \\}.", "6fd75e83f953b3c77d357edb8911cef2": " \\ f^{(k)} _c (z) ", "6fd78b9395a7022c6a62ea6d12fa1696": "\\rho_i' = \\frac{P_i \\rho P_i}{\\operatorname{tr}[\\rho P_i]}.", "6fd85e623df0284d9f0bfdab071b2b1e": "b\\!", "6fd8a5bd5d54deb88cc79caa7006bf78": "\\gamma =\\infty", "6fd9217a1d3803f429196113c8255a89": "\\! \\frac{e^{it\\mu}}{1 + b^2t^2}", "6fd9849d1835ba03c317c723c40b343b": " {\\dot p_i} , {\\dot q _i} ", "6fda27ca5aa96451936871832c803d1f": "\\Bbb C^n", "6fda707f6d3237ca9f9908257b2c6bc5": "\\scriptstyle \\hat T_j", "6fda765560bee66401c7096bc750e933": "\\overline{a} \\langle x \\rangle.P", "6fda880884ab286d50b1269f5f07c7a2": "\\lim_{t\\rightarrow \\infty} P(M_t\\geq x)=1, x \\geq 0. ", "6fdad2f0fd1650b31bc3d954f7cad3d0": "\nZ = V \\left(\\prod_{i=0}^{N+1} M(i)\\right)\\tilde{V}\n", "6fdb7ed74cf5d5ffad507b21541df322": "\\tau_{ac}", "6fdba289fc678478d42f4ca51b92232f": "1+\\xi(x-\\mu)/\\sigma>0", "6fdbd910cdca25206b3a0ea3ad1d8e36": "\\scriptstyle T=\\frac{|\\mathcal{N}|-1}{2} ", "6fdbda4566cb0a4561cfca6cd75a7fcc": "(t',\\epsilon')", "6fdc1e6a4a66a51e6e45876511b5d2a7": "a = \\frac{dv}{dt}.", "6fdce2803e883b7ca6ade4b234537b62": "\\sigma^\\delta\\Gamma(1-\\gamma \\delta)\\Gamma(1+\\gamma \\delta)", "6fdd18ae8fdde9ac279e8353bd069be7": "f_2(x) = 1 + \\sum_{k \\geq 1} (-1)^k \\frac{\\cos(k \\pi / 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2", "6fdd9b990daf28f221e43108509a54d9": "T_v = \\log_2 ", "6fddbd374c43e8e8392dfb8dad74b983": "X\\rightarrow X+s I ~.", "6fddbf12b3b4965c24145c2c2d5c291b": "\\langle 0|\\mathcal{T}\\{{\\phi}(x_1)\\cdots {\\phi}(x_n)\\}|0\\rangle", "6fddcaa069cc31974bff5674dfb44821": " V^e ", "6fde25eeed4a0d41e19b13d33b8b6e83": "N\\pi \\leq t < (N+1)\\pi.\\,", "6fde333c9a1e58b57a9503315a3fc725": "v=\\sqrt{2gr\\,}=\\sqrt{\\frac{2GMr}{r^2}\\,}=\\sqrt{\\frac{2GM}{r}\\,},", "6fde33c15f2894f6138dbddbc1b38a84": "h_\\mu(T,Q) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} H\\left(\\bigvee_{n=0}^N T^{-n}Q\\right).\\, ", "6fde48d077991bd7774bf55448e2784b": "\n\\begin{align}\n\\frac{dc}{dt} & = \\frac{4 mi \\cdot (-80 mi/hr) + 3 mi \\cdot (60) mi/hr}{\\sqrt{(4 mi)^2 + (3 mi)^2}}\\\\\n& = \\frac{-320 mi^2/hr + 180 mi^2/hr}{5 mi}\\\\\n&= \\frac{-140 mi^2/hr}{5 mi}\\\\\n& = -28 mi/hr\\\\\n\\end{align}\n", "6fde4cac6e919040963dcd881dc28477": "c' |A|\\,", "6fde6a90c7ba1197d6d516b41ac0071d": "A^d", "6fde70889efe9b7b46689663bc4e1911": "DT/ds", "6fde9f47bb3a4c960937891a538716f9": "\nF(\\mathbf{k})=F(k)= 2\\pi\\int_0^\\infty f(r) J_0(kr) r\\operatorname{d}\\!r\n", "6fdea4b9b9ec1fb6827f1519073fe85e": " \\xi \\geqslant 0 \\,", "6fdebc60d672fad76458fa172b1f3c69": "\\begin{align}\n\\sigma_1 &= \\sigma_\\mathrm{avg}+R \\\\\n&= 70 \\textrm{ MPa} \\\\\n\\end{align}", "6fdecf79525d2eedcd2a463a2e1bcacd": "R(1) = 0", "6fdf033ceb306b6898318d3c162f37a1": "z^\\star = [3 + \\eta - (\\eta^2 + 2\\eta + 5)^{1/2}]/(2\\eta).", "6fdf15bd060028f5f0e591f91bbd82ac": "f_0 = f|_{X\\times\\{0\\}}", "6fdf3c84543ca4b83dde25d50d3b2e11": "g\\le^* f \\text{ if } B\\subseteq A \\text{ and } g(X)\\le' f(X) \\text{ for every } X\\in[B]^{\\omega}", "6fdf3dab500db3b1108c5128c8564ade": "\\alpha + \\sum_{i=1}^n x_i,\\, \\beta + \\sum_{i=1}^nN_i - \\sum_{i=1}^n x_i\\!", "6fdfabdf44b3cccb51530816859eac88": " \\mathcal{D}_\\eta (\\rho) \\equiv \\mbox{Tr}_C [ V \\left( \\rho \\otimes |0 \\rangle_C \\langle 0| \\right) V^{\\dagger}]", "6fdfb7910197cd08af70024cebeb62fc": "{L_{th}}^2", "6fdff76fdd0f30a455ea768331932f49": "~\\mathrm{PuF_4+2Ba \\ \\xrightarrow{1200^\\circ C} \\ 2BaF_2+Pu}", "6fe053bb339ced7e3ea1ead75890ee8a": "\\overline{2m-n}=\\sum_{m=\\frac{n}{2}}^n (2m-n)P_{m,n}=\\frac{n n!}{2^n \\left [ \\left (\\frac{n}{2} \n\\right )! \n\\right ]^2}.", "6fe05479260fb3628db04d789855981e": "\\scriptstyle D_F(4\\rightarrow 2)= 4(0)-1+1-0=0", "6fe084252b225fcfcc24f34f0525899c": "Q[\\phi(x)]=x^\\mu\\partial_\\mu \\phi(x)+\\phi(x). \\!", "6fe089f7ca47df6bf59018a61e711ca6": "\\in(x,y)", "6fe0cd9dc8ed02e01a5751f75b421cf4": "N/V = \\int_{-\\infty}^{\\infty} g(E) f(E)\\, dE", "6fe110a428c1e47e2b85f05a8dfc29e5": "\\Phi(\\vec{r},t;\\vec{r'},t)=\\frac{c}{[4\\pi Dc(t-t')]^{3/2}}\\exp\\left[-\\frac{\\mid \\vec{r}-\\vec{r'} \\mid ^2}{4Dc(t-t')}\\right]\\exp[-\\mu_ac(t-t')]", "6fe12e3cb6d3f9b91eaa5d2435f66e9e": "[A^-]/[HA]", "6fe136738e72f1b97e361d43569b6911": "W_{\\mu}W^{\\mu}=-m^2 \\vec{J}\\cdot\\vec{J}.", "6fe15790038d456905e3ce514cf61329": "CAPE_b", "6fe15b3139707a5c8d2d140a27ac1684": "M_n", "6fe18638c342f5982a16e5f3a128a6b2": "\\tfrac{1}{\\sqrt{N}}", "6fe19ea9ef3dadef3b7b8906e8e2b467": " t_e=0", "6fe1a4fdf7099604cb589201c6ecf183": "\\{p_1, \\ldots, p_k\\} \\subset M", "6fe1bccbd188abce650a5fc1b2c9f809": "\\gamma>\\frac{4}{3}", "6fe1e6389ff5ec002a5d2cdf6bad5337": "\\begin{array}{rrl}\\forall M_1, M_2 :\\; & M_1\\ {=_L}\\ M_2 & \\land\\\\\n& (P,M_1) \\rightarrow^* M_1^\\prime & \\land\\\\\n&(P,M_2) \\rightarrow^* M_2^\\prime &\\Rightarrow\\\\\n&M_1^\\prime\\ {=_L}\\ M_2^\\prime\\end{array}", "6fe1f56de3384d1c62ed3f99001055a3": "\\mathbf{J}_b = \\nabla\\times\\mathbf{M} + \\frac{\\partial\\mathbf{P}}{\\partial t}.", "6fe200df07e62ecd3f3062e5b6a2a38d": "= \\text{decimal } 494", "6fe2218ed3606ada7a651c79e2f4049d": "U_{\\alpha_1 \\ldots \\alpha_n}", "6fe2ca21743bc09bb6777f443956bb71": "[K_0] \\delta\\mathbf{x}_i+[\\delta K] \\mathbf{x}_{0i} = \\lambda_{0i}[M_0] \\delta \\mathbf{x}_i + \\lambda_{0i}[\\delta M]\\mathrm{x}_{0i} + \\delta \\lambda_i [M_0]\\mathbf{x}_{0i}. \\qquad(3)", "6fe2ca7041de33a739f5383c0b1c4a1c": "T_{2x}=T_2 \\cos(\\beta)\\approx T.", "6fe31c07c3c40968e52c5f7ea28b06b8": " E_\\text{k} = \\int \\mathbf{F} \\cdot d \\mathbf{x} = \\int \\mathbf{v} \\cdot d (m \\mathbf{v}) = \\int d \\left(\\frac{m v^2}{2}\\right) = \\frac{m v^2}{2}. ", "6fe364ae72d7489b2bc881556dbe4cc1": "\\pmod{\\beta}.", "6fe3b6f31bde4601e1595868674f09f8": "\n K_{\\rm Ic}^2 = K_{\\rm I}^2 + K_{\\rm II}^2 + \\frac{E'}{2\\mu}\\,K_{\\rm III}^2\n ", "6fe411e8bcb4e4f7dd11802c9e5311fb": "d=\\int_{}^c S(c,c)\\text{ or }\\int_{}^\\mathbf{C} S.", "6fe4822c40485353208f8ce60cca3b53": "\\beta _2", "6fe509605a6a5ec70274d7844e526605": "N \\times M", "6fe5711cf66f4f39098466142d879d5f": "b_0 = \\frac{Ze^2}{4\\pi\\epsilon_0} \\, \\frac{1}{m_e v^2}", "6fe5f0aac35233d4e41f0962ca361cb9": " (\\lambda x.f\\ (x\\ x))\\ (\\lambda q.f\\ (q\\ q)) ", "6fe69f564fa0bed177d6c0bbec895117": "\\boldsymbol{\\ddot{\\rho}} = - {\\mu \\over \\rho^3} \\boldsymbol{\\rho}", "6fe6a31993ddbc1eb47f097e0538b89b": "f = \\frac{\\Sigma_a^F}{\\Sigma_a} ", "6fe6d683be7ab04acd455b78b825d860": "\\displaystyle \\epsilon ", "6fe7c97cf1b5f4351fa5178f9248aae4": "\\left\\{\\, p: \\forall S\\ p(S)\\leq \\operatorname{pos}(S)\\,\\right\\}.", "6fe8450cf4125e532e8025b35f4423db": "i\\,\\frac{\\partial u}{\\partial t} + \\frac{\\partial^2 u}{\\partial x^2} + |u|^2 u = 0,", "6fe87e1e193b3f4a9da87a820a842144": "\\Delta \\Pi = \\left(-\\frac{\\partial V}{\\partial t} - \\frac{1}{2}\\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2}\\right)\\Delta t", "6fe8916fe9acecce3f16193e5ed40e59": "X_w(a,b)", "6fe894283ffb69ed15ed4eaf529c9a0d": "\\Omega_X^0", "6fe8da1b0f822278151ae2b6e08e7e51": "0\\to \\Bbb{Z} \\;\\xrightarrow{2\\cdot}\\; \\Bbb{Z} \\to \\Bbb{Z}/2\\Bbb{Z}\\to 0", "6fe918a54d9f09fe53eb15007b91a3f6": "min(k,j)", "6fe92796133f2102a75c1b97ff3e293a": " \\dot{q} \\,\\!", "6fe968b6c4701719af3598ef9b80cb8f": "A^+ = A^*\\,\\!", "6fe968d2ee20e0ec58889881c9485ab8": "\\sqrt{1-e^2}\\!", "6fe97b358b528edc477ba63d50b652af": "p_2", "6fe9a78d26f73e61612495f2b4b58f9b": "-167\\pm 6", "6fe9b209343c06b476121a056f89bb0a": "X \\times Y \\overset{\\pi_X}{\\to} X", "6fe9c79658ec12fdca8ed60626e26745": "a_n=\\Lambda(n)", "6fe9e2023d7af4ca284f3772ec2f9d2d": "u(\\mathbf{y}^1)>u(\\mathbf{y}^2)", "6fea13297045dce4e1e9909fc48893b9": "\n\\begin{align}\n\\lambda(s)=(1-\\frac{3}{3^s})\\zeta(s) =\n(1+\\frac{1}{2^s})-\\frac{2}{3^s}+(\\frac{1}{4^s}+\\frac{1}{5^s})-\\frac{2}{6^s}+\\ldots\n\\end{align}\n", "6fea4f4deb5a9e50bbf1d2cd88ba733d": "V(q) = q^{-1} + q^{-3} - q^{-4}, \\, ", "6fea668f36498d134c5734423f42a574": "\\scriptstyle n =\\pm\\sqrt{\\epsilon_\\mathrm{r}\\mu_\\mathrm{r}}", "6fea66e7113e7b8bcdebd4a3f865153d": "\\gamma = \\frac{1}{\\sqrt{1 - v^2 / c^2}} \\equiv \\cosh \\varphi", "6fea75ea13c78d7f511499f4f2d72d0e": "R\\cong \\mathrm{End}_B(P)", "6fea8153d5e7b67e5cdd4aa43d436907": "\n\\sqrt{\\bar {v_{n}^2}} = 0.13 \\sqrt{R} ~\\mathrm{nV}/\\sqrt{\\mathrm{Hz}}.", "6fea8aec01bcae126f7207b3c2b66a51": "\n\\Gamma_{21} = -\\Gamma_{12} \\, ", "6fea92123f010fe0e4920ce4ba8d3bb1": "\n\\begin{align}\n& \\Pr(F^{-1}(U) \\leq x) \\\\\n& {} = \\Pr(U \\leq F(x)) \\quad &\\text{(applying }F,\\text{ which is monotonic, to both sides)} \\\\\n& {} = F(x)\\quad &\\text{(because }\\Pr(U \\leq y) = y,\\text{ since }U\\text{ is uniform on the unit interval)}.\n\\end{align}\n", "6feaefcc6bd3a846041eadade64a47e3": "Scott's Pi = \\frac{0.333 - 0.369}{1 - 0.369} = -0.059 \\!", "6feb089e02298affaec2a9ca623e96d2": "\\tfrac1{(n-1)!}", "6feb1e2850b94ed500413c6410319aa5": "\\frac{L}{T}\\sum_{k=-\\infty}^{\\infty} X\\left(f-k\\cdot \\frac{L}{T}\\right),", "6febbe25ca403de27420608593f461eb": " \\left[ \\begin{array}{c} \\dot{x}_1(t) \\\\ \\dot{x}_2(t) \\\\ \\dot{x}_3(t) \\end{array} \\right] = \\left[ \\begin{array}{c} x_2(t) \\\\ m^{-1} \\left[ f(t) - c x_2(t) - a k_i x_1(t) - (1-a) k_i x_3(t)\\right] \\\\ x_2(t) \\left\\{A - \\left[\\beta\\operatorname{sign}(x_3(t)x_2(t)) + \\gamma\\right]|x_3(t)|^n \\right\\} \\end{array} \\right] ", "6febc0c58d8fe6084c346553814777a7": "m=\\int_0^2 -4x^2-8x+32\\,dx", "6febc5acdbe632e37b79b777fe88e7d7": "h_1\\,", "6febcbfc9660df8d074bc568e2add425": "\\mathrm{E}[\\nu]", "6febdc8409d5ea5521cb820b1f946470": "\\lim_{n\\to\\infty} x_n = - \\infty.", "6fec00e367753d6b660db30bcebf9a2e": "k \\leq \\log_2 n", "6fec2f7464cbc363f8edffc23a81e60c": " v=\\frac{m_a u_a + m_b u_b}{m_a + m_b}", "6fec4b4d88e2907194d22e3d043a9577": "(\\mathbb{R}^n)^T", "6fecae68d9a8bd1410fd767890715ca9": "\\mathcal{L}:B \\to B", "6fecda87de8f7c75155a9cb9fb9bbecd": "B_n(x) = -\\Re \\left[ (-i)^n \\beta(x;n) \\right] ", "6fed70fb5333fd31b0a76becf73e60af": "S_A", "6fedc5f097661b29c65d325e4d3f4d19": "y \\;", "6fedf4c67269d5e49c18a4fb4e5cff0d": "\\tilde{g}= e^{2f}g", "6fee1cb53c906ce21435aebb373ce49d": "\\bar V \\times V \\to \\mathbf{C} ", "6feed1a0d8f7832705d456a844cf9f5a": "\\Delta Q(P(t_1,t_2))\\, ", "6fef1df992344736532b86700d80963c": "f(x)={n \\choose x}\\exp\\left(x \\log\\left(\\frac{p}{1-p}\\right) + n \\log(1-p)\\right),", "6fef29f23ea42ecba803cca49b5d2c7a": "S^1 \\to SO(m-1)", "6fef5a9a8023de8d8d32aa5f4787c38f": "~^{\\circ}", "6fefcefbd4e8e63598c6e4b29d96855d": "43:61 \\approx 1:1.4186", "6fefdfeff7997e2413ed3840cb969fee": "\\phi(y)=\\frac{1}{8} \\text{ln}||\\Omega||+\\text{constant}", "6fefec3b494b52578459b1b794cb1c97": " X_k = \\sum_{n=0}^{N-1} x_n \\omega_N^{nk} ", "6ff0076671f794d2e0881dc7030da8ab": "R(x)", "6ff009d76c8bf6d535b930fbbbf3862b": "\\frac{1}{2\\pi}\\vartheta\\left(\\frac{\\theta-\\mu}{2\\pi},\\frac{i\\sigma^2}{2\\pi}\\right)", "6ff0519b35df9a12f482a29a5128793b": "T_2(x)", "6ff092651c84f2b27c470d24acfab141": "u \\equiv \\frac{1}{r}", "6ff094e637a6aac7c697bc9f45483e13": "u_{i}=t", "6ff09d65bac7da7053787d7d8a2675c0": "\\displaystyle{d(a,b)}", "6ff0b7b6aaad81a1ad38a0aa5d07241f": "x_j=0", "6ff0f23910d3fc6451111d61d3ca9b6c": "\\Phi = 2\\sqrt[4]{125}\\varphi(\\tau)\\psi^4(\\tau)x\\,", "6ff16557375cac8024f64a86f8dfdedd": "f(-0.54719,-1.54719) = -1.9133", "6ff18d5c3c7f41f10e95414e103a9d8c": "\n \\mu_c=m_{1,c}\n \\ \\ \\ \\ \\ \\sigma^2_c=\\theta_{2,c}\n \\ \\ \\ \\ \\ \\alpha_{3,c}=\\frac{\\theta_{3,c}}{\\sigma_c^3}\n \\ \\ \\ \\ \\ \\alpha_{4,c}={\\frac{\\theta_{4,c}}{\\sigma_c^4}}-3\n", "6ff1a608a953ff04dddd9844d988488c": "\\mathcal{P}_n(z) = 0\\;(\\mbox{mod}\\;n)", "6ff1c8123861b55e80031ef08863025a": "u(t)/v(t)", "6ff2216b2261ceb5cadb0cc739672a58": "\\nu (x) = \\int \\theta (x) dx ", "6ff241b7b18f322bce436b4fe4c49098": "P=-\\frac{dE}{dt}", "6ff275a7899615151476158d26f1e727": " dy_{ }^2 = dx^2 ", "6ff2829232d4d0aa2811ed5892ac15b2": "a_n^-", "6ff2970a930f07d7af47f5dc22bd8382": "\\{-1,\\ 1\\}", "6ff29977f47c77690c4170ef6bc6fbb7": "L [\\rho_{\\infty}] = 0 ", "6ff2b8c9e25f368c1406ef7b984cd346": "\\sum_n c_n \\lambda^{-n}\\,", "6ff2e658f9421959ce31a6e52c69fad6": ") \\land (", "6ff30509c43915a59aba5cc6027c1c06": "N_{F_s/F}(\\alpha):=\\alpha\\cdot\\alpha^q\\cdots\\alpha^{q^{s-1}}.\\,", "6ff3090cb71551332e29743e5a85e8fc": " u=\\frac{gr^2}{3\\eta_2}(\\rho_2)\\!", "6ff30eba0536d13f639ed80257291295": "\\chi_- = \\begin{bmatrix}\n 0\\\\\n 1\\\\\n \\end{bmatrix} \n", "6ff3168b198b9405dc0c6b73687ef2a5": "\\rho = R \\frac{A}{\\ell}, \\,\\!", "6ff34ecc2851d3a2450889c2dad940f9": " \\mathfrak{T}", "6ff35340c8fb16f54caf0605f01dda31": "\\operatorname{nec}(U) \\leq \\operatorname{pos}(U)", "6ff37dfe6440010ad19d5997c413edd6": "\\hat{x}_2", "6ff37fdbee6410e8acf85257f5b91b78": "f^{\\star} \\left( x^{*} \\right) := \\sup \\left \\{ \\left. \\left\\langle x^{*} , x \\right\\rangle - f \\left( x \\right) \\right| x \\in X \\right\\},", "6ff38ee3f8d4e02aca94cffedc0a9dfb": "\\operatorname{tr}(\\widehat{a}_j\\rho e^{i\\mathbf{z}\\cdot\\widehat{\\mathbf{a}}}e^{i\\mathbf{z}^*\\cdot\\widehat{\\mathbf{a}}^{\\dagger}}) = \\frac{\\partial}{\\partial(iz_j)}\\chi_P(\\mathbf{z},\\mathbf{z}^*).", "6ff3da4464d18024c1a6a57e893598b7": "\\scriptstyle -\\boldsymbol \\nabla V \\;=\\; S \\boldsymbol\\nabla T", "6ff4e85aa7a164e8cc99a2f342660da8": "dV = \\sqrt{\\det g}\\, dx^1\\cdots dx^n", "6ff4fc2c78a2986dc32131425bcf6896": "\\text{Spec }L // G", "6ff4fd6807c8725e58b0742056336215": " R_c= r*\\frac{V_S}{A_m} ", "6ff51dddbcfa782b339d828ed99de134": " \\frac{1}{o(G)} \\sum_{g \\in G} g \\otimes_K g^{-1}", "6ff558739b8846b9f25a9d63f5a734c1": "U^{ij}", "6ff568d956a0eb87ae44534bb78cb9c0": "\\pi_3,", "6ff570bcc7d48ab36043d559943da66b": "E(p,t)(r) = D(p,t)(r) + S(p,t)(r).", "6ff57a692419f36a0a31209d775bd28a": "T_i = -I\\ddot{\\theta}", "6ff5a113a65dbde982bf1ea5871c8fd7": "\\rho_1 \\simeq \\rho_1 ' \\oplus (\\sigma_1 ' \\oplus \\rho_2) \\quad \\mbox{where} \\quad \\rho_2 \\simeq \\rho .", "6ff5d076615c770b2942e76e5dcccab4": "f(A) \\leq D", "6ff5f604436b5d249dd29a5034fc54ab": "{\\color{Blue}~2.26}", "6ff5ffcac301b879100821dc91233341": "2\\rightarrow3\\rightarrow2 = 2\\uparrow\\uparrow3 = 2^{2^2} = 16", "6ff619ec38c99403c157b224763791aa": "\\Lambda(n) = -\\sum\\limits_{i=1}^{\\infty} n^{\\rho(i)} ", "6ff624ea2624f94c5b278c0ecaee3c94": "\\overline{J_{\\phi}}", "6ff63e5341bbe990dd4ec1e3517a0c20": "({\\and}R)", "6ff68d1558d5241729ae91031b7e7855": " \\mathbb{Z}^n_p ", "6ff6a791b712e5ca890c0f802d022303": "x\\rightarrow\\frac{x-\\xi}{\\omega}", "6ff6bfb86614c2c763b867f14f52c9b1": "\\theta = 2 \\cdot \\operatorname{arg}\\left(\\sqrt{1-e} \\ \\cdot\\ \\cos \\frac{E}{2}\\ ,\\ \\sqrt{1+e} \\ \\cdot\\ \\sin\\frac{E}{2}\\right)+ n\\cdot 2\\pi", "6ff7226454a8ce32a3ddb3825cf29fc7": "x_0, ...,x_{n+1}", "6ff737088980062a674d035d3476a1e7": "\n\\begin{align}\nx(u,v) & = uv \\\\\ny(u,v) & = u \\\\\nz(u,v) & = v^2\n\\end{align}\n", "6ff785d10afb9b064343c27e2b348867": "\\scriptstyle \\lor", "6ff78c2cfbafc6442035e983a8fcda39": "\\bullet\\rightrightarrows\\bullet", "6ff7fec31ed71b3bea6d02434a0f2d81": "f(V_{i+j})\\subseteq W_j", "6ff8072d38dce83c8844e691d749869d": "E(v) = h\\nu_0 (v+1/2) - \\frac{\\left[h\\nu_0(v+1/2)\\right]^2}{4D_e}", "6ff80c7a0c6d340bef035636b33d13e7": " \\mathbf{e}_{12} + \\mathbf{e}_{34} + \\mathbf{e}_{56} ", "6ff8144ce592440cb6d48623559fca67": "s\\in S_{-p}", "6ff836a1fbd5050268178616b1ae5363": "~~{\\rm noise} =c^2\\!-\\!1.", "6ff8de36e84be436a4af0dc0812b69c7": "u^6 - v^6 + 5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0\\,", "6ff9133b26ed00fc1d289e7251173442": "\\alpha=\\{(i_1,j_1),(i_2,j_2),\\cdots,(i_n,j_n)\\}", "6ff93922c7f77c7dd0b72c348819babb": " {}^\\mathrm{N}\\mathbf{v}^\\mathrm{R} = {}^\\mathrm{N}\\mathbf{v}^\\mathrm{Q} + {}^\\mathrm{B}\\mathbf{v}^\\mathrm{R}", "6ff954c13c529239c7a9be496fb190dc": "(I - P)\\,\\!", "6ff96eb0f88775f628dbbc8397e2c6ca": " \\lambda h.h\\ x ", "6ff96f33e7a881cf4b0c04ec8bbe504a": " \\ln \\left ( \\frac{4}{\\pi} \\right ) = \\int_{0}^{1}\\int_{0}^{1} \\frac{x-1}{(1+x\\,y)\\ln(x\\,y)} \\, dx\\,dy = \\sum_{n=1}^\\infty (-1)^{n-1} \\left( \\frac{1}{n}-\\ln\\frac{n+1}{n} \\right). ", "6ff9739aeaf19a1d4efe392ce4478b8d": "q\\epsilon", "6ff9d9caa420009bef58c7c08beb6f8e": " ds^2 = - f(r) \\, dt^2 + {dr^2 \\over f(r)} + r^2(d\\theta^2 + \\sin^2\\theta \\, d\\phi^2) \\,", "6ffa1c41a85b306b619ea0ed63d38c3f": "S^1.", "6ffa2f383ebd4ae50a8d8d15b3cfd6cc": "\\sigma_f\\sqrt{a} = \\sqrt{\\cfrac{E~G}{\\pi}}.", "6ffa4a32d1231e4b67ef2f5c559c1364": "y(\\hat\\theta) : \\Theta \\rightarrow S(\\Theta) \\rightarrow Y ", "6ffa733cd5db2b7cbed3f1a33433f5f0": "b(s_1,s_2)", "6ffae1b1bcab8bf60bddb501c133dca2": "E(X)=\\frac{1}{2}(a+b)", "6ffaf4bafdda0b12674d7cc2ab718cbc": "p(X^2) \\approx 0.1824671", "6ffb5fc51251918fcf67d54c1f1e4f82": "\\,\\! U(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m)) ", "6ffb748eb4cfd36a537dcaeec1d51c55": "\\left. \\frac{d\\Phi_B}{dt}\\right|_{t=t_0} = \\left( \\int_{\\Sigma(t_0)} \\left. \\frac{\\partial\\mathbf{B}}{\\partial t}\\right|_{t=t_0} \\cdot d\\mathbf{A}\\right) + \\left( \\frac{d}{dt} \\int_{\\Sigma(t)} \\mathbf{B}(t_0) \\cdot d\\mathbf{A} \\right)", "6ffba30bb04bd03fcdc00f5b952159a8": " \\sigma_{min} = -130 \\, \\mathrm{dBm} = 10 ^{-130/10} \\, \\mathrm{mW} = 10 ^ {-16} \\, \\mathrm{W} \\,\\!", "6ffc0ec625c8adb292ef6eb01b48fa2e": "\n \\frac{p}{k_BT} = \\rho + B_2(T) \\rho^2 +B_3(T) \\rho^3+ \\cdots,\n", "6ffc3634c736cb0806f328014c823895": "{-y {\\partial f \\over \\partial y} \\over {\\partial f \\over \\partial x}}.", "6ffc6d183472491e5cc60afc3bb7c1fd": " \\hat{\\mathbf{k}}", "6ffc803da61f44a5beb0f13669828203": "\\Delta F = -k_{B}T ln (q_{1}/q_{0})", "6ffc8d1ef2a70a03640beb03ceba2f25": "(F\\downarrow G)^{op} \\cong (G^{op}\\downarrow F^{op})", "6ffcad02445701f52a7454a480c19134": "n^{1-\\epsilon}", "6ffce096e592dc495445c225c8b242f6": "\\scriptstyle \\mathcal{L}^p(S,\\, \\mu)", "6ffceb4c465d3a028a916e09f2f8058d": "x_{2m + 1}", "6ffcfc8d14e764c0ddcc95e041d28afe": "\\mathrm{Ra}_H = \\frac{g\\rho^{2}_{0}\\beta HD^5}{\\eta \\alpha k}", "6ffd5b5bed027953b34600917c7246e0": "\\ r_3", "6ffdfd972d1754a62e0923b926181be5": "x_{k-2}", "6ffe6cd1c9f73db8b01eabb9d855e444": "\\mbox{Spec} \\; \\mathbb Z [x, y, z] / \\langle y^3-x^4+z \\rangle \\rightarrow \\mbox{Spec} \\; \\mathbb Z", "6ffed6080d131fc1e32e342902ab9a99": "\\scriptstyle b_i", "6ffeebc91af15368f2396e901f340c60": "P(1 + r)^t", "6fff047b90b9045a4608fe7c1c054231": " d_m \\left( \\sin{\\theta_m} + \\sin{\\phi_m} \\right) = M \\lambda", "6fff0bc8ef55d19c805e9d41e2d8dc26": "a_r = \\frac{(r + c + \\alpha - 1)(r + c + \\beta - 1)}{(r + c)(r + c + \\gamma - 1)} a_{r - 1},", "6fff4a9138b25594a9e7bf9a8e563a98": " |a(u,v)| \\le \\|A\\|\\,\\|u\\|\\,\\|v\\| \\, ", "6fff4c52d3574d1a79bade999d5717e8": " (\\lambda p.(p\\ f)\\ (p\\ f))\\ (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "6fff656707f0fc20faf8b2444b6f7ea4": "y=x^2/(1+2a)", "6fff6f4f6892f0fb4479602f0506e226": "y=\\begin{cases}0 & \\{ft\\} < 0.25 \\\\ a & 0.25 < \\{ft\\} < 0.5 \\\\ 0 & 0.5 < \\{ft\\} < 0.75 \\\\ -a & \\{ft\\} > 0.75 \\end{cases}", "6fffa9e6028c84e188a95fa874ca39bf": "\\log\\ D_{\\rm oct/wat} = \\log\\Bigg(\\frac{\\big[\\rm{solute}\\big]_{\\rm octanol}}{\\big[\\rm{solute}\\big]_{\\rm water}^{\\rm ionized}+\\big[\\rm{solute}\\big]_{\\rm water}^{\\rm neutral}}\\Bigg)", "6fffe6f982a7b2e6116b7d09264f5605": "B(r) = \\left\\{ x \\in \\R^n \\,:\\, \\sum_{i=1}^n \\left|x_i\\right|^p < r^p \\right\\}.", "7000016d3bd72425e834863e9b7e8247": "H(A : B | \\Lambda)\\geq 0", "700067b22f52ae0c771c0575c3f1c341": "i_{\\text{1}} = i_{\\text{F}}", "7000c1c3d969f9dfecc5687cc87bee57": "N_o \\ = \\ N(0).", "70011a93b8aec61d8365ed9eb556ac52": " p = 1, \\ldots, 12; ", "70014fd2d60d841446b5c73a1f76a112": "\\mathrm{2S_2F_2 + 5O_2 \\ \\xrightarrow{NO_2}\\ SOF_4 + 3SO_3 }", "70016814a43bd256bdb0126dadfbe22d": " \\frac{\\partial}{\\partial t} \\left( - \\frac{\\partial Z}{\\partial p} \\right) + V \\cdot \\nabla \\left( - \\frac{\\partial Z}{\\partial p} \\right) - k\\omega = \\frac{R}{C_p \\cdot g} \\cdot \\frac{q}{p} ", "70017dd7b979fd7efd0b7335dc211705": "\\text{Pythagorean wins} = \\frac{\\text{Points For}^{2.37}}{\\text{Points For}^{2.37} + \\text{Points Against}^{2.37}}\\times 16.", "70021370e64735f58031f2b1ab672632": "Kw ", "70022e7135cdb080d170343867a9e1e9": "D_{2}", "700247282fcd0257d3ad44ac5be94021": "\\hat f(3)", "70025c316d14602cbbfa666bf58ea35e": "C \\cdot D = C' \\cdot D", "700276278bd5bc85fdda4194551c1fb6": "\\Box(H,V)", "70029aad2f14da79a25411d812ade574": "\\alpha_{P}(X)", "7002ba9298010b089e1ca12a1b25e24f": "E*G", "7002e7626f0553fe5b0217fff321d599": " a\\wedge b = \\varphi(a)\\wedge b , b\\wedge a = b\\wedge \\varphi (a)", "70030a4e05ce732c16c9159922a80783": "|\\triangle M|", "70030a58ed2611385cec5e95266eae16": " i \\in E ", "7003337c1a34da5271659c8cab00d33c": "{ds_B}^2", "700356a0a7aa489770c0a3a5303b960f": " \\boldsymbol{\\mathsf{a}} \\cdot \\boldsymbol{\\mathsf{b}} =\\eta (\\boldsymbol{\\mathsf{a}} , \\boldsymbol{\\mathsf{b}})", "7003772c3c163113785a6440766c9c12": "V\\cup\\{F-A;\\,A\\in V\\}", "7003a3848a4a23118c86a0e768548c65": "Y_{12} = {-2 S_{12} \\over \\Delta_S} Y_0 \\,", "7003b6b04d235c35e945c484a1895fa2": "2\\beta + \\gamma = \\pi \\ . ", "7003c9194b6a8d00088e2403d6756fbd": "\\scriptstyle\\hat\\ell", "70043fad081c00f30f9b0eaf8815a79f": "\\nabla \\times \\nabla \\phi(x,y,z)=0,", "70047c98e4163674e78bb42d0cf4aea8": "\\frac{dy}{dx} = \\frac{dy}{du} \\frac{du}{dx}", "700485942a317917ef4a136d891de023": " S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3} \\leq S_{2m+1} ", "7004c43493dd4653e1b0161a89cacb03": "\\ \\Phi(\\mathbf{r}) = R_l(r) Y_{lm}(\\theta,\\phi)", "700514311a7c6bce4bc3c3c06a132298": "\\Delta_r G", "700582f0e252e42e0959303813155bf5": "\\mathrm{Va} = \\frac{\\phi\\, \\mathrm{Pr}}{\\mathrm{Da}}", "7005ac5f8ae8bd7cfb0461e07155dd84": "\\frac{1}{\\infty}", "7005b4bf8c233fb051527747b2593997": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & 0\n \\end{Bmatrix}\n = \\frac{\\delta_{j_2,j_4}\\delta_{j_1,j_5}}{\\sqrt{(2j_1+1)(2j_2+1)}} (-1)^{j_1+j_2+j_3}\\{j_1,j_2,j_3\\}.\n", "70061e542860d7a8a19b9843bc21bf02": "L(\\lambda):=W(\\lambda)/\\operatorname{rad}(\\phi_\\lambda)", "70068f327f81f50a434790df89c7df2e": "f(x,y) = x^2 + xy + y^2.\\,", "7006a3088fa189c8e908c8e28a976f8f": "\\lambda:S \\rightarrow Y^\\phi ", "7006b5d2997f3ac48168c931ac98ae2b": " t \\mapsto U_{t} ", "700726d98dc8c2d3d0daf8d8b071da7f": "\\Delta(X_k,Y_k)", "70075aa148d9ce436cb9511393355641": "\\kappa_6=\\mu_6-15\\mu_4\\mu_2-10\\mu_3^2+30\\mu_2^3\\,.", "70078941b81bee9e78070624f1cacb57": "\\ w_j", "7008407ceb19b8903e9055c1de780a7f": "\\begin{matrix}2&2\\\\3\\end{matrix}", "70085576ee536343d93d40843d4a40de": "f_d \\approx 2v \\frac {f_t}{c} ", "700878c2184e40b7a2364186e7dc67c6": " K(x,x) \\;=\\; \\langle K_x, K_x \\rangle \\;\\geq\\; 0, \\quad \\forall x\\in X. ", "7008799edbe4675989781229646b6a33": "Gen(W)", "700888b8a6c100e08732ba22afd7b67b": "x_1,\\dots,x_n", "70088a290c836673c9f8e0b2e3c0df94": "h(p)", "70091912e1d40d77c910e17a24929884": "p_G\\left(z=\\eta\\right)-p_L\\left(z=\\eta\\right)=\\sigma\\eta_{xx}.\\,", "70092a8bc8133f8853c14d32222db198": " \\{\\{f,g\\},h\\}+\\{\\{g,h\\},f\\}+\\{\\{h,f\\},g\\}=0, ", "70096d274b1db62a821d60c78a411667": "\\text{im}\\ f", "7009976d0f796199b50ae416bc146c37": "\\mathcal{A} (\\omega)", "700a185891701d249d77de079d10147f": "2^{131104 }", "700a2a1737a21e7089e42c37308068b0": "a_{1}+b_{1}+c_{1}", "700a2ca65274062c836df713be8533e0": "\\mu_p", "700a3025909b65dd72ea6a9d2d8bb7dd": " du = 2x\\,dx, \\quad v = \\frac12 e^{x^2}. ", "700a42e9cff075180162aad8855ad6cc": "\\operatorname{d}V/{\\operatorname{d}t}", "700a45043b085b943f0ae952f5043142": "~\\frac{p}{V}\\gg 1~", "700a6aae346a7752e58a2550081d03d1": " \\begin{align} x_{n+1} & = \\frac{5}{s_n} - 1 \\\\\n y_{n+1} & = (x_{n+1} - 1)^2 + 7 \\\\\n z_{n+1} & = \\left(\\frac{1}{2} x_{n+1}\\left(y_{n+1} + \\sqrt{y_{n+1}^2 - 4x_{n+1}^3}\\right)\\right)^{1/5} \\\\\n a_{n+1} & = s_n^2 a_n - 5^n\\left(\\frac{s_n^2 - 5}{2} + \\sqrt{s_n(s_n^2 - 2s_n + 5)}\\right) \\\\\n s_{n+1} & = \\frac{25}{(z_{n+1} + x_{n+1}/z_{n+1} + 1)^2 s_n}\n \\end{align}\n", "700a7e7364c784ab4dc3c16e1d16eb61": "\\frac{\\partial}{\\partial\\bar{z}_i} \\left(f\\circ g\\right) = \\sum_{j=1}^n\\left(\\frac{\\partial f}{\\partial z_j}\\circ g \\right) \\frac{\\partial g_j}{\\partial\\bar{z}_i} + \\sum_{j=1}^n\\left(\\frac{\\partial f}{\\partial\\bar{z}_j}\\circ g \\right)\\frac{\\partial \\bar{g}_j}{\\partial\\bar{z}_i}", "700a8c440114ba883c4c4e0df1db0c8e": "K_\\alpha(x) = \\int_0^\\infty \\exp(-x\\cosh t) \\cosh(\\alpha t) \\,dt.", "700a92d68adb443d803704e5d403a64c": "\\frac p q \\le x < \\frac r s.", "700a93dba48e53fe9e875fd486e3a011": "\\sqrt{10} (4 \\rho^4 - 3\\rho^2) \\sin 2 \\theta", "700b37b05c8a4b4db2d730d9dfc92314": "\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n+1} & F_n \\\\ F_n & F_{n-1} \\end{pmatrix}.", "700b3f7608d84a7be1469baf199eb6d6": "\\mathcal{F}^{n}", "700bd689f990a4e93b30135417007cec": "e_1=(1,0,0)\\,,\\qquad e_2=(0,1,0)\\,,\\qquad e_3=(0,0,1)", "700be9e1462d42cbd0d0b3d5fa89d828": "Y_{7}^{6}(\\theta,\\varphi)={3\\over 64}\\sqrt{5005\\over \\pi}\\cdot e^{6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot\\cos\\theta", "700c1c01814b98f25ec6b1e4c9bc4961": "\\left(\\frac{n-3}{n-1},2\\frac{n+1}{n-1}\\right)", "700c5bd8451f93ab56d6d3ccb04805bc": "\\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)", "700c82f9eff5caf9ac661f331218917c": "a = \\frac{r_1 + s_1}{2} = \\frac{r_2 + s_2}{2} \\quad (6)", "700c9200405e61a4a13d8ac9c8ca1bc1": "S= a_o/Error(a_o)", "700ca6ae2c2234a6316c97713cfcc81e": "2^4\\cdot 3\\cdot 5", "700cfada121e317ba068bd51116eccbf": "(m,q)\\mapsto mq=\\mu(m,q)", "700d3607b4b4267179c1d5e1d55cdb3d": "P_{ij}", "700d6f05c5d9ebe89c657e3f8b5f51af": "S \\frac{\\partial h}{\\partial t} = \\nabla \\cdot (k H \\nabla h) + N. ", "700d90618b55b66a0ff83e47df62d7fa": "\\frac{1}{\\Lambda_{(\\mathbf n)}}=\\frac{dX}{dx}\\,\\!", "700dc8e08137bd20a14bc70eede8672d": " T_n = \\frac { a + 1 } { D^2 - \\frac { b - 1 } { n } } ", "700e0d667603291ce60556f9ccecc33a": "B(u,v)=B(v,u) \\ \\quad \\forall u,v \\in V", "700f6fa0edb608ee5cc3cfa63f1c94cc": "dh", "700f78a53582a42719d1a56aac2a2ac1": "k_Br_B=0.1,1.0,10", "700f8a40ab922bb3e8ad5980e633e337": "\\operatorname{CAT}(0)", "700f9f91c845c94f025facd5c12b4f4e": "F_{n+m} = F_n F_{m+1} + F_m F_{n-1}=\\frac{F_nL_m+F_mL_n}{2} \\,", "700fb2686dee8b2d05bf8323160e546b": "Y(t) = P(t)^{-1/t} -1. ", "700fb960e4888d7d69096b5ee5b1949f": "q'", "700feb9450c920ede15e28b7d3528327": "s(i) = \\frac{b(i) - a(i)}{\\max\\{a(i),b(i)\\}}\n", "70104a4235c1147ae0e9756bbb1f1fbe": "|\\psi\\rangle=\\sum_{n=0}^{\\infty} a_n|\\psi_n\\rangle", "701067ed5d646af1c269d1bb85bd3e69": "2^1 = 2", "70107853de3c94e57fd835226bcade01": "F[\\cdot]", "7010a6098496f74dfb8503339dce9c2e": "\\mathcal{F}_c", "70112bc326f64ab603ce5659bca1b8dd": "A^+P=A^+=QA^+\\,\\!", "701136cbbafcb14884e79f107a854d5d": "\\text{If }", "701168f621faac0641a082ae3886ba8b": " \\nabla_\\mathbf{B} \\left( \\mathbf{A \\cdot B} \\right) = \\mathbf{A} \\times \\left( \\nabla \\times \\mathbf{B} \\right) + \\left( \\mathbf{A} \\cdot \\nabla \\right) \\mathbf{B} ", "701179ff83fce9b7b984c7ec2e09d8df": "({v_0+v_i})10^{b_1E_{i}} \\text{ vs. } v_{i^{ }}", "7011c5277b250171efb9315088cf5c76": "\\overline{p \\lor q}", "701230ada5a697202acab23117581f65": "\nL =\n\\begin{pmatrix}\n 1 & & & & & 0 \\\\\n-l_{2,1} & \\ddots & & & & \\\\\n & & 1 & & & \\\\\n \\vdots & & -l_{n+1,n} & \\ddots & & \\\\\n & & \\vdots & & 1 & \\\\\n-l_{N,1} & & -l_{N,n} & & -l_{N,N-1} & 1 \\\\\n\\end{pmatrix}.\n", "701254111b96e9042b50ec05c04f3e30": "\\lambda(n)\\mid \\phi(n)", "701264aebb650c73dcf16009c952e12f": "C_{XYZ} \\,", "7012772fedc8ee49b6dc269c8933ab7a": "p(\\vec{r},t)", "70128dba109b9e0ba1e6abe059448513": " FV \\ = \\ PV \\cdot (1+i)^n ", "7012dd2a302df017379f868fa15a04ab": "r<3r_s/2", "7012e52f3f3477488aecd3107ec409c5": " e^{rx} \\, ", "7013009c64e6789b3c3f88bdd2b8cafe": "1 - 0.999^{10000} = 0.99995 = 99.995 %", "7013dcd33e720bc10a672bd2fb86863e": "\\,M_s", "701421c3a6de076512bd34c6674d3aa4": " 2 > \\beta \\ge 1.645", "7014a1748fc425e04b013e4a6ba525c1": "\n\\log p(\\mathbf{X}|\\boldsymbol\\theta) - \\log p(\\mathbf{X}|\\boldsymbol\\theta^{(t)})\n= Q(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)}) - Q(\\boldsymbol\\theta^{(t)}|\\boldsymbol\\theta^{(t)})\n + H(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)}) - H(\\boldsymbol\\theta^{(t)}|\\boldsymbol\\theta^{(t)}) \\,,\n", "701529bab267537e02ecb6ceea6863ec": "T:A\\rightarrow\\Gamma", "70153123d9d315b0f5f92ad0d7994da7": "\\mu _{Y} (y) =\\displaystyle \\sup_{\\displaystyle \\sum_{k =1}^n \\bar {w}_i a_{\\sigma (i)} = y }\\left({\\begin{array}{*{1}l}\\mu _{W^1 } (w_1 )\\wedge \\cdots \\wedge \\mu_{W^n } (w_n )\\wedge \\mu _{A^1 } (a_1 )\\wedge \\cdots \\wedge \\mu _{A^n } (a_n )\\end{array}}\\right)", "7015440590904d38e35615ff0672d21c": "+1", "7015a682c3383496d82c5e5d63338fbe": "x:U \\subset \\mathbb{R}^{2} \\longrightarrow S', \\qquad p \\in x(U)", "7015a9ce3111106b438b66f02730b735": " <_\\mathcal{O}", "70163103c81512d959632cf88ac860cf": "d\\ll n", "701642da51a3053881302012691e39fb": "\\mathbf{M} = \\mathbf{U}_r \\boldsymbol{\\Sigma}_r \\mathbf{V}_r^*", "701653d734e23ac51c25137fb8831ba8": "n = 2", "70165a09edf9ebda60603cd13420210b": "A(T,V,N)=-NkT\\left[1+\\ln\\left(\\frac{(V-Nb')T^{3/2}}{N\\Phi}\\right)\\right]\n-\\frac{a' N^2}{V}.", "70170af5c0775bc096dfcbeb1f40008a": "1/n=0", "70182b561265f051243a24424a74a3bc": "\\tilde{f}(s) = \\frac{a_N}{s-N} + \\cdots. \\, ", "70183204f4a3eacb2d3160982ab997d4": "S_\\text{BH} = A_\\text{BH}/4 = 4\\pi m^2_\\text{BH}", "70184dbdc680a4cba81c9386517dc522": "2x^2 + 3x", "70188ccc97126de982a3a3f3b0201c2e": "\\scriptstyle g''(x)g'(x)^2 \\;", "7018c2bd3d935de3c2100b77b7dc5845": " \\delta\\boldsymbol{\\Pi} = \\delta(\\mathbf{U} + \\mathbf{V}) = 0 \\qquad \\mathrm{(2)}", "7018d471bd6dd0d36e3fb2170a538e79": "K\\subseteq_e M", "701925abfd618f53b7463b35b8ba3fa0": " [ABCD,E] = ABC[D,E] + AB[C,E]D + A[B,E]CD + [A,E]BCD", "70196278739833a6e13463c3876c7bec": " \\Psi = \\sum_k c_k \\psi_k ", "7019bc954d0f3abecef813b1fb885eb3": "\\mathrm{proj}_\\theta F", "701a289b88afb81777b83b152e7ebf14": "\\mathbf{x}_{k+1}=\\mathbf{x}_k+\\alpha_k\\mathbf{p}_k", "701a4b4938ea37b5e8fa7c182594251d": "\nB \\rightarrow \\langle E (B) x, y \\rangle \n", "701aab3b362f914670326b42c5d0292c": "X=(X_v)_{v\\in V} \\sim \\mathcal N (\\boldsymbol \\mu, \\Sigma)\n", "701b3e0df7cf6ccbf04f8059c0202746": "\\ge 50", "701b5d8ccfdd3ffcb127e615881d7dde": "x=10^6", "701b752b876fca7c2454eec8bb6d83dd": " x \\oplus \\lnot 0 = \\lnot 0,", "701babf0d3fbc692f75447ecff291a31": "{\\rm det} \\begin{bmatrix} \\lambda - a & -b \\\\ -c & \\lambda - d \\end{bmatrix} = \\lambda^2\\, -\\, \\left( a + d \\right )\\lambda\\, +\\, \\left ( ad - bc \\right ) = \\lambda^2\\, -\\, \\lambda\\, {\\rm tr}(A)\\, +\\, {\\rm det}(A).", "701bbaf2f1d677caf43191ad72ef18ed": "f\\in C^k", "701bc2f019e4706194085e43537a297d": "\ns \\ = \\ j \\omega\n", "701bf1cbbdecc87f2b12a1e1595e3042": "x' = 0 0 \\ldots 0 ", "701c1571cc4922fe41c1d8268cbb589c": "\\mathbf{rank}_q(x) = |\\{ k \\in [0 \\dots x) : B[k] = q \\}|", "701c5573afd3368042297175b0a062c6": "0 \\le V_+,V_- \\le V_{cc}", "701c5acf8fface9d8436a276594daf1d": "y_a=y(x_a)", "701c6532ed586e90581bd61b9dedba78": "A = \\frac{x + y + z + 2 p}{5}", "701c9a513a653499e58460394c37201a": "\n\\left|\\frac{d\\lambda\\left(x\\right)}{dx}\\right|\\sim 1\n", "701c9e38fb4c65e0bd4fa9adab715062": "\\begin{cases}\n\\frac{\\theta \\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \n\n\\frac{(x - a)^{\\alpha-1} (b-x)^{\\beta - 1} }{(b - a)^{\\alpha + \\beta + 1}}\n\n + \\frac{1 - \\theta}{b-a}\n& \\text{for } x \\in [a,b] \\\\\n 0 & \\text{otherwise}\n \\end{cases}", "701d85838bf3e9f28172914962f83b67": "p_1=\\frac {q_4(q_1+q_2)}{2m_2}\\ ,", "701d86d8276a35e719ab66fc47f7568a": " \\varphi(t;\\mu,c,\\alpha,\\beta) = \\exp\\left[~it\\mu\\!-\\!|c t|^\\alpha\\,(1\\!-\\!i \\beta\\,\\textrm{sgn}(t)\\Phi)~\\right] ", "701d93e0fc59e075ae5dc465de9a2ed6": "K_i(R)=\\pi_i(BGL_\\infty(R)^+)", "701dacb27e8d40388147510f36d94939": "\\sigma^{2}_y", "701dd89f40bf02bc8d5c815939fdf3ce": "\\Omega_{0}e^{-\\beta U_{0}}", "701ddca5dc1b3330f1b0ab69513917c6": "\\mathcal{A} = (Q,\\Sigma,\\delta,q_0,F)", "701de20ad27d435bb4b8862cb7f9a61a": "5n^7 + O(1)", "701e28460d86333491e9aecc6cf16408": "\\mu,\\nu", "701e37dd4c47c8555af4742ba67bb63c": "\\Gamma = {\\omega \\over 2\\pi}e^{{-\\Delta E}\\over kT} ", "701ec8502cfd73c63e520c0f3d5b01b0": "\\mathbf{A}_{\\text{Electric quadrupole}}(\\mathbf{x},t) = \\frac{-k \\omega \\mu_0}{8 \\pi} \\frac{e^{i k r - i \\omega t}}{r}\\frac{1}{3}\\mathbf{Q(n)}", "701f50fafebca127e2c10b0065c66f51": "Z={S \\over j\\omega}", "701f5dedc1ca48c642a14238086053b9": "K[X_1, \\dots , X_n]", "701f63e100474c6346b554e222b56b5c": "A\\backslash B", "701f8272148e16066c01e6f89c9bd952": "J(u)", "701f8c854f458e89a34179cca6b7a1f7": "\\theta^*_k, k=1,2,...", "701fc0d326976b7c99f37b35c32930f6": "\\mathbf{a}_{31}", "70207a4b318e80fe85a965c3bbd606ad": " = k_Dn_An_B/R_{TOT}", "702087fdab2cd0474137c13c0480180f": " \\| \\bold{p} \\|^2 = \\begin{bmatrix}p_1 \\cdots p_n\\end{bmatrix} \\begin{bmatrix}p_1 \\\\ \\vdots \\\\ p_n \\end{bmatrix} = \\bold{p}^T \\bold{p} . ", "7020d9d09a7e8c0cef7758767ad1435c": "C_F=\\frac{3h^2}{10m_e}\\left(\\frac{3}{8\\pi}\\right)^{2/3}.\\ ", "7021064269189f46aebda3837ecc3d58": "\\textstyle \\lambda_{PP}", "702128aa4978f7b014f084b5d1613d37": "\\,\\hat{m}_h\\,'s", "702161c1e9a5332920828a02dfa55616": "\\scriptstyle A_{\\ell k}", "7021a38e63c24fb894a843db3b39e45e": "\\begin{cases}\\sigma^2\\,(g_2-g_1^2)/\\xi^2 & \\text{if}\\ \\xi\\neq0,\\xi<\\frac12,\\\\ \\sigma^2\\,\\frac{\\pi^2}{6} & \\text{if}\\ \\xi=0, \\\\ \\infty & \\text{if}\\ \\xi\\geq\\frac12,\\end{cases}", "7021a89ff831fe45307a1fa1c1a279a4": "S(b) = (y-Xb)'(y-Xb) \\,", "7022028763e141f19fd836818a261467": "\\{e_i : i\\in I\\} ", "70220f9e969125e6bff3a41e12f5c835": " M(q,t)\\ddot{q}(t)=Q(q,\\dot{q},t),", "7022551baf45e8b0dc97ab7860cb42f0": "F_{x1}=F_{x2} \\,", "7023309d73726e858eef0127c99efd59": "B = 2X + 4X^3 + 6X^5 + \\cdots,", "70233aef862d710f5d1129f66f9b034a": "\\psi_- \\approx \\frac{1}{2mc} \\sigma\\cdot \\left(p - \\frac{e}{c}A\\right) \\psi_+", "70239dc9201fec3035d8019a28372c1b": "\\varepsilon_{ijk}\\sigma_{jk}=0\\,\\!", "7023c9657fbe430c87dc8e5e11f9fc49": "\\gamma_{a3}^2 \\ll \\gamma_{aa}\\gamma_{33}", "7023f05e2e27a853643094354ec694d6": "\n\\kappa = \\begin{cases} 3 - 4\\nu & \\qquad \\text{plane strain} \\\\\n \\cfrac{3 - \\nu}{1+\\nu} & \\qquad \\text{plane stress} \\end{cases}\n", "70243a2e9570cca247e5cd1c2e175198": "c\\in \\mathbb Z^{*}_{n^{2}} ", "70246c86a0e5f336652365f5f38c4931": "\\lambda n.\\lambda f.\\lambda x.n\\ (\\lambda g.\\lambda h.h\\ (g\\ f))\\ (\\lambda u.x)\\ (\\lambda u.u) ", "702486b6f3558715a04f9433f93b6a51": "-\\nabla^2 \\phi_1 = \\rho_1 / \\varepsilon_0", "70248a6cc450dcb5c9e077881acffdd1": "6-2d(f)", "7024b25af80791051c9a5bf94903c7fa": "\\phi_i = \\phi_{DL}-\\phi", "7024bad168eb812e851d95764f6f1f9f": "\n\\bar{h}^{i j} (t,\\vec{x}) \\approx\n-\\frac{4}{r}\\, \\frac{M_1 M_2}{R}\\, n^i(t-r) n^j(t-r)\n", "70253d3edb2a08943983605763ab98ce": "\n\\frac{\\partial \\mathbf{w}}{\\partial t} + \\mathbf{\\Lambda} \\frac{\\partial \\mathbf{w}}{\\partial x} = \\mathbf{0}\n", "7025536d9f2619363576863ec6abd9f2": "S_q=S_q^{FD}+S_q^{BE}", "70255c75350eb97537ffa5165240dddb": "u(B)", "7025e557609b50abeeae695bedf5974c": "V_{ij} = V_{unsel} - V_{on|off}", "70260b17b3569e1ce1f9f9ad8e29508b": "\n \\Phi_{\\langle\\cdot,\\cdot\\rangle} : V\\to \\overline{V}^*.\n ", "7026371d564ce433b9fd9d146bb09300": "\\begin{align} & \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\frac{\\partial \\rho(\\mathbf{r},t)}{\\partial t} dV + \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\nabla \\cdot \\mathbf{j}(\\mathbf{r},t) dV = \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\sigma(\\mathbf{r},t) dV \\\\\n& \\frac{d}{d t} \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\rho(\\mathbf{r},t)dV + \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\nabla \\cdot \\mathbf{j}(\\mathbf{r},t) dV = \\Sigma(t) \\end{align} \\,\\!", "70264cf8c8399ff7bc4a0f8546559287": "\\left\\{ q \\right\\}", "70268427cbe1516a578e295287c3b679": "p(\\mathbf{y}|m)=\\int p(\\mathbf{y}|\\mathbf{X},\\boldsymbol\\beta,\\sigma)\\, p(\\boldsymbol\\beta,\\sigma)\\, d\\boldsymbol\\beta\\, d\\sigma", "7026913a9b77255e876c391698d7f7d8": "\n\\begin{align}\n R(u,v) &= \\frac{ \\mathbf{G}_a \\mathbf{G}_b^*}{|\\mathbf{G}_a \\mathbf{G}_b^* |} \\\\\n &= \\frac{ \\mathbf{G}_a \\mathbf{G}_a^* e^{2 \\pi i (\\frac{u \\Delta x}{M} + \\frac{v \\Delta y}{N}) }}{|\\mathbf{G}_a \\mathbf{G}_a^* e^{2 \\pi i (\\frac{u \\Delta x}{M} + \\frac{v \\Delta y}{N}) }|} \\\\\n &= \\frac{ \\mathbf{G}_a \\mathbf{G}_a^* e^{2 \\pi i (\\frac{u \\Delta x}{M} + \\frac{v \\Delta y}{N}) }}{|\\mathbf{G}_a \\mathbf{G}_a^*|} \\\\\n &= e^{2 \\pi i (\\frac{u \\Delta x}{M} + \\frac{v \\Delta y}{N}) }\n\\end{align}\n", "7026e2d5b673263f14653830f76453ed": "A=[a_1, \\ldots ,a_n]", "7026fb79c3eea20589e43c480ebdd2a2": "m_e: E\\to E'", "702721bc722e59d477a7410083ca3ca3": "\\sum \\vec{F} = T - mg \\ne 0", "70274a113644e1835dd847067ffba158": "m=(jmd)\\cdot (jd)^{-1} \\;mod\\;n^s", "702750ba76344391954607f20a7e1253": "(\\mathcal{X},\\mathcal{Y})", "70277f97092c4067632a14aeb559cb4d": "Zz\\,", "7027f815633a758b8c14604b02875614": "\\mathbf{s} \\in \\mathbb{Z}_q^n", "702810bb3efdea99971e0b5aecc08032": " \\left| \\bigcup_{S_i \\in S^{'}}{S_i} \\right| ", "70282df3c4312a8b2f71259e59037243": "H=\\frac{1}{2m} \\left(p-\\frac{qA}{c}\\right)^2 +q\\phi", "702836580dc51df4ff0b62b0851f9f80": "\\int\\langle x-a \\rangle^n dx = \\begin{cases} \\langle x-a \\rangle^{n+1}, & n\\le 0 \\\\ \\frac{\\langle x-a \\rangle^{n+1}}{n+1}, & n \\ge 0 \\end{cases}", "70284e4b7927bd1dbd809403d01d3654": " \\sum_{\\chi} \\chi(a) = 0 \\ ", "7028b4384af589f25c164afc286df37e": " z \\to -\\infty ", "7028db162915d18a7dc8fca39524434c": " S ", "7028e22d70a3112fdc19e7a603d46a52": "M_{enc}(r)", "7029058633b9d8b68ca8d13b402b428b": " \\int_{-\\infty}^{\\infty} \\exp\\left( {1 \\over 2} i a x^2 + iJx\\right ) dx. ", "702933680c64d78cf65652bac82baff8": "V = V(x,y)", "702936d27997f9d23034be6afabcab5c": "\\tbinom{p+q-1}{q-1}", "70294687040c16932827de7e9c5a52c3": " f \\circ \\gamma([0,1])", "70294d5a48cd91408e727f0864f83499": " \\gamma_1 = - \\frac{1}{3} \\frac{2 m_0}{\\hbar^2} (A_0 + 2B_0), ", "7029a7469c064be4f4993adb4255a738": "\\mathbf{\\rho}", "702a003d20c2a0974e2584b285e26b45": "a, b \\in N_O", "702a40f4a70647ff004f16472b553202": " \\frac{1}{1-x} = 1 + x + x^2 + x^3 + \\dots ", "702a5487d6a5489252439edbcaac3994": "\\vec{\\textbf v}\\,", "702aa0c3037d261dab2205ec7cbc642a": "\\scriptstyle j\\beta", "702aa3a1bd6154b3ea7815abb7f18538": "y^2 = x(x - a^\\ell)(x - c^\\ell).\\ ", "702abcc0b817a165a04ee3442070777f": "{1,2,3}\\,\\!", "702af294bb517b1f48b955844ee56246": "x\\wedge y = 0", "702b08e6f4e82f4bbfc0538dbe1676b0": "h+R_1+R_2", "702c7ec36a2e8896018332ba9b37e8cb": "\\mathbf{u}_i", "702c8eef93ddb5d60ef2c0b1209715d1": "\\psi_0(x) = \\lim_{\\varepsilon \\rightarrow 0}\\frac{\\psi(x-\\varepsilon)+\\psi(x+\\varepsilon)}2.", "702ca35c3a5c8742397bb3985525df46": "IG(Ex,a)=H(Ex) -\\sum_{v\\in values(a)} \\left(\\frac{|\\{x\\in Ex|value(x,a)=v\\}|}{|Ex|} \\cdot H(\\{x\\in Ex|value(x,a)=v\\})\\right)", "702ca3792aa9cc7d0335f81bc0c4652d": "(\\sqrt{G})_{\\rho \\rho} + K\\cdot \\sqrt{G} = 0", "702d0752a6e48c5a2be400efe5f53de8": " \\mathbf{y}_n= \\mathbf{B}\\mathbf{z}_n +\\mathbf{c}+\\mathbf{e}_n ", "702d197830ccd2d880f3f705f7ee9c7d": "{E}_{8}^{(1)}", "702d328a3929f2e0a75e2b7b93bc8897": "Pr[M^w(1^k)=1]\\leq1/2", "702d7410364cfd39ceea7774a40c3e3a": "gx=\\phi_\\alpha\\circ\\phi_\\beta^{-1}(x)", "702dc30eeaa3b22059732cbae5a28c27": "\\log_{b} a = 1", "702dda2f3c3ce2f1b94458d6e548a1b2": "\n \\begin{bmatrix}\\sigma_{\\rm xx} \\\\ \\sigma_{\\rm yy} \\\\ \\sigma_{\\rm xy} \\end{bmatrix}\n = \\cfrac{1}{1-\\nu_{\\rm xy}\\nu_{\\rm yx}}\n \\begin{bmatrix} E_{\\rm x} & \\nu_{\\rm yx}E_{\\rm x} & 0 \\\\\n \\nu_{\\rm xy}E_{\\rm y} & E_{\\rm y} & 0 \\\\\n 0 & 0 & G_{\\rm xy}(1-\\nu_{\\rm xy}\\nu_{\\rm yx}) \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{\\rm xx} \\\\ \\varepsilon_{\\rm yy} \\\\ 2\\varepsilon_{\\rm xy} \\end{bmatrix} \\,.\n ", "702de5cc93cbff8355f79b398d76a13a": "\\mathrm{C_2H_5OH + 3\\ H_2O \\to 12\\ H^+ + 12\\ e^- + 2\\ CO_2}", "702e36f1d9667dcd61357cd6795e3d7c": "c \\,\\ ", "702e6dcc3ab39086292e1ea7a66fb131": " a=\\frac{a''}{p''} ", "702e9335ba75cd55e55186eba9273a07": "b \\,", "702eaf212783845d15eac6aafc712d22": "\\forall a \\forall b [ \\forall x (x \\in a \\leftrightarrow x \\in b)\\rightarrow a = b ] \\,.", "702ecf15cf1c48504a1826924ea4d27a": " \\sin\\left(a - b\\right) ", "702efdbaa64fe1471e5c8ea0d2c978ba": "-C\\frac{d^2\\beta}{dt^2} = N_\\beta \\beta - N_r \\frac{d\\beta}{dt} + N_p p", "702f0c3f3073f59998c733a4011db533": " \\phi (\\mathbf r , t) = \\frac{1}{4\\pi\\epsilon_0}\\int_\\Omega \\frac{\\rho (\\mathbf r' , t_r)}{|\\mathbf r - \\mathbf r'|}\\, \\mathrm{d}^3\\mathbf r'", "702f5c2e53982b05470a0a67ce4379c6": "h = \\sqrt { x^2 + y^2 } ", "702fa36d5c998225accb1a1ef8b55a63": "\\mathbf{\\nabla} \\times \\mathbf{H} = \\mathbf{J}_{\\text{f}} ", "703007b59b4b92ce00d1850a81b6ca41": " \\begin{align}\nK_{\\frac{1}{3}} (\\xi) &= \\sqrt{3}\\, \\int_0^\\infty \\, \\exp \\left[- \\xi \\left(1+\\frac{4x^2}{3}\\right) \\sqrt{1+\\frac{x^2}{3}} \\,\\right] \\,dx \\\\\nK_{\\frac{2}{3}} (\\xi) &= \\frac{1}{ \\sqrt{3}} \\, \\int_0^\\infty \\, \\frac{3+2x^2}{\\sqrt{1+\\frac{x^2}{3}}} \\exp \\left[- \\xi \\left(1+\\frac{4x^2}{3}\\right) \\sqrt{1+\\frac{x^2}{3}} \\,\\right] \\,dx.\\end{align}", "70303e8650494f1c7ae0e1870385c2f8": "\\mathrm{d}{\\star\\eta}=\\left(\\frac{\\partial A}{\\partial x}+\\frac{\\partial B}{\\partial y}+\\frac{\\partial C}{\\partial z}\\right)\\mathrm{d}x\\wedge \\mathrm{d}y\\wedge \\mathrm{d}z", "703040d811d1f10135076ab28d5a81aa": "\\frac{\\partial (\\mathbf{x} \\cdot \\mathbf{x})}{\\partial \\mathbf{x}} = \\frac{\\partial \\mathbf{x}^{\\rm T}\\mathbf{x}}{\\partial \\mathbf{x}} =", "70304d3bc5639e1e4e869778016ca4e6": "w(x)=1", "70306dd9bee1c3c119e774735bfc41b0": "f^{-1}=\\exp", "7030737a41f0833d3f4559512e3c7553": "a=\\sqrt{mkT}", "703083acdf2b52d01b7c37e44a397b67": "T_{ON} = T_B + T_{Sync}", "7030bf19b70714a1fc533fecea67c136": "x^{q^{n_i}}-x \\pmod f", "7030c6d37d2c802990c7c28db8aba4cb": " \\mathbf{e}_{k} ", "7030cbf895ab7cdf7544ede0e57ccada": " \\sigma_C", "7030e5d5e7384bb45efafc7078214a30": "c_2=1.432 \\times 10^{-2} \\text{m·K}", "70311630d609df0cf826984302545bb6": "\\text{Velocity}\\; \\begin{cases} Vx = (X_1 -X_0) / \\Delta T \\\\ Vy = (Y_1 -Y_0) / \\Delta T \\\\ Vz = (Z_1 -Z_0) / \\Delta T \\end{cases} ", "70313c333cf4cc4bbb1a063051ce0f36": "M(x) = \\frac{1}{2 \\pi i} \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty} \\frac{x^s}{s \\zeta(s)}\\, ds", "703157efcc5c5822cea8844afa138f16": "\\alpha =\\frac{a}{b}", "70316c71655efce4849dca6e593aa582": "\nx_0 = x_{step},\n", "703179cd3c96b032a9e9a27d51a89460": "\\mathcal{P}_p(X)", "703184f308809925a850ba86a221edeb": "r_{k+1} \\equiv r_k \\,\\bmod{p^k}, \\quad r_k^2 \\equiv a \\,\\bmod{p^k}. ", "70318c2bffa3326616fc66b949d0d46d": "(x^2+y^2)(x^2+y^2-a^2-b^2+c^2)^2+4a^2y^2(x^2+y^2-b^2)=0.\\,", "7031a571be205e628e8a4420e8052f7c": "\\ [A]_e = [A]_0 - x = \\frac{k_{b}}{k_f+k_b}[A]_0 ", "7031b7e434833de52a0a919a125567f0": "\\hat{H}_{\\textrm{qp}}=", "7031d6ad214fe461d53593ab7816437b": "= \\frac{\\partial \\phi^{\\alpha}}{\\partial x^{i}}\\, dx^{i} + \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}}\\, (\\theta^{k} + u_{i}^{k}dx^{i}) + \\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}_{i}}\\, du^{k}_{i} - \\chi^{\\alpha}_{i}dx^{i} - \\,", "7031d8c95e34b0d6f71ced285417d490": "\\Delta f= f_r - f_n \\approx -\\frac{f_n} {2k} \\frac{\\partial F_z} {\\partial z}\\,\\!", "70320be866fded2817a59624c40fd462": " V_{max}^{app} = \\frac{V_{max}}{1+\\frac{[I]}{K_I}}", "7032b62a019a8d564d1dc7d65840204d": "{A}^{2} + (\\mu-\\lambda){A} + (\\mu-k){I} = \\mu {J}", "7032bc2f069c22658a0829849179d4e9": "O(\\varepsilon n)", "70333afcee496c4bacb7d6519b01c3c6": "1/11 = 0.0\\ 0\\ 2\\ 0\\ 5\\ 3\\ 1\\ 4\\ 0\\ 10_!", "703366bce743a57c1b1cf857c346f019": " h(b) = \\sum_{i=1}^{m}\\alpha_i \\centerdot b_i", "70339c1dade5aec165c195810f557ffd": "f'(x) = 3x^2-1 ", "7033be83ed5707a23d33d893de1a526e": " k_{b_2} ", "70340cbb8e12f75ee74a49e58ccdd778": "\\textstyle D^\\alpha", "70343d0d72cf09973aa6dad7d3ef1118": "E''(t,t_1) = [E(t_1) - R(t,t_1)]/\\varphi(t,t_1)", "7034494d1a59bd5068c446136bcf2849": "H=TS +\\sum_i \\mu_i N_i\\,", "7034662fca9ed2d80a82d935f4f91ece": "\\kappa_q : \\ell^q \\to (\\ell^p)^*.", "7034a780239e7a8eddbc888a86df3e30": "a_{\\overline{n|}i}", "7034b6c52ecaac385ce0c449d227bcd5": "\\ d[\\mathbf{x}(1), \\mathbf{x}(2)]=\\max_a |u(a)-u^*(a)|=|u(2)-u(3)|=9>r=3 ", "7034fd34df9adfbe938eff9ba80f18bc": " P = F \\times v \\;", "7034fdb35b4713878ff726d1748f70e2": "A(k)", "703504d43feb4c20ae6f5e57f146c660": "\\Gamma(x_\\mathrm{min}) = 0.885603194410888\\ldots\\,", "7035242790f4df952f3d23f50197906e": "(\\phi \\to \\chi ) \\to ((\\phi \\to \\lnot \\chi ) \\to \\lnot \\phi )", "70356b177c86395aaa8efdf63073f81e": " \\langle N_1 E \\rangle - \\langle N_1 \\rangle\\langle E\\rangle = k T \\frac{\\partial \\langle E \\rangle} {\\partial \\mu_1},", "70359490cf1d2d9bd50d7857606f7f45": "\\Gamma_{e}(x=0) \\ne 0", "7035a363b9c12af62bb4c4308766b589": " \\frac {p(r) - P} {P} \\approx \\frac {R_{critical}} {r} ", "70365efc27a15d4fafb4ffe6a5277549": "\\operatorname{ker}(p) \\cap A", "70366029d73ac14e6de9b05a26b6cb9f": "\\sigma_3(n)=\\sum_{d|n} d^3", "7036700b5586fb4adb8f9ac960f02871": "\n\\mu _z \\,\\, = \\,\\,\\,\\int_{\\,\\,0}^{\\,\\,\\infty } {z\\,{\\rm PDF}_z } \\,dz\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\sigma _z^2 \\,\\, = \\,\\,\\int_{\\,\\,0}^{\\,\\,\\infty } {\\left( {z - \\mu _z } \\right)^2 \\,} {\\rm PDF}_z \\,dz", "7036eecbced7fdce7296e268ee3526fe": "\nU\\, a_i^\\dagger \\,U^\\dagger = \\sum_{j=1}^N a_j^\\dagger\\,U_{ji}\\quad\\hbox{for all}\\quad\nU \\in U(N),\n", "7037201e63a30d26e011a8492fd77e2e": "x\\,R\\,y\\iff\\forall a\\in A\\,(\\Box a\\in x\\Rightarrow a\\in y)", "70375345bab4b3cc274f13008ce4e273": " B=G \\qquad \\mbox{(7)} ", "70375a4d40726b96afa7aeb991bbedd8": "E^\\ominus {\\rm (H^+/H_2)(abs)} = \\phi^{\\rm{Hg}} + \\Delta ^{\\rm{Hg}} _S \\psi^\\ominus_{\\sigma=0} - E^{\\rm{Hg}}_{\\sigma=0}\\rm{(SHE)}", "703791c21377a6ee17bf9d9de4cf7093": " q_1 = \\mathbf{v} \\cdot \\mathbf{e}_1; \\qquad q_2 = \\mathbf{v} \\cdot \\mathbf{e}_2; \\qquad q_3 = \\mathbf{v} \\cdot \\mathbf{e}_3. \\, ", "70379a6f897006bd6d76e9cc29c9dec3": "S^1=\\{z \\in \\mathbb{C}, \\; z\\overline{z}=1\\}", "7037a6e92cf3c3ed7c87cd2e016d998e": "\\overline {R}=1-(1-{1\\over T})^n=1-(1-p(X\\ge{x_T}))^n", "7037efddf7f002b685c95a3806f024e2": "X^2\\backslash\\left\\{ y~\\backepsilon~x\\succ y\\right\\}=\\left\\{ y~\\backepsilon~y\\succcurlyeq x\\right\\}", "70381ec40ccc7e7d7ddc8c543d43d553": " {n \\choose k} ", "70388b88d768d9281a01af97e0a728ce": "\\mathbf{L} := \\mathbf{L}_{1} \\mathbf{L}_{2} \\dots \\mathbf{L}_{n}.", "7038cd1dfdcd049f919018c43acb0cb1": "<_s", "7038f08ff5a0cbf7bd6d9d18aa9ad49a": "\\sigma_{0,\\,j}", "703947eb1eb13e3b9d01a40bb5f8d548": " \\vec{\\Omega} = -a \\, \\exp(a^2 r^2/2) \\, \\vec{e}_1 ", "703990123d776aebbc1891001a800c00": "i=1,2,\\ldots,k,", "7039c71bf73a80a9dc643fc5d2bcc45a": "fragile", "7039dada18b9909e9853e800fe144c23": "x^3+\\frac{3cd}{b-c}x^2+\\frac{3(a+c)hd^2}{(H-h)(b-c)}x=\\frac{6Vd^2}{(H-h)(b-c)}", "7039fda789c6b4fd36eee29bbff15067": "n(p;365)\\approx \\sqrt{2\\times 365\\ln\\left({1 \\over 1-p}\\right)}.", "703a042a09a379a6d969d96e4fbac4cc": "\\alpha_i(x,\\xi) = \\tfrac{\\partial L}{\\partial \\xi^i}(x,\\xi)", "703a19e182189abcd77fa7273e1e7d13": "\\underline{\\lambda}^{(l+1)} \\leftarrow \\underline{\\lambda}^{(l)} - \\mathbf J_\\sigma^{-1} \\underline{\\sigma}(t+\\Delta t)", "703a273fa12a9a52b709e72898be4305": "X\\times (Y \\times Z)\\simeq (X\\times Y)\\times Z\\simeq X\\times Y\\times Z", "703a589b70b0db663c191144f9aac9a2": "F_i = \\mathbf{F} \\cdot \\hat{\\mathbf{e}}_i", "703a5f3b4ad7bd2583da0c5c807847c6": "\nP(2 | \\vec{y}) = \\int^{\\infty}_{q_0} P(q | \\vec{y}) \\, dq\n", "703abe66b53c278cfa14e59305809222": "\\hat f_s(\\xi)", "703af0112b2466e19b3004fc70109010": "d > 0", "703b376cbed5a9b79fa5db228f35285f": "x/a", "703b5b559bec0cc69071c2bb14e46cb1": "p_n(k)=\\frac{n!}{(n-k)!k!} p^k (1-p)^{n-k}. ", "703c0d46fbc96320ce42e94d4289b201": "\\iint\\bold{g}\\cdot{\\rm d}\\bold{S} = -4\\pi G m \\Rightarrow \\bold{\\nabla}\\cdot\\bold{g}=-4\\pi G\\rho_m", "703c6abf5c691592eeb937a1c175d0cb": "F(t+a)=F(t).", "703c7073123bbe4cf8093a5124609868": "F_\\otimes(x,y) = \\max\\{0, x + y -1 \\}", "703cd4741f8e6d1654be7adb6b65a1bb": "\\begin{matrix}\\text{If } y(t)=x(t-t_0)\\text{ then }\n\\\\ W_y(t,f)=W_x(t-t_0,f) \\end{matrix}", "703ce42fcf27af0ad5c39e00808ffd42": " \\frac{force}{area}=\\mathrm{\\frac{N}{m^2}} ", "703d09496e012597323f24e3173efc1d": " \\tilde{\\mathbf{x}} ", "703d161ae754dae6b138844fede2ca7c": "F_\\lambda \\ \\hat{\\lambda}\\ = -J_3\\ \\frac{1}{r^5}\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2\\lambda\\ -1\\right)\\ \\cos\\lambda\\ \\hat{\\lambda}\\ =\\ -J_3\\ \\frac{1}{r^5}\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2\\lambda\\ -1\\right)\\ (\\sin i \\ \\cos u \\ \\hat{t}\\ +\\ \\cos i \\ \\hat{z})\\,", "703d30dab73ee453c210562a9d390ed7": " n \\mapsto n + 1, \\quad z \\mapsto z^2 + c \\quad \\hbox{ while } \\quad |z|<2 ", "703d42f708c01cdde59e20bc41cb0510": "u(x,t_n) = \\sum_i m_i \\frac{u_i^n}{\\rho_i} W(|x-x_i|)", "703d6f3cc054d29888dec0e0341befd5": "h_v \\ge 0", "703d7e7da69a957c727a4fa68f18cfe6": "Q\\,", "703daaf93740ac59d333d396b8ff65ad": "u^*\\ = \\frac{u}{U}\\,", "703e00444fc764d9b734270f41cffc14": "\\displaystyle{ \\Pi(f)=\\int_{-\\infty}^\\infty f(t)\\Pi(t)\\, dt,}", "703e01ed7816e3743b89dec3bb4241d1": "(\\alpha,\\delta)\\,\\!", "703e1472c8ed6cef4b69b88e1a12093e": "\n\\dot{\\mathbf{P}} = -\\frac{\\partial K}{\\partial \\mathbf{Q}}\n", "703e1729cb1bcab8ec043af1eb6f53f1": " |\\cdot\\cdot\\cdot\\cdot|\\cdot\\cdot\\cdot\\cdot\\cdot\\cdot| \\, ", "703e31367cc907af34c807889d59d159": "{} + 162 (y^4 z^2 - x^4 z^2) + 27 (y^6 - x^6) + 9 (x^4 z + y^4 z) + 48 (x^2 z^3 + y^2 z^3)\\ ", "703e341e19e3fee01dd4271699698e3d": "Z\\ne 0", "703e4fd328b0f4a10e4f041ba39baa84": "\\nabla C(n)", "703e856e46f7360dc8ae73679151aa74": "\\psi(\\Omega+\\alpha) = \\varepsilon_{\\zeta_0+\\alpha}", "703f02129969f7ef1c8ffae1d3a86650": "\\left \\vert p+q-1 \\right \\vert < 3 \\sqrt{N}", "703f1b80e0664a64ea27e587fb0180c8": "\\lnot \\phi \\to (\\phi \\to \\bot )", "703f259d2f1a609d112d61c642e0f17d": "u,v", "703f3020407d0e6a097012d4da64beec": "I=\\left(\\begin{array}{ccc}\nI_{1}&0&0\\\\\n0 & I_{2}&0\\\\\n0 &0 & I_{2}\n\\end{array}\\right)\\,,", "703f68fa40c74d36ead45ffaab839283": "K_\\mathbf{C}^*(\\mathbf{CP}^n) = K_\\mathbf{C}^0(\\mathbf{CP}^n) = \\mathbf{Z}[H]/(H-1)^{n+1}.", "703f729652e87b582cec1ba1e04e486d": "\\partial_\\mu \\phi", "703f7f8d2fcd4b19ddfcac04b013649b": "\\neg P\\or P", "703f886e886d2915a1874b181724c0bb": " a.b = (1*a).(b*1) = (1.b)*(a.1) = b*a = (b.1)*(1.a) = (b*1).(1*a) = b.a \\,", "703fc7537fb0b0ff8b9352bd9819c7ff": "\\quad\\quad\\quad\\displaystyle \\frac{d^2}{dt^2}T=-\\omega^2 T.\\ ", "70404be91b1a1b8eb21259a8308d6951": "f_{QH}(D)=\\beta \\times \\delta^D,\\,", "7041147353ce0ed546b0ac3eee56b475": "\\sum_{k = 0}^{n} \\frac{16^{n-k} \\bmod (8k+1)}{8k+1} + \\sum_{k = n + 1}^{\\infty} \\frac{16^{n-k}}{8k+1}. \\!", "70415e03e05343a62a34100d4a3456b3": "\\beth_n(T)", "7041668b8d4d53d0bb35aa0e269cc92b": "\\psi(h, g)N + \\frac{1}{8}\\sqrt{\\frac{\\pi}{c}}(c-1)N^{3/2}+O((c-1)gh^{1/2}N) + O(c^2g^3h^2)", "7041d411574ddfb7190b3d730d2355f9": "V_F", "70429b04d92a33143bb20ca272bbdf2b": "\\textbf{let}\\ x = e_1\\ \\textbf{in}\\ e_2", "7042a3b7eb0be5c16dc2e128133823d2": "h(f) \\le d(f,v_1) + h(v_1) \\le d(f,v_1) + d(v_1,v_2) + h(v_2) \\le \\ldots \\le L(P) + h(g) = L(P)", "7042f809da011719a4874672dcc34af3": "-q/2< h_j< q/2", "7042fca709125b37b659f9c7a4c66e83": "\\aleph_d", "70431496ac3e7b05191c23d43027854f": "k = R/(f_1, \\dots, f_n)", "70431aaef278a3ea6ea46eb07309f4b0": "\\bar\\psi(x_i)", "70443bae0d35c5b9a71cafe423c33045": "\\sqrt{\\tfrac{613t+203}{9(35t-62)}}", "7044b12fce9c03d2f25fc20136ab7998": "H(\\omega)\\triangleq|K(j \\omega)|=\\frac{1}{\\sqrt{1+\\left(\\frac{\\omega_0}{\\omega}\\right)^2}},", "70450dc47915a15d75b85ae28fd647c5": "\\sum_{n=1}^\\infty \\mu(n)x^n = x - \\sum_{a=2}^\\infty x^{a} + \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty x^{ab} - \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty x^{abc} + \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\sum_{d=2}^\\infty x^{abcd} - \\cdots", "704511d6a605eeeb59b815f9aa76f207": "\n\\mathbf{j=}D_{\\alpha }\\left( \\psi (\\widehat{\\mathbf{p}}^{2})^{\\alpha /2-1}\n\\widehat{\\mathbf{p}}\\psi ^{\\ast }+\\psi ^{\\ast }(\\widehat{\\mathbf{p}}^{\\ast\n2})^{\\alpha /2-1}\\widehat{\\mathbf{p}}^{\\ast }\\psi \\right). \n", "70455a8f2d52a94b012cf191e0b8b398": "v={MV \\over d N_A \\ell e f \\rho_0 (1+\\alpha_0 T)}", "70456bfcb9e9073fd03a82d434ebfd7b": "M_1,...,M_t\\,", "7045facbec26f69a5f16634c9250fcfb": "K^\\ominus=Q^E \\times \\Gamma \\times 1 = Q^E\\Gamma", "7046835d4207e373a602cd10121f3661": "w/h", "70470a716d11891572fb2dffce576b1f": "\\rho^{XA}", "704736cecbd183a8b68aed19c7955780": "L_1(x)=x \\,", "70473f4b5374f5be4265280399538a09": " \\frac{1}{PM} = \\frac{AB}{AP \\times BP}", "70475cf5b01216ddaac54cd497834b38": "w \\mapsto y^*(w)x.", "7047822382ec39c98e96a7d815c13603": "f_k(x,y)", "7047a1e82a2194ca4bf84c3b3c540172": "M_i=\\sum \\alpha_{ij}E_j + \\sum \\beta_{ijk}E_jE_k+\\ldots", "7047e29fe47f98156de523f119f3f38a": " (X_t)_{0\\leq t\\leq T}", "7047e53bb4a606f28b7cc80a5033a9c0": " v_{\\mathrm{max}}", "7047fc71ddea59b42157f2b72304c982": "\\varepsilon_0 = {10^{-9}\\over 36\\pi} \\;\\; \\mathrm{C^2\\ N^{-1}\\ m^{-2}}\\approx 8.854\\ 187\\ 817 \\times 10^{-12} \\;\\; \\mathrm{C^2\\ N^{-1}\\ m^{-2}}. ", "704803384bec66dd492170ee012504d1": " \\angle BCA \\cong \\angle EFD.\\, ", "7048170d385869efa59756aab1af3d09": " \\left( \\bar{T}_\\mathit{in} = \\frac{\\int_2^3 T\\,ds}{Q_\\mathit{in}} \\right)", "70481caf85d3a7150cf284b278c01271": " 2^3 = 8 ", "70485f077b4c4c726548c8c79da37bb7": "n m= \\frac{dM/dt}{4\\pi r^2 v}", "70487ec9eda4306bc894c45c2721abe0": "\\mathcal{F}^{-1}(\\mathbf{x}) = \\mathcal{F}(\\mathbf{x}^*)^* / N", "70489fa919144bcf705fa0aa6e4c80f1": "f\\colon M \\to \\mathbf{R}", "7048dc650d2be70e529573e5ca5acdd2": "\\mathbf{\\gamma}(t_0) = \\mathbf{p}_0", "70493a1048aedbad58991f032271e871": "\\scriptstyle \\vec S = \\vec J - \\vec L", "70493a34b7e1c358335006102dd26c12": "\\vec{x}(i)", "7049433740131aaa55656213cf88128e": "\nA = \\bigoplus_{\\ell:\\;\\ell\\not=i\\;\\mathrm{and}\\;\\ell\\not=j}{D_\\ell} = \\mathbf{P} \\;\\oplus\\; \\mathbf{D}_0 \\;\\oplus\\; \\mathbf{D}_1 \\;\\oplus\\; \\dots \\;\\oplus\\; \\mathbf{D}_{i-1} \\;\\oplus\\; \\mathbf{D}_{i+1} \\;\\oplus\\; \\dots \\;\\oplus\\; \\mathbf{D}_{j-1} \\;\\oplus\\; \\mathbf{D}_{j+1} \\;\\oplus\\; \\dots \\;\\oplus\\; \\mathbf{D}_{n-1}\n", "70494adfbebc6faade5716cb62bdb706": "\\chi_{\\text{e}} \\mathbf{E} = N \\alpha \\mathbf{E}_{\\text{local}}", "70497b300deaf69d5269533e32a00f14": "\\arctan x", "7049c52bd1506a0845ab98fd09f39cc3": "_i", "704a2b697a8b1159b632b302b51de38b": "\\lim_{t\\to\\infty}\\varepsilon = \\frac{\\sigma_0}{E},", "704a38507240be20df80bba1957914ff": "E = \\frac{W_p S + W_s P}{W_p + W_s}.\n", "704a7a5ffb668db768a577f10e1c6601": "a = 1/2 \\textrm{ln}\\frac{W_+(true)}{W_-(true)}", "704b13710b13b704b6816acb12485946": "k=1,2,...n_E", "704baef979670c3aa40107d79f26b944": "\\psi ={{\\psi }_0}\\left(\\frac{\\sin \\left(\\frac{{\\pi a}}{\\lambda }\\sin \\theta\\right)}{\\frac{{\\pi a}}{\\lambda }\\sin\\theta}\\right)\\left(\\frac{\\sin\n\\left(\\frac{N}{2}\\big(\\frac{{2\\pi d}}{\\lambda }\\sin\\theta + \\phi \\big)\\right)}{\\sin \\left(\\frac{{\\pi d}}{\\lambda }\\sin\\theta +\\phi \\right)}\\right)\n", "704bf3e16adf5c8be07f6cdcba15f19e": "\\rho(u) = 1-\\log u", "704c099e795a33d835560ed320c85490": "X_1 (i+\\alpha,j+\\beta)", "704c1aa10c70869b8b2c7ddeeb1c03ff": "(i\\omega_1,i\\omega_2)", "704c69d72b79a7d2ef613b322a879911": " \\langle \\xi \\mid T \\xi \\rangle \\geq 0 \\quad \\xi \\in \\operatorname{dom}\\ T ", "704c9a7eee2b2e9a855dfed5d19c621a": "\\quad \\frac{dr_{ff}}{d\\tau}=\\frac{v}{\\sqrt{1-v^2}} \\ge 1\\,\\!", "704ca877b8f477bf87e8475bcdfb5e21": "\\tilde f_0\\colon X\\to E", "704d604e546d1bdff28fec9d04c9644f": "S_{\\rm s} = k \\log \\frac{dv}{dE} = S_{\\rm B} - k \\log \\omega", "704d6899ddec3b7aebdff4b9d4f05e40": "\\phi_{12} : A \\otimes A \\to A \\otimes A \\otimes A", "704d6d9372439077ec8086d7358ed167": " R_{\\|} = -{8 \\pi G \\over {3 c^2 } }\\rho (r) ", "704dbcf1b9cd9379f512b13ef654292f": "H(s)\\ \\stackrel{\\text{def}}{=}\\ \\mathcal{L}\\{h(t)\\}\\ \\stackrel{\\text{def}}{=}\\ \\int_{-\\infty}^\\infty h(t) e^{-s t} \\, \\operatorname{d} t", "704dbf47f5d8ce478da08341f39ca7d8": "\\chi(K)=\\sum_{i=0}^\\infty(-1)^ib_i(K,F), \\,", "704e301ca926300bcfff19e228c102ad": " \\frac{1}{(1-t)^{\\chi(X)}},", "704ee1032a35557f41baf9b2cfcae6cf": "p(y\\overline{\\|}x) = p(y|x)", "704f4ac05f865fbf375d31405acff525": "\\ddot{a}_{\\overline{n|}i}^{(m)} = \\frac{1-v^n}{d^{(m)}}", "704f880eb76811ed183ec7d867b13062": "A_n = \\bigcup_{m\\geq n} \\left \\{ \\left |X_m-X \\right |>\\varepsilon \\right\\}", "704fc1252ef79bbfa122224d0d3dbdd6": " 2^{n-1} ", "70501277b3f94135d1cb5e7f7ed80e1a": "\\left(x+\\tfrac{b}{4a}\\right)^4.", "70505757561ba3f190f02c7748220e1f": "\\widehat{U}(R)", "70509fb6445668d3964f7a21f33131ad": " P(O^t \\mid S^t) ", "7050ac1d459b8ada0c8c57b9cc888f32": "T_W", "7050beff8777f4a3816c6d032848b6b6": "\\sqrt{\\frac{5}{9}}\\!\\,", "7050e1c88b36478b433de50a240c9fda": "\\alpha=[X]/K_R", "705104b5db755b81f95cfd71b04a0fb9": "\\Psi_l^k = V_l^k \\Psi_m^{(0)}", "70516fbf2ccf576bc07af5872912e5ac": " v_c ", "70523320308f52aca5e4bf9531bfeb45": " \\theta_m = \\frac{ M_m }{ \\sigma_m } ", "70527fe0102cd98fb67342b5abb751e5": "Ascendant =\\arctan(\\frac{y}{x})= \\arctan (\\frac{-\\cos A}{\\sin A \\cos E + \\tan L \\sin E} )", "70529b767c556a1e8dd11e2ed4e4ad5e": "\\scriptstyle b^0", "7052c9d9a180d9f16040ac4594c2abaf": " \\gamma_{a'} ", "7052f05832e89cead95476aa3592e033": "2N + 2", "7053a748927c3927292009e7995d5e60": " \\text{Rate of profit }p = {{s \\over v} \\over {{c \\over v} + 1}}", "7053c8270071da283e87f17173130754": "\\Delta G_{v+\\frac{1}{2}}", "7053eec7425a96af0633813abdbdff75": "\\mathcal{E} = \\frac{d\\lambda}{dt}", "70541819afd54b0923eb76158f155560": "\\Lambda:=\\lbrace 0,\\ldots,n-1\\rbrace", "705425c1223fb8baa55f14d9e806689a": "\\Gamma_x", "70548ba3741e17ac72ec38cdd9aa1418": "-\\log (K_{a})", "7055225f08f52599f7941138354f6748": "\\frac{e^{i\\phi} - e^{-i\\phi}}{2i} = \\sin(\\phi) ", "70552792f78c9cb26275f743b3fefa6c": "L= -\\partial_r^2 - 2 \\coth r \\partial_r.", "70553c8a0102505d59d19d79acfb765d": " \\frac{d^{\\alpha_i}}{dx_i^{\\alpha_i}} x_i^{\\beta_i} = \\frac{\\beta_i!}{(\\beta_i-\\alpha_i)!} x_i^{\\beta_i-\\alpha_i}", "705590cd1965dbaec1b5d44ea8b1e13e": "\\ast", "7055a284f183917c921e036424e4e16f": "\\mathbf a^1\\land\\cdots\\land\\mathbf a^k=\\mathbf 0\\ \\Rightarrow\\ f(\\mathbf a^1)\\land\\cdots\\land f(\\mathbf a^k)=0.", "7055bc6d853248eeecece13675bdd5bc": "\\epsilon_H", "7055db5a956fdaaaab7e2bdf13e3f458": "\\widehat{\\beta}_{FE}", "7055fa0666151531c0da1aa2482dcba7": "\\psi=\\cos\\alpha+\\frac{\\sin\\alpha \\tan\\phi}{F}", "7055fbcfe19ed8fc14d847e853db56cb": "\\alpha: S \\times S \\to A", "70561621bf20f3e6f426e16672c3e934": "\\scriptstyle n_f\\,", "70561e49c0486888077b730d32bf2203": "\\phi(p) = \\left(\\frac{2 \\ell^2}{\\pi \\hbar^2}\\right)^{1/4} \\exp{\\left( -\\frac{\\ell^2 p^2}{\\hbar^2}\\right)}", "70563067a97fb4fc35a4f94a3be42838": " \\frac{d{\\mathbf r}(s)}{ds}={\\mathbf T}(s) ", "7056fca8ce36aceac48fbb375b49547b": "k=0,1,2,\\ldots,\\lfloor i/2 \\rfloor", "7057158ce350b79b0b20df4e6adb902b": "\n\\sum_{k_1+k_2+\\cdots+k_m=n} {n \\choose k_1, k_2, \\ldots, k_m} = m^n\\,.\n", "7057296746375723cc7a8f89f111fa37": "X\\,\\sim\\,\\textrm{Inv-Gamma}(\\tfrac{1}{2},\\tfrac{c}{2})", "7057389d14f462d504d60db719f350cf": "n\\lambda=2d\\sin \\left(90^{\\circ} -\\frac{\\theta}{2} \\right),", "705743820f38467b38de2f39b060496f": "S=\\int w dx", "70574f0d38e95df156e847228e3cb32c": "aX_n+bY_n\\ \\xrightarrow{a.s.}\\ aX+bY", "7057941a4eb51d8dacb8ec60a58d189e": "\\Rightarrow \\Delta p = \\frac {1}{2} C_L \\rho w^2 (\\frac{l}{s}) \\frac {\\sin(\\beta - \\phi)}{\\cos\\phi} = \\frac {1}{2} C_D \\rho w^2 (\\frac{l}{s}) \\frac {\\sin(\\beta - \\phi)}{\\sin\\phi}", "7057a84a30118572c5f357807ae754ba": "\\leq max_{x\\in X^{2m}}(Pr[\\sigma(x)\\in R])\\,\\!", "7057be771e6bed87a44243614d944d38": "K_{00} = (K_0 \\cap K_+) \\oplus (K_0 \\cap K_-)", "7057e724f7a128748cddc6d6fcd51bc4": "B= Ff", "7057e8ce179f78cfb7388547150710a0": "D_t = \\sum d_i", "70584a759e82db265df46dd14945a791": " G_{\\alpha\\beta} = \\pm \\frac{1}{g_s} [D_\\alpha, D_\\beta]\\,,", "7058724f89c8f8a15aa645f55d74ca0f": "\\ slope=-\\frac {\\partial F(K,L)/ \\partial K} {\\partial F(K,L)/ \\partial L}", "70593cae6d824de1d91c3e7281b77826": "\\,e^{iS[\\sqrt{\\left(1 - \\frac{r_s}{r} \\right) c^2 dt^2 - \\left(1-\\frac{r_s}{r}\\right)^{-1} dr^2 - r^2 \\left(d\\theta^2 + \\sin^2\\theta \\, d\\varphi^2\\right)}]}", "70593f70ba96a367f72269c1f13b9322": "(s,a,b,d,c,t)", "7059511f949b4e6d3af94c4a472b1156": "f_c(z)=z^2+c\\,", "70595bf730308e2262616915d2d42958": "x_1[n]x_2[n]", "70596662367966c9904d04d1c6ffdc9a": "\\sigma_\\mathrm{tr}", "7059c4f319872ebf0d49d9d317a1af2e": "L_z", "7059dd5ddb65315d1955dd5342aa6694": "\\left(a \\csc\\left(A - \\frac{\\pi}{6}\\right), b \\csc\\left(B -\\frac{\\pi}{6}\\right), c \\csc \\left(C - \\frac{\\pi}{6}\\right)\\right)", "7059f208e13675220a69e15a4ae7b2e4": "|init\\rangle", "705a27284a7672b5a793d380223dce7d": "\\nu:\\Gamma \\times \\mathbb{H} \\to \\mathbb{C}", "705a37b682d60e52f4f4ed3e1ec358fa": "I_\\mathrm{out} = (V_\\mathrm{in+} - V_\\mathrm{in-}) \\cdot g_\\mathrm{m}", "705a80496316f0c2305690679ae3a019": " i,j \\in \\left\\{ {1,2}\\right\\} ", "705afe3c39fc024c3507cb0dc47f0206": "F (D, N, \\dot{m}, p_{01}, p_{02}, RT_{01}, RT_{02}) = 0 ", "705b2878e1ecd61cb63430a8ad40700b": "x(t-t_0) \\rightarrow W_x(t-t_0,f)\\,", "705c45f4dc3c7bbca769d1e18fb824ef": "y=ax^2+bx+c", "705c4a0de84b439d732afd86bb97c3c0": "s = -j \\omega_0\\,", "705cdd74ee9fc7fa3296811a4a7d0b84": "\\frac{(t_2-t)(s-t_1)}{t_2-t_1}.", "705d1df0b46afb7083ae2f734c9af024": "KU_*(X)", "705d22d813b334721f23eec4f486c4ab": "\\eta_{\\mu \\nu}", "705d368f709c3e4591b0af81063c12cf": "\\lim_{|\\alpha|\\to\\infty} \\sqrt[|\\alpha|]{|c_\\alpha|\\rho^\\alpha}=1", "705d5153248f38d104595e572af07ac3": "\\hat{\\textbf{x}}_{k\\mid k} = \\hat{\\textbf{x}}_{k\\mid k-1} + K_{k}( \\textbf{z}_{k} - \\hat{\\textbf{z}}_{k} )", "705d57396401c54e6fb960fda960abf3": "\n\\begin{align}\nk_1 &= f(t_n, y_n),\n\\\\\nk_2 &= f(t_n + \\tfrac{1}{2}h , y_n + \\tfrac{h}{2} k_1),\n\\\\\nk_3 &= f(t_n + \\tfrac{1}{2}h , y_n + \\tfrac{h}{2} k_2),\n\\\\\nk_4 &= f(t_n + h , y_n + hk_3).\n\\end{align}\n", "705d6f9fe7dd0548bab39ba7f4816285": "\n y(t) = \\cos(t) \n + \\varepsilon \\left[ \\tfrac{1}{32} \\cos(3t) - \\tfrac{1}{32} \\cos(t) - \\underbrace{\\tfrac38\\, t\\, \\sin(t)}_\\text{secular} \\right] \n + \\mathcal{O}(\\varepsilon^2).\n", "705d9cf5220b93288d854cdc637f44c6": "iM", "705dd72e902c2dba4708a30cb435d0cd": "\\mathcal{E}(\\pi)\\left[\\phi(x_1)\\cdots \\phi(x_n)|\\Omega\\rangle\\right]=\\phi(x_{\\pi^{-1}(1)})\\cdots \\phi(x_{\\pi^{-1}(n)})|\\Omega\\rangle", "705ddadd675177f09d4e28d0c51489c0": "\\xi_1^{2^i}", "705ddbf7d821c782baa151d7e8952423": "Fr = \\frac{q}{\\sqrt{gy^3}} = \\frac{5}{\\sqrt{32.2(3^3)}}= 0.17\\text{ (subcritical flow)}", "705de42093e5cab3333dd2f28e1b4c3a": "\\beta=\\frac{(L_r N_p - L_p N_r)}{(L_p N_\\beta - N_p L_\\beta)}\\frac{d\\mu}{dt}", "705e12b741039d330dbbc41787472d12": "\n\\operatorname{Li}_2(z) = \\frac{\\pi^2}{6} - \\int_1^z{\\ln(t-1) \\over t} \\,\\mathrm{d}t - i\\pi \\ln z\n", "705e3d30346dd62948d2e71feb06bfc2": " \\left\\{ u_1 , u_2 , \\ldots , u_n , \\ldots \\right\\} \\in \\mathcal{V} ", "705e4d424c8aaa2fcd366e8111571ab6": "\\textstyle \\sum 1/{n^2}", "705e7c49d3aa32f33c0b0f55511b359f": "\\delta ^{18}O = \\Biggl( \\frac{\\bigl( \\frac{^{18}O}{^{16}O} \\bigr)_{sample}}{\\bigl( \\frac{^{18}O}{^{16}O} \\bigr)_{standard}} -1 \\Biggr) * 1000\\ ^{o}\\!/\\!_{oo}", "705eb95d2d29151cd96d800601218228": "\\Gamma:\\partial_\\lambda\\to \\partial_\\lambda\\rfloor\\Gamma=\\partial_\\lambda +\\Gamma^i_\\lambda\\partial_i. ", "705ebe92af7da691c455b00c113ea794": "{{Q}_{y}}=hA\\left( {{T}_{s}}-{{T}_{\\infty }} \\right)", "705f1531ec52a8be15e78a17fdc33b5b": "\n\\delta'_{2s}(n)=\n\\frac{(\\frac12\\pi)^s}{(s-1)!}\\left(n+\\frac{s}4\\right)^{s-1}\n\\left(\n\\frac{c_1(4n+s)}{1^s}-\n\\frac{c_3(4n+s)}{3^s}+\n\\frac{c_5(4n+s)}{5^s}-\n\\dots\n\\right).\n", "705f3a3c949674de2fbdd45398d0c611": "\\, \\! V_+", "705f7f6cdbeb8bcc4b310efa2b259fc6": "\\vec{X} = ( x^2 - y^2 ) \\, \\partial_x + 2 \\, x y \\, \\partial_y", "705f9cd412edc8a0a27b9585f67e99f6": "R \\to X \\times_{X/R} X \\,\\!", "705fceb0c032dcd0d0a0d9531af189a1": " LPM/\\sqrt\\text{bar}", "705fd3016c96d94771a9615d662220fe": "\\Omega \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathbf{a}_{1} \\cdot \\left( \\mathbf{a}_{2} \\times \\mathbf{a}_{3} \\right)", "705fdfcfeeac79d113b808356575f5aa": "\\binom{n}{2k}", "705fe3a84a0d86de7fbd21284ee4ea7d": "x_6\\,\\!", "705ff9569651a95c25943d48bd73a695": "\\int_{0}^{\\infty} \\frac{\\cos px - \\cos qx}{x^{2}}\\ dx=\\frac{\\pi (q-p)}{2}", "70600bb5fc6a55dc86f64b1a56c7ff9f": "J_\\nu(z)=\\frac z {2 \\nu} (J_{\\nu-1}(z)+J_{\\nu+1}(z));", "7060d360505265c78ee93da8b9fe6587": " u' - (q_1 + 2 \\, q_2 \\, y_1) \\, u = q_2 \\, u^2, ", "7060ff99121e5a1da8033c0539d970ab": "S_{n+1} = \\bigcup_{i=1}^N f_i[S_n] ", "7061229e5d957d5b1fd276e990571970": "r > 3m", "7061451f5450e54e7fd5891142199e1f": "\\mathbf{f}=\\mathbf C^{-1}=\\mathbf F^{-1}\\mathbf F^{-T} \\qquad \\text{or} \\qquad f_{IJ}=\\frac {\\partial X_I} {\\partial x_k} \\frac {\\partial X_J} {\\partial x_k}\\,\\!", "706147204f35ca44a72fd767aab6fa6e": "E_{a}", "70617eb4f317a68af4e29aada3a615c7": "H_{\\mathbf{k}} u_{n,\\mathbf{k}}=E_{n,\\mathbf{k}}u_{n,\\mathbf{k}}", "7061b41456224ce59a4e0c0ff9da9714": "L_e = \\frac{\\mu_0\\lambda_0}{2\\pi}. \\ ", "7061bb952d4307a57b656338c2d7ce95": "\\chi_i \\in C^{n-i}([a,b],\\mathbb{R}^n) \\mbox{, } 1 \\leq i \\leq n-1", "7061c8e502a2ed9b934ed687e15b0fb1": "t_{0}<0", "7061f0fb6b507c80e5bfe9859311084a": " x<0,0L ", "7062482bf816f2179054e16aa88581f5": "Violates(x=a, y_i=b)", "70626327ab1e56cf65b93901bf489793": "y=b \\cosh v \\sin\\theta ", "7062a2c547659785ec514ea4cd03214b": "\\sigma_i = -\\sigma_i", "70639beb660bc39e066b6f1431e2257c": " \\lambda_c= c \\lambda, ", "7063a69b2aa671272afdf3f6972df2b7": "\\mathbb{E}[X] := \\rho(-X)", "7063d07b09961de15f8aa1a5b744d2a2": "C_{\\mathrm{Katz}}(i) = \\sum_{k=1}^{\\infin}\\sum_{j=1}^n \\alpha^k (A^k)_{ji}", "7063fc0d2bb763431b6c4ebc9b47bedd": "\\mathbf{P}_0", "70645aa3ef21c194b8fd2f682637bc62": "X_1,\\dots ,X_m", "7064663a175d7d63db8abaa6ee9d7346": "\\forall x\\, Ux", "70646e9ea3863cd6b01ae7771c34b500": "\\{s_i\\}", "7064776e1602d669cba11b996606818a": "\\,G(V,E)", "7064813cce9f3b3339c7b55348a6e45f": " C(n) = E\\left\\{|e(n)|^{2}\\right\\}", "70649a465a8a66209afc094da7a4bd74": "F(x)^2 = ax+(n+a)^2 +x\\sqrt{a(x+n)+(n+a)^2+(x+n) \\sqrt{\\mathrm{\\cdots}}} \\, ", "706531068a4b0272e6dfaf3e883a9d5f": "\\mathcal{E}(2^{x})=\\mathbf{Q}", "706541f81390771a5eab15f42dd05acc": "\\frac {d \\vec r}{ds} = \\begin{pmatrix} dx/ds \\\\ dy/ds \\end{pmatrix} = \\begin{pmatrix} \\cos \\varphi \\\\ \\sin \\varphi \\end{pmatrix} \\quad \\text {since} \\quad \\left | \\frac {d \\vec r}{ds} \\right | = 1 ,", "706551890277fd718643887defdb7465": "\\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 \\end{pmatrix} ", "706571907a343118f1d5fc2c1c990c18": "\\rho\\in A(G)", "70659210367eb182f6e2d6f7c179b4b2": "GapSVP_\\beta", "7065acf168fb1e7a9974007d9706a138": "\n\\begin{matrix}\n ( \\; A \\; A \\; | \\; A \\; A \\; ) \\\\\n ( \\; B \\; B \\; | \\; B \\; B \\; ) \\\\\n ( \\; A \\; A \\; | \\; B \\; B \\; ) \\\\\n ( \\; A \\; B \\; | \\; A \\; B \\; ) \\\\\n\\end{matrix}\n ", "7065b7824125e43776d47a99488e9e8c": "\n C_{16} = C_{26} = C_{36} = C_{45} = 0 ~.\n ", "70661c814aef060c2f929444c11ab510": "p(v) = \\frac{\\operatorname{add}(v, \\pi_i)}{\\sum_x{\\operatorname{add}(x, \\pi_i)}}", "7066297f854af316271599d32ecb8e9c": " c[ x(t),t] ", "70664169d1bc6295401283e5520056e1": "O(\\log N)", "70667048f270f3e2d5f1a973ac8e193c": " x^{(1)}= \n \\begin{bmatrix}\n 0 & -1/2 \\\\\n -5/7 & 0 \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 11/2 \\\\\n 13/7 \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 5.0 \\\\\n 8/7 \\\\\n \\end{bmatrix} \n\\approx\n \\begin{bmatrix}\n 5 \\\\\n 1.143 \\\\\n \\end{bmatrix} .", "7066b0b446ee88d57c0ccb2d6e373e0c": "R_{\\odot}", "7066c3904cadca7244740d806bd4bde5": "C=\\frac{e^2}{\\Delta E}.", "7067720fedaf1538513271bfc8b0b60a": "L_n(x)=\\sum_{k=0}^n \\binom{n}{k}\\frac{(-1)^k}{k!} x^k .", "7067bd8a86a414833253321e04a4ee36": "G_{formation} \\, = \\, 53.88 + \\sum {G_{form,i}}", "7067e3b0dc97361abff4bae3a65bc6cf": "\\bar{f}=\\frac{1}{b-a}\\int_a^bf(x)\\,dx.", "7067e55c263a7701ac3d2bbcf098539e": "\\operatorname{tr}(a\\!\\!\\!/b\\!\\!\\!/c\\!\\!\\!/d\\!\\!\\!/) = 4 \\left[(a\\cdot b)(c \\cdot d) - (a \\cdot c)(b \\cdot d) + (a \\cdot d)(b \\cdot c) \\right]", "7067e81fc98a32636b807c761005a90f": "T \\square F \\square = \\frac 12 ", "70681962905bcf959a1003e3117909d1": "\\Delta G_{37}^\\circ (\\mathrm{total}) = \\Delta G_{37}^\\circ (\\mathrm{initiations}) + \\sum_{i=1}^{10} n_i\\Delta G_{37}^\\circ (i)", "706865056194abad5558b25a828fa673": " \\lim_{x \\rightarrow c} m(x) \\geqslant \\lim_{x \\rightarrow c} \\frac{f'(x)}{g'(x)} ", "7068aed0f04e8500f6c8c900b2c8df11": "(E,\\mathcal E^*)", "7068bf092a60ab370af815d6b97749fd": "P_{ } ", "7068d1fbc0f9bb8f71a345cc178ea482": "\\mathbf{j}=\\int_{t_0}^{t_1}\\mathbf{f}dt", "7068e50bd2968326cf36cdcda5fdc8b8": "\\sum_{i = 0}^{n - 1} d(f(t_i), f(t_{i+1}))", "7068f7e7447c92ea4db95443235ad076": "F^{\\mu\\nu}", "7068f7fd32cbf53828c6b47a6475329c": "N(0,1).", "70694497d85b13521a3687cbf99f3870": "+\\infty\\,", "70696acec2349072d9274aaa165573ab": " a_3(S,H) = \\frac{\\ln(S/H) - (r-\\frac12\\sigma^2)\\tau}{\\sigma\\sqrt{\\tau}} = a_1(S,H) - \\frac{2r\\sqrt{\\tau}}{\\sigma},\\text{ with }H>0, S>0,", "70698f06bb73eb767ff38cc0fc92c9f7": " \\boldsymbol{\\nabla}\\boldsymbol{T}\\cdot\\mathbf{c} = \\left.\\cfrac{d}{d\\alpha}~\\boldsymbol{T}(\\mathbf{x}+\\alpha\\mathbf{c})\\right|_{\\alpha=0}", "70699a4f613af1dddf58cd72b804f5ab": " \\left(\\frac{\\omega}{c}\\right)^2 = k^2 + \\left(\\frac{m_0c}{\\hbar}\\right)^2 \\,.", "7069c903beb7f70611bd9dbcfc5b445c": "\\| \\mathbf{x_a} - H\\cdot \\mathbf{x_b} \\| < 1.5 ", "7069d1c81c8eadcfaddeadfaca66df9a": " k = m^2 / ( s^2 - m ) ", "7069f4face224883d3d27f8cb78bb531": "\\Sigma^0_{n+1}", "706a0bfaf11152b057e5d636829153ef": "s(1)=t(1)=1", "706a3f930e802dbc2f1bf8e390637184": "\n \\mathbf{u} = u_r~\\mathbf{e}_r + u_\\theta~\\mathbf{e}_\\theta + u_z~\\mathbf{e}_z\n ", "706a75c83b561a8401d351f79cce2202": "\n\\Pr \\{X_{ni}=x\\} =\\frac{\\exp{{\\sum_{k=0}^x (\\beta_n} - ({\\delta_i-\\tau_{k}}))}}{\\sum_{x=0}^m \\exp{{\\sum_{k=0}^x (\\beta_n} - {(\\delta_i-\\tau_{k}}))}}\n", "706a796a2b848387a74b2c51f07ec011": "0 \\le y \\le \\pi", "706aad1970f6f7d54eb21297b27320d6": " \\frac{r_o}{r_\\pi} + g_m r_O + 1 ", "706ae7a707c32436e0ed96fa2f9e12a6": " M = \\sum_{i=1}^j\\ f_i, ", "706b1ba9298970601a0f544a6eef8b20": "\\mu_t = \\mu_s = \\mu \\,", "706b40f1f7a291283b6c4e7347b01a32": "\\mathbb{Q}[\\sqrt{5}]", "706b571ca560939f14718848a5f41d54": "r_t = R_t - 1\\ge \\delta", "706baf751532abfad8c434da11a19ece": "F(\\bold{x}, t;\\nu) = \\int_{\\Omega^{^+}} I(\\bold{x}, t;\\bold{\\hat{n}},\\nu) \\,\\cos \\,(\\theta(\\bold{\\hat{n}})) \\,d\\omega(\\bold{\\hat{n}})", "706bda4719f37bdc519080ce526ee33b": "W_n \\sim \\sqrt{\\frac{\\pi}{2\\, n}}\\quad", "706be2c0c637f0fd1b6bcb365c95e08f": "(4\\pi/3) a^2c \\approx 4.19\\, a^2c", "706c35cd0448b412f802af3fdfdc1eca": "V=\\sum_{i=1}^t c_i G_i ,", "706c716ceb9aa724886dfab39e2d04b9": "\\{s, 1, 3, t\\}", "706cb87701bc2e34076be932bd75c5c7": "\\mu (p,T)=\\mu (0,0)+\\int_{0}^{p} V_{m}(p^\\prime,0)\\mathrm{d}p^\\prime\n\n-\\int_{0}^T S_{m}(p,T^\\prime)\\mathrm{d}T^\\prime.", "706cd7d918e812899da178272eedd140": "\\mathcal{L},\\mathcal{G}", "706d11c4a1f372a21098e45b8272c9c0": "\\sigma(h) = h", "706d3e7284ac8bc0a16e100b923367b5": "r(u,v)", "706d7153882f571a4f203d3ece6b7d7e": "2 \\cdot\n\n\\begin{bmatrix}\n1 & 8 & -3 \\\\\n4 & -2 & 5\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n2 \\cdot 1 & 2\\cdot 8 & 2\\cdot -3 \\\\\n2\\cdot 4 & 2\\cdot -2 & 2\\cdot 5\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n2 & 16 & -6 \\\\\n8 & -4 & 10\n\\end{bmatrix}\n", "706d9aa4754f9a1c1f95d51a1143e4ae": "\\frac{1}{2}\\begin{bmatrix}1+c&a-ib\\\\a+ib&1-c\\end{bmatrix}", "706e3f70d4d3fb5421292130f2d0f9c2": "F^{-1}(p;\\lambda) = \\frac{-\\ln(1-p)}{\\lambda},\\qquad 0 \\le p < 1", "706e650ceb4a4e67b3901ccef22680eb": "X_G = X/G", "706e72640c295c88a6ea52c399468b2e": "\\alpha = \\int^{\\infty}_{G_{0}} N(0,\\frac{N_{0}E}{2})dG \\Rightarrow G_0 = \\sqrt{\\frac{N_0 E}{2}} \\Phi^{-1}(1-\\alpha)", "706ed9c99b3e66fadaa90fa01c585000": "\\Delta^n\\rightarrow X", "706f60601c0f77ddd57cf0a7785a7edd": " \\mathbf{F}=m\\mathbf{a} \\Rightarrow { ke^2 \\over r^2}={mv^2 \\over r}", "706f956dc2478c66e7b66adbad72480a": "\n \\delta U = - \\int_0^T \\left\\{\\int_{\\Omega^0} \\left[N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta} \n + M_{\\alpha\\beta,\\beta\\alpha}~\\delta w^0\\right]~\\mathrm{d}A\n - \\int_{\\Gamma^0} \\left[n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta} \n + n_\\alpha~M_{\\alpha\\beta,\\beta}~\\delta w^0\n + n_\\beta~M_{\\alpha\\beta,\\alpha}~\\delta w^0\\right]~\\mathrm{d}s \\right\\}\\mathrm{d}t\n", "706f9bf7619022c6dde5e9d95cf2bfc0": "\\big\\|\\mathbf{\\zeta}\\big(\\mathbf{\\rho}-\\mathit{x}\\mathbf{\\rho'}\\big)\\big\\|_{1} = \\big\\|\\mathbf{\\rho}-\\mathit{x}\\mathbf{\\rho'}\\big\\|_{1}.", "706fbed7b1cdacb1c5792ee4a4e1fe25": "A_{nr}=\\frac{4(2-\\delta_{n0})}{a^2}\\,\\,\\frac{\\sinh k_{nr}(L+z_0)}{\\sinh 2k_{nr}L}\\,\\,\\frac{J_n(k_{nr}\\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}.\\,", "706fd3a477338f837755ab50b3fe2be2": "p_s(t)", "70701154d2518f6aa14dbfa7c5ac120e": "\\sum_{i = 1}^{n} \\frac{\\log_2 {M_i} }{T_i}", "7070b1c1162cb62baed02883a5b9fa06": "A(i\\omega) \\to A\\left( i\\frac{\\omega}{\\omega_c}\\right)", "7070f37ae1d38fcf1a26786dbb02d0a3": "\\int x\\sin ax\\;\\mathrm{d}x = \\frac{\\sin ax}{a^2}-\\frac{x\\cos ax}{a}+C\\,\\!", "70712a53dfec479912ca25e5df4a0e29": "E\\colon \\mathbf{A} \\hookrightarrow \\mathbf{B}", "70713210b1fa843268e66d92daacbd92": "u\\mapsto \\varphi_u", "7071580c58bc6e41c5523eac883b7279": "\\nabla^2 \\left(-\\frac{\\partial Z}{\\partial p} \\right) + \\nabla^2 V \\cdot \\nabla \\left(-\\frac{\\partial Z}{\\partial p} \\right) - \\nabla^2 k \\omega = \\frac{R}{C_p \\cdot g} \\cdot \\frac{\\nabla^2 q}{p}", "70716795a7d690c0161f2b21b9c0e9f1": "(\\mathbb{Z}_{p})^\\times", "7071841103f7a2125bd1745f6de7a468": "Af\\zeta", "70719cd069456f496fced72282b98035": "k(\\mathbf{x_i},\\mathbf{x_j})=(\\mathbf{x_i} \\cdot \\mathbf{x_j} + 1)^d", "7071a27a2d918252846639ec8debf4fc": "\\begin{pmatrix}\nF_1(x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n)}) \\\\\nF_2(x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n)}) \\\\\n\\vdots \\\\\nF_m(x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n)}) \\\\\n\\end{pmatrix}=\\begin{pmatrix}\n0\\\\\n0\\\\\n\\vdots\\\\\n0\\\\\n\\end{pmatrix}", "7071fbb42d0223918c19711056017690": "d_r^{p-r,q+r-1}", "707205e4912ce75b66f76f96ed8f0e99": " Z_t = \\frac{(2 \\pi m k_B T)^\\frac{3}{2} V}{h^3} ", "70723c02e12a928fff53270c8ae2b0d1": "(5 \\times 5) + 5 + 5 + 3", "7072864a3629c972f2b707dc15d3fe43": "\\Delta V_{BE}=\\frac{KT}{q}\\cdot\\ln\\left(\\frac{I_{C1}}{I_{C2}}\\right) \\,", "7072a0835967efb15b48d27c620e6b9d": "\\mu_4=12(k+2\\lambda)^2+48(k+4\\lambda)\\,", "7072b304459b8f346ecddecd31309ff2": "25^{1,312,000} \\approx 1.956 \\times 10^{1,834,097}", "7072c0b79dfc90fe1567b37ff83774e1": "{t}={0}", "7072caae035ef8f0733a1889a63638c2": " g(x) = a_0+ \\sum_{0 y \\geq 0 \\\\\n\\frac{x}{y}-x & \\text{if } 1 \\geq y > x \\geq 0 \\\\\n1-x & \\text{if } x=y>0 \\\\\n0 & \\text{else}.\n\\end{cases}\n", "7083f424c75eaffb678956b16be8efc1": "\\textstyle [f]([\\mathbf{x}]) \\supseteq \\bigcup_{i=1}^k [f]([\\mathbf{x}_i])", "70844049cacb397e3a4f876ab7585056": "d\\omega", "7084d520256624cae9098eaf35d3fb10": "T(n) = T(1) \\left(B + \\frac{1}{n}\\left(1 - B\\right)\\right)", "7084d7af57900a96e96a526d4d982df7": " \\left[\\frac{\\lambda}{2 \\pi x^3}\\right]^{1/2} \\exp{\\frac{-\\lambda (x-\\mu)^2}{2 \\mu^2 x}}", "708514c5020d4c3a6c89c1c42f99c8da": "\\tilde{\\mathbb{P}}", "7085a1413394fff11afdc9366bc27e43": "k = \\frac 1 s ", "70864ade9ec1d28ccd9a65922a6a9938": "r z_1", "708673420174dfbc55e2513c0cd1d559": "(\\mathbb{Z}/2^k\\mathbb{Z})^\\times ", "70870b52a35745f68db5ed9eb924ba6d": "\n\\ x(t) = \\begin{cases} 1, & |t| < T_1 \\\\ 0, & T_1 < |t| \\leq {1 \\over 2}T \\end{cases}\n", "70872fcd7158e5a037d92461d221956e": "\\sigma: {\\mathbb C} \\to {\\mathbb C}\\,", "7087338678c59fa2282936be54010390": "\\theta (z) = 0", "7087388f44335cbf099ad7ef29cdcfd8": "b^2-4\\,a\\,c=0", "70873dd4722354f07afd67adb2950373": "A^TC = C^TA", "7087987cea96603d6ae103b7f096daf5": "\\vartheta = l(\\theta)", "7087e823b53691b8be93f0bcb5b63822": "r = \\frac{D_1}{P_0} + g.", "70883e148d1600aa2bf5378a5666f17f": "\\langle *,e,(-,-),L,R\\rangle", "70883fa62a388d055f64809ccb184712": "b = \\frac{R}{1-\\kappa+\\gamma}", "70887ad6bf88a01cd1ba17bfed26fe3c": "g(x) = \\sum_{m=1}^\\infty \\alpha(m)\\frac{f(mx)}{m^s}\\quad\\mbox{ for all } x\\ge 1\\quad\\Longleftrightarrow\\quad\nf(x) = \\sum_{m=1}^\\infty \\alpha^{-1}(m)\\frac{g(mx)}{m^s}\\quad\\mbox{ for all } x\\ge 1.", "7088986ecbb070e8ffc59546a57ac417": "\\delta t=-1.9\\pm3.7\\ (\\mathrm{stat.})", "7088be41339ef7ed97a56acc040b6c30": "S_{T}-K<0", "7088d2b16892f6770f0a6f12db3b27e7": "\\begin{align} \\pi(x) &= \\sum_{n=1}^{\\infty}\\frac{\\mu(n)}{n}\\Pi(x^{\\frac{1}{n}}) \\\\ &= \\Pi(x) -\\frac{1}{2}\\Pi(x^{\\frac{1}{2}}) -\\frac{1}{3}\\Pi(x^{\\frac{1}{3}}) -\\frac{1}{5}\\Pi(x^{\\frac{1}{5}}) \\\\\n&\\ \\ \\ \\ +\\frac{1}{6}\\Pi(x^{\\frac{1}{6}}) -\\cdots, \\end{align}", "7088ddae4a05d81e0893015f43c67d03": "\\textbf{v}_i = \\sum_{j=1}^{d_i} v_{ij} \\textbf{e}_{ij}\\!", "7089948f18fdf660329156e61903ae40": " |\\psi ' \\rangle - |\\psi\\rangle = i\\hat{H} |\\psi\\rangle. ", "70899937c9eba65d1bef1f2624f1ee9b": " n = 0 ", "70899d4b467899bfb250c08eeeb5f7f3": "V_e=V-v_e", "7089ca925a91190718bb1a7c6a3c9a65": "f_0(z) = \\,_2F_1(a,b;c;z)", "7089cb759d850550cce2cc2785166d03": " \\tau_n", "7089e9dcd89bb953ee392cbb8b23f480": "\\beta = \\alpha^k", "7089f279101a49c6daf2dad625da3722": "{\\tilde{C}}_9", "7089fef551a07560cdc4dffc0016b4f2": "c_m\\,", "708a8226346860386235c1e36b1995f7": " {d^2 X^\\mu \\over dT^2}={d^2 x^\\nu \\over dT^2} {\\partial X^\\mu \\over \\partial x^\\nu} + {d x^\\nu \\over dT} {d x^\\alpha \\over dT} {\\partial^2 X^\\mu \\over \\partial x^\\nu\\partial x^\\alpha}", "708a8d50183f2fa178f07d412a6bf457": "a+b+c=0 \\bmod v", "708ac4e17368b509fe83dbe33fb492de": "\\scriptstyle A <_\\alpha C", "708ae827e513863a259d50dd8093c546": "\\delta = \\rho(\\psi(M)\\psi(P_A)^k) \\in \\mathbb{F}_q^m", "708afcadfecf14998ad5b758a13c2f20": "c(\\zeta,\\tau) \\ \\stackrel{\\mathrm{def}}{=}\\ e^{-\\zeta/2} T(\\tau) P(\\zeta)", "708b100aa12b7e481f6bb00187a24463": " \\phi_{mc} (r) = \\max \\left[ 0 , \\min \\left( 2 r, 0.5 (1+r), 2 \\right) \\right] ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{mc}(r) = 2", "708b2744f53d3337b485def943128473": "\\operatorname{Tr}_{M/K}=\\operatorname{Tr}_{L/K}\\circ\\operatorname{Tr}_{M/L}.", "708b37ae8a0d09d15645da61b553460b": "x^{-5} \\cdot 2^3 = \\frac{8 \\sqrt{5}x}{25} ", "708b68a4ea6fff6415bcf14a3f5c38ed": " \\mathcal{O}_k / \\mathfrak{p}\\;:\\;\\;\\; \\mathrm{N} \\mathfrak{p} = |\\mathcal{O}_k / \\mathfrak{p}|.", "708bcf3cac457ec8c5bb860d4b635962": "A^\\mathsf{T}\\mathbf{x} = \\begin{bmatrix} \\mathbf{v}_1 \\cdot \\mathbf{x} \\\\ \\mathbf{v}_2 \\cdot \\mathbf{x} \\\\ \\vdots \\\\ \\mathbf{v}_n \\cdot \\mathbf{x} \\end{bmatrix},", "708c42047ad321164dd2890ca4617523": " \\det(1-t\\sigma_3) = (1-t^3)", "708c5350ddd300abbcaea01bb707f1b6": " \\mathrm{d}f = \\frac{\\mathrm{d}f}{\\mathrm{d}x}\\mathrm{d}x ", "708c63ec3e22d4991068ae955ac4d760": "x = a\\theta - b \\sin(\\theta) \\,", "708c8b92835a191d8eb709f4d9fe5321": "C_p = (A_8 \\times A_{12})\\, \\rtimes \\mathbb Z_2.", "708c95d7d1e160032097d31df821990b": "|mg| > |T|", "708ca7fab362d15ac971b4fd4659074a": "h = \\sqrt{b^2 - (\\frac{b}{2})^2} = \\sqrt{b^2 - \\frac{b^2}{4}} = \\sqrt{\\frac{3}{4}b^2} = \\frac{\\sqrt{3}}{2}b", "708cc4537e02baa2c65bc03c3ba6f2ad": "\\phi\\colon C\\nrightarrow D", "708cf6ef31179a7d13d74d0dd650a71f": "\\mathcal{B}_{\\epsilon}", "708d03f509869fd7401a78dc3d61885e": "\\rho_f = \\frac{C\\rho_i C^\\dagger}{\\text{Tr}\\left[ C\\rho_i C^\\dagger\\right]}", "708d1ec890ab147e6483b3e6f9ec856f": "\n\n\\boldsymbol{\\Omega}=\\left(\n\\begin{array}{cccc}\n 1 & \\sqrt{\\mathrm{abs}(\\rho_{12})} & \\cdots & \\sqrt{\\mathrm{abs}(\\rho_{1k})} \\\\\n \\sqrt{\\mathrm{abs}(\\rho_{12})} & 1 & \\cdots & \\sqrt{\\mathrm{abs}(\\rho_{2k})} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\sqrt{\\mathrm{abs}(\\rho_{1k})} & \\sqrt{\\mathrm{abs}(\\rho_{2k})} & \\cdots & 1\n\\end{array}\n\\right), \n", "708d2dfeb3a964708d85bf0660d960a4": "p^{-2}", "708d5b04b777dfa6998d33f97bc2f193": "\\textstyle P(H|E)", "708d69c47ab500f1ea0c1e777af13e18": "\\mathcal{S}(\\sigma) =", "708dd619e0058b2e89ed3df2171ee03f": "F(t)=\\int_{0}^{t} f(\\tau)\\, d\\tau. \\!", "708dec8f6e72403b4549d53ec55a7e5a": "\\int_{X}|f-f_n|d\\mu\\le \\int_{E}|f|d\\mu+\\int_{E}|f_n|d\\mu+\\int_{E^C}|f-f_n|d\\mu", "708e158d264d4937842931eb2d854524": "HJD = JD - \\frac{r}{c} \\cdot cos(\\beta) \\cdot cos(\\lambda - \\lambda_{\\odot})", "708e227f7ba0aec4a9ee5bbc2642f92d": " W_{PPT}=I_{PPT}W(\\mathbf{E}, \\omega)=|C_{n^{*}l^{*}}|^{2}\\sqrt{\\frac{6}{\\pi}}f_{lm}E_{i}(2(2E_i)^{\\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}(1+\\gamma)^{2})^{|m/2|+3/4}A_{m}(\\omega, \\gamma)e^{-(2(2E_i)^{\\frac{3}{2}}/F)g(\\gamma)} ", "708e8b81aeef0d536c7a2fb1bbb76b07": "\\kappa^{-1}\\rho^T d", "708e9927be7f4e09313fac6dfa2f8ebd": "\\beta_i(x) ", "708ea32aa58de2479a00037eefafe74c": "\n\\begin{align}\nh(t) = \\mathcal{F}^{-1} \\{ H (f)\\} & = 2B \\frac{\\sin(2\\pi Bt)}{2\\pi Bt} \\\\\n & = 2B \\, \\mathrm{sinc}(2 B t)\n\\end{align}\n", "708f0d7e1b635477b0ee0b2553f962da": "\\Phi_{22}\\neq 0", "708f14b1d6d2df960e0487fec3f98e84": "0 \\le \\beta \\le\\pi", "708f36127e5f60b42c5c17d21c6313e6": "(0\\leqslant t 4 \\\\\n \\end{cases} \\\\\n f_{2}\\left(x\\right) & = \\left(x-5\\right)^{2} \\\\\n\\end{cases}\n", "70955e3c3fdb70141847fb42bd099e97": " \\int_{-\\infty}^{\\infty} \\exp\\left( {1 \\over 2} i a x^2 + iJx\\right ) dx = \\left ( {2\\pi i \\over a } \\right ) ^{1\\over 2} \\exp\\left( { -iJ^2 \\over 2a }\\right ). ", "709580b3e0d56aeb4062090c39979f60": "S^6 = \\left\\{ x \\in \\mathbb{R}^7 : \\|x\\| = r\\right\\}.", "7095d87d511cf48614b053ca9b4ba17a": "p(x)=x^3 -7x + 7", "7095e687ed0bcfc3c4779c05f7766750": "P(k_{in})=k^{-(2-p)/(1-p)}", "7096195c6bac5a3b2dc22ad9df3a889e": "E_e^0=J", "7096219408158dd77827a76491473aac": "\n \\begin{bmatrix}\n 1 & \\mbox{coat}\\\\\n 10 & \\mbox{lb. of tea}\\\\\n 40 & \\mbox{lb. of coffee}\\\\\n 1 & \\mbox{quarter of corn}\\\\\n 2 & \\mbox{ounces of gold}\\\\\n 1/2 & \\mbox{ton of iron}\\\\\n x & \\mbox{commodity A, etc.}\\\\\n \\end{bmatrix}\n=\n20 \\mbox{ yards of linen}\n", "7096409a55a9a3f1fdc20baae15db22a": "S_{\\mu\\nu\\alpha}=-S_{\\mu\\alpha\\nu}=\\frac12(T_{\\nu\\mu\\alpha} +T_{\\nu\\alpha\\mu} + T_{\\mu\\nu\\alpha}+ C_{\\alpha\\nu\\mu} -C_{\\nu\\alpha\\mu}), ", "709658b52ed4adaf8f8ffe47a01cca1f": "V^1 \\oplus \\mathbb{R}^2", "709658df5fadcf2a2b1aec1c5cd2084e": "28_{11} \\ ", "70967c69e65b7423bc7a24b4bace1660": "w_A", "7096c5d961701b9578e43078049346a4": "\\chi_\\alpha=\\chi_{[\\alpha,1]}=\n\\begin{cases} 0 & \\mbox{if } x \\notin\\; [\\alpha,1] \\\\ \n 1 & \\mbox{if } x \\in [\\alpha,1]\n\\end{cases}\n", "709706468b48faa974c67e3abddb0465": "E \\left[ \\widehat\\sigma^2 \\right]= \\frac{n-1}{n}\\sigma^2.", "70971b87b5d4e89308619153e3ff09f0": "\nZ_n(z) = \\sum_{k=0}^{n-1} \\operatorname{Li}_{n-k}(e^{-z}) \\,{z^k \\over k!} \\qquad (n = 1,2,3,\\ldots) \\,.\n", "70974ec6ba3e1bf31fe7fd3a1f5acac5": " S_{x,y} = \\{ r \\in \\bold{R} : x < r < y \\} . ", "70975b04a713dbcc7507b1cae0f31950": "\\dfrac{\\mbox{b.hp.}}{\\mbox{i.hp.}} \\times 100", "7097944d6f359af495b25254a6b94b05": "\\scriptstyle 1 / (1 \\,-\\, P)", "70979bcc9cd2d4f148d350391e174149": "S^{r-1} \\times D^{n-r} \\subset \\partial W", "7097ae351ad5ccb7143bdf22eb97e82b": "f\\; x", "7097dfcd0fda24ee7082d6a1510a3c75": "\\displaystyle h", "7097fbb13b32e3c101a0386c171272a1": "C_{8} = G_4 + G_0 \\cdot P_4 + C_0 \\cdot P_0 \\cdot P_4", "7098323b7308771f5dafdd865c89a15b": "1/[y_1, y_2] = [-\\infty, 1/y_1] \\cup [1/y_2, \\infty]", "70984ceb76201249219b5593cbc4cee4": "\\exists\\lambda\\in\\sigma(A): |\\lambda-\\tilde{\\lambda}|\\leq\\frac{\\|\\mathbf{r}\\|_2}{\\|\\mathbf{\\tilde{v}}\\|_2}", "709874b2891f9d9652f754b3f4949443": "f(x) = \\log(1 + e^x)", "70988a38a2a0172e4d067cdeafc66456": " r > 0 ", "7098a14a7c07066eadc1b97ff6832882": " V(u,U):=\\sup\\left\\{\\int_\\Omega u(x)\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x\\colon\\boldsymbol{\\phi}\\in C_c^1(U,\\mathbb{R}^n),\\ \\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\\le 1\\right\\}", "7098c36b1bbbaa8b85cda9f7af82b38c": "{\\mathbf{}}A(t), B(t), Q(t), R(t)", "70990ff2890f5162748692a688f64559": "\n \\rho_t = \\sup_{\\phi\\in Z:\\,\\|\\phi\\|_2=1} \\| \\mathcal{E}_t\\phi \\|_2.\n ", "709960ac3b085407d9b9a14d0009bc3c": "KA", "7099649b4fb1062d34e33c99f81fc5f1": "\\left(\\frac{n}{m} = 4\\right)", "7099669815abde6ba0b07232bdcff6e5": "\\Delta_q", "70997645a5392c97a87e4195b55a1d5f": "f_a(x)=\n\\begin{cases}\n0 & x\\neq a, \\\\\nf(a) & x=a, \\\\\n\\end{cases}\n", "709981376108670085cc0bf6f4da5fee": "T_3 =(S-T_1-T_2)\\ ", "709985ca5905819dbd49125a0edc00b3": "\\Omega(x) = -\\lambda_0\\sum_{j=1}^v e_j\\alpha^{c\\,i_j} \\prod_{\\ell\\in\\{1,\\dots,v\\}\\setminus\\{j\\}} (\\alpha^{i_\\ell}x-1).", "70999f3fd7291cfcd55c95c9d055d868": " \\det A_{33} = 0 ", "7099ac4efece0c5cf33f0b4d023fd5c2": "( \\{ (n, m) \\}, (n+1, (n+1)\\cdot m))", "7099f02d6a43a37431510987c55a31d8": "F(\\eta) = \\sum_{i = 0}^{\\infty} c_i (\\eta - 1)^i", "709a2bae653015c14a763d4c04b582ac": "\\mathcal P ", "709a4230bf83cc0e8547772ab9e6b243": "\\limsup_{\\delta \\downarrow 0} \\delta \\log \\mu_{\\delta} (F) \\leq - \\inf_{x \\in F} I(x), \\quad \\mbox{(U)}", "709b1b020c4d82f5ab26ea47894a821a": "\\frac{\na_1 \\lor \\ldots \\vee a_i \\vee \\ldots \\lor a_n, \n\\quad b_1 \\lor \\ldots \\vee b_j \\vee \\ldots \\lor b_m}\n{a_1 \\lor \\ldots \\lor a_{i-1} \\lor a_{i+1} \\lor \\ldots \\lor a_n \\lor b_1 \\lor \\ldots \\lor b_{j-1} \\lor b_{j+1} \\lor \\ldots \\lor b_m}", "709b3850f1eb77937bda46cef598f572": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 1}{4 \\choose 2} = 3,744", "709ba663c80d98fb01ce4e785bd14a61": "\\min(\\lambda x^*_s, 1)", "709bbb5189bb49dea6187d69b10b7d44": "\\sqrt{\\varepsilon}x", "709bd521a90b83a99e83cb7e9af954f2": "\\ [A]_e = [A]_0 - [B]_e ", "709c1c481944fe877b2cd351b01f1638": "\\tau=t/4M", "709c2d1901feb2c7d6d95c87b97c1f9a": "{\\left(\\frac{\\mathrm{e}^{t}-1}{t}\\right)}^n", "709c2e8704c7d0fdc58e711144c710a8": "\\mathbf{s}_{u}=-\\sin\\varphi\\cdot{\\mathbf{x}}_{u}+\\cos\\varphi\\cdot{\\mathbf{y}}_{u}", "709c433d1edae7907b9c3dbb5c45ac11": "A \\oplus 0 = A", "709c6f50a1257ca287a1027142ab9fd1": " M_{EGB}(Z)=\\frac{e^{\\delta t}B(p+t\\sigma,q)}{B(p,q)}{}_{2}F_{1} \\begin{bmatrix}\n p + t\\sigma,t\\sigma;c \\\\\n p + q +t\\sigma;\n\\end{bmatrix}.\n", "709c94bfa3181708ccf4df8b35dd3992": "\\delta_B = \\frac {F L^3} {3 E I}", "709ccf6253098ccf7f2c5b6a8872a748": "\\bar{y}\\equiv\\varphi_i(x,y)", "709dacb0f4c80adcbad1284fbedf7740": "10^{-33}", "709e068bba23da1a8f375a56ca33982b": "\\underset{c}{\\square}", "709e10ca39f1c25a0a25150ba951f034": "\\int_{A} e^{- \\theta \\varphi(x)} \\, \\mathrm{d} x \\approx \\exp \\left( - \\theta \\mathop{\\mathrm{ess \\, inf}}_{x \\in A} \\varphi(x) \\right).", "709e421b53b288867ac307f4b272e361": "\\phi_B^{-1}:F^n\\to V", "709e850bf5ab043869753fbe81bf102a": "\\sin A = \\frac{h_1}{c}\\text{; } \\sin C = \\frac{h_1}{a}", "709eb80c5206b84490f331a6e3cb3e4b": "\\begin{array}{rll}\nA & = & (a_{ij});\n\\\\\na_{ij} & = & ||x_i - x_j||_2^2\n\\end{array}\n", "709ee47e5b33ca67916ab479d44b4f07": "g_k^i", "709f0f4dfa59b8d5989ddb0d3da5378d": "\\delta_{mn}", "709f69407689ccc726359a7f8c72b610": "i^* R^q f_* \\mathcal{F} \\to R^p \\widehat{f}_* (i'^* \\mathcal{F})", "709fb2c87c72b7e7dbf03f88428441f5": "C_{4,n} = n^2 + (n - 1)^2.\\,", "709fd69c651b698efeeab7c0e3c7e9b5": "X=(X_1,\\dots,X_n)", "709ff95a7781cbf374ef9202154e9459": " x_2-(x_1+y_1)\\cdot x + x_1\\cdot y_1=\\theta", "70a00cabfd1188d1567e553ca32c9e6c": "\n \\frac{\\omega^2}{c^2} = \\left(\\frac{n \\pi}{a}\\right)^2 + \\left(\\frac{m \\pi}{b}\\right)^2 + k_{z}^2,\n", "70a04967c00bb9c3ebead9a161a0242f": "| \\phi \\rangle", "70a0ad778600a60028df2ec59b1697b4": "x' = x A_{1 1} + y A_{2 1} + b_{1}\\,", "70a0ca63b6a0f9b5fa05781d28dcebf6": "x_1^{k_1}+\\cdots+x_n^{k_n}=0", "70a0de29d269f76d5ec016775ee24b34": "\\int \\tanh^n ax\\,dx = -\\frac{1}{a(n-1)}\\tanh^{n-1} ax+\\int\\tanh^{n-2} ax\\,dx \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "70a0e8b5154daef588d248272b14f66a": "|V \\times V| = T^2(|V|) = |P_1^2(V)|", "70a0fa7a422c6230b9f1347b30f41dda": "\\left(\\frac{\\partial^2S}{\\partial y\\partial x}\\right) = \\left(\\frac{\\partial^2S}{\\partial x\\partial y}\\right)\n:\\left(\\frac{\\partial^2V}{\\partial y\\partial x}\\right) = \\left(\\frac{\\partial^2V}{\\partial x\\partial y}\\right)", "70a116f827f7980ad9ff9fae9b3c3f58": " \\mathrm{Vol}_n(K + tB)\n = \\sum_{j=0}^n \\binom{n}{j} W_j(K) t^j.", "70a1b60f594b030b37c57287563d05bb": " \\psi (0) = e^{ik \\cdot 0} u(0) = e^{ikL} u(L) = \\psi (L) \\,\\! ", "70a1cad98a224164df6a080f8dc51d29": "\n\\det(LoG(\\mathbf{x}, \\sigma_I)) = \\sigma_I^2 \\det(L_{xx}(\\mathbf{x}, \\sigma_I) + L_{yy}(\\mathbf{x},\\sigma_I))\n", "70a21e3e66aeacbc18be562b4cd7cb8a": "F(x)|\\nabla_S T(x)|=1,\n \\,\\, \\mbox{for the surface} \\,\\, S, \\, \\mbox{and} \\,\\, x\\in S.\n", "70a24f5afebe72c79465f48ae8aa9638": "u_i,i=1,...,m", "70a2523359d9caa80339422c0b395488": "T_\\text{new}=\\frac{A \\times T_\\text{old}}{A+T_\\text{old}}", "70a26a0cd7587140cb16f85414f2aa44": "V_\\delta", "70a2e464cf16ba89e0e98e6e0d0a4cee": " \\mu = 0 \\ \\text{and} \\ C=0 ", "70a31e4fbd8df7d90007910048537c3d": "\\tfrac{X}{X+Y} \\sim \\Beta(\\alpha, \\beta).", "70a38892273e82d4556e3002662d1daa": "F \\mapsto \\pi_0 F", "70a3af31fbdbc094b9a80163c7b5d3ab": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi(\\bold{r},t) = E \\Psi(\\bold{r},t)", "70a3e4c3937f04f091fc831670244ad8": "P = \\frac{2}{7}", "70a454839ffbafaee4391a5aee3862d8": " \\theta/2 - \\sin(\\theta)/2", "70a4870358fc61c68225548bd6fa0ccf": "N\\left( \\mathcal{S}\\right) \\backslash\\mathcal{S}\\in N\\left(\n\\mathcal{S}\\right) ", "70a4a61dc163e3fccb0861b7f483db2d": "\\scriptstyle r_i", "70a4fc01f5c870cfd4cc076c5e5a1f6d": " \\mu = \\mu' - j \\mu'' ", "70a5a25bba53b74faf47d392ad87ecee": "V_i= R_{\\text{L}}\\frac{-V_o}{(1-D)R}+ (1-D)(V_i-V_o)", "70a6691ad11ce755e39e01a5f85f4158": " v_j(x) = \\sqrt{\\frac{2}{L}} \\sin\\left(\\frac{j \\pi x}{L}\\right) ", "70a669a5ce687ad573dad521e70d65ea": "Q_{vib}=\\prod_j{\\sum_i{e^{-\\frac{E_{j,i}}{kT}}}}", "70a6e75807a0631c562ce8ae562ae749": "A P_1", "70a7452a21a49b106397d34961c42fba": "\\mathcal{A} = \\mathfrak{C}\\{\\mathcal{B}\\}", "70a7a163d75098e603cfa9d39176a7bf": "{\\operatorname{d}N\\over N}=2{\\operatorname{d}F\\over F}", "70a7fcc01d39bea01dec47ce4c26d69c": "\\rho_i = \\frac {m_i}{V}.", "70a8085b18cceb8c8a710ad7f11bced5": "w_{dn}", "70a8341c570fafdb4273e7167d434b70": "y, r, r_v \\in G", "70a893b77a3bb5cf78923e6cb13087a3": "\\phi = \\frac{ C_b V_b }{C_a V_a} = \\frac{\\alpha_{A^-} - \\frac{[H^+] - [OH^-]}{C_a}}{1 + \\frac{[H^+] - [OH^-]}{C_b}}", "70a9547570fd50b65b5cfcf59df19735": "Y=g(x)", "70a98516e1dc872a439e5cb216998d32": "(F,G,\\eta,\\varepsilon)", "70a98f96c8fd01a0fc93bb768ede6f35": "(b/e)^2", "70a9b3686be3f639045246c4a208b2b8": "\\boldsymbol{ a}_C = -2 \\, \\boldsymbol{ \\Omega \\times v}", "70a9b8cda49a71e8313075161feea2b0": "\\, \\pi r^2 + E(r) \\, ", "70a9fc008684ce17480ed15c59fdfa70": "{D} ", "70aa244d777eba7921839d2d229d968e": " \\arctan \\left(\\sqrt{2}\\right) = \\arccos \\left(\\sqrt{\\tfrac13}\\right) \\approx \\textstyle {54.7356} ^{ \\circ } ", "70aa2da73096190ad090d80fa7a0a884": "f_n : A_n \\rightarrow B_n", "70aa3ccac4e77c25fe1603679c055366": "M \\in B^{\\ast}\\,", "70aa4cb2f298647fc1aa75e4bb8100d7": "\\delta F_{\\theta}", "70aa60f3bb84d732ef581dbf6650d93e": "\\beta=\\frac{2\\pi}{\\lambda}\\ ,", "70aa90b2f3039fd0ce68868438493298": "\\rho(x,y)", "70aa9aa06aaad5f7082b181c0dfecc15": "\n\\omega = c |\\mathbf{k}|\\quad\\hbox{and}\\quad \\omega' = c |\\mathbf{k}'|.\n", "70aada1454923181c19160dc3e0c8a65": "\n\\mathbb{E}_\\theta\\left[\\frac{\\mathbf{(\\theta-X)^T X}}{|\\mathbf{X}|^2}\\right] = \\sum_{i=1}^n \\mathbb{E}_\\theta \\left[ (\\theta_i - X_i) \\frac{X_i}{|\\mathbf{X}|^2} \\right]", "70aaf0cccc37773f649112aab18892b7": "\\tau(1)=z", "70aafe6386897ad96ae79cab4047ad4e": "\n\\left\\{\\begin{matrix} \\ln\\ G_{12}=-\\alpha_{12}\\ \\tau_{12}\n\\\\ \\ln\\ G_{21}=-\\alpha_{21}\\ \\tau_{21}\n\\end{matrix}\\right.", "70ab14bd7508bb573cb37406401f36b1": "h(a)+ 2b=0", "70ab1e1549a4bf949a1f8738bc37c408": "h = \\left( 1 - \\frac {1} {2} d \\right) \\sqrt{\\frac {T} {1-T \\cdot \\widehat {\\operatorname{Var}}(\\widehat\\beta_1\\,)}},", "70ab1fe34ca4a7baf545ede7f18e65fc": " H_e = \\mathrm{d} Q/\\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) \\,\\!", "70ab3c419e33333f5f2bc8fdbba9828f": " dS_t = \\mu S_t\\,dt + \\sigma S_t\\,dW_t \\, ", "70abd73aec2ed1e8297b41dd82841cff": "\\langle 1 \\rangle", "70abeca51a0b4ffd0cddb9eb3482a893": "(\\alpha_i)", "70ac01845f190f02bb981f31931f43fa": "\\cos x = \\frac{e^{ix} + e^{-ix}}{2}\\quad\\text{and}\\quad\\sin x = \\frac{e^{ix}-e^{-ix}}{2i}.", "70ac54fec2c49d7eb43793b0069e0cc0": "\\gamma(t):= (\\gamma_{1} \\oplus \\gamma_{2} \\oplus \\gamma_{3} \\oplus \\gamma_{4})(t) ", "70ac58e43162de6674de3d1d7ca8f0c3": "\\sigma_j(t + \\Delta t) := \\left\\| \\hat{\\mathbf x}_{j\\alpha}(t+\\Delta t) - \\hat{\\mathbf x}_{j\\beta}(t+\\Delta t) + \\sum_{k=1}^n \\lambda_k \\left(\\Delta t\\right)^2 \\left[ \\frac{\\partial\\sigma_k(t)}{\\partial \\mathbf x_{j\\alpha}}m_{j\\alpha}^{-1} - \\frac{\\partial\\sigma_k(t)}{\\partial \\mathbf x_{j\\beta}}m_{j\\beta}^{-1}\\right] \\right\\|^2 - d_j^2 = 0, \\quad j = 1 \\dots n", "70ad26afdc722d6f726b54d0a92ed93c": "\\begin{align}\nG(x, y, x_0, y_0) =\\dfrac{1}{2\\pi} &\\left[\\ln\\sqrt{(x-x_0)^2+(y-y_0)^2}-\\ln\\sqrt{(x+x_0)^2+(y-y_0)^2} \\right. \\\\\n&\\left. + \\ln\\sqrt{(x-x_0)^2+(y+y_0)^2}- \\ln\\sqrt{(x+x_0)^2+(y+y_0)^2}\\right]\n\\end{align}", "70ad31ef4f65cfba30e577994399e675": " r_i > 0,\\ r_i > 1.0 \\sigma ", "70ad5e897f0f611e2ef496edc7fb5c9f": "N = \\sum_{i = 0}^k F_{c_i},", "70addc7912eb506e2d654cf2b79e605e": "\\frac{9}{4 \\sqrt[3]{2}}", "70ae1e1294708d7f97bb6b8c2712c8f0": "R_{i-1}", "70ae3b9a922c958ace07d2cb239e801f": "\\Re\\left\\{\\cdot\\right\\}\\,", "70ae901dc106620b4dc0e86213a17640": "\\mathbf{Q}(\\sqrt{5})", "70ae9f8efded220d0f722fca28e787e2": "\\scriptstyle\\mathrm{Vol}(a,b,c) = (a\\times b)\\cdot c.", "70aece052e3c242f555545c270f434bb": "S_1(q) = \\sum_{n\\ge 0} {q^{n(n+2)} (-q;q^2)_n \\over (-q^2;q^2)_n}", "70aeeaf34edd324fa8b31f908cdf4351": "R_{J}(x,y,y,p) = 3 \\int _{\\sqrt{x}}^{\\infty}\\frac{1}{(u^{2} - x + y) (u^{2} - x + p)} du =\n\\begin{cases}\n \\frac{3}{p - y} (R_{C}(x,y) - R_{C}(x,p)), & y \\ne p \\\\\n \\frac{3}{2 (y - x)} \\left( R_{C}(x,y) - \\frac{1}{y} \\sqrt{x}\\right), & y = p \\ne x \\\\\n \\frac{1}{y^{\\frac{3}{2}}}, &y = p = x \\\\\n\\end{cases}", "70aefa7919279af40fa4bb88e8881a28": "c = \\sqrt {\\frac {g}{2 \\pi} l }", "70af50666745a472ba53ded31b4e8806": "(\\phi \\vee \\psi) \\to (\\neg \\phi \\to \\psi)", "70af8239cc23f80b631b8cb0caeaca6d": "\\mathbf{X^TM^{-1}X\\Delta \\boldsymbol\\beta=X^T M^{-1} \\Delta y} ", "70afe3e89e980c54fdd11e8d944371e3": " A \\xleftarrow{n+\\mu-1} B \\xrightarrow[T]{n\\pm i-1} C", "70b04a8dbe40405260d13741a40e8d32": "f(z) = \\sum_{k=0}^\\infty a_k z^k", "70b0f8266052f00d65416cfbd9703bd7": "f(n)=\\tilde{O}(g(n))", "70b122ee551b3389c906fe4d82a1024b": "N= \\, ", "70b1661412908b91f99289312fff6a88": " M_X(t) := \\mathbb{E}\\!\\left[e^{tX}\\right], \\quad t \\in \\mathbb{R}, ", "70b1a917d54f6b915d7011f9dd7e7404": "4^d", "70b1ac10c70b5b89b72bbc1e2cbc868b": "\n\\bar{h}^{\\alpha \\beta} (t,\\vec{x}) =\n-4 \\int\\, \\frac{\\tau^{\\alpha \\beta}(t-|\\vec{x}-\\vec{x}'|,\\vec{x}')}{|\\vec{x}-\\vec{x}'|}\\, \\mathrm{d}^3x'\n", "70b1c515f2ebecdde9e53e5a5b762c7f": " M_{X_1} = I - X_1(X_1'X_1)^{-1}X_1'. \\! ", "70b277e5cd579c156efa27a59f1e26fc": "N\\omega", "70b2d8f37f51ee3db81aa84739ebc0e4": "Z_n(s)=v_n(s)/i_n(s)", "70b2e90a78e06d485ee318b313c4bcb4": "J^k_pf", "70b2f887449cbe6bad02912285ac294a": "\\lim_{x\\to -\\infty}F(x)=0, \\quad \\lim_{x\\to +\\infty}F(x)=1.", "70b301d6395c83b474b0124dca03ad23": "f(a,b) = 1 - (1 - a)(1 - b)", "70b31697ba8201adf04292bf5cd2e135": "\n\\mu_c = kT \\ln \\frac{[\\mathrm{Ox}]}{[\\mathrm{Red}]}.\n", "70b3b492b0f799caba178a49720f5893": "p^0 = q^0 = p^2 = q^2 =~0", "70b41a88bf2e358cadaf34818b8756c8": "\\hat{I}_{S}\\big(\\hat{I}_{B}\\big)", "70b4422f771fa424d6f364a95eb4af14": "\\prod_{i = 1}^k p_i^{(n_i +1)}", "70b44aca9914bca2dd848a019c5911ce": "f = {1 \\over 2 \\pi} \\sqrt{k \\over m}", "70b499e034d87a637d20532d545fac5a": "2 S^{IJ}", "70b4bb0f84bf5c99491df4da8b173439": " A-\\text{vertex}= 0 : \\sec^2 \\left(\\frac{B}{2}\\right) :\\sec^2\\left(\\frac{C}{2}\\right)", "70b4f37a629df67c50ef39300e60ae13": " S_2 ", "70b4ff108ecef11b1f5e8c4d194c3343": "S' = \\{T_s | s\\in S\\}", "70b52861afa1b589da451864d0ca78d2": "\nAL_i = \\left( e_i^t-h_i^t \\right) \\left( g_i-G_i \\right)\n", "70b53f15c40fff1cd550457ddd79a0e1": " \\cot \\theta = \\frac {\\mathrm{adjacent}}{\\mathrm{opposite}} = \\frac {b}{a}", "70b557a81e1125c34a87f67578129f9a": "\\,(h_0,\\;h_1,\\ldots,\\;h_n)\\,", "70b56a18171487d0b576f97c3c6192cd": "\n w_i = \\frac{2}{n(n-1)[P_{n-1}(x_i)]^2} \\quad (x_i \\ne \\pm 1).\n", "70b57bf026a44703b4db86b48aea3422": "\n \\quad\\quad\n A \\begin{bmatrix} 1 \\\\ \\lambda_3 \\\\ \\lambda_2 \\end{bmatrix} =\n \\begin{bmatrix} \\lambda_3 \\\\ \\lambda_2 \\\\ 1 \\end{bmatrix} =\n \\lambda_3 \\cdot \\begin{bmatrix} 1 \\\\ \\lambda_3 \\\\ \\lambda_2 \\end{bmatrix}\n ", "70b588c8f9783c1acf8cd75a40301ec7": "[E,E + dE]", "70b59b6f21e9076f92dc36aedfed6918": "\\left.\\frac{\\partial}{\\partial z_1}g\\right|_{z_1=\\bar{z};z_2=z}", "70b656f55f10e10dca1e49d9a40d059e": "x=[0;a_1,a_2,a_3,\\dots]\\,", "70b661a6ed29ce88826245393be189ff": "1< a\\in F", "70b68b1b9b75f3d36f2ffce4af4171b4": "\\eta = \\left(1 - \\frac {\\sigma T_H^4 }{IC} \\right) \\cdot \\left( 1 - \\frac{T^0}{T_H} \\right)", "70b6f7dee934e6bf4d5db900314f51f2": " K_a \\times K_b = {[H_3O^+] [NH_3]\\over[NH_4^+]} \\times {[NH_4^+] [OH^-]\\over[NH_3]} = [H_3O^+] [OH^-]", "70b796eb02c35a3f95f502ea860a6777": "y = \\frac{\\cos^2(\\theta) \\left(2+\\cos(\\theta)\\right)}{3+\\sin^2(\\theta)}", "70b802b31cc6c7e2d4395e75c36476b9": "C(x_1,...,x_k) = C(x_1)C(x_2)...C(x_k)", "70b82e73a42c60af1ea6db32f92aa070": "0\\notin S", "70b830c28b000a006307bf02df8134a6": "B_\\nu, B_\\omega, B_\\tilde{\\nu}", "70b83f727e7bcaa3d58efd4272bd5963": " X_i \\sim Exp( \\lambda_i ) ", "70b8e2fd3577f7ef1ad541870ab27888": "\n\\frac{a(\\omega)}{\\sigma_{\\rm e}(\\omega)}=\n\\frac{\\omega^2 v(\\omega)}{\\pi^2c^3}\n~~~~~~~~~~~~~~~~~~(\\rm comparison)(av)\n ", "70b90c8a1bbfcf5f50ad4a91381376ba": "d{\\lambda}=\\cfrac{3}{2\\bar\\sigma^2}dw_p(H)", "70b9b3088a19ab7a09da191e8275880c": "P(h|v) = \\prod_{j=1}^n P(h_j|v)", "70b9e9bf45006c2c15be6925f4317077": "f(n)=n+\\left\\lfloor\\log_2\\left(n+\\log_2 n\\right)\\right\\rfloor.", "70ba546c24c94395c965ed6ffb3bb0fb": " \\mathbf{R} = \\big( \\sum_{e} \\mathbf{k}^e \\big)\\mathbf{r} + \\sum_{e} \\big( \\mathbf{Q}^{oe} + \\mathbf{Q}^{te} + \\mathbf{Q}^{fe} \\big) ", "70ba7f39771bbe12752d878d8b4e8dde": "p_z = N_1^c \\frac{z}{r} = Y_1^0", "70ba89e37b7b1a004f155af4929f124d": "{Y_n \\sqrt{n} \\over {\\sigma}},", "70bb1db0f435c8e75460fde575c76319": "\\alpha(p_{S_{i}})", "70bb651d5a44bb853893d097f1d97ec8": "x^jb'(x)", "70bb68700bcb73349a8f745d619051a7": "= \\frac{t}{i}", "70bc09e358e3502046ccab105c211807": "\\textstyle \\lambda_{\\diamond P}=\\lambda(a_\\diamond\\mid A)", "70bc1823692a15828c774adaade3d9cb": "|c-\\gamma(s)|=|c-\\gamma(t)|=r.\\,", "70bc3d26cfcfee3754a0ec486e1cc482": "F_{out 1}=F_{in 2} \\,", "70bd17ce67d699392bb69445d4a4eb90": "\\alpha_{g}=\\frac{m^{2}G}{\\hbar c}=\\frac{m^{2}}{m_{P}^{2}}", "70bd51410ca772ba12676c8b7d4c482b": "y \\in \\{ INT\\_MIN , ... , INT\\_MAX \\}", "70bd9acc510457a6a67b47edde4d3548": " \\langle \\cdot, \\cdot \\rangle ", "70bdb341d8f34c88aaf04900e10657be": " = \\lim_{p \\to \\infty} \\left( \\sum_{i=1}^n \\left| x_i - y_i \\right|^p \\right)^{1/p}", "70be1936ac46c67622a04296f0599985": "\\left\\{ L^x(T) \\colon x \\in [0,a] \\right\\} \\stackrel{\\mathcal{D}}{=} \\left\\{ |W_x|^2 \\colon x \\in [0,a] \\right\\} \\,", "70be1deafac2108bc83af6e755a815d5": " (0, t_1) ", "70be4f5b34a9355c9f561636cf788474": "N(2) = 4", "70be62938fde747dbbad70f871cdc171": "\\ \\Box A^{\\mu} = 0", "70bea00405efa214271ff337d0f66544": "c_{l+1}, \\ldots, c_{r-1}", "70bebf1af617164a2e0d722b779f530f": "C[p/q]", "70becdfc6398398071c886bf287a71f7": "\n\\begin{align}\n&\\hat{\\varphi}=-\\sin \\varphi \\hat{x} + \\cos \\varphi \\hat{y} \\\\\n&\\hat{\\theta}=-\\sin \\theta\\ (\\cos \\varphi\\ \\hat{x} + \\sin \\varphi \\hat{y})+ \\cos\\theta \\hat{z} \\\\\n&\\hat{r}= \\cos \\theta\\ (\\cos \\varphi\\ \\hat{x}\\ +\\ \\sin \\varphi\\ \\hat{y})+\\ \\sin\\theta\\ \\hat{z}\n\\end{align}\n", "70bef7eb8358d8c20e505920e2fc6858": "\\dfrac{J^2}{N}", "70bfaeb9f4a51859d43d23187bcedd7f": "E_n=\\omega\\varepsilon_n", "70bfd023363901afac670e77d76944ec": "u \\in C^{2,\\alpha}(\\Omega)", "70c03cd0e5a8587c2cb4003dcd16ec2c": "\\widetilde{K}_0(\\mathbb{Z}[\\pi_1(X)])", "70c06310fcee88f1ca51d10ada90df2f": "{}^2 E_{pq} = H_p(G, H_q(\\widetilde{Ff}, \\mathbb{Z})) \\Rightarrow H_{p+q}(Ff, \\mathbb{Z}) = H_{p+q}(*, \\mathbb{Z})", "70c06dd13c255460c2c0a42e2ffd5cd3": " f ( t ) = z e^{i\\omega t} \\,", "70c08e355ad0202e9e091c9595abfe05": "2^{58}+1 = 536838145 \\cdot 536903681. \\,\\!", "70c0f66806a1c7fde24e1a49193b9bc5": "\\mathcal{O}_X^m|_U \\to \\mathcal{O}_X^n|_U", "70c1766e06029897eab63b66f37d3187": "r^2+h^2=d^2", "70c17b1d6a8faf2f2022bd186bde7c1e": "\\int_M{\\mathrm{d}\\omega} = \\int_{\\partial{M}}\\omega", "70c17e944b8e7caaff67c01ec550887f": "\\scriptstyle Z=B", "70c1807cdf6cd7d78b6748980205f549": "x\\,\\Phi_2(x)\\,\\Phi_3(x)\\,\\Phi_6(x) = x\\;(x+1)\\;(x^2 + x + 1)\\;(x^2 - x +1)", "70c19d5e4d0801733dc0185308767cbb": " \\varphi(r) = { 1 \\over 4 \\pi \\varepsilon } \\frac{Q}{r}\\,\\mbox{erf}\\left(\\frac{r}{\\sqrt{2}\\sigma}\\right)", "70c1abb2a2b2c76c6629f12949966816": "\n \\boldsymbol{\\tau} = \\boldsymbol{F}\\cdot\\boldsymbol{S}\\cdot\\boldsymbol{F}^T ~.\n", "70c1e810bc5cd8a2df7f61a78ce17214": "A_1 \\rightleftharpoons A_2", "70c202b4afa2e3fe75eee0abab03bb1b": "y=\\tanh^{-1}(\\sin \\phi).\\,\\!", "70c253b6143378dfcdbed95ce15117b1": "\\tau_\\mathrm{max}=\\frac{1}{2}\\left|\\sigma_1-\\sigma_3\\right|=\\frac{1}{2}\\left|\\sigma_\\mathrm{max}-\\sigma_\\mathrm{min}\\right|\\,\\!", "70c26d86f0ad0b378849ed05df262981": "IV(\\textrm{sector}) = E_s(\\textrm{sector}), \\quad\\textrm{where}\\;\\; s = \\textrm{hash}(K)", "70c2e875e18c3b1dc08da9e09bfd3609": "\\mathcal{L} = \\mathcal{L}\\ (\\Alpha,\\ \\Omega,\\ \\Zeta,\\ \\Iota)", "70c2ec027ced83df0f8ad46cd0ef58d1": "\\lambda+ \\tfrac{2G}{3}", "70c3354fcb335f64e9c53666574befea": "\\alpha,\\lambda,\\rho", "70c36d6c890837f26615ac4aeb36d655": "H(Y'|Y)=0", "70c3741967caa96a7a65bb38a3a67eaf": "\\|\\tau_t f - f \\|_p \\rightarrow 0", "70c3b08d81033159a722ccfce31ee221": "d_{k,n}", "70c3f8e82a30895e08318b3dae54b071": "f(x,\\theta)=u(x)", "70c402aee1bfd53d188ba281d9e24690": "I= \\frac{V_S-V_D}{R} ", "70c40c5137b6c67d7f27c9a78ab4e557": " |A| = b_{\\{1,2\\}}c_{\\{3,4\\}}\n -b_{\\{1,3\\}}c_{\\{2,4\\}}\n +b_{\\{1,4\\}}c_{\\{2,3\\}}\n +b_{\\{2,3\\}}c_{\\{1,4\\}}\n -b_{\\{2,4\\}}c_{\\{1,3\\}}\n +b_{\\{3,4\\}}c_{\\{1,2\\}} ", "70c47003f4d328dfedaedeebb78d8fa0": "\\displaystyle{\\sum_{x\\in \\mathbf{Z}/2m\\mathbf{Z}} e^{\\pi i x^2/2m} = \\sqrt{m}(1+i).}", "70c4c807ecfa3fa26b9a179b31286a96": " f(x) = ax + c", "70c4e11c5423be55f30af295a45d1a9b": "\\iint_D \\left( -\\frac{\\partial v}{\\partial x} -\\frac{\\partial u}{\\partial y} \\right )\\,dx\\,dy = \\iint_D \\left( \\frac{\\partial u}{\\partial y} -\\frac{\\partial u}{\\partial y} \\right )\\,dx\\,dy =0", "70c5106e61e023a248881ab41148f70e": "z_{\\mathrm{diff}} = z_1 - z_2", "70c519f39bfdae4b92ba49ca132470d2": "\\mathbf R(T)=\\mathbf R(0)", "70c54136f2cf38221c7d1a0800ba9f53": "\n\\begin{cases}\na+c=b+d\\\\\nA+C=B+D=\\pi.\n\\end{cases}\n", "70c5aa436e811b3a14037905e9a8b788": "{\\sqrt{LOA}+Beam \\over 16}", "70c612060bb4c336ea559881305cfcaf": "\\angle ABC", "70c623ddc6fac629538d612049e8d6a3": "a_2,b_2", "70c62fd60f7281c91269092e5c355e98": "\\scriptstyle\\pi_\\ell", "70c65b7436db9f9baa0a214877c274f0": "\\frac{\\sin(\\pi x)}{\\pi x} = \\prod_{n=1}^\\infty \\left(1 - \\frac{x^2}{n^2}\\right)\\,\\!", "70c693f4135b4ff356d5bffec0075f18": " \\vec{v_s} ", "70c6ac67b72b0731fa1d1d1876ec947a": " T^{ik}", "70c6db3eec32062cf95fd95033155186": "\\Gamma(-\\tfrac52)\\,", "70c6e4d31644eb38176139a728b61689": "\n \\mathbf{u}\\cdot\\mathbf{v} = u^i~v_i = u_i~v^i = g_{ij}~u^i~v^j = g^{ij}~u_i~v_j\n ", "70c6f581038cb2f9d8d12d2eaef7f819": "p\\ddot r =nab \\varepsilon \\cos \\theta \\,\\dot \\theta\n=nab \\varepsilon \\cos \\theta \\,\\frac{nab}{r^2}\n=\\frac{n^2a^2b^2}{r^2}\\varepsilon \\cos \\theta . ", "70c70943d2c6a6f36ba9930f4b9d1a16": " J_0 = z_{\\mathrm{S}} d_{\\mathrm{F}} D_{\\mathrm{F}} \\int_{-\\infty}^{0} \\mathrm{exp}(\\epsilon /\nd_{\\mathrm{F}}) \\; \\mathrm{d} \\epsilon \\; = \\; z_{\\mathrm{S}} {d_{\\mathrm{F}}}^2 D_{\\mathrm{F}} \\; = \\; Z_{\\mathrm{F}} D_{\\mathrm{F}}, ..........(23) ", "70c733f8d4d4d4b1c156f75ed3241304": "\\mathbf{F_C} = \\mathbf{V_C} - s_C\\mathbf{V}", "70c76a65a74625f48b2a1880c1846f93": "\\left(\\sum_i a_iX^i\\right) + \\left(\\sum_i b_iX^i\\right) = \n\\sum_i (a_i+b_i)X^i", "70c77fb296568ef520c9d2822e42b31b": " a_y = -g ", "70c7888877cafa4e1904f3b988798f07": " \\delta\\mathbf{r}_i=\\sum_{j=1}^m \\frac{\\partial \\mathbf{r}_i}{\\partial q_j}\\delta q_j =\\sum_{j=1}^m \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}_j}\\delta q_j.", "70c7c656b23e43d719644bf703a2f64a": " a\\,", "70c8509ad2b14dc97f217548c46e0958": "L:S\\to2^A", "70c85e1b0b2c75ebf267fe8965254c6a": "s, h \\models \\mathbf{e}\\mathbf{m}\\mathbf{p}", "70c8b76281d8b282aaaafe614ed90ca1": " x(t) \\approx x \\left( t + T(t) \\right) \\ ", "70c8ed77669f252b4268bbe5c69bc14a": "F: S \\times S \\rightarrow \\mathbb{R}", "70c9202db39aae5639b0d78f9b3770b8": "\\tfrac{dR}{dT} = \\gamma I - \\mu R", "70c93ffaa0035c31d4d7b5a496283e19": "\\|x(0)-x_e \\|< \\delta", "70c9776e01e5290d672013dbd334f01a": "x\\in G-H", "70c98379c359f04257ab60c0c89de5bf": "R_C = \\frac{R_B}{T_{ON}+T_{OFF}}", "70c98f6a05fd48a3475f7f602be0f116": "P'(Q/P)=", "70c9c4ce8514d0a1b46218ef5c3be747": "|\\gamma(s, |u|) - \\gamma(s, u)| \\le \\int_u^{|u|} |z^{s-1} e^{-z}|\\,{\\rm d}z = \\int_u^{|u|} |z|^{\\Re s - 1}\\,e^{-\\Im s\\,\\arg z}\\,e^{-\\Re z} \\,{\\rm d}z", "70ca0dcd47280ce7602b7b718c4681e5": "(100-28%)\\cdot m", "70ca2dbd42e151bba81a3b9530ff2c03": "=\\frac 1{\\pi Z_q}\\,\\frac{-1+i}{\\sqrt{2}}\\sqrt{\\rho \\omega\n\\left( \\eta ^{\\prime }-i\\eta ^{\\prime \\prime }\\right) }=\\frac i{\\pi Z_q} \n\\sqrt{\\rho \\left( G^{\\prime }+iG^{\\prime \\prime }\\right) }", "70ca4ed10c8534d77ce983fd9d6e01e2": "\\left(\\!\\!{u\\choose k}\\!\\!\\right)={u+k-1\\choose k}\n=\\frac{(u+k-1)!/(u-1)!}{k!}\n", "70ca755d48f586a3c1e093b8ec3b7849": " r^2\\nabla^2 Y_\\ell^m (\\theta, \\varphi ) = -\\ell (\\ell + 1 ) Y_\\ell^m (\\theta, \\varphi ).", "70ca8100eff261507e196901856d8371": " e ^{1 / 2}", "70ca9f16accd33019d766fac516fea4c": "2^{-n(H(Y) + \\varepsilon)} \\le p(Y_1^n) \\le 2^{-n(H(Y)-\\varepsilon)}", "70cac0c805bfa77eaf79f9ee98164552": "\\bar{C}\\neq\\bar{\\bar{C}}", "70cbbc0dc0ab17575bb0c22d8b44ada0": "G_{\\mathrm{Newton}}", "70cbee1b3f9a43a54cad495c7a8ce5d2": "R'(W) = 0", "70cc0d9f129abf48559ff3b389f63670": "\\pm e^i \\wedge e^j", "70cc76b090231ba7b1e33654e30c9e75": " b \\triangleleft a = a^{-1} b a", "70cc8bfcbffdcf330ee44c2ae7b1edae": " \\mathbf{1}_{\\bigcup_{k} A_k}= 1 - \\sum_{F \\subseteq \\{1, 2, \\dotsc, n\\}} (-1)^{|F|} \\mathbf{1}_{\\bigcap_F A_k} = \\sum_{\\emptyset \\neq F \\subseteq \\{1, 2, \\dotsc, n\\}} (-1)^{|F|+1} \\mathbf{1}_{\\bigcap_F A_k} ", "70ccdcb6461b88a432561354b57d44b9": "H_C(s)= {k_C \\cdot s^2\\over(s+129.4)^2\\quad (s+76655)^2}", "70cd49fe230770a0fe0307a3f8d3c6ff": "A_4 \\times Z_5,", "70cd66f271d04ef0bbd2920456fedc11": "\n+V=\\left(\\frac{\\partial H}{\\partial p}\\right)_{S,\\{N_i\\}}\n =\\left(\\frac{\\partial G}{\\partial p}\\right)_{T,\\{N_i\\}}\n", "70cd7327f5a90ae1d3afdd5ae31bdca7": " 1\\leq i \\leq k", "70cda0aeb46c195b32239355439ad1a7": "Q(x,y) = [(y - \\beta) - 4(x - \\alpha)^2)] [(y - \\beta) + 6(x - \\alpha)^2)]", "70cdc596d18efad4f585139cb150cbde": "\\mathbb S^1", "70ce2a452606925aea40df28ba0c5905": "T_{0} \\!", "70ce49bca3dc68218fa6f8fe36e7ba5b": "{\\frac{P_y}{P_x}=\\frac{2\\gamma}{\\gamma+1}M^2_x-{\\frac{\\gamma-1}{\\gamma+1}}}", "70ce624909cc34e888baef352c06d9b1": "ta:S \\rightarrow \\mathbb{T}^\\infty", "70ce788afdaa4dca5239c0cb0605ca1e": " S = \\int {1\\over 2} \\partial_\\mu \\phi \\partial^\\mu \\phi d^dx \\,.", "70cec74d5c322a59f5d401e411fe4dea": "\\sum_{n=0}^\\infty a_n(x-c)^n.", "70cec80aa348131663f9375eb38fbb7f": "\\Gamma(\\tfrac52)\\,", "70cec9de1952ae5fba574d3ec6edcb39": "\n\\int (d+e\\,x)^m\\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{e(m+2 p+2)(d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{(p+1)(2p+1)(2 c\\,d-b\\,e)}\\,+\\,\n \\frac{(d+e\\,x)^{m+1}(b+2 c\\,x) \\left(a+b\\,x+c\\,x^2\\right)^p}{(2p+1)(2 c\\,d-b\\,e)}\\,+\\,\n \\frac{e^2m(m+2 p+2)}{(p+1)(2p+1)(2 c\\,d-b\\,e)} \\int (d+e\\,x)^{m-1} \\left(a+b\\,x+c\\,x^2\\right)^{p+1}dx\n", "70cee4f0fb1dd72a88b46e4804996176": "v_{3}= \\begin{pmatrix}0.0049651 \\\\ -0.7075770 \\\\ 0.7066188 \\\\\\end{pmatrix}", "70cee737b3962966ec8f15ccdfadb899": "\\kappa \\neq \\omega", "70cefb77e90d51d654c544a3ff79b49f": "0=f^k\\left(y\\right)=x", "70cf02e2cbad7199b62d277501a5e99d": "L/qL", "70cf0e0b697ac583ba0322006cc5fc72": "i_\\gamma(t)=0", "70cf14c5e071285a96624ee40cccecec": "\\mathbf n^e=\\left[ \\begin{matrix}\n \\cos (\\mathrm{latitude})\\cos (\\mathrm{longitude}) \\\\\n \\cos (\\mathrm{latitude})\\sin (\\mathrm{longitude}) \\\\\n \\sin (\\mathrm{latitude}) \\\\\n\\end{matrix} \\right]", "70cf25dc0a167e11e0c5302821e9c4b3": "\\widehat{O1QP2}+\\widehat{O2QP2}=\\pi", "70cf3688a1b173a56b12c156b61aaca4": " V_1, V_2, I_1, I_2, x \\,", "70cf7fe083f7bb7997fcbf8967897edf": " a=a(x,y), \\ \\ b=c(x,y), \\ \\ c=c(x,y), ", "70cfd11f8ffe90f449ae2729c1ba246e": "a^{n - 1}\\equiv 1 \\pmod{n}\\ \\text{ if and only if } \\text{ ord}(a)|(n-1).", "70cfd1f76aa73cb137e0d184172ff13f": "(2,\\mathbb{Z})", "70d013ba0536e7dee703627d9ca8cc77": " D_2\\frac{dC_2}{dx}=\\frac{1}{N_A}\\frac{dF_2}{dx}B_2C_2", "70d01694cec886762eb0088bd735c9fa": "|M|^2 \\leq \\mbox{Im}(M)", "70d07522c96a783047b6ddd70e0b134a": " C_p = ", "70d0d89e90b15db57c8b5f8ecca5a82f": " \\frac {N}{t} = \\frac {I}{e}", "70d10c10bf36441292c7a78acf75a353": "red(v) := d^{-1}(v - f_v)", "70d123d8660eae0737d64442211f6ff4": "ln(2) = 0.693", "70d126017b23c77426b1cf515b85fb62": "u^{0}_{z} = \\frac{e_i - e_s}{e_s} \\chi N_w \\bar{r_w} \\,", "70d13173652f5f0afff6068ad394cdaf": " t \\leq 0", "70d15960e4ca7b30e5a09e4cb7bf69b2": "\\{l_a, n_a, m_a, \\bar m_a\\}", "70d1849fd969b9396f33ec48825b8461": "x^2 + px = q", "70d1d1883f1e84f5cb4426d07921e155": "\\begin{align}\n\\sigma_{rr}(R) + P_B - \\frac{2S}{R} & = -P(R) + \\left.2\\mu_L\\frac{\\partial u}{\\partial r}\\right|_{r=R} + P_B - \\frac{2S}{R} \\\\\n& = -P(R) + 2\\mu_L\\frac{\\partial}{\\partial r}\\left( \\frac{R^2}{r^2}\\frac{dR}{dt} \\right)_{r=R} + P_B - \\frac{2S}{R} \\\\\n& = -P(R) - \\frac{4\\mu_L}{R}\\frac{dR}{dt} + P_B - \\frac{2S}{R} \\\\\n\\end{align}", "70d216ecfc53462ac83bbeea93cc9862": "P(T,\\Delta)=\\cos^2\\left(\\frac{\\Delta T}{2}\\right)=\\cos^2\\left(\\frac{\\Delta L}{2v}\\right)", "70d227fc5642e099c25bf49c1efdb133": " B_{noise} = \\frac{1}{|H(f)|^2_{max}} \\int_0^{\\infty} |H(f)|^2 df.", "70d232c1f62d9f165efe7c9b3d629642": "H_0,", "70d249e849b17ceca9161278396984ee": "k_{\\perp} \\rho_{i} \\sim 1", "70d265c14b1dc6415f48ab7676012eb6": "\\pi = s_o, s_1 \\dots", "70d2e5cc3248b5b532cff6c5a1355d61": "A_i \\subseteq A_{i+1}", "70d2eee7d0e75a77d0d77d7968980d8b": "c_k\\epsilon^{1/k}\\,", "70d3178ae9130972b8d05016a364b11c": "\\sum_{i=0}^\\infty a_i", "70d33ac0126b8cccd0fe2d1435fcf77b": "\\dot{x}(t) = A(t)x(t) + B(t)u(t)", "70d355de0f1bb91b7aeb47e8804a3b13": "\\mathbb{E}\\left[g(X_1,\\dots,X_d)\\right]\\approx \\frac{1}{n}\\sum_{k=1}^n g(X_1^k,\\dots,X_d^k)", "70d3745b6bda0e19bbb63a5021e79a1b": " \\int f \\, \\mathrm{d} \\mu = \\sup \\left\\{\\int g \\, \\mathrm{d} \\mu \\,|\\, g \\in SF, \\ g\\leq f \\right\\}, ", "70d3cf5218c32eb684870d052e7b6d75": "h(z_0)", "70d402347e5df39dc9f4a4b2d1422b86": "n = p+q", "70d409161a14f2968565a532251221b7": "\n \\boldsymbol{w} = \\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T + \\frac{1}{2}~\\boldsymbol{R}\\cdot(\\dot{\\boldsymbol{U}}\\cdot\\boldsymbol{U}^{-1} - \n \\boldsymbol{U}^{-1}\\cdot\\dot{\\boldsymbol{U}})\\cdot\\boldsymbol{R}^T\n", "70d419735a099c61a939c56509399429": " (40, 46);", "70d442ff0e7292a3a5d840a5b8f49b82": "\\frac{\\partial s}{\\partial t} + \\frac{\\partial }{\\partial x} u = 0", "70d44fc2426e9ca6fcee5b7ab31938ae": "t(1,n) \\leq \\lceil\\log{n}\\rceil", "70d44fc5219d0b779092d9176ae84fe8": "\\Phi(\\epsilon_i) = \n\\begin{cases} \n e^{\\beta(\\epsilon_i-\\mu)}, & \\mbox{for particles obeying Maxwell-Boltzmann statistics } \\\\\n e^{\\beta(\\epsilon_i-\\mu)}-1, & \\mbox{for particles obeying Bose-Einstein statistics}\\\\ \n e^{\\beta(\\epsilon_i-\\mu)}+1, & \\mbox{for particles obeying Fermi-Dirac statistics}\\\\\n\\end{cases}", "70d45b3d7acc7fbb4a2ae09dd1bc76c5": "\\operatorname{arsech}(z)", "70d483da58b45b2c614de9031ff3b67e": " \\operatorname{E}(R_a) > R_f ", "70d507a6c4a6eea637fbaebf5ea5a5c8": "L[u]=\\delta.\\,", "70d53d1e381ba9e96f2fc0e8b8d0c043": "V\\setminus S", "70d5f2fd2533851da65b4cbdb21360b1": "K_{12}, K_{21} \\ge 0", "70d655d5d52935c8a3b1410ccb0abdad": "P_A= \\rho(\\psi(g)^a) \\in \\mathbb{F}_q^m", "70d65b660124146e529feab201ed6f99": "CB^\\alpha_\\alpha dt", "70d66405361ae8eb85f26dfc6885862a": "[X+f,Y+g]=[X+f\\frac{\\partial}{\\partial\\theta},Y+g\\frac{\\partial}{\\partial\\theta}]_{Lie}", "70d668debcad264a54b6473e4a74b861": "\\sqrt[5]{2}\\,", "70d6aae7a51d21596f5a0f8422ce5c31": "\n \\boldsymbol{\\sigma} = -p^{*}~\\boldsymbol{\\mathit{1}} + 2 C_1~\\boldsymbol{B} - 2C_2~\\boldsymbol{B}^{-1}\n ", "70d6c9dc201fdcf2bdc8b64a747da798": "\\mathit{8415}\\, ", "70d6ca1534365679a475fe45fa72f5ed": "n_{\\rm max} = \\sqrt[3]{N}\\,.", "70d6d43e80e85b743e99e09b0acace0e": "X\\colon \\Omega \\to \\mathbb{R}", "70d6d79b9dc16844893be754819dc880": " \\mathbf{E} ( \\mathbf{r}, t ) = \\frac{1}{r} \\mathbf{E}_0 \\sin( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} + \\phi_0 ) ", "70d7bca33399904c7d2dfc99b169eafb": " K_\\mathrm{sat}^{(2)} = \\frac{K_\\mathrm{mineral}}{ \\left[ {\\frac{K_\\mathrm{sat}^{(1)}}{K_\\mathrm{mineral}-K_\\mathrm{sat}^{(1)}}-\\frac{K_\\mathrm{fluid}^{(1)}}{\\phi (K_\\mathrm{mineral}-K_\\mathrm{fluid}^{(1)})}+\\frac{K_\\mathrm{fluid}^{(2)}}{\\phi (K_\\mathrm{mineral}-K_\\mathrm{fluid}^{(2)})}} \\right]^{-1} + 1} ", "70d8371c70667e6541f5ddd1b70f78cb": " \\nabla_{\\mathbf v} {\\mathbf u} = \\nabla_{v^i {\\mathbf e}_i} u^j {\\mathbf e}_j = v^i \\nabla_{{\\mathbf e}_i} u^j{\\mathbf e}_j = v^i u^j \\nabla_{{\\mathbf e}_i} {\\mathbf e}_j + v^i {\\mathbf e}_j \\nabla_{{\\mathbf e}_i} u^j = v^i u^j \\Gamma^k {}_{i j}{\\mathbf e}_k+v^i{\\partial u^j\\over\\partial x^i} {\\mathbf e}_j ", "70d86b1906cfc6720ad3991351ea6719": "\\mathcal{O}(M^2)", "70d872efdd001518090c52446222bf89": "\nc_{\\mathrm{ideal}} = \\sqrt{\\gamma \\cdot {p \\over \\rho}} = \\sqrt{\\gamma \\cdot R \\cdot T \\over M}= \\sqrt{\\gamma \\cdot k \\cdot T \\over m}\\,\n", "70d8b86c06793d988f381233f32b93d4": "k = \\sqrt{\\frac{Z}{Y}} ", "70d8d704ecea8831e8ec29db3f873e8d": "R_N^k(n) = \\sum_{i=0}^n r_N^k(i)= n^{N/k}", "70d8f94427bed18f0ffe99bed9254911": "p_z=m\\dot{z}", "70d91c44bf68a6d71c82df619ba80e35": "F_i + \\mu F_n - F_w \\sin \\theta = 0 \\,", "70d9492a6b151d53f85ab493ff4c0624": "Q = \\frac{dh}{dl}KA", "70d9787bc6a802eac0e319a334c0cb39": "(i_1, i_2, \\dots, i_k).", "70d99b57a9197d0cf03397f6e5ae1a79": "\\mathrm{VO_2\\; max} = {d_{12} - 505 \\over 45}", "70d9b16237e8caaef2d86634a6bded20": "\\oint_C \\mathbf{B} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = \\mu_0 \\iint_S \\mathbf{J} \\cdot \\mathrm{d}\\mathbf{S} = \\mu_0I_\\mathrm{enc}", "70d9cd46466ef5950ac043f39127b333": "a=\\infty:", "70d9d9e8d14bdc8743ba9dede323f731": "\\text{base} \\times \\text{height} = \\pi r^2 \\cdot r = \\pi r^3", "70d9e75c10da7ef3ca000d17ec176ed1": "\\textstyle \\alpha > \\gamma > \\beta", "70d9ef5b6cf8996e4897c1e182851fcc": "\\frac{dy}{dx} = y\\quad \\Rightarrow \\quad \\frac{dy}{y} = dx \\quad\\Rightarrow\\quad \\int \\frac{1}{y}\\,dy = \\int 1\\,dx \\quad\\Rightarrow\\quad \\int \\frac{1}{y}\\,dy = x + C = \\ln y + C.", "70d9f8ca1121f7b704f4f399f77ecaf8": "\\forall j \\in S", "70d9fee89b2d20eee41e274e79600a20": "\\hat{f}_i^\\dagger \\, \\hat{f}_j^\\dagger = -\\hat{f}_j^\\dagger \\, \\hat{f}_i^\\dagger ", "70da72ebd653e07e2c4a30b0325e111e": "\\mathbf{e} = \\theta \\mathbf{\\hat{e}}", "70da99fdf6cd6b2c051b0d4cb701f788": "f_1(\\text{Aa}) = pq + qp = 2 pq = 2 f_0(\\text{A}) f_0(\\text{a})", "70dbedeefa786340ceba1645dd486cf6": "\\frac{\\Phi(x)}{x} \\to \\infty,\\quad\\mathrm{as\\ \\ }x\\to \\infty,", "70dbf4dcb6484f26df6028235597e251": "y \\in Y = \\{-1, +1\\}", "70dbf96567025389d8854f1aea4e62ad": "\\vec{n}", "70dc182a93e15ced4f6569568bc3b5c3": "\\iint\\limits_D \\, dx\\,dy", "70dc3a389a2f46e96a35875f48662c04": "(g,f)=-(-1)^{|f||g|}(f,g) ", "70dc4b127a5fa11e4677d1b3fd60366a": "U_{Cp}(t) = \\frac{1}{R_pC_p}\\exp (-t/R_pC_p)\\left(\\int\\limits_{0}^{t}U_g(t_1)\\exp (t_1/R_pC_p)dt_1 + C_{int}\\right)", "70dc7844d0c4daa579b6b569b41568a5": "\\dot\\gamma_V(0)=V", "70dcefced79c2003bf0c782f979c6ad3": "\n\\psi^\\dagger(x) |N;x_1 ... x_n\\rangle = |N+1; x_1, ...,x_n, x\\rangle\n\\,", "70dd1f82e1526d27705952955042c245": "\\begin{array}{rl}\n h=&-a_0=\n \\frac{\\frac1{(d-1)!}(1/f)^{(d-1)}(x)}{\\frac1{d!}(1/f)^{(d)}(x)}\\\\[1em]\n =&d\\,\\frac{(1/f)^{(d-1)}(x)}{(1/f)^{(d)}(x)}\n\\end{array}", "70dd32a54506f7419ea12379c1a2e70e": " y_{n}\\equiv m_{1}y_{n}^{(1)} + m_{2}y_{n}^{(2)} + \\dots + m_{r}y_{n}^{(r)} \\pmod m. ", "70dd6e68296b4e7e6595a5579394d0e3": "(X, \\le)", "70dda5964718ab1561070da649cc34d4": "\\gamma_s(h)=\\gamma(0,0+h)", "70ddd0520a851eac704c0623a67882b6": "\\mathcal{O}(h^2)", "70de6f9774035da2e29a3f30e792b9a4": " S_x(t,f) = \\int_{-\\infty}^{\\infty} x(\\tau)|f|e^{- \\pi (t- \\tau)^2 f^2} e^{-j2 \\pi f \\tau} \\, d \\tau ", "70de894c00e18c919745ce80dd309e8e": "b_{ijkl}", "70de8ef7c901ba57746895743be5d9d4": " \\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{a} \\oplus \\mathfrak{n},", "70de9612377f6870496c10d79b4a6c0c": "I(met : N^r \\cdot S \\cdot N^l) = [x] \\cdot met(x,y) \\cdot [y] : E^r \\cdot T \\cdot E^l", "70dec3df63e89bd58de3fe430f53281d": "\\tilde g\\,", "70ded21d8138a58abddbffe20df22549": "\\tau H^{n-i} M \\equiv \\mathrm{Ext}(H_{n-i-1} M; \\mathbb Z) \\equiv \\mathrm{Hom}(\\tau H_{n-i-1} M; \\mathbb Q/\\mathbb Z)", "70def96437b0bd0224878974e29864a4": "\\mathcal{E}(x_1) \\cdot \\mathcal{E}(x_2) = (g^{r_1},x_1\\cdot h^{r_1})(g^{r_2},x_2 \\cdot h^{r_2}) = (g^{r_1+r_2},(x_1\\cdot x_2) h^{r_1+r_2}) = \\mathcal{E}(x_1 \\cdot x_2)", "70df0ece98816826b30bb05e6edfb5b9": "t < 0\\,", "70df440a14517d2ef273d181c238ee6e": "K(a,n) = \\{[f]: f: K(G,n) \\to K(G',n), H_n(f) = a\\},", "70dfe8dda7c1091b09f6c74b1af9ae8f": "\\scriptstyle (n-1)", "70e01fe528e401d36e251bc8a575bde5": "\\mathrm{d}P/\\mathrm{d}T", "70e02b1d84a50d610f8e2624aa58639b": "P : C_n(X) \\rightarrow C_{n+1}(Y)", "70e050f9e3ef03fea119f1c5ce0bf897": " \\sgn(x) = \\begin{cases}\n -1, & x < 0 \\\\\n 0, & x = 0 \\\\\n +1, & x > 0 .\n\\end{cases}", "70e0560d017d888cce4dda6cab588ef3": "s_4 = E(h_4, K_4)", "70e0ba3d1f03dcc13469ba4ae436c446": "\\textstyle H_2\\left( e\\left(d_{ID}, u\\right) \\right)", "70e0bf4657fd3c0026f34e9a12a255df": " C_{\\alpha \\beta} =\\begin{bmatrix}\n C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\\\\n C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\\\\n C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & C_{44} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & C_{44} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & C_{44} \n\\end{bmatrix}.\n\\,\\!", "70e0cbcedd0731e640a6885f9e9f517e": " R(M,x) = \\frac{x^T M x}{x^T x} = \\lambda \\frac{x^Tx}{x^T x} = \\lambda.", "70e1c31f68833eac0d98b8d838c40ee2": "h: M' \\rightarrow M", "70e1db248b939a2e275badb768ec8bb0": "RP(2)", "70e1e400a6357a984ef83d6b8b7973a3": "\\langle\\mathbf{p}(t)\\rangle = q \\tau \\mathbf{E} + \\mathbf{C} e^{-t/\\tau}", "70e1f7417729f745417597fcf23278f5": "\\textstyle \\underset{i\\in e}{\\sum }w_{i}^{t}>\\theta_{t}", "70e2091afcea8cfeaf3d11d1f1a911dd": " \\lambda_{\\mathrm{e}} \\ ", "70e22fd44643604b65c54c12de801ff8": " v = \\alpha _1 b_1 + \\alpha _2 b_2 + \\cdots + \\alpha _n b_n .", "70e230a7beafe0343baea9d820884f58": "\\{ \\omega_1, \\omega_2 \\}\\,\\!", "70e2471a6fda4fab63ffdc4022a10ecb": "f(x)^0 = 1 = \\sum_{k=0}^{\\infty} M[f]_{0,k} x^k = 1+ \\sum_{k=1}^{\\infty} 0* x^k ~.", "70e297a0d8707ce7ba592004192cea9f": "\\varphi^n/\\sqrt5", "70e29ac78e4dbe7af98b12e9e0e88b29": "10^n \\equiv 1^n \\equiv 1 \\pmod{3}", "70e2c5dcd221a6947bfe81c3a3dfd547": "V_{out,2nd order} = k_{2}[A_{1}\\cos(w_{1}t) + A_{2}\\cos(w_{2}t)]^{2}", "70e2dfd8dc3481fa815f3ea5587a0dc9": "\\mathcal U= \\{U_\\alpha\\}_{\\alpha\\in\\mathcal A}", "70e31be24da2c16592cbf70f26070cd3": "{x=0}", "70e3854f875bb8fc9af46cee94910c67": " a_P T_P = a_E [ \\frac {T_E + {T_E}^0} {2}] + a_W [ \\frac {T_W + {T_W}^0} {2}] + [ {a_P}^0 - \\frac {a_E} {2} - \\frac {a_W} {2}] {T_P}^0 + b ", "70e3c2300a24d086d33978afaf57f966": "i=\\emptyset", "70e3c68102bf73f49130c16c243e30a7": "\\log |f(z)| \\le \\frac{1+|z|}{1-|z|}(7+\\max(0,\\log |f(0)|))", "70e3d2cbac9fe4f8979b9ff33613502e": "B_6(\\theta)", "70e40fd8d7b0dc07d326cd14773004b6": " \\frac{\\partial z}{\\partial x} = y(2x +y)(y+1) ", "70e4312961472ee1d488ffb85e3de67d": "\\ln \\left |\\sin x\\right | + C", "70e46ac81a049b84a602e8e55a5e326f": "\\frac{\\partial V}{\\partial T}\\ ", "70e4c7b983e3bc5849c2cfd6af4a431e": "\\vdots ", "70e4cfd067cf16576f5f819c29b9fbd3": "\n1=\\det(\\mathbf{I})=\\det(\\mathbf{R}^\\mathrm{T}\\mathbf{R}) = \\det(\\mathbf{R}^\\mathrm{T})\\det(\\mathbf{R})\n= \\det(\\mathbf{R})^2 \\quad\\Longrightarrow \\quad \\det(\\mathbf{R}) = \\pm 1.\n", "70e4d86aaa19fb96234570223bed697f": "{}\\vec r=(x,y,z)", "70e4ea75be9169f389e9be0b1cf4bfb1": "|\\psi\\rangle|\\phi\\rangle \\,,\\quad |\\psi\\rangle \\otimes |\\phi\\rangle\\,,\\quad|\\psi \\phi\\rangle\\,,\\quad|\\psi ,\\phi\\rangle\\,.", "70e50025f27f220f4c0be1feb99fe12e": "P=\\sum_{\\vec{k}, \\vec{\\ell}} c_{\\vec{k} \\vec{\\ell}} \\,\\, \\partial_{x_1}^{k_1} \\partial_{x_2}^{k_2} \\cdots \\partial_{x_n}^{k_n} x_1^{\\ell_1} x_2^{\\ell_2} \\cdots x_n^{\\ell_n}", "70e5352f992f389694d15b5eb2003852": " \\textbf{N}", "70e568ee0c196b99a46caf7b0fba0b7f": "\\hat{R}.", "70e59a996bd69a0c21878b4093375e92": "\\mathbf{x}", "70e5ac8fe939d51cf0abc18824d48937": "\\Sigma_R = B \\Sigma_L B^T", "70e5b8f06be7a32e17f2522d7cc3c90f": "d_1\\times d_2", "70e5be4a1923b1e8d9feb26977a4ac3e": "K_a^i = K_{ab} \\tilde{E}^{ai} / \\sqrt{det (q)}", "70e63033e694f80d6d762f43a33b8920": "\\log_{1/p} n\\,", "70e719b361156aef336e3dfe748d3702": "(\\phi \\rightarrow \\psi)", "70e764bb2734b442f8ac5c1ffed499c1": "f^*=f,~g^*=g\\,", "70e77cb0c86811685963b4255e542b1d": "\\|.\\|_2", "70e796ed88a1c0a46c9dd8f65774f776": "(\\Omega, \\mathcal{F}, p)", "70e7b8d328ff0ead9a482bf10033674b": "\\hat{H}_6 = 2\\mu_B \\sum_{i} \\left[ \\mathbf{H}(\\mathbf{r}_i)\\cdot\\mathbf{s}_i + \\frac{q_i}{m_ic}\\mathbf{A}(\\mathbf{r}_i)\\cdot\\mathbf{\\hat{p}}_i \\right] ", "70e7ce7e7a29d936813eb18b00c05bc2": "\\left\\langle R(t)R(t') \\right\\rangle = \\delta(t-t')", "70e7d5a73a633468b4cf4561e10d1cf7": "\nh(x)=x^{\\left\\{m\\right\\}'}Hx^{\\{m\\}}\n", "70e7e5203ab281e9853f823604bc74d1": "\\mathcal{G}_t", "70e824ed1a5359f2cab5224744d782a1": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{vercosin}(x) = -\\sin{x}", "70e88093f97d3c4bc108b27484a41db9": "\\Delta t,", "70e899339f2e6673e77812f950eb5ea9": "C(u\\otimes v)=ND(N\\otimes N)(u\\otimes v).", "70e89cda89834c2e2fae5e9c18992730": "2(\\mathbf{X}^{+})^{\\rm T}", "70e8af89b9a43512bf9546c291bf2623": "\\frac{4,200,000\\ \\mbox{MW·h}}{(365\\ \\mbox{days}) \\times (24\\ \\mbox{hours/day}) \\times (2,080\\ \\mbox{MW})}=0.23 = 23%", "70e8e0c9ade019301fe3e99a0b81fc9f": "\\begin{bmatrix} a \\\\ b \\\\ d \\end{bmatrix}", "70e8e7b87f0574f17b924b0f827e4dbb": "\\scriptstyle T^* = 0.71, \\; n^* = 0.844", "70e91d935effa7fdce6a74b33f5c371a": "\\beta_N", "70e990db4fa66ecc60293059d2618fff": "\\frac {\\operatorname{d} \\ln K} {\\operatorname{d}T} = \\frac{{\\Delta H}^{\\ominus}} {RT^2}", "70e9e4dfc151ee0a77fc135c13c2674f": " u^S = u_1 \\cdots u_n, \\quad \nu_i=a \\text{ for } i\\notin S, u_i=b \\text{ for } i\\in S. ", "70ea282e150aec0090b0c230f548c94e": " H_0^{(1)}(kr) \\simeq \\sqrt{\\frac{2}{\\pi kr}}e^{i(kr-\\pi/4)}", "70ea292dd77bc28c76894f6d1b574b91": " 2^n ", "70ea82215547e01ed209938af774c954": "n=0,1,\\ldots,N", "70eaae23e49c44f3ac2b831f3c94488e": "V_F = A \\exp(j \\omega t)\\exp(\\gamma l)\\,", "70eabcd9008740a202f6e9850b008509": "\\left(\\frac{\\partial x}{\\partial y}\\right)_z\\left(\\frac{\\partial y}{\\partial z}\\right)_x\\left(\\frac{\\partial z}{\\partial x}\\right)_y = -1", "70eb0b590ff5f9f1a6d4c45cc32aa0c1": "h^A = \\mathrm{Hom}(A,-).", "70eb1ea9aeac1bb7d8e539bdea7e8bb3": "\\ \\frac {1} {(1+j \\omega \\hat {\\tau_1 })(1 + j \\omega \\hat { \\tau_2} )} \\ . ", "70eb7cb6b9d3726a376943e87b281aeb": " w_i = \\frac {\\pi} {n+1} \\sin^2 \\left( \\frac {i} {n+1} \\pi \\right). \\,", "70eba7d44a200c5042f286d6e43e3106": " T/2", "70ec06378222af0e9af278e3be7fe08b": "(e',e'')", "70ec1031c8b5c007f4ff32e6e3136d43": " \\bold{0} = (\\lambda I - Q) \\bold{v} . \\,\\!", "70ec2be5049b28f9e0daa856c77f6c54": "f=\\log", "70ecaf4bc89f0b45d28a79d9cec01507": " \\nabla \\times \\mathbf{v} = 0, ", "70ed2e6bd238c7d29645731d6edf97bf": "\\displaystyle u_{xy}+\\alpha u_x+\\beta u_y+\\gamma u_xu_y=0", "70ed38fbf7ed86c09d00d95c9d8bbb42": "r = \\frac {a^2 + h^2}{2h}", "70ed63dbcf52bd9189ec83caeae2a20c": "\\langle x\\rangle", "70ed8383b6229e32a2ae273a7367ce12": "\\iiiint\\limits_F \\, dx\\,dy\\,dz\\,dt", "70ede12f807ce7d551cf619b247e8905": "\\!\\mathrm{I}(x,y) = x^T\n\\begin{pmatrix}\nE & F \\\\\nF & G\n\\end{pmatrix}y\n", "70ee0688bb3f007406a2508f9606c1ad": "{n,r,w}", "70ee6301d1f442d83355937e13c5ae5f": "e(m,n_1+n_2)=e(m,n_1)+e(m,n_2)", "70ee70fa5c6da9d37a1ee00785e7505b": "0.1 < \\lambda < 5", "70ee804a42455c4d809b5167015c8491": "(w+g+a-c)\\ s\\ r=m\\,", "70ee958beaa85cb8d02aa5a05360c54b": "(x,1) \\sim f(x)\\,", "70eed03fc274fc8cffcb35c85a7196d6": "c \\simeq d", "70ef0978c50ae4e10f439f164f936394": "\\mathfrak{Sin}", "70ef782fa032821696dc805f21f966f1": "G = \\frac{y_\\text{out}}{y_\\text{in}} = \\frac{ \\sum_{k=1}^N {G_k \\Delta _k} }{ \\Delta\\ } ", "70efc4cc99912e4741ef08b51863d587": "H_i (X) \\otimes H_{\\text{dR}}^{2n-i}(X) \\rightarrow \\mathbf{C}", "70efd81c0b19631c9695d2782ff8b6f7": " P=\\frac{r-q}{c} \\quad \\text{and} \\quad H=\\frac{m+q}{ac}", "70efdf2ec9b086079795c442636b55fb": "17", "70f02567c75bff6a9926409cf00d6daf": "\\mathbb{E}[2e' + s'] < D", "70f02bdddbe864357737e928a0bd9271": "T x = \\lambda x, \\; i.e. \\; (x_2, x_3, x_4, \\dots) = \\lambda (x_1, x_2, x_3, \\dots),", "70f0446feba0eaface2d16d73dc3ec6b": "\\Delta(C^*(m_1),C^*(m_2)) \\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k \\cdot T \\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k \\cdot (1-R-\\varepsilon) \\cdot N", "70f05078e049c2bf189cd9379f87f9f2": "T_{p}", "70f07b306c8e7cca3d9bb230b0817cf3": "\\sum_{j=0}^l \\sum_{k=0}^{m+(l-j)d} q_{kj} x^k y^j = 0", "70f0c5c80a60cffeab786b41c382f71f": "x=\\theta, y=(\\theta,1), z=((\\theta,1),1)", "70f0f78ddfc73a96cb18beaa5cf83e9c": "(v_0v_1), (v_1v_2), \\dots, (v_{d-1}v_d) \\,", "70f107e63b5c99f613e7effd3f2d74c1": "x,y \\in \\mathfrak g", "70f10fdcafb19d441a23f42d048de4ee": "\n F(x) = \\begin{cases}\n 2x^2 & \\text{for }0 \\le x < \\frac{1}{2} \\\\\n 1-2(1-x)^2 & \\text{for }\\frac{1}{2} \\le x \\le 1\n \\end{cases}\n", "70f171d4c9fc3603155299f74d770d00": "\\tilde{f}(\\lambda)=\\int f \\varphi_{-\\lambda} \\,dA,", "70f17703f1fbc706ddb3ec28a6b8cebe": "r \\,,", "70f1aff1058729b649d1e789ea140489": "\\scriptstyle V_R", "70f1e2c0b0bd01b9e65134891e6a7c9c": " x(t)= \\left | 2 \\left ( {t \\over a} - \\left \\lfloor {t \\over a} + {1 \\over 2} \\right \\rfloor \\right) \\right | ", "70f20cf1ae36e6a3ff48db4684376e4a": "\\frac{\\mathbf{d}\\varepsilon_1}{\\sigma'_1}=\\frac{\\mathbf{d}\\varepsilon_2}\n{\\sigma'_2}=\\frac{\\mathbf{d}\\varepsilon_3}{\\sigma'_3}=\\mathbf{d}\\lambda", "70f2385e41836d1ee48be87cb8e83e2a": "\\left(\\sigma_1, \\sigma_2, \\sigma_3 \\right)", "70f24e0a0828783be786764e89631eee": "f(z) = \\frac {1}{z + 1/2}, \\quad f:U_1 \\to T ", "70f262347c1ba9de0808aa63cac94273": "{{\\sigma_0}/{E}}\\,", "70f2647e9042fb87d077a425411c0029": "A-\\lambda B \\, ", "70f2a6cab1915baf9b75bd7aedd17e82": " Cl_{1,3}(\\mathbb{C}) = Cl_{1,3}(\\mathbb{R}) \\otimes \\mathbb{C}. ", "70f2c66824907ecd7586f03bd638700c": "\\langle K\\rangle = \\bigcup_{n \\in \\mathbb{N}} (K \\cup K^{-1})^n = G.", "70f2e6785e387a5c94c4d3cf2d892e7c": "\\sum_\\nu|\\nu\\rangle\\langle \\nu|=\\bold{\\hat 1}", "70f2faa664a522c5d53608828da94d59": "Score_{TA}=Score_{GS}*\\sqrt{T_{game} \\over T_{ach}}", "70f33d5760c18d73943c804995fac05f": "v = e^{i \\theta_2} \\sinh r \\,\\! .", "70f34688aacd8ac697ff4d4a9a1a0b08": "\\sigma=|\\Sigma|", "70f354d23075333bc12ee59cc2f09a16": "x \\mapsto x^k", "70f36bed4bee4da7867f606763deef43": " \\vec{H_k} ", "70f3a979ec6093857a9424e8c4b7b7c5": "x \\mapsto (y \\mapsto \\langle x , y\\rangle)", "70f3b52e11ef2d93cd2a5b0b33d7dc31": " \\mathbf{J}dy = dx", "70f3fbd7d233599442b992016533f2cf": " {d^2 \\mathbf{r} \\over dt^2} = - {GM \\over r^3}\\mathbf{r} ", "70f4414fdaabfd7411d35872ea58bdc6": "dp=2e^2/bv", "70f45af1ec6b6b89240a919688830e22": "\\mathbf{p}=\\nabla S", "70f4ab4d99cdceb7ee1de5a9aca7b84a": "{dy \\over dx}(1+\\tan^2 y)=1", "70f4ab9ce5039beab5b4b282f68a113d": "\\sqrt{2} \\times \\sqrt{5}", "70f4cca72c3cd797acc55ada042083d1": "I=\\int\\vec J\\cdot d\\vec A", "70f512176e345e90e89a86ba10093ef6": "a = b = c = d, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = \\zeta, cos \\alpha = -\\color{Black}\\tfrac{1}{4}", "70f5881e5621592c62c4c0f7c6d96791": " (P_k)_{k\\in K} ", "70f5cd4aeeadb9697bde849e8878ad12": "P_n(x) = \\,\\frac{1}{2^n n!} \\ \\frac{d^n}{dx^n}\\left([x^2-1]^n\\right).", "70f5ef611f7f38c0e99edf4f4df737fc": "Q(x,y) = \\pi(x) K(x,y) ", "70f613a7d9fe880c969c89ee078bfcf6": " \\, S=0 ", "70f6b972a19c7942a48ce841e9b6019f": "E_n(z) = \\begin{cases} (1 -z) & \\text{if }n=0, \\\\ (1-z)\\exp \\left( \\frac{z^1}{1}+\\frac{z^2}{2}+\\cdots+\\frac{z^n}{n} \\right) & \\text{otherwise}. \\end{cases} ", "70f6dacf742ac7468448ec21bea7a75c": " \\exists xFx \\rightarrow (\\exists x(E!Fx))", "70f7356de3a1238f3a75c585c876c870": "x^{x^{\\dots}}", "70f740c43b6879071aed0878c2a3f1da": "1/(d_i d_j)^{1/2}", "70f75af2ae6987975e7d68e9ede84a4b": "f(x; x_0,\\gamma) = \\frac{1}{\\pi\\gamma \\left[1 + \\left(\\frac{x - x_0}{\\gamma}\\right)^2\\right]} = { 1 \\over \\pi } \\left[ { \\gamma \\over (x - x_0)^2 + \\gamma^2 } \\right], ", "70f764d32c16a4f5e127d237413b5e56": "1\\rightarrow0\\quad\\mbox{at rate }1,", "70f777be584bbf977c8aa30ee4299e84": "\nBn_P(Cl_t^{\\leq}) = \\overline{P}(Cl_t^{\\leq})-\\underline{P}(Cl_t^{\\leq})\n", "70f7aee223b60b4d46e9f782923a3c3d": "(Q;\\Sigma; \\delta; q_0; Q_\\mbox{acc}; Q_\\mbox{rej})", "70f840d3e019bc61d9b36f23fa4a28dc": " Q(z)=(z-1)^d P\\left({{z+1}\\over{z-1}}\\right)\n", "70f8815538f34e0eec021c4b86458d4a": "H\\gg \\ln\\ln T", "70f89e52563e8533ab92456b68251495": " G(1+z)=(2\\pi)^{z/2} \\text{exp}\\left(- \\frac{z+z^2(1+\\gamma)}{2} \\right) \\, \\prod_{k=1}^{\\infty}\\left(1+\\frac{z}{k}\\right)^k \\text{exp}\\left(\\frac{z^2}{2k}-z\\right) ", "70f8be037e5eeee6352d49a514d4e7ee": "-\\frac{S N'(d_1) \\sigma}{2 \\sqrt{T - t}} + rKe^{-r(T - t)}N(-d_2)\\,", "70f8c2b41b4caea43b086317a5b432f5": "SO(m,\\mathbb C) ", "70f8d8651b149ca53872d63a5fe4dfd2": " P = X \\times_Z Y.\\, ", "70f92f3f6e408488cfef5ac6b843efd3": " \\displaystyle{ r^{1/N} R^{1-1/N} } ", "70f958256d15d226356a526588f3d054": "\\left\\{\\begin{array}{ll}2 m! n! & n = m\\\\ m! n! & \\text{otherwise}\\end{array}\\right.", "70fa5a9f5012422e984f3df36010b784": "D = \\{ (x,y) \\in \\mathbf{R}^2 \\ : \\ 2 \\le x \\le 4 \\ ; \\ 3 \\le y \\le 6 \\}", "70fa60ebe0de4b3cb11088487322b89d": "\\kappa \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{e}{m \\omega x_0}", "70fa747f3d95fa56056220faf1f2e3c9": " H=k\\circ h^{-1} (z) ", "70faa3ecab5e581abc9de99776081b87": "m_A(x)=I(x-\\tau_1)", "70facb5cba855273f986fb608718090f": "U \\rightarrow \\neg O", "70facf7f8384e0da0e343cee84b819fb": "(1-a)S = 4 \\epsilon \\sigma T^4", "70fad41ef5912ab1f6b9b9419109705c": " \\operatorname{build-param-lists}[f, D, V, T_2] \\and \\operatorname{build-param-lists}[p\\ p\\ f, D, V, K_2] ", "70faed9a63c3238e55b3bec8cb2dbcfa": "\\cosh c = \\cosh a\\, \\cosh b - \\sinh a \\,\\sinh b \\,\\cos \\gamma.", "70fb00cf098d58f0dcdbc64154934f3b": "4(\\lambda^2 - \\lambda\\ + 1)^2 = 2(2\\lambda^4 - 4\\lambda^3 + 6\\lambda^2 - 4\\lambda + 2)\\,", "70fb3184b9189c956a47230603802e16": " \\varphi := E^{-1} \\circ \\exp_p^{-1}: U \\rightarrow \\mathbb{R}^n ", "70fb31a598879eeada75206341a780e4": "\\begin{align}\n (x^i x^j) x^k &= x^i (x^j x^k) \\quad\\text{(power-associative property)} \\\\\n x^{m+n} &= x^m x^n \\\\\n (x^m)^n &= x^{mn}\n\\end{align}", "70fb438d3aa370bdf9840eb675c9865d": " \\mu_1' = \\kappa_1 = \\mathrm{E}[X] = K'_X(0) = A'(\\theta)\\, .", "70fb5eecacfff3b5a2e1bcccb3aaba72": "\\partial D \\in C^{1,\\alpha}", "70fb8cb71c181d6d185672f2cd10894d": "\\mathfrak{b}^{ce} \\subseteq \\mathfrak{b}", "70fbb20a82a302137015e1fca8e230d9": "\\mathrm d U_0 \\,=\\, \\delta Q\\, -\\, \\delta W\\, +\\, \\sum_{j=1}^n \\mu _j \\, \\mathrm d N_j\\, \\,\\,\\,\\,\\,\\, \\text {(suitably defined surrounding subsystems, quasi-static transfers)}\\,. \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, (3)", "70fbbfe3b807106ea3b1ba3603b7b07c": " L>0 ", "70fc5b6b528ec2e83fae781d2dfa426b": "f\\circ \\phi(T) = \\phi(U) \\circ f", "70fc710e5c17d5a9d94addd7a71293cf": "\\vartheta_{s}^{-1} (E) \\in \\mathcal{F}", "70fc94407471a8597564de8bca71b81b": "|N+Q\\rangle ", "70fcbeb796eacedc5d37b544401288d8": " \\|\\eta_n-\\eta_m\\|^2 \\le (\\eta_n,\\xi) - (\\eta_m,\\xi),", "70fcc967c92a427268d9c5c63967f1a7": "{\\tau}_F", "70fcfbd49573ced5b17e572bc127d580": "\\rho_s = \\cfrac{\\partial^2}{\\partial \\theta^2}\\cfrac{E_0(\\theta)}{N}|_{\\theta = 0} ", "70fd3f388413505934da60b43afc4088": "[0,2]", "70fd420c3aabe90c30835fd973fa1250": "||x||>\\gamma_2", "70fd54d5d2cf00fb27cdff488d36c308": "\\ k_b \\, dM = \\sum_i (E_i \\, dI_i).", "70fd737d60256868705131f66e5f2720": " \\sigma = \\begin{pmatrix} \n1 & 2 & \\cdots & n \\\\\n\\sigma_1 & \\sigma_2 & \\cdots & \\sigma_n\n \\end{pmatrix},", "70fe9d772a079349d29efa0b4d6878e9": "F_{drag}=fs=(6 \\pi \\eta a)s", "70fec6bb854296b40a5758398a540a64": "\\Phi_G = \\int_S \\mathbf{g} \\cdot \\mathrm{d}\\mathbf{A} \\,\\!", "70ff033ea9a6eabd94a5e953c11577d1": " D\\{x[n-m]\\} = x[n-m-1] = x[(n-1)-m] = D\\{x\\}[n-m]\\,", "70ff35246b7c891e55ab38c8690a7e75": "N(l)_k = N_{k+l}", "70ff446af024255cc3335ff5ef551317": "f(x) = \\sum_{n=0}^\\infty a_n (x-c)^n", "70ff48a9fbd2e892a019e4555bfe2618": "\\left(\\frac{p}{q}\\right)_3", "70ff4af639c53b88f96f332997a03fe5": "S = k \\sum_{i=1}^N \\left( p_i \\log{\\frac{1}{p_i}} \\right) = - k \\sum_{i=1}^N \\left( p_i \\log{p_i} \\right)", "70ff524831c82e03b16fd2d5cd670ecc": "rR", "70ff82f97fbab879eaf014bc18409722": "A= d^2", "70ff83728fb2701cbc52fac8df762bf1": "p-1", "70ffab6dd532a15abc8090b343dfb5eb": "x_1 0 ", "7138955ecec132095b1f4178946d05c7": "\\{2,2,3\\}.", "7138baf89aff0a284da09a28683514b5": "b \\equiv y^2 - x^3 - ax \\pmod{N}", "71390f85baadb24cff0ffd63d43e0459": "{mv^2\\over r}= \\mu_s N\\cos \\theta +N\\sin \\theta ", "71391a83d305710cae67867955d788c5": "Y_{iy}", "713932a29eda4e7770d478f194d3ac5e": "{M^2_y=\\frac{{\\frac{2}{\\gamma-1}}+M^2_x}{\\frac{2\\gamma}{\\gamma-1}M^2_x-1}}", "713935fe318d997915fb41ff28e07bb7": "\\text{for all }b \\in B\\text{ and }b' \\in B,\\text{ and for all }i,\\quad\\tilde{e}_i b = b'\\text{ if and only if }\\tilde{f}_i b' = b.", "71397116c56c939e2a4baa021c635509": "\\{U_{ij}\\}_{ij} \\in \\mathbb{C}^{nm^2 \\times nm^2} \\quad \\text{such that}\n\\quad A_i = \\sum _{i = 1} U_{ij} B_j.\n", "71399dd23c9c4663f8b2a28a11374d6a": "A(L(G))", "7139d56e75552da7ad61a7235238b61a": "\\scriptstyle(1.4\\pm1.4\\times10^{-13}", "713a20619395cb30c08b4f86a893b77c": "X_1 \\vee \\cdots \\vee X_k \\vee \\cdots \\vee X_n", "713a6fdf9b638c17aafb54018ec640fc": " S_{(3,1,0)} = e_1^2 \\, e_2 - e_2^2 - e_1 \\, e_3", "713a80fbe83412120afd2deff913f1b4": "1/2\\,", "713b0e125ac5f7b265694790dbaea7ad": "B_{2}= \\langle b_{2}^{*}, b_{2}^{*} \\rangle = \\frac{2}{3}", "713b2ce316a485c4e8a297b3711dadc1": "\\oint \\frac{\\delta Q}{T}=0", "713b90f4fcb55e263f55cbf4748446f5": "V_{n-1}(R) \\cdot R \\cdot \\int_0^1 u^{(n-1)/2}(1-u)^{-1/2}\\,du", "713bb38a659327998f74e8c7bef57649": "[1..m] = abcbdab", "713bb8147c269262d151046e04643c12": "\\frac{V_\\mathrm{out}^2/50}{V_\\mathrm{in}^2/50} = \\frac{V_\\mathrm{out}^2}{V_\\mathrm{in}^2}=\\frac{10^2}{1^2}=100\\ \\mathrm{W/W}.", "713c1b7205f8eb2352b808251101df0e": "\\lambda > 0\\,", "713cdb43026919339f3d4dc1ffcdd103": "\\sum_{i=1}^n A_i \\preceq \\sum_{i=1}^n B_i", "713cdb9d351b470fbaad7d73256d89da": "0\\le k\\le 1", "713d1c7504998b8f836c7fd8338898a6": "p_n(x+y) = \\sum_{k=0}^n {n \\choose k} p_k(x) y^{n-k}.", "713d35c4c9e1b2af696f5097c2f540ef": " \\begin{align}\n(\\mathcal{L}_X \\Lambda)^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} = X^\\gamma \\Lambda^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s , \\gamma} & - \\, X^{\\alpha_1}{}_{, \\gamma} \\Lambda^{\\gamma \\alpha_2 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_s} - \\cdots - X^{\\alpha_r}{}_{, \\gamma} \\Lambda^{\\alpha_1 \\cdots \\alpha_{r-1} \\gamma}{}_{\\beta_1 \\cdots \\beta_s} \\\\\n& + \\, X^{\\gamma}{}_{, \\beta_1} \\Lambda^{\\alpha_1 \\cdots \\alpha_r}{}_{\\gamma \\beta_2 \\cdots \\beta_s} + \\cdots + X^{\\gamma}{}_{, \\beta_s} \\Lambda^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_{s-1} \\gamma} \\\\\n& + \\, wX^{\\gamma}{}_{, \\gamma} \\Lambda^{\\alpha_1 \\cdots \\alpha_r}{}_{\\beta_1 \\cdots \\beta_{s}}\\,.\n\\end{align}", "713d48b33f61f81c7b5ef314d3789ce9": "\nS = -p_{t} t + p_{\\varphi} \\varphi + S_{r}(r) \\,\n", "713d4bb6b8318802c2be2ef03b1db061": "SNR = \\frac{E_r}{\\sigma^{2}} = \\frac{K^2 A^2 T}{\\sigma^{2}}", "713da541a189afb37da6b42a40e2a78e": "v_\\lambda(p) = \\begin{pmatrix}\nv_{-1}\\\\\nv_{+1}\n\\end{pmatrix} = \\begin{pmatrix}\n-\\lambda \\sqrt{E+\\lambda |\\vec{p}|} \\chi_{-\\lambda}(\\hat{p}) \\\\\n\\lambda \\sqrt{E-\\lambda |\\vec{p}|} \\chi_{-\\lambda}(\\hat{p})\n\\end{pmatrix} \\,", "713de2fdb0f53fe1ce68eeba056a5e56": "C\\subset\\mathbb{F}_q^n", "713dfa4ec3bac2c5adabb4f987e8fe09": "B \\otimes B' = \\{ b \\otimes_{\\Bbb Q} b' : b \\in B,\\ b' \\in B' \\}", "713e15959f39fdd0a478e4b8f1bcb966": "\n\\mathbf{v} = \\omega r(1 + a') \\hat{\\mathbf{\\theta}} + v_{\\infty}(1 - a)\\hat{\\mathbf{z}}\n", "713e8429d62ffa0cdc7c71b016990cad": " 0 > \\mathrm{NPV}_{n} > \\mathrm{NPV}_{n-1} ", "713e9bc9885edc7b2df0930f9cf3d0a5": "K(\\pi,n) \\to K(G,q)", "713eaa1ef0bf75ea8728d287ea3c29e3": "\\textbf{Q}_{k}", "713eb8db0a468aa702fb2d731d36763d": " Z_0 =0", "713ebc794565f784d6cf11e91f892dcd": "X_4", "713ed7544e11b171404e87819355fdf2": "\\frac{R_s} {R}=(1-\\cot \\theta \\tan \\alpha)\\sec \\alpha ", "713ed9fa62127099d803fcc32bac8841": "\\zeta(z)=\\exp (\\sum_{n=1}^\\infty \\left|\\textrm{Fix} (T^n)\\right| \\frac {z^n}{n}),", "713f0ea8e0a3f16cbf28b9ef8443bd4c": "\\gamma^c", "713f65f1804f775a2d3a22f1d059607e": "z=\\eta", "713f6b2ee38f2bacafeee691c382066f": "|J^{(3)}(u)| < a_3 < \\infty ", "713fd5bf68f310218bb99bb8d510637b": "u_{p+1}=u_p +K * e_p", "71401679a9e680168e6a275c0aef032d": "f(a,z)\\,", "71402aa1a127c1428be204c6bfc2b4da": "\\dot{{\\tilde{{\\mu }}}}-D\\tilde{{\\mu }}=-\\partial_{\\tilde{\\mu}} F(s,\\tilde{\\mu})", "7140c69c133acf908457a57f548b4f86": " \\eta_\\varepsilon(x) = \\varepsilon^{-1}\\max \\left (1-|\\frac{x}{\\varepsilon}|,0 \\right) ", "7140decf674473c07a1534848acb8ca7": "\\frac{\\mathbf{v \\cdot a}}{|\\mathbf{a}|}", "7140dfc9c83362df59984525399f938b": "g(\\epsilon)\\rightarrow 0", "71412044942ad501294fdc78d8dc5e80": "\\rightarrow \\infty", "7141597e135e8c0750122a88b0feaaed": "f (X \\setminus K) \\subseteq X' \\setminus K'.", "71415dcf27f9310d709615fdfa7797fe": "F(4)", "71419a3cedea0a5f3cf37f45abf65950": "Q \\equiv \\frac{1}{4} F_{ab} \\, {}^*F^{ab} =\\frac{1}{4}\\epsilon^{abcd}F_{ab}F_{cd}= \\vec{E} \\cdot \\vec{B}", "7141a04b0cc96ca6908907d0a28732bc": "\\mathbf U(\\mathbf x,t) = \\mathbf b(\\mathbf x,t)+\\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad \\text{or}\\qquad U_J = b_J + \\alpha_{Ji}x_i - X_J \\,\\!", "7141c618c337776a6cb38d325d781c31": "\\scriptstyle l", "71420de5436b4f6bf9701a014c7cb5b9": "\\alpha(i)=h(i)=w(i)", "71423cffb040f9f213afcc0607ef624c": "t_1\\le t\\le t_2", "714250a7bb37128857f750b02a960f99": "\\nabla S(x^0) = 0", "71425903ba2de415a5828540c2bb5b96": "= \\mathcal{L}_{V^{1}}du^{\\alpha} - (\\mathcal{L}_{V^{1}}u_{i}^{\\alpha})dx^{i} - u_{i}^{\\alpha}(\\mathcal{L}_{V^{1}}dx^{i}) \\,", "714274d9604b1bdf32d06a64c2103d9b": "T_{abcd}=T_{(abcd)}", "7142a4b433175fec2b31c7c3fe7c959c": "\n d\\boldsymbol{\\sigma}:\\frac{\\partial f}{\\partial \\boldsymbol{\\sigma}} < 0 \\,.\n ", "7142b803bab844c22d62f275b227e77b": "{\\mathrm{h}} \\ = \\frac{k}{L} \\left(0.68 + \\frac{0.67 \\mathrm{Ra}_L^{1/4}}{\\left(1 + (0.492/\\mathrm{Pr})^{9/16}\\right)^{4/9}}\\right) \\, \\quad \\mathrm{Ra}_L \\le 10^9 ", "7142c8040fcc55d0d96fc34b710d9b42": "\\operatorname{Categorical}(\\boldsymbol\\phi)", "7143496033a43f1745abbf744576ff18": "b^2>0", "71435475255ca8ac683213eb2c439a8e": "\\varphi = \\frac{1 + \\sqrt{5}}{2}.", "71437a22c698c4b7a70d1be46a27eea2": "M(n)/n < 2/5", "714389db96a7feab4440529d702c6a50": "U_{out}", "7143ddb660a3ea1d2c48275b1361711d": "C(f) = 1/(F(f) + k)", "714461fb16edb38961d223781e73ff16": "2.5GeV g^{-1} cm^{2}", "714467cb2dda4df59db6b9cb78851449": "\\Delta_r G^\\ominus", "714483e47a1d5aee6b9bc7a81968f0e8": "H^2(a) = \\left(\\frac{\\dot{a}}{a}\\right)^2 = H_0^2\\left [ \\Omega_m a^{-3} + \\Omega_r a^{-4} + \\Omega_k a^{-2} + \\Omega_\\Lambda a^{-3(1+w)} \\right ]", "7144c87dedb68aa7dea7180bf07e3df2": "\\frac{\\pi}{4} = 12 \\arctan\\frac{1}{49} + 32 \\arctan\\frac{1}{57} - 5 \\arctan\\frac{1}{239} + 12 \\arctan\\frac{1}{110443}\\!", "7144ca4ac93f658e69047658ceff269b": "\n\\Pr \\{X_{ni}=x\\} =\\frac{\\exp{{\\sum_{k=0}^x (\\beta_n} - {\\tau_{ki}})}}{\\sum_{x=0}^m \\exp{{\\sum_{k=0}^x (\\beta_n} - {\\tau_{ki}})}} \n", "7144f5aeae71ba29a7a4be4ba257ffaf": " r^{-1}~\\cos\\theta \\,", "71456a23a7d91bd2dd9ac181eafafeb6": "\\frac{d\\rho}{dt} + \\rho\\nabla\\cdot v = 0", "71457dd2c512c5fd34546115c352ae44": "(J = 0)", "7145cfe085b17a7990535a310565623c": "(\\overline{A} \\vee \\overline{B} \\vee \\overline{C}) \\wedge (A \\vee B \\vee \\overline{C}) \\wedge (A \\vee \\overline{B} \\vee C) \\wedge (\\overline{A} \\vee B \\vee C)", "7145e1e955837f91baf744dff551d00b": "G_0(k,i\\omega_n)", "71466439fcfb2cea04fcd9a69f545fea": "b_{15}", "71468e21d40bba4c2a4272a9d15a91fb": "+\\lambda\\sum_{m\\neq n}\\frac{\\langle m|V|n\\rangle}{E_n-E_m}|m\\rangle\\langle n|", "714691c6571f72a37151511649207ce7": "\\begin{bmatrix}\n0 & 2 & -1 \\\\\n-2 & 0 & -4 \\\\\n1 & 4 & 0\\end{bmatrix}.", "7146fb124d6a48687ad0b58b46e277c3": "f(P,w) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ w\\neq1\\ \\mbox{in}\\ G \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ w=1\\ \\mbox{in}\\ G\n\\end{matrix}\\right.", "7147037d5937f68e49f938364c6b80af": "P(d)=\\log_{10}(d+1)-\\log_{10}(d)=\\log_{10} \\left(\\frac{d+1}{d}\\right)=\\log_{10} \\left(1+\\frac{1}{d}\\right).", "7147976de86119e926bc9ceef802a462": "J_- |j,m\\rangle = \\hbar \\sqrt{j(j+1) - m(m-1)} |j,m-1\\rangle", "714807918c720a436c6873f631cf7114": "e_1, e_2,\\ldots,e_{n-1}", "71481d1fe3aa313481a5e3eddec13fa1": "g_{\\mu \\nu} = e_\\mu^I e_\\nu^J \\eta_{IJ}", "714877e884aa9b43467904211a0be030": "\n\\begin{array}{ccl}\n{\\rm var}(\\bar{D}) &=& {\\rm var}(\\bar{Y}_2-\\bar{Y}_1)\\\\\n &=& {\\rm var}(\\bar{Y}_2) + {\\rm var}(\\bar{Y}_1) - 2{\\rm cov}(\\bar{Y}_1,\\bar{Y}_2)\\\\\n&=& \\sigma_1^2/n + \\sigma_2^2/n - 2\\sigma_1\\sigma_2{\\rm corr}(Y_{i1}, Y_{i2})/n,\n\\end{array}\n", "7149585a54000b2cbcd8817aac728b35": "\\displaystyle{ \\|S_Z\\|<1.}", "71496a5d11e6af77494e3e1d81f4cc21": "\\mathrm{V\\ m^{-1}=C^{-1}N=kg\\ A^{-1}m\\ s^{-3}}", "71497fffdad6a49e3f728bfd8f9271e9": "\\frac{\\left( x-h \\right)^2}{a^2} - \\frac{\\left( y-k \\right)^2}{b^2} = 1.", "7149c1658ae8121963e3b500d4aee2e2": " m \\in M ", "714a43baa551f76372cbaa34eb700c90": " A = A(h)+ a_0h^{k_0} + O(h^{k_1}). \\,\\!", "714a55c98b8af29226d96fb32893dfd7": " u(c) = \\frac{(c-c_s)^{1-R}}{1-R}", "714a88ea312be48570684128af3236be": "f = \\sum_{n\\in\\Z} a_n X^n", "714a9e83a87f0e0a640ae021bceecb0e": "1 \\rightarrow 3;", "714ac5c4f2b1958b0d0c770763ffd279": "\\displaystyle \\mu^* \\big( f(\\Omega) \\big) = 1, ", "714adbd104e55a738f476c0d27b1f34e": "\\rho_\\infty", "714adc48ba2ba5b37fa90dee481809d5": "\\Gamma, e", "714b045dba2f570dc022f27aeeb822b8": " (\\frac{N}{N_0}) \\!", "714b2daaba34aa1f2823a558067db0d8": "a^2 - 61b^2 = 1", "714b76530ea64e43f9d1a106631be569": "L^{p_0} \\cap L^{p_1}", "714bd7b204a2012205ca80d3f58b807e": " y_c = \\sum_{j=1}^n C_j e^{\\alpha_j x} \\,\\!", "714c1d5f1c5563b3e10a44bc6efbcfcc": "{\\sum_{w\\in W} (-1)^{\\ell(w)}w(e^{\\lambda+\\rho}S) \\over e^{\\rho}\\prod_{\\alpha \\in \\Delta^{+}}(1-e^{-\\alpha})}.", "714c1f6efa8623dd0fe8dd9ced1cdfd1": "\\mathbf{r} =", "714c24184abac44a1fcaa8013998cbde": " F(x) = 10^{mx + b} = (10^{mx})(10^b). ", "714c528d2dbc07fbc38fe2856c076f58": "\\mathbf{Y}(s) = C((s\\mathbf{I} - A)^{-1}B\\mathbf{U}(s)) + D\\mathbf{U}(s). \\,", "714c81b3ee7bef79ff3e8a980ae2225c": "x_t=x_0\\cos\\omega t\\,\\!,", "714c9537ec43b8d9283e6e50a3ad5568": "k> \\frac{h \\sqrt{2}}{d}:", "714cd2fd86447a75337fb43f64d93077": "h^*", "714cda142fd5ebad4c259062a196bed5": "S = k_{B} \\ln \\Omega,\\!", "714d41ca2d905bc5aa667cd5368c1fff": "\\begin{align}\n\\Delta\\theta \n &= \\left|\\frac{1}{2}(60 \\times 2 - 11 \\times 20)\\right|\\\\\n &= \\left|\\frac{1}{2}(120 - 220)\\right|\\\\\n &= 50\n\\end{align}", "714d7d148008c9ccd1b05e737c374aa0": "\\frac{\\pi}{2} = \\frac21\\cdot\\frac23\\cdot\\frac43\\cdot\\frac45\\cdot\\frac65\\cdot\\frac67\\cdots", "714df09fa7596ef12852def2cc386eac": " c \\cdot (x + iy) = (c\\cdot x) + i (c \\cdot y)", "714df9735545539cc696627fb86969c4": "y\\in V(S)", "714e82e24e7704b926b01016d2533d9d": "e = E(s)", "714ea5486f7e9cb9a30e33a04f753f01": "[x,y,z] = (xy)z - x(yz)", "714ed93b865af8d8f5eb9b4c530e88c6": "\\lim_{y \\to c} f(y) = f(c) ", "714edd8efecd8ad4d979526a3886076c": "\n\\hat{\\gamma}_{ij} = Y_{ij} - (\\hat{\\mu} + \\hat{\\alpha}_i + \\hat{\\beta}_j),\n", "714ee8b0d6fb8369b97522832d9088ce": "\\rho(x)=\\underset{\\varepsilon\\rightarrow 0^+}{\\text{lim}}\\frac{S_{\\rho}(x-i\\varepsilon)-S_{\\rho}(x+i\\varepsilon)}{2i\\pi}.", "714f54af2f1464599a22560c68822b85": "F(x) = x \\int_x^\\infty K_{\\frac{5}{3}}(t)\\,dt", "714f98ac9e715f2b94af7beddf62fb55": "F(r) = \\frac{Gm \\rho}{8r^3} \\int_{R-r}^{R+r} { \\frac{(r^2+s^2-R^2)\\sqrt{2(r^2R^2+r^2s^2+R^2s^2)-s^4-r^4-R^4}}{s^2} } \\, ds", "7150f5f4966137a48a01a1ec211e4916": "\\rho\\frac{\\partial^2u_i}{\\partial t^2} = \\partial_j\\tau_{ij}", "7151555f777b0f0cb2f1d57cdecfefa3": "h(x) - h(y)", "7151cf013df88ef3d70a9d1f7f1f2a63": "\\Delta\\Pi", "71529e8aa9cf983a95cd0bae4fc4b027": "\\theta(a) = M\\cap C_G(a)", "7152b3ed3ecb85a18f81811cebcb6446": " \\lambda_1 = \\lambda_2 = -\\frac{b}{2 a}.", "7152f82449d4a12ac8cf4601c9128ea2": "E_{tgu} = 0.5 \\cdot [\\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} { 1000 } ]^2 / m_{gu}", "7153074f8eca5f8e185db130f414c020": "B = \\begin{cases} B' &\\text{ if }m\\text{ divides }\\deg(f), \\\\ B'\\cup\\{\\infty\\} &\\text{ otherwise.}\\end{cases}", "7153688acf636d2ea37689be8f714d2b": "(r'(s),\\ r\\theta'(s)) = (\\cos \\psi,\\ \\sin \\psi)", "7153a0e3d90cff1d7ff66aa5946f2e12": "ab'c", "7153a28856b2a4b29a862712777e0724": "A\\subseteq A\\oplus B", "7153b2752070ad4b4e22dba6327b6a1b": "H^i(X, \\mathcal F \\otimes \\mathcal L^{\\otimes m}), \\ i \\geq 1", "7153bbc7346b5d449508d55638103e46": " z = \\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix}.", "71544f398d604bbf7d689e206d6230e7": "\\pi (\\frac{9}{2})^2 \\approx 8^2", "7154553abc1f0c4d6eafd87619071411": "\\alpha=\\frac{\\Delta\\Omega}{\\Omega_{0}}", "715497828cdcae54018bd1ffeaf940c3": "\\mathbf{x}(i)", "71549baaf4143e36b6e2b1eab2887b31": " \\frac{dS(t)}{S(t)} = r\\ dt + \\sigma dW(t) ", "7154d9f3dd1ae6f4aee4f2e747d250b6": "\\psi_{+}(\\mathbf{r})", "7154ddd44c78f0cb460857aca624cee4": "Z_{21}={v_2\\over i_1}", "7155e5564c7946cc0b8065f456f9c8c5": "\\Delta R", "7155f4dceffce72ab442405317baa81e": "U_\\epsilon X\\subset E", "7156879cb4c18851041138d528877dc8": "b=\\sum b_n, \\qquad b_n =(f - \\mathbf{Av}_{J_n}(f))\\chi_{J_n}.", "7156decdd68d98a2ecb693ddf1ede6a5": " \nG^3-PG^2+QG-R=0\n", "71571e85b7b57f1b0891e3896c3d9aa4": "{\\Delta p}_D \\,,", "71574b516655038ba789aa2a7272b98e": "\\aleph_{\\beta} < \\operatorname{cf} (\\aleph_{\\alpha})", "715797a0f57349d1f625d1e19d832b1b": "\\{\\ |\\ \\} \\!\\,", "71579dda79b5aa5bca44bfd374092db9": " v = \\frac{D^2*\\Delta P}{32*\\mu *L} ", "7157ab0b4bbb1b0411cefcce73c5ca35": "c_b", "7157e4051fa2400cd90679664618e419": "M\\left[\\mathbf{q}(t),\\mathbf{\\dot{q}}(t),t\\right]=0,\\quad \\tilde{M}\\left[\\mathbf{p}(t),\\mathbf{\\dot{p}}(t),t\\right]=0", "71580b3d74f3cae7ff53dfb9b423cc96": "\\phi: G\\to \\mathrm{GL}(V)\\,", "7158440ff01f91d3383cf5a629d616fe": "T_E=\\varepsilon / k", "71585760dd086ff8464884ff8101ee80": "P(recalling~m_{ab})~=~\\frac{similarity(a,m_{ab})}{SS(a,M)+error}", "7158b275e0a4cce68017bf7426e8a825": "L_z|E_n, l, m\\rangle=m\\hbar|E_n, l, m\\rangle", "7158d00171210f6ee2e9e323f4e650bd": "a_{i1},a_{i2}", "71592bebe39c7b32a545abb38ff838ef": "(a \\rightarrow b) \\rightarrow ((b \\rightarrow c) \\rightarrow (a \\rightarrow c))", "7159d3caac5992999fba4d35040da9b0": "j = \\min \\left\\{ j' \\in \\{1,\\dots,k\\} : \\sum_{i=1}^{j'} p_i - X \\geq 0 \\right\\}.", "7159fe7eafa4356d26ea4d4a2d990137": "0\\leq y", "715a4803cb26d53823389f46a7994b7f": "N(0^*,n) = \\{ (x,y) \\in {\\bold R}^2 : x^2 + y^2 < 1/n^2, \\ y < 0\\} \\cup \\{0^*\\} . ", "715a56507f69be30ee828b50402a7804": "\\mathrm{R}_{i+1} \\,", "715a9667e19ebd2f95c09335df94c8d2": " \\alpha(t) = \\frac{ 14 K_1 \\text{k}_{3(2)} [63 \\ {_2^0}S + 31 \\ {_2^1}S] }{ 31 [29 K_2 \\text{k}_{3(1)} \\ {_2^0}S + 14 K_1 \\text{k}_{3(2)} \\ {_2^0}S ] }", "715aadf4281319fa7905217b7f72e039": "\\mathrm{ad}\\colon \\mathfrak g \\to \\mathrm{Der}(\\mathfrak g).", "715ac82c6c50a9ddbb8ab4f4efcd3557": "\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} + m~\\cfrac{\\partial^2 w}{\\partial t^2} = 0\n ", "715ad6dedd6ff8cb4df12691557f6dc2": "[-U,0]\\cup[U,2U]", "715ae5b25a3e217a8ca8ad132a0f59d4": " \\mathcal{L} \\, = \\, - \\frac{1}{4 \\mu_0} F^{\\alpha \\beta} F_{\\alpha \\beta} - A_{\\alpha} J^{\\alpha}_{\\text{free}} + \\frac12 F_{\\alpha \\beta} \\mathcal{M}^{\\alpha \\beta} \\,.", "715b6e0fe24563ec87266258ba5bf716": "\\mathrm{root} \\simeq \\frac{a^2 + x}{2a}\\,\\!", "715bdf4a9cffc168d5eb42d13b93a57d": "H( K \\oplus L , X \\sqcup Y)", "715c272827423616374ffeca33e3180e": " \\ln \\Omega = - \\Omega.\\,", "715c2d51c5cc9ce1140823b536612542": " \\frac{\\delta J}{\\delta f(x)} = \\frac{\\part L}{\\part f} -\\frac{d}{dx} \\frac{\\part L}{\\part f'} \\, . ", "715ca05badf978701f41bee9b3e2d754": "\\,V(\\vec{r})", "715d22af89c6424c2b7950b0d889c06f": "\\begin{alignat}{3} F = \\frac{1}{|\\nabla \\theta|}\\cdot \\frac{\\partial \\theta}{\\partial x}\\left \\{ \\frac{1}{C_p} \\left ( \\frac{p_\\circ}{p} \\right )^\\kappa \\left [ \\frac{\\partial}{\\partial x} \\left (\\frac{dQ}{dt} \\right ) \\right ] - \\left ( \\frac{\\partial u}{\\partial x} \\frac{\\partial \\theta}{\\partial x} \\right ) - \\left ( \\frac{\\partial v}{\\partial x} \\frac{\\partial \\theta}{\\partial y} \\right ) - \\left ( \\frac{\\partial w}{\\partial x} \\frac{\\partial \\theta}{\\partial z} \\right ) \\right \\} \\\\\n+ \\frac{\\partial \\theta}{\\partial y}\\left \\{ \\frac{1}{C_p} \\left ( \\frac{p_\\circ}{p} \\right )^\\kappa \\left [ \\frac{\\partial}{\\partial y} \\left (\\frac{dQ}{dt} \\right ) \\right ] - \\left ( \\frac{\\partial u}{\\partial y} \\frac{\\partial \\theta}{\\partial x} \\right ) - \\left ( \\frac{\\partial v}{\\partial y} \\frac{\\partial \\theta}{\\partial y} \\right ) - \\left ( \\frac{\\partial w}{\\partial y} \\frac{\\partial \\theta}{\\partial z} \\right ) \\right \\} \\\\\n+ \\frac{\\partial \\theta}{\\partial z}\\left \\{ \\frac{p_\\circ^\\kappa}{C_p} \\left [ \\frac{\\partial}{\\partial z} \\left (p^{-\\kappa} \\frac{dQ}{dt} \\right ) \\right ] - \\left ( \\frac{\\partial u}{\\partial z} \\frac{\\partial \\theta}{\\partial x} \\right ) - \\left ( \\frac{\\partial v}{\\partial z} \\frac{\\partial \\theta}{\\partial y} \\right ) - \\left ( \\frac{\\partial w}{\\partial z} \\frac{\\partial \\theta}{\\partial z} \\right ) \\right \\}\\end{alignat} ", "715d8d0ac95734f35016a4205956fee9": "A = \\left\\| \\mathbf{a} \\times \\mathbf{b} \\right\\| = \\left\\| \\mathbf{a} \\right\\| \\left\\| \\mathbf{b} \\right\\| \\sin \\theta. \\,\\!", "715d9720dd2ee3ebcfff6e164d4112e6": "\\frac{\\left(\\ln(N\\sqrt{30})\\right)^3}{6\\ln 2 \\ln 3 \\ln 5}+O(\\log N),", "715e16f5117f226e20f6e56898df5055": "R_G\\,", "715e524a31bb7f940fd4b85a8d68bc6d": "\\displaystyle{[D,v_n]=-nv_n.}", "715e548241eb50ae3a7d5945078554c4": "\\tfrac{mX}{n(1-X)} \\sim F(n,m)", "715e596a729702c5e825c41fe18a130f": "\\langle \\overline\\psi\\psi\\rangle \\simeq (-0.23)^3 {\\rm\\ GeV}^3", "715e804ef30a39dcc7ebfe96e68bf3df": "\\hat{\\eta}_i", "715eb97096073d45b3259829f7a717c3": "A = \\begin{pmatrix}\na & b \\\\\nc & d \\\\\n\\end{pmatrix}", "715eec9fc248614c6e95c75493c5a72a": "\\alpha_1 \\wedge \\alpha_2", "715f07fff78505147bec746b62cb0293": " (\\partial_t^2-v^2 \\partial_x^2 ) \\Psi(x,t) = 0 .", "715f23cf517d78b5581c365022115961": "V(\\mathbf{x})", "715f248b4f0037ee7cb5b5460c53bdd3": "\\textstyle w_{i}=1/N", "715f3e3406245670c52d566390672e81": "y_{11}-y_{12}", "715f454a697c1f0fd12d1dd514578cf9": " m c^2 = h \\nu ", "715f735aaaf806d988447c389423b52c": "\\ \\frac{\\operatorname{d}E'}{\\operatorname{d}y'}=1 - \\frac {1}{y'^3} = 0 ", "715faadd15583e2753d06f1fc5f0161a": "\\sigma^2 > 0.\\,", "715fb9848106ae2f3878c965f6664b57": "r'(A) = r(A \\cup T) - r(T).", "715fe18e9c8dd61ae9e5a658c65ff7d8": "\\int_{\\Lambda^n}\\frac{\\partial f}{\\partial\\theta_{i}}\\,\\mathrm{d}\\theta=0,\\ i=1,\\dots,n", "716008c7719f258fd4eaad52fdb5d2a1": "E(t-\\tau)", "71601416118f47dfced2a25e117a958c": "b \\propto Q^{0.5}", "71605b15fd8c38f235064fa7ce8be9d0": "\\rho(\\mathfrak{H}(G))V=V", "7160cc013902a64d84d6f4a20333b151": "1/2^{O(T^2 \\log T)}", "71611f6543bfedae1cb92fa2aaec04d4": "y=\\Phi^n\\left(x\\right)", "71616a1eb189735acf47c63d59c1041c": " \\csc \\theta = \\frac {1}{\\sin \\theta}", "7161703ad5ef58e56697d13e023530ea": "\\frac{\\mbox{Price per Earnings}}{\\mbox{Annual EPS Growth}}", "716171cd00237fe88707ac46e0483abc": "\\mathbf{R}^{*}", "71617323f98f42a5afba5e49fbc89ed6": " \\lambda\\, I - T ", "716190b0123987491d207eb63cf14bc0": "\\textstyle{1+\\frac{\\log 2}{\\log 3}}", "7161930585150034f09e4490242bcac6": "\\omega_j^k = \\frac{ {\\displaystyle \\sum_i x_i \\phi_{ij} } }{ {\\displaystyle\\sum_i \\phi_{ij} } }", "7161b79be07f19d300e9581c0bc916e1": " \\forall A \\in \\mathcal{A}: \\Pr[A] \\leq p ", "7161d1a0ec6429404c6a8b9d8370eddf": "\n \\begin{align}\n \\operatorname{E}[S^2]\n &= \\operatorname{E}\\bigg[ \\frac{1}{n}\\sum_{i=1}^n (X_i-\\overline{X})^2 \\bigg]\n < \\operatorname{E}\\bigg[ \\frac{1}{n}\\sum_{i=1}^n (X_i-\\mu)^2 \\bigg] = \\sigma^2.\n \\end{align}\n ", "7161d716efd39912e36d999773c98564": " R(1) = \\frac{1}{2\\pi} e^{i\\,\\omega_0} ", "7161faa22a9e597e25f96eea1cfbbbe7": "T_t S_{(\\Delta x, \\Delta_y)} f = S_{(\\Delta x, \\Delta_y)} T_t f", "71620d7a26efdd23cc5b2139d15a09c5": " \\hat{x}_T(\\omega)", "716222a46de0df42d9029c20f4ce7bf1": "f^k\\left(x\\right)\\in\\mathrm{im}\\left(f^k\\right)=\\mathrm{im}\\left(f^{2k}\\right)", "71625772e3167f2c0427d39650caaa67": "f,g:A\\to B", "71628b8002fd500e3144678bac86203f": "NH_{3(gas)} + H_{2}O \\rightarrow NH^{+}_{4} + OH^{-}", "71634e633f463479fa776cbaf1330fa5": "GT = - D \\times \\ln ( SF )", "71637aa9b3a0cc5a59c23d8cddfea5bf": "e_3 = R\\, \\sin{(\\alpha)} \\sin{(\\theta)} d\\phi", "7163b976b48e69bdb5ab8d0ae27723b9": "d\\mu = e ^{i \\theta}d |\\mu|\\,", "7163c6943e65a419175c50d8e0d52a68": "t + C_2 = \\int \\left((2 - n) \\int f(x) dx + C_1\\right)^{\\frac{1}{n - 2}} dx", "71640badf52aa29bdf5b6dd3e6f6617f": "-cZ^2/A^{4/3}", "716411240fc671fe1f89ce44f4b66b9e": " \\theta_{12}^{PMNS}+\\theta_{12}^{CKM}\\simeq 45^\\circ \\ ,", "716423d84b961ec6f4f0e9c3abc89523": "B:Y\\to X_E^*,", "7164699cf4367254415b816311ee2acc": "l^2", "7164dea4bcbaf0c122dd20ccfb49b582": "k,l=0,1,2,\\dots", "7164faf37b547a41b82f17c4e3c72ac0": "[Y] \\cdot [Z] = [Y \\cap Z].", "7165031702df0cb6164b770fdcfa1489": "\\frac{\\partial T}{\\partial x} < \\frac{m C_0 (1-k) v}{kD}", "71650c476450f404e1fee7bc00e6ae2b": "\\mathbf{r}=\\mathbf{X}^+ \\mathbf{y}", "716549096b58a07cd57b2815b814fa28": "c_1=10^{-4}", "71658b7eaf64a927722c99281112c305": " \\Lambda(X_k^{-1}) = 0 ", "716598a134c746a245d3c38f6ad11863": "A_1, A_2, \\dots, A_t", "7165cc03e2a9b65fbc74701d79cf3c84": "\nT_0\\exp_p(v) = \\frac{\\mathrm d}{\\mathrm d t} \\Bigl(\\exp_p\\circ\\alpha(t)\\Bigr)\\Big\\vert_{t=0} = \\frac{\\mathrm d}{\\mathrm d t} \\Bigl(\\exp_p(vt)\\Bigr)\\Big\\vert_{t=0}=\\frac{\\mathrm d}{\\mathrm d t} \\Bigl(\\gamma(1,p,vt)\\Bigr)\\Big\\vert_{t=0}= \\gamma'(t,p,v)\\Big\\vert_{t=0}=v.\n", "716620a778dc0b545c480c7ab3ec3470": "P(H|D)=\\frac{P(D|H)P(H)}{P(D)}", "716652476355ca77b3a222f979a39cfe": "\n\\mathcal{L} = T_{ij}^r - \\hat{\\tau}_{ij}^r\n", "7166c48d0e299d0aabecc51246a08119": "T^{\\alpha \\beta} = \\, -\\frac{1}{\\mu_0} \\left( F^{\\alpha}{}^{\\psi} F_{\\psi}{}^{\\beta} + {1 \\over 4} g^{\\alpha \\beta} F_{\\psi\\tau} F^{\\psi\\tau}\\right) ", "716702eeb0961c8be27456336c3d8ac3": "U_{j,k}=n^{-\\frac{1}{2}}\\cdot z^{j\\cdot k}", "71670c23e35f502e1f1b2b580b77452c": "f(w_1,\\dots,w_n) \\mapsto \\frac{1}{n!} \\sum_{\\sigma \\in S_n} f(x_{\\sigma(1)},\\dots,x_{\\sigma(n)})", "71678125804070b652e94142d2808b1c": "\\phi(x_n) \\to \\phi(x)", "71679db25df90df39159e6f3bcd6bc04": "\n\\epsilon_\\mu^1( p ) = {-1 \\over \\sqrt{2}} [u^{-1}_{-1}(\\mathbf{p})]^\\dagger\n \\gamma_0 \\gamma_{\\mu} u^{+1}_{+1}(\\mathbf{p}), ", "7167a6cfd8efdd3cc11808c617d058a8": "C_q = {{2 \\sqrt{\\pi} \\Gamma\\left({1 \\over 1-q}\\right)} \\over {(3-q) \\sqrt{1-q} \\Gamma\\left({3-q \\over 2(1-q)}\\right)}} \\text{ for } -\\infty < q < 1 ", "7167c10814eda181e7ac7443eb58c11e": "\\operatorname{mwnchypg}\\left(\\mathbf{x};n,\\mathbf{m}, (\\omega_1,\\ldots,\\omega_{c-1},\\omega_{c-1})\\right)\\, =", "7167d6360e0b3bc522b9d9478e1f5b4a": " S_{j_1 j_2 \\dots j_r} = \\frac{V_{j_1 j_2 \\dots j_r}}{V} ", "7167f55431ae446e5bdeb625b3ca70ba": "{{\\mu }_{L}}", "71684a1b4a1ffb0c4b4a7483285fb67b": " MNDif = 1 - \\frac{1}{ N( K - 1 ) } \\sum_{ i = 1 }^{ K - 1 } \\sum_{ j = i + 1 }^K | f_i - f_j | ", "7169120ea393c3e670d90e4a33872f6e": "h=\\frac{P_a-P_o}{g \\rho}", "716913108ec67c8ad36d84436fb745ce": "P_{\\lambda}^{\\mu}(z) = \\frac{1}{\\Gamma(1-\\mu)} \\left[\\frac{1+z}{1-z}\\right]^{\\mu/2} \\,_2F_1 (-\\lambda, \\lambda+1; 1-\\mu; \\frac{1-z}{2}),\\qquad \\text{for } \\ |1-z|<2", "716932582ceaee7abc3ed24252c3610a": "y_n = \\frac{1}{N} \\sum_{k=0}^{N-1} X_k \\cos \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}+\\frac{N}{2}\\right) \\left(k+\\frac{1}{2}\\right) \\right]", "7169e0ce1e44e6b69f86b50d743e7831": "u_\\mathrm{axial} = \\frac{8 \\pi n}{\\lambda} \\sin^2(\\frac{\\alpha}{2}) z", "716a4fb063b02e7f004f2cc9910d3614": "q_i = \\mathbf{v}\\cdot \\mathbf{e}_i = (q^j \\mathbf{e}_j)\\cdot \\mathbf{e}_i = q^je_{ji}", "716ad67da55357d3205844023a2c7057": " b,c,d,e ", "716ad73fe9be1462bf392bec81fcc977": "\\{\\mathbf{\\hat p}, H\\}", "716b1aeec2acebb2f5dd687e46878545": "y_d=\\sum_{j=1}^{j=n} X_{dj}\\hat\\beta_j ", "716b3a914580772ca238d8c05a1790bd": "\\sigma_{tot} = \\sigma_1 = \\sigma_2", "716b548816e8b196bd3fc44f85b02346": "|E(a, b) - E(a, b^\\prime)| \\leq \\int [1 \\pm \\underline {A}(a^\\prime, \\lambda)\\underline {B}(b^\\prime, \\lambda)]\\rho(\\lambda)d\\lambda", "716b9034787d14c1c707234a59ccb267": "\\displaystyle{L_0(a,b)c=L(a,b)c}", "716bacad70c0d6a2feda7cba7c9ba835": " \\phi \\to \\pi^0 \\eta \\gamma ", "716c31ed626cc9ed22489ddb101c52a0": " \\sigma_g = \\exp \\left( \\sqrt{ \\sum_{i=1}^n ( \\ln { A_i \\over \\mu_g } )^2 \\over n } \\right). \\qquad \\qquad (1) ", "716c3a821c85f1a27bec475cfa49259b": " |0 \\rangle , |1 \\rangle ", "716c77e97fce73dda4cbd36fa0b65165": "E = s/t", "716ca4ec53a4cf897ef7b81a666528d1": "r:X\\to\\mathbb{R}^+", "716cd186de5bb8e8f8f48d940c940fec": "\\sqrt{n} / \\ln(\\sqrt{n})", "716d09b9f4d37dfc0cc480ceb96af5a0": "\\frac{\\mu_o I^2}{4 \\pi} \\left( \\frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \\right)^2 \\ln\\frac{r_{o2}}{r_{o1}} - \\frac{\\mu_o I^2}{8 \\pi} \\left( \\frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \\right) - \\frac{\\mu_o I^2}{16 \\pi}", "716d6a86df816ed6acda26117b019f02": "q=2,", "716e18ca55904aa6abb66d169aae7728": "[P] = \\begin{cases} 1 & \\text{if } P \\text{ is true;} \\\\ 0 & \\text{otherwise.} \\end{cases}", "716e7f5593b015feb09af35557f31065": "\\mathbb{R}^8", "716e965caa0d8dc5476adbf7188ef827": "~E=\\frac 12 \\left(\\frac{{\\rm d}x}{{\\rm d}z}\\right)^{2}+\\Phi(x)~", "716edb2eb95d91512291c61e6ac915ef": "\\scriptstyle 1 + \\sqrt{-5}", "716f59b0aaa66558b6ac34b752c6ac76": "\\bigl(\\pm \\tfrac12,\\pm \\tfrac12, \\ldots \\pm\\tfrac12\\bigr)", "716f7b91045b27a81368e3ccbc559a2a": "D_{even} := \\oplus_{n \\, even} \\, D_n", "71709fb73f93771f492aca18e4e963d9": "c^{-1} ED(c\\mu,\\sigma^2c^{2-p}) \\rightarrow Tw_p(\\mu,c_0 \\sigma^2)", "717111814fa701f33b3cf5cadd842a98": " p+1-N_p=2\\sqrt{p}\\cos{\\theta_p} ~~ (0\\leq \\theta_p \\leq \\pi).", "717130f41e9d16d72b4f244b32f9c0f0": "v \\in M_n(A)", "7171a2b6fbfbacbe8bdbb9840f11f526": "\n \\begin{align}\n \\sigma & = n_1^2 \\sigma_{11} + n_2^2 \\sigma_{22} + n_3^2 \\sigma_{33} + \n 2(n_1 n_2 \\sigma_{12} + n_2 n_3 \\sigma_{23} + n_3 n_1 \\sigma_{31}) \\\\\n \\tau & = \\sqrt{(n_1\\sigma_{11} + n_2\\sigma_{12} + n_3\\sigma_{31})^2 +\n (n_1\\sigma_{12} + n_2\\sigma_{22} + n_3\\sigma_{23})^2 +\n (n_1\\sigma_{31} + n_2\\sigma_{23} + n_3\\sigma_{33})^2 - \\sigma^2}\n \\end{align}\n ", "7171ee16808e0b5f341f6b3f2fe12153": " P^*_i \\propto 1/{K^*_i}^{\\beta^*} ", "7171efa01d1464044c380ab12d9482fb": "x^{(\\infty)} = \\left( 1, \\frac1{2}, \\dots, \\frac1{n}, \\dots \\right),", "7172ef439e904573afb99505ea53dff6": "\\frac{\\alpha}{\\beta-1} \\text{ if } \\beta>1", "717346857ee3a79c12b311469935ea0f": "v_1, \\dots, v_n", "7173ea110c2a58a51031b86f47adfde4": " \\psi (x, t) \\equiv {1\\over \\sqrt{\\lambda} } \\psi_j (t) ", "717409a153199909956e3f6af5c73a35": "\\scriptstyle f_\\ast", "717437fa1006c6d0b21a8406b6af3558": " f : \\Omega \\to \\mathbb{R}^\\infty \\,", "717447f906712a3b5c8172bf28199e1a": "\\Sigma^{1,A}_n", "7174cbd6aeaaa56e37102b72386bb2b9": "\\eta ", "7174ceda9686505fe581844a4ac13160": "\\bar{x}=\\frac{\\sum{f\\,x}}{\\sum{f}} .", "7174df1b82435fe251a0c8832d6035cd": "a^\\bar{\\alpha} = a^\\beta L^\\bar{\\alpha}{}_\\beta ", "7174df25935e16126d47644d930b6dbc": "M_r\\approx\\sqrt{\\frac{a^2+b^2}{2}}\\,\\!", "71750120c18739c295604f342b3419c1": "C = P \\times \\frac{F_L}{F_n} ", "71751d0cf2fde379316feeca319a0c5f": " G_\\mathrm{pixel} \\ge G_\\mathrm{objective}", "7175465e6aeeb244d1c9b379cbf3b1bf": "\\left(\\sqrt{1/55},\\ \\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "7175473d1cc2bc83d0e94819efac12fc": "y \\mapsto \\frac{1}{4} y^2 + \\frac{1}{4} \\sqrt{\\frac{3}{2}} y^{-1} - \\frac{11}{8} y^{-2} + \\frac{5}{8} \\sqrt{\\frac{3}{2}} y^{-3} - \\frac{1}{8}.", "71754c778d021fe9a424526427198294": " -2 p_1 \\cdot p_4 \\approx \\,", "71759ff58bf1b45ad5587e05019cab4f": "\np_x \\frac{\\partial H_{\\mathrm{kin}}}{\\partial p_x} = c \\frac{p_x^2}{\\sqrt{p_x^2 + p_y^2 + p_z^2}}\n", "717609735408a94e2304fd0b5eca914c": "2\\delta(s)=\\sup_{t\\geq0}\\left\\{\\omega(t)-st\\right\\},", "71763d4c6dafae2d9695de7d621a1893": "|\\int e^{ -i\\omega t }E_{ 1 }(t)|E_{ 2 }(t-\\tau )|^{ 2 }dt|^{ 2 }", "7176617cbd12942a4530e543b10eef77": " a_o = \\left( \\frac{h}{2 \\pi m e} \\right)^2 ", "7176cf9a663c24fa88dd74fc1bff403b": " ~\\epsilon_{t-1} \\le 0 ", "7177217b262d168ac45ed581c0c1d783": "Tr (A) Tr (B) = Tr (AB) + Tr (AB^{-1})", "71777326b0716d7b883b8f679bb3f7f0": "\\mathbf{x}_0 \\!", "7177b14053c9cbc62c7351492db38efb": "\\scriptstyle T\\colon Q\\times Q\\times Q\\to Q \\,", "7177c88d2af4a623634d7cd8beb6151e": "g=1-\\frac{1}{2}(|Z(\\sigma)|-|Z(\\theta)|+|Z(\\sigma\\theta)|)", "717802666440730d7b00a1e5e3c24863": "\\lim_{t\\rightarrow\\infty}\\overline{x}(t)", "71780b661c1b3647ddd29ec0f4dfb059": "p=\\infty", "71789de565a0b15628b8ab974d7f5f3e": "\\|T(t)\\|\\leq M", "7179c3514109b6384450a4e646b6abea": "\\Delta_{N-1} = \\left \\{ x_i : x_i \\ge 0, \\sum x_i = 1 \\right \\}", "717a4f902d3a7a636fa367dd8c33c23c": " f(x,y,z) = \\left(R - \\sqrt{x^2 + y^2}\\right)^2 + z^2 - r^2.", "717a7d5dc9df29a30fdbe287c2510347": "I_{\\text{D}}", "717b1d213cee2c577149828e9fe3fd05": "k[y]", "717b38db253876b5b328d13815fd4fda": "\\gamma=\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}", "717b7640c83fbd8bd219823397803965": "N' \\times M", "717b96eef40b6134d6d2e0e1c99d075b": "\\boldsymbol{\\omega} = \\nabla \\times \\boldsymbol{v} = \\nabla \\times \\nabla \\times \\boldsymbol{\\psi}", "717c1b7c1b840ec8c830c4d610568096": " \\csc\\left( \\pi/2-\\theta\\right) = \\sec \\theta", "717c439de19534b3781d28d549e36ab8": "t + \\delta t", "717c543df67a20a3544d30720d9ad325": " C_\\bullet(X) \\otimes C^\\bullet(X) \\overset{\\Delta_* \\otimes \\mathrm{Id}}{\\longrightarrow} C_\\bullet(X) \\otimes C_\\bullet(X) \\otimes C^\\bullet(X) \\overset{\\mathrm{Id} \\otimes \\varepsilon}{\\longrightarrow} C_\\bullet(X) ", "717c6e62c7b347037e4bcc4cf8cf155e": "k_r", "717c8776e866aac536ca146fcecaa8cb": "\nb_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cfrac{a_4}{b_4 + \\ddots\\,}}}} =\nb_0 + \\cfrac{c_1a_1}{c_1b_1 + \\cfrac{c_1c_2a_2}{c_2b_2 + \\cfrac{c_2c_3a_3}{c_3b_3 + \\cfrac{c_3c_4a_4}{c_4b_4 + \\ddots\\,}}}}\n", "717c9d51ada9623073cefdd3fa11b665": "\nI(R) \\propto e^{-kR}\n", "717cb47a3e972e33f95a0bd20d14929e": " \\mathbf{ \\hat T} (t) \\mathbf{ \\hat U} (t)|\\psi(0)> ", "717d08dc10795960336944bfb4382fa6": "\\mathrm{SO}(12,\\mathbb C)", "717d1aa152931266e9e9231f0fa60c97": "2D_k", "717d1c1d8745139fd54c0cf8d1647b58": "\\frac{dV}{dr}", "717d3cd254d475581e7f98126166efcf": ") - \\pi(", "717d3fec84ade1523771aef3708f2ef1": "\n\\frac{\\partial g}{\\partial A} = 2A \\sum_i \\left ( \\sum_k p_{ik} x_{ik} x_{ik}^T - \\frac{\\sum_{j \\in C_i} p_{ij} x_{ij} x_{ij}^T }{\\sum_{j \\in C_i} p_{ij}} \\right )\n", "717d80442c684cfbe32ec5b969a343d2": "M <_{DM} N", "717e17e1d864dd626f87055bb0e71db7": "\\varphi\\colon E\\to F", "717e2718b92f0941cf404f3f400d4d56": "\nR_{\\mbox{out}} = 4\\pi r^2 \\epsilon \\sigma T^4\\frac{}{},\n", "717e54dc6c51da577c4171b368505bc2": " R = 2.35 \\sqrt{\\frac{F w}{E}}, ", "717e6ba35e34fbd22cc0838fcd9a41cb": "\\psi_\\nu(\\bold{r})= \\langle \\bold{r}|\\nu\\rangle", "717e75eb26dc08854cfe0671bea37b41": "\\left(Im(\\Gamma(\\omega)) > 0\\right)", "717e94893e8481c284fc9e8a10155b7a": "(A)", "717eadaa814d89249d73e3fa9aa8e53c": "P_o", "717f0f45d5d6e45b8d6153bfb0523c11": "x^3-x=x(x-1)(x+1)", "717f1265ba9150429315cf7a45d344d4": "\\forall x (Mx \\to x \\in V).", "717f791a123773443ae1237fb7024332": "M_3(\\tau+1) = e^{-2\\times 121\\pi i/168} M_3(\\tau).", "717fcba464242eb67a13fe08e7509795": "P^n \\times P^n", "717fcc84036259a62083e13b891bdeae": "\\hat{W}_T", "717fdb3c373d624e85a8a8c76ce80778": "(0,2)", "718016d047ce36ffaef1c1a14d3d134e": "D_B(p,q) = -\\ln \\left( BC(p,q) \\right)", "71806b44ccad1056303e31163899a356": "47+\t31+\t2+\t63+\t34+\t18+\t15+\t50\t=\t\t260", "71807da952b7fbd20d622dc2511e41a6": "\\xi^L_t = R_{\\exp tX}", "718086c3031f806fe52361a8a119da35": "\\mu_{21} = M_{21} - 2 \\bar{x} M_{11} - \\bar{y} M_{20} + 2 \\bar{x}^2 M_{01}, ", "7180d8f74a436db3b0a7fb206be915d7": "\\cos x = \\prod_{n = 1}^\\infty\\left(1 - \\frac{x^2}{\\pi^2(n - \\frac{1}{2})^2}\\right)", "7180f99a7a8b96051b5cb97b7efead57": " H_n (x) = \\sum\\limits^{n}_{k=0} a_{n,k} \\ x ^k ", "7181213c078406322e538520da3aab08": "\\mathfrak{so}(m-1,\\mathbb C)", "7181284f0ac55c660f8e8abb41497cc6": " P = 10 \\ kN ", "7181a3d065b7e742f6c7f8e4d6c11ea5": "\\sigma^2_X", "7181bde3ac09ed973ba32d7d52426816": "u(x,t) = c \\, e^{-|x-ct|}", "7181c44ae005f36beedc76901f46aa4a": "C_N(L) = L", "7181e27d9aca0f51b1728432e93c88fd": "\\gamma(V,T)=\\frac{C^{(T)}_p(p,T)}{C^{(T)}_V(V,T)}", "71820aed4aa6da5cb472df731c0b9066": "{\\rm Inj}(M_i)<\\varepsilon(n)", "71825b56d116c1f76a09001b6ece7f89": "dx' = \\gamma \\left ( dx - v dt \\right )", "718268f00a66470c9ceeb7aa10ba53da": "L_p(\\Omega,\\mathrm{loc}),", "7182a3f626c7135740b93a721a9b9c14": "\\sigma_{v_x} = \\sigma_{v_y} = \\sigma_{v_z} = \\sqrt{\\frac{kT}{m}}", "718302fbdca1fbbfb635738b03943b95": "\\ \\psi_0(q)", "71830862e92b6a542048f38b9e8881b7": "O(|V|^3)", "71835739283bae3c8d8df3e07bac6270": "G(x) = x^5 + x^2 + 1", "7183b35dfc7a793f217c57a4bc40f96f": " \\displaystyle M^*\\Delta_c= 4\\Delta M^*.", "7183c863ddc5b7737a7dc8a91d5ff135": "f_i \\colon M_i \\rightarrow X", "7183ca9e3cc240052279d8673de55ecd": "(P_1 \\uparrow G_1 \\Leftrightarrow P_1 \\uparrow G_2) \\land (P_2 \\uparrow G_1 \\Leftrightarrow P_2 \\uparrow G_2)", "7183d216ed31678282a457546a75d9eb": "n=2^m+l", "7183ef90cc153d01f6c00d467674c8a0": "\n\\begin{align}\nn\\neq 6: \\mathrm{Out}(S_n) & = 1 \\\\\nn\\geq 3,\\ n\\neq 6: \\mathrm{Out}(A_n) & = C_2 \\\\\n\\mathrm{Out}(S_6) & = C_2 \\\\\n\\mathrm{Out}(A_6) & = C_2 \\times C_2\n\\end{align}\n", "718462077a6950e5f1e53b56dac66e00": "\\sigma\\ge 1-\\frac{1}{57.54(\\log{|t|})^{2/3}(\\log{\\log{|t|}})^{1/3}}.", "7184a79a747df8449045d32e0422dd72": "L \\subseteq \\Sigma^* \\times \\N", "7184b6a50c27589711848ed940038ff9": "R(h) = \\mathbf{E}[L(h(x), y)] = \\int L(h(x), y)\\,dP(x, y).", "7184c5c361e84541b63fda3ac71f7f11": " G = C_o \\rtimes C_p. \\, ", "718551ac381fbe89a8d37e69e84f445d": "IRR = \\left( \\frac \\text{InvoicePrice} \\text{PurchasePriceOfBond} -1 \\right)\\left( \\frac \\text{dayBase} \\text{daysToDelivery} \\right)", "7186554c7c80311560f7e6d6c3092acb": " |\\psi\\rangle = \\begin{pmatrix} \\cos\\theta \\\\ \\sin\\theta \\end{pmatrix} \\exp \\left ( i \\alpha \\right ) ", "718656e21bc6c7d097e1330d4649c5f4": "\\vec{a}_A=d^2\\vec{x}_A/dt^2", "718694b5953c8738dc87b6df8aa3b047": "f(n) = \\sum_{i=0}^n {1 \\over i!}, \\quad g(n) = n.", "7186a5979655ada97d673044b3a55cae": "\nh^{\\mu}(x) = M^{\\mu \\nu}x_\\nu + P^\\mu \n", "7186b6b7ba443c943fc8cbad4ea60f75": "[A : B : C : D : E : F]", "7186ebbe0dcb4f9546ab7f3a48c2a8fc": "\\frac{s}{s+1}\\left(1-\\frac{mR}{m-s+1}\\right)", "7187453a4f66e4b3cfac867d65dd2e4c": " \nc(E) = 1+c_1(E) + c_2(E) + \\cdots \\,\n", "7187cba0919b46758671a8db53171d6d": "dW = \\mathbf{N} \\cdot \\delta \\boldsymbol\\phi", "7188be5536e9ddfd8859faa303906b88": "0= \\oint_c\\rho(x;\\alpha,\\beta)y_n(x;\\alpha,\\beta) y_m(x;\\alpha,\\beta)\\mathrm d x", "71892e4515144e60ea8f1bbdfc34ebf8": "\\!\\mathcal A \\models_{X[A/x]}^- \\phi", "71897bb6ab6acb6a6d35cd11ff6bb495": "ax^2+bx=c", "7189917f5e2ea7f96416deb06021e267": "w_n\\in\\mathcal{P}", "7189985f7b815aa48e08637b30fe1038": " \\zeta < 1 ", "7189b1f76a13e63cb93ee26dbbf0e171": "w^0", "718a1c44ff1ccdba5d35e44270ef4143": "N(0)", "718a342c1e32a72b8e616a787ddb9dad": "{1+\\sqrt{2}}", "718a3a448f068c3726873beec9fffda4": " \\rho_e ", "718a876dbafdaf43bffd7974d6f9c776": "g=g_t", "718a9c633df2d2c98fc4e1f8b3ca11cd": "\\mathbf{F} ", "718add90d7cb23997a6c009c4a44dd70": "(c_1 A+c_2 B)\\cdot X=c_1 A\\cdot X + c_2 B\\cdot X\\,", "718ade5700a0cdf931653f63958c0fd2": "Q\\in E(\\bar{K})", "718b2efea414a41d1d0aaeaf598298b6": "p(z) = p_0 + g \\int_{0}^{z} \\rho(z) \\, dz", "718b612a34721cf1913e0502f57eb646": "gl_n \\to k", "718b854ceb50364cc27683d6ec79a1ad": " \\lim_{a\\to 0}\\phi = \\tfrac{1}{2}\\pi\\quad\\text{ and }\\quad\\lim_{a\\to\\infty} \\phi = 0. ", "718c0ddc969d3c3e245e987e481e7ed7": " k\\geq 1 ", "718c86792c66690997d8d155ca57a29c": "A = (L,G,c) . ", "718c926fbe17ce865dbc01791f2a6747": "S_w(p)=\\int w(r) S_0(p-r)\\,d r", "718ca348bc200d2c4e60d60d14c2b9fe": " \\tfrac{1}{2} m_x v_0^2 = \\tfrac{1}{2} m_x v_1^2 + \\tfrac{1}{2} m_y v_2^2 \\,\\! ", "718d41975b5aa05a0d13099ddbd601b7": "s\\in\\mathbf{C}", "718d5d8f22958ba71b01873ed1fc77e6": "\\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B, \\psi_{l})", "718d8822d7b4f8fec3e6160b8aff0dfc": "T_F", "718de1a2ec5d6eb4e37a17ebe28ac9a0": "x \\wedge y \\wedge x = x", "718e172f2d1e3d15c736946051fbf31b": "x \\oplus x = x", "718e225fb51ac47a37d1f65ae6b1c34c": "L^\\infty(\\mathbb{R}^d)", "718e7621829c54ea33e612614953cd48": "- - -", "718ea463c79fc1909bfc1d78f83b9b81": "\\Delta loc \\approx \\frac{\\Delta}{\\sqrt{N}}", "718ef6aac2d7b8df4f14d24e3d0ad60e": "F_b", "718f19e43714b8af75e231c0b078e131": "a_M \\le +\\infty", "718f1f268d268da69a36e92beb4bea92": " \\frac{n^np^k (1-p)^{n-k}}{\\left(n-k\\right)^{n-k}e^kk!} \\rightarrow \\frac{n^k\\left(\\frac{\\lambda}{n}\\right)^k (1-\\frac{\\lambda}{n})^{n-k}}{\\left(1-\\frac{k}{n}\\right)^{n-k}e^kk!}=\\frac{\\lambda^k \\left(1-\\frac{\\lambda}{n}\\right)^{n-k}}{\\left(1-\\frac{k}{n}\\right)^{n-k}e^kk!}\\rightarrow\\frac{\\lambda^k \\left(1-\\frac{\\lambda}{n}\\right)^{n}}{\\left(1-\\frac{k}{n}\\right)^{n}e^kk!}", "718f85c19e47b88758f161513315d974": "Sing(F)", "718fc494cf0f26253030bc8a85c5e636": "\\lim_{x \\to p^{-}}{f(x)} = L", "718ff50558fa3d10fdae36870954f7d6": "m \\times l", "719007b9620bea8a7e73c43b560893cf": "T_g = \\frac{H_d}{S_d+ R \\ln(\\frac{1-f_c}{f_c})},", "71905cc43e8f945d46beaa48c12d1822": "\\mathbb{CP}^{3|4}", "71912f620520dcd58beddfd35b90f3cb": "P(x) = e^{-x /\\lambda} \\!\\,", "7191411086b40be957f1949885ded8b4": "H^1_{dR}(\\mathbf{R}^2\\setminus\\{0\\}) \\cong \\mathbf{R},", "71918c12e0b0a84fdb028b5e030bc702": "\\lim_{x \\to a^-} f'(x) = {-\\infty} \\quad \\text{and} \\quad \\lim_{x \\to a^+} f'(x) = {+\\infty}\\text{,}", "7191c3688d0208909eabfdd3f56c86d2": "a_e = a(1+{ f \\over 3} )", "7192023fbbb27936af9ff12b0de34c8c": "\\tan[\\frac{1}{2}(\\alpha-\\beta)] = \\frac{a-b}{a+b} \\tan[\\frac{1}{2}(\\alpha+\\beta)]= \n\\frac{a-b}{a+b} \\cot[\\frac{\\gamma}{2}]", "719206e7d1585078f1e9cae26e56e0ec": "\\mathbf{r}_{ik}", "7192943e61368b70ca83688a938c51a2": "P = {(4x - 6) \\choose 4} \\div {50 \\choose 4}", "71929820a3972d30560d8b01fc9b6dfb": "\nN(\\mathbf{p}) = Q^\\dagger(\\mathbf{p}) Q(\\mathbf{p}).\n", "7192d8d9aee7e5e94fc9a277cb96b8e1": "D_\\mu e^I_\\nu = 0.", "7192ee192e945d038546edee65415783": "x + I^n M \\quad\\text{for }x\\in M. ", "7192ff94c75c7ddcef1afb409e98165e": "f[x_0,\\dots,x_n]", "71934d28ee2205f627bb901fbfe82166": " Q = \\oint \\vec{B} \\cdot \\vec{ds} ", "7193873f3c92d482730f81204e2cf851": "\n\\left|\\sum_{i=1}^{n}a_{i}b_{i}\\right|^{2}+\\sum_{i \\max\\{ |a_{11} e_1 + A_{12} \\mathbf{e}_2| \\}.", "71985f4fbdae56ddb98ac53c00f8fd55": " \\lfloor \\frac{w_\\min - 1}{a_\\max} \\rfloor ", "71988fc91a9f061a6268e3009f9ece12": "\\sqrt{{f(x)}} < g(x)", "719912a9b08e8cbac812b4db40b89087": "\\Sigma^x\\,", "71992dadd3762d9c90b98715878a081d": "F = \\frac{\\cos \\phi_1 \\tan^{n} (\\frac14 \\pi + \\frac12 \\phi_1)}{n}", "7199326deb68b02d284f0cc984ad5477": "f(s_k)", "71994b1f795a64bafee9559ae3abcd61": "{D_B}", "7199ab16319bb177f2c714f83d89fff3": "B=S_0", "7199ba343e7cfc1e339d035ce0f70afd": "\nH(C) = H(A) + H(M(A) B)\n", "719a14b62d3d979b620de180e9a346bf": " i \\in v ", "719a38f95100f5827308d0693b77dfdf": "\\vec{K}", "719a44132c874cb3b8179a0b54b57e23": "y_3 = \\frac{y_1 y_2 - x_1 x_2}{1 - dx_1 x_2 y_1 y_2} = 0", "719a4609ea715dcdb37fa677dc3718ad": "\\frac{d^2\\beta}{dt^2}+(\\frac{4k}{MV}+\\frac{2k(a^2+b^2)}{VI})\\frac{d\\beta}{dt}+(\\frac{4k^2(a+b)^2}{MV^2I}+\\frac{2k(b-a)}{I})\\beta=0", "719a660fc82f2da9ca68b5a95170098f": "d_1 \\cdots d_n.", "719a69a973f1a41b99096c6b2f241c02": "\\pi^{-1}(\\{y\\})", "719aa8ee8766571da93eadca1f316ed8": "7^2+2^2+1^2 + 7^2+6^2 + 11^2+6^2 + 10^2+7^2+3^2 + 5^2+3^2+4^2 + 11^2+4^2", "719ababcfb5f0d3127cdfb795bf3e3d8": " \\langle A \\rangle_\\psi = \\langle \\psi | A | \\psi \\rangle ", "719abf9cda0307de7fa97fcc50cf44cd": "\\nu_e = (2.8\\times10^{10}\\,\\mathrm{Hz}/\\mathrm{T})\\times B", "719ac76cc48893ad9252183f405ed664": "\\mathcal{S}[\\mathbf{q}(t)]", "719b80cb08741e9b3280f3485fd01955": "\\alpha = 180^\\circ - \\beta - \\gamma ", "719bdbbed6464c6d05dd5a5626edf94e": " \\|r_n\\| \\leq \\left( \\frac{2\\kappa_2(A)-1}{2\\kappa_2(A)} \\right)^{n/2} \\|r_0\\|. ", "719c204048a1a404ca45a1dc391c88f6": "\\|P_n-P\\|_\\mathcal{C}=\\sup_{c\\in\\mathcal{C}}|P_n(c)-P(c)|\\to 0,", "719c87ae88f42664dbd60e4f353e12eb": "\\omega=\\ldots A^{k_1}B^{k_2} \\ldots B^{n}", "719ce4d08daf3126438fd60141c1caea": "\\tilde{u},", "719d3db893f321363ce61229529009da": "S=\\{\\alpha\\in E|\\alpha \\mbox{ is separable over } F\\}", "719d6c84d9c41c25c2b0436509c572e6": "x \\div 3", "719d93ec924b282c532c20e69f4a67a5": "\\frac{1}{2}\\chi^2\\left( -\\frac{1}{\\mathfrak{M}^2} + 1 \\right) \\ge 0", "719e0139e658162df114be76a4884c67": " a_0=Vol(M,g),\\ \\ \\ \\ a_1=\\frac{1}{6}\\int_MS(x)dV ", "719e209d0f399b2185651a9f1b0ac65c": "\\mathcal{\\tilde{H}^{\\bot}}_{S}", "719e4b45a9069566c3818058bd7262c5": "X = C_0({\\Bbb R})", "719e79f7996bd62c8995b0385c22a6d0": "\\ (1/(1+r)^t) ", "719e8c24669cd80244e09995a88f22a4": "w = f(z) = u(x,y) + iv(x,y)\\,", "719ea1274ae44a6a5902d2eaf827b3c4": "f^{-1}(s)=\\{w\\vert f(w)=s\\}", "719eb50895eca491d8495a33f8c991e1": "|\\hbar k_{max}/m|", "719f0315fbf77f6fddc55b552743ba34": "f(-1)=f(j)= f(-j) = 1. \\ ", "719f4684f404f8af3bb4433aa896999e": "\\phi ( \\bold{r} )=\\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac {1}{|\\bold r - \\bold{r}_0|}\\ \\bold{p} \\cdot d\\bold{A_0} \\ , ", "719f4c96cf97befa73fc4502cc9a3246": "\\bar{x}yz", "719f5b6a83ac9fb38823066444bac7d9": "\\Psi\\propto\\begin{pmatrix}\n\\rho^{\\gamma-1} e^{-\\rho/2}\\left(Z\\alpha\\rho+(\\gamma-1)\\frac{\\gamma\\mu c^2-E}{\\hbar cC}(-\\rho+2\\gamma)\\right)z/r\\\\\n\\rho^{\\gamma-1} e^{-\\rho/2}\\left(Z\\alpha\\rho+(\\gamma-1)\\frac{\\gamma\\mu c^2-E}{\\hbar cC}(-\\rho+2\\gamma)\\right)(x+iy)/r\\\\\ni\\rho^{\\gamma-1}e^{-\\rho/2}\\left((\\gamma-1)\\rho+Z\\alpha\\frac{\\gamma\\mu c^2-E}{\\hbar cC}(-\\rho+2\\gamma)\\right)\\\\\n0\n\\end{pmatrix}", "719f7bc5f46794ebf879843b97e3d7ca": " \\Phi = \\iint E_\\lambda \\mathrm{d} \\lambda \\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right )", "719f945579ea8bd004e2e7a8003c41af": "\\mathbf{e}_1 \\times \\mathbf{e}_2 = \\mathbf{e}_4, \\quad \\mathbf{e}_2 \\times \\mathbf{e}_4 = \\mathbf{e}_1, \\quad \\mathbf{e}_4 \\times \\mathbf{e}_1 = \\mathbf{e}_2,", "719fbb8f4537b9c23d17c9f0ce1e5389": " E=\\frac{mc^2}{\\sqrt{1-\\frac{v^2}{c^2}}} = m c^2 \\mbox{cosh}(s)", "719fd70100ebd422e5efafdf1c8f69f7": "B(\\mathbf{v}, \\mathbf{w}) = x^\\mathrm T Ay = \\sum_{i,j=1}^n a_{ij} x_i y_j. ", "71a02fa3e10668258bf54d70b5f12ef8": "TM_{011+\\delta}", "71a061257466e8971de1885c6fa22a57": "|U_1|^2,", "71a06817ebe0e5f6c9e474e6f24cab33": "\\mbox{A} \\rightarrow \\mbox{products.}", "71a0c1562b2cc87244a1c2b21092be1b": "\\cos_k(i)\\equiv \\cos_i(k),", "71a0cee70f846c886f9d3d784000cf35": "\\mathcal{T}(G)", "71a1089e302e5af15a90df066ec15825": "\\textstyle (\\mathbb{R},\\gamma) ", "71a110e30c2f555dbf42e5cd0565a247": "M_{sup} (R,T)=\\sup \\frac{H(\\rho)^2}{A(\\rho)}", "71a142bce708aca4a126cdeb9e83f61a": "- \\!\\,", "71a17b76ac64c243ef9a3482a2c726da": "\n\\tau_{eq} = \\left( \\frac{1}{k_{B}T} \\right) \\frac{F_{ax}F_{eq}}{F_{ax} + F_{eq}}\n", "71a2049fc1d98ff2d5d0babebef6be39": "\\mathbf{A}_{\\text{Electric quadrupole}}(\\mathbf{x},t) = \\frac{-k \\omega \\mu_0}{8 \\pi} \\frac{e^{i k r - i \\omega t}}{r}\\int d^3\\mathbf{x'} \\mathbf{x'} (\\mathbf{n}\\cdot\\mathbf{x'})\\rho(\\mathbf{x'})", "71a2e546d2f62cf05364b24b92db617a": "g(K_1 \\# K_2) = g(K_1) + g(K_2)", "71a3005cdd70cbbfd90bb0e7f5bf63c2": "\\Lambda_{GUT}", "71a324b6b0b1d1d80ed39ed95034ebfc": " \\iint_D \\left[ -v \\nabla \\cdot \\nabla u + v f \\right] \\, dx \\, dy + \\int_C v \\left[ \\frac{\\part u}{\\part n} + \\sigma u + g \\right] \\, ds =0. \\,", "71a332f2981a67cd11b1b314ddd47c99": "E_o=100{\\left [ \\frac{F(1-f_x)}{O} \\right ]}=100(1-o_x)", "71a33d7e7d44bcb8b8f56c73edff9378": "a^*(m)", "71a34201c337bf62a0cbbc01c638cbb5": " \\langle \\psi| \\psi_i\\rangle \\langle \\psi_i \\mid \\psi \\rangle = | \\langle \\psi \\mid \\psi_i\\rangle | ^2. ", "71a3bb622f85c1cc1f9b072a3c0cda8d": "\\varepsilon^{\\frac{1}{2}}", "71a3e03f0a1811dd748a8c5f44e5f074": "\\phi\\ \\stackrel{\\mathrm{def}}{=}\\ \\theta_2-\\theta_1", "71a4080e443fb899d98962b41ea88745": "\\frac{2\\sqrt{\\pi}(\\pi - 3)}{(4-\\pi)^{3/2}}", "71a432b8143396cbcad737d6524ffa25": "\\scriptstyle \\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)", "71a446da4b5bc589551e04f4dbf0359b": "u=u_q e_q", "71a49467eda1cf665b86e4f31fe33beb": "2=2^1", "71a4b21a406fe68786c29e35057c74fa": "ekt/V = spKt/V \\cdot \\frac {t}{t+C}", "71a4dadd39000f255d63847b0ae8757b": "\n\\Pr[R(x,y) = 1] > \\frac{1}{2}, \\textrm{if }\\, f(x,y) = 1\n", "71a51dc52d40a218a6ac93f2488db913": "\\Phi_E=\\phi_D^{-1}\\circ\\phi_E", "71a521746c548ab8adf7aa5f1f6689bd": "\\displaystyle \\sum_{s \\in S}W(y|x, s)U(s|x') = \\sum_{s \\in S}W(y|x', s)U(s|x)", "71a545d6c612cfe1f8f38bfe4dd9f7ed": " s \\mapsto \\bigg(\\frac{d \\mu}{d \\nu}\\bigg)^{1/2} s ", "71a54d88ca9ce32855f8dad5e4af999a": "f_j(x_{ij})", "71a56ebb450f12a20189b02565b99e5f": "z=1089", "71a58995475a216e93e393672450caa9": "\\left(\\frac{p}{q}\\right) \\left(\\frac{q}{p}\\right) = (-1)^{mn}=(-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}.", "71a58d345414ffea4bcdc11123f93e50": "\\textstyle\\log_a'(x)", "71a5c26bbcb69eb3da4ba1192c794aa2": "Y_{6}^{0}(\\theta,\\varphi)={1\\over 32}\\sqrt{13\\over \\pi}\\cdot(231\\cos^{6}\\theta-315\\cos^{4}\\theta+105\\cos^{2}\\theta-5)", "71a602f0eea364abb4652bb11ac11ab8": "\\boldsymbol{v}", "71a63833380e22da15116197642bbe3b": "V_i=\\mathbf{X}^\\mathrm{T} U_i/\\sqrt{\\lambda_i}", "71a6b0c1a5fef2ce00b99db46e77300e": " \\bar q = \\frac{\\sum_{k=1}^n d_i(B)}{nTL} ", "71a7082a0e70a39ebe6f7847afbf90a0": "C_{XX}(\\tau) = E[(X(t) - \\mu)(X(t+\\tau) - \\mu)]\\,", "71a73dfa1bc052b01cfd7a1be5123b99": " A^{(i)} = F \\ast a^{(i)}", "71a7588fcd162baa8bcbfb626e2c54eb": "\\zeta(2)=3\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}\\binom{2n}{n}}.", "71a7896a6b5e09899b98b0a8cf096e7b": "E/\\rho_s^2 = \\frac{9P_{cr}\\sqrt{3(1-\\mu^2)}}{2\\rho_a}", "71a7b3cb346c20c7aac71c8079df6d1b": "\\Phi_{\\text{Euler}}(\\ h,t_{n-1},y(t_{n-1})\\ )\\ y(t_{n-1}) = (1 + h \\frac{d}{dt})\\ y(t_{n-1})", "71a7cbdac923dcb5ade5cea601459097": "\\sin(67\\tfrac12 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2}};", "71a7d9c5a790aad3c4eea59be8a6021d": "y_{ist} ~=~ \\gamma_s + \\lambda_t + \\delta D_{st} + \\epsilon_{ist}", "71a7da584b255d1d6dc24b693862e683": "x \\leftarrow x + r", "71a87790707dde2eca353188f600f712": "\\int\\frac{\\sin ax}{x^n} \\mathrm{d}x = -\\frac{\\sin ax}{(n-1)x^{n-1}} + \\frac{a}{n-1}\\int\\frac{\\cos ax}{x^{n-1}} \\mathrm{d}x\\,\\!", "71a8838d081399d4c5b5b0f93e60a689": "\\tau_{sX}^{-1}\\tau_{tY}^{-1}\\tau_{sX}\\tau_{tY}Z.", "71a89ea76122ec0f7f41f675c89ee9ad": "U \\subseteq V", "71a8b62f56fe95e4e4a452fce1ba9730": "h(L)", "71a8d0b8410525182a5d71b87091633f": "\\Omega^4", "71a926512824ddc2f1bf59697acd6622": "B = -5.775 \\times 10^{-7} \\; {}^{\\circ}\\mathrm{C}^{-2}", "71a99965fd2e4065cbdc34e2e8ff1d47": "v_{bullet}\\sin(\\delta\\theta)=\\frac{1}{2}g t=\\frac{1}{2}g \\frac{x}{v_{bullet}\\cos(\\delta\\theta)}", "71a9e5961be6b8b1040e64a737083695": "\nf_{U+V}(x) = \\int_{-\\infty}^\\infty f_U(y) f_V(x - y)\\,dy\n= \\left( f_{U} * f_{V} \\right) (x)\n", "71a9e5be3d07185f0ba49bf63bb7dc28": " \\hat{I} = \\sum_i |\\phi_i\\rangle\\langle\\phi_i|", "71aa00c6fe8a13b4abf03de2936a2ecb": " \\frac{d\\omega}{dz} = \\frac{\\omega}{1 + \\omega}", "71aa5f86a4320786dd851ac97e6aa432": "S_- = \\hbar \\sqrt{2s} a^\\dagger \\sqrt{1-\\frac{a^\\dagger a}{2s}} ", "71aa68fa94f503edc1ff3409c9a4e3bd": "FR_n = 1 - (1-\\beta)^n ", "71aa6c90b5e35870b57abe38ee9aa56a": "j = \\frac{y}{m}", "71aa72a2e9ae37f051ff52b5c3fd81d7": "(0,1,2,3,\\dots,22,23,24; 70)", "71aab29da8d3a8d13a559c3125e33994": "\\alpha_S", "71aac6b58d02c49aae0791d63bc6a327": "\\scriptstyle PG(3, q) ", "71ab14e828645d43bbc01dd45b11aaa2": " F/F_G = 10^{-7} ", "71ab1d7c87d70c21283d1b84098a9fcb": "y = z = 0", "71ab4417ce0b486bd1d9ad6dfdceb96e": " j_{\\mathrm{F}} ", "71ab8bf9628c3fc8d04e10fa7abf4a54": " \\dot{V}(x_1,x_2) = \\frac{g}{l} \\sin x_1 \\dot{x}_1 + x_2 \\dot{x}_2 = - \\frac{k}{m} x_2^2 ", "71ab950b39d5645d84e7b31d40e05acc": "\\frac{1}{\\sqrt{-K}}", "71abdd9e0c27cc4103d3b6a635279eee": "\\operatorname{and} = \\lambda p.\\lambda q.p\\ q\\ p", "71abde8df8d9eff170068e0245a72922": "\\quad Q = - \\frac{\\hbar^2}{2m} \\frac{\\nabla^2 R}{R}", "71ac359d8fef4a5b59ba628a625d0312": "\\{ \\tilde{E}_i^a (x) , A^j_b (y) \\} = 8\\pi G_{\\mathrm{Newton}} \\beta \\delta^a_b \\delta^j_i \\delta^3 (x - y)", "71ac93f5abf0c2954b6337eba4600d34": "H(u)", "71ac981837386e9a16080ae6d44752e1": "\\lim_{x\\to a}e^{-i\\xi x}v_{2}-\\lim_{x\\to b}e^{-i\\xi x}v_{2}=-\\int_{a}^{b}e^{-i\\xi x}\\,q_{1}^{*}\\,v_{1}\\,dx", "71acbc669f8bff095de8d39cf0abdb84": " \\frac{ ( \\mbox{10.5} \\times \\mbox{124} )}{113} = 12 ", "71ad0afd20027204bc80bd2b1eb6b5c7": " \\operatorname{de-lambda}[p\\ f], \\operatorname{de-lambda}[M_1\\ N_1], \\operatorname{de-lambda}[M_2\\ N_2], ", "71ad0e0f8d436207c58d67f76276d370": "y = ax^2 + bx + c", "71ad2036c3144875d447960c9e1dd54b": "\\Omega(\\mathbf e\\, g) = g^{-1}\\Omega(\\mathbf e)g.", "71ad27df9b603275fa6bbde9a48f3163": "A_\\alpha(x) \\frac{dB_\\alpha}{dx} - \\frac{dA_\\alpha}{dx} B_\\alpha(x) = \\frac{C_\\alpha}{x},\\!", "71ad38c1fdad54b9a33a43a5ab4f1842": "SD_k=\\frac{\\sum_{t=0}^{T}|h_{k-1}(t)-h_k(t)|^2}{\\sum_{t=0}^{T} h_{k-1}^2 (t)}.\\,", "71ad4561fe54d08ea838db8dba78277f": "\\text{P}(x_1,x_2,x_3,\\ldots,x_n) = B \\prod_{i=1}^n \\text{P}(x_i)", "71ad8af3e3d4068d11d042d6a19bcfdf": "[x, s]", "71add061862ed6784bb1075bb9dbb533": " \\int f \\, d \\mu \\geq \\lim_k \\int f_k \\, d \\mu ", "71ae67e1c2c740c377212dcfd54b19aa": "A^+ + B \\to A^- + B^{2+} ", "71aed7a44eea9c5626a106bed93bec6d": "\\int_{a}^{x_1 + \\Delta x} f(t) \\,dt - \\int_{a}^{x_1} f(t) \\,dt = \\int_{x_1}^{x_1 + \\Delta x} f(t) \\,dt. ", "71aeecd1800d92bca57103a1d626e64f": " f_*(x)=\\frac{1}{a} f \\bigg( \\frac{x}{a} \\bigg)\\,", "71af133e5e5e9cb8f0a415d9a6056834": "|f|^p", "71af3167b02ddcfe27579084d510078f": "\nF_1(a,b_1,b_2,c;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a)_{m+n} (b_1)_m (b_2)_n} {(c)_{m+n} \\,m! \\,n!} \\,x^m y^n ~,\n", "71af51f9d46e22602de22887332f1a91": "0\\le t \\le 1", "71af542df1a4acd181f427727b0eb42e": "p_1+1,c_1,sid_1", "71af5cbecd4f6caaf240b3b1af34509c": "C_p = {p - p_\\infty \\over \\frac{1}{2} \\rho_\\infty V_\\infty^2}", "71af838a5068d75c4fdb71390487de94": "k \\, d \\, \\cos{\\theta} = \\beta \\left ( f \\right ) = 2 \\, \\pi \\, \\frac{c}{\\lambda_0} \\, \\Delta \\tau", "71b030b9629d51f7f8522a0995eefd10": " x \\mapsto ax+b ", "71b048e71dbea0f4b2d64cb6044f3407": "x y-1 = 0", "71b070401dba68ef72531ad1525bee4b": "g = {x_k}g, \\alpha = \\alpha^{x_k^{-1}}, k = v[\\alpha]", "71b08e2c647f0fbcc7872fdab19ddc78": "10 + (-3) + 2 \\,", "71b09079bda8f8ce4b8bd3cd3cc42a0f": "m_{\\mathrm{e}}", "71b0c2883f0d756d7bb8c608ef15032e": "\\mathbb{D}_6 = \\mathbb{Z}_3\\rtimes\\mathbb{Z}_2", "71b0daa3d2fee4272dc7d5dead72f371": "\n \\sigma_1 = \\sigma_3 + \\sqrt{A\\sigma_3 + B^2}\n ", "71b0ecb56f15239ab0513a35dc720a3a": "|E^{(+)}|^2=|\\mathcal{E}_1|^2+|\\mathcal{E}_2|^2+\\lbrace\\mathcal{E}_1^*\\mathcal{E}_2exp\\lbrack i(\\nu_1-\\nu_2)t\\rbrack+c.c.\\rbrace", "71b0ee6f47436d572e0d9d3e3606bc9d": "\\sigma_k^{(o)*}(n) = \\sum_{{d\\mid n \\atop d\\equiv 1 \\pmod 2} \\atop \\gcd(d,n/d)=1} \\!\\! d^k.", "71b10f87f30f5eedf9b81fee55153837": "\\frac{a + bi}{i} = -i\\,(a + bi) = -ai - bi^2 = b - ai. ", "71b1258ae68c0e8b86d6b248d917cf5d": "k : \\mathcal{X} \\times \\mathcal{X} \\rightarrow \\mathbb{R}", "71b191b0e81b526a30b2d72868ba4ea7": "\\mathrm{U} = \\frac{H\\, \\lambda^2}{h^3}", "71b1f45fc91d58868591cba92ebb49b7": " x = {1 \\over 3} + {(-23 + 3i \\sqrt{237})^{1/3} \\over 3 \\cdot 2^{2/3}} + {11 \\over 3 \\cdot (2 \\cdot (-23 + 3i \\sqrt{237}))^{1/3}} ", "71b1f5288b4e3898533b6830eb6e05ab": "\\tilde{G}_{1,n} \\cong S^{n-1}", "71b1f56c577913ff87669cd3e79e8350": "n=\\frac{\\Delta_{1}-\\Delta_{2}}{\\lambda}\\approx\\frac{2Lv^{2}}{\\lambda c^{2}}", "71b26fa13bf0f3dc5e61afc828d6cc77": "\\ b_{2p+1}\\equiv 0 \\ mod \\ 4", "71b2bb22dd7636822af187a0406bac42": "\\|x+y\\| \\le \\|x\\| + \\|y\\|", "71b329d343aef455c0d70c366833017e": "v=e^{-(\\theta/2)(e_i \\wedge e_j)}u e^{(\\theta/2)(e_i \\wedge e_j)}", "71b3329d1096799941c58bc776f081c2": " n \\choose k", "71b336618c597dc0fd535c86370a797f": "D_c \\approx 956", "71b33fede79cbbee66160e170affc798": " |\\cdot| ", "71b364d0286b8212837fd0017fa83127": "{v^2 \\over 2}+gz+{p\\over\\rho}=\\text{constant}", "71b3c413220fb6fd8d0958b1fbef9b47": "\\chi_{r,n} = 0", "71b488d1f4b60ce0745a574cbed1438a": "(a,b)=\\begin{cases}1,&\\mbox{ if }z^2=ax^2+by^2\\mbox{ has a non-zero solution }(x,y,z)\\in K^3;\\\\-1,&\\mbox{ if not.}\\end{cases}", "71b49025fa0f9b4513f8b00ee17e29a4": "\\alpha = {k \\over {\\rho c_p}}", "71b493cd0e7b995d8fb863abfa90a9a6": " t = \\frac{\\bar{x} - \\mu}{s/\\sqrt{n}}. ", "71b51707c71e3f724b173191968be2a7": "\\beta(g)=\\frac{3}{16\\pi^2}g^2+O(g^3)", "71b5a0fbf50e0a1838be482b1b467eb5": "\\liminf_{n\\to\\infty} x_n = \\infty", "71b5b88131d85a2c91d1d66766f6d2f1": "e^{{-iE_n t}/\\hbar}, ", "71b5e2f569242bf0cfae55bace4f16ef": "\\varnothing \\leq A", "71b643717e218b454f0283526ff5de87": "S_{spw}", "71b66bf15521c7a81521b99458fd8794": "\\mathrm{D}^i f(\\mathbf{y})", "71b6eb90cce702c04995b5703f3a9781": " {P = - K \\cdot S} ", "71b79cf9ed17f55df17abae324e5f658": "f(x)\\sim g(x)", "71b7c79f46ee19841bfcd320ce757d3b": "|n|^\\alpha", "71b7f7d87cd8db4e8a3bc9f22779e012": " {\\rm Airglow}/{\\rm S}_{10} = 145+108(S-0.8)", "71b7fc0d18092b1e9d16eb3c997fc1b0": "\\textstyle \\sum_{n=0}^\\infty c_nX^n", "71b860e78e5a9a55e5c2f6435b321543": "r_1=Do", "71b86fb60347b911645e91039628ec90": "\\mathbf{v}_1 = \\begin{bmatrix}1\\\\-1\\end{bmatrix},\\quad \\mathbf{v}_2=\\begin{bmatrix}1\\\\1\\end{bmatrix}.", "71b87efb765bca806e6816a9e95a878a": "h_2(n)", "71b89dd824c3bdda491cf948577b7831": " \\frac{u_j^{n+1} - u_j^{n}}{k} = \\frac{1}{2} \\left(\\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2}+\\frac{u_{j+1}^{n} - 2u_j^{n} + u_{j-1}^{n}}{h^2}\\right).\\, ", "71b8d95a1d7298d4ec9757bc9c160251": "2^{10} = 1,024 \\approx 1,000 = 10^3", "71b8ddec7be5d32a384c56d034fbd62f": " \\Delta T ", "71b989551ff85a1020782a626bfab456": "(\\Omega, P)\\,\\! ", "71b9b934a01239b995a89fa4f26b746f": "L^n M = {\\rm Map}(S^n, M)", "71b9d17616660d24cc38e192e52afbcb": "\\,S_c |c+\\rangle = \\frac{\\hbar}{2} |c+\\rangle", "71ba5460eb11f8c4dc48ff83cb9909f3": " \\sim \\!", "71ba586e7265a6211f90923f0e9e25fa": " \\delta \\langle B|A\\rangle = i \\langle B| \\delta S |A\\rangle,\\ ", "71baabd4e2826557794e3416b62f17dc": "10^{18}cm^{-3} ", "71bb41d2bf3b185b6a90e1a5b41a49b5": " (\\neg L_1 \\or \\cdots \\or \\neg K_1 \\or \\cdots \\or \\neg K_m\\ \\or \\cdots \\or \\neg L_n)\\theta ", "71bb45ae3d969f8454b34d08be19d6b5": "\\begin{bmatrix}Y' \\\\ U \\\\ V \\end{bmatrix} =\n\\begin{bmatrix}\n 66 & 129 & 25 \\\\\n -38 & -74 & 112 \\\\\n 112 & -94 & -18\n\\end{bmatrix}\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n", "71bb5a25c5faaac75083edbb0ebf00b0": "\\overline 3", "71bb662d3ecda264d683d3fcd7961142": "\\tfrac{\\lambda}{M+\\lambda}", "71bb9727bb1d98b59ad6575d5b8b75cb": "\\frac{d^\\alpha}{dx^\\alpha} f \\overset{def}{=} \\frac{d^{\\lceil\\alpha\\rceil}}{dx^{\\lceil\\alpha\\rceil}} I^{\\lceil\\alpha\\rceil-\\alpha}f", "71bc08fd4621610499e1b7139d1323c4": "f(y,x)=ay^2+bx^2-r^2.", "71bc102ae7ab0469e8573e9f170ccda2": "D = 0", "71bc6acc6b254aaf1f00da503689770c": "\\sum_{k=1}^{n}\\mathbf(2k-1) = n^2.", "71bc6d5f3964923f1aaea392b995e7e9": "\\frac{\\partial}{\\partial{z}}\\frac{\\partial}{\\partial{z}}P_c^p(z_0)", "71bc86f255731676ea877766ac8ee0e4": "\\text{what you took out} + \\text{what's left over} = \\text{original thing}", "71bcd093bb9a7ea6f45a278c1133474c": "\n\\text{Maximum Profit} = \\left[ K_u - 2 \\times \\left( K_u - K_l \\right) + C_n \\right] \\times N\n", "71bd38738239a669364a22df666eb672": "AP_+=\\alpha P_+ \\quad AP_-=-\\alpha P_-", "71bd4cd5370ab1f22d5a81ce848b5dc9": "A \\otimes A", "71bd64bb2c8e67d11b4efac03c3b22bd": " V_\\mu \\otimes V_\\nu = \\bigoplus_\\lambda g_{\\mu \\nu}^{\\lambda} V_\\lambda. ", "71bd6fabb424081593916ffa6820d9c8": "\\forall x. \\lnot (Sx = 0) ", "71bd799b73386abbbaa2015b8bfb536a": " \\omega_m =\\omega'_0\\sqrt{\\frac{-1}{Q^2_L}+\\sqrt{1+\\frac{2} {Q^2_L}}} ", "71bdcaeac6f49e05c5da8c5379b11562": " \\left[ \\frac{\\pi}{2^j}, \\frac{\\pi}{2^{j-1}} \\right]", "71bdcffe117d643f965d2635cbcc7e97": "\\mathcal{S}\\subset\\Pi^{n}", "71bdda8b35a934812cd71a98f49129d6": "(P \\or P) \\to P \\, ", "71bdf4d7e5f30a109ce56a57583bd280": "\n\\nabla_{\\hat{\\mathbf{h}}^H} (e(n))= \\nabla_{\\hat{\\mathbf{h}}^H} \\left(d(n) - \\hat{\\mathbf{h}}^H \\cdot \\mathbf{x}(n)\\right)=-\\mathbf{x}(n)\n", "71be09d714e8905a6d930463918fe973": "\\textstyle B(\\mathbf{c}_j)", "71be21140558b021776bdc0af357750a": "f(x) = x^4 + \\sin (x^2) - \\ln(x) e^x + 7\\,", "71be4d81c2a17ef07898711517f0235c": " \\langle \\operatorname{T}_\\pi(f) \\xi \\mid \\eta \\rangle = \\int_X f(x) d \\operatorname{S}_\\pi (\\xi,\\eta)(x) ", "71be9f56791e8e0d9de7f4ff2e6faa1f": "u_{mf} = \\frac{(\\rho_p-\\rho)gd_p^2}{150\\mu}\\frac{s^3}{(1-s)}", "71bec6905271edcd19c0ff0d69006cea": " P_k ", "71bec9a71bc9d2f49946d4b85baadfec": "\\mathrm{adj}(\\mathbf{A}^{m}) = \\mathrm{adj}(\\mathbf{A})^{m}.", "71bf652ea1cbb27a49fb837808c95e5e": "{\\bar{P}}_5", "71c097a22d5861730d5e82eee07b5099": "\\infty \\times 0 = 0", "71c15fe6e62d85ebe2a50d63a8503777": "\\frac{d^2 \\Delta z}{dt^2} = - g^\\prime = - g (\\rho_0(z_0)-\\rho_0(z_0+\\Delta z))/\\rho_0(z_0) \\simeq - g \\left(-\\frac{d\\rho_0}{dz} \\Delta z\\right)/\\rho_0(z_0)", "71c1ea4dbfb4fbb004a80659f39133a2": "\\begin{align}\n f(k; r, p) & = \\int_0^\\infty f_{\\text{Poisson}(\\lambda)}(k) \\cdot f_{\\text{Gamma}\\left(r,\\, \\frac{p}{1-p}\\right)}(\\lambda) \\; \\mathrm{d}\\lambda \\\\[8pt]\n & = \\int_0^\\infty \\frac{\\lambda^k}{k!} e^{-\\lambda} \\cdot \\lambda^{r-1}\\frac{e^{-\\lambda (1-p)/p}}{\\big(\\frac{p}{1-p}\\big)^r\\,\\Gamma(r)} \\; \\mathrm{d}\\lambda \\\\[8pt]\n & = \\frac{(1-p)^r p^{-r}}{k!\\,\\Gamma(r)} \\int_0^\\infty \\lambda^{r+k-1} e^{-\\lambda/p} \\;\\mathrm{d}\\lambda \\\\[8pt]\n & = \\frac{(1-p)^r p^{-r}}{k!\\,\\Gamma(r)} \\ p^{r+k} \\, \\Gamma(r+k) \\\\[8pt]\n & = \\frac{\\Gamma(r+k)}{k!\\;\\Gamma(r)} \\; p^k (1-p)^r.\n\\end{align}", "71c20aed96af0a13a407f9d6a75b9488": "0\\leq P(E)\\leq 1\\qquad \\text{∀} E\\in F.", "71c234cc7154e5d52ac02083d3dedfe9": "S_{mk}^{}", "71c25337de3a20f3c5900ab33ef5306c": "\\mbox{eGFR} = \\mbox{170}\\ \\times \\ \\mbox{Serum Creatinine}^{-0.999} \\ \\times \\ \\mbox{Age}^{-0.176} \\ \\times \\ {[0.762\\ if\\ Female]} \\ \\times \\ {[1.180\\ if\\ Black]} \\ \\times \\ \\mbox{BUN}^{-0.170} \\ \\times \\ \\mbox{Albumin}^{+0.318}", "71c2a7a8fa6b264c9add81a36345eb69": "{\\mathbb C}", "71c2c7b10b5ff11cef5d2e92b4c65e2d": "k \\approx aF^b(T_{2lm})^c", "71c3322ba7a83877536c4e8461f4f677": "h_{\\text{out}}(G) = \\min_{0 < |S|\\le \\frac{n}{2}} \\frac{|\\partial_{\\text{out}}(S)|}{|S|},", "71c33db4f991bb78d960b4fcab09773e": "\\scriptstyle E(x,y)", "71c372dd1855db27170e1a2af5595586": "\\mathbb{U}/P", "71c37e2de6284bf6bc4b2abd7d4193d0": "\\int \\frac{x}{x^2+a^2}\\ln(x^2+a^2)\\; dx = \\frac{1}{4} \\ln^2(x^2+a^2)", "71c380a58c57515058dfb87e20888add": " S_x(t) = e^{-\\int_x^{x+t}\\mu(y)\\, dy\\,}. ", "71c39a3b6e072f298064c42a7e8bf354": "1.2903", "71c43def16efeb540aec9ad2b647e6ee": "\\lim_{t \\rightarrow +\\infty}X(t)=K", "71c45d86b52b25deb456ece7b97694a0": "P = (1,v(1))", "71c4aca38f40719d8089981c2b142a41": "\\operatorname{diag}(1, ..., 1, \\det(\\mathbf{UV}^T))", "71c51a029815a94b8f803080d9d084a8": "\\int_{0}^{T} X_{s-} \\circ \\mathrm{d} Y_s = \\int_0^T X_{s-}\\,\\mathrm{d}Y_s+ \\frac{1}{2} [X,Y]_T^c,", "71c531da6e7bd3cbda8f230cd09962ab": " \\mathbf{R}^o = \\sum_{e} \\big( \\mathbf{Q}^{oe} + \\mathbf{Q}^{te} + \\mathbf{Q}^{fe} \\big) ", "71c549e21c4e71cec0f9202e5339dcbb": "z_w ", "71c5f5a36c6f5b5df8601b753fe15153": "Z=\\sum _{i=1}^m n_i \\bar{Z_i},", "71c621d8c563a76f4504c70717127b66": "M_p \\times_G EG \\to M_p/G = N", "71c62e46278bffddd4effdc615608c9f": "(f*\\Delta)^\\wedge = \\hat{f}\\widehat{\\Delta} = \\hat{f}\\Delta", "71c66542c4fb0dc8a5c6871155dfb8f3": " A + S \\leftrightharpoons A - S ", "71c6a3fcac6705bec2ddef27188ab623": "M = \\left( \\begin{array}{cc} \\cos kL & \\frac{1}{k} \\sin kL \\\\ -k \\sin kL & \\cos kL \\end{array} \\right),", "71c6c0b5735ce03f9a66574e5f0e0c1b": "P e ^ {rt}", "71c6ee74718b12c750e5ad144597e4a2": "p_c,", "71c6f5004e852df73487754b0dda7bb1": " \\int f(x) d_q x = (1-q)x\\sum_{k=0}^{\\infty}q^k f(q^k x). ", "71c72eef8ab79e946de4a830f577d701": " \\mathbf{N}(s) = \\frac{\\mathbf{T}'(s)}{\\|\\mathbf{T}'(s)\\|}, ", "71c7370cfa450555c5fc7287addb78d8": "j_0=f^2\\partial_0\\theta(t)", "71c738fc93d28b652a89cc7c1211a6f4": "\\operatorname{E}(\\mathbf{Y}) = \\boldsymbol{\\mu} = g^{-1}(\\mathbf{X}\\boldsymbol{\\beta}) ", "71c749e14749773898806af556988183": "\\rArr ", "71c7f3018b372068413945a03879c2fe": "\\begin{align}\n PV &= D_1 e^{-(r)(\\frac{\\Delta t_1}{m})} + D_2 e^{-(r)(\\frac{\\Delta t_2}{m})}\n\\end{align}", "71c838cdc17592e47d6691e8b4e060be": "W=0", "71c8810b2e3708f2f3290a9686a8af5f": "f(\\beta,0)=-\\frac{1}{\\beta } \\ln\\left[e^{\\beta J}+ e^{-\\beta J}\\right]. ", "71c89bd52539e07e40d964282a1f3e0e": "\\epsilon_0^1(p)", "71c8b38659b72c3bfa632a4d2d62fae1": "C^{jm}_{kqj'm'}=\\langle j'm';kq|jm \\rangle", "71c8baab4405f390852403b811e89aef": "\n\\sum(X_i-\\mu)^2=\n\\sum(X_i-\\overline{X})^2+\\sum(\\overline{X}-\\mu)^2+\n2\\sum(X_i-\\overline{X})(\\overline{X}-\\mu).\n", "71c8d872c91cb1458afca072d9a6fe9b": "\\begin{align}\n{}_2F_2(a,b;c,d;x)=& \\sum_{i=0} \\frac{{b-d \\choose i}{a+i-1 \\choose i}}{{c+i-1 \\choose i}{d+i-1 \\choose i}} \\; {}_1F_1(a+i;c+i;x)\\frac{x^i}{i!} \\\\\n=& e^x \\sum_{i=0} \\frac{{b-d \\choose i}{a+i-1 \\choose i}}{{c+i-1 \\choose i}{d+i-1 \\choose i}} \\; {}_1F_1(c-a;c+i;-x)\\frac{x^i}{i!},\n\\end{align}", "71c8e9384b1599785c5cfe6525eddbf9": " v = v_L = v_C \\,", "71c925e7db29293fdeed99ebacd2a34a": " \\chi=n_{+}-n_{-}=1\\,", "71c967853b4ca64e81b296e1d81ac58d": "E_{SOFC} = \\frac{E_{max}-i_{max}\\cdot\\eta_f\\cdot r_1}{\\frac{r_1}{r_2}\\cdot\\left( 1-\\eta_f \\right) + 1}\n", "71c97891f2c920127c3d643a4078f0ed": "d\\ln W=\\alpha\\,dN+\\beta\\,dE", "71c984531d8a8ed60d5ab18cf02d3a63": "\\gamma(t')\\in U_t", "71c9fa475b9d5e17088d1f8a83aaac71": "\\begin{align}\\dot{x} &= f(x) + g(x)u \\qquad &(1)\\\\\ny &= h(x) \\qquad \\qquad \\qquad &(2)\\end{align}", "71c9ff25f0c09814847f7b68f43f55ef": "\\approx \\!\\,", "71ca3034f38690317ae3fdbd3c69569b": "t=256", "71ca518572278af1db1e8339879aa6d6": "\\mathrm{m^3}", "71ca59f9e32aafaa4c283cd4437cb435": "Q = \\frac {currentChips} {startingChips} \\times \\frac {currentNumPlayers} {startingNumPlayers + numTotalRebuys + numTotalAddOns}", "71cacb67c0cbd2b30196428e76b29547": "p(D) = p(D | drunk)\\,p(drunk)+p(D|sober)\\,p(sober)", "71cb6d9d13ef8f420bea632c589fc30b": "B+Z \\rightleftharpoons BZ ", "71cb837ffa6d694cf6ed97e0b12d47ab": "z_3 = \\frac{1}{12} (X^2Y + XY^2 - 2XYX + Y^2X + YX^2 - 2YXY)", "71cb939d2655393929b198c904c43b03": "x^2 = px + q", "71cbf7e9e1bb769f84e620c1e0457d85": "d = \\lceil {(\\ell -1)}/2\\rceil \\log n +1 = \\lfloor \\ell/2 \\rfloor \\log n +1", "71cc0d484529afeadfb2934c1aebea5a": "U_G=\\frac{1}{2}\\mu \\sum_{(W_i,W_j,W_l) \\in G} \\|W_i -2W_j+W_l\\|^2 ", "71cc1b7911801f7ed13c6354a808197f": "\\vec\\rho = (A^T A)^{-1} A^T \\vec{p}", "71cc668606a3880a476c387ddd48a1e3": "\\{x \\mid \\phi\\wedge \\psi\\} = A \\cap B", "71cc6982ebacce408517b7710bae82ee": "\\scriptstyle4\\sqrt{\\frac{2}{3}}", "71ccaccfde849fe19b54c76c56991355": "g_\\lambda", "71ccb17f13d13c36b537dd549558ad07": "d_1, d_2, ..., d_n", "71cd0363666c8a3330c5a6967634907a": "S^2(k)=\\frac{1}{m_k-1}\\sum_{i=1}^{m_k} (\\bar{X}_i(k)-\\bar{X}(k))^2", "71cdc6e3d3d8bb1b10de9a66a529a422": "z \\mapsto {{az+b}\\over{cz+d}}", "71cdf6b026e714076b05e23b84b3c576": "\n{{d\\dot C_{FF} } \\over {dt}}\\,\\,\\, = \\,\\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,Q(t)\\,\\, - \\,\\,\\lambda \\,\\dot C_{FF}", "71cdfd42def9da9ad2050fa9f3a4c1e0": "\\sum F_i=m\\frac{dV}{dt}+v_e\\frac{dm}{dt}", "71ce182b23ea82e97ef99950534a709b": "\\mathrm{Hi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\exp\\left(-\\frac{t^3}{3} + xt\\right)\\, dt.", "71ce2442afe5d02e7bc43e8515ca4b20": "\\alpha = \\tfrac{2}{\\pi} \\min_{0 \\le \\theta \\le \\pi} \\tfrac{\\theta}{1 - \\cos \\theta}", "71cf7cba30b743a6b18638902b56eba9": "\n E(\\theta, \\sigma^2) = -\\sum_{j=1}^N \\log \\sum_{i=1}^{M+1} P(i)p(s|i)\n", "71d029b9b59385b58e133907153074b0": " U(x, v) = \\ln(x) + v ", "71d03d3a70d33363c44d7f02d65f468b": "\\mathbf{1}\\{x_i \\sim y_i \\in a\\}", "71d09f1f1bfb42251c86ab9dc18592e8": "\\mathrm{SL}(n,\\mathbb C)\\,", "71d10eacc39473fe9f77070f41274f23": "R^{(1/p)} \\cong A[X_1, \\ldots, X_n] / (f_1^{(1/p)}, \\ldots, f_m^{(1/p)}).", "71d1744c3c7a8477b903c32a16d970c7": "L_{uu}", "71d18c91edd778e59ef745570da62ccc": "r* = \\Sigma r_i", "71d1971641519560d66283a9d792a4d5": "\\mathbb{I}(\\tau<\\infty)", "71d1a004a98d8b73539b0728e35061f3": " \nV^0_{\\textrm{th}}=\\frac{\\Delta G^0}{n\\cdot F}=\\frac{285.9 \\ \\textrm{kJ/mol}}{2 \\times 96,485 \\ \\textrm{C/mol}}=1.48 V\n", "71d1b0eb2db4634151c8d4cf92977385": "E(\\gamma_n):=\\left \\{(x,v)\\in\\mathbf{P}^n(\\mathbf{R}) \\times\\mathbf{R}^{n+1}:v\\in x\\right \\}.", "71d1c773b67cb4e6a360cc81a8c6ef08": "\\mathcal{C}=", "71d20cb32c7af4c213269132dcd4154b": "\nCU(C,F) = \\left [p(c) \\sum_{i=1}^n p(f_i|c)\\log p(f_i|c) + p(\\bar{c}) \\sum_{i=1}^n p(f_i|\\bar{c})\\log p(f_i|\\bar{c}) \\right ] - \\sum_{i=1}^n p(f_i)\\log p(f_i)\n", "71d23350e00e336175df494082f27503": "f(x) = x^3 + x^2 - 5x + 11\\,", "71d2941565bf0b50681e06face5e0c8c": "\\int_0^{\\infty}\\frac{e^{-\\alpha x}}{\\Gamma(x+1)}dx = e^{e^{-\\alpha}}-1+\\int_0^{\\infty} \\frac{1-e^{-x}}{(\\ln(x)+\\alpha)^2+\\pi^2}\\frac{dx}{x} \\qquad \\qquad \\forall \\alpha \\in \\mathbf{R}", "71d2a9cda7116c9502e6ae05ae6a2744": "\\hat{G}(z)", "71d2b50290923673938c3a767c83abc4": "\n R^\\gamma_{\\alpha\\beta\\rho} = 0 ~;~~ g_{\\alpha\\beta} = C_{\\alpha\\beta}\n", "71d2c46af01feeea54a0f541243e297b": "T2", "71d2ff5d95802075a47d2256fd0d64a8": "\\{1, 2, 4, 5, 7, 8\\}", "71d30b435abd419546a46e3c15b85ea3": "\\gamma_{SG}", "71d33a97933381e3ab0e04e6b724d49f": "k>\\frac{n}{p}", "71d35b56689b083e8241b6fc9748150b": "x^5 + px + q = 0", "71d38814b7eb99c2fc6d53a3e0c32af0": "\\scriptstyle d, \\mathcal{N} ", "71d3ab99014a3d9a0aac5457f4c7fb07": " F_{2,B}(x,Q^2) = f_B(x)\\log{\\frac{ Q^2}{\\Lambda^2}} + ... ", "71d3aee1a3c89680f780651eea2c6574": " S/r \\equiv L/r \\,\\!", "71d3cf6e1887f938f7dcd2e24fbeb7bb": "\\bar{\\psi}\\gamma^\\mu\\gamma_5\\psi", "71d3d0a8d3dfa12e2de48a05aafe337b": "\\mathrm{0123456789} \\!", "71d41e456c7069d1701deabb248e78b5": "(1+\\epsilon_i)", "71d434e7a559520aeed69f6a485141dd": "\\frac{4 \\sqrt{2}}{\\sqrt{5}} = \\frac{4 \\sqrt{2}}{\\sqrt{5}} \\cdot \\frac{\\sqrt{5}}{\\sqrt{5}} = \\frac{4 \\sqrt{10}}{5}", "71d44b349fe62a69d6067a5996748206": " ADF = 100 {(p-m) \\over p} \\approx 100 {(OG-FG) \\over (OG-1)}", "71d4825400b7c084b044598b427841d9": "\n\\left(\\frac{\\alpha}{a }\\right)_m =\n\\left(\\frac{a}{\\alpha }\\right)_m. \n", "71d4beabab730f61690080311a0e79ce": " \\begin{align}\n\\mu = \\operatorname{E}[X]\n &= \\int_0^1 x f(x;\\alpha,\\beta)\\,dx \\\\\n &= \\int_0^1 x \\,\\frac{x^{\\alpha-1}(1-x)^{\\beta-1}}{\\Beta(\\alpha,\\beta)}\\,dx \\\\\n &= \\frac{\\alpha}{\\alpha + \\beta} \\\\\n &= \\frac{1}{1 + \\frac{\\beta}{\\alpha}}\n\\end{align}", "71d4c44d6017d1decd23ff7d46dd0e0f": "Q_2(X)E_2(X) \\le {2e + k - 1}", "71d52d904208df638f111f0077b52637": "({\\color{blue}d_1}, {\\color{blue}d_2}, {\\color{blue}d_3}, {\\color{blue}d_4})", "71d53190b4e327b0240d527583b136db": "\n\\sum_{d|n}\\lambda(d) =\n\\begin{cases}\n1 & \\text{if }n\\text{ is a perfect square,} \\\\\n0 & \\text{otherwise.}\n\\end{cases}\n", "71d60524d5d69540accfd7276f2b54bf": "A Q = Q D ", "71d6bb1d17abe82a589a75cd6aa886a6": "d \\equiv 2 D \\tan \\left( \\frac{\\delta}{2} \\right)", "71d6dc157615f0fc97e3e4c5e64e4f9d": " v(x,\\tau)=\\exp(-(1/2)x-(9/4)\\tau) u(x,\\tau) ", "71d6fffe8440f266e4242c8693ec9233": "\\left \\{ b ( f^n ( s ) ) \\right \\}_n", "71d7133347ced18786b1bd12665c0e71": "\nRCA_ij = \\dfrac{\\dfrac{x_ij}{X_i}}{\\dfrac{x_aj}{X_a}}\n", "71d72213fe9094ad8ec04a14b09bf476": "M := \\mathbb{R}^{2} \\setminus \\{ 0 \\}", "71d72bf2c89c1ebab981475f8f5a5961": "H_{rot}=B\\mathbf R^2=BR(R+1)", "71d74c985772f978cf31d5727998659e": "\\mathbf{Y} = [y_1,\\ldots,y_n]^\\top", "71d75221f6b884d21f6599b9c28b3b31": "\\dot{\\mathbf{p}_i} = m_i \\dot{\\mathbf{v}}_i = m_i \\mathbf{a}_i", "71d78f148b2e162fd2612beb5bac030f": "H(U)=-\\sum_{i=1}^R P(i)\\log P(i)", "71d7d0b5213d55e4423153a9b87a1c01": "y = c_1 x^\\alpha \\cos(\\beta \\ln(x)) + c_2 x^\\alpha \\sin(\\beta \\ln(x)) \\,", "71d80861e72c4ff83710fdf6afcbd387": "\n\\mathbf{w} \\sim \\mathcal{N}(0,\\mathbf{I}) \\propto \\exp(-\\|\\mathbf{w}\\|^2).\n", "71d815ce022bc52c9933dd0691b0bbe3": "\\dim(U_1 + U_2) = \\dim U_1 + \\dim U_2 - \\dim(U_1 \\cap U_2)", "71d83554ca5c775865ead96edddec0e5": "s = m'^{-1} * (h + xr)", "71d843a14e982ca4b852039787c183b2": "=(r_{1})^2", "71d849fe1cf8ae5a81227e168a2c7d62": "\nJ_z = \\frac\\hbar2\n\\begin{pmatrix}\n3&0&0&0\\\\\n0&1&0&0\\\\\n0&0&-1&0\\\\\n0&0&0&-3\n\\end{pmatrix}.\n", "71d86e3696204755411d8ec06df5afaa": "\\Delta(x) = x \\otimes x;", "71d89c746b6e94ea9aeb0a3524898abe": "(f \\circ T^n)_{n \\ge 0}", "71d8c06470e49f3f798be203e64e8c4f": "\\mbox{normal}(g(v)) = \\frac{g(v) - \\min(g)}{\\max(g) - \\min(g)}", "71d8fb762fdcac308298e5ebc1c66046": "\n\\Phi(t;\\alpha,\\beta,\\lambda,\\mu) = \n\\exp\\left[~it\\mu\\!-\\!|\\lambda t|^\\alpha\\,(1\\!-\\!i \\beta \\operatorname{sign}(t)\\Omega)~\\right] ,\n", "71d8fe23b712032ae497caf0673d9dc1": " \\neg (\\operatorname{def}[F_2] \\and ...) ", "71d94c72b53f2d51b32f8c9faf9a305f": "\\cdot : M \\times M \\to M", "71d9bd62d24dc18517495bffedf622ac": "a = 2b", "71da223d34eef543431d000980da2813": "ax + b \\ (a \\ne 1)", "71da4b482b4260de7bb5c91a6215016b": "\\oint_{\\partial \\Sigma} \\mathbf{E} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = - \\frac{d}{dt} \\iint_{\\Sigma} \\mathbf{B} \\cdot \\mathrm{d}\\mathbf{S} ", "71da85e8c124a056172560097399f9db": "0 = T^{\\mu \\nu}{}_{,\\nu} = \\partial_{\\nu} T^{\\mu \\nu}. \\!", "71da8b996d10129d4fca14e6dd4cd100": "(X_0,X_1,...)+(Y_0,Y_1,...)=(X_0+Y_0,X_1+Y_1+(X_0^p+Y_0^p-(X_0+Y_0)^p)/p,...)", "71dab793f6d5c6730a8a7355c51d8d97": "\n \\begin{align}\n K_{\\rm I} & = \\frac{F_x}{2\\sqrt{\\pi a}}\\left(\\frac{\\kappa -1}{\\kappa+1}\\right)\n \\left[G_1 + \\frac{1}{\\kappa-1} H_1\\right] \\\\\n K_{\\rm II} & = \\frac{F_x}{2\\sqrt{\\pi a}}\n \\left[G_2 + \\frac{1}{\\kappa+1} H_2\\right] \n \\end{align}\n", "71dabe781cbb6070cb9307456cb00f60": "d = \\frac{v_2^2 - v_1^2}{2a}", "71daceac9d397543ee32131777bfa4bf": "\\kappa_j'-i+\\alpha(\\kappa_i-j+1)", "71daeb8535369d842adba6222f771351": "\nv = \\frac{1}{\\hbar} \\nabla_k E(k)\n", "71daeefde7037d90f22d5539cd1f1ee6": "P(\\{H,T\\}) = 1", "71db33dbdbd194cc7efa5672bf493cf8": "{\\mathbb R}", "71db50b9dd0c8632859b71be9d0333d8": "\\int_X \\! |f(x)| \\, d\\nu (x) < \\infty. ", "71dc488341d4de8c4f34069eccae87b5": "\\mathit{PPV} = \\mathit{TP} / (\\mathit{TP} + \\mathit{FP})", "71dc85dbedacf975b48e2f94856e251f": "F(0)=f,\\,F(1)=g", "71dcb6c12920511e3b9f31df6b460f47": "x_i\\otimes x_j \\mapsto q_{ij}x_j\\otimes x_i", "71dcee4f4f2184fe7214081d7c3981a7": "H_{(1)} \\ldots H_{(k)}", "71dd185d8e8aaa28dd8ae94f4a00671a": "\\frac{-\\alpha \\pm \\sqrt{\\alpha^2-4}}{2}", "71dd9ae96019852f0226ad0c82c21919": "s_n = \\sum_{k=0}^n (-1)^k {n\\choose k} a_k.", "71dddb5c70fbf148aafd751ef0315b07": "C^r", "71ddec029f31b67acbcd40d5572a978c": " L^2_{\\operatorname{P}}(\\Omega;M) \\rightarrow L^2_{\\operatorname{P}}(\\Omega;N). ", "71ddf3813d7751f7ae1e029bf7142c4e": "\\tfrac{1}{2}\\pi - \\mathrm{gd}\\,x", "71de1897c83a04d34282e44e0f635a2b": "x'\\in\\bigcap_{n=1}^\\infty C_n", "71de1ad43ee9759d85d7559c4ce69442": "\\|L_x\\|_{(\\ell^p)^*} \\stackrel{\\rm{def}}{=}\\sup_{y\\in\\ell^p, y\\not=0} \\frac{|L_x(y)|}{\\|y\\|_p} \\le \\|x\\|_q.", "71deb16417df7ae33e43fb9e5a566d9d": "H = \\sqrt{{\\frac{5-\\sqrt{5}}{10}}}\\,a \\approx 0.5257\\,a.", "71defbd88e70f681e98f899603416db4": " f(m|n) = \\frac{r(m) l( \\vec X(n|m)) }{ \\sum_{m'=1}^M { r(m') l( \\vec X(n|m')) } } ", "71df1fc4dd83e2d215b46a3b629bfad7": "B^n y^n \\le B^n x + \\alpha", "71df33d5efa744d019768e981adbf4a3": "\\{R_a:a\\in\\Sigma\\}", "71df6337b07806058a820ddf38d75812": "H_2 O", "71df802206705cca2047d525473fde7a": "q_4 \\ge 0", "71dfc83b056925769bb6bf619829edb2": "x, 1/(1-x), (x-1)/x,", "71dfd9827dce994acc3d15382a9c2ebe": "A \\rightarrow B: A", "71e0032667cbc6ed307ec6d48784d285": "Q_{CA} \\,", "71e009a1d62c9eccb99e3002cc6b8a25": " W_{t_2} = ( W_{t_2} - W_{t_1} ) + W_{t_1} ", "71e0148bec947d5493f33a09af5b3a3f": " \\forall l>0", "71e0a64292f4f30aee90374d5bfa8e12": "O\\left(\\sqrt{\\frac{d}{n}}\\right)", "71e0a94316b1e85fea800e2609fb9272": "\\big.\nU\\left( \\{r_i\\} \\right) = \\sum_{i=1, i K \\, ", "71ed6cd9a4526354ace77827d766c539": "\\dot{f}(s)", "71ed90b05829e6556a5d5f2435eeb438": "\\mathfrak J^k(a)_n=b_n=(\\mathbf E-k)^na_0", "71edadaa6f463fc71340668eb4b00876": "[n]_q!(q-1)^n q^{n \\choose 2}.", "71edef95a6eab5d7f200cfb1b54a9eb7": "Y=f(\\eta_1^\\top Z,\\ldots,\\eta_k^\\top Z,\\varepsilon)", "71ee22242708ad2538eece9976de68f4": " \\dot{\\boldsymbol{v}} = (\\sigma - \\sigma_0)\n\\boldsymbol{v} - |\\boldsymbol{v}|^2 \\boldsymbol{v}", "71ee2f7ceaa9cf2d25c3ee1dc78f9438": "\n\\phi=\\sqrt{\\varphi_x^2+\\varphi_y^2+\\varphi_z^2}.\n", "71ee5abafdd46cf79a7824fb294d1bf0": "\\{\\phi_i\\}_{i\\in J}", "71ee642f9e92aaa7da62ba7f051eb227": "\\frac{1}{3} = 0.333\\dots = 0.\\overline{3}", "71ee755df667c2c8b6c6b55731d1264f": "{\\upsilon}_{out} = A_v \\ {\\upsilon}_{in} \\begin{matrix} \\frac {R_{L}}{R_L + R_{out}} \\end{matrix}", "71ee9dc85ea59bc9c85d8fbb0551f30b": "A\\rtimes K", "71eed60fe638b7d761a333d0dfdf80fc": "\\frac{1 + {\\scriptstyle\\frac{1}{4}}z}\n{1 - {\\scriptstyle\\frac{3}{4}}z + {\\scriptstyle\\frac{1}{4}}z^2 - {\\scriptstyle\\frac{1}{24}}z^3}", "71ef21e381788d4f121513c63c661835": "\nKZ_{m,k=1}[X(t)]= \\sum\\limits_{s=-(m-1)/2}^{(m-1)/2}{X(t+s)} \\times \\frac{1}{m}\n", "71ef374c89055ef500e1439d3f2d3d16": "\\lambda = \\frac{\\nu^2}{2\\sigma^2}.", "71ef567492699837dadc52038c027e4c": "\\sum_{j=1+n_1}^{n-n_2}{\\frac{R_{(j)}^T}{n-(n_1+n_2)}-K})^{+}.", "71ef5c014150e028e8c11e82285417a2": "\\begin{Bmatrix} \\infin \\\\ \\infin \\end{Bmatrix}", "71ef835258ad9a77f1c9fe907330d57e": "0 k. \\end{cases}\n", "71f0bccd64e234bda115c039566fe944": " W = \\int_{\\theta_1}^{\\theta_2} \\tau\\ \\mathrm{d}\\theta,", "71f0edbccbf8c8fe940df368972a94e3": " \\mbox{displacement} = \\mbox{bore}^2 \\times 0.7854 \\times \\mbox{stroke} \\times \\mbox{number of cylinders}", "71f14c6cc9e718353dc613f6f06b0623": "B^2 - 4AC .\\,", "71f1d596b76b7ede834c0af169ad128b": "C_{\\alpha \\beta}", "71f1d789742570787e1c22825311b1d8": "L \\subset k'(x_1^{1/q}, ..., x_d^{1/q}).", "71f1d7f55b1c7f0fdc76bd2c5b81957e": "\\int_S f\\,d\\mu \\le \\liminf_{n\\to\\infty} \\int_S f_n\\,d\\mu\\,.", "71f1f17fcfa35a74886c38d8ce0a24ba": "I = \\frac{\\dot\\gamma d}{\\sqrt{P/\\rho}},", "71f224992c42f1561cf88156ad0faf1a": "a \\to 0", "71f22c67bfbf16ab1b6fa04d3c3682c8": "\n C_5 = -\\frac{625}{12EI}(-5675 + 8 M_c + 240 R_a) \\quad \\text{and} \\quad\n C_6 = \\frac{250}{EI}\\left(3R_a -70\\right) \\,.\n ", "71f24ce7eb0623931581acaf738e9602": "\n\\begin{bmatrix}\nt' \\\\ x'\n\\end{bmatrix} =\n\\frac{1}{\\sqrt{1 + \\kappa v^2}}\n\\begin{bmatrix}\n1 & \\kappa v \\\\\n-v & 1\n\\end{bmatrix}\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix}.\n", "71f2a8bf41471c1018abc867414773e9": "\\operatorname{div}(\\mathbf{F}\\times\\mathbf{G}) \n= \\operatorname{curl}(\\mathbf{F})\\cdot\\mathbf{G} \n\\;-\\; \\mathbf{F} \\cdot \\operatorname{curl}(\\mathbf{G}),", "71f2d8051452130f54d4631f9c4a7843": "a = 2mn,\\quad b=m^2-n^2,\\quad c=m^2+n^2", "71f2d91885ba54027a9c0e760cc340c2": "{\\varphi^\\prime}_u = \\varphi_u + 2\\beta\\sin\\left(\\frac{\\varphi^\\prime + \\varphi}{2}\\right),", "71f34483c9305df077977b0bf5f79a4c": "\\ {\\mathcal C}^\\infty", "71f3ac54a28c0a76241483cb78dc8f59": "-\\frac{1}{\\omega^{2}}", "71f412dc18681e5251287cec0108145c": " K(0) = 1 ; \\, ", "71f41eebe4ec8f72822ac12ce3e4f9da": "|\\beta|^2 = \\hbar^2j(j+1) - \\hbar^2m^2 + \\hbar^2m = \\hbar^2(j+m)(j-m+1).", "71f43a4746a9a3ad5c8dcdfe6adc7930": " (-1)^{\\text{sign}}(1 + \\sum_{i=1}^{52} b_{52-i} 2^{-i} )\\times 2^{e-1023} ", "71f4e15140095b12d23357bda40e81e9": "\n[S_x, \\, S_y] = i \\hbar S_z \\quad\\hbox{and cyclically}\\quad x\\rightarrow y \\rightarrow z \\rightarrow x.\n", "71f55a67747e8ba99da38b40094e60fb": "\\phi_t", "71f575636c507a2673ea3f520920e65d": " \\mathcal{H}_C ", "71f5915a09070976b2ff97f2162b4a58": "t^{[k]}=t(t+1)\\cdots(t+k-1),", "71f5d9dd311b73e658da6168118b4c42": "\\hat{H}^{i}", "71f5dda68e0b4941f9ec9e3d531aff93": "\\left(\\frac{\\part S}{\\part T}\\right)_P=\\frac {C_P}{T}.", "71f5e49fb655ac77e0f15d73e8e6e59c": "V = \\operatorname{Var}[x_i\\varepsilon_i] = \\operatorname{E}[\\,\\varepsilon_i^2x_ix'_i\\,] = \\operatorname{E}\\big[\\,\\operatorname{E}[\\varepsilon_i^2|x_i]\\;x_ix'_i\\,\\big] = \\sigma^2 M_{xx}", "71f5e5918054bc064820aecd985c2195": "\\Pr(X>n/2)\\le \\frac{2}{n} E[X] \\le \\frac{1}{n} \\sum_{i=3}^{g-1}p^i n^i\\le gn^{-1/g}=o(1)", "71f62a636c223526e4acb86eac6a22f1": "P_{4} \\ne 1", "71f6317de007b0f7d483a5638c2f4b00": "\\sqrt{1-x^2}P_\\ell^{m+1}(x) = (\\ell-m)xP_{\\ell}^{m}(x) - (\\ell+m)P_{\\ell-1}^{m}(x)", "71f6df2a7443c27b6de4aa407f6f2ad3": "j + \\lfloor js/r \\rfloor", "71f6e4f5a387f83421d979503a35f87c": " \\nabla^2_H ", "71f72403d81f1179636acdbc31566843": "\\displaystyle{|F^{(m)}(z)|\\le K_{N,m} (1+|z|)^{-N}}", "71f88dee52337b5ce2b52f08775ebd21": " Z(E_n) = a_1^n.", "71f8a554bdc3db153f9b2345e76937e7": "\\ f(x) = x^4", "71f8fc328a525f1c47b27aeba6bfcdf5": "\\definecolor{red}{RGB}{255,0,0}\\pagecolor{red}e^{i \\pi} + 1 = 0\\,\\!", "71f956438c96ed304ff32eb5e91552a1": " e^{it \\Delta}. \\quad ", "71f97541a7fd4d563d26621e22e5adb0": "\n \\begin{align}\n I_1 & = \\mathrm{tr}(\\boldsymbol{\\varepsilon}) \\\\\n I_2 & = \\tfrac{1}{2}\\{\\mathrm{tr}(\\boldsymbol{\\varepsilon}^2) - [\\mathrm{tr}(\\boldsymbol{\\varepsilon})]^2\\} \\\\\n I_3 & = \\det(\\boldsymbol{\\varepsilon})\n \\end{align}\n ", "71f9aa3c12d35d276e58f2f44975f2bb": "L^2(\\Omega)", "71f9d6b563dd5071ab6fe2fbed8ee8e0": "\\mathbf{x}_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathbf{x}(t=0)", "71f9e59ae47d14eb49a3def9f2f382bd": "\n\\text{SuperMix}(U) = \\text{ROL} \\left( M \\cdot U +\n\\begin{pmatrix}\n\\sum_{j \\ne 0} U_j^i & 0 & 0 & 0\\\\\n0 & \\sum_{j \\ne 1} U_j^i & 0 & 0\\\\\n0 & 0 & \\sum_{j \\ne 2} U_j^i & 0\\\\\n0 & 0 & 0 & \\sum_{j \\ne 3} U_j^i\n\\end{pmatrix} \\cdot M^T \\right)\n", "71fa303837f5e4bc5f2892cea734267d": "y \\equiv \\frac{g_\\text{F} M_\\text{F}}{g_\\text{GT} M_\\text{GT}}", "71fa40e0b2b8103bf9cfc7b16a923f07": "t_o\\,", "71fa56c98c5dfa180eb2e0caa76cb91f": "\\sum_{n=1}^N x_n^2.", "71fa6364b8882c34d42ffbfd513c24f6": "\\frac{\\nabla S}{m}.", "71fa923a081b8e8af5f868b1ff5cd603": "\n \\begin{bmatrix}Q_1 \\\\ Q_2 \\end{bmatrix} = \\kappa G h\n \\begin{bmatrix} w^0_{,1} - \\varphi_1 \\\\ w^0_{,2} - \\varphi_2 \\end{bmatrix} \\,.\n", "71fb51f1e7e94af011fa53f28ee78caf": "[\\mathbf{v}]_\\times", "71fb5be8db3d0910db52ec8a83818f2b": " \\Gamma_q(x) =[2]_{q^{\\ }}^{\\frac 12} \\Gamma_{q^2}\\left(\\frac 12\\right)(1-q)^{\\frac 12-x}e^{\\frac{\\theta q^x}{1-q-q^x}}, \\quad 0<\\theta<1.", "71fb9ec8c95dea06762e9c6b6880522a": "v_{p1},v_{p2} \\in \\mathbb{R}^3", "71fbe67b0573b1e38770d6a5ea639e34": "A^{*}=I_{2}\\otimes A_{2^{m-1}}=\\left[\\begin{array}{cc}\nA & 0\\\\\n0 & A\\end{array}\\right],", "71fc375e02dd1830b312dd469c3d1310": "c = \\sqrt{a^2 + b^2 + 2ab\\cos \\alpha},\\,", "71fc68f212906486a6deed804eeae9e7": "\\sqrt{K}", "71fc8ddfc1eaa0599f1b285387f4bd6d": "\\Theta \\mapsto \\left|{\\mathbf I} - 2i\\,{\\mathbf\\Theta}{\\mathbf V}\\right|^{-\\frac{n}{2}}.", "71fc9c554049ea10151df10a951250cd": "P(Y_i=0)=\\left(1+\\sum_{n=1}^\\infty \\frac{\\lambda_i^n}{M_i(n)}\\right)^{-1}", "71fcdde9051ba62b92706daf8676c3ac": "H(w_j) = jw_j", "71fce020f9d8bf76723f56f48fd75c19": "H^2-2P^2=-1 \\,", "71fd133c1745858985661dcf08b62bd1": "\\delta(x)=(-1)^x. \\, ", "71fd5068d664cad7708627b7d50f55b6": "\\pi^2/3-(\\log_e(4))^2=1.368\\ldots", "71fd7aa085b2b3c4e582bdcf632782e1": " l = v \\, dt ", "71fdcc92bf4ed645154e02ec56e109e8": "{\\mathcal L}_{xy}^7", "71fdff34e11f31fa3ccddf4c55883313": "G = \\bigl({V,E}\\bigr)", "71fe2c9b40c1dfaf4472618cfe71909a": "f = \\tfrac{J}{1+J}", "71feb8039d17421f7f4b4b14f5d58da8": "D(S)=\\bigcup(R^n(S)\\mid n<\\omega)", "71feb8117dce7c43b2b666546c8ca7fe": " Z=W^N ", "71ff32da449b0109873b0f9a5608ca14": "c_{2r}", "71ff8c4552f820e8a2059eab70f3b152": "\\text{Range Resolution} = \\left ( \\frac { C } { \\text{PRF} \\times (\\text{Number of Samples Between Transmit Pulses}) } \\right)", "720007131e14de1ab8b87cdd9f529bd9": " a=|\\mathbf{E}|\\sqrt{\\frac{1+\\sqrt{1-\\sin^2(2\\theta)\\sin^2\\beta}}{2}}", "7200252221a0cbd97b50464c0d60df01": " \nu{\\partial u \\over \\partial x}\n+\nv{\\partial u \\over \\partial y}\n=\nc^{2}m x^{2m-1}\n+\n{\\nu}{\\partial^2 u\\over \\partial y^2}.\n", "720061abebb2ea158fec7288cff883a9": "\\text{for } f(\\Theta)\\text{, } \\exists i \\in I \\text{ such that } u_i(x,\\theta_i) \\geq u_i(x',\\theta_i) \\ \\forall x' \\in X", "72014d1262ca4bdb6529075b12f9ee39": "B : [0, 1] \\times \\Omega \\to \\mathbb{R}", "720157746d5fff94f830c2d03f745350": "F_n(K,\\theta,\\phi)\\propto\n\\left | \\int_{-\\lambda_u/2\\bar{\\beta} c}^{\\lambda_u/2\\bar{\\beta} c}\\hat{n}\\times\\left ( \\hat{n}\\times\\vec{\\beta} \\right )e^{i\\omega(t-\\hat{n}\\cdot\\vec{r}(t)/c)}dt\\right|^2", "7201ef45e09b3815763cc9cfa80093bd": "r_t \\,", "7202c91f4a91d4bb2620ab88ce16c7ac": "f \\colon X \\rightarrow M", "7202e45be51a326ba7fa1a5169c00272": "p_{surv}\\in[0,1]", "7203754492e4d3cb33e539c609260c13": " \\textbf{A}_P = \\alpha\\times\\textbf{R}_{P/O} + \\omega\\times\\omega\\times\\textbf{R}_{P/O} + \\textbf{A}_O,", "7203b98b9a569922dff672d3eb68a9db": "\n\\begin{matrix}\n\\,\\,23\\\\\n4\\overline{)9^15^30}\\\\\n\\end{matrix}\n", "7203c031fec802dffd2ab9508ce74206": "S_3, S_2, S_1,", "7203ef312a98ada99aa28bd31353f3a9": "1\\le p\\le q\\le\\infty", "72045e6e462eadeefc6976f51c73a96a": "f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'CD' + AB'CD + ABC'D' + ABCD,", "72045f702dd18c34821e145713a7d5e1": "L=2", "720484bc9df4ac5f9fed3e2d0f1527c0": "A_{\\alpha\\beta}=g_{\\alpha\\gamma}g_{\\beta\\delta}A^{\\gamma \\delta}", "72048838dfc455ae93b1738f04b195c6": "u(x,t) = A(x - v_g \\ t)\\sin (kx - \\omega t + \\phi) \\ , ", "72049b6b58359a4ec2ef9f7735f999da": "\\gamma = \\dfrac{\\sum_i \\gamma_i c_{V,i}}{\\sum_i c_{V,i}} = \\dfrac{\\alpha V K_T}{\\tilde{C}_V}", "7204a5994344701fa045baf4be1286b1": " \\tilde{a}_1, ... , \\tilde{a}_{m/n} ", "7204a8dfdb61b3496295b04c057e337d": "X_0", "7204e02277d4eb44ea42ccd84941587a": "{d_{mx}} = {r_m} + {r_x}", "72052d58bfe8576ee3194e191302316f": "\\tilde c_n = \\sum_{m=0}^N V_{nm} c_m", "720587cdefc3a5f65142bf8ad6d69e2d": "f(x) > 0 \\quad \\text{ for } \\quad a < x < b.\\,", "7205966d20fbfdc3971a4923f5c67b38": "\\lambda\\frac{\\partial g_{\\mu\\nu}}{\\partial\\lambda}=\\beta_{\\mu\\nu}(T^{-1}g)=R_{\\mu\\nu}+O(T^2).", "7205dac5484dd02505ddb6ff8491755b": "Ass(A/fA)", "7205f350021ab51159547fd495b1a968": "\nN_i(c) = (s_1, \\ldots, s_l) \\text{ with } s_j = S_i(I_c(j)) \\, \n", "7205f705cac797a425ce8fd6b5da14f4": "M(q)=\\frac{\\mathrm d\\Phi_m}{\\mathrm dq} ", "7206282d3c464efcab3feb8e56c81489": " \\operatorname{lambda-anon}[M\\ N] = \\operatorname{false} ", "72062e38bfb8742807c8b6cbc55c13b5": " \\boldsymbol{ v} = \\begin{pmatrix} v_e \\\\ v_n\\end{pmatrix}\\ ,", "720647a11c055baed6d814a1be93838a": "\\Beta = \\frac{\\varepsilon}{3.7D}Re + 2.51\\alpha", "7206c4ddc569cbdce78d683e1a6c6ee0": "dU\\left(S,V,{N_{i}}\\right) = TdS - pdV + \\sum_{i} \\mu_{i} dN_i", "7206ec763f4e62d43b7916606a50eb58": " \\textstyle f_{j+1}[n] = U(f_j^e[n];f_{j+1}[n]) ", "7206fb63d75438b08393c15076d191e5": "E_2 - E_1 = \\hbar \\omega,", "72070fcc544214d3a2f5bc7657e2a923": "\\scriptstyle \\forall x,\\eta(x)\\leq\\zeta(x) ", "72075a37f35e29118d6328acddd2f201": "G_A = \\frac{P_\\mathrm{load,max}}{P_\\mathrm{source,max}}", "7207782cd1eb15f64a3de6fd3646126c": " \\operatorname{E} ", "72077a903d420e3ee536903c5e537c43": "f(g,h,\\frac{a\\tau+b}{c\\tau+d})", "720781e72cc19ae4357b1f48cb2dd2a4": "b'=\\frac{1}{r}\\mathbf{r}\\cdot\\mathbf{Q}c_0+\\alpha_j'\\big[asc_1+bs^2\\bar{c}_2-\\gamma s^3(2\\bar{c}_3-c_1c_2)\\big]", "72079be4871ca23e4aa7d0404789456f": "\\Vert T_j^\\ast T_k\\Vert=0", "7207cad60d0b18b8c8b4627c87469565": "\nK(x,y) = K(x) + K(y|x) + O(\\log(K(x,y)))\n", "7207ef4d06a5a40ddc3141545aaaed0b": " T_{\\parallel}=\\sqrt{T_x^2+T_z^2} ", "7207f1dd3b685300f4afa9413f98332e": "T_A", "72087da566b6c6c2d7a97469cbf31e8f": "C_\\phi = S^n \\cup_\\phi D^{2n},", "72088a53ab957e953dd7fe2e3545300f": "\n\\lambda=1\n", "7208be78d04d98e4e8d8594ef3080864": " \\sum _{v \\neq v _{0}} (q _{v} - q _{v \\cap w}) ", "7208ea6c55f86705f08f95b3b234cd40": " \\mathbf{m} = \\frac{1}{2}\\int_V \\mathbf{r}\\times\\mathbf{J} \\mathrm{d} V ", "7209474de8d35ee28b4920ca11e5faf8": "\\mathbf{H}_n^+", "72098874bb4aab60e43a92331430622c": "f : S^1 \\times D^2 \\to S^3", "7209be72e5ed6fde68f316dc2b801548": "\\lim_{x\\to\\infty}\\log x=\\infty", "7209cf5c45d6c447e077320d5ff24e36": "A_\\mathit{eff}", "720ae59f63e54708926ee7743a413241": "\\langle | \\rangle", "720b377eefc80885cd994f4cb1e21300": "\n\\begin{bmatrix}\n17 & 23\\\\\n11 & 7 \\end{bmatrix}", "720b4d5beb975ddfdb0b472b4eb39ad8": "\\vec f^{-1}", "720b8115bf61ed679b4b556292b75648": " \\Psi_{ij} (r_e, r_h) = \\psi_{ik_e}(r_e) \\psi_{jk_h} (r_h)", "720b965b7c879c14796ff0c27034dddb": "\\frac {d\\epsilon_\\mathrm{Total}} {dt} = \\frac {d\\epsilon_D} {dt} + \\frac {d\\epsilon_S} {dt} = \\frac {\\sigma} {\\eta} + \\frac {1} {E} \\frac {d\\sigma} {dt}", "720be45e38cec1fcaf0fd6c54a0960f7": "\\alpha = \\nu/2", "720bfb428feef8748fc34e606989ba1e": "\nD_{N}^{*}(x_1,\\ldots,x_N)\\leq C\\frac{(\\ln N)^{s}}{N}.\n", "720c31303f1f65f1d312ce1309fa8582": "p( \\theta )\\!", "720c54411ceab86f1a2889813ebac2f6": "y = \\hat y y_\\text{scale} + y_\\text{shift}.", "720c571975bfbdf5d7af0083cd96e86d": " P = e^{-r_{DOM}T}\\Phi(-d_2) \\,", "720c95b7455eb0052287ea1a3020ddec": "P_{ref} = 1\\,atm", "720cd52ec7e93ebcd1f6f9f9bc7600bc": "y \\prec x", "720d147d1564de935833aec26f66e97f": "\\Pi_2 : a_2 x + b_2 y + c_2 z + d_2 = 0\\,", "720dce5db72fb0714aef875758c4b17a": " dE + \\delta w_u \\le 0 \\, ", "720de1f3d573e63e1b9fc7b2dca3275a": "a \\in S", "720df4a90bdb698f9c6d3ae88970e119": " W_j(K) = V(\\overset{n-j \\text{ times}}{\\overbrace{K,K, \\ldots,K}}, \\overset{j \\text{ times}}{\\overbrace{B,B,\\ldots,B}})", "720e58bdc8cc47a42b6382dfc8fe6cbd": "s_1 - s_2 = \\frac{h_1 q_1 - h_2 q_2}{f} +f \\ln \\left(\\frac{h_1 + q_1}{h_2 + q_2}\\right)", "720ea61bbdd9f1dd17219963523322ea": "f\\left( \\hat{\\mathbf{k}} \\cdot \\mathbf{x} - c_0 t \\right)", "720ee3a73427cb25aa1e053dd68de1aa": "P_3 = B,\\ P_4 = Q, D = M;", "720f0ad34e26d8bec3e4247b54660b68": "U_i U_j = U_j U_i", "720f18aad48914844bacfba430778954": "\\frac{ax + b}{\\left((x-c)^2+d^2\\right)^n}.", "720fb7c624310c7b8a2eaae9bf74a22e": "0.5^\\circ/\\sec^2", "72104b4e25eb11e38e926d19ea5871af": "V_A", "72105e543b222dce5ea7f86ee3b2eab9": " \\mathbf{B} = \\frac{\\mu_0}{4\\pi} \\oint\\frac{I d\\mathbf{l} \\times \\mathbf{\\hat r}}{r^2}", "7210c1f10710ae888ea2b7ccaeca5772": "t(G)=2^{2^n-n-1}\\prod_{k=2}^n k^{{n\\choose k}}", "7210ef23d6a405b421e2e1a7eadad64e": "\\partial \\bar{\\partial}", "72110cca17927c53f40fa8d309ccdeb1": "\\psi : \\mathbf{Gr}(r, V) \\rightarrow \\mathbf{P}(\\bigwedge^r V).", "7211203163a1d4ae2a093bb40338bedb": "\\frac{a}{2}", "721142881efdbdfb083a541dd1b322e6": "t_s ", "72116a9c39be6fd33e452d28fd090402": "\\mathrm{Out}(S_n)", "7211a2ab827246f7cff95cf758d70a5b": "\\sum_{k=1}^n \\frac{1}{k} - \\ln(n),", "7211c2fa4ea74200d14e81d44376b8c3": "\\Psi", "72122a5270595a2d4be312239b749d79": "t_i=\\alpha^{k_1}s_i-s_{i+1}.", "721276b0c763b88aaf9de42163981f62": "\\mu dN", "72129c1b65e39dbf788c82a171c53681": "\\tilde{R} = \\frac{R}{g\\, h_c},", "7212a459b07fbae1cd3198294113cdb1": "e^{2 \\pi i\\mathbf{K}\\cdot\\mathbf{R}}=1", "7212b74251311f9d4e4128e266e2c9fc": "R = {c\\over\\Delta v}", "7213019200a32164b06dbf4d3f6d3a7b": "\\rho \\vec{u} = \\sum_i f_i^{eq} \\vec{e}_i", "721338aa27f381fcda1ed35148e130c3": "= \\frac{\\alpha_i + c_i}{\\sum_i \\alpha_i + n}", "7213558b35c8f34f773bc4e70fce1aa3": " I_xI_yu + (I_y^2 + \\alpha^2)v = \\alpha^2\\overline{v}-I_yI_t", "72135fe47ae3cc638d2a8f0151189585": "\\Phi_e =(\\Phi_P + \\Phi_E)/2", "721387e4fe22984d18cc47deebbef895": "\\langle \\phi | \\psi \\rangle^* = \\langle \\psi|\\phi\\rangle ~.", "7213b89654ace5aa823d172db46d4da4": "_{q \\nleftarrow p}\\!", "7213c2515c0bd29bafcef6e82d0e8932": "R \\ \\stackrel{\\mathrm{def}}{=}\\ [\\mathrm{R}]", "7213da83457e4305cc363ca71a603005": "4 X_1^2X_2^2 +X_1^3X_2 + X_1X_2^3 +(X_1+X_2)^4", "721424e471b4764bf354789b4554f851": "\\partial_{n-1}(a) \\in \\ker \\alpha_{n-2} = \\{0\\}", "721433a766948b5d544ff7d2c7b19414": "\\operatorname{E} \\left [x \\right ] \\,", "72144f3d1148b1b270b9f6dd9e837565": " f \\leq g \\Rightarrow f^* \\leq g^* ", "72144fb91afe38b157bc30f22072ef5e": "\\frac{1 \\,\\mathrm{AU}}{2953.25\\,\\mathrm m} \\left( \\frac{2 \\pi \\,\\mathrm{AU}}{\\mathrm{light\\,year}} \\right)^2 = \\left(50 655 379.7 \\right) \\left(9.8714403 \\times 10^{-9} \\right)= \\frac{1}{2}.", "7214639323ed05db46835e98a0eb29d6": "V_t=V_0=\\sqrt{\\frac{\\mu}{p}}=\\sqrt{\\frac{\\mu}{\\frac{{(r \\cdot V_t)}^2}{\\mu }}}", "7215005600bb4981e6410ba34d7d3d34": "k^k", "72154a5b24f6d483a592e2c62b2e8c00": "\\aleph_0 < \\aleph_1 < \\aleph_2 < \\ldots . ", "721583c46171aaeda436478c7788277e": "r_{gr}(t)=Re\\{\\frac{\\lambda \\Gamma(\\theta) \\sqrt{G_{gr}}}{4\\pi}\\times \\frac{s(t-\\tau) e^{-j2\\pi (x+x')/\\lambda}}{x+x'} \\} ", "72158ce8141b5183eaed3408e3450aba": " f_0 ", "7215bfb4a1e7b83a7a5ea1b1e4ea0412": "Y_m.", "7215ee9c7d9dc229d2921a40e899ec5f": " ", "7216352890b718f05d1680568f369cb4": "V_{\\!\\text{out}} = A_{OL} \\, (V_{\\!+} - V_{\\!-})", "72163796f84600fd318935ac4a503451": "\\pi_{k+l-1}(X). \\, ", "7216426588b13592dd6a2150de574478": " {n \\choose k_1, k_2, \\ldots, k_m}\n = \\frac{n!}{k_1!\\, k_2! \\cdots k_m!}.", "721657b9dcda9268198ef6a5ed0d255b": "p\\in[0,1]", "72166b173b755ad6bac4a2c9052e2a75": "p_n(x) = a_0 + a_1T_1(x) + a_2T_2(x) + \\cdots + a_nT_n(x).", "72166bdea22987a0b49aeec05ba54601": "g=e^{2f(t,x)}[(dt+\\phi_{\\alpha}(x)dx^{\\alpha})^{2}\n-\\hat{g}_{\\alpha\\beta} dx^{\\alpha} dx^{\\beta}]", "7216994bd58c7e0253fb3378f815663c": " \\frac{\\delta v_1}{v_1} = \\varepsilon^{1}_{E_1} \\frac{\\delta E_1}{E_1} + \\varepsilon^{1}_S \\frac{\\delta S}{S} ", "7216bb1e18d173933bc065da72b3d169": "C_{4,2} = 1 + 4 \\times 1", "7216d07c2502777bca148dc4de954570": "\\sin(R)", "7216fe89e72e79da653be4ee74e1f6b7": "\\int_{t1}^{t2} L \\, dt", "72175d00a7003cfbf8a65c6e978cfb02": " \\mathbf{f} ", "721792df0dc0c149edd60fa061857c28": "1\\{A\\}", "7217977ac9f9792b64787c67931280bd": "\n {\\det}_q A = \\bigl|A\\bigr|_{11}\\,\\left|A^{11}\\right|_{22}\\,\\left|A^{12,12}\\right|_{33} \\,\\cdots\\,|a_{nn}|_{nn} , \n", "7217d781b0f814767a8b85d26aa7dea9": "\\frac{K_i}{K_d}=\\omega_0^2", "72185419194dd399a19bb474654259c1": " y_i ", "721911bc627c6e8a6f2c4aa29531759e": "H(t) = -\\sum_{i} P(X(t)=i) \\log \\left( \\frac{P(X(t)=i)}{\\pi_i}\\right) , ", "721939cf01518fef74844ef26b67b9ed": " I[y] = \\int_0^a \\sqrt { {1+y'^{\\, 2}} \\over y } dx \\, . ", "7219436b4a0538a95bc2177d1ed0e244": "|r\\rangle = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix}\\,,\\quad |g\\rangle = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix}\\,,\\quad |b\\rangle = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix}", "721961514a94482ec5e17f40fe5a846d": " Y(i)Y(-i) Q_B \\Psi = 0 \\left.\\right. \\ .", "7219a06e3d0c786469be9a0a0949a92e": "S^{max} = \\sqrt{\\frac{1}{1+m+n}} (1 + \\sqrt{3m} + \\sqrt{5n})", "7219c5c11bdcc2926d343b25a562612f": "\n\\operatorname{Li}_{n}(z) + (-1)^n \\,\\operatorname{Li}_{n}(1/z) = -\\frac{(2\\pi i)^n}{n!} ~B_n \\!\\left( \\frac{1}{2} + {\\ln(-z) \\over {2\\pi i}} \\right) \\qquad (z ~\\not\\in ~]0;1]) \\,,\n", "7219e0530b84df41db9b060cd7e80d7d": " \\Phi^{-1}(p) ", "7219f4cf34dd7a07ca3a0cf99b1a116a": "L=4 \\pi R^2 \\sigma T_{\\rm eff}^4", "721a26d4bc73401962f1d636149923fe": "a > b", "721a3c5381dd73c281c718a81ca71cfa": "\\scriptstyle z=\\eta", "721a77b0cf95f4cd36098773161b8021": "F(\\gamma)", "721a80929a2821e6a877e1000bcd57c4": " \\Psi ", "721ac321f7255fe5c7c6fd2122216dba": "\\text{st-non-connectivity}=\\{\\langle G=\\langle V,E\\rangle,s,t\\rangle\\colon \\text{there is no path from }s\\text{ to }t\\text{ in graph }G\\}.", "721ac53bf9f2cf886c877d899ecc3ce2": "\\forall x, y \\in U, ~ x\\ne y: ~~ \\Pr_{h\\in H} [h(x) = h(y)] \\le \\frac{1}{m}", "721ad8da07ee97b509bab6ca234d34ad": "a((bc)d)", "721add16b03ecd75892fa13e0bd5a62e": "(x-r_i)", "721b75309ed38bb52450a401b2a5b822": " u^{\\prime\\prime}(z) + \\tfrac{1}{2} p(z)u(z)=0.", "721bda251f662447cea523b87fed2407": "f_X ", "721be5c850e2accb84808649427f983d": "PU(\\mathcal H)", "721c4378ec5eb8c12f6136a3dd49d674": " m = \\frac{4\\pi\\rho_m r^3}{3}", "721c77b810d537330a89c53272c579c2": "Slope= -\\frac{\\Delta H}{R} > 0", "721ced4c4280f0591cb79a1108cf321f": "p_{\\nu}", "721ddcfd36d77654350988e2b2f7abec": " a(\\epsilon_n, v_n) = a(u,v_n) - a(u_n, v_n) = f(v_n) - f(v_n) = 0.", "721e054df3ca9926fa867ad254802ef7": " {P V \\over T} = \\operatorname {constant} = {{10^5 * 10^3 } \\over {300} } = 3.33 \\times 10^5 ", "721e1ddd0482a10400ae9a4627cc87d3": " B_{n1} = B_{n2}, ", "721e6a55bf9d75221e83d4d683ce0165": " \\mathcal{C}_{XY}^\\pi ", "721f863762d59caca85f316a458932be": "\\mathbf{A= \\left( \\left( C^TC \\right)^{-1}C^TA \\right) C}", "721fe0d3a13c9c9fe75b86e09965315d": "\\alpha \\mathbf x = (\\alpha x_1, \\alpha x_2, \\ldots, \\alpha x_n).", "722031e9edc88eb744c69c4b4afcfc65": "\\mathbf{r} = \\mathbf{x}", "7220480f56bd0267a97ff75285a1f800": "f(x) = {1 \\over \\pi \\left \\vert \\sigma \\right \\vert (1 + \\frac{(x-\\mu)^2}{\\sigma^2} ) }\\,,", "7220c842597720d0e6fa0c197da7108c": "(a+b)(a-b) = a^2+ba-ab-b^2\\,\\!", "7220da744e9f847aeb95a12f4a4c8643": "\\mathcal{M}_k \\subseteq \\mathcal{K}", "7220ff7f68782177c01ad6261d6b2df5": "\\, y_{n+1} = y_{n}(1.5-\\frac{xy_n^2}{2}) = \\frac{y_{n}(3-xy_n^2)}{2}", "72216e9a29a465308490f52f26b25974": " (a \\cdot c - b\\cdot d) + (a\\cdot d + b\\cdot c) \\,\\mathrm i = r\\cdot s \\cdot ( \\cos(\\varphi+\\psi) + \\mathrm i \\sin(\\varphi+\\psi) ) = r\\cdot s \\cdot \\mathrm e ^{\\mathrm i (\\varphi+\\psi)} ", "722187334ddbcebd71bb3d03a149b3d0": "L_n^{(\\alpha')}(x) = (\\alpha'-\\alpha) {\\alpha'+ n \\choose \\alpha'-\\alpha} \\int_0^x \\frac{t^\\alpha (x-t)^{\\alpha'-\\alpha-1}}{x^{\\alpha'}} L_n^{(\\alpha)}(t)\\,dt.", "7221a887e6cf8bfd675ccd52ef8fa59f": "\\beta_i^{(j)} := \\beta_i^{(j-1)} (1-t_0) + \\beta_{i+1}^{(j-1)} t_0 \\mbox{ , } i = 0,\\ldots,n-j \\mbox{ , } j= 1,\\ldots,n", "722233f07a22298592ad526a473c4bb7": "\n\\begin{array}{rcl}\nR(i+1,j) = A_1R(i,j) + A_2S(i,j) + B_1u(i,j) \\\\\nS(i,j+1) = A_3R(i,j) + A_4S(i,j) + B_2u(i,j) \\\\\ny(i,j) = C_1R(i,j) +C_2S(i,j) + Du(i,j)\n\\end{array}\n", "722259b05bf82fb7dfd1f1c7cfaf6f9a": "r_4(n)=\\begin{cases}8\\sum\\limits_{m|n}m&\\text{if }n\\text{ is odd}\\\\[12pt]\n24\\sum\\limits_{\\begin{smallmatrix} m|n \\\\ m\\text{ odd} \\end{smallmatrix}}m&\\text{if }n\\text{ is even}.\n\\end{cases}", "72226d487b696c08582d3a2575e9c928": "\\mathbf{x}_{k+1} := \\mathbf{x}_k + \\alpha_k \\mathbf{p}_k", "7222812b0524a2e403a8e141a40e99ff": "\\tfrac{1}{2}\\|\\mathbf{w}\\|^2", "722309983f7839b9ad334d677a243437": " MRP_L = 40(90 - 2L) ", "7223148707621eba5440731e8998e5b1": "(1/n)", "72239f03e4ce0134d92eaf07d84cf006": "q^*(\\mathbf{Z}) \\propto \\prod_{n=1}^N \\prod_{k=1}^K \\rho_{nk}^{z_{nk}}", "7223a992536591c08472edccb2077d19": "\\begin{align}\nk(\\theta) &= \\pi^{-\\frac{1}{2}} \\sin^{-\\alpha-\\frac{1}{2}} \\frac{\\theta}{2} \\cos^{-\\beta-\\frac{1}{2}} \\frac{\\theta}{2}~,\\\\\nN &= n + \\frac{\\alpha+\\beta+1}{2}~,\\\\\n\\gamma &= - \\left (\\alpha + \\tfrac{1}{2} \\right ) \\frac{\\pi}{2}~,\n\\end{align} ", "7223fcfb7c426ab8b2e2d54ef0cacf5b": "q_{and}=k_1 \\land^p k_2 \\land^p .... \\land^p k_t ", "722448c9faedd7eab4e2b3af1a5cdb8f": "G_\\mathrm{lpf}=R_{2}/R_{1}", "72249a0ba9af3f1d23477390e7bae82e": "\\ln\\left(\\frac {[A]_{\\infty}}{[B]_{\\infty}}\\right) = \\ln\\ K_{eq} = -\\frac {\\Delta G^\\circ}{RT}", "7224ec486099c78cee2cc1399795dc19": " rank_u(n_i) = \\overline{w_i} + \\max_{n_j \\in succ(n_i)} (\\overline{c_{i,j}} + rank_u(n_j)) ", "7224f2d55f255d2fe36545228c20b1af": " \\liminf_{z\\rightarrow x}\\frac{\\phi(z)}{\\log |z-x|^2}.", "7225190a2b52ddfa1fac2f2993e223eb": "a < b < c < ...", "72251a910b778b280f670909236c7bd6": " \\Lambda(x,y,\\lambda) = f(x,y) + \\lambda \\cdot \\Big(g(x,y)-c\\Big), ", "7225240755b19c4f6c4b3e341243936b": " x' = c ", "7225761585affc454b25169bc01b3a16": "I_0^f", "72259f13210e7bf0db3c11c22496272f": "\\rho v_\\infty x / \\mu", "7226216f97d8d9e3a046502500b2c24c": " \\hat{X}=\\{\\hat{x}_1,\\hat{x}_2,\\ldots,\\hat{x}_j\\} ", "7226552a1c1620104a4aed9f40aeddb8": "T(constructor_\\Pi(u, u'))", "72268ae9e9e089bc0ae3c61b7435f434": "I(X;Y)", "7226aea7e0e541f23fd6846cb5e4243e": "\\sigma_m \\to \\infty", "722715840455b4aebeb91f09caa9e6c9": "\\vec{E}(\\vec{x},t)", "72275df4e449895b3b3ebdf1add173e3": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\nabla^2 + \\frac{m\\omega^2}{2} r^2 ", "722789a8a3b648008a44f3ca0be5015a": "H_{p}x(n)", "72278dc42d99df8ca9220a7d3960c8f3": "\\alpha \\ge 5.9", "7228637dcb3307930c2991e6f8d7ab1f": "\\mu=\\lambda^{1/\\nu}", "72287c99c60444c9189babca2d7bf4fa": " \\min_{x\\in \\mathcal{H}} F(x)+R(x),", "722888dcc8250339e1b0b06f0b1178b5": "\\ell \\in \\{0,1,2,\\ldots\\}", "722893aedbc909bb81ad0c6544b19ca4": "\\bar \\psi (\\partial + ieA) \\psi", "72294369b8cb8f1e43c1b16f1a2afcc2": "e(u)=m(\\mathsf{i}_0)m(\\mathsf{i}_1)\\cdots m(\\mathsf{i}_k)", "7229516c3f5b9fd4a6b89f2e6c67329c": "\\mathbf{BA} = \\mathbf{I}. \\ ", "7229bf9b89080151610a7760c3866115": "3\\uparrow\\uparrow n", "7229e734fd7eb87d97af524d7f2cf5ba": "Happens(a,t_1) \\wedge Initiates(a,f,t_1)", "722a24f9f9b867618e055f2d1d18ac83": "\\int_0^1 \\frac{\\ln(1+x)-x}{x^2}\\,dx=1-2\\ln2.", "722a4904959b2b89a608d9d50f7d47d1": "e^{i\\mu t-\\sqrt{-2ict}}", "722a557c08eeb1ceae3b915a4c73448b": "(\\forall x, P \\Rightarrow Q) \\Rightarrow (\\forall x, T(P) \\Rightarrow T(Q))", "722a5ec531df0c9a200f40d54ba9faab": "rl_.", "722a748f711a2b3357b005e77cb91e55": "\\widehat{R_j f}(t)={it_j\\over |t|}\\widehat{f}(t).", "722a7a935fe6d167a5556fe827a8954c": "c_1 = 0.363445176, \\,\\!", "722a863723d599c0cf0f31b59bc23fa8": "Q(i,j)", "722a900f38209ba61cbd5bbc557a9026": "\\Re \\left[ \\mathrm{Ai} ( x + iy) \\right] ", "722a96577bd00a085477aa96b4f692cc": "\\tau_{k}", "722affd112f663e00612041f4cef1285": " v = {V / \\Delta t \\over {A}} ", "722b3cccc49f9a3d7e70523049c29ae7": "M = \\frac{1}{1-|\\frac{V}{V_\\mathrm{BR}}|^n}", "722b54fcf651d798e14b3f56ceded2f1": "f_{1}(e^{i\\theta})", "722b82647aa1f795103cf070686b8aca": "\n\\mathbf{C}\\,\\, \\equiv \\,\n\\begin{pmatrix}\n {\\sigma _1^2 } & {\\sigma _{12}} & {\\sigma _{13}} & \\cdots & {\\sigma _{1p} } \\\\\n {\\sigma _{21}} & {\\sigma _2^2} & {\\sigma _{23}} & \\cdots & {\\sigma _{2p} } \\\\\n {\\sigma _{31}} & {\\sigma _{32}} & {\\sigma _3^2 } & \\cdots & {\\sigma _{3p} } \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n {\\sigma _{p1} } & {\\sigma _{p2} } & {\\sigma _{p3} } & \\cdots & {\\sigma _p^2 }\n\\end{pmatrix}", "722bb08b83d09ad894eec2e2f3dae984": "\\,[p_x,H]=0", "722bd0517f394d9b7fc1b726ef9c4b6a": "\\vec R ", "722bd6007f18cea6db79efe67a5a0884": "x_{t+2} - \\alpha x_{t+1} + \\beta x_t =0. \\,", "722be3e95d74173f3614ab3a9cc607c2": "Z(s)=\\frac{| \\mathbf Z|}{z_{11}}", "722bee2508989297eb428da3c5a1f328": "B_m", "722c15e18b76bb5da67b00f547c0c24b": " \\mathrm{ IMC } = m + a^{ -1 } m^{ 1 - b } - 1 ", "722c34c6cd6f55f0901fb906c5f64b20": "g \\in A", "722c582f8abfe357d18d2bc3cc07929e": "\\ (1+(g_m+g_{mb})r_O)R_S + r_O ", "722c793dd13f1c95fc0ed12d1768adff": "V(y_k) = \\sum_{i=1}^{n}PV_i = \\sum_{i=1}^{n} \\frac{CF_i} {(1+y_k/k)^{k \\cdot t_i}} ", "722c7b7d35ccb41d11775075e260a457": "\\tanh x = \\frac{\\sinh x}{\\cosh x} = \\frac {e^x - e^{-x}} {e^x + e^{-x}} = \\frac{e^{2x} - 1} {e^{2x} + 1} = \\frac{1 - e^{-2x}} {1 + e^{-2x}}", "722cbf42a901c458c26e11a1a1edbf1d": " y=E(x)=(5x+8)\\pmod{26}", "722cd4839dea68d1cb97508476f607ed": "B(x, y) = \\operatorname{tr}(\\operatorname{ad}(x)\\operatorname{ad}(y))\\text{ where }\\operatorname{ad}(x)y = [x, y] = xy - yx", "722d2789552a3a868e484b89417465e2": "f_1, f_2, ..., f_k", "722d9ff41640ff21f41389968f52b56c": "y^4+py^2+qy+r=0,", "722df84147b6d3e8aa4f19b816995df2": " \\sum_{i=1}^n | \\varepsilon_i \\rangle \\langle \\varepsilon_i | = 1 \\,,\\quad \\int d\\varepsilon \\, | \\varepsilon \\rangle \\langle \\varepsilon | = 1 \\,. ", "722e05126fd1e5164537c2a3ab6b916f": "\\left\\{ p \\right\\}", "722e528a4cd68678c5f089946e2da2d7": " \\gamma_{a} ", "722e69727f89da4651b50c0a9494badf": "\\mathfrak{h}_n", "722e922493fc7fc32ead9ad5a25fb032": "p\\in M", "722ebaaae6a9158808df96d4414ad2ba": " {d \\over dp} H_{\\mathrm b}(p) = - \\operatorname{logit}_2(p) = -\\log_2\\left( \\frac{p}{1-p} \\right). \\,", "722edd8476bdb0cac9fe6e50dc713812": "f*g = fg+\\sum_{n=1}^\\infty\\left(\\frac{i\\hbar}{2}\\right)^n \\sum_{\\Gamma \\in G_n(2)} w_\\Gamma B_\\Gamma(f\\otimes g).", "722f58c297308e2c1634fe3beeb1d3c7": "E \\subset B", "72300d6e3614c35b3ebd0267b72db84c": "\\gamma(1,x) = 1 - e^{-x},", "72300f48a0725fecdf6f68536541245e": "\\mathbf{u}=\\nabla\\varphi+\\mathbf{v}", "7230306edfd454fe2988652bd38bf752": "\\rho_i", "72304dd06a56340ade6fe017681561d0": "F(x) = \\sum_{1 \\le n \\le x}\\alpha^{-1}(n)G(x/n)\\quad\\mbox{ for all }x\\ge 1.", "72304ec8abb912c77afed4483fb9da60": "\n\\overline{z}=\\frac{1}{N}\\sum_{n=1}^N z_n\n", "723051cf0f687bfcfa103b91e16e216e": "\\sum_{n=0}^\\infty (-1)^n a_n = \\sum_{n=0}^\\infty (-1)^n\n\\frac {\\Delta^n a_0} {2^{n+1}}", "72308f69009b1c6fab397d3a64c5461a": "\\mathrm{Ad}_{gh} = \\mathrm{Ad}_g\\mathrm{Ad}_h", "72312551e859a16dcc71052b1ebb67d4": "f_+(f_-(f_-(0))) = 1 + (1-w)w = 1 + 1w - 1w^2", "72319118e61ae5aca40a9176113e02c6": "\n\\cosh \\mu = \\cos i\\mu\n", "7231fa806691800f095133f6fb720d82": "\\sum", "72322a529b1254bdd1b0cdc63ca73ffe": "\\displaystyle{B(a,b)Q(a^b) = Q(a^b)B(b,a)=Q(a)}", "723251050006238132710f020940e59c": "x=x_0+sh", "7232de9c80ee9652d4e385cf86b565b5": "UT2 = UT1 + 0.022\\cdot\\sin(2\\pi t) - 0.012\\cdot\\cos(2\\pi t) - 0.006\\cdot\\sin(4\\pi t) + 0.007\\cdot\\cos(4\\pi t)\\;\\mbox{seconds}", "72331710d060df724d8dd70a57f4c14c": "r \\in \\{0, \\ldots, q-1\\}", "723325c7354d9b35c627ae180de5c662": "B_{j_k}", "723342896fabe058053287c4a15fd99a": "\\phi_i(v)", "72336d1dcdf6dd623ee30d108b90f9f0": "{\\rm{sn}} u\\,", "7233e39d64ea474eeac37d83f2ab2e6e": "\\Lambda x\\in J. K(x)", "7233f04cae0745d8766b6375b989e9d8": "\\sum_{d=1}^{\\infty}\\frac{2^{d-1}n}{4^d}=n/2", "7234950289beec8711a5bbd6aefb7289": "h(x,1) \\in A", "7234c87ffa8ba8dce012e339edbdd96f": "S \\otimes_R S\\to S \\otimes_R R", "7234cb382be83887a0ff8790deba1e39": "\\dot x=X(t,x).", "7234d646ab17d87d331a89c959d9ae1a": "\\bar{\\dot{R}_n}", "72350c401d453be07e7385d510be694a": "\\Gamma_2(a)=\\pi^{1/2}\\Gamma(a)\\Gamma(a-1/2)", "7235138bf53896ed61cc395eba5c83b4": "\\,_2F_1 (\\alpha, \\beta; \\gamma; z) = \\frac{\\Gamma(\\gamma)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\sum_{n=0}^\\infty\\frac{\\Gamma(n+\\alpha)\\Gamma(n+\\beta)}{\\Gamma(n+\\gamma)\\ n!}z^n,", "72353c6a742664400945ca640d745c72": "\\sum_{\\nu=1}^nc_\\nu\\theta_\\nu=0", "72353d0c104e34009e77323ab41d8c73": "\\left\\lfloor\\frac{(r - 1) n^2}{2 r}\\right\\rfloor", "7235734ecc87aefe5927bc839288d3ac": "\\mu\\left(\\bigcup_n B_n\\right)=\\lim_n \\mu(B_n) \\leq \\lim_n (1 - \\epsilon)^{-1} \\int f_n \\, \\mathrm{d}\\mu. ", "7235d65086ea1d78cb2dfb412f9cfc42": "\\textstyle S(r )", "7235ecd3580917292e7bd821edbe93d6": " \\frac{dN}{dt} = r N \\left( \\frac{N}{A} - 1 \\right) \\left( 1 - \\frac{N}{K} \\right),", "723605d9475700fa0d06b2e5a0263c79": "\\mathbf{Set}\\downarrow D", "72376937b469b10fb1b72f3bf8912a5e": "\\ log_2n", "723793f1abe318e4670f363fa24f5bc7": "(0,0,-a)", "7237dbbb7e87ca4d05adfd6964036cdd": "k=2l+1", "7237eb59c09c755d2155e5ed63e9e5b1": "\\mathbb{P}(A) = \\frac{|A|}{37}", "723812569fa03f184f1ed18e6f6878e0": "j_2", "723840faca6813e3e9b11db8549fad04": "v'=(yq_2)'= y'q_2 +yq_2'=(q_0+q_1 y + q_2 y^2)q_2 + v \\frac{q_2'}{q_2}=q_0q_2 +\\left(q_1+\\frac{q_2'}{q_2}\\right) v + v^2.\\!", "7238c3ac3be9f5540f5c46d026c39cbe": "\\pi^{-1}(V \\setminus Z)", "7238dfb0e300278ee1b58691ee95aa15": "g_2 = g_2(x_3,x_4,x_5)", "7238f34138c6b3cf85a3fcb3a6693900": "A_i\\subseteq\\Bbb{R}", "723902a096a2f7a417ebb8f3116bd002": "\\ S(t)=m{ \\frac{(p+q)^2}{p}} \\frac{e^{-(p+q)t}}{(1+\\frac{q}{p}e^{-(p+q)t})^2} ", "72392ab605270ea48814dd486ea773d1": "\\frac{T_s}{T_o}", "7239401ea865db35e350dcf15d5234b7": "\\ln(x) = \\ln\\left(\\frac{1+y}{1-y}\\right)", "72397eb8c36d936e0e4379c9d20c1778": "\\alpha \\mapsto \\Xi(\\alpha)", "7239f6df3e0748db9946986fc629c311": "D_{t+1}(i) = \\frac{ D_t(i) \\exp(\\alpha_t (2 I(y_i \\ne h_{t}(x_{i})) - 1 )) }{ Denom }, ", "723a057252158c7587034484c36993fe": "\nRGE_i = h_i^t \\times \\left( g - G \\right)\n", "723a0576e2ce4b4cc043e1d80624ea36": "p f(\\textbf{x}_{n})", "723b8610fe2e96c200eb843cd71248bd": "Y_{8}^{-6}(\\theta,\\varphi)={1\\over 128}\\sqrt{7293\\over \\pi}\\cdot e^{-6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot(15\\cos^{2}\\theta-1)", "723c69db90f5926ba744b228572c0ae2": "\\overline{\\Lambda}_n - \\underline{\\Lambda}_n \\ge 0.", "723c6ddc0c25600fcc3dc05d50bcf5d2": "O(n^{3+\\epsilon})", "723c6eea56d7a3fd0a4a1d3b9731038b": "\\mathfrak{B}", "723d0425cb663169e56292aa47c1cf41": "p(z_i = k|\\mathbf{z}_{1,\\dots,i-1},\\alpha) = \\lim_{K\\to\\infty} \\frac{n_k + \\alpha/K}{i - 1 + \\alpha} = \\frac{n_k}{i - 1 + \\alpha}", "723d23d4810ba40586b752a202aeed8f": "|1-a|<1", "723d3fefc289f603827088bee838dd62": "\\mathbb{P}(X = x)", "723d632aea20630dc11e39e5446aff8c": "\\Phi (z)=\\frac {2\\pi}{\\lambda} \\int_0^z \\! \\delta (z') \\, \\mathrm{d} z'", "723d9b5bf8f7bc242fecf343104522f6": "\\forall x_1 \\forall x_2 ... \\forall x_n", "723db928282da165673c76baabfd99a7": "m \\left( x, y \\right)", "723ddf31476062b6b81a64b1d25adba4": "{v_4(t),v_5(t),v_6(t)}", "723e7c614934c3099583f58b445940be": " L_3 = \\frac {-K_C R_S} {sL_M} \\, ", "723ed66501e3b785d8b760a87f63d43e": "\\{\\neg\\tau, \\rho_2, \\rho_3, \\rho_4, \\dots\\}", "723ed91fb71c6422c5df8106a4775a68": "\\tau_{B}", "724003b7c1cd51ead3c4b196a0f96cc6": " \\dot{x} = -x^2 + \\varepsilon ", "724055ec51fbdbfb0a7b09ce82de3401": " [\\theta]= 100 \\,\\Delta \\varepsilon \\left( \\frac {\\ln 10}{4} \\right) \\left( \\frac {180}{\\pi} \\right) = 3298.2\\,\\Delta \\varepsilon \\,", "7240a8389ce7afe931c33058dad339fd": "t(0)=0\\,", "7240bd0ccf4b2245b096fd93814ff702": "\\frac{\\text{d}v}{\\text{d}t} = \\frac{L'I^2}{2m}", "724105340cebb49b323f588d4984ca13": " (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x).\\,", "72413020bdc8460cac0401adb9a9af2e": " \\ln(\\Gamma(z)) \\approx \\tfrac{1}{2} \\left[\\ln(2 \\pi) - \\ln(z) \\right] + z \\left[\\ln \\left( z + \\frac{1}{12z- \\frac{1}{10z}} \\right)-1\\right]. ", "72413c47b2a1ae03e1ca9cddf0268539": "\\tfrac{1}{q}\\begin{pmatrix}r & p \\\\ p & r \\end{pmatrix}", "72415fb2e22e573b7f6b32e5cc63ca51": "\\frac{P_x}{P_y}", "7241738509376acddc41dbad8f13fe12": "T_c = \\frac{\\Delta H_p}{\\Delta S_p}", "72418aad0c2f8738d7df1a3a0da36582": "\n\\frac{\\partial \\Delta E}{\\partial P_x}=\\alpha_0\\left(T-T_0\\right)P_x+\\alpha_{11}P_x^3+\\alpha_{111}P_x^5=0\n", "72418fe836da9ca63bcd34ac9cbe32f0": "x^{*n}/\\sigma\\sqrt{n}", "72419ca227288f0381119742bfd1dc53": "\\Omega \\in \\mathbb{R}^{N \\times N}", "7241fd63b7461cc9961c7623170a174a": "\\mathrm{slog}_a {^x a} = x", "72426d8545df600a3cd1406c7c12bfa1": " ( \\gamma, \\alpha ) > 0 \\, \\forall \\gamma \\in \\Delta", "72427a970166bb4bd4104c81e46f72bb": "\nT_-(x) = 576\\int^\\infty_0 \\frac{dt}{t^3}J_4(xt)[J_4(t)]^2\n", "72429e78e98afc5948acd59363f637b9": "E_a=m_1(r^\\ddagger-r_1)", "7242acc05f09838b7ef22ccf618a6392": "(T_{M\\;\\sigma }^{\\;\\;\\sigma })", "7242c3effa406e3b1f9ce5646c40ffa5": " \\psi (z) , {\\frac {1} {m^*} }{\\partial \\over {\\partial z}} \\psi (z) \\,", "7242fb78ad7e712035d56b804ed27045": "\\mathbf{\\ddot A} = \\mathbf{\\hat r} (\\ddot A_r - A_\\theta \\ddot\\theta - 2 \\dot A_\\theta \\dot\\theta - A_r \\dot\\theta^2)\n + \\boldsymbol{\\hat\\theta} (\\ddot A_\\theta + A_r \\ddot\\theta + 2 \\dot A_r \\dot\\theta - A_\\theta \\dot\\theta^2)\n + \\mathbf{\\hat z} \\ddot A_z", "7243093635757342e40e0b2242a84c9a": "\\sup_{t > 0} \\mathbf{E} \\big[ \\big| M_{t} \\big|^{p} \\big] < + \\infty", "72432e0a8e3a31cd423ba8b4bfded2a2": "(0 + 1 + 2 + \\cdots + k )+ (k+1) = \\frac{(k+1)((k+1) + 1)}{2}.", "72435834ca6ed48a79d91e39100e392f": "C^* = \\{ a \\in A \\colon \\forall b \\in C \\langle a,b\\rangle > 0 \\}", "724370472baaa9fab00a82e2975d38ae": "y_i^\\prime = MLD_{C_\\text{in}}(y_i)", "7243a054208b6dd19bcb8fb7602b2280": "\\mathrm{SCL} : R_{i,t} - R_{f} = \\alpha_i + \\beta_i\\, ( R_{M,t} - R_{f} ) + \\epsilon_{i,t} \\frac{}{}", "7243dc28c3a85b51e6e4d8a9614a1b9f": "\\textstyle p,q,\\,p \\equiv q \\equiv 3 \\mod 4", "7244e7a9047bf41c6053b0ad7616fe56": "\\delta=\\lim_{R_{s}\\rightarrow \\infty}(R_{e}-R_{s})=z_{e}-z_{s}", "72450886f7c155bac99a783335fdeb5c": " | \\gamma_\\mathrm{e} | = 1.760\\,859\\,708(39) \\times 10^{11}\\, \\mathrm{\\ \\frac{rad}{s\\cdot T} }", "724547c6d63204a6d765d3dc584b6a6d": "A \\hat{\\otimes}_\\alpha B", "7245795061c938fff0ded1dd7b726330": "u_G'(x)=A(x)u_1'(x)+B(x)u_2'(x).\\,", "724580bdec582bf0421655d6e25c33a4": "\\mathbf{Z} \\wr \\mathbf{Z}", "724588f9ece02094050fa5c190e4f771": "t = C[C'[u]]", "7245eb11f59c8fc4afc9c06c07f78ebe": "f\\in C^\\infty({\\mathbb R}^n,{\\mathbb R}^m)", "724613ba98be650b431e6e1cc090900a": "z=(x-\\mu)/\\sigma\\,", "724616a3690a0f52c58e3dbe46bb4b0a": "C=B/A\\cong\\{\\pm 1\\}", "724631ef3379b2b64d2bc3e614e61018": "A \\in \\mathbb{F}^{m \\times n}", "72468afe253d7be996800f46f6d9fb6a": "\\ddot{r} - r {\\dot{\\theta}}^2 = - \\frac {\\mu} {r^2}", "7246bf801b597336ad53bb35ee710853": "N_{eq} =( D / l_0 )^{n_d}\\Psi", "7246cab3bbbaec5e18326e67504de101": "\\text{Regional strength multiplier} = \\frac{\\text{Team 1 regional weighting} + \\text{Team 2 regional weighting}}{2}", "724766e144b96ba6128729a3f24ad501": "=\\int_{-\\infty}^\\infty p(x)\\,e^{-2\\pi ixk_x} dx\n", "7247767256f5aef95a2147aceb81abcc": "P^*", "7247ac286735c7c6ab6d984a686e35d5": "r = r_o", "72481586ffd96c0a7a32e387e8f8d204": " {\\partial^n \\over \\partial a^n } I = \\int {x^{2n} \\over 2^n} e^{-ax^2/2}dx = {1\\cdot 3 \\cdot 5 ... \\cdot (2n-1) \\over 2 \\cdot 2 \\cdot 2 ... \\;\\;\\;\\;\\;\\cdot 2\\;\\;\\;\\;\\;\\;} \\sqrt{2\\pi} a^{-{2n+1\\over2}}", "72483e14b0c3ad0121983ce2af7008ec": "w-1", "7248525467d0158cc285091e2cb89abd": "\\overline{(zw)} = \\overline{z}\\; \\overline{w} \\!\\ ", "7248934257f50a226c9449cf7a775177": "\\int\\limits_{-\\infty}^\\infty |x(t)|^2\\, dt = \\frac{1}{2\\pi}\\int\\limits_{-\\infty}^\\infty |\\hat{x}(\\omega)|^2\\, d\\omega.", "724929d872745604e99ff7cedc40716a": " \\scriptstyle |P|:=\\max_{0\\le i0. \\end{cases}", "724ddda1e7e31717638b808d21678132": "10+1+7+16=34", "724dfdb6c941e41b9549ea8d7ee542ff": " Z=\\begin{pmatrix} z & 0 \\\\ w & -z\\end{pmatrix}.", "724e3297165bed011e5f706f5e1c81b6": "\\scriptstyle C_i \\;=\\; E_K(C'_i) \\,\\oplus\\, 2^i L", "724e4f2786b29ee0bacf8eadf4cab4ab": "R \\sim \\frac{1}{M^{1/3}}, \\,", "724ea2736ee2d78ef2cd9c0cb1653025": "\\frac{\\Delta x^2}{\\Delta x^2}", "724eb02ab8e3ad47afa6cf207d1cafbd": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{(m+n(2 p+1)+1) x^{m-n+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{b\\,n^2 (p+1) (2p+1)}\\,-\\,\n \\frac{x^{m+1} \\left(b+2 c\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{b\\,n (2p+1)}\\,-\\,\n \\frac{(m-n+1)(m+n(2 p+1)+1)}{b\\,n^2 (p+1) (2p+1)} \\int x^{m-n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}dx\n", "724ef55d4a37be53ef3995b99d1d0995": "P_{FA}", "724f0cf4d5a867c1fa4688878c10c012": "P(x,y)=\\beta A_{ji}/k_i", "724f36941325b105f230f5452c4babdf": "C = U_1 U_2...U_i....U_w ", "724f55e124a5dc982bc5cc4d23f12865": "\nY=\\sum^p_{j=1} X_j\n", "724f55eb57b1f41384ce81a59748715f": "q_{max}\\equiv C_{min} (T_{h,i}-T_{c,i})", "724f5c69844aa78249fe36e0bab707fd": "Q = A_2\\;\\sqrt{\\frac{2\\;(P_1-P_2)/\\rho}{1-(A_2/A_1)^2}}", "724f64edf804f667768448a71fc99dcd": "\\mathbf{C} = \\mathbf{R}[x]/(x^2+1),", "724f83befcf2f4ef58ef2874d3612af9": "X_k = b_k^* \\sum_{n=0}^{N-1} a_n b_{k-n} \\qquad k = 0,\\dots,N-1. ", "724f87c1ba797c43fb6bd686324c732d": "B_1 = 1/2", "724fc2598319a2ba5074d417935275ae": "x = f_2(p),\\ y = px", "724fdb6f7b4f42cb84e4e02379f9f338": "k = \\begin{matrix} \\frac {E A_0} {L_0} \\end{matrix} \\,", "724febfe7970528f8c8d5c457a05f27c": "\\Phi_{12}(r) = A \\exp \\left(-Br\\right) - \\frac{C}{r^6}", "72503093eec2004ae61685312c56fe62": "du/dt=F(u)", "72507d63ded0e41f1ba03f38f846d255": "i_k: A_k \\mapsto A_1 \\oplus \\cdots \\oplus A_n ", "72507e129f477afefc7691aafa03a2bc": "\\log( \\gamma ) = \\sum_{i=1}^{s} \\nu_i \\log( \\gamma_i ).", "725082fd96b024c1357020359b7b4e0c": "Pwo = 1 - \\dfrac{2}{3} = \\tfrac{1}{3}", "7250a362c50df8a20c3f09043ff4bdd2": "\\displaystyle{Lf_n=\\lambda_nf_n,\\,\\,\\,\\,\\, \\lambda_n=\\mu_n^{-1} -1.}", "7250b47f649106f0127edf460d061193": "\\;\\deg(G)<\\deg(H)", "7250e2b5827171439ce461d81b07013d": "\\Omega(d)", "725102d1a10723e5d608171b1d23c257": "\\overline{P}_+", "72511892984f1c0219b60e9b0ce6bc35": "\nD = \\frac{k_{B} T}{f}\n", "725145074312c9415246a8758bd80022": "= \\lambda_n ", "725147d23f772d5442cb2c8c27cbd78e": "g:\\mathcal{O}_{X,f^{-1}(P)}\\rightarrow \\mathcal{O}_{Y,P}", "7251c514aaf04ce69470d64e77572584": "\\displaystyle{[L(a),L(a^2)]=0}", "725299ebfd05d7f410209a7ff43c83b8": "g_{12}(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iint \\dfrac{\\phi_1(\\theta,\\tau)}{\\phi_2(\\theta,\\tau)}e^{j\\theta t+j\\tau\\omega}\\, d\\theta\\, d\\tau", "7252ed029501fd7ad7d0626b98eebcdc": "\\hat{C}_a |\\psi \\rangle = 0", "72532e264a4ed4ced340bc083850ec42": "\n\\text{If }p \\equiv 1 \\pmod 8 \\text{ is prime, then there exists an odd prime }q <2\\sqrt p+1 \\text{ such that }\\left(\\frac{p}{q}\\right) = -1.\n", "725353f47815bfc2c22dd83d651a2eaf": "10^{-1}", "7253e482104654ab8f2f55f6f89b0c8a": "x = x_1 \\cdots x_n", "72542727db629d5dcef865b1aef6ff08": "u(0,x') = u_0(x'),\\text{ for } x = (x_1,x')", "725459b4f5d509876cf9b3e72b493ffb": "I=- \\iint\\limits_S\\mathbf{J}\\cdot d\\mathbf{S}", "725461f4b9940fb0593d69089284680f": "w^T \\phi (x) + b", "7254641a85a73596c08b3b55b2a917e3": "3 \\le \\max(3, -\\infty)", "7254a217faf8e77c23d95f32cc6b089b": "\\mathbf{v_{av}} = \\frac {\\Delta x}{\\Delta t} = \\frac {x_2 - x_1}{t_2 - t_1} ", "7254d2a7128a7186db7abfbd027f2adb": "E=\\sum_{n=1}^N \\frac{\\mathbf{p}_n\\cdot\\mathbf{p}_n}{2m_n} + V(\\mathbf{r}_1,\\mathbf{r}_2\\cdots\\mathbf{r}_N,t) = H \\,\\!", "7254e300193b79306cb7be4fc52638bc": "r - 4", "72551506fc2663677795f0a322edcde0": "\\ 4y'^2 - 2y'^3 - 1 = 0 ", "7255197ec12d23b9fc1d66d226a1f3db": "Q = (7)\\mathbf Z[i] + (i^2 + 1)\\mathbf Z[i] = 7\\mathbf Z[i].", "72551b1705808d2e2e7456a509597c34": " \\frac{\\partial I}{\\partial t}=\\mathrm{div} \\left(c(|DG_{\\sigma} * I|)\\nabla I \\right) ", "7255b17f662e7287db1c781dba4959c6": "\\sum _x f(x)=\\sum_{n=0}^\\infty\\left(f(n)-f(n+x)\\right)+ C.", "7255e55df35edc5fc3dfbed671fd3ee7": "\\scriptstyle \\omega = \\{ S_* | \\} = \\{ 1, 2, 3, 4, ... | \\}. ", "7255ef03f01030b455ce97a0867b1fa4": " \\Bigl| \\left \\{ x : (f-g)^*(x) > \\alpha \\right \\} \\Bigr| \\leq \\frac{A_n}{\\alpha} \\, \\|f - g\\|_{L^1} < \\frac{A_n}{\\alpha} \\, \\varepsilon,", "72560e8a313beaa13743f0ceca5f2931": "\nn_i = \\frac{g_i kT}{\\varepsilon_i-\\mu} ", "72567e02e72ab68085b5300dffe60644": "I:\\; g \\rightarrow g", "725719cd6ed9bfee6d6fc88bec86eee5": "\n \\zeta=\n \\begin{cases}\n 1 & \\mbox{if }n\\ne 0 \\\\\n 1/2 & \\mbox{if }n=0\n \\end{cases},\\quad\n \\xi=\n \\begin{cases}\n 1 & \\mbox{if }m\\ne 0 \\\\\n 1/2 & \\mbox{if }m=0\n \\end{cases}\n ", "725729fee8e55c04c4b569d1043edf6a": "nM^2", "72572ee64c591726ceafe913c5078fd6": "\\varepsilon_{\\zeta_0+1}", "72574808221c0bbcdedaaaed7dfa6a5e": "\n f a \\approx 3.26~ \\rm m/s\n ", "72575555412d5c93c92c3368eac01aa2": "X_{ii}", "72575ee846b1e54e6b5cd5ce9b0a54d9": "\\sigma_0=\\frac{n_ee^2}{m_e\\nu_c}", "7257c771da8b0a4fb39d6c3120d29935": "A \\cup B = B\\,\\!", "72586b50e26c796f9f722b7c9294a4db": "r \\equiv x_1 \\pmod{n}", "7258f01c8453706f3e55b215bc2f1696": "h= {0.023} {k^{1-n} j^{0.8} c_p^n \\over \\mu^{0.8-n} d^{0.2} }", "7259160156e3dc89f47eff7d70e861dd": "P = L \\cdot \\frac{c\\,(1 + c)^n}{(1 + c)^n - 1}", "725946e62c10b41ba43eace93ae03f67": " [(x,x'),(y,y')]=([x,y],[x',y']), \\quad x,y\\in\\mathfrak{g},\\, x',y'\\in\\mathfrak{g'}.", "7259abd849f1b806bc61bdf757bb1122": "Q > 0.5", "7259c04bef3ff32715ef1998e19649c9": "\\scriptstyle{g \\rightarrow e}", "7259c2b206134631f2517a6aa66bd1bb": "a > 1", "725a3acfe69f66379692f0abc02d623e": "f(z)=\\int_{\\gamma}\\! g(\\zeta)\\, d\\zeta", "725a53868660b5c02e20ce5c844d826e": "\\text{tf}_{t,d}", "725aeb5bf46a4c5b8a16cdb94799bc68": " \\left(\\sec(x)\\right)' = \\left(\\frac{1}{\\cos(x)}\\right)' = \\frac{\\sin(x)}{\\cos^2(x)} = \\frac{1}{\\cos(x)}.\\frac{\\sin(x)}{\\cos(x)} = \\sec(x)\\tan(x)", "725bb9fe0fa0196147339c68f5d73ade": "d=1:", "725c5c2d28ab4f648851d042fc77ee22": "\\Upsilon_1", "725c767f18c746e8d959b98e71fb3e34": "4\\log_5(x - 3) - 2 = 6\\,", "725c833af9a5ee5264e1e7627c47c9c7": " \\varphi _j^{n + 1} \\ ", "725c9423eedc679a0d8ebdb5b2e80a5c": "\\ f_c(z)=z", "725cad8803b4b1085040806d5416cf72": "s=1+it", "725cc7404f873d0f90e2dbbe9b4a699e": "e^I_\\mu", "725cc8d62128b838f4c9ffc464db7452": " (u_1 + u_2) \\smile v = u_1 \\smile v + u_2 \\smile v ", "725d0f8a2466691059f2ca2d74cab638": "_{2}F_{1}(-n,\\alpha;\\alpha+\\beta;1-e^{it})\\!", "725d863e0c6364a7df06e5dd6fbaac36": "mac \\leftarrow AXUHash(k_{AXU},lastBlock,\\Bigl[e\\Bigr]_i^{i+r}) \\in W^4", "725e0298a1e51a06143571df1ce37c5c": "A=-k_\\mathrm{B}T \\ln Q", "725edd1c3b2d70f3ee2f4e42dc060e94": "\\theta_k^\\dagger, \\ \\theta_k", "725ef43d0d4862a28c07d3e93d0bffa5": "\\tfrac{1}{2^x}", "725f7a6f1a678a932a1dd3b27db34451": " y \\in C, \\ \\ \\ \\|x - y\\| = \\mathrm{dist}(x, C) = \\min \\{ \\|x - z\\| : z \\in C \\}.", "725fadd07f93af2abcb1cfd047c6447b": " T_4(x) = 8x^4 - 8x^2 + 1 \\,", "72610de5bfb9459db4791a5afb5c99de": "\\textrm{pH} = \\textrm{pK}_{a}+ \\log \\frac{[\\textrm{Ind}^-]}{[\\textrm{HInd}]}", "726146c7187017e210114928680267b8": "\\varepsilon_{ijkl } =\n\\begin{cases}\n+1 & \\text{if }(i,j,k,l) \\text{ is an even permutation of } (1,2,3,4) \\\\\n-1 & \\text{if }(i,j,k,l) \\text{ is an odd permutation of } (1,2,3,4) \\\\\n0 & \\text{otherwise}\n\\end{cases}\n", "726157a52ec7ceb2b61d8869417d4e3a": "E_\\mathrm{heat} = V_\\mathrm{eff} I_\\mathrm{eff} \\; t ", "72624d6b5fd9a981c4aa06797d1f9b62": "f(\\mathbf{v}, t)d\\mathbf{v}", "72628d313d0122b8c28adc1a13fbafa2": "\\lambda\\|\\nabla\\beta\\|_1", "7262e363d1c5eedad35d819164faefd6": "\\mathbf{Q} = -k \\boldsymbol{\\partial} T = - k\\left( \\frac{1}{c}\\frac{\\partial T}{\\partial t}, \\nabla T\\right) ", "7262e41ef8b66b81c7950ab1b47a85c9": "\\qquad{\\it (Mem)} \\ \\frac{\\displaystyle M \\ \\rightarrow \\ M'}\n{\\displaystyle [\\;M\\;]_u \\rightarrow [\\;M'\\;]_u}; \\qquad \\qquad{\\it\n(Struc)} \\ \\frac{\\displaystyle M\\equiv_{mem} M'\\quad M'\n\\rightarrow N'\\quad \\ N' \\equiv_{mem} N} {\\displaystyle M\n\\rightarrow N}", "72638742ccb0d9bde36bda3e1c16b1dd": "\\frac{13}{8}+\\sum_{n=0}^{\\infty}\\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}", "7263a1aec7bd904092a5a65121437e3a": " \\langle \\cdot, \\cdot \\rangle : V \\times V \\rightarrow \\mathbf{F} ", "72644366827f93a460ebd9621d9e8f82": "\\begin{align}\n x &=& Q_{zy} - Q_{yz}\\\\\n y &=& Q_{xz} - Q_{zx}\\\\\n z &=& Q_{yx} - Q_{xy}\\\\\n r &=& \\sqrt{x^2 + y^2 + z^2}\\\\\n t &=& Q_{xx} + Q_{yy} + Q_{zz}\\\\\n \\theta &=& \\mbox{atan2}(r,t-1)\\end{align}", "7264458f5245512ed11d2b431bbd4507": "A \\subset \\mathbb{R}^2", "726451de3c4b3efc8b245525a9f482f3": "Q\\ ", "72646dd422b69903f2ddb0a513f67780": "{f_n}", "72648c5ee6fe7e2f849b9cd9c1c0a365": "Y \\sim \\operatorname{Log-\\mathcal{N}}\\Big(\\textstyle \\sum_{j=1}^n\\mu_j,\\ \\sum_{j=1}^n \\sigma_j^2 \\Big).", "7264ab0c7df00504783f7ee6689dc3a8": "((a),[\\infty])\\in I", "7264fb25cd2d2380ba00d16d1c362cfd": "log\\text{P}(x, \\hat{\\pi}|\\theta)", "72654224adf416663fa15229d918c894": "c < K_{\\varepsilon} \\cdot \\operatorname{rad}(abc)^{1+\\varepsilon}", "7265522ff5a28e58df9ca316d3be5af8": "\\tan\\,(n{+}1)\\theta = \\frac{\\tan n\\theta + \\tan \\theta}{1 - \\tan n\\theta\\,\\tan \\theta}.", "7266123f5a3b1e5e0793aa628b32cd29": "E_K = hf - \\varphi = h(f - f_0).", "726624b92774fcf03ae4d8ac315851b2": "D \\gg \\delta", "72664407ba733788b9a409a28b55215e": "f \\cdot dx", "7266b28e167006eb7451d8cc1cf66388": "N\\cos \\theta =mg", "726705c0cfa3d99d80793d7e573a5933": "(a_1,a_2,\\dots,a_n)", "726730eb8a3bcbfa8079c10f43230a90": "\\exp\\left\\{ -\\beta u_2(r_{ij}) \\right\\}=1+f_{ij}", "7267324b2f1f30c20a0336058a11cfaf": " \\phi(q)=\\sum_{n\\ge 0}{q^{n(n+1)/2}\\over (q;q^2)_{n+1}}", "72674b0957dc9a90f95ef49a9dff38cb": "V(x,0) = \\max_{\\alpha B_0 + \\beta S_0 = x} \\mathbb{E}[1 - \\exp(-.1 \\times (\\alpha B_T + \\beta S_T))]", "7267afea10120212a8d25f2981b6f1f6": "\\phi_4=102.24^\\circ", "7267ff1bc81f55f54284e75a9f01579b": "f: A \\rightarrow A^{\\prime}", "7268c757672b4c6dcf46ed7039d3eb3b": " 4 \\pi / c ", "7269a0e005b7655bfcdaba5f55e1cf4b": " N^2 = -\\hbar^2 [ \\frac{1}{sin\\Theta} \\frac{\\partial}{\\partial \\Theta}(sin \\Theta \\frac{\\partial}{\\partial \\Theta})+ \\frac{1}{sin^2\\Theta} \\frac{\\partial^2}{\\partial \\Phi^2} ] ", "7269a6a634ce0560f081f5a694ec9fdd": "O(|A|^3\\ell)", "7269a89c4d0ab7db2ba65abe21e85d81": "\\widetilde{C}_{\\alpha\\beta\\gamma}=C_{\\alpha\\beta\\gamma}+(n-2)\\partial_{\\lambda}\\omega {W_{\\beta\\gamma\\alpha}}^{\\lambda}", "7269cd19bd44d8caa5e9ad441b864fa4": "\\eta_{\\rm Rx}", "7269e42273e90592fff775386d8e75b2": "\\mathbb R / \\mathbb Z", "726a45bf57afa085436dd2ddb9f648b0": "\\Gamma;\\phi \\vdash \\psi", "726a49340cc7246e51a23ace92d74b4f": "\\mathbb{O}P^2", "726a5095c8ca429c0626ea87e4c11e5a": "\\boldsymbol{\\mathcal{B}} = \\left( \\boldsymbol{U} + \\boldsymbol{c}_g \\right) \\mathcal{A},", "726a51fc873dc4ac80eb513322dad9e5": "\\sqrt{2p}", "726a74124da7fd127c6598243c45b755": "S1\\ \\delta^a\\ S2", "726a87a6a20cefc5009568053c70705e": " \\sin\\theta = \\frac{e^{i \\theta} - e^{-i \\theta}}{2i} \\;", "726a95082f67d103e7c9dc74f93adb30": "\\rho(x) \\neq 0", "726aaa776bc8108b798c93a375809723": "0=\\varepsilon_0<\\varepsilon_1<\\varepsilon_2<\\ldots\\;", "726ade4686c90bfebb61b69b6664ceb9": " \\frac{100}{2+1} + 1 = 34 \\frac{1}{3} ", "726af6ba5fbaf21a5ac31ce78b519ec2": "= \\langle I(t)^2R \\rangle \\,\\!", "726b345e4c682c8c441fb52af5e6a81e": "\\textstyle \\sum_i 1 \\cdot \\pi_i=1", "726b3f22ce0493a4e51e06141e079545": "\\lambda = 0", "726b4db3e017aa963db799a75f650b9e": "A \\longmapsto P", "726b794c4ac4cec9b84a751385636f80": "\\mbox{d}Q = r\\mbox{d}M = \\frac{1}{2}\\rho C_L \\frac{V_a^2(1+a)^2\\sin(\\varphi+\\beta)}{\\sin^2\\varphi\\cos\\beta}br\\mbox{d}r", "726bc44420d892e1a8f62ecbbc402714": "\\Delta x \\Delta p \\ge \\frac{\\hbar}{2},\\, \\Delta E \\Delta t \\ge \\frac{\\hbar}{2} ", "726befc50ecb16ccaf2e9e2433cd6a61": " \\frac{K \\cdot t}{V} = \\ln \\frac{C_o}{C} \\qquad(10)", "726c10cd61a56100e88e25dc15776564": "\n\\log \\frac{\\bar{Y}}{1-\\bar{Y}} = n\\cdot{}\\log [X] - n\\cdot{}\\log K_d\n", "726c351a9be9761968ea9732c611f582": "\\chi=\\alpha-\\beta=\\left(\\frac{2A}{\\sqrt{\\xi}}\\right)\\sin \\left(\\xi-\\xi_0\\right),", "726c8faa89ee67842c09dc0bf9f824a8": "\\theta_n(\\xi_1) = 0", "726ca52f26433c14a0a3e7d1a352425d": "\\omega _d = \\omega _n \\sqrt {1 - \\varsigma ^2 } ", "726cb11f5eb3da70d506fa4c5fdfefb3": " v(p_1,w-CV)=u_0", "726ccfa056bf05bde7feb27f190b3be9": " F_{(h(i))} ", "726cf4b013355b706c89ce64c507fd8e": " \n\\begin{bmatrix}\n a & b \\\\\n c & d \n\\end{bmatrix}\n= \n\\begin{bmatrix}\n 1 & 0 \\\\\n \\frac{-1}{\\lambda f} & 1 \n\\end{bmatrix}.\n", "726d183fa332785e440b03e5013c2461": " \\text{im } H^{q+1}f^\\bull : H^{q+1}(A^\\bull) \\rightarrow H^{q+1}(B^\\bull) \\cong H^{q+1}(A^\\bull) / (\\mbox{im } d^1_{0,q} : H^q(C^\\bull) \\rightarrow H^{q+1}(A^\\bull))", "726d1d37e943c7222d051b85b78483bf": "S_C(n)<2^n", "726d6f0d4f05c86bc85e0cd2b28a7ba3": "C_{t+1} = C_t -1 ", "726da4b11f674f37397c2d30326c5881": "\\bar{S}_i(c) = S_{2i+1}(c)", "726dd2dc7b8b01a281b89f07a0473e16": "\n\\lambda=1 \\quad \\lambda=-1\n", "726de781e55d4162b92101dcfdef2d0f": "\nR = cs_1^{i_n-i_{n-1}}s_2^{i_{n-1}-i_{n-2}}\\ldots s_{n-1}^{i_2-i_1}s_n^{i_1}\\,\\!\n", "726e6b05048d41595c28174af2ada04b": "f:F\\rightarrow G", "726e8c3e12aba0368c064e31ddb4a1bc": "R(X) = \\mathbb{E}[-X]", "726e974771e01b64a07a107b1ddfabf0": " Z_\\phi(s,\\chi) = \\int_{K^n} \\phi(x_1,\\ldots,x_n) \\chi(ac(f(x_1,\\ldots,x_n))) |f(x_1,\\ldots,x_n)|^s \\, dx ", "726f246a28a557dd3762dc4b543c5e21": "T^n(\\Omega)", "726f347617434d92b6a3330613e37a3b": "\\mathbf{\\hat{e}}_3", "726f430fc86aeaf752ee3856c1033514": "\\sigma_{ik}= - \\frac{\\partial W}{\\partial \\varepsilon_{ik}} \\qquad\\forall i,k=1,2,3", "726f6b4b6f91c6d328204af3422e14fc": "\\omega_{ij}", "726f70007cc7d79f7c21a3584897c59a": "\\sin^{2n+1}x \\le \\sin^{2n}x \\le \\sin^{2n-1}x, 0 \\le x \\le \\pi", "726fbba9932455ab287c17a4ab192cb5": "F_n(x)=F_n(y)", "726fc9eb7b31d19489017bc287755bc0": "\\tan (\\arctan x) = x", "727015464b6348a9c9c398e0996bf47d": " [u,v] = \\sum_{x \\in X} c_{u,v,x}\\; x. ", "72703f140e8091da5b46981fd7ac3d23": "\n i\\begin{pmatrix} \n i & 0 \\\\ \n 0 & j \\\\ \n \\end{pmatrix}\n= \\begin{pmatrix} \n i^2 & 0 \\\\ \n 0 & ij \\\\ \n \\end{pmatrix}\n= \\begin{pmatrix}\n -1 & 0 \\\\\n 0 & k \\\\\n \\end{pmatrix}\n\\ne \\begin{pmatrix}\n -1 & 0 \\\\\n 0 & -k \\\\\n \\end{pmatrix}\n= \\begin{pmatrix}\n i^2 & 0 \\\\\n 0 & ji \\\\\n \\end{pmatrix}\n= \\begin{pmatrix}\n i & 0 \\\\\n 0 & j \\\\\n \\end{pmatrix}i\\,,\n", "7270559ce7f1b91bd298274f76e705e3": " \\Re\\, {h(w)\\over w h^\\prime (w)} \\ge 0.", "7270711c4bb3ae2fe702a9a7ed2c5fcf": "\\begin{matrix}\np \\to q \\to r = \\text{hyper}(p,r+2,q) = p \\!\\!\\! & \\underbrace{ \\uparrow \\dots \\uparrow } & \\!\\!\\! q = p\\uparrow^r q.\\\\\n& \\!\\!\\! r \\text{ arrows} \\!\\!\\!\n\\end{matrix}", "72708747232e98e1b9f02e2a19c37d31": "\\widehat{QP1A}=\\widehat{QP2A}", "72709a95ce5e9e1d80f0a07de3585c6a": "n > 2 \\cdot N \\cdot C(\\mathcal{N})", "7270e9962d1bebd964635b4a94aafd45": "(1,0) \\rightarrow (0,t) \\rightarrow (\\frac{t}{2},0)", "7270f09fe43687817f618ba99940067c": "I_\\mathrm C = \\frac{V_\\mathrm{in}}{R_1}", "7270fe3414be61d2566cdac07e16246e": " H_{(h)}=M_{h,k} H_{(k)} M_{k,h} ", "7271527beb10334e4e4db0da56260b9f": "P = \\frac{E}{K} = \\frac{D}{K}", "72718ac44b80a9180e7dd20bddce01f2": "\\sum_{j=1}^n w_{ij} x_j \\leq W_i,", "727192e91d57bdd8fa9601cfcd2a2bf9": "F\\subseteq X", "7271a51d80d25017a6320123f3410519": "0 \\subsetneq \\mathfrak{p}_1 \\subsetneq \\cdots \\subsetneq \\mathfrak{p}_m", "7271c3b28c06a36420e4ced8fb991914": "Y_{7}^{-5}(\\theta,\\varphi)={3\\over 64}\\sqrt{385\\over 2\\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(13\\cos^{2}\\theta-1)", "7271ca36e1289059f34298b0e4402977": "(-\\infty, \\lambda]", "7271d5cc538e23b5d5438bb311458ce4": "(9) \\ \\nu (\\alpha) = \\nu_{110} \\sqrt{\\cos^2\\alpha + \\left(\\frac{\\nu_{001}}{\\nu_{110}}\\right)^2 \\sin^2\\alpha}", "7271e9bbdb144921bba70c3ce8a18f45": " 2S\\tau(j) = (1-1/4)/(1-j)^2 + (1-1/9)/j^2 + (1-1/4-1/9)/j(1-j). \\, ", "727227e5151c948f2c57e3dc0630c01d": "F_1(G_1)=F_2(G_2)\\,", "72723dcbfc22d43a4c66be19b56fa114": "e^{i 2\\pi k/n}", "727259b7f9b2061ac690df6111662edc": "\\Delta H_S^\\circ = \\Delta H_A^\\circ + \\Delta H_B^\\circ - \\Delta H_{AB}^\\circ ", "72725cac3c6efc155e8ddfe1945606fd": "\\chi^2_{\\nu_{1}}", "727292bd2a45289668544762aa799461": "(A \\rightarrow B)^*", "7272acc98e0c05abd4710c18e097afb8": "R_F(x,y,z)", "7272c8c6d1421fd5e755469cf68fb7ff": " |\\mathbf{r}_{12}| = |\\mathbf{r}_2 - \\mathbf{r}_1 | \\,\\!", "7272ca5a19786e43450cd9b7e74b3d2e": "\\frac{dy}{d\\mathbf{X}} = \\mathbf{A}", "7272ce280b47d3d0f8d9ea91144d81e4": "\\dfrac{Y(s)}{X(s)} = \\dfrac{G(s)}{1 + G(s) H(s)}", "7272dd97500aa4fe6870e5af0047c88b": "\\left( \\frac{3}{2} \\right)", "7272fb1e08142fa0f8ed7119d719de77": "Powe{{r}_{memory}}={{\\beta }_{1}}\\left( IFetc{{h}_{miss}} \\right)+{{\\beta }_{2}}\\left( DataDep \\right)+{{K}_{memory}}", "72730d7d0542013710b1b0c7883f35a9": "\\mu_L=-g_L\\frac{\\mu_B}{\\hbar}\\langle\\Psi_{n,\\ell,m}|L|\\Psi_{n,\\ell,m}\\rangle=-\\mu_B\\sqrt{\\ell(\\ell+1)}.", "7273833e7116309ec4f2da8fd46ae2f8": "\\mu\\uparrow\\lambda", "7273cf0de5416d19801d9133409822a0": "R(X,Y)Z = u\\left(2\\Omega(\\pi^{-1}(X),\\pi^{-1}(Y))\\right)(u^{-1}(Z)),", "7273d6ef3d93c0436831299ea3572d2b": "s,t,u,v\\in M(\\lambda)", "72745acdea745d0e0e1a440868850d7b": "\nRD_{H_2O} = \\frac{\\rho_\\mathrm{Material}}{\\rho_\\mathrm{H_2O}}\\ = SG,\n", "7274cc1abe500d09edef2347eeecdf8f": " \\bar{v} = - \\sum_{i=1}^N sin(\\theta_i) + \\sum_{j=1}^M cos(\\theta_j) ", "72759420c0da10814e1e280944e285ab": "\nV=V(0)e^{-\\kappa t}\n", "7275f6dd3b85e1169e773dea705c1fe7": "p = (A \\to w, R) \\in P", "72760d63cff71bfd436f773f23a41f10": "p(w,b|D,\\log \\mu ,\\log \\zeta ,\\mathbb{M}) = \\frac{{p(D|w,b,\\log \\mu ,\\log \\zeta ,\\mathbb{M})p(w,b|\\log \\mu ,\\log \\zeta ,\\mathbb{M})}}{{p(D|\\log \\mu ,\\log \\zeta ,\\mathbb{M})}} .\n", "727676af825eb8456ad99ec794527b3b": "\\overline{R_F}", "727683aa7c82c48f01be13864291cc60": "\\{b(f),b^*(f):~f\\in H\\}", "7276e53b67c5aa1e94d1c2b610b1cef4": "10\\log(|H_{NEXT}(f)|^2)=\\begin{cases} -66 + 6\\log(f) dB & f < 20 KHz \\\\ -50.5 + 15\\log(f) dB & f >= 20 KHz \\end{cases}", "72770eecde2f6041d04094dae8a08804": "\\Phi \\propto {\\frac{k_{CT}}{(k_{CT}+k_R)}}", "7277251c32eb45b86cd64a3c84e9f24e": "p_1/\\pi_10} (a+b-s-1)!\\Big[\\frac{Z_a(a+b-s,s)}{(a-s)!(b-1)!}+\\frac{Z_b(a+b-s,s)}{(b-s)!(a-1)!}\\Big]", "72881686e2d95104abfd84d646a6f79e": " \\text{(2)} \\quad \\hat{\\textbf{v}}_{k} \\leftarrow \\hat{\\textbf{v}}_{k-1} ", "7288a75e8df53c6ef8eb13154f207b62": "I=\\int d\\tau \\mathcal{L(\\tau )=}\\int d\\tau ^{\\prime }\\frac{d\\tau }{d\\tau\n^{\\prime }}\\mathcal{L(\\tau )=}\\int d\\tau ^{\\prime }\\mathcal{L(\\tau }^{\\prime\n}\\mathcal{)}.", "7288b9037f30cfd42b4fa4a58cc47769": "\\Pi_B", "728914270e4df3c31d98ae7252313ac6": "x=f^k\\left(y\\right)", "72893abb4faa09993e3923cc3e649e3b": "Pm = Pmf Pwo + Pmo Pwf", "72894f88e4bd2ebccfe6b49c92769187": "\n\\begin{align}\n&\\frac{d v}{dr} =\\frac{v}{r}= \\Omega \\\\\n\\end{align}\n", "728950a8c6b66ca4234142cd29fcd37b": "A \\vdash \\neg \\neg A", "72895329535c178a189d2de203cf0e98": "W(\\lambda)", "728955734d9e7ead73cf5e98116eef51": "\\lambda_n=n", "7289561f496569daeb7e3516c01736f2": " \\int_E (\\alpha f + \\beta g) \\, d\\mu = \\alpha \\int_E f \\, d\\mu + \\beta \\int_E g \\, d\\mu. ", "72896aade52e856a6a74974c61ce9f02": "\\int^{2\\pi}_0 |f(re^{i\\vartheta}) |^2 \\, \\mathrm{d}\\vartheta = 2\\pi \\sum^\\infty_{k = 0} |a_k|^2r^{2k}. ", "7289a3632cfabcfb6a4bfc1218b8836c": "\n\\begin{bmatrix}\n 1 & 2 & 3 & 4 \\\\\n 2 & 1 & 4 & 3 \\\\\n 3 & 4 & 1 & 2 \\\\\n 4 & 3 & 2 & 1 \n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 & 3 & 4 \\\\\n 2 & 4 & 1 & 3 \\\\\n 3 & 1 & 4 & 2 \\\\\n 4 & 3 & 2 & 1 \n\\end{bmatrix}\n", "7289bc95f7103ce85cae92b31372056d": "\\displaystyle \\, \\phi_{tt}- \\phi_{xx} + \\sin\\phi = 0", "7289d5108aa931088fd3de798faf8238": "S(p) = -\\mathrm{Tr}(p \\ln p)", "728a2a2c116f7d5dbb1c6b389c479ac5": "f(c) \\ge f(x) \\ge f(d)\\quad\\text{for all }x\\in [a,b].\\,", "728a313a0e2c34b7b1fc27751f87a9a5": "d = \\log{(n-k)}+2\\log \\left(\\frac{1}{\\varepsilon}\\right) +O(1)", "728a7f9d196fad4797f67dcd71baa81f": "\\mathrm{kg\\,m^2\\,s^{-1}\\,{rad}^{-1}}\\,", "728af8902223e70f5f69f552b9739716": "x(0) = x_0 \\in \\Omega_0, \\quad x'(s) = V(x(s)), \\quad T_s(x_0) = x(s), \\quad s \\geq 0 ", "728b09ef45144e062a3fce2ffe7835d2": "\n\\begin{cases}\nf_1(x_1)= 0\\\\\nf_2(x_1,x_2)=0\\\\\n\\cdots\\\\\nf_n(x_1, x_2, \\ldots, x_n)=0\n\\end{cases}\n", "728b8712012ffbdf19b0ab2101048ae4": "(X,\\mathcal{A},\\mu)", "728b9d89137aa6cc1cc4b4ac317474ff": "l_\\text{S} = \\sqrt{\\frac{G e^2}{c^4 (4 \\pi \\epsilon_0)}}", "728bc6a002a894abc9f934984ba3bde1": "1 \\over 2*3*4", "728c5e12211f8b588ac2746416ce1c7d": " f(z) = \\sum_{n\\ge 0} a_n z^n", "728c6564b2065d5055a5246ea42c5524": "F_{\\alpha,\\nu_1,\\nu_2}", "728c70b21e041a70ec40c0915a88a290": " \\cdot \\,", "728c79edd6913cfa519ce5392610ef3e": "F(s) = \\frac{1+s C R_2}{1+s C (R_1+R_2)}", "728c94ec2412341238c222df9b2216e1": "\n\\sigma_I^{(k)} = \\underset{{\\sigma_I = t\\sigma_I^{(k-1)}\\atop t \\in [0.7, \\dots, 1.4]}}{\\operatorname{argmax}} \\, \\sigma_I^2 \\det(L_{xx}(\\mathbf{x}, \\sigma_I) + L_{yy}(\\mathbf{x},\\sigma_I))\n", "728c9cfaf2d8bd53a4ad23d9add76a3d": "Y_{2,1} = \\gamma_e", "728cf888d6adce331f44fcd1495958bf": "\nV_R \\approx RC\\frac{dV_{in}}{dt}\n", "728d14ee143d92e3ced501c013f22516": "\\mathbb{P}\\left( X=x|Z=z \\right)", "728d171ab23f7f56e6d7412148e0ce55": "|p-1/2|", "728d7e9ce1fbd9a0e29b9c5fc5eab860": "F=m\\cdot a", "728d9ab8b09eb4686be81031d50249b7": "C(\\Psi, \\mathbb{C}^2)", "728dbf3c2c04311fd314e88fc80d165a": "||r_1| - |r_2||", "728dca9373344bdad232107320400500": "HoldsAt(on(box,table),t)", "728e72af7df437291b84e3bcf2e84320": " \\mathbf{T} + \\lambda_1 \\stackrel{\\nabla}{\\mathbf{T}} = 2\\eta_0 (\\mathbf{D} + \\lambda_2 \\stackrel{\\nabla}{\\mathbf{D}}) ", "728e864de41dca948bf3493d6e0ee420": "\\frac{\\rm d}{{\\rm d}t}\\psi(t)=f(\\psi(t),\\phi(t-\\tau))", "728ec92c476a504ac46c90fb1145f10b": "\\exp(1)*x+\\frac{1}{6}*\\exp(1)*x^3+\\frac{1}{30}*\\exp(1)*x^5+\\frac{31}{5040}", "728f0e8831aa9c89e3e31eac819953a7": "\\sigma_\\text{monthly} = 0.1587 \\sqrt{\\tfrac{1}{12}} = 0.0458.", "728f2deb5d8909dff2d100284f7f10a0": "(x, z)_{p} \\geq \\min \\big\\{ (x, y)_{p}, (y, z)_{p} \\big\\} - \\delta.", "728f5f796b476df7351f7941d3708401": "\\tau_1,\\dots,\\tau_{n-1}", "728fba2022c141ea1da1d3d210907bd7": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1=x^2_2\\left[\\tau_{21} +\\tau_{12} \\right]=Ax^2_2\n\\\\ \\ln\\ \\gamma_2=x^2_1\\left[\\tau_{12}+\\tau_{21} \\right]=Ax^2_1\n\\end{matrix}\\right.", "728febdf913fd1118fb2ff8c4711a077": "\\scriptstyle H_{m0}=4\\sqrt{m_0}", "72901d1984c3e38c762deeda76502882": "\\frac{p+1-q}{p+1}.", "72903360c963f97a34aebcbe520c13b1": "[dw]=0", "72903518d67a653d464828d9af770700": "g^{(n)}(\\mathbf{r}_{1}\\, \\ldots, \\, \\mathbf{r}_{n}) = \\frac{V^{n}N!}{N^{n}(N-n)!} \\cdot \\frac{1}{Z_N} \\, \\int \\cdots \\int \\mathrm{e}^{-\\beta U_N} \\, \\mathrm{d} \\mathbf{r}_{n+1} \\cdots \\mathrm{d} \\mathbf{r}_N \\, ", "72905c531031e33beb9bc9331d6bd244": "T_{\\phi \\,\\mu \\nu }", "72909d80962652bb2fa74b1e5db777f4": "\\nabla^2\\omega + \\frac{f^2}{\\sigma} \\frac{\\partial^2\\omega}{\\partial p^2} = \\frac{1}{\\sigma} \\left[ \\frac{\\partial}{\\partial p} J(\\phi,\\eta) + \\frac{1}{f}\\nabla^2 J \\left(\\phi, -\\frac{\\partial \\phi}{\\partial p} \\right) \\right] - \\frac{f}{\\sigma} \\frac{\\partial}{\\partial p} \\left( \\frac{\\partial \\omega}{\\partial y} \\cdot \\frac{\\partial u}{\\partial p} - \\frac{\\partial \\omega}{\\partial x} \\cdot \\frac{\\partial v}{\\partial p} \\right) - \\frac{f}{\\sigma} \\frac{\\partial}{\\partial p} \\left( \\xi \\frac{\\partial \\omega}{\\partial p} - \\omega \\frac{\\partial \\xi}{\\partial p} \\right) \\frac{R \\cdot \\nabla^2 q}{C_p \\cdot S \\cdot p}", "729100b22fa35ba2d1721b2962ce4002": "\\frac{[A]}{S_0} = \\frac{c_B \\, x_B}{(1-x_B)\\,[1 + (c_B - 1)\\,x_B]}", "729146601b352689c0e2bfd90c0240e3": " f_s=p \\cdot A \\ ", "729181eb93f5087f991df8bbe2d5f73c": " \\underline{\\mathsf{f}}(A + B) = \\underline{\\mathsf{f}}(A) + \\underline{\\mathsf{f}}(B)", "729283830b860af8bf40047b34e84a93": " \\mathrm{DCG_{p}} = rel_{1} + \\sum_{i=2}^{p} \\frac{rel_{i}}{\\log_{2}i}. ", "7292c1090f6a26d91e5ce239ded0b83e": "m_k", "7292ec297060c0d3e3b1d04397d1ad23": "\\le^{+}", "729323e889936fc4d6b7169b8e38b966": "K(x) = x \\cdot K'(x)", "729327858774a790aa6bb422b8fcb90a": "-B = \\bar{B} + 1", "72935aaf4644a5cd6c2ee08d6f803011": "\n \\Delta = \\frac{\\delta_1 (\\bar{\\lambda})}{V_1} + \\frac{\\delta_2 (\\bar{\\lambda})}{V_2} \\ ,\n", "72938ccab477253afeed060c36fb3f21": "\n\\left[\\begin{matrix}-\\lambda_{1}&\\lambda_{1}&0&\\dots&0&0\\\\\n 0&-\\lambda_{2}&\\lambda_{2}&\\ddots&0&0\\\\\n \\vdots&\\ddots&\\ddots&\\ddots&\\ddots&\\vdots\\\\\n 0&0&\\ddots&-\\lambda_{k-2}&\\lambda_{k-2}&0\\\\\n 0&0&\\dots&0&-\\lambda_{k-1}&\\lambda_{k-1}\\\\\n 0&0&\\dots&0&0&-\\lambda_{k}\n\\end{matrix}\\right]\\; .\n", "72945078b647a672581fe52e3c23ab0e": "(M, +)", "729462aff1ea418c82c41ee1b63b64c7": " x e_\\lambda \\rightarrow x ", "72946df8a13387f37be03cd75d1e43d5": "{g} = \\det ({g}_{\\mu \\nu})", "7294722c6b29460c0cddd295b3c238a8": "m\\frac{dv}{dt}=\\frac{1}{2} \\rho C_{\\mathrm{D}} A v^2 - mg \\, ,", "729477e776b3c125563cc665ffa89b7c": "\\mathbf{B} = (\\mathbf{H} + 4 \\pi \\mathbf{M} )", "72947b669cbb5a77430e2b1d7e1dd8a7": " S_t = S_0\\exp\\left( \\left(\\mu - \\frac{\\sigma^2}{2} \\right)t + \\sigma W_t\\right).", "72948f0159539b1673f027c5e0764ebc": "8\\tfrac{1}{4}", "7294bda6c8db34c42a39f0ad22501abe": "u(i,x) := \\varphi_i(x) \\qquad i,x \\in \\mathbb{N}", "7294bfdcee55cd1effaf27a7088b22be": " x_g,\\, y_g,\\, z_g ", "7294d011e4d315b2c4279979d1735929": "\\sum_{s_1\\cdots s_N} \\operatorname{Tr}(A^{s_1}\\cdots A^{s_N}) | s_1 \\cdots s_N\\rangle", "7294d2826868295607007c9603d228c3": "c_4(N)\\,=\\,\\sqrt{\\frac{2}{N-1}}\\,\\,\\,\\frac{\\Gamma\\left(\\frac{N}{2}\\right)}{\\Gamma\\left(\\frac{N-1}{2}\\right)}.", "7294e0960969d8c1c7c877f9f56b74cc": "\\frac{46}{45}", "7294f732c0602bd915d9ff81be7ea2af": " f(z) dz \\otimes dz ", "7294f95eac61f1ccc6c7b21dcc8621e8": "\\int\\sin^3 {ax}\\;\\mathrm{d}x = \\frac{\\cos 3ax}{12a} - \\frac{3 \\cos ax}{4a} +C\\!", "7295225266b92542dc80139dbd82112e": " \\begin{align} \nz = \\frac{[Z]}{[Z]_0 } = G(v_1, v_2, J_1, J_2) &= \\frac{ 2 v_1 J_2}{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\\\\n\\end{align}", "72954ab2c6fd45a05780f77d6b64e7ad": "R_E = 6.378 137 \\times10^{6}m", "72954b4a44376285ca76ba7c1551937c": "\n\\mathrm{E}\\left[ \\hat{A}_2 \\right]\n=\n\\mathrm{E}\\left[ \\frac{1}{N} \\sum_{n=0}^{N-1} x[n] \\right]\n=\n\\frac{1}{N} \\left[ \\sum_{n=0}^{N-1} \\mathrm{E}\\left[ x[n] \\right] \\right]\n=\n\\frac{1}{N} \\left[ N A \\right]\n=\nA\n", "72956bf2eeb3a97de32b3552f0d74fc1": "v_p = \\frac{\\omega}{k},", "72958eface164fa55421035db4b751fc": "\\forall (y_1, \\dots, y_k) \\in [m]^k", "7295d0b146884f6353e575bdac33ec4d": " \\theta_2 = \\angle EOC, ", "729603d6d0636ab91d6cfddd46eb40aa": "\\mathbf{D_p x}(\\mathbf{p}, w) = \\mathbf{D_p h}(\\mathbf{p}, u)- \\mathbf{D_w x}(\\mathbf{p}, w) \\mathbf{x}(\\mathbf{p}, w)^\\top,\\,", "729621cdaf942996e5ae9c422d6f4e01": "v_1^* B v_2 = 0", "729680fecb6eac5539c493748a9900fd": "A \\to \\alpha_1 \\And \\ldots \\And \\alpha_m \\And\n\\lnot\\beta_1 \\And \\ldots \\And \\lnot\\beta_n ", "729681679d7be49c26613ae1d6af05ef": " 3 \\times 3 ", "72969e196b44ed040ea9afbf611d6988": "f_4(z)= z+a/c \\quad", "7296d64e73ad4e0a50d5fe9701c02e35": "AMA = \\frac {F_{out}} {F_{in}},", "72974a8b3ff4261878a9ec106886e0da": "g^a", "72979103710d3fa97f2a5b0f3581ea18": "\\omega r_{\\mathrm{axle}}", "7297b5576c9771c5daeb70fb8806a74c": "\\beta_1\\log \\alpha_1+\\cdots+\\beta_{n-1}\\log\\alpha_{n-1}=\\log \\alpha_n", "7297f834c57f0a367cec4db1cc09ca91": "\\begin{matrix} \\frac{1}{13} \\times \\frac{1}{17} = \\frac{1}{221} \\end{matrix}.", "7297fa128ed585e529556b5610f31fa5": "(uv)", "729848b9717a590fd7160ab3e0699ec8": "\\frac{n + 1}{4n + 2} h", "7298db848934dbae09065631effd12bd": "K = 5", "7298ea6e642608bf00174a44fd4d477d": "a_{12} x_2", "7298eb793c87cdeba52b4ce45d6170c1": "\\|C_h U^*f\\|^2 \\le \\|U^*f\\|^2 = \\|f\\|^2 - |a_0|^2,", "729933ee5ca4f39529b30eab92b074c5": "{E}_{\\rm i}\\,", "72993a42be92952dd51584341f7c56a8": "X\\times Y", "729989b82d11c524120c2e2034472a87": "\\hat\\psi:\\hat A\\to\\hat A", "7299da039e5a2c550ee88e9a8166313e": " \\xi=(\\xi, p, X, \\theta)\\,", "729a2ca0f56530bd2fb11fda642d3c78": "|\\psi\\rang = c_1 | 1 \\rang + c_2 | 2 \\rang + c_3 | 3 \\rang + \\cdots", "729a8c17bc6f6f3e18ff0c29933ebd56": "\\gamma_1=\\cfrac{\\mathrm{d} w}{\\mathrm{d} x}\\,", "729a9a5ae10105d1a540c5868ab7f031": "\\begin{cases} - \\Delta u(x) = 0, & x \\in D; \\\\ \\displaystyle{\\lim_{y \\to x} u(y)} = g(x), & x \\in \\partial D. \\end{cases}", "729abcfbd42360ee3cd4162300b71541": "p=(x,y)\\ ", "729b24cc3e9c411caba2019295b23fe7": "\\textstyle(0,1,u,v,1,0)", "729b250693fd7974f34ef2037efc862a": "\\Lambda = \\frac{1-\\exp(-\\beta|m_i-m_j|^2/2)}{\\beta} \\cdot I(|i-j|\\le W)", "729b632b0fdcb6444d2dae9af2e1237c": "\\Psi(\\mathbf{r}_{1},\\mathbf{r}_{2}) = \\frac{1}{2\\sqrt{8\\pi^{5/2}+5\\pi^{3}}}\\left(1+\\frac{1}{2}|\\mathbf{r}_{1}-\\mathbf{r}_{2}|\\right)\\exp\\left(-\\frac{1}{4}\\big(r_{1}^{2}+r_{2}^{2}\\big)\\right).", "729c10a596c0de922984d85c4e313a6f": "\\sum_{r=0}^{n_0} p_rM^r = 0", "729c1959f1cc5d087bf856c655f734cf": " S_z = S_{xy} =\\int_{\\partial \\mathcal{V}} [(x - x_\\text{com}) T^{0y} - (y - y_\\text{com}) T^{0x} ]dxdydz ", "729c2b1c96b3e6e2305292c99658753b": " H_f \\in \\mathbf{TIME}(f(m) \\log f(m)) ", "729ce7c831730712faf27c80c7e8b059": "\\mathbf{w}=\\mathbf{v}\\oplus \\mathbf{u}=\\frac{1}{1+\\frac{\\mathbf{v}\\cdot\\mathbf{u}}{c^2}}\\left\\{\\mathbf{v}+\\frac{1}{\\gamma_\\mathbf{v}}\\mathbf{u}+\\frac{1}{c^2}\\frac{\\gamma_\\mathbf{v}}{1+\\gamma_\\mathbf{v}}(\\mathbf{v}\\cdot\\mathbf{u})\\mathbf{v}\\right\\}", "729cf6399d4bf1bb8595454928becb5e": "D = \\frac{1}{\\alpha + \\beta + \\gamma}", "729d0372d73fbcd9deae5ab7c787d27c": "\\beta_k", "729d23951a55035f9683be45b6a3b038": "R^{(E)}", "729d61cb8b79bfacfb898110df2ac623": "_{q=qp+qp'\\,}\\!", "729debd9d2382a38b4840aa915b42c51": "{\\boldsymbol{\\hat{\\mathbf{\\boldsymbol\\alpha}}}=\\mbox{minimize}\\;\\Vert \\mathbf{\\boldsymbol\\alpha} \\Vert_1 \\;\\;\\;\\;\\mbox{s.t.}\\;\\;\\;\\; \\mathbf{y}=\\mathbf{\\Phi x}=\\mathbf{\\Phi \\Psi} \\mathbf{\\boldsymbol\\alpha} = \\mathbf{A \\boldsymbol\\alpha}, \\;\\mbox{where} \\;\\;\\mathbf{A}=\\mathbf{\\Phi \\Psi}}", "729e2fb014962184cf72c8a227372caa": "{N \\choose 2}", "729e560f8b6d3b7455c6bf3b6fe837df": "(2\\lambda -1)!/(2^{\\lambda -1}(\\lambda -1)!)", "729e65a618cd2726f63fb041e632a2e8": "\nR_\\mathrm{eq} = \\left( R_1 \\| R_2 \\right) + R_3 = {R_1 R_2 \\over R_1 + R_2} + R_3.\n", "729e6afaeedbd365ff2dcd29ebb0f5f8": "(b,j')=\\mathrm{Rot}_{H}(b',j)", "729e7dd0eaf7c2b93f613f72004c9d4f": " m = \\frac{ \\sum_{ i = 1 }^K X_i }{ N } ", "729ebe758d89446c09b130e0315919b2": "\\mathbf{e}_4", "729f02aeb271e90d336739b4a9a8513e": "\\,n_0", "729f6a3f3c8c1a7353749d52aee0aa33": "\\widehat{P}_{t+1}(z^{\\tau})", "729f74c9fb988ad041c4be36e971e4d6": "g(x) = \\sum_{m=1}^\\infty \\frac{f(mx)}{m^s}\\quad\\mbox{ for all } x\\ge 1\\quad\\Longleftrightarrow\\quad\nf(x) = \\sum_{m=1}^\\infty \\mu(m)\\frac{g(mx)}{m^s}\\quad\\mbox{ for all } x\\ge 1.", "729fb413dbf2a59ef83035114daded76": "\\sum_{j=2}^k \\mu_j (\\mathbf{x}_j-\\mathbf{x}_1)=\\mathbf{0}.", "729ffd7c7488fc4afd62153f4a5fe646": "\\sum_{i=1}^n F_i = F_{n+2} - 1", "72a018f1ff50dc686e329238a7f1e07b": "\\left\\lceil\\log_k (k - 1) + \\log_k (\\mathit{number\\_of\\_nodes}) - 1\\right\\rceil.", "72a05ef3c07202997483efed6e474925": "f(T)=1+\\prod_{\\alpha \\in F}\\left(T-\\alpha\\right)", "72a08dc97f6e891583ac24005e50f822": "x_1\\ge x_2\\ge\\cdots\\ge x_n", "72a0c46ccdb7b0e3060b8df6f86a9edc": "V(x, t)", "72a10acf542f7f7824f001627a9f31be": "k_1<1", "72a173f25b1755fcc7364113531fd3c7": "\\{\\vec e_1,\\ldots,\\vec e_n\\} ", "72a17c82d84684f2e1e49e122a10685d": "\\vec x \\in \\mathcal V(K)", "72a1859dcc2206de8c710ac07a2f6593": "\\operatorname{M}_n(F)", "72a1b8259bbc6eca78f7f9fe71658541": "P_{TOT}=\\frac{3 V_P^2}{2R}", "72a1c5792f41468bfd434950099033e5": " L[G := S] ", "72a22bcfffd146e9361e3e1f38b94c84": "Q_1 Q_2 / (4 \\pi \\epsilon_0 \\Delta r)", "72a242ce37e01f58cbb35b91f8624bc5": "\n \\sigma_e = a + b~\\sigma_m + c~\\sigma_m^2 ~;~~ \\sigma_e = \\sqrt{3J_2} ~,~~ \\sigma_m = I_1/3 ~.\n ", "72a274c2c37c4514267c33cece8dce5c": "\\gamma^3 m", "72a2ae64b5eed5e965a007cf5b6133ed": " A\\, ", "72a31044786afb42ab6927ef373e946c": "\\left( \\max\\left\\{ u^{-\\theta} + v^{-\\theta} -1 ; 0 \\right\\} \\right)^{-1/\\theta}", "72a379a3cc28ca669c0ec2fd1cd14416": " {S_3 \\over S_1} = {{16\\over15} \\div {25\\over24}} ", "72a3be8954f8014be92678598798bdaa": "\\psi^{\\mathrm I}(t,\\cdot)", "72a3cbc7a90e03ebbcffd2cb01597c8b": "\\begin{align}\nr &= \\sqrt{\\rho^2 + z^2} \\\\\n\\theta &= \\arctan{(\\rho/z)}\\\\\n\\phi &= \\phi \\end{align}", "72a41804551a50e99ae7611241026657": "\\frac{dy}{dx} = \\frac{dy}{du}\\cdot\\frac{du}{dv}\\cdot\\frac{dv}{dx}.", "72a44f9d32e1841cf74f9de38ce3951f": "\\aleph_0,", "72a4aa2322ae519e250c04a7bd93a426": "mp^{e}+nq^{e}=1", "72a4aff7cccdf9825f6d910fee51d811": "\\mathbf{B}_\\text{el}^s = \\dfrac{\\mu_0}{4\\pi r^3}\\left(3(\\boldsymbol{\\mu}_\\text{s}\\cdot\\hat{\\mathbf{r}})\\hat{\\mathbf{r}}-\\boldsymbol{\\mu}_\\text{s}\\right) + \\dfrac{2\\mu_0}{3}\\boldsymbol{\\mu}_\\text{s}\\delta^3(\\mathbf{r})", "72a506a3da2dfc9a67db958201f295f7": "\\mathbf{p}\\in [0,1]^n", "72a530b2433264cef373d8fcca5662c1": "DX", "72a53af1de2499a369cfdea0cc5e90f0": "\n \\lambda_1 = \\lambda ~;~~ \\lambda_2 = \\lambda_3 = \\sqrt{\\tfrac{J}{\\lambda}} ~;~~ \n I_1 = \\lambda^2 + \\tfrac{2J}{\\lambda}\n ", "72a55da9c52fdd761f39751b493f4d34": "\\Lambda_{\\mathrm{m}}^\\circ", "72a5dde99a1ecf270a697ba6c2f54146": "\\scriptstyle \\boldsymbol \\nabla S", "72a61e83a52ed1d68f77f1ca70f4f15b": "r/s", "72a621bfc609c1f293827a3761cff423": "\\|X_n\\|\\geq \\frac{2}{\\pi} \\log(n+1)+C.", "72a643cb5c0e5fd3e08dd96142bf36b3": "{f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \\geq 0}.\\,", "72a6cbd971a4ecc84f9b59ae6fd3e997": "A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1} \\cdots A_n B_n A_n^{-1} B_n = 1", "72a6d781673b4c9ebec6aeebcf2adb22": "B = \\frac{V}{F_D} = \\frac {C_c}{3 \\pi \\eta d}", "72a6e07323262cfc0d6f2710698d899e": "\\pi_1(T,t_0) \\approx \\mathbf{Z} \\times \\mathbf{Z}", "72a6ec305780c21e3e6787b5da17d89b": "(f\\cdot g)(t)=[-t+\\cos (2\\pi t)]+i[t+\\sin(2\\pi t)]", "72a770b7b9a14bb3c3f5306753c3e7a3": "{a \\over (a+b)}{(a-1)-b \\over (a+b-1)}+{b \\over (a+b)}{a-(b-1) \\over (a+b-1)}={a-b \\over a+b}.", "72a7b5bcdad3701d3d039b5a1554bf00": " S = \\int d^dx \\; \\bar\\psi (\\partial\\!\\!\\!/ + m) \\psi \\;.", "72a810049a8226732f57c9c9c6934948": "x + 2 = 3(x - 2) - 6x\\,.", "72a818fa00ca0b2d7a0c4a317c1615a2": "\\operatorname{d}U = T \\operatorname{d} S - p \\operatorname{d} V", "72a8771cda02b6fc7517fa0f3c04538c": "\\delta L= \\delta_i\\mathcal{L} \\, dy^i\\wedge d^nx,\\qquad \\delta_i\\mathcal{L} =\\partial_i\\mathcal{L} +\n\\sum_{|\\Lambda|}(-1)^{|\\Lambda|} \\, d_\\Lambda\n\\, \\partial_i^\\Lambda\\mathcal{L},", "72a8b0b3998d267c4daa09b4671e1ae4": "(\\rho,\\varphi,z)", "72a8f0cda7fe224bbec286e58c1f57ab": "t_0=({x_1 \\cdots x_n})^{\\frac{1}n},", "72a8f6cccc35d4e1a0c9f018e9cb1e0e": "\n \\eta(x)~\\cfrac{\\partial w}{\\partial t}\n ", "72a91ffcdd1854d33ea7e729dfd42a20": " k \\geqslant 2 ", "72a926d07959a7eaf3cf6b89dcf1c325": "\\deg x < \\deg q", "72a9812c84cf87e2a80a0531e6281408": "\\left[\\mu- z\\sigma,\\ \\mu + z\\sigma \\right]. ", "72aa0416f7786c07de9eff4fff8534c9": "f: \\kappa \\to \\mathcal{P}(\\kappa)", "72aa11dbfba1c0cdad2a7c3fc8b64e10": "\n\\; C_\\Phi = \\sum _{i = 1} ^{nm} b_i b_i ^*. \n", "72aa19bd17c554bb6b2c479187fe9792": "\\langle x,y \\mid yxy^{-1}xy=xyx^{-1}yx\\rangle", "72aa2c19f4c6495c050d71b8cc9a070d": "share_i", "72aa77c7d3ceed9495c8cc5664faef3d": " = a_1\\lambda_1^k\\mathbf{u}_1 + a_2\\lambda_2^k\\mathbf{u}_2 + ... + a_n\\lambda_n^k\\mathbf{u}_n ,", "72aa87121403fd73c75a57c53be96f0d": "p_1 = -p_2 + p_3 + p_4 \\quad \\quad \\quad \\quad \\quad \\quad \\quad (2)\\,", "72aad05e9990a1e571dd58b93d2e676f": "\n \\mathcal{M} := -D\\nabla^2 w^0 \\,.\n ", "72aae12cc31e663e184f4bcef99af73e": " z^T A^T A z = \\| A z \\|_2^2 > 0, ", "72aafc663b2d5021d9fe0dad5709e5e7": "\\Psi_0=\\Psi_1=\\Psi_3=\\Psi_4=0\\,,\\quad \\Psi_2=-\\frac{M}{r^3}\\,,", "72ab12b73eec22b8b23975f7078266f4": "\\operatorname{VaR}_q", "72ab8af56bddab33b269c5964b26620a": "pi", "72aba70a60f3c781b4bf6d0c271c8fd9": "\\phi_{1..N}", "72aba8e1fe6544e5c940773cc2a7cd38": "\\rho(v) = \\sum v_i \\otimes e_i", "72abb72bf267ec1843c375f2d32f4e09": "C_k", "72abd7703cbd0a5c8511fdae531022ef": "X \\to M_f \\to Y", "72ac01d1bcce79fbbf63662378daf5df": " H=X(X'X)^{-1}X'", "72ac18d2e0292a966a666cf2c254dbbd": "K \\ll D", "72ac691fbe739f6e3fb3719fb429d80f": "ST_x(\\neg \\varphi) \\equiv \\neg ST_x(\\varphi)", "72accf3292eebec881ba4050db720533": "J\\ \\xrightarrow{p}\\ \\infty", "72acd3f701be2b9e522490c328ddfc43": "\\mathbf{v}\\cdot(\\nabla\\mathbf{v}),", "72ad629b238f2ace32c9df6a0b312e72": "e^{x+y} = e^{x}e^{y} \\,", "72adbd597ee0443ef41a19c6f1bd0ef8": "\\textstyle(\\prod_{i \\| f(t_0) - f_e \\| ", "72b16f07c81ba1bc7f82a96fdec0025e": "\\textstyle u(z)", "72b19ca31f1a5054f759103fdb9356aa": "\\mathrm A (M) = \\frac{9,700 + 100}{9,700 + 150 + 50 + 100} = 98.0%", "72b1a0ad03930f1229d9c52d8e369a15": "-x^2", "72b1bd960e6c193ceaf5b751fe112575": "{\\alpha + 1 \\over \\alpha}", "72b1ddbe8ad8a96b5776080c06143e4e": "\\bar{f}(s)=\\int_0^\\infty f(t) e^{-st} \\, dt", "72b23c509b9fafb1a2c7301745dbe943": "-\\frac{M(u)_{,\\,u}}{4\\pi r^2}", "72b260b78a4c39cad7baf2285b64ebf3": "\\ ax^2+bx=c", "72b27b59c691107077f2fce3519312e0": " D ", "72b28bc00d78f85ed83fc9a35a8ca576": "x = bt + A_x\\,", "72b28fcba5874af007a63f49a5f5becf": "-\\frac{1}{3}\\cdot B_2(t)", "72b29c96534b9732d6b393859ea143db": "J = \\frac{V_a}{nD}", "72b2e738672ca692e5ed08f318bb8c36": "\\text{A} \\mapsto 1, \\text{B} \\mapsto 2, \\text{C} \\mapsto 3,", "72b31c72e9ef5cdc1836c0026fc84e28": "5x^2+8xy+5y^2=1.", "72b32a1f754ba1c09b3695e0cb6cde7f": "57", "72b37139c2882fe4db0d57ab98431246": "\\omega_0\\,", "72b3d727d1fbdc3576ef6c0328f59716": "\n\\theta_j=\\theta_j'+\\theta_{j-1}\n\\qquad\nj\\ge 2\n", "72b3edf6e9b0ed215d73ef5e8a3289cf": "a/c < b/d ", "72b40375bf03d60c480966067148f04b": "\\phi (\\omega)=\\arg K(j \\omega)=-\\arctan \\frac{\\omega}{\\omega_0}.", "72b429b93867b885b6fc8ac4b9e98eb2": "\\rightsquigarrow y^{-1} \\cdot x^{-1}", "72b478c3ae8437104ff72efec512c953": "\nv_{12} = {z_1 + z_2 \\over \\sqrt{2} }\n", "72b4e01933ea83637c665d87f15907f7": "\\nu_s/2", "72b5065a81e26e287902f5e73967e44b": "\\Pi_{q,{q'}}(\\partial_i)", "72b50b4cfcc2c896ca0525e96ada7a51": "\\langle i,j\\rangle ", "72b50db50026cc7ffc8e6d67c21f01fe": "dA=\\tfrac 1 2\\cdot r\\cdot r d\\theta", "72b517e8bf5036a07758e5e1b663def5": "v_{Te} = (kT_e/m_e)^{1/2} = 4.19\\times10^7\\,T_e^{1/2}\\,\\mbox{cm/s}", "72b56ef56788d8088bc53fc106f96717": " X \\in \\mathfrak g ", "72b59ddcb54456a19bb8b31685d6190f": "\\|\\nabla P(Z)\\|_E \\leq d \\|P\\| \\, \\|Z\\|_E^d", "72b5d8e22dae024ec3b0fd9fe2efb1d8": "PSL(2,\\mathbf{Z}) \\simeq B_3/\\langle c\\rangle.", "72b5fae2f1ace03779d790f7b2b3b6bd": " \\ln \\,\\operatorname{var_{GX}}=\\operatorname{var}[\\ln X]= \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta) ", "72b60c6602c0a009171e57ccf11ae2c5": "\\bar{\\nu}_{\\mu}\\rightarrow\\bar{\\nu}_{e}.", "72b62eae139a81a60017fb176a7d71ee": "\\{X \\subset J \\; | \\; P_2 \\uparrow S(X,J)\\}", "72b631b41463a9f6d67e50bf4fcced7f": "\\alpha \\circ \\beta := \\pi_{XZ*}(\\pi^{*}_{XY}(\\alpha) \\cdot \\pi^{*}_{YZ}(\\beta)) \\in Corr^{r+s}(X, Z)", "72b63a66ea3901c5df296f8daf0f38eb": "4(\\frac{8}{9})^2 = 3.16049...", "72b65e855982e61eb46918ac90f451da": "\n\\mathbf{G_x} = \\begin{bmatrix} \n\\quad~ & \\quad~ & \\quad~ \\\\[-2.5ex]\n1 & 0 & -1 \\\\\n2 & 0 & -2 \\\\\n1 & 0 & -1 \n\\end{bmatrix} * A =\n\\begin{bmatrix} \n 1 \\\\ 2 \\\\ 1 \n\\end{bmatrix} *\n\\begin{bmatrix} \n +1 & 0 & -1\n\\end{bmatrix} * A\n", "72b69b51ae893a3074fe77a30b018260": "L_\\mathrm{dB} = 20 \\log L \\,\\!", "72b6a47e467ea64e89507d1cb01e12d6": "D+dD", "72b6b1f38e92d35082b50153fc108776": "\np_n \\geq \\prod_{i=0}^{n-3} \\Bigl(1-\\frac{2}{n-i}\\Bigr) =\n \\prod_{i=0}^{n-3} {\\frac{n-i-2}{n-i}}\n = \\frac{n-2}{n}\\cdot \\frac{n-3}{n-1} \\cdot \\frac{n-4}{n-2}\\cdots \\frac{3}{5}\\cdot \\frac{2}{4} \\cdot \\frac{1}{3}\n = \\binom{n}{2}^{-1}\\,.\n", "72b6c3a0db39b79fcffb08e0841ec74b": "\nT_{\\mu\\nu}(\\tau)=P_{\\nu-1/2}^\\mu(\\cosh\\tau)\\,\\,\\,\\,\\mathrm{and}\\,\\,\\,\\,Q_{\\nu-1/2}^\\mu(\\cosh\\tau)\n", "72b71167f2329743e5d7eb79566d0d2b": " \\operatorname{head} \\equiv \\lambda z.\\operatorname{first}\\ (\\operatorname{second} z) ", "72b71c36ff2e2cc84157a85de0cdd97b": "\\psi^*", "72b73fa92567a351560fec3a5303c71f": "\n\\varrho_{A, B, \\Lambda}=\\sum_\\lambda \\varrho_A^\\lambda \\varrho_B^\\lambda w_\\lambda |\\lambda\\rangle\\langle\\lambda|\n\\,", "72b75d576f0322e230ac6fa8343c9642": "x_i < \\frac{p-1}{2}", "72b7e7774ce5523027369576b2252ebe": " \\textbf{K}_{k} = \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} \\textbf{S}_k^{-1}", "72b7ea778846dc005e5cf1367bffd931": "\\alpha + \\beta + \\gamma = 180^\\circ ", "72b7f2ea16d6f2cd6ff324c5a2da0842": "\\mathfrak{m}/\\mathfrak{m}^2", "72b82247ebb3c79d565642b6c21cb43e": "n = 3.4 \\times 10^9", "72b83e92f4c41b3ed7005f9122f0a7fc": "\nJ_y = \\frac{\\hbar}{\\sqrt{2}}\n\\begin{pmatrix}\n0&-i&0\\\\\ni&0&-i\\\\\n0&i&0\n\\end{pmatrix}\n", "72b89a55dd13a736616421578c4dbf45": "\n\\begin{array}{lll}\n&\\\\\n \\frac{\\sin{\\textstyle\\frac{1}{2}}(A{+}B)}\n {\\cos{\\textstyle\\frac{1}{2}}C}\n=\\frac{\\cos{\\textstyle\\frac{1}{2}}(a{-}b)}\n {\\cos{\\textstyle\\frac{1}{2}}c}\n&\\qquad\\qquad\n&\n \\frac{\\sin{\\textstyle\\frac{1}{2}}(A{-}B)}\n {\\cos{\\textstyle\\frac{1}{2}}C}\n=\\frac{\\sin{\\textstyle\\frac{1}{2}}(a{-}b)}\n {\\sin{\\textstyle\\frac{1}{2}}c}\n\\\\[2ex]\n \\frac{\\cos{\\textstyle\\frac{1}{2}}(A{+}B)}\n {\\sin{\\textstyle\\frac{1}{2}}C}\n=\\frac{\\cos{\\textstyle\\frac{1}{2}}(a{+}b)}\n {\\cos{\\textstyle\\frac{1}{2}}c}\n&\\qquad\n&\n \\frac{\\cos{\\textstyle\\frac{1}{2}}(A{-}B)}\n {\\sin{\\textstyle\\frac{1}{2}}C}\n=\\frac{\\sin{\\textstyle\\frac{1}{2}}(a{+}b)}\n {\\sin{\\textstyle\\frac{1}{2}}c}\n \\end{array}\n", "72b8d21fdfc7b1c80bc835001984135b": " S_i \\subseteq \\mathbb{R} ", "72b9c9bf38436cadc5542bfdb0c94d87": "\\langle{\\underline P}X,{\\overline P}X\\rangle", "72ba3ff4c78c94b5140beb0b0cdc0ecf": "\\left\\{ z \\in \\mathbb{C} \\left|\\ \\left| {1+{1\\over 2}z \\over 1-{1\\over 2}z} \\right| < 1 \\right.\\right\\}.", "72ba44ce53328723eef91a36bc0a7fde": " \\nabla \\times \\left( \\nabla \\times \\mathbf{A} \\right) = \\nabla(\\nabla \\cdot \\mathbf{A}) - \\nabla^{2}\\mathbf{A}", "72ba6f144d2ac47642314bd67f7842bc": "\\displaystyle{\\|S\\varphi\\|_\\infty \\le C^\\prime \\|\\varphi\\|_\\infty,}", "72bae8a53c819a959e1b296088e1a3c7": " \\langle j | s \\rangle = \\Psi(r_{j,s}) e^{-i \\theta _j}", "72bb068e2350ec9e8675873c1b075d1b": "\\mathcal{F}\\{g(x P)\\}\\ = \\frac{1}{P} \\cdot \\hat g\\left(\\frac{\\nu}{P}\\right)", "72bb120d7d1fcbb3ca8ae10434338df5": "\nQ_k = {1\\over\\sqrt{N}} \\sum_{l} e^{ikal} x_l", "72bb49fd0bbe80d91428abd9dcce9174": "\\theta=\\tfrac{\\pi}{2}", "72bb5b4b1f2afc9d17a18518f314af0e": "g^{00}", "72bb7fe94c47d7e6bea1ef4b83dc4e9d": "\\zeta(2n) = \\sum_{\\ell=1}^{\\infty}\\frac{1}{\\ell^{2n}}=\\eta_n\\pi^{2n},", "72bb931f14701df501d2b6935fd6a55d": " a=b=0 ", "72bbe6ada5f1ec13cb1fb3c7a475adc0": "\\text{who} : N \\circ W", "72bc1c1a57d060ec6f2a94b90c1598d8": "V\\oplus W", "72bc45c11a9cd90b1ac9dc2e18aa9b16": "\\alpha_i \\in N", "72bc85275ad2142d8343b47f6ca44aca": "\\Delta G_{\\text{em}}=\\gamma{3V\\ \\over R}", "72bca34f44eaecee9962adc5b421db42": "\\displaystyle{\\psi(a)\\psi(b) =\\psi(a\\circ b),}", "72bcb2665c5ea62cad511212d84392af": " \\lambda f.\\operatorname{sink}[(\\lambda p.(p\\ f)\\ (p\\ f))\\ (\\lambda f.\\lambda x.f\\ (x\\ x)), X] ", "72bcc72f98380aaa532d9a06278e3d60": "S_\\text{BH} = \\frac{A_\\text{BH} k_\\text{B} c^3}{4 G \\hbar} = \\frac{4\\pi G k_\\text{B} m^2_\\text{BH}}{\\hbar c}", "72bccb57801423fe8238b3c263c25dbb": "Y^{\\prime}(t)=A(t)Y(t),\\qquad\\qquad Y(t_0)=Y_{0}", "72bcefdc28a13bd0db34eec17556e9a0": "G(x, t) = \\frac {1}{\\sqrt{2\\pi t}} e^{-\\frac{x^2}{2t}}", "72bd2814aec46568848bb6cda32afbe5": "\nx_1 \\in [-\\infty ,5], \n", "72bd875d128edd4fd440ab21afe1f49d": "\\displaystyle{Q(y)R(c,Q(y)d)x=2Q(y)Q(c,x)Q(y)d=2Q(Q(y)c,Q(y)x)d=R(Q(y)c,d)Q(y)x.}", "72bdb6806e78df03ad9c93e5a1608546": "v(f({\\mathbf A})) = f(v({\\mathbf A})).", "72bdc810925b35fff12d60eaace12038": " [T_K \\varphi](x) =\\int_a^b K(x,s) \\varphi(s)\\, ds. ", "72bdda82106d79da8dfb6e9c58f47970": "M(x) \\cdot x^n + \\sum_{i=m}^{m+n-1} x^i = Q(x) \\cdot G(x) + R (x)", "72be2d4cb49dd177732339f920da95ae": "\n\\begin{align}\n\\mathbb E|\\langle z,Z\\rangle|^2 \\geq\\kappa(A)^{-2}{\\lVert z \\rVert^2} \\qquad\\qquad (3)\n\\end{align}\n", "72be5370c8ceb23bf77d6d62a79badff": "\\wedge \\, \\sum_{i=1}^c \\mu_i = n \\, \\wedge \\, \\forall\\, i \\in [0,c]\\, :\\, 0 \\le \\mu_i \\le m_i\\,.", "72be5796af26ef94d89352994ae10e80": "\\int x^2\\cos^2 {ax}\\;\\mathrm{d}x = \\frac{x^3}{6} + \\left( \\frac {x^2}{4a} - \\frac{1}{8a^3} \\right) \\sin 2ax + \\frac{x}{4a^2} \\cos 2ax +C\\!", "72be67a3a179d1f6d8e8857b82785e63": "R_{\\alpha\\beta}", "72be76d8c70aa1b5737441f4255fc4e2": "\nx(n)=(g_{b_1}(n),\\dots,g_{b_{s-1}}(n),\\frac{n}{N})\n", "72beaa1b4071974f15560c3bba19c563": "\\int_0^t(\\sigma_s^2+|\\mu_s|)\\,ds<\\infty", "72bed6c25ae52775f5e7dac5d7ad3d93": "f(x) = \\int_0^{f(x)} \\mathrm{d} t.", "72bef1bb545c56f761f82039fd65ea0c": "y\\sim e^{S(x)}\\,", "72bf0e0168e698dea3306f1706369c11": "h(x)=\\alpha e^{\\beta x} + \\lambda ", "72bf1706d8efd65e8ec839eedfa47ca9": "C_n = {2n\\choose n} - {2n\\choose n-1} \\quad\\text{ for }n\\ge 0,", "72bf49a2ca55cc43edc6abd1f433f8e6": "\\omega\\wedge\\eta = (-1)^{pq}\\eta\\wedge\\omega.", "72bf572d6d87f4034305e65349a043ef": "\\frac{d}{d\\rho}(2{\\cos \\alpha_1-\\rho^2 }(1+kc)) = 0 ", "72bf948691da3a15003e4a362759b411": "e^\\frac{-ik\\left(x^\\prime -jd\\right)^2}{2z} \\approx 1", "72bfecdeb15f7d277fe22883fc65fa1c": " \\mathcal{O} (n^3) ", "72c029b60ecad1434535d8240dc6ccfc": " \\Sigma=\\frac{\\sum_{i=1}^{N}w_i}{\\left(\\sum_{i=1}^{N}w_i\\right)^2-\\sum_{i=1}^{N}w_i^2}\n\\sum_{i=1}^N w_i \\left(\\mathbf{x}_i - \\mu^*\\right)^T\\left(\\mathbf{x}_i - \\mu^*\\right). ", "72c081edf05762fa949c245c43977439": "ax^2 + bx + c", "72c094fafb1fd8ca4e93eedadca06466": "D_\\mu\\Big.", "72c0b06d1f567eed2921acaeef9ac71d": "A = u^{-1} (0)", "72c0e88d8090ede36a6c4c83a2e92a93": "\\vec z=(z_1,\\dots,z_n)", "72c10fe53c5757ae65683d803a8f0b93": "v_{-i} = (v_1, \\dots, v_{i-1}, v_{i+1}, \\dots, v_n)", "72c179a78b81c08c8efb9327b220698c": "\\{(x,y) \\mid \\exists A \\in P\\,(x \\in A \\wedge y \\in A)\\}", "72c1b3c72daa509f48e9f90156888c00": "u_{2}(\\mathbf{q}) ", "72c1b4950a16124fccfbc3992beee653": "|(a,b)| = |a| + |b|", "72c20f341753e0bf925adf85858077cd": "\\mathbf{R} = \\log_2 \\mathbf{R}^+", "72c2418fd26d2ed064dac5fc43c56218": "K_1 \\times (\\Sigma_1 \\cup \\{ \\epsilon \\}) \\times g(\\Gamma^*)", "72c29c948fa149102eafadc2210e7203": "a_0 = 1", "72c2de6dada513c3d289a176500f29c8": "g\\in G", "72c329b9e554a1eb6ad72df593f400ee": "= \\arctan \\frac{120}{119}", "72c3757ed582724c411c02b92cdda1f0": "\\begin{align}\nS = \\ln\\Omega_{E,\\ell} &\\approx n\\ln\\ell + n \\ln\\sqrt{E} + const.\\\\\n&= n\\ln\\ell + n \\ln\\sqrt{E} + f(n)\\\\\n\\ln\\Omega_{E,\\ell,n}&\\approx n\\ln\\frac{\\ell}{n} + n \\ln\\sqrt{\\frac E n} + const.\\\\\n\\end{align}", "72c37b9a35c1a8e858fac3f4186bfa18": "E_l\\,", "72c3b2d964dec1d027f804418747a2ff": "f(k;p) = p^k (1-p)^{1-k}\\!\\quad \\text{for }k\\in\\{0,1\\}.", "72c3d40ee7576f314778388be3b931f4": "E(t) = \\Re(E_0 e^{i\\omega t});", "72c3e3d9c3dd5ace7c2c34b1868effff": "L_1(x) \\in L_1(B)", "72c424d62f90e66cf24c676312a981e1": "SRM = 1.3546 \\times Lovibond - 0.76", "72c424ee17352dfa64217e5b5838e571": "\\Phi=\\varphi-\\int_{t_0}^t f(\\tau)\\, \\operatorname{d}\\tau,\\text{ resulting in }\\frac{\\partial \\Phi}{\\partial t} + \\tfrac{1}{2} v^2 + \\frac{p}{\\rho} + gz=0.", "72c430aa57c0a0a49f457e5a96b23246": "s_1\\,\\!", "72c4434c5bc13b9f26f516604b4a4b10": " \\frac{\\delta J}{J} = \\frac{\\delta E_1}{E_1} + \\varepsilon^{1}_S \\frac{\\delta S}{S} ", "72c45af757d5f90884b24a6a559005b9": "\\gamma_{ijk}", "72c4c10517de11771fd70139a801acd6": " \\mathbf{x}_j = 0 ", "72c4c4dccf02441295b8cd4e269a4287": "= \\frac{\\Delta t}{\\Delta i}", "72c4eda29d6b2bf081e20a65dcf24a80": "f_Y(y) = 2f_X(g^{-1}(y)) \\left| \\frac{d g^{-1}(y)}{d y} \\right|.", "72c4fdc6281af6975d1fd2c602f2a86e": "\n \\mu_0 = \\cos\\varphi~\\cos(\\alpha-\\lambda) ~;~~ \\mu = \\cos\\varphi~\\cos\\lambda ~.\n ", "72c5331676427bf34879f32c45da7671": "\\Pr(|S - \\mathrm{E}[S]| \\geq t) \\leq 2\\exp \\left( - \\frac{2t^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right).\\!", "72c5660a47ab4978704525746f81190a": "H(\\boldsymbol{q},\\boldsymbol{p},t)", "72c56de169ddc6ce4e131b9cd0e4cea3": "\\{x_0,x_1,\\ldots,x_N\\}", "72c5cc0e2586935d16539f31a2a4fec4": "ACD", "72c625b62690b56b8b6a62584e13e035": " |\\textbf{G}(\\pm j\\infty)| < \\infty ", "72c72ff82c3ae9cee8b8602085746989": "\\varphi_e \\simeq \\varphi_{F(e)}", "72c75c2c7f1b30c91e54e4f89efa54c2": "x_2 = \\frac{1}{2} \\left(x_1 + \\frac{S}{x_1}\\right) = \\frac{1}{2} \\left(404.457 + \\frac{125348}{404.457}\\right) = 357.187.", "72c76e6424fa938bf57a05b0ca71380c": "N_\\mathit{o}", "72c7cef49898ccfe81a1e588a7a81922": "<: \\!\\,", "72c88be4d31dca911601d331f0d841e2": "\\Gamma_\\tau\\subset H_\\tau", "72c88d9295c4086717f63756ec75c06a": "\\pi_t", "72c8c31c96f0420121ffaed82c41f8ed": " \\leq2\\left[ \\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{\n\\left( I-\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\right) \\Pi_{\\rho\n,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\}\n+\\sum_{i\\neq m}\\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left( i\\right) },\\delta\n}\\Pi_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}\n^{n}\\right\\} \\right\\} \\right] ^{1/2},\n", "72c97487524715607505aefb24895da2": "\\stackrel{\\delta\\phi\\pi\\beta}{\\Mu}", "72c9b43b7dd51a52befe7710997838ef": "\\frac{\\pi^2}{6}\\,", "72c9d6b32648fec35d2a74b6b5ac7722": "\\text{Amplitude (or height)} = \\frac{1}{\\pi\\gamma}. ", "72c9e4c116b7df9869477dcf7aba3840": " K g_2 L = \\coprod_j b_j L", "72ca16c7f743dfbcae05adb9f1dc0f3a": "v_y=0", "72ca296df138f349a49e229edbb7aafa": "\\sgn(x) = \\begin{cases}\n-1 & \\text{if } x < 0, \\\\\n0 & \\text{if } x = 0, \\\\\n1 & \\text{if } x > 0. \\end{cases}", "72cae59d8fb818409f0e3bca20bc63a5": "\\beta_i \\equiv ", "72cb08d7fae1e297fb5e8ffd2f45b3ba": "\\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1) x^{r + c - 1} -\\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1) x^{r + c}+\\gamma \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} -(1 + \\alpha + \\beta) \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c} -\\alpha \\beta \\sum_{r = 0}^\\infty a_r x^{r + c} =0", "72cb4bed8f29ed954f72e45e2c3fe672": "\\scriptstyle I_0", "72cbc0a7f751a4a429ed2d1366d043b6": "H=\\frac{1}{2}\\int \\left[\\sum_i\\left(\\frac{\\partial \\mathbf{S}}{\\partial x_i}\\right)^{2}-J(\\mathbf{S})\\right]\\, dx\\qquad (1)", "72cc6f974978ebecb256a53e6d0d504c": "Y\\to X,", "72cc7883d62a2ce5d24d01c91b2b9325": "f\\cdot(c\\cdot\\gamma) = (f\\cdot c)\\cdot\\gamma", "72cc83ca2a36a38a9f50b61c3f9c9746": "v:2^N \\to \\mathbf{R}", "72cc8abf3e5644daf0a95bfb609df538": "\\le_+", "72cca49d7ddc404d73a126b2cc4fa4b9": "{E'_y \\over E'_x} = {E_y \\over E_x\\sqrt{1 - v^2/c^2}} = {y' \\over x'}", "72ccaabb74d2b160850b2e43abf8c29e": "-A_v", "72cce0189fe01ccc02c40ad2d885632b": "A = 2\\mu_0 M_S H_C(0)", "72ccf09bbb977b8ba6120ef29efaca6e": " {dx(s)\\over ds} = f(x(s)),\\,\\,\\, x(0)=x_0,", "72ccfa9987c396d1bd120c7975ebfc50": "\\frac{w+n_w}{w+b+n}", "72cd13e93c474f53487dd4890c8ce7dd": "\n \\underline{\\underline{\\boldsymbol{K}}} = \\begin{bmatrix} K_{11} & 0 & 0 \\\\ 0 & K_{22} & 0 \\\\\n 0 & 0 & K_{33} \\end{bmatrix}\n ", "72cdb4be5650e656e414c2d6e8034eaa": "|(a,b)| = |\\mathbb R| = |\\mathbb R^n|.", "72cdd4c155e658c4968c18251a323b86": "\\phi^{(j)}", "72ce63a397ae6848e358522b82a898f2": " \\hat{S} \\ \\stackrel{\\mathrm{def}}{=}\\ |R\\rangle \\langle R | - |L\\rangle \\langle L | = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}. ", "72cea41a1bc4887b501ae3a3e79e7409": "a^* = \\frac{L^*} {T^{*2}}", "72cee2b2a5655ac725cdddb213cd6205": "\\phi(1) = 1\\,", "72cf15c6723f9e889a83821b57c58993": "\nE_n = \\frac{\\hbar^2n^2\\pi^2}{2m^*L^2},\n", "72cf39c597d02485c0304b67ef88ae89": "\\mathrm{SE} = \\frac{\\sigma}{\\sqrt n} = \\frac{12}{\\sqrt{55}} = \\frac{12}{7.42} = 1.62 \\,\\!", "72cf434eec960fc36dee6ae444028554": "P_1 \\oplus IV", "72cf47142567fce0ba2cee52dbc656e3": "\\left \\langle F^\\sharp S,\\varphi \\right \\rangle = \\left \\langle S, \\left |\\det d(F^{-1}) \\right | \\varphi\\circ F^{-1} \\right \\rangle.", "72cf5a42780eedc4393cd22fef5815b0": "\\frac{d^2 x}{d t^2} = f(x)", "72cf5e53967acef689eed0ae7fba09eb": "A\\scriptstyle \\overline{D}", "72cf8a520c20539591697e5a03fd4d51": "f(u)=F(z)+e", "72cfaa35accab2f23256df9f773ced50": "n_2-n_3", "72cfc3ff0005faf1fab32b857655d98c": "x = a \\sin(\\theta)", "72cfcd696663b4c02836ae66e90e3351": "\\,\\overline{a}_x = \\int_0^\\infty \\overline{a}_{\\overline{t|}} f_T(t)\\,dt = \\int_0^\\infty \\overline{a}_{\\overline{t|}} \\,_tp_x\\mu_{x+t}\\,dt.", "72cff63e4a498149841a0bf943b8f7f5": "u_1 '", "72d068b6d043ef33c9283c8de0f0c4e6": "W_i^* = \\frac{R g q_{bi}}{F_i u*^3}", "72d094fecd9a25e8b5d615ae34094e96": "K_i \\equiv \\frac{y_{ie}}{x_{ie}}", "72d0a9d23b31c79bc5a3cb87ae7806f4": "\\textstyle\\bigoplus_{n=0}^\\infty \\mathcal{I}^{n+1}", "72d1489d8e35962bae53fb8c0aec738c": "T_2^{*}(O)", "72d1844219538c3e686fc61525bed291": "a_T", "72d19bdb1e6a602635701ec7655ad246": "\\delta' = \\lim_{h\\to 0} \\frac{1}{h}(\\tau_h\\delta - \\delta)", "72d21189e22da424431834009facef7f": "h = -\\lambda \\pi*g", "72d21e908d4f20285b1532258f8f89a7": " \\Sigma(W,S) ", "72d23c36dca83418872be1ba0b53b135": "\\alpha=1/2", "72d2daec3fb7d4400b65331037367df6": "\nT = k^{-1} \\ln 2\\frac{\\tau_\\mathrm{2}}{\\tau_\\mathrm{1}} \\left(E - E_{F} \\left(1+\\frac{3}{2N}\\right) \\right),\n", "72d35866cdaed773d44c4da821adfb25": "\\scriptstyle\\ p \\leq \\alpha ", "72d3859ca466b37be4f1f63e76c10ae9": "\\sum_i M_i", "72d38fc76485f090564988ddcb071b1e": "\\scriptstyle \\geq6.9\\times10^{15}", "72d3a894aa2599f11ca329db8a9aaf0e": "\n\\begin{align}\n\\psi &= \\ln \\left[\\tan \\left(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\right) \\right]\n -\\frac{e}{2} \\ln \\left[ \\frac {1 + e\\sin\\phi}{1 - e\\sin\\phi} \\right].\n\\end{align}\n", "72d3dafd6b429dd43163ed6a2ec0c391": "{(1,\\tfrac{3}{2})}", "72d3e70551401740124f2fa123e9e202": "(C, j)", "72d3f277e73f0123d0ed41c7e60aadd8": "\\varphi_{j} \\circ \\varphi_{i}^{-1} : \\varphi_{i} (U_{i} \\cap U_{j}) \\to \\varphi_{j} (U_{i} \\cap U_{j})", "72d4380eada29dcafdf3ef68717844cf": "\\scriptstyle\\Psi=Ae^{k z}", "72d45a7bc3e1e11583e25a6f772bf0e8": "x_{i_1}^{\\alpha_1} x_{i_2}^{\\alpha_2}\\cdots x_{i_k}^{\\alpha_k}", "72d4617c0e74b3e13473e4a8b5e8cd35": "\\int_{-\\infty}^x [F_B(t) - F_A(t)]dt \\geq 0", "72d4ea1ac04700392ca24c0b6f9756be": " \\scriptstyle |\\psi\\rang = \\sum_n c_n |\\psi_n\\rang ", "72d5038dac73b8751381d278bc4c7f38": "\n\\begin{align}\n\\sum_{i=1}^{n}2(\\hat{y}_{i}-\\bar{y})(y_{i}-\\hat{y}_{i})\n\t& = \\sum_{i=1}^{n}2\\hat{b}\\left((y_{i}-\\bar{y})(x_{i}-\\bar{x})-\\hat{b}(x_{i}-\\bar{x})^2\\right) \\\\\t\n\t& = 2\\hat{b}\\left(\\sum_{i=1}^{n}(y_{i}-\\bar{y})(x_{i}-\\bar{x})-\\hat{b}\\sum_{i=1}^{n}(x_{i}-\\bar{x})^2\\right) \\\\\t\n & = 2\\hat{b}\\sum_{i=1}^{n}\\left((y_{i}-\\bar{y})(x_{i}-\\bar{x})-(y_{i}-\\bar{y})(x_{i}-\\bar{x})\\right) \\\\\n & = 2\\hat{b}\\cdot 0 = 0.\t\t\t\t\t\t\t\n\\end{align}\n", "72d5a30ca04a3b87e2fb23094bd3746e": "\nx \\mapsto \\left( \\sum_{n=1}^\\infty ||x\\xi_n||^2 \\right)^{1/2},\n", "72d643deb5993b3a074de505a6ff3865": "I = \\bigg(\\frac{0.1299}{12}\\cdot $2500\\bigg) \\cdot 3 = ($27.0625/month) \\cdot 3=$81.19", "72d68a0c72e867d8f139498c16d3301b": "0<\\theta<\\frac{\\pi}{2}", "72d6aabf3f5e921d4baf1f3dc7f3e955": "H_{2k+1}(M;\\mathbb{Z}_2)", "72d6b86626ca552685192d7004e90a3b": "\\zeta(s,q) = \\sum_{k=0}^\\infty \\frac{1}{(k+q)^s}", "72d6d70d90a9da8224b2eb78a19b5c17": "dy = \\left(\\frac{\\partial y}{\\partial x}\\right)_z dx", "72d6dddc43dc566b5ac18691be575efa": "H_2(x)=4x^2-2\\,", "72d6ed245aadfcff140f00682ceff73d": "A_1, \\ldots, A_n", "72d6fe9ab4fd2d89e59016eaba93190b": "\\scriptstyle |\\mathcal{N}| ", "72d73393c53cfcd58ee4daf13f8752e5": "\n\\begin{bmatrix} |V_\\mathrm {ud}| & |V_\\mathrm {us}| & |V_\\mathrm {ub}| \\\\ |V_\\mathrm {cd}| & |V_\\mathrm {cs}| & |V_\\mathrm {cb}| \\\\ |V_\\mathrm {td}| & |V_\\mathrm {ts}| & |V_\\mathrm {tb}| \\end{bmatrix} \\approx\n\\begin{bmatrix} 0.974 & 0.225 & 0.003 \\\\ 0.225 & 0.973 & 0.041 \\\\ 0.009 & 0.040 & 0.999 \\end{bmatrix},", "72d7ba8a94216d862d0080380d9733cb": "\\displaystyle\\begin{matrix}\n0 \\le x_1 \\le 1 \\\\\n\\vdots \\\\\n0 \\le x_n \\le 1\n\\end{matrix}\n", "72d7e2a1e4dd9e26306b409d05ef3323": " T(h, 1) = \\frac{1}{2} \\Phi(h) \\left(1 - \\Phi(h)\\right) ", "72d7f4f6ee4f2b6247784437a208c681": "T^*M \\otimes \\varphi^{-1} T M", "72d84552880a85f7f95c73ced181123f": "\\varepsilon' < \\varepsilon", "72d8475ab108970c8a2685e67ac5c056": "(\\mathbf{AB})^*=\\mathbf{B}^* \\mathbf{A}^* ", "72d8ed98c13dae5c526c42f7201ab952": "(\\mathbb{C}^2)^{\\otimes N}", "72d9af9dd3b6c3398d43e0c901247bc7": "A_n(k)", "72d9eabb843baeedd621dd41e2a2bdf8": "I(P) = (X_1 - a_1, \\cdots, X_n - a_n)", "72da121b8d2f46800cf275683ad01c46": "\\pi \\sqrt{r/g}", "72da64b97716a97767e1824d7a255a4b": " \\text{variance} \\le (M - \\mu)(\\mu - m). \\, ", "72da7290cfc23c061e058c3cf572ff13": "\\begin{align}\n \\theta(\\sigma_e) & = \n \\theta_0 [ 1 - F(\\sigma_e)] + \\theta_{IV} F(\\sigma_e) \\\\\n \\theta_0 & = a_0 + a_1 \\ln \\dot{\\varepsilon_{\\rm{p}}} + a_2 \\sqrt{\\dot{\\varepsilon_{\\rm{p}}}} - a_3 T \\\\\n F(\\sigma_e) & = \n \\cfrac{\\tanh\\left(\\alpha \\cfrac{\\sigma_e}{\\sigma_{es}}\\right)}\n {\\tanh(\\alpha)}\\\\\n \\ln(\\cfrac{\\sigma_{es}}{\\sigma_{0es}}) & =\n \\left(\\frac{kT}{g_{0es} b^3 \\mu(p,T)}\\right)\n \\ln\\left(\\cfrac{\\dot{\\varepsilon_{\\rm{p}}}}{\\dot{\\varepsilon_{\\rm{p}}}}\\right)\n\\end{align}", "72da81e9f332d845ba937c7497e76e56": "{I_\\mathrm {v}=\\frac{I_\\mathrm {d}}{3}- \\frac{I_\\mathrm {ac}}{2} }", "72dab007e0f34c4d1bcbaab56b6178d4": "S \\otimes T : X \\otimes_\\alpha Y \\to W \\otimes_\\alpha Z", "72dae088c8fac718de27a76ce5bdbe60": "\\lim_{t\\to 0}y = a\\lim_{t\\to 0}{\\sin t \\over t}=a\\cdot 1=a.", "72db3056f59447f4c54fd58d45e92b55": "\\ \\Psi = c_1 \\phi_1 + c_2 \\phi_2", "72db5124e4e4c3a81a6aa6afa2cc4bc6": " \\sum_{A=1}^N M_A\\; \\mathbf{d}_{A} = 0 \\quad\\mathrm{and}\\quad\n\\sum_{A=1}^N M_A\\; \\mathbf{R}^0_{A} \\times \\mathbf{d}_{A} = 0.\n", "72db5a0e27f2248a2d37722228fa0ccc": "\\Pi_2 = \\bigg(a - b(q_1+q_2)\\bigg) \\cdot q_2 - C_2(q_2).", "72db7a40532bd9b309d430be0964210f": "T_1=\\left(1.02\\times 10^{10}\\left[\\frac{ 5\\times 10^{-12} }{1 + (3.2\\times 10^{-5} )^2} + \\frac{ 4\\cdot 5\\times 10^{-12} }{1 + 4\\cdot (3.2\\times 10^{-5} )^2}\\right]\\right)^{-1} ", "72db9d289f344440765736e0a7c5f8c6": "p(U) = \\frac{1}{\\sqrt{2\\pi}} e^{-U^2/2}", "72dbbfee3fa939c95fb5b7b24188f482": "\\frac{v_{\\text{in}}-v_x}{Z_1}=\\frac{v_x-v_{\\text{out}}}{Z_3}+\\frac{v_x-v_-}{Z_2}.", "72dc25ada41c9376a2026d24303b0812": "h:A \\to D", "72dc6454ba37b4d3818cf9b3a17db7af": "\\mathbf u(\\mathbf X,t)=u_i\\mathbf e_i\\,\\!", "72dca82df600641d97863675409d56ea": "\\Delta w", "72dd8836e40859273379c936acd97cfa": "\n\\begin{matrix}\nY_{1} &=&\\{X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7}\\} \\\\\nY_{2} &=&\\{X_{4},X_{5},X_{6},X_{7}\\} \\\\\nY_{3} &=&\\{X_{5},X_{6},X_{7},X_{8}\\} \n\\end{matrix}\n", "72dd8fc2bf52655dc2e920d090eb8d4a": "0\\leq x", "72dd905e8cfb3f195ba34eac7382cda4": "2R_{1212} \\,= S \\det (g_{ij}) = S[g_{11}g_{22}-(g_{12})^2].", "72dd9844e725066051b7e2a87c5e4db9": "|\\mu(x)|<\\epsilon\\,.", "72ddc8a177b13cf94d45a10a5e4667f1": "t_{i}-t_{i-1}", "72de35f89e0a0dd8787080eb4aacc75c": "L = nW", "72df007e1951a3787465446deb1b25e8": "\\hat{L} = - i\\frac{\\partial H(x, p)}{\\partial p} \\frac{\\partial}{\\partial x} + i\\frac{\\partial H(x, p)}{\\partial x} \\frac{\\partial}{\\partial p},", "72df348891cf784157a1f635fe9e8e6c": " f(7) = 0.055296 \\, ", "72dfe86bad064ea9050d4cd79b95be33": "(\\Omega, \\mathbb{P},\\mathfrak{F})", "72dff5066d074ba8e460284607a11ba0": "r^n = x,", "72dff7d7fbdc93ec646c6e0e60004c22": "\\tbinom n0=\\tbinom nn=1", "72dffdf2b082ffd9a07f740d94acbfd3": "\\displaystyle s=3\\sqrt{3}r", "72e015e721043c7585aa3f7c3f0b2db9": "g(x) = e^{f(x)}", "72e042dcfd4f04b37b24a930637b8094": "\nJ_\\kappa^{(\\alpha )}(X)=J_\\kappa^{(\\alpha )}(x_1,x_2,\\ldots,x_m).\n", "72e047c0b45cf5dfa1dcb5eee6d621de": "P_{\\mu}=\\frac{1}{c}\\int T_{\\mu\\nu}\\sqrt{-g}\\,dS^{\\nu}", "72e07933327508be8d2ca29128a878d7": "n \\in \\N", "72e0cbea1b351d7eda76b55481337e3c": "\\frac{\\delta L}{\\delta g^{ab}} = 0", "72e12377bd5727e2cefb8c2eae926e96": "dist_{oneway} = \\frac {c' t_r}{2}", "72e16ec91665531730530b8dd71f1900": "\n\\begin{cases}\n\\frac{V_1 - V_\\text{B}}{R_1} + \\frac{V_2 - V_\\text{B}}{R_2} + \\frac{V_2}{R_3} = 0\\\\\nV_1 = V_2 + V_\\text{A}\\\\\n\\end{cases}\n", "72e1ae03605a23f0a407ed48cf1a91f2": "C=(A-\\arctan(\\tan(B)/\\cos(23.44)))/180", "72e1b5b1883130388556e4f26feda2f2": "\\mathfrak{P}^{96}", "72e1dae254608e8296d3f6f277f144fc": " d\\tau^{2} = \\left(1-\\frac{2M}{r} \\right) \\, dt^2 -\\frac{dr^2}{ \\left(1-\\frac{2M}{r} \\right)}- r^2 \\, d\\theta^2-r^2\\sin^2\\theta \\, d\\phi^2\\,", "72e208db56a175cc762b6bf6909c22bc": "\\mathrm{d} \\colon \\mathcal{C}(M) \\to \\mathrm{T}^*(M) : f \\mapsto \\mathrm{d}f", "72e2404bd01f99608e738518709e46ea": "\\Phi: U \\subset T \\times M \\to M", "72e257703039bd24e46b67272521d3da": "\\textbf{A}_P = \\frac{d}{dt}\\textbf{V}_P = \\frac{d}{dt}\\big([S]\\textbf{P}\\big)=[\\dot{S}]\\textbf{P} + [S]\\dot{\\textbf{P}} = [\\dot{S}]\\textbf{P} + [S][S]\\textbf{P} .", "72e2f7c5a5eff0955685c60b067edc69": "Z(x)-Z(x)=0", "72e31316405262289520cdb350f6bef3": "W^\\bot = \\left\\{\\,x\\in V^* : \\forall y\\in W, x(y) = 0 \\, \\right\\}.\\, ", "72e34d4e5a3898bcbaa24f670f0ff130": "\\mathcal{O}(\\log n)\\,", "72e357faf0fb24e62cfdfca59f61fdd9": "H_{out} = {\\psi}^{61}(H_{in} \\oplus \\psi(m \\oplus {\\psi}^{12}(S)))", "72e38e0595aaefd169fe4ad9c4f4bed9": "g(z) = \\frac{1}{z - \\gamma}", "72e3ba4765b0929dfff7c17c35fd009b": "\\forall x, y \\in A \\; [x \\neq y \\rightarrow \\neg\\exists z \\in X \\; [ z \\leq x \\land z \\leq y]]. ", "72e3e6e01100a178b0fd4cd2b4246c91": "S(P) > S(Q)", "72e407ffe7a8beb8a654ec40a565fa12": "F_{[\\mu \\nu ; \\lambda]} \\, = \\, F_{[\\mu \\nu , \\lambda]} \\, = \\, \\frac{1}{6} \\left( \\partial_\\lambda F_{\\mu \\nu} + \\partial _\\mu F_{\\nu \\lambda} + \\partial_\\nu F_{\\lambda \\mu} - \\partial_\\lambda F_{\\nu \\mu} - \\partial _\\mu F_{\\lambda \\nu} - \\partial_\\nu F_{\\mu \\lambda} \\right) \\,", "72e4344d19620305bb083ba5ac75544c": "E(\\mathbf {r}, t)", "72e48e2990757e5ace41319a76e3408c": "\\sum_k r_{l k} n_k = m_l.\\;", "72e4d1712e50ef3cb7dbae6aecaa07a5": "e d - 1 = k(p-1)(q-1)", "72e4e74004484a7485180ee5bbda370a": " = -(1+a_{n-1}\\,u+\\cdots+a_1\\,u^{n-1}+a_0\\,u^n)\\cdot(t_1+t_2\\,u+t_3\\,u^2+\\dots+t_n\\,u^{n-1}+\\cdots).", "72e50658677f7f735793f046c0a9a7eb": "b_\\theta(t) = f_\\theta(t)", "72e516b5759f5d6e27701e7639a16190": "a = b\\ \\frac{\\sin\\alpha}{\\sin\\beta}", "72e52626c8b3e7860b3a83b00a1b0f44": " Re^{-1/2} ", "72e59e2fb4fa9f10e5ad165b1f9dd9d1": "\\tilde{t}_1 = e^{+i\\phi} \\cos(\\theta) \\tilde{t_L} + \\sin(\\theta) \\tilde{t_R}", "72e5a4d60f8b806de6811a69f219a082": " K_x = K_1 = i\\left.\\frac{\\partial \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_x)}{\\partial \\varphi}\\right|_{\\varphi=0} = i \\begin{pmatrix}\n0 & 1 & 0 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{pmatrix} \\,,", "72e6008543c5ffd0c16528069e15161e": "K^A_{eq}\\,p_A = \\frac{\\theta }{1-\\theta}", "72e614cf8c78cbe7585a50ffa0aa0db1": "N = N[k] = Q \\frac {f_s}{f_k}", "72e651e9e38dcc6cce8ba7ed23bf503e": "d _ 2 = d _ 1 - \\sigma \\sqrt { n/365 }", "72e66023c55f4a354f37d5975b141650": "I / I_0 = e^{-\\tau}.\\,", "72e69ce32ddca0b57ea39ad99f458e15": "\\boldsymbol\\theta_{1 \\dots V}", "72e6ba178cf37bef974f763febe837de": "\\operatorname{SL}(2n+1, \\mathbf{R}) \\overset{\\sim}{\\to} \\operatorname{PSL}(2n+1,\\mathbf{R})", "72e6f6e0f08ca88f02b1480464afd55b": "1_0", "72e82d3a0c5b9f725242d47bff2386f5": "h[n] = \\frac{1}{3}\\delta[n] + \\frac{1}{3}\\delta[n-1] + \\frac{1}{3}\\delta[n-2]", "72e894ac81513cf4edeaf37c2107ef34": " \n\\begin{array}{lcl}\n\\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{u} \\cdot \\nabla \\mathbf{u} & = & -\\nabla p + \\mathbf{B} \\cdot \\nabla \\mathbf{B} + \\nu \\nabla^2 \\mathbf{u} \\\\\n\n\\frac{\\partial \\mathbf{B}}{\\partial t} + \\mathbf{u} \\cdot \\nabla \\mathbf{B} & = & \\mathbf{B} \\cdot \\nabla \\mathbf{u} + \n\\eta \\nabla^2 \\mathbf{B} \\\\\n\n\\nabla \\cdot \\mathbf{u} & = & 0 \\\\\n\\nabla \\cdot \\mathbf{B} & = & 0.\n\n\\end{array}\n", "72e8b5f884cb7eec8636fa2e921d57ec": "s=1, \\dots, S", "72e8c36095a043749d4dfea0b60cb6c1": "C(S, t)", "72e91137c92b163a1280ab88245c74a9": "\\Theta(|V|+|E|)", "72e97f42a152efb5c0e2c7363ec2295b": "(\\underbrace{+,\\cdots,+}_{k},\\underbrace{-,\\cdots,-}_{n})\\,", "72e98f20e89e54a84b4ec0ea213e839c": "\\Phi : X \\to X \\,", "72e9d9da0e44b44ea94b7d952c6a403b": "f_i:\\mathbb{R}^2\\to \\mathbb{R}^2.", "72e9ee8286cb6c5a537989f0666d076d": " I_{16} ~,~ \\Gamma_{a} ~,~ \\Gamma_\\text{chir} ~,~ \\Gamma_\\text{chir}\\Gamma_{a_1 a_2 a_3} ~,~ \\Gamma_{a_1 \\dots a_4} ", "72e9f10be3bd25f58a497be4cdf4ad20": "[CF_i]", "72ea1f0ef3fa165c6cad6840cfc89c18": " f'= {\\frac {df(r)}{dr}}", "72ea4f0616641fb4bff7b209ad182181": "{\\rm length}({\\mathbb B})", "72eaa3646391041449f2e8a5434a76fa": "sig(m_1\\ldots m_k)=(y_{1,m_1},\\ldots, y_{k,m_k})=(s_1,\\ldots,s_k)", "72eaf8d80e755c754f78feaa8db48eac": "\\text{1 point} = \\frac{1}{72}\\text{ inch} = \\frac{127}{360}\\text{ mm} = 352.\\overline 7\\text{ micrometer}. ", "72eafdfa999ac75466831fb89175ae35": "k = \\sum^{m-1}_{i=0} {k_i 2^i} = k_{m-1} 2^{m-1} + k_{m-2} 2^{m-2} + \\cdots + k_1 2 + k_0", "72eb1731ddd6c85b22efb8ed2d67ebb4": "x_i\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{n_i}{\\sum_j n_j}", "72eb50c72721d662bc2294ec43e24c9c": "\\lim_{\\Delta x \\to 0} \\frac{F(x_1 + \\Delta x) - F(x_1)}{\\Delta x} = \\lim_{\\Delta x \\to 0} f(c). ", "72eb994648fd6e709d386a0fc2179321": "z\\left(x,y\\right)=\\left(\\left(-1\\right)^zx,y\\right)", "72ebb0153293d77df18b3e48b068e586": "p(x)=c \\exp\\left(\\sum_{j=1}^n \\lambda_j f_j(x)\\right)\\quad \\mbox{ for all } x\\in S", "72ec3c4af5ed376fac5fdf0e2da65b1b": "S = \\lim_{\\Delta x_i \\to 0} \\sum_{i=1}^\\infty \\sqrt { 1 + \\left(\\frac{\\Delta y_i}{\\Delta x_i} \\right)^2 }\\,\\Delta x_i = \\int_{a}^{b} \\sqrt { 1 + \\left(\\frac{dy}{dx}\\right)^2 } \\,dx = \\int_{a}^{b} \\sqrt{1 + \\left [ f' \\left ( x \\right ) \\right ] ^2} \\, dx. ", "72ecbe963f102b79fbac8bd6a265fea8": "{z_1}^{z_2} = e^{z_2 \\operatorname{Log}(z_1)} ", "72eccb865449b99cc0fa9138b8a93e65": "\\mathbf{I}_k", "72ecf2c8ab68b19d6c8b84a8b750eb25": "w\\in\\,G", "72ed1a531dd0e05fec172ce095d5adea": "{v^2 \\over 2}+ w = w_0", "72ed4343842b339a0201259946a131da": " u(x,0) = e^{-x^2 +ik_0x},", "72ed725f1d51fb5dcd05f4687953e3eb": "f(u,v)>0", "72eda00172d6faae31e54b43d368e02e": "4 \\pi\\epsilon_0 = 1", "72edcb2377c2c40a6dcf1ac388b3c5ab": "\n\\frac{\\mathrm d \\vec I(\\hat n, \\nu)}{\\mathrm d s}= - \\mathbf K \\vec I + \\vec a B(\\nu, T)\n + \\int_{4\\pi} \\mathbf Z(\\hat n, \\hat n^\\prime, \\nu) \\vec I \\mathrm d \\hat n^\\prime\n", "72edd15b78c9fd80bb2cf442c495b70f": "X^\\text{ét}", "72edf02db3b0a624182f32240d1fefca": "\\begin{pmatrix}\n1 & 0\\\\\n1 & -1\n\\end{pmatrix}", "72edf155bf21e74e5bbca3aaf7477970": "\\langle a, t \\mid (a t^n a t^{-n})^2, n \\in \\mathbb{Z} \\rangle", "72ee11d649a684adfc6d03aee5d5f520": "\\mu (A) \\geq b \\geq0\\, ", "72ee24c13ddae8804d42c5faa25fc6de": "K=N", "72ee9643545fa55fc742ed3e719e05e1": "p_{N} = p_{U} = 1/2", "72eee0ab676e8b1059382fc9f066a332": "Z_{h}", "72ef4d97d42aff15f461e04586de2cd4": "\n\\frac{\\partial f}{\\partial t}\n+ \\frac{1}{\\hbar} \\nabla_k E(k) \\nabla_r f \n+ \\frac{qF(r)}{\\hbar} \\nabla_k f\n= \\left[\\frac{\\partial f}{\\partial t}\\right]_\\mathrm{collision}\n", "72ef5800ec7fad312587759a1304ffad": "P^{-1}(Ax-b)=0.", "72ef831bb2fac020e7188e3671505476": "k = 2 \\pi \\xi = \\frac{2 \\pi}{\\lambda}.", "72efad85acaec4bc4b3339830344f69c": "X^+(\\sigma, \\tau) = p^+ \\tau", "72efd843093f410fc143ec16e41e0951": "\\psi=(1/r)P(r)S_\\ell(\\theta,\\phi)", "72efe08b4e84a84a6acbc3591ab8fbbb": "\nRS_i = \\sum_{k=t+1}^{t+n} \\left[ e_i^{k-1} \\left( g_i^k - G_i^k \\right) \\right]\n", "72f0050a182be9ab5ce63069651d78d8": "\\nabla \\times \\mathbf{H}_0 = \\frac{-i\\omega}{c} \\mathbf{E}_0(\\epsilon + i\\frac{4\\pi \\sigma}{\\omega})", "72f0a777cc3be00d17faebb6c0e1c5a6": "\n\\mathbb{E}[\\log(Z_\\pi(T)] \\leq \\mathbb{E}[\\log(Z_\\nu(T))].\n", "72f0cf0811c7b1dd9c17ddbca5213f5d": "U_t", "72f0dd775d520a53871bacecbbf4ac0d": "\\bar{\\Phi}(s;\\tilde{L})=1", "72f0fcc3af486954604cf7a62ee6cf76": "\\tbinom{-1}{-1}=0", "72f10674600c152391de868851bb92fe": "M(2,1,M(2,1,5))", "72f176ce3217768762a7411a084bbfad": " \\frac{c_g}{c_p}\\,", "72f1ed346be8cf14558db3b833fd11b8": "\\omega_\\infin", "72f212386f18d1719a30912cb7ca0ed8": "\n p - p_0 = \\frac{\\Gamma}{V} (e - e_0)\n ", "72f212f27e6deb394e00686200851782": " \\mathcal{M}_{\\rm Tot} \\,", "72f22c9f054a829a63d4f058c3996cfd": "S_3 = (X_1)^3 + (X_2)^3", "72f2bd075e6bb22f38a339d1823288cf": "\\sin(x) x + \\sin(2x) x^2 + \\sin(3x) x^3 + \\cdots \\,", "72f315faebdd4f617b94fd6dd2a3f9c3": "\\frac{1}{7}", "72f33c5da4132ba6070093fadef34c14": "H_2O^{+\\bullet} + H_2O \\to H_3O^+ + OH^{\\bullet}", "72f396d4db96679a98f72dd7f74b417b": " A < N < K ", "72f3bc3da217347c9812386af430a27a": "\nE= h f = \\hbar \\omega\n\\,", "72f406cbc790579640c53eb35d17a8f2": "\\mbox{NADPH} + \\mbox{H}^+ + \\mbox{RH} \\rightarrow \\mbox{NADP}^+ + \\mbox{H}_2\\mbox{O} +\\mbox{ROH} \\, ", "72f412bdbc2f68fcc653a96454432d92": "\\scriptstyle {n-r\\choose r},\\ 0\\leq 2rn", "72fa6af90581a518f3f98d9be109ad00": "u_{ij}", "72fa79773602ab0efb08595224d3ca9b": "\\scriptstyle \\mathbf{\\hat{e}}_r, \\mathbf{\\hat{e}}_\\theta, \\,\\!", "72fa7dd82364878a1d688863d87f50b2": "\\omega_{m,n}=k_{m,n}*c", "72fac94b87484d9440fe09001ddae793": "V \\oplus K", "72fc043546c438f3e965e3acea3f8b53": "\\frac{3EI}{L}", "72fc0ffc9d0985ca14f4eea1bf7c4975": "( X \\in X ) \\iff ( ( X \\in X ) \\to Y ) ", "72fc379c85feb47b95eb0b42bd07fd16": "\n\\begin{cases}\nAB+CD=BC+DA\\\\\nA+D=B+C=\\pi.\n\\end{cases}\n", "72fc62676bd48b6cc1f7bf7f38b61bc6": "\\mathbf{m} = \\prod_{\\mathbf{p}} \\mathbf{p}^{\\nu(\\mathbf{p})},\\,\\,\\nu(\\mathbf{p})\\geq0 ", "72fca5dbca21e51243dbf10475e1094a": "x = \\frac{a\\sqrt{2}\\cos(t)}{\\sin(t)^2 + 1}; \\qquad y = \\frac{a\\sqrt{2}\\cos(t)\\sin(t)}{\\sin(t)^2 + 1} ", "72fcd67a722bc95f8030a727f90ddb14": "\\sigma(T)-\\mathcal{A}'", "72fcde3d787bdbfad1522e8a17913c21": " \\int\\!\\!\\!\\!\\int_{S} \\mathbf{j}_1(\\mathbf{r},t) \\cdot d\\mathbf{S} = \\Sigma(t) - \\frac{dq(t)}{dt} ", "72fd22454c7eaa67eaca09d5a60ecb7a": " n_B= \\frac{\\epsilon_0 m {\\Omega_c}^2}{2 q^2}=\\frac{B^2/(2 \\mu_0)}{m c^2}", "72fd5d0aea123d4bbc5bb07a093208e6": " x \\rightarrow {x \\over \\sqrt{a}} ", "72fd64437b6f43934ee602bb904645d7": "M,N,M',N'\\in K", "72fd6e7487596158ee301e1498523aec": "[M+Na]^+", "72fd9c87f4de90bc23e46a48ea4d64a1": "{d^2 X \\over dt^2} + 2 \\cdot \\zeta \\cdot \\omega \\cdot {dX \\over dt} + \\omega^2 \\cdot X = {d^2 z \\over dt^2} ", "72fdc9b72bc31b0529123f4b6fe14309": "k+1\\leq i\\leq n-k", "72fdcc3e3bf24c5d84e486c39aa62728": "\\tau_1 + \\tfrac{1}{2}(\\tau_2 - \\tau_1) = \\tfrac{1}{2}(\\tau_1 + \\tau_2)", "72fde3e28a927bd179f91cdea533b773": "\n \\frac{d}{dt}\n \\frac{\\partial T}{\\partial \\dot{\\theta}}\n- \\frac{\\partial T}{\\partial \\theta}\n+ \\frac{\\partial V}{\\partial \\theta} = 0,\n", "72fdeb7252bce701be941c718174eaef": "P_B= \\rho(\\psi(g)^b) \\in \\mathbb{F}_q^m", "72fe5e8a4b84abccb1463bbad4b2d119": "mc_p/hA", "72fe63986d04ed5bd343d094c5989730": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = u_{a1}\n\\end{cases}", "72fed9a46ab36313069122d112b7e309": "f : \\mathbb{N} \\rightarrow \\mathbb{R}^+", "72fee442ae0d4ac16fd61c28c7d80187": " |x-y|", "72fefa011fbb646c48732698fcfc9e95": "I+1\\,", "72ff169ba15a6b80e5f2a774a9b41dea": "p(\\lambda) = \\exp(-\\lambda/k)/k\\,", "72ff2e3ea61acdc2793ba65b32842b98": "\\Omega_X^* (E)", "72ff51424bccc67ce7bbb9f690b52925": " \\int \\exp\\left( {i \\over \\hbar} S\\left( q, \\dot q \\right) \\right ) Dq ", "72ff65b7b1644999edc16754b0b651c2": "I = \\begin{bmatrix}A & B\\end{bmatrix} \\begin{bmatrix}(A^\\mathrm{T} W A)^{-1} A^\\mathrm{T} \\\\ (B^\\mathrm{T} W B)^{-1} B^\\mathrm{T} \\end{bmatrix} W .", "72ff9c6c10e6a1e1ff429e9c430e927b": "\\pm\\frac{R_1}{R_2}{V_s}", "72ffbd78688ac42e5a820a058f1f8c30": "T_K=\\cfrac{\\frac{\\Delta H}{R}}{-\\ln \\left ( \\frac{e^ \\left ( \\frac{-\\Delta H}{RT_1}\\right ) + e^ \\left ( \\frac{-\\Delta H}{RT_2}\\right ) + \\cdots + e^ \\left ( \\frac{-\\Delta H}{RT_n}\\right )}{n} \\right )}", "72ffeb3e8a8d2e7073d1c64ec7b525be": "\\displaystyle{Q(a,b,c)=abB(u,v)+bcB(v,w) + caB(w,u).}", "730023d7720c7d594ccacbe7d459e782": "(HA + ECT_0 + M \\vdash \\phi) \\leftrightarrow (\\exist n \\; PA \\vdash (\\bar{n} \\Vdash \\phi))", "730067aab452010800cb52df74131443": "\\lambda\\left(\\Sigma+\\frac{1}{m+1}\\cdot K+\\frac{1}{n+1}\\cdot L\\right) -\\lambda\\left(\\Sigma+\\frac{1}{m}\\cdot K+\\frac{1}{n+1}\\cdot L\\right)-\\lambda\\left(\\Sigma+\\frac{1}{m+1}\\cdot K+\\frac{1}{n}\\cdot L\\right) +\\lambda\\left(\\Sigma+\\frac{1}{m}\\cdot K+\\frac{1}{n}\\cdot L\\right)", "7300bef281ff6c8f68306e087f6c07a1": "\\begin{align}\n R_1(x) & = x + 1 \\\\\n R_2(x) & = 2 x^2 + 4 x + 1 \\\\\n R_3(x) & = 6 x^3 + 18 x^2 + 9 x + 1 \\\\\n R_4(x) & = 24 x^4 + 96 x^3 + 72 x^2 + 16 x + 1.\n\\end{align}", "7300d4b1b6d8c3f875eff5a55be42ab2": "\\scriptstyle \\psi^\\dagger(x) \\psi(x)", "7300d683b65b31c29ea585ac2beb6b44": " \\epsilon y'' + (1+\\epsilon) y' + y = 0,\\,", "7300d9a4d1cb50e6ae4fdcb7b5da7d9a": "BT = \\frac {(CP-AP)*CS} {BR*AP}", "7300ede257c65c20afe007ea365b2605": "1-S", "7301350354efa1ff06afdfef6cb78df2": "G_{imp}^0(\\tau)", "7301683c54eb619ca7f9e867f37b23e5": "[\\text{Ca}^{2+}]_\\text{max} = \\frac{K_\\text{sp}k_\\text{H}} {K_\\text{h}K_\\text{a1}K_\\text{a2}} \\frac{[\\text{H}^+]^2}{P_{\\text{CO}_2}}", "7301b6f87e5f47d62bb6def869f0eb36": "f:\\mathbb{R}\\rightarrow{}\\mathbb{R}", "7301eec4008f18dd90a725c0faa25e23": "i^\\mathrm{th}", "73020351d4bf634bfa91687977b2fe2c": "\\int_c \\omega = \\int_{\\sum_i k_i c_i} \\omega = \\sum_{i=1}^N k_i \\int_{c_i} \\omega", "73022411a2ee58724852d5b98941282c": "p_n(0)=0\\quad{\\rm for\\ }n\\geq 1,{\\rm\\ and}", "73026c6e7eb5ed4d993effe5c41b3283": "\\mathcal{L}(H_i, H_j)", "7302986b7b70af978ece9b9c2fd81831": "Ratio_i = \\text{if}\\; sw_i \\;\\text{then}\\; \\alpha^{2^{i-1}} \\;\\text{else}\\; 1", "73034c096155c7e6f6aac24b4a30a270": "/VA", "730357b6d1c395a3c78503916858d2ba": "K+1", "7303aab615eb1d2eb7dd1bae825237d8": "X_2,\\ldots,X_k", "7303c9187aac12f0a8f0b4db94df53ac": "\\sigma_n(\\Delta^n)", "7303d25d6943730b3038acab48d3f628": "\\ E \\,", "73040cbdd6fab74a6b2eb2f30d621482": "P(a1,a2,a3,...)=(a1,0,0,...)", "730455bc941f0394ab99e15fa2f9940b": "(y^m u^r)^{(p-1)(q-1)/r} \\equiv y^{m(p-1)(q-1)/r} u^{(p-1)(q-1)} \\equiv y^{m(p-1)(q-1)/r} \\mod n", "7304b97f2dbe46c1cf9288b8624057e4": "\\displaystyle{B(a,b)=Q(a)Q(a^{-1}-b);}", "7305065cd6e22f7b223b9bd970a29929": "\\mathbf{x}^{Worst}", "7305223923afa53f685f81460c55b65d": "\n\\sigma_{rr} = \\frac{2C_1\\cos\\theta}{r} + \\frac{2C_3\\sin\\theta}{r} ~;~~\n\\sigma_{r\\theta} = 0 ~;~~ \\sigma_{\\theta\\theta} = 0\n", "7305503ebbaaff292135471fac06c05e": " \\operatorname{build-param-list}[L, D, V, \\_] ", "73056a209d6b2c8165d0ae806f28b11e": "c_{2n+1} = c_{n+1} \\cdot c_n - c \\cdot c_n^p + c_{n-1}^p", "730588cf376ca8433d32a356d5facb36": " u_n =\\sum_{m=0}^{N-1} a_m T_m (x_n) ", "730595fa267b63049e7f9d2e51ff115c": "\\Pi\\sigma_{Products}", "7305d2e319a6938e40ac3e4269b7d2e8": "R_{\\rm E} \\,", "73062596e0ce75fcd6a0276f214b9d59": "\\liminf_{n\\to\\infty} x_n = -1", "730670484c9b72b50db992b308b2d343": "T_G(x,y)= T_{G \\setminus e}(x,y) + T_{G/e}(x,y), \\qquad e \\text{ not a loop nor a bridge.} ", "7306978b1787037d1e3e29654a13d45d": "\\varphi\\circ{\\theta}_{D}", "7306dfce2afbdee119edd043ad9fbdb9": " \\text{cat}(X \\times S^n)=\\text{cat}(X) +1, n>0 \\,\\!", "73075f9e4654c49f1b9d946430821cbc": "E_\\infty^{p,q} = \\mbox{gr}_p H^{p+q}(C^\\bull)", "73078f64a2172e0119ba105a18e1b0ec": "b \\mid a", "7307a9ed415c26ed0601f6e595bf4797": "\\mathbb{Z}_m", "7307b871276aaaf43c1ab3ae3d8d32f1": "{df^\\star(p) \\over dp} = {d \\over dp}(xp-f(x)) = x + p {dx \\over dp} - {df \\over dx} {dx \\over dp} = x~.", "7307e693ae63b32b5192d58869b26f44": " 1,1,1,1,1 ", "7308bc9b72ccde16f7038692ec466709": "\\Box_{S_\\ast(B)}", "7308ebed8df8535c331aa55352cca5ba": " \\Delta = (a_3(a_1^3 - 27a_3)) = a_3^3 \\delta, ", "7308f852d2039c275789121627271b81": "\\lim_{b \\to 0} {a \\over b}", "7308fb45f2098b277d9ab8b27ae88069": "2 + 10 + 50 + 250 = \\frac{2(1-5^4)}{1-5} = \\frac{-1248}{-4} = 312.", "73095dfb24642fea5c1d7e87c858a8c9": "E \\equiv \\frac{q}{q_{max}}", "7309a509c9f789a34d0e392925a3e745": "{\\boldsymbol \\eta}", "730a24034f3ab09371c71e4db4c5def0": " Y_t =\\alpha_0+\\alpha_1 Y_{t-1}+\\alpha_2 Y_{t-2}+\\cdots+\\alpha_p Y_{t-p}+\\varepsilon_t\\, ", "730a55c863e5363553c0beab1da86f17": "u_i\\colon \\Omega \\times A \\rightarrow R", "730a8d4f3753364167e9fd304dbe28d1": "f:L_\\bullet\\to M_\\bullet, g: M_\\bullet\\to N_\\bullet,", "730b0e85d5dcf13dea40873e6835781c": "\\rho_g: L^p \\to \\mathbb{R}", "730b0ea65ec0b5f5d42a2c31548c4572": "\n\\vartheta_{1,1}(x) = \\sum_{n=-\\infty}^\\infty (-1)^n q^{(n+1/2)^2} \\exp (\\pi i (2 n + 1) x/a)\n", "730b66ce5386ce84922c33c68c021865": " f(E;\\beta)=\\beta\\frac{(\\beta E)^{m-1}}{\\Gamma(m)}e^{-\\beta E}", "730be3fe016ae9b55107846d2c0adb80": "\\Delta S_{\\text{bath}} +\\Delta S=-\\frac{\\Delta A+ W}{T} \\,", "730be737651ff1a1cf6998fa535dc4fc": " F = \\frac{d[M_1]}{d[M_2]} \\,", "730c2f45ae551e84193c95607fd8f5ca": "\\phi_{i=1 \\dots N}", "730c44f80a466dbef05ba27182edbe3e": "(r^2-a^2-b^2+c^2)^2+4a^2(r^2-a^2-b^2+c^2)\\sin^2\\theta+4a^4\\sin^4\\theta=4a^2\\sin^2\\theta(c^2-a^2\\cos^2\\theta),\\,", "730c7babc8c63dfafaa72f63059e0d05": "\\textstyle \\sum_{i}u_i (g)\\ge \\textstyle \\sum_{i} u_i (g')", "730c94b84897dce6ba0bd4ddff8db588": "R_c = R\\left[1 + \\left(\\frac{H}{2 \\pi R} \\right)^2\\right]", "730ca410c4c1f16e16180b25935ea0fb": "3 X^6+5 X^4-4 X^2-9 X+21,", "730ca5332922d82b0fef93a1eda2b0c1": "\\partial u/\\partial z>\\gamma_0", "730cdd0bc61ca70d9911fe37ad39af10": "O(2^nn^2)", "730cf28ae4343c73557ae80c9d52c0f6": "p \\circ T = a_{T} (p) p \\mbox{ for all } p \\in \\mathrm{Ext} (X),", "730d1c81e4d63ba4ea194868359c63ff": "{11 \\choose 4} \\cdot \\left[ {1 \\choose 1} \\cdot 36,864 + 28,160 \\right] = {11 \\choose 4} \\cdot 65,024 = 21,457,920", "730d449858518dec35cd071bfe65dc43": "V(a) = 1, V(b) = 0", "730e74f3e88f4f6bc55b24da97cd52e3": "x = \\sum_{g\\in G} a_g g", "730ea4979800cc154b3a4ff24bd59b68": "V = \\frac{1}{2} bhl", "730eacfd6f4f2e02c649b2f3fac08d50": "I(X;T)\\,\\,I(T;Y)", "730ef1850e0ae676eb3f1787fac2e423": "b \\cdot d \\equiv 1 \\pmod{m}", "730fce226d07f8072049398ab5d504c6": "1/[0, y_2] = [1/y_2, \\infty]", "730fda690bf992bd9aafe67c13469056": " \\Psi = X \\times \\Theta \\times \\Pi ", "730fe532f8e4c6da8486da2a05151f06": "\n \\frac{\\partial \\boldsymbol{A}}{\\partial \\boldsymbol{A}} = \\boldsymbol{\\mathsf{I}}^{(s)}\n = \\frac{1}{2}~(\\boldsymbol{\\mathsf{I}} + \\boldsymbol{\\mathsf{I}}^T)\n", "73100e21cdf1f88700a1a3e88b97a3d8": "Ext(w;i)", "7310278d380f2f8265a99a999750be8c": "\\mathcal{D}\\psi\\mathcal{D}\\overline{\\psi} = \\prod\\limits_i da^i db^i = \\prod\\limits_i da^{\\prime i}db^{\\prime i}{\\det}^{-2}(C^i_j),", "73103ab58053923050cd03df1f4077c0": "f^h_{\\mathbf k}", "7310dea089687c1c7d57f043b0bcc66b": "\\mathcal{H} = \\{(x_0, x_1, x_2)\\colon x_0^{q+1} + x_1^{q+1} + x_2^{q+1} = 0 \\}.", "7310fea972d33993658e21692eadd353": "(4\\,", "731120dfd536c61c5f9df43ae4421fcb": " P_2(y)=-c,", "7311526795a153b7705a22ed8e3b655d": "\\begin{align}\nb_n & := a_n \\\\\nb_{n-1} & := a_{n-1} + b_n x_0 \\\\\n& {}\\ \\ \\vdots \\\\\nb_0 & := a_0 + b_1 x_0.\n\\end{align}", "73115a25a2f13e2cc9485ff766b9ef77": " = 90 ^\\circ - \\arctan \\left( \\frac {\\beta A_0 f_1} {\\alpha \\beta A_0 f_1 } \\right) ", "731168061fd2d4338603304035195fa5": "ELA\\,\\!", "7311a1f1692ba7e6935d28790c3f9a42": "e^{ix} = \\cos x + i \\sin x \\;", "7311eadac874d3eb683dd147ce424084": "x\\in V", "7311f1233893f186c299871518ba5db5": "\n \\vert \\alpha\\rangle = {\\mathcal N}(\\vert \\alpha\\vert^2)^{-\\frac 12}\\; \\sum_{n=0}^\\infty \\frac {\\alpha^n}{\\sqrt{\\varepsilon_n !}}| n\\rangle\\, .\n", "7311f2ff36a2621c924a1012c3c64baf": " (n,2m) ", "7311f5ae9a92a2cb73362d63ecaa9977": "\\nabla\\cdot{\\mathbf\\Omega}=0", "7312507861feb04dcd0fc911c1499f69": "aB \\subset A", "73125cfc95e7de4bf8c7a9e70e7c7ce8": "\\limsup_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{W}_{\\varepsilon} (F) \\leq - \\inf_{\\omega \\in F} I(\\omega)", "7312883facbe0e4861c30c008ac90583": "(z+1)^{d}\\;p\\left(\\frac{x_1}{z+1},\\ldots,\\frac{x_k}{z+1}\\right)=0", "7312da64494fe79355914663717269d4": "\\angle KAP_1", "7312ee6e83568f280b16503657227bbe": "\\{w | w \\in L_1 \\land w \\in L_2\\}", "7313293c19e507e6e20e40c0410f612b": "\\mathrm{lights}", "7313371a9bf04b295cdea1f5a33f8440": "1 - \\log_2\\left(\\frac{k+2}{k+1}\\right)", "73138276499afb1dde7c117d6f945236": "y\\in\\mathbb{Z}/2\\mathbb{Z}", "73139b4de06ce3a067e59d665a473fc5": "(p_1 + \\cdots + p_n)^2\\,", "7313e8f1b3fc973f354e2ebfe35540d8": "\n7.000 \\mbox{ metres} = \\frac{L + 0.25G +2d + \\sqrt{S} - F}{2.5}\n", "73145ee4c866d7cbd6e875da01b35810": "\\textbf{x}_{c}", "731476dd558f0dc663a61cf9209aa01b": "u=(u_1, \\ldots, u_n)", "73148ccbd876d2cf425fa59dadf24042": "\\mathrm{GL}_n(\\mathbb{R})/\\mathrm{GL}_n(\\mathbb{Z})", "73148ccdfcdc25d8c1f3f3efbe60af9c": "f(x)=(x-a)g(x)", "73152692b973e5e17e8fc85518c058e1": "\n\\Phi_{G} \\ \\stackrel{\\mathrm{def}}{=}\\ U - T S - \\mu N\n", "73152eb3f58be3642cb692ef7eb75e67": "k_{2H}", "7315b5b30d2c0a4c7aea8361c38b7d52": "G^* + S \\to G + S + e^-", "7315ceed71aed8d1fe2e0d0513c3b17e": "B'_w", "7315d6fc0882ff4afa24bd96970e2bf7": "q_{max}", "731607bc4d0a637812c31c8305f510a7": "c_{ij} = \\min_{k=1}^n \\{a_{ik} + b_{kj}\\}", "73164647334d9684bcd8653dcbccf002": "\\Delta=\\bar{\\partial}^*\\bar{\\partial}+\\bar{\\partial}\\bar{\\partial}^*", "731648d1cb99b8efe0775d92e319905c": "w(\\gamma^1_1)= 1 + a \\in H^*(\\mathbf{P}^1(\\mathbf{R}); \\mathbf Z_2)= \\mathbf Z_2[a]/(a^2)", "731651cfd361c56721228ad378deba56": " S = \\frac{ ( \\alpha - 1 )( \\alpha - 2 )( 1 - ( \\alpha - 1 )( 2^{ 1 / \\alpha } - 1 ) ) }{ \\alpha^{ 1/2 } } ", "73165ebb1b4569b09c6a63b578511349": " M \\Gamma^1{}_{11} + N \\Gamma^2{}_{11} + L_v = L \\Gamma^1{}_{12} + M \\Gamma^2{}_{12} + M_u ", "7317209daedf61f671a8eb15547259b0": "\\frac {\\mathrm {x}\\ miles} {7\\ hours} = \\frac {90\\ miles} {3\\ hours}", "731733ef1b55b18d6a0cf77fdc749e6b": " \\left(f \\Box g\\right)(x) = \\inf \\left \\{ f(x-y) + g(y) \\, | \\, y \\in \\mathbb{R}^n \\right \\}. ", "73173ebc3e82dd49576c4ea6cf96e6ac": "p_i(r)", "731757837ceda2540ee521bb16ea7b03": "A_{12}", "731770ae63a3bb32138e72ff017a2f54": " s_{\\lambda}(1,t,t^2,\\cdots) = \\frac{t^{n\\left( \\lambda\\right)}}{\\prod_{(i,j)\\in Y(\\lambda)}(1-t^{h_{\\lambda}(i,j)})} ", "7317e009edcd211186133ba9df7ca268": "|\\Psi\\rangle=\\mathcal{T}|\\tilde{\\Psi}\\rangle", "73185cb31a90d235fd311cbef6eaf606": "=\\{ 2k: k \\in \\mathbb{Z} \\}", "73187d132cadb8c2fa7670e3b86dae8c": "\\sin 15^\\circ = \\cos 75^\\circ = \\dfrac{\\sqrt6 - \\sqrt2}{4}\\,\\!", "73188a7153eeea1b2299b53fb50252e3": "\\mathbf{}\\begin{bmatrix}\n\n4&-3&0&0 \\\\\n-3 &10/3 &-5/3&0 \\\\\n0 & -5/3 &-33/25& 68/75 \\\\\n0 &0 & 68/75&149/75 \\end{bmatrix},", "731899cd6f936993fe9f2f7f15a6e25f": "\\Pi=\\Pi_{1}+\\cdots+\\Pi_{p}-I. \\, ", "7318e412b51aa2804aa4564b40d02ccb": "\\gamma^{-1} \\, ", "731946654737b3478162940c1b570e22": "a_i > T", "731973cdc202ebf22fc49ebb2d3af7af": "A + B = \\{a+b : a \\in A, b \\in B\\}.", "7319b56b5d32205e11ebe74c84da127c": "s_\\lambda", "7319e263aad0c2580bc4a93245546f5d": "b_n>0", "731a0a7228d4d553401a86a751aa26ac": "\\frac{a_{rel}}{a_0} < 1", "731a99cc73a28548fc27d8a622363357": "x_\\mu", "731a9f1d43ee48e2e8df85b084dd2c68": "\\langle nlm_l|z|n_1l_1m_{l1}\\rangle\\ne0", "731abdd5a24434d1c2ebea3030c9db19": "\\cos\\varphi=0", "731b266be9b4b23883043bc15d6c3f93": "y = \\pm \\ell x. ", "731b3cf766f4490bb8cbdde1cc54a737": "= 2Q\\left( \\sqrt{\\frac{E_s}{N_0}} \\right) - \\left[ Q \\left( \\sqrt{\\frac{E_s}{N_0}} \\right) \\right]^2", "731b4ba519c5ab2c1c74389b0ff6f6af": "\\mathcal{F}_{Ch}=k_2(\\mathbf{\\hat{n}}\\cdot\\nabla\\times\\mathbf{\\hat{n}})", "731b59391ad0e7601574d63b662d3141": " L | f\\rangle = | k_1 \\rangle \\langle b_1 | f \\rangle ", "731b7a72989078edc2783ed5148b3106": "\\boldsymbol{\\mu} = -g \\mu_B \\frac{\\mathbf{L}}{\\hbar}.", "731b886d80d2ea138da54d30f43b2005": "join", "731bb31417e33d609cd2d53cef930f52": "h'(a)=\\lim_{z\\to a}\\frac{(z - a)^2f(z)-0}{z-a}=\\lim_{z\\to a}(z - a) f(z)=0", "731bb725284edcc4eca9fc6113bfd649": "\\mathcal{R}^K _0", "731bdfc0f0908bfb8a9369e8bde81376": "|p_{k,S}^{\\mathcal M}-p_{k,U}^{\\mathcal M}|\\geq\\frac{1}{Q(k)}", "731c194e1ba6362f5d69f33ae666fa2b": " ijk = \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_3 \\mathbf{e}_1 \\mathbf{e}_1 \\mathbf{e}_2 = -1.\\!", "731c361fc506fb63ffc6c16523ff19d4": "[S:T]", "731c7d2f5e2adbda6be9d2f03efe9dbc": "\\mathrm{(L)Fe^{n}N_{3} \\longrightarrow (L)Fe^{n+2}N + N_{2}}", "731c9b031fcdfb29394256d4518ef22f": "RQD = \\left(\\frac{l_{\\mathrm{sum~of~100}}}{l_{\\mathrm{tot~core~run}}}\\right)\\times 100 ", "731caed8a411ff4240a3ff69e78e8f04": "\\int\\limits_0^\\infty f(v)dv = 1", "731cf5e71f2ac95f8f70f5e2e139155b": " K_n(R) = \\pi_n(BGL(R)^+),", "731cf87c923b91497191714898fb5ef4": "V(S) = \\{x \\in \\mathbb{P}^n \\mid f(x) = 0, \\forall f \\in S\\}.", "731d00d72c1dc09c53ded9d42e4880e9": "\n|z| < \\frac{1}{4M}\\,\n", "731d1e197a555ac64e5a24fb7bb8b92d": "\\mathfrak{a}^{ec}=\\mathfrak{a} ", "731d31c80f5514f478a0aa4d265d0bd2": "\\det(I-\\lambda K) = \\exp \\left[\n-\\sum_n \\frac{\\lambda^n}{n} \\operatorname{Tr}\\, K^n \\right]", "731dd4cd05bb3f154c5b1da55e908112": "\\begin{align}\nY(z) = \\frac{1}{2} \\Re \\Bigg\\{ \\Big(\\frac{i}{kz(z^k-1)}\\Big) \\Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k) \\\\\n&{}+(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k)\\\\\n& {}-kz^k+k-z^2-1 ) \\Big] \\Bigg\\}\n\\end{align}", "731e088cdc7a85dace75f6220f9193e3": "\\operatorname{PSL}(2,\\mathbf{Z})", "731e2a0e6501eb993a16c30742c8bc0b": "\\operatorname{ad}_{\\{f,g\\}}=[\\operatorname{ad}_f,\\operatorname{ad}_g]", "731e586afb8727c05c4040bc67723c0f": "\\sin\\frac{\\theta}{2} = \\pm\\sqrt{\\frac{1-\\cos \\theta}{2}} \\, ", "731e8b9e3b797b71e6f3e94bb3e3e01a": "\\theta_{I}^{\\alpha}", "731ea64506dd3475b3c7cf58f1754f1b": " S =\\frac{k_B c^3 A}{4G \\hbar} ", "731eb1850ee8611ac61e6319d0e05348": "(1,3,\\bar{3})_H(3,\\bar{3},1)(\\bar{3},1,3)", "731ef54f2b151cbfd638d8b9838900e5": " (F'\\cdot\\sin - F\\cdot\\cos)' = f\\cdot\\sin\\!", "731ef989523f18a4ee5e15db1a564d36": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 0\\\\\n0 & 1 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 1 & 0 & 0\n\\end{array}\n\\right] .\n", "731f234e6f225e32355bd25857f3823e": "(\\hat{x}, \\hat{y}; \\hat{t}) = \\operatorname{argmaxminlocal}_{(x, y; t)}(\\nabla^2_{norm} L(x, y; t))", "731f479a172eb7b3c9248a5d89ff6453": "\\eta(\\theta)=-\\eta(180^\\circ-\\theta)", "731fe7b96abe821766d6941a92c7e565": "\\frac{d^n H_{x,3}}{dx^n} = (-1)^{n+1}\\frac{1}{2}(n+2)!\\left[\\zeta(n+3)-H_{x,n+3}\\right].", "73200453d393f69e1c57d517c51ea27c": "\nq\\, +\\, \\rho\\, g\\, h\\,\n =\\, p_0\\, +\\, \\rho\\, g\\, z\\,\n =\\, \\text{constant}\\,\n", "7320f0d873c106a465102d6cc458a82f": "\\mathbb P(n_1,n_2,\\cdots,n_M) = \\frac{1}{\\text{G}(N)}\\prod_{i=1}^M \\left( X_i \\right)^{n_i}", "73213de995f21710ff31a54d814283da": "\\begin{bmatrix}\n0 & 1\\\\\n0 & 0\\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\n1 & 2\\\\\n3 & 4\\\\\n\\end{bmatrix}=\n\\begin{bmatrix}\n3 & 4\\\\\n0 & 0\\\\\n\\end{bmatrix}\n.", "7321442c7798514cecb987aa3ad925a9": "\\max_{x_{1}^{1},x_{2}^{1},x_{1}^{2},x_{2}^{2}} u^1(x_{1}^{1},x_{2}^{1})", "73216a140f8e00fc01804b4e5b77f649": "\\Gamma (p,m)", "73218e46f4ddf2d5c983a91162f203e8": "2\\uparrow(2\\uparrow1\\times1+0)\\times1+(2\\uparrow1\\times1+0)", "73219842a54856538e7968642a156754": "di = e\\Delta Sdn(v, \\vartheta)v\\cos \\vartheta", "7322562c6d194b37f96ecdd726f17f46": "so(n)", "73227cc0f7a5788f04173f2b1ef3fb39": " c_w = {\\omega \\over k}\\ ,", "7322a97e399a5c3dbfaf9cced7c81535": "{\\partial\\!\\!\\!\\big /}", "7322b074fd8f68c833e08687f60c318a": "{}\\sim{}", "73237a011c4b6697689efc9c3364b1ee": " \\beta = 1/ k_B T", "73239709f6951d352cbe9bfbf7afa756": "A_R(R)\\to A_K(K)", "7323ac8f61c908aaa9083cdb4f431294": "\\rho _{\\alpha -} ^{i_0 } \\ge A_{\\alpha - }^{\\sigma (i_0 )} ", "7323df138a81e849264fc4cf40d7c6b9": "e \\notin S", "7323e0c975b043c46036276df060e68e": "\\operatorname{let}p, q : p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p ", "7324297321f2a99ba1ff487010c14d82": "Z^{I}_{i,j}", "73245082854a5cf63e468600093994cb": " \\underline{0} \\in g", "732471bbcff50f434babbf82626584ee": "\\mu = 1 * \\frac{DU^3}{TU^2}", "7324b353faa61e8f2d8e402f8fa5d8a8": "\\omega(t)=\\omega", "73250921adfb97d24d0f66e6eafc4234": "f(x-a)", "732558d0dcd2666d545423b5dc775e0c": "\\lambda=\\mu_1-(a_2-a_1) \\frac{m_1+1}{m_1+m_2+2}-a_1", "7325b5cb542d1751a0d215e0bef928f0": "\\operatorname{GKdim} = \\sup \\limsup_{n \\to \\infty} \\log_n \\dim_k M_0 V^n", "73260184d8bc9cdda2d8722fedd0bd91": "= \\left(\\|x\\| + \\|y\\|\\right)^2", "7326162e87775666aa42021dfe67ab6f": " e^{\\gamma} = \\left ( \\frac{2}{1} \\right )^{1/2} \\left (\\frac{2^2}{1 \\cdot 3} \\right )^{1/3} \\left (\\frac{2^3 \\cdot 4}{1 \\cdot 3^3} \\right )^{1/4}\n\\left (\\frac{2^4 \\cdot 4^4}{1 \\cdot 3^6 \\cdot 5} \\right )^{1/5} \\cdots ", "732631c70c6c7fc28248b01677a7304a": "\\frac{\\text{physical capacity}-\\text{user capacity}}{\\text{user capacity}} = \\text{over-provision}", "7326aeac2b7608809cebe6cc61573fb4": "\\!\\sigma_D = \\sigma_1 - \\sigma_3", "7326b3266a81663eacdbeec2dcb1f029": "\n \\lim_{n \\to \\infty } \\left(\n \\sum_{p \\leq n} \\frac{1}{p} - \\log \\log(n)\n \\right) = M\n", "732705acb19cf9528293cfe8ce08a4d7": " \\pi(i)=s_j", "73272b6a4c38843064d58c109414d157": "p_i^2+q_i^2 \\rightarrow \\infty", "732750709bbfc04eaea7a36b55449b09": "\\frac{f'}{f} = \\frac{u'}{u} + \\frac{v'}{v}.", "7327def338f3047c995ca6884a1c3c5e": "\\, \\sinh 2K^* \\sinh 2L = 1", "7327f480dff344195257372c8f39b252": "\\rho \\left[ \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right)}{\\partial t} + \\left( \\overline{u_j} + u_j' \\right) \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right)}{\\partial x_j} \\right] = -\\frac{\\partial \\left( \\bar{p} + p' \\right) }{\\partial x_i} + \\mu \\left[ \\frac{\\partial^2 \\left( \\overline{u_i} + u_i' \\right)}{\\partial x_j \\partial x_j} \\right].", "73283178b6a5d8fc2e00f5e86178b194": "10,000A=7\\,162.162\\,162\\,\\cdots .", "73284a6ed99e51fbd3f81245fe775dc6": "t_r = \\frac{L\\cdot(360-4\\alpha+229.2\\cdot\\cos\\alpha\\cdot\\sin\\alpha)}{48\\cdot{n}\\cdot{r}\\cdot\\tan\\beta\\cdot\\cos\\alpha\\cdot(\\sin\\alpha)^2}", "7328a6a43628a8ee44ed41464807bf38": "f_L", "7328d3ba8263daaf8c3fe221c65e272e": " f = f^{**} ", "7329562f0c71e35e48951c33321f8d74": "\\mathrm{tr}\\varepsilon", "73297c94bed1811ec127743f164d3517": "p=\\frac{(a-d)^2}{4}+bc", "732984409fb2daa155d0fa16e8bb2b1f": "p(0)=0", "7329be501bbc8477a50f044d7492e4e4": "\\alpha_*^{ }", "7329feaf80b1079570dab67fe3ceb7ff": "B_4(x)=x^4-2x^3+x^2-\\frac{1}{30}\\,", "732a11be523a89e50c229cbe8c587551": "e^{\\pi \\sqrt{163}} \\approx 640320^3 + 744", "732a1a169f04dca35687c83370ed24ad": "\\mu_k=\\int_0^\\infty x^k f(x)dx=\\int_0^\\infty t^{-\\frac{k}{\\alpha}}e^{-t} \\, dt,", "732a4b318d14be328f9685b207978220": "\\phi(^8B) \\propto T^{25}", "732a9dff92278cbe919001b94006e2a8": "\\scriptstyle \\Lambda \\rightarrow \\infty", "732aad92e238f46597cd078d1f5b210c": "\n v = \\hat{u} \\hat{u} v\n= \\hat{u} ( \\hat{u} \\cdot v + \\hat{u} \\wedge v )\n", "732ab924dac8342b3529c271c1014471": "\\int F(x)\\delta_\\alpha(x) = F(0)", "732af6dddd67529eb1397558faff89be": "{{V}_{CC}}", "732b5944e9860c2bd69955992c58b4c6": "GapCVP_n", "732b89db62f905f4382b734a81998f86": "\\tilde{H}_{BS}=e^{i(H_S+H_B)t} H_{BS} e^{-i(H_S+H_B)t} ", "732bbd7b21eb14590a85edf5d0382422": "L = (L/T)^{a+b} (L/T^2)^c\\,", "732be111f273a40c6207e5788f19aae9": "q_{d/s} = \\frac{Q}{W_{d/s}} = \\frac{20}{6} = 3.33\\text{ ft}^2/s", "732c1bef00b4daf557084b454c1c17f3": " \\mathcal{V} ", "732c31730e0d13ccb7d057b09f30de88": " \\sum_{i=1}^n \\text{MRS}_i = \\text{MRT} ", "732c3e1820ccbe5da992a003d2971b38": "\\longrightarrow^*", "732c5507c2a01ff0d4f059cd343cc717": "\\Theta \\mapsto \\operatorname{E}\\left [ \\mathrm{exp}\\left (i \\mathrm{tr}(\\mathbf{X}{\\mathbf\\Theta})\\right )\\right ] = \\left|{\\mathbf I} - 2i{\\mathbf\\Theta}{\\mathbf V}\\right|^{-\\frac{n}{2}} ", "732c90ccbd28dfb9485fc43132b5f7c4": "m = \\Omega(n^{2\\gamma})", "732ca76b691f1e10accd96b0dc0ffb58": "(a,b)\\in D", "732cfca3c9b85433f5d0823f124dddc0": "K=\\cfrac{E}{3(1-2\\nu)}", "732d0f1bc827ce19a029e26d7e9cebee": "\\mathfrak{P}^{85}", "732d336e7963cbfcec88004342e7a011": "q''\\,", "732e0f7b99a92ebcd7754653f43020ed": "C(B,C)=B", "732e18a7a995ebc88c0b155867474bae": "FV = C/i + FVA", "732e27b81858fbe19da8b4c226e5cafb": "\\{V_0,V_1, \\ldots, V_{k-1} \\}", "732e2fc27472ab940cd929d0fffc6b9f": " f(z) = \\frac{e^z}{z} \\text{ and } f(z) = \\frac{\\sin{z}}{(z-1)^2} ", "732e63dc7a234511a8f21570e947ebde": " M > 1 ", "732e7e60a6896d0d8f75f47962deb6dc": "{D_{KA}} = {d_{pore} \\over {3}} u = {{d_{pore}\\over{3}}} \\sqrt{{8 k_B T}\\over {\\pi M_{A}}}", "732e95c3d3136d02207021b82e385611": "| \\alpha | = \\alpha_1 + \\alpha_2 + \\cdots + \\alpha_n", "732ed8168674ac5baa9b65bef94ab12c": "c_2=1.4387752(25) \\times 10^{-2} \\text{m·K}", "732f10657c704c7a0306b9dec044ada8": "z_P-z_0=R_{13}(X-X_0) + R_{23} (Y-Y_0) + R_{33} (Z-Z_0)", "732fc5928f5fe4cda402dc9fcc3c61ee": "\\varphi_{AC} = \\frac{1}{\\hbar c^2} \\int_P (\\mathbf{E}\\times \\boldsymbol \\mu) \\cdot d\\mathbf{x}", "732fee04463413f11fa101ac8e40f44e": "E\\left[e^{tX}\\right] = \\int_0^\\infty e^{tx} \\frac{k}{\\lambda}\\left(\\frac{x}{\\lambda}\\right)^{k-1}e^{-(x/\\lambda)^k}\\,dx.", "7330a1dbce665c4a7198c3ff1eac23e0": "f(\\sigma) \\leq f(\\tau)", "7331241ecbe7506056e2af605b4afe3d": "dp \\wedge dq", "733133217d83ebff23a7e409b4cb0421": "\\mathrm{E}\\left((\\hat{\\theta}-\\theta)^2\\right)\\geq\\frac{[1+b'(\\theta)]^2}{I(\\theta)}+b(\\theta)^2,", "7331a6e4fcd24c2a97050f017cfc3d42": "Y[k] = H[k]\\cdot X[k]", "73323de58b4f110e3958d181743b8e84": "\n\\Phi_\\alpha \\left( {A_\\alpha ^1 , \\ldots ,A_\\alpha ^n } \\right) =\\left\\{ {\\frac{\\sum\\limits_{i = 1}^n {w_i a_{\\sigma (i)} } }{\\sum\\limits_{i = 1}^n {w_i } }\\left| {w_i \\in W_\\alpha ^i ,\\;a_i } \\right. \\in A_\\alpha ^i ,\\;i = 1, \\ldots ,n} \\right\\}", "73326a432e6d3d0d7f6c722ceaf0fa66": " \\qquad \\qquad \\mathrm{monatomic\\ ideal\\ gas}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ c_{v,f} = \\frac{\\partial e_f}{\\partial T}|_V = \\frac{3R_g}{2M},", "7332b1289958a894ac8a986953af17a5": "g_2, g_3", "733329d4b8851b9cc61cd1ce1c60532f": "(y_1, \\dots, y_m, x_1, \\dots, x_n)", "73336215d4e0de31e13c0e34bf6089a9": "1.9126", "7333967c2e8768e72a717f1543f147ba": "\n\\mathbf{A}^{-1}=\\frac{1}{\\det (\\mathbf{A})}\\left[ \\frac{1}{2}\\left( (\\mathrm{tr}\\mathbf{A})^{2}-\\mathrm{tr}\\mathbf{A}^{2}\\right) -\\mathbf{A}\\mathrm{tr}\\mathbf{A}+\\mathbf{A}^{2}\\right]. \n", "7333b9c4b29f4945b2661b9a4ba27f03": "\\boldsymbol{\\mathsf{A}}", "7333fb8b9a68d0a70843f4a98a819cd9": " \\dot x = u ", "73342a874eaa9daf399cbcc7a71b5697": "R_1(t_1) \\wedge \\cdots \\wedge R_n(t_n)", "73342f569ceaa28862cd8e5bb0296729": "q < 2 ", "73343995de31dfd17e84eae789094cad": " \\sum_{k=1}^\\infty \\frac{(-1)^{k-1} H_{2k}}{2k+1} z^{2k+1} = \\frac{1}{2} \\arctan{z} \\log{(1+z^2)}, \\qquad |z|<1 ", "7334865a6a99080d813de031eb106fdc": "g=g_{\\mu\\nu}dx^\\mu\\otimes dx^\\nu", "7334fda48ccd3c75e4d63eeaa7a61ad4": "S_t = \\displaystyle \\sum_{k=1}^n | \\Phi_k^t \\rangle \\langle \\Phi_k^t | , \\quad |\\Phi_k^t\\rangle = |\\phi_k\\rangle^{\\otimes t} ", "7335243abb0ed094ccc8bf7cabc052db": " \\operatorname{build-param-lists}[p\\ p, D, V, T_3] ", "733544fa5808a84d060d4906b60be681": " C(k) = \\frac{K^{+}(k)\\hat{g}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}. ", "733594d4ff64751c9abe407cfd434db4": " \\left| \\int f(x) e^{-izx}dx\\right|=\\left|\\int \\frac{1}{iz} f'(x)e^{-izx}dx\\right| \\leq \\frac{1}{|z|}\\int|f'(x)|dx \\rightarrow 0 \\mbox{ as } z\\rightarrow\\pm\\infty. ", "73360aa09e001f7a22023137f5075cec": "|m_1 m_2 \\cdots m_N; S/A \\rang", "73360f72c9f42ac5628b43de27a5afa1": "2^3 = 2 \\times 2 \\times 2 = 8. \\,", "7336a2c49b0045fa1340bf899f785e70": "vw", "733704388156fff8711db31870a9684e": " {}^\\mathrm{N}\\mathbf{v}^\\mathrm{Q} = {}^\\mathrm{N}\\!\\mathbf{v}^\\mathrm{P} + {}^\\mathrm{N}\\boldsymbol{\\omega}^\\mathrm{B} \\times \\mathbf{r}^\\mathrm{PQ}.", "733734cc34717ee91ecbb8d482bac108": "x = 3 + 4s", "7337b8d2a46b964363a58e38c5d1ab89": "\ng_{kl} \\Gamma^k{}_{ij} = \\frac{1}{2} \\left( \\frac{\\partial g_{jl}}{\\partial x^i} + \\frac{\\partial g_{li}}{\\partial x^j}- \\frac{\\partial g_{ij}}{\\partial x^l}\\right).\n", "7337dab7c95ed12c8bb6ac0a973a299a": "X \\perp Y \\,|\\, W", "7337db67ee6deab5898bdaf00dbbbcff": "16:9 = 1.77\\overline{7}", "733852d969daf55ff61f8b617530a1ed": "\\scriptstyle\\overline{OM}", "73388f41c04e2467cf187e566e673305": "L_{\\mathrm{I}}(x,y,z) = -2\\cdot\\frac{x\\cdot(\\ln y-\\ln z) + y\\cdot(\\ln z-\\ln x) + z\\cdot(\\ln x-\\ln y)}{(\\ln x-\\ln y)\\cdot(\\ln y-\\ln z)\\cdot(\\ln z-\\ln x)}", "7338a388ae3579f0260be4e620ab3362": " \\sigma_x>0 ", "7338c47f5c629b66849e8de3691a793e": "\\displaystyle{\\pi(g_1)f(x)=\\pm \\det (A)^{-1/2} f(A^{-1}x),\\,\\, \\pi(g_2)f(x) =\\pm e^{-ix^tBx} f(x),\\,\\, \\pi(g_3)f(x)=\\pm e^{in\\pi/8} \\widehat{f}(x)}", "7338d6089ea4430074aa1626d8258b39": " K = \\lim_{r\\to 0^+} 3\\frac{2\\pi r-C(r)}{\\pi r^3}", "7338fd89cd0eebcc6f6ddd4beb69b822": "\\psi_0(z)", "73394d9ff44aa2d3aa244868efd7e678": "\\varepsilon'_{ij}\\,\\!", "733989c973c6650d1c60fff62221dd5a": "\\mu_Q = \\sum_{i=1}^n \\lambda_i m_i(Q)", "73399d1966c3682570831f18ca80d2dc": "m(m-1)", "7339e9d695ad3acb7fa5342ab086dd5c": "\\psi_{u,v}", "733a4eeac70e6e28cd3e4b9269f96642": "\\pi: \\mathbb{R}^2 \\to T^2", "733a7e608e6de475c2f6275b4d75eaf5": "(p_c,", "733b106884b5d16478b4dbb99a62738d": "\\mathrm{height}(u)=\\max(\\mathrm{height}(u), \\mathrm{height}(s) + 1)", "733b12f1a97f9b620e41acd95107e186": "\nL = \\frac{m}{2}(\\dot{x}^2 + \\dot{y}^2) + \\frac{qB}{2c}(x\\dot{y} - y\\dot{x}) - V(x, y) ~,\n", "733b369d28e039566416fa981c3746cc": "MSD\\sim \\mid MAD\\mid^2", "733b4990ed3fb9d75076ddcb6cf0c5b8": " l = \\int d\\tau = \\int {d\\tau \\over d\\phi} \\, d\\phi = \\int \\sqrt{{(d\\tau)^2 \\over (d\\phi)^2}} \\, d\\phi = \\int \\sqrt{{-g_{\\mu \\nu} dx^\\mu dx^\\nu \\over d\\phi \\, d\\phi}} \\, d\\phi = \\int f \\, d\\phi", "733b4a84b964417dc45941017def1e1f": "\\displaystyle{W^\\prime(z_1)W^\\prime(z_2)= B(z_1,z_2) W^\\prime(z_2)W^\\prime(z_1),}", "733b6588bc573b11bf33540f9c9e8820": "\\displaystyle \\sqrt{\\frac{\\pi}{a}} \\cos \\left( \\frac{\\pi^2 \\xi^2}{a} - \\frac{\\pi}{4} \\right) ", "733b851ba31240d2a9515e377e1b7d1b": "\\approx -3.5449077018110320546\\,,", "733b9c5a4aaa668799c1860a6b48f8b3": "p(n, t) = e^{-t} \\frac{t^{-n}}{(-n)!}", "733c56a5ecfd0617ea79662f25a9ff25": " M=m", "733c6a470f9d286e03c143daef0ddc88": "(0, \\infty] \\times [0, 2\\pi) \\times [-\\infty, \\infty]", "733cac635fabf38931fa16ccc267174f": "\n{\\rm E}[z]\\,\\,\\, = \\,\\,\\,\\mu _z \\approx \\,\\,a\\mu ^r \\,\\,\\, + \\,\\,\\,{1 \\over 2}a\\,r\\left( {r - 1} \\right)\\,\\,\\mu ^{r - 2} \\,\\,{{\\sigma ^2 } \\over n}", "733cdf133444467585f38f3ab67b9c4f": " \\ \\alpha_0 ", "733d21be2075f82fc9e38092b4dba33f": "\nf(z^2) = f(z) - z \\qquad f(z^4) = f(z^2) - z^2 \\qquad f(z^8) = f(z^4) - z^4 \\cdots\\,\n", "733db05a2685982f0101aeaa97d986b5": " E[h(x,X_1)]\\equiv 0", "733dcb8d023d4e4ab5ad57f5e7a4989d": "Q(a, z) = \\frac{\\Gamma(a,z)}{\\Gamma(a)} = \\frac{1}{\\Gamma(a)} \\int_z^\\infty u^{a-1} e^{-u} du", "733deeb9cfe0525e43755bdec98a9a35": " \\tau_g(\\omega) = - \\frac{d \\phi(\\omega)}{d \\omega} \\ ", "733e00eda5763bee6d59f1cb8ae797b4": "\\scriptstyle E_s", "733e0370f6cde6513755584c6a1a6cad": "Y:T\\times\\Omega\\to S", "733e0cad6af461861c263866a785fb2d": "K_{isb} = 0.59 \\left(\\frac{E}{H}\\right)^{1/8}[\\sigma (P^{1/3})]^{3/4}", "733e1011cc772a6e8ac569df08ca8d8b": "\\omega + W_a", "733e43890d289c8103c9cf83602e3113": "C_x = PC_x", "733ea21c4b2b05838a4c5f524829e444": "{\\mathbf{}}J", "733ecd2b842453a401f9a1e897db4842": "k_e = \\frac{1}{4\\pi\\varepsilon_0}", "733ee6c23e450f29a73833c6f5c0b1c9": "\\sum_{n_1,\\dots,n_m\\ge 0}\\frac{1}{L_1^{s_1} \\cdots L_k^{s_k}},", "733f76b0282f70a4b21ee064eeefe8aa": "\nh_{\\sigma} = a\\sqrt{\\frac{\\sigma^{2} - \\tau^{2}}{\\sigma^{2} - 1}}\n", "733fb32a8ea9e5b70f2cd3a778faee5e": "(\\phi \\to (\\chi \\to \\psi )) \\to ((\\phi \\to \\chi ) \\to (\\phi \\to \\psi ))", "7340255065c87c5bd71c30d53fe0d004": "G_{X_i}", "734091d59eda78229eb41be37073cd9c": " L = 44.625 ", "73409b9e4512d72c55574da0d98a495e": "\\,P_v(v)dv", "734123fca8368a667c8c59399ac6afc1": "\\int_a^b G(t) \\, dt=\\ G(x)(b - a).\\,", "734134a02eec920dbd44b47200434703": " AvgPrec(r_{f(q)}) \\geqq {1 \\over R} \\left[ Q+ \\binom{R+1}{2} \\right ]^{-1}(\\sum_{i=1}^R\\sqrt{i})^2", "73416d199288409128524c16838f110c": "10^{15}-10^{20.5}", "73417a3a5b0b09b1e50de57f9bd29feb": "h*g^*", "734184c771f29073c8fd11c53d14215f": "\\nabla = \\left\\{ \\frac{\\partial\\quad}{\\partial q_{1}},\\quad \\frac{\\partial\\quad}{\\partial q_{2}} , \\ldots, \\frac{\\partial\\quad}{\\partial q_N} \\right\\}", "7341aed74f73b4d40c88563645084898": " M = \\int d^3x {H(x)^2 \\over \\sqrt{det (q)} (x)} = \\int d^3x ({H \\over [det (q)]^{1/4}})(x) \\int d^3y \\delta (x,y) ({H \\over [det (q)]^{1/4}}) (y)", "7341b52b87c394070273b15fb0d694d3": "\\mathcal{O}(p^4)", "7341e1c9064c096a6cc97c8a25e89f7c": "\\scriptstyle \\epsilon = \\{ S_- \\cup S_0 | S_+ \\} = \\{ 0 | 1, \\tfrac{1}{2}, \\tfrac{1}{4}, \\tfrac{1}{8}, ... \\} = \\{ 0 | y \\in S_* : y > 0 \\}", "734205fb3470d2588480bd4003273e36": "h = \\mathcal{L}_X g = 0", "73429eaeafd4ff2516d2f6a3f30f320a": "|f_n(x) - f_n(y)| \\le K|x-y|", "7342b03bb2862a8a154417004cc35a44": "1 + {1 \\over 2} + {1 \\over 4} + \\cdots", "7342b7128a2df69c73f3a7e8326763a4": "u_t = \\alpha \\nabla^2 u \\ ", "7342fa2c24a8ec7c272a33d2ac3e8161": "L(O) = 1", "73433631263b4a73fddd6d650158357f": " F(e^{\\lambda x}\\Phi) ", "734362f46b30fce4a2a3623e7e5fe08a": "B(\\mathbf{r},i) = \\sum_{j=1}^C (y_j-r_j)^2 ", "73437434061c324838a8182b4afbf92e": "f(1)=5", "73439967a0fab7656655b129cce82d9d": "\\begin{pmatrix} \\cos\\alpha & -\\sin\\alpha \\\\\n\\sin\\alpha & \\cos\\alpha \\end{pmatrix}", "734418a9fbe8ef7a5404805c9e927bc0": "D = 1.496 \\times 10^{11} \\ \\mathrm{m},", "73442f1ed2a44abec85bec671ae59e0d": "p_{1}y_{1} + p_{2}y_{2} = m", "73445eb8c3a758d663f2e6d5c6c5467f": "\nH = \\frac{L + V}{\\left| L + V \\right|}\n", "7344f05b8ada4faadfaf8f91bb0b8f2d": "\\sum_{n=0}^{\\infty} a_n ", "7344f7e4badd15f81e9d16afd8bfc318": "\n \\varepsilon_{\\alpha 3} = \\cfrac{1}{2}~\\kappa~\\left(w^0_{,\\alpha}- \\varphi_\\alpha\\right)\n ", "7345ae113af783eca9d62acfbe931a5e": "D(\\theta,\\phi) = \\frac{U(\\theta, \\phi)}{P_\\mathrm{tot}/\\left(4\\pi\\right)}.", "7345c43728ff11da2c66ca466edadcf3": "\\begin{align}\nr & = \\sqrt { x^2 + y^2 } \\\\\n & = \\sqrt { x^2 ( 1 + (y/x)^2) } \\\\\n & = |x| \\sqrt {1 + (y/x)^2 }\n\\end{align}", "7345dfce6789a6d36d4e5fc1355ebb89": "x_0 = \\alpha \\sinh(t/\\alpha) \\cosh\\xi,", "7346453d67e10f62087de22fbcafbfd6": "O_j = \\mathrm{old}O_j", "7346a7d512c1e2a453f6040efad9d41a": " \\theta : \\xi \\otimes \\xi \\rightarrow \\xi ", "7346c443f200896bae1635e1341963e1": "c^{2}(\\mu)<\\infty", "7346c5f96e98a4ef27c34694eaa5922d": "\ni{d\\over dt} \\psi = \\left(\\frac{\\nabla_1^2}{2m} + \\frac{\\nabla_2^2}{2m} + \\cdots\n+ \\frac{\\nabla_N^2}{2m} + V(x_1,x_2,\\dots,x_N) \\right)\\psi \n\\,", "73470a3db83982a462befe2ece221fcc": "{}^2\\!D_n, {}^2\\!E_6, {}^3\\!D_4, ", "73470c3f7881256ab06902f9aa7a7970": "a\\hat{x}, a\\hat{y}, a\\hat{z}", "73472a6f3048954125ea8559c39155bc": "r \\approx a (1-e) \\sqrt[3]{\\frac{m}{3 M}}.", "734764b1c86c6075a9bafc9627916a7d": " \\vdash \\left( (\\vdash p)\\rightarrow(x \\pmod 2 \\equiv 0) \\right)", "734773eb9bf0b196e21108da09a13415": "\\left( \\left| 0 \\right\\rangle + \\left| 1 \\right\\rangle \\right)/\\sqrt{2}", "7347840e7c7025b0a4dfdb24e9cfaf4b": "x-M_2", "7347af8421d101292e8596296581d724": "\\Big( (\\mathcal{M}, s) \\models EF\\phi \\Big) \\Leftrightarrow \\Big( \\exists \\langle s_1 \\rightarrow s_2 \\rightarrow \\ldots \\rangle (s=s_1) \\exists i \\big( (\\mathcal{M}, s_i) \\models \\phi \\big) \\Big)", "7347ed3014d55008442d170285aca68e": " \\lim_{x \\rightarrow c} m(x) \\leqslant \\lim_{x \\rightarrow c} \\frac{f'(x)}{g'(x)} \\leqslant \\lim_{x \\rightarrow c} M(x) ", "7348299bc093299949edad6f6099e5ed": "\\Sigma_g", "73484e9cbb1dd8aa3fa117de2ab49fc1": "\\textstyle\\int_0^\\infty \\left|f(x)\\right|dx = L.", "73485908cebc63433f7bf64ba6328286": "\\theta_B = \\frac {K^B_{eq}\\,p_B}{1+K^A_{eq}\\,p_A+K^B_{eq}\\,p_B}", "73486a6d420ad43256bf0b5f17f0bbab": "Misfit = \\sqrt{\\sum_{i-\\sigma}\\frac{(x_{di}-x_{ci})^2}{\\sigma^2_i n_F}}", "734871fe9d17cb554a08d0e6c61f78bd": " \\theta = T \\left(\\frac{P_0}{P}\\right)^{R/c_p}, ", "73488a74470c22426803a5c4b6a8a3b6": "S_{1},...,S_{n}", "734917493ec482e76c97f631678c24bf": "{GM \\left (r^2 - 2z^2 \\right ) \\over \\left ( r^2 + z^2 \\right ) ^ {5/2}} \\times 10^9 \\; \\left [ \\text{E} \\right ]", "7349af0b4fd7cb0b6f11a2b045e38230": "g^{r0}", "7349c07587a5e28e7748d6ddc7e0a927": "f:\\mathbf{C}^n\\longrightarrow\\mathbf{C}", "7349eb8ade9af66d337fd492399aa3ce": "\\sum_{i_0+\\cdots+i_n=k} P_{i_0\\dots i_n}(x_0,x_1,\\dots,x_k) \\, dx_0^{i_0} \\, dx_1^{i_1} \\cdots dx_n^{i_n}=0", "734a2501ea74dc54c40e320d86305f0c": " u_z = 0", "734a3520286d43a955051dcf36bb5440": " \n\\mathcal{P}_2,\n", "734a3f2f2b8cae146f072ce0dde3fccd": " \nQ=\\left[\n\\begin{array}{c}\nR \\\\\n0 \\\\\n... \\\\\n0\\\\\nP \\\\\n\\end{array}\n\\right]\n", "734a7c5d5837e0198b83f756919160a4": "\\text{EM}=\\frac{g'}{g}=\\left[\\left(\\frac{a_\\text{f}^2}{g^2} +1\\right)\\right]^{0.5}", "734a9a858e336235450d6b70ad6ce8ff": " \\boldsymbol{\\mathsf{P}} = m \\boldsymbol{\\mathsf{U}} \\,\\!", "734aa72b483fa6382347ded015ec6fd2": "\\frac{\\hbar}{e a_0^2}", "734adc87e0e2548bd695cd9be35150a9": "\\mathfrak{a}~ \\mathfrak{b}~ \\mathfrak{c}~ \\mathfrak{d}~ \\mathfrak{e}~ \\mathfrak{f}~ \\mathfrak{g}~ \\mathfrak{h}~ \\mathfrak{i}~ \\mathfrak{j}~ \\mathfrak{k}~ \\mathfrak{l}~ \\mathfrak{m}~ \\mathfrak{n}~ \\mathfrak{o}~ \\mathfrak{p}~ \\mathfrak{q}~ \\mathfrak{r}~ \\mathfrak{s}~ \\mathfrak{t}~ \\mathfrak{u}~ \\mathfrak{v}~ \\mathfrak{w}~ \\mathfrak{x}~ \\mathfrak{y}~ \\mathfrak{z} ", "734ae1b514a78ca52d1eb172e972a477": "S^{n - 1}", "734aee4709d6be50b425c948b74cea2b": "R(\\theta,d) = \\mathbb{E}_\\theta[ |\\mathbf{\\theta - X}|^2]", "734b5b4a056603029deb7f17bfc10458": "\\omega_n = \\sqrt{k/m}", "734b8000dedea13d78e7f9b98b39df72": "\\psi(\\xi_x,\\xi_y,\\xi_z)=C_x C_y C_z Ai(-\\xi_x)Ai(-\\xi_y)Ai(-\\xi_z)", "734bafbcb1d20486cfec4d35e2a58ba0": " \\mathbf{r_0}=(x,y,z)", "734c0b1325f80c3072b70a1173240ee0": "\\bigcup_i A_i", "734c1c1455aa124942dd2e930366fd88": "\\begin{bmatrix}\nX\\\\\nY\\\\\nZ\\\\\n\\end{bmatrix}\n= \\begin{bmatrix}\n -\\sin\\lambda & -\\sin\\phi\\cos\\lambda &\\cos\\phi\\cos\\lambda \\\\\n\\cos\\lambda & -\\sin\\phi\\sin\\lambda & \\cos\\phi\\sin\\lambda \\\\\n0 & \\cos\\phi& \\sin\\phi\n\\end{bmatrix}\n\\begin{bmatrix}\nx \\\\\ny \\\\\nz\n\\end{bmatrix}\n+ \\begin{bmatrix}\nX_r \\\\\nY_r \\\\\nZ_r\n\\end{bmatrix}\n\n", "734c4ea950c54a342997fdc990e4e6c4": "\\mathrm{d}U_{cv}=\\mathrm{d}U_{in}+\\mathrm{d}(P_{in}V_{in}) - \\mathrm{d}U_{out}-\\mathrm{d}(P_{out}V_{out})+\\delta Q-\\delta W_{shaft}\\,", "734c8675b9e2006f9076229be91f24df": "[e_i,f_j] = \\delta_{ij} \\frac{k_i - k_i^{-1}}{q_i - q_i^{-1}}", "734c951fa5dff94b458eb8aafb25a31a": "{6\\choose 0}{43\\choose 6}\\over {49\\choose 6}", "734c9a50d4532aefcb2cd1dc9d377665": " M(t) = \\frac{1}{|G|} \\sum_{g\\in G} \\frac{1}{\\det(I-tg)} ", "734ca3fb03e5357bcd6c60ab007c8d00": "w={{A}^{-1}}u", "734cc34864d1a96afd41b38c8cd6177f": "\\pi^{-1}_{r}(p)\\,", "734d0980963bf19799c976115f7285f4": " \\displaystyle{PE_0=\\lambda E_0},", "734d3ee0691b0ab92ba6d238922240c7": "\\Phi: D \\to \\mathbb{R}, \\qquad D \\subseteq \\mathbb{R}^n \\smallsetminus \\{ \\bold{0} \\},", "734d434020ec539ba0e684e77a5e3508": "y + x + 5 = 0 \\,", "734d62b4748a88ee71f10046972d77d7": "\\beta_\\pm(t)", "734d82329a73f44e295cce2a33fd01e3": " y[k] = y(k T_s) = \\sum_{n = -\\infty}^{\\infty} x[n] \\cdot h[k - n] ", "734daf014631d25536709186e7e05b1d": "-\\frac{1}{\\pi}\\int_\\epsilon^\\infty \\frac{u(t + \\tau) - u(t - \\tau)}{\\tau}\\,d\\tau\\to H(u)(t)", "734e2966e8b70de429f04ba37f6a985d": "\\mathbf{x}_1=(1, 0, 0, \\ldots, 0), \\mathbf{x}_2=(0, 1, 0, \\ldots, 0), \\ldots, \\mathbf{x}_n=(0, 0, 0, \\ldots, 1)", "734e6658b084aafc3a8170a39e359f2f": "h_{Preucil\\ circle} = 60^{\\circ} \\cdot \\left( 2 + \\frac{B - R}{G - R}\\right)", "734e6a0330f54b6ee91e8b65105fc131": "\\| (\\epsilon_i x_i) \\|_X = \\| (x_i) \\|_X ", "734e6c86f9776758e6d188611d6c22ed": "\\binom{d+n-1}{n-1}=\\binom{d+n-1}{d}=\\frac{(d+n-1)!}{d!(n-1)!}.", "734eb00888ce02df0f00d0a1d21b7c13": "\\kappa_3=\\mu_3\\,", "734ef0e56fa1f29e34b4d9a39da0d19e": "VY", "734f14030bf82ad18abfa7d7e8146277": " \\frac{\\delta S}{\\delta h^{ab}} = T_{ab} = 0 ", "734f41f718889d00aa2b32d2deb6fbf8": "n=0,\\dots,p-2", "734fa9f47bc14f4a633f798e2be92ca4": "\\exists x_1 \\cdots \\exists x_n \\phi(x_1, \\ldots, x_n)", "734fadaded3737bfbdf9351f14024f74": "\\ln(f_{WN}(\\theta;\\mu,\\sigma))= \\ln\\left(\\frac{\\phi(q)}{2\\pi}\\right)+\\sum_{m=1}^\\infty\\ln(1+q^{m-1/2}z)+\\sum_{m=1}^\\infty\\ln(1+q^{m-1/2}z^{-1})", "734ff13957b527ae412980daa6b89f5c": "\\textrm{VD}=\\frac{\\textrm{32}}{\\sqrt{\\left(\\frac{\\textrm{1920}}{\\textrm{1080}}\\right)^2+1} \\cdot \\textrm{480} \\cdot \\tan{\\frac{1}{60}}}=112.36", "73504efffce58858aa59a4f58481b721": "\\phi(q)= q^{-\\frac{1}{24}} \\eta(\\tau)", "73505266f40bad3b191a5a7913e31e0a": " {H}_{2n+1} ", "7350b1d423d221f24fc3fc4180a77511": "F_2\\rightarrow S_n,", "73511766723ec47d90b6425836d8575f": "1-\\left(1-\\exp(-t)\\right)^{1/\\theta}", "735133e2f42d40d4a52b9a6bdb1600e3": "\\begin{cases}4x + 2y &= 14\n\\\\ 2x - y &= 1.\\end{cases} \\,", "73516cea7b9ebc7ef021e18fe8d3a565": "H_{10}(x)=1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240\\,", "735177ac4fbb2fd2dbc2bb35bf26da23": "T(z_1,z_2)", "7351a7b45ec262ef408d9b3992584d83": "\n\\begin{align}\n\\frac{d}{dz} \\arcsin z & {}= \\frac{1}{\\sqrt{1-z^2}}; \\quad z \\neq -1, +1 \\\\\n\\frac{d}{dz} \\arccos z & {}= \\frac{-1}{\\sqrt{1-z^2}}; \\quad z \\neq -1, +1 \\\\\n\\frac{d}{dz} \\arctan z & {}= \\frac{1}{1+z^2}; \\quad z \\neq -i, +i \\\\\n\\frac{d}{dz} \\arccot z & {}= \\frac{-1}{1+z^2}; \\quad z \\neq -i, +i \\\\\n\\frac{d}{dz} \\arcsec z & {}= \\frac{1}{z^2\\,\\sqrt{1 - z^{-2}}}; \\quad z \\neq -1, 0, +1 \\\\\n\\frac{d}{dz} \\arccsc z & {}= \\frac{-1}{z^2\\,\\sqrt{1 - z^{-2}}}; \\quad z \\neq -1, 0, +1\n\\end{align}", "7351db2521527d77e1936ca2b9c2f5b8": "T(2,s,x)=\\Gamma(s,x)/x", "73522bd8f3c00e65706f112eecd51831": "v^\\mu = \\left(\\sqrt{\\frac{r}{r-2M}},0,0,0\\right)", "735240daebf8efb9c6e836ef0fbf1a6a": "x \\in \\mathbb{R}^m", "735299f29bb9dc97e5f6a97f7b0a7cac": " g^{\\mu \\nu} ", "7352b4cf7b7b08ea54f7a9e82c917abd": "2P_{\\frac{1}{2}}", "73530a0d2eec47d5d9edf443b6131ce3": "| \\alpha \\rangle_R = D(R) |\\alpha \\rangle", "73530c641372f68e6e75589c09025b6c": "D = \\frac{\\delta^{2}}{\\tau}=v^{2}\\tau=\\delta\\,v,", "73534a1e20dc9c66e7d880961aac8036": " 16r^2y^4(a^2 - 1) + 1 - u^2 ", "7353c064711d366926ee9d638cc259a3": "\\lambda = \\frac{i_cL}{\\Phi_0}", "735427fc33fa7c06217df1c5455d90d3": "\\begin{align}\n\\sum\\limits_{i=1}^\\infty \\sum\\limits_{j=i}^\\infty P(X = j) &=\\sum\\limits_{j=1}^\\infty \\sum\\limits_{i=1}^j P(X = j)\\\\\n &=\\sum\\limits_{j=1}^\\infty j\\, P(X = j)\\\\\n &=\\operatorname{E}[X].\n\\end{align}", "73542b6a1fcbe33bc4f28636fb44e53c": "n^{-1+p(\\log n)^{-1/2}},", "7354385658d8283aac40eeb0cd09aebc": "\\begin{align}\\frac{\\partial \\mathcal{L}}{\\partial (\\partial_{\\beta}A_{\\alpha})} & = - \\ \\frac{1}{4 \\mu_0}\\ \\frac{\\partial (F_{\\mu \\nu}\\eta^{\\mu\\lambda}\\eta^{\\nu\\sigma}F_{\\lambda \\sigma})}{\\partial (\\partial_{\\beta}A_{\\alpha})} \\\\\n & = - \\ \\frac{1}{4 \\mu_0}\\ \\eta^{\\mu\\lambda}\\eta^{\\nu\\sigma}\n\\left(F_{\\lambda\\sigma}(\\delta^{\\beta}_{\\mu}\\delta^{\\alpha}_{\\nu} - \\delta^{\\beta}_{\\nu}\\delta^{\\alpha}_{\\mu})\n+F_{\\mu\\nu}(\\delta^{\\beta}_{\\lambda}\\delta^{\\alpha}_{\\sigma} - \\delta^{\\beta}_{\\sigma}\\delta^{\\alpha}_{\\lambda})\n\\right) \\\\\n & = - \\ \\frac{F^{\\beta\\alpha}}{\\mu_0}\\,.\n\\end{align}", "7354bc6d1ef9296d785308f5b4dd8913": "x^3 - 2x^2 - 5x + 6", "7354c182e0fe7cb433db330df2a1521a": "S^2|s,m>=s(s+1)\\hbar^2|s,m> ", "7354defbe901cc3431e34f7fe6ec0e26": " \\mathbf{e} = {\\mathbf{v}\\times\\mathbf{h}\\over{\\mu}} - {\\mathbf{r}\\over{\\left|\\mathbf{r}\\right|}}", "7354eece98de059a1b9cbcce470652f6": "G = \\sigma\\sqrt{2a}", "735513ff01cd9a118b9ba31701250c40": "g_{\\vec z}(X)=f(X)", "73556706ec7e708285ccf6cc1f4705da": "\\phi(x) \\leq \\phi(x_0)", "73557068b5c14a2436cd91f0c817119f": "\\psi_{II}(x) = B_I e^{i \\rho ' y} + B_{II} e^{-i\\rho ' y}\\quad |y|<1 ", "73557d1a9ace004e6d8a05175b4ec4fc": "\\sqrt{\\frac{ G( M_\\star \\! + \\!m ) } {a^3}} \\,\\delta t", "73558fc722ec35e878f6a92127468fce": "\\sigma_{i j} = \\epsilon_0 \\left(E_i E_j - \\frac{1}{2} \\delta_{ij} E^2\\right) + \\frac{1}{\\mu_0} \\left(B_i B_j - \\frac{1}{2} \\delta_{ij} B^2\\right)", "7355abbf199a61f0a8c856156b35356c": " \\overline{F}(x_1,x_2) = \\left(1 + \\sum_{i=1}^2 \\frac{x_i-\\mu_i}{\\sigma_i} \\right)^{-a}, \\qquad x_i > \\mu_i, i=1,2,\n", "7355bf6b8ed54b79768988ec4694fc75": "exp[i(px-Et)/\\hbar]", "7355e878c1a9824309942387e8965aa8": "M=\\begin{pmatrix}\\lambda_{11}&\\lambda_{12} \\\\ \\lambda_{21}&\\lambda_{22}\\end{pmatrix},", "7356677f8e6e4c2b7d31e41b39e9f370": "A_x\\left(\\eta,\\tau \\right)", "73573b01c65409d7152012e4e3f821b6": "g_S = -2", "7357428d3ee47ac4a3b2737d26c871df": "y(r)", "73574e893da0eb5dd815d80f6687b6ff": "T_v\\exp_p(v)=v\\ ", "7357638fed990a26ac8135e8e4cde6e6": "\\langle, \\rangle", "73577bf31e166a768177abf4991267d0": "\\mathcal{G}= G + \\sum_{i=1}^k\\lambda_i\\left(\\sum_{j=1}^m a_{ij}N_j-b_i^0\\right)=0", "735785fa5686ceeaa0bfc3131e6c814b": " q = \\Pr( X < \\operatorname{ E }( X ) ) ,", "7357ed35a27594ff83cb1e9025c7da14": "\\frac{d\\beta}{dt}=-\\frac{4k}{MV}\\beta+(1-2k)\\frac{(b-a)}{MV^2}\\omega", "7358037a5941e593f0956023a40ce224": "\\psi(\\bold R, t) = \\int \\psi(\\bold R^0,t=0) G(\\bold R - \\bold R^0,t) dR_x^0\\,dR_y^0\\,dR_z^0,", "7358a76b048e92984b756f82bb8d26a5": "\\left(\\pm 1,\\ \\pm 1,\\ \\pm 1,\\ \\pm(1+\\sqrt{2})\\right)", "73590d28abef222dddc4d9e79d1db775": "\\displaystyle{\\int_{\\partial\\Omega} A\\, dx + B\\, dy =\\iint_\\Omega (B_x-A_y)\\, dx\\, dy.}", "7359780f0f419d2897a46ff3e07b62cd": "(19)\\quad k^c\\nabla_c \\hat\\omega_{ab}=-\\hat\\theta\\hat\\omega_{ab}\\;.", "7359eabd53c5006e17818319ad09f6f4": "\\dot J(t, t')", "7359fd8a83b5c715c3d545141e3e4de9": "\\langle x, B^*By\\rangle = \\langle Bx, By\\rangle_A = \\langle[x], [y]\\rangle_A = \\langle x, Ay\\rangle", "735a07614492280eeca8e4840ebe0753": "\\max_{a\\in A}C(a,f(a))", "735a158729c4a95e521ff9193cd1c870": "\\boldsymbol{y}_t \\sim P_{z_t}(\\boldsymbol{y}_t)", "735a3371e33f6bdff41f46f25b83dce3": "\\sinh^{-1}{x} = \\ln(x + \\sqrt{x^2 + 1})", "735a56bdd5f235c93af14e40a406bc3e": "\\boldsymbol{v}=\\boldsymbol{v}_T+\\boldsymbol{v}_R", "735a8a0e6b1230bd64130447daa66129": " \\underline \\sigma = \\sigma (\\underline \\varepsilon) =E\\underline \\varepsilon = \\frac{\\underline P}{A} ", "735aa49bb3f884fe5ff9076670ea529a": " \\begin{bmatrix} \\theta_1 \\\\ \\theta_{2} \\\\ \\dots \\\\ \\theta_{N} \\end{bmatrix},", "735ac1f43d0654e64b4a1928febb761d": "TSI(c_0,r,s) = 100 \\frac{EMA(EMA(m,r),s)}{EMA(EMA(|m|,r),s)}", "735ad20523c61f4b80c935752dc80c6c": "b\\mapsto\\lambda\\cdot b", "735b05e6097f98da56f2ca14b8005d36": "x\\in X", "735b59a09bda45204ff2af5dd139874f": "\n \\Beta(x,y) = \\Beta(x, y+1) + \\Beta(x+1, y)\n\\!", "735b704fc3e866f1372bf2f1f3d86319": "{D}_{9}^{(2)}", "735bdcd4e94ba7860e33a72cfebab3bf": "f_x(y) = x^2 + xy + y^2.\\,", "735c11a9f104e1db3a69937229629c69": "(5.b)\\quad \\gamma_{,\\,\\rho}=\\rho\\,\\Big(\\psi^2_{,\\,\\rho}-\\psi^2_{,\\,z} \\Big)\\,,", "735c76f20cbd536b4ffab680de31ebda": "\\mathcal{L}_Y = -\\sum_f \\frac{gm_f}{2m_W}\\overline ffH", "735ce1044e4f99ba6e189c5b1b733acd": "\\nu(dx) = C \\left(\\frac{q_\\alpha(\\lambda_+ |x|)}{x^{\\alpha+1}}1_{x>0} +\\frac{q_\\alpha(\\lambda_-\n|x|)}{|x|^{\\alpha+1}}1_{x<0} \\right) \\, dx,\n", "735d4bd0fb9763db09db26346338067c": "\n\\begin{align}\nH_0 &:& \\theta=\\theta_0 ,\\\\\nH_1 &:& \\theta=\\theta_1 .\n\\end{align}\n", "735d69b43d23a4b9712e2439766a07eb": " \\Gamma = W J - J W^t \\, ", "735d75b60dcf0791886b5ec087fd66c4": "\\begin{align}\n \\rho \\left(\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} + v \\frac{\\partial u}{\\partial y}\\right)\n &= -\\frac{\\partial p}{\\partial x} + \\mu \\left(\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2}\\right) + \\rho g_x \\\\\n \\rho \\left(\\frac{\\partial v}{\\partial t} + u \\frac{\\partial v}{\\partial x} + v \\frac{\\partial v}{\\partial y}\\right)\n &= -\\frac{\\partial p}{\\partial y} + \\mu \\left(\\frac{\\partial^2 v}{\\partial x^2} + \\frac{\\partial^2 v}{\\partial y^2}\\right) + \\rho g_y.\n\\end{align}", "735de346b0540f73fcb8f6e23eb82203": " \\frac{d}{dt} y(t) = Ay(t) + z(t), \\quad y(0) = y_0. ", "735e25305a593a611cc997170e285701": " (1-1)^t= \\binom{t}{0} - \\binom{t}{1} + \\binom{t}{2} - \\cdots + (-1)^{t}\\binom{t}{t}", "735e25782e6fe5f70ad32228ba83b66f": "x_0, x_1, x_2,", "735e28d9959f38a57026803a139cdbb6": " \\ln \\frac{k}{T} = \\frac{-\\Delta H^\\ddagger}{R} \\cdot \\frac{1}{T} + \\ln \\frac{k_\\mathrm{B}}{h} + \\frac{\\Delta S^\\ddagger}{R} ", "735e445b0f2eb70b773511937f9c169b": "t = -1,1", "735e55ec72cc36e8f6002f7d996baf67": "B_n(mx)= m^{n-1} \\sum_{k=0}^{m-1} B_n \\left(x+\\frac{k}{m}\\right)", "735e880f6ab9b24c37e0c4c35feffda4": "\n \\frac{\\sqrt 2 (a\\!+\\!b\\!-\\!2c)(2a\\!-\\!b\\!-\\!c)(a\\!-\\!2b\\!+\\!c)}{5(a^2\\!+\\!b^2\\!+\\!c^2\\!-\\!ab\\!-\\!ac\\!-\\!bc)^\\frac{3}{2}}\n ", "735e931e764ffd365a6de00a9596779e": "f : G \\rightarrow G", "735eaee653efbb4d9ab680ec110e3bed": "\\begin{align}\nx'_{A}& = \\gamma\\left(x_{A}-vt\\right)\\\\\nx'_{B}& = \\gamma\\left(x_{A}+L-vt\\right)\\\\\nL'& = x'_{B}-x'_{A}\\\\\n& =\\gamma L\n\\end{align}", "735ec8d765a22500fad36e935bc1272a": "H(\\rho,\\sigma)", "735f074426c2a2f7b210b7762e2da766": "\\beta_k \\le 3", "735f316137b586ab081bd5b3377d5071": "D_i \\oplus D_j = A", "735f448e5b56d2b63855d83625d64b06": "b_\\mathrm{in}(t)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty\\mathrm{d}\\omega\\,e^{-i\\omega(t-t_0)}b_0(\\omega)\\,,", "735f662e428d677fdff397bb2546b53d": "a \\frac{x^n - 1}{x - 1} = p", "735f85fbccb948988806f41eb92be857": "(RvR^{\\dagger})^2 = Rv^{2}R^{\\dagger}= v^2RR^{\\dagger} = v^2 ", "735f921287a0b2760d22210278403cb8": "u=1", "735f943cd729c88ebc4c58a9768b7bd1": "\\operatorname{recc}(A) = \\bigcap_{t > 0} t(A - a)", "735fce06be749015236a86542f48aa29": "a^\\smallfrown b", "735fe62d2db167013bd86475466b35f8": "{D \\over p} = .0003 \\times {\\sqrt[3]{n \\over p}} ", "735ffd6c1bc33bb9acb224a1e4f89d02": "\\begin{pmatrix} E_x(t) \\\\ E_y(t)\\end{pmatrix}\n= \\begin{pmatrix} E_{0x} e^{i(kz- \\omega t+\\phi_x)} \\\\ E_{0y} e^{i(kz- \\omega t+\\phi_y)} \\end{pmatrix}\n=\\begin{pmatrix} E_{0x} e^{i\\phi_x} \\\\ E_{0y} e^{i\\phi_y} \\end{pmatrix}e^{i(kz- \\omega t)} ", "73601f7ef69b151c5a414dac489739d7": "T_4\\;", "736060c55cfe17e4f5accafc7cb61a33": "\\|u\\|^2 = \\int_a^b u^2(x)\\, dx \\le C \\int_a^b u'(x)^2\\, dx = C\\,(Bu|u)", "7360c95d503c182f56ea79aa2aceb30d": "r_{g,i}", "7360c9b70f141b5d7befc2354ad61ae5": "S_1(x) = -\\frac{1}{4}\\ln Q(x) + k_1,", "7360e4d6a96024661a649155b0ba085d": "J\\tilde{P}=x\\tilde{P}", "73611ebe3d57c94da0a05e5dfb3068b9": "h_C", "736214a38fcf761a267a87394974789b": "\\xi_{[T-1]}", "73628747a9b9bd709f8adad6f5f9375f": "\\mathcal{}\\pi_*M", "7362b57caeb2f946ee05c8315c7d80e7": "\\sqrt{ \\left({dx \\over dt}\\right)^2 + \\left({dy \\over dt}\\right)^2 }", "7362ee9ddc4c4225dbf4df26cfbfaac7": "\\tau_j", "7363212f8e5a9d7ace25c6991a33c6fe": "t_{1}", "7364069f201c57d492b40451a23407fa": "\\ U=X\\setminus F ", "73646199f6741793f01b7751be0cb459": "\\rho_{l}", "73649270cbe7aa3d4cd09af8338f31c6": "K / \\mathbb{Q}", "7364afa196a12b6910f8988bdfc4a0af": "\ng^{\\alpha \\beta} \\, = \\left( \\begin{matrix}\n - c^{-2} & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 \n \\end{matrix} \\right)\n\\,,", "7364c09ff2722fb12feeb8c67f86013c": "U = W V^*\\,", "7364d9bdfaec619af7dee34da95f8868": "g_5^{-1}g_7\\ne 1", "7364e3262bb067e1829d31944ac6ba64": "u (x) = \\frac{1}{(2 \\pi)^n} \\int e^{i x \\xi} \\hat u (\\xi) d\\xi = \n\\frac{1}{(2 \\pi)^n} \\iint e^{i (x - y) \\xi} u (y) \\, dy \\, d\\xi ", "7364f94bd88bd45865dd4c496b90c45f": "\\sigma\\ ", "7365049bfc3925dc86a72cd996718c42": "\nl = \\frac{1}{\\beta} \\left[(n+1)\\pi - \\arctan \\left(\\frac{1}{\\omega C Z_0}\\right) \\right]\n", "7365b5d74d77dd65f10297406cdf5013": "\\scriptstyle\\hat{m}(\\theta_0)\\;\\approx\\;m(\\theta_0)\\;=\\;0", "7365c765a498ddf259138cab96b32f7d": "\\frac{x^2}{2}-\\frac x2\\,", "7365e5e7b92273135cd9bca02f8ad12d": "u\\mapsto DF(u) \\,", "736609c5a6917aa37944fbdd35e3dd75": "P_1\\,", "73665bbeaa7b5c84b4760a91b2ae3d3c": "\\frac{d\\sigma}{d\\Omega}=\\left(\\frac{1}{n}\\right)\\frac{dN}{d\\Omega} = \\left(\\frac{\\alpha}{4E}\\right)^2 \\csc^4\\frac{\\chi}{2}", "736664f15facd29c6b86c35cf4367466": "\n\\int_{0}^{+\\infty} \\frac{W(x)}{x\\sqrt{x}}\\mathrm dx = 2\\sqrt{2\\pi}\n", "736696406283454097d270fcb9c958a1": "E= E_{0}\\exp[i(k_{x} x + k_{z} z -\\omega t)]\\,", "7366e74fe3bc0add07a0bc98a2e68a6b": " a \\uparrow \\uparrow \\uparrow \\uparrow b = \n \\left. \\underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \\underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \\underbrace{\\vdots}_{a} }} \\right\\} b", "7366ed22c8d9582583e97ef158e6ea0a": "A \\otimes_R B", "736737be4d5f33e6eeed804d696387d3": " \\gamma = \\frac{1}{2} V_\\infty c C_{l_\\alpha}(\\alpha_\\infty + \\alpha_{geo} - \\alpha_0 - \\alpha_i) \\qquad (4)", "7367823d5b080dc84220c545ac56c1da": "\\varphi^{\\mathop{\\rm MC}}(f)=\\frac 1n \\sum_{i=1}^nf(x_i),", "7368310f604c791a3f2707edc76f53ef": "X=X_1\\times\\cdots\\times X_n", "7368318dd3647eb6bbf6afaf6d26c48d": "\\nu ", "7368338e8a014176c57027442e757169": "a_{ijs} \\in \\mathrm{GF}(p)", "736876f942aa303b18422282bdad82d8": "(p,f)\\cdot g = (p\\cdot g, \\rho(g^{-1})f)\\, .", "73688a27f446bf04c7480fdfbcfd7cdb": " \\int_0^1 \\Bigl( \\sum |c_k h_k(x)|^2 \\Bigr)^{\\frac{1}{2}} \\, \\mathrm{d}x.", "7368d7deee3d0a5f58527fdae201ee55": "f(2j)=2f(j)-1\\;.", "7368e421a30a4ccd4ae4fa49ccdbb87b": " r = \\rho +\\theta g", "73692769fc3554c9af0b57422a7e805d": "\n1 - \\frac{\\lambda_\\min(\\mu_i^{(k)})}{\\lambda_\\max(\\mu_i^{(k)})} < \\varepsilon_C\n", "736935f5f33ae2607cc80dc83de2741d": "\\alpha(k) = \\operatorname{Cor}(z_{t+k} - P_{t,k}(z_{t+k}),\\, z_t - P_{t,k}(z_t)),\\text{ for }k\\geq 2,", "73693a81b2bb71ac474e4dbeacfa0c4a": "\\ (y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2.", "73695fa4f0ef2bce5e34189856ae20fa": "\\max\\{\\deg(a),\\deg(b),\\deg(c)\\} \\le \\deg(\\operatorname{rad}(abc))-1,", "736962cf3b5395f793a50871b268843b": "\\!(\\mathcal A, X, \\phi) \\in \\mathcal C", "736964c91d34feae03e1525dfaaba999": " \\phi_0 (z) ", "736a2b4dcc55c751c0903c67856b4117": "e(TS^n) = \\chi(TS^n)[S^n] = 2[S^n] \\not =0", "736a2daa6bf80fd1f798f7a7b728b94c": " \\gamma_2 \\,", "736a303d9ee31e3884a26fa741d39be9": "|z| < 1/(t+\\epsilon)", "736a8c170aceea22a540731dca4b1177": "| \\psi \\rangle = \\sum_{i} \\sqrt{p_i} |i \\rangle \\otimes | i' \\rangle.", "736ae9ac1176f3a636f0bb22bd41325a": "S_i \\geq 0", "736b3cf3a2f02b11edf2cac6e5c0cda2": "\\widehat{a}:\\Phi_A\\to{\\mathbb C}", "736b4f8d3bda52a5e8ed90e1cf083e81": "u_\\epsilon", "736b531b98306bcf4dda0d4067fe0a09": "0= \\Delta E_1 - \\Delta E_2 \\,", "736bdeb74f2a6990f06a5dcede3e3e0a": "g_{ab} \\equiv \\eta_{ab} = \\langle X_a, X_b\\rangle", "736c4eecd1b75c40fc1c5362779d4a85": "L_P^\\sigma=L_P\\cdot{(1-k)}", "736c5cefe3bcb16d8188f95454c6eb39": "\nL_\\Sigma = 10\\,\\cdot\\,{\\rm log}_{10} \\left(10^{\\frac{L_1}{10}} + 10^{\\frac{L_2}{10}} + \\cdots + 10^{\\frac{L_n}{10}} \\right)\\,{\\rm dB}\n", "736c7f4311c5533e5ce8c79bd41d2f2f": "I : \\mbox{STRUC}[\\sigma] \\to \\mbox{STRUC}[\\tau]", "736ca63829427cf2b514c7b261dd6c89": " \\delta W = \\mathbf{F}_1\\cdot\\mathbf{V}_1\\delta t+\\mathbf{F}_2\\cdot\\mathbf{V}_2\\delta t + \\ldots + \\mathbf{F}_n\\cdot\\mathbf{V}_n\\delta t", "736cacb26346a7071e699b1f826a0ae2": "\n\\ E_{net} = { \\ E_{in} - \\ E_{out} }\n", "736ceb9a7ff429d71ab26b6e09c6eaf8": "c^{1/m}", "736cfe559d8fef9be90206e55632cf5e": "t_0,\\dots,t_{n-1}", "736d08b02ea820071a42970a065a84f1": "\\beta_{k+1}=\\beta_{k}-\\lambda_{k}A_{k}\\frac{\\partial Q}{\\partial \\beta}(\\beta_{k}),", "736d4df96400620f2cf8c3e1c0c8089a": "\\mathbf{1}_{n \\otimes A}(x) = n \\times \\mathbf{1}_A(x). \\,", "736d5d99ac9589376284f9121c3c4e24": "(\\mathbf{T}\\oplus\\mathbf{T}^*)\\otimes\\mathbb{C}", "736dc283ec69f9139af2d41d93ca1518": "((x_1-\\bar{x})/s,(y_1-\\bar{y})/s)", "736de28075c07fe319697dacf24c38f7": "\\scriptstyle{F^-=F\\setminus \\lbrace \\langle x_0,\\emptyset \\rangle \\rbrace}", "736de485cec6ac666f52dcd236273f14": "\\frac{\\sin \\theta}{\\theta} < 1\\ \\ \\ \\mathrm{if}\\ \\ \\ \\theta \\ne 0\\,", "736eb91c9dea24e08a932cbace67f50f": " \\mathit BV = \\mathit P \\,", "736ef9820ebd7d4cb18ad1e50b1530b2": "g^{}", "736f56f1d9ab4e2dac285e05d0f727dc": "x \\sim \\mathcal{N}(\\mu, \\tau)", "7370133ebbe50fed9aa692ca2b7ae476": "\\mathrm{_8^{16}O} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{10}^{20}Ne} + \\gamma + Q", "73702c4d905afa719969704ab26a8415": "E(x, y; u) := \\sum_{n=0}^\\infty u^n \\, \\psi_n (x) \\, \\psi_n (y) = \\frac 1 {\\sqrt{\\pi (1 - u^2)}} \\, \\mathrm{exp} \\left( - \\frac{1 - u}{1 + u} \\, \\frac{(x + y)^2}{4} \\,-\\, \\frac{1 + u}{1 - u} \\, \\frac{(x - y)^2}{4}\\right),", "73706bc96b9bb80fa1ef8e7157d98445": "\\Sigma^T\\Sigma", "7370cdf0b44c9b3100ca5d28dac3bcd1": "(0,u)", "7370d7e98d36bc79c41110938633d86b": "\\,I_D = k((V_{GS}-V_{tn})V_{DS}-(V_{DS}/2)^2)", "7370de037029727c66245bc0714e59cc": "\\pi r^2 \\left (\\frac{4}{3}r + a \\right )", "7370e06d3956fc278c5b0d6c37b10787": "\n\\lambda(t|X) = \\lambda_0(t)\\exp(\\beta_1X_1 + \\cdots + \\beta_pX_p) = \\lambda_0(t)\\exp(\\beta^\\prime X).\n", "73712166116b70efef5cb1be2c0bfec4": "d_{\\lambda} = c_{\\lambda} q^{(\\lambda,\\nu)}", "73712da29b2707b2aa901496af353738": "(q)_n = \\frac{\\Gamma(q+n)}{\\Gamma(q)} = q\\,(q+1) \\cdots (q+n-1) ~.", "73713a06e0ca95217f4d8cf1f0d4179d": "\\alpha_1, ..., \\alpha_d", "737144453b9be2d19c158a0c5dff6491": "R_{\\alpha \\beta \\gamma}^{\\;\\;\\;\\;\\;\\; \\delta} = e_\\gamma^I R_{\\alpha \\beta I}^{\\;\\;\\;\\;\\;\\; J} e_J^\\delta , \\quad R_{\\alpha \\beta} = R_{\\alpha \\gamma I}^{\\;\\;\\;\\;\\;\\; J} e^I_\\beta e^\\gamma_J \\;\\; and \\;\\; R = R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} e_I^\\alpha e_J^\\beta", "737146eb19a7f316e7624b70dbdbfc43": "\\ P_{ij\\ldots}(\\cdot)", "73718672b5183c37c67f61ad88b54201": "dU =\\delta Q - \\delta W ", "7371d686ca15d3d172376c2d9ee7e509": "N,", "73723f147a6eb605351fc7f278dcecef": "B(z)", "737257da14ff5629bd39f244fd2e827a": "2\\pi \\chi.", "737270296a2a6bd91cfe90457f99bca9": "\\textstyle \\eta(\\varepsilon)", "7372804c2d2af6c9b6765c37690ff0c2": "\\text{Prim}\\,\\!", "7372846d880d9e3aa7f976d8c6e2450f": "\\hat{H} | \\Psi_i \\rangle = \\epsilon_i \\hat{S} | \\Psi_i \\rangle", "73729d04b8e6f0a5d694ba46a855be7e": "[M]", "7372dd5b25d586e14456f08d3534da35": "\\Omega(k^{2/3})", "7372df30c30eb59db2df908f529b81e1": " (\\mathbf{AB})^\\mathrm{-1} = \\mathbf{B}^\\mathrm{-1}\\mathbf{A}^\\mathrm{-1} ", "7372e95c8a4d8b868c972752c372fbf7": "\\sum_{x,y \\in C} d(x,y)", "7372fefe58fd5a67c6a38f4537910fc8": " g_m = \\frac{{2I_{DSS} }}{{\\left| {V_P } \\right|}}\\left( {1 - \\frac{{V_{GS} }}{{V_P }}} \\right)\n", "7373dd233cf93d889e28c9e532e1c939": "\\textstyle P(A\\mid [x]) \\geq \\alpha", "7374050c6241687beeb56a1c785194cb": " \n\\leq B + C + VE[P(\\alpha^*(t), \\omega(t))] + \\sum_{i=1}^KE[Q_i(t)Y_i(\\alpha^*(t), \\omega(t))] \n", "73742d95339d7449b93b237e8cd4b56e": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "7374756aff5fb1a26300327e3934f3f0": "{\\mathbf a} = a_1{\\mathbf e}_1 + a_2{\\mathbf e}_2 + a_3{\\mathbf e}_3", "737477a875c5cb320e2f01ea1e79c2a3": "1, 4, 9, 20, 46, 103, \\ldots", "7374cd002a45e827b67150ede49e3a31": "\\mathfrak{P}^{97}", "73751250fc3fdc0be3515b26a2d3201e": "\\ \\frac{S}{C} = \\frac{1024(\\log2)G\\sigma}{(4\\pi)^3R^2c\\tau\\eta}", "737554ca4511bf760d3882baf0c3b530": "\\displaystyle{\\|P_rf\\|^2=\\sum |a_n|^2 r^{2|n|},}", "737599c2fc4f35f195ab3d3adabff295": "g_{\\vec z+\\vec w}(X)=f(X)", "73760363a713accb37bc6ba9a271bd20": "\\|\\varphi\\| = \\sup_{\\|x\\|=1, x\\in H} |\\varphi(x)|.", "73762b92b72e586b7c6ef7ff7cb401f0": "r(s,o) \\subseteq R", "7376555afdd1804a4a9b8321d0458763": " \\mu_\\pm^{(0)} ", "7376699800b7884122a9b972213de37b": "d(1/z) = -\\frac 1{z^2} dz.", "737672155933144e15ca7223fb8ebcdb": " \\frac{S_n(t)}{S_0(t)} = S_n(0)\\exp\\left(\\int_0^t \\sum_{d=1}^D \\sigma_{n,d}(s)dW_d(s) + \\int_0^t \\left[b_n(s) - \\frac{1}{2}\\sum_{d=1}^D \\sigma^2_{n,d}(s)\\right]ds )\\right), \\quad \\forall 0\\leq t \\leq T, \\quad n = 1 \\ldots N. ", "7376d22f6af6b8330c3ef9401dbe7ec9": "(n-b,k-b)", "73779b3b39186a4ba730e38b780b8a52": "\\sum_{i \\ne n + 2}^{2n} 2^{i - 1}.", "7378050ab50b11ae40bda991103c1113": "(c \\mathbf{A})^\\mathrm{T} = c \\mathbf{A}^\\mathrm{T} \\,", "73781386b32eb266381878e6229cf868": "\\scriptstyle\\hat\\theta_{(1)}", "73783352385755b3ee36e766db56a8e9": "\\alpha(v_i)", "73783f7fa79186ee6da8b51b45b4ef6f": "\\text{EA}_\\text{X}", "7378648d38a91321b57043392ef90866": "S=\\begin{pmatrix}\n0 & 0 & \\dots & 0 & c_0 \\\\\n1 & 0 & \\dots & 0 & c_1 \\\\\n0 & 1 & \\dots & 0 & c_2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\dots & 1 & c_n\n\\end{pmatrix}.", "737868acb3bf737aa4187985a89626b3": "(h^2+k^2)^\\alpha", "73786e3567064497469ae990121c860d": "\\gamma _p^{} < 0 \\ ", "73795951d26e2cb6b88fd380cf5ac3a3": "aa^{-1},\\,a\\in\\,S^{\\pm}", "73797abe592798ba9fb3a917d11bd9f0": "\\frac {\\frac{ \\partial Y}{ \\partial t}}{Y} = \\alpha \\frac{ \\frac{ \\partial K}{ \\partial t} }{K(t)} + (1 - {\\alpha})\\frac{ \\frac{ \\partial L}{ \\partial t} } {L(t)} + (1 - {\\alpha})\\frac{ \\frac{ \\partial A}{ \\partial t} } {A(t)} ", "7379ea44876e1bc10c643e8c305a0334": " \\and (S_9 \\implies (\\operatorname{equate}[A_9, y] \\and V[F_9] = A_9) \\and K_9 = D[F_9] ", "737a76a1542c0dc475ec21972e3d77ef": "\\circ E", "737af797de47b77cd3b14d2607db52fe": " A(x_1, \\ldots, x_n) = \\frac{1}{n}(x_1 + \\cdots + x_n) ", "737b0013ef85624d9e6a86fdf77297b2": "\\hat{\\mathbf{S}}", "737b30d46a5484c9b9cb27f9c080ae60": "\\alpha= 2 \\cos^{-1} w = 2 \\sin^{-1} \\sqrt{x^2+y^2}", "737b80f1abe84a76ea54aa9595ade696": "\\bold{\\bar{3}}", "737c1e040c6862ed3ff0d9eab2c5a57f": "\\int_{-\\frac{a}{2}}^{\\frac{a}{2}} x^2\\cos^2 {\\frac{n\\pi x}{a}}\\;\\mathrm{d}x = \\frac{a^3(n^2\\pi^2-6)}{24n^2\\pi^2} \\qquad\\mbox{(for }n=1,3,5...\\mbox{)}\\,\\!", "737c30510d0130a89613b5a1c0c89c54": "E_N(\\rho \\otimes \\sigma) = E_N(\\rho) + E_N(\\sigma)", "737c6649997fa6fb9ea96652f047ba8c": "g_i(x) \\leq 0", "737c8fe6c62028a33b77f6510a9891db": " x\\ ", "737ca699e583ded399f95784fc24e26c": " H = H_A \\otimes H_B. ", "737d17b9e645167fd9efb0fc501c057d": " \\frac{\\partial C_2}{\\partial t}=\\frac{\\partial}{\\partial x} [D_2 \\frac{\\partial C_2}{\\partial x} -C_2 \\nu]", "737d557f0a5b95cb6f1b8e03da25ce1c": "\\ \\displaystyle \\ \\mathcal{U}(\\alpha^{*},\\tilde{u}) \\subseteq \\mathfrak{U}\\ ", "737d690b8a1c4d95a4834094b82a5d1c": "\\chi_i\\xrightarrow{n_{ij}}\\chi_j", "737dcc389932e09f75e54c90b742c7d0": "F \\in D(H(\\xi))", "737e6f3d78b94777f434f7e41859fb8c": "\\mathrm{tr}(\\boldsymbol{\\varepsilon}) = 0", "737e787bd7d8365618c97cce34092718": "f(1) = 1", "737ec6fb9f9111196d3b4c274f4b3fd5": "\\frac{dN}{dT} = B - D = bN - dN = (b - d)N = rN,", "737eeda7ad54524f2c344e7f7d24a466": "\\int \\sec^2 x \\, dx = \\tan x + C", "737efbab7808df60ce7b7b5303500fd0": " p_\\theta (x) = \\frac{ \\theta e^{-x} }{\\left(1 + e^{-x} \\right)^{\\theta + 1} } ", "737f6da9e402946c295401017d63ec74": "\\mu=\\exp{(\\mathbf{X}\\boldsymbol{\\beta})}\\,\\!", "737f897dd85af549983ab9ce31f1a2c1": "\\exp(-t)", "737fba89f57d77ddfa4b514c11a79ede": "f(t) = \\mathcal{L}^{-1} \\{F(s)\\}\n = \\lim_{k \\to \\infty} \\frac{(-1)^k}{k!} \\left( \\frac{k}{t} \\right) ^{k+1} F^{(k)} \\left( \\frac{k}{t} \\right)", "73801ccfb3c467c69db50788d8fc5840": "I_3 = 1", "73803aa989332cf9cfdc0461e1309ac5": "\\{v_i\\}_{1\\leq i \\leq k+1}", "73809ef1650383fb157f389d331fb501": "O_j = O_{1j} + O_{2j}", "7380b58d376f12a68a0938bb370e0db5": "t_A=G/v\\approx14.14\\mbox{ ns}", "7380c9b9f45531a7e2da3355a5cd7429": "\n\\sum_{i=0}^n (B_1^*)^i h_i = h_0+ B_1 ^* h_1 + (B_1^*)^2 h_2 + \\cdots + (B_1^*)^n h_n \\quad \\mbox{where} \\quad h_i \\in H.\n", "7380e78537890adfd0dfb94ea415e477": "f^n(x)=h^{-1}(h(x)+n)", "73810e78b000d8bf6fcdd5f07199645c": "x_3 \\le x \\le L", "73818c01cfe07eb94e374d95b1c1e11b": "\\kappa=\\frac{\\sqrt{(z''y'-y''z')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}", "7382367fcd68ae5ccb03b9dcfbb8e4fa": " a'_{\\ell\\ell} = a_{\\ell\\ell} + t a_{k \\ell} \\,\\! ", "73825aaad6bc5026d53490c44dbca989": "d \\approx 2{.}423 \\cdot R \\cdot \\sqrt[3]{ \\frac {\\rho_M} {\\rho_m} } \\,. ", "7382d24f345267777d1a35a72465b449": " \\frac { w_2 } { ( 1 - w_2 ) } = \\frac { s^2 - m } { m } ", "73830a872ca36440f7cf5b16b55d72b3": "\\bar{\\sigma} = f'_t", "73836943fccf9c214f0d5365452912f6": "\\chi_{T}(G)", "73836b7fa5611e264a02a39fb75723aa": "D_i: \\mathcal{C} \\to \\mathcal{C}", "7383ad94cc403d9f94ff9a675c4f2459": "p_{10}/(p_{11}+p_{10})", "7383eb47aef5e2073c9849a46d2baf4c": "\\mathrm{S\\scriptstyle ET}", "73847fa28423a8c8dd20951faedfca92": "j(\\alpha)<\\alpha", "7384b5cd8b5ea259a48c7379957ab576": "C_n\\,", "7384e1a6a1617d5c93432b5a883f7f74": "\\color{Black}\\tfrac{2}{m}", "738504a21945dd1dbfae658fbdbdef46": "u_{01}", "738511b7664777f6aeae6e8deb6ff526": "z_j", "738534fe7cfa717c5862acbd17335630": "\\delta p = \\frac{p}{2}", "73854bf9daa90a77842b2c4a2fd88e40": " \nH_E = \\frac{(l_1)^2}{2I_1}+\\frac{(l_2)^2}{2I_2}+\\frac{(l_3)^2}{2I_3},\n", "73855998debccf172285392e2163d4ac": "\\forall A\\subseteq X\\colon A\\in U", "7385a415df694114575d0884086365ed": "n R^2\\,\\sim\\,\\chi^2_p, \\, ", "7385b2daa7d6c2cafba8ab94775e9257": "|j_2m_2\\rangle", "7385d7cb269f32185e9c51d1ccbf67b8": "\\lambda_u", "7385e545ee889d872ba7387eddad319b": "\\sin c", "7386c93f98ba19df079677a2bcf5be5d": "f^{**} \\leq f", "7387331f249876938630d4fd97bfc9d5": "\\{1,\\,i,\\,j,\\,k,\\,-\\!1,\\,-\\!i,\\,-\\!j,\\,-\\!k\\}", "738774c94a141c84e779f155351c9336": "\\frac{\\pi}{4} = 2 \\arctan\\frac{1}{3} + \\arctan\\frac{1}{7}", "7387a7673acb2545123a697c7315c765": "\n \\mathrm{IND}(P) = \\left\\{(x,y) \\in \\mathbb{U}^2 \\mid \\forall a \\in P, a(x)=a(y)\\right\\}\n", "7387c57ddef692d18451f2a3463329d0": "G: C_1^\\mathrm{op}\\rightarrow C_2", "7387edcae850910b0f075ae2f8ba54e1": "\\mathcal F_3 ", "738836f8d1c36580730694e0b3865721": "\\frac{a+b+c}{b+c}+\\frac{a+b+c}{a+c}+\\frac{a+b+c}{a+b}-3\\geq\\frac{3}{2}.", "738839734ba54ee9379c7100eda9337f": " V_3 ", "7388b19b2f9a3680cd38be54da693723": "\\frac{dn}{dx}=\\frac{e}{kT}\\left (\\frac{j}{e\\mu}-nF\\right )", "7388c69d55f855a0b82bcee0961f6392": "(c_0, \\dots, c_{L-1}), y", "7388cd4c572d49d7c4126354b61aaf35": " n = p_1^{a_1}\\ldots p_k^{a_k} ", "73892b7fd2d93db1c70c7a04d408cddc": "\n A_{\\infty}=\\lim_{c \\rightarrow \\infty} A_c = \\lim_{c \\rightarrow \\infty}\\frac{1}{c}\\int_0^c A(t)\\,dt,\\quad c > 0.\n", "738940982fdb0e508b315d5eb607de05": " h_{ii} \\geq h_{ii}^2 \\implies h_{ii}\\leq 1 ", "73897d2809f9dc33b7f06b7c7750e460": "\\mathbf V", "738991f2ebd5fbbadba697b22f0e7c1c": "P= \\lim_{N\\rightarrow \\infty} \\sqrt{N} P_N", "7389a3d10e44a3c642e01ded4e17c2ca": "\n\\dot{\\mathbf{q}} [t] \\rightarrow \\dot{\\mathbf{q}}' [t'] = \\frac{d}{dt} \\phi [\\mathbf{q} [t], \\epsilon] = \\frac{\\partial \\phi}{\\partial \\mathbf{q}} [\\mathbf{q} [t' - \\epsilon T], \\epsilon] \\dot{\\mathbf{q}} [t' - \\epsilon T]\n.", "738a0bed23382faea5abd23c9375329d": "Y_0\\,", "738a192bbde97d80bd69fc30019990fa": " F_e = Q E = Q {\\lambda \\over 2 \\pi \\epsilon _0 R} = {Q q v^2 \\over 2 \\pi \\epsilon _0 c^2 R l} = {Q v I\\over 2 \\pi \\epsilon _0 c^2 R } ", "738a55b98a74e22823ff7cb18683ab16": "\\epsilon_c = \\epsilon_f = \\epsilon_m", "738a5c18c617b5b19e4eb7f0e80010ff": "k = 0.", "738ac2d04c7e49778caac08d7cd1e370": " N_\\text{ZC} = \\text{length of sequence.}\\,", "738ad79241b9e799c650fd7748b03e14": "\\frac{166047}{(1+0.10)^{12}}", "738b3f1ceab3ae8a0c1fcf60d7e5565c": "n=1, 2, \\dots,", "738bd2cd2db0c1ecea3922f13961a8a9": "\\dot B", "738c700a70afcb0cfc8c9555d6595fa0": "b\\in J", "738c7c77f09e3156ad132d76a32eaae8": "\n\\mathrm{var} \\left(\\widehat{\\theta}\\right)\n\\geq\n\\frac{1}{I(\\theta)}\n=\n\\frac{1}\n{\n -\\mathrm{E}\n \\left[\n \\frac{\\partial^2}{\\partial\\theta^2} \\log f(X;\\theta)\n \\right]\n}.\n", "738d33cc3afb7f9d3c23491ad52b80a2": "\\textstyle\\frac {6}{3-1}=3", "738dcbfbdd6a1a8ba83f99d5900beaed": "\\sigma_m=E\\epsilon_m", "738dcc1aec35d39ed156d7c0d289f23e": "n_i -1 ", "738e2c8fe287f66cf9915b9565603fc1": "Ob(\\mathbb{C})", "738e36228e20048e4bd50363f3c13b7f": "\\sum^{\\infty}_{k=0} \\frac{2^{2k} (k!)^2}{(k+1) (2k+1)!} z^{2k+2} = \\left(\\arcsin{z}\\right)^2, |z|\\le1 ", "738e4b2fa420bff62af872111a60303d": "\\ |y(t)| \\leq B \\quad \\forall t \\in \\mathbb{R}", "738e5db89d803fff905cbc356dd93dc9": "\\mathfrak{p}=\\oplus_{j \\geq 0} \\mathfrak{g}_j", "738edceca2d8a64a72cd3969cacdd7b6": "\\mu_i,\\sigma^2_i, i=1,2", "738edd964341a3b1dab5ab96f074f765": "\n\\omega_c =\n{ a_1 B \\over \\sqrt{4 \\pi} m_1 c}\n ", "738f334c6bde7907e483ee1b5dbc633f": "\\Rightarrow x = ae^{kt}\\, ", "738f96a4cb37df3457c637024d23a488": "b_2", "738fad65bc540965d21c63bf80e4bdc8": "I_1(\\lambda)", "738fd17ce84a874c854db48c9f8bd61e": "\\scriptstyle(\\Omega,\\,\\mathcal F,\\,\\mathbb P)", "73900e74268ceed8731db9aaee177c94": "2^{\\aleph_0} = \\aleph_1", "7390152c0e7167a1eff2aa88bc196cc3": "\\,l_z", "73902e8d66a5c2356ab74dc24d8116ec": "\nP \\mathbf{E}(\\mathbf{x},t) P^-1 = -\\mathbf{E}(\\mathbf{-x},t), ", "739041747c8a273fdecff0f843bf1a1e": "\\gcd{(S_i,p)} = 1", "7390523b0f0e4a710aa86c71daa41365": "\\tau^-\\,", "7390630de2672a693576e4e99ff88b68": "E=E_\\lambda=\\{x\\,:\\ |p(x)|\\leq\\lambda\\},\\ \\lambda>0", "7390e82d8f5d98b25c75dd9c4ed42d8c": "p_{cv}", "7390f510234a5d86e5425db154c43ee4": " \\alpha^{p^{m}-1}=1", "739103c556c541f204c111be63734f54": "X\\setminus S", "73910827855be6f5869d43a5faa8da0f": "v_{hull} \\approx 1.34 \\times \\sqrt{L_{WL}}", "73910c2dc9baed50ed59754b7ff643c7": "~g=\\beta+\\theta~", "739166dd514b6920c99479e58a721bea": " = \\nabla \\cdot \\mathbf{A}", "7391db74064e8c6712d1a7455b4c3e9d": "\\log 3.", "739221b3d4feb01ff8197a34256cf6e8": "D = 2 \\alpha h \\tanh (\\alpha h/2) \\,", "7392a3f4889e61e6a15ffaaf0da06238": "\\displaystyle 16T^2R^2=l^2m^2n^2+9V^2. ", "7392d03ea1e89639989b350cea2d7fc0": "\n \\kappa = \\begin{cases}\n 3 - 4~\\nu & \\rm{for~plane~strain} \\\\\n \\cfrac{3 - \\nu}{1 + \\nu} & \\rm{for~plane~stress} \\\\\n \\end{cases}\n ", "739321b76eaeb4ada6a711d1bde04584": "x_2=1", "73933ab20b4cb416a65f8388350c88bb": "T_{\\mu\\nu} = \\frac{\\partial \\bar{x}^\\kappa}{\\partial {x}^\\mu} \\bar{T}_{\\kappa\\lambda} \\frac{\\partial \\bar{x}^\\lambda}{\\partial {x}^\\nu} \\,,", "73937b3a0634bf69d8e66680ca83f7ba": "\\mathfrak{so}(3)\\times\\mathbb{R}", "73939d9ef74b21d4cf9566f9779771b2": "X = G/H", "7394324ec48e978df7264b85b4991c93": "\\bold{B}_\\perp", "739449703c66912e2869f44ff40b7bb4": "T_2=\\sum_{i,j}\\ \\frac{1}{2}m\\frac{\\partial \\mathbf{r}}{\\partial q_i}\\cdot \\frac{\\partial \\mathbf{r}}{\\partial q_j}\\dot{q}_i\\dot{q}_j\\,\\!,", "7394c70c02d68e46da26ad8f73485bba": "\\scriptstyle\\boldsymbol{\\Pi}^1_1", "7394d91116c2c38529b9f9ed696c4306": "10\\uparrow\\uparrow\\uparrow\\uparrow 4=(10 \\uparrow \\uparrow\\uparrow)^4 1", "7394e9b1400b880790ae42a0ee8d62e5": "U^{(0)}", "7394fe8a615828debb94d31c8d538d05": "\\mathbf{x} = [0, 0, \\ldots, 0]^{\\text{T}}", "739546a8691de8be996af482355dbed2": "W_{LC} = W_C + W_L, \\ ", "739603a0acd8e508aa963c16824da46b": "b^1", "7396403c545aa4482d3c5865c25d20dd": " \\|C_h f\\|_{H^p} \\le \\|(C_hf_i) (C_h f_o)\\|_{H^p} \\le \\|C_h f_o\\|_{H^p} \\le \\|C_h f_o^{p/2}\\|_{H^2}^{2/p} \\le \\|f\\|_{H^p}.", "73964eb927876ae4f0a1da3e8c717467": "i = 1,...,n", "73968da021f99091b48f6f1986c47377": " [AB, C]=A\\{B, C\\}-\\{A, C\\}B", "73971c21a8f43d84048eb908e2b07622": "R' \\approx_\\bar{x} R", "73972ea7f5d4ea592124713bb514d0bb": "{v_\\mathrm{RMS}} = {\\sqrt{3RT \\over {M}}}", "739746ee5cfb836346c00d270cd55f7d": "Q(\\mathbf{x},\\mathbf{y})", "7397d1395a772cd75865b9de363652f0": " \\left( z^{-1} \\right) \\ ", "73981006d58c19c5ac9e6b836638b173": "\\mathbf{X} = \\{x_1, \\dots, x_N\\}", "73981b1169dec5762da05dc2d57ee649": "(x_{i1} = x_i, x_{i2} = x_i^2, \\ldots, x_{ip} = x_i^p)", "739851988dfbcb42054bb2afdf0ffaf4": "\n\\begin{array}{c|c|c}\n\\hline & \\text{Tonsillectomy} & \\text{No tonsillectomy} \\\\\n\\hline\\text{Hodgkins} & 41 & 44 \\\\\n\\hline\\text{Control} & 33 & 52\n\\end{array}\n", "739896ecf679a38d719bd98783659472": "-\\frac{\\Delta E_i}{T} = \\ln\\left(\\frac{1}{p_\\text{i=on}} - 1\\right)", "7398f1ec3e75a3aef47702a94942ff6e": "g_{Sun}", "73992c98ac4d57a87f30b7ae1f93e316": "z_i=0", "7399483c497603262fd50db2f528c7d7": "H_{cr}/2K_u", "7399527504c08375ae1b58e624bd8782": " \\left\\langle H_{SO} \\right\\rangle = \\frac{E_n{}^2}{m_e c^2} \\left( n \\frac{j(j+1)-l(l+1)-\\frac{3}{4}}{l \\left( l+\\frac{1}{2}\\right) (l+1) } \\right)", "73998471435a9df0319d829c2d8cac2b": "b_{t+1}>\\eta", "7399bff7f6d4b233fa716aef6a0a7d4e": "\n\\begin{align}\np(0) & {} = m_0 + m_1(0) + m_2(0)^2 = m_0 \\\\\np(1) & {} = m_0 + m_1(1) + m_2(1)^2 = m_0 + m_1 + m_2 \\\\\np(-1) & {} = m_0 + m_1(-1) + m_2(-1)^2 = m_0 - m_1 + m_2 \\\\\np(-2) & {} = m_0 + m_1(-2) + m_2(-2)^2 = m_0 - 2m_1 + 4m_2 \\\\\np(\\infty) & {} = m_2\n\\end{align}\n", "7399f84f3c75715046d48a5b2dd86d22": "y:[x_0,x_1]\\to V", "739a375f72ddf9bca076f9694271d750": "\n\\frac{1}{r} = \\sqrt{\\left ( \\frac{\\mathrm d^2 x}{\\mathrm d s^2}\\right )^2 + \n \\left ( \\frac{\\mathrm d^2 y}{\\mathrm d s^2} \\right )^2}\n", "739a78d57f82e2b6399af0d6063e295c": "1/A(z)", "739b4dbadf9f5f58f8062b1c7389b8b3": "\\le40", "739b5242ea30c2f9d31ba844d03d5ae0": "4-d", "739b677de61f94384d21fd9f014da218": "\\mathbf{V}\\sigma^2", "739bc1edefcb71b84dfca84d09afc4df": "q(x) = x_1^2+ x_2^2 + x_3^2-x_4^2, ", "739bcb6002b4c30c7cfc371cec961391": "[(i-1)w , iw)", "739c00ed13e458132557fa0531ca26ef": "H(X) = E[I(X)] = E[-\\ln(P(X))].", "739c31c41ed0dc9237afac79e434581c": "\\tilde{H}_i(X)=0, \\quad \\forall i\\ge 0.", "739cc034c8efb7eb29b59dec8127d6be": "\\, n-1 \\, ", "739ce5dda2b7a0ca6af45d3d2876d617": "r(x) = \\sum_{i=1}^{n}r(x_{i})\\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}\\frac{x-x_{j}}{x_{i}-x_{j}}", "739ce7feab38615b92ffd60d2cff68d2": "(0, \\tfrac{1}{2})", "739d9c2303171a15f82e325b695a90e2": "{{({\\nabla^2 + {{\\partial \\over \\partial p}({{f_o^2 \\over \\sigma}{\\partial \\over \\partial p}})}}){\\chi}}=-{{f_o}{\\overrightarrow{V_g} \\cdot \\nabla}({{{1 \\over f_o}{\\nabla^2 \\Phi}}+f})}-{{\\partial \\over \\partial p}({{-}{f_o^2 \\over \\sigma}{\\overrightarrow{V_g} \\cdot \\nabla}({\\partial \\Phi \\over \\partial p})})}}", "739db1b5bcd7cffa1b46fb71b9c125d3": "d_0 = 1", "739dbb34d923bc2078b35c74ac1d608f": " \\frac{\\mathrm{mol}}{\\mathrm{L} \\cdot \\mathrm{atm}}", "739dc2c7a9af3a8394bb51a8af21e11b": "\nI_{yy} \\alpha_{y} - \\left( I_{zz} - I_{xx} \\right)\\omega_{z} \\omega_{x} = N_{y}\n", "739e31476c5df344061f4592f3cc15ce": "\\alpha=5, \\quad\np_{01}=p_{02}=0.5, \\quad p_{03}=0,\\quad ", "739e6d7611da60d59e849b31caadd9a7": "{\\Pr}_{h \\in H}[h(a)=b] = \\frac {1}{\\left\\vert B \\right\\vert} ", "739e8bda7928c56f75e2feab8de18a35": "A = \\begin{bmatrix}2 & -2 \\\\ -2 & 1\\end{bmatrix}", "739eb6ea403a6cab381bba0dfe2aaf0a": "g_\\alpha^* \\neq 0", "739eb7166db41de2482868fa4ed3bfcb": " v=v_x+iv_y", "739ec12ed2421c48e188fbbe8b884013": "\\int\\arccos(a\\,x)^n\\,dx=\n x\\arccos(a\\,x)^n\\,-\\,\n \\frac{n\\sqrt{1-a^2\\,x^2}\\arccos(a\\,x)^{n-1}}{a}\\,-\\,\n n\\,(n-1)\\int\\arccos(a\\,x)^{n-2}\\,dx", "739ee8487f6cca8fc2452347efe97176": "G(Y) \\oplus R", "739ef181ef38fce7319882d4a69fabde": "S = \\int d^dx dt \\left[ \\frac{1}{2}E^2 - \\frac{1}{4}B_{ij}B_{ij} - \\frac{m^2}{2}A^2 + \\frac{m^2}{2}\\phi^2\\right]", "739f08e64d4a42cf2ca59148f32fe6c4": " i, 0 \\leq i < d", "739f0a64f6e64b6a89e8ccbba76d170a": "{{\\beta }_{3}}", "739f1b4f38bfd9e8cde1d8cd011e32b1": "\\deg(f_ig_i) \\le 2d_s\\prod_{j=1}^{\\min(n,s)-1}d_j.", "739ff21fbc72c66b0d27afadc277315b": "t = (p-c) \\cdot v = (p_x - c_x)v_x + (p_y - c_y)v_y,", "73a0779ca31442946ee400ba323a7c25": "u_x = 0", "73a090a59dd93159443e3ca275222725": "\\lambda_0 - \\ ", "73a0c491dcebe3aed97a3342fb34d99d": "t_{ii}(x) = 1\\,", "73a120bc94225d9dea22d95facfdbc69": "\\left\\{{3'\\atop4'}\\right\\}", "73a1e935e68a3cd367360f7264127ce9": "VAS(x^3-x^2-2x+1,\\frac{x+2}{x+1}) ", "73a294d789fb0e9e66aa1e0055b35a7c": "\n\\frac{d^3 W}{d\\Omega d\\omega}=\\frac{e^2\\gamma^2N^2}{4\\pi\\varepsilon_0 c} L\\left ( N\\frac{\\Delta \\omega_n}{\\omega_\\text{res}(\\theta)} \\right ) F_n (K, \\theta, \\phi) \\qquad (15)", "73a2b55ba61502e73bba50f472cbaa68": "\n\\operatorname{CTF}(\\vec{s}) \\; = \\sqrt{1 - A^2 \\,} \\cdot \\sin{ \\left( \\gamma(\\vec{s}) \\right)} \\, + \\, A \\cdot \\cos{ \\left( \\gamma(\\vec{s}) \\right)} \\; = \\; \\sin{ \\left( \\gamma(\\vec{s}) + \\varphi \\right)}\n", "73a2ec887cc3398ce938e902bf33f7c5": "Z(E(\\mathbf{F}_p)) = \\frac{1 - a_pT + pT^2}{(1 - T)(1 - pT)}", "73a319acdb788b28973c8c5c1e9d0735": "S_L(2) = 71.4\\%", "73a32c4bb6066387b93b9a4a53bb7fb6": "p^2+x^2(ix)^\\epsilon", "73a36427d06cfedf7e6e4332dfbbc82f": "\\vec e_\\text{e}", "73a3d76adc96ed39f872181198fc46e7": "S = k_{\\mathrm{B}} \\ln \\Omega \\, ,", "73a4083ee923cf69310897bd8d9f8400": "D^k f: \\mathbb{R}^n \\to L^k(\\mathbb{R}^n \\times \\cdots \\times \\mathbb{R}^n, \\mathbb{R}^m)", "73a41cfb216d76a575241108f4342b25": "\\mathbf{v}_b=0", "73a42c538f4c609a7b5d06546828bdfd": " |p|= p_\\text{T} \\cosh{\\eta}", "73a44db09d5d8aa4e10794218b14a12e": "\\Omega \\!", "73a45a3fc2066f80f15f3e9fdf4135aa": "P(E=G \\bar D|C=c) = (0.01 + 0.16(c-11))(0.5 + 0.09(c-11))", "73a45c648d78e934a9ec3602072495b4": "A, B\\in\\mathbf{H}^+_n ", "73a472b5e8ed07ed31aaf55fe45696a4": "\\sum_{n=1}^{p} a_{n}^{+} + \\sum_{n=1}^{q} a_{n}^{-} = a_{\\sigma(1)} + \\cdots + a_{\\sigma(m_1)} + a_{\\sigma(m_1+1)} + \\cdots + a_{\\sigma(n_1)},", "73a4ad36d7a376ae1996e8ef55b9f2d1": "\\boldsymbol B\\ ,", "73a4b2449e9c9b8e9fa7f1178e13ba35": "\\overline{\\mu_{k,i+1}}", "73a4e362f52f85b14b1d0c02c143c273": " \\langle \\mathcal{M}, f , \\mathcal{T}\\rangle ", "73a4fb278a3b249cdbae22edc667b331": "y(t) = A \\cdot e^{-\\lambda t} \\cdot (\\cos(\\omega t + \\phi) + \\sin(\\omega t + \\phi))", "73a5014cad35e15a311933096548dbb8": "\\overline{A} = \\frac{\\sum_{g=1}^G A_g}{G} ", "73a56d00ad36a4d4e609d8884c36e6ed": "Z_i \\vee \\neg X_i \\vee \\neg Y_i", "73a570f8d45fe84f3859a35ab9b6398b": "[0, f(x)]", "73a5c2eaa4ce0bc070ff2c0a1950dd15": " \\frac{f(k)}{k} = A ", "73a6123aa1148172dc9e4de349aaa5b7": "(a\\,\\bmod\\,n)\\,\\bmod\\,n = a\\,\\bmod\\,n", "73a6567b13185332aed52de81fc96d37": "\\begin{array}{rl} \\min\\limits_{d} & f(x_k) + \\nabla f(x_k)^Td + \\tfrac{1}{2} d^T \\nabla_{xx}^2 \\mathcal{L}(x_k,\\lambda_k,\\sigma_k) d \\\\\n\\mathrm{s.t.} & b(x_k) + \\nabla b(x_k)^Td \\ge 0 \\\\\n & c(x_k) + \\nabla c(x_k)^T d = 0. \\end{array}", "73a666ea0e6d20d7a2baebd40701315e": "r=n/2", "73a68c3c09fc895e3e9784d1242bc5e5": "P(\\omega)", "73a6bbb8b40ede8b5a226bcf4e784f45": "(U \\downarrow X)", "73a6c9cf15eeb200e46a9a732dfe3e46": "Arf(H_k(M;\\mathbb{Z}_2);\\mu)", "73a74b4945d23618e9d8c6310d63eff3": " 0 \\to \\mathcal O_{\\mathbb P^{n}} \\to \\mathcal O (1)^{\\oplus (n+1)} \\to \\mathcal T_{\\mathbb P^n} \\to 0 ", "73a7766bf7581a53cc165e03f1e6032f": "\n=\\langle i_1i_2\\rangle -\\langle i_1\\rangle \\langle i_2\\rangle,\n", "73a78898e996c2957faccbbcc14b6b0a": "\\alpha = 1 - \\frac {D_{within~units~=~in~error}}{D_{within~and~between~units~=~in~total}}", "73a78e40ce39ce7d2c15741c0fd55caa": "\\scriptstyle{\\{\\emptyset\\}}", "73a7af2cd4c4f1d85b094e63870d757d": "b_1 = x_{11} a_1 + x_{12} a_2 + \\cdots + x_{1n} a_n ", "73a7f740fd6e4006ae617dc0cb94a1e7": " \\mathbf{L}_{i} =\n\\begin{bmatrix}\n 1 & & & & & & & 0 \\\\\n 0 & \\ddots & & & & & & \\\\\n 0 & \\ddots & 1 & & & & & \\\\\n 0 & \\ddots & 0 & 1 & & & & \\\\\n & & 0 & l_{i+1,i} & 1 & & & \\\\\n\\vdots & & 0 & l_{i+2,i} & 0 & \\ddots & & \\\\\n & & \\vdots & \\vdots & \\vdots & \\ddots & 1 & \\\\\n 0 & \\dots & 0 & l_{n,i} & 0 & \\dots & 0 & 1 \\\\\n\\end{bmatrix}.\n", "73a826e64c50328da54322ea49b31738": "c = \\Bigl[ 1 - \\tfrac16\\, \\left( \\kappa h \\right)^2 \\Bigr]\\, \\sqrt{g\\,h},", "73a84d9fd2b911469b4623a87a109364": "F(sx) =s F(x)", "73a85e33fae0a794dc33031b55f7ba6b": "G^{(n)}=\\{1\\}", "73a8a72f39ad23cf649ae34534a325ac": "\\tilde{f}(\\theta) = \\sum_{n=1}^\\infty \\left(a_n\\sin n\\theta - b_n\\cos n\\theta\\right).", "73a951c8399eea8e74d492c76b148efa": "\\sum_{p\\le x}\\frac1p=\\log\\log x+M+O(1/\\log x)", "73a970f5ea301fd76396f3ae2734572a": "\n\\ (1 - \\alpha z^{-K}) Y(z) = X(z) \\,\n", "73a97f0c1e38c1e5ce519e3158b4a413": "W =\\frac{\\gamma_r H^3}{K_D \\Delta^3\\cot\\theta}", "73a9bb48a25e667ff59bc39a85029342": "\\frac{1500}{1542} = 97.28%", "73a9c415878b8c5d04f03befaf9af010": "M=(1- (\\sinh 2\\beta E_1 \\sinh 2\\beta E_2)^{-2})^{\\frac{1}{8}}.", "73a9c510b2315b01b9056ac66d500a3a": " y \\rightarrow \\infty ", "73aa390d29261aa39211631ddd48352d": "Z=\\sum_{n=0}^{\\infty } \\left ( \\frac{4}{85} \\right )^n \\frac{(133n+8)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{6} \\right )_n \\left ( \\frac{5}{6} \\right )_n} {(n!)^3}\\!", "73aa494bcf2b458546a6896abb7bfab4": " r = \\frac{R}{R+G+B}", "73aa8bf92b3d0f50cf57274da9b2e751": "n > j \\geq 1", "73aa959c09496ea3c87db1e93e22ffcc": "\\tilde{\\rho}(\\rho)=\\frac{1}{2}(\\zeta_1\\wedge\\zeta_2\\wedge\\zeta_3-\\bar{\\zeta_1}\\wedge\\bar{\\zeta_2}\\wedge\\bar{\\zeta_3})", "73aaa137d9c1b31eba43be9fa72927d0": "\\psi(\\mathbf{x},\\tau)", "73ab122ebb19570bbbb6a80560cea4ce": "\\frac{F(z)^\\nu-z^\\nu} {\\nu}= \\sum_{n=1}^{\\infty} a_nz^{\\nu+n}", "73ab15c7da9287cac92799bba1eb15c5": "\\frac{\\lambda }{2 \\pi}", "73ab637825e67bbb54583efd31fc8de9": "4 a^2 x^2 + 4abx + 4ac=0", "73ab63a3e80b658db8b3c5409e9d22e9": "F^\\lambda := \\frac{DP^\\lambda }{d\\tau} = \\frac{dP^\\lambda }{d\\tau } + \\Gamma^\\lambda {}_{\\mu \\nu}U^\\mu P^\\nu ", "73ac31c21ec28c3b5f6124801af5a984": "\\sigma(x) = \\sigma(y)", "73ac674accb72e44f9932634e29398c3": "\\tan[\\theta]=\\mathfrak{Im}[f(x)]/\\mathfrak{Re}[f(x)]\\,", "73acbc57c0c9f2931693d153ab679d3e": "\\mathbf{d_2} \\cdot \\mathbf{q}", "73ace8b6098c3a4f4179d40386ce9757": "L_{[\\omega]}^i : H_{DR}^{n-i}(M) \\to H_{DR}^{n+i}(M).", "73acf3850a2e41b8305e6ebee8456d0d": "\\begin{align}\nw(c_1\\mid c_1+c_2) & = 2w(c_1) \\\\\n& \\geq 2w(C_1)\n\\end{align}\n", "73acf960ee663b89bc6e933ac0dcf67e": "\\scriptstyle A(S)", "73ad38b86bd9299e504e82d234d428b6": "\\mu(\\rm Mulliken) = -\\chi(\\rm Mulliken) = -(E_{\\rm i} + E_{\\rm ea}) /2 \\,", "73ad9b3c4fdc1cf21216ad6023a42ed4": "F_c = (n+mg)\\,", "73ae0e8055e9f0680878bc3a74fc0c47": "A \\to (B \\wedge \\neg B) \\vdash \\neg A", "73ae1b2b28220dddae6007699fe308ef": "p = \\hbar k", "73ae3c685cd4c07fa5f00589d1f2ba27": "\\delta G_r=\\left(\\frac{\\partial G}{\\partial \\xi }\\right)_{T,P}; \\delta G_r(Eq)=0", "73af50eac6c3cce228f18fc7d0235337": "\\;\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!|}}\\;\n\\dot{T}\\dot{F}\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!|}}\\;\n\\qquad \\text{or} \\qquad\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;\n\\dot{T}\\dot{F}\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;\n", "73af6e42a8fdac13df7ea9a2cb194970": "z_k (k = 0, 1, 2, \\dots, z_0 = z)", "73afc8ece59fd84af1177a2cc75c38f9": "B \\in M(m,r;\\mathbb{K})\\,\\!", "73b017ef6356d2c66b286c577ee14a49": "(y_n/x_n)X", "73b05d34b58994d35ee7fbc2add4a062": "o_W", "73b08c8fae69ec918614d97ed64372e4": " S_k=\\{-1,+1\\} ", "73b0cde1b5570d47b22ef3e647565f9f": " dW_t \\,", "73b0d663bff262574fa4a30ac80428f6": "~\\omega~", "73b25ea5be4a21d33aaeefad0e46a146": "V_q(\\mathbf{R})=\\frac{k}{|\\mathbf{R}|^3} \\sum_{i,j} Q_{ij}\\, n_i n_j\\ ,", "73b346891acedcc8ee8f91dee8098359": "\\Delta p_m = \\Delta p_{ref} - 1/R\\times\\Delta f", "73b353d5cd7205e52ba44440eab3262c": " \\frac{1}{2}v_a^2 = GM \\left( \\frac{r_p}{r_a(r_p+r_a)} \\right) ", "73b3678266a9735a0a35c069e92c9cde": " U = \\frac{1}{2}\\kappa\\theta^2", "73b391d1daef79f4247d479d28bf9456": "\\mathbf{N} = m\\mathbf{x} - \\mathbf{p}t = \\frac{E}{c^2}\\mathbf{x} - \\mathbf{p}t = \\gamma(\\mathbf{u})m_0(\\mathbf{x} - \\mathbf{u}t) ", "73b39eee11b31100be88a5917e8b4448": "\n\\hat{f}(\\mathbf{x}') = \\sum_{i=1}^n c_i k(\\mathbf{x}_i,\\mathbf{x}') = \\mathbf{k}^\\top \\mathbf{c},\n", "73b3b4fbe7914624b1d18c348d4b797d": "0\\rightarrow R^{\\binom{d}{d}} \\rightarrow\\cdots \\rightarrow\nR^{\\binom{d}{1}} \\rightarrow R \\rightarrow R/(r_1,\\ldots,r_d)\n\\rightarrow 0", "73b449b0da0895e9828a58855dadc400": "[(1+2)-3]-(4-5) = [3-3]-(-1) = 1. \\, ", "73b4527d578d8ffba256f7b5b87749cd": "\\hat n_{i}=\\hat a_{i}^\\dagger \\hat a_{i}", "73b4f561f940a0ff7f0cef1ac039b4a9": "\\tilde{f}_i : M \\to M", "73b5437942acf48dac148d13b3b81ac9": "4\\pi", "73b59a3b996e1ee5342515fb09c847b9": "\\alpha_{min}=f_1(d_m)", "73b5b5516f5fc1e21bebfbaad6c88a33": "C = \\{ (x,y) \\in K^2 | y^2 + h(x) y = f(x) \\} \\cup \\{ O \\} ", "73b5fba6b155b2e8289658e460d311ee": "\\frac {1}{c(w)} =\\frac {1}{c_r} |\\frac{w}{w_r}|^{-\\gamma} \\quad (1.5)", "73b6150c2d5820bb4f368acaa7f290c7": "x^{ 18 }+x^{ 11 }+1", "73b647c2b8bfbf7466bbadc92787678f": "\\mathfrak{B}(M_X)", "73b6d13c66cace9d430612914628a32a": "\n\\begin{array}{lcl}\n2 \\left [ (x_i^2 + y_i^2 - r^2) + (2 y_i + 1) \\right ] + (1 - 2 x_i) & > & 0 \\\\\n2 \\left [ RE(x_i,y_i) + YChange \\right ] + XChange & > & 0 \\\\\n\\end{array}\n", "73b731741de9397ecd842697ebaf81e6": "\\mathbf{D} = \\varepsilon\\mathbf{E}\\,,\\quad \\mathbf{H} = \\mu^{-1}\\mathbf{B} ", "73b789147c191d4a4d07d99321b9e3a1": "\\sin m\\pi \\equiv 0, \\quad \\, \\cos m\\pi \\equiv (-1)^m \\quad \\Leftrightarrow m=0,\\, \\pm 1,\\, \\pm 2,\\, \\pm 3,\\, \\cdots ", "73b7ab642b8661d37d7a123ecdefd912": "\n\\begin{align}\n& Y_k X_k^{j+\\nu} \\Lambda(X_k^{-1}) = 0. \\\\\n\\text{Hence } & Y_k X_k^{j+\\nu} + \\Lambda_1 Y_k X_k^{j+\\nu} X_k^{-1} + \\Lambda_2 Y_k X_k^{j+\\nu} X_k^{-2} + \\cdots + \\Lambda_{\\nu} Y_k X_k^{j+\\nu} X_k^{-\\nu} = 0, \\\\\n\\text{and so } & Y_k X_k^{j+\\nu} + \\Lambda_1 Y_k X_k^{j+\\nu-1} + \\Lambda_2 Y_k X_k^{j+\\nu -2} + \\cdots + \\Lambda_{\\nu} Y_k X_k^j = 0 \\\\\n\\end{align}\n", "73b7c54519858722494712ea1d19e87b": "c_n = a_n + i b_n.", "73b7d78810ef15d5b6d4a643fbfe27a7": "D_H = \\{\\cdot, H\\}", "73b7d8a26e7394e1d9f021a0603ecce6": "Z_f=m\\frac{dw_f}{dt}=m\\frac{dU}{dt}\\sin(\\theta-\\alpha)+mU\\frac{d(\\theta-\\alpha)}{dt}\\cos(\\theta-\\alpha)", "73b826c07cb0a312f2e138e5cedb170d": "u \\cdot \\hat{n} < 0", "73b956c5c2d489c31e7af9339c648bad": "\\alpha_c = \\frac{1-\\sqrt{1-4c}}{2}", "73b9a43fb4fc99ef9c26bddefa0b2511": "(v^i,v^i_j)", "73b9e89e57bd825e73af04962530de55": "S_1, \\dots, S_n", "73b9f09c6e545109679341e81fd2894f": "\\vec{B}[\\vec{x},t] = \\nabla \\times \\vec{A} [\\vec{x},t] ", "73ba1eb9da7919a9b71a63e6a25e6ed8": "{R^\\prime}(r) = a", "73ba45386f3919de7ca5eecc8703e578": "|\\lambda-\\mu||\\lambda|^{-1}", "73ba509c7aec9f5d9635227408396fde": "a_1 = \\frac{1}{4}", "73ba605cab2a8291fc1c6249386f2061": "E(1) = 1 \\,\\!", "73ba8c076a1b833f34a06e0445b89c9a": " r = \\cos(2\\theta)", "73ba9fa8f27405bbd6257a718c1ad0c1": " f\\prec g\\iff f\\in o(g); ", "73baa54622b0f1976344ce47c936d492": "\\Delta(t) = \\begin{cases}\n\\frac{n+1}{2}t - n + \\frac{n+1}{2}t^{-1} & \\text{if }n\\text{ is odd} \\\\\n-\\frac{n}{2}t + (n+1) - \\frac{n}{2}t^{-1} & \\text{if }n\\text{ is even,} \\\\\n\\end{cases}", "73bac96b5487a46acac64264d2cc5f18": "\\int_{-1}^{1} \\frac{P_\\ell ^{m} P_\\ell ^{n}}{1-x^2}dx = \\begin{cases} 0 & \\mbox{if } m\\neq n \\\\ \\frac{(\\ell+m)!}{m(\\ell-m)!} & \\mbox{if } m=n\\neq0 \\\\ \\infty & \\mbox{if } m=n=0\\end{cases}", "73bada245be81479f66641cafa291ee6": "x=\\xi(\\xi_0,t):", "73bae15e254427bbddc9fc242d19f853": "4/m\\bar{3} 2/m", "73bb3690ce5df05f48d7b87300b798a6": " a_{31} ", "73bb3a61f5542c94ea522669a31a88b7": "\n\\phi(x)\\equiv(x;q)_\\infty=\\prod_{n=0}^\\infty (1-xq^n),\\quad |q|<1\n", "73bb4319792d7249dc1aaaefc0a39332": "\n\\exp \\left (-{a } (q^2+p^2)\\right ) ~ \\star ~ \n\\exp \\left (-{b} (q^2+p^2)\\right ) = {1\\over 1+\\hbar^2 ab} \n\\exp \\left (-{a+b\\over 1+\\hbar^2 ab} (q^2+p^2)\\right ) ,\n", "73bb5803a7d8faf64e76fced5900567a": " r B_\\theta ", "73bb6733da3df3c0a7b63d9aea13c055": "\\beta\\ \\stackrel{\\mathrm{def}}{=}\\ 1/\\left(k_\\mathrm{B}T\\right).", "73bba1b83a02083e098717c94026911c": "\\Omega_1,\\Omega_2,\\ldots,\\Omega_\\omega", "73bba935939db4db898e0c42d5552daf": "W_{mn}(x,y,t) = A_{mn}\\sin\\left(\\frac{m\\pi x}{L_1}\\right)\\sin\\left(\\frac{n\\pi y}{L_2}\\right)\\cos\\left(\\omega_{mn}t\\right)", "73bbb17b9c3879bb944b07c8e2c4dadb": " \nE + S \\, \\overset{k_1}\\underset{k_{-1}} \\rightleftharpoons \\, ES \\, \\overset{k_2} {\\longrightarrow} \\, E + P\n", "73bbe012edfb61eca43444d61fefe937": "s=1", "73bc5502622fe2a26d7f848e8a5bc6e8": "\\begin{align}\n P &= 0.5\\rho AU_2(U_1^2 - U_4^2)\\\\\n C_p &\\equiv \\frac{P}{0.5\\rho AU_1^3}\n\\end{align}", "73bc6714fdf33f70bb371adb0eac45f9": "\\varepsilon\\sum_{\\beta = 1}^{3} \\gamma_{00|\\beta|\\beta} = -\\kappa \\, \\rho_0", "73bc89d19b73baeb91627028f051c49e": "y'(t) = L y(t) + \\mathcal{N}( y(t) ), \\qquad y(t_0)=y_0, \\qquad\\qquad (1)", "73bc8da42aaca89ccd3b6d353ae9f9c5": "\\varphi: G \\rightarrow G^{\\operatorname{ab}}", "73bc9851270421c3a7e7dd37621d0dda": " B ", "73bcbbb9c6641f7e59220d9530c67f3b": " \\{O_i,O_j\\}=\\{O_i,E_j\\}=\\{E_i,E_j\\}=0 ", "73bcdb5db8ef88044683c584c7531939": "\\textstyle \\mathrm{length}(y)", "73bd36e6ba25d37d8941241235d2703c": "T_q", "73bd3c6b8820c14ec85763dcf8427cf0": "\\Delta (-1)^F=(-1)^F \\otimes (-1)^F", "73bd68432a02c7cc3d58bbb51f7b7873": "\\exp(\\beta Q_{ij})", "73be36e8c4b8323351caf40783d4d095": "\\frac{1}{k!} \\frac{d^k}{d x^k} x^\\alpha L_n^{(\\alpha)} (x)\n\n= {n+\\alpha \\choose k} x^{\\alpha-k} L_n^{(\\alpha-k)}(x),", "73be77b738b5bcaf8fdf904f3de2c79c": "\\alpha < \\beta \\Rightarrow \\alpha+\\gamma \\le \\beta+\\gamma", "73be89b662e05829515dcf913a9b8e33": " \\quad 0", "73bef4b3dec43d3c115fe220325fabbd": "-45^{\\circ}", "73bf1de655134bb0c504d7f850f194b1": "{\\widehat{AB}}_3", "73bf510bfbd59acacc8b44e2a36917c6": "(1,2)_{-\\frac{1}{2}}", "73bf9715f704892119b98aeed586dbe7": " \\langle x^{2n} \\rangle = (2n-1)\\cdot(2n-3).... \\cdot5 \\cdot 3 \\cdot 1 (\\langle x^2\\rangle)^n ", "73bf98450e020205edb756d85e81a181": "\\|Y-X\\beta\\|_1", "73bfd65fdf8bbb90f66607ab730fafa0": "\\scriptstyle L \\;=\\; I \\omega_s", "73bfdae58f613018d79544440657bc60": "u=0.5", "73c0300fdedae27c4963ca485e7a2247": "\\Im\\{W\\}", "73c031d68efa3d2d99e100d9b1a53636": "A x = b , \\,", "73c06e1d8c654ced7a2b76af6036cb76": "\\forall\\alpha.\\alpha\\rightarrow\\forall\\alpha.\\alpha", "73c192115d2e0e11a8e6e65f07b30865": "\\frac{\\partial{H_z}}{\\partial{y}} = C'\\frac{\\partial{E_x}}{\\partial{t}}", "73c1bccebafda891fe33434335077e4e": "\\Gamma(t;\\gamma=1/\\nu,\\lambda=1/\\nu)", "73c1d870764fa062184fbc63d9bbc106": "\nE\\left[X_1^{r_1}X_2^{r_2}\\cdots X_k^{r_k}\\right]=\n\\prod_{j=1}^k\n\\frac{\n \\Gamma\\left(\\alpha_j+\\beta_j\\right)\n \\Gamma\\left(\\alpha_j+r_j\\right)\n \\Gamma\\left(\\beta_j+\\delta_j\\right)\n}{\n \\Gamma\\left(\\alpha_j\\right)\n \\Gamma\\left(\\beta_j\\right)\n \\Gamma\\left(\\alpha_j+\\beta_j+r_j+\\delta_j\\right)\n}\n", "73c214261e558740a94aff3da43f1838": "\\begin{pmatrix}\nA_x \\\\\nA_y \\\\\nA_z\n\\end{pmatrix} = (\\mathbf{U}^\\mathrm{*})^{-1} \\begin{pmatrix}\nA_+ \\\\\nA_{-} \\\\\nA_0\n\\end{pmatrix} \\,,\\quad (\\mathbf{U}^\\mathrm{*})^{-1} = \\begin{pmatrix}\n- \\frac{1}{\\sqrt{2}} & + \\frac{1}{\\sqrt{2}} & 0 \\\\\n- \\frac{i}{\\sqrt{2}} & - \\frac{i}{\\sqrt{2}} & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\\,.\n", "73c214e944a909e7fba26c36a5f1e620": " X \\stackrel{\\mathcal D}{=} Y", "73c22a61edb0a076c1d327371354ce0e": "Z'\\,", "73c22c44e214dec505f089a00b0a0bb6": "\\frac{a}{b} = a \\times \\frac{1}{b}", "73c24a203070c2507f9a27c7a11a31ec": " r \\leq t ", "73c28eee64dfbdcc708b4bb2cc8db990": "10^{600}", "73c2a2dba4bce2ab2166e9375a125ec6": "k_{\\lambda} k_{\\mu} = k_{\\lambda+\\mu}", "73c2cb309596224325aca42f3800143e": "\\nabla \\mathcal{F} \\in L^2(\\partial \\Omega_0)", "73c2e603df3a18a4003955c072d14ef3": " \\phi > 0", "73c33568dae826242c048bc3b551719d": "\\frac{\\partial a}{\\partial \\mathbf{X}} =", "73c33666171c2ae8da519903d83c69f7": "\n\\frac{\\partial}{\\partial t}\\begin{bmatrix}\\text{Re}\\,u\\\\ \\text{Im} \\,u\\end{bmatrix}=\nA\n\\begin{bmatrix}\\text{Re}\\,u\\\\ \\text{Im} \\,u\\end{bmatrix},\n", "73c33e61b27e9ec5ae14ce87627fdbc6": "\\scriptstyle V \\, \\overset{\\sim}{\\to} \\, V^{**}", "73c35f00269fbd55b359867a90064500": "\\textstyle \\Omega", "73c3849e15587a9912edf5ecee3b5409": "<\\epsilon/3", "73c3c0a3b980ac6e2e07a1606f5bdffa": "\\mathit{SS}_\\text{total} = \\mathit{SS}_\\text{regression} + \\mathit{SS}_\\text{residual}.", "73c3edf7c5f081fe54de26faf970b18d": "(\\Delta x)^2", "73c3f3f73dd97453f629bd8bc8b44b4d": "U = \\sum_i w_i(Y_i-y_i)^2 ", "73c43d9a3832a764832af809e037cc4c": "\\sum_{k=0}^{n}A_{k}B_{n-k}.", "73c4562aadf386977bb4bae5fa867c3a": "\\sigma A + \\sigma B \\ge 1", "73c46c47d80949f66df47b459a8e9e8d": "\\text{Aortic Valve Area (cm}^2\\text{)} = {\\text{LVOT diameter}^2 \\cdot 0.78540 \\cdot \\text{LVOT VTI}\\over \\text{Aortic Valve VTI}}", "73c4aa7303f2d496846556c07b50cdc5": "m_1 m_2", "73c4bfd7bddb16838a85471e4c9a4787": "\\mathrm{dim}_\\mathbf{Q}(\\Gamma \\otimes_\\mathbf{Z} \\mathbf{Q})", "73c4ea1485c7ebcdd13f1b2eacbe5f53": "\\hbar- \\ ", "73c50c67a2255356b6ca08755cb6c455": " I_\\mathrm{min} ", "73c52abe970b02794ea8257fd5a27613": "\\R^m", "73c5472312534bedbd125e5c1f10d64a": "\np_{HB} = (p_{HB}+p_{LB})(p_{HA}+p_{HB}).\n", "73c5667a7f85de9c88b260213fb3963b": "\\begin{align}\n\\boldsymbol{P}_i&=\\boldsymbol{V}_{i}\\boldsymbol{U}_i^{-1}\\text{,}\\\\\n\\boldsymbol{z}_i&=\\boldsymbol{L}_i^{-1}(\\lVert\\boldsymbol{r}_0\\rVert_2\\boldsymbol{e}_1)\\text{.}\n\\end{align}", "73c573cdf9be0ce28947c6e6c10460b4": "\\Delta = \\frac{1}{1+m+n} \\left [ 1+m \\cos \\alpha + \\frac{n}{2}(3 \\cos ^2 \\alpha -1) \\right ]", "73c5b3dcc6cf2663c045eb8d8488e12c": "\\int_0^\\infty \\frac {x}{e^{x}-1}\\ dx=\\zeta (2)= \\frac {\\pi^2}{6}", "73c5cc9e20eb68805db488bf4793f51b": "\\frac{b_n}{a_n}", "73c6c5d71263b61112c8c58328bfab78": "\\sqrt{4-2\\sqrt{2}\\ }", "73c6cd59c43f406213a04a08ea4f9e7f": "\\ (x_1,y_1)\\parallel_- (x_2,y_2) \\ ", "73c6e5986cd1a72499819d755dbf2b04": "r = 1.1019 t^2.\\,", "73c7070fbc887c38daea22738774c01c": "x\\neq y", "73c74cb242332ab8159f6377ee6c1ab6": "\\varphi(k)", "73c792853330bedc4aa706c6965fe238": "(b_P + b_T) = b_{PT} + 2(c - 2) + 2.\\,", "73c7bce118f44fc623fc17277b369994": "{q}_{1}={q}_{2}+{q}_{3}+G", "73c7cda7fb40afe01731e9282563536b": "A = B^* B", "73c7e851ca2215338e4e9a237b6a5521": "\\begin{matrix} {13 \\choose 1}{12 \\choose 1} \\end{matrix}", "73c7ee43ef5fdb6aa52a8a749c9cb564": "\\begin{align}D^{k+1}(e^{ax}y)&\\equiv\\frac{d}{dx}\\{e^{ax}(D+a)^ky\\}\\\\\n&{}=e^{ax}\\frac{d}{dx}\\{(D+a)^k y\\}+ae^{ax}\\{(D+a)^ky\\}\\\\\n&{}=e^{ax}\\left\\{\\left(\\frac{d}{dx}+a\\right)(D+a)^ky\\right\\}\\\\\n&{}=e^{ax}(D+a)^{k+1}y.\\end{align}", "73c83cb0a1287c9daa36454ce839e334": "\\eta(\\xi) = \\eta_1\\; \\cos^2\\, \\psi(\\xi) + \\eta_2\\, \\sin^2\\, \\psi(\\xi),", "73c840aa89e9af7be6537fce0619429d": "\\frac {10 \\cdot 4^5 - 40} {2{,}598{,}960} = \\frac {10{,}200} {2{,}598{,}960} \\approx 0.39\\% ", "73c864d56797332023f78902cdf4f7a4": "\\Delta\\sigma=\\sqrt{\\Delta x^2 + \\Delta y^2 + \\Delta z^2 - c^2 \\Delta t^2}", "73c88cc79862e46f2748d749e62058e6": " \\operatorname{E} \\operatorname{tr} e^{\\mathbf{H}+\\mathbf{X}} \n\\leq \\operatorname{tr} e^{\\mathbf{H} + \\log( \\operatorname{E} e^{\\mathbf{X} })} ", "73c8b48107c42752f0d923932d0e1298": "Y^{\\prime}(t) = \\alpha \\left(1 - \\left(\\frac{Y}{K} \\right)^{\\nu} \\right)Y ", "73c8ba9a8cf6a60d27a1dff1e97c95f1": "p(x|\\theta) = f(x-\\theta)", "73c8bc2dcba6eef703822ef1d7807e88": "\\, x_{2}", "73c8c3f027796198fdfcd02f2e26a252": "B(x)", "73c8e2f86a68cd4af7ead39209188a64": "\\zeta_3=\\beta_3-1", "73c8f63c0e4b9868bb5b6bb620c10137": "\\mathit{MS}", "73c9130cc3f0065c5a9ae240d52d85a9": " AdS_n ", "73c93824118f9d6d6f44b4d3a6c98fd9": "F_r(\\mathbf{a}_{0,k})=\n\\begin{cases}\na_0, &k=1\\\\\nf(a_0, F_r(\\mathbf{a}_{1,k})), &k>1\n\\end{cases}.", "73c952f5230ac988e66e825e834ee537": "COP = \\frac{ Q}{ W}", "73c9c9671dfc278acf5cf830d3753126": "\n\\langle\\psi| ((|x\\rangle + |y\\rangle)\\otimes(|x\\rangle + |y\\rangle))\n\\,", "73ca1dbf1d2dc6288928989677283b03": "\\ K \\,", "73ca5b658c58ad701849fb803a9ddd0b": "(a + b)^{(n)} = \\sum_{{j=0}}^n {n \\choose j} (a)^{(n-j)}(b)^{(j)}", "73ca988e9aa081b0da776f8ea225eee1": "10\\uparrow\\uparrow\\uparrow n=(10\\uparrow\\uparrow)^{n-2}(10\\uparrow)^{10}1<10\\uparrow\\uparrow\\uparrow (n+1)", "73cabfc8d9a6701f3aedc2d7d445c793": "\\Box = \\left(\\frac{\\partial^2}{\\partial t^2}\\right)-\\nabla^2", "73cae05e3e673366fc2bf8c142be6fbc": "f(x, x^+) > f(x, x^-)", "73cb062506eaf1b9fac307638792b181": "Y \\prec X", "73cb3d9517a93273f95b891cff3493fb": "{v \\choose k}", "73cb5c1a39139d4c65f44de0c7cc31ad": "C_{GD}=\\frac{C_{oxD}\\times C_{GDj}\\left(V_{GD}\\right)}{C_{oxD}+ C_{GDj}\\left(V_{GD}\\right)}", "73cbd3d741b19dc7f1185582e4b784b5": " \\R^3 ", "73cc12f7e6392ac024cecb356e30eeca": "\\varphi_{\\ell r}(\\mathbf{r})", "73ccc7c9edd822049f244a678095de0b": "\n w^- = \\operatorname{max}\\left\\{0, \\frac { 2n\\hat p + z^2 - [z \\sqrt{z^2 - \\frac{1}{n} + 4n\\hat p(1 -\\hat p)+(4\\hat p - 2)}+1] }\n { 2(n+z^2) }\\right\\}\n", "73ccc9b03a7126efc23bb1ea3af7a0b6": "= d(V^{1}u^{\\alpha}) - V^{1}u_{i}^{\\alpha}dx^{i} - u_{i}^{\\alpha}d(V^{1}x^{i}) \\,", "73ccfb151ed860b8f095d9ed1bade27a": "\\langle\\mathbf{v},\\mathbf{w}\\rangle = \\mathbf{v}^* \\mathbf{w}", "73cd2752a1e97cdde3714d8f397fa7b8": "=\\frac{d}{dt}\\left(\\frac{dy}{dx}\\right)\\cdot\\frac{dt}{dx}", "73cd3b462f75706872211da76588acc2": "c_{d,\\epsilon}", "73cd3f2025f1bdec3a12b4994e7754f0": " \\alpha N_{vs} << 1", "73cd6def7906561ee1ccc251c9f43639": "Q(N,V,T)=\\frac{V^{N}}{\\Lambda^{dN}N!}\\int_{0}^{1}\\ldots\\int_{0}^{1}ds^{N}\\exp[-\\beta U(s^{N};L)]", "73cd831dac5b53e38ca4146c920b51d9": " Z = \\sqrt{n}\\left(\\frac{1}{n}\\sum_{i=1}^n X_i\\right) ", "73cdcdb165b17159f0c74ddf64a7fa72": " \\deg(\\textbf{N}(s)) = 4 \\leq \\deg(\\textbf{D}(s)) = 4 ", "73cef503ec1fb2bf9f9084ed960b05b9": " K_\\mu = -i(2x_\\mu x^\\nu\\partial_\\nu - x^2\\partial_\\mu) \\,. ", "73cf479a9491fac63081bdb850382a39": "~L~", "73cf6f80607019438014f2b40ada121f": "X_t = X_t + x_t^{[i]}", "73cfc3d8844212426638947b738c7b79": "\\psi(\\psi(0))", "73cfd211fc516cda5d4754d447adc276": "\n\\begin{align}\n T(s,\\mathbf{x})&=\\frac{1.74C_2s}{6.6C_1 C_2 s^2+0.66C_2 s+0.33} \\\\\n \\mathbf{x}&=[C_1~C_2]\n\\end{align}\n", "73cff4637f11a2c6c5ef3eb0186dd3a3": " Z_N(K^*,L^*) = 2 e^{N(K^*+L^*)} \\sum_{ P \\subset \\Lambda_D} (e^{-2L^*})^r(e^{-2K^*})^s ", "73cff527cf4130ce340dfb068b568cd8": "t \\sigma", "73d0462f9f0e893e6367e5ddbab46ac7": "k_{\\text{inh}}", "73d0fdaa79e3486b356ce3be8183ee90": "\\tilde{\\psi}_{jk} = \\psi^{jk}", "73d123b1626656f675a1bda3dd6d485b": " V_n^{(a)}(x;q) = (-a)^nq^{-n(n-1)/2}{}_2\\phi_0(q^{-n}, x;;q,q^n/a)", "73d13ceb7574c84b19677b0308d3a971": "\\textbf{P}_{0\\mid 0} = \\begin{bmatrix} L & 0 \\\\ 0 & L \\end{bmatrix} ", "73d1d5d850d732a2a5deebed9befc93a": "{\\mathcal I}_\\eta", "73d251021249c7df4ffddf84ce7d56ba": "\\sum a(n) q^n, \\qquad q = \\exp(2 \\pi i z)", "73d2cb0dedb445be45467caa42cd3f29": "\\Phi_\\Lambda: S^\\Lambda \\longrightarrow \\R\\cup\\{\\infty\\},", "73d338f94400863c7eaf8bb62ceb0fe9": "X \\sim \\textrm{Levy}(0,c)\\,", "73d35a3bc5802a0798327bbcca578c66": "f(x) \\equiv 0 \\pmod{p^k}", "73d43cd7e9521d5cf6d58a75094c8064": "\\exp(\\pm i k_z z)", "73d46ea7769f7f3507f7f3533b7385c8": "p(x|\\mu,\\sigma^2) \\propto e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}", "73d4ae2308c2e29370264f465d812d08": "caa = c,\tcab = a,\tcac = b,\tcba = b,\tcbb = c,\tcbc = a,\tcca = a,\tccb = b,\tccc = c.", "73d4cbbdab0d0e7de3ebe470ca11f205": "\n\\Bigl[1-\\binom{n}{2}^{-1}\\Bigr]^T\n \\leq \\frac{1}{e^{\\ln n}} = \\frac{1}{n}\\,.\n", "73d5195170bebb9ef8d44a6887e5d9f2": "\\{0, (\\epsilon,g);\\}", "73d5419eb6432ee6f5e12b6e4c14640d": "\\mathbf{n} = (a,b,c)", "73d567438e82240ca11cecafec04d3b8": "\\forall x \\in I, \\forall r \\in R: \\quad x \\cdot r \\in I.", "73d59165404019dabfec697f47842471": "\\bar{G}_i \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial G}{\\partial n_i}", "73d59f63648f0a629c7e6e10c60537ba": " (n \\ge 0)\\,\\!", "73d6253f5bf4d87d99d0ca118d5f5b35": "\\nabla f \\cdot \\nabla g = 0", "73d636c66e7e7fa4c4ae3438fd02a25c": "B_{t}=B_{t-1}+1", "73d6708ef88827c963743371823b6c2a": "\\Phi(\\varphi)", "73d685adcd10596b3847259b5f704ee0": "\\mathbf{s_2}=\\mathbf{H_2}\\mathbf{y}", "73d71c54f44202da29c1d6bd536aac18": "{v} \\,\\!", "73d72a75920060e6e4bcaa3a7ae170f7": "n^{a_1}", "73d78945f28bbca84328ba531a1918ed": "\n\\lim_{|x| \\to \\infty} x \\sin \\frac{1}{x}.\n", "73d78c06ba8a3df0c259f8bf72118bc0": "E=pc\\,", "73d7b1451c75cace6e327a3ad503cfc6": "\\omega_r(\\phi,\\lambda) = \\left|\\left| {\\partial (x,y,z) \\over \\partial \\phi} \\times {\\partial (x,y,z) \\over \\partial \\lambda} \\right|\\right| = r^2 \\cos \\phi \\ .", "73d7c2c4b5ae43de2313c26796f9122d": "\\sigma(\\Gamma)\\vdash\\sigma(A),", "73d7f8a7d7a0cbfa390e983aab9d52b4": " P(Z \\ge k) \\le \\frac{ 1 } { 1 + k^2 }.", "73d83d55083b025023c146ecf9c01d63": "\\omega = \\omega'", "73d856fa9df0bde949a4d3e405202571": "a^{\\mu}", "73d89de3405cd8aa6d6722bac0793fc4": "\\sigma^{2}", "73d8b5c2c4dc9b35d123c65784847da6": " \\tilde{f}: M \\otimes_R N \\to Z", "73d92dc09ea677a95319f74a86aa187e": "\n\\mathcal{A} \\equiv \\frac{1}{N!} \\sum_{P \\in S_N} (-1)^\\pi \\hat{P} .\n", "73d9360fae861f79d99f8d4294942151": "(\\forall n\\in\\mathbb{Z}_+):A_n(x)=\\frac{x^{2n+1}}{(2n+1)!!}-\\frac{x^{2n+3}}{2\\times(2n+3)!!}+\\frac{x^{2n+5}}{2\\times4\\times(2n+5)!!}\\mp\\cdots", "73d93ccd118ba83bc2c636c1fe5bfdb8": "\\mathrm P(X_1=x_1, \\ldots, X_n=x_n) = \\prod_{v=1}^n \\mathrm P (X_v=x_v \\mid X_j=x_j ", "73d950ac974f08def985f147e5c6e0a1": "O(2^{0.386n})", "73d9bc2526e46dea251e3d8c6d513002": "\\mathcal{M}(k; p_1 \\cdots p_n; q_1 \\cdots q_n) = \\epsilon_{\\mu}(k) \\mathcal{M}^{\\mu}(k; p_1 \\cdots p_n; q_1 \\cdots q_n)", "73d9be5bc8cdc3510b63e9b9cd41dc05": "\nm \\approx \\sqrt{2}\\Delta \\phi.", "73d9bf584be81b41f523099b236bde06": "f_1(x,y,z)=0,\\ f_2(x,y,z)=0", "73da033be2413b947f9a533b4c49cba5": " B^* = -N \\ln \\left[ 1 - B / N \\right]/k", "73da044a7b79076ffc80a95d15034faf": "E = \\frac{k_{ET}}{k_f+k_{ET}+\\sum{k_i}}", "73da2ca9119f360fdcbc2e26f01a320b": "m=\\tfrac{v^4-24v^2-25}{48},\\; k=v^2,\\; F(m,k)=\\tfrac{v^5+47v}{48}", "73daa69ed6457c1f2a46957f70947ad3": "y\\left( m \\right) = a_0 + a_1 m + a_2 m^2", "73dad9a19c8d79698312b1923864cd9a": "\nu(z) = \\int_{a=-1}^{a=1} f_a(z) \\,da = \\int_{a=-1}^{a=1} {1\\over z-a} \\,da = \\log \\left({z+1\\over z-1}\\right)\n", "73db90dceb41aee5c88de92d6468253d": "\\tau_G(v)", "73dbbadc957c0a6b4dad293229e4bca1": "T_{a}", "73dc9fda49a128d90f68fa51063924f6": "\\textstyle -\\sum_{i,j} p_{i,j}\\cdot \\log(p_{i,j})", "73dce5a6070ee9731d9d114a0e4aabc3": "G_I", "73dcf7fd428baaf82502817ba71c99a7": "R=\\lambda R_1+(1-\\lambda)R_2", "73dd0d35c36e9fe70745fa14c30b70de": "C(x)_{q_1}, \\ldots C(x)_{q_k}", "73dd148dcafbf142253d74e9ad7939d3": "f(z) = \\frac{3}{z}", "73dd5336a5b4558adb8ab2fc692c2d53": "P_{rr}", "73dd85a53bce5e8d3efd0d3c528c37d5": "\\mathbf{R}^*_{N \\times 1} ", "73ddf62d2a17661c56f31e6c9fd1b021": "\\operatorname{Li}_2(2)=\\frac{{\\pi}^2}{4}-i\\pi\\ln2", "73de06580dd0988f66764c144bc0d038": "\\textstyle {C_{ox}} \\rightarrow \\infty ", "73ded00b666dd3f7ad893f1fcd8a0d37": "w_y(x) > 0", "73def3f527623cf426154ed690e81953": "\\gamma^5 \\gamma^\\mu + \\gamma^\\mu \\gamma^5 = 0", "73df1cfcce86d4405a92549199016119": "f(n) = c/n", "73df43bfebc6a1e074b195c4f91dd8fb": " \\psi \\le \\frac{ 3 }{ \\sqrt{ 3 } + 1 } ", "73df58544125130f8c5f9c7a7f1ce82b": "\\frac{(p-1)+\\sqrt{(p-1)^2+ab}}{a}", "73df6bdf903df0dc640845194a6f30e2": " \\frac{\\partial I}{\\partial t} = \\mathrm{div} \\left( c(x,y,t) \\nabla I \\right)= \\nabla c \\cdot \\nabla I + c(x,y,t) \\Delta I ", "73df8d1e303386f63e22fd295bb8765e": " \\tanh \\frac{x}{2} = \\sqrt \\frac{\\cosh x - 1}{\\cosh x + 1} = \\frac{\\sinh x}{\\cosh x + 1} = \\frac{\\cosh x - 1}{\\sinh x} = \\coth x - \\operatorname{csch}x.", "73df9214c84dcee483c29c80e1af2fce": "\\scriptstyle w_i=\\frac 1 {\\sigma_i^2}", "73df999a1516b2486e912bfea71aacb6": "r = \\frac{\\sum_{jk}jk(e_{jk} - q_{j} q_{k})}{\\sigma^2_q}", "73dfb15d60750ee7f723d0b621fb2408": " \\and (S_2 \\implies (\\operatorname{equate}[A_2, g\\ m\\ p\\ n] \\and V[F_2] = g\\ m\\ p\\ n)) \\and D[F_2] = K_2) ", "73dfbf0ef9005ca0794ac13a2e485594": "S(-1)^F=(-1)^F", "73dfe0992d1869c7c9c4299bd67df7b8": " N=N(\\rho) ", "73dfe95d216ca25177f1b49e709c70fe": "\\, a = amm^{-1} = (a\\cdot m + a \\wedge m)m^{-1} = a_{\\| m} + a_{\\perp m} ", "73dff2ad536adc03957fb0c66fa0c415": "\n\\sum_{n=1}^{\\infty}(\\zeta(2n)-1)=\\tfrac34\n", "73e0b682352488db38667b73810d9a9d": " \\bar{\\mathcal{M}}^i_j \\mathcal{M}^j_k = \\delta^i_k ", "73e0b7c9c888704bad8292c595ad4acc": "\\sigma_I^{(k+1)}", "73e0cf2e3d99e9a0ae7bfba78dd3c7cd": "m \\frac{v^2}{r} - T_{centripetal}=\\Sigma F_{rot} = m a_{rot} = 0.", "73e108db542676f1d201aa62bd9aa17b": "s_{i+1}", "73e1a9e2b7f2d92fc613e9dda06a7ea5": "\\scriptstyle{\\frac{\\log{\\left (\\sum_{k=1}^p n_k^a \\right )}} {\\log{p}}}", "73e1bcdd1e4fbef20fd9dfecf676725e": "\n(\\pi\\ominus\\sigma)(i) = \\begin{cases} \\pi(i)+n & \\text{for }1\\le i\\le m, \\\\\n\\sigma(i-m) & \\text{for }m+1\\le i\\le m+n,\\end{cases}\n", "73e1e12f5f1026fd731c7ec3db26f2e8": "\\{ s, v_2, v_3, t \\}", "73e213367f71fe07889bcadc446e95c7": "X,Y\\subseteq S", "73e2531c82661f2088a3c764cc528d6a": "\\left(\\frac{p^*}q\\right) = \\left(\\frac{-p}q\\right) = \\left(\\frac{-1}q\\right)\\left(\\frac pq\\right) = \\begin{cases} +\\left(\\frac pq \\right) & \\mbox{if } q = 1 \\text{ (mod }4), \\\\ -\\left(\\frac pq\\right) & \\mbox{if } q = 3 \\text{ (mod }4)\\end{cases}", "73e3072b9eeb080936a68d4e4b9402c4": "\\mathrm{Ref}_{a,c}(v) = v - 2\\frac{v\\cdot a - c}{a\\cdot a}a.", "73e312442f442f7c4460d75c062553a8": " \\operatorname{st}(1/x) = 1 / \\operatorname{st}(x) ", "73e35490335e9c4dd87da9b78769757b": "\\int_{- \\infty}^\\infty dx \\psi^* (x) \\psi (x) = \\int_{- \\infty}^\\infty dx e^{ikx} e^{-ikx} = \\int_{- \\infty}^\\infty dx = \\infty", "73e37825e5eda709956b3b21d559ee0a": " \\rho(\\mathbf{r}) ", "73e395ac97ceb6490d88626c7f268765": " w_+ = \\frac {c}{n} + v \\left(1 - \\frac{1}{n^2} - \\frac{\\lambda}{n} \\! \\cdot \\! \\frac{ \\mathrm{d} n }{ \\mathrm{d} \\lambda } \\right) ", "73e3ab2e6892574b0e0fe870608a689a": "\\sqrt{4} = 2", "73e3b5776bed3793b0eabb5fe26d5c01": "\\alpha_{0}=\\frac{1}{\\rho_{0}}\\left [ \\frac{\\delta \\rho}{\\delta T}\\right ]_{T=T_{0}}", "73e45049923768a6db3e74e01dd7127a": "\\gamma_2 = -\\frac{6\\pi^2 - 24\\pi + 16}{(4 - \\pi)^2} \\approx 0.245", "73e456dde33166623b892102ca46dc0e": "R_i(\\lambda)", "73e46ec1e722364d943404206b8aa1f0": "\\!\\tau_b = S_0 + \\mu(\\sigma_n - P_f)", "73e4b689270d152feac8f8f7d6448d0e": " \\left\\{\\boldsymbol{\\beta}^m, \\sigma_m, \\omega_k^m, k=1,\\ldots,d+r\\right\\} ", "73e515c356a820286139b8db061cfb13": "2 + \\cfrac{1}{4 + \\cfrac{1}{4 + \\cfrac{1}{4 + \\cfrac{1}{4 + \\ddots}}}}", "73e55061250cb4bb2a824d6a94631b8d": "h_\\gamma [A]", "73e564d74546b8e72b7dc2d50da5910a": "\\langle u, v\\rangle", "73e5c53844669c602c491c3fb20f6b30": "\nR(a_i\\mid [x]) = \\sum_{j=1}^s \\lambda(a_i\\mid w_j)P(w_j\\mid[x]).\n", "73e5ce41af84edeea2b187ea29499183": "1 = \\sum_{k=0}^{\\infty} a_k x^{k+2} - \\sum_{k=0}^{\\infty} a_k x^{k+1} + 2\\sum_{k=0}^{\\infty} a_k x^k.", "73e6071500350b290ab8ed04bf221832": "A_{21} >0 ", "73e63b190c09ab86b04a45b6d2dd9c47": "\\scriptstyle x_i \\,\\ne\\, 0", "73e66497804fc50a260e1b4675a6a0d4": "(A \\cap B)^c \\subseteq A^c \\cup B^c", "73e685d4f3247591868a3262a7e36280": "-\\hat{\\alpha}'_i=(1-\\lambda_i\\sqrt{\\rho^{-1}-1})\\left(\\frac{1}{i-1}\\sum_{j=1}^{i-1} ln\\left(\\frac{Y_{D_j}}{D_j^d}\\right)-\\beta^T{X}\\right)+\\lambda_i\\sqrt{\\rho^{-1}-1}\\mu_\\alpha({Z}) ", "73e6886058218df517297fa809961fe4": "\n\\begin{align}\n& f_r^+\\text{ is a function of }\\dfrac{\\phi_P-\\phi_L}{\\phi_R-\\phi_L}. \\\\[10pt]\n& f_r^-\\text{ is a function of }\\dfrac{\\phi_R-\\phi_{RR}}{\\phi_P-\\phi_{RR}}, \\\\[10pt]\n& f_l^+\\text{ is a function of }\\dfrac{\\phi_L-\\phi_{LL}}{\\phi_P-\\phi_{LL}},\\text{ and} \\\\[10pt]\n& f_l^-\\text{ is a function of }\\dfrac{\\phi_P-\\phi_R}{\\phi_L-\\phi_R}\n\\end{align}\n", "73e68f87aec47a6bc89e18106f9347a3": "\\sum_{n=1}^\\infty 2^n = 2 + 4 + 8 + \\cdots", "73e70b472b0427351a6f27fb85d8272b": "x \\in y \\land y \\in V \\rightarrow x \\in V", "73e70ea14918d502240362bd78f91690": "\\textstyle 3.\\ Differentiating\\ it,\\ we\\ obtain", "73e7256cc8d0944762879b01c7713bde": "(x-y)(x+y)= 0", "73e738e734680b7d652e0cbdd29fe509": "\\rho = {1\\over 2} \\begin{pmatrix} \n1 & 0 \\\\ \n0 & 1 \\end{pmatrix} ", "73e749dd67a9018744d44b94d1f0e98a": "\\frac{d}{dx}\\left(\\frac{1}{x^3 + 4x}\\right) = \\frac{-3x^2 - 4}{(x^3 + 4x)^2}.", "73e7523a8547611f207f1668533f7ace": "S_1,\\ldots ,S_k", "73e7578b2b342a9b4ec2fc702cf40f11": "\\sqrt{a_r^2+a_t^2}=a", "73e7a77006927b607da3d1b4615b0ab4": "\\rho = \\rho_{min}", "73e7e53c84809e22c4bf673608b97bbd": "\\delta w = -V\\sigma_{ij}d\\varepsilon_{ij}", "73e88e08a79a4e3cece8883ce269b179": "\\begin{matrix}\n 245 & 239 & 249 & 239 \\\\\n 245 & 245 & 239 & 235 \\\\\n 245 & 245 & 245 & 245 \\\\\n 245 & 235 & 235 & 239 \n\\end{matrix}", "73e8916df78f526a377ff67097a88ae1": " \\mathcal{E}_c = \\frac{\\mid \\mathbf{E} \\mid^2}{8\\pi} ", "73e94875c8b1a7538fe317eeacfa5f1e": "\\int_\\Omega F \\, d\\Omega ", "73e958b2e361cd19ecfe044e0ba02237": "S_2 \\wr S_n", "73e96d238034d98cd5a559145ce1b5ea": "\\mathfrak{R}\\otimes _{K}\\bar{K}", "73e96e3c8fb73cabde62eb04d3643bef": " \\scriptstyle \\mathcal{P} =\\left\\{P=\\{ x_0, \\dots , x_{n_P}\\}|P\\text{ is a partition of } [a, b] \\right\\} ", "73ea04e8a3ba3b019f2dc8cd2df13b84": " \\text{Combined PAR} = 1 - (1-\\text{PAR}_1)(1-\\text{PAR}_2)(1-\\text{PAR}_3)\\cdots. \\, ", "73ea47d97e689b0767abadc200bff18b": " M2 = \\frac{ K }{ K - 1 } \\left( 1 - \\sum_{ i = 1 }^K p_i^2 \\right)", "73ea4962aa143149251e4869d584fd1d": "\\begin{align}\n & L(a_{i},d_{i},h;\\ i=1,..C)=-\\log \\prod\\limits_{n=1}^{N}{\\frac{\\lambda _{n}^{x_{n}}\\exp (-\\lambda _{n})}{x_{n}!}\\frac{\\mu _{n}^{y_{n}}\\exp (-\\mu _{n})}{y_{n}!}}=-\\sum\\limits_{n=1}^{N}{\\log \\left( \\frac{\\lambda _{n}^{x_{n}}\\exp (-\\lambda _{n})}{x_{n}!}\\frac{\\mu _{n}^{y_{n}}\\exp (-\\mu _{n})}{y_{n}!} \\right)} \\\\ \n & =\\sum\\limits_{n=1}^{N}{\\lambda _{n}}+\\sum\\limits_{n=1}^{N}{\\mu _{n}}-\\left( \\sum\\limits_{n=1}^{N}{x_{n}\\log \\left( \\lambda _{n} \\right)} \\right)-\\left( \\sum\\limits_{n=1}^{N}{y_{n}\\log \\left( \\mu _{n} \\right)} \\right)+\\sum\\limits_{n=1}^{N}{\\log \\left( x_{n}! \\right)}+\\sum\\limits_{n=1}^{N}{\\log \\left( y_{n}! \\right)} \\\\ \n\\end{align}", "73ea8d784893242065e4f2bc0f85c4b0": "c = \\left ({\\text{TD} \\over \\text{ATT}} \\right ) \\times 20", "73ea984cd6f46d5a6d72fc4fb47b56e1": "[X,Y,Z]=1", "73eb048876d3fea9661aad4c57cc75b2": "\\mathbf{S}(\\mathbf{p}(t))", "73eb1845c54fc44c39e750f701d277dc": " \\boldsymbol{\\tau} = \\int_V \\mathrm{d} \\mathbf{p} \\times \\mathbf{E} ", "73eb26623b0ecdaab5c3fefccdad049e": "C_1 + \\frac{C_2}{1+r} = Y_1 + \\frac{Y_2}{1+r}", "73eb42cf3ef653899e17a4a9bb484585": " \\psi(x-\\varepsilon) \\sim \\exp(-(x-\\varepsilon)^2) ", "73eb69f0cad3b88f13a9a2e089642c20": "\\tau(i)=j", "73eb8bac23a2e73ca37fffd2274194bc": "\\big\\{ \\omega \\in \\Omega \\big| d(Y_{\\varepsilon}(\\omega), Z_{\\varepsilon}(\\omega)) > \\delta \\big\\} \\in \\Sigma_{\\varepsilon},", "73eb96f0c56a7503fa4cada4f27c30ed": "(5 \\; 6) \\; (9 \\; 13)", "73eb9be0301dd595b503deecae800dde": " \\begin{align}\n& y_{n+s} + a_{s-1} y_{n+s-1} + a_{s-2} y_{n+s-2} + \\cdots + a_0 y_n \\\\\n& \\qquad {} = h \\bigl( b_s f(t_{n+s},y_{n+s}) + b_{s-1} f(t_{n+s-1},y_{n+s-1}) + \\cdots + b_0 f(t_n,y_n) \\bigr),\n\\end{align} ", "73ebdcf914f7aa221957875218c1e2e1": "r_{jk}", "73ec3a94025c1d679da1505105e21e40": "\\|f*g\\|_p\\leq \\|f\\|_1\\|g\\|_p.", "73ec4e22477a6febf20ce4ba08d79057": "1782^{12} + 1841^{12} = 1922^{12}", "73ece6e921de616336f7bb9198b008e1": "H = N_G(D)", "73ecea744836f57d1b306a3642fe3591": "f(\\boldsymbol{A}) = \\det(\\boldsymbol{A})", "73ed58fc27230e5584aa1c07a9f60921": "\\frac{L_n^{(\\alpha)}\\left(\\frac x n\\right)}{n^\\alpha}\\approx e^\\frac x {2n}\\cdot\\frac{J_\\alpha\\left(2\\sqrt x\\right)}{\\sqrt x^\\alpha},", "73ed7ec59144c57cd55cf64b4c076646": "\\textit{fem}: \\textit{female}", "73eda362ea0605c44f393afd15084fb7": "\\begin{align}\n\\int_{L_1} \\left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \\right ) &= \\int_{L_1} \\left ( c u_x(x,t) dx + c u_t(x,t) dt \\right)\\\\\n&= c \\int_{L_1} d u(x,t) \\\\\n&= c u(x_i,t_i) - c f(x_i + c t_i).\n\\end{align}", "73edc937aa8d3ca9c0db06827e78a0dc": "a:~a\\in (-\\infty,\\infty)", "73edcb3f6659b5881c9af536cec730f7": "a_i a_j + 1", "73ef6e1fafdafef6909ec9fda169543f": "| \\psi \\rangle = \\alpha |H \\rangle + \\beta |V\\rangle,\\,", "73ef90915f01ab22db4a01346d8add95": "(0.8 \\cdot 1 + 0{,}2808 \\cdot 0.4 \\cdot 1) \\cdot (1+0.0961) = 1", "73efdc496f1a3c84e7f5ed251cc0a8de": " Y\\rho_wg+c\\rho_cg+b\\rho_mg ", "73efe189d866b534c0a20ce11ddf65cc": "\nV_{L} = V_{U} = V_{A}=E_{U} = E_{L|T=0}+D\n", "73efec3cab7fc20c5cf5d17b9188bf8c": "0\\to \\mathbf Z_2^\\infty\\to MCG(\\mathbf{T}^n) \\to GL(n,\\mathbf Z)\\to 0", "73efffd39ecad6c304e13bc623fcd395": "q\\in {\\mathrm {Mp}}(n,{\\mathbb R}).", "73f066fb7e1e72fdaf2efb3daa157e11": "G\\left(b\\right)-G\\left(a\\right)", "73f0a8006cbde67a298a1d6bae90a870": "R_Y=7", "73f177dbcc4617bed1ddce84c0c516cc": " \\Box p \\rightarrow p ", "73f2199976a15251838ef1600af45fe4": " H_4^{T}H_4 = \\frac{1}{2}\\begin{bmatrix} 1&1&\\sqrt{2}&0 \\\\ 1&1&-\\sqrt{2}&0 \\\\ 1&-1&0&\\sqrt{2} \\\\ 1&-1&0&-\\sqrt{2}\\end{bmatrix}\n\\cdot\\; \\frac{1}{2}\\begin{bmatrix} 1&1&1&1 \\\\ 1&1&-1&-1 \\\\ \\sqrt{2}&-\\sqrt{2}&0&0 \\\\ 0&0&\\sqrt{2}&-\\sqrt{2}\\end{bmatrix}\n= \\begin{bmatrix} 1&0&0&0 \\\\ 0&1&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{bmatrix}\n", "73f21b9adf94bc840dda48363ef88546": "\\frac{ \\text{d}B }{ \\text{d}t } = Y \\sum_h \\sum_{d_h}\n\\sum_{\\gamma_h} \\frac{\\text{d}[{^{d_h}_{c_h}}P^{\\gamma_h}_h]}{\\text{d}t} - \\mu B \\qquad \\qquad (3e) ", "73f24c49e797c8b20e151db0de5b70ac": "\\theta (z+a+\\tau b, \\tau) = \\exp 2\\pi i \n\\left(-b^Tz-\\frac{1}{2}b^T\\tau b\\right) \\theta (z,\\tau)", "73f2a11740f306f8cc0082e99df0345d": "\\scriptstyle A \\;=\\; \\bigcup_{n=1}^{\\infty}F_n", "73f3129dbcfa86cef9f1cff33f09991e": "N(t) = \\sum^{\\infty}_{i=1} N_i \\Phi_i(t), 0 1.\n", "74015e8e014031135a01941f874d0940": "\\begin{matrix} {2 \\choose 1}^2{11 \\choose 1}{4 \\choose 2}{40 \\choose 1} \\end{matrix}", "7401a41d62f8a885c48169389ccabcae": "|S\\rangle", "7401b1e42322c8024d0468e45c52f19e": "\\frac{SS_{Treatment}}{DF_{Treatment}}", "74021d2c44b69218f40e1f3160f20fc6": "x(t) = \\sin\\left[\\phi_0 + 2\\pi \\left(f_0 t + \\frac{k}{2} t^2 \\right) \\right]", "74022aa009243478e5a876d7d8bf919d": "v_1, v_r", "7402646449a5ae775f10e59269c1caf8": "\\scriptstyle f(A){\\sim}f(A+E)", "740269f70c142624b55476db54d873aa": "22 \\times 10^{-9} ", "74028857a9c415a4df21f814971b8743": "x=\\frac{(a+1/2)(d+1/2)}{(b+1/2)(c+1/2)}", "740296f922d2a9cd37812aa50102de85": "{{}^\\star R}^{abcd}", "7402bc0ee28a98a4fb428f0d8dbe652e": "mp=n", "7402c49e8e82ec45fb79a8727eda7e42": " \\acute{T}_{\\alpha \\beta} = C \\left ( \\acute{R}_{\\alpha \\beta} - {1\\over 2} \\acute{R} \\, g_{\\alpha \\beta} \\right ) ", "740309cc6cf11c7000ac8695cc30fb99": "(A \\in H(A_n)) \\land (A \\subset A' \\subset Y) \\Rightarrow (A' \\in H(A_n))", "74035bafa79d42318a677f1b7b1b3a61": "\\frac{\\partial I}{\\partial y}(x, y) = \\frac{\\partial J}{\\partial x}(x, y).", "74035fa22394be9787ba7f65372be849": "F_0. ", "740379887cb98be8875418731e690537": " H = - \\frac{d^2}{dx^2} + U_\\alpha,", "74037b9b3e1555063914993f7cec7858": "\\frac{y - y_0}{x - x_0} = \\frac{y_1 - y_0}{x_1 - x_0}", "7403f4520806437e8ef248b0b991d01a": " k_s = hf(t_n+c_sh, y_n+a_{s1}k_1+a_{s2}k_2+\\cdots+a_{s,s-1}k_{s-1}). ", "740413d2162a0f91e63c559cf5cf0596": "\\limsup_{x \\to +\\infty} x^2 q(x) = \\liminf_{x \\to +\\infty} x^2 q(x) = \\frac{1}{4} - a", "74045077267dee78131746495710be5e": "\\varphi_g, \\psi_g, \\rho_g", "74051ed68710120c16ec59719393dcb1": "F = \\frac{B^2 S}{2 \\mu_0}", "74052fd6dd1a22e4586e4cb622a74dee": " \\pi(A) = U^* \\rho(A) U \\quad ", "740563b71aa0fade691e1958150e5816": "\\begin{align}\n & q=\\underbrace{{{{{v}'}}_{y}}\\rho {{c}_{P}}{T}'}_{\\text{experimental value}}=-{{k}_{\\text{turb}}}\\frac{\\partial \\overline{T}}{\\partial y} \\\\ \n & \\tau =\\underbrace{-\\rho \\overline{{{{{v}'}}_{y}}{{{{v}'}}_{x}}}}_{\\text{experimental value}}={{\\mu }_{\\text{turb}}}\\frac{\\partial \\overline{{{v}_{x}}}}{\\partial y} \\\\ \n\\end{align}", "7405814e70c27d6c47926533e76f2662": " \\boldsymbol{F} = - \\int_A p\\, \\boldsymbol{n}\\; \\mathrm{d} S ", "740590376b65bd5973282e266d1840c2": "u_0 N} |\\hat{f}(n)|", "7429eb17c08ab7dd9d463424420e66a9": "W'=\n\\begin{vmatrix}\ny_1 & y_2 & \\cdots & y_n\\\\\ny'_1 & y'_2 & \\cdots & y'_n\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ny_1^{(n-2)} & y_2^{(n-2)} & \\cdots & y_n^{(n-2)}\\\\\ny_1^{(n)} & y_2^{(n)} & \\cdots & y_n^{(n)}\n\\end{vmatrix}.\n", "742a01e38612ef861259c6108a079c35": " \\sin \\theta =\\!", "742a22aa4c35feea8311da8996c05c19": "\n \\qquad \\cos(k_m \\Delta x) = \\frac{e^{ik_m \\Delta x} + e^{-ik_m \\Delta x}}{2} \\qquad \\text{and} \\qquad \\sin^2\\frac{k_m \\Delta x}{2} = \\frac{1 - \\cos(k_m \\Delta x)}{2}\n", "742ad0257901879443722d9b68d8d42c": "- \\sigma_{xz}\\sigma_{xy} + \\sigma_{yz}\\sigma_{xy} - \\sigma_{yz}\\sigma_{xz}", "742aecbba95ee01ead71f4ed6af373f2": "\\begin{align}\n\\sigma & = H_{soil}\\gamma_{soil}\\\\\nu & = H_w\\gamma_w\n\\end{align}\n", "742b46bfaa39c521083cc6055f6ba116": "\\left[J_a, \\widehat{T}^{(2)}_{q} \\right] = \\sum_{q'} {D(J)}^{(2)}_{qq'} \\widehat{T}_{q'}^{(2)} ", "742b6b7eb14f623f21e6c5493d6ba731": "M = 6n - \\sum_{i=1}^j (6 - f_i) = 6(N-1 - j) + \\sum_{i=1}^j\\ f_i,", "742b758d965f5468a775c8d078b211d5": "\\hat{H}_{\\textrm{int}}", "742b7917ffe064c2160183c2ca2b8d3c": "f_z(x)", "742be8547e3a55590072623704b59740": "\\pi = max_{\\{K,L\\}} F(K,L)*P - rK - wL\\,", "742c0eadc71468d7a0e7f57e551105d1": "S_+=\\wedge^{\\mathrm{even}} W", "742c4c0a390da87d0594d354eeb2c36a": "\\operatorname{Cl}_{2m+1}\\left(\\frac{\\pi}{2}\\right)=-\\frac{1}{2^{2m+1}}\\eta(2m+1)=-\\left(\\frac{2^{2m}-1}{2^{4m+1}}\\right)\\zeta(2m+1)", "742c8e6ceea3ae0793ef80193fdf217a": "X(x) = Bx + C.", "742d54c8ea18c2b41ab2eb2277597acd": "\\mathrm{e}^{-\\lambda x}", "742d695f597614d82af2e8f5f218c75e": "\\gamma^{-1} ", "742d98d83e16b652cabe005845f27e32": "\\scriptstyle (\\cosh\\,a,\\,\\sinh\\,a)", "742db51f15397dd94c4cc04bd024a662": "(a_2b_1-a_1b_2)^2+(a_1a_2+b_1b_2)^2=(a_1^2+b_1^2)(a_2^2+b_2^2),", "742db90db00454cf91599014d8c35f48": "(N,\\psi:N\\to \\mathbb{R})\\ ", "742e4966de14209d67e62fe66d1481d0": "\\mathcal{U}(0,{\\tilde{u}})", "742e4a70e8ae7dd599a7e8bda9928459": "\\textstyle x\\in {N},", "742e812f2081aef4fd44a370c4ec1ce9": " \\varphi_k = \\underset{\\Vert \\mathbf{\\varphi} \\Vert = 1, \\langle \\varphi, \\varphi_j \\rangle = 0 \\text{ for } j = 1, \\dots, k-1}{\\operatorname{arg\\,max}} \n\\left\\{\\operatorname{Var}(\\int_\\mathcal{T} (X(t) - \\mu(t)) \\varphi(t) dt) \\right\\}, ", "742ea3164d9edca4f7171de2f44c2780": "C \\cap \\mathcal{H} ", "742eac105e13989a96b8b3d4139da0ef": "{}^2E", "742eb6a15d5d1504beb4ee16512cff02": "\\int_{\\tau_1}^{\\tau_2} \\mathbf{F}_\\mathrm{rad} \\cdot \\mathbf{v} dt = \\int_{\\tau_1}^{\\tau_2} -P dt = - \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2}{6 \\pi c} \\mathbf{a}^2 dt = - \\int_{\\tau_1}^{\\tau_2} \\frac{\\mu_0 q^2}{6 \\pi c} \\frac{d \\mathbf{v}}{dt} \\cdot \\frac{d \\mathbf{v}}{dt} dt", "742ef8133428cd584c70e318718adc5d": "MRTS_{12} = \\frac{a}{1-a} \\frac{x_2}{x_1}", "742ef90ead370fa9f531d56bbc39a173": "\\pi\\approx\\frac{428224593349304}{136308121570117}=3.14159265358979323846264338327(569...)", "742f74629904b2fb13756f9099c5369f": "c_7 = 1.22874 \\times 10^{-3}, \\,\\!", "742f942410c08cf7d22bf3341c0a7855": "Y_t(u)", "7430903f021bae87a9f7f2994e6b5dd7": "\nP \\int_x \\psi(x) |x\\rangle = \\int_x - i \\psi'(x) |x\\rangle\n\\,", "7430b173f8c94b3000f2a192971f19df": "t_{12}t_{34}+t_{14}t_{23}=\\frac{1}{R^2}\\cdot \\sqrt{R-R_1}\\sqrt{R-R_2}\\sqrt{R-R_3}\\sqrt{R-R_4}\\left(\\overline{K_1K_2}\\cdot \\overline{K_3K_4}+\\overline{K_1K_4}\\cdot \\overline{K_2K_3}\\right)", "7430e4ac202b9795efd073e67e58885b": "a_3a_2a_1a_0", "7430f8854af60257afabe8e970d60654": " \\operatorname{sink-tran}[(\\lambda p.(\\lambda q.q\\ p)\\ (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f)))\\ (\\lambda f.\\lambda x.f\\ (x\\ x))] ", "7430fecdf6389c9dc4e42e5f6cbbdbc8": "| \\mathbf Z| = (R_1+R_2)(R_2+R_3) - {R_2}^2 = R_1R_2 + R_1R_3 + R_2R_3 \\ ,", "743105be9673e6e6a0c3149aca5334c8": "1-4x^4-8x^5-4x^6+2x^8+8x^9+12x^{10}+8x^{11}+2x^{12}=0", "7431470a4f6360a944e727ea0eee1009": "n_\\text{clause}", "743163c316cadd47dcfc46c9288e1e68": "\\omega = \\omega_1/\\omega_2", "74317f5b9546ef44342744e1f20e6bae": "x^4=x^2+y^2", "743180a6ee2d2b1eaa4152e07db22f44": "\\sigma_y^2(M, T, \\tau).\\,", "7431837b98c6044256e88a5a8bbbcd91": "T_{yy}", "7431b62dc822d54c371428ed18931417": "\nu(x,t) = \\sum_{i=1}^n m_i(t) \\, e^{-|x-x_i(t)|},\n", "7431b76204ad2d83b386724282355e8c": " f_{lm}= \\frac{(2l+1)(l+|m|)^{!}}{2^{m}|m|^{!}(l-|m|)^{!}} ", "7431e81f5ee3462e41738dc810c697ae": "c_n = \\langle f, \\phi_n \\rangle = \\sum_{n=0}^{N} w_i f(x_i) \\overline{\\phi_n(x_i)}", "743201a39160a9f6a2fc706363af3cf5": "O(n\\log n)=O(\\log n!)\\,", "74320e81c0af632da9bcbe87bc6a246e": "(F\\otimes G)_{i_1i_2...i_{m+n}} = F_{i_{1}i_{2}...i_{m}}G_{i_{m+1}i_{m+2}i_{m+3}...i_{m+n}}.", "74323d8d3871e0a377b4122997ef8962": "\\varphi_H", "7432ba375e2d5d3f4d7d66377f05faec": "\nm = \\frac{\\langle \\Delta I^2 \\rangle^{1/2}}{\\langle I \\rangle}.\n", "7432ee0a81b939703317d6bef5a1a874": "q^i = \\psi^i(\\mathbf{x})", "7433344ded27b48e9769cb4bee706dc4": "\\tbinom{n}{2}/( 2^{n-1} -1 )", "743340dbde4a8884dcd34c3c288c0758": "\n\\Delta_{\\mathrm{ret}}(x)=i\\theta\\left(x^0\\right)\n\\int \\frac{\\mathrm{d}^3k}{(2\\pi)^3 2\\omega_k}\n \\left(e^{-ik\\cdot x}-e^{ik\\cdot x}\\right)_{k^0=\\omega_k};\\quad\n\\omega_k=\\sqrt{\\mathbf{k}^2+m^2}\n", "74336602a3bae18ddbfac1eae983ff20": "- \\leftarrow", "74336f970a457dc3f0c070aa11a17b80": "f^-=g_0^-\\circ T", "7433a851b360af95028b3a7dc076c89a": "a_0=0.62;\\quad a_1=0.48;\\quad a_2=0.38\\,", "7433ea8f40a92518b9b09eb71a826acf": "(P \\and P) \\to P \\, ", "743400cf7a812e774b834dce292aeead": "\\sigma_1=\\sigma_2=\\sigma_{b12}, \\sigma_1=\\sigma_3=\\sigma_{b13}, \\sigma_2=\\sigma_3=\\sigma_{b23}", "74341d404e0a3aace53fd3f7aaa64a71": "\\zeta_0 = \\psi(\\Omega)", "7434ab38e15b581bfbb4c9e275d9e1a1": "\\tilde{u}_{k}(\\mathbf{q}) = u_{k}(\\mathbf{q}) + (\\hbar^{2}/2m)\\tau_{12}^{2} ; k = 1,2,", "7434e0b5d3dbbaa438440dc75be32095": "\\omega r = v = \\beta c", "7434eca55c15a1108ffa4f21541fcaef": "r = a \\frac{\\sin (\\theta+\\alpha/2)}{\\sin (\\alpha/2)}", "74357985d181c85081f05679ae3bb649": "k = 0, 1, \\ldots", "74358f1e88045b904bf135095b9b90fb": "R(X,Y) = v\\left([(\\mathrm{id} - v)X,(\\mathrm{id} - v)Y]\\right)", "7435934f05c0515646b2e8bc930a62a2": "\n\\delta_{ext}(s, t_s, t_e, x) = \n\\begin{cases}\n(s',t_s - t_e) & \\text{if } \\delta_x(s,x)=(s',0) \\\\\n(s',\\tau(s')) & \\text{if } \\delta_x(s,x)=(s',1) \\\\\n\\end{cases} \n", "7435dcc3bfa8c7a2224f2803ecd958c0": "O(N_1^3N_2^3)", "743605d90781c00e5c1c5fd366f6d2bd": "\\rm \\ V + 5F_2 \\xrightarrow{300C} VF_5", "7436191dc3588057b7e8008d23daebcd": "X_c = \\frac{1}{2 \\pi f C}", "7436218fe16af4804c6feacf748ce4f8": "a(\\cdot, \\cdot)", "743639d1beb78a02effd6698552f2585": "f(z)=a\\overline z+b", "7436659d6f445e7dbf25b1b457c54589": "H^*(X,\\mathbb{Z}) = \\bigoplus_{i=0}^{\\infty} H^i(X,\\mathbb{Z}),", "74369989fe6eed74c5053160976e1999": "(1.059463094359295264561825)^{84} \\approx 127.999999999946", "7436afd42963110bd5c572e55d2c005c": "\\mathrm{D \\cdots X{-}A}", "7436c5cdc51b64dcafcfe2460457e540": "a^2 \\le 0, B(a,a)>0 ", "7436d542c7f6e1c66aa73b5fa2b1fa57": "\\alpha_l", "74370d6a9a7abd2af19a72b4cf649cef": "0=u'_1y^{(n-2)}_1+u'_2y^{(n-2)}_2+\\cdots+u'_ny^{(n-2)}_n", "743714b57cefdf4ec6714e38aa14f696": " (x(\\theta)\\,, y(\\theta)) = (\\sin(5\\theta)\\,, 2\\sin(4\\theta)). ", "74375e9089b451a0f4c2be7c9d5b28fc": "T_Z[T_Z(w)]\\,", "74377f5b65ad33bc82fe7c10b6118e63": " L \\, = \\, -k_{B}T\\ln(\\mathcal{Z})\\, = \\, -k_{B}TN_{S}\\ln(1+x) ", "74378b6386576af169e7b97f02b28c78": "\\displaystyle{u(z_n) \\mp {1\\over 2}\\varphi(z) - T_K\\varphi(z)=\\int_{\\partial \\Omega} (K(z_n,w) -K(z,w))(\\varphi(w)-\\varphi(z))\\,|dw| = D(\\psi)(z_n) - D(\\psi)(z),}", "7438e9a90ae89535a804df3eadcc2214": "\\rtimes \\!\\,", "74392d02712aec9d7a47cbaef61f3e88": "\\left|\\tau_{ind}(\\omega)\\right|^2", "74396db40cab1537229f29fa0b877c36": " \\frac{s}{s_0}=[1+\\frac{K\\theta}{1-\\theta}]^{-1} ", "743976a746f4340ef9135d494dbb945d": "K_n(z)", "74399b6b527b864d731aca1ae2d308af": "P(m) = P(m+3) - P(m+1),", "74399d8aa0d2c9ea0882ba3326667ed8": "\\lfloor q \\rfloor", "7439b782a644f5f2fa4251147eb8c551": "g^{\\sigma\\nu}R_{\\sigma\\nu} = R", "743a0aece8c91ddea369d2becee76fec": "f(x) = \\sum_{n=0}^{\\infty} a_n \\phi_n(x)", "743aaaaad8c222a48e02872c4fc12180": "P \\lor A", "743bf2ebca04238fbac460ce5b5c9c25": "RC\\frac{\\mathrm{d}i(t)}{\\mathrm{d}t} + i(t) = 0", "743bfe91cfbb6082e0e621c7e997e079": " [ A_\\phi : A_\\phi^p ] = q^{e_\\phi} . ", "743c07c571ba5acafad2f63c8cc6e912": "N_\\alpha(t)=N_0 \\exp(-t/\\tau_\\mu) (1+\\alpha A)", "743c8aef2baa681cfb4a3f0b0a03e331": "L_e = \\frac{2 \\times A_t}{0.5 \\pi \\left( D - 0.64952 p \\right)}", "743c995ad158826dad47a804a41ed6b3": " \\begin{align}\nk_i = N \\left[ \\sum\\limits_{j=1}^{i} \\frac{N-i}{N-j} + \\sum\\limits_{j=i+1}^{N-1} \\frac{i}{j} \\right]\n\\end{align}", "743cb2c8192a213f99cf9d6c1c6c0d9a": "\\frac{L}{L_{\\odot}} \\varpropto \\frac{M}{M_{\\odot}} \\qquad (M > 20M_{\\odot})", "743cbf242d6e549980eeb34836894c68": "\\mbox{P} \\subseteq \\mbox{NP} \\subseteq \\mbox{MA} \\subseteq \\mbox{QCMA} \\subseteq \\mbox{QMA}\\subseteq \\mbox{PP} \\subseteq \\mbox{PSPACE}", "743ce75bff7d4d0a1700d49d8c258705": " Spin(14,\\mathbb C)", "743d1b3b48d030add7caa51e1d4d4cb9": "\\tilde{y}_{i+1} = y_i + h f(t_i,y_i)", "743d63b71ceb419ec4b34b394b9c8210": "\\scriptstyle \\hat B \\;=\\; \\hat x \\hat y \\hat z \\hat v \\;=\\; i \\hat v", "743d79d3a62980cab6fd12897fb909e1": "p_{i+1} = 2p_i + 1", "743dd86257bb41f890825bc718292731": "\\frac{1}{2}\\langle u_i u_i \\rangle = \\int_{0}^{\\infty}E(k)\\mathrm{d}k", "743e3195a513ec5e862254be2d3bf49b": "g_3(\\lambda \\omega_1, \\lambda \\omega_2) = \\lambda^{-6} g_3(\\omega_1, \\omega_2).", "743e445527ff93a2416356c656054a1f": "f_s = {R \\over N}", "743ea2793e0e10e10cfe6a3d6cf80e12": "p : P \\to X", "743eddca6df7296d87c66ca3f462ced9": "\\Sigma Work = \\mathit{W}_{1-2}+\\mathit{W}_{3-4} = \\left(\\mathit{u}_1 - \\mathit{u}_2\\right) + \\left(\\mathit{u}_3 - \\mathit{u}_4\\right) = -4 + 5 = +1", "743ef0cd314087cbc350c953639309b8": "\\sqrt{gk}=\\sqrt{\\frac{2\\pi\\,g}{\\lambda}}", "743f0c4a0d5116d3de1b68b6903314be": "S ,", "743f1323bdef5b219d3f4dfcc93be9d7": " \\frac{100}{2} = 50 ", "743f22aa971cf92f2aa5c1a871d6b46e": "F[y]=\\int_a^t |y'| \\,dt ", "743fa978e2bb438162a5b1bdb9aa2b77": "i_\\mathrm{b}", "743fbf47c619b394fbec65a1b4d386ed": "\\sigma_{\\mathrm c} (\\varepsilon, t, T)", "743fe257bcd2ab4f15b3d577088779eb": "d_1+\\cdots+d_n", "7440311cd9f6f74a296a14cf55f053c5": "(\\lambda,Q,P)", "74404356b595bcb9aba8b606684b2fa0": "F_i(k_{xy})=H_i(j_{xy}): H_i(M_x) \\rightarrow H_i(M_y)", "744058f42ce65d587c80331f0ba84e73": " R(0) ", "74406469430f552545f30dc9a7a90ff8": "\n\\frac{\\partial S}{\\partial p_{\\varphi}} = \\varphi + \\frac{\\partial S_{r}}{\\partial p_{\\varphi}} = \\mathrm{constant}\n", "74407572a1d83c08e21563161c3dff11": "f(r + tp^k) = f(r) + tp^k\\cdot f'(r) + O(p^{2k}).", "7440cd1c6e6ed9793b318f855946761e": "T_{em}=\\frac{3V_{TE}^{2}}{(R_{TE}+\\frac{R_r^{'}}{s})^{2}+(R_{TE}+X_r^')^{2}}.\\frac{R_r^'}{s}.\\frac{1}{\\omega_s} (N.m)", "7440fc7d0d7a5ecdc19bc45d5ba46dce": "(x,y)=x+iy'", "7441080da01511aa2cc63b67cf6b86e9": "A=X\\setminus U", "7441a2b6c04a3eaab676bf6d06cec2fa": "\nu_{1} + u_{2} = \\frac{2}{A \\left( 1 - e^{2} \\right)}\n", "7441ab1f370142f2d75ed697efb36434": "\\lambda_n=\\max\\bigl\\{1,|\\alpha_n|,\\|\\psi_n\\|_\\infty,\\|\\psi_n^{(1)}\\|_\\infty,\\ldots,\\|\\psi_n^{(n)}\\|_\\infty\\bigr\\},\\qquad n\\in\\mathbb{N}_0,", "7441c6ed26caaad0ef475a717ba27f2e": "\\mathbf{p} = d\\mathbf{l} + \\mathbf{l_0} \\quad d\\in\\mathbb{R}", "7441dba4aabae04e2529e6d3208eb54c": "h_N[n]\\ \\stackrel{\\text{def}}{=}\\ \\sum_{m=-\\infty}^{\\infty} h[n - mN]", "744221543584f406d3aa61aec3e7a9df": "x = \\mu - \\sigma/\\xi", "7442622e3ba9e98f202353a4a54d0455": "\\scriptstyle p \\;=\\; 5", "7442725f624850310007c8a6701a9171": "\\gamma(s, z)", "7442a8c3774c1476b1d91a5952d400a0": "F_\\textrm{r}", "7442f42e0022d4ebecedaa9edc7bf02d": "J=j_1+j_2", "744315f2abae3ba9a84649bb7da32fcb": "\\mathbf{E}[X_T] = \\int_{\\Omega} X_{T} (\\omega) \\, \\mathrm{d} \\mathbf{P} (\\omega)", "74439eebb60aab71668dc0a18ca5d021": "D_{\\mathrm{KL}} \\big( p(\\cdot\\mid y_1,y_2,I)\\big\\| p(\\cdot|y_1,I) \\big) + D_{\\mathrm{KL}} \\big( p(\\cdot \\mid y_1,I) \\big\\| p(x\\mid I) \\big)", "74448c7fb70021cf67497504bf09a9d7": "u_2 =\\left(P_1^\\beta-P_3'^\\beta\\right) \\sqrt{\\frac{(1-\\Gamma^2)P_1^{1/\\gamma}}{\\Gamma^2 \\rho_L}}", "7444c523cb5fa3831fdbef43c274fb23": "{\\mathrm{I}}_1", "74452cdd84c1feeff1c0e512a18e95eb": "\\operatorname{p.v.}\\,\\, K[\\phi] = \\lim_{\\epsilon\\to 0^+} \\int_{|x|>\\epsilon}\\phi(x)K(x)\\,dx", "74454b324ec883b12434c24140df2d15": "dev (B) = \\{B_i+g: i=1,...,s, g \\in G\\}", "74454f814b9badc05f6cc3779d4bcec8": " \\rightarrow 0 ", "7445585b3c551ea8f180a173c1e015c4": "P(q | \\vec y)", "74456dfee96adeffddb87541c6baecd4": "y_1(t) = t\\, x_d(t) = t\\, x(t + \\delta)", "7445a73e1a1aa0ef5bba942679390a11": "f(n) = \\Theta\\left( n^{c} \\right)", "7445fc31cd050c0fbd276948c934db4d": "E_1\\triangle E_2", "744610d37950221c24f5da83c58a0687": "\\frac{4}{3\\pi}", "74463264607a6317a79b5a4d124fa366": "p(x+e)", "74466d0a1c0006ba989b9a91e7e0672c": "X:=\\{X_n\\}", "744673aa80bbdb2b797d5ffc161a1679": "\\nu = \\mu \\circ f", "74468318f31792a2ac59efd8f59faf3b": "m, n > N", "7446d371e98a86979efa4a7f8333122c": "\ne = 1 + \\frac{1}{1!} + \\frac{1}{2!} + \\cdots + \\frac{1}{(m-1)!} + R_m.\n", "7446e3244f5ddc90f0e979a82dd302ca": "(2/3)^3 + 3(1/3)(2/3)^2 = 20/27", "7446efc878e77fb0b9f36a53896706d0": "\\frac{\\partial z}{\\partial x} = 3", "744721137f62bfabfcc8f1c8de596284": "d(\\mathbf{X}^{\\rm T}) =", "744740ea099968c8c6e53748d2607249": "\\,(s,x\\$)\\rightarrow_M^*(q,\\epsilon).", "744742b0f15eb30e3926acb5355be0c3": "\\begin{align}\n &\\left( 1 + \\sum_{n=1}^{p-1} \\text{bit}_n\\times 2^{-n} \\right) \\times 2^e\\\\\n = &\\left( 1 + 1\\times 2^{-1} + 0\\times 2^{-2} + 1\\times 2^{-4} + 1\\times2^{-7} + \\dots + 1\\times 2^{-23} \\right) \\times 2^1\\\\\n = &\\; 1.5707964\\times 2\n\\end{align}", "74474477ec8ce9d54f35c67a7726ea1f": "(I, S, S_0, s, T)", "7447ace28da31616cd94a58909c9ca39": ">p", "7447bef29647e7da4f4deb950a4f1d4a": "h(x) = \\alpha f(x) + \\beta g(x)", "7447e282fe5b1f8caefabf2fc2888d90": "\\theta\\in(0,1)", "74480a882e5a73ec28d4074e0081a758": " \\Theta = \\alpha \\cos n\\theta + \\beta \\sin n\\theta, \\,", "744814d8e015d2ef39b968ba1b93a9ca": "\\varphi = \\log r,", "74481a54d1691d5409ae9d83e78b3795": "f(z) = w \\,", "744832d30d8126eddd6b3a0faedd8669": "\\frac {F_A A} {Y}*g_A", "7448498d477aa87ef7608f235feffa05": "\n \\begin{align}\n \\varepsilon_{\\alpha\\beta} & = \\frac{1}{2}(u^0_{\\alpha,\\beta}+u^0_{\\beta,\\alpha}+w^0_{,\\alpha}~w^0_{,\\beta})\n - x_3~w^0_{,\\alpha\\beta} \\\\\n \\varepsilon_{\\alpha 3} & = - w^0_{,\\alpha} + w^0_{,\\alpha} = 0 \\\\\n \\varepsilon_{33} & = 0\n \\end{align}\n", "7448e506d78a4ab0f33cc1a7f1ae176a": "p(e|\\mathbf{\\theta})", "74492ce810f649675af936b7b3b01582": "M = -\\frac{f_2}{f_1},", "7449ed150936102b675e07d41b56620d": "mr^2\\dot{\\theta}", "744a1d5d9b175d621995371d932f0fa0": "\\omega \\mapsto \\inf_{b \\in K(\\omega)} d(x, b)", "744a322399dad505877e7de6afaf1d49": "O\\;\\mid\\;\\sigma\\mid\\tau\n\\;\\mid\\; a.\\sigma", "744a9d45619605487b3255c82611f804": "[0.131,0.376,0.371,0.123]", "744ac73f068e5fc57986371e64de9400": " \\hat{L}=f(x) \\frac{\\mathrm{d}^2}{\\mathrm{d} x^2}+g(x) \\frac{\\mathrm{d}}{\\mathrm{d}x}+h(x), \\qquad \\hat{M}=\\frac{\\mathrm{d}}{\\mathrm{d}t} +k(t)", "744afabfbb25473c7478566a31b26a0d": "\\aleph_{\\alpha+1}=2^{\\aleph_\\alpha}", "744b4529390be4cae205f93abf8c428b": "z_1(t)=y(t), z_2(t)=y'(t),\\ldots, z_N(t)=y^{(N-1)}(t)", "744b8f6261a7c110b2ea5c420c52d8d4": "P(w_i|w_{i-2}, w_{i-1}, t_{i-1})", "744ba083503a12c9ed87b21d114fa666": "f(\\cdot,y)", "744ba59e8e6131127b6dd1e0da92fe4b": "y(t) = \\int_{0}^{T} x(t-\\tau)\\, h(\\tau)\\, d\\tau", "744bbfae191bbbbfcdb11a53fb46a82e": "n + 1 \\times n + 1", "744bc7972748bd4115e652fae79a223b": "\\mathbf{DTIME}\\left(o\\left(\\frac{f(n)}{\\log f(n)}\\right)\\right)\\subsetneq \\mathbf{DTIME}\\left (f(n) \\right).", "744becf60e266492909bfa9a7d16eb38": " q = Bq^ \\ast /d ", "744bf0a085ddf0d11dff99f06d128fa2": "\\vec{\\Gamma}", "744c155b5bbe60bce1955f210d5e9eec": "\\lambda <\\kappa <2^{\\lambda}.\\,", "744c214c91ecd5f770c4443f4db58df8": "|H|", "744c857ada25c7b0bfa627aa1507fe70": "2 \\log_2 N", "744ca045134074ebb89ef414fbdfc6d1": " \\theta_2", "744cc42d7adb3da1d4ac27ae88b48d58": "G=\\Delta_x", "744cfa65c205dd1e915c11f8cd2c83e7": "a ^ b = a \\cdot (a ^ {(b - 1)}),\\,\\!", "744d2f37c966fba71a228a20145bdca4": " N = \\sum_{n=1}^k b_n q_n \\ ", "744d978ccd754c49c2a06b9473a9bce7": "\nK_a = {{x^2} \\over {F}}\n", "744da0d4104b81a1b0289230e383ba6b": "\\frac{A_{21}}{B_{21}}=F(\\nu)", "744df7f9802ec887263ab6a4df2f5c3f": "\\lambda(x,t)", "744e5c181e650ca3de8124f343da6562": " \\hat{h}_i = -\\frac{1}{2} \\nabla_{r_i}^2 - \\frac{Z}{r_i}, i=1,2 ", "744ec93086ebbb9279e0ae2fe3bfffc5": "\\mathbf T^n \\vec f^n = \\hat e_1.", "744eddcddce14c92a0f43a7fae1e6a1c": "Q(BC)", "744f439ae2b172896984684866fef972": " \\mathbf {n} ", "744f8c23cde407f1292a215e04a34831": "2, 3, 2^t, 3^t.\\,", "744f9e132cb0e433dedcad8a5e0a6198": " n(n+1)~r^n~\\sin(n\\theta) \\,", "744fc6552221c605211ea064eb4e4231": "\n\\gamma _1 \\,\\, \\equiv \\,\\,1\\,\\,\\, - \\,\\,{2 \\over {n\\,\\, - \\,\\,1}}\\,\\,\\sum\\limits_{k\\, = \\,1}^{n\\, - \\,1} {\\,\\left( {1\\,\\,- \\,\\,{k \\over n}} \\right)} \\,\\rho _k \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\gamma _2 \\,\\, \\equiv \\,\\,1\\,\\,\\, + \\,\\,2\\,\\sum\\limits_{k\\, = \\,1}^{n\\, - \\,1} {\\,\\left( {1\\,\\,- \\,\\,{k \\over n}} \\right)} \\,\\rho_k", "744fdf3e7099400e2f78fa413d2ac58b": " \\psi_t(x) = \\int_{y} \\psi_0(x) {1\\over \\sqrt{2\\pi it}} e^{-i (x-y)^2 / 2t} \\, .", "7450f8a128e1a34c5f9116318bafb928": "Rt_1 \\ldots t_n", "74518c43824c043dc9c4a8531f13c626": "\\tau \\equiv \\int_0^z (\\alpha) dz = \\sigma N", "7451d02ead7d19bdce3f445c657262b9": "\\int\\frac{x}{ax^2+bx+c} \\, dx = \\frac{1}{2a}\\ln\\left|ax^2+bx+c\\right|-\\frac{b}{2a}\\int\\frac{dx}{ax^2+bx+c} + C", "7451fc73ba899cf3f6abc02eb5997267": " f(a,b) =\n \\begin{cases}\n 2 a b, &\\mbox{if } a < 0.5 \\\\\n 1 - 2 ( 1 - a ) ( 1 - b ), &\\mbox{otherwise}\n \\end{cases}", "74526649480ee3041b7bfa80d4df3431": "\\mathrm{vol}_M = \\frac{\\omega^n}{n!} \\in \\Omega^{n,n}(M)", "74527cd3de9cec7b7ca5cdda8f355b57": "\n \\begin{align} \n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} + [N_{\\alpha\\beta}~w^0_{,\\beta}]_{,\\alpha} - q & = 0 \n \\end{align} \n", "745281d30ea1d85ab0e7926a51402670": "\\hat{b} = (\\hat{b}_1,\\ldots,\\hat{b}_J)^T = \\lim_{\\eta\\rightarrow\\infty} \\hat{b}^{(\\eta)}", "7452edbf40c3ae31b5de535e2650c23e": " \\operatorname{Q}(\\xi, \\eta) = \\langle \\xi \\mid T \\eta \\rangle + \\langle \\xi \\mid \\eta \\rangle ", "7452f255a31bda55d93476dad49a4e01": " \\chi(\\lambda) = \\det(\\lambda I - A) = \\lambda^4 - 11 \\lambda^3 + 42 \\lambda^2 - 64 \\lambda + 32 = (\\lambda-1)(\\lambda-2)(\\lambda-4)^2. \\, ", "74531ee8bce8badc8d68fd29955c111e": "Homeo(F) \\to M_p \\to N", "7453bdeef6d6098892dcef6977d241a7": "\\displaystyle{B(a,b)R(a^b,c)=R(a,c) -2Q(a)Q(b,c)}", "7453e38dea61d7bdd3161ba67e44f921": "\\ q=0.4 ", "7454afd3a2c08dbc00328ede4042bf1f": "T_x X", "745518573cce1b6a00f48c8778650694": " \\lim_{n\\to\\infty} x_n \\frac{\\log n}n = 1, ", "74551f7effb4d87eacea90515275df96": "k+jd \\le m+ld", "74555cba7447d29a9be59660c3d78520": "b_m = (a*1)(m) = \\sum_{n\\mid m} a_n\\,", "74557063a568ad7db464a22ac2544af6": "\\phi=2\\pi n", "7455c710b0a584f9eddb7ab657500c01": "G_1 = 1 - \\sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})", "74565ed12e16f0545fffcd26a05934ea": "\\mathcal{L} = \\frac{1}{2}\\left(\\mathbf{E}^{2}-\\mathbf{B}^{2}\\right)+\\frac{2\\alpha^{2}}{45 m^{4}}\\left[\\left(\\mathbf{E}^2 - \\mathbf{B}^2\\right)^{2} + 7 \\left(\\mathbf{E}\\cdot\\mathbf{B}\\right)^{2}\\right]", "74569cec3789f5a2798192c333d2718b": "dX^{\\mu'}=\\Lambda^{\\mu'}{}_\\nu dX^\\nu", "7456c3803a13faa82b9bb3f7499581d4": "\\left|\\int_0^\\delta e^{-xt}\\phi(t)\\,dt\\right| \\leq \\int_0^\\delta e^{-xt}|\\phi(t)|\\,dt \\leq \\int_0^\\delta |\\phi(t)|\\,dt", "7456ce7ded41558da00bd29417e8d9f6": "E_e^-=X", "7456ec8a8deb5176c32beb3bb6f999e4": "\\rho A \\operatorname{d}x \\frac{\\operatorname{d}v}{\\operatorname{d}t}= -A \\operatorname{d}p ", "745712de330dc68dc8bdaf1ef17a3750": "{_1^0}\\text{S} + {_1^1}\\text{S} \\rightarrow u_\\beta {_2^1}\\text{P}^\\beta + u_\\gamma {_2^1}\\text{P}^\\gamma, ", "7457261e76347390e72f5c4724bb1b31": " \\langle\\phi(x)\\rangle = {1\\over Z} {\\partial \\over \\partial h(x)} Z[h] = {\\partial\\over\\partial h(x)} \\log(Z[h]).", "745780325c7a4f10a48c6c4ade5866e2": "(14)\\quad \\gamma(r,\\theta)\\,=\\sum_{i=1}^\\infty \\sum_{j=0}^\\infty a_i a_j", "74578d2772dcd5eaf0bc725c2a9be4a2": "\\lambda\\in K", "74578d739506de762fdcee3a46849457": "[X]^\\lambda", "7457bc20ce023eb54afd113e49e3d533": "dE = \\frac{q^2}{4 \\pi} \\mu(\\omega) \\omega {\\left(1 - \\frac{c^2} {v^2 n^2(\\omega)}\\right)} dx d\\omega", "74584e49cf40789f194de6394da90112": " \\lambda^2 + \\frac{b}{m} \\lambda + \\omega_0^2 = 0. ", "7459004a25e47f98350aaf4f900cb67f": "Q_i = \\sum_{j = 1}^n c_{ij}\\phi_j \\mbox{ (i = 1,2,...n)}", "7459140ec01dd79b529ae62b230c3d6e": "\\lim_{\\Delta x\\rightarrow 0}\\frac{\\Delta y}{\\Delta x} = \\lim_{\\Delta x\\rightarrow 0}\\frac{f(x + \\Delta x)-f(x)}{(x + \\Delta x)-x},", "74593c73c25f4e49040326e20e97af78": " N_\\gamma = \\frac{ \\tan \\phi ' }{2} \\left( \\frac{ K_{p \\gamma} }{ \\cos ^2 \\phi ' } - 1 \\right) ", "74593f4bd73bce45c5d70762fda9d178": "\\omega_M", "7459574a6bd8df19e45b89f75e58971f": "\\partial^\\mu J_\\mu^B = \\sum_j \\partial^\\mu(\\bar q_j \\gamma_\\mu q_j) = 0. ", "745988782650804cdbdd45c92cb170c4": "(1,d)", "7459c2c6530d68fb0856bd933aa5cced": "y(x) = 4", "7459c942f815640fc3c3ce0a0d8b8f73": "n=1500", "7459ece43896be67f393a51b673aee33": "H^p", "745a18ecfaa095278943968ee3bd1572": "(\\hbar m_j)", "745a2d7154cc1621a45e87c3caeddbed": "H^2 = H", "745a37b9a10c06b372da3378f5b6fa36": "\\lnot (\\lnot \\phi \\land \\lnot \\psi)", "745a722199028b36303383f2d244dcee": "\\chi_\\ell:G_\\mathbf{Q}\\rightarrow\\operatorname{GL}_1(\\mathbf{Z}_\\ell)", "745ab1a9feac6f147b01d9e234df2737": "L_1^{*} = \\{\\epsilon\\} \\cup \\{w \\cdot z | w \\in L_1 \\land z \\in L_1^{*}\\}", "745b2afdffee5dd189b5b9b23cdc97b9": "\\nabla_{\\bold{v}}{f}(\\bold{x}) = \\lim_{h \\rightarrow 0}{\\frac{f(\\bold{x} + h\\bold{v}) - f(\\bold{x})}{h}}.", "745c07ceacbe11909a8f0bb75529c5d3": "\\scriptstyle(1.5\\pm1.5\\pm0.2)\\times10^{-13}", "745c1693dba53d8672e8d23af9085f66": "Qd = Qs + Qm", "745c89f2486ea89782b80618c8d6f3ae": "\\displaystyle \\mathbb{E} X_t \\to \\infty ", "745c93a3d809f05d68269d75a76bb363": "\\omega=\\eta=\\epsilon=\\tau=0\\;", "745cc2013b9a40dc46a9e447dad0c586": " c_{t+n}=(1-R^{-2} b^{-1}) A_{t+1} - \\frac{u_1}{u_2} \\frac{(R^{-1} b^{-1} L^{-1})} {1 - R} + \\frac{(1-R^{-2} b^{-1})} {1-L^{-1}R^{-1}} E_{t+n} y_{t+n} ", "745ce51e7d337a176b4b3d5e538634b4": "\\tilde{\\kappa_{norm}}(x, y;t) = t^{2 \\gamma} (L_x^2 L_{yy} + L_y^2 L_{xx} - 2 L_x L_y L_{xy})", "745d123a4270ef06c63143087d79a3d9": " n = 1.2 \\sqrt { \\frac{ c }{ 2j + 1 } }", "745d26be9e442b22b14409cca429fd19": "M_{\\perp}", "745d419a37df39fe06c80e695034d9a4": "Q = (2)\\mathbf Z[i] + (i+1)\\mathbf Z[i] = (1+i)\\mathbf Z[i].", "745d435e1b6f131087ca3eaac1a8d69b": " T = \\dot m\\, w = \\dot m\\, (2 v) = 2 \\rho\\, A\\, v^2.", "745d61daf9d0bd617e2643ba637ca92d": " \\langle f, v \\rangle = f(v) ", "745db14dd69934e839847d8f5e3e6a70": "\\varepsilon_{22}\\,\\!", "745db4e5e8c53a365f86c389b661b00a": "S = S \\cap C", "745dcc7543fc042a14775286803d036c": "V/T", "745ddb3398979ce4152231015f366200": "\\chi=\\pi - (r_0,s)", "745e0c7b0b3fa181f9c519d5f3319536": "\\Delta \\theta \\simeq \\frac{1}{\\frac{L}{\\lambda _{0}}cos\\theta _{m}}", "745e1a94478f42e6cb48900b4b8c9d6d": "G_{xx}=2 \\frac{e^2}{h}", "745e1ffe1ad26a8f1a4f9a17ff6e1a5f": "w^\\delta \\in A_p", "745e524258300879930350e1daa591a1": "{R^k}_i", "745edd626e4d4a514e70458540258d67": "K_0", "745ef6b71e6b4b272f3d673b75050d88": "\n g = \\begin{pmatrix} a & b \\\\ 0 & 1 \\end{pmatrix}\\; ,\n", "745eff023b9fb09a2995e9ef07e04ad2": "m = \\sqrt{ 6 \\sqrt{2} - 8} \\frac{\\sqrt{2}+1}{2}", "745f0356209b0301553bc985afe3a0f6": "x_{2n_2}", "745f05aab7528b8288dd55daf09a4cd3": " \\Leftrightarrow V_{Ba}'' + V_{Ti}'''' + 3V_O^{\\bullet \\bullet}", "745f7fcd95d1992272cd7f23eb267932": "B'=B(b', \\lambda)", "745ff760f8bbd4f3321d08c45f9ed952": "\\dot m_s = v_s \\cdot \\rho = \\dot m/A ", "74600831b53e6805fb7ae4414650dee1": "\\sigma_e^2", "74601e30bcd30284cbdd38fc38d30949": "\\mathbf{st}(f^*(x_{i_0}))= f(\\mathbf{st}(x_{i_0}))=f(c)", "746053c87ead1ae9805b7c4df642577d": " \\mathbf{u}(\\mathbf{r}) = \\mathbf{F} \\cdot \\mathbb{J}(\\mathbf{r}), \\qquad\np(\\mathbf{r}) = \\frac{\\mathbf{F}\\cdot\\mathbf{r}}{4 \\pi |\\mathbf{r}|^3} ", "746065ae1b2f5be140573385310497bd": "P = {n \\cdot V_{stroke} \\cdot \\Delta p \\over ~\\eta_{mech,hydr}}", "74608811e3dba3f4abb8eeb3f794e859": "g_m = \\frac{i_\\mathrm{d}}{v_\\mathrm{gs}}\\Bigg |_{v_\\mathrm{ds}=0}", "7460c7bc19a741162f008c6ee31e0666": "\\begin{align}\\operatorname{MSE}(\\hat{\\theta})\\equiv \\mathbb{E}((\\hat{\\theta}-\\theta)^2)&=\n \\mathbb{E}\\left[\\left(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta)+\\mathbb{E}(\\hat\\theta)-\\theta\\right)^2\\right]\n\\\\ & =\n\\mathbb{E}\\left[\\left(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta)\\right)^2 +2\\left((\\hat{\\theta}-\\mathbb{E}(\\hat\\theta))(\\mathbb{E}(\\hat\\theta)-\\theta)\\right)+\\left( \\mathbb{E}(\\hat\\theta)-\\theta \\right)^2\\right]\n\\\\ & = \\mathbb{E}\\left[\\left(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta)\\right)^2\\right]+2\\mathbb{E}\\left[(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta))(\\mathbb{E}(\\hat\\theta)-\\theta)\\right]+\\mathbb{E}\\left[\\left(\\mathbb{E}(\\hat\\theta)-\\theta\\right)^2\\right]\n\\\\ & = \\mathbb{E}\\left[\\left(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta)\\right)^2\\right]+2(\\mathbb{E}(\\hat\\theta)-\\theta)\\overbrace{\\mathbb{E}(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta))}^{=\\mathbb{E}(\\hat\\theta)-\\mathbb{E}(\\hat\\theta)=0}+\\mathbb{E}\\left[\\left(\\mathbb{E}(\\hat\\theta)-\\theta\\right)^2\\right]\n\\\\ & = \\mathbb{E}\\left[\\left(\\hat{\\theta}-\\mathbb{E}(\\hat\\theta)\\right)^2\\right]+\\mathbb{E}\\left[\\left(\\mathbb{E}(\\hat\\theta)-\\theta\\right)^2\\right]\n\\\\ & = \\operatorname{Var}(\\hat\\theta)+ \\operatorname{Bias}(\\hat\\theta,\\theta)^2\n\\end{align}", "7460d1788fac6cac7b8aa7968b86e8ee": "\\vec{E}\\times\\vec{B}", "7460d3174ebaf56a835686e2bfe31765": "|T|\\geq\\gamma'q", "7460dea1720644091b9c9dc2eaaa9dc6": "\\mathrm{azeq}_y", "7460deccbdab4cc76095a8261c98e763": " g_i(M) = 0 ", "746122016ce980c815639c3487d6771b": "\\begin{matrix} \\frac{rotational \\;velocity \\;vector \\;at \\;a \\;given \\;latitude} {rotational \\;velocity \\;vector \\;at \\;North \\; Pole} \\end{matrix}", "7461a418eab1d3c55b55758743685290": "Y \\leftrightarrow XY", "74620c63877dabc2e3090061bb5a3850": " m = \\pm m_0 ", "746236e0391b53b3dd41ba612765918b": "\\displaystyle{2Q(Q(a)b)=2R(a,b)Q(Q(a)b,a)-R(b,a)Q(a)R(a,b) +2Q(a)Q(b)Q(a)=2Q(a)Q(b)Q(a).}", "746257ff6a26864a06fb85adff740fe6": "a, b \\in \\widehat{\\mathbb{R}}", "74627f79bb5e6166ff59b7398e87c7e2": "(\\lambda_{1}, \\lambda_{2}, \\lambda_{3})", "7462d7fa751a34040500106376e4b169": "\\omega:e\\ddot\\to S", "746357f506879c9a234bee5e0dab56c1": "G^2(\\omega)=|H(j\\omega)|^2=\\frac{1}{1+\\omega^6},", "7463c691f1c7b68c7ca6964214e341d9": " (b_0 - \\tilde{b}_0) |\\Psi\\rangle = 0 ", "7463f0d95136a98a962971c189178cad": "R_1 = 1~ \\mbox{since}~0.0362\\ >= 0.02 ", "74641308c76e437c86bcd253e06a46cc": "\\psi\\left(\\nabla\\varphi\\cdot\\hat{\\mathbf{n}}\\right)dS = \\iiint _{V}\\left(\\psi\\nabla^{2}\\varphi+\\nabla\\varphi\\cdot\\nabla\\psi\\right)dV", "746452a58b12f45f269797a5970a3896": "P \\and P \\vdash P \\, ", "7464cf1b92c8fd0b791ba19c1e8d186d": "\\; - \\sum_j p_j \\log q_j.", "7464fb0d6f593d415557e5ba93b50f04": "(\\gamma^\\mu)^\\dagger\\gamma^0 = \\gamma^0\\gamma^\\mu \\,", "74653d7272bf388896921b9ffa7d07c5": "\\boldsymbol{m}_k = \\frac{\\boldsymbol{p}_{k+1} - \\boldsymbol{p}_{k-1}}{t_{k+1} - t_{k-1}}", "7465454e14527af168d91cb4b456fe47": "z \\in K", "74654f3a36fd0f4a8820d783a89ba19a": "X_t = \\frac{N_t - \\lambda/\\mu}{\\sqrt{\\lambda/\\mu}}", "74659b027fee7a7a0e4afd007f360cb7": "\n\\frac{1}{2}\\sum^n_{i=1}F^P_id^P_i = \\frac{1}{2}\\int_\\Omega \\sigma^P_{ij}\\epsilon^P_{ij}\\,d\\Omega\n", "7465d77d7c8a66201942fc5b9a8cc130": "H_r", "7466471289a7e9a385517673900d12bf": " \\operatorname{Var}(x_i) = \\sigma^2", "74668fda9857739140db14420d7668e2": "b = a \\times 2 ^ {n/1200}", "74669bf6a34def3534ffa45f3b215f62": "\\mathbf{v} \\cdot \\nabla", "7466ee932d43397a0f3c3a6ddc056438": "{\\sin \\gamma}", "7468178343effad1293c1b7721f9d736": "as_i+bt_i=r_i", "74681a80e488b979504a2d246079eeec": "N_{T} = N_{W_{\\alpha}XY}(\\hbar \\omega) + N_{B}(\\hbar \\omega),", "74682ffafc9e4142b3be273e990d4c4b": "\\mu_{\\ell,z} = -m_\\ell\\mu_B\\,\\!", "74688d0f933b6f244d0d8a01f4a4761d": "\\mathfrak{sp}(6,\\mathbb C)", "7468b8b8839ed2403303b01b6460ecf3": " \\bold A = \\left( \\begin{array}{rr}\n0 & 1 \\\\\n-1000 & -1001\n\\end{array} \\right), \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad (11) ", "7468c375c89dbe828200d58a4cb86b0f": "{1} + {1 \\over 3} + \\cdots + {1 \\over 2 a - 1} - {1 \\over 2} - {1 \\over 4} - \\cdots - {1 \\over 2 b} + {1 \\over 2 a + 1} + \\cdots + {1 \\over 4 a - 1} - {1 \\over 2b + 2} - \\cdots", "7468ce862e8748486af632cae7581f2a": "\nR_{se} = \\frac{\\gamma_e^2 B^2 T_{se} }{2 \\pi} \\frac{1}{2}\\left( 1-\\frac{(2I+1)^2}{Q^2} \\right)\n", "7468e2b3f7b2e6b121a603789a7546d1": "\\alpha \\wedge (\\beta + \\gamma) = \\alpha \\wedge \\beta + \\alpha \\wedge \\gamma. ", "74699fe03032243496d5500eddbe2906": "\\textbf{H}_{k}", "7469b9a4450c7f716c5cd403297138d0": "(\\delta_k)", "7469ce9776bfe0a8ebc03dfe011a1528": "f^2(x) \\equiv f(f(x))", "7469d66cd2f7ad75287f8a63bb13716e": "\\frac{d^2 f}{d z^2} + R f^2 = -1; \\quad f(-1) = f(1) = 0.", "746a008921aba87c539accf515e54b5a": "V \\otimes_R U", "746a76156b03aafd878fe64f2a91d11f": " t = t^n ", "746a7efa1a98de1a98cfc7d710b8f201": "B > 0", "746ae6c3e7cbc0c79d6c2a8e369e84af": "\\{ \\, |\\mathbf{r}\\rangle \\,\\} ", "746b62311c9533dacf9edda4d7cf3f88": "\\deg(f) = \\deg(g)", "746b63e977717533c8d1290d223ed188": " 2 T_{d,ave}/0.8", "746b663453614f321faa203df3833366": "\\ W_n > 0", "746b6d6c109758e280874b17da4e7ae7": "\n\\begin{align}\nf^{**}(x)\n&{} = {\\left(x\\cdot p_s - f^{*}(p_s)\\right)}_{|\\frac{d}{dp}f^{*}(p=p_s) = x} \\\\\n&{} = g(p_s)\\cdot p_s - f^{*}(p_s) \\\\\n&{} = f(g(p_s)) \\\\\n&{} = f(x),\n\\end{align}", "746b8c7b17b62508aba0e6ff527ea382": "p_0^* \\leftarrow r_0^*M^{-1}\\,", "746b9c22830a9f234db1acd35d5dfc5a": "R_{ij}=u_{i}^\\prime u_{j}^\\prime=-\\tau _{ij}/\\rho", "746b9c8c412a10062fc6efa3e0cea4c1": "\\{\\mathbf e_B \\mid B \\text{ is a basis of } M\\} \\subseteq \\mathbb{R}^n.", "746c4ae53cc132cd7496e852c01ae0de": "e^{-\\lambda} \\sum_{i=0}^{\\lfloor k\\rfloor} \\frac{\\lambda^i}{i!}\\ ", "746c896f681c3d25aac5841d227044e5": "\\mathbf{n} = \\mathbf{e}_r", "746cf19fd18f959f4f597c6218bc4696": "(a_n)_{n\\ge1}", "746cff3ec955ea4387f78b1938480c94": "z(t)", "746d03845bcb0136d5cf4016bc631c09": "(h'_e)_{\\alpha \\beta} = U_{\\alpha \\gamma}^{-1} (x) (h_e)_{\\gamma \\sigma} U_{\\sigma \\beta} (y)", "746d3a03bdfc0087da40fca520e4d9f3": "\n\\begin{array}{c|cccc}\nc_1 & a_{11} & a_{12}& \\dots & a_{1s}\\\\\nc_2 & a_{21} & a_{22}& \\dots & a_{2s}\\\\\n\\vdots & \\vdots & \\vdots& \\ddots& \\vdots\\\\\nc_s & a_{s1} & a_{s2}& \\dots & a_{ss} \\\\\n\\hline\n & b_1 & b_2 & \\dots & b_s\\\\\n & b_1^* & b_2^* & \\dots & b_s^*\\\\\n\\end{array}\n", "746d3c60e742d2d0bb2435a9e14b5c76": " \\rho = \\frac{ D - 1 } { n - 1 } ", "746d41aa557b7c3977995d3911f2cd01": "{{\\mathbf{k}}}^d", "746e093d70b5f1812dc6ce4f5e78c068": "f(x)=g(x)-h(x).\\,", "746e0992a194e240ff3f46eac3032ecf": "10log_{10}\\left[\\left(\\frac{d_2}{d_1}\\right)^2\\right] = 20log_{10}\\left[\\frac{d_2}{d_1}\\right]", "746e8d76efefec6c5939be2279d5936f": "v_{i}=\\sum_{j} L_{i,j}\\frac{di_{j}}{dt} ", "746f1432b47527841067ee6c03ca9dd5": "\nf(x|a,b,c)= \\begin{cases}\n 0 & \\mathrm{for\\ } x < a, \\\\\n \\frac{2(x-a)}{(b-a)(c-a)} & \\mathrm{for\\ } a \\le x \\leq c, \\\\[4pt]\n \\frac{2(b-x)}{(b-a)(b-c)} & \\mathrm{for\\ } c < x \\le b, \\\\[4pt]\n 0 & \\mathrm{for\\ } b < x,\n \\end{cases}\n", "746fafd95a63e15ff6be2dd01da0469f": "\\begin{align}\np(\\mathbf{X}|\\mu,\\tau) &= \\prod_{i=1}^n \\sqrt{\\frac{\\tau}{2\\pi}} \\exp\\left(-\\frac{1}{2}\\tau(x_i-\\mu)^2\\right) \\\\\n&= \\left(\\frac{\\tau}{2\\pi}\\right)^{\\frac{n}{2}} \\exp\\left(-\\frac{1}{2}\\tau \\sum_{i=1}^n (x_i-\\mu)^2\\right) \\\\\n&= \\left(\\frac{\\tau}{2\\pi}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{1}{2}\\tau \\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right)\\right].\n\\end{align}", "746fb5e09fcd9f4c1d7dddcd94a98478": "V_\\text{x}", "746fe7b4d3e87b7ec1d1bbb09c43db53": "\\!t_1 = t_2", "747056a43bf75d4f9047d46c5148204e": "\\begin{matrix}\nC^\\infty(M,TM)\\times C^\\infty(M,TM) & \\rightarrow & C^\\infty(M,TM)\\\\\n(X,Y) & \\mapsto & \\nabla_X Y,\n\\end{matrix}", "74712242e905c4f0e68f0ac7b39949d7": "\\mathbf{[Z']} = \\mathbf{[T]}^T \\mathbf{[Z]} \\mathbf{[T]} ", "7471a39c59bf294f6a62f1b9253cd5e0": "\\sin \\left[(kp+j)\\frac{q\\pi}{p}\\right]=\\sin\\left(kq\\pi+\\frac{qj\\pi}{p}\\right)=\\sin kq\\pi \\cos \\frac{qj\\pi}{p}+\\cos kq\\pi \\sin\\frac{qj\\pi}{p}", "7471ec673ad383ca94c3102cb92a59fd": "P(z) = \\binom{z+n}{n}", "74725c16c49e5d3c0f72da1351d9a91f": "\\zeta(z+w;\\Lambda)-\\zeta(z;\\Lambda)", "747267a43bc156c0d6a053febb833416": " \\partial ^2 \\rho =0 ", "7472f5cceb1aeeca3f86788293ca132d": "n^{k/4+O(1)}", "74735466555cb48d666eee23e680aff6": " DEF = \\sqrt{DSH^2 + DST^2} ", "74735527625bb5fcc0f3a8f786974692": "\\sec\\phi_1=1/0.9996=1.00004", "74739f983bc59aa41b1c5769368d1f1a": "E - m_0 c^2 = \\frac{1}{2} m_0 v^2 + \\frac{3}{8} \\frac{m_0 v^4}{c^2} + \\frac{5}{16} \\frac{m_0 v^6}{c^4} + \\cdots ;", "7473b0f70ca821f533722dbb99a5c42a": " \\mathbf{F} = - \\nabla U", "7473f663b0bb15661554d0fba72b2592": "\\frac{q^d-1}{2q^d} \\sim \\tfrac{1}{2}.", "7473f7a5d2cd9b0d59f502c8249e32e8": "f : X \\to \\mathbb{R} \\cup \\{ + \\infty \\}", "747418caeb9d03665955bd33f095640a": "id_\\mathcal{D}", "7474534872b9683c845b8039f69dd6a9": "\\sigma_{zz} - \\frac{\\sigma_{yz}\\sigma_{xz}}{\\sigma_{xy}}", "7474770cf48b9fa237af84df615e2e59": "\\theta_1>\\theta_0", "747490932c3830dd3e78c1d315dcbd37": "x \\in H^{n-4i} (BG;\\mathbb{Q} ).", "7474949c958817f6f546c7467c259417": "{}_2F_1 (a+m,b+n;c+l;z),", "7475065312f0702ab59f67f1ccade036": "\\Phi_{10}=D\\alpha-\\bar{\\delta}\\varepsilon-(\\rho+\\bar{\\varepsilon}-2\\varepsilon)\\alpha-\\beta\\bar{\\sigma}+\\bar{\\beta}\\varepsilon+\\kappa\\lambda+\\bar{\\kappa}\\gamma-(\\varepsilon+\\rho)\\pi\\,,", "7475309b6c49c1c0a6454bece69dc1a5": "\\Delta_{2}^P", "74755c623872a9103de7e2b0a33eb27b": "H_\\bullet(X)", "747586e0a31da2d8554cd5e6a192c0b6": " \\int_{0}^\\infty \n \\left|\\frac{ \\omega^2_p ( f^{(n)},t) } {t^{\\alpha} }\\right|^q \\frac{dt}{t} < \\infty ", "7475dcc6f443920db4bb67653b1974d7": "| E | \\geq \\prod_{j = 1}^{d} \\frac{| E |}{| \\pi_{j} (E) |}.", "7475ebc188b94ba21cce23dca81e0372": "p(x_j|x_1^{(i)},\\dots,x_{j-1}^{(i)},x_{j+1}^{(i-1)},\\dots,x_n^{(i-1)})", "7475f593723e550ba35c66c614793aa5": "\\mathbb{E}_\\theta [ (\\theta_i - X_i) h(\\mathbf{X})]= - \\mathbb{E}_\\theta \\left[ \\frac{\\partial h}{\\partial x_i}(\\mathbf{X}) \\right].", "7476806f221e2247ab2bb0929af8345c": "\\mu \\in \\mathbb R, \\sigma, \\gamma > 0, \\alpha", "7476dcba4edfd849a07cb5dcfb90ac01": "\\varphi(p\\cdot g) = \\varphi(p)g.", "74773a2294e70b5a97142cc3801e06c2": "\\frac{B_3}{{v_0}^2}", "74774b9b7746bfd073e5a83821b2502e": "s_1,s_2,\\dots,s_n", "7477735c462d472ebae345dc41fcb75a": "\\mathbf{v} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{d\\mathbf{r}}{dt}", "74783b522e2a394397cd092ad4ccc741": "\\begin{pmatrix}a & b\\\\b & c\\end{pmatrix}", "74784cadc14b6b84f9463c705b7d52c3": "\\mbox{Glass Property} = b_0 + \\sum_{i=1}^n b_iC_i", "74786e8af70f8fb19b5af8fcf9e5ee8f": "\\varphi:X\\times \\R\\rightarrow X", "7478c84ff2e127f7a25047bc16cd9950": "|S(\\alpha)|\\ll\\frac{N}{\\log^A N}", "7478fb1c0ee27ab6e5e1d71267fdfa24": " - \\log P(x)", "7479299541293d3ee57564450bbca767": "\\kappa_j'-i+1+\\alpha(\\kappa_i-j)", "74795081bf177d01c1b9c1bea752fd67": " \\alpha^*VCYC ", "7479764ea680c72ddc45e5a484f4c4ce": "(a,\\lambda).P", "7479a806f48587cec5f8aefae8e83299": "f(z,\\bar{z})", "7479af0ffccb5361ea8286ec5b85f053": "z_n, \\overline{z_n}", "7479ce9693791c2fb44015985476b673": " \\alpha (w)= \\frac {a_2|w|}{1+a_3 |w|} \\quad (2.1)", "747a0c81abe40ac3bddb38cfff6144a0": "\\operatorname{supp}(f) := \\overline{\\{x \\in X \\,|\\, f(x) \\neq 0 \\}}", "747a6e8d1c7c14492b2339cb05082d4f": "\n\\gcd(a,b) = \\gcd(b,c) = \\gcd(c,a) = 1.\\;\n", "747aa15df488fc70476c88effffd4793": "1+\\tan^2(\\theta) = \\sec^2(\\theta).", "747acc1d17b2483070b79218a6e1d298": "\\text{ Theorem (Lyapunov Optimization):}", "747b9838514557fff02aad7e52c250bb": "\\mathrm{O}_2 + \\mathrm{2H}_2\\mathrm{O} + \\mathrm{4e}^- \\longrightarrow \\mathrm{4OH}^-", "747c45a0adac14c9adb04aa9ea39ffd8": "\\scriptstyle v_\\mathrm i", "747c63f190fc68db10be20fe422cebc0": "\\operatorname{let} E \\operatorname{in} L ", "747c65e03d6f0e0593f4a516942382a0": "\\operatorname{vec} (X)", "747d02def9619fa2728937b826d0ef9c": "B^i_{jk} = T^i_{jk} + \\frac{1}{n-1}\\delta^i_ja_k-\\frac{1}{n-1}\\delta^i_ka_j ", "747d123a26ce8b37327d213e8ce9aeb7": "K = \\frac{QL}{Aht}", "747d2d627e4175c5967fc88f8392da21": "\n3.530972829\\ldots", "747e01dfa48bbb8e35745ddcc12d8271": "M(\\Theta, p) = p\\sqrt{2} W + (1-p)(\\cos\\Theta X + \\sin\\Theta Y)", "747e2385280cbf7935e281daa925aa69": "f(x,y) = -\\cos \\left(x\\right)\\cos \\left(y\\right) \\exp\\left(-\\left(\\left(x-\\pi\\right)^{2} + \\left(y-\\pi\\right)^{2}\\right)\\right).\\quad", "747e7fa5f6dad5fa15f0f2ab0a15353a": "p_{n,k}", "747e85a54f136ec04223a680961eba12": "y(r,\\theta) = \\Re ( 4i \\tan^{-1}(re^{i \\theta})) = \\frac{1+r^2-2r \\sin\\theta}{1+r^2+2r \\sin \\theta}", "747ee1ae2f1f2b07872843b46caf6bac": "M_{BC} = \\frac{2EI}{L} \\left( 4 \\theta_B + 2 \\theta_C \\right) = 0.8EI \\theta_B + 0.4EI \\theta_C", "747f3220e8d4299eaf257706d45e4c66": "E_{n,n-1/2}=\\frac\\gamma n\\mu c^2=\\sqrt{1-\\frac{Z^2\\alpha^2}{n^2}}\\,\\mu c^2", "747f9e7186b6e7dd5e49e12649df893a": "X^{(\\omega)} = \\{p_i^k \\mid k \\in X^{(i)}\\},\\,", "747fa3cdc8b988a203dc8c56149a5d6f": "\\mathbf{v} = \\nabla \\times \\mathbf{A}.", "747fb6eb6b07361d7dcbf7d9d79385ea": "{D\\boldsymbol{U} \\over Dt} = -2\\boldsymbol{\\Omega} \\times \\boldsymbol{U} - {1 \\over \\rho} \\nabla p + \\boldsymbol{g} + \\boldsymbol{F}_r", "747ffe0c8d3c77b3707c49988e631fa8": "S\\underset{\\mathrm{Spec}(R)}{\\times}\\mathrm{Spec}(k_t)", "74805fca1733949970793c5b115fb8be": "x \\div 1", "7480b740b9805e2ddc5fd6ddb8d3bf09": " \\mathbf{v} = \\nabla \\times \\boldsymbol{\\psi}", "7480f8d3c60f41decd1d31fc92cdd421": "y=Ae^{-t}\\,", "7480ff0a46bd2e8277f1ee1f568ed3a7": "\\tfrac{S(2)}{S(1)} = \\frac{\\big(\\tfrac{Q(2)}{Q(1)}\\big)}{\\big(\\tfrac{P(2)}{P(1)}\\big)}", "74813616fc31966c110e7fe9657ef8a5": "\\textbf{F} = d\\textbf{A}.", "748161bd5b83d74e1ee0072d2fb8d5b2": "\\cup \\{ k \\mid k \\text{ circle of } {\\mathfrak A}(\\R)\\} ", "74817e216e880a0a469a09380b307659": "f_1(x) = x^\\pi \\ ", "74825ca9b2f88df56bbb019019c51ffe": "c_{3}", "74827bb976f8191b34bb2dc9fc65b8a6": "\\mathrm{R}_\\mathrm{m} \\gg 1", "74828251a26f12b2365ac8b47e8b38df": "\\mathit{SPC} = \\mathit{TN} / N = \\mathit{TN} / (\\mathit{FP} + \\mathit{TN}) ", "748313066baf5e80e875d513a18604cd": "\\scriptstyle c", "74831c4fca7c634cab14b80e4b44839d": "|F_n| = 1 + \\sum_{m=1}^n \\varphi(m).", "7483761560363ec1de70da740b69e7ec": "Z^{-1}_1", "7483a7dd030c9e408eff4a2056a7e558": "x_1<\\cdots j = 1}^N u(\\left | \\mathbf{r}_i - \\mathbf{r}_j \\right |)", "74956b3b39ee4f0e7f9e782ce99ba9ff": "G^{\\mu\\nu a}\\tilde{G}_{\\mu\\nu}^a = \\partial^\\mu K_\\mu", "74960fc85b17e41b63ef9673042f1b92": " \\textbf{r} = r \\cos(\\omega t) \\hat{x} + r \\sin(\\omega t) \\hat{y} ", "7496254c5bc195a2881bb0011fb9de30": "\\frac{d}{dx} \\left( A x e^x \\right) = A x e^x + e^x", "749626ef39293993cf1ed177216219e9": "(n-1)! \\times n =: n!", "74963b0604f06d5ad0763b3dbc4bd04f": "(\\mbox{fuel-to-oxidizer ratio based on number of moles})_{st} = \\left(\\frac{n_{\\rm C_2H_6}}{n_{\\rm O_2}}\\right)_{st} = \\tfrac{1}{3.5} = 0.286 ", "74969e9db5c25350cbc15e3973ac887b": "\\int_I\\varphi^2(x)\\rho(x) \\, dx = \\frac{4\\pi^2}{3}\\int_I\\rho^3(x) \\, dx.", "7496d0623d0920164fd022838a5926b8": "O(\\sqrt V \\log^* V + D)", "7496f8e8be84ecc5c144afacc8c511df": "\\ \\chi(\\cdot)", "7497173ad509a4b287733ce90d63d41d": "\\cdots", "74972e38679ce44ea60bb7c47eab5215": "\n \\cfrac{\\Gamma, A \\vdash B, \\Delta}{\\Gamma \\vdash A \\rightarrow B, \\Delta} \\quad ({\\rightarrow}R)\n ", "7497503eb8ef9f15c6bc854ad6d46ef7": "(\\forall)", "7497d1363daa40ee5ee647f04d797e98": " P_\\mathrm{proj}=\\frac{1-|\\lang\\phi|\\psi\\rang|^2}{2}.", "74981aa3ed6355b7526c250936449915": " \\mathbb{Z}_q", "7498494daf38ec24d36a11dd30add8a8": "O(\\log n/P(n))", "74984a5e49e99bdd87105cbbe52020ad": "\\beta=b", "7498fe274f8216cb5d5113aa2e811b70": "A_m(0,r){{=}}\\binom{r}{m}", "7499886838bc50000f8ba557bcecca18": "a_0+h", "749990fccde070c494a64b3606b8c8a1": "\\begin{array}{cc}\n \\begin{array}{rrr} \\\\ &1& \\\\ 2&& \\\\ \\\\&&/3 \\\\ \\end{array}\n \\begin{array}{|rrrr} \n 6 & 5 & 0 & \\text{-}7 \\\\\n & & 2 & \\\\\n & 4 & & \\\\\n \\hline\n 6 & 9 & & \\\\ \n 2 & 3 & & \\\\ \n \\end{array}\n\\end{array}", "74999dc303a04331e78857dad6f03e75": " \\frac{1}{\\sqrt{\\varepsilon_0}}\\left(q, \\rho, I, \\mathbf{J},\\mathbf{P},\\mathbf{p}\\right) ", "7499a9893df380d7fcfd0e84dad42ddc": "\\psi(x)\\,", "7499dd583e7f47ada6e26bf037df9236": "\\{ 1, 2, \\dots, n \\}", "7499efeab32cd877ea37fab78270ad17": " \\Delta x_{\\rm meas}(t_{j+1})", "7499f3610b43ae1ff63c35cbac99dfd0": " p(.|\\theta) ", "749a140407ee93b863fa3214633214f3": "i = j= 0", "749a80d96a270e6f427963e5b751ea57": " \\mathbf{r} \\cdot \\mathbf{r} \\equiv r^2 \\equiv x_1^2 + x_2^2 + x_3^2 \\,\\!", "749a866861684d9b7b4cae8603179174": "p_{ik} ", "749ab53b6d83a2e1609ce1fde73e73a8": " \\mathbf{u}=\\mathbf{u}_1\\times\\cdots\\times\\mathbf{u}_n= [\\underline u_1,\\overline u_1]\\times \\cdots\\times [\\underline u_n,\\overline u_n] ", "749abd77848a314d9d89ba25dd25d491": "s_2, s_3, s_8", "749ad12d4f90ae81bac622dd306d24c5": "\\{(x-h, y), (x, y), (x+h, y), (x, y-h), (x, y+h)\\}, \\,", "749b1630f5add65ea4de25efa6693354": "e_{ijl}\\, ", "749b4de2fd73333a732f5a1d97647e16": "\n\\begin{align}\n& {}\\qquad P(\\text{malignant}|\\text{positive}) \\\\[8pt]\n& = \\frac{P(\\text{positive}|\\text{malignant}) P(\\text{malignant})}{P(\\text{positive}|\\text{malignant}) P(\\text{malignant}) + P(\\text{positive}|\\text{benign}) P(\\text{benign})} \\\\[8pt]\n& = \\frac{(0.80 \\cdot 0.01)}{(0.80 \\cdot 0.01) + (0.10 \\cdot 0.99)} = 0.075\n\\end{align}\n", "749b53474a45ed4495e58635dae12aae": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon} \\frac{1}{R} =\n\\frac{q}{4\\pi\\varepsilon} \n\\frac{1}{\\sqrt{r^{2} + r^{\\prime 2} - 2 r^{\\prime} r \\cos \\gamma}}. \n", "749b54d83afb919daf8ac95df7b0eebe": "y = \\rho_0 - \\rho \\cos[n (\\lambda - \\lambda_0)]", "749b6b18eb295d828f5563c7b1f6b5a9": "\\left\\{ p_{X}\\left( x\\right) ,\\rho_{x}\\right\\}\n_{x\\in\\mathcal{X}}", "749ba23ac8493e67121e4810ddd5155d": "G \\sub \\mbox{GL}(n,\\mathbb{R})", "749bbac1a4db178b9fde92e9a1c56090": "\\ S=\\frac{P_tG^2\\lambda^2}{(4\\pi)^3R^4}\\sigma", "749c60207adb3082980fa76d60983d1e": "q = \\left\\lfloor {n_M} / {n_N} \\right\\rfloor", "749c662b44bd83c54e15b67f547a121a": "\\Delta_rG", "749c7961493c7a400890412a2d28ae7b": " 1-\\alpha ", "749c93a45d9a299c8ec9b7ff75cf0543": "p:\\tilde{X}\\to X", "749dc8318b99d45c4ba32d97bc440c55": " (y- f(x))", "749df8e477a24a9c51c39fb59b1b6622": " s^{t}", "749e754390e9522202e565872abdcfab": " \\mathcal{F}(t) \\triangleq \\sigma\\left(\\mathcal{F}^\\mathbf{W}(t) \\cup\n\\mathcal{N}\\right), \\quad \\forall t \\in [0,T] ", "749e83d5fffc571e8849ea4f681a21a2": " \\mathrm{tf}(t,d) \\neq 0", "749edd4be866597b2c1feaaf3a4f99a8": "M_\\alpha", "749ef18c454231f193108c21536fd827": " U_n = \\frac{1}{\\sqrt{n}} F_n, \\quad\\text{where}\\quad F_n = (f_{jk}) \\quad\\text{with}\\quad f_{jk} = \\mathrm{e}^{-2jk\\pi\\mathrm{i}/n}, \\quad\\text{for}\\quad 0\\leq j,k \\sqrt{2(n+1)d} - \\frac {d}{2} -1 ", "74cb56a80b4e3a92776c3cb3a66e8d97": "\nS(x,y) = \\sum_u \\sum_v w(u,v) \\, \\left( I(u+x,v+y) - I(u,v)\\right)^2\n", "74cb5d219d18eff7d6140db08044d28b": "s(x)=\\sum_{k=0}^{\\infty} F_k x^k.", "74cc4ffab78203dd31299e6934f65bd2": "\\mathbf{\\nabla}\\cdot\\mathbf{H}(\\mathbf{x})=-\\mathbf{\\nabla}\\cdot\\mathbf{M}(\\mathbf{x})", "74cc83f2a9a2ab1642d39173c7708a4c": "\\Phi_*(v_1\\otimes v_2\\otimes\\cdots\\otimes v_r)=\\Phi(v_1)\\otimes \\Phi(v_2)\\otimes\\cdots\\otimes \\Phi(v_r).", "74cc844bb4a9c119dfc1d119c99e1135": "3n/\\lg \\lg n + O(\\sqrt{n} \\lg n \\lg \\lg n) = o(n)", "74ccb8a03a984d349e733362e04ee1e0": "{\\partial P(x,p,t) \\over \\partial t} = - \\{\\{ P(x,p ,t) ~,~ H(x,p)\\}\\}~,", "74cccd6aad3571eefd7a498a6e7b51af": "= \\lim_{N \\to \\infty} \\sum_{i=1}^N b \\log_2 \\left( \\frac{A/N}{W} + 1 \\right)", "74ccdc33fbdc75eea42ae4768f0803fd": "V_Y^2=1.4742Y-0.004743Y^2", "74ccde2e4d7d8694d59dd868cf898107": "x^d-1 = \\prod_{m|d} \\Phi_m(x)", "74ccf762235ff3c6f2ebb4f6b2d7f20e": "\\left|E_0\\right|^2", "74cd029cec498b21ad943a21d8fa48a2": "\\operatorname{E}[g(X)] = \\sum_x g(x) f_X(x), \\,", "74cd3cf4d7c72eda89c2e7c893b5ded2": "\\psi(x_n)", "74cd45b5e99d239f70fb692b6ac2bb8e": " R_{xyz} ", "74cd46df1b7e2e423ed0df65501c5014": "10^{10^{10,000,000,000}}", "74ce4855bfc7490fbc2d03e0e48ab993": "Y^TA_iY", "74ce6a00748ef318248ce3959c48c3c5": "\\frac{dQ^S/Q}{dP^S/P}<\\left|\\frac{dQ^D/Q}{dP^D/P}\\right|,", "74ce727ef35378c4f109295b72054fc7": "f_{\\ell m}(k r)=A_{\\ell m}^{(1)} h_\\ell^{(1)}(k r) + A_{\\ell m}^{(2)} h_\\ell^{(2)}(k r)", "74ceb5c78af0f31e088010d57549a145": "E_ \\mathrm {FSR} = V_ \\mathrm {RefHi} - V_ \\mathrm {RefLow}, \\,", "74cede822a892237b78fad33c491254e": "\\Delta w_{ij} \\propto \\langle v_ih_j\\rangle_\\text{data} - \\langle v_ih_j\\rangle_\\text{reconstruction}", "74cee1c60e3b237e20e7012551cfdaf2": "\\sigma_x(t) \\sigma_p(t) = \\frac{\\hbar}{2} \\sqrt{1+\\omega_0^2 t^2}", "74cf27ad50eb2d5e3b95b47b79b5f940": "(P \\land Q) \\vdash P", "74cf91bcfe210983f12447cf5103c505": "n \\ ", "74d007a652f1fb7067c2795529d44b0c": "{\\sigma}(\\mathbf{X}_t,t)", "74d01491e82d9ded16f71d34936db9b7": "x_{\\lfloor h \\rfloor} + (h - \\lfloor h \\rfloor) (x_{\\lfloor h \\rfloor + 1} - x_{\\lfloor h \\rfloor})", "74d0548c10191d178d17eadaf4393e3a": " \\textstyle \\pi_i ", "74d0b282f76a33830311c655fb344f23": "R_\\infty = \\frac{\\alpha^2 m_e c}{4 \\pi \\hbar}.", "74d0fb3a322b7be85ddacbf354615a30": "x,y \\in B, x \\neq y", "74d0fb44de9b737d12758e981c67f711": "a_{n+1} = \\frac{a_n + h_n}{2}, \\quad a_0=x", "74d11d577748888a85bbc4644d4696d6": " (2\\pi r)(2\\pi R) = 4\\pi^2 Rr", "74d142d8f8035013ee5eba4cef2deffc": "D=S_1,S_2,\\dots,S_d", "74d177ec6a130bb184d55b72d85f6bb6": "P_{\\text{total}} = p_1 +p_2 + \\cdots + p_n", "74d1782ddf738595930fa06be1cc2480": "\\mathbb{S}_5\\;", "74d18ef98d2e75d252959cea2413ed5e": "\\sigma>0 \\,", "74d19c7ceb3a46f56d71f7e025d45147": "\n\\left(\\sum_{i=1}^{n}a_{i}b_{i}\\right)\\left(\\sum_{i=1}^{n}\\overline{a_{i}b_{i}}\\right)-\\sum_{i 1 \\mid N(t+\\Delta t) - N(t) \\geq 1)=0 ", "74da13eb4a816bcb107f0baa2cc42066": " \\langle \\; \\cdot \\; \\rangle^\\theta", "74da977ee82507a0515eb5205535a4be": "X=\\left[\\begin{matrix}1 & x_1 \\\\ \\vdots & \\vdots \\\\ 1 & x_n \\end{matrix}\\right]", "74dad2a57e1039418305846b0d4e952b": "(TM \\to M) \\times (\\mathbf{R}^n \\times \\mathbf{R}^n \\to \\mathbf{R}^n).", "74db419a9a12691df2ee3b7970e1d5de": "\\phi(x+y) =\\phi(x)+\\phi(y)\\,", "74db5d27537ba8ada68cc9a9cb5c1972": "v=\\left( z^{\\prime}|x^{\\prime\n}\\right) ", "74db6baecaa8bc4a1b1bb53dd38bb90a": "\\mathbf{v} = \\frac{ \\mu}{1+(\\mu B)^2} \\left( \\mathbf{E} + \\mu \\mathbf{E \\times B} \\right), \\ ", "74db9bded28f3c86fb877e73995e650c": "r_b=r_f", "74dc3a03162e4a79e06cd3f07b234cf0": "N(d_+) ~ F", "74dc4ddf088b115a33f03ac3844eb0cc": " z = \\omega", "74dc9595fe71d0b70ac22a5b7b6bca70": "H_7(x)=128x^7-1344x^5+3360x^3-1680x\\,", "74dca6d5883930bf733c6ff9eaacf9b3": "\\mathbb O", "74dcc4e609a14114716904a7322beade": "(a_{k_0},a_{k_1},\\dots)\\,", "74dceba62b0df9b7e991970fb761a81e": "\\operatorname{Cov}(g(X),Y)=E(g'(X)) \\operatorname{Cov}(X,Y).", "74dd29ecd31b6aae89b70fdf8350f2c3": "\\rho(x,y)=x^2+xy+y^2", "74dd34adc0c5a21dca943de1271b2745": "\\hbar\\mathbf{k}", "74ddb10c7c8e27a7e2cd587a2e398434": " \\vec{E}(z,t) = \\begin{bmatrix} e_{x} \\\\ e_{y} \\\\ 0 \\end{bmatrix} \\; e^{i(kz - 2 \\pi f t)} ", "74de06ceb1e27973f41e559c1206a0e5": "d\\rho=0", "74de1ccbb9b6f917de71c966cabf4836": "\\Delta\\varphi^*=2\\pi\\frac{2eV}{h}t.", "74de423681929a3b10e6eee2e3cc4246": "\\lambda_2 = 1000,", "74de570717f3e0e2b6ad2d3fd2cb0246": "\n0 = \\frac{d^{2}r}{dq^{2}} + \\frac{1}{2v} \\frac{dv}{dr} \\left( \\frac{dr}{dq} \\right)^{2} - \\frac{r}{v} \\left( \\frac{d\\theta}{dq} \\right)^{2} - \\frac{r\\sin^{2}\\theta}{v} \\left( \\frac{d\\phi}{dq} \\right)^{2} + \\frac{c^{2}}{2v} \\frac{dw}{dr} \\left( \\frac{dt}{dq} \\right)^{2}\n", "74de5eb7a49e75ad68d67ced2c92c698": " h^{ab} ", "74de67d69e847d800cbddc59b1114d2e": "|0\\rangle = \\left(1,0\\right)", "74de7ba22399ea9f75776157d0f1d557": "\\scriptstyle{\\Vert\\cdot\\Vert_{p,\\omega_k}}\\to\\mathbb{R}^+", "74de958c7a659d291fdbd0449d38b7a8": "b_i(x)", "74df0022c392de4e464007ea6470824e": "I(C, T) = \\sum_{c\\in {0, 1}}{ \\sum_{t\\in {0, 1}} {p(C = c, T = t)log_2\\left(\\frac{p(C = c, T = t)}{p(C = c)p(T = t)}\\right)}}", "74df0b2d64cf5f7f87ddc3278631bacb": "\\int \\operatorname{li}(x) \\, dx = x \\operatorname{li}(x)-\\operatorname{Ei}(2 \\ln x) ", "74df1528022268233ffecb1a9a526fac": "\nZ = \\sum_{i=1}^W B(E_i),\n", "74dfa9514f27475bf85084685dd8013b": "|\\lambda|^k\\|\\mathbf{v}\\| = \\|\\lambda^k \\mathbf{v}\\| = \\|A^k \\mathbf{v}\\| \\leq \\|A^k\\|\\cdot\\|\\mathbf{v}\\|", "74dfb2480c8a9dc11b849eb6a548dab7": "\\,\\!\\theta = \\alpha", "74dfb3d40b1a8e3c615a0bdcca5d652a": "s_j = R(\\alpha^{j}) = C(\\alpha^{j}) + E(\\alpha^{j})", "74e05c51b01954930ef3401087a38d45": "x:\\mathbb{R}^{2} \\longrightarrow S'", "74e095a5fc055ec99c29bc679e1cdbd3": "y(n) = f_m(n) \\, ", "74e0afa8537cc31b47f2c048a23f8f4a": "\\langle\\overline{z}\\rangle=\\frac{I_1(\\kappa)}{I_0(\\kappa)}e^{i\\mu}.", "74e10116eef94a8113a253552b5dc5f2": "Deflection = \\frac {Force}{Stiffness} = \\frac {F_{static}}{2k} ", "74e1138c9e6f8a9a47c95169ef47ad9a": " P(\\partial_t,D_x)u(t,x) = F(t,x) \\,", "74e12c16e6f095e219fb616b1be8031b": "\\begin{pmatrix}1/\\sigma^2&0\\\\0&1/(2\\sigma^4)\\end{pmatrix}", "74e13d49311adfd0d9dcbe165571a2fb": "-412''\\sin(2F)", "74e1474aca73cb32699df93537331748": "x \\in [0.1, 0.8]", "74e163c8e7cf3569a4e76f84f75df48a": " 1 \\ \\mbox{ft} = 0.3048 \\ \\mbox{m} \\ ", "74e1654a27b0ebf38e76301d61f7b18b": "k = \\kappa\\frac{k_BT}{h}e^{\\frac{\\Delta S^{\\Dagger }}{R}}e^{\\frac{- \\Delta H^{\\Dagger }}{RT}}", "74e1acb1705102ca01d2effa16eaeb4b": "\\textbf{z}_{k} = \\textbf{H}_{k} \\textbf{x}_{k} + \\textbf{v}_{k}", "74e1afdf34e2cdd6945ec34ef51d8579": "K(t,x,y) = \\frac{1}{(4\\pi t)^{d/2}} e^{-|x-y|^2/4t}\\,", "74e1b467d9bad1b30b5258cd362df9c5": "+\\left(y-1\\right)^{2}\\left(1+\\sin^{2}\\left(2\\pi y\\right)\\right).\\quad", "74e1b951567aca4ac0cd0ab82cde2127": "\n\\begin{bmatrix}\n A_{11} & A_{12} & A_{13} \\\\\n A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \n\\end{bmatrix} =\n\\begin{bmatrix}\n \\langle\\hat e_1 | \\mathbf{A}\\hat e_1 \\rangle & \\langle\\hat e_1 | \\mathbf{A}\\hat e_2 \\rangle & \\langle\\hat e_1 | \\mathbf{A}\\hat e_3 \\rangle \\\\\n \\langle\\hat e_2 | \\mathbf{A}\\hat e_1 \\rangle & \\langle\\hat e_2 | \\mathbf{A}\\hat e_2 \\rangle & \\langle\\hat e_2 | \\mathbf{A}\\hat e_3 \\rangle \\\\\n \\langle\\hat e_3 | \\mathbf{A}\\hat e_1 \\rangle & \\langle\\hat e_3 | \\mathbf{A}\\hat e_2 \\rangle & \\langle\\hat e_3 | \\mathbf{A}\\hat e_3 \\rangle \n\\end{bmatrix}\n", "74e1c760730f3bd4c58699ee4b2ccae6": "\\theta(x)\\Big|_{-j\\infty}^{j\\infty}\\,", "74e205b6a7e5e891cc4fe10f0b102155": "1/p + 1/p' = 1", "74e233739e3deddfbf68ccabb4da53a3": "w = \\sum_i y_i \\alpha_i", "74e23dd95fe256a1e180507454402431": "p\\equiv 3", "74e2568ca3a35073441ab2b2a1f5db14": "a_{i,j}=b_{i+j-2}", "74e32d036c20bf9a257575fa7798d719": "\\sum_{q}\\left[v\\right]_{q}=1", "74e344f2035faa26ef34972b45628a0a": " d\\xi~", "74e35484114161922a642cb8a4b1f737": "\\frac{\\left|\\tilde{X} - \\bar{X}\\right|}{\\sigma} \\le (3/5)^{1/2}", "74e3b2b6e9abaac34e6cec5d7f3e67ab": "\\scriptstyle \\mathbf{\\hat{r}}", "74e44f81d4ad6fdd9c3be527aefdc0f3": "\nS = 1 + \\frac{1}{1!} + \\frac{1}{2!} + \\cdots + \\frac{1}{(m-1)!} =\n\\sum_{j=0}^{m-1}\\frac{1}{(m-1-j)!} ,\n", "74e48ba8fac554e0e78848fec502e0a6": "\n N_{FP} = 0.8 {F_{Dn} \\over 2 T_{d,ave}}\n", "74e4946836ee56496e6cb921158c6021": "\nS(t) = \\begin{cases}\n(t+1)^2-1 & -2 \\le t < 0\\\\\n1-(t-1)^2 & 0 \\le t \\le 2\n\\end{cases}\n", "74e4fb60554fb19ca5ee4bc673e5a9e8": "a^{(A-1)/2}\\equiv +1 \\pmod A\\;", "74e50cd1b9eae465c0a2e670ed4d4e47": "\n \\begin{align}\n F_1 & + 2\\int_{\\alpha}^{\\beta} \n (C_1\\cos\\theta + C_3\\sin\\theta)~\\cos\\theta~ d\\theta = 0 \\\\\n F_2 & + 2\\int_{\\alpha}^{\\beta} \n (C_1\\cos\\theta + C_3\\sin\\theta)~\\sin\\theta~ d\\theta = 0 \n\\end{align}", "74e541979740d3264237d036253b5154": "\\vec x(t)=(\\vec x_1(t),\\ldots,\\vec x_N(t))", "74e58e825acb648faaa8ff845d0b6969": "\\Delta T= -\\text{tr}\\;\\nabla^2 T,", "74e5a238a1c0105719cdfa2ac1ffaa99": "\\mathbf n_1", "74e6221e2dfc409db0ed317f33343583": "\\cdots\\to L_2G(C) \\to L_1G(A) \\to L_1G(B)\\to L_1G(C)\\to G(A)\\to G(B)\\to G(C)\\to 0", "74e6992b1506538121839877460777fa": "P^{1-\\gamma}T^\\gamma = C", "74e6c79d729609f9c1852849ce522b6e": "SSE=\\sum_{i=1}^n e_i^2. \\, ", "74e700bdb7b66b2a42444cce416f5c03": "Q = \\frac{\\Delta P \\pi r^4}{8 \\eta L}", "74e753fdd72ca83715e15c692a9a120b": "H_{2x,2}=\\frac{1}{2}\\left(\\zeta(2)+\\frac{1}{2}\\left(H_{x,2}+H_{x-\\frac{1}{2},2}\\right)\\right)", "74e7ae58725135688d56f07c7fbb7085": "P(n) = O\\left(\\frac{1}{\\ln n}\\right)", "74e7daec3e32f9302f349094f9155586": "\n{\\Delta f \\over f} = {\\Delta E \\over E}\n", "74e8be1b2e4891b5b82fb7078924c9eb": "a=K_a\\left(\\frac{X/X_n-Y/Y_n}{\\sqrt{Y/Y_n}}\\right)", "74e930d0c328051679a83dbd139e85b6": "(x_{1},x_{2}) \\in X", "74e9c6b2480086618327e3770855a69f": "qx+(1-q)y", "74e9ecacc2cd75f79766c88f58bc8b80": "u(k)", "74e9fe7c527c835a46dba06da94812e9": "L = \\int_0^{t_0} {\\sqrt {\\mathrm{d}x^2 + \\mathrm{d}y^2}} = \\int_0^{t_0}{\\mathrm{d}t} = t_0 ", "74ea5a87e9cd5cec6b0e397b7e0f036e": "\nu_{pm}(\\rho,\\theta;\\zeta)=\n \\sqrt{\\frac{2^{p+|m|+1}}{\\pi\\Gamma(p+|m|+1)}} \\frac{\\Gamma(1+|m|+\\frac{p}{2})}{\\Gamma(|m|+1)}\n \\,\\,i^{|m|+1}\\zeta^{\\frac{p}{2}}(\\zeta+i)^{-(1+|m|+\\frac{p}{2})}\\rho^{|m|}e^{-\\frac{i\\rho^2}{(\\zeta+i)}}e^{im\\phi}{}_{1}F_{1}\\left(-\\frac{p}{2}, |m|+1;\\frac{r^2}{\\zeta(\\zeta+i)}\\right),\n", "74ea7451de5c12eebd42ec5cb675aeec": "W_u(x)", "74ea8382743282302527dddbb2e7c22e": "d=25", "74eac41e2692566d11c6cf1ebbbb9729": "k\\ge\\exp(c\\sqrt{\\log d})", "74ead709844ff4ef00ff3359e0150db3": " M(t,0) \\simeq (-t)^{\\beta}", "74eb368dd24485b6867cf518e14ccd34": "x_{-M}\\le x_{-M+1}\\le\\cdots\\le x_M", "74eb4e994c81baa2147ffee2492c5adc": " N=[n_i + n_\\mathrm{osc}] ", "74eb962d07050b1aab728790c295b694": " D = \\left\\{ A \\in \\sigma(I) \\colon \\mu_1(A) = \\mu_2(A) \\right\\}. ", "74ec7a7c0d96d170341094ac66f9471f": "\\scriptstyle \\log_{10} {P_{mmHg}} = 7.87863 - \\frac {1473.11} {230.0+T}", "74ec89aed56525ceb5643ab6bad2470a": "\\frac{dg}{d \\ln \\mu} =\\beta(g)=\\beta_2 g^2+\\beta_3 g^3+\\ldots ", "74ecb46b19869bd4c5a2dd59c1caa65f": "f:M \\to \\mathbb{R}", "74ecbd3cca8bd557a58fb876e5d92d7d": "x'=l'\\|r'", "74ecc9736ce1d10eeced660d4a100782": "\\gamma\\subseteq\\Gamma", "74ecd606520bfa969f8cfbf4c30ba570": "\\mathrm{I\\!I}(v,w)=(\\nabla_v w)^\\bot, ", "74ece100747bbf66b451a37884f1c797": "\\mathbf P^\\mu", "74ecf2ce6cf204cddf93a394377d6876": "\\sigma:G \\to G", "74ed040c88c777ac93f8a1770e1752b0": "{\\scriptstyle a\\to -\\infty}", "74ed27e504577cdd64744356d71d570d": "x\\leq10", "74ed68a9c048f8368a773df965a4ab57": "x(x(x(xx)))", "74ed7dc6a5439552efc39abd058dc272": "{{P}_{OS}}={{K}_{1}}*\\text{ }IP{{C}_{OS}}+{{K}_{0}}", "74ed8ebd8556314d8fd3db05cb0645f4": "O(n^2 \\log |G| + tn)", "74ed8fd436cc7fa3036317cb0f642ade": "\\vec{v}_R = \\frac{2\\epsilon_\\|}{qB}\\frac{\\vec{R}_c\\times\\vec{B}}{R_c^2 B}", "74edc679810805b44e818a032cbe3390": "p \\ge 3", "74ee0ab6dea81428208a4bc72fb29502": " \nP( T \\le t ) = 1 - \\Phi( \\frac{ \\omega - t \\mu }{ \\sigma \\sqrt{ t } } ) \n= \\Phi( \\frac{ t \\mu - \\omega }{ \\sigma \\sqrt{ t } } ) \n= \\Phi( \\frac{ \\mu \\sqrt{ t } }{ \\sigma } - \\frac{ \\omega }{ \\sigma \\sqrt{t} } )\n= \\Phi\\left( \\frac{ \\sqrt{ \\mu \\omega } }{ \\sigma } \\left[ ( \\frac{ t }{ \\omega / \\mu } )^{ 0.5 } - ( \\frac{ \\omega / \\mu }{ t } )^{ 0.5 } \\right] \\right)\n", "74ee253b0336ce5eed058046c1095eee": "J(u) = \\int |\\nabla u|^p \\,dx", "74ee2ca6bf6e7de29fa9ba4116b50ea9": "q_1 \\cdots q_n", "74ee9bc01f6dd49a0c328c9ede55593c": "\\scriptstyle U\\subset X", "74eee2c733761cb5468396bb5bf79506": "(\\operatorname{gr}_I R)_0 = R/I", "74eee4bf044b86af78e5dfaf08337aed": " \\mbox{subject to: }\\ ", "74ef881238f68fc76e7171d0fe68a288": "E(X^2)-(E(X))^2=\\mathrm{var}(X).\\,", "74ef9c567b735bdd357a7628fb270d74": "\\;\\text{Activity} = f(\\text{physicochemical properties and/or structural properties})+\\text{Error}", "74f0041c102e6937acfa868f64e32cf1": "T=\\sum_{j\\in\\mathbb{N}}T_j", "74f0cb401a0fd5ef1a0b3c53f32a5d12": "Q(u) = u_1^2 + u_2^2 + \\cdots + u_n^2", "74f1ab642f8b81db3fcb87db83f7d4d8": "\\scriptstyle \\sqrt[3]{31}", "74f1e460f7d9e3ff088a2457b15ce585": " \n\\begin{align}\n & \\sum_{m=0}^n (-1)^m (z - a)^m \\left[\\phi^{(n - m)}(1)f^{(m)}(z) - \\phi^{(n - m)}(0)f^{(m)}(a)\\right] \\\\\n {} = & (-1)^n(z - a)^{n + 1}\\int_0^1\\phi(t)f^{(n+1)}\\left[a + t(z - a)\\right]\\, dt.\n\\end{align}\n", "74f1fe7d695c974585a7f181aa3ed942": "\\Vert g \\Vert > N", "74f211c76efab65898e6d44608165e62": "\\vdash E \\rightarrow F ", "74f24f00831bebf3cebfdbd094fb74aa": "\n{\\rm cov}(Y_{i1}, Y_{i2}) = {\\rm var}(\\alpha_i).\n", "74f260d5569ba5e0181f7923cad43df3": "\\rho (x_1,x_2,x_3)", "74f2d243d8d7df8ad32258353cba18d2": "\\mathbf{A} \\cdot \\mathbf{B} = A_{\\nu} B^{\\nu} = A^{\\mu} B_{\\mu} ", "74f342cdc9614e29ad9c8f93451b792e": "V=8\\pi r^3", "74f3b8420ee471fd9b949f8541793dd0": "\\mathbf{i}=2, 3, \\dots n", "74f3ba0ed1aee3a95629b59a8120872e": "a_1x_1 \\oplus a_2x_2 \\oplus \\cdots \\oplus a_nx_n \\oplus \\!", "74f3da5e2328857b9ec9a9631c7c228a": " [ u_i, u_j ]_{p,q}, \\quad 1\\leq i,j\\leq 2n ", "74f402d4991a87df13baf6e0563b277c": "\\Leftrightarrow (x + 3c/4)^2 + y^2 = 9c^2/16 ", "74f40e2b3114bc7bdad73fd8056c3c8d": " \\delta \\mathcal{S} = \\delta\\int_{t_1}^{t_2} L(\\mathbf{q}, \\mathbf{\\dot{q}}, t) dt = 0 ", "74f4be70f0996372ff78938ca64eb212": "S_i \\subseteq [n] (1 \\leq i \\leq t)", "74f4e17306c68275c3d2368e43c985a2": " T \\perp Y(0), Y(1) \\,|\\, X ", "74f51c24ecb1ba5dd1e67262baea9936": "\\mathbb{S}:(\\nabla\\boldsymbol{U})", "74f52c10e3a37f042a15d484c858c2a6": "1+(p-1)!", "74f54601ccb427d775fa8807408e6b56": "\\int x^m\\,\\operatorname{arcosh}(a\\,x)dx=\n \\frac{x^{m+1}\\,\\operatorname{arcosh}(a\\,x)}{m+1}\\,-\\,\n \\frac{a}{m+1}\\int\\frac{x^{m+1}}{\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}}\\,dx\\quad(m\\ne-1)", "74f556662e6bfaaac3b698f04c673975": "m_E=-\\frac{m_1m_2 + m_1m_3 + m_2m_3 + 3}{m_1 + m_2 + m_3 + 3m_1m_2m_3}.", "74f58e44b813e3cb74664e8b9d261bf6": "x^2 + y^2 + z^2 = 3xyz,\\,", "74f65b239e6139cfa80ea0bc9a28ffec": "a_1 \\mid a_2 \\mid \\cdots \\mid a_m", "74f692c8b6cd25f79dbd0b79e307e9cf": "X_0 = {48 \\over 17} - {32 \\over 17} D \\,.", "74f6acf8fe54bae2acaf3dc00279258c": " MD(f,z)(x)=\\lim_{r\\rightarrow 0} \\frac{d_{X}(f(z+rx),f(z))}{r} ", "74f703e7b0fc18a0e10051c9a3546be1": "\\mbox{f(t)} =sin(\\omega t+\\theta)", "74f725ac10fd07911d173c277b58814f": "\\alpha \\in [0,1]", "74f747ea0705ae9526e9c834f684d6d0": "g_1 \\neq 0", "74f75690b9b60f9cc82d2741e26b44d0": "(w_2, w_3)\\in I_D", "74f7711c0ae4dff9914fc5cd69516a1c": "\n S_{IJ} = F_{Ik}^{-1}~\\tau_{kl}~F_{Jl}^{-1}\n", "74f784f81e1d56277fdecd55cbfe3395": "A^*=\\bigcup_{i=1}^\\infty A^i \\in\\mathrm{RAT}(N)", "74f78f000bbe9f2e9180cacc41b24f24": "{^{131}_{54}\\mathrm{Xe}^*} \\rightarrow {^{131}_{54}\\mathrm{Xe}} + \\gamma ", "74f819435b0d8f299fd08b1491455c49": "\\frac{\\Gamma_e}{\\delta x_{PE}} A_e", "74f83fbbaca01e6f4dd6dd9a76060d89": " \\mathbf{W}^{-1} ", "74f86f736b508674c066a2c7ef047766": "\\rho(x,y)=x^2-2y^2", "74f8b3bc8906cb4ff86c98f6074c100c": "\\epsilon^{\\mu\\nu\\rho\\sigma} \\,", "74f8e8399b2cff4489d86c181e69ce2a": " SK_3 = \\frac{ \\mu - 2 Q_2 }{ E | y - Q_2 | } ", "74f8f1780394b2f06097a8d5642fb2a7": "B^2=4AC \\,", "74f99c7cc7b07a03b924790db7259818": "\\vDash \\varphi", "74f9b56d4c461291af35b53d26d774f0": "a \\times b = \\frac{\\left(\\left(a + b\\right)^{2} - a^{2} - b^{2}\\right)}{2}", "74f9f733eddd4ad592b3d1acbf92fd6d": " \\phi = \\phi_0 e^{-cz} ", "74f9fa7a03cde4401320ebde8594f631": "f_i: Y_i \\to X", "74fa0cc8099dbf9ae72051efe2c2f2f3": "pred[S]", "74fa243a19c947e2ced8c9bc4efe47b0": "c_{d_2}>0 ", "74fa472d024c8199f34909634b30ae40": " D(f) ", "74fa4cdf8bdfed979070a95bdd24c07a": "K_i(\\mathbf{F}_1)=\\pi_i(BGL_\\infty(\\mathbf{F}_1)^+)=\\pi_i(B\\Sigma_\\infty^+)=\\pi_i^s.", "74fb383af935ea72655772c00819f3a8": "\\begin{smallmatrix} m = 4.83 + 5\\cdot((\\log_{10} 1.834) - 1) = 1.15 \\end{smallmatrix}", "74fb6152ed4ac1c9769783de68334629": "r\\left\\{\\begin{array}{l}p\\\\q,r\\end{array}\\right\\}", "74fba8207ff49bf147f17f1066a8bcaa": "\\Omega = D\\omega = d\\omega + \\omega\\wedge\\omega", "74fbbe1b0c215471b3df36365c842d67": "1 \\times x", "74fbee53d08c46c391a28a7e59010442": "\\zeta = 0", "74fc03dd0dd14fd0f55a384f278fbeca": "n\\equiv 2k+l ~.", "74fc21998d22b22d4975d8d21aca2d55": "\\{ z : \\lVert z \\rVert = -a^2 \\}", "74fc3781a86364e886edef5bea84cfb8": "a_m = a_1 = 1", "74fc47b67a2238da0dfa8f3eb52f2a5b": "(1-x^2)\\,y'' - 2x\\,y' + {\\lambda}\\,y = 0\\qquad \\mathrm{with}\\qquad\\lambda = n(n+1).\\,", "74fc4e9dbde53451dea9ec42c923ee8a": "a^n + c_1 w(a)a^{n-1} + \\cdots + c_n w(a)^n = 0 ", "74fc7be7403d9b7a4d9be38d28cc6ce9": "a=\\frac{p}{q}<1", "74fc9982b960fbcaf0d2ce0330c84d8c": "\n \\langle F | \\exp\\left({- {i \\over \\hbar } \\hat H T}\\right) |0\\rangle =\n\\langle F | \\exp\\left( {- {i \\over \\hbar } \\hat H \\delta t} \\right) \\exp\\left( {- {i \\over \\hbar } \\hat H \\delta t} \\right) \\cdots\n \\exp\\left( {- {i \\over \\hbar } \\hat H \\delta t} \\right) |0\\rangle\n", "74fcb594bdd93c0f956682ae1ea013e6": "|V|", "74fcee97b7ac7f60b852d70858d1ab2b": " u = \\langle v, v \\rangle^{-1} \\, \\overline{\\varphi (v)} \\, v.", "74fd22ca4f9458f85c2c2ec87727a102": " V_D ", "74fd35f76f82fe5ccad56cba8a1072e8": "\\mathrm{FWHM} = 2 \\sqrt{2 \\ln 2}\\ c \\approx 2.35482 c.", "74fd465d314b483d545e6b01c9bf359e": "\\mathbf{ \\left(J^T W J +\\lambda I\\right)\\delta p=J^T W r }", "74fd53da17c4fa1c7f7cbe5dd9fa1916": "B_i (2,0)", "74fdae78711fcdb0929fa222b125a145": "GTC \\geq b^d", "74fdd0aeb19e9dcf8c432639ed49275a": " \\mathcal F^\\mathrm{an} \\cong \\mathcal R ", "74fdef2e9055aa0d49c8a0afddfeae49": "w \\;\\stackrel{\\mathrm{def}}{=}\\; \\operatorname{E}(w_i) = \\frac{1}{n}\\sum_i w_i n_i = \\frac{1}{n}\\sum_i \\frac{n_i'}{n_i} n_i = \\frac{1}{n}\\sum_i n_i' = \\frac{n'}{n}", "74fe3783aad6649f339cdf77e91985e3": "\\nabla^2_{norm}(x, y, t)", "74fefff863932e36d648250e9e4dec35": "10^{7\\times 2^{122}}", "74ff03cc14086e83f1ceace9efa37669": " \\prod_{p} (1-p^{-s})^{-1} = \\prod_{p} \\Big(\\sum_{n=0}^{\\infty}p^{-ns}\\Big) = \\sum_{n=1}^{\\infty} \\frac{1}{n^{s}} = \\zeta(s) ", "74ff280d98d69dbe6c859c19401b3a68": "R_{\\mu \\nu} - {1 \\over 2}g_{\\mu \\nu}\\,R - g_{\\mu \\nu} \\Lambda = -{8 \\pi G \\over c^4} T_{\\mu \\nu}.", "75000c350b0dc205542bc1c89b2d1ba0": "\\scriptstyle A \\;\\Leftrightarrow\\; B", "75008f92206cfa78323fec083199cbff": "u\\le v", "7500ac1be02dc495ef0a1d278fd71e1c": "z_T=\\frac{1+\\Gamma}{1-\\Gamma}\\,", "7500b7080a2f17cf5cafea8f2a33decd": "\\scriptstyle\\varphi(x)\\,=\\,\\log(x)", "7500fbc9685194187ac1e4a5079ae9b6": "H_{s,2}=-\\sum_{k=1}^\\infty \\frac {(-1)^k}{k} {s \\choose k} H_k.", "75012bff4fc80882d7e91016e0da587d": " P\\left( \\bigcap_{ i = 1}^2 \\left[ \\frac{ | X_i - \\mu_i | } { \\sigma_i } < k \\right] \\right) \\ge 1 - \\frac{ 1 + \\sqrt{ 1 - \\rho^2 } } { k^2 }.", "750173f6804f20b6fccb06a0398cec7b": "\\mathcal{O}_{\\widehat{S}}", "750199e41ded8d652372bd3f288d22ae": "\\sum_{i=1}^{n} a_i^k = b^k", "7501a3a81105db91c05c323aa343017e": "\\lambda (t)", "7501b62e510f48af45c403988548aacd": "~dU = \\delta W + \\delta Q.", "7501bf4c71f8f280f4ab250209d83be4": " t_1=u+v=\\sqrt[3]{-{q\\over 2}+ \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}} +\\sqrt[3]{-{q\\over 2}- \\sqrt{{q^{2}\\over 4}+{p^{3}\\over 27}}}", "7501f16f44fd8757aed055c566b6e43a": "\\Delta z = \\int 1 \\; dz", "75020639a4d0394294f2b3825ca2183b": "\\Delta S = \\int_a^b \\frac {dQ_{rev}}T ", "75021634d3501cbdc9239e71643809cb": " a_0 + a_1 x + \\cdots + a_{n} x^n ", "750216b006a9b3175dc7f68a0d3e4189": "R = -\\Delta f/\\Delta p_m = -slope", "75021e9e123bf454a365c34a59613df7": "k \\in [2,n-1)", "75022cc983bbf09ddda7bc41bcf64271": "i_1(t)\\,", "75023572a75335c861a3ccce051a6da7": "\\frac{dy}{dx} = \\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}}.", "7502512c361a1d8ed5d5a3ed159f4d65": "A = \\sum_{k=0}^n \\alpha_k \\rho^k", "75029ee5629b954665a230a59e0fa5ee": "\\circlearrowleft", "7502a04b31cba68bb535b48a8e34ad1a": "\\mathrm{Hom}_C(F(i),G(j))", "7502d8d551dc8ec5e7d550b7cec79ad7": "a^2+b^2", "7502d9d31e4ba3dad70a3eae2569118a": "\\frac{M_1}{(R-r)^2}+\\frac{M_2}{(2R-r)^2}=\\left(\\frac{M_2}{M_1+M_2}R+R-r\\right)\\frac{M_1+M_2}{R^3}", "7502ea5cb31d584016c78829971dde1c": "\\frac{1}{x} \\sum_{n \\leq x} \\lambda (n) = \\frac{x}{\\ln x} e^{B (1+o(1)) \\ln\\ln x / (\\ln\\ln\\ln x) }", "7502eff82bf05b388d71ae8e2508d6b3": "z = r \\sin(u).", "7502f0086a234637bd4f83726943c758": "\\Theta(E)", "7503030c573da1ac89f2d73b3e7394ed": "(\\tanh x )'= {\\operatorname{sech}^2\\,x}", "7503410616b94485415b31ff590aef5f": "\\left(1/6,\\ \\sqrt{1/28},\\ -\\sqrt{12/7},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "75035a7a06a0e9093d49c899447d8781": "\\begin{align}f_1^\\prime &= P_1(x,f_1,f_2,f_3)\\\\\nf_2^\\prime &= P_2(x,f_1,f_2,f_3)\\\\\nf_3^\\prime &= P_3(x,f_1,f_2,f_3).\\end{align}", "7503701cd92b65fc176bcd06cb0cd123": " b(\\lambda) =C\\cdot d(2\\lambda)^{-1}\\cdot \\prod_{\\alpha>0} \\tanh {\\pi(\\alpha,\\lambda)\\over (\\alpha,\\alpha)},", "7503cfacd12053d309b6bed5c89de212": "1853", "7503d1d5903f320499e3920d5ba232da": "(\\mu_n)", "7503d896351ff1da54c5c41bb9c8427d": "X_{a;bn} - X_{a;nb} = R_{ambn} \\, X^m", "7503ddda86f2a16681099471bbaa663d": "{log_2 3}", "75041a8c49f3e03555573541a9f4675f": "\\int e^{cx}\\sin^n x\\; \\mathrm{d}x = \\frac{e^{cx}\\sin^{n-1} x}{c^2+n^2}(c\\sin x-n\\cos x)+\\frac{n(n-1)}{c^2+n^2}\\int e^{cx}\\sin^{n-2} x\\;\\mathrm{d}x", "75041e842cf05bc8371f3a8533536ff9": " \\exp S = E^T \\mbox{Diag} (\\exp e) E ", "750428c77fc75a5dc1872e81708ce069": "\n\\frac{\\displaystyle 1}{\\displaystyle 65}\n\\begin{bmatrix}\n1 & 49 & 13 & 61 & 4 & 52 & 16 & 64 \\\\\n33 & 17 & 45 & 29 & 36 & 20 & 48 & 32 \\\\\n9 & 57 & 5 & 53 & 12 & 60 & 8 & 56 \\\\\n41 & 25 & 37 & 21 & 44 & 28 & 40 & 24 \\\\\n3 & 51 & 15 & 63 & 2 & 50 & 14 & 62 \\\\\n35 & 19 & 47 & 31 & 34 & 18 & 46 & 30 \\\\\n11 & 59 & 7 & 55 & 10 & 58 & 6 & 54 \\\\\n43 & 27 & 39 & 23 & 42 & 26 & 38 & 22 \\\\\n\\end{bmatrix}\n", "750438b3bdbe50d879ec9e2b62473303": "%\\;\\text{Methylation} = 100 \\times \\frac{\\text{Digested Fragments}}{\\text{Undigested Fragments + Digested Fragments}}", "750454dacafae6605b2f2519f5e826f5": " k X + b \\sim \\textrm{Levy}(k \\mu + b ,k c) \\,", "75046270fe7ae17eadb29142f727179d": "s\\tbinom{n+s-1}{s-1}", "75047bec51963381f07fd25a1b94d3ef": "L_P^\\sigma=L_P-a{M}", "75050768fbdef5a2be94793e38667dd3": "\\frac{V}{F}=\\frac{1}{c_T W} \\sqrt{\\frac{W}{S}\\frac{2}{\\rho}\\frac{C_L}{C_D^2}}", "75050c5a87bbe0f1f692b6f54a2c5f70": "y(0)=1.", "750554f9c5fcaf3bd0263a51c081632c": "\\displaystyle{\\|b\\|_1 \\le 2\\|f\\|_1.}", "7505b562526958293d2c03fc66ddf08e": "AH = \\frac{b^2}{2a}", "75060581d59f8c0cb5b401f304898b3e": "Z_n(c) = e^{-c\\beta} |\\mbox{Fix}\\, \\tau^n| = e^{-c\\beta} \\mbox{Tr} A^n", "750613ea4dd471824296353b1096ad6d": "A \\vee B = \\{ x \\in A\\cup B \\mid \\not\\exists y\\in A\\cup B\\mbox{ s.t. }x < y\\}.", "750668c4749cf71768719757ae713338": "\nu^{+1}_{-1}(\\mathbf{p}) = \\sqrt{ {E + p_3} \\over 2 E}\n\\left( \\begin{array}{c}\n0 \\\\\n0 \\\\\n{{-p_1 + i p_2} \\over {E + p_3}} \\\\\n1\n\\end{array} \\right).\n", "750699bf72af5cd7d5f6a993d6ce8b9f": " d\\left( \\gamma + (\\dfrac{\\lambda}{d}) \\left( 1-\\gamma\\right)\\right) < \\gamma d", "750744d8ad0858fbdafe4db6c06fc25e": "x' = {x / L}\\,\\!", "75074defe061264ac7b6d07510abbfbe": "\n\\begin{array}{lcl}\nP\\left(\\frac{\\bar{Y}_2 - \\bar{Y}_1}{\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}} > 1.64\\right) &=&\nP\\left(\\frac{\\bar{Y}_2 - \\bar{Y}_1}{S} > 1.64\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}/S\\right)\\\\ \n&=& P\\left(\\frac{\\bar{Y}_2 - \\bar{Y}_1-\\delta+\\delta}{S} > 1.64\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}/S\\right)\\\\\n&=& P\\left(\\frac{\\bar{Y}_2 - \\bar{Y}_1-\\delta}{S} > 1.64\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}/S - \\delta/S\\right)\\\\\n&=& 1 - \\Phi(1.64\\sqrt{\\sigma_1^2/n + \\sigma_2^2/n}/S - \\delta/S),\n\\end{array}\n", "7507529b4fba1772edae1b1bd8e319e9": "\\mathcal |a|= \\begin{cases} a, & \\textrm{if } a^{(p-1)/2}\\equiv 1 \\bmod{p}, \\\\ -a, & \\textrm{if } a^{(p-1)/2}\\equiv -1 \\bmod{p}. \\end{cases}", "75076953cb8ce98a0642c8e2a5f02f0f": "s\\,\\!", "7507b9d5519f6735ff979b40025e4b15": "\\mathbf{J}_{12}", "7507cee424d8873f16e6cbe2927396b5": "\\mathrm{LE}(\\gamma)(j)=\\gamma(i_j).\\,", "7507d999089df28f498572e05d3fe8f8": "L^p(d\\omega)", "75080242772d7e9a2ad6ba496cc65e8d": "k = 3, 4, 5, \\ldots", "750819ab69260fffea545b3838a04286": "\nx = \\frac{r_1^2-r_2^2}{4c}\n", "7508381eff9a30d17ec0be0290d45e2c": "U = 2 \\pi r N", "75084fbc619f6ee3640a329e16cf4017": "k = \\left(2a\\right) \\sinh^{-1}\\left(\\frac{L}{2a}\\right) ", "75085d67d1645141765c8b78de09e168": "\\alpha(J_n)=J_n +{c\\over 6}\\delta_{n,0}", "7508699004dbe407175c6b5a7fff4da2": "d = 0", "7508fcbef2201984709d3d8617962efa": "\ndV(t) = \\sum_{i=1}^{n} h_i(t) dS_{i}(t).\n", "75090b8c0ca333f71bf1418b90ff3c95": "\\left(\\cos\\tfrac{2\\pi}{n} + i \\sin\\tfrac{2\\pi}{n}\\right)^n = \\cos 2\\pi + i \\sin 2\\pi = 1,", "7509701bb5c766aba6169ea870294c78": "P=\n\\begin{pmatrix}\n | & & | \\\\\n \\mathbf{v_1} & \\cdots & \\mathbf{v_n} \\\\\n | & & | \\\\ \n\\end{pmatrix}\n", "7509b845722526b2fd208e8d0da48a05": " \\hat{\\Sigma}_{Y_i} = \\tilde{G} + \\hat{\\sigma}^2 \\mathbf{I}_{m_i} ", "750a2d314f9427eafb69434af8fb5c3f": "T_1^2 - T_2^2\\,", "750a592b904730e5d0df866f2ae60e72": "\\det A_Q = 0", "750a79fc152e4925245613c2ccf52900": " u(0) = u^0 \\mbox{ in } \\Omega", "750ae55496a680415fccee4269e98988": "L^1([0, 1]),", "750af0709ba98387db296826e6cdd89f": " \\tau = \\frac{(\\alpha \\Delta t )}{\\Delta x^2} \\, ", "750af35e0097f9fb4f0be522b714467d": "F' \\circ F : \\mathcal{C} \\leftarrow \\mathcal{E}", "750b1c0ca0f7f934ca9b84df7eee47ed": "\\delta_{ab}={\\rm\ndiag}(+1,\\ldots,+1)", "750be93bb58312fc68dd22a1be865d8d": "\\Delta S_{mix} = -\\left(\\frac{\\partial \\Delta G_{mix}}{\\partial T}\\right)_P", "750c0f3f6c1f9c2c334fabd18ccc04fa": "\n[k_\\pm (a \\longrightarrow j)] = \\int_{0}^{\\infty} k_\\mp (a \\longrightarrow j) d \\omega - {\\pi^2 \\over \\hbar} (N_a - N_j)({\\alpha_2 \\over n}) \\Big| \\big\\langle a \\Big| m_\\pm \\Big| j \\big\\rangle \\Big|^2\n", "750c676beedc9fe3014e862e578e4299": " -\\gamma -\\frac{\\pi}{2} - 3\\ln{2} = -\\gamma+\\sum_{n=0}^{\\infty}\\left(\\frac{1}{n+1}-\\frac{1}{n+\\tfrac14}\\right)", "750c8ceb291324598555e50bc06e2240": "\ndx\\;dy\\;dz=\\det{\\frac{\\partial(x, y, z)}{\\partial(\\rho, \\phi, \\theta)}} d\\rho\\;d\\theta\\;d\\phi =\n\\rho^2 \\sin\\theta \\; d\\rho \\; d\\theta \\; d\\phi \\;\n", "750cc8c98b2dac75c277e3330220f475": "V^2", "750cced7001cd354dd71b47056a4acc5": "\n c(x,\\eta)= \\left\\{\n \\begin{array}{l}\n \\sum_y p(x,y)\\eta(y) \\quad \\text{for all}\\quad \\eta(x)=0 \\\\\n \\sum_y p(x,y)(1-\\eta(y)) \\quad \\text{for all}\\quad \\eta(x)=1 \\\\\n \\end{array} \\right.\n", "750cdc0c02032e8fbe16be3f3c672794": " d_M(x_n, y_n) < \\frac {1}{n} \\wedge d_N ( f (x_n), f (y_n)) \\ge \\varepsilon_0.", "750cf01ecb8ddea0ed9f8f8cade5a07d": "f=\\exp_{b}.", "750de71494dce7e1d2c0f1cf1231cd4d": "J^1Q", "750e2c3f7cb50dd9eb454662a33a2123": "d \\neq 1", "750e7ed2a4f833bb3ac9e7fc4957d444": "C_n = I_n - \\tfrac{1}{n}\\mathbf{1}\\mathbf{1}^\\top", "750ea1171d49465aa5f60182d5f779a5": "\\mathcal{K}_a P \\rightarrow \\mathcal{K}_a Q", "750eb89abb649994bbd4996128c34f76": "\\left[K\\right]=\\begin{bmatrix} 3\\frac{EI}{L} + 4\\frac{2EI}{L} & 2\\frac{2EI}{L} \\\\\n2\\frac{2EI}{L} & 4\\frac{2EI}{L} + 4\\frac{EI}{L} \\end{bmatrix}", "750ec8effc77587fc818f19b18f7c76f": "\\tau_g", "750f228cbaada8afd5abfb755527e398": "\\vec{r}_v,", "750f49eb53173dc8e1052ea8d447df55": " F_m = 2 k_A \\frac {I_1 I_2 } {r}", "750f607914be190a303c30b9cb2297ee": " \\int_H \\operatorname{E}\\left[X|\\mathcal {H} \\right] (\\omega) \\ \\operatorname{d} \\operatorname{P}(\\omega) = \\int_H X(\\omega) \\ \\operatorname{d} \\operatorname{P}(\\omega) \\qquad \\text{for each} \\quad H \\in \\mathcal {H} ", "750f77b882c6eef51d0160576754b749": "H(2n+1,q^2)(n\\geq 1)", "750f8b186742f674be5fe70681b7b377": "|\\varphi + \\mu\\theta|^2", "750f9b3b9a934441424c6013dcb8348b": "Q_k = \\begin{pmatrix}\n I_{k-1} & 0\\\\\n 0 & Q_k'\\end{pmatrix}.", "750fa1a904a9de70d946c2f8fbb4e171": "w = \\mathbf{x}^H\\mathbf{v}\\mathbf{/}\\mathbf{v}^H\\mathbf{x} ", "750fd1001c7b264ec04b66e32da8f17e": " I_y", "750ff757d51302a394e380b36439d9fc": "(\\bold{3};\\bold{3};\\bold{3})", "750fffdf8b86fc0d632094c1a67ae31b": " \\approx 4.3 \\times 10^{369,693,099} ", "751029b6dee8040789db4df95b90b45f": "A \\subseteq A^*", "75109a6614f3a44a697bd8d09a33e349": "r - \\frac{r^2}{2}", "7510ac46e1047c4dfe97a0344c74371a": "\\Delta t' = \\frac{\\frac{2L}{c}}{\\sqrt{1-\\frac{v^2}{c^2}}}", "7510fb8ef8aab2f34faa17b4145d50ed": "(X, \\mathcal{T}')", "75110b7b2cf199b20614f255b0a5af99": "N_{i,1}", "7511150e041ecaa85e8d6c497947b001": "\\Phi(x,t):=\\frac{1}{\\sqrt{4\\pi kt}}\\exp\\left(-\\frac{x^2}{4kt}\\right),", "7511be525935c344b3547cdb44bc0204": "\\hat n_i", "75125fdb0d2ee9fd65d936ad48ba107c": " F(t) = g(t) F(0) g(t)^{-1} \\,", "751293de9bd2fcd0da3053a377a7e72b": " \n {A'\\,}^{i j} = \\frac{\\partial {x'}^i}{\\partial x^l}\n \\frac{\\partial {x'}^j}{\\partial x^m} A^{l m}\n", "7512b7e39b92c15c08afb7dea587f036": "G_0^\\pm", "7512c3fb906adcade44348091b2f3616": "n=\\prod_{i} {p_i}^{e_i}", "75130dbf20b5481d6554339ea203d4d6": " \\Delta p", "751324f10ab1cdd53c0035cb6e971b1d": " = \\frac{q^2}{16\\pi^2\\varepsilon_0c}\\,\\frac{|\\hat{\\mathbf{n}}(t')\\times\\{[\\hat{\\mathbf{n}}(t')-\\vec{\\beta}(t')]\\times\\dot{\\vec{\\beta}}(t')\\}|^2}{[1-\\vec{\\beta}(t')\\mathbf{\\cdot}\\vec{\\mathbf{n}}(t')]^5} \\qquad \\qquad (4)", "751328e772e5476ea21c3acc955dbba0": "\nd\\nu^1_t = \\kappa^1(\\theta^1 - \\nu^1_t)\\,dt + \\xi^1 \\sqrt{\\nu^1_t}\\,dW^{\\nu^1}_t \\,\n", "751333a7b204ea2f9c11042b3fb491e9": "d_0 = \\frac{d(t)}{a(t)}", "751333fda16815598a338386136a0f86": "T(\\mathbf{M}) = \\mathbf{M}\\mathbf{B}^T + \\mathbf{1}\\mathbf{t}^T", "751389df747759aeb90119a905b2798f": " W_t = \\sqrt{2} \\sum_{k=1}^\\infty Z_k \\frac{\\sin \\left(\\left(k - \\frac{1}{2}\\right) \\pi t\\right)}{ \\left(k - \\frac{1}{2}\\right) \\pi}. ", "75139f08a73bbd0283d7e16c7f76611a": "\\mod 0;", "7513bd27cb4b39a587aec924b88a0abf": "\\operatorname{ad}_gX = \\frac{d}{dt}g\\exp(tX)g^{-1}\\bigl|_{t=0}", "751440161e09cae887f50a95031bb369": "\nV=\\frac{U}{\\sqrt{\\rho}}\n", "75145c8ad7d8ebcbff06074fcfd74f82": "\\overline{P}(A)", "75146fe0be8f26b5ad141f41f11b0d60": " H=k ", "751471f42ca31084c3e5cc249d988920": "\\delta^*_\\omega h = \\omega^2 h", "7514be1a22d6ffead01e71577ef085f1": "\\phi(x,y,t):", "7514dce7b388f1d7c6545101a1ff0d34": "\nV_n-\\langle n\\rangle=\\langle (a^\\dagger)^2 a^2\\rangle -\\left(\\langle a^{\\dagger}a\\rangle\\right)^2.\n", "7515bb5ef6267bc79e3d3ec52fd0556e": "\\frac{\\mbox{Operating Income}}{\\mbox{Net Sales}}", "7515ccf1ef8f878812ccca284cee4adf": "(B + C) ", "7515d4710d2181ba0a687c632a8284d4": "\\alpha_j\\,", "751618ba1d3ae27c11a6444763e53c99": "B_{\\lambda k}^+=e_{\\lambda}(k)\\cdot B_k^+", "75162e78d51d36d82c88d75ac9e20893": " y_i = \\beta_1 x_{i1} + \\beta_2 x_{i2} + \\cdots + \\beta_p x_{ip} + \\varepsilon_i, \\, ", "7516728b024d1d23b3155c0e5a695d9e": "p_1 = A\\rightarrow aA", "7516b96678349ed002f1931a294f577c": "d_k", "751720c4317192abd6394d3438455cda": "1 + 0.98x - 0.98x = x - 0.98x", "7517877fad6456f42d0f540ca30da2b5": "f \\colon [0,\\infty) \\rightarrow [0,\\infty)", "7517ab360a8a05475dd93cf0fc4cc8a9": "8 |E|", "7517bb3087dd9ba0afd641221abd4b6d": " N_P \\propto 1/P^\\nu ", "7517cb51d649e5f1bc0cca63fdc3660b": "\\textstyle \\alpha = \\beta = \\gamma", "75180dcf3878ea48eed497ffc912df5d": "\\displaystyle{\\pi_z(g) = (I-zT)^{-1}\\lambda(g)(I-zT)}", "75181a9bff02af059fac646a777a9b9a": "E X_i = 0", "751884af6a2d8587325a007f656492bd": "T = 2\\pi\\sqrt{I/\\kappa}", "75188d18feee82281f70ecaade5d78d1": "Z_a(s,t)=\\zeta(s,t)+\\zeta(\\bar{s},t)-\\frac{\\Big[\\zeta(s,t)+\\zeta(s+t)\\Big]}{2^{(s-1)}}", "751919e58eb68a644b67d01f44a97e7e": "E = 2 \\hbar \\omega_0 |\\cos(\\pi/2 - ka/2)| ", "75191b1bc5f159445c5bfd3d1d023ef5": "\\lambda n_{0, t} = \\left(f_0 + f_1\\frac{s_0}{\\lambda} + \\cdots + f_{\\omega - 1}\\frac{s_0\\cdots s_{\\omega - 2}}{\\lambda^{\\omega - 1}}\\right)n_{(0, t)}. ", "75194b6494be1b37c37bd214c0cdd475": "\\scriptstyle |x|^k |\\nabla^k m|", "7519803e6d214b21396e7ea46f4e615b": "x_i=true", "7519ad0492881dd3129ee4c8b863f09e": "T=|t|^2= \\frac{1}{1+\\frac{V_0^2\\sin^2(k_1 a)}{4E(E-V_0)}}", "7519d4c30cb84bf8684d857a0d33776c": "H_TB", "7519f852177980df597c3c5340291f38": "MTTF=A(J^{-n})e^{\\frac{Ea}{kT}}", "751a159543659a98d5c88c51d8949ec2": "\\displaystyle{B(a_1,b_1-b_3)=B(a_1,b_1-b_2)B(a_2,b_2-b_3),}", "751ade154241ed38c3d140629c507cbc": " j^{\\star} = \\varepsilon\\sigma T^{4}.", "751b9c711b25028d1ccdd0e70f126694": "C_V = 3Nk\\left({\\varepsilon\\over k T}\\right)^2{e^{\\varepsilon/kT}\\over \\left(e^{\\varepsilon/kT}-1\\right)^2}.", "751bbee7bddc5c8f46fb50d04268a808": " R_{ik} - {g_{ik} R \\over 2} + \\Lambda g_{ik} = {8 \\pi G \\over c^4} T_{ik}", "751c7af7d26d1f5beada8c153c59e37f": "\\nu:\\mathbb{P}^2\\to \\mathbb{P}^5", "751cdaed95df32b8d2c55092a790cbad": " a_i\\wedge u,", "751d758a95fb9cb2bb29b9f7d08c0e62": " D_e = \\frac{D\\varepsilon_t \\delta} {\\tau}", "751d8001c2dfc0973c5d4c531abbd9c6": "\\Omega = \\frac{1.5}{\\phi} ", "751db6c8059519410e3e6092679aa27c": "m^{\\prime} \\in \\{0,1\\}^k", "751dc4ab3241de4c8895459046f14206": " (s_d - s_s) \\le reg + path_{min} - H ", "751dd072b068329224be75c25581c073": "{\\rm blanc}(x) = \\sum_{n=0}^\\infty {s(2^{n}x)\\over 2^n},", "751ded5bba3175cda615447fd4e81ffe": "E = \\lbrace e_i | i\\in I_e, e_i \\subseteq X \\rbrace,", "751dff0deda9f7740690be6ff5f70f61": "\\rho=\\limsup_{r\\rightarrow\\infty}\\frac{\\ln(\\ln\\Vert f \\Vert_{\\infty, B_r} )}{\\ln\\, r},", "751ea7f77044053c75c81653690d16c7": "|\\phi_i\\rang", "751f05de07223ae438bf804a5f81e497": "L_{k} \\ \\stackrel{\\mathrm{def}}{=}\\ I_{k}\\omega_{k}", "751f7bae01d19615bf15e90c2c53d7ea": "2\\pi\\nu t", "751fad7b53944d582d739399773d4bda": "m_i = |\\{ j : \\delta_{k_j} = i, 1 \\leq j \\leq L \\}|", "752021d2dc82c6a2086d33439f976228": "\nk^{2} = 1 - \\frac{\\mu}{h^2}\n", "75208585bbef46dac029c74ab74f14a6": " K(k). ", "75218b576d1f4e5397e69d1163632e30": "F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1}]} = \\int{ F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1} \\AA^{-1}]} f(\\lambda) d\\lambda}", "752191fdeb4e8bfdcf554d66340a8f13": "f(-d/c) = \\infin \\mbox{ and } f(\\infin) = a/c;", "7521bd8acc0c83baedc7ee2a29d8a42c": "\n\\Pr[X \\geq E(X)+\\lambda]\\leq e^{-\\frac{\\lambda^2}{2(Var(X)+\\sum_{i=1}^n a_i^2+M\\lambda/3)}}\n ", "7521f3b4fd0fca49976985a7faa623f9": "I(\\lambda) d\\lambda = K \\left( \\frac{\\lambda}{\\lambda_{min}} - 1 \\right)\\frac{1}{\\lambda^2} d\\lambda", "752274c133502b4b37cac8b4d3c5ec8a": "0 \\in \\operatorname{core}(A)", "75229ef88e16c51f450dc1e1d567b31a": "y:=x_n+x_{n+1}-\\alpha\\ge x_n-\\alpha>0\\,,", "7522d49e409e81a0dde9b17dac84abb9": "\\{\\land, \\lor\\}", "7522dc27414c249b709d9162aafb586e": "D\\nabla^2\\psi", "75230e53737745856abc42aa4495181a": " Y = f(X_1, X_2, ... X_k).", "75231d36e27922180f10ac2823175133": "R'(1) = Q'(1) - P'(1) = 0", "75232cebf412d67ec46facfd89398140": "X_0 = S^{-1},", "7523779c6e85c7a991f80493f18b9bdd": "\n R^2 = \\frac{\\sum(\\hat y_i-\\overline{y})^2}{\\sum(y_i-\\overline{y})^2} = \\frac{y'P'LPy}{y'Ly} = 1 - \\frac{y'My}{y'Ly} = 1 - \\frac{\\rm SSR}{\\rm TSS}\n ", "7523d444dfad1d48a8c3b3f136835b86": "\\cos\\, 0 = 1", "75245535562b1b55c51819aa2313ae49": "X_3 = D^2-G-2H-a_3E", "75246b41fb340d1da317dc73ec01dccf": "\\mathbf{(3)}", "75247d65ff9f48801b6ec30ce79cacaa": "\\scriptstyle \\in", "7524ce52a0ddebc7aaf727e0c1d313bf": "\\ \\sigma_1 - \\nu(\\sigma_2 + \\sigma_3) \\le \\sigma_y. \\,\\!", "7525324fc2028bff9cfec616d66fa0eb": "Y=\\frac{\\partial}{\\partial y} + \\frac{1}{2} x\\frac{\\partial}{\\partial z},", "752564f677bbb346083d7103c419e4ac": "\n\\hat{f}(x,y) = n^{-1} h^{-2} \\sum_{i=1}^{n} K\\left(\\frac{x-x_i}{h}\\right) K\\left(\\frac{y-y_i}{h}\\right)\n", "75258b752151042dbf2dc6e541814a14": "\\gamma/\\alpha.", "7525b2f2655d9aa0bafc0394083ea640": "c\\to 0^+", "75260fa46d6fccceaddbaf9c85b075ce": "+|e|", "752639b29de75de96d280abc53557aa4": "\\nu,\\, \\sigma_0^2\\!", "75263f1c9d179db995ea3d3c4a6acae6": "\\frac{5n^2 - 3n}{2}", "75266c75e11c45277c14aee2c52665be": "f=F", "75266e6a02651f467d7d4c3ab504022c": "m\\in M", "75267f48da5c3e9abecc983d46007a8a": "=2x\\,", "7526e2d137e41689104ae91db4e0da3b": " E_\\mathrm{sig} ", "7526ebeedd6ba8151dd02f7516299443": "k_3^+k_4^+/k_2^+=k_3^-k_4^-/k_2^-", "752746301c3197081440623969f13b28": " \\mathcal{S} = \\int \\bar\\psi(i \\hbar c \\, \\gamma^\\mu \\partial_\\mu - m c^2 ) \\psi \\, \\mathrm{d}^4x", "75276e5ef991e623587b42b4b08f7665": "\\frac{\\sqrt{3}}{4}", "75282dcc88593ccc1f3b09261f2f1eff": "\\tilde{\\Gamma}", "75284159cfa8a26be129b330a48472cb": "2^{2^{6}} + 1 = 2^{64} + 1 = 274177 \\times 67280421310721", "7528e44da43b64f5a0d47ec6e95a9be9": "\\Phi_3(x)", "7528e6d7263d156765c26836f3cea19c": "\n\\sin\\theta=\\frac{2}{N}-\\frac{2\\phi }{\\pi }\n", "75290e20557399b145db6c49cd792dce": "A(n,d,w)", "75293163fe63a20c300efff3e2683d31": " \\left \\| A \\right \\| _1 = \\max \\limits _{1 \\leq j \\leq n} \\sum _{i=1} ^m | a_{ij} |, ", "7529700812ca9bdd7f65ae3fa3eaaf76": "x^{AA'}=\\sigma^{AA'}_\\mu x^{\\mu}", "75299b912b6bd96ef52c726388ccb616": "D^{(\\alpha)}=\\frac{4}{1-\\alpha^2}(1-\\sum p^{\\frac{1-\\alpha}{2}} p^{\\frac{1+\\alpha}{2}}).", "75299c2520ca389119694b3da7cc7a84": "X^2", "7529e229497ad65b229f162fb655b115": " \\left(\\arctan(x)\\right)' = \\frac{1}{x^2+1}", "752a24e3401b4e17d7d0d6ee7eea04c5": "d(x,y) = 0", "752a5c6d265219ee007114854507ed66": "P_t = D \\Delta P. \\quad ", "752ab821062f0fd9faf6f9320a50a3ea": "\\textbf{x}_{k\\mid n} = \\textbf{x}_{k\\mid k} - \\textbf{P}_{k\\mid k}\\hat{\\lambda}_k", "752ad754b96a76276f226c50a272247f": " GB2(y;a,b,p,q) = GB(y;a,b,c=1,p,q). ", "752af1bbcc6ecf9a3a280adc252db0ff": "\\begin{align} e^x &= (c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \\cdots)\\cos x\\\\\n&=\\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \\cdots\\right)\\left(1 - {x^2 \\over 2!} + {x^4 \\over 4!} - \\cdots\\right)\\\\&=c_0 - {c_0 \\over 2}x^2 + {c_0 \\over 4!}x^4 + c_1x - {c_1 \\over 2}x^3 + {c_1 \\over 4!}x^5 + c_2x^2 - {c_2 \\over 2}x^4 + {c_2 \\over 4!}x^6 + c_3x^3 - {c_3 \\over 2}x^5 + {c_3 \\over 4!}x^7 +\\cdots \\end{align}\\!", "752babc87a24c930016a97ec93ac599f": " x_1 \\ x_2 = \\frac{c}{a}.", "752bff032802668e644d57e47ee6d655": "0 < \\alpha < \\tfrac{1}{2}", "752c264266abc15e65ca7daa509e7d78": "\n\\tilde{\\phi} = \\frac{ \\overline{\\rho \\phi} }{ \\overline{\\rho} }\n", "752c2fdff64b750f007285f719949f24": "\\scriptstyle (3.7\\pm3.0)\\times10^{-7}", "752c55d785230def5a9b3ba1a4ac2d6e": "\\{0,1,3,4,5,9\\}.\\ ", "752c6445c67b0a50297731269960e8f6": "U\\subset M", "752cd1bafc9252807b01f93fe9b5bba6": "\nv_\\alpha(\\mathbf{R}) \\equiv\\left( \\frac{\\partial v(\\mathbf{r}-\\mathbf{R}) }{\\partial r_\\alpha}\\right)_{\\mathbf{r}= \\mathbf0}\\quad\\hbox{and} \\quad\nv_{\\alpha\\beta}(\\mathbf{R}) \\equiv\\left( \\frac{\\partial^2 v(\\mathbf{r}-\\mathbf{R}) }{\\partial r_{\\alpha}\\partial r_{\\beta}}\\right)_{\\mathbf{r}= \\mathbf0} .\n", "752cef88e8627fcba17f473d7f241161": "262144 \\cdot \\left(\\textstyle{\\frac 2 1}\\right)^1 = 524288", "752d1bfad757034aab8f99e2f01f3638": "n=p", "752d6295ec09c76d087c0480fa17b14d": "\\mathbf{v_1}", "752d817538efa7adc303cede34b8e98c": "2^{19937}-1", "752eb3cccad62d9c385c98b7ac79a5de": "x_i^{(t)}", "752eee845ae5486c601ee333199b9ab0": "R =", "752f336b33fa2c4bbb667ccd9efb6160": "(x, y, z) = \\left(\\frac{2 \\mathrm{Re}(\\zeta)}{1 + \\bar \\zeta \\zeta}, \\frac{2 \\mathrm{Im}(\\zeta)}{1 + \\bar \\zeta \\zeta}, \\frac{-1 + \\bar \\zeta \\zeta}{1 + \\bar \\zeta \\zeta}\\right).", "752f5a63e3bcfeb936c963dd4492091c": "D_A", "752f6f9adf7072f8f74e56691c98574b": "A_1A_0X_0,", "7530040f2a76fb37ae1136328a848788": "\\begin{align} \\mathbf{L} & = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times\\mathbf{v}_i \\\\\n & = \\sum_{i=1}^n m_i \\Delta r_i\\mathbf{e}_i \\times(\\omega \\Delta r_i\\mathbf{t}_i + \\mathbf{V}) \\\\\n & = (\\sum_{i=1}^n m_i \\Delta r_i^2)\\omega \\vec{k} + (\\sum_{i=1}^n m_i \\Delta r_i\\mathbf{e}_i)\\times\\mathbf{V}. \\\\\n\\end{align}", "7530a1ad5657f36289bf66e54cbd9937": "(xy)^{\\lambda}x = y^{\\lambda}", "75312c3978b37e468d7a41426ea65059": "H^1_{\\mathbf{R}}(\\mathbf{C}, \\mathcal{O}) = \\left [ H^0(\\mathbf{C}^+, \\mathcal{O}) \\oplus H^0(\\mathbf{C}^-, \\mathcal{O}) \\right ] /H^0(\\mathbf{C}, \\mathcal{O}).", "75312de553ccadd93771d8e2599b8e05": "p_s \\leftarrow \\min(\\lambda x^*_{s},1)", "7531313f700b8c3999f0e5efa12eba22": "\\tilde{\\rho}(\\mathbf{k})", "75314b1079e6843856288ac82d804bf7": "\\begin{align}&\\left((1+x)^y+(1+x+x^2)^y\\right)^x\\cdot\\left((1+x^3)^x+(1+x^2+x^4)^x\\right)^y\\\\&\\quad=\\left((1+x)^x+(1+x+x^2)^x\\right)^y\\cdot\\left((1+x^3)^y+(1+x^2+x^4)^y\\right)^x.\\end{align}", "753246b75256ba733ae03b5dc2d07218": " V_+\\otimes V_- \\to \\mathfrak{gl}(V_+)\\oplus \\mathfrak{gl}(V_-)", "753250b94f950a5fed8f0d2bc2f9bf58": " T_s(t)=c H \\cos \\theta ", "753253996d12b1ecf49f48c33e675ac3": "1 < \\gamma < 2", "7532b8b661d563a0e953505a80ebd6ac": "A_a^i = \\Gamma_a^i - i K_a^i", "7532ce8d750affaaa175706b0e689e1c": "\\dfrac{PA'}{L1} + \\dfrac{QC'}{L2} = \\dfrac{\\Delta A -\\Delta B}{L1} + \\dfrac{\\Delta C -\\Delta B}{L2}", "7532fd7cf9533a6ffe89fc997f5642d6": "\n\\begin{align}\ny^{\\prime}=y-\\phi(1-z)^{\\mu}y\n\\end{align}\n", "753300367abf417542fab337fd03d0d6": "m\\frac{d\\mathbf{v}}{dt} = q\\mathbf{E} + q\\mathbf{v} \\times \\mathbf{B}", "7533202bcaad1a01d3f567ff1e9e2b8c": " U,\\mu", "753346f2b248a48e7ecfd8eefc6e7a40": " \\bar{n}", "753364f30961e246aafe75a390a54260": " x_2(x_1,\\lambda) ", "7533bd985a067c238c43a23cfbce5c1f": "(x'', f'') \\in fRep_{red}", "75343c9bb09a06591407cea10543d748": "L_{g}L_{f}^{r-1}h(x_0) \\neq 0", "75343d1b9f9adec9459f673433c5d748": "p(y_1^n|x_1^n)", "75344d2d81357a8022f7dfbe2697b3ea": " \\int d^2\\theta \\; (2 \\lambda f + \\mu) H_{\\bar{3}}H_3 + (-3\\lambda f +\\mu) H_{\\bar{2}}H_2", "7534cec804c64a325a0327772b0e4e63": "S_k = \\sum_{n=0}^{k}a_n = a_0 + a_1 + \\cdots + a_k.", "7534d01af3db4acbe077a885b12080a3": "Y(\\omega)", "75352b77d3cc77ab8b55e1164dc38f0e": "R_1 = (A, B, C)", "7535cbe4b47334a6924947637556572d": "B = q(A) \\oplus u(C) \\cong A \\oplus C", "7536248a91acf0b341bc7c572ffae019": "(x, v) \\mapsto (x, v).", "75363fb0138373d02a1b3b24fe40e558": " B^{N \\lambda}", "75364a46e7e060672bb12d1e611dcecb": "c(x) \\in x", "753669f959b6191c140edeba216c6dcf": " m = 27 ", "75366e4252757333c968d18e2b24203b": "E_k(t)", "753675332088eafac7bfefb620361e19": "\n\\alpha_{ij} = \\alpha_0 \\delta_{ij} \\Longrightarrow E^{(2)} = -\\frac{1}{2} \\alpha_0 F^2,\n", "753679bfbec8d7688db7c9ba952e745f": "I_1 = I_2", "7536967a1a93fa45525117159ad0b668": "B \\to b | a", "75378ea934d3857175c247f09f73704a": "dr^2", "7537e4bdf377a6c73bd6044bb9f7587e": "\\sum_{i=0}^2 p_i = 1 \\quad \\mbox{and}\\quad 0\\leq p_i \\leq 1.", "75385e4c2f82710eab5627b8d313ead2": "M_4 = -\\frac{1}{32} \\, S^{ag} \\, S^{ef} \\, {S^c}_d \\, \\left( {C_{ac}}^{db} \\, C_{befg} + {{{}^\\star C}_{ac}}^{db} \\, {{}^\\star C}_{befg} \\right)", "7538634d5a4a8ceba8ef3d30a93579d5": " I \\propto V \\rho_s (W, E_f) ", "7538ad161e529bc240f661ed89c13ea7": "\\mathrm{Ric} \\geq \\rho \\,", "7538bd19251b69816b45e16f986626da": "y = -2x + 6", "7538e9c296d0bfc3868ea3995bf6af75": "[k: k^p] < \\infty", "75396e304db096f7e41e3e0aa8fe2d27": "(x)_k=x(x-1)(x-2)\\cdots(x-k+1)", "7539c37b47eb6648e7d3eea5b6735586": "A f = i \\frac{d}{dx} f.", "7539e33204294651f954f8db705bd0f5": "k^* \\neq p", "7539f1052837c4f6dce0b83191a902f4": "\n\\int_a^b f(x) \\, dx\\approx\n\\frac{h}{48}\\bigg[17f(x_0)+59f(x_1)+43f(x_2)+49f(x_3)+48 \\sum_{i=4}^{n-4} f(x_i)+49f(x_{n-3})+43f(x_{n-2})+59f(x_{n-1})+17f(x_n)\\bigg].\n", "7539f34c802d314d87695714237619f0": "\\Delta_t \\rightarrow \\infty", "753ac60e9e5caaf041252ed578860db2": " (AB - BA)[0,k] =\\sum_{r'} \\left(\\; {dB\\over dJ}[r']i{dA\\over d\\theta}[k-r'] - {dA\\over dJ}[k-r']i{dB\\over d\\theta}[r']\\right)\n\\, ,", "753aec99ca956da74deeaed89f8ece0c": "V = \\frac {-Gm_S} {\\sqrt (x-d)^2 + y^2} ", "753afa14315dcbeaa23099d1561eefb8": " \\log {f(z)\\over z} = \\sum_{n>0} d_{0n} z^n", "753b11c9b06f266175656b7df68e2de3": "\\omega_N^{k+N/4} = -i \\omega_N^k", "753b19c6bd230a11dbc36ef1a2c5de2b": "\\mathrm{ker}\\left(f^k\\right) \\cap \\mathrm{im}\\left(f^k\\right) = 0", "753b4542c2ecb4e995f24a892aa31937": " K_\\text{OU}(x,x') = \\exp \\Big(-\\frac{|d| }{l} \\Big)", "753bca6ab776703350ebab78d9758a95": "i>j.", "753bdbab12089972a9ba0c33458d158d": "||f||_{W,p}=\\lim_{r\\mapsto\\infty}||f||_{S,r,p}", "753bdcd9385342513fab2c0dbee568ae": "n > 75", "753c4632efbd5d073365c45bae0a50fe": "\\left |\\frac{\\partial(x,y,z)}{\\partial (\\rho,\\theta,\\phi)}\\right| = \\rho^2\\sin\\phi", "753c6e4f11346f140f42622e626c703a": "|f(x)-f(y)|=\\int_0^1 |\\chi_{[0,x]}(t)-\\chi_{[0,y]}(t)|\\,dt = \\int_x^y \\, dt = |x-y|,", "753d1110daa4283041816d93fa6dc952": "NaN = NaN", "753da29851e3f71ea94bfc5f465c79b0": " S_{i,j}^t \\in \\{0,1,2\\}.", "753db35a0fda38554a132c5a7dc55af9": "\\boldsymbol{J}_1 = diag[J_{1xx},J_{1yy},J_{1zz}] = diag[0,J_{1},J_{1}]", "753db87f5900e20c3b80e900fb537d8e": "P_B = \\frac{(u!)^2(2-q)^{2u-m}q^m}{m!(2u-m)!}", "753dc1a8492fb19cf9069b2829c78377": "E_{sig}(t,\\tau) = E(t) E(t-\\tau)", "753dc1b3fdbf7f3e15e84212410597c6": "p\\in S^{2n+1}", "753dd995ccf39174b9f3d3024a61aca4": "k=e^{i\\,\\theta}. \\, ", "753e2a787708a862cd2e5aafcab20aeb": "\\vec {U}", "753e72ad716d56b516485d46f43e9e98": "\\mathbb{A}^n(k)", "753e8fdfdd2a87a10eeee9d70db14613": "T^t := S^1 \\times \\cdots \\times S^1,", "753e97fc5ff71bc9e7843a0cde831322": "\\alpha _T", "753e987feb33ea352d5e226a4e2c88ec": "\\mathbf{v}' = R\\mathbf{v}R^{-1}.\\,", "753eb657993792f3fd55207d36b58fe4": "L(p;q_2)", "754019d01fc889b161fee0f02b4309cd": "T(a,x,y)=b,T(c,x,y)=d \\,", "754044ef76686870923e9f36ab588e89": "L\\!> \\lambda_2 \\approx \\lambda_3", "7549e63bbdd31253e52403774a04d37c": "\\text{Hom}_R(V,R)", "754a29402d6d0314f0e58460c4cefb9a": "z_0 = u(x_0, y_0).\\qquad\\qquad (2)", "754a6e206ae4d9e931113726cddfd4e1": "{b \\over a}={{1+\\sqrt{5}}\\over 2}.", "754aae2423c97a0773aea7e1ba5779ab": "\\rho>0", "754aae2612e555e2343efbdcd8848d61": "\\dot Q_a", "754b0b8f3f42e2e13c3c0c179ed82ce8": "\\omega_m\\ ", "754b365426c47e9705f785d86f8eafa0": "U_s (J_0 \\tau_s = \\pi) \\equiv U_{sw}", "754b5dbceb0f91795a9ae8c6f45b85e3": "\\mbox{SC}= \\sin( \\omega_{s} t)", "754ba835472d3acdfa81a35f3c27355d": "|R|", "754bb6c5a43ae2ef7362ef32b01e4bdd": "\\bigstar\\bigstar\\bigstar\\bigstar \\mathbf S", "754c02831402cff9848b17953dd782ae": "1 \\leq j < i \\leq n\\colon \\left|\\mu_{i,j}\\right|\\leq 0.5", "754c054683d1a204a4142317d99351b2": "n = N", "754c938f8e62743082940830c3c1ab16": "\\begin{align}\n\\frac{\\partial F}{\\partial n_1} = n_1\\sigma_1^2-2n_1\\sigma_1\\left(\\sigma_1 n_1^2+\\sigma_2 n_2^2+\\sigma_3 n_3^2\\right)+\\left(\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\\right) n_1 &= 0 \\\\\nn_1\\sigma_1^2-2n_1\\sigma_1\\sigma_\\mathrm{n}+\\left(\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\\right) n_1 &= 0 \\\\\n \\left(\\sigma_1^2-2\\sigma_1\\sigma_\\mathrm{n}+\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\\right) n_1 &= 0 \\\\\n\\end{align}\\,\\!", "754cae0cf261e828f847539a2505a7e3": "[\\sigma_1]", "754cca3f699ccd0d72694c8e89cd06e7": "\\mathbf{v} \\cdot \\, d\\mathbf{s} = 0 ", "754cdbbe9ac93964a2fa22c84c0327a5": "ST_x(\\varphi \\wedge \\psi) \\equiv ST_x(\\varphi) \\wedge ST_x(\\psi)", "754cf97a677cdfbba0e15a0a8194db5c": "\\mathbf{B}(\\mathbf{c},\\mathbf{e}) =\n\\begin{bmatrix}\ne_1^1 - c_1^1 & e_2^1 - c_2^1 & \\cdots & e_T^1 - c_T^1\\\\\ne_1^2 - c_1^2 & e_2^2 - c_2^2 & \\cdots & e_T^2 - c_T^2\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ne_1^{n_T} - c_1^{n_T} & e_2^{n_T} - c_2^{n_T} & \\cdots & e_T^{n_T} - c_T^{n_T}\n\\end{bmatrix}\n", "754d94b40f9aae0ad322458e9844ee62": "\n{\\alpha\\over 2} (\\partial^\\mu A^\\mu)^2\n", "754df575de778283babfce4ae186311e": "\\scriptstyle 10\\log(2^{2N+1})\\,dB ", "754e00569577ac23b7460a25e0c8eafc": "1/(1-e^{-i \\omega})", "754e32aef0f48c6572b8b2dab8a7627f": "\\mathfrak{J}^\\mu", "754e3c72468b5765e2082e3562c6758c": "=\\mathbf{a}_\\mathrm{B} + 2 \\boldsymbol{\\Omega} \\times\\sum_{j=1}^3 v_j \\mathbf{u}_j (t) + \\frac{d\\boldsymbol{\\Omega}}{dt} \\times \\sum_{j=1}^3 x_j \\mathbf{u}_j + \\boldsymbol{\\Omega} \\times \\left[\\boldsymbol{\\Omega} \\times \\sum_{j=1}^3 x_j \\mathbf{u}_j (t) \\right].", "754e58c05c33ebfdc84e1a5ee87b2df4": "\\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+1)\\frac{q\\pi}{p}\\right]}{(kp+1)^{2m}} + \\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+2)\\frac{q\\pi}{p}\\right]}{(kp+2)^{2m}} +\n\\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+3)\\frac{q\\pi}{p}\\right]}{(kp+3)^{2m}} + \\, \\cdots \\, ", "754e6dfa913a7078e3eda2b3e1f2b893": "(X,\\mathcal A)", "754f2a5d156cd277a74308bc85b5c525": "\nH ={1\\over 2}(X^2 + P^2)\n", "754f812a3bcac00b4eac6329a1995b39": "X(.)", "755004c4a1a5c5f0408dec3f423fe7cc": "\\scriptstyle{\\left(0<\\frac{dB}{dt}<\\infty\\right)}", "75509417bfa20b31851ba67a0fc262ae": "f=\\breve{f}\\circ m", "7550a13d95cc0311d58c8dad407b098f": "\\textstyle \\frac{282.481}{x}", "7550b332d2cd7487e46afcb4e6f0f69c": "y = 100000845^{4096}", "755100fc7695a29ece527e3962efacaa": "\\left(\\frac 1{2\\pi}\\right)^{d/2} \\int_{\\mathbb R^d}\\cos(\\|x\\| )e^{-\\| x\\|^2}\\,dx,", "755123733de42d9948b3a27f9fd2c2f2": "-3x_1+x_2=-2", "7551e135ebabd4b88a70ff3b91dfdd28": " \\hat{l}_z = \\hbar \\hat{S}_d. ", "755251b28f6e464a911e55e3e20816cb": "e(t)=r(t)- y(t)", "75527e3e788634ba119350b1be27a3df": "\\frac{\\pi^2}{3}", "7552bc5d10bde5c52095e065827565a6": "\n\\mathbf{T} = \\left(\\begin{matrix}\nx_1-x_3 & x_2-x_3 \\\\\ny_1-y_3 & y_2-y_3 \\\\\n\\end{matrix}\\right)\n", "7552cae211b9415cd494dc6560379ed3": "H(Y|X)", "75532e962c8b3cbda3276b17b41b52ac": "u(t)=S(t)u_0", "755337d0f10b2e6de2c065e2b287d085": "\\textstyle \\varepsilon > 0 ", "755348255ea0d7b075e22e19082c3859": "{\\psi}_n(x) = \\frac{2^{1/4}}{\\sqrt{n!}} \\, e^{-\\pi x^2}\\mathrm{He}_n(2x\\sqrt{\\pi}),", "755354dc4e0109ddf8131fbd8647ecfb": "\\delta(t) - 1", "7553dc1d921d25f8b60ef2431e8c6512": "s_5=\\alpha^{5},", "75540bcb805a80373030eb24af7bba88": "T(n) = T\\left(\\left\\lfloor \\frac{1}{2} n \\right\\rfloor \\right) + T\\left(\\left\\lceil \\frac{1}{2} n \\right\\rceil \\right) + n - 1", "7554218248180b54e3a67f1b0f671731": "[T(u)]_\\gamma=[T]_\\beta^\\gamma[u]_\\beta", "7554423e30a7be0600bd83b551e3adf6": "\\beta:= -\\frac{1}{2}\\big(n^a\\delta l_a-\\bar{m}^a\\delta m_a \\big)=-\\frac{1}{2}\\big(n^a m^b\\nabla_b l_a-\\bar{m}^am^b\\nabla_b m_a \\big)\\,,", "75546413847ed388b1ce670f8915abc0": " f = (1\\ 3)(4\\ 5)=\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\ 3 & 2 & 1 & 5 & 4\\end{pmatrix} ", "7554712e25638361dc532bfa6fe0f03f": "\nq_{k} = \\sum_{s_{1}=-\\infty}^\\infty \\sum_{s_2 = -\\infty}^\\infty \\ldots \\sum_{s_N = -\\infty}^\\infty A^k_{s_1, s_2, \\ldots, s_N} e^{i2\\pi s_1 w_1} e^{i2\\pi s_2 w_2} \\ldots e^{i2\\pi s_N w_N}\n", "7554720480e960dd5cf4ffeb68932a91": "2\\rightarrow\\left(3\\rightarrow2\\right) = 2^{(3^2)} = 2^{3^2} = 512", "75548a9518e74d02e301c85c9c25d0fa": "A = (\\Sigma, D, Q, \\delta, q_0, F )", "7554bd2522fe4038fd7a4095b6b34e0b": " \\mathbf{L} = n \\cdot \\hbar = n \\cdot {h \\over 2\\pi} ", "7554eb7cd9e49ba813c49eb80d33f4ff": "\\mathrm{FillRad} \\leq C_n \\mathrm{vol}_n{}^{1/n}(M),", "75551a741005032705bdb7fd8bd219d5": "\\mathcal{H}\\Psi =0", "75559a2723eaf88d6eaaf6daab2d3c0c": " T(0) = I ", "7555bab333eb1bc8fced18787aad0f71": "\\omega^n= x^*_1\\wedge y^*_1\\wedge \\ldots \\wedge x^*_n \\wedge y^*_n.", "7555f593df02a593176c5da3894600ca": "\\partial_\\mu A^\\mu=0 \\!", "75560b9d08c7029bd6d69f2c4b7220db": "\n\\dot C_{FF} (t)\\,\\, = \\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,\\,\\exp \\left( { - \\lambda \\,t} \\right)\\,\\,\\int_0^t {Q(\\tau )\\,\\,\\exp (\\lambda \\tau )\\,d\\tau \\,\\,\\, + \\,\\,\\,\\dot C_0 \\exp \\left( { - \\lambda \\,t} \\right)}", "75565270da0c2b034ed41c4b4bf3c3ee": "W_t", "7556c02f89ccbba295f813007b5ddc3d": " \\sum_{k=1}^\\infty H_k z^k = \\frac{-\\ln(1-z)}{1-z}, |z|<1", "7556c4d817506fc810657b7c53138ef3": "\\cdots\\to H^1(\\mathcal O_M)\\to H^1(\\mathcal O_M^*)\\to H^2(2\\pi i\\,\\mathbb Z)\\to \\cdots.", "7556d011f32f131d31f36d55db855908": "\\Phi=\\frac{0.0312+0.087(c-1)+0.008(c-1)^2}{1.000+2.455(c-1)+0.732(c-1)^2}", "75573fd17ff9eefe945a2a05b6a38752": "b - 1", "75583dc57d1a19cbf7b1d25ec6fe1061": "b=(b_{-N},\\dots,b_0,\\dots,b_N)", "755867fa99b1829f01b526c2bbad1b8b": "p_Y(y\\mid X = x)=P(Y = y \\mid X = x) = \\frac{P(X=x\\ \\cap Y=y)}{P(X=x)}.", "7558abd27c7d521efab8a7b34546abc4": "= (\\sum_{i} (v_i^{T} x) v_i)^{T} M (\\sum_{j} (v_j^{T} x) v_j)", "7558b327714ce8566dbe624679cee3f5": "S^3 \\setminus V", "7558bde5e8ba82d9ad360526c12be4dd": " m = \\int_V \\left( 2 T_{ab} - T g_{ab} \\right) u^a \\xi^b dV", "7558ceb0fd895e076400a5d90aa23631": "p_\\text{sat}(T)", "7558dc8ef57546e57d907bf5c80f4afb": "T \\subset N", "755915b5730ff4fc9b9ff95eb96621a9": "B_4 = \\left(\\frac{1}{5} + \\frac{1}{7} + \\frac{1}{11} + \\frac{1}{13}\\right)\n+ \\left(\\frac{1}{11} + \\frac{1}{13} + \\frac{1}{17} + \\frac{1}{19}\\right)\n+ \\left(\\frac{1}{101} + \\frac{1}{103} + \\frac{1}{107} + \\frac{1}{109}\\right) + \\cdots", "75596b9ba501c0ffd85b74d1b2e16028": "1 + {1 \\over 2} + {1 \\over 4} + {1 \\over 8} + {1 \\over 16} + \\cdots=\\sum_{n=0}^\\infty{1 \\over 2^n}.", "75596ba0d0b892f229040df3cbbe639b": "U_{r}(z)=\\sum_{k=0}^{\\infty}A_kz^{k+r}", "755987fe533be2b4b948c6419b6f9856": "\\rho_2=\\rho_3 = \\frac{x^2H(x)Q(x)H(x^9)Q(x^9)}{Q(x^3)^2}", "755995e756770d7ddc7cd1a4a884f482": "\\sum_{k=0}^n f_k t^k = Z(G; 1+t, 1+t^2, \\ldots, 1+t^n),", "755a1732ee062bcdbcca8fbb0664f02a": "\\varphi(x,y,t)", "755a2eae4f161f0d65cd09c1545aaa25": "InDeg(p_k) = S", "755a32b1523ef1cf147cc94781c66e94": "b(s)=\\prod_{j=1}^r\\prod_{i=1}^{n_j}\\left(s+\\frac{i}{n_j}\\right).", "755a894bfee424c95b067fc0d6c9f3d0": "x\\equiv 2\\mod 9", "755b5f73c50d7b24c7cfaf305da944d1": " \\psi(x) = C \\sin kx + D \\cos kx.\\!", "755b8f8586f157a7425ffb271159ecb0": " G_\\mathrm{eff}=\\frac{1}{8\\pi F}\\frac{1+4\\frac{k^2}{a^2R}m}{1+3\\frac{k^2}{a^2R}m},", "755bd1fac81644cfbcad2844615c9bcc": "(x,y,z) = \\left(r\\cos\\phi,r\\sin\\phi,R\\cos\\frac\\phi2\\right).", "755c27e528782b0685522aa3605722b5": "\\scriptstyle d\\vec{S}", "755c2f54f010ea57e145af9458c8cce8": "\\operatorname{E}(TX^2)=\\frac{1}{\\lambda} + \\mu^2 \\frac{\\alpha}{\\beta}", "755c3878b8b97542b814c4ea32b7b233": "m - 1 \\le d - 1", "755c7983483e0419b2ea18957115fa6b": "\\, S_3", "755c9eb85adb0d0d95ef29978b012275": "\nBn_P(Cl_t^{\\geq}) = \\overline{P}(Cl_t^{\\geq})-\\underline{P}(Cl_t^{\\geq})\n", "755ca87c28d8cb59468f2e71bb0db752": "\\, =[10(x+y)-10(x+y)] + [100-100] + xy", "755cc64108d666a486b44e5a47dfac1d": "m_i \\times m_i", "755d057aa948bc52320671939f4aa69d": "\\scriptstyle{\\sqrt{2} \\hbar}", "755d561e912f11ca04b78eb9c42ac002": "V = \\frac{1}{3} \\pi R^2 H. ", "755d6c75c25fb62c372b5cbd569d1051": "(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z)", "755da44de52191540a632f23fd1d7004": "\\mathrm D_{\\mathsf C}", "755dd6a7ed0ac147e31c8429b4b9898b": "(\\rho\\sigma)^{1/\\rho} =e^{1-1/\\rho} \\limsup_{n\\to\\infty}\\frac{|f^{(n)}(z_0)|^{1/n}}{n^{1-1/\\rho}}.", "755e09654d2b145f68ed8fcf0808f405": " [v]_B = (\\alpha _1, \\alpha _2, \\cdots, \\alpha _n) .", "755e4ed1139fac6fea74f73deb4ad123": "k \\leftarrow ESHash(k,L_B(P),s,u_1,u_2,\\tilde{h_1},\\tilde{h_2}) \\in W^8\\,", "755ea14d2609a62285dbdfefbef26e36": "\\|\\mathbf{a}\\| = \\sqrt{g(\\mathbf{a},\\mathbf{a})}", "755eaefa7c578980bbfc68ff713a0983": "\\chi_\\pi = \\chi_\\lambda", "755eb5305fe976c90e66514d234266ed": "P_{em} = \\frac{3R_r^{'}I_r^{'2}(1-s)}{s}", "755ec75b03da09bfa62c2a11f5ce5686": "8x+4x^2=0", "755eceba2d1c23fb97b1d4eb3a880058": "\\Longleftrightarrow", "755ef4c72ea624d50e8d6d5c9c1f2041": "\\ E_{bonded} = E_{bond} + E_{angle} + E_{dihedral}", "755ef928859a6823c0822c18e7e844ea": "L(\\alpha) = \\int_0^1{\\|\\alpha'(t)\\|\\, \\mathrm{d}t}.", "755f71bc8bb24343a241b5c4f009c82d": "Pf(PXP + \\lambda(1 -P))P \\leq Pf(X)P", "755f7f97ac1765c83ed2e1fe588e22a9": "\\{a^n b^n c^n | n \\geq 1\\}", "755fc2eb2d6e7d0ea69c499547e51dc5": "e^{i0} = 1", "755fe12c886c58bc5bc509518a289602": "r' \\ge 0", "7560032697564033969cec85a60b7f7f": "2^{k - 1} + k", "7560268fc3a77169576b61fbca215c50": " \\ell(|x-y|){{=}} |x-y|^{\\alpha}", "75606253cb6fc32fa5f0d8c51eccbbb6": "f: \\{0,1\\}^{n} \\rightarrow \\{0,1\\}^m", "7560670c364031423a35903cce7ad42e": "4^n+2^n+1", "7560d4c045aab89875af32c7c9d93990": "\\ x=(d/2) \\xi \\eta, ", "7560f6d484b5adcc85406791f670466f": "G_{imp}^n = G_{imp}^{n+1}", "7561137b65b092f15f75acccaddeb314": "\\left[J_+, J_-\\right] = 2\\hbar J_z.\\quad", "7561309bd41a74e99d1ea16e4a550db3": "G_X(t)=M_X(\\log(1+t))", "7561679067d39d0423fd6031753fd7c6": "V_{j+1} = \\left( A \\times V_j \\right) \\bmod P", "756176bf0ca291701033a7eb2eadb18a": "Y_{7}^{4}(\\theta,\\varphi)={3\\over 32}\\sqrt{385\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(13\\cos^{3}\\theta-3\\cos\\theta)", "7561d5d9468a99cee909b59788057d5f": "x^2 - \\left ( \\left (90581 - 89964 \\right ) + 1 \\right )x + 90581 = 0", "75620886ddd35318e4d96ce0d87c1db5": "\n\\phi(q) = \\sum_{n\\ge 0} {q^{n^2}\\over (-q^2;q^2)_n} = {1\\over \\prod_{n>0}(1-q^n)}\\sum_{n\\in Z}{(-1)^n(1+q^n)q^{n(3n+1)/2}\\over 1+q^{2n}}\n", "75629126ce539ea243591a0c2654e693": "F = \\frac12 \\times 1.2 \\times 10 \\times 0.9 \\times 8.3^2 = 372 \\ newton", "7562a489a67c4e0dd3ade54c501ba2a1": "\n{\\mathbf P} = \\varepsilon_0\\chi{\\mathbf E} = (n_r+in_i)^2{\\mathbf E},", "756374060dc489c359a383f525e9451e": "(T^{\\ast}N,\\theta)\\, ,", "7563aa031fa40d77f5faf24ae9ee66b4": "l/k", "7563daefde9ade20a0cf9c283add459d": " \\left(\\frac{dP}{dT} \\right) = \\frac{\\gamma_D}{V}C_V, ", "756447cdddea927d703dbd8c15af66f1": "\\sqrt{S^2-P^2}", "756471f376731843a482283dbffd16b0": "I_1\\cup I_2", "7564749da3ec104b604c0a016b24825c": "g(0)\\neq 0", "7564c0121dbb825b16194845465a5c18": "(2,d,1)", "75655583ab4641726bd3580a7875ef48": "q_\\beta(Tv)\\le Mp_\\alpha(v).", "7565715076322dacae04b7ee4926f03b": "t_-", "7565f6d2e40c9cf710d9bcb4369d50fa": " \\cfrac{\\Gamma \\vdash p \\qquad \\Delta \\vdash q}\n{(\\Gamma - \\{q\\}) \\cup (\\Delta - \\{p\\}) \\vdash p = q}\n", "7566090544a0d05b57d9b9b3fa9988e2": "P \\cup L", "75660a2b0fd7f503fb79a44bf6279f8a": "|\\triangle M| = \\sum_{l=1}^{n} (-2)^{l-1} \\sum_{i_{1} \\ne i_{2} \\ne \\ldots \\ne i_{l}} |M_{i_{1}} \\cap M_{i_{2}} \\cap \\ldots \\cap M_{i_{l}}|", "756622ce4179d8bd7d6b58f6d7bbf5d4": "\nq\\sigma \\xrightarrow\\alpha q'\n ", "756672a2e7911590b6ed6ee38250cbf1": " S^{\\sigma} ", "75669318f4bde648391fb0fce0187a97": "x \\;=\\; \\frac{2t\\sqrt{c}+b}{t^2-a}.", "7566ca7ca1c27d7297e831c5dae48785": " |\\mathbf{V}| = |\\dot{\\mathbf{P}} | = \\frac {d s}{d t},", "75670004255a2abb9f4b2995a41afc78": "x_j\\notin A_2", "756714061551bb8b93ff27f48eea3215": " \\Phi_B = B A \\cos(\\theta),", "7567457f11bcc27635de887dcc1419a4": "\\ d = G(m).", "75677452989b3c41d093f30ab476d39f": "x^2+q=px", "7567c0be587398c1cbc1a708ffba0e88": "h_{\\alpha \\beta} = g_{\\alpha \\beta} - \\eta_{\\alpha \\beta} \\,.", "7567fcd788dfcd284af9aedc3048b8e2": "y_k = g_{k+1} - g_k\\,\\!", "7568b31316db5011bc602290ab12e7c3": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 2}{40 \\choose 2} \\end{matrix}", "7568b5a0959582033085fc66d8d8b1b4": "\\bigwedge^nT^*M", "75695b46abca7ce53dfa3b4e984a45ca": " R ", "756964da96024351c6f1cfb0ae99432c": "{\\mathbf W}=\\frac{1}{n-1}\\sum_{i=1}^n (\\mathbf{x}_i-\\overline{\\mathbf x})(\\mathbf{x}_i-\\overline{\\mathbf x})'", "75698ca31a2c391c728c179981586895": "\\mathrm{d} G = V\\mathrm{d}p - S\\mathrm{d}T + \\sum_i \\mu_i \\,\\mathrm{d} N_i \\,", "756a22a74ef5c8b19429276dd77eac72": "\\int\\limits_{-1}^1 \\sqrt{1-x^2}\\,dx = \\frac{\\pi}{2}\\!", "756a6d4bbc3d3f288d1ace0e898123ff": " y = \\frac{1}{1+e^{-z}}", "756aaa9df8adc5d8a0a3b089614937a6": " |\\zeta \\rangle \\equiv \\begin{pmatrix} \\zeta_x \\\\ \\zeta_y \\end{pmatrix} = \\begin{pmatrix} \\cos\\theta \\exp \\left ( i \\alpha_x \\right ) \\\\ \\sin\\theta \\exp \\left ( i \\alpha_y \\right ) \\end{pmatrix} ", "756ab044cab75b686f226c1fb301dc48": "\\scriptstyle \\Lambda(d_k)", "756b8f03e2a41aa367477ef0430f156d": "\\frac{q_C}{q_H} = f(T_H,T_C) = \\frac{T_C}{T_H}.\\qquad (3).", "756bb031a004cf374f0f843df94b959e": "+j \\frac {1}{\\sqrt 2}", "756c02427f7cb361146a64f2cc1bdbd6": "f,g\\in H^1(\\Omega)", "756c10277b66b9a49529f286c6f8a129": "\\Omega_0 = 0", "756c3f7890e81adec8550b20663ab499": "2^{|x|^c}", "756c68785915252fe8cbefb2886c5f9c": "x_0 \\in U", "756cac88dab62c67a5adc165be9248da": "H\\left(z_1,z_2\\right)", "756cdb0d9c8a6fdf7dd2a4ae437b5033": "\\zeta_a", "756d244e127bc6f6517b4afc80496085": "\\phi_m\\!", "756d329d18b10b4168c3a4aded3f10b5": "r(x) = r'(x)\\,", "756d3a73a364908bbb24bdceed8f2df0": "\\beta > -1", "756d45bc146fb29c8f552e5a42c5b930": "\\frac{!n}{n!}", "756d70f4210e2889ed409e25036ddd35": " r_c ", "756daa4deaf322e1856af058f186ff56": " S_h' = \\sum_{n=1} {{h^n} \\over {(n-1)!} }D^{n-1} = h \\cdot S_h. ", "756dcc13b60cb379ea80a627155b70b5": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{N} \\langle \\sum_{k=1}^{N} \\left( \\mathbf{r}_{k} - \\mathbf{r}_{\\mathrm{mean}} \\right)^{2} \\rangle,\n", "756dd0912fe8ed8880e6839f0e9a1ff6": " PSN = \\frac{2*HR*SB}{HR + SB}", "756dd0d5d6a872955e5b57f9ce59d216": "3^5+11^4=122^2\\;", "756e309a92e9700d398cdbcdd9d56c23": "\\Omega_{\\mathbf k}", "756eee5c238be2565f8a0324886325a9": "\\sigma_x(n) = \\prod_{i=1}^{r} \\frac{p_{i}^{(a_{i}+1)x}-1}{p_{i}^x-1}", "756efbcb50a4b221c21ec9a34b968a36": "ds^2 = \\gamma_{33} \\left ( d\\xi^2 - dz^2 \\right ) - \\gamma_{ab}dx^adx^b - 2\\gamma_{a3}dx^adz. ", "756f769518a0e061c689bd3a8007a5f5": "S\\;", "756f8f263b941cd21d05b54d36073866": "\\nabla \\cdot \\mathbf{E} = \\frac{\\rho}{\\epsilon_0},", "756f92bba077150c19a9d4e1a76d272c": " \\lambda \\cdot (\\vec{a}+\\vec{b})=\\lambda \\cdot \\vec{a}+ \\lambda \\cdot \\vec{b} ", "756feb7c7e1a0677fbd0738ed2276343": "a \\in A_i", "757002a3e3388d7b348025b428f3307e": "q_{\\theta} = \\cos \\theta + \\bold{k} \\sin \\theta", "75703ec49748a3cef69b0dfc1d29f639": "(\\Gamma_1 \\cup \\Gamma_u)^*", "75705d2debc1ed720400af9b59929f11": "\n \\mathbf{t} = \\boldsymbol{\\sigma}\\cdot\\mathbf{n} \\quad \\implies t_r = \\sigma_{rr}, ~ t_\\theta = \\sigma_{r\\theta} ~.\n ", "7570733b82e9907c73194ab5b0eabcd6": "F = \\mathrm{d} A", "7570853314d86254dcd8f9380f87065c": "x_1, \\cdots, x_K", "7570f0760e041488a08a91639210dc0d": "\\displaystyle{D=-{d^2\\over dx^2} + x^2.}", "7571535aecde52a1d482a2df75dd32c7": "H = \\sqrt{det (q)} (K_{ab} K^{ab} - (K_a^a)^2 -\\;^3R)", "75715c38d2c7603600b7476c1a580bc1": "\\mathrm{I}_\\mathrm{P}\\,", "757216e2b48232f749cfd0783566298e": "x_i=X_i/X_0", "757253855343b685287db4b97c5ca677": "\\Longrightarrow \\quad \\,", "7572576a20a5e722dafb572a5e39bf9e": "\\dot\\gamma(t)\\in H_{\\gamma(t)}(M)", "7572a975df013325ecbcf4bd58884220": "E_\\mathrm{p,e} = \\frac{1}{{4\\pi\\epsilon_0}}{{Q_1Q_2}\\over{r}}", "7572b21b20356ead4bc6354ca099ee9e": "\\hat{\\mathbf{h}}(0)=\\operatorname{zeros}(p)", "7572b2edf5ae7b2075602c35416462c6": "\\begin{align}\nc &=& \\cos \\theta\\\\\ns &=& \\sin \\theta\\\\\nC &=& 1-c\\end{align}\n", "7572e688de7f1788f205ddcdded98fd0": "\\frac{1}{\\sqrt{N}}", "7572fdb7929eac9d9d11ec9cef163306": "\\le 2", "7573569a402426b840983f3d0221ab6c": "j/j_c", "7573b61f41715cf70c3119505f33c7af": "T_{ko}", "7574d336b4c5cf1f91875ff9329659fe": "\\mathrm{GL}_{5}(\\mathbb{F}_{2})", "7574d9e6a830c89f275c7fbcb54ae408": "B(v,w)", "7574f3adfbc38f0d07d7c6cbb76e6554": "c\\sigma^2", "757526b43f2b9659ca216bcfc0c0fa0f": " i\\not=j ", "75757894bd802567b1cf23dcd6f9728e": "(1+x)(1 - x + x^2 - x^3 + ... + (-x)^n) = 1", "75759434f34b0ef620de8613c19dac6c": "dY=e^{- \\int_t^s V(X_\\tau)\\, d\\tau}\\sigma(X,s)\\frac{\\partial u}{\\partial X}\\,dW.", "757598e2c0e83eda985c3bc0359f88a4": "\\ P_o ", "7575b77b94cfb6f68b0af53b7942315a": "[f(x + \\Delta x) - f(x)]/f(x).", "7575be2720f6084c2b2b370a3ed5a1e7": "f: I_1\\to I_2", "757608bc3dfb07daf3c1d0e95e68b90e": "\\alpha \\leq 1", "757660e60dd0476f51351864fae067c3": "f(t)=\\sum_{k\\ge 1} \\frac{1}{b}\\binom{-(b+ak)/b}{k-1}\\frac{t^k}{k!}", "75766c64dbe96317a7f23d970e24c245": "y_1,\\dots, y_T", "75767d9db33b737a71b61895f7ec612c": "f\\colon x\\to x'", "757696af7978517db3a281bb16f90795": "x^5+x=-a\\,", "7576b0f8db20fabe1cb8361745230731": "|x_1 x_2 \\cdots x_N; S\\rang = \\frac{\\prod_j n_j!}{N!} \\sum_p |x_{p(1)}\\rang |x_{p(2)}\\rang \\cdots |x_{p(N)}\\rang ", "7576b28f593a51e37771c166130bac88": "f:c\\to c'", "7576ced2d525fa9d31423ed536573351": "\\{ \\hat e^i: 1\\leq i\\leq n\\}", "757761dc4993b0c3182370cb310d513d": "\\frac{\\lambda-\\lambda_{0}}{\\lambda_{0}} \\approx -\\frac{f-f_0}{f_0}", "757781250cfc6ef8f1bcde24168f37cf": "\\ln \\,\\operatorname{var_{GX}} (\\Beta(\\alpha, \\beta))=\\ln \\,\\operatorname{var_{G(1-X)}}(\\Beta(\\beta, \\alpha)).", "75778aed12a16702ed2db7cd29ce1995": "H^{0}_{\\mathrm{dR}}(M) \\cong \\mathbf{R}^n. ", "7577c5800eda6853a4d752d248905a06": "\\pi_i \\Pr(X_{n+1} = j \\mid X_{n} = i) = \\pi_j \\Pr(X_{n+1} = i \\mid X_{n} = j)", "75784cb259bee5420953266a19938487": "\\left(\\theta\\right)", "75786ac2cc9182c4dd8b2cd7dca427b6": "\\ a : b : c", "75786d6189e6998caf15bd8744299c69": " \\int f(x) \\,\\mathrm{d}x ,\\,", "75787adf8974c39ef41f1c735f40d729": "\\langle\\psi '| \\psi '\\rangle = \\langle\\psi |\\hat{U}^{\\dagger}\\hat{U}|\\psi\\rangle = \\langle \\psi|\\psi\\rangle = 1.", "7578aa8daec314cdf3ad84deeb81b39b": "(\\epsilon \\otimes 1) R = (1 \\otimes \\epsilon) R = 1 \\in H", "7578ca0be84007a7fd6ff5a1b4836ef1": "\\frac{r_\\max} b =\\frac b{r_\\min}", "757907ddc54802be96c264ff86d7cfa8": " \\mathbf{1}_E \\le S_{\\mathbf {G}} := \\sum_{B \\in \\mathbf{G}} \\mathbf{1}_B \\le b_N.", "75790cea974920ed0f0ccc0a8afd2130": "y(t) = A_3 \\sin (tf_3 + p_3) e^{-d_3t} + A_4 \\sin (tf_4 + p_4) e^{-d_4t}. \\,\\!", "7579210c390b6acfcc703f97fb0ad212": "Y = A + b\\,\\mathbf{C}X", "75793bda98e7969a1f055e66c80d2817": "\\mathrm{ROE} = \\frac{\\mbox{Net Income}}{\\mbox{Shareholder Equity}}", "757992a085a63735f371092d473685ae": "\\ \\varphi", "757a30a5ae52c3612d441a02d6e2fcae": "\\Pr_x[C(x, y_0) = f(x)] \\ge \\rho", "757a5156fa982700542fa6f65f229bbb": " {-{\\pi}\\alpha (\\tau-t)^2} ", "757a58c29dd253f4e5e906e5efd717bb": "\\begin{matrix}\\frac12\\end{matrix} (3x^2-1) \\,", "757a7feaa5761a35a520ad95a406a00d": "\n \\boldsymbol{K} = \\boldsymbol{A}^{-1}\\cdot\\boldsymbol{K}\\cdot\\boldsymbol{A} = \\boldsymbol{A}^{T}\\cdot\\boldsymbol{K}\\cdot\\boldsymbol{A}\n ", "757a8090bdfd7dd1c0c2af8507879270": "\\overline{\\mathbf{e}_2}(t) = \\mathbf{\\gamma}''(t) - \\langle \\mathbf{\\gamma}''(t), \\mathbf{e}_1(t) \\rangle \\, \\mathbf{e}_1(t).", "757a89fb6cfe1892b73850c00d53ca44": "V(\\phi)=\\frac{1}{2}\\mu^2\\phi^2 + \\lambda\\phi^4", "757af36545cbbb8f5d8ef1d2af80c180": "b_1x-b_0", "757b10a1c48e6fba1ec5abf1462571fd": "\\langle x,y \\mid x^2=y^3 \\rangle \\, ", "757c5361f4d7ea7155212164a3d09a74": "(S,T,W,C,M_0)", "757cb5062586d0afef61697ee0834220": "\\sigma(A \\oplus B) \\ge \\sigma A + \\sigma B.", "757d123ef8c5ec0f43044732a43c1382": "ar+ar^2+ar^3+\\cdots = \\frac{ar}{1-r}.", "757d379bad9897f73409308bccf2d2ec": "{dy \\over dx} = -\\frac{1}{\\sqrt{1-x^2}}", "757e252b61c8f93039952ce04e0c072b": "\n f(\\mathbf{y}) \\geq f(\\mathbf{x}) + (\\mathbf{y} - \\mathbf{x})^T \\nabla f(\\mathbf{x})\n", "757e309d403b19f67206333262f986c5": " (1-t^2)\\frac{d^2y}{dt^2} - t\\, \\frac{d y}{dt} + (a + 2q (1- 2t^2)) \\, y=0.", "757eaccf591398611cee41b077021b16": "x_{1}(t)", "757f364c3fcc96e88da2f839bb09aaa9": "(r, \\theta, z)", "757f49caa99f7b2f95b02fa29225f5dc": "P(t) =\\ I(t)V(t) =\\ I^2(t) M(q(t))", "7580780538c89b4bb4d288aee305f005": "''sliding-angle'' =\\frac{Rl * Rs * Im * Ka}{0.0113}", "75808b4018f334669c578acc071b7f98": "\\left( \\begin{matrix}\n 1 & 0 \\\\\n -\\frac{1}{f} & 1 \\\\\n\\end{matrix} \\right)", "75809831fda2b1ae78df97abf26d53ee": "\\frac{dB_{i,k}(x)}{dx}=(k-1)\\left(\\frac{-B_{i+1,k-1}(x)}{t_{i+k}-t_{i+1}}+\\frac{B_{i.k-1}(x)}{t_{i+k-1}-t_i}\n\\right)", "7580afe66dda4dbc281d8fb078e51eaf": "0 + 120 sin(0^\\circ) + 240 sin(120^\\circ) = 120\\sqrt{3} = 207.8", "7580f22f591be0b0faa0e9f335868a6f": "{\\mathbf{}}N", "75816890e63292e53165ae69d1c1637c": "\\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{q_i}} -\\frac {\\partial L}{\\partial q_i} = 0 ", "7581d4baa29af4509b5cd3d5cdcc610f": "\\bar{X}_4", "758202b53386e0153743d2b85cffe07a": "\\mathbf{E_2}(\\mathbf{r},t)=\\mathbf{E}_{02}\\cos(\\mathbf{k_2\\cdot r}-\\omega t + \\epsilon_2)", "7582748f3b07cc06a07eab33e07c0478": "{\\sigma_i}^2 = 1,", "75831632c31ca26de989125b304600da": " \\frac{0}{0} = NaN ", "758398dba8a9b1e1826cfe10ff181458": " \\theta_f \\approx \\lambda / d ", "7583a86529ef2cd416cc798558b22688": "2\\pi R^2 (\\cosh \\frac{r}{R} - 1) \\,.", "7583dda2f4f3e97ccbfa5f6f6fca6c99": "n,d,t", "7584b3a8d3a24b11395b0ffe26ca1409": "\\hat{a}_p^\\lambda \\,", "7584dfd12bdaf1083afa555b0ab5a2ca": "c : V \\to S", "7584fb47ca680015ddd9fefce2832395": "w'\\,\\!", "7584fc85457ed3ae4693586ef3b4521a": " \nPV = pf(T) \\exp(-rT) \\cdot {(1 + \\exp(-rT) + \\exp(-2rT) + \\cdots) } = \\frac{pf(T)}{\\exp(rT) - 1}.\n", "7585275bdcd8ce72f9c0245d2f146c46": "S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\\epsilon(c)1\\qquad\\mbox{ for all }c\\in H.", "758540b4cefee3744c431f4c801b779d": " Nx: Fx ", "7585782a401df8a95bdafa2a07e24115": "X_i: \\Omega \\to S", "758589db76bd1d485f5ad1f92a33d338": "\\star \\mathrm{d}y=\\mathrm{d}z\\wedge \\mathrm{d}x", "7585e675512b4559178d8f8643e6f581": "1/f:(x_1,\\ldots,x_n)\\mapsto 1/f(x_1,\\ldots,x_n),", "7586247b3a1eb272f315a3b110fb2e8a": "{\\bold \\ h_f}", "75865752269a5d210982ced056de9eda": "\\xi^1,\\xi^2,\\dots,\\xi^N", "758663ee39367ead6933dad0967538e2": "\\cdots\\rightarrow F_j \\rightarrow\\cdots\\rightarrow F_0\\rightarrow M\\rightarrow 0", "75870910497452370a27e02a543977ff": "\\scriptstyle m", "75876be9fbe6a8e3334c0e1423823113": "D\\subset \\mathbb{N}", "7587753300cfedd4813d1b2591aad157": " E_i = \\int_{\\theta_i}^{\\theta_f} \\frac{1}{\\theta_i-\\theta_f} \\psi(\\theta)\\, d\\theta", "7587bb718bd177f08bdc529b4d0e929d": "\\overline{H_1 X}", "7588043c6de188052dfc54ba09bf29f4": "{L\\over{\\delta}_1}=L\\sqrt{{\\rho}{\\omega}\\over {\\mu}} =L\\sqrt{{\\omega}\\over {\\nu}}=N_W \\,\\!", "75886198ceacf0f60751996e88044c5b": "\\sigma(\\mathbf{x}) \\triangleq \\hat{\\mathbf{y}} - \\mathbf{y} = \\mathbf{0}", "75887722c7016951aa4d3106fa7f829b": "\\mathbf M = {d\\mathbf L \\over dt}", "7588a10137f2d7cbd2886ca73328f511": "~\\sigma_{\\rm e}(\\omega)~", "758951cac5e62129e654698c8ca0333b": "CF", "7589735fbc2001e0af43d5cf014beeff": "L^c", "7589a6f4afa498fb4912b18f786c0f4e": "\\text{Tp}(\\text{Prim})\\,\\!", "7589d58ee929991183f15f968dee4ac6": "t = (q_1 - p_1)(q_2 \\cdots q_n),", "7589e883a03ee12017b394261123afb0": "\\rho (m_{n}, m) \\to 0 \\Rightarrow m_{n} \\rightharpoonup m,", "7589e889f07d52d08722aaba76afd4ae": " \\mathbf \\lambda_j = - \\mathbf J^{-1}\\left[ \\left\\| \\hat{\\mathbf x}_{j\\alpha}(t+\\Delta t) - \\hat{\\mathbf x}_{j\\beta}(t+\\Delta t)\\right\\|^2 - d_j^2\\right].", "7589e8b8598bde2e6d19fe5b40e8433a": "()^H", "758a282f50a7dd9486a0b820d3d7ff65": "k\\leq10^{-6}", "758a5a6d36fd274f2fd9cb048ba8bb4b": "\\hat u_n(t)=\\hat v_n(x)", "758a625d59cdb0c922959b6ca3db7937": "\\mu_i^\\circ", "758a645fef8c642231f9af6ed2d6d6ad": "\\int_a^x E_n(t)\\,dt =\n\\frac{E_{n+1}(x)-E_{n+1}(a)}{n+1}", "758a81532e63d00a8a9bc971cab92430": " f(x) = f(x_1,\\dots,x_N) = \\sum_n f_n(x_n). ", "758abcd035365a01e8e0c329be0ba020": " E/V = K_1 \\left(\\alpha^2\\beta^2+\\beta^2\\gamma^2+\\gamma^2\\alpha^2\\right) + K_2\\alpha^2\\beta^2\\gamma^2.", "758ad890d2b783a25160492f68c477d5": "\\theta_n(x;\\alpha, \\beta):= \\frac{n!}{(-\\beta)^n}L_n^{(1-2n-\\alpha)}(\\beta x)=x^n y_n\\left(\\frac 1 x;\\alpha,\\beta\\right).", "758ae227a8e2f5bb8beeb05e41d3150c": "{V_p}", "758b4c22c22dd679d96b173f6e3d508c": "\n W = U - (P_q + P_n + P_t) \\,.\n", "758ba26bb1fada1aefe72ef99f93ef79": "\\rho_{\\xi,\\eta}(\\lambda)=(E(\\lambda)\\xi,\\eta).", "758bc4bc7db2dc6bb97cd1a8e2195cce": "2^{(e-127)} = 2^{124-127} = 2^{-3} ", "758bdab1874a43840c71b10e833b65b3": "\\sum_{ij \\in A} w_{ij} x_{ij}", "758bef0d54380ed776b64676e853bd39": "\\begin{align}J_n(x)&=x^{2n+1}\\int_{-1}^1(1-z^2)^n\\cos(xz)\\,dz\\\\&=\\int_{-1}^1\\bigl(x^2-(xz)^2\\bigr)^nx\\cos(xz)\\,dz.\\end{align}", "758cad1fa142eb07f849abde40a86354": "S_{ij} = \\delta_{ij}\\,", "758cb04b5133046c5f86e3dc9b5aebda": "\\boldsymbol\\phi_{i=1 \\dots N}", "758cb48f9881c59ed5af1e7fd3206aed": " \\int \\delta(\\partial_\\mu A^\\mu - f) e^{-{f^2\\over 2} } Df. ", "758cc021e208eda6337e3b7e1cc039f2": " W^s ", "758cc6937dfb1a533d3f8d85ce0be642": "x \\in \\mathfrak{g}", "758ccb1eae82cd2de7a0dc5a1bfe1103": "\\scriptstyle{\\theta \\, = \\, \\pi/2}", "758cda45b482cee1753ee496accff562": "{p_e}", "758ce53bb5fe37806ed99fdc024056e3": "\\mathbf F_C", "758d6620f13b38ffed0a907a195fd500": "\\mathbf{A}\\bar x\\ =x_1\\mathbf{A}\\hat e_1+x_2\\mathbf{A}\\hat e_2+x_3\\mathbf{A}\\hat e_3", "758d9dfb6fff0ab56611ee4afcb8eedb": "\\biggl|\\int_{C_R} f(z)\\, dz\\biggr| \\le \\frac\\pi{a}\\max_{\\theta\\in [0,\\pi]} \\bigl|g \\bigl(R e^{i \\theta}\\bigr)\\bigr|\\,.", "758dc6d71d37a6db6e8d2cd263751620": " \\|\\cdot \\|_X ", "758dc7f89b57e9ea54bc291cd2f13f3a": "\\begin{align}\nS &= \\frac{2}{n} \\times \\left(\\frac{2}{n}\\right)^2 + \\cdots + \\frac{2}{n} \\times \\left(\\frac{2i}{n}\\right)^2 + \\cdots + \\frac{2}{n} \\times \\left(\\frac{2n}{n}\\right)^2 \\\\\n &= \\frac{8}{n^3} \\left(1 + \\cdots + i^2 + \\cdots + n^2\\right)\\\\\n &= \\frac{8}{n^3} \\left(\\frac{n(n+1)(2n+1)}{6}\\right)\\\\\n &= \\frac{8}{n^3} \\left(\\frac{2n^3+3n^2+n}{6}\\right)\\\\\n &= \\frac{8}{3} + \\frac{4}{n} + \\frac{4}{3n^2}\n\\end{align}", "758dd37f0c3b88a08e76f6aba8053eed": " \\cos \\theta \\,", "758e2336d995cfcaf8390e923d2e3ead": "D_0(n)", "758e3ad75ff1e677528f3284c6294855": " 6 \\div 2", "758e4d9fc7fb18eb2019e61f92d271dd": "for\\, each\\, a \\in Attributes", "758e583cdd333b4227593a5cdd3fc12d": "i_c = c_m \\frac{\\partial V}{\\partial t}\\ ", "758e86f4ff9f23f7e0e53502d0212efd": "\\lambda_0\\,\\!", "758e8c12e5e5bb5cee1afd0cb05a9655": "\nq_{xy} = (a^2-b^2)\\sin \\theta \\cos \\theta\\,\n", "758f364bff3d2232c7a5eec505fd517b": "\\mathcal{O}_P", "758fed847ad9c1c5a2f1019847776bce": "1/e^z = e^{-z}", "758ff7ccaf4a28769d17fc593fa5eb5a": "SQ = \\log_{10}\\left(\\frac{I}{M}\\right)", "75902e9ead84834c2de7fc8e54020474": "H^\\prime(\\mu)", "75903971fafc9e1e8da68585bfa52ae9": "\\theta(s) = e^{-s^2}", "75905bc59dea11f07f4b268f8ef18eae": "\\displaystyle{Q(e^a) = e^{2L(a)}}", "75916747fd620d86b11b19b200326dc6": " \\frac{z}{n} \\cdot \\frac{z'}{n'} = \\frac{z\\cdot z'}{n\\cdot n'}", "7591b6490053c6e814e781524229d5f8": "a_1,\\ldots,a_m\\in M", "75922fd601844bfdc34918534657b2ad": "\\{a \\vee b, \\neg a \\vee c, \\neg c \\vee d, a\\}", "75926ac4e0f9256cfd2f5f0fd2253bfa": "2^{2^n}", "759293a8051482d706583218c4bad4dd": "\\ A_i", "7592a539255b42e61224f6c536c1cb7a": "\\lambda(T,W)", "7592b3b0aabb6b3306eea0583cdbf834": "\\underset{x, \\; y}{\\operatorname{arg\\,max}} \\; x\\cos(y), \\; \\text{subject to:} \\; x\\in[-5,5], \\; y\\in\\mathbb R,", "7592f0374df31bb92c5f124b9f486d72": " x = \\ln\\left(\\tfrac{S}{K} \\right) ", "75930d388285296254e385b5ed2d38c3": "\\mathsf{BP} \\cdot \\oplus \\mathsf{P}", "7593363312a748c2af246a7f541c6f73": "y = \\left\\{ \\begin{matrix} 1 & \\mbox{if }u \\ge \\theta \\\\ 0 & \\mbox{if }u < \\theta \\end{matrix} \\right.", "7593d90807615632b57cb954b4cc14a7": "\\frac{\\partial f(\\boldsymbol{u}(\\boldsymbol{x}))}{\\partial x_i}=\\sum_{k=1}^p\\frac{\\partial\\bar{f}(\\boldsymbol{u}(\\boldsymbol{x}))}{\\partial u_k}\\frac{\\partial{u_k(\\boldsymbol{x})}}{\\partial x_i}\n\\qquad\\forall i=1,\\ldots,n", "7593dd3501b4d0b1aca384fdcdf7fb5b": "b_{k,\\ell}", "7593e02cb5d0bbfb9d944febeed2e0d1": "(a + \\sqrt{3}b)(c + \\sqrt{3}d) = [ac + 3bd \\pmod{q}] + \\sqrt{3}[bc + ad \\pmod{q}].", "7594306821fb0e9ea4c44f7139144b2e": "\nB_{k-1}w^2 + (B_k - A_{k-1})w - A_k = 0.\\,\n", "75946cab19388c95808c8c85fd35a436": "t\\in \\left( 0,1\\right) ", "7594f3c6eff6aa0916cb04e42e451561": "{52 \\choose 3} = 22,100", "75951fa4ef31b6e83d89f4c35d1706bc": "\\Phi\\approx U", "7595235d78ee91c6f02c6840b5e35667": "\\mathcal{ALCNIO}", "759529abf612fa4f5169e7079e8a7d35": "A_I = \\bigcap_{i\\in I} A_i.", "7595a68aac7b1040145c1e51753ec88e": "X_C - X_L", "7595c6618fa22d7fc16a2d1302f9d8dc": "\nf(x;\\lambda) = \\left\\{\\begin{matrix}\n\\lambda e^{-\\lambda x}, &\\; x \\ge 0, \\\\\n0, &\\; x < 0.\n\\end{matrix}\\right.", "7595fb6097010c626ed5b652c52cfc5d": "f(n) = O(n^{-1})", "75965e6b637db6137a3d9ebab99d5d7b": "\\lambda+\\Delta \\lambda", "75966006fcd8e9da21f0ebcefa753540": "CCI = s_1 + \\sum_{i=2}^N s_i^2(2 - s_i)", "759667608eeb2f476023bda3265d3f6a": "\\mbox{The radius of the circle would be }\\sqrt{\\frac{4}{\\pi}}\\simeq 1.13", "75967374cca0a5d2d0898a6e94e6ab35": "\n J_{\\Gamma_2} = -\\alpha\\,K^{n+1}\\,r^{(n+1)(s-2)+1}\\,I\n ", "759697906912b0a6a2bf68065c410369": "E=\\gamma(\\mathbf{u})m_0c^2", "75969d1a5a3483bfa8f463753b773efd": "\\forall a,b,c,d \\in R", "75969d875c55bfd8969c755453672c8a": "p = t + \\mathbf v = t + x \\ \\mathbf i +y \\ \\mathbf j + z\\ \\mathbf k.", "75969f1b4d1cc03ba50a5fab5f2afc8c": "\\gamma_{yy}", "7596ebe76e6a43e0f363a5cfc8ecb07c": "p ( \\xi\\mid\\omega_s )", "7596fd7434acbc0af8d505119595cc7f": "\\sin \\theta \\approx \\theta ", "7597d86b073989c60d29c501fcf728ec": "\\{0,1\\} \\cap \\{0,1,2\\} \\cap \\{0,1,2\\} = \\{0,1\\}", "759815ea05964cd6fa8bd32a8646326a": "1 + i, 1 - i", "75982347626f1daeec0afbd7cb4493c8": "\\Delta_y.", "75983ea556d4f8c37bdccfbce1045f33": "{}E[X_{n+1}|X_1,\\ldots,X_n] \\le X_n.", "759844070af2b2c2af93bd7e0e272ee6": "\\mathcal{F} : \\mathcal{J} \\rightarrow \\mathcal{C}", "7598518e35df41bb4ef78114e933f9ef": "t(x_,x_2,\\dotsc) = \\lim_{i\\to \\infty} \\tilde x_i^{p^i}", "7598b5568b809b7c3bbe99b1563647eb": "b_{k}:= b_{k}- rb_{l} ", "7598ee2d36ef7798f2ef0bb3ffee55e9": "\\mathbf{q}_{2}", "7598f63a803d5128e8179d1dd6eef58a": "k = 4", "7599051cbcb8020a241362d6c130c243": "Y_l^m(\\theta,\\varphi)", "759926a3e6859114f4fcbcc2e76f828c": "\\mathcal{E}(g) = \\int_M R_g \\, dV_g = \\int_M 2K_g \\, dV_g, ", "75994181c7108946479606eeabd1a1c1": "\\textstyle E\\left( \\left\\vert x_{i}\\right\\vert ^{2}\\right) <+\\infty", "75997cb02862e5b7cac678a54d5a96f9": " m\\equiv\\frac{RF_{,R}}{F}.", "7599aa48b6d1bc7e860f9f02f3f9d641": "\\mathbb{C} [x_{ij}] = S(\\mathbb{C}^n \\otimes \\mathbb{C}^m) = \\sum_D \\rho_n^D \\otimes\\rho_m^{D'}. ", "7599bcc74c3a214a7037680d59e44850": "(A - 2Z)^{2}", "7599bd64d83ccdffab372d9c4f96ff8d": "1.15^{12} = 5.3503", "7599d13c0437b9ea0642045c92ca6acf": "\\mathcal{N}^{-2}=\n\\sum_{m_{i}}\\sum_{n_{j}} \\frac{(z_{i}/2)^{m_{i}}(z_{j}/2)^{n_{j}}}{m!n!}\n\\langle \\mathrm{MFT}|\n\\prod_{i\\,j}\n(\\tilde{\\mathbf{q}}_{i})^{2 m_{i}}\n(\\tilde{\\mathbf{q}}^{\\dagger}_{j})^{2 n_{j}}\n| \\mathrm{MFT}\\rangle\n", "7599f0909196143f60d5e5663b7c3192": "S_\\eta=\\frac{1}{\\beta}\\sum_{i\\omega}g(i\\omega)", "759a1f95741cbfec66913c78bfd1bb91": "E_x(t)", "759a1fabfe9f19100597032a7f413b28": "\\overrightarrow{r_1} + \\overrightarrow{r_2}", "759a7c1b5a72b825b5ff2d1ef90af498": "\\int_{\\hat{s}\\cdot \\hat{n}<0}L(\\vec{r},\\hat{s},t)\\hat{s}\\cdot \\hat{n} d\\Omega=\\int_{\\hat{s}\\cdot \\hat{n}>0}R_F(\\hat{s}\\cdot \\hat{n})L(\\vec{r},\\hat{s},t)\\hat{s}\\cdot \\hat{n}d\\Omega", "759ac72443636653353514d6ffe174c5": " \nH \\left (X \\right ) = H(Y) = 1, I \\left (X;Y \\right ) = 0\n", "759ad61ee38924a34d6aeb1c6c01fccc": "y = 0.08", "759ae3c47567b0c37a18a6beb29d5bc0": "d=v_i\\Delta t+\\frac{1}{2}a\\Delta t^2", "759af5c44faf1d94497101e3b57223ee": "N=a^\\dagger a\\,,", "759b16ef4ce43691c38ad5ce28fe50ee": "{a^n b^n c^n d^n : n \\geq 0}", "759b266ef0ae055530d274f19dc12b42": " (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "759b92621b22ee7981234398a323e9af": "R(t, \\hat{t} )", "759b9c755c516b050dd3ed07bda82c02": "U_1,\\ldots,U_L", "759bfaf8b4e02f0ae06487c605bb3a33": "\\textstyle \\langle I(\\mathbf{q}) \\rangle", "759c0ddce43acf607caf11cfdf56128e": "P(y) \\land \\exists x Q(x,z)", "759c7ecf093da93a99938b68ba99997b": "e^{i \\pi} = -1 + 0 i,\\,\\!", "759c8203d43b539fc9202393946b2ec8": "\\textstyle f([\\mathbf{x}]) = \\bigcup_{i=1}^k f([\\mathbf{x}_i])\\mbox{.}", "759cac91ac1497600571bf16b40c376b": " \\tilde{\\gamma} = -(\\sqrt{\\nu^2 + 2r} + \\nu) / \\sigma ", "759d07ea9b56470e062b8fc12f156c67": "\\scriptstyle \\int_0^\\pi \\sin^nxdx", "759d36603e582519888eea54be8cfb21": "\\mathbf{X}'", "759d4ecc6c7ceb597d809d4dc6daf0a7": "\\forall x \\forall y \\exist w \\forall z [ z \\in w \\leftrightarrow (z \\in x \\or z=y)].", "759d61be5441681e4ba74a6116f8fd34": "(A.6)\\quad n^a\\nabla_a n_b=\\kappa_{(n)}n_b\\;.", "759d896c1209d9e326b1d2553d5db4bc": " \\left| \\log {g^\\prime(\\zeta)g^\\prime(\\eta) (\\zeta-\\eta)^2\\over (g(\\zeta)-g(\\eta))^2}\\right|\\le \\log {|1-\\zeta\\overline{\\eta}|^2\\over (|\\zeta|^2 -1 )(|\\eta|^2 -1)}.", "759daba70ba9ee4f057997e3da44bb63": "\nf_{t2} = C_{t3} \\exp\\left(-C_{t4} \\chi^2 \\right)\n", "759dc9b352ee9121f308dbc2d8540a1b": "f_a(x) = x^{-a}", "759ddae4e4ac9580285dee45ca84342d": "\\mathfrak{gl}(1,\\mathbb C)\\times\\mathfrak{gl}(1,\\mathbb C)", "759ddb0d2c81772a6c05e162c4b1666c": "\nHa=E(1-v)/[(1+v)(1-2v)]\n", "759e293c64305488f51a88aeb0fd8ac6": "X_L = R^T (X_R-T) ", "759e91ed2d8c11d9d08cef455fa58dcb": "\\sum_{i=1}^m b_i y_i \\leq \\sum_{j=1}^n c_j x_j", "759eaba85f40e97252a8d15af9c6f3ee": "f(L)", "759eaf75e5fb9972af29e0b9494567b0": "\\hat{A}|\\psi_\\alpha\\rangle=\\alpha|\\psi_\\alpha\\rangle", "759eb7b2cbab76454da165fa8f23a801": "\\lambda C + \\mu C' = 0\\ ", "759eedde9520aa893e036233ebcbe5af": "[L]_D=\\{[w]_D \\vert w\\in L \\}", "759f29fb64d71797853f5bf3c15d434f": "\\Delta A= A_{2} - A_{1}\\leq 0", "759f57bb828205a4f46f10a4d35faf4a": "\\scriptstyle \\mathcal K \\;=\\; T\\, \\frac{dS}{dT}", "759f6454e0fe2103ea226b96c61ea106": "\\boldsymbol\\alpha=(\\alpha_1,\\ldots,\\alpha_K)", "759fe27f06319baab668c5404ad2a914": "\\begin{alignat}{3}\nV_{LJ}& = 4 \\varepsilon &\\left[ \\left( \\frac {\\sigma} {r} \\right)^{12} - \\left( \\frac {\\sigma} {r} \\right)^6 \\right]\\\\\n& = \\varepsilon &\\left[\\left( \\frac {r_{m}} {r} \\right)^{12} - 2 \\left( \\frac {r_{m}} {r} \\right)^6 \\right]\n\\end{alignat}", "75a00041d3067e3446329c1c38186012": "(16)\\quad ds^2=-\\Big(1-\\frac{2M}{r} \\Big)\\,dt^2+\\Big(1-\\frac{2M}{r} \\Big)^{-1}dr^2+r^2d\\theta^2+r^2\\sin^2\\theta\\, d\\phi^2\\,.", "75a02230db2eaf2beaf20dd333dca1df": "r(n) = \\frac{n^2 \\hbar^2}{Zm_e e^2}", "75a02fffe5cf58dd835d2b70c5f73974": "\\displaystyle{D=P^2+Q^2=-{d^2\\over dx^2} + x^2}", "75a0510f1e9628bd913c176559680f40": "\n\\begin{align}\nO_n\\{x[m-k];\\ m\\}\\ &\\stackrel{\\quad}{=}\\ y[n-k]\\\\\n&\\stackrel{\\text{def}}{=}\\ O_{n-k}\\{x\\}.\\,\n\\end{align}\n", "75a05472e2c7762c72b0864887aafa2a": "GL(n,\\mathbf{R})", "75a06ee2854f306b07709ef488528385": "G(a,\\chi)=\\sum_{n=1}^{p-1}\\left(\\frac{n}{p}\\right)e^{2{\\pi}ian/p}. ", "75a139ab8ae9f2c75a178bc20969cb3b": "\\text{(2)} \\qquad \n \\sigma_y(\\varepsilon_{\\rm{p}},\\dot{\\varepsilon_{\\rm{p}}},T) = \n \\left[\\sigma_a f(\\varepsilon_{\\rm{p}}) + \\sigma_t (\\dot{\\varepsilon_{\\rm{p}}}, T)\\right]\n \\frac{\\mu(p,T)}{\\mu_0}; \\quad \n \\sigma_a f \\le \\sigma_{\\text{max}} ~~\\text{and}~~\n \\sigma_t \\le \\sigma_p\n", "75a14f5f2eaa2dca2ac11eb8a072d681": "100\\cdot$1\\cdot2/38\\approx$5.26", "75a1fe953e05f543284a817e1650b845": "\\quad z=1-x-y . \\,\\!", "75a243271ab8096ef9859615a02eb6c0": "\\mathbf{r_i}", "75a29d156d6b8e33e3a92179dc8cf1f2": "\\Omega(U/O) \\simeq Z\\times BO", "75a2a09000ea9c01b3f4c1eb3705d7be": "\\pi^2(R-r)^2", "75a2b5d0c49f68e5a2e9e210bbad6b6c": "p^0\\Psi=0", "75a2d721e189cb5b4e3496583df3e2f7": "2\\cdot 3", "75a30cdfc684a716d4ec0c894673a849": " \\alpha = {a\\over r},", "75a372029a494b98d16235e0b14f8930": "\\mathrm{Sp}(1) \\to \\mathbb{S}^{4n+3} \\to \\mathbb{HP}^n.", "75a3c55ce090b6291d9bba66a53e8db4": "0= N_1\\,\\mathrm{d}\\mu_1 + N_2\\,\\mathrm{d}\\mu_2 \\,", "75a40b120f83d9cf416166604ebde635": "a=6378.137", "75a41a92b5c9bc59ebfe4f679bcdb517": "\\pi_1 \\big(PSO(2k+1)\\big) = \\pi_1\\big(SO(2k+1)\\big) = \\mathbf{Z}/2,", "75a41d51d9ffb140b5df70bb4f2eeb3b": "\\psi_0(q) = \\sum_{n\\ge 0} {q^{(n+1)(n+2)/2}(-q;q)_{n}}", "75a44d021746bdb587c2c7681cfd9275": "\\text{Pressure} = \\text{weight density} \\times \\!\\, \\text{depth}", "75a4608c2bd2339c423f0d11d5fea27d": "s_0 \\cdots s_N", "75a4612fc956762034944ccf7d6572f2": "\\scriptstyle\\{u_n\\}_{n\\in\\mathbb{N}}", "75a47303027a1a505e4eae1c9f0905ce": "1,x,x^2,.\\ldots,x^n", "75a4bd8bc9d0dd8b0058588ca3556029": " g = e^\\varphi (dx_1^2 + \\cdots + dx_n^2),", "75a4d5e8f17c61ebc12f39bb7eacd2f0": " \\frac{x_3 - x_1}{x_2 - x_1} + \\frac{x_2 - x_3}{x_2 - x_1} = 1 . ", "75a4db4579fd26bc372bce3b4beb2f78": "\\mathbf{A},\\mathbf{B} \\in M_{mn}(F)\\,,\\quad \\mathbf{A} + \\mathbf{B} \\in M_{mn}(F) ", "75a501a1ab0727979ad5e143a31e0835": " \n\\prod_{i=1, i2 \\cdot E_i ", "75b7dea0dc8685bba4373b816556d2f1": "\\frac{\\partial \\mathcal{L}}{\\partial \\psi} = -e\\bar{\\psi}\\gamma_\\mu (A^\\mu+B^\\mu) - m \\bar{\\psi}. \\,", "75b825f8ff2e8378bf94201abfdb5817": "A\\in \\mathbb{M}_{m\\times n}", "75b83f88d182b974d556fe1171007f40": " \\begin{cases}\ns^3+4c^3-3c&=0\\\\\ns^2+c^2-1&=0\n\\end{cases}\n", "75b85826a15607f238debae369a5571c": "ABD", "75b88a1987b33f78f5a8b7ebdf5f2618": "\\omega_r", "75b9740f0761a79369fba2fa30769332": "f^{(n+1)}(c)>0 \\Rightarrow c", "75b97c0a29ccd9d70c92aaff3ba58aaa": "\\int_{c_0} \\mathbf{F} d{c}_{0}=\\int_{c_1} \\mathbf{F} dc_{1}", "75b9a229bb1873d751a1e0f775ceb2aa": "RC", "75b9ad61b4508854cebfe96e6527b9aa": "f^{-1}(U) \\simeq U", "75b9fc02e3dbd057d6b94045b8976ad3": "\\sum_{k=0}^n {n \\choose k} d_k d_{n-k} = \\frac{2n+9}{3n+6}d_{n+2}", "75ba1a1edb44209774892db17afc8636": "y_p(x) = u_1(x) y_1(x) + u_2(x) y_2(x) + \\cdots + u_n(x) y_n(x).", "75ba4a6e3bca8ce453bb05d0938114df": "\\vec{F}\\!", "75ba57c65875bfdfb676f38e8c03f275": "\n\\begin{align}\nb_0& = \\frac{1}{L} \\left(\\frac{x_{L+1}-x_1}{L} + \\frac{x_{L+2}-x_2}{L} + \\ldots + \\frac{x_{L+L}-x_L}{L}\\right)\n\\end{align}\n", "75ba73e4f85c1bab752ca60556fd8d9b": "X_C^' = \\frac{1}{Y \\sin\\ \\theta }", "75ba96281b2678e8f89ed9329cad094d": "I - T^{-1/2}A T^{-1/2}", "75bab9d6e18c757cedae81cbafba5dcb": "\\Delta{}_{v}H", "75bae05326e3292dc7371d9a561cf1a7": "L_3", "75bae88a3467e5e8b55e383d7f6e66c7": "(q_f, q_r) \\in F \\times Q", "75baf3511583f1d701ab4979a90d757b": "F'(x) \\approx \\dfrac{F(x+\\Delta x)-F(x)}{\\Delta x},", "75bb47bb9659df45d3f1b43362638d7c": "\\beta = 1", "75bb4bd3804777d956622c919698ca59": "\\sigma_x^2 \\sigma_p^2 \\ge |\\langle f | g \\rangle|^2 \\ge \\left(\\frac{\\langle f|g\\rangle-\\langle g|f\\rangle}{2i}\\right)^2 = \\left(\\frac{i \\hbar}{2 i}\\right)^2 = \\frac{\\hbar^2}{4}", "75bb765b8f24c91159c458cbedd3f7b1": "1/\\Lambda \\ll \\epsilon \\ll 1/\\Lambda'", "75bbb3f5f4838d1b9d5b267033b75a3e": "\n\tS = \\begin{pmatrix}0.58464 & 0.0505 & 0.6289 & 0.2652 & 0.6857 \\\\ 0.0505 & 0.19659 & 0.2204 & 0.3552 & 0.0088 \\\\ 0.6289 & 0.2204 & 0.44907 & 0.1831 & 0.5086 \\\\ 0.2652 & 0.3552 & 0.1831 & 0.21333 & 0.272 \\\\ 0.6857 & 0.0088 & 0.5086 & 0.272 & 0.49667 \\end{pmatrix}\n", "75bbbdc5ff69d76a2e87cb768af88ff6": "h_{ii}", "75bbf2b97d086fc8bb2b030c4e8484c7": " \\ldots \\left(1-\\frac{1}{11^s}\\right)\\left(1-\\frac{1}{7^s}\\right)\\left(1-\\frac{1}{5^s}\\right)\\left(1-\\frac{1}{3^s}\\right)\\left(1-\\frac{1}{2^s}\\right)\\zeta(s) = 1 ", "75bc2989c8ece961680508e4f156751a": "x^2+ y^2+ z^2- c^2t^2", "75bc3ecc157d04acc54d7b79784f5b79": "P(q_1+q_2)= a - (q_1+q_2)", "75bc99d5679891dd93c6ff4942fa9133": "P=r_k,\\;Q=v_k", "75bcc6c0d36bf1bfd579e4439d5668d0": "Aff/k ", "75bd3d82ba8198af1e167a96b6b3eb42": "C^{op}\\to Set", "75bdb0259307cb463e25943e6897171c": "\\sum_{i=0}^{k-1} {\\tbinom{n}{i}} ", "75bdc6b6967415474adb7de548cd8c64": "{-\\psi_f}", "75bde64f621589d390922cee5269333f": " n_s \\,", "75bdf34685d222ba742cc757613b8047": "|n\\rangle = |k_2, k_3, k_5, k_7, k_{11}, \\ldots, k_p, \\ldots\\rangle", "75be667ec14b703405643c908ead456a": "\\textstyle{I = \\int 1\\,d\\Omega}", "75be791cde3049f516ee1b6dca80f354": "\\approx 2.6 \\times 10^{42}", "75bf1cbc33ce4efb20a952ed0779df3f": " (\\rho_2-\\rho_1)gz=\\gamma\\left (\\frac{-1}{R_3}+\\frac{-1}{R_4}+\\frac{1}{R_1}+\\frac{1}{R_2}\\right)\\!", "75bf2b306f85a2699096f9b8aeb07d24": "2\\,\\frac{I_1(R\\,t)}{R\\,t}", "75c065adc853e4dfebf9f5623f86e5b4": " i^* \\in D ", "75c09f4b2adb91d110854614971028cd": "f(r)a", "75c10bbcf1ce8e2103bb67f7e03bac84": "V^\\infty \\wedge V^\\infty \\wedge X", "75c114370cbf09a36d39d1c5f70eea8d": "n = \\frac{T}{\\Delta t} = T(2\\omega) = 2\\omega T", "75c142a1a1677651ffe404b1a1d02cc2": "u = m_1 \\, e^{-|x-x_1|} + m_2 \\, e^{-|x-x_2|}", "75c1525e6ee21d197cecc4c601243e29": "H_2(X) = H_2(X, \\mathbf{Z}) / \\mathrm{torsion}", "75c16c3c54c1eedad9fb533bb34e76af": "\\left \\vert \\frac{e}{N}- \\frac{k}{d} \\right \\vert = \\left \\vert \\frac{ed-kN}{Nd} \\right \\vert ", "75c1731640fd1e6295e409a2efdac4a0": "a_1b_0", "75c209294b63f1830029081993bc5315": "I = \\frac{R_t}{R_o} = S_w^{-n}", "75c249aed0225a76b1a17d9a5d0b5ced": "\\cos(\\theta'-\\theta)=\\cos\\theta'\\cos\\theta + \\sin\\theta\\sin\\theta'", "75c277f2baa6030f97954bcba7155d43": "(y_{1},y_{2}) \\in X", "75c2838dfbc8d30c797b9cf27ecad88d": " c=(c_1,...,c_n) ", "75c2999b740c47c9c6c8a43df6e1c2d8": "U(a,1)\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix} = U(1,a) \\thicksim U(a^{-1},1).", "75c2eeff7f83aa3bcf5ebc8c502e02f4": " \\ker f_x (x_0,\\lambda_0)=X_1 ", "75c3073394c3186a9265462eb46b1586": "\\mathbb{R}_{++}", "75c3a2e2181e308dee09f197ae6ed396": "\n X \\approx \\sqrt { \\frac {\\pi R} {2 H}}\n \\exp {\\left ( \\frac {R \\cos^2 z} {2 H} \\right )} \\,\n \\mathrm {erfc} \\left ( \\sqrt {\\frac {R \\cos^2 z} {2 H}} \\right ) \\,.\n", "75c43c5f82b6a404a6c6b17672715c3c": "x_i = y_i + \\sum_jA_{ij}x_j", "75c4d755e8391887ad8962969d36b508": " \\mathcal{L} = { 1 \\over \\omega } \\mathcal{E}_c \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right ) = { N\\hbar \\over V } \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right )", "75c4fb12a4e34e2b4b01aa1b03038863": "=\\partial_x^2 + \\partial_y^2", "75c541c3151e3d346a64ff07c62358ae": " (\\mathbf{u}\\cdot\\mathbf{w})\\mathbf{v}_y-(\\mathbf{u}\\cdot\\mathbf{v})\\mathbf{w}_y", "75c5aca6307badd5cc11916fc859c5e4": "\n\\mathcal{P}(t)=\\sum^n_{k=0}a_{n-k,k}t^{n-k}, \\quad\n\\mathcal{P}(\\omega)=0.", "75c5ea03767c7ad7a6866821043e34b5": " \\langle O, \\mathbb{F} \\rangle ", "75c68a7462cecbfa72543c97f0ff2d3c": "\\sin \\theta \\approx \\tan \\theta \\approx \\theta", "75c69d79a0008d170b301a7b21ee0f4e": "\\sigma\\mathbf{E}_1^{(e)}\\mathbf{E}_2^{(e)}", "75c6e65ac83054c3e8306469dac78fa6": "w_-=\\frac{c}{n} - v \\ , ", "75c70618055da4939e3efc39282bfd3c": "\\bot\\left(R'(X,Y)Z\\right) = (\\tilde{\\nabla}_X\\alpha)(Y,Z) - (\\tilde{\\nabla}_Y\\alpha)(X,Z).", "75c76ccab36c642b8749eb86dccf9cf8": " 2 n - 2 s_2 (n) - e_2 (n) ", "75c7783044cf236944460722eb944ca6": " \\operatorname{perm}(X)=(-1)^n\\sum_{S \\subseteq \\{1,\\ldots,n\\}}(-1)^{|S|}\n\\prod_{i=1}^n \\sum_{j \\in S} x_{i,j} ", "75c79634dee71ca841840b5aad37af81": "\\neg (B \\vee C)", "75c83993ff081bd306bda1291454eb5e": "X_t=N_tD_t^{-1}", "75c8ac14e27093a4f5324238f77f76e1": "\\bar{P} - \\bar{P_e}", "75c8f63b9c2eb7217fe37fa380bf1c9a": "N_{B/A}(xB) = N_{L/K}(x)A.", "75c9157ef1585e97186a5c3d1754923f": "\\int \\sec{x} \\, dx = \\ln{\\left| \\sec{x} + \\tan{x}\\right|} + C", "75c98f4e28e8a7c33d28e003c7354189": "X, Y \\subseteq \\Omega", "75c99d0c1f40daf1721738d3e86780c0": "\\mathbf{p}_0 := \\mathbf{z}_0", "75c9d3a08f245d22e920b9f8e595a811": "R(\\pi / 4)", "75c9da4b2d5a7465b6c3a0227d0a9772": "\n \\mathcal I = \\begin{pmatrix} \\frac{1}{\\sigma^2} & 0 \\\\ 0 & \\frac{1}{2\\sigma^4} \\end{pmatrix}\n ", "75c9ec18c8eefd357aa025ba2a0f3d36": " \\lim_{x \\to 0} \\left( \\frac{a^x - 1}{x} \\right) = \\ln{a}, \\qquad \\forall~a > 0", "75c9f27ad20f901218d811540fa9a208": "\\lim_{k \\to \\infty}\\left | \\frac{q_{k+1}}{q_k} \\right \\vert \\quad \\scriptstyle \\text {where:} \\displaystyle \\;\\; Q(x)=\\frac{1}{P(x)}= \\! \\sum_{k=1}^\\infty q_k x^k ", "75c9fd69c8226a0b7170e20e89d88d8c": "w_0 \\ge 2\\lambda/\\pi", "75ca298588544328c5e4c0551f5b4e3b": "\\Gamma = \\{(s_1,t_1), (s_2,t_2), \\dots, (s_k,t_k) \\} \\subseteq (V \\times V)", "75ca4d7a7bf3135515d1dfadc3fccfda": "\\bar{ }", "75cad8d0b3b3b217efb403ada1c7e747": " v = x - y", "75caec696863eaf3992200a790747e76": "\\langle\\mathbf{v},\\mathbf{w}\\rangle = \\mathbf{v}^\\top \\mathbf{w}", "75cb2cd74b072a5dbcd438a6b4a1eff6": "\\frac{\\Delta f^{*}}{f_f}=\\frac{-1}{\\pi Z_q}Z_{\\mathrm{F}}\\tan \\left( k_{\\mathrm{F}}d_{\\mathrm{F}}\\right)", "75cb37fe2fde600f97dc812e53fe60f6": "(x_1x_2 + Ny_1y_2 \\,,\\, x_1y_2 + x_2y_1 \\,,\\, k_1k_2).", "75cb9f811d92fc0603f5077c53a6e136": "(x+y)+z=x+(y+z)", "75cc11cb448a31ded4931cf25243a537": "\\left( 2,0\\right)", "75cc12451fe324f16cf33b81803f1c1f": "\\left\\langle A\\right|", "75cc424d79c4af2d7a013f89ebb31c83": "U_g = U_p + \\Delta U", "75ccaa418dd99c3faa425e053126e0a8": "q(d,r)\\,\\!", "75ccb4e18c56aaae49cf3e9bfbdee714": " w' \\in \\mathbb{M} ", "75ccb812bb96f4f16f58841da89c5017": "\n m_{n,c}=\\frac{\\gamma_{n,c}}{\\gamma_{0,c}} \\quad \\textrm{for} \\quad n=1,2,3,4\n", "75cd063929964414770b038221f33ea3": "\\varphi \\,\\!", "75cd3b543091895646a8905f5c69e6d6": "n < 1", "75cd3bae881091427f4524a57810a1db": "\\beta = |\\boldsymbol{\\beta}| = \\sqrt{\\beta_x^2 + \\beta_y^2 + \\beta_z^2}\\,.", "75cd6a0219ccfe08a8e89e6f1952b181": "\\sigma_{p_1} \\leq \\sigma_{p_2} \\leq \\cdots \\leq \\sigma_{p_N}. ", "75cdc43c5f6453692d24e9ae622372b8": "10^{303}", "75cde9459abfa1ae79c8127e00a36937": "\n \\mathbf{u}\\times\\mathbf{v} = [(\\mathbf{b}_m\\times\\mathbf{b}_n)\\cdot\\mathbf{b}_s]~u^m~v^n~\\mathbf{b}^s\n = \\mathcal{E}_{smn}~u^m~v^n~\\mathbf{b}^s\n ", "75ce4468a96957d6fa1f4d5bc3c22252": "\\mathbf{X}_{AB} = R \\left( \\cos ( \\omega t) , \\ \\sin (\\omega t) \\right) \\ ,", "75ce47520e56854848b7ee337a81b507": "X^T X", "75ce63171f7b81d2e5f170276805efb6": "D\\subset X^{nr}", "75ceb90363f7ac8bad77f72cb5ea20b9": "\\frac{\\partial u}{\\partial x}-\\frac{\\partial v}{\\partial y} = \\alpha(x,y)", "75cf880978ba30b985a6d750b282a3cb": "B \\to G", "75cfd47ac9e59854a079135b4f31ad50": "\\cot(x)=\\sum_{n=-\\infty}^{\\infty}\\frac{1}{n\\pi+x}", "75cfd52eceec0c42f4f1c0e81709076a": "P_n=P_0(1+i)^n-\\dfrac{x[(1+i)^n - 1]}{i}", "75cfeb6db35084763e29a682ce35681d": "\nS[\\gamma]=r\\int_a^b\\sqrt{\\theta'^2+\\phi'^{2}\\sin^{2}\\theta}\\, dt.\n", "75d01dafad575048cf16cc0029406e7c": "m=N - h\\,\\!", "75d07bbf5fa1fd78eef918f0783e1536": "{\\mathbf{}}\\tau(t)", "75d0a5010db0ce68d54819310e935bae": "\\bold{k}", "75d0bdd76fd96324e682e13fc241b67d": "z = \\frac{A + a}{2} + \\jmath \\ \\frac {A - a}{2} \\ ", "75d0e9260e366433371e4d1ab321e679": "-2 \\log \\Lambda = 2\\sum_{i, j} k_{ij} \\log \\frac{n_{ij}}{m_{ij}}.", "75d0fa7c8b6734da71039287f2250c25": "dp/dt+Ap+Pf(p+q)=0", "75d12e1874a60e8cb8b351987dbb6890": "(T)(\\rho \\wedge \\rho')", "75d132b29a69839c21cfd20af08447b1": "\\lim_{x\\rightarrow \\infty}(x+1)R_n(x)=\\sqrt{2}", "75d1462586150c48071a8d9e94995422": " Mc^2 ", "75d1637239a8f57d0d2d80dcf30a2116": " \\mathbf{\\hat P} ", "75d1c9e669af6c87fabf493db88383cb": "-B^{-1}C", "75d1d7f734184b24ba305f1ce8ff5bd9": " u_E\\,\\!", "75d1fc5c251dd90afc9c653ff084ed60": "x = \\frac {bc} {a}.", "75d215871dc932a37caa9e1546bbc7cc": " <_\\mathcal{O} ", "75d27bfa218aad5657a86edd4640ab8c": " i_\\text{real} = \\frac{(i-r)}{1+r} \\,\\!", "75d28bc298e99845cc9f7952bc9271e5": "F(z) = F'(z)/F''(z)\\,", "75d2a5f7f75e272771db25cd0c00450f": "(A.1.c)\\quad \\frac{1}{\\rho}\\,\\gamma_{,\\,\\rho} =\\,\\psi^2_{,\\,\\rho}-\\psi^2_{,\\,z}-e^{-2\\psi}\\big(\\Phi^2_{,\\,\\rho}-\\Phi^2_{,\\,z}\\big) ", "75d2f55559921f6d884e0d6eb3773d6a": " (q_1,\\ldots,q_N)\\cdot (r_1,\\ldots,r_N) = \\sum r_i^*q_i", "75d2f589e01a0ab42da0a9ed03b2285a": "\\scriptstyle >7.6\\times E_{\\mathrm{Pl}}", "75d35ac39acd61382e23ddf6cacb46c8": " {}_{1}F_{1}(a,b;x) ", "75d35adffb61e123f54dfbb5c507f7d6": "\\frac{\\mathrm{N}_{\\mathrm{O},\\mathrm{P}}}{\\mathrm{N}_\\mathrm{P}+\\alpha}\\,", "75d371614b37bdec48aee339f68afe26": "\\dim\\pi_\\lambda = \\frac{10!}{7\\cdot5\\cdot 4 \\cdot 3\\cdot 1\\cdot 5\\cdot 3\\cdot 2\\cdot 1\\cdot1} = 288.", "75d375c871fe682081669b2315b2e73c": " \\mathbf{v} ", "75d37ca5c742494727bc499b99cfd76e": " |g;1_{ks}\\rangle ", "75d3898b277ccd4965cf703c42b2b8b5": "D_3", "75d40b295f237eeb9eac822aa9f9576b": "1+kF(s)", "75d4497b2dea09eac45acaaa47f9f956": "\n\\mathbf{A} \\cdot \\mathbf{r} = Ar \\cos\\theta = \n\\mathbf{r} \\cdot \\left( \\mathbf{p} \\times \\mathbf{L} \\right) - mkr\n", "75d46c94152c46402c3d941e46c11ca7": "X\\colon (t,Y) \\mapsto Y_t", "75d4b4f92d7283996ba1aef8b9d492b1": "E_\\text{k} = m c^2 \\left( \\sqrt{\\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \\right) \\,.", "75d4be57ae2b784734cf7ecd203cc700": "J=\\begin{pmatrix} 0 & I \\\\ -I & 0\\end{pmatrix}", "75d60885debd89af59719ae5eaf95102": "\\sin\\left (\\frac{\\pi}{6}\\right )=\\frac{1}{2}\\!", "75d642e9c84c22d9e5b2d8ee6d772980": "\n\\int_a^{+\\infty}f(x) \\, dx =\\int_0^1 f\\left(a + \\frac{t}{1-t}\\right) \\frac{dt}{(1-t)^2} ", "75d665d8afe981228b8cbaf2e0485e18": "\\text{DET} = \\frac{\\sum_{\\ell=\\ell_\\min}^N \\ell\\, P(\\ell)}{\\sum_{i,j=1}^N R(i,j)},", "75d6d0d1a78aa399bc3a7b79c5ffd964": "\nSh = \\frac{K_c L}{\\mathcal{D}}\n", "75d6e6c3ae78c2230b159ca664dd14ac": "Z_\\alpha(e) = -\\infty", "75d72ddfc71a7d2172ec17b92d7a6f11": "\\sum_{n=1}^\\infty \\frac{H_n^{(b)}(-1)^{(n+1)}}{(n+1)^a}=\\zeta(\\bar{a},b) ", "75d7711442d5d2e0c9c5219f8b8e29b4": "(k_n)_{n\\in\\mathbb{N}}", "75d801d79f42048401629875d1ea4d60": "r \\in Q, ~ r \\ne t", "75d8547540c54bfad870a7f22e9e4633": "|V|\\cdot |V| = |V|", "75d86e629b26c008c1c36fe9f024cb40": "\nL \\rightarrow \\lambda L, \\qquad E \\rightarrow \\frac{1}{\\lambda^{2}} E ~,\n", "75d8850097e973ce45f6b5483ca59fa7": "\\omega_{ij,kl} = \\omega_{ij,lk}", "75d8a63629e79be4879b40f48045d2af": "A = 5 \\left(\\sqrt{3}+6\\sqrt{5+2\\sqrt{5}}\\right) a^2 \\approx 100.99076a^2", "75d8e3a88f3b3e3608e107a44fa07bb2": "|V|=\\infty", "75d8ef87d97f0f5e94b9716ddc82cdaa": "\\begin{cases} \nv_{t}=kv_{xx}+f, \\, w_{t}=kw_{xx}, \\, r_{t}=kr_{xx} & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ \nv(x,0)=0, \\; w(x,0)=g(x), \\; r(x,0)=0 & IC\\\\ \nv(0,t)=0, \\; w(0,t)=0, \\; r(0,t)=h(t) & BC\n\\end{cases}", "75d973419e9da77622f1e033b57cb587": "F_N(J)=N/N=1", "75d9977463a0a9318f1bbd0f83b42e2e": " \\partial_{x_n}f = A_1(x,f) \\partial_{x_1} f + \\cdots + A_{n-1}(x,f)\\partial_{x_{n-1}}f + b(x,f)\\, ", "75d9a125389190379a00f08d706b2ce4": " \\rho(\\phi)\\langle \\chi,\\psi\\rangle= \\langle [\\phi,\\chi],\\psi\\rangle +\\langle \\chi,[\\phi,\\psi]\\rangle .", "75d9c655ace7c6b73b1f32dfb0c48fcb": "53^2", "75da1d68424e0999418e54dc89de2dbf": "\n \\begin{align}\n X &= \\mathrm{AGTACGCA},\\\\\n Y &= \\mathrm{TATGC},\\\\\n \\operatorname{Del}(x) &= -2,\\\\\n \\operatorname{Ins}(y) &= -2,\\\\\n \\operatorname{Sub}(x,y) &= \\begin{cases} +2, & \\mbox{if } x = y \\\\ -1, & \\mbox{if } x \\neq y.\\end{cases}\n \\end{align}\n", "75da2cc426ee05caaffb1f225fd262de": "R(\\lambda)(1 \\otimes T(\\mu))(T(\\nu) \\otimes 1) = (T(\\nu) \\otimes 1)(1 \\otimes T(\\mu))R(\\lambda) ", "75da747f5a0df573509eba08a310e8b6": "f^\\#", "75daa0103dd96ba888dee6cc5f80dbaa": "\\lambda x.~x", "75dab7c2a18c7adcc880a21b3869bd37": "A_p=\\sigma_{e_1 e_2} \\sigma_{e_2 e_3} \\dots \\sigma_{e_n e_1}", "75dae1e6d2bc5e4d8f698a7ca7270cbc": " \\bold{A} = \\nu \\frac{f(\\xi)}{\\xi}\\hat{\\bold{r}}\\times\\bold{\\sigma} \\, , \\, \n\\phi = \\frac{\\nu}{\\sqrt{2}}h(\\xi)\\hat{\\bold{r}}\\cdot\\bold{\\sigma}\\phi_0", "75db7ee31afe9b018e33740c186a6be8": "\\xi_2", "75dba202e99b63df8c67f1508b963533": " \\mathbf{e}^1 = \\frac{\\mathbf{e}_2 \\times \\mathbf{e}_3}{\\mathbf{e}_1 \\cdot (\\mathbf{e}_2 \\times \\mathbf{e}_3)} ; \\qquad \\mathbf{e}^2 = \\frac{\\mathbf{e}_3 \\times \\mathbf{e}_1}{\\mathbf{e}_2 \\cdot (\\mathbf{e}_3 \\times \\mathbf{e}_1)}; \\qquad \\mathbf{e}^3 = \\frac{\\mathbf{e}_1 \\times \\mathbf{e}_2}{\\mathbf{e}_3 \\cdot (\\mathbf{e}_1 \\times \\mathbf{e}_2)}.\n", "75dbd4bb721caae58a4dc96dda080084": "F^\\ast", "75dc1381e753282e80627a2127726736": "a=x_00", "75e57bd13b35946625963fcace3e0eea": "\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = -1", "75e581ac7cfe1fddf6911679eed5e9ad": "\\hat {\\boldsymbol \\theta}_1", "75e609d6202fac25c3243588ddce9436": "y_1(x)=j_{-2}(x)=-\\,\\frac{\\cos(x)} {x^2}- \\frac{\\sin(x)} {x}", "75e61e35d2cf831a333adbd847732033": " \\frac{dr}{d\\tau}, \\; \\frac{d^2r}{d\\tau^2}, \\; \\frac{d\\theta}{d\\tau}=0 ", "75e660d518accf3d0e2f4aed2465b95d": " \\overline{\\frac{\\partial \\left( \\bar{u_i} + u_i^\\prime \\right)}{\\partial x_i}} = 0 ", "75e711adceb8c806390afcaa7fdbd907": "\\|L(u,v) - L(x,y)\\| \\le K \\big( \\|u-x\\| + \\|v-y\\| \\big)", "75e73b983d78a1567166b8b57a034267": "\\lim_{p\\to0+}p\\log (p) = 0", "75e7627c527c1fc440508e0d55d9cbe1": "2Q", "75e76cc7c4fb45ae5f6d293622fe144b": " l \\isin \\left \\{ l_0, l_0 + 1, ..., l_0 + n \\right \\} ", "75e7c14373da3f021551822d192e8ae0": "Q_A=-\\int\\limits_{\\text{Voronoi cell }A}\\Big(\\rho (\\mathbf{r})-\\sum_B \\rho_B (\\mathbf{r} )\\Big) d \\mathbf{r}", "75e7c271ffb0fd3f22a55c32f26811ec": "\\chi_e(\\Delta t) = \\chi_e \\delta(\\Delta t)", "75e81d6475974ded853ae933ea16ef51": "\\left(\\vec{S}_1 + \\vec{S}_2\\right)^2", "75e84f387e007e8d5a7b7b0619d677b9": "p \\cdot z \\leq r", "75e8658ffa9ab21994b665626ba19151": "{{i}_{C1}}", "75e882cac524db3803f75853111a5ec4": "\\mathrm{FWHM} = 2 \\sqrt{-2 \\ln 0.5}\\ c", "75e893439bff1aed3df8a27b75419f9b": "M(T) = M(0)\\left(1-(T/T_c\\right)^{3/2}),", "75e8fa593ed7e2a459c7cf4354ceb8b2": "s_i \\in \\Sigma_{k_i}, t\\in \\Sigma_n", "75e90ed5d916a42dea43d017465eb91c": " \\sigma(E) = b E^{\\alpha -1} exp(\\beta_o E) ", "75e91ddb925cae56448f4d4f1edc3a44": " x_n ", "75e9470812fdc9fa0a8d9c3eb21f4ddb": " REI = \\frac{S1}{S2} 100", "75e96b1216584e524f05b4d5ea943d47": "g_i(\\vec{x}+\\vec{e}_i\\delta_t,t+\\delta_t)-g_i(\\vec{x},t) + G_i=\\Omega(g)", "75e9ab57e5a23a9b6d35c264f6857809": "F(u) \\leq \\varepsilon + \\inf_{x \\in X} F(x).", "75ea1535db60b0e5f8541d7b2e28ac5f": "\n\\sigma _\\mu =\\overline{n_\\mu ^2}-\\overline{n}_\\mu ^2=\\overline{n}_\\mu + \\overline{n}_\\mu ^2\\left\\{ \\frac{\\mu B(\\mu ,\\frac 12)}{2^{2\\mu -1}} - 1\\right\\},\n", "75ea20bc7410b74d47ca3edf90569235": "v = (\\ldots, v_{-2},v_{-1},v_0,v_1,v_2,\\ldots)", "75ea24eaee623b91dd7e7900089e5fff": "mX+b \\sim \\textrm{GEV}(m\\mu+b,\\,m\\sigma,\\,0)", "75eaf21ae1e191411e5f8f8ed5bc2942": "f^*(t)=\\mu \\left (\\{x|f(x)>t\\} \\right ).", "75eb7051e99bf81849a337653588fadd": "\\textstyle{\\sum_{k=0}^n a_k}", "75eb8a3e1f29ff2877c70eeb00b05aa1": "\\texttt{AX}_{\\kappa}.\\,", "75eb9158c7fd61db5c4bf40e92d5f649": "d=\\sum_{r+s=p+q+1} \\pi_{r,s}\\circ d=\\partial + \\overline{\\partial}+\\dotsb.", "75eba624133b71c24fa67283b985a6ed": "\n\\begin{bmatrix}\n U_x & U_y & U_z \\\\\n V_x & V_y & V_z \\\\ \n W_x & W_y & W_z \\\\\n\\end{bmatrix}\n", "75ebc485e1e9ef3d6e056505b346e8c6": "(a+b)(a-b)=0 \\pmod n", "75ec0b79d087b93baa789413d5d3a050": " x(t) = A\\sin\\left(\\omega t +\\varphi'\\right)", "75ec3b9ef48ae499019a9247343f948d": " q_{ij} = \n\\delta_{ij} + (\\delta_{ik}\\delta_{jk} \n+ \\delta_{i\\ell}\\delta_{j\\ell})(c-1) + (\\delta_{ik}\\delta_{j\\ell} \n- \\delta_{i\\ell}\\delta_{jk})s . \\,\\!\n", "75ec6c2410271b67c06e374bc7b30f85": "w(S,T) = |\\{\\, uv \\in E \\colon u\\in S, v\\in T\\,\\}|\\,.", "75ec8ac924e5c555869ef516437939ad": " \\scriptstyle{\\mathrm{tan}\\left(\\frac{\\theta}{2}\\right) = \\frac{\\mathrm{radius~of~moon}}{\\mathrm{distance~from~surface~of~asteroid~to~center~of~moon}}}", "75ecb761cefe39d323775c01caa51068": "\\scriptstyle\\boldsymbol{\\mu}", "75ecbe7d19db961707ba743257ece51d": "\\mathcal{M}( \\phi (p) + ... \\rightarrow ...)=\\mathcal{M}( ... \\rightarrow ... + \\bar{\\phi} (-p) ) ", "75ed19cd26a4a7293512154a6e1aced6": "\\bar{x}=\\lambda^{-1},", "75ed4b7124e49cb82cd28a175c060f99": "E(f|\\mathcal{C})", "75ed7d24327a3eee051a68d8ca4776a6": "\\frac{1}{1+x^2}=\\frac{1}{2} \\left(\\frac{1}{1-ix}+\\frac{1}{1+ix} \\right) \\ .", "75edf71e8a107e7b18eafd1a08ebe00b": "d \\times d ", "75ee0560810782206fda2f0e515ba29f": "-1\\leq p \\leq 0", "75ee0f33ecdee13bddade1759b84e613": "{\\mathbf e}'_i = {\\partial}/{\\partial {x'}^i}", "75ee158d40b2fd3513a68302159d2dc1": "\\frac{ P^m_n(\\sin\\theta) \\sin m\\varphi}{r^{n+1}}", "75ee3c22940f8d3789b6365c9b7091dd": " v_1=\\begin{bmatrix}1 \\\\0 \\end{bmatrix}", "75ee7a7b4e1ed6215d1547db65f01811": "\\Omega(n^2)", "75eea141dbb86e909bc5e16e361d93d5": "\\mathrm{B}(\\boldsymbol\\alpha) = \\frac{\\prod_{i=1}^K \\Gamma(\\alpha_i)}{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i\\bigr)}", "75eedbd76af8c40fd468a5e4feeb5136": "\\prod_{i=1}^{p-1} i^{p-1} \\equiv (-1)^{p-1} \\equiv -1 \\equiv +1 \\pmod p.", "75eedf1e947cd06b6b6b5856f8354cb5": "\\mathrm{NapLog}(x) = \\log_{\\frac{10^7}{10^7 - 1}} 10^7 - \\log_{\\frac{10^7}{10^7 - 1}} x", "75ef24a6ace794b8eca8f8ef5b6d52e6": " \\delta w_u \\le -d (U - T_R S + p_R V - \\sum \\mu_{iR} N_i )\\,", "75ef9c19c31d705c3166edbe8a277b59": " \\Delta S = \\Delta S^\\prime+\\Delta S^{\\prime\\prime} .", "75efa20199491863c472f03c1c4dd6df": "\\textstyle l_m", "75efac2260c90fe120db31403ee54fd6": "\\sum_{(T,\\theta)\\in \\kappa, \\bmod G^F} {\\epsilon_G\\epsilon_TR_T^\\theta\\over (R_T^\\theta,R_T^\\theta)}", "75efda817fdeb4fd7151534c942b6919": "M_\\mathrm{L} = \\log_{10} A + 1.6\\log_{10} D - 0.15", "75eff24ace6df1dc37e036586fd53dcc": "\\frac{\\mbox{Gross Profit}}{\\mbox{Net Sales}}", "75f00b766878c4263bf2abebb3113d6e": " ( \\tau_1 , \\dots, \\tau_m, \\sigma) ", "75f03b28201ea80096d552b57aeff5a2": "\\widehat{c_v}^*=\\bigg(1+\\frac{1}{4n}\\bigg)\\widehat{c_v}", "75f05d59a7dd920073e15af0a6940378": "\n\\nabla^{2} \\Phi = \\frac{1}{\\sigma^{2} + \\tau^{2}} \n\\left( \\frac{\\partial^{2} \\Phi}{\\partial \\sigma^{2}} + \n\\frac{\\partial^{2} \\Phi}{\\partial \\tau^{2}} \\right)\n", "75f0c06bdc60eb44a40cd8281566118a": "nN^{O(log nN)}", "75f113d8d5070627264783cf0261d16e": "\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{pmatrix}\n\\quad\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{pmatrix}\n\\quad\n\\begin{pmatrix}\n-\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\n\\frac{\\sqrt{3}}{2}& \\frac{1}{2} \\\\ \n\\end{pmatrix}\n\\quad\n\\begin{pmatrix}\n-\\frac{1}{2} & -\\frac{\\sqrt{3}}{2} \\\\\n-\\frac{\\sqrt{3}}{2}& \\frac{1}{2} \\\\ \n\\end{pmatrix}\n\\quad\n\\begin{pmatrix}\n-\\frac{1}{2} & \\frac{\\sqrt{3}}{2} \\\\\n-\\frac{\\sqrt{3}}{2}& -\\frac{1}{2} \\\\ \n\\end{pmatrix}\n\\quad\n\\begin{pmatrix}\n-\\frac{1}{2} & -\\frac{\\sqrt{3}}{2} \\\\\n\\frac{\\sqrt{3}}{2}& -\\frac{1}{2} \\\\ \n\\end{pmatrix}\n", "75f1202527de5144bd29b89bd4e319cc": "Y_{7}^{3}(\\theta,\\varphi)={-3\\over 64}\\sqrt{35\\over 2\\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(143\\cos^{4}\\theta-66\\cos^{2}\\theta+3)", "75f122392cf65ef8f0267b674b0a39b1": " \\left(\\mathbf{A}\\times\\mathbf{B}\\right)\\cdot\\left(\\mathbf{C}\\times\\mathbf{D}\\right)=\\left(\\mathbf{A}\\cdot\\mathbf{C}\\right)\\left(\\mathbf{B}\\cdot\\mathbf{D}\\right)-\\left(\\mathbf{B}\\cdot\\mathbf{C}\\right)\\left(\\mathbf{A}\\cdot\\mathbf{D}\\right) ", "75f148bc17b31d792f8b31c2dbc438eb": "\\displaystyle{\\pi_s(L_0)f_m=mf_m, \\,\\, \\pi_s(L_{-1})f_m=-(m+1/2+s)f_{m+1},\\,\\, \\pi_s(L_1)f_m=-(m-1/2-s)f_{m-1}.}", "75f14fcd54821b5ece59682383ddd475": "\\sqrt { \\frac{(V1-V2)^2 + (H1-H2)^2} {10} }", "75f1527932f85f8b3a1e95d002f033ad": " \\qquad \\qquad \\alpha_\\mathrm{S,mix} = \\frac{1}{q}\\frac{\\partial S_\\mathrm{mix}}{\\partial N} = \\frac{k_\\mathrm{B}}{q}\\mathrm{ln}(\\frac{1 - f_e^\\mathrm{o}}{f_e^\\mathrm{o}}),", "75f16378841d33a06bb6fc309bfffb0c": "\\displaystyle{U_\\theta f(z)=f(e^{i\\theta}z).}", "75f16d071d68caa8b156185cbe6cc741": "\n \\langle \\left( x-x_0\\right)^{2}\\rangle =\\langle x^2\\rangle+x_0^2 - 2x_0\\langle x\\rangle,\n", "75f17c9aff7458e2bdfd05feb937040c": "U_e = \\{ a \\in \\ker w : ea = a/2 \\}", "75f1afc7d9c396b8929cc1993f924417": " \\nabla \\vec{H} ", "75f1bd86030bf5356809756258652a03": "\\underset{i}{\\overset{1}{x_j}}(t_0)", "75f1f3c4b3a6aef446dac2aa5a52094b": "[\\,f\\,]", "75f2155f308f37496fc21af09bc193d4": "\\textstyle \\int", "75f2797c717e06d1db84b128bdaad27b": " \\Delta E = \\frac{e\\hbar B}{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})", "75f302b26a1a3a7d0c8554b64702b71f": "L + 1", "75f31851b10a383a0f0b22084988f80c": "A^{-1}JA = J", "75f340b9468f9a45632168c94e96308c": "x_0=x_1=1", "75f35969a12d7df468a15132558dd214": "\\frac{\\partial V}{\\partial c} = \\frac{ -{c}^{\\,2}+n }{ 2{c}^{\\,2}n }.", "75f382585bd7400674b8d9014caba5d4": "\\Rightarrow c_{a} \\isin \\Gamma(W) ", "75f3d5bf0774eafc3b6706259cec063a": "\\cos \\psi = \\frac{{R}_{\\text{E}}{\\mu}_{0}}{\\left( {{R}_{\\text{E}}}+h \\right)\\mu } \\,,", "75f3e411a3dd783a785144e5f3c8e36c": "\\left|V_o\\right| = \\frac{V_o}{V_i}", "75f401087baa8548337e7c5c94f6710b": " S_{conf}/k_B = -\\theta_A \\, ln(\\theta_A) - (1-\\theta_A) \\, ln(1- \\theta_A) ", "75f40432095eaa314c8b9af6ec2ce55f": "\nV_\\mathrm{rms} \\cdot I_\\mathrm{rms} = \\frac{V_\\mathrm{rms}^2}{R} = \\frac{V_\\mathrm{peak}^2}{2R} \\,\n", "75f44615a517e1c9641025da185d4733": "(P_{t})_{\\ast} (\\mu) = \\mu \\mbox{ for all } t > 0.", "75f4681a02e8a64bf3956c67b79ca44c": "{\\mathcal L}^2_0,\\ldots{\\mathcal L}^2_3", "75f46f28d30cc662da4b659e27012550": "\\pi : \\mathbf{C}^n \\times \\mathbf{P}^{n - 1} \\to \\mathbf{C}^n", "75f4b0c92c557736c0e8738df470238c": "B_{eq} = \\frac{e \\, E}{D}", "75f4cea2a6f7ffb0084492e077cafb04": "{\\binom {k}{2}} = {{k(k-1)}\\over 2}\\,.", "75f521d72dbcd1c4022784990d4cf8b5": " = \\sum_i \\sum_k \\operatorname{tr}(T_i V_k V_k^*) S_i ", "75f5510b3f9dae37c83d69cbe79f8cbd": "z^i\\alpha:=\\sigma^i(\\alpha)z^i", "75f5ded74a6222fd701f062648cfd4a8": "r=\\alpha\\xi", "75f634ca25334ca6c072bff584029f42": "R_{tot} = Q \\Delta \\nu", "75f6511d049280a6ebc23eea87d8411f": "\nv_A = \\beta _0 + \\beta _1 \\left( c_A - c_T \\right) + \\beta _2 \\left( t_A - t_T \\right) + \\beta _3 I + \\beta _4 N\n", "75f6591e8dc6e844cd9a25d37e859ad9": "G_{S_1}", "75f676ad7f81e2b41f8227a607714402": " F_{1}(\\alpha_{k-1},\\alpha_{k})|_{\\alpha_{1},\\alpha_{2},...,\\alpha_{k-2} \\, fixed}=0 ", "75f75daed3373b39ee67e33c84afc37d": "OD", "75f767d1e8806c87853f58c593033ca3": "\\boldsymbol{\\tau}", "75f7b65b16e47818ca92f0ff5a8f901b": " Q_N=\\{ \\bar{s} \\not \\in S \\}", "75f7c0f176e60ec45f91019321c88766": "d = \\mathbf{e}^T\\mathbf{A}\\mathbf{e}.", "75f7c9cc8ec103dc6418de4fbdf24852": "f(z) = \\frac{1}{\\Gamma(z)},", "75f7ece2829d5cccb77e1bd2fb9193f7": " {d \\mathbf p_1 \\over dt} = \\nabla_1 L\\left( \\mathbf r_1 , \\mathbf v_1 \\right) ", "75f81ca413ab9141b18acf0dc9229095": "\n\\begin{align}\nY_i\\mid x_{1,i},\\ldots,x_{m,i} \\ & \\sim \\operatorname{Bernoulli}(p_i) \\\\\n\\mathbb{E}[Y_i\\mid x_{1,i},\\ldots,x_{m,i}] &= p_i \\\\\n\\Pr(Y_i=y_i\\mid x_{1,i},\\ldots,x_{m,i}) &=\n\\begin{cases}\np_i & \\text{if }y_i=1 \\\\\n1-p_i & \\text{if }y_i=0\n\\end{cases}\n\\\\\n\\Pr(Y_i=y_i\\mid x_{1,i},\\ldots,x_{m,i}) &= p_i^{y_i} (1-p_i)^{(1-y_i)}\n\\end{align}\n", "75f84ecced74f7993e9d4fd5e9e0a202": "f^*(x) > \\lambda", "75f97ebdbd28e17384c505e7752830b1": "\\displaystyle{ |f(z)| =\\left|\\sum_{n\\ge 0} a_n z^n\\right|\\le \\|f\\| e^{|z|^2/2},}", "75f98e965f4db01e430b33f5c3704ae0": "m = \\frac{m_0}{\\sqrt{1-\\left(\\frac{v}{c}\\right)^2}} = \\frac{m_0}{\\sqrt{1-\\beta^2}} = \\gamma {m_0}", "75f9ba31d29ce1f588795db6197f0370": "Y_t = \\sum_{k=-\\infty}^\\infty a_k X_{t+k}\\,", "75f9eef1cfcf6af283df91de013e3751": "\\delta'=\\arg\\max_{\\delta\\in{SC(A,t)}}\\{U_{A}(\\delta)\\}", "75fa4e1e6b92a56ef6581d7b28ba512d": "M_4' = M_4 + \\frac{\\delta^4 (n - 1) (n^2 - 3n + 3)}{n^3} + \\frac{6\\delta^2 M_2}{n^2} - \\frac{4\\delta M_3}{n}", "75fa967937d39b794c4cd5dfd06c21c9": "(L_i,f_i^j\\mid i\\leq j\\text{ in }I)", "75fa973dc5c5c7514ee39f813cd5f876": " \\frac{\\partial \\phi_i}{\\partial t} = \\sum_{j = 1}^R S_{ij} f'_j (\\mathbf{\\phi}). ", "75faaaf69cf6208cd116173152b254d4": "u=0 \\text{ on } \\partial \\Omega", "75faae10990687264817dfdbdb1fc56e": "\n w^0_{,1111} + 2\\,w^0_{,1212} + w^0_{,2222} = -\\cfrac{q}{D} \n ", "75faf2efaa3f5d53d6cf25a23b50642f": "|x-\\alpha| = H^{-n\\omega^*(x,H,n)-1}.", "75fb0c92c7fc14f40d3e0b912fbaff7e": "\\varepsilon_Z : \\operatorname{Hom}_S (X, Z) \\otimes_R X \\to Z", "75fb1aab2750cd76a0bf8bc9d1e985c5": "Y_t = \\exp\\Bigl(X_t-X_0-\\frac12[X]_t\\Bigr),\\qquad t\\ge0.", "75fb7f7af82eac92d1dcd1ade32c5c93": "\\omega = \\sum_I f_I\\; \\mathrm{d}x^I", "75fbd2eb9cf4cf5ae3413148d035e60a": "\n \\hat{\\boldsymbol\\beta} = (\\mathbf{X}^{\\rm T}\\mathbf{Z}(\\mathbf{Z}^{\\rm T}\\mathbf{Z})^{-1}\\mathbf{Z}^{\\rm T}\\mathbf{X})^{-1}\\mathbf{X}^{\\rm T}\\mathbf{Z}(\\mathbf{Z}^{\\rm T}\\mathbf{Z})^{-1}\\mathbf{Z}^{\\rm T}\\mathbf{y}.\n ", "75fbd657af77fc3d72b1caf320df16e0": "\n 3b=0 \\;", "75fcc4c99f9f9a765c1ea0038f2f3f45": "\nN (a^\\dagger)^n |\\,0\\,\\rangle = n (a^\\dagger)^n |\\,0\\,\\rangle.\n", "75fd10e6169e4f68a2ef5b2c1a179d3c": "f(\\mathbf{v}) = f_1(f_2(\\mathbf{v}))", "75fd380f61697bfc24c5b2606dfddc52": "\\int \\frac{dx}{x\\ln x} = \\ln \\left|\\ln x\\right|", "75fd86fa1157c22d6fa8bef713cf9dbb": "\\frac{\\mathrm{D} \\rho}{\\mathrm{D}t} = 0 \\, ,", "75fe6a81ea0e92a902d3af54decdb4eb": "a\\in Attr", "75fea3932a5093e6e1c46b3aa0a8ecd6": " V := \\bigcup_{i} V_{i} \\! ", "75feac1edcf8c8090b60da3f19248b1c": "\\psi (\\mathbf r)", "75feb7615079d514823beb7b46dd1b51": "f^\\leftarrow", "75ff0882dd29782eaff47748c75117ef": "\n\n\\phi(e^{-8\\pi})=\\frac{e^{\\pi/3}\\Gamma\\left(\\frac14\\right)}{2^{29/16}\\pi^{3/4}}(\\sqrt{2}-1)^{1/4}\n\n", "75ff3325ebb8f750c86e10ae593399b2": "p^2=0.01", "75ff3ce4267faff68bd50c6b3baa1c52": "S_N(f) = \\frac{N_0}{2(1+(\\frac{w}{B})^2)}", "75ffa207bfd5b3dde73769db13490e65": "B_{r''} (x; d_{2}) \\subseteq B_{r} (x; d_{1}).", "75ffb36cae1910dac4b3b5d06e2c532b": "\\begin{align}\nw\\left(nT\\right) \n= \\frac{1}{N} \\left[ C^{\\mu}_{N-1}(x_0)+\\sum_{k=1}^{\\frac{N-1}{2}} C^{\\mu}_{N-1}(x_{0}cos\\frac{k\\pi}{N})cos\\frac{2n\\pi k}{N} \\right]\n\\end{align}", "75ffdc6d7fcd064c6bb037269b7bfb28": "x_{max}", "75fff97a65040a23622163f480ceae36": "y_t = \\frac{ r_n(\\rho - R)}{\\rho - r_n}", "760050cff3fffa31998430d4d7e1259f": "\\alpha>1", "760063650ea848a03189cec2d9040488": "m \\ll 1", "76008c60fcafdba3f860a8aa49b785aa": "\\sum_{k=0}^n c_k D^k", "7600ab805e661b8755521a6e6761b282": "x^8 M(x)", "7600e3063a6f4d9cc2611af6bccb32b2": "\\mathbf{t} = N \\mathbf{1},", "76012291ab982e3ce81808bcc96908ad": "\\left\\Vert e\\right\\Vert ^{2}", "76016ccdd8f83b440bcf4dc4feb58046": "\n\\frac{\\left|{\\mathbf\\Psi}\\right|^{\\frac{\\nu}{2}}}{2^{\\frac{\\nu p}{2}}\\Gamma_p(\\frac{\\nu}{2})} \\left|\\mathbf{X}\\right|^{-\\frac{\\nu+p+1}{2}}e^{-\\frac{1}{2}\\operatorname{tr}({\\mathbf\\Psi}\\mathbf{X}^{-1})}\n", "7601a6d24a2c430c3137453f65be649c": "\\frac{2^{\\alpha+\\beta+1}\\,\\Gamma(n\\!+\\!\\alpha\\!+\\!1)\\,\\Gamma(n\\!+\\!\\beta\\!+\\!1)}\n{n!(2n\\!+\\!\\alpha\\!+\\!\\beta\\!+\\!1)\\Gamma(n\\!+\\!\\alpha\\!+\\!\\beta\\!+\\!1)}", "7601b62a3a7c4ba55b693442601f5e4f": " \\mu_0 ", "7601eb29db6974a8111f90454eecd9ce": "\\sigma(x) = 1/(1+e^{-x})", "7601fb3b888470e1128f1ad5a227f32e": "x(v)\\cdot P", "760202d89ab1bd69428c688d3aab33f9": "\\mathrm{S}_{\\mathrm{O},\\mathrm{P}} > \\beta\\,", "760228a02a83425268fa8846ba8d5bc7": "M = M_0\\oplus M_1\\qquad N = N_0\\oplus N_1.", "7602297644b103abf205b00e8264622a": " \\sqrt{\\varepsilon_0} \\left(\\mathbf{E}, \\varphi\\right) ", "76023837d013816c69c15e710a01a9d4": "\nm(i) = \\begin{cases} 1 - m(n) & i = 1\\\\ \\\\\n \\dfrac{i - 0.3175}{n + 0.365} & i = 2, 3, \\ldots, n-1\\\\ \\\\\n 0.5^{1/n} & i = n.\\end{cases}\n", "76025d083e4db7f65691841f433bbaeb": " {d \\over dt} A(t) = {i \\over \\hbar} H e^{iHt / \\hbar} A e^{-iHt / \\hbar} + e^{iHt / \\hbar} \\left(\\frac{\\partial A}{\\partial t}\\right) e^{-iHt / \\hbar} + {i \\over \\hbar} e^{iHt / \\hbar} A \\cdot (-H) e^{-iHt / \\hbar} ", "76029b7e58436632af86670b7bd63798": "\\nabla^{2}(A\\left( x,y,z \\right) e^{ikz}) + k^2 (A\\left( x,y,z \\right) e^{ikz}) = 0", "7602a34118412ea1a9861a5847b52ced": "k_\\pm + 2 \\lambda k_0", "7602f4abc35b02c70a262f27061b6c9a": " Q = \\int_0^t I(\\tau) \\ d \\tau ", "7603007f00dfc55944fc3d31c993cf7c": "10^{10^{64}}", "76030c20d76c02849efa7362cae25448": "X \\cap Y = \\emptyset", "76032e77f8e01e09a4c1ac64460cee56": "d_1=c_1-b_1'", "760378d8beb08ebe634c955d5c7f0b41": "A_0 A = (A_0 A)^*", "76038790b280cb641d8c51089eae4af3": "P_N = P_0(1+r)^N - c ((1+r)^{N-1} + (1+r)^{N-2} .... + 1)", "76038e58f59ed1c815ecf302e3c9e37f": " u ( x ) ", "7603a7695a90e385e2fee6fb504207c8": "\\mathcal F, \\mathcal G", "7603d0b8a439f67df6e38d3fae94887b": "\\mbox{high} - \\mbox{low}", "7603e25467de37822a8f6d87a0ffed94": "d\\log{\\frac{n}{d}}", "76045f202471116a33f6fa09de711b55": "\n\\kappa = \\cfrac{10(1+\\nu)}{12+11\\nu}\n", "76047107a72c5bdf45879d7d0956d3ea": "\\operatorname{sin}(x) \\approx x - \\frac{x^3}{6} + \\frac{x^5}{120} - \\frac{x^7}{5040}", "7604a0b947e76c01bb51f4f18ee4bbe6": "R = kN_A", "76051729a77fe95fea192d98b65f64c0": "f\\colon {\\mathbf P_1}\\to {\\mathbf P_2}", "76052f8f3c2abc68fc3541c27202648a": " -n \\ln (1-p)", "7605afbcd3b58da0d1b75310307e12a4": "\\omega^\\mu_\\nu\\;", "7605d97da4084001d59eb60b8b72ea04": "g(N_j)=\\sum_{i=2}^j c(N_{i-1},N_i)", "760605f3c09619449d6024e1c5fa2362": " B^n = \\{(x_1,\\ldots,x_n) | x_1^2+\\ldots+x_n^2 < 1\\}", "7606505111c56bb3e4de370eb7cce1b3": "\n\\begin{align}\n& \\underbrace{1 + \\cdots\\cdots\\cdots + 1}_{n\\text{ terms}} \\\\\n& \\underbrace{2 + \\cdots\\cdots + 2}_{n-1\\text{ terms}} \\\\\n& \\underbrace{3 + \\cdots + 3}_{n-2\\text{ terms}} \\\\\n& {}\\qquad\\vdots \\\\\n& \\underbrace{{}\\quad n\\quad {}}_{1\\text{ term}}\n\\end{align}\n", "760699ff60b954fc035e80f513073c50": "u(x-x_{\\alpha})", "7606a9db28cd87f53f6aad6175816588": "{f_o + \\beta y}", "7606b3f1fcddd3b5e819219562c28cd6": "(5.b)\\quad \\big(\\sqrt{-g}\\,F^{ab}\\big)_{,\\,b}=0\\,,\\quad F_{[ab\\,,\\,c]}=0", "7606b8ff6c80a860be56ce762abf6445": "\\hat{C}_I", "7606f53bbb5fec797a7133dd33488c4a": "\\frac{1}{r_\\min}-\\frac{1}{p}=\\frac{1}{p}-\\frac{1}{r_\\max}", "7607109cd81cffd675bfeb2bee29aba2": "\\chi(\\theta) = \\exp \\left [ \\frac{2\\pi i }{p} \\left ( \\theta+ \\theta^p + \\theta^{p^2}+ \\cdots + \\theta^{p^{r-1}} \\right ) \\right ],", "76072e123c598dd641c73b684efcd65b": "v=2 \\mu g t_{p-r}", "76073ff99dd132571eaf0c6d0d03e570": " h_{k,k-1} \\leftarrow \\|q_k\\| \\, ", "7607454bf2d683206b895abd1370127b": "a > b > 0", "7607967f6fa6c80696574d058793fb48": "\\int \\mathcal{D}\\phi\\, q(x)[F][\\phi]=0.", "7607ba09a54d6e5586b5f47d819e1468": "\\alpha = 0.05", "7607f765ea587379f6f2b9468419226c": "=Q", "7608796076ae3e5e0bbcc0ebd05e8cae": "\\pm\\alpha_i", "76088b1ffcedcc2351e51c430d137e77": "\\sum_x \\arctan a x = \\frac{i}{2} \\ln \\left(\\frac{(-1)^x \\Gamma (\\frac{-i}a) \\Gamma (x+\\frac ia)}{\\Gamma (\\frac ia) \\Gamma (x-\\frac ia)}\\right)+C", "7608a465d4c509b2b572b9a701e503cc": " H(D^{\\alpha_1,\\alpha_2,\\ldots,\\alpha_m})(y_1,y_2,\\ldots,y_l) = G(D^{\\beta_1,\\beta_2,\\ldots,\\beta_n})(u_1,u_2,\\ldots,u_k)\n", "7608a95ac1e181265e963beba0c410b8": "\\begin{align}\nv_x & = u_x + 2a \\sin \\Bigl( \\frac{u+v}{2} \\Bigr) \\\\\nv_y & = -u_y + \\frac{2}{a} \\sin \\Bigl( \\frac{v-u}{2} \\Bigr)\n\\end{align} \\,\\!", "7608d4f437461cb91349d5ed1b923cd6": "F_i = M_i ^2", "7609577c2b1febf4fcdc37512f9cb8d8": "\n\\displaystyle\n\\text{Performance rating} = \\frac{2000 + 400 \\times (2)}{2} = 1400", "7609746d96531eaa59f11e6f2b798059": " \\deg\\, P_n = n~, \\quad \\sup_{0 \\leq x \\leq 2\\pi} |f(x) - P_n(x)| \\leq \\frac{C(f)}{n^{r + \\alpha}}~,", "760997354eda75226c217e4e91828c54": "f\\in L^2([-\\pi,\\pi])", "7609cc11275792334efa1ebd1bdf731a": "\\tau\\ ", "7609eb83fb0f4ed29c50b68463e4696a": "\\scriptstyle t\\,=\\,\\beta/i", "7609fac041fac617a5097f2d5d8c6b26": "\\omega_\\omega", "760a36aa160770e676008520e6fb0998": "\\langle 1, f^\\vee\\rangle = f(0) = \\langle\\delta,f\\rangle", "760a4329c591aa05c4d868f1b5e231a5": "B_n(x_1,\\dots,x_n)=\\sum_{k=1}^n B_{n,k}(x_1,x_2,\\dots,x_{n-k+1})", "760afa713753fc2c8fac00d8eaa1ebf0": "vxy = a^j", "760afb3275ec1fbff7d262fea233c1a5": "U_k(\\omega) \\to \\left(\\frac{\\omega_c}{i\\omega}\\right)^2", "760b07110b0b5159a2b74e00750fac58": "| SA | : | SB | =| SC | : | SD | ", "760b0be0f7282b718d1d5ff4c3106ad6": "E^0 = \\Lambda^0", "760b0be7781d0487aa46ce43b6c61c26": "E_{\\mathrm tS} (\\varepsilon, T)", "760b3ad797ee238b98d15c840851d76d": "~K=\\exp\\left(\\int G {\\rm d} z\\right)~", "760b414cc1ac00b0de2d3fc18c7b6c67": "v=q^a\\mathbf{e}_a+p_a\\mathbf{f}^a+tE", "760ba11cb2d830f9d8bb0448fb2420e8": " \\mathbf{\\hat r} ", "760bb80dabecc0936754e2149dbd37da": "{\\mathcal M}B(z^1)=B(z^2) {\\;} {\\mathrm {or}} {\\;} {\\mathcal M}{\\;}{\\mathrm {red}}\\subseteq{\\mathrm {green}},", "760c15d87a6f89a8e3520509dd93659f": "X_{2\\pi}(\\omega) = 1", "760c35c36e42d37ee655899f3c06dde4": "\\frac{h(y)}{h(x)} \\times \\frac{{m^{\\star}}^3}{m^3}\\ ", "760ccd83e36120a4d98ec80433ebc515": "~\\vec K~", "760cf2665086833dc52514bb61a1abff": " \\delta q = T_R (-dS_R) \\le T_R dS ", "760d86e87e5d8893c3a7170c226558fd": "p(v)=\\sum\\xi^i p_i(v)=\\xi^i p_i", "760d94c7f35cf94b08fb90d412dedc3e": "(h=2\\sqrt{2}r)", "760d9d045f2aaf5ca28a55c0de7578b1": "u\\in H^p", "760de6131fc9093b863bc6e1666cccf2": " -\\delta\\ \\mathbf{q}^T \\sum_{e} (\\mathbf{Q}^{te} + \\mathbf{Q}^{fe}) \\qquad \\mathrm{(17a)}", "760e15337e0ec854c6ebdd4750a1f308": " \\rho(\\sigma^{2}) \\propto (\\sigma^2)^{-(v_{0}/2+1)} \\exp\\left(-\\frac{v_{0}s_{0}^{2}}{2{\\sigma}^{2}}\\right).", "760eedf97224c0973cdbe07b7b58039c": " \\int_{-\\infty}^\\infty \\exp({a x^4+b x^3+c x^2+d x+f}) \\, dx\n= e^f \\sum_{n,m,p=0}^\\infty \\frac{ b^{4n}}{(4n)!} \\frac{c^{2m}}{(2m)!} \\frac{d^{4p}}{(4p)!} \\frac{ \\Gamma(3n+m+p+\\frac14) }{a^{3n+m+p+\\frac14} } ", "760f1fa37e831597b00a2898c8cc78db": "\\sigma_w^2.", "760f6dc796317b1e7614802e37030e5a": "C(\\epsilon) = \\epsilon", "760fb990f1096d364960b10770f53b81": "=\\frac {G_{\\infin}T}{1+\\frac{1} {G_{\\infin}} G_{\\infin} T} \\ . ", "760fbfc5337726036d49d20716b1b095": "0=(1-\\tau)\\int_{-\\infty}^{q_{\\tau}}dF_{Y}(y)-\\tau\\int_{q_{\\tau}}^{\\infty}dF_{Y}(y).", "760fc7eb34654217aecd35bfee9084ce": "y_i = \\phi\\left( \\sum_j w_{ij} x_j \\right)", "760fcca38a6a0986a592f5090541a95e": "\\|P_n-P\\|_\\mathcal{C}\\to 0", "760fe1ef71bd6dd2e78b61d6c4217244": "B+\\frac{P L}{2}\\,\\!", "760ff8c8f8d540b70897dbf4c050cdb1": "\\partial q_x /\\partial z=0", "760ffaedd028fe1c64f8b7a35e4a60e6": "Q_m = p(K_m)", "761012c50e9536d07deaec89b763a8d2": "\\, \\hat{C} = \\begin{bmatrix}0 & C_{ro} & 0 & C_{\\overline{r}o}\\end{bmatrix}", "761029f477c24bcc03e101edfa3b09e9": "p\\ne q", "76102ddcc5f9b827ea2791b5a7818c6f": "\\sigma_X^2 = x_0^2/3.", "7610419c5cbc829157587a79a12c8551": "(|a|^2, |b|^2, ..., |h|^2)", "7610609152203c7fa18aa639381635fe": " [I_R] = [I_C] - M[d]^2,", "7610b9a0e404d05051590d99d310b865": "\\begin{align}\n F &= \\frac{2}{\\pi}\\arccos\\left[e^{-\\frac{B (R - r)}{2r\\sin\\phi}}\\right]\\\\\n a &= \\frac{1}{\\frac{4}{C_n\\sigma}F\\sin^2\\phi + 1}\\\\\n a' &= \\frac{1}{\\frac{4}{C_t\\sigma}F\\sin\\phi\\cos\\phi - 1}\n\\end{align}", "761190b09906a697d55cc1c37aeeed64": "\\hat{\\sigma}^2(g^*) = \\frac{n_1+n_2}{n_1 n_2} + \\frac{(g^*)^2}{2(n_1 + n_2)}.", "7611b072e467b69b75dd51bbf8628de0": "P_{1}^{1}(x)=-(1-x^2)^{1/2}", "7611cfa7c2d08173a612d79addc0ad81": "\\tau_\\text{N} = \\tau_0 \\exp \\left(\\frac{K V}{k_\\text{B} T}\\right)", "761207a6a6bbb7feff7f125d866d273c": "T_{11}-T_{22}", "76125a77af1819f34355c85ee79e1845": "k[t]/(p_i^{k_j})", "7612dddb1102163319cdf023a9bd2e85": "m_{\\text{e}}/M,", "7612e5af4178956d5b15e0d482409773": "\\frac{\\partial {\\rm tr}(\\mathbf{AX^{\\rm T}})}{\\partial \\mathbf{X}} = \\frac{\\partial {\\rm tr}(\\mathbf{X^{\\rm T}A})}{\\partial \\mathbf{X}} =", "761379ca7a5c6d695759b69fed9f917d": "L \\propto \\log N", "7613d2113a45094e2c9bf599f08abdd0": "\\alpha^m > x_1 x_2 \\cdots x_n \\alpha^{m-n}", "7613da59fe2856edf10b99269971fa05": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=\\text{constant}>0.", "7613ee6ae8df9f346bc7a6c6eb3095d1": "\\textstyle \\mu; ", "76147873dee6418067c6ae146c9f6e96": "D_{\\mathrm{out}}", "7614b51b168d4e2204ec36b85ba25db9": "I(X_1; \\cdots; X_{n-1}|X_n) = \\mathbb E_{X_n}\\big(I(X_1; \\cdots; X_{n-1})|X_n\\big).", "7614d5abf6c00e7060d65a6e7f3a3f24": "(g_1,\\ldots,g_k)", "7615436aae1f7125e9aa80ba8d9e84e5": "nn(i)", "76159971f684651f5b2d9ae7c5f4b3da": " \nK=1", "7615e3c74229f6c66a6edadebedb9df4": "\\bar{f}=\\frac{1}{\\hbox{Vol}(U)}\\int_U f.", "7615ee55d8b80c3b1e4a616392476ced": "K = \\bigcap_n E_n", "7615fe84bb69a6f9e68994fd518a0778": " \\frac{1}{|H_n(\\mathbf{K})|} \\sum_{g \\in H_n(K)} |\\chi(g)|^2 = \\frac{1}{|K|^{2 n+1}} |K|^{2 n} |K| = 1. ", "76160334f0ab3e0948f5aa5dd9296072": " \\langle a, b, c \\rangle ", "76161a898da01dd4e2fb2b3df67deddf": "\\mathsf{NEXP} \\not \\subseteq \\mathsf{P/poly}", "761621d304d4db9eb0398a301f6b4d69": "\n\\hbox{Years per added zero of the price } = \\frac{1}{\\log_{e} \\left(1+ \\frac{\\hbox{inflation}}{100}\\right)}\n", "76163223362b34ebe731c108657f49fe": "E_{line}= I(x,y)", "761665b1355d1ca1630fc4658a228327": "z_2 = \\frac{1}{2} (XY - YX)", "7616a25771a6636c5a05e47360ebb4a8": "i_0 = 1", "7616aacf412333e47862a591bc200e47": "\\omega^{A}_{y\\|x}=\\omega^{A}_{x}\\circledcirc (\\omega^{A}_{y|x},\\omega^{A}_{y|\\overline{x}})\\,\\!", "7616ed39b94d8c2cf0702b53597b44b5": "e,", "7616f36ad51f562b20113fc523656f44": "\\gamma_{xy}=\\gamma_{zx}=\\tau_{xy}=\\tau_{xz}=0", "76172dbcf888d18c7a0dffad85393f7e": "K(k) = \\int_0^{\\pi/2}\\frac{d\\theta}{\\sqrt{1 - k^2\\sin^2(\\theta)}} ", "761754e5af1501e1f14d18af666a3a33": "= \\text{NAND}_2^{\\otimes |U|} \\text{EQUAL}_3^{\\otimes |V|},", "76175872c5b99d754829d33b922c14fb": "m_\\text{aver}", "7617a78b6ca2ec6cfa7214e79ebeb970": " \\mathbf{J} = \\mathbf{J}_f + \\nabla\\times\\mathbf{M} + \\frac{\\partial\\mathbf{P}}{\\partial t}", "7617d735d20d991058df4279176cc384": "\\mathcal{D} = \\mathbf{Ab}(Y)", "7617de70417ef9cdf89529f410437307": "\\operatorname{Exp}(k \\theta) \\,", "76180233874a4aec91fea12b8577ec48": "(f_0, f_1, f_2, ..., f_{d+1})", "761803786923d84846853a7c5587a3ef": "\\nu_\\mu\\rightarrow\\nu_e", "76183d23325bbb17f951ee8f1e27dffc": "t \\rightarrow t' = t + \\epsilon T \\!", "76183e3e8c35a3ef3e61b86b240172a7": " \\mathbf{a} \\wedge \\mathbf{b} = (a^2b^3 - a^3b^2) \\mathbf {e}_{23} + (a^3b^1 - a^1b^3) \\mathbf {e}_{31} + (a^1b^2 - a^2b^1) \\mathbf {e}_{12} \\ . ", "7618dcaab7e6e456df8eedf95db28a88": "=E[(y_i-f_i)^2] + E[(f_i-g_i)^2] + 2E[(f_i-g_i)(y_i-f_i)]", "761900132cd13565933d86c6318fa287": "\\Phi(0)=0", "761903f397a2ab346edbfa578bd99844": "\\begin{alignat}{7}\n2x &&\\; + && y \\;&& - &&\\; z \\;&& = \\;&& 8 & \\\\\n&& && \\frac{1}{2}y \\;&& + &&\\; \\frac{1}{2}z \\;&& = \\;&& 1 & \\\\\n&& && && &&\\; -z \\;&&\\; = \\;&& 1 &\n\\end{alignat}", "761939137a47a1333ea68965f0c1a2dd": "(\\{0\\} \\times \\{0,1\\} ) \\cup (K \\times [0,1]) \\cup ([0,1] \\times \\{0\\}) ", "76197941015ee569974dd71f6070e3f3": "\\xi_1 = \\xi_1(x_1,x_2,\\ldots,x_n), \\quad \\xi_2 = \\xi_2(x_1,x_2,\\ldots,x_n), \\ldots \\xi_m = \\xi_m(x_1,x_2,\\ldots,x_n),", "76197da381b859ba7a11cd23c3635fb0": "\\varphi (0, \\omega) = \\mathrm{id}_{X} : X \\to X", "7619a3d342e79774fb5e356386187f5a": "\\displaystyle A^{(k+1)} = P^{k}A^{(k)}P^{k}", "7619e829c072e9309cdd905d7dae721c": "\\begin{array}{lcl}\n M = \\sqrt{\\alpha^2+\\beta^2} & \\cos (\\theta) =\\tfrac{\\alpha}{M} & \\sin( \\theta) = \\tfrac{\\beta}{M} \\\\\n C,D = E \\mp F i & & \\\\\n G = \\sqrt{E^2+F^2} & \\cos (\\delta ) = \\tfrac{E}{G} & \\sin (\\delta )= \\tfrac{F}{G}\n\\end{array}", "7619f95515306b1b747f3296c8bb5026": "0\\to L^G\\to M^G\\to N^G\\overset{\\delta^0}{\\to} H^1(G,L) \\to H^1(G,M) \\to H^1(G,N)\\overset{\\delta^1}{\\to} H^2(G,L)\\to \\cdots", "761a1875860472ab61a9395fe71656c0": "c \\leq z < 1", "761a2bcf9c88cb96e21212ad1a180bd3": "\\sin(2 \\theta)=2\\sin(\\theta) \\cos(\\theta),", "761a782321c6bb1f456a28c4ee05ce09": "L = \\sqrt{1 + [ f'(x) ]^2} \\, . ", "761ab3f00754ed02be0d7a730f4c27fa": "\\frac{B}{3}+\\frac{C}{3}+\\cdots+\\frac{Y}{3} = \\frac13(B+C+\\cdots+Y).", "761b0570e37f25a799a11307376e12c6": "Q_3 = Q_2 Q_1 = Q_1 Q_2. ", "761b2412ee4630bfb462f6dd6dbc0f27": "(a, b) (c, d) = (a c - b d, a d + b c).\\,", "761b485e082c8ba10c91c7b5fa61c867": "\\frac{\\partial c}{\\partial t} = \\nabla \\cdot (D \\nabla c) - \\nabla \\cdot (\\vec{v} c) + R", "761c306e040ab5226e2946a158ea1dee": "L = n_{\\ell} - n_{\\overline{\\ell}}", "761c75c4242dd00c5895c3da5a4f3ff8": "m\\in\\{m_1,m_2,\\ldots\\}.", "761caa50823f54a68e27069340d6da1d": "\\hat y=y", "761cdc845c06c51053ddb1732b347f39": "T = \\sum_{i_1,\\dots,i_k=1}^N T_{i_1i_2\\dots i_k} e^{i_1} \\otimes e^{i_2}\\otimes\\cdots \\otimes e^{i_k}", "761d1f0db5cea0634edf00198b11fdec": "(C+INF+IF+F)/(C+INF)", "761d2077eb4283640679938b3bf8ada5": "\\nu', {\\sigma^2}'", "761d5824168a0e10d736bbabe394491b": "\\mathbf{T}q=\\sqrt{qKq}\\,", "761d72d90ed4fa0da6700159f51f0f50": "\\textstyle \\bar{\\sigma}_k = \\bar{M}_{\\mathrm f} R^k (\\Delta)", "761d7f116ed6d2d99a14a3a5fc038199": "\\textstyle B_1 \\cap B_2 \\cap \\dots = B ", "761d80df43d23b9606180facc9cc2104": " P^{(n)}(\\mathbf{r}_1,\\ldots,\\mathbf{r}_n) =\\frac{1}{Z_N} \\int \\cdots \\int \\mathrm{e}^{-\\beta U_N} \\, \\mathrm{d} \\mathbf{r}_{n+1} \\cdots \\mathrm{d} \\mathbf{r}_N \\, ", "761dae2b237fa7ad990004e197d357ee": "K_0, K_1", "761dc3d7a2f169ef3b8f0992e37d3a95": " (G\\times SL(n), V\\otimes\\mathbb F^n) ", "761de7d0ea5405b0ba46a48adea896c3": "0 < \\alpha \\leq 2 .", "761df2342c4f9bc7c673bfabbf113e6c": "e^{-H}", "761e195b5e8196bddef5841e9451e2a6": "C\\,", "761e2e0d061b3166d9dd077d9cb4db28": "\nc (\\tau) = \\lim _ {T\\rightarrow\\infty} \\frac 1{2T} \\int_{-T}^T x(t) x(t+\\tau) dt.\n", "761eece47dfc0b6a9f7882ad0c3003a6": "E(\\omega) = \\sqrt{S(\\omega)}e^{i\\phi(\\omega)}", "761f13e535f20ce4fb6883472571e059": "\\textstyle\\beta=1", "761f6baaf2bf7f49969398b813a62747": "\\textstyle \\equiv (", "761f9891ac1fd225c20bd6c21377dd90": " \\mathbf{f} = \\nabla F .", "761f9de52948abd8f15968af4c59f91a": "Y_3 = C(F-X_3)-8E = -72", "761ffadc0f6b520b3862b889e9b27dc9": "j\\omega \\to s", "76200c5d897de3681df42f48ce7d8c65": "A_f(j_f)", "762013b21a1a561fccef2d5eeb6dae98": "(x_{(k)} + x_{(k+1)})/2", "7620418e6fa9a8c50fd07aab842406db": "\\begin{bmatrix}\\begin{bmatrix}A\\end{bmatrix}-\\lambda\\begin{bmatrix}I\\end{bmatrix}\\end{bmatrix}\\begin{Bmatrix}X\\end{Bmatrix}=0.", "7620a8fe3fa768aec55179122a2abf58": "A\\subseteq \\mathcal{R}\\,\\!", "76213bb4de36c1991579af799c7a1d2c": "A_n=A", "76219f18de62e51471069f54100007d5": "J_2 \\simeq J_1 \\times J_2/J_1 .", "7621a33f600d7245ebb02349798c34d2": "(c,d,b)", "7621b606501f91bc363c11934704752b": " f(n_1,n_2,\\dots,n_m) = \\left(\\frac{1}{2 \\pi}\\right)^m \\int_{- \\pi}^{\\pi} \\cdots \\int_{-\\pi}^{\\pi} F(w_1,w_2,\\ldots,w_m) e^{j w_1 n_1 +j w_2 n_2 + \\cdots+j w_m n_m} \\, dw_1 \\cdots \\,dw_m ", "762207042a74c03cf1c08d83e583b202": "q_\\text{P} = e/\\sqrt{\\alpha}", "762208c3c4d5456f3eeaaf21da4690f2": "t_2 = 1213", "7622c711bccf47c786869d9995d14425": " \\sum_{i=1}^{n} \\sigma_i(x)", "7622ca5c310b61445020998d2fd6edf9": "\n(-\\gamma^\\mu \\hat{P}_\\mu + mc)\\Psi = 0 \\,,\n", "7622ec77d21b4548e2744c597df3d8b9": "\\Phi(\\omega)=\\arg(H(j\\omega)).\\!", "7622ed25ff36cfda64be1227ea91ec29": "\\mathbf{A}e^{x\\mathbf{A}} = e^{x\\mathbf{A}}\\mathbf{A}", "76230c7bba7ad00912912835ba974f75": " A = \\begin{bmatrix}\n 0 & 0 & i \\\\\n 0 & I_{n-2} & 0 \\\\\n -i & 0 & 0\n \\end{bmatrix}. ", "76230e5bd66e0795c375c702624c4a6f": "6 < z < 10", "76232626d0d0ba3fa608975ed9587300": "m((v_x - u_x)^2 + (v_y - u_y)^2 + (v_z - u_z)^2)", "76246d808cbe8cac2e5868d7a4dc73e0": "\\scriptstyle C \\;=\\; E_{K}(P \\,\\oplus\\, X) \\,\\oplus\\, X", "76249228e6f43dd9684a5abe2d766aaa": "\\text{TVC} = \\text{V} \\times \\text{X}", "7624f6f13d77daf4db5405c7dadd4b33": " s < 1 ", "762531a5e7d2dd8ac3d7db27f0c9a213": " \n\\{ e(t) : \\ {0\\le t \\le 1} \\} \\ \\stackrel{d}{=} \\ \\left \\{ W_0 ( \\tau_m + t \\text{ mod } 1) - W_0 (\\tau_m ): \\ 0 \\le t \\le 1 \\right \\} .\n", "762553c5368168da468f5b5df11f0bb2": "p = \\mathbf{v},\\quad q=\\mathbf{w}.", "76255b4b2003d03a2d92f1f6f56dde6e": "\\displaystyle \\partial_tu+\\partial^3_x u+6u\\partial_x u=0", "76259cd39cb8d296954718965ae21bde": "(E(\\ln(X))", "7625e8f499a64429d5200d2ee004debb": "T_M(d)=T_{MB}(1-(\\frac{c}{d})^2)", "7625f9f0bde9709440bf7cc224ede0f0": " \\langle e_s^\\vee, v \\rangle = 2 \\frac{B(e_s,v)}{B(e_s,e_s)},", "7626165ffc2bab6061ba669ab5535277": "N_{M_i} V", "76263613ce4fe8c4fefdec0f3f34af9c": "\\{d_i\\}", "76264612203eba4a8ce5fbb246737501": "i \\neq 4 ", "7626568c160c7691097500b4f8797b07": "M_{\\mathfrak{p}}(V)\\simeq \\mathcal{U}(\\mathfrak{g}_-)\\otimes V", "762686a9ce92303d4dabc0808c0e5411": "[e_i,f_j] = h_i\\ ", "762695fc809edd246e35acd10bab0b4a": "w[n] \\sim \\mathbb{N}_N \\left(\\theta {\\boldsymbol 1}, \\sigma^2 {\\boldsymbol I} \\right).", "76269c809ed95f49bdba4964a309a176": "\\cos(t)", "762707200e672d904291404f99a844c7": "a_{1,0}b_{0,1} - a_{0,1}b_{1,0} \\neq 0,", "7627242241663df9af5352f299def9ac": "\\left(1-\\frac{2M}{r}\\right) \\dot{t}^2 - r^2 \\dot{\\phi}^2 = 1", "76275c5541ad3dd697cb4548aedc578a": "\\varphi:D\\to D'", "762785d56d351be8577fd7015aaf08ec": "\\beta_{i_1} \\cdots \\beta_{i_k}", "7627e3217ab8a7e56ef92e2e94d32298": "A_{\\alpha\\beta\\cdots} = g_{\\alpha\\gamma}B^{\\gamma}{}_{\\beta\\cdots}", "762800ec411838254b6ae68ee36a41b6": "\\textstyle \\alpha \\neq \\beta", "76280e56c149e8671d669cd26456a746": "x(t) = e^{-\\alpha t} \\left[\\cos{(\\omega t)}u(t)+\\left(\\frac{\\beta-\\alpha}{\\omega}\\right)\\sin{(\\omega t)}u(t)\\right].", "76281d596295ffe153e2934e3242950b": " \\ln 2 = \\cot (\\operatorname{arccot} 0 -\\operatorname{arccot} 1 + \\operatorname{arccot} 5 - \\operatorname{arccot} 55 + \\operatorname{arccot} 14187 -\\cdots). ", "762826242fa8bc06a01bfcfdef076ee5": " V(\\vec{r}) ", "76282f4d4b79d644d51210b06c99f7cf": "L_3(\\xi)=\\xi^3\\left(\\frac{1-x_p^2}{\\xi^2-x_p^2}\\right)^2", "76285bcce427d9f4f8b9d80bf51b90a5": "-\\mu(-A) \\leq {\\frac {x^{\\mathrm T}Ax}{x^{\\mathrm T}x}} \\leq \\mu(A),", "76285c46150e260e54d664747f72abc0": "w\\, =\\, W(\\eta)\\, \\varphi, ", "762865de3e75bdf034d4197fbbd97ef5": "\\frac{\\nu\\mu_0+n\\bar{x}}{\\nu+n} ,\\, \\nu+n,\\, \\alpha+\\frac{n}{2} ,\\, ", "762988be75607f341cf2dade2c76b271": "s=\\lceil l^{1/d}\\rceil", "762998e315a42be504191718800a0704": "\\operatorname{EG}(a_n;x)=\\sum _{n=0}^{\\infty} a_n \\frac{x^n}{n!}.", "7629a8e8e1339b571049981b750bec21": " {u}_{0} (\\mathbf{q})", "7629f9616f9f41166d7bb5913626c823": "l_a (x) = \\|ax\\|, \\; r_a(x) = \\| xa \\|.", "762a0cf9fb49d244c4e64e692d746e91": "\\frac {R}{t}", "762a0e4e09a9bc4d04da3da24d86d6e9": "\\begin{align}\n\\varepsilon_{ijk}\\varepsilon_{lmn} & = \\begin{vmatrix}\n\\delta_{il} & \\delta_{im}& \\delta_{in}\\\\\n\\delta_{jl} & \\delta_{jm}& \\delta_{jn}\\\\\n\\delta_{kl} & \\delta_{km}& \\delta_{kn}\\\\\n\\end{vmatrix}\\\\\n & = \\delta_{il}\\left( \\delta_{jm}\\delta_{kn} - \\delta_{jn}\\delta_{km}\\right) - \\delta_{im}\\left( \\delta_{jl}\\delta_{kn} - \\delta_{jn}\\delta_{kl} \\right) + \\delta_{in} \\left( \\delta_{jl}\\delta_{km} - \\delta_{jm}\\delta_{kl} \\right). \n\\end{align}", "762a29ad0c6af6e4b990826666ce5edd": "\\frac{a^2+b^2+c^2-ab-ac-bc}{18}", "762a2b455a07b56fb549367eb8b16d83": "c^2R(r)T(t)", "762a319d41c5fb33c341f884ca13976c": "(a_1,a_2,\\dots,a_n)\\in R \\implies (h(a_1),h(a_2),\\dots,h(a_n))\\in R", "762a571a44677a91b37bfd5e12605eb8": "D_{BA} = \\frac{\\frac{3EI}{L}}{\\frac{3EI}{L}+\\frac{4\\times 2EI}{L}} = \\frac{\\frac{3}{10}}{\\frac{3}{10}+\\frac{8}{10}} = \\frac{3}{11} = 0.(27)", "762b0d0f96c8dd5009a4f8194d555257": "x \\leq x \\land y \\leq x", "762b2ebd5d109186b1beb3e2ec79bf2e": "RP(g)=(\\Pi_{i=1}^kr_{g,i})^{1/k}", "762b3f344e242de517d9766c60c4fe46": "S:= S\\cup{(f^*,\\deg f^*)}", "762bbe6549fefc26078433b70a253088": " \\mathfrak{g} = \\mathfrak{u} \\oplus i\\mathfrak{u}.", "762bdefccb77c15df134affec7c05c15": "H (N)_\\varphi = \\int d^3x { N \\over \\sqrt{det (q)} } \\Big( \\tilde{\\pi}^2 + \\tilde{E}_i^a \\tilde{E}^{bi} \\partial_a \\varphi \\partial_b \\varphi + det (q) V (\\varphi) \\Big)", "762c07d8f56ca2ba3c3929c0ab2b2442": "f^2 = {\\omega^2 \\over 1 - \\omega^2}", "762c0f38c71ffa6e93f0a1daab55ab04": "Y_i", "762c1445dccdc497a6a60278163dbf9b": "f^{-1}(U)", "762c2faac65bf36be8fb85fb53d1e1e1": "C = 1.9148 \\sin(M) + 0.0200 \\sin(2 M) + 0.0003 \\sin(3 M)", "762c33c16fe4da59c1fae921f576250d": "a\\,A + b\\,B \\leftrightarrow c\\,C + d\\,D", "762c7543a4792e5214fdbb3dfe734f22": "\\displaystyle{([b^2,a,b],c)=(b^2(ba)-b(b^2a),c) =-(b^2,[a,b,c])}", "762d098b4be5c3df8077a5c7787f39b3": " Z_i(N)/Z_i(N-1)", "762d41e96e0f310951bd3b03694bf715": "2ax = -b\\pm\\sqrt{b^2 - 4ac}", "762da712e3a09f12d33b4ac6aeb3e697": "\\theta_A = \\frac{p_A}{\\frac{1}{K_{eq}^A} + p_A}", "762db8c7a5611413ce75a9b6a12ac6f0": "\\langle \\hat{B}_\\omega \\rangle", "762e026781dc720d364542e57bde23d2": "(f_j)_{j\\in J} \\in \\prod_{j \\in J}\\operatorname{Hom}(X_j,Y)", "762e0f1d9fb15762fc3f375585a1363a": "\nV_\\mathrm{out} = \\frac{1}{2} \\cdot V_\\mathrm{in}\n", "762e3786914e1ad7a67da51b8647bf2e": "\\Phi:X\\rightarrow \\mathbb{C}", "762e6f55417bde3afff4b6bf02a58373": "\\mathrm{SCl_4 + 2H_2O \\ \\xrightarrow{}\\ SO_2 + 4HCl }", "762e801e10488c7895d0b824d76c039f": "\\mu_i = i\\mu\\text{ for }i\\leq C \\, ", "762ee03083c02d78f4512c0ad2c97e9d": "d(X)=0", "762ee4c0d00dbd0e5505ddea41ffc224": " \\begin{align}\nP_{0,0}&=1\\\\\nP_{i,i-1} &= \\frac{g_i (N-i) }{f_i \\cdot i + g_i (N-i)} \\cdot \\frac{i}{N}\\\\\nP_{i,i} &= 1- P_{i,i-1} - P_{i,i+1}\\\\\nP_{i,i+1} &= \\frac{f_i \\cdot i}{f_i \\cdot i + g_i (N-i)} \\cdot \\frac{N-i}{N}\\\\\nP_{N,N}&=1.\n\\end{align}", "762eeac9d71c129043ef281e92888c8b": " \\mathrm{d}S =\\mathrm{d}S_{\\mathrm e}+\\mathrm{d}S_{\\mathrm i}\\,,", "762f496655ecdd2a757779f22f7467ca": "a_n a_{n-k} = a_{n-1} a_{n-k+1} + a_{n-2} a_{n-k+2} + \\cdots + a_{n-(k-1)/2} a_{n-(k+1)/2}", "762f542a494e2fc1125a15852024d864": " \\left\\vert \\ell\\left( u\\right) \\right\\vert \\leq C\\left\\Vert \\ell\\right\\Vert _{W_{p}^{m}(\\Omega)^{^{\\prime}}}\\left\\vert u\\right\\vert _{W_{p}^{m}(\\Omega)}. ", "762f74c01672d4b51c5d4f477d749765": "op_2", "762f98c3843195d8c4d2b3098aebac08": "\\mu_{0} \\vec{J} = \\frac{1}{r}\\frac{d}{dr}(r B_{\\theta}) \\hat{z}", "762f9fc5b26251a3590c25eeff1e6140": " H_2 \\le 2H_\\infty ", "762fa02b86fb1ae43b6bca8fdc5cc1cd": "p(x|\\theta,M)", "762fcdcda3aa6def98c58a99c6385a89": "\\textit{mother}: \\textit{plant} \\longrightarrow \\textit{plant}", "762fd7b1e0dd9df7cb2afcdb24ad5c36": "z_1 \\ne z_2", "76300bb65eb9234f02e39bf98dc6f736": "(s-i)h", "76302f6c6893a8511120bdbab7a516f5": "\\frac{V_o}{V_i}=\\left(\\frac{-D}{1-D}\\right)", "7630470fbfabb26c9bc3403ff4b54d1f": "l\\,", "76306913f545b973170dfc58539ff69f": "\n\\begin{align}\n\\{ {\\left\\langle B \\right\\rangle} & \\mid {\\exists A, C \\ \\left\\langle A, B, C \\right\\rangle \\in \\mathrm{Enterprise} } \\\\\n & \\land \\ {\\exists D, E, F \\ \\left\\langle D, E, F \\right\\rangle \\in \\mathrm{Departments} } \\\\\n & \\land \\ F = C \\ \\land \\ E = \\mathrm{Stellar \\ Cartography} \\} \\\\\n\\end{align}", "76312efe6db89fd4fd5cad991add0caf": "\\tau = t/\\epsilon", "7631b7ee7c8066f250957a4885fb29ce": "\\operatorname{arg}(x\\ ,\\ y)", "7631fd67b9c07139b82cd1c1a1f8732f": "f_\\text{circ} = \\frac {4 \\pi A} {P^2}", "7631ff64588d39d7195ebac8144a832c": "\\mbox{eGFR} = \\mbox{141}\\ \\times \\ \\mbox{min(SCr/k,1)}^{a} \\ \\times \\ \\mbox{max(SCr/k,1)}^{-1.209} \\ \\times \\ \\mbox{0.993}^{Age} \\ \\times \\ {[1.018\\ if\\ Female]} \\ \\times \\ {[1.159\\ if\\ Black]} \\ ", "763202f958fecfa32f48f36679ae6bd0": "\\mathrm{Diff}_k(P,Q)", "76320c14158d7f680ad87451dbfc7a0e": "(n-1)\\mathbf W", "763247b052b2f069510c704a108ad17a": "M_{23,2}\\,", "7632697f601acf00a369af8d720fb14b": "[M\\rightarrow Fred(\\mathcal H)].", "76327683749a41c209ccef83af1b43bc": "\\mathbf{a}_{-i}=(a_1,a_2,\\ldots,a_{i-1},a_{i+1},\\ldots,a_N)", "7632bd6cf0e6634b7a553adf14de0550": " f_*(x)= f(x-c), \\quad c>0 \\,", "7632c7635809949c945fa61f93af4fc6": "\\left(H^*\\right) ^* = H", "76330b9168aae091bc84bb42a74ac8c1": "g:2^\\Omega\\rightarrow \\mathbb{R}_+", "76331235cbc3e318b0f26499f844b838": "\\frac{1}{M M_{SUSY}}", "763360040945883e64039509f52a7fae": "v_o \\approx \\frac{v_e}{\\sqrt{2}}", "7633915fd89782e37dca5b160b1055b1": "\\mathbf{x}=\\sum_{j=1}^k \\lambda_j \\mathbf{x}_j", "7633b8dd96d2af0c6939a7f943f28b74": "P_1\\parallel P_2", "7633d84f4534e9270e47067628311899": "H=H_0 + \\epsilon x", "7633d94d8f7cc466eeeeb4211f852c5e": "g_k(t)=kt", "763407c443bdd8ed67cb002180109110": "z \\in x ", "76341b8654c1bd6eaa1ee581a7ae90b2": " \\mathbf{x} \\in \\mathcal{D}", "763434e8026031664d11e9ae507a2103": "\nj^\\nu_r = \n- \\left( \\frac{\\partial L}{\\partial \\boldsymbol\\phi_{,\\nu}} \\right) \\cdot \\boldsymbol\\Psi_r + \n\\sum_{\\sigma} \\left[ \\left( \\frac{\\partial L}{\\partial \\boldsymbol\\phi_{,\\nu}} \\right) \\cdot \\boldsymbol\\phi_{,\\sigma} - L \\delta^{\\nu}_{\\sigma} \\right] X_{r}^{\\sigma} \n", "7634b4299361bb2d3c97f40a8568b70e": " (\\Omega^\\cdot_{\\operatorname{vert}}(M_0), d_{\\operatorname{vert}}) ", "763567c9f2e21fd2bb9a237ad95e00ea": "L^{SCC}[\\tilde{g},\\tilde{\\phi }]=\\frac{\\sqrt{-\\tilde{g}}}{16\\pi\nG_{N}}\\tilde{R}+\\tilde{L}_{matter}^{SCC}[\\tilde{g}]+\\frac{3\\sqrt{\n-\\tilde{g}}}{8\\pi G_{N}}\\tilde{\\square }\\tilde{\\Phi }_{N}\\left( \n\\tilde{x}^{\\mu }\\right), ", "76357e304cc0f5cf307ef19a45f20e84": "t)", "7635b95b1fb5925d3380789457a5078e": "\nx^2 + bx + c = 0\\qquad (b\\ne0)\\,\n", "7635db2e7d1cf3ed5bce7cbaae91246a": " P_\\mu x_\\perp^\\mu =0. \\, ", "7635e56d29b269c4b6d5e5bc5fcb03c2": "\\Phi_{i+1}=[\\Phi_{i}:\\phi_{i+1}]", "7635eeb2a5e6911bb3d8c051fd292128": "n_f < {33 \\over 2}.", "7635f23d87b6c8aaedbc0a8477707367": "\\rho_{S}^{\\prime}(t) = \\frac{-i}{\\hbar}\\big[\\mathbf{\\tilde{H}_{S}},\\rho_{S}(t)\\big] + L_{D}\\big[\\rho_{S}(t)\\big]", "763628c0293d5a104a3429f194e6a21a": "\\scriptstyle \\{y\\}.", "7636966fe0d36ab3a687cb1b25e34ca0": "\\sigma \\upsilon", "76376390e69c58965477698b8b239c0b": "\nX\\odot Y\\cong Y\\odot X,\n", "763765ff2c798c7a92477220e81022d4": "L^2 (G, d\\mu )", "763777c1182466d4c710518d08e53310": "(g,g^a,g^b)", "7637b13271a7577f0e61b6c2ed848efb": " 4x^3 - 3x - \\cos (\\theta) = 0.", "7637ea5f4ac9da3593d01e1eee4e343f": " k_{b_{n+1}} ", "7637efd6e57e10d893e5d1963b4e46c8": "xx' = \\gamma^2 \\left(1 - v^2/c^2\\right) xx'. ", "76382e76169af154776a207732985bcc": "\\sum_{m=0}^n \\tbinom m j \\tbinom {n-m}{k-j}= \\tbinom {n+1}{k+1}\\,,", "76384e9a293299cb76bc294c7cbbc9c7": "g\\in V(\\Gamma)", "7638657fc61fa4bf9f374c1a95f2dbee": "color(x)\\neq color(y)", "76386fe85dce262b4d2aeb0be9259807": " D = 10\\log_{10} {\\left ({101.5\\over {HPBW - 0.00272(HPBW)^2}}\\right )} \\;\\; dB.", "763960715c4931e7cf099eda94b6b397": "\\displaystyle(\\mathbf r,\\mathbf v)", "7639af1e6f3d8ff87cbe45c4f41f5463": "B(u, v) = \\langle f, v \\rangle \\mbox{ for all } v \\in V.", "7639c57c1ebcccf9b36269c1ff3a03c8": "n(t) = n_{0} - [ (n_{0}-n_{\\infty})(1 - e^{-t/\\tau_n})]\\, ", "7639ef6076fc1497a1632192aec0399b": " |a| ", "763a1b3b5bfe2cf744466ddf45ac279c": "\\quad n=2\\,p", "763a3ca1e53fc21bd6af092b561dc220": " U \\sim \\mathrm{Rayleigh}(\\sigma) ", "763a4e83364143b15adc98075f4dd2ec": "A, B, ... \\equiv P (t_{1}, ..., t_{n}) \\ | \\ A \\wedge B \\ | \\top | \\ A \\vee B \\ | \\perp | \\ A \\supset B \\ | \\ \\forall x. A \\ | \\ \\exists x. \\ A ", "763a8a1cc87027c8d133a0a64a67aa40": "b \\mid c", "763ab751ef7d063d48d7a79277040f3d": "\\Delta\\left(\\,P-P_e\\right)+\\frac{\\rho}{2}\\,\\left[\\frac{\\,f}{\\,D}-\\left(\\,1-\\frac{W_0}{W}\\right)\\frac{f_e}{D_e}\\left(\\frac{F}{F_e}\\right)^2\\right]\\,W^2\\Delta\\,X\n+\\frac{\\rho}{2}\\left[\\left(\\,2-\\beta\\right)\\,-\\left(\\,2-\\beta_e\\right)\\left(\\,1-\\frac{W_0}{W}\\right)\\left(\\frac{\\,F}{F_e}\\right)^2\\right]\\Delta\\,W^2\\,=\\frac{\\rho\\,f_e}{\\,2D_e}\\,W_0^2\\left(\\frac{F}{F_e}\\right)^2\\Delta\\,X", "763ad555a2d6c9ff4cba06f5dd852fbe": "1+\\ln\\left(\\frac{\\beta-1}{\\lambda}\\right)-\\frac{\\beta}{\\beta-1}\\ln(\\beta)", "763ae8e31d2877a7d79df9bf285ce650": "\\qquad \\sum_{j \\in J} w_j\\,x_j \\ \\le \\alpha\\,w_i", "763b3ef5643488e2fb160cf6309b09d3": " l = \\int \\sqrt{\\left|g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu \\right|} \\, ds\\ .", "763b41adf17024ce5987e4e512fe8495": " \\rightarrow y(4k+4) = a^{4}y(4k) + a^{3}u(4k) + a^{2}u(4k+1) + au(4k+2) + u(4k+3)", "763b90e3b677c215c8f97efcada4a278": "\\lambda=1\\,", "763bb06c7877f90e4eaf2f4fa14044fd": "S = \\{ (x_i,f_i) | f(x_i) = f_i \\} ", "763c8ef4d4fcefbc839fd7917240ac5d": " E[R] = \\frac{1}{3} \\cdot 1 - \\frac{2}{3} \\cdot 0.5 = 0.", "763cb2657dbb7ae886a33a680a979a38": " f = \\sum_{S \\subset [n]} \\hat{f}(S) x_S \\!", "763ceae7c0e42e50f872cac6ea6a238f": " \\chi : (\\mathbb{Z}/k\\mathbb{Z})^* \\to \\mathbb{C}^* ", "763d1116916d1d23b788cf7c39bc9f38": "+\\sum_i \\mu_i \\,\\mathrm{d}N_i\\,", "763d26e27f2f155800f653e179edec53": "\\eta(\\boldsymbol{x},t)\\,", "763ddc2f7e8309e090819cc2dec12ddb": "\\textstyle \\overline{a}_{..}", "763df73dae0a05684fcc95d9396b146e": "x_+^\\alpha = \\begin{cases}x^\\alpha&\\text{if }x>0\\\\ 0&\\text{otherwise}\\end{cases}", "763e157e46e6fb5de8ed7b2d53733784": "\\delta_{i j}", "763e229bacc9ec96bfd7d295454f8dab": "\\|{\\boldsymbol\\theta - \\boldsymbol\\nu}\\|", "763e689f4bcd68abda77ce8900067511": " \\mathrm{d}f = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{(d_p - \\bar{d_p})^2}{2 \\sigma^2} }\\mathrm{d}d_p", "763edf0a79753e133329923cf0f5a3e5": "u > |\\dot{x} + a(t,x,\\dot{x})|", "763f1c04dffeedf32d10860bbe4a8a96": "y^{\\prime\\prime}=R(y^{\\prime},y,t)", "763f4f9c49f911e0c9b1ede099c74c6e": "D = c \\beta_1 \\beta_2", "763f8e24bbda538ad903369f3846921d": "a_5=2", "763fba280f027abfce9d99556797d591": " \\underbrace{ \\overbrace{\\sigma^{\\text{T}}}^{\\tfrac{\\partial V}{\\partial \\sigma}} \\overbrace{\\dot{\\sigma}}^{\\tfrac{\\operatorname{d} \\sigma}{\\operatorname{d} t}} }_{\\tfrac{\\operatorname{d}V}{\\operatorname{d}t}} \\leq -\\mu ( \\mathord{\\overbrace{\\| \\sigma \\|_2}^{\\sqrt{V}}} )^{\\alpha}", "7640021f07aec65823a12e443e28a021": "\\psi(\\tau_n) = \\tau_n\\otimes 1 + \\sum_{i=0}^n \\xi_{n-i}^{p^i} \\otimes \\tau_i", "764029bfd1bb009eaca2ff35570b6e01": "(\\alpha x+\\beta y)^2 + \\gamma x + \\delta y + \\epsilon = 0 \\,", "76406855f7f5401f2d9304f66ef1b192": " \\nu_r = \\int{ ( x - \\mu )^r f( x ) dx } ", "7640d6e7140ba6fb1b3654efa39e0350": "\\Gamma(\\varepsilon - n) = (-1)^{n-1} \\; \\frac{\\Gamma(-\\varepsilon) \\Gamma(1+\\varepsilon)}{\\Gamma(n+1-\\varepsilon)},", "764106aa32f21fa68b5f154d87f0cba1": "i_n = \\sqrt{2 I q \\Delta B} ", "76411c2175559d538b4156395f32a002": " \\begin{align} \\rho = I | \\rho ) &= \\displaystyle \\sum_\\alpha \\left[ (d+1)\\Pi_\\alpha - I \\right] \\frac{ (\\Pi_\\alpha|\\rho)}{d} \\\\\n&= \\displaystyle \\sum_\\alpha \\left[ (d+1)\\Pi_\\alpha - I \\right] \\frac{ \\mathrm{Tr}(\\Pi_\\alpha\\rho)}{d} \\\\\n&= \\displaystyle \\sum_\\alpha p_\\alpha \\left[ (d+1)\\Pi_\\alpha - I \\right] \\quad \\text{ where } p_\\alpha = \\mathrm{Tr}(\\Pi_\\alpha\\rho)/d\\\\\n&= \\displaystyle -I + (d+1) \\sum_\\alpha p_\\alpha |\\psi_\\alpha \\rangle \\langle \\psi_\\alpha | \\\\\n&= \\displaystyle \\sum_\\alpha \\left[ (d+1)p_\\alpha - \\frac1d \\right] |\\psi_\\alpha \\rangle \\langle \\psi_\\alpha |\n\\end{align} ", "764150424a0b0cd889d2d1918ba9575b": " (\\alpha_h)^2 \\mathrm{M}(a,b,c) =\\mathrm{M}(- a, -b, c). ", "7641857365b63edfb4831f7c3a4c9bdf": "g_i = \\frac{1}{\\sqrt{\\sum_j \\mathrm{tf}_{ij}^2}}", "7641858891d1acb6a26c5bf4e2f6ca20": "\\operatorname{Cov}[\\mathbf{Y},\\mathbf{X}]", "76419fc420da30da9fc16709c935567b": "S_{n} \\subseteq \\Omega", "7641b23ff3ab5c3db241e6ecf1a91d5e": " E = \\{ \\sqrt x \\le 4 \\} ", "7641c695f3810474d1cb1c26246ffeb7": "\\begin{align}e_1e_2\\cdots e_r &= e_1 \\wedge e_2 \\wedge \\cdots \\wedge e_r \\\\\n&= \\left(\\sum_j [\\mathbf{O}]_{1j}a_j\\right) \\wedge \\left(\\sum_j [\\mathbf{O}]_{2j}a_j\\right) \\wedge \\cdots \\wedge \\left(\\sum_j [\\mathbf{O}]_{rj}a_j\\right) \\\\\n&= \\det [\\mathbf{O}] a_1 \\wedge a_2 \\wedge \\cdots \\wedge a_r \\end{align}", "76421c7176b796e8e8d0796c561762f1": "\\left\\langle T \\right\\rangle", "76423d7b7c7d059579bd0d95d8bbcefd": " U[\\boldsymbol{\\mu}]", "764245d4b7e4b6a7ebafa542270a7467": "\\ \\mathbf E_J \\cdot \\mathbf e_i = \\delta_{Ji}=\\delta_{iJ}", "76431f7541c08053a9b7b540a832bfb1": " I = P_0/(\\Omega r^2)\\,\\!", "76432cc83ba654bf45d0b001829dafcd": "\\mathbb{T}_{ij}", "7643564cd369cff3ba06a95d696a173a": " a_0x^n + a_1x^{n-1} + \\ldots + a_{n-1}x + a_n = 0 ", "76436af2732669eb21cf6b9e1a541ed1": "\\tau_M(\\rho:\\mu)\\in\\mathbf{R}^+", "7643d507466031566cc5fdc4524ec601": "e(p, u^*) : \\textbf R^L_+ \\times \\textbf R\n \\rightarrow \\textbf R", "7643dc79f8b4095598e31b1305ba3f8c": "\\mathrm{d}G = V\\mathrm{d}p\\,-S\\mathrm{d}T\\,+\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}n_i\\,.", "7643eb82b9939ec461a450afe92aeb12": "x=\\cos^3\\theta,\\qquad y=\\sin^3\\theta.", "7644a9e3088df17bacbf50509e9bddf6": "\\pi_*:H^*(E)\\longrightarrow H^*(M) ", "7645331798b3f549699d026e98cc2c7c": " \\Pr( O_n ) = |\\lang n | \\psi \\rang|^2 = | c_n |^2 ", "76454932064e6bf57f0b5fed670453c8": " r+g+b = 1", "7645bc0b6b1a32ccf2ef2b42c2d6403e": "\\textstyle l \\le (n+1)/2", "7645ce18cfc536316b99a90ffd0aeb36": "{\\partial f \\over \\partial x}\\hat{\\mathbf x} + {\\partial f \\over \\partial y}\\hat{\\mathbf y}\n+ {\\partial f \\over \\partial z}\\hat{\\mathbf z}", "7645d4b3fce8da9f35cae886456e9fa2": "\\bar{y}_i = \\frac{1}{\\tau} \\int\\limits_0^\\tau y(i\\tau+t) \\, dt.", "764673fe39bc9d49c4f24f252b123a49": "(x1\\vee x2\\vee \\overline{x3})", "76467939cc697cc1ad0791037d6e86c4": " D_0 f = -(p_0f^\\prime)^\\prime + q_0f", "764697ed438e5202aca975bc2b301eda": "\\frac{}{}_S \\bold{F}\\cdot{\\rm d}\\bold{S} = \\Sigma ", "764743b75212212af3093dc1c2aadc68": "\\theta \\in \\mathbb{R}", "7647966b7343c29048673252e490f736": "89", "7647d9f04ded9cb2c8cbde0dbdde25f6": "g_{\\Omega}(\\hat{a},\\hat{a}^{\\dagger})", "7648051991c00db5e7b4d4ac62d3c76e": " \\frac{ b - y^{-1} }{ b - a } ", "764819737e6110b1ccfa66e5e77abd79": "Q=\n\\begin{bmatrix}\n 16 & 11 & 10 & 16 & 24 & 40 & 51 & 61 \\\\\n 12 & 12 & 14 & 19 & 26 & 58 & 60 & 55 \\\\\n 14 & 13 & 16 & 24 & 40 & 57 & 69 & 56 \\\\\n 14 & 17 & 22 & 29 & 51 & 87 & 80 & 62 \\\\\n 18 & 22 & 37 & 56 & 68 & 109 & 103 & 77 \\\\\n 24 & 35 & 55 & 64 & 81 & 104 & 113 & 92 \\\\\n 49 & 64 & 78 & 87 & 103 & 121 & 120 & 101 \\\\\n 72 & 92 & 95 & 98 & 112 & 100 & 103 & 99\n\\end{bmatrix}.\n", "76481c80eeab81b9dee39e1a69c04e6f": "dij", "764826e91df63a201e2e17b923aed9af": "\\tau_m", "76482f12851b9bbacca9b4e80ab7662c": "p(w,b|D,\\log \\mu ,\\log \\zeta ,\\mathbb{M})", "764847fdcd6cf3681efa8f73c3bf4b8f": "\\gamma^4 \\equiv \\sigma^2 \\equiv 2 (\\bmod(2^h - 1))", "764875f0336193d912bf307b002c9e37": "= \\frac{1}{10}", "7648b0c6c4e93ad10f3991270be7c595": " \\underline u_i \\approx u_i(p_0)-\\left|\\sum_j\\frac{\\partial u(p_0)}{\\partial p_j}\\right|\\Delta p_j ", "7648c3f3a79335c03c66e4acc2b4419c": " \\overline{\\rho}(g) := \\rho(g)^{\\dagger}", "7649200e6e04923229ff8d8d4ae2fbf9": "f(x) = e^x", "7649333174346d37f6971a899d0f9161": "\\begin{cases} 1, & \\text{if tf}_{t,d} > 0 \\\\\n 0, & \\text{otherwise}\n\\end{cases}\n", "7649913f7ef8a4075b963c061931b4e9": "\\,\\eta_i=\\Sigma^{1/2}_{xx} \\beta_i", "764a846ff80021d0e3fd99c5d55ef0b5": "P_{\\mathrm{error}\\ 1\\to\\mathrm{any}} \\le M^\\rho \\left(\\sum_{y} \\left(\\sum_{x} Q(x)[p(y|x)]^{\\frac{1}{1+\\rho}}\\right)^{1+\\rho}\\right)^n. ", "764a8f214b41a7c4f73f343286c2c1e2": "\\hat{Z}", "764a935dcc3b5e8536b22d3c99c1a964": " H(\\ln T)^{\\frac{1}{3}}e^{-c\\sqrt{\\ln\\ln T}} ", "764b392e132eed28f6c61cf04c9c7336": "{\\alpha'}_j=\\alpha_j+y_j", "764bdeafd30c2f163205e889f9d841e9": "\\tfrac{17,804}{9,933}", "764c12ee9fbd8ebe1e193118453af363": "\\rm ADP + P_i + 4\\; H^+_{intermembrane} \\rightleftharpoons ATP + H_2O + 4\\; H^+_{matrix} \\! ", "764c2eb22021153e0d6019bac7e2ab56": "[H_n^{-1}(\\alpha_1), H_n^{-1}(1-\\alpha_2)]", "764c7c30a4389e1a452b5c5e827591ac": "\\begin{align}\n y & = (2x+8)^3 \\\\\n \\sqrt[3]{y} & = 2x + 8 \\\\\n\\sqrt[3]{y} - 8 & = 2x \\\\\n\\dfrac{\\sqrt[3]{y} - 8}{2} & = x .\n\\end{align}", "764c93abe611796abf22ee4ce8fa2c15": "(a+b)x\\,\\!", "764ca8d50af6ea094a061e7e90b898c5": "O(\\infty)", "764d023013e1dd0e29d4b994f061b660": "\\mathbf{R}\\mathbb{P}^5", "764d806fe67c5a0861c2b8772d0c54ef": "\\frac{1}{n-1}\\sum_{i = 1}^{n} (X_i-\\bar X)^2\\,\\!", "764df9a6324ceaff414d970620f882cf": "\\textstyle\\dot s", "764e021ed81bb6cf9c76c26e089925ff": "\\partial_x=\\frac{\\partial}{\\partial x}", "764e3667169025a44728dd54268aa6d1": "\n\\Gamma_{12} = {Z_2 - Z_1 \\over Z_2 + Z_1}\n", "764e6472539c6d03313a62a19f5ee76d": "\\rho=\\frac{1}{2}r\\left(1-r\\right),", "764efb0b8bbe75e8836179a56eab5d55": "N \\in \\N", "764f3d96225ec302e124bcd74a361491": "\n E=\n- E_0 \\; \nI_1 \\left( \\mu \\right)K_1 \\left( \\mu \\right)\n", "764f4a54c16f1e4cc3f49bbe18ee0a26": " U \\bigg(\\int_X^\\oplus \\psi_x d \\mu(x) \\bigg)= \\int_Y^\\oplus \\sqrt{ \\frac{d (\\mu \\circ \\eta^{-1})}{d \\nu}(y)} \\ U_{\\eta^{-1}(y)} \\bigg(\\psi_{\\eta^{-1}(y)}\\bigg) d \\nu(y),", "764f52f0e05a2a84c0a4e46cdbefbbc1": "T[v] \\neq v", "764f784577950c5a65a2a43ce0192751": " \\lambda(n) = \\frac{\\phi(n)}{\\Gamma(1+n)} \\!", "764fabcf3f4ae0bcd54807fd91cc5382": "\\mathbb{U}/\\mathrm{IND}(P)", "764fb0c5c52a346c6ab47fbf9e4f4131": "\\textstyle F_\\gamma(z) = F_0 + \\int_\\gamma f dz", "76503726ee7c5e42ab4573ea6d05a8bf": "\\begin{align}\nh_1&=h_2=\\frac{a}{\\cosh v - \\cos u}\\\\\nh_3&=\\frac{a\\sinh v}{\\cosh v - \\cos u}\n\\end{align}", "765075f2b0c0b2d7f8941fedb250c1ab": "\\Sigma_{cr}\\,", "76507e86cab1324cb69ddf9f6bad6e7e": "\\Delta E_{2,1,m_l}=\\pm|e|(\\hbar^2)/(m_e e^2)E", "7650a7d3ecf8394d25384be231bfbb43": "P_{i,j}=\\exp (- \\lambda d_{i,j} ) \\,", "7650b64a61bccf384a492c247a6c89f3": "P(x|\\theta)\\,\\!", "7650badb84abf0ff28390def2f15bc0a": "\\tau\\sim", "76510380cbadbd3f59db4d005bd2c384": "\na:b:c:d=\\operatorname{snh}(\\eta-u):\\operatorname{snh} (\\eta +u):\\operatorname{snh} (2\\eta): k\\operatorname{snh} (2\\eta)\\operatorname{snh} (\\eta-u)\\operatorname{snh} (\\eta+u)\n", "7651229706f7cf5227c82ae63806f223": "\n \\delta U = \\int_L \\left[\\left(\\frac{\\partial M_{xx}}{\\partial x} - Q_x\\right)~\\delta\\varphi - \\frac{\\partial Q_{x}}{\\partial x}~\\delta w\\right]~\\mathrm{d}L \n", "76516f19136bfaba51631466f4b5f1fd": "P_\\theta^A(B) = P_\\theta (A^{-1} B) = \\int_{T(X)} P_\\theta(A^{-1}B | T=t) \\ P_\\theta^T (dt) \\,", "7651a79445f2c86f3b7fd6ecee4ed7ac": "S = X\\,C_n\\,X^T", "76521216e22270288e93657b200ced89": "\\Phi = 1 / \\varphi = \\varphi - 1", "7652713b3995805d9f2dabc1f4579129": "P\\left(\\omega\\right)", "7652825b979098d7d21b2e10ebbd2a35": "\\text{true} \\land", "7653383da94b756c3a5d9c52a1568c5a": "p(\\epsilon_{kk})", "76536c53ba8d1976ca21afc9034a285a": "\\frac{\\exp\\left( -\\frac{1}{2} \\, \\mathrm{tr}\\left[ \\mathbf{V}^{-1} (\\mathbf{X} - \\mathbf{M})^{T} \\mathbf{U}^{-1} (\\mathbf{X} - \\mathbf{M}) \\right] \\right)}{(2\\pi)^{np/2} |\\mathbf{V}|^{n/2} |\\mathbf{U}|^{p/2}}", "7653c2b15c6d9d9ca1930b229000c0ca": "\\mathcal{A}^n_\\alpha\\rightarrow e^{i\\bar{\\theta}(\\mathbf{r},t)} \\left(\\mathcal{A}^n_\\alpha + \\frac{i}{g_s}\\partial_\\alpha\\right)e^{-i\\bar{\\theta}(\\mathbf{r},t)}", "7653cf43386976885d58988569040aa3": "W = x'r + (W_0 - x'k) \\cdot r_f", "7653d7b48e38523edd35f4776ce65f0d": "{ 1 \\over x\\pi } \\left[ { \\sigma \\over (\\ln x - \\mu)^2 + \\sigma^2 } \\right], \\ \\ x>0", "7653e151c174a60345efe8ac5afcc7d4": "\\sum_{i=0}^{N-1} y_i |i\\rangle", "7653ff6e16c8a8fb9f9558082fbbe100": "\n d'=d_I \n +\\lambda_1 \\coth\\left(\\frac{d_1}{\\lambda_1}\\right)\n +\\lambda_2 \\coth\\left(\\frac{d_2}{\\lambda_2}\\right),\n", "765425a68a90756e2ef7fceb57b8471d": "Z_B = Z_0\\,\\!", "76543ab821e5f237eabfe161a0b46c1d": "-1.0113", "765449134504156f016058e0d18267c2": "\\frac{d^2 x}{dt^2} + \\frac{k}{m} x = 0, \\,", "7654af58b5644514dee06ef3d944d384": "A\\triangle B=A^c\\triangle B^c", "7654e46354f462b28bed81c8b75974b4": "\\exists A^{n+1} \\forall x^n [ x^n \\in A^{n+1} \\leftrightarrow \\phi(x^n) ]", "7654eb43fd12ca0fafc1d5617af57c13": "\\begin{pmatrix}1 & a \\\\ 0 & 1\\end{pmatrix}", "76557b97bd1d30ff1e9ec6879876bc30": "\\,\\delta_i=\\theta_{i+1}-\\theta_{i}\\,", "765581990075cdafd1e4f34b7013e09c": "^*\\mathbb{Z}", "76558c4e1970f45075c378e5382e63ae": "\\sin\\theta\\approx\\theta\\,", "7655a698f3d16c611c6212531ee93fe7": "\\frac{3\\pi}4\\!", "7655c1d8bfd6c5494759308cce48a1f2": "^{b}a", "7655c1eaf124a0563e92fb2baf638377": "w_g", "7655eac4439787da2ad4b6bb8184edc1": "L_\\bullet, M_\\bullet, N_\\bullet", "765607ab250028f4ab5cf15acac88ff6": "z \\bar z - z \\bar \\gamma - \\bar z \\gamma + \\gamma \\bar \\gamma = r^2", "7656252045f3d1eb0a6de9db5a5ef4f2": "\nx=a\\,x_{frac} + c\\,z_{frac}\\,\\cos(\\beta)\n", "76565ea73a128f0cbae41b2473248538": "y_1, y_2,\\dots, y_m,", "7656b4102a4ad07e3c0d0b85a876bc15": "\\lim_{x \\to c}f(x)=\\lim_{x \\to c}g(x)=0, \\text{ or } \\lim_{x \\to c}f(x)=\\pm\\lim_{x \\to c}g(x) = \\pm\\infty, \\text{and}", "7656cf4dcb08ae38422ee7e307fdb0a4": "V = \\frac{V_{\\max} [S]}{K_m + [S]} ", "7656f33f61e4460fc86fcc93e5dd600d": "F(\\underline{x}_0,\\underline{u})", "7656f946ee8a2812076ac090b89cc137": "\\textstyle M_{\\mathrm f} R M_{\\mathrm f}^{-1} = T", "76573e826a14d357d667d01bfa643fd5": "h:X\\to[0,1]", "76575c062fa920b741daff14d23022e3": "x - a,\\ ", "7657f81f90b4dc1cac4e04c6b20bdb15": "\\{\\widetilde{y}_1, \\dots, \\widetilde{y}_{\\widetilde{n}} \\} ", "765802398bb0f758e103f65d9562fcc3": "u_\\beta = \\eta_{\\beta\\alpha } u^\\alpha = \\eta_{\\beta\\alpha} \\frac{dx^\\alpha}{d\\tau}", "76583a05aa8f134d3c2f35e1399a13e9": "f:B_n\\to B_{n+1}", "765842f5f212751fb7a4cba03745bbf4": " \\vec{\\theta}_i ", "7658aa328835dfbeeb5c7aaa79f809b9": "M(0 ", "76606c08434f1936d5637e5ca689f797": "R[S^{-1}]", "7660e02fed9cb91a0e9461ecd0eaa66c": "(0, 1, -1, \\dots, 0),\\ ", "7660e515435387104de2da7956d2257d": " \\int^\\oplus_X H_x \\ d \\mu(x) \\cong \\bigoplus_{k \\in \\mathbb{N}} H_k ", "76610d1b3388c1ca6a99f7cf08d5578d": "b_3=\\frac{r_2,c_4 - b_2\\times a_2 - b_1\\times a_3}{a_1}\\mbox{ with remainder }r_3", "766177ee088a72dc02228a22d4e93062": " \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty W_x(t,f)\\,df\\,dt = \\int_{-\\infty}^\\infty |x(t)|^2\\,dt=\\int_{-\\infty}^\\infty |X(f)|^2\\,df ", "766206ca8d382fdfc03defdf8dd1cb86": " \\lambda(s) ", "766210dff276ea4ea445d4b66ab3748b": " \\Phi \\lbrace \\cdot \\rbrace \\, ", "76623dc823061ac829c561254daa95f0": "g(z) = h(z) \\sqrt{\\frac{1+z}{1-z}},", "76627a3debdc7cb6f501fd429bcdc719": " \\text{square} \\subset \\text{quadrilateral} \\subset \\text{polygon} \\subset \\text{shape} \\, ", "7662a9f4799b5cd9e5399b5fc62be924": "g:A \\to Type", "7662c61cb363a19f7b7191c8c0187f7b": "k = \\frac{\\pi}{\\textstyle 180^\\circ}.", "766330db064263429e84f15efd1579e5": "m_k = \\frac{\\Delta_{k-1}+\\Delta_k}{2}", "7663973093af6a211b0826c8b4ac2f48": "m(n)=M(n)/\\sqrt{n}", "7663b14195c8a0279690cefc12ac0026": "x^{21} \\pm 1", "7663eb17bb35a904eb7487bd59bee229": "L_2(x)=x^2+2 \\,", "766401a9955dfe79ee29dadb9fe6a69f": "H =\\frac1N+N V", "76640fb75d127cc61e1e7ca3049ff3a8": "V_\\mathit{Au} = \\frac{M_a \\times \\tfrac{kt}{24}}{19.32}", "7664403cd68598176eb3b5f085c53a9d": "U = -G\\int_0^R \\frac{m(r) 4 \\pi r^2 \\rho}{r}\\, dr", "7664731b8d76bf048da93209affeebdd": " H(x) = \\mathbf{1}_{[0,\\infty)}(x).\\,", "766478ebed04122b69e91701bb6c3fe6": " \\nabla^2 ", "76648417f220581a92ac045a00ecda8e": " r = 1 ", "76649710ea48b1c143993942b8ee7c3f": "{\\partial^2 f \\over \\partial x^2} + {\\partial^2 f \\over \\partial y^2} + {\\partial^2 f \\over \\partial z^2}", "7664a1998f8d7b4d2fc7626ff0545ccd": "\\textstyle\\ x_{i}=x_{i-1}+\\sigma _{i-1}d_{i}\\begin{bmatrix}\n\\cos (\\theta _{i-1}+\\phi _{i})\\\\ \n\\sin (\\theta _{i-1}+\\phi _{i})\n\\end{bmatrix}", "7664a798f134f89aab1f4f99e82e1bfa": "\\sum_{\\rho} \\frac{x^{\\rho}}{\\rho} = O(\\sqrt x \\log^2 x).", "766535480313e1d7efdde83e29f99ebc": "{\\tilde{A}}_8", "76653b943a77b1b79b293ea995714c73": "a = c", "766546ad448b45ae158d1bfa17ab3e74": "a,b,c,d\\in F", "766585ca32d3e295dde2df4f57f9054f": " 1 + \\frac{1}{\\varphi} = \\varphi. ", "7665952765474e8e84a9c96cc84e8d05": "\\operatorname{GL}(V) < \\operatorname{Aut}(V)", "76660518fad21849f1361282315b14dc": "r = rank([A^*,B^*])", "766629c56aedebe3f5fca80ee81f2ed2": "x_i-y_j = i+j-1. \\;", "7666399e9ca910320779b30fb04952a0": "\\frac{\\partial S}{\\partial x_i}", "766697f42a14eda4281ee50bbcf0bb9c": "PHM_{\\infty} = 0", "76671702af9b9c5935c48ee2aa4ab3a5": "(P \\uparrow Q)", "76674d59d65f427408cf1e75d628c732": " f(0) = 0. \\,", "76677d441420b7745b4d4335f47bf941": " \\Phi^{n}(a_{1},\\ldots,a_{n}) := \\underbrace{[[\\ldots[\\Delta,L_{a_{1}}],\\ldots],L_{a_{n}}]}_{n~{\\rm nested~commutators}}1 . ", "7667b35858066e2320c659122a85cebb": "\\gamma\\colon \\mathbb R \\to G", "7667c02207cfa8777fb3b10a7cfdcff5": " a, b \\in S", "76684dfb829ee7e63fbcc668b0a7ea87": "-z", "7668692ab3c6974a6ee73f2ce98fd278": "a^{(n)}b", "7668e19cbae866def3c982a2f24e1c39": "\\{ z \\in D : |f_j(z)| < 1, 1 \\le j \\le N \\}\\,", "7669695dc672d8940855d69216bf35cd": "a = \\frac{T_0}{\\lambda g}\\,", "76697bb8e9ea6cdb2ee4d5184dda9127": "a_0=v_0/2", "7669d26eb29886d83cabf004c395ccdb": "\\sum_{m=0}^n P(5m)=P(5n+1).", "7669e4e174ac46d0f3685d7c5eabc8c8": " (x'\\mathbf{e}_1 + y'\\mathbf{e}_2) = (x\\mathbf{e}_1 + y\\mathbf{e}_2)e^{i\\theta}.", "766a20c7ce18d13ac791f8431e9cb93b": "\\sum_{k=0}^\\infty ar^k = \\frac{a}{1-r} - 0 = \\frac{a}{1-r}", "766a690ec62f6185220dd6928a92ff28": " D\\wedge F = 0 ", "766a95d5513ff9ec5b9e1d4b6aea8117": "\n\\left[A,B\\right] \\equiv \\frac{\\partial A}{\\partial x}\\frac{\\partial B}{\\partial y}-\\frac{\\partial A}{\\partial y}\\frac{\\partial B}{\\partial x}.\n", "766ad59530cf7318578d13edddee2fcc": " \\delta(a,f)=\\liminf_{r \\rightarrow \\infty}\\frac{m(r,a,f)}{T(r,f)} = 1 - \\limsup_{r \\rightarrow \\infty} \\dfrac{N(r,a,f)}{T(r,f)}. \\, ", "766ae22a54012e69079b7507e92e3fec": "\\ r = k[A][B] = k'[A]", "766afe44610c3cb5837b9a13ff365575": " X_{t+2}", "766b2937d39b665ebd15fdc6a49b1b66": "(s, t_s, t_e) \\in Q", "766b2d85ddc47f8529bb8f9e67c3f242": "-S_{T}+K", "766b652815ad6e5b934e159f39f656ab": "a_1 x_1[n] + a_2 x_2[n]", "766b775dcac93fa1cf6f6b365212665f": "N_p", "766b99f274fca8eed50cfc56b71e6db3": "C=u_1 \\times G + u_2 d_A \\times G", "766bc228003d5dbf9405e1e719e4d4f4": "\n\\mathcal{L} = \\frac{1}{2} \\left(\n \\varphi_t^2 - a^2(\\varphi) \\varphi_x^2\n\\right),\n\\qquad\na(\\varphi) := \\sqrt{\\alpha \\sin^2 \\varphi + \\gamma \\cos^2 \\varphi},\n", "766bd6042463cecf7f01d84dd6aa4dc1": " h_i ", "766c08f2dfea738f51f323bd97b50ec5": "p_\\pi(\\boldsymbol\\eta|\\boldsymbol\\chi,\\nu) = f(\\boldsymbol\\chi,\\nu) g(\\boldsymbol\\eta)^\\nu \\exp \\left (\\boldsymbol\\eta^{\\rm T} \\boldsymbol\\chi \\right ), \\qquad \\boldsymbol\\chi \\in \\mathbb{R}^s", "766c15d59bf4b74ec16c9f28c015777e": "({v_0+v_i})10^{-b_1E_{i}}", "766c1fb9e5b4d7480c6bef0437ec25e6": "r = 2m", "766c2be8b25800ec687cfa3ff92a2a4d": "\n\\sum_{n=1}^{\\infty}(\\zeta(2n+1)-1)=\\tfrac14.\n", "766cb6fe1e408450ce86b3ec5e388ef6": "f(T_2,T_3) = \\frac{T_3}{T_2},", "766cf63c360f42d6898dff8302c6444c": "z^'", "766cfdc5729ad3e600299b0889dd2f27": "X_L = {16 \\over \\pi} f L", "766d06a970ac96eb8de0148381261f70": "r \\leq g", "766d7d651496047f3137e2c7c2fcc8ea": "a_1x+b_1y=c_1, \\ a_2x+b_2y=c_2 ", "766d852e2f486e016747e98b592be5a7": "\\sqrt{\\pi} r", "766dcbe7911e15a1544d8d4b03338238": "u(0,x)= u_0\\ \\left(1-\\left(\\frac{x-x_1}{x_1}\\right)^2\\right)", "766dcd089d86b00b2a957dccc7037a0d": "\\; M_v = U_1 \\Sigma V^T .", "766e1303c8d5eb412454340b5509bb58": "\\,^{z_{11} = x_{11} y_1 - x_{12} y_2 + x_9 y_3 + x_{10} y_4 + x_{15} y_5 + x_{16} y_6 - x_{13} y_7 - x_{14} y_8 + x_3 y_9 - x_4 y_{10} + x_1 y_{11} + x_2 y_{12} + x_7 y_{13} + x_8 y_{14} - x_5 y_{15} - x_6 y_{16}}", "766e7bc76fed66aad821cbde6f6e134f": "V_j\\;", "766e93046aec19ff02ba47b878c34936": "H^*(BU(n))=\\mathbf{R}\\lbrack c_1,\\ldots,c_n\\rbrack,", "766e9aa95860c429a7a480a66bba28c7": "\\mathrm{Ext}^1(H,F)", "766ec22826eed082280e0cbfcd82d78c": "t_r = t - \\frac{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}{c}", "766f52f94b253a86071741410790f9b9": "\\frac{n^2(n+1)}{2}", "766faf35719de5ba27eb3112983c04e9": "Frac_{13/12 (sample)} = \\frac {(^{13}C/^{12}C)_{wood}}{{(^{13}C/^{12}C)_{sample}}}", "766fb3174d6cf1b7fc5f8feb62a89db6": "\\; Y_i = \\operatorname{Ker} (\\lambda_i I - A)^{\\nu(\\lambda_i)}.", "766fbf5a3d612476184e2924da97193d": "\\mod q", "76702b66443b21bcc963d0ae9202756d": "y_{ist}", "76703ca76f48245e8843c2021aeaa986": "D\\Delta^2\\ w=P", "767095ccffccf54ad82541cadeb11034": "\\scriptstyle D^2 \\times I", "7670cc1132c28ea7034c26ebf4b064f5": "\\bold{F}=d\\bold{A}+\\bold{A}\\wedge\\bold{A}", "7670e6460e92f3e586bfcb78023650c0": "h_k = \\lfloor2.25 h_{k-1}\\rfloor", "7670fce961e060e2001b915b4c27bd93": "\nY = X + Z, Z_i \\sim \\mathcal{N}(0,\\tfrac{N_0}{2} I_n)\n\\,\\!", "767106e928b8ef241f2a2e5bdada0b9f": "f(x_1,\\dots,x_n)=\\bigvee_{i\\in I}x_i", "76718019035e386170ae5546b3441d61": " \\mathrm{conv}(X\\Rightarrow Y) =\\frac{ 1 - \\mathrm{supp}(Y) }{ 1 - \\mathrm{conf}(X\\Rightarrow Y)}", "767195d62ff0032052fcc2a6686037ae": "\n\\frac{1}{k_{1}} \\frac{dR}{dt} = -R^{2} + 2ER + K_{d}R_{tot} =\n-\\left( R - R_{+}\\right) \\left( R - R_{-}\\right)\n", "7671cb199b798e23053e4dc2b5057ed8": " S_1,\\ldots, S_k\\subset S,\\quad k\\in\\mathbb{N}", "7671cc1603f5a2abba4c525f83e5db61": "\\! v_{rec} = \\dot{a}(t) \\chi(t)", "76722cdb6e7b21c4be20dc907569a6b1": "k(\\phi)=\\sec\\phi.\\,", "76722f31d058d725b62884699c430a5f": " \\hat{V}(t) ", "767260f92af9993f157889e54060d5e3": "[\\mathbf{e}^1\\ \\mathbf{e}^2] = [\\mathbf{e}_1\\ \\mathbf{e}_2]\\begin{bmatrix}1/2 & -1/\\sqrt{2}\\\\\n-1/\\sqrt{2} & 2\n\\end{bmatrix}.", "7672726dec35027740740588d37a1f9e": "S = I - \\left( \\operatorname{diag}(Q) \\right)^{-1} Q", "76729d0de5dff349f855ac6dd671cd47": "M^{-1} = \\eta^{-1}M^T\\eta.\\,", "7672a70b31bd70bc1fbf8d4c819124ea": "S = DF", "7672b22f122f2903cc36cbe98a46789e": "\n \\Delta p_{c}\\propto k_c/R^3\n", "7672c82214266030556fab2c784ae4e3": "x_1 \\left. \\frac{\\partial \\ln \\gamma_1}{\\partial x_1} \\right |_{p,T}\n=-x_2 \\left. \\frac{\\partial \\ln \\gamma_2}{\\partial x_2} \\right |_{p,T} \\,", "7672d625e9a2492987c50d3b87c04349": "\\theta_1", "767322b767fe51153132090efea3c629": "2a\\Phi+3b\\Phi^2=\\lambda \\mathbf{1}", "7673894bf0a42a23faea23c77c5aa5f4": "F_b\\,", "76738c06c9dbd54defa11268a6a04520": "a \\times (b \\times c) + c \\times (a \\times b) + b \\times (c \\times a) = 0\\quad \\forall{a,b,c}\\in S.", "76740f282c6fbe097ffcaddf8f8665c5": "\\int_{-\\infty}^{\\infty} h(t - \\tau) A e^{s \\tau}\\, \\operatorname{d} \\tau", "767451eace187ae61a3f4b37f89fc7c2": "\\alpha^m / \\alpha^n = \\alpha^m \\cdot \\left( \\alpha^n \\right)^{-1} = \\alpha^{m - n}", "767456870c9a6ba32a506fc05320fa6b": "\n\\begin{align}\n\\int_{x_1}^{x_2} \\left.\\frac{dL}{d\\varepsilon}\\right|_{\\varepsilon = 0} dx \n & = \\int_{x_1}^{x_2} \\left(\\frac{\\partial L}{\\partial f} \\eta + \\frac{\\partial L}{\\partial f'} \\eta'\\right)\\, dx \\\\\n & = \\int_{x_1}^{x_2} \\left(\\frac{\\partial L}{\\partial f} \\eta - \\eta \\frac{d}{dx}\\frac{\\partial L}{\\partial f'} \\right)\\, dx + \\left.\\frac{\\partial L}{\\partial f'} \\eta \\right|_{x_1}^{x_2}\\\\\n \\end{align}\n", "767481082cd097ee7f52373bf97abe1c": "\n\\eta = k_z z_0.\n", "7674aba7be70474f8a94a4cf97cc72c0": "\\mathrm{MASE} = \\frac{1}{n}\\sum_{t=1}^n\\left( \\frac{\\left| e_t \\right|}{\\frac{1}{n-1}\\sum_{i=2}^n \\left| Y_i-Y_{i-1}\\right|} \\right) = \\frac{\\sum_{t=1}^{n} \\left| e_t \\right|}{\\frac{n}{n-1}\\sum_{i=2}^n \\left| Y_i-Y_{i-1}\\right|}", "76758227fc25c9eb8c9c5a3d050da9bc": "6.6\\times 10^{-6}/\\,^\\circ\\text{C}", "767591af468b2449855681c9f34ebdd8": "\n\\rho_{\\alpha\\beta}(\\omega) = \\frac{1}{\\mathcal{Z}}\\sum_{m,n} 2\\pi \\delta(\\xi_\\alpha-\\omega)\n\\delta_{\\xi_\\alpha,\\xi_\\beta}\\langle m |\\psi_\\alpha|n \\rangle\\langle n |\\psi_\\beta^\\dagger|m \\rangle\n\\mathrm{e}^{-\\beta E_m}\\left(1 - \\zeta \\mathrm{e}^{-\\beta \\xi_\\alpha}\\right),\n", "7675ace726be0dc09820b5571558b25e": "\\int_E f(t) \\, d\\mu(t)", "7675e36e5e652c08237c20adf2e111ad": "\\textstyle {N}(B),", "767674860edfd420f1b9a74dd8af88c3": "x \\to 1", "7676985bdeb50f381df8fce2fbfb3c83": "\\scriptstyle\\lesssim5\\times10^{-9}", "7676dc05e28b6ca2c505721f1a4531ae": "\\lim_{n\\rightarrow\\infty}\\; \\frac{1}{n} \\sum_{k=0}^{n-1} f\\left(T^k x\\right)=E(f|\\mathcal{C}),", "767710a16fac9788a349224bb0af4e07": "N\\mu", "7677236b9a9f809d61863a0f14a1c8fa": "x \\in [0.5, 1].", "76772bbfff25c3a3d6a773b9006dfc62": "O\\left((\\log n)^2\\right)", "76773060cc76dd60ec91cefc72a5f0e0": "I_s(\\phi) = I_{c1}\\sin(\\phi) + I_{c2}\\sin(2\\phi)", "76773431d0f82e86c058f27f2bab5663": "\\sum F_y=0=-10-F_{AD}\\sin(60)-F_{BD}\\sin(60)=-10-\\left(-\\frac{10}{\\sqrt{3}}\\right)\\frac{\\sqrt{3} }{2}-F_{BD}\\frac{\\sqrt{3}}{2} \\Rightarrow F_{BD}=-\\frac{10}{\\sqrt{3}}", "76773a7cfcc1d1963e6669861526d8c9": "t = 4 \\pi X /( g T) ", "7677d8ee1a7a0d689c5680d50db900cf": "k \\neq 0,1,3", "767803efb7ae09af1ef4a2527467e4e9": "y \\in [1,2)", "7678839aec6612b92ac1a616102d1dc2": "? \\,", "7678b6d5d540ddc11b63ace79d27ada2": "\\left( \\begin{smallmatrix} 1 & -6 \\\\ 12 & -2 \\\\ \\end{smallmatrix} \\right)", "7678cf621196130b72481efd9e50e5a9": "I = I^*", "76794fd57eaf677792984c30633fb021": "r(\\phi) = a+b p(\\phi), \\, ", "767953a191436189e97689871eb5b096": "h = \\pi_* \\circ f^*", "767983580741a76c7947b545b556e835": "\\, {\\hat{D}} = D", "767a22dbebf06ae77eea864b6e7d31b7": "Ei(y) \\sim e^y \\sum_{n=1}^{\\infty} (n-1)! y^{-n} ", "767a241906ca3c4e031abc42332783a6": "x \\in \\bigcup\\mathbf{M} \\iff \\exists A \\in \\mathbf{M},\\ x \\in A.", "767a4b3f8d7eb71d1cbadbb83f55000d": " g \\equiv \\lambda f.g\\ f ", "767a9a60314fd9298166c7ab617fb574": " f^{*} = \\frac{b_2 - 1}{2b_2} + \\frac{b_1 - 1}{4}(\\frac{1}{f_1} - \\frac{1}{f_2}). \\! ", "767aa9207a585644ae206c93514263d4": "\\|\\nabla f(\\mathbf{x}_k)\\|", "767acc4e5fd84552660591123791d596": "\\int_{\\nu_0}^\\infty \\frac{L_\\nu}{h\\nu} d\\nu = \\int_0^{r_1} n_p n_e \\alpha_B dV", "767afea239fc0e3c2f3f34e4b5ab94f2": "v\\mapsto cv", "767b2944566b4ae751a9247850fb58ba": "= a \\left(3 \\tan \\frac{\\theta}{3} - \\tan^3 \\frac{\\theta}{3} \\right) = at(3-t^2)", "767b41a1c71c8f3ae91c0678a2eace4b": "\\frac{v_1^2}{g}\\left(\\frac{1}{y_1 y_2}\\right)=\\frac{1}{2 y_1^2}(y_2+y_1)\\qquad\\text{recall }F r_1^2=\\frac{v_1^2}{g y_1}", "767b86f248bba2e754ca9babfc70d299": "\\scriptstyle E_r \\,=\\, K^2 A^2 T", "767ba26200d38ed2c8aae26a351cb362": "\\begin{align}\nx &= R\\,\\cos\\varphi \\sin\\left(\\lambda - \\lambda_0\\right) \\\\\ny &= R\\big[\\cos\\varphi_0 \\sin\\varphi - \\sin\\varphi_0 \\cos\\varphi \\cos\\left(\\lambda - \\lambda_0\\right)\\big]\n\\end{align}", "767ba3c6a02aa4167409ccc826d7f0fa": "\\mathbb{E}[f(S)]\\geq \\sum_{R\\subseteq [l]} \\Pi_{i\\in R}p_i \\Pi_{i\\notin R}(1-p_i)f(\\cup_{i\\in R}A_i)", "767bb0d1a4ea524687f397b1abf42823": "\\tilde Z(s) = \\tilde Y_1(s) \\cdot \\tilde Y_2(s) \\cdot \\cdots \\cdot \\tilde Y_n(s)", "767bdc476eb2959a0ebbf27b408f1ebe": " \\pi^0\\,", "767bde7dcd2d512d171dc57ad687af1a": " \\nabla_{{\\mathbf e}_i} {\\mathbf e}_j = \\Gamma^k {}_{i j} {\\mathbf e}_k,", "767c3f3432bd3ea9846d7a01fd5313f2": "A\\,=\\,(a_1,a_2,\\dots,a_n)", "767c6781398e3095944d7042793cfb7c": "\\begin{align}\n 0 &= \\left(\\lim_{k \\to \\infty}A^k\\right)\\mathbf{v} \\\\\n &= \\lim_{k \\to \\infty}A^k\\mathbf{v} \\\\\n &= \\lim_{k \\to \\infty}\\lambda^k\\mathbf{v} \\\\\n &= \\mathbf{v}\\lim_{k \\to \\infty}\\lambda^k\n\\end{align}", "767c6f5de2c72478bfb413ed1a33bf26": "\\{x_N\\}\\subset X", "767c7eefeb259aa52299a93c2c9f1689": "S(\\rho_{ABC}) + S(\\rho_{B}) \\leq S(\\rho_{AB}) + S(\\rho_{BC}).", "767cf3f8bbcb70624864bc0136259590": " x_{\\alpha} \\in U ", "767d3758ccabf519cbbe4954d7ce94bd": "m_\\infty", "767d46e67751eb1e24e1d4b8befc1423": "\\lambda= v/f", "767da91173a8ef9df9488e018d9d4b5e": " \\{ n_1 ,\\; n_2 \\}\n\n", "767e3e999312a38cc6bfe7d6b7e19601": "m_{\\mu} \\approx 3 m_s", "767e5910a14a64793b2a533c95225d9b": "(i\\gamma^\\mu\\partial_\\mu-m)\\psi=0", "767e7ee5d1963bc7e6551ed4803f7911": "H^{p,q}(X)", "767e91b2899c0e72d7d698fffcf79af7": "\n2\\int_0^\\infty uH(u)\\,du - \\Big(\\int_0^\\infty H(u)\\,du\\Big)^2.\n", "767eb18a52807dc54c1b1755ffd7586a": "K_2=R_2+3R_3", "767eca9136df9e11f33ba9c21c183bf2": "\\Phi_E(G,k)=\\min_{S\\subseteq V} \\left\\{|E(S,\\overline{S})| : |S|=k \\right\\}", "767edf0fc64a42407bf3481f30a24867": " (y, m_0, \\dots , m_L) ", "767f398d304da272231a2f65d922faa8": "f = \\sum_{i=1}^m (f_i)", "767f6065b39a2d94bb7c940d5cdb4b21": "\\scriptstyle \\Gamma > M", "767f708cf830f98b77a288d8f0e089a5": "Q(\\beta, V)= \\int {\\mathrm d} x_1 \\int_{x_1}^{x_1} Dx(\\tau)e^{-S[x(\\tau)]}", "767fa791ea0eec9f2e93ca37110eee2a": "\\tau : \\pi^{-1} (U) \\to U \\times X", "767fb80db56195e7946d4e075ad75d1d": " z \\in \\mathbb C^n . ", "767fe969f00015e334645a5c34889f14": "C_i ^n", "7680a4adfff78f062d020883061a8276": " \\Delta f =\\frac{\\partial}{\\partial \\xi^j}\\left(\\frac{\\partial f}{\\partial \\xi^k}g^{ki}\\right) + \\frac{\\partial f}{\\partial \\xi^j} g^{jm}\\Gamma^n_{mn} =0,", "7680a4ae7c117db2d8e021971df3c220": "(P_{i0} + P_{i1} + ", "7680e1e982282f06c7a037e8c4348431": "\\nabla\\to\\nabla^*", "7680edc609e3fa36b49b988d6b9dece3": "\nf = \\frac{x} {1 - x - x^2} .\n", "76812edcd9d7788a42309558972c7f6c": "\\operatorname{Ind}(f)", "768166c8bdb18ea9a5f4e5ec3ee0d0ac": "q\\lesssim1\\text{ nm}^{-1}", "7682275858651836c7ef9e0266a36a68": "(a+b)\\,\\bmod\\,n = ((a\\,\\bmod\\,n)+(b\\,\\bmod\\,n))\\,\\bmod\\,n", "76822faa443768a208ff562cf7142b3b": "p \\equiv q \\equiv 3\\pmod{4}", "76828020a09dd8f9bf388b8b4526fd8b": "\\cos \\left(x-y\\right)=\\cos x \\cos y + \\sin x \\sin y. \\,", "76828369f2f3317e7624833d655439fb": "R_{\\mbox{in}} = R_{\\mbox{out}}", "7682999189e1d1c3d880ff7991f0ec6a": "\\sim_o \\subsetneq \\sim_l \\subsetneq \\sim_e", "7682ae1202bf66bf04804d713b370a78": "\\hat x_{\\mu}", "768300fa7c2b00247ab7f1470c09409e": "\\left(\\frac{7}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "76832ce94c3a4fbdacf307ac9df66a30": "S(x)\\Lambda(x)=S(x)\\lambda_0\\prod_{\\ell=1}^v (\\alpha^{i_\\ell}x-1)=\\lambda_0\\sum_{j=1}^v e_j\\alpha^{c\\,i_j} {(x\\alpha^{i_j})^{d-1}-1\\over x\\alpha^{i_j}-1}\\prod_{\\ell=1}^v (\\alpha^{i_\\ell}x-1).", "76834020001d2268ec9994ff7796e5ee": "\\rho(x)=\\langle \\xi, \\pi(x) \\xi \\rangle", "7683df27231aeaf2aa52d27b7457dcf2": "\\operatorname{lcm}(a, a) = a,\\;", "7683f8afc49c375a05f98ed7fb05a34f": "GL_\\infty = \\cup GL_n", "76842573459a3dc14c24d6d36193e0a6": "u_{2,1}^{0}", "7684382938be1de8218991ed0b8d9cc7": "H(w)", "76844172f14620a7c9e48a17ccccf1b0": " \\operatorname{drop-param}[(g\\ q\\ p\\ n), D, \\{p, q, m\\}, \\_] ", "7684671ee6c407e5e7903c91ac1b3bbf": "\\begin{pmatrix} 1 & 2 & \\cdots & n\\\\\nn & n-1 & \\cdots & 1\\end{pmatrix}.", "76849f53b55f02b3f5c973ced977d6cd": "k = k^{\\Dagger }K^{\\Dagger } =\\kappa\\frac{k_BT}{h}e^{\\frac{- \\Delta G^{\\Dagger }}{RT}}", "7684d3c296ad248da159edc4e1cf6921": "x^a(\\tau),\\; a=0,1,2,3", "7684fb238784114cdfa3fc3724f3e764": "A_\\text{ellipse} = \\frac{b}{a}A_\\text{circle} = \\pi ab.", "7685020f4d5eba65d0d60fceafeef153": "\\lambda_0=\\frac{2\\pi c}{\\omega}, \\;\\; \\lambda = \\frac{2\\pi}{k} = \\frac{2\\pi v_p}{\\omega}, \\;\\; n=\\frac{c}{v_p}=\\frac{\\lambda_0}{\\lambda},", "76851dd6dceb63252fb786bf0fb7b353": " v_{xx} + v_{yy} + k^2 v =0.", "76852c275b0e6bff4794deb2978ca650": "w\\Vdash A\\to B", "768547fe70ca7680f50e302bbf332bc4": "\\mathbf{y} =\n\\begin{bmatrix}\ny_1 \\\\\ny_2 \\\\\n\\vdots \\\\\ny_m \\\\\n\\end{bmatrix}\n", "76859490d28ab8fd0e11b197d1fef0f3": "{}_RQ_P - {}_RW_P = U_P - U_R ", "7685a08081d8bcf8f4b825c2cb058537": "\n\\left [\n\\begin{array}{c}\nI \\\\ Q \\\\ U \\\\ V\n\\end{array}\n\\right ]\n= \n\\left [\n\\begin{array}{c}\n|E_v|^2 + |E_h|^2 \\\\\n|E_v|^2 - |E_h|^2 \\\\\n2 \\mathrm{Re} \\\\\n2 \\mathrm{Im}\n\\end{array}\n\\right ],\n", "7685e3400abe79b32debf59b2b677783": "\\scriptstyle{R_J}", "768660c2f35f461203e111a76f92a1ce": "\\sin z = \\sum_{n=0}^\\infty \\frac{(-1)^{n}}{(2n+1)!}z^{2n+1} = \\frac{e^{i z} - e^{-i z}}{2i}\\, = \\frac{\\sinh \\left( i z\\right) }{i} ", "7686a20ca50a08a57db598ff6088a765": "R(\\theta )=\\left(b^2-a^2\\right) \\cos (2 \\theta -2 \\varphi )+a^2+b^2", "7686f0a2dab140b2ea7a744094401e0c": "A S \\mathrm{e}^{\\mathrm{i} \\mathbf{k}_{out} \\cdot \\mathbf{r}_{\\mathrm{screen}}}\n\\int d\\mathbf{r} f(\\mathbf{r}) \\mathrm{e}^{-\\mathrm{i} \\mathbf{q} \\cdot \\mathbf{r}} =\nA S \\mathrm{e}^{\\mathrm{i} \\mathbf{k}_{out} \\cdot \\mathbf{r}_{\\mathrm{screen}}} F(\\mathbf{q})", "768706e462e337db79233832d43c19fb": "(W_1,W_2,W_3,...)", "76870faeeb14d915c3ec7af88848db11": "\\sigma_2(n)=\\sum_{d|n} d^2", "76871b0dacbfc003348dc52f36acdb31": "r\\cdot x = \\eta(r)x", "768786e2fc71d3d75e5c04f31097c851": "gk = \\sigma", "7687a7bd221fce6514496c2051b31a40": "nB=\\underbrace{B\\oplus\\ldots\\oplus B}_{n\\mbox{ times}}", "7687aee12d65e2559cbf442962bafdc0": "M_x[a + x] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&1&0& \\cdots \\\\\na^2&2a&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "7688563d5b8c171c2ab100c760c03cd5": "I_z = I_x + I_y\\,", "76887ba65f2d80df9f060167a6b85ac8": "C(1) = X", "7688d545416c2b21a99f5ed26843f1b5": "\\left|\\left(\\frac{p}{q}+\\frac{1}{q^n}\\right)-\\left(\\frac{p}{q}-\\frac{1}{q^n}\\right)\\right|=\\frac{2}{q^n}", "768911820734272caed41c545afd219a": " \\frac{r}{r_s} \\left( \\frac{v}{c} \\right)^2 \\left(1 - \\frac{r_s}{r} \\right) = \\frac{1}{2} ", "76891472aad91557b6f24aaa1e5a9f2b": "\\mathbf{A} = A \\mathbf{\\hat{n}}", "76892ae9649c28fe34f2b75aec9a97f9": "\\lim_{\\varepsilon \\to 0} \\frac{1}{\\varepsilon} L = 6t(t-x) \\ . ", "76892ca412b842bcf43025dd269bd131": "\\ V_b ", "76897b5388ff341dd6db2924b1e16653": "\\gamma^{*}=(\\gamma(0)+\\gamma(1))\\beta - \\gamma ", "768a0c5a5bf57ce0689bc61689899412": "c_i \\equiv t_i \\mod vF^+", "768a1962f0560d887e81810d6fd59c62": "\\left[\\Lambda^{-1}\\right]_{ii}=\\frac{1}{\\lambda_i}", "768a75504a3fe5ba874434e4169eee2b": "\n(\\mathbf{V}_{\\mathrm{int}})_{kl} = \\langle \\psi^0_k | V_{\\mathrm{int}} | \\psi^0_l \\rangle =\n-\\mathbf{F}\\cdot \\langle \\psi^0_k | \\boldsymbol{\\mu} | \\psi^0_l \\rangle,\n\\qquad k,l=1,\\ldots, g.\n", "768a9f84e534bcc5cae46158d97d8ab8": "\\frac{\\exp(\\mu\\,t)}{1-b^2\\,t^2}\\,\\!", "768ac06aaf2c0283adf42ed8194ebe95": "H(x) \\approx \\frac{1}{2} + \\frac{1}{2}\\tanh(kx) = \\frac{1}{1+\\mathrm{e}^{-2kx}},", "768acd15ecf38ebeab9544a71b316710": "\nK(p) = {i \\over p_0 - \\sqrt{\\vec{p}^2 + m^2} + i\\epsilon} + {i \\over p_0 - \\sqrt{\\vec{p}^2+m^2} - i\\epsilon}\n", "768ae4cb4fac29fb31a12ff023620cd6": "R_X\\geq H(X|Y), \\,", "768b4168e74c8a51f07b907c765fbcb3": "\\scriptstyle{t'=t-{vx}/{c^2}}", "768b4437c1d2285aacd38049e6efec19": "\\frac{\\part^2\\ln \\Beta(\\alpha,\\beta)}{\\partial \\alpha^2} = \\psi_1(\\alpha)-\\psi_1(\\alpha + \\beta) > 0", "768b68f4ca9f25e5a3388cbc15827334": "B_{\\hat{m}\\hat{n}} = -a^3 \\, \\exp(a^2 r^2) \\, \\left[\\begin{matrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{matrix} \\right] ", "768b7b075ce34b95d014a1d92bf2378c": " U_P ", "768bbe16b32478d5742e328dfd6d82a1": "\\exp(iS)", "768c212dcf44fad53681678e24ce4f42": "\\mathbb{C}^{n \\times n} \\otimes A.", "768c4e404da131faa9da7cf4d31a4bc0": "I_\\mathrm{v} = 683 \\cdot \\overline{y}(\\lambda) \\cdot I_\\mathrm{e},", "768c73b0c14483f0bdcff0bf7fdb65cb": "\nN \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{6}{a b^{2}}\n\\frac{\\left( a^{2} - b^{2} \\right)}{a^{2} J_{\\alpha} + b^{2} J_{\\beta}}\n", "768d130a3556a36e4420f8958931b8df": "z=(z_1,\\ldots,z_n)", "768d498f08abc15dd6e69909b8b19dd0": "\nE\\left(X_j\\right)=\\frac{\\alpha_j}{\\alpha_j+\\beta_j}\\prod_{m=1}^{j-1}\\frac{\\beta_m}{\\alpha_m+\\beta_m}.\n", "768d71e07fcb5159b10ddf1558b95402": "\\left(-\\frac{2}{3},\\frac{1}{3},\\frac{1}{3}\\right),\\ \\left(\\frac{1}{3}, -\\frac{2}{3}, \\frac{1}{3} \\right),\\ \\left( \\frac{1}{3}, \\frac{1}{3}, -\\frac{2}{3} \\right)", "768d97d9a3ccd02c1c898d5f139b25bf": "f(k) \\cdot |x|", "768dc368a2e1c656dca1f72701d311a5": "\n P_{n}^{m}(x)\\ = (1-x^2)^{\\frac{m}{2}}\\ \\frac{d^n P_n}{dx^n} \\quad 1 \\le m \\le n\n", "768decba252d5602ddca990ebb230dfb": "\n M = M_1(x) + P_1\\langle x - a_1\\rangle + P_2\\langle x - a_2\\rangle + P_3\\langle x - a_3\\rangle + \\dots\n ", "768e06801de0e52f87e6920ea01288ac": "(P \\to (Q \\to R)) \\leftrightarrow (Q \\to (P \\to R))", "768e2b3f39093be4d31ce34aeb7ac69d": "\\frac{{3(\\sqrt{3}-\\sqrt{5}) }}{\\sqrt{3}^2 - \\sqrt{5}^2} = \\frac{ 3 (\\sqrt{3} - \\sqrt{5} ) }{ 3 - 5 } = \\frac{ 3( \\sqrt{3}-\\sqrt{5} ) }{-2}", "768e5512c26ff8bd4b0c20c59d9e4f4c": "k=\\frac{2\\pi}{\\lambda}=n\\frac{2\\pi}{L}", "768e67dc11de718307c5219b388ceb0b": "W_{U}\\,", "768e927e97d2b99a22d43818285bbddc": "\\varepsilon_{ij} \\varepsilon^{ij} = 2. ", "768ea7f1064b7fe35e6ef710d4dbcbb4": "L_{[\\omega]}^{n}: H_{DR}^0(M) \\to H_{DR}^{2n}", "768ed77f70904e3bd9dca26ce3854d9c": " Z", "768f3e9c865e62b0c6f4e6e57ebbf26a": "\\frac{k}{\\sqrt{\\pi}} x^{-1/2} \\exp(-k^2 \\log^2 x)", "768f45ba57d902de46d7cdc4ede16073": "m \\mu(a/a_0) a = GMm/r^2", "768f5fa88e4653bd96ad9479b13bd575": "\n\\begin{bmatrix}\nf_1(P_1) & ... & f_1(P_n) \\\\\n... & ... & ... \\\\\nf_k(P_1) & ... & f_k(P_n) \\end{bmatrix}\n", "768f7d3c61fda337b28e4494834822a7": "\\Omega \\,\\! ", "768f99d332efc957aa4444c56630ab66": "\\max_x \\min_{i=1,\\dots,N} R_{\\delta_i}(x). \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)", "768fe1d50bcbb330e2443f616d1a6928": "R_{HOxI,-19} = R'_{HoxI}\\left ( \\frac {1 + \\frac {-19} {1000}} {1 + \\frac {\\delta^{13}C_{HoXI}} {1000}} \\right )", "76900327fbd8155d97095679729581f1": "\n \\int_{C} \\left| f(x) e^{\\lambda_0 S(x)} \\right| dx < \\infty,\n", "76908ef2ab7a368f2e59e0a135a1fc90": "\n\\begin{align}\n\\langle \\partial_{t} u , e^{i k x}\\rangle &= \\langle \\partial_{t} \\sum_{l} \\hat{u}_{l} e^{i l x} , e^{i k x} \\rangle = \\langle \\sum_{l} \\partial_{t} \\hat{u}_{l} e^{i l x} , e^{i k x} \\rangle = 2 \\pi \\partial_t \\hat{u}_k,\n\\\\\n\\langle f , e^{i k x} \\rangle &= \\langle \\sum_{l} \\hat{f}_{l} e^{i l x} , e^{i k x}\\rangle= 2 \\pi \\hat{f}_k, \\text{ and}\n\\\\\n\\langle\n \\frac{1}{2} u^2 - \\rho \\partial_{x} u\n ,\n \\partial_x e^{i k x}\n\\rangle\n&=\n\\langle\n \\frac{1}{2}\n \\left(\\sum_{p} \\hat{u}_p e^{i p x}\\right)\n \\left(\\sum_{q} \\hat{u}_q e^{i q x}\\right)\n - \\rho \\partial_x \\sum_{l} \\hat{u}_l e^{i l x}\n ,\n \\partial_x e^{i k x}\n\\rangle\n\\\\\n&=\n\\langle\n \\frac{1}{2}\n \\sum_{p} \\sum_{q} \\hat{u}_p \\hat{u}_q e^{i \\left(p+q\\right) x}\n ,\n i k e^{i k x}\n\\rangle\n-\n\\langle\n \\rho i \\sum_{l} l \\hat{u}_l e^{i l x}\n ,\n i k e^{i k x}\n\\rangle\n\\\\\n&=\n-\\frac{i k}{2}\n\\langle\n \\sum_{p} \\sum_{q} \\hat{u}_p \\hat{u}_q e^{i \\left(p+q\\right) x}\n ,\n e^{i k x}\n\\rangle\n- \\rho k\n\\langle\n \\sum_{l} l \\hat{u}_l e^{i l x}\n ,\n e^{i k x}\n\\rangle\n\\\\\n&=\n- i \\pi k \\sum_{p+q=k} \\hat{u}_p \\hat{u}_q - 2\\pi\\rho{}k^2\\hat{u}_k.\n\\end{align}\n", "7690fd3d71c0442274ff1c4cd67ec5ce": "f_1(x)=\\cdots = f_k(x)=0,\\quad\\forall x\\in V.", "76910aac62ea40701a5e0ef587889848": "k_U=1", "769118f729d187c195c900f15abc7fd6": "u_*=\\sqrt{\\left(\\frac{\\tau_b}{\\rho}\\right)}", "76912a50322bf49773629bdaac264e76": "\\begin{align}\\mathbf{x} \\times \\mathbf{y}\n = (x_2y_4 - x_4y_2 + x_3y_7 - x_7y_3 + x_5y_6 - x_6y_5)\\,&\\mathbf{e}_1 \\\\\n {}+ (x_3y_5 - x_5y_3 + x_4y_1 - x_1y_4 + x_6y_7 - x_7y_6)\\,&\\mathbf {e}_2 \\\\\n {}+ (x_4y_6 - x_6y_4 + x_5y_2 - x_2y_5 + x_7y_1 - x_1y_7)\\,&\\mathbf{e}_3 \\\\\n {}+ (x_5y_7 - x_7y_5 + x_6y_3 - x_3y_6 + x_1y_2 - x_2y_1)\\,&\\mathbf{e}_4 \\\\\n {}+ (x_6y_1 - x_1y_6 + x_7y_4 - x_4y_7 + x_2y_3 - x_3y_2)\\,&\\mathbf{e}_5 \\\\\n {}+ (x_7y_2 - x_2y_7 + x_1y_5 - x_5y_1 + x_3y_4 - x_4y_3)\\,&\\mathbf{e}_6 \\\\\n {}+ (x_1y_3 - x_3y_1 + x_2y_6 - x_6y_2 + x_4y_5 - x_5y_4)\\,&\\mathbf{e}_7. \\\\\n\\end{align}", "76913266e57c28ce9f1762f92e8c1106": "\\mathrm{2 \\ S^0 + 3 \\ O_2 + 2 \\ H_2O \\longrightarrow 2 \\ SO_4^{\\,2-} + 4 \\ H^+}", "7691332444e92f87d7999b0621d66a3c": "{}_2F_1(1,b;c;z) = \\cfrac{1}{1 + \\cfrac{-b z}{c + \\cfrac{(b-c) z}{(c+1) + \\cfrac{-c(b+1) z}{(c+2) + \\cfrac{2(b-c-1) z}{(c+3) + \\cfrac{-(c+1)(b+2) z}{(c+4) + {}\\ddots}}}}}}", "76914aca5b40a4649adca6258d788c24": "Q_{ij}", "76914c99069c6a09b4709aa6876425ad": " d(n) = \\sum_{m p\\\\\n0.5 & \\text{if } q=p\\\\\n1 & \\text{if } q 0)", "76cb005fa17ad521914fe4b542d2a490": "Q = I_\\text{3} + \\frac{B+S}{2}", "76cb0d128525df1ecd0ad436a66be43c": " A[[x]] ", "76cb1787e43df96e85d033f019818681": " E_{ij}^\\text{dual} ", "76cb39ca92c52f3c16c2f58e9ee7cc99": "O(k n \\log (n/k) + z)", "76cbae19f123f60862750fb6c5f0773c": " |B| = -2 \\cdot \\begin{vmatrix} 4 & 6 \\\\ 7 & 9 \\end{vmatrix} + 5 \\cdot \\begin{vmatrix} 1 & 3 \\\\ 7 & 9 \\end{vmatrix} - 8 \\cdot \\begin{vmatrix} 1 & 3 \\\\ 4 & 6 \\end{vmatrix} ", "76cc23f24f0161fe9e69ff983d553ab4": "\\|Df^n v\\| \\le c\\mu^{|n|} \\|v\\|\\text{ for all }v\\in T\\Lambda\\text{ and }n \\in \\mathbb{Z}.", "76cc26bebc6e617928591aea45110a3c": "\\ \\frac{p_2}{p_1} = 1 + \\frac{2\\gamma}{\\gamma + 1}\\left[M_x^2 - 1\\right].", "76cccfbdf13e5caf46e6e8e52a6bd3aa": "d \\left( \\varphi(t_{n}, \\omega) x_{0}, a \\right) \\to 0", "76cce0a5592aaf4393553753b8296d59": "H_*(\\mathcal{X},\\partial) \\simeq H_*(\\mathcal{A},\\Delta)", "76ccee43836ec17e2c068238fba9ebd8": "\\scriptstyle \\sum k^3 = (\\sum k)^2", "76cd0a66c436d33856b17808b817d7d8": "\\beta N (S/N)I = \\beta SI", "76cd2073fbb8fd41916148433c5cd335": "x_{n_2} \\geq x_{n_1}.", "76cd57ace6bccb75f823611b195780e7": "{{\\text{ }\\!\\!\\varepsilon\\!\\!\\text{ }}_{4}}", "76cd781d757162f417678d085679214a": "0\\to F(C)\\to F(B)\\to F(A)\\to R^1F(C) \\to R^1F(B) \\to R^1F(A)\\to R^2F(C)\\to \\cdots", "76cda74a50f6a2e7749896c03f6fff1a": "\\begin{align} \nA_x & = - \\frac{1}{\\sqrt{2}} A_+ + \\frac{1}{\\sqrt{2}} A_{-} \\\\\nA_y & = + \\frac{i}{\\sqrt{2}} A_+ + \\frac{i}{\\sqrt{2}} A_{-} \\\\\nA_z & = A_0\n\\end{align} ", "76cdde9850cb22a859c14757f5e572a1": "\\left\\langle\\mathtt{halt},\\ v\\right\\rangle", "76ce06e4c04a06e31fb96e2a2fe7ce8a": "\nI = \\int dE | \\Psi_{E}\\rangle \\langle \\Psi_{E} |\n", "76ce8530c6a64ea415b8150609a4a4e3": "\\widehat{\\mu}_\\pi=\\widehat{\\mu}_m, ", "76ceb11be7c821f93ca57f87bfb485c0": "\\tfrac{P_{n-1}+P_n}{P_n}", "76cf0f8dfefde73acd1c1421aa09a3cc": "\\begin{pmatrix}r + 1\\\\2\\end{pmatrix}", "76cf32180bfd9d667461cfb2adb1d63d": "u = 0.995", "76cf653b2c63c39d0aa2f14417032d7b": "f_\\beta (t) = e^{ -t^{\\beta(t)} }", "76cf92e17fd763300607035269a2e032": "\\varphi = \\sqrt{1 + \\varphi}", "76cf9f9eafb2aca2915cdcd30c8f6c36": "\nA = -kT \\ln Q.\\,\n", "76cfc44a948bbe2576b3fd1540880f67": " \\mathsf{T} = (\\vec{\\omega},\\mathbf{d}\\times \\vec{\\omega} +\\mathbf{v})=(\\mathbf{T},\\mathbf{T}^\\circ),\\quad\\mathsf{W} = (\\sum_{i=1}^n \\mathbf{F}_i, \\sum_{i=1}^n \\mathbf{X}_i \\times\\mathbf{F}_i)=(\\mathbf{W},\\mathbf{W}^\\circ), ", "76cfcd2250eba6eb5b5a80037462e72e": "\\scriptstyle R_{xs}", "76cff78f43fb0e0e965dda7ebe8f0115": "\\forall x \\forall y (\\forall X_{\\in B} (Xx \\leftrightarrow Xy) \\rightarrow \\forall Y_{\\in A} (Yx \\leftrightarrow Yy))", "76cffbd6ac3a4b77cdf9ce08aab0f3e9": " \\varphi\\circ d\\colon \\Omega^0(P,V)\\rightarrow \\Omega^0(P,V\\otimes \\mathfrak g^*).\\,", "76d01ffecff78c91b0ddcafad0396602": "\\Psi(x) = e^{\\Phi(x)}, \\! ", "76d032c00b1ece1d5ffffe53386b02bf": "\\alpha_G", "76d04de18f6f13505fffbebd81216f86": "n \\approx \\sqrt{ 2 \\times 365 \\times 0.5} = \\sqrt{365} \\approx 19", "76d06fd0ffb3d1e3acbe26d80b95806b": "\\sqrt{3}\\times\\sqrt{3}", "76d07e503a507f6fdd9dc4c50fec51d3": "N_r < 10", "76d0db7c6857fcd1b75fb4caa4c5b44f": " (E - e\\phi) \\psi_+ - c\\boldsymbol{\\sigma}\\cdot \\left( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right) \\psi_{-} = mc^2 \\psi_+", "76d103ed1db0099f478984f1878b5911": "G = \\pi_1(Ff)", "76d15268f9e16aac83538c2d0d96154b": "(D,V,s,R)", "76d1534c5ebb1cb6eee8f1dab7688c06": "~\\Phi(T)=[T]_\\beta^\\gamma", "76d1920fb59c4e9740d8dd507314cc64": "2 \\pi r\\over\\lambda ", "76d1b1d48f1cab4aeec2b6688bc71305": "\\prod_{j=1}^n\\frac{\\Gamma(1+j\\gamma)}{\\Gamma(1+\\gamma)}.", "76d1f1af19f92b62873df23df230da63": "{<}{\\cdot} \\!\\,", "76d1f43387311a890b5cd709d35c0462": "D_0 = \\left(\\frac 12_A\\right) \\sum_{a, \\lambda} \\left( \\left| \\left\\langle A_\\alpha \\left| m_- \\right| J_\\lambda \\right\\rangle \\right|^2 + \\left| \\left\\langle A_\\alpha \\left| m_+ \\right| J_\\lambda \\right\\rangle \\right|^2\\right)", "76d1fca3467a82ffe977e101e4c29d2b": "a a_1 + b b_1 + c c_1 = 2K.", "76d21407487401bcd7b704829a5a0c3e": " f_b(r)=b, ~~r< w, f_b(r)=0, ~~r>w ", "76d2329b2aedb2c6d53870d341ccb25b": "\\left(a_1 b_1 - a_2 b_2 - \\frac{a_4 u_1}{b_1^2+b_2^2} - \\frac{a_3 u_2}{b_1^2+b_2^2}\\right)^2\\,", "76d2730cc8f455db0d1a381ec962cae8": "\n \\eta = a(x_0,t_0)\\; \\cos \\left[ k_0\\, x_0 - \\omega_0\\, t_0 - \\theta(x_0,t_0) \\right],\n", "76d29556d1b58758aa90babe25d76143": "w_2\\,\\!", "76d2d45782b217a1e36a57d7bcd101e8": "\\left(\\frac{-1}{n}\\right)=\\left(\\frac{-1}{p}\\right)\\left(\\frac{-1}{q}\\right)=(-1)^2=1", "76d2dd9334b1153462ca38d78fc114f8": " k(x,x') = \\exp\\left(-\\frac{1}{2\\sigma^2} ||x-x'||^2 \\right) ", "76d2f15ce0154b98818dfb764a51c6da": "g-1", "76d313be8cea5cb2022d7f487e42b54e": "i = \\frac{V}{A}", "76d3273123c05c33b7c4c5d111ee61d7": " P(E) = \\sum_i P(E|S_i) P(S_i) \\leq \\sum_i P(X_1^n(i)) \\left ( \\frac{1}{M} \\sum_{i'} \\left ( \\frac{P(X_1^n(i'))}{P(X_1^n(i))} \\right ) ^s \\right ) ^ \\rho \\, . ", "76d336a769d60f47bfa6067ace4ca2be": " M_i(n)=\\prod_{j=1}^n \\mu_i(j) ", "76d3e4fd726f0795f7b5e445827e37cc": "f : X \\to \\mathbb{R}^{k}", "76d3fe1eab6b3fe3fd6c324feed2bf51": "(g,a)(h,b)=(g\\phi(a)(h),ab)", "76d43982395fce40dcfd0767789a1c10": "Px^a + Qx^b + Rx^c", "76d44dba0ef46e05647a0996491cbbe0": "\\mathbf{w} =\n \\begin{bmatrix}\n \\alpha & 0 & 0 & 0 \\\\\n m\\alpha & (n - 3m)\\alpha & m\\beta & m\\beta \\\\\n 0 & 0 & \\beta & 0 \\\\\n m\\alpha & m\\alpha & m\\beta & (n - 3m)\\beta\n \\end{bmatrix}\n", "76d4660b9de537032355bbcd177605c9": "\\Omega\\sim(t_0-t)^{-\\frac{1}{6}}", "76d512879d96482003067c47c99ec60e": "n!+1 = m^2,", "76d539d2089b6818e937299e96c15f80": "\\pi\n(t,p_{-i})-\\pi (0,p_{-i})", "76d55d9d9373c78a12e379a7a1f4b90a": "w \\models \\Diamond P", "76d5929182d502c60593426ede0b6bc5": "\\mathcal{L}=-\\frac{1}{4}(\\partial^\\mu A^\\nu-\\partial^\\nu A^\\mu)(\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu)+\\frac{1}{2}(\\partial^\\mu \\phi+m A^\\mu)(\\partial_\\mu \\phi+m A_\\mu) ", "76d59609d3a5a03f8fd12e5e62447dd5": "p_k=\\begin{cases}[kp_{k-1}-(k+1)p_k]\\frac{m}{m+1} & \\text{for } k\\geq 1 \\\\1-p_0\\frac{m}{m+1} & \\text{for } k=0\\end{cases}", "76d59a8df53c05507cfb57c0c027e464": "\\frac{(2N)!}{k!(2N-k)!} p^k q^{2N-k} ", "76d5f434bc4d7a84ffed9bb95da5bf75": "\\exists T \\subseteq X\\times X ", "76d64ebbd63a1a4822bceedbfb78b5b3": "\\operatorname{Der}_k(F, F) = 0", "76d67507b89329825b83f21f973b62ef": "\\alpha_1,\\dots,\\alpha_n", "76d69d3f68b4705bb19692b72300310b": "\\Delta x \\Delta p \\ge \\frac{\\hbar}{2} ", "76d6a32801d3ca51b87440d091a87464": "\n \\left.M_{rr}\\right|_{r=a} = \\frac{qa^2}{8} ~,~~\n \\left.M_{\\theta\\theta}\\right|_{r=a} = \\frac{\\nu qa^2}{8} ~,~~\n \\left.M_{rr}\\right|_{r=0} = \\left.M_{\\theta\\theta}\\right|_{r=0} = -\\frac{(1+\\nu) qa^2}{16} \\,.\n", "76d6ca2553243995cae7d43bdc94f41d": "H_i = E_{m_i}{(H_{i-1})} \\oplus {H_{i-1}}.", "76d6e071f46d0b8f71055f8cd07a500b": "T_{JMAX} = 125 \\ ^{\\circ}\\mbox{C}", "76d6e7842bd41df0832ecb94e3ac2463": "\\boldsymbol{r_{21}}=\\boldsymbol{r_1-r_2}", "76d71a1a207d15d9e608d5126fe5de81": "\\scriptstyle \\leq1.3\\times10^{-6}", "76d72600ecb3cbecfd6320cbe7eb200b": "\\tilde{U} \\tilde{V} = \\left(1-\\frac{r}{2GM}\\right)e^{r/2GM}.", "76d74046162a546a333c0adbb32508d2": " H(x,p) ", "76d755ee91b5e7a279e654843bac3c74": " T_m(0) \\neq 0 , \\quad m=n, \\ldots , 0 ", "76d75ffa50b6623809a8ca289527a534": "C_p(p,T)\\ ", "76d7645081f98c6e3d582e2098a48f2f": "F\\, =\\, \\int_{-h}^0 f^2\\; \\text{d}z", "76d7d80721ae9a2fb41ff087c49fd731": "\\displaystyle{W(F)W(G)=W(F\\star G),}", "76d7d8f10ba0d435dd85331020910601": "r = \\rho\\ - %dMU*\\dot c \\,", "76d7e353b34db618e4645fcef033e5ab": "AB=T_{ab}P_{ab}^{'}+E_{ab} \\,", "76d7f75586138c751349618d44d5430e": " \\begin{bmatrix} x_1, x_2, \\dots, x_m \\end{bmatrix}^{\\rm T} ", "76d80dcecce90ad8bbb51ce80204ebb0": "a_{\\mathit{wf}}", "76d82b95abb2207b97a1e3b15ab578a7": "\\mathrm{CaSO}_4(s) \\rightleftharpoons \\mbox{Ca}^{2+}(aq) + \\mbox{SO}_4^{2-}(aq)\\,", "76d851b9eacc4ee46b1f5913e701f992": " r\\ddot{r} + \\dot{r}^2 \\ge 1.", "76d85b546c9994e48fe0a26fc422dc65": "s(g\\sigma_0)=\\exp(F) ", "76d8b7648be0421c359894199714dfce": "\\left( \\gamma^0 \\right)^2 = I_4 \\,", "76d8df2870289c53cd34155a94273798": "l_1=S7(r_0)\\oplus LS7(r_1)\\,", "76d906e1c73ce785fafc5b9f6dcb9119": " \\frac{G^{ex}}{RT}= \\frac {A_{12} X_1 A_{21}X_2}{A_{12} X_1 +A_{21} X_2} ", "76d90d77f5ffc4787a4d26c6dd0eb37b": "\\ v(x,t) = \\dot{x} = \\dot{Q}(t)^T[x^*-c(t)] + Q(t)^T[v^*-\\dot{c}(t)]. ", "76d970095c93d5011b2dbfab01a9b25b": "y_i = \\beta_1 x_i + \\gamma_1 z_i + u_i ", "76d974a271d41e4508d69bfece894862": "X=(X_0 , X_1 ,\\dots , X_{M-1})", "76d9c4175b9568f3e91aa1e736027dfe": "k=1.0004", "76d9fc6ea7434fa372bbf25df77e042b": "M_2=\\frac{0.0300-31.4424x+30.0717y}{0.0241+0.2562x-0.7341y}", "76da0849018bcada43662fe8eb3bfbf8": "g_{ij} = \\frac{\\partial \\bold{r}}{\\partial q^i}\\cdot\\frac{\\partial \\bold{r}}{\\partial q^j}", "76da186cd42f0d273179483684e49dc9": " z(x,y) = ax + by ", "76da1dc8ec86443db4b4276bae60aa0a": "\\hbar/m_{\\mathrm{e}} c", "76db24da6668cbaddb2de4633985e6d6": "\\ dU=(\\partial U/\\partial x)dx + (\\partial U/\\partial y)dy ", "76db2682ca7fdf59969783cfd698e596": "k^{O(1)}\\log |V|", "76db344752317e68404a685b095bd2eb": "\\displaystyle x_0+x_1+x_2+x_3+x_4 = 0", "76db36fae5024597adfec14848a9f3cc": " \\hat{A}\\hat{C} = (A, B)(C, D) = (AC, AD+BC).\\!", "76db58a3a94c80c1357c559b32d1c871": " \\bold{p}_1^\\prime + \\bold{p}_2^\\prime = m_1\\bold{u}_1^\\prime + m_2\\bold{u}_2^\\prime = \\boldsymbol{0} ", "76db6b803fd08acbc82b0ebd17dd6435": "\\mathrm{A}(\\theta,\\Phi) = \\frac{\\lambda^{2} \\mathrm{G}(\\theta,\\Phi)}{4 \\pi}", "76db80b9acf6fbf9d4b21a888e957ca5": " \\alpha + 1 ", "76dba58fbd144dde10b505c21192d0f2": "\\frac {1}{1+\\beta^2} \\ll 1", "76dbd05db778b1022795198c0e0f67f2": "g:U\\to \\mathbb C,", "76dbd312c41176128675c4ea626328ce": " (1+z)^{\\alpha} = \\sum_{n=0}^{\\infty}{\\alpha\\choose n}z^n = 1+{\\alpha\\choose1}z+{\\alpha\\choose 2}z^2+\\cdots.", "76dbf7045c779e5d2c7f9e5a987b7825": "\\textstyle (C,\\; r)", "76dc4268c16765befdf265d9ff5b6daa": "r = 1 + e \\cos \\theta,\\,", "76dc46939ae3ed49d4a9e2fd96940fb8": "S^6", "76dc5baabda86400ba97781f7f2b4179": " \\delta(x) = \\frac{\\sigma(x)-x^p}{p} ", "76dc5f4377a191feec587348654c099e": "z \\mathbf{X}(z) = A \\mathbf{X}(z) + B \\mathbf{U}(z)", "76dca455c041d1c07d7509983d0b7cb7": "\\arctan \\left(\\frac{d_2-d_1}{d_1}\\right) - \\arctan \\left(\\frac{d_2-d_1}{d_2}\\right).", "76dd3ccd5b6423375c8e515f7e3cbc2a": "\\int\\frac{S}{x}\\,dx =\n\\begin{cases}\n 2 \\left( S - \\sqrt{b}\\,\\mathrm{arcoth}\\left( \\frac{S}{\\sqrt{b}}\\right)\\right) & \\mbox{(for }b > 0, \\quad a x > 0\\mbox{)} \\\\\n 2 \\left( S - \\sqrt{b}\\,\\mathrm{artanh}\\left( \\frac{S}{\\sqrt{b}}\\right)\\right) & \\mbox{(for }b > 0, \\quad a x < 0\\mbox{)} \\\\\n 2 \\left( S - \\sqrt{-b} \\arctan\\left( \\frac{S}{\\sqrt{-b}}\\right)\\right) & \\mbox{(for }b < 0\\mbox{)} \\\\\n\\end{cases}", "76dd8c26f9f3d5e9cc10dea5f148425e": "C_\\mathrm{dry\\, basis} = \\frac{C_\\mathrm{wet\\, basis}}{1 - w}", "76ddae094c0586e403c873040622415e": "\\alpha_p = p_\\infty/\\tau_p", "76de0074a67d4971ef15c6e295ea8d1b": "\\mathrm{im}(\\partial_{n+1})", "76de06cf0ef2d77534fa3652b8e6d9da": "M: S \\to \\mathbb{N}", "76de90326fb52c0db68755bd24dd898c": "(x_1-a_1), (x_2-a_2),\\ldots,(x_n-a_n)", "76dece1f520d9dba5fd0bee97c21e7ea": "c_1,c_2 \\subset V", "76ded0710557304ca3ee2987cb205332": "U = -m/\\sqrt(x^2+y^2+z^2)", "76deee94008f39f81260f7885151bdaf": " \\quad T(au+bv)=T(au)+T(bv)=aT(u)+bT(v) ", "76df02997eda55c9c7aaf2fd54d8188a": " (\\lambda x.f\\ (x\\ x))\\ (\\lambda q.\\operatorname{de-let}[f\\ (q\\ q)]) ", "76df105d5628729366e115caa27020b2": "\\int x^3 r \\; dx = \\frac{r^5}{5} - \\frac{a^2 r^3}{3}", "76df1c9b7cd093ee6ea81c81521b4b6b": "\\mathrm d \\varphi_x:T_xM\\to T_{\\varphi(x)}N\\,", "76df30cb2f3c2675e057e25429917f27": "h[n]", "76df82e96b31f14e1995350066520b2d": "t _1[\\alpha] = t _2 [\\alpha] = t _3 [\\alpha] = t _4 [\\alpha]", "76dff99274fd90f3969316026b348d1b": " \\begin{align}\nu(x) &= f_0(x) \\qquad && \\text{for all } x\\in S; \\\\\n\\frac{\\part^k u(x)}{\\part x^k} &= f_k(x) \\qquad && \\text{for } k=1,\\ldots,\\kappa-1 \\text{ and all } x\\in S,\n\\end{align} ", "76e0d946a573dc3004c20d2adaedd6a1": "y = C\\cdot e^{\\frac{x^2}{2}}", "76e117c80d7cad8574f7f24ac8367829": "a^{n/m} + b^{n/m} = c^{n/m}", "76e13a2b2737b957329010db35f89c96": "\\ln \\left( \\frac{ P_2 }{ P_1 } \\right) = - \\frac{ \\Delta H_{ vap } }{ R } \\left( \\frac{ 1 }{ T_2 } - \\frac{ 1 }{ T_1 } \\right)", "76e162566fcd466c0d157b8a0cdfcdce": "m_{\\pi}^4 \\pi^2 + (\\partial \\pi)^6", "76e1ab2e9429405162f3e09fbc0488b3": "p \\times q = \\textstyle\\frac{1}{2}(pq - q^*p^*).", "76e20abd9e0ced1c7054dcf14dd694c4": "\\Phi_{02}:=\\frac{1}{2}R_{ab}m^a m^b\\,, \\quad \\Phi_{20}:=\\frac{1}{2}R_{ab}\\bar{m}^a \\bar{m}^b=\\overline{\\Phi_{02}}\\,,", "76e23157b6bdd7176fd1c3699d9b0215": "D = {\\sigma^2 \\over \\mu }.", "76e254115d9a4c3dfd97422271628a8b": "\\Delta C_p", "76e2f022635a61f5b9b82ffa9ee52ece": "U(s,p)", "76e2f12cf88bc97e371134fe3cc2bea0": "\\frac{\\sin(B-C)}{x} + \\frac{\\sin(C-A)}{y} + \\frac{\\sin(A-B)}{z}=0. ", "76e2fd57e4ff7d660366a709ec5e2300": " c^{2s}(H^{s}|_{E})=\\infty", "76e37a2fe056da842f700256ca22bff4": "\\nabla\\cdot\\mathbf{v} = 0", "76e3bf6c215f5dfa7dbfc9ccffffd07c": "S=\\partial D ", "76e3fbe6f9b34e6d5f9f917d5bc0d846": "\\gamma_{\\pm}", "76e3fe6a0cccb7578ee6a405c5dec983": "\\bigstar \\bigstar |||\\bigstar", "76e42ef94c9c59a3a713496d174d0201": " D = 2 \\sum_{ a = i }^K \\frac{ d_a }{ K - 1 }", "76e4ab84762aa04634ac80a019cb62fa": " \\displaystyle{f(x) ={1\\over (2\\pi)^{n/2}}\\int_{{\\mathbf R}^n} \\widehat{f}(t)e^{ix\\cdot t}\\, dt}", "76e4b8be2a4a30a35e12172db97bdcf3": "\\scriptstyle n\\equiv a_1 \\pmod m_1", "76e4c8a46a6a153261cbfe772f972998": "L(\\bar{x}; \\Sigma_t) = \\int_{\\bar{xi}} I_L(x-\\xi) \\, g(\\bar{\\xi}; \\Sigma_t) \\, d\\bar{\\xi}.", "76e4ee122064a61338872f9286ca8015": " \\Delta_r G = \\sigma \\mu_S^\\ominus + \\sigma RT \\ln a_S + \\tau \\mu_T^\\ominus + \\tau RT \\ln a_T -(\\alpha \\mu_A^\\ominus + \\alpha RT \\ln a_A + \\beta \\mu_B^\\ominus + \\beta RT \\ln a_B)=0", "76e4ef80bf37aeefcc128d9326d2d97c": "0=u'_1y'_1+u'_2y'_2+\\cdots+u'_ny'_n", "76e4fd4cd65f941500899d2874d1d605": "\\mathbf{B}(\\mathbf{x},t)=\\mathbf{\\nabla}\\times\\mathbf{A}(\\mathbf{x},t)", "76e584ad23a6346b9f76ab35ab1c3dd2": "\n\\left(\\frac{a}{p}\\right) = \\left(\\frac{b}{p}\\right).\n", "76e59f309d3161d88fe3558c3c2acabb": " \\left \\{ 1, e^{ix}, e^{-ix}, e^{2ix}, e^{-2ix}, e^{3ix}, e^{-3ix}, \\ldots \\right \\}.", "76e5d080a638d6ef664f1cc159e8b019": "\\beta_{ij} = 0", "76e6c16c56765a741a59f00701a33613": "10^{60},", "76e7205c0be12cb039dc4bde362cc878": "u \\rightarrow v", "76e777eb44a1ceeddea68decb3379dee": " H(s) = \\frac{s^2}{s^2+\\underbrace{2\\pi\\left(\\frac{f_0}{Q}\\right)}_{2 \\zeta \\omega_0 = \\frac{\\omega_0}{Q}}s+\\underbrace{(2\\pi f_0)^2}_{{\\omega_0}^2}}, ", "76e79935f6ced130ac75fa90a4365ed5": "Z_{CO}^2", "76e7fe6e1938d6e46a5fba50cf0e8bae": "\\ G_{u,v} =\n \\frac{1}{4}\n \\alpha(u)\n \\alpha(v)\n \\sum_{x=0}^7\n \\sum_{y=0}^7\n g_{x,y}\n \\cos \\left[\\frac{(2x+1)u\\pi}{16} \\right]\n \\cos \\left[\\frac{(2y+1)v\\pi}{16} \\right]\n", "76e80a73d7d563243884efbad3c70c68": "\nP/\\alpha ", "76e81f7b40c645c79dfa100ea6eaa7bf": "(S, \\bar S)", "76e8325b8ddd89977f727695e6c7009f": " f^-(x) = \\left\\{\\begin{matrix} -f(x) & \\text{if } f(x) < 0 \\\\ 0 & \\text{otherwise} \\end{matrix}\\right. ", "76e83b1042dd33b2fef4a1b5b521e7fc": " P_3 = \\frac{1}{2}( 1 + \\mathbf{e}_3), ", "76e87bb5155bcb65da0df7da8102cf67": "|D\\chi_E|(A) = \\mathcal{H}^{n-1}(A \\cap \\partial^* E)", "76e896819bbf7a52ea6934cee71771d1": "\\psi^{0}", "76e8994a88833097a30f0d85aaa320a9": "Sales = \\frac{A}{Y}", "76e8ab439090a769c325bfbbf98dccc4": "\\forall z\\in M_2:\\exists x\\in M_1: d_2(z,f(x))\\le C.", "76e9a5c890ac7d3f1782766da0aa50fc": "V_1 \\, ", "76e9e79747bd5e62f8653d72d79e5588": "\\Delta H_\\text{L} =-\\boldsymbol{\\mu}\\cdot\\boldsymbol{B}.", "76e9eba9f7e80024a285a0260843e495": " \\mathcal{F}\\{f * g\\} = k\\cdot \\mathcal{F}\\{f\\}\\cdot \\mathcal{F}\\{g\\}", "76ea2cd4433c5c2e514fb2ae1d7e9ccc": "\\nabla_\\perp=\\nabla-\\frac{\\mathbf{B}}{B}\\cdot \\nabla", "76ea61235c08eb06eda37961a9c70108": "\\mathbf{x}_j ", "76ea84a26c7a9a2ca0b7d3ab2b3c8de8": "W = \\Delta G =NkT_o \\Theta(V/V_o)", "76ea8870f50ef452c529565bba92f60e": "2\\arctan\\frac{1}{\\varphi}\\approx63.43495", "76ea8ea094a5fe968889e11895433ea1": "R_{\\sigma\\nu}", "76eaac3a89723c12fb401b80a66bc5d3": "\\widehat{A_f}", "76eaee90c92b6481fcb7e2548b09c657": " (W/m^2) ", "76eb8855dbf453a59e6ee36c1da98758": "f(n,k) = \\sum_{d\\mid k} g(n,d)", "76eb8b3d20e612cd5436feb152eb0420": "\n\\begin{align}\n\\mathbf{\\pi} & \\sim \\operatorname{SymDir}(K, \\alpha_0) \\\\\n\\mathbf{\\Lambda}_{i=1 \\dots K} & \\sim \\mathcal{W}(\\mathbf{W}_0, \\nu_0) \\\\\n\\mathbf{\\mu}_{i=1 \\dots K} & \\sim \\mathcal{N}(\\mathbf{\\mu}_0, (\\beta_0 \\mathbf{\\Lambda}_i)^{-1}) \\\\\n\\mathbf{z}[i = 1 \\dots N] & \\sim \\operatorname{Mult}(1, \\mathbf{\\pi}) \\\\\n\\mathbf{x}_{i=1 \\dots N} & \\sim \\mathcal{N}(\\mathbf{\\mu}_{z_i}, {\\mathbf{\\Lambda}_{z_i}}^{-1}) \\\\\nK &= \\text{number of mixing components} \\\\\nN &= \\text{number of data points}\n\\end{align}\n", "76eba06a4684137390858eb8f836d6ab": "U_k(\\omega) \\to \\left(\\frac{i\\omega}{\\omega_c}\\right)^2", "76ec053788165461d126052ee9d40e33": "x\\ast x=0", "76ec0c59a47c1c8a34b7f62078f93081": "C_1,\\ldots,C_n", "76ec9b1e03a4d00dd843bc390b55dce6": " n\\ge m", "76eca7417361409b06661e26640215cd": " { \\alpha_{1},\\alpha_{2},\\alpha_{3}.....\\alpha_{k} } ", "76ecb389e5224ea8b60a515c4765be66": "AP \\cdot AQ \\, ", "76ecd2cf55f36c74b86b6b9d6341a905": "j \\sqrt{\\frac{1}{2}}", "76ed164a0d43b0abf4716d25e1d673c8": " f(x).", "76ed7c2588ad9cdfdb6f1d656523c0a7": "Z_t=\\sum_i e^{-\\beta\\epsilon_i}k_i(\\epsilon_i,t,t_i).", "76edab26640d3001a5d684b6906a7578": "C \\to B", "76eddf7ca1b1fc14f4b58577d547a89e": " |r|_{\\infty} \\prod_{p} |r|_p = 1", "76eec61db3f7c550becd09a98b18cab9": "\n\\log(1+u)=\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} u^n\n= u - \\frac{u^2}{2} + \\frac{u^3}{3} - \\cdots \\,\n", "76eedfbcb2bcef2243ea325bf820666b": "\\mathcal{STUVWXYZ} \\!", "76ef297b29b71dca57af59355d3bcf2a": "N_{\\text{corr}}", "76ef6391bdcfa4f52aa22a54f4cb9ac9": "u\\colon U\\to \\mathbb{R}", "76ef858d9f5dc3f911a05639e3ff5071": "p_1,...,p_r \\le p", "76f019f9e4c5d42b0f7a21f3381ccf95": "f(x) = 0^x", "76f0473706198bc1f8d79bfeaf3f2840": "(n+1)^2 = n^2 + O(n)\\ ", "76f0516b8bf86de4ee2e8558bf90f5c0": "\\bar {x} = \\sum_{i=1}^n {w_i x_i}", "76f069595e4e7a7d59cc5f24bb32d3c9": "\\textstyle||\\mathbf{x} - \\mathbf{x'}||_2^2", "76f093e9f53dff11102af7cd1ced7c6c": "\n\\operatorname{E}(T) = n \\cdot H_n = n \\log n + \\gamma n + \\frac1{2} + o(1), \\ \\ \n\\text{as} \\ n \\to \\infty,\n", "76f0c2bd9df997354b9c8b2b535e0ddb": "A = 0.05", "76f0c63a7494e7baa61713d8345b7c4c": "\\gamma \\not= \\pm i", "76f2c510f92f5a2a592807d7bbf8fede": "2^{-j}", "76f305c4650450d6450f4154df5e6f70": "{\\tilde{G}}_2", "76f308a5f75dfbfc9b479f3e65923ec9": "2.1620", "76f38ff0089ee998bd84f20dc3eba9ea": "11g_{k_2+1}, \\cdots, 11g_{k_3}, 01g_{k_3}, \\cdots, 01g_{k_2+1}, 00g_{k_2+1}, \\cdots, 00g_{k_3}, \\cdots,", "76f39663d4057d35bf37048c696ae5aa": "\\mathrm{\\frac{Q}{L^2 t T}}", "76f451f3fad7b92e65f18426495c9e2b": "T_0^1(M)=T(M) =TM ", "76f4a9539ef417c2461d18967241f675": "\\beta\\!", "76f4b2153e0e2203af141a277678a28b": "\\det(\\mathsf{f} \\circ \\mathsf{g}) = \\det\\mathsf{f} \\det\\mathsf{g}", "76f4d742274acd059eb7f1acfb52157e": "\\boldsymbol{x}_{k} = f(\\boldsymbol{x}_{k-1}, \\boldsymbol{u}_{k-1}, \\boldsymbol{w}_{k-1})", "76f4dd8cadeed77f3c3713830bf56bbc": "\\sigma_y\n= -2\\frac{\\partial^2 B}{\\partial z \\partial x}", "76f4edcb62fd052773ee9243f5e11ad5": "\\theta^i", "76f57867f47773534d482417d763bb98": "e(n)=d(n)-\\mathbf{x}^{T}(n)\\mathbf{w}_n", "76f59835f72c8c26acef2870b26a3a94": "\\scriptstyle ( \\frac {1} {3} \\pi R^2 D ).", "76f5b25c00e86693db0f377d732c124e": "\\gamma(t)\\rightarrow t^{-\\beta}L(t)", "76f6277086ac346b9b6e3ca315b9a73d": "\\sigma = (U_i)_{i \\in \\{ 0 , \\ldots , q \\}}", "76f62cf5f8ae7885e4b38e7e741458b6": "2, \\sqrt{5}, \\sqrt{6}, 3", "76f63f2eb5de68b244c43ba12ba62c83": "\\scriptstyle\\frac{1}{\\zeta}", "76f647d069fddafbe4ff80192f949496": " L_u=2L_v=2L_w=1750 \\text{ft}", "76f65b378f5c644d7fdb5dabc18c1fa1": "r_e = v_{Te}/\\omega_{ce} = 2.38\\,T_e^{1/2}B^{-1}\\,\\mbox{cm}", "76f6ae7bd7191dc917996d34ad38229b": "\\lambda\\colon H_k(M;\\mathbb{Z}_2)\\times H_k(M;\\mathbb{Z}_2)\\to \\mathbb{Z}_2", "76f6d03cfd71531380a9f5bb74d12829": "\\mbox{dist}\\big(\\gamma(t_1),\\gamma(t_2)\\big) = A|t_1 - t_2|.", "76f72bf24c87052a626fdb8e90a8b04c": "s \\log_2 a\\leq t \\log_2 b \\leq (s+1) \\log_2 a \\, ", "76f75cc60fe0810eba16b196e732ec6a": "\\int_{[0, t)} \\Delta (\\mathrm{d} z_{n_{k}}) \\to \\delta(t);", "76f7805dd1b6e1bc679bce893eafe82b": "f'(x_0) > K/2.", "76f79e3dadaec1b256d1f8dcb4226574": " y = y_1 + u ", "76f7c2aa4d4d28a4f5527ed5cf8a4d4a": "2d = N\\lambda,\\qquad\\qquad N \\in \\{1,2,3,\\dots\\}", "76f7dc71a1e5ac55c240c327f5a0d145": "L/F", "76f888f6cf082c62e43e6e9e37093f1e": "K\\subset X", "76f88ba010d8c28007b6dc5631ad0a77": "\\Delta E_1 = \\left[\\frac{1}{2} \\rho_1 v_1^2 + \\Psi_1 \\rho_1 + \\epsilon_1 \\rho_1 + p_1 \\right] A_1 v_1 \\, \\Delta t", "76f8d4552e60ed888ee2cbe90e455060": "\\eta(2n) = (-1)^{n+1}{{B_{2n}\\pi^{2n}(2^{2n-1} - 1)} \\over {(2n)!}}. ", "76f93876279d88af145ee2b800a8d6bd": "f(x) = \\mathbf{E}^{x} \\left[ f \\left( X_{\\tau_{D}} \\right) \\right] - \\int_{D} A f (y) \\, G(x, \\mathrm{d} y).", "76f96611d090c0b77cd9ee51338560d7": "s > 1", "76f9adf5c056c090c3baf7ec9d1873a6": "\\Delta v\\ = V_e \\ln \\frac {m_0} {m_1}", "76f9dbd233967600752a116893ee9e37": "\\displaystyle \\hat{f}(\\omega - 2\\pi a)\\,", "76fb040e5a0c58a2450363fa13470178": "\\scriptstyle{T_\\alpha^0}", "76fb15c6f118fc5b3962d5adc521b4a0": "\\frac{B_{21}}{B_{12}}=\\frac{g_1}{g_2}", "76fb56ad50d4cb3385b182bf2f10831a": "G(z) = E[z^X].\\,", "76fb60e774fe17f32a0a19c734f840c7": "x_3 = \\frac{x_1y_2+y_1x_2}{1-(x_1x_2)^2}", "76fb6c6addcba590df37a952a43d8ce8": " (X(t))_{t \\in \\mathbb R^+} ", "76fba9eda41d92705ae721b12c61ffc2": "\\mathrm{rot}\\,\\mathbf{A}\\,", "76fbd72e339a27f1e6782d4699107b92": "y=\\sqrt{1-x^2}", "76fbdda561e34d086968fe0280191c07": "v'(t) = \\beta(t)\\,v(t),\\qquad t\\in I^\\circ,", "76fc5029572c09b24993da9aca0a89bb": "\n\\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^m (b+2 c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^p}{2c (m+2p+1)}\\,+\\,\n \\frac{m(2 c\\,d-b\\,e)}{2c (m+2p+1)} \\int (d+e\\,x)^{m-1}\\left(a+b\\,x+c\\,x^2\\right)^pdx\n", "76fc8725a21ba35342934e121242d50a": "q > 0", "76fcc0c1b5cd12da10710727d9077826": "\\pi(\\mathcal{A})", "76fce83b215db93ac3ae553821a3d5b1": "\\theta \\sim G(\\theta|\\alpha)", "76fd156a90b4e2543a69d7566e80918e": "\\dot{V}_1\n= \n-W(\\mathbf{x}) + \\mathord{\\overbrace{\\frac{\\partial V_x}{\\partial \\mathbf{x}} g_x(\\mathbf{x}) e_1\n- e_1 \\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})}^{0}} - k_1 e_1^2\n= -W(\\mathbf{x})-k_1 e_1^2 \\leq -W(\\mathbf{x})\n< 0", "76fd452e3a651f49f53e29cde3e758fd": "\\mathcal{L}(n) = [n \\ge m]\\frac{m - 1}{\\binom{n}{2}}", "76fd6e358314591202c2f971703cdaf6": "A = \\frac{0.75a-0.25}{5.5a+b+1.5}", "76fda57b72ff2f055eb5cec796606ad5": " c_H ", "76fda5c2037384799da264aceb310d47": "X^N", "76fdc225d2f2211e1519863b802d6709": "E(X)=\\mu", "76fdde9e82afaef8cf7564268c1af0c9": "\\tilde{\\varphi}:\\mathbb{S}^\\lambda E \\to M", "76fde016456ef850d409a7881c5da52e": "T_R=\\frac{R_0+v_i[R]}{v_0+v_i}", "76fdf49119e394063c0afd2a8a856de7": "\\frac{dx'}{dt'}=\\frac{dx}{dt}-v", "76fe6a25d1dda80aa96b8982bb014132": " D^\\alpha = \\partial^\\alpha_{x}/i^{|\\alpha|} ", "76fe6acae119149d77a9455299ac29e5": " f(x) = 0 ", "76fef6f9a387983c35015e7d2602bb7b": " v\\cdot v > 0", "76feffd816ca680b63a036b22ff88cf5": "E( R_i ) = R_f + \\beta( RP_m ) + RP_s + RP_u", "76ff2030d3132b3f08ddadeaa74822cd": "Y>m_N", "76ff3142b4a11f4aa4dc518164a74d63": "F_t\\ \\hat{t}", "76ffac0095d8955f9b17246cee8d7c5f": "TM= \\frac{1}{2}\\left(Q_2 + \\frac{Q_1 + Q_3}{2}\\right)", "76fff42bddebaf8aef22f5876615ecaa": "\\lambda:S \\rightarrow Y^\\phi", "76fffb2905acd6b7032705d404149a08": "\\,\\! \\text{mag}_{AB} = -2.5 \\log_{10} f_{\\lambda} - C", "77000675e460d047b316bd01b4593b57": "H=G A_+ G,", "77005b905790829e0df948ba1e64298d": "ln Y=ln A + \\alpha ln L + \\beta lnK + ln\\varepsilon", "77009e276e949b9feca323e421dba795": "\nR^2 = 1 - 2V\n\\,", "7700f3794790c4541f9f3541f3a3c9e6": " |x| \\leq \\|x\\| \\leq (1+\\epsilon)|x| \\quad \\text{for every} \\quad x \\in E.", "77018b0dff67a48fb96b0154d1546dc1": " P\\approx 170", "7701a39268e06966e2dbd6b72bcf2971": " \\begin{align} \n&\\lim_{\\alpha\\to 0} \\gamma_1 =\\lim_{\\mu\\to 0} \\gamma_1 = \\infty\\\\\n&\\lim_{\\beta \\to 0} \\gamma_1 = \\lim_{\\mu\\to 1} \\gamma_1= - \\infty\\\\\n&\\lim_{\\alpha\\to \\infty} \\gamma_1 = -\\frac{2}{\\beta},\\quad \\lim_{\\beta \\to 0}(\\lim_{\\alpha\\to \\infty} \\gamma_1) = -\\infty,\\quad \\lim_{\\beta \\to \\infty}(\\lim_{\\alpha\\to \\infty} \\gamma_1) = 0\\\\\n&\\lim_{\\beta\\to \\infty} \\gamma_1 = \\frac{2}{\\alpha},\\quad \\lim_{\\alpha \\to 0}(\\lim_{\\beta \\to \\infty} \\gamma_1) = \\infty,\\quad \\lim_{\\alpha \\to \\infty}(\\lim_{\\beta \\to \\infty} \\gamma_1) = 0\\\\\n&\\lim_{\\nu \\to 0} \\gamma_1 = \\frac{1 - 2 \\mu}{\\sqrt{\\mu (1-\\mu)}},\\quad \\lim_{\\mu \\to 0}(\\lim_{\\nu \\to 0} \\gamma_1) = \\infty,\\quad \\lim_{\\mu \\to 1}(\\lim_{\\nu \\to 0} \\gamma_1) = - \\infty\n\\end{align}", "7701a81402c80ae88539da3042196ef5": "f: \\mathbb{R}^n \\to \\mathbb{R}", "7701b9985a7a8b73451532a04ab64f4c": "\\forall y (F(y, z_1, \\dots, z_n) \\rightarrow y \\in V) \\rightarrow \\exists x (x \\in V \\land \\forall y (y \\in x \\leftrightarrow F(y, z_1, \\dots, z_n))).", "7701f777a02327b15a0f8020461c2352": "\\{a^n b^n c^n d^n | n > 0\\}", "7701f921802cea736935ed3e2cea1dd5": " p_6 =\\frac{ (a_{20}\\omega+a_{11})(a_{10}-\\mathcal{L}a_{20})-a_{20}(a_{01}\n -\\mathcal{L}(a_{20}\\omega+a_{11}))}{2a_{20}\\omega+a_{11}}.", "7702421aa088d49a8d65b873b933f6e6": "P(N=k)_{k=1,2,\\cdots} ", "77030eb03f7ab796158c136f46bc2eba": "[\\partial a_\\lambda b]=-\\lambda[a_\\lambda b], [a_\\lambda \\partial b] = (\\lambda + \\partial)[a_\\lambda b],", "77031744c26968012db8aa98155b714c": "B = \\frac {\\mu_0 I_D}{2 \\pi r}\\,", "77032c2304afd358ba3a26f169850c04": "{\\sum_{w\\in W} \\varepsilon(w)e^{w(\\rho)} = e^{\\rho}\\prod_{\\alpha \\in \\Delta^{+}}(1-e^{-\\alpha})}.", "7703481cbf2591f7d160ab66d0da884a": "\\bar{\\bold{3}}", "7703d8f23786ff6b4576d4a63123df81": "\\hat{1}", "7703eefa9e37d1b02800191c9f6ebea0": "n_0\\,u_0 = n_\\mathrm{i}(x)\\,u(x)", "7703f31cc059161cb783749bb9a7c739": " \\mbox{diff}(M) \\times \\mbox{diff}(\\mathbb{R}),", "7703f59dd3565d04c3d2ed50758dcc5d": "a-x_1", "7703fc7020d8e257e12bc8a31fcb55bd": "\\Delta I_{\\text{L}_{\\text{On}}} + \\Delta I_{\\text{L}_{\\text{Off}}}=\\frac{V_i \\, D\\, T}{L}+\\frac{V_o\\left(1-D\\right)T}{L}=0", "77044e0bc12b6b42bd1513c3f9814b0c": "X <_{HYP} Y <_{HYP} \\mathcal{O}^X", "77046b62c2a801c5ece06bf7f9f32add": "k < n", "7704981ba739bbda792525bd734de5a5": "i = \\cos\\left (\\frac{\\pi}{2}\\right ) + i\\sin\\left (\\frac{\\pi}{2}\\right )", "770519ba9e6870d2095a45e3ae5ceb63": "UH |a\\rangle = U E_a|a\\rangle = E_a (U|a\\rangle) = H \\; (U|a\\rangle). ", "770522b2c42673603c76238523baef67": "\\{3,4\\} ", "7705317553a276ee082489dd681bf4cf": "\n R = \\cfrac{\\epsilon^p_{\\mathrm{xy}}}{\\epsilon^p_{\\mathrm{z}}}\n ", "77054c4d089da2a2a4e4be77ae55e1f5": "\\frac{(\\tau^2\\nu/2)^{\\nu/2}}{\\Gamma(\\nu/2)}~\n\\frac{\\exp\\left[ \\frac{-\\nu \\tau^2}{2 x}\\right]}{x^{1+\\nu/2}} ", "77056022c0ae96a92559eb648f854784": "\\forall a, b \\in X,\\ a R b \\; \\Rightarrow \\lnot(b R a)", "77057e4443fae7691dbd0eafea416cdf": "E[F|x^{(t-1)}]", "77058a6c313b551fa2f75f1993186cfb": "f(a) = 0 ", "7705a93ba753242bf32f359c16c6fe7d": "\\ +(j \\omega) ^2 C_C C_L (R_A//R_i) (R_O//R_L) \\ . ", "7705ff1ed602a106f61e10b53a06ad32": "y(5)", "770615ba4504b5420cdec89ff4f547db": "f\\colon X \\to \\mathbb P^N, ", "770683e5f27d461f38b061a7775a59d2": " {{\\partial \\varphi } \\over {\\partial t}} + c{ { \\partial \\varphi } \\over {\\partial x}} = 0 , \\quad t > 0, \\quad x \\in \\mathbb{R} \\quad \\quad (10)", "770710ee52b01751439c0e08b6297496": "0 = 0^2", "770761b98698d2812d89ca8de4b4449a": " \\beta = \\tfrac{v}{c} ", "7707663698585a51eb60693ca9a72647": " \\epsilon _{h,l} = - {{1} \\over {2}} \\gamma _{1} k^{2} \\pm [ {\\gamma_{2}}^{2} k^{4} + 3 ({\\gamma _{3}}^{2} - {\\gamma _{2}}^{2} ) \\times ( {k_{x}}^{2} {k_{z}}^{2} + {k_{x}}^{2} {k_{y}}^{2} + {k_{y}}^{2}{k_{z}}^{2})]^{1/2}", "7707e04036cb2314edc7bdd738e47fe9": "\\Gamma, R", "7707f17f034f582e9e7cb2148b8baac0": " G = \\bigcup_{t \\in T} HtK ", "770805bc9427dfeba6e1adf5a8f8b2b4": "(\\Omega,Pr)", "7708b0807c2180bcdbdfd6168920a6d5": "\nn\\# = \n\\begin{cases}\n 1 & \\text{if }n = 1 \\\\\n n \\times ((n-1)\\#) & \\text{if }n > 1\\ \\And\\ n \\text{ is prime} \\\\\n (n-1)\\# & \\text{if }n > 1\\ \\And\\ n \\text{ is composite}.\n\\end{cases}\n", "7708ccb5d48ccd18fea0d16437094d7d": " \\frac{\\mathrm{d}^2 x}{\\mathrm{d}t^2} = -\\left(\\frac{k}{m}\\right)x,", "770908f3d178caa5ce732be4a22f7a0f": "\n\\begin{align}\n-\\mathbf{q}^2&=-|\\mathbf{p}_+|^2-|\\mathbf{p}_-|^2-\\left(\\frac{\\hbar}{c}\\omega\\right)^2+2|\\mathbf{p}_+|\\frac{\\hbar}{c}\n\\omega\\cos\\Theta_+ +2|\\mathbf{p}_-|\\frac{\\hbar}{c} \\omega\\cos\\Theta_- \\\\\n&-2|\\mathbf{p}_+||\\mathbf{p}_-|(\\cos\\Theta_+\\cos\\Theta_-+\\sin\\Theta_+\\sin\\Theta_-\\cos\\Phi),\n\\end{align}\n", "770976aaffef06cc779d07d65c455dbe": "\n \\frac{\\partial\\tilde{\\rho}}{\\partial t} +\n \\langle\\rho\\rangle\\nabla\\cdot\\tilde{\\mathbf{v}} = 0\n ", "7709f31137122fd67003d910a9bc9c8a": "multiplier = \n\\frac{1+r\\frac{delivery term days}{currency basis}}{1-y}", "770a8cdc453f979afb6a797aff76edaa": " k_B T", "770a9fe760bd9666bfee5daafa743804": "\\begin{cases}\ny = mx+b_1 \\\\\ny = -x/m\n\\end{cases}", "770aea2aa2c66d9d62d8331c207d851b": "\\begin{align}\n &m = Y^\\prime_{601} - (.30R_1 + .59G_1 + .11B_1) \\\\\n &(R, G, B) = (R_1 + m, G_1 + m, B_1 + m)\n\\end{align}", "770b3847d33860f37bc39c56c2bae3a6": "A^{\\ast}", "770b51f9cb96e266d924b97900595e45": "\\{x\\in\\mathbb{C}^2:|x|=1\\} \\, ", "770b7908a539820ed843de2fb900a1b2": "J^{\\alpha\\beta} = 2X^{[\\alpha} P^{\\beta]} + \\frac{1}{m^2}\\varepsilon^{\\alpha \\beta \\gamma \\delta} W_\\gamma p_\\delta \\quad \\rightleftharpoons \\quad \\mathbf{J} = \\mathbf{X}\\wedge\\mathbf{P} + \\frac{1}{m^2}{}^\\star(\\mathbf{W}\\wedge\\mathbf{P})", "770bb3c34a266880187a0532f4989858": "P_3=(x_3,y_3)", "770bc7eded1762180d2d3390cf2f6692": "D_{5}", "770bd7482e75da82d20eb07cf74ce1ee": "B_k^+={1\\over{\\sqrt{N}}}\\sum\\limits_{A=1}^N\\exp(ikr_A)a_A^+,", "770bd7e53366c716777204681bfefeff": "+\\frac{1}{8}(2g^{\\mu \\alpha }g^{\\nu \\beta }-g^{\\mu \\nu}g^{\\alpha \\beta })(2g_{ \\sigma \\rho }g_{\\lambda \\omega}-g_{\\rho \\lambda }g_{ \\sigma \\omega})(\\sqrt{-g}g^{ \\sigma \\omega}),_{\\alpha }(\\sqrt{-g}g^{\\rho \\lambda }),_{\\beta })", "770bf6d546ff66d297cbadb323036747": "C_\\mathrm{perfect-CSI} = E\\left[\\max_{\\mathbf{Q}; \\, \\mbox{tr}(\\mathbf{Q}) \\leq 1} \\log_2 \\det\\left(\\mathbf{I} + \\rho \\mathbf{H}\\mathbf{Q}\\mathbf{H}^{H}\\right)\\right] = E\\left[\\log_2 \\det\\left(\\mathbf{I} + \\rho \\mathbf{D}\\mathbf{S} \\mathbf{D} \\right)\\right]", "770c059f79af074b1cfffd87fdf9d47c": "w \\neq 0", "770c48ad66a6fdad99c65da38e77aa29": " \\eta_{a \\mu \\nu} = \\epsilon_{a \\mu \\nu 4} + \\delta_{a \\mu} \\delta_{\\nu 4} - \\delta_{a \\nu} \\delta_{\\mu 4} ", "770ce6fbded2643e8ab2356c339973c3": "f^{-1}[y, +\\infty)", "770cf990471b55b72f7e30a0bf93e8c1": "c^i,b_i", "770d0122aa93550ad7956843048c7ba1": "\\Delta(X^n) = \\sum_{k=0}^n \\dbinom{n}{k} X^k\\otimes X^{n-k},", "770d0a2417951be517aebc7b56bcafdf": "\\ m,n ", "770d296d3f6cdf5eea2944013a67830e": "\n\\int_{\\phi(a)}^{\\phi(b)} f(x)\\,dx\n", "770d353fdf3453229260ae430d08ffea": " \\cos \\theta = \\frac { \\bold{x} \\cdot \\bold{y} } { \\left\\| \\bold{x} \\right\\| \\left\\| \\bold{y} \\right\\| } = \\frac { 0.308 } { \\sqrt { 30.8 } \\sqrt { 0.00308 } } = 1 = \\rho_{xy}, ", "770d8315c61880815e3ebf0c154b34e0": " a_W + a_E - S_P ", "770daf9dde6ff43f4dda76a3e4366db3": "\\mathrm{Im}(z)=\\frac{z-\\bar z}{2i}\\;", "770dfdcae185fe25eab0afec88703155": "Y'=\\exp(-X)", "770e1583794d2cf79155082b1c422e61": "F_X(a) - F_X(a-0) = \\lim_{T\\to\\infty}\\frac{1}{2T}\\int_{-T}^{+T}e^{-ita}\\varphi_X(t)dt", "770e30ed472103268d75f5e4b267c44b": "c=\\mathrm{st}(x_{i_0})", "770e546f65682a0895d1f2ba10a5aaab": "\nY_i(x(t)) = \\sum_{n=1}^Na_{in}x_n(t) - b_i \\text{ } \\forall i \\in \\{1, ..., K\\} \n", "770eb76f77dc972482b97914463b805d": "\\Sigma_k^{\\rm P} = \\Sigma_{k+1}^{\\rm P}", "770edea1cb4cf645c0f53afec5eb2bac": "(10-x)^2=81 x", "770f2fb0a6eb94fbdc8d2775228c2523": " P( x ) = P( x - 1 ) \\frac { k + x - 1 } { x } \\frac { m k^{ -1 } } { m k^{ -1 } - 1 } ", "770f30364d8797d4b5ecbcb4e7a995d1": "V_q(\\mathbf{R})=\\frac{1}{4\\pi \\epsilon_0} \\frac{1}{|\\mathbf{R}|^3} \\sum_{i,j} Q_{ij}\\, n_i n_j\\ ,", "770ff191c16c231ff685d9d47d183572": "\\exp(-Dz/2)w(z)", "771001d4a770f89109bfee21457ce36e": "(s_1, s_2) \\in \\to", "77107d79770907505fa5c10f5a61712e": "\\phi_1 / \\phi_2 = g_1 / g_2 ", "7711fd8d60ba9c48f3debd1f06873fc8": " X_{1,2}X_{3,4} - X_{1,3}X_{2,4} + X_{2,3}X_{1,4} = 0", "77123c1041dfb0d44b1e2e3d33e73944": "\\frac{\\pi}{\\sqrt{m}\\, K(m)}\\, \\operatorname{sech}\\, \\left( \\frac{5\\, \\pi\\, K'(m)}{2\\, K(m)} \\right) = \\tfrac{1}{256}\\, m^2 + \\cdots.", "7712d8bc807e6cfd474a2ed42ea567c4": "p_{n+1} = p_n^2 + 2 q_n^2 \\, ", "7712fa2b485f57eee0475416707e431f": " P=\\frac{1}{N-1}HA\\left( HA\\right) ^{T}+R, ", "7712fe330e29db4540ee772c86fdb470": "\\partial A = A \\cap \\partial M", "77130caeb454c7aac8ec02d90677f490": "(1 \\; 2) \\; (2 \\; 3) \\; (3 \\; 4)", "7713ec3941e6a9c69686bdf569b67553": "\\scriptstyle k \\;=\\; \\frac{1}{ab}\\left(a^2 \\,+\\, b^2 \\,+\\, 1\\right)", "771403ad48fc4b6fa1470e74ddf144d6": "S_{\\phi}", "771412d875da4e6e6e4b2e53f701a92d": " w ( u \\wedge v) = w \\cdot ( u \\wedge v) + w \\wedge u \\wedge v", "771445d67c4f145c924d92e657618bb1": "|{\\psi_{{T}}}\\rangle=e^{-iH_nT}|{\\psi_0}\\rangle", "77146f766784b3cc4ba5da08fcf280aa": "r_m = \\frac{R_m}{2 \\pi a\\ }", "77147f228e44539aab5c9adaae4f7da4": " \\begin{align}\n\\frac{dx}{dt} & = Ax + By \\\\\n\\frac{dy}{dt} & = Cx + Dy \n\\end{align}", "7714831af07ae2aab34c71b2de068433": "\\chi - i \\kappa = \\frac{\\xi }{\\sqrt{\\varepsilon \\mu}}", "77149147779f60aa51dbbe73d993f0b7": "x^2=4ry\\sin{\\theta}", "7714fa1f6b09841f8ca6fde1f1e2918f": "{\\log}\\circ f", "7715146c09107fe06ce3c8db9c4a4d20": " C(V) = \\{ f\\in \\hat T(V): \\hat \\Delta(f) \\in \\hat T(V)\\otimes \\hat T(V)\\}. ", "7715c74dbf699f1e4dd889bc568412a4": "\n \\left| x- \\frac{p}{q} \\right|>\\frac{c(x, \\varepsilon)}{q^{2+\\varepsilon}}\n", "7715e44e7595b521e5df743aa24860dc": "\\begin{matrix}{n-1 \\choose l-1}\\end{matrix}", "7716484366d45dbe64077a6daaa2513e": "\\underline{3.2898}41960364(17) \\times 10^{15} \\text{Hz} = R_\\infty c", "7716487a2ad44c3431c6a3f220df7bde": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A+\\alpha \\mathbf{I}} \\right) =\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( A\\right)+\\alpha.", "77164eec547d090905ac64f23d9d3e5a": " \\left(r^2 {\\partial\\over \\partial r} \\right) R(r) = \\left[(n - 1) r - \\zeta r^2 \\right] R(r) ", "771658c24808335dbe512164da6acfbc": " \\Delta f = \\nabla^2 f = \\nabla \\cdot \\nabla f ", "77167d0a696183fc3bc26668dcea3d15": "\\prod_{1\\le i 1\n\\end{cases}", "7719da226f6935e141e8c65cbe088d83": "\\frac{\\partial S(\\mathbf{x},t)}{\\partial t} = -\\left[ V + \\frac{1}{2m}(\\nabla S(\\mathbf{x},t))^2 -\\frac{\\hbar ^2}{2m} \\frac{\\nabla ^2R(\\mathbf{x},t)}{R(\\mathbf{x},t)} \\right]. ", "771a1e3512dd6c76c73e56d898ae3706": " X(t) ", "771a3c45f33970e3edbc5f446c8ba015": "\\ell^2(\\Gamma)", "771a9353e8719692503c3b641f7f736e": "x+y+z=0.\\,", "771aa2765ddce51423756a2633b0b1e9": "\n\\mathbb{A} = \\begin{pmatrix}\nATM_{vega} & RR_{vega} & BF_{vega} \\\\\nATM_{vanna} & RR_{vanna} & BF_{vanna} \\\\\nATM_{volga} & RR_{volga} & BF_{volga}\n\\end{pmatrix}\n", "771aa868b0a28fd0f2db224f97da796c": "(yx)x = y(xx)", "771aabedbb74b6086928b61402fd871d": "\\Lambda^k(V\\oplus W)= \\bigoplus_{p+q=k} \\Lambda^p(V)\\otimes\\Lambda^q(W).", "771aad3a2e0d146cf7c16761115ac4ca": " \\mathbf{u}_2 = \\mathbf{v}_2 - \\mathrm{proj}_{\\mathbf{u}_1} \\, (\\mathbf{v}_2) = \\begin{pmatrix}2\\\\2\\end{pmatrix} - \\mathrm{proj}_{({3 \\atop 1})} \\, ({\\begin{pmatrix}2\\\\2\\end{pmatrix})} = \\begin{pmatrix} -2/5 \\\\6/5 \\end{pmatrix}. ", "771b4b4b3873ed2f59cb15481b02a375": "\\eta \\ge 0.95", "771b6481a22eb2d88ff218721cb91bcf": " \\frac{r(1-p)}{p} \\, ", "771b666611f1f718a7c6805c99e0d1da": "\\sum_{m=-\\infty}^{\\infty} h[m] \\, z^{(n - m)} = z^n \\sum_{m=-\\infty}^{\\infty} h[m] \\, z^{-m} = z^n H(z)", "771ba53ec5ebbd2ade675f239ab1b49e": "f(0) \\le f(0+y) - f(y) = 0", "771bb9b17fe0597cf7fde53d6fdc0549": " \\tilde{a}_{\\alpha}=\\sum (-1)^{|\\alpha+\\beta|}\n\\binom{\\alpha+\\beta}{\\alpha}\\partial^\\beta(a_{\\alpha+\\beta}).\n", "771c004a329692ef7a3f0007f9023043": " (T_0/2,T_1,\\cdots,T_n) ", "771c26c01eabf5641740002c0f909072": "\\ C_{rr} = 0.01 ", "771c4c987f0a0582bc3953769a315e02": "I = mr^2", "771c8048ea584064bbf88c340de5c54d": "\n\\mu", "771c939d5061e333fc47747eda450d79": "\\sin(x)=\\frac{e^{ix}-e^{-ix}}{2i}", "771e07b49a3f79fd02c1f5e6bd18d824": "\\sum_{i=1}^{m}{a_i}=\\sum_{j=1}^{n}{b_{m+j}}", "771e93f21124514c6892db2225ed943f": " A(\\epsilon )=\\frac{\\Gamma \\left( \\frac{3+\\epsilon }{2} \\right) }{\\Gamma \\left( 1+\\frac{\\epsilon }{2} \\right) } ", "771eb2784caaa55f798bbc4382ae2a1a": "(x^3-6x+2)^6(x+2)^4(x-1)^4(x-4).\\ ", "771edf473b6704b15e6c8b5a28e7a8cc": "f((x + y)/2) = (f(x) + f(y))/2\\,\\!", "771f01a0343e5de54f2f79d55371750e": " \\text{MTTFd} \\approx \\frac{B_{10d}}{0.1n_{op}},", "771f3a93e959a905d2cf1c68e84e226e": "\\sqrt{g/L}", "771f4366df8e413789001be8432e4594": " \n \\begin{align}\n \\mu &= \\frac{\\alpha}{\\alpha+\\beta} \\\\\n M &= \\alpha+\\beta\n \\end{align}\n ", "771f8f458864ec6caa6e4fc601234feb": "\n\\ddot{y} + 2\\Omega_0 \\dot{x} =f_y", "771fe64b00e985ac47886b895540940e": "\\begin{align}\nx &= \\gamma \\left( x' + v t' \\right)\\\\\n\\end{align}", "77202677233bdf17fb950cc8fd5af10a": " \\begin{align}\nH_X &= \\frac{1}{\\operatorname{E}\\left[\\frac{1}{X}\\right]} \\\\\n &=\\frac{1}{\\int_0^1 \\frac{f(x;\\alpha,\\beta)}{x}\\,dx} \\\\\n &=\\frac{1}{\\int_0^1 \\frac{x^{\\alpha-1}(1-x)^{\\beta-1}}{x \\Beta(\\alpha,\\beta)}\\,dx} \\\\\n &= \\frac{\\alpha - 1}{\\alpha + \\beta - 1}\\text{ if } \\alpha > 1 \\text{ and } \\beta > 0 \\\\\n\\end{align}", "77203ad2d75f817f2a0ad3a0a7986d5a": " H_n = Q_n^* A Q_n. \\, ", "77205f3caf98f8fd10739cf0eb966bc0": "u\\in \\R^n ", "7720707f9fd4ffef7308ee55ede312f3": "\\nabla^2 \\varphi_f = 0 \\ . ", "7720e9e5a8210b7d968cb4270f8a96ec": "(10\\uparrow)^n a", "7721605cc2a078b2b5a6553ae32d49e8": "\n \\gamma = \\sum_{k=2}^\\infty (-1)^k \\frac{ \\left \\lfloor \\log_2 k \\right \\rfloor}{k}\n = \\tfrac12-\\tfrac13\n + 2\\left(\\tfrac14 - \\tfrac15 + \\tfrac16 - \\tfrac17\\right)\n + 3\\left(\\tfrac18 - \\dots - \\tfrac1{15}\\right) + \\dots\n", "7721b64fc7477ba619209da33daf82de": "p(x) = a_0 + x(a_1 + x(a_2 + \\cdots + x(a_{n-1} + a_n x)\\cdots)). \\, ", "7722b94a8ef7b8eb0870fe36c009bb57": "\n\\left(\\nabla \\times \\mathbf F\\right)_k =\n\\frac{h_k \\hat{ \\mathbf e}_k}{H}\n\\epsilon_{ijk}\\frac{\\partial}{\\partial q^i}\\left(h_j F_j\\right)\n", "7722dcc23aa731d20ab240e4dbf343b9": " 0 = a_{11} \\, x_{2k} \\, y_{1k} - a_{21} \\, x_{1k} \\, y_{1k} + a_{12} \\, x_{2k} \\, y_{2k} - a_{22} \\, x_{1k} \\, y_{2k} + a_{13} \\, x_{2k} \\, y_{3k} - a_{23} \\, x_{1k} \\, y_{3k} ", "7722e9a3457cf68e04b75505a4a48f21": "\\{F_i;\\,i\\in I\\}", "77237ada6cc5233bdca18af25351d9ed": "\\mathbf{A}=\\begin{pmatrix}1 & m\\\\ 0 & 1\\end{pmatrix}", "7723a8dba790267ba69780e447980804": "\n \\begin{matrix} X_k & =\n& \\sum \\limits_{m=0}^{N/2-1} x_{2m}e^{-\\frac{2\\pi i}{N} (2m)k} + \\sum \\limits_{m=0}^{N/2-1} x_{2m+1} e^{-\\frac{2\\pi i}{N} (2m+1)k}\n \\end{matrix}\n", "7723d5a15e6ea0015fa7a60a0b325a5f": "g_{\\alpha\\overline{\\beta}} = g\\left(\\frac{\\partial}{\\partial z^\\alpha},\\frac{\\partial}{\\partial \\overline{z}^\\beta}\\right).", "7723dfc8c57e3e5cef3c77f6952924a4": "f_\\alpha\\,\\!", "77242e0fd4bca5cd8f55e1ddab74666b": " y = \\cos(4\\pi t)\\sin(2\\pi t), 0\\leq t \\leq 1 ", "7724536a082c10314846d0c0305f3437": "\\sin(z)/z,", "772472b506293d414cb28ebc049eaa6f": "E_{gate}(t-\\tau) = E(t-\\tau)", "77247bb2d1c980d68507576adc90272e": "\\sum_{r} \\tilde{P}_{r}\\log\\left(\\frac{\\tilde{P}_{r}}{P_{r}}\\right) \\,", "7724b0f1600c10fe8d78a977cc0eb998": "C \\in \\{ 2^{-5}, 2^{-3}, \\dots, 2^{13},2^{15} \\}", "772546c996d158d25b54774fd0617e66": " \\mathrm{Hom} (U \\otimes V, W) \\cong \\mathrm{Hom} (U, \\mathrm{Hom}(V, W)).", "7725630af661de60e6c451f2ac85c09d": "\\ F=k\\cdot x^{1.5}", "7725f8f56f0a2007fb617b62a784803b": " \\frac{p^2}{a^2} + \\frac{z^2}{b^2} =1 .", "77262f8a1128c4fde74e404f8aa1303f": "x^4(x^2+y^2)-(ax^2-b)^2=0, \\,", "7726303c3a1798b542add262af014c0d": "p(x,y|\\theta)=p(y|\\theta)p(x|y,\\theta)", "772642129f53ed712da59ea12ea4bcaf": "F(t) = m\\frac{d^2x}{dt^2}.", "772694656c1d19450ac1012929b07f6e": " s_i", "7726999efab1e71eba52eb6fd7bc8d09": "\\mathcal{H}_Q", "7726f0fbb2bf62e0618a51e43d43462c": "\\vert \\psi \\rangle \\approx \\sum_i \\vert A_i \\rangle_\\mathcal{S} \\vert R_i \\rangle_\\mathcal{R}", "77272ff279680c9bea935289e04e23ed": "a\\rightarrow 0", "77278440836d8994598f778b09ea92d3": " n! = \\int_0^\\infty e^{n\\ln x-x} dx = e^{n \\ln n} n \\int_0^\\infty e^{n(\\ln y -y)} dy.", "7727a4c76a6ec6cafe016215af3f2db0": "g(x)=c+\\sum_{j\\in Q}x^j", "77283decd2cf0f1398a00ab5f8b4b754": " \\mathcal{N}\\left( \\boldsymbol\\mu_n, \\sigma^2\\boldsymbol\\Lambda_n^{-1} \\right)\\,", "77286af979877e0420e845c17888b931": "M=m_{0}+t", "7728d05d95ba2b1ef1a49601d536eef3": "\\text{sample variance(X)} =\\bar{v} = \\frac{1}{N-1}\\sum_{i=1}^N (X_i - \\bar{x})^2", "77291872a007d0416c4d7c2f18eff727": " I_1,\\, I_2,\\, \\dots ,\\, I_n ", "772988dd63969d722e04a614761965e5": "T = \\frac{2}{s}\\left\\langle K\\right\\rangle _{t}", "7729a0f0eb16ea4600a0a791df53b379": "F^m \\times G^n", "7729a176b65b6f765e02c0946a7fe185": "\\Omega_n = \\aleph_n", "7729a5f887459aad4e4a1ca5c5697b38": "\n\\text{span}\\{ u_1, \\ldots, u_k \\}\n", "7729f7de69acf681c362575dc2b374af": "p_i=1", "7729fd5b6a4e21192654c59dc7619b40": "\\textstyle \\sum_{i=1}^k \\pi_{i} = 1", "772a03e778b5e185845ab74c3b7002cc": "J^1_1Q", "772a3c42a0e0fc97bbe5a4ebdf6373b2": "RG(x,y,z) =\\sum_F x^{r(G)-r(F)}y^{n(F)}z^{k(F)-bc(F)+n(F)}", "772a7ada0f53787ef962662a023f1e77": " \\Phi=\\Phi_0+\\delta \\Phi \\,", "772b6972c20cd8e0354ed605269792d9": "^{N}{\\!\\!\\mathbf{\\omega}}^R", "772b94581a36ba6f0b59997175e44424": "R\\!", "772ba6b73554d867e8d5a238d2784a67": " g(z, u) =\n\\exp \\left( z + \\frac{z^2}{2} + \\frac{z^3}{3} + \\cdots + \nu \\frac{z^{n+1}}{n+1} + u \\frac{z^{n+2}}{n+2} + \\cdots \\right)", "772bcbaa30baff2a6c299e537773f850": "\\mathcal{N}(\\mu_2, \\sigma_2^2)", "772bd459943e3bdc3ece327742da3507": "O_f", "772c42aac1237a34232d4f315e7f61cc": "x(t)|_{t=0} = 0: C_1+C_2+\\frac{Q_0.1-R_0.1}{P}=0", "772cce7ea8a90fcf847e43276256e1ad": " f(\\mathbf{x}) := \\int_{\\gamma[\\mathbf{a}, \\mathbf{x}]} \\mathbf{F}(\\mathbf{u}) \\cdot d\\mathbf{u} ", "772e0d7ab9ffca0f9e30f0e19509ef85": "X_k e^{-i 2 \\pi k\\ell/N} \\,", "772e2f9d36a7e3b37a28cfef9c962135": "\\tilde V_x = | V_x |", "772e397427fd061ffff0e028650cc138": "\\forall x \\forall y [ \\forall z (z \\in x \\Leftrightarrow z \\in y) \\Rightarrow x = y]", "772e615f782c40ec03d83499bfd8fe97": "min \\, max_{0\\le i \\le n} \\, g_i(F_i)", "772e84794159589e3c0e9c1e5f0b6353": "L = \\frac{N \\Phi}{i},", "772f1fcba5c4a3224f3e1164082aaecd": "D e_\\alpha = \\sum_{\\beta=1}^k e_\\beta\\otimes\\omega^\\beta_\\alpha.", "772faf41ab66a7a33945aa138f4df1d3": "B_n(p,q)", "772fcc5e8412528600f335e91d72962e": "\\delta(f(t')) = \\sum_i \\frac{\\delta(t' - t_i)}{|f'(t_i)|}", "772fd3c2b67de5e4f1b928976205b1e2": "(f(t_{11}, \\dotsc, t_{m1}), \\dotsc, f(t_{1k}, \\dotsc, t_{mk}))", "7730025fc0893f7cd7b04f4121e31de9": "\\alpha = e/{\\gamma}", "7730422ebf5087dd5c4d125c3cafddf2": "A_n \\simeq A \\otimes_k k_n", "773057c4af83589b91832c5ff8a0b4ac": "\\begin{align}\nA & = \\left ( 20 \\cdot \\frac32\\sqrt{3} + 12 \\cdot \\frac54\\sqrt{ 1 + \\frac{2}{\\sqrt{5}}} \\right ) a^2 \\approx 72.607253a^2 \\\\\nV & = \\frac{1}{4} \\left(125+43\\sqrt{5}\\right) a^3 \\approx 55.2877308a^3. \\\\\n\\end{align}", "7730d170047632f49a4386b0e0ce3bc3": "\\Phi' = (\\forall x')(\\exist y') Q(x',y') \\wedge (\\forall x)(\\forall y) (\\forall u)(\\exists v)(P) ( Q(x,y) \\rightarrow \\psi )", "77310e9983cf250a3f891df877f47727": " \\frac{\\beta N}{\\gamma} > 1 \\Rightarrow \\lim_{t \\rightarrow +\\infty}I(t)=\\frac{\\beta N - \\gamma}{\\beta} ", "77315fff3f3a7cc28d0bf858739eab3f": "l=\\frac{y_2-y_1}{x_2-x_1}", "7731ea23b00bc3ddd2f169ed257a93f9": "\\omega_f", "773240ac7a48359875b1c7fc9f5230a3": "F \\subseteq 2^X", "7732475b4d46b49b9d166a3e11b4d485": "\\theta \\in [q_i, q_{i+1}]", "773247a6b010f77ace718740b9c89c79": "\\left[ \\frac{\\tau_{I1}\\tau_{I2}}{1-e^{-2\\gamma}r_{I1}r_{I2}} \\right].", "77324c6163c5e99b5b4670e36a236e60": " \\mathrm{Tr} \\, \\sigma_{\\mathbf{f}}(U) = {\\mathrm{det}\\, z_j^{f_i +N -i +1 } - z_j^{-f_i - N +i -1}\\over \\prod (z_i-z_i^{-1})\\cdot \\prod_{i \\quad{ \\ll }\\quad <(\\phi)^2> ", "773e444bc52e22f03eabfeae8d7a1d4c": "a \\vert a \\rightarrow c", "773e68493c8065b1d9eb1e190d09c084": "z(t) = bt.\\,", "773eea30980968332ae35789592bd14b": "\\ d[\\mathbf{x}(1),\\mathbf{x}(j)]\\le r ", "773f5b7fdede06037c0f9694ad2e9fad": "S \\subset M", "773f5cd4ad710b160e845f206e40328c": "\\mathcal{P}_x\\otimes L^2_y", "773f6574d21a500d243f0a78bd102b8a": "\\mathbf{F_r} = \\mathbf{F}_{\\mathrm{imp}} +\\mathbf{F}_{\\mathrm{centrifugal}} +\\mathbf{F}_{\\mathrm{Coriolis}}+\\mathbf{F}_{\\mathrm{Euler}} = m\\mathbf{a_r} \\ . ", "773f65c2cd34160cf752ebea1110a132": "S_n,C_n", "773f722b1f30805bdf9221fea8c683bc": "\\sqrt{8} (3 \\rho^3 - 2\\rho) \\cos \\theta", "773f740649428507c8e8441892f6784c": "({\\mathcal O}(n+m))", "773fb62564e94097b38eab9d68136c36": "\\delta_i ", "773fbe096e820180a6f10264b57d6352": "\\phi_y(x) = \\delta(x-y).\\;", "773fd0d81f243a0bd90056edcaa659f0": "\\Delta = \\mathrm{div} \\circ \\nabla", "773fdeb33f0d64a9a8c3ee2cd295d7ec": "L_a^b(c) := \\int_a^b \\sqrt{g(c'(t),c'(t))}\\,\\mathrm d t = \\int_a^b\\|c'(t)\\|\\,\\mathrm d t.", "773ff25bb3a85e13890f11de903aae5b": "W_\\mu^a", "773ff4838c6dd7cb529e2a4dee393ab9": "\n\\exp_p: T_pM\\supset B_{\\epsilon}(0) \\longrightarrow M,\\quad v\\longmapsto \\gamma_{p,v}(1),\n", "7740cab55bf179a3bc1543c599131f83": "d(q,r) \\le d(q',r').\\,", "7740ed88a26711d729fa64f5a1864322": "\nA =\n\\left(\\begin{matrix}\n\\alpha_0 & \\alpha_1 & \\cdots & \\alpha_n \\\\\n\\bar{\\alpha_1} & \\alpha_0 & \\cdots & \\alpha_{n-1} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\bar{\\alpha_n} & \\bar{\\alpha_{n-1}} & \\cdots & \\alpha_0 \\\\\n\\end{matrix}\\right)", "7740fa4ec4140a455ff4f196362edd3a": "_{s.4.left=s.6.right \\,}\\!", "77416a0b2c8adcdaf85461356153d109": "\\{g(q_j)\\}_{j\\in \\N}.", "7741abd75a7bdbe92ed22e6d4556fb7d": " S \\approx \\frac{ \\sqrt{ 2 } - 0.6745 \\sqrt{ \\pi } }{ \\sqrt{ \\pi - 2 } } \\approx 0.36279 ", "7741bd0c29b3f8e374faa7758a02f638": " \\text{length} (\\gamma|_{[t_1,t_2]})=|t_2-t_1|. ", "7741c5571ebee53b1f6269d1830cf3fc": "x \\in C'(B',\\succeq)", "7741f628a05ee2cdf9e9f4bf82441bf5": "E\\{y\\} = A\\bar{x},", "77423a462690d9248ee8272382650daf": "E = {\\mathbf F} \\cdot ({\\mathbf x}_2 - {\\mathbf x}_1).", "774263754815843a93e6d20e585212d0": "2d = \\frac{p^2}{\\delta p}", "77428a95d36c3bf5828a4a75619b8bd9": "\\ln(k)= \\frac{-E_a}{R}\\frac{1}{T} + \\ln(A)", "77430a9018a47061c37b5b5e07688029": "T\\colon X \\to \\mathbf{R}^2 \\times I", "7743253d3a9aa758e5bd617c4ca181be": "\\sqrt{n_1 n_2/(n_1+n_2)}\\theta", "7743370abbb04e8a17efa2f753cde9ff": " S_n=\\frac{1}{2}c_0(f)+\\sum_{i=1}^n c_i(f) T_i ", "774345a9298175afb5cf5a532554c368": "TW \\cong W \\times W,", "774366d740a77312cec620ade5a62824": "-x-\\frac{1}{2}x^2-\\frac{1}{3}x^3-\\frac{1}{4}x^4-\\cdots\\!", "7743a31bf0085ca46d6cf36866e844e1": "k : = \\nabla_x \\varphi", "7743e0a76a034576c6aa08ee5357d3be": "U(x_1, x_2, \\ldots, x_n) = x_1 + \\theta (x_2, \\ldots, x_n)", "7743ea2c3461fbb0146590c32b4d6335": "B : A \\to Type", "77444a786386ffab303510265098f8b4": " \\sum_i B_i B^*_i \\leq 1. ", "774454d531da5221052fd38a9a1a44be": "\n \\begin{matrix}\n a\\uparrow^n b & = & \\mbox{hyper}(a,n+2,b) & = & a\\to b\\to n \\\\\n \\mbox{(Knuth)} & & & & \\mbox{(Conway)}\n \\end{matrix}\n ", "7744590ef9441f4c91a73e262ad2fe99": "\\Phi(\\mathbf{x})", "774507b185ae429dbb0e4e05fe6652dc": "c=\\varepsilon(c_{(1)})c_{(2)} = c_{(1)}\\varepsilon(c_{(2)}).\\;", "7745b64a31c218896f1c97cc68cc943f": " Q = \\frac{\\text{gap}}{\\text{range}} ", "7745be17bea706c1970af331e805ba44": "\\Omega=\\log A_1 \\log A_2 \\cdots \\log A_n", "7745de53235517b97fe3f830f8a797dc": "a=\\exp\\left(\\frac{\\mu-\\mu_0}{RT}\\right)", "7745fc198c00c6f35d13146d2ef26cc8": "A = B = k", "77462b9b5931b8d0030e2d3c3c3692b2": "\\tilde{e}_i \\tilde{f}_i", "77469a867644a103ad1e4bedbe81778e": "\n \\hat\\phi(t) = \\int_{-\\infty}^\\infty\\! f(x)e^{itx} dx = e^{\\mathbf{i}\\mu t} e^{- \\frac12 (\\sigma t)^2}\n ", "774734f12e1b85fef992aaac4e39d180": "\\mathit{SS}", "77477c49059406e7ab235240177526ae": " \\ \\frac{1}{2} N ", "77479e668f50f6d1c53612a56146f7e6": " \\left|\\Theta\\right\\rang = \\left|1,V\\right\\rang \\left|2,V\\right\\rang ", "7748ce98a25a9eb94a0116ce352553c0": "\\int \\left| \\csc{ax} \\right|\\,dx = \\frac{-1}{a}\\sgn(\\csc{ax}) \\ln(\\left| \\csc{ax} + \\cot{ax} \\right|) + C ", "7748d108439668863dd895e5e3eda90d": "\\mathbf{a}\\perp \\mathbf{v}", "7748e09f9c7775a2af9c8eff37e89a8e": "{\\color{Blue}~2.3}", "77497f03a74f3093ecf56125969f4e4a": "\\mbox{sgn}(\\rho_n) = (-1)^{n+1}\\mbox{sgn}(\\rho_{n-1}).", "77498cb57b24d5408dbb94da978c913d": "\\eta_{IJ} = diag (-1,1,1,1)", "77498e0b56a14b434bc45046c5bfb355": "b(\\cdot)", "7749922f040402a6fae6a934c7d35c1e": "b\\ne 0~ mod~ p_i", "7749f50313e12e091780975363bfc087": "k = 1 / [MPS+MRT+MPM] = 1 / MPW\\,\\!", "774a25b88135e2cb06ec1c671358961a": "K_\\alpha (u, x) = \\begin{cases}\\sqrt{1-i\\cot(\\alpha)} \\exp \\left(i \\pi (\\cot(\\alpha)(x^2+ u^2) -2 \\csc(\\alpha) u x) \\right) & \\mbox{if } \\alpha \\mbox{ is not a multiple of }\\pi, \\\\\n\\delta (u - x) & \\mbox{if } \\alpha \\mbox{ is a multiple of } 2\\pi, \\\\\n\\delta (u + x) & \\mbox{if } \\alpha+\\pi \\mbox{ is a multiple of } 2\\pi, \\\\\n\\end{cases}", "774a7935acbcbffb15d50aa93f770c24": " f_X (\\,\\cdot\\, |Y=y)", "774a84bfd9207ee8752384bc5dadf9d1": "x^3(x^2-x-1)+(x^2-1)", "774a97a48d2faf336d209ba9ca369ee2": "\\frac{\\ln(1-pz)}{\\ln(1-p)}\\text{ for }|z|<\\frac1p", "774ab34c8bfd6bdeadb18a65c586ed3f": "n + \\frac{2 (\\lambda,\\alpha_i)}{(\\alpha_i,\\alpha_i)} \\ge 0", "774abba65364ed65958e4e68c0e766a9": "z_{n+1}=(z_1-(y+\\sum_{0\\sigma^2", "775ca941ee8aef7c09cfcb97cb76fb34": "\\sum_{i=1}^p x_i^2 - \\sum_{j=1}^{q+1} t_j^2 = -\\alpha^2", "775cabf65ff8ce3fe7cc871e22d15fe7": "1/0.54", "775ce7f10c51447079ae845dba506a79": " \\mu_{k,l}:= \\mu_{k,l} - r ", "775d02cb78c20e4702f951df48772aa1": "d_{ij}(Z)\\ne 0, i \\ne j", "775d2e879f1cb9acbb384c0e037d4f1d": "\\epsilon_r'-1=\\frac{f_c-f_s}{2f_s}\\frac{V_c}{V_s}\\,", "775d50a7ee964bcb87cd835af4e29309": "S(N) = \\frac{1}{(1-P) + \\frac{P}{N}}", "775da8229456b6cf82e2cb14aa8ea174": "1=\\frac{v_en\\hbar 4 \\pi \\epsilon_0}{Ze^2} ", "775db08723f4639a4911ab6e4f4cd627": "\\int_U \\left( \\psi \\nabla \\cdot \\mathbf{\\Gamma} + \\mathbf{\\Gamma} \\cdot \\nabla \\psi\\right)\\, dV = \\oint_{\\partial U} \\psi \\left( \\mathbf{\\Gamma} \\cdot \\bold{n} \\right)\\, dS. ", "775df4053d82bb9bf281d8de7616887d": " a, b \\in {\\mathbb R}^2", "775df4bdae6d502fd86d7d7149f07295": "\\textstyle \\{ q (p \\alpha - q) : p,q \\in \\mathbb{Z}, q > 0\\}", "775e1c54851c9c9edfee9e4db362d716": "E_{1,3} \\!", "775e4c69ac25ad559ee2ec8a72c64dce": "b(4p+2)=(2k'+1)(4p+2)=4(2pk'+p+k')+2", "775e4fa1c2bf8652944a74cafd316ebb": " DO_s ", "775f4b9147d832dc2257fb97561d2a8e": "b_0 = -\\frac{1}{6 \\pi \\mu a}", "775f51c82c1ac015627597116a086800": "m_n = \\frac{b\\Gamma(1+n/a)\\Gamma(b)}{\\Gamma(1+b+n/a)} = bB(1+n/a,b)\\,", "775f684d9c2581dba7c349f9a57e3dc4": "\\begin{align}\nY' &=& 16 &+& \\frac{ 65.738 \\cdot R'_D}{256} &+& \\frac{129.057 \\cdot G'_D}{256} &+& \\frac{ 25.064 \\cdot B'_D}{256}\\\\\nC_B &=& 128 &-& \\frac{ 37.945 \\cdot R'_D}{256} &-& \\frac{ 74.494 \\cdot G'_D}{256} &+& \\frac{112.439 \\cdot B'_D}{256}\\\\\nC_R &=& 128 &+& \\frac{112.439 \\cdot R'_D}{256} &-& \\frac{ 94.154 \\cdot G'_D}{256} &-& \\frac{ 18.285 \\cdot B'_D}{256}\n\\end{align}", "775f8132612c525d43bab9c6610b10d1": "m=1,2,\\ldots,M", "775fc6ef9cbf96667226d08507afc5c9": "f\\ x = y \\iff f = \\lambda x.y ", "775fdcb08f753a559ec54d097dad906a": "\\alpha = \\left( \\frac{e}{q_\\mathrm{P}} \\right)^2.", "77601afaad27dcff6bffe04130d13450": "\\|\\hat{f}\\|_{l^{p'}(\\mathbb{Z})} \\leq \\|f\\|_{L^p(\\mathbb{T})}", "77601c14143ba4da166a06b4c230536d": "dy_i{\\and}dx_i", "77609a383590cc8f8d529e3dbc9671a0": "R^{d\\times d}", "7760c9f2ab68a4d41cc8903e31b136b6": "\\mathcal E_{{\\mathbf Gr}(r, \\mathcal E)}", "77612c78f2994b5f00ee62c076214044": " \\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} g(s)e^{st} \\,ds ", "776170b9dbb3ae1df3678e90131f45c5": "\\pi_1(A \\cap B)", "7761d176ea24cc9f68e6b519bcc6d368": "\\hbar\\mathbf{q}", "7761df7c69b2e4b6a370f29e535e7a84": "X = \\{a, b, c\\}, N = \\{p, q\\} \\text{, then } \\left\\vert\\{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)\\}\\right\\vert = 3^2 = 9", "7761e291124180e1fe49a4c34c53608b": "Y=Z/\\lambda \\sim ED(\\mu,\\sigma^2)", "7761e3123a3ffe5fe18dacea1f85492e": "\nL_\\mathrm{dB} = 10 \\log_{10} \\bigg(\\frac{A_1^2}{A_0^2}\\bigg) = 20 \\log_{10} \\bigg(\\frac{A_1}{A_0}\\bigg). \\,\n", "7761e37860f35340efb76335fd8bfcec": "\\frac{1}{\\sqrt{1-x^2}}P_\\ell^m(x) = \\frac{-1}{2m} \\left[ P_{\\ell+1}^{m+1}(x) + (\\ell-m+1)(\\ell-m+2)P_{\\ell+1}^{m-1}(x) \\right]", "7761ebf529297d55378510d0026d8a2e": "\\mathbf{u} \\oplus_M \\mathbf{v}=\\frac{(1+\\frac{2}{s^2}\\mathbf{u}\\cdot\\mathbf{v}+\\frac{1}{s^4}|\\mathbf{v}|^2)\\mathbf{u}+(1-\\frac{1}{s^2}|\\mathbf{u}|^2)\\mathbf{v}}{1+\\frac{2}{s^2}\\mathbf{u}\\cdot\\mathbf{v}+\\frac{1}{s^4}|\\mathbf{u}|^2|\\mathbf{v}|^2}", "776214f9df6bc43a3561080a56b76f8a": "y_{0}\\equiv m_{i} (y_{0}^{(i)})\\pmod {p_{i}} ", "776218d1f87ad1c8cb5412638cf6829d": "\\sum\\limits_{r=0}^{k(m-1)} {z^{r} c_{r-k(m-1)/2}^{k,m}=(1+z+\\dots+z^{m-1})^k}", "776331a8bf785018bdd5f34ee2b92886": "\\qquad A::= 0 \\quad \\mid \\quad A\\;\\mid\\;B \\quad \\mid \\quad C.A \\quad\n\\mid \\quad n[\\;A\\;]", "77633ddcc471ce722e8559103b11124f": " \\Lambda^\\cdot {\\mathfrak g} \\otimes C^{\\infty}(M) ", "776351afbcfe76141393465abb3c04ad": "\\mathbf{e}_6 \\times \\mathbf{e}_7 = \\mathbf{e}_2, \\quad \\mathbf{e}_7 \\times \\mathbf{e}_2 = \\mathbf{e}_6, \\quad \\mathbf{e}_2 \\times \\mathbf{e}_6 = \\mathbf{e}_7,", "7763da75b2fba251bee72b9c8e304638": "\\displaystyle{\\mathfrak{n}=\\bigoplus_{i y", "777bae4cde07feb9bf40d80baf08c7ba": " {\\mathbf{a}}^2 = {\\left |\\mathbf{a}\\right|}^2", "777bbb7ce57c3cc9e2d0f1905c9ca116": "r0}\\,{x^n}{n^{-2}} = x \\; {}_3F_2(1,1,1;2,2;x)", "778f3abb9109d3e694a4af227acd62c7": "\\mathcal{N}^{TOP} (X) \\cong [X,G/TOP].", "778f430abeb4865371ad957d1bc806c1": "\\vert{\\Psi_{\\mathbf{p}}^{1}}^{(\\pm)}\\rangle", "778f6981832d15b4d31c71c5b1b6d8f6": "\n[\\hat{x}, \\hat{y}] = -i\\tfrac{\\hbar c}{q B}\n", "77903973b3d1348f6978392a994c0ea4": "n=2^k-2", "7790550f9ea4e66fc3f2e608cb140b7f": "f_P(\\langle u,v \\rangle) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ \\langle u,v \\rangle \\in S \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ \\langle u,v \\rangle \\notin S\n\\end{matrix}\\right.", "77906ce80e83a507671978859db1a479": "y_{n+1}^{(1)} + \\tau_{n+1}^{(1)}=y(t+h)", "7790b2bcace551a607b2369a0aea49ef": " Q(Y \\mid x) = \\frac{P(x \\mid Y) \\pi(Y)}{Q(x)} ", "7790b689c9d042eae6de484197be2f53": "f_i: M_i \\to X", "7790dc5c5c64683354a4426bf807adad": "\\begin{align}Q(AB) & \\equiv (B_x - A_x)^2 + (B_y - A_y)^2 \\\\ & = b^2 + (-a)^2 \\\\ & = a^2 + b^2\\end{align}", "7790e2247c568c071c770b8ad0cdcbc2": "\nP_L\n=\\frac{\\sum (p_{c,t_n}\\cdot q_{c,t_0})}{\\sum (p_{c,t_0}\\cdot q_{c,t_0})}\n=\\frac{\\sum (p_{c,t_n}\\cdot \\frac{E_{c,t_0}}{p_{c,t_0}})}{\\sum E_{c,t_0}}\n=\\frac{\\sum (\\frac{p_{c,t_n}}{p_{c,t_0}} \\cdot E_{c,t_0})}{\\sum E_{c,t_0}}\n", "77911caba5254099bab5a416a68c1bce": "Z_\\mathrm B=Z_0+2Z_2\\ .\\!", "7791a3e2a518913c8dbcab71a974ab5a": "\\begin{pmatrix} P_x \\\\ P_y \\\\ P_z \\end{pmatrix} = \\varepsilon_0\n\\begin{pmatrix} \\chi_{xx} & \\chi_{xy} & \\chi_{xz} \\\\ \\chi_{yx} & \\chi_{yy} & \\chi_{yz} \\\\ \\chi_{zx} & \\chi_{zy} & \\chi_{zz} \\end{pmatrix}\n\\begin{pmatrix} E_x \\\\ E_y \\\\ E_z \\end{pmatrix}\n", "7791cb4d4c94254e1636966c1ee5d356": "H_\\Delta", "779201afd1fff9f8b5af806777356356": "x[n_1,n_2]", "77920f490caccdd6ce605d039b3d79b7": "y_0,\\;y_1,\\ldots,\\;y_n,", "77922a3584dc13ac8d79c428dd968c14": "t\\rightarrow -t", "7792c5f6c44522d73c15dc29a112ab65": "\\tfrac{1}{M}", "7792d20df47cbeded719d1673be3b385": "T_{i+1}\\not\\in L_i", "7792e2ae724e61be7d3fa3979c365fcf": "\\mathbf{\\dot{p}} = - \\frac{\\partial H}{\\partial \\mathbf{q}}\\,,\\quad \\mathbf{\\dot{q}} = + \\frac{\\partial H}{\\partial \\mathbf{p}} \\,,", "7792f6d8f7c6d9fe343702003452e283": "P_E=IV\\,", "7793414fecf1beb6bfc1a62af9123430": "\\gamma(n)", "77934752e2dc1eae0c9146d8fdcc95ee": " u_\\varepsilon (y) = \\int_\\Omega \\int_0^\\infty p_\\varepsilon (x,t y) \\, dt \\, dx ", "779361d8cea9b76d1bcfc4ba2dff2d67": "[u, v]_{q,p} = \\frac{\\partial u}{\\partial q_i} \\frac{\\partial v}{\\partial p_i} - \\frac{\\partial u}{\\partial p_i} \\frac{\\partial v}{\\partial q_i}\\!", "77937c64fe05263181c1c8ed20a8f917": "K(u) = \\frac{1}{e^{u}+2+e^{-u}}", "77938b916ba47588da85075bd375b425": "F^*\\circ f", "77942142623576fbf0a77a03f30aa7b8": "2\\Gamma(3/2) = \\sqrt{\\pi}", "779468ee4cbfc1b194c46b2949b7ca02": "p_{1},p_{2},p_{3},p_{4},r_{1},r_{2},r_{3},r_{4} \\in \\{0,1\\}", "7794730c9866f9d2f74058e2ae6133b3": " \\Rightarrow(y_2 + \\frac{q^2}{2gy_2^2} = 4.04ft)", "779474bbb6c5346a7a748b67f08d438a": "\\log_a x^x-\\frac{x}{\\ln a}", "77947dbfdc6bc937d35499899eacb146": "\\int_0^1 f(x) dx = 0", "7794a6b16bc28d9eb8171104de73383e": "\\tfrac{\\pi}{\\sqrt{3}}", "7794bc467d9cc8d9c8021a07ebd5c54f": " y_k(x) = f(x_{k-1}) + (x-x_{k-1}) f[x_{k-1}, x_{k-2}] + (x-x_{k-1}) (x-x_{k-2}) f[x_{k-1}, x_{k-2}, x_{k-3}], \\, ", "7794e7a527db72ebf15778eff59943a2": " -\\ln \\alpha - \\alpha \\ln x_{\\mathrm m}", "779518c3cb7ba012f8c232f02afa9a1e": " L(x) = \\begin{cases} \\operatorname{sinc}(x)\\, \\operatorname{sinc}(x/a) & \\text{if}\\;\\; -a < x < a\\\\ 0 & \\text{otherwise} \\end{cases} ", "779523a4e2cfe62478f251928929214c": "g_N \\ge 0", "779527bf26741b54bf368883a917644b": "\\boldsymbol\\Lambda_n=(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\mathbf{\\Lambda}_0), \\quad \\boldsymbol\\mu_n = (\\boldsymbol\\Lambda_n)^{-1}(\\mathbf{X}^{\\rm T}\\mathbf{X}\\hat{\\boldsymbol\\beta}+\\boldsymbol\\Lambda_0\\boldsymbol\\mu_0) ,", "779557b18a4818da4d7ab9d7664e0ae2": " \\mathbf{\\ddot{r}} = \\left( \\ddot{r} - r \\dot{\\theta}^2 - r \\dot{\\zeta}^2 \\cos^2\\theta - R_0 \\dot{\\zeta}^2 \\cos\\theta \\right) \\mathbf{e}_r ", "77958f518990d96829c378d6ed183087": " M^1_1 = - \\sqrt{\\tfrac{1}{2}} \\sum_{i=1}^N e Z_i \\langle \\Psi | x_i+iy_i | \\Psi \\rangle\\quad \\hbox{and} \\quad\nM^{-1}_{1} = \\sqrt{\\tfrac{1}{2}} \\sum_{i=1}^N e Z_i \\langle \\Psi | x_i - iy_i | \\Psi \\rangle. ", "779591ca2f4e3b48cc98b77d9291f543": "\\boldsymbol{p}(t) = (2t^3-3t^2+1)\\boldsymbol{p}_0 + (t^3-2t^2+t)\\boldsymbol{m}_0 + (-2t^3+3t^2)\\boldsymbol{p}_1 +(t^3-t^2)\\boldsymbol{m}_1", "7795b029b6c926da4fa22c604aea7e40": "Q=\\begin{pmatrix}\n-\\lambda & \\lambda \\\\\n\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&2\\mu & -(2\\mu+\\lambda) & \\lambda \\\\\n&&3\\mu & -(3\\mu+\\lambda) & \\lambda \\\\\n&&&&\\ddots\\\\\n&&&&c\\mu & -(c\\mu+\\lambda) & \\lambda \\\\\n&&&&&c\\mu & -(c\\mu+\\lambda) & \\lambda \\\\\n&&&&&&&\\ddots\\\\\n&&&&&&&c\\mu & -(c\\mu) \\\\\n\\end{pmatrix}", "7795c4c6f216e6ce9da9f869f15d7dde": "\n I_0(\\lambda) = \\exp[{\\lambda S(x^0)}] \\int\\limits_{I_w} f[\\boldsymbol{\\varphi}(w)] \n\t\\exp\\left( \\lambda \\sum_{j=1}^n \\frac{\\mu_j}2 w_j^2 \\right)|\\det\\boldsymbol{\\varphi}_w'(w)| dw.\n", "7795c9c66b0ebd4a3daee549d3467cc8": "|\\phi,\\phi,\\ldots \\phi\\rangle", "7795e31a365dad5c0f9ee67d4a07ebc6": "\\lim_{n\\to\\infty} a_n^{1/n}=\\infty.", "77966019e0ddd835ac676336ec86154b": " P = - \\left( \\frac{\\partial E}{\\partial V} \\right)_S \\qquad (1)\n", "779687dbc51885469625a5db7b9f4942": "= 2 p_1p_2(1 - \\cos\\theta). \\,", "779691160d5f449d409c962b29f74ab2": "E_r^2 - |\\vec{p} \\,|^2 c^2 = m_0^2 c^4", "7796e4d52618eb24af5090628bb5be82": "\\vec{u} = \\vec{R}_1 - \\vec{R}_0", "7796f5331c15f41d053f8aaa9fb92761": " rec = \\frac{m_c W_c}{m_f W_f} ", "77971eaf2dacdff6323f499fdbec0402": "E=m g l+\\frac{\\hbar^2 a_n(q), \\, b_n(q)}{2 m l^2}", "779738f8bf468e12f338edf66d42f371": " \\bold{X} : U \\to \\mathbb{R}^{n+1}", "7797581d879b750ef84deeb12e03e577": " \\frac{\\zeta(s)}{\\zeta(2s)} = \\sum_{n=1}^\\infty \\frac{|\\mu(n)|}{n^s} \\equiv \\sum_{n=1}^\\infty \\frac{\\mu^2(n)}{n^s}. ", "77979593a6c823ef64d0b6c6b4f6e72e": "T(w) = 2^{|w|}", "7797ab2e861cb26e3725135a2ef0af66": " r_n = r_{n-1} - \\frac{f(r_{n-1})}{ (r_{n-1}-p_n)(r_{n-1}-q_n)(r_{n-1}-s_{n-1}) }; ", "7797fa8336816398a65f573f211371c7": "d \\ne \\textrm{MAC}(k_M; c \\| S_2)", "779803261222d9b26dcd11d9b99ccec1": "\\Delta_1(k_\\lambda) = k_\\lambda \\otimes k_\\lambda", "7798136c7f19ebc1bcfce64cd6faec2d": " \\int_X^\\oplus H(x) \\, d \\mu(x) ", "7798611692f3a0d24e0087d0177c2759": "(1 + x + x^2)", "7798cd0b77d693e89b96adc2f8695426": " INT\\_MAX = x + y ", "7798ed51caabd1d4db9a23946acf7d72": "A_0, A_1, A_2,\\dots,A_m", "77996cccda3eb4a72ba360e2c60caf82": "\\textrm{erfc}\\left(\\sqrt{\\frac{c}{2(x-\\mu)}}\\right)", "77997c72143325bfbc3174846720a977": "\\Theta_{D}", "77997d2ebcde14b4016355e8f4e7b415": "S = \\pi R^2 = {A \\over 4}.", "779990adfe4d2f5d1cb47e4518d2cc66": "PWV = \\sqrt{\\dfrac{E_{inc} \\cdot h}{2r\\rho}}", "779995f0582420b13fd89806e3e45e55": "\nA_{n} = \\frac{1}{n!} \\frac{\\mathrm{d}^{n}}{\\mathrm{d}\\lambda^{n}} f(u(\\lambda))\\mid_{\\lambda=0},\n", "77999a348a385fda6b13d256b3517fa0": "\\theta = \\theta(t),\\quad \\phi = \\phi(t),\\quad a\\le t\\le b", "77999d4f1db872f3704ee1c7e932cd5a": "\\scriptstyle \\left(b\\, \\mid\\, a a^*b\\right)^*a a^* ", "779a421c78e08018711f5211cc6927e0": "L^{(N)} = 0", "779a4fa3c939817dd48c4a62b8c197ac": "\\omega_d = \\omega_\\mathrm{nat} \\,\\!", "779a8ba8d10f1905adc51c3729e6ba2c": "(t, u)", "779b3992492edab23a89c10a385e9541": " 0\\in \\partial (F+R)(x), ", "779b40b0e5333dc9db2d4e12c1b33407": " \\forall f : A \\rightarrow \\mathbb{R} \\dots \\, ", "779b427b51acb1e9af18d40d4ee834e7": "\\Gamma (X, \\underline{\\mathbb Z}_X) = \\mathbb Z^{\\pi_0(X)}", "779b7a754bf07eaa667523da937e5dab": "\n\\left(\\frac{\\partial T}{\\partial p}\\right)_H\n\\left(\\frac{\\partial p}{\\partial H}\\right)_T\n\\left(\\frac{\\partial H}{\\partial T}\\right)_p\n= -1\n", "779b850a4bdf282894b703722cdae2ac": "M\\cdot V_T = P_T\\cdot T", "779bd327338a8fd93fb884a038662e31": "r=2,...,R", "779be48222f4b095b3053fc210c56f84": "X \\rightarrow Y \\in S^+ \\land Y \\rightarrow Z \\in S^+~\\Rightarrow~X \\rightarrow Z \\in S^+", "779c0004e426f6514a9dabf3b610e1b6": "M \\mapsto M^{\\mathfrak{g}} := \\{ m \\in M \\mid gm = 0\\ \\text{ for all } g \\in \\mathfrak{g}\\}.", "779c0b7c87f0adb2791f4fa1e656bd13": " A_{ABC}^2 = A_{\\color {blue} ABO}^2+A_{\\color {green} ACO}^2+A_{\\color {red} BCO}^2 ", "779c1031f1f5d55b4abcf6107ee76cea": " X(Y,Z)=(\\nabla_X Y,Z) + (Y,\\nabla_X Z) ", "779c721d0fdfd7338a780ec04c9245f7": "CE(1,1,x)", "779cc5e27086ddee4fedda52893f3a87": "\\frac{\\partial \\mathrm{net}}{\\partial w_i} = x_i", "779ce165c58917f88f43aca1e54e8a34": "1\\over \\sqrt[4]{\\pi}", "779dc69a5e6db1541dc0a5a1b9e926e7": "\n\\det J_F(x,y)=\ne^{2x} \\cos^2 y + e^{2x} \\sin^2 y=\ne^{2x}.\n\\,\\!", "779e436bb9dacbc12b17fb8f7cf18ff9": "p_AO_2=F_IO_2(P_{ATM}-pH_2O)-\\frac{p_aCO_2(1-F_IO_2[1-RQ])}{RQ}", "779e8117b652e37591534f1cc87a93dd": "00", "77af7e6a059beb0defe0186866949ab3": "D = \\left\\{ \\frac{Bird(X) : Flies(X)}{Flies(X)} \\right\\}", "77af915cff652326b57a2adc5e04cab9": "\\displaystyle{\\gamma^2=\\alpha.}", "77b0164479aba80cf5dd0beb0ad69d76": "\\begin{align}\n f_{mnl} &= \\frac{c}{2\\pi\\sqrt{\\mu_r\\epsilon_r}}\\cdot k_{mnl}\\\\\n &= \\frac{c}{2\\pi\\sqrt{\\mu_r\\epsilon_r}}\\sqrt{\\left(\\frac{m\\pi}{a}\\right)^2 + \\left(\\frac{n\\pi}{b}\\right)^2 + \\left(\\frac{l\\pi}{d}\\right)^2}\\\\\n &= \\frac{c}{2\\sqrt{\\mu_r\\epsilon_r}}\\sqrt{\\left( \\frac{m}{a}\\right) ^2+\\left(\\frac{n}{b}\\right) ^2 + \\left(\\frac{l}{d}\\right) ^2}\n\\end{align}", "77b020dff2f666b4ea9090ed9b0b8735": "F_y", "77b05f67973c74a30707e14d0475a95f": "q=\\frac{1}{2R}", "77b066d8a1ca106ae192ac23aa9d82ba": "1 \\rightarrow \\mu_n \\rightarrow \\mathbf{G}_m \\xrightarrow{n} \\mathbf{G}_m \\rightarrow 1.", "77b07a76e76c7a3d8e51bdab05079854": "\\frac{1}{2}bhl", "77b0efa47fe8947bdc0d5ff3f19ef41c": "\\,\n{\\mathbf{U}}_{||} = \\mathbf{U} \\ , \\quad {\\mathbf{U}}_{\\perp} = \\boldsymbol{0} \\ , \\quad \\mathbf{V} \\cdot \\mathbf{U} = \\pm V U\n", "77b0f0e847bf0856693bca9c7e805c4a": " 1 \\, ", "77b1689b8b860593e096891883e78116": "K=\\frac{1}{4}\\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}.", "77b1778fbad59d6b782885cd7963f55d": "{\\sigma_x}(y,z) = -\\frac {(M_z~I_y + M_y~I_{yz})} {I_y~I_z - I_{yz}^2}y + \\frac {(M_y~I_z + M_z~I_{yz})} {I_y~I_z - I_{yz}^2}z", "77b189f924bba5156570b6862fe7a07f": " h \\to 0 ", "77b1f3b021cd2d2e35fae83d4a1c0b23": "= \\langle\\Psi|\\hat{A}\\hat{B}\\Psi\\rangle-\\langle\\Psi|\\hat{A}\\langle \\hat{B}\\rangle\\Psi\\rangle\n-\\langle\\Psi|\\hat{B}\\langle \\hat{A}\\rangle\\Psi\\rangle+\\langle\\Psi|\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle\\Psi\\rangle", "77b2b02291d38b7c759871995923533c": " (x - 1)( x + 1) \\equiv 0\\pmod{p}.", "77b3004517a1ae9af16fd0cb08966db7": " V_{fb}(t) = \\frac {Q_s(t-T)}{C_{fb}} + V_{fb}(t-T)\\, ", "77b32b09a6cbf394532a3dc116692249": "\\text{OPL}=nd", "77b346aedcaad919aa5e856a5b14c00f": "\\operatorname{P}(T \\geq c \\, n H_n) \\le \\frac1c.", "77b3738d7dda60f29d52a2d745f5fa0f": "\\phi_c = e^{-\\eta_c} ", "77b38199c44e7677c1ca7be19bc3725d": "H_{\\lambda X} = d(S_\\lambda)_X H_X\\,", "77b3f3c3a7fc29b0b31b9a76a9208f62": "\\frac{d}{ds}T(s)\\cdot\\mathbf{s}_u = -t\\cdot\\mathbf{s}_u - n\\cdot\\mathbf{n}_u", "77b4131ad12ec7375e46f89d63fb0838": "\\nabla^2\\omega_2 + \\frac{f^2}{\\sigma} \\frac{\\partial^2\\omega_2}{\\partial p^2} =\\frac{R \\cdot \\nabla^2 q}{C_p \\cdot \\sigma \\cdot p}", "77b4389c29c4cf54111dd0e46489df24": "g(E)=\\frac{L^3}{(2\\pi)^3}\\iint\\frac{dk'_x\\,dk'_y}{|\\vec{\\nabla}E|}", "77b45ae5e15470b96f951cdc03f181d4": " \\mathbf{B}(\\mathbf{r}) = \\boldsymbol{\\nabla} \\times \\mathbf{A}(\\mathbf{r}) ", "77b4c8ce4e1a51614249caf6bd019670": "T_{1B}=W_{1B}(y)", "77b4ec7e475ac2281e598b6282110f03": "EQ = \\operatorname{Brain weight}\\over\\operatorname{0.12}\\cdot\\operatorname{Body weight}^{0.66}", "77b52e4268176ab2ce34cb876aab3a4c": "\\scriptstyle 0 \\,\\oplus\\, 1 \\;=\\; 1", "77b570204d96f890b800410b04b86ae1": "B_t", "77b573619cbad0f9ce513e0f3daa0065": "mL_c = \\sqrt{\\frac{2h_f}{k t_f}}L_f", "77b578214fb6393a8c00971d101b2e5b": "v = \\sum a_i v_i = \\sum b_i v_i\\text{ where } (a_i) \\neq (b_i).", "77b59e6abaf1b048f4a885c280ffc531": "P_X(z)=b_0(X)+b_1(X)z+b_2(X)z^2+\\cdots , \\,\\!", "77b5d464d9db2591a13e9ff6f78ab45b": "f(T)v=f(T)w", "77b5f0b50478ac2a9e3b1c3b35dfbab4": "g(\\vec{r},\\vec{r'})", "77b62e34087396e4e465dd169fd49d6f": "\\frac{dy}{dx}=a_i(x,y)", "77b63a46cfd947e30cdbee29e3e337b6": "\n\\begin{alignat}{2}\n\\epsilon(0,\\omega) & \\simeq 1 + V_q \\sum_{k,i}{ \\frac{q_i \\frac{\\partial f_k}{\\partial k_i}}{\\hbar \\omega_0 - \\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m}} }\\\\\n& \\simeq 1 + \\frac{V_q}{\\hbar \\omega_0} \\sum_{k,i}{q_i \\frac{\\partial f_k}{\\partial k_i}}(1+\\frac{\\hbar \\vec{k}\\cdot\\vec{q}}{m \\omega_0})\\\\\n& \\simeq 1 + \\frac{V_q}{\\hbar \\omega_0} \\sum_{k,i}{q_i \\frac{\\partial f_k}{\\partial k_i}}\\frac{\\hbar \\vec{k}\\cdot\\vec{q}}{m \\omega_0}\\\\\n& = 1 + \\frac{V_q}{\\hbar \\omega_0} 2 \\int d^2 k (\\frac{L}{2 \\pi})^2 \\sum_{i,j}{q_i \\frac{\\partial f_k}{\\partial k_i}}\\frac{\\hbar k_j q_j}{m \\omega_0}\\\\\n& = 1 + \\frac{V_q L^2}{m \\omega_0^2} 2 \\int \\frac{d^2 k}{(2 \\pi)^2} \\sum_{i,j}{q_i q_j k_j \\frac{\\partial f_k}{\\partial k_i}}\\\\\n& = 1 + \\frac{V_q L^2}{m \\omega_0^2} \\sum_{i,j}{ q_i q_j 2 \\int \\frac{d^2 k}{(2 \\pi)^2} k_j \\frac{\\partial f_k}{\\partial k_i}}\\\\\n& = 1 - \\frac{V_q L^2}{m \\omega_0^2} \\sum_{i,j}{ q_i q_j 2 \\int \\frac{d^2 k}{(2 \\pi)^2} k_k \\frac{\\partial f_j}{\\partial k_i}}\\\\\n& = 1 - \\frac{V_q L^2}{m \\omega_0^2} \\sum_{i,j}{ q_i q_j n \\delta_{ij}}\\\\\n& = 1 - \\frac{2 \\pi e^2}{\\epsilon q L^2} \\frac{L^2}{m \\omega_0^2} q^2 n\\\\\n& = 1 - \\frac{\\omega_{pl}^2(q)}{\\omega_0^2},\n\\end{alignat}\n", "77b69c74392d139b706cd4ae1b580fb7": "\\neg A \\cdot \\neg B \\cdot \\neg C", "77b6de38eed994fafcf3069c5096a759": "M_{\\mathbf{\\Xi}} = M_{\\xi_1} \\ast M_{\\xi_2} \\dots \\ast M_{\\xi_N}. ", "77b77f2bba8e7fac02b6b9d4eac8f7f6": "(a_1, a_2, \\ldots, a_n)", "77b7fd79be3ceba13cd9e8a1be7b7d91": "E(\\tilde{m}\\tilde{x}_i)", "77b813e14f06dcb916b530ff0197df8b": "E_\\text{B} = -\\dfrac{Rhc}{n^2}", "77b896497fa9e044cb9bb3136e6e03cf": "\\nabla^{2}\\psi=-\\nabla\\cdot\\mathbf{H}=\\nabla\\cdot\\mathbf{M}.", "77b956fa1d1af9487777f17abd0cdba0": "\\begin{matrix} {10 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "77ba1154425fa2f8ccacde6f6059aa80": "N_G(D)", "77ba5a27ef0408f95cbba79ed107afd1": "\\ S_{3,t} = F(t_3) (m_3 + F(t_2) (m_2 + F(t_1) m_1 )) ", "77ba7641d5fe37578e6dcd8a55581e9c": "MV = PQ", "77ba8b5890d6b78574f77b26713dbc1b": "1/e", "77bacc2a4f886e77bc53e8c23ea803e4": "\\displaystyle \\mathbf r_1,\\,\\ldots,\\,\\mathbf r_N", "77bae4aca72a09ee39e0bba30a032619": "(c+1-\\alpha )_{\\alpha -\\beta } =(c+1-\\alpha )(c+2-\\alpha )\\cdots(c-\\beta ).", "77baef7365d6d2634fa6c6d8575ac29b": "\\mu = \\sqrt{2+\\sqrt{2}}", "77bb35981ef1be6276ebf42af2efbd6f": "\\mu = a_1+2a_2", "77bb35def84627b0a684a44e07c4f357": " A = [a_{ij} ] ", "77bb754ab5c61e673aaf1baaacca9b12": "1 \\le \\mu_w \\le \\mu", "77bb7c389bc3e9e16f17de4aeaf9521d": "axy+bx^2+cy^5+d", "77bc0c89d29ee1a8425dd689430494f6": "c=\\frac{{\\text {Damping Coefficient}}}{{\\text{Critical Damping Coefficient}}}", "77bc6e0405da3018aea404591a228de9": " A(h) ", "77bc6e413fe0b93ba103a50cab9b554b": "Square numbers end on 3", "77bcc27324b7ac65a1e46d0e2a3a36df": "n \\geq m", "77bd021cec74c71d5e1f8c028ccc2cfa": " T \\rightarrow T_C ", "77bd111202a61c35cb6b52563f23e78a": "(Q(p;\\lambda)\\,,Q'(p;\\lambda)^{-1}),\\, 0\\leq\\,p\\,\\leq\\,1", "77be6692846ac3fea5b91c9d72895150": "r = a", "77be6d2ae970da7241394dc7b3c24971": "\\mathbf{B}\\cdot\\mathrm{d}\\mathbf{S} = 0", "77bec904cfaf0919fe2bbf418f233c10": "e^{-\\frac{E_a}{RT}}", "77bed58b17def2013093b4d4a7d1579c": "\n\\dot{v}=v-\\frac{v^3}{3} - w + I_{\\rm ext} \n", "77bee7239dfaf9b86f94caf12c8f371b": "d <0", "77bf68e0ca5f2c9447350f4f93ccd99a": "{\\tilde L}_v^2 = L_x^2 \\, L_{xx} + 2 \\, L_x \\, L_y \\, L_{xy} + L_y^2 \\, L_{yy} = 0", "77bf954948d6dda50187a055b2d07e5e": "(M_1 \\setminus V) \\bigcup_{N_1 \\setminus V = N_2 \\setminus V} (M_2 \\setminus V)", "77c04a47a4f06dc762fb4c8b8ade5f83": "{{P}_{T}}f(u,\\xi )={{\\left| \\left\\langle f,{{\\phi }_{\\gamma (u,\\xi )}} \\right\\rangle \\right|}^{2}}", "77c09b0716bfe098f3a883236f0fc48a": "\nA_{ij} = {\\partial \\beta_j \\over \\partial \\theta_i} = {\\partial \\tau \\over \\partial \\theta_i \\partial \\theta_j } = \\delta_{ij} - {\\partial \\psi \\over \\partial \\theta_i \\partial \\theta_j } \n= \\left[\\begin{array}{ c c } 1-\\kappa -\\gamma_1 & \\gamma_2 \\\\ \\gamma_2 & 1-\\kappa +\\gamma_1 \\end{array}\\right] ", "77c0ab82182a676a01498a9844869e9a": "\\mathrm{FB107E70}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathrm{bvtdll}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{0EE80890}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathbf{kiebgt}", "77c0b1164f7f871a2da62c91eae81ceb": "\\alpha=\\frac {(w\\tau_r)^2}{c_0 Q_c \\tau_r[1+(w\\tau_r)^2]} \\quad (3)", "77c0c5ba6988a4aa1fbfafc697f7760e": "(\\ x\\ ,\\ y\\ )", "77c15259e2c8b41ad88639d816b21491": "K(X_1,\\ldots,X_n)=\\mathrm{CH}\\{ P(X_1,\\ldots,X_n): P(x_1,\\ldots,x_n)=\\prod_{i=1}^n P(x_i\\mid\\pi_i) , P(X_i\\mid\\pi_i) \\in K(X_i\\mid\\pi_i) \\}", "77c15da9527cfbfd476449896aa0da81": "{n-1\\choose k} + {n-1\\choose k-1} = {n\\choose k}\\quad\\text{for }1 \\le k \\le n ", "77c1a74a8f085dc5aba695c6750db90c": " \\tan^2 \\theta + 1 = \\sec^2 \\theta\\!", "77c1d9d38e2af63f15deb3bfab0a75e8": "\\intercal \\barwedge \\veebar \\doublebarwedge \\between \\pitchfork \\!", "77c1e50788df654146f75774f0dc01b2": "\\mathbb{HP}^2", "77c244aac04583321f438f32173738f7": "i_{th}", "77c287e47349363b6e910ed3c58d2287": "u = \\Phi_A(\\mathrm{id}_A)", "77c2d7748a35b8849b29ab1c88294012": "P=X^2-1,", "77c2f6d40109b203c0a19f55e3f70d75": "f = b \\sin\\vartheta \\cos \\vartheta", "77c3291e62d0fb25e08821a8bfbe2acf": "\\lambda + \\rho \\in \\mathfrak{t}^*", "77c337a8416826e165d9a72bdaf83a45": "\\| \\cdot \\|", "77c35b9d5c9fda598bd79b27982f3c8a": "T = (1 / 2 \\pi) \\sqrt{\\Lambda / 3}", "77c35c39788007350c67a4feede553d5": "e^{\\ln x} = x", "77c3853ae30074512e64573704596c09": "\\Delta = 7.5625", "77c3be504e58a63961b190c2299cd69e": " \\mathrm{sys}\\pi_1(\\Sigma_g) \\geq \\frac{4}{3} \\log g,", "77c425e678c9733b41e38058d8a6e67c": "U(\\theta) \\propto \\frac{J_1(\\pi W \\sin \\theta/ \\lambda)}{\\pi W \\sin \\theta / \\lambda}", "77c4393d8a20ff929dadfd70f3579c17": "U\\to R^n", "77c4597468fa2719cc43cafc5dd615af": " c_H(t,0) \\simeq \\begin{cases}\n\t(t)^{-\\alpha} & \\textrm{for} \\ t \\downarrow 0 \\\\\n\t(-t)^{-\\alpha'} & \\textrm{for} \\ t \\uparrow 0 \\end{cases}\n ", "77c4f3877aafa4b01f5c30256ec644d8": "-\\arccos {\\left(-\\frac{1}{e}\\right)} < \\theta < \\arccos {\\left(-\\frac{1}{e}\\right)} ", "77c5864b8ffda0b84c6c7e917d9cfd9c": "\\ z_2 (x,y) = x y ", "77c5a529d0a0a2f31c08fe599b6a26cc": "px = xq", "77c5ae8f438dfcbff7b1cb3e21426aa9": " \\int\\!\\!\\!\\!\\int_S dA = 4 \\pi r^2 ", "77c60be5305f586defd976e857e1cfcf": "r_g = \\frac{m v_{\\perp}}{|q| B}", "77c62a5858c2db8509fc564a0fcd514a": "\\ \\kappa ", "77c648a3b6addf5db0a45af1f034249a": "\\textstyle K", "77c6b02efbea9a354d23bebd9abfa3de": "\\pi \\approx 3.1410319509 +0.0005605826", "77c712fff9cea47c204f5a20271eaa30": "\\chi_{0,0}", "77c72ef0da734ad0463efb1391812ad3": "g (x) = F(x) - dx", "77c7593a95602a87bdb9c46d1b070bec": "\\Leftrightarrow P(A|B) \\ = \\ P(A)", "77c77b41f2cab8dc8fe216f2acf85119": "\\chi_{\\left[0,1\\right]},\\;\\chi_{\\left[0,\\frac12\\right]},\\;\\chi_{\\left[\\frac12,1\\right]},\\;\\chi_{\\left[0,\\frac14\\right]},\\;\\chi_{\\left[\\frac14,\\frac12\\right]}", "77c78cdb5643f9043d14efbc6f9eb14f": "\n\\begin{bmatrix}\n A_{11} & A_{12} & A_{13} & 0 & \\cdots & 0 \\\\\n & A_{22} & A_{23} & A_{24} & \\ddots & \\vdots \\\\\n & & A_{33} & A_{34} & A_{35} & 0 \\\\\n & & & A_{44} & A_{45} & A_{46} \\\\\n & sym & & & A_{55} & A_{56} \\\\\n & & & & & A_{66}\n\\end{bmatrix}.\n", "77c7b343e679717cde65ed7f78d76687": " \\mathbf{f}^{T} \\, \\mathbf{Y} = \\mathbf{0} ", "77c7db6cd9d9ddfb3cf8b232568c5510": "A(a)\\to A(\\varepsilon(A))", "77c80e2817a264cd291d52ee3ce6c368": "{120 \\choose 1} = {16 \\choose 2} = {10 \\choose 3}", "77c844607a2eebef9dd357bc51a2f71d": "c=1\\,\\mathrm{km}\\,\\!", "77c86d32a2276035ba191c7ce9e66110": "N_{\\mu \\nu}", "77c87fb210188161fd03729c4914cfa0": "0<\\epsilon<\\frac{1}{2}", "77c8ab9f91e070148a359c7ea117b521": " \\sqrt[5]{34} = \\sqrt[5]{32 + 2} \\approx 2 + \\frac{2}{5 \\cdot 16} = 2.025. ", "77c8acb015f3bcc738c241d5ef20f7f0": " \\frac{R[u]}{2} V_1 = \\int_{x_1}^{x_2} v(x) \\left[ -(p u')' + q u -\\lambda r u \\right] \\, dx + v(x_1)[ -p(x_1)u'(x_1) + a_1 u(x_1)] + v(x_2) [p(x_2) u'(x_2) + a_2 u(x_2)]. \\,", "77c8b54d0dbc4a825e0667e66207cb39": "v\\in H^1_0(\\Omega).", "77c8bf31789b542b035c028b53818a3c": "P(a+b)=P(a)+P(b)\\,", "77c8d903bf0fa0165c7d063e38dba67c": "E = Z \\times \\mathbf{P}^{n - 1} \\subseteq \\mathbf{C}^n \\times \\mathbf{P}^{n - 1}", "77c92527149f570cbf0ba1443d0e8ea9": "\\theta_0 /2 + \\pi/2", "77c94e301f220ad97ca22c18124dbd03": "x_M = z_M \\sin \\phi", "77c9686fc3b281d1001a958d745fa2fc": "S_{act}", "77c97ef745663f729d781d352d5c6e1c": "\\left({6 \\atop 2}\\right) = 15", "77c9b96f76252e0587a327c918e2b042": "v,v'\\in T", "77c9c82abc7b0d04ed7814653df77761": " \\operatorname{let-combine}[\\operatorname{let} F : \\operatorname{de-lambda}[F = L] \\operatorname{in} E] ", "77c9e1306d4be61f9fbbcb42378670c8": " x=a+bi", "77ca3339a24a71d340bba3374ed491a0": " n_2 < n_1", "77ca38b58544cb063267dd49f8c538ca": "A = \\{ i | a_i(x) = 0 \\}, ", "77ca6c8562ca88683db8ed761536ce4e": " \\int", "77ca8db303ec75f1db434950724ab133": " \\sigma(R) ", "77cab73f797c1832704f3624b20d7376": " \\chi^2 = \\sum_{i=1}^{r} \\sum_{j=1}^{c} {(O_{i,j} - E_{i,j})^2 \\over E_{i,j}} .", "77cae504b23dc6f9ad519592cf64fbd8": "A=V\\Lambda V^{-1}", "77cb4ab9fbafaeda306fdb39cf1e83bf": "y'(\\phi)=a", "77cb56cc9f3a1362c5574a7b9d3c49b6": "\\mbox{Aut}\\,G \\to \\mbox{Aut}\\, G/H", "77cb84e69e34562403990b122ee083f3": "\\mathrm{N\\,m\\,{rad}^{-1}}\\,", "77cbb9a8c1acfd58f63426017d009938": "\\frac{dv}{dt} = - (1/\\rho) \\nabla p - g(r/r) + f_r", "77cbe10b7b636647b1f693d35f827fb3": "u(t) = K_P e(t) + K_I \\int e(t)\\text{d}t + K_D \\frac{\\text{d}}{\\text{d}t}e(t).", "77cc46855d1fddbffc8d3b0c9ee0e36e": "\\lfloor n\\phi\\rfloor", "77cc4a60263d24a2b2bd90c18c198641": "\\mathbb{R}^S", "77cc5ba6c19a970eb2f75ff230fd4242": "\\begin{align}\nd\\mathbf{x}^2 - d\\mathbf{X}^2 &= d\\mathbf x\\cdot d\\mathbf x-d\\mathbf x\\cdot\\mathbf c\\cdot d\\mathbf x \\\\\n&=d\\mathbf x\\cdot (\\mathbf I - \\mathbf c)\\cdot d\\mathbf x \\\\\n&= d\\mathbf x \\cdot 2\\mathbf e \\cdot d\\mathbf x \\\\\n\\end{align}\\,\\!", "77cc72d3568eb94132a5814ce92aa6a5": " E_{\\omega} = E + \\omega \\left[ Q - \\frac{1}{2i} \\int d^{3} x(\\phi^{*} \\partial_{t} \\phi - \\phi \\partial_{t} \\phi^{*}) \\right], ", "77cc978a4ed258879bd53d438a972da1": "f_{\\rm{best}}^{(k)} = \\min\\{f_{\\rm{best}}^{(k-1)} , f(x^{(k)}) \\}.", "77cd20d23f50dd02c59268a6a884a29b": "\n\\min_u \\left\\{ \\mathcal{A} V(x,t) + C(t,x,u) \\right\\} = 0,\n", "77cd52bbc4aadbf7476c0c0320ab7c74": "<\\Delta R_i \\cdot \\Delta R_j> = \\frac{3 k_B T}{\\gamma}[U\\Lambda^{-1}U^T]_{ij}=\\frac{3 k_B T}{\\gamma}\\sum_{k=1}^{N-1}\\lambda_k^{-1} [u_k u_k^T]_{ij}", "77cd61e41b1c19ab9a3e0a080272a74d": "\\begin{align}\n&d(\\omega+\\eta) = d\\omega + d\\eta\\\\\n&d(\\omega\\wedge\\eta) = d\\omega\\wedge\\eta + (-1)^p\\,\\omega\\wedge d\\eta\\qquad(p=\\deg\\omega)\\\\\n&d(d\\omega) = 0.\n\\end{align}", "77cd9c6ec919fbf436fa4631e75bf41a": "L_1(Q)", "77cdcdb55d194e086a460c8a7cd0bc58": "m = 1,2,\\ldots,M", "77cde2c6b85e399cc071bf7a07b0a02b": "g(X_n)=g(\\theta)+g'(\\tilde{\\theta})(X_n-\\theta),", "77ce1804b88303057d9a24a55aef3c95": "i + \\frac{b}{2} - 1", "77ce3b7e05597e6301e6d01627724792": "y^{(n)} = \\sum_{i=0}^{n-1} a_i(x) y^{(i)} + r(x)", "77ce3e7c23ed79f2733ff8ed3f51dcf1": "K < 0", "77ce471d90aa1dfd0481fb5f16f14c31": "\\mathbf{V}_d(\\mathbf{r}) = \\nu\\left( \\dfrac{\\mathbf{I}_2(\\mathbf r)}{I_1(\\mathbf r)} + \\dfrac{\\mathbf{I}_3(\\mathbf r)}{2\\pi\\varepsilon^2 - I_0(\\mathbf r)} \\right)", "77ce64c040540403462db7192b79b9de": "\\dot C_{FF} (t) = \\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 {{1\\,\\,\\, - \\,\\,\\exp ( - \\lambda \\,t)} \\over \\lambda }", "77cfdc19bea37a5b82a63ef8b2d362eb": "\n \\boldsymbol{D}^p = \\tfrac{1}{2}[\\boldsymbol{L}^p +(\\boldsymbol{L}^p)^T] ~,~~\n \\boldsymbol{W}^p = \\tfrac{1}{2}[\\boldsymbol{L}^p -(\\boldsymbol{L}^p)^T] \\,.\n ", "77cfef3d1ee5b80902b23fb751b79095": " C \\cap D ", "77d01b0673a0bf00022ae4c2b6720ea7": "D(s)=1+kG(s)=0", "77d028c8a057912d4ecfa885178d11ec": "g=e^{-B/ln[c]}.", "77d0678c0ec7ca2b3e14338dd98714c1": "\n\\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{\\tilde{e}}_{k} \\rangle \\mathbf{e}_{k} =\n\\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle \\mathbf{\\tilde{e}}_{k} = \\mathbf{v}\n", "77d0a8f2ff4a045d1398f4d8fdaa7e14": "U/D=\\pi", "77d0d616b2ffe2cb3d32ba51bbf940e0": "P_{\\text{ps}} = { 1.52\\left( \\phi+d\\right)d\\sqrt{f_{\\text{c}} }\\left(100\\rho_{\\text{e}} \\right)^{0.25} } ", "77d1ec52fb5d70fc3bca5c9562035a6b": "C = \\sum\\frac{(-1)^i}{s_i-1}=\\frac11 - \\frac12 + \\frac16 - \\frac1{42} + \\frac1{1806} - \\cdots\\approx 0.64341054629.", "77d231458244b8b764ca36df746f597f": "{\\rm as}_5(1,5,3,2,4) = 4, ", "77d24d8c2c30e46f43898c0cf5763d19": "0 \\leq x \\leq y", "77d24f41dc211dc014b59259709afafd": "\\mathbf{Y}_{lm} = Y_{lm}\\hat{\\mathbf{r}}", "77d27aec73a04b9941bf6f855157dbf0": " {f (x^{*}) \\leq f (x), \\qquad \\forall{x}\\, \\in X} ", "77d2aad1f622802fd6aedba5b8f7312d": "1-\\sin^2(\\theta) = \\cos^2(\\theta).", "77d2dead6a7324c611c0c25d61d39fcf": "\n\\langle T_v\\exp_p(v), T_v\\exp_p(w)\\rangle_q = \\langle v,w\\rangle_p.\n", "77d33476b457b7add5d9c9d2e7211e06": "2^{p-1} \\equiv 1 \\pmod {p^2}", "77d3600f4dbb227427393381570c4a3c": "A_2=\\int_{x_1}^{x_2}y(x)dx", "77d36eb3ebcfa4d3fbc7d6d1be8f398b": "A=\\partial x/\\partial y", "77d3ca7c2c991a1590debc4ed2191c4c": "\\exists \\ u_{1}, u_{2}, \\cdots u_{k}\\in (V\\cup\\Sigma)^{*}, k\\geq 0", "77d4139b2d7d6259e04b2db105dac326": "V_{dd} = -V_{ss}", "77d470cf1469041a4e31cc9b1a6470b0": " 0 \\le \\mu_i \\le \\nu_i ", "77d494f77d69a63d80b5d6b02c16f725": "{x_M}^q", "77d4cc7496904de22407963adf21f858": "P_n \\xrightarrow{\\mathrm{D}} P \\Rightarrow P_n \\xrightarrow{\\mathrm{TV}} P", "77d50766c590c989c9898ffd5ec77304": "RE(x_i,y_i) = \\left\\vert r^2 + 0^2 - r^2 \\right\\vert = 0", "77d53502b675d8c07b8cd806ff37ba7b": "E = pc \\,\\!", "77d57931701df5419c1ce2cc617f4c97": "2p_1(E\\oplus F)=2p_1(E)+2p_1(F),", "77d6585b1ac62bf7cafa0c732397473e": " \\nabla^2 \\mathbf{A} = \\nabla(\\nabla \\cdot \\mathbf{A}) - \\nabla \\times (\\nabla \\times \\mathbf{A}) ", "77d7480cce3b1b91eab6073da993c64a": " C_{(+)}= \\sigma_1 = C_{(+)}^* = s_{(2,+)} C_{(+)}^T = s_{(2,+)} C_{(+)}^{-1} ~~~~ s_{(2,+)}=+1 ", "77d7b719bb3c602b72d9c2c3c000ee1d": "G_6", "77d7c8cc9fb8cbed493f183422d38bdb": "\\exists y(y \\frac{3}{2}|\\tau(M)|.", "77e051c17e0ebe18278aa7a0b39f0f5a": "a = p_1^{\\alpha_1} p_2^{\\alpha_2} \\cdots p_r^{\\alpha_r}", "77e0b7132d93e7a86551c239a2368234": "\\left(\\mathbb{Z}^{(A)}\\right)^n", "77e106c595979b780494dcb3ef423e90": "\\pi_T = a \\frac{n^2}{V^2}", "77e109ea483f26392c434add73fdbbd5": "\\log f_X(x)= \\log \\left(\\alpha\\frac{x_\\mathrm{m}^\\alpha}{x^{\\alpha+1}}\\right) = \\log (\\alpha x_\\mathrm{m}^\\alpha) - (\\alpha+1) \\log x.", "77e1334b25d6864f8120b593987f6111": "\\epsilon=\\frac{a+b\\sqrt{\\Delta}}{2}", "77e14fae01493bceed978a3af78c3bd1": "R_2={(1.523-1) \\over -7.0\\ \\mathrm{dpt}} =-0.0747\\ \\mathrm{m}", "77e18fe91b21587204904091070e19a4": " S= \\int \\partial^\\mu A^\\nu \\partial_\\mu A_\\nu ", "77e1af127d135878b60113296dbe0f32": "4 \\sqrt{q}", "77e2134ffabaea1f577e964b1575bd2c": "\n\\begin{align}\ne^{2 \\pi i} &= 1 \\\\\n(e^{2 \\pi i})^{i} &= 1^{i} \\\\\ne^{-2 \\pi} &= 1 \\\\\n\\end{align}\n", "77e2862d4c661903add9a5061cf8cee6": "|\\beta|^2", "77e2eccb43ef776aa10fe90d51bc70b1": "\\scriptstyle P_c", "77e33932672ca5975b2994703d9a85ee": "U_n(x)= \\frac 1{2{n+\\frac 1 2 \\choose n}} P_n^{\\frac 1 2, \\frac 1 2}(x)= C_n^1(x).", "77e3dd9bd3549d1d44984314072d51dc": "\\delta = \\frac{\\partial \\bold{U}}{\\partial P}", "77e3de814d06ba9f81c78a0c89646725": "1^{p-1}+2^{p-1}+ \\cdots +(p-1)^{p-1} \\equiv -1 \\pmod p", "77e3f591834d167fe312dcdb4d3b60e7": "\\int_a^b f(x)\\,dg(x) := -\\int_a^b f(x) \\,d (-g)(x),", "77e401a15edc8d5e01bad8939bb8a4c6": "p\\in H, p' \\in S'", "77e42d7e72ef76560292d4885e1ba024": "H^i(X/W) = H^i_{DR}(Z/W) \\quad(= H^i(Z,\\Omega_{Z/W}^*)= \\lim_{\\leftarrow}H^i(Z,\\Omega_{Z/W_n}^*))", "77e46402d2d2c500807f314c16b5cd75": "p\\in\\operatorname{cl}(A) \\Rightarrow f(p)\\in\\operatorname{cl}(f(A))", "77e48305824e9bb54606edc126826ec8": " V = \\frac{1}{3} (a^2 + ab + b^2) h ", "77e4b9dc83a48501c7e6ed3f06839e30": "\\vartheta = \\frac{\\alpha(T_o-T_u)}{e}\\,", "77e4ba71e43feb4957dcbda377aa433e": "\\hat{\\boldsymbol\\varphi}=(-\\sin(\\varphi),\\cos(\\varphi)) = \\hat {\\mathbf{k}} \\times \\hat {\\mathbf{r}} \\ , ", "77e510a71a1a746ea375127f24e6438f": "(x)^{(3)} = x(x + 1)(x + 2) = 1 \\cdot x^3 + 3 \\cdot x^2 + 2 \\cdot x", "77e51918208190b4b02d7f93c4c005d6": " \\mu(B_n) (1 - \\epsilon) = \\int (1 - \\epsilon) 1_{B_n} \\, \\mathrm{d} \\mu \\leq \\int f_n \\, \\mathrm{d} \\mu ", "77e521fbeb914b82c9fa7c3be7e0d4bc": " p_k=p \\,\\!", "77e5282412c2f126b2c33635027b9a44": " v' = v - V_c, ", "77e5fb4d58cc3c7ece813f6de3a3f2f4": " \\frac{1}{5}\\left(2051 + 2053 + 2055 + 2050 + 2051\\right) = 2052", "77e6136362fd6b2c93158fae6bcbeb75": "\\boldsymbol\\beta = {\\rm vec}(\\mathbf{B}), \\hat{\\boldsymbol\\beta} = {\\rm vec}(\\hat{\\mathbf{B}})", "77e61462d1adabba326f954b07b72122": "\\blacksquare", "77e636763321f232b384c1bb9b66d616": " \\vec p", "77e6400cce2ccc026a100efad0fbf7ae": "m_S=\\sqrt{\\frac{e^2}{4 \\pi \\varepsilon_0 G}} = \\sqrt{\\alpha}\\, m_P ", "77e66c75fddd29bc55a1c3c082cd882f": "({b^{(\\dagger)}}_{\\nu})", "77e6ad2c66611e2bd3f30e7a04b78e93": "\\begin{align}\n& \\varphi(\\mathbf{r},t) = \\dfrac{1}{4\\pi \\epsilon_0} \\int \\dfrac{\\rho(\\mathbf{r}',t_r)}{|\\mathbf{r}-\\mathbf{r}'|} \\mathrm{d}^3 \\mathbf{r}'\\\\\n& \\mathbf{A}(\\mathbf{r},t) = \\dfrac{\\mu_0}{4\\pi} \\int \\dfrac{\\mathbf{J}(\\mathbf{r}',t_r)}{|\\mathbf{r}-\\mathbf{r}'|} \\mathrm{d}^3 \\mathbf{r}'\\\\\n\\end{align}", "77e6ec4104f30a0569e6a90be237dadb": " Y_i = \\begin{cases} 1 & \\text{if }Y_i^\\ast > 0 \\ \\text{ i.e. } - \\varepsilon < \\boldsymbol\\beta \\cdot \\mathbf{X}_i, \\\\\n0 &\\text{otherwise.} \\end{cases} ", "77e6f6785a81f8db8b580fc0e951a408": "u=\\sqrt{{p\\over 3}} \\qquad \\text{and} \\qquad v=-\\sqrt{{p\\over 3}},", "77e7595a48601d9284d1f6cb23307bbb": "O(m \\alpha(n)\\log^2 n)", "77e7af77ea527fb45ef930880fa08ed6": "\\lambda_\\mathrm{op,ems}", "77e7d8d25daed58837c8bc13f63e1e7c": "f_n^{(k)}(0)=\\begin{cases}\\alpha_n&\\text{if }k=n,\\\\0&\\text{otherwise,}\\end{cases}\\qquad k,n\\in\\mathbb{N}_0.", "77e85cd4d738e5686a229ca35ac1eebf": "\\mathcal{F}(R_jf)(x) = i\\frac{x_j}{|x|}(\\mathcal{F}f)(x)", "77e883ddc6b54b071ab802b13f93e171": "\\int_x \\int_y f_{X,Y}(x,y) \\; dy \\; dx= 1.", "77e889c7161de8224d0e04f493c1590c": "\\displaystyle\\mathbf r_1,\\,\\ldots,\\,\\mathbf r_N", "77e89d850fd7620c3b1acf5536a84531": "h={R\\tan\\theta \\over 4}", "77e93c64a20f1cbc6f223b14628bac6a": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{T}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "77e964d3f7397f4c8df31cb307aa5bc5": "2\\pi-\\nu\\ ", "77e9741f340bda72d0c30e3a95a034b0": "u'(c_{t})=Eu'(c_{t+1})", "77e9851479cd67aca184f2f6b00a23e0": "k \\gg 0", "77ea0ec8e2407960bcffd9ef38eb63d4": "\\begin{matrix}\\Delta L = 0, \\pm 1, \\pm 2 \\\\ (L = 0 \\not \\leftrightarrow 0, 1)\\end{matrix}", "77ea19bc59e1aa9e6dbf6913d883978a": "d(A_x)= x", "77ea943eaa067888aa16d66b7a057ae0": "F(x;k) = u;", "77eaecde2835ee469f870bb6d0d0fc58": "\\scriptstyle \\langle 1 \\rangle \\oplus \\langle -1 \\rangle ", "77eb2ffee14855ba40349aa0cc5a6718": "\n \\hat{S}_{ij} = S_{ij}~\\sqrt{g^{ii}~g^{jj}}\n ", "77eba71c50aa0f35e1bdecf0f6480c64": " \\{ \\zeta \\in \\mathbb{C}: \\operatorname{Im}\\,\\zeta > 0 \\} ", "77ebadf470f1e13972aaf97ab9c858e6": "H^{p,q}(X,Z/\\ell)", "77ebc881727baea86ef0140e6b240460": "\\vec a_t", "77ebdf39b0b93395fd1488066249522f": "\\mathbf{u} = (u,v,0)", "77ec8309956ce9d008a6db528f8a13d2": "h \\approx 0", "77ecb831cc39844638f9bd4b6466efe7": " \\mathbf{T} = 2\\eta_s \\mathbf{D} + \\mathbf{\\tau} ", "77ece71cf2e205843ee3acefdbdfd711": "\n\\begin{align}\n\\sum_{R=1}^{N-n}{\\prod_{j=1}^{n-1}(N-R-j) \\over R} & \\approx \\int_1^{N-n}{(N-R)^{n-1}\\over R} \\, dR \\\\\n& = N\\int_1^{N-n} {(N-R)^{n-2}\\over R} \\, dR - \\int_1^{N-n}(N-R)^{n-2} \\, dR \\\\\n& = N^{n-1}\\left[\\int_1^{N-n}{dR\\over R}-{1\\over n-1} + O\\left({1\\over N}\\right)\\right]\n\\approx N^{n-1}\\ln(N)\n\\end{align}\n", "77ecfbb69e598dcb1f720c6980cb592d": "u_2(r)", "77ed084809ff1f4caa840ef394ee428e": "\\lbrace \\Delta \\sigma_i \\rbrace", "77edc71d1f1786054f5a90fd7a972a65": "m_{h^0}^2 \\le m_{Z^0}^2\\cos^2 2\\beta + \\frac{3}{\\pi^2} \\frac{m_t^4 \\sin^4\\beta}{v^2} \\left(\\log \\frac{m_{\\tilde{t}}}{m_t} + a^2 ( 1 - a^2/12) \\right)", "77ee1196712d897cf126b22c1f3e62b5": "L = \\sum_{i = 1}^{n} b_{i} (x) \\frac{\\partial}{\\partial x_{i}} + \\sum_{i, j = 1}^{n} a_{ij} (x) \\frac{\\partial^{2}}{\\partial x_{i} \\, \\partial x_{j}},", "77ee4574788e667094d77dfa33fc9be7": "ODF", "77ee6b0ee0715f53baa677bf02d50c3c": "e_k \\in L^2", "77ef55963e2e80d3a022982f1055c570": "\\hat\\sigma_t^{-2}", "77efc928fd076c682e996a5cece6b6fe": "r_0:\\tilde Z_0\\to \\tilde{\\mathbf Y}", "77eff3e8a08db666685078275e175aa7": "\\vdash (\\exists y_1,\\ldots,y_n)F(y_1,\\ldots,y_n)", "77eff95a64e70cc677fc2e2eaba5883e": " \\pi - 2 \\epsilon ", "77f0108b9d6974c3932a3d7dd34f7f07": "\\scriptstyle{ct\\sqrt{-1}}", "77f0c67d1ab1b22d5f295b2c10a23b8e": "a,b=2,3", "77f0d236eee84831f24276bbcf21d231": "\\begin{align} \n\\psi_r(e^{i\\theta}) &=1+\\frac{1-r}{1+ r} \\cot \\left(\\tfrac{\\theta}{2} \\right ) K_r(e^{i\\theta}) \\\\\n&\\le 1+ \\frac{1-r}{1+r} \\cot \\left (\\tfrac{1-r}{2} \\right ) K_r(e^{i\\theta})\n\\end{align}", "77f188223e1a7cc578043ad0082da915": "\n\\frac{1}{c^{2} \\left(1 - \\frac{r_{s}}{r} \\right)} \\left( \\frac{\\partial S}{\\partial t} \\right)^{2} - \n\\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{\\partial S}{\\partial r} \\right)^{2} -\n\\frac{1}{r^{2}} \\left( \\frac{\\partial S}{\\partial \\varphi} \\right)^{2} = c^{2}\n", "77f1bc401fba7ee262250ddfbcfcacd5": "\\scriptstyle \\mathbf{E}[ X] = k \\theta ", "77f1f8727602d89ef418f793ef3f9896": " f(x) = C \\exp(-|x|^2/2) \\int_{R^n} (Bf)(x+iy)\\exp(-|y|^2/2) \\, dy, ", "77f23b7c842939ffbcd2e7e00db3a39b": "\\phi\\left[W_s(x)\\cap E_i\\right] \\subset W_s(\\phi x) \\cap E_j ", "77f291882ea0fbbcba4d24f64dd89754": "S(P)= \\alpha + P\\cdot(1-\\alpha) = P - \\alpha\\cdot(P-1) .", "77f2c31976376267bf0f0afb55e50f06": "\\{f^n(x)\\}_{n\\in \\mathbb{N}}", "77f2ca3a9781a1ff4a48cae2d84983d5": " -\\infty < M^-_\\infty = M^+_\\infty < +\\infty,", "77f2e8bd51d6c5d33bd9b2dec137810e": "\\pi^e", "77f319ae0e543425a1fff0683e8b6ddb": "L_\\rho(\\gamma)\\ge\\inf\\{h(s):s\\in[0,r]\\}\\,\\mathrm{length}(f\\circ\\gamma)=\\infty", "77f32dda297e00f75d6ec5df0845d012": "\\mathcal{J}(\\theta)", "77f35676c5f8483faa40b5b6568def9b": "\\mathbb{Z}_{20}^\\times", "77f3573d41f320bbcd4e8e7d39e7c55a": "T_\\theta : [0,1] \\rightarrow [0,1],\\quad T_\\theta(x) \\triangleq x + \\theta \\mod 1, ", "77f375b48523c8f7b164989cad583791": "d\\left(f(x), a\\right)\\le M", "77f378e02220bdafac1bdff3155dff4f": "\\frac{1}{\\sqrt{-g}}\\frac{\\delta \\sqrt{-g}}{\\delta g^{\\mu\\nu} } = -\\frac{1}{2} g_{\\mu\\nu} .", "77f3e83675e243e2e27f93fd94950873": "\\Big( \\pi \\models G\\phi \\Big) \\Leftrightarrow \\Big( \\forall n\\geqslant 0: \\pi[n] \\models \\phi \\Big)", "77f403f97d0712d04489c9ab5c679c79": "p\\in \\Sigma, p\\neq \\hat{p}", "77f455f320b9513c4cff74737b841704": " \\mathbf{P} \\mathbf {X_2} \\boldsymbol {\\beta_2} = \\mathbf{P}\\mathbf {y}", "77f468c3960c07e929c0819b63c772ad": "\\tfrac {11}{15} \\pi^2 + \\tfrac {1}{2} \\ln^2(-1 / \\phi) \\,", "77f4826c75c3a23dc884b061cced0dd0": " m = m'", "77f491f402172fd6fa3dbaec68f2c86e": "e^{e^{e^{79}}}<10^{10^{10^{34}}}.", "77f4e315e5375df95edb0104c6cc6edf": "\\langle a_0,\\dots a_{n-1} \\rangle", "77f504e1a4b160558a329ef9bef2859c": "N=\\int d\\vec{r}|\\psi(\\vec{r})|^2", "77f51c4e54924b0d76c29aa16ab611c4": "\\Delta_{vap}{H}^\\ominus", "77f52ee415b5bf4ba1bad7816bdd0389": "1 = [1,1,\\ldots,1]^T", "77f535055b223b467ec54dbc2fffab05": "h(i)=\\sup_\\pi \\left\\langle \\sum_{t=0}^{\\tau-1} \\beta^t R[Z^\\pi(t)] \\right\\rangle_{Z(0) = i}", "77f5e3aec6b1cb261d2e2305559e642a": "\n\\frac{d\\xi}{d\\tau} = \\frac{d}{dt} \\left( \\frac{x}{y} \\right) \\frac{dt}{d\\tau} = \\left( \\frac{\\dot{x} y - \\dot{y} x}{y^{2}} \\right) y^{2} = - h\n", "77f61fb16b569ec65453a74c8d9e5f41": "\\mathbf{q} \\cdot \\mathbf{p}", "77f651fc904ace30f075feb6f365cccd": "\\omega+3", "77f6a52d5f7058eda3af199a1ca775eb": " 1 \\to \\{\\pm 1\\} \\to \\mbox{Pin}_V(K) \\to \\mbox{O}_V(K) \\to K^*/K^{*2},\\,", "77f6b08dc8b6d3c83d8b809f70622350": "s = 1/2", "77f6b1767f0c70a04930598e3b6a0184": "\\mathbf{a}_2 = \\mathbf{a} - \\mathbf{a}_1 ", "77f6fc022192c8d92da423ffab3e9db1": "\\nu_0\\ll\\mu", "77f72bea40dcc1200b81059a54744448": "\\hat{\\bold{H}}_{\\operatorname{SCV}}", "77f73d403cd19df3d6fd67655ef21478": "X_0 \\prec X_1", "77f77b69ff879a1f8b6c40863f2974ff": "\nf_z =\\ -J_2\\ \\frac{1}{r^4}\\ 3\\ \\sin i\\ \\cos i\\ \\sin u\n", "77f77bbc23114683cacb9c49fc789a39": "\\begin{align}\n & \\min_{\\alpha (X_0),\\beta (X_0)} \\sum\\limits_{i=1}^N {K_{h_{\\lambda }}(X_0,X_i)\\left( Y(X_i)-\\alpha (X_0)-\\beta (X_{0})X_i \\right)^2} \\\\ \n & \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\Downarrow \\\\ \n & \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hat{Y}(X_{0})=\\alpha (X_{0})+\\beta (X_{0})X_{0} \\\\ \n\\end{align}", "77f7bde857d783fcc00355ca8e02b0b6": "T^l_d", "77f876f4236ad721aa5429cd86713cad": "{\\frac {|BD|} {|DC|}}={\\frac {|AB| \\sin \\angle DAB}{|AC| \\sin \\angle DAC}}. ", "77f887c3d1a21d681c74689072de8e92": "\\tfrac{A}{100} + 45", "77f8c543abeb862d414911edddb1ed6e": "F(R)", "77f94c062f19ca9a2e193b90e6e151ca": "\\det(B)", "77f96738a9b28aa481ff3aadd416b98f": "=E(W^4)\\,", "77f979be5848fdaf97b618c6a2b23e74": "g=p+0i", "77fa0b0556acfc1c2c363bc9a1ba068c": "\\pi(j)", "77fa36a01e8e6ae9ed2468c0146a20f6": "{B}_{9}^{(1)}", "77fa52860180a36ee43d1e5c367a44bf": "J_{\\varphi_1}(\\varphi_1^{-1}(P)) = J_{\\varphi_0}(\\varphi_0^{-1}(P))\\cdot J_{\\varphi_{01}}(\\varphi_1^{-1}(P)). \\, ", "77fa53ae7661197d31fb67fcbcf774e7": "A(x,y)\n", "77fa669f471ce0ec261a653d6f49c809": "U_{NI,J} = U_{NI-1,J}\\frac{M_{in}}{M_{out}}\\,", "77fa6ee9e2d9dd250a00bee10251a1d3": "\\begin{align}\n\\boldsymbol{x}_i&=\\boldsymbol{x}_0+\\boldsymbol{P}_i\\boldsymbol{z}_i\\\\\n&=\\boldsymbol{x}_0+\\boldsymbol{P}_{i-1}\\boldsymbol{z}_{i-1}+\\zeta_i\\boldsymbol{p}_i\\\\\n&=\\boldsymbol{x}_{i-1}+\\zeta_i\\boldsymbol{p}_i\\text{.}\n\\end{align}", "77fa7597d45d3f49c7e67687552cdb45": "G_nf", "77fa8291689f096468eaa6ea9a39a72d": "\\frac ab.", "77faaa561a9b849fe2e415b84dca53b2": "(X, Y) = \\left(\\frac{x}{1 - z}, \\frac{y}{1 - z}\\right),", "77fadbc824a15975328f76b444a0faff": "T_{\\text{D}} ", "77fb081c9f0c9f01390ad387b4288bb2": "\\overrightarrow{\\mu}", "77fb232d9624d9ae29de97db79c1ac7d": "\\scriptstyle \\mathbf{X}", "77fb5e22e684d4419a9f377362979404": "\\oplus_U", "77fb618c5d38ab7f04203f1683af0af1": "\\{(v,1) | v \\in V\\}", "77fbae376df01d98b8d68b15cd04115a": "P(\\omega)/Fin", "77fbc66c9d58e5578c712b2276decd0c": "Z, X", "77fc238e6d05b913b7fe7db18ed21b9f": "K[[w,x,y,z]]/(wy,wz,xy,xz)", "77fd274741b0e437343ef4faea9953dc": "c_3=\\sqrt[3]{199+3\\sqrt{33}}", "77fd2a03c39c778ffddc6399f9593192": "E(k)\\!", "77fd3b9a65a3aec7458c63ed49297924": "v\\in V\\,", "77fd7504ba036698fe8a19e98e9af84e": " \\frac{1}{6} ", "77fda0f1b5252c36fb517f0f08e8efd8": "(\\Theta\\subset\\mathbb{R}^r,S)", "77fdb0c8e6fdd3afca158aa512408ca4": "d_{(ij)k}", "77fe1f439e5f64b8e82dea53495e4db3": "V_v = 0.286 \\cdot V_r ", "77fe479b7e737afb05981dec6aa16754": " \\langle q|\\mathbf{\\hat T}(\\lambda)|\\psi\\rangle = \\langle q-\\lambda|\\psi\\rangle = \\psi(q-\\lambda) ", "77fe855048a05aa7b94ba5e409befb8b": "\\varepsilon_2 = \\frac{1}{E}(\\sigma_2-\\nu(\\sigma_1+\\sigma_3))", "77fea35b967dc192409f95d5b2d92475": "\\mathit{a(V^*)}", "77fefcc3e115194ab07dc7f70421a463": "925 \\times 10^3", "77ff0b4384ca584fa616e047c9180a84": "I_{\\nu} = \\frac{2 \\nu^2k T}{c^2}", "77ff3b73d812f17376ab78b903589872": " (\\forall{\\alpha<\\kappa^{+}})V_{\\alpha}\\setminus\\bigcup_{\\xi<\\alpha}V_{\\xi}\\,", "77ffbb85236fc981b5b46810c07cae18": "\\nu_{\\rm zx}", "77ffbe60bb65652a923dd8eaf40ea31b": "\\sin((N+\\tfrac12)t) / \\sin(t/2)", "77ffd36e7308837e3af1d77ce91cdb07": "\\alpha_p \\ll \\alpha_n", "77fff29d192368da8dbb73af59ba7898": "(A + B)^m(A + B)^n", "78002e43039036a654c419b2d9c0fbd4": "f: {\\Bbb R} \\to {\\Bbb R}", "780043b3e48cd96218ebd4c450c73e55": " |\\bold{k}|^2 = k_x^2 + k_y^2+ k_z^2. ", "78006d0fda2dbe638eb06f8273557868": "y_3=y_2+h(\\tfrac14k_1 + \\tfrac34k_2)=\\underline{1.227417567}", "78008c541f7a26e18b7d0fc3a7eda3c1": "\\textstyle\\int \\tan x\\,dx,", "780138d43d77903843e8bd4b815d3adb": " g_{\\rho 0}g^{\\rho \\sigma} = \\delta_{0}^{\\sigma} \\,", "78014a10d90cfdb88068956a12ed89d6": "= \\operatorname{E}(\\operatorname{cov}(X,Y \\mid Z))+\\operatorname{cov}(\\operatorname{E}(X\\mid Z),\\operatorname{E}(Y\\mid Z))", "780154eea7556df040ab28e558031e67": "\n\\left[\n\\begin{array}{cc}\np_1 & p_2\\\\\np_2 + 1 & p_1 \n\\end{array} \n\\right]\n\\left[\n\\begin{array}{cc}\nu_1\\\\\nu_2\n\\end{array} \n\\right]\n=\n\\left[\n\\begin{array}{c}\n\\frac{p_1+6p_2}{5.0} \\\\\n2p_1-6\n\\end{array} \n\\right],\n\\ \\ \\ for \\ \\ p_1\\in[2,4], p_2\\in[-2,1].\n", "7801836b968544b7edbad32a6b928dbf": "2^t \\geq Vol_2(d,n)", "780188722b9d655c7668d62a8c0d2427": "\\neg A\\or\\neg C", "78018c5a5362e2c287a39aef68792bbc": "\\| \\varphi \\|_{\\alpha} = \\max_{x \\in K_i} \\left |\\delta^{\\alpha} \\varphi \\right | ,", "7801b81afc18c2a55b6f49dcc43b9ede": "S_{11}", "7801eef112c89e32c6750b481f4f54c8": " z = \\sqrt{a}\\,x + i \\sqrt{b} \\,y . ", "7802395fc63640b0ffb7a42f9e5d19c0": "\\exp(\\omega t)\\,", "78027e8a0f734c852197e13768d9ec4c": "\\partial_v(L_v) = 0", "780290a07f6cdc169c610f2e703e6787": "\\eta\\rightarrow \\infty", "7802942d2178e5e64e17ca38502e69ca": "A_{\\beta,s}=((\\lambda-\\beta)I+A_{\\lambda,1})^s=\\sum_{m=0}^s\\binom{s}{m}(\\lambda-\\beta)^{s-m}A_{\\lambda,m}", "7802a9f8b6472f4fbea0473bca6103c4": "\\frac{n + 2}{3}", "7802d864bd25b6025e4b2f713984b506": "\\frac{d\\theta}{d\\zeta}", "7803388fd2c7be963e470d0454107ad5": "{{i}_{IN}}-{{i}_{B}}", "780339bf32cc54eb266893f864e57bad": "x^2-ay=0,", "780355dfc079286a8216cb807b84e8b0": " \\nu = (r-\\delta)/\\sigma - \\sigma / 2, \\quad b = \\gamma K / (\\gamma - 1). ", "78036d14c98084aaa8546e900b2465b6": "F \\left( u_{i + \\frac{1}{2}} \\right) = f^{low}_{i + \\frac{1}{2}} - \\phi\\left( r_i \\right) \n\\left( f^{low}_{i + \\frac{1}{2}} - f^{high}_{i + \\frac{1}{2}} \\right)", "78038ada14e87814740152d969fb022d": "\\textstyle \\{ \\sum_i \\lambda_i \\mathbf{v}_i : (\\forall i) \\; 0 \\leq \\lambda_i \\leq 1 \\}", "78038ef2da60fbfbf0ad3592f4a57e4b": "f = F \\cdot ( 1 + m )", "78039fba422ea491d767322de68a3450": "g = g_e. \\,", "7803b8a49c868ab2de7dc1610d8ff917": "R_2 = \\sqrt{R^2 + x_2^2 + y_2^2} \\, ", "7803d34c169f3e0e47bcc44c6551e7e9": "(A,\\cdot)", "7803d8312f98b71ba38d31c13df92537": "R_1\\,\\!", "78041439faa6ab5bcf105159c0e7ebe6": "min[\\frac{vitual delivery_{p,DW}}{virtually committed order_{p,DW}},1]", "78041f7ea22c460dcd5567d8f47fa7f4": "K_1,K_2,\\dots", "7804202a55c888bbc03100f10c274c9a": "L_n[\\alpha,c],\\;0 < \\alpha < 1=\\,", "7804701774d01e5817efd687f38b1454": "R_2 =(57-12) \\times 0.0519 ", "780491ea4a379c1ef7a223ca0b6e6fe8": "{\\textbf{x}}_3 = \\scriptstyle\\sigma_3^{2} (\\scriptstyle\\sigma_1^{-2}{\\textbf{x}}_1 + \\scriptstyle\\sigma_2^{-2}{\\textbf{x}}_2)", "7804960ac407fe3c047f3771abc66ab4": " \\geq \\int_{\\min_x \\in X}^{x_1} f_{\\theta_1}(x_0) f_{\\theta_0}(x_1) \\, dx_0", "7804aeca69b8cfa911f45899e99b47f5": "T_p N \\to T_p M \\to T_p M / T_p N", "7805079efc09a420652639effd3a1b85": "dA_3= \\left(\\mathbf{n} \\cdot \\mathbf{e}_3 \\right)dA = n_3 \\; dA,\\,\\!", "780513c4b0e4330b78db5d97f614a535": "\n\\nu = \\frac {\\eta} {\\rho}\n", "780528fc1a587e7faba7a4d8f1dde4fe": "K: \\mathbf X \\times \\mathbf X \\to \\mathbb R", "78056f44df1dc625165172caef5f9007": "\\Delta A_p = A_{p+1} - A_p.", "7805a611740bd881418657f7a074da6d": "I_{rim}=KI_{flywheel}", "7805ae9392310a98e82697b969deabb4": "W\\cdot \\operatorname{sinc} ( W n)", "7805af785578354794aff2d757e3721c": "s = \\gamma n\\,", "7805bbd9652edbed20a48376f4e76793": "0 < \\varepsilon \\ll 1, ", "7805d653b3f7510dc6bf09d9eac9a706": "\n\\begin{vmatrix} a & b\\\\c & d\\end{vmatrix}\\,=1", "7805eba586f1593f610d48d02c5d5ecb": "n=m", "7805ee38b7f48c63fe7668397ca90ea8": "T < 0", "780615ae873778a8a379d2e93a754cb4": "P(V) = \\lim_{n\\to\\infty} \\frac{1}{n} \\log Z_n(V)", "7806902fe682741452268aac06438f64": "\\tan\\frac{2\\pi}{15}=\\tan 24^\\circ=\\tfrac{1}{2}\\left[\\sqrt{2(25+11\\sqrt5)}-\\sqrt3(3+\\sqrt5)\\right]\\,", "7806c5b03aa820bd472b3d22dad8061d": " [p_0 ff^\\prime]_x^y = \\int_x^y (q_0 -\\lambda)|f|^2 + p_0 (f^\\prime)^2 .", "7806c6806c6d69f69fe8b1e8234d2b09": "O_{8}", "7806ccf71bc5e0403cb9a822af567742": "G(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iint \\dfrac{e^{-j\\theta t-j\\tau\\omega}}{\\phi(\\theta,\\tau)\\phi(-\\theta,\\tau)}\\, d\\theta\\,d\\tau", "7806e8618de1351b3384e408ff2bd80d": "\\Sigma_L", "7806f983420034735ddd02a2b9f34b1e": "\\displaystyle\\underbrace{(a_1 \\ldots a_{j_1})(a_{j_1+1} \\ldots a_{j_2})\\ldots(a_{j_{k-1}+1} \\ldots a_n)}_{ k\\ \\mathrm{cycles}}.", "7807066d4a42e5fa3ead6e273b483dda": "v \\in V_i", "78074f0d51ac15b7cd2098a19b39ed21": "e^{i\\bold{k}\\cdot \\bold{r}} = \\sum_l (2 l + 1) i^l j_l(kr)P_l(\\cos\\theta)", "7808398e389a0e4df77d23b7e9c6a8a0": "\\Delta (G) \\geq n/3", "78089209070e031b4f245144b03ee055": "n' = n \\times (M_{3x3} \\times V_{3x3})^{-1T} = n \\times M_{3x3}^{-1T} \\times V_{3x3}^{-1T}", "78092a2c9eba60ec0e678e8eb543bcea": "\\tau\\mapsto\\frac{a\\tau+b}{c\\tau+d}", "7809d5b9a85daf9de422c37eac3dd83a": "YPR = (\\theta_3,\\theta_2,\\theta_1)", "7809ed3274ce5fa32e5e59f6bd95fea9": "\\varphi^{n+1} = \\varphi^{n} + \\Delta t F( \\varphi^n ),", "7809f600e1c0e875f00a26e8018eaef8": "\\varrho(\\eta(m))=m", "7809f9d582254ca1b3717391c9552676": "\\mathcal{F}(\\mathbf{c}) = \\mathcal{F}(\\mathbf{a})\\mathcal{F}(\\mathbf{b})", "780a1130686576cb43de8ac99f90785a": "f(a) = \\{a\\}", "780a3110687a17872d3903d601c4ef36": "\n\\begin{align}\nU(x,y,z) \n&\\propto \\iint_\\text{Aperture} \\,A(x',y') e^{-i \\frac{2\\pi}{\\lambda}(lx' + my')}dx'\\,dy'\\\\\n&\\propto \\iint_\\text{Aperture} \\,A(x',y') e^{-i k(lx' + my')}dx'\\,dy'\n\n\\end{align}\n", "780a3fab5590fa4b5b88c08037d3dc16": "\\varphi(\\mathbf{x}) = \\sum_{i=1}^N a_i \\rho(||\\mathbf{x}-\\mathbf{c}_i||)", "780aa5fea3d44ec3c60563ae91bd8f0f": " HA=HX-\\frac{1}{N}\\left( \\left( HX\\right) \\mathbf{e}_{N\\times1}\\right) \\mathbf{e}_{1\\times N}, ", "780af14297d0bc616667d3b52390ff56": "x_1 \\in B_1, x_2 \\in B_2, \\ldots,x_m \\in B_m", "780af77a200fb1adbf356d005474e424": "x(t).\\,", "780b48905ae0762a19141ca914a181b4": "\\scriptstyle (1+\\sqrt 2)", "780b6491cbef7b54badcdd5a387fc051": "\\mathrm{Y}_c", "780ba7569a4f8eda5c02f37592046431": "\\mathrm{Bm} = \\frac{ \\tau_y L }{ \\mu V }", "780bb0ef014cdeabdc7c1f46a1887929": "m-n \\times n", "780bf4dc828521483a9bb4408e066279": "x_k[m] = \\delta[m-k].\\,", "780c494d1588f4d962cb411890f28535": "{^{(4)}}g_{\\mu \\nu}", "780c5398ef30d726709a8676e88e8af6": "G = (\\{S, A\\}, \\{a\\}, S, \\{r_1,r_2,r_3\\})", "780d15c0a7cb449419f8abb10f4ecc2c": " \\sum_\\text{cyclic} S_A^2 = S_\\omega^2 - 2S^2 \\quad\\quad \\sum_\\text{cyclic} S_BS_C = S^2 \\quad\\quad \\sum_\\text{cyclic} b^2c^2 = S_\\omega^2 + S^2 \\, ", "780d1eed6e92b54f812b5cef74ee12e7": "Y^* = X\\beta + \\epsilon \\ .", "780d577d587e4190926364b898d52a0b": " \\%N ", "780d806d4349c6d4e85d066632b066c6": "(V+W)_x = V_x + W_x\\,", "780e17b255f829db1135d2bd9be84003": "\\left [\\begin{smallmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{smallmatrix}\\right ]\n", "780ec8038f485872522a971c05ac05e1": "a = b = 0.", "780ed375e7441878be9d1a86e8aceb6f": "\\frac{q\\Gamma(\\alpha+\\tfrac{1}{p})\\Gamma(\\beta-\\tfrac{1}{p})}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\text{ if } \\beta p>1", "780f5dcc5abf57ef1ba631098f8f644d": "X\\subset{}^\\ast X", "780f5ece2bbbdcd2a8c775e9dda78bea": "\\scriptstyle \\sum_{i=1}^k{X_i} \\;\\sim\\; \\mathrm{Erlang}(k,\\, \\lambda)\\,", "780f64f85a4b784a5d2f87d799cd256d": " CLV = { (close - low) - (high - close) \\over high - low } ", "780fa8d11c186b05e38ca3be557d4125": "\\mathit{AIC} = 2k - 2\\ln(L)\\,", "780faefc802d66c95bf640dc43392738": " [X_i]=\\operatorname{Trans}_{X_i}(a_{i,i+1})\\operatorname{Rot}_{X_i}(\\alpha_{i,i+1}).", "780fc125085ab1cc911bb6325eddd83e": "\\Delta w=0^*", "780fc3085b5556d2624e60f45d0727c8": "D^*=\\frac{\\mathfrak{R}\\cdot\\sqrt{A}}{S_n}", "780fc81c1ecc4f2d5cd6fc2ae00bf30d": "\\left\\lfloor \\left( \\frac{m}{r} - \\left[ \\frac{m}{r} \\right]_1 + 1\\right) r \\right\\rfloor = m", "780fd400f6b1a0782598b81755a0a357": "e = \\sum_{k=0}^\\infty \\frac{(3k)^2+1}{(3k)!}", "780ff34eebde502c2a57455de5f5700f": "\\frac{\\mathrm{rad}(z,D)}{4} \\leq \\mathrm{dist} (z,\\partial D) \\leq \\mathrm{rad}(z,D), ", "781001200a1661e7b1f4186b254651e4": "M_p \\times_G EG \\to EG/G = BG", "781002526d71742b5c9bb61d6d7be0e6": "\\frac{2,279,000\\ \\mathrm{N}}{(3,526\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=65.91", "781066ad6e24dc4a5d4fc033cecf702c": "\\varphi_{\\overline{X}_n}(t)= \\left[\\varphi_X\\left({t \\over n}\\right)\\right]^n = \\left[1 + i\\mu{t \\over n} + o\\left({t \\over n}\\right)\\right]^n \\, \\rightarrow \\, e^{it\\mu}, \\quad \\text{as} \\quad n \\rightarrow \\infty.", "7810d148c01ad02b233ffa8e144aa57f": "\\nu_b \\,\\!", "7810d50c7cb28653bed19e435fc3f770": " P", "781119c6f1a1ac5a362d93a62ab7ab4f": "M = m/\\sqrt{1 - v^2/c^2}", "781162a83787c724f243fb9e2707d87b": "\n\\sum_{k=1}^\\infty\\frac{\\mu(k)-\\varphi(k)}{k}\\log\\left(1-\\frac{1}{\\phi^k}\\right)=1.\n", "7811b2fa4adf1d71e3e26f49052b5444": "\\bigstar\\bigstar\\bigstar", "78120ac9d05b4da5ea0e4591b96c6b07": "\\begin{vmatrix}\n 0 & 5 & 0 \\\\\n 8 x_1 & -2 x_3 \\cos(x_2 x_3) & -2x_2\\cos(x_2 x_3) \\\\\n 0 & x_3 & x_2\n\\end{vmatrix} = -8 x_1 \\cdot \\begin{vmatrix}\n 5 & 0 \\\\\n x_3 & x_2\n\\end{vmatrix} = -40 x_1 x_2.", "78121f4fa9e33c4131738854263bd777": "126217 = 7 \\cdot 13 \\cdot 19 \\cdot 73\\,", "781270214cb3e87c6db2220071b78af8": " W(\\mathbf{Q}) \\rightarrow W(\\mathbf{R}) \\oplus \\prod_p W(\\mathbf{Q}_p) \\ . ", "78127a5431b4adf2988d616193b54c9f": "\\frac{\\zeta(s)\\zeta(s-k)}{\\zeta(2s-k)} = \\sum_{n\\ge 1}\\frac{\\sigma_k^*(n)}{n^s}.", "78128267e2e5e0b07ccd6821a45cd7d0": "{\\mathit l \\over \\mathit l^*} ={1\\over 3}, {2\\over 5}, {3\\over 7}, \\mbox{etc.,} ", "7812c282b5d90e1a939dc54544b631e9": "\\langle C_i\\rangle", "78134440b91b2f68dc210f07eceef646": " E\\left \\{ \\widetilde{\\mathbf{x}} \\widetilde{\\mathbf{x}}^{T} \\right \\} = \\mathbf{I} ", "78134e0709c2ccf1d74ce4c5b9d1ec34": "\\frac{p_{n+1}-p}{p_n-p}\\approx\\frac{p_{n+2}-p}{p_{n+1}-p}", "78135b06e5e6cbb54ea739222bda6457": " F_\\mathrm{f} = \\mu_\\mathrm{k} F_\\mathrm{N}, ", "7813a87285ee11975d432ef3784b7bf0": "\n\\begin{align}\n\\sum_{k=1}^n k^3 &= 1 + 8 + 27 + 64 + \\cdots + n^3 \\\\\n&= \\underbrace{1}_{1^3} + \\underbrace{3+5}_{2^3} + \\underbrace{7 + 9 + 11}_{3^3} + \\underbrace{13 + 15 + 17 + 19}_{4^3} + \\cdots + \\underbrace{\\left(n^2-n+1\\right) + \\cdots + \\left(n^2+n-1\\right)}_{n^3} \\\\\n&= \\underbrace{\\underbrace{\\underbrace{\\underbrace{1}_{1^2} + 3}_{2^2} + 5}_{3^2} + \\cdots + \\left(n^2 + n - 1\\right)}_{\\left( \\frac{n^{2}+n}{2} \\right)^{2}} \\\\\n&= (1 + 2 + \\cdots + n)^2 \\\\\n&= \\left(\\sum_{k=1}^n k\\right)^2.\n\\end{align}", "7813b0203cb513312b7b73a87a3e3275": "... \\,", "7813c02abb7b7028bd53d68abb9d6d2d": " 0.1_2\\leq a<10_2", "7813f2af04c131b4cbf77518fb517a39": "(\\Sigma _{11} )^{ - 1} \\Sigma _{12} \n", "78142dcc962f15db95107f42abcbfc86": "b^2", "781489bd305359745cbd402cd67ad4ca": "P^{opt} = \\frac{(\\text{E}a)(y_d - \\text{E}u)}{(\\text{E}a)^2 + \\sigma^2_a}.", "7814a0e7fffecda9cebb60d2bc160257": "\\bot", "7814d4e068840cd36b82a4728e07bac9": " \\phi(x,t) = e^{i (kx -\\omega t)} ", "7815511e00837facdc703d49660a84eb": "T(t)=A\\cos c\\lambda t + B\\sin c \\lambda t.\\, ", "78155d2ec4b025a5ac903d6f8611756c": "f_0", "7815a8c6c789aa7f875272ac7b32a9df": "a_{i1}=\\frac{P(x_{i})}{Q'(x_{i})},", "781626bb134e615bd7e80f48c7ce13ef": "\\mathcal O(n^3)", "78163e1552efbf47e6d28144e2cf6891": "\\mathbf{L}_X \\, dz = f(z) \\, dz", "781668af87e61098383cb29ac776c25a": "f_X(\\mathbf{x}|\\boldsymbol \\theta) = h(\\mathbf{x})\\exp\\left(\\sum_{i=1}^s \\eta_i({\\boldsymbol \\theta}) T_i(\\mathbf{x}) - A({\\boldsymbol \\theta}) \\right)", "78167645f06639b52e4bd2a62ffe58da": "E = - \\frac{1}{2} \\sum_{\\langle i,j \\rangle} J S_i S_j - \\frac{1}{2} \\sum_i (4 J - \\mu) S_i", "78169e3d2c51d27e5240cd9d67c7c439": "\n\\vec J_1\\left( \\vec k \\right) = a_1 \\left[ 1 - \\hat k \\hat k \\right ] \\cdot \\vec v_1 \\exp\\left( i \\vec k \\cdot \\vec x_1 \\right)\n+ a_1 \\left[ \\hat k \\hat k \\right ] \\cdot \\vec v_1 \\exp\\left( i \\vec k \\cdot \\vec x_1 \\right)\n.", "7816c4e5891f0886aee85fad14edff1c": "\\mu (X)<\\infty", "781710fa5a31cd715045084579a7e068": "\\vec{F}(\\vec{r}) = \\vec{\\nabla}_{\\vec{r}}\\bigg(\\frac{-1}{(n-2)A_n}\\vec{\\nabla}_{\\vec{r}} \\bullet{} \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|\\vec{r}-\\vec{r}'|^{n-2}}\\vec{F}(\\vec{r}')d\\tau'}\\bigg)", "7817d6236e7a59086f752b14afc69ebf": "[u, v]_{q,p}", "78180a3c93bc6011d2bba15213ece074": "R^{\\times}", "78181f4a30ae5995ff1bb79a852c124e": "N_{8}", "78182ec41d7cc0d44cee33eab464741d": "h_{ij}\\,", "781844c76d4e216f91633a2611c10e4c": " \\text{PoS} = \\frac {\\text{value of best Nash equilibrium}} {\\text{value of optimal solution}},\\ \\text{PoS} \\geq 0.", "78184e10e8348a1f55e30fb5c0204610": "\\Phi_a", "78186f3b036738c9a3f9c8b7c82f2983": "i_p = 268,600 \\ n^{\\frac{3}{2}} AD^{\\frac{1}{2}} Cv^{\\frac{1}{2}} ", "78187053fc1a1f8d7edd78bbb888c3f4": " n = max(X) ", "78187c115b49bccff6677ecdf6571c8a": "\\displaystyle f(x) = A_0x_1^2+2A_1x_1x_2+A_2x_2^2", "78189b2f384e5736a568d85cfffc8d64": "p(x) = \\frac{\\left|\\frac{\\Gamma\\!\\left(m+\\frac{\\nu}{2}i\\right)}{\\Gamma(m)}\\right|^2}{\\alpha\\,\\mathrm{\\Beta}\\!\\left(m-\\frac12, \\frac12\\right)}\n\\left[1 + \\left(\\frac{x-\\lambda}{\\alpha}\\right)^{\\!2\\,} \\right]^{-m} \\exp\\left[-\\nu \\arctan\\left(\\frac{x-\\lambda}{\\alpha}\\right)\\right]. ", "7818c8de0c4c7a5711f66842e8488ef6": "V_o = V_i\\frac{1}{\\frac{2LI_o}{D^2V_i T} + 1}", "78193f233ab07ab71bb918e0dafa8304": "L(Ux)=b", "78194c6f570eb38db448991f84ccf59a": "\\scriptstyle \\gamma_\\mathrm{ls} - \\gamma_\\mathrm{sa}", "781984d72a092a7052ab6e30ce681930": " v_g= \\frac{d\\omega}{dk} ", "7819ad1e6dfda9d3efb1892423ba2ca7": "\\frac{V_1}{T_1} = \\frac{V_2}{T_2}", "7819fe1fc3f215ad26eabe657c05ae51": "C=\\mathbf{a}\\mathbf{b}^\\mathrm{T}", "781a35f50ca0d02a14c5475e1a428dd6": "(x_{i:\\lambda} - m_k)/\\sigma_k", "781a4f87cc18ce281f16f197e7747061": "\\mathfrak{sl}_n(F).", "781a9d5246af373c5a43637a13b0fe02": "\\mathbf{Gr}(r, V^{\\vee})", "781ad2a2558c717beb71f9434a0283ef": "\\mathrm{V}=\\mathrm{C}_2 \\times \\mathrm{C}_2", "781aefc8354e3eb3c5c4b67dcd23c1ea": "P(0), P(1), P(2),\\dots", "781b100b757c487a061c47084691b390": "\\theta_{n+1} = \\theta_n + p_{n+1}", "781b158d19b220385d018b248034b47e": "k < \\pi", "781b18a155bd6772fef27f1e3c460e62": "\n\\Gamma^i{}_{jk}=\\Gamma^i{}_{kj} \\,,\n", "781b834195efc1a8c5e3f4d5b714edd6": " V_i(x)= [V^-_i(x)-V^+_i(x)] ", "781bbf761f3a50312ed3eb61959f6bb0": "L(s,\\chi)", "781c16e6a5cc09ea029e963fc4c85e88": "a + \\frac{t-t_1}{t_2-t_1}(b-a)", "781c5211115354ec33462163302e4076": "t \\approx \\,", "781c685c52bba18bd63b721f33bad0dd": "\\mathcal{L}_\\mathrm{H} = \\left|\n\\left(\\partial_\\mu + {i\\over2} \\left( g'Y_\\mathrm{W}B_\\mu\n+g\\vec\\tau\\vec W_\\mu \\right)\\right)\\varphi\\right|^2 \\ - \\ {\\lambda^2\\over4}\\left(\\varphi^\\dagger\\varphi-v^2\\right)^2\\;.", "781c7442800125091398d86e29f1421d": " \\eta = 1 - \\frac {T_1}{T_2} = 1 - \\left(\\frac{P_1}{P_2}\\right)^{(\\gamma-1)/\\gamma} ", "781c981f936468dd1c23f8b17bead890": "\\pi=\\mu-bY+bY_{-1}+h(\\Delta i^W+\\Delta \\epsilon^e)", "781ca85795026f9286d9c08ff8a8e2cf": "\\lambda(G) \\geq \\lambda_1", "781cbb9e722e30507ddd5012c4fa7665": "x_1(t)", "781cc1c9aa61244d9b65a5fc5b380adc": " x^i ", "781d2fbef7d0fd9a2e35c37d700d926a": "b^{-2} ", "781d661963acc657c85c1bd24f2f739e": "\\displaystyle u(x,t)=\\sum_{i=1}^n m_i(t) e^{-|x-x_i(t)|}", "781ea92e3666f6810125a5cbf4782493": "({R_1})", "781eb41c4693f96e22c00141b8c02f49": "\\frac{-d[A]}{dt} \\equiv r = k[A]", "781f0a2df1c0346b4537253d844c5abe": "P=\\langle\\vec p\\rangle \\in {\\mathcal P}", "781f2c98e22a4939824d166fc2fb8691": "a \\in kG", "781f2dabb982cf2d41015d3900012daa": "T(n) = \\left(\\frac{n}{e}\\right)^{n/2} \\frac{e^{\\sqrt{n}}}{(4e)^{1/4}}\\bigl(1+o(1)\\bigr).", "781f4067572f7657b3ac852764ad52e0": "\\frac{11}{6}", "781f707ed11cdee09e76105a91b66555": " y \\in \\{0,1 \\} ", "781f8d04374b2270f7b314681b5bb234": "(n Z_1 \\cdots Z_n) = n(Z_1 \\cdots Z_n)", "781fc75c62c82dff83ea947d27f2f729": "x \\in A^c \\cup B^c", "781fc840736c4e0eb3d11b646b74a5df": "(x,y)\\in R^2", "781fccfd4ecea070e9f1bdfaa0bfc230": "\n L = e^{\\frac{1}{2}W}\n", "781fdd52bedb4fc5c127811585c4f834": "\\dot{x} = f(t,x)+G(t,x)[u+\\delta(t, x, u)]", "781fe7b123ed150bfd2c5bfdc65869a5": "(1+c)\\mathrm{OPT}", "781ff4289c6cc5fc2973b7a57791e0e2": "\\Lambda", "78204c86dee364d964b2cd5bfddc32d6": " h(t^\\star)/h(t)\\ ", "782054b7fd937966915bae681103a3fa": "\n\\qquad \\sum_{n=1}^\\infin \\frac{1}{n^2} = \\frac{\\pi^2}{6},\n", "782074b0a1f072bf6082b3fa7cb2a312": "d_\\sigma=\\infty", "78208529e46cc3ab79452c1b5527905b": "|\\ell_1 - \\ell_2|", "7820aea97dcd0614b3e3f7681f65e028": " T=\\left | \\frac{\\bold{j}_\\mathrm{trans}}{\\bold{j}_\\mathrm{inc}} \\right | , \\qquad R=\\left | \\frac{\\bold{j}_\\mathrm{ref}}{\\bold{j}_\\mathrm{inc}} \\right | , ", "7820e878fcc9ff7bff73d91810f859b9": "f(x;\\mu,\\sigma,\\xi) = \\frac{1}{\\sigma}\\left[1+\\xi\\left(\\frac{x-\\mu}{\\sigma}\\right)\\right]^{(-1/\\xi)-1} ", "7820ef69dde7a9ede96c63da2f7c9190": "\\langle\\Delta\\eta,\\eta\\rangle \\ge 0.", "782111037df7eb2dbb9465b7efa69124": "\nU = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} Q_{1lm} I_{2lm}^{*}\n", "78211d1fdae80ab2767a3f275d1bb400": "[(x_0-x_1)^2+(y_0-y_1)^2+z^2]^{1/2}", "7821233e6835831d33f131b3f09194ef": "\\varphi(t)\\ge 0", "7821545e125a0e069e71e4a08e248033": "\\mathfrak{P}^{75}", "78219bd7bf450d5c7fe10f89fa6cd762": "\n\\mathrm{SINADR} = ENOB\\cdot 6.02 + 1.76\n", "78223561cb136d8dfb3bcac127235bb5": "= (p^{2})^{2} \\cdot 2p^{1}p^{2} - 2 \\cdot p^{2} \\cdot p^{1}(p^{2})^{2} \\,", "782327bc4029959665e7ca501030028c": "S_w", "7823cf7cfdbb5d50f553612b51fd0896": "B(z)=\\lambda a(e^{z/a}(z+1-a)+a-1)", "7823fba1de6fbac4012bb6732186e88e": "f(x,y) = f(0,y) \\oplus x(f(0,y) \\oplus f(1,y))", "782412f83b0c54d6a9f9c94c1368adae": "\\star\\omega \\cong (I^{-1} A)^{\\dagger} \\cdot \\mathrm{d}^kX,", "7824741c7c7b46582315bf15da5d5d57": " V(r)= - \\left[\\frac{(V_{2}-V_{1})}{(R_{2}-R_{1})}\\right]\\cdot\\frac{(R_{1}R_{2})}{r} + const.", "7824962e4f41646adfe5814a457a75ac": "\\tau' = f(\\tau)", "7824cecf5f4f8431f13e8df961e53f8c": "\\|f\\|_p = \\sup_{\\|g\\|_q = 1} \\int |fg| d\\mu, \\qquad 1/p + 1/q = 1", "7824fe7b651fa001a76013f6bf7d9a2e": "m\\to \\infty", "78251ebfc814c88f6a3cc7c5fea4df80": "O(\\lambda_\\delta(n)\\log n)", "782550aed48731057b12ce0311bb8189": "\\Phi_i\\ \\,", "78259457e13133eb2916e6aed2453b72": "\\big[n]_q!", "7825c713e7caf53da69c4ade94a95939": "m_2 = [-1.47, -1.20] + [12.3, 7.6] = [10.8, 6.40]", "7826216a74c0dae077410945f6f604be": "H_{1,I}=-\\hbar\\left(\\Omega e^{-i\\Delta t}+\\tilde{\\Omega}e^{i(\\omega_L+\\omega_0)t}\\right)|\\text{e}\\rangle\\langle\\text{g}|\n\n -\\hbar\\left(\\tilde{\\Omega}^*e^{-i(\\omega_L+\\omega_0)t}+\\Omega^*e^{i\\Delta t}\\right)|\\text{g}\\rangle\\langle\\text{e}|,", "7826235016ffbb2784e51294f2855250": "\\exists k>0 \\; \\forall n_0 \\; \\exists n>n_0 \\; g(n)\\cdot k \\leq f(n)", "78262dc2a925f17eaaeee0dd3ac8273d": "G = \\{G_1, G_2, \\ldots, G_k\\}", "7826f31a1245c0c86a689e1fcd34b2f3": "Z_{1,t}", "78273b5bcf573ca9e079897adf6f3a79": "r_o", "78276cc19aff1d6ac4fa3a330a99de84": "RV^{(n)} = \\sum_{i=1}^n r_{i,n}^2,", "78277dc519923be702f69796f2f8c10d": "X=X_2 \\leftrightarrow Y=Y_2 ", "7827c4ddd802c0995870e8951db19a93": "+_w", "7827c8a13c9fa4422e0176482908cd5c": " \\mathbf{r}_k = \\mathbf{b} - \\mathbf{Ax}_k. \\, ", "7827d3857baa94c70c1f2588cff13869": " g\\left(\\theta\\right) = \\exp(\\xi\\theta) g\\left(0\\right),", "78281c09fa26b6cb2d1289f2dd3eb016": "D_\\mathrm{KL}(q(x|a)u(a)||p(x,a)) = \\mathbb{E}_{u(a)}\\{D_\\mathrm{KL}(q(x|a)||p(x|a))\\} + D_\\mathrm{KL}(u(a)||p(a)),", "7828aed9004822a057f4ba170fb6ff46": "K_{\\mathrm{J}} = 2 e / h \\,", "7828b28381cb77f6365e5645244cde71": "K=\\frac{[S]^\\sigma [T]^\\tau \\dots } {[A]^\\alpha [B]^\\beta \\dots}\\,", "7829185eef1d2de4da1f90ff10f1e4be": "\\text{Lower endpoint} = \\bar X - 1.96 \\frac{\\sigma}{\\sqrt{n}},", "78295448fb49406836509520e0a590f6": "\\displaystyle u_t - u_{xxt} + 2\\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx},", "7829b6d3c846d4a0bfbd3f2cfb9c4558": "U_1 + U_2 = R_1 - {n_1(n_1+1) \\over 2} + R_2 - {n_2(n_2+1) \\over 2}. \\,\\!", "782a240bd027dfd821f90deb63a49f3a": " V \\times V \\to V", "782a95a74280750a10764a2034e2283e": "\\frac{\\nu}{2} = \\frac{\\bar{x}}{\\bar{x} - \\tau^2}.", "782aa239a4c19204fe0beae763b28949": "\\vec{h}_1, \\; \\vec{h}_3", "782aca9dbd5c530ccb499dc26d8e9681": "D^{1/2} = \\bigl( \\begin{smallmatrix}\\\\ 9&0\\\\ 0&3\\end{smallmatrix} \\bigr)", "782ae98d94198ee29b1e5a338122fa06": "\\scriptstyle \\sqrt{1}, \\sqrt{2}, \\sqrt{3}, \\sqrt{4}, \\sqrt{5} ", "782aeb2d39017595462a71da8a69df96": "\\alpha = \\sum_{\\mathbf{x'} \\in M_{\\mathbf x}} q(\\mathbf x,\\mathbf{x'}) = \\sum_{\\mathbf{x'} \\in M_{\\mathbf x}} q'(\\mathbf x,\\mathbf{x'})", "782b149283ef86570b198eb3af9a28ce": "C_Y", "782b8ff6055bc07f85abe9abd993374a": "x^2 + K_a x - K_a C_a = 0", "782bcb5b746dc4368a77bfadb3e0a9ed": " \\boldsymbol\\psi(t,\\mathbf{r}_0) = \\mathbf{a}(t,\\mathbf{r}_0) - \\boldsymbol\\omega(t) \\times \\mathbf{v}(t,\\mathbf{r}_0) = \\boldsymbol\\psi_c(t) + \\boldsymbol\\alpha(t) \\times A(t) \\mathbf{r}_0", "782bd502203d4953aaa290c8aa7c0597": "SU(5)", "782bf85d7707bc3ac9e332e2e14cf95e": " \\mathbf{e} = \\begin{pmatrix} e_{11} \\\\ e_{12} \\\\ e_{13} \\\\ e_{21} \\\\ e_{22} \\\\ e_{23} \\\\ e_{31} \\\\ e_{32} \\\\ e_{33} \\end{pmatrix} ", "782c24870d4cc3fd40beb3f6b718f124": "K(u) = (1-|u|) \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "782c380a62cac960c2bb01fc229e082e": "c \\to d", "782c483379e3e0ffcafa3a3843bf4a6e": " \\hat{M} T(t) =\\lambda T(t) \\, ", "782c55c4a434c2329cb3174f0b68ccd9": "r_g \\in G", "782c61eb2d411bb02d1fe764967efc21": "B_1, ..., B_n", "782c6b350f7b43e95be70dddf28a4eda": "\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}=\n\\begin{bmatrix}\n 1 & \\lambda z \\\\\n 0 & 1\n\\end{bmatrix},\n", "782c8288ff23671a5029cf51bf455c63": "P_1 \\subseteq P_2", "782c8e0586f26ab4a69b5323488352c0": "Ly=(D+a_2)(D+a_1)y=0", "782cbe5226d6bd3964b20d8398099841": "\\bar{u}_S\\, =\\, \\frac12\\, \\sigma\\, k\\, a^2\\, \\frac{\\cosh\\, 2\\,k\\,(z+h)}{\\sinh^2\\, (k\\,h)}\\,", "782d23a93c2632b7718c1e7b3ca86d3e": "\n\\mathbf{C}_{\\alpha \\beta} = \\sum^s_{\\gamma=1}\\mathbf{A}_{\\alpha \\gamma}\\mathbf{B}_{\\gamma \\beta}. \n", "782d2b0340821cd48eab446ef8982e64": "(x,b),", "782dbc1af3d405ab5595d5e1dabe691e": "C_k^{\\;-1/2} = \\sqrt{C_k}^{\\;-1} = \\sqrt{C_k^{\\;-1}}", "782dcf8f981476b5a547aa5559959cb5": "\n\\lambda^{[l]}_{{\\alpha}_l}{\\sim}e^{-K\\alpha_l},\\ K>0.", "782debda36aa8248e7f89e484616bbad": "\\sum_{n=0}^{\\infty} {\\Delta u_n} a_{u_n} = \\sum_{n=0}^{\\infty} (u_{n+1}-u_n) a_{u_n}", "782dfa67ad245ae874648932aa6a2b79": " f \\in \\mathbb{Z} ", "782e1893bc88737b2f6d166c5ae75bd4": "Q(X) / E(X)", "782eccb65701b1f61af465c533a9f1c6": " A_2 ", "782ee2784cf16269dffee17a29beb167": "Su_k = \\lambda_k u_k, k=1,2,\\dots,", "782f19b590d0692c9c2f702474eecac6": " \\pi_3 = L^k \\mu^l k^m \\beta^n g^o \\Delta T", "782f4bdd2667e051af7ce31a7fd41fb7": "R\\to k", "782f594355a87d7859c933d907cdbb43": "W^{\\mathfrak{p}}\\subset W", "782f8c7c6f72a21d63c9efd71a36f39e": "\\rho_p", "782f8dbf50545504aad4cb2c2d4dee6f": "x = 0, y = 2r\\sin t\\,\\!", "782fa9c79fb01a76ed2f61c3b847eb2e": "[p?]q \\equiv p \\to q\\,\\!", "78311533207971287cbe0559b83e02db": "A = 4\\pi r^2", "78313a7d599ddb9c8e4c2d06790c8230": "\\frac{}{}\\omega_p", "783143f0b3e3926b4a42495439aa78ae": "E^\\mathrm{tot}(\\mathbf{x},t)=\n\\sum_{n}\\frac{E_n^\\mathrm{ret}(\\mathbf{x},t)+E_n^\\mathrm{adv}(\\mathbf{x},t)}{2}+\n\\sum_{n}\\frac{E_n^\\mathrm{ret}(\\mathbf{x},t)-E_n^\\mathrm{adv}(\\mathbf{x},t)}{2}\n=\\sum_{n}E_n^\\mathrm{ret}(\\mathbf{x},t).", "783156e662b59ec1b0f50e0d803a4f01": " \\frac 1 \\varphi = \\varphi - 1;\\; \\varphi = \\frac{1 + \\sqrt{5}}{2} \\approx 1.618 ", "7831d262f5beec2057f63a590209ee64": "r_{xy} = \\frac{ \\overline{xy} - \\bar{x}\\bar{y} }{ \\sqrt{ (\\overline{x^2} - \\bar{x}^2) (\\overline{y^2} - \\bar{y}^2 )} } ", "7831ef1b29669d7db2303b8cc697f989": "\\begin{bmatrix} c & -s & 0 \\\\\n s & c & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}\n \\begin{bmatrix} 6 & 5 & 0 \\\\\n 5 & 1 & 4 \\\\\n 0 & 4 & 3 \\\\\n \\end{bmatrix}", "7831fc6814a693477f41c070f218d3de": "2t", "7832162bb78a3ca74987f958c5262098": "\\delta \\to 0", "78322beac51a7f7da292db6a596b7264": "\\int_\\gamma f(z) \\, ", "7832a86ae1076dd6e7187ea7a30e45b7": "m = 3.4~\\mathrm{a.u.}", "7832b00dde4fc419c8d8c6cd5976cf92": "\\rho(x)=\\frac{2}{\\pi}\\sqrt{\\frac{1-x}{x}}.", "7832d85a569e280de18e10ecea7f18fe": "a=-0.5.", "7832e1c8cfc85b91f7477bdd48fe7a8b": "\\ell p (1 - d_j)", "7832f9bb8be4623fdf7846b0140e4546": "\\operatorname{E}[S_N]\n=\\operatorname{E}[T_N].", "78331f96b1aa4401067a735841284ba6": "\\chi = c t", "783372a3667ba439d723e239c18724a8": "\\pi=\\frac{\\sqrt{2}}{4Z} \\!", "78339237ce464160d3bb5ee808331315": " L_{expected}=-\\tau_{P(f)}=-\\int \\tau(r_{f(q)},r^*)dPr(q,r^*)", "78339499216fccc331b21f64bb5a9edc": "\\int_X \\langle\\phi| x\\rangle\\langle x|\\psi\\rangle\\; d\\nu (x) = \\langle\\phi|\\psi\\rangle\\;.", "7833f582088813d7daed87eee79cf8ad": "~x_{\\max}~", "7833ffa95fe0aaaea0a4cbcb3355c3e2": " H(s) ", "783420e24ef35c512fc2502952192d8c": "\\Gamma^{\\alpha}_{\\beta\\gamma}-\\frac12 J^{\\alpha}_{~\\delta}\\nabla_{\\beta}J^{\\delta}_{~\\gamma}", "7834454f1333048fed9f378a1cd0378b": "\\langle y, x \\rangle = \\overline{\\langle x, y \\rangle}, \\ \\ \\langle x, x \\rangle \\ge 0, \\ \\ \\text{for all}\\ \\ x, y \\in H, \\text{and} \\ \\ \\langle x,x \\rangle = 0 \\ \\ \\text{if, and only if,} \\ \\ x = 0.", "78345827823193dafc4f26e2ad36cb3d": "3^\\frac{1}{13}", "78353d082c8168cd0e7de5ff99035eb8": "\\text{CA} = \\text{NS} - \\text{NI} \\,", "78354b4d0cdd4e2dec9c094a916bbb95": "\\scriptstyle \\tilde{B}_0(x) \\;=\\; 1", "78355f1a455c7afa2f9c02ab52dcaf54": "\\displaystyle{|f_z|^2 - |f_{\\overline{z}}|^2 = u_x v_y - v_x u_y >0,}", "7835da86478c854ccdd604fa3e2e96ad": "\\left(\\frac{{T}_{2}}{{T}_{1}}\\right)=\\left(\\frac{{V}_{1}}{{V}_{2}}\\right)^{(\\gamma-1)}={r}^{(\\gamma-1)}", "7835f09c41a624fa5c2cf29d320977a9": "a_{3}+(5/6)b_{2}", "78361702b190b03c04743386e1be5d20": "\\mathbf{B}=\\mathbf{\\nabla}\\times \\mathbf{A}. \\, ", "78364bf674c109a3adc749de47fc300c": "\\begin{align}\n & \\operatorname{E}[\\,X\\,|\\ X > \\alpha \\,] = \n \\mu + \\sigma \\frac {\\phi\\big(\\tfrac{\\alpha-\\mu}{\\sigma}\\big)}{1-\\Phi\\big(\\tfrac{\\alpha-\\mu}{\\sigma}\\big)}, \\\\\n & \\operatorname{E}[\\,X\\,|\\ X < \\alpha \\,] = \n \\mu + \\sigma \\frac {-\\phi\\big(\\tfrac{\\alpha-\\mu}{\\sigma}\\big)}{\\Phi\\big(\\tfrac{\\alpha-\\mu}{\\sigma}\\big)},\n \\end{align}", "78364d5ea187ab858d96b84b576a3f7f": "{\\bar{O}}_3", "783661f972bb93a4ba6513356e776c52": "|\\psi(q,t)|^2", "7836c536e85fcb25e67ce17c6a8cc5e3": "\\mathbf v=\\frac{m_a \\mathbf u_a + m_b \\mathbf u_b}{m_a + m_b}", "7836e793b86a7dedfe33d09aaebd1a18": "(v-k-1)\\mu = k(k-\\lambda-1)", "783752a318d007828d29407e3dabb185": " \\|x\\|_2 = 1 ", "78377b439b4f516c760fdf858ae214c4": "p_F", "78378328323de7855d7877fb951f5f87": "I_{\\mathrm{out}}=K_{i}\\int_{0}^{t}{e(\\tau)}\\,{d\\tau}", "7837d54c99543d360727c0ffe66e907c": "f(x)= F'(x)\\,", "7837f0b6d93b2e2a8728e17f9bdf2fce": "\\scriptstyle(-9.0\\pm11)\\times10^{-17}", "78381befee65ecdbe80066e20a599a01": "\\frac{\\partial \\sigma_x}{\\partial x} + \\frac{\\partial \\tau_{yx}}{\\partial y} + \\frac{\\partial \\tau_{zx}}{\\partial z} + F_x = \\rho \\frac{\\partial^2 u_x}{\\partial t^2}\\,\\!", "783874baf49cfa1ebb108c416cfa813a": "q_1: X \\times_S P \\to X, q_2: X \\times_S P \\to P", "7838cbb538b595ab20d7df40737085ed": "a\\circ b= \\sum_{n\\ge 0} {i^n\\over n!} \\left({\\partial^2\\over \\partial x_1\\partial y_2} -{\\partial^2\\over \\partial y_1\\partial x_2}\\right)^n a\\otimes b|_{\\mathrm{diagonal}}.", "7838e5241d6aa4b7f94309dc8af93e03": "a^*_0", "783957736eec66e9250571d7ea004912": "x^2+x+1", "783975c9f86c75292ea0c1adefaae411": "i^{k+1}\\left.\\frac{\\partial^{k+1}}{\\partial t^{k+1}}\\big(\\mathbf j(\\mathbf r,t)-\\mathbf j'(\\mathbf r,t)\\big)\\right|_{t=t_0}=i\\rho(\\mathbf r,t)\\nabla i^k\\left.\\frac{\\partial^{k}}{\\partial t^{k}}\\big(v(\\mathbf{r},t)-v'(\\mathbf{r},t) \\big)\\right|_{t=t_0},", "783a5c1310a7d221d1edfa2a58ede25e": "r = a \\frac {\\sin (-\\theta + \\theta_0)}{\\sin (-2\\theta + \\theta_0)}", "783a63a7c536aabd57e3a721bbb6610d": "\\mathrm{Var}(\\delta(X)) \\ge \\sup_\\Delta \\frac{\\left[ g(\\theta+\\Delta) - g(\\theta) \\right]^2}{E_{\\theta} \\left[ \\tfrac{p(X;\\theta+\\Delta)}{p(X;\\theta)} - 1 \\right]^2}.", "783b374c03193be2d5818160c484aa04": "\\tan\\frac{11\\pi}{60}=\\tan 33^\\circ=\\tfrac{1}{4}\\left[2-(2-\\sqrt3)(3+\\sqrt5)\\right]\\left[2+\\sqrt{2(5-\\sqrt5)}\\right]\\,", "783b3f8a4df7dd3b5f6c2ab18c084d04": "\\Lambda= \\frac{ch}{2 \\pi^{1/3} k T}", "783bcd6e300efb0d2d46079bd77f578f": "\ny = \\frac{B+\\sqrt{B^2-4A}}{2}\n", "783bf3cfd483d717084ef39e1325beb5": " S = \\sum_{i=1}^m r_i^2", "783c1687a4a3d88256cd766dcd0c0bce": "\n \\hat\\beta = \\frac{\\hat{K}(n_1,n_2+1)}{\\hat{K}(n_1+1,n_2)}, \\quad n_1,n_2>0,\n ", "783c4de1ea89ac88504ed14207931160": "\\int d^dx\\, J(x)Q[\\phi(x)]\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]=0.", "783c5f2be65ac21b253a1a260d0b849f": "V = \\begin{bmatrix} 1 & 139 & 23 & 1 & 61 & 647 & \\cdots & 17191 \\end{bmatrix}", "783c6e54ead2869caec29d578514180f": "S = k_{\\mathrm B} \\ln\\left[\\Omega\\left(E\\right)\\right]\\,", "783cc658a9a9237061983a383b467aae": "m=\\pm 1", "783cf4b552f67dd944a84c01629fdfbf": " \\ln({\\sigma_0 T_m}) ", "783d0c03443a79ba331916a149d03933": "f \\in I^{n-k} M_k", "783d92a7318db2f415edc1b1d5b2de18": "\n\\mathcal{G}(\\omega_n) = \\int_0^\\beta \\mathrm{d}\\tau \\, \\mathcal{G}(\\tau)\\, \\mathrm{e}^{\\mathrm{i}\\omega_n \\tau}.\n", "783e635f62280af8b2254417d5099cc0": "x^n + y^n = 1.\\ ", "783ea4533900cf3c14f45b16ad9687ba": "v\\mapsto Av+b", "783ec146d0c075c97cc5277a4f6677f9": "\\displaystyle \\times J_{n/2+\\delta}(2\\pi|\\boldsymbol \\xi|)", "783f1c66c2295517fcc22e1f5433b2b3": "\\begin{align}\n\\left(\n \\begin{array}{*{3}{r}}\n 4 & 12 & -16 \\\\\n 12 & 37 & -43 \\\\\n -16 & -43 & 98 \\\\\n \\end{array}\n\\right)\n& =\n\\left(\n \\begin{array}{*{3}{r}}\n 1 & & \\\\\n 3 & 1 & \\\\\n -4 & 5 & 1 \\\\\n \\end{array}\n\\right)\n\\left(\n \\begin{array}{*{3}{r}}\n 4 & & \\\\\n & 1 & \\\\\n & & 9 \\\\\n \\end{array}\n\\right)\n\\left(\n \\begin{array}{*{3}{r}}\n 1 & 3 & -4 \\\\\n & 1 & 5 \\\\\n & & 1 \\\\\n \\end{array}\n\\right)\n\\end{align}", "783f2703eb8b2184c8d5533f72e2fb90": "\n(5-3)-2 \\, \\ne \\, 5-(3-2)\n", "78400fb5ab991061d43d9accb5d679f2": "\\int_{0}^{\\infty }\\frac{\\sin px \\sin qx}{x^{2}}\\ dx=\\begin{cases}\n \\pi p/2& \\text{ if } 0 t\\right\\}", "7842de604ee7e5e547b318abe3fc0974": "\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}", "7842eb7c0f52da050245a09fbdf164c4": " (r,\\theta,z)\\mapsto (r,2\\theta,z). ", "78432a1c7fa5ff0e4c87494d5875fd92": "U_t \\pi(a) U_t^* = \\pi(\\sigma(t)(a)),", "78433ecd536a6cdd963d54fbc705d299": "\\frac{\\delta f( F[\\rho])}{\\delta\\rho(x)} = \\frac {df(F[\\rho])} {dF[\\rho]} \\ \\frac{\\delta F[\\rho]}{\\delta\\rho(x)} \\, . ", "784351dc176f1d9399ce38aaeed1cdf8": "\n\\operatorname{Var}(x_{ii}) = \\frac{2\\psi_{ii}^2}{(\\nu-p-1)^2(\\nu-p-3)}.", "78435858e28831659c850958c2166d41": "M _{BA} ^f = \\frac{P a^2 b}{L^2} = \\frac{10 \\times 3^2 \\times 7}{10^2} = 6.3 \\mathrm{\\,kN \\,m}", "7843800670d347fa6d90e4781d328184": " \\frac{1.3 \\times 10^5}{\\sqrt{1.3 \\times 10^7}} = 36", "7843a25a78a06ca8bd07c8a4fbd37d7f": "\\tau_{H} (\\omega) = \\inf \\{ t \\geq 0 | X_{t} \\not \\in H \\}.", "7843fedd5bc5bd198a937e825e18fb88": "(z = 0)", "78442dcbfc1ff4807b31d160016c82fe": "\\mathcal{L}_{WWV}=-ig[(W_{\\mu\\nu}^+W^{-\\mu}-W^{+\\mu}W_{\\mu\\nu}^-)(A^\\nu\\sin\\theta_W-Z^\\nu\\cos\\theta_W)+W_\\nu^-W_\\mu^+(A^{\\mu\\nu}\\sin\\theta_W-Z^{\\mu\\nu}\\cos\\theta_W)]", "78444b0e13a00d66367d5182b0f870e5": "a=x_0< x_1 < \\cdots < x_{n-1} < x_n = b", "78445a109d7f17d95540c35be44fd090": "\\sin A=\\frac{\\textrm{opposite}}{\\textrm{hypotenuse}}=\\frac{a}{\\,c\\,}\\,.", "784490997b5d062cfa14038a0d6dcbf2": " \\alpha = \\arctan(\\cos \\epsilon \\tan \\lambda)", "7844988b14218afe4b9663a0c435cbb8": " \\langle N_1 \\rangle = -\\frac{\\partial \\Omega} {\\partial \\mu_1} , \\quad \\ldots \\quad\\langle N_s \\rangle = -\\frac{\\partial \\Omega} {\\partial \\mu_s},", "78449a382b777d360e6735b9c74c595d": "vm_{pq\\mu\\gamma}", "7844e7b03d637c6808ecdfed9f50c4d9": " y(t_k) ", "78450b6bbbeb08f45d26ad02d3e55df6": "10^{10^7}", "78453f771955c34ac4349a1127dfae34": " \\ 2c > \\frac{q}{2} ", "7845c4e2e572707afd81cb8a0617c954": "\\mathbf{F}=-\\boldsymbol{\\nabla}\\Phi+\\boldsymbol{\\nabla}\\times\\mathbf{A},", "7845ef4c6f6254343388df4b6b573c82": "F,G \\colon \\mathbf{C} \\to \\mathbf{D}", "784620fc6649b46159000602cde7fc66": "\\sum_{n=0}^\\infty V_{2n} = e^\\pi. \\,", "78468a5db965a15fb40844d7672a6f5f": " y_i \\in Y_i", "7846f9b0250a1a836f6407d51d005d75": "z = r_1\\, \\cos(\\theta) \\dots", "78475164dceba653184084a1951a9a96": "\\beta = \\frac{y - x}{\\delta}", "7847593d40472cedbaeecc8341e63f73": " \\eta_z^{(0)}(y) = G_z(c,y), \\,\\,\\,\\, \\eta_z^{(1)}(x)=\\partial_x G_z(c,y), \\,\\,\\,\\, (z \\notin [1,\\infty)).", "7847769fc400f4db4333e1d4e4674932": "f(x_1,x_2,\\dots,x_n) = \\phi(g_1(x_1,x_2,\\dots,x_n), g_2(x_1,x_2,\\dots,x_n), \\dots g_m(x_1,x_2,\\dots,x_n))", "7847cbd5297bf0b7b120fa4fd8864ea0": " -V (m z - p_z t) = V_z N_x - V_x N_z = \\left(\\mathbf{V}\\wedge\\mathbf{N}\\right)_{zx} ", "7847d56e000ea03629a328b442cf6d8f": "s_0-S", "7847e530741d018a93873f433baf9006": "S_i = \\int^T _0 S(t)\\Phi_i(t)dt, 0= x ", "787258d1661a666bc0150b329db0ec28": "T = +\\infty", "78729c0929c2d646db3545348f7cbfb1": " |\\langle \\psi_s^b | \\psi_{s'}^{b'} \\rangle|^2 = k>0, s,s'\\in \\mathbb{R} ", "7872b190bc425143a43939345275938d": "K(\\mathbf{x}, \\mathbf{x'}) = \\exp\\left(-\\frac{||\\mathbf{x} - \\mathbf{x'}||_2^2}{2\\sigma^2}\\right)", "78730b8380140b4bee5e5bc3932e6fd7": "(F)-(H)", "787339f6ec30225e241a2f93db8c33b2": " \\varepsilon > 0, ", "78735a570e7601ec9a5a310623521da2": "\\left(1-e^{-t}\\right)", "78735f3c34f7fece930c01b086b138b7": "\\mbox{Aut}(\\widehat{\\mathbf{C}}) \\cong \\mbox{PGL}(2,\\mathbf C).", "7873cf0e4e1b6ddcc5a26209789c5332": "L \\leq \\begin{pmatrix} x \\\\ g(x) \\end{pmatrix} \\leq U ", "787435aa3489726a9cd3c3c12d4d3c12": "\\omega_p = -\\frac{3}{2}\\frac{6 378 137^2}{(7 178 137(1-0^2))^2}\\ 1.08262668\\times10^{-3}\\ 0.001038\\ \\cos 56^{\\circ}", "7874bb213a495c2030599dd50dce9299": "\\mathfrak{g}_2^{\\mathbb C}\\times\\mathfrak{g}_2^{\\mathbb C}", "7874d1e6a5381cb281e521bf95302ec6": "w:A \\wedge B", "7874daa8dfba2c293772c4bcd76e65a5": " {\\rm {}}_{}^{2}\\Sigma_{\\rm g}^{+}", "7874f7fb09cc0a0a67096eaff15d46a5": "\\scriptstyle s_{j,0}", "78751247e5e4ac1f110a855a2f7e0b79": "-\\sqrt{\\frac{2}{35}}\\!\\,", "787519ea0233a135c104aca7d4df936f": "\\sum_{(u,v) \\in E} \\left( a(u,v) \\sum_{i=1}^{k} f_i(u,v) \\right)", "787530459f1aa95e7973d9c39c02b872": "{\\rm d}:A\\to\\Omega^1", "78759ca6d034000a555cbf4f28b36f3e": "N_k=|\\operatorname{dom}(C_k)|", "78760ddb609dbea739587046522b329e": " \\mathrm{Eu}=\\frac{\\Delta{}p}{\\rho V^2} ", "78763486618a41dc8ba28761bf68a9cb": " L y = P b ", "7876684ee1b5cd4d2da92603e169d09c": "\\mathcal{O}^\\times\\supseteq U^{(1)}\\supseteq U^{(2)}\\supseteq\\cdots", "787691c774f5effca82d2565506bd6aa": "\n(6.1)\\quad\n\\left|E(f'(W)-Wf(W))\\right| \n\\leq ||f''||_\\infty\\sum_{i=1}^n \\left(\n \\frac{1}{2}E|X_i X_{A_i}^2|\n+ E|X_i X_{A_i}X_{B_i \\setminus A_i}|\n+ E|X_i X_{A_i}| E|X_{B_i}|\n\\right)\n", "7876af4a0356f1fe36be5e8ca8c99ed0": "x^5+5ax^3+5a^2x+b = 0\\,,", "7876c110169a60096717ef4246139851": "u(r,\\theta)=R(r)\\Theta(\\theta)", "7876def4720a773270646f03b76c6484": " \\mathcal{L} = \\mathcal{L}_0 + \\mathcal{L}_\\mathrm{I} ", "787705e22f3be76ecb906122288d1cf5": "z_{i}=\\frac{\\lambda}{2 \\pi} \\left(\\frac{|\\varepsilon_1'|+\\varepsilon_2}{\\varepsilon_i^2} \\right)^{1/2}", "787706b07e6df29f26122355bcfeb360": "R_{r} \\approx 80 \\pi^{2} \\left( \\frac{\\ell}{\\lambda}\\right)^{2}", "78772402b85275f976002f00a3923b00": "X_3 = X_1Z_1Y_2 + Y_1X_2Z_2", "7877504dd7f53fa99047af8e6df265a4": "\n \\mu := \\cfrac{d_c}{z_0} \\approx \\left[\\cfrac{R(\\Delta\\gamma)^2}{{E^*}^2 z_0^3}\\right]^{1/3}\n ", "7877972e3a37a8e92ffd300a35e0efff": "l,", "7877ff364fc0ecc697a186c5dc9ee124": "\\frac 12", "7878157afd5fc1ec85b6b76ac7302933": "t \\in \\Bbb{R}", "78785bcbf886f596c56442723c7f453b": "G(y_p=0)=1", "78787dc95c9826dfe46634da7c1c8c0b": "K=\\frac{\\phi_{\\text{N}_2\\text{O}_4} p_{\\text{N}_2\\text{O}_4}}{\\left(\\phi_{\\text{NO}_2}p_{\\text{NO}_2}\\right)^2}", "787887f4e0d3fc98ff345bdc9411886b": "a,b \\in \\mathbb{Z}^n", "7878b2a50a762a70025a094c048a9c24": "G_{abcd}", "787905381b9ce7722e88955d44332ebb": "\\sum_{n=0}^{\\infty} |f_n(x)|", "78790bdf4a473b25874c69309b7f2e00": " [max(r_1, r_2), max(g_1, g_2), max(b_1, b_2)] ", "78792dbfdf32e14812b2cc4c015d3e9a": " f_\\mathrm{3dB}=\\frac {1}{2\\pi R_\\mathrm{A} (C_\\mathrm{M}+C_\\mathrm{gs})} =\\frac {1}{2\\pi R_\\mathrm{A} [C_\\mathrm{gs} + C_\\mathrm{gd}(1+g_\\mathrm{m} (r_\\mathrm{O} \\| R_\\mathrm{L}))]}", "78792edb73cdff58795c52150f6862a0": "\nR(\\theta,d') < R(\\theta,d).\n", "787a4a1cd717fe835750f639ffc476a8": "\\overline{2} + \\overline{3} = \\overline{1}", "787a7f25f4d6174a81c92e2b726215bd": "(f) = 2nP - 2nO", "787a7f2add8980b67312003cf67fd45c": "\n\\biggl|\\bigcap_{i=1}^n \\bar{A_i}\\biggr| = \\biggl|S - \\bigcup_{i=1}^n A_i\\biggr| = \\left| S \\right|\\; - \\sum_{i=1}^n\\left|A_i\\right|\\;\n+\\sum_{1 \\le i < j \\le n}\\left|A_i\\cap A_j\\right|\\; \n-\\; \\ldots\\ +\\; \\left(-1\\right)^{n} \\left|A_1\\cap\\cdots\\cap A_n\\right|.\n", "787a9d074cca883af3ce481c85cec9fe": "\\textstyle \\int_{-N}^{N} e^x\\, dx", "787abaef47ee341b4d9451b0cad0c6fd": "e^{\\frac{k}{m}t}(\\frac{dv_y}{dt} + \\frac{k}{m}v_y) = e^{\\frac{k}{m}t}(-g)", "787afdcf9d7b59e296409df1525af611": "x_{k_n}>\\Lambda\\quad \\forall n.", "787b0d4869503220ea48a1f0ea0ceeb4": "\\textstyle{\\int_1^0 dx = - \\int_0^1 dx = -1}.", "787b2a62c637afb7f45cef60ccaed0db": "rh =", "787b5c4c72a47356f3607b24adf72cd7": "s''\\in X", "787b78d4c2ee800d15e0d3df782bddf4": "P_1 ; P_2 \\equiv \\exists v_0 \\bullet P_1 [ v_0 / v' ] \\land P_2 [ v_0 / v ]", "787b7c21d336e575460262dcbaf79442": "\\mathrm{Hol}_y(\\nabla) = P_\\gamma \\mathrm{Hol}_x(\\nabla) P_\\gamma^{-1}.", "787baae1fa15e403564b9f6683a5a1f3": "A=2(\\sqrt{2}-1)S^2", "787bc0352284b0424ce26a7d35dc390a": "dl^2 = \\left ( -g_{\\alpha \\beta} + \\frac{g_{0\\alpha}g_{0\\beta}}{g_{00}} \\right )\\, dx^\\alpha \\,dx^\\beta.", "787c3fdf7362a8d68844556322342187": "\\mathcal{O}_p \\to k[\\epsilon]/(\\epsilon)^2", "787cb79a349d1a816f7e606bf2c96252": "i_{exc}(t)", "787d0b6e5d9e7525a7054c6f96c377ea": "s=0", "787d258de9f45ff2bb8f7770fc6ed668": "\\frac{\\part f_i^{(eq)}}{\\part t_2} +\\left ( 1-\\frac{1}{2 \\tau} \\right ) \\left [ \\frac{\\part f_i^{(1)}}{\\part t_1}+\\vec{e}_i \\nabla_1 f_i^{(1)}\\right ] =-\\frac{f_i^{(2)}}{\\tau} ", "787e70fcb3328f091e4308a0a07c31df": "\n\\frac{n*\\sum_{i=1}^n (x_i-Q)^2(1-u_i^2)^4 I(|u_i|<1)}{(\\sum_i(1-u_i^2)(1-5u_i^2)I(|u_i|<1))^2} ,\n", "787e80723dd1076058e1892306013be6": "\\epsilon=\\pm 1", "787e824c7cb89b2531118cd68ca2e020": "\\cos(a) = \\mathbf{u} \\cdot \\mathbf{v}", "787e994a9587687c4ce6acbb2432d2f7": "Fb=I\\frac{d\\omega}{dt},", "787f1106b6189d57c6cb776897a3d52c": "D\\, T + \\delta \\, T=T", "787f731d696c5d9c23d442c7200d88e3": "W \\subset X", "787fc77c79b47a5cb9359d16b693feac": "\\Pr(A \\mid C) = \\sum_n \\Pr(A \\mid C \\cap B_n) \\Pr(B_n \\mid C) = \\sum_n \\Pr(A \\mid C \\cap B_n) \\Pr(B_n) ", "787fed294e4041d2fd565d03dc97de62": "\\textstyle n \\le 2^{r-b+1}-1", "78804110e6eeeaa2f5ce88e0e8a32b07": "{\\mathcal L}_{xx}^i", "788041ce2a9077ae28a6975c6d2391b8": "\\delta_\\odot = - \\arcsin \\left [ 0.39779 \\cos \\left ( 0.98565 \\left (N + 10 \\right ) + 1.914 \\sin \\left ( 0.98565 \\left ( N - 2 \\right ) \\right ) \\right ) \\right ]", "788053b7472b2a449c2e1a946fce37e3": "\\mathbf{X}_n=[x_n, x_{n+\\tau}, x_{n+2\\tau}, \\ldots, x_{n+(M-1)\\tau}]", "788056b1e714a1cd07153b464f6e6ba0": " c_0 = 1/ \\sqrt{\\epsilon_0\\mu_0} \\,", "78806f8378c14af32d4241cb03f73138": "\nc \\delta \\frac{d\\tau}{dq} = \nc \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\delta \\frac{dt}{dq}\n= c \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\frac{d \\delta t}{dq}\n\\,.", "788118bbdc62d9dc5bb3bb341a4d4452": " f(\\theta_{m+1})", "7881433ec2a3e5015ef2a38fa44f491f": "\\frac{d \\theta}{dt} = \\omega = 2\\pi f.\\,", "78814409cefcbc1a4e283600eb51cdb5": "f(z)=1, g(z)=z", "788167b0cb6156bb87753f9ae48221ba": " 3 - 1 = 2 ", "788173b5510317f2ab693120e8793525": " \\hat{H}", "7881a9f1036433de0f5ebe05555745b1": "1,\\ s\\ c_1(\\alpha s^2),\\ s^2\\ c_2(\\alpha s^2),", "7881e1cd7318399c47075eedef15d530": "\\mathbb{Z}/r\\mathbb{Z}", "788239fece2e2722673a2eb23c4f84cd": "\\Omega\\times v\\sim U\\Omega", "78823b7694b9492306d04d36d62d1559": "\n \\boldsymbol{F} = \\boldsymbol{F}^e\\cdot\\boldsymbol{F}^p\n ", "7882cc792fe85be64690e380af6dc129": " DI_m = \\underset{ 1 \\leqslant i \\leqslant m}{\\text{min}} \\left\\{ \\underset{ 1 \\leqslant j \\leqslant m, j \\neq i}{\\text{min}} \\left\\{ \\frac{\\delta(C_i,C_j)}{ \\underset{ 1 \\leqslant k \\leqslant m}{\\text{max}} \\Delta_k} \\right\\} \\right\\} ", "7882d41268a3e21ad078079d4c3e7b80": "\\delta = T_e \\eta^R\\,", "7882d4fcae2b358f63b24def6f56e7d4": "x-3", "7883357100bdab388f988f331cb1590b": "F_n=\\bigg[\\frac{\\varphi^n}{\\sqrt 5}\\bigg],\\ n \\geq 0.", "78833d3c53e189383ac80f25d9bbd982": "f = \\sum_{n=1}^\\infty \\alpha_n \\varphi_n. ", "788356a075428979d1d13c084981130c": "B = \\frac{4}{3} B_0 \\ln\\left(\\frac {R_o} {R_i}\\right)", "78839dcca3395c17001b05731f6e37cc": "\\scriptstyle\\lim_{x \\to c} f(x) \\;=\\; 0^+\\!", "7883ea64ea23949908e1fdbddaef6abe": "n_1n_2n_3\\cdots n_d", "7883eb8f499be1e9d53f7e1ccb4daba2": "E = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}{X(e^{i \\omega}) \\cdot Y^*(e^{i \\omega}) d\\omega} \\!", "7883f253448e7065dbb1dabd6e6235e7": "n^n\\approx 10^n", "7883f8b7e25b0c772094d449426a08dc": " J[f] = \\int \\limits_a^b L[ \\, x, f(x), f \\, '(x) \\, ] \\, dx \\ , ", "7884197a61a7670ce1d65f5e6477892f": "\\mu^0(l,p)", "78849a6d2cb98e74a9aaca52f942bb28": "k_{ij}.", "7884a16d34f4104ee8b9755e60c46126": "(\\Sigma, A, R)", "7884f730b44a4b829b5aaec20dee71fe": "v(x)\\mod g(x) = (a(x)g(x) + e(x))\\mod g(x) = e(x)\\mod g(x)", "7885126e7a07d84d780e916909dc4dca": "(\\mathit{n} - 1) \\times (\\mathit{n} - 1)", "788555fcbc6d2fab1a35743584327164": "\n\\begin{align}\ne^{i \\theta} & = \\cos \\theta + i \\sin \\theta \\\\\ne^{-i \\theta} & = \\cos \\theta - i \\sin \\theta \\\\\ne^{2i \\theta} & = \\frac{e^{i \\theta}}{e^{-i \\theta}} = \\frac{\\cos \\theta + i \\sin \\theta}{\\cos \\theta - i \\sin \\theta} \\\\\n & = \\frac{1 + i \\tan \\theta}{1 - i \\tan \\theta} \\\\[8pt]\n2i \\theta & = \\ln \\frac{1 + i \\tan \\theta}{1 - i \\tan \\theta} \\\\[8pt]\n2i \\tan^{-1} x & = \\ln \\frac{1 + ix}{1 - ix} \\\\[8pt]\n\\tan^{-1} x & = - \\frac{1}{2} i \\ln \\frac{1+ix}{1-ix}\n\\end{align}\n", "788597e091606961e53771fbfe4e02b4": "v_S(t) = V_P\\cdot \\cos(\\omega t + \\theta),\\,", "78859a82584aadb6a1dba2b0bc933a33": "x\\;(x + 1)\\;(x^2 + x + 1)\\;(x^2 - x + 1)^2 = x + x^3 + x^4 + x^5 + x^6 + x^8", "7885c543e79b22ba182acdcd64da50c0": "\\mathrm{\\frac{1}{T}}", "7885ea7670e24a5ce0198a907f25a34e": "\\mu = \\rho \\neq \\nu", "788618f6b38a43fa8fad0dc11677ebf8": " g(z)= 1 +b_1 z + b_2 z^2 + \\cdots.", "7886334bc3b9e4810c3b2d9bd73d93a3": "G= \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{4^{k+1}} \\left(\\frac{1}{(4k+3)^2}+\\frac{2}{(4k+2)^2}+\\frac{2}{(4k+1)^2}\\right)+\\tfrac{\\pi}{8}\\ln(2)", "788656101da1afb8f758ce80c6b88277": " \\mathbf{A} \\equiv A\\mathbf{\\hat{n}}, \\quad \\mathbf{S}\\equiv S\\mathbf{\\hat{n}} \\,\\!", "788665198e9dc3f8706cc490565ab46b": "\\{p_2\\}", "788674ded542e43bb766534f42741d48": "j\\textrm{'th}", "7886798da057820d635f743e5dd4e494": "\\int\\limits_{cv} \\!\\!\\!\\int_t^ {t+\\Delta t} (\\rho c \\frac{\\partial T} {\\partial t}\\,dt)\\,dV = \\rho c(T_P - {T_P}^O) \\Delta V ", "7886d9451576e0006625bc6b1bfbbb8a": "\\frac1{4g^2}{\\rm Tr}\\,G_{\\mu\\nu}G^{\\mu\\nu}", "7886ed9415d1a4a140388c780fbf18ef": "\\,(1-d) = v = {(1+i)}^{-1}", "7886f8bbd099511de3c02944c1bcc82b": "\\lambda (\\sqrt[\\alpha]{2} - 1)", "788715e589ecb3269f572e3234c849ef": "\\theta = Pi/2", "7887e0a8e2fae6df681fb3e476515f46": " i,j.", "7887ef0ee20e4761c3462e949e5d337c": "\\left \\langle \\nabla_{\\partial_i}\\partial_j, \\partial_k \\right \\rangle.", "788836bab47a8f444c9a6e8f68785294": "\\begin{array}{c}\n\\bar{x}_j = x_i\\frac{\\partial\\bar{x}_j}{\\partial x_i} \\\\\n\\upharpoonleft\\downharpoonright\\\\\nx_j = \\bar{x}_i\\frac{\\partial x_j}{\\partial\\bar{x}_i}\n\\end{array}\n", "788863c3d4b5884a2fd469c75ea86f43": "\\rho_{1234}", "788873365e9f8d8c309d218007d6a05e": "\\mathcal{C} = \\mathbf{Ab}(X)", "78887dc75bd531d3c8527aa7e75f81d1": "\n U_s = C_0 + s\\chi U_s \\quad \\text{or} \\quad U_s = \\frac{C_0}{1 - s\\chi} \\,.\n ", "78888440916069fdce194504757bb66e": " \\cos \\angle (v,w) = \\frac{v\\cdot w}{\\|v\\| \\cdot \\|w\\|} ", "78889fd8ddd1c2a3d460804dcf23df27": "\\int_{c(r)}R(z)dz=\\sum_{k=0}^{\\infty}\\int_{c(r)}\\frac{dz}{z^{k+1}}A^k=2\\pi iI_n", "7889a80079c57e40e831e7feadc7b3bf": " Y = \\log\\left(\\frac{X}{x_\\mathrm{m}}\\right) ", "788a024434a019e812e2f52479650357": " E[X]_{rr} = -2m/r^3/(1-2m/r), \\; E[X]_{\\theta \\theta} = m/r, \\; E[X]_{\\phi \\phi} = m \\sin(\\theta)^2/r", "788a8056d3df7923945fa0c3564ef5eb": "\\phi(1)=-\\ln 2", "788a9182c39e7dc978713873349b2a17": "I_{SC} \\approx I_L.", "788a95269f164c65a299336113ac1d3e": "-2\\sum_{i=1}^{n}J_{ij} \\left( \\Delta y_i-\\sum_{k=1}^{m} J_{ik}\\Delta \\beta_k \\right)=0", "788aa8a372810997124396dfe7f5f02b": " \\mu_{Y \\mid x} = \\mathcal{C}_{Y \\mid X} \\phi(x) = \\left(\n\\begin{array}{c}\nP(Y=1 \\mid X = x) \\\\\n\\vdots \\\\\nP(Y=K \\mid X = x) \\\\\n\\end{array}\n\\right) ", "788acaaa159cd260cfe3e98ff2a941a4": "x, y\\in S_X", "788ad7b5400b1668429cb6f67a3b22c0": "2.6510", "788b44fa1e2df93b394a2a62b84cb69c": "(c_{low},c_{hi})", "788b9d654917c66c5d74bbf657b17fa1": "\n\\begin{align} \n& V_{\\text{obs, r}}=\\left(\\Omega-\\Omega_{0}\\right)R_{0}\\sin\\left(l\\right) \\\\\n& V_{\\text{obs, t}}=\\left(\\Omega-\\Omega_{0}\\right)R_{0}\\cos\\left(l\\right)-\\Omega d \\\\\n\\end{align}\n", "788c53a810609a057294cc6eb231cb03": "-\\mu(-A) \\leq \\real\\, \\lambda_k \\leq \\mu(A)", "788c540b6194bdd1b64ebb61db869a9b": "F_\\alpha(\\sigma^\\gamma)=\n\\delta^\\gamma_\\alpha +\n\\frac{1}{12}(c^\\beta_{\\alpha\\mu}c^\\gamma_{\\beta\\nu} - 3 c^b_{\\alpha\\mu}c^\\gamma_{\\nu\nb})\\sigma^\\mu\\sigma^\\nu, \\qquad I_a(\\sigma^\\gamma)=c^\\gamma_{a\\nu}\\sigma^\\nu, ", "788c918479548ce20fcd4df8f0be5e51": "z_0=0", "788c99263745893b50ed24b0dc638b4a": "f \\longmapsto (f(P_1), \\dots ,f(P_n))", "788cad5feb4a8b7b7993cdaa96d5cc21": "\\psi(\\psi(0)) = \\omega^{\\omega^{\\omega^\\omega}}", "788cb62b3ae1f89886963d46b0751863": "\\mathbf X\\,\\!", "788d2acd81b8c1ba269c01aa66bc5139": "\\det (A-sI)", "788db96f0e841d472ce7594a492c7226": "r < r^*", "788de6c9708f73792677e021e08da8da": "\\left|\\frac{p_n}{q_n} - \\sqrt{S}\\right| < \\frac{1}{q_n^2 \\cdot \\sqrt{S}}.", "788dee5bfab262d34c01504da6cbafd9": "\nE =\n-{4\\over 3}{a_1 a_2 \\over 4 \\pi r } \\exp \\left ( -m r \\right )\n", "788e21edd8b3602789053331bacdfafb": "E_z=0", "788e4793328e235fa4023efc416821f1": "\\boldsymbol{x}_{k} = f(\\boldsymbol{x}_{k-1}, \\boldsymbol{u}_{k-1}) + \\boldsymbol{w}_{k-1}", "788e84122f38f59a051452dd457b6828": "\\scriptstyle{26.1%}", "788ee744955843cca2bcba1d9ac35492": "(N-1)(N-2)/2", "788ef92517edcba28dabcef6fbed55db": "a+b=0", "788f16bf57592ff951d0d61468237db3": "z(\\mathcal{N}):= \\max_{x\\in X}\\ \\{f(x): g(x,u)\\le b, \\forall u\\in \\mathcal{N}\\}", "788f51eee988d0bc654ff8857a47c68f": " \\lim_{n \\to \\infty}\\ _{interval}\\alpha = r_{ii}", "788f7e505059972921a7b4c9f554cb1c": "\n\\begin{array}{lcr}\n \\Delta X_t & = X_t - X_{t-1} \\\\\n \\Delta X_t & = (1-L)X_t ~.\n\\end{array}\n", "788fac0165ab43aea784bc755afacb7e": " B_k = \\{x \\in A: f_k(x) \\geq 1 - \\varepsilon \\}. ", "788faf94903393a765d2466e84a409f2": "x^l\\equiv a\\pmod{p}", "788fbd02dd8a48cff148eaf664affff3": "N(1-R) \\delta_1\\,", "78906f0b72090ebea338ba67861c27ba": "\\mu_1-\\mu_2\\,", "7891164bb1249f54cf273af2c5ed2d18": "v^{(jam)}", "7891fa1c2293f9c8b0796c28c083c500": "h_j", "78922428cb02d5b31687d30009c8c5c2": " \\ddot {\\mathbf r } = (\\ddot x, \\ \\ddot y ) = \\ddot r (\\cos \\varphi ,\\ \\sin \\varphi) + 2\\dot r \\dot \\varphi (-\\sin \\varphi ,\\ \\cos \\varphi) + r\\ddot \\varphi (-\\sin \\varphi ,\\ \\cos \\varphi) - r {\\dot \\varphi }^2 (\\cos \\varphi ,\\ \\sin \\varphi)\\ = ", "7892273e8f8dad244fa39267df139170": "{\\rm SQNR}= 20\\log_{10}{2^N} = N\\cdot(20\\log_{10}2) = N\\cdot 6.0206\\,\\rm{dB}", "7892aad4e64618570eef9b5546571520": "P(\\operatorname{deg}(v) = k) \\to \\frac{(np)^k \\mathrm{e}^{-np}}{k!} \\quad \\mbox{ as } n \\to \\infty \\mbox{ and } np = \\mathrm{const},", "789324ee75c4c00bd9bed5ab23103302": " \\chi_3(x) \\sim \\mathrm{Maxwell}(1)\\,", "789351eed1c78fd7e1252263984cd02f": "\\ \\displaystyle u \\in \\mathcal{U}(\\alpha,\\tilde{u})\\ ", "7893ceba9ca09696acc9b8918f818ca6": " \\binom nk = \\binom n{k-1} \\frac {n-k+1}k,\\text{ for }k>0 ", "7893d18584364ba342c0675adbe2fb34": "\\vartheta(z+a+b\\tau;\\tau) = \\exp(-\\pi i b^2 \\tau -2 \\pi i b z)\\,\\vartheta(z;\\tau)", "7893eea9929d640c7c740f9ab133fa9e": "y\\in \\lbrace 0, ..., N-1\\rbrace", "78942a88ab7eba117ae4240cfeb11a54": "fmep={imep-bmep}", "7894abf0839b01813792875aba66a980": "\\rho_{x^{n}\\left( m\\right)\n}", "7894b648b477abc64813dc2c30c38ad3": " \\phi \\, \\dot{u}_a = -\\phi_{,a} - \\dot{\\phi} \\, u_a ", "7894d812384bad54f18f61d278b86bbe": "T(r[f]+s[g])(p)=rT[f](p) + sT[g](p), \\qquad \\forall p\\in X, r,s\\in \\mathbf R;", "789507332418188f3309872e9f40333d": "w(x) = \\mathrm{e}^{-x^2/2}\\,\\!", "789564218daac5427ef9a75da96178eb": "M_{n+1} := \\left\\{\\begin{matrix}\n(x,y,z)\\in\\mathbb{R}^3: & \n\\begin{matrix}\\exists i,j,k\\in\\{0,1,2\\}: (3x-i,3y-j,3z-k)\\in M_n\n\\\\ \\mbox{and at most one of }i,j,k\\mbox{ is equal to 1}\\end{matrix}\n\\end{matrix}\\right\\}.", "7895a949ce1a60b7d4d53f840aee1e8a": "t=\\frac{T}{1-Re^{i\\delta}}", "7895b15da8950afa88c33dcfc440ce35": "\\tilde{H}_n(K\\vee L)\\cong \\tilde{H}_n(K)\\oplus\\tilde{H}_n(L)", "7895c2396b4ace91ffaf7b9dfed8535a": "(x^2+y^2-1)^3+27x^2y^2=0. \\,", "7895e96553399f2d87fc5aae9792d0fc": "~ d \\theta_n = n \\lambda,~ n=0,1,2.....", "789604b94ff7afb5f59f6794e27d3f74": "R = P \\circ Q", "7896f160503a2f83eb6c1b57fdc866b8": "\nE_\\text{out} = \\sum_{n=1}^\\infty {K_n E_\\text{in}^n}\n", "789705513bd29a165de311c22120f137": "z=\n\\frac{- \\mathit{far} \\cdot \\mathit{near}}{\\frac{z'}{S}\\left(\\mathit{far} - \\mathit{near}\\right) - {far}} \n=\n\\frac{- \\mathit S \\cdot {far} \\cdot \\mathit{near}}{z'\\left(\\mathit{far} - \\mathit{near}\\right) - {far} \\cdot S } ", "789732e82290bcb7331ee141497145f4": "AX = \\Omega_{e^x} + \\begin{matrix}\\frac{a}{b+\\frac{c}{d}}\\end{matrix}", "78973538e9712d7663a961fa1db4ea5a": "\n\\ln(t^r) = \\int_1^{t^r} \\frac{1}{x}dx = \\int_1^t \\frac{1}{w^r} \\left(rw^{r - 1} \\, dw\\right) = r \\int_1^t \\frac{1}{w} \\, dw = r \\ln(t).\n", "789758e7b773d353f3f7a1eacba308f8": "x \\mapsto f_x,\\,", "7897a033c74f7bcf792c5c974cb6217d": "V^{\\prime }\\left( t\\right) ", "7897b375a6f690d398815dd754265378": "S \\cdot T = \\mathrm{Tr}[S^\\dagger T] = \\vec{S}^\\dagger \\vec{T}", "7897e9c69298b1740f6c7380e187d91e": " \\tau_2 = \\frac {\\tau_1 \\tau_2} {\\tau_1} \\approx \\frac {\\tau_1 \\tau_2} {\\tau_1 + \\tau_2}\\ . ", "78980849e7915440cc4089d7852c0aec": "L_1=(1+i) L - P", "789866aefeec1a744360f38934c17647": " \\left\\{\\frac{p}{q}\\right\\}^n=\\left\\{\\frac{p^n}{q}\\right\\} ", "78986963afd33745c48f16ca8a9979b9": "\\sigma_p(t) = \\hbar / x_0 \\sqrt{2},", "78986c4ec77d9893141db464d11c4a14": " \\textbf{W}^{\\centerdot} f_A(y) \\bmod \\ q ", "789882809432d979dd7571eddfae0bf0": " Q: \\{Q_1(X_0,X_1,X_2,X_3)=0\\} \\cap \\{Q_2(X_0,X_1,X_2,X_3)=0\\} ", "78989706dbb587f8f3e296e13a5fc178": "\\color{OrangeRed}\\text{OrangeRed}", "78989c8529606cf0d28a1e95476faa81": "\n1 =\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{\n\\rho_{X^{n}\\left( m\\right) }\\right\\} \\right\\} ", "7898e777f1260e40e54630e029f897d5": "\\displaystyle c_n(x;a;q) = {}_2\\phi_1(q^{-n},x;0;q,-q^{n+1}/a)", "789905d83d1d9dd2c5deda725a8882b5": " (x , y)", "78990f6a0647efe2473534c2266d5d02": "g^z", "78991de1411093cdc6ef63f876d8cd1d": "\\mathbf F = (f_{s,t})", "789a3443ed2c4e3d3f4f99ca4d6450c5": "M \\propto N", "789a51f5e78b3bb169e1788215bfce18": "\\mathbf{E} = \\{e_1, \\dots, e_n\\}", "789a7b30b1cf3e91709cd7174d798765": "\n\\begin{align}\n\\frac{a}{2} [ 1 0 \\overline{1} ]\\rightarrow \\frac{a}{6} [ 2 \\overline{1} \\overline{1} ]+\\frac{a}{6} [ 1 1 \\overline{2} ]\n\\end{align}", "789ab7f029c4230949037367f6be76f1": " \\Delta\\nu ", "789ad8df090f98234fa991b0ac75728c": "\\mu(Z; V, W) = \\mu(Z; W, V)", "789aec8ff72d8c1cf2643d5abc16e17d": "-\\tfrac{b}{4a}", "789b5d76808fc838cb7149608672b52a": "\\scriptstyle r(t)", "789bcd6aef8f8b2bb24b8f8c6e5513de": " X \\sim \\mathrm{Benini}(0,\\tfrac{1}{2\\sigma^2},1), ", "789bce68c21f7faadc589160b013d4c1": " \\lambda_i \\rightarrow \\alpha \\lambda_i ", "789d44ce74d459686271f4a2cd806d78": "dI/dz = \\beta I ", "789d88135d444b8d67f2450e4a9b9ec5": "\\textrm{Im}\\left( \\operatorname{Li}_s(z) \\right) = -{{\\pi \\mu^{s-1}}\\over{\\Gamma(s)}} \\,.", "789d94a441312e77656488e4e26737a3": "Blind \\ Range = \\left (\\frac {C}{2 \\times PRF} \\right)", "789d95ac852169ae37f08241ad86150e": "\\hat{r}=\\cos u\\ \\hat{G}\\ +\\ \\sin u\\ \\hat{H}\\,", "789e111bb38849a9842bdd4114fbc7cd": "y[n] = \\begin{cases} +1 & x[n]\\geq e[n-1] \\\\ -1 & x[n]x_c", "789f455b85c727b115c545fb0de29958": "C_v = \\frac{\\partial \\langle E\\rangle}{\\partial T} = \\frac{1}{k_B T^2} \\langle (\\Delta E)^2 \\rangle.", "789f880a9e2176142bd7ae90dadada46": "a\\text{ and }b", "789fdc20848185d7cabcba1484769ef2": "\\begin{align}\n & \\operatorname{E} [\\,x_1x_2\\cdots x_{2n}\\,] = \\sum\\prod \\operatorname{E}[\\,x_ix_j\\,], \\\\\n & \\operatorname{E}[\\,x_1x_2\\cdots x_{2n-1}\\,] = 0,\n \\end{align}", "789fe8c5a02d1f14e196e76467eb5493": "\\epsilon^\\mu(q,x) \\,", "78a02933c66a8feb7dd612e89f363a80": "S(z)-S(z^0)", "78a02978f43e2e05ed8476b1760fb3d0": " U = U(S,V,\\{N_i\\})\\,", "78a043de091e25024fdb32b04f02b26c": "\\partial_k: C_k \\rightarrow C_{k-1}", "78a05a1baf37637f18ce39b7a6d9521f": "\\mathcal{A}.", "78a06b6a628ae5a3676a45b01d506ef8": "\\sum e(x_n),", "78a08ed5f7763e9769b50692583be836": "\\displaystyle{R^kf(w)=\\lim_{\\varepsilon\\rightarrow 0} \\int_{|z-w|\\ge \\varepsilon} M_k(w-z)f(z)\\,dx\\, dy,}", "78a12bac8c5bc513745e47b2f154b316": "{\\phi_{wk} = \\phi_{Wk}}", "78a1fb7950da84d8a0576429f65f6a97": "\\oint_S\\vec{E} \\cdot\\mathrm{d}\\vec{A} = \\frac{1}{\\varepsilon_0}\\,Q_{enclosed}=\\int_V{\\rho\\over\\varepsilon_0}\\cdot\\operatorname{d}^3 r,", "78a239faa2a9b3234c76b42b9b336e6d": " R_C \\| \\{[1+g_m (r_{\\pi}\\|R_S) ]r_O +(r_{\\pi}\\|R_S)\\} ", "78a25e083e30895e8b8d889d4b124553": "A=\\begin{bmatrix}0&L_0\\\\-L_1&0\\end{bmatrix},", "78a282756c24dd81dddc96bb90ecbc1a": "c^{2 \\alpha} (\\mu) = \\iiint_{\\mathbb{R}^{2}} c(x, y, z)^{2 \\alpha} \\, \\mathrm{d} \\mu (x) \\mathrm{d} \\mu (y) \\mathrm{d} \\mu (z).", "78a29677a26496674395be05d34339c4": "\\displaystyle{A(r)=\\pi \\sum_n n|c_{-n}|^2 r^{2n}.}", "78a2f0f198ddc152759fff8cef66c5dd": "2^{56}", "78a3161f92c42a26a098b88199d7b21d": " \\chi_{i,j} = {\\vec{d}_{i,j}\\cdot\\vec{E}_0 \\over \\hbar}", "78a333aaef84ee0660814fc9a6b65abc": "\\boldsymbol{\\psi}(\\boldsymbol{\\theta})", "78a33abf39e9aaaf4f1660e24b6169d5": "H_z", "78a33cc8bdb4e05af0a82180b2e8f68d": "\\textrm{change}(v,f(t))", "78a350b5e24ae92c44288f81066033ba": "\\frac{dQ^S/Q}{dP^S/P}>\\left|\\frac{dQ^D/Q}{dP^D/P}\\right|.", "78a36c10c17a82ab3b2ae09360dc55bd": "\\mathit \\Gamma = \\frac {Z_{02}-Z_{01}}{Z_{02}+Z_{01}}", "78a3a46d84793b122318db8ef1c64e32": "\\Pi_{i=1}^n(X_i+1)", "78a3b95747c9991fc59866989fcdfb4e": "O(Z)", "78a3d4e2dd776152e22a44addfd34495": " a = \\sqrt{k R \\theta} \\equiv ", "78a463335360107bb177c2cc7d756dde": "h=i\\partial\\bar{\\partial}F|_L.", "78a4d36338bc8c49b71e86c6d89a470b": "[0,\\tau]", "78a52178545ab634bb0b3afdc83ffdba": "R_{\\mathfrak{m}}", "78a53129784b54c76f6df4aa52a5a420": "\\displaystyle{UH_{\\mathbf{T}} U^* = H_{\\mathbf{R}}.}", "78a5440a91035f881117b0d5f54fb645": "\\text{If } p\\ne q, p'\\ne q', p \\equiv p' \\pmod 4, \\text{ and } q \\equiv q' \\pmod 4\\text{ then }\n\n\\left(\\frac{p}{q}\\right) \\left(\\frac{q}{p}\\right)\n=\\left(\\frac{p'}{q'}\\right) \\left(\\frac{q'}{p'}\\right).\n", "78a58ca1cb062981ca29264071547124": "z = h", "78a5de2c79e257d7422ad134cd158484": " \\sigma_0/E", "78a6038595e399d21edc29b9ecda0c6f": "\\overline{x}(t)", "78a61534c6d80b6c06d25f9c13a57972": "{\\color{Blue}~2.7}", "78a6264f4a2ed73c6493c3dcce5ad0f4": "r:x=k\\ ", "78a62822ecb5719c3d0bc74a96199c3a": "\\left[ \\begin{array}{ccc|c}\n2 & 1 & 0 & 7 \\\\\n0 & 1 & 0 & 3 \\\\\n0 & 0 & 1 & -1\n\\end{array} \\right] ", "78a6591bee0ff9dbe0bbe6d7455e41ae": "Expr \\rightarrow Expr\\,+\\,Expr\\,|\\,Int\\,|\\,String", "78a6e81c0c44e937d5bc34cc96f6f916": "\\,m=0,\\pm1,\\pm2,...", "78a700200ea34b29e5197a874667a853": "D_{NE}/H_{NE}\\,.", "78a71276f13c2257f4fbca40a13e198f": "\n\\left(\\frac{\\alpha}{\\mathfrak{p} }\\right)_n= \\zeta_n^s \\equiv \\alpha^{\\frac{\\mathrm{N} \\mathfrak{p} -1}{n}}\\pmod{\\mathfrak{p}}.\n", "78a77117ba1811742724e7362954dca2": "PG(2,q^2)", "78a81fcd63a7cb9ad49b07a70beb0811": "\n\\prod_{i}\\sum_{m_{i}} (z_{i})^{2 m_{i}} {1/2 \\choose m_{i}}=\\sqrt{\\det(1-|Z|^2)}\n", "78a858ae060041937331a8c37e74e33c": "\\mathcal{L}_\\mathrm{ghost} = \\partial_\\mu \\overline{c}^a\\partial^\\mu c^a + g f^{abc}(\\partial^\\mu\\overline{c}^a) A_\\mu^b c^c. ", "78a85fbdbe974b2d37905afa2ae619c2": " \\mathrm{CVA} = E^Q[L^*] = (1-R)\\int_0^T E^Q\\left[\\frac{B_0}{B_t} E(t)|\\tau=t\\right] d\\mathrm{PD}(0,t) ", "78a870acbaa2bc43ab8420310738fc2a": "H_i M \\otimes H_j M \\to H_{i+j-n} M", "78a8b9a2a1f8c0c41afa4fedb166f708": "x\\sin\\phi_0 - y\\cos\\phi_0 = 0", "78a8cc574710ea1f15013c6fff453600": "(\\mathbb Z/n\\mathbb Z)^\\times", "78a8e1284acbdbf590c78f147f6a865a": "F(x;a,b,p)= {\\left( 1+{\\left(\\frac{x}{b}\\right)}^{-a} \\right)}^{-p} ", "78a8f10484fc516e72a12f448f3f569e": "[H]_o = [HG]_{eq} + [H]_{eq}", "78a8f269ad1d00d3fe5ba822b22ca1d8": "E_4 = \\frac{i \\omega l}{2 n c} \\chi^{(3)} E_1 E_2 E_3^* ", "78a8f7e109889150dc4b288f3b913f7d": "\\mathrm{Ox} + e^- \\rightarrow \\mathrm{Red}", "78a92323cbfc7ce94e06e3844345a2d4": "\\sqrt{\\frac{121}{315}}\\!\\,", "78a9556d70b4d131b43f93222cc6b0c6": "\\mathbf{Top} \\times \\mathbf{Top} \\to \\mathbf{Top}", "78a9748383f1b0547ed320ed3791523f": "L=\\mathcal{L}(x^\\lambda,y^i,y^i_\\Lambda) \\, d^nx,", "78a9850eae5b5acae76711c19f0b3e24": "q_0 =1", "78a9b02f6a23a499594e1b1536f413ea": "\\psi^{(m)}(z+1)= \\sum_{k=0}^\\infty\n(-1)^{m+k+1} \\frac {(m+k)!}{k!} \\; \\zeta (m+k+1)\\; z^k \\qquad m \\ge 1", "78a9c7ddb881e661dcb72ba69d7cdeb3": "\\mathrm{ind} (U \\circ T) = \\mathrm{ind}(U) + \\mathrm{ind}(T).", "78aa20d935666174a907c9b76a75e1a2": "alive(0) \\equiv alive(1)", "78aa9ae886999ad3a7e7c9e224618f1b": "|\\psi\\rang = \\int_a^b c(x) | x \\rang \\, dx", "78aac77644e5ebd8b75f5bb30ffad43b": "ax^2+bx+c", "78aae2f62bff940193ee0b4f009638c1": "\\left( x',y',z' \\right)", "78aaf3d0a2f609d0b23fdb290797a726": "f: \\mathcal{A} \\to \\mathcal{B}", "78ab09aac37a1e097150111645b687f0": " \\lim_{n \\to \\infty} a_n = L ", "78ab14d5c59d6da16731c29b90c3ce45": "\\vec y=\\vec x/\\epsilon", "78ab3b57f5fcebe09bbbceeadca7e41d": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x) = f(x).", "78abb00d101cfb0f7f438d54c010b4c1": "PoA = \\frac{\\max_{s \\in E} C(s)}{\\min_{s \\in S} C(s)}", "78ac37ea2a399c684126e078a8063156": "a \\mid b,\\, a \\mid c \\Rightarrow b=ja,\\, c=ka \\Rightarrow b+c=(j+k)a \\Rightarrow a \\mid (b+c)", "78ac43dfa3770a4c5859fcf7c214fc69": "\\begin{align}\n\\Delta\\theta\n &= \\left|\\theta_{\\text{hr}} - \\theta_{\\text{min.}}\\right| \\\\\n &= \\left|\\frac{1}{2}(60H + M) - 6M\\right|\\\\\n &= \\left|\\frac{1}{2}(60H - 11M)\\right|\n\\end{align}", "78ace938a37208fef4216822bab9f4e6": "\\scriptstyle \\mathbf{f} = (f_x, f_y, f_z) ", "78ad1f3e9492599aeb62de8dd6f4beaa": "\\frac{1}{T}\\int_0^T U_t\\,dt", "78ad3e44d34aa203a6d6863eda4f9cff": "k^\\mu = \\left(\\frac{\\omega}{c}, \\vec{k} \\right) \\,", "78ad4c90f6d8404277bad119d1ad9508": "\\Gamma^{k}_{ij}", "78ad64b0be69aa304d54cb8ffffae673": "N = \\left(\\frac{Vf}{\\Lambda_c^3}\\right)\\zeta(3/2).", "78ad7b89837d9f0b5e7fa1cd1a3f0e4c": "X \\rightarrow Y \\in S^+", "78ad8335a1031d9ef812da583f06b950": " \n\\begin{align}\nT & \\equiv R_{pq} H R^\\dagger_{pq}, & & \\\\[6pt]\nT^\\dagger & = (R_{pq} H R^\\dagger_{pq})^\\dagger = R^{\\dagger^\\dagger}_{pq} H^\\dagger R^\\dagger_{pq} = R_{pq} H R^\\dagger_{pq} = T\n\\end{align}\n", "78ada7aa6fd3941bcb4fa049d0c4db46": "\nt = -RC \\times \\ln\\left(\\frac{V_{\\text{BE}\\_\\text{Q1}} - V_\\text{CC}}{V_{\\text{BE}\\_\\text{Q1}} - 2 V_\\text{CC}}\\right)\n", "78adb77466aaa69725936ed12dafe904": "\\mathrm dA", "78addf77600d8144d0526fd7c83c31dd": "\\Rightarrow -q\\xi_y + qv_xB_z = 0", "78ae1bcd119ffe2b085cd01a4c139d10": " \\frac{ TP \\times TN - FP \\times FN } {\\sqrt{ (TP+FP) ( TP + FN ) ( TN + FP ) ( TN + FN ) } }\n", "78ae29eabd3b1349fad658ae80399039": "CVP_\\gamma", "78ae617df133bdf39e60a35ff5b11166": "f(x_1, x_2) = (1-x_1)^2 + 100(x_2-x_1^2)^2 .\\quad ", "78af2348ae0b53d36ad22da14ef75f35": "s_1+\\ldots+s_{\\min(N_t, N_r)}=N_t", "78af348ccff6014e00f99fda51c55be8": "L^+", "78af3de77e6f66183e498c7e42b605f3": "\\operatorname{relint}(S) := \\{ x \\in S : \\exists\\epsilon > 0, N_\\epsilon(x) \\cap \\operatorname{aff}(S) \\subseteq S \\},", "78af4c2d4530bb7a66c1d2c428f1c3ad": "\\mathbf{\\ddot r}_{Moon} = G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}\\hat{\\mathbf{r}}_{{Moon},{Earth}}", "78af53d1b2e65fe6ddd14c4fb8c34d4a": " M_1", "78af623ba59afadb9ae583e02a6613b2": "u: B \\to C/N", "78af8ef2902a38dc9598f47b6d7b2646": "\\hat{\\theta} = \\theta + \\varepsilon", "78b040b5737549131868b0b22da8b71f": "\n\\begin{align}\nS_S & = -\\int d^4x\\,\\sqrt{-g} {1\\over G^3} \\left( \\frac12g^{\\mu\\nu}\\,\\nabla_\\mu G\\,\\nabla_\\nu G -V(G) \\right) \\\\\n& {} \\qquad\\qquad + {1\\over G} \\left(\\frac12g^{\\mu\\nu}\\,\\nabla_\\mu\\omega\\,\\nabla_\\nu\\omega -V(\\omega) \\right) +{1\\over\\mu^2G} \\left( \\frac12g^{\\mu\\nu}\\,\\nabla_\\mu\\mu\\,\\nabla_\\nu\\mu - V(\\mu) \\right)\n\\end{align}", "78b08064ac220141ff831ac018e97bff": "{\\langle}s_i{\\rangle}", "78b0bdb794d1059a888e361b9c321622": "I^\\pm", "78b0f5ac1670bb3efe64d3b20d8ba3f7": "[P_\\mu, P_\\nu] = 0 \\,", "78b11cd7deb752eed8d754b3c66fdcbc": " dG_{T,P} = -\\sum_k\\mathbb{A}_k\\, d\\xi_k + W'.\\,", "78b125e769f65467be2f2dddbc3ad5dd": "k=p/\\hbar", "78b15166e22c485352b7e4d1d7ce7159": "D-2", "78b166cd369e3b2964f1eb4dfa4eb893": "\\sum_{n=-\\infty}^{\\infty} |a_n|^2 = 2\\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi x^2 \\, dx", "78b16b9eab2a839946769c82312203c5": " -\\ln (1-p)", "78b198dba2e6473be6cdc71cb7ff9966": "\\nabla^2 \\Psi = 0", "78b1bd27779f6e078364e52ee7ba1702": " Q_B l_A a_B ", "78b1ea0096df561ccda487ef3ce49b59": "\\mathcal O (i)", "78b231c2c2f121192b6b08372596b612": "s=k^{-1}\\left(H\\left(m\\right)+xr\\right)\\bmod\\,q", "78b23983fe78bd5bc7a4d3edba97a468": "d_{LS}", "78b240f1e2e784cae59fa05701d11dfa": "\\mathcal L^{\\otimes n}", "78b25665fdd040219dda90cfb429948e": " F =\\frac{\\tanh(ht \\cdot \\sqrt(1-\\alpha)) \\cdot \\sqrt(1-\\alpha)}{\\tanh(ht)} ", "78b2f48887d811732983c3ad8edbb560": "\\sum_{i=1}^n \\left ( \\frac{Y_i - \\hat\\mu \\left (x_i \\right )}{\\delta_i} \\right )^2 \\le S", "78b3307cdc7c05913bbc490622681896": "g(x)= P(A\\mid X=x)", "78b340356680a411d9865b941e2f7b29": "A^+", "78b3cf92c4d2ac74a408dccf8231525b": "\n \\gamma = \\lambda - \\cfrac{1}{\\lambda} ~;~~ \\lambda_1 = \\lambda ~;~~ \\lambda_2 = \\cfrac{1}{\\lambda} ~;~~ \\lambda_3 = 1\n ", "78b3d749598557cadc96e8112e89db3e": "2.6153", "78b43eb760247182e0e5f5d933c4146a": "\\displaystyle{f_-(e^{i\\theta})=f_+(f(e^{i\\theta})).}", "78b452969571e267d2c2dbd8d45d26d8": "f^{(j)}(a)=P^{(j)}(a)", "78b45571a60f991c41a69929d3dbb73b": "\\ L", "78b4cf4c9bbdd5bce8f13c7c38d1de02": " 0 \\leqslant n_i < h", "78b50dc4100f07b4bfcec4982fd414d6": "\\Omega_t \\subset M", "78b52f4a591d5d4a71b4ecd8026505b9": "\n\n\\left[\\frac{m}{n}\\right]_3 =\n\\begin{cases}\n&+1 \\mbox{ if }m\\mbox{ is a cubic residue }\\pmod{n}\\\\\n&-1\\mbox{ if }m\\mbox{ is a cubic nonresidue }\\pmod{n}\n\\end{cases}\n", "78b57f7c5be060774738cb3a842bc5ae": "\n\\boldsymbol{\\nabla} \\times (\\boldsymbol{\\nabla} \\times \\mathbf{f}) = \\boldsymbol{\\nabla} (\\boldsymbol{\\nabla} \\cdot \\mathbf{f}) - (\\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\nabla}) \\mathbf{f}\n", "78b59cf446cbe193d249db243eb3e1a3": "X:=X-\\mu_X", "78b5a10bc6bbb53f58a08f3c29487338": "\\displaystyle{C^*=\\{X: (X,Y) > 0\\,\\,\\mathrm{for}\\,\\,Y \\in \\overline{C}\\}.}", "78b5b5471e73823de084e23c5861ce6a": "\\displaystyle V_o = D V_i ", "78b5e3581f816da2eb85db07f8dcdd11": "{px+q}\\,\\!", "78b607d0e94fc099f818576a58add3ea": "P_\\mu(n)=\\frac{1}{\\mu}\\int_0^{\\mu}P_\\lambda^*(n) \\, d\\lambda.", "78b61649a4efa8c5ec087c28a38f9c13": " 0\\le x\\le 1 ", "78b67d14049c6ae5129991d6ba8cb389": "4 p_0 p_2-p_1^2=0.\\ ", "78b69f08362ff7abc0846fbcf0f81184": "a + mb + nc + \\ldots", "78b6da12b1a83b4b5c9257423c90b667": "0-1\\neq 1-0", "78b6da7ea43e85631a1b96523b7066ea": "f''", "78b6e2df95c05c28604b777741a687b9": "0\\leq r\\leq n-1", "78b6e70e5ba83c5e25438e9cfb7d9087": "\\frac{\\partial c}{\\partial t} = \\nabla \\cdot (D \\nabla c) - \\nabla \\cdot (\\zeta^{-1} \\vec{F} c) + R", "78b6f16ee049deec5c4e397f209f015e": "C_{rr}", "78b70da0fb6369f45abaccaaef4cabe9": "x,y,z", "78b74563cca859d92f44d3aba33fc511": " IAt \\equiv SAt \\,\\!", "78b74dcdde3defe1f7de73448961c35f": "(-7)^2 \\equiv 6^2 \\equiv 10 \\pmod {13} ", "78b787532b55e73196e30f929a1f1f1c": "t_1=t_2", "78b7a3a07319ed685b3f8a32c043bcc9": "{1\\over2}mv^2 = mgh", "78b7a580567f488faf88b3ffe6e2d8ca": "E = (E_0 + 2 \\Delta) / (1 - 2 S)", "78b83774d3a907fbea42783d58a95204": "ji", "78b897dda2d9d36bda02361665ded044": "V_t = Var(X(t)|X(0) = i),", "78b932db1000269b5d6b72881c8564fc": "\n \\nabla^2 w \\equiv \\frac{1}{r}\\cfrac{d }{d r}\\left(r \\cfrac{d w}{d r}\\right) \\,.\n", "78b93d0b8f40d5404f38aa67383c78f8": "z \\in C\\,\\!", "78b93f026c37534aa25f510afb868980": "\nS_vp =\n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & \\frac{1}{s} \n\\end{bmatrix}\n\\begin{bmatrix}\np_x \\\\ p_y \\\\ p_z \\\\ 1 \n\\end{bmatrix}\n=\n\\begin{bmatrix}\np_x \\\\ p_y \\\\ p_z \\\\ \\frac{1}{s} \n\\end{bmatrix}\n", "78b94a7ec52177975bdeaefae7ef7c1c": "\\begin{align}E\\|\\mathcal{N}(0,I)\\| &= \\sqrt{2}\\,\\Gamma((n+1)/2)/\\Gamma(n/2) \n \\\\&\\approx \\sqrt{n}\\,(1-1/(4\\,n)+1/(21\\,n^2)) \\end{align}", "78b980f0bdf1498795f3de136c104286": "F= {}_2F_1(a,b;c;z), F(a+)={}_2F_1(a+1,b;c;z)", "78b98fd08cab83fe122c9601e803f1e7": "\nF(x) = \\frac{\\pi\\mu_0}{4} M^2 R^4 \\left[\\frac{1}{x^2} + \\frac{1}{(x+2t)^2} - \\frac{2}{(x + t)^2}\\right]\n", "78b9928359fd84a3a1ca78274000f728": "\\varrho_P", "78b9d7653846348bea8f9639f4d126ab": "\\{A : \\exists C,D,E\\in\\Gamma ( A=C\\setminus(D\\setminus E))\\}", "78ba248d82b85a7e4f1f6ac2608e2bef": "[\\ ,\\ [ \\!\\,", "78ba659dcccf8a2f1f564461dd69af4f": "\\Delta N_\\lambda", "78ba7b546ad156be5f0986cd20422ece": "A_{\\mathrm{Head}}=\\left(\\frac {2}{3}+\\frac {\\pi}{4\\sqrt {2}}\\right)a^2", "78bab63d6d7a87fac669f617e37298d1": "K=2\\sqrt{S(S-B)(S-C)(S-D_1)}", "78bacf8965544e3f239f4836a8349077": "K = \\frac{k}{\\sum_{i=1}^g \\frac{1}{n_i}}", "78bb04ef0da0883b099385c666f4fc8a": " \\bar D = {4 \\bar {B}^3 \\over \\bar{\\boldsymbol\\omega}^2}", "78bbb669169b5b5db7dfc3903667d1a7": "\\scriptstyle O(\\sqrt{\\log y})", "78bbd959fe7a6955a3d9ae3ccec3ee22": "(s,t)", "78bbf96612dd48263e7bf4f416d28331": " {X_{G}}", "78bc14bf95bc0841c529b0eb0e9ca5f1": "L(E):={E_1,E_2,\\dots,E_{m/2}}", "78bc7bc603f8529b9eec776d3188996d": "P_\\text{eff}=\\frac{1}{8}\\cdot\\frac{V^2}{R_i} ", "78bcdeb9a22b1ec82aba5a6dff68d836": " {T_P}^0", "78bdaa7ebae0c163c023918181f5b100": "(\\mathcal{F}f^{(k)})(\\xi) = (2\\pi i\\xi)^k \\mathcal{F}f(\\xi),", "78be405f570fa50e45e9035afc9444e2": "F^{\\alpha \\beta} = \\begin{pmatrix}\n0 & E_x/c & E_y/c & E_z/c \\\\\n-E_x/c & 0 & B_z & -B_y \\\\\n-E_y/c & -B_z & 0 & B_x \\\\\n-E_z/c & B_y & -B_x & 0\n\\end{pmatrix}\n", "78be4ea882b910619b0ee81996d23059": "\\sum_{n=1}^\\infty a_{\\sigma (n)} = M.", "78be5c3ca8e124034a864731544d17c6": "W = \\{x \\in I:\\forall J(\\Phi(J) \\to x \\in J)\\}", "78be65732713e93f4b0eb5a775d5780b": "\\mathbf{C}=d\\mathbf{B}", "78beee54d129ac6a7bd65f49ecf000c4": "\\operatorname{Del}(x)", "78bf03bbce87751ed67640a7d941fbd0": "\\langle p? \\rangle\\,\\!", "78bf37695e1ebe030425a91b979353d1": "\\scriptstyle \\mu", "78bf529ee540c4d33deca19f829d57d3": "Z_1, Z_2, ... , Z_n", "78bf604ddda8c5416440212a5c82fd13": "\\sqrt{\\frac{2}{\\pi}} \\sin(ut)", "78bf8bcf872c17d07c73c0a8154028a6": "\\text{s}^{-1}", "78bf8d475914ed62d4dac927206f6139": " f(\\zeta_1,\\ldots,\\zeta_n) = f(\\zeta,\\ldots,\\zeta). \\, ", "78bfbc3ec70ff25d010e625c52aa0104": " \\tfrac{X}{Y} \\sim \\textrm{Cauchy}(0,1)\\,", "78bfc3317ede3cebe1fda8ec2a111d7e": "\\chi \\, ", "78bfdf6be43eeb3e7b1d8374b46aa3f2": "A \\vee B", "78bfdfec045398a4e22e37c06da176dc": "\\epsilon_{k\\,\\ell}", "78c0087e09704a7f6bc9decd579ca8fc": "dx = \\frac{\\partial X}{\\partial u} du + \\frac{\\partial X}{\\partial v} dv", "78c07ff94618fbdcfe961606931809a3": "H(H(\\kappa))", "78c0adad15303ccaa35750a2bfbffe2e": " \\boldsymbol{\\tau} = \\mathbf{I} \\cdot \\boldsymbol{\\alpha} ", "78c0db613d230bd56c0369bdd6e47725": "T_s(t)", "78c116f68b8ae4e8c315872f1b8dda5e": " \\pi _T = 0 ", "78c132e6d9fd2edb66b4c453ac7fdac0": "(V_\\kappa,\\in,U)", "78c1482d5c0b893eba38d569760dd1fa": "4 \\dot{A} B^2 - 2 r \\ddot{B} AB + r \\dot{A} \\dot{B}B + r \\dot{B} ^2 A=0", "78c17cd0c8bd9f05897d35ee0da14807": "f(V^*XV) \\leq V^*f(X)V", "78c1b86cb5f17f5817f9baa36c05b6bd": "\\det[H_\\mathrm{eff}(z)-z]=0 \\, ", "78c2127c0f8d499e155300f34e6608a9": " e^{iaP} ", "78c225471c64c352ca08415ce8d3675b": "MRS_{GS} = \\frac{2\\text{ goat}}\\text{sheep} \\neq \\frac{1\\text{ goat}}{2\\text{ sheep}} = \\frac1{\\left(\\frac{2\\text{ sheep}}\\text{goat}\\right)} = \\frac1{MRS_{SG}}", "78c265223c9ff5c914499877503df800": "x^d p\\left(x^{-1}\\right),", "78c292f7a1a8bd8c88ea38e002ebc769": "B(t)", "78c2bf0e745ef66ed23a53269b5435ad": "A=B", "78c34cb98e8e2e8086efdd8041ea68ce": "r_{e} = P_{1}/P_{0}", "78c373014ddf5d71ae8ef1807bd5fc96": "\\frac{B}{r+1}", "78c398872bddc2db7f84a630c3cc5217": "5_2 = (5, 5)", "78c3a024854e3306bd52cb84309b8909": "\\mathrm{CMRR} = \\left (\\frac{A_\\mathrm{d}}{|A_\\mathrm{cm}|} \\right) = 10\\log_{10} \\left (\\frac{A_\\mathrm{d}}{A_\\mathrm{cm}} \\right)^2 dB = 20\\log_{10} \\left (\\frac{A_\\mathrm{d}}{|A_\\mathrm{cm}|} \\right) dB", "78c47a39eb3dec5f1f3a48d14a2471fc": " \\det (\\bold{1-c})=0 ", "78c49e897824edf12c0bfd1ad179c770": "I_{\\text{b}} = V_{\\text{R}_{\\text{b}}} / R_{\\text{b}}", "78c4db00693669212b6e855735b3772c": "\\mathbf{u}_n", "78c5908d8f9a329b02435d4fdbf946f0": "G_{\\mu\\nu} = R_{\\mu\\nu} - {1\\over2} g_{\\mu\\nu}R.", "78c5edb33510f0fb7f4793aa649c5256": "PPI = \\frac{d_p}{d_i}", "78c64aa44d98491a39632522e23bde74": "\\textbf{diag}(Ax+b)\\geq 0", "78c680126e369142289e52e604a14b3f": "g(n)\\ge\\log(n)", "78c6df19dc7f82d603d952beed836ea7": "V_\\mathrm{rms}=\\frac{V_\\mathrm{peak}}{\\sqrt{3}}.", "78c7359f68888fc2c041dbf48f3e24bd": "\\tbinom X k", "78c7d8c793b29e1063aee0c5f8713dcf": "G_M = G_N;", "78c7f77f7a29e6e1360a2b6b28766398": "_{rp=0}\\!", "78c821c8e7a1ab636950581cb60825a4": "g^n \\in GF(p^6)", "78c86e1262469ea20f4a9906d093022e": " \\partial_\\nu \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_\\nu A_\\mu )} \\right) - \\frac{\\partial \\mathcal{L}}{\\partial A_\\mu} = 0\\,. ", "78c8d814491814fe1e2cb499e10c6be9": "\\ v_a =\\frac{ \\sqrt {2gd}}{2} \\ ", "78c8eef24abad442ba1720c4c3eb711d": "f = 0.25 \\left[\\log_{10} \\left(\\frac{\\varepsilon}{3.7D} + \\frac{5.74}{\\mathrm{Re}^{0.9}}\\right)\\right]^{-2}", "78c92fa60c0a86864174434c4cee3db0": "\\log_b \\!\\left(\\frac x y \\right) = \\log_b (x) - \\log_b (y) \\,", "78c9d190d9e4449da8229353bc9e25fa": "(k+p)^2= \\,", "78c9edcb89a3a16abcf780207c5fb077": "SOD(n) \\mbox{ is the sum of all digits in } n", "78ca28bc33f2c638f9bac9f88d1aeb3d": " \\mathbf{k}_F ", "78ca2e96cacd0447a06ba9eb3fba5de6": "\\exp\\left(W_k[J]\\right)=Z_k[J]=\\int \\mathcal{D}\\phi \\exp\\left(-S[\\phi]-\\frac{1}{2}\\phi \\cdot R_k \\cdot \\phi +J\\cdot\\phi\\right)", "78ca34f85d8b5d0e88506e6ee6f34ce5": "\\scriptstyle +\\infty", "78caae70a99fc0d0ca4c39d006b2829c": "i^{p-1} = (-1)^{\\frac{p-1}{2}}", "78caaf93e44304d95f79f8dacf9f7f2a": "(CA,BD) + (CA,BD') = (CA,BS) .\\ ", "78cad89b5b8f78ca58052c832fbac830": "\nQ_1^{(\\text{red})}(t) - Q_2^{(\\text{red})}(t) = 1 \n", "78cb482fd4e7224e5417687de4f762ba": "n_1\\sin\\theta_1 = n_2\\sin\\theta_2\\ .", "78cbc7501dd3109831634ca210b4b6fe": "u[6] := 2*atan(\\sqrt(a1^2+b1^2)*(cosh(\\sqrt(a1^2+b1^2)*\\eta)+1)/(a1*sinh(\\sqrt(a1^2+b1^2)*\\eta))+b1/a1)", "78cbdfbf027f93e61654f314d1462d98": "\\lambda^2 = \\begin{pmatrix} 0 & -i & 0 \\\\ i & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}", "78cc1d27dd9e3532e7d0949550e3887c": "Q_{i,j,k}>0.0\\,", "78cc1edded6de50e1bfb01893b1b9ad5": "\\left({\\mbox{20 votes cast} \\over {\\mbox{3 seats to fill}+1}}\\right) +1 = \\mbox{6 votes required}", "78ccc889696daec926dcee6b8b8cedf3": "\\!\\rho g H sin\\beta", "78ccfc6ac61422b2cadca7ce6a28fbf2": "\\scriptstyle \\ddot{R}", "78ccfd55bac3bf18cc4f423513bb33a1": "= \\frac{1}{T} \\int_{-T/2}^{T/2} \\mathrm{III}_T(t) e^{-i 2 \\pi n t/T}\\, dt \\ ", "78cd14a6a27d4e3596c0a7b785e7c1c4": "\\ SF^n =_{def} \\{x_2...x_n : F^n x_2x_2...x_n\\}.", "78cd29bac8ae70c32d9672e9fbb525b9": "p(F_i\\vert C)", "78cdccedb9b4202d48af066dae007e57": "\\displaystyle 15", "78cdd593305f47552e04addfa3b9400a": "C=C_1\\ ||\\ C_2\\ ||\\ \\cdots\\ ||\\ C_n", "78cde33e9f053c95f4c0a30d0ab9d883": "\\left(\\frac{p^*}q\\right) = \\left(\\frac pq\\right)", "78ce1a67cf0b0c0f83eb94d17682416b": "\\Delta A =2 L \\Delta x", "78cea9accdb900d214fef6178f82532b": "W(2, k)", "78ceddfce19fd117b324f09cb3a72f32": "E_0 = 0.000 - 0.059*pH", "78cf2556870776787ad0e6ef8c66ccb2": "r=\\frac{L^2}{8v}+ \\frac{v}{2}.", "78cfaf6f2d9aa5f7361a23c055b43714": "\\displaystyle \\pi_n E = [\\Sigma^n S, E]", "78d018f5d1b6e7e5885a3578a9c80abb": "\n\\begin{align}\n\\mathbf{x}\\cdot\\mathbf{a} &= \\mathbf{x}\\cdot\\mathbf{b} = 0\\\\\n\\mathbf{a}\\cdot\\mathbf{a} &= \\mathbf{b}\\cdot\\mathbf{b}=1\\\\\n\\mathbf{a}\\cdot\\mathbf{b} &= 0\\\\\n\\mathbf{x}\\cdot (\\mathbf{a}\\times\\mathbf{b}) &> 0,\n\\end{align}\n", "78d0347a94d745173832c5bbcbaa88a7": "v(t)=V_\\mathrm{peak}\\cdot\\sin(\\omega t)", "78d0c79db13ddf66cf3a24c831508dd1": " ds^2~=~d\\tau^2~+~dx^2~+~dy^2~+~dz^2 ,", "78d103068e44f009a8346106eaf85acd": "q_{ab}", "78d138791efb742810e357b6dbf35e65": "\\mathrm{ROO{^{\\cdot}} + RH \\ \\xrightarrow {H-abstraction} \\ ROOH + {^{\\cdot}}R}", "78d1a94634238b927546b02255ae064c": "\\rho_{t+1}(X) \\leq \\rho_{t+1}(Y) \\Rightarrow \\rho_t(X) \\leq \\rho_t(Y) \\; \\forall X,Y \\in L^{0}(\\mathcal{F}_T)", "78d2008db19ae9b00263c5399e73a0c9": "\\frac{\\ln 2}{2\\pi}, \\frac{\\ln 3}{2\\pi}, \\frac{\\ln 5}{2\\pi},\\ldots,\\frac{\\ln p_N}{2\\pi}", "78d20d3377b200c339fcace145a84b61": "10^{(5\\cdot2^{103})}", "78d2738f5d90ee8be98a3b328cb9ffa8": "g:C\\rightarrow D ", "78d2981cf27ab5f5a33fe87112185caf": "\\scriptstyle \\overline{\\epsilon^2\\cos^2(\\omega t)}=\\epsilon^2/2", "78d29de50f8413fa9830da19c823a14f": "E^{(2)}_{lm}", "78d346bb740f9eb90301c5a12a5b198b": "\\tau_{ij} = e^{-\\Delta u_{ij}/{RT}}", "78d36d44940efefcf4771f588ef7518c": "v_T", "78d38022945245923f02698613921557": "\\langle\\bar{\\psi}|\\widehat{V}_a|\\bar{\\psi}\\rangle = \\langle \\psi | {U(R)}^\\dagger \\widehat{V}_a U(R) | \\psi \\rangle = \\sum_b R_{ab} \\langle \\psi | \\widehat{V}_b | \\psi \\rangle ", "78d3ef217bb8172fd13ecf738297a1d6": "\\mathrm{_{16}^{32}S} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{18}^{36}Ar} + \\gamma", "78d3f6d54e038f5e8d45da7215f8af39": "\n\\begin{align}\n & {} 2x^6-4x^5+5x^4-3x^3+x^2+3x \\\\\n & = A(x-1)^2(x^2+1)^2+B(x-1)(x^2+1)^2+(x^2+1)^2+(Dx+(A-B))(x-1)^3(x^2+1)+(x-1)^3 \\\\\n & = A((x-1)^2(x^2+1)^2 + (x-1)^3(x^2+1)) + B((x-1)(x^2+1) - (x-1)^3(x^2+1)) + (x^2+1)^2 + Dx(x-1)^3(x^2+1)+(x-1)^3\n\\end{align}\n", "78d4165d77badc83b8c283e164e32ef3": "s=\\alpha_1+\\alpha_2\\,e^{2^k(r_1-r_2)}.", "78d45234ca689ed17e625e0f762e3685": "\\begin{cases}\n\\frac{\\partial}{\\partial t}\\eta(t,x) = A\\eta(t,x), \\quad t>0 \\\\\n\\displaystyle\\lim_{t\\to 0^+} \\eta(t,x) = \\delta(x)\n\\end{cases}", "78d47a522fb4938b2e735026d235d931": " G := \\mbox{diff}(M_x) \\times \\mbox{diff}(N_y)", "78d4f3066d0c2ebecdfdf3aced833e22": "R_m \\leq 2^{-n-1}", "78d4fe65d181ad5c789d941f6e5cbb81": "\\psi' \\circ f(\\alpha) = g(\\beta) \\circ \\psi", "78d52512c395473e88c06be4a746f965": "\\displaystyle{S\\circ \\mathrm{Ad}(\\sigma(h))= \\mathrm{Ad}(h)\\circ S.}", "78d5356e2e314b9cb631392ce6d2b9fc": "\\hat{H}_{1} = -\\frac{1}{8c^{2}}\\sum_{i}\\frac{\\hat{p}_{i}^{4}}{m_{i}^{3}}", "78d690541bf13b7c5eb92b364be6cd5c": "W=0.1658 Y e^{-0.04Y}", "78d6cdf48e01023999f2780523a195cf": "i_n = i_r + p_e + rp + lp\\,\\!", "78d70030bf5e49c733e1778a37cdd64f": "\nf^{}_X(x)=\\frac{\\lambda^r}{\\Gamma(r)}\\,e^{-\\lambda x} x^{r-1}~~~~~~(x>0;\\,\\lambda,r>0)\n", "78d78bcd333f94fc5aee3464903c7cf7": "\\Omega(n^2/2^p)", "78d79a847053f68421d9267bdfa4ec0d": "\\operatorname{E}(Y | X) = m(X)", "78d7bfd21d4f391ab445da23b6a3ff0c": "S^m", "78d7d21f2930d19c3539c04e0fefeae6": "\\mathrm{R(T)}=\\mathrm{span}(T(\\beta\\mathrm{))}=\\mathrm{span}({T(v_1),T(v_2),\\ldots,T(v_n)})", "78d83e814e70c76b992887a8b9cfbba6": " \\left\\{ \\frac{a_1 + a_2 + \\cdots + a_n}{n} \\right\\}_{n=1}^\\infty \\in c_{00} ", "78d871266705b7da2f2496e0175cdf62": "\\epsilon(k)", "78d8c5efcc05bebf4001fd6df3ab95b3": "b(n)=\\frac{\\sqrt{2\\pi} n}{2m}", "78d8f088609adc55186add88c408ccc7": "\n \\begin{align}\n EI\\dfrac{dw}{dx} &= \\left[\\dfrac{Pbx^2}{2L} -\\cfrac{Pb}{6L}(L^2-b^2)\\right] - \\cfrac{P\\langle x-a \\rangle^2}{2} \\\\\n EI w &= \\left[\\dfrac{Pbx^3}{6L} -\\cfrac{Pbx}{6L}(L^2-b^2)\\right] - \\cfrac{P\\langle x-a \\rangle^3}{6} \n \\end{align}\n ", "78d92a95f37fe6ceff3bb66cbc67c0ba": " L = (L^x(t))_{x \\in \\mathbb R, t \\geq 0}", "78d9507fe7ca1a154d46b1d013586a42": "(b-a)\\inf_{x \\in [a,b]} f(x) \\leq L_{f,P} \\leq U_{f,P} \\leq (b-a)\\sup_{x \\in [a,b]} f(x)", "78d96d473f198b56723d0105dfdf303a": "1/\\sqrt{1+\\epsilon^2}", "78d9a63c0003f2d698276d9bc85d65ce": "\\frac{1}{(x - 1)(x^2 + x + 1)} = \\frac{A}{x - 1}", "78d9dc6c665f9a9e2b81c280a13771b7": "\\longrightarrow", "78da48f5399329a5055f8c417f000b50": " w(m,n) \\sum_{y \\mapsto x} \\phi(y) = \\phi(x) ", "78da4e2a19d2356849ea33ec18fb218b": "R(q,u)", "78dab0ad12a8dcbd53ee33c0b95b25c4": "a_1b_3", "78dab5a83650438e5ec8da6489af2daf": "E_{ij}\\in\\mathbb{C}^{n\\times n}", "78db09f69f2916301bd29fabf9fe8585": " E_M = ", "78db0ca0b2486106939effa61a924fa1": "\n \\hat{f}_0(a) = 2^n,~\n \\hat{f}_1(a) = -2^n.\n", "78db236c0bcdd2554e9a9981eb971e9c": "x^5-\\frac{22}{5}x^3-\\frac{11}{25}x^2+\\frac{462}{125}x+\\frac{979}{3125} ", "78db37e5980014fd74669eb9c7f35df3": "S_{12} = {2 Z_0 Z_{12} \\over \\Delta} \\,", "78db498500cab8d4e6802774543a19db": "v = {d \\over t}.", "78db614e8d60af195d78d8bbe9f3d895": "[a_{ij}]", "78db7445be46ec06b4abf848592b0eb9": " \\epsilon y'' + (1+\\epsilon) y' + y = 0,", "78db93befcc5f110e95b5e8153293322": "f^{-1}(V_i)", "78dbb34d86a1e641e8672a9db6e552b6": "I_M(\\tau) = \\int_{-\\infty}^{+\\infty}|E(t)E(t-\\tau)|^2dt = \\int_{-\\infty}^{+\\infty}I(t)I(t-\\tau)dt", "78dbb6b3e97e42e9dc1a71b227ac6207": "\\mathbf{L}_\\parallel' = \\gamma(\\mathbf{V})\\left(\\mathbf{L}_\\parallel + \\mathbf{V} \\wedge \\mathbf{N} \\right) ", "78dbc0028d22b0f58dba00053873843d": "(A\\,\\triangle\\,B)\\,\\triangle\\,(B\\,\\triangle\\,C) = A\\,\\triangle\\,C.\\,", "78dbfcc9bd6fbea432e146e0eaca3262": "GS({\\rm SI}) = (1+L_{\\rm B}) GS({\\rm TDB})\\,", "78dcb14d228c8a1af47d4fbdb65bdbdf": "(x-27)(x-9)^7(x+1)^{27}(x+3)^{21}. \\, ", "78dcdc5454a5ea8dda9f5535b1bd227e": "k_{\\mathrm{H,pc}} = \\frac{p}{c_\\mathrm{aq}}", "78dcf73852a012741bab034cfc044602": "\\Psi(x,t) = c_{0}(t)\\Psi_{0}(x,t) + c_{1}(t)\\Psi_{1}(x,t)", "78dd166f6c0d736c3b43d9ab7d3d3a32": "\\operatorname{Var} \\hat{f}(\\bold{x};\\bold{H}) = n^{-1} |\\bold{H}|^{-1/2} R(K) + o(n^{-1} |\\bold{H}|^{-1/2}).", "78dd1c9d3171c76c71fd239f20403fe2": "V_0 >\n\n0", "78dd3dd890402dd782c4645dc4ace4db": " \\quad 0 < \\alpha < 1 ", "78dd43f7ab33ace0d6b4400667c14c9c": " A=2 ", "78dd618802f6620c2ede5f9e4913f81c": "H(e)=-P(e)\\log P(e)-(1-P(e))\\log(1-P(e))", "78dd79e3b58616e160ae87f5467e65cf": "T = \\pi \\sqrt{\\frac{k'^2}{g h}}", "78dd860da8246a49de361c6f11b69553": " \\bar{H_0} = -\\frac{1}{2} \\nabla^2_{r_1} + V(r_1) - \\frac{1}{2} \\nabla^2_{r_2} + V(r_2) ", "78dd9363a812f3ec01444c18d364f650": "\n\\left[\\mathcal{J}_1, \\, \\mathcal{J}_2\\right] = i \\mathcal{J}_3, \\qquad \\hbox{and}\\qquad\n\\left[\\mathcal{P}_1, \\, \\mathcal{P}_2\\right] = -i \\mathcal{P}_3 \n", "78dda268c762099f96748fcdc563f797": "P(y)", "78ddd34de72036ff7b2cc774ed80b639": "x \\in cl(Y) \\Leftrightarrow (\\exists Y' \\subseteq Y) Y' \\text{ is finite and } x \\in cl(Y').", "78ddf425db836fc7dc49ca6047faaaae": "1 - \\cos^2 x = \\sin^2 x", "78de2385f87ff06fb9650f2a632210b4": "\\begin{pmatrix}\n 1 & 0 & z\\\\\n 0 & 1 & 0\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}", "78de59ec053b2352efe2b62a2717d7a8": "\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon} = \\boldsymbol{\\nabla} \\mathbf{w} = -\\boldsymbol{\\nabla} \\times \\Omega", "78de81ea161132aa123cd262b7362d14": "\\pi_*^{S}", "78deb5ad700048b3e68292e7961ca669": "S' = ", "78df12334cd6b3fedbcb35afbb9fb1fd": "f'(t)=\\lim_{h\\rightarrow0}\\frac{f(t+h)-f(t)}{h}.", "78df693d63d466a14576b7d3c2766333": "B_re^{iak_1}+B_le^{-iak_1}=C_re^{iak_0}+C_le^{-iak_0}", "78dfffcdb1822950d57ef0ba2780cef3": "u - u_{Wall} = \\beta \\frac{\\partial u}{\\partial n}", "78e00a5482296d63db0eb1dd3b57b127": "\\int f\\,dx \\not= \\lim\\int f_n\\,dx .", "78e0360c48095f5d50f6c088431428ea": "Z\\!", "78e04a95c215907a2396943ec60d9e6e": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t) = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t) + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t)\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t) ", "78e0622786cf04fb940c2bde3192973a": "N(x, y, z) = \\frac{1}{\\sqrt{1 + P^2 + Q^2}} (P, Q, -1).", "78e06f07e0a2c5616ba966c2bc3918b6": "v^2=Q(v). \\, ", "78e0ce249a53ad16400083e8761344d5": "p^{\\mu}=\\left(E/c,p_x,p_y,p_z\\right)", "78e0ea908a32279f1a4697092c774d36": "F(r)=m(\\ddot{r}-r\\dot{\\theta }^{2}).", "78e0f07096d63b38bee05c21af1a8e8f": "\\sum_{k=0}^\\infty \\frac{z^k}{k!} = e^z\\,\\!", "78e1142297d2e46543e41ec714c6e7f4": "t_{p-1} \\equiv 1, \\quad u_{p-1} \\equiv 0", "78e141764cf751c1f5b68ebb116a0f10": " \\textbf{y}(t) = \\begin{bmatrix} 1& 0\\end{bmatrix}\\textbf{x}(t).", "78e183116d496fa39a90503eb4a1c2cc": "C = 2\\pi r", "78e196a9f6c72fb30054b10c11fd474e": "O(V E)", "78e19d2ce739427aab29a3c2b97db074": "m_i,k_i \\in \\big\\{0,\\ldots, 2^w-1\\big\\}", "78e1c0430f4b90f1e1454e6d7c55e30b": "L^{2} ([0, T]; \\mathbb{R}^{n})", "78e20d417767fa188a6d553e0ddae9dd": "x^i,\\ i=0,1,2,\\dots", "78e26057720d754ebdca48672e70a163": "\\operatorname{ri}", "78e27ba9cb3eaeff0d58848d933de1a4": "\\vec{V} = [u(x,y),v(x,y)]^\\top", "78e2b60df65feea469437143c0f39a72": "PR(p_i;t+1) = \\frac{1-d}{N} + d \\sum_{p_j \\in M(p_i)} \\frac{PR (p_j; t)}{L(p_j)}", "78e2be85cdb11efd88e0df9a5e3d2d55": "Y \\sim \\mathrm{Pois}(\\lambda \\cdot p)\\,", "78e2cb4e95a3b6b6ff6ed2b3c0051722": "S^{q-1}\\rightarrow S^{q-1}", "78e2da38307cf42756f56516733e37a6": "E_{k} = \\frac{1}{2} m v^{2}", "78e3af73a540ac137e5a617f5283dccb": "\\text{1. }\\omega \\in B : P(\\omega|B) = \\frac{P(\\omega)}{P(B)}", "78e449b84b31bb96cafd0461c8c69a9d": "\\bar{g}_{F}", "78e4ad1218b59ee7e16bcfed6a942024": "E_1^{p,q}=H^q(X,\\Omega_X^p)\\Rightarrow \\mathbf{H}^{p+q}(X,\\Omega_X^{\\bullet})=:H_{DR}^{p+q}(X/k)", "78e4bfb5c3eaa11cbae915973bace42c": "g = |f|/I(|f|)", "78e4c6312e1a755976e5e73060a6d1c9": "\\forall c \\exists r, \\mbox{fell}(r,c)", "78e5060af090d3056d9c431d0c9b138a": "t_1 = 0", "78e50922d04fcc1b21aaf0b29cd33161": "f_0=f", "78e5508fef913284577f0ea2fe7716ed": "\\delta(v)=av", "78e56d1e3ba3cee2aed8d746c7e1d888": "g_{ij}A^j=A_i.", "78e587731b3db872a56afd387fc70b6d": " \\beta=12 ", "78e5b690a2281690cf20f3ce49f2caab": "v\\,\\!", "78e5cb6c411601f0e4aaed6f42f07f5e": "p(x) = \\sum_{n=0}^{N} a_n T_n(x).", "78e5f81a23359600cb348f29543e5ed5": "O(\\log n/\\epsilon) = O(\\log n)", "78e6004f0c76d869a6f8916bdbfdc04d": "\\mathcal{H}_{SB}", "78e601513d5adaf342155001454f3f8b": "\\eta_\\varepsilon(x) = \\varepsilon^{-1} \\eta \\left (\\frac{x}{\\varepsilon} \\right). ", "78e6155536f861e6950a1920672157d2": "\\frac{GVM}{GVD}\\gg TBP", "78e63912c076d55fd6c75e8f515bd7ee": "...d_{5}d_{4}d_{3}d_{2}d_{1}d_{0} . d_{-1}d_{-2}d_{-3}...", "78e699f62cc1896f6e51e9bf8fa4118b": "\n x( t ) = \\sum_{n=1}^{ \\infty } \\left(\n\\lim_{ r \\to 0 } \\left(\n {\\frac{ w^{n-1} p^{ n }}{ n! }}\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{ d } r ^{\\,n-1}} \\left(\n r^n \\left( \\frac{ 3 }{ 2 } ( \\arcsin( \\sqrt{ r } ) - \\sqrt{ r - r^2 } )\n \\right)^{ - \\frac{2}{3} n }\n \\right) \\right)\n \\right) ", "78e6c90e154bb48b57a3ca2035506728": "\\left(\\frac{300\\times300}{2}\\right)", "78e6e208adbcceb107d04cd53b72b6d9": "g(f(x))=h(x)=\\sum_{n=0}^\\infty{c_n \\over n!}x^n,", "78e70c92fcd16e5f7466885405233e61": "d\\vec B=\\frac{\\mu_0}{4\\pi}\\frac{Id\\vec l\\times\\vec r}{r^2} \\, .", "78e71586c955464cddacd05aeb8a462d": "g(x) = \n\\left\\{\\begin{array}{cc}f(x), & x \\in F, \\\\\n\\frac{1}{m(Q_j)}\\int_{Q_j}f(t)\\,dt, & x \\in Q_j,\\end{array}\\right.", "78e739b7cb96e35411cb021bc754269b": "\nk=1,2,....,n_c\n", "78e7990344b3bb13b0a06fbf78258bd1": " N_{-} = \\operatorname{ran}(A - i)^{\\perp} ", "78e7c86a30bd38186b641852c395a6d3": "\\frac{d\\bigl(f(x)\\bigr)}{dx}\\,.", "78e7cce33cbbcb91f7bbcb0c48ec22fd": " \\psi_1 ", "78e808fd63b9b21fd7b3091a2e6a52e6": "\\displaystyle \\partial_t u+u\\partial_x u=0.", "78e8363d0ca88546d1542a22a0d5b69e": " \\sigma_{V, W} : V \\otimes W \\rightarrow W \\otimes V : v \\otimes w \\mapsto w \\otimes v ", "78e869d1cf733618b266fc058ebed6ec": "\\alpha^{\\frac{N\\pi - 1}{4}}\\equiv i^k \\pmod{\\pi}", "78e87f454916456733d09251bfe256ca": "\\aleph_1 ", "78e89bf3a8a5e6a1973cb57de348b49f": "f^{\\mathrm{H}}_p(x) = \\begin{cases}\n \\frac{1 - x}{x} & \\text{if } p = 0 \\\\\n \\log\\frac{p + (1 - p)x}{x} & \\text{otherwise.}\n\\end{cases}", "78e8e51be78955c31adb34151cf8328c": "\n\\frac{x^{2}}{\\nu - A} + \\frac{y^{2}}{\\nu - B} = 2z + \\nu\n", "78e8eaa33b1930e23d6b57bb4827fd70": "Ms", "78e9439bc4d685c1e8814613a3fbaae6": "Q(I)", "78e9519975679c2dcab5150e4453fa05": " R_{\\rm specific} = \\frac{R}{M} ", "78e95f980a7d880e5f63e9268681f1fc": " r = L/2", "78e972eab93276233e812ab418d4a43b": "0^{\\circ}", "78e986e8f4b86856bb2185a11ae0767e": " \\|\\vec{n}\\|=\\sqrt{n_1^2+\\cdots + n_d^2}. ", "78e9a10bd3acc02bfc9230095417f0df": "\\frac{\\alpha-1}{x_\\min}", "78e9d43fc68e0270f3da83c7fe8c862f": " P() ", "78e9ebf3327446be9018ec77ce07a9da": "r_i(n) \\sim\\ N(0,\\sigma\\ ^2)", "78e9f623169070d64d68869067c7f73b": "R;\\ \\Zeta;\\ \\Chi", "78e9f76fbdde9e11dc6b6ea5bf5d80aa": "\\text{Area}=mn(m^2-n^2)(m^2+n^2)^2 \\, ", "78ea1269ba8f57573b7f74623202d44a": "[S_0] = [S] + [A_{ad}] + [B_{ad}]\\,", "78eaa18ac5eb898703ed16af5606b9b5": " 1\\leq p,q \\leq \\infty ", "78eaa70eafc29d2c73bfe06bd02cc8bd": "2^{O(k)}", "78eae079802e431330461b851869f6a4": "2\\Delta \\theta=\\frac{\\Delta n L2\\pi}{\\lambda}", "78eb5dd75a668ad3f8fb2921031751c7": "\\delta_1,\\dotsc,\\delta_{k+1}\\in\\left\\{+1,-1\\right\\}", "78eb6a9982f001236c2b725c4f7003be": "\\Phi(\\mathbf{x_i})^T\\Phi(\\mathbf{x})", "78eb875bb4d1b3e95c2f98de722d6f15": "C_\\bullet(Y),", "78ebbeb60798dbf719077553398c8ada": "\\frac{n'}{n}", "78ec1e89a2c6bc21e3a58361b2cbcfa2": "\\{w^{2^{k-1}+1}|w\\in\\{a,b\\}^*\\}", "78ec33ed5673ca333fad771f16e4b415": "\\scriptstyle m(t) = \\int_0^{t} \\lambda (u)\\text{d}u", "78ec42eb60dce04fcc40c14bf0cf5319": " \\frac{ \\log_e( x ) - \\log_e( a ) }{ \\log_e(b) - \\log_e( a ) } ", "78ecba65849a9030f544e6a5398842d6": "a_1\\times a_2", "78ecd2f4faec37f2873cbba502bf7d48": "\\sum_{i=1}^n x_i^2.", "78ecd69995fe6eb96974fe10e468b194": "\nL_{\\rm Np} = \\ln\\frac{x_1}{x_2} = \\ln x_1 - \\ln x_2. \\,\n", "78ed0864796237bcfbe70c110d9b80de": "q = e^{-\\frac{\\pi K'}{K}} = e^{{\\mathrm{i}} \\pi \\tau}\\,", "78ed15d79708750eaeccf7fb5da37b34": "I_{G}(f) = \\sum_{i=1}^{m} f_i (1-f_i) = \\sum_{i=1}^{m} (f_i - {f_i}^2) = \\sum_{i=1}^m f_i - \\sum_{i=1}^{m} {f_i}^2 = 1 - \\sum^{m}_{i=1} {f_i}^{2}", "78ed3c774ea92c03ce3ae470e48cb93d": "A_{1} \\subseteq A_{2} \\subseteq \\dots \\subseteq A = \\bigcup_{n = 1}^{\\infty} A_{n},", "78edb824d4b0275d713637cbd017f89b": "r = v/i \\qquad\\qquad\\qquad (2) \\,", "78ede3f6674c9bad8411ac950d8f07b8": "\\rho(X_i,X_j) = \\frac{\\operatorname{cov}(X_i,X_j)}{\\sqrt{\\operatorname{var}(X_i)\\operatorname{var}(X_j)}} = \\frac{-p_i p_j}{\\sqrt{p_i(1-p_i) p_j(1-p_j)}} = -\\sqrt{\\frac{p_i p_j}{(1-p_i)(1-p_j)}}.", "78edf2e5839c3e106e5157c275a454c2": "f(x;b,s,\\beta) = \\frac{bse^{bx}\\beta^{s}}{\\left(\\beta-1+e^{bx}\\right)^{s+1}}", "78ee13f06e06f197ec4edbd34cb1343a": "1 / \\pi = 12\\sum_{n=0}^\\infty \\frac{ (-1)^n (6n)!\\, (A+nB) }{(n!)^3(3n)!\\, C^{n+1/2}}\\,\\!", "78ee28c8444be4a9966870cb1ebac055": "V(x) = x^TMx", "78ee9fd2d95548ba523ca8766c179f68": " p_z = p_\\text{T} \\sinh{\\eta}", "78eebb8176f5b3ff56e7fc430be66c5b": "D^q(B/A, M) = H^q(\\operatorname{Hom}_B(L_{B/A}, M)),", "78ef3ba49a35fee6a314062e893104c9": "f(0) \\geq 0", "78ef9f58a6bcdf7a0a0a8d9673b16955": "A \\in S", "78efa35052d6c55470f4e5518bdecd95": "a_n\\ge a_{n-1}^2-a_{n-1}+1,", "78efd1e8148c68a2a0f5e116bfc3bdbe": "X_{ij} = (v_i, v_j)", "78f014a801403045a6d7f0ad0c534893": "\\,\\omega_s", "78f033cea31f1d9e226fe05d78e260ea": "\\lambda_i ", "78f093147ee894d453b7d1b248d72bf0": "Z_0^2 = L/C", "78f0bc264d501541f7495ff09e504307": "\\Pi = \\sum_{t=0}^{T-1} \\left[ pu_t - \\frac{u_t^2}{x_t} \\right] ", "78f11646db7b3289c5d55df9677e5e8b": "M_n(\\mathbb{F}_q)", "78f1932fbdb3f482495354bcde2850b9": "X\\subseteq \\R", "78f1bb505bc26f49bfb0002bce0f10a1": "\\epsilon(\\rho)=\\int_\\rho^\\infty -\\frac{{\\rm d}\\phi(H)}{{\\rm d}H}\\epsilon_{\\rm dry}(\\rho,H)", "78f1df7aab8bd2bba295738d8d02dbcf": "\\alpha = 2 \\arctan \\frac {d} {2 f}", "78f1e32a29f19d51227bddbfab03700b": "R_2(x)=\\frac{x^2-6x+1}{(x+1)^2}\\,", "78f1e64f905d2277c04ad9adae15a21c": "d(S \\upharpoonright n)", "78f22a2f1a70bb29d11504a3603c9529": "[c] = \\mathrm{\\tfrac{J}{kg \\cdot K}}", "78f23b1aeac9a2712bfa127befc693e6": "a\\nleq b,", "78f25a2110692073e94a421800a29135": "\\sqrt{3} + \\sqrt{7}", "78f286754108e7179e9fae99fdee8762": "\\sum_{n=1}^\\infty \\frac{(-1)^{n}}{(n+2)^a}\\sum_{n=1}^\\infty \\frac{H_n^{(c)}}{(n+1)^b}=\\zeta(\\bar{a},b,c) ", "78f2871eacbbb252af537ddc0a8fcd8b": "\\alpha_{t_2}", "78f2971512a5fdd1f50965fe9d84e9b0": "H[p]", "78f2d406beb6db2b0863704b1442ba7d": "P(X,x)", "78f301cbb636981a7772e94d431656cb": "p_{1\\infty} \\leftarrow x^3+6x^2+5x+1, M_{1\\infty}\\leftarrow x+2", "78f3b81618cddc238bb3cb3a48db1f85": "Pb(NO_3)_2 + 2 KI \\longrightarrow PbI_2 + 2 KNO_3", "78f3ce51a95d0c546158d01de8e58ead": "H^\\infty(\\mathbb{C}_+;L(X))", "78f49e86a97cef82abfade455209eff9": " \\{\\,x \\in {^*\\mathbb{R}} : 0 \\le x \\le 1 \\,\\},", "78f4ca7d08652bba7ba7d0e5f9f9df3f": "c_1,c_2,\\dots,c_k", "78f527c3e13ec8878d47f7b13c76c958": "(V_t)", "78f54732a7c01c0c50d2682fdaea3a3e": " \\mu = p \\mu_1 + ( 1 - p ) \\mu_2 ", "78f5ba919db461b73d0e88fb7672adb8": " D_\\mu \\psi ", "78f694276fa57da53bfd534c87bbef37": "levol", "78f69dee98e3a2ce1bf0c994407a979a": " d = R\\left( 3\\;\\frac {\\rho_M} {\\rho_m} \\right)^{\\frac{1}{3}} \\approx 1.44 R\\left( \\frac {\\rho_M} {\\rho_m} \\right)^{\\frac{1}{3}} ", "78f6cae8e5315189c22b0a0b3cfb522c": "\\operatorname{dom}(\\varphi) = \\{ (x,t) \\ | \\ t\\in[a_x,b_x], \\ a_x<0N\\}\\right)=0.", "78fde28e051a15198a2dda315a61d6ce": "\\sum m()", "78fe013604e361d09396aaad6f5f6069": "2^{s^2}", "78fe1f7bce4cd3d8349390dc7934ebae": "{an \\choose bn} \\equiv {a \\choose b} \\pmod{n^k}.", "78fe6e9544ae7d853a90c496de390abb": "I_a = 0.05S", "78fe88488d302870574a5e3c4456c64f": " k \\ge \\frac{ \\theta_3 + \\sqrt{ \\theta_3^2 + 4 } }{ 2 } ", "78feabda7ed6626c0ac4a49841e9bb57": "\\textstyle a(x) = b(x)", "78fed4d2355d3f3c2e96f2c17987cefd": "0 \\over 1", "78fee4eb068bad2e9e6cae0caf3d24f3": "\\begin{align}\nX[k] \\ &\\stackrel{\\text{def}}{=}\\ \\frac{1}{NT} \\int_{NT} \\left[\\sum_{n=-\\infty}^{\\infty}x[n]\\cdot \\delta(t-nT)\\right] e^{-i 2 \\pi \\frac{k}{NT}t} dt \\quad \n\\scriptstyle {\\text{(integral over any interval of length NT)}} \\displaystyle \\\\\n&= \\frac{1}{NT} \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\int_{NT} \\delta(t-nT)\\cdot e^{-i 2 \\pi \\frac{k}{NT}t} dt \\\\\n&= \\frac{1}{NT} \\underbrace{\\sum_{N} x[n]\\cdot e^{-i 2 \\pi \\frac{k}{N}n}}_{DFT} \\quad \\scriptstyle {\\text{(sum over any n-sequence of length N)}} \\\\\n&= \\frac{1}{N} \\underbrace{\\sum_{N} x(nT)\\cdot e^{-i 2 \\pi \\frac{k}{N}n}}_{DFT},\n\\end{align}", "78fee5e728e82d456a8ca75a7c73a6c3": "X_i = \\{x\\in X\\colon T_x X", "78ff381d800d85cde62cc2ffb764b0ba": "\n (\\sigma_y)_{n\\rightarrow\\infty} = \\max \\left(|\\sigma_2-\\sigma_3|, |\\sigma_3-\\sigma_1|,|\\sigma_1-\\sigma_2|\\right) \\,.\n ", "78ff80cf00131dd1b99e3b2c6d7f0272": "\n \\frac{\\partial f}{\\partial x} = +k\\, \\frac{\\partial f}{\\partial \\theta} \n \\qquad \\text{and} \\qquad \n \\frac{\\partial f}{\\partial t} = -\\omega\\, \\frac{\\partial f}{\\partial \\theta}.\n", "78ffc9279968e8e07c1204020226d895": "\\scriptstyle h = r - \\sqrt{r^2 - a^2}", "79001a254d8dc5161cf8ef160fddb0bc": "x^1\\in X", "790024d6064e5934deabe66e20fea1e0": "C_A(B)", "7900775a21d22d25d36332f682636abd": "\\operatorname{Ob}(C)", "7900a983562ba5f3492ecab83e91d85d": " \\frac{2\\pi}{z} ", "79011b46f504109f61f86028a28ab6a5": "\\tau \\ll {\\hbar \\over \\Delta\\bar{H}}", "790122e53088d38940c09740b9e8bdc1": "\\mathtt{union}", "790143fc9ba5e18860283608a44faaae": "\\bar{g_i}=\\mu _i=g_i^\\mathrm{gas}+RT\\ln \\frac{f_i}{p^u}=g_i^\\mathrm{gas}+RT\\ln \\frac{f_i^*}{p^u}+RT\\ln x_i=\\mu _i^*+ RT\\ln x_i", "79014c8ac50f69a8fb8a2c740edcdd13": "x_i/x_j \\mapsto s_i/s_j", "7901bd610d9829f5fff8889f6aefcbd4": "q:=\\frac{p_n}r", "79023a3144e1462f92d15aeb46e370d0": "10 \\uparrow \\uparrow \\uparrow \\uparrow 2 = 10\\uparrow\\uparrow\\uparrow 10=(10 \\uparrow \\uparrow)^10 1", "7902677cdc27694e234952c2499b0ed0": "J_{a}(\\lambda) = \\lambda I + N \\,", "7902c0489ced1291bbdc68aa92b438d6": "1.295 \\pm 0.006", "79030bf4de67909a1f6ac1a3b7a3a4cc": "1 \\leq t(d, n) \\leq n", "79035077a352f1d06a2b13f90ee67e7c": "s \\longrightarrow_R t", "790362645eee1ae0866cda907fdee66d": "P_n(x^2)\\sin(x)+xQ_n(x^2)\\cos(x)", "79038b04d9e1d0d383493003c32a82fa": "a \\Leftarrow b.", "790394d155c0d51ea5f3bd39441b8b9a": "I(0,z) =\\lim_{r\\to 0} \\frac {P_0 \\left[ 1 - e^{-2r^2 / w^2(z)} \\right]} {\\pi r^2} \n = \\frac{P_0}{\\pi} \\lim_{r\\to 0} \\frac { \\left[ -(-2)(2r) e^{-2r^2 / w^2(z)} \\right]} {w^2(z)(2r)} \n = {2P_0 \\over \\pi w^2(z)}. ", "79039c17b381034819ad16e04d8aa4ba": "\\mathfrak{L} = (w^{1})^{2} D a_{11} + 2 w^{1}w^{2} D a_{12} + (w^{2})^{2} D a_{22}", "7904888f566bc9d46137fc772cb9785e": "gF_i\\ ", "7904d09549ad423513d48045a00dab62": "p^k", "790543e4ec2d51b54b12c779f7c17829": " (\\langle M \\rangle, 10^k) ", "79059dca2ce46f4d96b4763524d983bd": "R_{se}", "7905b7291328b4758a84c3b770797089": " \\sigma^0 = -\\sqrt{1-2m/r} \\, dt, \\; \\sigma^1 = \\frac{dr}{\\sqrt{1-2m/r}}, \\; \\sigma^2 = r d\\theta, \\; \\sigma^3 = r \\sin(\\theta) d\\phi", "790690479fb21d7fce0fbd41d391b959": "d\\Phi", "7906bfae7486baf777c081b31b2ea66c": "0 \\leq \\alpha_1, \\alpha_2 \\leq C,", "7906d9d0f38056936773a25aaaba7d2f": "\\lim_{t \\to \\infty}p_k(t)", "7907305bdda9e2bbccf06e0d208fef15": "W_1 ", "790738d512fae95a2a1ea54880e99d72": "x=[0;a_1, a_2, a_3,\\cdots]", "790751be8fff25669e699dd66319c97e": "\\lambda_{\\bold{k}} = \\frac{\\hbar^2 k^2}{2m}", "790751d7c271b538491ab80f33529a3b": "B_4(f,g)=\\left[\\begin{matrix}-1 & 0 & 3 & 0\\\\0 &8 &0 &0 \\\\3&0&15&0\\\\0&0&0&0\\end{matrix}\\right].", "7907ae4af7e83c1be9a61b65ff67c414": "\n\\mu \\ddot{\\mathbf{r}} = {F}(r) \\hat{\\mathbf{r}} \\ ,\n", "7907b99d0f2c0e2e1aeba433b2edf057": "\n\\Pr(B_n = B) = \\dfrac{\\prod_{b\\in B} (|b| -1)!}{n!}\n", "7907e809794b3a5c49f7791953f53ffa": "\\mathcal{F}_x := \\varinjlim_{U\\ni x} \\mathcal{F}(U).", "790802f00bc83c2f8a6596278b7cf34f": "\\sigma(E) \\sigma(t) \\ge \\frac{\\hbar}{2} \\,\\!", "790823bc3c79ad2cc14a31005f47eef7": "/(n+1)", "79082c62e294264216c802005a71b274": "\\vec S", "79083893f5b127ba6d1a1cf8645d897b": "(\\mu, \\nu, \\phi)", "79085bf6da1b6f425700f20bda804b6d": "\\Delta f = {1\\over H} (\\partial_x {G\\over H} \\partial_x f - \\partial_x {F\\over H}\\partial_y f -\\partial_y {F\\over H}\\partial_x f + \\partial_y {E\\over H}\\partial_yf),", "790873ac735d31e21adfd06013d2127f": "\\left(t_i\\right)_i", "79089fbb20632c0c57767e25b5368bd3": "e_1 = \\frac{1}{2}", "790900f74135b999f5643944ca5c1370": " \\int_{S} \\mathbf{u \\cdot T} dS + \\int_V \\mathbf u \\cdot \\mathbf f dV = \\int_V \\boldsymbol\\epsilon : \\boldsymbol\\sigma dV ", "7909017d3c919ee8e4b011ecaa5e1378": " 0 = (x^2+y^2+z^2 + 16)^2 - 100(x^2+y^2). \\,\\! ", "79095058fdce3df41d0175acee6dc2a0": "\\phi ( \\bold{r} ) =\\frac {1}{4 \\pi \\varepsilon_0}\\int \\bold{\\nabla_{\\bold {r_0}}\\cdot} \\left( \\bold{p} ( \\bold{ r}_0 ) \\frac {1}{|\\bold r - \\bold{r}_0|} \\right) d^3 \\bold{ r}_0-\\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac {\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 )}{|\\bold r - \\bold{r}_0|} d^3 \\bold{ r}_0 \\ . ", "79099b8939dd21a40a6f993b0d446d4a": "F_V(t,T)", "7909ea7018bccd93313f8083775f8581": "q(\\xi ,\\tau_1 + \\tau_2 ) = q(\\star q(\\xi ,\\tau_1 ),\\tau_2)", "7909f16ca55868dc8134a89ce17e28cf": " \\frac {dR}{dz}= \\frac {iw}{c_0 Q} RD+T(D-R) \\quad(2.8.a) ", "790a130709d7eeed0784fe96fc46a5b5": "\n\\begin{align}\n & \\mathbf{H(s)}={{\\left( \\mathbf{X}-\\mathbf{K\\tilde{N}} \\right)}^{-1}}\\left( \\mathbf{Y}-\\mathbf{K\\tilde{D}} \\right) \\\\ \n & =\\left( \\mathbf{\\tilde{Y}}+\\mathbf{DK} \\right){{\\left( \\mathbf{\\tilde{X}}-\\mathbf{NK} \\right)}^{-1}} \n\\end{align}\n\n", "790a2bc05b28b07121261629568685c9": "(\\Omega, \\Gamma)", "790a3ef2c37e39f91fd2ec2a028965fc": "G \\ge B > R", "790a4be287d3676518bbb50967d89c2e": " \\varphi(x, y)", "790ab1c677ba93cf3d3d9023b2c7cce8": "\\dot{\\rho}", "790bb3b1649c948afc0055f0b83fb3fb": "\\int\\cosh^2 ax\\,dx = \\frac{1}{4a}\\sinh 2ax + \\frac{x}{2}+C\\,", "790bc5fb8506d363ce2e369788eb51d6": "\\widetilde{\\Gamma}^{\\alpha}_{\\beta\\gamma}=\\Gamma^{\\alpha}_{\\beta\\gamma}+S^{\\alpha}_{\\beta\\gamma}", "790c0fd10c76047c4a4daad5a423ba8b": "(1/b^2)(\\mathrm{E}\\{[\\ln(X)]^2\\} - (\\mathrm{E}[\\ln(X)])^2)\\,", "790c2e709771a8b96d8d8dbed0168a02": "\\sigma(A) = -\\sigma(-A)", "790c45c5acda1091dfbd65e0e7e2f505": "\\tau_b\\propto h S", "790c75239e3f2638cd5b1969765a9a19": " A_i \\cup A_j ", "790c76ceb13e928d08edc53d7ac4bb5c": "\\otimes", "790cd6a209a3cd8912417a7f8cb73d2a": "lcm(1, 2, ..., n) \\ge e^{n-o(n)}", "790cddd5ad4fe7845da3d639dc4fa137": " \\Delta t' = \\gamma \\, \\Delta t = \\frac{\\Delta t}{\\sqrt{1-\\frac{v^2}{c^2}}} \\,", "790ce6ec982f786cbbc30583e1b4fe42": "\\langle R_N(u,v)w,z\\rangle = \\langle R_M(u,v)w,z\\rangle+\\langle \\mathrm I\\!\\mathrm I(u,z),\\mathrm I\\!\\mathrm I(v,w)\\rangle-\\langle \\mathrm I\\!\\mathrm I(u,w),\\mathrm I\\!\\mathrm I(v,z)\\rangle.", "790d28dd3aa2fcbfcf094701655c5267": "|p|_{\\ast}^{e},|q|_{\\ast}^{e}<1/2", "790d2e59792aa62d69d686c9a310139b": " \\ddot r = -k/r^{2} + l^{2}/r^{3}", "790dbdab313313d54bf804c29615317b": "q_\\text{S} = e \\ ", "790dd175d3eab4bd7acb5f51e842f557": " \n| Re( \\overline{\\lambda} ) | \\geq\n| Re( \\lambda_t ) | \\geq\n| Re( \\underline{\\lambda} ) |, \\qquad\nt = 1, 2, \\ldots , n \\qquad (8) ", "790df860e506f25b04b2d49b3f10d2e2": "{[f(x)]}^{g(\\theta)}", "790e0b8e16783898d66dc884fbfbd7c4": "\\mathbf{y}^{\\prime\\prime} = (y_1^{\\prime\\prime}, \\ldots, y_N^{\\prime\\prime})", "790e43302b2ae19ac517dce1bd5e4c75": "(X_0, X_1)_{\\theta, 1} \\subset (X_0, X_1)_\\theta \\subset (X_0, X_1)_{\\theta, \\infty}.\\,", "790e7395ba3dac7f619f4b9a525da497": "e_{j,t-1}", "790e73b071d0ef1c45357e3735f48d94": " T_{\\mu\\nu} ", "790e8b7a56e4369a9185b83120f34b24": "R_{TOT}", "790f0ec90ca7c19015bf687ca575228c": "v = v(\\lambda).\\,", "790fc53ac0f087abacdd95b602109dda": " c - h \\le X \\le c + h ", "790fd2714505810930ab5bfa68ec89c8": "\\nabla_{\\!\\theta} \\ln p(x|\\theta) = \\frac{\\nabla_{\\!\\theta} p(x)}{p(x)} ", "790fd6df3c7d4df1d71b53d5a2d49035": "\\operatorname{Var}(X)\\ge 0.", "79105bcc3bae8620c78184a1f115729a": "\\operatorname{pos}(U) = \\max_{\\omega \\in U} \\operatorname{pos}(\\{\\omega\\})", "79111bf3643e865931cc167e7b04c18f": "(a_1,a_2)", "7911266b5210b0eb6cc6c408ac54586a": " \\text{E}(\\xi_k) = 0, \\text{Var}(\\xi_k) = \\lambda_k \\text{ and } \\text{E}(\\xi_k \\xi_l) = 0 \\text{ for } k \\ne l.", "791190c5f9a7fb44f77000331ef3fbcb": "\\begin{align}\n R_3 &= \\frac{R_1 \\cdot R_4}{R_2} \\\\\n L_3 &= R_1 \\cdot R_4 \\cdot C_2\n\\end{align}", "79119ecf9be1c71a628b855df672fa1e": "k=\\frac{1}{3}nvcl. ", "7911c2228eb4612ff414a4884981b265": "G H^T = P-P = 0", "79122a8c8bf24f277a220c25f132272f": " i [L(\\hat{x}, \\hat{p}), \\hat{p}] = 0 = -U'(\\hat{x}) ", "791292129f40bc0897600f7e951623ce": " Q = \\int \\lambda d \\Omega_Q(\\lambda).", "79129e654380b8a3f87ef9e568faf06c": "\\nu_0(A):=\\nu(A)-\\int_A g\\,d\\mu", "7912cce3558a0b161c317c2c8dbb0782": "S(\\mathbf{q}) = \\frac{1}{N} \\left \\langle \\sum_{jk} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_j - \\mathbf{R}_k)} \\right \\rangle", "7912dbb94ecad3fe18cc3774c09df38a": " \\left(\\frac{M L^2}{T^3}\\right)", "7912ee8c470a6640bea867643ce69da5": "{\\mathbf{}}n=n_r", "791330a570e7d859f46f47a941c0c734": "\\int d^3\\mathbf{x'}\\mathbf{M}_{\\text{effective}}(\\mathbf{x'}) = \\mathbf{m}", "791339076da2aaeb31a30fb7c25842b7": "f : \\mathbb{R}^n \\rightarrow \\mathbb{R}", "79134112e000fd858fcf3205baab2cb5": "{7N(n) + 3 = T(7n - 3)}.", "79136bf64194fc5d72bd222bfdb09e72": "c=\\pm i \\sqrt{\\frac{g\\mathcal{A}}{\\alpha}},\\qquad \\mathcal{A}=\\frac{\\rho_G-\\rho_L}{\\rho_G+\\rho_L},\\,", "7913740d7a9ed37b395de50b6d43b5de": "2 \\geq k", "7913771486a2d225b873c8e75644dd26": "f := 2 \\cdot k^2/\\cosh(k \\cdot (x - 4 \\cdot k^2 \\cdot t))^2", "791387e9cf343c6dc4cc6b9c5663142f": "\\zeta_{\\mathrm{Ai}}(s)=\\sum_{i=1}^{\\infty} \\frac{1}{|a_i|^s}.", "79139797ae8318d397aac0044cb12122": " y(k)= \\sum_{m=0}^ {M-1} \\sum_{n=0}^{N-1} C_{nm} \\cdot g_{nm} (k) ", "7913a48eb5acc72e366c4db42a62fb1e": "v_0 = 0", "7913d062e9b4903a1588944635d576b7": "\nn!\n\\left(\n1 - \\frac{1}{1!} + \\frac{1}{2!} - \\frac{1}{3!} + \\cdots \\pm \\frac{1}{n!}\n\\right)\n=\nn! \\sum_{k=0}^n \\frac{(-1)^k}{k!} \n", "7913d4449af44fa3a749596e7e9826a1": "\\scriptstyle\\text{VII}_h", "7913f2399923a73e485507a4217710d7": " \\frac{1} {1 + \\frac{v_\\text{s}}{c}} \\approx 1 - \\frac{v_\\text{s}}{c}", "791440eb775570dee17c64e6b8454b2a": "\\textstyle [0, 1]", "7914626dd76814cefc5dfa22acdb5186": "\n\\begin{align}\nT(x_1,\\dots,x_d) &= \\sum_{n_1=0}^\\infty \\sum_{n_2=0}^\\infty \\cdots \\sum_{n_d = 0}^\\infty \n\\frac{(x_1-a_1)^{n_1}\\cdots (x_d-a_d)^{n_d}}{n_1!\\cdots n_d!}\\,\\left(\\frac{\\partial^{n_1 + \\cdots + n_d}f}{\\partial x_1^{n_1}\\cdots \\partial x_d^{n_d}}\\right)(a_1,\\dots,a_d) \\\\\n&= f(a_1, \\dots,a_d) + \\sum_{j=1}^d \\frac{\\partial f(a_1, \\dots,a_d)}{\\partial x_j} (x_j - a_j) \\\\ \n&\\quad {} + \\frac{1}{2!} \\sum_{j=1}^d \\sum_{k=1}^d \\frac{\\partial^2 f(a_1, \\dots,a_d)}{\\partial x_j \\partial x_k} (x_j - a_j)(x_k - a_k) \\\\ \n&\\quad {} + \\frac{1}{3!} \\sum_{j=1}^d\\sum_{k=1}^d\\sum_{l=1}^d \\frac{\\partial^3 f(a_1, \\dots,a_d)}{\\partial x_j \\partial x_k \\partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \\dots\n\\end{align}\n", "7914693654f51d534364dabc2b19a327": "l^a=(1,\\frac{G}{2},0,0)\\,,\\quad n^a=(0,-1,0,0)\\,,\\quad m^a=\\frac{1}{\\sqrt{2}\\,r}(0,0,1,i\\,\\csc\\theta)\\,,", "79148975791a59355ef6662e1b13151c": "\\int_{-\\infty}^\\infty (1 + x^2/\\nu)^{-(\\nu + 1)/2}\\,dx = \\frac { \\sqrt{\\nu \\pi} \\ \\Gamma(\\nu/2)} {\\Gamma((\\nu + 1)/2)}", "79148d27488249098243b08338304275": "\nF = \\mathbf{E} + i \\mathbf{B}^{\\,},\n", "7914b984f04718091ef5f3b9ab0928a2": " K : S \\times S \\to \\mathbb{R} ", "7914d01c58ebd655e6daea9b8fcff0e3": "I_1 = \\left( \\frac{Z_2}{Z_1 + Z_2} \\right)I", "7914e863aeecebfce5604349f7c52ab6": "O(mn/L)", "7915149c0303176cc35e76f90530060f": "\\varphi(p)= p_1\\cdot p_2 \\cdots p_n", "7915167f21eea172dd5f1d93613b4566": "(A \\wedge B) \\vee C", "7915391524bbba820dde542855bde79e": "G = \\bigcup_{n \\in \\mathbb{N}} K^n.", "7915985fb7f78b3891be3525d98fc499": "\\lambda_{s+2}(q)", "7915da983c607d4312441614b94f982a": "\\frac{d\\mathbf{y}}{d\\mathbf{x}} = \\mathbf{A}", "791642a83c89289bf991d23e7f459597": " g(a:b:c) = (a:-c:b),", "79164f71a2962459e798b477f19cc5c5": " \\frac23 n^3 ", "7916cc89926b629d9002e1ab8e78cc17": "a(P)", "7916ef50651d77b60f89e843decb35d2": "(x_1,\\dots,x_n)", "79171192f8ba35da2a9cccb44f5ed31d": "J_0 \\tau_s = \\pi (mod2\\pi),", "791761bc8c6067289e7995592ad4b551": "\\mathfrak{P}^{46}", "79179570fc2a8bc07d8f0ee1a9c32747": "k \\in \\N", "7917a75b2ba3a42e6ea63406ff4e188d": "0''", "7917b5d20fa53f708da3bdce3859c237": " U(t) = e^{-iHt / \\hbar} ,", "7918174bb74e4cd1dc2e204843b50a12": " M_j ", "79186f254c2cc4d7a4567abc094ad59f": "g_{\\alpha\\beta}", "791899104f3fc13c61f3e7e7144129fb": "C_2 = ", "7919ead26747447264e0d98154e2a7d6": "(P, V_\\infty)", "7919fbd40db0cf34b510e53e0135c8d5": " \\delta Q = \\epsilon = \\frac {hc}{\\gamma} =\\frac{6.62 \\times 10^{-34}J\\cdot s * 3 \\times 10^{8} m/s}{0.01 m}=2 \\times 10^{-23} J", "791a471e9c19097ebb3466708f9a54bc": " A_{ij} x^i x^j \\equiv x^T A x = x^T \\left( O O^T\\right) A \\left( O O^T\\right) x = \\left( x^T O \\right) \\left( O^T A O \\right) \\left( O^T x \\right) \n ", "791a8e87efde6dc4c2706e1dd2540b27": "P = {\\left \\{ \\{2k-1,2k\\},k\\in\\mathbb{N} \\right \\} }. ", "791aec615bbc0dff3d13c6971564009b": "\\sigma = \\arctan\\frac{\\sin\\sigma}{\\cos\\sigma}\\,", "791b3708bf7fb6dd7c28e6ec538c4d07": "S = \\int \\left[ {1 \\over 2\\kappa} \\left( R - 2 \\Lambda \\right) + \\mathcal{L}_\\mathrm{M} \\right] \\sqrt{-g} \\, \\mathrm{d}^4 x ", "791b7b483a19707d987fb85d858c9786": "c\\sqrt{n}", "791b8309d991c71c521b034caa0bf943": "\\mathbb Z[e_1(X_1,\\ldots,X_n),\\ldots,e_n(X_1,\\ldots,X_n)].", "791be9da8be0533a03f9cf003135eee4": "\\rho>4\\,", "791bf703a08b3db8daaf523f688ca819": "\\mathcal{B}^k \\rightarrow \\mathcal{B}", "791c1247de90b2529080288f3ce19c9b": "x\\in\\operatorname{club}(\\kappa)", "791c4cb39e6f2e79f11ae6e4a6f9079e": "r \\leq N", "791c5533eca168e1d1c8d9aa0c7cee2e": "\\mathit{q}^R", "791d11dc55b4ceca9d970b71723f6ead": "\n\\tilde{\\textbf{y}}_k = \\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{x}}_{k\\mid k-1}\n", "791dd56dc74f29ee401e58ced935a4e8": " T_s ", "791e0fc3d75ed3703222dc7a82c9678c": "{{^\\text{206}\\,\\!\\text{Pb}^*}\\over{^\\text{238}\\,\\!\\text{U}}}=e^{\\lambda_{238}t}-1", "791e8d457e1c9f95ae53af4525596403": "\\cos\\frac{7\\pi}{30}=\\cos 42^\\circ=\\frac{\\sqrt2\\sqrt{5+\\sqrt5}+\\sqrt3(\\sqrt5-1)}{8}\\,", "791e9ef3af29fb4f0a1374779139676d": "\\frac {n}{2}", "791ee4dc5558a8b54343907369949a78": "x_t = x_o - \\sqrt{r_n^2 - y_t^2}", "791f19991c6d42dbd89088e75b2c675e": " K_p = \\tan ^2 \\left( 45 + \\frac{\\phi}{2} \\right) \\ ", "791f32dd9ded22a03087f86aef458364": "\n\\Phi_{fl} = (k_BT_e/e)\\, ( 2.8 + 0.5\\ln \\mu_i )\n", "791f4b71e727ee7f76e3688083bfc723": "p_3=\\textstyle \\frac{1}{9}\\ .", "791fae9afcb4c51fc67c539057a28231": "\\hat {\\boldsymbol \\theta} = {\\mathbf X}. \\,", "791fe6849f847b2cdbb9b3e1298a74f4": "Q = \\frac{2\\Delta m F}{63.546}", "791fee8fc50186d9daf9452b031ae755": "\\big| c (t, x) \\big| + \\big| \\gamma (t, x) \\big| \\leq C \\big( 1 + | x | \\big)", "79204401f1c627fc95010a0f00900653": "i=1,...,k", "792074a0891922672378c38b5a246eff": " \\begin{pmatrix}\na_1 & b_1 & 0 & \\ldots & 0 & 0 \\\\\nc_1 & a_2 & b_2 & \\ldots & 0 & 0 \\\\\n 0 & c_2 & a_3 & \\ldots & 0 & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n 0 & 0 & 0 & \\ldots & a_{n-1} & b_{n-1} \\\\\n 0 & 0 & 0 & \\ldots & c_{n-1} & a_n\n\\end{pmatrix} . ", "79207c82cf23a060ba62e332bb6a87dc": "\\{ (\\mathbf{x}, g(\\mathbf{x})) \\mid \\mathbf x \\in U \\} = \\{ (\\mathbf{x}, \\mathbf{y})\\in U \\times V \\mid f(\\mathbf{x}, \\mathbf{y}) = 0 \\}.", "7920956fa184b2aa72a7eb221e6c46e3": "\n u_i^{n+1/2} = \\frac{u_i^n + u_i^{\\overline{n+1}}}{2}\n", "7920c56eb6ac75639eaab7a3b086bd90": " p = \\frac{\\partial L}{\\partial \\dot q}", "792110cd3866ca9f4a9560019be8f424": "Q^{s}", "792136fca0474427bd2cd4cc433b8bbe": "\\sigma_c=-\\infty", "792158698f5651960e4e34d8052f942d": "x=ny", "792199758c87b7f2daa90686ee3b8c18": "\\vec j_s", "7921b1880dfaad086203eafeb2e5c81f": " r \\le \\min(m,n)", "792200454e8b1fa1f3f98d592897230c": " Q (\\psi)(x) = x \\psi (x) ", "7922664e41a5c9b993dff92bec2ffef3": "\\lim_{i \\to \\infty} a_{i,j} = 0 \\quad j \\in \\mathbb{N}", "7922a8f08328980376edb5ffb69ff547": " \\underbrace{ \\overbrace{\\sigma^{\\text{T}}}^{\\tfrac{\\partial V}{\\partial \\sigma}} \\overbrace{\\dot{\\sigma}}^{\\tfrac{\\operatorname{d} \\sigma}{\\operatorname{d} t}} }_{\\tfrac{\\operatorname{d}V}{\\operatorname{d}t}} < 0 \\qquad \\text{(i.e., } \\tfrac{\\operatorname{d}V}{\\operatorname{d}t} < 0 \\text{)} ", "7923b5c2ff0478b6a3f968bf2aa39b97": "\\frac{g_{\\mu\\nu}}{k^2-m^2+i\\epsilon}.", "792418abdc43f660d47d9e715744ed1d": "\\frac{d\\rho}{dt}=\n\\frac{\\partial\\rho}{\\partial t}\n+\\sum_{i=1}^n\\left(\\frac{\\partial\\rho}{\\partial q_i}\\dot{q}_i\n+\\frac{\\partial\\rho}{\\partial p_i}\\dot{p}_i\\right)=0.", "79242106249c14383992b6a2d304a48e": "\\vec{e}_0=\\partial_t, \\; \\vec{e}_1=\\partial_z, \\; \\vec{e}_2=\\partial_r, \\, \\vec{e}_3=\\frac{1}{b(r)} \\, \\left( -a(r) \\, \\partial_t + \\partial_\\phi \\right)", "7924863bc3f62ff6092bdc76e32e8ff1": "\nG_\\mu (s,t)=\\sum\\limits_{n=0}^\\infty s^nP_\\mu (n,t). \n", "79248c5478da10b709a225d5548be8e5": "\nx = b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cfrac{a_4}{b_4 + \\ddots}}}}.\\,\n", "792511800855fc78b68666817e65aab1": " x^{q^{s_i}+q^{t_i}}", "79253fe2e19552dc9e3391edcfeeae8c": " \\sigma_P (\\theta, \\dots, \\theta)", "79256210a2400dbe7529d97a6c81d106": "\\top'", "7925866ae7fb55f1271a7457b5aff51f": " m_\\textrm{b}\\cdot v = (m_\\textrm{b}+m_\\textrm{p}) \\cdot \\sqrt{2\\cdot g\\cdot h} ", "7925e2bcce865d6d9b8d378a9f5a4380": "y^n+r=x B^k", "7925f4be6f57eca2fab64ca6f988ade6": " T_s\\colon X \\to X", "792681ea0ed11965ec4a81926dbc697b": "\\csc A = {\\sec A \\over \\tan A} ", "79268e12c9c38d30bb7b96b19d4d1ccb": "p(s, \\psi | m)", "7926d860e5382b9edbe1ab0278fb765a": "\\left\\langle V_\\text{TOT} \\right\\rangle", "79271fd613b2d89e89bf22a0b3050c0b": "X\\Vdash A\\iff A\\in X.", "79272b396d5706bb5fea5906e32a3dc6": "\\frac{\\partial U/\\partial x_i}{p_i}=\\frac{\\partial U/\\partial x_j}{p_j}\\,\\forall\\left(i,j\\right)", "7927309cca04b9f3059b46e3fb92cfa7": "10^{-12}\\frac{m}{s}", "79275e29bc374ae2ca302a48aea71f61": "\\hat{\\mathbf{B}}= B^r(r)\\mathbf{Y}_{lm}+B^{(1)}(r)\\mathbf{\\Psi}_{lm}", "7927655b2a946982f27420de19e03587": "r_i = \\frac{\\sqrt{6}}{3}a \\approx 0.8164965809a,", "79277ebc82649b6efdba4a45b2a20e00": "\\scriptstyle\\delta(\\tilde x -\\tilde x')", "7927817d967a4e41d425a645e7d426b7": "=\\lim_{x\\to\\pm\\infty}-\\frac{1}{x}=0.", "7927a629ab52880a9a6e71de22c5fba2": "x_{1}^{0}", "7927b304545acc9984b6222955f00475": "e^{i\\theta} \\cdot (x, z) = (e^{i \\theta} \\cdot x, e^{-i \\theta} z)", "7928229b807846d50572e25165fb05f1": "\n\\epsilon^* = a V_b + b\n", "792827835ccf51b3da95937ce6ff8313": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {c_2} {c_1} \\left ( \\frac {m_1} {m_2} \\right )^2 = \\frac {l_2} {l_1} \\left ( \\frac {l_1} {l_2} \\right )^2 = \\frac {l_1} {l_2} \\,,", "792853700dec042358003d3df376d534": "V_q = \\frac{2 \\pi e^2}{\\epsilon q L^2}", "792863e27e043d916883ad5225cdfce2": "\\mathbf{R} = \\begin{pmatrix}\n 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix} ", "79289a4979b5b1706d37879341a879e0": "\\left ( \\tfrac{\\partial Q}{\\partial P} \\right ) \\times \\tfrac{P}{Q}", "7928a7c4358234c66b125a4984d25a7a": " S = \\left\\{ z \\, \\big| \\, \\alpha < \\arg z < \\beta \\right\\} ", "79291eef9ab072d238bb6755ef0144d5": "R_f(N) = \\frac{\\varepsilon_p(N)}{\\varepsilon_m}", "7929343207209d5dc1e36831332690eb": "\\frac{1}{2} \\sqrt {2}", "7929563547802be89dbd1b12e645c6d8": " \\rho_B(t)= \\mbox{Tr}^{(B)} [ U(t) (\\rho_A \\otimes \\sigma_0) U^{\\dagger}(t)] ", "79297181476a70606223cd41c4b20bd4": "i \\partial_t^{} u + \\nabla^2 u = un", "792993d5cb1415c5a68c9d23b1134c5b": " a^{n+1} - b^{n+1} = (a - b)\\sum_{k=0}^{n} a^{k}\\,b^{n-k}", "79299f3cb2523e1b09f333f6f36212d4": "\\displaystyle \\beta", "7929c940d05d5c98e7f0cb6ab3f0e7c9": " {1 \\over c^2 T } { d^2 T \\over dt^2 } = -k^2", "7929da44ffd00c1cd6d68bcabce1e2b9": "\\int \\rho_1 \\phi_2 dV = \\int \\rho_2 \\phi_1 dV.", "792a029a5b6c5564dd9f12a746189089": "\\| a - b \\| \\leq 2 \\tanh \\frac{d(a, b)}{2}, \\qquad \\qquad (1)", "792a204aa5a2f000fb5c23630e948c1f": " m := \\cfrac{c}{a}", "792a82539e7f855dfc0cc9e0cb7d3bf1": "m_J(E,a_E,a)_{lJ}(E,a_E,a) \\int dL \\frac{dF}{dL}LQ_{trial-J}(L)X_{L-J}", "792a8401633ff6ad7a5ec1f55daa4b3b": "e_I^\\alpha e_J^\\beta R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "792ac35c0fe5b74b58ffabb45eadedd7": " F_a(z) = \\exp(\\overline{a}\\cdot z) ", "792b4ded82f5f5b4eb55f0bac356d8bb": "y_{ij}^2=a_3x_{ij+1}+a_4x_{ij+2}+b_1y_{ij+1}^2+b_2y_{ij+2}^2", "792b6daf8a1af8da5b223a8157cd2579": "\\tilde{A}=(\\rho(A)+\\epsilon)^{-1}A.", "792ba703b34e3e2ad88435733b00f534": "\\neg \\Box x \\vee x", "792bd3ed1504341fc0a3603175b0e3fc": "\\Rightarrow \\frac{dx}{x} = k\\, dt", "792cba6362f45b29e868163f65e51a1d": "M_{M_2} = M_3 = 7 ", "792cc0471d946d8f915d067b4fc6b271": " r(z) = \\textrm{e}^z + O(z^{p+1}) ", "792cd7cdde272bd65f1e15b6b77a387e": "y_{4N-n}=y_n", "792ce2ce838a75cb7d39f1b2ce4494bb": " x_1,\\dots,x_N \\in\\mathbb{C}", "792cfaea67819f329bd2302cb6b2fb25": "\\textstyle {365 \\choose n}", "792d2cf1744091b725c015e6f8ff9543": "\\scriptstyle M\\, = \\,2y\\,", "792d771f38cba13d757e6410a4d6bfb8": "\\frac{\\partial \\mathcal{L}}{\\partial \\psi} = 0 \\,", "792db628d5527b91a16ae0b7600cb1d4": "\\scriptstyle\\rangle", "792dc6a114fbffeacdf748339b9626dd": " \\int \\frac{\\delta J}{\\delta\\rho(\\boldsymbol{r})} \\phi(\\boldsymbol{r})d\\boldsymbol{r} = \\int \\left ( \\int \\frac {\\rho(\\boldsymbol{r}') }{\\vert \\boldsymbol{r}-\\boldsymbol{r}' \\vert} d\\boldsymbol{r}' \\right ) \\phi(\\boldsymbol{r}) d\\boldsymbol{r} ", "792dc8f4fab6c12af0685b404d83ac33": "f \\left(t,T\\right),t\\leq T", "792e0a55b6268c4e31c0519d2f8513f6": "p = (A \\to w, \\sigma, \\phi) \\in P", "792ed8d7a080bd1b59972d0226ef019c": "\\frac{k_i}{n \\Delta}", "792f781531fe52058f5800f594dafdf1": "5 \\log_{10}(d) - 5 = \\mu ", "792fac01391114cf1d73f9e66252bbfc": "\\frac{d^2}{dt^2}y^i(x(t))=\\frac{\\partial^2 y^i}{\\partial x^j\\partial x^k}(x(t))\\frac{dx^j}{dt}(t)\\frac{dx^k}{dt}(t)+\\frac{\\partial y^i}{\\partial x^j}(x(t))\\frac{d^2x^j}{dt^2}(t)", "792fd381188ff44d8b7aaaf2a21147ef": " \\frac{\\part F(\\lambda)}{\\part \\lambda} = \\left\\langle \\frac{\\part U(\\lambda)}{\\part \\lambda} \\right\\rangle_\\lambda ", "7930269aeb523969eae31b6901cbbc7f": "\\frac{\\nu \\tau^2}{\\nu-2}", "793049178fddf55d74eeb46523f03c32": "Commit(x,open)=Commit(x',open')", "79305d76596e7be974067d6d581239c9": "y_{D}(x) \\, ", "793066b25d9b43c4cc18eccb20a737fa": "R = \\left|\\frac{n_1-n_2}{n_1+n_2}\\right|^2", "79308924d48fdeeefc09e34fb89371b6": "v[\\mathbf{f}]\\longleftarrow v[\\mathbf{f'}]", "79309ef36639c27714d5d26dda514b81": "x \\le p \\le 2x", "7930b9c48887c1d43d0a59b3bf40fae7": "\n \\vec{v}_j = \\frac{\\nabla_j S}{m}\\; .\n", "7930d59c32d56aadab41eac95371bbed": " H = \\bigoplus_{W \\in X} W. ", "7930e16827063329ec8974a1563ae9a3": "\\mathcal{F}\\subset\\{f:\\mathcal{S}\\to \\mathbb{R}, f \\mbox{ is measurable}\\,\\}", "793122a508c1db938843636ec5e229b8": " \\Pr( \\sum_i X_i a_i > t || a_i ||_1 ) = 0 ", "79313a3127e31977208da2fad35e0fc8": "\n\\begin{align}\n & 0011 & 1001 &\\quad\\text{39} \\\\\n+\\; & \\underline{0100} & \\underline{1000} &\\quad\\text{48} \\\\\n & 1000 & 0001 &\\quad\\text{81, intermediate result} \\\\\n+\\; & & \\underline{0110} \\\\\n & 1000 & 0111 &\\quad\\text{87, adjusted result}\n\\end{align}\n", "793162282327d71450aecce05bd6ee74": "{\\Pr}_{\\theta,\\phi}(u(X) < Y < v(X)) = \\gamma\\text{ for all }(\\theta,\\phi).\\,", "7931894ea7ae06d95e9696fabfa74774": "((\\phi(0) \\land \\forall x.\\,(\\phi(x) \\to \\phi(Sx))) \\to \\forall x.\\phi(x)", "7931b7e9759dcee07b18bf2898a597f5": "\\sin^2\\theta \\cos^2\\theta = \\frac{1 - \\cos 4\\theta}{8}\\!", "7931c017c1fde12dfeab43a141551e2c": " \\{1, 2, \\ldots, n \\}", "7932198f0e9d0f236cda03d3d5a23f1a": "\n\\frac {6}{\\pi^2} < \\frac{\\varphi(n) \\sigma(n)}{n^2} < 1,\n", "7932348f21d76e623734b05349f5c7e5": "\\mathfrak{a}^* + i\\mu t", "793254ca9a9b60c4b6de4844152f7393": "VIS_{Op}", "793255f682191c5ef9e24dd906bf600a": " \\displaystyle{X_k}", "7932deaa08bedbb8606f98319fcf8c0d": "m_\\alpha=\\sum\\nolimits_{\\beta\\sim\\alpha}X^\\beta.", "7932fdfc9fd62cd965d5097973252fc9": "\\textstyle R(a_N\\mid[x]) = \\lambda_{NP}P(A\\mid[x]) + \\lambda_{NN}P(A^c\\mid[x]),", "79332515ff65d1e1ec79eed1eb2b84ad": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 1}{4 \\choose 3}{40 \\choose 1} \\end{matrix}", "79333e3baaf02600326bd4bc6605d43c": "n \\choose k", "79334b44669512e68cb60ee9f26c328d": "\\left\\{ {a_1 , \\cdots ,a_n } \\right\\}", "793367c39c4a117c6ed7082cf9fa61cf": "|d-d_1| \\, ", "7933bd3a5693e87dc6cbf92a6d8fe5d4": "\\tau=1\\;", "79345b7fe7d764557aeb73a1639915c6": "p V\\;", "7934733651f3f672baceb5765b0826ac": "i\\ne j", "7934941f1f4e37f223746845d28ffee5": "\np_i(t) = \\sum_j \\frac{A_{ij}}{deg(v_j)} p_j(t-1),\n", "7934a66691458a087dde74f89f56063d": "{x\\!:\\!\\sigma \\in \\Gamma}\\over{\\Gamma \\vdash x\\!:\\!\\sigma }", "793522977a691f0cdd0c3d012c9e0be7": "i\\colon A \\to X", "79355e0462fd905c0c2604f2e018dd7b": "p_{\\mathrm{downstream}}", "7935bd6fb94283e0c36d875c23492265": "\\Gamma^c_{ab} - \\Gamma^c_{ba} = 0,", "7935c7d74154e10754e6944d5c0438f8": " u = \\frac{U}{V} ", "7935d4965e84441dea49c81a1b29bc31": "\\scriptstyle r<1+\\sqrt{2}", "7935fb826fa04d0b622793c63f8bb9b3": "\\sigma L = \\rho c^2 h \\, ", "79367611835e1f1f347289e30353fac8": "l\\|\\cdot\\|", "79368eb21ec57ca3471224ef368bc872": "C = L_*^{-1} \\mathbf{b}.", "7936ea55c7ef321f510119398fa6629e": "\\nabla ^{2} V(\\bar{r}) =\\frac{-e\\delta (\\bar{r})}{\\varepsilon \\varepsilon _r}= -\\sum_{\\bar{q}}\\bar{q}^{2}V_{\\bar{q}}e^{i\\bar{q}\\bar{r}} = \\frac{-e}{V\\varepsilon \\varepsilon _{r}}\\ \\sum_{\\bar{q}}e^{i\\bar{q}\\bar{r}} \\;\\; (8)", "7937188b73e071578c0fe2ce0ff5ea72": "c_\\text{l} = \\frac{l}{\\frac{1}{2}\\rho v^2c},", "79371ffb7e6df7d4208322a94deedd06": " I_K ", "79375534151312d9c15fe2ac736c21f8": "PV = nRT\\,", "793764027f05f6608d016ae37ef63889": " f(z) = z + a_2 z^2 + a_3 z^3 + \\cdots ", "79380e9512234b0619b413d7624a8dfa": "(T, \\;\\, r\\cos(\\omega T),\\;\\, r\\sin(\\omega T), \\;\\, 0)", "793815116660245cb49970136ad958fc": "(f * g)(x) = \\int_{-\\infty}^{\\infty} f(y)g(x-y)dy.", "79385302da9963daaf0d74f62445df6e": "0 = f'(c) = \\frac{f(b)-f(a)}{b-a}.", "793857e8639dad24d794896f47f70892": "\\{\\varnothing,\\{0\\},\\{0,1\\}\\}.", "7938719ce3e5b7f9746583eaf80a73f5": "g_1,... g_r", "7938c9e05cf03a5ebbfb06142f2f9a32": "\\displaystyle{a^b = B(a,b)^{-1}(a-Q(a)b).}", "7938d0f53e7539f74619954ea6a37eac": "B_{p_1, \\ldots, p_n} = \\{ x = (x_1, \\ldots, x_n) \\in \\mathbf{R}^n : \\vert x_1 \\vert^{1/p_1} + \\cdots + \\vert x_n \\vert^{1/p_n} \\le 1 \\}.", "7939194e3ed6ffa4d3d7f943d243c2ec": "a_n a_{n-5} = a_{n-1} a_{n-4} + a_{n-2} a_{n-3}\\,.", "793946aeb85fb7139aa6084c54b1cc4e": "J_i = \\sigma_{ij}E_j \\equiv \\sum_{j} \\sigma_{ij}E_j ", "7939a63b3f87b3f2b15ad207b059029d": "R_{\\alpha}^{\\beta}=-\\left ( \\frac{1}{2}\\sqrt{-g} \\right ) \\frac{\\partial}{\\partial t} \\left (\\sqrt{-g} \\varkappa_{\\alpha}^{\\beta} \\right )-P_{\\alpha}^{\\beta}=0,", "7939d5de9927140c34daaadfc9fa5c7f": "K_{1/2}(z)=\\sqrt{\\frac{\\pi}{2z}} e^{-z}", "7939e19bfc1eace3c7f1d6babcce39a3": "\n\\beta \\,\\, \\approx \\,\\,{{a\\,r\\left( {r - 1} \\right)\\mu ^r } \\over {2\\,n}}\\left( {{\\sigma \\over \\mu }} \\right)^2", "793a6fa67cb3c2096928cce27ccdb272": "0\\leq j", "79402738bdbf483229f7b2a49df3e43e": "u^R_{i + \\frac{1}{2}} = u_{i+1} - \\frac{\\phi \\left( r_{i+1} \\right)}{4} \\left[ \n\\left( 1 - \\kappa \\right) \\delta u_{i + \\frac{3}{2} } + \n\\left( 1 + \\kappa \\right) \\delta u_{i + \\frac{1}{2} } \n\\right], ", "79404689eeea21dd3e909f731b024ee2": "\\lambda^{[2]}_{{\\alpha}_2}", "79406c03e781dddd984e84acc50cb6ca": "\\max(x_1,\\ldots,x_n)", "79418779aa1e5bef51726b4d070b3f6a": "\\boldsymbol{S}_{k} = {\\color{Red}\\boldsymbol{H}_{k}}\\boldsymbol{P}_{k|k-1}{\\color{Red}\\boldsymbol{H}_{k}^\\top} + \\boldsymbol{R}_{k}", "7941b125fec243e3962433e6d9f771e6": "(x, y, \\theta)", "7941b68b4dab02c141b1ea69728c52bf": "D\\!\\!\\!\\!D", "7941dd0f57dd883fd2311f2fc8b2199e": " \\begin{align} \nK_M^{\\prime} \\ &\\stackrel{\\mathrm{def}}{=}\\ \\frac{k_3}{k_2 + k_3} K_M = \\frac{k_3}{k_2 + k_3} \\cdot \\frac{k_{2} + k_{-1}}{k_{1}}\\\\\nk_{cat} \\ &\\stackrel{\\mathrm{def}}{=}\\ \\dfrac{k_3 k_2}{k_2 + k_3}\n\\end{align} ", "7942098f91decdb225b9f1417b3b280f": "x^2 - y^2 = n\\,\\!", "7942143105c4bb01e2b6876293833fb9": "U(\\theta,\\phi)", "794329847090b78d0ff121c7aea0812f": " \\int_{B} X(\\omega) \\ d \\operatorname{P}(\\omega) = \\int_{B} g(\\omega) \\ d \\operatorname{P} (\\omega) ", "79432bb1c5078864ecfcd2748c4f16de": "a(\\xi,\\zeta = 0) = N \\operatorname{sech} (\\xi) ", "79434a62463064ac8a017da3cf544c48": "\\mathbf{F}_{\\mathrm{thrust}} = \\mathbf{v}_{\\mathrm{rel}}\\frac{\\mathrm{d}m}{\\mathrm{d}t}", "7944138212f6c8778d4a9c4f53296b6e": "2^{2^{\\kappa}}", "79444d11b53ff33ee0fcd217bbf79d9d": " \\delta \\approx {5.0 * x \\over \\sqrt {Re}} ", "79447b53f3df11e9ec2b4a43230a50e6": "S'\\subseteq S", "79448827eff631e21e3a08fc69cd6b25": "({x_1 \\cdots x_n})^{\\frac{1}{n+1}}t_0^{-\\frac{n}{n+1}}=1.", "7944f1586e8628873d3c953da603df4f": "\\cdots - \\frac{1}{3} \\left( \\dot{\\theta} + \\frac{\\theta^2}{3} \\right) \\, h_{ab} - {h^m}_a \\, {h^n}_b \\, \\left( \\dot{\\sigma}_{mn} - \\dot{X}_{(m;n)} \\right) - \\dot{X}_a \\, \\dot{X}_b", "79451258462117d89797d57518abb040": "\\sigma^2_y", "794585948a3db0ca832661f885f31582": "|\\alpha\\rangle=e^{\\alpha \\hat a^\\dagger - \\alpha^*\\hat a}|0\\rangle = D(\\alpha)|0\\rangle\\,", "794612e6963a99c3c5a75d0b7e0d92f9": "=\\left\\langle\\begin{bmatrix}\ne_1 e_1^* & e_1 e_2^* \\\\\ne_2 e_1^* & e_2 e_2^*\n\\end{bmatrix} \\right\\rangle", "79464602bff36e82f4b9c57da07e5af0": "f^{-2}", "79465fb4e9abdd61e20351cb2768f3ef": " \\mathcal{G}=", "79471f18c84462a5f05207a0dccee8d5": " \\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4+ \\cdots\\text{ for }|x|<1 ", "794725a9cfda9c8759bf2269dd606506": "F_s = P A = P \\pi d^2 / 4", "794747b3443beda9c901d224b2e737d0": "\\nabla_\\theta \\log\\pi(x_k | \\theta)", "794781c722220d6ea649b440f6f295de": "= \\gamma^\\mu \\left(2 \\eta_\\mu^\\nu I_4 - \\gamma_\\mu \\gamma^\\nu \\right) \\,", "794793fb43a1187d6e02038fa2c0f36f": "V^{\\otimes n}=V \\otimes V \\otimes \\cdots \\otimes V", "7947d16b3bc9a3d1cbaeaeec03a9a6d0": " = 118.7\\,\\mathrm{mm} (= 4.67\\,\\mathrm{inches})", "794817b2f9b92386c452a5b10bc2f862": "\\int_a^b f(x)\\,\\mathrm{d}x = F(b) - F(a).", "7948515ec781b2aa7c577db8963f9e8f": "R_s[0 \\dots l/s)", "7948835da25a3dc41ec8e99ffd5bf19d": "\\lambda^{\\mathrm{state 1}}_{\\mathrm{observed}} > \\lambda^{\\mathrm{state 2}}_{\\mathrm{observed}}", "7948e9c70fe22e5f633488110cfe550e": "\\sin\\left(\\text{half top angle}\\right) = \\frac {\\text{center distance}} {2 \\times \\text{radius}}", "794909c95ef83afadc10827834019949": " ?x +y^2 \\times \\frac 12 ", "79496abd924860ebb51f002386785809": " \\int \\sec^n x \\, dx = \\frac{\\sec^{n-2} x \\tan x}{n-1} \\,+\\, \\frac{n-2}{n-1}\\int \\sec^{n-2} x \\, dx \\qquad \\text{ (for }n \\ne 1\\text{)}\\,\\! ", "794973cc491a1a53a89855eb9bbf8b59": "[{\\mathfrak g}_i,{\\mathfrak g}_j]\\subseteq {\\mathfrak g}_{i+j}.", "7949be057f99c65f8a14ea2feb7bea9b": "\\mu \\ge 35", "7949c41231dfbd394e292e40c2a9c8bd": " \\frac{\\sigma_t}{\\rho} = v^2 ", "794a319783808987f26090a69bcbcd91": "p(z) = z^n - 1", "794a443dcaf37c8ad9168300f19b238d": "V \\subset U, p \\in V", "794aa5d7a042f28125e9c7d08b1d3267": "M_n=\\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{n}{2k} C_k.", "794b4faa02660a6d9e5bdb927935d75c": "\\{e\\} + B' \\subset A' ", "794b6996c87fa64f0945762124489abf": "klhm' = kmhl'", "794bd63d7da080717dc6ae860323c5cc": "Attr", "794bf2df59fffb5588536086a0cf22f6": "\\sigma_x = 2\\mu\\varepsilon_x + \\lambda(\\varepsilon_x + \\varepsilon_y +\\varepsilon_z)=2\\mu\\frac{\\partial u_x}{\\partial x}+\\lambda\\left(\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}\\right)\\,\\!", "794c4813d68d92d2d43aec316a23b2fd": "\\overline{A_1 A_2 A_3},", "794c5863a835b282841244eda310d05d": "\n\\begin{align}\nTr(h)^{p^2} &= h^{p^2} + h^{p^4} + h^{p^6} \\\\\n &= h + h^{p^2} + h^{p^4} \\\\\n &= Tr(h)\n\\end{align}\n", "794c83628e60e9a0f6eefe763f35320b": "\n\\sum \\psi_n^*\\psi_n = 1\n", "794c95f330c36f8c8a08e29b8ca7a151": "S^{2n}", "794ca669531c380230997b3f02a46371": "r \\le 0", "794cc20b6024b516f412ddc3cb5e6c62": "v(i,k)=\\sup_{\\tau>0} \\left\\langle \\sum_{t=0}^{\\tau-1} \\beta^t R[Z(t)] + \\beta^t k \\right\\rangle_{Z(0) = i} ", "794cc26f79990e553fca99d1bec92ff3": "(x-a)p'_x(a,b)+(y-b)p'_y(a,b)=0", "794cd80757019d4a8cb57fdcf34de99f": "(\\lnot P) \\land (\\lnot Q)", "794d4c7ab7cd3d61a64d6194d46aa04c": " |\\epsilon|", "794d5de7284b8baeeb96cadb78334a61": "h:X'_\\beta\\to {\\mathbb F}", "794d623b188958ec4db202b8548dbc06": " 2^{79,300,000} -1", "794d6b2ed42619c9f81efccfda761e49": "v_g \\ \\equiv\\ \\frac{\\partial \\omega}{\\partial k}\\,", "794d7ef6ad32947bd9f4508f95e2c53f": "\\begin{matrix} {11 \\choose 1}{4 \\choose 3}{44 \\choose 2} \\end{matrix}", "794df380af4414df79ab33bb0b721b11": "\\ a_i ", "794e083fdd8cf72b8969d8a831fed1f1": "t^* \\ \\approx \\ D_{cl}/\\hbar^2", "794e475083e0f742f2c50a58b3c30ca7": "\\textbf{v}", "794e8f88e5ad94841cb53558c7aef70c": "\n\\begin{align}\n f(t) = & \\frac{1}{2\\pi} \\int F(\\omega) \\mathrm{e}^{i\\omega t} \\mathrm{d}t \\\\\n x(t) = & \\frac{1}{2\\pi} \\int X(\\omega) \\mathrm{e}^{i\\omega t} \\mathrm{d}t.\n\\end{align}\n", "794ea658e90f5dafc6c4939e80bc3b9d": "\\delta=\\lim_{h\\rightarrow0}\\sqrt{h/(24\\alpha)}", "794eaad358b039a4ede0a48ef9ca535f": "\n\\eta_{inh} = \\frac{\\ln \\eta_{rel}}{c}\n", "794ec9e64b11b83279bc1bce34c998c1": "r = \\left(\\dfrac{k}{n}\\right)", "794ee5d1e856a026879433639ac29a12": " F_B = - \\oint_S \\nabla p \\cdot \\, d\\mathbf{S}. ", "794f0442e387a77ab97cbeb71cc0e0f3": "\nL = \\frac{\\pi t}{6}\\left(2^{n}+4\\right)\\left(2^{n}-1\\right),\n", "794f28f0972120bd7e60c504b8a8fd9e": "\\{s\\cap A|s\\in C\\}\\subseteq P(A)", "794f29bd8eca16348e121e79ae649ccb": "D = D_1^{-1} D_2=\\begin{bmatrix} L_2/L_1 & 0 & 0 \\\\ 0 & M_2/M_1 & 0 \\\\ 0 & 0 & S_2/S_1 \\end{bmatrix}", "794f478611fa8d33fa5a5e321863211f": " \\displaystyle{\\{e_r,e^*_s\\}=\\delta_{r,s},\\,\\,\\,\\, \\{e_r,e_s\\}=0.}", "794fc6663bb38ac99d5931a9f56d381f": "Clock = tick \\rightarrow Clock", "7950298a6a8348e4cf994e4f5d6dfff4": "\\scriptstyle P=\\frac{R^2}{2D})", "79508e7ab9a23a237a38a8e6994048f3": "\\scriptstyle \\tilde{t} \\;>\\; t^*", "79510d99d2b953a70d52f67c3c5e97e0": " \\phi. ", "795112fea0213959de40c4130f26e724": "\\displaystyle \\sum_{k} | f(y_k) - f(x_k) | < \\epsilon.", "79514b238e6760819e4f96e108cbe22b": "{\\scriptstyle b\\to +\\infty}", "79515ca56c62e4c93e7ae29ac8ecccc2": "\\begin{matrix}{5 \\choose 3}{3 \\choose 1}\\end{matrix}", "79522d790f422140004953f2c406f3bc": "g(\\beta)= 1 + \\frac{\\psi(1/\\beta)}{\\beta} - \\frac{\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta} \\log|x_i-\\mu| }{\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta}} + \\frac{\\log( \\frac{\\beta}{N} \\sum_{i=1}^{N} |x_i-\\mu|^{\\beta})}{\\beta} ,", "795274de7d75fcf9407d5567880011c4": "g(x)=\\sum_{n=0}^\\infty {b_n} x^n", "7952960829b13882651f7b279d9fb20c": "| \\ldots | \\!\\,", "7952bee9f44de146ce3cf0185162c7c0": "2^4\\cdot 3^4\\cdot 5\\cdot 7", "7952da8149002ca0c6e9273ac3a25520": "\\mathsf {Ca(HCO_3)_2}(aqueous) \\longrightarrow \\mathsf {CaCO_3}\\downarrow + \\mathsf {CO_2}\\uparrow + \\mathsf {H_2O} ", "7952f26fa35311f81c3b97044f951332": "Q=\\frac{{D_{dir}}}{{D_{{dif}}}}", "7953286cc5c95f114f09e770338a9972": "0\\le\\phi\\le2\\pi", "79532f2f8637ef8c27922f293f16041f": " p_{1,3}(x) \\, ", "795375b25e627b7a766cf0383e583228": "\\tau=0", "7953979383fa4c02a57aada61fc667e6": "J^2=-1. \\,", "79539b7af77845fb62e84d4ba089dd9f": "N_{n}", "7953a391ab11ea2abb4ce60eb930f3d5": "J=\\{f(\\cdot ; \\theta):\\theta\\in\\Omega\\}", "7953aed24e1ed234cc0b2e011f767e67": "(p \\to (q \\to p))", "7953c90a8dc72b475a18e411df372e0f": " 1 - p_{ij} ", "7953d43d11127a10e0b901a118bd573c": "\n\\dot{M} = \\frac{2}{5} \\frac{L \\mu m}{kT},\n", "795499ba4653c517bbde17a17058c777": " U = - J \\coth(2 \\beta J) \\left[ 1 + \\frac{2}{\\pi} (2 \\tanh^2(2 \\beta J) -1) \\int_0^{\\pi/2} \\frac{1}{\\sqrt{1 - 4 k (1+k)^{-2} \\sin^2(\\theta)}} d\\theta \\right]", "7954ab9f52509ea633c8e1caf755d2a5": " arg(z) \\,", "7954af6cfe7e1d0a70ae8306c0949c5d": "(m \\ll p). ", "7954f3df8b6dd0aeda32d3813a9569ed": "\\scriptstyle \\rho ", "7955435581c547cc211e9cdbcb978dc6": "\\nabla_X Y = P((X\\otimes I)Y)", "7955e13f379566fe3fef0168c903d966": "\n\\lim_{|\\mu|\\rightarrow 0} \\operatorname{Li}_s(e^\\mu) = \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} \\qquad (\\mathrm{Re}(s) < 1)\n", "7955f065ad41ccbee30bf1b1d0346da0": "\n\n\\mathbf{J}_{\\rm init} = \\mathbf{L}_{\\rm orb} + \\mathbf{S}_1 + \\mathbf{S}_2.\n\n", "7955fa748e019e30eeb513f9197ad415": "\\theta = \\arcsin \\left( \\frac{\\text{opposite}}{\\text{hypotenuse}} \\right) = \\sin^{-1} \\left( \\frac {a} {h} \\right).", "795628c2caeb9a6272f6e1f6c8a76871": " \\operatorname{E}[X] = \\mu = \\sum_{i = 1}^n w_i \\mu_i ,", "79568864fb36b5bc80427bed372fcdaf": "(xp_r,p_{r-1})=(p_r,xp_{r-1})=(p_r,p_r)", "7956a1451f08f8235a7ddc312f4ef040": " \\vec w^T \\Sigma_{y=i} \\vec w ", "7956af2ea10202a210ffdfcebe800b08": "\\beta^{2} \\equiv 1 - J^{\\prime}(u_{0})", "7956c5d1973ea16f4503dcd7009a4972": "\\ x(t) = M_a \\sin(2 \\pi f_a t + \\phi_a) + M_b \\sin(2 \\pi f_b t + \\phi_b) + M_c \\sin(2 \\pi f_c t + \\phi_c)", "7956d508f48535d3bc871c82a0369981": "X''(x) = - \\lambda X(x).", "7956de450c09fa40683b734b07e4b044": "\\oint{dl}", "7957531ec72bdaeae572adfda7cb3377": "\ne^{ - \\lambda _i } = \\frac{{T_i }}\n{{\\sum_j {e^{ - \\lambda _j - \\beta C_{ij} } } }};e^{ - \\lambda _j } = \\frac{{T_j }}\n{{\\sum_i {e^{ - \\lambda _i - \\beta C_{ij} } } }}\n", "79575e7c054db38436bde63866e8489c": "r_p=a(1-e)", "79577562663ccd550d1d6f8f524fbe50": " \\Delta Y= \\frac{\\Delta I}{(1-q) \\cdot (1-\\alpha)} \\,\\ ", "79582baf04792df0bee9788be2c7b57c": "t : V \\leftarrow (D \\in \\mathfrak{D}) \\cup \\{\\emptyset\\}.", "79583d27c5c7b1b8441e9408da603701": "\\nabla\\times\\left(f(r)\\mathbf{\\Phi}_{lm}\\right) = -\\frac{l(l+1)}{r}f\\mathbf{Y}_{lm}-\\left(\\frac{\\mathrm{d}f}{\\mathrm{d}r}+\\frac{1}{r}f\\right)\\mathbf{\\Psi}_{lm}", "795844932f2b694aa73ba72902a8bf25": "\nK_{11,reduced}=K_{11}-K_{12}K_{22}^{-1}K_{21}\n", "7958aed2549cd731ffe808e4d38ec70b": "x + n", "7958e36285d4133bca8dee5bcc4d83d2": "\\mathfrak{g}_\\lambda = \\{x\\in\\mathfrak{g}:\\forall h\\in\\mathfrak{h}, [h,x] = \\lambda(h)x\\}", "7958e86e16f1198bf75d78540aa1db04": "y = x \\tan(\\theta)", "795917408a5f8af9dada99884b93acfc": "S=\\mathbf{p}+W,\\,", "79591efc90fc40ba8de9499bbfd8db52": "p_0, p_1, p_2, p_3,", "7959307a4fbe7b8ed15702850fb44388": "\\psi(x;q,a)=\\sum_{n\\le x\\atop n\\equiv a\\bmod q}\\Lambda(n),", "7959f62926e8d81b39523c0d22980fbd": "MRS_{AB} \\neq \\frac1{MRS_{BA}}", "795a02f66c7e8ddf793a962bb85b5d1d": " \\{ x+iy \\in \\mathbb{C} \\, | \\, x^2+y^2=1 \\} ", "795a921fcd6a90f26ea4b59028ac8136": "dy = iy\\,dt + F\\,dt", "795a98d154254a0612d19d46be3a173a": "dln\\gamma_1/dx_1", "795b1be15ba928611443c15533e8ea0d": "A(z) = \\sum_{k = 0}^\\infty k!\\left(-1 \\cdot z\\right)^k,", "795b6aa8e056c86c70800522580d99b0": "\\scriptstyle \\leq10^{-23}", "795bb991d3321f44de4f1f09203939b9": "x(F)=\\frac{x_\\mathrm{m}}{(1-F)^{\\frac{1}{\\alpha}}}", "795c09a22835cb42fe4b543083423a2b": "\\langle G,S \\rangle", "795c1a21212d76a9a3e0ac2660dbc8c6": " \n (25) \\qquad p_2 - p_1 = u_s^2\\left(\\rho_1 - \\frac{\\rho_1^2}{\\rho_2}\\right)\\,\n ", "795cab2f530e8286eb4eedc185b0961e": "\\int_N^\\infty f(x)\\,dx", "795d018d375de9845a4997b30eca99ec": "S/N", "795d9f2d6240e29881a2afc8c0dfcf7e": "\\begin{cases}\\frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^{2}}-3 & \\text{if}\\ \\xi\\neq0,\\\\ \\frac{12}{5} & \\text{if}\\ \\xi=0.\\end{cases}", "795e11844f7a1429b1c8e72fd4190e55": "[\\varepsilon_1, \\varepsilon_2]", "795e2b01bedd7bec8c0b83350806034a": "x_i=x_k", "795e67d4a8702668c4f0e741ae08d9b7": "x_0 \\ ", "795e6c968db8da2f5d4931b7ec18cd74": "x=vu, y=v, z=u^2.\\,", "795e8776c977cda723afd429e685c2cd": " \\bigotimes ^k TM ", "795e970317df647b90df94353bfc9e32": " \\langle T_1, T_2 \\rangle = \\sum_n \\langle T_1 e_n^*, T_2 e_n^* \\rangle, ", "795eda9ea64db039282f8e9c903a7c97": "{\\bar{H}}_4", "795ef5ce1778941404e4a952be2cb7d6": "\\alpha\\leq\\Gamma_1", "795efae74aab97fe8d90d79ab0f3ea25": "b_\\gamma(M)", "795f1b2c35ea90c9cfa4ffaedfb08b73": "\\ln\\left|\\frac{x+s}{a}\\right|\n=\\mathrm{sgn}(x)\\,\\operatorname{arcosh}\\left|\\frac{x}{a}\\right|\n=\\frac{1}{2}\\ln\\left(\\frac{x+s}{x-s}\\right)", "795f47b52c10ceaec47dae25601a9b96": " t \\not\\equiv 1 \\pmod {13}", "795f82151431433a31745499fa075ca0": " CI = e^{ m \\pm 1.96 s } ", "795f90301f9a1eab2d25ba07623477a8": "\\beta_p = -\\frac{dV_t}{d\\sigma_e}\\frac{1}{V_t}", "795fe0ad7434ff0b219b0dbd6e741e6f": "\\frac{a+b}{a} = 1 + \\frac{b}{a} = 1 + \\frac{1}{\\varphi}.", "796002f8a48ff53bb187b061da4bab84": " v( S \\cup \\{ i \\} ) = v( S ), \\forall~ S \\subseteq N \\setminus \\{ i \\} ", "79603546aa8e9f2cc7f4d4e3a8c0b958": "\\mathbf{B}_\\text{el}^l = -2\\mu_\\text{B}\\dfrac{\\mu_0}{4\\pi}\\dfrac{1}{r^3}\\dfrac{\\mathbf{r}\\times m_\\text{e}\\mathbf{v}}{\\hbar}", "796042516253cdc76f370a760b59b361": "P_{{\\mathrm{O}}_2}", "79605011601d9669c31d65b4a94f25e2": "={1 \\over 2}(uu' + vu' + uv' + vv') - {1 \\over 2}(uu' - vu' - uv' + vv')", "7960d598e270f630f7331157593e8aa2": "\n\\vec{S}_i \\cdot \\vec{D} = \\sum_{A=1}^N \\; M_A \\vec{s}^{\\,A}_i \\cdot \\vec{d}^{\\,A}\n=\\sum_{A=1}^N M_A d_{Ai} = 0,\n", "7960e7402c420dc1ba0324867807f1f6": "m_b", "79611694d87508f623b3af1e9e4f5824": "Y_t", "7961514b0c466d41566a8040e38af779": " b(\\lambda)=M^*\\Phi_{2\\lambda}(1)=\\int_K \\Phi_{2\\lambda}(k)\\, dk. ", "79616975cc0b037ead524154d4fc225a": "\\frac{D}{Dt} = \\frac{\\partial}{\\partial t} + u_j \\frac{\\partial}{\\partial x_j}.", "7961adc3b140838d7c05b22bf8ef040c": "e^{\\pi}-\\pi=19.999099979189\\cdots\\,", "7962011d1934adf2abd68e3ac817ba52": "I_x = \\ker \\delta_x = \\{f \\in C(K) : f(x) = 0\\}, \\ \\ x \\in K.", "796212aa62d36c7e2367c755cd3da118": "\\hat{B}(\\xi)", "796229432719f6ab648858883a04a33c": " \\min_{x \\in \\Theta}\\; f(x) = \\mathbb E[F(x,\\xi)] ", "79624b855de89a10e767e930ac300ff0": "\n\\begin{align}\n\\mbox{Var}[X_t]&=\\sum_{k=0}^\\infty e_k(t)^2 \\mbox{Var}[Z_k]=\\sum_{k=1}^\\infty \\lambda_k e_k(t)^2 \n\\end{align}\n", "79625cae5556b43918d17ec491295368": "\\sum_{n=1}^\\infty |\\rho_n| < \\infty", "79625ec3f10df1ed5669d4b06ed4bd8c": "\\omega = -\\gamma B", "7962a7061d1ede634a92601313185d4d": " t_2 = t_1^2/4, \\; t_3 = t_1^3/16, \\; t_4 = t_1^4/64 ", "79632153193bb1cff9aff907159ad3ec": "B_{r} (p) := \\{ x \\in \\mathbb{R}^{n} | \\| x - p \\| < r \\}", "796381ebe9fdf36b69309c88a0845444": "A = 4 \\pi r^2", "7963c7d78575796d43214e290bc7ec47": "= \\frac{e^4}{(k-k')^4} \\Big( \\left(\\bar{v}_{k'} \\gamma^\\mu v_{k} \\right) \\left( \\bar{u}_{p} \\gamma_\\mu u_{p'} \\right) \\Big) \\Big( \\left( \\bar{v}_{k} \\gamma^\\nu v_{k'} \\right) \\left( \\bar{u}_{p'} \\gamma_\\nu u_p \\right) \\Big) \\,", "796400f4e7acf80e3a5fe4e5da4a0ad3": "\\epsilon_{LR}^{-1}", "796401639d7f93bb55dfdbd1926721f6": "\\mathbf C=\\mathbf F^T \\mathbf F\\,\\!", "79642df53c0ef2695c424fc52574eabd": "\\epsilon(q,\\omega) = 1 - V_q \\sum_k{\\frac{f_{k-q}-f_k}{\\hbar(\\omega+i\\delta)+E_{k-q}-E_k}}.", "79644a055e56ecf1c37d7afc59a28923": "\\alpha = n T_0", "79644de5ae90bdcc93dea28a3ea2b6a9": "\\sqrt[7]{3/2}", "79645be7731263811deaf9c2559d4791": "\\displaystyle X_t = \\mathbb{E} ( f(W_1) | F_t ) = \\begin{cases}\n f_{1-t}(W_t) &\\text{for } 0 \\le t < 1,\\\\\n f(W_1) &\\text{for } 1 \\le t < \\infty;\n \\end{cases} ", "7964938c811c64fcf53c8e8f2a324f60": "\n \\boldsymbol{U} = \\left[\\begin{array}{ccc}\n\\lambda_{X}\\\\\n & \\lambda_{Y}\\\\\n & & \\lambda_{Z}\\end{array}\\right]\n", "7964ba15800306a379e42a8bc986119d": "\\scriptstyle \\lim_{x\\rarr 0} f(x)^{g(x)}", "7964c4fa3481b0fec42331ae081502f2": "r_h", "7964c52e4ca718de8499205bae907045": "\\textit{dau}(x_{me},x_{ht}) \\leftarrow \\textit{par}(x_{ht},x_{me}) \\land \\textit{fem}(x_{me}) \\land (\\text{all background knowledge facts})", "7964c6a339acf2ddea25a5ef0552b97e": "\\frac{1}{3}", "7965031e9a68100e48fb08d12c9e24b5": "\\sum_{k=1}^n {f_k}^* (g_i) f_k (g_j) = n \\delta_{ij}", "796556913c486062667b8f37b9698369": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_{V\\,\\subset\\mathbb R^3}\\left(\\nabla\\cdot\\mathbf{F}\\right)d\\mathbf{V}=", "796563eaa4eb05c812941754ea4ee828": "(-\\infty, H_n^{-1}(1-\\alpha)], [H_n^{-1}(\\alpha), \\infty)", "79656e86c155058ffb223804a384c944": "M_{unit,2}= 10 \\quad ft^2", "79658c0182aa962afb7751f6ad4a367d": "(d,r)\\,\\!", "7965c513e15618e96a1e1c8d2d142ccf": "R_{\\rm hyd}", "7965dc62db46c2f12276dac894c11740": "B_r^{p,q} = (\\mbox{im } d_0^{p-r+1,q+r-2} : F^{p-r+1} C^{p+q-1} \\rightarrow C^{p+q}) \\cap F^p C^{p+q}", "79660fe21f4498a7ab9677fac2f0ce62": "C = \\pi \\epsilon_0 \\epsilon_r /\\ln {( 2D / d )}\\,", "79667804dd885fb9dbfb533b9ceea0c3": "\\mathrm{3Mg_2SiO_4 + SiO_2 + 4H_2O \\rarr 2Mg_3Si_2O_5(OH)_4}", "7966985d637425d895b08e5892bb2b02": "U_3(8): 6", "7966b7e473e9b9643170145859d1f117": "F(f,truthvalue,max)", "7966b7fed6ef2655869504044ce1d634": "a_t^n", "7966cc9ef538683d11b37383600c148c": " a=Tr(g^x)", "7966d52cbd335d4efb80c85af5861610": "Hb", "7966e02a52f7967616d3b116023fc6ad": "y|A^T\\textbf{x}", "79680f5010f9cf1d981338fc162a8c16": "\\Phi_{00}:= \\frac{1}{2}R_{ab}l^a l^b\\,\\hat{=}\\,\\frac{c}{2}\\,l_b l^b\\,\\hat{=}\\,0", "796834e7a2839412d79dbc5f1327594d": "NL", "796836430f1e2323b0fc452cc905c778": "\\mathbf{F}_q", "7968746f2ad65842a1e1ad6556ddadc9": "{s+t \\choose n}=\\sum_{k=0}^n {s \\choose k}{t \\choose n-k}", "7968883565ab0ead7143d8b3141cd34c": "\ne\\epsilon(t)\\left(c_{1}(t)\\langle \\Psi_{1} |x|\\Psi_{1}\\rangle + c_{0}(t)\\langle \\Psi_{1} |x|\\Psi_{0}\\rangle\\right)=\ni\\hbar c_{1}'(t)\n", "7968c0b75da1c5d2cdb777383c65adae": "\\langle X, \\leq \\rangle", "7968fce07c55de699b08574ccdcc7ee6": "{BSA}= \\sqrt{W \\times H} / {60} ", "79692fef84b37aace35898825213fb8a": "\\text{ Call } + \\text{ Present Value of Strike Price } = \\text{ Put } + \\text{ Underlying security } \\,", "79697125bd31d038fb044f811e82a63e": "\\mathit{I}", "7969dbb7f3b73b76cb8b78d26437ecb0": "|n_1 n_2 \\cdots n_N; S/A \\rang", "796a2dd2bc0b274a24d1c41efc225f9a": "typical\\ price = {high + low + close \\over 3}", "796a2e73a8b3ba100350212c91e441a6": "\\bar{V_1}=\\bar{V_2}", "796a4eb0c44819904f75ff13faa7df95": "d^\\prime = \\cos \\theta_\\mathrm{c} d + \\sin \\theta_\\mathrm{c} s . ", "796a7f532ccffdc9eada942cff96db09": "\\Psi_0(x)", "796ad3c1d1aab7078e253bd313a42570": "\\varepsilon_M\\delta_{ij}\\,\\!", "796ad53a9e526028329dc4ba1038389e": "\\boldsymbol{p}_0,\\boldsymbol{p}_1,\\boldsymbol{p}_2,\\ldots", "796b0c71742ac6f52403dfbf02d1a451": "o(\\sigma\\to\\tau)=\\mbox{max}(o(\\sigma)+1,o(\\tau))", "796bc29d209c4b6f64a751317a4f4c82": "\\scriptstyle 32\\times\\log_2(32) = 32\\times5 = 160", "796c5b253b21503ba8dd08b75417a488": "\\textstyle \\alpha > \\beta", "796c6178a90224c3a9af87813ef13ac9": "\\scriptstyle\\operatorname{Var}[\\ln X] = \\psi_1(k)", "796ca5eda9acdfafe9d106193648eb93": " A(r)=\\iint_{\\Omega_r} |f'(z)|^2\\, dx dy = {1\\over 2i}\\int_{\\partial\\Omega_r} \\overline{f(z)}f^\\prime(z) \\,dz={1\\over 2i}\\int_{|w|=r} \\overline{f(g(w)))} f^\\prime(g(w)) g^\\prime(w)\\, dw.", "796ca716a2287d3fc34a6f092729d7c7": "\\displaystyle{T={1\\over 4} (H +JHJ),}", "796ca7b82cb2dc216d20a7ad7c9c6fa0": "1/\\Omega", "796d268abd983584a1739bff5ea6e1b6": " \\operatorname{tr}(A) = \\operatorname{tr}(A^{\\mathrm T})", "796d2bbc1546780bfa56d72656de3b4c": "f:2^\\Omega\\rightarrow \\mathbb{R}_+", "796d424ce1df526ff87b1c9d3af97422": "F_{eq} = F_1 + F_2 ", "796d6e5b1f64b9ad223f401c290fc074": "p_f + p_p = 0\\,", "796dbcf8d852f776056c11cb42a95ed9": "0 \\le \\alpha \\le \\pi", "796dfcc0a030e7ae0e546066f27a9f5e": "\\dot{X}", "796e0742f663f21d127522be72ec5306": " E_{2ss} = \\frac{3}{2} \\times y_{2ss} = \\frac{3}{2} \\times 4.27 = 6.40 \\text{ ft}", "796e4c318be30d7190e38b59cc21e33f": "\\theta_D = \\frac {K^D_{eq}\\,p^{1/2}_{D_2}}{1 + K^D_{eq}\\,p^{1/2}_{D_2}}", "796e69ab0020fe43c2eae581becd0b79": "B_e", "796eff4955d099d445a8b8cab3376afc": "\nr_{b} = \\frac{M_1 - M_0}{s_n} \\frac{n_1 n_0}{n^2 u},\n", "796f024ac30f6dcdfc65214f7a320f52": "\\scriptstyle c_{e} - f(e)", "796f0b3a301df86f23d33aaa3ffee178": "\\scriptstyle \\psi(x)", "796f3d4a98ea01103e7671b356ebd079": "y (\\theta) = (R + r) \\sin \\theta - r \\sin \\left( \\frac{R + r}{r} \\theta \\right),", "796f4ae59512431e38c850272ea9bfe2": "F_n(a, 0) = a", "796f5108c2ab5bbed11224709e9aaca3": "\n\\begin{align}\n\\sigma_7(n)\n&=\\frac{1}{20}\\left\\{21(2n-1)\\sigma_5(n)-\\sigma_1(n) + 504\\sum_{0\n \n \n \n a\n \n \n x\n 2\n \n \n \n \n b\n x\n \n c\n \n", "797513f534321fe82a42ce011db2e62b": "\\operatorname{tr}(X^{\\mathrm T}Y) = \\operatorname{vec}(X) \\cdot \\operatorname{vec}(Y) = \\operatorname{vec}(X)^{\\mathrm T}\\operatorname{vec}(Y)", "79754f95247d3f83bda924fe58a084e3": "D[\\Lambda(\\theta,\\hat{\\mathbf{n}},\\varphi,\\hat{\\mathbf{a}})] = \\exp\\left[-\\frac{i}{\\hbar}\\left( \\varphi \\hat{\\mathbf{a}} \\cdot D(\\mathbf{K}) + \\theta \\hat{\\mathbf{n}} \\cdot D(\\mathbf{J})\\right)\\right] ", "79756f83cbda7eb228aa6e6054bbc31f": "E(\\tilde{m}\\tilde{x}) = p, E(\\tilde{m}\\tilde{R}) = 1.", "7975991b16a865f194090821f4e946af": "\\chi_e(\\Delta t)", "7975deb63f55c71be4bffb9ab44ebccb": "\\phi_{N, \\alpha}(z) =\n\\left\\{ \\begin{matrix}\nz^{2N} - 2 \\cos (2 \\pi \\alpha) z^N + 1 & \\mbox{if } 0 < \\alpha < 1 \\\\ \\\\\nz^{2N} - 1 & \\mbox{if } \\alpha = 0\n\\end{matrix} \\right.\n", "7975f000a080af52f6a424b0217fe44e": "\\mathbb{E}(X_t \\mid S_1=s) = \\mathbb{I}_{\\{t \\geq s\\}} \\left( 1 + \\mathbb{E}[X_{t-s}] \\right). \\, ", "797629901e6acf50ff4181c5a2319ed9": "S(T,V) = S(T_0,V) + \\frac{3}{2}R \\ln \\frac{T}{T_0}.", "7976365e6c93bfee1e70682dd1240513": "\n C_0^N=(1+r)^{-N}\\sum_{n=0}^{N}\\frac{N!}{n!(N-n)!}q^n{(1-q)}^{N-n}\n {[S_0{(1+b)}^n{(1+a)}^{N-n}-K]}^+\n ", "7976809f41f56b4db37e2728fedd000d": "~x_\\max/x_\\max=2\\gamma~", "7976cf2ece471ae771a966cdf6c15966": "8.4 \\times 10^{-17} \\ \\mathrm{seconds} \\,", "7976d36a4b55fb98e1d8081beee0da3d": "n_{AB}=0", "79771231519f0aa63b3c50b3d7e619fc": "2\\mbox{proj}_nd-d=\\frac{2n}{\\sqrt{n\\cdot n}}\\frac{n\\cdot d}{\\sqrt{n\\cdot n}}-d=2n\\frac{n\\cdot d}{n\\cdot n}-d=\\frac{\n(av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2)\n}{v'^2+u'^2}", "79772ea0a557e0b12ab233b4c5c210da": "\\scriptstyle b^3", "797739e41bc0c515a0cd30ac236407d0": "T_{\\mu \\nu} \\approx \\mathrm{diag} (T_{0 0}, 0, 0, 0) \\approx \\mathrm{diag} (\\rho c^4, 0, 0, 0) \\,.", "79773fa0c907feae54bfd3f73f998620": "\\rho\\frac{D\\mathbf{v}}{D t} = \\nabla\\cdot\\mathbb{P} + \\rho\\mathbf{f}", "797753a7c10b54d891fda09211b9b100": "p_n(x) = a_0 + xb_1(x) - b_2(x).", "797759a203b363817ffa472512e7d6a4": "1.13862 < \\liminf \\frac{\\log Q(x)}{\\log\\log x} \\le 1.44 \\ ", "79775d18d3753491358180586361d0bd": "{\\mathbf{x}}_r(0)", "79778a1fbfe5a584f7af7595480d1692": "{n1}", "797791aff34cab0fed4f86cc4c360beb": "g(a_1, a_2, a_3) \\geq \\sqrt{3a_1a_2a_3} - a_1 - a_2 - a_3", "7977cbfaccc8c0a65f7262afe61b9603": "g(n,k) = \\begin{cases} >0 &\\text{if } k\\mid\\lambda(n), \\\\ 0 & \\text{otherwise}. \\end{cases}", "7977f796123f67e71c5f2cad8d01cf72": "1/s_{zm} = -j\\cos\\left(\\frac{\\pi}{2}\\,\\frac{2m-1}{n}\\right)", "797808cd07ce04616703b1a1f99446ba": " R( p \\in I_0)=\\sum_m I(\\min_{ q \\in I_m}{\\lVert H_m(p)-q \\rVert}_2 < \\epsilon), ", "7978213681366263c0fb4b6684d8a3ec": " \\displaystyle{f_t(z)=f_s(\\varphi_{s,t}(z))}", "79786bf5b79d01b42240f477f22cdbfb": "\\ S_b ", "79786d335b37f3a63ad6f59ee0af6c91": "\\mathfrak{q} \\cap A = \\mathfrak{p}", "7978fff02ad87c96012e1b08fe6014ef": "Q_D\\left(n\\right)=O\\left(Q_S\\left(n\\right)\\right)", "797921e7521164550e628741f39d116f": "x \\sqsubseteq y", "7979625b2e00fa9457450391e8d0810d": "c_1 = 0.73761", "7979690ecf7d73ecbe86d6aae4639aef": "q\\not\\in\\{p,s\\}", "797a0440dc713b9880dce73f509e41eb": "V = \\frac{k}{n} {R_h}^{2/3} \\, S^{1/2}", "797abb90e8d8c7b6fbc3bf0d799b4808": "\n\\mathbf{s} = \\mathbf{L} \\mathbf{Q}.\n", "797b030b585b737fa35bb5af01ec9a35": "auth_{0}, ... , auth_{n-1}", "797cb3d6fe61ea1965afe80f20b81739": " \\nu = \\frac {[\\mbox{M}]_o} {[\\mbox{I}]_o} ", "797cca177e5898ea58fe741a76790489": "\\begin{bmatrix}-1&0\\\\1&0\\end{bmatrix}:\\mathbf a", "797d2105e16547d9b87b69778a801513": "\\mu \\frac{d^2Q}{dt^2} + k Q = 0", "797d27f620e6632366847a30a6aa0dfb": "(n-k)/2", "797d8b99f5a868a1f9df0a0ec51372db": "x = \\frac{X}{Z}", "797db02c95c390a0bb41ccc92ca373e3": "\\begin{matrix} {10 \\choose 1}{4 \\choose 4} \\end{matrix}", "797db02d5cfaa384909ad052ccb81597": "{Q_u}={Q_r}-{L}={Q_r}-{Losses}", "797db9c6135354a7b75482ee2003e1d0": "Re[U(\\mathbf{r},t)]= A (\\mathbf{r}, t ) = A_o \\cos (\\mathbf{k} \\cdot \\mathbf{r} - \\omega t + \\varphi )", "797e7cd9a6ac4e671db656bb0e3fe8d0": "\\alpha<1/2", "797ea9c63429ff2c9deac9fd4cf3b664": "7\\, ", "797eee6537282c7ebca98e35b58f40ea": "a^{\\frac{x^2-x}{2}}\\,", "797f4229d448290f15dd4911cee12e08": "1-e^{-1}", "797f4cdad56ae7cd49be6ba12bee012c": "E_{u/p} = y_1 + \\frac{q_1^2}{2gy_1^2}= 3 + \\frac{5^2}{2(32.2)(3^2)}=3.04 ft", "797fe19ed0e58d591f00beb867fe6582": "\\Gamma,\\ x:\\sigma \\vdash e_1:\\tau'", "798037143a1d4982e62e758a29e29d19": "I \\subset \\mathbb{Z}[\\eta]", "798047253129ca15df807a7eb4c9fd46": "e_i \\otimes f_j", "7980bdf6607a588d9978d50cac595516": " \\mathcal{N}(\\hat{O})", "7980d3b6c243531186a64a537e775f9b": "\\dim(\\operatorname{Hom}(M_\\mu, M_\\lambda))\\leq 1", "7980eff251e7d489a0e0a48ddb184639": "\\det(\\boldsymbol{F}) = 1", "79818ebf4624e51272754e8e5ef26f21": "S_v=-{K\\over 32\\pi G}\\int[g^{\\alpha\\beta}g^{\\mu\\nu}U_{[\\alpha,\\mu]}U_{[\\beta,\\nu]} -2(\\lambda/K)(g^{\\mu\\nu} U_\\mu U_\\nu+1)]\\sqrt{-g}\\,d^4x\\;", "7982223c056696bd146d5871dbd9c00e": "\\langle a \\mid a = 1\\rangle.", "798281593a6504d7e9a1bb7b01fb00c8": "\n\\begin{cases}\n\\frac{2}{\\pi} &:\\ 0 \\le \\theta \\le \\frac{\\pi}{2}\\\\\n0 &: \\text{elsewhere.}\n\\end{cases}\n", "7982971ff7d3dd6089974978cb3f0dc5": "{9\\over2}=4{1\\over2}", "7982a07d22626751ed49f9a51d1db037": "\\varphi = -\\frac{c^2}{2} \\varepsilon \\nabla \\gamma_{00}", "7982e3501002a129d3be46449ae95129": "T_{s}", "79831eb4a56b5cab858697b1f62e75dc": "g^{ik}", "7983abc1239ea6095fd8ffd0714457c6": "\\textstyle z_u", "7983cc37f1bd91340f4e41fae0f95b2d": "\\mbox{mex}(\\left \\{0, 1, 4, 7, 12\\right \\}) = 2", "798493a108b292aed4408e0cc0e7db24": "Y\\not\\Vdash B", "7984b5f80ac13d3051fefd83e2958b92": "\\kappa_{n+1} = \\mu(\\mu+1) \\frac{d\\kappa_n}{d\\mu}.", "7984bebbfac4df01bb5676fbcb582786": "failures_\\perp\\left(P\\right) = failures\\left(P\\right) \\cup \\left\\{\\left(s,X\\right) \\vert s \\in divergences\\left(P\\right)\\right\\}", "7984c39ec18b749c05e0655d7bc14e8b": " \\tau_n ", "7984d9fc59019e58ea3f4c2882845406": " \\Delta_r G = \\sigma \\mu_S + \\tau \\mu_T - (\\alpha \\mu_A + \\beta \\mu_B) = 0\\,", "79850715708bc517dacef1ede68d6fca": "H_c^q(X,F)", "79850b4f5e9af0fbb2b8379cca687f6b": " I_r ", "7985acce73b92427b64b9ac9b9115909": "P_{B}", "79861e9af5b532b0040828d9fc9b8f1f": "W=im(T)", "79863439d25a9564ff520b4ebb35f1f6": "A = \\iint_{D}dA.", "79865f326fd29eb5d9984ea76ae6ce05": "(z/2,z/2)\\,", "7987459bc87fe53605e087ccc6f31dfe": "\\scriptstyle R^4 \\rarr R", "7987b0f917e0f675a03c3072ac85911e": "w_{ij} = \\exp\\left(-\\frac{r_m^2}{\\kappa}\\right), \\, ", "7987fa5bb5596b669297d9035f22e311": "\\ \\gamma, \\ \\eta, \\ \\lambda", "7988213289a02e7ff1d2ae64b846c1f2": "d(x, z) \\le \\max \\left\\{ d(x, y), d(y, z) \\right\\}", "7988b778aeb9de3122a31ab8d1cf2291": "I_n = \\int \\frac{1}{(ax^2+bx+c)^n}dx\\,\\!", "7988c0daaedd3abfec958ff395ccc672": "a_m = \\frac{1}{(n^2-(2\\cdot m-1)^2)\\cdot a_{m-1}}", "7988ec59da1193cb028db6cafb1a975f": " \\gamma_\\mathrm{S,p} ", "7988f26d11cb581f100d01ee8ef04da3": "u_1 = \\begin{bmatrix}1\\\\0\\end{bmatrix}", "798975a1cd9e02f180a86883eb7d6d5b": "V_k = Y \\setminus f(X \\setminus \\cup_{\\lambda \\in \\gamma_k} U_{\\lambda})", "7989772a438bd6e73d37627e0a414ba4": "\\frac {1} {D_{L_{CO}}} =\\frac {1} {D_M} + \\frac {1} {\\theta * V_c}", "7989927ec35a604ef0f979249bac8aee": "\\Delta y \\stackrel{\\rm{def}}{=} f(x+\\Delta x) - f(x)", "7989934306f012aaaf407dfec42a127b": "p_3=1+m_2\\ ,", "7989976b58e03a7709890273406492db": "HA^+ \\rightleftharpoons H^+ + A", "7989ee54c04abab3962b60260e0159b5": "\\mathbf{E}_x = \\mathbf{e}_x, \\mathbf{E}_y = -\\mathbf{e}_z, \\mathbf{E}_z = \\mathbf{e}_y", "798aa997b14fa1b521cb46c7844f59e3": "\\nabla f={\\partial f \\over \\partial r}\\boldsymbol{\\hat r}\n + {1 \\over r}{\\partial f \\over \\partial \\theta}\\boldsymbol{\\hat \\theta}\n + {1 \\over r\\sin\\theta}{\\partial f \\over \\partial \\varphi}\\boldsymbol{\\hat \\varphi},", "798b3dde8183ac816234b374f6e9f0b8": "\\begin{align}\n\\log \\Lambda(x)&=\\log \\left[ \\frac{\\theta_1^{-1}\\exp\\left(-x/\\theta_1\\right)}{\\theta_0^{-1}\\exp\\left(-x/\\theta_0\\right)} \\right] \\\\\n&=\\log \\left[ \\frac{\\theta_0}{\\theta_1} \\exp \\left(x/\\theta_0 - x/\\theta_1 \\right) \\right] \\\\\n&=\\frac{\\theta_1-\\theta_0}{\\theta_0 \\theta_1} x - \\log \\frac{\\theta_1}{\\theta_0}\n\\end{align}", "798b989aa46283053dd28be96d300bc9": "\\mathcal{L} := -\\partial_t + r(t).", "798b9ad962126ab0c5b2ba58322e43ab": "\\mathbf{U}= (u_1, u_2, \\ldots, u_n) ", "798c15b36ac6398126e904ffc4391c82": "\n \\boldsymbol{S} = \\boldsymbol{N}\\cdot\\boldsymbol{F}^{-T} \\qquad \\text{and} \\qquad\n \\boldsymbol{S} = \\boldsymbol{F}^{-1}\\cdot\\boldsymbol{P}\n", "798c38c5855bcf7c35fce04786211e5a": " C_{t}", "798c3a4c3d9d363980ee46c49a47bae7": "\\phi_M =", "798ced61b00603bbe8f9ae8ef0e13645": "T_i^j", "798d8b010d2d185c3af52fb525577db0": "e_n^{\\alpha}", "798db03824a6025c30214527fff7ff05": " P_{sIsPositive} = (s > 0)", "798e0d266a7138bbcc3050e41fb11b92": "\\le 10", "798e30a6ad3ef8bfcd109189b2f78b1f": "10\\uparrow\\uparrow\\uparrow 2", "798e4bc7a60cd6e070dd1bb693914cd7": " \\gamma_{B,A} \\circ \\gamma_{A,B} = \\text{Id}", "798eafe74e91bc4e5c844485dacb1364": "W(\\boldsymbol{F})", "798edda0650606c5074826babd1d5f58": "X^G = F(*,X)^G", "798f0402540a225e495fe7b13e87903f": "\\kappa_t(\\cdot)", "798fdf8003e6a6e56fb57fcb7a4d3d91": "n\\ f\\ x = f^n\\ x ", "798fee77a1a39aeefa62141e48adccfb": "P_K T \\vert_K : K \\rightarrow K ", "798fef2324a236d8fd44a09e5e5bf162": "p(x) = \\frac{1}{a} | \\phi(x) |.", "798ff3b8eab2535832c26625f2e9ffb2": "R={\\sqrt{6}\\over4}a=\\sqrt{3\\over8}\\,a\\,", "798ffc35588a6690b52efdfc4bcf7776": " P,Q \\in G ", "79902121a7dc18941f80d49c3e360aa8": "L= K[x]/p(x)", "7990972abab1a380e9563b6393c32997": "p(\\theta|\\mathbf{X},\\alpha) = \\frac{p(\\mathbf{X}|\\theta) p(\\theta|\\alpha)}{p(\\mathbf{X}|\\alpha)} \\propto p(\\mathbf{X}|\\theta) p(\\theta|\\alpha)", "7990a9408a7cf939935ff598eebfad8f": " \\mathbf{n} = { \\mathbf{k} \\over k }. ", "7990ab817849691d8d2d3545c7f85379": "\\frac{ \\partial V}{\\partial r} = \\frac{ 2 \\pi r h}{3},", "79910f0b1cbda2380f4b6be88a525994": "(x_0 - \\delta,x_0 + \\delta) \\subset (a,b)", "7991d89ac432498ece0ad56ced154c40": "\\theta_{1}, \\theta_{2}, \\dots, \\theta_{k}", "7991e13149a6a2183f0726c1f2d45cd7": "g = {\\frac{GM}{r^2}}", "7991f1b37bc08a940e394abc9137c5e9": " H_f \\notin \\mathbf{TIME}(f( \\left\\lfloor \\tfrac{m}{2} \\right\\rfloor )).", "79923247b07ea88cccab44c04236087b": " \\zeta=e^{2\\psi}=\\Phi^2+\\tilde{C}\\Phi+B", "7992947b3dc43c7c33c399e9f8cb67d5": "\\mbox{linking number}\\,=\\,\\frac{1}{4\\pi}\n\\oint_{\\gamma_1}\\oint_{\\gamma_2}\n\\frac{\\mathbf{r}_1 - \\mathbf{r}_2}{|\\mathbf{r}_1 - \\mathbf{r}_2|^3}\n\\cdot (d\\mathbf{r}_1 \\times d\\mathbf{r}_2).", "79929ef28bd8f969154ac7cb32cfba3a": "\\dfrac{\\alpha : X}{\\alpha : T/(T\\backslash X)}T_>", "7992c7e9950f170ee6a16af148a1adc7": "c_{1}(t') = \n\\dfrac{\\mu_{01}\\epsilon_0}{i\\hbar}\\int_{0}^{t'}\\mathrm{d}t\\exp\\left(-i\\left(\\dfrac{E_{0} \n- E_{1}}{\\hbar} - \n\\omega\\right)t\\right)", "79930f3a849bc72faba22edbd8567a92": " \\{ \\sigma \\} ", "7993c2a115fa59df21be851e4ee9394c": "\nW_{ij} = \\alpha A_{i} A_{j} + \\beta B_{i} B_{j} \\, .\n", "79944824b775e45b5f3ffdb7e2585d92": "N_{l}", "7994790a2764148b07a3c696fa4b4baa": "E_N(\\rho_1)/n_1, E_N(\\rho_2)/n_2, \\ldots", "7994d2bf9816cf32d8d0426beaa1e1c4": "E^3", "7994dda4e6f67e478e419fc3d4c784d1": "\\theta^a", "7994e56a45ea768620689aeea8b8335c": "\nD=(R/100)\\sqrt{(K_1\\Delta\\phi)^2+(K_2\\Delta\\lambda)^2};{\\color{white}\\frac{\\big|}{.}}\\,\\!", "7994fd5e0e86d881a92d8c9252a3de03": "i,j=1,\\dots,n,", "799531ba2e3e9c833d0c448c68333295": "\\leftrightharpoons", "7995e101d9b25a5c6fbe5029d7508c9c": "x_i = \\alpha \\sum_{j =1}^N a_{ij}(x_j+1). ", "79961557e53defd6116931fa0226e659": "\n\\bigcup_{\\lambda\\in{\\mathbb F}}\\lambda\\cdot A^\\circ=Y.\n", "7996670ccfbcca9181b7b905b2db07de": "\nH_{R}=\\alpha(\\boldsymbol{\\sigma}\\times\\bold{p})\\cdot \\hat{z}", "799668cbf1af09e2d8177eb0903a1ab9": "{\\nabla_r}^2", "79967a55fdd8d50fda6dd4d3103188fd": "\\tan A = {\\sec A \\over \\csc A} ", "7996a1390567dc60c66534b3ae83bbc3": " \\ e ", "7996b408482204f5c4b04f3d8a3ebe90": "\\Phi (\\gamma + \\sigma \\sinh^{-1}z)", "7996d9798d23c4f9a86bd66caa215dad": "\\varphi = \\sqrt{1 + \\sqrt{1 + \\sqrt{1 + \\sqrt{1 + \\cdots}}}}.", "7996f1a1225e0070ce1505ff041d835e": "C \\to RC ~|~ L", "79972d5ef3e3d6ac363d35f1c06e5926": "q^Ty", "7997481a45d5248a72039c8827495d5f": "\\mathbf{\\mu}=I \\mathbf{S}.", "799769c42c084571c8e119842cf864f7": "\\frac{(\\log g)^2}{g}.", "7997a8627117916093796ddd925d2b9a": "y(t) = M(t) \\cdot \\cos(\\omega_c t),", "7997afe89fad4bd0eef5b7c97d922dd3": "{}_e\\langle X \\rangle_N", "7997d7a94dfd7fa7aab20708b7c14c73": " \\exists x \\, ( \\mathrm{inf}(x) \\land \\forall y \\, ( \\mathrm{inf}(y) \\rightarrow x \\approx y ) ) ", "7997f1efd38b5e2d5c46f7b8644b50f6": "{13 \\choose 1}{48 \\choose 3} = 224,848", "79981e74f23502913a109e51e42d19ee": "S_n \\wr Z_2", "799840bc7252125b5e10ec6d65a078a2": "p, q \\in R", "79986103e59d9f974465d2130d314415": "\\,(b + c)^2 = a^2 \\left( \\frac{d^2}{mn} + 1 \\right).", "7998830f939578222fbcba5c76c3044d": "{a_1}^2+{a_2}^2+\\dots+{a_7}^2 = 1", "7998886b28d5485087b8ee11678bf592": "H(X) < \\log|\\mathcal{X}|.", "79989e39db5df7a96b5d488853ce34e2": "\\mathbf{P}_{Z_i,z^{n\\backslash i}}", "7998a30adbf552d3e3b82377ef1b7a99": "c^k\\,", "799904b20f1174f01c0d2dd87c57e097": "ix", "7999116a174f7aa411192c73ee2ecb00": "MSO_{\\mathbf Q} = H(\\pi_*(MSO_{\\mathbf Q}))", "79994b712433b3d175a2a5a0e7cd198c": " C^J_{v_2} = 0 ", "799961cc38af06403b509d86745d4057": "\\begin{smallmatrix}L_\\odot\\end{smallmatrix}", "7999a23af25a521c89002f8c53476f58": "(X, \\Sigma),", "7999bc29b8bd4ce24cb3b165dc4122f0": "\\alpha\\mapsto\\alpha^\\vee", "7999c3c413c298b40711cd9be6291ab4": "\ndQ = C_{P} dT-V\\,dP.\n", "7999c707939eb472ff336c6914dfd9a0": "i_1, i_2, \\dots, i_n", "7999d7a79fdf1add2dd42f9a8476b80c": "R_2 =1", "7999e60bd0ce68e18499122bc522c938": "(n_0, n_1)", "799a465cccffeb3efbd8a491cefd0438": "b\\quad", "799a643e8b23177c28914ec9ce65b53e": "\\sigma^2_S", "799aab0c65605c7bde444d205b95f440": "\\lambda_0 = h/m_0c - \\ ", "799acd3d54bf41d634099b7fd3db994c": " |\\psi\\rangle \\in H_1 \\otimes H_2", "799ad56398600cad901db08e3bba1c4b": "\\left( \\frac{3}{\\sqrt{10}},\\ \\sqrt{3 \\over 2},\\ 0,\\ \\pm2\\right)", "799b566a51242a2f54aeb41492927935": "\\sum_{n=0}^\\infty H_n^{(r)}\\frac{t^n}{n!}=e^t\\left(\\sum_{n=1}^{r-1}H_n^{(r-n)}\\frac{t^n}{n!}+\\frac{(r-1)!}{(r!)^2}t^r\\, _2 F_2\\left(1,1;r+1,r+1;-t\\right)\\right),\n", "799b773c38f87e227d1c5ed86dbafb6e": "\\int x r\\;dx=\\frac{r^3}{3}", "799b9a6feb9afad9e623b9691d137c64": "\n\\min\\limits_{x\\in X}\\{ \\hat{g}_N(x) = f(x)+\\frac{1}{N} \\sum_{j=1}^N Q(x,\\xi^j) \\}\n", "799bb6c5f043491c1561e7fa4cc74595": "\\varepsilon_k", "799bd7d5c489df220dcb3f4d67e80235": "x(t)=\\frac{1}{t_c-t}", "799be516dbc5df4308641265228a26d1": "I : (x_0, \\dots, x_n) = I", "799c18186833b3d916bc48da18a64103": " nk ", "799c71319ae729457e517ce2e12534df": "A\\ge 1", "799c7f4c31079dcb7894f5f58688b91d": "\nS = \\int_x \\psi^\\dagger \\left(i{\\partial \\over \\partial t} + {\\nabla^2 \\over 2m}\\right)\\psi + \\lambda (\\psi^\\dagger \\psi)^2. \n", "799ccb664ff7fc863c6b4a060150bd3b": " \\lang A \\rang _t = \\lang \\psi (t) | A | \\psi(t) \\rang.", "799cd0f96275870fcb53e92e305d9b25": " {d \\over d\\tau} \\left( {-g_{\\mu \\nu} {\\partial \\dot x^\\mu \\over \\partial \\dot x^\\lambda} \\dot x^\\nu - g_{\\mu \\nu} \\dot x^\\mu {\\partial \\dot x^\\nu \\over \\partial \\dot x^\\lambda} \\over 2 \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\right) = {-g_{\\mu \\nu, \\lambda} \\dot x^\\mu \\dot x^\\nu \\over 2 \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\qquad \\qquad (1)", "799cf068f706101db146cb85e3f41f65": "\\displaystyle{R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b).}", "799d12f729b6d420fa0d30cc8509e971": " \\int_0^1 x^2 \\, dx, ", "799dbcb018a93aeb98147a002321ab7b": "\\neg \\neg P \\rightarrow \\neg \\neg Q ", "799e30b1340d127c6c82411704027560": "415_{11} \\ ", "799e890aa3e1d8a52cae6877ae209d67": "\\mathrm{UT} (M) := \\coprod_{x \\in M} \\left\\{ v \\in \\mathrm{T}_{x} (M) \\left| g_x(v,v) = 1 \\right. \\right\\},", "799f07a29f98f273803d17829459bc11": "e_i(t)", "799f0d6dd55cbe7e9d3f7555045d0934": "J \\setminus X", "799f3b84f2395ff5f00619e7bab82e1f": " -\\textbf{c}_N(-\\textbf{A}_N)^{-2}\\textbf{b}_N + \\textbf{c}_P(-\\textbf{A}_P)^{-2}\\textbf{b}_P", "799f7e3b01226637f86c8ce3a0cc77b4": " e = \\left( \\frac{q}{p} \\right)^{\\left( \\frac{q}{p} \\right)} = \\sqrt[p]{\\frac{q^q}{p^q}} ", "799f9e698a41cbc69a298ba5ae6d9aea": "a = b = c = d, \\alpha = \\zeta = 90 ^\\circ, \\beta = \\epsilon = 120 ^\\circ, \\gamma = \\delta \\ne 90 ^\\circ", "799fe1c2cad781dba3dd3d3415f0b9f5": "\\lambda_{1}\\,", "79a0236f074473473067662ad21e7f4f": " \\mathfrak{Q} ", "79a04c6394d27f6b8a11739b61b0d20e": " \\mathfrak{p}", "79a0513e778f440038afee63dba7fc26": "\\mu_{\\rm B} = \\frac{e \\hbar}{2 m_{\\rm e}} = \\sqrt{\\frac{c_0 \\alpha^5 h}{32 \\pi^2 \\mu_0 R_{\\infty}^2}}", "79a060111daa5434edff6d8698596955": "d^2y/dt^2 = (1/m)[-mg+N(t)]", "79a088f975f76ea8a37473b5345960f5": "E^a", "79a0bdd15b036c53271c367517267a71": "M=\\frac{2\\pi}{t_Y}n\\mbox{ day}=M_D+\\frac{2\\pi}{t_Y}D\\mbox{ day}=6.24004077+0.01720197 D", "79a0e541049d91d022572b2fb7ab748c": "F(1)", "79a1a0583fd6b0f5662ff86da170a126": "L=\\frac{1}{2} \\mu \\left(\\dot r ^2 +r^2 \\dot \\theta ^2 \\right) - U(r), ", "79a1ac55733cdc2d72053d0f9c44020f": "\n w(x,y) = w_x(x) + y\\,\\theta_x(x) \\,.\n", "79a1e9f39c815c2bc041c96718a5d2e2": "\\Pr(N) = \\frac{k}{N^\\alpha}, 0 < \\alpha < 1.\n", "79a1fda3656c39ff91bba3f5f8fe4a88": "\\langle n_i\\rangle", "79a221013d846ae466183ced65f02a50": "?\\left(\\frac{p+r}{q+s}\\right) = \\frac12 \\left(?\\bigg(\\frac pq\\bigg) + {}?\\bigg(\\frac rs\\bigg)\\right)", "79a2365432301fadccbcc47868b53073": "n_1,\\, n_2,\\, ... \\;,", "79a244953cd16ab01d8cd69da3acc068": "f_2,", "79a2a24f29aacca480c9c888e0c4e409": "\\frac{R_s}{R}=(1-\\cot\\theta\\tan\\alpha)\\sec\\alpha", "79a2b51d8bc277833d1fda2a1455f7c2": " R_1 , \\ldots, R_N ", "79a2c72f816595c091e8324b61ff9691": "h:G\\rightarrow H", "79a30186f090719694e99c5eea537bcc": "\\Pi(p(^k M_0(mech,\\mu_1)))", "79a309f278302846b4b6ad86e27adea8": "x^{\\prime}=x-vt,\\quad y^{\\prime}=\\frac{y}{\\gamma},\\quad z^{\\prime}=\\frac{z}{\\gamma},\\quad t^{\\prime}=t-x\\frac{v}{c^{2}}", "79a3593f61f7e953fbbf68a396d16bd8": "= \\left( \\frac{e^{i \\pi fT} - e^{-i \\pi fT}}{i 2 \\pi fT} \\right)^2 \\ ", "79a3c796177342ce57f5518ee73fc44c": "q(q+1) \\equiv \\frac{p(p-1)}{2} \\mod p.", "79a427fc458d527066b26d1a490372da": "x,t\\,", "79a436dcca6075f2fa9cf9f88a13dba6": " Z_3 = (T_1Z_1)^2 - (T_1Y_1)^2 + (Z_1Y_1)^2 = -12", "79a4691b2b3541d2750660d90984db78": "\\mathrm{rem}\\left(a_i,m_i\\right) = a_i", "79a477994fb85b927873ba1009a1d3cc": " \\frac{dI}{dt} = \\beta(t) \\frac{I}{N} S - (\\nu +\\mu ) I ", "79a493865cb4a49923058e4af13dafff": "\\begin{align}\n\\frac{\\pi^2}{\\sin^2 \\pi z} = \\sum_{-\\infty}^\\infty \\frac{1}{(z-n)^2}.\n\\end{align}", "79a4b942bd1d4a2ff24bcbe9ed4cb114": "U_e", "79a51b79d4556678060aa05036ce33ee": " \\mathrm d\\omega + \\omega\\wedge\\omega,", "79a5287e127289144e6f84443e443485": " u\\dfrac{\\partial u}{\\partial x}+v\\dfrac{\\partial u}{\\partial y}={\\nu}\\dfrac{\\partial^2 u}{\\partial y^2} ", "79a56371cef406de4415fe12c5988629": "\\frac{1}{2} V_\\text{esc}^2", "79a58aed75b66a43d8e73759e5c87d84": "g=\\int_{4\\pi}(\\hat{s}'\\cdot\\hat{s})P(\\hat{s}'\\cdot\\hat{s})d\\Omega", "79a5922c55813322f6675eba9c6ace02": "e_2 = - \\frac{1}{2}", "79a59d6d1c8205bcda2731c9dc6643a0": "R(0,0) = \\frac{1}{2} (b-a) (f(a) + f(b))", "79a6065a1e6d6f38ff19bce150e85200": "\\hat P_{MN}(e^{j \\omega}) = \\frac{1}{|\\mathbf{e}^H \\mathbf{a}|^2} ; \\mathbf{a} = \\lambda \\mathbf{P}_n \\mathbf{u}_1", "79a6b64442bcde698e0dc48306479536": "t \\mapsto 0, t \\mapsto \\epsilon^2", "79a6e5cd2cfa707ea0f04c242a9bf20c": " \\frac{V_1}{V_2} = \\frac{W' \\left ( 1 -W' + K_1 \\right )}{\\left ( 1 -W' \\right ) \\left ( W' + K_2 \\right )},", "79a6e751da8fe9a828fe099738da59b6": "\\rm \\ 3 CO + 1.5 B_2H_6 \\xrightarrow{LiBH_4} (CH_3BO)_3", "79a74dbb3df9abd7dba2a00b4ba4cf03": "Y_{n+1} = (3 - b_n)/2", "79a78b70aa86dac92997bc3d1bb360ec": "(p + q)^c\\,", "79a7a45fade05f8676d9aa5992aa4f0d": "\\mathrm{SU}(2)", "79a7ac051c7e19d16a484c88255fc0d7": "D_-(x) = \\frac{\\sqrt{\\pi}}{2} e^{x^2} \\mathrm{erf}(x)", "79a7c20f55482d22410dbfb0e0dea102": " \\vec{F} = \\int_0^L I(L)m \\, d\\vec{L} \\times \\vec{B}. ", "79a8b640507f91c75de1d971c25fcd65": " \\eta \\in E ", "79a91c2f2633bfb705e2ea24eb9fcc35": "\\int_X f\\, d\\mu = \\lim_{n\\to\\infty}\\int_X s_n\\, d\\mu.", "79a91d0f6c385228b71a4377e35c837e": "\\ y[n]=b_0 w[n]+b_1 w[n-1]+b_2 w[n-2],", "79a93e9eb08ccc5a40cbf3f68a608897": "\nX^{\\{3\\}}=[2,4] \\cup [6,7],\n", "79a9e7750f1defb822c9a56f8799aa36": "-\\frac{b}{2a}.", "79aa9ac687a1fb37248895920a096ca7": "\\partial \\Omega_N", "79aa9f8a9a733a6efbf5254d60603089": "\\frac{d^2 Q}{d p^2} = H(Q) \\left(\\frac{d Q}{d p}\\right)^2 ", "79aaa1c3a08272f637d219f024dbabda": " a_n>0,a_{n-1}>0,a_{n-3} >0, \\ldots,\\, \\Delta_3>0,\\Delta_1>0", "79aab21ca5777e0decca89eb6d326c88": "\\frac{dy}{dx} = f(x)", "79aad403620a6e0f2c96eacd69202b2f": "\\underline{P}(A)", "79aae9a881793d478cf1846b5daf4751": " E \\xi^n(\\cdot)", "79ab0f4db8ed8f8be8df866cefc8bc54": "e''^2 = \\frac{\\mathrm{a}^2 - \\mathrm{c}^2}{\\mathrm{a}^2 + \\mathrm{c}^2}", "79ab39197a375e100797ef86dc6f4507": "E=-{2.303RT \\over F}pH - {RT \\over 2F}\\ln {p_{H_2}/p^0}", "79ab63f6da7dc417a5a06be4e465f0cd": "xt \\ge 1-(1-x)^t. ", "79abad3e73110bbee35064b6ca4424ed": "PV = NkT \\,", "79abb07009a917167be2f769c1a2c98d": "P^{(2)} (\\mathbf{x}, t) \\propto E_1^{n_1} E_2^{n_2} e^{i ((\\mathbf{k}_1 + \\mathbf{k}_2)\\cdot\\mathbf{x} - \\omega_3 t)} + c.c.", "79ac2e2b622017c853a17559b0f782d2": " \\operatorname{extract}\\ k ", "79ac305baa9477fb7b65e8a42b3f2d85": "{ E = m c^2} \\ ", "79ac3bdf1b3b8de0eabc19b68c68df04": "L_2(9) \\cong A_6,", "79ac4ebd37849367dd4621f204f7f9c5": " M = (m, c) ", "79ac57b9275f4dabb71a5e815554b469": "sgn", "79ac668bc16a5dea31e6ca369fef8117": "\\varphi=2\\pi", "79acd6de4b4a0b0fdbb9399eef752fd0": "F=B_IP-E-T-ET-T-I_A-B_O", "79acdf3811403f03e0d520ececc7f895": "\\mathbf{F}_{jk}", "79ace10bd338fd32453a13a2b59bee97": "\n\\begin{align}\n\\operatorname{cov}(x_s,x_t) & = E[(x_s - E[x_s])(x_t - E[x_t])] \\\\\n& = E \\left[ \\int_0^s \\sigma e^{\\theta (u-s)}\\, dW_u \\int_0^t \\sigma e^{\\theta (v-t)}\\, dW_v \\right] \\\\\n& = \\sigma^2 e^{-\\theta (s+t)}E \\left[ \\int_0^s e^{\\theta u}\\, dW_u \\int_0^t e^{\\theta v}\\, dW_v \\right] \\\\\n& = \\frac{\\sigma^2}{2\\theta} \\, e^{-\\theta (s+t)}(e^{2\\theta \\min(s,t)}-1).\n\\end{align}\n", "79aceeffd7ea3989e3ac5033ef3bf64d": "FP(\\mu, \\sigma, \\gamma, 1, \\alpha) = P(IV)(\\mu, \\sigma, \\gamma, \\alpha).", "79ad009df767ce3a3e5aaf68e8b6861c": " J(x,t)=-D\\nabla\\rho(x,t) ", "79ad2eb4bcbfb5febc1f6160acc57b3e": "\\begin{align}\nP_\\mathrm{n} &= 0.85f'_\\mathrm{c}(A_\\mathrm{g} - A_\\mathrm{st}) + A_\\mathrm{st}f_\\mathrm{y}\\\\\n\\end{align}", "79ad30dbe6ea589bea56ec3166063e7c": "k_\\text{grating}", "79ad6e3db5904ce085881f6bbd4b2490": "\\Pr(\\text{sun will rise tomorrow}) = \\frac{d+1}{d+2}", "79ad7c0cd6f7067ec1eb06a6ca811a81": "\\frac{{81 \\choose 2}}{3} = \\frac{81 \\times 80}{2 \\times 3} = 1080", "79ad84e00e87f1c78ac99977c2bf28fd": "b = \\frac{u^2-v^2}{2},", "79ada97ba946374a86b31be4af8e25ab": " \\delta = {1 \\over h} (S_h - 1),", "79adc624b59b7b92e1298dae47be46d0": "\\det(I_p + AB) = \\det(I_n + BA),\\ ", "79adecf0d2e71e2b6942ff5d87d3417c": "\\psi_{2S}(0)=\\frac{1}{(8\\pi a_0^3)^{1/2}}", "79adf46d8fcc0eca7f54d0311bad4c6d": "\\ \\phi\\,\\!", "79ae5da63533e2533e04a0f6a980919a": "\\mathbf{p} = \\hbar \\mathbf{k}\\,,\\quad |\\mathbf{k}| = \\frac{2\\pi}{\\lambda} \\,. ", "79aec01cf25b0558efd92c21eb4acc97": "I_{h}", "79aec40bf0319711f293aca32d848aaf": "\\begin{align}\\hat{H} & = -\\frac{\\hbar^2}{2}\\sum_{j=1}^N\\frac{1}{m_j}\\nabla_j^2 + \\frac{1}{8\\pi\\varepsilon_0}\\sum_{j=1}^N\\sum_{i\\neq j} \\frac{q_iq_j}{|\\mathbf{r}_i-\\mathbf{r}_j|} \\\\\n & = \\sum_{j=1}^N \\left ( -\\frac{\\hbar^2}{2m_j}\\nabla_j^2 + \\frac{1}{8\\pi\\varepsilon_0}\\sum_{i\\neq j} \\frac{q_iq_j}{|\\mathbf{r}_i-\\mathbf{r}_j|}\\right) \\\\\n\\end{align}", "79aee80f8e767c19e14c4c2d0c7eb0c4": "\\frac{\\Gamma(1+3/k)\\lambda^3-3\\mu\\sigma^2-\\mu^3}{\\sigma^3}", "79af044b9068019ca766efcd487fee82": " \\epsilon_f ", "79af32139c6ebca792cb12c0e6016691": " \\Pr(\\varnothing)+\\Pr(\\mathrm{Head})+\\Pr(\\mathrm{Tail})+\\Pr(\\mathrm{Head,Tail}) =1.", "79af4780f104d3552cd89729880ef171": "\\displaystyle{B(a,b) = I - R(a,b) + Q(a)Q(b).}", "79b01c6c35e2e17f10ceb63b280dedfe": "\\lim_{x\\to\\infty} \\frac{Q(x)}{x} = \\frac{6}{\\pi^2} = \\frac{1}{\\zeta(2)}", "79b0426d4b6192c9508189484a06583d": "d\\times2", "79b0b2a89a02b9eb06bbba20c945788d": "\\psi_{-}(\\mathbf{r})", "79b0fd5fdb80798e865bdbce6e84ccb3": "[X,Y]_t=\\tfrac{1}{2}([X+Y]_t-[X]_t-[Y]_t).", "79b17543c1b038867aee1455bfc363f1": "\n\\tilde{\\mathbf{e}}_{k} = \\mathbf{S}^{-1} \\mathbf{e}_{k}\n", "79b19c2cb4f63bf6ba3ad98056e9f203": "\\operatorname{dVar}(X) = 0", "79b2036cdab85e0feb2ddf082bc9b699": "\\mbox{2.15 dBi} = 1.64", "79b209a484004c8054288c1fd20ff4bf": " D_{\\mathrm{KL}}(P \\| Q) = D_{\\mathrm{KL}}(P_1 \\| Q_1) + D_{\\mathrm{KL}}(P_2 \\| Q_2).", "79b21b2c3d32219af6e628d7ac126a34": "\\displaystyle{ds^2=|f_z dz + f_{\\overline{ z}}d\\overline{z}|^2= |f_z|^2 |dz + {f_{\\overline{z}}\\over f_z} d\\overline{z}|^2.}", "79b24814550979b6a9a29af51c717832": "\nL_\\mathrm{p}=10\\, \\log_{10}\\left(\\frac{{p}^2}{{p_\\mathrm{ref}}^2}\\right) =20\\, \\log_{10}\\left(\\frac{p}{p_\\mathrm{ref}}\\right)\\mbox{ dB}\\,\n", "79b2809d6145b62c9a8f61ffad11e7bc": "{\\rm Inc}({\\mathbb B})=\\sup\\big\\{ |A|:A\\subseteq {\\mathbb B}", "79b2c77ab68338d8eababa8e9cc0f93d": "f(x)=\\alpha_i\\,", "79b2e37d9ea993931f130c84e95e29c6": "\\varphi_{x,y}", "79b33be10cd02a2482e1132495f41d8d": "x^{\\underline n}", "79b34e42443f8ad21a01ed736468159f": "A f = 0", "79b36ca2723977244e8c5db1a44c87c9": "x=\\sqrt{R^2+r^2-r\\sqrt{4R^2+r^2}}.", "79b395047a8bcbea9826eb9e64e5e209": "\\Delta C= 50", "79b3bb53dbda6f4e0bd6971b89d5c1e6": "\\displaystyle a^2+b^2+c^2=ab+bc+ca.", "79b3cec87f68857bcddb53f8cb16b3a5": "\\; \\Phi^* (A) = \\sum_i R_i(A) F_i.", "79b3ea455f10120df56fbf7e352b55c3": "\\approx 0.551286", "79b455ba07bfd07294cd6cabc6336e0f": "g(\\boldsymbol{x})=\\frac{1}{f(\\boldsymbol{x})}", "79b4852033fdaaf74d69b804ad7b2e96": "[0,0,1]^\\mathsf{T}", "79b489a3519cbf39ca272f75fe5ec028": "a_{ii} = 2", "79b48fad7eb49d2c98bc5b4556463d93": "2x^{7/22}", "79b4de7cf79777bf4af9e213ede350af": "wx", "79b5482175ef43d8b7d7bb3e927e5e20": " \\Sigma_1^{\\rm P} = {\\rm NP}, \\Pi_1^{\\rm P} = {\\rm coNP} ", "79b5dfcbd71fba3f783440a8fe956f88": "(-\\eta)^{\\frac{1}{k}}", "79b6692edeb27250e80235be802b0832": "\\Delta x \\Delta p = \\frac{\\hbar}2", "79b68136807e6248f4df0f1cd47a976b": "\\vec{F} = - \\frac{G m_1 m_2 \\vec{r}}{r^3}", "79b68c89064e95d7e9ad15561e2c474c": "V_o\\,", "79b69f49c2f6040b904312e6f65833c0": "SO_n", "79b6b7b51b097ed57d15509dec2cc4b5": "L_2=(1+i) L_1 - P", "79b722c4535839bf550a3dc15ea9e6d2": "\\frac{y_2}{y_1}=\\frac{1}{2}\\left(\\sqrt{1+8 F r_1^2} - 1\\right).", "79b756b31b5303aa581dd39d396f52a2": "m_1 u_{1} + m_2 u_{2} = m_1 v_{1} + m_2 v_{2}.", "79b75c7810e1d5f90ed92010a31b41ae": " H |\\psi> =E |\\psi> \\ ", "79b795fca653f79a735d7800e57dd2f8": "f = AB\\,", "79b7c3d539e61ed0acfab5a98c040542": "\n \\boldsymbol{\\varepsilon} = \\tfrac{1}{2}\\left[\\nabla\\mathbf{u} + (\\nabla\\mathbf{u})^T\\right] \\,.\n", "79b801882ac982bf3807075ac9e00ca9": " \\hat \\epsilon_t^2 = \\hat \\alpha_0 + \\sum_{i=1}^{q} \\hat \\alpha_i \\hat \\epsilon_{t-i}^2", "79b85a0d282eccd099f007df3d0351ae": " G = H - TS = U + PV - TS ~.", "79b87da94b0fe75341f3c15371090dc5": "d^* = \\max_{\\lambda \\ge 0, \\nu} g(\\lambda,\\nu) = \\inf f_0 = p^*", "79b885ca71d7c3e631b5fd00991d2c07": "f^\\text{pmi}(t_i, w)>0", "79b8de4393e5863475f58ba8cf117da5": "\n[P_i, P_j ] = 0\n\\,", "79b8e714fcca8bc8852279ad60cdfd87": "2 A t + B = t^2 - (A t^2 + B t + C)", "79b8f1ecf2c8ba3eacabf9fed8f3e2ca": "S_3= \\{ e, (1\\ 2), (1\\ 3), (2\\ 3), (1\\ 2\\ 3), (1\\ 3\\ 2) \\}", "79b9128ae421db553d288264e211519f": "W_{U}=\\bigcup\\{A\\in\\mathcal{V}:\\bar{A}\\subseteq U\\}\\,", "79b926273d02355b589668f7d9c67ff4": "\n\\mathrm{[C_5H_5NH]SO_3CH_3 + I_2 + H_2O + 2 C_5H_5N} \\longrightarrow \\mathrm{[C_5H_5NH]SO_4CH_3 + 2 [C_5H_5NH]I}\n", "79b9299641eff1bd4672ae0d170331b2": " N_B' ", "79b952ad6888837cedfab66c0981d360": "= \\frac{a}{b} \\cdot \\frac{c}{d} + \\frac{a}{b}\\cdot \\frac{e}{f}\\text{,}", "79b95f66759b0267b1a1244638c2a830": "\\scriptstyle{1 \\ll K \\ll N}", "79b9c3e1f31c808c7124894f44e3d76f": " V= \\sum_{i>j} \\frac{q_i q_j}{r_{ij}} ", "79b9e48440f309b382920164acd4dc76": "X = \\frac{\\omega_0^2}{\\omega^2}", "79ba45d15085672b43758d513e6fed48": "M = E - e \\cdot \\sin E", "79bb6fb2214b8d68c2d64532fd282551": "\\dot{C}_t = \\tau_t^0\\dot{x}_t,\\quad C_0 = 0.", "79bb8abf153e76168c1e0f188c4c0f55": "\\sum_{k=1}^n|b_{2k+1}|^2\\le n.", "79bbb83a91b2fa84f7f92d32b10af887": " y = f^{-1}(x).", "79bbea13c5df8a5e55e78f15ba527332": "\\frac{d X(e^{i \\omega})}{d \\omega} \\!", "79bbf584659a183d3477da4c871b2963": "\n\\left[ k^2 + \\lambda^2 \\right] G(\\mathbf{k}) = 1.\n", "79bc085c13915df43c88a3d43ea34cae": " q_1 q_2 \\ ", "79bc4a599017871d546db2b7e4544c85": " \\lambda(n) ", "79bce40d7396dbb301d1c2f410a9172b": "\\bar{\\mathbf{e}}{}^j = \\bar{\\mathbf{e}}{}^j\\left(\\mathbf{e}^1,\\mathbf{e}^2\\cdots\\right) \\quad \\rightleftharpoons \\quad \\mathbf{e}{}^j = \\mathbf{e}{}^j \\left(\\bar{\\mathbf{e}}^1,\\bar{\\mathbf{e}}^2\\cdots\\right)", "79bd2890d78accb5ff8df077bc497966": "A_x = [HA]\\epsilon^x_{HA} + [A^-]\\epsilon^x_{A^-} ", "79bd2f7c76731b59012440612655c20f": "P = {(4x - 6) \\choose 3} \\div {50 \\choose 3},", "79bd741f4d523eb5183fa2510f7ff33a": "X=n(s\\otimes s)^T+s(s\\otimes n)^T+s(n\\otimes s)^T+n(n\\otimes n)^T,", "79bde24815cb1e8b5ca61eef5b4c0bdb": " u_{tt} - c^2 \\left( u_{rr} + \\frac{2}{r} u_r \\right) =0. \\,", "79bdf01d18e27f8adfd4c947be4ec087": "\\frac{\\partial (-v)}{\\partial x} - \\frac{\\partial u}{\\partial y} = 0.", "79be2be833fef0a73c721968602114d3": " \\boldsymbol{\\mu}_L = -\\frac{g_L \\mu_\\mathrm{B}}{\\hbar}\\boldsymbol{L}", "79be4e49736695875e541a26e4f5600b": "\\cos(i)=(2^{-1}\\bmod{p})\\cdot(\\epsilon^{i}+\\epsilon^{-i}),", "79be8b4a2030db77093258dec35c6773": "\\omega_{\\Lambda_2}", "79be8ee4f4f32e5bb70486f34edd8909": "\n\\theta_2(q) = \\sum_{n=-\\infty}^{\\infty}q^{(n+\\frac{1}{2})^2}\\qquad\n\\theta_3(q) = \\sum_{n=-\\infty}^{\\infty}q^{n^2}\\qquad\n\\theta_4(q) = \\sum_{n=-\\infty}^{\\infty}(-1)^n q^{n^2}.\n", "79becef6f8330f5e337dcba99fcf5535": "3\\le seqs \\le60", "79bf5e36f7f99fbb9d86d8270b38a1e2": "\n\\begin{align}\n& {} \\quad -\\sqrt{(l_3\\mp s_3)(l_3\\pm s_3+1)} \n\\begin{pmatrix}\n l_1 & l_2 & l_3\\\\\n s_1 & s_2 & s_3\\pm 1\n\\end{pmatrix}\n \\\\\n& = \\sqrt{(l_1\\mp s_1)(l_1\\pm s_1+1)} \n\\begin{pmatrix}\n l_1 & l_2 & l_3\\\\\n s_1 \\pm 1 & s_2 & s_3\n\\end{pmatrix}\n+\\sqrt{(l_2\\mp s_2)(l_2\\pm s_2+1)} \n\\begin{pmatrix}\n l_1 & l_2 & l_3\\\\\n s_1 & s_2 \\pm 1 & s_3\n\\end{pmatrix}\n\\end{align}\n", "79bf71e78b5acfab988cdffb7b4185e2": "P = \\{p_0, p_1, p_2, p_3, p_4, p_5, p_6\\}", "79bfc1ff19f113c55b78e8adb1ac6455": "[J_m,P_n] = i \\epsilon_{mnk} P_k ~,", "79bfd0f7ddbb7b61775a23067d239e33": "x^2-3x-1", "79c00d03c3e3daa53b639b8c4f46f8d0": " \\mbox{EXPTIME} = \\bigcup_{k \\in \\mathbb{N} } \\mbox{ DTIME } \\left( 2^{ n^k } \\right) . ", "79c00f05f2d297119b1932af8b546fd0": " {(\\sigma_1 - \\sigma_2)^2 + (\\sigma_2 - \\sigma_3)^2 + (\\sigma_3 - \\sigma_1)^2 = 2 {S_y}^2 }\\!", "79c0266beb75ddf3efd3672bdb048f6a": "\\scriptstyle\\left(\\tfrac{P}{40}\\right)^{1.6} + \\tfrac{U}{45}", "79c0a69bd0991f3075d4f31b6b2f44eb": "\\{0, 2\\pi, -2\\pi, 4\\pi, \\dots \\}.", "79c115768093d6fd8fd57c542930c870": "\n\\begin{cases}\n\\mathrm{out}_A = \\mathrm{src}_A + \\mathrm{dst}_A (1 - \\mathrm{src}_A) \\\\\n\\mathrm{out}_{RGB} = \\bigl( \\mathrm{src}_{RGB} \\mathrm{src}_A + \\mathrm{dst}_{RGB} \\mathrm{dst}_A \\left( 1 - \\mathrm{src}_A \\right) \\bigr) \\div \\mathrm{out}_A \\\\\n\\mathrm{out}_A = 0 \\Rightarrow \\mathrm{out}_{RGB} = 0\n\\end{cases}\n", "79c15f207c8a32cd7596b2a0fffea7cc": "P\\,|\\!\\!\\!\\sim_L B", "79c16abcd55c3e5239dc437eea89102e": "\\Gamma = {V_r \\over V_f}.", "79c196695eea789f3401c14397c796e8": "\nu^{-1}_{+1}(\\mathbf{-p}) = u^{+1}_{-1}(\\mathbf{p}).\n", "79c1f4ba651ad17ab0701d18f40b8ae9": "\\arcsin x = \\sum^{\\infty}_{n=0} \\frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\\quad\\text{ for }|x| \\le 1\\!", "79c207a2f9b345fc158acc182a3d6605": "f(q, E) = \\sum_{n=1}^\\infty a_n q^n.", "79c22529077a0271d79cc0f5b48f8052": "lastBlock \\leftarrow 1", "79c23baecb00e28ecec20e6a5a76ef7b": "\\sqrt{7+\\sqrt{6+\\sqrt{5}}} = 3.1416^+", "79c27fddb6439d94014b64eb21ad2066": "A \\wedge B = \\{ x\\in L_A\\cap L_B\\mid \\not\\exists y\\in L_A\\cap L_B\\mbox{ s.t. }x < y\\}.", "79c2bc718e15527525f2bc4cd21034d3": "N=D", "79c2c8a18e7aeadd8b226bf3ebeaec40": "F_{\\mu\\nu}=(\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu)", "79c2d25bfd20d8a67af9e59f602b928d": " \\left ( \\frac{-\\Delta\\ G'}{RT} \\right ) ", "79c32f6cf713d63853a6d8b1ad2480ee": "\ny(t)=\\begin{cases}\n(at/3)^{3} & t<0\\\\\n0 & t\\ge0\n\\end{cases}\n", "79c3890f21ff36d86216e2d39b40ccc9": "\\Delta = H - L ", "79c3a6de7cb71aee88742e81c26d3180": "f(rs) = \\begin{pmatrix}\n rs & 0 \\\\\n 0 & rs\n\\end{pmatrix} = \\begin{pmatrix}\n r & 0 \\\\\n 0 & r\n\\end{pmatrix} \\begin{pmatrix}\n s & 0 \\\\\n 0 & s\n\\end{pmatrix} = f(r)\\,f(s).", "79c3a83e2be1b67145ffbb8d837de920": "|f| = \\alpha|g|^{-p/(p-1)}\\,", "79c3bfb094a57fcc4cc23c874cfbdc28": "\\mathcal{A} = \\{ ((x_n),(\\phi_n)) : x_n\\in X, (\\phi_n)\\in \\Delta, \\phi_m x_n =\\phi_n x_m \\text{ for all } m, n\\in \\mathbb{N} \\}", "79c3c74830537454e043749ecb26de00": "\\left (\\frac{V_1}{V_2} \\right )^{\\gamma}", "79c40dd2b43f3c03eaf88b5fc4c199b8": "\\chi ", "79c412ee186bf0008c2b5634ad39772a": " S_i \\, ", "79c442d893d325da2fc4de9ec1bfedfd": "\\ |x[n]|", "79c457bd9cb61320d43c2b19ab95fd3d": " \\rho = \\rho_c \\theta_n^n ", "79c4d07d6233e47e29ed597456107ed9": "y_T", "79c4e13f2767b907c961d4c0cca8d38d": "\\left [\\begin{smallmatrix}\n\\cos 2\\pi/p & \\sin 2\\pi/p \\\\\n\\sin 2\\pi/p & -\\cos 2\\pi/p \\\\\n\\end{smallmatrix}\\right ]\n", "79c4fc87fc18033d3e34d32efaef8ae0": "P(x) = P_\\ell(x) + |x|^2P_{\\ell-2} + \\cdots + \\begin{cases} \n|x|^\\ell P_0 & \\ell \\rm{\\ even}\\\\\n|x|^{\\ell-1} P_1(x) & \\ell\\rm{\\ odd}\n\\end{cases}", "79c5196c7fc61d3cb0ff51e140452224": "\n\\frac{\\partial S}{\\partial t} = p_{t} = \\frac{ac^{2}}{b}\n", "79c59b18308aa40a5c10018abdfa7dd2": " a = \\frac{n}{\\gcd(k,n)}.", "79c5c4310f0d0c2aa0c980e87fc4f705": "\n \\boldsymbol{F} = \\boldsymbol{\\nabla}_{\\mathbf{X}}\\mathbf{x} = \\frac{\\partial \\mathbf{x}}{\\partial \\xi^i}\\otimes\\mathbf{G}^i = \\mathbf{g}_i\\otimes\\mathbf{G}^i\n", "79c5d02953a250087f3b71320d5a707a": "s\\Vdash p", "79c620742815b16b51d2def5f84d5025": "\\Lambda\\,", "79c657b7ed103fd700da7c71e77bdf8b": "\\{y|\\Phi(y)\\}", "79c67b455bdfffc81142974e9232ac5a": "\\mathbf{\\rho}_{i},\\mathbf{\\rho}_{j}\\in\\mathit{S} \\big(i\\ne j\\big)", "79c6a53000cb3e54cba86190f65dfe7f": "x x y^{-1} z y z z z x^{-1} x^{-1} \\,", "79c6da56bca67c4b9dd63f08acb43b14": "\\begin{align}\n& \\operatorname{E}[1-X] = \\frac{\\beta}{\\alpha + \\beta } \\\\\n& \\operatorname{E}[X (1-X)] =\\operatorname{E}[(1-X)X ] =\\frac{\\alpha \\beta}{(\\alpha + \\beta)(\\alpha +\\beta + 1) } \n\\end{align}", "79c72e47fb2eec74b05478e3b8f6a6d7": "\\,\\!\\lambda_j", "79c77e38be97c380ead5a85d1ff3459b": " t_1 < t_2 < \\dots < t_N", "79c78f88c103424366d517be212681fb": "a_{i,j}", "79c8621e8b94d42e80bafac8c5798ee0": "(A[1])^n = A^{n + 1}", "79c875c45998073f801f4b57b85f1619": "\\alpha, ", "79c9131320dfee87aaf4fb092edaaf9f": "\\gamma^{\\prime}_{\\rm p} = \\frac{K_{\\rm J-90} R_{\\rm K-90}}{K_{\\rm J} R_{\\rm K}} \\Gamma^{\\prime}_{\\rm p-90}({\\rm hi}) = \\frac{K_{\\rm J-90} R_{\\rm K-90} e}{2} \\Gamma^{\\prime}_{\\rm p-90}({\\rm hi})", "79c9677bd13b17f15d81037bb6db9ceb": " \\Phi^{oxidation} = \\frac {1} {t_c} \\int_0^t exp \\left[-\\frac {1} {2} \\left(\\frac{(\\dot{\\epsilon_{th}} / \\dot{\\epsilon_m}) + 1} {\\dot{\\zeta}^{oxidation}} \\right)^2 \\right] dt", "79c9cf97ce1aa29850e54c043639743a": "\\textbf{d}_j", "79ca0256fc42694836516bce8ab6037c": "\\scriptstyle \\frac {a x + b} {c x + d} ", "79ca47eec142457f27c74cde8b1c870f": "\\Theta( \\cdot )", "79ca54779342062cadf44828f8f13dc1": "A \\wedge b \\wedge A", "79ca65714cb3e6bfce275dbce9e3ae58": "\\{ e_i \\ | \\ i \\in \\mathbb{N} \\}", "79cac5dfc8b2346be160edbd64faec33": "Y(z_1,z_2) = \\sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q}X(z_1,z_2)", "79caf5b99e44f1180a8cacff8cc98fc0": "x\\cup \\mathop{\\rm tc}(x)\\subseteq \\{y|\\Phi(y)\\}", "79cb9af4c4a43c9b16453e19d01e05ac": "\n\\begin{align}\ny[n] = O_n\\{x\\}\n&= O_n\\left\\{\\sum_{k=-\\infty}^{\\infty} x[k]\\cdot \\delta[m-k];\\ m \\right\\}\\\\\n&= \\sum_{k=-\\infty}^{\\infty} x[k]\\cdot O_n\\{\\delta[m-k];\\ m\\},\\,\n\\end{align}\n", "79ccc2a06888dc062cb4b4d6891fd091": "\\rho_0\\frac{\\partial u}{\\partial t} + \\frac{\\partial p}{\\partial x} = 0", "79ccd48be82b106a0a4a972f78f51fc4": "0 \\;\\xrightarrow{}\\; A \\;\\xrightarrow{f}\\; B \\;\\xrightarrow{g}\\; C \\;\\xrightarrow{}\\; 0", "79cd5a96c936ea209883054b7a7f10be": " \\partial_v ", "79cd7888607e227d88f85c8d53f8185a": " N \\left ( ka \\right ) = I_1 \\left ( ka \\right ) \\Delta_1 - \\left \\{ ka I_0 \\left ( ka \\right ) - I_1 \\left ( ka \\right ) \\right \\} \\Delta_2 ", "79cd9b72549fbd8e59f3960967698f4d": "U^{(i)}", "79cd9b899412beb7df76bff65e9ba8d6": " V_D(R) ", "79cdbc4e8fec4ef1fcf7f81982ef5e65": "x_1=a_1", "79cdd7a2aeaeebf3394fa42015228c4b": "f(T,c)", "79cde883e94820e4e1ec9dbe2576846f": "Z(L) := \\{x \\in \\mathfrak{g} | [x, \\mathfrak{g}] = 0 \\}", "79ce095d93e44ef2cbbe34f845e25154": "\\psi(x) = \\int \\frac{d^{3}p}{(2\\pi)^{3}} \\frac{1}{\\sqrt{2E_{p}}}\\sum_{s} \\left(\na^{s}_{\\textbf{p}}u^{s}(p)e^{-ip \\cdot x}+b^{s \\dagger}_{\\textbf{p}}v^{s}(p)e^{ip \\cdot x}\\right).\\,", "79ce436ef39915208bed688221ad9cbc": " = E[X(t) X(t+\\tau)] - \\mu^2\\,", "79cee56ee8353906d0650e67626a8176": "u_i^*=\\bar{u_i}", "79cf057464ee5973ca65e002d57f323e": "\\begin{matrix}\n&&&&&2&&&&\\\\\n&&&&2&&2&&&\\\\\n&&&2&&1&&2&&\\\\\n&&2&&0&&0&&2&\\\\\n&2&&6&&5&&6&&2\\\\\n&&&&&\\vdots&&&&\\\\\n\\end{matrix}", "79d01bee367686fe9ea0f785793971c6": "\n\\Omega_{\\theta\\phi}=\\partial_\\theta\\mathcal{A}_\\phi-\\partial_\\phi\\mathcal A_\\theta={1\\over 2}\\sin\\theta.\n", "79d05ec17f964f3bff557467dc682576": "y(0)=y_0", "79d07c927f8a1ca2c6b4f16de65e13d0": "\\theta=\\pi/2 ", "79d082a84759adb16be8f4b01d4e15d1": "d\\colon \\Omega^k(P,V)\\rightarrow \\Omega^{k+1}(P,V)\\,", "79d0ea28287ceccbc54ef391e2131509": "\\langle -, -\\rangle", "79d1108b5850e9da812503c3a0134b2b": "w_3", "79d144fe1ce099483afce640f7e5b68c": "M(a,b,z)=\\sum_{n=0}^\\infty \\frac {a^{(n)} z^n} {b^{(n)} n!}={}_1F_1(a;b;z)", "79d14658a20202b8655247845a3f2a5f": "\\lambda f.f\\ ((x\\ f)\\ (x\\ f))[\\lambda f.f\\ ((x\\ f)\\ (x\\ f)) := q\\ x] ", "79d154023c619070fd2fbbd3d675e60a": "y\\in P", "79d1589659df04fea5fc09e4bd9e3d1e": "\\operatorname{Var}\\left[f(X)\\right]\\approx \\left(f'(\\operatorname{E}\\left[X\\right])\\right)^2\\operatorname{Var}\\left[X\\right]", "79d17483ff0f1101a8600a9799519761": "\\boldsymbol{\\nabla}\\cdot\\boldsymbol{S} \\neq \\operatorname{div}\\boldsymbol{S} = \\boldsymbol{\\nabla}\\cdot\\boldsymbol{S}^\\mathrm{T}.", "79d175b64da0c72733d18bb5eaf8e9eb": "\\overline{\\widetilde{c}_i}=\\widetilde{c}_i", "79d19cf790138f4edf7391c00839cb23": "\\langle G, R, v \\rangle", "79d237dabc255620c626ad30b4ae08c7": " \\begin{align}\n & \\operatorname{minimize}_{x \\in \\mathbb{R}^N} & & f_1(x) +f_2(x) + ... + f_{n-1}(x) +f_n(x) \n\\end{align}", "79d284fa5f72978671dbe338103ed3a4": "\\langle x_{n} - x, x_{n} - x\\rangle = \\langle x_{n} , x_{n} \\rangle - \\langle x_{n} , x\\rangle - \\langle x, x_{n} \\rangle + \\langle x, x\\rangle, ", "79d29480a2e21e6bae10935c6777c4cd": "\\scriptstyle\\pi^0 \\rightarrow \\gamma + \\gamma ", "79d33ac7faa0ebecefaa92fb6cef8e0e": " \n a+2b=0. \\,", "79d3627f79814f672a5abaf5de46aaa3": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 1\\end{pmatrix} ", "79d3b6e208ddf6d1b9706b88a15442bd": "{Q} {x} = {A}^{T} {\\lambda} - {c}\\,", "79d3bd131ff7f41fe77f584deb6f9141": "\\delta ^{13}C_\\text{Sample} = \\left(\\frac{^{13}C/^{12}C_\\text{Sample}}{^{13}C/^{12}C_\\mathrm{PDB}} - 1\\right) \\cdot 1000", "79d4121f90fb3d8a0c44d800b706b683": "\\operatorname{ess\\,supp}(f) := X \\setminus\\bigcup \\left\\{\\Omega\\subset X \\,|\\, \\Omega\\,\\text{is open and}\\, f = 0\\, \\mu\\text{-almost everywhere in}\\, \\Omega \\right\\}", "79d443e7dd9e848ea13ea8c157c694b1": "x_iy_i\\geq 0", "79d468633e7c71d467eba339c8cdffb5": "f^{-1}(K)", "79d46c6d2583369c6d9d869c1cf5b7db": "\\,!(A \\& B)\\equiv \\,!A \\otimes \\,!B", "79d4855869c79606873fa63d6a39771e": "\n\\int (d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{e\\,m(d+e\\,x)^{m-1} \\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{2c (p+1) (2p+1)}\\,+\\,\n \\frac{(d+e\\,x)^m(b+2 c\\,x)\\left(a+b\\,x+c\\,x^2\\right)^p}{2c (2p+1)}\\,+\\,\n \\frac{e^2m(m-1)}{2c (p+1) (2p+1)} \\int (d+e\\,x)^{m-2} \\left(a+b\\,x+c\\,x^2\\right)^{p+1}dx\n", "79d4b6061b3fab0327c31361d7edfafc": "\\exp(t+s)X = (\\exp tX)(\\exp sX)\\,", "79d5112b7c852f59d9a54180bd90f549": "w^{(0)}", "79d5265e06e74b70dbddcae7734ea9df": "\\omega(X,Y) = g(JX,Y), \\, ", "79d53734be879f97fab2a2a3ed306e2f": "\\mathcal{C}\\in\\mathbb{Z}[\\mathfrak{A}]", "79d58427729ba8a8f86f6afd655c17db": " var[ \\log( t_a / t_b ) ] = var[ \\log( t_a ) ] + var[ \\log( t_b )] - 2r ( var[ \\log( t_a ) ] var[ \\log( t_b ) ] )^{0.5} ", "79d5f238dfde0bea083e4843a5f31477": "\\omega\\left ( t-\\frac{\\hat{n}\\cdot \\vec{r}(t)}{c} \\right )\n=\n\\omega\\left [ t-\\frac{\\rho}{c}\\sin\\left ( \\frac{\\beta c t}{\\rho} \\right )\\cos\\theta \\right ]", "79d61975257fadccddb45567af57f054": "\\tilde{c}(\\Pi)=(-1)^{k-l}c(\\Pi)", "79d679d77d8f929bbc8b866fd7bfaf9d": "f \\colon G \\to {\\mathbb{R}} \\cup \\{ - \\infty \\}", "79d72a986f1bf8f7bde1a70598352d57": "T_{ff}=\\frac{\\pi}{2\\sqrt{2}}\\sqrt{r^3\\over{\\mu}}", "79d7b0e0cd5d55afdae3ee64e1b09902": " \\mathbf{A}(\\mathbf{r},t) = \\dfrac{1}{4\\pi \\varepsilon_0} \\nabla\\times\\int \\mathrm{d}^3\\mathbf{r'} \\int_0^{|\\mathbf{r}-\\mathbf{r}'|/c} \\mathrm{d}t_r \\dfrac{ t_r \\mathbf{J}(\\mathbf{r'}, t-t_r)}{|\\mathbf{r}-\\mathbf{r}'|^3}\\times (\\mathbf{r}-\\mathbf{r}') \\,.", "79d8846754cd7d7078db7eaae84d72b2": "\\Box \\Box p", "79d8b96e3e8c522273afa3a61d0dd233": "2 \\Sigma = RD + DR'", "79d98e4957e2b53f4b776dc777f2505e": "(\\dot{V}_{O_{2}})", "79d9a49461445d1a0232df823c191f67": "K_c = \\sqrt{E G_c}\\,", "79d9d197e6e0b6167dc406541bea3bdf": "3^2+4^2 = 5^2", "79da2721e844be8e24961709e193c9d8": "(1)\\Leftrightarrow(5)", "79da382424538237e0586e411662c9d4": "\\bigcap{}_{i=1}^n", "79dad68f1d5d15e0b8232d1f2f2adcfb": "\\langle B_E u | v \\rangle_E = (u|v)_E = (Bu|v) = \\langle u|B|v\\rangle", "79daecc86a62a0fcc28410e1ec004894": "\nP_0=\\frac{$120,000}{360}=$333.33\n", "79daf97a77b11c5efeb2c1a6984cfdfe": "n_{k\\ell}=k^{-3}\\ell^{-3}", "79db021139157e984bd7e84f29f0263f": "\\textstyle \\mathbf{R}_{j}, j = 1, \\, \\ldots, \\, N", "79db5415166f1d4c2ce1cb171d84be84": "\\mathbf x_{k\\alpha}", "79db5c2e844e66d082d536f44f9bea7c": "\\left( y' \\right)^m = f(x,y),\\,", "79db6ab8c187f3c28df241c909bc6d61": "e^{X}\\in H", "79db8ec9f58aacdbaf4dec19a59889d1": "G(T,P,N)=NkT\\left(\\hat{c}_P-\\ln\\left(\\frac{kT^{\\hat{c}_P}}{P\\Phi}\\right)\\right)", "79dbcae0043dbdf16d587dcc1e91a889": "\\mathcal{V} = \\{g(y)| g:\\mathbb{R}^m \\rightarrow \\mathbb{R}, E\\{g(y)^2\\} < + \\infty \\}", "79dbfe520b85e755ff3b429500522751": " k_{act} = A \\cdot e^{- \\frac{ \\Delta {G_{in}^{\\ddagger}}+ \\Delta {G_{o}}^{\\ddagger}}{kT}}", "79dc317f921b635309d63377f4501f1a": "\\Sigma^\\hat{\\alpha} = \n\\begin{pmatrix}\n\\chi^{\\alpha} & \\bar{\\psi}_{\\dot{\\alpha}}\n\\end{pmatrix}\n", "79dc348b6735cc1766b09560fbe3e014": "\\nu >4\\,", "79dc9f5e29d93b99f3a3800144b88fc7": "\n \\cfrac{d}{dt}\\left(\\int_\\Omega \\rho~\\eta~\\text{dV}\\right) \\ge\n \\int_{\\partial \\Omega} \\rho~\\eta~(u_n - \\mathbf{v}\\cdot\\mathbf{n})~\\text{dA} - \n \\int_{\\partial \\Omega} \\cfrac{\\mathbf{q}\\cdot\\mathbf{n}}{T}~\\text{dA} + \n \\int_\\Omega \\cfrac{\\rho~s}{T}~\\text{dV}.\n ", "79dd0b534c3a92204f1f7bc7b1918f5a": "\\gamma_i(g) = \\delta_{ij} g", "79dd847bd167a0b0c73586bf89cf1fc0": "(\\;1)\\quad \\quad\\frac{\\partial\\rho}{\\partial t} \\;\\; = -\\frac{\\partial}{\\partial x}\\left(\\rho u\\right)", "79dd9720ffa5bbe026e23afc9ab4df3c": "m\\,", "79ddd3a864c94b6ad4cbbbd60591bbc4": " \\angle ZHA = 180^\\circ - \\angle OHA ", "79de23ad4dc88863f59a2798d28635da": "b_i(a_{-i})=\\arg\\max_{a_i\\in A_i} \\Phi(a_i,a_{-i})", "79de7cb892122aac806ebabe792c9d70": "x, y \\in V", "79de9e4250c27ac9645f507629792a53": "\\sum_{n=0}^{\\infty}u_{n}=\\sum_{n=0}^{\\infty}\\frac{p(n)}{q(n)}", "79defd5ec1189f7ef52ca085502118fb": "S= \\int_{\\mathbf{A}}^{\\mathbf{B}} n \\, ds= \\int_{\\mathbf{A}}^{\\mathbf{B}} L \\, dx_3", "79df114fc938ad7b7330df042d732b9e": "[T^i_j,S^k]=\\delta^k_j S^i", "79df12fe5cae03ed9b3c7c0670fa3cc8": "f(x,y) = -0.0001 \\left( \\left| \\sin \\left(x\\right) \\sin \\left(y\\right) \\exp \\left( \\left|100 - \\frac{\\sqrt{x^{2} + y^{2}}}{\\pi} \\right|\\right)\\right| + 1 \\right)^{0.1}.\\quad", "79df2aa649e197bd959e5c7d57ea0ab9": "D_{ac} ", "79df3e540cb900671a01045cc93c63b7": "\\epsilon_v\\approx \\ell^2\\pi n\n\\frac{\\hbar^2}{m}\\ln\\left(\\frac{b}{\\xi}\\right)", "79df3ef2159ed0e0efaf7b2aac97ad57": "F_{i_1} \\cap \\dots \\cap F_{i_k}", "79df446c254400ede75fd8a118640d69": "\\Pi_2 = P(q_1+q_2) \\cdot q_2 - C_2(q_2)", "79df7d792c55f9d45d9b4170335b4a16": "V=\\frac{1}{2}(3+2\\sqrt{2}).", "79dfdf1915c12577f711d0b53d448c34": "*[*F,G]^{IJ} = - *[G,*F]^{IJ} = + [G,F]^{IJ} = - [F , G]^{IJ} .", "79dffd1878603c19715a62ed6bb4d26e": "\\sqrt{\\frac{3}{20}}\\!\\,", "79e0237353a337724302d255f5746d9c": "\nN(\\lambda,h)\\sim (2\\pi h)^{-d} \\omega_d \\int _{\\{ |\\xi|^2 + V(x)<\\lambda \\}} dx d\\xi \n", "79e030fb6e996592b2a0e8152d210393": "B_{OFDM}", "79e04807d9ae8e04b44f168284c00fd3": "\\lim_{n\\to\\infty}\\frac{1}{n}\\chi\\left(\\Psi^{\\otimes n}\\right)", "79e06320a9695b4bbe614c912436920a": "Y_t(u) - Y_c(u)", "79e085e82eff5eeddcbde917f643e961": "\n V_{\\mathbf{k}-\\mathbf{k'}}^{\\mathrm{eff}} \\equiv (1 - f^\\mathrm{e}_{\\mathbf{k}} -f^\\mathrm{h}_{\\mathbf{k}}) V_{\\mathbf{k}-\\mathbf{k'}}\n\\,,\n", "79e0a455b9c506a7d8c70f30dad420c6": "b_{2} = b_{2}- 4b_{1}= \\begin{bmatrix}4\\\\5\\\\4\\end{bmatrix}- \\begin{bmatrix}4\\\\4\\\\4\\end{bmatrix}=\\begin{bmatrix}0\\\\1\\\\0\\end{bmatrix} ", "79e0a4cefb011fb17b95005608ff6d88": "O(KN(M + N\\log N))", "79e0ac9244dec49cd99b59859004e802": "g_{ij} = \\mathbf{e}_i\\cdot\\mathbf{e}_j .", "79e0c8c861f2d68f15be7cb1e1515846": "d_{A[1]}^n := - d_A^{n+1}", "79e1143801a7856fb2163211d14ce258": "\\frac{e^{it}-1}{i}\\mbox{ , }i = 0\\,", "79e124413160985260468b4744f3960a": "(q;q)_n", "79e14885367693dc018d92876e399c9f": "x\\preceq y", "79e1b0e42ffc0a7574f705548a357289": "G = G_{0} + \\frac {1} {\\beta} \\frac {-T} {1 +T} ", "79e1b2bea806fddebdd6c83660d63e5b": "p\\times M", "79e1ee9852a2e52e09d234cede5dc1f6": "-\\hat{\\alpha}'_i", "79e2511cb29ea486b514e3179982cee1": "\\displaystyle{U(f_- \\oplus f_+) =(\\partial_{z} f_-)\\chi_\\Omega + (\\partial_{z} f_-)\\chi_{\\Omega^c}.}", "79e26971c6350c13d045a06b955c289f": "\\begin{align}\n\\sum_{n=1}^\\infty \\frac{1}{n(n+1)} & {} = \\sum_{n=1}^\\infty \\left( \\frac{1}{n} - \\frac{1}{n+1} \\right) \\\\\n& {} = \\lim_{N\\to\\infty} \\sum_{n=1}^N \\left( \\frac{1}{n} - \\frac{1}{n+1} \\right) \\\\\n& {} = \\lim_{N\\to\\infty} \\left \\lbrack {\\left(1 - \\frac{1}{2}\\right) + \\left(\\frac{1}{2} - \\frac{1}{3}\\right) + \\cdots + \\left(\\frac{1}{N} - \\frac{1}{N+1}\\right) } \\right \\rbrack \\\\\n& {} = \\lim_{N\\to\\infty} \\left \\lbrack { 1 + \\left(- \\frac{1}{2} + \\frac{1}{2}\\right) + \\left( - \\frac{1}{3} + \\frac{1}{3}\\right) + \\cdots + \\left( - \\frac{1}{N} + \\frac{1}{N}\\right) - \\frac{1}{N+1} } \\right \\rbrack = 1.\n\\end{align}", "79e26ee036e8e298cc22d10089d9d8d5": "-\\log_2(P(H \\and E)) = -\\log_2(P(H)) + -\\log_2(P(E|H))", "79e275b5805dfb588b3330bb33625dfb": "PQ = \\sin \\beta\\,", "79e2905ea0b6cccfad671cbb12808a97": "\\nu_s = R \\left(\\frac{Z_{3p}^2}{3^2} - \\frac{Z_{ns}^2}{n^2}\\right) n=4,5,6,...", "79e29f969c327723837895ba9fabb4e5": " 2 \\times 2", "79e2a14300567f34f9ebdf65e866c8f8": "\\simeq H^1 (X, \\mathcal L (D))^\\vee", "79e2fefeb3bcc099d0f95de3dadb6076": "d,e,n\\in\\mathbb{Z}^+", "79e3145a0fb5a952c472d9c61bc9fc36": "i \\in I ", "79e31c5170c731dfa4a03c2c722100d8": "\\begin{align}\n\\sigma_2 &= \\sigma_\\mathrm{avg}-R \\\\\n&= -30 \\textrm{ MPa} \\\\\n\\end{align}", "79e42af4c55378abb91a50b9720c03eb": "\\begin{matrix}{13 \\choose 5} - 1,208 = 79\\end{matrix}", "79e4ad4f42297fcb26e4fb047f5c0ab3": " \\phi_{mg}(u,\\theta)=\\textrm{minmod}\\left(\\theta\\frac{u_{i}-u_{i-1}}{\\Delta x},\\;\\frac{u_{i+1}-u_{i-1}}{2\\Delta x},\\;\\theta\\frac{u_{i+1}-u_{i}}{\\Delta x}\\right),\\quad\\theta\\in\\left[1,2\\right], ", "79e4c26eb7f8d08cc94bbda164145969": "ed = 1\\bmod \\varphi (N)", "79e50ec061d923415bd966e7b41faa25": "e_a \\colon \\bigl(P \\in \\operatorname{Spec}(A)\\bigr) \\mapsto \\left(\\frac{a \\; \\bmod P}{1} \\in \\operatorname{Frac}(A/P)\\right)", "79e5c4719b671a25957b79bed96929b6": "\\textstyle\\mathbf{V}", "79e66d009750f4831508478a6ec6c6e2": "\\int_{\\mathbb{R}^{d}} \\prod_{j = 1}^{d} g_{j} ( \\pi_{j} (x) ) \\, \\mathrm{d} x \\leq \\prod_{j = 1}^{d} \\| g_{j} \\|_{L^{d - 1} (\\mathbb{R}^{d - 1})}.", "79e676e9558e8b797ca0bce11ac5e419": "2^{A}", "79e6b252d5e6019a9efd3827f7c7d890": "S(P_0) = - P_0\\,", "79e6c884faf89c063888c2106e90ac29": "\\sum_{n}^1Em_i", "79e6e5fe2f96a62514fb68f0221ba11a": "\\mathrm{div}(F)=\\mathrm{div}(G)-\\mathrm{div}(H)", "79e745ce8ba9a1e71f77f83784102b3c": " L^\\%(M)\\to L(M)", "79e7cd271631a4dc78c85758ce40844f": "D^{(r)} = \\frac{1}{r!} \\left(\\frac{\\mathrm{d}}{\\mathrm{d}X}\\right)^r \\ . ", "79e7e56b0d01cdadc4259136b72fe576": "g(w)=t", "79e81d3e6429ef015ce2e406a191ac33": "(a+c) = hm'", "79e8238f6fc9767bf4fdaf7253abca25": "G(z, u) = \\exp \\left( u \\log \\frac{1}{1-z} \\right) =\n\\left(\\frac{1}{1-z} \\right)^u =\n\\sum_{n=0}^\\infty \\sum_{k=0}^n \n\\left|\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right]\\right| u^k \\, \\frac{z^n}{n!}.\n", "79e8271740808e73b1011e62c7a0fee7": " \\Phi(e,x)= [[e]](x) ", "79e9001bd3dd5421a9e7b20553b0d18b": "a_i=\\sum_{j=1}^Cn_{ij}", "79e93289242cbc1e305b563ab4008493": "\\begin{align} \n a_1 &= \\frac{(c + \\alpha)(c + \\beta)}{(c + 1)(c + \\gamma)} a_0 \\\\ \n a_2 &= \\frac{(c + \\alpha + 1)(c + \\beta + 1)}{(c + 2)(c + \\gamma + 1)} a_1 = \\frac{(c + \\alpha + 1)(c + \\alpha)(c + \\beta)(c + \\beta + 1)}{(c + 2)(c + 1)(c + \\gamma)(c + \\gamma + 1)} a_0\n = \\frac{(c + \\alpha)_2 (c + \\beta)_2}{(c + 1)_2 (c + \\gamma)_2} a_0 \\\\ \n a_3 &= \\frac{(c + \\alpha + 2)(c + \\beta + 2)}{(c + 3)(c + \\gamma + 2)} a_2 = \\frac{(c + \\alpha)_2 (c + \\alpha + 2)(c + \\beta )_2 (c + \\beta + 2)}{(c + 1)_2 (c + 3)(c + \\gamma)_2 (c + \\gamma + 2)} a_0 = \\frac{(c + \\alpha)_3 (c + \\beta)_3}{(c + 1)_3 (c + \\gamma)_3} a_0\n\\end{align}", "79e9481e518f7e42b00028b00420161f": "w/2", "79e95b4f1931f9fe3ef84e43c7cd17bd": "(\\mathcal{F}f)(\\xi):=\\int_{\\mathbb{R}^n} e^{-2\\pi iy\\cdot\\xi} \\, f(y)\\,dy.", "79e9e40e923616dc6b324a5c9e312e29": "\\delta_t = \\frac{\\delta_x}{C_s}\\,\\!", "79e9f1ab5713e9540c08d5a71a7f625b": "\\scriptstyle \\frac{1}{\\sqrt{2}} = 0.70710678...", "79ea0247ef6f246121dba5a151e3e26e": "h_1^{-1}g_1\\,=\\,1", "79ea59d46a8abacd9b2e15ec2686e706": "C_{AB} = \\frac{M_B}{M_A}", "79ea685f5101d6bfc0255b41937baf0e": " \\frac{d^2}{dx^2}U=k", "79ea9bb6b30843bbdcb032c6bfca3d9e": "d_0 .. d_m", "79eb860e6e9e7867f1bb77b94ef265f1": "v(\\mathbf r,t)+c(t)\\rightarrow e^{-ic(t)}|\\Psi(t)\\rangle\\rightarrow\\rho(\\mathbf r,t).", "79eba83cb1dcf5f7dd0aabc024d5d200": "\\mathbf{F}=q_{\\mathrm e}\\left(\\mathbf{E}+\\frac{\\mathbf{v}}{c}\\times\\mathbf{B}\\right) + q_{\\mathrm m}\\left(\\mathbf{B}-\\frac{\\mathbf{v}}{c}\\times\\mathbf{E}\\right)", "79ebd5133893904c3af90bd083dc4598": "x^{\\mathrm{lcm}(k_1, \\dots, k_m)/k_l}", "79ec10d6ba1ba0aeee243f0dae22fd5e": " g (x_0)\\equiv 0 \\bmod \\prod N_i ", "79ec3c9e8a401542d52d2d2ba0b7cbed": " k\\ \\lambda u.u ", "79ecb1ea9046eac4225de4902dd5d2fd": "m_{\\mathrm{eff}} = \\int_m\\frac{1}{2}u^2\\,dm", "79ed9212f3194cf72833e2fc01cbccf9": "1 \\; \\mathrm{C} \\text{ corresponds to } 3.7673 \\times 10^{10} \\; \\mathrm{statC}", "79ee5295d0c48af2544a50d4e042190a": "(~ | z_k| < 1 ~) ", "79eeb83b574cd2bec66e7cb1d449fc95": "\\displaystyle F(x;\\theta)", "79eebae50c06104a470c91b62595d946": "d_1 = \\frac{\\ln(F/K) + 0.5 \\sigma^2t}{\\sigma\\sqrt{t}}", "79eeeef543629b4a7b96b8fac28d9472": "E_m = \\sum_{i=l}^{l - \\left ( \\beta\\, - 1 \\right )} \\log_2 \\left (i \\right ) : \\beta\\, \\ge \\, 1, l \\ge \\, 1", "79ef726f3f5d115b10c208d7a9480195": "\\ldots,\\;w-4\\pi i, \\;w-2\\pi i, \\;w, \\;w + 2\\pi i, \\;w+4\\pi i, \\;\\ldots,", "79ef9991eaefc82394db21c121aeb025": "\\bar{h} (s,i;L)", "79f03a199fb631d79e2f89a3b39df758": " \\tilde{S}_F(p) = {(\\gamma^\\mu p_\\mu + m) \\over p^2 - m^2 + i \\epsilon} ", "79f06b33c19fd335d702ef628120851d": "\\mathcal N", "79f0ec3bc98448b83225a0cfd4eca47d": "\n\\cfrac{dN}{dt}=r_1 N\\left(1-\\cfrac{N}{K1}+\\beta_{12}\\cfrac{M}{K_1}\\right)\n", "79f1058993004340818eabeba4fba772": "(\\hat\\alpha + \\alpha_{LO})", "79f11f7c689a415bfaa6954f934630ca": "\\textstyle(x, y\\pm1)", "79f168a842505b90582ff1f6ad3a638c": "\\textrm{HDI} = \\sqrt[3]{\\textrm{LEI}\\cdot \\textrm{EI} \\cdot \\textrm{II}}.", "79f1a2d55c3aed6aba996116abe73bd1": " \\mathbf{x} \\wedge \\mathbf{B} = 0.", "79f1c488476b3d550452691b1e2e78f1": " \\and (S_7 \\implies (\\operatorname{equate}[A_7, p] \\and V[o] = p)) \\and D[o] = D[p] ", "79f1c98b23457cc405d841ecf13fce34": "dG= - SdT + Vdp \\,\\!", "79f1ce125a8b2744c8b88abce6835659": "V_c = \\sqrt{\\frac{2\\sigma}{\\rho}}", "79f2113926aec993332c13b59b3e5bae": "f'(P) \\cdot \\frac{P}{Q}", "79f2205401e608db5342a0e113a84e9b": "\\frac{\\sin A}{\\sinh a} = \\frac{\\sin B}{\\sinh b} = \\frac{\\sin C}{\\sinh c} \\,.", "79f24dc2822858a4ea8cce4ebc3881f1": "f(x) = x^3 - 2x + 2 \\!", "79f252adde594aed741c8e8ddeefff0f": "b_i=k_iI_i", "79f28f5795ad312b2db280542a8a2859": "\\operatorname{tr}=\\operatorname{det}'_I", "79f2b3017eb7d2f736d602feb7611e00": "F(\\R^m,n)", "79f2e54eac50d9f18f48f17d5cfca1a0": " \\sup_{z \\in S} e^{-k|\\mathrm{Im} z|} \\left| \\int (T_zf)g \\, \\mu_2 \\right| < \\infty", "79f30ebe21e551b113f897d59bc77bd6": "V=\\frac{\\lambda}{4!}\\phi^{4}+\\frac{\\lambda^2\\phi^4}{256\\pi^2}(\\ln{\\frac{\\phi^2}{M^2}}-\\frac{25}{6})", "79f30ffdec2249f3326febe89e1d6518": "T(n) = n + T \\left(\\frac{1}{2} n \\right)", "79f32742516efb9fa376e80488d53eed": " DTFT(f) = F_2(w)", "79f339e9c4d901e6a4ab611949e97ca4": "x \\to y\\;", "79f3553c79054dd434f8b77729a9bdc9": "1/ \\epsilon", "79f36c04a175e55e91540947d8751175": "\\mathbf{f}=(f_{e_1}(0),f_{e_2}(0),\\dots,f_{e_{d}}(0))^T , \\qquad \\mathbf{f}'=(f'_{e_1}(0),f'_{e_2}(0),\\dots,f'_{e_{d}}(0))^T.", "79f39556124db5b36a61d578d7048239": "\\mathbb{R}^m\\ ", "79f39912b28305e4ee2683896c47060e": "\\frac{2n}{\\widehat{\\lambda} \\chi^2_{1-\\frac{\\alpha}{2},2n}} < \\frac{1}{\\lambda} < \\frac{2n}{\\widehat{\\lambda} \\chi^2_{\\frac{\\alpha}{2},2n}}", "79f3d54a405a4eacc734c626b44cf014": "\\displaystyle \\sum_{k\\geq2} (k-3) t_k \\leq -3.\\,\\! ", "79f3e0666f2a62d53540f559c1b2afe4": "t(\\theta)", "79f3fd27530ae6ff30cc47b4487f2217": "T_{n}(z)", "79f41d0f757ca8b64a70ec53cfb8337d": "\\scriptstyle{R_2}", "79f4ee905b9c43b2d59648813f1c3845": "\\boldsymbol\\Omega", "79f530a36b5936960bbefaf3c1fee385": "r(A) = \\sup \\{ |\\lambda| : \\lambda \\in W(A) \\} = \\sup_{\\|x\\|=1} |\\langle Ax, x \\rangle|.", "79f56573b682eb4403f0b0c23ddbb5d5": "F(\\mathbf{x}) = \\tfrac{1}{2} G^\\mathrm{T}(\\mathbf{x}) G(\\mathbf{x}) ", "79f575f5d4bc2ede6f6c36e4da946ca9": "\\mathcal{E}(\\sigma)", "79f5a0d4287d853f97edcb36612aa411": " \\Delta Q = \\frac{ \\sum{\\scriptstyle\\text{head loss}_c} - \\sum{\\scriptstyle\\text{head loss}_{cc}}}{n \\cdot (\\sum\\frac{\\text{head loss}_c}{Q_c} + \\sum\\frac{\\text{head loss}_{cc}}{Q_{cc}})},", "79f5a7f1e72747c6b2fceeb53d4d1bdd": "f(x)=\\Omega_L(g(x))", "79f5c2d39a56b569235505b7a20ae681": "{P\\over S}={c\\varepsilon_\\circ\\over2}{E_\\theta}^2={1\\over 2} {{E_\\theta}^2\\over Z_\\circ}\\,\\!", "79f5cb96cef356362f52172054e9e623": "A \\mapsto {\\rm Tr} f(A)", "79f5cf9f28ee52f80bd373b57faa9ab4": "\\,M_A,\\ M_B", "79f5e2d2f2123e4a72c7357cb28aaa72": "\n \\begin{align}\n & M_{\\alpha\\beta,\\beta}-Q_\\alpha = 0 \\\\\n & Q_{\\alpha,\\alpha}+q = 0\n \\end{align}\n ", "79f629b0b038b48423011d92a9cd6257": "\\mathbf{r}_1 = (a/4)(\\hat{x} + \\hat{y} + \\hat{z})", "79f63f5cb309fd47878405bc53bb672e": "\nR_{m,n}(z) = \\frac{P_m(z)}{Q_n(z)} = \n\\frac{a_0 + a_1z + a_2z^2 + \\cdots + a_mz^m}{b_0 + b_1z + b_2z^2 + \\cdots + b_nz^n}\n", "79f651196f80868bcabde11ad660366d": "(X-1)^n \\equiv X^n - 1 \\pmod{n, X^r - 1}", "79f665b907fa2b7aa1c0b2fcab05ea61": " UDU^{-1} g= -g^{\\prime\\prime} + R g^{\\prime} + Q g,", "79f6748164f00225beae72e75b828086": "O(|R||S|)", "79f709b28f421e9c94c29d27b1e78551": "g \\in G_2", "79f7289ff8c3d2c183de0866f5349324": "{m1,m2,k}", "79f7f48725520664c38512cc6fed8086": "T_m = \\frac{\\Delta H^\\circ}{\\Delta S^\\circ+R\\ln([A]_{total} - [B]_{total}/2)}", "79f85abfab2857bbb18559176af3bc9f": "\\mathbf{R^n}", "79f87bc93284d3848268529e53f549bc": "\n\\Delta(\\mathbf{p}^{\\prime},\\mathbf{p}) =\n\\delta(\\mathbf{p}^{\\prime}-\\mathbf{p})\n\\sum_\\mathbf{k} \\left| F(\\mathbf{k}) \\right|^2\n\\left[ a^\\dagger(\\mathbf{p}/2-\\mathbf{k})\na(\\mathbf{p}/2-\\mathbf{k}) \\right. ", "79f8b3b190038f421cdd868ef9264643": "x'(t)", "79f93db6b28dc27596c8931dbfa39895": "f/8", "79f962733f3ce932f0d72619fb9e9631": "\\mathbf a = \\mathbf A + \\mathbf a'", "79f985f1694c75b81d18c616c4293026": "\\mathcal C \\, |\\psi\\rangle = \\eta_C \\, | \\bar{\\psi} \\rangle", "79f9b95f643781d9d2d041db98fc4c39": "C^1\\,", "79fa0bdef0873195086e144d0b1361d4": "\\bar{r} = \\bar{E} - \\bar{C} + \\gamma (\\bar{E}\\bar{C} + \\text{Cov}(E(t),C(t))) \\, ", "79fa10879e2bba30449d955992fa3120": "\\mu_i=-k_B T \\ln \\left (\\frac{\\mathbf{B}_i}{\\rho_i \\lambda^3} \\right ) ", "79fa2ea7128e9d8e28f1bd910822a6c5": "\\partial_\\mu {S^\\mu}_{\\nu\\lambda}=T_{\\lambda\\nu}-T_{\\nu\\lambda}.", "79fa58ab38c8b1437e673358dc145ee5": "\\displaystyle r=\\frac{\\sqrt{abcd}}{a+c}=\\frac{\\sqrt{abcd}}{b+d}.", "79fa6ef266ad9dc33c5024443e996cd0": "\\tfrac{5625}{243}", "79fa7bb6899e1b42663743f7c96f6c43": "|f(z)|", "79fa8dbf3d3c826d6142c206e2981b6b": "d\\boldsymbol{\\sigma} = 0 ", "79fad285432ea30662d32dfc768c1d08": "\nL(X^o|\\Theta) = \\sum_{i=1}^I \\log \\sum_{h_i} \\int p(X_i^o,x_i^m,h_i|\\Theta)dx_i^m\n", "79faf4f3df56ba5d645812ee1b3f2fc9": "t_{orbit} = \\frac{2 \\pi}{\\sqrt {G(M+m)}} \\left(\\frac{R}{2}\\right)^{3/2}=\\frac{\\pi R^{3/2}}{\\sqrt{2 G(M+m)}}.", "79fb51d0cf67d180fb09d227f7c0eec0": "He^{S}", "79fbab42fd7998323448fecc58d7c780": "H_{ext}", "79fc5e62e22abbe0399970e701c5122d": "y_\\text{DM} = \\int \\operatorname{Quantize}\\left( u - y_\\text{DM} \\right).\\,", "79fc7f403252ff5a5257ced31ecc365e": "d(|\\mathbf{X}|) =", "79fc980146c66fb97c7cfa9609ac5ffa": "\\operatorname E (R_b) \\leq \\frac{\\operatorname E (X^2)}{\\operatorname E(X)}.", "79fcbf1b24f2b0337b3856f9b3c8e79a": "{\\scriptstyle \\partial S}", "79fd1b95b1b704c09e7cc4c65e99e0b0": "\n \\varphi_1 \\ne w_{,1} ~;~~ \\varphi_2 \\ne w_{,2}\n ", "79fd87fc6e1b38c55ac9ea78596709f9": " = 2 \\eta^{\\rho \\sigma} \\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\nu \\right) - \\operatorname{tr} \\left( \\gamma^\\mu \\gamma^\\nu \\gamma^\\sigma \\gamma^\\rho \\right) \\quad \\quad (1) \\,", "79fdaf6f0b1a5809374b580090adb394": "\\scriptstyle \\{x:\\eta(x)=0\\} ", "79fddf1768109887ea77a114e07294c1": "\\delta x=\\frac{v_{Ar}}{\\omega}\\sin(\\omega t-kx)+\\frac{v_{Al}}{\\omega}\\sin(\\omega t+kx)", "79fde8a36e6800f787de00833e2b467e": "p(\\tilde{x}=1) = \\frac{\\alpha'}{\\alpha'+\\beta'}", "79fe7553e45b9019e0386086125b46bd": "C_{rs}", "79fec2da12263ab066d1de006e1ef729": "\\! h_m(x)", "79ff27d18378e867104cadb080e45f45": "3g-1", "79ff61aef4d08ff2bdfd1ddc221dbb5c": "f(x) \\ge g(x)", "79ff6bf45fbc5fe922155329b1d9a112": "3 Pmf = 1 - Pmf + \\tfrac{1}{3}Pmf + 2 Pmf", "79ff8d218efd008c40189744f28e167b": "k\\cdot b=33", "79ff9bbf381fcbaed1be859ac5f2a418": "\\begin{matrix} \\frac{7}{5} \\end{matrix}", "79ffa9e2a5e2499fe560b1fd9c21ce12": "\\beta - 1", "79ffaa339185dccc000a9571f1f35e6b": "I_\\text{sp}", "79ffc4abd5fe57827aefb8b0e1ee5989": "\\phi=\\pi", "7a00009b74b3d2f2ecc861ad761a2082": "A(x)>\\sqrt[3]{x\\log x}.", "7a002cbb3ffc3f950bbff469603c2593": "p[T]=\\{\\vec x\\in X^{\\omega} | (\\exists \\vec y\\in Y^{\\omega})\\langle \\vec x,\\vec y\\rangle \\in [T]\\}", "7a0048ea90d9265c30468d2a0b1799fa": "z = -\\frac{w}{\\gamma}+\\frac{1}{2},", "7a0055c37a49a19f3de0401432f7a0a6": "g(X) = 1", "7a0073afaeb4599b2d931d55a913fc59": "f * s", "7a01b01ca66693686f1ca40d4bb9188a": " r(t) = e^{-\\alpha t}r(0) + \\frac{\\theta}{\\alpha} \\left(1- e^{-\\alpha t}\\right) + \\sigma e^{-\\alpha t}\\int_0^t e^{\\alpha u}\\,dW(u)\\,\\!", "7a01bdd8c3e5e9b51315a084b8202e80": " \\sigma(cX) = |c| \\sigma(X). \\, ", "7a01c509d4fad1e2389f8b240100bcbc": "\\begin{align}\n &\\scriptstyle{ (Q_{xx}-M_{xx})^2 + (Q_{xy}-M_{xy})^2 } \\\\\n &\\scriptstyle{ {} + (Q_{yx}-M_{yx})^2 + (Q_{yy}-M_{yy})^2 } \\\\\n &\\scriptstyle{ {} + (Q_{xx}^2+Q_{yx}^2-1)Y_{xx} + (Q_{xy}^2+Q_{yy}^2-1)Y_{yy} } \\\\\n &\\scriptstyle{ {} + 2(Q_{xx} Q_{xy} + Q_{yx} Q_{yy})Y_{xy} . }\n\\end{align}", "7a01ca6e843c48f1786b3e76f20e7489": "U/\\mathrm{Sp}", "7a02142a752644fdef5e8d0f5363580f": "\\min(1000-0, 0-(-1), 1000-0)=1", "7a0240e684b0c431d090b7521273488a": "\n\\displaystyle y = \\nu^2\n", "7a024780aea7911e7978f9fea1015188": "curry(T)", "7a0298f50f11b99700b7bba917551472": "\\hat\\mu(T)=\\mu^*(T)=\\mu_*(T)", "7a02eb3ebb69ad1b7f678fd456bb59ba": " \\bar{X} = \\lim_{T \\to \\infty}\\frac{1}{T}\\int_{t_0}^{t_0+T} x\\, dt.", "7a0307907ec5bd8a19f173ce9cd8d9ee": "\\mathrm{CAGR}(t_0,t_n) = \\left( {V(t_n)/V(t_0)} \\right)^\\frac{1}{t_n-t_0} - 1 ", "7a0332a9c88742218c13c13e57814f69": "x/y/z=(x/y)/z\\qquad\\qquad\\quad\\mbox{for all }x,y,z\\in\\mathbb{R}\\mbox{ with }y\\ne0,z\\ne0.", "7a0339559fc4e5c2be304b490d0cf939": "\\frac{\\partial\\bar{e}}{\\partial V_r} = -\\frac {1}{V_0} \\hat{t}", "7a033ea6df48f3f1eeabb52527eb6c86": "H(f)(\\mathbf x)", "7a03c5d841c1ca3e3b407cf15d97ee62": "b\\mathbf{i} + c\\mathbf{j} + d\\mathbf{k}", "7a03cd07e8b6a1c8e1b485228e544ec0": "M \\subseteq M'", "7a03d309aa9469c25c30f6bae282b968": "t = q^2 \\approx (kd)^2 \\leq (d \\log n)^2", "7a0405b10aab6ff80caa1cb826174296": "p(x_1,x_2, \\ldots, x_n) \\, ", "7a046cf6c316021ee86af7291d9b8882": "f(x), \\tilde{f}(x) = V(x)f(x)", "7a0516bf355e1b1f129ee8001e03bbf2": "\\lambda(n)\\;", "7a054719e241ac571748676f38812029": "\\mbox{L} \\subseteq \\mbox{AL} = \\mbox{P} \\subseteq \\mbox{NP} \\subseteq \\mbox{PSPACE} \\subseteq \\mbox{EXPTIME}.", "7a05d1358cfdf729507eaeb72db8fff9": "\\widehat{T}^{(2)}_{\\pm 1} = \\frac{1}{\\sqrt{2}}\\left( \\widehat{a}_{\\pm 1} \\widehat{b}_0 + \\widehat{a}_0 \\widehat{b}_{\\pm 1} \\right)", "7a05ef94a9219a1ade384153ee4a790c": "\\cos\\frac{\\pi}{4}=\\cos 45^\\circ=\\frac{\\sqrt2}{2}=\\frac{1}{\\sqrt2}\\,", "7a05fe35399402d3c05e9733103369bc": "\\left(\\frac{1}{2},0\\right)\\otimes V\\oplus\\left(0,\\frac{1}{2}\\right)\\otimes V^*", "7a0612c7283f4b2e197bdc4190ee3960": "T\\chi(w) = -\\varepsilon^2 \\frac{1-\\chi(w)}{(w-z)^2}.", "7a06a84ad2fb74316fe2e56cf8c78a1c": "x \\in [0,n]", "7a06b38865cff508150d76c85a2cd6f6": "f\\in L_{1,\\mathrm{loc}}(\\Omega).", "7a06f0c7dec61a6ca6e6387e038d70d2": "\\mu_{k,j}", "7a07021d7d73906b3fb009cc18456e94": "A_1 \\cos(\\omega_1 t+\\phi_1)", "7a07339c9a413fdf910a1563d65a93c2": " p_E ", "7a0758a90dce22a1c2ce00e822d2dec2": "\\mathrm{rect} \\left(\\frac{t-T/2}{T} \\right)", "7a078570196d69b92cc82c0b88a9ae4d": "p(\\sigma) = \\frac{4\\sigma}{\\sigma_{av}^2} e^{-\\frac{2\\sigma}{\\sigma_{av}}}", "7a078d9620d4ef74b48702b2f6ef76fc": "|010\\rangle = \\left(0,0,1,0,0,0,0,0\\right)", "7a07ad98d2839aa6585391569f7f5f5b": "m_n=m_t \\cos \\beta \\, ", "7a07b414b69bf6e03cb893cec830862f": "MMOS_m(phago(r),exo(s)) = RE", "7a07c902e8e044f6a6273673eea71257": "\\scriptstyle (X,\\tau_1,\\tau_2)", "7a08260a1957dbb186753ab9c8f9e84d": "\n\\begin{align}\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr \\rangle &=\n\\frac{1}{\\Gamma} \\, \\int_{H \\in \\left[ E, E+\\Delta E \\right]} x_{m} \\frac{\\partial H}{\\partial x_{n}} \\,d\\Gamma\\\\\n&=\\frac{\\Delta E}{\\Gamma}\\, \\frac{\\partial}{\\partial E} \\int_{H < E} x_{m} \\frac{\\partial H}{\\partial x_{n}} \\,d\\Gamma\\\\\n&= \\frac{1}{\\rho} \\,\\frac{\\partial}{\\partial E} \\int_{H < E} x_{m} \\frac{\\partial \\left( H - E \\right)}{\\partial x_{n}} \\,d\\Gamma,\n\\end{align}\n", "7a083ee4399f6ec0db6bd31620ee5bf9": "\\begin{align}\nR &= Y +0.000092303716148 D_B -0.525912630661865 D_R\\\\\nG &= Y -0.129132898890509 D_B +0.267899328207599 D_R\\\\\nB &= Y +0.664679059978955 D_B -0.000079202543533 D_R\\\\\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix} &=\n\\begin{bmatrix} 1 & 0.000092303716148 & -0.525912630661865 \\\\\n1 & -0.129132898890509 & 0.267899328207599 \\\\\n1 & 0.664679059978955 & -0.000079202543533 \\end{bmatrix}\n\\begin{bmatrix} Y \\\\ D_B \\\\ D_R \\end{bmatrix}\\end{align}", "7a08471ee72bb960d529964249a83e11": "r=\\frac{mr\\cos\\theta+b}{\\sin\\theta},", "7a0864198d70b5c8983e604f7476db85": "F_n(x)=\\sum_{k=0}^n F(n,k)x^k,\\,", "7a088137a3054675476ae6bb84b4b2a9": "S^{2}\\,", "7a0888a352d5ba47bcb8a62b8914bd91": "\\mathbf{w}\\cdot\\mathbf{x} - b=-1.\\,", "7a08dbaca7baa414545dbe61ec76af17": "(\\omega_1+1)\\times(\\omega+1)", "7a09122266225d4e9d9ec42fedbca967": "-1.5", "7a0986c9cb90b7c5e2fd0402dcd8bcfd": " \\displaystyle\\mathcal{L}^{-1}", "7a098e9372669801beafe7c0cc8fb826": "\\nabla f({\\mathbf x}_0) \\cdot {\\mathbf \\gamma}'(0)=0.", "7a099a2c648d779db168f1853b347c1b": "M_{\\mathrm{left}}^{\\mathrm{fixed}} = \\int_{0}^{L} \\left \\{ - q_0 \\frac{x}{L} dx \\frac{x (L-x)^2}{L^2} \\right \\} = - \\frac{q_0 L^2}{30}", "7a09c5645920ad045042cccf29d1c0c8": "\\left(-\\frac{in (e^{\\tfrac{ibt}{n}}-e^{\\tfrac{iat}{n}}) }{(b-a)t}\\right)^n", "7a0a2d7c80a7db81625453029412a1fc": "\\zeta_{\\mathbf P^n(X)}(s)=\\prod_{i=0}^n \\zeta_X(s-i)", "7a0a694920da03ecc1d54f3edb3183c2": "H^0_{\\mathrm{DR}}(M, F) = \\ker d_0 =", "7a0ac4ced6c205c979dde5e1f5afe81d": " | \\psi_i\\rangle \\langle \\psi_i| \\,", "7a0ac4eca8e392c852bf14d392239b4c": "\\sigma_{A}(R)=\\sigma_{A}\\sigma_{A}(R)\\,\\!", "7a0b5730fd48f587d7f34ef77eddd3a4": "\\mathbf{T} = T_{ij} \\mathbf{e}_{ij} \\equiv \\sum_{ij} T_{ij} \\mathbf{e}_i \\otimes \\mathbf{e}_j \\,,", "7a0b6bfb7c8862c41cecb5cc0f928dc4": "TE_i = \\exp \\left\\{ { - u_i } \\right\\}", "7a0b7bf0fa544346f8f890b3bd6c7035": "\\sum_{n=1}^{\\infty} \\beta_n<\\infty.", "7a0bafd30d6f1343a310a3e0635516b1": "\\ln(f(\\theta;\\mu,\\kappa))=-\\ln(2\\pi I_0(\\kappa))+ \\kappa \\cos(\\theta)\\,", "7a0bc5bbbb768ed4f18eaae2cec53257": "\\boldsymbol\\tau = [I_C]\\boldsymbol\\alpha + \\boldsymbol\\omega\\times[I_C]\\boldsymbol\\omega,", "7a0be35f716390eb0672e3f9bbe9a5bd": "\\frac{1}{9!}", "7a0c03b5b99083701d03c1f76ccb150b": "\\lim_{m,n\\to\\infty} \\mu (A \\cap T^{-m}B \\cap T^{-m-n}C) = \\mu(A)\\mu(B)\\mu(C)", "7a0c081ff1f942bdbfba03d98a48a065": "\n\\ x(t) = \\sgn\\left(\\mod\\left(\\frac{x}{\\lambda},1\\right)-1\\right)\n", "7a0c3d170cdb2c0e0902da931cbfa580": "\\scriptstyle R \\;=\\; \\exp\\left(\\frac{-\\hat B\\theta}{2}\\right) \\;=\\; \\cos \\frac{\\theta}{2} \\,-\\, \\hat B \\sin \\frac{\\theta}{2}", "7a0c686db8ee4defa0d6c8df9bffe758": "(1 + o(1)) \\frac{\\sqrt{2} s}{e} 2^{\\frac{s}{2}} \\leq R(s,s) \\leq s^{-\\frac{c \\log s}{\\log \\log s}} 4^{s},", "7a0c87897c5ae8e2411da108a11302dc": "C_{ij}=\\frac{r_i s_j}{x_i-y_j}.", "7a0cbd553c158a734cc08e0b87e10953": "\\frac{U_e} {A_0 l_0} = \\frac {Y {\\Delta l}^2} {2 l_0^2} = \\frac {1} {2} Y {\\varepsilon}^2", "7a0d1d9a3e2a28e8188ca4da97aa5bed": "n_{4}'", "7a0d2594082a21f88c6a7566dae12907": "q \\in \\mathbb{R}", "7a0d2d6a5ae4b88eca185642ffe805fa": "p_n\\ ", "7a0d6cfdd663ebaf66f29b2283ec1369": "D^2\\psi = \\nabla^*\\nabla\\psi + \\frac{1}{4}\\operatorname{Sc}\\psi", "7a0da57504f71e552ddff098e35a5625": "v \\not\\in X", "7a0dab986e245c049b4eb579154472f1": "a=\\sum_j u_j a_j\\,", "7a0db1b03015476919bacdfe75622ae2": "\n E=\n \n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_{B}^2 }\n\\; M \\left ( \\mathit l + 1, 1, -{k^2 \\over 4} \\right) \\;M \\left ( \\mathit l^{\\prime} + 1, 1, -{k^2 \\over 4} \\right) \\;M \\left ( n + 1, 1, -{k^2 \\over 2} \\right)\n", "7a0dbb80d2a057190094cb04606b0d93": "\\mathbf{b} = \\mathbf{B}-\\mathbf{C}.", "7a0dc5f332d61f10b6398b00c22a3034": "\\boldsymbol{w}", "7a0e51d1d14552151dbfcdc88f21159c": "F_{preload}", "7a0f1aa2a7c8aff05d2f4ad772bbb316": " \\boldsymbol{F} = - \\int_A p\\, \\boldsymbol{n}\\; \\mathrm{d} S = \\rho \\int_A \\left(\\tfrac12 \\boldsymbol{u} \\cdot \\boldsymbol{u} - \\boldsymbol{v} \\cdot \\boldsymbol{u}\\right) \\boldsymbol{n}\\; \\mathrm{d} S, ", "7a0f304793c5a8af1dac0cd4d913cca3": "\\sum_{j\\in local} k_{j}=M\\left \\langle k_{i} \\right \\rangle = m\\left\\langle k_{i}\\right\\rangle\\approx mk_{i}", "7a0f8924fc6caf39285789b59a78b7bb": "\\forall (\\tau ,\\gamma )\\in {{\\mathbb{R}}^{2}},\\hat{\\theta }(\\tau ,0)=\\hat{\\theta }(0,\\gamma )=1", "7a105854b77d5d2eb94098ce9b2d26c8": "~(\\cos(x))^3~", "7a10704b2835b0ec8baacc60d272fd7f": "p(x|\\mu,\\sigma^2,\\nu) = \\frac{1}{\\sqrt{\\nu\\sigma^2}\\,\\mathrm{\\Beta}\\!\\left(\\frac{\\nu}{2}, \\frac12\\right)} \\left(1+\\frac{1}{\\nu}\\frac{(x-\\mu)^2}{\\sigma^2}\\right)^{-\\frac{\\nu+1}{2}}, ", "7a10b39eba3f785961dad6e9d32222d6": " y(t) = e^{L t } y_0 + \\int_{0}^{t} e^{ L (t-\\tau) } \\mathcal{N}\\left( y\\left( \\tau \\right) \\right)\\, d\\tau. \\qquad (2) ", "7a10ba55cd6f80239e8e6ec709506048": " 12\\cdot(283)^{3/2}\\zeta_k(2)(2\\pi)^{-6} = 0.9812\\ldots", "7a10f3c5b494437aead1831bb89a9f6c": " f_c^{k} ", "7a113c75f6e59c93c70712a0ded1728c": "\\|f\\|_2^2 = |\\langle TSf \\mid f \\rangle_2| \\le \\|S\\| \\|f\\|_2 \\|T^*f\\|_1", "7a1161b8c11f010c7fab2b5f1050cacf": "\\mbox{P} \\subsetneq \\mbox{EXPTIME} ", "7a11bbe46d231eac84ca7170bcd496c1": "K^{*}=\\{y\\in\\mathbb{R}^{n}:x\\cdot y \\leq 1 \\text{ for all }x\\in{K}\\}", "7a124a4286fc74fedb42f15e53cbff84": "\\sum_{k=1}^m f(P_k)\\, \\operatorname{m}(C_k)", "7a1295e8ecb887d151bdb4401b23ebfc": "H_0(x)=1\\,", "7a129e334f3f141b053b4eea2cb464d8": " 5 > \\lambda \\ge 3", "7a12c68ccd22ab1f25ab3afc7f9e2cc4": "\nk \\equiv \\sqrt{\\frac{e_{2} - e_{3}}{e_{1} - e_{3}}}\n", "7a12cfef468a0b601817bce4bfbe5495": "x \\in [-X_{max} , X_{max}]", "7a12d0c9b4dcf9a4debdfbdf8792378e": "y(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\\,", "7a132c40015557198acd8988489305e8": " H_n^{(r)} = H_{n-1}^{(r)} + H_n^{(r-1)}. ", "7a13821be0a7d7ba4812c4d9a1039c0c": "I = O + \\frac{\\Delta S}{\\Delta t}", "7a1390cb93fb1ebbaa2d1643c53e2e2e": "\\begin{matrix} \\frac{16}{9} \\end{matrix}", "7a13ab13cb758e5909c45c7335921872": "\\xi_{ijl}", "7a13cfc79bdc3d29d69b47db5b10fe7d": "Tv-\\lambda v=(T-\\lambda \\, \\text{I})v=0,", "7a13eecb24e0688fba3fd1fa7e752ca5": "\\sum_j \\mu(E_j) = \\infty,", "7a13fceaad5428a44e0d4e6a5fa9c8e7": " B \\subset \\mathbb R.", "7a14457669688c7332dcda55b334f880": " +\\frac{1}{q}\\oint_{C}\\mathrm {\\mathbf{effective \\ chemical \\ forces \\ \\cdot}} \\ d \\boldsymbol{ \\ell } \\ ", "7a144628d56c5fd238a60f7dafc138e3": "\\frac{\\partial C}{\\partial t}", "7a14536c1a074b612f4ceca64cb26efb": "\\,\\beta", "7a14a553909d4919cd584c1ff1f6b373": "\\sum_{n=1}^\\infty\\frac{1}{n^s}", "7a14c45758a97b21b5c7c7bb425b46d6": " \\hat T(V) = \\prod_{k\\in\\mathbb{N}} T^kV ", "7a14d1ae71dd7328ef2a42f315e9e179": "\\mathrm{V=J\\ C^{-1}=kg\\ A^{-1}m^2s^{-3}}", "7a150d4a6e2aaa5902e3e5dec5fca5fd": "A+A = \\{ a+b: a,b \\in A \\}", "7a15316452323775ad4285373fd70315": "\\displaystyle{}_{r+1}e_r(a_1,...a_{r+1};b_1,...,b_r;\\sigma,\\tau;z) = \\sum_{n=0}^\\infty\\frac{[a_1,...,a_{r+1};\\sigma;\\tau]_n}{[1,b_1,...,b_r;\\sigma,\\tau]_n}z^n", "7a1543322d86684984326fff6bec0be3": "\ne^{-} e^{-} \\longrightarrow e^{-} e^{-}\n", "7a155eb8010dfae34bd094ba1b21a327": "Gx = \\left\\{ g\\cdot x \\mid g \\in G \\right\\}", "7a15815254f54ef1086d98c96daa1f0f": "E_{up}=\\frac{4}{3}\\pi\\lambda^{1/4}Q^{3/4}\\sigma_0", "7a15db733250c6368b58f7c58bf57f08": "(c_{V,W}\\otimes \\mathrm{id}_U)(\\mathrm{id}_V\\otimes c_{U,W})(c_{U,V}\\otimes \\mathrm{id}_W)=(\\mathrm{id}_W\\otimes c_{U,V}) (c_{U,W}\\otimes \\mathrm{id}_V) (\\mathrm{id}_U\\otimes c_{V,W}):U\\otimes V\\otimes W\\to W\\otimes V\\otimes U.", "7a15edf7b9b0205257361ca27a990206": "r=\\frac{\\sqrt{3}}{6} a", "7a161d1fc41144cd3668235e6707b638": "1.00086", "7a161fefd484d33f77bc8d1f76a533ba": "\\|L\\|_{\\mathrm{op}} \\,", "7a162cb4a7f8b6b917fe8d4730463c77": " P_n(x) = B_n \\left(x - \\lfloor x\\rfloor\\right)\\text{ for } x > 0", "7a163e351f4805867d0ce88812fd7c3c": "K^1\\,\\!", "7a164543c685cdc662a36d47ce1b86b8": "x = e^{\\gamma t}\\,", "7a1683c7a11d73b4200af69ea444cc08": "\\langle\\phi|e^{-i\\theta} A e^{i\\theta}|\\phi\\rangle", "7a16a44a10654108d6a333dd11083e4d": "\\begin{bmatrix} 1 & 0 \\\\ 1 & 3 \\\\ \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\\\ \\end{bmatrix} = \\begin{bmatrix} a & b \\\\ c & d \\\\ \\end{bmatrix} \\begin{bmatrix} x & 0 \\\\ 0 & y \\\\ \\end{bmatrix}", "7a16c6f06a3a4796d61cbf9963357f77": "\\mathbf{\\Sigma^p=(J^TWJ)^{-1}J^TW \\Sigma^y W^TJ(J^TWJ)^{-1}}", "7a1719b587abdbeb36e5b2a32f448b4a": "f_{-1},\\dots,f_{-n}", "7a1723bc404d7d47950b64d451f8eee4": "\\operatorname{Supp}(M) = \\operatorname{Supp}(M') \\cup \\operatorname{Supp}(M'').", "7a1732484f7199913e2f0608bf7a0d64": " \\eta_{ab} = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right) ", "7a1740855da27caf0edf9126d82c4274": "\\alpha + \\sum_{i=1}^n x_i,\\, \\beta + n - \\sum_{i=1}^n x_i\\!", "7a175a729acc8d23cfa6722b85200f64": "s_3 = c_2", "7a181dff24e59888cc5393773065dfd1": "\\scriptstyle |e\\rangle \\rightarrow |e_n\\rang ", "7a183817eb0e9bcf9b54d4f413abec7d": "Q^\\mathrm{adynamic}_{A\\to B}=\\Delta U\\,.", "7a1842500d54b119da0540e11ac44f72": "- n^2 a^n u[-n -1]", "7a18501aeba3a251664d2c59e0e305ba": "P_{j+1} = R\\times P_j - q_{n-(j+1)}\\times D\\,\\!", "7a1896290b20f61ebc4be44502a5718e": " \\underbrace{x+\\cdots+x}_{n\\text{ terms}} < y. \\, ", "7a18fbf2f52498548550785a31c9c0d4": "\\delta = q \\overrightarrow{a} p ", "7a19d6e4ab4adab507f60a3afc341e01": " u_2 ", "7a1a67d8c6db05ca26e870abbb7258c8": "-3 < \\lambda \\le -2", "7a1aca87e6cedfd47d8c5b3ee58e7ec6": "\\Delta,", "7a1b029c7c5a8eb7ad97399bda364b37": "\\mathrm{C_6H_6 + \\,2\\ CH_3Cl \\rightarrow C_6H_4(CH_3)_2 + 2HCl}\\,", "7a1b22123e15c97989d6a22ef729a285": "m=q_m d \\,", "7a1b2932803b76f0e54a53607cd9b05f": "S_i Y", "7a1b480155ca6ea077184b6dd3e3ad54": "\\scriptstyle G(n)=\\lfloor\\sqrt{n-1}\\rfloor+n.", "7a1b70bb185d9578a590b204b56605d2": "E(S_{t+k})", "7a1b97777d4e1458473334e4aefdc717": "t ", "7a1bb084d680bb1522cab882935aacfe": "M_2 \\cup M_4 \\cup \\cdots ", "7a1c22282ce8550bc3f8b0b41c258245": "\\mathbf A_+", "7a1c4df9dc092e3cbe0465004bec27a1": "(x^2-a^2)(x-a)^2+(y^2-a^2)^2=0 \\,", "7a1c7e5c732dda413f96e2da82e69bac": "c_{nr}", "7a1cafbd8705e31d2f109bc06b8c9060": "= \\frac{\\pi\\sqrt{3}}{8} \\approx 0.68.\\,\\!", "7a1cdaf0a8d41f79a3f1b350d817e312": "\nP_{\\mu }(n,t)=(\\frac{(-z)^{n}}{n!}\\frac{d^{n}}{dz^{n}}E_{\\mu }(z))|_{z=-\\nu\nt^{\\mu }}, \n", "7a1cf54b4b2ac2cd79e905f46ec93b84": "q\\in M\\,", "7a1d045a85873b49c937dbab22489d2d": "\\frac{\\pi}{2} \\sum_{\\alpha} \\frac{c^2_{\\alpha}}{m_{\\alpha}\\omega_{\\alpha}} \\delta(\\omega-\\omega_{\\alpha}) \\ \\delta(x-x_{\\alpha}) \\ = \\ J(\\omega)", "7a1d654f1f284971b0f702ada2cc4898": "E_\\text{i}", "7a1db578e31a89bd1bcc879f9b0cab25": "V_{outsensmin}", "7a1dceca04010b541e9c8e5b925c0738": "z = \\sin\\left({u \\over 3} - 2v\\right) + 2\\sin \\left({u \\over 3} + v\\right)", "7a1e8e181fa1ac88ed9ec4c4b5a5e527": "\\sigma_{\\rm a}(\\omega)", "7a1e9143f0f9182131feba053954036f": "B_i (1,1)", "7a1ea4ea2e1cbc3e59b3e94891e2424b": "p=\\frac{h}{\\lambda},", "7a1ead3314d31e2906df510bb07072bb": "\n\\operatorname{rank}(\\widehat D) \\leq r \n\\quad\\iff\\quad\n\\text{there are } P\\in\\R^{m\\times r} \\text{ and } L\\in\\R^{r\\times n}\n\\text{ such that } \\widehat D = PL\n", "7a1ee06f77579bf4c5c47d251893ba77": "W(z)", "7a1f011d8e8d7d6fe34e4b137891ce0c": "\nQ^* = Q_E + Q_H + Q_{LW} + Q_{SW}\n", "7a1f1b5446a80bbc79d672bb2c144479": "2\\sin^2(A/2)=1-\\cos A", "7a1fa47356c8decae0a3bcb4529cdc1f": "\\bar F(x) \\leq \\frac{\\mathbb E(X)}{x} .", "7a1fa47f85210bcc8e72577b3f4ee3c4": " F[n]=T[n]+U[n] ", "7a1fdcc49b3041f28e4ca28830080803": "(gate8\\vee \\overline{gate6})\\wedge (gate8\\vee \\overline{gate7})\\wedge t(gate6\\vee \\overline{gate8}\\vee gate7)", "7a1fe79366462d69a51c6a763c00ff39": "\n\\begin{align}\n \\theta &= \\arccos\\left(\\frac{1}{2}[A_{11}+A_{22}+A_{33}-1]\\right)\\\\\n e_1 &= \\frac{A_{32}-A_{23}}{2\\sin\\theta}\\\\\n e_2 &= \\frac{A_{13}-A_{31}}{2\\sin\\theta}\\\\\n e_3 &= \\frac{A_{21}-A_{12}}{2\\sin\\theta}\n\\end{align}\n", "7a2001603fd554c27624889e8202cc69": "u(x,t) = \\int \\Phi(x-y,t) g(y) dy.", "7a2017a4b3168df96344661812952cb4": "\\mathsf{P} \\cdot \\oplus \\mathsf{P} \\subseteq \\mathsf{P}^{\\sharp P}", "7a20948a7b48505bf4b8938f7db22566": "C = W \\log_{2} (1 + \\tfrac{S}{N}),", "7a20c52c5ea169ad03c25fb15a84c3e7": "172_{11} \\ ", "7a20d6588273a23ef8066380afcf455a": "\\vert a, m \\rangle", "7a20e45816d3cf47f5b90b6613172b8b": "D(f) \\leq R_2(f)^3", "7a21edaebce3c8942ce9f36a4f3b8e48": "\\underset{x}{\\operatorname{arg\\,max}} \\, (1-|x|) = \\{0\\}", "7a21f7001ee0c8a140b2f05707220b10": " \\Leftrightarrow |PA|^2 = 9 |PB|^2 ", "7a21fa720d7c265978e4506575ccd043": "S(u,v) = \\sum_{i=1}^k \\sum_{j=1}^l R_{i,j}(u,v) P_{i,j} ", "7a225be31ebdc324256204521fa83f9b": "t \\in \\{0,1,...,T-1\\}: \\rho_{t+1}(X) \\geq \\rho_{t+1}(Y)", "7a2270d8894e66f2cb2608a964787184": "w = \\epsilon + \\frac{p}{\\rho}", "7a22757c6a6b4f404dd76cd8bd8fa40a": " OA \\to \\Diamond A. ", "7a22ed7ffe4f78e147686ffc08dcdc3b": "\\mbox{C}_n\\mbox{H}_a\\mbox{O}_b\\mbox{N}_c + \\left( n + \\frac{a}{4} - \\frac{b}{2} - \\frac{3}{4}c \\right)\\mbox{O}_2 \\rightarrow n\\mbox{CO}_2 + \\left( \\frac{a}{2} - \\frac{3}{2}c \\right)\\mbox{H}_2\\mbox{O} + c\\mbox{NH}_3", "7a230ff8eba8cf188ab6026ff6af595d": "g_7^2g_2g_8g_1=1", "7a231e1bdfbe97a99d5269099e13d211": "860 mm \\times 1220 mm", "7a23cc23b65606e19c69337fb85d8ce2": "1-\\left(\\frac{\\sin(\\pi u)}{\\pi u}\\right)^2 +\\delta(u),", "7a23d93b4f1799cd39c11648b52f601a": "KC", "7a2406b3eb06ccf9a230a9b66ee522ac": "\n\\mathbf{y}(\\mathbf{x}_t) = \\left(\\mathbf{A}^{*}\\right)^{\\text{T}}\\mathbf{K}_t,\n", "7a2409fab82fb6eabe2e284955211d47": "P_N = Q", "7a2477cdeaf8e583c289d9eaf3ffad08": " \\Psi = \\Psi(x,t) ", "7a24af546dbd28ab6386f884f6997f37": "\n\\begin{align}\nb^2 & = 1 + a^2 - 2(1)(a)\\cos\\left(\\alpha - \\frac{\\pi}{2} + \\frac{\\beta}{2} \\right)\\\\\n & = 1 + 4\\sin^2\\left(\\frac{\\beta}{2}\\right) \n - 4\\sin\\left(\\frac{\\beta}{2}\\right)\\sin\\left(\\alpha+\\frac{\\beta}{2}\\right)\\\\\n\\end{align}\n", "7a251de98c0efdcef50217e193f4ab59": "\\Gamma(t) = \\oint_C \\boldsymbol{u} \\cdot \\boldsymbol{ds}", "7a2554504865e3febcb08a7207fe1e1c": "V_\\mathrm{out} = I_\\mathrm{out} \\cdot R_\\mathrm{load}", "7a256aaa45b5b62d98c89d881ec7a074": "b_n(t) = r(t)", "7a259bf56db252c10f885152e90653f9": "E_{POT} = mgy", "7a2640ef89c3367f21caba2644fcdb10": "\\scriptstyle z,", "7a26b16ef21c8983706661f0f4e03d0c": " t\\notin E", "7a26d4d2e72d59183e5f4a974a61a8cf": "E(x_0,x_m) = \\frac{P(x_0,x_m)}{F(x_0,x_m)}", "7a2710ada8b7afdcc751767b076a398a": "(1-e^{-2\\gamma})", "7a273433e851222880ac6aa1ec7b9df4": "\\left(\\frac{\\partial T}{\\partial V}\\right)_S = -\\left(\\frac{\\partial P}{\\partial S}\\right)_V", "7a27744908345591adb1e6909ab9185d": "\\textit{NOUNPHRASE}", "7a27f41e3b879cf3d314044c7dc10825": "a_1,a_2", "7a284a8a9336d7e35a4d9b096d5bd5a3": " y_1^{k_1} y_2^{k_2} \\cdots y_\\ell^{k_\\ell} ", "7a2867cdae986983a075c3e7afb307c0": "\\left\\|\\phi_k\\right\\|_d=1.", "7a2920b39036fb4ece285dffeff90d01": "\\rho \\left[ \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right)}{\\partial t} + \\frac{\\partial \\left( \\overline{u_i} + u_i' \\right) \\left( \\overline{u_j} + u_j' \\right) }{\\partial x_j} \\right] = \n-\\frac{\\partial \\left( \\bar{p} + p' \\right) }{\\partial x_i} + \\mu \\left[ \\frac{\\partial^2 \\left( \\overline{u_i} + u_i' \\right)}{\\partial x_j \\partial x_j} \\right].", "7a292473834153a3892d02d9868b94c1": "X : \\Omega \\to \\mathbb{R}", "7a293db15bdf2eb2575ca594b3d59596": "\\Delta \\omega_1\\ =\\ -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\,", "7a294f98b1abfa665e606dda94e3e0b3": "(-,y_j)", "7a296190badd4b8a1623f9b72d002640": "i<\\omega", "7a29826f40bf25738bbad0c7873f35b7": "g(x)=\\int f(x,y) dy", "7a29c8c19d913502598ae55f94f837f6": "dp(x,t) = L[p(x,t)] dt + p(x,t) [h(x,t)-E_t h(x,t) ]^T \\eta^{-\\top}\\eta^{-1} [dz-E_t h(x,t) dt].", "7a29d8dbf6644ca462487b611c665738": "a \\rightarrow b \\rightarrow a \\rightarrow b \\rightarrow \\cdots", "7a2a3dc600cef3d9309c5aaf9dad9458": "\\textstyle X_r", "7a2a3eda6b005f76c1335d14f0dcadd4": "\\frac{dy}{dt} = \\delta xy - \\gamma y.", "7a2a6f3ad3150b326c8b0877bc1ec106": " G:A\\times C^\\mathrm{op}\\times C\\rightarrow D ", "7a2af7f2d6d39b89a9450f93407bff61": "\\scriptstyle 1/2+iH", "7a2b66ec3a1b23d105ddf1647cfeeb2c": " \\begin{align} K &= \\text{area of } \\triangle ADB + \\text{area of } \\triangle BDC \\\\\n &= \\frac{a d \\sin \\alpha}{2} + \\frac{b c \\sin \\gamma}{2}. \n\\end{align} ", "7a2bacf7bf416f6143595a75413ea5c0": "\\rho_c = \\frac{3H^2}{8\\pi G}", "7a2bb63188d1fe193f4e1a3e79d94ed4": "x' = f(x), t \\in \\R", "7a2bfe9432905f0e9d29c62abdedb7db": "id_{X}", "7a2c1e69f0135a172dabdefbe86f6e83": " \\langle R \\rangle ", "7a2c2f5beb80ea0ce6b2bcbf925ddb88": "T>0,", "7a2c6e5bc8db3b8a536855e264af860a": "E^*(\\omega) = \\frac 1 {1/E - i/(\\omega \\eta) } = \\frac {E\\eta^2 \\omega^2 +i \\omega E^2\\eta} {\\eta^2 \\omega^2 + E^2} ", "7a2c701f8ed951511e8331e39a4441b0": "\\{\\phi(w) : w\\not=v\\}", "7a2c9a93a3aa6095381d3df710938142": "r_A", "7a2c9f719c235d8775df0f9ea8066810": "R=\\frac{2h}{3} ", "7a2ce0ec74323e6c5ca96c4c00674a05": "F'= \\frac{F}{(1-v^2/c^2)\\gamma}=\\gamma F", "7a2d11438f75d27d0b87627274f3c45c": "\\mathrm{Gr}=\\frac{g \\beta(T_s-T_o)L_c^{3}}{\\nu^2}", "7a2de0144372a021ac45e7168db4cd94": "A_\\alpha ^i=[A_{\\alpha-}^i, A_{\\alpha+}^i], W_\\alpha ^i=[W_{\\alpha-}^i, W_{\\alpha+}^i]", "7a2e03cf0654885f56ba0b611f3c0009": "V^+", "7a2e3b46a03b615e07438677258f61bf": "F_Y(y; \\mu, \\sigma) = \\int_0^y \\frac{1}{\\sigma\\sqrt{2\\pi}} \\, \\exp \\left( -\\frac{(-x-\\mu)^2}{2\\sigma^2} \\right)\\, dx\n+ \\int_0^{y} \\frac{1}{\\sigma\\sqrt{2\\pi}} \\, \\exp \\left( -\\frac{(x-\\mu)^2}{2\\sigma^2} \\right)\\, dx.", "7a2e507cb631be6b4dccf3fc43d2e57f": "p_{n,k} (x)=0", "7a2e6939bb75604731e2dd92ed881390": "N\\to\n\\infty", "7a2e86361482994d7eae10b3e62a52c3": "\\scriptstyle \\, s", "7a2ecb6fa8c1b586a655a6d9cf1ed2db": " (\\boldsymbol\\mu,\\boldsymbol\\Lambda) \\sim \\mathrm{NW}(\\boldsymbol\\mu_0,\\lambda,\\mathbf{W},\\nu) .\n", "7a2f666b60cf0c53f60befb4d65e307a": "\\Phi_c = \\{ \\mathbf x : f(\\mathbf x) =c \\}", "7a2fd5582c7468f81d74e17bcd7a2c65": "(n,m) [T_+,T_-]", "7a3034ad95ded59dfe121304367050c0": "\\max_\\alpha\\{\\mathsf{H}((1-\\alpha)(1-p)) - \\mathsf{H}(Y \\mid X = 1) \\mathsf{Prob}\\{X = 1\\} \\}", "7a306f95a65071b03668cf78c4acf16a": "\\,\\ \\sec x\\tan x", "7a307e97eaa9a541a3a56807f0474df2": "E=nh\\nu", "7a3080f5a57a137642f4ca46198d7931": "F_{eachAnchor}=F_{load}\\frac{Sin(\\alpha )}{2Sin(\\alpha)Cos(\\alpha)} \\,", "7a30e0b832d71e9dfc42bb95627d4acd": "k = 1 - x^2.\\,", "7a315811a698dfd16fcadddb619440e9": "f(x;\\mu,s) = \\frac{a} {4s \\cosh^2\\left[a \\sinh^{-1}\\left(\\frac{x-\\mu}{2s} \\right ) \\right ] \\sqrt{\\frac{(x-\\mu)^2}{4s^2}+1}}", "7a316830f52956f78257eb30a1e196ac": "\\sqrt{2} \\approx 1.41421", "7a31a9236a7240164237a5bff73575a4": "1\\le\\sup_{t\\in[0,2\\pi]}|p(e^{it})|<2^{A-1-n}", "7a31ba68cc748d2c2b41891c1cf41e08": "Q = \\{0,1\\} \\cup \\{{2\\omega_1 \\over d}, \\ldots,{2\\omega_N \\over d}\\}", "7a31c53a730a8ef5f84d86140073fa81": "\\mathfrak{so}^*(3,\\mathbb H) \\cong \\mathfrak{su}(3,1).", "7a31e9fc8883e4bfe19dc72e74f024be": "\\operatorname{gr}A = \\oplus A_i/{A_{i-1}}", "7a31ea30f7be214091c4e8cc67c10974": "1/\\lambda_1,1/\\lambda_2,\\dots,1/\\lambda_n", "7a3201be641fd57fd1971fd4d312d64f": "\\,\\mbox{T}(a)", "7a3207e98ece97e6c4143b0d61395bd9": "P\\cdot p\\Psi =(-P^0p^0+\\vec P\\cdot p)\\Psi=0. \\,", "7a3229adcaac538e9aa3400f0a18e605": "SUV(t) = \\frac {c(t)}{injected\\ activity(t) \\quad / \\quad body\\ weight}", "7a322fd490f78e9ea00afd56508a46f3": " |\\langle x,y\\rangle| ^2 \\leq \\langle x,x\\rangle \\cdot \\langle y,y\\rangle,", "7a3261a6953b313345cedc0b9c9730b6": "\n\\mathrm{E}[Z_k]=0,~\\forall k\\in\\mathbb{N} \\quad\\quad\\mbox{and}\\quad\\quad \\mathrm{E}[Z_i Z_j]=\\delta_{ij} \\lambda_j,~\\forall i,j\\in \\mathbb{N}\n", "7a32876285b227d456714e44d04e0cf4": "area ~ OACBO = area ~ OABO + area ~ OACO \\;", "7a32e5a99bf0764121600b62f9a28d6b": "\\delta_{\\theta_k}(\\theta_k)=1", "7a32f64edc45244fd41116b06564ca27": "S_g=-(\\bar c^a\\partial_\\mu\\partial^\\mu c^a+g\\bar c^a f^{abc}\\partial_\\mu A^{b\\mu}c^c)", "7a33869396a2406b9c45a3191d8a480e": "eRPF = \\frac{U_{PAH}}{P_{PAH}} V", "7a33d029e3ba3300e752c1fadd5437ec": "\\varphi(x) \\leq h(x)", "7a33d89ea5497ddd01f0f06eaa722540": "\\frac{dP}{dt}\\frac{1}{P}=K\\left(1 - 2\\frac{Q}{URR}\\right) \\qquad \\mbox{(3)} \\!", "7a343b04007833f9d3ef224499f6c197": " m_\\mathrm{s} = + \\tfrac{1}{2} ", "7a34e45da307c53891b75d544e43cb0f": "(x1,x2,x3) = (0,0,1)", "7a34f0b59126c70a6ce62778a5a9e369": "\\tbinom {13}4", "7a3552686ebf2738773cddff558bd357": "\\frac{n!}{(m-1)!}.", "7a3578b7887442dfbbb7bdf942947b80": " G = \\int_0^{\\infty}{\\frac{dx}{\\sqrt{\\cosh(\\pi x)}}} ", "7a35d4fdd34a4ed671a10aa0cbd23809": "radius_s", "7a35f0990a3147a3e484354700457150": "C = \\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|a_n|},", "7a35fe3bd82dfaa3375fc2541df25055": "\\{1,2,3,4,5,\\dots\\}", "7a3611b3f474fcbf527f51e59664a320": "\n \\ Z_t \\ = e^{\\int_0^t X_s\\, dW_s -\\frac{1}{2}\\int_0^t X_s^2\\, ds},\\quad 0\\leq t\\leq T\n", "7a362df02122539e277c68de1c6157b5": "\\sum \\chi(X(X-1)(X-2)\\ldots (X-k))", "7a366999ac6bcec934e01e48875d028d": "X\\to\\mathbb R", "7a36b6729c232d951a937ef0608c8a50": "v(\\{1,2\\}) - v(\\{2\\}) = 0 - 0 = 0\\,\\!", "7a36ea36e78a053a95492313526a05f0": "(t,q^i,p,p_i)", "7a3758b6fe57168a2b6ad5635f916d5d": "(2g-2,0,2g) \\sim (g-1,0,g)", "7a379bf23a9091034900ad6b461db1ea": "\\dot s", "7a37dff2491b6ea10ca8510caaa2f78b": "\na_x =\n\\begin{bmatrix}b_y\\\\b_z\\end{bmatrix} \\times\n\\begin{bmatrix}c_y\\\\c_z\\end{bmatrix}.\n", "7a3816548f0330d184180c0454448d41": " Q_o = \\frac{\\omega U} {P_d} ", "7a38a777484d609a0014e651b9501daf": "\\omega(X,D)(E)=|\\phi^{-1}(E)|/2\\pi", "7a38bf04f873c57683dbccb7770d9394": "R_{x_c x_c}(\\tau)=R_{x_s x_s}(\\tau) \\qquad \\text{and }\\qquad R_{x_c x_s}(\\tau)=-R_{x_s x_c}(\\tau).", "7a38ceec8b70c0f7a9898c962b72092a": "m_{em}", "7a38dd477dbcf7b0660af0aea65fb5e2": "\\dot{V} = \n \\frac{\\dot{m}}{\\rho}\n ", "7a38f37eeb8f1b648f3639ef2785d036": " = a", "7a38f9b5930ef4bb0c5a702a019a82c4": "C_{i+1} = G_i + \\left( P_i \\cdot C_i \\right)", "7a395b2024fa3316ffcd6210ee875239": " a_{s2} ", "7a396a5384af0c9227a580fc20f9bbee": " \\int_{\\phi(U)} f(\\mathbf{v})\\, d \\mathbf{v} = \\int_U f(\\phi(\\mathbf{u})) \\left|\\det(\\operatorname{D}\\phi)(\\mathbf{u})\\right| \\,d \\mathbf{u}.", "7a39a9f353e499b6ba520834e8cf3e13": "\\| Du \\|_{L^\\infty}", "7a39da5f61d8008243bd5d8d3183a1b2": "\\phi = 1-\\frac{\\rho_{\\text{bulk}}}{\\rho_{\\text{particle}}}", "7a39f182f7c25e6d5cf5ceffb0da2a1d": "\\overline{X}e^{-qV}X", "7a39f48f58fc92aea38a2eedb6f574e5": "M = \\frac {f}{d} = \\frac {650}{8} = 81.25", "7a39f5b5361fd01cb83a54dc6abd3ff7": "P'(q)=0", "7a3abac30fcc8cbd2aeddfaa631f1c07": "e^{-\\alpha(\\lambda) d_i}", "7a3ac7cbfd1c72e3853f0405505c9fad": "x = \\frac{\\displaystyle 1+\\frac{\\displaystyle 1+\\frac{\\displaystyle 1+\\cdots}{\\displaystyle a_3}}{\\displaystyle a_2}}{\\displaystyle a_1}.", "7a3af0462b775e5c1589853d18966f55": "s : I \\leftarrow \\mathbb{N}", "7a3b041d4b373bc91de454fd2c4bde63": "{\\text{A}_x \\text{B}_y}_{(s)} \\Longleftrightarrow x \\text{A}_{(aq)} + y \\text{B}_{(aq)}\\,", "7a3b3cfe26b33e62e1130ba1f05668b8": "T = d_\\Gamma \\sigma = (\\partial_\\lambda\\sigma_\\mu^i + \\Gamma_\\lambda^j\\partial_j\\sigma_\\mu^i -\n\\partial_j\\Gamma_\\lambda^i\\sigma_\\mu^j) \\, dx^\\lambda\\wedge dx^\\mu\\otimes \\partial_i. ", "7a3b42ce738db3dba595beb6f4a5a310": "f(X) = \\sum_{n=1}^\\infty a_n X^n = a_1 X + a_2 X^2 + \\cdots", "7a3b4997b445c2b7b6d2df06e1abe4a7": "\\left(\\sum_{l\\in\\mathbb{Z}}q^{l^2/2}z^l\\right)\\left(\\sum_{n\\geq0}p(n)q^n\\right)=\\left(\\sum_{l\\in\\mathbb{Z}}q^{l^2/2}z^l\\right)\\left(\\prod_{n>0}(1-q^n)^{-1}\\right)", "7a3b6431203a14a7de56242e1cf6d931": " F_{\\mathrm{m}} \\propto \\frac {I^2} {r}. \\; ", "7a3bb57c5175ec3b941a8cc0d7c027c2": "\\phi'(x)", "7a3bc2be4647c6c6f26c77996a88810c": "Q_{Fan}\\,\\!", "7a3bcb8df4e9f9fdd2ee8d3039f8213e": "s=\\lbrace 1,2,3,4\\rbrace", "7a3bfc42949190bbecbaa0e80506a0fb": " h_{22} = \\left. \\frac{I_{2}}{V_{2}} \\right|_{I_{1}=0} ", "7a3bfebc4c4ff6c2f83a22a32dd3840d": "\\mathrm{Re} z = \\theta", "7a3c83a79272c26fea343e633b517188": "\\! e^{it^T\\mu - \\frac{1}{2}t^T\\Sigma t}", "7a3c8f00e08aacba4620b762319b578d": "\\xi=-\\alpha^{-1}<0", "7a3cc000ca794d711ab835c99d3c2692": "D_{fan}", "7a3cedf57b82b99b45631143ecfddc4f": "\\beta=-v\\gamma \\,", "7a3d13e084ad36fa443ded1e157ea7d7": " \\langle \\pi_i(x) \\xi \\mid \\eta \\rangle \\rightarrow \\langle \\pi(x) \\xi \\mid \\eta \\rangle \\quad \\forall \\xi, \\eta \\in H_n \\ x \\in A. ", "7a3d1972faec4e69179b053f8dd95855": "BA=\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix}", "7a3d60d95faace305914a524bb0e1578": "\\eta_{\\alpha \\nu} { T^{\\nu \\beta } }_{,\\beta} + F_{\\alpha \\beta} J^{\\beta} = 0,", "7a3d950dd8ae2cd8c653339672a40a25": "F^a_{X/S} = 1_X \\times F_S.", "7a3da255e54a9904f0d1cd3619889e88": "\\left\\{ f\\left( x,\\cdot \\right) \\right\\} _{x\\in X}", "7a3e142cc4d69c8bd2aafcd66e775080": "g(x)= \\phi(x) - \\lambda \\int K(x,y) \\phi(y)\\,dy", "7a3e39e31ab9bc983f990b1e1ae7cac4": "\\boldsymbol S: \\mathsf C\\cup\\{\\emptyset,\\mathfrak X\\}\\mapsto 2^{\\mathfrak X}", "7a3e3d6ce38b0f6fda9371c8ac6e5fa1": " \\, h \\, \\sim \\, 2\\, \\sqrt{ \\kappa t } \\,", "7a3e934a5f222ee6c2bd01722e43e7cb": " \n(Eq. 4) \\text { } E[Y_i(\\alpha^*(t), \\omega(t))] \\leq 0 \\text{ } \\forall i \\in \\{1, ..., K\\} \n", "7a3e947b48709becfeadcdd6c39efff1": "O(\\sqrt{V}E)", "7a3ed90f954073e9c5a5e9272b032f19": "\\left\\{ {W^i} \\right\\}_{i =1}^n ", "7a3ee966c8961f573821d8ecad69f80e": "k = \\frac{{\\rm d}\\left [ {\\rm Y} \\right ]/{\\rm d}t}{\\prod_{i=1}^{r}\\left [ {\\rm X}_i \\right ]^{\\sigma_i}}", "7a3f5cffc25c2be178b054819394262c": "EER=(135.3-(30.8*Age))+PA*((10*wt)+(934*ht))", "7a3f9a73aa33087ac47e01fb378164f9": "\\mathbf{E}~", "7a3fc22ad482dc4ae51bd6dafc50937e": " \\phi '", "7a3fef3d04180c0c25cc61c81f4dcb79": "V_n(R) = \\frac{R^n}{n} \\bigg(2\\int_0^{\\pi/2} \\sin^{n-2}(\\phi_1)\\,d\\phi_1\\bigg) \\cdots \\bigg(4\\int_0^{\\pi/2} d\\phi_{n-1}\\bigg).", "7a402621c61a6e15f2f8314540c67abc": " \\langle \\psi \\vert L \\vert \\psi \\rangle \\to \n\\langle \\psi^\\prime \\vert L^\\prime \\vert \\psi^\\prime \\rangle = \n\\langle \\psi \\vert L \\vert \\psi \\rangle + \n\\frac {q}{\\hbar c} \\langle \\psi \\vert r \\times \\nabla \\Lambda \\vert \\psi \\rangle \\, .\n", "7a404b460b29a923cf1b636e98840035": "U_n := X_n \\lor \\mathbb{E}^{\\mathbb{Q}}[U_{n+1} \\mid \\mathcal{F}_n]", "7a407bc684f80fa7d3f7a5ec9e5a0906": "2^3 = 8", "7a4081e4417d6589939c26a3273137d7": "\\frac{q^2}{g}\\left(\\frac{y_2-y_1}{y_1 y_2}\\right)=\\frac{1}{2}(y_2-y_1)(y_2+y_1)", "7a40c9b4c5573fe514854dd00b0f3b1f": "0.15K_p T_u", "7a40db289f6e11bb8172d9e4939f6600": "m(x) = 1", "7a40e7f98dc69c2015f8df016cec2890": "4.7 \\times 10^{7} \\text{J/kg}", "7a40fb126f1d45742203207eb36df979": "Z'=\\frac{1}{Z}", "7a41245e0ee9110d325074622a64052c": "(16/9)10^{6}", "7a412bee297b007a2a21a743178d6c04": "H=\\frac{4}{3}\\,U", "7a42260a1fe651f3961db65d4bfb8445": "\\square = \\frac{1}{c^2}\\partial_t^2-\\Delta", "7a422ccf88af697fc5d9ea8f422a0e04": "\\textstyle \\frac{\\partial C_1}{\\partial t}", "7a424ad6096c93417572b50dff9bdaf8": " {\\rm ad} (x) ", "7a429dc28b094f9d8b5205d572824f06": "\\mathbf{v} \\left(\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v})\\right) + \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = \\mathbf{b}", "7a4304f8894a54a9f2674a0836fd5dd1": "\\scriptstyle 0.9 - 1.0 = -0.1.", "7a432f8fa700698a315c36d752f662b3": "\\mathbf{S}= \\begin{pmatrix}\nS^{11} & S^{12} & S^{13} \\\\\nS^{21} & S^{22} & S^{23} \\\\\nS^{31} & S^{32} & S^{33} \\\\\n\\end{pmatrix} = \\begin{pmatrix}\n0 & S_{xy} & S_{xz} \\\\\nS_{yx} & 0 & S_{yz} \\\\\nS_{zx} & S_{zy} & 0\n\\end{pmatrix} = \\begin{pmatrix}\n0 & S_{xy} & -S_{zx} \\\\\n-S_{xy} & 0 & S_{yz} \\\\\nS_{zx} & -S_{yz} & 0\n\\end{pmatrix}", "7a43fab3f835012fe50f9a3eedc46e5f": " \\mathbf{m} = \\sum_{i=1}^N \\mathbf{r}_\\mathrm{i} m_i \\,\\!", "7a4442838d250866553f50ff7c2dc24e": "\\scriptstyle\\mathcal{C}", "7a447a644142f9d96a1e3f135f408146": "\\mathcal Q = (Q, \\sigma_f, I_{\\mathcal Q})", "7a44a3613f8856b7806b48e7d56dfc83": "f=(f_1,f_2,f_3,\\ldots)", "7a44f364fedd3a2f41fbb64884bea2ad": "Q_1=f_a(S_a)", "7a45376ef42cc17894d2007aaec6dc84": "\\omega_2(t)", "7a457aee8295600e513d7221f083f256": "\\theta^{\\prime}=\\left(\\theta_{2},...,\\theta_{n}\\right)", "7a45c7478d16285ecfc19db66340c1b8": " a_3a_2a_1 > a_4a_1^2 + a_3^2a_0 ", "7a45e4e02aefad60eac818f8a712484b": "x % y", "7a45f2cf3994b488f76c21aa9367bc4f": " X(\\omega) = \\frac{F(\\omega)}{6\\pi i R \\eta \\omega + \\frac{F}{l}} ", "7a461bf3d5a3f8f56c922338370a7203": "\\frac{\\partial }{\\partial \\tau} \\hat{f} = -\\frac{i}{\\hbar}[\\hat{f},\\hat{H}],", "7a4705b3dab5eb797393a99255bcc7c2": "1 \\over 9", "7a47789d6d3551579934c3e568e263f1": "f(1),f(2), f(3),\\dots", "7a4782280d56630832d8432e0098a49b": "E(B|A=a)>a", "7a47d5892b3ccd63d9652e0f2dfe6286": "Y_{t+1} - Y_t", "7a483b910546729e8390cddbf40bbd5f": "F(f : X \\to Y) = \\eta_Y \\circ f", "7a4871484bb2d41c0f9772e3030c7200": "n_u", "7a48ba667c19e094e0c8ac3863cc23c7": "E = \\frac{1}{2}m_1\\left(e^{\\Delta v\\ / v_\\text{e}}-1\\right)v_\\text{e}^2", "7a48c61bc39d52cc26b7dcacb44e6380": "(x-x_0)^{2k}", "7a4927fa010806acbcf3c69389824ff6": " f^* s=s\\circ f", "7a492b12f486db1fd5f8851f8e4fa206": "\n\\Gamma := \\left[ \\hat{S} ~ \\vdots ~ \\hat{M} ~\\vdots~ \\hat{S} \\times \\hat{M} \\right]\n", "7a49e7b5374a39e3882f7e2fafb71267": "K_P \\Delta + K_I \\int \\Delta\\,dt", "7a4a175ba15731da1d56cd0661080d25": "v_b", "7a4a2624e7c14d221c8211828e467235": " 2C(\\cos\\,kd-1)U_k = m{\\operatorname{d^2}U_k\\over\\operatorname{d}t^2}", "7a4a4e5ad57fd89ffd43b492bf6a5d6a": " \\mathbf{J} = \\int_{\\Delta t} \\mathbf F \\,\\mathrm{d}t .", "7a4a5bd0d7831d514bd651f700037681": "I_D = (2a) \\frac{W}{L} q N_d {{\\mu}_n} \\left[1 - \\sqrt{\\frac{V_{GS}}{V_P}}\\right]V_{DS}", "7a4a608d8a165d7c1ccc556b9d8060fa": " f(x) = \\frac{x^2 - 1}{x - 1} ", "7a4b769c0b07b74ee567b0078b8cf258": " x = a(Y, D Y, \\ldots, D^\\beta Y) ", "7a4c18cf1dac003961934be897571e64": "(R,PQ)", "7a4c266b000d6e412cfec3dfa6064931": "\n \\vec k^2 + \\vec k_D^2\n=0\n", "7a4c42679761c46b33249c7bd9e25ef4": "\n\\sqrt[12]{2^7} = 1+\\cfrac{7} {12+\\cfrac{5} {2+\\cfrac{19} {36+\\cfrac{17} {2+\\cfrac{31} {60+\\cfrac{29} {2+\\cfrac{43} {84+\\cfrac{41} {2+\\ddots}}}}}}}} = 1+\\cfrac{2 \\cdot 7} {36-7 - \\cfrac{5 \\cdot 19} {108-\\cfrac{17 \\cdot 31} {180-\\cfrac{29 \\cdot 43} {252-\\cfrac{41 \\cdot 55} {324-\\ddots}}}}}.\n", "7a4c5e5fc64b477f00934de7127b3a15": "K(u,v)={\\langle R(u,v)v,u\\rangle\\over \\langle u,u\\rangle\\langle v,v\\rangle-\\langle u,v\\rangle^2}", "7a4d018173de0a73d6b3a19b3ca7ede3": "{\\rm trig}(M)=(0,0,h)", "7a4d6446c903831aa37cc09e36c57463": "-\\overline{u'v'}", "7a4db8fc17eaa773e2cfa7dc769cea21": "\\mathbf{x} = \\mathbf{a}_1 = (12, 6, -4)^T", "7a4df3625c483421c6c8081b7a964534": "f/|f|", "7a4e34a26ce4f6d4daac804af448423c": "E_1 = k \\cdot E_0, ", "7a4e40c440fd38104aca0b02d7e34e9b": "\n G\\left(t;\\mu\\right)= e^{\\mu(t-1)}.\n ", "7a4e5a8ae4f5e0a40134b0631ff4fd6c": "A \\begin{bmatrix} 0\\\\0\\\\1 \\end{bmatrix} = \\begin{bmatrix} 0\\\\0\\\\3 \\end{bmatrix} = 3 \\cdot \\begin{bmatrix} 0\\\\0\\\\1 \\end{bmatrix}.\\quad\\quad", "7a4e73b9fa9dd489258b73c7d5be09fd": "\\frac {Imports+Exports} {GDP}", "7a4e79e5ce1c7aae68109494e67f6134": " \\psi(x) \\equiv \\frac{x \\log x}{x-1}= 1- \\sum^\\infty_{n=1}\n{(1-x)^n \\over n (n+1)} ~, ", "7a4e9ad305baaeab8e7c1d458d0a05d7": "\\nu : X \\to [0,1]", "7a4eb4cf45c44d1f42a37f1167780960": "\\mathrm{PS}_1 \\rightarrow \\mathrm{TT}_1 \\rightarrow \\mathrm{EE}_1 \\rightarrow \\mathrm{PS}_2. \\, ", "7a4efffbee99c165c333523dbebabab7": "\\mathrm{ord}(f')=\\mathrm{ord}(f)-1.", "7a4f3710d2814ec6dd47631f3a0a627f": "T^2= \\frac{4\\pi^2}{G(M+m)}a^3\\,", "7a4f51497ea01f6117f489e49ede6e74": "L \\cdot \\frac{8 f}{g \\pi^2 d^5} ", "7a4f89e182d85590b548456d5a539e4d": " R_1 = R_2 = R\\,", "7a4f9c0e34d7facb78f79ecf7db7da98": "A_n(V)=-kT \\log Z_n(V)", "7a50280a360b2789496b010dbf18d0e1": "a^{(p-1)/2}\\equiv -1 \\mod{p}\\,\\!", "7a503d6eb5e068dea12f4d11058f23ec": "j \\in [r(i),r(i+1))", "7a5068dc4086e71f7d581fb760bd2f59": " \\rho_G ", "7a507141d46c0ceb00aec0569b8aa0f3": "(m/P_y)", "7a50c5e42ad597da40aff03a15171016": "(22)\\qquad \\mathcal{L}_{\\bar{m}}m=[\\bar{m}, m]=\\bar{\\delta}\\delta-\\delta\\bar{\\delta}=\n(\\bar{\\mu}-\\mu)D+(\\bar{\\rho}-\\rho)\\Delta-(\\bar{\\beta}-\\alpha)\\delta-(\\bar{\\alpha}-\\beta)\\bar{\\delta}\\,,", "7a50c75cb7eeda6a78493fbcfe15d436": "\\mathbf x=\\mathbf s+t\\mathbf d,", "7a515a527383507e730e25fa4e6b5dc3": " \\langle \\mathbf{v} , \\mathbf{w}\\rangle ", "7a515ff683578b1b00d316a5dac8f221": " g_{ab} ", "7a51651643f7512d4a43bc33aad15193": "\\mathcal{Z}(a^n-b^n)\\subset\\{1,2,6\\}", "7a516ddee617bb61c3c326e5b4eafd83": "F_\\text{anis}(\\mathbf{m}) = -K m_z^2", "7a517cd9c1e8ee8955c11fa584234185": "f(s) \\equiv 0 \\pmod{p^{k+m}}", "7a519138544a9929a0b916dd16fa92db": "g[j] = \\sum_{k=1}^j D[k,k] \\qquad \\mathrm{for} \\qquad j = 1,\\dots,p ", "7a524290636843ec26fc40957bd38e98": "LS(K)", "7a5268aed60d7696aeebaa79b9befa4f": "\\displaystyle{\\alpha={1\\over 2\\pi}\\int_0^{2\\pi} \\kappa(\\theta)\\, d\\theta,\\,\\,\\, h(\\theta)=\\kappa(\\theta) -\\alpha.}", "7a52b64bc3b6fc5b02ec5112e4bd6b8a": "\\mathbf{u}", "7a540b5e017443b75eab34a78f045b3d": "\\psi \\in U (\\mathcal H)", "7a54415a2a92702b80ce992b739d5edb": "\\min_x f(\\mathbf{x})", "7a55284a615cce5560a2b042c04e39d2": "y=\\pm \\sqrt{\\frac{a^2b^2-b^2x^2}{a^2}}", "7a55364c18ef29fb686c6c4b69a7f5d9": "g(r) = 1 ", "7a556180944e6fa45abf27efdb89ab57": "A \\subseteq A \\cup B\\,\\!", "7a559a7c098df998968f14d81dae5d1c": " y_i = \\alpha + \\beta x_i + \\varepsilon_i", "7a55ac3d559308480327bda36deecfd4": "f:S\\rightarrow \\mathbb{R}", "7a55bd7782b9c5f524f2feddb530d754": "\\textstyle T = \\sum _{i=1}^N \\lambda_i E_i", "7a55ee2954975835beaa864a7ee73699": " \\langle q | q'\\rangle = \\delta(q-q')", "7a563df04727f10516c1cb73b56067b7": "p(\\theta|x)", "7a56bb86d110c7ad026d956ffdab7c14": "f(x;\\alpha,\\beta)=\\frac{1}{B(\\alpha,\\beta)}\\frac{\\exp(-\\beta x)}{(1+\\exp(-x))^{\\alpha+\\beta}}, \\quad \\alpha,\\beta > 0 .", "7a56d29de96a05fc48fe56b1f3413f43": "x = r \\cos \\theta, \\qquad y = r \\sin \\theta,", "7a5745a9287acf1c9be2206819256cdd": "\\lim_{x\\to\\infty}N/x=0 \\text{ for any real }N ", "7a576abdaaa1d9dda4acbf055a597497": "\\mathcal{L}_f = \\overline{Q}_i iD\\!\\!\\!\\!/\\; Q_i+ \\overline{u}_i iD\\!\\!\\!\\!/\\; u_i+ \\overline{d}_i iD\\!\\!\\!\\!/\\; d_i+ \\overline{L}_i iD\\!\\!\\!\\!/\\; L_i+ \\overline{e}_i iD\\!\\!\\!\\!/\\; e_i ", "7a577a4d03549543e08db388f57d37eb": "S'=\\{ s'_n \\}_{n\\in\\N}", "7a57cf31fbab49855bd329e6111fc5ed": "t \\leftarrow", "7a57f4d86ede9095e532f49d15b0047d": "\\{w | w \\not\\in L_1\\}", "7a5809b13f4d8cc85f2303ca9ca2496b": "\\dot{\\omega}_\\mathrm{MOON}=-0.00169(4-5\\sin^2(i))/n", "7a5846e4ce191f48dc88aae9a4318fac": "H = \\int_0^t \\exp\\left(43.2-16115/T\\right)\\,dT \\,", "7a584bc41faf28092d26f708407d0f69": "\\text{Hypothesis }H_0:\\text{ No leak}", "7a584e6ae86becd3a5a44178486a0788": "t_o", "7a588a297ab2d1d02be1e1443d333517": "\\scriptstyle F_\\sigma\\subseteq G_\\tau", "7a589c864d4e67b166f0a9e6e09d9bd0": "I_0 = \\left( \\theta -\\sin \\theta \\right) \\frac{r^{4}}{8}", "7a58a8bebddc9b4a9b71204e91a65fc3": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.8562 & 0.3372 & -0.1934 \\\\\n-0.8360 & 1.8327 & 0.0033 \\\\\n0.0357 & -0.0469 & 1.0112\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "7a58aa77028af4429d8880dbd7fbcd66": "F_0(x)=0 \\,", "7a58aed82e032fe19a0aa4e05e6a3cc0": "\n\\begin{align}\n\\lambda &= a + b\\omega \\\\\n\\omega\\lambda &= -b + (a -b)\\omega \\\\\n\\omega^2\\lambda &= (b-a)-a\\omega \\\\\n-\\lambda &= -a-b\\omega \\\\\n-\\omega\\lambda &= b + (b -a)\\omega \\\\\n-\\omega^2\\lambda &= (a -b) + a\\omega\n\\end{align}\n", "7a58dcb77c7fcd813d43b3569ba5fb49": "a + a + \\cdots + a", "7a595d9cd95d40aa7adf7f4046f2e346": "\\boldsymbol{L}_{xy}f(x,y)=0,", "7a5972319333294cce72caf58acaa9a0": "n_i/m_\\mathrm{solvent}", "7a59a8393d52abc4f11112ee9ec52dc0": "O(n\\sqrt{\\log n})", "7a59d19ae9b3d071830b1c286a0318fb": "\n R = \\cfrac{1}{4}\\left(R_0 + 2~R_{45} + R_{90}\\right) ~.\n ", "7a59d489c752aa58036307b69fb11597": "s_i = \\left(\\mu - \\frac{1}{\\rho d_i^2} \\right)^+, \\quad \\textrm{for} \\,\\, i=1,\\ldots,\\min(N_t, N_r),", "7a59febe6b078d3dca7829742ea83f59": "\\operatorname{mr} (G)\\leq |G|-1", "7a5a2e0c722eab0d58ab21c3ee1cbb46": "\\sum_{i=0}^{\\infty} \\, f(i) = \\infty", "7a5a5458472a0bf0f5b57ee7d909bec5": " \\mathrm{flattening} = f =\\frac {a - b}{a}.", "7a5a89dacaa687aa2a688b5d014a54c2": "{\\mathbb E}\\bigl[|X|^r\\bigr]\\le\\bigl({\\mathbb E}\\bigl[|X|^s\\bigr]\\bigr)^{r/s}.", "7a5aa3a19351ac5c9c9f7b9b0b1759aa": "\\sum_{i=1}^k p_i = 1", "7a5ae156fc2a13ef13f9446e8ffa6478": "\nS= N \\big( 3/2 log(E)- 3/2 log(3N/2)+log(V)-log(N)\\big ) ", "7a5b301ec3c689c76cee851901a44661": "\n \\delta G = -P~\\delta l + \\gamma~\\delta A = 0 \\qquad \\implies \\qquad \\gamma = P\\cfrac{\\delta l}{\\delta A} \n ", "7a5b4aa509882a9f5a5b1ff1a9668a18": "\\sqrt{\\frac{k-1}{k}}", "7a5b9ad4587ea12c1bbee0c70cba1dfc": "F(A_1, A_2,\\ldots, A_n)", "7a5c2167320b2e97569d16f8249cc0a7": "\\omega(t) = \\omega_0 + \\frac{4 \\pi L n_2 I_0}{\\lambda_0 \\tau^2} \\cdot t \\cdot \\exp\\left(\\frac{-t^2}{\\tau^2}\\right).", "7a5c411383568187fd6109830918963e": "\\sigma^2/\\sum(x_i-\\bar{x})^2,", "7a5c7b2df547e728efeac8f1006bbadb": "\\begin{align} a_1 &\\leftrightarrow& \\text{ 1st element} \\\\\na_2 &\\leftrightarrow &\\text{ 2nd element } \\\\\n\\vdots& &\\vdots \\\\\na_{n-1} &\\leftrightarrow &\\text{ (n-1)th element} \\\\\na_n &\\leftrightarrow &\\text{ nth element} \\\\\na_{n+1} &\\leftrightarrow &\\text{ (n+1)th element} \\\\\n\\vdots& &\\vdots\n \\end{align}", "7a5c95cf284a565453ac5f45b9e2843e": "||\\cdot||_\\mathcal H", "7a5d1220805b006dee1ef1a9f0c97349": "X_k \\in \\{X_1, \\dots, X_t\\}", "7a5d13026609fb75837d23a0d3cbc97e": "f(\\vec{x})\\rightarrow g(\\vec{x})", "7a5d164f3df0329a8032cda67d95d9d4": "\\textstyle y_{i}", "7a5d313be2ea111f6f4c67aa845525f9": "\\mathbf x_{k\\beta} + \\lambda_k \\frac{\\partial \\sigma_k(t)}{\\partial \\mathbf x_{k\\beta}}", "7a5d3e841d89f1de953f594bc5135341": "R_{\\rm l}^{\\rm pp}(r)=R_{\\rm nl}^{\\rm AE}(r).", "7a5e3bc9783c2d535fea122762c824ce": "\nx = \\frac{B+\\sqrt{B^2-4A}}{2}\n", "7a5e52c37956f4c46ddde5b033e42d88": "\ns_\\mathrm{a}(t) = A\\cdot e^{i (\\omega t +\\theta)}\n\\,", "7a5f091fb77d0705ff16528bbdc7a1d4": "W_m=\\operatorname{span}(\\psi_{m,n}:n\\in\\Z),\\text{ where }\\psi_{m,n}(t)=2^{-m/2}\\psi(2^{-m}t-n).", "7a5fb2bff2e88ff6cd91eb4bcb7ed544": " \\frac{\\sin{z}} {z(z-1)} = {\\sin{1} \\over z-1} + (\\cos{1}-\\sin1) + (z-1) \\left(-\\frac{\\sin{1}}{2!} - \\cos1 + \\sin1\\right) + \\cdots.", "7a5fd34de309e5455abd5e3b5103e80e": "\\int_V \\left( \\frac{\\partial |\\Psi|^2}{\\partial t} \\right) \\mathrm{d}V + \\int_V \\left( \\bold \\nabla \\cdot \\bold j \\right) \\mathrm{d}V = 0", "7a5fd48bbdc41c120789848c5b7a56c7": "\nMD = (\\lambda_1 + \\lambda_2 + \\lambda_3) /3\n", "7a60213af4a9e31737e633a559980795": "h[n]=4\\delta[n-3] + 3\\delta[n-2] + 2\\delta[n-1] + \\delta[n]", "7a60276760ae6d46a36a724b764b912d": "|S| = d+1", "7a6047136cd155e681432bd065adae64": " \\mathbf{G} ", "7a60645e52985fccc4632ecb50c4af42": "{CE}_{6}", "7a607433155a88c81e23ec6612b43fac": "\\tau(n)|n", "7a6089fa359efbaf00217bf8e79426da": "\\scriptstyle H", "7a608f8e97f628547eab7d7b5aaf5c6b": "\\varphi(n)=\\lambda(n).", "7a6186adb005d4411895baeee49572fa": "P - R = ET", "7a61a22cedf4bb79e77625d07aa8477b": "\\operatorname{E}[S_N]=\\operatorname{E}\\!\\biggl[T_N\\underbrace{\\sum_{i=1}^\\infty1_{\\{N=i\\}}}_{=\\,1_{\\{N\\ge1\\}}}\\biggr]=\\operatorname{E}[T_N].", "7a61ca9bc5226a383b1dfa3c6cb4181e": " \\hbar \\equiv \\frac{h}{2 \\pi} \\ ", "7a623e3b8e7ef00540e8c0356c7b53d3": " x(t_0) = x_0, \\qquad v(t_0) = v_0. ", "7a62b254a129a9b8172b9d2322824ec5": "(2n+1) P_n(x) = {d \\over dx} \\left[ P_{n+1}(x) - P_{n-1}(x) \\right].", "7a62d1ed7a671d5f750d70141dbe934e": "\\sigma^2_a", "7a62d2d0508785f53985bd74951f8ae3": "\\Delta t=\\frac{\\left(R+A\\right)l\\Delta h}{1-h}", "7a633b17dd58d10a4c9ad44a6b062401": " \\int_{S(r)} B(r) d^3r = -\\frac 2 3 \\mu_0 \\boldsymbol{\\mu} ", "7a637b7205f414a0177e98754748f084": " z' + (q_1 + 2 \\, q_2 \\, y_1) \\, z = -q_2 ", "7a637ed6ea0393959d94016836b2aa35": "\nr_{inner} = \\frac{r_{s} + \\sqrt{r_{s}^{2} - 4\\alpha^{2}}}{2}\n", "7a638474c4979a59896a1ec4c6523a5e": " t_l \\leftarrow t;", "7a639669e9efb5ba2e9f5509bf2ed776": "\\langle,\\rangle = \\lambda x \\,\\lambda y \\,\\lambda z.\\, z x y", "7a63e0bbed75e9a213c1a73b800adeda": "(\\mathbf h_k)_k", "7a63fb16f1f1a45ea9fe61e0f6d1c450": "\\scriptstyle{R_l^n}", "7a640afa159db8bfda965260bafb35bc": "x_{i}\\left( y,\\xi\\right) ", "7a64ba51543d0e3335e3294505e14f14": "(\\mathbb{F}_2)^d", "7a64d5239362d9fb4c20ac5e9010643e": "\\mathrm{slog}_b(1) = 0", "7a6533de128a7bc03bc24638d6b1a65f": "x=\\sum_{k=1}^{n} s_k a_k", "7a6578387c5b96b630c309ba04c76fbb": "|1\\rangle = \\left(0,1\\right)", "7a65be6eb04c8b3ed7f2abd668e3d4a7": "\\Delta f = \\frac{\\partial^2f}{\\partial x^2} + \\frac{\\partial^2 f}{\\partial y^2}", "7a65c77503ab8295d1d57c615d942555": "b \\to s l^+ l^-", "7a660bef0896e72440147aba3c8233c9": "O(kN)", "7a6630de906894ff94fbf71acd405b89": "l u =1 + \\varepsilon \\cos\\theta.", "7a6665e965a0769933db8235dd2b0f6a": "N_L", "7a6672ebeece1a1f4d102209c8d19b86": "\\sigma_c=\\infty", "7a6682401a995aaa7ee3597470f62741": "\\lambda\\rightarrow 0", "7a66988dc1a7f27c20c1062b57410783": "C = \\frac{1}{P}\\frac{{\\partial ^2 P}}{{\\partial y^2 }}", "7a66a409a1f31ec4579d8197e58cdba4": "1\\varepsilon + 2(1-\\varepsilon) = 2-\\varepsilon", "7a66bcc55175bbe25ec375f6fbb7eaaf": "\\displaystyle \n\\phi\\left(\\frac\nt{1-t}\\right)\\exp\\left(-\\frac{xt}{1-t}\\right)=\\sum_{n=0}^\\infty\\pi_n(x)t^n\n", "7a66c9e264e24f23463adbd420b43029": "\n D\\left( k \\right) \\mid_{k_0=k_B=0}\\;\n= \\;\n-\\left( {1\\over \\vec k^2 + k_X^2}\\right) \n", "7a6729b2fb4792154ca791608eef6bd5": "y(t+h/2)", "7a67336d269bac9abfad89fb6de44400": "V(+\\infty) + V(-\\infty) = n\\,", "7a677cb5a3ca6290fe59250d32ab28e8": "\\begin{matrix}\n\\ddot \\varphi_0 = - \\frac{g}{l}\\sin \\varphi_0 - \\frac{1}{2}(\\frac{a\\nu}{l})^2\\sin \\varphi_0 \\cos \\varphi_0 \\;.\n\\end{matrix}", "7a677cc9c7a399086a9548210dd2b90f": "W_\\min = \\frac{Q}{q_\\max} = \\frac{20}{16.4} = 1.22\\text{ ft}", "7a67a3431302c2c58c93bba250757ce1": "e=(e_0,e_1,\\dots,e_{n-1})", "7a67b0ec60f0685a58e13b422b6cd3cb": "d_s = \\sqrt{\\frac{A_p}{\\pi}}", "7a682b23cc203045a012e747e9dd9a4c": "\\scriptstyle \\frac{c}{2} \\frac{4\\sqrt{2}}{2\\sqrt{2}}", "7a68676442bd604307dfe294ab2c251f": "\\ \\cos(\\omega t + \\phi) = \\Re \\Big\\{ e^{j(\\omega t + \\phi)} \\Big\\}", "7a68f6c67913389ee5ff8572435ebd60": "p^r", "7a68f6d5885a70e8d7be52fec747b783": "l^2 \\ll A", "7a69582ec9938c92d7ebde7161ed5001": " G^{ab} = -\\Lambda \\, g^{ab} ", "7a69683a4a0e41bfcee9c8482ea68d9c": "0.\\overline{09}", "7a6a372c22e58d8c9d8deb1e408fb111": " = 8*10000 + 2 (10000/300) + 0.16 (300/2) = $80091", "7a6a4ab56b5659de4e43ac09913c342a": " \\mathbf{P} = \\mathbf{U\\Sigma U}^{-1} .", "7a6aa8531ecc2075b1456bac60b1c8a3": "\\mathbb E[\\bar v_N] \\to \\lambda", "7a6af1b2d41d5c593f4e410c8088a64d": "\\frac{D}{P} = r - g", "7a6b0d43955161ec6df8788f0c952ee9": "{}314\\frac{64}{625}<100 \\times \\pi <314 \\frac{169}{625}", "7a6b45cd430df158c3e1d3a8b9cd1cf7": "y = g_1(x)", "7a6b740fcdf2ea7866b5364b8f6434b3": "f(x;k,\\lambda)=\\frac{e^{-(x^2+\\lambda^2)/2}x^k\\lambda}\n{(\\lambda x)^{k/2}} I_{k/2-1}(\\lambda x)", "7a6ba4ecc5c1ff5937ea622e2736ceba": "Scenario \\quad I: \\qquad R_B = {\\left ( {\\frac {{0.05660} \\cdot 197.0}{2}} \\right )} = 5.582 AU = 5.6 AU", "7a6bacd4ca4a55a492ec46abf598cc45": "\\Lambda(n) = \\begin{cases} \\log p & \\mbox{if }n=p^k \\mbox{ for some prime } p \\mbox{ and integer } k \\ge 1, \\\\ 0 & \\mbox{otherwise.} \\end{cases}", "7a6c6688b84df2caee2d5acdb2c979d8": "\\phi(q) = \\prod_{n=1}^{\\infty} \\left(1-q^n\\right),", "7a6c83477d63945ab044bc5fc563904b": "\\alpha_k=1", "7a6cee92b27c1df4764b297b675cc71f": "\\ U_a (z) = \\,", "7a6d60c91652ef5067840cc0b2ead187": "\\langle B\\alpha^\\mu_{\\tau+i\\beta}(A)\\rangle", "7a6d9c3842cbdc928c8b54f72dc83d0f": "\nN_{tot} = \\int_{z_{b}}^{z_{a}} dz \\ c(z, t)\n", "7a6da08575af022664f20de663c181d8": "Y( x_{n_1+1}x_{n_2+1}x_{n_3+1}...x_{n_k+1}, z) \\equiv \\frac{1}{n_1!n_2!..n_k!}:\\partial^{n_1}b(z)\\partial^{n_2}b(z)...\\partial^{n_k}b(z) ", "7a6e428bb8a82480c148e29dce4bba7f": " \\omega_c t = {\\omega_0 \\over \\gamma} ( \\gamma t_0 ) = \\omega_0 t_0", "7a6e5e2d4cabdc473854689bc24f84dc": "I(t) = C_s \\frac{dU}{dt}", "7a6e71ddbdd546af1caa40447b1166a0": "\n\\Delta g\\ =\\ \\frac{P}{2\\pi}\\ \\int\\limits_{0}^{2\\pi} \\bar{V} \\bar{h}d\\theta\n", "7a6ef958310ddb7bd5f1cdb464523846": "N_{B/A}(\\mathfrak q) = \\mathfrak{p}^{[B/\\mathfrak q : A/\\mathfrak p]}, \\mathfrak q \\in \\operatorname{Spec} B, \\mathfrak q | \\mathfrak p.", "7a6f24e49b136f02da0d1a44f5676c0f": "f(\\vec{x},\\vec{v},t)", "7a6f45686151105770643bff887f6671": "\\varphi^n\\begin{pmatrix} A_L \\\\ A_S\\end{pmatrix} = \\begin{pmatrix} 2 & 1 \\\\ 1 & 1 \\end{pmatrix}^n\\begin{pmatrix} A_L \\\\ A_S \\end{pmatrix}\\, .", "7a6f6490da490733b4a54ee4cd7244ee": "\\ln(\\Delta l/\\Delta t) = ln(A') - \\Delta H/(R \\cdot T)", "7a6fd5a0b622c024a99d0af243d31643": "(\\mathbf{F}, \\mathbf{G}) = \\int \\mathbf{F} \\cdot \\mathbf{G} \\, dV", "7a6fe1b998fa3c4fddb3f311f5b97585": "E_K(IV_1 \\oplus P_1)", "7a706250d54238db51ddf23fef9aacce": "\\mathbb{D}^q(af)=a\\mathbb{D}^q(f)", "7a708d1929dd670172207060abae6fe3": "m_{i,j}=x^i \\cdot x^j", "7a7099fe845388fedc546319e90466ca": "\\mathbf{J}\\times\\mathbf{B}", "7a70e6dfac8555eda99201210ec5069a": " R' = \\frac{dR}{dQ} ", "7a713ca667c761b1c8c9767e34e5d068": "\\bar{r}_s = \\frac{1}{K} \\sum\\limits_{k=1}^K r_k,", "7a7162efc6240fec4d1565b1593983d1": "\\mathbf{Z_0}", "7a71733a4b789348250135130f9ce27f": "\\frac{(0,1,1)\\cdot (1,1,0)}{ \\Vert (0,1,1)\\Vert \\Vert (1,1,0)\\Vert }= \\frac{1}{2}", "7a71c7d23173b17d7395a743f5ff3be2": "f(x)=\\frac{1}{x^2-1}", "7a71c8a8dd8fa46b7ddf34eb1d2093df": "\n\\begin{align}\n W'&= -y_1(py_2'+qy_2)+(py_1'+qy_1)y_2 \\\\\n&= -p(y_1y_2'-y_1'y_2)\\\\\n&= -pW.\n\\end{align}\n", "7a71dcc70dfa1f5f5875a9a4034bb7a7": "\\sigma_{yy} - \\sigma_{yz} - \\sigma_{xy}", "7a71e8e156ac74fc48e8deeda9ee2e99": "\\mathrm{NS}(A)\\otimes\\mathbb{Q}", "7a720be92d46f9c210072703a62539ed": "\\psi - k\\beta = \\theta \\mod \\pi. \\, ", "7a720ea101a8968dd0f7780c10d62285": " \\{a\\}", "7a722b1887e58539dba1c481bad897cd": "|\\Pr(X_t \\in A) - \\pi(A)| \\leq 1/4 ", "7a725e1f95a78852326137c280d2854c": "\\alpha = \\sum_{i_1 0 \\,", "7a7c254ff8392aece81cfd4463e99fb2": "R(\\theta) = R_0 + (1 - R_0)(1 - \\cos \\theta)^5", "7a7c88034a741666b43a236b52e14d32": "EE(i)=(e^A)_{ii}=\\sum_{j=1}^{n}[\\varphi(i)]^2 e^{\\lambda _j}", "7a7cadeef922a8c44998e3995fbdc7e8": "R(Q)", "7a7d27655b10943ca46ae275a320ee20": "H \\otimes H", "7a7d38d8112138e62918fa3a02bf57eb": "\\overline\\partial", "7a7d5bbb4f6cb389c654e78c22e26b40": "\\scriptstyle\\{X_k\\}", "7a7d8f2a4639c94e247f4189711e803d": "\n\\begin{align}\n2^t &= 5t\\\\\n1 &= \\frac{5 t}{2^t}\\\\\n1 &= 5 t \\, e^{-t \\ln 2}\\\\\n\\frac{1}{5} &= t \\, e^{-t \\ln 2}\\\\\n\\frac{- \\, \\ln 2}{5} &= ( - \\, t \\, \\ln 2 ) \\, e^{( -t \\ln 2 )}\\\\\nW \\left ( \\frac{- \\ln 2}{5} \\right ) &= -t \\ln 2\\\\\nt &= -\\frac{W \\left ( \\frac{- \\ln 2}{5} \\right )}{\\ln 2}\n\\end{align}\n", "7a7daac122b5db67ca62f9fdc4a0e6b2": " \\sum_{\\ell} T_{\\ell \\ell} ", "7a7db5525c041ce8956b5886732997a6": "\\operatorname{dom}\\{x \\in X: f(x) < +\\infty\\}. \\,", "7a7df707c3a8aee09cc29602f33769b2": "F(s)=p_1 +s(p_2 - p_1).", "7a7e621451ba05577daa081a03536c17": "\\mathbf{p} = \\left( p_{1}, p_{2}, \\ldots, p_{N} \\right)", "7a7ea7a5e07dcd9b80db9647808410de": "(\\mathcal{F}f)(\\xi):=\\int_{\\mathbb{R}^n} e^{-2\\pi iy\\cdot\\xi} \\, f(y)\\,dy,", "7a7f3a9b31ddbb220f1622d57e56877f": " N ", "7a80502a79080e1b28542932b92d2eae": "d_i(p_{(n-1)})=d_{(i+1)}(p_0)", "7a813d1b6257b12f9da829350e18febe": "Z_{CO}", "7a817c9a43273df387768190400ca1a7": "K_n=2^{n-1}(n-1)!(k+n\\lambda).\\,", "7a8219bc75aea001d959c2619406d85b": "c_1,c_2,\\ldots \\in k", "7a82a35b44d586635ea59983a994b4db": " \\and (S_4 \\implies (\\operatorname{equate}[A_4, p] \\and V[F_4] = A_4)) \\and D[F_4] = D[p] ", "7a82a3bcc82fefd038d9800243be2367": "\\sigma>5 \\quad \\Rightarrow \\quad \\kappa_{\\sigma}<10^{-5}\\,", "7a82cfbd39c598e3ce8915aa4e8bbbd2": "f(z)\\in H_2 ", "7a82d2f830a1f7bbbb1cb03ca91a37d1": "T(n) = 1*0.4 + 2*0.3 + 3*0.1 + 4*0.1 + 5*0.1 = 2.2", "7a8343de29102998a1917c7eb8becff2": "\\mathbf{\\mathit{(X,A)}}", "7a83466a432395b96ac15d16d1e99bd9": "\\psi(\\Omega^{\\Omega2})", "7a83b62e67ed407ae8a4eecb5c33dd49": "\\displaystyle{H_\\varepsilon f \\rightarrow if}", "7a83cf8ee8d53e43a33d83bc08fec157": "m+n \\ge 1", "7a83e41b6fb1781b8a95811412667b28": "P=V\\cdot I =\\frac{V^2}{R} = I^2\\cdot R", "7a83f67058258eab5d7228c377fc28f3": "a_2(1320)", "7a8439644b4439514a7e06e6e84b75ae": " e^{-\\beta E}\\Omega(E)", "7a84473b376f6e8f1286f04a21a59106": "C_1 , C_2", "7a845655187a0429049603842335810e": "\\rho(\\boldsymbol\\beta,\\sigma^{2}|\\mathbf{y},\\mathbf{X}) \\propto \\rho(\\boldsymbol\\beta |\\sigma^{2},\\mathbf{y},\\mathbf{X}) \\rho(\\sigma^{2}|\\mathbf{y},\\mathbf{X}), ", "7a84806fc33e429ea0d629dd2b0e758a": "P_{\\nu_b\\rightarrow\\nu_a}^{(0)}", "7a84a37c864d0e0d6441e38883718d72": "\\mathcal{I}(\\alpha) = - \\operatorname{E} \\left [\\frac{\\partial^2}{\\partial\\alpha^2} \\ln (\\mathcal{L}(\\alpha|X)) \\right].", "7a84bf5b73c8b12e0421c9aba91e037d": "\\operatorname{Tr}(M \\operatorname{diag}(a) N \\operatorname{diag}(b)) = \\operatorname{Tr}(M^{1/2} M^{1/2} \\operatorname{diag}(a) N^{1/2} N^{1/2} \\operatorname{diag}(b)) = \\operatorname{Tr}(M^{1/2} \\operatorname{diag}(a) N^{1/2} N^{1/2} \\operatorname{diag}(b) M^{1/2})", "7a8509baec9e0c0c87d65cccf6cb9d7c": " \\mathrm{ESR} = \\frac {\\sigma} {\\epsilon' \\omega^2 C} ", "7a8578588c96a4a3624bad77564545ea": "\\ddot{x}^\\mu + \\Gamma^\\mu_{\\nu\\sigma}\\dot{x}^\\nu\\dot{x}^\\sigma = 0", "7a859c28ddd2d88e6d16166f0374c836": "\\begin{align}\n\\int_\\Omega [u(x)+v(x)]\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x & =\n\\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x +\\int_\\Omega v(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x = \\\\\n& =- \\int_\\Omega \\langle\\boldsymbol{\\phi}(x), Du(x)\\rangle- \\int_\\Omega \\langle \\boldsymbol{\\phi}(x), Dv(x)\\rangle\n =- \\int_\\Omega \\langle \\boldsymbol{\\phi}(x), [Du(x)+Dv(x)]\\rangle\n\\end{align}\n", "7a85b2d492413802d54ae96fbad13ea3": "\np \\overset{\\alpha}{\\rightarrow} q. \\,\n", "7a86071c0b599b899b7fedf638f118f7": " I_3 = \\sqrt{3}I_{31} \\angle (phase_{I_{31}}-30^\\circ) = \\sqrt{3}I_{31} \\angle (120^\\circ-\\theta) ", "7a862df000dc5235b5cb6f0d976efdb7": "f'(x) \\ge g'(x)", "7a86431ac91bdab9c43954a66b60c058": "S_l", "7a8650f48e963b69708a66ddf17074e8": "x[n]=\\mathcal{Z}^{-1}\\{X(z)\\}", "7a8656f182964354f54c8690c54cd16f": " i \\neq j ", "7a86b6f82324a9a3e4608bacd0afbbfb": "[t_0,t_1]", "7a8727f4a77cf731591691d0cb8ae6b6": "I(U) = \\{ f \\in K[X_1,\\dots,X_n] : f(x) = 0 \\mbox{ for all } x \\in U\\}.", "7a874046f9d3b5df8f1b2bc07f553fba": "F_1, F_2, \\ldots", "7a877e3cc3f4c16b4cbbd8cd2349a3a3": "G_{\\alpha \\beta}(\\mathbf{x},\\mathbf{x'})=\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell i k h_\\ell^{(1)}(kr) j_\\ell(kr') X_{\\ell m \\alpha}(\\theta,\\phi) X_{\\ell m \\beta}^*(\\theta',\\phi')", "7a87a4fc8e453118e832e23ff18bc9e7": "\\begin{bmatrix}\nc_2 & s_1 s_2 & c_1 s_2 \\\\\ns_2 s_3 & c_3 c_1 - c_2 s_3 s_1 & - c_3 s_1 - c_1 c_2 s_3 \\\\\n-c_3 s_2 & c_3 c_2 s_1 + c_1 s_3 & c_3 c_2 c_1 - s_3 s_1\n\\end{bmatrix}", "7a87dde9beef7660f8c999abb704f57e": "d(x,\\ z) \\le d(x,\\ y) + d(y,\\ z) \\ , ", "7a881000da6bbc7b34d40a92f7262b34": "c_{i,T_0}", "7a885aa9d8402bc64cd33baf34ab3515": "m_e", "7a888d2ce511d63133ebd383c63b4be8": "w_2\\in\\Sigma", "7a88be5f83a47cc4f0445e138ce7ddc2": " d = \\frac{v^2 \\sin(2 \\theta)}{g} ", "7a88fc514089033087c69d1c48393ee8": "[L_{ij},L_{kl}]=i\\hbar [\\delta_{ik}L_{jl}-\\delta_{il}L_{jk}-\\delta_{jk}L_{il}+\\delta_{jl}L_{ik}]", "7a89257b7e5be47f0a0f17748bc62301": "p=(e_1,e_2,\\dots,e_n)", "7a89670339af4f04990a990956d73e21": "e^{-\\frac x 2} \\, {}_2F_2 \\left(a, 1+b; 2a+1, b; x\\right)= {}_0F_1 \\left(;a+\\tfrac{1}{2}; \\tfrac {x^2} {16}\\right) - \\frac{x\\left(1-\\tfrac{2a} b\\right)}{2(2a+1)}\\; {}_0F_1 \\left(;a+\\tfrac 3 2; \\tfrac {x^2} {16}\\right),", "7a89b3821557393e2ef9c1d93f2de718": "\nNew average rate = \\frac{\\text{Total balance in company currency } + \\text{new funds in company currency}} {\\text{Total balance in voucher currency } + \\text{new funds in voucher currency}}\n", "7a89b67b978608ee90a102a4cbf28006": "\\gamma(\\mathbf{h})", "7a8a17c3f3af186aba34f147e3af6636": " D_\\tau = \\partial_\\tau + \\frac{1}{2} \\omega ", "7a8a75b5a5fa01455a699a52bed51988": "\\bigcap_{j=1}^n X_j\\ne\\varnothing.", "7a8b38e532a6895b01cdc48349e9f8bc": " f(1/z) ", "7a8b84e9d0e21a54d4826bee0a51cd06": "\\prod_p f(p)\\;", "7a8beb7401541be1feb55449d8017676": " dt \\approx \\frac{1}{\\sqrt{r^2-R^2}} \\; \\left( r + m \\, \\frac{R^2}{r^2} \\right) \\; dr ", "7a8bed611525a0058fa0ad5e09a88ca3": " \\scriptstyle \\sin (\\theta - 120^ \\circ)", "7a8cd1ec3739e71d108397da16df03ca": "f(x) = x^m", "7a8ce39f651a81781924ea0c17c0399a": "\\mathbf{F}_N =\n\\begin{bmatrix}\n 1 & 1 & 1 & \\ldots & 1 \\\\\n 1 & \\omega & \\omega^2 & \\ldots & \\omega^{(N-1)} \\\\\n 1 & \\omega^2 & \\vdots & \\ldots & \\omega^{2(N-1)} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 1 & \\omega^{(N-1) } & \\omega^{2(N-1)} & \\ldots & \\omega^{(N-1)^2} \\\\\n\\end{bmatrix}\n", "7a8da66f8347d50cbc2e500ba1d3f5e5": "\\boldsymbol\\psi_c(t)", "7a8dcebc3c4b26ada4b9629324ca6573": "3 \\uparrow \\uparrow \\uparrow 3 \\ = \\ 3 \\uparrow \\uparrow (3 \\uparrow \\uparrow 3).", "7a8e2cbcd528e2637b38da78fa962361": "\\mathcal{L}_{\\bar{m}}m", "7a8e3ec59d67076bc6dae09e814fa2dc": "C^I", "7a8e4546224c5d1cffa431654cdd6ce4": " \\Pr(X_{n'} = i | X_0 = i) > 0.", "7a8e88ff66a14f30f91a94c96a955a94": "x \\in H", "7a8ef289c064ecfd21fd58b17eb2a5cb": "V_{n,m}:=\\frac{x^{m+1}}{m+1}\\,_1F_1\\left(\\begin{array}{c}1\\\\1+\\frac{m+1}{n}\\end{array}\\mid -ix^n\\right)", "7a8ef29f6da19d45bc63aba0e537b779": "J_M=\\frac{f}{2} = J_H = \\frac{h}{c_p\\, G}\\,{Pr}^{\\frac{2}{3}}= J_D = \\frac{k'_c}{\\overline{v}} \\cdot {Sc}^{\\frac{2}{3}}", "7a8f08c1ca8036cd553a44f15d4788d2": "R = \\cap_{i \\geq 0} M_i = \\cap_{i \\geq 0} N_i.", "7a8fb11f95ecdf3c97ded5633de7879e": "\\bar{r}_{\\cdot j} = \\frac{1}{n} \\sum_{i=1}^n {r_{ij}}", "7a900b0303fb6c12222de9f9bb55dde1": "\\sup |\\Phi(x_n)|<\\infty", "7a905197cf247331d914e1732c3ce211": " COP_{heating}=\\frac{Q_{hot}}{Q_{hot}-Q_{cold}}", "7a9083a74022c1070398616571c84453": "(\\{T, F\\}, \\wedge)", "7a9087c8a56be8cf83cc2819b20e43a3": " H_{\\frac{1}{2}} = 2 -2\\ln{2}", "7a908cf2566950167069c006a7f93478": "\\left ( i\\hbar \\frac{\\partial}{\\partial t}- q\\phi\\right) \\psi = \\frac{1}{2m}{(\\widehat{\\mathbf{p}} - q \\mathbf{A})}^2 \\psi \\quad \\Leftrightarrow \\quad \\widehat{H} = \\frac{1}{2m}{(\\widehat{\\mathbf{p}} - q \\mathbf{A})}^2 + q\\phi.", "7a90c272ca8a9f8b4a64974ad485da88": "\\mathcal{N} \\to \\mathcal{N}", "7a90c29e01fbee7c2047b76cf303d8d8": "i=1,\\ldots,n-k", "7a90f6f51b9c3cb26373236b97285916": "\\mathcal{B} \\circ \\mathcal{C}", "7a90febb936c98522cf67c8b38a7b253": "\\frac{\\partial\\mathbf m}{\\partial t} = - \\frac{|\\gamma|}{1+\\alpha^2} \\mathbf{m} \\times \\mathbf{H}_\\mathrm{eff} - \\frac{\\alpha|\\gamma|}{1+\\alpha^2} \\mathbf{m}\\times(\\mathbf{m}\\times\\mathbf{H}_\\text{eff})", "7a91750f6875454996c37344adb505f6": "U_{-1}(x) = 0\\,\\!", "7a917618d7ac597dea48becb5b15b7f0": " [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \\quad ", "7a92112eac912e03ccb8314d24318858": "0=c^{T}\\left(\\left(\\sum_{i=1}^{t}\\lambda_{i}x_{i}\\right)-x^{\\ast}\\right)=c^{T}\\left(\\sum_{i=1}^{t}\\lambda_{i}(x_{i}-x^{\\ast})\\right)=\\sum_{i=1}^{t}\\lambda_{i}(c^{T}x_{i}-c^{T}x^{\\ast}).", "7a922d37e6db586c7aab59e504efff08": "\\widehat{\\theta}(x)=\\frac{\\sigma^{2}}{\\sigma^{2}+\\tau^{2}}\\mu+\\frac{\\tau^{2}}{\\sigma^{2}+\\tau^{2}}x.", "7a92cc376fec225a7326ceef945f84b8": "{d \\over dx} p_n(x) = p_{n-1}(x)", "7a92e091c608ebdbaf07cf8b081d77bc": "a_1,a_2 \\in \\mathbb{N}", "7a92eba150a2a2e1cb7c56f8d4b4aaf7": " \\mathbf{v}\\mathbf{w} + \\mathbf{w}\\mathbf{v} = -2 \\,d(\\mathbf{v}, \\mathbf{w}).\\!", "7a93077e20c487efb7237097a0a73489": "F \\colon A \\to C", "7a932276e50641b8e25c81844c75c866": "f(x_k+s_k)=f(x_k)+\\nabla f(x_k)^T s_k+\\frac{1}{2} s^T_k {B} s_k, ", "7a934f9180886ce28676e3c340d1ddda": "c = (1-p)(1-q), \\, ", "7a936f60716af563966a89aad24e87eb": "m(f) = 2\\pi^{-n(n+1)/4}\\prod_{j=1}^n\\Gamma(j/2)\\prod_{p\\text{ prime}}2m_p(f)", "7a93ed53b2cb48727197924bf1620900": "\\tilde{\\mathcal{H}}_{S}\\subset\\mathcal{H}_{S}", "7a9405be01e49762ef77bd9a63613163": "e^{-\\chi}", "7a9438660c709270414992de2fc915f0": "\\mathbf{B_{1}} = \\mathbf{B_{2}} ", "7a947be9369b31450940a07c89d7a253": " r_0 = \\infty \\,\\!", "7a948d3fa8000483bce8d5773f242711": " \\operatorname{E}(X) = \\alpha b / \\sin b , \\quad \\beta>1,", "7a9497ddabe3ad6bf3b4a61aedc6dc5d": "\\mathit{Vs}(a)=\\mathit{CT}_v(a)-\\mathit{CT}_f(a) \\,", "7a94bd8b8356c1eec431be85afb01c4c": "D_{\\rm AB}=\\frac{1}{3}\\sqrt{\\frac{k_{\\rm B}^3}{\\pi^3}}\\sqrt{\\frac{1}{2m_{\\rm A}}+\\frac{1}{2m_{\\rm B}}}\\frac{4T^{3/2}}{P(d_{\\rm A}+d_{\\rm B})^2}\\, ,", "7a94ca8e4efdc58247b67a60400e4312": "P(P'(Q/P)+1)=MC", "7a94e6e86afd297626fd3a98a8451507": "v(i) = a^i + a^{-i}", "7a9515a084f92ad3e7b882552507ee6b": "\\displaystyle{Q(a,c)={1\\over 2} (Q(a+c) -Q(a)-Q(c))=L(a)L(c) + L(c)L(a) - L(ac).}", "7a953e0b6a2163f580bf6df861eea7d4": "\\phi_{\\gamma_n ,e}(\\alpha)=\\alpha + [\\alpha(1)-\\alpha(0)](\\gamma_{n}-1)e. \\ ", "7a9545ea369cbf61539fc5c99f3fd84a": "\\{0,\\ldots, \\kappa\\}^S", "7a9547e103c7940f4a52d55765874481": "\n\\log \\Lambda = \\sum_n^N \\beta_n r_n - \\sum_i^I \\delta_i s_i - \\sum_n^N \\sum_i^I \\log(1+\\exp(\\beta_n-\\delta_i))\n", "7a955ddbc098d467090413c37fc44632": "Y_{1},Y_{2},...,Y_{N-1}", "7a9571df789c0580d9230c1df0934bfc": "S_{ij} \\,", "7a95797b39a7eb927739cb7683119328": " r_1 = 1/g_{1,2} ", "7a957fb758dbc3fffaa09e430c4db12c": " \\sum_i C^{S_n}_i \\varepsilon^i_{S_m} = -1 \\quad n = m ", "7a95be9f7df35fdae2d9ff31049a50f7": "A \\subseteq V(G)", "7a95fed7d9185a2d1a0ec0433cc3863f": " f(x) = m x + b \\qquad \\text{with} \\ \\ m = \\frac{y_2 - y_1}{x_2 - x_1} \\quad \\text{and} \\quad b = \\frac{x_2 y_1 - x_1 y_2}{x_2 - x_1}", "7a96291aadd1098b1ebd811a0c3285b9": "A = \\{ a\\in\\mathbb{Q} : a^2 < 2 \\lor a < 0 \\},", "7a962d109600af551ab5a60aa2206a65": "\\log \\left\\vert \\tan \\theta \\right\\vert = -0.2706462 \\text{ or } 0.2706462", "7a96633741489d7e17f60fd0b8c00be0": "f(a+h) \\approx f(a) + f'(a)h", "7a96891eb104a91b772c297e0747bee2": "M^{-\\bullet}", "7a9698905ae6306937fec6674179ce3c": "\\int_{X} f \\, \\mathrm{d} \\mu = \\beta(f).", "7a969f9f6967e460aa5d7c1add51d4e3": " G = G_m^3 ", "7a96a921b3115b8cf8a7008c2009d540": "\\xi^i \\rightarrow \\hat{\\xi}^i", "7a96aad8847f2b26884971b4f83e55e3": "n\\cdot m", "7a96cb1b31840322992f21aa871e62fb": " LP_t = -S\\Phi(-a_1(S,M)) + Me^{-r\\tau}\\Phi(-a_2(S,M)) + \\frac{S\\sigma^2}{2r} ( \\Phi(a_1(S,M)) - e^{-r\\tau}(M/S)^{\\frac{2r}{\\sigma^{2}}}\\Phi(a_3(S,M))).", "7a96d29cf6f8c095481712d5e0ae7bee": " \\displaystyle{f_s(D) \\subsetneq f_t(D)}", "7a96ee8588a98a6ba6aa30fe52e03454": " \\hat{C} = \\cos\\frac{\\hat{\\gamma}}{2}+\\sin\\frac{\\hat{\\gamma}}{2}\\mathsf{C}\n=\n\\Big(\\cos\\frac{\\hat{\\beta}}{2}+\\sin\\frac{\\hat{\\beta}}{2}\\mathsf{B}\\Big) \\Big(\\cos\\frac{\\hat{\\alpha}}{2}+\n\\sin\\frac{\\hat{\\alpha}}{2}\\mathsf{A}\\Big).\n", "7a976f03bf72c0006b645cc30c212bce": "r\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "7a97caaf7b15f4fe263b4e4346539d4b": "d(x_n, x_m) \\to 0", "7a97e350d1093af3b9bf84e07c768af1": "a_0 f + a_1 \\frac{d f}{d x} + a_2 \\frac{d^2 f}{d x^2} + \\cdots + a_n \\frac{d^n f}{d x^n} = 0", "7a9804ac49e7d04360d45983a58d1ea1": "\\mathbb{H} \\, \\tilde{x} = \\tilde{y}", "7a98452438effb239af7c6127b4c930b": "\\displaystyle F_0(q) = \\sum_{n\\ge 0}{q^{n^2}\\over (q^{n+1};q)_n}", "7a985351a480fa5a8912ee0246f8fd25": "|v\\rangle", "7a98e177892ec59e08a8f69884cf483a": "v_c", "7a99402c4b3792cf2a73099fcc566892": "\\operatorname{E}(T) = \\operatorname{E} \\left( \\operatorname{E}(T|R_1,R_2) \\right) = \\operatorname{E} (0) = 0.", "7a995fc060fe03bc59690eb901916379": "H = \\begin{bmatrix}\n \\sigma & \\varphi \\\\\n \\varphi & -\\sigma \n\\end{bmatrix}.", "7a9a2bdacb1c33540642e4431d489d28": "\\Gamma= (dy^i -\\Gamma^i_\\lambda dx^\\lambda)\\otimes\\partial_i, ", "7a9a55bf5b3dd743ac2a9cce401ab71b": "\\mu = {1\\over 2}({g^{(s)}}_p + {g^{(s)}}_n) = 0.879", "7a9a6baab6add3460181f93770e7bfdd": "\\mu(H)\\leq\\mu(G)", "7a9adbc30dbfa8fcd21d7f74ebfe29c6": "\\displaystyle n=3N", "7a9b49f8ee9d484a33b633739df51354": " (length)\\cdot(width) = \\mathrm{m \\cdot m }= \\mathrm{m^2} ", "7a9b79612693d35bda47197a3afb88d4": "\\exists {x}{\\in}\\emptyset \\, P(x)", "7a9b8ecafdcb250b3953fae2f3181b85": "\\;p(0) = P", "7a9bc56e479cde41489dd0031a86b3a9": "T_j = \\text{target for car } j \\, ", "7a9bf84c0e84420b750bb5ebe0cb2bc6": "U_\\mathrm{wall}", "7a9c1a1e443c8c0fbc9ae88b01680525": "H(x,\\nabla u(x)) = 0", "7a9c21ea4c47667a7e84a49aa6c367f7": "C_{P,m}", "7a9c62f524af405da73e14c649674720": "Y \\sim \\mathrm{Geometric}(\\tfrac{1}{1+\\lambda})", "7a9c7215125ab45b224d14a05477accb": "P = D_1\\times\\frac{1}{1+r}\\times\\frac{1+r}{r-g}", "7a9c8568d6bf9cffb3401e312d2c4a17": "T < T_\\textrm{inv}", "7a9c990b4495a5bdf227decf7012a789": "V \\frac{\\partial S}{\\partial V} = - \\sum_i \\frac{V}{\\omega_i} \\frac{\\partial \\omega_i}{\\partial V} \\;\\; k_B \\left(\\frac{\\hbar \\omega_i}{k_BT}\\right)^2 \\frac{\\exp\\left( \\frac{\\hbar \\omega_i}{k_B T} \\right)}{\\left[\\exp\\left(\\frac{\\hbar \\omega_i}{k_B T}\\right) - 1\\right]^2} = \\sum_i \\gamma_i c_{V,i}", "7a9d0042bd9572dd5902865fbcd0a919": "\\theta_{r_i}(x) = \\angle(x-r_i) \\quad (8)\\,", "7a9d09a9054e6fde9ac57d19f7df8a1a": " P_{warm} = \\frac{P_{cold}} {\\eta_C \\ \\eta_{p}} ", "7a9d3cb6ccbe99e2d5af744fd69d4f26": "n = e_1", "7a9d409c8241b56490c850739a83f5aa": "N/4", "7a9dccd2182637032d6d7cbd4b91fe07": "\\dot{\\hat{x}} = ( A - L C ) \\hat{x} + B u + L y \\, ", "7a9de071691d17e208680618e3766d2c": "n = a^2 + b^2 = c^2 + d^2", "7a9de6c99a315dfa814f9246a9ad8fe5": " cov[u_i,X_i]=0 .", "7a9e0064d98490e053c2310188c67813": "\\mu_r = \\frac{\\mu}{\\mu_0}", "7a9e5ef6a10cf9c2024de5bbccce8bcd": " h(r) =h(0) + \\frac{1}{2g} \\left( \\mathit{\\Omega} r \\right)^2 \\ ,", "7a9e60b328618e0907bcc07bcbd94d5c": "H(x,p)=\\frac{1}{2}g^{ab}(x)p_a p_b.", "7a9e61f6ded4b11647852c05edcaa3bd": "\nf = xf + x^2 f + x . \\,\\!\n", "7a9e737b325967ac404abd6e03d3650b": " \\gamma_{x}^{+} \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\{ \\Phi(t,x) : t \\ge 0 \\} \\,", "7a9f0e606d8b289221b418693c94b8e1": "\\|u\\|_{BMOH}=\\sup _ {|a|<1}\\left\\{\\frac{1}{2\\pi}\\int_{\\mathbf{T}}\\frac{1-|a|^2}{|a-e^{i\\theta}|^2}|f(e^{i\\theta})-u(a)|\\,\\mathrm{d}\\theta\\right\\}", "7a9f2b4232e922472db399733541e4ee": "A_{21} + A_{12} > 4 ", "7aa01d13c60b28b0bf31684984f36003": "\\lim_{x\\to a} f(x)=L", "7aa042f7e4073ea3bb92374c25c1004a": "1 - a_i z^{-1}", "7aa07e20e3b9188c49fd964f767e43ae": "ncp_t", "7aa0a75972716ef1865e06c5fc14b7f6": " \\varphi : U \\subseteq \\mathbb{R}^n \\to \\mathbb{R}", "7aa0cc5b76475cf0abfdacf6cfd2e121": "|\\alpha-\\frac{a}{q}|<\\frac{1}{q^2}", "7aa0f72579aa595b7f0a5f1550442459": "t_5", "7aa1a09a2c404debb17ca7bef97d7ab5": " \\boldsymbol{a}=\\frac{\\operatorname{d}^2\\boldsymbol{r}}{\\operatorname{d}t^2} \\ , ", "7aa1b6a0908704c725c7570b334e1986": "S_B(2) > S_L(2)", "7aa1c804693fa3a3ee16bcfc84f182f4": " \\frac{ Y / m }{ Z / n } = F_{ m, n } ", "7aa20604e9c3b21f8c754a88adb9d8d5": "\\boldsymbol{F_2}=-\\boldsymbol{F_1}", "7aa212a947f54bea64f0551a18d5edcf": "T_{a,r}(x)=\\frac{x-a}{r},", "7aa237a9153fec44f57f24140a6d00ca": "\\textstyle X^+", "7aa2621dfb0341ea3de0e4a1d7f89bfa": "C_\\mathrm{no-CSI} = E\\left[\\log_2 \\det\\left(\\mathbf{I} + \\frac{\\rho}{N_t}\\mathbf{H}\\mathbf{H}^{H}\\right)\\right].", "7aa357bf074bd2cbaaad52b1a455fe50": "\\operatorname{Tr}\\, K = \\int K(x,x)\\,dx", "7aa392d023b961ca9611047609931bdc": "\\left\\{ i \\in I: a_i = b_i \\right\\}\\in U \\, ", "7aa3b8aa03017a5deed74651d306c000": " \\frac{\\partial(x,y)}{\\partial(s,t)} ds \\wedge dt ", "7aa3f2a25e0c74d886fbfe6254299778": "\\sigma_Y \\neq 1", "7aa41487a1a40b0077afa0c3331ba111": "\\Delta p", "7aa41f3062ebea02b1ae89b9d36da3c9": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} I_{l} r^{l} P_{l}(\\cos \\theta)\n", "7aa430f193adf4c75b059a1f90ec2c40": "\\,s_{\\overline{n|}i}", "7aa52e7dfa25b8b0195f1da0c838323b": " k = | \\mathbf{k} | = { \\omega \\over c } = { 2 \\pi \\over \\lambda } ", "7aa563740c76a18f90c637699e603375": "R_n(x)=\\frac{1}{(x+1)^n}\\sum_{m=0}^{n} (-1)^m{2n \\choose 2m}x^{n-m}\\,", "7aa56713849e5d11bec54b4ab2628dd7": "E_\\text{P} = \\sqrt{\\frac{\\hbar c^5}{G}} \\approx", "7aa587488381f95e00ad0c0554d6932e": "\\textstyle(x\\pm1, y\\pm1, z\\pm1)", "7aa59df92d94027a05d3eb55d618a566": "d_n (f,g) = \\sup_{x \\in K_n} \\delta(f(x), g(x))", "7aa59e457dfef84ab572a9cd758f2ffa": "V_{out1}", "7aa5cd211eb9078ad4979b1584338669": "f(x) = \\frac{1}{(2\\pi)^n} \\int_{\\mathbf R^n} \\hat{f}(\\omega)e^{i\\omega \\cdot x}\\,d\\omega.", "7aa5e1c0c8e2d16a7ee4df2f7592f742": "y = x' w x''", "7aa60f975b7406f7d7beafaef9074ccf": "C \\subset \\mathbb R^2", "7aa62b060c551b741dc82f612f6d7b3a": "V^{2}+D^{2}=1", "7aa62c6f01a8f162ca4d08e7dceb7324": "\\beta_1 \\ge \\beta_2 \\ge \\ldots \\ge \\beta_k \\ge 0", "7aa669e3a2533f0fa4531823154f3b84": "\\int_{-\\infty}^{\\infty} e^{-x^2}\\,dx = \\sqrt{\\pi}.", "7aa682b3328d4c15c3e2cac44bbe0995": "\n S(\\boldsymbol{\\phi}(y)) = y_1^2 + \\cdots + y_{r-1}^2 + \\sum_{i,j = r}^n y_i y_j H_{ij} (y),\n", "7aa6c550af9d17bb09a240f7383abdd8": "n = [m_1,...,m_k]", "7aa6cca1ad30ee46ae02166ba489ade7": "K\\ge0", "7aa74b10d1432766fd267d14f7ca5ec2": "k_3 = 1.90476.", "7aa7b3aed26ca9529789dffd9b016bdb": "\\mathbf{r}_i", "7aa7b8e6086af154767f8aec38f4090b": " \\theta = 0,\\,", "7aa8282ec65818972dc641a51762650e": "\\mathbf{e}_j", "7aa865157a59a9db261ff32200c89ad3": "J = - \\frac{1}{2} \\left[\\frac{ N(x + \\Delta x, t)}{a \\Delta t} - \\frac{ N(x, t)}{a \\Delta t}\\right]", "7aa86883cc3afaaefb77205049c614ad": "\\textstyle{4(\\frac{5}{3})^n}", "7aa8f1d0eff7eda919f9874e0803f289": "\nY^2=\\frac{N(N(r_1+2r_2)-R(n_1+2n_2))^2}{R(N-R)(N(n_1 + 4n_2) - (n_1 + 2n_2)^2)} \n", "7aa8f8b716a9da1d43c65c37c6b059c7": "\\mathcal{E} =", "7aa94a67921b355676ffeb35f77a1342": "\\arccos\\alpha \\pm \\arccos\\beta = \\arccos\\left(\\alpha\\beta \\mp \\sqrt{(1-\\alpha^2)(1-\\beta^2)}\\right)", "7aa99d06072650d3bb845221887395eb": "T(b_1)", "7aa9bb62bbe58fdac8359f08dd718c4b": " g(z) \\leq f(z, w), \\forall z, w ", "7aa9de8d38c2d3c32dc59f1ff700dd90": "200MeV g^{-1} cm^{2}", "7aa9f558abda6b744498e7699174b1a5": "\\frac{t e^{xt}}{e^t-1}= \\sum_{n=0}^\\infty B_n(x) \\frac{t^n}{n!}.", "7aaa11dd013cdf8cdba664ffca7de862": " x(t) = \\sum_{i = 1}^n a_i \\exp(t e_i) E_i ", "7aaa1a5a8e00649006e4e839da1cc758": " 180^{\\circ} < \\Theta < 360^{\\circ} \\, , \\,\\, \\Theta = \\left( 360 - \\frac{180L}{\\pi R} \\right) ^{\\circ}=2\\pi-\\frac{L}{R}", "7aaab63f5ffd6bab40aa56a5b6ab5c3e": "\\Pi_f=\\frac{m_\\text{fuel}}{m_\\text{initial}}", "7aab0bd65046beadb192a0a8ccc9623d": "v_{e0} = 0", "7aab1c45586466e19e87eb0723ae6c79": "h\\sim{2 \\times 10^{-20}/\\sqrt{\\mathit{Hz}}} ", "7aab1ec7a17d2f01177f37a1c4d42fa0": "x W_1 y \\leftrightarrow f(x)W_2f(y)", "7aab2aa40647128e052b1899bc812b44": "c_1 \\mid \\uparrow \\rangle + c_2 \\mid \\downarrow \\rangle", "7aab775dc09d0e54c7a29ba3dc04d594": "g=\\frac{v^2}{r}\\frac{(h_a+h_b)}{G}", "7aaba2b91dc7a1da1b997635aef5d4a1": "e = \\frac{{TL}}{{EA}}", "7aabedac6c6a137732aefedfd6d3fd83": "r_{i+1} = \\operatorname{Mut}(f,r_i,s,t)", "7aac3a98223e91a8b6057f670d0c1851": "\\mathcal{E}'(\\mathbb{R}^n)", "7aac5e91f424e0da29cd18cbdc35a574": "\\# X (\\mathbf{F}_q) = q^{\\operatorname{dim}X} \\sum_{i \\ge 0} (-1)^i \\operatorname{tr}(f; H^i(X, \\mathbb{Q}_l)),", "7aacb0c532c5f1c886f29f24edcd790f": " U(i,\\mu,\\gamma) = (\\overline{x_i}-\\overline{x_{i-1}})^\\mu (\\overline{y_i}-\\overline{y_{i-1}})^\\gamma ", "7aacbcdf7afe6e1424e25289988b32c8": " a_{10} = \\mathcal{L}(p_4) + p_3p_4+p_1p_6, ", "7aacc92af341bc5ffd16fbccbc252312": " [n,k_1 - k_2, d]", "7aacdea33128e1318c71fc6329034364": "\\begin{matrix}\nr = p \\land q & \\Leftrightarrow & r = p \\cdot q \\pmod 2 \\\\\n\\\\\nr = p \\oplus q & \\Leftrightarrow & r = p + q \\pmod 2 \\\\\n\\end{matrix}", "7aadaf55971624a6e92a78a2817d56ec": "s \\in \\operatorname{Pref}_L(t)", "7aae0559cdecad7bfae1ac4e7eb9174d": "w_i = g(\\Theta_i|Z)", "7aae515211d6e191b92fd346ec026b04": " \\overbrace{\\smile \\smile \\smile-}^{\\mathrm{Foot 1}} | \\overbrace{\\smile\\smile\\smile-}^{\\mathrm{Foot 2}} \\star \\overbrace{\\smile\\smile\\smile-}^{\\mathrm{Foot 3}} | \\overbrace{-\\smile-}^{\\mathrm{Foot 4}}", "7aaec31c9ffebb03565865a062306c0b": "(10 \\uparrow^n)^{p_n}", "7aaee147f8604d50c829e9a13ede260e": "{b}\\,", "7aaee5e688f04fd76616870c4c55e9d9": " H_n = \\int_{X_n}^\\oplus H_x \\ d (\\mu | X_n) (x) ", "7aaef92d1d4f2675acb55d1a1724b6b9": "v^2 = u^3 + \\frac{A}{B}u^2 + \\frac{1}{B^2}u", "7aaf66840bada82bf68bd9bee8d25084": "S^7\\hookrightarrow S^{15}\\rightarrow S^8 \\,\\!", "7aafc8d8be268be55fc79997e20746fb": "t\\mapsto c(t)+{ c'(t) \\cdot (P-c(t))\\over|c'(t)|^2} c'(t)", "7ab07394e3c40fbd09b591d0a376b8c8": " d=(d_1,...,d_n) ", "7ab0d5faebbd4cf222cd39370878ebfd": "u(\\C) ", "7ab0d7e79984a30dbca130fca529834e": "S_i-S_{i-1}", "7ab0fceeaba84e30cf956ab8f5b1cd47": "d_m=p_0+p_1d_{m-1}+p_2(d_{m-1})^2. \\, ", "7ab14c44190451b6ed02a1067c65b09c": "\\left\\lfloor n/3 \\right\\rfloor", "7ab17262fca9567a2c70c9217d8ce5b1": "0 = 0 \\times c", "7ab17fdb73554eb37d58503c692ecccd": " \\partial W = M_0 \\cup M_1", "7ab18a56d78ff83b9750190f92afd149": "f \\rightarrow f + mF", "7ab23bc49c489b660316117627b068d2": "\\begin{align}\n f(\\mathbf{x}) & \\approx f(\\mathbf{a}) + (D f)(\\mathbf{x}) + (D^2 f)(\\Delta(\\mathbf{x-a})) + \\cdots\\\\\n & = f(\\mathbf{a}) + (D f)(\\mathbf{x - a}) + (D^2 f)(\\mathbf{x - a}, \\mathbf{x - a})+ \\cdots\\\\\n & = f(\\mathbf{a}) + \\sum_i (D f)_i (\\mathbf{x-a})^i + \\sum_{j, k} (D^2 f)_{j k} (\\mathbf{x-a})^j (\\mathbf{x-a})^k + \\cdots\n\\end{align}", "7ab242e1f593324f9ed4625df12e4d77": "\\dim_\\text{upper box}(A+B)\\leq \\dim_\\text{upper box}(A)+\\dim_\\text{upper box}(B).", "7ab287f591a72d0db2b59cec21b3bb75": "Q/(2\\pi)", "7ab2d94165bf5d90ee48dc2f45c428d4": " HRR = \\frac{a^'}{b^'} * \\frac{d^'}{c^'}", "7ab3043641f757dfed4bc239a49d8f09": "(\\phi \\leftrightarrow \\chi) \\to (\\chi \\to \\phi)", "7ab30c010e3ab1738df7636270d93418": "A \\in\\ \\beta", "7ab407b4fc095bfb43b3f79bb007087c": "\nP(w_i|w_{i-(n-1)},\\ldots,w_{i-1}) = \\frac{count(w_{i-(n-1)},\\ldots,w_{i-1},w_i)}{count(w_{i-(n-1)},\\ldots,w_{i-1})}\n", "7ab4124e65f52cc2b12f2699715a19e8": "\\forall V \\in \\mathcal{V}(x) \\quad \\exists B \\in \\mathcal{B}(x) \\mbox{ with } B \\subset V", "7ab42354fdb774bcbae380f26f842aa8": "t \\equiv u", "7ab443eda2df1e9c3ec2a3944b8c1a48": "c_{7}*c_{9})", "7ab4e19ee5fc788b3f6280202cd9443e": "(\\Omega,\\Sigma),", "7ab4e1c6da2a5c758e43948b6134d29b": "W_B = cW_{TOT}", "7ab4fba0f43367cedd2d2e8ace5385eb": "{\\it{N \\times 1}}", "7ab5130e10a2625b1e8be48adcd60406": "\\mathrm{\\Beta}(n,m)= {(n-1)!(m-1)! \\over (n+m-1)!}={n+m \\over nm{n+m \\choose n}}", "7ab5306af61abbc42cf96312ba9d90ac": "= I + uv^T A^{-1} - {u(1 + v^T A^{-1}u) v^T A^{-1} \\over 1 + v^T A^{-1}u}", "7ab53661ae5750a24ef156bc3e511a96": "\\frac{\\phi \\not\\vdash \\neg\\theta \\quad \\phi \\vdash \\psi}{\\phi \\wedge \\theta \\vdash \\psi}", "7ab53ac13408b22dcba7511fbece3811": "\n \\frac{\\partial \\boldsymbol{A}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\left[\\frac{\\partial }{\\partial \\alpha} (\\boldsymbol{A} + \\alpha~\\boldsymbol{T})\\right]_{\\alpha = 0} = \\boldsymbol{T} = \\boldsymbol{\\mathsf{I}}:\\boldsymbol{T}\n", "7ab583b51a4de8ef4469752b02854307": "\\left \\{ - {\\hbar^2 \\over 2m_0 r^2} {d\\over dr}\\left(r^2{d\\over dr}\\right) +{\\hbar^2 l(l+1)\\over 2m_0r^2}+V(r) \\right \\} R(r)=ER(r).", "7ab5a9130d84ee5780402bc53f321bca": "L = \\langle \\in \\rangle", "7ab5e69c19644554c297ea4cc509a120": "\\frac{\\alpha(\\alpha+\\beta-1)}{(\\beta-2)(\\beta-1)^2}", "7ab5eddfef71e90ddfb3c72b774fcd1a": " t \\rightarrow \\infty ", "7ab62673984b6049f605081852905872": "\\{\\mathbf{v}^K\\}", "7ab62799c0360779c811453a9ded7876": "\\begin{align}\n \\dot{\\mu} & = \\mu' \\\\ \n \\dot{\\mu'} & = -\\partial_\\mu F(\\tilde{s},\\mu )-\\kappa \\mu'\n \\end{align}", "7ab63bed730d191c37d7dafa481ddbe9": "\\psi \\left({\\mathbf r}, t \\right) = A \\cos \\left(2\\pi({\\mathbf k} \\cdot {\\mathbf r} - \\nu t) + \\varphi \\right)", "7ab6bc75ead20c25496fe82b16463496": "C_s = (\\frac {1 +2a}{1 + a})C", "7ab703b4b431309c6774792863b7cdb2": "f_{a}(k) = g^{a_{2}^{x_{2}} ...a_{n}^{x_{n}}}", "7ab7455a90409ff150a36c5870e48266": "U |\\phi\\rangle_A |e\\rangle_B = |\\phi\\rangle_A |\\phi\\rangle_B \\, ", "7ab773ada516d04617eed8377f41cd3d": "\\mathfrak{C}_{\\operatorname{even}}", "7ab7b3b89813a1da9e7d9cf7e083084d": " Q_{Corrected} = Q_{Measured}*{\\rho_{In} \\over \\rho_{Out}}\\,\\!", "7ab821c6df20c9c8b42fbba8b39da69c": "\\min_{d\\in D}\\max_{s\\in S} r(d,s)", "7ab84d99994425a14740aed1a6ed34e3": "\n \\nabla I(x,y) = G(x,y) = \\sqrt{ G_x^2 + G_y^2 }.\n", "7ab862331ac2d6b684333993c872132c": "\n\\mathrm{area}(D)=\\int_\\gamma x\\,dy=-\\int_\\gamma y\\,dx\\,,\n", "7ab86ac12ecc3f336cfdb0d0d530aec2": "\\mathbb{R}\\times \\mathbb{R}", "7ab89435eccf78608747fd60c1274a97": "\\text{Var}(Y_1) = \\text{Var}(Y_2) ", "7ab8a3bb54ba69d041d651fc6481a984": "=\\sum_{k=1}^{d} \\dot q_k \\ \\boldsymbol{e_k} + \\sum_{j=1}^{d} q_j \\ \\dot{\\boldsymbol{e_j}} , ", "7ab90654f6bafc648d65073164dbd195": "\\cos A = {\\sin A \\over \\tan A} ", "7ab917205ee30ad137f0bbc204c9400b": "\\sum_{n=0}^{1 - a_{ij}} (-1)^n \\frac{[1 - a_{ij}]_{q_i}!}{[1 - a_{ij} - n]_{q_i}! [n]_{q_i}!} f_i^n f_j f_i^{1 - a_{ij} - n} = 0.", "7ab98b3dfa75b81bf2773e2e5f9cc127": "Z_{P^n} \\times_{K(l)} \\mathbb{C}_i \\longrightarrow M^{2n}", "7ab9d8a97f2ad3481916af24e83aefc9": "h = \\left \\| \\mathbf{h} \\right \\| ", "7abaa13d67e2df8ed423fad898c7e832": "x_3=\\frac{x_1-y_1^3\\cdot x_1}{a\\cdot y_1\\cdot x_1^3-y_1} ", "7abaa63b40cac875f6bd417f102f0679": "P = (\\frac {D}{D_{p}})^2 = (\\frac {254}{7})^2 \\approx 1316.7", "7ababa8311941d5b80d5c310cb27949f": "\\overrightarrow{OA}", "7abb05f61f8375973b897986f5a82bcf": "C_\\text{M} = C(1 + A) \\,", "7abb11d7a562ac89ffbe5b97af5f7e72": "\\delta_w", "7abb1a9475f7f4951c69ca6688f23697": "n \\geq p", "7abb2f586bd276aea259701a17936229": "\\forall A \\, \\forall B \\, ( \\exist X \\, (X \\in A) \\Rightarrow [ \\forall Y \\, (Y \\in A \\iff Y \\in B) \\Rightarrow A = B ] \\, ).", "7abb5eca88eb4ef0364e8325593d414c": "G_{ab} + \\Lambda g_{ab} = {8 \\pi G \\over c^4} T_{ab}", "7abbcf0ce3c8e0797da5ba0684e82a50": "C_{p}", "7abc3091a407cdb0aa36bb03d81ee207": "A \\in \\mathbb{B}_{b}(S)", "7abc72e752576c9a9222a4a063f49379": "LLLL", "7abc7702fcaab2f6d3bdfe7807d89273": "\\mathrm{proj}_{\\mathbf{e}}\\mathbf{a} \n= \\frac{\\left\\langle\\mathbf{e},\\mathbf{a}\\right\\rangle}{\\left\\langle\\mathbf{e},\\mathbf{e}\\right\\rangle}\\mathbf{e}\n", "7abc79371d14255899b8e1c9cb7e6639": "L=2+\\frac{2}{d_n}", "7abc7c5c2ef335e04f16f8293762b63c": "L_2 + L_3 = 0.177\\lambda + 0.152\\lambda = 0.329\\lambda\\,", "7abca79a54a4c843a80c7c5755433eaf": "d(w)\\geq d(v)", "7abccab1740c3133946788df541dfe92": "(M', g')", "7abcec3163150c72fa44066ed0751122": "\\log\\zeta(s)=\\sum_{n>0} \\frac{P(ns)}{n}", "7abd30dee40707019015fafc98dbef7b": "{TL}_0", "7abd41e0899abe3d55c4eb90a90e7a72": "\n\\lim_{n \\to +\\infty} \\left( \\frac{\\int_a^b e^{nf(x)} \\, dx}{\\left( e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}} \\right)} \\right) \\ge 1\n", "7abd4c6c3bf551d6588f2b887017a6e1": "\\text{Informedness} = \\text{sensitivity} + \\text{specificity} - 1=2*\\text{balanced accuracy}-1", "7abd9d95d243357b92ffbc61d5d488a4": "\\partial P(\\sigma) = f_{\\sharp}(\\sigma) - g_{\\sharp}(\\sigma) + P(\\partial \\sigma).", "7abd9e5eed3cbd2c5c1a023e6808f46a": " Fd \\,\\!", "7abdac088b3dfd90e7d7db70e92f985f": "\\|u-u_h\\|\\le C \\|u-v\\|", "7abdbc035db62c55e90d831d7d83ff1f": "\\frac{\\partial^2 u}{\\partial x_1^2}+\\cdots+\\frac{\\partial^2 u}{\\partial x_n^2}-\\frac{\\partial^2 u}{\\partial y_1^2}-\\cdots-\\frac{\\partial^2 u}{\\partial y_n^2}=0.\\qquad\\qquad(1)", "7abdbeafb8abd8d79808a30381f5edfa": "\\left(1 -\\frac{t}{N}(1 - \\cos \\theta) \\right)^N = F_N(\\theta, t),", "7abdc069838a324488d58e7c4845ae45": " (a_1, b, 1) \\sim (a_2, b, 1) \\quad\\mbox{for all } a_1,a_2 \\in A \\mbox{ and } b \\in B.", "7abdc15e44d772f18b8ba87c662b649f": "d\\omega+\\omega\\wedge\\omega=0,", "7abdca834217e2e0fb88f041372a2f99": "V_g", "7abde3e599a6589f17c492ef4a0b46a4": "\\rho(T) = \\rho_0 + aT^2 + c_m \\ln\\frac{\\mu}{T} + bT^5,", "7abe62849f32c7875bccdb13abe07d27": "\\frac{\\partial C}{\\partial \\sigma} \\,", "7abe741f5dcec159e835605a7c033726": "\\left(\\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "7abe8c26cef4352fd0233496cc6c11d0": "\\rho_\\alpha(x)=e^{-\\pi(|x|/\\alpha)^2}", "7abeca8d14cb1fa95cfdef28192f58e8": " (1 - B)^d X_t= \\varepsilon_t, ", "7abeee465e938ab1ecc13e8eefbf96ea": "x_\\text{min}", "7abf37cb47d60debd7d8de5f7b52702e": "C_{456}=\\frac{1}{4}A", "7abf4755efbcbec5eab442732b7ad803": " \\frac{\\Delta{y}}{\\Delta{x}} = \\frac{\\Delta{u}}{\\Delta{x}} + \\frac{\\Delta{v}}{\\Delta{x}}. ", "7abf7613b4807281f11a00200d9ad40e": "4^4=256", "7abf9fc0426fa0da3cff27cef2bcb5c1": "\\frac{P}{P_0} = 0.066 \\left( \\left( \\tau - \\tau_s \\right)^{-0.2} - \\tau^{-0.2} \\right)", "7abfa98ab43522cfd5b136204c8e975a": "g(x,s)= \\begin{cases}\n c_2 \\sin kx, & \\text{for }x 0, \\\\[8pt]\n\\ge 1 - \\frac{\\sigma^2}{\\sigma^2 + \\lambda^2} & \\text{if }\\lambda < 0.\n\\end{cases}\n", "7ade2e51097ad847343b3cff738416e1": "k \\cdot \\int_a^b \\sin(k \\cdot t) \\; dt", "7ade2e7940649d8e7b1a273d8172dc61": "\\bar H", "7ade35a74a08b95d5f115bcb6706946c": "\\mu_0, \\mu_1", "7ade9b410e46e0d1d18d63b6225c9822": "(\\log_{2}{N})/b", "7adeacef4aebdb96648a5dcfa41c351e": "\\| \\mathbf{L}_k \\|^2 \\le \\| \\mathbf{L}_k \\mathbf{L}_k ^* \\| = \\| \\mathbf{A}_k \\|.", "7adeb0f4764b4bdc3cb8d5614e4c0535": " \\beta = \\frac{p_1 c}{E_1 + m_2 c^2} ", "7aded0c7455dd2a0a7f840a1c50a44ab": "E^\\star ( \\omega ) = E + i \\eta \\omega. ", "7adef364dcd5f6f7406566468b22eba8": "\\mathcal{N}=4", "7adf68f54f650f475b1681c6294b7669": "T=|T'|=\\frac{4k_1k_2}{(k_1+k_2)^2}", "7adfbb449f8b78548209b8804ecf461c": "\\bigcap_k A_k \\neq \\emptyset ", "7ae03fd30d0702a35d895722e0c8c6a9": "h(k_1)", "7ae094f7cf4da8c5f464e7a3d1531787": "f(z) = \\frac{e^{2 \\pi i \\tau} z^2(z - 4)}{1 - 4z}", "7ae0b00fce141e98eeab4d56173ce718": "\\mu(f) = 4", "7ae0ed61ff9e2a60ecd0c45d50a250c7": " \\nabla^{2}(\\nabla\\cdot\\mathbf{A})=\\nabla\\cdot(\\nabla^{2}\\mathbf{A})\n", "7ae188b4524ea7a2269b48b82eb81ba9": "Tr(g^x)", "7ae1a06441413b75d2ebc13b35139fd7": "\\Phi_{\\text{P}}", "7ae1b4435345e848cfc6552dc0ece2a9": "\\text{subject to }x_i\\ge x_j~ \\text{ for all } (i,j)\\in E.", "7ae1b662ee78d47d89247e72f020d94a": "LL(0) \\subset LL(1) \\subset LL(2) \\subset \\cdots", "7ae1d6544bfdbad924580b1d202f76b7": "\\epsilon'", "7ae1dec14ee18736b7bbe2dc6370942a": "T_0 =\\,", "7ae210127cc2e3f876078ea7e6bb1d5d": "\\displaystyle \\operatorname{lcm}(\\{\\omega_i\\}) \\displaystyle =\\prod_p p^{\\sup(v_p(\\omega_i))}", "7ae22e9dc07127af30544e0dedc9943b": "\\mathcal{P}_p(\\mathbf{R})", "7ae233bbf154a5274d7acdd3d78ccf2d": " f(x; p, \\beta) := \\left( \\frac{1}{-\\ln p}\\right) \\frac{\\beta(1-p)e^{-\\beta x}}{1-(1-p)e^{-\\beta x}} ", "7ae24b18c32d368c011ce951baeb96d2": "(-1)^p", "7ae27185b63b8e6b2fbf43a471da9dbe": "X\\mapsto X^\\vartriangle", "7ae3060512f80d97e81faa778341ff2f": "\n\\begin{align}\n\\sigma_s(n)\n&=\n\\zeta(1-s)\n\\left(\n\\frac{c_1(n)}{1^{1-s}}+\n\\frac{c_2(n)}{2^{1-s}}+\n\\frac{c_3(n)}{3^{1-s}}+\n\\dots\n\\right)\\\\\n\n&=\nn^s\n\\zeta(1+s)\n\\left(\n\\frac{c_1(n)}{1^{1+s}}+\n\\frac{c_2(n)}{2^{1+s}}+\n\\frac{c_3(n)}{3^{1+s}}+\n\\dots\n\\right).\\\\\n\\end{align}\n", "7ae337e77c4205eaa9896566b54ed5b8": " \\vec h, M ", "7ae372982e5ece24ed864fadf841d52b": "\\kappa_{}\\,\\hat{=}\\,\\rho\\,\\hat{=}\\,\\sigma\\,\\hat{=}\\,0", "7ae378c05374cb80e07e93a305eb3ef9": "d_j = \\lfloor\\beta r_{j+1}\\rfloor, \\quad r_j = \\{\\beta r_{j+1}\\}.", "7ae384852097a9256e0382f7a91588bb": "\\tfrac{7}{2}", "7ae39850e6450b417fdeb1769e5fec8d": "\\varepsilon_{i_1 i_2 \\cdots i_n}", "7ae41b61a3de0deebd912cfe05ba83a3": "V_r=\\frac{\\partial \\phi}{\\partial r} = U\\left(1-\\frac{R^2}{r^2}\\right)\\cos\\theta", "7ae46dcc200bc8a49f9251ce53b684cd": " I = \\sigma V.\\, ", "7ae49a6240125f3a6317502f8d01a23e": "(X_t,Y_t)", "7ae49c8e69db1a2794184f8bffd3aa81": "(Ry')' = R\\,y'' + R'\\,y' = R\\,y'' + \\frac{R\\,L}{Q}\\,y'.", "7ae4d6ec7eb86b9f3199ee453afa3aac": "{BE}_{9}", "7ae596d5bbe72eb86529e4a4d3aa19b5": "g''(x) = -\\frac{1}{4 x^{1.5}}", "7ae5df68d259ca8748fc63cc9942aa4b": "\\exist x Lxx", "7ae5ffe3bc88d6b154d0706578929760": "\\Delta w''=0", "7ae64d1fee3d5c7e2e385286d5a7fb74": "A_1 A_2 =0 , \\qquad A_1+A_2= I ~. ", "7ae676a6a9c58ed636c3a95eb609bd66": "\\phi (x,y) = X(x)Y(y) \\ ", "7ae6a7cb6eca343cfe242caffc8af3a9": "\n\\left[ \\vec{R}_1 ~\\vdots~ \\vec{R}_2 ~\\vdots~ \\left(\\vec{R}_1 \\times \\vec{R}_2 \\right) \\right] = A \\left[ \\vec{r}_1 ~\\vdots~ \\vec{r}_2 ~\\vdots~ \\left( \\vec{r}_1 \\times \\vec{r}_2 \\right) \\right]\n", "7ae6d81bc378809113c1371017fe26e5": "I_C = \\beta I_B = \\beta \\frac \n{ \n \\frac \n{V_{CC}}{1+R_1/R_2}\n - V_{be}\n}\n{( \\beta + 1)R_E + R_1 \\parallel R_2 } \\approx \\frac \n{ \\frac {V_{CC}}{1+R_1/R_2}- V_{be}}\n{R_E} , ", "7ae6e7d31a554a6dafa05fd0ae30678c": "S_{\\bar{m}}(c)", "7ae718003a9127c7323184f08c78dc5f": "\\mathrm{hom}_{\\mathcal{C}}(FY,X) \\cong \\mathrm{hom}_{\\mathcal{D}}(Y,GX)", "7ae7530171beb148c11c8aec0e9aeac3": "U_y", "7ae7666feff1b74d771f1638e6afc7fe": "j^{*}(\\omega)=d\\alpha", "7ae7bcf81ea42c39831750ac1ece262e": "2R\\left|\\Gamma\\right|^2", "7ae7c53d25bb5913ba7d5799d2215c02": "Lu=f \\text{ in } \\Omega", "7ae7ea923c40ac71e3f8f521d06aecf8": "F:\\mathbb{R}\\rightarrow\\mathbb{R}", "7ae7ecd2b21a4a699aa3b1970c6c975d": "\\partial_{x_i}g_j=\\partial_{x_j}g_i", "7ae7fe1a4cb6952019d82f8cf33982f8": " (\\lambda V.\\operatorname{de-let}[L])\\ \\operatorname{get-lambda}[V, E] ", "7ae82a91db0369923269af090dee1f04": "\\sin 2\\pi x + \\ln e", "7ae8325b1b5ff20e63edd26d144bed72": " \\begin{pmatrix} \\frac{d y_{1}}{d t} \\\\[2mm] \\frac{d y_{2}}{d t} \\end{pmatrix} = \\begin{pmatrix} \\frac{d m_{1}(x_{1}, x_{2}, x_{3})}{d t} \\\\[2mm] \\frac{d m_{2}(x_{1}, x_{2}, x_{3})}{d t} \\end{pmatrix} = \\begin{pmatrix} \\frac{d m_{1}}{d x_{1}} & \\frac{d m_{1}}{d x_{2}} & \\frac{d m_{1}}{d x_{3}} \\\\[2mm] \\frac{d m_{2}}{d x_{1}} & \\frac{d m_{2}}{d x_{2}} & \\frac{d m_{2}}{d x_{3}} \\end{pmatrix} \\, \\begin{pmatrix} \\frac{d x_{1}}{d t} \\\\[2mm] \\frac{d x_{2}}{d t} \\\\[2mm] \\frac{d x_{3}}{d t} \\end{pmatrix} ", "7ae83d86192bee4ddfb64dde07ceccb2": "(\\text{Re}(z_n), \\text{Im}(z_n))", "7ae88d10be18d845865604981a7f9917": "\\cot\\frac{7\\pi}{30}=\\cot 42^\\circ=\\frac{\\sqrt{2(25-11\\sqrt5)}+\\sqrt3(3-\\sqrt5)}{2}\\,", "7ae892a5a29b9d67587c644a8576bffb": "\\;_2F_1(a,b;c;z) =\nP \\left\\{ \\begin{matrix} 0 & \\infty & 1 & \\; \\\\ \n0 & a & 0 & z \\\\\n1-c & b & c-a-b & \\;\n\\end{matrix} \\right\\}", "7ae94f5819edce30b7550513ca0b2ad9": "\\langle f\\rangle", "7ae98c335cf009531f1025b2cfdc8652": " k_{y}=\\frac{m\\pi }{a}", "7ae9fbf1044aa6707dfa42eb085d896a": "P=\\frac {1} {1+\\eta} MC", "7aea6e9d4b112715be9fc23ad884e8d6": "S_{n,m}", "7aea7992a259a9d4fb9df8e0cc5e44dc": "\nU= \n\\begin{bmatrix}\n\\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} \\\\\n\\frac{\\omega}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{\\bar{\\omega}}{\\sqrt{3}} \\\\ \n\\frac{\\bar{\\omega}}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{\\omega}{\\sqrt{3}} \n\\end{bmatrix}\n\\Rightarrow (|U_{i\\alpha}|^2)=\n\\begin{bmatrix}\n\\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \\\\\n\\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \\\\ \n\\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3} \n\\end{bmatrix}\n", "7aeaa9cf5700d2d39813e6e0806885ce": "\\sigma = - 2/3 x^{3/2}", "7aeabac20cd1c7b9adc2f0cc033f5a09": "S=\\int \\left( \\mathcal{L}_\\mathrm{G} + \\mathcal{L}_\\mathrm{M} \\right) \\, d^4x ,", "7aeb7587de2d3f2e5c991db3f6503e3d": "\\dot{\\mathbf{A}} = \\dot{A}_x \\hat{\\mathbf{x}} + \\dot{A}_y \\hat{\\mathbf{y}} + \\dot{A}_z \\hat{\\mathbf{z}}", "7aeb796c92ab34acb2be3a4dec988b87": "S_{(1-i)}", "7aebad03e2c44ac129f68bae4a0c6024": "\\frac{\\partial A}{\\partial z}+i\\beta_2\\frac{\\partial^2A}{\\partial t^2}=i\\gamma|A|^2A", "7aebf5afeae4e9ae6ec72b8c6dffb11b": " z_1+w_1,\\dots,z_n+w_n ", "7aec2975269cf00a669cf35c4ded20e6": "y^2=\\frac{-(\\frac{b^2}{4a}+\\frac{-b^2}{2a}+c)}{a}=\\frac{b^2-4ac}{4a^2}", "7aec5d3dbf4adbb47248a7e8f186f962": "\\omega _t", "7aec6cb5e4e6b93033a79aafa362962d": "\\Rightarrow_{k} aSSS \\Rightarrow_{k} aaSS \\Rightarrow_{k} aaaS \\Rightarrow_{k} aaaa", "7aed0b513d255b1a0d463ca0b4c3cc27": "e_{ij} = \\frac{1}{2}\\left( \\partial_i u_j + \\partial_j u_i \\right)", "7aed16a6abc56df36913868ac9088cd8": "s_{mv} = 2(c_v + \\frac{s}{10}) + 0.25k_1\\frac{d_{bv}}{\\rho_{v}}", "7aed6eb475383f181012e4ffd6147ef6": "(p \\to q) \\vdash (\\neg q \\to \\neg p)", "7aedcc896a59d0b6aa6676cb9b2e31d8": "G(P(n);x)=\\frac{3-x^2}{1-x^2-x^3}.", "7aedfa97ff52f8103e0534839fe75fc7": "M_{T}^2 = m_1^2 + m_2^2 + 2 \\left(E_{T, 1} E_{T, 2} - \\overrightarrow{p}_{T, 1} \\cdot \\overrightarrow{p}_{T, 2} \\right) ", "7aee2c44ea83e2fe9fc37aa487564bee": "\\Gamma_k(x) = \\lim_{n\\to\\infty} \\frac{n!k^n (nk)^{x/k - 1}}{(x)_{n,k}}. ", "7aee3fe5d8d2075c803bc6d3313ea3b4": "\n\\text{SE}(\\theta) = \\frac{1}{\\sqrt{I(\\theta)}}.\n", "7aee838379254e3ee4f6db40688e0591": "obs_B", "7aee97b1e4e996551933df2c7be59197": " A = \\begin{pmatrix}\n2 & 0 \\\\\n0 & 1\n\\end{pmatrix},", "7aeec5deaddea09c42183b74fb6d71a9": " ax + by + cz + d = 0, \\text{ where } d = -(ax_0 + by_0 + cz_0).", "7aeed07a763f3a639c236549b945a3d6": "E_{2,1}-E_{1,0}=0", "7aeed20313b1190bb36af4dec65ebc5b": "\n\\begin{align}\n\\boldsymbol{\\nabla}\\cdot \\boldsymbol{S} & = \\frac{\\partial S_{rr}}{\\partial r}~\\mathbf{e}_r \n + \\frac{\\partial S_{r\\theta}}{\\partial r}~\\mathbf{e}_\\theta\n + \\frac{\\partial S_{rz}}{\\partial r}~\\mathbf{e}_z \\\\[8pt]\n & + \\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta r}}{\\partial \\theta} + (S_{rr}-S_{\\theta\\theta})\\right]~\\mathbf{e}_r +\n\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta\\theta}}{\\partial \\theta} + (S_{r\\theta}+S_{\\theta r})\\right]~\\mathbf{e}_\\theta +\\cfrac{1}{r}\\left[\\frac{\\partial S_{\\theta z}}{\\partial \\theta} + S_{rz}\\right]~\\mathbf{e}_z \\\\[8pt]\n & +\n\\frac{\\partial S_{zr}}{\\partial z}~\\mathbf{e}_r +\n\\frac{\\partial S_{z\\theta}}{\\partial z}~\\mathbf{e}_\\theta +\n\\frac{\\partial S_{zz}}{\\partial z}~\\mathbf{e}_z\n\\end{align}\n", "7aef13dca5fe2417334b3811f8dae4cb": "\\ln \\left |\\sec x + \\tan x\\right | + C", "7aef2b1eee27072418154fe69846b141": "0\\to C_\\bullet(A) \\to C_\\bullet(X)\\to \nC_\\bullet(X) /C_\\bullet(A) \\to 0", "7aef3733a2ea63727008d4922224cdc1": "\\lim_{x \\to +\\infty}\\left[ f(x)-(mx+n)\\right] = 0 \\, \\mbox{ or } \\lim_{x \\to -\\infty}\\left[ f(x)-(mx+n)\\right] = 0.", "7aef50ba396a1fc8e03d9d87780a1380": "\\mathbf g=2\\mathbf{J_r}^\\top \\mathbf{r}, \\quad \\mathbf{H} \\approx 2 \\mathbf{J_r}^\\top \\mathbf{J_r}.\\,", "7aef84280dead9d110931d572ff981a6": "\\textstyle \\lambda = e^r", "7aeffff235941bc6f6f3f57c96aa8885": "\\langle x,\\ y \\rangle \\,", "7af054110a55aca6ef73275af099ea43": "\\frac{L^\\alpha}{1 - \\left(\\frac{L}{H}\\right)^\\alpha} \\cdot \\left(\\frac{\\alpha}{\\alpha-1}\\right) \\cdot \\left(\\frac{1}{L^{\\alpha-1}} - \\frac{1}{H^{\\alpha-1}}\\right), \\alpha\\neq 1 ", "7af10d5a59b7c556fb4b5d0e6f150709": "W(n)", "7af1354fe6b2979c78d1e5d6f35bd5ff": "~ \\beta_{T \\text{ or } S} = -{ 1\\over V } \\left ( {\\partial V\\over \\partial p} \\right )_{T,N \\text{ or } S,N}", "7af19a0bb18e86a585c4d35ab0421bae": "\\mathbf{A}(n\\times n)", "7af1cccfc8e1f60670b4a8b881e911d8": "\\arccos{\\left(-\\tfrac{1}{3}\\right)}", "7af1cecd43b9192729bf76fad6948e8a": "n_c - n_d", "7af212e8294444d2f798dc745e3ab619": "\\Delta \\overline{Y}/\\overline{Y}", "7af223987d7a24ba0012d3bfbd98be7e": "\\mathcal{L} = y_\\text{t} h q u^c \\rightarrow \\frac{y_\\text{t} v}{\\sqrt{2}}( 1 + h^0/v) u u^c", "7af27247e96c6d536391f2d10163d5b6": "\\left| e \\right|", "7af27d11966178be733a4363c09e3e3a": "Y=Y^{d}(\\tfrac{M}{P}, G, T, Z_1)", "7af301426edf70df75f252c6b0fc4ed3": "\\mathcal{H}^{(1)}", "7af3328757bdf90c7177cfce119b4d2b": "\nn = \\frac{1}{\\Gamma - 1}.\n", "7af375c27816eefd837e9c9edac034bd": "\\sin\\left( x \\right) = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots \\pm \\frac{1}{(2n+1)!} x^{2n+1} + \\cdots. ", "7af3d7036dc3fc4b8ee79f4595259d9b": " \\frac{\\mathrm{d} J_\\varepsilon}{\\mathrm{d} \\varepsilon} = \\int_a^b \\left[\\eta(x) \\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon} + \\eta'(x) \\frac{\\partial F_\\varepsilon}{\\partial g_\\varepsilon'} \\, \\right]\\,\\mathrm{d}x \\ . ", "7af43ad9d76ff2ef14a74f0d0baf129d": "\np(x_j,t|x_i)=M^t_{ij} \\,\n", "7af4c6ab706d0c6b11b6023ab399bf8d": ":\\phi(x)\\chi(y):=\\phi(x)\\chi(y)-\\langle 0|\\phi(x)\\chi(y)| 0\\rangle", "7af4e9572270bf1b1e40d0566b4618e1": "K_{r,n}", "7af50caac3d48c843aa5c8b70fd44ae2": "\\Omega_{1}, \\Omega_{2}", "7af5122f4470b3d31f3fcb238e827086": "\\, \\delta", "7af54ce537cb298e202e4d652c7de19c": "\\frac{\\partial \\kappa(\\theta)}{\\partial \\theta}=\\tau(\\theta)", "7af54e02150e17c3a4d85d47da87b153": "(\\pm P_{i+1},\\pm P_i)", "7af57676c7571c86342cb36bc84865e5": "\\sum F_y=0=R_{Ay}-F_{BD}\\sin(60)-10=5-F_{BD}\\frac{\\sqrt{3}}{2}-10 \\Rightarrow F_{BD}=-\\frac{10}{\\sqrt{3}}", "7af578f408cdfffe3976a05e22ca8a90": " [ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC", "7af5a68c680db73ca466464291cfe979": "z^{-1}", "7af5e37ed0e4bab4dcb146b959954f08": "\\gcd(a,b)\\,\\operatorname{lcm}(a,b) = a\\,b.\\;", "7af6173e04c82fee633e53fad7b9e73a": "\\scriptstyle\\sqrt{n}", "7af6204330b3672e7e18cdcca3c051ce": "I_n=\\int_{t_0}^t{dt_1\\int_{t_0}^{t}{dt_2\\cdots\\int_{t_0}^t{dt_nK(t_1, t_2,\\dots,t_n)}}}.", "7af6300f3ca8f879ddecb2d366307870": "n^2 - n + 41, \\, ", "7af6331ffe24d282a1afed505cc47024": "{\\rm si}(x)", "7af6607a193dd5b56517e46f765baeb1": "d \\approx 1.41 \\cdot\\sqrt{h} ", "7af687f8f29c49f78785e8ebb8395df2": "m_F = m_J + m_I", "7af69a5c4a0858eac660eed20289a17a": "S ", "7af6c25a375da475d33cf0d79623d135": "\\mathbf{\\mathit{\\phi}}", "7af6ee37571ad250a5c57f72abc7401b": "\\mathbf{A}^{-1} = \\det(\\mathbf{A})^{-1}\\, \\mathrm{adj}(\\mathbf{A}).", "7af7273db093d2eb0dca971a5ebc8b3b": "\\displaystyle{Tf_n=\\mu_nf_n,\\,\\,\\mu_n >0,\\,\\,\\mu_n\\rightarrow 0.}", "7af7287e622f9040c19111fdd6ac7891": "H(x,0) = x", "7af74672c092961b53f51f6a42e8d9ac": "k_0=\\frac{2\\pi}{T}", "7af7506190fed734fce1e3871ac55f30": "x(t) = A_1 \\sin (tf_1 + p_1) e^{-d_1t} + A_2 \\sin (tf_2 + p_2) e^{-d_2t}. \\,\\!", "7af756a61034d19041a3ff8362ac065b": "\\begin{pmatrix}\n\\bar{x}_1\\\\\n\\bar{x}_2\\\\\n\\bar{x}_3\n\\end{pmatrix}=\\begin{pmatrix}\\bar{\\mathbf{e}}_1\\cdot\\mathbf{e}_1 & \\bar{\\mathbf{e}}_1\\cdot\\mathbf{e}_2 & \\bar{\\mathbf{e}}_1\\cdot\\mathbf{e}_3\\\\\n\\bar{\\mathbf{e}}_2\\cdot\\mathbf{e}_1 & \\bar{\\mathbf{e}}_2\\cdot\\mathbf{e}_2 & \\bar{\\mathbf{e}}_2\\cdot\\mathbf{e}_3\\\\\n\\bar{\\mathbf{e}}_3\\cdot\\mathbf{e}_1 & \\bar{\\mathbf{e}}_3\\cdot\\mathbf{e}_2 & \\bar{\\mathbf{e}}_3\\cdot\\mathbf{e}_3\n\\end{pmatrix}\\begin{pmatrix}x_1\\\\\nx_2\\\\\nx_3\n\\end{pmatrix}=\\begin{pmatrix}\\cos\\theta_{11} & \\cos\\theta_{12} & \\cos\\theta_{13}\\\\\n\\cos\\theta_{21} & \\cos\\theta_{22} & \\cos\\theta_{23}\\\\\n\\cos\\theta_{31} & \\cos\\theta_{32} & \\cos\\theta_{33}\n\\end{pmatrix}\\begin{pmatrix}x_1\\\\\nx_2\\\\\nx_3\n\\end{pmatrix}\n", "7af7acbc37c7040d14571dcd1d447a6e": "\\theta_V", "7af80e68015e30b5e7313b104c489f87": " \\dim_{\\operatorname{box}} \\left\\{0,1,\\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots\\right\\} = \\frac{1}{2}. ", "7af8387722b36addb252bbd1dc375ef7": "x(t)=\\begin{cases} \\cos(2\\pi t) & t\\le-2 \\\\ \\cos(4\\pi t) & -2 < t \\le 2 \\\\ \\cos(3\\pi t) & t>2 \\end{cases}", "7af8a3c6bfe81afa42dfadee6717e3c0": " \\sup_{\\mathrm{Re} z = 0 \\mbox{ or } 1} e^{-k|\\mathrm{Im}z|} \\log\\|T_z\\| < \\infty", "7af8bc7888f5cadcf635a50b3219baec": "S(0) \\approx 1", "7af939ac3452b6e522f51c22e52fa3f5": "e \\!", "7af98db0e5eabac0ff448e1d6ce47650": "\\rho:\\R^2\\to[0,\\infty)", "7af9a09d8c1019735a5ffb2407faa5a6": "\n \\begin{pmatrix}\n \\displaystyle \\frac{\\partial\\Phi}{\\partial{x}} \\\\[2ex]\n \\displaystyle \\frac{\\partial\\Phi}{\\partial{y}} \\\\[2ex]\n \\displaystyle \\frac{\\partial\\Phi}{\\partial{z}}\n \\end{pmatrix}\\,\n \\approx\\,\n \\begin{pmatrix}\n \\displaystyle f\\, \\frac{\\partial\\varphi}{\\partial{x}} \\\\[2ex]\n \\displaystyle f\\, \\frac{\\partial\\varphi}{\\partial{y}} \\\\[2ex]\n \\displaystyle \\frac{\\partial{f}}{\\partial{z}}\\, \\varphi\n \\end{pmatrix}.\n", "7af9b91cf2d585805f223167d19e3a6a": "(R^\\gamma{}_{\\varepsilon} \\, - \\frac{1}{2}g^\\gamma{}_{\\varepsilon}R)_{;\\gamma} \\, = 0", "7af9c65e27896233bde5d84e2854b967": "F(s) = \\int_0^\\infty f(z)e^{-sz}\\,dz", "7afa13c84920a60a17e8bf6a230a6fb0": "m_{\\rm star}=m_\\odot-2.5\\log_{10}\\left[\\frac{L_{\\rm star}}{L_\\odot}\\left(\\frac{d_\\odot}{d_{\\rm star}}\\right)^2\\right]", "7afa59febc3d241f9e74cfb1f90ab8ed": "F_L=j\\left [\\left(\\frac{SC}{60}\\right) (25(ERC))\\right ]^{0.46}", "7afa80abd12b17199ca34b7377fe5766": "{\\textstyle \\alpha^4}", "7afa88c2877d79e6a8a190b360edfcd6": "\\subset ", "7afa8ae5ee6dbe4bf26a71a9a0487e12": "\\frac{f_{\\theta_2}(X=x_1,x_2,x_3,\\dots)}{f_{\\theta_1}(X=x_1,x_2,x_3,\\dots)} ", "7afab976e915256eb431ead2ec1b0275": "L(x) = -(2\\alpha+1)\\, x", "7afaf09d009c49771dbbe629c9565dcf": "g^{n-i}", "7afaf4efc3100a27040fcf86abc3800e": "\\tbinom N2", "7afb475bc6076df138d49fc2a71b941b": " \\hat{T}_{n_1,n_2,n_3} \\!", "7afb54b68eb83746c9197ed1297950bf": "R_{\\mu \\nu }-\\frac 12g_{\\mu \\nu }R = \\frac{8\\pi }\\phi T_{M\\mu \\nu }-\\frac\n3{2\\phi ^2}\\left( \\nabla _\\mu \\phi \\nabla _\\nu \\phi -\\frac 12g_{\\mu \\nu\n}g^{\\alpha \\beta }\\nabla _\\alpha \\phi \\nabla _\\beta \\phi \\right)", "7afb6517d1fc46ecf8ab007fc3f91728": "S(t)\\subset S(t'),", "7afbd5b539632379886206769077ddd0": "{\\rm Tr} B \\rho_c", "7afc18c72f051ddf13bcb15f84066a94": "H = T^{a + \\varepsilon}", "7afc190f90d87fac746386f31f215d44": "\n\\bar{h}^{\\alpha \\beta} (t,\\vec{x}) =\n-16\\pi \\int\\, G^{\\alpha \\beta}_{\\gamma \\delta} (t,\\vec{x};t',\\vec{x}')\\, \\tau^{\\gamma \\delta}(t',\\vec{x}')\\, \\mathrm{d}t'\\, \\mathrm{d}^3x'\n", "7afc4dacd13b39297e045c0c7210e46b": "wOBA = {(\\alpha_1 * uBB + \\alpha_2 * HBP + \\alpha_3 * 1B + \\alpha_4 * 2B + \\alpha_5 * 3B + \\alpha_6 * HR + \\alpha_7 * SB - \\alpha_8 * CS) \\over (AB+BB-IBB+HBP+SF)}", "7afc881602d5e72a57a3a3e01878eb2b": " \\begin{matrix}\n\\mathbf{\\sigma}_0 &=& \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} &\n\\mathbf{\\sigma}_1 &=& \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix} \\\\\n\\\\\n\\mathbf{\\sigma}_2 &=& \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} &\n\\mathbf{\\sigma}_3 &=& \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}\n\\end{matrix} ", "7afc9fee7144aeb789b9c027bbb0ccb7": "\\frac{\\sum{\\sum{ L(r,c) \\cdot R(r,c) }}}{\\sqrt{(\\sum{\\sum{ L(r,c)^2 }}) \\cdot (\\sum{\\sum{ R(r,c)^2 }})}}", "7afcc2caaad1690cdd405d6c4bb2ac44": "VSWR = \\frac{Z_\\text{0}}{R_\\text{L}} ", "7afd0f3f7de7782481f414430758f255": "\n V(r,\\phi,\\theta) = a \\sum_{\\ell=1}^L\\sum_{m=-\\ell}^\\ell\n \\left(\\frac{a}{r}\\right)^{\\ell+1} \\left(g_\\ell^m\\cos m\\phi + h_\\ell^m\\sin m\\phi\\right) P_\\ell^m\\left(\\cos\\theta\\right)\n", "7afda79559d0ba674ff56a19d07732a5": "P=VI", "7afdd2ecbc8d5a7dd625b26068d8210b": "\\textstyle A=\\{a_P,a_N,a_B\\}", "7afe245b7c9cd58169fd495296fcef7c": "C_1: (x_1(t),y_1(t)), \\ C_2: f(x,y)=0.", "7afe262575fd653549fcfa46bb25138b": "f_r=|\\mathbf{f}_r|", "7afe5ee7545e98e62fa91de0208ffe4c": "Y\\to X ", "7afe6b85b473cd87a56c05c186b6b03c": " {h_1 \\over h_0} =\\frac{-1 \\pm{\\sqrt{1+{\\frac{8v_0^2}{gh_0}}}}}{2}. ", "7afe999a1ddb14e2e0232ffc9545c3e9": "b_{\\nu, n}(1) = \\delta_{\\nu, n}", "7afefd3a8b5b787d43ecc1c827056837": "z = \\frac{T_2 - T_1}{\\log D_1 - \\log D_2}", "7aff8df410c8cb4e5a4dc6505e6cdbff": "p(S\\vert D)={p(S)\\over p(D)}\\,\\prod_i p(w_i \\vert S)", "7aff8ec48afac85680a202b95d3d9039": " (-1)^k\\sum_{|\\alpha| = 2k} a_\\alpha(x) \\xi^\\alpha > C |\\xi|^{2k},\\,", "7affbe82d04b4f7b7e993127dc327eab": "\\{H\\}^\\perp", "7b001ef6a48b0c9ce7183c915bc92a9f": "\\scriptstyle y \\in Z^d ", "7b003d900d24d5c9563fc2be69279f9d": "\\psi(\\mathbf{r})=\\frac{1}{(\\sqrt{2\\pi})^3} \\int_{\\mathbf{k}{\\rm-space}} \\phi(\\mathbf{k}) e^{i \\mathbf{k}\\cdot\\mathbf{r}} {\\rm d}^3\\mathbf{k} ", "7b005942cba87da5f3963296965c396f": "\\text{inj}_2 : B \\to A+B", "7b006cfe4f402fa1b79eef17a09366b4": "r=\\sqrt{x_1^2+x_2^2+\\cdots+x_n^2}.", "7b00f0fe50b04b28fae61138d4544b14": "\\left[1 - \\frac{L}{G}\\right]^N \\sim \\exp(-NL/G),", "7b01275bfa74594db7db2e2a1744721a": "z \\neq x \\pm i \\pi", "7b012eb41f680ed750347c64bcae90d3": " S(i_1,i_2,\\cdots,i_k)=\\sum_{n_1\\geq n_2\\geq\\cdots n_k\\geq1}\\frac{1}{n_1^{i_1} n_2^{i_2}\\cdots n_k^{i_k}}", "7b018ffba07dd4e58df0ce882d954671": "U_n=a \\varphi^n + b \\psi^n\\,", "7b01d76f87b69b655caa103b53af8797": " [t_l, t_u] ", "7b0219399de993bd2723e250035e60ca": "G^t(V)=\\dot\\gamma_V(t)", "7b0236f3bec4f53a99c742f5a16e3dc9": " \\left|\\sum_{m=1}^N \\sum_{n=1}^N \\alpha_m\\alpha_n \\log{g(z_m) -g(z_n)\\over z_m -z_n}\\right|^2 \\le \\sum_{m=1}^N\\sum_{n=1}^N \\alpha_m\\overline{\\alpha_n} \\log{1\\over 1-(z_m\\overline{z_n})^{-1}}.", "7b0239b25faeb00ba8a0f19c687b9ac8": "\n\\mathbf{A}(\\mathbf{r}, t) = \\frac{\\mu_0}{4\\pi} \\int \\frac{\\mathbf{J}(\\mathbf{r}', t_r')}{|\\mathbf{r} - \\mathbf{r}'|} d^3\\mathbf{r}'\n", "7b028ff5e2fbaa6c4ca463e9a7a83d09": "u_1(\\mathbf{x},z_1)", "7b03508c96c75900d4e86cefb144596b": "\\frac{\\partial M}{\\partial \\alpha} ", "7b037901e5b91234a0830f311d757621": "Y - X", "7b03798e28254784fdd2e7df5b8b261c": "\\sin \\theta \\le \\sqrt{n_o^2 - n_c^2}", "7b03a59136bc5b48874c36519b297e34": "BS^1 \\times X \\to X", "7b03edc094cda0caad6afb0a5b5822d2": "\\delta \\mathbf{u}(r) = \\mathbf{u}(\\mathbf{x} + \\mathbf{r}) - \\mathbf{u}(\\mathbf{x})", "7b03f33472c0289dd1a2d489244f9797": "\\gamma^{\\langle2\\rangle}=\\gamma", "7b04105300c23a641699cb907b932a7e": " S = - \\mathrm{tr}(\\rho \\ln \\rho),", "7b0439830ef61523fea81809efcdc076": " 10^{30/10} = 10^3 = 1000", "7b044be44060cd50fa96e9c3512e7cda": "f(x, y) = g(y, x)", "7b0466156587e7817cdabb9a62b874ed": "\\pm\\left(0,\\ 4\\sqrt{\\frac{2}{3}},\\ \\frac{5}{\\sqrt{3}},\\ \\pm1\\right)", "7b046741e52ba07897a2becbb047c159": "G^{\\hat{a}\\hat{b}} = 8 \\pi \\sigma \\, \\left[ \\begin{matrix}-1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{matrix} \\right] ", "7b04c5869094065d98d28a4b70736821": "\\lambda: A \\longrightarrow A^\\vee", "7b053a1379caa67c56816de6548b99a8": "V'_x=\\frac{ V_x - v }{ 1 - \\frac{V_x v}{c^2} }", "7b0551464bc05b5fff6800baa907bce0": "\\begin{align}R_{J}(x,y,z,p) & = \\frac{3}{2 A^{\\frac{3}{2}}} \\int _{0}^{\\infty}\\frac{1}{\\sqrt{(t + 1)^{5} - (t + 1)^{4} E_{1} + (t + 1)^{3} E_{2} - (t + 1)^{2} E_{3} + (t + 1) E_{4} - E_{5}}} dt \\\\\n & = \\frac{3}{2 A^{\\frac{3}{2}}} \\int _{0}^{\\infty}\\left( \\frac{1}{(t + 1)^{\\frac{5}{2}}} - \\frac{E_{2}}{2 (t + 1)^{\\frac{9}{2}}} + \\frac{E_{3}}{2 (t + 1)^{\\frac{11}{2}}} + \\frac{3 E_{2}^{2} - 4 E_{4}}{8 (t + 1)^{\\frac{13}{2}}} + \\frac{2 E_{5} - 3 E_{2} E_{3}}{4 (t + 1)^{\\frac{15}{2}}} + O(E_{1}) + O(\\Delta^{6})\\right) dt \\\\\n & = \\frac{1}{A^{\\frac{3}{2}}} \\left( 1 - \\frac{3}{14} E_{2} + \\frac{1}{6} E_{3} + \\frac{9}{88} E_{2}^{2} - \\frac{3}{22} E_{4} - \\frac{9}{52} E_{2} E_{3} + \\frac{3}{26} E_{5} + O(E_{1}) + O(\\Delta^{6})\\right) \\end{align}", "7b0578b2f728e7aa384b25fcb73a39a7": "\\frac{\\partial}{\\partial t}\\left(\\rho\\mathbf{v}\\right) + \\nabla\\cdot(\\rho\\mathbf{v}\\otimes\\mathbf{v}) = -\\nabla p + \\nabla\\cdot\\sigma.", "7b058b85fdc4b2564986befb97ca20be": "\\alpha=\\frac{1}{\\left(1-u^2/c^2\\right)^{3/2}}\\frac{du}{dt}", "7b05986c4bbf2bcac46e3f225eb39a9d": "-(4n+3)", "7b05dfcd9041a80fee51eecab5f897e8": "\\left [\\begin{smallmatrix}\n1&-1&0&0&0&0&0&0 \\\\\n0&1&-1&0&0&0&0&0 \\\\\n0&0&1&-1&0&0&0&0 \\\\\n0&0&0&1&-1&0&0&0 \\\\\n0&0&0&0&1&-1&0&0 \\\\\n0&0&0&0&0&1&1&0 \\\\\n-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}&-\\frac{1}{2}\\\\\n0&0&0&0&0&1&-1&0 \\\\\n\\end{smallmatrix}\\right ].", "7b068ddf85226877899f9f8b9d005cc8": " f_n:[0,1]\\rightarrow [0,1] ", "7b06fa08ea381c475e65ac8e00d93740": "23^\\circ = 23 \\cdot \\frac {\\pi} {180^\\circ} \\approx 0.4014 \\text{ rad}", "7b07bbb955ccbcecbf9f6b6b632ae61b": "\nRe \\left( \\frac{\\partial \\mathbf{u^*}}{\\partial t^*} + \\mathbf{u^*} \\cdot \\nabla \\mathbf{u^*} \\right)\\ = -\\nabla p^* + \\nabla^2 \\mathbf{u^*}.", "7b0809d6505c74ff2fce98784e4cf023": "\\scriptstyle\\mathbb{E}|\\langle z, X \\rangle|\\,<\\,\\infty ", "7b0821048e10755b694fc147d4686302": "\\scriptstyle((1/2) a,\\, \\infty)", "7b0884d16736bba5f6e69f134685ccd4": "\\vartheta_{s} : \\Omega \\to \\Omega", "7b08fe24c973ebee29193f03b087315c": "\\mathbf{M} - \\mathbf{1}\\mu_s^T", "7b090547a45e761454c39c588809fcb4": " \\left( \\frac{p}{1 - x(1-p)}\\right) ^r \\,", "7b093d5bd46e7312cf1dea238f2799cb": "{\\mathbb C}^2", "7b09b26dd37b3c1d57d3849684fb0cc2": "N_{W_{\\alpha}XY}(\\hbar \\omega) = (4 \\pi)^{-1}\\psi_{W_{\\alpha} XY}[1-\\kappa]\\int_\\Omega \\int_0^\\infty \\ \\rho_{\\alpha}(z)\\,\\,P_{W_{\\alpha}}(\\hbar \\omega; z)\\exp\\left[\\frac {-z}{\\lambda(W_{\\alpha} XY)}\\cos\\theta\\right]\\ dzd\\Omega,", "7b09fea775960abce9cd821eb6ee9ef7": "\\R", "7b0a285f257be6dc1bd5aaada583d1e9": " \\hat{u}", "7b0a294932746d65352306e52541e9a9": " \\rho_t(x) = {1\\over \\sqrt{2\\pi t}} e^{-x^2 \\over 2t} ~,", "7b0a366b0b8457cf0d09dedcc659af52": "(q_3, \\omega_2, q_4) \\in \\Delta ", "7b0a842be6b7ebe08a7ff451756120bd": "\\lim_{a\\to\\infty} I(a)", "7b0af073ade5303fa87d16c279db66ed": "\\frac{a}{2^b}+\\frac{c}{2^d}=\\frac{2^{d-b}a+c}{2^d} \\quad (d\\ge b)", "7b0af766668b5aed410cd834a9ab6a33": "\nR_\\mathrm{eq} = 1\\,\\mathrm{k}\\Omega + (2\\,\\mathrm{k}\\Omega \\| (1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega)) = 2\\,\\mathrm{k}\\Omega.\n", "7b0afb7fdd5313f265ccd4193e656964": "P = (1,8)", "7b0b38bdc89b0f03dd164f83c8d9d4ef": "\\sup_F\\|F_n(x)-F(x)\\|_\\infty\\to 0", "7b0b60ad7805c0be106fb2468d840f62": "\n\\begin{align}\n\\mathbf P_A^\\perp & = \\mathbf I - \\mathbf A (\\mathbf A^T \\mathbf A)^{-1} \\mathbf A^T, \\\\ \\mathbf P_B^\\perp & = \\mathbf I - \\mathbf B (\\mathbf B^T \\mathbf B)^{-1} \\mathbf B^T.\n\\end{align}\n", "7b0baa90161529c98d5cceef35e14963": "-\\frac{1}{\\log (1-a)}", "7b0bc503e5dfdf68c21b68620e6d19e8": "K = F_{0}", "7b0bca831d5c408c8c5be6356f54b565": " \\mathbf{A}+(\\mathbf{B}+\\mathbf{C})=(\\mathbf{A}+\\mathbf{B})+\\mathbf{C} ", "7b0bf329d2cc1c4fe8def41d02ce54b6": "\\perp \\cup ", "7b0c0d7cac586f8517015ea532e7d6b9": "\\pi = {3927 \\over 1250}.", "7b0c4895182a7cc905ea70ecff9c2f19": " f(r)= 1- {2a\\over r} - b r^2 \\,", "7b0c5ecdccbef5c85c0296b40c95d6dd": "\\mathbf{T}^T\\mathbf{T}\\mathbf{u}_i = \\lambda_i \\mathbf{u}_i", "7b0c7fd1f6b4bbdb88f6a64c20717beb": "W_i(X) = 1/k", "7b0ccf19aea823e575e02ab11fc72fc3": "E_\\text{K} = \\frac{1}{2}\\omega^2\\sum_{i=1}^n m_i \\Delta r_i^2(\\mathbf{t}_i\\cdot\\mathbf{t}_i) + \\omega\\mathbf{V}\\cdot(\\sum_{i=1}^n m_i \\Delta r_i\\mathbf{t}_i) + \\frac{1}{2}(\\sum_{i=1}^n m_i) \\mathbf{V}\\cdot\\mathbf{V}.", "7b0ccfd886f330f52ddce886d6ab4ec1": "2^{-1} \\times 0.100_2", "7b0d06a4a3498852a912cdc707348ac4": "2\\pi\\, \\sin(\\phi)", "7b0d06bebb14a9fd4eb48d4fff1eede0": "\\sqrt[4]{2},\\quad i\\sqrt[4]{2},\\quad -\\sqrt[4]{2},\\quad\\text{and}\\quad -i\\sqrt[4]{2}.", "7b0d0d340ebbf3b5649c6f7e3db6c8c9": "J=i", "7b0d543bdb1a6c020de55093130421b0": "\\$ 3.54 > \\$ 3.50", "7b0d925bdc9a94d68a605ac7966357a5": "G = (\\{S,A,B,C\\}, \\{a,b,c\\}, S, P)", "7b0d9ddab9ececff267e713e150ac212": "\\tau = 0", "7b0ec044538d3de7186d466df7f4e9f6": "\\omega\\bar{\\omega}=1", "7b0f28fdfdeb7a8c95fdf66bee23d02c": "n\\geq 5,", "7b0f29f1c8bfd962c34654b071403b2f": " \\delta \\mathbf{r}_1 = (L_1\\cos\\theta_1, L_1\\sin\\theta_1)\\delta\\theta_1, \\quad \\delta \\mathbf{r}_2 = (L_1\\cos\\theta_1, L_1\\sin\\theta_1)\\delta\\theta_1 +(L_2\\cos\\theta_2, L_2\\sin\\theta_2)\\delta\\theta_2", "7b0f6870abe623690fe497a5d88b5769": "k_{2T}", "7b0f73a8f419b44dc990a68b44f72e43": "{c_0}", "7b0fd586379d7a633a28cbadfdc454fa": "\\bold{r}_{uu}=\\Gamma^1{}_{11} \\bold{r}_u + \\Gamma^2{}_{11} \\bold{r}_v + L \\bold{n}", "7b0ff9fc53b3db34f7049efe899f0450": "a^2_{0}", "7b104ef0b7cbe40b4754cf81e9458b92": "\nq_t \\equiv \\sum_{i=1}^N\\sum_{\\alpha=1}^3 \\; Q_{t, i\\alpha} \\rho_{i\\alpha},\n", "7b10ab0888dbf0ede76a732c14284be6": "n/\\log^{1/3}n", "7b11b0de45180d184700861b7cfea876": "\\underline{E}=-\\triangledown \\times \\underline{F}\\ \\ \\ \\ \\ \\ , \\ \\ \\ \\ \\ \\ \\underline{H}=-j\\omega \\underline{F}+\\frac{\\triangledown\\triangledown \\cdot \\underline{F}}{j\\omega \\mu } \\ \\ \\ \\ \\ \\ \\ (9) ", "7b11bd4d520aa8eba426fdc300b953a6": "t=t_n+h/2,", "7b11eafd4e4f1f35b42d4fc1747f665e": "\\frac{\\mathrm{d}}{\\mathrm{d}x} \\int_{\\sin x}^{\\cos x} \\cosh t^2\\;\\mathrm{d}t.", "7b1216a02fb8940585b23a5356a52ff0": "\\sqrt[x]{2}", "7b126705921d993bad7843ac07d640e0": "J_+|j\\,m\\rangle = \\hbar\\sqrt{(j-m)(j+m+1)}|j\\,m+1\\rangle = \\hbar\\sqrt{j(j+1)-m(m+1)}|j\\,m+1\\rangle,", "7b129a592dcbfaec679f05f351b31ff4": "\\lambda x.f\\ (x\\ x) ", "7b131c8e2bca8f7d14b65bd989b3a2d6": "2^3\\cdot 3\\cdot 5\\cdot 7", "7b1344b76bfa71daaa9229b408dce5b8": "10^{-25} \\ \\mathrm{seconds} \\,", "7b134b4deb825bddae64cf05f13d5129": "-\\mathrm{d}\\gamma = mRT \\Gamma_i^W\\, \\mathrm{d}\\ln C_i\\,,", "7b134c5583b3c7bc34ae626c25489d52": "\\textstyle (1 + x + \\ldots + x^{k-1})", "7b1362241079c9779d82171ac56ff665": "\\displaystyle{A\\xi=\\lim_{t\\downarrow 0} {1\\over t}(T(t)-I)\\xi,}", "7b13c45a26b4076ce4e7ac0568e806db": "0\\ < e\\ <\\ 1\\,", "7b13df3be45d4478956a12b4c0365494": "\\begin{align}\n \\prod_{i=1}^nx_i^{w_i} &\\leq \\sqrt[q]{\\sum_{i=1}^nw_ix_i^q} \\\\\n \\sqrt[q]{\\sum_{i=1}^nw_ix_i^q} &\\leq \\prod_{i=1}^nx_i^{w_i}\n\\end{align}", "7b13ee0fbbac02ea2bb139ee68012080": " s U(s) - u(0) + 2U(s) + \\frac{5}{s}U(s) = \\frac{1}{s}. ", "7b1430927de1ca154ceca1e5d9d9cf91": " Q_n(x;a,b;q) = \\frac{(ab;q)_n}{a^n}{}_3\\phi_2(q^{-n}, ae^{i\\theta}, ae^{-i\\theta}; ab,0; q,q)", "7b1473347b0127098f7047cd6ad7948d": "\\langle \\hat{A} \\rangle ", "7b147e30b703f7931e81c3b22acdead9": " {G_{I}} (s)", "7b148182fb268d725d7420d7779a81e5": "\\theta_\\mathrm B = \\arctan \\left( \\frac{n_2}{n_1} \\right), ", "7b14bc102c824e0a4dd55c871ec80302": "\\arccsc x = -i \\ln \\left(\\tfrac{i}{x} + \\sqrt{1 - \\tfrac{1}{x^2}}\\right) \\,", "7b14cccf676ce3d6e0499725ac577150": "(x_1 z + y_1)^2 + \\cdots + (x_n z + y_n)^2 = \\left( \\sum ( x_i^2) \\right) \\cdot z^2 + 2 \\cdot \\left( \\sum ( x_i \\cdot y_i) \\right) \\cdot z + \\sum ( y_i^2) ", "7b1509b42ef4aa3913037b08451950fa": " 0 = \\lambda^3 - \\lambda^2 - \\lambda + 1 = (\\lambda - 1)^2(\\lambda + 1),", "7b1553422e0c18f840000df4620f4cd1": "k_{Ds} ", "7b15649a415c8b1069371e518840c8dc": "S_{z}", "7b158011e6d3ccfbfbc422fad00011d5": "\\!g_x(m)", "7b15d7988dc80fca7e550ea9cb38f0c2": "X_{G}\\,", "7b15fe4265e03cfb83aa1d7c6823bd21": "t'=\\gamma\\left(t - \\frac{vx}{c^2}\\right) ", "7b1635aabd332aa8eb71f2c15b0fe577": "\\Psi_1 = -2\\log(\\frac{\\varepsilon}{3.7D} + \\frac{12}{Re})", "7b163e10e2578dabab47cc69c7385764": " \\mathbf{0} = [\\mathbf{x}_{k}]_{\\times} \\, \\mathbf{A} \\, \\mathbf{y}_{k} ", "7b166d6a0d68814fab0dea2c81e2fb15": "0< n \\leq N/2", "7b166fb6152cadc8ad3e24417bd61b0e": "\\textstyle P(A\\mid[x]) > \\alpha", "7b16aa81a60e42ea4ca04ed5854b7086": "82A_{11} \\ ", "7b16c30335902251b0c00844452ec6da": "c = c(\\varepsilon) > 0", "7b16d37aaaab5a06acca070c17146c6d": "\\delta g^{\\mu\\nu}", "7b1714a7e5e0f52d9a9698a7a33605fb": "\\mu_s\\,", "7b171e47c2049951935c5af1926f405f": "\\operatorname{ad}(x) = \\operatorname{ad}(s) + \\operatorname{ad}(n).", "7b18c01b3b05709ebaec9dd0c3c89ddd": "h(k,i) = (h(k) + i + i^2) \\pmod{m}", "7b192d4d21a58d865669e17e4a3ebdc5": "\\frac{64}{45}", "7b193b3d33184464106f41ddf733783b": "a-b-c", "7b193e313938b9c616e971404f8cad1a": "v_\\text{K}", "7b19bc07f0c124094c231690fde58a46": "a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4 ", "7b19d742e7d43c0850dce6c1e5a9c789": "(S\\vec v=0)", "7b19d8163c7dac58c33ca7bb9f4cbc04": "G^{p^k}", "7b19e982d2a24e3a11052d200602c08e": " Cov(Y_1,Y_2)", "7b19fed9f7f329663823ac3529a55d6f": "A = -1 + \\sum_p \\frac{\\log p}{(p-1)^2} \\approx 0.2269688 \\ . ", "7b19ff346cb8cd73788a3ddb9b5f35b8": " T : \\mathbb{R}_+ \\to L(X) ", "7b1a1b8d9d1c682c7e99c94379259da2": "f\\mapsto\\overline f", "7b1a33f8e14ee2e26289a58d892d69db": "\\underline{\\mathbf{C}}", "7b1a83ef1f11a56dd3fc3550294a1325": "v,v'\\in\\Gamma(T)", "7b1a924904c8c69e5298fb55d31d3239": "X_1^{(4)} = x\\partial_y + \\partial_{y'}", "7b1a93df40775aad329753fd00e996a6": "C(X_1,X_2,\\ldots,X_n)= \\sum_{x_1\\in\\mathcal{X}_1} \\sum_{x_2\\in\\mathcal{X}_2} \\ldots \\sum_{x_n\\in\\mathcal{X}_n} p(x_1,x_2,\\ldots,x_n)\\log\\frac{p(x_1,x_2,\\ldots,x_n)} {p(x_1)p(x_2)\\cdots p(x_n)}.\n", "7b1ad1a76cdc3de42f7552af0be64316": "\\operatorname{nil} \\equiv \\lambda c.\\lambda n.n", "7b1b71bcc9086348c2a3c5071f43e160": "\\int e^{x}\\;\\mathrm{d}x = e^{x}", "7b1baa624aec5a0eb46ea9f8277d5a7d": "V = -\\delta \\, x + \\gamma \\, \\log(x) - \\beta \\, y + \\alpha \\, \\log(y)", "7b1c75bba512aa13c851552ccfca6808": "\\begin{pmatrix}\n1 & 2 & 3 & \\cdots & n \\\\\nx_1 & x_2 & x_3 & \\cdots & x_n \\\\\n\\end{pmatrix},", "7b1c9d475e368e6e2c89dfc5e8341674": "\\scriptstyle 1.2m", "7b1cd7734ff4fd7c6106f91a08a3730e": "\\theta_{ab} = {h^m}_a \\, {h^n}_b X_{(m;n)}", "7b1ce4359b951d9f3893e4acc57b0c81": " \\gamma = \\frac{1}{\\sqrt{1- \\boldsymbol{\\beta}\\cdot\\boldsymbol{\\beta}}} \\,,", "7b1cf413fa8f7157b0615a6cb61c9e98": "Q_i=\\{(s_i,t_{si}, t_{ei})| s_i \\in S_i, t_{si} \\in \\mathbb{T}^\\infty, t_{ei} \\in (\\mathbb{T} \\cap [0, t_{si}])\\} ", "7b1cfb8005ab6dc4fb6f16c71caa8edf": "f^{\\Delta}(t)", "7b1cff0d23868c60a1a1a2eb4f94319f": "\n\\left(\\frac{\\partial T}{\\partial V}\\right)_S =\n-\\left(\\frac{\\partial p}{\\partial S}\\right)_V\\qquad=\n\\frac{\\partial^2 U }{\\partial S \\partial V}\n", "7b1cffd4f64ce98939551c37826df9be": "e^{[a}_M e^{b]}_N \\delta^M_{[I} \\delta^K_{J]}", "7b1d39599225c609e88c9f79716cdfc3": "L/\\sqrt[3]{N}", "7b1d55f140cbde01708d2a2610d33e3e": "\\left\\{{ 0 \\atop 0 }\\right\\} = 1\n\\quad \\mbox{ and } \\quad\n\\left\\{{ n \\atop 0 }\\right\\} = \\left\\{{ 0 \\atop n }\\right\\} = 0", "7b1ddd943b4c2eb5d47dc14af1860445": " y_1(x)=4x-1 ", "7b1e04d987c3c5bb8954300b6e0c782d": "\\lambda_\\alpha", "7b1e3b05364b49a56f6ec3fa9ba8e7c4": "|G| |X/G| = \\sum_{g \\in G}|X^g|.", "7b1e3e8db8e127efbb40d2e87ea569d5": "\\left [\\begin{smallmatrix}2&-3\\\\-3&2\\end{smallmatrix}\\right ]", "7b1e56943691bcc6e95cd1d86b61f5cc": "s_{iw} (i = 1, 2, . . ., m)", "7b1eccf53a38dd77a6a1c9c8c0150a9a": "x = g^{-1}(y) = \\sqrt{y}", "7b1f0b61e70e940d6c3f120db46daaef": "g(r) = -GM/r^2", "7b1f38cd7ad24e8faa1a729f352b3404": " p_i = 2^{i-1}p_1 - (2^{i-1}-1) \\, ", "7b1f72d0ae26ad7efff7b51b326a9d18": "\\frac{1}{Z_\\text{in}} = \\frac{1}{Z_0 + Z'} + \\frac{1}{Z + Z_0}", "7b1f7ce9ed690c0eee9f2aa1f5a99c5a": "\n\\Omega_i^j(\\mathbf e) = d\\omega_i^j(\\mathbf e)+\\sum_k\\omega_k^j(\\mathbf e)\\wedge\\omega_i^k(\\mathbf e).\n", "7b1fd0c9f507e4a8fdfde517217b4400": "1/f\\,", "7b20095eb0ad82e3b0094f6ba21f475a": "(E)-(F)", "7b20b6e96ed129931a35e161367ce8f8": "\\frac{\\partial \\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y) }{\\partial \\beta} = \\sum_{i=1}^N \\ln (c - Y_i) - N(-\\psi(\\alpha + \\beta) + \\psi(\\beta))- N \\ln (c - a)= 0", "7b20c343d9facc1dad620c7ededd4344": "X = \\{ (Q, \\ell) \\mid P,\\,Q \\in \\ell\\} \\subseteq \\mathbf{P}^2 \\times \\mathbf{G}(1,2).", "7b211bbae2288689a3d4d61622528cd0": "\\textstyle\\frac{4}{3}\\tan(t/4)", "7b2151e442da6d2891aa908144744c77": " \n\\beta + \\tfrac{1}{2} \\sum_{i=1}^n (x_i - \\bar{x})^2 + \\frac{n\\nu}{\\nu+n}\\frac{(\\bar{x}-\\mu_0)^2}{2} ", "7b21c0673c38a931bbaeb46db5a0c686": "g=\\binom{d-1}{n} , \\,", "7b21df57f51ee3ba734526a3fcb760d6": "\\Delta_S = (1 + S_{11}) (1 + S_{22}) - S_{12} S_{21} \\,", "7b21e879348227a8bfc6858998057ed5": "\n C_1 := \\frac{\\rho_d L_d a b^2 \\nu}{2 w^2}; \\quad\n C_2 := \\frac{D}{\\rho_d b^2}\n", "7b21f65a323bc46ea3b1de05cb09d32f": "q^{ - \\varepsilon n} \\ll 1", "7b221fb94e9b53865536b22e2a527eea": "U+PV\\,", "7b224c6d9d9551054ffa1d097b20e94c": "1, i, \\varepsilon, i_0", "7b225c33155b9eb264ad289f5f8c14d7": "f(x)\\,=\\,a_n x^n + \\cdots + a_1 x + a_0", "7b225e0e00031e1254b26e8ce459aa73": "27^9", "7b22aacfc76b5423f7bab151f19fe66c": "\nL_x(x^*,u^*)=\\begin{bmatrix} \\frac{\\partial L}{\\partial x_1}|_{x=x^*,u=u^*}\n& \\cdots & \\frac{\\partial L}{\\partial x_n}|_{x=x^*,u=u^*}\n\\end{bmatrix} \n", "7b22ac8ca3ecc82ac175860afac21ed3": "\\ln(C) = - \\frac{K \\cdot t}{V} + \\mbox{const} \\qquad(2b)", "7b22bbc0b18433654267c7c916a7fafe": "\\Box(A\\equiv\\Box\nA)\\to\\Box A", "7b239d25609390a6c0724b22f32c3388": "Z = 4.4172 \\,\\! ", "7b23a044768151828198e379d574a940": " {} = {(PM-XM).(MQ+XM) \\over (PM+MY).(QM-MY)}, ", "7b23afdab310a2e7d5e5bb9f80b995a2": "\\mathfrak{L}\\,", "7b2436b5311ee256659a8bed8f1940ae": "\n\\kappa^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{b^{2} + (3/4) c^{2}}{R_{g}^{4}} = \\frac{3}{2} \\frac{\\lambda_{x}^{4}+\\lambda_{y}^{4}+\\lambda_{z}^{4}}{(\\lambda_{x}^{2}+\\lambda_{y}^{2}+\\lambda_{z}^{2})^{2}} - \\frac{1}{2}\n", "7b245c4c3304a2e7381c8eb61ea93460": "P_\\theta=\\frac{\\partial L}{\\partial \\dot \\theta}=mr^2\\dot \\theta", "7b247bc6dca18034f0e0d886de776a17": "(x^{q^{2}}, y^{q^{2}}) = +\\bar{q}(x, y)", "7b248432caab4436d827559313f2fc42": "i=k,\\ldots,n-1", "7b248493fde5cf7dda69123b02e66602": "\n X^{VG}(t; \\sigma, \\nu, \\theta) \\;:=\\; \\theta \\,\\Gamma(t; 1, \\nu) + \\sigma\\,W(\\Gamma(t; 1, \\nu)).\n", "7b24bd57640bf9294a8281216478a481": " \\bar{P} = \\frac{1}{t_2-t_1}\\int_{t_1}^{t_2} v(t) i(t)\\, dt .", "7b24cc6ed9d638a08a12007535ce2a99": "\\left[\\frac{d+1}{2}\\right]", "7b2512d4a172f8726949e633ac96aeaa": "G: \\mathbb{F}^\\ell \\rightarrow \\mathbb{F}^n", "7b25364e433fc986e1d5aa955651e565": "Y_{t} = Ytd", "7b25394af28186d91542136e60e0e4a4": "\\mathfrak{P}^{8}", "7b255dd38c475b34223c7018fe445c72": "\\mathbb{C}^{p+q}", "7b25f0c6a41780f68379d4be275f6a74": "\\scriptstyle \\dot{\\phi}", "7b2682423811efbab87f4ca5be02b516": " \\frac{X / d_1}{Y / d_2} \\sim \\mathrm{F}(d_1, d_2)", "7b26910a0bbd5420467bfb4957000e72": "\\{H_k(x_1,\\cdots)\\}", "7b26dfad9e60a2d8e50ba0586b53e262": "[x, y^{-1}] = [y, x]^{y^{-1}}", "7b27316ece7a16a3c3157c15ffc0e027": "\n\\partial_\\theta U_\\epsilon(t_1,t_2) = \\epsilon g \\partial_g U_\\epsilon(t_1,t_2) = \\partial_{t_1} U_\\epsilon(t_1,t_2) + \\partial_{t_2} U_\\epsilon(t_1,t_2). \n", "7b277701c97fea60fad12f4e6fd12b22": "\\nu_m < \\nu_c", "7b27be150de12a453beee93ba3bec9ad": "\\ \\frac{d}{dt}[p_{ij\\ldots}(\\mathbf x,t)]=\\frac{\\partial}{\\partial t}[p_{ij\\ldots}(\\mathbf x,t)]+ \\frac{\\partial}{\\partial x_k}[p_{ij\\ldots}(\\mathbf x,t)]\\frac{dx_k}{dt}", "7b27d0db69568c5af40d98ae2d66d7d3": "V_{out} = -\\dfrac{V_{in}}{RC}t_{int} + V_{initial}", "7b27eee156f2b25c8287c45e8ab41236": " - \\partial_t^2 \\psi + \\nabla^2 \\psi = m^2 \\psi", "7b27f4772000b1e4d0f478d681d27f6d": "\\text{Per-unit amperes}=\\frac{\\text{amperes}}{\\text{base amperes}}", "7b27fdb4f8a0a2d40da0f9bcbcd06bec": "(E-H_0)", "7b281bbd9285fcf17d61cbb3582eeef9": " M_2 (\\vec {\\rm E}) = \\left[ {\\begin{array}{*{20}c}\n 0 \\\\\n { - \\Sigma ^{ - 1} } \\\\\n\\end{array}} \\right]\n", "7b28529fff5ee3b678e4a4519f324eb1": " Y_t ", "7b287707553e7b568401be03bc0837a0": "\n(f_1\\ ,\\ f_2\\ ,\\ f_3)\\ = \\ - \\frac {\\mu} {r^3}\\ (x_1\\ ,\\ x_2\\ ,\\ x_3)\\ +\\ (h_1\\ ,\\ h_2\\ ,\\ h_3)\n", "7b28777ad0944cb9b64e0ae81423f442": "a_m = 1.", "7b28a3128112f66a614e1864a4cabeca": "x_{m}=(E/q^{2})^{1/\\beta }", "7b28aa6f8cf850022fd9269ddebbf504": "z \\;\\stackrel{\\mathrm{def}}{=}\\; \\operatorname{E}(z_i) = \\frac{1}{n}\\sum_i z_i n_i", "7b28dfe29e51ccecd1a9c6dadef7ef6f": "C=\\sqrt{\\frac{\\chi^2}{N+\\chi^2}}", "7b29364cb3fc1500eed21beb2e5299fa": "\\mathbf{H}_{\\alpha}(z) = \\frac{(z/2)^{\\alpha+1/2}}{\\sqrt{2\\pi}\\Gamma(\\alpha+3/2)}{}_1F_2(1,3/2,\\alpha+3/2,-z^2/4).", "7b297c233106e3f36a277aef9575ba83": "R=\\rho{l \\over A}", "7b2984ef6c8ed1ced9a3234111e3e9e0": "\nP_\\mathrm{L} = I^2 R_\\mathrm{L} = {{\\left( {V \\over {R_\\mathrm{S} + R_\\mathrm{L}}} \\right) }^2} R_\\mathrm{L} = {{V^2} \\over {R_\\mathrm{S}^2 / R_\\mathrm{L} + 2R_\\mathrm{S} + R_\\mathrm{L}}}.\n\\,\\!", "7b29d56c2e67a0f14d573c7eba1ea08f": "\\vdash \\neg A_1 \\lor \\neg A_2 \\lor \\cdots \\lor \\neg A_n \\lor B_1 \\lor B_2 \\lor\\cdots\\lor B_k", "7b29e4f7e1d845b8cb71fbb90e56f388": "\\overline{AB}\\cdot \\overline{CD}+\\overline{BC}\\cdot \\overline{DA} \\ge \\overline{AC}\\cdot \\overline{BD}", "7b2a33cb46f5635e9ae587838fd0b25b": " \\dot{x} = f(t, x), \\quad x(t_0) = x_0. ", "7b2a68ba186421bf2c52bc9ca76156cc": "\\scriptstyle \\frac{f(i)}{i}=\\frac{1}{2}+O(\\sqrt{\\frac{\\log\\log i}{i}})", "7b2ab8f85a7c748185b71f03c65b83ff": "R = p_{\\theta}\\dot{\\theta} - L = \\frac{p_{\\theta}^2}{2mr^2} - \\frac{1}{2}m\\dot {r}^2 - \\frac{k}{r}", "7b2ac5645e798f09ce619c99d5e01cff": " N = \\frac{K}{2}", "7b2b6091b3d631da8bdf8bbd47cdd036": "\\theta_{C_1 \\otimes C_2} = c_{C_2, C_1} c_{C_1, C_2} (\\theta_{C_1} \\otimes \\theta_{C_2}).", "7b2be3bc44a57ba18caa570468059cb4": "1-\\tfrac{1}{\\sqrt{2}}< \\text{median} < \\tfrac{1}{2}", "7b2c06875b2f5d2c5d7a734fe3d5de95": "\\tan \\left( \\frac{1}{2} \\Omega \\right) =\n \\frac{\\left|\\vec a\\ \\vec b\\ \\vec c\\right|}{abc + \\left(\\vec a \\cdot \\vec b\\right)c + \\left(\\vec a \\cdot \\vec c\\right)b + \\left(\\vec b \\cdot \\vec c\\right)a}\n", "7b2c21c3ed84db162399b3d94aa62a60": "A = i\\beta \\alpha_1, B = i\\beta \\alpha_2, C = i\\beta \\alpha_3, D = \\beta \\, , ", "7b2c5a4225f42ab3ba207917e565f6bd": "\\sqrt{\\frac{2}{5}}\\!\\,", "7b2c835d075415250f126860ba672747": "V_1y_1", "7b2d1652e9c30c4e45416b05d28d8694": "\\psi (\\boldsymbol{r}) \\approx \\frac {1} {\\sqrt{N}} \\sum_{m,\\boldsymbol{R_n}} e^{i \\boldsymbol{k \\cdot R_n}} \\ \\varphi_m (\\boldsymbol{r-R_n}) \\ .", "7b2d23ab613c349c4c74ddeda966227c": " \\sigma: R\\to R ", "7b2d305fbc0a174b28ba022a48ec9dfd": " \\frac{2 x^3 + x y^2}{a^2 y - x^2 y} = \\frac{dy}{dx} ", "7b2d8b7d678b0e31299b377604b54004": "A_{\\mathrm{Pa}} = A_{\\mathrm{mmHg}} + \\log_{10}\\frac{101325}{760} = A_{\\mathrm{mmHg}} + 2.124903.", "7b2e5aed63cead3b875074ab920361f5": "(((x_\\text{int} / 2^n - b) / 2) + b) \\cdot 2^n = (x_\\text{int} - 2^n) / 2 + ((b + 1) / 2) \\cdot 2^n.", "7b2e927bba5c48069a292de4c6f78a05": "(1,2t)\\cdot(1,2\\tau)=0 \\,", "7b2e9cbdde7dd4878fac342ff4aec9f5": "\\textstyle 2^k", "7b2ec2469723c799f2fe49fcfd87b1c1": "\\begin{array}{rcl}\n \\dfrac{d V}{d t} &=& V-V^3 - w + I_\\mathrm{ext} \\\\ \\\\\n \\tau \\dfrac{d w}{d t} &=& V-a-b w\n\\end{array}", "7b2ec2c843be8ceb8955064c49bd7bee": "(k^2B^2/\\mu_0\\rho)=15\\Omega^2/16\\ .", "7b2ec8b1ecd1e0bf9219823ca133b7da": " \\mathbf{X}^{-1}", "7b2f33533634507d6e60d2e3e8c6cfc5": " \\mathbf{r} = \\mathbf{r}_2 - \\mathbf{r}_1 \\,\\!", "7b2f6d7c901f445c9f5f58cf78f681e5": "G=(G_t)_{t\\ge 0}", "7b2f953ce193e35c35433e89f2ff5b15": "\\mathbf{R^3}", "7b30095cebdfbd41e84eadb432f616b2": "f(\\alpha)", "7b302882144e898568fcd9598329fc15": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\n = \\cfrac{E}{1-\\nu^2}\n \\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & \\cfrac{1-\\nu}{2} \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ 2\\varepsilon_{12} \\end{bmatrix}\n ", "7b3036f2a633315b011cfecff68a8153": "\\log\\rho", "7b3049f86047b2af15e8e6e6787407cb": " cos \\phi_{S} = \\frac {W_{S}} {{\\sqrt{3}} {V_{SL}} {I_{SL}}} ", "7b30563a3a7ecae7b5d20aca3fb94b88": "Q(t,x,y) = \\frac{1}{2\\pi} \\, \\int_{(x-x')^2 + (y-y')^2 < t^2} \\frac{q(x',y') \\, dx' dy'}{ \\left[ t^2-(x-x')^2-(y-y')^2 \\right]^{1/2}} ", "7b306237cfa8562e53bf0fa4086e14d4": "\\psi(\\Omega^{\\Omega^2 (\\omega^2 6 + 1)})", "7b3085d98fea67ce6fceeeea6dba3a1c": "\\displaystyle{f_z-g_z=T(\\mu(f_z-g_z)).}", "7b309ba5b1cbb23af2136888a94224ee": " (-M-)_n (polymer) + M (monomer) \\rightarrow (-M-)_{n+1}", "7b30bc720935d492fa723d1af39f89e3": " g_i(x) \\le a_i , h_j(x) = 0", "7b30f47eac12c4bde00079943574dcb4": "w_{\\mathrm{max}}/w(L/2)", "7b30fcbf94907b66c0fd6f5b7007d145": "y=ax+bx^2", "7b30fdc088deab98a4dca881198b4064": "\\mathcal{H}_L", "7b3136147077353586284f0d67691621": "x\\left(\\frac{az + b}{cz + d}\\right) = x(z)", "7b314653855c4bb311ae9d54c6fd8c25": "M \\otimes_{S} N \\cong R", "7b3152e159c0a440ebaeb054b03cd478": "a_{i,m}=a_i/\\pi^m", "7b316d358038cde03f323a2c1b886520": " \\Delta\\mathbf{r}_i ^\\perp= (\\mathbf{r}_i-\\mathbf{R}) - (\\mathbf{S}\\cdot(\\mathbf{r}_i-\\mathbf{R}))\\mathbf{S} = [[I]-[\\mathbf{S}\\mathbf{S}^T]](\\Delta\\mathbf{r}_i),", "7b319a17e3484174d068205fc8f8a396": "1\\,R_{\\odot}", "7b31b8a0c63c2815e89a1b870ae748ca": "\\vec{\\xi}_1 = \\partial_v", "7b31daa32caa49465619b2166182d76d": "\n \\sum_{\\alpha\\in A} f_\\alpha\\mathbf{e}_\\alpha\n ", "7b31ee2fcda9336edf467bbeca4eb32e": "\n R_{2}\\in [2.584 \\Omega,3.355 \\Omega].\n", "7b32642cb5481f8b32215833a7348d81": "y\\alpha>x_nx_{n+1}\\,,\\qquad({*}{*}{*})", "7b327af63e7c8eee3ba86fe1a4c7cff0": "\\in \\!\\,", "7b327e8df70d9eb7a12271bc7f135876": "\\mathcal L=|\\partial_\\mu \\Phi|^2-U(|\\Phi|) \\, ", "7b336af06cf62a5b8d596f1199c16984": "0 < a < 2", "7b33da032527ca34cc4f26a6f1113369": "\\theta \\log{\\tan \\theta} - \\frac{1}{2}\\int_0^{2\\theta}\\log\\left(\\frac{\\sin (x/2) }{\\cos (x/2)}\\right)\\,dx=", "7b348a994ab8651ee485dba44f13d633": "\\varepsilon^2T_n^2(-1/js_{zm})=0.\\,", "7b34d77bbcdb959dd88b07724505ac31": "x^2 = y^3", "7b34f28f91cf6cbabc0ec297cf8e9fe9": "\\Rightarrow T\\mathbf{v} = a_1 T\\mathbf{u}_1 + \\cdots + a_m T\\mathbf{u}_m + b_1 T\\mathbf{w}_1 +\\cdots + b_n T\\mathbf{w}_n", "7b35085a38b068930ec3c49760499cd9": "\\mathbb{E}\\left[\\sup_{f \\in \\mathcal{F}} |\\hat{R}_n(f) - R(f)| \\right] \\leq 2 \\sqrt{\\dfrac{\\log S(\\mathcal{F},n) + \\log 2}{n}}", "7b351b64a77914b0c3a8cf3afea2d18d": "g_{ij} = \\delta_{ij}+ O (|x|^2).\\,", "7b352a0204982f8f51b0385e04a7ec48": "(A\\rtimes G) \\rtimes \\Gamma", "7b35480b041844284055662886ba0c81": "f(\\bar{z},z)", "7b355c626e809e72746a1e51daa5d87f": "v+k_{1}=k_{2},\\,", "7b35645102c2b666b565d09357b17f39": " B_n = \\frac{n!}{2 \\pi i e} \\int_{\\gamma} \\frac{e^{e^z}}{z^{n+1}} \\, dz. ", "7b35c04e877d4eda3afa127bbcb64401": "AH = t\\ \\text{Crd}\\ \\angle AOH = t\\ \\text{Crd}(\\varphi_H - \\varphi_A)", "7b36002b8909879f17e207a8a6d3fc0d": "\\mathrm{1\\; psi/ft = \\frac{1\\; ft}{12\\; in}\\times \\frac{1\\; lb/in^{2}}{1\\; psi}\\times \\frac{231\\; in^{3}}{1\\; US\\; Gal}=19.25000000\\; lb/gal}", "7b360ae6b700030834f7e092a400b19f": "|S| \\cong S^1", "7b364c92d7aca0a201c129ab5b9c3651": "\\vec{r}.", "7b365dce7715dd4a5802f9a2208cdc3d": "K = \\oplus _{i = 1} ^{nm} C_i ^n.", "7b3667d63fa98fc943020faa3408d311": "\\cos(2k+1)\\frac{\\pi y}{2}", "7b369e87d4aea2c6ce22ac28191807c6": " {n^2-3n+4 \\over 2} ", "7b36cd33ec1fd4fea4f08bb5c1adb1ca": "y = at^3. \\,", "7b36e0955913031f062d259968fdce30": "t\\mapsto(t^3,1-t^2),", "7b36e87cf378a2e32d62d4e40fd48ba5": "r = \\sqrt{\\left(x - x^\\prime\\right)^2 + y^{\\prime2} + z^2}", "7b370f854cb17c957932b53ead014aca": " f^{-1}(x_i)", "7b37117f2325f18d29ec98fd5f37aeaf": "10_{75}", "7b37928614261cba5e6c8c88f19cc413": "\\overline{\\operatorname{cone}}(\\cdot)", "7b37f3450be94bbd07e88633dbb27a68": " \\int_{\\mathfrak{H}^3} (M_0f)\\cdot F \\,dV = \\int_{\\mathfrak{H}^2} f\\cdot (M_0^*F) \\,dA", "7b385114353d43765058c09cdaa149ef": "{n+1 \\choose k+1} = {n \\choose k+2},", "7b389ed2683beee91e569dd26e930da3": "\\int_{A\\times B} |f(x,y)|\\,d(x,y)<\\infty,", "7b38cc864a346879d67899d547fed72c": "\\scriptstyle \\boldsymbol R_{new} \\;=\\; \\exp([\\boldsymbol\\omega(\\boldsymbol R_{old})]_{\\times} dt) \\boldsymbol R_{old}", "7b38ec04350c999d70a787a1e381607f": " \\ R_k(x) = f(x) - P_k(x),", "7b396da2a21828de195bff7174bbdb8e": " d \\in D ", "7b39767aeab8b3762ca70e684d9a398b": "C=\\{C_k\\}", "7b39783a3f1b192489d3d828c95c0d41": "\\oint_C \\mathbf{B} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = \\iint_S \\left( \\mu_0 \\mathbf{J}+ \\mu_0 \\epsilon_0 \\frac{\\partial }{\\partial t}\\mathbf{E} \\right) \\cdot \\mathrm{d} \\mathbf{S}", "7b39d5af3a670cfcf9fc5bb57badae4c": "G = X_v \\cdot X_v = 1", "7b39e4b1797efc27065ad52cfd612e3f": "\\{S_k\\}_k", "7b39fd92f41de78e04021bf5b4dd3a74": "GL_n R\\,", "7b3a22e01c1eedd620280c96e44d4a9b": "D(X)", "7b3b02571511a447bd7f37516f7c26ed": " k\\in K", "7b3b2d29bbc15370326216535f5d4624": " \\text{Ext}: \\{0,1\\}^n \\times \\{0,1\\}^d \\rightarrow \\{0,1\\}^m \\, ", "7b3b44cf5115909e1407ba1c21c33fde": "F(t, x, y) = F(u, x, y) = 0\\,", "7b3b84c813cdf31a673a05a0fa44a0b9": "\\mbox{for all } x\\geq1, \\sum_{n\\leq x}d(n)=x\\log x+(2\\gamma-1)x+O(\\sqrt{x}),", "7b3b862e3d3b22756a129b2e14ac7008": "f_x(1,1) = p_x(1,1) = \\textstyle \\sum_{i=1}^3 \\sum_{j=0}^3 a_{ij} i ", "7b3bd657fae06d7dc514f5a2e466e485": "O(N/\\log N)", "7b3be43ba4e0cb01befe8e0a472ccfb4": "\\sqrt[3]{1.80 \\times 1.166666 \\times 1.428571} = 1.442249", "7b3c44c0cfd73b5f5fa80aa1f90edbd7": "J_-|\\psi\\rangle", "7b3c567d45d3a79b1afd92c81be88855": " \\frac{a_{ph\\mathrm{-}e\\mathrm{-}p,a}(\\hbar\\omega-\\Delta E_{e,g}+\\hbar\\omega_{p})^2}{\\mathrm{exp}(\\hbar\\omega_p/k_\\mathrm{B}T)-1} ", "7b3c5710d0252161577a4d475f523748": "Z_L=-iZ_q\\tan \\left( \\pi \\frac{\\Delta f}{f_f}\\right)", "7b3c777f227ea554fd9402a5e9bf8d50": "(\\textbf{A}_N,\\textbf{A}_P)", "7b3c7b867d671794de9f9afd9f71181f": "\\begin{align}\\frac{k}{2}&+\\ln(2\\Gamma(k/2)) \\\\ &\\!+(1-k/2)\\psi(k/2)\\end{align}", "7b3c83d1f0351b7d4748a443b51b9129": "c_{\\mathrm{air}}", "7b3ca70deccd4f604f7c2443fc9e73e5": "\\ell = 0, \\ldots, k-1", "7b3cc248c56485985a5e5f3dad66b5d5": "\\scriptstyle D", "7b3cc94f7cb845fea7cb764ed56cc37c": "t \\cdot \\int_a^b \\sin(k \\cdot t) \\; dt", "7b3cd22596b66a993a0cb07f73ae1eff": "N_A \\approx 6.022 \\times 10^{23}", "7b3cf2a1913a62348babbbe34d4dcd0b": "\\textstyle{\\frac {\\log(8)} {\\log(3)}}", "7b3d3c809757a08c177c78d0286877b4": "\\Psi(x,y)=e^{ik_y y} \\phi_n(x-x_0). \\, ", "7b3d9f0bb15e4d9a78ac268230cbf692": " \\boldsymbol{\\tau} = \\mathbf{m} \\times\\mathbf{B}", "7b3dce230ec071994313b2d7e4efd0a1": "E_{n+1}", "7b3e12d364929765e2dea0db9063a922": " M(p) = \\alpha^{-p}\\int^1_0(1-z)^{\\gamma-p-1}z^{p-1} dz ", "7b3e2e9078602e70a793e5f16ef43272": "\\forall x [ \\exists a ( a \\in x) \\Rightarrow \\exists y ( y \\in x \\land \\lnot \\exists z (z \\in y \\land z \\in x))].", "7b3e7475776e4110a845928b76b559c7": " \\alpha_s(k^2) \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{g_s^2(k^2)}{4\\pi} \\approx \\frac1{\\beta_0\\ln(k^2/\\Lambda^2)},", "7b3e80d3f2d79bf4383f754b0f6de6a7": "\\sum_{i=0}^n i = \\sum_{i=1}^n i = \\frac{n(n+1)}{2}", "7b3e8ce75801fef65fdd0587b0abfd96": "Z_L = j \\omega L = j 2 \\pi f L = j32.7 \\ \\Omega\\,", "7b3efa2b54e64ab95c2b0957486a758c": " \\hat{a}|\\alpha\\rangle=\\alpha|\\alpha\\rangle \\,", "7b3f21f3abb7e062de2e08217b00c640": "\n (n\\mid m,k) = \n \\begin{cases}\n \\frac{(m\\mid n, k)}{\\sum_{n=m}^{\\Omega - 1} (m\\mid n, k)} &\\text{if } m \\le n < \\Omega \\\\\n 0 &\\text{otherwise}\n \\end{cases}\n", "7b3f4269a64d27760a7575aa037bf579": "P_n^{(\\alpha, \\beta)}(1)=\\frac{\\Gamma(n+1+\\alpha)}{n!\\,\\Gamma(1+\\alpha)}\\,", "7b3f99d5be225f62773c9a7f38765fac": "\np_{HA} = (p_{HA}+p_{LA})(p_{HA}+p_{HB})\n", "7b3f9ff995fd3be8543c8d9735c5f365": "\n\\mathrm{DR_{dB}} = \\mathrm{SNR_{dB}} = 20 \\log_{10}(2^n) \\approx 6.02 \\cdot n\n", "7b3fe8df0f005e5a3bf2ee9549b45b51": "\\langle P,A,\\mathit{IC}\\rangle", "7b400de9a77dcac24b840830bce023ff": "5\\zeta(3)\\zeta(5)-\\tfrac{147}{24}\\zeta(8)-\\tfrac{5}{2}\\zeta(6,2)", "7b4058ed2838a59ca715fa46d83b4ad5": " \\Sigma_g", "7b406fe953851c444d90f171fa52fffc": "p_0(x)=p(x)=x^4+x^3-x-1", "7b40f29aa807b39992fa950de12a608f": "\n M_4 = M_c = -937.5 + 40 R_a + 25 R_b \\,.\n ", "7b413fb1e7085b01b4b6e4225a00f5e9": " T_P ", "7b416606b70fc35602119091d50a7ae1": "0\\rarr M_1\\rarr M_2\\rarr M_3\\rarr 0", "7b41b18d5c649fc15e9740fe0737aec0": "W_1 = m_1 g", "7b41b28775ad01822c58a89abbc5b9cc": "\\Omega_p\\;", "7b41dbd9651cc5f427a05f17390dcf9a": "M^T \\Omega M = \\Omega.", "7b424dfb1bd3daaa4e065c55c8db64ff": "\\{R_i;\\,i\\in I\\}", "7b427b0b7b8062414b8bf3f33d899b72": "\\operatorname{\\bar{F}}(\\bar{r},\\dot {\\bar{r}},t)=-\\mu \\cdot \\frac {\\hat{r}} {r^2}+\\operatorname{\\bar{f}}(\\bar{r},\\dot {\\bar{r}},t)", "7b42b4f2dd1e3ebce5707e8c76eefbf5": "\\nabla f(x_k+\\Delta x)=\\nabla f(x_k)+B \\, \\Delta x,", "7b42d202c06fab06ee1b245c2a7b1d96": "k x^n \\mathrm{,}\\; n = 0, 1, 2,\\cdots\\!", "7b432bb818e371304538b63ad10f49b3": " SG_\\text{true} = {\\rho_\\text{sample} \\over \\rho_\\text{water}}", "7b435610a8cb4902737f348011316170": "|{\\Psi}\\rangle=\\sum\\limits_{i_1,{\\alpha_1=1}}^{M,\\chi}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}|{i_1}\\rangle|{\\Phi^{[2..N]}_{\\alpha_1}}\\rangle\n", "7b435eb1650b5c110be0425e21ef1fdf": "0\\leq s(x)\\leq f(x)", "7b436ca74442efd1a88a89829e198412": "x = AN \\pmod{r^n-1}", "7b441ea8f427038c8b3a5d36be8c1eca": " \\text {where Ei}\\left(z\\right)=\\int\\limits_{-z}^{\\infin}\\left(e^{-v}/v\\right)dv", "7b445a98d22c5d4643f61a5b0108ffa1": "\\delta_x(y) = \\begin{cases}1 \\quad x = y \\\\ 0 \\quad x \\neq y\\end{cases}", "7b447a7b312cb95604d8af74dccec8d1": "\\Pr_R[\\exists (x,y):\\, |p'_R(x,y) - p(x,y)| \\geq 0.1] \\leq \\sum_{(x,y)} \\Pr_R[|p'_R(x,y) - p(x,y)| \\geq 0.1] < \\sum_{(x,y)} 2^{-2n} = 1", "7b449d4b72bc0c690c6cc3924c1476a0": " p = p_0 cos(\\omega t - kx)", "7b4518b5cf19514c4631aa165badea30": "\\tilde X = X \\cup e^2 \\cup e^3", "7b452f2b5f33ce882591fecfe2e4d561": "|f| > B.\\,", "7b454aa761ebc8c53c94dc9fc8556eac": "D_a e_b^i = 0", "7b457762d79a7b150935c639016a487e": "\\displaystyle V(x) = \\lambda\\delta(x)", "7b4595c0d3483c344f74cbe0f547349a": "O(n^{d-1})", "7b45a1834607e644424e856dc6aeb8a1": "s:=\\frac{\\text{lc}(r)}{c}x^{\\deg(r)-d};", "7b45d39106e35e23318c2fc189e2e258": "\\scriptstyle \\sum_n a_n\\varphi_n(t)", "7b45fed99c5f7836f49e440676aa3410": "\\left \\| x \\right \\|_Q^2", "7b460d1eadd8f04697f020121f18f721": "\\frac{\\partial L}{\\partial t}=\\frac{du}{dt},", "7b470947a9c91d35800cd59be043ca25": "\\textstyle {\\sum(\\frac1{p}+\\frac1{p+2}+\\frac1{p+4}+\\frac1{p+6})} \\scriptstyle \\quad {p,\\; p+2,\\; p+4,\\; p+6: \\text{ prime}} ", "7b4719792fa3e27cc65a2b0071cc481c": "\\dot\\gamma(0)=X", "7b47270b2d55bfbd561f2232ff5964e0": " C_{N0}^{j}", "7b472f1c7d326d1b8f854e4f110e6795": "F(x,y,z) = (x^2 + y^2)(\\cos\\theta)^2 - z^2 (\\sin \\theta)^2.\\,", "7b47767e9c40b729eb50b93e3ca2ea99": "S^2\\mathbb C^m", "7b47a84a6f0680575c9d05cc4a380929": " (4)\\,", "7b47e3d3b8c529e323525579f8821202": " \\partial_t^+ \\omega_{n,i} + \\partial_x^+ \\kappa_{n,i} = 0 \\quad\\text{where}\\quad \\omega_{n,i} = \\mathrm{d}u_{n,i-1} \\wedge K_+ \\, \\mathrm{d}u_{n,i} \\quad\\text{and}\\quad \\kappa_{n,i} = \\mathrm{d}u_{n-1,i} \\wedge L_+ \\, \\mathrm{d}u_{n,i}. ", "7b48443026f30cffb0ab48bc1299dd13": "m_i\\,", "7b4861b31c0f20594de27eaaa5405c07": "u_0 \\ge (k_\\mathrm{B}T_\\mathrm{e}/m_\\mathrm{i})^{1/2}", "7b4880922a942fb7c557eb4875da386b": "d(f(x), f(y))", "7b4894394d01fd1b34c496531b5090a7": "f(\\gamma_i)", "7b48af2349e1812fa955fcdbcc347759": "L \\left( x, y, k\\sigma \\right) = G \\left( x, y, k\\sigma \\right) * I \\left( x, y \\right)", "7b48c3b44a12b3fee3a96376a4538faf": " y = \\alpha + \\beta x, \\,", "7b48d1b4e2cd5dba9881e55f54ab4686": "\n\\boldsymbol\\nabla \\cdot \\mathbf{q} =\n\\frac{\\partial q_{x}}{\\partial q_{x}} + \n\\frac{\\partial q_{y}}{\\partial q_{y}} + \n\\frac{\\partial q_{z}}{\\partial q_{z}} = 3,\n", "7b496e675cf190bef509020c7c77d989": "\\mathrm{^{237}_{\\ 93}Np\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{238}_{\\ 93}Np\\ \\xrightarrow[2.117 \\ d]{\\beta^-} \\ ^{238}_{\\ 94}Pu}", "7b4973f9709ea88d9f5617aa7512bce6": "\\boldsymbol{F_1}", "7b49a4651a3f49032926cf80ecced15b": " T_a = T_a^\\dagger ~.", "7b49bb4af7302d63e8b4abf9f291815a": "\\frac{d\\rho}{\\rho}=-3\\frac{da}{a}\\left(1+w\\right).", "7b49e2833d41dc3b0e862afa32955aff": "c=1,\\ G=1 ", "7b49f3cc79d8ab404e4d2a7988b85422": "\\begin{align}\n\\mathbf{u}\\times\\mathbf{v}=&u_1v_1\\mathbf{0}+u_1v_2\\mathbf{k}-u_1v_3\\mathbf{j}\\\\\n&-u_2v_1\\mathbf{k}-u_2v_2\\mathbf{0}+u_2v_3\\mathbf{i}\\\\\n&+u_3v_1\\mathbf{j}-u_3v_2\\mathbf{i}-u_3v_3\\mathbf{0}\\\\\n=&(u_2v_3-u_3v_2)\\mathbf{i}+(u_3v_1-u_1v_3)\\mathbf{j}+(u_1v_2-u_2v_1)\\mathbf{k}.\\\\\n\\end{align}", "7b4a03673d86c0f91026867b762286d2": "\n \\tilde{f}^{*} = \\operatorname{arg min} \\left\\lbrace E\\left( (x_1, y_1, \\tilde{f}(x_1)), ..., (x_n, y_n, \\tilde{f}(x_n)) \\right) + g\\left( \\lVert f \\rVert \\right) \\mid \\tilde{f} = f + h \\in H_k \\oplus \\operatorname{span} \\lbrace \\psi_p \\mid 1 \\le p \\le M \\rbrace \\right \\rbrace, \\quad (\\dagger)\n", "7b4abd8e5b7997fcfa54d0a2f73f65c0": "\\rm{dIBB=0.9926 \\ IBB} \\,", "7b4b33d70daf8b49c02ed99df710e48f": "\\langle v_1,\\ldots,v_n \\rangle \\in I(P)", "7b4b9e68a4e0f6d49d22ab8d1f7120d8": "R_J", "7b4be1c50dd0a109ce22daedb6e3b777": "q= \\sqrt{\\frac{ab^2-a^2b-ad^2+bc^2}{b-a}}", "7b4c12cdc0a7ccf3ea5a4fb07e6b80af": " \\gamma_{txx} ", "7b4c165a883754e6d90f3773274c63b9": "U_{ix}\\times\\{y_0\\}", "7b4c47a5c79b2c9bbdaa406c40ba7ff4": "\\epsilon n", "7b4c59a5959f1074b4a6ba41a58ab966": "P_{\\mathrm{s, max}} = R \\frac{L^2}{h}\\,", "7b4c5d8034ea4bf3c7fe3849e9b62f77": "|q|", "7b4c5fbab1ffdc3281eefef2bf749bb2": "p \\mapsto (u + v) \\oplus (u - v) , \\quad q \\mapsto (w + z) \\oplus (w - z).", "7b4d1e72b5b011683b872747e7c42355": "i^2 = -1 \\ . ", "7b4d281df3b50d7ea7050731dfda7f78": "\n\\begin{align}\n\\underset{\\boldsymbol{u}}{\\text{minimize}} \\quad & \\underset{\\boldsymbol{x} \\in \\mathbf{S}_0}{\\operatorname{sup}} \\frac{f(\\boldsymbol{x}) - \\boldsymbol{u}^T \\boldsymbol{h}(\\boldsymbol{x})}{g(\\boldsymbol{x})} \\\\\n\\text{subject to} \\quad & u_i \\geq 0, \\quad i = 1,\\dots,m.\n\\end{align}\n", "7b4d7f716733608ab034508d2457549c": "\\lim_{n \\to \\infty}{ E\\left( \\left| f\\left( \\frac{K}{n} \\right) - f\\left( x \\right) \\right| \\right) } = 0", "7b4d875dd53cc45991fa2c25ce21d44d": "f\\in V_k", "7b4d8d3bbdd3c25449e210b4350c7cd2": "\\scriptstyle L_k", "7b4e04c307333d3b7f4c302822383ef7": "g_{ij}[\\mathbf{f}] = g\\left(X_i,X_j\\right).", "7b4e8c2aa42f2f5ed6383a6357505f8c": "[O,Y]", "7b4ee5affd308dcd829821b4d9ee83eb": "h_{-2}", "7b4f0ac8996042f85ed21dc602eadd00": " \\langle B\\rangle \\,=\\, Tr (\\rho \\, B) ~.", "7b4f1029ac16898199e37e8efc4dcc72": "(K,\\,\\nu)", "7b4f28a3c8640712aa86cb361bb5c69b": "1/d= 1/l * (\\tfrac{1}{\\tan \\alpha} + \\tfrac{1}{\\tan \\beta})", "7b4f3706a6bb123e98147304da89b0fa": "U\\subset V", "7b4f65cf47f9509db61c89b908b21031": "\\mathcal{A}, \\mathcal{B}, \\mathcal{C}", "7b4fa528be3916e5acec00d41f487b49": "\\Phi''(x) + \\left[\\Phi'(x)\\right]^2 = \\frac{2m}{\\hbar^2} \\left( V(x) - E \\right),", "7b500be3aaf3870112b75b4cd9cb545d": "\\cos \\boldsymbol{\\Phi}_r =\\frac{dr_r}{dt_r} = \\frac{ \\sqrt{\\dfrac{2M}{r}}+ \\cos \\boldsymbol{\\Phi}_s } {1+ \\sqrt{\\dfrac{2M}{r}}\\ \\cos \\boldsymbol{\\Phi}_s },\\,", "7b50a107cdfae22fcf03c6f676d5de2a": "x{\\partial f\\over\\partial x}(p, q, r)+y{\\partial f\\over\\partial y}(p, q, r)+z{\\partial f\\over\\partial z}(p, q, r)=0.", "7b50ce052f289d9e7aee7cc9c4cd6b4f": "t \\equiv TR^{-1} \\mod{N}", "7b511d02e832d0558baab2821cce8988": "R=\\frac{I_{r}}{I_{i}}=\\left \\lbrack \\frac{n_{1}-n_{2}}{n_{1}+n_{2}} \\right \\rbrack^2={r_{12}}^2", "7b513849b0151a4f18fd58b937cffdb5": "\\textstyle {N}_c", "7b514da3db68d7c7f7c8c22a6cd99561": "W \\subset \\mathbb{C}^n", "7b516481fdbb361ca8f57b3caec9dfa3": "= {p'}^{\\rho} p^\\sigma \\operatorname{Tr}\\left( \\gamma_\\rho \\gamma_\\mu \\gamma_\\sigma \\gamma_\\nu \\right) + m^2 \\cdot 4\\eta_{\\mu\\nu} \\,", "7b518d03f39f136d576c781afd16a6fe": " \\sum_{\\, e_j \\in S_i} x_i \\geq y_j ", "7b519c0c864aff0176a65be339de34e2": "y_i=\\beta_0 +\\beta_1 x_i +\\varepsilon_i,\\quad i=1,\\dots,n.\\!", "7b51c1d5843d4aa9fe30f1cc8f86a597": "\n \\quad \\min \\limits _{D, X} \\{ \\|Y - DX\\|^2_F\\} \\qquad \\text{subject to } \\forall i, x_i = e_k \\text{ for some } k.\n", "7b51db1784a73f734cabafae17408587": "-\\mathbf{e}_{123}", "7b5282cab939a69bd580481be9ee69ae": "J/{m^{-2}}", "7b528bab84738e8eb46c182534f34280": "\\overline Q \\not= 0", "7b530f18a99afcaf9212bed977db95b2": "x = (p - h) t + h\\,", "7b53af70fae34888d557e8084084f6b2": "({v_0+v_i})10^{b_1E_{i}}", "7b53f8c5eef8ce11ef1f8a12de4ee1bb": "\\Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6", "7b54a69b34332ee48543339f87dc7ab1": "V_n(R) = \\int_{-R}^R V_{n-1}(\\sqrt{R^2 - x^2}) \\,dx,", "7b54b98cede11d006245c12515f1f4ff": "f'(x)>g'(x)", "7b54bc7d2c24e715716131e2a5487a54": "\\tfrac{n(n+1)(2n+1)}{2}", "7b557be02a7c05f61c510302ebc01ac6": " \\sqrt{4\\pi} \\left(q, \\rho, I, \\mathbf{J},\\mathbf{P}, \\mathbf{p}\\right) ", "7b55837edf32f18ac1d64f104d3e6db3": "~V_4~", "7b55c31b8ad112624438b0a005ab094f": "s_2 \\setminus s_1", "7b55c3253d2c9d63d6cc2ac49d3611e6": "\\boldsymbol{Q}(t)", "7b55d8221157b63abad5ea90da580074": " | \\Delta x_k | < \\epsilon", "7b55e666d5e19769d9d00db4aa0c10ae": "B = \\{-1, 0, 1\\} \\,", "7b55ede9154f488a46414e160482d33c": "0 \\to \\Omega^0_{\\mathrm c}(X) \\to \\Omega^1_{\\mathrm c}(X) \\to \\Omega^2_{\\mathrm c}(X) \\to \\cdots", "7b55f2d6e21f08be5055261baad75fcb": " \\sgn x = \\left\\{ \\begin{matrix} \n-1 & : & x < 0 \\\\\n0 & : & x = 0 \\\\\n1 & : & x > 0 \\end{matrix} \\right. ", "7b5637294292d0bdd6a91f3a5a40f389": "{x^2 \\over a^2} + {y^2 \\over b^2} + {z^2 \\over c^2} = 1 \\,", "7b569a138a0dfb206e7795895e4da83f": "[P_\\mu,K_\\nu]=-2M_{\\mu\\nu}+2\\eta_{\\mu\\nu}D", "7b56bfc7208760bfcb19cc3637f0163f": "((a_n x+a_{n-1})x+a_{n-2})x,\\ \\dots", "7b56f4cd2652a41bd2f612507fb168ab": "J_1", "7b5706c92a949a258e3cad097464d1cd": "\\leftarrow", "7b574551a6c1e174ea09768303d99f17": "\\mathbf{AA} = \\begin{bmatrix}\nAA1 & AA3 & 0\\\\\nAA3 & AA2 & AA3\\\\\n0 & AA3 & AA1\\end{bmatrix}", "7b579c80e7e6e595705452c53272e9a4": "Y_{10}^{2}(\\theta,\\varphi)={3\\over 512}\\sqrt{385\\over 2\\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(4199\\cos^{8}\\theta-6188\\cos^{6}\\theta+2730\\cos^{4}\\theta-364\\cos^{2}\\theta+7)", "7b57db233529517935c453305278be6f": "\\epsilon={v^2\\over2}-{\\mu\\over{r}}=0", "7b57f3c88b2e3310e1e2167038dbce14": "f(u) = \\int_{-1}^{+1} u'(x)^{2} \\, \\mathrm{d} x.", "7b5827337c22b6ccc3c1f1d10863b7fd": "\\displaystyle -i\\pi\\left(\\delta(\\nu-a)-\\delta(\\nu+a)\\right)", "7b5844c42e1bf95101404ffa1f7cfd23": " S \\ (\\hbox{or } S_1) = {8 \\over { 5^{5/4} }} \\approx 117.1 \\ \\hbox{cents} ", "7b587e172f41ce50f594e4a3b7f1d9b0": "Pr\\{X_{ni}=1\\}", "7b58c203fa4a6d769e4da1d45a8b2b8b": "(0,\\tfrac{1}{2},..)", "7b58d83372f0c2660215645120f16500": " T_{\\alpha \\beta} {}^\\lambda \\, \\delta_\\lambda {}^\\gamma = T_{\\alpha \\beta} {}^\\gamma ", "7b591b26093422de8556efe6d309f85a": "\\{ \\{ x_\\alpha : \\alpha \\in A, \\alpha_0 \\leq a \\} : \\alpha_0 \\in A \\}\\,", "7b5977ffba1dc78df3a814262cb6bdb0": "\\nu_1 \\ge \\cdots \\ge \\nu_n\\, ", "7b597fb2debc701f0e704a3897bfd18c": "\\widehat{p}_i, n_i, i=m,f", "7b59a4d8c6cc28616e3ef4a193757a6d": "S_{RBB}(n)= S_{RRB}(n)", "7b59e3df9837c3a883dd1a523b842427": "(a,b,k) = (41,5,6)", "7b59fc64e2147cceeaf84a150fd2c443": "\\frac{1-x}{1+x}", "7b5a09ebbba60c6fe4c301c7e63867af": " |f(b) - f(a)| \\leq \\sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \\leq \\sum_{k=1}^{N-1} \\left\\{|f'(x_k)| + \\epsilon_k\\right\\}|x_{k+1} - x_{k}|.", "7b5a3d15081e8f76ee2e6cc269d5a7ff": "\\hat{\\beta}_0 ~=~ (y ~|~ T=0,~ S=0)", "7b5a6ffb486f90cca2924127327318b3": " P_n = 2v ", "7b5adc234fbb2ffc8c50ac63bc836b61": "{\\boldsymbol \\theta} = \\left (\\theta_1, \\theta_2, \\ldots, \\theta_d \\right )^T", "7b5bdc2cf25e35a6bfc382ff29943bcc": " \\{ \\lambda_1, \\lambda_2, \\ldots , \\lambda_n \\} \\,, ", "7b5bf75682aaea728726194eea73c68e": " \\alpha = \\pi d \\omega ", "7b5bfd8b75732d611d3a0dde9faea0bc": "\\sigma_1 = \\sigma_y\\,\\!", "7b5c25f5ae6c5f774fb88046b0e6acec": "\n\\begin{align}\n& {} \\qquad \\frac{f(x)+f(y)+f(z)}{3} + f\\left(\\frac{x+y+z}{3}\\right) \\\\[6pt]\n& \\ge \\frac{2}{3}\\left[ f\\left(\\frac{x+y}{2}\\right) + f\\left(\\frac{y+z}{2}\\right) + f\\left(\\frac{z+x}{2}\\right) \\right].\n\\end{align}\n", "7b5c465baf78eee9f50f9c1374c83ebb": "\\frac{(\\sqrt{3} + x) + (\\sqrt{3} - x)}{(\\sqrt{3} + x) - (\\sqrt{3} - x)} = \\frac{2 + 1}{2 - 1}", "7b5d9feda58ffea2b946ee30bdee224c": "Z_\\mu(0) = X(0),", "7b5da56da43e07855cccc2a27d8ecd3e": "S^+", "7b5dc2061074819edbd3308ce964a3ad": " p_i = f_i / N ", "7b5dd317a480822c11698057ff45bff1": " f (t)", "7b5de4578597841bcd75bd654cafbd0f": " \\frac{d^2 A}{dt^2} = 0 ", "7b5dfe0c6efc943e3356e62e3f4cd1ec": "\n\\operatorname{probit}(p) = \\Phi^{-1}(p) = \\sqrt{2}\\,\\operatorname{erf}^{-1}(2p-1) = -\\sqrt{2}\\,\\operatorname{erfc}^{-1}(2p).\n", "7b5e38887216156295337eda99d527a8": "x^{\\mathrm{T}}Ax + x^{\\mathrm{T}}b + c = (x - h)^{\\mathrm{T}}A(x - h) + k \\quad\\text{where}\\quad h = -\\frac{1}{2}A^{-1}b \\quad\\text{and}\\quad k = c - \\frac{1}{4}b^{\\mathrm{T}}A^{-1}b", "7b5e3cf1d7f8e1bbedd9c5f9a5b5522c": "\\Phi_{ij}=\\,\\text{Tr}\\,\\big(\\,\\digamma_i \\,\\bar{\\digamma}_j \\,\\big)", "7b5e4d4cd41969dcd4e53cba34321f70": "\\displaystyle M", "7b5e75f8dc7d1b6c68249d4a3575cd60": "v = (v_0, v_1, ...., v_{n-1})", "7b5e82c87b6f306103b4895a6b94b1b4": " (x^\\mu,y^i) ", "7b5e9cb01c8c76c935ba40e1d1042fd6": "\\lambda\\rightarrow\\infty", "7b5e9d75d3b63101fb261bec4a8f1996": "O\\left(\\exp\\left(\\left(\\begin{matrix}\\frac{64}{9}\\end{matrix} b\\right)^{1\\over3} (\\log b)^{2\\over3}\\right)\\right).", "7b5ee1d58827b08e8788c2c0314c3e31": "\n \\hat{x} |x,p\\rangle \\approx x |x,p\\rangle \\qquad \\qquad \n \\hat{p} |x,p\\rangle \\approx p |x,p\\rangle \n", "7b5f1b0ebb4187fab39a80035a2c98a0": " E_\\mathrm{mod} = \\sigma / \\epsilon \\,\\!", "7b5f7dcd1849497a9c6a8dd69d1e59c8": "\\{1: a \\vee b, 2: \\neg a \\vee c, 3: \\neg c \\vee d, 4: a\\}", "7b5fa1c9f792126936a98ff9242c9921": "\\{p_1,p_2\\}", "7b5fd388091bcdf064dbc36dd6b926ea": "\\omega = \\sum_{i=1}^n df_i \\wedge dg_i.", "7b5fe71392e72cc9b95881ae977e1ded": "\\operatorname{End}_{\\mathbf{Grp}}(G)", "7b5fee738a8cc68569124340a1ffeb01": "\n\\left| \\frac{1}{N} \\sum_{i=1}^N f(x_i)\n - \\int_{\\bar I^s} f(u)\\,du \\right|>D_{N}^{*}(x_1,\\ldots,x_N)-\\epsilon.\n", "7b601994c72aab117bf74ba4b6af3ec4": "\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\cfrac{1}{h_1 h_2 h_3}~\\sum_i \\frac{\\partial }{\\partial q^i}\\left(\\cfrac{h_1 h_2 h_3}{h_i^2}~v_i\\right)\n", "7b6050c21dcc145e8bf94726a085c7b9": " \\mathbf{A}\\lambda = \\begin{pmatrix}\nA_{11} & A_{12} & \\cdots & A_{1m} \\\\\nA_{21} & A_{22} & \\cdots & A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n1} & A_{n2} & \\cdots & A_{nm} \\\\\n\\end{pmatrix}\\lambda = \\begin{pmatrix}\nA_{11} \\lambda & A_{12} \\lambda & \\cdots & A_{1m} \\lambda \\\\\n A_{21} \\lambda & A_{22} \\lambda & \\cdots & A_{2m} \\lambda \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n1} \\lambda & A_{n2} \\lambda & \\cdots & A_{nm} \\lambda \\\\\n\\end{pmatrix}\\,.", "7b606497785fde9d9cc20349da9ddfb3": "\\sum_i p_i\\,\\mathrm{d}q^i", "7b60809c076f0d5ce8f921d51427d22f": "v_C(t) = V_0 + {1 \\over C}\\int_{0}^{t} i_C(\\tau) \\, d\\tau \\iff i_L(t) = I_0 + {1 \\over L}\\int_{0}^{t} v_L(\\tau) \\, d\\tau ", "7b609f2b5182031fffe62ebe29ba60cb": "\\omega\\in\\Omega", "7b60a39fc2a49bbac1b3426abb5ada4b": "BO", "7b60b25c5e48916a3428bf4e321d51d3": "U = U_{pot} + U_{kin} \\!", "7b60b4e048f5e840a26baf7c88a2cb6f": " \\frac{\\Omega_Q(B) \\psi}{ \\|\\Omega_Q(B) \\psi \\|} ", "7b60e1c38f18fb7839b523b5c8b0bce4": "\\left(0,\\frac{q_k}{p_k}\\right)", "7b60e8c5a324dfa9be680bcb5c8fcdb5": "\\rightarrow\\,", "7b6104bd6a7089d4357ac67be734cab0": "\n\\left(\\frac{ab}{p}\\right) = \\left(\\frac{a}{p}\\right)\\left(\\frac{b}{p}\\right).\n", "7b61538f3b75f6700d605cbe7a1f190f": "T\\mu", "7b61ab638aae7662472642b324a09a5b": "\\psi * \\alpha \\not\\models \\neg \\mu", "7b61e96f452ae89c13804e74c422eaea": "(1,-1)^{n+1}", "7b620c67a28cf733f7078f100a3e7e1c": " A_0 = \\left\\langle \\mathbf{e}_0, \\mathbf{A} \\right\\rangle = \\left\\langle \\mathbf{e}_z, \\mathbf{A} \\right\\rangle = A_z ", "7b627266038b26b89e8daddec0efa641": "\\chi_{\\textrm{e}}", "7b6275bf475a4042e550e6a8d432cc28": "\\scriptstyle \\boldsymbol\\omega", "7b628e2b9cb80f59f4ec9137412b7b84": "\\,w_{ij}=x_ix_j", "7b633205117edf9ae809a3d2834f78fe": "-i\\frac{d}{c}", "7b6347178a0578b1c8ca05a57ddc363b": "k=1,..,\\operatorname{deg}(p)", "7b636c0209cb869f58cdc0023a031694": "\\scriptstyle s(t)", "7b637986949791c82bc6e8b3a5908381": " H(x)= \\{y\\in\\mathbb{R}^n: y\\cdot x = h_A(x) \\}", "7b638bbf759915205b82e685842cf441": "\\frac{dX}{dt} = - \\kappa X", "7b63977a204e85f4f2ae7947423c63e8": "\\Delta f=\\frac{\\Delta v}{c}f_0", "7b63a513e56394cfbc1a7c5d8f8b858b": "\\mathbb{N}\\begin{Bmatrix} 1 & \\longleftrightarrow & \\{4, 5\\}\\\\ 2 & \\longleftrightarrow & \\{1, 2, 3\\} \\\\ 3 & \\longleftrightarrow & \\{4, 5, 6\\} \\\\ 4 & \\longleftrightarrow & \\{1, 3, 5\\} \\\\ \\vdots & \\vdots & \\vdots \\end{Bmatrix}P(\\mathbb{N}).", "7b63b4adcc3d579f2eb988e323ec0af8": " L_{x} : f \\mapsto f(x) ", "7b63c83666feed8ecfb10a1b22ed4bdb": "\\alpha_{n-2} \\partial_{n-1}(a) = \\partial_{n-1}' \\alpha_{n-1}(a) = \\partial_{n-1}' \\partial_n'(b) = 0,", "7b63d15ab407e130a6b47bde8a335734": "\\gamma_{ij}=\\delta_{ij} \\,\\!", "7b63f4a57476a90535ceedf22d0832fb": "f\\,:=g^{-1}", "7b64057fc8bb6b0584a100ae86a44309": "\\displaystyle{(P\\psi)_{\\overline{z}} = \\psi, \\,\\, \\, (P\\psi)_{z} = T\\psi}", "7b645327526717b5c31a54a789bf792e": "{m \\choose k}", "7b64c3c71c27070fccee06966611d9f8": "\\bar{f} = f(c)", "7b64cc2e4bf9adb167e1e8aecfaebcff": "V \\otimes V^* \\to K", "7b6549881317aa86d09927d48787283e": " S_\\mathrm{pooled}^2 = \\frac{\\sum_{k=1}^{n_1}(X_{1,k}-\\bar X_1)^2 + \\sum_{k=1}^{n_2}(X_{2,k}-\\bar X_2)^2}{n_1+n_2-2} = \\frac{(n_1-1)S_1^2 + (n_2-1)S_2^2}{n_1+n_2-2} ", "7b6553ad31aa385f2ca6c52260132f38": "BC_n", "7b655eb0c435c8dd1a7fdc40af8c2ecc": "D\\left[ \\Gamma \\right]=\\hat{P}e^{\\oint_{\\Gamma }{\\tau ^{\\mu }dR_{\\mu }}}", "7b656a08d2243e97ae46f709975c5950": "(\\mathbf{X}^{-1})^{\\rm T}", "7b65917c8de1377f640464f85d6fbde1": " t = \\sigma n ", "7b660014ec5eb9695b4a867531a0cb25": "\n G := -\\cfrac{\\partial (U-V)}{\\partial A}\n ", "7b6634c303d6c1e8540c6764201c9dcd": "R = \\left.{1 \\over s+\\alpha}\\right|_{s=-\\beta} = {1 \\over \\alpha - \\beta}.", "7b6664dba6649e267034057b9030b0f5": "B\\supsetneq A.", "7b66a47308671123d143eec7be69298a": "\\left\\{\\,k 2^n + 1 : n \\in\\mathbb{N}\\,\\right\\}", "7b66ffb7eb82a7c53eabe84662ae0bc6": "m>n", "7b6738e85ce2034d1f3373e15838c4ce": "a_n=a_0+\\frac{n}{2} ,", "7b6757e2920260b40b8cf49a5819128e": "\\frac{x^2}{2a}\\pm\\frac{ay^2}{2b^2}=x", "7b675886bedb25286853c89cf22c43c3": " {[S] \\over v} = {1 \\over V_\\max }[S] + {K_m \\over V_\\max }", "7b677f53fbeada7485c793b6ad40ffef": "y_{n+1}=y_n+{1\\over 2}h\\left(f(t_n,y_n)+f(t_{n+1},y_{n+1})\\right). \\qquad (3) ", "7b6787ac146705f01d921f262e7f8c6d": "4^4", "7b678cfb13579e210d4c6f2f21af5fb1": "D(\\hat{e}_2,\\hat{e}_1) = -D(\\hat{e}_1,\\hat{e}_2) = -D(I)", "7b679b0038a3b29cb3dcb9fa78524a7a": "\\beta\\in\\mathbb{R}_+", "7b67a19f0c27370861a6a2e6f98ebb7f": "2^1 + 2^0", "7b67a2c7e91c6a707aa2ff8a26d957e2": " \\prod_{p} \\Big(\\frac{p^{4}+1}{p^{4}-1}\\Big) = \\frac{7}{6} ", "7b67dbaf6d32a59a1ff61ecb3596f764": "\\frac{c^2k^2}{\\omega^2}=1-\\frac{\\omega_p^2/\\omega^2}{1-(\\omega_c/\\omega)}", "7b68198f0a1e86acd48bfda969521df2": "\n\\frac{d}{d t}\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}}=\\frac{\\partial \\mathcal{L}}{\\partial \\theta}\\Rightarrow\n", "7b6872547bd1634d11a2275d1f653797": " \\lambda q.\\operatorname{de-let}[f\\ (q\\ q)] ", "7b698c31e057672b50e8e84b875fa6e8": " h {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} k", "7b699616a57f7e6c9a2f304ff36af45f": " I_R = \\frac{ p_{ xx } - p_x } { 1 - p_x } ", "7b6a050f4358a151d4e80841fc9d1003": "p_i (k)", "7b6a4ee2cba52caee6245bc66015c604": "P_1,...,P_8 ", "7b6a767fc51fffcd9a9aa57dd53045f2": "\\textstyle \\chi _T", "7b6aa7dec241a5d2b51268476e4528af": "\nW_0 = \\sum_{i:h_t(x_i)=0} D_t(i) \n", "7b6af2c164fe5b2492551adae7b6cf87": "\\frac{z}{1-\\exp(-z)} = 1 + \\frac{1}{2}z+ \\sum_{i=1}^\\infty (-1)^{i+1}\\frac{B_{2i}}{(2i)!}z^{2i} ", "7b6b00f710c18e84f591eb832745ba42": "S_{ARC} = \\frac{AR}{AD} S_{ADC} = \\frac{AR}{AD} \\frac{DC}{BC} S_{ABC} = \\frac{x}{zx+x+1}", "7b6b57d4a7d28d1c7f90a6ca1f766b61": "1,2,4, \\ldots,2^n,\\ldots", "7b6b7ab5a5e64f5f65cc32f33d1ad37a": " \\mbox{If } \\frac {dB}{dt} \\begin{cases} >0, & \\frac {dB}{dt}=\\mbox{ maximum power generated} \\\\ <0, & \\frac {dB}{dt}=\\mbox{ minimum power required} \\end{cases} \\qquad \\mbox{(8)} ", "7b6bbdf655a9d27652639f6fb614a4b4": " E_{in}-E_{out}= \\Delta E_{system}", "7b6bd7500827a266d4a814bd0af02fa7": "(A.5)\\quad l^a\\nabla_a l_b=\\kappa_{(\\ell)}l_b\\;,", "7b6be866ba71fdefe6be96d55e6d7c1f": "\\mathbf{f}_0, \\mathbf{s}_0", "7b6bea62f75f9b6c0c8081382c402d89": "\\begin{matrix} {2 \\choose 1}{2 \\choose 2}{3 \\choose 2}{3 \\choose 2} \\end{matrix}", "7b6bed494e4656dc550ad324e04ee66e": "M_0 M_1 M_2 M_4", "7b6bf4bd0d1d823a844c08967d59f6bd": "\\scriptstyle \\zeta(\\bar3, \\bar1) ~= \\,\\sum_{m>n>0} \\,(-1)^{m+n} m^{-3} n^{-1}", "7b6bf9f4e83df50bdaa32e46eed79ae9": "E = \\frac{v^2}{2g}+y", "7b6c0334386d92ad0aeae9958be0c146": "F_{in}", "7b6c3b57ec4f4029dc81acb4f821a784": "p(v_i,c_j)", "7b6cb35c1e235ec9e899292f83894620": "BF = 3AF", "7b6cb96d1f1613acbc480ac942fa539e": "H(X_2) \\le H(X_1, X_2)", "7b6ced9fd42e64c38b0dd362ded62fe6": "e^{X} e^{Y} = e^{\\exp (s) ~Y} e^{X}~,", "7b6d1dcc41e92c04a62cda4a1c98837c": "|v|", "7b6d1dd7a502ee97da0a2a18321ed077": "\\begin{align}\n\\mathbf{A}\\,\\Delta \\vec{p} &=\\Delta\\vec{F}\\\\\n\\mathbf{A}^T\\mathbf{A}\\,\\Delta \\vec{p} &= \\mathbf{A}^T\\Delta\\vec{F}\n\\end{align}\\!", "7b6d51d2258972a366576f2b6193dd03": "\\sigma = \\frac{2 \\pi^5 R^4}{15 h^3 c^2 N_{\\rm A}^4} = \\frac{32 \\pi^5 h R^4 R_{\\infty}^4}{15 A_{\\rm r}({\\rm e})^4 M_{\\rm u}^4 c^6 \\alpha^8}", "7b6daa2107a6c98394befeceb3794381": "\n\\begin{align}\nN^{(\\mu)}(\\mathbf{k})\\; |\\, \\mathbf{k}',\\mu'\\,\\rangle &= \n{a^\\dagger}^{(\\mu)}(\\mathbf{k})\\, a^{(\\mu)}(\\mathbf{k})\\; {a^\\dagger}^{(\\mu')}(\\mathbf{k'})\\, |\\,0\\,\\rangle\n= {a^\\dagger}^{(\\mu)}(\\mathbf{k})\\,\\left(\\delta_{\\mathbf{k},\\mathbf{k'}}\\delta_{\\mu,\\mu'} + {a^\\dagger}^{(\\mu')}(\\mathbf{k'})\\,a^{(\\mu)}(\\mathbf{k})\\right) \\, |\\,0\\,\\rangle \\\\\n&=\\delta_{\\mathbf{k},\\mathbf{k'}}\\delta_{\\mu,\\mu'} \\,|\\, \\mathbf{k},\\mu\\rangle,\n\\end{align}\n", "7b6e0e0841be28a86a17e9f8ff345d5a": "e^{iz \\sin \\theta} = \\sum_{n=-\\infty}^\\infty J_n(z) e^{in\\theta},", "7b6e5a8ac91b6c71553f916854a9a511": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\sqrt{\\frac{3}{2}},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "7b6e91e615144b18caa657ce210350c3": "n!/e", "7b6ea5f48d5769f0a912791d4c02894a": "\\textstyle (n-2)", "7b6ec5fb653015a17f633adb20f49cf1": "T^n,", "7b6ec9491b30a33b4dfca6a341d04bb5": "\nD :=\\frac{Eh^3}{12(1-\\nu^2)}\n", "7b6ef851958b155661f1de8258fc820c": "\\phi_\\alpha(v)= \\Phi_{\\alpha,\\beta}(u)(\\phi_\\beta(v))", "7b6f0473b6de5b3975bca86c7aca2e3f": "\\|\\mathbf{p}_A\\|=n_A", "7b6f12ad8def09d6a0b7fdc4273d66bc": "\\beta < \\alpha ", "7b6f4c393cabc8ddb800aa8fa666e005": "W^\\ast(s) = \\frac{\\mu(1-\\rho)}{\\lambda^+}\\frac{s+\\lambda+\\mu(1-\\rho)-\\sqrt{[s+\\lambda+\\mu(1-\\rho)]^2-4\\lambda^+\\lambda^-}}{\\lambda^--\\lambda^+-\\mu(1-\\rho)-s+\\sqrt{[s+\\lambda+\\mu(1-\\rho)]^2-4\\lambda^+\\lambda^-}}", "7b6f85b027c9dfeca1d68a8290861906": "|N\\rangle ", "7b6fbc0ca8bfb6a1f7b8622e81ee5ac3": "(1-\\alpha)", "7b6fe15c448f4c0f872da939ae92630f": "\\Delta (x)_{k} = k\\ (x)_{k-1},", "7b7026ed59c46aae3b4b34496dd64b43": "0=-PA+mg\\,", "7b704921fb14e9b5a11eb1e140256d7d": "p^4=4m^2(H^0+e^2/r)^2", "7b70de460a73aa61caa083b7abd43d8c": "A[\\phi U \\psi] \\equiv \\psi \\lor (\\phi \\land AX A [\\phi U \\psi])", "7b7166cbb3d0252097181e51fa3b0b45": "SL(3,\\mathbb{R})", "7b717d0b48a2cdbe1913dadffd167c8c": "a,b,c:", "7b71d6cbfa42f59f920bc4f4108f4bfd": "\\psi(x) = \\frac{x}{x^2 + \\nu}", "7b71e37a76ccb685de831412b0ae07f6": "b\\mapsto B", "7b71f1e840ed3a9b3ec91384e844e00e": "{d_0}", "7b723934f9b3d57fd347519450e24a08": "R = r^{\\infty}e^{\\frac{B}{T}} = R_{0}e^{-\\frac{B}{T_{0}}}e^{\\frac{B}{T}}", "7b723fcfafe555a5c41b9549cdd91da6": "\\Delta G = \\Delta H - T \\Delta S", "7b726d44a05029102a34fcbbd3856dac": "8 * 2 = 16; 16 - 13 = 3", "7b72b46588d29e5a32c1ad963fe35b2b": "\\frac{1}{5 + 1} = \\frac{1}{6} = 0.1666...", "7b72dd41929a5663a34e88ffe858f96b": "\n d\\mathbf{a}^{T} \\cdot \\mathbf{F}\\cdot d\\mathbf{L} = dv = J~dV = J~d\\mathbf{A}^{T}\\cdot d\\mathbf{L}\n\\,\\!", "7b72f6d445b8226da933bdfb0826728f": "f\\star g = f^*(-t)*g.", "7b72f7a6287d12d4e6c30e5f6a8e7438": "C_{out} = (A \\cdot B) + (C_{in} \\cdot (A + B))", "7b731469e21d2164930fc8d4ca0cce12": "\\part^2+m^2", "7b7340aeaef148046072882c382d0298": "f(x)\\approx f(c)\\approx f(y),\\,", "7b73809d40e074d27acb188a5f39886f": "\nV_{\\rm f1} = V_{\\rm f2} = V_{\\rm f}\n", "7b73b22d27a0cf89cb85663013bde0cf": "p_* \\circ \\tau = \\chi(F) \\cdot 1.", "7b73d4e3649c654c3de416a3036c484a": "\\sigma\\cdot\\sigma_f=\\frac{1}{2\\pi}", "7b740eb544b1e364745545edf718ac4f": "\\arctan x = \\sum^{\\infin}_{n=0} \\frac{(-1)^n}{2n+1} x^{2n+1} = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\frac{x^7}{7} + ...", "7b745f06b22847469f8d9ddc1f2ac729": " \\beta(M) = (-1)^{r(M)} \\sum_{X\\subseteq E} (-1)^{|X|}r(X). ", "7b74acf3db10bfdae226e462e358b253": "P(v,u\\in K)= p^{2n-k(v,u)}.", "7b74adcae623867d43b34fae1d7670b1": "\\textstyle g(x_i) \\leq \\Gamma", "7b74b5bf6492cbe52d7b2b7589e23d09": "\\vec{c}_4", "7b74b7fd96c817d332107b932cbd5f70": " \\delta x_{PE} = \\delta x_{WP} = \\Delta x ", "7b74fa1cbe9523b7dccdf45f255c7f15": " a \\star b = a \\circ b + b \\circ a ", "7b7513ffe47452d642c97877862a345f": "rol(x,t)", "7b75377f77843c7489099cae8b51e177": "\\omega_m = \\sqrt{\\left(\\frac{1}{2 R_0 C}\\right)^2 + \\frac{1}{LC}}", "7b7584c35637747f2790d6aeade7f61f": "k_\\mathrm{spec} = \\left [ \\cos(\\angle(L, T)) \\cos(\\angle(V, T)) -\\sin(\\angle(L, T)) \\sin(\\angle(V, T)) \\right ] ^n=\\cos^n(\\angle(L, T)+\\angle(V, T)),", "7b7592f9d2c5cee8189d117fa54d52b4": "\\mathcal{L}(\\Phi,\\zeta)", "7b759968274f2f43cfaab3ce5672da74": "\\scriptstyle S", "7b759f4c5025485f2c44db723d8b4910": "[\\hat{x},\\hat{p}]=i \\hbar.", "7b75aabf8ffa571d707e3ac4ecf24f36": "2r^2+x^2\\le R^2 \\le 2r^2+x^2+2rx.", "7b75cb91160b1d884bcb58d2e43668a0": "\\langle i,x \\rangle", "7b764334fd4f71b3478258d21c985877": "T = \\boldsymbol{\\sigma} n", "7b76ac1fa912b6b0891bb2f85ca5c10a": "\\begin{cases}\n\\dot{\\mathbf{x}}_2 = f_2(\\mathbf{x}_2) + g_2(\\mathbf{x}_2) z_3 &\\qquad \\text{ ( by Lyapunov function } V_2, \\text{ subsystem stabilized by } u_2(\\textbf{x}_2) \\text{ )}\\\\\n\\dot{z}_3 = u_3\n\\end{cases}", "7b774effe4a349c6dd82ad4f4f21d34c": "u", "7b781a7c9af0bfccdbedf3178cab71c3": "\\displaystyle{I-P(T)^*P(T) =\n4 (T^* -iI)^{-1}[\\mathrm{Im}\\,T] (T+iI)^{-1}}", "7b783c20ba20036b9ca4d82fb00cdd91": "a_x = \\text{(women with at least } x+1 \\text { children ever borne) }/\\text{ (women with at least } x \\text{ children ever born)}", "7b7887b547cb57027fbbaab23db16494": "p_k^\\prime(t)=\\lambda_{k-1} p_{k-1}(t)+\\mu_{k+1} p_{k+1}(t)-(\\lambda_k +\\mu_k) p_k(t) \\, ", "7b78dcd44d0f189234bf201da13ae42c": "\n-\\left(\\frac{1}{\\pi^2} + \\frac{1}{4\\pi^2} + \\frac{1}{9\\pi^2} + \\cdots \\right) =\n-\\frac{1}{\\pi^2}\\sum_{n=1}^{\\infty}\\frac{1}{n^2}.\n", "7b7917018bbe25723d4260b77a0040d1": " \\sum_{n=1}^\\infty a_n =1-1+1-1+1-\\cdots ", "7b792da6be2ab68bda59b3430ff1a4b3": "k\\to\\infty", "7b79437f96becdc372f0388cd30e97b3": "S(\\omega) \\propto X(\\omega)X^*(\\omega)", "7b79a511f97a2bf81f68d2c5ad08d3df": "\\Psi_\\pi = - MiRT", "7b79d61d83d76436dabbd99002c49220": "a_B", "7b7a0c325951f5c80e6ce9d70238bb8b": " \\vec{J}\\left(\\vec{r}\\right) ", "7b7a8af8f0ddb586e153340c5c93e72c": "w_i>0", "7b7acba7dc124f8d9a9f005ee4177395": "\\lambda = (\\lambda_1,\\dots,\\lambda_k)", "7b7adff3a40fb5e20edeb212ed746d2c": "\\gamma'=\\gamma^3/108", "7b7ae1ddc32ada9742f34480ecb09c37": "y^2 - x^3 = 0", "7b7b058a0864ef97ef6c08fe44ed07e3": "\\bar B", "7b7b5d089043e1e806c47cf207225ee8": "\n I_\\theta = \\int \\!z_\\theta z_\\theta' \\,dP_\\theta\n ", "7b7bc9d4c0f8f460d05a3997f5baf7f4": "R\\left( t_{n} \\right)", "7b7beef627e34eb0b976bf3130c11b79": "\\begin{align}\n i &= 1 \\cdot e^{\\frac{1}{2} i \\pi} \\\\\n i &= 0 + 1i\n\\end{align}", "7b7c325e6bb6fdf396745bb5bd6db2cc": "T'_{ij}=Q_{ip} Q_{jr} T_{pr}", "7b7cb36792c245022ad55484bfdb0ef2": "\\phi_{13} : H \\otimes H \\to H \\otimes H \\otimes H", "7b7cb7dbb320dc15dfb32b35c6892cf8": "n_i(x,t)=\\int_c f_i(x,c,t)\\, dc", "7b7cdd055abf812dc1be6ecbbda6edf0": "\\frac{1}{1+e^{-\\frac{x-\\mu}{s}}}\\!", "7b7cea5dcd8ef8d3f341dd10463cc9a2": "\\begin{matrix} {3 \\choose 1}^2{10 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "7b7cf92b80ac90690091418bcf5bb8e3": "P(AB)=P(A|B)P(B)=P(B|A)P(A)", "7b7d1699ebef97f89c027a4649fdc52d": "\nj_{elec}^{sat} \n= en_e \\sqrt{k_B(\\gamma_eT_e+T_i)/m_e}\n= j_{ion}^{sat}\\sqrt{m_i/m_e} \n= j_{ion}^{sat} \\left( 42.8 * \\sqrt{\\mu_i} \\right)\n", "7b7d8e24020aae877187d55e0b4f4dba": "\\lambda_\\text{res}(\\theta)=\\frac{\\lambda_u}{2\\gamma^2}\\left ( 1+\\frac{K^2}{2}+\\gamma^2\\theta^2 \\right )", "7b7d8f32cacdb5b2b05dca4306b803f7": "{\\mathcal L}^2_{xx}: z_i(x,y)={\\mathcal E}_i(x,y)F_i\\big(\\varphi_i(x,y)\\big), i=1,2; ", "7b7dded57e9e343e8243f42e8c0dd216": " k_{1}, k_{2}, ... , k_{n} ", "7b7e1ddb3215ca1a4913080452a834d4": "\\begin{alignat}{2}\n\\text{right} = & \\quad \\text{last(L + 1)} -2j\\\\\n = & \\quad (2^{L + 2} -2) -2j\\\\\n = & \\quad 2(2^{L + 1} -2 -j) + 2\\\\\n = & \\quad 2i + 2\n\\end{alignat}\n", "7b7e8abaa75735ac78eb4df4b5aee8c9": "\\pi(x) = \\frac{e^{\\beta_0 + \\beta_1 x}}{e^{\\beta_0 + \\beta_1 x} + 1} = \\frac {1}{1+e^{-(\\beta_0 + \\beta_1 x)}}.", "7b7e8d0698f4e984fa97f70fc4a3741b": "p(V,T)\\ ", "7b7ed2e120a93fa60467cd112de88951": "J_{\\lambda_k,m_k}", "7b7eddcb44005862c6d159cfc78ac5e5": "\n\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\n\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}\n=\n\\begin{pmatrix}0&0\\\\0&0\\end{pmatrix}\n=\n\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}\n\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}", "7b7f09c72de1f8f8ab4a9c1d0ce47a88": "f_j ^T", "7b7f0c126db8925e8a8991c45af01f01": "\\tfrac{9}{2}\\zeta(5)-2\\zeta(2)\\zeta(3)", "7b7f9dbfea05c83784f8b85149852f08": "\\alpha", "7b80511f188e7c08c5ca58879a1da616": " F = c_1 e^{r(T - t_1)} + \\cdots + c_n e^{r(T - t_n)} ", "7b80b396afeba4fabef7774fd2db083a": "\\begin{Bmatrix} x \\end{Bmatrix}= \\begin{bmatrix} \\Psi \\end{bmatrix} \\begin{Bmatrix} q \\end{Bmatrix}. ", "7b80ebccd4420d9579e7d488396b7f5c": "A\\,", "7b81343a114cb2033d8e04ddfb320bcd": "E_{in}^{L} = \\eta_M Q + \\eta_M Q \\left(\\frac{1}{\\eta_{L}}- 1 \\right )=\\frac{\\eta_M}{\\eta_L}Q = E_{out}^{L} ", "7b813c04ab5b185e149ec8790ed47e45": "(p,ax,A\\gamma) \\vdash_{M} (q,x,\\alpha\\gamma)", "7b818bee8d9e74844c67f82412544469": " (X_{t_1},\\ldots, X_{t_n}) \\stackrel{\\mathcal{D}}{=} (Y_{t_1},\\ldots, Y_{t_n})", "7b8198f3e0d90a909cef946eaf782357": "\\tau=1 ", "7b8199811fcee9d0e3d23c02afbe87a4": "d\\tau =0\\,\\!", "7b821d76c00b3007558c446bd38c5c9c": "3 \\times 4 = 4 + 4 + 4 = 12", "7b8228d4240521bde73c97b4b9267f9b": "(-a;q)_\\infty = \\prod_{k=0}^\\infty (1+aq^k)\n = \\sum_{k=0}^\\infty \\left(q^{k\\choose 2} \\prod_{j=1}^k \\frac{1}{1-q^j}\\right) a^k\n = \\sum_{k=0}^\\infty \\frac{q^{k\\choose 2}}{(q;q)_k} a^k", "7b82464af9acfe507b092b44a4460b90": "f(a)+\\frac {f'(a)}{1!} (x-a)+ \\frac{f''(a)}{2!} (x-a)^2+\\frac{f^{(3)}(a)}{3!}(x-a)^3+ \\cdots. ", "7b827a1791c9bb4187e7979bd9f07a31": "f_n = n\\chi_{\\left[0,\\frac1n\\right]}", "7b82c6830e6600c14f5ea9653f0bfd01": "\\{x_1 \\mapsto t_1, \\ldots, x_k \\mapsto t_k \\}", "7b831ffd92afec78c3380858e058571d": "k \\psi^{(0)}(kz) = k\\log(k)) + \\sum_{n=0}^{k-1}\n\\psi^{(0)}\\left(z+\\frac{n}{k}\\right)", "7b83333eff61ee5508779b0034f55993": "A(x)=\\Omega(x^{2/7})", "7b8382d1ac532bdedfbc69b7356878a7": "H(T,q) = -\\sum_{i=1}^N \\frac{1}{N} \\log_2 q(x_i)", "7b83d4432cfa171203487be2e8c78e31": "L_1=0", "7b847443703e3553cfd42fe33a7840c1": "y = a \\cosh \\frac{x}{a}.\\,", "7b84bd818391a6950b10de49068d3c64": " K(1) = a_1 ; \\, ", "7b84c4b3cbd736503d7099adaa551b1d": "E^i_a", "7b853ed8fac5762c4ca1c66c31e941dc": "E[Y] = \\mu = \\frac{1}{\\theta}", "7b8543a88e3419beb258c7c122049e45": "\\int_{-\\infty}^\\infty a e^{- { (x-b)^2 \\over 2 c^2 } }\\,dx=ac\\cdot\\sqrt{2\\pi}.", "7b85c37307f27300b6ea333f5305d6f7": " X(\\omega) = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) \\, e^{-j \\omega t} \\, dt \\, d\\tau ", "7b860458f541207e90ce89e9838d8f10": "\\Omega_{Z,[t_l, t_u]}", "7b8654ce1fd9436f67935ca0bd9191bd": "(y_1, \\ldots, y_n) = h(x_1, \\ldots, x_n)", "7b867a3304f1ed1230ce4569eed848fc": "B_N", "7b86d2b279f7ee833092a5f2838ea839": "\\nabla \\wedge F = I \\, \\operatorname{curl} F.", "7b8742b87d34e442837c71aa7ca3ead1": "\\mathbf{e}_2 \\triangleq (e_2, e_3, \\ldots, e_n) \\in \\mathbb{R}^{(n-1)}", "7b87674d928dc558927c73d9dc150822": "d\\left(f_n(z),f(z)\\right)\\, ", "7b8787c112504173958606259b8d6363": "\\textstyle\\log(1-tS)=-\\sum_{i>0}\\frac1i(tS)^i", "7b87adbca82d29e2937f24ec115b463c": "\\mu'_1=\\kappa_1\\,", "7b87f9292c75542c511f9387581bbabb": " M = A \\rtimes \\Gamma", "7b881041b1cd3579831051bf0acb6e7b": "p\\dot r = nab\\,\\varepsilon\\sin \\theta. ", "7b882243baa2a28a2ce083379dc56534": "x_{12}=p_1q_2-D", "7b885fb901dc041159edcfdb9270c0fb": "dx\\,", "7b88ad975cfea6941c71a6e3ce0521c1": "k_1 ", "7b88ee1ddc4968f40551eb160af2634a": " x_{n+1}=f(x_n), \\quad n=0,1,2,\\ldots.", "7b8919be726fbe5f0e029208f0cd72c6": "\\left(\\sum_n E_n \\right)^2 = \\left(\\sum_n E_n \\right)\\left(\\sum_k E_k \\right) = \\sum_{n,k} E_n E_k = 2\\sum_{n 1", "7b9f32a8c5f0d13e90ac1b122d01fa89": "|\\cdot |", "7b9f50e8abfd5a60fa5033e3a448d701": "GE(\\alpha) =\n\t\t\\frac{1}{N}\n\t\t\\sum_{i=1}^N\n\t\t\\left[\n\t\t\t\\frac{y_i}{\\overline{y}}\n\t\t\t\\ln\n\t\t\t\\left(\n\t\t\t\t\\frac{y_i}{\\overline{y}}\n\t\t\t\\right)\n\t\t\\right],\n\t\t\t\\quad\\quad\n\t\t\t\\text{ for } \\alpha = 1,\n", "7b9fd7e85c2a15348c5063f1738da44a": " \ny^2 = x^3 + Ax + B. \\, \n", "7ba07463235f33a6c719d90818718be0": "\\mathbb C^N", "7ba0ca87f02bb99bad5c0426c4a566b7": " \n\\int_\\Omega \\langle\\boldsymbol{\\phi}, D\\chi_E(x)\\rangle = \n - \\int_A\\chi_E(x) \\, \\operatorname{div}\\boldsymbol{\\phi}(x)\\, \\mathrm{d}x = 0\n", "7ba1b1596853d0d4de08ff9381ecddde": "\\mathbf{R}_k", "7ba1c13d6e71ab750396408aa32a1ec6": " ( \\cdots ( A_N \\otimes A_{N-1} ) \\otimes \\cdots ) \\otimes A_2 ) \\otimes A_1) ", "7ba2cac0caaf051bd5d6e65067c2d004": "T(V^*)", "7ba30c0d0f7adcfe0314d6b0b72eb114": "\\begin{pmatrix}u+hv & w+hx\\\\-w+hx & u-hv\\end{pmatrix}", "7ba314ab3e6e65eecdc8217f4b90bbf1": "I_p=\\sqrt{|L|^2+|V|^2}", "7ba3cbeab870e1cfdd3bc2d2a7164efb": "hev=(\\lambda+2)ev", "7ba3d26982896a4a45fd0e7924536699": "\\mathbf{A}^* = (\\overline{\\mathbf{A}})^\\mathrm{T} = \\overline{\\mathbf{A}^\\mathrm{T}}", "7ba4845526f539669a4e92e58c8b1528": "\nN^2 = \\begin{bmatrix} \n 0 & 0 & 2 & 7\\\\\n 0 & 0 & 0 & 3\\\\\n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 \n \\end{bmatrix} \n\n;\\ \nN^3 = \\begin{bmatrix} \n 0 & 0 & 0 & 6\\\\\n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 \n \\end{bmatrix}\n\n;\\ \nN^4 = \\begin{bmatrix} \n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 \n \\end{bmatrix}.\n", "7ba4877f16f47b8816a8d3116fc02ff9": "\\mathbb{E}[\\tau] = \\mathbb{E}[X^{2}]", "7ba4a893b44a0f6c69f61418749fb435": "M=a=J/M", "7ba53a2e5389fd0b3aa44ce0406ddba6": "\\text{MMD}(P,Q) = \\sup_{||f ||_\\mathcal{H} \\le 1} \\left( \\mathbb{E}_X[f(X)] - \\mathbb{E}_Y[f(Y)] \\right) ", "7ba55e7c64a9405a0b39a1107e90ca94": "O(n)", "7ba5b4c97d0f130050d574a79b5ba65d": "J_{\\hat{n}}", "7ba5ea63fa135d85d0b368dc3280f31f": " X_i(s) = x_0 +\\sum_t {s^{|t|}\\over |t|!} \\alpha(t) t! \\sum_{j=1}^m a_{ij} \\varphi_j(t)\\delta_t(0),\\,\\,\\,\\,x(s) = x_0 +\n\\sum_t {s^{|t|}\\over |t|!} \\alpha(t) t! \\varphi(t)\\delta_t(0), ", "7ba5f56caaf1bc28aef8df5cd4bc6dd2": "\\Theta_j", "7ba658e93d1a87dec982068581809e4c": "U = -\\mathbf{M}\\cdot\\mathbf{B} = -M_x B_x - M_y B_y - M_z B_z.", "7ba6744abee22968d0d6dc5dbcc7fb85": "(\\#\\phi)", "7ba699b079a7c5f67a900abfae46553a": " \\sin(x) = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\frac{x^7}{7!} + \\cdots. ", "7ba6a2448da4345e3e060b3ec99af5c1": "e^2/kT=1.44\\times10^{-7}\\,T^{-1}\\,\\mbox{cm}", "7ba6c6d98ddca7e6ef12eb3a8f930257": "\\boldsymbol{r}\\in \\Omega", "7ba6ee09a037f0a9d361965b63a80264": "R_{ik\\ell m}=\\frac{1}{2}\\left(\n\\frac{\\partial^2g_{im}}{\\partial x^k \\partial x^\\ell} \n+ \\frac{\\partial^2g_{k\\ell}}{\\partial x^i \\partial x^m}\n- \\frac{\\partial^2g_{i\\ell}}{\\partial x^k \\partial x^m}\n- \\frac{\\partial^2g_{km}}{\\partial x^i \\partial x^\\ell} \\right)\n+g_{np} \\left(\n\\Gamma^n{}_{k\\ell} \\Gamma^p{}_{im} - \n\\Gamma^n{}_{km} \\Gamma^p{}_{i\\ell} \\right).\n\\ ", "7ba6f510eba5a77c1dea221df49d4d6b": " = \\begin{bmatrix}\\sum {c_i\\,\\lambda_i^{a+n-1}} \\\\ \\sum {c_i\\,\\lambda_i^{b+n-1}} \\\\ \\vdots\\end{bmatrix} = \\begin{bmatrix}\\lambda_1^{a+n-1} & \\lambda_2^{a+n-1} & \\cdots & \\lambda_n^{a+n-1} \\\\ \\lambda_1^{b+n-1} & \\lambda_2^{b+n-1} & \\cdots & \\lambda_n^{b+n-1} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\end{bmatrix}\\,\\begin{bmatrix}c_1 \\\\ c_2 \\\\ \\vdots \\\\ c_n\\end{bmatrix}.", "7ba7068002b85626e9203174b23e9ea1": "D^2 f : U \\to L\\big(V, L(V, W)\\big). ", "7ba7134896b5389243f9f71d4feb24af": "\n \\tilde{r}^2~\\cfrac{d^2R}{d\\tilde{r}^2} + \\tilde{r}~\\cfrac{dR}{d\\tilde{r}} + (\\tilde{r}^2-\\alpha^2)~R = 0 ~;~~ \\cfrac{d^2Q}{d\\theta^2} = -\\alpha^2~Q\n ", "7ba789108832b1c0db1e112d21b83882": " \\mathbb{Z}/4\\mathbb{Z}, \\mathbb{Z}/8\\mathbb{Z}, \\mathbb{Z}/12\\mathbb{Z}, \\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/4\\mathbb{Z}", "7ba7c6cd0b907abf03b6e4e888fdfe61": "\nh(z,y) = \\frac{i}{\\pi} y e^{- \\nu \\pi i} \\left[ e^{\\pi y} A(\\nu + i y, \\nu - i y \\,|\\, z e^{i \\pi} ) - e^{- \\pi y} A(\\nu - i y, \\nu + i y \\,|\\, z e^{i \\pi} ) \\right],\n", "7ba84f1c9acc90277a04965379ffe275": " v_1=\\frac{2 m_1 m_2 c^2 u_2 Z+2 m_2^2 c^2 u_2-(m_1^2+m_2^2) u_2 u_1^2+(m_1^2-m_2^2) c^2 u_1} {2 m_1 m_2 c^2 Z-2 m_2^2 u_1 u_2-(m_1^2-m_2^2) u_2^2+(m_1^2+m_2^2) c^2} ", "7ba8dda649c0ab799c16bcf100281c6b": "D = \\frac{d\\ log{L(k)}}{d\\ log{k}}", "7ba8e7d959820814958d2324369ca8b4": "H[a\\xi+b\\eta]=|a|E[\\xi]+|b|E[\\eta]", "7ba9155e6534d5cfea4e8a32edaa7552": " Q^2 = Q\\,\\!", "7ba91c6df0c147c6f9ba3365bc18dbd3": "M/{\\asymp}", "7ba9932881dbbfe8f68f53adcb167bb5": "Var(\\epsilon(x_0)) \\overset{*}{=} \\begin{bmatrix}W^T&-1\\end{bmatrix} \\cdot \n\\begin{bmatrix}Var_{x_i}& Cov_{x_ix_0}\\\\Cov_{x_ix_0}^T & Var_{x_0}\\end{bmatrix} \\cdot\n\\begin{bmatrix}W\\\\-1\\end{bmatrix}", "7ba9a4fde69f3411a6fb85dcbf99e948": "\\textstyle S_{d}(r) \\rightarrow O(2^{d}r^{d-1}/\\Gamma(d))", "7ba9f1977dacbfb626ae6869ad02a00a": " MN_{p\\times q}(\\cdot,\\cdot) ", "7baa2ff76fe9c552266985c30f06dadc": "\n\\psi(\\vec{\\theta}) \\approx {1 \\over 2} \\kappa_{\\rm smooth} |\\theta|^2 + \\sum_i \\theta_E^2 \\left[ \\ln\\left( { |\\vec{\\theta}-\\vec{\\theta}_i |^2 \\over 4} { D_d \\over D_{ds} } \\right) \\right]. \n", "7baa3f218770979b01dd9dc833d1da25": "\\delta V\\ ", "7baa7bca814e67c42079ba241f352408": "(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2 = \\bigl([ \\tilde{t}_\\text{r} + b - t_i]c\\bigr)^2, \\; i=1,2,\\dots,n", "7baaa67970f22b69bd1d599543e5e435": "P(s)f(x)^{s+1} = b(s)f(x)^s. \\, ", "7baae0dbe997694d6c5ce31a63d5a467": "\\begin{cases}\n L_2 (w,b,e,\\alpha )\\; = J_2 (w,e) - \\sum\\limits_{i = 1}^N \\alpha _i \\left\\{ { \\left[ {w^T \\phi (x_i ) + b} \\right] + e_i - y_i } \\right\\} ,\\\\\n \\quad \\quad \\quad \\quad \\quad \\; = \\frac{1}{2}w^T w + \\frac{\\gamma }{2} \\sum\\limits_{i = 1}^N e_i^2 - \\sum\\limits_{i = 1}^N \\alpha _i \\left\\{ \\left[ w^T \\phi (x_i ) + b \\right] + e_i -y_i \\right\\} ,\n \\end{cases}", "7baae9800edf3bf65f8532a6626cadbc": "x-y+b^n", "7baaeae7592dd38dd016cc760e74a94e": "F(X) \\in D", "7bab20d651d7c1205e513be5649c0ea8": "\\langle P_\\lambda, P_\\lambda\\rangle = \\prod_{\\alpha\\in R, \\alpha>0} \\prod_{0\\varepsilon} \\nabla N(w-z) \\cdot \\nabla S(\\psi)(z) \\,dx\\, dy=-\\int_{|z-w| =\\varepsilon} \\partial_n N(z-w)\\,S(\\psi)(z)=-S(\\psi)(z),}", "7bb0ac143d370b6fb8a14c7ad2a90508": "(i, j) \\in E", "7bb107a4fd6b20059908ae795a780188": "t\\to\\infty . \\qquad \\qquad \\qquad \\qquad (2) ", "7bb125c3d79a53604f468f84c730df4c": " i = 1, \\ldots , k", "7bb1329988efb1264395b4018ff991cd": "\\Pr(A)= \\operatorname{E}[\\Pr(A\\mid \\mathcal{F}_X)],", "7bb133183bc53d8efc2c85f004623e32": "\\gamma_n = 2 \\cdot 4 \\cdot .. \\cdot n", "7bb15624e2a3e58f22b892ab138e1180": "\\nabla^2\\phi=0", "7bb191ecdbfc4cfd917003142be3cd42": "\\mathcal{B}(M) = \\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\}\\}.", "7bb19bcacd989d1f0e2fe57ae7693c57": "83^2", "7bb1b180b7f632f91853a70a02d7a74f": "2c_{ij}/c_{ii}\\ ", "7bb1b48a5b002029cbfc43efdc26d31b": "\\hat{a}", "7bb1d0139b51bf55edce4de81d1363e7": " \\lceil x \\rceil = \\sum_{n=-\\infty}^{\\infty}n[n-1 < x \\le n].", "7bb1eb6069a2098f529bb6cd4bc013e3": "c = \\frac{\\rho\\, b}{1+ b M},\\ b=\\frac{c}{\\rho-cM},", "7bb264bf29ddf02569b77f36f330377e": " \\Delta \\sigma = B \\sin \\sigma \\Big\\{ \\cos(2 \\sigma_m) + \\tfrac{1}{4} B \\big[ \\cos \\sigma \\big(-1+2 \\cos^2(2 \\sigma_m) \\big) - \\tfrac{1}{6} B \\cos(2 \\sigma_m) (-3+4 \\sin^2 \\sigma) \\big(-3+4 \\cos^2 (2 \\sigma_m)\\big) \\big] \\Big\\} ", "7bb2b6717fc18e6ee41148f7ed9d291e": " P^* ", "7bb2ee94ba44013bd21c505f0d72cddc": "\\frac{dT}{dx} \\Bigg|_{x=L}=\\frac{dT}{dx} \\Bigg|_{x=0}e^\\left(AL \\right)", "7bb349aa03626678ba5923e497a0b5f3": " \n\\mathrm{Ri} = \\frac{\\text{potential energy}}{\\text{kinetic energy}} = \\frac{gh}{u^2} \n", "7bb354f3542f98187efa57110ee7bd67": "\\textbf{V}_P = [\\Omega](\\textbf{P}-\\textbf{d}) + \\dot{\\textbf{d}} = \\omega\\times \\textbf{R}_{P/O} + \\textbf{V}_O,", "7bb3604c3217ce604cd38e5fbceb2cfe": "M \\propto a^3/P^2", "7bb3724202272ca27f9dd0643a7f9d1d": "a_n=1\\,", "7bb37d32034e592c690da3be26aea42b": " ({\\mathcal F}f)(t) = (2\\pi)^{-n / 2} \\int_{{\\mathbf R}^n}f(x) e^{-ix \\cdot t}\\, dx.", "7bb384291a510f63442f4041f51b6bce": "\\mathcal{E}= \\int \\mathbf{E}\\cdot\\operatorname{d}\\mathbf{l}", "7bb3b4425fbb36d5477dba78cd735a6f": "\\frac{1}{2.303RT/F}\\ ", "7bb3b57d854adb0b074931b85ae9fef5": " d_i^6 = \\frac{1024 Q^2 \\mu }{ \\pi^2 k [ \\rho_{\\text{tube}} (c^2+2c) + \\rho_{\\text{fluid}} ] } ", "7bb3db830902ece9f92d184773aafa78": "dx \\wedge dy", "7bb405d6e771839f1bf9182c92a5653c": "\n f(x;\\mu,\\sigma^2) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{ -\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2 }.\n", "7bb40839a2e68736abe37d320b7b2283": "I \\subset \\Omega", "7bb450ffbcb8b2c73b5f66663190353c": "ik", "7bb5b927f1e0958630a4ca43e9300add": "|S(\\alpha)|", "7bb5d93641ce107e8b359ec5f77c4237": "C_{out}(m_2) = (c_1^2,c_2^2,..,c_N^2)", "7bb62905f51038fda615cf60f6e2573d": "A_{|\\alpha \\beta\\gamma|}{}^{|\\delta\\epsilon\\cdots\\lambda|} B^{\\alpha \\beta\\gamma}{}_{\\delta\\epsilon\\cdots\\lambda|\\mu \\nu \\cdots\\zeta|} C^{\\mu\\nu\\cdots \\zeta}=\\sum_\\alpha \\sum_\\beta \\sum_\\gamma \\sum_\\delta \\sum_\\epsilon \\cdots \\sum_\\lambda \\sum_\\mu \\sum_\\nu \\cdots \\sum_\\zeta A_{\\alpha \\beta\\gamma}{}^{\\delta\\epsilon\\cdots\\lambda} B^{\\alpha \\beta\\gamma}{}_{\\delta\\epsilon\\cdots\\lambda\\mu \\nu\\cdots\\zeta} C^{\\mu\\nu\\cdots\\zeta} ", "7bb650b753994c784461d0b001372352": " k_-(x,y) = J_{x-\\frac{1}{2}}(2\\sqrt{\\theta})J_{y-\\frac{1}{2}}(2\\sqrt{\\theta}) + J_{x+\\frac{1}{2}}(2\\sqrt{\\theta})J_{y+\\frac{1}{2}}(2\\sqrt{\\theta}) ", "7bb674476a2cb09396a3c4a15e9c6af2": "\\mathrm{CINT}_x(p_{-1}, p_0, p_1, p_2) = \\frac 12 \\begin{pmatrix} -x^3 +2x^2 - x \\\\ 3x^3 - 5x^2 + 2 \\\\ -3x^3 + 4x^2 + x \\\\ x^3 - x^2 \\end{pmatrix}\\cdot \\begin{pmatrix} p_{-1}\\\\p_{0}\\\\p_1\\\\p_2 \\end{pmatrix} = \\frac 12 \\begin{pmatrix} x ((2-x) x-1) \\\\ x^2 (3 x-5)+2 \\\\ x ((4-3 x) x+1) \\\\ (x-1) x^2 \\end{pmatrix} \\cdot \\begin{pmatrix} p_{-1}\\\\p_{0}\\\\p_1\\\\p_2 \\end{pmatrix}", "7bb67cec7e1a8a0172fb6f34b1c63f17": "\\eta_{t,d} \\sim N(\\alpha_t,a^2 I)", "7bb681a859e553856e683fd44bbd2bb1": "\\nabla \\times (E + iB) = i\\frac{\\partial}{\\partial t}(E+iB)", "7bb6876cc773eb8e4b9e25c576a36dad": "360L=1440", "7bb69429f19dffe75c5eec81781d5e6b": " \\log {g(z) -g(\\zeta)\\over z -\\zeta}=-\\sum_{m,n\\ge 1} c_{mn}z^{-m}\\zeta^{-n}", "7bb73b8dfed8cbca2ddf255a5151f87e": "a\\cdot b=1", "7bb8691c8152eb6e22537c483a769e80": "\\dot{y}_{i}", "7bb86fff3fd5805391616094c3b8cf71": "Please do not write below this line or remove this line. Place comments above this line.-", "7bb8b3256068aa2f5e07d0b0df08cc8b": "(A=B) = x_3x_2x_1x_0", "7bb8c94db4cf96fe5fa6d77ebc84558f": " \\frac{dI}{dt} = \\beta \\frac{I}{N} S - (\\mu+\\nu) I ", "7bb904bd38418cd82ed338a7cfc3032f": "Re = \\frac {R} {n ^ 2 \\left( 1 - X \\right) ^ 4} ", "7bb941a5fe647555e8cdc408b723f6a5": "\nS_0[p] =\n\\begin{bmatrix}\n(I_x[p])^2 & I_x[p]I_y[p] & I_x[p]I_z[p] \\\\[10pt]\nI_x[p]I_y[p] & (I_y[p])^2 & I_y[p]I_z[p]\\\\[10pt]\nI_x[p]I_z[p] & I_y[p]I_z[p] & (I_z[p])^2\n\\end{bmatrix}\n", "7bb954f09c57ca3664f7c944cad158b3": "\\phi(Hp_n)|\\prod_{i = 1}^{n - 1} {Hp_i}^a\\mbox{ and }Hp_n > Hp_{n - 1}", "7bb9604f0c7c2d4af7df31ac022d28f1": "\\mathbb{Z} \\to \\mathbb{Z}\\left\\lbrack i \\right\\rbrack", "7bb9befd10c0f5eb5c1c9287aa9c22c9": "P_1=(1,\\sqrt{3},0,2)", "7bb9d941d22396f1ddf635a5cc656c77": "0=0", "7bba01e48bb5b16d9d3d9fc26ccb141f": "p = 7", "7bba0d5723891da47c5dbda68051ca4b": "\n\\begin{align}\na_{j,k} & {} = 2^j \\int_{-\\infty}^\\infty f(t) \\cdot w^*(2^j t - k) \\, dt \\\\\n\\tilde{a}_{j,k} & {} = 2^j \\int_{-\\infty}^\\infty f(t) \\cdot w(2^j t - k) \\, dt \\\\\na_k & {} = \\int_{-\\infty}^\\infty f(t) \\cdot \\varphi^*(t - k) \\, dt \\\\\n\\tilde{a}_k & {} = \\int_{-\\infty}^\\infty f(t) \\cdot \\varphi(t - k) \\, dt.\n\\end{align}\n", "7bba276b88af4d9272370656af0dfa16": "\\scriptstyle x_{i}", "7bba35722e1d571cc9501192a8295585": "w^2+x^2+y^2-z^2=1.", "7bba3fe42216b2b65c2e499590073dd1": "C_2(\\dot{u}(t),u(t),z(t),\\beta_4,\\beta_5,\\beta_6) = \\beta_4 \\operatorname{sign}(\\dot{u}(t)) + \\beta_5 \\operatorname{sign}(z(t)) + \\beta_6 \\operatorname{sign}(u(t))", "7bba62b0e2789af901ac1d76550a22aa": "j_1+j_2+\\cdots = k\\quad\\mbox{and}\\quad j_1+2j_2+3j_3+\\cdots=n.", "7bba9231ff3e871abf5d826671d7b310": "V(\\sigma)", "7bbafcb09a07bffc0dd743f6432063fb": "[\\text{ }]", "7bbb43dfcb99fc0e7e03f9350a63c745": "B_2 \\cong C_2.", "7bbb9ef5b45dbc280e0970c6b1746f0f": "\n\\begin{align}\n\\frac{d}{dx} \\operatorname{arsinh}\\, x & {}= \\frac{1}{\\sqrt{1+x^2}}, \\text{ for all real } x\\\\\n\\frac{d}{dx} \\operatorname{arcosh}\\, x & {}= \\frac{1}{\\sqrt{x^2-1}}, \\text{ for all real } x>1\\\\\n\\frac{d}{dx} \\operatorname{artanh}\\, x & {}= \\frac{1}{1-x^2}, \\text{ for all real } |x|<1\\\\\n\\frac{d}{dx} \\operatorname{arcoth}\\, x & {}= \\frac{1}{1-x^2}, \\text{ for all real } |x|>1\\\\\n\\frac{d}{dx} \\operatorname{arsech}\\, x & {}= \\frac{-1}{x\\sqrt{1-x^2}}, \\text{ for all real } x \\in (0,1)\\\\\n\\frac{d}{dx} \\operatorname{arcsch}\\, x & {}= \\frac{-1}{|x|\\sqrt{1+x^2}}, \\text{ for all real } x\\text{, except } 0\\\\\n\\end{align}", "7bbbc15e77a759482dcbdaa1a1a1b192": "T( \\vec x ) = \\mathbf{A} \\vec x", "7bbbf0384cce9f2d6e3afe77ce3a6437": " Y_{r-1}", "7bbbf9e8841e68a8722fea4934ba7602": "\\liminf_{n\\to\\infty} \\frac{|C(S \\upharpoonright n)|}{n} < 1,", "7bbc33c490c6a91b35a85f688c6fe3df": " PVPV = k_v T k_p T k_t \\,\\!", "7bbc59b0d316c881f793b3dfd550d305": "_nC_k", "7bbcb481bbae921a2334bdec9b453d08": "|\\psi(x) - x| < \\frac{1}{8\\pi} \\sqrt{x} \\log^2(x), \\qquad \\text{for all } x \\ge 73.2, ", "7bbcfb9606bc445d10ca0b0c289538b8": " A_{drag} = \\frac{1}{2m}\\rho_{air}v^2 \\cdot 0.47 \\cdot \\pi \\left(\\frac{D}{2}\\right)^2", "7bbd4e8a8426cb9f857ab9621edebfc8": "p(x + a + 1)", "7bbd6621a652d3624ac0ec7bcb395dc5": "f(z) = \\bar{z}", "7bbd7e50903e2e5ed65bd259142317df": "\\infty_1", "7bbd86856df4c561c6501d1f1237f813": "\\mathcal{L} \\in \\left\\lbrace \\alpha \\; P_\\theta: \\alpha > 0 \\right\\rbrace, \\, ", "7bbd907455cd9ace49984a2929e05250": "f\\in {\\mathcal F}", "7bbd9de052edac5d5bf40d7ad293a114": "l \\neq p", "7bbdabbc900ddf615a55c42e49969e32": "m_2 = 0", "7bbdb6e93f4f8f01531519c786054715": "S(x) = \\sqrt{R(x)} = e^{\\int \\frac{L(x)}{2\\,Q(x)}\\,dx}.\\,", "7bbde46117ed6bd75297453fe2379482": "z^{*}=\\langle g|f\\rangle", "7bbe5d4ba7f20d0025f5a87397f6f4d4": "\\kappa := \\min \\{ \\dim K_{+}, \\dim K_{-} \\} < \\infty", "7bbe7ca10b2d660a24fdb69ffcca3f44": "Wins = 52.7 + 0.97*fWAR", "7bbea786f04e4b5ea5b0737e35c0f8e3": "c_2\\cap{\\boldsymbol S}(c_1)=\\emptyset", "7bbf1530d358a47c31086535e8ca63a5": "Position = (-1.2,0.6)", "7bbf5a72330d1fd17d81f028c8c39c86": "\\omega\\cdot\\omega=\\omega^2", "7bbf63ec3cca10a74fc01cd8aeaa4db9": "| \\phi_k \\rangle", "7bbf75bbf44c95f4dacfc103ba85b1c1": " \\sum_{ k = 10^{ n - 2 } }^{ 10^{ n - 1 } - 1 } \\log_{ 10 } \\left( 1 + \\frac{ 1 }{ 10k + d } \\right )", "7bbf9390da7326fb8d4fb41a0ebabad0": "\\begin{matrix}\nx' &=& 2x&-y&+z \\\\\ny' &=& &3y&-1z \\\\\nz' &=& 2x&+y&+3z \\end{matrix}", "7bbfd69b25a3bac6a37aa42904c8f33c": "\\cos(\\alpha)\\cos(\\theta-\\alpha)", "7bc049b3aee74cf5d617f4ea0426043a": "i^2 = j^2 = k^2 = ijk = -1.\\,", "7bc06653e876cb6bdb882d7ad57d5c81": "\\ \\frac{1}{v}=\\frac{K_m}{V_{max}[S]} + {1 \\over V_\\max}", "7bc0813e7d9f4a33ae119485ad55ef7e": "V_{out} = a \\cdot (\\mathrm{CodeValue} + b ) \\cdot V_{in}", "7bc0815fdeaa0ad2e70dbb8ccb919a02": " \\tau^2 = {4\\pi^2\\mu \\over kZe^2}a^3 ", "7bc08622cb408cd967053c1860fe1894": "\\phi(-\\theta,\\tau)\\phi(\\theta,\\tau) = 1", "7bc105ed686514caa00cb0ad2fd72a48": "W^TAW=D_A", "7bc10daa347f0670e7b67da426332f3a": "(\\nabla \\times \\mathbf{F} ) = \\boldsymbol{e}_k\\epsilon^{k\\ell m} \\partial_\\ell F_m", "7bc118ce9872211692fb21daff3da795": "R(3,3: 2) = 6.\\,", "7bc131510ea169c3411c4b6a31ee9611": "\\mathrm{_{22}^{44}Ti} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{24}^{48}Cr} + \\gamma", "7bc1691097ec14bc3462a085b1b2f1eb": "\\phi^*(T^*N)\\to T^*M", "7bc1827715139c47dfde28b8fd482be7": "\\,E_R = \\frac{B}{r^n}", "7bc1bc4f2675f52f32f768b78f9c8eb4": "E=F-T\\left(\\frac{\\partial F}{\\partial T}\\right)_V,", "7bc1c043dbfd40d9b117c8d8718cc64f": " K_0\\supseteq K_1 \\supseteq K_2 \\cdots", "7bc1c43759255d8068f71b996aa5a5f3": "r\\geq0", "7bc24cc274b8256b48b74b737c64f381": "x^2+\\frac{b}{a}x+\\left( \\frac{1}{2}\\frac{b}{a} \\right)^2 =-\\frac{c}{a}+\\left( \\frac{1}{2}\\frac{b}{a} \\right)^2,", "7bc2de49c08be6383cf659031cf981a5": "\\rho=\\frac{\\exp(-\\beta H)}{\\mathbf{Tr}( \\exp(-\\beta H))}.", "7bc30c7bda20e6113b7cf15ad6beb888": "ax^2 + bx + c = 0 ", "7bc34a831335bf8dfe149d1a23ae8927": "\\scriptstyle{R_1}", "7bc34df632cab84442d3a95bacc57117": "I_{y}", "7bc35d81cadc0b8656640c98c9e29fd0": "a = 0", "7bc377c3976a4fcc5101f0a01c5ea050": "P(1) \\equiv 0 \\pmod p", "7bc3c9a05ef638a5a9f432aee8e6ba5a": "\\cos(\\omega_0 n) u[n]", "7bc3d5cd97415e9333977b1dfa0b675c": "\\left( \\bold{v}_3\\times\\bold{v}_2\\right)\\times\\bold{v}_1 ", "7bc40b1506ed19e55010b2e7f6623250": " \\sigma = \\frac{I_\\text{channel}}{V_\\text{Hall}} = \\nu \\; \\frac{e^2}{h}, ", "7bc41024f28eb16a10992acac69f34fa": "dx_i", "7bc4475fbf54a4accbc6e2effe9f6266": "|\\Phi^-\\rangle = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_B - |1\\rangle_A \\otimes |1\\rangle_B)", "7bc4899868e1582eeaf5a325caa1e9cc": " \\ddot{T} = 0, \\; \\; \\ddot{Z} = 0, \\; \\; \\ddot{R} - R \\, \\dot{\\Phi}^2 = 0, \\; \\; \\ddot{\\Phi} + \\frac{2}{R} \\, \\dot{\\Phi} \\, \\dot{R} = 0", "7bc49423e2c0366fd49329851360f739": "x(x+1)(x+2) \\cdots (x+n-1)", "7bc4b8d0690e3b63486734107e9206cc": "I(z) = I_0 e^{-\\alpha_{abs} z}", "7bc540de8b4a50197cea749dffa77f60": " I_z = \\int_V d^3 r \\, \\rho(\\mathbf{r})\\,r^2", "7bc572d7b05cf09886152555b73e7ce0": "K = (N-1)\\frac{\\sum_{i=1}^g n_i(\\bar{r}_{i\\cdot} - \\bar{r})^2}{\\sum_{i=1}^g\\sum_{j=1}^{n_i}(r_{ij} - \\bar{r})^2},", "7bc5b417571431a69e79982d7c359f20": "V(r) = -\\frac{a}{r} + br", "7bc5ceb1175de9d81421847969033fb2": "\\mathcal{I}_S", "7bc5d680c1cfee0ad6b7a39227553fc2": "(A-\\tilde{\\lambda} I)^{-1}", "7bc5fb5a4449e32b7b1c692bc39f222d": "(u,v,z)\\in[0,2\\pi)\\times(-\\infty,\\infty)\\times(-\\infty,\\infty)", "7bc661a80c761d3325a18363117f4657": "\\theta = 0", "7bc6bae9524ce8d885749b5fa6a965e0": "K=\\frac {\\epsilon^{5.5}}{5.6}d^2", "7bc6e15708d1bdded27e6628481d0826": "x= g_{(a,k)}=(1 - u)^{-\\frac{1}{a}} k", "7bc6fc864b7deb70f1ad15efdfa80074": "\\sum_{k=1}^n\\frac{1}{\\varphi(k)} = \\frac{315\\zeta(3)}{2\\pi^4}\\left(\\log n+\\gamma-\\sum_{p\\text{ prime}}\\frac{\\log p}{p^2-p+1}\\right)+\\mathcal{O}\\left(\\frac{(\\log n)^{2/3}}n\\right)", "7bc7165b4d6b514376254fc4a9aba8a8": "b:B(a)", "7bc73a77e4a664be62148ccd52b3f0a9": "\\psi: B \\rightarrow C", "7bc79cd2436cdf52884c819ad087549e": "x_i, y_i", "7bc79e00d18bc63a7e3c17ea6bb048d5": "E_{ij} = \\sum_{a=1}^n x_{ia}\\frac{\\partial}{\\partial x_{ja}}.", "7bc7b655048c684aa447386bb54412c7": " \\int_B \\operatorname{E} (X | Y=y) \\operatorname{P}(Y=y) \\ \\operatorname{d}y = \\int_B \\sum_{x \\in \\mathcal{X}} x \\ \\operatorname{P}(X=x,Y=y) \\ \\operatorname{d}y. ", "7bc7c6bc8d1e84fafcdb161a1423307d": "p_k \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial L}{\\partial \\dot{q}_k}", "7bc7d36caf67a15d0e7c401e98dd0894": "\\vec{\\tau} = \\frac{\\mathrm{d}\\vec{L}}{\\mathrm{dt}},", "7bc7ddf9ad7d26cad406d4b632ff2877": "X \\sim \\mathrm{Gamma}(\\alpha, \\beta),", "7bc809399b5adcff3a4c28ba6732e2f3": "\\tilde{A}(\\boldsymbol{U}_i,\\boldsymbol{U}_{i+1}) = A(\\boldsymbol{U})", "7bc82a0afdef81a5ab32563712b6af25": "\\exp(-\\lambda\\cosh(S))\\,dS", "7bc84de2cb918e4feb3456c6c98b7d78": "\\text{D} \\boldsymbol{u}_f / \\text{D} t", "7bc87d0a79f6c1e3b43876e24b73d4f4": "\\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\begin{bmatrix}\n0 & 0 & \\varepsilon_{13} \\\\\n0 & 0 & \\varepsilon_{23}\\\\\n \\varepsilon_{13} & \\varepsilon_{23} & 0\\end{bmatrix}\\,\\!", "7bc89210d24c55217f6f4bef66708e2a": "\\mu(A)\\in\\ [0, \\infty] \\mbox{ whenever } A \\in \\mathcal{A}.", "7bc8a7cc67e54184731b3e33619ad250": " A(S) = A(S_1) + \\cdots + A(S_r). ", "7bc8f1ca60dc7611b8f85fd813a08b08": "U_i \\subset N", "7bc8f2817b823a9d66ec10f07b99a9ca": "G_{ij}(t)", "7bc94671629007b47908e904365fef4a": "y_3-y_1", "7bc9a918f5f8309515efe93c40155f95": "S(\\omega, \\Theta)", "7bc9ce731b3af7d0fafb46f890a5af42": "u <^\\mathrm{d} v", "7bc9eb8a21eb3653a97b0771800b6fef": "\\lambda < g", "7bc9f4811a4b13e181097c8e689cbf62": " r(\\nu) = \\frac{a(1-e^2)}{1+e\\cos(\\nu)} ", "7bca23661aaf4e4e1a2eb48678dd2442": "\\|x+y\\|^2= \\langle x, x\\rangle + \\langle x, y\\rangle +\\langle y, x\\rangle +\\langle y, y\\rangle =\\|x\\|^2+\\|y\\|^2, ", "7bca2e3386b28b6d079262d408b9d39a": "\n D_i(\\theta) = F(x_{(i)};\\,\\theta) - F(x_{(i-1)};\\,\\theta), \\quad i=1,\\ldots,n+1.\n ", "7bca2e5adf4149f9db7c1cfc3e57ac8f": "\\, \\frac{(1-p)^r}{(1-pe^{it})^r}", "7bca37a006e4a9e69ace1c39335a3992": "\\nabla\\colon \\Omega^0_M(\\mathbf V)\\to \\Omega^1_M(\\mathbf V),", "7bca3b8075d4e0bdd17844edb133fab4": "\\begin{align}\nd\\mathbf x_1 \\cdot d\\mathbf x_2&=dx_1dx_2\\cos\\theta_{12} \\\\\n\\mathbf F \\cdot d\\mathbf X_1\\cdot \\mathbf F\\cdot d\\mathbf X_2&= \\sqrt {d\\mathbf X_1 \\cdot \\mathbf F^T\\cdot\\mathbf F \\cdot d\\mathbf X_1}\\cdot \\sqrt {d\\mathbf X_2 \\cdot \\mathbf F^T\\cdot\\mathbf F \\cdot d\\mathbf X_2} \\cos\\theta_{12} \\\\\n\\frac{d\\mathbf X_1\\cdot \\mathbf F^T\\cdot\\mathbf F\\cdot d\\mathbf X_2}{dX_1dX_2}&=\\frac{\\sqrt {d\\mathbf X_1 \\cdot \\mathbf F^T\\cdot\\mathbf F \\cdot d\\mathbf X_1}\\cdot \\sqrt {d\\mathbf X_2 \\cdot \\mathbf F^T\\cdot\\mathbf F \\cdot d\\mathbf X_2}}{dX_1dX_2}\\cos\\theta_{12}\\\\\n\\mathbf I_1 \\cdot \\mathbf C \\cdot \\mathbf I_2&= \\Lambda_{\\mathbf I_1}\\Lambda_{\\mathbf I_2}\\cos\\theta_{12}\\\\\n\\end{align}\\,\\!", "7bca5943a504e0bf34f43c5d9aa854ff": "(p \\land q)", "7bca6b64dd493383c8a4e291404f2198": "a_{12}=0.4", "7bca7a0fbfb12fce0ece4f70a19b5776": "F(d,a,n) = y^3", "7bcb321e6a0daf0010c0da3913524e8b": "U_{i+1}", "7bcb679078af2c23e792d2911bd1a35c": "\n\\pi(1)\\approx 1-\\Phi(1.64-\\sqrt{n}/\\hat{\\sigma}_D) >0{.}90\\ ,\n", "7bcba58494c2adc784f57630afe028c3": "\\int (\\ln x)^n\\; dx = x\\sum^{n}_{k=0}(-1)^{n-k} \\frac{n!}{k!}(\\ln x)^k", "7bcc3ae70a5e783b7f9c4a65e4c48043": "\\omega_r=\\omega_n \\sqrt{1-\\frac{1} {k} \\frac{\\partial F_z} {\\partial z}} \\approx \n\\omega_n \\left (1-\\frac{1} {k} \\frac{\\partial F_z} {\\partial z} \\right )\\,\\!", "7bcc7d44aeda18951293201d1f30130b": "x \\in I", "7bcc9d6cc1124c82c9ec50d4fb98463a": "\\Gamma_{k\\to\\Lambda}=S", "7bcca404eda66eaa21adac773bd80fde": " y_i \\approx y(t_i) \\quad\\text{where}\\quad t_i = t_0 + i h, ", "7bccc4406d919349e15958574a0ed4a0": "y_i = x_i - x_{i-1}", "7bcd121ac3bf1d848e976787ebafb13a": "K_{SP}=[Na^+]^2[SO_4^{2-}]", "7bcd191d55c4f24d3f9e842cd9276cbd": "v^3", "7bcde2251dc1f7f1afed1cc4609895d0": "\\varepsilon_0 + 1", "7bce0da8e0ae66ae76568590899ad2b2": "I=\\cup_{n=1}^\\infty I_n.", "7bce5fff136d56293f149279a48ac342": "X = (x_1, x_2, x_3, \\dots, x_n)^T", "7bcec419c3e816443a26f067150e5866": " S = k_B (\\ln \\Omega) ", "7bcef9594588d671f1736b3d8f7ba8cb": "x^2+y^2=h^2\\,", "7bcf24d749d2c37b78a8fa07ef3c4485": "r^* = -\\frac{2 \\sigma T_m}{\\Delta H_v} \\frac{1}{\\Delta T}", "7bcf8ac391ad169136e1404bf60fd294": "\\bigoplus_{-\\infty}^\\infty \\mathbb{Z}/2\\mathbb{Z},", "7bcfd89dff06e0f16587a36059b19329": "{y})", "7bcfeb7124b2da328effb9ba29d5c923": "T = \\tau \\, ", "7bd02cc550a748e6e7b24f1e052fbce4": "{D_g \\overrightarrow{V_g} \\over Dt} = {-f_o \\hat{k} \\times \\overrightarrow{V_a} - \\beta y \\hat{k} \\times \\overrightarrow{V_g}}", "7bd04092d4b0308553b47fabbee6b7fc": " f: \\mathbb{R} \\rightarrow \\mathbb{R} ", "7bd04b95d4287d012aed9d894db0e13a": "f^1(\\theta^1(t))", "7bd0568acfecfab3fe3407a6abe53a43": "\n(f \\otimes g) (t) := \\inf_{0 \\leq \\tau \\leq t}\\left\\{f(\\tau) + g(t-\\tau)\\right\\}\n", "7bd0597d542fde6214c1d467f50983f1": "x\\in \\mathrm{ker}\\left(f^k\\right) \\cap \\mathrm{im}\\left(f^k\\right)", "7bd06b10daf973959a05319563783b4b": "L_{x}\\equiv\\frac{\\partial \\mathcal{L}_{1}}{\\partial\\dot{\\psi}}=I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta\\cos\\alpha)=\\mathrm{constant}.", "7bd084b3df5c61f200109a1929f91e61": "\\mathfrak{P}(\\mathfrak{C}_{\\operatorname{odd}}(\\mathcal{Z}))\n\\mathfrak{P}_{\\operatorname{even}}(\\mathfrak{C}_{\\operatorname{even}}(\\mathcal{Z})).", "7bd094c1e09823ceaf82808d3c0cd3d1": " \\mathcal{E}_c = \\frac{1}{8\\pi} \\left [ \\mathbf{E}^2( \\mathbf{r} , t ) + \\mathbf{B}^2( \\mathbf{r} , t ) \\right ] .", "7bd0d7a87903af343cd191dff3330a72": "\\Bigg(\\frac{\\alpha}{\\pi}\\Bigg)_3=\\Bigg(\\frac{\\beta}{\\pi}\\Bigg)_3", "7bd10815c56853fd7ae24ae55db09069": "\\boldsymbol{\\beta}=a\\mathbf{v}_0", "7bd113f981a292283adc1c6ef20deb42": "P_e = \\left( 1 - \\frac{1}{L} \\right) \\operatorname{erfc} \\left( \\frac{A g(0)}{\\sqrt{2} (L-1) \\sigma_N} \\right) ", "7bd1178ffe35d5b1157c43d08aedf12e": " x = \\pm \\int^y \\frac{ d \\lambda}{\\sqrt{2 \\int^\\lambda F(\\epsilon) \\, d \\epsilon + C_1}} + C_2 \\, \\! ", "7bd12835988d3f6a92ac8c3fa6c521da": "F = 2 \\times \\frac{\\text{precision} \\times \\text{recall}}{\\text{precision} + \\text{recall}} ", "7bd13a26112abe7b0fa7e79bdb69159b": "\\frac{1}{\\sqrt{|G|}}\\sum_{x\\in G}{|x\\rangle \\otimes |f(x)\\rangle}", "7bd16aa22f021ce29bb85d44c7f78eef": "x_{n+1}=x_n-\\frac{1}{f'(x_n)} f(x_n) .", "7bd19eb5a05760f98f9c77c90a9cfbd0": "\\cos x = \\mathrm{Re}\\{e^{ix}\\} ={e^{ix} + e^{-ix} \\over 2}", "7bd1b4936c9b980674991d3f1633613a": "B_k", "7bd1cfd2fb1bf4ef625cb00b2a6319c2": "\\scriptstyle{Z_a}", "7bd1f82aa79312dfecc6a1a5f76c6838": "\\tfrac{3}{9}", "7bd21293b8f6e92c11b92a8fcc06e0dc": "r = v \\ge 2", "7bd21fc462e8c9002385da3c6c5d7b4f": "\\scriptstyle a_1=2", "7bd23338ca9ec6a5bb8e7bc7e0a6da07": " {h_1 \\over h_0} > 1 ", "7bd234282b9edcc6a467e7f6cf8b0518": "a\\in\\mathbb{Z}", "7bd237f01b4e1a4315ea28026becbe07": " \\begin{pmatrix} y_1 \\\\ y_2 \\end{pmatrix} = -\\frac{f}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} ", "7bd28a7d6163590bc3ced9d82475e657": "\\sigma_{K} \\colon \\ell \\mapsto \\sup_{x \\in K} \\langle \\ell, x \\rangle,", "7bd29a45732a030ee7bf07406ac352fb": " \\text{Relative change}(x, x_{reference}) = \\frac{\\text{Actual change}}{|x_{reference}|} = \\frac{\\Delta}{|x_{reference}|} = \\frac{x - x_{reference}}{|x_{reference}|}.", "7bd29a85730052d4a36ff1ff31040047": "\\bar T T", "7bd2be4c2e5a5051d83eeb338df62ae5": "|X| \\times_{Ke} |Y|", "7bd2dc7649c107d903ff90c4028c1738": "h_A(x)", "7bd337e0ab4d08902609f35a9c5e9b95": "\\tfrac{a}{c} \\cdot \\tfrac{b}{d}", "7bd3916509ddedba626af48c40669a56": "T_0^1(V)", "7bd3ba134ccb1163df9c8ae4eab578fb": "\n I(\\lambda) = \\left( \\frac{2\\pi}{\\lambda}\\right)^{n/2} e^{\\lambda S(x^0)} \\left[ \\det (-S_{xx}''(x^0)) \\right]^{-1/2} \\left[f(x^0) + O\\left(\\lambda^{-1}\\right) \\right],\n", "7bd3bd68beb3361fdef9049c8b1468d9": " \\tfrac {d \\sigma}{d \\Omega} ", "7bd3d5f5b3cdb13aed632121206e729c": "alive", "7bd3df974219dcf8a390bfa9b79565d1": "\\psi \\,", "7bd45836fbefb4e3cab11a9d6d6695b9": "\n\\mathbf{x}_1 = \\mathbf{x}_0 + \\alpha_0\\mathbf{p}_0 = \n\\begin{bmatrix} 2 \\\\ 1 \\end{bmatrix} + \\frac{73}{331} \\begin{bmatrix} -8 \\\\ -3 \\end{bmatrix} = \\begin{bmatrix} 0.2356 \\\\ 0.3384 \\end{bmatrix}.\n", "7bd45ad9ef6d72d0d58672730b1c5479": "\\psi_2(x) = (\\sqrt{2} \\, \\pi^{1/4})^{-1} \\, (2x^2-1) \\, \\mathrm{e}^{-\\frac{1}{2} x^2}", "7bd4dc7574e07f60237fab25d2d04467": "\\ A,\\sim A", "7bd56c6b03d5aa54bf0f7ee102ce8b87": "\\pi(g)=\\pi_a\\pi_b(g)", "7bd591dd0379eab6d146b159e5f161d7": "N(t) = N_0 \\left(\\frac {1}{2}\\right)^{t/t_{1/2}}", "7bd59f978456ce41b0c80e9130f3424f": " R = \\frac{T_B}{T_A},", "7bd5a409c569808c39afaa5859b868f1": "v_{k+1}= - \\sum_{i=1}^k v_i.", "7bd5cdadf23e19740922db69f7cdec2d": "\\gcd (ab,n)>1", "7bd6633a158c2950043a6f33caf9b9d1": "{s_1}, {s_2}\\in\\{1,\\dots,S\\}", "7bd6c4a7870a2f678c368b8cd6552f9e": "\n\\bold {A}_1 = \\begin{pmatrix}\n \\cos \\alpha & -\\sin \\alpha & 0 \\\\\n \\sin \\alpha & \\cos \\alpha & 0 \\\\\n 0 & 0 & 1 \n\\end{pmatrix}\n", "7bd6d63cc2a679280ebbf11c0101f433": "\n\\begin{bmatrix}\nf_{x1} \\\\\nf_{y1} \\\\\nf_{x2} \\\\\nf_{y2} \\\\\n\\end{bmatrix}\n=\n\\frac{AE}{L}\n\\begin{bmatrix}\nc^2 & sc & -c^2 & -sc \\\\\nsc & s^2 & -sc & -s^2 \\\\\n-c^2 & -sc & c^2 & sc \\\\\n-sc & -s^2 & sc & s^2 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{x1} \\\\\nu_{y1} \\\\\nu_{x2} \\\\\nu_{y2} \\\\\n\\end{bmatrix}\n\n\\begin{array}{ r }\ns = \\sin\\beta \\\\\nc = \\cos\\beta \\\\\n\\end{array}\n", "7bd6e3faca93ba51118a6ca42473496a": " b_0, \\ldots, b_s ", "7bd70e0baa0b2d7f033f00f4e4773470": " {}= \\begin{vmatrix} x_{i} & y_{i} \\\\ x_{j} & y_{j}\\end{vmatrix} ", "7bd749d6e7f1018c2fc6a43fa69d3232": "n_{i,j}", "7bd76df2daeafc00fa84ab6154f79c7e": " (x \\mathbin{:} A)B", "7bd7c0dde4b0b4e935897dff32cb80ac": "\n \\Delta\\gamma = \\gamma_1 + \\gamma_2 - \\gamma_{12}\n ", "7bd7f3cab5eb59cc93cff70aa6d66c92": "\\operatorname{var}(X)", "7bd80074554ffc4ece2e65eb09301ca7": "R s = \\text{constant} = R_c s_o\\,", "7bd86f1939d5d3a766ff9b7fe179ed7f": "\n\\operatorname{Li}_s(z) = \\tfrac{1}{2}z + {z \\over 2 \\Gamma(s)} \\int_0^\\infty e^{-t} \\,t^{s-1} \\coth{t - \\ln z \\over 2} \\,\\mathrm{d}t \\qquad (\\textrm{Re}(s) > 0) \\,.\n", "7bd871a1561a6770e136dadf581c43fa": " P_{absorb} = \\frac{W}{c R^2} cos \\alpha", "7bd96754108cb5808a51292ee67525a8": "\\beta =17", "7bda19cb7b3d7ff24cb4f40e295abd04": "z\\in M", "7bda33cf9c0e65d73341ee5154d76d45": "\\lambda\\|\\beta\\|_1", "7bdaafacefe55b782a3af60c8a745df9": " P(x,p,t) = P(\\star (x_{-t}(x,p),p_{-t}(x,p)),0)", "7bdb06bc95964f2bce4badd249a0db4e": "\\mathbf{E}^{a} [\\tau_{K}] = \\frac1{n} \\big( R^{2} - | a |^{2} \\big).", "7bdb5450086c4bea43f5c48b87a770db": "\n \\Delta\\langle\\left[\\hat{X}^{\\dagger}\\right]^J\\hat{X}^K\\rangle=\\eta^{\\frac{J+K}{2}}\n\\Delta\\langle\\left [ B^ { \\dagger\n}\\right]^JB^K\\rangle\\,,\n", "7bdb802caa00f3322fd5dcc65dcfd835": "Ha^2 = {{k_{2} C_{A,i} C_{B,bulk} \\delta_L} \\over {\\frac{D_A}{\\delta_L}\\ C_{A,i}}} = {{k_2 C_{B,bulk} D_A} \\over ({\\frac{D_A}{\\delta_L}}) ^2} = {{k_2 C_{B,bulk} D_A} \\over {{k_L} ^2}}", "7bdbaf57d552ab698e43a65375b6bfb5": "\\mathrm{Bi}_m", "7bdbd562269e7f3d9b9914d211340fc6": "\\zeta\\left(\\frac{1}{2} + it\\right) = O(t^\\varepsilon),", "7bdbf0b210504976843fd77ee243003d": "\na = x - x_1 \\,\n", "7bdc31e84af14a28dcda977476ff81e4": " \\sigma^2 > 0 \\;\\; ", "7bdc332e3fad07db251054dc8b38c5ec": "{z} = \\left[\\begin{array}{c}{x}\\\\ {\\lambda}\\end{array}\\right]\\,", "7bdc68738d2e60eace8307d2c92f919d": "\\mathbb{R}^\\times_{>0}", "7bdc6fa094f1666f8a1966af73b28677": " g(\\tilde{x}) ", "7bdcca98daff7e0a7e4ffae1da9caf6a": "n_G", "7bdd1335a362244581cdf2b9a4dfa26a": "\\rm\\frac{1}{M \\cdot s}", "7bdd18cafe87368d49df6e3552c62f7c": "\\boldsymbol{J}_1", "7bdd42f2efda98eeb1b2b833c6fc7996": "\\mathfrak{S}_p", "7bdd9f1c8ae8c0dfc164aee95e9166b0": "\\tau_1=L_U/v_f", "7bdda362e3c3a6fec4f5632ab8973e71": "Z_n = \\frac{n\\overline{X}_n-n\\mu}{\\sigma \\sqrt{n}} =\\sum_{i=1}^n {Y_i \\over \\sqrt{n}}", "7bddba963d0700d6a93a954cd53d0d23": " \\cup_{i} D_i", "7bddbb45b7cbaa64221cf34ccc9f3909": "x = \\begin{pmatrix} 1 & 2 \\\\ 2 & 3 \\\\ 3 & 1 \\end{pmatrix}", "7bde2d8f6b738c32409b02ca9a92327e": "\\triangle^{(0)}y_{i} := y_{i}", "7bde5b94e35f8f88ad4f7b007e8f3919": "X_k=X_{N-k}^*\\,", "7bdf0d47fcc7a90ce7d8ea240e589d45": "\\int_0^t H_s \\, dX_s=\\int_0^t H \\, d^-X.", "7bdf117618b9324284e5d4b7a6e0eabe": "\n\\frac{dq^s}{dt}=w^s,\\qquad\\frac{d w^s}{dt}+\\sum^n_{i=1}\\sum^n_{j=1}\\Gamma^s_{ij}\\,w^i\\,w^j=F^s,\\qquad s=1,\\,\\ldots,\\,n", "7bdf22f8ab54df62b86846efa77524b9": "B = \\dots \\to 0 \\to B^0 \\to 0 \\to \\cdots,", "7bdf252a5cbc327455cef76763542b4f": " W_{ij} = 0 ", "7bdf3c6fefa6c3fbc8dc0c8ac8dc5cb7": "p_1,\\ldots,p_{k-1}", "7bdf95c492305fe1a9a0815d3877a928": "f:U\\rightarrow\\mathbb{R}", "7bdfd50769f788a47ca9c6209d2c58b0": "\\int\\limits_0^{1} \\!\\frac{x^{n-2} \\ln\\ln\\frac{1}{x}}{\n1+x^2+x^4+\\cdots+x^{2n-2}}\\,dx \\, =\n\\int\\limits_1^{\\infty}\\!\\frac{x^{n-2} \\ln\\ln{x}}{1+x^2+x^4+\\cdots\n+x^{2n-2}}\\,dx =\n", "7bdfde699d1228d18f66e93d8a8d5201": "n \\log_2(n) - \\sum_{i=1}^A f_i \\log_2(f_i) ", "7bdfe1927b183a1d8afab0439b9eab5e": "c=\\lambda (z_P-z_0),", "7be02eacbebf16f0d498eabe948a878d": "\\tilde{a} \\rightarrow \\tilde{b}", "7be041d4788aea5c68a02029f88812f7": "n_\\mathrm{obs} - 1", "7be05d73a9a5cc351761593ca97c1003": "\\textstyle (\\Lambda - \\Lambda_0(P))", "7be05f06d4d2f22e337fcf06f86b6159": "a^2 + b^2 + c^2 \\geq 4\\sqrt{3}\\, \\Delta. ", "7be17091b6ef66ad96ecec76f515d235": "{\\rm Tr}_{i}f(\\xi_{i})", "7be1cf6dd36f537f52eb4c57f2707b0b": "I \\big( \\stackrel{\\circ}{S} \\big) = I \\big( \\bar{S} \\big),", "7be209ced7759fde2d90a5f98cd877eb": "\n\\frac{100 - \\text{Clean price}}{\\text{Maturity in years} } + \\text{Spread}.\n", "7be2700b9124351a609acce8e09b0c85": " k=k_{ii}", "7be2743d3c5f2e3266ebbdf116d4514d": "g - \\frac{1}{n}x - \\frac{1}{n}h", "7be2a3b5ad1ed0eef984f2bb1b32c44b": "RIR = \\frac{NIR - I}{I }\\times 100", "7be2cabf75202a13695228f49b8c450b": "1 < \\frac{x}{\\sin x} < \\frac{1}{\\cos x}", "7be2cfd1a0cf4e4f31a622f4aea7bc84": "\\displaystyle{R-r \\le |x-y| \\le R+r,}", "7be2d473ef4d145a290fbbb6c3233aab": "C_G(P)\\leqslant P", "7be2f167736292fbb999f081e90c1316": "k = N/8", "7be30caa9f7d89744ada3cbe3253b39c": "A \\circ B = B \\circ A,", "7be30cbc9943ae0c14220f232b52f21b": " u_i< u^{max}_i, \\ \\ \\ u_i\\in [\\underline u_i, \\overline u_i] ", "7be341849660006ff315f6942c99d812": "k_BT < \\frac{e^2}{C},", "7be36417e987f977cea4431a7bbfeeaf": "1-\\gamma_1 \\gamma_2 \\neq 0 ", "7be36a4756b0572a9979d1e99eef6062": " [H, G] = 0 \\,", "7be4243a900d35b62c524cb508ea8726": "X = (X_1, \\ldots, X_K)\\sim\\operatorname{Dir}(\\alpha)", "7be43082258e9a25df2fd94d2da0616b": "(c_{1}-b_{1})+(b_{1}-a_{1})+(c_{1}-a_{1})", "7be46e128a299c59e1c829380951e6b3": "(x_0\\lor\\lnot x_3) \\;\\equiv\\; (\\lnot x_0\\Rightarrow\\lnot x_3) \\;\\equiv\\; (x_3\\Rightarrow x_0).", "7be4ec5519dcdff2a8f054a2e285ba5d": "{{z}_{\\text{max}}}=180{}^\\circ -{{\\sin }^{-1}}\\frac{{{R}_{\\text{E}}}}{{{R}_{\\text{E}}}+h}\\,.", "7be50f1cfbd45689b4bb9a559ba21dc8": "\\vec \\nabla \\times \\vec \\nabla \\phi = 0 ", "7be59753a4fd77d0116edf661758b754": "F_{mean}", "7be5ea619bccad047b1ab41daf8f261b": " T_s = ", "7be617e9a606434c6ce98c1ff91c3bea": "\\displaystyle f'(x_0)", "7be63572d21bfbbcc71cd688c811ad6b": "U_i(x)-w_i", "7be65883e94459c0f84c1c239dcd3803": "\\dot{y}=q_2z-q_6y-q_3xy", "7be6ea71fa6ad73337b54747f806588a": "\\omega = f_I\\mathrm{d}x^I=f_{i_1,i_2\\cdots i_k}\\mathrm{d}x^{i_1}\\wedge \\mathrm{d}x^{i_2}\\wedge\\cdots\\wedge \\mathrm{d}x^{i_k}", "7be70cf6acc40d781dda9d3c640b5d4d": "\\coth(x)", "7be715d363e00b6378c6d6776cffc155": "\\boldsymbol{\\omega}=\\nabla\\times\\mathbf{u}.", "7be75e99db1c55eac5ab8e4e8df7947c": "|k\\rangle", "7be7796e300dfd3861a9f73671b8e8e0": "\n\\mbox{Var}\\left[F\\right]=\n\\begin{cases}\n2\\frac{(\\nu_1+\\lambda)^2+(\\nu_1+2\\lambda)(\\nu_2-2)}{(\\nu_2-2)^2(\\nu_2-4)}\\left(\\frac{\\nu_2}{\\nu_1}\\right)^2\n&\\nu_2>4\\\\\n\\mbox{Does not exist}\n&\\nu_2\\le4.\\\\\n\\end{cases}\n", "7be802d3ee3c77d8e623c94da5c53f28": " y_{t\\mid t-1} = z(t) - H\\hat{\\textbf{x}}_{t\\mid t-1} ", "7be85e2488a474bc6d8e3ceca3eb49df": "x_1=2", "7be8837f05aad77632b37c6a65fabf87": "\\max\\limits_{z\\in I_x} \\Re[S(z)] = \\Re[S(x^0)]", "7be88f60b84703e8ce319ce56277fcf5": "\n \\sum M_A = R_a (10) - (1)(x-10)\\frac{(x + 10)}{2} - V_2 x + M_2 = 0 \\,.\n ", "7be89793b7c98b2fcfbaa0d4b2f0b3fd": "K: X_i = N_i + S_i, i = 1,2...n.", "7be8e46fd0b1f6e2fb2b0dfeb96a1a2e": "\\delta t = L\\left(\\frac{1}{v_1}-\\frac{1}{v_2}\\right)\\approx \\frac{Lc}{2p^2}(m_1^2-m_2^2)", "7be98416b2c185e79f48aaf37a3448bd": "\\widehat S^{2n}", "7be9870c51a3debd7ff3bea4c2fc5f82": "\\scriptstyle V_S = V_i - V_o", "7be98b3e40fdf4da74e00345f1c22704": "\\{U_1, \\dots, U_d\\}", "7be994992c916dc4c34d45e9ced83f72": "\\frac{ \\partial Y}{ \\partial t} = \\frac{ {\\alpha}Y }{[K(t)]} \\frac{ \\partial K}{ \\partial t} + \\frac{ (1 - {\\alpha})Y }{[L(t)]} \\frac{ \\partial L}{ \\partial t} + \\frac{ (1 - {\\alpha})Y }{[A(t)]} \\frac{ \\partial A}{ \\partial t} ", "7be9a5af4b2fb331bb8f37767390a16a": "\\sigma=ne\\mu_e", "7be9b1717d47b457f6f854402cc50af1": "\n\\begin{align}\np(x_0) & = a_0 + x_0(a_1 + x_0(a_2 + \\cdots + x_0(a_{n-1} + b_n x_0)\\cdots)) \\\\\n& = a_0 + x_0(a_1 + x_0(a_2 + \\cdots + x_0(b_{n-1})\\cdots)) \\\\\n& {} \\ \\ \\vdots \\\\\n& = a_0 + x_0(b_1) \\\\\n& = b_0.\n\\end{align}\n", "7be9c6d73fed396587c3e9e7c073cdbc": " T(t) - T_{\\mathrm{env}} \\ , \\quad ", "7be9fd97bbbf144b5189f62af87dba2b": "\\nabla_X Y = \\top(\\nabla'_X Y),\\quad \\alpha(X,Y) = \\bot(\\nabla'_X Y).", "7bea3485e20e4d253010887df8e70696": "\\varphi\\left(h(x\\otimes y)\\right)=\\varphi\\left(x\\otimes S(h_{(1)})h_{(2)}y\\right)=\\varphi\\left(S(h_{(2)})h_{(1)}x\\otimes y\\right)=h\\varphi(x\\otimes y)=\\varepsilon(h)\\varphi(x\\otimes y),", "7bea5e9ee1e6980f27f0d3a81d78c9a3": "{{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}=1-{{(z_1-z_2)(z_3\n-z_4)}\\over{(z_1-z_4)(z_3-z_2)}}", "7bea646b10d5029bad7e6b0abae34124": "\\scriptstyle \\frac{1}{p_i - 1} \\,<\\, \\frac{1}{p_{i - 1}}", "7bea87c7556294078c82ce815ef1b140": "i = 0, 1, 2, ..., N-1", "7bea99aed60ba5e1fe8a134ab43fa85f": "a\\,b", "7beaabad18565100deff8860cf5c07e3": "\\scriptstyle T_\\mathrm{g}", "7beab6f51aaabb413bf16a6a9da5eb49": "E_A + E_B = 1", "7beb85b29d61a206bc36670b463e5473": " 2r\\sin\\left(\\theta/2\\right)", "7bebb75b4269a1ad4590641533284df6": "B (d' \\setminus f) \\to B(d \\setminus f)", "7bec0d47ec90d4474300aaf23722b147": "E_p = \\sqrt{\\frac{\\hbar c^5}{G}} ", "7bec1fda65c989dd6a6233169de8450d": "f_3(x)=0", "7bec2502696d8d8603337a0ea08c40d7": "\\begin{align}\n u &= \\sin^{n-1}x \\\\\n \\Rightarrow du &= (n-1) \\sin^{n-2}x \\cos x dx \\\\\n dv &= \\sin x dx \\\\\n \\Rightarrow v &= -\\cos x\n\\end{align}", "7bec46bd735379aa13be3c97e1fb4c5a": "|\\phi^{(k)}(x)|\\ge 1\\,", "7bec7cdf50d4c97f164a4d31285651e0": "s_{mv}", "7becf419ca4290e38585809f5624b28d": "I_v", "7bed0982093d5b9a1d5643835995e692": "\\frac{\\text{d}\\omega}{\\text{d}k} = \\frac{c}{n} - \\frac{ck}{n^2}\\cdot\\frac{\\text{d}n}{\\text{d}k}.", "7bed635a441a637b09c740f66117e661": " [T \\varphi](x) = f(x) \\varphi(x). \\;", "7bed657a775c37c2570786d0cbeefd88": "ij", "7bedcdf20866aa958a612c053dada312": "365 = 13^2 + 14^2", "7bedea1b2f4e7e75e1061a9ff3cdbd82": "\\theta\\approx1", "7bedf2249c2ec2c173a0aefaa552b01a": "_k\\mathbf{a}_{l,m,n} = \\frac{1}{2\\sqrt{Z_F}}{_k\\mathbf{E}}_{l,m,n}+\\frac{\\sqrt{Z_F}}{2}{_k\\mathbf{H}}_{l,m,n}", "7bee001de8d9fcbd6e13650a88e97b50": "\\begin{align}\nL_z &= -i\\left(x\\frac{\\partial}{\\partial y} - y\\frac{\\partial}{\\partial x}\\right)\\\\\n&=-i\\frac{\\partial}{\\partial\\varphi}.\n\\end{align}", "7bee27034b019846cb9d0847d9f44645": "\\mathcal{O}^{3+}(x)", "7bee3f1ffd4646e061d11b8f9c0b6839": "n_{min} = 2\\left\\lceil\\frac{b}{2}\\right\\rceil^{h-1}", "7bee4fcfdafa8abe3f77c544c47626af": "x\\rightarrow\\lambda x,", "7bee65da8ed7e588f9d815ebbb1440b3": "M_\\odot", "7beed71f0de654fc5a6acea5c04b4a55": "\\sqrt{y^TQ^{-1}y}", "7bef0167af5a538e55fa031e2c2ce7cc": "\\phi'(b)=f'(b)-y 0 ", "7c0abeac444ff490cb1e1834eaa2ca56": "\\left ( -\\frac {M_x} {T_2}, -\\frac {M_y} {T_2}, -\\frac {M_z - M_0} {T_1} \\right ) ", "7c0b26dfc7e2f3ca9989d4bb04bd6649": "{\\models_{\\Sigma}}\\subseteq|{Mod(\\Sigma)|\\times sen(\\Sigma)}", "7c0b8f2aa39c4dfa17dc6104da28dce2": "320^2+410^2+416^2=102,400 + 168,100 + 173,056 = 442,556 = 666^2", "7c0ba96914558977becf08acbffb36f1": "\\textstyle X = \\prod_{\\alpha} X_{\\alpha}\\,", "7c0bfe911702e3c072a73a05b42b9e24": "a = \\tfrac{27}{82} = \\tfrac{1}{3} -\\tfrac{1}{246},", "7c0c0986d497630a76f122f5f9c71612": "\\operatorname{Li}_2\\left(-\\frac{1}{2}\\right)+\\frac{1}{6}\\operatorname{Li}_2\\left(\\frac{1}{9}\\right)=-\\frac{{\\pi}^2}{18}+\\ln2\\cdot \\ln3-\\frac{\\ln^22}{2}-\\frac{\\ln^23}{3} ", "7c0c17e7e892973599407fcf75512a91": "dx=\\sqrt{\\frac{y}{D-y}}dy", "7c0c7bd9dd1678fa9987c936e62ef4aa": "\\sigma=\\begin{pmatrix}1 & 2 & 3 & \\cdots & n\\\\ \\sigma_1 & \\sigma_2 & \\sigma_3 & \\cdots & \\sigma_n \\end{pmatrix}", "7c0cc9c6c0ad2d86328eee4fc2ed22ec": "p({\\rm label}|\\boldsymbol{x},\\boldsymbol\\theta) = f\\left(\\boldsymbol{x};\\boldsymbol{\\theta}\\right)", "7c0ce32b76d6b7f2f4dc3ddcee074ac1": "y^\\prime=2|y|-1=-2y-1, y<0", "7c0d38b9530286893e5f634f6459d2df": "\nP = 1-\\left(1-\\frac{1}{N}\\right)^n = 1 - \\left(\\frac{999}{1000}\\right)^{100} = 0.0952\\dots \\approx 9.5\\%\n", "7c0d5294f313c45f566500c3722d289e": "H^p(X,\\Omega^q)\\simeq H^{p,q}_{\\bar{\\partial}}(X)\\simeq\\mathbf{H}^{p,q}", "7c0d6d56ab86f8e4a2d7156efdde55d7": "\\begin{align}\nA^* x = 0 &\\iff\n\\langle A^*x,y \\rangle = 0 \\quad \\forall y \\in H \\\\ &\\iff\n\\langle x,Ay \\rangle = 0 \\quad \\forall y \\in H \\\\ &\\iff\nx\\ \\bot \\ \\operatorname{im}\\ A\n\\end{align}", "7c0daa10617092cc9720e0568289aaca": " E_F^n=m_n c^2 \\,", "7c0db4a9025f110cd85dbfe3d3044b51": "(1-p)k", "7c0de6df33aac6d26ae1bc59e5bc65f2": "\\tfrac{d\\mathbf{T}}{ds}", "7c0dfcc61a033371359680923a0e4e64": "\nz = g(w) = \\frac{-a + cw}{b - dw}\\,\n", "7c0e25f400491e14731332b72c249f45": " \\theta_k", "7c0e2ce180d3bb0f14fd3f9b7863f656": "\\ln z = W(\\ln z) e^{W(\\ln z)}\\,", "7c0e38296137e826c2a8e07e68db7682": " Q = \\begin{bmatrix}\n 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\\\\n 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\\\\n 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2\n\\end{bmatrix} . ", "7c0e44bf9eb0edbc25d78f340caed4d9": "\n\\begin{align}\n\\operatorname{E}\\left[f(X)\\right] & {} = \\operatorname{E}\\left[f(\\mu_X + \\left(X - \\mu_X\\right))\\right] \\\\\n& {} \\approx \\operatorname{E}\\left[f(\\mu_X) + f'(\\mu_X)\\left(X-\\mu_X\\right) + \\frac{1}{2}f''(\\mu_X) \\left(X - \\mu_X\\right)^2 \\right].\n\\end{align}\n", "7c0e688008952b8629d51501961bd253": "\\psi _{beta}(t|\\alpha ,\\beta )=(-1)\\frac{dP(t|\\alpha ,\\beta )}{dt}", "7c0ed34c5d2ae0a0a2bd4b93e27b0955": "\n\\langle k, \\chi(T) k \\rangle = \\int_{\\sigma(T)} \\chi(\\lambda) \\cdot \\lambda^2 d \\mu_{h}(\\lambda) = \\int_{\\sigma(T)} \\chi(\\lambda) \\; d \\mu_k(\\lambda).\n", "7c0ee1bcfe16fbb5b5c0e7078c3b372d": "n = \\sum_{k}a_kp^k", "7c0ee937d328eae8531e977bd6f9308c": "2 \\pi ", "7c0f30629adcccf9b359efcd4dd26a0d": "\\boldsymbol\\phi_{k=1 \\dots K}", "7c0f717f23b16b987461892f5be7735c": " \\mathcal{I}_{\\mathcal{C}} = \\{1_C: C \\in \\mathcal{C} \\}", "7c0f72041c8526760e37f410b06d9f83": "\\scriptstyle \\vec \\tau ", "7c0f7e898beceff9249d45085e832143": "{y}=\\rho \\, \\sin\\theta \\, \\sin\\phi \\quad ", "7c0f853b035888614415491de677b659": "P^\\mu_{-(1/2)+i\\lambda}(x)", "7c0fd6a57bad9b4154bebc53a74a6fbe": "\\sum_{k=2}^\\infty \\frac{\\zeta(k) - 1}{k} = 1 - \\gamma", "7c0febdf165e78eccd1aa1870a12750b": "h_p(t)=\\left( \\frac{\\alpha}{\\pi} \\right)^\\frac{1}{4}e^{ -\\frac{\\alpha}{2}(t-T_p)^2}e^{jt\\Omega_p},", "7c102a39f995c3dc5c2fe2cef518bfa2": "\\mathrm{DCF} = \\frac{\\mbox{FCF}}{\\mbox{WACC}} - {\\mbox{BVD}}", "7c102c53377f4897dec06721491064c0": "e_ie_0 = e_0e_i = e_i;\\,\\,\\,\\,e_0e_0 = e_0,\\,", "7c1062740a70b80882ce12ce83b97989": "\\left|f\\left(\\frac{p}{q}\\right)\\right| \\leq c(\\alpha)\\left|\\alpha-\\frac{p}{q}\\right|.", "7c10637a069f6c6142946684e9f4352b": " z = A^+b ", "7c107dd10b67f3cc929e20c12665ea61": "{\\frac {\\mathrm{d}\\mathbf{u}_\\mathrm{\\theta}}{\\mathrm{d}t} = -\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t} \\mathbf{u}_\\mathrm{r} = - \\omega \\mathbf{u}_\\mathrm{r}} \\ , ", "7c10b62bf7c1da4f113d52eb4417e763": " _{j} ", "7c10db2c2371e543ed11801fc6e2dbed": "\\sigma_{ij}=C_{ijkl} \\varepsilon_{kl}", "7c10efc4dfae0422b810713addd35346": "\\begin{align}\n\\Delta^r(\\Delta^s(\\phi_{1,1,1}))\\,&=\\,\\Delta^r(\\phi_{1,2,1} - \\phi_{1,1,1})\n &=\\,\\Delta^r(\\phi_{1,2,1}) - \\Delta^r(\\phi_{1,1,1})\n &=\\,(\\phi_{2,2,1} - \\phi_{1,2,1}) - (\\phi_{2,1,1} - \\phi_{1,1,1})\n\\end{align}", "7c11044625eeeed43a057b9e51a40cc3": "\\alpha \\colon d_{1} \\to d_{2}", "7c111f5da4dd3191b3d7db247e3d7477": "r = hA/C", "7c115c97324a95b22d9808b85deff9f6": "d = \\operatorname{gcd}(b-1, N)", "7c11605dda354fcff944944bd13eacf8": "Z_\\Lambda[J]=\\int \\mathcal{D}\\phi \\exp\\left(-S_\\Lambda[\\phi]+J\\cdot \\phi\\right)=\\int \\mathcal{D}\\phi \\exp\\left(-\\frac{1}{2}\\phi\\cdot R_\\Lambda \\cdot \\phi-S_{\\text{int}\\,\\Lambda}[\\phi]+J\\cdot\\phi\\right)", "7c11a988cd105134de4d35a92f384fd3": "\n\\begin{align}\n(a + b) + c & = a + (b + c) \\\\\na + b & = b + a \\\\\n(a \\cdot b) \\cdot c & = a \\cdot (b \\cdot c) \\\\\na \\cdot b & = b \\cdot a \\\\\na \\cdot \\infty & = \\frac{a}{0} \\\\\n\\end{align}\n", "7c11efd8ac7641278e412c26f98dae8f": "\\frac {\\Delta I} {I} = k,", "7c122921808b8bc0c51893d4dfcb814d": "\\varepsilon_A:A\\otimes A^*\\to I", "7c124572828cf1518fcf0921056178d9": "{e^2 \\over L_B}", "7c1286d4398218a18365ca450b93153d": "F:C\\to D", "7c128e5b12c8e3f59668d23f0aa0f053": "\\scriptstyle \\leq2\\times10^{-19}", "7c12d020280d950669a0c1d9d9007555": "b_{k} = e^{i \\phi_{k}} v_{1} + r_{k}", "7c13c72f572c37d2d85418b8c3baa5b2": " Z = R \\sin \\epsilon \\sin \\lambda ", "7c13f1e6b9130d68bbfd187cc1627aa8": " \\tfrac{w-\\mu_w}{\\sigma_w},", "7c13fbb441fbdb1d17675043d6886c18": "|q| < 1", "7c13ff2f95b263531f8f298fe95f35c7": "ds^2= \\frac{\\operatorname{Re}(dz \\otimes d\\overline{z})}{\\left(1+|z|^2\\right)^2}\n= \\frac{dx^2+dy^2}{ \\left(1+r^2\\right)^2 }\n= \\frac{1}{4}(d\\phi^2 + \\sin^2 \\phi \\,d\\theta^2)\n= \\frac{1}{4} ds^2_{us}\n", "7c14177ac62f0950e4edd42b3d522223": "\\,\\Pi ", "7c142d87b0c1bfcbb1cfd22cdc52b3e6": "F(x)=0", "7c1431d0378e9b9e30d9c771e33ee80e": "\\phi^\\alpha_{v,w}(g)=\\langle v,\\pi^\\alpha(g)w\\rangle ", "7c14555aff343c345674611fd3b147e1": "f(s^*,s)=d_s\\;(d_s=\\sum_{(s,u)\\in E}f(s,u))", "7c1485e1f2b52febae9191d61bf89872": "\n\\mathrm{F^{-} \\approx SO_{4}^{2-} > HPO_{4}^{2-} > acetate > Cl^{-} > NO_{3}^{-} > Br^{-} > ClO_{3}^{-} > I^{-} > ClO_{4}^{-} > SCN^{-}}\n", "7c14bfb99aa4f3279aead54d098a05ca": "x^2+h=a^2", "7c14e1eedb69587da69abc4f8d415cc8": "\\lim_{x\\rightarrow c}\\left(\\liminf_{y\\in S_x} \\frac{f(y)}{g(y)}\\right)=\\liminf_{x\\rightarrow c}\\frac{f(x)}{g(x)}", "7c152988334c08cedebdd8284c25ebd8": "c_1,c_2,\\ldots,c_r", "7c155234a4035eb7accb8337246f0b61": "\n \\begin{align}\n I_1(\\boldsymbol{A}) & = \\text{tr}{\\boldsymbol{A}} \\\\\n I_2(\\boldsymbol{A}) & = \\frac{1}{2} \\left[ (\\text{tr}{\\boldsymbol{A}})^2 - \\text{tr}{\\boldsymbol{A}^2} \\right] \\\\\n I_3(\\boldsymbol{A}) & = \\det(\\boldsymbol{A}) \n \\end{align}\n", "7c15ab3d70920c58a2165504211f4805": "\\left(\\frac{b}{a}\\right)^2=\\frac{b}{a}+1", "7c15e1c02e044614abbdad3f32761ed0": "\\gamma _{lkm}\\, ", "7c160cfa8fccba28406cf2f38dd353b4": "R_G^{(f)}(Y)", "7c1624c5b75ef7331956beef0a768348": "H^*(G;M)=\\bigoplus_n H^n(G;M)\\,", "7c165352d2ae05a504ae94cad1b0d6f5": "y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6", "7c166215de7fffebb8f98431e23a8a10": "C=A\\cdot B^{-1} \\mod M", "7c16bb018a7ee165104543ab31119c9a": "\\frac{a^2}{18}", "7c16ef242c5a6009928e2a8222f5ad58": "N(\\bar{X},\\sigma^2)", "7c170871aa453dce952aa16380c464cf": "|\\dot{x}| + |a(t,x,\\dot{x})| \\geq |\\dot{x} + a(t,x,\\dot{x})|", "7c17748c806d67232c635516949e394c": "L = K(\\theta).", "7c177992cf75f4a4861b0cc4ede3c313": "[(X+E) \\; (Y+F)] \\begin{bmatrix} -V_{XY} V_{YY}^{-1} \\\\ -V_{YY} V_{YY}^{-1}\\end{bmatrix} = [(X+E) \\; (Y+F)] \\begin{bmatrix} B\\\\ -I_k\\end{bmatrix} = 0 ,", "7c17cba4629e479d4a8790440de8f8bb": "\\frac{3}{4}V_g + V_e + r({\\frac{1}{4}V_g})", "7c1836025b87af690cbba2aa188c76f0": " SAD(x, y) = \\sum_{i=0}^{T_{\\text{rows}}}\\sum_{j=0}^{T_{\\text{cols}}} {\\text{Diff}(x+i, y+j,i,j)} ", "7c18b4de0c4052084358596ac87abcf6": "R\\rightarrow\\infty", "7c18b4f62c1da2efaa34cbdd11f454be": "X\\to X\\times X", "7c18d435350f36b3632f057305d51ef2": "\\hat{x}(\\omega)", "7c18ec0a876a20470059cf754f0a0704": " \\operatorname{E}_A(U) = \\int_U \\lambda d \\operatorname{E}(\\lambda), ", "7c1912dfe4c8eed90ed6a4714fa9d784": "S_{\\mathrm{f}4}", "7c195e972725f0100bb8974067699e29": " \\mathbf{u}=\\nabla\\mathbf{\\phi} ", "7c197c56ae322bb25e15809d5d5c4605": "\\mathfrak{L}=(X,S,\\{ f_s \\}_{s\\in S})", "7c19a28da3f35a0517eb08c78492e328": "Lg(x)=f(x)", "7c19ab3e2e7454bf32a107e433591364": " \nM^{\\ell_A}_{m_A} \\equiv \\langle \\Phi_0^A | Q^{\\ell_A}_{m_A}| \\Phi_0^A\\rangle\n\\quad\\hbox{and}\\quad\nM^{\\ell_B}_{m_B} \\equiv \\langle \\Phi_0^B | Q^{\\ell_B}_{m_B}| \\Phi_0^B\\rangle .\n", "7c19cc1379da1874f6613d2a78d30cf7": "0 < \\delta < 1", "7c1a1471fe55c0a6d02b56138c0c1d74": "ax^2+1=y^2,", "7c1aa54501828e29c9079c41d3e66385": "\\forall \\vec{y}\\,\\exists x\\,\\!\\,\\varphi(x,\\vec{y})", "7c1b1b18b87d82cef389d2276b391e18": "\\rho_0 = \\sqrt{\\frac{L_0}{C_0}} = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} = 2\\alpha \\frac{h}{e^2}", "7c1b2d4fc69750ae41d623f367a30e30": " t = (\\sqrt{2nd})", "7c1b7a4a1fe01a1fa2204d2c658d508f": "D=\\sum_{j=1}^{n}e_{j}\\frac{\\partial}{\\partial x_{j}}", "7c1ba170eea43535986016b2b83b9c5c": "\\frac{\\left(ax+by+c\\right)^2}{{a}^{2}+{b}^{2}}=\\left(x-u\\right)^2+\\left(y-v\\right)^2 \\,", "7c1ba8065800981b2510fd46617b70dc": "x*(1,-1)", "7c1ba8a52078ecffa9bde310a3a98464": "G \\cong (\\mathbb{Z}_n, +_n)", "7c1be8cf830e9e0cad9d5b701f086863": "t_{2}=\\frac{AB}{c-v}+\\frac{BC+DA}{\\sqrt{c^{2}-v^{2}}}+\\frac{CD}{c+v-\\Delta v}", "7c1c20133c7da868e1e1500988e37067": "(x, y, z) = x r + y p + z s", "7c1c3ada408f6eb57b4716fc7d044f13": "\\sum_{i=1}^n i^c \\in \\Theta(n^{c+1})", "7c1c3fa970d7f8cb7b36979b66deee30": "do(a',s')", "7c1c4cdba5409e9ff8c0370eee292d75": "L(n) = an + b", "7c1c9491ba7c6e8d6d2cfa82e39b22ca": "y=f(x)", "7c1cbfb279653e5cdb5aa6da96a4f971": "s \\leftarrow \\frac{2}{T} \\frac{z - 1}{z + 1}.", "7c1cec0da9304bf1a6ca636e0ed1e2fa": "f(x,p) \\star g(x,p) = \\frac{1}{\\pi^2 \\hbar^2} \\, \\int f(x+x',p+p') \\, g(x+x'',p+p'') \\, \\exp{\\left(\\tfrac{2i}{\\hbar}(x'p''-x''p')\\right)} \\, dx' dp' dx'' dp'' ~.", "7c1d0c1f95b8aecba416a9259dacd73a": "\\sqrt{\\frac{2mV_0^{2}L^{2}}{\\hbar^{2}}}", "7c1d4aaad433c9be58c5fd5bbd1e7de8": "B(N)", "7c1d4bb43d33a27b5f42938058606e31": "2\\uparrow\\uparrow\\uparrow 6", "7c1d5c84fa93ad5ee4e4c21b8247d45e": "\n\\Phi(\\rho, \\theta) =\n\\frac{-Q}{2\\pi\\epsilon} \\ln \\rho^{\\prime} +\n\\left( \\frac{1}{2\\pi\\epsilon} \\right) \\sum_{k=1}^{\\infty} \n\\rho^{k} \\left[ I_{k} \\cos k\\theta + J_{k} \\sin k\\theta \\right] \n", "7c1e3e29bd87aef7138c3704771e5681": "\\hat{\\mathbf{h}}(n)", "7c1e48680db41846835005814b93bab6": "f_{RP}", "7c1ef83fd29ed3c11fd8f873cbea09b5": " |f(b) - f(a)| \\leq \\sum_{k=1}^{N-1} (M + \\epsilon)(x_{k+1} - x_{k}) = M(b-a) + \\epsilon (b-a)", "7c1efd9eac85b70057e3d69436b2ef60": "n = h(e)", "7c1f2f1a5ec7bd6de76b0dbdea2f0528": "a^{\\lambda(n)} \\equiv 1 \\pmod{n},", "7c1f3474ca889cee55a5cfd897d3adc3": "{{\\phi }_{\\gamma (u,\\xi )}}(t)=g(t-u).{{e}^{i\\xi t}}", "7c1f69bdbe0135eb077730d02838fe1e": "\\overline{NE(X)}", "7c1f6a115948d5a2fbe9c61bf77081ad": "\\pi_2 \\pi_3=\\frac{\\beta g \\rho^2 \\Delta T L^3}{\\mu^2} = \\mathrm{Gr}", "7c1fb130c84925d63204b23dab240d39": " (U^t a)(x) = a(\\Phi^{-t}(x)). \\, ", "7c200acc4ec343df45edf2353ae438f0": "\\Delta_M(t)", "7c200c7d794e56701de40aee74bd6ae8": "a_{3,j}={1\\over12}(-y_{j-2} +2y_{j-1} -2y_{j+1} +y_{j+2})", "7c20226917269ffe695111df9d037309": "{\\pi\\over 5}\\ {\\pi\\over 3}\\ {\\pi\\over 2}", "7c2042058430b6ba566cf7cac7ab9d90": "H(s)= \\frac{1}{2^{n-1}\\varepsilon}\\ \\prod_{m=1}^{n} \\frac{1}{(s-s_{pm}^-)}", "7c20adc9f8c45b11fc6c98f521bf995b": " \\mathbb{E} | M_t^{\\tau_k} - M_t | \\to 0; ", "7c20cb4a80a0558952f64211023af86e": " T = 2 \\pi \\sqrt{\\frac{\\ell}{g}}", "7c21168b87c6e5f986ea646e59961bb8": "\n\\begin{align}\ny & = c_1y_1+c_2y_2+\\frac i{2(k^2+4ik-5)}y_3+\\frac i{2(-k^2+4ik+5)}y_4 \\\\[8pt]\n& =c_1y_1+c_2y_2+\\frac{4k\\cos(kx)-(k^2-5)\\sin(kx)}{(k^2+4ik-5)(k^2-4ik-5)} \\\\[8pt]\n& =c_1y_1+c_2y_2+\\frac{4k\\cos(kx)+(5-k^2)\\sin(kx)}{k^4+6k^2+25}.\n\\end{align}\n", "7c211e2249d9d3f784b0cd1ea76cc31a": "\\Delta E_d", "7c214886f7d2e2740905af721bda5f28": "\\ \\beta = \\frac{v}{c}", "7c21b676ca3e799def708a14853f1a27": "F_D = 2 \\times F_T \\times \\left (\\frac {V_R}{C} \\right)", "7c21d4c24e0489333776a4f1f4347357": "X_n Y_n\\xrightarrow{p}\\ XY", "7c21dba61f5fa412ca6fed9af68d89d8": "a b = \\frac{1}{4}(P^2 - D) = Q\\, ,", "7c21f4ee7531a3e722d6ed2d6367b862": " (x_0,t_0) ", "7c221c04ede7f5fd20faea73ec6f696e": "r_\\mathrm{sh} =\\frac{2GM}{c^2} \\approx 2.95\\, \\frac{M}{M_\\mathrm{Sun}}~\\mathrm{km,}", "7c223eeac49c67705a68e69401dd3432": "\\mathbf{u} \\oplus (\\mathbf{v} \\oplus \\mathbf{w}) = (\\mathbf{u} \\oplus \\mathbf{v})\\oplus \\mathrm{gyr}[\\mathbf{u},\\mathbf{v}]\\mathbf{w}", "7c225e62e8098acdb5ee05d84c0d9a9b": "0'", "7c228e9a5632d6e9bbd980a4fa527ec7": "\\frac{i}{p^2 - m_0^2 + i \\varepsilon} \\rightarrow \\frac{i Z}{p^2 - m^2 + i \\varepsilon}", "7c22b67b6b201da080e2f17312b7140a": "\n[x,p ] ={\\rm i}\n\\,", "7c22d935d6414c2676408e146eb7dda3": "f_Y(y) = \\sum_{i} f_X(g_{i}^{-1}(y)) \\left| \\frac{d g_{i}^{-1}(y)}{d y} \\right|. ", "7c22f584ef0d3f7b9d994e69d69abef8": "\\scriptstyle (R+j \\omega L)", "7c22fc597d2af5a39189a2e30b478a19": "F_k=H_1\\ast\\dots H_m\\ast U", "7c2325e86ed5f8763c11fad00f4115e6": "\\dot{y} = L_{f}h(x) + L_{g}h(x)u", "7c23616b34b9c873f34f7e80e4f81525": "f(\\alpha)-f(\\tfrac{p}{q}) = (\\alpha - \\frac{p}{q}) \\cdot f'(x_0)", "7c2394fb80b88f3a1d7140544a9b8bda": "\\textstyle{(AB)_{ik} = \\sum_j a_{ij}b_{jk}}", "7c239f16013b0ee6b59eee8ffdc3824d": " F = 1 - \\alpha ", "7c242ff3cc1fb6aff00a446ddba4cad5": "\\sum_n\\frac{1-\\operatorname{Im} z_n}{|z_n|^2}<\\infty", "7c24891fc98ba2fd6e2fc3bce1c94797": "k_{TOF}\\,", "7c2546b9f32af7a7c6988b222a4bc40d": "a + ar + a r^2 + a r^3 + \\cdots + a r^{n-1} = \\sum_{k=0}^{n-1} ar^k= a \\, \\frac{1-r^{n}}{1-r},", "7c25c4fb3797c4c7699a7daacd23f4fb": " E\\left[\\sum_{x\\in {N}}f(x)\\right]=\\lambda \\int_{\\textbf{R}^d} f(x)dx ", "7c25e92571d681d10cdd96b529183cfe": "\\textstyle m, n \\in \\mathbb{N}", "7c2615e52a28857448efff8a44df06b1": "\\mu_{ex}", "7c262d4e84f4db0bd34a44d3388625ed": "D_\\mathrm{min} = D_\\mathrm{maj} - 2\\cdot\\frac58\\cdot H = D_\\mathrm{maj} - \\frac{ 5 {\\sqrt 3}}{8}\\cdot P = D_\\mathrm{maj} - 1.082532 \\times P", "7c26cfc3ebe54089fe3c3444ee684026": "J^{\\mu}(x)", "7c26def8d5d28613b13e64e62e6155a7": "\\pi = Y^{S} - \\frac{w}{p} \\cdot L^{D} = Y^{S} - \\omega \\cdot L^{D}", "7c271ec345e28d65364e50eee22853d8": "\\displaystyle Z(s) = \\int_M Z(P,P,s)dP", "7c272110e116a6629f790523029d14c9": "2 \\, f(x) = 1 + \\tanh \\left( \\frac{x}{2} \\right).", "7c2725ecc29d37841255d4b129c2313f": "\\{a, c\\}", "7c2738498564c910b7cc64c7c3ef1294": "X_1 = x_1", "7c277ec67f76e7a8e08fef3b4c57f737": "\\{P_1,P_2,P_5\\}", "7c27aba05018eb26d9a974bc5f9d9713": "\\left\\vert \\Phi_{n}^{+}\\right\\rangle", "7c27e2bf9f73339cdb46ca92647ccd0d": "u \\rightarrow S(\\Lambda)u = e^{i\\pi(\\omega_{\\mu\\nu}M^{\\mu\\nu})}u;\\quad u^\\alpha \\rightarrow [e^{\\omega_{\\mu\\nu}\\sigma^{{\\mu\\nu}}}]^\\alpha{}_\\beta u^\\beta.", "7c2823f9fe3e7bc1070408661708d960": "\\bold{v}_1\\times\\left( \\bold{v}_2\\times\\bold{v}_3 \\right)", "7c28e19eeeb23d5c6bb25b3a24cf1918": "{ f=\\frac{[{sm}_{fu}-m_{ox}]-{[{sm}_{fu}-m_{ox}]}_0}{{[{sm}_{fu}-m_{ox}]}_1-{[{sm}_{fu}-m_{ox}]}_0}}", "7c294fd7a65fc3a29035f29a12d6e855": "M(x) = P(x-L)", "7c2953ee90461010efd72cf84145a7cd": "\\frac{d^2T}{dx^2}=\\frac{hP}{kA_c}\\left(T-T_\\infty\\right).", "7c2975f98ccc16cb91be11508c8cc862": "\\rho_2 \\otimes I - \\rho \\geq 0", "7c29c7278a1229429569cf917cb97e43": "\\mathbf{F} = m \\mathbf{a} \\, .", "7c29d8f183b8405eb4861d5c907f2b31": "l\\equiv x", "7c2a222eb88ed0e73b628542698dda34": "\\begin{align} 2\\cdot R_*\n & = \\frac{(142\\cdot 1.08\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 33\\cdot R_{\\bigodot}\n\\end{align}", "7c2a28eecf60bbaac7907ad78d176120": "f(x,i,j)", "7c2a89d0eecaf38bcf1451084e19282d": "E \\approx m_0 c^2 + \\frac{1}{2} m_0 v^2 . ", "7c2ac1b18a0a19aacfa0d59f126c187f": "\\sum 1/L_j = \\infty", "7c2b34db0c811c9cba6b0c2825810af4": "A_e", "7c2bbeca1d3c58102957bd9a4c3ef6ef": " P(E) \\leq \\frac{1}{M^\\rho} \\left ( \\sum_i P(X_1^n(i))^{\\frac{1}{1+ \\rho}} \\right ) ^ {1+ \\rho} \\, .", "7c2bf28680d7e75c9821614d6969a062": " \\mathbf{J}= \\left[ \\begin{array}{rrrr} I & 0 & 0 & 0 \\\\ 0 & c & s & 0 \\\\ 0 & -s & c & 0 \\\\ 0 & 0 & 0 & I \\\\ \\end{array} \\right],", "7c2c27d31dd888dce16daeeae3359937": "A \\cup B = U\\,\\!", "7c2d619268d1b926d941178eb8ec0ba0": "\\hat{a}_i^{(\\eta)} = \\frac{x_{i+}}{\\sum_j \\delta_{ij}\\hat{b}_j^{(\\eta-1)}},", "7c2d9aca553c6ff2ce50e2e03d95d234": "\\boldsymbol{r_i}", "7c2daf7f198130579f0afdac40c30104": "\\alpha = \\omega^{\\beta_1} c_1 + \\cdots + \\omega^{\\beta_k}c_k", "7c2dc0d3c244eaab0cc0361bbe772fb1": "15=\\binom{6}{2}", "7c2dc27c93ee1cc599739295c8c2d347": "T=2\\pi\\sqrt{r^3\\over{\\mu}}", "7c2dc6260b49730091d58d8442653e52": "1\\leq m,n\\leq k", "7c2dd815078ea38e1d5e6161e52f5bff": "\\text{DOR} = \\frac{\\text{sensitivity}\\times\\text{specificity}}{\\left(1-\\text{sensitivity}\\right)\\times\\left(1-\\text{specificity}\\right)}", "7c2de7fbe3f0ee3fe6f9f49d08843d0a": "\n\\begin{align}\np_H(x|\\boldsymbol{\\chi},\\nu) &= {\\displaystyle \\int\\limits_\\boldsymbol{\\eta} p_F(x|\\boldsymbol{\\eta}) p_G(\\boldsymbol{\\eta}|\\boldsymbol{\\chi},\\nu) \\,\\operatorname{d}\\boldsymbol{\\eta}} \\\\\n &= {\\displaystyle \\int\\limits_\\boldsymbol{\\eta} h(x)g(\\boldsymbol{\\eta})e^{\\boldsymbol{\\eta}^{\\rm T}\\mathbf{T}(x)} f(\\boldsymbol{\\chi},\\nu)g(\\boldsymbol{\\eta})^\\nu e^{\\boldsymbol{\\eta}^{\\rm T}\\boldsymbol{\\chi}} \\,\\operatorname{d}\\boldsymbol{\\eta}} \\\\\n &= {\\displaystyle h(x) f(\\boldsymbol{\\chi},\\nu) \\int\\limits_\\boldsymbol{\\eta} g(\\boldsymbol{\\eta})^{\\nu+1} e^{\\boldsymbol{\\eta}^{\\rm T}(\\boldsymbol{\\chi} + \\mathbf{T}(x))} \\,\\operatorname{d}\\boldsymbol{\\eta}} \\\\\n &= h(x) \\dfrac{f(\\boldsymbol{\\chi},\\nu)}{f(\\boldsymbol{\\chi} + \\mathbf{T}(x), \\nu+1)}\n\\end{align}\n", "7c2e2011f825fedec81c7edbe5eeb914": "\\left(\\prod_{i < n} a_i\\right)+1\\,.", "7c2e5ad23c9764586c99292acc6bf955": "T_w(x)", "7c2e823f084175d708fbabd884aa133c": " P(1) = \\frac{n!}{\\prod_{(i,j)\\in \\lambda}h_{(i,j)}}. ", "7c2ecc35e82e53219fb91fdd0eee53d4": "\\vec v_{B|C}=\\vec v_B", "7c2eedb587b9a61a50fde8f557917efd": "\\ltimes", "7c2f1569d488822546bcf81fa9fcff94": "x_1y_2-x_2y_1,\\,x_1z_2-x_2z_1,\\,y_1z_2-y_2z_1", "7c2f371998b6d5ed1e252d270c4cd854": "H^i(X/W)=\\lim_{\\leftarrow}H^i(X/W_n)", "7c2f74e06fe31b430803174d1d070be1": "\n\\ x(t) = \\sgn(\\sin[t])\n", "7c2f9caa01c06ce335a0a9786f49c51a": "\\ln \\frac{n_1}{n_2} B\\left(\\frac{n_1}{2},\\frac{n_2}{2}\\right) + \\left(1 - \\frac{n_1}{2}\\right) \\psi\\left(\\frac{n_1}{2}\\right) -", "7c2fbf4ad0efb28e4dc4faa945130ff3": " \\operatorname{Var}(X) = \\mathbb{E}[(X - \\mathbb{E}(X) )^2]. ", "7c300c98a7d6346ac470dd4e7f11ab3b": "e^{i \\phi_{k}} = 1", "7c300f81dd3ee10c56aaa1716220348f": "v=m\\cdot B'", "7c3033d873fea1df9c46565495b9744a": "1, a, a^2, \\ldots , a^{n-1}\\in H^*(\\mathbf{P}(E))", "7c30591e513c2e88585c08ca6f1c54d4": "p_{\\operatorname{interp}}(r) = \\operatorname{max}_{\\tilde{r}:\\tilde{r} \\geq r} p(\\tilde{r})", "7c3073a5b2385c7665803114cc4a78e4": "N \\cup T", "7c30a07c6f8c3d2ba69de4aa83f90ac8": "F_{k}", "7c3127fd3e32cb682eb67da213c17c6b": "\\{\\lambda_4, \\lambda_5, x\\}", "7c317d0fe2d88f7004a7922ee5c73500": "\\left(-\\sqrt{7/4},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "7c31badb4a832534d6de6e9f0db59890": "\n \\lambda_1\\cfrac{\\partial{W}}{\\partial \\lambda_1} = 2C_1\\lambda_1^2 + 2C_2\\lambda_1^2(\\lambda_2^2+\\lambda_3^2) ~;~~\n \\lambda_2\\cfrac{\\partial{W}}{\\partial \\lambda_2} = 2C_1\\lambda_2^2 + 2C_2\\lambda_2^2(\\lambda_1^2+\\lambda_3^2) ~;~~\n \\lambda_3\\cfrac{\\partial{W}}{\\partial \\lambda_3} = 2C_1\\lambda_3^2 + 2C_2\\lambda_3^2(\\lambda_1^2+\\lambda_2^2)\n ", "7c31eef3252a13faa1896d877eb8c629": "t, \\tau", "7c32065aaf6562d71a366e235b7328dd": "\n\\begin{align}\nP\\left [ T > \\beta n \\log n \\right ] = P \\left [ \t\\bigcup_i {Z}_i^{\\beta n \\log n} \\right ] \\le n \\cdot P [ {Z}_1^{\\beta n \\log n} ] \\le n^{-\\beta + 1}\n\\end{align}\n", "7c327810b15fd6a22bb38ef2ec5b63a5": " N(t) = \\frac{K}{1+ C K e^{-rt}} ", "7c329faa8aad44601c4c6b3caa8438a2": " F^{\\alpha \\beta} = \\frac {\\partial A^{\\beta}}{\\partial x_{\\alpha}} - \\frac {\\partial A^{\\alpha}}{\\partial x_{\\beta}} \\, ,", "7c32aab82d95eb6cfacf9559b3b1f14f": "A[i+1]", "7c32d768ded0ec28075ea4550b909e0a": " W_M = \\sup \\{ W_s \\, \\colon \\, s \\in [0,1] \\}. ", "7c3301b22ac35ed55815cfdcb2e07c3e": "\n\\begin{pmatrix}\n\\boldsymbol{\\alpha^3}'\\\\\n\\boldsymbol{\\alpha^2\\beta}'\\\\\n\\boldsymbol{\\alpha\\beta^2}'\\\\\n\\boldsymbol{\\beta^3}'\\\\\n\\boldsymbol{\\alpha^2\\gamma}'\\\\\n\\boldsymbol{\\alpha\\beta\\gamma}'\\\\\n\\boldsymbol{\\beta^2\\gamma}'\\\\\n\\boldsymbol{\\alpha\\gamma^2}'\\\\\n\\boldsymbol{\\beta\\gamma^2}'\\\\\n\\boldsymbol{\\gamma^3}'\n\\end{pmatrix}=\\begin{pmatrix}\n1&0&0&0&0&0&0&0&0&0\\\\\n{1\\over 2}&{1\\over 2}&0&0&0&0&0&0&0&0\\\\\n{1\\over 4}&{2\\over 4}&{1\\over 4}&0&0&0&0&0&0&0\\\\\n{1\\over 8}&{3\\over 8}&{3\\over 8}&{1\\over 8}&0&0&0&0&0&0\\\\\n0&0&0&0&1&0&0&0&0&0\\\\\n0&0&0&0&{1\\over 2}&{1\\over 2}&0&0&0&0\\\\\n0&0&0&0&{1\\over 4}&{2\\over 4}&{1\\over 4}&0&0&0\\\\\n0&0&0&0&0&0&0&1&0&0\\\\\n0&0&0&0&0&0&0&{1\\over 2}&{1\\over 2}&0\\\\\n0&0&0&0&0&0&0&0&0&1\n\\end{pmatrix}\\cdot\\begin{pmatrix}\n\\boldsymbol{\\alpha^3}\\\\\n\\boldsymbol{\\alpha^2\\beta}\\\\\n\\boldsymbol{\\alpha\\beta^2}\\\\\n\\boldsymbol{\\beta^3}\\\\\n\\boldsymbol{\\alpha^2\\gamma}\\\\\n\\boldsymbol{\\alpha\\beta\\gamma}\\\\\n\\boldsymbol{\\beta^2\\gamma}\\\\\n\\boldsymbol{\\alpha\\gamma^2}\\\\\n\\boldsymbol{\\beta\\gamma^2}\\\\\n\\boldsymbol{\\gamma^3}\n\\end{pmatrix}", "7c330df7a1cbfab5c41a3d81e78481cb": "\\alpha < \\varepsilon_{\\Omega+1}", "7c334d12bd5681db512b6cdbe03b3e68": "e = h^k m \\,", "7c3380f52ff60d1facdec374c31db243": "\\Delta t = k_\\mathrm{DM} \\times \\mathrm{DM} \\times \\left( \\frac{1}{\\nu_{\\mathrm{lo}}^2} - \\frac{1}{\\nu_{\\mathrm{hi}}^2} \\right)", "7c33a76c03aaa44762af08272b5a49de": "\\varnothing \\to \\varnothing", "7c34b4bd660a49bfcb7d5ba5c0390831": "\\displaystyle{\\Phi(g)=PU(g)P,}", "7c34c5ad468e90b56a13d1113f29e739": " \\left\\{ D_{j,k} |\\phi \\rangle \\right\\}_{j,k=1}^d ", "7c34d38fbf13d627b7b288d2fb4ef1ac": "\\tbinom nn = \\tfrac{n!}{n!0!} = 1", "7c34ec78f99f1e0d7ec2ac2f0aa991c6": "\\alpha = 2 \\arctan \\frac {d} {2 S_2}", "7c350c0dfd7d8403cbced4bfd6ea8721": " \\phi = {{e_w} \\over {{e^*}_w}} \\times 100% ", "7c352675f238dfac3f19e330cf2ec17c": "\\eta_c = (4/3) \\pi r^3 N / L^3", "7c35347b933cef627040627422623455": "\n\\sigma^*_i = \\frac{g_i(\\sigma^*)}{\\sum_{a \\in A_i} g_i(\\sigma^*)(a)}\n\\Rightarrow\n\\sigma^*_i = \\frac{\\sigma^*_i + \\text{Gain}_i(\\sigma^*,\\cdot)}{C}\n\\Rightarrow\nC\\sigma^*_i = \\sigma^*_i + \\text{Gain}_i(\\sigma^*,\\cdot)\n", "7c353cc41726dda599e90b4f1e9d31b9": "S=-\\left(n_\\text{s}-n_\\bar{\\text{s}}\\right); ", "7c3597618f7e00577f46a4a1b3dbb80a": "T^{liq}\\,", "7c35f3f823f3bd43745f2c2818e588df": "X_{\\sigma}", "7c361a4c3ad2a29a55b606a8c145c7ae": "\\textstyle b^{dp} = (b^{dp_1},\\cdots,b^{dp_m})\\in\\mathbb{N}^m,\\ p\\cdot k={\\sum_{i=1}^m{p_i k_i}}\\in\\mathbb{N}_0", "7c365e1144ef4307dfa2d7a3af5b7f25": "\\lambda=e^{i\\alpha}", "7c366557e8ea19574cd4713c152fb5d4": "e := \\det e_\\mu^I", "7c36d7646602d19de86d0e0e169935af": "\\hat{\\vec B_j}", "7c373e5228a932d89bf15edcab2e312b": "x = D/R", "7c3743301bae8b24790fed7040877e16": "\\mathbf{e}_1 \\wedge \\mathbf{e}_2 \\wedge \\mathbf{e}_3", "7c37c0d9b1f41176b00a75d2ae27f935": "(0.8 \\cdot 1 \\cdot 1.78 + 0.1 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.0961) = 1 \\cdot 1.78", "7c3800d927ec5633ef75139a02731ddf": "\\alpha = \\left(\\frac{T_3}{T_1}\\right)\\left(\\frac{1}{r^{\\gamma-1}}\\right)", "7c3880385d6bebb8de0bd308b8ee9ba3": "\\sigma(n)=\\min \\left\\{\\,k \\mid n \\le \\binom{k}{\\lfloor k/2 \\rfloor} \\,\\right\\}", "7c388f2d6afbe8d316114de92ed9a07d": "c_1 T\\mathbf{w}_1 + \\cdots + c_n T\\mathbf{w}_n = 0 \\Leftrightarrow T\\{c_1 \\mathbf{w}_1 + \\cdots + c_n \\mathbf{w}_n\\}=0", "7c3895f52fe42a81d3b8092a6937d4e6": " \\vec y = \\mathbf M \\ \\vec x. ", "7c38cb5613464e7d6827f204f14729b4": " \\Delta d ", "7c38d7a4510cb8040fdae6c4f6c0439a": " \\Delta_2^{\\rm P} = {\\rm P^{NP}} ", "7c391a1fe8ce020ded1cfd65146387bf": " \\hat{J}_x \\,\\!", "7c391aa8a3a5530a957735306343b61a": "\\scriptstyle \\Delta_T", "7c398322f1ad01c3c5841ca7ee735323": "\n \\langle j_1, j_1; j_1, {-j_1} | J, 0\\rangle = (2j_1)! \\sqrt{\\frac{2J+1}{(J+2j_1+1)!(2j_1 - J)!}}.\n", "7c39acbfa2eabfc1b333e50be0d19c08": "Y_{10}^{0}(\\theta,\\varphi)={1\\over 512}\\sqrt{21\\over \\pi}\\cdot(46189\\cos^{10}\\theta-109395\\cos^{8}\\theta+90090\\cos^{6}\\theta-30030\\cos^{4}\\theta+3465\\cos^{2}\\theta-63)", "7c39d4210cc454bccfbaa6e22131795f": "S_i = \\Sigma_{j=1}^i \\; f(y_j)\\,y_j\\,", "7c39e7a57cac86385079c13ebe502d09": "E[v(x)] = \\int_\\Omega \\left(\\frac{1}{2}|\\nabla v|^2 - vf\\right)\\,\\mathrm{d}x", "7c3a15fb3fa958a61cb3798474ee443c": "E(x,y,z)= \\sum_{k=1}^M {(a_k e^{(i \\beta_k z)}+ b_k e^{(-i \\beta_k z)})E_k(x,y)}", "7c3a755a175438e179abbb51c4f2160e": "\\scriptstyle \\frac{p+2q}{p+q}", "7c3afb46a16f336a5c5d313916ce5043": "(A\\land B)\\to C \\vdash (\\Box A\\land\\Box B)\\to\\Box C.", "7c3b2f778b744045cf41f48d01462e1e": " \\text{(4)} \\qquad P V^\\gamma = \\text{constant} = P_1 V_1^\\gamma ", "7c3b380d8f1ab90669cc6f11b1d58929": "\\bar{q} ", "7c3b82a8979a2e256f214d6614901c21": "T(x,y) = (0,y).", "7c3bd6025203612910cfbd31109021bb": " \\frac{\\dot{r}^2}{ \\left( 1+m/r \\right)^4} = E^2 - V", "7c3bdbf01baa008dc1ee542b9da7e5d6": "C_*(M, (f_0, g_0))", "7c3c07086f77e11995d5ef00c9ae43de": "\\psi_2 = A \\sin(kx) + B \\cos(kx)\\quad", "7c3c4f5baa756ea3f3e9c66f19d783ac": "\\begin{align}q^2 &= q_1^2\\\\\n&= \\left[\\left(p + q\\right) \\left(q + r\\right)\\right]^2\\\\\n&= \\left(p + q\\right)^2 \\left(q + r\\right)^2\\\\\n&= p_1 r_1\\\\\n&= p r\\end{align}", "7c3c76f87ed833e45a121dedd0fa2a67": "N(t_{1/2})=\\left(\\frac {1}{2}\\right)N_0", "7c3cb5db3efe80debd440ca88e36eace": "\\sqrt{\\langle \\vec{R^2} \\rangle} = \\sqrt N \\, l = \\sqrt{L \\, l}~", "7c3cc6008a396aed9e0c874f5d559b50": "\\mathrm{AgNO_3 + H_2N\\text{-}C(O)\\text{-}NH_2 \\longrightarrow}", "7c3d5f8761560a9794c1cbb1ab903126": " i\\hbar \\partial_{t_1} U_\\epsilon(t_1,t_2) = H_\\epsilon(t_1) U_\\epsilon(t_1,t_2)", "7c3d890c71b76c51edd8e08382bab2cd": "\\Re(z)", "7c3da18c4a15839e103f789ca74c5afb": "\\begin{align}\n z &= \\frac{1}{n}\\sum_i z_i n_i \\\\\n z' &= \\frac{1}{n'}\\sum_i z_i n'_i\n\\end{align}", "7c3df308fe7e9ac74da0a684f4caa2b0": " 0 \\to P\\times_G \\mathfrak g\\to TP/G \\to TM\\to 0.", "7c3e1ef481fbb6fc751acd2d21e8c33e": "K_1,K_2,\\dots,K_k", "7c3e4d077cb268f195f200e289e63919": "\\lbrace\\mathcal{M},\\mathcal{S}\\rbrace", "7c3e5ea427bdc2045eae691e3c68290f": "\nSSNR (r)\n = \n\\frac{\\displaystyle\\sum_{r_i \\in R}\\left|\\sum_{k_i}{ F_{r_i,k} }\\right|^2}\n{\\displaystyle \\frac{K}{K-1} \\sum_{r_i \\in R}\\sum_{k_i}{ \\left|{ F_{r_i,k} - \\bar{F}_{r_i}}\\right|^2}} -1\n", "7c3e71b62e10986b1b0cac37e4bf1724": "\n\\hat{f}(\\mathbf{x}') = \\mathbf{k}^\\top(\\mathbf{K} + \\lambda n \\mathbf{I})^{-1} \\mathbf{Y},\n", "7c3e78ec97b45ed36b0dc35d9a4792aa": " \\frac{1}{101} x^{101} +C", "7c3eb88194033a80e54fb823b719f37e": "{1 \\over 2}\\ln\\left({{1+\\rho} \\over {1-\\rho}}\\right),", "7c3f12964b1f4cfe99877920a5a72ac3": "\\partial_{u}", "7c3f3cd25ea696d06a51f6769b8d10b1": "\\operatorname{dim} R[x_1, \\dots, x_n] = n.", "7c3f9477f753e0bad5c52baa7ecc716f": "a_n=\\frac{H_{2n+1}-1}{2}", "7c3f96c89b0acc61fd6e9467e5c35b0a": "=\\left ( \\frac{3}{331}\\right ) \\left ( \\frac{5}{331}\\right ) \\left ( \\frac{823}{331}\\right )", "7c3fcb4c1701f3631a238c2eec241693": "\\omega_D \\otimes i^*\\mathcal{O}(-D) = i^*\\omega_X.", "7c404b334b2e0f2a244ae59a67909187": " \\langle x, x \\rangle = q(x)", "7c4073ca34bcc95361750a3f1fddc7a8": "h\\,", "7c408885d78578505416a987ad166755": " E(\\mathbf{r}) = E_o e^{-i ( i \\alpha y + \\beta x ) } = E_o e^{\\alpha y - i \\beta x } ", "7c40ac0adf50bff435059891d4dddbb4": "\\sqrt{\\frac{16}{9}\\times\\frac{4}{3}} \\approx 1.5396 \\approx 13.8:9,", "7c40bf1939c40b4f9a63ddcc932c8ab3": "T\\theta", "7c40d2571d38878acfdc851c58eb05fe": "\\lambda=[\\lambda _1, \\lambda _2,\\dots,\\lambda _n]", "7c410e618c39640ff6d4bb27c14c8491": "\n\\mathbf{M} = \\begin{pmatrix} m_{11} & m_{12} \\\\ m_{21} & m_{22} \\end{pmatrix} =\n\\prod_{i=0}^N \\begin{pmatrix} q_i & 1 \\\\ 1 & 0 \\end{pmatrix} =\n\\begin{pmatrix} q_0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} q_1 & 1 \\\\ 1 & 0 \\end{pmatrix} \\cdots \\begin{pmatrix} q_{N} & 1 \\\\ 1 & 0 \\end{pmatrix}\n", "7c41394c12fb93c886cb756d585af522": "R_O(x)", "7c419b2f3a4fdc9430bc5560b48a4866": " \\lambda f.(p\\ f)\\ (p\\ f) ", "7c41a3ceb27dc25869bb5505a7918186": "\nL \\propto 4\\pi\\left(\\frac{LG}{\\sigma^2}\\right)^2B\n", "7c41d5efa504204bd0f01d35e2f600a8": " 0 < | x - 5 | < \\delta \\ \\Rightarrow \\ | (3x - 3) - 12 | < \\varepsilon . ", "7c42c1558a5bf7e23d0cf936f5f5eac4": "\n\\sinh\\left(2L^{*}\\right)\\sinh\\left(2K\\right)=1\n", "7c42c2797a1a3854066a178ac322a3a3": "> 0", "7c4415ce1d3467d70ee3c5aa9549f231": "\\,N_2", "7c4417ad1af1e22f8718f69908159dc0": "E_{ann}", "7c44261a911ecb833f102bf828deb7db": "13\\,\\!", "7c448bc29ce4067c5d0511430b620883": " \\sum_{j}h_j \\delta O_j + \\sum_{j}O_j\\delta h_j= 0 \\,\\!", "7c44937b97d41fbea185fc20e5fc27a6": "F_Z", "7c44a168ce24eec1f2f623ef018d1327": "I_u", "7c44bf79418c0ca3cd1fbaf0a84ad6d4": "x = T t + U\\,", "7c44c4fd9ee64f79d37dc97e3ceb3c17": "\\alpha\\rightarrow\\infty", "7c45dcc50ba7f9d7b1c1edeeaeaa2a45": "\\scriptstyle f_s", "7c45f5a71358f20cd2225a60cc7f2ef3": "\n\\text{S} = \\frac{ TP + FN } { N }\n", "7c464465201b266ec9db92aa0f7d7c61": "V = \\frac{23}{12}\\sqrt{2}a^3 \\approx 2.710575995a^3.", "7c4658bc4d09c6912a6981b6e027dada": "\\begin{bmatrix}\n\\mathbf{e}_{a_1} \\\\\n\\mathbf{e}_{a_2} \\\\\n\\vdots \\\\\n\\mathbf{e}_{a_j} \\\\\n\\end{bmatrix}", "7c4663f768ec4951877be725c628544f": "\\mathbf{L}^2\\Psi = \\hbar^2{\\ell(\\ell+1)}\\Psi", "7c46d710156c363aed5f6ed4294a8aec": "a\\,\\bmod\\,b \\!", "7c46e7d54c6109d94d8982aa60d87b4a": "A\\;", "7c475e434f563c0e8bdb5dba1b4c9d78": "{}_{\\ 88}^{224}\\mathrm{Ra} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 86}^{220}\\mathrm{Rn}\\ \\mathrm{(3.6\\ d,\\ 0.24\\ MeV)}", "7c477670495296c99a5898d116e275fc": "x = -1", "7c4789c3a3ef41840e259513fb8c0113": "S = k_B \\, \\ln W,", "7c47a0e25759adb5a0ae43f92642ab24": " \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] ", "7c47ab0a5c67b00a2e3b25fa8d7960cd": "\nJ_{k} = \\oint p_{k} dq_{k}\n", "7c47bbd3c691286a05f28860fb85bfd8": "\\textstyle x\\in R^{N}", "7c47c728c30a69d58f878ef51ee6a063": " |\\psi\\rang \\rightarrow \\sum_n |c_n|^2 |\\psi_n\\rang \\lang \\psi_n|", "7c47e0cae777e860fcedf0294ff45abe": "\n\\begin{align}\nf(x,y) &\\approx f(a,b) +(x-a)\\, f_x(a,b) +(y-b)\\, f_y(a,b) \\\\\n&\\quad {} + \\frac{1}{2!}\\left[ (x-a)^2\\,f_{xx}(a,b) + 2(x-a)(y-b)\\,f_{xy}(a,b) +(y-b)^2\\, f_{yy}(a,b) \\right],\n\\end{align}\n", "7c47f9e3315f37047da3dc24cc9dac6d": " \\int { d^4 k \\over \\left ( 2 \\pi \\right ) ^4 } \\; {\\exp \\left ( ik \\left ( x-y \\right) \\right ) = \\delta^4 \\left ( x-y \\right ) } ", "7c483eb9ba8b935b1e267d28cb57aed3": "J=\\pi L_e", "7c48496bb1d6546511dc2fc557c3e1cb": " I_{ LG } = \\frac{ K }{ K' } ", "7c4890c7c6f202be88e945ae25dccf2c": "\\mathcal(Q),", "7c48d22c5c00dfdadb8eb00416d7af32": "\\scriptstyle n \\hat F_n(t)", "7c48db4aa8398d39817bac5798bb54d3": "\\widehat{\\mu}=\\frac{\\sum X_i}{n},\\,", "7c4932307e8110fa66db9b1946e738b0": "(2x)\\frac{dx}{dt}+(2y)\\frac{dy}{dt}=(2h)\\frac{dh}{dt}", "7c49384144eb0a26e3d90dd5804edf11": " N_C = \\frac{\\lambda_C}{\\lambda} N_{A0} \\left ( 1 - e^{-\\lambda t} \\right ).", "7c494b4186fe655ab93db4a4ba1c17b4": "dE^\\prime", "7c49756f5282ac343aa798be0c30c29a": "srg((s+1)\\frac{(s t+\\alpha)}{\\alpha},s(t+1),s-1+t(\\alpha-1),\\alpha(t+1))", "7c4a135bd6bf381cde4af2c5ff8b91c4": "\\|A\\|_{1}= {\\rm Tr}|A|:=\\sum_{k} \\langle (A^*A)^{1/2} \\, e_k, e_k \\rangle ", "7c4a3d32669eaa19094da4c274179e0c": "\\sum_{n=0}^{\\infty}", "7c4a94b007f0fdfa10ce6e0e39719b85": "\\gamma_1,\\ldots,\\gamma_k", "7c4ad178115a836402b175755defe457": " g = 1+(M_{5} +2 M_{6}-M_{3})/8, \\ ", "7c4ae735548fe1d0683e36802a3de0c7": "{\\boldsymbol J}^{\\mathrm T}", "7c4b2eea02d7bb149a62e5ff192e845c": "T_0,T_1", "7c4b3b45f4e5f9c7734d36c7ad083ecd": "\\frac{p-q}{p+q}", "7c4b51371513a738c7d2b2d2284d62ed": " [\\operatorname{E}(R_a) ] = R_f + [\\operatorname{E}(R_m) - R_f] * [\\sigma_{am}] / [ \\sigma_{mm}] ", "7c4b6789e752808426c3280a8137231c": "[X]^{<\\kappa} = \\{Y \\subset X \\; | \\; |Y|<\\kappa\\}", "7c4b8046de0af448844d833f396db8ac": "40+\t24+\t9+\t56+\t41+\t25+\t8+\t57\t=\t\t260", "7c4bd5eebe82b9be1bc35417005f6368": "\n C(x) = \\int_0^x \\frac{dt}{1-t} = \\log \\frac{1}{1-x}.\n", "7c4c8f78fe89912c48e41b34f0c6fff3": "g = \\frac{R(\\frac{\\pi \\times \\mathrm{rpm}}{30})^2}{9.81}", "7c4c935343c738e216f64bf5b6047959": " accdist = accdist_{prev} + volume \\times CLV ", "7c4cdc9d32f1ba8d869b633eaba3edbf": "\n\\begin{align}\nI(x) &= \\int_0^x{\\rm blanc}(x)\\,dx,\\\\\nI(x) &=\\begin{cases}\n1/2+I(x-1) & \\text{if }x \\geq 1\\\\\n1/2-I(1-x) & \\text{if }1/2 < x < 1 \\\\\nI(2x)/4+x^2/2 & \\text{if } 0 \\leq x \\leq 1/2 \\\\\n-I(-x) & \\text{if } x < 0\n\\end{cases} \\\\\n\n\\int_a^b{\\rm blanc}(x)\\,dx &= I(b) - I(a).\n\\end{align}", "7c4d095b0ca45a7d24a022a8c7b7a29d": "V_i = \\mbox{Spec} \\; B_i", "7c4d1e634d2b5aea684295dbdcf7e24e": "\\frac{1}{1-V_\\beta/V}dV_\\beta = dV_\\beta^e \\,\\!", "7c4d996e4623f4e2bda5f09d8d5bc644": " \\boldsymbol{\\mathsf{A}} \\cdot \\boldsymbol{\\mathsf{U}} = 0 \\,\\!", "7c4db5f2d92bbe538c340bf30729041e": "\n G_1 = \\frac{k_3}{k_2^{3/2}} = \\frac{\\sqrt{n\\,(n-1)}}{n-2}\\; g_1,\n ", "7c4dd23b6324f5ee4c0c2cdb40d3f7a1": "\\begin{pmatrix}0 & 0 & 1 \\\\0 & 0 & 1 \\\\1 & 1 & 0 \\\\\\end{pmatrix}", "7c4df11705f3628aa8e0f0413b42a470": "2^s=2", "7c4e0857d1527c2c8e8cb62b8171c840": "\\ m", "7c4e0d01cfd17e8450b6c64b2162535f": "A= \\pi \\times 230^2 = 52,900 \\pi \\approx 166,190.25 ", "7c4e1cdf406fb0e0d1893e4163175977": "\\simeq 28.4652\\,t^2.", "7c4e271ae25ad92b85c9a0bbf4855717": "\\mathbf{j}_s =-\\frac{n_se_s^2}{mc}\\mathbf{A}, ", "7c4e35c6b2894ed705ff8270a51ff97e": "\\lim_{x \\to \\infty} \\frac{x^{1/2}+x^{-1/2}}{x^{1/2}-x^{-1/2}} = \\lim_{y \\to \\infty} \\frac{y+y^{-1}}{y-y^{-1}} = \\lim_{y \\to \\infty} \\frac{1-y^{-2}}{1+y^{-2}} = \\frac{1}{1} = 1.", "7c4e3f55a55dff3921a915f43a87ce7f": "EX\\to X", "7c4e5dd05567602f395f4608c66bc6ce": "\\sigma^2_f= \\sum_i^n \\sum_j^n a_i \\Sigma^x_{ij} a_j= \\mathbf{a \\Sigma^x a^t}", "7c4e5e3cd4d059ec83f27121ba19fac0": "p \\equiv 3 \\pmod 4", "7c4eb00aaa92b7d1f554d023f9333c18": "\\left\\{{n\\atop x}\\right\\}=p_x(n)=0 \\quad\\hbox{whenever either } n>0=x \\hbox{ or }0\\leq nu) = O(u^{-k}),", "7c50151e1e8d74dc7136743d9cc17a59": "x_{min}=\\max(0,n-m_2)", "7c502a5325105d06fdb1c6908c8faacc": "\\alpha_n / H", "7c505c34a5c5657d4329e0966249d0dc": "\n\\Delta G = \\Delta H - T \\Delta S = \\Delta G^0 + kT \\ln Q, \\,\n", "7c5081abe6c2100f0e44396b6ac51661": "H_{0}", "7c508c58d4429eac5e0dbd19a7f2dffd": "\\begin{align}\ndN_\\mathrm{coll} & = \\left(\\frac{\\partial f}{\\partial t} \\right)_\\mathrm{coll}\\Delta td^3\\mathbf{r} d^3\\mathbf{p} \\\\\n& = f \\left (\\mathbf{r}+\\frac{\\mathbf{p}}{m}\\Delta t,\\mathbf{p} + \\mathbf{F}\\Delta t,t+\\Delta t \\right)d^3\\mathbf{r}d^3\\mathbf{p}\n- f(\\mathbf{r},\\mathbf{p},t)d^3\\mathbf{r}d^3\\mathbf{p} \\\\\n& = \\Delta f d^3\\mathbf{r}d^3\\mathbf{p} \n\\end{align}", "7c51a53d44d3ed39b06ddfca5436365f": "\\dot{f}\\le 0", "7c51ab09e17f17d24b7bb96d1434398c": "r_\\mathrm{D} = \\frac{1}{di_\\mathrm{D}/dv_\\mathrm{D}} \\approx \\frac{nV_\\mathrm{th}}{i_\\mathrm{D}} , ", "7c51ca68efb480751ef3322e1cb7b2e3": " g(n) ", "7c51dee4cd461ebcbbecf8fcee8d236a": "\\alpha >0", "7c52084bb327a25b3a754c1c2414d381": "\n\\gamma = \\lim_{\\varepsilon\\to 0} \\frac {\\ln f(\\varepsilon)} {\\ln \\varepsilon}\n", "7c52157af042ba2f5811be423e0bdba7": "u^{}_{\\alpha}= \\frac{ (1 - \\alpha ) \\frac{ u_0 }{ z_0 } + \\alpha \\frac{ u_1 }{ z_1 } }{ (1 - \\alpha ) \\frac{ 1 }{ z_0 } + \\alpha \\frac{ 1 }{ z_1 } }", "7c5237d48fa208a2e2165ccdfaf943a1": "\\mu_M (A) = \\frac{1}{M} \\# \\left\\{ \\lambda_j \\in A \\right\\}, \\quad A \\subset \\mathbb{R}. ", "7c5281a07aaa6c07ae1cfab1ae09fa0d": "w^6 + qw^3 - \\frac{p^3}{27} = 0", "7c52b8db87bad476b2b44df0b67be1d6": "\nP(\\mathbf{x},t\\rightarrow\\infty\\mid x_0) = \\Pi_{j=-M}^M \\delta(x_j-x_{0,j}). \n", "7c52d6d99ef1c59976a2fda2788b5020": "\\left(-\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "7c52e57a098f0d8eebd3b4fc1f8188d3": "t = r\\sqrt{\\frac{n-2}{1 - r^2}}", "7c53780f13ab17a5589183052ecaf2b0": "f(t) = \\sum_{n=-\\infty}^{\\infty}x_n{\\sin \\pi(2Wt-n) \\over \\pi(2Wt-n)}.\\,", "7c539dce932d519d3f149ad6c33fd714": "Tr[\\hat{B}(\\xi )\\hat{B}(\\xi ^{\\prime })] = (2\\pi \\hbar )^{n}\\delta^{2n}(\\xi -\\xi ^{\\prime }).", "7c53ac39c6ccb769f9a425b3931a1c37": "wp(S,.)", "7c53b2e004a00cb56b9e4f503c587e79": "n_1 + N_1 \\cdot (n_2 + N_2 \\cdot (n_3 + N_3 \\cdot (\\cdots + N_{d-1} n_d)\\cdots)))\n= \\sum_{k=1}^d \\left( \\prod_{\\ell=1}^{k-1} N_\\ell \\right) n_k\n", "7c53f9b59f66724eefac452349e18065": "\\Pi_p \\subset T_p M", "7c5455611a89db5e2cac1d9450678369": "K_M \\subseteq R(0) \\; \\mathrm{and} \\; R(0) \\cap -\\mathrm{int}K_M = \\emptyset", "7c5456396236123a72fd0ac753d3ee44": "p_\\infty", "7c5495663b86d209aef9b95d475112e2": " F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \\times 6700417. \\; ", "7c54cd2b8a696a045b3bd1a25f205849": "E_{E,a,d}", "7c554265cd49bccc458d247023c3aa33": "{\\mathbf{\\lambda }}_{\\mathbf{0}}^{\\mathtt{KED}}", "7c557d7650512105d88da4b909930473": " A \\in M(m,n;\\mathbb{K}) ", "7c5583da2b8b688acc7cc43f3114386b": "N \\oplus \\bigoplus_{i=1}^d K'_{i(b_i+1)}=rol(\\bigoplus_{i=1}^d K_{i(b_i+1)},u) ", "7c55be08503e66efd67e490c214e1660": "\\scriptstyle\\frac{1\\,\\,2\\,\\,4}{2\\,\\,3\\,\\,5}", "7c55c064b7bbe9bfa621c70d05148b57": "\\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}}", "7c55c95586c65f7c453ea503b851a9d0": "\\frac{1}{b}.", "7c563ec7367ca1b0c0a3b7fa26890891": "\\int_0^1 x^2\\ln(1+x)\\,dx=\\frac{2}{3}\\ln 2-\\frac{5}{18}.", "7c5674d1f43a15db39666ba050fb1fd1": "y_0.", "7c56c2c9aadb5d4a79e701521830cee8": "\n\\gamma_2 = \\dots = \\gamma_p = 0.\n", "7c56ec21b4bd3d4e924d79f03ea221e8": "Clipped", "7c570747607aced3dc7c3925404b8fcc": "\\left(\\begin{array}{cccc} 5 & 2 & 6 & 0\\\\ 4 & 7 & 3 & 8\\\\ 5 & 9 & 0 & 4\\\\ 3 & 1 & 0 & -3\\\\ 9 & 0 & 2 & 1\\end{array}\\right)", "7c579fb2ffb6482dd5746a29c6395975": "\\nabla^{0,1}=\\bar\\partial", "7c582b34a3cd228c9cf41d64d26a79e3": "\\sum_{jk}{e_{jk}} = 1\\,", "7c5850a360c33c91a6b125bfd0c05b64": "\\displaystyle\\frac{\n\\Gamma, \\Gamma' \\vdash e_1:\\tau_1\\quad\\dots\\quad\\Gamma, \\Gamma' \\vdash e_n:\\tau_n\\quad\\Gamma, \\Gamma'' \\vdash e:\\tau\n}{\n\\Gamma\\ \\vdash\\ \\mathtt{rec}\\ v_1 = e_1\\ \\mathtt{and}\\ \\dots\\ \\mathtt{and}\\ v_n = e_n\\ \\mathtt{in}\\ e:\\tau\n}\\quad[\\mathtt{Rec}]", "7c58535e573e641f1bb587182dfa1fa8": "A_{\\infty}", "7c58706b5ea16d2a0e9dc57be28c14d2": "\\frac{ \\sqrt{2 + \\sqrt{2}} } {2}", "7c5891a187c670c267185cb911b89e19": "\\pi:EG\\longrightarrow BG", "7c58b0b9422bed7fdedbe3eca51a7456": "\nd_1 = d_3 = \\frac{1}{2-2^{1/3}},\\ \\ d_2 = -\\frac{2^{1/3}}{2-2^{1/3}},\\ \\ d_4 = 0.\n", "7c58b70a6ccef0f06960a760ca1a28c7": " \\lambda=1 ", "7c58deeed801f493f251da921cadd107": "Z=\\frac{X-\\mu}{\\sigma}", "7c58e955bafcdd5123c11e9036500dfb": "{\\tilde{B}}_4", "7c58f71ad9e0e421904d134ea6e98361": "\\mathcal{H}_\\psi", "7c58f98643968c5bfdde701bfcf77acf": "+1/2", "7c590fc7c38f71154e6ed9d110df17e8": "{O}(n^3)", "7c5931001a3a1f0fd228ae4e8242ac67": "[Rp_!k_X,k]\\cong H^0(\\mathrm{Hom}^{\\bullet}(\\Gamma_c(X;I^{\\bullet}_X),k))=H^0_c(X;k_X)^{\\vee}.", "7c5940437ba5fbed50581372af0949d0": "\\left\\{\\mathcal{C}^{i}\\right\\}_{i\\in\\mathbb{N}}", "7c5942000463d8e2c1c86cf970ead5ad": "|\\langle x_i, x_j \\rangle|", "7c59a183ecd81916ca8c2ac4ffca4588": "K = \\{ x \\in X : | \\phi(x) | \\leq a \\}. ", "7c5a1f5b0764cc449bded38485335f69": "f = 1 - {2\\over N+1} = {N-1\\over N+1}", "7c5a287d36cb2594aaffaf5d9dc3a56c": "\\frac{\\partial E(\\omega)}{\\partial z}=-\\frac{i\\omega}{n_{\\omega}c}d_{\\text{eff}}^*E(2\\omega)E^*(\\omega)e^{-i\\Delta k z}", "7c5a40cea7ce36076e7032815ccce04e": "t \\in Q", "7c5a4bc6b6fcae9a58aa32d7c32d41eb": "D(n, m)", "7c5a5b4425fbd2fa0a033e5ea544d9cd": "\\theta (x) = \\frac{1}{EI} \\int M(x)\\, dx ", "7c5a7871b23d46dae4e22b06db57a826": "\\alpha_i \\boldsymbol{\\in}", "7c5accee088fafd265660d6a099c4b48": "Y = S+C+B^\\prime+T+B ", "7c5ad3a4c540093721b2249a98c1c382": "m(r)=4\\pi\\alpha^3\\rho_c\\phi(\\xi)", "7c5b1394602f0c1167092039c164a0a7": "\\int_a^b f(x) \\, dx\\approx\n\\frac{h}{3}\\bigg[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\\cdots+4f(x_{n-1})+f(x_n)\\bigg].", "7c5b1e9c767b4fd80755a4128bfd8856": " \\prod_{k \\in I} ( 1 - \\mathbf{1}_{A_k}) = \\mathbf{1}_{X - \\bigcup_{k} A_k} = 1 - \\mathbf{1}_{\\bigcup_{k} A_k}.", "7c5b3702051c3a1319c8fd3ba426116b": "\nP(Spam \\land W_0 \\land \\ldots\n \\land W_{N-1}) = P(Spam)\\prod_{n=0}^{N-1}[P(W_n|Spam)]\n", "7c5c149ed6da2a98ec23cac8c6ce1fc4": "\n\\eta(s_n) = (1-\\frac{2}{2^{s_n}})\\zeta(s_n)\n = \\frac{1-\\frac{2}{2^{s_n}}}{1-\\frac{3}{3^{s_n}}} \\lambda(s_n) = 0.\n", "7c5c4abd05e79bb806e442941ca03f5e": " N=285509 ", "7c5c4f4abe5d26fd457c65536b7b8cb8": "a_1, a_2", "7c5c90377f2cb72d3aad1dfbfeaf9e74": "(n-1) + (n-2) + \\cdots + 1 = n(n-1)/2", "7c5cc750c838094d161562e10d191dd3": "\\omega^{ry}", "7c5d1e5ec4199f6ed8135981ddded06c": " {\\nabla \\cdot \\left(\\rho_{o}\\mathbf u\\right) = 0} ", "7c5d23c76fa4362ca6937e7a93921e88": "A\\,\\triangle\\,B = (A \\cup B) \\smallsetminus (A \\cap B),", "7c5d3244436edc310ab848366005b705": " g(g^{-1}P) = gL ", "7c5db8b1d361a12ab31936d31b71ecd0": "\\cos \\left(x+y\\right)=\\cos x \\cos y - \\sin x \\sin y, \\,", "7c5dc1c2a419b035cc505fd0e9028553": "1 \\leq k < n", "7c5de39a9aaffd681b38c96c2bbbaa61": " \\text{If} \\, \\oint_K \\! \\kappa(s) \\, \\operatorname{d}s \\le 4 \\pi \\ \\text{then} \\ K \\ \\text{is an unknot}. ", "7c5e155fdf44dc3972c9400e3ca32678": "D=\\frac{\\partial}{\\partial \\theta}-i\\theta\\frac{\\partial}{\\partial t}\\quad \\text{and} \\quad D^\\dagger=\\frac{\\partial}{\\partial \\theta^*}-i\\theta\\frac{\\partial}{\\partial t}", "7c5e35ac71e0fa7b3fc1b1f0e58d9de4": "D= x_{11}-p_1 q_1", "7c5e3832b23b29592c8f4c7b313e62d1": "e(H') = \\bigcup_{h \\in H'} e(\\{h\\})", "7c5e64689968c9048db88109b2dd30b5": "A = \\Delta y", "7c5e81c849f2c4121bf4de5081f9a67b": "\\ M_{heel} = - D_{heel} \\times drag ", "7c5ef869331d29e8c0490daee18e36cd": "A_1=\\{30.02,\\ 29.99,\\ 30.11,\\ 29.97,\\ 30.01,\\ 29.99\\}", "7c5f20739c0fb9c75e72110880c3a1ee": "\\mathbf{x} = \\left[\\begin{array}{c}x\\\\ y\\end{array}\\right]", "7c5f89f09ec0cad46e7d9a3c796716fb": "\\lambda_{23}=11.6198", "7c5f8e64ae8a4e29baafe65daa071c04": " m \\ell ^2 \\ddot \\theta= m g \\ell \\sin \\theta\\,\\!", "7c600d6d023a24b17faa265325a7c85c": "\\int_a^b\\! e^{M f(x)}\\, dx\\approx \\sqrt{\\frac{2\\pi}{M|f''(x_0)|}}e^{M f(x_0)} \\text { as } M\\to\\infty. \\,", "7c60711e61f8fba700bc214ff4dcf7de": "-x-y\\omega", "7c60ceaa174f3269a4b42da03985ee32": "R=A^{1/3}r_0", "7c60e53edbe1cba71eb6d7f98e33c6ae": "\\kappa(\\sigma,~\\tau)", "7c61122ee108230c18c75ca8b4d9e4e1": "\\lambda_{\\hat{x}}", "7c6127516c6f3d3fc483b5075cdaad01": " P_{max} ", "7c612d6bf28f0f89c0fb433dd08125e2": "d\\lambda > 0", "7c6199c41526caa1419dfd1adb37bdf0": "F(x) = \\frac{2}{\\pi}\\arcsin\\left(\\sqrt x \\right)", "7c619f0d97d7f54f145af0682e0e7df0": "F = \\frac{dW}{ds} \\propto \\text{Mileage}", "7c61bd35808a2017266d0c064be6c013": "(Total)\\ Dependency\\ ratio = \\frac{(number\\ of\\ people\\ aged\\ 0-14\\ and\\ those\\ aged\\ 65\\ and\\ over)} {number\\ of\\ people\\ aged\\ 15-64} \\times 100 ", "7c624d4543c4f68f19de389ec8f30078": " \\psi_n(x) = \\frac{1}{\\sqrt{2^n\\,n!}} \\cdot \\left(\\frac{m\\omega}{\\pi \\hbar}\\right)^{1/4} \\cdot e^{\n- \\frac{m\\omega x^2}{2 \\hbar}} \\cdot H_n\\left(\\sqrt{\\frac{m\\omega}{\\hbar}} x \\right), \\qquad n = 0,1,2,\\ldots. ", "7c6266a8592c456187613df3e936492a": " \\mathrm d\\theta + \\omega\\wedge\\theta", "7c6270537cb2cf1c86fd46bbc6975dd3": "r=1", "7c62810d6d5f721e3813815d9eddb6fc": " \\mathbf u \\, \\mathbf u = \\mathbf u \\cdot \\mathbf u + \\mathbf u \\wedge \\mathbf u = \\mathbf u \\cdot \\mathbf u", "7c62e4dd22b48d7eb6a1385ef12b0301": "g_m = \\frac{i_\\mathrm{c}}{v_\\mathrm{be}}\\Bigg |_{v_\\mathrm{ce}=0} = \\frac {I_\\mathrm{C}}{ V_\\mathrm{T} } ", "7c62e947cf9b9e2039875ea7e45d12f0": "\\sum_{i}(U_i\\cap V) = \\left(\\sum_{i}U_i\\right) \\cap V.", "7c633af94830b16a16be5da8815177b7": "\\{92, 19, \\mathbf{101}, 58, \\quad 91, 26, 78, 10, 13, \\quad \\mathbf{101}, 86, 85, 15, 89, 89, 25, \\mathbf{2}, 41\\} \\qquad (N = 18)", "7c6393ecc63bce2a32f599b33376c66c": "X_1,\\dots,X_N", "7c63dafacf9c7ea037ff39bda9978aea": "\\operatorname{dn}(u,k)", "7c63f1c19e54004f595000ad2ccb2cc7": "\\frac{R^2 + Z_o^2\\Omega^2}{(1+R)^2+Z_o^2\\Omega^2}", "7c64315401dae2da4af1425bcb7e814e": "\\Bigg(\\frac{q^*}{p}\\Bigg)_4= 1 \\mbox{ if and only if }\n\\begin{cases}\n b\\equiv 0 \\pmod{q}; & \\mbox{ or } \\\\\n a\\equiv 0 \\pmod{q} \\mbox{ and } \\left(\\frac{2}{q}\\right)=1; & \\mbox{ or } \\\\\n a \\equiv \\mu b,\\;\\; \\mu^2+1 \\equiv \\lambda^2 \\pmod{q}\\mbox{, and }\\left(\\frac{\\lambda(\\lambda+1)}{q}\\right)=1.\n\\end{cases}\n", "7c64370eb34999b7c6f84e0e43dc1064": "L=\\bigcup [L]_D", "7c645258320ee181761a1ddb60205192": "u_1{\\Delta}^{*}=u_2{\\Delta}^{*}", "7c6489118717040e8188c1374f9cd021": "w^+_r=w^{\\rm eq}_r \\exp \\left(\\sum_i \\frac{\\alpha_{ri}(\\mu_i-\\mu^{\\rm eq}_i)}{RT}\\right); \\;\\; w^-_r=w^{\\rm eq}_r \\exp \\left(\\sum_i \\frac{\\beta_{ri}(\\mu_i-\\mu^{\\rm eq}_i)}{RT}\\right);", "7c64b086870026d0d61f26864a6d8243": "6^5", "7c64e3708b88043dfecde05826111761": "T_{t}=\\frac{2L}{\\sqrt{c^{2}-v^{2}}}=\\frac{2L}{c}\\frac{1}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}\\approx\\frac{2L}{c}\\left(1+\\frac{v^{2}}{2c^{2}}\\right)", "7c65093ba6fe4b8aae4e1e215661145c": " M^n \\mathbf{e}_2 = M^n (\\mathbf{v}-\\mathbf{u}) = b^n \\mathbf{v} - a^n\\mathbf{u} = (b^n-a^n) \\mathbf{e}_1+b^n\\mathbf{e}_2.", "7c65348321835145709f53b4e5d656b6": "Z(\\mathbf{s}) = m(\\mathbf{s}) + \\varepsilon '(\\mathbf{s}) + \\varepsilon ''", "7c6536369fa751437cac119a4dc34fb3": "i\\in t,j\\in s", "7c65a42b9ff1171bc335d9e80b5a43a6": "Z_\\mu = \\frac{\\dot N}{\\dot \\Phi} = \\frac{\\dot {N}_m}{\\dot {\\Phi}_m} = z_\\mu e^{j\\phi}", "7c65b23f59b75ce52603222558449d66": "F_{ab} \\, = \\, \\partial_a A_b \\, - \\, \\partial_b A_a \\,", "7c65c3419398f12f76f0654e621a8237": " B_P = \\{ 1, x, x^2, x^3 \\} ", "7c660d5fc8adbb79f10e37fefc4ab97b": "\\int L(a-\\theta) f(x_1-\\theta) d\\theta = \\int L(a-x_1-\\theta') f(-\\theta') d\\theta'.", "7c662cf96559edd21d247911dc9aa11e": "ax^2+bxy+cy^2", "7c668c8039e7a8d72eac4dabe1820b04": "\\langle \\cdot, \\cdot \\rangle _K", "7c6696494942f01caf6162d97a757e2e": "\\int_0^\\infty{x^3 e^{-a x^2}\\,dx} = \\frac{1}{2 a^2} ", "7c66a6cd357c033b1a035a13beb84324": "L' \\to L= \\frac{\\omega_c'}{\\omega_0 Q}L' \\,\\lVert \\,C= \\frac{Q}{\\omega_0 \\omega_c'}\\frac{1}{L'}", "7c66ac832c75ed0bb817f265b1585acd": "Z_p^{*}\\,", "7c66bafc9be560a8e66d751c387aa58a": "\\textstyle W_{p}^{m}(\\Omega)", "7c66f96ca530272d1bd4dd44ffd7ca55": "\\|\\sigma(\\mathbf{x})\\|_2", "7c66feb87fa0f4ba11df4a83c9167a77": "\n[x^{\\mu}, x^{\\nu}]=i \\theta^{\\mu \\nu} \\,\\!\n", "7c675b64dada4c44471f67b48c022010": "\\beta + \\gamma", "7c678a5bb6fbd47e7592460914df075c": "\\scriptstyle 2(C'_0 \\,\\oplus\\, C'_{k-1})", "7c67a8cc8eb1307604e3a6f28306e588": "\\tilde f_{i-1}(r_1, \\dots, r_{i-1}) \\neq f_{i-1}(r_1, \\dots, r_{i-1}),", "7c680cab6f519217c550e1bc0a15b426": "\\text{Spec} \\; k", "7c682817cde06ed2f642cf3ac47e8b63": "V_{\\text{Yukawa}}(r)= -g^2 \\;\\frac{e^{-mr}}{r}", "7c6891fd7946f6279630f0fe93a7dc52": "\\textbf I\\;=\\;\\int_0^{\\frac{\\pi}{2}}\\,\\frac{1}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^2}\\;\\mathrm{d}x=\\frac{\\pi}{4\\sqrt{ab}}\\left(\\frac{1}{a}+\\frac{1}{b}\\right),", "7c691518bb2001b9b7d63f3639145362": "(18)\\qquad \\Phi_{00}\\,\\hat{=}\\,0\\;\\;\\Leftrightarrow\\;\\;2\\,\\phi_{0}\\,\\overline{\\phi_0}\\,\\hat{=}\\,0\\;\\;\\Rightarrow \\;\\;\\phi_0=\\overline{\\phi_0}\\,\\hat{=}\\,0\\,.", "7c692719ce1b4ecd3a0f6128b07e30d4": "6n^2", "7c696fccb8db97def94ae4a442d7c4ca": "-\\wp(z)", "7c697ea15719d5364c4721ebf6423f53": "A:\\mathcal{U}", "7c69a8e7a10ec9538b9b4712387c81af": "p(\\varphi)", "7c69edec436c1e60947741bc640a88ff": "X \\lessdot Head^+(Y)", "7c69fae858483bb9db9247fa68ddec28": "\\mathcal{Q}_{Hur}=\\mathbb{Z}[\\eta][i,j,j']", "7c6a7a038e95b2621e77fdbacbcf5878": "a_r =\\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r(c + \\gamma)_r} a_0, \\text{ for } r \\geq 0", "7c6a8ec9fe4c859633a393853f1a48ba": "V_{S} = \\frac{4}{3}\\pi r^3.", "7c6b566ba035f16148cba9cc2eb6545e": "D(l,m,n)=\\langle x,y \\mid x^l,y^m,(xy)^n\\rangle.", "7c6b6c6d214213e8e9ec68211662853c": "\\tau = - m g L \\sin\\theta\\,", "7c6be894a8e0dc7564625592c0a05f81": " f(nx) = n f(x) \\ ", "7c6bef9ccdca418402592c1a76651cbf": "\\mathfrak{P}^{113}", "7c6cb21ebc41580b53426b7a5bd7b0ba": "\\varphi(n)\\;", "7c6ceafebc5a232f4ce55a61cefb4244": "\\{\\lambda_1, \\cdots , \\lambda_n \\}", "7c6d4c9c09a12535c7c49ac59608a386": " H :P_n \\rightarrow R^+ ", "7c6d55eb64cf597d24ffcdf807c22a2f": "\n \\sigma_{k+1} = \\sigma_k \\times \\exp\\bigg(\\frac{c_\\sigma}{d_\\sigma}\n \\underbrace{\\left(\\frac{\\|p_\\sigma\\|}{E\\|\\mathcal{N}(0,I)\\|} - 1\\right)}_{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{unbiased about 0 under neutral selection}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n}\\bigg)\n ", "7c6d68bf0cd67faac4e4c28961d301ec": "k\\geq \\max\\{n-m+1, 1\\}", "7c6d6e723c743498adae196a685fb8b1": " \\int (a f + bg) \\, d \\mu = a \\int f \\, d\\mu + b \\int g \\, d\\mu. ", "7c6dae63a692798f4af4f20f5e241fa4": "\\hat f(k):=\\int_{[0,1]^n} e^{-2\\pi iy\\cdot k} \\, f(y)\\,dy,", "7c6df1a017af9ac1d5eeb7ec84d04919": "|f(x) - P_{n-1}(x)| \\leq \\frac{C(r)}{n^r}, \\quad 0 \\leq x \\leq 2\\pi, ", "7c6e18695f5877d9c6d21ccdc487f020": "T_vT_pM\\cong T_pM\\cong \\mathbb R^n", "7c6e560068f5fac003b80aa9da635d24": "\\begin{matrix} \\frac{velocity \\;vector \\;at \\;latitude \\;A} {velocity \\;vector \\;at \\;latitude \\;B} \\end{matrix}", "7c6e6dc2157e670a8c60ce29927b4d71": "h\\sim{2 \\times 10^{-17}/\\sqrt{\\mathrm{Hz}}} ", "7c6edf799b7e412a10fc2d5d684f3f82": "D_\\mathrm{p} = D_\\mathrm{maj} - 2\\cdot\\frac38\\cdot H = D_\\mathrm{maj} - \\frac{ 3 {\\sqrt 3}}{8}\\cdot P = D_\\mathrm{maj} - 0.649519 \\times P", "7c6ee42a0dbd6d1debf62278c867466c": "\n\\left.\n\\begin{matrix}4^{4^{\\cdot^{\\cdot^{\\cdot^{\\cdot^{4}}}}}}\\end{matrix}\n\\right \\}\n\\left.\n\\begin{matrix}4^{4^{\\cdot^{\\cdot^{\\cdot^{4}}}}}\\end{matrix}\n\\right \\}\n\\dots\n\\left.\n\\begin{matrix}4^{4^{4^4}}\\end{matrix}\n\\right \\}\n4,\n", "7c6f2e24de5454e643eed8ec7698c8d6": "s = r/(r-1)\\,", "7c6f872380791e44fed48b83000c0201": " 0 = \\mu_0 < \\mu_1 \\leq \\mu_2 \\leq \\cdots ", "7c6f8c159912b2e40c84cf7e4d34a8fe": "W(x,p)", "7c6fbaade541a68d28955684959312dc": "2^{0/12} = 1", "7c707945d42377aa6248f623c0e787d3": "arcsin", "7c7083801908586d47736d9bd079090f": "|\\Psi^+\\rangle = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |1\\rangle_B + |1\\rangle_A \\otimes |0\\rangle_B)", "7c7088a604dce20fa60cb2277dda5abb": "\nH = \\frac{1}{2m} \\left( p_{r}^{2} + \\frac{p_{\\phi}^{2}}{r^{2}} \\right) + U(r) \n", "7c70ad7ec1feb2a201eb1e0c885f5362": "p \\vee q", "7c70bd93e6103cd4e115e936ddb71f91": "P(X\\mid\nA)", "7c70e85550edeea40e0452fe549bcccb": "\\{\\max cx \\mid Ax \\le b\\}", "7c7134bb58495221eff52de5da384184": "\\scriptstyle\\ell^p", "7c716d9675fd10988102a8fef3a7c499": "f(0,\\dots,0)=0", "7c718d41e594aede6f6b26400bd45136": "2^{n} a_{2^{n}}", "7c7197c66ec960a613cff4b03060bbaf": "\\begin{align}456_{10} = 456 / 8 = 57\\text{ with a remainder of }(0)\\\\\n57 / 8 = 7\\text{ with a remainder of }(1)\\\\\n7 / 8 = 0\\text{ with a remainder of }(7)\\\\\n&= 710_8\\end{align}", "7c71ada520bc916d45924fd57039a3c0": "\\Lambda(A_1:A_2|B) \\triangleq \\frac{P(B|A_1)}{P(B|A_2)},", "7c7204e73edf72fb4b9b9a83dc774da3": "t_{\\nu'-p+1}\\left(\\tilde{\\mathbf{x}}|\\boldsymbol\\mu,\\frac{1}{\\nu'-p+1}\\boldsymbol\\Psi'\\right)", "7c7210cb866d2fe5f5ff6cb0e9d88133": "[x:\\cdots]", "7c72a8d04298f0a675862a84130862f8": "X=2", "7c72bc796bc6838714514804b053c073": "p(\\vec{r},t)=\\frac{1}{4 \\pi v_s^2} \\frac{\\partial}{\\partial t} \\left [\\frac{1}{v_s t} \\int d \\vec{r'} p_0(\\vec{r'}) \\delta \\left (t-\\frac{|\\vec{r}-\\vec{r'}|}{v_s} \\right) \\right] \\qquad \\,(3), ", "7c731373d2c6e5005742654ceafff28a": "|x|^n=\\sum_{|k|=n}{\\binom{n}{k}x^k};\\ \\ x\\in\\mathbb{R}^m,\\ k\\in\\mathbb{N}^m_0,\\ n\\in\\mathbb{N}_0,\\ m\\in\\mathbb{N}", "7c7323b9c70cedee32a56ad914fec137": "\n\\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\cdot \\mathbf{Q}_{r} = \n\\mathbf{p} \\cdot \\left( \\mathbf{n} \\times \\mathbf{r} \\right) = \n\\mathbf{n} \\cdot \\left( \\mathbf{r} \\times \\mathbf{p} \\right) = \n\\mathbf{n} \\cdot \\mathbf{L}.\n", "7c7363afa013fcc2850f187cab3d74b2": "\nP_A = e^{v_A } - P_A e^{v_A } \n", "7c736f23bb9cbc7a125af391b14d18c1": "m\\geq\\frac{2}{\\epsilon^{2}}\\,\\!", "7c7389b1ab8d8ac1dabafe6d22fc38a9": "\\arccos x ={\\pi\\over 2}-\\arcsin x={\\pi\\over 2}- \\sum^{\\infty}_{n=0} \\frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\\quad\\text{ for }|x| \\le 1\\!", "7c73a447a062a31f3c47a9b78b389ced": " S_1S_2S_1=S_2S_1S_2 \\, ", "7c73b67cdefbba0f25a3363f95edf24e": "A =_c B", "7c73e59562e65cd9c14bfff8e54a213f": "f^{-1}(\\left\\{1,4,9,16\\right\\}) = \\left\\{-4,-3,-2,-1,1,2,3,4\\right\\}", "7c741086e1c9b1bbe9d813439c4d261c": "N\\equiv 0", "7c741c467ffacd3820ef74415cb09793": "\n \\begin{align}\n \\frac{\\partial X^\\mu}{\\partial x^m}\\,_{(x)}\\Gamma^m_{ij} & = \\frac{\\partial X^\\mu}{\\partial x^m}~\\frac{\\partial x^m}{\\partial X^\\nu}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\nu_{\\alpha\\beta} + \n \\frac{\\partial X^\\mu}{\\partial x^m}~\\frac{\\partial x^m}{\\partial X^\\alpha}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j} \\\\\n & = \\delta^\\mu_\\nu~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\nu_{\\alpha\\beta} + \n \\delta^\\mu_\\alpha~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j} \\\\\n & = \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\mu_{\\alpha\\beta} + \\frac{\\partial^2 X^\\mu}{\\partial x^i \\partial x^j} \n \\end{align}\n", "7c751077ae37efcbe2c8ffa438deb358": "H(P) = -\\operatorname{E}(\\ln p(x))", "7c7517a76b4839854073bb042157a50e": " a(n,t) = \\frac{1}{2} {\\rm e}^{-(q(n+1,t) - q(n,t))/2}, \\qquad b(n,t) = -\\frac{1}{2} p(n,t) ", "7c751ca66cfc52fb01b0f5b813e7e416": " f=f_{z}\\, dx \\wedge dy + f_{x}\\, dy \\wedge dz + f_{y}\\, dz \\wedge dx ", "7c756426266ff76b9b9b2ce8b410c188": " D^\\prime", "7c75650a43f1c67fd553804f52242e80": "q\\in\\left\\{1,2,3,\\dots\\right\\}", "7c758a5206995bb560b0b0f575234673": "\\cos A + t \\cos B \\cos C : \\cos B + t \\cos C \\cos A : \\cos C + t \\cos A \\cos B\\,", "7c7628b171dc73e604293d13800d425a": "f_i(\\vec{x}+\\vec{e}_i\\delta_t,t+\\delta_t) =f_i^t(\\vec{x},t+\\delta_t) \\,\\! ", "7c7670a3d6ad5742ff513fbd524f2d77": "\\alpha < \\kappa \\,,", "7c769d8d889d587bbcfb8c88bf31f380": "B:P \\times L", "7c76dfc81cd7317be79c2570e4358bc1": " I_{23} = \\frac{V_{23}}{|Z_\\Delta|} \\angle (-90^\\circ-\\theta) ", "7c77066e65f9e538be790ff27a60a970": "S(\\boldsymbol \\beta) = \\sum_{i=1}^m [y_i - f(x_i, \\ \\boldsymbol \\beta) ]^2", "7c776afdc7c49e110f1bf39b2edee0b1": "\\sigma_t", "7c776f2d1c2c490a7eb61aa02d9cb748": "(\\theta) \\frac{A_1;\\ldots;A_n;B_1;\\ldots;B_m}{A_1;\\ldots;A_n}", "7c779194ed3e59e5254b0b302e0102ff": "\\phi(t)=c_1 e^{\\lambda_1 t} + c_2 t e^{\\lambda_1 t}.", "7c77a489d2056cc592b52780f6236bbc": "Percentage\\ protonated = {molarity\\ of\\ HB^+ \\over\\ initial\\ molarity\\ of\\ B} \\times 100\\% = {[{HB}^+]\\over [B]_{initial}} {\\times 100\\%}", "7c77b8b2f369dc1189261a9320b8f4e4": "\n\\frac{1}{2}\\rho W^2Nc(c_l\\sin\\phi - c_d\\cos\\phi)r\\delta r = \\rho(2\\pi r\\delta r)U_{\\infty}(1 - a)\\times(2\\Omega a'r^2)\n", "7c78412032454d21f4b6db674ae2d5d9": " C_A = [A^{3-}]+\\beta_1 [A^{3-}][H^+] +\\beta_2 [A^{3-}][H^+]^2 +\\beta_3 [A^{3-}][H^+]^3", "7c7843adaff8748eb06cf275b333902b": "f_\\mathrm{s}", "7c78448a9b74f132425be21e71f00456": "\\frac{(\\operatorname{observed} - \\operatorname{expected})^2}\n {\\operatorname{expected}}", "7c784cbf9f909a54435ff709c4bc958f": "[X, Y]", "7c78b969566d04758e2f6cbf928340ec": " x_{k+1} = \\mathcal{P}_C( y_k + q_k ) ", "7c78d4bf0e14c111c1467ac1decc613a": "\nF_N = \\frac{1}{\\sqrt{N}} \\begin{bmatrix}\n1&1&1&1&\\cdots &1 \\\\\n1&\\omega&\\omega^2&\\omega^3&\\cdots&\\omega^{N-1} \\\\\n1&\\omega^2&\\omega^4&\\omega^6&\\cdots&\\omega^{2(N-1)}\\\\ 1&\\omega^3&\\omega^6&\\omega^9&\\cdots&\\omega^{3(N-1)}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&&\\vdots\\\\\n1&\\omega^{N-1}&\\omega^{2(N-1)}&\\omega^{3(N-1)}&\\cdots&\\omega^{(N-1)(N-1)}\n\\end{bmatrix}.\n", "7c7903810ee80a4c15ae000781c0d084": "\nE^2 - (pc)^2 = (mc^2)^2\\,,\n", "7c793011b22c07d4f5f077511ed94d33": "T_m : \\varphi\\mapsto m\\varphi", "7c796bd447be51a0259a46e97cab6aac": "\\varphi_{2}", "7c799b9f11504a996a3a1c8add05307c": "\\frac{p}{p_{eq}} = \\exp{\\left(\\frac{R_{critical}}{R}\\right)}", "7c79cf5ab52ed4c855bef00769471803": "1 \\in C", "7c7a1a6f1b2d5e8a947d9b2819a03e91": "F(t)-kx-c\\frac{\\mathrm{d}x}{\\mathrm{d}t}=m\\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2}. ", "7c7a5019f5c3c20de9c52fcc7506b428": "\\circ, \\bullet", "7c7a6854c1011390d6dc8c3942fd441c": "F(z, m)", "7c7a86c268c5a521d730733ab037f9a8": "C\\to|K_C|^*", "7c7a92deadf14b92f32657e365ef0805": " g(r) 4\\pi r^2 = -4 \\pi GM \\,", "7c7ace0368660821ade2fcc814e8e571": "\\Gamma= I ", "7c7ad18cdd3c992d4bcc7234452af94b": "i \\in D, M_i", "7c7b4a603eda3cd9294a151b8d6014d8": "D_1F(x,y)\\cdot(s_1,s_2) + D_2F(x,y)\\cdot(F(x,y)\\cdot s_1,s_2)", "7c7b4de419426017ca6d6a18525d4586": " \\nabla_{\\dot{c}} v = 0", "7c7b544137595cccefab8f306cfff170": "k \\varphi (N) 0 ", "7c7ef79e2e18b1d9770afc8d5a8332e7": "\\mu_n^+(r)=\\int_r^\\infty (x - r)^n\\,f(x)\\,dx.", "7c7f161dcc2359602bb31834521ffb3e": "\\tau_1 =1\\Big.", "7c7f49ba10580656286610cd7f80a704": "\\mathrm{d}W(t)", "7c7f688a22a58de817011dadcbb50999": "\\tfrac{1}{\\infty}", "7c8051fd80124eb26517830932d02dc8": "\\sigma^{n}(A_{r}) \\geq \\sigma^{n}(B_{r}),", "7c808dfae105fc879f1a5bb95f58eedf": "1 + (c + 2)*\\,X + (c^2 + 2*c + 5/2)\\,X^2", "7c808fa098f29171441a0b24c2b00573": "\\forall z \\forall w_1 \\forall w_2\\ldots \\forall w_n \\exists y \\forall x [x \\in y \\Leftrightarrow ( x \\in z \\land \\phi )].", "7c80c6e3966ee73e1beaf6b1c20b8aae": " r^{n+2}~\\cos(n\\theta) \\,", "7c80ec9018348ef0dd5e3b2784c5142c": " \\mathbf{B}_{2} = \\frac {\\mu_0} {4 \\pi} I_2 \\oint_{C_2} \\frac { (d \\mathbf{l_2} \\ \\mathbf{ \\times } \\ \\hat{\\mathbf{r}}_{21} )} {r_{21}^2} \\ ", "7c811960d76c393ef7b1bd949b707cf2": "\\sigma_p^2 = \\frac{\\hbar m \\omega}{2}.", "7c8120cac01f18304bd92ace40f835f2": "\\mathbf{\\Omega} = \\mathbf{\\Sigma}^{-1}", "7c8126422f0f4bdddecf2e1f9b7bb8e5": " {dx \\over dy} = {\\sqrt{1-4y}\\over 2\\sqrt{y}} \\,", "7c81377ca9d20bef10b0ad3693d0d90b": "\\alpha_{P}(X) = 1", "7c814aa0d685368dbec8995b0657a973": "\\phi_D:\\Sigma^*\\to \\mathbb {M}(D)", "7c814fcce37dad031f9f2bb0d57a86db": "r \\rightarrow r_c", "7c81704b4ad90ec2f90b153a4358e3a9": "\\frac{\\partial E}{\\partial w_i} = \\frac{dE}{dy} \\frac{dy}{d\\mathrm{net}} \\frac{\\partial \\mathrm{net}}{\\partial w_i}", "7c81a66dbc6ae3503c880a0e1cccce5c": "u\\wedge v\\wedge w", "7c81e58f05b88cac5bcb390608c210de": "\\mathcal{Z}(R_R)\\neq\\mathcal{Z}(_R R)", "7c81f973231d0022d3a2de1d33d7daba": "x \\mapsto x_+^0", "7c8258a500bc5ac2550d3bc71e05b4e6": "d(42) = 8 = 2 \\times 2 \\times 2 = d(2) \\times d(3) \\times d(7)", "7c82793db4be582004b0a18bd09a1e16": "X=\\mathbb R \\times M", "7c82ba976a9b428a622459618052ad05": " \\prod_{k=3}^\\infty \\cos\\left(\\frac\\pi k\\right) = 0.1149420448\\dots. ", "7c82d066ae939d64e5c6399e1bc9670f": "\\scriptstyle \\mathbf{R}^n", "7c8344e47f1a67fd61d970e2e6cc5893": "\\sum^{\\infty}_{n=1}a_n b_n", "7c835735abf742bb3b6c80e41a0762c4": " \\delta W = \\sum_{j=1}^m Q_j \\delta q_j, ", "7c8373755d0aa3c08fbafc94cb2e80e6": "\n\\begin{array}{lrclr}\n\\max\\limits_{x_{T-1}} & E[U(W_T)] & \\\\\n\\text{subject to} & W_T &=& \\sum_{i=1}^{n}\\xi_{iT}x_{i,T-1} \\\\\n &\\sum_{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\\\\n\t\t & x_{T-1} &\\geq& 0\n\\end{array}\n", "7c83a9ef0b4707117b57f946f0c20951": "\\frac{\\partial u}{\\partial t} -\\alpha\\left(\\frac{\\partial^2u}{\\partial x^2}+\\frac{\\partial^2u}{\\partial y^2}+\\frac{\\partial^2u}{\\partial z^2}\\right)=0", "7c83ecc314303883b9d3fbdb6797f9be": "\\ A = R(c\\tau/2)\\theta\\sec\\psi", "7c8418c53ca4bf15178bd056b407e375": " I \\propto\\int_0^{eV} \\rho_S\\left(E_F-eV+\\epsilon\\right)\\rho_T\\left(E_F+\\epsilon\\right)\\,d\\epsilon\\ ,\\qquad\\qquad (3)", "7c845a3faed596b9814a1e109a04c013": "\\Delta{}H_{\\mathrm{vap}} = \\Delta{}U_{\\mathrm{vap}} + p\\Delta\\,V", "7c84620e601eb45645c04fee32cd69a0": "\\kappa_i", "7c84b03a02f5fd2bce54ff574adfc4ec": "H = \\begin{pmatrix} E_{+} & 0\\\\ 0 & E_{-}\\end{pmatrix}; ", "7c84bc3bf786ca30ffebabe28842020d": "-x=\\gamma\\left(-x' - vt'\\right)", "7c84edc7075fd5a665e39828c0eb3223": "|\\mathbf{v_3}'| = |\\mathbf{v_1}'-\\mathbf{v_2}'|", "7c8509706fc395ce83b1dbbca1f87255": "\\rho_r(x)", "7c8570a6fb92e57415584693fadeae90": "0.42 \\lambda N", "7c85918a40c4fd1fa67f89e96521fc7f": "\n\\begin{align}\ng(x,y,z) & =\\frac{1}{r^{n+1}} P^m_n(\\sin \\theta) \\cos m\\varphi \\,,&\\quad 0 \\le m \\le n \\,,&\\quad n=0,1,2,\\dots \\\\\nh(x,y,z) & =\\frac{1}{r^{n+1}} P^m_n(\\sin \\theta) \\sin m\\varphi \\,,&\\quad 1 \\le m \\le n \\,,&\\quad n=1,2,\\dots\n\\end{align}\n", "7c85a4978a6589d0cf5f72b4b25e56d5": "\\ f ", "7c85d3899c50fbc292900ad3401dc88c": "NA=n\\sin\\alpha", "7c86571b4e251228985d514d1f6c0d7f": "v=\\sqrt{2}\\cdot v_o", "7c8694e65c63b8472ab65376ae9f895a": "T(\\mathcal{M})", "7c86ada9fda8478af3e0afaa96cd8ac3": " \\delta W = \\bar{W}-W = \\int_{t_0}^{t_1}(\\mathbf{F}\\cdot \\epsilon\\dot{\\mathbf{h}})dt.", "7c86fa5887b6c575f994b2cbb61effee": "\\max\\{\\,| f(v_i) - f(v_j)| : v_iv_j \\in E \\,\\}", "7c87428b05b743e8c6458bc098a26d2d": "T_{\\theta}: \\boldsymbol{x} \\mapsto T_{\\theta}\\boldsymbol{x}", "7c875c1293057597bad43f41881f5409": "\\operatorname{erf}(z) = \\frac{2}{\\sqrt{\\pi}}\\int_{0}^z e^{-t^2}\\,\\mathrm dt.", "7c878351c6fb258e01374e7d81b60e70": "(|S|-1)", "7c8796c689ba0ac51790d7d4c1d77c00": "U(1)_{B-L}", "7c87c726d18aff87fadaf396910a4984": "\\sin^2{\\theta}+\\cos^2{\\theta} = 1\\,", "7c8829aedb6af3e63a44aae9a7a6f2ed": " \\frac {\\partial p}{\\partial y} = \\frac{ P_p - P_s} { \\partial y_v} ", "7c88542b207054e6c1bbb37649982523": "\\mathrm{d}^3 s_{\\nu}", "7c888909df4fd192929bc2a355620073": " s_{12} =\n\\int_{\\lambda_1}^{\\lambda_2} L(\\phi, \\phi')\\,d\\lambda,", "7c88b30213fbdc9f978b41d661e84781": "Z(x)", "7c88fed9efb38b4cbfa727ba405df1d0": "F_{BH} \\; = \\; \\left( \\; \\frac{h_B}{30.48} \\; \\right)^2", "7c895831e050f237e3e60457e36f9bf5": "\\scriptstyle{\\vec E}", "7c898ef7d9b148053f6b16185156134c": "R = I + 2 [\\omega]_\\times^2 = I + 2(\\omega \\otimes \\omega - I) = 2 \\omega \\otimes \\omega - I", "7c8990929b604ca812c0fd814ee2efb4": "\n\\begin{align}\n \\langle 3\\rangle &= \\langle 3\\rangle_\\mathrm{S} + \\langle 1\\rangle\\ \\Delta \\langle 2\\rangle +\\Delta \\langle 3\\rangle \\,, \\\\\n \\langle N\\rangle &= \\langle N\\rangle_\\mathrm{S} \\\\\n &\\quad+ \\langle N-2\\rangle_\\mathrm{S}\\ \\Delta \\langle 2\\rangle \\\\\n &\\quad+ \\langle N-4\\rangle_\\mathrm{S}\\ \\Delta \\langle 2\\rangle\\ \\Delta \\langle 2\\rangle +\\dots\\\\\n &\\quad+ \\langle N-3\\rangle_\\mathrm{S}\\ \\Delta \\langle 3\\rangle \\\\\n &\\quad+ \\langle N-5\\rangle_\\mathrm{S}\\ \\Delta \\langle 3\\rangle\\ \\Delta \\langle 2\\rangle +\\dots\\\\\n &\\quad+ \\Delta\\langle N\\rangle\\,,\n\\end{align}\n", "7c89a02d96a201a288e5a3e3c313fbcd": "P(k) \\sim \\int\\rho(\\eta)\\frac{C}{\\eta}(m/k)^{C/\\eta +1}\\,d\\eta", "7c89eb83688300e0e7b9fc989dff2e15": "i^{4n} = 1\\,", "7c8a22ae0ee8f3a4a0cb0a115f11a7be": "O(x^{n+1}).", "7c8a2320ba08ff74e783b46a8513dfa1": "Ax+b", "7c8a437e91ef893a9a205463966253ee": "{P(N)_{t+1}}", "7c8a49cc904e7a2c23515c8ea01a0b9c": " x^2 y^2 + y^2 z^2 + z^2 x^2 - r^2 x y z = 0. \\,", "7c8a606132d7029010ed9d7818faae89": "V^{\\mathbb C} = V\\otimes_{\\mathbb{R}} \\mathbb{C}.", "7c8a676f25297c329ffb2c01c5757bc3": "\\mathbf{A}\\left( \\mathbf{q}|\\Gamma \\right) = \\hat{P} \\exp ", "7c8ad54ebb1282b26608c8a47dad3650": "n=3, \\quad I_3 = \\tfrac{1}{3} \\cos^2 x \\sin x + \\tfrac{2}{3} I_1, \\,", "7c8aec0987140fef15d51e6b98840b60": "D^+ f(t)\\,", "7c8aed341ac7f68b9b28d56f2ef64ccd": "\\hat{\\mathbf x}_i(t+\\Delta t) \\leftarrow \\hat{\\mathbf x}_i(t+\\Delta t) + \\sum_{k=1}^n \\lambda_k\\frac{\\partial \\sigma_k}{\\partial \\mathbf x_i}", "7c8afe9095b2273343cd4783d92a11dc": "P(\\alpha)", "7c8b16ff939c8b0550c0af7e65f56b17": "\\textstyle x^2 y^2 = x^2 + y^2.", "7c8b74bd9cd34b49dc0a745c711a6b80": "\n- \\Gamma^{\\lambda}_{j\\alpha}]-\\Gamma^{\\lambda}_{j\\beta}[g^{hi}g_{h\\lambda},\\alpha - g^{hi}g_{m\\lambda}\\Gamma^{m}_{h\\alpha} -\\Gamma^{i}_{\\lambda\\alpha}]\n", "7c8bb5ed2374925bbca4375ca58824b8": "e^{ct}", "7c8bce60059d5c3a1d28e30ba05d5963": "d\\to u+ e^- + \\bar\\nu_e~", "7c8be40f911cc36aeff27a87ed974413": "S=\\sum_{i=1}^n (R_i- \\bar R)^2 ,", "7c8beaed72d18e9906333721542789b7": "\\scriptstyle{Y}", "7c8bf1b0c7f6b7e742ecc842eba9919e": "N_{\\rm A}", "7c8c10236ec012f0dc804ba57389c418": "\\nu(W)= +\\infty", "7c8c42a5b1801e4f4b24a2f0b92ff60e": "\n\\begin{align}\n\\boldsymbol{b_1}\\rightarrow\\boldsymbol{b_2}+\\boldsymbol{b_3}\n\\end{align}", "7c8cb751f8e83784b68083fa56d903c4": "X \\perp\\!\\!\\!\\perp Y \\Rightarrow Y \\perp\\!\\!\\!\\perp X", "7c8d30fc61f7eae5066eee908eedb071": "{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}.", "7c8d4a870ccc0e2af3433e29d2e6afb1": "\\displaystyle{G_{\\overline{z}}=\\mu G_z.}", "7c8d4e66286d201a1f96b8c158a38439": "\\scriptstyle\\vec{x}", "7c8db424831c7d3b094e8ba721351531": "\n\\begin{align}\n\\int_{C} L\\, dx & = \\int_{C_1} L(x,y)\\, dx + \\int_{C_2} L(x,y)\\, dx + \\int_{C_3} L(x,y)\\, dx + \\int_{C_4} L(x,y)\\, dx \\\\\n& = -\\int_a^b L(x,g_2(x))\\, dx + \\int_a^b L(x,g_1(x))\\, dx.\\qquad\\mathrm{(4)}\n\\end{align}\n", "7c8dc6555edbc34145ce0d3edc8682cc": "1 = f(0)= f(x-x) = f(x) f(-x)", "7c8dc6d7385cb95b89b71f9dc9feb1c0": "\\left(\\frac{\\Delta}{n}\\right).", "7c8deb7e584ad9746f3eee5ef4fbf5f7": " 1^2 + 2^2 + \\cdots + n^2 = \\frac{1}{3}\\left(B_0 n^3+3B_1 n^2+3B_2 n^1 \\right) = \\frac{1}{3}\\left(n^3+\\frac{3}{2}n^2+\\frac{1}{2}n\\right).", "7c8df5bb96776e9afc10bf85cf4b514e": "\\sqrt{\\frac{2}{\\pi}} \\frac{x^2 e^{-x^2/(2a^2)}}{a^3}", "7c8dfd28c972e12eb16df29f36d652d1": "\\scriptstyle X \\to Y \\to Z", "7c8e10fafdd4d11684381755761715f9": " \\operatorname{G}(A) = \\{(\\xi, A \\xi): \\xi \\in \\operatorname{dom}(A)\\} \\subseteq H \\oplus H .", "7c8e2d521e47b782b0a1404d90e3e3d4": "R_1 C_\\text{F}", "7c8e3b0510d4bff46b4d2e44e5b38cc1": "h(x) = f(a(x))", "7c8e505d3fc42f3103613a8f40ecf72d": "\n\\frac{\\Gamma\\left[(\\nu+p)/2\\right]}{\\Gamma(\\nu/2)\\nu^{p/2}\\pi^{p/2}\\left|{\\boldsymbol\\Sigma}\\right|^{1/2}\\left[1+\\frac{1}{\\nu}({\\mathbf x}-{\\boldsymbol\\mu})^T{\\boldsymbol\\Sigma}^{-1}({\\mathbf x}-{\\boldsymbol\\mu})\\right]^{(\\nu+p)/2}}", "7c8e7a35515ca3aa82b6b6c7ae787ad6": "\\left[S_x, S_z\\right] = -i\\hbar S_y \\ne 0", "7c8ea7592fd67cb0a35db940e95c87de": "a:H\\times H\\to \\mathbb R", "7c8f0da19667b53c0b922e9314200a4b": " a(x + h)^2 + k\\, ", "7c8f27b07ce68ff7ce525adf8e28aea5": "\\zeta(3)=\\sum_{n=1}^\\infty \\frac{1}{n^3}=\\frac{1}{1^3}+\\frac{1}{2^3}+\\frac{1}{3^3}+\\ldots = 1.2020569\\ldots", "7c8f28db7109f3bbbe0048d7f8ec5e56": "1% < Y < 98%", "7c8f4c06c5c5576296faedfcf6acdb9c": " W^{k+1,p}(\\Omega) \\hookrightarrow W^{s',p}(\\Omega) \\hookrightarrow W^{s,p}(\\Omega) \\hookrightarrow W^{k, p}(\\Omega), \\quad k \\leq s \\leq s' \\leq k+1 ", "7c8fc1e86449f36eeab2c826be474071": "\n\\frac{s}{D} = \\frac{m_{b}}{k_{B} T}\n", "7c8fccfa1806d0985c8c39c5aa4df9b9": "\\alpha(0)=\\beta(1)=1, \\mbox{ the f-product } \\alpha \\cdot \\beta = \\beta + \\alpha -e", "7c9084d9d8f788791d143ec6f1fe6480": "\\textstyle (x_1,\\dots,x_N)", "7c90aebd7e624b88f9005abefcd22f7f": "s,s' \\in S", "7c90f3e67dae78637220dc67c6561e6d": "V=V_{\\mathbb{R}} \\oplus iV_{\\mathbb{R}}\\,", "7c9135fae369c22d4a7241bf6dc86865": "\nE = - p_{t} = m \\, c^2 \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau}\n\\,.", "7c913dd86a40193ab4aa5b98fc2642f3": "\\operatorname{GL}(V \\oplus K)", "7c913e757e40b090b073cacd4511a152": "t_m", "7c914d1a06d3c9c5b577358b9be147ad": "f''(x_n)/f'(x_n)", "7c91549eac20153b55e56faaac6f2679": "L(k_1)=20l^\\prime+64", "7c91949e9752ed47f5f689a5a8e0f9f9": "Q=\\int d^dx q(x)", "7c91c918af61b941f9580811c2f9d56a": "a_i = \\langle v,v_i \\rangle", "7c91e427270e0f6f85d3909e107acde4": " (x-1)(x^{n-1}+x^{n-2}+\\cdots+ x + 1)", "7c91e5b69bfb8161665e733e1e836d29": "\\langle x,\\,y \\rangle = 0", "7c92b0bd513b5dd3d7f2189a3ea1b9e1": " V = W_{H,i} + W_{H,p} - W_{g,p} + W_{\\sigma} ", "7c92f953cb09bd2da7f7c59dfb6929e5": "\\Pi_{(n:{\\mathbb N})}\\; {\\mathbb R}", "7c9303480c06312622efafffdabcac28": "W_T,", "7c9329fc417a662fb1aae810e4758244": "3.2 U_p < I_p", "7c9372b083490b548ce5208071ff08e4": "K = D + E - M \\, ", "7c9381e41afe116bf62bafd10a5c6d3f": " \\mathbf {spectrogram} (t,f) = \\left| \\mathbf{STFT} (t,f) \\right|^2 ", "7c93fa3f63f3d8ce95a175c5eb6c88fc": "1.175=\\frac{1}{1}+\\frac{1}{1\\cdot6}+\\frac{1}{1\\cdot6\\cdot20}", "7c94491497d24c15c550f33a85dd75be": "4 \\lceil n/3 \\rceil", "7c94f4819176418244957c788fc37763": " \\frac{(\\Pi_2)^{ 0.25}}{(\\Pi_1)^{ 0.5}} = ", "7c953047ca4dd8e6ab7d575807946c40": "d(z,m)^2\\le \\tfrac12d(z,x)^2 + \\tfrac12d(z,y)^2 - \\tfrac14d(x,y)^2.", "7c9546fa2bcde50a2022a37db6f92044": "\\Pi = \\int_0^T \\left[ pu(t) - \\frac{u(t)^2}{x(t)} \\right] dt ", "7c955174bce232eb9a1a56d1ab68787d": "r_e = \\frac{\\lambda_0}{2\\sqrt{2}\\pi} - \\ ", "7c9581b15bf6fdbfde568e23162f2896": "(-1)^\\text{signbit}\\times 10^{\\text{exponentbits}_2-6176_{10}}\\times \\text{truesignificand}_{10}", "7c9634f11d83621f784c95c9cf9263d8": "{}_{\nx_{N}=-\\frac{a}{2b}\\sqrt{\\left(\\frac{c}{b}\\right)^{N-1}}{}_{N-1}F_{N-2}\n\\begin{bmatrix}\n\\frac{N+1}{2N},\\frac{N+3}{2N},\\cdots,\\frac{N-2}{N},\\frac{N-1}{N},\\frac{N+1}{N},\\frac{N+2}{N},\\cdots,\\frac{3N-3}{2N},\\frac{3N-1}{2N};\\\\[8pt]\n\n\\frac{N+1}{2N-4},\\frac{N+3}{2N-4},\\cdots,\\frac{N-4}{N-2},\\frac{N-3}{N-2},\\frac{N-1}{N-2},\\frac{N}{N-2},\\cdots,\\frac{3N-5}{2N-4},\\frac{3}{2};\\\\[8pt]\n-\\frac{a^2c^{N-2}}{4b^N\\left(N-2\\right)^{N-2}}\n\\end{bmatrix}\n+\\sqrt{\\frac{c}{b}}{\\rm{i}}{}_{N-1}F_{N-2}\n\\begin{bmatrix}\n\\frac{1}{2N},\\frac{3}{2N},\\cdots,\\frac{N-4}{2N},\\frac{N-2}{2N},\\frac{N+2}{2N},\\frac{N+4}{2N},\\cdots,\\frac{2N-3}{2N},\\frac{2N-1}{2N};\\\\[8pt]\n\n\\frac{3}{2N-4},\\frac{5}{2N-4},\\cdots,\\frac{2N-3}{2N-4};\\\\[8pt]\n-\\frac{a^2c^{N-2}}{4b^N\\left(N-2\\right)^{N-2}}\n\\end{bmatrix}\n}", "7c96a260d8b4da60dedf5f9caec59616": " \\theta_y = R \\theta_x \\, ", "7c96cc3b761101157bd89d5856af246a": "(y_i)_{i\\ge 1}", "7c9738400b03542f13709c4f969bc091": "\\left[{n \\atop 1}\\right] = (n-1)!, \n\\quad \n\\left[{n\\atop n}\\right] = 1,\n\\quad\n\\left[{n\\atop n-1}\\right] = {n \\choose 2},", "7c97893b1a9d34d632e7b063a552d210": " g_1(x) = \\begin{cases}\n 1 &\\text{for } 0 < x < 0.5,\\\\\n 0 &\\text{for } x = 0.5,\\\\\n 1/3 &\\text{for } 0.5 < x < 1.\n\\end{cases} ", "7c98274c63a1588cfa35ffb5c0adc833": "Rc", "7c98558e557dc39f8628b662478624f3": "f: U \\subset \\mathbb{R}^n \\to \\mathbb{R}^n", "7c986a2c140d2d69ba59fa2e010f2301": "\\overrightarrow{X}=X_1,X_2,\\ldots,X_n", "7c986a64bc9b1a8314472349ffa419a9": " f(x) = \\frac{ 1 }{ 2 } \\sqrt{ ( 1 - x^2 ) } ", "7c98c73b80d6bc3af17cb59a1a8460c4": "M=\\left[\\begin{matrix} A & B \\\\ C & D \\end{matrix}\\right]", "7c98e2067dcf6630800a3dceb8be619e": "\\lambda=4\\pi k_{\\rm C}\\cdot\\epsilon_0", "7c9913c50a1218f2809894c3e0776803": "\\lim_{\\Delta x\\rightarrow 0}\\sum_{i} f(x_i)\\,\\Delta x,", "7c9923f22335ad48b180173af81e3ab1": "5\\times 5\\times 15\\mu ", "7c9985024e9afd6180247ae9a134ae3c": "N_{t+1}=rN_t[(K-N_t)/K]", "7c99b12727bceefb70f93939fb353016": " X + m \\sim \\textrm{Landau}(\\mu + m ,c) \\,", "7c99d49144cac79bf58d5ed336bdfc0c": "\\frac{\\rho\\sin(\\alpha)}{\\alpha}", "7c99e0c7a7d18a4ab3a5de308394aaa3": "\\alpha(t)=e^{-t}", "7c9a09e711314fcc68bd803082ece9cf": "\\mu_0 = \\mu_1 - \\mu_2.", "7c9a22d347f5e09a0919fd08a666c5fa": " where \\quad s(a,b) =\n\\begin{cases}\n 1, & \\text{if } sign(a) = sign(b) \\\\\n -1, & o.w.\n\\end{cases}\n", "7c9a2d37980b6e0c57b76a44e7f950b8": "G(t_n;x)=\\sum_{n=0}^{\\infty}t_nx^n \\, .", "7c9adaf5d11d79f399e743e717423459": "U_w", "7c9af70af2406fc51c0005737830ddef": "{\\beta'}_j=\\beta_j+\\sum_{i=j+1}^{k+1}y_i", "7c9afd3c57d9dd4e0892814ec0232ba6": "V = (V_t)_{t \\in [0,T]}", "7c9b43d6cce322c732afb60e0506bcb3": "\\Pr(Y = y) = (1-p)^n p\\, ", "7c9b48c5a954eff17fc0fdc8a1518da3": "s(k) \\approx k^\\beta \\,", "7c9b4fe109f25c4d027839ac6ca674f4": "(c_1 + 1) \\times (c_2 + 1) \\times \\cdots \\times (c_k + 1).\\qquad (2)", "7c9b9e84e2a1cc4ea3f6d96cc95160bb": " \\sum_{i=0}^{n} {n \\choose i} = 2^{n} ", "7c9c41bbf1c9b543440742d892df11d0": "J = \\frac{d}{dt} \\frac {1}{4 \\pi \\mathrm E^2} \\mathit E = \\frac{d}{dt} \\varepsilon_r\\varepsilon_0 \\mathit E = \\frac{d}{dt} \\mathit D \\ . ", "7c9c516758e61e5db7557e9c62d436ff": "\\mathbf{n}_{t} = \\mathbf{L}^t\\mathbf{n}_0", "7c9c70a09d6790bcc0d8725d1cd6ceb8": "f(z_1, z_2) = z_1 / z_2", "7c9c8f655efbd4eaed1260d43716ff5a": "(V_n, i_{nm})", "7c9cc6d4bc96fdaf883da4e18045b485": " B_n = n! \\begin{vmatrix}\n1 & 0 & \\cdots & 0 & 1 \\\\\n\\frac{1}{2!} & 1 & & 0 & 0 \\\\\n\\vdots & & \\ddots & & \\vdots \\\\\n\\frac{1}{n!} & \\frac{1}{(n-1)!} & & 1 & 0 \\\\\n\\frac{1}{(n+1)!} & \\frac{1}{n!} & \\cdots & \\frac{1}{2!} & 0\n\\end{vmatrix}", "7c9ce666cd951371ff2bbef21c4cf742": "\\begin{align}\n s_1 &= a_1,\\\\\n s_2 &= a_1 s_1 - 2 a_2,\\\\\n s_3 &= a_1 s_2 - a_2 s_1 + 3 a_3,\\\\\n s_4 &= a_1 s_3 - a_2 s_2 + a_3 s_1 - 4 a_4, \\\\\n & {} \\ \\ \\vdots\n \\end{align}", "7c9d2f72df88e6e9dbd092c2c8db0b1a": "\\operatorname{spin}(10)", "7c9d5cf5fdac8561139b67913600cd83": " K^* ", "7c9d7042f167264fdc539885e3a01f1b": " \\{ \\mathtt{Map^2,\\ Set^1,\\ string^0,\\ int^0} \\} ", "7c9da147bc5dbe6d89a8308180a94f5e": "1+z=(1+z_{\\mathrm{Doppler}})(1+z_{\\mathrm{expansion}})", "7c9db14ba69999f282efe3cb24a686cf": "JM^\\prime J\\subseteq M", "7c9dcb0212882d85f272ecfb3f361f6e": "\\{\\dots, 6, 8, \\dots, 16, 18, \\dots, 66, 68, 69, 80, \\dots \\}", "7c9e571b736fded9cf516033839fcf76": "f'(a) = \\lim_{x\\rarr a} \\frac{x^n-a^n}{x-a} = \\lim_{x\\rarr a} x^{n-1}+ax^{n-2}+ \\cdots +a^{n-2}x+a^{n-1} ", "7c9e65b9c0841d5456f0d8b88c28de9e": "[L^2,L_x] = [L^2,L_y] = [L^2,L_z] = 0~.\\,", "7c9e9fbcab66e6fcb0bf996d7fa04b61": "SHA(message || key)", "7c9ed35d326f1ebbd0c33fbdaf78e4c9": "\nu(\\mathbf{r})_{(\\text{Poisson})} = \\iiint \\mathrm{d}^3r' \\frac{f(\\mathbf{r}')}{4\\pi |\\mathbf{r} - \\mathbf{r}'|}.\n", "7c9ef1e4ff762c7d3b28615c3c930721": " \\mathbf{v}=\\nabla\\varphi.", "7c9efc8ee196a4dcc2cb63e6d4a607d6": "0 \\cdot P := \\emptyset \\qquad P \\in \\mathcal{E}(G)", "7c9f3579ed02afbc90cefde39b38451c": "AUC_{\\tau, \\text{ss}}", "7ca0269e39360003e89eb4a8a04e0c5f": "S\\sqrt{\\rho}v", "7ca02a71e84838f6898fe63595968ed0": "\\sum_i r_i < 5H\\,", "7ca03377feddef84c8bc081d53b8023b": "Y=X\\sqcup X^\\dagger", "7ca0531600b0623adc9a064cf675e644": "\\frac12(1+Z)^2\\,Z", "7ca05aeb2e0421aa34b8e4abbd946d64": "R_p\\;", "7ca05b65647e76211b80e9474fe226b4": "\n\\begin{align}\n\\mathbf{y}_p(t) & {} = e^{tA}\\int_0^t e^{-uA}\\mathbf{b}(u)\\,du+e^{tA}\\mathbf{c} \\\\\n& {} = \\int_0^t e^{(t-u)A}\\mathbf{b}(u)\\,du+e^{tA}\\mathbf{c}\n\\end{align} ~,\n", "7ca070976532db3c9455060940c238a9": "\\gamma^0", "7ca11e2dafa9ee31aa5ffa7b7049be2c": " d={\\lVert v(\\mathcal{N}_i)-v(\\mathcal{N}_j)\\rVert}_{2,a}^2 .", "7ca179bc08aa1bb9d1a16e8b3c18eaa1": "R = \\rho \\ell / A", "7ca1a4d005ded48fdf7dc6c0880331e0": "\n\\mathbf{ v} = -\\nabla\\psi\\times\\mathbf{\\hat z}.\n", "7ca1c8858f910dc2c4713b90c51c9283": "p_0,p_1,\\ldots,p_r", "7ca1d193cf6e380b266b973ee697a268": "\n\\begin{align}\n&\\text{VaR}_{1-\\alpha}(X):=\\inf_{t\\in\\mathbf{R}}\\{t:\\text{Pr}(X\\leq t)\\geq 1-\\alpha\\},\\\\\n&\\text{CVaR}_{1-\\alpha}(X) := \\frac{1}{\\alpha}\\int_0^{\\alpha} \\text{VaR}_{1-\\gamma}(X)d\\gamma,\\\\\n&\\text{EVaR}_{1-\\alpha}(X):=\\inf_{z>0}\\{z^{-1}\\ln(M_X(z)/\\alpha)\\},\n\\end{align}\n", "7ca1fc2bca050cae2c74bb7df7ad9ccb": "\\mathbf{s_1, s_2, ...}", "7ca207b008f767a7254d66881fa0aa9d": " \\mathrm{ROV} = \\frac{A(\\mathrm{Pos})}{A(\\mathrm{Pos})+A(\\mathrm{Neg})} \\times E[A_+]", "7ca217262ef14cab3f9a0ad7339dae4d": "O(D+L\\log n)", "7ca21a2ca03e2d8d9dc52a96a4a2074c": "S_r(r) = 0", "7ca22304a1a8e27c6597e2797a1f44af": "\\chi(U)", "7ca2547eeb894594ecbbd1cc0f4e0771": "\n\\Omega_{\\mathbf{k}}\n=\n\\mathbf{d} \\cdot \\mathbf{E} + \\sum_{\\mathbf{k}' \\neq \\mathbf{k}} V_{\\mathbf{k} - \\mathbf{k}'} P_{\\mathbf{k}'}\n", "7ca264a1525ad69977c9c5102574b771": " YY=Y\\cdot Y ", "7ca2ede109fe0a5cbaf1b7fdb837d5f8": "\\beta = {2\\pi \\over a}.", "7ca3533bb6e9190ac71bf6206f898168": "\\! J = 3", "7ca36fa6d1031a85e92a2515d5609f3f": " 4\\pi r^2 = \\pi d^2\\, ", "7ca38140f7f63bad566ef0c5d6fe9e2c": "F(\\mathbf u) = 0", "7ca3a315fb05f287937c03a65ae8d6cd": "\\Pi(n,k) = \\int_0^{\\pi/2}\\frac{d\\theta}{(1-n\\sin^2\\theta)\\sqrt {1-k^2 \\sin^2\\theta}}.", "7ca3bddaed254ac2e9262c1934ae1e64": "\\gamma\\colon L\\rightarrow L", "7ca3f17deae3dd8666f0dc07067269b8": "a_0\\,", "7ca467728da194abbcbfb073fc3320f6": "MC_{HK}=r+\\delta\\,", "7ca47eba0d57e3af2bd80061920c6b44": " R = N \\, r = N \\, \\Phi \\, \\sigma ", "7ca4a9be5706c828286c195a2e32d7f3": "G\\setminus C", "7ca4d56f66f008762d52e205e0286f31": "\\displaystyle \\sigma_\\lambda= \\det(\\sigma_{\\lambda_i+j-i})_{1\\le i,j\\le r}", "7ca527fb513b60bcac75b42c8ef40243": " \\gamma = \\frac{1}{\\sqrt{1-\\frac{u^2}{c^2}}} ", "7ca52c5e3703747d6430c3645889b07d": "(k, \\epsilon)", "7ca609cb74ab46d133b93d4eeac3517a": "T_{0,r\\#}\\nu \\in \\mathrm{Tan}(\\mu,a)", "7ca669fac35afe2499f74a8678de70fa": "\\mu_n(t) = E(X_t^n)", "7ca67222d83333a00abcbf0ce7dc58a2": "x\\sim y,", "7ca683196b56ef4972264d50bf881489": " \\xi_k = \\langle X - \\mu, \\varphi_k\\rangle, k = 1, 2, \\dots ", "7ca7202e129ac942522ab81b09e9a7de": "r^2=\\sum_i y_i^2", "7ca79a044d123f87411ab72c2a18a5e5": "{0\\choose 0}_2=1", "7ca7dbbe2bd3588cc3d1d0599a659f6f": " \\frac{\\delta\\Pi}{\\delta\\alpha} = 0 ", "7ca7e425658ad1e729e05dfb5d0f41cb": "h(\\theta; f) = \\underset{r\\to\\infty}{\\limsup}\\, \\frac{1}{r} \\log |f(r e^{i\\theta})|\\,", "7ca8343724c0ec7dae9861c57cb88929": "\\varphi = \\frac{1}{4\\pi\\epsilon_0}\\iiint\\frac{(\\bold{r}-\\bold{r}')\\cdot\\bold{P}}{|\\bold{r}-\\bold{r}'|^3}d^3\\bold{r'}", "7ca879b274d4c7c5907906ac20e3ff0f": "V \\ge LC_{50}", "7ca88ee24110b4d78030e329c7353ff9": "f:x\\mapsto h(\\alpha x)", "7ca8b8291d68cfe57b1aaa97cd39072b": "t/2", "7ca9279f9675d4a14b96364f637c885a": "R_X+R_Y \\geq 10", "7ca93caf2a7509a541ce061170464171": "\\mathrm{C}^0 \\; \\mathrm{A}^{2-8} \\; \\mathrm{G}^{\\overline{(2-4)}}", "7ca976022353f8f53611a952302af809": "F(x) = x^n+a_{n-1}x^{n-1}+\\ldots +a_1x+a_0", "7ca97b2ac74c46c7408896b36ff29cf1": "y\\sim f(x)=x^2-x+3\\,", "7ca99792bb13ad1c6a8688f929bea072": "G(n)=\\sum^{n+1}_{i=1}D^2_i ,", "7ca9eb57115e38c40e6e8d948232d62d": "V \\subset U", "7caa19facdfbd3060ee803b38bb5830c": "f(t_i)", "7caa24e2c97cb4d8e9dfb6cacc7728cf": "K = \\frac{L_H -L_L}{L_H}", "7caa5a3ecf43f7d83ee0e4fff735e2b8": " PLEASE ADD THIS TEMPLATE'S CATEGORIES AND INTERWIKIS TO THE /doc SUBPAGE, THANKS", "7caa8af80129e75cf22ee399730348d8": "T_n", "7caabff7ee450bacc63234c57a2d2824": " M_{-j}", "7caadadf201e2ea9918bba2ef3577492": "x_{n+1}, \\dots, x_m", "7cab6727a6ec74dd19ea1fea16590ebb": "r > 1 \\,,", "7cabd41cd6e23262e194e47541dca8c2": "\\pi(w)\\mapsto (\\pi_1(w), \\pi_2(w), \\ldots , \\pi_n(w)). \\,", "7cabf9392a9cca92746c36be161a33cd": " N_e ", "7cac156f832c5f332a51f88224caf069": "-169\\pm 15", "7cac5adf5949df0ce57cac8f3bbcdce3": "\\frac{|x|^p}{p}", "7cac5cce47a951b8a31fd47f38839d1c": " (c - \\varepsilon)b_n < a_n < (c + \\varepsilon)b_n ", "7cac8090b27f879876d0b731d08c2509": "L^{q_\\theta}(\\mu_2)", "7cacc25441a31ef27bfac47eaffde528": "f_{4a}", "7cad0e01b2063881abb342754f7b1661": "\\left(\\frac{1}{x}\\right)\\left(\\frac{1}{y}\\right)", "7cad6e8207668ff0f780150ab0beb484": " \\frac{Q_{SP}}{Q_T} = \\frac{CcO_2 - CaO_2}{5+(CcO_2 - CaO_2)}", "7cad92a44358e78847ff33096a07e799": "\n R_O = \\left(1 - \\frac{x_A}{L}\\right) F \\quad \\text{and} \\quad R_B = \\frac{x_A}{L}\\,F \\,.\n ", "7cada739b604018911c7f8c74eb6fd1f": "ax + by + cz = 0", "7cadec2db16fe945244cab72a2bae236": "P_\\mathrm{dBm} = 10\\ \\log_{10}(k_B T \\Delta f \\times 1000)", "7cae24c61a4c89ce5d584b5472e0cdb6": "a+R = r", "7cae4777ceaa2d1ae22af11601660e57": " \\pi (p)=\\max_{x\\in X}p\\cdot x=p\\cdot x^{\\ast }\\left( p\\right) \\text{.} ", "7cae9da3a782e9cd5fd774102c75a48f": " 0 < \\sigma _c < 1 ", "7caed8d37e1dca7f60655dc556818958": "\\{A_i\\}_{i\\geq 0}", "7caf2c67048a9c0f8a5abce388bee3ed": "H_1(X)", "7caf79a5b472723da027310ed335df30": "O(1.9129^n)", "7caf88482991af28febcc8a81e7602d2": "\\,{}_pF_q \\left[\\begin{matrix} \na_1 & a_2 & \\ldots & a_{p} \\\\ \nb_1 & b_2 & \\ldots & b_q \\end{matrix} \n; z \\right]", "7caf89c95d8cfe9f0db3a08a99755404": "\\operatorname{P}(X=b) = F_X(b) - \\lim_{x \\to b^{-}} F_X(x).", "7cafe1cb231274bc22b2ca3993299e68": "{1 \\over 1 - e^{-Ts}} \\int_0^T e^{-st} f(t)\\,dt ", "7caff9ab5cc1a172e2b5cff00c99b0c8": "0*\\,\\!", "7cb0353d19d66ea429bd48a4456d6d40": "\\int x\\arcsin(a\\,x)\\,dx=\n \\frac{x^2\\arcsin(a\\,x)}{2}-\n \\frac{\\arcsin(a\\,x)}{4\\,a^2}+\n \\frac{x\\sqrt{1-a^2\\,x^2}}{4\\,a}+C", "7cb0a9a069256de326a9c50e082ebb2e": "\\log\\, m(s)", "7cb1029b758275f31e108018b80a0aed": "t^2_i = 16", "7cb11772f12fd1d150c40e65b5e6b3d1": "(f(p_{1}, p_{2}, p_{5})", "7cb17b2a7044c775000636ea338788c5": " ds^2 = dx^2 +dy^2 + dz^2 - c^2 dt^2 \\equiv g_{\\mu \\nu } d \\acute{x}^{\\mu} d \\acute{x}^{\\nu} ", "7cb1837c599ede26336242dceee67cce": "\\tbinom{5}{0}=1", "7cb1a7256425d34f5abd725f2469847a": " F(a,m)=\\int_0^\\infty \\frac{dx}{(x^4+2ax^2+1)^{m+1}} ", "7cb1bdaedd16a3bcfbac6701548e84f3": "\\phi(\\mathbf{k})=\\frac{1}{(\\sqrt{2\\pi})^3} \\int_{\\mathbf{r}{\\rm-space}} \\psi(\\mathbf{r}) e^{-i \\mathbf{k}\\cdot\\mathbf{r}} {\\rm d}^3\\mathbf{r} ", "7cb268bdb73f3a72ac0f4b2caf8dcc9d": "\nm\\frac{d{V}}{d{t}} = -\\Upsilon(V - \\Gamma{u}) - \\nabla \\Phi[X] + \\xi + F_{thm}\n", "7cb2a2703fcfbb2b29891e34ef5dbb39": "0 < \\left|x- \\frac{p}{q}\\right| < \\frac{1}{q^n}\\, .", "7cb2bb83706e779b93b5dee813fd6fc6": " \\psi \\mapsto e^{i \\theta} \\psi", "7cb2c7b5049638ad42e3bdf0eb892f5e": " t_\\text{score} ", "7cb2cff696362ebc3d244826cd2f7c53": "f : S\\to \\mathbb{R}", "7cb32837c4182ea712c74396dc736b27": "\\int_0^1 \\frac{\\ln x}{(1+x)^2}\\,dx = -\\ln 2.", "7cb38b70dc01f28637d2d64ac19341ef": "\\sum_i e_i \\wedge e_{n+i}", "7cb3bcf4d54ec5baf82d6f2f42eb1c6b": "H=\\int {d^3k\\over (2\\pi)^3}\\frac{1}{2} a^\\dagger(\\vec{k}) a(\\vec{k}) , ", "7cb3e1381266aaa454d7d46a6a3c737e": " o(f(\\varepsilon)) ", "7cb3e57f0e129e953bba96b10af0d4c8": "\\,D(t)\\,", "7cb3ffd4e6320bbc110c25f43eddfaf5": " t( x, w ) = \\lim_{ u \\to w } \\frac{ \\arcsin( \\sqrt{ u \\, x } ) - \\sqrt{ u \\, x \\ ( 1 - u \\, x ) } }{ \\sqrt{ 2 \\mu } \\, u^{3/2} } ", "7cb4a4ae456c7d364e0ed6dc2df73b9b": "(x_2-x_1)(y_3-y_1)-(y_2-y_1)(x_3-x_1)", "7cb50cf64066fccd63f242b1a5cdec89": "\\max\\left|x_i-m\\right|", "7cb536690ab7891789daf1e3dbda0c56": "62.6~", "7cb5397e649e439a45bab1970aaf3c13": " \\operatorname{Diff}^1(M) ", "7cb5ef7315f9c006786d45809b96452e": "\\sqrt{\\sigma_S^D}\\sqrt{\\sigma_L^D}", "7cb65d09089f1b2a7dfb5ae3f4ef82ba": "P=i\\partial_\\mu", "7cb67c4b0e8387a9db6a9076fabb71eb": "d_{p}", "7cb79a453ac6b982d4a9454eb40dcbd4": "D = - \\frac{\\lambda}{c} \\, \\frac{d^2 n}{d \\lambda^2}. ", "7cb7a6f365fea9a81ad2af916f8da705": "\\{w_j\\}", "7cb7ae78f9a60729c36a2a048ef39625": "\\zeta \\left(1-s,\\frac{m}{n} \\right) =\n\\frac{2\\Gamma(s)}{ (2\\pi n)^s }\n\\sum_{k=1}^n \\left[\\cos\n\\left( \\frac {\\pi s} {2} -\\frac {2\\pi k m} {n} \\right)\\;\n\\zeta \\left( s,\\frac {k}{n} \\right)\\right]\n", "7cb7ea6a2dd136494381d74438256cb0": " cf \\beta < \\kappa ", "7cb80c56878145dab0b5fcdbcaa6cedb": " \\mathcal{G}(A)\\times \\mathcal{G}(A) \\to \\mathcal{G}(A)", "7cb829ca3fc7c9041a14f872a202224f": "\\scriptstyle R_i= \\sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}", "7cb83b83e7602ed56ce440eaac109fdf": "I_0", "7cb888905866a8fbc3b0c50cd8489d47": "\n\\begin{align}\n-1 &= i \\cdot i \\\\\n&= \\sqrt{-1} \\cdot \\sqrt{-1} \\\\\n&= \\sqrt{-1 \\cdot -1} \\\\\n&= \\sqrt{1} \\\\\n&= 1\n\\end{align}\n", "7cb8a2cf5ec5604213d8fdb8a9f42246": "\\phi_p = -\\frac{1} {4 \\pi} \\iint\\limits_S\\left(\\frac{\\sigma}{R} - \\mu \\cdot \\mathbf{n} \\cdot \\nabla \\frac{1}{R} \\right) dS", "7cb95ac151f635558194804999e4a5c5": "\\frac{\\pi^4}{24} R^8", "7cb98e5757237660b914d71557aeef2a": "Id(1-d)^2", "7cb98e5932fed00406a49e0760882dfc": "\\gamma_p", "7cb9a811c59fffb8f197e96037c09979": "\\{\\theta: \\theta = \\varphi_n \\mbox{ for an } n \\in A\\}", "7cb9aa289385656c41b0c5112152e94b": "\nN(t) = \\frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}.\n", "7cba522db8098e9d15a56ed08a577966": "Tr(g^b)\\in GF(p^2)", "7cba5563e97819a77243a13d8a5df59f": "n_\\mathbf{p} = {2 \\over u+(u+2)/ \\Omega + \\sqrt{u^2(1+2/\\Omega) + (u+2)^2 /\\Omega^2}}, ", "7cbaa25119e64e613537d0b132aed102": "y^2 = f(x) \\,", "7cbabb8ede8e0636b40ee2484be167d6": "x\\in\\mathcal{O}_{[g]}\\; y\\in\\mathcal{O}_{[h]}", "7cbad964914a6b87e7248c705d674895": "r_0 A^{1/3}", "7cbaeca5fd22c7ece998ea5572d33610": "\\Phi \\left(t,\\Theta,\\Theta^* \\right)=\\phi(t)+\\Theta\\Psi(t)-\\Theta^*\\Phi^*(t)+\\Theta\\Theta^* F(t)", "7cbb1ae6e7648b84f22ffd9133b51580": "n=1, d=3,", "7cbb21ccca11f61f806a4f620a0c2271": "q(x)[S]\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]+J(x)Q[\\phi(x)]\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]=\\partial_\\mu j^\\mu(x)\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]+J(x)Q[\\phi(x)]\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]=0.", "7cbb84be69adf0ebb50f87018bd43398": "L_{\\text{YES}} \\cap L_{\\text{NO}} = \\varnothing", "7cbbb154b24f024da51d4060d1ce856b": "\\frac{\\partial \\mathbf{U}^{-1}}{\\partial x} =", "7cbbba28d00ea184fddf5c5b1047adc8": "\\frac{\\Delta\\rho}{\\rho}=\\alpha\\Delta T-\\beta\\Delta S", "7cbbc224a2ea48f1f12d90557c8c87c1": "\\tbinom{n}{r}=\\tfrac{n!}{r!(n-r)!}", "7cbbc409ec990f19c78c75bd1e06f215": "70", "7cbbfa6f6385b4b486033c9c881d6a8f": "\\psi(\\mathbf{r})=\\int_{\\mathbf{k}{\\rm-space}} \\phi(\\mathbf{k}) \\psi_{\\mathbf{k}}(\\mathbf{r}) {\\rm d}^3\\mathbf{k}", "7cbc18a7dc2eb5fafb16b9ebbe9c9a1f": "({X\\cup X^{-1}})^+", "7cbc39f7398257cd460f6220566ddf0f": " M_X(t) = E\\left( e^{\\langle t, X \\rangle}\\right) ", "7cbc3a9c4384eb64a876bbce347cd6a7": "W_{ijk} = P_i L_{jk} + P_j L_{ki} + P_k L_{ij}", "7cbc48b37a2437e5604e8553a279f760": " = \\frac{1}{2bx}\n \\left\\{\\begin{matrix}\n \\exp \\left( -\\frac{\\mu-\\ln x}{b} \\right) & \\mbox{if }x < \\mu\n \\\\[8pt]\n \\exp \\left( -\\frac{\\ln x-\\mu}{b} \\right) & \\mbox{if }x \\geq \\mu\n \\end{matrix}\\right.\n ", "7cbc4e37bc72647ae2959a7822fb11ea": "\nf(x) = \\sum_{j=0}^\\infin \\frac{1}{j!}\\left(\\frac{\\lambda}{2}\\right)^je^{-\\lambda/2}\\frac{x^{\\alpha+j-1}(1-x)^{\\beta-1}}{B(\\alpha+j,\\beta)}.\n", "7cbcd85dc1816e47a0e5868cf3717a27": " (1-\\alpha), ", "7cbd0b3cde6b73b34fd8bd01ef69ffcc": "2\\chi", "7cbd2d63490c3ad67f46d5e6b3a20e61": "~|{\\rm initial}\\rangle~", "7cbd53498d0c0238dd34e962d5e1ef75": "\\pitchfork", "7cbd94973e0bebf9cc7d2818bae5941d": "e_{j,t}", "7cbe08e1d7cc16cfcb81cebc6289fa5d": "S = R - \\mathfrak{p}", "7cbe0ce8617252c7f6c3e608dc6722dc": "\\sigma_{11}(n)", "7cbe1050979d579fcb1ae7ba4f3d5cd5": "\\ 1-\\delta ", "7cbe262c16e22709eceb19816d0fed1e": "n! = n!! \\times (n-1)!!.", "7cbe3fe3031d7bd3f1502236bdbb1e8b": "\n2x_{2} = D - \\frac{r_{1}^{2} - r_{2}^{2}}{D}\n", "7cbe8b301677c9ba322af06f00efa5d9": "\\forall x\\ \\|P\\cdot x\\|_2 \\in \\left[\\|P^{-1}\\|_2^{-1}\\cdot\\|x\\|_2, \\|P\\|_2\\cdot\\|x\\|_2\\right]", "7cbec2160590dcf0641fc9f38efe2ed8": "x=1/2", "7cbefaf39da793b0813f90f77afaaf35": "<\\overline{16}_{-1H}>16_1 \\phi", "7cbf0f285214bc261bdf95940777f317": "\\gcd(a, b) = 1\\;", "7cbf1e9a1b667e4282ee27e10c2faff7": " \\bar{E_V} = E_V - \\frac{1}{3} \\Delta_{so} ", "7cbf1f91d5ddae9cb48dd126e383e8cf": "A=\\begin{pmatrix}a&b&c\\\\d&e&f\\\\g&h&i\\end{pmatrix},", "7cbf22275bb15fea701eeadafa2fd575": "\\{\\tau\\in S_n\\colon\\tau(i)=j\\}.", "7cc0b665e0f002fa3539e499e641d5d9": "v = u_1, \\ldots , u_D", "7cc1f8f2e5900816a89e7d63a4693706": "\\operatorname{Var} \\left( \\frac{a}{b} \\right) = \\left( \\frac{a}{b} \\right)^{2} \\left( \\frac{\\operatorname{Var}(a)}{a^2} + \\frac{\\operatorname{Var}(b)}{b^2}\\right).", "7cc23254193ed82f2a1ff4d434be921c": " t = T, \\; \\; z = Z, \\; \\; r = R, \\; \\; \\phi = \\Phi - \\omega \\, T", "7cc268248444fafb46ec60338dea3dc9": "\n \\boldsymbol{S} = 2~\\cfrac{\\partial W}{\\partial \\boldsymbol{C}}\n ", "7cc26b7e8e9c03c330e1fe722a74df75": "v_\\text{s} \\,", "7cc27414a100971baa6c503e788d372d": "\\R^d \\to \\R", "7cc28332706f2ce4c18ac43d03f5c52f": "f(T)v", "7cc2ad83a486730b894eae986fd91ae2": "\\psi' = S_1 y_1\\dots S_k y_k[\\varphi], \\qquad \\text{ where }S_i \\in \\{ \\forall ,\\exists , R\\}, \\ y_i \\in \\{ x_1,\\dots,x_m\\}", "7cc2d645ff2c9310366618614acb83e5": "\\mathbf{S}\\cdot d\\mathbf{A}=\\int_V\\nabla\\cdot \\mathbf{S} dV .", "7cc2dc5a0d351d6e6e0929b4a4f03dc5": "10\\times 10", "7cc38cd4cfb6ead0938e311700118086": "f_k[n] = \\frac{1}{T} \\int_{(n-1)T}^{nT} f_k(t)\\ dt \\,", "7cc38d147d163a4793447348a901f5b9": "x^2+b_{2}x-b_{2}b_{11}=0", "7cc398f70d4f7b37ddc2dd969d8c3696": " m \\ge 1 \\, ", "7cc399ceb284b38ae30fa98398e3dbe9": "\\textstyle \\int\\limits_{{V\\!ol}_I} {k^{ion}}{n_e}{n_0}\\, d{{V\\!ol}_I}={\\frac{n_e}{\\tau_n}}{V\\!ol_I}", "7cc3df9ed0ca4d0ddcc8b4e4da2fa534": " k^2 ~ = ~ \\omega^2 (\\mu \\epsilon) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (1.3) ", "7cc41a8e3ff743c3e95de8a9a8b5e8fb": "\\sum_{i=1}^k p_i e^{t_i}", "7cc41db92c85a4df95cea2bbfbbe14d3": " \\textbf{y}(t) = \\begin{bmatrix} 1& 0& 0& 0 \\end{bmatrix}\\textbf{x}(t)", "7cc4647cb583c1da0f38adf358e6b164": "E= h\\nu = E_i-E_f=\\frac{m_e q_e^4 (Z-1)^2}{8 h^2 \\epsilon_{0}^2} \\left( \\frac{1}{1^2} - \\frac{1}{2^2} \\right) \\,", "7cc482a6104c7b92addad51a3c6fd63e": "P\\parallel P'", "7cc49ade9bca137b5d34ed6d2b58d463": " \nE_{\\alpha}\\triangleq\\{x_{\\beta} : \\beta \\geq \\alpha \\}.\n", "7cc4c5a6dba6d24245fa5455237151b5": "\\phi_l", "7cc4c96250e9537bdae49494600226e6": "\\operatorname{atan2}(y, x) = \\begin{cases}\n\\arctan\\left(\\frac y x\\right) & \\qquad x > 0 \\\\\n\\arctan\\left(\\frac y x\\right) + \\pi& \\qquad y \\ge 0 , x < 0 \\\\\n\\arctan\\left(\\frac y x\\right) - \\pi& \\qquad y < 0 , x < 0 \\\\\n+\\frac{\\pi}{2} & \\qquad y > 0 , x = 0 \\\\\n-\\frac{\\pi}{2} & \\qquad y < 0 , x = 0 \\\\\n\\text{undefined} & \\qquad y = 0, x = 0\n\\end{cases}", "7cc51b8e3ae4edb85c527c3e07992841": "x \\cdot x^{-1} ", "7cc556b5fa0035e9ca24ab5f9d354ff2": "\n EI\\dfrac{d^2w}{dx^2} = \\dfrac{Pbx}{L}\n", "7cc5c1841915de8dc734d32aa64eaf30": "\\frac{dh}{dx} = \\frac{df(g(x))}{dg(x)} \\frac{dg(x)}{dx}.\\,", "7cc603a11654583901c45a55cc10fdd0": "x\\ \\vdash\\ y", "7cc6076a8fc1559e9ba5e27d9d3d48c6": "\nG(\\mathbf{r}) = \\frac{1}{2\\pi^2 r} \\; \\int_0^{+\\infty} \\mathrm{d}k_r \\; \\frac{k_r \\, \\sin k_r r }{k_r^2 + \\lambda^2}.\n", "7cc615e8f6b078b88e5bcd113403bf85": " Var[c_n^*] = \\frac{\\zeta(2) - 2(\\log 2)^2}{n^2} + O\\left( \\frac{(\\log n)^2}{n^{7/3}}\\right).", "7cc62684b8f7af1c260d7f7186d1fc65": "\\psi = \\omega e^{-i p \\cdot x}", "7cc62c60616c9beca012d44a186d52b5": "R_{ij} = {R^k}_{ikj}.", "7cc639bb1969ab20b5898fd9df696c51": "\n\\begin{pmatrix}-(\\alpha^4+\\alpha^{-5}x)&\\alpha^{-3}+\\alpha^{5}x+\\alpha^{7}x^2\\\\\n\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2&-(\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3)\\end{pmatrix}\n\\begin{pmatrix}S(x)\\Gamma(x)\\\\ x^6\\end{pmatrix}=\n\\begin{pmatrix}\n\\alpha^{-3}+\\alpha^{-2}x+\\alpha^{0}x^2+\\\\\n\\alpha^{-2}x^3+\\alpha^{-6}x^4\\\\\n\\alpha^{-4}+\\alpha^{4}x+\\alpha^{2}x^2+\\\\\n\\alpha^{-5}x^3\\end{pmatrix}.\n", "7cc6e2c1d270615d1322cb58fdcc24ba": "\\text{NIG}", "7cc6fb1d9d7fc19fecabd0ec87b1d9bd": "N = pq", "7cc742a0a043bd49a75b1c0b356926e4": "\\frac{T_p}{p}", "7cc78e04f5e8eed4bc3a40d9417f14d2": "\\tilde{A} = - i( \\tilde{U} + 1)( \\tilde{U} - 1 )^{-1} ", "7cc79cb29afc2f89b8208d08dd91e315": " y_1 = y_0 + h f(t_0, y_0). \\quad", "7cc7e0626ab4ae9c32d2a3473320bb87": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{4}{\\sqrt{3}},\\ \\pm2\\right)", "7cc7f94418da95dbece8f230e52c1e92": "\\mathfrak{sl}_3(\\mathbf R)", "7cc818544aaea1c5a7f5a121a47209ab": "\\mbox{pOH} = -\\log_{10} \\left[ \\mbox{OH}^- \\right]", "7cc83ec04e69e7c25c2f37b5477c1aae": " \\frac{\\partial c}{\\partial t} + \\nabla\\cdot \\left(\\vec{-D\\,\\nabla c + \\vec{v}\\, c}\\right) = R. ", "7cc8531270e078049171cc0de4472de6": "k_g", "7cc8a942db64d8249e0bc9682a8e3741": "a(n+100) \\equiv a(n) \\pmod{1000},", "7cc9af4318721a252974165049128c9a": "\\Phi(\\sqrt{\\frac{\\pi}{8}}x)", "7cca364e2ac4d46c90aefaad72a7516f": "(u_j,f_j)\\in U_j \\times F", "7cca7ba03b7b2f4d8f9d45e9447c8cfd": "m= (\\frac{-4x^3}{3}-4x^2+32x)|_0^2", "7ccb82e456fa312ff95fcf9d6b4f6410": "\\pi(s(x))=x", "7ccb92d075d7dcf91c759d9a7559b80a": "\n\\begin{bmatrix}\nn\\\\\nk\n\\end{bmatrix}_q\n=\n\\frac{[n]_q!}{[n-k]_q! [k]_q!}.\n", "7ccbe45463d53bf1a3350c8aeba49048": "\\{x_n|n\\in \\mathbb{N}\\}\\;", "7ccc025e2f375b602a554f3376a66941": "D = \\frac{1}{2}", "7ccc3e4bab00ea0a33023e7bab50dbf9": " M_n ", "7ccc6e58fadcf8fa6fe32161b1ea594f": "dz_3(t)", "7ccc788a87d5a0b3c2c55a52974c0df8": "a=b=(m-1)", "7cccc0e8089f83b85b8c7195c947c07d": "(\\Sigma \\cup\\{\\varepsilon\\})", "7ccd2f53b84f5520e4c36419d7fef241": "\\begin{matrix} 64 \\times {4 \\choose 1}{3 \\choose 1}{3 \\choose 1} = 2,304\\end{matrix}", "7ccd43e3e429d58827b53a93bcec149e": "\\mathbf{L}\\mathbf{x} = \\mathbf{b}", "7ccd4565687ec66e48eee1457f9afa41": "-\\pi/3", "7ccdaead937da59ac96f5a21ee86c218": "j=1,\\dots,k, \\delta=\\det(\\boldsymbol{\\Omega})^{\\frac{1}{k-1}},", "7ccddb5d7c725861291010f07ae3f7fb": " VarNC = 1 - \\frac{ 1 }{ N^2 }\\frac{ K }{ ( K - 1 ) } \\sum( f_i - \\frac{ N }{ K } )^2 ", "7cce357d35cb4992d18acb29a412f769": " u - u_0 = p(x-x_0) + q(y-y_0), \\,", "7cce6c385f4e16a5a29296cf3c839034": "u_x = v_y, \\quad v_x = u_y.", "7ccea8f7791b74bebbbedee867fe7d64": "P \\propto |\\psi_n (0)|^2 e^{-2 \\kappa W} ", "7ccef3da74a14882911f3f9f5c8498a4": "\\scriptstyle \\rho_l", "7ccf01dd83a831f7bb83ed312eea4097": "J_k(n)", "7ccf4d61905c7dcf650d2e69887fbbce": " I_{t-1} = 0 ", "7ccf98528d989c2629eed07cfb8b7892": "\n\\frac{\\mathrm{d}\\mathrm{Power}_{ext}}{\\mathrm{d}a} = 2v_{\\infty}^3\\rho A_D \\times \\left((1 - a)^2 - 2a(1- a)\\right)\n", "7ccfca654c068532d95dda3e75f95f67": "\\scriptstyle \\arctan(a/b)", "7ccfda5d07f8f10551ab041729154079": "\\frac{g(n+1)}{g(n)}=1-\\frac{1}{n+2}\\,", "7ccfe61f8843cf47c14050e9e3301052": "a_c=0.2", "7cd022a4264b96be55f50147af400b27": "\\,\\!z", "7cd0343dec72b59a37b07d5dd560cc02": " Y ( \\sin( C + \\theta), - \\sin \\theta , \\sin( A + \\theta) ) ", "7cd048be53a77c57ae752d6a958b74c7": "D_0 = r_1 r_2 A(Q_1,Q_2,Q_3)/r_3", "7cd04c4283c256121c464fdce207a733": "\\vec{F} = m \\vec{a}", "7cd082fcbfbb4f986cff17a120763abc": "(D/p)=-1", "7cd0b462f2469b180883fa05693581b7": "\\textrm{pH}_{new} = \\textrm{pK}_{a}+ \\log \\left ( \\frac{[\\textrm{A}^-] (2)}{[\\textrm{HA}](1/2)} \\right )", "7cd0ca845d9caed756efa6ec56b0f34f": "\nN(T) \\sim e^{-\\gamma T}\n", "7cd0f99f5d3b76017b35b2752ee2c5e3": "(3)\\; h_j=\\frac{y_1}{2}\\left(\\sqrt{1+8 F r_1^2}-1\\right)-y_1", "7cd182efd684f22425ef2a9841b37cb3": "\\frac{1}{|G|}\\sum_{g\\in G}\\rho(g)", "7cd1948b5ca5491f2ff869c0c71da387": "\\gamma_\\mu^*\\geq h > 0", "7cd1c92b95156c468126760c509a9a6b": "\\displaystyle{\\mu^\\prime(w) =-{\\overline{g_w}\\over g_w} \\cdot \\mu(g(w))}", "7cd1dc0161f8fba9ce5889c3bd00834d": "\\{\\textbf{b}_1,\\ldots,\\textbf{b}_d\\}", "7cd202cdc2c7085cb4b8791ad514291f": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=\\frac{r_s c^{2}}{2 h^{2}}+\\frac{3 r_s}{2}u^{2}", "7cd243bb6905d32ff05a506cec9858df": "\n \\mathbf{A}^k = \\begin{pmatrix} \n A_{11} & 0 & \\cdots & 0 \\\\ \n 0 & A_{22} & \\cdots & 0 \\\\ \n \\vdots & \\vdots & \\ddots & \\vdots \\\\ \n 0 & 0 & \\cdots & A_{nn}\n \\end{pmatrix}^k =\n\\begin{pmatrix} \n A_{11}^k & 0 & \\cdots & 0 \\\\ \n 0 & A_{22}^k & \\cdots & 0 \\\\ \n \\vdots & \\vdots & \\ddots & \\vdots \\\\ \n 0 & 0 & \\cdots & A_{nn}^k\n \\end{pmatrix}\n", "7cd27edb1d345c6b48dcd5f7a1931650": "C\\subset \\mathbb{P}^n", "7cd2b8a3426bb39d07b03c70573db4bf": "\n \\boldsymbol{\\nabla}\\times(\\boldsymbol{\\nabla}\\mathbf{u}) = \\boldsymbol{0}.\n ", "7cd2c712c0bae3e66e6327e1c7ef7a4c": "P(x\\mid\\theta)", "7cd2f89016196ec598c755c4ca4e774e": "\\int \\psi_e'^* \\boldsymbol{\\mu}_e \\psi_e \\,d\\tau_e", "7cd30aa91d762a6ef6edb60e9c3a7c15": "\ny = k_{y1} \\cdot \\frac{I_2 - I_1}{I_0 - 1.02(I_4-I_3)} \\cdot \\frac{0.7(I_4+I_3) + I_0}{I_0 + 1.02(I_4-I_3)} \n", "7cd338e01cb5b9294b3e086e64f33843": "A_\\mathrm{cm}", "7cd368b922e0bfb9dd7af94f06f43ebd": "[fg,h] = f[g,h] + [f,h]g", "7cd40f2040570a73c996000ae352d64f": " \\{\\cdot,\\cdot,\\cdot\\}_+\\colon V_-\\times S^2V_+ \\to V_+", "7cd42c5990c99de4b515f29d8e273ae8": " y_b=b_0\\sum_{r=0}^{\\infty }{\\left( \\frac{(c-\\beta )(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}s^{r+c} \\right)}=b_{0}s^c\\sum_{r=0}^{\\infty }{M_{r}s^{r}}", "7cd43d260a4a96237c117092be946f13": " \\omega", "7cd44bce06e699c31eaa392fc037aae4": " p(x,y) = 0", "7cd51473a986a18c0d63bca0508b6edb": " \\mathrm{He} = { \\rho {\\ D^2 } {\\ \\tau_o} \\over {{\\mu}^2}} ", "7cd579d6d6d8e489ee63d1059617ddb3": " a^5 - b^5 = (a - b)(a^4 + a^3 b + a^2 b^2 + a b^3 + b^4).\\,\\!", "7cd5b4d209fb42d457a37d7c687488bf": "D(O_j)", "7cd5c075a953e0079ed7d1190cff0f09": "R_\\mathrm{h}", "7cd640dc98b2f241de63e88cc6b5bb83": " [\\mathbf{AB}]_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + \\cdots + A_{i,n}B_{n,j} = \\sum_{r=1}^n A_{i,r}B_{r,j}", "7cd64324f6900644c64beebc06b1df2f": "k_{m,n} = R_{m,n}/a", "7cd659e8f77eaed0612b999ba71b6f60": "w^2+A^2=b^2", "7cd67128152ea569a625b1f8e35e0b3d": "\\widetilde{M}", "7cd67378c2d1fa86a5a9f79d6f68b441": "P(t)= \\frac{1-t Z(t)}{1-t}.", "7cd67d61ad3d42dd2e63eb65991750f1": "\\tfrac{3\\sqrt{2}}{4}", "7cd705a30211e17401015155088b9c62": "\\frac{\\partial}{\\partial x}F(x)", "7cd74886ad0a0d9bd5a988a6ba900370": "200 + 22 t = 30t \\quad", "7cd7686b501c5b41450c6796ae821e43": "f(\\sigma, \\cdot) \\,", "7cd7cd1765e3e98c74c37bb35152c807": "\\frac{\\partial S}{\\partial \\beta_j}=2\\sum_i r_i\\frac{\\partial r_i}{\\partial \\beta_j}=0 \\quad (j=1,\\ldots,n).", "7cd8767f783d6bfb002029a241512994": "R(z;A)", "7cd8aba3f27a64c5629e57b1e14bca0a": " E_1 = \\frac{ 1 - D }{ 1 - \\frac{ 1 }{ K } } ", "7cd8bf2ca95131981dfb1642cf1d84fe": "\\tilde \\partial", "7cd9335852bd1b1770ebf092c85fa206": "C_k = \\left\\{\n\\begin{array}{lr}\n\\mathbb Z & k \\in \\{0,n\\} \\\\\n0 & k \\notin \\{0,n\\}\n\\end{array}\n\\right.", "7cd996af9a63497575429dfb5b85c29b": "E(Q) ", "7cd9ac9d5695dfc52de092142532cb78": "\\operatorname{BER}=\\frac{1}{2}\\operatorname{erfc}(\\sqrt{E_b/N_0})", "7cd9c1703c08464b3ce04eea2ff1f130": "\\ \\Delta S = \\frac{\\Delta s}{c_p} = \\ln\\left[\\left(\\frac{1}{H} - 1\\right)^\\frac{\\gamma - 1}{2\\gamma}\\left(\\frac{2}{\\gamma - 1}\\right)^\\frac{\\gamma - 1}{2\\gamma}\\left(\\frac{\\gamma + 1}{2}\\right)^\\frac{\\gamma + 1}{2\\gamma}\\left(H\\right)^\\frac{\\gamma + 1}{2\\gamma}\\right] ", "7cd9d8ef332009ea407cd5c0ad567afa": " b\\,", "7cda06fd1deb5087f103d364e67958d1": "C_{p~max}", "7cda5d1c096bb7bfd3d80b860d2f6625": "\\approx \\frac {18904}{17843} \\approx 1.05946309477106 ", "7cdab26111f0bd6a0e277cc67f74146c": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n-{1 \\over \\vec k^2 + \\omega_p^2}\n.", "7cdabbdb6e7d15191141e472227861f0": "{\\mathbf{}}F_i^r, K_i^r, L_i^r ", "7cdb13f95b41b79506f5b7bf973fa5a2": " \\mathbb{R}^{n} ", "7cdb1a3d4beedfd1cfe7b28782e88c16": "\nF = B_{x} \\xi + B_{y} \\eta + C\\,\n", "7cdb53aceca8a316b5ed6b053c0ed4d3": "\\sum_{0}^{19} $50,000=$1,000,000", "7cdb6573cc4002e2ac8494fc543fa10c": "\\lim_{n \\to \\infty} Cov (f \\circ T^n, g) = 0", "7cdb658f5583cd559fe07d8f06408618": "f(G) = \\Omega", "7cdbd70007c61d47b350630b45132ff0": " R(\\beta) = - \\frac{M}{N_\\nu} (f '' + 2\\eta Y + 2k\\beta^2)\\beta^2", "7cdbf5093e71bac7d573e55c65048c03": "\\mathrm{FCF} = {\\mbox{NOPAT}} - {\\mbox{Change in NOA}}", "7cdc2ed65a20fc81c372d5ab73d22dc6": "(N^{\\prime}-1)!", "7cdce149d426b2a44098501fb253ca1e": "\\pi({\\mathbb B})=\\min\\big\\{ |A|:A\\subseteq {\\mathbb B}\\setminus \\{0\\}", "7cddbf0d882b7af318f14e51737d2878": " \\langle u , v \\rangle = \\frac{1}{2} \\left( Q(u+v) - Q(u) - Q(v) \\right) ", "7cddbfec93f61c6ffa0f384b721224e1": "\n \\begin{align}\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} & + N~\\cfrac{\\partial^2 w}{\\partial x^2} + m~\\frac{\\partial^2 w}{\\partial t^2} - \\left(J+\\cfrac{mEI}{\\kappa AG}\\right)~\\cfrac{\\partial^4 w}{\\partial x^2 \\partial t^2} + \\cfrac{mJ}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial t^4} + \\cfrac{J \\eta(x)}{\\kappa AG}~\\cfrac{\\partial^3 w}{\\partial t^3} \\\\\n & -\\cfrac{EI}{\\kappa AG}~\\cfrac{\\partial^2}{\\partial x^2}\\left(\\eta(x)\\cfrac{\\partial w}{\\partial t}\\right) + \\eta(x)\\cfrac{\\partial w}{\\partial t} = q + \\cfrac{J}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial t^2} - \\cfrac{EI}{\\kappa A G}~\\frac{\\partial^2 q}{\\partial x^2}\n \\end{align}\n", "7cdddfe9513dc3c8088b6295efeb5b76": "h(n)=3+1+0+1+2+3+3+4+3+2+4+4+4+1+1=36", "7cdde47a306521c4391675d40fde38c2": "\\gamma_d", "7cde0b3d7c6ae6c6b1f947c194dcd6a1": " sA\\,", "7cde1d665782328835d91b85950cc60c": "\n\\widehat{\\mathbf{v}}=\\alpha D_{\\alpha }|\\widehat{\\mathbf{p}}^{2}|^{\\alpha\n/2-1}\\widehat{\\mathbf{p}}\\,. \n", "7cde5106a4ccce72e71c13547d8abfae": "K_b = RMT_b^2/\\Delta H_{\\mathrm{vap}}", "7cde86e066bcbb1a2f3979dca1fd8215": "\\hslash {\\omega }_{1}=\\hslash {\\omega }_{2}+\\hslash {\\omega }_{3}", "7cded230cd33ef5cc0b1adb36e3b6209": " \\bar{\\alpha} \\le n \\cdot \\alpha_\\mathrm{\\{per\\ comparison\\}},", "7cdf316392f86e4018773478430ee063": "e^{\\frac{\\boldsymbol{\\Omega}\\theta}{2}} = \\cos\\left(\\frac{\\theta}{2}\\right) + \\boldsymbol{\\Omega}\\sin\\left(\\frac{\\theta}{2}\\right) ", "7cdf5592f718adf0769f887433beac7c": "(t_0,\\cdots,t_n) \\mapsto \\sum_{i = 0}^n t_i v_i", "7cdfa5eaefda939b0f11368ab8dea77e": "X_i \\cap X_j \\subseteq X_k", "7cdfc5d900b036b0e49e2a9c59f8cdff": " \\arcsin x = y \\, ", "7cdfdd38d0ece8325bb74b544005c87b": "x=a\\tan(\\theta),\\quad dx=a\\sec^2(\\theta)\\,d\\theta, \\quad \\theta=\\arctan\\left(\\tfrac{x}{a}\\right)", "7cdfe3398e127ced45967a23d1324719": "\\ell_0 = [L_0:L_1:L_2]", "7ce057a646c5f0f535f0ccca6cee50ad": "\\scriptstyle v\\in T_xM", "7ce06eec896c0a784b33fc371bb49cda": "\\mathcal{L}(\\phi,\\partial_\\mu\\phi,x)", "7ce07b3742db81988d72c2b1f3c66653": " P_{i+1} = A_i \\left( P_i - P_i C ^\\mathrm T _i \\left( C_i P_i C ^\\mathrm T _i+W_i \\right)^{-1} C_i P_i \\right) A ^\\mathrm T _i+V_i, P_0=E \\left( {\\mathbf{x}}_0{\\mathbf{x}}^\\mathrm T_0 \\right). ", "7ce09ee151273fd0d98509970099ddc1": " b_+ \\le 2 e + 3 \\sigma + 5 \\, ", "7ce0bbc8fe376ebbc6ef1cb3ad84cfc9": "u\\in H_0^1(\\Omega)", "7ce0dbff53217d2960f8dec2cf663958": "\\Omega_n - \\Omega_0 = \\left [n(n-1) + \\frac{nf(u_{n-1})}{u_{n-1}}\\right ] \\delta_0\\Omega_0. ", "7ce117acd373cd5a4bb782f4f43367a8": " H_{\\frac{1}{4}} = 4-\\tfrac{\\pi}{2} - 3\\ln{2}", "7ce13ca0a45a130657a56a3a83c827c3": "D_{\\rm Chebyshev}(p,q) := \\max_i(|p_i - q_i|).\\ ", "7ce1412915d9e2889898fbaf60a116d1": "\\tilde{\\epsilon}_{\\mathbf{k}}", "7ce1645b26241955b5af16d8cd6aaa7c": "w = x \\cdot p_1 \\,", "7ce1c64a3ab2b59058538477dce41f0e": "\\alpha _{n}", "7ce224819137b9650f30586504f488dd": "p = m \\gamma v", "7ce256e00d0e5b11b06f73999a3f506f": " \\hat{\\mathbf{T}} ", "7ce269c3486b78299778636a633346e4": "A \\in \\mathcal{A} \\supseteq \\mathcal B", "7ce2851a46c56095322b796123f345f7": "\\delta W = \\sum_{i=1}^n \\mathbf {F}_{i} \\cdot \\delta \\mathbf r_i", "7ce285453c995488d080e018df73c87a": " s^*", "7ce28dbcdff9f5472624eb34ea736ca3": "\\{-\\varepsilon_1,\\dots,-\\varepsilon_M\\}", "7ce2918b2fa9ccd4ec484bf663c4a1c1": "\\displaystyle{[T,R(c,d)] = R(Dc,d) + R(c,Dd).}", "7ce2bcc86376645262eb4c60bb770395": "\n[{\\mathbf x}^']_{\\times} \\left( \\sum_i x_i {\\mathbf T}_i \\right) {\\mathbf l}^{''} = {\\mathbf 0}\n", "7ce2cbe6aebe6088f9ed6b90a183e3ca": "w_1, w_2, w_3, w_4", "7ce2d18f80b3173a506b8506e990115d": "\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{bmatrix} \\qquad (\\text{reflection across }x\\text{-axis})", "7ce30fe4c5311cd401500e2ff4ee33af": "\\triangle ABC\\sim\\triangle DEF \\, ", "7ce32cde3388f4aa59ed31abda8b96cd": "\\mathbb{H} \\, \\mathbb{T}_r \\, \\tilde{x} = \\mathbb{H} \\, \\tilde{x}_r", "7ce335a3cf25cdefb9b68da889a0956c": "M_4, \\, M_5", "7ce338d2d9adccd6eb33721a1bde8b59": " (X_t)_{t \\in T}, (Y_t)_{t \\in T}", "7ce33efd7b322bcfa5c9902cbcb51a77": " \\frac{{\\rm d}a}{{\\rm d}N} = C \\Delta K^m =C(\\Delta\\sigma Y \\sqrt{\\pi a})^m ", "7ce35a502a04508217c8070101f47e66": "\\boldsymbol{\\alpha}(itI-\\Theta)^{-1}\\Theta\\mathbf{1}", "7ce3cada94bc0a6ca9fd5f4c6028b931": " B_{ EP } = \\min\\{ 1, k^{ -2 }, 2 B_E \\} ", "7ce466f37bf4e027c314883bf66cd4b8": "\\mathcal{O}_k", "7ce4bc0e1c485dc1b67ed9e60df82bcf": "\\overrightarrow{s}\\cdot\\overrightarrow{j} = {1\\over 2} \\left(\\overrightarrow{j}\\cdot \\overrightarrow{j} - \\overrightarrow{l}\\cdot \\overrightarrow{l} + \\overrightarrow{s}\\cdot \\overrightarrow{s}\\right)", "7ce4c7a0bc5f48a55dc7eb883d8b642a": "\\mathbf{E}^p(G) \\supseteq \\mathbf{A}^p(G) \\supseteq \\mathbf{O}^p(G).", "7ce4e8794142016d78f24cd3e86e2d57": "\\operatorname{var}\\left[\\ln \\left (\\frac{X}{1-X} \\right )\\right]=\\operatorname{var}\\left[\\ln \\left (\\frac{1-X}{X} \\right ) \\right]=-\\operatorname{cov}\\left [\\ln \\left (\\frac{X}{1-X} \\right ), \\ln \\left (\\frac{1-X}{X} \\right ) \\right]= \\psi_1(\\alpha) + \\psi_1(\\beta)", "7ce5314aa2aa96f0b1644d69cbc293b6": "AUC = (G+1)/2", "7ce539d63e7aaa9ce5d9f98ac93c5a4d": "p \\leftrightarrow p", "7ce55367d3d177f2a774c1b36b9daf4c": "{\\partial^n \\over \\partial x_1\\,\\cdots\\,\\partial x_n} (uv)\n= \\sum_S {\\partial^{|S|} u \\over \\prod_{i\\in S} \\partial x_i} \\cdot {\\partial^{n-|S|} v \\over \\prod_{i\\not\\in S} \\partial x_i}", "7ce59bb71bc20319db792904306c55c5": "\\hat {\\mathbf Z}", "7ce5e52638358f56b6196e7cf168f468": "K[\\theta_1, \\cdots, \\theta_N]\\otimes V.", "7ce60387367ab052771b804b28a0dd79": "0\\rightarrow \\pi_i(S^3)\\rightarrow \\pi_i(S^2)\\rightarrow \\pi_{i-1}(S^1)\\rightarrow 0 . \\,\\!", "7ce63b9ca99a3d35d9d1929ffa4c56eb": "\n p := -2D_1~J(J-1) ~;~~ p^{*} = -2D_1~J(J-1) + 2C_1\n ", "7ce6d1ae637ae2d40060b0812027e005": "X \\equiv \\sqrt{15347} - 124 \\pmod{p}", "7ce72075691dd74f8962f87d18dc5fca": "=\n-\\frac{T_2(n)(2\\gamma\\log2-\\log^22)}{2}\n-\\frac{T_3(n)(2\\gamma\\log3-\\log^23)}{3}\n-\\frac{T_4(n)(2\\gamma\\log4-\\log^24)}{4}\n-\\dots\n,\n", "7ce74d817cb966cada68808f1140fff7": "\\mathcal C(x_1, x_2)=\\left \\langle 0 |\\mathcal T\\phi_i(x_1)\\phi_i(x_2)|0\\right \\rangle=\\overline{\\phi_i(x_1)\\phi_i(x_2)}=i\\Delta_F(x_1-x_2)\n=i\\int{\\frac{d^4k}{(2\\pi)^4}\\frac{e^{-ik(x_1-x_2)}}{(k^2-m^2)+i\\epsilon}}.", "7ce79645cbfb7e22532f4deb46115a4d": "ln \\mathbf{L(W)}", "7ce7b3fadf748f077f6c854d5fcddc0a": "\n\\mathcal{A}_\\phi=\\langle u|i\\partial_\\phi u\\rangle=\\sin^2{\\theta\\over 2}\n", "7ce7e57bfd44fbe790acebcb5b7aaffa": "\\textstyle\\frac {2}{2-1}+", "7ce837d05c58b5fc24d84f45262e9c50": " \\ddot{q} = f_1(q,\\dot{q},t) + f_2(q, \\dot{q}, t)u ", "7ce90203beb727113a4b07528bf53ec7": "=(1\\text{eV}^{-1})\\hbar ", "7ce9318f63173df42cdfcdcc9d2c9073": "(1 + p^2 m^{-2}c^{-2})", "7ce95ea4a133e87122a19e8613470e68": "S = \\bigoplus_{n=0}^\\infty S_n.", "7ce96e5b033e3ca45fe2fc45eab816f6": "\nT_0 + T_0 \\rightarrow S^{*} + S_0 + \\text{phonons}\n", "7ce980d7e9b52a3121a2abc7070e0b8d": " {G_i}^2 = m(G_i+l^{-1}E_i)-1. ", "7ce990bd1926b0212615fb07220be280": " W_{PPT}", "7ce9ec859cbf0e6f2341dd1ab484347f": "t_2=0", "7cea40d7a6c13db2d4908472bcd7b8f6": "\\left(\\mathbf{a}\\mathbf{b}\\right)\\cdot\\left(\\mathbf{c}\\mathbf{d}\\right) = \\mathbf{a}\\left(\\mathbf{b}\\cdot\\mathbf{c}\\right)\\mathbf{d}= \\left(\\mathbf{b}\\cdot\\mathbf{c}\\right)\\mathbf{a}\\mathbf{d}", "7cea42b00df9e2ff6c5c53c857e00d02": "p, q", "7cea4c99b3b091fec15d47a499f5aa3e": "\\overline{3} + \\overline{3} = \\overline{2}", "7cead12c9d83bef16684309291b391da": "k \\to \\infty", "7ceb0a09a5166e99f60902d0eab2c616": "p(\\mu|D, I) \\propto \\; A^{-\\frac{\\nu + 1}{2}} \\int_0^\\infty z^{(\\nu-1)/2} \\exp(-z) dz", "7ceb64f3e3a134ae0d8d8205be0c1218": "\\sum _x \\operatorname{sexp}_a (x) = \\ln_a \\frac{(\\operatorname{sexp}_a (x))'}{(\\ln a)^x} + C \\,", "7ceb6badbca25731a0f8638180074480": "L_0 \\; = \\; G_B \\; + \\; G_M \\; + \\; 20 \\; (\\log \\lambda \\; - \\; \\log d) \\; - \\; 22 ", "7ceb8ed61c6c481db7e8eb8c472a618d": "4_3", "7cebc528e37d39c2054c985f54354a06": "g_{obs}=0", "7cec2060b72da390a7df42a22c421ad9": "\\mbox{R} = \\mbox{RE}\\cap\\mbox{co-RE}.", "7cec4a6fec6589e637a4667e5ce9a5c9": "x = a\\sqrt{(1+\\zeta^2)(1-\\xi^2)}\\,\\cos \\phi\\,", "7cec793564b007d5efb5f9fffc3fee9a": "B_H(\\cdot)", "7ced590c2919c328da75241de277f8fe": "d_1 = \\gcd(u_1,u_2) = x-1", "7ced8be3b45cac8a5db2c59d804379b5": "{}+ c_1a_2j + c_1b_2ji + c_1c_2j^2 + c_1d_2jk", "7cedd417151aff6048bcd40b294ddb0f": " [E] = \\frac{K_m[ES]}{[S]} ", "7cee177a13fabe2351515ecb8f6eeb4f": "\\langle x_ix_j \\rangle", "7cee20ca13769e9ed4dab25d62036eed": "| \\Psi(t+\\delta) \\rangle = e^{{-i\\delta}\\frac{F}{2}}e^{{-i\\delta}G}e^{{-i\\delta}\\frac{F}{2}}|\\Psi(t) \\rangle,", "7cee30f730897fd9f6d2af70b42cc7b9": " I(z) = I_0 \\propto |\\mathbf{E}_0|^2", "7cee5b92c1cb1b4cc3768a216ebe18d5": "T^{\\mu\\nu} =\\begin{bmatrix} \\frac{1}{8\\pi}(E^2+B^2) & S_x/c & S_y/c & S_z/c \\\\ S_x/c & -\\sigma_{xx} & -\\sigma_{xy} & -\\sigma_{xz} \\\\\nS_y/c & -\\sigma_{yx} & -\\sigma_{yy} & -\\sigma_{yz} \\\\\nS_z/c & -\\sigma_{zx} & -\\sigma_{zy} & -\\sigma_{zz} \\end{bmatrix}", "7cee7874e9725835eb8565f525f83947": "2^{2^{23}} + 1", "7cee809ca7801271e72c788be4c81b25": "\\beta_j = \\min_{x \\in S_{m-j+1}, \\|x\\| = 1} (Bx, x) = \\min_{x \\in S_{m-j+1}, \\|x\\| = 1} (P^*APx, x)= \\min_{x \\in S_{m-j+1}, \\|x\\| = 1} (Ax, x) \\leq \\alpha_{n-m+j},", "7ceed1b7bbcc3f7837cb3dc2b5376a0a": "a_1, a_1\\zeta_n, a_1\\zeta_n^2, \\dots, a_1\\zeta_n^{n-1}", "7ceedd74188035686c1392797b950cc3": "(\\log^{10.5}(n))", "7ceeec45f75b915db2c9faed6e7e49e5": "\\left(\\rho u A \\right)_{e} - \\left(\\rho u A \\right)_w = 0 \\, ", "7cef25bb51a087146c7cda0e31b09cba": "f=F(x+{\\rm i}y)", "7cef8a734855777c2a9d0caf42666e69": "open", "7cefb8ac8a2eb13f1a2ca1e117935e80": "{\\tilde{A}}_{kn-1}", "7cefd6916dbf70e05e3e548e1eb9729f": "\n \\begin{bmatrix} \\sigma_1 \\\\ \\sigma_2 \\\\ \\sigma_3 \\end{bmatrix} = \n \\tfrac{1}{\\sqrt{3}} \\begin{bmatrix} \\xi \\\\ \\xi \\\\ \\xi \\end{bmatrix} + \n \\sqrt{\\tfrac{2}{3}}~\\rho~\\begin{bmatrix} \\sin\\left(\\theta-\\tfrac{2\\pi}{3}\\right) \\\\ \\sin\\theta \\\\ \\sin\\left(\\theta+\\tfrac{2\\pi}{3}\\right) \\end{bmatrix}\n= \\tfrac{1}{\\sqrt{3}} \\begin{bmatrix} \\xi \\\\ \\xi \\\\ \\xi \\end{bmatrix} + \n \\sqrt{\\tfrac{2}{3}}~\\rho~\\begin{bmatrix} -\\cos\\left(\\tfrac{\\pi}{6}-\\theta\\right) \\\\ \\sin\\theta \\\\ \\cos\\left(\\tfrac{\\pi}{6}+\\theta\\right) \\end{bmatrix}\n \\,.\n ", "7cf02dfcd27b498f5a27e9b34dc6fc65": " i = 1 + \\alpha (n - 1) ", "7cf086e6ee03a121914c4b3d91f9c681": "\\nabla_2 := (\\nabla \\otimes \\nabla) \\circ (id \\otimes \\tau \\otimes id) : (B \\otimes B) \\otimes (B \\otimes B) \\to (B \\otimes B) ", "7cf12e3bf928c8e7e1d53ec877462284": "\\ r=a\\cdot\\frac{1-\\varepsilon^2}{1+\\varepsilon\\cdot\\cos \\theta}.", "7cf1353fec4684211bd1f2ddec1ab0ea": "A_{\\alpha \\beta} \\in L^{\\infty} (\\Omega) \\mbox{ for all } | \\alpha |, | \\beta | \\leq k.", "7cf144516fd735e8a4ba340920cde7ac": "\\frac{\\left \\Vert v_{Target} \\right \\| }{ \\sin(\\theta_{Deflection}) } = \\frac{\\left \\Vert v_{Torpedo} \\right \\| }{ \\sin(\\theta_{Bow}) } ", "7cf15f00ebd0d0bfb2bd920cb0fc7cea": "\\alpha = \\frac {d\\omega}{dt}", "7cf19b8e8b3c7b73f78c3f51504774bb": "R = \\left( \\frac{n_0 - n_S}{n_0 + n_S} \\right) ^2", "7cf1d643423fd75c66d366e9af58b261": "S = cA^z", "7cf2828f62b86622ae452e0453bc39a0": "\\omega = \\tfrac{L_r}{L} - \\tfrac{C}{C_l}", "7cf28e33b9ca92c6614874885dcab1ff": "f_1,f_2,\\ldots,f_{n-r}", "7cf2a263097b823e9cfc3190debf7b45": "\\{\\sigma_{j}^{+},\\sigma_{j}^{-}\\} = 1", "7cf2ae448ab63be557262c8b366630ee": "\\chi_k\\times\\psi_l", "7cf2bd54852bda369758f010e9ae4bb6": "\\begin{align}\n\\mathbf{T} & = \\mathbf{X} \\mathbf{W} \\\\\n & = \\mathbf{U}\\mathbf{\\Sigma}\\mathbf{W}^T \\mathbf{W} \\\\\n & = \\mathbf{U}\\mathbf{\\Sigma}\n\\end{align}", "7cf2db35307260510b1a57c69b069ef7": "\\vec{v} = \\vec{v}_{\\parallel}+\\vec{v}_\\perp", "7cf341dacc767a5e9e4b8612d16640f3": "\\scriptstyle 1.05m", "7cf37433a3927c60e5b84ee6252c4e2f": "d(\\cdot,\\cdot)", "7cf376d9d38116e2ebc3f3d8703cc33d": "A, s \\in \\mathbb{C}", "7cf37dd7311883cfbbdfc98ae6c49c2b": "z_3 = x_3 y_1 - x_4 y_2 + x_1 y_3 + x_2 y_4 + u_3 y_5 - u_4 y_6 + u_1 y_7 + u_2 y_8", "7cf3904c6e8b2e29fbc2167cb3fd5b19": " d' = 3 - d ", "7cf39293a77f17e4a8ed97ff388e6db5": "\\bigcup M_i", "7cf39e78f657a837062d1724a0ccbd80": "\\sigma_x(\\tau) = \\frac{\\tau}{\\sqrt{3}}Mod.\\sigma_y(\\tau).", "7cf4345831f772247738b5502ba057cb": "\\displaystyle W\\left( i\\right) =\\tfrac{1}{2}\\sum \\limits_{m,n=1}^{K}i_{m}L_{m,n}i_{n}.", "7cf480679e5f0d34a53332716ce2e01e": "\\Delta P = \\alpha\\left(\\frac{1}{r_t}+\\frac{1}{r_b}\\right)", "7cf4b11570b861a4e20298258115fde4": "~v^{}~", "7cf4d46ff7222391aee12f4042054aa9": "\n \\left\\{\\begin{matrix} -2x+ {\\ }y &{}=0\\\\+6x-3y &{}=0\\end{matrix}\\right.\n", "7cf502ec1e6f03df442e602b1a663250": "2 \\dim \\ker (T - \\lambda_i I)^j - \\dim \\ker (T - \\lambda_i I)^{j+1} - \\dim \\ker (T - \\lambda_i I)^{j-1}", "7cf50632ad5baa86d5138c7742ed22a6": "F_\\Theta(\\theta)", "7cf51d02bbaae582c8b9616c2a26a947": "g(\\chi\\psi)", "7cf52462e9757a541c0e5ae252e7ca7f": "|\\mu (A \\cap T^{-n}B) - \\mu(A)\\mu(B)|", "7cf52bc9d11d56495e31246dee5be133": "\\varphi\\colon E\\to F,\\quad f\\colon M\\to N", "7cf59273303e6a05ac23480b9fb4aee0": "h = 1", "7cf5b4749de4942aad8bee92498ed61c": "\n\\mathbf{X}_k \\succeq \\mathbf{0} \\quad \\text{and} \\quad \\lambda_{\\text{max}}(\\mathbf{X}_k) \\leq R\n", "7cf5d6489889a0ebe316cd670bc56f31": "Z(n) = \\frac{\\sum_{j=1,\\ldots,n} Z_j}{n}", "7cf61e85a75951d6107ece7c54ab641a": "\\Phi = \\frac{P V}{T} + \\sum_{i=1}^s (- \\frac{\\mu_i N}{T})", "7cf6294b8bc2ff1ac93e1e1112c14513": "{{\\rho}}", "7cf67adb0e2c23e87b3ad0f5b2fd4e22": "\\rho^2{d^2J\\over d\\rho^2}+\\rho{dJ\\over d\\rho}+\\left[\\rho^2-\\left(l+\\frac{1}{2}\\right)^2\\right]J=0", "7cf67c4cb2f1645189af32d459ac4507": "\\alpha=\\frac {|w|}{(2 c_r Q_r)} \\quad (1.3)", "7cf68ca5292cb60233fc42cd2fcf36e1": "\\operatorname{sech}(z) = \\frac{1}{\\cosh(z)}.", "7cf6d7a1ff18696889796731d30a82d3": "\\scriptstyle{R_1 : R_2}", "7cf71f76bff57fcc70ba9e0074ceb35e": "{52 \\choose 4}{4 \\choose 3}{48 \\choose 2} = 1,221,511,200", "7cf74ef0b85539b4c6e0758f8e4a47dd": "v \\equiv \\frac{1}{\\sqrt{u^2-b}}", "7cf756a5d4c40a7bac792c0e31bbf02f": "|\\zeta(\\sigma)^3\\zeta(\\sigma+it)^4\\zeta(\\sigma+2it)| = \\exp\\sum_{p^n}p^{-n\\sigma}\\frac{3+4\\cos(t\\log p^n)+\\cos(2t\\log p^n)}{n}", "7cf76cfa59037e07f23d3d306ebf419e": "p: R \\to R/I", "7cf77bed0160e5f8df4d84bd4a639009": "\\mathbf{B} \\rightarrow \\mathbf{B} + d\\mathbf{\\alpha}", "7cf7890e36d879ede92451e852a4102f": "S \\to W~~~~~~~~~~~\\text{start with a probability of 1}", "7cf7a202a9c86de25964bbec572f2cc7": " \\Pi_A ", "7cf7f017c4a5b91e3d734f654886b00e": "\\Psi_0=\\Psi_1=\\Psi_3=\\Psi_4=0\\,,\\quad \\Psi_2=-\\frac{M(v)}{r^3}\\,,", "7cf80a2cc55de43e9514341f69225cc4": "L \\approx 0.007975 {d^2 N^2 \\over D}", "7cf81093f393b25e7072f260d61f1ead": "-|k|x^2 + |k|^2y^2 = 1.", "7cf8635484031fb5489385ab5b13854b": "\\begin{cases}\n\\overbrace{ \\begin{bmatrix} \\dot{\\mathbf{x}}\\\\ \\dot{z}_1 \\end{bmatrix} }^{\\triangleq \\, \\dot{\\mathbf{x}}_1}\n= \n\\overbrace{ \\begin{bmatrix} f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 \\\\ 0 \\end{bmatrix} }^{\\triangleq \\, f_1(\\mathbf{x}_1)}\n+\n\\overbrace{ \\begin{bmatrix} \\mathbf{0}\\\\ 1\\end{bmatrix} }^{\\triangleq \\, g_1(\\mathbf{x}_1)} z_2 &\\qquad \\text{ ( by Lyapunov function } V_1, \\text{ subsystem stabilized by } u_1(\\textbf{x}_1) \\text{ )}\\\\\n\\dot{z}_2 = u_2\n\\end{cases}", "7cf9531ccd3c6a799b0fbbda2a0ba81d": "P(\\boldsymbol{r}) = \\frac{1}{\\rho(F(\\boldsymbol{r}))}", "7cf9f71ffcfb8b64ea76d45b7fa1fcd7": "\\mathcal{O}_{x, X}", "7cf9f7cf855b89ffbc31f121e3336fc4": " x^2 + y^2 = 1.\\ ", "7cf9f9be99fcc76822fc05bf011f26b8": "~\\gamma = 180^\\circ - \\alpha - \\beta.", "7cfa523cc58f63674ebdab00332bb97b": "\\Sigma_1(L_\\alpha)", "7cfa5433d3cc0d6367cfd794d8c02733": "P =\n \\begin{bmatrix}\n 1/2 & 1/2 & 0 & 0\\\\\n 0 & 1/2 & 1/2 & 0\\\\\n 1/2 & 0 & 0 & 1/2\\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix}.\n", "7cfa8676c4e5e6cd9a20dd32b9455415": "\\langle f, g \\rangle = \\int f(x) g(x) \\; d\\alpha(x).", "7cfaeba471b3980aeb6e2d1ec03d7827": "\\forall m \\in n[\\forall k \\in m(\\bot) \\or \\exist k \\in n(k \\in m \\and \\forall j \\in k(j \\in m) \\and \\forall j \\in m(j=k \\lor j \\in k))])).", "7cfb1dd82af286130da614553e1d4d83": " 0 \\longrightarrow L_n \\stackrel{f_n}{\\longrightarrow}\nM_n \\stackrel{g_n}{\\longrightarrow}\nN_n \\longrightarrow 0 ", "7cfb1ea884f5509de1043c73a4d7cbda": "\\mathfrak{g}\\otimes C^\\infty(S^1)", "7cfc0d6216c447d62150f7c1cf38c7c0": "f(n) = Ln/250", "7cfc7d53cf5f9146cdf00c142dbdd264": " \\frac{1}{\\lambda_{1}} J_{i} ", "7cfcad7eb640f4fce69da6f6b7217c42": " ~\\epsilon_{t-1} \\ge 0 ", "7cfcddd6b2d07adcc07a6ed1ddc26f24": "dG/dr=0", "7cfd11688d9e96bc05802d5b2aec1c20": "0\\le x_{1} \\le 1", "7cfd1579e5a10c19d1f2e4c2ffa171d1": "H^0 = \\tau_{\\leq 0} \\tau_{\\geq 0} = \\tau_{\\geq 0} \\tau_{\\leq 0}", "7cfd71f209833be56f7caccc8aea8037": "b=\\tfrac{1}{2}\\pi-\\delta \\, ", "7cfd7d399fc9783095dfa50b310f045f": "\\sqrt{after_{and}another}", "7cfd7f1b3417fd4bf483b4722a08fb37": "\\lfloor \\log_k (n+1) +\\log_k(k - 1) \\rfloor", "7cfd9fede77f84d9ef3f62776ffc77f4": "\n0< \\mathcal{N}(x) := \\sum_{n} \\vert \\phi_n (x)\\vert^2 < \\infty \\, \\quad \\mathrm{a.e.}\\, . ", "7cfe3c8717b338df5ad8fe09a8818882": "y_t = y_{t-1} + c + e_t", "7cfe64ea44dc3bbeb63b29ff3039a481": "heavy", "7cfe7fbb56c749571be15b57f91ebaa5": "\\boldsymbol{\\Omega} = \\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T", "7cfe9e72698fd7ccee5e1aa50f25de6a": " M \\frac{d^2}{dt^2} X(t) = - \\frac{\\partial V(X)}{\\partial X} - \\eta \\frac{d X(t)}{d t} ", "7cfeee1452eaeb58fbe93711560b11f0": "\\mu_2,", "7cff32d28f7428bd45e89a1d5057ed8d": "\\textstyle d", "7cff60f68955bced662b51b24bfc93a9": "\\varphi_{AC}= \\varphi_{AB} + \\varphi_{BC}", "7cff91714421a302a95f99e911d964f9": "A_0(R) = 2,", "7cffa9b8f26f29a2b557503b85b84624": "Y_\\ell^m (\\equiv Y_{ACN})", "7d002ae48bbefe5eb6c38cbc7be98568": "\\scriptstyle\\vec{l}", "7d004f29036836ef8590076d1f21649f": " \\frac{1}{N(t)} = \\frac{1-e^{-rt}}{K}+ \\frac{e^{-rt}}{N(0)}. ", "7d007a174c6041555f81581227f39c2d": "f(X\\cup \\{x_1\\})+f(X\\cup \\{x_2\\})\\geq f(X\\cup \\{x_1,x_2\\})+f(X)", "7d00b2eeabdf9d3e7bf8a53d9d62a3c4": "\\alpha, \\beta \\in \\mathbb{R}", "7d00c1d356950225cd944ddd757178fb": "\n{\\sqrt{1- s^2/c^2} \\over (1- s/c)} =\n{ \\sqrt{1-v^2/c^2} \\sqrt{1-u^2/c^2} \\over (1- v/c)( 1-u/c)}\n", "7d00d1ed90a0844211407051e826366c": "P(\\lambda) = Q(\\lambda)", "7d00ec0c7cfaaa31f14e97829f5701b2": "= \\int_V \\left[ \\mathbf{E}_1^{(e)} \\cdot \\mathbf{J}_2 - \\mathbf{J}_1 \\cdot \\mathbf{E}_2^{(e)} \\right] dV.", "7d010443693eec253a121e2aa2ba177c": "||", "7d0237ec86934e5346607efdb72536c5": "\\left\\{\\begin{matrix}ax + by + cz&= {\\color{red}j}\\\\dx + ey + fz&= {\\color{red}k}\\\\gx + hy + iz&= {\\color{red}l}\\end{matrix}\\right.", "7d0258aaa57e0a5caf84a6b87a858ac2": " D^{}_{}", "7d025fbbe1ce53889d410cb39fe56116": "\\displaystyle{h_z(x)=\\widehat{g_z}(-x)={i\\over \\sqrt{2\\pi}} (x + z)^{-1},}", "7d02717d73c83990e2a68c9f9c2de718": "E = \\{ z; |z| < 1 \\}", "7d027194613a44bcba5c9ce7b400aa90": "\\mu (p){\\stackrel{{\\rm def}}{=}}\\min _{\\alpha >0 ,\\beta>0} \\ell _{({M},\\varphi )}(x+\\alpha ,y-\n\\beta)-\\ell _{({ M},\\varphi )} (x+\\alpha ,y+\\beta )-\n\\ell_{({ M},\\varphi )} (x-\\alpha ,y-\\beta )+\\ell _{({ M} \n,\\varphi )} (x-\\alpha ,y+\\beta )", "7d02dfe9e3e76bd7e62a36711b4c96df": "v_i=C_i-V", "7d03368a440dfc87daa07d1b8dd97bf8": "B=0.", "7d0347ecb262be5a569c55c2a52f26de": "\\hat{H}_{\\text{JC}} = \\hbar \\omega \\hat{a}^{\\dagger}\\hat{a}\n+\\hbar \\omega \\frac{\\hat{\\sigma}_z}{2}\n+\\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{\\sigma}_+\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_-\n+\\hat{a}\\hat{\\sigma}_-\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_+\\right),", "7d035c68f484516c0224a0346006baa8": "S^n=\\partial D^{n+1}\\approx \\partial (D^{p+1}\\times D^q)= S^p\\times D^q\\;\\cup\\;D^{p+1}\\times S^{q-1}", "7d037a58a4ab1ebfcec2c0ca7b343402": "x_\\perp ", "7d037ad1349ed0c04f823cffa985bf59": "\n2\\pi J = n h\n\\,", "7d03b5ee91823aa35647c86da4c514f3": "t = \\sqrt{\\frac{30m}{sd^3l(1+l^2)}}", "7d03c690d1df3e04c98c4a0174d657f1": "z=-h", "7d03f4499e789964d65d5c37101346a4": " v = k\\ S_1^{n_1} S_2^{n_2} ", "7d03fe34a58761ddb88f82d8dbc1e67f": "\\textstyle -\\sqrt{1-x^2} < y < \\sqrt{1-x^2} ", "7d041d5fa14a17e801f9ac9cb1060049": "\\int_{0}^{\\infty}f\\left(x\\right)dx=\\int_{0}^{\\infty}f\\left(x\\right)e^{x}e^{-x}dx=\\int_{0}^{\\infty}g\\left(x\\right)e^{-x}dx", "7d04237d8d89a6f2b0848a89d4dc8fc9": "\\mathrm{^{239}_{\\ 94}Pu\\ \\xrightarrow {2(n,\\gamma)} \\ ^{241}_{\\ 94}Pu\\ \\xrightarrow [14.35 \\ yr]{\\beta^-} \\ ^{241}_{\\ 95}Am}", "7d0426227293b54f8fbf9089cd047444": "0 = V_2 = Z_{21} I_1 + Z_{22} I_2 ", "7d048957c2302af01f696f1fde76f23b": " \\operatorname{sink-tran}[M\\ N, X] = \\operatorname{sink-tran}[M, X]\\ \\operatorname{sink-tran}[M, X] ", "7d048d635fa2c3ada72bfdf5bc06764a": "D=(f_1+f_2)/2", "7d04937bbf2f0fffeb0e98a07a22b5eb": "g_0^-(T(\\omega))=f^-(\\omega)=0", "7d04bcb0049722ec11e24baa05cff1aa": "RE(x_i,y_i) = \\left\\vert x_i^2 + y_i^2 - r^2 \\right\\vert", "7d0518e89ebd8434a83d74342a37297d": "x = \\cos \\theta \\ \\mathrm { and} \\ y = \\sin \\theta \\ .", "7d0566b0d6669ed27d885ff987b16777": "E_{k_{max}} = \\frac{1}{2} m v_m^2 ", "7d0577d2d9ba73d0eba33fe12f683799": "2\\theta\\,\\!", "7d05989990a26135151e64a2e44d3e87": "u(\\boldsymbol\\beta^{(t)})", "7d05ba4c8c421c161840fec37aa85d88": "n\\cdot 1", "7d05ca92f4a182cbe078466a6c1c0e08": "H(s)=\\frac{G_\\mathrm{lpf}{\\omega_0}^2}{s^{2}+\\frac{\\omega_{0}}{Q}s+{\\omega_0}^2}", "7d05ebf552b3cf72be7a7b0334e42767": "(p_{r+1},p_s)=(xp_r,p_s)-a_{r,r}(p_r,p_s)-a_{r,r-1}(p_{r-1},p_s)\\ldots-a_{r,0}(p_0,p_s)", "7d0611cbb574bf5eb3249bb762868325": "n=2^{n_2} 3^{n_3} 5^{n_5} 7^{n_7} \\cdots = \\prod_p p^{n_p},\\;", "7d06169b7ee32e0833c9b6775ba79bad": " E=f(a)=f(-a) ", "7d0617df08b72cee3f09e6a037c9af4f": "F(M)=F(I)D(M).\\ ", "7d066d7b612cdc00a70a435d1c5b1a09": "\\text{SD}_\\bar{x}\\ = \\frac{\\sigma}{\\sqrt{n}}", "7d06850851ab2977c74d41e8ff681751": "\\ell_3 = \\tfrac{1}{3} {\\tbinom{n}{3}}^{-1} \\sum_{i=1}^n \\left\\{ \\tbinom{i-1}{2} - 2\\tbinom{i-1}{1}\\tbinom{n-i}{1} + \\tbinom{n-i}{2} \\right\\} x_{(i)}", "7d06936f2935a6a9b3655324562a413e": "\nS= \\int {\\dot{x}^2\\over 2} dt\n\\,", "7d06e4d8e0fa3e488cb34c8f648a71e0": " \\theta_m = \\arcsin{ \\left( \\frac{m \\lambda}{d} - \\sin{\\theta_i} \\right)} ", "7d071257e11df73ed86b7938f54e8c96": "\\operatorname{lambda-lift}[(\\lambda V.E)\\ S, L] \\equiv \\operatorname{let} V : \\operatorname{de-lambda}[G = S] \\operatorname{in} L[(\\lambda V.E)\\ F:=E[V:=G]] ", "7d071f4df0d89dd9101b81a6523fb521": "U(\\theta)= 2 \\cos \\frac {\\pi S \\sin \\theta} {\\lambda}", "7d0725fc4ade2994844566bd2f27e2af": "\\mathbf{e}_1 \\times \\mathbf{e}_2 = \\mathbf{e}_3 =-\\mathbf{e}_2 \\times \\mathbf{e}_1", "7d0736f0197a0d7db8c1680e87745c87": " w(x,y)=\\frac{1}{4} \\left(1 + \\cos {\\frac{x \\pi}{N}}\\right)\\left(1 + \\cos {\\frac{y \\pi}{N}}\\right) ", "7d0745e5dd6f6e0a2f20087e78c9e5bb": "\n\\frac{d^k}{d x^k} L_n^{(\\alpha)} (x)\n= (-1)^k L_{n-k}^{(\\alpha+k)} (x)\\,.\n", "7d074b09e059dccc4a4e901fdd677362": "\\mathcal{F}_{*}^{W}", "7d07b42b017e39b71794fb0382126fd1": "\\hat K_j (x_1)", "7d07cc743d676c7ff598b04678e0ef9b": "\\beta \\in \\mathcal{B}", "7d07f17aa5032ae51c382c58b8b93f41": "W_{1-2} = P_1 V_1 \\ln \\frac{V_2}{V_1}", "7d07f43a01c9df441cee39418b266283": "\\frac{10^6}{5700} - \\frac{10^6}{3200} \\approx -137\\ \\mbox{MK}^{-1}", "7d082d3adb0fba6c403f6d052a1cbe26": "(a, b) = 1", "7d083826ea495335e75462250047e250": "\\mathbb{P}(A\\mid E) = \\frac{1%}{99% + 1%} = 1%=\\mathbb{P}(A) ", "7d084bbffb632810ac2db9114923dd0c": "\\Diamond\\varphi", "7d089fccca22102ab83ac72770200f33": "B_v", "7d08d6023f5155862c575a5b599853a3": " x<0.", "7d08ea22d9adc83be5eb1a5620e7056a": "\nA_3^2 = A_1^2 + A_2^2 - 2 A_1 A_2 \\cos(180^\\circ - \\Delta\\theta),\n = A_1^2 + A_2^2 + 2 A_1 A_2 \\cos(\\Delta\\theta),\n", "7d0903163dad483b2a0b2b89bbbe6111": " \\underline{\\mathbf{e}}(\\ell) = \\underline{\\mathbf{d}}(\\ell) - \\underline{\\hat{\\mathbf{y}}}(\\ell) ", "7d094b90989930f742890fa10911c48f": "\\nabla_\\mathbf{X}^2 f = \\boldsymbol{\\nabla}_\\mathbf{X}\\boldsymbol{\\nabla}_\\mathbf{X}^\\mathsf{T} f", "7d09d2d3c98923a705293efc109bb4b3": "j\\sqrt{\\frac{1}{3}}", "7d0a0dfc38b26d1fe660e7a29ff26fc7": "~W = P_{\\rm p}/({\\hbar\\omega_{\\rm p}})~", "7d0a567817e72ddc9674696b535e45fb": "\nx Q x^T + P x^T + R = 0\\,\n", "7d0a7b991cbad8bf4cc2ab55f3797b9c": "W_n(x;a,b,c,N;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&cq^{x-n}\\\\\naq&bcq&q^{-N}\\\\ \\end{matrix};q;q\\right]", "7d0ab8fde227931c7e02de2f71305a20": "\\frac{\\pi}{4}", "7d0ac0cfc631c45587e76c0c2345e8d4": "\\ v_{10}", "7d0ade468d1ab148c74f375480c2ce57": "\\sqrt{ax+b}\\,\\!", "7d0b00e64bcc78df89829c966d036e02": " \\psi \\mapsto f \\psi. ", "7d0b50985ee9d6aa55c25883de51e296": "\\!X", "7d0b972c3b6cc5e3766e037903b673bb": "\\scriptstyle A\\, \\rightarrowtail\\, B", "7d0bae878bcb178eb00599bae5e34738": " R = e^{\\frac{\\mathbf{B}_1 + \\mathbf{B}_2}{2}} = e^{\\frac{\\mathbf{B}_1}{2}}e^{\\frac{\\mathbf{B}_2}{2}} = e^{\\frac{\\mathbf{B}_2}{2}}e^{\\frac{\\mathbf{B}_1}{2}}", "7d0bb802067e79d1921622c982f80b27": " b = (1/3) a ", "7d0be661d27dc396b45aeae8ea4d9816": "\\nu_\\mu", "7d0bf11978fb046a202d1c452081801c": "\\langle x,y,z \\rangle = \\langle z,x,y \\rangle", "7d0c32367747d9ae732d8e4058df5e15": "\\det(A) = \\bigl( (\\mathrm{tr}A)^2 - \\mathrm{tr}(A^2)\\bigr )/2, \\, ", "7d0cab212e1c6253e6e6695e7e851099": " D_i \\equiv \\max_{j : i \\neq j} R_{i,j}", "7d0cdef5493ee4f7310651be4b94e81b": "x_0\\,\\!", "7d0d21d2090d9f7db20b1bb6ca84dbf6": "\\alpha_{\\iota} < \\alpha_{\\rho}\\!", "7d0d464138b4fe2c8ae9df1095a48e9a": "U(P,f,g)-L(P,f,g) < \\varepsilon.", "7d0d4ff7d153c184ad64df934ebe1eb8": "\\frac{4 \\pi}{3} (nx)^2 \\beta_2 R_S^3 = S_*", "7d0d81b320131ac0cbadbbff5f9cc1c6": "\n \\cfrac{dp}{d\\rho} = \\cfrac{\\gamma~p}{\\rho} ~;~~ \\gamma := \\cfrac{c_p}{c_v} ~;~~ c^2 = \\cfrac{\\gamma~p}{\\rho} ~.\n ", "7d0d8ee79c292505f8b092ab090eabf0": "\\operatorname{mr}(G)=|G|-1", "7d0d9f3957578ae13f319d22802db043": "B \\frac{dB}{dz} \\approx 375\\ \\mathrm{T^2/m}.", "7d0e116f2afbe229258d0d9b53ad4873": "(\\pm 2, 0, 0)", "7d0e812713987576236b3cb98c71c480": " y_i \\in \\{0,1\\}", "7d0ed3c7f27fb920301f2cd5b7ac25e7": "j^{1}_{p}\\sigma \\in S", "7d0ee68ea06b52bf8837a5e4c5927503": "(\\Psi_g)^{-1} = \\Psi_{g^{-1}}", "7d0f2c7e698f89227117b03f5de3ebba": "\\rho : \\mathbb{R} \\rightarrow \\mathbb{R}", "7d0f52b512ad23a26703e1058574c01c": "\\scriptstyle \\frac{64!}{32!{8!}^2{2!}^6}", "7d0f792c546690fc756a123b37102300": "\\sum_nD_n(x,\\alpha)z^n = \\frac{2-xz}{1-xz+\\alpha z^2} \\, ", "7d0fae427178ba91124e99a959452bd0": "O(\\eta*\\log{n})", "7d0fe4eba06177cf2fe5f364def93ba6": "\\frac{\\operatorname{d}v}{\\operatorname{d}t}= \\frac{\\operatorname{d}v}{\\operatorname{d}x}\\frac{\\operatorname{d}x}{\\operatorname{d}t} = \\frac{\\operatorname{d}v}{\\operatorname{d}x}v=\\frac{d}{\\operatorname{d}x} \\left( \\frac{v^2}{2} \\right).", "7d1031ffb97f62ace8e6cfadde80dbe0": "L^\\infty(X)\\rtimes Z", "7d104871fbcfe20fc2bc0a42291db0cb": "(ax)(ya) = a(xy)a", "7d1051442aa5fe3f5c477615b18ebab5": "p_{e'}^{\\, 2}c^2 = p_{\\gamma}^{\\, 2}c^2 + p_{\\gamma'}^{\\, 2}c^2 - 2c^2 p_{\\gamma}\\, p_{\\gamma'} \\cos\\theta.", "7d1088b876060c9f484eefc926884304": "\\sigma_\\text{annual} = {0.01 \\over \\sqrt{\\tfrac{1}{252}}} = 0.01 \\sqrt{252} = 0.1587.", "7d10b93b99c9e484b89b6b2fb949038d": "IDCG_p", "7d10ba44c915cc05cc7f79bf4eb32b48": "z=a^2+b^2", "7d110555b15cee329f7d69eae49d097c": "C = \\frac{1}{T}W+ \\frac{R}{T}Y.", "7d1128e526cd17c05bc0bef52a5bb920": "\\displaystyle \\ln(W)", "7d11f27b3634c33b00a3c4269d667051": "\\textstyle{1+\\frac{\\log 3}{\\log 5}}", "7d127861b19efca2c004bc8af45f3cea": " f = f_1 \\circ \\mathbf{b}. ", "7d12ac2c31f7f79672658332a16f2dd6": "d_0 = c_2 -c_1^2, ", "7d12f5322bef7e6c95f59eb3491aec93": "\\mathcal{Z}", "7d12f79362ea378c5417ba9669a795f5": "MU_*(*) = MU_* \\cong \\mathbb{Z}[x_1,x_2,\\dots]", "7d130f5d8b3afbde4045c1783571aeb5": "a\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}} + \\frac{\\partial a}{\\partial \\mathbf{x}} \\mathbf{u}^{\\rm T} ", "7d132e019a0e0fb00166e12f79797729": "\\lambda(y)= \\int h(x,y)L(dx)", "7d14355a0745fb84e66202df326e07ae": "\\nabla_{\\bold{v}}{f}(\\bold{x}) = \\nabla f(\\bold{x}) \\cdot \\frac{\\bold{v}}{|\\bold{v}|} .", "7d146b69a17dd19dde46610f09faa8ac": " 0 = p\\wedge q ", "7d146c8205593618fe7f7500352754f1": "(e_ie_j)e_k = -e_i(e_je_k) \\neq e_i(e_je_k)\\,", "7d149ed907461978708806bff3e201f8": "f(x)=\\frac{1}{x}\\sin\\left(\\frac{1}{x^3}\\right).", "7d14c470daca3084f9800e936bfcffc3": "\\log(1-x) = - \\sum^{\\infty}_{n=1} \\frac{x^n}n\\quad\\text{ for } |x| < 1", "7d14f737c27a5b099fb636d9afc46b10": "\\scriptstyle |\\phi\\rangle_B = \\sum_{j} c^B_{j} |j\\rangle_B.", "7d15018b2c21f83d0d1641a90463e5b8": "E_c", "7d154f8119fd9e5f3e10adfc79dd658d": "\\Delta w = \\chi_r - \\chi_s\\;", "7d1561ace3735976ecc27add4517a61f": "q=\\sqrt{y_c^3 g}", "7d15cfe1bd42af65c26cea44ace9d578": "K = K'\\cap K''", "7d16011902530227b788ec62b9ad00e3": "Z(E(\\mathbf{F}_p)) = \\exp \\left(\\sum \\mathrm{card} \\left[E({\\mathbf F}_{p^n})\\right]\\frac{T^n}{n}\\right)", "7d163f2cd5aef7cc37b3235e118bd575": "\\mbox{If }X \\subseteq_f x \\mbox{ then } X \\in Con", "7d1737ffc2d080f5fda3e4fe3556a338": "\\alpha_i \\in \\mathbb{R}", "7d17701d49e5e89683cca4d08046677c": " 0 < n < d ", "7d17b86aff9ab13ce0f3aa3fda511759": "\\sum_{n=2}^{\\infty}S(n)/n!\\approx 1.71400629359162\\ldots", "7d17e034aa3e35ed36c1052e2e35c3fe": "\\bigcup_i K_i = \\mathbb{R}^n", "7d17e4ab3c2ae69f4ef765d9860d4551": "x(nT) \\ ", "7d17e750be0d6d622181f7eab96548cb": "\\langle G,s,t\\rangle", "7d17f51ad783c150e02f3c0f81419a16": "\\lfloor \\log_{b}{k} \\rfloor + 1 = \\lceil \\log_{b}{(k+1)} \\rceil ,", "7d1847e84daf4cbbe555e37a22795aa6": "c=\\frac{a}{2x}", "7d187cf99f6fa88e01aa18726416c619": "m = \\operatorname{tr.deg}_k F", "7d18989b94eba73fca444bf1c21222d5": "w(x) ", "7d18c5e3338d5a5894a45f33f9e97c7a": "[x_b,y_b]", "7d18e9f9902ee5843b0d024fe3eaee40": "\\tilde{E}^a_i (x)", "7d192705f30b267114ae7e25f3a40be8": "\n\\begin{pmatrix}\n0 & 1 \\\\\n-1 & 0\\\\\n\\end{pmatrix}\n", "7d193f3a9e894b197ec03a49a2c7b7bd": "Q^{-1/2} \\sum_{x=0}^{Q-1} \\left|x\\right\\rangle \\left|0\\right\\rangle", "7d197927542523af1e63c6f89b2837c0": "EER=(354-(6.91*Age))+PA*((9.36*wt)+(726*ht))", "7d19a91ccdc31cf3533fde8c0e13afef": "\\Delta\\rho/\\rho", "7d19fa7dc613535eef0cedc71e67a35f": " \\hat{L}_z, \\hat{S}^2 ", "7d1a03ca3f392596ba0aa842acbf3659": "P_c: z\\mapsto z^2 + c,", "7d1a60058fa1279d0f849d7e98d00006": "\\lambda_{Te}", "7d1a6dc0e6796df0d951f747155daf33": "Z=\\frac{V}{I}", "7d1a7a3d9ec693dcc2ec25fb45bec253": "= -\\frac {8}{15} \\sqrt{\\pi}\\,", "7d1ac67c8f90af9d252fc7b81f227b17": "K_{2,2}", "7d1afd19b6a99551b25dd078dc77b796": "SO(n)/(SO(r)\\times SO(n-r)).", "7d1ba58ecea3853aa04caafbe8c36904": "\\begin{align}\n\\text{FVU} & = {VAR_{\\rm err} \\over VAR_{\\rm tot}} = {SS_{\\rm err}/n \\over SS_{\\rm tot}/n} = {SS_{\\rm err} \\over SS_{\\rm tot}} = 1-{SS_{\\rm reg} \\over SS_{\\rm tot}} \\\\[6pt]\n & = 1 - R^2,\n\\end{align}", "7d1ba8a23fc999a32b216649c08b7307": " l_\\text{P} = c t_\\text{P} ", "7d1bd15c19aaa5ac2acf4240f7ce3c2e": "f(t) \\mapsto \\sum_{k \\in K} \\begin{cases}\n 0& r(t,k) \\leq c(k),\\\\\n r(t,k)-c(k) & \\text{otherwise},\n \\end{cases}", "7d1bd5686555fab9047c069cf047be57": "0 \\leq v_1 \\leq v_2 \\leq 1", "7d1bd98f248f796197033f36a6e11175": "\\mathbf{B} = \\frac{\\omega^2}{4\\pi\\varepsilon_0 c^3} \\hat{\\mathbf{r}} \\times \\mathbf{p} \\left( 1 - \\frac{c}{i\\omega r} \\right) \\frac{e^{i\\omega r/c}}{r}.", "7d1ca62ef1bc4c1d1f4eff3e347bf7d2": " \\Delta z = H(z) \\Delta \\chi ", "7d1ce6a13a0d308cbd97ca7be027ccca": "\\mathrm{~^{238}_{92}U}\\rightarrow\\mathrm{~^{234}_{90}Th} + {\\alpha }", "7d1d37000598c599ded01c4817451540": "\\begin{align}\\Box H_{abc} = & J_{abc}\\\\\n& {}- 2{R_c}^d H_{abd}+{R_a}^d H_{bcd}+{R_b}^d H_{acd}\\\\\n& {}+ \\left( H_{dbe}g_{ac}-H_{dae}g_{bc} \\right)R^{de}+\\frac{1}{2}RH_{abc},\\end{align}", "7d1d816745c73d6de772bf12fdfca181": "-2 v-\\frac{\\partial v}{\\partial\\tau}+\\frac{\\partial v}{\\partial x}+ \\frac{\\partial^2 v}{\\partial x^2}=0.", "7d1d95078a57be65f348365d2000c3c9": " y(t_{k+1}) - y(t_k) \\approx h f(t_{k+1}, y(t_{k+1})). ", "7d1e096402928952df26d695434ee2ca": "x_1,\\ldots,x_I \\in \\mathbb{R}^N", "7d1e125eb52dc089e497dfcd5f4c5543": "(f^{-1})'(f(z)) = \\frac{1}{f'(z)}", "7d1e266f849cd4aee267c92910ae0e78": "U = \\hat{c}_V nRT ", "7d1e9e29029e80121fe1a5952d533aeb": "a - e X", "7d1e9ff1c372797aca165b7e69b6e88c": "\\mathbf{U}^{-1}=\\mathbf{U}^*", "7d1eb21d020866be99713a1fe9079852": "{r \\over \\sin \\theta_1} = {a \\over \\sin \\psi}\\! ", "7d1ee50325a8fa9ba520a9720878e0dc": "g_7=-x+56x^2-105x^3;", "7d1f07cb41199944163ec5f9bd736afb": "Z_3", "7d1f3bdb511249f2aa46b8bd913773be": "F: \\Gamma \\rightarrow \\bigcup\\nolimits_{G \\in \\Gamma} 2^{S_G}", "7d1f7024b70c3182c3eaf569fb877c38": "\\cup \\{1 \\pm \\sqrt{n}\\}", "7d1fb1bce4f3a725cdc77fcccf248379": "\\int_0^\\infty e^{-t}\\mathcal{B}A(tz) \\, dt = \\int_0^\\infty e^{-t} e^{tz} \\, dt =\\frac{1}{1-z}", "7d202148ee351a9f595eefc564668035": "\\boldsymbol{q} = x", "7d2034f61d51bc3e9dfdf1026f89840f": "\\mathbf{q}_{M \\times 1} = \\mathbf{f}_{M \\times M}\n\n\\Big( \\mathbf{B}_R \\mathbf{R}_{N \\times 1} + \\mathbf{B}_X \\mathbf{X}_{I \\times 1} + \\mathbf{Q}_{v \\cdot M \\times 1} \\Big)\n\n + \\mathbf{q}^{o}_{M \\times 1} \\qquad \\qquad \\qquad \\mathrm{(6)}", "7d20bfdfd95e2353731820640eed012b": "\\alpha\\in\\mathrm{Sper} A", "7d20fa6f23ddf47aaa487b629d2ec50d": "\\displaystyle{A(f_n(U)) = \\iint_U J(f_n)\\, dx\\, dy =\\iint_U |\\partial_z f_n|^2 -|\\partial_{\\overline{z}}f_n|^2\\, dx\\,dy\n\\le \\iint_U |\\partial_z f_n|^2\\, dx\\, dy\\le \\|\\partial_z f_n|_U\\|_p^2 \\,A(U)^{1-2/p}.}", "7d210779cf17a17beae49e4d7980a4b8": "G(t) = \\Pr(X > t).\\,", "7d2109522c29e57f38040d9daf82666d": "i:X\\rightarrow M", "7d210ecdd968e83a6254a8038f88d96f": "\\scriptstyle \\leq10^{-17}", "7d212510f1cba0b126957c56424ba0f5": "V = 100 \\cfrac{{(L-U)}}{{(L-H)}}", "7d212a0991b303163221da701f4c34a2": "\nF = \\frac{\\mathrm{d}p}{\\mathrm{d}t} = \\dot{m}(v_{\\infty} - v_{w}) = \\rho A_Dv_D(v_{\\infty} - v_{w}) = \\rho A_D(1 - a)v_{\\infty}(v_{\\infty} - v_{w})\n", "7d215146b84016f4eacf5fb26c987ac7": " a\\times b ", "7d21b8e1d73134d0d8afa77d971ec5e0": " \\sum_{2m+n=k}{m \\choose n}=P(k-2).", "7d21bcb6ec9d1c8f72d7bebd8146cd0f": "A^*A = (A^*A)^{\\frac{1}{2}} (A^*A)^{\\frac{1}{2}},", "7d21deb8cab2ac1896723c743823bdd2": "\\mathcal{H}_\\mbox{non-halting}", "7d2223065e0cda347c39606896f753be": "c_g.", "7d22290955c6e3fa16a03f5b84b09431": "CFM = \\frac{3.16 \\times 500 W}{(130 - 100)} = 53", "7d2277caeeeb726fc281c3d54bce37b6": "\\displaystyle{(0)\\rightarrow H^1_0(\\Omega) \\rightarrow H^1(\\Omega) \\rightarrow H^{1/2}(\\partial\\Omega) \\rightarrow (0)}", "7d22819b591718444008e435f5d842ec": "2^p - 1 \\equiv 7 \\pmod{12}", "7d235b0d680f38b7bcb86f3a9f64f834": "\\Phi =S-\\frac{1}{T} U", "7d23b1b4f85423be5c4be1fc4bc4220c": " Y^2=ZX^3+aZ^2X^2+16aZ^3X ", "7d244e7b09ce70089e0a195e8e111d83": "S=(S_k)_{k\\in K}", "7d24578a0fef99dd5445b2b741a7f08e": "\\frac{d \\chi}{d \\tau} + \\chi = F(\\tau).", "7d2471ff8ab674be9491afc91bca68ff": "\\textstyle{\\frac {\\log(9*0.75)} {\\log(3)}}", "7d247b43ecafa22547d9fcc2c6721366": "\\frac{d}{dt} \\det e^{tB} =\\mathrm{tr} \\left(B\\right) \\det e^{tB} , ", "7d2493616a0d2bf33243eb6ed8c44e6f": "\\frac{\\part}{\\part x}=K\\frac{\\part}{\\part x_1}", "7d24b68f9f801a71e7eff3b519a7b6d5": " \\pi_1 (G) \\subset Z(G'), ", "7d24c915003614b233167495e6a1a2c3": "\\sum_{k=0}^a \\left[{n\\atop k}\\right] = n! - \\sum_{k=0}^n \\left[{n\\atop k+a+1}\\right].", "7d24f6939d4157f09151fd70b9006893": "\\Omega^k_M(\\mathbf V)", "7d250fb2d25c3315dc4749c2379b565c": "|\\psi_{E}\\rangle", "7d252e274ab54ed8019e4ff1086e71e8": "\\forall x,y: \\neg\\;(x < y \\;\\wedge\\; y < x)", "7d254dde3bd81e91b277743f6f960111": "m + 1", "7d257bb29805f7e1c04527dbecc8fd7c": "X_{AC} = X_{ref} + c{dC_m\\over dC_z} + c{dC_n\\over dC_y}", "7d259384298c3ed1649a3d3ebe6d0d12": "\\mathit{i}^2 = -1 \\ . ", "7d25d0e19e2723c0f935b98ff840f502": "\nR_\\mathrm{S}^2 / R_\\mathrm{L} + 2R_\\mathrm{S} + R_\\mathrm{L}\n\\,\\!", "7d25eb78d7c8ccb116b2a6b5e8addbd3": " \\log{ \\left( \\frac{p(x,y)}{p(x)\\,p(y)} \\right) } = \\log 1 = 0. \\,\\!\n ", "7d262f6126b2cdfa3059bd26995199ad": "\n\\epsilon = \\frac{\\epsilon_s+g}{1+g^*\\epsilon}\n", "7d2688f58e8d63be20c29cfcede4bc62": " \\mathbf{E} = \\begin{pmatrix} e_{11} & e_{12} & e_{13} \\\\ e_{21} & e_{22} & e_{23} \\\\ e_{31} & e_{32} & e_{33} \\end{pmatrix} ", "7d2701ea05aa51ca3ba7743b35dbef89": "\\operatorname{E}(S) = (n - 1)\\sigma^2. ", "7d27a4ec766e02308a2740959ba6c28c": "\\mathcal S:=\\mathcal R\\cap\\mathcal Q", "7d280809b1a5999a02db689e0e98e205": "\\mathcal{F}_{\\mathsf{Anon}}", "7d28217918ea2910b65d9d7a3992f7a6": "s=E^2", "7d284b266377c810f4317990de46839e": "\\zeta = \\zeta(\\xi_1,\\xi_2,\\ldots,\\xi_m) ", "7d2882af21077aa9cfa5a09da9f9da97": "\\scriptstyle \\{X_n\\}_{n=1}^\\infty", "7d2885a07ebf0520e97c6b46170ea64a": " (x_1...x_k) < (y_1...y_k) ", "7d28e3efc53bc37e9d5993c12371950d": " \\frac{d N}{d D} \\propto D^{-q}", "7d28f59f09b12c02c76be6236fd0e13d": "\ny_{n+1} = \\frac {\\left( 2-\\frac{5 h^2}{6} f_n \\right) y_n - \\left( 1+\\frac{h^2}{12}f_{n-1} \\right)y_{n-1}}{1+\\frac{h^2}{12}f_{n+1}} \n", "7d290886cc81faa64c9d0968af8dc432": "\\bar{v_i},", "7d293902aecd0a78c8524529fc5a2b96": "(f+g)\\circ h=(f\\circ h)+(g\\circ h)", "7d29913334282129f1b9052e30d6df80": "\\mathrm{inv}", "7d29c5140898a3d87899aa25d817b4b5": " X = f_1({\\bold x})\\,\\frac{\\partial}{\\partial x_1} + \\cdots + f_n({\\bold x})\\,\\frac{\\partial}{\\partial x_n} . ", "7d2a4959a0514dc7849bf864abed7088": "\\left[\\begin{matrix}0&A\\\\A^T&0\\end{matrix}\\right],", "7d2a5157cdcd84487f18a01f0e1f7e3a": "N(d_-) K", "7d2ad325eda43b82f80d9ae371db0fed": "G/H", "7d2b253222578931658705cbd6aadd7b": " apk=k^{(a-1)} ", "7d2b2e7bf6799b99a30b6bbcdf86f92b": "y(N)", "7d2b7c4f19ac5fb8f8814c29f15b1d25": "\\Gamma=\\mathbb{Z}^2", "7d2bb40568fedb8533d34725460c52d5": " \\frac{2}{n} = \\frac{1}{3} \\frac{1}{n} +\\frac{5}{3} \\frac{1}{n} ", "7d2be21a47f746d060d33c9cca8efaae": "\\begin{bmatrix}x & y & z\\end{bmatrix} . \\begin{bmatrix}1 & 0 & 0\\\\0 & -1 & 0\\\\0&0&0\\end{bmatrix} . \\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix} = 0 ", "7d2c41eaaa7c1bc75dac827eb7284307": "\\beta = \\frac{\\partial f}{\\partial y} ", "7d2c74ce13754f1ee080273cc378b54f": "(a + \\mathbf{A})(b + \\mathbf{B}) = ab + a\\mathbf{B} + b\\mathbf{A} + \\mathbf{A} \\cdot \\mathbf{B} + \\mathbf{A} \\times \\mathbf{B}.", "7d2c7c18078bb8c194aec90122ba4bd2": "\\mathfrak{e}_8 \\oplus \\mathfrak{e}_8", "7d2c8fd97b79b3463b8dcf362d8b86cd": "\\ln \\frac{Y_i-a}{c-a},", "7d2ca6857c7c350a3ecc10ee9ee8be74": "\\scriptstyle \\hat{r}", "7d2ce599e2fd936dca4889a4c1239dd8": "\\mathrm I_s^{2 \\omega}(\\gamma)=C |s_1 \\sin {2 \\gamma}\\ \\chi_{xxz}|^2(I^{\\omega})^2", "7d2d2c842f913f4d972bf4b48db81d15": "\n\\langle H_{\\mathrm{grav}} \\rangle = \\frac{H_{\\mathrm{grav}}}{N} = - \\frac{3G M^{2}}{5RN},\n", "7d2d615dee1373ab9582b7fcf076a4ab": " \\frac{E}{B} = v", "7d2d7b3efadb4f94acce85be9aaa3f2e": "b_0 = 1", "7d2d7dcb0b1307e9cfc03bb979bc5bca": "\\int_{s=a}^{b+1} f(s)\\ ds \\le \\sum_{i=a}^{b} f(i) \\le \\int_{s=a-1}^{b} f(s)\\ ds.", "7d2d912a37ce1d72ad546ba750c5fad2": "c_s = \\frac{(S_{11} - \\Delta S_{22}^*)^*}{\\left|S_{11}\\right|^2-\\left|\\Delta\\right|^2}\\,", "7d2da398c2144b964c5d4c87067740c2": "\\mu F_\\mathrm{n}\\,", "7d2dde7144b2cafe16d50aebc1317cc2": "t_{zerocross}", "7d2e08ee5fbc840b79a41a6f13bd41f6": "\\mathcal{N}\\left(-A,\\frac{N_0}{2T}\\right)", "7d2e390f3d46a3944a50592235abb342": "z' \\cdot x' = 0", "7d2e66e81c096d080020447a9738ccb4": "\\frac{dy}{ds} = \\sin \\varphi = \\frac{\\lambda_0 gp}{T},", "7d2e6c56aef31ab87c5008cd658427f8": "(\\lambda,\\mu(\\lambda))", "7d2f86b299c74f4aff8022245eab0c87": "\\sum_{m=0}^{n-1}(-1)^m\\frac{A(n,m)}{\\binom{n-1}{m}}=0 \\text{ for }n \\ge 2,", "7d2f88006a328877158bb87fa2abaf99": "\np_i = \\left \\{\n\\begin{array}{ll}\n(\\frac{b-1}{b})^{b-i} \\frac{1}{b(1-(1-(1/b))^b)} & i \\leq b \\\\ \n0 & i > b\n\\end{array} \\right . ,\n", "7d2fa893c3035093a80ab83e01b39791": "x^2+y^2=r^2 \\,", "7d303c74c895c491edbf47866af574cc": "s(n,k) = \\sum_{j=0}^{n-k} (-1)^j {n-1+j \\choose n-k+j} {2n-k \\choose n-k-j} S(n-k+j,j)", "7d30469559bc96df1d651ad2d07abd7b": "\n\\psi(\\{\\mathbf{r}_i\\},t) = \\exp[-iEt/\\hbar]\\phi(\\{\\mathbf{r}_i\\},t)\n", "7d305ed898db56d3d6d46d8a8fc88408": "\\lim_{k \\to \\infty}r(k)/k^{-d} = (2-d)(1-d)/2", "7d306dcdfc112b65335a41b9c2bba63c": "\\int \\frac{\\mathrm{d}u}{\\sqrt{a^2-u^2}} =\\sin^{-1}\\left( \\frac{u}{a} \\right)+C", "7d309e08b21e77f1bc4cfa5d2ecf6812": "\\scriptstyle (1 \\,-\\, k)\\psi(k) \\,+\\, \\ln\\left[\\frac{\\Gamma(k)}{\\lambda}\\right] \\,+\\, k", "7d309ef15202934ecdd2f5508c4c7337": "\\frac{\\sin A}{\\sin a}=\\frac{\\sin B}{\\sin b}=\\frac{\\sin C}{\\sin c}.", "7d30ae7658642b8aaa349987601f68bc": "s f(a) \\leq t f(b) \\leq (s+1) f(a),\\,", "7d311649ef0302c635ed45ec92d1282c": "\\;2\\mathbf{k}_i\\cdot\\mathbf{G}=G^2", "7d3123c3c0376adea91cc01fbe110000": "aX_n+bY_n\\ \\xrightarrow{L^r}\\ aX+bY", "7d317c8cecb115df0d268a1e5dfdddf4": " \\mathrm{M}(a,b,c) = \\begin{bmatrix} 1 & a & c \\\\ 0 & 1_n & b \\\\ 0 & 0 & 1 \\end{bmatrix}. ", "7d317de80cc1d4c16e769ed083345f0d": " a(n,m) = m^2 + mn + n^2 - 1 \\mod n+m+1 \\, ", "7d31a5166930bacba1dade2eaef53939": "a_0, a_1, \\dots, a_m, b_1, \\dots, b_n", "7d31cf5da08226803ce2b0bce9c03ce3": " \\begin{align}L(x) &= {1\\over 243}\\Big(f(x_0)x (2x-3)(4x-3)(4x+3) \\\\\n& {} \\qquad {} - 8f(x_1)x (2x-3)(2x+3)(4x-3) \\\\\n& {} \\qquad {} + 3f(x_2)(2x+3)(4x+3)(4x-3)(2x-3) \\\\\n& {} \\qquad {} - 8f(x_3)x (2x-3)(2x+3)(4x+3) \\\\\n& {} \\qquad {} + f(x_4)x (2x+3)(4x-3)(4x+3)\\Big)\\\\\n& = 4.834848x^3 - 1.477474x.\n\\end{align} ", "7d31e3bab371d6b03988bc0cf50a4a67": "\\frac 1 {D_\\mathrm F} + \\frac 1 v_\\mathrm F = \\frac 1 f\\,;", "7d3203b1be01acda48326a3e0d8e1bef": "A_0,\\dotsc,A_m\\in \\mathbb{R}^{n\\times n}", "7d3214417f3c70de784fbb668baddc5e": "\\scriptstyle \\boldsymbol \\nabla \\,\\cdot\\, \\mathbf J \\;=\\; 0", "7d321b43a4bab8226f8ee2c85260c32a": "- [F , G]^{IJ}", "7d3221a108780bf3f7234b8b8dfe4023": "f_{*} = g_{*}:H_n\\left(X\\right) \\rightarrow H_n\\left(Y\\right)", "7d3236a7c148d8beef473beb27583306": "\\Vert \\mathbf{s}\\Vert", "7d32703ec7f6b799a904bb1443611bcd": "1 = A + 2B. \\,\\!", "7d3372bdc83356f1b1f9fafcd7670a7c": "\\lim_{x \\to 0}\\left ( \\frac{\\sin x}{x} \\frac{\\sin x}{1 + \\cos x} \\right ) = \\left (\\lim_{x \\to 0} \\frac{\\sin x}{x} \\right ) \\left ( \\lim_{x \\to 0} \\frac{\\sin x}{1 + \\cos x} \\right ) = \\left (1 \\right )\\left (\\frac{0}{2} \\right )= 0 ", "7d33c61d172a217352ba3733a71fc295": "~\\sgn(0) = 0~", "7d3428df652590dd5766e8e4b1d6e776": "1 = \\chi(1)", "7d343604c1fafdbda4655b8cd1877b4f": "[Rf_!\\mathcal{F},\\mathcal{G}] \\cong [\\mathcal{F},f^!\\mathcal{G}] . \\,\\!", "7d34870d1353747d920c3a329b55507b": "_{q.1\\,}\\!", "7d34bfece33468e8b52bd8685d315c34": "\\mathrm{P_1} = \\begin{bmatrix} x_1 \\\\ y_1 \\\\ z_1 \\end{bmatrix}, \n\\mathrm{P_2} = \\begin{bmatrix} x_2 \\\\ y_2 \\\\ z_2 \\end{bmatrix}, \n\\mathrm{P_3} = \\begin{bmatrix} x_3 \\\\ y_3 \\\\ z_3 \\end{bmatrix}", "7d34e185d7a839dc80f460398b02bd54": "m_\\mathrm d = \\frac {v_\\mathrm s} {D} = \\frac { \\left ( m_\\mathrm s + 1 \\right ) f } { D },", "7d3577c05e2a9477e6071c785bcef495": "U(V,T)\\ ", "7d3577cf0ab99f37a8f80fb60752314e": "k_1 \\in [1.2,2.0]", "7d35a5be0b360085e9c5d0531ce960f8": "y.\\;", "7d35b2f0a04bba78650334ab52f6c710": "\\prod_p p^{n_p}\\cdot\\prod_p p^{m_p}=\\prod_p p^{n_p+m_p}", "7d35d739d84b7e945af632094dff435d": "\ne^3 \\gamma^\\mu \\gamma^\\alpha \\gamma^\\rho \\gamma^\\beta \\gamma_\\mu \\int {d^4 q \\over (2\\pi)^4}{q_\\alpha q_\\beta \\over (r-q)^2 (p-q)^2 q^2}\n", "7d368f79d9127179cca3dfb8bb96be9d": "\\theta,\\theta", "7d36f99433778ca5212b8640d0ca5ed1": "70/256", "7d36fbaf6dba730b9520ab1645285fca": "f(\\bold x)", "7d37d40d493f40aa0009cc5fd360f37a": " d + H ", "7d37e4d682bdb39d5e47335e8984556d": "\\langle u, A u \\rangle = \\int_{0}^{1} u(x) u'(x) \\, \\mathrm{d} x = - \\frac1{2} u(0)^{2} \\leq 0.", "7d37ff8bb44b24fd60c2f8833c9f93eb": "c=c(v)", "7d38176457234c826d1c0e9d901a058b": "S: [a,b]\\to \\mathbb{R}", "7d38c14af81185fbd5cde9979ccb2da2": "\\scriptstyle{r_1 < r_0}", "7d38c16367c722e60cc917e6e0e445e8": "(5)~~ ~~ u = \\omega(-t) ", "7d38ebb97abc3176bcd6036dd4d2f478": "Xn = {\\text{rate(propagation)} \\over \\text{rate(termination)}} = {k_p[\\text{M}] \\over k_t} ", "7d38ebc7916ee1e6d4c3282965808539": "\\nabla \\cdot \\bold{E} = \\nabla \\cdot ( - \\nabla \\varphi ) = - {\\nabla}^2 \\varphi = \\frac{\\rho_f}{\\varepsilon},", "7d39172e66dff2ab96bfa252887759e4": "V = \\frac{1}{8\\pi\\varepsilon_0}\\sum_{j=1}^N\\sum_{i\\neq j} \\frac{q_iq_j}{|\\mathbf{r}_i-\\mathbf{r}_j|}", "7d395997213954d3743bfc402434c1c5": "((\\lnot A \\to B) \\land (\\lnot A \\to \\lnot B)) \\to A", "7d397c794d4ec860532ac10aacd6db2a": "\\frac{2}{1}", "7d398d5776959e65bf8ddf7d45dc1f33": "(d\\Psi_g)_x : T_x G \\to T_{\\Psi_g(x)}(G) ", "7d39db85d045989347402f4aee5c5e70": "0 \\leq y_j \\leq 1", "7d39fcf11ad9e12b946610512e4d0786": "\\sum_{i=0}^{i=n} N_{i,p}(u)w_{i}P_{i}", "7d3a11b42a07c4b6353e12cbc384ab12": "c_{j,k} = \\frac{1}{\\lambda_k^{j+1}} \\operatorname{Res}_{z=\\lambda_k} f(z)", "7d3a4ddf38abeefb1a8f932a6ee441e4": "\\cos\\delta=\\frac{x_1}{-c\\,t_1}", "7d3a58d5b794780552e17890073f061a": "x \\mapsto \\mu_{x}", "7d3a63defd27a6a7f776cb932b325a70": "k=\\frac{1}{3}\\frac{y''''}{(y'')^{5/3}}-\\frac{5}{9}\\frac{(y''')^2}{(y'')^{8/3}}", "7d3a86cae52b92c76de08b7dcba1de20": " \\phi(x,\\xi) ", "7d3a89fa279c29c0d8a6e7c038ae8369": "\\epsilon\n\\left(\n \\begin{array}{c}\n k_1\\\\\n 2 \\\\\n \\end{array}\n\\right)", "7d3a919ba987af2f9cd67fbc30a42484": "\\left( -2\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ 0 \\right)", "7d3ab337a21805c694476cf3e28f2d36": "a/r", "7d3ae441b5c316aa098dbbb82b3ae31f": "\\Psi : \\mathbb{C}^{n \\times n} \\rightarrow \\mathbb{C}^{m \\times m}.", "7d3b16bb2b3fdce6b837a87eed37b61d": " Y = \\pi (X) \\oplus R", "7d3bb62437ed566cbdfecc7b0a5035a0": "X^*_1=X_2/(1-X_1),X_3/(1-X_1),\\ldots,X_k/(1-X_1).", "7d3c3cfc419fd5b062f21f17a49e033a": "r\\ =\\ p", "7d3c552c43101a42f5bc208dc5b94c44": "-\\frac{\\xi}{2 t} \\frac{\\partial c}{\\partial \\xi} = \\frac{1}{4 t} \\frac{\\partial}{\\partial \\xi} \\left[ D(c) \\frac{\\partial c}{\\partial \\xi} \\right]", "7d3cdad22066e4e46c3c91f933249b11": "(\\hat{x}, \\hat{y}; \\hat{t}) = \\operatorname{argminmaxlocal}_{(x, y; t)} \\tilde{\\kappa}_{norm}(x, y; t)", "7d3cdb2ab5b0752cdd9a860ed57c148e": " F_p ", "7d3cf0eeebf317f1c592b25ed9532c61": " T_2 - T_1 = [\\frac{V_{r1}^2}{2c_p} - \\frac{V_{r2}^2}{2c_p}]\\,", "7d3d24a87b800ef8564bbfecf8cbf63d": "|\\psi \\rangle = a_{\\psi} |\\uparrow_z \\rangle + b_{\\psi} |\\downarrow_z \\rangle", "7d3d2d4b2d10ab690f53ccb46ea65961": "F = m r \\frac{4\\pi^2}{T^2}.", "7d3d8a356f7837e0fecc214096fb976a": "P=(x_P,y_P)", "7d3d903b817f86449fd7e8cb7312b15b": "0=\\left(x+f'\\left(\\frac{dy}{dx}\\right)\\right)\\frac{d^2 y}{dx^2}.", "7d3d9b9f8d8c511da542a38a24317fb7": " t( x,w ) = \\frac{ \\sqrt{ (|w|x)^2+|w|x} - \\ln (\\sqrt{|w|x}+\\sqrt{1+|w|x}) }{ \\sqrt{ 2 \\mu } \\, |w|^{3/2} }", "7d3dde36c84cfbe5f02099c42f204c14": "\\frac {d}{dt} \\left ( \\iint_{\\Sigma (t)} \\mathbf{F} (\\mathbf{r}, t) \\cdot d \\mathbf{A} \\right ) = \\iint_{\\Sigma (t)} \\big(\\mathbf{F}_t (\\mathbf{r}, t) +\\left( \\mathbf{F \\cdot \\nabla} \\right)\\mathbf{v} + \\left(\\mathbf{ \\nabla \\cdot F } \\right) \\mathbf{v} -(\\nabla \\cdot \\mathbf{v})\\mathbf{F}\\big) \\cdot d \\mathbf{A} - \\oint_{\\partial \\Sigma (t) }\\left( \\mathbf{\\mathbf{v} \\times F }\\right)\\mathbf{\\cdot} d \\mathbf{s}. ", "7d3e32ea512f0fdf969d0a523e5041bc": " F_1 ", "7d3e4d04fef04d102a26e4d4c945fc53": "(G; c)", "7d3e7eee0cde57f0fbc52b22db209b8a": "V^\\infty \\wedge V^\\infty \\wedge Y \\wedge Y", "7d3e98638af64956788b78c481a251f9": "A_1,A_2,A_3,A_4, B_1,B_2,B_3,B_4 ", "7d3e9bb2c059325df0cfb6c5a1e37f67": "V_{m}=0", "7d3f0e224757308173a1c5bfcfc6ac10": "\\alpha \\le \\beta", "7d3f423aafbdb2ba8ea9805df1b80852": "\\operatorname{SR}(n) = \\sum_{d\\,|\\,n}\\operatorname{SP}(d).", "7d3f6f5697abc75c218ad00f20c13ba2": " \n\\begin{align} \nP_{ni}& = Prob(\\, y_{ni} = 1 \\,) \\\\\n & = Prob(\\, U_{ni} > U_{nj}, \\quad\\forall j \\not= i \\,) \\\\\n & = Prob(\\, U_{ni} \\, - \\, U_{nj} > 0, \\quad\\forall j \\not= i \\,) \n\\end{align}\n", "7d3f7b1facb130205f4be4ac3372f648": "\\{ \\mathcal{F}_{t} | t \\geq 0 \\}", "7d3f81d81c7fd07c358eec0ec17673d7": "\\upsilon:\\Gamma\\to S^1", "7d3fa379394e2726c37c93e80e7b3cf5": "\\varepsilon(q^k-1) = \\varepsilon N", "7d3fca16cd081bc7c1786a941584a539": "\\hat{f}_\\text{effect} = {\\sqrt{(df_\\text{effect}/N) (F_\\text{effect}-1)}}.", "7d40159d93b3d5d3b91f3ff4c51cb8dc": " dq v = I dl ", "7d4080daaff9afda7d1abe40db134907": "(\\dot{m}_{air} + \\dot{m}_f) V_j", "7d40a04d4c3e237479fead1c2b221e8f": "w\\in \\Bbb{R}^n", "7d40ae4420e64d2032bd7b7333102de7": "k\\left[\\psi(kz)-\\log(k)\\right] = \\sum_{n=0}^{k-1}\n\\psi\\left(z+\\frac{n}{k}\\right).", "7d40d4a7d82058e9c1b04e92a3973bd7": "\\mbox{width}(B_n) = {n \\choose \\lfloor{n/2}\\rfloor}.", "7d40f870a1d28b289663034198fad30d": "f'(x) = 2x", "7d41180c334582c5a348dedb9e88e2f8": "{f_{n-1} \\choose f_{n}} = \\begin{pmatrix} 0 & 1 \\\\ \\pm 1 & 1 \\end{pmatrix} {f_{n-2} \\choose f_{n-1}},", "7d41f68933de7824bafad79d214d7557": "k = { 2\\pi \\over \\lambda } ", "7d41fb1f710dbf879f05b1c308fb5d13": " n_{\\infty} = e_- + e_+ ", "7d420b8711684597d6df3c1953ae85b3": "C_q(n,d)", "7d4230f9764f4f8507ff8ecd75fa1d4f": "q_i, q_{ij} \\in \\{0,1\\}", "7d426bf0fddea9d03c81ec8aa7079f54": "b = a \\sqrt{1-e^2}\\,\\!", "7d42a08c8ff75306b1ec3487504db1d1": "\n\\Sigma=\\frac{2\\rho_s/\\rho-1}{3}-\\frac{\\cos\\theta}{2}+\\frac{\\cos^3\\theta}{6}", "7d42a8efbdcc2157c2eb148c323f37f7": " -2\\log{(X)} \\sim \\chi^2(2)\\,", "7d42ceb40f3b21b20839ff6566f35c99": "F(e^\\lambda g) := \\varphi - d\\lambda", "7d42e54b24fde7299bff079244f2bc8e": "P(|1\\rangle|2\\rangle)=(P|1\\rangle)(P|2\\rangle)", "7d43219025779e4f200d7d4797ef6548": " -\\mathrm{d}\\gamma = \\Gamma_{\\text{RNaz}}^W\\, \\mathrm{d}\\mu_{\\text{RNaz}}\\, + \\Gamma_{\\text{NaCl}}^W\\, \\mathrm{d}\\mu_{\\text{NaCl}}\\,.", "7d4330903a84ea96f0540d18c6eec7ea": "d < \\frac{1}{3} N^{\\frac{1}{4}}", "7d43443bed14e2107071f7c563d7fffc": "\\begin{align}2 H & = -\\nabla \\cdot \\left(\\frac{\\nabla(z-S)}{|\\nabla(z - S)|}\\right) \\\\\n& = \\nabla \\cdot \\left(\\frac{\\nabla S}\n{\\sqrt{1 + |\\nabla S|^2}}\\right) \\\\\n& = \n\\frac{\n\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial y^2} - \n2 \\frac{\\partial S}{\\partial x} \\frac{\\partial S}{\\partial y} \\frac{\\partial^2 S}{\\partial x \\partial y} + \n\\left(1 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial x^2}\n}{\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right)^{3/2}}.\n\\end{align}\n", "7d43ea4b82fa57b30fe202b7fdbfc3ca": "\\rm \\ 2Li_2NH \\rightarrow LiNH_2 + Li_3N", "7d441f4f33caf92f9f903a7004f4561d": "\\hat{x}_2 = \\hat{x}_1 + K_2(y-A\\hat{x}_1),", "7d442fed4509c63e7dd6715c619cc8b7": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi \\epsilon_0} \\int q\\frac{\\delta(t' - t_r')}{|\\mathbf{r} - \\mathbf{r}_s(t')|} dt'\n", "7d44d36bbf36d4f92d7b69a4ac646a09": "L = L_{1} [+] L_{2}", "7d4514626b4252132d1c64866e60db3f": " [AB_\\mathrm{r}]^{\\ddagger} = [AB_\\mathrm{l}]^{\\ddagger} = \\frac{1}{2}[AB]^{\\ddagger}", "7d452b053d759ff901575705c76a1328": "\\mathbf{c}_I^j", "7d457463dafdcf3ec91f25109f39d6e2": "v_e [OH^-]_{0^{ }}/v_e^{\\prime}", "7d457d22ccd2d77c0815ce0d0fd16078": "\\rho \\frac{D u}{D t} + \\frac{\\partial P}{\\partial x} = 0", "7d45bb9de30d3cd52bccf0113f3805b7": "\\ [A] = [A]_0 e^{-kt}", "7d45bd96b18180d35926fa9eed803867": " 3 < 4", "7d45d050cd2bbfe8af576d18e63d9001": " \\hat {\\mathbf{r}} ", "7d46b10fa9cb3571c7050cfd1ae7b6d9": "\\int\\frac{dx}{\\cosh ax} = \\frac{2}{a} \\arctan e^{ax}+C\\,", "7d46c5c2e1604c80beeb4dbad40a2691": "u = u'", "7d46d8a4a1e213746828a28b3dd9e5a1": "-\\eta^2=\\eta_0^2-\\eta_1^2-\\eta_2^2-\\eta_3^2-\\eta_4^2", "7d46fc3b6f47deccf24a6eb205ed346b": "\\delta\\mathcal{S}=0", "7d47121561773a4d2339bcd144eee776": "D_w", "7d4713a1a04f8a04f791f3e285bbea88": "n_\\mathrm{B} = n_\\mathrm{A*} \\frac{R_\\mathrm{A*}-R_\\mathrm{A*B}}{R_\\mathrm{A*B}-R_\\mathrm{B}} \\times \\frac {x(^{j}\\mathrm{A})_\\mathrm{A*}}{x(^{j}\\mathrm{A})_\\mathrm{B}}", "7d472255b23fee6d02c384d582fc0a2d": "A;B \\subseteq \\N", "7d472f21609a726da9046384e37b27ca": "\n g_{ij} = \\mathbf{b}_i \\cdot \\mathbf{b}_j = g_{ji} ~;~~ g^{ij} = \\mathbf{b}^i \\cdot \\mathbf{b}^j = g^{ji}\n ", "7d478af04832893fef64a72326debc85": "\\alpha\\rightarrow\\varepsilon", "7d47d31326149376423e18ee0abdb650": "\\vec E = \\frac {1}{4 \\pi \\epsilon_0} \\iiint \\frac {\\vec r - \\vec r'}{\\left \\| \\vec r - \\vec r' \\right \\|^3} \\rho (\\vec r')\\, \\operatorname{d}^3 r'.", "7d47f68ee275195c7aed867d159f0fca": "\\phi(x^p) = \\phi(x)^p.", "7d4815e0d14db257f418fbc71b6c67cb": " \\mathbf{U} = \n \\begin{bmatrix}\n \\rho \\\\\n u \\\\\n v \\\\\n p \\\\\n \\end{bmatrix} \\ , \\ \\mathbf{F} =\n \\begin{bmatrix} \n \\rho_0 u + \\rho u_0\\\\\n u_0 u + p/\\rho_0 \\\\\n u_0 v \\\\\n u_0 p + \\gamma p_0 u \\\\\n \\end{bmatrix} \\ , \\ \\mathbf{G} =\n \\begin{bmatrix} \n \\rho_0 v\\\\\n 0 \\\\\n p/\\rho_0 \\\\\n \\gamma p_0 v \\\\\n \\end{bmatrix},\n", "7d483592096bffff079542698127a78a": "\\mathbf{A} = \\begin{pmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{pmatrix}", "7d483da7f2a97bc122d964b04f3dad4f": "\n \\sigma_{11} - \\sigma_{33} = 2(\\lambda_1^2-\\lambda_3^2)C_1 ~;~~\n \\sigma_{22} - \\sigma_{33} = 2(\\lambda_2^2-\\lambda_3^2)C_1\n ", "7d48823d1fbcfd381b01ff3373418356": "\\left(\\frac{\\partial H}{\\partial T}\\right)_P = C_P = c_P N", "7d48fc18e78ee2a6a21cd9401e2b522a": "B^{13}_p=\\operatorname{diag}(z_1^{p_1},\\dots,z_1^{p_5})\\backslash U(5)/\\operatorname{diag}(z_2A,1)^{-1}", "7d4901e551016133f567f7b84ef0692f": "\\mathrm{Res}(f'/f)=\\mathrm{ord}(f),\\qquad \\forall f\\neq0;\\,", "7d497bf25a1a178da8b070600c0ed32c": "R = |x-x'|. \\, ", "7d49a7f3c3d4b2c839458ddd1dc7467c": " y_{t} ", "7d49e47d7f25d5b8e16313d9c0bd7634": " dS_t = \\mu S_t\\,dt + \\sqrt{\\nu_t} S_t\\,dW_t \\,", "7d4a4f28f7a1d06ba8af765d292df1d2": "\\frac{\\alpha L^\\alpha x^{-\\alpha - 1}}{1-\\left(\\frac{L}{H}\\right)^\\alpha}", "7d4a7c2d432ef1e3511f60215f222876": "a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n < b ", "7d4a8aba468b55c889708d8326e1e276": "\n{log \\left( \\frac{s - a_v}{\\Omega \\cdot a_c} \\right)} \n\\over \n{log \\left( \\frac{100 + a_c}{100 + a_v} \\right)} \n", "7d4abbb119f5022b3005a79ed4295605": "D(h_n) := D(\\ h_n,t_n,y(t_n)\\ )", "7d4ac2f016c821cfc4840fe2b689ba4f": " g_{ik}g^{kj} = \\delta_i^j", "7d4b3e75c846e19e7a107986f10253c6": " \\lim_{n \\to \\infty} {}^n i = \\lim_{n \\to \\infty} \\underbrace{i^{i^{i^{\\cdot^{\\cdot^{i}}}}}}_n = \\underset{ W:\\; Lambert \\; function}{\\frac {2}{\\pi}\\,i \\;W\\left(-\\frac {\\pi}{2}i\\right) }", "7d4b5b9fa0c1a10e701df014b7523a02": "\\mathbb{Z}/0\\mathbb{Z}", "7d4b91bb63d303106dd748c3fe1c15df": "\\mathcal O_{X, z}/I \\otimes_{\\mathcal O_{X, z}} \\mathcal O_{X, z}/J = \\mathcal O_{Z, z}", "7d4ba7b0ca08869e3985566a809cfe21": "m =\n \\begin{matrix}\n \\sum_{k=0}^t \\binom{n}{k}(q-1)^k\n \\end{matrix}", "7d4bf4f712c97b47d3a78ab48574ae62": "N_0", "7d4bfbb6dab0305f2fd847ba0e93ec06": "\\alpha |0\\rangle + \\beta|1\\rangle", "7d4c3080e6698d0b167ae5de81b1b303": "a \\land b", "7d4c5ecfff33359d4fece27e4bea5e2d": "B^{\\tau}", "7d4cb0c651b3769b8a6f3aa5bed55476": "I > 0", "7d4cbac1c8b8916707afff346aaa281e": "\\operatorname{mr}(G)=|G|-2", "7d4cd8b39569b92e63b30ee139e4e6f9": "\\mathbb R^{+}", "7d4d05f75a8c411fbe3fc6b1c414e1e4": "\\frac{2n+1}{n+1}\\,", "7d4d1112f6d943668e7d0c5c30f29e9e": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "7d4d118803ff8086d5ca502249c05bc0": "\\frac{1}{2\\,b} \\exp \\left(-\\frac{|x-\\mu|}b \\right) \\,", "7d4d26eb71d9150602d0d2f310401c47": "\n\\pi \\rightarrow \\pi+C\n\\,", "7d4d4262d868e3a6f86268bf03aea32c": "\\frac{8}{7}", "7d4d5a9dd19197b14f35e4ef9cc020ea": "E[q] = \\left(1 - \\frac{1}{m}\\right)^{kn}", "7d4d7513a760760b5c19b3bfd18fada1": "R_2 = (A, D)", "7d4d7bf98caef5878a1dbd39d10564a3": "f(\\lambda)", "7d4da12e2b6758ea5f56fc56f8152d11": "K^*(X)\\otimes\\mathbf{Q}\\to H^{2*}(X,\\mathbf{Q})", "7d4dca68bd291ba95a4feba970113337": "a_0,\\dots,a_{d-1}", "7d4e024f29c38952eeaff72c4ffd7ccc": "P_\\text{threshold} = K(\\,L - \\ln R_\\text{OC}\\,)", "7d4e4a6f876c5b8902fe88fe28076a90": "y'(t) = f(y(t)), \\qquad y(t_0)=y_0, ", "7d4e6cfc134b6dda0e09e37e9ac7a076": "t \\approx 2Nc \\,.", "7d4e7b0ff84a8a3ac2298764b8e47830": "\\mathrm{classify}(f_1,\\dots,f_n) = \\underset{c}{\\operatorname{argmax}} \\ p(C=c) \\displaystyle\\prod_{i=1}^n p(F_i=f_i\\vert C=c).", "7d4f28771fcac36b7a7746e43a67e71c": " g\\cdot (v,w) = (g\\cdot v, g\\cdot w).", "7d4f5bb9155e17bce3b63279919f4850": "\\ E[M] \\leq \\frac {mln(1/\\beta)+ln(n)}{1-\\beta}", "7d4ff76c0e6cc0909b8fa4e32519c0c0": " P(X > k\\sigma) \\le \\frac{ \\kappa - \\gamma^2 - 1 }{ (\\kappa - \\gamma^2 - 1) (1 + k^2) + (k^2 - k\\gamma - 1) }.", "7d50252c3dfcc6d5316746fb4930c3ed": "\\vec{x}(t + \\Delta t) = \\vec{x}(t) + \\vec{v}\\left(t + \\tfrac12\\,\\Delta t\\right)\\, \\Delta t\\,", "7d502dbaf801526433b3bfffde6a7d3a": "\\operatorname{cov}(\\mathbf{X}, \\mathbf{Y}) = \\mathbf{0}", "7d5082a274e72df5ef1a0e2a4e753e32": "[-1/\\mu_1,-1/\\mu_2]", "7d50ab2e3ba8027f2d872b0a018cd9ba": "F_Y(y)=\\frac{F(y)-F(a-)}{F(b)-F(a-)} \\,", "7d511200dd01e9b95027e94216447f7b": "\\lambda x.e", "7d5117100c8c71f954a2fcb7b36bc94d": "\\operatorname{tr}(cA) = c \\operatorname{tr}(A)", "7d511b5d0a9cab2a26b002fa987e13bb": "2P^2=H^2+1", "7d5124f019de1a2c78ef96598ed3931a": "{f_o \\over \\sigma}", "7d5148e8b6145f110207633169fdc018": "\\mathcal{A}_\\phi=-\\cos^2{\\theta\\over 2}", "7d514af09a497a02cb825c1255840301": "\\dot{\\rho}=\\mathcal{D}[c]\\rho-i[H_\\mathrm{sys},\\rho]\\,.", "7d51a9df28dc06a010e452728aec2601": "P^T AP", "7d51ad10edeeff6de4f0d656a5d77d56": "\\sum_{j=1}^n \\sum_{k=1}^n z_j \\bar z_k C(x_j - x_k) \\ge 0.", "7d51c5d0ec0f3209347151fdb4ad3ecf": "\\mathbf{a}\\cdot\\mathbf{b} = 0", "7d51c95fef7ad7f4a2e470c67956c9e9": "\\Delta_d=2\\Delta_{\\bar{\\partial}}=2\\Delta_\\partial . ", "7d51cb7a8f63d367115074e1d74a5dec": "A\\equiv r(mod f)", "7d51eef2a72d5b2ed543c4a480d27f8b": "\\Delta{H}", "7d52456451169a3275a682fe39310495": "\n \\mathbf{b}_i\\cdot\\mathbf{b}_i = g_{ii} = \\sum_{k} h_{ki}^2 =: h_i^2\n \\quad \\Rightarrow \\quad \\left|\\cfrac{\\partial\\mathbf{x}}{\\partial q^i}\\right| = \\left|\\mathbf{b}_i\\right| = \\sqrt{g_{ii}} = h_i\n ", "7d52622284c47946799310fd029ac194": " v_2=\\frac{2 m_1 m_2 c^2 u_1 Z+2 m_1^2 c^2 u_1-(m_1^2+m_2^2) u_1^2 u_2+(m_2^2-m_1^2) c^2 u_2} {2 m_1 m_2 c^2 Z-2 m_1^2 u_1 u_2-(m_2^2-m_1^2) u_1^2+(m_1^2+m_2^2) c^2} ", "7d52810477fda489ffb41d8882fd4d38": "\\vec \\omega_{vorticity}=\\frac{v_{\\theta}}{r}+\\frac{dv_{\\theta}}{dr}=0", "7d52ea5caa67a3a7a17dc8bc7132cdd5": "\\hat{\\mu}^1_{ij}", "7d539f930d1370f42bd8ee669cbab8c9": "=-\\frac{{\\pi}^2}{15}+\\frac{1}{2}\\operatorname{arcsch}^2 2", "7d53ba02db250ad43ee837c739baa09a": "\\omega\\colon V\\times V \\to \\mathbb F", "7d5430f2e67f9ab1d5e46445c982c1cc": "\nP(W_1|Spam \\land W_0) = P(W_1|Spam)\n", "7d5437c950cf80201d194b4750c30829": "V[a\\xi+b]=a^2V[\\xi]", "7d54584ccfcf544501bcd27d07ce2164": "\\mathcal{D}_{\\Gamma} = Ran( D_{\\Gamma} ) \\quad \\mbox{and} \\quad \\mathcal{D}_{\\Gamma^*} = Ran( D_{\\Gamma^*} ).", "7d546b53c50b1312fae35517701da649": "h_a(x)=h_a(y)", "7d54a913ccbc240031686e59c1ac8790": "0\\geq t_2\\geq t_1", "7d54b40a742036da9f3b25c3a039c2af": "H \\cdot t = a \\cdot b \\cdot ( E - e \\cdot \\sin E)", "7d54e86551504510ad1c6c0d9b6a6b11": "\\eta_f", "7d55595a705b4cb83feb840d102f264e": "Z = 1", "7d55978b12a6b5a584bf272e3e1bd78f": " W^{2} =M^{2}c^{2}S^{2}", "7d55c128ca67147362cc43923fff0d88": "{2}/\\sqrt{3}", "7d56314582931f6009cfd3d26c6a5940": "P={^hp}=p_d+zp_{d-1}\\cdots+z^dp_0", "7d564ded0ac9b244f8aa90b1b975d5fd": "X^2-X+1\\ \\text{mod}\\ q", "7d56720d2a9c647db205ffc738913971": "k_i=\\frac{2 \\pi n(\\lambda_i)}{\\lambda_i}", "7d56ab40e5669d98ae939e3f56262d51": " y_{n+1}\\equiv a y_{n}^{\\varphi(m)-1} + b \\pmod m \\text{, }n \\geqslant 0 ", "7d56d3fa14e191d89b5a4d52f3f3b558": "\\eta=0.\\,\\!", "7d56e93fec57c1da6243b17051c81e09": "A \\smallsetminus B \\in \\mathcal{R}", "7d57a93316ef0c0724dc9070437bafad": "C_1 = 40 \\ \\mathrm{pF}\\,", "7d57c43156b24de56fc7b299610ee84d": " \\vec{\\Omega} = \\frac{\\omega}{1 - \\omega^2 \\, R^2} \\; \\vec{p}_1 ", "7d5846aecd2bd91d966129f500077727": "x^2+ux+v=(x-s)^2+t^2", "7d58480ada77a26d73cc16beca2b585e": "\\Pr [r \\notin S] = \\Pr [r \\notin A(x) \\oplus t_1] \\cdot \\Pr [r \\notin A(x) \\oplus t_2] \\cdots \\Pr [r \\notin A(x) \\oplus t_{|r|}] \\le { \\frac{1}{2^{|x| \\cdot |r|}} }.", "7d58680a77ed448c84a1fa543f8815bc": "\\int_0^\\infty \\frac{\\sin{x}}{x}\\,dx = \\frac{\\pi}{2}", "7d5895ee4497605a4ce895c8128766b0": " P(S) = q^{N_\\text{sample}(S)} \\times (1-q)^{(N_\\text{pop} - N_\\text{sample}(S))} ", "7d58efe2189af967b6658456ed163fd4": "b=rt^m", "7d59c4d1c67a52440efff527e1df5da3": "s_k = 6s_{k-1} - s_{k-2},\\text{ with }s_0 = 0\\text{ and }s_1 = 1;", "7d59dbfbe4fc5642d80ec02672de03e4": "\\mathcal{B}(m,n) + O(n t^t / \\lg^t n + n^{3/4})", "7d59e9f57090f648f0c133be6703728a": "{\\rm tr}(\\mathbf{X}^{-1}d\\mathbf{X})", "7d59eb36bebdf838fa06358a39bd2b6e": " rate= -\\frac{d[R 1]}{dt} = k_1[SH^+] [R 1] [R 2] + k_2[AH^1] [R 1] [R 2] + k_3[AH^2] [R 1] [R 2] + ... ", "7d59eb63b8eaac55a75a082b370abdae": "\\frac{\\partial \\rho}{\\partial t}=-\\nabla \\cdot \\mathbf{J} ", "7d59f0afa1af58ee1de695769fdde938": " \ne(n) = d(n) - \\hat{y}(n) = d(n) - \\hat{\\mathbf{h}}^H(n) \\cdot \\mathbf{x}(n) \n", "7d59f871b0cfd83d81921423c95db1db": "r (s m) = (r s) m", "7d5a4ee847e3fb1a28e17fe2a153846f": " z_k ", "7d5a54165d7d57f82495aca6333df0c8": " \\, s_{j+1,k} = \\frac{s_{j,k}+s_{j,k+1}}2 ", "7d5ac4520e9c9bfa9df02450012847ea": "\\, a", "7d5ac612116f44ff2cbb1f21576a6184": "I_\\mathrm{v} = 683 \\int^\\infin_0 \\overline{y}(\\lambda) \\cdot \\frac{dI_\\mathrm{e}(\\lambda)}{d\\lambda} \\, d\\lambda.", "7d5ad75505d47210de618e473f23be3d": " S^2 | s, m_s \\rangle= {\\hbar}^2 s(s+1) | s, m_s \\rangle ", "7d5af42f61d784e66bf52fe45494ca9c": " p_2,", "7d5b35beed00804ad9afa606c27e1fe4": "\\sum_{n=0}^\\infty 2^{-2^n}", "7d5b7cd5795e32bb3a44c318341af49d": " {\\Delta}P\\,\\!", "7d5baebf80dc2b359b57ce95c4a8c465": "\\scriptstyle \\mathbb{R}^n \\setminus K", "7d5bf013f81c25206ccedd6d426e6407": "K = 100(W/L^3)\\!\\,", "7d5c0694b5b9245c8858e0ba88c1986b": " g: \\mathbb{R}\\rightarrow \\mathbb{R} ", "7d5ca3ad142059808f02183e9b023e20": "\\{(m,r_m):\\;m=0,\\dots,n\\}", "7d5cb2b9f8929906fd8cba74350b9b71": "(a \\times b) \\mod{9}", "7d5cdaf7260fb0261efaff4a94c9d062": "(A_i , U_i)", "7d5ce304fed3c6314fcb61bb1c858409": "O(n \\cdot \\log \\log n)", "7d5ce95e088e5f09c1e26bd26ede6755": "\\nabla L = (L_x, L_y)^T", "7d5d465fdfd3b46ddb918ec7301c6e67": "\\frac{ax^2+2bx-ax}{2}\\,", "7d5d8854a5ff50996f602ef52f71746d": "d\\epsilon_{i,j}^e=\\cfrac{1}{2G}(d{\\sigma}_{i,j}-\\cfrac{3\\nu}{1+{\\nu}}{\\delta}_{i,j}dp)", "7d5d936af9bfd2d47fa1a4b16060b5f1": "\\bold j = \\frac{1}{2m}\\left[\\left(\\Psi^* \\bold{\\hat{p}} \\Psi - \\Psi \\bold{\\hat{p}} \\Psi^*\\right) - 2q\\bold{A} |\\Psi|^2 \\right]\\,\\!", "7d5e0dcb2481da58f1bde2d2b234db61": " y_{n+1}= (x-c_n)y_n - d_n y_{n-1}", "7d5e2186d843f1bdacf875bbe9d46503": "PV \\,", "7d5e5febf9808533f3c34c600c06ae44": " A\\xrightarrow{\\cong} A\\otimes I\\xrightarrow{A\\otimes\\eta}A\\otimes (A^*\\otimes A)\\to (A\\otimes A^*)\\otimes A\\xrightarrow{\\epsilon\\otimes A} I\\otimes A\\xrightarrow{\\cong} A", "7d5e747d1c67f0c665e38f01bf809942": "\\phi\\neq\\theta ", "7d5e7acd3d3a26143f62b1e3c2f3241f": "\\displaystyle{L(a(bc) -(ab)c)=[[L(a),L(b)],L(c)].}", "7d5f77f336dc7586d3c94095d2d6087b": "-\\Gamma^{d}{}_{b_ic}", "7d5fb1b9980ce5778fddb71d41206a03": "\\cos\\beta", "7d603652a058553e4571c0882eb4c2e3": "\\neg \\phi", "7d603e4df81f115afa89195ae7cff1a2": "S = c + z log(A) = log(cA^z),", "7d60695f24499ebaa3b928776aa202a2": "M_1 \\equiv f(M_2)\\, \\bmod\\, N", "7d6097b14f4faf9ff68e2d52684b4fb6": "Y(a,z)Tb - TY(a,z)b = \\frac{d}{dz}Y(a,z)b", "7d611ac55296c60142f3e4cafec791a6": "\\dot{k} = 0", "7d6186ceffc5708c51dcc706f01ada83": "x_1*x_2*\\dots*x_n = \\mathfrak{0} ", "7d61f9c4510c4b5411e8963d6f84a2bd": " g^{-1} h ", "7d622de97c1b0057b9795b2eb4bbc211": "h \\not\\equiv 2 (\\bmod 4)", "7d626407e41037a5b0d6e413b832b43c": "W_{magnetic}=\\int_{0}^{H}(-M_\\perp\\sin{\\theta}-M_\\parallel\\cos{\\theta})\\, dH=-\\frac{H^2}{2}(\\chi_\\perp+\\Delta\\chi\\cos{\\theta}^2)", "7d62bcac6363e4a782521475f513ebfa": "\\begin{align}\nh'(x_0) & = \\lim_{\\Delta x\\to 0} \\frac{\\Delta h}{\\Delta x}\\\\\n& = \\lim_{\\Delta x\\to 0} \\left ( \\frac{\\Delta f}{\\Delta x}g(x_0) \\right ) + \\lim_{\\Delta x\\to 0} \\left ( f(x_0) \\frac{\\Delta g}{\\Delta x}\\right ) + \\lim_{\\Delta x\\to 0} \\left ( \\frac{\\Delta f \\Delta g}{\\Delta x} \\right ) \\\\\n& = f'(x_0)g(x_0) + f(x_0)g'(x_0) + 0 \\\\\n& = f'(x_0)g(x_0) + f(x_0)g'(x_0) \\\\\n\\end{align}", "7d63643765fb9941710fc84be2b4288f": "\nk(z,y) = 2 \\gamma \\; (zy)^{\\gamma - 1/2} \\; G_{p+q,\\,m+n}^{\\,m,\\,p} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p}, \\mathbf{b_q} \\\\ \\mathbf{c_m}, \\mathbf{d_n} \\end{matrix} \\; \\right| \\, (zy)^{2 \\gamma} \\right),\n", "7d638f555c957441253a2b4f63953eea": "\\begin{align}Instrumented \\\\ Range \\end{align} \\begin{cases} Minimum \\ Sample \\ Width = \\left( \\frac{Duty \\ Cycle}{PRF} \\right) \\\\ \\\\ Maximum \\ Distance = \\left( \\frac{Pulse \\ Spacing}{Sample \\ Width}\\right) = \\left( \\frac{C}{Sample \\ Width \\times 2 \\times PRF^2}\\right) \\\\ \\\\ Samples \\ Per \\ Transmit \\ Pulse = \\left( \\frac{1}{Minimum \\ Sample \\ Width} - 1 \\right) \\end{cases} ", "7d639ef1bd23f5b0723186dede2e5560": " i < 0", "7d642443e77ae52502a3ae6c136c40b4": " \\textbf{D}^{\\rm T} \\textbf{C}^{\\rm T} (\\textbf{A}\\textbf{x} + \\textbf{b}) + \\textbf{A}^{\\rm T}\\textbf{C}(\\textbf{D}\\textbf{x} + \\textbf{e})", "7d644146a2e368669af87762f4e4ba44": " [T \\phi](x) = -\\sum_{i,j} \\partial_{x_i} \\{a_{i j}(x) \\partial_{x_j} \\phi(x)\\} \\quad x \\in U, \\phi \\in \\operatorname{C}_0^\\infty(U), ", "7d644fbfab008f41c7cfbcbacaf69870": "I_A", "7d649fa9cc361084b9c75eaca4f9923b": "l(x^1)=1", "7d64b5196fc9fb441a40499149f7541f": "s_m = 2(c + \\frac{s}{10}) + k_1k_2\\frac{d_b}{\\rho_{ef}}", "7d64c4ccea03184ef6aacc9448a4b91c": "\\ \\gamma=1", "7d650d70f37c5063d9336599341f4598": "S=S_G+S_F+S_{FM}+S_M\\;", "7d65b2ab5b3bce9deca25dacfe669989": "c\\eta=\\Psi.\\,", "7d65b9a264a28325a43c6b8926d7d039": "ACBED", "7d6627680f95997f7527a10ebc48dac2": "sim(q_{and},d)=1-\\sqrt{\\frac{(1-w_1)^2+(1-w_2)^2}{2}}", "7d663b4350abe0077763cccc9d6fd500": "P \\setminus \\{x\\}", "7d666dea0fc5ab8b43d76ac43c567bfd": "D_n = 4n^2 - 3n.", "7d667b0ef0f35070385b3c5e276937c3": "g_o", "7d668b8b6cdf6f4ca4181593f6139312": " {} + \n*(\\partial_i f \\, \\mathrm{d}x^i \\wedge \n\\varepsilon_{jJ} \\sqrt{|g|} \\partial^j h \\, \\mathrm{d}x^J + \n\\partial_i h \\, \\mathrm{d}x^i \\wedge \n\\varepsilon_{jJ} \\sqrt{|g|} \\partial^j f \\, \\mathrm{d}x^J) ", "7d669fd28f03845796a67eb739e3e4b2": "r_4 = (S \\to A, \\emptyset, \\{X\\})", "7d67611541c0af2e63fec81bc593b7d9": "p=\\exp(\\sigma)", "7d676531d0476d821e8c59472f82b2b6": "{\\Bbb C}P^3\\backslash {\\Bbb C}P^1", "7d6774ade191488268831fe319436d4f": "(-1)", "7d67b69bc504c07e81a4ec73f8978968": "d_\\nabla: \\Omega^p(M,E) \\to \\Omega^{p+1}(M,E)", "7d67d80fc160dc40c453615b5f20cbd7": "\\frac{t}{1-t-t^2}.", "7d6811412af92280ddde2d3d48de4419": "k_\\mathrm{E}=\\frac{S_\\mathrm{E}}{k_\\mathrm{ES}.S_\\mathrm{D}}.", "7d68e67823e06b743a5307e437b52026": "\\|u\\|_{C^{0,\\gamma}(U)}\\leq C \\|u\\|_{W^{1,p}(U)}", "7d6943f0c52e654afb26650b8e4f6cfc": "p \\colon A \\times B \\to A", "7d69e7090d0671bae83a11d8d54a0edd": "\n\\lambda(t|\\theta)=\\theta\\lambda_0(\\theta t)\n", "7d69ef49eab4845f079355031c72d241": "\n\\begin{align}\nY_{00} & = s = Y_0^0 = \\frac{1}{2} \\sqrt{\\frac{1}{\\pi}}\n\\end{align}\n", "7d6a1d95e0c7ba64677bba0e31979471": "M _{AB} ^f = - \\frac{P a b^2 }{L ^2} = - \\frac{10 \\times 3 \\times 7^2}{10^2} = -14.7 \\mathrm{\\,kN \\,m}", "7d6a2bc7d5540366716774c695d4ac8b": " e^{-as} F(s) \\ ", "7d6a37ce91f5031272f584162cbe5b12": " \\frac{ 1 } { 1 + \\sigma^2 }.", "7d6afd5bec31e90c8938ab765bb71df3": "(g, h)", "7d6b512cd0df0592daaed1466fe61efd": "\\{D=l^a\\partial_a\\,, \\Delta=n^a\\partial_a\\,,\\delta=m^a\\partial_a\\,,\\bar{\\delta}=\\bar{m}^a\\partial_a \\}", "7d6b6ded5632518d5a1f2f82a31df03f": "k=\\frac{k_0 A}{a}\\sqrt{\\left\\{1+\\left(\\frac{1-n}{1+n}\\tan\\varphi\\right)^2\\right\\}\\frac{\\cos^2\\xi'+\\sinh^2\\eta'}{\\sigma'^2+\\tau'^2}},", "7d6b8dcfbc59bcd723aa216ac437ec1d": "P=P_n", "7d6b9b2611715be8dc3a611e9729053f": "\\chi ^2 _{0.50}", "7d6bd5431462a9b524bd7c125680e87a": "k\\cdot v", "7d6beedd4f8fc6fc95d67f8c38405357": "f(z) = \\sum_{k=0}^{\\infty} (\\operatorname{PP}(f(z); z = \\lambda_k) + c_{0,k} + c_{1,k}z + \\cdots + c_{p,k}z^p),", "7d6c03a0007c86d428f66831acf80afc": "X_\\theta\\subseteq X", "7d6c28f41df6bdbbb9af3e3ecbebfb5e": "\\mathrm{III}(t) \\quad\\stackrel{\\mathcal{F}}{\\longleftrightarrow}\\quad \\mathrm{III}(f)", "7d6c495acf21e687254d2ff676ae77ae": "W_E = \\frac{1}{2} \\displaystyle \\sum_i \\sigma_i\\epsilon_i ", "7d6c584cf68ddafd0c5eb6c8b01f0b3e": "\n\\begin{align}\n{\\mathbf{A=LDL}^\\mathrm{T}} & =\n\\begin{pmatrix} 1 & 0 & 0 \\\\\n L_{21} & 1 & 0 \\\\\n L_{31} & L_{32} & 1\\\\\n\\end{pmatrix}\n\\begin{pmatrix} D_1 & 0 & 0 \\\\\n 0 & D_2 & 0 \\\\\n 0 & 0 & D_3\\\\\n\\end{pmatrix}\n\\begin{pmatrix} 1 & L_{21} & L_{31} \\\\\n 0 & 1 & L_{32} \\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix} \\\\\n& = \\begin{pmatrix} D_1 & &(\\mathrm{symmetric}) \\\\\n L_{21}D_1 & L_{21}^2D_1 + D_2& \\\\\n L_{31}D_1 & L_{31}L_{21}D_{1}+L_{32}D_2 & L_{31}^2D_1 + L_{32}^2D_2+D_3.\n\\end{pmatrix}\n\\end{align}\n", "7d6d1a9287a835588c3a7a3dd20a0fab": "Q(\\mathbf{r},i) = 2r_i - \\mathbf{r}\\cdot \\mathbf{r} = 2r_i -\\sum_{j=1}^C r_j^2 ", "7d6dc127b0e7775588a6ff1979a5f0af": " m_1\\bold{u}_1 + m_2\\bold{u}_2 = m_1\\bold{v}_1 + m_2\\bold{v}_2 = (m_1+m_2)\\bold{V}", "7d6e1f622c66f82fbf3bd8f3a263c196": "\\frac{\\alpha_\\text{G}}{{m_\\text{e}}^2} \\,", "7d6e25b39fe28bd1716dd8e857de6c57": "\\lambda_0e_0 + \\cdots + \\lambda_{n+1}e_{n+1}=0\\,.", "7d6e409ba9fcdf6cda2a905dab134e7f": "(\\forall k)({a^n_1}...{a^n_k}) < (n-1)m + 2", "7d6e819bd824ccdbb05d794dc7990f89": "(s_j-j)^2", "7d6f06782db6f190724310d2436cec12": "\\subset G\\}", "7d6f10dd02467f44efb406551a77ecd8": "x_{n_1} \\leq x_{n_2} \\leq x_{n_3} \\leq \\ldots", "7d6f405f9ddd13d04f9e5bc7e9257422": "b=d", "7d6f46b5c5aa6ffb78b1017874a28c3a": "\\textstyle {N}(B_i)", "7d6f8959e0fb003b9e4ab9bb740a39c3": "M \\leq n\\,\\!", "7d701523ee16b87ab745a76d29154085": "V(A \\lor B,1) \\Leftrightarrow V(A,1) \\ or \\ V(B,1)", "7d701a1a87328d0360e3f6fe65e6e834": "Alt^k\\ {\\mathbb C}^n", "7d704f964d8a2a19857051d4ad4f3658": "u_{|\\partial \\Omega}", "7d7059f0d1d5ed989992dc13c28ee43f": "\\int^y P(\\lambda)\\,{d\\lambda} + \\int^x Q(\\lambda)\\,d\\lambda = C\\,\\!", "7d70ef4172a2d896b228dfb7ff5f31c5": "\\le 1 ", "7d7115f5950fed3e10a75cf701504395": "\\gamma\\cos\\,{\\theta^*}= f_{1}({\\gamma_\\text{1,sv}}-{\\gamma_\\text{1,sl}})-(1-f_{1}){\\gamma}", "7d71169da22b8be91feffb004423f3c7": "\ndet \\left( A \\right) = 1.\n", "7d711a4dfd6ffec718a444ff5ff29452": "\\left[\\lambda_a, \\lambda_b \\right] = 2i f_{abc}\\lambda_c", "7d713d624e6dcb04e4896d706147ba7e": "f_k^m(n)", "7d71da17ad781b0cbfcc985e975526fd": "E(x)=k,\\,E(\\ln(x))=\\psi\\left(\\frac{k}{2}\\right)+\\ln(2)", "7d7227afa565d58d9eb26a479cf461a2": "u\\neq v", "7d7228b478fa0324e7893c75b0b84ba1": "g_1(x_i,y_j) = g_0(x_i,y_j) + \\sum( f(x,y) - g_0(x,y)) \\exp\\left(-\\gamma \\kappa \\frac{\\pi}{\\lambda}^2\\right). \\,", "7d72d75062f2421225b5ae573f0f3fe5": " a y''(x) + b y'(x) + c y(x) = 0, \\;", "7d72da4bfd4f0b7f119847cdfb45858d": "\\mathcal Q_n", "7d7340d0d995e46e6269091552a20582": "z_1 = \\sin\\eta\\,e^{i\\varphi}", "7d73a2d7d7088ee40b789584292aa176": "\\sin x \\approx \\frac{16x (\\pi - x)}{5 \\pi^2 - 4x (\\pi - x)}, \\qquad \\left(0\\leq x\\leq\\pi\\right).", "7d740c05ada92fbdd097ea2fc01515f8": "\\hat{x} = hy + c", "7d74183bff8a6a28cd3ea24cc4ce5f39": "AA^*", "7d742b1e50b498ac228c6c7da7211442": "\\, T = \\begin{bmatrix} T_{r\\overline{o}} & T_{ro} & T_{\\overline{ro}} & T_{\\overline{r}o}\\end{bmatrix}", "7d74648d113449151b23c8c064898ef8": "A f(t, x) = \\frac{\\partial f}{\\partial t} (t, x) + \\theta(\\mu - x) \\frac{\\partial f}{\\partial x} (t, x) + \\frac{\\sigma^{2}}{2} \\frac{\\partial^{2} f}{\\partial x^{2}} (t, x).", "7d748acacdecb0cd7ae43449f12e680d": "B = D + b", "7d749ad3277c9928ac3dd58f521f6ab2": " \\mathbf{C}(n,k) = \\mathbf{C}_k^n= {_nC_k} = {n \\choose k} = \\frac{n!}{k!(n-k)!}.", "7d74e0c9a8fa26a10cbef4f9436291b5": "\\alpha\\,G \\,", "7d753ac65aa0141f2433f338be638dc6": " \\alpha(n) = d(n)-\\mathbf{x}^T(n)\\mathbf{w}(n-1)", "7d75b55bc90accd15aea0cf4e1dbe48e": "p(S\\vert M) = 0.1 \\times 0.6 \\times 0.7 \\times 1.0 \\times 1.0 \\times 0.6 \\times 0.7 \\times 0.2 \\times 0.2 = 0.0007056.", "7d76587d05357bc7851151e1f612edd9": " \\mathbf{B} = \\frac{\\mu_0}{4\\pi} \\int_C \\frac{I d\\mathbf{l} \\times \\mathbf{r}}{|\\mathbf{r}|^3}", "7d765c04755e5e806b764615399f1c7b": "\\vec a \\cdot \\vec b", "7d769161e69dce31fda98b079cf711fd": "V_1,\\ldots,V_n", "7d7720115b405d4c2e837c6b68a82b8d": "\\displaystyle{F_{f\\circ g}=F_f \\circ g.}", "7d77822d84720f4ff384bdb02af74140": "g(t) = t+b_1t^2+b_2t^3+\\cdots\\in R[[t]]", "7d779d5a00d4515b5fa6e84487074a92": "\\omega(t)=\\inf_{s\\geq0}\\left\\{2\\delta(s)+st\\right\\}.", "7d78151697936f2dbd41f02a7afaca4e": "u'' + p(x)u' + q(x)u=f(x),\\quad x>a", "7d781d12e1ae3d15d492131ca46aa54c": "\\beta=1/kT", "7d7850b0a1cccaa2d50f361b2b4d646d": "\\psi_p^{(n)}(a) = \\frac{\\partial^n \\log\\Gamma_p(a)}{\\partial a^n} = \\sum_{i=1}^p \\psi^{(n)}(a+(1-i)/2).", "7d78934640dfcb9c87215448ddf28f97": "s>t", "7d793314da8f59c29f534b07927f4821": "\\sum_{b=1}^B m_{vb} m_{wb} = \\Lambda", "7d7953dbb24aaa26d6374a0c504c8d24": "\\beth_1 \\ge \\aleph_1.", "7d7981a4af813a5f869881c11668dc31": " {v_M} ", "7d79a3a56ab7b49b87515c2080b6d759": "F(x; 0, \\sigma, -\\alpha)", "7d79a644ba5a12a974877b7283d0adb1": "(X-X_0,Y-Y_0,Z-Z_0)", "7d79d5dce068422a676fab468a38ce9e": "s_n(w)", "7d7a477756280bfdf8f1601b5ae5ca2b": "\\mathrm{4 \\ FeO + O_2 \\longrightarrow 2 \\ Fe_2O_3}", "7d7a913c155a321ba4c75a7cf84815d2": " \\lim_{k \\to \\infty}( \\bold T^k)_{ij} = \\bold 0, \\quad (1) ", "7d7aba3cb240e6c084bcd6f652323506": "\\tilde{V} = V + U,", "7d7b271826e31fb9d845ae87aa7c1fe7": "\n[\\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c})] \\mathbf{a} = \n(\\mathbf{a}\\times \\mathbf{b})\\times (\\mathbf{a}\\times \\mathbf{c})\n", "7d7b56f5a7511f528da7c25d697c0dce": " (\\sigma_i)_i \\geq 0 ", "7d7b587455c1134b7d98cdd5ab3f0eab": "\\lim_{x \\to 0^+} \\ln x = -\\infty. \\, ", "7d7bb3563d64e47afd31b3b98fa34e5f": " H = x_m \\left( 1 + \\frac{ 1 }{ \\alpha } \\right) .", "7d7bd89cb12b710d2c149bd522df6da5": "f^{\\mathrm{SS}}_p (x,y) = \\begin{cases}\n -\\log x & \\text{if } p = 0 \\\\\n \\frac{1 - x^p}{p} & \\text{otherwise.}\n\\end{cases}", "7d7bfcf004f96c1a2ed83eb55891e5f1": " r(t) = r(0) e^{-a t} + b\\left(1- e^{-a t}\\right) + \\sigma e^{-a t}\\int_0^t e^{a s}\\,dW_s.\\,\\!", "7d7c34ffdf8783de17ec16958c0e5f7e": "\\operatorname{dom}(f^*)", "7d7c825a38c79fbfca3fd607c1fdff11": " A \\otimes_B A ", "7d7c83b17971444d57909b5aca4b4ff7": "r_d =", "7d7c9c65912e790ba3d053d32444a1f4": "\n\\frac{1}{1+\\frac{1^2}{2}} = \\frac{2}{3} = 1 - \\frac{1}{3}\n", "7d7dd78b9f76bd4cb5fcf5a1ea593746": "F_{i0} = d*i", "7d7de434210ff54a06b217df39417c18": "g(z) dz^{2}", "7d7de99c37507b220fd977a50d804178": "P = \\{ 0, \\dots, n-1 \\}", "7d7df5e819dddcb18cd2160b240fe4b2": "a \\in B\\wedge a\\wedge B", "7d7e0295b30189e765f408cc1fe1e1a8": "(D + w w^{T})q = \\lambda q", "7d7e1293e54464e522c93fe0b6df51f8": "{\\mathcal W}_{\\mathcal P}", "7d7e69a99ddaa50e81a0f35f030af3db": "\\xi^{(a)}_{i}", "7d7e7428d24990cd86c537e8da9add21": "\\psi_{\\nu_\\tau}", "7d7e74f53fba0ca685b8ed7641e5b0d3": "Q = 1/\\sqrt{3}", "7d7e92b494befa9a3e2a0d8e9fc2ade3": "(646, \\ldots)", "7d7ea7421f9be8ed042d6100248f36dc": "r_{i+1}:=-\\text{rem}(r_{i-1},r_{i}).", "7d7ef27a1c9651a5a6cc9185e4b2a007": "T_w", "7d7ef322fbafd2be5ab57b67b52ad884": "N=\\frac{1000W_{KHP}}{204.23V_{eq}}", "7d7f6edf32c2fac5f2f4481181abdc6c": "\\rm{g}(x)=\\langle x, y\\rangle", "7d7f7df10694fc1213969559444b15fd": "P(X=115|M_2)=\\int_{0}^1{200 \\choose 115}q^{115}(1-q)^{85}dq = {1 \\over 201} = 0.004975...\\,.", "7d7fadfdc594cedddf3349eb1ba87c88": "\\Gamma_q(n+1)=[n]_q!\\frac{}{}.", "7d7ffcdf8170cafd56b73027034d719d": "q_{jk} = 0", "7d80e02c0d08264e51910af738354e87": " \\tan \\varphi' = \\left(\\frac{c_1}{c_2}\\right)", "7d80e9e01a23ca3bcf429d62344483f2": "a \\not\\equiv b", "7d80ef5514edbd9122748fccdc747d4b": "\n\\psi(\\vec{\\theta}) \\approx \\sum_i \\frac{2 GM_i D_{is} }{D_s D_i c^2} \\left[ \\ln\\left( { |\\vec{\\theta}-\\vec{\\theta}_i |^2 \\over 4} { D_i \\over D_{is} } \\right) \\right]. \n", "7d8132af22d9d63bf94d3017a89ca08d": "H = -\\int_\\Gamma f_w(\\theta)\\,\\ln(f_w(\\theta))\\,d\\theta", "7d8134b6df2a763f29c398e3ff5406e5": " \\mathrm{K} = 0.13", "7d81357ae9f4cbf336a20544a1e709f5": "\\delta^\\alpha_\\beta", "7d81381d86de92b218fc7ea62f823f10": "u^0_\\alpha,~ \\alpha=1,2", "7d815b4ab112b78e01f627be820af9bc": "{\\mathcal Q}", "7d81ab53cb9cd649595adeb4486c6531": "\\; \\pi_{. j} = {\\sum_i O_{ij} \\over N} \\;", "7d8219ac657db9836a7455904bddba6a": " \\text{confidence} = \\frac\\text{signal}\\text{noise} \\times \\sqrt{\\text{sample size}}.", "7d824095b95fb05992ad1ae0b4e29923": "\\frac{256}{243} \\sqrt[4]{2}", "7d8251d4f2f9045254921ebfa9b1d979": "\\|x\\| \\le \\|x+y\\|", "7d825f476ea0f57a9427e0434aea4db3": "b<_sce(ab)", "7d82fc3b440e967f04af52554b5ba5cc": "\\kappa_4(X)=\\kappa(X,X,X,X)\\,", "7d83041b10efad6d0ede7d7410a6af0d": "[-1,\\ 1] \\times I", "7d834d13c99f7e99f889f45c3cd58510": "(y_k)", "7d8387f424c2a376de12296172748352": "\\theta = \\frac{2\\pi}{\\phi^2} n,\\ r = c \\sqrt{n}", "7d843e9a58db4c78e751bb1c989a2ce3": "f_1=g_1=1", "7d8482445d9fdf4b5ecab08dee8dfbc6": "\\langle 2 \\rangle_\\mathrm{S}", "7d84987651236d1f0891e1d72ba15f35": "C(x)", "7d84a3f40dad63bdb2a47e2c86d4da89": "Q_f", "7d84aad7de1050b11fbbf761a0fe8af3": "p = {\\tbinom {n + 1} 2} + 1 = {\\tbinom n 2}+{\\tbinom n 1}+{\\tbinom n 0}. ", "7d851e9be244431cbe0a7c7ebb40b544": "p \\leftarrow \\hbox{not } p", "7d857cc210b84061f43046e4a8b1dc89": "X_i \\sim \\mathrm{Pois}(\\lambda_i)\\, i=1,\\dots,n", "7d858d7fb7bc7089e1241073635e4d19": "y_{2ss} = \\left ( \\frac{q_2^2}{g} \\right )^\\frac{1}{3} = \\left ( \\frac{50.0^2}{32.2} \\right )^\\frac{1}{3} = 4.27 \\text{ ft}", "7d85a9f5eab6bdd1af43eff12455633b": "q(T)", "7d85b41936c263b1cebc130bef3095e4": "\\left( \\begin{array}{c|c} M & v\\\\ \\hline 0 & 1 \\end{array}\\right) ", "7d85d12b23d91564becd212d03c41d8a": "f(u\\mathbf{x} + v\\mathbf{y})(u\\,dv-v\\,du)", "7d863de0cd7e68dfbfafb9448eb58792": "\\hat{x} = hy+c", "7d8677b38fdc69e701d72ac875c86c33": "\\scriptstyle\\boldsymbol{g}(\\boldsymbol{x})=\\left(g_1(\\boldsymbol{x}),g_2(\\boldsymbol{x}),g_3(\\boldsymbol{x})\\right)", "7d86a3ebe0ac150d0c9b70aa4d3b63b6": "cT_1", "7d86fd6828b6b58f62d7d010cb9ba888": "h_i(\\mathbf{p}, u) = \\frac{\\partial e (\\mathbf{p}, u)}{ \\partial p_i}", "7d87375b1c21bf53b8584bfc7429e2c7": "\\displaystyle P(x|z,c)", "7d8757166ed39e56fa69616ee639b2b2": "\\scriptstyle \\dot{\\boldsymbol{r}}_i \\;=\\; \\dot{\\boldsymbol{r}}_i (t_i)", "7d876bfe7c04757a9334ee31e4bb43af": "[[w]]", "7d8813d0ead712c76054bdebbc15abc4": "A_{12} < 0 ", "7d8828cb24dd32ef8e25bc664d0c4544": "t \\in [t_0,t_1]", "7d8829a75ca4ed3bdf1db201978c1b90": " \\mathbf{A}\\mathbf{A} = \\mathbf{A} \\cdot \\mathbf{A} + \\mathbf{A} \\wedge \\mathbf{A}. ", "7d886fc4dc2307f57bcd1a457a494cba": "g:Y\\rightarrow Z.\\,\\!", "7d889c4f75855df58a406022ece4bf8c": "r_\\mathrm{min}", "7d889fa3e7422f62f3be30f3629f6c01": "\n= \n{1\\over 2} {e^{ - m r } \\over 4 \\pi r } \\left[ \\mathbf 1 - \\mathbf{\\hat r} \\mathbf{\\hat r} \\right]\n+\n {e^{ - m r } \\over 4 \\pi r } \\left\\{ 1+ {2\\over mr } \n- {2\\over \\left(mr\\right)^2 } \\left( e^{mr} -1 \\right) \\right \\}\n\\left\\{ {1\\over 2} \\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right] \\right\\}\n\n", "7d88e139d7c3b7e029fa2a76768b39ff": "\\sin A = \\sin (\\pi - A') = \\sin A'", "7d8917f64656830995c759538fac78e0": "\\alpha = n-2,\\,", "7d899d9c044bcd21f9ff75c34c691a23": "X_i = {1 \\over D} (1 + \\epsilon_i) \\,", "7d89ecb58dd1dbff7a0f667cfbe429dd": "2^{s_n}=2", "7d8a149bd2b12b229bbc658cbbd465b5": "\\det(M) \\cdot \\det(M')", "7d8a220d2262f9d6c658d549ee12cf2c": "ABS", "7d8a28dcb4bcca950c6119756d364305": " A_{m_j} ", "7d8a3f10ba87a256408ca57d08ea0b7b": "\\Sigma^0_\\alpha \\setminus \\Pi^0_{\\alpha}", "7d8a442191c4a3a2f437887db04bbf76": "=\\sum_{i=1}^J[\\pi(\\mathbf{x}-\\mathbf{e}_i) \\alpha p_{0i}+\\pi(\\mathbf{x}+\\mathbf{e}_i)\\mu_i(x_i+1)p_{i0}]+\\sum_{i=1}^J\\sum_{j\\ne i}\\pi(\\mathbf{x}+\\mathbf{e}_i-\\mathbf{e}_j)\\mu_i(x_i+1)p_{ij}.\\qquad (2)", "7d8a71477cf946aeff9a92a1ae783942": "I_{xx}", "7d8ae1f96f1dadda5df02c89ab333ab3": "\\frac{1}{2}e^{-(x+\\lambda)/2}\\left (\\frac{x}{\\lambda} \\right)^{k/4-1/2}\n I_{k/2-1}(\\sqrt{\\lambda x})", "7d8b1667478a0132d2a004dbefc214dd": "(\\phi,x) \\otimes (\\psi,y) = (\\phi \\otimes \\psi,x \\vee y)~~~~\\,", "7d8b1b35cfdc9a8f45db357e8169d40b": "\\textbf{P}_{k\\mid k-1}^a = \\textbf{F}_k\\textbf{P}_{k-1\\mid k-1}^a\\textbf{F}_k^T + \\textbf{Q}_k^a ", "7d8b2bb40401fe29756353b7da0c195a": "f = 0, \\frac{1}{K}, \\frac{2}{K} ...", "7d8b64c2f848805f9312525eb2786c37": "a \\oplus b = \\sqrt{a^2+b^2}.", "7d8bc0586daafa9b14161ca8d1097958": " \\bar X = \\frac{1}{3} \\sum_{i=1}^{3} X_i ", "7d8c56fae89c0d543ed0560b785a2073": "Z_1 = R \\sin(\\Theta) = \\sqrt{-2 \\ln U_1} \\sin(2 \\pi U_2).\\,", "7d8c8019a301f6385932fb41375503e5": "I_4", "7d8c8f482e3e702a871458d78da0fa40": "\\rho \\ ", "7d8ca4678aef0d862085d5daf92531c4": "\\tau_{ix} : \\pi^{-1} (x) \\to X_{i}", "7d8ca670a4231a3f332d36ea79271295": "\\gamma^0 = \\beta \\,", "7d8cb7af8b278013e71ee5042f102584": " \\widehat{\\mathbb{Z}} = \\prod_{p} \\mathbb{Z}_p. ", "7d8cb7bca6e729a3204bde5002d12fe6": "x_0, -x_0 \\in \\mathbf{F}_p", "7d8cfdba71d45ddc21f5d6ac79fb9fcd": "k=\\pm\\pi/a", "7d8d3e1294e485282ad3867d5bc0b601": "\\sum_{i \\mathop =3}^6 i^2 = 3^2+4^2+5^2+6^2 = 86.", "7d8d3f45406b9069e85661d0eaecf875": "a(\\theta-b)", "7d8d86cec4b060c1a04710e02fcb57a0": "L(x,y;t)", "7d8d9e5f13a745f8d3a96e4cbdf243ba": " \\mathbf{a}\\ \\stackrel{\\mathrm{def}}{ = }\\ \\frac {\\mathrm{d} \\mathbf{v}} {d\\mathrm{t}} = \\mathbf {\\Omega} \\times \\frac{\\mathrm{d} \\mathbf{r}(t)}{\\mathrm{d}t} = \\mathbf{\\Omega} \\times \\left[ \\mathbf {\\Omega} \\times \\mathbf{r}(t)\\right] \\ .", "7d8e22c48c23ca4cc2de7f23642fea83": "I_{sp} = \\frac{dp_e}{dm_{fuel}} = \\frac{(1 - \\eta) \\ v_e}{\\sqrt{1 - \\frac{v_e^2}{c^2}}}", "7d8e745dc4ea9486784f1dab3bed3d29": "g_\\mathrm{ds} = \\frac {1} {r_\\mathrm{O}} ", "7d8e8823ea741e37eb22df3917007682": " d \\times q ", "7d8ea7893b5384ea6e5b03ca3dfd37da": "P(x_1,x_2)= (x_1 + 7x_1x_2 + x_2)^2\\,\\!", "7d8eac6426cf5747eb23c09e23efd4e7": "H(x^*(t),u^*(t),\\lambda^*(t)) \\equiv 0.\\,", "7d8eafab1fdd743c5afefccaa8e2af01": "\\int_0^\\infty \\frac{x^2}{e^x-1}\\,dx = 2\\zeta(3) \\simeq 2.40", "7d8ee0475efce255d8883387b147cd4a": "\\textbf{G}(\\infty)", "7d8f26f62f3bdddf3a8fbe6ce35976d0": "\\mathbf{u}(\\mathbf{x},t)", "7d8f324a735693452a437f0a98b0ab0c": "|c_v| \\le 1\\,", "7d8f3a810055daaabce2b3fa6202ec68": "I_N=(I-\\text{Min})\\frac{\\text{newMax}-\\text{newMin}}{\\text{Max}-\\text{Min}}+\\text{newMin}", "7d8f5a30d16254d9944d310de7e23095": "BSC", "7d8f756594f77ab9dcdd47b66ed58f31": "\nm_n(x_n)=I(x_n-\\tau)=\n\\begin{cases}\n 1 & x_n > \\tau \\\\\n 0 & x_n\\leq \\tau\n\\end{cases}\n", "7d8f7ca17e2ca6cbfed7fa9ee07e4f1b": "S^* = S \\pm y^* \\sum \\Delta x", "7d8f85e1ff14d29bfbef61160c5a5353": "\\mathbb{R}^{n-p}", "7d8f8a915e9c39a28f76ab0c19c88202": "\\omega^{\\omega^{\\varepsilon_1+\\varepsilon_0+1}}", "7d8fd24f912dba3cc57fc9143a0a29dd": "\\varepsilon_1 = \\frac{1}{E}((1+\\nu)\\sigma_1-\\nu(\\sigma_1+\\sigma_2+\\sigma_3))", "7d900d2ed031abed4341df5b3be98a5e": "\\lfloor x + 0.5\\rfloor.", "7d9092145fa21fa63ba3c09ffcb2956a": "Z = a^{(p-1)/2} \\left(1 \\cdot 2 \\cdot 3 \\cdot \\cdots \\cdot \\frac{p-1}2 \\right)", "7d90cc23ceb7780bbd4bf9daa2cb14ff": " \\beta = \\|b-Ax_0\\| \\, ,", "7d911f76373e2056f78aee40321d52e5": "|E(x+iy)|", "7d913d7f71a832a24634ea2086eb7406": "\\operatorname{E}(T) = k\\sigma^2 + n\\mu^2.", "7d9172ab5e8779a8a8ec6f0655800876": "\\mathbf{\\Psi}_{00}= \\mathbf{0}", "7d917dbb5773203866316183a189eb0d": "\\mu_{3,2}= \\mu_{3,2} - r\\mu_{2,2}= \\frac{13}{14}-1= \\frac{-1}{14}", "7d918f3d29abb4b0b368b6b3c5970276": "X \\sim \\Gamma(k \\in \\mathbf{Z}, \\theta), \\qquad Y \\sim \\mathrm{Pois}\\left(\\frac{x}{\\theta}\\right),", "7d91d777d15f6a2f2df897156cc720ad": "\\lambda=\\frac{U}{V}", "7d922c814c22c9cbadbb5029c70ba166": "(r,1]", "7d9277a00c80be8ec4548b15e6119e3f": "\\frac{dP}{dt}=\\lambda \\cdot f(P,...)", "7d927b7d68c23ba85317d97bd896120c": " y_i = A_{ij} x_j\\,,", "7d929293bac47cfca15d5a7b56863966": "\\frac{9}{3}", "7d92981112c89068abb9bff13cb40c7e": "\\alpha \\approx \\frac{LG+RC}{2\\sqrt{LC}} = \\tfrac{1}{2}\\left(Z_0 G + \\frac{R}{Z_0}\\right) \\approx \\frac{R}{2Z_0}", "7d929e5de2240a705b8752b4c8af1315": "\\psi(x) = \\frac{d}{dx} \\ln\\Gamma(x)=\\frac{\\Gamma'(x)}{\\Gamma(x)}", "7d92a5bc8cc4720601fcf63ad370898a": " c_{t+1} - c_t = y_{mt+1} - y_{t+1} - y_{mt} + y_t - \\frac{(R-1)} {R} y_t ", "7d92b1e891173b282c7b2733ff220210": "\\frac{\\pi}{3\\sqrt{3}}", "7d931d8e38bf5ecb142ad134675d3869": "\nS (\\omega) = 2e\\vert I \\vert \\ ,\n", "7d935268fb38658e6f1307994e9d82da": " \\nu = \\sqrt{\\left( \\mu^{'~2}_1 + \\left(\\xi\\left(\\theta^{*}\\right) - 2\\right)\\sigma^2 \\right)}. ", "7d93bc129a1c42c9eea1a859429b89ab": "\\bigl(\\tfrac{z}{p}\\bigr)=-1", "7d9420f6c30fa44ade79a58b432b84a7": " T = \\frac {70 \\times \\text{patient's weight in kg}} {4900} ", "7d94338e087a5ac881c5d634ab4110ae": "X \\preceq Y \\Leftrightarrow Y - X", "7d943d2f586a31af13330883c4282146": "\\mathbf{R}_1^0,\\ldots, \\mathbf{R}_N^0", "7d9441366d6be9d03d2561423f82f0e6": "\\pi_j = \\sum_{i \\in S} \\pi_i p_{ij}.", "7d947ead9513aaff3b024912641d6cf5": "\n\\begin{align} |Tf(x)|^2=\\left|\\int_Y K(x,y)f(y)\\,dy\\right|^2\n&\\le \\left(\\int_Y K(x,y)q(y)\\,dy\\right) \n\\left(\\int_Y \\frac{K(x,y)f(y)^2}{q(y)} dy\\right)\\\\\n&\\le\\alpha p(x)\\int_Y \\frac{K(x,y)f(y)^2}{q(y)} \\, dy.\n\\end{align}\n", "7d94a871884c8f16e0deadca55e2a681": "T_{\\pi^{-1}, \\lambda'}", "7d94c887dfd40e14f9837daa9fbad550": "\\tan{\\frac{\\alpha}{2}}\\tan{\\frac{\\beta}{2}}+\\tan{\\frac{\\beta}{2}}\\tan{\\frac{\\gamma}{2}}+\\tan{\\frac{\\gamma}{2}}\\tan{\\frac{\\alpha}{2}}=1,", "7d94d9b37cf4825e8b0e7131f10d3aa0": " r^2\\nabla^2 Y = -\\ell (\\ell + 1 ) Y", "7d94fa02e5dfee5963f190cc354f45c4": "k-FWER", "7d9524bfa14c26ce3b0e3fd0e4098d6e": "\\sum_{k=1}^n \\tilde{I}_k = 0", "7d95429c79e85359e8117668df0567be": "\\scriptstyle A\\,=\\,2.5dt", "7d955e09db73ceb5dbf7cf86536e3eb1": " Y \\sim \\operatorname{Log-\\mathcal{N}}(\\mu, \\sigma^2)", "7d95dc2c07e9048561517d8754b61a57": "w_0^{-1} = w_0", "7d95e1c84632f7a7f1854fba37a4afd8": "\\nu = \\frac{2a \\kappa V^{1/3}}{3 k_BN} = \\frac{2a \\kappa}{3 k_B\\rho^{1/3}N^{2/3}},", "7d960575199dc1c7b4237728a056bc14": "K = \\tfrac{1}{2}|\\tan \\theta|\\cdot \\left| a^2 - b^2 \\right|.", "7d96ab99433710027e3636275df1429a": " p(x) = \\sum_{i=0}^n c_i x^i ", "7d96d1be4155d3f87385a55d3d0354cd": "S=\\{1,x,x^2,...\\}", "7d974ce0109d1f30850cec26e0ec041f": "(C^{\\textbf{.}}(\\mathcal{U}, \\mathcal{F}), \\delta)", "7d97a050965a3b2163adeb6f092b64a6": "{e}'(v)F(v)+e(v)f(v)=vf(v)", "7d97f864841e157dbf61e576f2079025": "H^G_*(E_{VCYC}(G),L^{\\langle-\\infty\\rangle}_R)\\rightarrow H^G_*(\\{\\cdot\\},L^{\\langle-\\infty\\rangle}_R)= L^{\\langle-\\infty\\rangle}_*(RG)", "7d9847d8c7025bfe49b2ad2ea7834bc6": "\\{v_1, \\ldots, v_n\\}", "7d98fb1d30ef58f488b42ac44c24dff9": " i\\pi = \\log(-1) = \\log\\left[(-i)^2\\right] \\neq 2\\log(-i) = 2\\left(-\\frac{i\\pi}{2}\\right) = -i\\pi", "7d990b54f39e3e92352b91f7a1c42d4a": " n = 0, 1, 2, \\ldots", "7d9927cfd70a887845f2e6cf0165246a": "2i\\sqrt{t}", "7d99afb09ccd1c5fa630dc7e4ab490c4": "(\\pm 2,\\pm 2,0,0,0,0,0,0)\\,", "7d99cc3ce7842ea8e9408592d83a3b9e": "\\Omega = U\\Lambda V^T\\text{ with }\\Lambda = \\operatorname{Diag} \\big ( \\lambda_1 \\le \\lambda_2 \\cdots \\lambda_N \\big ) \\,", "7d9a0bd2bb60637f12bb119b0ab775e2": "A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\\cdots A_n B_n A_n^{-1} B_n^{-1}=1", "7d9a272c56ff10531cdedc3c9a10aaf0": "\\left({l-1/2,k+1/2}\\right)", "7d9a4480cb250cc3f443d7ccd6eeec14": " S_1(a,b,c)=(S(a,b),c)", "7d9a5455fbb0071641472518ee344ec5": " \\left(\\mathbf{a}\\mathbf{b}\\right)\\cdot \\mathbf{c} = \\mathbf{a}\\left(\\mathbf{b}\\cdot\\mathbf{c}\\right) ", "7d9a5f002d2ea8d402deb11627a20706": "d x^n \\in A", "7d9a65ab3ccf6a8f779f5f8b392c725c": "X \\sim \\Gamma(k, \\theta) \\equiv \\textrm{Gamma}(k, \\theta)", "7d9a70964848361265b613b7cec62e6b": "\\left(\n\\begin{array}{llll}\n \\left\\{0,0,0,0\\right\\} & \\left\\{0,0,0,0\\right\\} & \\{0,0,0,0\\} & \\{0,0,0,0\\} \\\\\n \\left\\{0,0,0,0\\right\\} & \\left\\{0,0,0,0\\right\\} & \\{0,0,-r,0\\} & \\left\\{0,0,0,-r \\sin ^2\\theta\n \\right\\} \\\\\n \\{0,0,0,0\\} & \\left\\{0,0,\\frac{1}{r},0\\right\\} & \\left\\{0,\\frac{1}{r},0,0\\right\\} & \\{0,0,0,-\\cos \\theta \\sin \\theta \\} \\\\\n \\{0,0,0,0\\} & \\left\\{0,0,0,\\frac{1}{r}\\right\\} & \\{0,0,0,\\cot \\theta \\} & \\left\\{0,\\frac{1}{r},\\cot \\theta ,0\\right\\}\n\\end{array}\n\\right)", "7d9a74e4e10740a4bf2f1aecdfaf5113": "~p=\\sqrt{KL~}~\\theta~", "7d9a779668d0ad6438d1fed91067840e": "w'^2(z)=a_0^2+\\frac{a_0\\Gamma}{\\pi i z} +\\dots.", "7d9a84ed34404915f2ebabcf3dcb049a": " \\overline{p} = {F \\over A} -{\\overline{\\rho} \\overline{U}^2},", "7d9a8ba5b194928b0669ff75354c8b36": " U(1) ", "7d9b5619dabd93104618567c35b249a9": "\\displaystyle y=Q^{-1}(t)x", "7d9b6f341e875bb8d3e9c58af7b66b11": "\\mathcal{L}_X(B) = \\mathbb{P}\\left[ X^{-1}(B) \\right], \\qquad B \\in \\mathcal{B}(\\mathbb R) ", "7d9b9be764fbd419c2a8e7d4e2ab548a": "\\xi, \\eta\\in T_pM", "7d9badf6a838f9e26114dae8619d151e": "end\\, if", "7d9bf4d8a631b58ee3c3081be2c6a1c6": "\\left[\\hat{b}, \\hat{b} \\right]_- = 0", "7d9c01b8b01b0079bac34fcecad442ca": "\\langle x + y,x + y\\rangle = \\langle x,x\\rangle + \\langle x,y\\rangle + \\langle y,x\\rangle + \\langle y,y\\rangle.", "7d9c9f25e69ab3c1b540af21cbae44bb": "\\neg (\\overline{\\alpha_1} \\vee \\overline{\\alpha_2})", "7d9cccc5078647ded90078c9cc763168": "\\tilde \\delta_X(\\varepsilon) = \\sup \\{ \\varepsilon t / 2 - \\rho_{X^*}(t) : t \\ge 0\\}.", "7d9ce1620cf5ed7e8571edd4be33fc65": "\\frac{1}{2\\pi i}\\int_{\\Gamma}\\frac{d \\zeta}{\\zeta-z}\\,d\\zeta.", "7d9cf73ae0502bc81b6104e039b5747b": "Y^r = Y_{\\mathrm{ymid}+1:\\operatorname{length}(Y)}", "7d9d1640183db06415669f3e2e8c85f9": "P(H|E) = \\frac{P(E|H)}{P(E)} \\cdot P(H)", "7d9d1db1d27e90066a05133c6ecbd0f4": "\n\\lim_{n\\to\\infty}\\Pr\\left\\{\n\\left|\nt_n-\\theta\\right|<\\epsilon\n\\right\\}=1.\n", "7d9d84b31d825422b279f1d2ba1aa135": "z=c", "7d9dd9cd5f591c519ed546a5e512af16": "\\mathbf{M} = k{\\mathbf{B} \\over |\\mathbf{B}|},", "7d9e0a79736ac90133ac7f5caabb2288": "-0.25 \\le \\beta < 0", "7d9e19660ff8d60461b6b6291d48b49b": "\\textit{dau}(e,t)", "7d9e1a515f4b0f94f4e0f6e3dd02f15d": " \\langle \\chi | \\psi \\rangle = \\int\\limits_R d^3\\mathbf{r} \\, \\langle \\chi | \\mathbf{r} \\rangle \\langle \\mathbf{r} | \\psi \\rangle = \\int\\limits_R d^3\\mathbf{r} \\, \\chi(\\mathbf{r})^{*} \\psi(\\mathbf{r}) ", "7d9e1f1754248c08074bde779ca0045d": "\\sum_{k=1}^\\infty \\frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\\tanh z, |z|<\\frac{\\pi}{2}\\,\\!", "7d9e5b18bbbe3204da9ca26d61eaacf6": "\\left[ \\begin{array}{ccc|c}\n2 & 1 & -1 & 8 \\\\\n0 & 1/2 & 1/2 & 1 \\\\\n0 & 0 & -1 & 1\n\\end{array} \\right] ", "7d9e670b67e84b007d26514846c8270b": "\n\\xi = \\frac{v}{2 \\pi f} = \\frac{v}{\\omega} = \\frac{p}{Z \\omega} = \\frac{p}{ 2 \\pi f Z} \\,\n", "7d9e875c344dd9db8365baad7895ffcc": "V_t(\\mathbf{x})", "7d9ea9232087419cc72dd3ac205ad2ff": "\\displaystyle\\{E_i\\}", "7d9efedc2f0ca62e4287a7dd019e7355": " \\mathbf{r}_i = (\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{R}, \\quad \\mathbf{v}_i = \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{V}.", "7d9f0d9ab455f85648519ba6a9d046d6": " \\rho = 1- {\\frac {6\\times194}{10(10^2 - 1)}}", "7d9f11e5a5b85e9b4b7d74729090f21a": "p_1 \\equiv 1 \\pmod{2}", "7d9fdcb6468acb637830e752a75857b8": "E_{m} = \\frac{RT}{F} \\ln{ \\left( \\frac{ \\sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\\mathrm{out} + \\sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\\mathrm{in}}{ \\sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\\mathrm{in} + \\sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\\mathrm{out}} \\right) }", "7da0406e03fae8847023ef1b604412af": "R_1 = \\frac{1}{4} \\, {S^a}_b \\, {S^b}_a", "7da04ebd1e0eb7dd3575f0c21e1d3ea7": "\\alpha = \\sigma\\,\\sqrt{2\\,m-3}, \\!", "7da0c01f0bbf622b26152a444c7ee574": "\n \\sigma_{xx}^{\\mathrm{face}} \\equiv \\sigma_{xx}^{\\mathrm{face}}(z) ~;~~\n \\sigma_{zx}^{\\mathrm{core}} = \\mathrm{constant}\n ", "7da0d2bccaba571393d1d69bc9bf25bd": "i-\\lambda", "7da0faa1e3bb3d838a6070f4f96fdd25": "\\mathcal{V}(x) = \\{ X \\}", "7da1867e7764e4edb73ad7f65f1a098b": "\\forall t. \\textit{changeon}(t) \\leftrightarrow (\\neg \\textit{on}(t) \\leftrightarrow \\textit{on}(t+1))", "7da196a56c56d0f600c6aa01ce9ddfec": "{\\mathcal A}_p = \\left\\{\\mathrm{Log} (z) \\, : \\, z\\in \\left({\\mathbb C}\\backslash\\{0\\}\\right)^n, p(z)=0\\right\\}.\\,", "7da1b07094baf875b446ffd99635b3fc": " T_{11}(x) = 1024x^{11} - 2816x^9 + 2816x^7 - 1232x^5 +220x^3 - 11x. \\,", "7da1c2b150245e09c28749bfcb399343": "\\frac{\\left(m_1, n_1\\right)} {\\left(m_2, n_2\\right)} \\equiv \\left(m_1n_2, n_1m_2\\right).", "7da206321d73dd93ece0f41093f3ffcf": "\\text{Converting ohms from one voltage to another:}", "7da23f2515ee7500441aabd7377956ad": "\n\\mathbf{C} = \\begin{pmatrix}\n-3 & 6 & -3 \\\\\n6 & -12 & 6 \\\\\n-3 & 6 & -3\n\\end{pmatrix}", "7da24de39db37d28cab7ec3b95101308": "11\\cdot 13\\cdot 17 ", "7da287458ef16510a6a1683c9580d63c": "\\psi(n)", "7da29615e5f6cd1dc13c7d44a200e884": "s-b", "7da2d39d8636a3b2038f8b83443ac2b8": "\n _{(x)}\\Gamma^m_{ij} := G^{mk} \\,_{(x)}\\Gamma_{ijk} ~;~~\n _{(X)}\\Gamma^\\nu_{\\alpha\\beta} := g^{\\nu\\gamma} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} \n", "7da2e19996e848defb6f2f950d7f0e45": " [M]_{C}^{B} [M]_{B}^{C} = [M]_{C}^{C} = \\mathrm{Id} ", "7da3551144bb36415e028b52d3368d99": "U = \\operatorname{Spec}(R) \\subset X", "7da36e8f76453f2f05d4d01b27717474": "\\forall\\,n \\in \\N_0", "7da3793683ff884786953cca573051b2": "\\color{red}\\lnot", "7da3a53e17849764497091b742191741": " \\frac{1}{2^n}", "7da3ab15d58d66024eba15587d46d8af": "S = \\rho^\\nu_\\mu \\rho^\\nu_\\mu", "7da3f9417e349bc472da3b128413cd70": "V(x)", "7da40bc97ba6b376b528de3edfe16347": "X_{a;b} = \\sigma_{ab} + \\omega_{ab} + \\frac{1}{3} \\, \\theta \\, h_{ab} - \\dot{X}_a \\, X_b", "7da40cf9f3c9efa53e9e9377d91e1914": "\\text{Cl}_2(\\theta) = -\\int_0^{\\theta}\\log\\Bigg|2\\sin \\frac{x}{2} \\Bigg|\\,dx", "7da45f4d5c6f74d8e664298d35df526e": "(1 - \\beta\\lambda)", "7da48657a840463c22cdc07dbc991b58": " \\partial_\\mu \\partial^\\mu \\phi = 0\\,", "7da48c8f9be34725a9cea3c4e91139fe": " MA = \\frac{T_B}{T_A} = \\frac{N_B}{N_A}.", "7da4d862a3c362481aa0ff36959163c6": "(8) \\ f_i(\\varphi)=\\frac{\\nu}{\\lambda}\\left[n_\\varphi\\cos(\\varphi+\\alpha)\\pm\\sqrt{n_0^2 - n_\\varphi^2(\\varphi)\\sin^2(\\varphi+\\alpha)}\\right]", "7da5079a7553ad20f2b49d1de66c74ae": "B^{\\prime\\prime} ,", "7da51e77485b5549d80e314bd8328bde": " \\varphi(m,n,p) = \\begin{cases}\n\\varphi(m, n, 0) = m + n \\\\\n\\varphi(m, 0, 1) = 0 \\\\\n\\varphi(m, 0, 2) = 1 \\\\\n\\varphi(m, 0, p) = m &\\text{ for } p > 2 \\\\\n\\varphi(m, n, p) = \\varphi(m, \\varphi(m, n-1, p), p - 1) &\\text{ for } n > 0 \\text{ and } p > 0.\n\\end{cases}\\,\\!", "7da52fbcddcc8730a036c920c8034f58": "[X_i,X_j]", "7da5876135867f1587f16c59cb7f38fb": "\\int \\sinh (ax+b)\\sin (cx+d)\\,dx = \\frac{a}{a^2+c^2}\\cosh(ax+b)\\sin(cx+d)-\\frac{c}{a^2+c^2}\\sinh(ax+b)\\cos(cx+d)+C\\,", "7da5c8830c3dc2734b310f3e42f75563": "\\tau_y\\!", "7da5d417c5225beacfb8d1e9e628ab10": "\\left[J_a ,J_b\\right] = i\\varepsilon_{abc}J_c", "7da5db3aaf44be2a3fa79bf30208d95c": "\\mathbf{v}'_1", "7da6d8adb1aea4168a1b9c1a62e4b6b1": "S_p(0)=0", "7da6fd629e65c742ed20b02b4a86e148": " a = {{g (m_1 - m_2) - {\\tau_{friction} \\over r}} \\over {m_1 + m_2 + {{I} \\over {r^2}}}}", "7da7140b7c8ce39f83ccc8b67936dd30": " \\mathbf{F} + \\mathbf{R} =m\\ddot{\\mathbf{X}}. ", "7da75213150f65a6c8ff8a48d91440e8": "\\textstyle x_0, x_1, \\ldots", "7da77be12d32b60c00c806f68cfb0e0c": "\\begin{pmatrix}1 & 1\\\\0 & 1\\end{pmatrix} \\begin{pmatrix}x\\\\y\\end{pmatrix}\n= \\begin{pmatrix}x+y\\\\y\\end{pmatrix}.\n", "7da77e44705f34c7341b83a3c522039b": "\\textstyle\\frac{75}{272 - 92} = \\frac{4}{10}", "7da7e7e406f2f1e81d45f2a96d91684f": " L = \\lambda_1 | e_1\\rangle\\langle f_1| + \\lambda_2 | e_2\\rangle \\langle f_2| + \\lambda_3 | e_3\\rangle\\langle f_3| + \\dots , ", "7da828300e7430501147164d1a0b55b0": "a$", "7da963fae2e573957e6f9916e6716fb3": "\\mathbf{A} \\otimes \\mathbf{B}", "7daa0746a4f37e10dbb673fdac6e817d": " \\frac{d}{dx}\\cosh x = \\sinh x \\,", "7daa8f6a9b81e6f9463a86e67b252f6e": "\\mathcal{L}=p_{1}x_{1} + p_{2}x_{2} + \\lambda(u-U(x_{1},x_{2})) ", "7daad141b70b7c784c41722564e9bc82": "\\frac {V_\\mathrm i}{V_{xn}} = \\left ( 1 + \\frac {\\delta Z}{Z_0} + \\delta Z \\delta Y \\right)^{\\frac{x}{\\delta x}}", "7dab549e6651b751e9bb2ee862c90afa": "\\left[S_x,S_y\\right]=i\\hbar S_z", "7dab5cb09897bbf442c0ccf715ea01ec": "\\mbox{NEXP} = \\bigcup_{k\\in\\mathbb{N}} \\mbox{NTIME}(2^{n^k})", "7dab61cb9c570b7eba95e3ffc6f66ac2": "t^\\prime", "7dab6e01e11cd8cb4583f00760b7ca63": "x=\\tan y", "7dab7918284f90d6cf1c40b455fe5458": "\\ 1 \\le i \\le N-m+1 ", "7dabaa772fc37bc98148834073c35a38": "\\begin{align}\n\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ a\\cdot f(x) & =a \\cdot \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x) \\\\\n\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ (f(x)+g(x)) & = \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x) + \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ g(x) \\\\\n\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ (f(x)-g(x)) & = \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x)-\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ g(x) \\\\\n\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ (f(x)\\cdot g(x)) & = \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x) \\cdot \\lim_{x \\rightarrow x_{0}} \\operatorname{ap} \\ g(x) \\\\\n\\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ (f(x)/g(x)) & = \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x) / \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ g(x)\n\\end{align} ", "7dac181e55ffd85f64d4dd04a84b5c37": "\\{\\, uv \\in E \\colon u\\in S, v\\in T\\,\\}", "7dac2472d838699193178f0522c80b32": "\\{(x,y) \\mid x\\text{ is even}\\},\\quad \\{(x,y) \\mid y\\text{ is even}\\},\\quad\\text{and}\\quad\n\\{(x,y) \\mid x+y\\text{ is even}\\}", "7dac2d1253a7bdc4661e54b4e5d75aeb": " D_{blood} ", "7dacfa099753f24706a60491a5a5dc08": "\\phi(t) \\approx A \\sin \\omega t + B \\cos \\omega t", "7dad034cd37835382806edad81d63bfb": "X|Y \\sim \\mathrm{Rayleigh}(Y)\\,", "7dad34c053aeefca69d88d37c07a4eef": "\\frac{T_{now}} {T_{t}} = {(1+r)}^\\tfrac{t} {2}", "7dad4ef4418c6cc00ef8996e7e03166d": "h(x) \\in K", "7dad613c47cd017ab1442814f3fadf3a": "V \\times V \\to K", "7dada18ce64dac91d641b734875db028": "\n+T=\\left(\\frac{\\partial U}{\\partial S}\\right)_{V,\\{N_i\\}}\n =\\left(\\frac{\\partial H}{\\partial S}\\right)_{p,\\{N_i\\}}\n", "7dada21ce3b84a63d6d11f9534338d6e": "{E}\\left\\{\\mathbf{x}(n) \\, e^{*}(n)\\right\\} ", "7dadb7203b7f1234db16ee4b7a610675": "\\Delta z ", "7dadcac157399433928b5c660078b6b4": "f_j\\in (V^\\prime)^{\\otimes n}", "7dadd39e2e4408c8b93e1e410e3b69db": "n_3 \\neq 0\\,\\!", "7dae4439972b8af7c218e129e8d042a2": " A_i \\equiv \\prod_{j=1 \\atop j \\ne i}^k \\frac{1}{\\lambda_i-\\lambda_j} (A - \\lambda_j I). ", "7daf50369b9d8de88d5fb6865b3cf24d": "0 \\leq m < k-n/2", "7daf56263eb273ab5155bc09258b86aa": "\\|f_1\\cdots f_n\\|_r \\le \\|f_1\\cdots f_{n-1}\\|_{pr}\\|f_n\\|_{qr}.", "7daf83b7fa3f8a03e4e1d2e6143853c4": "p(m) = \\mathrm{Pr}(M=m)", "7daf8fd8afb360e1c726133e6ba77f6e": "A=\\frac{V_2}{E}=\\frac{2R_2}{R_1 + R_2}", "7daff58ba464b01cc223680269d1b4e1": "\\tfrac{t^{3}-q^{3}}{t^{2}-q^{2}}", "7db001163961a8cd50c7b26c0ed7b411": "\\mathbb{T}=\\mathbb{R}/\\mathbb{Z}", "7db01e7b3f184a40cac08f86339deccb": "a_r=-\\frac{n^2a^3}{r^2}. ", "7db024b67c74e3c462fbf3a51e7b3d24": "\\nu + \\bar{\\nu}\\rightarrow Z\\rightarrow \\text{hadrons}", "7db055c4b28b7147c5ebb9692fe29ce7": "\\sum a_\\alpha X^\\alpha \\mapsto \\sum a_\\alpha^{1/p} X^\\alpha.", "7db06955226846c73f8e20c49d7dd977": "\\frac{d\\bold{v}(t)}{d t}=\\bold{a}(t).", "7db0d52accfa1890068c3aeee2f2d61d": "R_3 = \\frac{1}{16} \\, {S^a}_b \\, {S^b}_c \\, {S^c}_d \\, {S^d}_a", "7db0e00dc54745368e40fff64b8a428b": "A + \\operatorname{recc}(A) = A", "7db118ccd37656a5e1174eb86777188c": "A \\vdash B", "7db14ede9938ed8904bbd18764355f01": " \\frac{rel_{i}}{\\log_{2}i} ", "7db18eb199e421f8c8f8f93729936608": "\\mathbf A\\cdot\\mathbf B = A_B\\|\\mathbf{B}\\|=B_A\\|\\mathbf{A}\\|.", "7db20ab69cec439fa727aabd6bf59e9e": "[0, +\\infty)", "7db24243a952791e88ae19dd16d98065": "\\sigma_{CC}=E(\\cos^2\\theta)-E(\\cos\\theta)^2 =\\frac{1}{2}\\left(1 + C_2 - 2C_1^2\\right)", "7db269f8529ff472116aeee0149b7e5e": "(a, b) \\mapsto Y(a,z)b = \\sum_{n \\in \\mathbf{Z}} a_n b z^{-n-1}", "7db26cf6b2bf7c23e8395ee04f394679": "e_P = v_P(t\\circ f)", "7db2d34d65259aea5623c3cd382055e1": "50i-8", "7db2df7e04ef5b5e74d7cac303b8ad5a": "\n\\dot\\gamma_{ij} = \\frac{\\partial v_i}{\\partial x_j} + \\frac{\\partial v_j}{\\partial x_i}.\n", "7db2eb78ae01a70b3144e641bbaa3ead": "F(x) = f(\\lfloor x\\rfloor)", "7db31a1d395b30f935e6a0232138f913": "k_{x}^2+k_{zi}^2=\\varepsilon_i \\left(\\frac{\\omega}{c}\\right)^2 \\qquad i=1,2", "7db37b248c77e140150a05f283dbc451": "C \\backslash \\{0\\} \\subseteq \\operatorname{int} \\tilde{C}", "7db38a231ec46c85f9253baec0576922": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n T(A-\\lambda_\\star I))\\mathbf{x}_n,\\ n \\ge 0.", "7db3dd9ddbc4ea9016e0dbfd4c54f7a4": "{D_{AA*}} = {{\\lambda u} \\over {3}} = {{\\lambda}\\over{3}} \\sqrt{{8k_B T}\\over {\\pi M_{A}}}", "7db3e3028aac24dca28c8a9b359ffdb2": "\\frac{d A}{d x}< 0\\Rightarrow ", "7db3f59c18984910127288b408bf885d": "a \\,", "7db40ed4df2dc942a3e0985d03fffb31": "F_4\\;=\\;G_M", "7db45aff2af4b2a67a20dfdcab19dc00": "(\\xi \\vee \\eta)_{sup}(\\alpha)=\\xi_{sup}(\\alpha)\\vee\\eta_{sup}{\\alpha}", "7db46f41d44bbb43e0d94c1178c5ffbf": "s(nT) = T \\int_{1/T} \\tfrac{1}{T}\\ S_{1/T}(f)\\cdot e^{i 2\\pi f nT} df\\,", "7db470d2bd098f6212dc27fe6baafa92": "v_{\\mathfrak D}", "7db47e98b5faa20eda60c6e368fe400e": "{\\mathfrak A}_P", "7db49cbbb359ab0934ee1c6dc542911c": "\n M = \\frac{Pbx}{L} - P\\langle x-a \\rangle\n ", "7db4b748a62ef247bcc7519c7aa3ea31": "\\mathrm{Le} = \\frac{\\alpha}{D}", "7db4cf142c70880e861a453fb7a3d538": "\\begin{align}\nx' & = \\cosh\\phi x - \\sinh\\phi ct \\\\\nct' & = -\\sinh\\phi x + \\cosh\\phi ct \n\\end{align}", "7db56d5467cb81b05af5865139d5d6e1": "M_{rs}/M_s", "7db5c67421e09abc259d071475b9ef9f": "V \\approx \\sqrt{ 20.9 d_{skid} } ", "7db5da3851e741df310097d883d46fd9": " \nc= \\sqrt{12 d^2} - \\frac{\\sqrt{12 d^2}}{3\\cdot 3} + \\frac{\\sqrt{12 d^2}}{3^2 \\cdot 5} - \\frac{\\sqrt{12 d^2}}{3^3 \\cdot 7}+ \\quad \\cdots\n", "7db62409e1bcb2d0873d3255c0b90d58": "\\ell=0", "7db6268d8087362b98476e7b9aeab73f": "\\frac{1}{(i\\omega-\\xi_1)^2(i\\omega-\\xi_2)^2}", "7db698c1c32f04610c16004b24b1ebfb": "[\\mathbb{R}] := \\big\\{\\, [x_1, x_2] \\,|\\, x_1 \\leq x_2 \\text{ and } x_1, x_2 \\in \\mathbb{R} \\cup \\{-\\infty, \\infty\\} \\big\\}", "7db6aab03994a67cda3d93e9591c3c2b": "{_uM_d}=\\frac{q^2}{gy_2}+ \\frac{y_2^2}{2}", "7db6b8cb25fc5492ecb43027f29cedcf": "p(X,f(X)) \\land \\lnot p(g(Y),Y)", "7db6c8e1e5a5b1971beb0490ead96f1f": " H_0: \\theta \\leq \\theta_0 \\text{ vs. } H_1: \\theta > \\theta_0 .", "7db6c938cdf560646c46d6beb76efd72": "\nf_v \\left(v_x, v_y, v_z\\right) = f_v (v_x)f_v (v_y)f_v (v_z)\n", "7db6d589ac5900ce14dcc21c14402510": "D(\\mathcal{F}) = R\\,\\mathcal{H}om(\\mathcal{F}, \\omega_X) . \\,\\!", "7db6ddc70e5cd2ccc96919186b57044f": "v = v^1\\mathbf{e}_1+\\dots+v^n\\mathbf{e}_n", "7db7348f82c9483c471bb24eab0bd39a": "\\begin{align}\n & er=\\sum_{j=i-ne}^i\n x^2_j/\\sum_{j=i}^{i+ne}\n x^2_j \\\\\n \\end{align}", "7db736f21bddc88278ac4fe1ce840aaa": "Z_\\alpha(-\\infty) = 0", "7db73fc3c48b5e4f5837eb89a8d35e61": "\\operatorname{Re}(\\log z)=\\log \\sqrt{a^2 + b^2}", "7db7c2af6c949f698713d8fe8a6d4ade": " K(x,y)", "7db7c9ad4b8ef5c4eeaa545aea446b25": "T_n \\to T", "7db7cb1b00bcd061797c0d5cedd741e9": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "7db7d7cc10acf533d942e4aec3c33ce1": "y_i=1", "7db7fd141bf184f011a3fc9fff98d5b4": "\n\\vec f_n = \\alpha^n_{f} \\begin{bmatrix} \\vec f_{n-1}\\\\\n0\n\\end{bmatrix} \n+\\beta^n_{f}\\begin{bmatrix}0\\\\ \n\\vec b_{n-1}\n\\end{bmatrix}\n", "7db832ff86676e218b287d83e89e16e4": "A^k", "7db87ff7da65443432b90ab976e1115e": " \\scriptstyle \\varepsilon_n \\equiv (\\varepsilon_{n1},\\ldots,\\varepsilon_{nJ}) \\sim N(0,\\Omega) , ", "7db883c820f4b51cf951c6e6c5286742": " (\\ln f)'= \\frac{f'}{f} \\quad", "7db885230e253da192d9aeefdf93a4d8": " G'_k(u) ", "7db8a6a860073a23bd9e31d0ca0e6725": "U = U(S,V,N_1,\\ldots,N_n)\\,", "7db927a15dd7aaab0571256da253f553": "r[k]=\\sum_{n=-\\infty}^{\\infty}h[n]s[k-n]", "7db9757ff2bebb423268bba4f954771c": "\\mathbf{c} = 2^{\\aleph_0} > {\\aleph_0}", "7db9dbdeaa322915990ca16087e02678": "\\pi \\approx \\frac{(a_{n+1}+b_{n+1})^2}{4t_{n+1}}.\\!", "7dba2cb263ed3a881b68b2a4f069a0ad": "\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n = 0,\n", "7dba2e1b9ffe08e8866fc1212ae624e1": "X:=1/2\\cdot(2-Z-Z^{-1})", "7dba2ea0d90d5429583abf34b938a8ec": "nx+1", "7dba5dcb94b0b549483691ba02cf079f": "\\displaystyle \\tau_g", "7dbad190382e9b95c91a27d99f0aaadf": "\n \\boldsymbol{\\mathcal{E}} = \\mathcal{E}_{ijk}\\mathbf{b}^i\\otimes\\mathbf{b}^j\\otimes\\mathbf{b}^k\n = \\mathcal{E}^{ijk}\\mathbf{b}_i\\otimes\\mathbf{b}_j\\otimes\\mathbf{b}_k\n", "7dbadc85ea790433e9a6afda15e60149": "\n\\varphi = \\operatorname{atan2} \\left( \\left([\\mathbf{b}_1 \\times \\mathbf{b}_2]\\times [\\mathbf{b}_2 \\times \\mathbf{b}_3]\\right) \\cdot \\frac{\\mathbf{b}_2}{|\\mathbf{b}_2|}, [\\mathbf{b}_1 \\times \\mathbf{b}_2] \\cdot [\\mathbf{b}_2 \\times \\mathbf{b}_3] \\right),\n", "7dbae0858c71362742e023e2e8a3b5c0": "\\mathcal{F}_{T}", "7dbb6cea3f28e46103cf66c35004a0eb": "a_i = \\frac{p_i}{p^{\\ominus}}", "7dbb88eaafaa9931770ea544f5aed126": "\\omega (x)", "7dbb9eb1a1a4c204419191f68e7be22d": "E_m = -mg \\mu_B B = -k_BTxm/J", "7dbbd8a74e9726b5e3132ef9e7048cdd": " B = \\beta_0 + \\beta_1 \\frac{u^2} {c^2} + \\beta_2 \\frac{u^2_r}{c^2}+... ", "7dbc3ba68a2df0ab0cd307227c23cb9c": "f(p)=p^3", "7dbc5dcbe37c39961fd949a7c639850d": "\\sigma_Y", "7dbc5e16edb8a64c07f3492ab991684a": "RfD (mg/kg/day) = {NOEL (mg/kg/day) \\over Uf_{inter} * Uf_{intra} * Uf_{other}}", "7dbc8ee2ee61bf7c1b1841901ff9280a": "\\int\\frac{\\sin ax\\;\\mathrm{d}x}{\\cos ax + \\sin ax} = \\frac{x}{2} - \\frac{1}{2a}\\ln\\left|\\sin ax + \\cos ax\\right|+C", "7dbcdcabeeef1b0f3b1e0fa284304ab3": "\\log(^*K_{A}) = \\log(^*K_{A \\to 0}) + \\frac{\\gamma A_m} {3.454RT}", "7dbd3b662dd437dba15b99aa0c0bba22": "\\frac{P(x)}{Q(x)}", "7dbd4aa15e2c2d2708df0ce51a4974d5": "H_{cr}/H_c", "7dbdd6f5e9f5c8b1f639a6ae35506e3d": "2\\pi rh", "7dbe058f2baa11e5101e00db3e55c92c": "B_n=B_n(0).", "7dbe3ebad99da9340bdbc18f4ecaec8d": "LH_2^- +H^+\\leftrightharpoons LH_3:[LH_3]=K[LH_2^-][H^+]", "7dbe5ea972bdf1a75e621b555b41b920": "\\widehat{\\beta_1}=\\frac{\\sum(x_i-\\bar{x})(y_i-\\bar{y})}{\\sum(x_i-\\bar{x})^2}\\text{ and }\\hat{\\beta_0}=\\bar{y}-\\widehat{\\beta_1}\\bar{x}", "7dbe8a6bd0e694b7b1d3ad7f81b7082d": "\\rho_3 = \\rho_1 \\left(\\frac{P_3}{P_1}\\right)^{1/\\gamma}", "7dbe987defe79a6060f6c045e47d66f7": "XY = YX = I", "7dbef586d3ab314f6e92be2a050aa2a3": "\\forall x, y \\in A, f(x)=f(y) \\Rightarrow x=y\\ ", "7dbf43a6a301d0d79390018a652b9e83": "= \\frac{x\\frac{dx}{dt} + y\\frac{dy}{dt}}{\\sqrt{x^2 + y^2}} ", "7dbf6e7dfa2a6bdae4b745c92795b7fd": "V_i = x_iV\\,", "7dbf88007667f4a1699fd193a00a3a93": " l \\le n-1", "7dbfd4df56d4d4921c5b5ba01066eab0": "\\boldsymbol{I}", "7dbfd69cdb139f0c34c5a5f4e706ce52": "\n\\langle T_v\\exp_p(v), T_v\\exp_p(w)\\rangle = \\langle T_v\\exp_p(v), T_v\\exp_p(w_T)\\rangle + \\langle T_v\\exp_p(v), T_v\\exp_p(w_N)\\rangle", "7dbfe08deda1f5f5eda345bd90472ae9": "\\frac{\\sigma_f}{f} \\approx b \\frac{\\sigma_A}{A}", "7dc0182d7a47ebb431cb692640479411": "w(a)=1", "7dc0ca309da89f3dd5be319584bc4d85": " \\partial, \\bar\\partial ", "7dc0dbafb09d20afd74ebd6d6d1c69e0": "A^{-1}=\\left[\\begin{smallmatrix}\n 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n -3 & 3 & 0 & 0 & -2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 2 & -2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 3 & 0 & 0 & -2 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -2 & 0 & 0 & 1 & 1 & 0 & 0 \\\\\n -3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & -1 & 0 \\\\\n 9 & -9 & -9 & 9 & 6 & 3 & -6 & -3 & 6 & -6 & 3 & -3 & 4 & 2 & 2 & 1 \\\\\n -6 & 6 & 6 & -6 & -3 & -3 & 3 & 3 & -4 & 4 & -2 & 2 & -2 & -2 & -1 & -1 \\\\\n 2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n -6 & 6 & 6 & -6 & -4 & -2 & 4 & 2 & -3 & 3 & -3 & 3 & -2 & -1 & -2 & -1 \\\\\n 4 & -4 & -4 & 4 & 2 & 2 & -2 & -2 & 2 & -2 & 2 & -2 & 1 & 1 & 1 & 1\n\\end{smallmatrix}\\right]", "7dc10e66da5549d351765bd940b81be9": "MD", "7dc16c2c2944809aab38f5aef2c9a6b8": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } { \\exp \\left ( i \\mathbf k \\cdot \\mathbf r \\right ) \\over k^2 +m^2 } = \n\\int_0^{\\infty} {k^2 dk \\over \\left ( 2 \\pi \\right )^2 } \\int_{-1}^{1} du {\\exp\\left( ikru \\right) \\over k^2 + m^2} \n", "7dc23890c54352e34d1b92239fabbf09": "\\cdots\\rightarrow A\\rightarrow A\\rightarrow A\\rightarrow\\cdots", "7dc286020905e103aa84208d9b6998df": "B_{k + 1} = \\big( 1 + r \\big) B_{k} - p", "7dc28f3982ff2217f4ccddd184310621": "\\textstyle k(x,x_{i})=\\phi (x)\\cdot \\phi (x_{i})", "7dc296bc9f6284e57bc262cca44b85ae": "\\textstyle B(\\mathbf{c})", "7dc2f195cb4edf362ceb457ebf6a62c8": "H_{y}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}} {k_{o}^{2}-k_{z}^{2}}[-j\\frac{k_{xo}}{\\varepsilon _{r}}(A \\ e^{jk_{x\\varepsilon }w}+B \\ e^{-jk_{x\\varepsilon }w})+\\frac{k_{z}}{\\omega \\mu }\\frac{m\\pi }{a}(C \\ e^{jk_{x\\varepsilon }w}+D \\ e^{-jk_{x\\varepsilon }w})]e^{jk_{xo}(x+w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ (49) ", "7dc2f39fc36a5a8bc7849036b86b9965": "\\omega = \\begin{cases} 1 - i\\tan\\tfrac{\\pi\\alpha}{2} \\beta \\, \\operatorname{sign}(t) & \\text{if }\\alpha \\ne 1 \\\\\n 1 + i\\tfrac{2}{\\pi}\\beta\\log|t| \\operatorname{sign}(t) & \\text{if }\\alpha = 1 \\end{cases}", "7dc315d795c6c76637902c61b6a71f4c": " V=\\{v_{i,s} | i=1,2,..,m; s=1,2,..,k_i \\}", "7dc34abcff1c579d76226d1f9d8c0b36": " \\begin{align}\nk_1 &= f(t_n,y_n), \\\\\nk_2 &= f(t_n + \\tfrac{2}{3}h, y_n + \\tfrac{2}{3}h k_1), \\\\\ny_{n+1} &= y_n + h\\left(\\tfrac{1}{4}k_1+\\tfrac{3}{4}k_2\\right).\n\\end{align} ", "7dc36e06d785697c94bfb7a45b95e53b": "1/E_r=(1-\\nu_i^2)/E_i+(1-\\nu_s^2)/E_s.", "7dc3725516049481f1198b03b9833d43": " e^{-i/365} \\approx 1 - \\frac{i}{365}. \\ ", "7dc3ce1eab02e4c60696aacdff23013d": "P(i,j,k,m) \\equiv d(c_i,c_j) \\leq \\frac{m}{k+1}", "7dc435921c31c263417d03a6110643c6": "\\left.x_i = y_i\\right.", "7dc47e00ff256706661dd45572c05c7e": "P : H^{p} (\\partial D) \\to L^{p} (D, \\mu)", "7dc4e239acd9937112ee8fa8a18b46fc": "\n\\left[\n\\begin{array}{rrrrrrrr}\n 52 & 55 & 61 & 66 & 70 & 61 & 64 & 73 \\\\\n 63 & 59 & 55 & 90 & 109 & 85 & 69 & 72 \\\\\n 62 & 59 & 68 & 113 & 144 & 104 & 66 & 73 \\\\\n 63 & 58 & 71 & 122 & 154 & 106 & 70 & 69 \\\\\n 67 & 61 & 68 & 104 & 126 & 88 & 68 & 70 \\\\\n 79 & 65 & 60 & 70 & 77 & 68 & 58 & 75 \\\\\n 85 & 71 & 64 & 59 & 55 & 61 & 65 & 83 \\\\\n 87 & 79 & 69 & 68 & 65 & 76 & 78 & 94\n\\end{array}\n\\right].\n", "7dc4f04af61370888b90f7f96394e6e2": "\\,^{238}_{92}\\mathrm{U} + \\,^{nat}_{32}\\mathrm{Ge} \\to \\,^{308,310,311,312,314}\\mathrm{Ubq} ^{*} \\to \\ fission.", "7dc53173c2a99c408e5cb84234d5f2c6": "V_{\\pi}", "7dc57a7ea643a7c5016c7ce571236aac": "e^{-iz} = \\cos(z) - i \\sin(z) \\!", "7dc593b24d79e1c4827b4ce6569eb1d2": "F^{\\alpha\\beta} = \\begin{pmatrix}\n0 & -E_x/c & -E_y/c & -E_z/c \\\\\nE_x/c & 0 & -B_z & B_y \\\\\nE_y/c & B_z & 0 & -B_x \\\\\nE_z/c & -B_y & B_x & 0\n\\end{pmatrix}\n", "7dc5cfdfe5a7b006eae7f93106cf4ba5": "\\text{Had}(x)", "7dc5f4dc5ac88945552ac305ae28eb8e": "\\mathbf{P} \\left (\\frac{1}{m}\\sum X_i \\ge p + \\varepsilon\\right ) \\le e^{-D(p+\\varepsilon\\|p) m}.", "7dc60ede2a762add2641453587c54c16": "\\mathbf{F}=q(\\mathbf{E}+\\mathbf{v}\\times\\mathbf{B})", "7dc6280a6394d0318716281abfe30a86": "A(n,m)=(n-m)A(n-1,m-1) + (m+1)A(n-1,m).", "7dc6621ae509055271ad544dee78d760": "g_M = \\frac{r_M}{z_{M}^2}", "7dc66577912cad5745fb8684a53b7be7": "\\left (\\frac{\\mbox{Inventory}}{\\mbox{Cost of Goods Sold}}\\right)\\mbox{365 Days}", "7dc6b731a0e74e0dc64faac8dea38bb7": "Y^4\\sin^4\\theta", "7dc6d3e5125fb6d51fd63d780bb3e926": "S = {K \\sqrt{p_N}}", "7dc6fa279b9b53c4a707f7cf0e695879": "\\begin{matrix}\\operatorname{Hom}(Y,Z) &\\rightarrow& \\operatorname{Hom}(X,Z)\\\\\ng &\\mapsto& gf\\end{matrix}", "7dc7a63a435584ba5585865b65f2dcd7": " y_{1}, y_{2} \\, ", "7dc7ac09c43e55c46e99a35c806f2816": "\\langle (x_1,\\ldots,x_n),R \\rangle", "7dc7de8fed13ab844e873fec647ea9ea": "S_B = \\begin{matrix}\\frac{7}{10}\\end{matrix}S_B(1) + \\begin{matrix}\\frac{3}{10}\\end{matrix}S_B(2)", "7dc7ffa8cc5745bd8897666e60d86a28": "r_e = \\frac {e^2}{4 \\pi \\epsilon_0 m_e c^2},", "7dc875ec31ccbd8cf0310c6f94670865": "\nCOP_\\text{heating} = \\frac{\\Delta Q_\\text{hot}}{\\Delta A} \\leq \\frac{T_\\text{hot}}{T_\\text{hot}-T_\\text{cool}},\n", "7dc8ef50b42dcc98cdd7a37fc6f4dd2f": "M^C_S", "7dc900d134cd199639e3fac71bb40e53": "{m}=\\left( {\\partial{\\log_{10} \\eta }\\over\\partial \\left (T_g/T \\right)} \\right) _{T=Tg}", "7dc93ed9b872bce14be60af128905145": "\n\\lambda_k = -\\frac{4}{h^2}(\\sin^2(\\frac{k \\pi}{2(n+1)})).\n\\,\\!", "7dc992ef53a89ca26e7caac7126505ff": "r_\\text{vir} \\approx r_{200}= r(\\rho = 200 \\cdot \\rho_\\text{crit})", "7dc9f2feb2221c7d754ae1eef8f11eb8": "V=1.34\\sqrt{L}", "7dc9fd26e2d9887011cf82d3853947fd": "D_{\\mathrm{KL}}(P\\|Q) = \\int_{-\\infty}^\\infty \\ln\\left(\\frac{p(x)}{q(x)}\\right) p(x) \\, {\\rm d}x, \\!", "7dca2020eeba1d145848f86b121f599f": "c(\\Pi)", "7dca2281d7ea7153462246a4c617208c": "\\vec{S} = 1", "7dca5aeb5eda81f89202a09b27b6fcb4": " {E'}_s(R)=E_s(R) - \\frac{\\Lambda^2 \\hbar^2}{2\\mu R^2} + \\frac{1}{2\\mu R^2} \\langle \\Phi_s |{L_x}^2 + {L_y}^2|\\Phi_s \\rangle ", "7dca5eb0b722da6575e50a7394dc92bf": "N-n-1", "7dcb5689cbcb57b2444ec2773004d197": " \\psi (0) = 0 \\text{ for } l > 0 ", "7dcb77049a83b0b5800aab2bdf8874ae": "\\mathbf x=\\{7.14, 6.3, 3.9, 6.46, 0.2, 2.94, 4.14, 4.69, 6.02, 1.58\\}", "7dcb826d778074fbb46a8c1dc2b99f2d": "\n\\langle p \\rangle_{IS} = \\langle p \\rangle_{SI} \\equiv \\langle \\langle p \\rangle_I \\rangle_S\n", "7dcb90960dcd5db794546737d5dcda9d": "x :> y", "7dcb938e767082ad9a626c61927606e7": "\\omega\\mapsto\\omega+\\delta\\omega", "7dcbad447cd5d1d3d0c108c74cef9e9b": "D=\\{A,B\\} ", "7dcbbd0ef8e68ff42e20518067ccd346": "c=(m^2+n^2)^2, \\,", "7dcbe2ad5492ba75927dd5b6644bd5ef": "\\boldsymbol{\\Delta}^0_2", "7dcc5e1df53c10c1a36de73d4e136453": " \\frac{d(af+bg)}{dx} = a\\frac{df}{dx} +b\\frac{dg}{dx}.", "7dcca6618098f23d8aa76aaf53c20f4c": "\\lim_{\\alpha \\rightarrow 0} \\frac{\\sin \\alpha}{\\alpha} = 1", "7dcccd955af9c3a4c9fcd53332a08943": "v\\in \\Omega-S", "7dcd0bbbd5ce1cba4c975210115adc54": "\\mathcal{D}'(\\mathbb{R})", "7dcd1d54150f1c98139254ff76131d5e": "\\mathbf{u}_{\\mathbf{k}}", "7dcd4f07d285e6c989c3a5be30c45188": "D+Cz(I-zA)^{-1}B", "7dce1511635bae4834c5fd9b37d9c0f3": "\\|x\\|_{bs} = \\sup_n \\left\\vert \\sum_{i=0}^n x_i \\right\\vert,", "7dce25735af3a37ea09758e1ffcd4af4": " \\mathbf{A} (\\mathbf{x}) = \\frac{1}{4 \\pi} \\nabla \\times \\int_{\\mathbb R^3} \\frac{ \\mathbf{v} (\\mathbf{y})}{\\left\\|\\mathbf{x} -\\mathbf{y} \\right\\|} \\, d\\mathbf{y}. ", "7dce31e3d1c74ce96934a9ba5a6a60ac": "SR=\\frac{Annual\\ Portfolio\\ Return - Annual\\ Risk\\operatorname{-}Free\\ Rate}{Average\\ Largest\\ Drawdown}", "7dce46ee4c66de2d020e03bf5bcf98e8": "T_2=T_1e^{\\mu_s\\beta} \\, ", "7dce653cd42e785760716844533e8992": "I_0\\,", "7dce79d3eed78ed5b367c8e2c9a1a934": "T_a = \\frac{\\lambda_a }{2}.\\,", "7dce86c8493bc31159fe5c42d44c1f1d": "T\\vdash_{\\mathcal{S}}\\phi", "7dceb07fe675d567ce1ff40df2324d1a": " \\hat{\\mathbf{x}} = (A^T\\!A)^{-1} A^T \\mathbf{b}. ", "7dcf3394ca3d4c461312dc86acea8607": "f_{radar}", "7dcf7735f6b12036c4ea140734cad281": "\\begin{bmatrix}\n1 & 0 & 0\\\\\n0 & 4 & 0\\\\\n0 & 0 & -3\\end{bmatrix}", "7dcff10f02eb40894b8b08a8047e17a1": "d_k(j)=e^{2 \\pi i j k \\over n}", "7dd04a669bd347dcae0ac71ab57c1de7": "\\alpha_j=\\bigvee(\\Theta_L(z_i,z_{i+1})\\mid it) = P(N(t)1; \\quad Var(X_i)=\\frac{a\\theta_i^2}{(a-1)^2(a-2)}, a>2; \\quad i=1,2,", "7df378066de926f39da8ea857b14b255": "\\textbf{G}_c\\textbf{G} = K\\textbf{G}", "7df397097fddca16626ffcb9eb150ac4": "y \\in F", "7df3ad0513aef993de053df6180ceed0": "\\frac {V_\\mathrm i}{V_x} = \\lim_{\\delta x \\to 0} \\frac {V_\\mathrm i}{V_{xn}} = \\lim_{\\delta x \\to 0} \\left ( 1 + \\frac {\\delta Z}{Z_0} + \\delta Z \\delta Y \\right)^{\\frac{x}{\\delta x}}", "7df3f0eeba607bc51c9487c336b10faa": "\\begin{pmatrix}\nu_1(x) & u_2(x) \\\\\nu_1'(x) & u_2'(x) \\end{pmatrix}\n\\begin{pmatrix}\nA'(x) \\\\\nB'(x)\\end{pmatrix} =\n\\begin{pmatrix}\n0\\\\\nf\\end{pmatrix}", "7df4651217764452f77e7db971542112": "\\alpha_V:B_\\ast\\otimes_ED_{B_\\ast}(V)\\longrightarrow B_\\ast\\otimes_{\\mathbf{Q}_p}V", "7df4eb5a1b4dd5cbd06d515eacefc09d": "\\check H^n(\\mathcal U, F)", "7df4fc459ffbe18f0ae4aeeb8b2a8f7a": "\\begin{cases} \\\\ \\\\ \\end{cases}", "7df5031ed3e1abbe8965da0563ebc848": "z[k] = n_r [k] + v[k] g[0] + \\sum_{n \\neq k} v[n] g[k-n]", "7df51ec137ca8eb4dde877868d2cf6b4": "\\scriptstyle{\\alpha_i(\\mathbf{n}-\\mathbf{e}_i)}", "7df52a4e71476f9391c1bd176f856e8f": "x_2^2 - K x_1^2 = \\mathrm{constant}", "7df598abc265876342068265cdc66dc6": " \\frac{n!}{k!(n-k)!} + \\frac{n!}{(k-1)!(n-(k-1))!}.", "7df59f9b1576f0dce8c1bdd75ec2a8c5": "\npad_l(x) \\stackrel{\\mathrm{def}}{=} \\begin{cases} \nx, & L(x) \\ge l \\\\\nx||0_{A^{l-L(x)}}, & L(x)0\\\\\n \\end{array}\n \\right.\n\\end{array}", "7e13632aff7138d561484c50d39278fe": " \\mathbf{F}_{ext} = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\big[m_{rocket}(t)\\mathbf{V}(t)\\big]", "7e13b44a3a9bcce834e90b3d98f718d9": "\\vert S \\vert = \\sqrt{a^2 + b^2}", "7e13d80757f4fa4fe380c60497cfd4b8": "\\mathcal{L}_{ij}", "7e13f761a39eed75432103f82fa6bb7a": "= {p_1q_1 + p_2q_2}", "7e1404d63f032f991a6e911bbc198f79": "f(x_1, x_2, x_3) = 2.7 x_1^2x_2^{-1/3}x_3^{0.7} + 2x_1^{-4}x_3^{2/5}", "7e143f1a03830747411bcba4442351c6": "\\zeta = A_z f_z(z) e^{i(\\omega t - k x)} \\quad \\quad (2) ", "7e1473cfa2fc5a01fc35b90d425ca360": "\\varphi_\\lambda=\\sum_{s\\in W} c(s\\lambda) f_{s\\lambda}.", "7e148ea60c43ab80043a8ba4ee8bd92e": "T+?T", "7e149c3bff35de6e198b47dc9bea2e0d": "\\mathrm{SINR}(x_i) {{=}} \\frac{\\ell(|x_i|)F_i}{\\sum_j^n [\\ell(|x_j|)F_j]-\\ell(|x_i|)F_i +N} ", "7e153e5e2204cc022d3920bea55dd3fc": "ax^3+bx^2z+cxz^2+xz = dz^3,\\,", "7e1540d4f2d492238c6d92f777e12909": " x \\to x \\or y = 1 ,", "7e157c4d661a698bbecad1d26df16746": "\\Pi^{4}", "7e15994e0b55ac0bb3864a96a1458ebb": "S + \\alpha.m_0", "7e159fef68efd5e2c5f451c9cb105429": "\n \\begin{pmatrix}x^\\prime \\\\y^\\prime \\end{pmatrix} = \n \\begin{pmatrix}x \\\\ m x + y \\end{pmatrix} = \n \\begin{pmatrix}1 & 0\\\\m & 1\\end{pmatrix} \n \\begin{pmatrix}x \\\\y \\end{pmatrix}.\n", "7e15f6cfeee4493fb19bb9a4187c0325": "\\sin^{(4n+k)}(0)=\\begin{cases}\n0 & \\text{when } k=0 \\\\\n1 & \\text{when } k=1 \\\\\n0 & \\text{when } k=2 \\\\\n-1 & \\text{when } k=3 \\end{cases}", "7e1647600d80aeb0cee0bf09d6cb0bc4": "(\\varepsilon,d)", "7e166ba2700fab85852e3859b44de4f1": "r_{d} = k_d \\, [A_{ad}]", "7e16851d97179976b752c8fee636fb5a": "n^{k_1}", "7e16bb1b142cd561e0924597987b0af8": "r(0) = 0", "7e16dc5003ce2d775a152707b313cff8": "V = \\frac{x^3}{3} - xy^2 + a(x^2+y^2) + bx + cy \\, ", "7e173a85a40b01c364d5df2c6832de02": " \\Delta U = Q + W_{mech} + W_{extra} \\, ", "7e175ee8f1a12043903d5c4d24adc587": "f'(x) = \\frac{-f(x+2 h)+8 f(x+h)-8 f(x-h)+f(x-2h)}{12 h}+\\frac{h^4}{30}f^{(5)}(c)", "7e176edc0af7d1e55fafa24f371780d1": "{\\operatorname{Var}(\\operatorname{E}(Y\\mid X)) \\over \\operatorname{Var}(Y)} = \\operatorname{Corr}(X,Y)^2.\\,", "7e178dd62e06fda44e52aa48531d0ad6": " \\left\\| f(x,\\theta) \\right\\| \\leq d(x) \\quad\\text{for all}\\ \\theta\\in\\Theta.", "7e17b0b6ec0e3da059bc3314ea404dbc": "\\and \\land \\wedge, \\curlywedge, \\bigwedge \\!", "7e17f496999c038f18e3121bfba5fbc6": "\\, T_{\\overline{ro}}", "7e180d199f4945a3ed1f288afab4486c": "m_b = \\sqrt {\\frac{2 a^2 + 2 c^2 - b^2}{4} }, ", "7e1815930e39ceaa3c74a4e58ba13f43": "\\Delta C^*_{ab} = C^*_1 - C^*_2", "7e181db67be0ae88e9de662fff16b95a": "a_{C} = \\frac{3}{5} \\left( \\frac{\\hbar c \\alpha}{r_0} \\right)=\\frac{3}{5} \\left( \\frac{R_P}{r_0} \\right)\\alpha m_pc^2", "7e182e2a33dde588f78d68aba3935847": "d/dx_i", "7e18c193e6f659bc166f3a4ffaafe198": "\\displaystyle \\sqrt{\\frac{2}{\\pi}} \\frac{1}{i\\omega } ", "7e193890baba035b0a057a1f1ad82658": " (\\partial T)_P=-(\\partial P)_T=1", "7e193cf90441c42350cfd550203e72d4": "2ds\\ d^2s=2dx\\ d^2x", "7e1970966a9a1a35ca88d1c563917c7f": "\\operatorname{relint}(C) := \\{x \\in C : \\forall {y \\in C} \\; \\exist {\\lambda > 1}: \\lambda x + (1-\\lambda)y \\in C\\}.", "7e197820957ac8690ed35908b14d6a6c": "\\rho'(1)=\\sigma(1)", "7e19e43d014391235d1fed65e291043a": "Q = (P_1-P_2)C.", "7e1a4d6ae6b3fac6f5212ec48f572ca1": "\\vec x_n\\approx \\vec x(t_n)", "7e1a6848cbe276b35c60a04663ed4118": "M=U\\Sigma V^*\\,\\!", "7e1a8512f323229a42f899d06f32992c": "[u,v]=0", "7e1a946c1a023540392a0e29e0c05409": "\\sqrt{0.6}\\,\\mathrm V\\, \\approx 0.7746\\,\\mathrm V\\, \\approx -2.218\\,\\mathrm{dBV}", "7e1af4789cf01923853fef995d6a07f3": "Z[J]=\\int \\mathcal{D}\\phi e^{-\\int d^4x \\left({1\\over 2}(\\nabla\\phi)^2+{m^2 \\over 2}\\phi^2+{g\\over 4!}\\phi^4+J\\phi\\right)}.", "7e1be0d7d19e94732a15bb92012970fb": "\\sqrt{4\\pi\\alpha}", "7e1bff6caeddd6c1569ed00b1333acdc": "s^2 = x^2 + y^2 + z^2 - (ct)^2 .\\,", "7e1c2c4c1ff73f06b86f21b958bec967": "X[x,y]=\\frac{(xy'-yx')y'}{x'^2 + y'^2}", "7e1c594309b271ea266ebae7f94d55a7": "\\frac{1}{2}N + O(\\sqrt q\\log q).", "7e1c84cbc02bd60bf34579071e109683": "11^{70}\\ \\equiv\\ 1 \\pmod {71}.", "7e1ccfa0a19df5f58daeba27b5aa04d8": "\\rho kT \\chi _T= kT\\left(\\frac{\\partial \\rho}{\\partial p}\\right)=1 + \\rho \\int_V \\mathrm{d} \\mathbf{r} \\, [g(r) - 1] ", "7e1d36b9f5872441eaa0779c279c73d6": "z_1=-1+i,", "7e1d4f48f83269cd38dd75c078a3e40c": "Q = aX_1^b X_2^c \\cdots .", "7e1d8fa2a45cafd4cc13d63e9b32099b": " I \\subseteq \\mathbb{Z}[x]/\\langle f \\rangle ", "7e1da42101953b01aefb78494273915f": "\\alpha^i", "7e1dd2353d0bf76e7d4458e1698c941c": "\\mathbb{A}", "7e1de379b3923f8c061f49044878f750": "\\sigma = | \\psi \\rangle \\langle \\psi |", "7e1df34224d27134e0dc5ef47a373aaa": "\\ (a,b)(c,d)=(ac-d^{*}b,da+bc^{*})", "7e1dfaa746c26ff281a49c742db52ab6": "\\text{bind} \\colon \\left( \\left( T \\rarr R \\right) \\rarr R \\right) \\rarr \\left( T \\rarr \\left( T' \\rarr R \\right) \\rarr R \\right) \\rarr \\left( T' \\rarr R \\right) \\rarr R", "7e1eac27704c17ff8bda2bad98a1a16c": "\\det(\\Lambda)", "7e1edb3f859af28c6684b0f9bd1ece71": "\\left (\\sigma_C\\right )", "7e1ef305e7c821973e82155b11eb88c2": "X\\oplus Y \\cong Y\\oplus X.", "7e1f237a20ba18ec6ef77a82a223d742": "X\\times_Y \\mathrm{Spec}\\, k(p)", "7e1f40363264370fb94f240a8ee6314d": "C_{in}^\\alpha (x) = (x,\\alpha x)", "7e1f478ce83da7998b3b1e2e1355c3e4": " \\rho =1/2+iE(n). ", "7e1f499a6edb3997434ec9bb1f6aa651": "B_H \\times_H G", "7e1f6d5ccff264aa666de1ae27fdf79c": "\\begin{bmatrix}a & \\mathbf v\\\\ \\mathbf w & b\\end{bmatrix}", "7e1f7542071b9e436efe853a3d72a776": "\\psi(\\mathbf{x},t) = R(\\mathbf{x},t)e^{i S(\\mathbf{x},t) / \\hbar}.", "7e1fdf43efea9190a891ec7b2735bb90": "q = \\exp(\\pi i \\tau)", "7e2081b861a40349bdca4285c763acab": " \\text{Books} = \\text{Knowledge} = \\text{Power} = \\frac{\\text{Mass} \\times \\text{Distance}^2}{\\text{Time}^3}.", "7e20c1ea3ed8043aaac47bb883f359ac": "\\begin{align}\nH(Y|X)=&\\sum_{x\\in\\mathcal X, y\\in\\mathcal Y}p(x,y)\\log \\frac {p(x)} {p(x,y)}\\\\\n =&-\\sum_{x\\in\\mathcal X, y\\in\\mathcal Y}p(x,y)\\log\\,p(x,y) + \\sum_{x\\in\\mathcal X, y\\in\\mathcal Y}p(x,y)\\log\\,p(x) \\\\\n=& H(X,Y) + \\sum_{x \\in \\mathcal X} p(x)\\log\\,p(x) \\\\\n=& H(X,Y) - H(X).\n\\end{align}", "7e20c233ce713221da7dcfdbab76a4e4": "\\rho \\le v_0 - v_2 = 2 ", "7e20d1aa5935c6046732e7cdc3fe267a": "\\mathbf{R}^{m+1}", "7e2141f812e6defeb73f07ce73453735": "\\pi(\\tau)\\approx 1-\\Phi(1.64-\\tau\\sqrt{n}/\\hat{\\sigma}_D).", "7e21647d6b80a39f7ca83d911ab31a8d": "g_5=-x+10x^2;", "7e21786a9c3c954e8cdc3854e4baf366": "\\Gamma(z) = (z-1)\\Gamma(z-1)\\,.", "7e219b39b484bf9980a5a9e6872e0c36": "\\scriptstyle 3 \\,\\times\\, 3 \\;=\\; 9", "7e21f1d2961d832c7e403e9004c7088d": "\\Psi(\\bold{r},t) = \\psi(\\bold{r}) e^{-i \\omega t} ", "7e2231e9db10d24f261bea95b20dd70b": "F_0(x) = \\ln(1+\\exp(x)).\\,", "7e2242765be19dac50000d5bbb1a8593": "\\Sigma^\\infty", "7e224a1349f79b4471c0bdeca56f5450": "M\\cup L", "7e22528d3fd2b6e429cbc5641f6e59c8": "\\ker(T_0-\\sigma)", "7e2278d6a8ee3e865a4731f79c1a5dd3": "C_\\kappa^{(\\alpha )}(X)", "7e22812087f9bb023d302a235295d7f3": "\\kappa_b(k,i) = \\frac{\\delta(k,i)}{\\xi^d_{f_{min}}(k,i)}", "7e22c2a7cd39496e3a92cce1e8c9f0f9": "P_b(E) \\approx 10^{-5}", "7e23379c7881ca0be2bfa419217f63e8": "u=-\\mathbf{F}(\\mathbf{p}(t))\\mathbf{x}(t)=-\\mathcal{F}\\boxtimes_{n=1}^N\\mathbf{w}_n(p_n(t))\\mathbf{x}(t),", "7e23585ddc3d4e3e7fe6ac208438499c": "A = U |A|\\,", "7e23c6f653246c209d62f5c3ca2eafe1": " Dy(x)e^{\\int f(x)\\,dx}+f(x)y(x)e^{\\int f(x)\\,dx}=g(x)e^{\\int f(x) \\, dx},", "7e240048317bdbd2de5f488339394170": "\\alpha_k \\leftarrow {\\hat{r}_k r_k \\over \\hat{p}_k A p_k}\\,", "7e240a38fa45971918df04f0c51ad01f": "c = 12", "7e241b95f7aadec701c786298f651128": " -e^{-q \\tau} \\frac{\\phi(d_1)}{S^2 \\sigma \\sqrt{\\tau}} \\left(\\frac{d_1}{\\sigma \\sqrt{\\tau}} + 1\\right) = -\\frac{\\Gamma}{S}\\left(\\frac{d_1}{\\sigma\\sqrt{\\tau}}+1\\right) \\, ", "7e2465e913a757fcd09f160bbe68f975": "{\\rm Th}\\mathcal{S}", "7e24688c881cc11485a0e9399247dc89": "\\begin{array}{ccc}\nH^*(F)\\otimes H^*(B) & \\longrightarrow & H^*(E) \\\\\n\\sum_{i,j,k}a_{i,j,k}\\iota^*(c_{i,j})\\otimes b_k & \\longmapsto & \\sum_{i,j,k}a_{i,j,k}c_{i,j}\\wedge\\pi^*(b_k)\n\\end{array}", "7e247165cc9c2f07b49f3d6145812af4": "\\int_0^{\\pi} \\frac{\\cos mx\\ dx}{1-2a\\cos x +a^2}=\\frac{\\pi a^m}{1-a^2} \\quad , a^2<1, \\ m=0,1,2,\\dots", "7e24d053c4fb2f07b64fda73be4099b9": "\\frac{d\\sigma}{dT}", "7e24d7d116f6a31fc6c215c83c5db5c4": "\\int_0^1 \\ln(1/x)^p\\,dx = p!\\;", "7e25410cbc85295764e2d65def0d5fdd": " \\mathbf{x}(n) = \n\\left[\n\\begin{matrix}\nx(n)\\\\\nx(n-1)\\\\\n\\vdots\\\\\nx(n-p)\n\\end{matrix}\n\\right]\n", "7e255bd4dce805fea5f3902946ac2dcc": " Q = a_D + b_D P + c Z\\, ", "7e2572627b1b5d2adf0b9c39019f30a6": "\\frac{a}{b} = \\frac{c}{d}", "7e2582720aa98aa363eb7528158a10cd": "\n \\mathbf{b}^i\\cdot\\mathbf{b}_j = \\delta^i_j \\quad \\Rightarrow \\quad \\mathbf{b}^1\\cdot\\mathbf{b}_1 = 1,~\\mathbf{b}^1\\cdot\\mathbf{b}_2=\\mathbf{b}^1\\cdot\\mathbf{b}_3=0 \\quad \\Rightarrow \\quad \\mathbf{b}^1 = A~(\\mathbf{b}_2\\times\\mathbf{b}_3)\n", "7e260c25b087432158464e21c1aafc5e": " \\kappa_{n} =\\frac{\\partial ^n}{\\partial t^n} g (t) \\bigg|_{t=0} ", "7e262250ec946136f6b13c42bf8db02a": " U = \\frac{\\varepsilon_0}{2} \\mathbf{E}^2 + \\frac{1}{2\\mu_0} \\mathbf{B}^2 ", "7e26398daf1af70b0cbfd2edb5395c93": "P_{11} = \\pi_1 \\cdot \\int_{R_1}p(y|H1)\\, dy ", "7e263eee9e10ede5f250deefe96346e7": "W=\\int_{1}^{2} A \\sigma\\,dx.", "7e26484de3507b8cce78e559a546aadc": "\\color{Peach}\\text{Peach}", "7e2663d56083499496fd56ec7dfd615a": "c(\\Delta x)", "7e267677b9a869f28ebe732b892569f9": "\nT_{X\\rightarrow Y} = H\\left( Y^t \\mid Y^{\\mathbf{t-1}}\\right) - H\\left( Y^t \\mid Y^{\\mathbf{t-1}}, X^{\\mathbf{t-1}}\\right),\n", "7e26a2d8f40be7e55884277f699ea098": "\\mu_{\\Phi | \\Lambda}", "7e26df5c0f06142957a2e1f69b3fd14d": "\\int_{-\\infty}^{+\\infty}(\\omega(t)-\\omega_0)^2 |x_\\mathrm{a}(t)|^2 \\, dt", "7e26e9e9a8c6b4c9321791a2aa3ba590": " f\\in H", "7e27171674adf0ded24d2fc073741c12": "v=x+yf(v)", "7e2746bbc95316e74c8f29aea567b277": "F[x,y] = x * y = \\int_E x(s) y(t - s)\\, ds", "7e27843b4cd8c0562faa12a8af2251cd": "\\vec u_o,\\vec u_w,\\vec u_g", "7e27abacba00da519ce19b924ec60c6f": " u(x,t) = E^Q\\left[ \\int_t^T e^{- \\int_t^r V(X_\\tau,\\tau)\\, d\\tau}f(X_r,r)dr + e^{-\\int_t^T V(X_\\tau,\\tau)\\, d\\tau}\\psi(X_T) \\Bigg| X_t=x \\right] ", "7e27fac61778e384b25e9c5c13928384": "p\\leq 73", "7e281bae7df78a9fb92e0fe056ada99e": " f(x) =\\frac{\\nu^{\\frac{\\nu}{2}} \\exp\\left (-\\frac{\\nu\\mu^2}{2(x^2+\\nu)} \\right )}{\\sqrt{\\pi}\\Gamma(\\frac{\\nu}{2})2^{\\frac{\\nu-1}{2}}(x^2+\\nu)^{\\frac{\\nu+1}{2}}} \\int_0^\\infty y^\\nu\\exp\\left (-\\frac{1}{2}\\left(y-\\frac{\\mu x}{\\sqrt{x^2+\\nu}}\\right)^2\\right ) dy.", "7e281bda894a527db1ee77dcbb4540cd": "u(\\mathbf{x}) = \\frac {-1} {4\\pi} \\int_S \\rho(\\mathbf{y}) \\frac{\\partial}{\\partial\\nu}\\frac{1}{|\\mathbf{x}-\\mathbf{y}|} \\,d\\sigma(\\mathbf{y})", "7e28604a2c893025f5656d97293e8b94": "\\operatorname{Bun}_G", "7e2876229c2addb40602b6b8f2a13bc1": " \\Phi (x,n,a) = G_{n+1,\\,n+1}^{\\,1,\\,n+1} \\!\\left( \\left. \\begin{matrix} 0, 1-a, \\dots, 1-a \\\\ 0, -a, \\dots, -a \\end{matrix} \\; \\right| \\, -x \\right), \\qquad \\forall x, \\; n = 0,1,2,\\dots ", "7e28fbd9b995da1d6a91c3a57060ee61": " b_i(\\theta) = \\frac{I-1}{I}(\\theta^{I-1})", "7e28fe6a4485a966a66b968510b7d306": " \\partial M", "7e296679db9166cbb75000a504487f39": "\\tbinom{n + k - 1}{k}", "7e29aba54b1b7bbbcaea2dd22d8d2643": "\n \\boldsymbol{\\sigma} = \\lambda~\\mathrm{tr}(\\boldsymbol{\\varepsilon})~\\mathbf{I} + 2\\mu~\\boldsymbol{\\varepsilon}\n = \\mathsf{c}:\\boldsymbol{\\varepsilon} ~;~~ \\mathsf{c} = \\lambda~\\mathbf{I}\\otimes\\mathbf{I} + 2\\mu~\\mathsf{I}\n ", "7e29af7eb740b288eb51d68b16e2459a": "\\nabla G(\\vec r, \\vec r') = \\delta(\\vec r- \\vec r')", "7e2a03eda9cce24403dedd4670baf481": "T=2 \\, R_0 \\, \\sin \\left( \\frac{\\Phi}{2} \\right), \\; Z=0, ", "7e2a30b7d2200ffbc44f5b39ffd1585b": "{n+k-1\\choose k-1}", "7e2a41ea46c780964a97128c12cce005": "\\int_0^\\infty e^{-tx} \\, dx = \\frac{1}{t},", "7e2a45104b27bf1e6f61a5680ab83032": "f(x) = \\pm \\sqrt x", "7e2a6461526f26b50e22a859c1727b17": "dQ = A(x) \\, dx", "7e2a698db47774a20b8ef9ab0b43b811": "\nv(\\mathbf{r}- \\mathbf{R}) \\equiv \\frac{1}{|\\mathbf{r}- \\mathbf{R}|} .\n", "7e2adb9cf4808cee3c1bf181ec1fde9c": "\\int_{-\\infty}^{+\\infty} \\left( \\phi(t)-(\\omega_0 t + \\theta) \\right)^2\\, dt.", "7e2aedd7335a2e9e7e052719ed348dcd": " \\dot{x}_1 = x_2 ", "7e2ba139aaf3d36227d5f54021205505": "V_{avg}=0", "7e2ba3acabdfeb1080687082ff286ce3": " ed > pq ", "7e2bd40de5c9c20c6964bb6a27265b9b": "l=B+C", "7e2bf8880d46cb84f91d8740b2659a88": "[A]", "7e2c2b1cb853e07655c55b2e1d498aef": "D_$ = DV01 = -\\frac{\\partial V}{\\partial y}. ", "7e2c3ea1bf83769c2e7435ceaa81a1a2": "g, h \\colon A \\to B", "7e2c6af0e6d34e5a019d6dda1f252601": "2H_2O \\longrightarrow 2H_2 + O_2", "7e2cf8a71daeffb7d245c4e13bb8873c": "f(x)=x^2-5x+6", "7e2cfd18b3e20fcd7c8368ca1fd24921": "<_P = \\bigcap\\mathcal R", "7e2d2a31d7fb434ebdfe4d5557686b37": " \\mathcal{L} = { 1 \\over \\omega } \\mathcal{E}_c \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right ) = { \\hbar \\over V } \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right )", "7e2da3c2806283544273ad82bab60681": "L\\,=\\,L(G)", "7e2dc114afd64a30406caca62e56ca4b": " e(P,Q) = e(g^p,g^q) = e(g,g)^{pq} = e(g^q, g^p) = e(Q,P) ", "7e2dcf7d984e13c3ccf63f7a36e1cccc": "T_G(x,y)= T_{G^*} (y,x)", "7e2def3409e4cd5120bd29e0f1be8774": "b_{k} ", "7e2e73818eaa051c9b9535b3ccdfa517": "\\bold{J}\\cdot\\bold{\\hat{n}}= J\\cos\\theta ", "7e2e7ee3094ee8020e17039298420937": "\\sum_{j=1}^n A_{j1} = \\sum_{j=1}^n f_j(e) = n", "7e2fc06e53c5b31ee8b0d881ab3aba75": "\\tfrac{\\pi}{6}", "7e3026bd39955e85e17ed84bf4b85c6b": "O(n^{1/2}\\ln^{3/2}n)", "7e303e429c5c2190bf064a7b3ee5fe8c": "P_W:\\,=\\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}", "7e304e00202081282770ed5f31911353": "\\frac{16}{15}", "7e306a35ad0dd7ccd784d2eefce481ed": "y_i=P^ir", "7e308fb10b2501662566ac93d8c8cbd0": " u\\left( {C,L} \\right) = \\frac{1}{{1 - {\\sigma _c}}}{C^{1 - {\\sigma _c}}}v\\left( L \\right) ", "7e30ba582f207dc9a6a4bb8cf39e9ccc": "h\\left(\\tau\\right)=h\\left(\\frac{a\\tau+b}{c\\tau+d}\\right)", "7e30c749b9a4932e2065313fafb3c644": "u_{\\rm r}^2 = \\left(\\frac{u(V)}{V}\\right)^2 + \\left(\\frac{u(m)}{m}\\right)^2 = \\left(\\frac{0.03}{22.45}\\right)^2 + \\left(\\frac{0.1}{781.4}\\right)^2 = (0.001336)^2 + (0.000128)^2", "7e30fe76c5ffb9417a06e33264b48b04": "P\\left( -z < Z < z \\right) = \\gamma", "7e3154ef3834faf257c1db9eabada81b": " (\\tan {\\beta_3} - \\tan{\\beta_2})", "7e31a5a22a2efe21ed1201e25879f3fb": "\n\\frac{\\partial f_s}{\\partial t} + \\sum_{i=1}^s \\dot{\\mathbf{q}}_i \\frac{\\partial f_s}{\\partial \\mathbf{q}_i} + \\sum_{i=1}^s \\left( - \\frac{\\partial \\Phi_i^{ext}}{\\partial \\mathbf{q}_i} - \\sum_{j=1}^s \\frac{\\partial \\Phi_{ij}}{\\partial \\mathbf{q}_i} \\right) \\frac{\\partial f_s}{\\partial \\mathbf{p}_i} = (N-s) \\sum_{i=1}^s \\frac{\\partial}{\\partial \\mathbf{p}_i} \\int \\frac{\\partial \\Phi_{is+1}}{\\partial \\mathbf{q}_i}\\cdot f_{s+1} \\,d\\mathbf{q}_{s+1} d\\mathbf{p}_{s+1}.\n", "7e31c2889431c9a30db29aa8b38ff419": "\\frac{\\partial n(x,t)}{\\partial t}=\\nabla\\cdot( D \\nabla n(x,t))=D \\Delta n(x,t)\\ , ", "7e31d20cf07d734d6b71920d850435a4": "\\sim \\int_{\\alpha}^{\\beta} \\left ( \n1- \\biggl ( \\frac{\\sin{\\pi u}}{\\pi u} \\biggr )^2 \\right ) du\n", "7e31f536d8fc66e5f9d37a235d17eaed": "\\dot{u}_x", "7e31f5bfc83a31a079df37461f6ad0ff": "C= \\frac{S}{\\Sigma}=n^2 \\frac{\\sin^2 \\beta}{\\sin^2 \\alpha} \\ ", "7e321322a0c2109042b5dad0540a1dd1": "x^{n}\\equiv x_{1}\\cdots x_{n}", "7e32602a3c98225bab235c99a8762928": "Q_\\mbox{rej} \\subset Q", "7e3295e6f56347bd119e846d94aa0790": " [\\cdot]", "7e32f665c15d36119e1adeb4e7100513": "\\rho_c = \\frac{3 H^2}{8 \\pi G}.", "7e330f292a98fa79bedba66f445dc608": "C_x \\to C^{\\infty}(S_i)_x", "7e3346bf6ca44e0c4eab09cf16488a3e": "dW = -P dV \\,", "7e34075cb983232f08460a4b564e8586": " H_s(r)=1-e^{-\\lambda \\pi r^2}. ", "7e344d8a96d498b330062f091f8b97f3": "\\tau_{x}(P)", "7e345f562542175a3758ddefc6a2690d": "\\scriptstyle \\gamma ", "7e347fd564c848c0bd09b36158e0c0fc": "\\log_b a=\\frac{\\log_d a}{\\log_d b}", "7e348bcf15476e769b88ed499a8e653d": "f'(r_1)=2r_1=6", "7e349c90d1a31e2656c2b72a97a78d2b": "s\\in X", "7e34b20b48ad47ab4fe20dd48f54bd1b": "RR = ( Rejected pieces / Processed Pieces ) * 100 ", "7e34d448f21076fd2a120b99983763a4": " Z_N(K,L) = 2e^{N(K+L)} \\sum_{P \\subset \\Lambda_D} e^{-2Lr-2Ks} ", "7e35c7a16411c9cd356447590dd88b51": "\\text{Gain}_i(\\sigma,a) = \\max \\{0, u_i(a, \\sigma_{-i}) - u_i(\\sigma_{i}, \\sigma_{-i})\\}.\\ ", "7e35ee06c39ffda2255afcfd6667d16b": "N(t + xi +yj + zk) = t^2 - ax^2 - by^2 + abz^2 \\ ", "7e3630ec68faad7259e6f8d9aa86bfc3": " a_{02} = p_2p_5. ", "7e36bc3ddedb08b3d5e0716ddf9398c6": " \\, k = 1, \\ldots, N. ", "7e3749b87fd3218e861b7cc4efe5abee": "\\vec J = \\vec L+\\vec S", "7e377b77782fb251ca9e063f0b2970d1": "1-f(x) = f(-x).\\,", "7e37a9d3384482f8388c222b4093355f": "T_0 \\,", "7e37c2ca4c3078baee60061302c28b47": "\n-ie^3 \\int {d^4 q \\over (2\\pi)^4} \\gamma^\\mu {i (\\gamma^\\alpha (r-q)_\\alpha + m) \\over (r-q)^2 - m^2 + i \\epsilon} \\gamma^\\rho {i (\\gamma^\\beta (p-q)_\\beta + m) \\over (p-q)^2 - m^2 + i \\epsilon} \\gamma^\\nu {-i g_{\\mu\\nu} \\over q^2 + i\\epsilon }\n", "7e37d80057fcce33b01cba8bcaa8ed40": "E_\\mathrm{translational} = \\frac{1}{2} m v^2 ", "7e37ff93a4cd618a97701441442cfb1d": "\\theta = \\frac{m_{\\text{wet}}-m_{\\text{dry}}}{\\rho_w \\cdot V_b}", "7e3836361e1272dc5a09cbd36a7cb010": "L_{\\text{NO}}", "7e3896341484fa51dcc5554aabd0add2": "\\Vert P\\Vert", "7e39006329c64d5f28fa84ce6f2e7cd6": "\\rho : S(K)\\rightarrow \\mathbb{R}", "7e39145b77ff1351a280efae2a2d905b": "f^b (t_i, w)>0", "7e3914bf400ee1b01c565ef899ab0959": "H=\\{0\\}\\,", "7e3915d617a90c4a3b5e8fae23445650": "L^2 (\\mathbb R, dx ) ", "7e392ce2791ef8d93108867acb821f5c": " y'(t) = f(t,y(t)) \\, ", "7e395d39a804bd07dfd00925e0951a18": "\\nabla f \\cdot \\mathbf{e}_1=0", "7e39621274aa2a7bd22ab257f984b98c": "\\rho_L", "7e39ad22e2637d51dbf949368c0e89fc": "m=2,", "7e39b8ea242708e57249e84c88e75493": "\\lim_{a\\rightarrow 0+}\\left(\\int_{-1}^{-2 a}\\frac{\\mathrm{d}x}{x}+\\int_{a}^1\\frac{\\mathrm{d}x}{x}\\right)=\\ln 2.", "7e3a27f734ae2c0cf912ef858d24162c": "\n \\int x^m\\left(c (A\\,b\\,n (p+1)+(A\\,b-a\\,B) (m+1))+d (A\\,b\\,n (p+1)+(A\\,b-a\\,B) (m+n\\,q+1)) x^n\\right)\\left(a+b\\,x^n\\right)^{p+1}\\left(c+d\\,x^n\\right)^{q-1}dx\n", "7e3a4067b40de44a091f6f05c8bf2228": "C^{LG}_{lp}", "7e3a95803382033b7f24a66ff7f95af7": " F\\sub F(\\sqrt{\\Delta})\\sub F(\\sqrt{\\Delta}, \\sqrt[p_1]{\\alpha_1}) \\sub\\cdots \\sub K\\sub K(\\sqrt[3]{\\alpha})", "7e3aa8a39b4d8432db270ae627bcfc8a": "= e ^{( V1 + V2) / V_e}", "7e3ab9118060fee92d7a975d1f80e81a": "T_p M", "7e3aedfa67c1509381e00b0db10acd57": "R{{e}_{b}}=\\frac{{{D}_{b}}{{G}_{b}}}{{{\\mu }_{L}}}", "7e3b2a3f1c0dbe7353d666f77dcb7593": "\nr_{c} \\le R(q,u) \\longleftrightarrow \\alpha \\ge \\psi(q,\\alpha,u)\n", "7e3b74064f5ab88446e1bce9dd525ed0": "\\Gamma_Y", "7e3b7a9a3a83bb7f88a46cdf680e7ad7": " \\mathbf{y}'_{2} ", "7e3b8132af7657fab37064208ffbeec1": "z \\geq y \\geq x", "7e3b8c753390efcc0ade209c390b6c83": "m\\Omega=n_pm=(m-1)n_s+\\dot\\Omega_s", "7e3b9b4d4cfbbbf7a7073a27cdab256d": "(A.2.c)\\quad \\gamma_{,\\,\\rho}=\\rho\\,\\Big(\\psi^2_{,\\,\\rho}-\\psi^2_{,\\,z} \\Big) ", "7e3c1d080226926cbe660b68b804289b": " \\widehat{\\Omega}_{OLS}", "7e3c2e63331a9bdb7cfa49241a6bfda6": "f(i)= \\boldsymbol\\beta^{\\mathrm T} \\mathbf{x}_i = \\mathbf{x}^{\\mathrm T}_i \\boldsymbol\\beta", "7e3c67ffc774e97ddf7fdf5b8211ae28": "H_v", "7e3c7f49a0a5def49787144cbecdaf82": "\n\\begin{bmatrix}\n\\boldsymbol{X}_j^{(t)}\\\\\n\\boldsymbol{X}_j'\\\\\n\\boldsymbol{X}_j^{(b)}\n\\end{bmatrix}\n", "7e3c911fb11c49829ba3ae9be898f544": "\\mathcal{G}\\,\\!", "7e3cce495b09ae091c3ff02adde12fb1": "(a-m^2)A_m-q(A_{m-2}+A_{m+2} =0", "7e3cd33070737e45c28bcd7c10c016ac": "\\int\\frac{x\\;dx}{s^7} = -\\frac{1}{5s^5}", "7e3ce9e7e20eddd70c3d0b773a1501cb": "\\Phi_l^{-k}", "7e3d369d40c425e91ec58973c2d0dcff": "3 \\mapsto (10-3) = 7; 7 \\mapsto (10-7) = 3; 1 \\mapsto (10-1) = 9", "7e3d7105ce740182609c34b1c8fc8ae5": "{\\hat\\theta_w}", "7e3d7e49bdfc54074571d5b7f06f1c80": "\\pi_1(\\mathbb R^3 \\backslash \\text{trefoil}) = \\lang a, b | a^2 = b^3 \\rang.", "7e3d8c7f407f8e7d47e900a85b1cc797": "A_1,A_2,\\ldots", "7e3e11370f66100ed55624a1817954d2": "L(f_m) = 10 \\log \\bigg[ \\frac{1}{2} \\bigg( \\bigg(\\frac{f_0}{2 Q_l f_m}\\bigg)^2 + 1\\bigg)\\bigg(\\frac{f_c}{f_m} + 1\\bigg)\\bigg(\\frac{FkT}{P_s}\\bigg) \\bigg]", "7e3e47344c158689d888b4a452a5137b": "\\Phi(0,x) = x\\,", "7e3e5e164d620c68d45540f85d132922": "1\\cdot m_1+2\\cdot m_2+3\\cdot m_3+\\cdots+n\\cdot m_n=n.\\,", "7e3ed09a7df93d504464996136a2d261": "\\begin{align} y_{n+1}^2 &= (y_n + 1)^2 \\\\ &= y_n^2 + 2y_n + 1 \\end{align}", "7e3f15fe3c3ae6322d3b3de06a3abf5c": "C_{XX}(t,s) = E[(X_t - \\mu_t)(X_s - \\mu_s)] = E[X_t X_s] - \\mu_t \\mu_s.\\,", "7e3f2eecd75652aefae79102f2728b44": "\\mu:\\Sigma\\to \\mathbb {R}\\cup\\{\\infty,-\\infty\\}", "7e3f370fd2bfdf63b726a8ced383fb27": " \\varphi \\left ( \\mathbf{x} \\right ) = \\int y \\, \\frac {P \\left ( \\mathbf{x} \\land y \\right )} {P \\left ( \\mathbf{x} \\right )} \\, dy ", "7e3f4e86ece7661194c00b570835837e": "E_8\\,", "7e400582779fe12eae8d66a100fac2f5": "Y_{3}^{-2}(\\theta,\\varphi)\n= {1\\over 4}\\sqrt{105\\over 2\\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot\\cos\\theta\\quad\n= {1\\over 4}\\sqrt{105\\over 2\\pi}\\cdot{(x- iy)^2 z \\over r^{3}}", "7e400a463accf3dce6b02c1079077267": "\\scriptstyle\\sqrt{2X}", "7e400ceb4a6585676d4af1a9f59c2ccc": "\\mathbf{K}_f", "7e403279eb2bb0ee58eb7cf694a7fd0f": "a_n^{2n-2}\\prod_{i 1 \\right \\} }.", "7e65be4545f8abc3acbc1f5be7f11634": "\n \\int_{-1}^1 {f(x) \\, dx} = \n \\frac {2} {n(n-1)}[f(1) + f(-1)] +\n \\sum_{i = 2} ^{n-1} {w_i f(x_i)} + R_n.\n", "7e65e2583db8fb3cfa98c157635511ea": "\\hbox{not}", "7e65e42dd25ae1a3471f41a569b1181b": "\\scriptstyle (1 \\,-\\, \\theta i\\,t)^{-k}", "7e66134cc4db2858f33c517f3077d0f2": "\\omega^{A}_{y\\overline{\\|}x}=\\omega^{A}_{x}\\;\\overline{\\circledcirc}\\; (\\omega^{A}_{x|y},\\omega^{A}_{x|\\overline{y}},a_{y})\\,\\!", "7e6637bf7cde959dea470d81807c937e": "\\frac{7}{3}", "7e664afade34c6d322e2ad2059f796a4": "\\operatorname{rank}(A) + \\operatorname{rank}(B) - n \\leq \\operatorname{rank}(A B).", "7e66822a35275751c571bf3a1fbcd94d": "Z' = \\frac {5}{4}\\left(1 - \\frac{9}{16}n^2\\right)(p' + q'i)^{8/33}", "7e668292685e414b73510666affb1eb5": "\\pi_1B", "7e66bdd6c65b7f6a7cccd5942f3814c7": "[D_{\\mu},D_{\\nu}]\\psi=\\frac{1}{2}\\mathsf{R}_{\\mu\\nu}\\psi", "7e66f4592221407a88157933cf916d3c": "\\vartheta\\, ", "7e670c394c1c4e04ac216cd3e98810b2": "r>0", "7e675c25aefbd3f39c2cd3187968f413": " L_z = \\sqrt{(1-e^2)} \\cos i .", "7e676f8ee1e7956da7c277395ca7bf02": "p : M \\to N", "7e677dff2d8181abe5dc8ca037d6f0c3": "\\; \\exp\\;[-\\,(z - H)^2/\\,(2\\;\\sigma_z^2\\;)\\;]", "7e67910194961b237c0c89339d759e7c": "M = {{wK_f}\\over{\\Delta T}}.\\ ", "7e67950a7c4fdc071ea54ec89feb01ec": "-2\\pi + 4\\pi^2 - 8\\pi^3 = -2\\pi + (-2\\pi)^2 + (-2\\pi)^3 = \\frac{-2\\pi(1 - (-2\\pi)^3)}{1-(-2\\pi)} = \\frac{-2\\pi(1 + 8\\pi^3)}{1+2\\pi} \\approx -214.855. ", "7e679fb7210106df9a6b76ba9e9af293": "\\mathfrak{g}\\otimes\\mathbb{C}[t,t^{-1}]", "7e67c5c1e628b42eaa87fbdff3e7f0bf": "a \\uparrow \\uparrow b = \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{b}", "7e67ca8d2da7b1075960e1b616c5856b": "~A \\leftrightarrow B \\leftrightarrow C", "7e67d1852ee71b869361d401f0104012": "\\bar{t}^{2}\\bar{q} \\equiv (2q)^{2} \\pmod l", "7e67d556b465c2fc5e3a5a83b9440236": "R=1-H\\left(\\mathbf{p}\\right)", "7e680a4230f173c7ccd125ea7aa3a41c": "f(t) = \\cos(\\omega t)", "7e681918507fd090aaf3e866086879b0": "\\xi^{-1} r = \\sum_{n=0}^\\infty (-1)^n (\\partial^n r) \\xi^{-1-n}", "7e682fb5ad77c686c6e4a496f87a0633": "\\gamma_c", "7e68b57ffbc5c82aed65297540e2a256": "\\operatorname{E}_t(X) = \\operatorname{E}_t ( \\operatorname{E}_{t+1} ( X )).", "7e6902901b5975a64cec209b1ee11468": "\\left\\langle \\frac{\\delta F[\\phi]}{\\delta \\phi} \\right\\rangle = -i \\left\\langle F[\\phi]\\frac{\\delta \\mathcal{S}[\\phi]}{\\delta\\phi} \\right\\rangle", "7e69427eea4cb2c8fce1a311ebec8dd2": "X_0 \\in \\mathbb{R}^p", "7e69465ae4b9b8f16305de6c12d81feb": " \\mathrm{d}\\,{\\star \\bold{F}} = \\frac{1}{6}{F^{\\alpha\\beta}}_{;\\alpha}\\sqrt{-g} \\, \\epsilon_{\\beta\\gamma\\delta\\eta}\\mathrm{d}\\,x^{\\gamma} \\wedge \\mathrm{d}\\,x^{\\delta} \\wedge \\mathrm{d}\\,x^{\\eta} = \\bold{J},", "7e69cd434cbcbecb6185a0cc02d932ba": "\nE_{CMI}(\\varrho_{A, B}) = \\frac{1}{2}\\min_{\\varrho_{A,B,\\Lambda}\\in K}S(A:B | \\Lambda) \n", "7e69d61e6d956a7d3a629574665991f3": " \\frac{1}{1-z^{-1} }X(z)", "7e69e5ecbc612263d33556f924af70fd": "\n \\begin{align}\n F|\\sigma_{2}-\\sigma_{3}|^m & + G|\\sigma_{3}-\\sigma_{1}|^m + H|\\sigma_{1}-\\sigma_{2}|^m + L|2\\sigma_1 - \\sigma_2 - \\sigma_3|^m \\\\\n & + M|2\\sigma_2 - \\sigma_3 - \\sigma_1|^m + N|2\\sigma_3 - \\sigma_1 - \\sigma_2|^m = \\sigma_y^m ~.\n \\end{align}\n ", "7e6a30c6c51f727be8968d53bfda6db6": "\\mathrm{d}X=Y\\circ\\mathrm{d}W + Z\\,\\mathrm{d}t.", "7e6a5b71d631f3459647c3050961540c": "20 \\log_{10}\\sqrt{1+\\varepsilon^2}", "7e6aa2d53f6ee2b1a34b017fa403cb76": "L2", "7e6abf450ebbd2e5536d58c3d965253e": "f_i = f_{i-1} + \\mathbf{g}(i) e(i)", "7e6adc1847c1e0901a78a181344d28f2": "R_{abcd}=C_{abcd}", "7e6aedfe15e29f3fff49efdb42e890e1": "\\vec{v}_\\mathrm{B|A}=-\\vec{v}_\\mathrm{A|B}", "7e6bce6e862fdb0389a0f638811e2d49": "(z_1,z_2;z_3,z_4) = (z_2,z_1;z_4,z_3) = (z_3,z_4;z_1,z_2) = (z_4,z_3;z_2,z_1).\\,", "7e6bfdc9e68d7948a8e7c7e15df1f3ef": "Hc_1 \\phi_1 + Hc_2 \\phi_2 = Ec_1 \\phi_1 + Ec_2 \\phi_2\\,", "7e6c4fa21d0739feb60510958a837203": "\\sum_{n=0}^{\\infty} \\|f_n\\| := \\sum_{n=0}^{\\infty} \\sup_S |f_n(x)| < \\infty.", "7e6c5a9305159d885d9b4d4e323ff19f": " b \\ge v.\\,", "7e6c6381d3dc98eef307b6017d876d23": " \\begin{pmatrix} 2 & -1 & 0 & \\dots & 0 \\\\-1 & 2 & -1 & \\dots & 0 \\\\ 0 & -1 & 2 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\dots & 2 \\end{pmatrix}.\\ ", "7e6c9bf731470975737131c5f8a22bee": "\\frac{1}{r^{3}}", "7e6ce4fb47f3bfa6bf744b2aacaeec1e": "\np(f)\n=\\sum\\limits_{k=0}^\\infty\\frac{e^{-\\lambda/2}(\\lambda/2)^k}{ B\\left(\\frac{\\nu_2}{2},\\frac{\\nu_1}{2}+k\\right) k!}\n\\left(\\frac{\\nu_1}{\\nu_2}\\right)^{\\frac{\\nu_1}{2}+k}\n\\left(\\frac{\\nu_2}{\\nu_2+\\nu_1f}\\right)^{\\frac{\\nu_1+\\nu_2}{2}+k}f^{\\nu_1/2-1+k}\n ", "7e6d1a9b9ef5c737331055786d7b45ed": "\\scriptstyle \\Phi_1,\\ \\Phi_2,\\ \\dots", "7e6d3b697937f37a8a2fe594f1f1055c": "\\sqrt{x^TQx}", "7e6d69f3585170adfad20473c5f899a3": " t_1,t_2,\\ldots,t_N ", "7e6d6e010fda7481c2c752fa448ec4a5": " T_\\mathrm{sol-air} ", "7e6d847ffeebc098cd994eb200e6af52": "M(x) = \\sum_{1\\le k \\le x} \\mu(k).", "7e6dbb320d8276da4daac7680664927d": "x^{i-u}", "7e6dbcfc97d84d19e0fbb412e04fc57a": "x'=\\gamma x + b t \\,", "7e6dc4f73d31b4b52624476a683b909e": "r(Q)", "7e6e4c385962e0af9cf14dc4659c8527": "y^2 (y^2 - a^2) = x^2 (x^2 - b^2).", "7e6e53b6122764f9a0b0af408c2d9a9c": "I = 100 \\times \\left [ n + ( N - n ) \\frac{log (t_{r (unknown)} ') - log (t_{r (n)} ')}{log (t_{r (N)} ') - log (t_{r (n)} ')} \\right ] ", "7e6e65e7a5ce18ac25180d39d4f96ee8": " \\lfloor\\frac{rB}{r+1}\\rfloor - \\lfloor\\frac{B}{r+1}\\rfloor", "7e6eb6439f8697cbb917650c13efa7c4": "\\frac{\\partial s^{\\ast }(p)}{\\partial p}\\overset{\\text{sign}}{=}U_{sp}(s^{\\ast }(p),p). ", "7e6ebd78bf3c7129052d27bd6e3a8ab2": "L_f = \\sup\\{L_{f,P} \\colon P \\text{ is a partition of } [a,b]\\} . \\,\\!", "7e6ecbc85122f9bd5ae946da29189e83": "P_{4}^{4}(x)=105(1-x^2)^{2}", "7e6eced38a084ebdbcefbf1fef680229": "t_{\\nu'}(\\tilde{x}|\\mu,{\\sigma_0^2}')", "7e6f7200c33928a7f8058d081a3fc4fa": "\n\\mathrm{SNR}_{in}^2(u) = q\n", "7e6fa2281b71b7282369f6c9861ea6ff": " {H}_{2n} ", "7e707f8db053608e90bf13b19e22986a": "\\frac{0.0016286 * P_{kPa}} {\\lambda_v}", "7e708bf806671db3c22b46219bf6c433": "L(x,-y^*) = \\inf_{y \\in Y} \\left\\{F(x,y) - y^*(y)\\right\\}.", "7e708d9daffbb244466c7245b5d6699d": "H_{0,I}(t) = e^{i H_{0,S} t / \\hbar} H_{0,S} e^{-i H_{0,S} t / \\hbar} = H_{0,S} .", "7e711cbae922b5a69fe39c61c9b27aab": "\\sigma\\,x^\\gamma\\, dW_t", "7e7120ea9f30feefa861b7f54fd70362": "\\frac{a_n}{\\prod_{k=0}^{n-1} f_k} = A_0 + \\sum_{m=0}^{n-1}\\frac{g_m}{\\prod_{k=0}^m f_k}", "7e7136db2d1700924263396f1e3b7387": "= (\\sum_{i} (v_i^{T} x) v_i)^{T}) (\\sum_{j} (v_j^{T} x) v_j\\lambda_j)", "7e7170a9b14950997919731464b4d7d1": " A(t)=\\langle x(t) x(0) \\rangle_0. ", "7e71cbf841a7deb134a8aa01aa87cf02": "\\omega(s)\\sim C s^{-\\rho},\\quad\\rm{as\\ }s\\to 0", "7e71cd9ae7ef256b05fb1ff2d9843950": "d \\ln \\Omega = \\frac{1}{k_B} d S .", "7e71e5d17f28cada43b78485ff70cd8e": "d\\neq1", "7e72e0fc3735bb74373d004ff92d0900": "|x_3 - x_1| < \\tau (|x_2| + |x_4|)\\,", "7e730a5b63e01914bd79df9f3739696d": "n \\rightarrow \\infty.", "7e734b4989f5b945047fe7e152f3df47": "u_A", "7e7395790299a29efba0ad852bd21e20": "( I_p(x)- C_a ) / C_d = L(x) \\cdot N(x) ", "7e73af9ab5c7142d10dedb553545ccda": " \\varrho_s ", "7e73e05bdf6982d1cccec1c588058cee": "~ n_1=\\frac{W_{\\rm d}}{W_{\\rm u}+W_{\\rm d}}.", "7e73ed0c72d6c229afd2bde6cbd45975": "dE", "7e73f848635874578aa65d4534ce1984": "\\operatorname{MISE} (\\bold{H}) = \\operatorname{AMISE} (\\bold{H}) + o(n^{-1} |\\bold{H}|^{-1/2} + \\operatorname{tr} \\, \\bold{H}^2)", "7e7400465113dbe5b3c41c0bb7299eef": "{n\\choose k}=\\frac{n!}{k!(n-k)!}", "7e74182c10b9516a8f8773213e47fe11": "\\{p\\in M:\\varphi(p)\\le y\\}", "7e742af663b1a6d6fce369a47fc4f935": "\\lang \\mathbf x | \\mathbf x \\rang", "7e742df984ccfc8d949cf9f2f7fae4cf": "v_n = \\prod_{1\\le i 4 \\end{cases} ", "7e74ee8ccca7314c06aa9e17e3946c1d": "u=2\\sqrt{-\\frac{p}{3}}", "7e7502eec9244b4fafbcd53efe9bd0d8": "dc/dt = -s", "7e751247ca0f37f49f595debca0cc967": "\\pi_1(SO(3))", "7e752686b1f6532f41e8a7923811d8bb": "\\ \\hat{x}(t)", "7e75487f33807a78f2158dc6ef6421de": "p(g(z))=\\left( \\sum_{n=1}^N n^{-1}\\lambda_n z^n\\right) +\\left(\\sum_{m=1}^\\infty \\sum_{n=1}^N \\lambda_n c_{nm}z^{-m}\\right).", "7e75ad9449e77e769eb2fa0d0aee272d": "K(k)", "7e75c0cf44cbdf6f1029c26015762cb2": " P = \\sum_{s_{zN}}\\cdots\\sum_{s_{z2}}\\sum_{s_{z1}}\\int\\limits_{\\mathrm{all \\, space}}\\cdots\\int\\limits_{\\mathrm{all \\, space}}\\int\\limits_{\\mathrm{all \\, space}} \\left | \\Psi \\right |^2\\mathrm{d}^3\\mathbf{r}_1\\mathrm{d}^3\\mathbf{r}_2\\cdots\\mathrm{d}^3\\mathbf{r}_N = 1\\,\\!", "7e75cbe75a707553b6fc8aaaeff105e8": "\\Theta = k_b T; \\beta = {1 \\over k_b T}", "7e769d19726d101e1aa235b1042a8806": "\nc^{2}d\\tau^{2} = w(r) c^2 dt^{2} - v(r) dr^{2} - r^{2} d\\theta^{2} - r^{2} \\sin^{2} \\theta d\\phi^{2}\n\\,", "7e76c0f2379695f7ee8a9a3b32cecfb0": "\\chi: \\mathfrak{N}_{2i} \\to \\mathbf{Z}_2", "7e76c5aad139e076e692e0d5361db595": "d^\\circ(Q)", "7e774e17edb195e4dd0f7330b50473ed": "[G] = M^{-1}L^3T^{-2} \\ ", "7e7761c55e4adf4ee2346b691540785b": " J_\\varepsilon = \\int_a^b F(x,g_\\varepsilon(x), g_\\varepsilon'(x) ) \\, \\mathrm{d}x = \\int_a^b F_\\varepsilon\\, dx \\,\\! ", "7e77860fbbd1835517ded46601ff0c43": "v\\in \\Sigma^*", "7e77ae2a495e0611b1d5656dbbda4a08": "a_{r}=\\frac{(r+c-1)(r+c-\\gamma )}{(r+c-\\alpha )(r+c-\\beta )}a_{r-1}", "7e77b1fbfe3fda7174db6703d96d6edc": "c^{2} d\\tau^{2} = g_{00} c^2 dt^{2} + g_{11}dr^{2} + g_{22}d\\theta^{2} + g_{33}d\\phi^2 \\;", "7e77d1a375971eee05042d0767c968df": "F:Y\\rightarrow X", "7e77d5209fb22a22475ac31eeb409b0d": "\\frac{dy}{dx} = y \\times \\frac{f'(x)}{f(x)} = f'(x).", "7e77f7850a99a413cfad6b24659103a7": " \\operatorname{E}[X] = \\sum_{i=1}^\\infty x_i\\, p_i,", "7e784353eb76176d8f10e9b4bfd0d318": " i=1,2,\\cdots,k ", "7e784b7de808d099559715b97b5f888c": " \\lim_{n\\to\\infty} x_n = \\lim_{n\\to\\infty} T(x_{n-1})", "7e788aaf8313c06e11fdac07c726059c": "Inverse Gamma(\\frac{v}{2}, \\frac{u}{2})", "7e78b6fae7d793567a0aaeae1078f1d8": "\\sigma(n) \\sigma(\\phi) \\ge \\frac{\\hbar}{2} \\,\\!", "7e791e8cc2cc94b936d5577f4e83dfe4": "\\Delta(L_+) - \\Delta(L_-) = (t^{1/2} - t^{-1/2}) \\Delta(L_0)", "7e792813eae6a2526902a9b3ff8fcfed": "iOPT-t\\}\\,\\!", "7e8e2e8efa0e83ebd22958f0d47436bc": " \\scriptstyle w_{nj}^k=s_n \\, d_j^k ", "7e8e391ae1bbc45d44a0707c0ae9dc32": "\\frac {\\alpha_c} {\\nu} = - \\left(\\frac {RT} {nF} \\right) \\left( \\frac {\\partial ln(|I_{red}|)} {\\partial E}\\right)_{T,p,c_{i,interface}}", "7e8e39d0e6568ffb38436998add1f548": "f^{-1}:\\mathbb{C}^n \\rightarrow \\Omega", "7e8e5e78b9d028702e080f9867b21c1b": "ay^2 + y(2am + b) + (am^2+bm+c) = 0", "7e8eac91a31227990e07486decad287b": "\\Omega\\subset\\mathbb{R}^2", "7e8f8519a284eee6835070a4405f7f7e": "MO_*(M)", "7e8fc48dbbe97aee798cf504a3135ea0": "\\textstyle s_i", "7e900bfce084f2521adcfc8709c9a4a0": "\\mu_1=Gm_1\\,\\!,\\mu_2=Gm_2\\,\\!", "7e90a7105263e03987128b58df36b947": "k,m \\in \\mathcal{N}", "7e911c3631b38f27c6bab906a3a9f9ef": "F(\\boldsymbol{v}) = \\int_A v_i f_i\\mathrm{d}x + \\int_{\\partial A\\setminus\\Sigma}\\!\\!\\!\\!\\! v_i g_i \\mathrm{d}\\sigma \\qquad \\boldsymbol{u},\\boldsymbol{v} \\in \\mathcal{U}_\\Sigma ", "7e916b1d5e4a4ac20dc1c596ad93a665": " \\varphi(m)=(p_{1}-1)\\dots (p_{r}-1)", "7e917abc35923d4918e807410890edff": "(y_{t} - y_{t}^*)", "7e919331a09b3909ce915872a87fca26": "\\beta = \\frac{\\Delta f}{\\left(\\frac{1}{2T}\\right)} = \\frac{\\Delta f}{R_S/2} = 2T\\Delta f", "7e91975b78a226b24a20c8390bd4a42f": " {} = p_{01}p'_{23} + p_{02}p'_{31} + p_{03}p'_{12} + p_{23}p'_{01} + p_{31}p'_{02} + p_{12}p'_{03} \\,\\! ", "7e919ab20db116b0619619d5ceb6d69d": "= \\langle x + y, x + y \\rangle", "7e91a0ca179fa9b86b837a435b745b56": "(\\wp, \\wp')", "7e922dc7b67c49cbfa5a45894f787f33": "n_c = (4 \\epsilon / \\pi)\\eta_c ", "7e9282cf8c01daac3b8943aeae1d3728": "\\vdash A \\Rightarrow \\ \\vdash \\Box A", "7e935c72506132e95325d9cc0974e24a": "w = 0.5\\ ", "7e93c2b734158fb8248844a51c9b2cb4": "X + 2Y_W = 5(B - L) \\,", "7e93df6940a236666032761106dbdfc7": "J = \\rho v_{\\it avg}", "7e93e9c9c75c1eb70f491c18c31e2de7": " \\begin{align} \nK_M \\ &\\stackrel{\\mathrm{def}}{=}\\ \\frac{k_{2} + k_{-1}}{k_{1}} \\approx K_D\\\\\nV_\\max \\ &\\stackrel{\\mathrm{def}}{=}\\ k_{cat}{[}E{]}_{tot}\n\\end{align} ", "7e93f7176e72e68304676893aae158d8": "\\frac{0.2}{0.4 \\times 0.4} = 1.25 ", "7e94942be75eb3e97417d1b52a71a390": "v^2=\\operatorname{Var}\\left [ m(\\vartheta) \\right ]", "7e953581e87514bc8441dadc1a21c627": "f(v)\\geq 0.", "7e957cacd1f2db94a8450d04a812dfb9": "\\epsilon > 1", "7e9597e101eb01722f65d246d6bb1e2e": "\\int_{-\\infty}^\\infty {e^{itx} \\over x^2+1}\\,dx=\\pi e^{-t}.", "7e95a27b42c3a68602db757e29eeb790": "2\\lambda c\\cdot x^*", "7e95b62bc9f61cc309b351939ddf432b": "\\alpha = \\arccos\\left[\\cos\\varphi \\cos \\frac{\\lambda}{2} \\right]", "7e963e95eae0894cacdff05f70206bbf": "(\\Z/n\\Z)^* \\simeq (\\mathbb{Z}/p^2\\mathbb{Z})^* \\times (\\mathbb{Z}/q\\mathbb{Z})^*", "7e9642f48204df755df1ec4fc78da107": "\\varepsilon=\\infty", "7e9668a2d09fe93554ec85603b20c618": "\\mathbb{F}_{p'}", "7e968762b60d7552e666c3255bd1e825": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}} \\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}}", "7e96940e7428c779807ba9d1fefe0c8b": "\\frac{f(z)-\\gamma_1}{f(z)-\\gamma_2} = k \\frac{z-\\gamma_1}{z-\\gamma_2}.", "7e969c843327e6488890f1802388dbeb": "K_1(A,I) = \\ker \\left({ K_1(D(A,I)) \\rightarrow K_1(A) }\\right) \\ . ", "7e96e8b2d83ff6842681ca102efbd65b": " \\varphi _j^{n } \\ ", "7e97220985e60593b20e06f0c8f711e5": "(*) \\quad |R_k(x)|\\leq M_{k,r}\\frac{|x-a|^{k+1}}{(k+1)!} ", "7e9775fe44f556522767a6822824de12": "F = 0.7tL(UTS)", "7e977b8c7a3d40f19ed785ed7da5c2ad": "n_{j,(\\cdot)}^i", "7e9785a4cd661ad2db4e25ba8e42ae31": "1/\\sqrt{1-x^2}.", "7e97b7ca7a70afafd9fd47013a79d976": "\\displaystyle{X_\\alpha=E_\\alpha + E_{-\\alpha}, \\,\\,\\, Y_\\alpha=i(E_\\alpha - E_{-\\alpha})}", "7e97c47802d0be5af017743a6e9469ce": "g = {1\\over 2}(h+\\bar h).", "7e97f8d53327f78a079db6119587f91c": "f(x):=\\lim_{n\\to\\infty}f_n(x)", "7e98033dd9a4a8c74c029d4d8edab9d3": "R\\varphi(\\xi,p) = \\int_{x\\cdot\\xi=p} f(x)\\,d^{n-1}x.", "7e98692dc425912eaaeb5ee638461ed1": " Q(t_2) ", "7e98997e156500e8a3f544582b90c4bf": "R_{\\theta HA} = 4 \\ ^{\\circ}\\mathrm{C}/\\mathrm{W} \\,", "7e990213067fee3e3312f0fcca325204": "w_t", "7e9911e4b9ecddde608adde2c0ed8861": "f(x;n,p)=\\binom{n}{x}p^x (1-p)^{n-x} I_{0,1,\\ldots, n}(x)", "7e992da2b4df9f81bfebfcad80a6dccf": " a \\ \\psi_0(q) = 0", "7e99689544235c9f3d3a770dd7a128ee": " f(z) = P_k(z) + R_k(z), \\quad P_k(z) = \\sum_{j=0}^k \\frac{f^{(j)}(c)}{j!}(z-c)^j, ", "7e998ae9c8d4a91c9d6df0227fd4448d": "\\scriptstyle \\mathcal K", "7e9a1301dbdc389400c9a409048dea53": "\\hat{J}", "7e9a4048e38254c48a0922ab257b0bfd": "R_V", "7e9a631189b274dfcf1e2d9d7dd0c405": " \\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{({\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x}) \\neq \\!\\!\\!\\!\\!\\!\\!\\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{(-{\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x})\n", "7e9a721b26c315d5d015d7dd2fd27d1d": "-\\pi", "7e9a8ca6fc33dd77a7a9bae3b512f795": "n = pq", "7e9abb2d903e0b61dd761d58dbd8409b": "\\kappa \\;", "7e9b15fe16fd46a4609a8393de450d34": "\\operatorname{haversin}(\\theta)=\\sin^2\\left(\\frac{\\theta}{2}\\right)=\\frac{1-\\cos(\\theta)}{2}", "7e9b7e1c92bb7b76db9966b42ae03c1a": "X_t(\\omega)=\\sum_{k=1}^\\infty A_k(\\omega) f_k(t)", "7e9b91714529331ab9487bdcc6eb196d": "\\textstyle A^c", "7e9c0c2e73549200787ca594574c26ce": "Y = \\exp \\left[-\\left(\\frac{0.191 \\omega T_1 -1}{2^{1/2}\\sigma}\\right)^2\\right]", "7e9c3ae1f49092a52d40c339a43bfc1d": "f^{**}", "7e9c546041ec03e8cb0c240467b6d2a1": "\\scriptstyle\\frac{d}{dt}", "7e9c9299eab395242f9e74856b120f9f": "\\phi_{23} : H \\otimes H \\to H \\otimes H \\otimes H", "7e9c9b082641ba1e4ddbef5bf00ddc3f": " f \\cdot g = L_1(f)g + L_2(g)f - L_1(f) L_2(g) e = f(a)g + g(b)f - f(a)g(b)e. ", "7e9cbad6b59d180de2a69ac14678b11d": "dx=\\Delta x", "7e9cd7195e63a9436c6cebe4351b094a": "\\mu_{ab}^{(c)}(t)", "7e9cf4cef568dc9a6c45de84ee27fe20": "2\\cdot x", "7e9d3951de09c1eb717242107851ef42": "\\mathrm{bei}(x) / e^{x/\\sqrt{2}}", "7e9d55ccfd933ed6026a5a4e1d6d6c9e": "\\frac{dy}{d\\mathrm{net}}", "7e9d634b30e1087acd867c2b056417bc": "\\operatorname{rank}\\,D_p f = \\dim M.", "7e9df02bc9ecef8c1182827385e2a747": "P[s(t)]", "7e9e3e20624d73ba392d94e999c7975a": "zf_{j+1}=\\frac{f_j(z)-\\gamma_j}{1-\\overline{\\gamma_j}f_j(z)}.", "7e9e5f9fa8eba3346537056ee8875668": "e_{(3)}=\\partial_3", "7e9eb27914b6c08f169fe51a82089162": "V(s) = -J\\delta(s_0,s_1)", "7e9ed5d6a721a8f4a99c2dd43449d243": "I(\\omega)\\propto|\\chi^{(2)}|^2I_1(\\omega_1)I_2(\\omega_2)", "7e9f2b698d9d4db3d658da3c9f0093f5": "K_{\\rm w} = \\frac{a_{\\rm{H^+}} \\cdot a_{\\rm{OH^-}}}{a_{\\rm{H_2O}}}", "7e9f41abc568fb391f0968dcc4dc87c6": "\\mathit{D}", "7e9f7caf140f67bb15e9187cc9fb6176": "1-\\frac{k}{|E(G_j)|}", "7e9fa57198b163310f6743b5f0667318": "\\phi \\leq u", "7e9fb86320eb896a1a245cb034984d94": "\\hat{p}_i", "7e9fc59354512d29715d6cfa3b01c8c2": " z = x + i y = r \\cos \\phi + i r \\sin \\phi \\ ", "7e9fe47eb7d46879cc1808635e4ba146": "\\text{L} = \\frac{\\sum_{\\ell=\\ell_\\min}^N \\ell\\, P(\\ell)}{\\sum_{\\ell=\\ell_\\min}^N P(\\ell)}", "7e9ff94ecf1a92756309d07d88ef6562": "I(Y;Z)", "7ea00c8f1dcaff0c519e68358ebfd492": "f(x_i)=y_i", "7ea04811d7c3ddefc0cb513a6bff9755": "\\min_{\\boldsymbol{w}\\in\\boldsymbol{W}, t>0}\\left\\lbrace\ng_0(\\boldsymbol(w))+t\\sum_{i=1}^m\\ln M_{g_i(\\boldsymbol(w))\\psi_i}(t^{-1})-t\\ln\\alpha\n\\right\\rbrace.\\,", "7ea0ecc7fc18e2cfa8fb03d85cdcd19b": "\\displaystyle AX\\cdot XC = BX\\cdot XD.", "7ea13c498c4145486a9df311ff2ba155": "\\mathbf{C}_{2,1} = \\mathbf{M}_{2} + \\mathbf{M}_{4}", "7ea165d125379497f807d0e6938c5345": " y\\ f = f\\ (\\ldots f\\ (y\\ f) \\ldots) ", "7ea16f1fa3646315fb7798c9b314d38b": "\\begin{pmatrix}{1 \\over 15}&0&0&0&0\\\\0&{1 \\over 15}&0&0&0\\\\0&0&{1 \\over 15}&0&0\\\\0&0&0&-{1 \\over 10}&0\\\\0&0&0&0&-{1 \\over 10}\\end{pmatrix}", "7ea17fcc41caff6b310c3b213e7d90f6": "\\sum_{i=0}^L q_i M^i x_{\\mathrm {base}}", "7ea1bdc7bcb585c039b4ce793358d9ac": "\n\\begin{array}{c}\n\\text{area under}\\\\\n\\text{curve}\n\\end{array}\n= \\int_1^\\infty\\frac{1}{x}\\,dx \\;=\\; \\infty.\n", "7ea1ed47d78267818031507422cff5ef": "\\sum_{m,n=0}^\\infty Q(m,n) z^m q^n = 1 + \\sum_{s=1}^\\infty \\frac{q^{s(s+1)/2}}{(1-zq)(1-zq^2) \\cdots (1-zq^s)}", "7ea1f8c67cfa9f5ba278d734022aed36": "L(y)", "7ea20286e72898e4bf8b63a2916620e2": "z^{2} + az^{4} + c", "7ea238abe9385d895ab2cf89a03cd7df": "\\{ 0,4,7 \\} \\times \\{ 0,2 \\} = \\{ 0,2,4,6,7,9 \\}", "7ea2583980fb3f5c0b00a255c18d138b": " K_\\text{P}(x,x') = \\exp\\Big(-\\frac{ 2\\sin^2(\\frac{d}{2})}{ l^2} \\Big)", "7ea290ea56c3846ca4369da675508578": "S(A|B)_\\rho \\ \\stackrel{\\mathrm{def}}{=}\\ S(AB)_\\rho - S(B)_\\rho", "7ea2ae50fc197e0110df59d89c3ed375": "f\\colon S^{n-1} \\to S^{n-1}", "7ea2b32edada9ca5035a0dc52af0be7e": " 4 \\sum_{k=0}^\\infty (-1)^k \\frac{1}{2k+1} = 4 \\left( 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots \\right) ", "7ea302f4059438e5f21298152c14ba74": "g_{A,B} = 1 - (4 5) + (2 4 5) + (4 6 5) - (2 4 6 5) + (2 5) (4 6)", "7ea32f5e1002cbf76d27d7026f664089": " \n{1 \\over c^2} {{\\partial \\varphi } \\over {\\partial t }} + \\nabla \\cdot \\mathbf{A} = 0\n\n", "7ea39ca7f369d0ddc4df9d90ddfa10b4": "\\overline{12}_{24} + \\overline{21}_{24} = \\overline{9}_{24}", "7ea3a080d41eb480cf5e6815da4655b2": "X_k =\n \\sum_{n=0}^{N-1} x_n \\sin \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) (k+1)\\right] \\quad \\quad k = 0, \\dots, N-1", "7ea3ac219f797d8712683f8440f7846d": "T = \\frac 12 \\mathbf{p} \\cdot \\mathbf{v}", "7ea3b4715970a508ce85e4bfa087d1d7": " q = 1 - \\frac {1} {R_0}. ", "7ea40a640c6ca0628e5c6367a826e339": "e^{-E_a/RT}", "7ea42f132a40f3d21654f1f6f225794c": "V(\\mathbf{x},N)\\equiv \\ell_f(\\mathbf{x}_N)", "7ea45111313b26f4b24035dca45924a5": "\\cos \\alpha=\\frac{\\langle\\hat{d}| \\hat{a}\\rangle+\\langle\\hat{e}| \\hat{b}\\rangle+\\langle\\hat{f}| \\hat{c}\\rangle-1}{2}", "7ea475e6c6bc1b934fe3d379c4d1f3bb": "K=(K-P)+P", "7ea47c9f7700aa272f7494c2d3ea7e6a": "\\scriptstyle \\boldsymbol v", "7ea483603117938601e6502688bf45c3": "V_j", "7ea4976c40e838fbc88b3cfb5f121d90": "\\bar{I}_1 = J^{-2/3} I_1", "7ea4cff42d92cead911227fa2de9fbec": "(x,y)=(0.4402,0.4031)", "7ea565f72f4d69c000f588e898662e0f": "g_{\\mu 4}", "7ea570e2ffdf3c9a2d63d0c921e4aa54": "{{h}_{rad}}", "7ea5803701b18f298dab9adeb38440d7": "\\omega_1\\to(\\alpha)^2_n.", "7ea58e53b0a4b433a89840f63978b925": "\\mathbf{j}(\\mathbf{r},t)", "7ea5bdec81e976382bacf3752123c76c": "m/n = \\sqrt{2}", "7ea5edfb7eb82e3ff0f05d935b94c04e": "\\gamma_{\\alpha \\beta} = -g_{\\alpha \\beta} + \\frac{g_{0\\alpha}g_{0\\beta}}{g_{00}}", "7ea5f761eedd06db3ada835e5f673f08": "\\left(x_n,y_n\\right)=\\left(\\frac {r(1+r)}{2[n^2(1-r)^2+r]}~,~\\frac {nr(1-r)}{n^2(1-r)^2+r}\\right)", "7ea60edcb22c9d5aab78033cade3682d": "I = Q_1 \\cap \\cdots \\cap Q_n\\ ", "7ea629689009af333833f3311c746010": "P\\, ", "7ea63512f446fe747ae064b34ccf0d55": "U_{es}", "7ea64875d465e815ef08431ccda8c20c": "\\dot{y} = \\frac{1}{c} \\overline{h}(0,y,0)", "7ea68199d5661867c148a8d481b8d58f": "f(k;N,q,s)=[\\mbox{constant}]/(k+q)^s.\\,", "7ea6b6f2879bc08fdc91fc2927960107": "E_{n_x,n_y,n_z} = \\frac{\\hbar^2 k_{n_x,n_y,n_z}^2}{2m}", "7ea721c764b520a7a772b15bf2618900": "d=\\left(1.708v-1.481u+0.404\\right)/v", "7ea723de319a02456f8b558768db5933": "\\frac{1}{3}\\sum_{i=1}^{3}\\frac{1}{2^{i-1}}=\\text{D}", "7ea72b4b293954ca4c181ccbdaf9888c": "e= \\prod_{k=1}^{\\infty} \\left(1-\\frac{1}{\\phi^k}\\right)^\\frac{\\mu(k)-\\varphi(k)}{k}. ", "7ea758fe0fd19dacff80355a515b90ff": "P(x_{1})P(x_{2})P^{*}(x_{1}+x_{2}).", "7ea7b7293845e995e047a6f906ed1e04": "\\Delta = {n_1^2 - n_2^2 \\over 2 n_1^2},", "7ea7ca6beee676c1c9aa6be2139b7d79": "T^m", "7ea84332e1707f5fd6d329ce946ddbff": "\\,q(x)>0", "7ea883c729b69ccbfaed9076b78f581a": "4\\alpha(1-\\alpha)=\\frac{1728}{N}", "7ea8ae2e5109865e9370135c6031060b": "\nT = W-U\n", "7ea8c3863dafeaa99e86b282456ac058": "G(r)\\sim\\exp(-r/\\xi)", "7ea8d96fcade5a62d79d9229a3b03a6f": "\n\\sum_{A=1}^N M_A \\mathbf{R}^0_A \\times \\mathbf{d}_A = 0,\n", "7ea8e798f90a9c34613e5d148c863fcc": "\n\\operatorname{ind}_H^G\\pi=\\{f:G \\rightarrow V|f(hg)=\\pi(h)f(g) \\text{ and } f \\text{ has compact support mod } H \\}.\n", "7ea8edb60889b5bd1f7cb693329bd4a7": "U \\sim -{3}/{(4\\omega)} \\times v_0 dv_0/dx,", "7ea9fbd1ca5c5bee5973ad19abe6e5dc": "v = c^k d^{k\\alpha} \\,", "7eaa466003e48c1c96824a2edf3de038": "\\textstyle \\alpha ", "7eab5b390aff59306f553fd0cff769f5": " \\{ |00 \\rangle , |01 \\rangle , |10 \\rangle , |11\\rangle \\} \\ as \\ ( \\hbar = 1)", "7eab9706ddf830cd29153c1d91cc94bb": " a \\lor \\lnot b ", "7eabb90d967d6d8d316ab46702334226": " I = T(I_L) ", "7eabe5d00b83a98373c4654acf769b3f": "a=x", "7eac19b1cede8c5f648f8d2ab3dadf8e": "3 > 4", "7eacbe08290112f99f2de95c5b0bd4db": " p_2 +\\ \\tfrac12\\, \\rho\\, v^2 =\\, p_0 +\\ \\tfrac12\\, \\rho\\, w^2.", "7ead3ab0d9ca765687f5aac2e80f162d": "\\frac{\\partial F}{\\partial n_2} = n_2\\sigma_2^2-2n_2\\sigma_2\\left(\\sigma_1 n_1^2+\\sigma_2 n_2^2+\\sigma_3 n_3^2\\right)+\\lambda n_2 = 0\\,\\!", "7ead89d93061254eb73536903ea4d732": "(1/y_1, 1/y_2)", "7eadcf64b4ed401883b9b7bfc1ea4914": "h = h(T)", "7eae19d3d1db9cec893e9e3a8b8fdb8b": "\\mathfrak{gl}(\\mathfrak{g})", "7eae25e3371ba789ab9afaed1bd9a8e8": "(\\lambda x.x)y", "7eae7c740e5c323a51308ae7fbf54792": "\\mathit{C_p}", "7eaec5324cda47faa9af10253d669cb9": " |\\psi (t+dt) \\rangle - |\\psi (t) \\rangle = - i\\hat{H} dt |\\psi (t)\\rangle ", "7eaef58d08a41eaf8bc86ae12f4966dc": "\n\\begin{array}{rl}\n{\\displaystyle\\min_{X \\in \\mathbb{S}^n}} & \\langle C, X \\rangle_{\\mathbb{S}^n} \\\\\n\\text{subject to} & \\langle A_i, X \\rangle_{\\mathbb{S}^n} = b_i, \\quad i = 1,\\ldots,m \\\\\n& X \\succeq 0\n\\end{array}\n", "7eaf4e81326a0d982410954a3b30e605": "\\text{excess kurtosis} =\\frac{6}{3 + \\nu}\\bigg(\\frac{(2 + \\nu)}{4} (\\text{skewness})^2 - 1\\bigg)\\text{ if (skewness)}^2-2< \\text{excess kurtosis}< \\frac{3}{2} (\\text{skewness})^2", "7eaf62f9b3921cb8310a9aed3dd4adb1": "\\min_{\\alpha} f(\\mathbf{x}_k + \\alpha \\mathbf{p}_k)", "7eaf6752bcb34da3a1f050dcb7e97843": "\\lambda_1 = \\lambda_2 = 0", "7eafae0e6aef12b219ce2e975e6508f2": "\\theta_{2}", "7eafc472161e062ec6283e8b80a96f5f": "\\lambda = \\lambda_0 + \\rho \\{ \\arctan [ x / ( \\cot \\varphi_1 - y)] \\} / \\cos \\varphi", "7eafeeeddca488535a246a98251f91ce": "\\overline V = \\{\\overline v \\mid v \\in V\\},", "7eb0182f9e7f3b32fa71ac6f478bfbb7": "(C)-(D)-(F)-(H)", "7eb033a4d7324622a60d61b1f73e3723": " p\\leq q/6+o(q)", "7eb04d7cb76f54ca5ada2dc198feb137": "r = \\sqrt{a^2+b^2}", "7eb05d2aa13e6699ac9379182576d48b": "f_{\\rm{best}}^{(k)}", "7eb06f3ce5d5bbaee5b2716033a06ab7": "|\\psi\\rangle = \\begin{pmatrix} a_\\psi \\\\ b_\\psi \\end{pmatrix}, \\;\\; \\text{OR} \\;\\; |\\psi\\rangle = \\begin{pmatrix} c_\\psi \\\\ d_\\psi \\end{pmatrix} ", "7eb0dd71904afe2a6e04548de4520323": "x_i = \\frac{2i}{n} - 1,\\quad i \\in \\left\\{ 0, 1, \\dots, n \\right\\}", "7eb0eef914942aa6b6af810a110c6b09": "\\begin{align}\n c^2\\Delta t^2 &= \\Delta r^2 \\\\\n s^2 &= 0 \\\\\n\\end{align}", "7eb1bd8f5a7c4f6087c59344c85da21f": "\\mathcal{U}_0", "7eb208589e314e6a020c30de5302422b": "p_n(x)=\\sum_{k=0}^n a_{n,k}x^k\\ \\mbox{and}\\ q_n(x)=\\sum_{k=0}^n b_{n,k}x^k.", "7eb220bb4d976c8190be68db3511379c": " \\displaystyle{f_0(z)=z,\\,\\,\\, f_1(z)=g(z).}", "7eb293b37441db00e40dc2015c5bd6d6": " n \\geq 1 ", "7eb2c25ca676888a08ac21b7c90fd4bf": "\n \\mathbf{v} = v_i~\\mathbf{b}^i = \\hat{v}_i~\\hat{\\mathbf{b}}^i \n ", "7eb32e72bcffd43a303648d1fb105ebb": "u^au_a = -1\\!", "7eb40ce7bc968c7e585a76360c35e792": "\\Gamma(W) \\Rightarrow", "7eb41ab71459112e28734166e0356975": "i\\neq k", "7eb41e67de4e1e8ed9173a7998d6344e": "\\sigma_y^2(\\tau) = \\frac{1}{2\\tau}h_0", "7eb4251d1dbf297faac8168dd74e3670": "\n\\mathbf{b_{t:T}}(i) = \\mathbf{P}(o_{t+1}, o_{t+2}, \\dots, o_{T} | X_t=x_i )\n", "7eb42695220f42000f3557775678044a": "\\sin^2 2\\theta=4\\sin^2\\theta(1-\\sin^2\\theta)", "7eb49d6a118ef3c04e42825d41250126": "2 \\pi^2 R r^2 = 2 \\pi^2 \\,r^3", "7eb4b52f48d8aa91afe58b5419281eec": " Q = \\begin{bmatrix} 0.36 & 0.48 & -0.8 \\\\ -0.8 & 0.60 & 0 \\\\ 0.48 & 0.64 & 0.60 \\end{bmatrix} ", "7eb4d29e7ea5c21964f4a8debb48428e": "\\mathfrak{P}^{93}", "7eb4de325aab638abecb9f6aa29cea7b": "2mn", "7eb513438cd963c2626d0b3cd5a9f9b4": "\\sigma^2I", "7eb5221ffb69572faa1afc3298ce5796": "\\scriptstyle{\\Bbb R}^n", "7eb57e5ef1da2b85707f844aad9f1fe3": "d \\varphi = -u\\, dx -v\\, dy,", "7eb5996c271ab02369f4aa631444da60": "\\textstyle P- \\sum_{j=1}^{r}Q_jA_j=0", "7eb5f53a955fed1653fecc204653776a": "\\ \\tau_D=\\frac{3\\pi\\omega_{xy}^2\\eta}{2kT}(M)^{1/3}", "7eb62668852ebd76bac99f2cad9c7bf2": "f(x) \\in \\mathbb{Z}[x]", "7eb637a02276bfa50b9756dbe3e6103e": "\n{\\mathcal L} = \\frac{1}{16 \\pi G} (R - 2 \\Lambda\n+ \\xi B^\\mu B^\\nu R_{\\mu\\nu} )\n- \\frac{1}{4} B_{\\mu\\nu} B^{\\mu\\nu}\n- V(B_\\mu B^\\mu \\pm b^2) + B_\\mu J^\\mu\n+ {\\mathcal L}_{\\rm M}.\n", "7eb6aecf4b37603f0603ad572aba6a71": "\\left|\\alpha+\\tfrac{1}{2}\\right|\\leq8.4\\times10^{-8}\\,", "7eb712a4a13f35574fed82778d42234b": "2^{c_1}+2^{c_2}+\\cdots+2^{c_k}", "7eb71d5ea3204519e8aaa60b0bb9fab8": "\\mathcal{A}/E", "7eb7f7d20a097c3c3839cf628dacf6b6": "n_k = \\sum_{c=1}^v o_{ck}", "7eb806f1b38ad4417fbb6f8ea8347abc": "P_\\sigma(P_\\pi(\\mathbf{g})) = P_\\sigma(\\mathbf{g}')\n=\n\\begin{bmatrix}\ng'_{\\sigma(1)} \\\\\ng'_{\\sigma(2)} \\\\\n\\vdots \\\\\ng'_{\\sigma(n)}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\ng_{\\pi(\\sigma(1))} \\\\\ng_{\\pi(\\sigma(2))} \\\\\n\\vdots \\\\\ng_{\\pi(\\sigma(n))}\n\\end{bmatrix}.\n", "7eb81f78049f5dac35e00ba01cceb02d": "s_{i} = \\sum_{j=1}^{N} a_{ij}w_{ij}", "7eb8b1e3f99bded3488e8a085e2c174a": "\\frac{(pressure) \\times (radius)}{2 \\times (wall \\ thickness)}", "7eb90c6a5ad8afa31d76f4880f8191a8": "df_{n} = {(1 - \\sum_{i=1}^{n-1} C_{n} \\cdot \\Delta_i \\cdot df_{i}) \\over (1 + C_{n} \\cdot \\Delta_n )}", "7eb924b4254ec304a4395b0a787b2b85": "\\scriptstyle \\frac{2}{3}", "7eb9eb93d4de3ff85573a7b1660ae600": "a, a' \\in A ", "7eba378d97f182b066a00503221f0135": "x_i = \\cos \\left( \\frac {i} {n+1} \\pi \\right) ", "7eba50f82287b39b99d374a2845799bf": "p p \\rarr n_\\text{jets}", "7ebac686aa975fa8af7aeaf9017eeb3e": "\\mathcal P(X,\\mathcal A)", "7ebb74ae701d9d8c05e8de58b78000ec": "x_n < K", "7ebb76fea0f091efeda3a1d1be7a9c87": "C^\\infty_Z", "7ebb7c563020b56e056a915fd2d9fda3": "code_0 = 0", "7ebbc1b2962f02e952029ab8888e08ae": "\\mathbf{P}_\\beta = \\hat{\\mathbf{a}} +\\beta \\mathbf{e}_\\infty", "7ebbcd5413c3542ebbffd388663e7502": "[g]\\subset G\\;", "7ebbf56c0582881152b71e874ec96f20": "\\boldsymbol{\\beta}_s", "7ebc0a7af5dafc3877819b549d4fc7a0": "Z \\subseteq Y", "7ebc3b4556379840f93531169782ef9d": "\\max_{x \\in [-1,1]} \\left| \\prod_{i=1}^n (x-x_i) \\right|. ", "7ebc572a3d6c3c057c14d3ee25a94029": "a_{8}*b_{7}=r^2", "7ebc7c54171bdf105080abc3eaef5d6c": "f=\\pm f_0,", "7ebc8faee69781c797a007d3e7d4bf91": "\\frac{ \\partial^{i+j+k} f}{ \\partial x^i\\, \\partial y^j\\, \\partial z^k } = f^{(i, j, k)}.", "7ebca976b0ea0b8f0245e255a7e78a2e": "L_1, L_2, \\dots , L_n", "7ebcac68688c198ee5a4e6dbab61e151": " \\overline{ \\nu } = s ( \\frac{ \\gamma_1 } { 2 } )^{ 1 / 3 } ", "7ebcbc4cd74ad557fe5715cff9898b4d": "\\textbf{S}_{k} = \\textrm{cov}(\\tilde{\\textbf{y}}_k)", "7ebcc3a14e7c254b22170a3def8f131d": "\n\\langle H_{\\mathrm{kin}} \\rangle = \\Bigl\\langle \\frac{p^{2}}{2m} \\Bigr\\rangle = \\langle \\tfrac{1}{2} m v^{2} \\rangle = \\tfrac{3}{2} k_{\\rm B} T.\n", "7ebce7c73d493e3536213957dd578ad8": "\\gamma_1 + \\gamma_2 = z_\\infty + Z_\\infty.", "7ebd00a821766e52032bbf91bf21b591": "\\begin{bmatrix} \\Psi \\end{bmatrix}=\\begin{bmatrix} \\begin{Bmatrix} \\psi_1 \\end{Bmatrix} \\begin{Bmatrix} \\psi_2 \\end{Bmatrix} \\cdots \\begin{Bmatrix} \\psi_N \\end{Bmatrix} \\end{bmatrix}.", "7ebd18932b4fe2c347f0d5cb548f5110": "\n D_1 J^{8/3} - D_1 J^{5/3} + \\tfrac{C_1}{3\\lambda} J - \\tfrac{C_1\\lambda^2}{3} = 0\n ", "7ebd3247282760db1607480ccd6e48ab": " 0 \\rightarrow R^m\\rightarrow R^n \\rightarrow R \\rightarrow R/I\\rightarrow 0", "7ebd62318b192989af084948d27f7d5e": " 2b = \\sqrt{(p+q)^2 -f^2} ", "7ebd63243c1b2f4aa063a15c2e67227b": "25-125 = -100", "7ebd7aea33fce083c26646887733dd0e": "{\\mathcal L}_{xy}^i", "7ebdd2425eb8d7893e08dea25b8cb48f": "h_1 = 1", "7ebdeb4c28e9930fbf1cc22b9408e9f9": "X'= \\{x \\mid \\varphi_x^X(x) \\ \\mbox{is defined} \\}.", "7ebe035870c2891a92c1a95cb0512226": "10.4\\pm1.2", "7ebec43f8401a874ec579923d4306415": "y_{01} = y_{02} = Y_0", "7ebeeaf1655509ce9d391fc16b92afc7": "{T_2}", "7ebeed492de4f86738b4ec1bdaa08948": "\\hat{x} = C_{XY}C^{-1}_{Y}(y-\\bar{y}) + \\bar{x}.", "7ebf2703b61f6b35ba784cce3c024d13": "B_V(v, g(w)) = B_W(f(v),w) \\quad \\forall\\ v \\in V, w \\in W .", "7ebfb1afca5aebb7d4f2061ae940eb8f": " f_1 \\ldots f_N ", "7ebfbb0066b1610ded0f1f837a995f0d": "\\rho(D^{-1}R) < 1. ", "7ebfea7b6532f6b9071a96764af5d0f7": "\\succcurlyeq", "7ebffb2dbe618e1039850b215110ae54": "T_{ach}", "7ec080be3e6fb891a85cd12d949e58ac": "cm^2/Vs", "7ec0f6914f9270728fb01752eead6422": "{x^2\\over a^2}+{y^2\\over b^2}+{z^2\\over c^2}=1. ", "7ec1093417bd1db710471e2da19c27ae": "v^T A^{-1}u", "7ec133ac6e1b481b2e2ae0609f9ee400": "\\langle u,v\\rangle=0", "7ec147b10f744b589ec7ebc97c59ca73": "\\Sigma^m_n", "7ec14b02d703576ed35a11c2f96dac3a": "\\ldots d_3d_2d_1d_0", "7ec1a6e14d74a64eab7246e327d14bd6": "\\text{sign}(p(-x))", "7ec1b89fd3c938bfb5970afcd717897c": "a=r_0", "7ec266e97e30b292a3b0306f8bc9e9dd": "\\phi:C\\rightarrow L", "7ec29f90c562b8cfc8deecc70ea88a35": "xy = 1", "7ec2c17e44a0e88609df4325186e8c98": "[(A\\to(B\\to A))\\to([(\\neg C\\to(D\\to\\neg E))\\to[(C\\to(D\\to F))\\to((E\\to D)\\to(E\\to F))]]\\to G)]\\to(H\\to G)", "7ec2ef958f8cf5ea1e8b87f0cf639c03": " u(-b j \\geqslant 1}\\left( z_i-z_j \\right)^n \\right] \\prod^N_{k=1}\\exp\\left( - \\mid z_k \\mid^2 \\right)\n", "7ec8b4cba3893e7b66ad9167c5c2663d": "\\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} e^{-ikt} d\\mu(t) = \\alpha_k", "7ec929852a94be37cdf8c5cc0ae42f1e": "w'\\Vdash'p", "7ec99f6a954a56c1775297e5bccfe0fd": "(k, h)", "7ec9b53750bc84514ebfcf25cdc7f88f": "\n \\operatorname{csgn}(z)= \\begin{cases}\n 1 & \\text{if } \\Re(z) > 0, \\\\\n -1 & \\text{if } \\Re(z) < 0, \\\\\n \\sgn(\\Im(z)) & \\text{if } \\Re(z) = 0\n\\end{cases}\n", "7ec9c592e9e74d3e551a11c161448c90": " 5 c_1(X)^2 - c_2(X) + 36 \\ge 12q ", "7ec9e839d1fd4b7796b8077155195f4e": " W(x,0)= \\frac{\\exp(-\\beta H(x))}{\\int dx' \\, \\exp(-\\beta H(x'))} \\;, ", "7eca27089d914a1d9869ecc8330a4ef8": "T\\cup S", "7eca2dc39e55f1514b2dfed7e50bbbda": "P_{j}H^{-1}", "7eca47a8bac848e8d4ca5c1490e26373": "Q=\\begin{pmatrix}-\\alpha & \\alpha \\\\ \\beta &-\\beta \\end{pmatrix}", "7ecad0dda5d1f132c028af12e5a5b10e": "\\hat{\\beta} = \\frac{\\bar{X}_P - \\bar{X}_N}{\\sqrt{\\frac{2} {K} ((n_P-1) s_P^2 + (n_N-1) s_N^2)}},", "7ecade293c739b19e441a9e183948ebc": "\\Delta : C \\to C \\otimes C", "7ecae2cb74531e572dae2569f704edb4": "x, y \\in X \\cup Y ", "7ecae6fa661168ce4cc147fc212fea5a": "M_{A}", "7ecb15754430943500d7d1aaa180f02a": "A^{[d]}", "7ecb2f8300d40d772827f0c127862a2f": " d_{m} ", "7ecb482372a13f9a76c565d2cb90cb0d": "HR/9IP = 9 \\cdot \\frac{HR}{IP}", "7ecb89bbe031a67abbcf95b11258b441": "(Q,\\Sigma,T,q_0,A)", "7ecbd78d74f2132324a6d059aa5a1bb2": "\\textstyle -a", "7ecbf0cff44024dfe7c30cc0cb6b4cb4": "Y^{(\\delta)}", "7ecc24f49d5b39549194558d8e401a8e": "\nn = 12 \\, \\log_2\\left({\\frac{f}{440 \\,\\text{Hz}}}\\right) + 49\n", "7eccc7ccae051c409f13c8806bcc44de": "\n= { {\\alpha (1-\\alpha)^k \\times {{1} \\over {1-(1-\\alpha)}}} \\over { { {\\alpha} \\over {1-(1-\\alpha) } } } }\n", "7ecccb6031c45e07f2b9b48e9be2582a": "g(x_1,x_2,x_3) := f(\\mathbf{r}) = f \\left (x_1\\frac{\\mathbf{a}_{1}}{a_1}+x_2\\frac{\\mathbf{a}_{2}}{a_2}+x_3\\frac{\\mathbf{a}_{3}}{a_3} \\right ).", "7ecda1e6e0b3c196e60573032907aca0": " f^{-1}(y) = \\arctan(y)", "7ece7f63f4c7d43cf3c8b553bcf5c4c0": "\\sqrt {\\Delta x^2 + \\Delta y^2} ", "7ecef43ca83424cbfe97a8b7b4ac411d": "\\operatorname{gr}_I R = \\oplus_{n=0}^\\infty I^n/I^{n+1}", "7ecf0a98482aa95758115230b1b9879a": "\\lim_{n\\to\\infty}x_n = \\infty", "7ecf2ae8779b2e0c822ead9d29c58fa5": "\\Box\\phi = \\frac{8\\pi}{3+2\\omega}T", "7ecf364093a8c7fd6d4aa99b425aa0e5": "(S_i)_{i \\in I}", "7ecf4e078ad036e8c1eb5e8831cb86d7": "Y^\\prime_{601} = 0.30R + 0.59G + 0.11B\\,\\!", "7ecfb3bf076a6a9635f975fe96ac97fd": "NaN", "7ed031a98ee97101f6bfec815af8e46c": "f(x_1)+f(x_2) = f(x_1+x_2)", "7ed05ea8f5ebeab3ae9bc9d7572b1d98": "\\textstyle \\phi", "7ed0989a5cbf5b99e71628aa701c8b7f": "\n\\; \\Phi(A) = \\sum_{i=1}^{nm} V_i A V_i^*\n", "7ed09c545c4995ed65e61bcdfe56298f": "\\mathbf{J}^{23} = \\left[\\begin{matrix}\n0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{matrix}\\right].", "7ed0c433c9d3e2b75e0310fca668fc3a": "k_L=\\frac{L_m}{\\sqrt{L_1L_2}}", "7ed0f8875ad751c776dcab8521f20088": "\\textstyle{\\frac {\\log(20)} {\\log(3)}}", "7ed1032eacecb2472a4cc0ca852b8c16": "\n(4) \\qquad \\lambda'(T)=\\Psi_x(x(T)) \\,\n", "7ed1094dc2fc718eb9773d7d7d3783ee": " R_0", "7ed117ce0a695d5da6b70395770703d7": "F=K_0, K_1, \\ldots K_{r-1}, K_r=K", "7ed13453e666d987ffdb7fdc798c7c3d": "\\displaystyle{T_K^*\\varphi +{1\\over 2}\\varphi = f,}", "7ed17b2f68b856bf28dae67628f900c5": "\\Lambda^k(A)", "7ed1bb5680dbcfb43d95047a072b679c": "\\|A^k\\|^{1/k}", "7ed1d4b9a0ac2d38100f944e5bc20853": "\\|f\\|_p \\equiv \\left({\\int_S |f|^p\\;\\mathrm{d}\\mu}\\right)^{\\frac{1}{p}}<\\infty", "7ed23dfd6639dcb9cdc7514cc47f5d0d": "(n_i)^2", "7ed242fd576a9555c0bdf0ff38cc9e7a": "L_1 \\backslash L_2 = \\{w \\ | \\ \\exists x ((x \\in L_1) \\land (xw \\in L_2))\\}", "7ed24d2e366434fcfab646684e9f9eeb": " Y_{i} ", "7ed2c572f037da0e7f2990b88e480f57": "EI \\frac{d^4 u}{d x^4} = w(x).\\,", "7ed2f1f095f7acb7bd4b612cd2c39d31": "k_{1s}, k_{2s}, \\dots, k_{Ns}", "7ed320c83d32745ff5a7bfb2f9c1f7c4": "-3/\\theta_0", "7ed41d5a8561b1d8c80659970c361a90": "\\|x\\|^2 = x^Tx = 1.", "7ed436cf40882217ca375acb737ab4cd": "-a^n", "7ed44befdc1b84e2ff744bf18965ecb9": "C_n^2(z)", "7ed45037db22de2e9e42a5d5d85eac02": "f^{-1}(J(f)) = f(J(f)) = J(f)", "7ed4aba1413fb1972fb0dc8e8ec0c6cd": " \\frac{24n(n-1)^2}{(n-3)(n-2)(n+3)(n+5)} ", "7ed50e6e5d3cff86ff4f35034ef4de9e": "\\frac{1}{3} \\pi\\, r^2 h ", "7ed54ae5dc85189315580dae6cbd2357": "S_{RBN(n)} = S_{RNB(n-1)} + 2\\cdot S_{RRB(n-1)} + 1", "7ed56e2831a40c7f89b9a160a5b8e692": "W_{2A}(x)", "7ed57784855af9d9705a0258ddd8a3d2": " y(t) \\ ", "7ed59e8311e74dbe167b234b5da7585c": "\\pi^{kl}", "7ed5ed55da835510e9923a5c025c95de": "\\hat{h}(\\xi) = \\hat{f}(\\xi)\\cdot \\hat{g}(\\xi).", "7ed5f0199b7658feb04b595b8e7ff206": " {\\pi\\over 4} = 4 \\arctan \\left({1\\over 5}\\right) - \\arctan \\left({1\\over 70}\\right) + \\arctan \\left({1\\over 99}\\right)", "7ed5fdc932540f7a680baf0eeef626fa": "[u,u']=u\\cdot u'-(-1)^{[u][u']}u'\\cdot u", "7ed647523c738c5edae2205ea7a028ab": "\\mathbf A_{1\\square 2} = \\mathbf A_1 \\otimes \\mathbf I_{n_2} + \\mathbf I_{n_1} \\otimes \\mathbf A_2", "7ed662fd97bc4c3bd6af10bf288276d6": "\\scriptstyle \\varphi(x)", "7ed71156c23af8ef91616324e1448a99": "g_x(u,v) = g_x(v,u) \\in \\mathbb{R}.", "7ed71b9e9007cb3e1bbe12430ada1881": "\\frac{dT}{dt} = \\frac{p}{c \\cdot {\\rho}}.", "7ed79fdf3f1b89cae19dc6c0462aa591": "\\frac{\\partial \\boldsymbol{k}}{\\partial t}\\, +\\, \\nabla \\omega\\, =\\, \\boldsymbol{0},", "7ed7c8da1a6bcec99c94721e9951ed15": "\nW = \\omega_{1}(\\varphi_{1}) + \\omega_{2}(\\varphi_{2}) + \\cdots + \\omega_{s}(\\varphi_{s}),\n", "7ed7c942bd7e65ece692a97955a83385": " I_\\mathrm{m} = \\iint \\mathbf{j}_\\mathrm{m} \\cdot \\mathrm{d}\\mathbf{S} \\,\\!", "7ed8055b3e91b84a54b96068c9bfc648": "R_1=\\frac{L_1}{k_1A}", "7ed82edab6d6629e7a3842d3fa70e949": "\nF(r) = Ar^{-3} + Br\n", "7ed896c95b834da06fbccfaed08ab028": "n_\\text{upper} / n_\\text{lower}", "7ed8bad55ce6ab8fc063b1b8450cebdc": "H_0(a, b) = b + 1\\,\\!,", "7ed8e04cce121bc7f70fa2682eff22c4": "\\frac{AF}{FB} \\cdot \\frac{BD}{DC} \\cdot \\frac{CE}{EA} = 1.", "7ed8ec072307a39699c889f9db069ef6": " \\xi_x\\, ", "7ed8eddac174e1ade04f05aa5cb45cb8": "\\frac{J_1 \\qquad J_2 \\qquad \\cdots \\qquad J_n}{J}\\ \\hbox{name}", "7ed9144de677b65844142d267648242e": "\n (Y^*-X^*b)'(Y^*-X^*b) = (Y-Xb)'\\,\\Omega^{-1}(Y-Xb).\n ", "7ed91de90440d9bdd0b1a057d90bc01c": "\\pi = 16\\arctan\\frac{1}{5} - 4\\arctan\\frac{1}{239}.\\!", "7ed974668de91e1c5af8c06b343b508b": "X Y", "7ed9abff4dafd78d08e616c899412e92": "\\infty", "7eda0593951309264e60d223b625287b": " [x,x]=0\\ ", "7eda09d262d6cd93844b52e3d76985b9": "d[V,W]", "7eda0c97d936464de6753ae0b52519ee": "a_{ij} = 1", "7eda2ad4d50a0a49eea86e526e0623e9": " \\lambda \\phi(x) = \\int_{-1}^{1} \\frac{\\phi(x)-\\phi(y)}{|x-y|} dy ", "7eda7939c4b49de78b55fa7e79a1cb17": "pg(t,s,\\alpha)", "7eda943fc0696f02ac40aa87708421a1": "m\\equiv n\\pmod{p-1}", "7edb061493143c5117cf6d31dd685819": "\\neg P(c)", "7edb3ef55c9cff8c7c12a77648800866": "\\left (\\frac{\\mbox{Accounts Payables}}{\\mbox{Purchases}}\\right)\\mbox{365 Days}", "7edb6c8ce7e770055e99682414604931": "\\displaystyle7.02", "7edb92c658133b3d01c58b31866a5cb5": "1/(2T)", "7edc13df7337fbf4210edbf86eb3d60d": " \\overline{BC} \\cong \\overline{EF}\\, ", "7edc803be909546d0b2a6582e4e0029b": "\\frac{p(M_1 |D)}{p(M_2|D)}=\\frac{p(D|M_1)}{p(D|M_2)}\\frac{p(M_1)}{p(M_2)} = B_{1,2}\\frac{p(M_1)}{p(M_2)}", "7edce3c7a77ae5dbc6305b7e7729e94f": "|n,-\\rangle= -\\sin \\left(\\frac{\\alpha_n}{2}\\right)|\\psi_{1n}\\rangle+\\cos \\left(\\frac{\\alpha_n}{2}\\right)|\\psi_{2n}\\rangle", "7edd2a98380330cf1e023c9d245daf7c": "f(x) = \\frac{(a/b)^{p/2}}{2 K_p(\\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}", "7edd76532992c57638e37cf3f5ea53dc": "k_a", "7edd974b0028ac7e4733b7517c4b1117": "x\\in\\mathbb C", "7edda8698c90012997425a763afdbde4": " \\mathrm{dim}[D(a)] = \\mathrm{dim}[D^{(1)}(a)] + \\mathrm{dim}[D^{(2)}(a)] + \\ldots + \\mathrm{dim}[D^{(k)}(a)] ", "7eddb8e9304e5a2574ad652c2083c6c1": "e x < -a\\,\\!", "7eddd91ec64662dc5c04eb46ac8561df": "\\Omega_{i,i+1}=\\Omega_{ii}=0", "7eddfcd7784f4400b82946e9e653cf3b": " D_{ijkl} = D_{jikl} ", "7ede2eafbe465bdedfc6ee4d8327fe85": "\\Omega^{framed}_2 \\cong \\pi_m(S^{m-2}) \\, (m \\geq 4) \\cong \\mathbb{Z}_2", "7ede5cee9f1adff36c74e44b5a2f4ce2": " \\hat{T}\\omega=\\sigma^{1/2}T^*\\Bigl((T\\sigma)^{-1/2}\\omega(T\\sigma)^{-1/2} \\Bigr)\\sigma^{1/2}, ", "7ede64d1404a725f12671ba5f638490e": "\nn_j=\n\\begin{cases}\n n_j^{(-n)}, & \\text{if }j\\not=k \\\\\n n_j^{(-n)}+1, & \\text{if }j=k\n\\end{cases}\n", "7edeb37f1dd4ce4fd6d41383c0e3bcb9": "h \\le \\left\\lfloor \\log_{d}\\left(\\frac{n+1}{2}\\right) \\right\\rfloor .", "7edee43e831204bb2df24b072b2943ec": "35^2", "7edf3b37873a019c873d8ef3752f3f16": "d(x,y) = w(x-y)", "7edf698e055b942a3948dc7a9639b46e": "\\varepsilon = -i,\\quad a = 2mn,\\quad b = -\\left( m^2 - n^2 \\right).", "7edf7ddcf11f1370927f89bd004f9573": "E=\\hbar\\omega,\\quad p=\\hbar k.", "7edfb6f7ff33f2b59e3ff72093407884": "3 \\times 10^{-7}", "7edfd02ad09de162ed19c6ed3dea93c7": "\\log M_{t}^{D}-\\log M_{t-1}^{D}=\\sum_{j=1}^{n}s_{jt}^{*}(\\log x_{jt}-\\log x_{j,t-1}),", "7edffe8e8702b55b57f9290f84d203bf": "\\boldsymbol{\\nabla} \\cdot \\mathbf{F}_t\\left(\\mathbf{r}\\right) = 0", "7ee0182305e45fe0a02f4662fd6a414b": "\n\\zeta(s) = \\frac{\\lambda(s)}{1-\\frac{3}{3^s}}\n", "7ee0379461c72b3830e37cfbd499db9a": "\\!\\gamma_{j-1}' = \\gamma_{j-1}\\ast\\gamma_j'", "7ee038165372654ac7887230cf7cc7fc": "E\\{\\hat{x}\\} = E\\{x\\}.", "7ee03cafcd6d24f2f952a9bf0a1f0e5d": "SL(m,\\mathbb C) ", "7ee0602bbda17e1c782d46e2a681efe6": " \\mathbf{\\hat{r}}_{12} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\mathbf{r}_2 - \\mathbf{r}_1}{\\vert\\mathbf{r}_2 - \\mathbf{r}_1\\vert} ", "7ee10602abd8b6c3df44a6c5809f45ab": "P_{total}=P_{static} + P_{dynamic} = P_s + \\frac{1}{2}\\rho V^2", "7ee135cd236881ab4eebb275dd0d4bd1": "z^\\star\\,", "7ee1401cea5286aaece44ff8d8b0570d": "v^2={v_{esc}}^2+{v_\\infty}^2", "7ee1503e996369827deb51c6e9ac02ae": "\\scriptstyle \\sigma_{xy}", "7ee1b5f0fa270b42c87462c785a900e9": "y = 3 x + 2 a - 2.5 \\sqrt{b}.", "7ee218225f5f7e9c6f0be39629226343": "P \\left ( { {a,b}{|}{A,B} } \\right ) = \\sum_{\\lambda} p(\\lambda) \\; P \\left ( { {a}{|}{A,\\lambda} } \\right ) \\; P \\left ( { {b}{|}{B,\\lambda} } \\right )", "7ee22fee97ba9816d38aae7be4936765": "\\eta_{\\mu\\nu} =\n\\begin{bmatrix} -1&0&0&0\\\\ 0&1&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{bmatrix}\\ .\n", "7ee2935277c31585077aae918a5fa752": " -2~r~\\cos\\theta \\,", "7ee2ed3ffcb08980e638555b24b4deef": "I({\\vec x}) = \\int \\mathrm{d}\\vec k \\ S( {\\vec k}(t) ) \\cdot e^{-2 \\pi \\imath \\ {\\vec k}(t) \\cdot {\\vec x} } ", "7ee33251939fe6984570f8d5dc377eba": "X:\\Omega\\to \\mathbb{R}", "7ee39d5f85707f39009cecba6639f2db": "\\theta = \\angle q", "7ee3cca34e20e397c46f309948a0b9ab": "\\frac{q^2}{4 \\pi \\epsilon_0 L_1^2}=mg \\tan \\theta_1 \\,\\!", "7ee3e587ff2066d4494abcebb9cde693": " pi \\approx 3.1416 ", "7ee41106323db98e1fc946644a63f3cd": "E[\\hat{m}(x,\\beta)]=m(x,\\beta)", "7ee423bcbf6476052185cb143942c9c1": "{\\gamma}", "7ee42d080c0ec5680d56bb50a75f87e1": "\\omega^{2^p} = 1.", "7ee49a1441f2a0f0beaf188649ed1e6d": "\\scriptstyle\\cap", "7ee4cf454a3ac5cd35aef7ad347f9d23": "A^\\circ := \\{y \\in Y : \\sup_{x \\in A} |\\langle x,y \\rangle | \\le 1\\}", "7ee4e0da212ed86f8189dba8127b049d": "m = (\\prod p_i^{\\beta_i})(\\prod p_i^{\\gamma_i}) = (\\prod p_i^{\\beta_i/2})^2(\\prod p_i^{\\gamma_i/3})^3", "7ee4e6dbb35122ecbc064b8cc76be0c3": "\\theta(t^{}_0,j)\\sim \\mathrm{Normal}(\\theta^{(m)},b a^{m-1} \\Phi)", "7ee4e957f84068f00ce423fc92dd1abe": "f, x_3g_0-g_3, \\ldots, x_ng_0-g_n", "7ee4f4ab25b0c433084ed889ce70300d": "1%-4%=-3%", "7ee5a1c9cce3de1bc96112d8cd9035bf": "\\int_{-\\infty}^t f(u)\\,du =\\tfrac{1}{2} + t\\frac{\\Gamma \\left( \\tfrac{1}{2}(\\nu+1) \\right)} {\\sqrt{\\pi\\nu}\\,\\Gamma \\left(\\tfrac{\\nu}{2}\\right)} {}_2F_1 \\left ( \\tfrac{1}{2},\\tfrac{1}{2}(\\nu+1); \\tfrac{3}{2}; -\\tfrac{t^2}{\\nu} \\right)", "7ee5a2b9ef314e8355e32a430d504078": "\\log |S|+O(1)", "7ee5d9a602095e406b8f5d99a23d5f58": "\\forall i < n \\; \\left(x \\equiv a_i \\pmod{m_i}\\right)", "7ee5edfd80d72701a92f0e86e57a1f82": "\\sin\\frac{\\gamma_1}{2} = \\frac{a_1}{2r_u}.", "7ee5f2538f4d47ce523f5d3483c25f07": "\\varphi(\\beta / \\alpha)\\,\\!", "7ee696abcb747b369f4b7ba21391b605": "\\begin{pmatrix} \\ \\ & \\alpha_{k+1,m+1} \\\\ \\ \\nearrow & \\ \\\\ \\alpha_{k,m} \\longrightarrow & \\alpha_{k+1,m} \\end{pmatrix}", "7ee6a78dfa6222939043a4f66bf98577": " {j\\choose m}={j+1\\choose m+1}-{j\\choose m+1} \\!", "7ee71342dc33e76066c0f6464c347483": " {1 \\over \\pi t}", "7ee715517e82c076f927d1eb5524c23e": "\\dot Q_L\\ge 0 ", "7ee769100bb5bbdf5dba1236610abb8d": "\\scriptstyle{s^2 = \\sum (x_i - \\bar{x})^2/(n-1)}", "7ee7bb68eb626b7b1d61f992b0cbccee": "\\Gamma_0(r) := \\left\\{\\begin{bmatrix} a&b\\\\c&d \\end{bmatrix} \\in \\Gamma : c\\equiv 0\\mod r\\right\\}.", "7ee844336b0aba11acd6e747614662b4": "x^2\\overline{y}zyz^3\\overline{x}^2. \\,", "7ee8a07778373f86a5d28da57420685d": "\\lambda_1 \\approx \\lambda_2 >\\!> \\lambda_3", "7ee9b2170cda764eb5f30eb6e05d4b23": " \\int_a^b \\left[ \\frac{\\partial F}{\\partial f} - \\frac{\\mathrm{d}}{\\mathrm{d}x} \\frac{\\partial F}{\\partial f'} \\right] \\eta(x)\\,dx = 0 \\ . \\,\\!", "7ee9c1b3e8051295193b0b4a7a1da2c9": "m(\\varphi) = a\\left(E(\\varphi,e)+\\frac{d^2}{d\\varphi^2}E(\\varphi,e)\\right),", "7ee9f9eaa689abe9ac59454748671e1e": "\\deg(v) < \\deg(u) \\leq g = 2", "7eea2ec737c6c3b4cd25676351822a73": "r_\\mathrm{O}", "7eea3f3da76dbabcd43e4afb8476fc57": "(x\\ ,\\ y)", "7eea886e7d7f41f736750801fc77f825": "(\\mathcal{K}, h)", "7eeaf609c176da9158977a97f5c9fb94": "\n\\Lambda =\n\\begin{pmatrix}\n\\lambda_1&0\\\\\n0&\\lambda_2\n\\end{pmatrix}\n, \\quad\nJ = i\\sigma_z =\n\\begin{pmatrix}\ni&0\\\\\n0&-i\n\\end{pmatrix}\n, \\quad\nU = i\n\\begin{pmatrix}\n0&q\\\\\nr&0\n\\end{pmatrix}.", "7eeb03c68544d8fcc617c753cfee25f6": "L(s,E)=\\prod_pL_p(s,E)^{-1}\\,", "7eeb7b7177b6ae7d907fc57bdbdff83a": "w_1 \\dots w_n", "7eebc4fa91ce3b08c6bb998ec8636459": "t_{bd}", "7eec6d4a22902ad011f974a9e711f8cf": " \\{\\phi_i\\}_i ", "7eecb89f248def30b5d8d6f5eb5605e4": "\\scriptstyle \\rho\\,", "7eeccc38782edd9bfcd2025a42cd2e0f": " (\\tfrac{ 1}{5},\\,\\tfrac{ 2}{5},\\,\\tfrac{ 3}{5},\\,\\tfrac{ 4}{5}),", "7eed093928b895e661ac51330bdcb825": "(A - 2 I) \\begin{bmatrix}\n0 \\\\ 0 \\\\ 1 \\\\ -2 \\\\ 0\n\\end{bmatrix} = \\begin{bmatrix} \n-1 & 0 & 0 & 0 & 0 \\\\\n3 & -1 & 0 & 0 & 0 \\\\\n6 & 3 & 0 & 0 & 0 \\\\\n10 & 6 & 3 & 0 & 0 \\\\\n15 & 10 & 6 & 3 & 0\n\\end{bmatrix} \\begin{bmatrix}\n0 \\\\ 0 \\\\ 1 \\\\ -2 \\\\ 0\n\\end{bmatrix} = 3 \\begin{bmatrix}\n0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0\n\\end{bmatrix}\n", "7eed43b31c64879b0d333685def40f45": "\\mathbf{g} = (g_x,g_y,g_z)", "7eed82e8a2367cbf498f4f495a4de141": "\\scriptstyle A \\circ . \\times B \\circ . \\times C", "7eed9d11be4262f95ca50861e2289f66": "a^2 = b^3 = (ab)^7 = 1,\\,", "7eed9e91618a1195df42cf0f6da702f3": " X = v_M/(\\sqrt{2} \\sigma) ", "7eedbd818609d4447f79ecf4894058da": "T_{2b}", "7eee638575ad856000ed6c69fe0375e7": " x \\in X ", "7eee6cb4494c0d74613489242047c9a2": "\n\\begin{bmatrix}\n-2 & -1 \\\\\n-1 & 2\n\\end{bmatrix}\n", "7eee6d9e164cda11ecf8eabf4eab7884": "d(a, b) = d(b, a) ", "7eee733b7e6e2b2a5e6a05c9420fbbf2": " \\delta_1 \\geq ...\\; \\delta_p \\geq 0 ", "7eee9070638d63b1e67539b613065d30": "\\Omega^2X", "7eeeacd7a46d1e7f725cba6fbb2bfec6": "\\mathbf{E}\\mathbf{R}=1", "7eefa71c8905f8bff227878c726d7b5a": "\n\\frac{\\langle C_1,s\\rangle \\longrightarrow s'}\n{\\langle C_1;C_2 \\,,s\\rangle\\longrightarrow \\langle C_2, s'\\rangle}\n\\quad\\quad\n\\frac{\\langle C_1,s\\rangle \\longrightarrow \\langle C_1',s'\\rangle}\n{\\langle C_1;C_2 \\,,s\\rangle\\longrightarrow \\langle C_1';C_2\\,, s'\\rangle}\n\\quad\\quad\n\\frac{}\n{\\langle \\mathbf{skip} ,s\\rangle\\longrightarrow s}\n", "7eefb034543efb5e816d1c2f31434e09": "H()", "7ef0881277daa1795257c775f3cf1c67": "K_{k-1} (\\tilde M)", "7ef0c47326ad2d1e32e99268af43b6f7": "\n\\begin{align}\nW & = {n \\choose 1} (n-1)! - {n \\choose 2} (n-2)! + {n \\choose 3} (n-3)! - \\cdots + (-1)^{p-1} {n \\choose p} (n-p)! \\cdots \\\\\nW & = \\sum_{p=1}^n (-1)^{p-1} {n \\choose p} (n-p)!. \\\\\n\\end{align}\n", "7ef16cac30fd19958cfff83f258d1967": "\n\\tan \\varphi = \\frac{y}{x}\n", "7ef16e639f2f545c27157a999d6d2ec5": " \\begin{array}{ccc} A & \\hookrightarrow & B \\\\ \\downarrow & & \\downarrow \\\\ K & \\hookrightarrow & L \\end{array} ", "7ef194865cd688c0756751d68b2d8a00": "a^2 - 67b^2 = k", "7ef1c19f19642b3fddcbb99eab6a5ec0": " \\operatorname{cov} (X_i,X_j) = \\operatorname{var} (\\operatorname{E}(X_i|F_\\bold{X})) = \\operatorname{var} (\\operatorname{E}(X_i|\\theta)) \\ge 0 \\quad\\text{for }i \\ne j.", "7ef1d2c0b591f12b3ddeaf7913d40528": "\\mathbf{r}_{12} = \\mathbf{r}_{2} - \\mathbf{r}_{1} ", "7ef1f4366d0a3d835b6c92e9ca3e71d7": "\\|x(0)-x_e\\| < \\delta", "7ef239218e1988a0ca57fc2bc1f5dd4f": "a(u, v) = L(v) = a(u_h, v)", "7ef24225708da0aef679313f4b4f35dd": " F_{perf} = exp (-b.D^*)\\,", "7ef2aa29a4d85826210b2cd47568e10e": "C = \\frac {u\\,\\Delta t} {\\Delta x}", "7ef2e6f2c6f412ed1bdfedd7a937cc5e": "\\,{}^{x}a \\approx a^{a^{(x-1)}}", "7ef312bf20c312c6caef7b1df6540542": "p_k = \\prod_{n=1}^{k} \\frac{2n}{2n - 1}\\frac{2n}{2n + 1}", "7ef319d14db5b0910f48097638ede119": "G \\in \\mathcal{G}(V,E,F)", "7ef34477386bf86b01b17f77c574e8b7": "\\hat{t}\\,", "7ef37f9c83582b1d1effba430265c2d7": "P-P_0=B \\frac{\\rho - \\rho_0}{\\rho_0}", "7ef3afa72e0f925827bf0e2956b3a987": "{D}_{7}^{(1)}", "7ef3b72e518ad46e9c802db5d7c9173c": "A^* = \\overline A^\\mathrm{T}", "7ef3c45969eb6ec9036b3be99d8f616b": "X = 100\\;\\frac{H_s^L - H_a^L}{H_a^V - H_a^L}", "7ef3c465ca216059e680e969e9b873b7": "\\frac{x^2}{(p+q)^2}+ \\frac{y^2}{q^2}= 1", "7ef3d4583591e9ea763fc01e36ba36b2": "D_1 = (u_1,v_1)", "7ef3ee4b179813af41eecaec8817293b": "B(\\mathbf{u}, \\mathbf{v}) \\le C \\|\\mathbf{u}\\| \\|\\mathbf{v}\\|.", "7ef41df8f970c04c5eec20ee8b669eb3": "\\begin{align} \\mathit{dr}(n) &=3 \\Leftrightarrow n=9m+3 & \\ \\text{for}\\ m=0,1,2,\\cdots,\\\\ \\mathit{dr}(n) &=6 \\Leftrightarrow n=9m+6 & \\ \\text{for}\\ m=0,1,2,\\cdots,\\\\ \\mathit{dr}(n) &=9 \\Leftrightarrow n=9m & \\ \\text{for}\\ m=1,2,3,\\cdots.\\end{align}", "7ef4244345786674120f978b9117d079": "f_L=0\\,\\Leftrightarrow\\,f_H=BW", "7ef48393db9737b87ab5eaa7a7193551": "\\sigma^2 = \\frac{t}{\\delta t}\\,\\varepsilon^2,", "7ef4b5285e553cd20ca743a19453e6b0": "\\beta_{w_{i-n+1} \\cdots w_{i -1}} = 1 - \\sum_{ \\{w_i : C(w_{i-n+1} \\cdots w_{i}) > k \\} } d_{w_{i-n+1} \\cdots w_{i}} \\frac{C(w_{i-n+1}...w_{i-1} w_{i})}{C(w_{i-n+1} \\cdots w_{i-1})} ", "7ef5100b00ad68398c001a2d9d4bcf29": " y_2'=2y_1-2y_2+\\sin(t)", "7ef54d91c352b9e58692392cb1af03a8": "a = 8, \\, b = 2, \\, f(n) = 1000n^2", "7ef5f92d285657af1bb5081fab3721ac": "\\forall m \\forall n [m \\cdot Sn = (m\\cdot n)+m].", "7ef6512eb2c4ae25fc1acf787afbcba2": " {p}_{n} \\neq p ", "7ef6a2ae6a14298952508c58824c89b8": "\nS \\ \\stackrel{\\mathrm{def}}{=}\\ k \\ln \\Omega,\n", "7ef6b88a680f80484db46ac231bc4f04": "\\frac{1}{2} \\, {C^{ab}}_{mn} \\, X^{mn} = \\lambda \\, X^{ab} ", "7ef6ded1558dba09b633f0d5e39bc9d0": "\\boldsymbol{\\lambda}", "7ef7049a33fe69cd50ca9ac64d4f8712": "X = -\\frac{dE_{r}}{dx}", "7ef746fec75a6c101e9f3b23f8b82aba": "F^{**}", "7ef76cd4e53ba60ed88b0d67c86e0ecb": "\\scriptstyle \\exp(1)", "7ef78ad19ddc12978faffb1ab2f34c94": "\\scriptstyle\\ll 1", "7ef7e67117f57235e1b36cb66ddf791a": "r \\ge 0", "7ef82a3c9ee61038b3a4b5736d3f638e": "\\frac{\\pi}{2}-\\alpha\\,\\!", "7ef8ac3b14947689be373476c0566e98": "\\ln(ab)=\\ln(a)+\\ln(b). \\,\\!", "7ef8c3691bebd53fe4d140152bbe9f97": "E^2=m^2c^4", "7ef8e9b2a040ba1f914e1a1c40eab094": "C_r \\ = \\frac{C_{min}}{C_{max}}", "7ef8f8a53e35537bb2fda04bd6732325": "\\ \\bar{\\Gamma}=q^2D_z\\,", "7ef911ef8c2382a8339fb1e4bfb02eee": "\\tfrac14", "7ef942548f0728b8e7ba73ce400ce79d": "y^2+x^2=1.\\,", "7ef9db6e43b9e21d28d0bcf4a4f4d5cc": "\n \\boldsymbol{\\sigma} \n = \\cfrac{2}{\\sqrt{I_3}}~\\left[\\left(\\cfrac{\\partial W}{\\partial I_1} + \n I_1~\\cfrac{\\partial W}{\\partial I_2}\\right)~\\boldsymbol{B} - \n \\cfrac{\\partial W}{\\partial I_2}~\\boldsymbol{B}\\cdot\\boldsymbol{B}\\right] + \n 2~\\sqrt{I_3}~\\cfrac{\\partial W}{\\partial I_3}~\\boldsymbol{\\mathit{1}}~.\n ", "7efa4dd9de2e1f931721050c91156baf": "{\\displaystyle}y_{3}=\\cos({\\alpha}_{1}+{\\alpha}_{2})=\\cos{\\alpha}_{1}\\cos{\\alpha}_{2}-\\sin{\\alpha}_{1}\\sin{\\alpha}_{2}=y_{1}y_{2}-x_{1}x_{2}.", "7efae0f070bd2f91e1ad49bbfd3e1da0": "\nN(\\lambda)= (2\\pi)^{-d}\\lambda ^{d/2}\\mathrm{vol} (\\Omega)\\mp \\frac{1}{4} (2\\pi)^{1-d}\\lambda ^{(d-1)/2}\\mathrm{area} (\\partial \\Omega) +o (\\lambda ^{(d-1)/2}).\n", "7efb94839900315410f43f79c786c577": "\\delta = \\omega - \\omega_{0}", "7efb964d7454d6bb78c9bb33523635b3": "''T''_C", "7efba618929c386ca74b11e3681aedcc": " \\begin{bmatrix} \\ln x \\\\ (\\ln x)^2 \\end{bmatrix} ", "7efbe2ac3dbe006abaf1004b414b2d3c": "\n \\frac{\\partial^4 W}{\\partial x_1^4} + 2\\frac{\\partial^4 W}{\\partial x_1^2 \\partial x_2^2} + \\frac{\\partial^4W}{\\partial x_2^4}\n = \\frac{2\\rho h \\omega^2}{D} W =: \\lambda^4 W \n", "7efc0ed510bb4f2a3527e81806f5a5a7": "L_w", "7efc9f832ef90731b528482c59050dfd": "r_\\mathrm{ap}=(1+e)a\\!\\,", "7efcbdcbafa78a8df5fe64c40e5b9d80": "t\\circ f", "7efd1ddab124aab5cab7457ec8690735": "\n\\nabla = \\mathbf{e}_1 \\partial_1 + \\mathbf{e}_2 \\partial_2 + \\mathbf{e}_3 \\partial_3,\n", "7efd2d3ddcff89f2081fa1461c63d357": "R *", "7efd316d8c830c8b5a966bc096ee538b": "T_0 Jac(C)\\cong |K_C|^*", "7efd365f52a7d29bcb07bb9a20727133": "\\varphi(t) = 1 - \\sigma te^{-\\frac{1}{2}\\sigma^2t^2}\\sqrt{\\frac{\\pi}{2}} \\left[\\textrm{erfi} \\left(\\frac{\\sigma t}{\\sqrt{2}}\\right) - i\\right]", "7efdaeed5b9036cdd8e9b7e525523b8b": "\n\\exp\\left(\n- \\int^T_0 L(\\phi_1(t),\\dot{\\phi}_1(t)) \\, dt\n+ \\int^T_0 L(\\phi_2(t),\\dot{\\phi}_2(t)) \\, dt\n\\right)\n", "7efdbd438fc739abcd7e41626b92ecab": "\\int x^2\\arcsec(a\\,x)\\,dx=\n \\frac{x^3\\arcsec(a\\,x)}{3}\\,-\\,\n \\frac{1}{6\\,a^3}\\,\\operatorname{artanh}\\,\\sqrt{1-\\frac{1}{a^2\\,x^2}}\\,-\\,\n \\frac{x^2}{6\\,a}\\sqrt{1-\\frac{1}{a^2\\,x^2}}\\,+\\,C", "7efe508673c7d5f3decdf7025be13a68": " \\langle \\sigma_i \\sigma_j \\rangle_\\beta \\leq C \\exp(-c(\\beta) |i-j|),\\,", "7efe5feabe3388468087cbb1fc513fd6": "=\\quad 0.971", "7efe964372b4432128db1ee4bcb225ad": "\\Delta n = \\pm 1", "7efeb52b4b4e91524cdfb989909f5317": "\\left(\\frac{\\partial U}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial U}{\\partial y}\\right)_x\\!dy = T\\left(\\frac{\\partial S}{\\partial x}\\right)_y\\!dx +\n T\\left(\\frac{\\partial S}{\\partial y}\\right)_x\\!dy - P\\left(\\frac{\\partial V}{\\partial x}\\right)_y\\!dx -\n P\\left(\\frac{\\partial V}{\\partial y}\\right)_x\\!dy", "7efeed826c51362ad2c6d4faaa11b9b3": "\\mathrm{z}^{-1}", "7eff3c828a69e0e3ffccfb5e9735f379": "J = \\tfrac{mI}{A}", "7eff3d8f9b3b46b60f367f7f13a3f59a": "\\frac{R}{R_\\odot} \\approx \\left ( \\frac{T_\\odot}{T} \\right )^{2} \\cdot \\sqrt{\\frac{L}{L_\\odot}}", "7eff907e01a6568a0eff390703e3b5c8": "T = \\boldsymbol{\\sigma}(n)", "7effadfaf57b727e770f5faea4c03c7c": "\\mu(g(z)){\\partial_{\\overline{z}}g(z)\\over \\partial_z g(z)}=\\mu(z)", "7effb9431f66d25a40a65a9ed4c26ad3": "p_n(z)", "7f0048a17d6baf9eb28c00ade2b15b38": "U(S) \\geq d(1-2\\varepsilon)s\\,", "7f00e823ebb1adadc9014cdb990939e2": "ax^2+bx", "7f01269d7d6477c41dc81c2ffd524a68": "u_y' = u_y", "7f01477373cc47d7c27240845db4d7ad": "\\sigma_\\mathrm e = \\sqrt{3~J_2} = \\sqrt{\\tfrac{1}{2}~\\left[(\\sigma_1-\\sigma_2)^2 + (\\sigma_2-\\sigma_3)^2 + (\\sigma_3-\\sigma_1)^2 \\right]}\n\\,.", "7f014de40e746ff06a8eb712146048eb": "\\mathrm F_{SO}(M)", "7f026505975f478eb1ab4c17542a79af": "d = \\sqrt{3 \\lambda L}", "7f02802c4308a7930b7087ed25bea195": "P_v(t)=\\frac{M_a}{r}(1-e^{-rt}).", "7f028463330408f4e3aa21a4afe7a970": "\\sigma_v^2 = \\tfrac{1}{2}[(\\sigma_{11} - \\sigma_{22})^2 + (\\sigma_{22} - \\sigma_{33})^2 + (\\sigma_{11} - \\sigma_{33})^2 + 6(\\sigma_{23}^2 + \\sigma_{31}^2 + \\sigma_{12}^2)]", "7f02c2fcf523236a86dbfb80ecb10877": "k_Bn_{AB}", "7f02cb0dee3208da0ac5f26a113e314c": "\\eta_q(n) = \\sum_{d\\,\\mid\\, q} c_d(n),", "7f02cc6ced98998c099f51d84a8719b0": "\\mathbf{i}^2 = \\mathbf{j}^2 = \\mathbf{k}^2 = \\mathbf{i} \\mathbf{j} \\mathbf{k} = -1.", "7f02fead3bb2a532fe248bee8a9ed1e8": "m_{h^0}^2 \\le m_{Z^0}^2\\cos^2 2\\beta + \\frac{3}{\\pi^2} \\frac{m_t^4 \\sin^4\\beta}{v^2} \\log \\frac{m_{\\tilde{t}}}{m_t}", "7f03126e1fa772d157c18f50a94a80fe": "p \\mapsto L_p(x)", "7f031526ebd30e1f937e8b35d55daf80": "\\operatorname{Symmetric-Dirichlet}_N(\\beta)", "7f0324b2bd49949eea354d42f8fa009c": " |1,\\psi\\rangle = \\begin{bmatrix} 0 \\\\ 0 \\\\ a \\\\ b \\end{bmatrix}", "7f03b69b269b21ef37afbc5861a53b05": " v_i= \\left( \\frac {R_i} {R_0}\\right)^{-6} ", "7f03feeeb72d5754a6151b38e88bcd95": "\\textstyle f : \\Omega \\to \\{0,1\\}^\\infty. ", "7f049ddb17e2b9d81d49adfb788ad2be": "\\mathrm{Hol}_x = \\{\\tau_\\gamma : \\gamma \\text{ is a loop based at } x\\}.\\,", "7f04be81334d398631c8011a1bb9bff7": "c_{\\lambda} c_{\\mu} = c_{\\lambda + \\mu}", "7f04c1b7208a1e057456026e50d978c6": "\\mathrm{sim}(d_j,q) = \\frac{\\mathbf{d_j} \\cdot \\mathbf{q}}{\\left\\| \\mathbf{d_j} \\right\\| \\left \\| \\mathbf{q} \\right\\|} = \\frac{\\sum _{i=1}^N w_{i,j}w_{i,q}}{\\sqrt{\\sum _{i=1}^N w_{i,j}^2}\\sqrt{\\sum _{i=1}^N w_{i,q}^2}}", "7f04d215c04a6c4f9afe86942faf7a47": "\\lambda = 0.62432 99885 43550 87099 29363 83100 83724\\dots.", "7f04e647323bd856c16f28bc79947417": "S_2 = I p \\sin 2\\psi \\cos 2\\chi\\,", "7f05195dbe8b2aaa5bfdad19d2c7bcd7": "q_1 = q_0 + \\frac{t_1 - t_0}{m} p_0 - \\frac{\\left(t_1 - t_0\\right)^2}{2m} \\frac{d}{dq_0} V\\left( q_0 \\right)", "7f053cdb7de79c82de53b9bc70c47025": "- + +", "7f0562b7361b94feb27ee472a1cbc253": "r(x)", "7f05a040acce474c44675e925787588a": "X_1^{n}(i)", "7f05a2a5c7d21c9d5f5ff3fe32b18bf5": " \\frac{\\mathrm{d}}{\\mathrm{d} t} \\left ( \\frac{\\partial L}{\\partial \\dot{q}_i } \\right ) = -\\frac{\\partial L}{\\partial q_i} ", "7f05d3bfc56e3590cae5ca53fdf000ac": "i^{\\mathrm{th}}", "7f05e537d4162b9432d3769c4ea4748d": "C a \\exp(-a^2/2) + 2P(\\xi>a)", "7f062b978e4feaf1a646c24bcbea32ba": "r \\leftarrow min(L(M)-i,m)", "7f06a1745d2c10cfead65cb4959ef73c": "\\{x_1,\\ldots,x_m\\}", "7f06ba53d9d9fba3aafe2a7a89e74669": "\\gamma = \\frac{1}{2}\\left(\\frac{m_{i}}{m_{f}} + \\frac{m_{f}}{m_{i}}\\right)", "7f06cedb53ebba6c0edd334f7eaa6dd8": "g = \\left[\\begin{array}{cc} 1 & 0 \\\\ 0 & \\sin^2 \\theta\\end{array}\\right].", "7f070de0d5775b692b1ce941477d813c": "\\mbox{vec}()", "7f0744faa8431c0b8d51f989317702d1": "\\Phi_c\\,", "7f084a3a12d7538fd906f35025abff7f": "\nh(Y_i) \\leq \\frac{1}{2}\\log{2 \\pi e} (P_i +N)\n\\,\\!", "7f085cd5926d4728b45d69e2c79ef60d": "\\begin{align}\nr_1 &= -p\\\\\nr_2 &= \\sqrt{p^2 - 4q}\n\\end{align}", "7f088735d363887ebd4e9d1a3ce201e8": "I = C_m\\frac{{\\mathrm d} V_m}{{\\mathrm d} t} + \\bar{g}_Kn^4(V_m - V_K) + \\bar{g}_{Na}m^3h(V_m - V_{Na}) + \\bar{g}_l(V_m - V_l),", "7f08a934b1481eec8ed33177b4047f23": " \\delta J = \\int_a^b \\frac{\\delta J}{\\delta f(x)} {\\delta f(x)} dx \\, . ", "7f09161fe9202210629f1292d7c64304": "h_r", "7f0926cb0725a8c6924dfc0ef148fbce": "2,3,5,11,17,41", "7f0958eb6f5d93502a13f33052bf8fa0": "R_v ", "7f09809c21f2ec54dd162af71de69cc9": "C_{exp}", "7f09a8444237f2efeded0b2507dd6e40": "\\pi_X :P^{(n+1)(m+1)-1} \\to P^n\\ ", "7f0a1f6c1d30799c16ed5fd76eed0882": "\\delta\\colon L\\rightarrow L", "7f0ab34712d956e06a709c1b81b9e06c": "V={1\\over3} A_0h ={\\sqrt{2}\\over12}a^3={a^3\\over{6\\sqrt{2}}}\\,", "7f0aba19f5791ebb07cdf4200d5f83e8": "h_{l} = hash(h_{l-1}\\; ||\\; l\\; ||\\; L\\; ||\\; A_{x}\\; ||\\; ts\\; ||\\; k_{js})", "7f0b8894ec1c491d191b44119fa88d05": "\\zeta_1,\\ldots,\\zeta_n\\in A", "7f0b9707811dcbb15c07814f40aee928": "t_1,\\text{ }t_2,\\text{ }...,\\text{ }t_n\\text{ }=\\text{ }\\{t_i\\}", "7f0bf546365c133b33fc55310f7aff62": "\n\\begin{align}\n\\text{jiva } \n& = s - \\frac{s^3}{R^2(2^2+2)} + \\frac{s^5}{R^4(2^2+2)(4^2+4)}- \\cdots \\\\\n& = s - \\left(\\frac{s}{C}\\right)^3 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^3}{3!} \n- \\left(\\frac{s}{C}\\right)^2 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^5}{5!}\n - \\left(\\frac{s}{C}\\right)^2 \\Big[ \\frac{R \\left(\\frac{\\pi}{2}\\right)^7}{7!} - \\cdots \\Big]\\Big]\\Big].\n\\end{align}\n", "7f0c1d152024044f19f8318ba59fa509": "(\\psi * \\mu) * \\alpha \\models \\mu", "7f0c78bb116466b86be06a84a949b891": "p_0 = 1 - \\sum_{i=1}^3{p_i}=0.5", "7f0cabbe8552331902273e72bf6a13ee": "P\\ ,", "7f0ce80346a50e94d3d4058502314710": "Z_o = \\frac{136190}{d \\cdot f}", "7f0d148666fda22c3bea54768b352dda": "\\scriptstyle\\kappa", "7f0d353f297f9a28f8adc585271ec92f": "((x, y) \\mapsto x \\times x + y \\times y)(5, 2)", "7f0d605bc7649dda120e6cb7c81848fc": " |L\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ {1 \\over \\sqrt{2}} \\begin{pmatrix} 1 \\\\ -i \\end{pmatrix} ", "7f0daab7696dd1fa0e0ab5908c481ecc": "\\partial_{i}", "7f0db8749397709938253bd74f425449": "\\langle m \\rangle = J B_J(x)", "7f0dc29f02bbadfcfdb1680ad47eb2e2": "\ndp = -{\\partial G \\over \\partial x} ds = \\{ G,P \\} ds\n\\, .", "7f0def5ba7dec9366dc920cbcb698c20": "x=\\left[\\begin{smallmatrix}f(0,0)&f(1,0)&f(0,1)&f(1,1)&f_x(0,0)&f_x(1,0)&f_x(0,1)&f_x(1,1)&f_y(0,0)&f_y(1,0)&f_y(0,1)&f_y(1,1)&f_{xy}(0,0)&f_{xy}(1,0)&f_{xy}(0,1)&f_{xy}(1,1)\\end{smallmatrix}\\right]^T", "7f0e1534900e129e4911dd18e824ee15": "L_d = X_0^2 k_0 n", "7f0e35044fa841eddfa8b34c6f949522": "f(x) \\!", "7f0e51a613362bdda808b263e2b8614f": "\n1-F(\\psi,\\rho)^2 \\le D(\\psi,\\rho) \\, .\n", "7f0e65b2db7579b46ea156a7a2237f18": "t\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "7f0ee0e9a0d9ce70de6c6c1ab8d99f3a": "\\Vert A \\Vert_{p} = \\Vert \\mathrm{vec}(A) \\Vert_{p} = \\left( \\sum_{i=1}^m \\sum_{j=1}^n |a_{ij}|^p \\right)^{1/p}", "7f0ef843e9fb8cfe841c67b8b62522ea": "U(\\lambda)^* = e^{\\bar\\lambda M - \\lambda M^*} = U(\\lambda)^{-1}", "7f0f339d3993f73c00f8bd91e65fd2e6": "r_\\mathrm{s}", "7f0f6b4d21ed1c295441c8b1f8bfe833": "\n\\gamma= 1- \\frac{ Expected~backorder~level~per~time~period}\n{Expected~period~demand}\n", "7f0f93961d947ad211421e21a2673fff": "h \\nu", "7f0faacffd87ca1f0f41edb83b965e45": "u\\in\\R^n", "7f0fb1e36e7ccdde99fcc515c0f2f044": "i=1,2,\\ldots,n,", "7f0ffa11fd22870ead767415e63cc784": "F'(x_1) = \\lim_{\\Delta x \\to 0} f(c). \\qquad (3) ", "7f1018ba7ed48415338024f03e0fe5d7": "\n{}^{p+q}f_{r+s}\\left(\n\\begin{matrix}\na_1,\\cdots,a_p\\colon b_1,b_1{}';\\cdots;b_q,b_q{}'; \\\\\nc_1,\\cdots,c_r\\colon d_1,d_1{}';\\cdots;d_s,d_s{}';\n\\end{matrix}\nx,y\\right)=\n\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty\\frac{(a_1)_{m+n}\\cdots(a_p)_{m+n}}{(c_1)_{m+n}\\cdots(c_r)_{m+n}}\\frac{(b_1)_m(b_1{}')_n\\cdots(b_q)_m(b_q{}')_n}{(d_1)_m(d_1{}')_n\\cdots(d_s)_m(d_s{}')_n}\\cdot\\frac{x^my^n}{m!n!}.\n", "7f101976069a3fa33ec62576cf08baaa": "(W_{ab}(t)) = \\left[ \\begin{array}{cccccc}\n 0 & 2 & 1 & 1 & 6 & 0 \\\\\n 1 & 0 & 1 & 2 & 5 & 6 \\\\\n 0 & 7 & 0 & 0 & 0 & 0 \\\\\n 1 & 0 & 1 & 0 & 0 & 0 \\\\\n 1 & 0 & 7 & 5 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 5 & 0\n \\end{array}\n \\right] ", "7f10a15e992db24af8e29fa0b4f91455": "\\mbox{HA} + \\mbox{H}_{2}\\mbox{O} \\rightleftharpoons \\mbox{A}^- + \\mbox{H}_{3}\\mbox{O}^+", "7f10a703229d1949b5734c1f65d6444a": " R_\\mathrm{int} = V_\\mathrm{ter}/I \\,\\!", "7f10a8f8de0ff53ff36ebf3a46be0778": "\\psi_{1}(x) = u(p)e^{-ip.x}\\,", "7f10d6cd51db2152d8b165c0eb04e3c7": "F(z) = \\tfrac12 a_0 + \\sum_{n=1}^\\infty (a_n-ib_n)z^n", "7f10dde0696f0e62a67e8648c7cdf9ae": "\\displaystyle[a_0; a_1, a_2, \\ldots, a_{k-1}, 1] = [a_0; a_1, a_2, \\ldots, a_{k-1}+1]", "7f11349f15a277860ab84c55eabb92e9": "b_i=\\left\\lfloor\\frac{\\lfloor\\sqrt{kN}\\rfloor+P_{i-1}}{Q_i}\\right\\rfloor,P_i=b_iQ_i-P_{i-1},Q_{i+1}=Q_{i-1}+b_i(P_{i-1}-P_i)", "7f11752a311acb1a6987748674c29b7b": "\\Phi = \\sum_{v} \\log{s(v)}", "7f11ad8acaf6a8eb678b2acdb99780a9": "\\tau\\leftrightarrow\\rho", "7f122c782b31623a687c6d5a74cb812c": "x^i \\to x^i", "7f1246191c1289fedf044c35b5899907": "Pr(U=a|X=b) = Pr(U=a)", "7f125bdf9898cbae44faa813c2370cf8": " \\pi_n(X,p) \\cong \\langle S^n,X \\rangle", "7f1296ba6cc9475a61cee2a3cddb573a": "e^{-\\frac{1}{2}t^2}", "7f129fe1feda5ae70bb9bafb2d839e29": " f(z)= \\sum_{-\\infty}^\\infty a_n z^n,\\,\\,\\,\\, g(z)=\\sum_{-\\infty}^\\infty b_n z^n", "7f1301b9c15a82ea1f893edfe2ed0c97": "u''-R(x)u' +S(x)u=0 \\!", "7f130a46f6a14759dedb790392c82c8f": "V \\oplus JV", "7f135fe0210da04fded18daf3355b998": "C_{abcd}", "7f139667890fa41ae6653d623454105d": "\\frac{1-a z^{-1} \\cos( \\omega_0)}{1-2az^{-1}\\cos(\\omega_0)+ a^2 z^{-2}}", "7f13b694a50e7584ad6d5ea181d9ceb7": " \\forall x ( \\forall z (( \\phi \\lor \\psi) \\rightarrow \\rho ))", "7f13c3a4040dfa8db4ca398b7a3d5228": "a x^2+y^2= 1+dx^2y^2", "7f13d8837f4015f9ca8f0f696fe37aab": "H(z)=\\frac{1}{1-a*z^{-1}}", "7f142a11f8259bb5ec64934b8dbf7654": "\\vdash q\\,\\!", "7f1442f6c34fab77ca5d0cbba26c9720": " V_{+} (x, a_1 ) = V_{-} (x, a_2) + R(a_1)", "7f145b88e07d75f341de29bdc2313198": "\\frac{x^{\\alpha-1}(1-x)^{\\beta-1}} {\\Beta(\\alpha,\\beta)}\\!", "7f147f3473db867c8211787b5d386205": "\\sum_{n=0}^{\\infty}\\frac{1}{an+b} ,\\!", "7f14c543e525cee888d3b20b4066e488": "\\,j\\omega_s", "7f155d7cc849c1d4e04df286c59fc517": " \\mathrm{Pr}(w(s+t),t|w(s),s) = \\frac{1}{\\sqrt{2\\pi D t}} \\exp\\left({-\\frac{\\|w(s+t) - w(s)\\|^2}{2Dt}} \\right)", "7f15743eac5846ef6b81c8cd2ec0502d": "d\\varphi\\wedge dp", "7f15e481f813d81587e046b9e5c5891a": " \\alpha = \\frac{ 2 k }{ b - a }", "7f1604e276d01cc94f6717566de32f19": "\\scriptstyle Z_{C}", "7f16cd9bca726cdc587f8ef21800af19": "v\\cong\\sqrt{2\\left(9.8\\ {\\mathrm{m}/\\mathrm{s}^2}\\right)(6.4\\times 10^6\\ \\mathrm{m})}= 11\\,200\\ \\mathrm{m}/\\mathrm{s}.", "7f16f44f96a197f08f83419741321cf4": "\\ln \\mathcal{L}(\\mu,\\Sigma) = \\operatorname{const} -{n \\over 2} \\ln \\det(\\Sigma) -{1 \\over 2} \\operatorname{tr} \\left[ \\Sigma^{-1} \\sum_{i=1}^n (x_i-\\mu) (x_i-\\mu)^\\mathrm{T} \\right]. ", "7f172a8faabc88fc19c8e9e3ab168ae9": "\nu(c) = \\begin{cases}\n\\frac{c^{1-\\eta}-1}{1-\\eta} & \\eta>0\\text{, }\\eta \\neq 1 \\\\\n\\log(c) & \\eta = 1\n\\end{cases}\n", "7f173d6fd84e1e1e2e15030900f675f8": "f(x_1, x_2) = (x_1 - x_2)^2/2", "7f1752ebf6ec49b60b706f6ff098db50": "\\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 2 & 0 \\\\ 0 & 0 & 2 \\end{bmatrix}", "7f175a20efea4cbe2bf7e158987d6257": " L_3 = L_1 \\cap L_2", "7f177646af9b7b4c8b16f6a9ac20b55b": "i=1,2,\\dots,k", "7f1784a3c927f98dffd60ec075bc0dc5": "\\Theta(k^2)", "7f17beff2132c7ed07a2ec4e00e8492a": "C_{pq}^-", "7f181ea7031837e075e6a3a8fe034ae3": "g_\\otimes(v\\otimes w) = g(v,w)", "7f186592f70e391b05ae6e4801e1c22d": "e_{ij} = \\frac{1}{2}\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right).", "7f188de84f8cec9c0613d9c94010201e": "p_S : S \\times \\mathbb{R} \\rightarrow \\mathbb{R}^3", "7f18a59e500485aac3cd22b428b0f77a": "\\tfrac{\\partial^2}{\\partial x^i \\partial x^j}", "7f1942cd9a156ce9e1c9c16368712482": " E_{21} =\\frac{d \\ln (c_2/c_1) }{d \\ln (MRS_{12})}\n =\\frac{d \\ln (c_2/c_1) }{d \\ln (U_{c_1}/U_{c_2})}\n =\\frac{\\frac{d (c_2/c_1) }{c_2/c_1}}{\\frac{d (U_{c_1}/U_{c_2})}{U_{c_1}/U_{c_2}}}\n =\\frac{\\frac{d (c_2/c_1) }{c_2/c_1}}{\\frac{d (p_1/p_2)}{p_1/p_2}}\n", "7f19499f2c8fa8a664c941d5771a6de8": "X/{\\sim}", "7f195e29bfd1afdf63b9ebec6af9f9ea": "F_{\\mu\\nu}", "7f19636495130bf8aea81a460b6b1043": "S(u,v) = \\sum_{i=1}^k \\sum_{j=1}^l R_{i,j}(u,v) \\bold{P}_{i,j} ", "7f1974da12a51ff3761945a3ec28c3c3": "= 2\\left[\\sum_{i=1}^{N} (x_{i2}-x_{i1})(x_{i2}-x_{i1})' \\right]^{-1} \\left[\\sum_{i=1}^{N} \\frac{1}{2} (x_{i2}-x_{i1})(y_{i2}-y_{i1}) \\right]", "7f19c1facc329b14a668978e1ccd0c41": " J^k = n^{k-1} J, \\mbox{ for } k=1,2,\\ldots.\\,", "7f19d640bb07c6503e8e5438c80f4008": " {1 \\over s} F(s) ", "7f1a0f7bdbc46b7839c4361ebb355502": " \\frac{\\sum |a_{nb}|}{|a'_{p}|} \\begin{cases}\n \\leq 1, & \\text{at all nodes}\\\\\n <1, & \\text{at all nodes at least}\n \\end{cases} ", "7f1a611bba2e54ea1f4b9b5b400369eb": "P[K=k] = P[M=N-k|M\\, 0", "7f29f181c4289da0251c13491765ea76": " S \\subseteq [n], |S| \\leq d, j \\notin S ", "7f2a0387445b12d87dcee72228189998": "\\mathbf{WH} = \\mathbf{WBB}^{-1}\\mathbf{H}", "7f2a13fc30f16b860d8d8b3e2ba033cd": "u_j", "7f2a394ecb419fb4dd047133fc270a1f": "\\implies", "7f2a635568ee857ad1d3cb339ae1b6af": "\n z = 1.48 + \\sqrt{n}\n", "7f2a99da0bff4d63f3d7dc1824a553a6": " {dw \\over dx} ", "7f2adbba5e692c70bbb76afbf26e905d": "175/1.25=140 ", "7f2b46c4f41bc7f1192718ce90467b97": "\\operatorname{artanh}(z)", "7f2b62cf2c65f9b066ea6ed7c9df563e": "\\beta_j(p,q)", "7f2bc67bed023df18eda12aa52ff3a6a": "k\\leq m", "7f2c594878d2079a3e5fe90f713c1f2e": "\n\\begin{array}{rclr} f_{ij} & \\leq & c_{ij} & (i, j) \\in E \\\\\n\\sum_{j: (j, i) \\in E} f_{ji} - \\sum_{j: (i, j) \\in E} f_{ij} & \\leq & 0 & i \\in V, i \\neq s,t \\\\\n\\nabla_s + \\sum_{j: (j, s) \\in E} f_{js} - \\sum_{j: (s, j) \\in E} f_{sj} & \\leq & 0 & \\\\\n- \\nabla_s + \\sum_{j: (j, t) \\in E} f_{jt} - \\sum_{j: (t, j) \\in E} f_{tj} & \\leq & 0 & \\\\\nf_{ij} & \\geq & 0 & (i, j) \\in E\\\\\n\\end{array} ", "7f2ccb84e1c3dd959317b0f719224e3d": " (1, . . . ,1) \\in \\mathbb{Z}^n ", "7f2ce6d5c3a700127312daa23335d943": "\\mathbf{x} = \\mathbf{X}/\\Omega", "7f2cf4bd886ee6151a642dfaa4b642f8": "(f'')'=f'''.", "7f2d2a576d6eb5550b69027725f31a00": " \\begin{align}\n& \\text{minimize} && \\sum_{i}{(s_i^+ + s_i^-)}\\\\\n& \\text{subject to:} && w_1x_{i1}+...+w_nx_{in}-t_r+s_i^+\\ge\\delta& \\text{for all reference alternatives in class } c_r (r=1,...,k-1)\\\\\n& && w_1x_{i1}+...+w_nx_{in}-t_{r-1}-s_i^-\\leq-\\delta& \\text{for all reference alternatives in class } c_r (r=2,\\ldots,k)\\\\\n& && w_1+...+w_n=1\\\\\n& && w_j,s_i^+,s_i^-,t_r\\ge 0\\\\\n\\end{align}\n", "7f2d4071724789c039f10b100012a090": "[0, \\infty]", "7f2d46621d0cce6d549b69a6971f65a1": "I = n \\cdot L", "7f2d48b1c6d42b0231724a4cdcee9622": "\\ \\boldsymbol{\\sigma} = \\mathcal{G}(\\boldsymbol{F}) ", "7f2d56f9d998866e39309816e264ac1a": "v=\\Omega r", "7f2da36a942a950e922b06bf6ae55ff5": "\\rho_{b,m} / \\rho_{FB}) = C_{up,dn}", "7f2dd5eb18dca398d082a146ad720efa": "W = \\int_0^Q V \\mathrm{d}q = \\int_0^Q \\frac{q}{C} \\mathrm{d}q = {1 \\over 2} {Q^2 \\over C} = {1 \\over 2} C V^2 = {1 \\over 2} VQ", "7f2e5d1a52aa6d9c0eb4ea5ac7ba4643": "\\textstyle\\frac{1}{3}", "7f2ea47cc74a1a00ebb1764f287c0e22": "\\begin{matrix} {47 \\choose 2} = 1,081 \\end{matrix}", "7f2ed8768b4107d3d88d0959bb5400d2": "F \\ge 1", "7f2f02119d307a5a87a682e505a0a52f": "S[A[i-1],n] < S[A[i],n]", "7f2f14962b7179ee0c112bc5afab8666": "\\left[\\gamma^\\alpha,\\gamma^\\beta\\right]_{+} = \\gamma^\\alpha\\gamma^\\beta + \\gamma^\\beta\\gamma^\\alpha = \\eta^{\\alpha\\beta}\\,,", "7f2f6dbff033fb2967af855a3d6b7257": "2 + 3 \\stackrel{?}{=} 8 + (-3)", "7f2ff9a0224f8da03170d6dce2fd9a6c": "R(N,M_0)", "7f2fff09ecdf89edd813163e71ee7965": "\\{w_k\\}_{k=1}^\\infty", "7f308c3089ec2633d5bae6c0e9228239": "\\alpha^\\prime_n= \\frac{(\\rho_1;q)_n(\\rho_2;q)_n(aq/\\rho_1\\rho_2)^n\\alpha_n}{(aq/\\rho_1;q)_n(aq/\\rho_2;q)_n}", "7f30a76c1de904f16f836535fecfb120": "W_E(1) = \\frac{3}{2} ( c_{11} + 2 c_{12} ) \\sigma^2", "7f30ec174a6714b672df07692d370a56": "n/p_k", "7f31b96a6708b6c60eb495b5fd465d11": " \\chi^2 = \\sum_{i=1}^{k} {\\frac{(N_i - np_i)^2}{np_i}} = \\sum_{\\mathrm{all\\ cells}}^{} {\\frac{(\\mathrm{O} - \\mathrm{E})^2}{\\mathrm{E}}}.", "7f32543d81c2fc2b2ff0a35691093aa7": "c_0 + \\underset{n=1}{\\overset{\\infty}{\\mathrm K}} \\, \\frac{1}{c_n} = c_0 + \\cfrac{1}{c_1 + \\cfrac{1}{c_2 + \\cfrac{1}{c_3 + \\cfrac{1}{c_4 + \\ddots}}}}\\,", "7f33e65b7c44689cd50982c5b742c02d": "\\langle a b \\rangle_0=a\\cdot b\\,", "7f3408c72246eece3d5542fc853ce417": "[a,b]\\,", "7f3447a68c313793b40041b5a6011b8b": "\\triangledown _{t}^{2}T+k_{t}^{2}T=0 \\ \\ \\ \\ \\ \\ (15)", "7f345c294911cd5826188a683dd26ff0": "v^\\prime = \\textstyle{\\frac{3}{2}}v\\,", "7f34a2c3ffac74d7551573349efda0ac": "T \\otimes U", "7f34b2ab3072a1ac2cf764639a1816fb": "L = L_{PE} \\quad \\mathrm{and} \\quad L = L_{CE}+L_{SE} \\;.", "7f34f55609db3c87bf12f5c376131295": "|R|=m\\,", "7f353d18833a0ebd163232a6a29db9e9": "\\langle g_1x_1-f_1,\\ldots, g_nx_2-f_n\\rangle.", "7f361231c4c9447d4c43a68c6a19eb69": "U_{++}=\\bigcup_{n\\ge 0}\\alpha^n(U_{+})", "7f36348ee6149ee4f5a414ec41e9ce0e": "\\mathbf{e}_{k} \\in V", "7f366c68673079e7c21fc70a83ac466a": "\\omega_{\\mathrm{f}} = \\omega_{\\mathrm{i}} + \\alpha t\\!", "7f36a64ebe73d673002ecb842009a812": "\\, u", "7f36bb7f207cb119583d642712cc3b82": "p=\\sqrt{a^2+b^2}", "7f36ded03cc9d4647389cc778e5cad58": " D_{fast, slow} ", "7f36f2618e0a86cd7de3ef63348eed78": "\\psi(z)=\\sum e_nz^{-n-1},\\,\\,\\, \\psi^\\dagger(z)=\\sum e_n^*z^n,\\,\\,\\,\\{e_n,e_m\\}=0,\\,\\,\\,\\{e_m,e_n^*\\}=\\delta_{m,n}I.", "7f3711f68fcc7e8d2263f47b80be9299": " \\Gamma_1 ", "7f37368315efeb016de46227baf30113": "1 \\leq i \\leq n-1", "7f374131481fe41d0fcb66ff79f5cb38": "\\hat{\\mu}_4", "7f374ffb705fddb052605cdd63884887": "E(Q_1)+E(Q_2)>E(Q_1 + Q_2) ", "7f3857f45faf1dd6866ee07d7d9fe489": "\n J^{+} = J^{-} = \\int_{\\Gamma} \\left(W n_1 - t_k~\\cfrac{\\partial u_k}{\\partial x_1}\\right) d\\Gamma = 0\n ", "7f386f1f582e48e8b9a0540867db5b19": "E_p = mgh", "7f3887fd7309e4487725052a6370cf19": "\\scriptstyle \\alpha", "7f38a16a08823bae74aab98e1e44234c": "Y \\sim \\mathrm{GEV}(\\alpha,\\beta,0)\\,", "7f38ab45dae10d76d6989ef26cae0c40": " \\mathbf{v} = \\frac{d}{d x^0}(x^1 \\mathbf{e}_1 + x^2 \\mathbf{e}_2 + x^3 \\mathbf{e}_3) ", "7f38cd1173573443ce401b9244d1dd4d": "\\alpha = \\frac{2 \\pi}{3}", "7f38f6175ba39bc1a9767e6a8541823b": "\\prod_\\varnothing{} = \\{ ( ) \\},", "7f397f3848eafba85eeef2fed7776e79": "\\frac{dy}{dx} = \\frac{-4x^3}{4y} = \\frac{-x^3}{y}", "7f3986de1588d0595c03f91fa81f0f2c": "j_{sat}^{ion} = en_ec_s", "7f399e0aec68bfce9a4bf9cabf28843b": "X_{a;b}+X_{b;a} \\, =0", "7f39b5d9781792c726a52c7996a14bdb": "\n\\begin{align}\nT(s,\\mathbf{x})&=\\frac{G_4 G_6 G_8 C_2s}{G_6 G_{11} C_1 C_2 s^2+G_1 G_6 G_{11} C_2 s+G_2 G_3 G_5 G_{11}} \\\\\n \\mathbf{x}&=[C_1~C_2~G_1~G_2~G_3~G_4~G_5~G_6~G_8~G_{11}]\n\\end{align}\n", "7f3a2a7e01e3bad368b2be4dfcfd84c8": "\\mathcal{O}_X^n|_U \\to \\mathcal{M}", "7f3a43274f7af9c0d03bf0cf516c139a": "b(x) \\neq 0, \\pm1", "7f3a823be6fd23f239b583d8ee1df2a6": "n\\,\\leftarrow\\,1", "7f3ab3a8db0031441876501685ddabdf": "\\|x\\|_\\infty = 1", "7f3b41bbb1441d3bc6d2a8ed7dfabd85": " i\\mathfrak{t}^*", "7f3b5007450bc6d6caafe8c930312f1c": "\\operatorname{rank}(A) + \\operatorname{nullity}(A) = n.", "7f3b6678372f4c004a7f4c9e4ee9cda3": "e\\ >\\ 1\\,", "7f3b67f37740cbb166d1f604ed3b6cdb": "\n\\psi(\\mathbf{r}) = e^{ikz} + f(\\theta)\\frac{e^{ikr}}{r} \\;,\n", "7f3bcee93dc8f077e84f4e0f9ee828df": "\\mathfrak{i}", "7f3beee6786bf92da4ec98eab790d0df": "\\operatorname{tr}(\\gamma^\\mu\\gamma^\\nu\\gamma^\\rho\\gamma^\\sigma)=4(\\eta^{\\mu\\nu}\\eta^{\\rho\\sigma}-\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\eta^{\\mu\\sigma}\\eta^{\\nu\\rho})", "7f3c3f96eb2c3bcb7fb1a8e98d2c87b7": "_{polar} \\delta_{ck}^2 = \\frac{(c-k)^2}{(c+k-2c_{min})(2c_{max}-c-k)} ", "7f3c7151b1507b569ab7df84d9c82d8e": "F:[g] \\rightarrow \\Lambda^1(M)", "7f3ca056c7bf8504843604697c290752": " E(x,y,z)={z \\over {i \\lambda}} \\iint_{-\\infty}^{+\\infty}{ E(x',y',0) \\frac{e^{ikr}}{r^2}}dx'dy' ", "7f3e02efd21cf3a9bc6585c41c95888c": "\\,v=dx^{(4)}/dt", "7f3e1ebaf0258e6fe9f57c12880197e8": " \\alpha_k = \\frac{\\mathbf{p}_k^\\mathrm{T} \\mathbf{b}}{\\mathbf{p}_k^\\mathrm{T} \\mathbf{A} \\mathbf{p}_k} = \\frac{\\langle \\mathbf{p}_k, \\mathbf{b}\\rangle}{\\,\\,\\,\\langle \\mathbf{p}_k, \\mathbf{p}_k\\rangle_\\mathbf{A}} = \\frac{\\langle \\mathbf{p}_k, \\mathbf{b}\\rangle}{\\,\\,\\,\\|\\mathbf{p}_k\\|_\\mathbf{A}^2}. ", "7f3e21ea4a0ea02ff21bae3ac0e28293": " {d E \\over d\\tau} = \\mathbf{F} \\cdot { \\mathbf{v} \\over c} ", "7f3e388fd11787fdf16fbdebc087cc72": "c_{n+m}", "7f3e55aa0a99e74c9aeeb63bcec61819": "x=x_0+h", "7f3eddaa4e2d62a848d125e6358c5de0": " \\begin{align}\nVar(Y) &= E[Var(Y|Z)] + Var(E[Y|Z]) \\\\\n&= E[(2Z/N)(1-Z/N)] + Var(Z)\\\\\n&= (2E[Z]/N)(1-E[Z]/N) + (1-2N^2)Var(Z).\n\\end{align}", "7f3fb9afa2a4e1932b193136278ab2c2": "f_{i} \\in L^{1} \\left( \\mathbb{R}^{n_{i}} ; [0, + \\infty] \\right)", "7f40157a85704097518281a171d8a210": " \\mathcal{S}^D=\\left\\{\\mathbf{x}=[x_1,x_2,\\dots,x_D]\\in\\mathbb{R}^D \\left| x_i>0,i=1,2,\\dots,D; \\sum_{i=1}^D x_i=\\kappa \\right. \\right\\}. \\ ", "7f4056f91b7f7ed0cc00001ecb76dc10": "J=-D\\frac{\\partial C}{\\partial x}", "7f40a3d022b1896bfd53b81488720e94": "(A,f') = (f\\to f')\\circ(A,f).", "7f40f14f6a3c2f5728dd0cc4f272c44f": "\\nabla F = \\nabla \\cdot F + \\nabla \\wedge F", "7f410d573e3e67580e080bd1054ad722": "\\Omega = \\tfrac{2\\pi}{N}", "7f4140abb715917bc167602363eb5b63": "\\textstyle{\\int f(xy^{-1})g(y) \\, d\\lambda(y)}", "7f41753d66a48b7896a2e8c8237bffad": "s(v) = \\sum_{u \\in V(G)} 2^{d(u,v)}", "7f41f7b9426b4956cc5573e40d817ff6": "T(\\mathbf{x})", "7f420f724994d13712032105e50f06a2": "\\forall u\\forall v\\,\\partial(uv) = u \\,\\partial v + v\\, \\partial u", "7f42720eb8edccf6d8c21a8d4e6e4698": "\\begin{matrix}\\frac{1}{2}\\end{matrix}MV^2", "7f4272fecd7e9b5a9e45df8c1260b38f": "(i + 1)", "7f42bc3ae9c6c0087f10423ccfc174e9": "f\\in C^\\infty(M)", "7f42d44ad61c6a73ddbbccf0cb2e7095": "\\partial^\\alpha \\ = \\left( \\frac{\\partial}{\\partial t}, \\nabla \\right)", "7f4323976becea2f834f125740c566f8": "(\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{\\hat{n}} \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\lim_{A \\to 0}\\left( \\frac{1}{|A|}\\oint_{C} \\mathbf{F} \\cdot d\\mathbf{r}\\right)", "7f4337ec5495083ec121191c97ed9347": "Q_h", "7f436e536e1ac6f2c79cbcfdaaf54f02": " (1+4t^2)^{1/2}", "7f43bc78430103419029eb6f9efa268c": "N=V\\cdot n_0", "7f4413155ffccf3e8f6a54f926df6737": "\\rho\\rightarrow 1", "7f442976b0c045389a1a660622456417": " (n-k+1)^2+(n-k+1)(n-k)+2 ", "7f44451c896c2a22eb90c232c76a03df": "m \\in N, s \\in R", "7f444b4e48e7a4af8bd42e530a9ada1a": "\\alpha^m", "7f444cf878bafe0c79749bd9dda0257c": "\\operatorname{Cl}_{2m}\\left(\\frac{\\pi}{2}\\right)=\\beta(2m)", "7f445009df3cab097b7c7fa66beeaf2b": "BV([0,1])", "7f445c33a7f81d537455e4acbbc0f3a3": " Q_{10} = \\frac{\\text{Shelf Life at temperature T°C}}{\\text{Shelf Life at temperature ( T°C + 10°C )}} ", "7f457a084bb513aa2c74417949ca6fdd": "G \\approx G'", "7f457b55557833f90abad53118715bc9": "\\ast=\\wedge", "7f45d22496c11c303a170c22db5b0c54": "r/\\xi\\gg1", "7f45dc4a6248df2841fd7693c4878267": "~\\dot x=\\frac{{\\rm d}x}{{\\rm d}z}", "7f4621d9f6947367822a2dcce0ec58dd": "\\Sigma_{model}", "7f463425e7c4d1ed60ffdad364ae13d5": "X_0=X_N, X_1=X_{N+1}", "7f4646b1ed5955e8abda46607d320eda": "\\mathcal{T}\\left\\{f(t)\\right\\} = s\\mathcal{B}\\left\\{f\\right\\} = sF(s) =\ns \\int_{-\\infty}^\\infty e^{-st} f(t) \\, dt.", "7f4652e80d5456b5397e759d502e1743": "L \\xrightarrow[-g]{} Lb ~|~ b", "7f466a75ae0489328b40e25724cbe457": "v_\\text{radial}=v_\\text{s}\\cdot \\cos{\\theta}", "7f4714224495bb4067264862187c9edd": "V_m - b", "7f47247f6337dd1cafb6ca479d3d50c2": "\\| f_{n} - u_{n} \\|_{\\infty} < \\varepsilon", "7f47b61c18c6698a47f3ff9bd51fc1ac": "G_{Eq} = \\frac{G_1 G_2}{G_1+G_2}.", "7f47ba3b8aa4511324c3dc0ea377bb09": "\\sigma_{i\\theta}", "7f47c7aa11e6506936fafc899a5d49b4": "\\begin{vmatrix} -1 & \\cos{(\\alpha_{12})} & \\cos{(\\alpha_{13})} & \\cos{(\\alpha_{14})}\\\\\n\\cos{(\\alpha_{12})} & -1 & \\cos{(\\alpha_{23})} & \\cos{(\\alpha_{24})} \\\\\n\\cos{(\\alpha_{13})} & \\cos{(\\alpha_{23})} & -1 & \\cos{(\\alpha_{34})} \\\\\n\\cos{(\\alpha_{14})} & \\cos{(\\alpha_{24})} & \\cos{(\\alpha_{34})} & -1 \\\\ \\end{vmatrix} = 0\\,", "7f47d7d3868e2cc42e0cf547151366bc": "\\{0,...,p+q \\}", "7f480b687d1908d993c92a9bfe669541": "K_{\\rm c}", "7f4844039471372f1269f29fd8d4f87a": "\\mathrm{AgNO_3 + CO_2 \\uparrow +\\ NH_4NO_3}", "7f484c81bb6c22c5a87353a45be9efc2": "\\sum_{w\\in W}\\epsilon(\\omega)Q(w(\\mu+\\rho_c)-\\lambda-\\rho_n)", "7f48692c0311aca37b37d2b936257ccc": "\\displaystyle{k(s,t)=K(v(s),v(t)).}", "7f486ad11db894e01668a4fa1340ad68": "\\sum F_x=0=F_{AB}+F_{BD}\\cos(60)+F_{CD}=\\frac{5}{\\sqrt{3}}-\\frac{10}{\\sqrt{3}}\\frac{1}{2}+F_{CD} \\Rightarrow F_{CD}=0", "7f48958d84bc78935b44be1d06773211": "\\; kT ", "7f489cd1ebe6129bcefa62dac748d41f": "f(x_k)", "7f48aacc5c9d754dc54c5afe4a207040": "\\operatorname{Out}(G)", "7f48c6a48f6f45cc0e105669059476d2": "\\mathcal{F} = \\mathcal{S}^{\\mathbb{L}}", "7f4907bebf41370456899d4befa08c8b": "G = \\dot{\\cup}\\ x_i H.", "7f493ea818d3c84c66a71b86d0469960": "B - L", "7f495d5f9b98ce65c84899999ec082ca": "3\\mu", "7f498fde53b0093a4f5779ada5c31759": " \\left (\\Omega, (\\mathcal{F}_t), \\mathbb{P}\\right) ", "7f49911426c2b7039c022392c0b8f57d": " r_\\mathrm{ corr } = r + c_x^2 \\frac{ m_y }{ m_x } - \\frac{ s_{ xy } }{ m_x^2 } ", "7f49bbe2f0af1edb6c6cee353d3e204b": "RL", "7f49be78fa84e8c466abcda098e4a85a": " a b x^{-a-1} e^{-b x^{-a}}\\!", "7f49ee0410cde6f8604d704188e332de": "\n \\begin{align}\n A(T,T)&=0\\\\\n B(T,T)&=0\n \\end{align}\n", "7f4a5c6469247c1b59220882b6b1ab02": "r(n) \\sim D\\alpha^n n^{-3/2} \\quad\\text{as } n\\to\\infty,", "7f4ac9f0be570a77e665358f029cbd93": "{\\mathfrak b}({\\mathbb P})", "7f4b15f9d80d16d88626ea74a1ac075c": "d_1, \\ldots,d_n", "7f4b4c5fde0afecf3ba9e5a7bd1f008a": "\\nu_{\\rm yz}", "7f4b6705c19705c8d227f9323d36bac6": "y(xz)=(yx)z", "7f4bafef37b19ee5437956da208f4365": "\n\\langle \\rho^2 \\rangle_{R_{vir}}=\\frac{\\rho_0^2}{c^3}\n\\left[1-\\frac{1}{(1+c)^3}\\right]\n\\approx \\frac{\\rho_0^2}{c^3}\n", "7f4be2919cd7cad9c17ffbb29bd993d6": "\\int_0^{\\infty} L_n^{(\\alpha)}(x)L_m^{(\\alpha)}(x)\\Gamma(x,\\alpha+1,1) dx={n+ \\alpha \\choose n}\\delta_{n,m},", "7f4bf4ce7889ad8ef3532a6d8d878211": "\\mu^D(f_{s_1}\\circ f_{s_2} \\circ \\cdots \\circ f_{s_n}(K))=(r_{s_1}\\cdot r_{s_2}\\cdots r_{s_n})^D.\\,", "7f4c6a0271ccff6d98108c52ed7990fa": "\\theta = 2 \\arctan \\frac{y}{x+r}", "7f4c88a6dc61e11363a66d7aaed0c7f5": "D = R/\\mathfrak{p}_0", "7f4c8c320866afed0f14147df7d15211": " x \\geqslant \\mu ", "7f4cd64c2d7f190123efb40a8c9433cb": "0 = \\int_a^b f(x) h(x) \\; dx = \\int_a^b r(x) f(x)^2 \\; dx.", "7f4cd90a2b25aa6cd33bdad9e8e08c2d": "\\hat{b}_i^\\dagger", "7f4cddc182a336f1681cf8e9412df2c9": "\\phi(u) := \\min \\{\\alpha \\mid u \\in \\mathcal{U}(\\alpha,{\\tilde{u}}) \\}", "7f4d0789d2e75ad9f89f12539b65cb55": "1 \\leq i \\leq m ", "7f4d11fef70a78e880f0c63564982db2": " \\mathbf{g} = \\frac{Gm}{\\left | \\mathbf{r} \\right |^2 }\\mathbf{\\hat{r}} \\,\\!", "7f4d4cef1d1e960e6cba12049433db2f": "\\alpha^{i,j}", "7f4d6dcce16fba856d951ebdd2c80033": "\\mu \\cos\\theta \\, ", "7f4dc3279c73e01e6c3b79d0936bf55d": " {\\frac{\\mathrm{d}z}{\\mathrm{d}u}} = A - \\left[\\beta\\operatorname{sign}(z(t)\\dot{u}(t)) + \\gamma \\right]|z(t)|^n ", "7f4e32af52b93c3c9383aaf9028a4eab": "B[f]_{jk} = \\frac{1}{j!}\\left[\\frac{d^j}{dx^j} (f(x))^k \\right]_{x=0} ~,", "7f4e74baf676f9dd7c429e7be8e5ba2c": "\\tau^2 + {\\kappa_\\mathrm I}^2 = 1 \\ ", "7f4f29b0932c59df66be4cdcc9ac8f3b": "B \\underline{A}", "7f4f54733803e2b1cb3e0c41985b81bd": "\\zeta(X, s) = \\exp\\left(\\sum_{m = 1}^\\infty \\frac{N_m}{m} (q^{-s})^m\\right)", "7f4f935bbda29fe88fef8383946acc70": "\\int \\sin^2 x \\, dx = \\frac{1}{2}\\left(x - \\frac{\\sin 2x}{2} \\right) + C = \\frac{1}{2}(x - \\sin x\\cos x ) + C ", "7f5032fb3e155574c5efa593e961e050": "\\wp(z;\\tau) =\\frac{1}{z^2} + \\sum_{(m,n) \\ne (0,0)}{1 \\over (z+m+n\\tau)^2} - {1 \\over (m+n\\tau)^2}.", "7f507ebad57b931b1bb5cc7ae1d9b916": "r \\cdot s = s \\cdot r = r", "7f5092c523638725fd62471787194a0f": "\\frac{D^2 X^\\mu}{dt^2} = {R^\\mu}_{\\nu\\rho\\sigma} T^\\nu T^\\rho X^\\sigma.", "7f50950885bd0e894360bdb9211845d1": " L = x_2 - x_1 ", "7f50e57b3beaac3e52010c4782fa7e55": "{\\partial \\mathbf{r} \\over \\partial x}=(1, 0, f_x(x,y))", "7f50f219fb79988843fb41626fb9aafb": "(p,q) \\in C^-", "7f50f426c9dfbff43845870f0be8c1a3": " p_n = p_{n-1} - \\frac{f(p_{n-1})}{ (p_{n-1}-q_{n-1})(p_{n-1}-r_{n-1})(p_{n-1}-s_{n-1}) }; ", "7f51244d268284c8764a15b816aa245a": "ds^2\\,=\\,-\\frac{\\Delta_K}{\\rho_K^2}\\,\\big(dt-M\\sin^2\\theta d\\phi \\big)^2+\\frac{\\rho_K^2}{\\Delta_K}\\,dr^2+\\rho_K^2 d\\theta^2+\\frac{\\sin^2\\theta}{\\rho_K^2}\\Big( Mdt-(r^2+M^2)d\\phi \\Big)^2\\,,", "7f515f147fc5340368bd244015aee67b": "= \\bar{x} + (\\text{SE}\\times 1.96) ,", "7f5174385cb4dcd06190dab326ffa5a7": "[\\Delta R_i \\cdot \\Delta R_j]_k = \\frac{3 k_B T}{\\gamma}\\lambda_k^{-1} [u_k]_i [u_k]_j", "7f51855817c1c159d3a0e9d2855971cd": "g(u)=\\log(u)", "7f51c50442ecbe98862fce3e1ed710c7": "0 < x < L\\,\\!", "7f51f0047be69ce3cbcdf978610b128f": "\n\\mathbb{E}\\,\\mathbf{X}_k = \\mathbf{0} \\quad \\text{and} \\quad \\mathbb{E}\\,(\\mathbf{X}_k^p) \\preceq \\frac{p!}{2}\\cdot R^{p-2} \\mathbf{A}_k^2\n", "7f5219426651ec92edfbfcd3bf7b524d": "\n \\begin{align}\n \\varepsilon_{rr} & = \\cfrac{\\partial u_r}{\\partial r} \\\\\n \\varepsilon_{\\theta\\theta} & = \\cfrac{1}{r}\\left(\\cfrac{\\partial u_\\theta}{\\partial \\theta} + u_r\\right) \\\\\n \\varepsilon_{\\phi\\phi} & = \\cfrac{1}{r\\sin\\theta}\\left(\\cfrac{\\partial u_\\phi}{\\partial \\phi} + u_r\\sin\\theta + u_\\theta\\cos\\theta\\right)\\\\\n \\varepsilon_{r\\theta} & = \\cfrac{1}{2}\\left(\\cfrac{1}{r}\\cfrac{\\partial u_r}{\\partial \\theta} + \\cfrac{\\partial u_\\theta}{\\partial r}- \\cfrac{u_\\theta}{r}\\right) \\\\\n \\varepsilon_{\\theta \\phi} & = \\cfrac{1}{2r}\\left(\\cfrac{1}{\\sin\\theta}\\cfrac{\\partial u_\\theta}{\\partial \\phi} + \\cfrac{\\partial u_\\phi}{\\partial \\theta} - u_\\phi\\cot\\theta\\right) \\\\\n \\varepsilon_{\\phi r} & = \\cfrac{1}{2}\\left(\\cfrac{1}{r\\sin\\theta}\\cfrac{\\partial u_r}{\\partial \\phi} + \\cfrac{\\partial u_\\phi}{\\partial r} - \\cfrac{u_\\phi}{r}\\right) \n \\end{align}\n ", "7f52b86d059721d04a58387137adccda": " \\left\\langle \\!\\! \\left\\langle {n \\atop m} \\right\\rangle \\!\\! \\right\\rangle = (2n-m-1) \\left\\langle \\!\\! \\left\\langle {n-1 \\atop m-1} \\right\\rangle \\!\\! \\right\\rangle + (m+1) \\left\\langle \\!\\! \\left\\langle {n-1 \\atop m} \\right\\rangle \\!\\! \\right\\rangle, ", "7f52d38f3764483099245b0b2a845e83": "\\exists z\\in\\{0,1\\}^{q(n)}\\,\\Pr\\nolimits_{y\\in\\{0,1\\}^{p(n)}}(M(x,y,z)=1)\\ge2/3,", "7f5316b8678f92878963549a95989268": "\n\\tau_k = \\frac {1}{L-k} \\sum_{i=1}^{L-k} \\, \\mathrm{J}_{i, i+k}, \\,\\,\\, (k < L)\n\\qquad \\text{(5)} ", "7f537948857b277405641d0dd88db6cd": "|x_n - x_m| > k", "7f541fb0231c2245fd434cce572cc1e4": "S^{-1} f", "7f54473f37c3c5b165057269132854ae": " \\operatorname{build-param-lists}[o\\ x\\ y, D, V, []] \\and D[g] = [x, \\_, \\_]::[o, \\_, \\_]::[y, \\_, \\_]::[] ", "7f547973f04f556c29d2d6c19db587b3": "d=\\frac{1}{2}(v_f+v_i)\\Delta t", "7f54a022c350f4d4b1be3daa673d1c15": "\\Psi_g(h) = ghg^{-1}\\,", "7f54a4be52a432241a81d4823f595f01": "x[S]z", "7f54b1514982ec1d495c811c87db4d03": "\\begin{align}\n\\frac 12 \\mathbf{p} \\cdot \\mathbf{v} &\n= \\frac 12 \\vec{\\beta} \\gamma mc \\cdot \\vec{\\beta} c\n= \\frac 12 \\gamma \\beta^2 mc^2\n= \\left( \\frac{\\gamma \\beta^2}{2(\\gamma-1)}\\right) T\n\\,.\\end{align}", "7f54d7bd322e18965597e6f714633dcb": " E_n^{(1)} = \\langle n^{(0)} | V | n^{(0)} \\rangle ", "7f552e6debf7589013d2492bfd850e5e": "|\\uparrow\\rangle", "7f55396b04cc5d9dc94677103ce43b4d": " \\mathbf{e}_1 ^2 = \\mathbf{e}_2^2 =\\mathbf{e}_3^2 = -1, \\,\\, \\mathbf{e}_4^2 =0.\\!", "7f5548fe527407ef2918fc20b6682e8a": "\\hat{\\mathcal O}_{X,x}/m_y\\hat{\\mathcal O}_{X,x}", "7f556354b024853312902b2bd1f3a7f3": "y*(t)", "7f56520ba01daa2e1e77501fd2a81b11": "P_C(t) - P_H(t) = (1+t)Q(t).", "7f56c92482097753bfb9c2e64caa9c2b": "\\begin{cases}\nt &= \\gamma \\left( T + \\frac{v X}{c^{2}} \\right) \\\\ \nx &= \\gamma \\left( X + v T \\right)\n\\end{cases}", "7f56c99376cf7ed405e0956fa758dc33": "\\theta \\in [0, 1]", "7f56d4dd0c5223636f83c92738f4bdd5": " b_n(t) \\ ", "7f56e4bd7d6d706356bd8d000f2ff571": "\\alpha = j \\frac{h_{15}}{2G}", "7f56f7eb1f3d1941dd4230dea6390477": "g_{ij}=D[\\partial_i\\partial_j||]=D[||\\partial_i\\partial_j]=-D[\\partial_i||\\partial_j]", "7f56f9ba076a371c52bb358c0c841a38": "{O}^\\times_{O}", "7f5709e173c0bfda65d1f5f92c422693": "R=2\\times T+14", "7f5717482e11b0d241b23216c68c0ff6": "l(\\gamma)=\\int_\\gamma \\lambda(z,\\overline{z})\\, |dz|", "7f57389a62dfdca76aa4e2bf108e8c1f": "\\lnot (\\forall x P(x))", "7f57478bc6741ab56e4eb541d8758eeb": " \\int_S J_i \\mathrm{d} A_i = I \\,\\!", "7f575180a9096a5f28e7d78b103ed0e4": "K(3,2) = 2K(1, 2) + 2K(2,1) = 4", "7f5788d56bd683ffba59eed8ad6ecbe6": "\\vdash", "7f57ce5c29b329529f4e3f9a3765b114": "P_j", "7f57ee9de0b1fc6ea6678da497a583ed": "\\frac{d}{dx} (x\\pm i0)^\\alpha = \\alpha(x\\pm i0)^{\\alpha-1}.", "7f58138119ac39b769505b4430b8d3d1": " = \\sqrt{{{a+b}\\over 2}\\cdot {{a^2+b^2}\\over {a+b}}} = \\sqrt{{{a^2+b^2}\\over 2}} = R(a,b) ", "7f58383240cbfdbb126ecc8d6a705148": "g\\cdot f(h)=f(g^{-1}h)", "7f583d4fe4f74c54f59a118318c740a2": "A=2R^2", "7f584552db987bc780aaae1c65b249bd": "\nL(x) = \\frac{x}{3+\\tfrac{x^2}{5+\\tfrac{x^2}{7+\\tfrac{x^2}{9+\\ldots}}}}\n", "7f58a214ff09126c1060a397bf72a81f": " h\\in\\mathcal{H} ", "7f58ae8ba29ab47b36f0ab714cac1099": "L^2 =Q^2 = \\frac{s^2}{\\sigma_0^2}.", "7f58b9fef748da17c1677d1765658e31": " 6 + 2.56 + 1.715 = 10.275 ", "7f59a51287fd1a84923c14024392192f": "\\Theta(\\vert\\mathbb{C}\\vert M_{a})", "7f59a8ca9f1b8ee85a2acf7c6e93cea0": " I_t = ( 1 - \\Gamma_{TL} ) I_i \\, ", "7f59cb23c35fc4358e4f5d4343c30284": "\\hat{n}=\\{m | m \\equiv n \\mod k \\}.", "7f5a0c1aa836580b7c14bcd90384f9c9": "\\Delta y", "7f5ae0f451704a327eb8cff2ae46790e": "H(X)+H(x) \\ge \\ln(N)", "7f5b0d1b1ff062687ae1d65e1b59fc55": "F_{drag} = f v_{term}", "7f5b428e689270b757a99a5972e15991": "\nm\\frac{d^2 r}{dt^2} = F_1(r) + \\frac{L_1^2}{mr^{3}} = F_2(r) + \\frac{L_2^2}{mr^3} = F_2(r) + \\frac{k^2 L_1^2}{mr^3}\n", "7f5b7bde2180d6d92ef6e3b4afdbf23b": "\\scriptstyle \\mathcal{N}=\\{-T,\\dots,T\\}, T\\geq 1 ", "7f5bf6cf842cd247708cdfb4d0c5135b": "\n\n\\hat \\sigma _{\\bar x} \\,\\, = \\,\\,\\,{{s\\,} \\over {\\theta \\,\\sqrt {\\,n} }}{{\\sqrt {\\,\\gamma _2 } } \\over {\\sqrt {\\,\\gamma _1 } }}", "7f5c0dae6e508d77245398edef411df9": "\\mathbf{a} = \\lim_{{\\Delta t}\\to 0} \\frac{\\Delta \\mathbf{v}}{\\Delta t} = \\frac{d\\mathbf{v}}{dt},", "7f5c27c6a08b4cb0de5d894713579181": "\\scriptstyle Q(x) \\;=\\; \\frac{1}{2}\\operatorname{erfc}\\left(\\frac{1}{\\sqrt{2}}x\\right)", "7f5c56f176ebbfa5397116a8751b73bc": "(R, +)", "7f5c6e10ff6cf5ac88989d629720884e": "E_AU(x) \\geq E_BU(x)", "7f5c97ee2f640c4bd1cf5af55c560f50": "\\mathcal O(Nc) \\in \\mathcal O(N)", "7f5cbf63d4e0035e511aa4b88df8f365": "e^{i\\pi} + 1 = 0\\,\\!", "7f5d7ceff72780f0ce88cc5084b09c0c": "\\tau_{\\lambda_1}\\,", "7f5dd41a5a180338788f6e2c83a21f60": " Q(X)", "7f5decf3f74f81c80e1db570a80a5303": " y_{j} = x_{j} + U \\pmod 1", "7f5e0ce75f4f139d1a81aa74cbcde869": "\\scriptstyle \\, T^{ab}", "7f5e5b9ef1e6b68dcada5687c997080c": "\\textstyle {7 \\choose 3} = \\left(\\!\\!{5 \\choose 3}\\!\\!\\right)", "7f5f289bea8093fd86d2f33edf2c5491": "(++-)", "7f605d32fc8b80959ebc734d6c0fba58": " k \\in \\{ 0, \\ldots, n-1 \\} ", "7f60607a949bb9ff898051b01bab42b1": "w^u = 0", "7f606e44c99b84fd258fcf85388d84c6": " = \\displaystyle{{{\\sqrt{R_aG_a}\\lambda\\cos\\psi\\over\\sqrt{\\pi Z_\\circ}}}} \\,", "7f6137eb4f2f4158e7eb447bed505d29": " \\Phi = \\frac{U}{q}", "7f613bd2f9bee99fff3773e062a897b2": "\n\\begin{align}\n \\left[-\\frac{\\hbar^2}{2m}\\frac{d^2}{dz^2}+V(z)\\right]\\Psi(z) &=& E\\Psi(z),\n\\end{align}\n", "7f62ab2e8e2d9167bfe7198024865480": "y_i=\\beta_0 +\\beta_1 x_i +\\beta_2 x_i^2+\\varepsilon_i,\\ i=1,\\dots,n.\\!", "7f62add4ece739fcd2b6808c2d010528": "\n A \\begin{bmatrix} 5\\\\5\\\\5 \\end{bmatrix} =\n \\begin{bmatrix} 5\\\\5\\\\5 \\end{bmatrix} =\n 1 \\cdot \\begin{bmatrix} 5\\\\5\\\\5 \\end{bmatrix}\n ", "7f62b937508bd831a54191614958b2bb": "\\exists i 0", "7f6750d3bc27d7acc5e9f9357c000483": "R(V,K)=R(V,K_V) \\,", "7f678afdac6bcf0fe72aad8445416b5a": "\\left\\langle{ a,b : a^A = b^B = (ab)^C = 1 }\\right\\rangle", "7f67ee6432c6e00464554ee2d33b02f1": "\\mathbf{f}(x,t)\\equiv 0", "7f680d3f7de09f0a92c3ba20565cdd55": "2^F\\,", "7f682977f25eb9d6f6d1fec73042e7de": "v_{7}", "7f6862e1413e74d94351abe315c259c3": "1-par", "7f688b7d0116333db2fdcfe2b0130b62": "V_\\mathrm{CB} \\ ", "7f68bb506cd6ed9f2c52f12c28d0b6cf": "\\beta = v/c", "7f68d85a9fbe3956c251c9a5c2090b97": "\n\\text{sign}(x) = \\begin{cases} \n +1 & x > 0 \\\\\n -1 & x < 0 \n \\end{cases}\n", "7f693e97b66c1f69901316a83d5f62e4": "\\Delta T = 2 \\pi A \\int_0^{\\infty} G(k) exp (-\\pi^2k^2(\\omega_0^2 + \\omega_1^2)/2) k dk", "7f694040325fe677954190988361835c": "v'(L \\; p) = \\text{False}", "7f69c02da634efb94245d4e3fcc8f67f": "\\scriptstyle x,y\\in X", "7f69c24857a2ea9d5449c46f4ea50b51": "q_\\text{P} = \\sqrt{ \\hbar c}.", "7f69d8e6acbbb975587e26623885d9f3": "\\beta(g) ", "7f69de9a02d6baf84076e0d42bb52fb1": "B_R^\\delta f", "7f6a2f209b9011ceebd3b14e26e6a6f9": "f = a'bc + abc", "7f6a952b179e0c28c9b9f3d1f66dd414": " \\phi_k \\, \\sim \\, \\mathrm{Dir}(\\beta) ", "7f6aba5c2dbc283c43aae1abfbcd9fbd": "Z_{m_n} = \\frac {Y_{m_n}} {k^2 + Y^2 - Y_{m_n}^2}", "7f6aea7ca683d84c671a8964d6f94da2": "p(7k+5)\\equiv 0 \\pmod 7", "7f6af822c4c5730eaa3b3d02eb695e9d": " t \\approx \\frac{69.3}{r} + 0.33", "7f6b30387fd1d9756db04cd82ebf4690": "\\phi : K^n \\mapsto \\mathbb{C}", "7f6b52e7dc16296b1493d411541fd778": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ 2 \\ \\sin u\\,\\ \\left(5\\sin^2 i \\ \\sin^2 u\\ -\\ 3\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right) \\ \\cos u\\right)\\ du\\ = \\\\\n&-10\\sin^2 i \\ \\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^3 u\\ du \\\\\n&+6\\ \\int\\limits_{0}^{2\\pi}\\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin u\\ du \\\\\n&-15\\ \\sin^2 i \\int\\limits_{0}^{2\\pi} \\hat{r}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^2 u \\ \\cos u\\ du \\\\\n&+3\\ \\int\\limits_{0}^{2\\pi} \\hat{r}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\cos u\\ du \\\\\n&+\\frac{15}{2}\\sin^2 i\\ e_g\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\ \\ \\sin^3 u \\ \\cos u\\ du \\\\\n&-\\frac{15}{2}\\sin^2 i\\ e_h \\ \\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\ \\ \\sin^2 u \\ \\cos^2 u\\ du \\\\\n&-\\frac{3}{2}\\ e_g \\ \\int\\limits_{0}^{2\\pi} \\ \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\ \\sin u \\ \\cos u\\ du \\\\\n&+\\frac{3}{2}\\ e_h \\ \\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\ \\cos^2 u\\ du\n\\end{align}\n", "7f6b7f1aebc39512b4cbfb1bd30e3b15": "\\mathbf v", "7f6ba95a36f178472df591765b31dc5a": "N_{M_j} = \\begin{cases}\n 0 & j = p\\text{ and }m_p = \\text{reject}\\\\\n 1 & j = p\\text{ and }m_p = \\text{accept}\\\\\n \\max_{m_{j+1}} N_{M_{j+1}} & j < p\\text{ and }j\\text{ is odd} \\\\\n \\text{wt-avg}_{m_{j+1}} N_{M_{j+1}} & j < p\\text{ and }j\\text{ is even} \\\\\n\\end{cases}", "7f6bb2b8a43890bc46ea8123c54412ce": "A\\cup B\\in \\mathrm{RAT}(N)", "7f6be9c8521e735f8b8cd8532d34499c": "\\mathbf{Q}=r^2\\mathbf{F}+2\\mathbf{r}(-V+\\sigma)", "7f6c3e8fb4217a2d25696bf549cb2d5a": " \\nabla \\cdot V=0 ", "7f6c5d66440780f301be3158b65a0d5f": "(-1)^{|V(G)|} P(G,-1)", "7f6d0de3cfc7a56b5d67ecb922ddd523": " T= \\frac{1}{2}m_1 \\mathbf{v}_1\\cdot\\mathbf{v}_1 + \\frac{1}{2}m_2 \\mathbf{v}_2\\cdot\\mathbf{v}_2 = \\frac{1}{2}(m_1+m_2)L_1^2\\dot{\\theta}_1^2 + \\frac{1}{2}m_2L_2^2\\dot{\\theta}_2^2 + m_2L_1L_2 \\cos(\\theta_2-\\theta_1)\\dot{\\theta}_1\\dot{\\theta}_2.", "7f6d3a77f78bd361b02fee38d780353d": "\\int \\operatorname{arcsch} \\, x \\, dx = x \\, \\operatorname{arcsch} \\, x + \\vert \\operatorname{arsinh} \\, x \\vert + C , \\text{ for } x \\neq 0 ", "7f6d5275918dfb1d5583665d355a7b70": " S_2 = \\{1\\} ", "7f6d551a010c961cd7f9a3a5e2563815": "\na_{\\varphi} = 2 \\dot{r} \\dot{\\varphi} + r \\ddot{\\varphi} = 0\n", "7f6db390bebc3fb80d719871587cd135": "-s,-s+1, \\ldots,s-1, s", "7f6dc21ba23575d1b51524f783a04950": "\\phi_R^n\\circ f:R^n(X)\\rightarrow S_R", "7f6e20578c7bfe69b31dab3c7add003e": " \t \t =\\mbox{ } \\frac{pq}{\\left[pq+\\left(1-p\\right)\\left(1-q\\right)\\right]}\\mbox{ } ", "7f6e2fa986f9fa671d7fd4a016c88191": " \\text{EVaR}_{1-\\alpha}(X):=\\inf_{z>0}\\{a_X(\\alpha,z)\\} ", "7f6e6d6ea1e6e4de2fe74921245820de": "A=\\frac{1}{2}\\int_a^b \n\\frac{\\partial\\theta^j}{\\partial t}\ng_{jk}(\\theta)\\frac{\\partial\\theta^k}{\\partial t} dt", "7f6e80c21e9288ee5a126750b2e94f0f": " \\psi_n(x) = \\sqrt{\\frac{1}{2^n\\,n!}} \\cdot \\left(\\frac{m\\omega}{\\pi \\hbar}\\right)^{1/4} \\cdot e^{\n- \\frac{m\\omega x^2}{2 \\hbar}} \\cdot H_n\\left(\\sqrt{\\frac{m\\omega}{\\hbar}} x \\right) ", "7f6e9627802017e75c59c75528797224": "A + B := \\{a + b: a \\in A \\and b \\in B\\}", "7f6eb1947a06b804ac4cf4877ccaba70": "{x^2 \\over a^2} - {y^2 \\over b^2} = 1 \\,", "7f6f5a357e8b9149f7787585d799facd": "c_i \\neq c_j", "7f6f642b502ced5d0c76c9ffb21836fc": "\\pm e_0\\pm e_a\\pm e_b\\pm e_c", "7f6fa2f86fb46143756485ce08e471c5": " A\\to A\\otimes I\\xrightarrow{\\eta^r}A\\otimes (A^r\\otimes A)\\to (A\\otimes A^r)\\otimes A\\xrightarrow{\\epsilon^r} I\\otimes A\\to A", "7f7048f58841e7a087c7716205c60977": "f_\\mathrm{oscillation}=\\frac{1}{2\\pi\\sqrt{R_2R_3(C_1C_2+C_1C_3+C_2C_3)+R_1R_3(C_1C_2+C_1C_3)+R_1R_2C_1C_2}}", "7f7063e5e6c8d81a2de05deea7988ca1": "\\Delta G^{mic} _m = \\mu_{micelle} - \\mu^{0} _{sur} = RTlnCMC", "7f7072f5fb0c2b3dfd882140506bb5bb": "\\lambda=d(2-d)\\,\\in (0,1)", "7f7087ebf52925f8a789ef87952d8605": "\\sum_{n\\in\\Z} a_n \\tilde a_{n+2m}=2\\cdot\\delta_{m,0}", "7f70cfe750cde8c3ecaea5d96f56b90e": "e_i e_j = - \\delta_{ij}e_0 + \\varepsilon _{ijk} e_k,\\, ", "7f70edb1a860333dfecc07eff1f920e3": "\\frac {-\\partial_\\mu[\\partial_\\sigma V_\\nu] + \\Gamma^\\rho{}_{\\sigma\\nu} \\partial_\\mu V_\\rho + \\Gamma^\\rho{}_{\\mu\\nu}\\partial_\\sigma V_\\rho + \\Gamma^\\rho{}_{\\mu\\sigma}\\partial_\\rho V_\\nu + \\partial_\\mu [\\Gamma^\\rho{}_{\\sigma\\nu} V_\\rho] - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma}V_\\rho - \\Gamma^\\alpha{}_{\\mu\\sigma}\\Gamma^\\rho{}_{\\alpha\\nu}V_\\rho}{\\partial_\\mu\\Gamma^\\rho{}_{\\sigma\\nu}V_\\rho\n - \\partial_\\sigma\\Gamma^\\rho{}_{\\mu\\nu}V_\\rho\n + \\Gamma^\\alpha{}_{\\sigma\\nu}\\Gamma^\\rho{}_{\\alpha\\mu}V_\\rho\n - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma}V_\\rho}", "7f711ac1314a21599dc5a9c5bb757eff": "\\overrightarrow{b}", "7f714844b315b52509cecf2f1f7de7ea": "\n\\begin{matrix}\n\\mbox{independent}\\qquad\\\\\n\\underbrace{\n \\overbrace{\n \\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix},\n \\begin{bmatrix}0\\\\2\\\\-2\\end{bmatrix},\n \\begin{bmatrix}1\\\\-2\\\\1\\end{bmatrix}\n },\n \\begin{bmatrix}4\\\\2\\\\3\\end{bmatrix}\n}\\\\\n\\mbox{dependent}\\\\\n\\end{matrix}\n", "7f717bec325b84eb3f9a5ced1dcdc3b9": "I=\\bigcap_{i=1}^t Q_i", "7f71a95a5a8a4f8b5182c9f2a2495ceb": " (r\\# h)(r'\\#h')=r(h_{(1)}\\boldsymbol{.}r')\\#h_{(2)}h', ", "7f7220a92796afcffeec442497a44294": "T=\\left [ a_1, b_1 \\right ) \\times \\left [ a_2, b_2 \\right ) \\times \\cdots \\times \\left [ a_n, b_n \\right ) \\subseteq \\mathbf{R}^n.", "7f7285160efc59bb93268ee3c6769cfe": "\\kappa=\\sqrt{\\left(1+\\frac{h}{R}\\right)^2-1}\\ .", "7f7288e247d469080320297dd0b162b8": " (R\\otimes R)\\times (R\\otimes R)\\to R\\otimes R,\\quad (r\\otimes s,t\\otimes u) \\mapsto \\sum _i rt_i\\otimes s_i u, \\quad \\text{and}\\quad c(s\\otimes t)=\\sum _i t_i\\otimes s_i. ", "7f7298e60ee162a30c9e01fe047fe93b": "Q_c^{(c)}(t) = 0", "7f72d9a3557782e5f25961aff257acda": "\\sum_{n\\in A} f(n) = \\sum_{n\\in \\sigma(A)} f(n)", "7f72de408da6e6ba38d5cede3cd99eb9": "\n \\mathbf{F}_1 =: N_{11}\\mathbf{e}_1 + V_2\\mathbf{e}_2 + V_3\\mathbf{e}_3 \n ", "7f72f4b806ea7407e44c3062e6e28715": " \\frac{\\frac{ K^- + \\bar{K}^0 }{2} + \\frac{ K^+ + K^0}{2}}{2} = 248~\\mathrm{MeV}/c^2", "7f7300eef3d0b12d304dfae815a0ca7a": "\\sin(\\omega_0 n) u[n]", "7f730e271ef35d2828cf281d4bccfc16": "1 + \\frac{a_1\\dots a_p}{b_1\\dots b_q.1}\\frac{cz}{d} + \\frac{a_1\\dots a_p}{b_1\\dots b_q.1} \\frac{(a_1+1)\\dots(a_p+1)}{(b_1+1)\\dots (b_q+1).2}\\left(\\frac{cz}{d}\\right)^2+\\dots", "7f732c5a345161a70a8609ee7ddd8bb2": "D(t)=\\frac{1}{T} \\sum_{1=t- \\frac{T}{2} }^{t+ \\frac{T}{2} } \\begin{vmatrix}R(i)-C(i) \\end{vmatrix}", "7f737aaadffb67aa4c3e8d743f109104": "{K_{ab}}^c", "7f73831832cad047624074a05d9d2516": "\nR = - V_{0}\n", "7f7383aed7b8a51eefcfb93ab53f6488": "\n M_c = 175 - 7.5 R_a \\,.\n ", "7f739b0965e1b776d2d313a57d99cdaa": "w((v|u)+r)\\geq d", "7f73d8feb3e961db5be8efd82a32841f": "S = \\sqrt{\\frac{9}{5}\\left( r'^2 + g'^2 \\right)}", "7f747e7aebb5ae16ebe1897241dc3434": "\\mathbf{\\alpha}", "7f74c4433339e091d0e4f5da598c2819": "\\scriptstyle \\pi /2", "7f74ec92112043213d120108b3cdc6e0": " \\bar{n}_i ", "7f753bd6fba45f8ee647711b4b609e88": "d\\bold{d}=\\bold{P}dV=\\bold{P}d^3\\bold{r} ", "7f757fce70e2327ab4bd6e3958983b76": "\n\\begin{bmatrix}\n\\begin{vmatrix} -2 & -1 \\\\ -1 & -2 \\end{vmatrix} & \n\\begin{vmatrix} -1 & -1 \\\\ -2 & -1 \\end{vmatrix} & \n\\begin{vmatrix} -1 & -4 \\\\ -1 & -6 \\end{vmatrix} \\\\ \\\\\n\\begin{vmatrix} -1 & -2 \\\\ -1 & -1 \\end{vmatrix} &\n\\begin{vmatrix} -2 & -1 \\\\ -1 & 2 \\end{vmatrix} &\n\\begin{vmatrix} -1 & -6 \\\\ 2 & 4 \\end{vmatrix} \\\\ \\\\\n\\begin{vmatrix} -1 & -1 \\\\ 2 & 1 \\end{vmatrix} &\n\\begin{vmatrix} -1 & 2 \\\\ 1 & -3 \\end{vmatrix} &\n\\begin{vmatrix} 2 & 4 \\\\ -3 & -8 \\end{vmatrix}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n3 & -1 & 2 \\\\\n-1 & -5 & 8 \\\\\n1 & 1 & -4\n\\end{bmatrix}.\n", "7f75906f7e39aec97ca4fd0d3555be6c": "T = tY ", "7f75ce4ba392a793cb5a3dba24edb211": "{}_{\\ 84}^{216}\\mathrm{Po} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 82}^{212}\\mathrm{Pb}\\ \\mathrm{(0.15\\ s)}", "7f75eddbe4c71e1e81d28963dcc68ad5": " \\forall x \\;(0(x) \\rightarrow P(x,y)) ", "7f76480fb9cd28decc5d266e33c5cfdf": "h = \\frac{c\\alpha^2 A_{\\rm r}({\\rm e})M_{\\rm u}}{2R_{\\infty} N_{\\rm A}}.", "7f764a116c5cfb89fa3741501dcac5c9": "\\frac{dv}{dt}=Mv", "7f7678fdf877f57b500a47a4ff4fc46d": "\\mathbf{C}_2", "7f76a400b558b5f04c0ff49a72f4fe60": "e^{itQ} e^{isP} - e^{-i st} e^{isP} e^{itQ} = 0.", "7f76f1d5ba8206d16ff882d0a2b0f1cd": "P_r(\\theta) = \\sum_{n=-\\infty}^\\infty r^{|n|}e^{in\\theta} = \\frac{1-r^2}{1-2r\\cos\\theta +r^2} = \\operatorname{Re}\\left(\\frac{1+re^{i\\theta}}{1-re^{i\\theta}}\\right), \\ \\ \\ 0 \\le r < 1.", "7f770ec90737cb58e66339dfdbd18b0e": "i\\ge 1", "7f775c400f826b5882a03bf4a5e72834": " e^{-||s_1 - s_1||^2/2\\sigma^2}", "7f7763f14d4732fcbe6c493f59a05f29": "\n\\begin{align}\n\\left(\\frac{\\partial T}{\\partial p}\\right)_H \n& = -\\left(\\frac{\\partial H}{\\partial p}\\right)_T\n \\left(\\frac{\\partial T}{\\partial H}\\right)_p\n\\\\\n& = \\frac{-1}{\\left(\\frac{\\partial H}{\\partial T}\\right)_p}\n \\left(\\frac{\\partial H}{\\partial p}\\right)_T\n\\end{align}\n", "7f778be12e6e17b59ffe48444ffd2b4d": " F_{X}", "7f77be3b96feeed409ef857b734b91bf": "f(x,\\theta)", "7f77c46c2f378d149d2432efd83b7c36": "F=F_0 - k_BT\\!\\ln\\left( Q \\right)", "7f77e8dce87f25ee6e59c31aac0c20ce": "C\\sum_{n=1}^{\\infty}\\beta_n0, {\\ } b,s,\\beta>0 \\\\[6pt]\n& = 1-e^{-bsx}, {\\ }\\beta=1\\\\\\end{align}", "7f8421d1899ead8ff57b59827ea3aeec": "f \\in\nL(G-D)", "7f8429043f1f45ad600ef0123751f712": "\\psi(\\Omega+1)", "7f8498e4a0e510cbf70c043b1f2b4d48": "p_0 \\in C", "7f84be6075d22e63737f6fc0fdcbbc7c": "y''_2=\\frac{y''_1}{2}\\left(-1+ \\sqrt{1+\\frac{8} {(y_1'')^3}}\\right)", "7f84bffb0b99955b3b082cb2716403f8": "\\omega(X,D)(E)=0", "7f84fd4a509ad0da2615cb307562b5a0": "\\|P_n-P\\|_{\\mathcal C}=\\sup_{c\\in {\\mathcal C}} |P_n(C)-P(C)|", "7f85075ee22c2fca49ded3e5a3f7dcae": "\\mathbf{i} = (1,0,0),", "7f854ad92759b60406deffc8dc86ddf3": "-\\ell \\leq m \\leq +\\ell", "7f855231946612242e0b639bec44372b": "\\epsilon_r''=\\frac{V_c}{4V_s}\\frac{Q_c-Q_s}{Q_cQ_s}\\,", "7f85879f60f0ddc7d0532a833fb46709": " F(z) = \\frac{1 + z}{1 - z}, \\quad |z| < 1", "7f85897f7cddaabc0089243d906f57ac": "\\sqrt{3}\\approx 1.73", "7f8668e7196d5b3d08c227467cc76ef8": "\\scriptstyle-\\pi/2 ", "7f8686513e37185327bbccb3a6e36e83": " E = {n h \\over 2\\pi} ", "7f86b338628a78e87d98bad3df3758c4": "r_g", "7f86b3a769371ccf38ad22423aede61c": "\\lambda_n=k_n^2-k_n", "7f86d24a6e4a5e2b65b1f2959e94832f": "\n\\mathbf{U}^\\dagger = \\mathbf{U}^{-1}.\n", "7f86ecfa01f30c420f64818d5c503f8a": " U = u(r_{12})+ u(r_{13})+u(r_{23}) +u(r_{12},r_{13},r_{23}),", "7f86f2aa73c28d25fe022b8792dabeb1": "x*(t)", "7f8717af4863d83e0891c1549b769a26": "d_{m', m}^j = (-1)^{m-m'}d_{m, m'}^j = d_{-m,-m'}^j", "7f878bfd26d9c403a095cf8ae28d1f0a": "\\lim_{r\\rightarrow 0^+}\\frac{1}{|B(x,r)|}\\int_{B(x,r)} \\!|f(y)-f(x)|\\,\\mathrm{d}y=0.", "7f8797bfd05b29b8a0ccce8d524d11c6": "N=\\left \\{S, A,B\\right \\}", "7f87d3bdbd59c67d0b244f56bdb62140": " \\mathbf{\\mu^*}=\\sum_{i=1}^N w_i \\mathbf{x}_i.", "7f88081dae30363ae070821498ddc423": "\\cos(\\theta) = u \\cdot s.\\,", "7f881063b880d97d38195d99d20100a6": "n = 137", "7f8881247a52a49c9c53f7927880513b": "\\scriptstyle \\gamma\\,(t)", "7f88bc9931e1d1bcfacda8c3bd749a76": " |x-\\mu| ", "7f88da6f007a5c6848e825736807d81a": "a_{\\mathrm{slow}}^2 = \\frac{1}{2} \\left[\\left(c_s^2 + V_A^2\\right)-\\sqrt{\\left(c_s^2+V_A^2\\right)^2-4c_s^2V_{A}^2 \\cos^{2} \\theta_{Bn}}\\,\\right],", "7f8904c65c66bba8e0f653e52fd66f22": "\\mathbf{e}_3 ", "7f891630a1f209a94fd9c9056436c314": "\\delta \\int 2T(t) dt=0", "7f891a45bcb956abeb06ade2b1923672": " \\ddot{\\bold{r}} = -\\frac{\\mu}{r^2}\\bold{u}", "7f892a13f323f685ac0fd7418c91516d": "\\varphi_{\\alpha\\beta}", "7f8936828e555aef4c61ef4469380d11": "X\\circledast Y:= \\text{Hom}(X,Y^\\star)^\\star,", "7f898cc1288b95c4da93188a7a12e1cf": "\\mathfrak{E}=[X]^{>\\omega}", "7f89a3abfab98d1817796ca9ab4c6adb": "\n \\begin{align}\n &1,\\, 1,\\, -1,\\, 1,\\, 2,\\, -1,\\, -3,\\, -5,\\, 7,\\, -4,\\, -23,\\,\n 29,\\, 59,\\, 129,\\\\\n &-314,\\, -65,\\, 1529,\\, -3689,\\, -8209,\\, -16264,\\dots.\\\\\n \\end{align}\n", "7f89c92d40e37b8e8d2aea99f6d32b48": " c = 2(nq-mp),\\,", "7f89d921063dd84a7c47c79fae9d2a1b": "a\\propto e^{Ht}", "7f8a0991f0f0b2d1d0c643719ab46419": "\\mathbf{y}_k = {\\nabla f(\\mathbf{x}_{k+1}) - \\nabla f(\\mathbf{x}_k)}.", "7f8a3577099e1e1da811505a40a4af4e": " q _{v \\setminus w} (d(v)-1)", "7f8a3678f241279db3f0039e1a9fb09d": "\\operatorname{ad}_g(\\cdot) = \\{\\cdot,g\\}", "7f8a8d59ce6df92920428355eb75326e": "N(a)", "7f8ac0febfca0e672b5ec4eac7f4ef91": " G' = \\frac {\\Delta T_s} {\\Delta d} \\cos \\phi", "7f8b458bee61bb6ddf6b7194df91cb38": "i = \\frac{dh}{dl} = \\frac{h_2 - h_1}{\\mathrm{length}}", "7f8bb2a6e101fa9a74750140a2657932": "\\Pi_2 : \\bold {n}_2 \\cdot \\bold r = h_2", "7f8bd4b7f731ddf449c88fece8cee4b4": "a(x)\\frac{d^2y}{dx^2} + b(x)\\frac {dy}{dx} +c(x)y +\\lambda w(x) y =0, ", "7f8c1165bdcbc6ead59a4f28bb470bd3": "\\frac{\\sqrt{\\alpha}}{\\pi} \\,", "7f8c22a710ee2d5c01d6ac92717ec40b": "e^C", "7f8c700084b9bbcc867ca15dd356f9f1": "b=m_1 + n - N -(m_1+n)\\omega", "7f8cc211b30ba13c1f23cb71635a3cb1": "|\\{x \\ : \\ (Mf)(x) > \\alpha\\}| \\leq \\frac{c}{\\alpha}\\int_{\\mathbf{R}^n} |f|.", "7f8cf2a606059999b734d993bb8488d6": "G_p\\cong H_q", "7f8cff71251135b43251aada11dc4e47": " \\displaystyle{b_n={1\\over n}\\sum_{m=1}^n ma_m b_{n-m}}", "7f8d15879422c008a202b0c57930ad2b": "\nS= \\int_t \\sum_i {1 \\over E_i} \\left({d\\psi_i\\over dt}\\right)^2 - E_i \\psi_i^2\n", "7f8d2febbe0f4be00ec71441b2c0d4e1": "\\operatorname{E}(1)=G(1^-)=\\sum_{i=0}^\\infty f(i)=1.", "7f8d32fbc0ef7ffa9d68a2c0481c2cbe": "\nD_\\alpha (-\\hbar ^2\\Delta )^{\\alpha /2}\\phi (\\mathbf{r})+V(\\mathbf{r})\\phi (\n\\mathbf{r})=E\\phi (\\mathbf{r}). \n", "7f8d44d1245ac0beb2dfa756701e7db4": "C_{1} = T_{1}", "7f8d74ff94fc4b3ee1fc74f52df9a128": "R=P+Q", "7f8d8b71a8200b992ad8366f8db5bcaf": "O\\left (M \\cdot N \\right)", "7f8db339ad0c0331f4befd3794105898": "\\operatorname{K}_3", "7f8e22c19b2513613769168973edab29": "M-1", "7f8e429e9351846a6243e7fb1961b6f3": "wR/L_{st}^2", "7f8e74083b021932c713e593597635b0": "\\textstyle i=1, 2,\\ldots, 6", "7f8e776a3ea61146beac41d90c00bc8c": "a_n = 2^{1/4 - n/2}", "7f8e7c3f670663818e45e98d94d03ec7": "\nU^T(\\hat{\\theta}_0) I^{-1}(\\hat{\\theta}_0) U(\\hat{\\theta}_0) \\sim \\chi^2_k\n", "7f8e8acfb9b1bef699944754fc33eb52": "\nH = {1 \\over {2g}} ( V_1^2 - V_2^2 ). \\,\n", "7f8eab9e1b7e66fb4c91f8be2136a3c5": "\\hat{x} = W(y-\\bar{y}) + \\bar{x}", "7f8ef812ec64febd57ad3aa6b92c8f53": "\\mathit{MPC}=\\Delta C/\\Delta Y= 50/60= 0.83", "7f8f1fdeae97c4bc0acf5b026c8eaba5": "\\mu(A ) = \\inf_{A\\subset U}\\mu (U)", "7f8f2e2b1f253e2deb44097c8e87fbea": "\\Omega^8BO\\simeq \\mathbf Z\\times BO ;\\,", "7f8f3b8a72dfeaa667dae1ec2a8f05cb": "D(a,b;c)=D(b,a;c),", "7f8f43c1a3b9800d41dd70f356c31c1e": " \\chi_T (c_H -c_M) = T \\left( \\frac{\\partial M}{\\partial T} \\right)_H^2 ", "7f8fb9b8203741dc534d899b655c0b6c": "\nE_{\\mathbf{k}} \\phi_\\lambda(\\mathbf{k}) - \\sum_{\\mathbf{k'}} V_{\\mathbf{k}-\\mathbf{k'}} \\phi_\\lambda (\\mathbf{k'}) = E_\\lambda \\phi_\\lambda (\\mathbf{k})\\,,\n", "7f8ff8a6e0e9cd947426296ac02c4792": "H_C", "7f8ffcc0ea42c51d07b674f450b7d20e": "\\int\\frac{dx}{xR}=\\frac{1}{\\sqrt{-c}}\\operatorname{arcsin}\\left(\\frac{bx+2c}{|x|\\sqrt{b^2-4ac}}\\right), ~ c < 0, b^2-4ac>0", "7f90e0b0bdaf6216ffde13af874e8fe6": "(u+v)_i = u_i+v_i", "7f910be237ec4acbe30cd8571cc08d69": "S^3(A_3)", "7f915c6a81679d424494ea673bdb20b4": "\\hat{a}\\ ,\\ \\hat{b}\\ ,\\ \\hat{n}\\,", "7f91794da974d842d151cf1fb770480a": "Y(X) \\equiv (X + \\lceil\\sqrt{N}\\rceil)^2 - N \\equiv 0 \\pmod{p} ", "7f91826d4868323e03c240c8184cbd63": "\\sqrt { \\Delta x^2 + \\Delta y^2 }=\\sqrt{ ({\\Delta x^2 + \\Delta y^2})\\,\\frac{\\Delta x^2}{\\Delta x^2}}=\\sqrt { 1 + \\frac{\\Delta y^2}{\\Delta x^2}}\\,\\Delta x=\\sqrt { 1 + \\left(\\frac{\\Delta y} {\\Delta x} \\right)^2 }\\,\\Delta x", "7f9191d15df7dd68f95c33c6ff4d5b4d": " x^2 - nx = 1,\\,", "7f91983cdac2c65478c47ff81005c51b": "\\begin{align}\n\\frac{k_H}{k_D} &\\cong e^{-1/2\\{\\sum\\limits_{i=1}^{3N^\\ddagger-7}(u^\\ddagger_{iH}-u^\\ddagger_{iD})-\\sum\\limits_{i=1}^{3N-6}(u_{iH}-u_{iD})\\}}\n\\\\ & = e^{1/2\\{ \\sum\\limits_{i=1}^{3N-6}\\Delta u_i - \\sum\\limits_{i=1}^{3N^\\ddagger-7}\\Delta u^\\ddagger_i \\}}\n\\end{align}", "7f91bc82381ce4cac2b3283d7b27e2f6": "\\mathcal{O}_k ", "7f91d39de7db4516f42818470ebe69c0": "{\\mathbf u}= u^je_j", "7f91d615acdc9ecc32f44db51601baa8": "\\textstyle|x|=\\sum_{i=1}^m{x_i},\\ |k|=\\sum_{i=1}^m{k_i},\\ x^k=\\prod_{i=1}^m{x_i^{k_i}}", "7f9277a6e171d43920768703610acfac": "\\operatorname{Hom}(X,-)", "7f930f9ca38221516ee86630d5b264da": "G' = \\sum_{v}{P^{-}(v)\\ln\\left({\\frac{P^{-}(v)}{P^{+}(v)}}\\right)}", "7f933b60f52d1c78d0d054a58343e4cd": "A'(0) + A'(0)^\\mathsf{T} = 0 \\, ", "7f936beae49978ff8d2452c5f9a0a331": "t_{1/2} = \\tau \\ln 2 = \\frac{\\ln 2}{\\lambda}.", "7f93d0aab07762b07290f6ef231566fd": " \\zeta(z + \\omega) = \\zeta(z) + \\eta \\ ", "7f93fef520d8a1f9c2a2e6a6a996c41c": "\\color{Periwinkle}\\text{Periwinkle}", "7f946245304b598e5a0e9ca9196334be": "\\| S_{T} \\|_{p} \\leq \\frac{e}{e - 1} \\left( 1 + \\| X_{T} \\log X_{T} \\|_{p} \\right)", "7f9466bb9fb4499d29253bfd8ae3f401": "E = \\sum \\gamma_S A_S = \\gamma_{LG} A_{LG} + \\gamma_{SG} A_{SG} + \\gamma_{SL} A_{SL}\\,", "7f94711159ccb51f014c7b5cf3e88fc5": "u_s(x)", "7f9495d93bde36faeabac26a6695bff2": "1 / \\epsilon", "7f94c7e9bc69de6553930888573f9bcb": "\\pi'=r_0, r_1, \\ldots", "7f9514e7f3597a3f8bdc99b46c5b8d30": " \\frac{k_t}{k_v k_p} = T^2 \\,\\!", "7f951ddafca22973c7ef13929a8908a9": " X = A^+ ", "7f953132ed003893186d0af6cc716d3e": "r_i = \\frac{n_i}{n_{tot}-n_i}", "7f95522d9f90e494b04bc29bb43282ea": "V_\\mathrm{S}", "7f956ca7a46e1f9298d1f41af5c0373f": "f_1,\\ldots, f_k", "7f959a3aa8daf9b25e89bd6bb0caf01d": "P(t)=P_se^{-\\frac{t}{t^*}}\\left(1-\\frac{t}{t^*}\\right).", "7f959a5dc60c55267825b950e9e622b8": "\n\n\\begin{bmatrix}\n1 & 3 & 1 \\\\\n1 & 0 & 0\n\\end{bmatrix}\n+\n\\begin{bmatrix}\n0 & 0 & 5 \\\\\n7 & 5 & 0\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1+0 & 3+0 & 1+5 \\\\\n1+7 & 0+5 & 0+0\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & 3 & 6 \\\\\n8 & 5 & 0\n\\end{bmatrix}\n", "7f95e2ddb60fd00ca3f691a34db56ff7": "W \\subset V", "7f9636ffe524a72a851500c5a7adbd7c": "c_1,\\dots,c_k", "7f9655477429e1216a086e3185ea0964": "{dx \\over d \\tau} = \\{ x , \\lambda C \\} , \\;\\;\\;\\; {dp \\over d \\tau} = \\{ p , \\lambda C \\} \\;\\;\\;\\;\\;\\; {dt \\over d \\tau} = \\{ t , \\lambda C \\}, \\;\\;\\;\\; {dp_t \\over d \\tau} = \\{ p_t , \\lambda C \\} ", "7f9658a8611d26374a7798b5cad44c84": "\\hat y_i", "7f965efc0184d5ee49df703f2eeed426": "{\\scriptstyle (z-1)!}", "7f97515e6ea7b1e843905a328b1ac259": "H_{(-1)^k}(R)^n", "7f9825669565810f1a1343408c76ab14": "v = \\Gamma_u \\circ w \\Leftrightarrow q_1 \\circ v = u \\circ q_2 \\circ v \\Leftrightarrow q_1 \\circ \\psi = u \\circ q_2 \\circ \\psi \\Leftrightarrow q_1 \\circ \\psi = u \\circ \\phi.", "7f984533aa08b1dbdeebafaab2b30eed": "G_q(x)= \\begin{cases} 0 & \\text{if } x < -\\nu, \\\\\n\\displaystyle \\frac{1}{2} + \\frac{1-q}{c(q)} \\sum_{n=0}^\\infty \\frac{q^{n(n+1)}(q-1)^n}{(1-q^{2n+1})(1-q^2)_{q^2}^{n}}x^{2n+1} & \\text{if } -\\nu \\leq x \\leq \\nu \\\\\n1 & \\text{if}\\ x > \\nu\n\\end{cases}", "7f98b0135c02379f9a2d61d100b5b11f": "\\sum_S \\exp\\biggl(\\sum_{ij} S_{i,j} S_{i,j+1} + S_{i,j} S_{i+1,j}\\biggr).", "7f98f8de859d49868e8f8fce6f0b81aa": "g(x, y)=f(x, y, 1).\\,", "7f98fb018d8697a977a7e833d05aff94": "(\\boldsymbol{\\nabla}\\times\\boldsymbol{T})\\cdot\\mathbf{c} = \\boldsymbol{\\nabla}\\times(\\mathbf{c}\\cdot\\boldsymbol{T}) ~;\\qquad (\\boldsymbol{\\nabla}\\times\\mathbf{v})\\cdot\\mathbf{c} = \\boldsymbol{\\nabla}\\cdot(\\mathbf{v}\\times\\mathbf{c})", "7f99678663adbc50f1096b4dafe70388": "\\begin{bmatrix}x_2 \\\\ x_3\\end{bmatrix} = \\begin{bmatrix}2 & -1 \\\\ -1 & 1\\end{bmatrix}^{-1} \\begin{bmatrix}1\\\\2\\end{bmatrix} = \\begin{bmatrix}3\\\\5\\end{bmatrix}", "7f99b36b81c32e5a37d7baa3d6b37c7e": "\\overline{ B_r(p) }", "7f99fd50a2eed0279c684e872d580059": "\\omega_1\\,", "7f9a039ac792fa3e7e789668b62459ea": "x(x-1)(x-2) \\cdots (x-n+1)", "7f9a1b98375ff9eb312520d81ef5d871": "\\approx 0.5772.", "7f9a7ef0199f04c743c3cb13e7eb9adc": "c^2_\\lambda = \\alpha_\\lambda c_\\lambda", "7f9a8433a77b6ed35dd4fe5805d4207f": "b_{x,y}", "7f9aa18bab3bd13136a32fe47c60d2f3": "F = \\frac{9}{5}C + 32", "7f9aa9eb9a67107babaca66b84eb04c4": "d = \\sqrt{a^2+b^2+c^2}.\\ ", "7f9aad1ef420e84bc53371dcd89c5685": "{n_N} = \\left( {n_M} \\mod {n_N} \\right)", "7f9ae45a6527edb23ff559b79efea6dc": " P_A O_2 = P_I O_2 - P_A CO_2 (F_I O_2 + \\frac{1-F_I O_2}{R})", "7f9afa486dc4814b91ed891f00298870": "{m^+}_3 = f(13.8, 6.40) = [15.18, 0.44]", "7f9bec28bc8902d45d905788d7aa59a1": "idx", "7f9c2b52ee1fc1bc7e58cc81660eaabf": "|\\psi(t)\\rangle = \\hat{U}(t,t_0)|\\psi(t_0)\\rangle.", "7f9c681baf5a19eb8e3445e244007a3f": "\\phi(t)=\\sin(2t)", "7f9cb10e74ebe714d617d45020a985c3": "H_3(x)=8x^3-12x\\,", "7f9cb9a6e9f5dd9111d8097b97c4c3bc": "-3\\le x,y \\le 3", "7f9cda3a956020c6510415a51254ff99": "f(x) \\ge \\textstyle{\\frac{y}{x+y}} f(0) + \\textstyle{\\frac{x}{x+y}} f(x+y)", "7f9d42cc92a5d160ca58f2dc74b877d7": "a\\le r\\le c", "7f9d707ddfbecf112904755d191703a0": "a^2-c^2\\,", "7f9d70d25883654158c03c9a74dfd65b": "\\exp_{10}^3(7.18045)", "7f9e016609743b5c81d94b4a892d60ed": "\\,f_{o}", "7f9e0824125299fb7c5ebde796598053": "\\nabla f(x_k+s_k)=\\nabla f(x_k)+B s_k,", "7f9e9b88aa1d2ae0b5538a58148397f4": "\\lbrack (id \\otimes id \\otimes \\Delta)(\\Phi) \\rbrack \\ \\lbrack (\\Delta \\otimes id \\otimes id)(\\Phi) \\rbrack = (1 \\otimes \\Phi) \\ \\lbrack (id \\otimes \\Delta \\otimes id)(\\Phi) \\rbrack \\ (\\Phi \\otimes 1)", "7f9ea5c9f66612f0beff45bdef858d86": "\\displaystyle{{R -r\\over R (R+r)^{n-1}} \\le {R^2 -r^2\\over R|x-y|^n}\\le {R+r\\over R(R-r)^{n-1}}.}", "7f9f765336c78a071f6c06077ff58444": " \\mathbb{R} \\times SO(3)", "7f9fa280b7fe65b261c07ce22ecbca26": "y = \\frac{x-\\mu}{c}\\,", "7f9fe8531bc51a983136c24dbfe66d93": "\\nabla_{\\mu} X_{\\nu} + \\nabla_{\\nu} X_{\\mu} = 0 \\,.", "7fa010e99ef747b4d40cc7668a195a45": "\\cos(\\phi)\\cos(\\lambda)", "7fa012d64d7c0e635f49c417f20248eb": "ax+by+c_1=0\\,", "7fa038effc72fddff98e162e4efe950c": "K\\otimes_{\\mathbb Q}\\mathbb R", "7fa083a1cde4bada28c2a6cc7fd68d4a": "\\mathbf{l}_b=(x_b, y_b, z_b)", "7fa0a433394024f69702a58b3d5505c5": "u_{| \\partial \\Omega} = \\lim_{n\\to\\infty} u_{n\\, | \\partial \\Omega}.\\,", "7fa10ccadf4fbc22f3bd5d2fdd60a4ed": "\n\\begin{align}\n\\langle X,Y,Z\\rangle\n=&XYZ\\\\\n&-\\operatorname{E}Y\\cdot XZ\\\\\n&-\\operatorname{E}Z\\cdot XY\\\\\n&-\\operatorname{E}X\\cdot YZ\\\\\n&+2(\\operatorname{E}Y)(\\operatorname{E}Z)\\cdot X\\\\\n&+2(\\operatorname{E}X)(\\operatorname{E}Z)\\cdot Y\\\\\n&+2(\\operatorname{E}X)(\\operatorname{E}Y)\\cdot Z\\\\\n&-\\operatorname{E}(XZ)\\cdot Y\\\\\n&-\\operatorname{E}(XY)\\cdot Z\\\\\n&-\\operatorname{E}(YZ)\\cdot X\\,\\\\\n\\end{align}", "7fa12ab5bd7897e304eea7121c96f13a": " | \\nu - \\mu | \\le \\sqrt{ \\frac{ 3 }{ 4 } } \\omega. ", "7fa12c7d23a9a5ae6525a19f142bb070": "S_k(\\beta)=\\sum_{j=1}^k |r_{(j)}(\\beta)|^2.", "7fa157c1e45240b0b5afe90150bc53f6": "d_2 = d_1 - \\sigma_P \\sqrt{S}.\\,", "7fa18ea9fc2a1b84073a7ac7aa9bd936": "\n\\tan \\theta = \\cos(\\lambda + \\chi) \\tan(15^{\\circ} \\times t)\n", "7fa1aa8bc38ef014221ee3f4fb4156c7": "\\tau=\\mu\\frac{du}{dy}", "7fa1cbd38ca7d05ebd6a6a402c843292": "\\mathbf{u}(\\cdot)", "7fa2aa07853855da918e21dd154f4df3": " \\mathbf{y} ", "7fa2d50d43adebd7c106a0aa0868e9dc": "C_2\\,", "7fa316897778367068ca8f92409df6e7": "S^o=S", "7fa3e9fd1cdc59fa91fe30f3ddd2547f": "\\sum_{i=0}^n i^3 = \\left(\\sum_{i=0}^n i\\right)^2", "7fa3fffb615958f7c7038816236dcd84": "T \\le 1/2e", "7fa40219e7c866e6649a89f220847767": "\\gamma=1-\\beta", "7fa46817acdd5164f7c44bf5cd729aee": "\\theta=2\\tan^{-1}\\left(\\frac{h}{2D}\\right)", "7fa4763b1c77d06caec6c60aedbefc82": " dS_U ", "7fa48a944655feee85ed20bb02c8cb93": "H^k(X_b, \\mathbf{C}) = F^pH^k(X_b, \\mathbf{C}) \\oplus \\overline{F^{k-p+1}H^k(X_b, \\mathbf{C})}.", "7fa4e7bde48392ba670320621196796c": "p,q \\in [0,1]", "7fa536eec0da1f9866ce9796c4fc2ae6": "\\tbinom{n}{k}=0", "7fa591f08e7e42f5f72c92fe7a88589c": "\\tilde n =n+i\\kappa", "7fa5948b8dce4f68e2eabfae93e4f270": " \\delta \\phi _r = \\phi_R -\\phi_P ;\\text{ and }\\delta x_r = x_{PR}; ", "7fa5bb2dd660ec16ac6afa7e7b17ab46": "\\mbox{std} \\frac{K \\cdot t}{V} \\ \\stackrel{\\mathrm{def}}{=}\\ \\mbox{const} \\cdot \\frac {\\dot{m}}{C_o \\cdot V} \\qquad(7)", "7fa5be6a98ca1537351ef745809fd270": "q_L (\\mathrm{or}\\,\\,\\ell_L) \\rightarrow T_L \\rightarrow T_R \\rightarrow q_R\\,(\\mathrm{or}\\,\\,\\ell_R)", "7fa5e3db5e1972180542a5267ef444ff": "\\frac{1}{10^3\\pi}", "7fa61b8f580848922b5f6ec56cdd3ae4": "S_1 = r(3^1) = 3\\cdot 3^6 + 2\\cdot 3^5 + 123\\cdot 3^4 + 456\\cdot 3^3 + 191\\cdot 3^2 + 487\\cdot 3 + 474 = 732", "7fa6225861db8f2ca96818f1ea9c4137": " \\{D_i \\}", "7fa65c474c4d5fad8d57e59a32e73174": "_\\odot", "7fa65f838f1019153b26c39d10450896": "\\alpha = \\varphi_{\\beta_1}(\\gamma_1) + \\varphi_{\\beta_2}(\\gamma_2) + \\cdots + \\varphi_{\\beta_k}(\\gamma_k)", "7fa67a7256221d1bcbda2507d5060f2f": "p_{sym}", "7fa6a9e47aa88aa626969eb0448b67a5": "c_i = \\frac {n_i}{V}", "7fa6b43a08dae433287d506a3db735e8": "0^2+1^2+2^2+\\dots+24^2 = 70^2", "7fa6b8b159c7a928d51c157d7764e172": "\\vec{\\nabla}E", "7fa6b9dcc30e70e70846cde460bfd86d": "F(x) = \\|x\\|^2", "7fa72288a2205757dc46da75377102ad": "\\begin{matrix}{8 \\choose 3} = 56\\end{matrix}", "7fa779b3b37b6156d37b21cd3a3200ff": "e^{-\\frac{x^2}{2}}\\cdot H_n(x) = 2^{n/2+\\frac{1}{4}}\\sqrt{n!}(\\pi n)^{-1/4}(\\sin \\phi)^{-1/2} \\cdot \\left[\\sin\\left(\\left(\\frac{n}{2}+\\frac{1}{4}\\right)\\left(\\sin(2\\phi)-2\\phi\\right) +\\frac{3\\pi}{4}\\right)+O(n^{-1}) \\right].", "7fa7b391eddadc0bcd55325b990a1177": "\\dot{x} = g(x + y^*) = f(x) \\,", "7fa7e2e1ab808fff8fe45bea3d6987fd": "\\le 1 \\,.", "7fa846eee4b98a9147b073fa6aea1a06": " (\\varphi^*\\omega)_x(v_1,\\cdots, v_p) = \\omega_{\\varphi(x)}(\\mathrm d\\varphi_x(v_1),\\cdots,\\mathrm d\\varphi_x(v_p)).", "7fa881821e1728b381bab077c7b9a4b2": "e \\in E, \\vert e\\vert > 1,", "7fa8a7f9fa47599f107070d4c2165e98": "t=t'-(q_{hash}+q_{sig}+1) \\cdot \\mathcal{O}(k^3)", "7fa9170804ef50b4b7d42cc3c37995ea": "\\frac{f_{\\theta_1}(x_1)}{f_{\\theta_0}(x_1)} \\geq \\frac{f_{\\theta_1}(x_0)}{f_{\\theta_0}(x_0)},", "7fa93aa93b3a5c3c58e0c602ed898af7": "f^{\\prime}\\left(w_1\\right)", "7fa950229331c91df52240deae3c713a": "L^f", "7faa75f53a533baeb77574233a22a479": "{\\partial \\rho \\over \\partial t} + \\nabla \\cdot (\\rho \\vec v) = 0", "7faa82c52deca0bb43d0a457e580306b": " g_m R_\\mathrm{E} \\gg 1 ", "7faa868a56c3fa14ab206f6842f6be29": "A_m(p,s+r) = \\sum_{k=0}^m A_k(p,r)A_{m-k}(p,s)", "7faabc116725287b21a12179fff0819f": "R(n)", "7faaca92bc1eb72cdb5a83b6b746e1f7": "\\mathrm{V}^{\\bullet\\bullet}_\\mathrm{O}", "7faae358991dd6944e893d94b30684c9": "{C_t}", "7fab0126822aedb2d21e0ed4310f411a": "(a_1b_2 + a_2b_1 + a_3b_4 - a_4b_3)^2+\\,", "7fab30da0f5b2a674a979a66d8d4d7cf": "z=r\\,\\exp(i\\phi)", "7fab4bda9525f6ab7286a0ed111bd65a": "\\displaystyle{\\mathfrak{g}=\\mathfrak{k}\\oplus \\mathfrak{p},}", "7fab5f2b5b01be86a3fbd3ecf6ceaf4e": "(a,\\,b)\\;", "7fab72035b24721b46fe0090a2571bdc": "L \\le H", "7fabe329142e25c78593e2b4c506f9de": "p \\approx 1", "7fabe523d5e451c78ceb61879ff9887e": "x\\langle y\\rangle \\cdot P", "7fac0ef310ff2db71446c6922f00176a": "A^* + B \\to AB^{+\\bullet} + e^-", "7fac59b066e9d372bd34895ae4bf3250": "h_{ab} = g_{ab} + u_a \\, u_b", "7fac60dadeb1698d2c8f4e4e7e3621a6": "1 \\times 10^{-5}", "7fac7f3049a2b2eaac18f136916f2e6c": "fH_i M = H_i M / \\tau H_i M", "7facaba145762a08cb41518b8fba8406": "m^2+1", "7face95fc28d82d2f88bc97b7fc9439e": " \\Psi = \\Psi(x_1,x_2\\cdots x_N,t) ", "7facfaf85d50bd7a18805434e74f991e": "\\frac{V_O}{V_{in}} = \\frac{R_0}{Z + R_0}", "7fad038a2672dc90b4af4e39cc8a5907": "G_{2k} \\left( \\frac{ a\\tau +b}{ c\\tau + d} \\right) = (c\\tau +d)^{2k} G_{2k}(\\tau)", "7fad1be31a60be31fd59f3dfbd066715": "{d F}/{d t}\\leq 0", "7fad354a9386c64a5cb8259fc588d375": " \\; x_1^2 + x_2^2 + x_3^2 \\;\\ldots \\;", "7fad35b0e69206896bfaf121a06425c8": " \\sigma_n ' ", "7fad8c360206475c943e295144b605dc": "(u_1\\ldots,u_m) := \\underline{u} \\in U", "7fadb32bc4f3a0f86fddc40c965db701": "V_f(x,t) = A \\sin (\\omega t - kx),\\,", "7fadb7dfe4b557c96fd31ca196f3c891": "\\{X_1,\\dots,X_N\\}", "7fadc8a2e46408ad94ee526f1d991f39": "E_K(P_1' \\oplus IV_1')", "7fadc988cbdc393363186bf795e6d069": "\\mathrm{C(NO_2)_4+CsCl\\ \\xrightarrow[]{DMF}\\ C(NO_2)_3Cl+CsNO_2}", "7fae3231ed7dfd9f93950ee8ed722222": " G = H - TS \\,\\!", "7fae7998f4fd82b58e7954777fe4f638": "p \\Rightarrow q \\,", "7fae8956a4ee671a3ecb7a49baaa525b": "d\\omega(\\bold{\\hat{n}})", "7fae9494f361f1c9ba4d6cc6d7e41abd": "\\!x = 2a \\tan \\theta,\\ y = 2a \\cos ^2 \\theta.\\,", "7faec103406f9a9a5fddc7fad0497788": "6r", "7faed0c7f611711d959b1bd03f6263f9": "e_{i,j} : v_i \\leftrightarrow v_j", "7faf18b1ee260ce284474f2b8212e50b": "\n\\begin{align}\n\\sum_{j=1}^{m}|\\langle z,a_j \\rangle|^2 \\geq \\frac{\\lVert z \\rVert^2}{\\lVert A^{-1} \\rVert^2} \\qquad\\qquad\\qquad\\qquad (1)\n\\end{align}\n", "7fafa7d74ae7f7492ccc8790c00e2957": "R = \\frac {u^2} {15(e+f_s)}", "7fafeb68cb3455440a281c0216815436": "\\mu_n(N_A) = 232 + \\frac{1180}{1+(\\frac{N_A}{8\\times10^{16}})^{0.9}}", "7faff870d7071cbb88ddbd914d80e1f0": "\\hat{x}[n]", "7fb03d9e09f81edc22f68625159baad8": "\\phi_{{\\Omega^\\omega}}(0)", "7fb09bdb7b611df9d9389661bea39bdb": "k^0, k^1 = k^0 \\cos \\theta. ", "7fb0a0be0090762d602e0cfc795f541b": "\\frac{\\left|\\uparrow\\right\\rangle+\\left|\\uparrow\\right\\rangle}{\\sqrt{2}}", "7fb0b47d52c85984c5ca2dc63a1f1e27": " \\phi \\to \\pi^+ \\pi^- \\pi^0,~~ K^0_S K^0_L ", "7fb0d328cb7be8a5d690a6feba31d998": "N \\}\\!", "7fb104433f9304af02425bb23070603b": "f^{1}(x)", "7fb16f338a1f2a83bdc98f1f9ed9b388": "D \\cdot D \\cdot D", "7fb189b9fb98c09095bb7343be96ca70": "p(z) \\bar p(\\bar z)", "7fb19415bdcc4c7d843ce9bebd32901e": "\\scriptstyle \\emptyset", "7fb1a6f7f8a0ce648d9296d04853fafb": " E = \\frac{ e^H - 1 }{ K - 1 } ", "7fb213c082507c84a0c577e8f4bea8d7": "\\gamma (\\psi, \\phi) = \\arccos \n\\sqrt \\frac {\\langle \\psi \\vert \\phi \\rangle \\;\n \\langle \\phi \\vert \\psi \\rangle }\n{\\langle \\psi \\vert \\psi \\rangle \\;\n\\langle \\phi \\vert \\phi \\rangle}\n", "7fb23eaeaf6b200a73757285586a1b49": "\\ V_\\text{eff}=\\pi^{3/2}\\omega_{xy}^2\\omega_z .\\, ", "7fb24db0196f8109be95e9739b34e347": "\\sum_{i=1}^n x_i (r_i j_i) = \\sum_{i=1}^n x_i", "7fb2d20fed1c640b613f428a15f246bd": "h(t) = 0 \\quad \\forall t < 0,", "7fb37455044b7bca4b82a09b30469a24": "\\frac{e^{i k \\|\\mathbf{x}-\\mathbf{x'}\\|_2}}{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}=\\frac{e^{i k r}}{r}(1 - i k(\\mathbf{n}\\cdot\\mathbf{x'}) + \\frac{(-i k)^2}{2}(\\mathbf{n}\\cdot\\mathbf{x'})^2 + ...) + O(1/r^2)", "7fb387e0585001db97f6f50db249f09d": "\\langle f, g\\rangle_w = \\int_a^b f(x)\\,\\overline{g}(x)\\,w(x)\\,dx", "7fb39ce110fe96ef68fc0d9893a61813": "\\underline{H}=\\triangledown \\times \\underline{A} \\ \\ \\ \\ \\ , \\ \\ \\ \\ \\ \\ \\underline{E}=-j\\omega \\mu \\underline{A}+\\frac{\\triangledown\\triangledown \\cdot \\underline{A}}{j\\omega \\varepsilon } \\ \\ \\ \\ \\ \\ (8) ", "7fb3eec76aae016968d8d5c92a773c51": " U(d) ", "7fb4938d61c2d3acd419eb52e3042390": "\\sigma_c=1, \\sigma_t=0.3, \\sigma_b=1.7", "7fb52682990616332343a2a6e1419753": "\\left(\\frac{ab}{n}\\right) = \\Bigg(\\frac{a}{n}\\Bigg)\\left(\\frac{b}{n}\\right)", "7fb5335ce238656f7fd4bbbc976c5518": "\\left(1+x+x^2\\right)^n= \\sum _{j=0}^{2n}{n\\choose j-n}_2 x^{j}=\\sum _{k=-n}^{n}{n\\choose k}_2 x^{n+k}", "7fb55ac635b8fc60dd2b94df97c2dd8b": "\\mathfrak{g} = T_e G", "7fb56f5138b93bdd01101bd245a3d100": "\\frac{dU}{d\\theta} = \\frac{\\partial u}{\\partial \\theta} = \\frac{\\partial V}{\\partial \\theta}", "7fb58d2ae466823038b08cc1a6c69788": "\\sigma_P=", "7fb5aac7f287eb30a236b720dbcc64da": "Q_j^*", "7fb611b6fc585d92abbfc63decf4b386": "h\\to +0", "7fb61ac07248ad617a32074222d762a0": "\\bar{\\bar{z}}=z", "7fb68b3534aeb73d658e2aaf77d1c3ff": "P(n_1,\\dots, n_c)", "7fb6d29a87f5a9308490517f47a45c5a": "77^2", "7fb6dfa763a141f6caac74be6db5807a": " \\frac{\\partial u}{\\partial x}", "7fb7086459391fe058239fcb96293574": "\nM^{-1} = \\frac{c}{q B}\n\\left(\\begin{matrix}\n 0 & -1\\\\\n 1 & 0\n\\end{matrix}\\right) \\quad\\Rightarrow\\quad M^{-1}_{ab} = -\\frac{c}{q B_0} \\epsilon_{ab},\n", "7fb7668e0b9dc3a98c4cc8ccf86194a8": "R(\\theta) = \n\\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\\,.\n", "7fb7bd7a3eb41e4e971c718bb3011a4f": " F_m= \\psi m_s g ", "7fb7cd9434c53eb1f5a8906d6b4725fb": "\\left(\\begin{matrix}a & c\\\\\nb & d\n\\end{matrix}\\right)\\in SL_{2}\\left(\\mathbb{Z}\\right)", "7fb7e6e568da8bcd4917979ea61dd128": "\\mathbf{u} - \\mathbf{v}", "7fb80cc1544adfdfba885f55971b5a44": "v_{1}= \\begin{pmatrix}0.0002422 + 0.1872055i \\\\ 0.0344403 - 0.0013136i \\\\ 0.9817159 \\\\\\end{pmatrix}", "7fb83f8b3d18dcfb7d589236ed06369c": "\\det\\begin{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} - 1 & -2 \\\\ -3 &\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} - 4\\end{pmatrix},", "7fb874403564d4d1e12a3c138012fc45": " K_{m1} ", "7fb8dac6e191e23481d6dfe84883496b": "\\mathrm{shoe~size ~({Paris~points})= {\\frac{3}{2}}\\times{{\\left[~foot~length\\left({cm}\\right)+1.5~{cm}~\\right]}}} ", "7fb8ddd57e5425474011cbfa6fcb961f": "= \\Gamma(x) \\left(\\sum_{n=1}^\\infty \\left(\\ln\\left(1 + \\dfrac{1}{n}\\right) - \\dfrac{1}{x + n}\\right) - \\dfrac{1}{x}\\right) = \\Gamma(x) \\psi(x)", "7fb8e2e8bef78a676cecfab58a80dd74": "\\operatorname{div} \\nabla u = \\Delta u = 0.", "7fb9a1b1cc051660b522aebd0d62b5f1": "\\iiint_\\text{cuboid} 1 \\, dx\\, dy\\, dz", "7fba3a9bfafaf869af460a2dda0868f5": "v(z)=z^{-c/2}(1-z)^{(c-a-b-1)/2}.", "7fba5f45be77b37ad55409c27cd3de9e": "\\operatorname{supp}M", "7fbab97ba85c04bc3e4e114c50ceb306": "s(A)\\le 2", "7fbacff525a5485ec1b6a0ff21075dc2": "\\phi_{\\text{Electric monopole}}(\\mathbf{x},t) = 0", "7fbb85c68e314f5d9d05d9196d6ef120": "R = (s-t) \\cdot (s-t) - (s \\wedge t) \\cdot (s \\wedge t).\\,", "7fbba7f11e35bfa404471f5c55d9abdc": "M\\ddot{q}=Q+Q^{c}(q,\\dot{q},t),", "7fbbd9884a112cdd48ceab1e650b1b06": "J := \\det\\boldsymbol{F}", "7fbbfd3110a61ea865557e9115d7d697": "\\eta(s)=\\sum_{n=0}^\\infty \\frac{1}{2^{n+1}} \n\\sum_{k=0}^n (-1)^{k} {n \\choose k} \\frac {1}{(k+1)^s}. ", "7fbc5be9403c138886bba79c51d52964": "M_{b_2}TM_{b_1}", "7fbcb87beee752642b087fcb6c095f94": "v_{LZ} = {\\frac{\\partial}{\\partial t}|E_2 - E_1| \\over \\frac{\\partial}{\\partial q}|E_2 - E_1|} \\approx \\frac{dq}{dt}, ", "7fbcc647e6a3d3816f5c6580c1ef8704": "\n \\ln \\frac{dP_{\\!n,\\theta+r_n^{-1}h_n}}{dP_{n,\\theta}} = h'\\Delta_{n,\\theta} - \\frac12 h'I_\\theta\\,h + o_{P_{n,\\theta}}(1),\n ", "7fbd0e56276c7807c69b5ab927fd267c": "\\delta(-x)=\\delta(x)", "7fbd3e01bfccebd224190e20d6df2e0b": "e^{2^k}", "7fbdf08c233011765071fa0aad009711": "\\begin{matrix} {12 \\choose 1}{4 \\choose 2}{44 \\choose 1} \\end{matrix}", "7fbe005f132671c3bc86df01174bb833": "\\Delta x_1 = 1", "7fbe2e7ec21740d040b8aee9424545f4": "m^*(E,\\hat B,k_{\\hat B}) = \\frac{\\hbar^2}{2\\pi} \\cdot \\frac{\\partial}{\\partial E} A\\left(E,\\hat B, k_{\\hat B}\\right)", "7fbe5531d7429b2b1b889a48b2c0783d": " \\textstyle x_0 ", "7fbed903f9fb8a07bc3c5028b2d7a66d": "X_t = X_{t-1} + \\varepsilon_t", "7fbf1c04998f53a5796e29b9918a1d51": "P[ \\{ t_i \\} | s(t) ]", "7fbf2eef31a7ee66a122fc9e09f6e7d5": "\\alpha I_E = \\beta I_B + \\beta I_{CBO} = I_C - I_{CBO} \\iff I_C = \\alpha I_E + I_{CBO}", "7fbf5f2d88eefa98f95c8d76d0cb08bc": "n\\geq\\frac{4}{\\epsilon\\alpha}\\left(km+ln\\frac{1}{\\delta}\\right)\\geq O\\left(\\frac{d\\cdot VCDIM(H)\\log(1/\\alpha)}{\\alpha^{3}\\epsilon}+\\frac{\\log(1/\\delta)}{\\alpha\\epsilon}\\right)\\,\\!", "7fbfa67e38467f0467346bb829b41dd8": "\\hat \\beta \\pm 2se_{\\beta}", "7fbfb6db105e80d9057b970570bc0ff5": "\\mathrm{Win} = \\frac{\\text{897}^{2}}{\\text{897}^{2} + \\text{697}^{2}} = 0.623525865", "7fbfcea6d5e7601ef7bc56ade0caf0d6": "{m^+}_1 = f(12.3, 10.0) = [15.85, 0.68]", "7fc055510cd52a906a5f06e8b86f5076": "A_0 (1 + \\cos (\\theta)) / \\sin(\\theta)", "7fc06c40b7a6e1ede5351e5c3e2a6281": "\\mathrm{EE}", "7fc0791cb0f7e32a0dbfbbe6810ce002": " P = P(\\theta) ", "7fc082ae54ab989cf9d68bb4b1516492": " \\delta\\left(\\left\\Vert P_{i}P_{j}\\right\\Vert \\right) ", "7fc0d1470feccdc75975f2b0b956ecdc": "\nW=\\frac{1}{\\sqrt{4}}\n\\begin{bmatrix}\n1 & 1& 1 & 1\\\\\n1 &-i&-1 & i\\\\\n1 &-1& 1 &-1\\\\\n1 & i&-1 &-i\\end{bmatrix}\n", "7fc11d72f61f8a9eff781f95b0d1d5ae": "g : Y \\to X, \\,", "7fc13d7af04367f93852375411286af2": "B_3", "7fc15b38a1278ba704f8033247a12398": "\\ln\\left(F/K\\right).", "7fc196c541a1e3850a8b86bf22584c82": "B_\\mathrm{linear}", "7fc1aa2899b85254bd799ffa783b9132": "e^{ L (t-\\tau) }", "7fc1b53a57471ee10617dda6cedba5f3": " 2q+1 \\geq N\\omega_{max}+1 ", "7fc1be662558860cd3d37c5ba28b8803": "T_X = X(k[\\epsilon]/(\\epsilon)^2)", "7fc1c7bed8f06000f52498bd168962a4": "x = \\left(\\sqrt[m]a\\right)^n", "7fc223948b0744fc7c40d1073439a045": "dp/dz", "7fc26f27f9c32a2bf019e58dbd63ee66": "\\lambda^3 = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}", "7fc2928aa929c9cfee72da13b9a60312": " X \\times I ", "7fc298e942c543cf05140dc3d39795e6": "\\mathbf{E} = - \\nabla \\varphi - \\dfrac{\\partial \\mathbf{A}}{\\partial t}", "7fc360f4043b1aecee5cf07a7eae4fc1": "\\ln{p(S\\vert D)\\over p(\\neg S\\vert D)}=\\ln{p(S)\\over p(\\neg S)}+\\sum_i \\ln{p(w_i\\vert S)\\over p(w_i\\vert\\neg S)}", "7fc38dd96548a306a5d74ba66c68646b": "0.7\\overline{3}", "7fc39588a037dda39a268448219b794a": "\\begin{bmatrix}C\\end{bmatrix},", "7fc3d4654c67fec44eff01770dd441f8": "\\hat{x}_i", "7fc3e19f7d10c3d8398f788798a84bcb": "T(\\rho,\\sigma)", "7fc3ee29eb3331f37a6a94ef8b4002b8": "_m^7", "7fc4ae136ccf75c5403e399195624e39": "r^n = a^n \\cos(n \\theta)\\,", "7fc4bfe518c3579eb82cefc3d277f63d": "\\hat{c}_V=3/2", "7fc4efbdf58738dfadbb7b42e3eea380": "B = \\Delta V + (\\Delta V )^T", "7fc52ce9ad3dbfa7abb5dd2e36d4b044": "A = 86\\,400 \\frac{c_{\\rm 0}}{c_{\\rm AU}}.", "7fc56270e7a70fa81a5935b72eacbe29": "A", "7fc57ef2821e5aed8b2099fd81abe60a": "q_{F}", "7fc5c217e28501073422be587771042c": "\\forall x\\in X\\,[(\\forall y\\in X\\,(y\\,R\\,x \\to P(y))) \\to P(x)]\\to\\forall x \\in X\\,P(x).", "7fc5f846e6f18d51fe124cb43e568aa8": "\\displaystyle{f_W(z)=e^{z^tWz/2}.}", "7fc6099fa96ca485ecba9f9635f4c5fc": "\\langle P,\\varepsilon\\rangle", "7fc61431a2b978c8d3f634e21683f819": "y_{ij} = g_{ij} + \\sqrt{-1} b_{ij}", "7fc67b821624d4a07c9cbc4e0a810c26": "(\\mathbf{T}q)^2 = \\mathbf{T}(q^2) = \\mathbf{T}q^2", "7fc693d78f4c031ed50714a8062aac8d": "\\chi_{\\bar{V}_i\\otimes V_j}(g)=\\chi_{\\bar{V}_i}(g)\\chi_{V_j}(g)=\\chi_{V_i}(g)^*\\chi_{V_k}(g)", "7fc6d45f64d392c9f5f1fdff7a1c9697": "T_\\sigma(q)=T(q,\\sigma)", "7fc6e0a2d5aed8f4cc2b2d2bd9f01a40": "\\Pi_{n \\mathbin{:} {\\mathbb N}} {\\mathbb R}", "7fc70f13f98c5f45edc8bd6110c66ac0": "\\{(t_1, \\dots, t_n) : \\sum t_i = 1\\}", "7fc744f4faea5b602440985751085ef3": " S_{\\lambda} = \\det_{ij} h_{\\lambda_{i} + j - i}. ", "7fc74dcea4aecfc71b81953be150c1b9": "p-f_0g_0=(f-f_0)g_0+(g-g_0)f_0+(f-f_0)(g-g_0)", "7fc75b72bd78ca1480dbd3cbc354700b": "\\widehat{\\vec{r}}", "7fc7a8b4dc0accc7a37027596101cbd2": "\\mathfrak{abcdefghijklm} \\!", "7fc7f973ab285c37f00d0cf6a276bdff": "E=(1,1)", "7fc82f1fdfaceeb02f821d79afdbce67": " \\alpha(\\rho,\\sigma) \\ge 60.8^\\rho 22.3^\\sigma ", "7fc86f8d595e66e024bc6f6baacd95d6": "\\begin{align}\n&{} D(X, Y) = \\int\\limits_{0}^L \\int\\limits_0^L |x-y||\\psi_m(x)|^2|\\psi_n(y)|^2 \\, dx\\, dy \\\\\n&{} = \\begin{cases} L\\left(\\frac{4 \\pi^2 m^2 -15}{12\\pi^2m^2} \\right) & m=n, \\\\ L\\left(\\frac{2 \\pi^2 m^2 n^2 -3m^2 - 3n^2}{6\\pi^2m^2n^2} \\right) & m \\neq n \n\\end{cases}\\end{align}", "7fc878aedcb39c628f689e99efe29da5": " \\nabla_X(fv) = X[f]v + f\\nabla_Xv", "7fc885ea639c79d3f8ba93dfc4c17cab": " P_A = \\sum_i \\langle u_i,\\cdot\\rangle u_i.", "7fc8870b1bfb440811e1f449a7fb2513": "\\omega^2=\\omega_p^2+k^2c^2", "7fc8bf434e19467cb65f82e38cdc2adf": "((Bxyw \\and xy \\equiv yw ) \\and (Bxuv \\and xu \\equiv uv) \\and (Byuz \\and yu \\equiv zu)) \\rightarrow yz \\equiv vw.", "7fc8e19affc1ab6fd7f7e21a96ed2946": "2=b^2,", "7fc8ee891fae5aafc360815910baf796": "B\\left(V\\right)", "7fc8fc586ab421fd1a1e760e11b3f571": "\\displaystyle{\\frac{\\left(w+400+x+400+y-400+z-400\\right)}{4}}", "7fc9157d54bc7dd5b9f5aec86a4129bd": "A^{-1}=\\begin{pmatrix}\n 1 & 0 & 0 & \\cdots & 0 \\\\\n 0 & 1 & 0 & \\cdots & 0 \\\\\n 0 & -a_{32} & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & -a_{n2} & 0 & \\cdots & 1\n\\end{pmatrix}", "7fc917560d7913bc779ea466228caba4": "\\beta = 1/(kT)", "7fc96400c15900b69edd141aa843120a": "y_{marker} = {y_{marker\\_orig} \\over y_{orig}} \\cdot y_{scaled} - {marker\\_size \\over 2} + y_{adjust}", "7fc9df02908fe2816e4c66283733a9c2": "f(\\zeta)", "7fc9e093b951b8a1970dba5b389e3b58": "r = k[NO_2]^2", "7fca060a57f3f6df37ef3995167ffc79": "i=\\lfloor x/M^{1/2}\\rfloor", "7fca18c6ad4f4349cb667ff6f73670ca": " \\frac{dx}{dt}=-A.e^{-At}.C_2-B.e^{-Bt}.C_1+\\frac{1}{P}.[\\dot{Q_t}.e^{-At}-A.Q_t.e^{-At}-\\dot{R_t}.e^{-Bt}+B.R_t.e^{-Bt}]", "7fca3f54f3294b08fb69d0b98b78eeaf": " PLEASE PLACE CATEGORIES AND INTERWIKI LINKS ON /doc, NOT HERE \n[[Category:United Kingdom politics and government templates]]\n", "7fca96b712e0eb6701eef59de0dbfef5": "V_L = V_{R_2}", "7fcafbff27edc72d60d95e7168cde6f3": "L_x/L_y=p/q", "7fcb423f6f8aa272845c03edb53d0aa9": "(3 \\times 5) \\times 7 = 3 \\times (5 \\times 7)", "7fcb58d311a231e7fe9e6ebab3334c20": "\\{ x \\in X \\mid \\text{for some open } N_{x} \\ni x, \\mu(N_{x}) > 0 \\}", "7fcb6faa64d91693639baa3fed3bcdfe": "\\begin{array}{ccc}\nH^* (F)\\otimes H^*(B) & \\longrightarrow & H^* (E) \\\\\n\\alpha \\otimes \\beta & \\longmapsto & s (\\alpha)\\cup \\pi^*(\\beta) \n\\end{array}", "7fcbfa9a8af68f3613ae4314191f4339": " -e^{-q \\tau} \\frac{\\phi(d_1)}{2S\\tau \\sigma \\sqrt{\\tau}} \\left[2q\\tau + 1 + \\frac{2(r-q) \\tau - d_2 \\sigma \\sqrt{\\tau}}{\\sigma \\sqrt{\\tau}}d_1 \\right] \\, ", "7fcbfab4ad7590b1675035cf92cddae9": "C=u_1 \\times G + u_2 \\times Q_A", "7fcca6608cd2aea6816d11aadeefaadf": "f(x)=1+2\\left(\\frac{1}{x}+\\frac{x}{x^2-4x+8}\\right)", "7fcd3111f62839e8f6a6fd08d912b7e2": "G = \\langle N,M \\rangle", "7fcd7f455b208f657f752daf6a86ba36": "m=30", "7fce8137ce7db2a512405e4ee503d96e": "\\frac{dH}{dt}=-\\frac{\\partial L}{\\partial t}\\,,", "7fce9505899769679dc6326c7cfb08dc": "\\kappa_{\\rm ff}(\\rho, T) = 0.64 \\times 10^{23} (\\rho[ {\\rm g}~ {\\rm\\, cm}^{-3}])(T[{\\rm K}])^{-7/2} {\\rm\\, cm}^2 {\\rm\\, g}^{-1}", "7fcecc22d0d37b35a5768c72e4ca6153": "\\chi_{m,n} = \\sum_{j,k} B^{-1}_{m,n,j,k} c_{jk}", "7fcee73e48eb8de1928fb4e3a73c3044": "E(C)=\\sum_i \\sum_{j N", "7fd42cc193f8f63db5451c8c6db7d048": "\\alpha/2", "7fd482efa55097af99b424c94346e0d3": "\\Omega^{(0,1)}\\mathbb{C}^n = \\mathrm{span}(d\\bar{z}_1,\\dots,d\\bar{z}_n).", "7fd4b1acdbad3426fce2d6417d912641": "I\\mathcal{Q}_{\\mathrm Hur}", "7fd4cf2db225d86b3bbf18a889e194e2": "P(z) = P_0 e^{- \\alpha z}", "7fd51c775b5028942a830ad6678ad394": "e = L - L_0 = \\frac{{L_0(T-T_0)}}{{EA}}", "7fd521b117272c2b19b715fedec8884b": "n^{\\Omega(1)}, 2^{n^{O(1)}}", "7fd521be12f3705785cd0e898fec90ac": "H^1(S, \\mathrm{Hom}_\\mathbb{Z} (X^\\bullet(T_1), X^\\bullet(T_2)))", "7fd534367d61b7cc0de9e902425cb1a6": "\\hat{H}=\\hat{H_0}+\\hat{V}", "7fd5348b36eecf9cc3b15f23ec1a9d96": " \\Box_1 P ", "7fd53b87e7e86869700df1244b94b8cb": " \\psi(r,\\theta,\\phi) = R(r)Y_\\ell^m(\\theta, \\phi) = R(r)\\Theta(\\theta)\\Phi(\\phi)", "7fd542359218e07bc9ad0d66a4829b96": "\\sum_{i = 0, j = 0} ^{i = m, j = p}", "7fd5eef0958a166180d508e235b1caf1": "s < \\gamma(1-2\\varepsilon)n\\,", "7fd5ff0fee011cb87e33b0f946cb8d4f": "\\mathrm F", "7fd60f8b06c21e343e638127a3792bac": "x_3 = \\frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2} = 0", "7fd61efc5759bfe1d733325b3a38325e": " L = \\frac{g}{\\omega_n^2}=\\frac{9.81 \\ \\mathrm{m/s^2}}{(3.14 \\ \\mathrm{rad/s})^2}=0.99 \\ \\mathrm{m}.", "7fd67e8a0d79adf0e46baf1ad6e8de47": "(p \\land q) \\to r\\ \\vdash\\ (p \\to r) \\lor (q \\to r)", "7fd6cbb97017c2abeaf233fe64c92487": "NOR(\\alpha,\\alpha')=1-OR(\\alpha,\\alpha')", "7fd81ab7a9afa238e3d7d02f38fe3fbf": "ITV \\to ran ~|~ danced ~|~ ...", "7fd82acaa2b031f2560641d7e391e3be": " \\overline \\Phi: \\overline Y\\to \\overline Y', \\qquad \\overline y'^i=\n\\frac{\\partial\\Phi^i}{\\partial y^j}\\overline y^j, ", "7fd85933c366172d5884c59a0970e9ae": "\\mathrm{C}_{\\mathfrak{L}}(S)=\\{ x \\in \\mathfrak{L} \\mid [x,s]=0 \\text{ for all } s\\in S \\}", "7fd8612820be5fcb49e4fbda04f9dc0a": "(g_\\gamma,\\nabla_\\gamma,\\nabla^*_\\gamma)", "7fd86fdd894caeedc20ab6abb86ec365": "p\\in \\mathbb R^2", "7fd8996cbff696ee61853c739366a517": "\\{T, F\\}", "7fd8ab3c22e205869a9fd131a32c1f89": "{QP - PQ = \\frac{ih}{2\\pi}}", "7fd8b321fabdc3c4155f4a6e2a5f29ab": "\\mathbb C^n", "7fd8cfdf5639661f5d751abf8a870d72": "q = 4n -1", "7fd911d02e8ec3a8863dd2a1e9157402": "m = -1", "7fd92b1bc37330b0a5800628992a03b6": "S_{t1}=S_{a1}+S_{b1}.", "7fd9577cf61fa16150e293d7e3bb1fcb": " {\\partial p}/{\\partial s} <0", "7fd998be86df2c7491ab7452e4e6fec0": " S = \\sum_{i=1}^{n} \\left(y_i - kF_i\\right)^2. ", "7fd9b33c0093839e0e770c72f803b44d": "\\boldsymbol{r}=(\\boldsymbol{q},\\boldsymbol{p})", "7fd9b606eaf50596ae1189b1bcfb2102": "\\operatorname{pf}(A)^2 = \\det(A),", "7fd9c0b741b2332ae45477ec58014afc": " P(\\vec{r},\\vec{p}) = \\frac{1}{(2 \\pi)^3} \\int \\psi^*(\\vec{r} + \\hbar\\vec{s}/2) \\psi (\\vec{r} - \\hbar\\vec{s}/2) e^{i\\vec{p} \\cdot \\vec{s}} \\, d^3 s. ", "7fda355f72136e6ef3b5035437fd1be8": "Z(T)=\\sum_i g_i e^{-E_i/k_BT}.", "7fdb4719e6167029153357f17ec3bae1": "D_{3,4} = I_{4,1} - C_{3,1} \\quad \\rm{dB}", "7fdb497018cee6e0b0dd9933f76de3a2": "a, b, c \\in \\widehat{\\mathbb{R}}", "7fdb6b5295446945761fc3f14871528f": "f(x) = \\sum_{n = 1}^{\\infty} D_n \\sin \\frac{n\\pi x}{L}.", "7fdbaec3e10dd287cf3c1fb9b4e6daa5": "\nx(t) = Ae^{\\gamma_+ t} + Be^{\\gamma_- t}\n", "7fdbc5120ee1f15689fd212883021cd7": "\\pm\\left(0,\\ 4\\sqrt{\\frac{2}{3}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "7fdbd09c5b4e95d9b34dd44a214b9ad5": "\\overline{MR_i} = \\frac {\\sum_{i=2}^m \\big| x_i - x_{i - 1} \\big| }{m - 1}", "7fdbe339f064e382cbbed41fe32a1897": "\\Delta n=n_e-n_o\\,", "7fdc3b4859ab20b41ea0f2658a464642": "\\psi(e^y)=\\sum_{n=0}^\\infty B_n ~ y^n/n! ", "7fdc4a3b09829335176e480afb19a3cb": "\\phi(\\overline{z}) = \\overline{\\phi(z)}.\\,\\!", "7fdc56a2ac3cc09a4f497e63282d6869": " BA = 0.005454 \\times DBH^2 ", "7fdc72baba6329c12972a5b9492e497e": "\\Gamma_{\\alpha}(\\theta)=D^{2}\\cap e^{i\\theta}(\\Gamma_{\\alpha}+1).", "7fdcc7cb6ccb3bf5f4bc5fdcfdea18ce": "r^{m}\\,", "7fdccaa94afa336c6e121e66b3c38937": "f(x) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{\\infty} \\ \\tilde{f} (k) e^{ikx} \\ dk \\ ; ", "7fdcf64a0ff42954d9847dc0cfadf9a4": "h^\\beta", "7fdd0665b0dae0725c6041a96c31a85b": "\\Phi_l(X,Y)", "7fdd73773220619e0e13d891d2dde6a4": "S(0)", "7fdd7ed8473b588ad7f332b988521232": "\n \\mu_{\\hat{j}} = \\frac{\\exp{(\\beta Q_{\\hat{j}})}}{\\sum_{j=1}^J \\exp{(\\beta Q_j)}}\n", "7fddb48a699b8bf52640853d37500cfb": "|g_2\\rangle ", "7fddc04800686d0568d254622829962d": "A=[0,5,2,5,4,3,1,6,3]", "7fddd1886a3325925001cbed532fa7a8": "(\\delta O_j)", "7fde23c69f04fca200f452398b6428c8": " \\sum_{n=1}^\\infty \\left(a_1 a_2 \\cdots a_n\\right)^{1/n} \\le e \\sum_{n=1}^\\infty a_n,", "7fde937a38f9e125ed1616b8584d72a0": "\\frac{\\partial f_i}{\\partial t} + \\frac{\\mathbf{p}_i}{m_i}\\cdot\\nabla f_i + \\mathbf{F}\\cdot\\frac{\\partial f_i}{\\partial \\mathbf{p}_i} = \\left(\\frac{\\partial f_i}{\\partial t} \\right)_\\mathrm{coll}", "7fde9a45d4629dc4647d6f6345e00b53": "wl", "7fdee52f2e3a8b685176450591233b63": "U=c_0I+c_1\\sigma_x+c_2\\sigma_y+c_3\\sigma_z", "7fdeede7c37fdfc333503caf8d7d61d1": " m^p - m^{p-1}a_1 - m^{p-2}a_2 - \\cdots - a_p = 0 ", "7fdf990c4105dc52207d0432c1ffe12e": "M ", "7fdfa776999e25a458ca011d774217f7": "\\begin{pmatrix}8\\\\4\\end{pmatrix}", "7fdfa851f6ae9de82856d7c48bbc3ec0": "4 b_2 b_0 - b_1^2 > 0", "7fdfb32c58e87b134dc70ba0717b7ed8": "\\frac{1}{0} \\cdot x^0", "7fdfb470aecb79419b40220ecb73d7b8": "\\Pr[p_i = 0 | y \\neq 0] \\leq \\Pr[r_j = 1] = \\frac{1}{2}", "7fe007ce10679ff0a91709e72fd0ddb9": " R^* = \\frac{4}{\\rho f^2} \\left| \\frac{\\partial p}{\\partial n} \\right| ", "7fe033a85edbe3a2b1a333324a116c72": "B_{\\mathrm{out},i} = \\frac{B_{\\mathrm{in},i+1}(d_{i+1}/d_i)^{3/2} }{ \\sqrt{R_{\\mathrm{m},i+1}/R_{\\mathrm{m},i}} }", "7fe057c16614b456d19dfdd565dde899": "y\\left(x\\right) = Cx^2.", "7fe0c85e1d714cacb15f7b80b259d8a5": "= 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 = 438579088", "7fe0e6c023cdc539fb6152f9326c55f9": "\n u(0) = 0, u(1) = 0 \\,\n", "7fe0ee73e98025c8e5c0a89d3722b494": "K_v\\subset\\mathbb R^2", "7fe10f34a24f7d6ee6978af76554fefe": "q = 8", "7fe15955e053d0abdfa73b3d40bdad7e": " v = \\mathbb{E} [H(x,y)]", "7fe15af4a072dbd8f0815fb09a6d9d71": "K_{ij}= -\\frac{1}{2}(\\mathcal{L}_{n}\\gamma)_{ij} =\\frac{1}{2}N^{-1}\\left(\\nabla_j\\beta_i+\\nabla_i\\beta_j-\\frac{\\partial\\gamma_{ij}}{\\partial t}\\right),", "7fe1bd31884cb4644fb7ea38e8abed47": "n^2 \\equiv 1 \\pmod r", "7fe1df739dd25ff10cc1514b846af50c": " v=\\sqrt{ \\frac{G M}{r} } ", "7fe2323e96cc8b286375470e14e17866": "c = 8.7", "7fe26f94a803c6315354ffafe28347c0": "\\{u_n\\}", "7fe2726b5aaf53a59a511f7c226b4152": "s(\\bold{u})=1", "7fe29c92c87ce9a4025f893954c17a4a": "x_2,\\dots,x_k", "7fe33b9234be848564c69d613542c38a": "k_1 = 0, k_{L-1} = -2", "7fe343c85a91486700edd61cf8e2252b": "d\\boldsymbol{\\varepsilon}_p:d\\boldsymbol{\\sigma} \\ge 0", "7fe3495fc599a6a825ceddb41a298d5e": " f : E \\rightarrow E' ", "7fe396ca2c872e53c1566bd64bd75a30": "(\\alpha,\\beta,\\gamma):L \\to K", "7fe3acb2d08e998ba91bbb178bd5073e": "\\sigma=\\sigma_{H_2O}\\times\\frac {m}{m_{H_2O}}", "7fe3ffee768d2af311882cd6b84b993f": " \\mathrm{Br} = \\frac {\\mu U^2}{\\kappa (T_w - T_0)}", "7fe40290c86c54c9c19bd419e0ac0789": "(\\forall n\\in\\mathbb{Z}_+):U_n(x)=\\frac1{(2n+1)!!}-\\frac{x^2}{2\\times(2n+3)!!}+\\frac{x^4}{2\\times4\\times(2n+5)!!}\\mp\\cdots", "7fe45c60991bcb6ec9daf9bb07edc8dd": "Q_{cold}", "7fe4ad9d9133d9abe5f0fa6578cff46f": "\\!\\,N(x,y)=(b-y,x),", "7fe5000a795c7514b7cd55fcb38cbc25": "\\frac {d M_z(t)} {d t} = \\gamma \\left ( M_x (t) B_y (t) - M_y (t) B_x (t) \\right ) - \\frac {M_z(t) - M_0} {T_1}", "7fe53b4f2a2571cf6ecba0bbf2292a5f": "I_{\\mathrm{RMS}} = I_\\mathrm{p}\\sqrt {{1 \\over {T_2-T_1}} {\\int_{T_1}^{T_2} {{1 - \\cos(2\\omega t) \\over 2}}\\, dt}}", "7fe578936a85916cecd76a11998aff44": "\\delta V^{eff}[\\rho](t)=\\delta V^{ext}(t)+\\delta V_H[\\rho](t)+\\delta V_{xc}[\\rho](t)", "7fe593ceb9b3520e2eca7d8e44548607": "A\\to B\\vdash(B\\to C)\\to(A\\to C)", "7fe599912283d22e98a46031b4595f5d": "\nP_C=\\frac{a}{27b^2}\n", "7fe5e9ce63e150d2395866f94ee892c7": "W^{\\perp}=\\{\\mathbf{v}| B(\\mathbf{v}, \\mathbf{w})=0\\ \\forall \\mathbf{w}\\in W\\} \\ . ", "7fe6279b75fef11e4b62944d853d604b": "\\Box = -\\partial_t^2 + \\Delta \\,", "7fe62a2a99cd6682dd3cc7c7a375b650": "\\binom{n}{0}_F = \\binom{n}{n}_F = 1", "7fe65c764608b1a8707a1d7f16c7307d": "\n[a_{\\mathrm{in}}(\\mathbf{p}),a_{\\mathrm{in}}(\\mathbf{q})]=0;\\quad [a_{\\mathrm{in}}(\\mathbf{p}),a^\\dagger_{\\mathrm{in}}(\\mathbf{q})]=\\delta^3(\\mathbf{p}-\\mathbf{q});\n", "7fe68d9c6fb3339f5727ebd4334f6a41": "\\alpha=\\sqrt{RG}\\,\\!", "7fe6e4ec3a60edd7bf208435a7e180bb": "P = 0.24 \\sqrt {D + 0.625} - 0.175", "7fe71cd89c0e554d338b9c1f4029f94e": "\\pi_i (X) = 0", "7fe744d89c7ee32c29be79d1ef8805d1": "\nPoss(a,s)\\rightarrow\\left[F(\\overrightarrow{x},do(a,s))\\leftrightarrow\\gamma_{F}^{+}(\\overrightarrow{x},a,s)\\vee\\left(F(\\overrightarrow{x},s)\\wedge\\neg\\gamma_{F}^{-}(\\overrightarrow{x},a,s)\\right)\\right]\n", "7fe75f54ce621d2e674b5bfc9f37b223": "2^p \\equiv 2 \\pmod{p}\\,", "7fe7e181ffc139105b78c82c67e320a2": "D_Y", "7fe816edb1e162600f195faeb7de5427": "\\ln \\,\\operatorname{cov_{G{X,(1-X)}}}(\\Beta(\\alpha, \\beta))=\\ln \\operatorname{cov_{G{X,(1-X)}}}(\\Beta(\\beta, \\alpha))", "7fe846a1a4a376a12c7975db15c9530a": "\\sum_{i\\in I}\\lambda_i = 1 ,", "7fe873d72146e5dcb6ac9b3e62b3bbcb": "d^\\prime = V_{ud} d + V_{us} s ; ", "7fe87bf269201b57b903052e8264dc68": "[\\mathbf{x}]", "7fe89406beb3eb3f6bea87a3d234b83e": "M_{i},E_{i}", "7fe8b57e8c74a58bbd0e9df9335d2264": " Q(a) \\to\\ \\exists{x}\\, Q(x)", "7fe8c71fbdddb03f604ee6f902237060": "D(r)=\\mathrm{true} \\Rightarrow \\exist s>r, D(s)=\\mathrm{true}.\\;", "7fe8e46f85491a0eccff857eab706cfd": "\\phi_e = \\phi", "7fe8fdade6f764086f9963f546f660b3": "{\\overrightarrow{V_g} >> \\overrightarrow{V_a} }", "7fe9133728d2f58f778e28e2978a99d5": "\\operatorname{red}(f,G)", "7fe940c886a2e50abd6ccb620ebc8a4f": "l(p-1)!d_i", "7fe97da5c0d4942f8a49a22f1d7a31ed": "p_{\\epsilon} \\in \\mathcal{M}_{\\epsilon}", "7fe988bce0223569986cf2cc2dfd96e2": "2^{360}", "7fe9d6a58e071edd013d852108d9b1ff": "d\\bar{\\psi}", "7fe9e2470c2a5e83c2707ec4f5862a3b": "X\\times\\{0\\}", "7fe9f43a43078b8180dec475d343371d": "\\partial/\\partial x^i", "7feab9254a7669c28be01b024bb0f9b9": "(c_1,\\ldots,c_n)", "7feb28317ab9e6c550981b3d3fd9c75d": "211,600", "7feb28cb226148f51b8621ebe9e21d94": " (a_1,b_1)\\cdot (a_2,b_2) = (a_1a_2 - b_1b_2,a_1b_2 + b_1a_2). ", "7feb28e7160102347dff8e11e18dd935": "\\varepsilon:F\\circ G\\Rightarrow 1_{\\mathcal D}", "7feb95220e2d8f8dd6a4b77bf985a125": "V:(s,w)\\,", "7fec3b8eef8698854d59c96fe61468fb": "\\frac{d}{dt}f(\\phi_x(t)) = X_Hf = \\{f,H\\}.", "7fec81947e76611d72ffc18fbc5b7973": "\\dot{V}(x,t)\\to 0", "7fec97275e15e7bb6722c7118dacfa53": " M_{X_1} Y = M_{X_1} X_2 \\beta_2 + M_{X_1} u \\!, ", "7feca0c45a2b4b341495c08d74867ac3": "P_5", "7fecc3d05a052ff55eb5a42b4f1aaf74": "E_{em}", "7fecc76b00397b723b25a439dc0e8dad": "2^{n-1} -1 ", "7fed89c20f1f9a2f363f8b9b50a20058": "\\tanh\\theta = \\frac{2t}{1 + t^2},", "7fee04ac4e5de2f20ea94ceaf9843728": "(4,2,1)\\rightarrow (3,2)_{\\frac{1}{6}}\\oplus (1,2)_{-\\frac{1}{2}}", "7fee339c2702da27d0a5b0de49d99eae": "G_1 = 2 AUC - 1", "7fee4d01f5877ad7a550631de376bc6a": "B = 0.2", "7feecc3b499a3cf1f3ebee412b35dedb": "\\displaystyle 2r^2(R^2+x^2)=(R^2-x^2)^2.", "7feef901c4416d505a85ff56ec65c4d8": "\\xi_m = 1 - \\ln {V_{3m - 1}}, \\ \\eta_m = V_{3m} e^{-\\xi_m}", "7fef26be3294cbc7341249259e2dc564": " Com.2", "7fef504ef7579fefb5e41eb69ed4091f": "\nF^{-1}(y) = \\inf_{x \\in \\mathbb{R}} \\{ F(x) \\geq y \\}.\n", "7fef94c5b56784d4757b474bc5f22083": "\\arg\\min_{\\phi_{X},\\phi_{Y}}\\mu\\sum_{i,j}\\left\\Vert \\phi_{X}\\left(X_{i}\\right)-\\phi_{X}\\left(X_{j}\\right)\\right\\Vert ^{2}S_{X,i,j}+\\mu\\sum_{i,j}\\left\\Vert \\phi_{Y}\\left(Y_{i}\\right)-\\phi_{Y}\\left(Y_{j}\\right)\\right\\Vert ^{2}S_{Y,i,j}+\\left(1-\\mu\\right)\\sum_{i,j}\\Vert\\phi_{X}\\left(X_{i}\\right)-\\phi_{Y}\\left(Y_{j}\\right)\\Vert^{2}W_{i,j} ", "7fefd945ccaffc7a99d83a0b39aab039": "P=\\frac{2 K e^2}{3 c^2} (\\frac{\\gamma^2 v^2}{r})^2 = \\frac{2 K e^2}{3 c^2} \\frac{\\gamma^4 v^4}{r^2}", "7ff03a0f7f65e2b2688d587255196fa9": " K(k) = \\int_0^{\\frac{\\pi}{2}} \\frac {d\\theta}{\\sqrt{1 - k^2 \\sin^2 \\theta}}.", "7ff0605cefefe271d1a6c5f8b4258455": "\\mathbf{c}_i", "7ff0f06b0d4eb02411fd6e884537596e": "r_{I}\\,\\!", "7ff0f81eb90995a82d5e674bb1b3721a": "\n g = g_{11}~g_{22}~g_{33} = h_1^2~h_2^2~h_3^2 \\quad \\Rightarrow \\quad \\sqrt{g} = h_1 h_2 h_3\n", "7ff140fff7dde71951767d28cb5304ac": "{\\scriptstyle S}", "7ff203bbd3242a6dae9e06a0d30f024a": "\\mathcal{L} = \\overline{u}\\,i\\displaystyle{\\not}D \\,u + \\overline{d}\\,i\\displaystyle{\\not}D\\, d + \\mathcal{L}_\\mathrm{gluons}~.", "7ff267caec21b3b07baa8f559101c6c7": "c_0 = { 1 \\over \\sqrt{ \\mu_0 \\varepsilon_0 } } \\,=\\, 2.99792458 \\times 10^8\\;\\textrm{m/s}", "7ff281aca13b442c1f1a9d2898965f4d": " rank(A)", "7ff30d5b517d42c125183bb83cfa29dd": "a = \\sqrt d", "7ff32fdf5361cfc3af72db5e27ea9c90": " p_i ", "7ff3340fcaff8c5086b5b129b8fa0da9": "\\lambda Z. \\phi", "7ff3934d6c152e450977bd5bce3eb96b": "\\dot{u}(t)", "7ff3d754defd51339fc0d555d35c11cf": "\\mathfrak{o}(n, F)", "7ff3e0ef1f21c8a90270ab9967f76e66": "i=1,\\dots,n,", "7ff3efadb173e7b71968ef8cb47a549e": "\\operatorname{E}\\!\\left[\\ln(1+X^2) \\right]=\\ln(4)", "7ff47e553bbe0278b3ae2b9b6e3e302e": "T([x])", "7ff499cf4c68e00e2992b997fe3197e0": "c^{\\epsilon}(x)", "7ff4cea3b132749416a61d32b5f81c07": " I(X;Y) = \\sum_{y \\in Y} \\sum_{x \\in X} \n p(x,y) \\log{ \\left(\\frac{p(x,y)}{p(x)\\,p(y)}\n \\right) }, \\,\\!\n", "7ff51da64c9d4d92abb7d77d0fe4d27f": "=\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)+\\mathbf{g}(n)\\left[d(n)-\\mathbf{x}^{T}(n)\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)\\right]", "7ff525f6557e318e738fc03f2797b0d3": "(i+1)", "7ff52daee2820d1ee3e88ff9f5e22401": "\\hat{H} \\left| \\psi_\\lambda\\right\\rangle = \\lambda\\left|\\psi_\\lambda \\right\\rangle. ", "7ff52ddb3569575fa4071aeaba53cbfd": "\\mathbf{v} = \\mathbf{r}_0\\ \\dot{f}(s) + \\mathbf{v}_0\\ \\dot{g}(s)", "7ff5b53400c69525ef5b04b4bcc17c70": "C_{P} - C_{V}\\,", "7ff5dabb4c1e2e500b0ff23f0aa4c2e7": "\n\\mathbf{J^TJ} = \n\\begin{pmatrix} m & \\sum z & \\sum z^2 & \\sum z^3 \\\\ \n \\sum z & \\sum z^2 & \\sum z^3 & \\sum z^4 \\\\ \n \\sum z^2 & \\sum z^3 & \\sum z^4 & \\sum z^5 \\\\\n \\sum z^3 & \\sum z^4 & \\sum z^5 & \\sum z^6 \\\\\n\n\\end{pmatrix}=\n\\begin{pmatrix} m & 0 & \\sum z^2 &0 \\\\ \n 0 & \\sum z^2 & 0 & \\sum z^4 \\\\ \n \\sum z^2 & 0 & \\sum z^4 &0 \\\\\n 0 &\\sum z^4 & 0 & \\sum z^6 \\\\\n\\end{pmatrix}=\n\\begin{pmatrix} 5 & 0 & 10 & 0 \\\\ 0 & 10 & 0 & 34 \\\\ 10 & 0 & 34 & 0 \\\\ 0 & 34 & 0 & 130\\\\ \\end{pmatrix}\n", "7ff65c6a64756e3702c043d64fc6adb4": "\\mathrm{If}\\; Z_1,Z_2 \\in \\mathcal{L} \\;\\mathrm{and}\\; Z_1 \\leq Z_2 \\; \\mathrm{a.s.} ,\\; \\mathrm{then} \\; \\varrho(Z_1) \\geq \\varrho(Z_2)", "7ff69d0d799745faf4d5049dce543be0": " g(z, u) =\n\\exp \\left( -z + uz + \\sum_{k\\ge 1} \\frac{z^k}{k} \\right)\n= \\frac{e^{-z}}{1-z} e^{uz}.", "7ff6a1a9fa147dcf45a94f1821434b57": "\nP(a, b, \\lambda) = P(a| \\lambda) P(b | \\lambda) P(\\lambda)\\,\n", "7ff6dedcce33cb3a90917df9469321e6": "\\omega_0 = \\sqrt{ k \\over m }", "7ff6dfee570c0b02016af6fdf34c0636": "a_\\theta=r\\ddot{\\theta} + 2\\dot{r} \\dot{\\theta}.", "7ff792f67548590906e72f1533135c3c": "\nY = \\alpha + \\tau D + \\beta_{1}(X-c) + \\beta_{2}D(X-c) + \\epsilon\n", "7ff795715b6092f0951792e82ff6a2a8": "\\mathbf{TR{\\tilde{T}}}", "7ff7c3d610271d4d25e712642c15b1dc": "M_\\mathrm{right}", "7ff7c85a42ed4867267d0e17717e1ad2": " \\$ 3.54 ", "7ff81075c632c0ee50fb6376db61861b": "n = N_1 n_2 + n_1", "7ff863a460fe37f34fd65ecd76ac134b": "{\\widetilde{R}^{\\lambda}}_{\\mu\\alpha\\beta}={R^{\\lambda}}_{\\mu\\alpha\\beta}+\\nabla_{\\alpha}S^{\\lambda}_{\\beta\\mu}-\\nabla_{\\beta}S^{\\lambda}_{\\alpha\\mu}+S^{\\lambda}_{\\alpha\\rho}S^{\\rho}_{\\beta\\mu}-S^{\\lambda}_{\\beta\\rho}S^{\\rho}_{\\alpha\\mu}", "7ff86823f68a0ba503a0f98e08b1f9c8": "E(m) = (g^b, m A^b)", "7ff87ccea7ecb8d2a0c2bca3b4029806": "Cl^{\\ge}_t", "7ff8a785e90556df3916b93fbb9cf54d": "\\omega_0 = \\sqrt{k/m_\\text{equivalent}}", "7ff8e25901d6fe15af60521d05f72b56": "{p(3)}", "7ff8e561d9df0943b6e2fe01fea93aae": " \\tfrac{1}{\\sqrt{w_1^2 + w_2^2 + ... + w_M^2}} ", "7ff97ba84cd38db493e2de1aaf83184b": "\\boldsymbol{v}_{i+1}", "7ff9e158ad3c24923cdf3c1d3a7ef1c0": "\\begin{alignat}{2}\nT & = 2\\pi \\sqrt{\\ell\\over g} \\left( 1+ \\frac{1}{16}\\theta_0^2 + \\frac{11}{3072}\\theta_0^4 + \\frac{173}{737280}\\theta_0^6 + \\frac{22931}{1321205760}\\theta_0^8 + \\frac{1319183}{951268147200}\\theta_0^{10} + \\frac{233526463}{2009078326886400}\\theta_0^{12} + . . . \\right) \n\\end{alignat}.", "7ff9f445b8be6fe22924252f0a5f7332": "\\omega^i_{\\ j}", "7ff9f5473c8d8d520ebb058fdfb5cf58": "F \\equiv \\sum_{k=0}^{\\frac{N}{2}-1}(H_{2k,2k+1}) = \\sum_{k=0}^{\\frac{N}{2}-1}(F_{2k}),", "7ff9feda68a46ad3c94c0c489050605d": "\\boldsymbol{\\Omega}_\\text{T}", "7ffa4b76bb748b4443ebfff56a9a36f8": " (11) \\quad\\quad f^{\\prime} \\left(w_1\\right) > u_s > f^{\\prime} \\left( w_2 \\right),", "7ffa8fb2b59557e5f5267da8f03dbe9d": "A \\varphi_{i} = \\lambda_{i} \\varphi_{i} \\mbox{ for } i = 1, \\dots, N.", "7ffaf8e94e2a4706d579c386de48de9e": "||\\cdot||", "7ffb226b679476817ff7460661c226b2": "(\\exists) \\frac{\\exists x . \\delta(x)}{\\delta(f(x_1,\\ldots,x_n))}", "7ffb3fbfdf0883de1f70fdd7cd90feff": "A_y = [HA]\\epsilon^y_{HA} + [A^-]\\epsilon^y_{A^-} ", "7ffb45f88a0ae7d4fa1ff8f8b85f46c4": "x = y^3 + 2z - 1", "7ffb8e7f707901e87c96882521ab0660": "L(\\sigma_n)", "7ffbeee366219ecdf47633b0dc622c29": "\\mu_a=\\frac{B}{\\mu_{0}*H}", "7ffcc617fa1b502947d466471fdd8cdb": "\nf_Y(y) = \n\\begin{cases}\n\\tfrac 12,& \\text{if }y=1,\\\\\n\\\\\n \\tfrac 12,& \\text{if }y=0,\\\\\n \\end{cases}\n ", "7ffd479fe501b8b6a1995527ce69055d": "\\bar{X}_f", "7ffd8d6ff78461fe9a5fe22275bf441f": "1=K\\frac{A_1 A_2 \\cdots A_n}{B_1 B_2 \\cdots B_m}", "7ffdadb41a167ac70aef7243b735ad20": " C^J_{v_1} \\varepsilon^{v_1}_S + C^J_{v_2} \\varepsilon^{v_2}_S = 0 ", "7ffde38e6ce07445b60639024bcefc77": "\\Omega_j\\!", "7ffdff694d34ae27a00dfbbdf931a5b7": "R(\\theta, \\delta)", "7ffe306f49f8d66b4480be67d70fde6a": "\\vec B_0=0\\ {\\rm or}\\ \\vec k\\|\\vec B_0", "7ffe968719ee9f83fc347b9b095c709f": "O(\\operatorname{min}\\{\\operatorname{length}(X),\\operatorname{length}(Y)\\})", "7ffe96ff648f9c0af24a43564acb0323": "3 \\cdot x \\equiv 2 \\pmod 6,", "7ffe9f261bc6630baf91a5a88a76d9e2": "=-\\frac{b^2-4ac}{4a}=-\\frac{D}{4a}", "7ffebdd78b1f0cada92a8e4ab1d2d804": "\\gamma^1 \\gamma^2 \\gamma^3", "7ffed06c950a4ddc1f8badeaab4f462e": "C_P=C_D \\left(\\lambda-2\\lambda^2+\\lambda^3\\right)", "7fff1a6f2d92a24efd378d4cf53bf6af": "q_\\text{avg}", "7fff69ac0cfc5be194e75032d31cf59f": "\\forall a, b \\in X,\\ R(a,b) \\and R(b,a) \\; \\Rightarrow \\; a = b", "7fff775d42ef0f6541138006fbfd5e08": "\n\\begin{array}{l}\n\\textbf{let}\\ \\mathit{bar}\\ [\\forall\\alpha.\\forall\\beta.\\alpha\\rightarrow(\\beta\\rightarrow\\alpha)] = \\lambda\\ x.\\\\\n\\quad\\textbf{let}\\ \\mathit{foo}\\ [\\forall\\beta.\\beta\\rightarrow\\alpha] = \\lambda\\ y.x\\\\\n\\quad\\textbf{in}\\ \\mathit{foo}\\\\\n\\textbf{in}\\ \\mathit{bar}\n\\end{array}\n", "7fff9af3618ed1edef0101453d2bb5af": "\\mathrm{var}(T)=\\frac{2(\\sigma^2)^2}{n}.", "800014db61360b46aec8e881d4d63dc9": "x=-2a", "80009ffccac22e835688a31108e3c6db": "f(\\vec r,\\vec v,t)", "80013616ab07e16f9f7feed10bc04921": "M_{\\mu\\nu}", "8001acdc1e37538ac912e0037005b3ad": "p_1= 1.78", "8001cce311c12e533e4f452557d8efe8": "\n\\begin{array}{rclllll}\n4s&=& &(1-2+3-4+\\cdots) & {}+(1-2+3-4+\\cdots) & {}+(1-2+3-4+\\cdots) &{}+(1-2+3-4+\\cdots) \\\\\n &=& &(1-2+3-4+\\cdots) & {}+1+(-2+3-4+5+\\cdots) & {}+1+(-2+3-4+5+\\cdots) &{}+(1-2)+(3-4+5-6\\cdots) \\\\\n &=& &(1-2+3-4+\\cdots) & {}+1+(-2+3-4+5+\\cdots) & {}+1+(-2+3-4+5+\\cdots) &{}-1+(3-4+5-6\\cdots) \\\\\n &=&1+&(1-2+3-4+\\cdots) & {}+(-2+3-4+5+\\cdots) & {}+(-2+3-4+5+\\cdots) &{}+(3-4+5-6\\cdots) \\\\\n &=&1+[&(1-2-2+3) & {}+(-2+3+3-4) & {}+(3-4-4+5) &{}+(-4+5+5-6)+\\cdots] \\\\\n &=&1+[&0+0+0+0+\\cdots] \\\\\n4s&=&1\n\\end{array}\n", "80020661a113cc7e6b116d9380070c45": "\\mu: \\Omega \\mapsto \\mathcal{H} ", "80020668f70523fb9ddcf7d22ca2c63f": "D_{4}", "80022dcd8d836151044c099c7fcb6a8a": "1^3+3^3+\\dots+57^3 = (41\\cdot 29)^2", "80023a2bc4809d7269ad84c6a5f9ca84": "\\gamma : [a, b] \\rightarrow \\mathbb{R}^n", "8002ed1a14fdd1e17e188266184d7f1e": "\n\\varepsilon_m = \\left(\\frac{\\delta_\\mathrm{acc}}{\\gamma\\sigma_{x'}}\\right)^2\n", "8002f0c20d3afb93ab7f370af8cef8ec": "(x^i,p_j)", "80034063aaae5b9d4ceceaa734c373e4": "\\frac{1}{\\Phi}", "800366704dbd757a5f045a2df71a2549": "\\mathbf{p} = \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\\\ d_4 \\\\ \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix}", "80039352a415e66e62f08aa869cb1c28": "\\text{PI}=100\\times\\frac{\\sqrt[3]{mass}}{height}", "8003b86ccaae94c0299d3f48af76887d": "Z_n(V) = \\mbox{Tr}\\, M^n", "8003bc25aa4ebff90872e75cb38b2576": "H_d^V", "8003d8dab8837d70d7b1ab3f55197d07": " r_\\mathrm{ corr } = \\bar{ r_g } + \\frac{ g ( m_y - \\bar{ r_g } m_x ) }{ m_x } ", "80040fa98a933e6f96abe067bc18ae41": "0 < c \\leq 2(p-1)", "8004348f1929e809c8895566525c1d5b": "\\mathrm{distance}_{\\textrm{moon}} = \\frac {\\mathrm{distance}_{\\mathrm{observerbase}}} {\\tan (\\mathrm{angle})}", "8004d283638cb2a6d879654e1f95ab2a": "M = |E|", "80054023e63e34cc862dd4084908bea0": "X_{N_2 k_1 + k_2} =\n \\sum_{n_1=0}^{N_1-1} \\sum_{n_2=0}^{N_2-1}\n x_{N_1 n_2 + n_1}\n e^{-\\frac{2\\pi i}{N_1 N_2} \\cdot (N_1 n_2 + n_1) \\cdot (N_2 k_1 + k_2) }", "8005dd97780c1f9a40a55f4fa7f88df9": "y=mx+d, m\\ne0", "80060ff9a3c3c20be551997de1532e8d": "C(u) = \\sum_{i=1}^{k} {\\frac\n\t{N_{i,n}w_i}\n\t{\\sum_{j=1}^k N_{j,n}w_j}}\n\t\\bold{P}_i = \\frac\n\t{\\sum_{i=1}^k {N_{i,n}w_i \\bold{P}_i}}\n\t{\\sum_{i=1}^k {N_{i,n}w_i}}\n\t", "800618943025315f869e4e1f09471012": "F", "80063cf0c606c0e4e63f26b94c69f5b9": "a(t) = r''(t) = (x''(t), y''(t), z''(t)) = (-a \\cos(t), -a \\sin(t), 0)\\,", "80063ef80d0cfc4549138fa843ff11f0": "(\\text{sample skewness})^2 = \\frac{4(\\hat{\\beta}-\\hat{\\alpha})^2 (1 + \\hat{\\alpha} + \\hat{\\beta})}{\\hat{\\alpha} \\hat{\\beta} (2 + \\hat{\\alpha} + \\hat{\\beta})^2}", "800670cb634ce913592592ccec51a703": "P(\\Lambda(X)\\leq \\eta\\mid H_0)=\\alpha ", "80071f0887c0fe0b79f68d507a69aa7b": "|2\\rangle\\leftrightarrow|3\\rangle", "800751969914b13701408b4ee434afcb": "|W|", "80077d0c665aa0b455d9dc95e1746e8e": "x_{k|k}^{(p)}", "80079409c52a2ebd322cfc7540d6ef5f": "\\bar{r}=\\tfrac{N+1}{2}", "8007b2b5bef3bc9ff2bb9e51083b2229": "HP_S(n)", "8007c5d6735ec95d2c439961515906ae": "f(x)=A_{i+1}", "8007df3b20de44123da27d0ce6d09ba2": "S_z = \\hbar(s - a^\\dagger a)", "800805b38e4c56c0ca894e9f36524003": "V_{\\mathrm{out}}(x,t) \\,", "80083f6dc029370053e039874fdfb47c": "\\mathbf{X} \\sim \\mathcal{MN}_{n\\times p}(\\mathbf{M}, \\mathbf{U}, \\mathbf{V}),", "80088e1245b574aa6493685459c3c00b": "\\infty_2 ", "8008b39258ae871c206f2ec12749f17d": "(\\tfrac{7}{5}) = -1: \\qquad \\tfrac{1}{2}\\left (5(\\tfrac{7}{5})+3 \\right ) =-1, \\quad \\tfrac{1}{2} \\left (5(\\tfrac{7}{5})-3 \\right )=-4.", "800967e2bf616752a17b1961dbf8a9d3": "\\frac{18}{(36+n)}", "80099553590e484ce4f7d8e1acd38ff1": "{\\mathbb L}_{x^m}(L)\\equiv{\\langle\\langle} L,{\\mathfrak l}_m{\\rangle\\rangle}", "8009b244f01075c9d630cb32f33b30e8": "\\displaystyle \\frac{1}{|a|} \\hat{f}\\left( \\frac{\\nu}{a} \\right)\\,", "8009be1dc104ff2c0a3c811a28e4dfec": " \\sum_{i=1}^{n}\\|\\mathbf{x}_i - V_{k}\\mathbf{x}_{i}^{k}\\|^2 = 0 ", "8009cb324872557c17d04befbabf2e0a": "2 c_1(A)", "8009ef129862907bc20b69304c06f84b": "\\varphi_1 \\ne w^0_{,1}", "800a14389fd3958a90474c72d66b4a25": "p = 2^{256} - 2^{32} - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1.", "800a43a75ffbf7dd94ee3c806ac5fee3": "A_{s} = \\left(36\\,\\pi V_{p}^2\\right)^{\\frac{1}{3}} = 36^{\\frac{1}{3}} \\pi^{\\frac{1}{3}} V_{p}^{\\frac{2}{3}} = 6^{\\frac{2}{3}} \\pi^{\\frac{1}{3}} V_{p}^{\\frac{2}{3}} = \\pi^{\\frac{1}{3}} \\left(6V_{p}\\right)^{\\frac{2}{3}}\n", "800a43d8d8390b8781d41837a4d2415b": "\\mathbf{r}_{1}\\,", "800a5bf7056994008f6e61c921dd27c1": " \\rho = \\sqrt {\\xi^2 + \\eta^2 + \\zeta^2} ", "800a97002815817aef1b9569c98bcb48": " V_0 = \\frac{-I_0 R \\alpha \\beta}{(\\alpha + \\beta)} \\left(\\frac{1}{\\beta}-\\frac{1}{\\alpha}\\right) ", "800a9bd147a00f8825c2f61c42b84417": "p^{*}", "800afaa098be3992c6ee495e24abfae1": " m \\frac{dV_r}{dt} = F_d + F_c + F_b ", "800b1dfe734c986c6a2f596b7cbaece6": "b_2 \\,\\!", "800b9e34bcf35d7aa6f05eadd9249138": "S = - k \\sum_i p_i \\ln p_i \\;", "800ba394e806c049c98cf4b570d04c72": "i-1", "800ba99bda2936c3995210153bcfe7e9": "(A \\wedge B)^*", "800c0ab8d517b2b394047e9afdc0cb14": "\n\\begin{alignat}{2}\nd\\log(S) & = f^\\prime(S)\\,dS + \\frac{1}{2}f^{\\prime\\prime} (S)S^2\\sigma^2 \\, dt \\\\\n& = \\frac{1}{S} \\left( \\sigma S\\,dW_t + \\mu S\\,dt\\right) - \\frac{1}{2}\\sigma^2\\,dt \\\\\n&= \\sigma\\,dW_t +(\\mu-\\sigma^2/2)\\,dt.\n\\end{alignat}\n", "800c10dd267afc18cb6e27311ee40ac2": "\n\\Bigg(\\frac{p}{q}\\Bigg)_4 \\Bigg(\\frac{q}{p}\\Bigg)_4 =\\left(\\frac{2}{q}\\right)^s.\n", "800c393aa3eabe9d6765246da1d8b867": "=\\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )\\operatorname E_{\\Theta}\\left \\{ \\operatorname E_X\\left [ \\left ( m(\\vartheta)-\\Pi\\right ) | X_{i1},X_{i2},X_{im}\\right ]\\right \\}=0", "800c3b566283d6e219b7c4b7f63248da": "\\mathbb{P}(A_1\\cup A_2)=\\mathbb{P}(A_1)+\\mathbb{P}(A_2)-\\mathbb{P}(A_1\\cap A_2),", "800c88aff17c3a4b0e0911f9afc4b7dc": "A = \\oplus_{n\\in \\mathbb{N}} A_n ", "800ccb7a57accd7b25a2b9141b5b6de2": "p = v(-i) = -vi \\,", "800d4e328cbb7d627bb6d42c27cd0b7a": "E[\\psi(\\theta)]=\\psi(\\theta)", "800da364c3edd391f5eb413e43cf84e0": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{p}) = \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) = \\sigma^2 \\sideset{}{}\\sum_{j = 1}^{p}\\frac{\\mathbf{v}_j\\mathbf{v}_j^{T}}{\\lambda_j} ", "800dc7a60ce2cb9cccc37b333c4da0ec": "\\frac{1}{M^{n-1} \\cdot s}", "800df8aed8fb3ae0ddb75095744a781f": "5 = 1^2 + 2^2, \\quad 13 = 2^2 + 3^2, \\quad 17 = 1^2 + 4^2, \\quad 29 = 2^2 + 5^2, \\quad 37 = 1^2 + 6^2, \\quad 41 = 4^2 + 5^2.", "800e15d8692557e3d5a3392403211a7b": "\\overline{T} = \\{U^k, SU^k,VU^k,SVU^k | k = 0,\\ldots, 5\\}.", "800e16c1b3d6fa6523d21a1e269920e7": " f_c", "800e26397147a616c66412035c43fed8": "\\delta \\Pi = \\delta U - \\delta V ", "800e2a1e69cf34f0180eb3e949780761": "\\Pi(z) \\Pi(-z) = \\frac{\\pi z}{\\sin( \\pi z)} = \\frac{1}{\\operatorname{sinc}(z)}", "800e329b23a07b3d888bdf92c61e0a5d": "\\theta_1 ", "800e67279f3558c61357ab20b4c749b0": "(y_1(t), ..., y_n(t))", "800e81926803a8714b6759568c2073a3": "\\operatorname{P}\\{X_{a,b}\\le x\\}=\\int_0^x ab t^{a-1}(1-t^a)^{b-1}dt=\n\\int_0^{x^a} b(1-t)^{b-1}dt=\\operatorname{P}\\{Y_{1,b}\\le x^a\\}\n=\\operatorname{P}\\{Y^{1/a}_{1,b}\\le x\\}\n.", "800e87ecca7d9069d11271599eb055b6": "V^{\\prime}", "800e9d50f5ac79c386a08823088a8bb4": "\\alpha \\approx 0.878", "800f0fdb6eec4b299c8aa5034359e89b": "\n{\\mathcal L}_\\phi=-\\frac{1}{4\\pi}\\omega\\left[\\frac{1}{4}B^{\\mu\\nu}B_{\\mu\\nu}-\\frac{1}{2}\\mu^2\\phi_\\mu\\phi^\\mu+V_\\phi(\\phi)\\right]\\sqrt{-g},\n", "800f4cdbaab74f0495430fcc4a7262f9": "f_{i-1}(r_1,\\dots)", "800fb50f8102722a8f8e7a8641811d6a": "\n\\sqrt{z} = \\sqrt{x^2+y} = x+\\cfrac{y} {2x+\\cfrac{y} {2x+\\cfrac{y} {2x+\\cfrac{y} {2x+\\ddots}}}} \n= x+\\cfrac{2x \\cdot y} {2(2z - y)-y-\\cfrac{y^2} {2(2z - y)-\\cfrac{y^2} {2(2z - y)-\\ddots}}}.\n", "800fcd7c7bf946f12524ba91ffa2f990": "\\omega_{n} f_{0}", "800fd0c41bb8b9c5e1c7bc0c704601ea": "y = \\frac{ax+b}{cx+d}\\, .", "801030ebd5d70e70461452343fdc56e4": "= 0\\,", "80104418623892db5ad23ead01c07b0e": " = \\vec{\\nabla}_{\\vec{r}}\\bigg(\\frac{1}{A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{{(\\vec{r}-\\vec{r}') \\bullet{} \\vec{F}(\\vec{r}')}}{|\\vec{r}-\\vec{r}'|^n}d\\tau'}\\bigg)", "8010968be76558a9a5ead04e44c8c332": "\\mathrm{Hom}(X\\times Y,Z) \\cong \\mathrm{Hom}(X,Z^Y)", "8010eb5d867e10e8317c45b5e0b2474a": " |a_i-b_i| \\leq 1/2 ", "80110a7bbe8945e5805c0aa740c6f5f1": "y = \\left(-{\\cos\\theta\\over\\sin\\theta}\\right)x + \\left({r\\over{\\sin\\theta}}\\right)", "801111377f738d578e79bc7eea5ce776": "\\begin{align} \\|f_1\\cdots f_n\\|_r &\\le \\|f_1\\cdots f_{n-1}\\|_r \\|f_n\\|_\\infty\\\\\n&\\le\\|f_1\\|_{p_1}\\cdots\\|f_{n-1}\\|_{p_{n-1}}\\|f_n\\|_\\infty.\\end{align}", "80111b31456fa5b5f5a0bb7dc675d30e": "e^{..r^2}", "80118b84aa5bc833636d83c5e6e16206": "u_x(0)\\,c(0) -D \\frac{\\partial c(0)}{\\partial x}=0\\,", "8011b15e0cdcdeeb5bfb0b50bbaf8ad7": "y^2=f(x)", "8011cecf84f514bc4d95d6fac8c26aab": " \\{ \\cdot,\\cdot \\}_{N} ", "8011ee9725c0afd74933061a44832cc6": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = 4x^{2} + 4y^{2} \\\\\n f_{2}\\left(x,y\\right) & = \\left(x - 5\\right)^{2} + \\left(y - 5\\right)^{2} \\\\\n\\end{cases}\n", "80125ffbba4905e979f034aae723c9b6": "\\dot{a}(t)", "801378194bff778d13caf570eb41a520": "M_{T}^2 \\rightarrow 2 E_{T, 1} E_{T, 2} \\left( 1 - \\cos \\phi \\right)", "8013e861487c5d9dea2921bebe76112c": "\\mathbb{Z}_{n}\\oplus\\mathbb{Z}_{n}", "801408d1a724a2532a2ea1105d3f2f49": " r^{(2)}\\in R. ", "801418052da20c3e01dd6a0d8c46fca0": "H^2(z)= H_0^2 \\left( \\Omega_M (1+z)^{3} + \\Omega_{de}(1+z)^{3\\left(1+w \\right)} \\right).", "80152dd4fac60327fbad0e96abeb9375": " \\mathbf{\\bar y}' = \\mathbf{T}' \\, \\mathbf{y}' ", "8015548db6152b4cab5c3a02139960e6": "\\left(\\sqrt{\\frac{2}{5}},\\ \\frac{2}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ 0 \\right)", "80158d8e3c0ae34c3fc0c48c7b020ccb": "R = a_1a_2....a_r", "8015ed861a2541f7b1f398808b5f1c33": "\\sum a_n ", "8015f2244383247721b553764d5d0cd6": "\\dot{\\sigma} \n= \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} \\overbrace{\\dot{\\mathbf{x}}}^{\\tfrac{\\operatorname{d} \\mathbf{x}}{\\operatorname{d} t}}\n= \\frac{\\partial \\sigma}{\\partial \\mathbf{x}} \\overbrace{\\left( f(\\mathbf{x},t) + B(\\mathbf{x},t) \\mathbf{u} \\right)}^{\\dot{\\mathbf{x}}}", "80160fefb353af913df9643007006652": "k\\ln(p/p_o)", "80163df38c88072d6f05c5d9f2467230": "\\mathfrak{n}_k", "801650fc88d006dd9629c5e5b97b093f": "(l_1 l_2)^\\theta f(x)=f(l_1 l_2 x)=l_1^\\theta f(l_2x)=l_1^\\theta l_2^\\theta f(x)", "8016771ec0b5c6a2a2cf98c2db327f36": "K = \\sqrt{\\frac{\\pi k_b T}{2}}\\left(\\frac{1}{m(x)}+\\frac{1}{m(y)}\\right)^{1/2}\\left(d(x)+d(y)\\right)^2.", "80168a9123e5e6d75c5b904076fdeb8f": "ij^{th}", "80168b3652f4f1f72ca5f828b56c2541": "\\{x\\}= \\frac{1}{2} - \\frac{1}{\\pi} \\sum_{k=1}^\\infty\n\\frac{\\sin(2 \\pi k x)} {k}\\qquad\\mbox{for }x\\mbox{ not an integer}.\n", "8016b68434871c8bb6863d39a48aed25": "f(x,[1,2,3],z) == [f(x,1,z), f(x,2,z), f(x,3,z)]", "8016ea17182d64b546ab62267df799f7": "\\boldsymbol{\\Sigma_i}", "8016fc0c6bc4fa732debb1baa0ee281e": "2^{k-1}\\cdot h", "80170966be5993fe203acafb0801f45b": "\\sim 10^{120}\\,\\!", "801783ad64a7ec8a639a166a2e1dc98a": "T_0 ", "80178bd27b0e318a4f030bb7e5bfb131": "\\pi = {-\\phi (\\operatorname{diag}(Q))^{-1} \\over \\left\\| \\phi (\\operatorname{diag}(Q))^{-1} \\right\\|_1}. ", "80180e2f21330b2fd543f260d0f1a3cb": "\\mathbf{M} = diag(\\begin{bmatrix} 1 & 1 & det(\\mathbf{U}) det(\\mathbf{V})\\end{bmatrix})", "80186217811236b6337d1254d341eaf8": "2^\\sqrt{2}", "80187af4f0041fe40b6fb0e2a16b0814": " N\\left| n \\right\\rangle =n\\left| n \\right\\rangle", "80187e4a32140ce1b83ae9d32dfbf566": "s_A=\\min_i x_i; s_B=\\max_i x_i", "801893ca264806df6e0742c4984cf1d5": "M^{\\alpha\\beta} = X^\\alpha P^\\beta - X^\\beta P^\\alpha = 2 X^{[\\alpha} P^{\\beta]} \\quad \\rightleftharpoons \\quad \\mathbf{M} = \\mathbf{X}\\wedge\\mathbf{P}\\,,", "80191156e02bab789e9fe4f8406a3813": "\\|\\cdot\\|''", "801918106b70317a5dd2f797372fbc08": "D=R\\sqrt{3N},\\,", "801aa73f0d5f82ecab0d18903cd3fcf1": "p \\equiv 3\\pmod{4}", "801af4137c03e0d8e514001be8352a09": "(\\lambda\\boldsymbol\\Lambda)^{-1}", "801b31861529a8cc11c57d6785c6b4b7": "e_j e_k = \\omega \\, e_k e_j \\,", "801b50f697196f1a86b45d264026a5a1": "y_j^*", "801b52850901103b4c5900ad840751fe": "\\psi_n (z) = \\psi_n (0)e^{\\pm ikz}", "801b55705502c1d1e19c9f8085a542bb": " = 7.4 ", "801bbde9be55d1f45ef5e0b5ff203b86": "W = K \\log m \\,", "801be3a109e8273b09c7e2a40ad4eb8a": "\n= {1\\over 2}\\int d^4x \\; A_{\\nu} \\left( \\partial^{2} A^{\\nu} - \\partial^{\\nu} \\partial_{\\mu} A^{\\mu} \\right)\n= {1\\over 2}\\int d^4x \\; A^{\\mu} \\left( \\eta_{\\mu \\nu} \\partial^{2} \\right) A^{\\nu} \n,", "801be6497012417d3e89d68ddfde1b16": "\n\\Psi (\\rho) = \\begin{bmatrix} \\rho (F_1) \\\\ \\vdots \\\\ \\rho (F_n) \\end{bmatrix}.\n", "801c0d01f67cff7ab5e856056176e7a2": "F(y) = \\int_{x_0}^{x_1}y(x)\\;\\mathrm{d}x", "801c298154eaf8ff839df93cf5a99414": "\\scriptstyle P_{ab} \\;=\\; -P_{ba}", "801c6095300aa9d0de25f99fc7161c36": "\\frac{dN_1}{dt} = r_1 N_1\\frac{K_1-N_1 - \\alpha N_2}{K_1}\\,", "801c83caee5f25a52a5a0eed50a44c2d": "M_\\sigma", "801cc375c6a91c159bf8d3994ee56aff": "\\sqrt{n}(\\hat\\sigma^2-\\sigma^2)\\ \\xrightarrow{d}\\ \\mathcal{N}\\big(0,\\;\\operatorname{E}[\\varepsilon_i^4]-\\sigma^4\\big). ", "801cc67b01de4530325f3e7491032c6a": "\n W = -\\cfrac{\\mu J_m}{2} \\ln\\left(1 - \\cfrac{I_1-3}{J_m}\\right) + \\cfrac{\\kappa}{2}\\left(\\cfrac{J^2-1}{2} - \\ln J\\right)^4\n ", "801cd1a81637937ea0b661a3d6228dac": "\\bar v_i \\leftarrow \\bar v_i-\\nabla E_{snake}(\\bar v_i)", "801ce1c818aea893432cc780a09d3b15": " \\int S = \\lim_{n \\to \\infty} \\frac{\\sum_{k=0}^n \\frac{1}{1+k} \\langle S e_k , e_k \\rangle}{\\sum_{k=0}^n \\frac{1}{1+k}} ", "801cf3fa99fea12ab661c5ffc0b4089a": "\\operatorname{pmi}(x;y)=\\operatorname{pmi}(y;x)", "801da42ca21c738a9d914ccf99ae75a5": "A, B\\in \\mathbb{M}_n", "801dba46ad4be88515a29311302f1c78": "\\textstyle\\beta", "801dd8e49c12855a8fb959ec5fe215ee": "E\\,\\!", "801e38d8a79ea7f1d27aca76b31c6c42": "|E(C)|", "801e5fa761aceb264a730042834fab7c": " g:S \\to \\mathbb{R} ", "801e89c3a5e1cfdfe9d229b56f65d413": "P_0 \\leqslant \\frac{M_a}{r} ", "801eb57a689504e96228755c8aa45389": "[S_y,S_z]=i\\hbar S_x", "801ef6baf40b10b94b53166f8446a577": "H(x) = \\sup_{y\\in K} \\langle x,y\\rangle.", "801f11b359f84f882e5b45e9bf5f1088": " t_{n+1} ", "801f872b8335e9a319e4334a5b6eff78": "\\mathrm{D} F := \\sum_{i = 1}^{n} \\frac{\\partial f}{\\partial x_{i}} (W(h_{1}), \\ldots, W(h_{n})) h_{i}.", "801fb036f118a8c5fe6c0f8c9de171b8": " J = \\left[w_i/y_j \\right] ", "801ffbe704591004debab7f79f777862": "\\tan x = \\frac{1}{1!}x + \\frac{2}{3!}x^3 + \\frac{16}{5!}x^5 + \\cdots = \\sum_{n=0}^\\infty A_{2n+1} {x^{2n+1} \\over ({2n+1})!}.", "802038116d74e86446bc2f20a6aa08b6": "y = W^Tx = W^Ts + W^Tv = p + z", "8020591a9fe215b352b0b65a04830b25": "M_{k-1}, M_{k-2},\\ldots , M_{1}, M_{0}", "802099b1aba3936e7a905298b290398f": "\\psi \\in A \\Leftrightarrow \\exists\\mbox{ a finite function }\\theta \\subseteq \\psi\\mbox{ such that }\\theta \\in A.", "80218973889da8e54dd1cb60aa37d40a": "y^2=(1-r)^2+x^2-r^2", "80219e1d3d501184778ba6ecacca213d": "\\mathcal{D}_0(S):=\\{D\\in\\mathcal{D}(S)|D\\cdot X=0,\\text{for all } X\\in\\mathcal{D}(S)\\}", "8021b37327dd626662432c231ce8074a": "|{\\tilde{\\psi}_{Tr}}\\rangle", "8022634598e3cff78552f482ce3669a1": " ID", "80226ba5aa068be17a1604945e0f0ce2": "2000\\ ", "8022f13bfcab1490ca60e539c16ffadd": "GE(\\alpha) =\n\t\t\\frac{1}{N \\alpha (\\alpha-1)}\n\t\t\\sum_{i=1}^N\n\t\t\\left[\n\t\t\t\\left(\n\t\t\t\t\\frac{y_i}{\\overline{y}}\n\t\t\t\\right)\n\t\t\t\t^\\alpha - 1\n\t\t\\right],\n\t\t\t\\quad\n\t\t\t\\text{ for real values } \\alpha \\ne 0, 1,\n", "8022fe06d9315bddf5b4433ccb2dfa94": "C_{V}=T\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}\\,", "80231ac31cc9a511f4b781569414157a": "[x_0:\\cdots:x_n]", "802344ecd2a2da57671ffcdb7a0097b7": "\\mathbf{Q}(t) \\equiv \\mathbf{q}(t+\\tau)", "8023fcf9fd829be928b1ff39415c1645": " Y = g\\left(X'\\beta_{0}\\right) + u, \\, ", "80240e4d3a2df43f5a6b94900df7850e": "{}^6_4", "8024229e894c6e9b2d0084eddc8db290": "L \\to s (70%) | dFd (30%)", "802427eb6b74053338aa8f53fb88e7b2": "\n(F \\circ \\phi)'(t) = F'(\\phi(t))\\phi'(t) = f(\\phi(t))\\phi'(t).\n", "80242a58fa5e5e056beed9bd9cb3fc0f": "\\mathbf W_\\delta", "80251121f64e490494225f97583a33aa": "\\frac{3054}{(1+0.10)^3}", "802515ef44cb021a7e65de4f880bddb2": "2^{6/12} = \\sqrt{2}", "8026130283efd4bdacffa1cbe3e4352d": "S_5 \\subseteq G(E/F)", "80261a1fd86bdd0b62ee078cab2eba32": "(Rf_!M) \\otimes_Y N \\to Rf_!(M \\otimes_X Lf^*N),", "80262e896c525e94970975c305910eb5": "a\\succ b\\;", "80267f9aaafe187e92c0d35e6ba71232": "D(f) = O(Q_2(f)^6)", "80268a4fb4cedd968da9627a9a09ad3a": " \\mathcal{F}\\{f\\}\\,", "80272e72a9c5a606b56cc7894deca383": " \\mathcal{E} = I K", "802744c416061cc27b9532b2e9bdfa1c": "R_\\mathrm{E}/2\\pi", "80275ef20e4b75147e4bea0455208791": "d'_i =\n\\begin{cases}\n\\begin{array}{lcl}\n \\cfrac{d_i}{b_i} & ; & i = 1 \\\\\n \\cfrac{d_i - d'_{i - 1} a_i}{b_i - c'_{i - 1} a_i} & ; & i = 2, 3, \\dots, n. \\\\\n\\end{array}\n\\end{cases}\n\\,", "8027d80894b3e8d44474fe0730f3bfe2": "\\sum\\nolimits_{n\\in \\mathbf{Z}} f(na)= \\frac{\\sqrt{2\\pi}}{a} \\sum\\nolimits_{n\\in \\mathbf{Z}} \\widehat{f} \\left (\\tfrac{2\\pi n}{a} \\right ),", "8027ebb74dd4aec9bfbfcc07162cc84d": " \\delta = \\begin{cases} \\frac{d+2}{d-2} & \\ if \\ 2 < d < 4 \\\\\n 3 & if \\ d > 4 \\end{cases}\n", "8028ae18c92af4539e951c54a04996f5": "\\theta(\\lambda)=\\sum_{r}\\varphi_{r}\\exp\\left(\\sum_i\\frac{(\\lambda \\alpha_{ri}+(1-\\lambda)\\beta_{ri}))\\mu_i}{RT}\\right)", "8028ebaf9aaaf35fd0ab433cb276db16": "J \\geq 0", "802914d74c5ec475b03d8e01114b4af4": "{x_1}", "802934b05c633a52c746de87b76bf6bc": "ax + by + cz = d", "8029362b179e1cf55756754d40e53ce4": "\\partial^2=\\bar{\\partial}^2=\\partial\\bar{\\partial}+\\bar{\\partial}\\partial=0.", "80293908e14fc5a39564dc7a19cdeb35": "\\mathcal{E} _{\\vec{k}}=(\\hbar ck/2\\epsilon _0 \\Omega)^{1/2}", "8029813ee8a5fd22fda13793bbe0812e": "{B}'(v)=\\frac{f(v|v)}{F(v|v)}(v-B(v))", "8029960abca70036925ffecf8af2243f": "0 + a = a,\\,", "80299adae059a80e3720233e75325598": "(I-\\gamma P)", "8029c8dc9b28f3ec224515d254cb796c": "A\\overline{C} + A\\overline{B} + BC\\overline{D}", "802a62be3c09fe2e9095d693f1f0d65c": "v \\in T_p M", "802ac88c37d6acb1cdc76fa16600b829": "\\Phi=e^{\\beta \\epsilon}/z-1\\,", "802b29c1c166e7dd16dd973386cf69fe": "0 = T^{\\mu \\nu}{}_{;\\nu} = \\nabla_{\\nu} T^{\\mu \\nu} = T^{\\mu \\nu}{}_{,\\nu} + T^{\\sigma \\nu} \\Gamma^{\\mu}{}_{\\sigma \\nu} + T^{\\mu \\sigma} \\Gamma^{\\nu}{}_{\\sigma \\nu}", "802b318cf8cceb3ce59b86e052de214f": "\\frac {1}{c(w)} =\\frac {1}{c_r} |\\frac{w}{w_r}|^{-\\gamma} \\quad (1.7)", "802bacabd03ab0a7c6c13855ad409b14": "f\\colon S\\rightarrow\\{1,\\ldots,n\\}", "802bda8e547964a7ac0ce8a2aca2e1fb": " \\lim_k \\int f_k \\, d \\mu \\geq \\int g \\, d \\mu.", "802c17bad6202f772e9b4ed382e1e8ef": "e^{1/r}-1=\\{1r, 2r, 3r, 4r, 5r, 6r, \\dots\\}\\;", "802cd00be6c18f833fd060ddae0621e5": "\\Xi(x)=\\Gamma(x)\\Lambda(x)=\\alpha^{3}+\\alpha^{-7}x+\\alpha^{-4}x^2+\\alpha^{5}x^3,", "802d42a8fc13f70cb612b79891007e47": "V_\\parallel", "802d6872c5e629306343b52209aa8eca": " | f(x) - f(y)| < {1 \\over n}", "802dacefd5a171415cabe425cee65d19": "f_0=f, f_1 , \\ldots, f_{s-1}", "802dd75db52d5128291782eb162ed1c9": "-1.1668", "802e26f9bfd6e9e3681ead3221f54f30": " h(i,k) = ( h_1(k) + i \\cdot h_2(k) ) \\mod |T|.", "802e2beed0c310d6ea9027374ae455dc": "A_{1, \\sigma_1} \\cdot A_{2, \\sigma_2} \\cdots A_{n, \\sigma_n}.\\ ", "802e71a1a52b382438f08e5e8cf87f3e": " W^{s,p}(\\Omega) = \\left (W^{k,p}(\\Omega), W^{k+1,p}(\\Omega) \\right)_{\\theta, p} , \\quad k \\in \\mathbb{N}, s \\in (k, k+1), \\theta = s - \\lfloor s \\rfloor ", "802e7a2825dab053f0382f8de4ea3cef": "N_e^{(v)} = {4 N - 2D \\over 2 + \\operatorname{var}(k)}", "802ea0c9803fe1552e10dd1f66d0551a": "\\boldsymbol{Q}(\\boldsymbol{N})", "802ec446c41fbc9e20c7ba9f34d9b9cb": "\\,\\! z=xy", "802ef8847f46e4021fb1f6ec65d8b764": "\\frac{L}{L_{\\odot}} \\approx 3200 \\frac{M}{M_{\\odot}} \\qquad (M > 20M_{\\odot})", "802f65ff76bdf7c9bd01996d31da160d": "\\delta = \\sum_{i=1}^{n} \\xi_{z_i}(Z) \\frac{\\partial}{\\partial z_i}", "802f9cdef84be563afdffc986f66c7cb": "\\beta^2 = \\frac{\\mu}{\\rho}", "802fceee9962c371dd7ab2054f82814f": "L[y]=\\frac{y\\circ t+y\\circ -t}{2}", "802fda344b8c0b80adad97d69b3a054e": "\nG_p(t) = \\alpha^2t exp(-\\alpha t) \\quad for\\ t \\geq 0\n", "8030888c9395aba06d9400e40fa14e04": " \\psi_{n\\ell m}(r,\\vartheta,\\varphi) = \\sqrt {{\\left ( \\frac{2}{n a_0} \\right )}^3\\frac{(n-\\ell-1)!}{2n(n+\\ell)!} } e^{- \\rho / 2} \\rho^{\\ell} L_{n-\\ell-1}^{2\\ell+1}(\\rho) Y_{\\ell}^{m}(\\vartheta, \\varphi ) ", "8030c0d5a592abc0dd0d201272cefa34": "f_{A_+} (x) = \\frac{2 \\sqrt{6}}{x^2} \\sum_{j=1}^\\infty v_j^{2/3} e^{-v_j} U\\left ( - \\frac{5}{6} , \\frac{4}{3}; v_j \\right ) \\ \\ \\mbox{with} \\ \\ v_j = 2 |a_j|^3 / 27x^2", "8030de7c87a510788c4adc5ce4b1c656": "\\langle E \\rangle = \\frac{\\hbar}{2} \\cdot 2\n\\int \\frac{A dk_x dk_y}{(2\\pi)^2} \\sum_{n=1}^\\infty \\omega_n ", "8031656267a7012568487e26fa59efd2": "W_{ij}=0", "8031a9bb95da0ab0e371e670ebf942f0": "\\sqrt{u_x^2 + u_y^2} = c", "8031d64f608e8fc88d776c4b332541f2": "K = [\\gamma(v) - 1]m_0 c^2\\,,", "8031f46909acb4035936f06c846d41fd": "b_k=(-1)^k\\,a_k^2+2\\sum_{j=0}^{k-1}(-1)^j\\,a_ja_{2k-j},\\text{ with }a_0=b_0=1. \\, ", "80327289bae1c736e8706a2f8db14be7": "\\alpha_j \\colon A_j \\to \\bigoplus_{i \\in I} A_i", "8032aedcd97c190e302b2a64dbfc1f60": "\\mu/\\mu_0 = 1+\\chi_\\text{m}", "8032b82fe12d9482bb4d1b0fd33152cf": " \\mathrm{Pr}[\\forall j ", "8032c0aa03a360e8a5c0920e33ef9f79": "K= C_{12}+C_{23}-C_{13}", "80331f5a2dc28b5bd99e4b8c13e6fd93": "GF(q)", "80332dd302b625f8e8402aaa6c9db865": "0.7080\\sqrt{N}+0.522,", "8033656957e80857f6f03c7fd0beb829": "(x_1,x_2,\\ldots,x_p)\\mapsto(x_2,\\ldots,x_p,x_1)", "803384698875304472fd37ba4812ff15": " \\begin{bmatrix} \\dfrac{d x_1^*}{d p_1} \\\\[2.2ex] \\dfrac{d x_2^*}{d p_1} \\\\[2.2ex] \\dfrac{d y_2^*}{d p_1} \\end{bmatrix}\n= \\begin{bmatrix} -1 & -1 & 0 \\\\ -1 & -3 & -1 \\\\ 0 & -1 & -3 \\end{bmatrix}^{-1}\n\\begin{bmatrix} -1 \\\\ 0 \\\\ 0 \\end{bmatrix}\n= \\frac{1}{5}\n\\begin{bmatrix} 8 \\\\ -3 \\\\ 1 \\end{bmatrix}\n", "80339117ba98ac98040c53affdfc675e": "N_{W-P} \\le 0.3", "8033be79159c046b6a9c0cacd017550e": "\\frac{n_3}{\\tau_{32}} = \\frac{n_2}{\\tau_{21}}", "8033cc80d3e02d4eec86b467da71a250": "a_{x,y,z}", "8033f1e0f3b1b01baf980b07d9c5e2ae": "M = \\sum \\mu_i m_i m_i^T", "8033fc2af6b68b4dde8dcff0b74c747f": "p_i=mg_{ij} \\frac{dq^j}{dt}", "8034615c89ae18d7f724f13d862953fd": "D(I) = 1", "8034cc3567e3317d94d0ea56fd9f04d0": " \\langle m_{\\mu},h_{\\nu} \\rangle = \\delta_{\\mu\\nu} ", "803511e5326220e1987a8035d9c149e0": "g_1=2a_1/\\gamma,", "8035b394c0f201ae09a3d10adadb1c14": " A_\\alpha B^\\beta \\rightarrow A_\\alpha B^\\alpha \\equiv \\sum_\\alpha A_{\\alpha}B^\\alpha \\,.", "8035b7a41ff3c26fd0a919e083f0266c": "u_x=v_y, \\quad u_y=-v_x,\\,", "80363bd1cb6853e028fe64760e3df7ac": " \\partial(...)/\\partial t = 0 ", "80367f0458be09e1ae96b555bfd7bfdc": "2^{16_{dec}}", "8036896237f2df9920b7ce4f53247158": "V_{out} = A_{OL}\\cdot V'_{in}", "8036969d1a6d7a67ee9f81feb391f065": " \\left(\\mathbf{A} \\circ \\mathbf{B}\\right)_{ij} = A_{ij}B_{ij}\\,,", "80369f009d0317b3c4a06ee2520990d9": "\\mathfrak{p}_iR[x]", "8036bf3aa70ca7b142dd511b33f954f5": "\\sqrt{x}+O(\\sqrt[4]{x})", "8036d49e16c38e259d98e0fa29919baf": "\\left[{n\\atop 4}\\right] = \\frac{1}{3!}(n-1)! \\left[ (H_{n-1})^3 - 3H_{n-1}H_{n-1}^{(2)}+2H_{n-1}^{(3)} \\right]", "8037161c370ad3c60e177aecfe45519e": "(\\rho_0 +\\rho_0 s)\\left( \\frac{\\partial }{\\partial t} + u \\frac{\\partial }{\\partial x} \\right) u + \\frac{\\partial }{\\partial x} (P_0 + p) = 0", "80371d7b51ee9fb89385bdea5b245eea": "\\Pr(X_{n+1}=x|X_1=x_1, X_2=x_2, \\ldots, X_n=x_n) = \\Pr(X_{n+1}=x|X_n=x_n)", "80378e42fa4ec89b8fb54c21772671c6": "\\begin{align}\n\\operatorname{Cov} \\left( {z',z'A'} \\right) &= E\\left[ {\\left( {z - \\mu } \\right)' (Az - A\\mu)} \\right] \\\\\n&= E \\left[ \\operatorname{trace}\\left[ A(z - \\mu )'(z - \\mu )\\right] \\right] \\\\\n&= \\operatorname{trace} \\left[ {A \\cdot E \\left[(z - \\mu )'(z - \\mu )\\right] } \\right] \\\\\n&= \\operatorname{trace} [A V]. \n\\end{align}", "8037aff223555a6a2d0cfacb418f4b87": "\\pi_\\lambda(g^{-1})\\xi(z)=|cz + d|^{-2-i\\lambda} \\xi(g(z)).", "8037bc946af1ec0d96f8adca5c393e4d": "{\\rm PV}=\\frac{$100}{(1+0.12)^5}=$56.74.", "8037cb5bd607b3635cb94b72b9fcb6a0": "<\\Psi , s> = < \\Psi , s'>", "8037e3ae2991867e102f8cea561c3894": "K(S)", "8037e930c5602678c5b6c1e6ad633807": "(x_i, \\;y_i)", "80382ddf08cf03883d62516fdd50b099": "\\sigma_D", "8038307f1fb7646f969a9434cfb2b915": "N = \\frac{MC}{R},", "8038753e9235227061f2f7e2d15cb44a": "\\epsilon_x(y) = \\frac {-\\epsilon_my}{c}", "80388615dcd334e93f66ad6a03e11d69": "\ng(y,x) = f(x) \\ , \\ \\forall x\\in X, y\\in Y\n", "8038b6c9909423c15d08e08900e16df7": "\n(T_h)_{j,k} = h_{2\\cdot j-k}.\n", "8038fdc4292e81dfdeb59d110dd10f1b": "|\\alpha|\\le q^{(d-1)/2 }.", "80391c68a70cf99ba116b3d01a403c93": "g_{\\rm spherical} = \\frac{GM}{r^2}\\,\\!", "80397f7582052a07566e51ddd9ac618f": "F(\\Omega_1,\\Omega_2,\\ldots,\\Omega_m) = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \nf(t_1,t_2,\\ldots,t_m) e^{-j \\Omega_1 n_1-j \\Omega_2 n_2 \\cdots -j \\Omega_m n_m} \\, dt_1 \\cdots \\,dt_m ", "8039d2f2a04b31ca014da5caae189349": "\\epsilon=-\\frac{\\mu}{2a}", "8039d78eef760cafb04de06031ecbcd4": "\\left|\\frac{x}{x_m}\\right|+\\left|\\frac{y}{y_m}\\right|+\\left|\\frac{z}{z_m}\\right| = 1", "8039ebb89377b123f7b7c3211c5f1055": "\\hat\\sigma_i^2 = \\hat u_i^2", "803a4b13d9d8fcd34e24d6e59d2fb8f6": " L ", "803a59a3f6a9c1abc5eada3b1eceb186": "|\\widehat{Q}_{r}(h)-\\widehat{Q}_{s}(h)|\\geq\\epsilon/2\\,\\!", "803a63675c1cc55a3d5a7bda5d2b0af8": "\\hat{\\bold{\\Psi}}_4", "803ad21d94eb2051cdb62143127c9048": "\\Delta{P}_t", "803afd9546b05fdcab4029dd496034a8": "T''\\,", "803b0a5b496f874cff249839a21168e2": "\\scriptstyle\\hat{m}(\\theta)", "803b4009d19e1d87ad0bff2fcdfedda0": "g > 1", "803b55ab6624a64463ff3eb4e0b9a093": "\\mathbb F_9", "803b815f4f0713805c33cecc8ff8147a": "\\int\\frac{\\sin^2 ax\\;\\mathrm{d}x}{\\cos ax} = -\\frac{1}{a}\\sin ax+\\frac{1}{a}\\ln\\left|\\tan\\left(\\frac{\\pi}{4}+\\frac{ax}{2}\\right)\\right|+C", "803b97674b0fb84b891df9538f989f55": " \\sqrt{z}, \\quad \\ln(z), \\quad \\tanh(z) ", "803b9be7bc647189d4c15df2f435236f": "\nd^\\alpha\\int_G \\phi^\\alpha_{e_i,e_j}(g)\\overline{\\phi^\\alpha_{e_m,e_n}(g)}dg=\\delta_{i,m}\\delta_{j,n}\n", "803bbd642c740a7053983b0a386b8676": "\\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1},", "803bfc179c3b9ca11ea89354ba01baae": "\\sum_{k=1}^n \\frac1{p_k}=\\frac1r.", "803c2a574a3ad82ac00c8c2a733f635b": "ProPoints = \\max \\left\\{ \\mathrm{round} \\left( \\frac{(16 \\cdot protein) + (19 \\cdot carbohydrates) + (45 \\cdot fat) + (5 \\cdot fiber)}{175}\\right), 0 \\right\\}", "803c34bbc9a3b86f7d58e860fe54f812": "\\{x\\in A:f(x)1 ", "8071eff79b7ffa63ddaa35187e8f6aa3": "A(x+h)-A(x)=f(x)h+(Red Excess)", "8071fdf1c24966a17e6aac1dd015cb65": "J_{i,j}", "80725129a7251b280ec28f43ab3d8469": "\\frac{V_t}{V_d} = \\frac{1}{1.2}", "8072697673c9c9758880bb7235202eab": "a=b=S", "80726bdfc3d8d5e85ca764eb72beccc7": "K[[x,y,z]]/(xy,xz)", "80728d38ea1d3e7ef487d9b5523f751f": " \\frac{1}{q(z)} = \\frac{1}{R(z)} - \\frac{i\\lambda_0}{\\pi n w(z)^2} ", "80732a701b270a5355be0019f59feb9c": "\\sqrt \\frac{150}{400\\times .005454}=8.29", "80736b32d5439efc25adf8ede4a1d9f0": "\n \\begin{matrix}\n 4\\uparrow\\uparrow 3 & = {\\ ^{3}4} = & \\underbrace{4^{4^4}} & \n = & \\underbrace{4\\uparrow (4\\uparrow 4)} & = & 4^{256} & \\approx & 1.34078079\\times 10^{154}&\n\\\\ \n & & 3\\mbox{ multiplied copies of }4\\uparrow\n & & 3\\mbox{ multiplied copies of }4\\uparrow\n \\end{matrix} \n ", "8073a0dae872e73b7e7464b56e772007": "E_{\\alpha ,\\beta }(z)", "8073bb1f8b64c924ec039b006c8d3e9f": "\n\\begin{align}\n\\sin x & = \\frac{2t}{1 + t^2} \\\\[8 pt]\n\\cos x & = \\frac{1 - t^2}{1 + t^2} \\\\[8 pt]\n\\mathrm{d}x & = \\frac{2 \\,\\mathrm{d}t}{1 + t^2}.\n\\end{align}\n", "8073fec1625f90270de3c6b2d1be1fd4": "\\boldsymbol{v}_{B\\text{ relative to }A} = \\boldsymbol{w} - \\boldsymbol{v}", "807429db6406c5e070b3313ce0550e94": "(1-1.5\\sin^2 i)", "807440dafc2e9ed56e8ab16f99b4589d": "\\mathsf{RCA}_0", "8074b8b2e23001ca581a366bf1a27231": "\\int xdy + \\int y dx = xy", "8074e2bcf089cecd66df1f46bb59b895": "\\,\\!C_x(t_1 - t_2, 0)\\,", "80757890c6b251d57e78017ed460b847": "0\\leq\\gamma< 1", "807599165759a2ca9bbc1903c6f51ca3": "\\mathbf{y}_k^{\\mathrm{T}} B_k^{-1} \\mathbf{y}_k", "8075ce550ab7d6dac63e955031420464": "K \\left(T_{0 0} - {1 \\over 2} T g_{0 0}\\right) \\approx K \\left(\\rho c^4 - {1 \\over 2} (- \\rho c^2) (- c^2)\\right) = {1 \\over 2} K \\rho c^4 \\,.", "807646023ed007557f0d1b5fec56b9dd": "DII = \\dfrac{average~inventory}{COGS/Days}", "80764b105d2b5f47819152a27aa6c8b4": "S=J|S|=J\\Delta^{1/2}=\\Delta^{-1/2}J", "8076758de3e647ff1c357d80f0e57e98": "10_{140}", "80768fc15c48649cc3b3329480fa6375": "\n- \\frac{\\partial S}{\\partial t} = \\frac{\\left(\\nabla S\\right)^2}{2m} + V +Q \\; ,\n", "8076a6ccb4bdacccc9e201774625ea28": " N > 1", "8076cbe8fd1cc085d174ef09aa60d474": "{R^*}", "8076d66fad11794e0a864b792bfa4b71": "\\nabla+T", "80771e8cb650f08bbf6275e767bde1d5": "\\Theta_2 = (\\frac{37530}{Re})^{16}", "8077cf5d109e10a51f4178a694418947": "gs : X \\times X \\to X", "8077d4adcd853191e6e340e82f0c6c35": "p(F^nG^m) \\leftrightarrow (pF^npG^m)", "807805253b14edf1b1014213a65e16a2": "\\beta - 1 \\ne 0", "807828e3919a0be21b180426a107415c": "[I_G(\\mu):N],", "80782efd2f7883683135ae4a58b6cde3": "\\displaystyle \\Delta = -\\partial^2_x -\\partial_y^2. ", "8078323cf77769bfe4740a7c301f73dc": "P_{a\\le x\\le b} (t) = \\int\\limits_a^b d x\\,|\\Psi(x,t)|^2 ", "80783449f9656ebbdae686098f54ac9f": "\n\\mathfrak{P}(\n- \\mathcal{Z} + \\mathcal{V} \\mathcal{Z} +\n\\mathfrak{C}_1(\\mathcal{Z}) + \n\\mathcal{U}\\mathfrak{C}_2(\\mathcal{Z}) + \n\\mathcal{U}^2\\mathfrak{C}_3(\\mathcal{Z}) + \n\\mathcal{U}^3\\mathfrak{C}_4(\\mathcal{Z}) + \\cdots)", "80783630f3ecec748daec9cd69f9a307": " s = (1/T) \\ln(z) \\ \\ ", "8078490d7b488262e3f399444124576a": "message'=message", "8078afa0ef559cc30a560d7cf97accc5": "\\phi:M\\to \\prod_{i\\in F}R\\,", "8078b37271678af007ed3ed34a5eb82b": "(a \\and b) \\rightarrow c", "8078cfc88ae5e9bc4a61261cc3f71764": "\\{U^\\alpha : \\alpha \\in (0,\\infty)\n\\}", "8079647a0a73fad84ad9246b7b9b9278": "\\ln z= \\ln |z| + i \\phi \\ .", "80796872b01e9877ba3537256e1ae269": "v_{2} '=-u_{2} '", "807a44d26897059013c4937b970a2d6c": " U_r ", "807aaaddb02c5503f081cbf1454ea2eb": "q = \\frac{2\\pi}{i} = \\frac{r \\cdot B_t}{R \\cdot B_p}", "807b592c1c16d3cc643ac302ded4ef76": "v_{g}", "807b7ff90abfb51e6b87c37fd8bf29ca": " \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\cos \\theta & -\\sin \\theta \\\\ 0 & 0 & \\sin \\theta & \\cos \\theta \\end{pmatrix} ", "807c2af35e45ccc93b25dc9235593f57": "S : \\mathcal{A} \\rightarrow \\mathcal{C}", "807c32d7bf2c4d3398fc3b7c1f7d19df": "A = \\frac{1}{2} | \\sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i) |\\,", "807c689204c4b741bb33d228681b4766": "\\det (A-\\lambda I) \\;=\\; \\det \\left(\\begin{bmatrix}\n2 & 0 & 0 \\\\\n0 & 3 & 4 \\\\\n0 & 4 & 9\n\\end{bmatrix} - \\lambda\n\\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{bmatrix}\\right) \\;=\\;\n\\det \\begin{bmatrix}\n2 - \\lambda & 0 & 0 \\\\\n0 & 3 - \\lambda & 4 \\\\\n0 & 4 & 9 - \\lambda\n\\end{bmatrix}", "807ca8c4d95e8bcd2f3c79f860d22122": "\\frac{\\partial a\\mathbf{U}}{\\partial x} =", "807cab6e0265e97a28b5e1fb0ebbbcc5": "f^{\\prime\\prime}(x)", "807cb7d37b4c223cffdadaf496ce0d2a": " M (Ext(W;I), I) \\approx_\\epsilon (U_l, U_r), ", "807d14ce00f704a762113dd49c500c4f": "W_x(t,f)= \\frac{\\sin (f(1-2|t|))}{\\pi f} ", "807d4cbf4a07fa48bd4a64f5b1baf5b1": "\\varphi_{\\alpha}(\\omega_{\\beta}) = \\omega_{\\beta} \\,", "807d654024a0b389f1ed56ae5adf01cb": "\n\\begin{align}\nY_i^{0\\ast} &= \\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i + \\varepsilon_0 \\, \\\\\nY_i^{1\\ast} &= \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i + \\varepsilon_1 \\,\n\\end{align}\n", "807d7a2c6d20b0afb4ed45951f49de74": "d\\Phi = L\\,dI", "807db591e81215cbfd4b10cdc6c2cd91": " \\int_{\\Bbb Z_p} f(x) \\, {\\rm d}x = \\lim_{n \\to \\infty} \\frac{1}{p^n} \\sum_{x=0}^{p^n-1} f(x). ", "807ddcbb078e7a7653a3ec73e4ac41ed": "f'(x_0) \\ge 0", "807de79717fda9b42cbd4405d4f3b632": "w_{RR}", "807ee2e30dde44407c611588e8dd550c": "A=C^{-1}JC", "807f217d7d455aca72d2317d286e18ad": "\\mbox{tr}\\mathfrak{H} = 0", "807fa27d1890b219e23abde9e3a0f962": "\\ (\\xi^2 +1) \\frac{d^2 R_{mn}(-i c,i \\xi)}{d \\xi ^2} + 2\\xi \\frac{d R_{mn}(-i c,i \\xi)}{d \\xi} -\\left(\\lambda_{mn}(c) -c^2 \\xi^2 -\\frac{m^2}{\\xi^2+1}\\right) {R_{mn}(-i c,i \\xi)} = 0 ", "807fc1ef64fbe24c6cd5193f04d0d0ec": "d(x,a)<\\varphi.", "807fca51b49a334f7ba24ebec3e22060": "\\beta =\\neg (\\phi_1 \\wedge \\dots \\wedge \\phi_n)", "8080307c0f41d08c829a53772b57b479": " p_{3,3}(x) = y_3 \\, ", "8080bd7a00f6da1392187523d4d859f3": "M(x) = \\sum_{n \\le x} \\mu(n)", "8081387348db3c334d0558792b89f270": "\\ (w' - w)\\cos w", "80816c1bee7396382596aa79e1780271": "\\textstyle F(B) = f^{-1}(B) ", "808189620ffb39fe6720469dad1fc4ea": "M_{s} ", "80819f3f8a69e157df65a546d86c3aea": "\\mathrm{Alg}_P^C", "8081c54da6296a1bf1f466d04912a058": "\\overline L", "8081dd2e2d05dfa7849628d4b8979004": "y = -2x + 6 \\,", "8081e397a7d05159a3f5a3e70851e689": "M = N g \\mu_B J", "8082267beaddcff6df70825136e8a429": "\\int \\frac{1}{e^x} \\left( \\frac{1}{x}-\\ln x \\right)\\;dx = \\frac{\\ln x}{e^x} ", "8083c44ad2e53cfa0c8a85f4ec7ca25a": "\\mathbf I_n", "80843d25e563a7df31286ca749ea654e": "n,\\,\\epsilon", "808456525548a263c00f33a9395e8d97": "\\exp(j\\omega t)", "80852eeb13e85b98514ce81cea53bf5d": "\nf_{\\mathbf x}(x_1,\\ldots,x_k) =\n\\frac{1}{\\sqrt{(2\\pi)^k|\\boldsymbol\\Sigma|}}\n\\exp\\left(-\\frac{1}{2}({\\mathbf x}-{\\boldsymbol\\mu})^T{\\boldsymbol\\Sigma}^{-1}({\\mathbf x}-{\\boldsymbol\\mu})\n\\right),\n", "808551c83479927e22f00102bbcf1be3": " \\int_0^1 f_{i_1 i_2 \\dots i_s}(X_{i_1},X_{i_2},\\dots,X_{i_s}) dX_{i_k}=0, \\text{ for } k = i_1,...,i_s ", "808588973dd821d318bc5b98c16b76a3": "z_{i,j} \\,\\sim\\, \\mathrm{Multinomial}(\\theta_i). ", "80863acc7cf2a7eb25ee8b4afc82838d": "\\tfrac 12 \\pi", "808672b8f7a08085ea4029ec5cd98004": "-\\sum_{k=1}^n w_k\\prod_{j\\ne k}(X-z_j)=f(X)-\\prod_{j=1}^n(X-z_j)", "8086886008b94ec335a9aea39d5fd1ca": "\\text{parent} = \\left\\lfloor \\dfrac{i - 1}{2} \\right\\rfloor", "8086c72b45384996bbb70d108626c084": "S_w(p)", "8086d4a230e8ae8b47c3bda654c1020c": "A f (x) = \\sum_{i} b_{i} (x) \\frac{\\partial f}{\\partial x_{i}} (x) + \\frac1{2} \\sum_{i, j} \\big( \\sigma (x) \\sigma (x)^{\\top} \\big)_{i, j} \\frac{\\partial^{2} f}{\\partial x_{i} \\, \\partial x_{j}} (x),", "8086d4c6a49da072a730c3ac1bcd659e": "\\gamma^k = \\gamma^0 \\alpha^k. \\,", "8086e0d867643f8fcebf2657b574184a": " \\min_{\\mathcal{C}: \\mathcal{H} \\mapsto \\mathcal{H}} \\sum_{i=1}^n || \\phi(y_i) - \\mathcal{C} \\phi(x_i) ||_\\mathcal{H}^2 + \\lambda ||\\mathcal{C} ||_{HS}^2 ", "8086fb74a8b3e2216b5858c6988e1852": "\\operatorname{dist}(a,b) = |\\ln(b/a)|", "80885a49daac2ec9a742138da1d0806f": "a_k(t) = a_k(0) e^{t/\\lambda_k}", "8088e28aa2a2b0717602f79d927180f4": "T = \\int_0^{\\infty} x e^{-rx} l(x) m(x) \\, \\mathrm{d}x", "8089a933c74c9f4381c638dde93486a3": "S_{n}R^{n}=\\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}}", "8089c36839261881ccc54e5d0787949a": "\\frac{\\gamma(\\theta)} {(2V)} = A_0 \\frac {(1 + \\cos(\\theta))} {\\sin(\\theta)} + \\sum A_n \\; \\sin (n \\theta))", "8089c5225f74e8dc325ccb095231827a": "\\sum{\\vec{F}}=0.", "8089e8218ae18baf7daf84ac5ce3b1ab": "\\frac{D}{E}", "808a99c2a3a7436c6b8a271c98780e92": "\\tfrac{1}{2}Q_5", "808aaac5dcd5ae7517ec145e8ed4dc79": "\\zeta(5)", "808ac7c8b7c9f2ac8815a2bfff76f8cd": "\\;V", "808ad738406d83bf94a7fddbd862474c": " 4 \\int_0^t W_s^2 \\, \\mathrm{d}s.", "808af51cad1488c866a111c1a60f49fc": "W_{k}", "808b6df8d028c97f451da1cb0d799acb": "g(X_1,X_2,\\dots,X_n) = \\prod_{j=1}^m f_j(S_j),", "808b72766eeb1cf6f10fa393b83c98f0": "4N\\log_2 N-6N+8", "808b9e128111e08615b1822e347846b4": "f_\\ast", "808ba712ea5c2e40edb133b7e47539af": "\\Theta(n\\, \\log\\, n)\\,.", "808bc3e42b517dafdd037fa6b9fddabb": "\\begin{align}J_n(x)&=x^{2n+1}\\int_{-1}^1 (1 - z^2)^n \\cos(xz)\\,dz\\\\&=2x^{2n+1}\\int_0^1 (1 - z^2)^n \\cos(xz)\\,dz\\\\&=2^{n+1}n!A_n(x).\\end{align}", "808bf9951cea3de6bba7ab02992a782e": " \\prod_{p} (1- \\chi(p) p^{-s})^{-1} = \\sum_{n=1}^{\\infty}\\chi(n)n^{-s} ", "808c1ef5ea329e30a32bd193cb248d7a": "a-b=47830.1", "808c4b250c1e5ace2c8ff7dd81ad2e05": "\\{x \\mid x\\not\\in x\\}", "808c9372e0db137ddce3739efe2517d1": "1+d\\sum_{i=0}^{k-1}(d-1)^i.", "808d0c2dc3ee367f0e033b539d50b2fe": "\\begin{align}\n\\sigma_{23}' = &a_{21}a_{31}\\sigma_{11}+a_{22}a_{32}\\sigma_{22}+a_{23}a_{33}\\sigma_{33}\\\\\n&+(a_{21}a_{32}+a_{22}a_{31})\\sigma_{12}+(a_{22}a_{33}+a_{23}a_{32})\\sigma_{23}+(a_{21}a_{33}+a_{23}a_{31})\\sigma_{13},\\end{align}", "808d4dfa1e618c0d79e5849cde6e0a0d": "\\sigma \\exp (-\\tfrac{X-\\mu}{\\mu \\sigma} ) \\sim \\textrm{Weibull}(\\sigma,\\,\\mu)", "808d5f6a328214954ac1059a3823fa9c": "\\frac{d\\omega}{d\\Omega} = \\left(\\frac{Z_1Z_2e^2}{4E_0}\\right)^2 \n\\frac{1}{\\left(\\sin{\\theta/2}\\right)^4},", "808d6669fc7e0648cff51f20e976c7b6": " W_{\\lambda} ", "808d874de681539bcb10e0359ec38ad6": "x^*=A^T (AA^T )^{-1} b", "808dcf2e1eb5c8c37a7753b6192125d5": "D^k f : T^k M \\to T^k N", "808eee5ff0c9dbc83ecccd8b82256e57": "\n\\operatorname{Li}_4(\\tfrac12) = \\tfrac1{360}\\pi^4 - \\tfrac1{24}(\\ln 2)^4 + \\tfrac1{24} \\pi^2 (\\ln 2)^2 - \\tfrac12 \\,\\zeta(\\bar3, \\bar1) \\,,\n", "808f2d681f715c494ef3a22ffe8f848b": "f'(r_k) \\not\\equiv 0 \\pmod{p}", "808f8b6fcf1313568f884b6c1a54cc92": "\\displaystyle{T={1\\over 2}I -{1\\over 2}\\int_0^\\infty e^{-t}T(t)\\, dt.}", "808fba806cc20e574b0b5d69a090959f": "(g^* q g^{\\star})(g^* q g^{\\star})^* = g^* q g^{\\star} (g^{\\star})^* q^* g = g^* q q^* g = q q^*.", "808fbd077dfe8dd54b1a29c8cff22812": "T_d(h) = e^{\\left(\\int_0^h g(\\iota) d\\iota\\right)/c^2}", "8090055a7eb12b30f2490db7aa8dddb1": "\\displaystyle{\\|f_r-f\\|_2 \\rightarrow 0.}", "80907ee352c2e05af63a7c89d5640f89": "TS% = \\frac{PTS * 100}{(2 * (FGA + 0.44 * FTA)) }", "809080d06175a62a7db2a477c0ebc0a1": "I_n = -\\frac{\\sqrt{ax+b}}{(n-1)x^{n-1}}+\\frac{a}{2(n-1)}I_{n-1}\\,\\!", "8090eb971002ee6048bc213044d8900e": "\\frac{\\part^2 \\varphi(r) }{\\partial r^2} + \\frac{2}{r} \\frac{\\part \\varphi(r) }{\\partial r} = \\frac{I q \\varphi(r)}{\\varepsilon_r \\varepsilon_0 k_b T} = \\kappa^2 \\varphi(r).", "80910b2b38fa5e808c13c54fd00b4799": " B = \\int \\Phi_b(r_2) \\Phi_a(r_2) \\, dr_2", "80913d937d5005c258b25a6472e5241f": "\\ell>k", "80914fc9e43cb874fe0e1e4fecb42d20": " \\mathbf{X}^{T}\\mathbf{X}", "80919996cd6ed74382b2109315376900": "M(a,b,z)\\sim\\Gamma(b)\\left(\\frac{e^zz^{a-b}}{\\Gamma(a)}+\\frac{e^{-i\\pi a}z^{-a}}{\\Gamma(b-a)}\\right)", "8091cf86425b6f6ce37ad6486a7a8bd2": "\\ 2S_n=n(a_1 + a_n).", "8091f84ee97306ea427491529e600b2f": "S_k = \\frac{\\sigma_k}{\\binom{n}{k}}", "809235df8ca4c662db7fe02ef50f42a8": "a \\mapsto \\biggl(\\sigma \\mapsto \\frac{\\sigma(\\alpha)}{\\alpha}\\biggr),", "80924a6d15c02de5e24b4efcfc4e82ce": " D=\\frac{3\\cdot (\\mu-\\delta)}{\\mu} ", "8092534d4649d14aaad917747fb14d86": "95^{14} \\approx 2^{92}", "8092b7f7d3a4560812be5e7ce9a2dc53": "= 2 e^4 \\frac{s^2 +u^2}{t^2} \\,", "8092e4f0888a93f7475048c519c69a7c": " F = \\cfrac{1}{1-k}", "809312268a14a3664a23a4e5c95dc9bd": " a F(s) + b G(s) \\ ", "80933509c433567443b9ac71f9fa3bdc": "d_k ", "80937c84e6256db626953463c8b8f66d": "i_G(s) \\ge i + 1 \\Leftrightarrow s \\in G_i.", "8093b2077dab66e43331c4ccdaf11cdf": "2x(m+\\lambda)B_{mk-1}^\\lambda(x;k) = mB_{mk}^\\lambda(x;k) +(m+2\\lambda)B_{mk-2}^\\lambda(x;k) ", "8093b7a592a9c43842e1465975a307b7": " T \\ ", "8093fd1268bb439dbb0d212e8175d0c9": "M_{PAW} = \\frac{(T_i * P_{IP}) + (T_e * PEEP)}{T_i+T_e}", "80941de09d4da6f0f03ac694e22adf0a": "C=\\{c\\in \\mathbb{F}^N :(c)_v \\in C_o\\}", "80942106c5f03ee145feb31f26602d24": "\\text{EXP} = \\bigcup_{c \\in \\mathbb{N}} \\text{DTIME}\\left(2^{n^c}\\right)", "809448bb57d1ee68665843f093aa8f18": "\\mathcal{E} = - \\frac {d \\Phi_B} {dt} = B \\frac{dA} {dt} = B\\ \\frac {R^2}{2}\\ \\frac {d \\theta}{dt} =B\\ \\frac {R^2}{2}\\omega \\ , ", "809448c12da6a61638ab2987bc9fcaeb": "c_m", "809449fcd99f252d304d99ae59e08e46": "\\text{pHad}(x) = \\Big(\\langle x , y \\rangle\\Big)_{y\\in\\{1\\}\\times\\{0,1\\}^{k-1}}", "80945780799d59bf115273a6b64c294d": "H_m-TS_m=G_m=\\mu", "809467383359b52b70ddf3b6d41d07fe": "\\theta \\colon \\mathcal{N} (X) \\to L_n (\\pi_1 (X))", "8094b0b429f26483fd74e5841b86c72f": "\\nabla_i\\nabla_j \\varphi = \\nabla_j\\nabla_i \\varphi\\ ", "8094bb80c5759a00941b9930b4bd42bd": "3 \\times x^2", "8095043db483a195bfce2fe625db1f98": "\\exists w (y \\not = w) ", "8095412225ff3185336f3b8b8f3f1eeb": "\n\\mathbf{w} = \\sum_{i=1}^l\\alpha_i\\phi(\\mathbf{x}_i).\n", "809544da1f1711269e95e30cbcc7a96c": "\\hat{a}_i \\,\\hat{a}_j^\\dagger \\, \\hat{a}_k \\,\\hat{a}_l^\\dagger", "80956b0e2c575c2354997fa2759455b4": "c_n = Q_n^{T} h_n", "809590159e74b2ee81c6bab0045d452e": "\\begin{array}{cc}\n \\begin{array}{r} \\\\ 3 \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & -12 & 0 & -42 \\\\\n & & & \\\\\n \\hline \n 1 & & & \\\\\n \\end{array}\n\\end{array}", "8095b7a8d4d6945b158733bd09ba200d": "\\big. Z_0 =\\frac{1}{N!h^{3N}}\\int \\prod_i d\\vec{p}_i\\;d\\vec{r}_i \\exp\\left\\{ -\\beta H_0(\\{r_i,p_i\\})\\right\\}\n=\\frac{V^N}{N!h^{3N}}\\left( \\frac{2\\pi m}{\\beta} \\right)^{\\frac{3N}{2}}.\n", "809657f4f731b495aa8adb530cc58858": "\n \\begin{align}\n 2R'_c L'_s & = 2 \\times \\tfrac{3}{\\sqrt{6}} \\times \\tfrac{1}{\\sqrt{6}} \\\\\n & = 1 \n \\end{align}\n", "80966321f2b7b38cab932f689ebcb496": "{1 \\over \\Phi} = \\Phi + 1.", "8096db5982a638d99b6e6fe4f87667f4": "\\rho_{i,i} = 1", "8096eabdbd852fa54e9c49dba02d5dbd": " \\exists . BPP", "8096f622090a0891c49219bf36d24909": " = 1- ", "80970c6ffd03d5914607ea3450eafccb": "C=C_0,C_1,C_2,\\dots\\ ", "809755c57105072fd570969c31494a25": " \\nabla_{x,y,\\lambda} \\Lambda(x , y, \\lambda)=0. ", "8097677a44178c6056c24256d317bb0c": "\\mathrm{im}(A)", "809785e6f01b2ff6e6d608afe6c6cdcd": "(\\boldsymbol\\mu,\\boldsymbol\\Sigma) ", "8097f4cd9ee7452675e9eb8b5da54b9c": "\\xi \\in \\mathbf{\\Xi}", "809837b7deb3e55626ef50f9e53b42c7": "E_{xc} = E_x + E_c\\ ,", "80986033aa91265ff99af048cbead971": "y=\\sum_{n=0}^\\infty a_n(x-x_0)^{n+\\sigma},\\quad a_0\\ne0\\,", "80988b7b8479cb6716771bbeaa195492": " dE = -p dV + T dS \\, ", "8098b3b1ab7ae7be1ccf53d92c33e128": "\\{1,2,4\\}", "8098c1783b2b68f709b4c74def298ebe": " F(t) ", "80997f3f3ca8dfb0c11837c5b63488c8": "S = \\frac{ln2 \\times {N_A}}{T_{1/2} \\times {m}}", "8099ac2b69179eba93c80299e4667e49": "c = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}} \\ .", "8099d43600f48487bf0822aaeb757ccd": "37_{11} \\ ", "8099e7e01053e60613687080516645fc": "\\ B = Q(1 - \\frac{T_o}{T_ {source}}) \\qquad \\mbox{(2)}", "809a148ae07c0f73494149ac7a4c581c": " z = x + iy \\,", "809a17734ee8390e1670cd46cc6ee44a": "\\rho+3+\\frac{11\\rho^3-49\\rho-22} {(\\rho-4)\\;(\\rho-3)\\;\\rho}\\,", "809a346f6f24a45995406d577b72e905": "F_{j}=U_{m+1}\\cdots U_{j}.", "809acdb91893f48fa845951180558d8f": "\\varphi = \\tfrac32\\, \\eta,", "809af5845029ce96e6384734ef3d231e": "p^2/(2 m)", "809b6a3c379bb6344bff09347c7a3d6b": "k: X \\rarr Y", "809ba0e17b04450f76d0b5ddc95dfc8c": "I = \\int_0^1 \\frac{1}{1+x} \\, \\mathrm{d}x.", "809bc86b838c678cd3942251024bba01": "\\sqrt{4.001}", "809bfd6b8cf7cf33e4df026faea3f0fd": "s(x^2+1)+t(x^3+1)=2x\\ ", "809c42cead66494329c1ac8137f69fec": " ~\\sigma_t ", "809c494f3e37375113cdd991fcf3a8a3": "z_0 = z_{cr} = 0\\,", "809c580bdce604f905011dbffadbdea4": "\\begin{align}Q(\\theta|\\theta^{(t)})\n&= \\operatorname{E} [\\log L(\\theta;\\mathbf{x},\\mathbf{Z}) ] \\\\\n&= \\operatorname{E} [\\log \\prod_{i=1}^{n}L(\\theta;\\mathbf{x}_i,\\mathbf{z}_i) ] \\\\\n&= \\operatorname{E} [\\sum_{i=1}^n \\log L(\\theta;\\mathbf{x}_i,\\mathbf{z}_i) ] \\\\\n&= \\sum_{i=1}^n\\operatorname{E} [\\log L(\\theta;\\mathbf{x}_i,\\mathbf{z}_i) ] \\\\\n&= \\sum_{i=1}^n \\sum_{j=1}^2 T_{j,i}^{(t)} \\big[ \\log \\tau_j -\\tfrac{1}{2} \\log |\\sigma_j| -\\tfrac{1}{2}(\\mathbf{x}_i-\\boldsymbol{\\mu}_j)^\\top\\sigma_j^{-1} (\\mathbf{x}_i-\\boldsymbol{\\mu}_j) -\\tfrac{d}{2} \\log(2\\pi) \\big]\n\\end{align}", "809c70f6424c6830babb4c426a66da05": "\\, n \\, ", "809ccf2c1a775e70c15741215f1820f5": "P(\\overline{\\zeta}) = 0", "809d3b4b4342a8e7a1903b73d5bd9c54": "\\mathcal{M}\\left\\{\\bigcup_{i=1}^\\infty\\Lambda_i\\right\\}\\le\\sum_{i=1}^\\infty\\mathcal{M}\\{\\Lambda_i\\}", "809da833fd6be65637b58a8d187e4b20": "\\langle u, v \\rangle = 4^{-1} \\sum_{k=0}^3 i^k\\|u+i^k v\\|^2.", "809db79aa32af98c30910a0ab6edbe31": "\\left[1,\\frac{n-1}{n},\\frac{n-2}{n},\\dots,\\frac{1}{n}\\right]", "809dbd04cee0ee05fa2d0470b87d6685": "\n\\frac{\\mathrm{d} \\ln I}{\\mathrm{d} \\ln R} = -(k/n)\\ R^{1/n} .\n", "809dc120145f14d4b8a9b7170e9f9690": "\\frac{s(\\frac{1}{10})}{10}=\\frac{1}{89}", "809e41c252b1d4458fdab6554714f2cd": " DO_0= \\frac{DO_s Q_s+DO_b Q_b}{Q_s+Q_b}", "809e47f4cfcd91a3804212a8005192b3": "c_n \\arctan \\frac{a_n}{b_n}", "809e8bdb492a20b398ed83ce63cf3c40": "\n F(x;\\,2) = 1 - e^{-\\frac{x}{2}}.\n ", "809ed11e9636538290a223b91a4f7b21": "T_G(x,y)", "809f3732f7925114f414e92cbe30bf24": "R_i = \\log_2(20) - (H_i + e_n)", "809f42b87ac9bae9d601b259bb74e80d": " \\gcd(54,24) = 6. \\, ", "809f70cda864b99fde425782e64cbf46": "\\{h_{t_1}(x,x),\\ldots,h_{t_n}(x,x)\\}", "809fce7dcfd308c5f60cf2f604ce58de": "a_n^{2n-2}\\prod_{i0,\\,", "80b0b8e7461709733536c1fdfdb5b57b": "g_2\\in G", "80b0be740581c7396771fe7b00b7cc1e": "k^{{\\rm th}}", "80b0cc83dd4ed2a185d183586e473897": " \\mathbb{Z}_{12} ", "80b0f1a24b561ae194b924b6fd5d230a": "\\overset{\\alpha}{\\omega}", "80b0f8524f95ab2db29e9dfcf628fe0a": "\\langle abbbba \\rangle", "80b127a5bb13cb4184ae63df17266872": "\\tfrac 12(1-\\gamma)=P(Z>z).", "80b16341a08556b7fae9bda17d35a7a2": "\\beta=B", "80b16f5de0519c44f654a44c26693a5a": "\\alpha = \\{\\beta \\in \\textrm{Ord}| \\exists i: \\beta \\hookrightarrow X\\}", "80b1ee811de47ed025c40458f8a74ed2": "f_{1/2}(x)=\\cos(2x)\\text{ and }f_{3/2}(x)=\\frac{\\sin(2x)}{2x}.", "80b228889559024b5b0fa3a60fe82f5f": "v_p(R)", "80b22c438d8d7228e830b7275cbee4c9": "\\operatorname{dCov}^2_n(X,Y)", "80b261372134a29479f929165bd387cb": "-9K_1", "80b275394fc9aac4d3247a53d17eee3d": "y'' - 2xy' + {\\lambda}\\,y = 0,\\qquad \\mathrm{with}\\qquad\\lambda = 2n.\\,", "80b285bef6d1c5f00c161b207fe72541": "v/\\nu = n \\lambda/\\sin \\theta", "80b29a74266793422144c43eda105a8f": "2k^2", "80b2f4e53e99b94ba02c917e0284f831": "((P \\and Q) \\to R) \\vdash (P \\to (Q \\to R))", "80b3096385ef102fea096ee8d5727118": " \\mu(A \\cup B) = \\mu(A) + \\mu(B). \\, ", "80b37fcfd525f056e9b7e1e525b127b2": " \\lnot \\exists x ((K(x) \\land \\forall y (K(y) \\rightarrow y=x)) \\land B(x)) ", "80b38d6ea30d555082ded62e06ea62c3": "\\operatorname{cov}(x,y;w) = {\\sum_i w_i (x_i - \\operatorname{m}(x; w)) (y_i - \\operatorname{m}(y; w)) \\over \\sum_i w_i }.", "80b3908ff78070954de1cef4571a657a": "E = \\alpha - \\beta", "80b39bdd4e87f71225fdaa53b308f299": "\\frac{\\partial \\Lambda}{\\partial x}\\approx\\frac{\\Lambda(x+\\epsilon,\\lambda)-\\Lambda(x,\\lambda)}{\\epsilon}", "80b3aed4bb525577163b6a34b7c0b9e1": "s \\equiv \\frac{\\alpha + \\beta}{2}=\\frac{\\operatorname{tr} A}{2}~, \\qquad \\qquad q\\equiv \\frac{\\alpha-\\beta}{2}=\\pm\\sqrt{-\\det\\left(A-s I\\right)},", "80b47e251d2d73a00464c00b82d83fac": "\\mathrm{Da} = k C_0^{\\ n-1}\\tau", "80b486bf504a941b3fe5da5992b7ecaf": "j_{XY}", "80b4b587d582c1329e3288e69a591bfb": "\\alpha\\in\\mathbb{R}\\setminus(-\\mathbb{N})", "80b4ca64cce2d5f684952a27c5ba1c78": "k* : S\\sim S/ L \\times _T S/R", "80b4e3d563a108c3bdf0ad48f72b008a": "\\,\\ \\sin x + C", "80b538d4e5e425b8f6946a9cddba2f14": "\\scriptstyle B < f_s/2.", "80b5908bfea015da5b71b3f27d3a933c": "C_D(G)= \\frac{\\displaystyle{\\sum^{|V|}_{i=1}{[C_D(v*)-C_D(v_i)]}}}{H}", "80b592514d7d55d457d73830979fc5d1": "f\\rightarrow 0", "80b63dd51e8ecf93e1434bbe68cccb35": "U(g(x)')| \\psi\\rangle = e^{i\\omega (h)}| x\\rangle", "80b649b3f272ff2b7a98c1f8cd652ad0": "\\mathrm{BMI}=\\frac{\\mathrm{kilograms}}{\\mathrm{meters}^2}", "80b69d2e4c8d6632f0e47a04bc5139ea": "\\tilde{\\pi}(\\vec{k})=\\int d^3x e^{-i\\vec{k}\\cdot\\vec{x}}\\pi(\\vec{x})", "80b6c530f6baf6c3ce2dcefce9967e72": "\\Psi_{xy}(m_x, m_y) + \\phi(D|m_x, m_y)", "80b7365cb15c6e73c4a9c8a25eb5a5a1": "|f(x)x^\\alpha|", "80b7442cc99d0db3fc619eb2368978ae": "\\Pr(X = c) = 1,", "80b76b1de507a0bc16c67b725df00d9b": " S_1 = \\frac{1}{2} + \\frac{1}{3} = \\frac{5}{6}", "80b7a6d1b7039e8634bbca74f3bec813": "dy = f'(u)\\,du = f'(g(x))g'(x)\\,dx.", "80b7c2da48bf62dff6de31ee7946536b": "\\tau(p)\\equiv -1\\ \\bmod\\ 23\\text{ otherwise}.", "80b7ce17d302b5adfe7f1f3cd9c4beeb": "I_\\nu (\\lambda z)= \\lambda^\\nu \\sum_{k=0} \\frac{\\left((\\lambda^2-1)\\frac z 2\\right)^k}{k!} I_{\\nu+k}(z);", "80b7d747ea0f20191e16de46487d8ce0": "[\\mathbf{P}^1] = \\mathbf{1} \\oplus L", "80b81a31cc4656596db1c7636c30b697": "r^2=\\sum_{i=1}^{n+1} (x_i - c_i)^2.\\,", "80b8336f1a282e61955e546712d680a9": "H(f_{LP}(t) f_{HP}(t)) = f_{LP}(t) H(f_{HP}(t))", "80b8682056bbadb15468c1c6b76ffaf1": "s=\\theta", "80b8f65b582a5e63b341894b7f11868b": "1+x+3x^2+7x^3+19x^4+\\ldots=\\frac1{\\sqrt{(1+x)(1-3x)}}.", "80b9304f312fc0f918c292dd6c68ce3b": "\\eta \\geq 1", "80b9309381ad1a69c2c06c9ad6457c72": "M_1 + M_2 = F_w + F_f + F_{P1} + F_{P2}", "80b9309fd22d6a3207fa4c812e9dd2be": "a_1 , a_2 , \\cdots , a_{18}", "80b978ac7277de110547f7202223932a": "p_{n-1}(x)", "80b9bda437933734478f7a253ab774cb": "A\\to\\Box\\Diamond A", "80b9d538faabe97de85afcd0aadeb925": "\\hat{\\sigma}_{OC}", "80b9deab724c104778b2c9d3b0f22f75": "P^{-1}_{ij} = \\frac{\\delta_{ij}}{A_{ij}}.", "80b9fe138a67839f098eaf79570006b7": "[t_{i-1}, t_i] ", "80bab1261cbcf00c4226a6e6b88937b8": "\\frac{\\mathrm{d}^2 \\theta}{\\mathrm{d}t^2} + b \\frac{\\mathrm{d}\\theta}{\\mathrm{d}t} + \\omega^2 \\theta = 0 \\,\\!", "80bb65349373189f44f9f0f876e5b0d7": " \\frac{\\partial V}{\\partial y} = - \\sum_{i=1}^{n} t_i \\cdot CF_i \\cdot e^{-y \\cdot t_i} = - MacD \\cdot V,", "80bb886609353e85383b479a1b7908d5": " y = -v(1 - v^2/3 + u^2)/3,\\ ", "80bb8ea077d50086e68b2dce78d39f31": "\nx_i \\sim N(\\theta_i,\\sigma^2),\n", "80bc10ae9e068ed07bb0e46f7d45620b": "\n\\Phi_E = \\mathbf{E} \\cdot \\mathbf{S} = ES \\cos \\theta,\n", "80bc3e3172b4700dc1cc63bd09c03796": "( a_1 * a_2 * ... a_n )", "80bc99a77ca8674b92446b554d948edb": "W=\\Omega^{-1}", "80bce97f811426ca2ae8ac9c97fad445": "\\gamma = \\nu (2 d_\\text{f} - d)\\,\\!", "80bcec7a25cb150b1511137115e0a0d6": " S^\\circ_T = A(\\ln T) + 2B(T) + \\textstyle \\frac {1}{2}C(T^{-2}) - D(T^{\\textstyle - \\frac {1}{2}}) + 1 \\textstyle \\frac {1}{2} E(T^2) + F'", "80bcf01d383665ab69eef19ad151ad3f": "\\begin{align} \nN_k &= {1 \\over 32} \\left( ( 1 + \\sqrt{2} )^{2k} - ( 1 - \\sqrt{2} )^{2k} \\right)^2 = {1 \\over 32} \\left( ( 1 + \\sqrt{2} )^{4k}-2 + ( 1 - \\sqrt{2} )^{4k} \\right) \\\\\n&= {1 \\over 32} \\left( ( 17 + 12\\sqrt{2} )^k -2 + ( 17 - 12\\sqrt{2} )^k \\right).\n\\end{align}", "80bd23d759ddf3e027282fec2c4f4958": "{[SU(2)_W\\times U(1)_Y]\\over \\mathbb{Z}_2}", "80bdebd5d9a51d49592dc1ef580fd114": "{\\rm C} + {\\rm O}_2 \\rarr {\\rm CO}_2", "80bdf3db59b921a29cee8855d70765fa": "F(y)=\\mathcal{O}\\left(\\sqrt{\\frac{1}{y}}\\right)", "80bdf5ace249c7d836bb025dca0789cd": "|D_t J(0)| = |\\widetilde{D}_t \\widetilde{J}(0)|", "80bebb3130c06c757fe5958bb58e6da9": "O(n^6 L)", "80bf0b70bdc5459fedaa65bbeae9e4a3": " \\{ f_{1},f_{2} \\}_{N} \\circ \\varphi = \\{ f_{1} \\circ \\varphi,f_{2} \\circ \\varphi\\}_{M} ", "80bf2fcb89b17d065ba6a9909103b0cc": "b_{n}=Ab_{n-1}+Bb_{n-2}+K", "80bf3d12f34867044d80f18534657e7e": "\\textstyle \\delta ", "80bf5633cfe93edca1c50422c1132847": "[\\alpha, \\beta]\\,", "80bfcc85b2bbb97cdad233de9114167a": "w_5 = w_3 + w_4", "80c01281a09990ea8edec347ee64c39f": "\\nabla \\times \\mathbf{g} = 0", "80c02aef1e68223a524780653800fd30": "V \\rightarrow P^1\\ ", "80c07ded45b8bc063f0f494d905d9691": " c_k = \\sum_{i+j=k} a_i \\cdot b_j ", "80c09c7ce7e2d8c07395438ab4e76fe2": "\\phi(z,w)=(\nz^2+w^3)/|z^2+w^3|", "80c0a1ba3e6b276aa8fb894e19325940": "\n\\pi \\cdot \\cot (\\pi x) = \\lim_{N\\to\\infty}\\sum_{n=-N}^N \\frac{1}{x+n}.\n", "80c0ba0724c64253ee3b22200fd03864": "\\R ", "80c0bb88430d8b28ca18c48e1f2419fb": "x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0\\,", "80c12a28594d340ef22da6226345ee71": "f(x) = \\int_{x-1}^{x+1} s(\\tau)\\, d\\tau\\ = \\int_{-1}^{+1} s(x + \\tau) \\,d\\tau\\,", "80c17abfb85351926365baa0265083ae": " \\;\\;\\frac{1}{2}", "80c17fc67b719732137893c9f1cb3367": "\n u_1 = \\cfrac{F_2}{8\\mu}(\\kappa-1) ~;~~ u_2 = \\cfrac{F_1}{8\\mu}(\\kappa-1)\n ", "80c19bb6d9296b8d90e7ca104a9d6204": "*(V)", "80c2156dff1d4cf336927fcf24f42ce0": "G(x) = \\frac{e^{ik|x|}}{4\\pi |x|}", "80c262dd11d7a0d04f8244997568c96b": "n!\\approx \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.", "80c26799bdab93e50e46f35c9c6d5545": "\\omega = \\gamma B \\,\\!", "80c282f47201fa171702deac72935ae5": "c_{8}=c_{9}", "80c2a61991a82e58c087c76758b8221e": "\\psi_{n\\mathbf{k}}(\\mathbf{r})=e^{i\\mathbf{k}\\cdot\\mathbf{r}}u_{n\\mathbf{k}}(\\mathbf{r})", "80c2b3f373dbe799e564b9e1e81924a9": "\\psi(\\vec{r})=\\phi(\\rho,z)e^{i\\ell\\theta}", "80c2b80cc955a13159d5b81abdba4d8f": "\\frac{1}{2} c_p \\left( 1 + \\frac{4\\pi h}{\\lambda}\\frac{1}{\\sinh\\left(\\displaystyle \\frac{4\\pi h}{\\lambda}\\right)} \\right)", "80c2e79aa5d974b28f50c3b35556f0bb": "\\mu = g J \\mu_B", "80c34080ca237effb36afd8356a378a8": "\\mathbf{\\Omega}", "80c3d8e2c63c3353d3a8bdda2d7bad69": "\\rm \\ FS-SF \\xrightarrow{KF} S=SF_2", "80c40ad91e5ddcb4e7c92e4a453e1d41": "cT_2 =L-vT_2", "80c42ce3ef40767bfde43169db96a29b": "e_s^\\vee", "80c4e486e54986f233ca93105437ba15": "P = \\left ( \\frac {13.397 m} {1 ~ \\mbox {kg}} + \\frac {4.799 h} {1 ~ \\mbox {cm}} - \\frac {5.677 a} {1 ~ \\mbox {year}} + 88.362 \\right ) \\frac {\\mbox {kcal}} {\\mbox {day}}", "80c4f4a2b62a2e7b1af4812de1275a5f": "z = r e^{j \\omega}", "80c50c7b510ffb4ee45f0eec76aef7d0": "\\mathrm{N} \\mathfrak{p} \\equiv 1 \\pmod{n}.", "80c5990f4f068a477ae4441057f04c24": "\\mathrm{Pr}\\left(m \\vert n\\right)", "80c5da8f58ea5cfe78ebe508ec577c57": "\\ a_C = \\gamma \\frac{[C]}{[C^{\\ominus}]}\\,", "80c61e0f5a2c1ad5b46de2a1d39e987c": " T_i = \\inf \\{ n\\ge1: X_n = i | X_0 = i\\}.", "80c65b8f9ca52d6f6b142027c68ad0b6": "f(x) = \\frac{1}{\\sigma \\sqrt{2 \\pi} } \\exp \\left[ -\\frac{(x-x_0)^2}{2 \\sigma^2} \\right]", "80c675042d33f8b1623b49cf36575ba0": "\\mathcal{O}_X(U) = \\bigcap_{x \\in U} \\mathcal{O}_x.", "80c6bc1499a09380477da8b4cb912e3b": " P = \\frac{n(X) \\dot X}{\\sqrt{\\dot X \\cdot \\dot X} }.\\,", "80c6be00397bd349a224646a326d5f8d": "\\dot{\\sigma} > 0", "80c6d9034bbd732541030f1241144f16": "s = 1 + 2\\pi in/\\ln(2)", "80c703b75e94afef10019f943f294f64": "H^* \\otimes H, \\, ", "80c81b819ae4eed8f558c027b3540c64": "\n\\begin{align}\n\\gamma(\\lambda,\\phi)&=\\arctan(\\tan\\lambda\\sin\\phi),\\\\\n\\gamma(x,y)&=\\arctan\\bigg(\\tanh\\frac{x}{k_0a}\\tan\\frac{y}{k_0a}\\bigg).\n\\end{align}\n", "80c82b502d3b3eb71e4b99e3c0eda2d4": "\\{q_n(x)\\}", "80c864fda9bfdde01f7e4dcd2c7173bd": "\\text{a. If system reactance is given in percent, use Eq. 16 to change from one kva base to another.}", "80c8e53ce21b91530dbf01130249244f": "B = \\vec{r}(T)", "80c948473cfa882e04e9c893b59d4619": "\\{M_{\\alpha} \\}", "80c984b611a81e414167086754eacc46": "q:[0,1]\\rightarrow [0,1]", "80c98ea34cb3ee830bee6af71e5c66e0": "q_u", "80c9a08cfd36639af5447269e0ecb162": "Q\\subset E\\,", "80c9a719c036d4f4d380d7ee166e5eed": "f(T_tw) = f(w)\\,", "80c9b4fe54f739d498db90dd33d0cf68": "C: S \\to T^*", "80ca105232a279ebe5338cd03e0795c0": "G^{\\alpha \\gamma} = \\kappa \\, T^{\\alpha \\gamma}~", "80ca816ebc45867439ac2d7edab9ea30": " X'", "80caa04bfe8ae5dcb42d980ff0277ba8": "U_{exp}", "80caa5fec16909c116b1f410fb9cd761": "D/dt", "80cac984708847a09fec94e76c77d9e0": " V_1\\oplus\\dots\\oplus V_t", "80cb212cbebcd49d9afda87d78f1c10d": "f^* : \\text{Mod}_S \\leftrightarrows \\text{Mod}_R : f_*.", "80cb5ce0f11513a0e5464bb46d31bff2": "R_{d} =", "80cb66237ae7d67c87e3b6c3820a9196": "\\mathrm {DOF} \\approx 2 N c \\, \\frac {m + 1} {m^2} \\,,", "80cc3f0dafaac1e89af729d11d83669f": "a \\in L(s)", "80cc957ee44201354e7dd220cd9973f8": " \\mathbf{\\hat T}(\\lambda)|q\\rangle ", "80cc986ab7b94e4ec468aee7a06fa2aa": "\\mathfrak{so}(n)", "80ccd4fdb267df6995ad1666747119c4": "\\scriptstyle \\operatorname{arctan2}(a,\\, b)", "80cd5c3d741e8e5c452dbf4bb72fdde4": " \\ln \\frac{ p_v}{p_{v,o}} = \\frac{ v_{liq} }{ R T } ( P - p_{v,o} ) \\!", "80cd5d33a0b808af4df363b6c436007b": "n_\\text{clause}^\\text{left}", "80cd5fa3adcbf14779e7bf15aae07dae": "\\frac{\\partial M}{\\partial \\alpha}=x_g\\frac{\\partial L_w}{\\partial \\alpha}-(l_t-x_g)\\frac{\\partial L_t}{\\partial \\alpha} ", "80cd9e3b258014171ab91ea19ce9d3d1": " T_j(x) T_k(x) = \\tfrac{1}{2}\\left( T_{j+k}(x) + T_{|k-j|}(x)\\right),\\quad\\forall j,k\\ge 0,\\,", "80cdc6249b73d6bee823c2360f8207ec": "\\phi(y,x_1,\\dots,x_n)", "80ce22b5d9fd123295cffaf275a783bb": "F(x_0, \\ldots, x_n)", "80ce4c34250d49694b0ad3e648cbd991": "G*_{\\alpha} = \\left \\langle S,t \\Big| R, tht^{-1}=\\alpha(h), \\forall h\\in H \\right \\rangle. ", "80cefda5d4027a56919ae95296d962b7": "= \\frac{4\\pi c G M_{BH} m_p}{\\sigma_t} ", "80cf626a27771471b386d4386fa5268f": "\\sqrt{6} \\rho^2 \\sin 2 \\theta", "80cfacb6cbc6cd368225ffb8347b6491": "\\displaystyle{F_f(re^{i\\theta}) ={1\\over 2\\pi} \\int_0^{2\\pi} f(\\varphi)\\cdot {1- r^2\\over 1 - 2r\\cos (\\theta -\\varphi) + r^2}\\,d\\varphi,}", "80d02e52949dc1f35c13f84095d7c9e5": "x_1,x_2", "80d05b1ac751365a2cfa8dfe4b423d45": "\\|\\cdot\\|_U ", "80d0691a4a5f381210183575f95c3864": "\\scriptstyle C^0(\\Omega,\\mathbb{R}^n)", "80d08d4d0bf9a7ecd93e429222bef8c8": "c \\in {\\mathcal W}_{\\mathcal P}", "80d0bf26f662de0052924e6a29c3cf83": "\\rho(A) = |\\det A|^{-sn}.", "80d0d32a562b0f98d144966bbf241f2b": "[X]", "80d0dd2630bd4835342f672e0d0529fe": "\\psi(\\mathbf{x})", "80d0ffcc260fc09b0a96bcfe2c570817": "4^{n}", "80d103bf05c84e0a00c9aec9ff1ee18d": "\n\\begin{alignat}{2}\n \\mathbf{j}^2 |j\\,m\\rangle = \\hbar^2 j(j+1) |j\\,m\\rangle & \\;\\;\\; j=0,\\frac{1}{2}, 1, \\frac{3}{2}, 2, \\ldots\\\\\n \\mathrm{j}_z|j\\,m\\rangle = \\hbar m |j\\,m\\rangle & \\;\\;\\; m = -j, -j+1, \\ldots , j.\n\\end{alignat}\n", "80d11efea84570ef24b144bf99eab801": "\\int_S f\\,dx^1 \\ldots dx^m.", "80d14394854c0d0caee34a2be5594115": "\\bigcup_{A\\in\\mathbf{M}} A", "80d17fb44b86c3de12e28765cc5b85a8": " r\\sqrt{2} = \\frac{a}{2}\\sqrt{2} \\!\\, ", "80d18c8e8fc68fe779ac4a5498dc008b": "(-1)\\cdot (-1) = 1", "80d20aeb00f984a41adef36cd54e0367": "\\Sigma r Q_0^n = -\\Sigma n r Q_0^{n-1} \\Delta Q", "80d20f88bd035ac18851cb07878cb6b4": " \\mathbf{A} _1 \\dots \\mathbf{A} _n", "80d23a61912bee08ddf501917a435c1d": "\\sigma_{xx} - \\sigma_{xz} - \\sigma_{xy}", "80d26298795f175fac6dbe48e507509f": "\\hat{U}_{en} = - \\sum_i \\sum_j \\frac{Z_i e^2}{4 \\pi \\epsilon_0 \\left | \\mathbf{R}_i - \\mathbf{r}_j \\right | }", "80d265d46052d1031d28559e523e661e": "\\frac{d A}{d x} = 0", "80d2ba9093e7867d9a60978b65bca0b2": "Prob_{slotted} k = e^{-G} ( 1 - e^{-G} )^{k-1}", "80d2dcf14fd7bb27a9095d1cde1acebd": "min(a,c)", "80d2e7982a6b134d447851b5da55d230": "\\sigma_k^2(x)", "80d2f411f81d72ce1759ba92a4750f62": "\\omega_f(t) = \\sup_{d(x,y)\\le t} d(f(x),f(y)). \\, ", "80d312f100cdb88355a203b7bad24de6": "P(\\hat{s}',\\hat{s})=P(\\hat{s}'\\cdot\\hat{s})", "80d34ba457fe957df12ac902d569af8b": "f[x_0,\\ldots,x_n,x]", "80d428f7f2f5e86d87e2e6013761f1df": "v>v_{i}", "80d44b82d9f3bbd107d85d43bfbc24d8": "A,B,\\Lambda", "80d45315acb3e29778b3963ec1689df5": "a_1x + b_1y= 0", "80d46e41fa876d99fe5cd6e0e59fcaa8": "T_0(x) = 1, \\quad T_1(x) = x.", "80d48211c22871584249fd2c9a53f9e1": " \\frac{d}{dx}\\left(u - v\\right) = \\frac{du}{dx} + \\left(-\\frac{dv}{dx}\\right) = \\frac{du}{dx} - \\frac{dv}{dx}. ", "80d4a4587c1a3f5d3340c5e75d0e603e": "r, d, C,", "80d4bd24cf1ce7806663f4c4a9907d19": "\\omega = \\frac{egB}{2m}", "80d4dae36907ff889ac1f7ea4a8d3702": "C_{P} - C_{V} = T\\left(\\frac{\\partial S}{\\partial V}\\right)_{T}\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}=VT\\alpha\\left(\\frac{\\partial S}{\\partial V}\\right)_{T}\\,", "80d4fa9d16ca82aff9f45b3a11afbb2f": " Y = Gen(W) = R, P ", "80d504001ab855661d540a6abcecffcb": "\\delta \\in \\mathbb{R}^d", "80d542b71df90c273155cc2fa2a7dd5f": "res_{U_i, U} \\colon F(U) \\rightarrow F(U_i)", "80d595751e3b7269447e642f4fe26b58": "\\left[ \\frac {\\text{result from specimen in test environment}} {\\text{result from specimen in inert environment}} \\right]", "80d5fe77d7609cb1eaf71ab5af8ddd1a": "A(x_1, ..., x_n) \\to \\beta", "80d6949da1be129da0fd7ac7b0b21255": "f_4(z) = \\,_1F_1(a+2;b+4;z)", "80d714328b7e608cee39f7979a676308": "I(x) = \\frac{I_0}{1 + \\beta c x I_0} \\,", "80d7173787ed090c07b1a5931b6cda0c": "\\Delta T = g^{ij}\\left( \\nabla_{X_i}\\nabla_{X_j} T - \\nabla_{\\nabla_{X_i}X_j} T\\right)", "80d763afbe452807cb717b6fe47ab5c9": "\\hat{P}\\left|x_1, x_2\\right\\rangle = \\pm \\left|x_1, x_2\\right\\rangle\\,.", "80d78565d5d0922b2e38295d94413db9": " Lq < s ", "80d799edbf32ba5e2464c802944e9474": "E = \\gamma(v)mc^2", "80d805c4768f1ce764cd3cbfd78a2f23": "Physics \\geq medium", "80d81f2a10c8a2b4ec63cc0283bef300": "A_{ij}=B(e_{i},e_{j})", "80d86662a450369b7cf277c63e51c380": "( (V_i,\\tau_i),f_i)_{i\\in I}", "80d88567bdf2b923c7e0d9b807a0a6d8": " f^\\downarrow( q_l , q_r ) = f( q_l ) \\quad \\text{ if } \\quad q_l = q_r, ", "80d8fdb85625177d1101466d66827b96": "x = \\log_2 3 + 1.\\,", "80d92f84fe4a21220a95d1238b506226": "p=u/3", "80d98877d8ca692cbdf69aa4d30d6412": "\\begin{align}\n a &= C_1 - \\textstyle{\\frac{1}{11}}C_2\n &= L^\\prime_a - \\textstyle{\\frac{12}{11}} M^\\prime_a + \\textstyle{\\frac{1}{11}} S^\\prime_a \\\\\n b &= \\textstyle{\\frac{1}{2}} \\left( C_2 - C_1 + C_1 - C_3 \\right) / 4.5\n &= \\textstyle{\\frac{1}{9}} \\left( L^\\prime_a + M^\\prime_a - 2S^\\prime_a \\right)\n\\end{align}", "80d99eaaef64d35e99462ce4b470b34b": "\\,2^7 - 1 = 127", "80da2e540f24022cf36b077f0e2d7598": "\\mathit {k}", "80da9219f444ea3a67f8868aa0199133": "U = x y, V = y z, W = z x,\\,", "80daaefe41c247241b08f6cd34962df8": "\\coprod_i U_i\\times Homeo(F)", "80dadf2c615e3ab63c234145c4a29616": "\\eta (a) = \\eta (b) = 0", "80daef8a508b4e6a7df29a8953d30d12": "Df(y)", "80db4badf7cd02e9f6f8e040e5122977": "S_{RRB}(n)", "80dbbe9a4f453ffbf966f946cc8b6de2": " = \\frac{1}{\\omega} \\; G_{p + \\tau ,\\, q + \\sigma}^{\\,m + \\nu ,\\, n + \\mu} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_n, -\\mathbf{d_\\tau} , a_{n+1}, \\dots, a_p \\\\ b_1, \\dots, b_m, -\\mathbf{c_{\\sigma}}, b_{m+1}, \\dots, b_q \\end{matrix} \\; \\right| \\, \\frac{\\eta}{\\omega} \\right) .\n", "80dbc01a67f21d81764a3278aa5ebe2c": "f(y_1) \\leq f(y_2) ", "80dbc4d21d690e4c7cc43adbc4846c2d": "(x(t),y(t)) = \\Big( (t\\sin\\alpha+s\\cos\\alpha), (-t\\cos\\alpha+s\\sin\\alpha) \\Big) \\,", "80dbe4c6591817984f812a315d1ba0bd": "q=\\frac{2Ta}{a^2+2T}.", "80dbe90cb160b49fdbc5ce0a5a8bc526": "\\mathfrak{P}^{5}", "80dc79ba59d56273d2d547357b2f4998": "\nF(x)=\\left(x^{\\{m\\}}\\otimes I_r\\right)'H\\left(x^{\\{m\\}}\\otimes I_r\\right)\n", "80dc7a513f888717d0561faf88af062a": "\\{w,uv\\}", "80dcb3d6e78de42246042b565c209603": " \\ F_p(z,f) = f^{(p)} _c (z) - z", "80dcb9fa8897b6dfe56ade7fbd313c70": "\\R^{k+1}", "80dcd782e6cbfabb8ff4df532f8df55f": "{ e}_i \\wedge { e}_j \\wedge { e}_k", "80dd483dc294063dad59105373815da3": "\\scriptstyle\\mathrm{res}_{VU}:\\mathcal{F}(U)\\to \\mathcal{F}(V)", "80dd50f8c9c09b98d14756530922e189": "\\vartheta(z)", "80dd6f332827bf2475b9e25c1bca0104": "T_{h}\\cdot x = \\lambda\\cdot x", "80ddcaef06dfbf5709f58527c52cccef": "k=k'", "80dde4c2af6ccf9afd64a1f6e526fa4c": "a,b,c,d\\;", "80ddec1d83343a59be27a84011c4e9aa": "a = \\frac{1}{2}\\, k (V + Z_{p} I)\\,", "80de4a499fa0bf7bf52695734b2dc8e8": "\\langle r, f \\mid r^8 = f^2 = (rf)^2 = 1\\rangle.\\,\\!", "80de78bda87716c173da6bf95e671e92": "(A\\to C)\\to((B\\to C)\\to((A\\lor B)\\to C))", "80dedd1e238f6e01178991643c07be79": "\\varepsilon_1(1-\\varepsilon_2)", "80df360c2206d17e05aea725f9045ab3": "1, ..., n", "80df6ad75120a3310841c0f9ce1f8f96": "n\\left(x,0\\right) = 0, x >0 ", "80e00b0c594948de0b66ab08eb48fadf": "x<0, 0a", "80e04c51f332b704cf63cbbbfec7dfe1": "[a^{\\,}_i, a^\\dagger_j] \\equiv a^{\\,}_i a^\\dagger_j - a^\\dagger_ja^{\\,}_i = \\delta_{i j},", "80e05b6bb2d015efe5a75a933cbb35af": "\\frac{n}{D}", "80e0aa749bcde0a4c068a78e6a159a40": " ds^2 = a(t)^2 \\, ds_3^2 - c^2 \\, dt^2 ", "80e0c0e549aafdd53f35cf46062acfc1": "\\gamma_i\\gamma_j + \\gamma_j\\gamma_i = 2\\eta_{ij}\\,", "80e0c785a413a04c30ea594493c0c4ef": "\\left| \\beta - b \\right| < \\frac{r}{2 \\left|\\delta\\right|}", "80e12f3f7cb4333c64473984d137b276": "\\sec(\\theta) - 1", "80e230c1407d55d959064abbe2c0c7cb": " \\dot{P}(t) = A(t)P(t)+P(t)A'(t)-P(t)C'(t)W^{-1}(t)C(t)P(t)+V(t)+\\tau_\\perp(t)\\Psi_1(t)\\tau'_\\perp (t),", "80e25c5413844f2f389ef90dcc025c39": " u\\in BV(\\Omega)", "80e25fb018fc726dcd2307a52940334c": "x\\otimes x\\rightarrow \\pm x\\otimes x", "80e2d608dc98071c9e18b39e2a46c1f2": " U_4(x) = 16x^4 - 12x^2 + 1 \\,", "80e2ebf207bc56a06352b4acfadc6f51": "\\scriptstyle{{\\pi}/2}", "80e2f7da3a70eb815c26366c10b95721": "_{nominal}\\alpha = 1 - \\frac{1+2}{\\frac{1}{26-1}(4\\cdot7 + 10\\cdot7 + 5\\cdot7 + 10\\cdot4 + 5\\cdot4 + 5\\cdot10)} =0.691", "80e2f878c20f839a68c68208626e79e5": "y^4+\\frac{xy}{2}=\\frac{x^3}{3}-xy^2+y^2-\\frac{1}{7}", "80e331e13d00374449837cd6c369e13e": " k_1[A][B]", "80e351d7eef0f7eb9aca02ec32e0ff88": "k = \\sqrt{d^2 + a^2 + b^2 + c^2}", "80e3a39038127adcb65d5f80103ce715": "\\begin{align}\n \\left(\\sum_{i = 0}^m B_i t^i\\right) (t I_n - A)&=(tI_n - A) \\sum_{i = 0}^m B_i t^i \\\\\n \\sum_{i = 0}^m B_i t^{i + 1} - \\sum_{i = 0}^m B_i A t^i &= \\sum_{i = 0}^m B_i t^{i + 1} - \\sum_{i = 0}^m A B_i t^i \\\\\n \\sum_{i = 0}^m B_i A t^i &= \\sum_{i = 0}^m A B_i t^i .\n \\end{align}", "80e3a7835fb3f6b48b9026adddf8f5da": "j_p=+\\mu_p p E-D_p \\frac{\\partial p}{\\partial x}", "80e3b4e50e631fa87c22980ed08b3d4e": "\\rho_{de}", "80e3c69cea3c3832554f5ec5a736e936": "Y_{9}^{-5}(\\theta,\\varphi)={3\\over 256}\\sqrt{2717\\over \\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(85\\cos^{4}\\theta-30\\cos^{2}\\theta+1)", "80e4076d95927f59e0e935f5b3e9b733": "\\Phi - \\ ", "80e45ce36f21bce321f7eafbc9ef2421": "f(x)\\times g(x)=\\sum_{i=0}^{n+m} c_ix^i", "80e475b1754978244b2e3552af8338db": "{{V_1^2} \\over {gy_1}} = {{y_2({y_2 + y_1})} \\over {2y_1^2}} \\Rightarrow {V_1^2 \\over gy_1} = F_{r_1}^2 = {y_2^2 \\over {2y_1^2}} + {{y_2y_1} \\over 2y_1^2} = {y_2^2 \\over {2y_1^2}} + {{y_2} \\over 2y_1}", "80e4942bfca76bc3512cb465f92a68ec": "\\mu^0_{ij}", "80e499b46f3db31f0e4ffbe7845a63ec": "q:=\\text{quo}(r_{i-1},r_{i});", "80e4e7925db5215c51a7c3fdd65e00ce": "\\mathbf{\\Psi} = \\left\\langle\\mathbf{e} \\mathbf{e}^\\dagger \\right\\rangle\\,", "80e5244e007f1810a392f2e04b71bdae": "\\partial^\\mu\\partial_\\mu f=0", "80e5b75e1265d912327c0a6179d6911b": "{\\mathbf e}_1 = \\begin{bmatrix}1\\\\0\\end{bmatrix},\\quad {\\mathbf e}_2 = \\begin{bmatrix}0\\\\1\\end{bmatrix}.", "80e5e506521d9911a318313a902692f6": "\\displaystyle Y_s(z)=\\begin{cases}\n \\phi^{X_s}(z) & \\text{if } \\phi^{X_s} \\text{ converges in at most } s \\text{ steps.}\\\\\n 0 & \\text{otherwise }\n \\end{cases} ", "80e5f3d32803c86c328f671cdc561f00": "\n\n\\langle M(r)\\rangle = {16\\over 3\\pi G} \n\\langle R\\left(2V_R^2 + V_T^2\\right)\\rangle.\n\n", "80e63272130fa86b32727b6728691329": "f_{|M} : M \\to \\mathbb R", "80e637121fdc0889a1d604e46a53d5cd": "-\\frac{1}{\\eta}", "80e72eb86d893f06143b775339c866d0": "\\begin{pmatrix}\n I | P^{-1}\n\\end{pmatrix}", "80e764dc1f2045ed160f4182ee4821b4": "\\approx failed \\ 30 \\ seconds/year ", "80e7708acad596d57ba86d8200824ff7": " \\int_a^b f(x) \\, dx \\approx \\frac{b-a}{6}\\left[f(a) + 4f\\left(\\frac{a+b}{2}\\right)+f(b)\\right],", "80e775284478f1ba5d3aa0618f2e15bc": "\\sigma : \\mathbb{R}^{n} \\times [0, T] \\to \\mathbb{R}^{n \\times m};", "80e778e588bf83a3ed2d1749676ac635": "R_{T_1}(x,y) = \\sup\\{z \\mid T_1(z,x)\\le y\\}.", "80e794b38a70c584183e158f92d76936": "x^5 \\cdot 2^{-2} = \\frac{5x}{4}", "80e7b29f7e46cb81700477bf91c54174": "\nt \\propto \\frac{1}{r^2}\n", "80e7f91c914b6a7a130713ae0af831e0": "deltaH", "80e7ff00e313a9878ea7490ce370b86f": "|\\psi_{E}\\rangle\\longrightarrow l|0\\rangle_{1}\\otimes e^{i\\phi}|1\\rangle_{2} + e^{i\\phi}m|1\\rangle_{1}\\otimes|0\\rangle_{2} = e^{i\\phi}|\\psi_{E}\\rangle.", "80e8c4db67fe49d36c31d8588da316f3": "\\displaystyle{H_\\varepsilon f(z) - {i(1-\\varepsilon)\\over \\pi} f(z)={1\\over \\pi i} \\int_{|\\zeta -z|\\ge \\delta} {f(\\zeta)-f(z)\\over \\zeta -z} \\, d\\zeta.}", "80e8e41a51a7b7f6baf9277467f15689": " 2 ( k_1 k_2 - 1 )^2 \\ge 2( 1 - \\rho^2 ) + ( 1 - \\rho )( k_2 - k_1 )^2 ", "80e925c1544dadba279c264d8ba834ca": "x\\in U\\setminus\\bar{W_{U}}\\,", "80e93cf981981ae38424e7b1e2e10b1c": " \\text{minimize}_{x \\in \\mathcal{R}^n} \\|x-r\\|^2 \\quad\\text{subject to}\\; x \\in C \\cap D ", "80e99a91f5d540774caf350cbd408d55": "DE=\\frac{1}{2}\\pi y^2", "80ea196d475a46f03d8cceb683b46a70": "x_1(0)=a_1,\\,", "80ea5fcb0128a32f18b846568f39d7cc": "\\,\\Gamma (TM)", "80ea7309b08df71fdce961b67b398c58": " (\\tfrac{ 1}{20},\\,\\tfrac{ 3}{20},\\,\\tfrac{ 7}{20},\\,\\tfrac{ 9}{20},\\,\\tfrac{ 11}{20},\\,\\tfrac{13}{20},\\,\\tfrac{17}{20},\\,\\tfrac{19}{20}).", "80ea85b0a47770e5eae37915f410e2dc": "x = y_1^{1/5}+y_2^{1/5}+y_3^{1/5}+y_4^{1/5}\\,,", "80ea8aef611409679a85f6d9ce3e3579": "\\int_W u(x) Q(x, \\partial) \\varphi (x)\\, \\mathrm{d} x = 0", "80ea90634d7d931374791d5f70d7cb46": "f(x, r)", "80ead2ac81e4022fb21dc4ea456a04c1": "B_\\text{g, Earth} = \\frac{G }{5 c^2} \\frac{m}{r} \\frac{2 \\pi}{T} = \\frac{2 \\pi r g}{5c^2 T},", "80eade55c07aea63b1b07fa6ce5cac89": "\\rho_2={}", "80eae4070241653e580db8d890cdb348": "(\\mathbb Z/2\\mathbb Z)^{s}", "80eb1d26290d117f22e18b3e6adda0ba": "- T(\\alpha_1, \\alpha_2, \\ldots, \\nabla_YX_1, X_2, \\ldots) \n- T(\\alpha_1, \\alpha_2, \\ldots, X_1, \\nabla_YX_2, \\ldots) - \\ldots\n", "80eb2720bf6be1ba19855429077a0f9f": " \\scriptstyle \\phi ' ", "80ebaded06b0cf45927307aaefa51423": "\\textstyle \\langle e_{\\alpha}, e_{\\beta}\\rangle=0", "80ec22b549965a71cda9ac68c72df0d6": "253=\\frac{23\\times(23-1)}{2}", "80ec33be707109ac8bede0072906f0be": "f(b_1, \\ldots, b_n) = a_0 \\oplus (a_1 \\land b_1) \\oplus \\cdots \\oplus (a_n \\land b_n)", "80eca0800a3e3568f0541d6f78cc9906": " a^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ {dv^{\\mu} \\over d\\tau} ", "80ecb3f688351bf6409aecdf903996c8": "P(S|N,n,s=0) = {\\prod_{j=1}^{n-1}(N-S-j) \\over S \\sum_{R=1}^{N-n}{\\prod_{j=1}^{n-1}(N-R-j) \\over R}}\n", "80ecb5610b57f8a55bf29b2f30a53379": "w_{jk} \\,", "80ecfb68c128fe9f1c5e328f9cfedc3a": "\\gcd (n^a-1,n^b-1)=n^{\\gcd(a,b)}-1.", "80ed0ddfd90ff9bb37c831d277493a34": "(0,0,a)", "80ed29a91166147cf23b2cdda78ab1d1": "\n\\frac{M}{10^8M_\\odot} \\approx 3.1\\left(\\frac{\\sigma}{200~{\\rm km}~{\\rm s}^{-1}}\\right)^4.\n", "80ed91f4947b90068e268d21d2280f6a": "D_1=6P_1+ 4P_2 ", "80ed9c50a6380850625ecb4ddf266c1c": "\\widehat{\\sigma}^2", "80ed9fe626f9e0feca36eb327f5e44b2": " \\mathbf{I}(t)", "80ee681b4e03b5192296c9862e4c8b1a": "Child\\ dependency\\ ratio = \\frac{number\\ of\\ people\\ aged\\ 0-14} {number\\ of\\ people\\ aged\\ 15-64} \\times 100 ", "80ee8c1daf19a69e68ec0713ceaf502b": " f^\\downarrow\\left( q_l, q_r \\right) ", "80eea8aa3224ddabb1e606dd45c14542": "u^0_1", "80eebe44c9ed073ac1dab5861a343a92": "E_c=\\frac{gy_c^3}{2gy_c^2}+y_c=\\frac{y_c}{2}+y_c=\\frac{3y_c}{2}", "80eedd389402b44bbf8cb17f37407a8d": "o_i = 0 , i = 1,..,M, i \\ne m", "80ef28ffe4da49fc08f372e108d66745": "(1-\\frac{1}{a})N_1 I_1 = \\frac{N_1}{a}(I_2-I_1)", "80ef637bb382f75a95ecc16a32881822": "\\beta_1,...,\\beta_n", "80efb773389ade6deffa7aefa57e28ef": " \\int_0^t C_p(\\tau) \\, d\\tau / C_p(t)", "80efb9aff4fcf6c46d015377069ffe82": "\\lambda(\\zeta)\\ d\\zeta", "80efba88eee33fae0dd75e3d70bb45bd": "i=0,\\ldots,n-1", "80f063c84d53f3749b801fee0699d1b6": "\\kappa_{12}r^{-\\nu}", "80f07976df84395ad990ef4ee362fb4b": " \\partial_{ \\bar z } = \\frac{ 1 }{ 2 } ( \\partial_{ x_1 } + i \\partial_{ x_2 } ) ", "80f08fb8829b69cc5b5389c3f843023a": " m_2 = m_1 - 2.5\\log(d_1^2/d_2^2) = m_1 + 5\\log(d_2/d_1) ", "80f0df8ddb4a74e4959f89bd3851ec2f": "\n\\begin{align}\n\\frac{dT_1}{dz}&=[T_2,T_3]\\\\[3pt]\n\\frac{dT_2}{dz}&=[T_3,T_1]\\\\[3pt]\n\\frac{dT_3}{dz}&=[T_1,T_2],\n\\end{align}\n", "80f0ec442f30dc0150e9219c23b4f2a8": " [A \\varphi](t) = t \\varphi(t). \\;", "80f27abdcc270d695d543ea245dadbdf": "\\left(1-\\tfrac{m}{n}\\right)\\,", "80f290577d3cf55f6725258c65ed6d55": "C_{XY}=C^T_{YX}", "80f29794ea3477c7ab77e82feb43eb75": "\\tfrac{3}{2}\\log_2 n", "80f2bbe3a135189799b704dff47d075b": "\\le \\deg F + \\deg G + 1.", "80f2e880b3b0dd428118d92b25a3f8d1": "\\theta = \\pi/6", "80f300a0c112f46e408c67f6f42db30b": "V_{s}/\\Omega", "80f3435af2b179e6dc2d7e03726f9522": "\\pi_0(A) \\to \\pi_0(X),", "80f3d62519832bc79c6e28dd59e2431a": "(n,q^2) \\notin \\{ (2,2^2), (2,3^2), (3,2^2) \\}", "80f42337bb1869d3bc09f3a920163c4d": " l^1(G)\\rightarrow C_r(G)", "80f45ecae6ae2133ea1b8ae545dcd0cc": " \\mathbf{\\phi}(g) = (\\psi(1),\\dots,\\psi(n)).", "80f47029388dd6829f2352f50d2270a9": "x=\\cot y\\,\\!", "80f52925db798314143977c477ab2457": "\\qquad \\qquad k_p = (48\\pi^2)^{1/3}\\frac{k_\\mathrm{B}^3T^3}{ah_\\mathrm{P}^2T_\\mathrm{D}}\\int_0^{T/T_\\mathrm{D}}\\tau_p\\frac{x^4e^x}{(e^x-1)^2}dx,", "80f536b7391a79cf99cca0fada077bf5": "Q_{2n+1}(n)", "80f5405458131be23e3f6bed12cd2801": "y=\\psi_i(x,\\bar{y})", "80f584f5cb71466654aed66258aea6d1": "\\Delta : \\mathcal C \\to \\mathcal C \\times \\mathcal C", "80f5a45f3b13ecd4ec84fba32e894410": "P_i^a P_j^b \\dots P_k^c \\to Q^d", "80f62d35af9938ac16dec78696c7de6a": "\\phi^{+}(a_i)", "80f652213971a1d7c27d53eff736aad4": " \\mu_{c} ", "80f68380ef683e163838811d4b4f6d76": " \\operatorname{inc}\\ (\\operatorname{inc}\\ (\\operatorname{inc}\\ \\operatorname{init})) = \\operatorname{value}\\ (f\\ (f\\ (f\\ x))) ", "80f6ff616854cb3e4f881c6e132879df": "e^- \\, / \\, e^+", "80f7189f0e7ab49d33e5c9359980b707": "dB(Ratio_i) = 20 \\log_{10} \\alpha^{2^{i-1}} = 2^{i-1} \\cdot 20 \\cdot \\log_{10} \\alpha", "80f71b61a261073e0833939ec4da39dd": " \\Delta G = \\Delta H - T \\Delta S.", "80f73e26478af9ff1dc87b8162478eea": "A(c)=2 \\alpha/(1-2 \\alpha c)", "80f77d27d4949aafd066b7f5d27cd4ac": "\\frac{64}{33}", "80f79c92cf3b60f76730fac457459e58": "\\, k", "80f84ecc6a93f66f0aaac64ae6061c29": "\n\\text{Index level}=\\frac{\\sum_i(\\text{Price of stock}_i\\times\\text{Number of shares}_i)\\times\\text{Free float adjustment factor}_i}{\\text{Index divisor}}\n", "80f85569b96c23afceea60a7ac926b32": " \\mathbf{A}(\\mathbf{r},t) = \\nabla \\times\\int\\frac{ \\mathbf{B}(\\mathbf{r'},t)}{4\\pi R}d^3r'+\\nabla \\psi(\\mathbf{r},t)", "80f86052b1e612b48f7154aeb589bb5f": " ds^2 = \\exp(2 \\psi) \\, \\eta_{ab} \\, dx^a \\, dx^b, \\; \\; \\Delta \\psi = 0 ", "80f896767216da19c1eaceea80482441": "\\mu_{s,z} = -e S_z/m_e = g_seS_z/2m_e\\,\\!", "80f8b0299d9070ea0d442a4777c4785d": "\n \\sigma_{11} - \\sigma_{33} = \\cfrac{2C_1}{J^{5/3}}\\left(\\lambda^2 - \\tfrac{J}{\\lambda}\\right) ~;~~\n \\sigma_{22} - \\sigma_{33} = 0\n ", "80f8e88fdc07e9e95ac64339b72ac691": "\\textstyle(x,y)\\in\\R_+^2", "80f8f8b6be2caaca1d970fa521e726bb": "{\\sum_{n\\le x}}'\\biggl\\{\\frac{an^{*}+bn}{m}\\biggr\\},\n{\\sum_{p\\le x}}'\\biggl\\{\\frac{ap^{*}+bp}{m}\\biggr\\},", "80f97ee318710ac5a0137aed68dbe6d2": "By^2 = x^3 + Ax^2 + x", "80fa49a1f5d20e0ac2feb37b3677f9f1": "\nT_{ij} = T_i\\frac{{A_j f\\left( {C_{ij} } \\right)K_{ij} }}\n{{\\sum_{j = 1}^n {A_j f\\left( {C_{ij} } \\right)K_{ij} } }}\n", "80fa73601c479771af27dbc3ebfefc3b": " \\frac{1}{2}X_2(g) \\Leftrightarrow V^{''}_A + X^{\\times}_X + 2h^{\\bullet} ", "80fab78b95336e4367ef375d23b7a6a5": "v[\\alpha] = 0", "80fadf70a7ddf08914d365927fc0e008": " \\begin{bmatrix} V_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} 0 & -r \\\\ r & 0 \\end{bmatrix}\\begin{bmatrix} I_1 \\\\ I_2 \\end{bmatrix}", "80faf9f383ecd07e35e742e06e834180": "x^2-ay^2-1=0,", "80fb30b91dfa5d0cc39dcbdaa9f5c51d": "w = \\frac{M_w}{M_s} = \\frac{W_w}{W_s}", "80fb49d51f56c7d980b71d87f32b2918": "g(z) = \\exp \\left( z^{-1/k}\\right)\\,", "80fbb3ce6db65f24493e0c95f263c97d": " f : \\mathbb N \\mapsto \\mathbb N", "80fbbd00944125348d6341f1c7803cf4": "\\langle (x_1,...,x_n),(y_1,...,y_n) \\rangle = \\langle x_1,y_1 \\rangle +...+ \\langle x_n,y_n \\rangle. ", "80fc318e844c15e0d45ec1597ea6dd0e": "\\sum_{i=1}^k p_i z_i\\text{ for }(z_1,\\ldots,z_k)\\in\\mathbb{C}^k", "80fc68f3eb6124179b51c8975a04127f": "(p \\land q) \\to p", "80fc732f1a9a915477b40c8dae3c94f1": "\\mathbf{h} = q' \\mathbf{B}_k,", "80fc9d5fdf15ce69554e84b0b0b80cac": "\n \\begin{bmatrix}a&b&c&d\\\\e&f&g&h\\end{bmatrix}\\begin{bmatrix}x_1x_2\\\\x_1y_2\\\\y_1x_2\\\\y_1y_2\\end{bmatrix}=\\begin{bmatrix}\\alpha\\\\\\beta\\end{bmatrix}\n", "80fcb8483886713ce3468df19af052d1": " \\frac{R}{4} \\sqrt{10+2\\sqrt{5}} = \\frac{a}{2}\\sqrt{5+2\\sqrt{5}} \\!\\, ", "80fd1b7240cb0aff6eb42435ae946804": "\n2^{bh(v')-1}-1 + 2^{bh(v')-1}-1 + 1 = 2^{bh(v')}-1\n", "80fd30acb849b04e99f5b4913237e56c": "\\frac{{\\tan\\theta_1}}{{\\tan\\theta_2}} = \\frac{{\\mu_{r2}}}{{\\mu_{r1}}}", "80fd34806aaa93ccd2c3ccd2ec11b615": "\\mathbf{\\lambda}=\\{\\lambda_i\\}_{i=1}^N", "80fd44d53d0f9efd72f39b4a798d5480": "I \\subseteq \\mathbb R", "80fd78440180c1a8ace8dccd1b122c6b": "\\frac{d\\ \\operatorname{Re} \\{V_c \\cdot e^{i\\omega t}\\}}{dt} \n= \\operatorname{Re} \\left\\{ \\frac{d\\left( V_c \\cdot e^{i\\omega t}\\right)}{dt} \\right\\}\n= \\operatorname{Re} \\left\\{ i\\omega V_c \\cdot e^{i\\omega t} \\right\\}", "80fdf6c9ad1fda0af0ff7b32ae42a472": " V_0 = \\frac{V_\\max\\,[S]}{K^\\text{app}_m + [S]}", "80fe0a58fd302f97702a238317799433": " \\hat{\\xi}_k = \\langle X - \\hat{\\mu}, \\hat{\\varphi}_k \\rangle. ", "80fe4d1520e09e1b10497242034d7c46": "\\big. \\frac{\\Delta Q}{\\Delta t} = U A\\, (-\\Delta T).", "80fe528efdf631638280221b3e518cf5": "{{i}_{IN}}={{i}_{C1}}+{{i}_{B3}}={{i}_{C}}+\\frac{{{i}_{C3}}}{\\beta }", "80fe550dd6cc0566a043f5bbd7a4a197": "g_\\mathrm{p}", "80fe8d2e1a2d0a911fad4a3431e4bc89": "(6,1,1)_H", "80feb4b63241b2de7b51cf21b4c89466": "(A-4I)", "80febcf40e34c898e430b7c24141b5da": "\\frac{1}{1+1} = 1 - ( 1 - ( 1 - ( 1 - \\frac{1}{1+1} ) ) )", "80fee28dc58eac3ced336b27e22e3686": "\\Gamma(Y, G) \\to \\Gamma(X, f^*G)", "80fef65844e5ab1335be77c95e54081f": "\\Rightarrow\\,", "80ff48885c2021320cc8fe752b09e84b": "t=1,\\dots,T", "80ff57028c6d7b2628fe24891d5f01d3": "\\frac{1}{(\\beta E_c)^\\alpha}=\\frac{f}{(\\hbar\\omega\\beta)^3}", "80ffc5c1b2c4e2376da3c10103f09aa8": "\\scriptstyle\\mathcal{M}_X(it)", "80ffe78b3114a88f9c10439c0473a156": "(\\deg P)^{-1}\\log\\mathcal{M}(P(x))", "810033ac002bd03418aebb0e96c9860a": "c_{1}(t)=H(t,0)=H({\\gamma}_{1}(t))={\\Gamma}_{1}(t)", "810060414b02b433f72a9b76f1aa7e40": "6 a_1^2 a_4.", "8100901dd7d6b778970ffc8c46352d80": "b_{7}-a_{8}", "8100c9839115d11c0dd71242ea1ba0d2": "\\quad \\sin^2(x) = 1-\\cos^2(x)", "81012b65e3486689c02a72a04a5efa65": "\\langle u_x \\rangle", "810137fdadcaa1e763fc5002bd6c6420": "\n\\mathrm{MSD} \\sim \\left(\\frac{\\alpha}{n_0}\\right)^2 \\ln^2(t).\t\t\t\t\n", "81015332bde58d864ae3561a23a4de24": "N' =N-M+1", "8101994bd6d89d0530e2a8b91718e2aa": "\\widetilde{x}_s", "8101c405f1f44c5da368d70d2e12b441": "dim\\,G \\le Edim\\,G", "81026c4a38f31708ac8875d890ba0055": "\n\\frac{d}{dt} \\left( \\mathbf{p} \\times \\mathbf{L} \\right) = \\frac{d\\mathbf{p}}{dt} \\times \\mathbf{L} = f(r) \\mathbf{\\hat{r}} \\times \\left( \\mathbf{r} \\times m \\frac{d\\mathbf{r}}{dt} \\right) = f(r) \\frac{m}{r} \\left[ \\mathbf{r} \\left(\\mathbf{r} \\cdot \\frac{d\\mathbf{r}}{dt} \\right) - r^{2} \\frac{d\\mathbf{r}}{dt} \\right]\n", "8102cda3ef2250fdf90a2aa5cba53b4c": "\\sin(k \\theta) = \\cos\\left( k \\theta - \\frac{\\pi}{2} \\right) = \\cos\\left( k \\left( \\theta-\\frac{\\pi}{2k} \\right) \\right)", "8103d5340857ee136c01b25f8573436c": "V_i, i = 1, 2, \\dots, w", "810415151e37ced4fb3978e71e8bf10a": "\\zeta_{i}", "81043f8c681575bca3e4eff6afbd9db9": "BW", "81044e8a1d8d30e92a76646ab71805b9": " y_1 + u ", "810467e331216b9b893bc4a58857b578": "\\omega := -\\varphi_t", "81046abd7abce1d078c0c67358f6177b": " n / 2 ", "81047d23d01c550b15f49a90883be234": "\\psi(x) = \\frac{1}{\\sqrt{2 \\pi \\hbar}} \\int_{-\\infty}^{\\infty} \\phi(p) \\cdot e^{i p x/\\hbar}\\, dp, ", "8104a4171b03a621a24dc03e044bc188": "\\vec k=\\vec k_0", "8104ccd0ad090cb2e3b0ed52964f38f1": "u_{13}", "810513ae9f4614502e560d032532e8bb": "\\nabla \\cdot (n_{a}\\boldsymbol{u}+\\boldsymbol{J}_{a})=0", "8105722d8b2bb3dae35e25d1b1e4f682": "\\tilde{J}_{il}", "8105864563ece8bed4acbb1c1af6f2e2": "\\Phi = \\Phi(\\frac {1}{T},V,\\{N_i\\})", "8105baff79ed90838444f110207ad478": "\n\\Phi_n(z) = U_n^2(z) - (-1)^{\\frac{n-1}{2}}nzV_n^2(z)\n", "810624e28159beeb6a6334d98f9b080f": "\\{(-\\infty, r]: r \\in \\R\\}", "8106398ed15434551473c93f2ba994fd": "\\mathbf{F}_\\text{surf}", "810662afcbfbcd0dcfcf6b8c82762310": "(\\mu_1 \\times \\mu_2)(E) = \\int_{X_2} \\mu_1(E^y)\\,d\\mu_2(y) = \\int_{X_1} \\mu_2(E_{x})\\,d\\mu_1(x),", "81068d94f1bd82bf9b79f0026c130963": "\\Pi_{(n:{\\mathbb N})} \\mbox{Vec}({\\mathbb R},n)", "81068ff2abc51c491067ce6754288599": "\\vec{h}_3 = \\frac{\\sqrt{1-2m/r}}{\\sqrt{1-3m/r} \\,\\sin(\\theta)} \\, \\partial_\\phi - \\frac{\\sqrt{m/r^3}}{\\sqrt{1-2m/r} \\, \\sqrt{1-3m/r}} \\, \\partial_t ", "81069e1f2800e8cc856ff57c91925e75": "|\\Lambda|^s_M", "8106c46e98e851199a90ac84839ee1d4": "\\phi(\\mathbf{r}) = \\sum_{j=1}^{N} f(\\mathbf{r} - \\mathbf{R}_{j}) = f(\\mathbf{r}) \\ast \\sum_{j=1}^{N} \\delta(\\mathbf{r} - \\mathbf{R}_{j})", "8107512a8db38070e96286c336f6edb4": "B\\rightarrow\\neg D", "810757cc43130e25e53c977f13a48c9f": " [K] ", "81076c0b50dec8a1d5b5436e16a18a38": "\\operatorname{Tr}\\left( (p\\!\\!\\!/' + m) \\gamma_\\mu (p\\!\\!\\!/ + m) \\gamma_\\nu \\right) \\,", "8108096fa97bda9f9585a09966a8681e": "\\textbf{D}_{R}", "81093345e51d24b486af941729b79a41": "\\textstyle \\mathfrak{H}", "810980db7ffad18605ac6d0828ef4755": "L_n^{(\\alpha+1)}(x)= \\sum_{i=0}^n L_i^{(\\alpha)}(x)", "810989f5a8da5bd723d6dd1cd9f65229": "\\mathrm{rect}(.)", "810991b6d469e6ddc91b4748c4194590": "\n{2\\ \\mathrm{k \\Omega} \\over 1\\ \\mathrm{k \\Omega} + 2\\ \\mathrm{k \\Omega} } \\cdot 10\\ \\mathrm{V} = {2 \\over 3} \\cdot 10\\ \\mathrm{V} \\approx 6.667\\ \\mathrm{V}.\n", "81099a209fff7880b5a04e1d6c086626": "V_{out,2nd order} = \\frac{k_{2}A^{2}}{2} + \\frac{k_{2}A^{2}}{2}\\cos(2wt)", "8109a95191d295d4ed3e5da478680b60": "\\Gamma > 1", "8109c5187bca78f291f61fa0797062d7": "y = x^3 - 12x^2 - 42", "810a0d4c13b2f8794c9827899456aa0a": "I(V) = \\{f \\in k[x_1,\\ldots,x_n] \\mid f(x) = 0 \\text{ for all } x\\in V\\}.", "810a5c3a901a00a12b5cf805e533e3a7": "C(S, T) = \\max\\{S - K, 0\\}", "810a742235e427d15daaeed7b5dc3432": "\n\\prod_{i=1}^{n}\\left(1-a_{i}\\bar{a}_{i}-b_{i}\\bar{b}_{i}+a_{i}\\bar{a}_{i}b_{i}\\bar{b}_{i}\\right)=1-\\sum_{i=1}^{n}\\left(a_{i}\\bar{a}_{i}+b_{i}\\bar{b}_{i}\\right)+\\sum_{i=1}^{n}a_{i}\\bar{a}_{i}b_{i}\\bar{b}_{i}\n+\\sum_{i", "810c6adbd394d42418773aa2e43b2fc6": "I \\in \\mathbb{R}^n", "810c8c141fd1f5dc125ee424be401fb5": "\\frac{128}{125}", "810cbaebba123acc3fe8c9afe7cc469d": "T [R] = \\bigcup_{t\\in T} R_t", "810cbf1ff4d5de438d11fb97759f82db": "E(K)[n]=\\{T \\in E(K) \\mid n \\cdot T = O \\} ", "810cdf429981a6c877c120dd7148a52c": "1\\leq i\\leq N", "810d7f7e710ea278467a497bc7541857": "e_i^2 =-1, e_i e_j = - e_j e_i. \\, ", "810df4a56f6be5aafe0ccc182a6c3c31": "M=(m_{i,j})_{i,j=1}^k", "810dfffe9e10a2ad8e0a2d6c7214adb5": "\\eta_{He3} = n_{He3}/n_e", "810e16c1608bdff662935cb5f919d0a0": " W^{\\mu } =\\frac{1}{2}\\varepsilon ^{\\mu \\nu \\kappa \\lambda }P_{\\nu}J_{\\kappa \\lambda } ", "810ed35ce96b515984a7018139b1054d": "B_{\\mathrm{st}}\\otimes_{K_0}H^\\ast_{\\mathrm{dR}}(X/K)\\cong B_{\\mathrm{st}}\\otimes_{\\mathbf{Q}_p}H^\\ast_{\\mathrm{\\acute{e}t}}(X\\times_K\\overline{K},\\mathbf{Q}_p)", "810f1e6108c0f9d840866baada521e98": "I / I_0 = e^{-m \\tau},\\, ", "810f3c80fe7cc4d5c91cc243fafa51da": "V_n(R) \\propto R^n.", "810f5580eb691d6499e779b0f1de5054": "r\\colon I\\to PX", "810feac96c3046a8c30f760a8d5a2962": "\\partial_{\\nu} {J^{\\nu}}_{\\text{free}} = 0 \\,.", "8110191b9dfcc182c47fa14772a680fe": "\nX = \\sum_{k=1}^n U_k.\n", "81101ebe041bc6decd92a73562b1fc85": "(fg)' = f 'g + fg' \\,", "81107f8a8b4a47ff111c8bc9332985a1": "\\sum_{j=1}^{D-1} p_j^2 = \\frac{1}{D-1} \\ne 1.", "8111216c0dcd271dc30050638ffe91be": "\n \\mathbf{M}_1 = \\int_{-t/2}^{t/2} [-x_3\\sigma_{12}\\mathbf{e}_1 + x_3\\sigma_{11}\\mathbf{e}_2] \\,dx_3\n \\quad \\text{and} \\quad \n \\mathbf{M}_2 = \\int_{-t/2}^{t/2} [-x_3\\sigma_{22}\\mathbf{e}_1 + x_3\\sigma_{12}\\mathbf{e}_2 ]\\, dx_3\n ", "81112211edfa5f5683c826bf684ae186": "\\frac{dM}{dx} = Q", "8111847237f9c4abd748e708a9580d04": "\\mu_{a}(cm^{-1})", "81123c1626e524fd3bdca7a645382adb": "v = \\sqrt{T \\over \\mu}.", "811284fdf95f3e76be2a86be46acdd14": " \\frac{dD}{dt} = -k_2 D + (R+P)_{avg} ", "8112a149eae31cc46564505823b62f19": "t_1^\\prime = 0", "8112a64dd7fd5eaaa1a4c2472e91cfee": "(x-3) x^{12} (x^2-6)^5 (x^2-2)^{12} (x^3-x^2-4 x+2)^2", "81131a91caaf384badd8efb95e475342": "\\sigma_y,\\!", "81133077026f407c559b0037ea5a60af": "i_j", "811331e4be6c9faffc679cc70b69f56a": " R = R(x)", "8113812c2c2128435b89b72a78437451": "F=-GMmu^{2}\\left(1+3\\left(\\frac{hu}{c}\\right)^{2}\\right)=-\\frac{GMm}{r^{2}}\\left(1+3\\left(\\frac{h}{rc}\\right)^{2}\\right)", "8114247d416c9b711b58a964c2098afe": " \\vec x_2 ", "8114555fe91df60d65867d0dcfa1d3e8": "P(t)=\\frac{1}{1+e^{-t}}.", "81146702f7c412b945d8a1baf15ef023": "\\overline{a}_0 = \\left\\{a\\right\\}", "81146814acf31a8864c1f8ac60833cd5": "TK_3", "8114a7d1f99a122e93163a4be7a74b54": "\\textit{mammal} \\subseteq \\textit{animal}", "8114c11e093304ae5944f0b5cacd69ee": "(X_\\infty, d_\\infty)=\\lim_\\omega(X_n,d_n, p_n)", "811542d59501304875838035f8424f2d": "T\\left(A,W\\right)", "81168afde2cdd93ce782aef6e65f5a7c": "D_x y \\;", "8116daa24d01faae1d4c0a2c9bec862c": "\\bigg|\\|x\\|-\\|y\\|\\bigg| \\leq \\|x-y\\|,", "8117157d9dbd9705e5bf357615d48713": "\\omega_{Y\\overline{\\|}X}=\\omega_{X}\\;\\overline{\\circledcirc}\\; (\\omega_{X|Y},\\omega_{X|\\overline{Y}},a_{Y})\\,\\!", "81173634e1159d9de8ff225d0cde0f42": "\n\\zeta(s_1, \\ldots, s_k) = \\sum_{n_1 > n_2 > \\cdots > n_k > 0} \\ \\frac{1}{n_1^{s_1} \\cdots n_k^{s_k}} = \\sum_{n_1 > n_2 > \\cdots > n_k > 0} \\ \\prod_{i=1}^k \\frac{1}{n_i^{s_i}},\n\\!", "8117602c8433e867757372e55d0e48ac": "\\mathit{C_{1}}\\,", "811775c1766743ea006ebf2d4fc09e9a": "|{\\Phi^{'[{{DK}}]}}\\rangle=\\sum_{j,\\gamma}\\Gamma^{'[{{D}}]j}_{\\beta\\gamma}\\lambda_{\\gamma}|{j\\gamma}\\rangle.", "8117786c066053ad584612e155162645": "times(x,y,z)", "811784feafd5a0d28b2609389576cf75": "\\{ x \\in S(B) \\mid b \\in x\\},", "81178548270cbd32757eb6fb26ce7076": "p = -u", "8117920ea1fd9ac48ac73f0b67d6cf4d": "\\frac{\\alpha}{c+vt}\\,\\!", "8117cdcdfcbec259987fbaa109f57172": " \\langle \\mathbf{u},\\mathbf{v} \\rangle_\\mathbf{A} := \\langle \\mathbf{A} \\mathbf{u}, \\mathbf{v}\\rangle = \\langle \\mathbf{u}, \\mathbf{A}^\\mathrm{T} \\mathbf{v}\\rangle = \\langle \\mathbf{u}, \\mathbf{A}\\mathbf{v} \\rangle = \\mathbf{u}^\\mathrm{T} \\mathbf{A} \\mathbf{v}. ", "8117f283bb42ddcb3e8cd2565520363b": "\\sqrt{\\frac{2}{3}}\\begin{bmatrix} \\cos(\\theta)&\\cos(\\theta - \\frac{2\\pi}{3})&\\cos(\\theta + \\frac{2\\pi}{3}) \\\\\n - \\sin(\\theta)& - \\sin(\\theta - \\frac{2\\pi}{3})& - \\sin(\\theta + \\frac{2\\pi}{3}) \\\\\n\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2}&\\frac{\\sqrt{2}}{2} \\end{bmatrix}\n=\n\\begin{bmatrix}\\cos(\\theta)&\\sin(\\theta)&0\\\\\n-\\sin(\\theta)& \\cos(\\theta)& 0\\\\\n0& 0& 1\\end{bmatrix}\n*\n\\sqrt{\\frac{2}{3}} \\begin{bmatrix} 1&\\frac{-1}{2}&\\frac{-1}{2} \\\\\n0& \\frac{\\sqrt{3}}{2} & -\\frac{\\sqrt{3}}{2}\\\\\n\\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}} \\end{bmatrix}\n", "8118020484ab8f120c22e8502d0bdbd4": " ep(D+1) \\leq e 2^{-k} \\frac{2^k}{e} = 1 ", "81183cc6a701811c03fd3d581e848a56": "\\frac{kPa.m^3}{Kmol.K}", "8118485743dae55492ce740514c4dae8": "\\mathrm{OPD}= d_1 n_1 - d_2 n_2", "811854e0b3847d5c6abd0b6a41ff2be7": " \\mathfrak{g}_0 ", "81189cee58c71bc5cea24eb5548c47fc": "|f_n(x)|\\le M", "8118ff4740bf6a7c6cc8658f97b9baf1": "\nf(x) = (x - y_1)(x - y_2)(x - y_3)(x - y_4)(x - y_5) \\in E[x].\n", "811902be8023e79808cc9562cf8d5083": " X_1, X_2, \\dots , X_n \\, ", "8119732a3299279dde79ac90fb25cfe5": "\\lambda_{jk}={ 1 \\over 1+\\pi^2 j^2+\\pi^2 k^2 }.", "8119bf65035e39c5373e7ed1f4b77665": "\\bar{q} = B \\bar{p}", "811a6218ae0c7de4d7b3f283fc61aad2": "f(f(u)) ", "811acd34da16cfc9e34c67c76b74aadb": "aA_2 = A_2 \\cup A_3 \\cup A_4", "811af3829d2a6b8ca7a4cf9643af42f7": " \\mu(a)=0 ", "811b57963c1a455be96bbd7ad5fee9b6": "(a_1{\\mathbf e}_1 + a_2{\\mathbf e}_2)+(b_1{\\mathbf e}_1 + b_2{\\mathbf e}_2) = (a_1+b_1){\\mathbf e}_1 + (a_2+b_2){\\mathbf e}_2", "811c3c22622a567de2cfd47d328af17c": "\\mathbf{f} = \\rho \\mathbf{E} + \\mathbf{J} \\times \\mathbf{B}\\,\\!", "811c4bf21d21983b30c5b8dba3c0682f": "q \\in R", "811c58f55dadebbf159213a6811e0d5e": "\\cot{ \\frac{B}{2 }} = \\frac{s-b}{\\zeta }", "811c98a3246e5fabc108a4f8b3914d37": "\\ invF^n =_{def} \\{x_1...x_n : F^n x_2x_1...x_n\\}.", "811cc5e7d456c2323d4ac2bf2e9f1092": "NER value=\\frac{N - E - R}{N} * 100", "811cd1c273b674fa7a1e52ad1c3f336d": "y_i = x_i", "811ce211c9df16403b82cf2f614b5569": "T = t_1t_2\\dots t_n", "811cfccf7c27cc43b64e60aa22327190": "z^i=\\langle z^i_1,z^i_2,..,z^i_n\\rangle", "811d0405b53d3cc59dce1137ba989f6e": "P(t) = -{\\nabla U} \\cdot \\mathbf{v} = \\mathbf{F}\\cdot\\mathbf{v}.", "811d04a5b16ddbf227ba0132e6492807": "\\forall y\\, q(y,x)", "811d089f8747ba7d69e08507816d7a69": "\\{(x-h, y), (x, y), (x+h, y), (x, y-h), (x, y+h)\\}. \\,", "811d13a17a9eef24c14a2d815a71fe66": "Kritm (i) = \\frac{1}{fin_i -st_i} \\sum_{n=st_i}^{n=fin_i -1} |S(n)-S(n-1)|, i = 1, \\dots, 5 ", "811d713ded1fdbf12ec764665bd81afb": "\\ddot{a}_{\\overline{n|}i} = 1 + v + \\cdots + v^{n-1} = \\frac{1-v^n}{d}", "811d7ff8435ac8035e46073a99ec5bbe": "\\vartheta_{11}(z|q) = -2 q^{1/4}\\sin(\\pi z)\\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 - 2 \\cos(2 \\pi z)q^{2m}+q^{4m}\\right).", "811d9cfc409bc3387757dfc20915195d": " d = |\\mathbf{n} \\cdot (\\mathbf{c} - \\mathbf{a})|", "811dadbdbd60e22ea8aab67fea34463b": "t_{r,\\alpha}", "811dbf22f13925f2919f8d1e35999d18": "\\neg(B_{1}\\land\\cdots\\land B_{m}\\land\\neg C_{1}\\land\\cdots\\land\\neg C_{n}).", "811dfca4f79e950cda0e3d6b9d7ff88b": "\n \\begin{bmatrix}\n \\sigma _{11} & \\sigma _{12} & \\sigma _{13} \\\\\n \\sigma _{21} & \\sigma _{22} & \\sigma _{23} \\\\\n \\sigma _{31} & \\sigma _{32} & \\sigma _{33} \n \\end{bmatrix}\n \\quad\\quad\\quad\n", "811e11456402f10513c9dc170a123b0a": "A_n=\\left[\\frac {1}{n+1},\\, \\frac{1}{n}\\right)", "811e1e108df8624b4a0c7fb78a0f49f1": " U = s^2 - m + m^2 / k ", "811e760cd05496b077be4208890e4357": "\\frac{m_{1}c^{2}}{\\sqrt{1-u_1^2/c^2}} +\n\\frac{m_{2}c^{2}}{\\sqrt{1-u_2^2/c^2}} =\n\\frac{m_{1}c^{2}}{\\sqrt{1-v_1^2/c^2}} +\n\\frac{m_{2}c^{2}}{\\sqrt{1-v_2^2/c^2}}=E", "811eb8a9e1f6fb839083ac37a8db43cf": "P_1 = A,\\ P_2 = S,", "811f1c8769bca768220c1fbee1feac68": "\\text{length}(\\gamma)=\\int_a^b \\text{speed}(t) \\, dt.", "811f2736a33932943a1d08b1d3dea733": "v\\mapsto v\\otimes 1.", "811f4b325fb5c67193ed0d9e6b614b6f": "\\|Hu\\|_p \\le C_p\\| u\\|_p", "811f722aa228cf375d340bf663399c98": "[f]([\\mathbf{x}]) :=\n f(\\mathbf{y}) + [J_f](\\mathbf{[x]}) \\cdot ([\\mathbf{x}] - \\mathbf{y})\n", "811f8a8b53be76d76367862a215c9d6e": "Z_t^j = \\sum_{i=1}^ne^{-\\hat{y_i}g_j^{t-1}(\\boldsymbol{x_{j,i}})}", "811f92f824dc5a7a88e0dbf226502161": "\\Bbb{Z}_3[\\rho]", "811f961c7ebf7074d908a20a350e4e1c": "I_n = \\frac{1}{n+1} \\int e^{ax} d(x^{n+1}) , \\!", "811fa419b4b9e632396d085e659aa61a": "\n\\begin{align}\n |z|^2 &{}= z z^*\\\\\n &{}= (\\sqrt{a}\\,x + i \\sqrt{b}\\,y)(\\sqrt{a}\\,x - i \\sqrt{b}\\,y) \\\\\n &{}= ax^2 - i\\sqrt{ab}\\,xy + i\\sqrt{ba}\\,yx - i^2by^2 \\\\\n &{}= ax^2 + by^2 ,\n\\end{align}", "811fd4cbad566f8f52d1b1d9fc5ac94b": " \\mathbf{E} ", "811fdea107445ef6579665e3482fb422": " v_p = a(M_\\mathrm{avg}) + b \\rho, ", "811fe03cd0fb64b13fbd06e9cde3c91f": "\n\\{p_i, p_j\\}_{DB} = x_j p_i - x_i p_j ~.\n", "811fe35b4cd7519c7f62104d29f2f2ea": "g^{i}", "811ff89900761bc3836d6ca006d23d54": "\\displaystyle{g=XDY}", "812011fc837510271fc33c84e9182328": " \n\\int_E \\phi \\, d\\mu_k = \\int_A \\phi \\, d\\mu_k = \\int_{A_k} \\phi \\, d\\mu_k + \\int_{A-A_k} \\phi \\, d\\mu_k.\n", "81207bbaf4f4f14474ddd59d7092dc98": "R[\\Delta^n]", "81208c5b3cca4043be20ab731d383424": "B\\cup\\{0\\}", "8120a6b49952c9c26cfe18b4e18524ac": "q=\\exp\\left( -\\pi\\, \\frac{K'(m)}{K(m)} \\right).", "812103062a2fed284f8210009f86b60c": "A^- + B \\to A^+ + B + 2e^-", "81213a87ae101a08f1dbbcf5f59399a1": "\\sigma A_1,\\dots,\\sigma A_n/\\sigma B", "81217e67b57df7c6c6e88ef6709f2585": "2^n\\approx 10^n", "8121a74e035e3be218af5670777db180": "\\mathbf{\\Rho}=I\\zeta", "8121b7dcceb30ceb5387c9a091f8c63a": "{\\mathit{He}}_n^{[\\alpha]}(x)\\,\\!", "8121d1ce22cc152199beca8217b8e46d": " \\omega = x_n ", "8121d6a44c74403426f0aa5bfe1d13b5": "\\rho(\\mathbf{x},t) = \\rho(\\mathbf{x}) e^{-i \\omega t}", "81220009fbaf8a67a729dfba06a0f578": "E=1/4", "812210299b951de241e2d253dba12124": "\\frac{d}{dt}\\int_{a\\left( t\\right) }^{b\\left( t\\right) }f\\left( t,x\\right) dx= \\int_{a\\left( t\\right) }^{b\\left( t\\right) }\\frac{\\partial f\\left( t,x\\right) }{\\partial t}dx+f\\left( t,b\\left( t\\right) \\right) b^{\\prime }\\left( t\\right) -f\\left( t,a\\left( t\\right) \\right) a^{\\prime }\\left( t\\right)", "81222870afdd3ade1ed9d6f440f9abd5": "+g^{\\mu \\alpha }g^{\\beta \\sigma }(\\Gamma^{\\nu}_{\\alpha \\rho }\\Gamma^{\\rho }_{\\beta \\sigma }+\\Gamma^{\\nu}_{\\beta \\sigma } \\Gamma^{\\rho }_{\\alpha \\rho } - \\Gamma^{\\nu}_{ \\sigma \\rho } \\Gamma^{\\rho }_{\\alpha \\beta } - \\Gamma^{\\nu}_{\\alpha \\beta } \\Gamma^{\\rho }_{ \\sigma \\rho })+", "81223aa0ea3e145a34695caa4689efd7": "u + v = \\int \\frac{du}{dx} \\,dx + \\int \\frac{dv}{dx} \\,dx \\quad \\mbox{(1)}", "81225fb36184267ffd3ba832b8e56f58": "\\sigma_b=\\bold{P}\\cdot\\bold{\\hat{n}}\\,,\\quad \\rho_b = -\\nabla\\cdot\\bold{P}", "81228567b66a6c04e23afa71c47dd2a8": " t_r \\cdot\\omega_0= \\frac{1}{\\sqrt{1-\\zeta^2}}\\left ( \\pi - \\tan^{-1}\\left ( {\\frac{\\sqrt{1-\\zeta^2}}{\\zeta}} \\right )\\right )", "8123451f681ea743e80111705b18571e": " W = E_{\\rm vac} - E_{\\rm F}", "81236b40ae270746fa889a5777f242f1": "\\sin\\beta=n \\Bigl(\\frac{\\omega_1}{\\omega_2}\\Bigr) \\sin(\\theta/2)", "812372ded18d4b9b43433095180f037a": "K\\subset S^3", "8123843a12c65f0dcb636520c8e4874c": " \\forall x \\exist y[Pyx \\and (Czy \\rightarrow Ozx) \\and \\lnot (Pxy \\and (Czx \\rightarrow Ozy))].", "8123e772f89c6f0717dd343689f6929e": "f(t) = \\mathcal{L}^{-1} \\{F\\} = \\mathcal{L}^{-1}_s \\{F(s)\\} := \\frac{1}{2 \\pi i} \\lim_{T\\to\\infty}\\int_{ \\gamma - i T}^{ \\gamma + i T} e^{st} F(s)\\,ds,", "81240b7acea083e3f99aab14a010b927": "\n(Eq. 1) \\text{ } Q_i(t+1) = \\max[Q_i(t) + y_i(t), 0] \n", "81243a9d273db4b5ef659ca685441e9d": "I_{i,k}\\to\\epsilon_i I_{i,k}\\epsilon_k ,\\quad S_i\\to\\epsilon_i S_i ,\\quad S_k\\to \\epsilon_k S_k\\,.", "8124fff2e9118ef44114a3636ce2f48b": "F_k(\\mathbf{a}) = \\sum_{i=1}^n\n\\frac{a_i}{m}\\log{\\frac{a_i}{m}}", "812526acbc4d35c2d05762d4887a61f7": "0 < k < log_2(n)", "8125273e7917f57c634787ab390b4244": "\\mathrm{d}\\boldsymbol{\\ell} ", "8125344f97a3151686b50bfbebc8a138": "\\begin{pmatrix}\\frac{1}{2}&\\frac{1}{2}\\end{pmatrix}\\begin{pmatrix} 0& 1\\\\ 1& 0 \\end{pmatrix}=\\begin{pmatrix}\\frac{1}{2}&\\frac{1}{2}\\end{pmatrix}", "81253874c2bd07dda52ed1ddb48d2747": "\\sum\\epsilon_n = 1 ", "81254290c45105823554715f6c7e6467": "\n\\frac{I+(C*(1-A)+G)+(A*F)}{E+(C*(1-A)+G)+(A*C)}\n", "81255e637cdfd98fa9a30181712c477b": "O(\\log^8 q)", "8125a2fb020b9b70861e05c4814804d1": "0\\leq x \\varepsilon \\sqrt n \\Big) \\to 0 ", "812c8edb0482fc681e35682dce2daccd": "(\\phi,\\pi)", "812cbef1675416b03c897010ce3aba0a": "_{q\\tilde{\\leftarrow}p=p\\tilde{\\leftarrow}q\\ \\Leftrightarrow\\ q=p\\,}\\!", "812d1e204f044a9f4fb8675487054e11": " \\alpha = \\pi - \\omega_2 t_H = \\pi\\left(1 - \\frac{1}{2\\sqrt{2}}\\sqrt{\\left(\\frac{r_1}{r_2}+1\\right)^3}\\right) ", "812d844675937513342cc675c0d0775e": "X^{c}=\\{x\\in E | x\\not \\in X\\}", "812d95b916f9815bb69a67bd093f4b27": "e=1.602 \\times 10^{-19} ", "812da549d8e31f60bdfb37550e135b08": "\n \\{ A_1, A_2, \\cdots, A_n \\}\n", "812de17fc3a0e8020199c4fa5f663983": "x\\le_2 y", "812e22a042b384f1b49a1c0ded659b01": "AJ = JA^T \\, ", "812e49a6ea271cc65a8588a89f7fcf4d": "|\\mathbf{z}+\\rangle", "812e5e307d50a48e74d97113309813c5": "\nV(r_{jk}) = \\alpha r_{jk}^n,\n", "812e6196621f74eaa1e4043c58b32899": "t_{a/2,n-1}", "812ec4701677501ac74f6fd5e7b42e95": " { \\vec x } ", "812f3af1d7aaa12531a09f514643c5d0": " G = 0 \\quad \\hbox{if} \\quad (x,y,z) \\qquad \\hbox{on } S.", "812fcf71a56e68cd4ae37c843983c980": "\\int \\sin (\\ln x)\\;dx = \\frac{x}{2}(\\sin (\\ln x) - \\cos (\\ln x))", "812fed930a71976b535c26aeb05e775e": "Ra,\\ Rb", "8130341a76bb0c56ed2208d4bcea5ccd": "\n\\begin{array}{lcr}\n\\begin{bmatrix} 2 & -1 \\\\ 1 & 0 \\end{bmatrix}, &\n\\begin{bmatrix} 2 & 1 \\\\ 1 & 0 \\end{bmatrix}, &\n\\begin{bmatrix} 1 & 2 \\\\ 0 & 1 \\end{bmatrix},\n\\end{array}\n", "813042913fb2326939f526ccf31bcaaa": "\\forall x \\in N", "81306508d9e2b9efeaeb99de157e353b": " p = ky ", "8130768addcef859c3cd0f6bbe30aab2": "\\bar{2} = m", "813079cfc2509446989faac957c2bb23": " |\\psi (x)|^2", "81309142f1418bd43ceb488f39dba764": "\\displaystyle \\frac{1}{a + i \\nu}", "813095b822cd039ce20ee8f6de6c6e8f": " \\langle A\\rangle=\\frac{\\mbox{Tr}\\, [\\exp(-\\beta H) A]}{\\mbox{Tr}\\, [\\exp(-\\beta H)]}", "813097e9efce01d8c8b1d3e5816594eb": "v_{g} \\approx -15 \\mbox{km/h}", "8130f4a8f45b26a5becca719a4c64b88": "\\langle\\omega, [M]\\rangle = \\int_M \\omega", "8130f5b06a041de3f90f898bbe7f03dc": "S = C_1 \\cup C_2 \\cup ... \\cup C_n", "813125853852a6d21232b3fdf926f39f": "E[\\vec{X}]_{11} = \\frac{f'' \\, g - f' \\, g'}{f \\, g^3}, \\; E[\\vec{X}]_{22} = E[\\vec{X}]_{33} = \\frac{f'}{r \\, f \\, g^2}", "81316242ae736dca93d90ff509efe7fe": "X_t = c + \\varphi X_{t-1}+\\varepsilon_t\\,", "8131a29f711d7c264717c9c2acc487ec": "\n\\begin{bmatrix}\n1 &0 &\\cdots &0 &0 &\\cdots &0 &0 \\\\\n-a(1,2) &1 &\\cdots &0 &0 &\\cdots &0 &0 \\\\\n\\vdots &\\vdots & &\\vdots &\\vdots & &\\vdots &\\vdots \\\\\n-a(1,i) &-a(2,i) &\\cdots &-a(i-1,i) &1 &\\cdots &0 &0 \\\\\n\\vdots &\\vdots & &\\vdots &\\vdots & &\\vdots &\\vdots \\\\\n-a(1,n) &-a(2,n) &\\cdots &-a(i-1,n) &-a(i,n) &\\cdots &-a(n-1,n) &1 \n\\end{bmatrix},\n", "8131b855a2c564c84b4dd7706fbad56c": "\\|x\\|_\\infty = \\sup\\nolimits_i |x_i|", "8131c76e01c78f100b77b46ae05df5e0": "E_{00}", "8131c88670960ffc41627a33c2108e4c": "\n\\begin{align}\ny(t+1) &= y(t)\\left(1+\\gamma-\\left(\\gamma+\\eta\\right)\\Big[1-z(t)\\Big]^{\\lambda}-\\gamma_0\\,\\frac{\\tau}{1-\\tau}\\right),\\\\\np(t+1) &= p(t)\\left(1-\\chi-\\chi_0\\frac{\\tau}{1+\\tau}\\right).\\\\\n\\end{align}\n", "8131d9ded78e3c52bd3aa17287f3975f": " (x, \\sqrt {1 - \\langle x,x \\rangle})\\in R^{n+1}", "81320ab317046b206c48bccc9072a638": "\\eta_{Y_1}\\circ f", "8132bae9b5d3b9fd40617792db669e1a": "w_i = \\Delta x", "8132f4fe7a09c22e5f00ef098d2e3b7d": "X^\\mathrm{opt}", "813301fb5c9bcea228b8014eff5f1a85": "D_B", "813337c13be5c36aaf4833c93bef70e0": "\nR= \\textrm{min}\n\\left\\{\n\\begin{array}{c}\n 3 R_{\\mathrm{T}}/q\\\\\n 2 k\\Delta T/q\n\\end{array}\n\\right.\n", "8133839d017cceaf2f65c593edf03888": "\\hat{A} = \\arg \\max \\ln p(\\mathbf{x}; A)", "813388ae5d6d810ba4654b59f6d2b7d2": " x \\ ", "8133acf0e18bbec6b6380edd7b8a1252": " I_0 = \\frac{\\pi \\Delta}{2 e R_n} ", "8133cedd0003560156f33019cbde0fad": "q_{\\ell}", "8133d4f5f85463140073761e8136e4b2": "~{\\rm besselj}(n,x) ~", "8133fd5c2b26db877dd55d4afc7cff2b": "\\ \\epsilon_2", "8134bf2b1222f7887747bfd5d0fe8748": "f\\, x = f(x)", "8134dbc938f577827dc2d1061c370086": "V = 2 (\\frac{1}{r_{12}} - \\frac{1}{r_{1a}} - \\frac{1}{r_{1b}} - \\frac{1}{r_{2a}} - \\frac{1}{r_{2b}} + \\frac{1}{r_{ab}})", "81353d0168d12bc78c65b126662e270d": "1 = \\frac{x^2+y^2}{r^2 + a^2} + \\frac{z^2}{r^2}", "81355b30e92479f06ad62b51e078b968": "-\\frac{1}{\\kappa_{\\lambda} \\rho} \\frac{dI_{\\lambda}}{ds} = I_{\\lambda} - S_{\\lambda} ", "81356924870ecf873d0c1877529f0928": " \\nabla \\cdot ( \\nabla \\times \\mathbf{H} ) = \\nabla \\cdot \\mathbf{J} + \\frac{\\partial (\\nabla \\cdot \\mathbf{D})}{\\partial t}, ", "81357bad2d80db0bdb3f7cdceb9855ba": "V = x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} \\,", "81358a92a476943481fbcabb6209e230": "x_i\\!", "813656e4ec0491486ccc0c96bdc0669a": " local.impact.of.prod=ext.impact.of.prod", "81365e313a0b6556cc964009fbacfa84": "\\sigma_\\theta = \\sigma_{\\rm long} = \\frac{pr}{2t}", "81366c771ed34172d93f19a62758e7c1": ": X_1, \\dots, X_k:\\,", "813683e36f15142abcdcb5c97402910d": "(i-1)+(n-j)\\leq\\dim\\partial W-1=n-1", "81368c31ec613878c382b61fd1b96c5f": " \\|f\\|_{L^p} = \\|f^*\\|_{L^p}.", "8136ec85bf203742019675e80f0c8366": "\\lambda_2^2=\\lambda_3^2=1/\\lambda", "813704a808ba3d6eb819b12049f5616f": "\\gamma_P(\\mathbf{Cl}) = \\gamma_C(\\mathbf{Cl})\\,\\!", "8137651c2844bf904f2026fb62b2a7e7": "\\textstyle\\left\\lfloor {{d-1} \\over 2}\\right\\rfloor", "8137796e540d1d8d1266ea03e22975a3": " u = A T^4 ", "8137c690c6af9274c7b350ab78ccf507": "S \\to NP \\quad VP", "81380371fa2bf18c4540be05305800b0": " L(\\mathbf{q},\\mathbf{\\dot{q}},t) = T(\\mathbf{\\dot{q}})-V(\\mathbf{q},\\mathbf{\\dot{q}},t) \\,\\!", "813839f67338b0fb0f4eaaa49caa0be7": "\\! e^{i\\omega}=\\cos \\omega + i\\sin \\omega ,", "81383e55bc9b32f1cb6807943f402c4a": "M \\times_{M''} N", "81384114677826856652de5505c46db4": "\\frac{x}{y}=\\frac{1}{d}+\\frac{xd-y}{yd},", "813853f2c07f2bdd968241dff2be7414": "x[-n]", "8138705d219acfe6215cfe9b3aeebbd9": " PaCO_2 = \\frac{0.863 * \\dot{V}_{CO_2}}{\\dot{V}_A}", "813889d8fc62577c4be98db21df3cad0": "T_j", "813896d2762792187d9f8ed1e690fdcb": "\n\\left(1+\\frac{\\Delta L}{L}\\right)^{-\\nu} = 1-\\frac{\\Delta L'}{L}.\n", "8138b7ab4daf096dd32d2cac409b85cd": "G=\\bigoplus_{1\\leq i \\leq n}C_{{p_i}^{m_i}}\\;", "8138c6f7ba1333b86857ab0e47d0c645": "y=\\arctan x\\,\\!", "81397616446af83683b35657ff07b252": "Z(s) = Ls\\, ", "8139b8865ea98475ffcf8e65d3260572": "\n\\frac{\\text{Output}}{\\text{Input}} = K \\frac{1}{1 + s \\tau}\n", "8139bbadc468a6b84ee0774eee39de42": "\\aleph_{\\alpha}", "813a430c267ddbce9e2d1f6b10deed10": " \\sigma = K \\epsilon_p ^n \\,\\! ", "813a66027d5fb7018c9945c7230bc396": " y_t = \\frac{t}{0.2}c\\, \\left[ 0.2969 \\sqrt{\\frac{x}{c}} - 0.1260 \\left(\\frac{x}{c}\\right) - 0.3516 \\left(\\frac{x}{c}\\right)^2 + 0.2843 \\left(\\frac{x}{c}\\right)^3 - 0.1015 \\left( \\frac{x}{c} \\right)^4 \\right],", "813a6acab5d15093930190c62740d136": "Re = \\frac{\\rho u L}{\\mu} = \\frac{uL}{\\nu} \\;,", "813aa3402e67067b59e2575c30e38927": "g^{ii} = g_{ii}^{-1}", "813aed378cd388275fe8fe9ade7b4b31": "j=1,..,n-1", "813b2474d22849cb23610ba90e0bb2a5": "G(s) = \\frac{1}{s} \\frac{6}{s+2}", "813ba4cb88ac800da6fc253247563f9f": "\n \\boldsymbol{\\mathsf{I}} = \\delta_{ik}~\\delta_{jl}~\\mathbf{e}_i\\otimes\\mathbf{e}_j\\otimes\\mathbf{e}_k\\otimes\\mathbf{e}_l\n", "813babf1b4fc39884a62e00df337f1f0": "M_n =\\max(X_1,\\dots,X_n)", "813bbbdddb16e5de3b639dff699dd404": "p_i=\\mathcal{N}(\\boldsymbol\\mu_i,\\,\\boldsymbol\\Sigma_i)", "813bbc7b55f8b32b010244b92992f216": "\\dot{\\sigma} = 0", "813bd92dca64663bf9f8fe87e6f9c65e": "w \\in \\mathcal{F}^n", "813bff01e23d0e14898d02e0293f7a97": "\\frac{\\delta \\mathcal{L}_\\mathrm{G}}{\\delta {T^{ab}}_c} -\\frac{1}{2}{\\sigma_{ab}}^c =0", "813c48a844b70343a92f9f1477ec3d7d": "\nS_{\\gamma \\delta}^{\\;\\;\\; \\alpha} = S_{(\\gamma \\delta)}^{\\;\\;\\;\\;\\;\\; \\alpha} .\n", "813c6b811d68e9b29c647f3f0a939571": "x^{-1/2}", "813d16a7e65301237bc0e8c4c0438e58": " V = x ", "813d4cc8a1dd09e6e3208fb72f611c75": "y' = \\nabla f(x)\\cdot x'", "813d96b50891c36c120e418d7a99b038": "V = k\\, C\\, R^{0.63}\\, S^{0.54}", "813d98f57eb6e18638da441e1b88d9a2": "{U\\over V} = \\frac{8\\pi^5(k_\\mathrm{B}T)^4}{15 (hc)^3},", "813dbc61a68ca66aef445472d4b1d2ee": " (l\\cdot x+m)^2+(a_1\\cdot x+a_3)\\cdot (l\\cdot x+m)=x^3 ", "813de7d73420f9190ab1f64587694f21": "2^{50}", "813e0d135ed01924521f97b62598aaa2": "{{z}_{out}}\\equiv \\frac{{{v}_{test}}}{{{i}_{test}}}", "813e45199d384da107be22e87bc82055": "q^k(y) = y^n + {a^k}_1\\,y^{n-1} + \\cdots + {a^k}_{n-1}\\,y + {a^k}_n \\, ", "813e4cf6072fbf309b0d373682f91d0f": "a_0 < a_1 < ... 0", "814a913d1cb76e573ab3666f33528249": " r_\\mathrm{corr} = gr - \\frac{ g - 1 }{ g } \\sum_{ i = 1 }^g r_i ", "814aa8fa34977432c74a2a82352fadf0": "\\mathit{E}", "814af76c2eadfbe6b93f3aeb369506ba": "G_X(t)", "814b7ab493ccac3ca23b07696b2022ef": "\\begin{matrix} \\frac {1} {n} \\end{matrix}", "814b8bae9d1539af2be4da91904acdb7": "w(z)=P \\left\\{ \\begin{matrix} a & b & c & \\; \\\\ \n\\alpha & \\beta & \\gamma & z \\\\\n\\alpha' & \\beta' & \\gamma' & \\;\n\\end{matrix} \\right\\}", "814b9f64b478b6caf94e723a0e6814c5": "z\\!", "814bf46e2c4627b0f48a46b5666c08c3": "E < mgl", "814c02b9b157c0627014f2b41ff9ac2b": "\\mathcal{O}_{C,P}", "814c44b7f6b7513e05687b8cf578cec1": "H^1\\subset L^2", "814c6663408142a10b454dcbc3148e77": "\\left(\\frac{d y}{d t}\\right)^2 = \\frac{6 A}{5} y^{5/3} + C_0.", "814c85d0ec8b579309b6a9ce490478a8": " x_1 + x_2 = - \\frac{b}{a}, \\quad x_1 x_2 = \\frac{c}{a}.", "814c901c78faae8531dbebff5edaa9f4": "O(|V||E|)", "814ca4f2eeebd11a7389cd5a70264328": "G_i(R) = \\pi_i(B^+\\text{f-gen-Mod}_R)", "814cc42670f4de066ec115a332cabec2": " u(t-\\tau) \\ ", "814d7a403d1a392e6588c9bdf2e7a0cf": "\\sigma_{A}^2\\sigma_{B}^2 \\geq \\left| \\frac{1}{2}\\langle\\{\\hat{A},\\hat{B}\\}\\rangle - \\langle \\hat{A} \\rangle\\langle \\hat{B}\\rangle \\right|^{2}+ \\left|\\frac{1}{2i}\\langle[\\hat{A},\\hat{B}]\\rangle\\right|^{2} ,", "814dede971cafb0c29ccfd268460d198": "\nU(r) = 4\\varepsilon \\left[ \\left(\\frac{\\sigma}{r}\\right)^{12} - \\left(\\frac{\\sigma}{r}\\right)^{6} \\right]\n", "814df2ae2ddd8c03318a6bc65c95f63b": "|\\hat{\\gamma}_3| = \\min(0.99, |(1/n)\\sum{((x_i-\\bar{x})/s)^3}|)", "814e13a60ce3afe7178600649ec67046": "(\\Delta = \\omega - \\omega_0)", "814e26a3e80b155ea86adcb8d62b17f7": "A=\\left(\\begin{matrix} 0 & 0 & 0 \\\\ 0 & 0 & \\frac{1}{\\epsilon} \\\\ 0 & \\frac{1}{\\mu} & 0 \\end{matrix}\\right),", "814e436ad1bcb427521c9bc43880bbaf": "\\omega^{\\mathrm{CK}}_1", "814e4f59ef9ba70b2129f68fd89b4bae": "\\mathbf{u},\\mathbf{y}", "814e57949e6adf6bf09acde6bb86feff": "\\chi_\\mathrm{red}^2", "814e783b1ba80bda5b20e43f77844376": " \\mbox{recall}=\\frac{|\\{\\mbox{relevant documents}\\}\\cap\\{\\mbox{retrieved documents}\\}|}{|\\{\\mbox{relevant documents}\\}|} ", "814e976f0e76ec87ffc8d6032000bbfc": "\\oint\\frac{\\delta Q}{T}\\leq 0", "814ed30f8a2fdba69c6fa5a2cbb2c2bf": "\\ p=0.03 ", "814eeaca0a6ea46ec26b445d7f7311c5": "l_i\\in S", "814ef29d7118abdb7b1db0b5ed593bb4": " {= 5 + 11 + 19 + 9 + 8 + 13 + 4 + 1 + 24 + 15 + 23 + 18}", "814f1455381adb135d057b211dee6f61": "\\begin{align}\n \\dot{\\tilde{\\mu}}_u^{(i)} & =D\\tilde{\\mu}^{(u,i)}-\\partial_u \\tilde{\\varepsilon}^{(i)}\\cdot \\Pi^{(i)}\\tilde{\\varepsilon}^{(i)}\n-\\Pi^{(i+1)}\\tilde{\\varepsilon}_u^{(i+1)} \\\\ \n \\dot{\\tilde{\\mu}}_x^{(i)} & =D\\tilde{\\mu}^{(x,i)}-\\partial_x \\tilde{\\varepsilon}^{(i)}\\cdot \\Pi^{(i)}\\tilde{\\varepsilon}^{(i)} \\\\ \n \\\\\n \\tilde{\\varepsilon}_u^{(i)} & =\\tilde{\\mu}_u^{(i-1)} -\\tilde{g}^{(i)} \\\\ \n \\tilde{\\varepsilon}_x^{(i)} & =D\\tilde{\\mu}_x^{(i)} -\\tilde{f}^{(i)}\n \\end{align}", "814f749f14b0b743fe24216da3c638bd": "\\mathcal{H}_{1}=(p_{1}-A_{1})^{2}+(m_{1}+S_{1})^{2}=p_{1}^{2}+m_{1}^{2}+\\Phi _{1}\\approx 0 ", "814fb7dfbe8ef7465fa34404826ecdea": "h_n^{(1)}(x) = j_n(x) + i y_n(x) \\, ", "815001562f3ac04e84117ac095e6de7f": "\\hat v_n(x)", "81502f86282493421ff2690e450f2cf6": "Q = 111\\bar{1}1\\bar{1}1\\bar{1}", "81503ac2bc5b5cbe3383978886aedef7": "+kT \\,{\\rm Tr}_{1,2,..,N}P^{(N)}_{0}(\\xi_{1},\\xi_{2},...,\\xi_{N})\\log P^{(N)}_{0}(\\xi_{1},\\xi_{2},...,\\xi_{N})", "81503ec96884f6ca56fe6942d08b7e95": "g^{\\uparrow \\downarrow}", "8150bdf7acee758a7d042535bf47b338": "A=\\frac{1}{2}(b+d)a\\, ,", "815111d9fca5a60611dd93569b9be4bb": "\\chi_k", "81515e46d0018bb1945c345f559e8b34": "\\boldsymbol{p} = p", "8151ca6dc2123d5b9acd155843ed9fc7": "\\mbox{Inventory Turnover}=\\frac{\\mbox{Cost of Goods Sold}}{\\mbox{Average Inventory}} ", "8151f43a7eca6607f790e198b2dba355": "y''=y-y'\\,", "8151fd09291d0a220862e862d73d3e30": "(\\tfrac{1}{4},\\tfrac{1}{2})", "81525b6410a3ee536d757fde97775710": "S:H\\to H", "815279f1d28a81d35771488c4ef82987": "\\vec j ,", "8152cc0b0213468480e58ec69e6033e1": "m = 1.0", "8152d035cde7aa960001e74ebb7472f8": "\ng'_{k+1} (x) \\equiv \\begin{cases}\n g'_k(u) & x \\notin \\gamma \\\\\n 0 & x \\in \\gamma\n\\end{cases}\n", "8152f69a20b278cbcd94142ad868cfb0": "\\sigma\\in {\\mathfrak G}", "8152f9246bafeccc1e8d7d7ff052fd1f": "1/8 = 1/25 \\times 25/8 = 1/5 \\times 25/40 = 1/5 \\times (3/5 + 1/40) ", "8152fcf7c8610bdfbddc79e359b29fb9": "g(\\alpha) = \\alpha^5 + \\alpha^2", "815393ca2c9bc5ebee6ecd9816e48163": "n\\log(p)", "81539a5241b77e468cb2ed46cba05b22": "\\Iota", "8153c1e6730064cfca151203a5b774ea": "X_1 = \\frac{\\partial}{\\partial x^1},\\dots,X_n=\\frac{\\partial}{\\partial x^n}.", "8153cb598ddb542d50441dbf21b3fbc7": "\\psi_{\\lambda} ", "8153d53dd73b217059813976a53f8326": "W_R \\cdot (W_R B+(W_R B)_R) =W_R \\cdot (W_R B+W B_R) = W_R^2 B+WW_R B_R", "8153e779d69fe6c35dc2bc4dfb0f2a11": "U(S,V)", "8154151388e603a3e5431b4dc2fed4d9": "\\Phi \\!", "815495851673ba4358f78971b19f9411": "v(\\lambda) = \\lambda\\ f(\\lambda).\\,", "8154bbd2348f896f45cd951e35b3fe5a": "Q + k(2mP)", "8154d68d5ce1ba8e66649e4630f1317e": "R_1(x)", "8154fac53bd4691095183dc69c058898": "j^a", "815517944ec85c7b4864cd157b3e6f81": "\\underline{\\varphi(\\beta / \\alpha)}\\,\\!", "81551a6b72c469dc975d9e5bcf415e19": "B_v = \\xi\\ B_\\text{crit} = \\xi\\ 2 \\sqrt{(A(\\omega) + M) C}", "81554526719e386c5a2d5820f6a11435": " \\frac{\\Pr(\\overline{\\Sigma}_{t}=A)}{\\Pr(\\overline{\\Sigma}_{t}=-A)}=e^{At}.", "81558ab6fe6441d790b1ff55697378de": "1645 \\cdot 990 \\cdot 36 = 58,627,800", "8155a90125357188c6eac94f3e3ff65a": "L f = \\frac {d}{dx} \\left( p(x) \\frac {df}{dx} \\right) + q(x) f. ", "815621a2d608ae591836e257407a2393": "G(a,b) < A(a,b)", "81562f537f2b99d776fe71dd1173dc3a": "\\scriptstyle{V =} \\tfrac{2}{3} \\scriptstyle{Bh}", "8156d4e5d0f26942ae842340d96f9b61": "l = 2Me", "8156ee6afb10ccaa8e32100435e778b4": "\nS = - {\\Delta V \\over \\Delta T}\n", "8157093e5a12518f966b98229d20a880": "F_n(K,0,0)=\\frac{n^2K^2}{1+K^2/2}\n\\left[J_{\\frac{n+1}{2}}(Z)-J_{\\frac{n-1}{2}}(Z) \\right ]^2\n", "815713a68eb1ba60ce258cc0a74c8b23": "\\tau\\frac{d\\bar{E}}{dt}=-\\bar{E}+(1-r\\bar{E})S_e[kc_1\\bar{E}(t)+kP(t)]", "81571ca30e53485de0b875b5df8883dd": "t(G)", "81573e424897cbaea0a8b4238f0c34bd": "\\delta_{ext}:Q \\times X \\rightarrow S ", "81580db2382b6a14395b4e48525ea5af": "f'(x+h) > 0", "81586d620fa40be1448b40cb07822b5a": "\\tau(t)", "81587d80600fe6cf8b62d53089f3d04c": " U_\\alpha (f) = S_\\theta (f) + \\alpha \\left\\langle f , \\frac{\\theta}{z} \\right\\rangle, ", "81590811ec60576277d5dc398c6420e2": " \ny\\in \\mathbb{R}_{+}^{K}", "81591a5c05356f8085f3017cce56d9fe": " R_2 = \\frac {{Z_0}^2 -{R_1}^2}{2R_1} ", "8159796e17723ab4c12824e55afa6c4b": " E(u)=\\int_\\Omega F\\left(u(x) \\right)dx ", "815989da8f16b645fda10e919cf0f0a2": " A = \\begin{bmatrix} 1 & 2 & 3 & 4 \\\\ 5 & 6 & 7 & 8 \\\\ 9 & 10 & 11 & 12 \\\\ 13 & 14 & 15 & 16 \\end{bmatrix}. ", "8159eb4dc734d019f88e358be4a979a5": "\\text{Gain}=10 \\log { \\left( \\frac { {I_\\mathrm{out}}^2 R_\\mathrm{out}} { {I_\\mathrm{in}}^2 R_\\mathrm{in} } \\right) } \\ \\mathrm{dB}", "8159f1246576cf4d82b99ddeddbf24ae": "\n W = C_{1}~(\\bar{I}_1 - 3 - 2\\ln J) + D_1~(J-1)^2 \n ", "815a6bf32bf8c57e8cc8f5122beec5f2": "X_0= {\\rm grad} \\frac{1}{T}\\ , \\;\\;\\; X_i= - {\\rm grad} \\frac{\\mu_i}{T}\\; (i >0) ,", "815a6e58bd785c920e42f59a5eb54bb8": "W_{\\tau_2}", "815a7d93c871e9e45d5358ada4012299": " s = |\\mathbf{a}| \\cos \\theta .", "815aa81c15ff46b9571ebd5b782e283f": "[0; d_1+1, d_2, \\ldots, d_n, \\ldots]", "815ab9b6012eae3a0dada7cdb57fc579": "dS_w=0\\,", "815b3751ccfdd0ce8941d664f6131ead": " S_{(2,1,1)} (x_1, x_2, x_3) = \\frac{1}{\\Delta} \\;\n\\det \\left[ \\begin{matrix} x_1^4 & x_2^4 & x_3^4 \\\\ x_1^2 & x_2^2 & x_3^2 \\\\ x_1 & x_2 & x_3 \\end{matrix}\n\\right] = x_1 \\, x_2 \\, x_3 \\, (x_1 + x_2 + x_3) ", "815bc36f657ab5f5737ef6b1f6363c08": "y_{n}\\equiv m_{i} (y_{n}^{(i)})\\pmod {p_{i}}", "815bcd7a21d4c5175b738711559a32fa": "Q=V \\cdot \\rho_{q,0}.", "815c250a17a7c063cfcb15b9fbd12681": "\\mbox{IF}\\cdot \\mbox{SC}= \\frac{1}{2}(1+f(t))\\cdot (\\cos(\\omega_{s} t-\\omega_{I} t)+\\cos(\\omega_{s}t +\\omega_{I} t))", "815c3bd0b4514edc5e36174adbe69248": "{\\Delta}A=\\epsilon^{HG}[HG]b+\\epsilon^{G}[G]b-\\epsilon^{G}[G]_0b\\,", "815c66c4fc3168fa0a94aa9b21d4a945": " \\begin{align} \n&\\lim_{\\beta\\to 0} \\operatorname{var}(X) =\\lim_{\\alpha \\to 0} \\operatorname{var}(X) =\\lim_{\\beta\\to \\infty} \\operatorname{var}(X) =\\lim_{\\alpha \\to \\infty} \\operatorname{var}(X) = \\lim_{\\nu \\to \\infty} \\operatorname{var}(X) =\\lim_{\\mu \\to 0} \\operatorname{var}(X) =\\lim_{\\mu \\to 1} \\operatorname{var}(X) = 0\\\\\n&\\lim_{\\nu \\to 0} \\operatorname{var}(X) = \\mu (1-\\mu)\n\\end{align}", "815c7f9ae6efe3124dc084203895458f": "\n{\\mathbf A} = \\left ( \\begin{matrix}\nw^2 & w^2 & 0 & 0 & 0 & 0 \\\\ \n1 & w^2 & 0 & 0 & 0 & 0 \\\\ \n1 & 1 & w^2 & w^2 & 0 & 0 \\\\ \nw & 1 & 1 & w^2 & 0 & 0 \\\\ \n0 & w^2 & w^2 & w^2 & 0 & w \\\\ \nw^2 & 1 & w^2 & 0 & w^2 & 0 \\end{matrix} \\right )\n", "815c833d3d8034fbbab04d755211e452": "I_1, I_2, I_3", "815cd95bec8ebec0b01fa3dd27243ebc": "\\scriptstyle 0\\, \\le \\,s\\, \\le \\,2\\pi", "815d30b29cdba1cea140149e09da464f": "\\beta_i \\overset{\\text{def}}= \\alpha_i^2", "815d92dde29d1575068545420a3fc89c": "Q(x|y) = Q(y|x)", "815d9ce9884c1f4f02d75ad1b8a98587": "\n\te_4 = 0.1666428611718905\n", "815df0fc30db53229fc553c845f2f998": "\\langle U^2x, U^2y \\rangle = \\overline{\\langle Uy, Ux \\rangle} = \\langle x, y \\rangle .", "815df16419f90f1de639b73e584f8b7a": "P^a = m_0 U^a = \\left[\\frac{E}{c}, p_x, p_y, p_z\\right]", "815df79f5eb5092f330b28ec30105aea": " a,b \\in \\mathbb{Z}_{m} ", "815e0b9a537648727a6923e21c6fe3ae": "2^\\ell \\leq \\text{poly}(2^k \\delta^{-1}) \\cdot \\log n", "815e0bebc59eda2d01638b89c72c3801": "\\,C", "815e3b581d8bbdc5df3853e2cbc757b7": "x = \\mathrm{laea}_x\\left(\\frac\\lambda 2, \\phi\\right)", "815e73e60b6758985fe86755c80bc0bf": " \\Gamma = \\gamma_1 + \\gamma_2 + \\cdots + \\gamma_n.", "815f00ebed697ab48922fe4f00714ab1": "\\hat{C}_1", "815f38a9455b7199daf53acfb1642292": "Q(x) = 1", "815fc468e7ceaac1ed0b5664278214de": "f(x) = \\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}(x-x_{j})^{2}", "815fe3c573599ddf7dfc983c4215b80a": "\\begin{align}\\varphi_Y(t)&=\\operatorname{E}\\left(e^{it\\sum_{k=1}^2 X_k}\\right)=\\operatorname{E}\\left(\\prod_{k=1}^2 e^{itX_k}\\right)\\\\\n&=\\prod_{k=1}^2 \\operatorname{E}\\left(e^{itX_k}\\right)=\\prod_{k=1}^2 \\left(1-p+pe^{it}\\right)\\\\\n&=\\left(1-p+pe^{it}\\right)^2=\\varphi_Z(t)\\end{align}", "816006badbcb49605ac41a0b1bde8143": "C_P \\,\\ ", "81607793c246e6be9f82bc707972ff75": "\\delta^c_a", "8160955fbd6d20e0cbcc9c7403213584": "\\mathit{i}", "8160d2cf6749d4fc0081ffc2c137af51": "\\Delta U_{system}=Q", "8161cca3ab960017b07d4bfcca8f25cc": "\\alpha|0\\rangle_{1} + \\beta|1\\rangle_{2}\\longrightarrow \\big(\\alpha|0\\rangle_{1} + \\beta|1\\rangle_{2}\\big)\\otimes|\\psi\\rangle.", "8161e95af224bee700fead7c692b21a2": "\\neg_{i,j}^u", "8161effb16bf608602d04c7e9e9ddbb3": "P_{j}(d_{j})=1-e^{-\\frac{d_{j}^{2}}{2s_{j}^{2}}}", "81623ba9ce21d5b4c1ef05760dd18dc4": "s=(a+b+c)/2,", "81625865670f43502fe6071c9d8ba1c7": "\\scriptstyle\\lim", "8162e6e09c78929bd72fe13fc603da99": "\\frac{df}{dz}=g", "816370d2dae0e58205e8b36aafed828b": "\ng(n) = \n\\frac {-1}{f(1)} \\sum_\\stackrel{d\\,\\mid \\,n} {d < n}\nf\\left(\\frac{n}{d}\\right) g(d).\n", "8163f7d2c1f754090c00e3b3e7c56e8b": "\\forall i\\in\\{1,...,m\\}, \\mathbb{P}_S\\{\\sup_{z\\in Z}|V(f_S,z_i)-V(f_{S^{|i}},z_i)|\\leq\\beta_{CV}\\}\\geq1-\\delta_{CV}", "81642020934cbd0158a8fe6f2f380609": "\\det\\left(\\begin{bmatrix} 3 & -4\\\\4 & -7 \\end{bmatrix} - \\lambda\\begin{bmatrix} 1 & 0\\\\0 & 1 \\end{bmatrix}\\right)", "81642b36a60f765517b04d737c357752": "c_1 = \\frac{\\hat X [1]-c_0}{1-z_0z_1^{-1}},", "81646511c2b14d9fb6733ec278d79617": "-\\frac{1}{2 t} \\int_{c_R}^{c^*} x \\mathrm{d}c = \\left[ D(c)\\left(\\frac{\\mathrm{d}c}{\\mathrm{d}x}\\right)\\right]_{c=c_R}^{c=c^*}", "8164969e99227e1a5f23afafdc53331a": "e^{2\\pi i \\frac{r}{p^n}}\\;", "8164a164eebdb09c737c5cbbfee584e6": "g(xP)\\ \\stackrel{\\text{def}}{=}\\ f(x),\\,", "8164a72e4e6faadaede1dfa5959f7ed5": "+\\frac{1}{4*3*2*1}n*(n-1)*(n-2)*(n-3)*d", "8164eab71e25f01b0acc2c0ad7027620": "C^d\\equiv (M^e)^d\\equiv M^{(ed)}\\equiv M \\bmod N", "8164ebb76407343a1061678d2361574f": " \\int_\\gamma \\langle V(x), \\mathrm{d}x \\rangle = \\int_\\gamma \\langle \\nabla f(x), \\mathrm{d}x \\rangle = f(\\gamma(1)) - f(\\gamma(0)).", "81650bc8bd3ea495a5c6a080123950e2": "\\frac{\\partial}{\\partial x_i} \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_j(\\vec{r}')d\\tau'} - \\frac{\\partial}{\\partial x_j} \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_i(\\vec{r}')d\\tau'}", "8165582ef7b132853232a0d60d32c406": "a^2 + b^2 = c^2.\\ ", "81657e830452f6884e1e9da08b893721": "y(x) := [z(x)]^{\\frac{1}{1-\\alpha}}", "8165b5c609095b669aa5eaaba04dbe30": "p_{d} =", "8165c225f0c8de6124d74bca26a29f7f": "b_i(a_{-i})", "8165e359323806bd996444b7d85e766f": "\\operatorname{E}[L] = \\operatorname{E}[v^{K(x)+1} - P\\ddot{a}_{\\overline{K(x)+1|}}]", "81662d01574fb840ac4fd6be6ed876cf": " = n \\ln(r(\\cos \\varphi + i\\sin \\varphi)) ", "8166e1f3f326ee2a5e5ee8cf3ca92541": " a_{02} = p_2p_5, ", "8167b76df8b3fe758708a3fda1fce303": "\nG^{\\mathrm{R}}_{\\alpha\\beta}(\\omega) = \\int_{-\\infty}^{\\infty} \\frac{\\mathrm{d}\\omega'}{2\\pi}\n\\frac{\\rho_{\\alpha\\beta}(\\omega')}{-(\\omega+\\mathrm{i}\\eta)+\\omega'}.\n", "8167de35e54923191834f287f1125716": "h : X \\to A\\Rightarrow B", "81684095786123efae292581b409c0da": "\nI = I_1 = I_2 = \\dots = I_n\n", "8168ac5250694536b8f8b574c4207f86": "\\sum_{p,q=0,0}^{m,n}a_{p,q}y(i-p,j-q) = \\sum_{p,q=0,0}^{m,n}b_{p,q}x(i-p,j-q)", "8168d8715a8282c0f0ffe6e440d39470": " \\tilde{a}_{jk}=a_{jk},\\quad j+k=2; \\tilde{a}_{10}=-a_{10}+2\\partial_x a_{20}+\\partial_y\na_{11}, \\tilde{a}_{01}=-a_{01}+\\partial_x a_{11}+2\\partial_y a_{02},", "81692443dadf123780ad3d1a2d7df663": "{2 \\choose 1}_q = \\frac{1-q^2}{1-q}=1+q", "816932cc02bd63bc706293af04d9971a": "\n \\quad (2) \\qquad \\epsilon_j^{n + 1} = \\epsilon_j^n + r \\left(\\epsilon_{j + 1}^n - 2 \\epsilon_j^n + \\epsilon_{j - 1}^n \\right)\n", "81693e0f7e11426eb8402efac3d3711a": "\\mathbf{F} = {1 \\over 4\\pi\\varepsilon_0}{q Q \\over r^2}\\mathbf{\\hat{r}},", "8169a3ad3157abbc70faa92278de576c": "\\{z \\in \\mathbb{C} | |\\phi(z)|< 1 \\}", "8169d128194ad22786360224c3215c23": "d_{ij} = \\min \\begin{cases} d_{i-1, j} + w_\\mathrm{ins}(b_{i-1}) \\\\ d_{i,j-1} + w_\\mathrm{del}(a_{j-1}) \\\\ d_{i-1,j-1} + w_\\mathrm{sub}(a_{j-1}, b_{i-1}) \\end{cases}", "816a09c803e4d063ecd1d392cdf72757": "\\rho_I (t)=e^{i H_{0, S} ~t / \\hbar} \\rho_S (t) e^{-i H_{0, S}~ t / \\hbar}", "816a1ef2d1a00580e55f85d3bee6e0f3": "x_i - x_{i-1}", "816a4431dac4f9afe420717f2ad34c04": "V_{\\rm C} \\propto V_{\\rm S}{}^{(1/\\gamma)}", "816a8d2187480f2a7b050e68a34c5526": "\\Omega_m h^2", "816af982cc5bcaf00fc79a682f40a477": " P = P_0 + \\varepsilon_0 \\chi^{(1)} E + \\varepsilon_0 \\chi^{(2)} E^2 + \\varepsilon_0 \\chi^{(3)} E^3 + \\cdots. ", "816b2be6746047164830b8a2d938135d": "\\uparrow\\uparrow", "816b91e0192ac562cbc84792ded759b9": "U(\\$100) > U(\\$0)", "816b988074c4577185761b548b8a2935": "\\begin{align}\ne^{\\pi \\sqrt{19}} &\\approx 12^3(3^2-1)^3+744-0.22\\\\\ne^{\\pi \\sqrt{43}} &\\approx 12^3(9^2-1)^3+744-0.00022\\\\\ne^{\\pi \\sqrt{67}} &\\approx 12^3(21^2-1)^3+744-0.0000013\\\\\ne^{\\pi \\sqrt{163}} &\\approx 12^3(231^2-1)^3+744-0.00000000000075\n\\end{align}\n", "816bc4a0914c63ec353dd89e031c8780": "\\mathit{n_q}", "816bd7e9bfff4cba1cf62af618366e9a": " T \\mapsto \\mathrm{tr}(T\\cdot ) ", "816c0481b7979c2b22f7975201c0b57e": "b=-1", "816c1fb81237bd28f8711937cefebda4": "\\nu_m = \\left(\\frac{3N}{ 4 \\pi V }\\right)^{1/3}v_s", "816c2bcbe4c72be9c2ca182430d646cf": "y(x) \\sim e^{S(x)}\\,", "816c2cb843a3f0e28704e6a863273dd5": "c_i+c_j=c_{i+j}", "816c46daca016b9f1c87cded829bfc9e": "W =\n\n (\\mathbf{E} \\cdot \\, \\mathbf{r})=\\mathbf{F_E} \\cdot \\, \\mathbf{r}", "816ca4594114b4198c766908784473fb": "M_j(-1/\\tau) = \\sqrt{\\tau/7i}\\sum_{k=1}^32\\sin(6\\pi jk/7)M_k(\\tau)", "816cc051440e47ab6271d3aa7950d4b2": "\\sum_{a=0}^{q-1} \\gamma(a,q)=\\gamma,", "816cc0c2a5626902e272b77ca830b73d": " G^{-1} = \\frac 2{2-K} + \\frac k{4\\pi}", "816cc87283be99b221cae1b2f7664f3c": " = 2(\\sinh 2K \\; \\sinh 2L)^{-N/2} Z_N(K,L) ", "816cf1404d2fee8238ad7d923a4eccf4": " 2_+^{1+8}\\cdot A_9", "816d200d8dd10048cfd3acda16f2187b": " IMM(s)=\\{i \\in D| t_{si} - t_{ei} = ta(s) \\} ", "816d238a6f601f68b0225b08eb74a005": "\\rho(X+Y) \\leq \\rho(X) + \\rho(Y)", "816d5af46500c47c3dd1a5313d3356f2": "y_n,y_m \\in C_n", "816d74034962edb5f1e9a4623adde409": "(\\Phi^Y_t \\Phi^X_s) (x) =(\\Phi^X_{s}\\, \\Phi^Y_t)(x)", "816d788ff404001b4bfb45bb77311990": "\\begin{align}\n\\mathbf{e}_+ & = -\\frac{1}{\\sqrt{2}} \\mathbf{e}_x -\\frac{i}{\\sqrt{2}}\\mathbf{e}_y \\\\\n\\mathbf{e}_{-} & = +\\frac{1}{\\sqrt{2}}\\mathbf{e}_x - \\frac{i}{\\sqrt{2}}\\mathbf{e}_y \\\\\n\\end{align} \\quad \\rightleftharpoons \\quad \\mathbf{e}_\\pm = \\mp\\frac{1}{\\sqrt{2}}\\left(\\mathbf{e}_x \\pm i\\mathbf{e}_y\\right)\\,", "816df83616b265994dcef6a55cfbbd61": " \n\\approx n_A\\ln\\left(\\frac{n_A}{\\ell_A}\\sqrt{\\frac{E_A}{\\ell_A}}\\right)+\nn_B\\ln\\left(\\frac{n_B}{\\ell_B}\\sqrt{\\frac{E_B}{\\ell_B}}\\right)+\nN\\ln N + const.", "816e3e7fe0026830dfdc1b442e421bb6": " \\begin{align} \n\\mathbf{\\hat{p}} & = \\bold{\\hat{P}} - q\\bold{A} \\\\\n & = -i \\hbar \\nabla - q\\bold{A} \\\\\n\\end{align}\\,\\!", "816e419f5640a49a84b9f38af67c4d60": "P_c^{(2)}(0) = P_c^{(3)}(0)", "816e77063d09b3990070eb5c977ce2ba": "Profit=PY-WL-RK", "816e834645a5f6e603d8368733dbf102": "\\Gamma_{i}(0)-\\Gamma_{i}(d) = \\Gamma_{e}(0)\\left(\\mathrm{e}^{\\alpha d}-1\\right)\\qquad\\qquad(3)", "816eb04af493ee7f0ea6d00b758c0d79": "\\rho(X) = \\rho(Y)", "816ec4afdc4045d9874af62fd8aaa46a": "x_1(s)=x_2(t)", "816ef635c7ab092335ab8e8a8d115276": "\\{1,1,1\\}.", "816fa1083d86f6a9fcd20eb4e9929b21": " \\langle\\Psi|\\hat{O}|\\Psi\\rangle, \\langle\\Psi|\\hat{O}|\\Psi_{m}^{p}\\rangle,\\ \\mathrm{and}\\ \\langle\\Psi|\\hat{O}|\\Psi_{mn}^{pq}\\rangle.", "816fa1e518a4ec5270dc0a85a42d8d76": "t=\\pm r", "816fd4c310129a0c9f94c1650ea6e22b": "[x,y,z]=x-y+z", "8170171d052ca2f095d54a2e9fc6f177": "C' : y'^2=\\bar{f}(x')", "81703af205b27aa6528f85d21b2b567f": "\\mathbf{Z}/p", "81704254f8c7a4d9aedfabe157f0f497": "\\left|\\alpha_i-\\frac{p_i}q\\right|\\le\\frac1{qN^{1/d}}.", "817061f62f5b73e27a0904ff7151eba4": "{a \\over R_{\\bigodot}} \\cdot {m_{planet} \\over m_{\\bigodot}} > 1 \\; \\Rightarrow \\; {a \\cdot m_{planet}} > {R_{\\bigodot} \\cdot m_{\\bigodot}} \\approx 2.3 \\times 10^{11} \\; m_{Earth} \\; \\mbox{km} \\approx 1530 \\; m_{Earth} \\; \\mbox{AU}", "817072bfc1e625ad1d10f81fece55b29": "R=UV^*.\\,\\!", "8170df6d3c1b997cfc00b938f8f36dd1": "\\text{abs}", "81714a3b2d96801ffd9f08036eef1ec4": "\\begin{pmatrix}\n0 & 1\\\\\n1 & 0\n\\end{pmatrix}", "81717876418fb58e17c949ea9b03cf3c": " \\sigma_E ~ \\frac{\\sigma_B}{\\left | \\frac{\\mathrm{d}\\langle \\hat B \\rangle}{\\mathrm{d}t}\\right |} \\ge \\frac{\\hbar}{2}, ", "81717e417d0975acd0225dc7bf791fc6": "\n{\\mathcal L}_G=-\\frac{1}{16\\pi G}\\left(R+2\\Lambda\\right)\\sqrt{-g},\n", "8171bb03d5bf81eca3d8011c9e2bd275": " s+\\sum_{l=1}^{n-1}lk_{l} = n - 1.", "8171ead9ec502f8e497cffa01aa5b4ba": "\\left|\\alpha-\\frac{p}{q}\\right| < \\frac{1}{q^2}\\,.", "8171f68a915f8e417792730441cf9f04": "\\mathbf{I}=\\int_V \\rho(\\mathbf{r}) (\\hat r)^2 \\, dV,", "8172109704020f1c13e6043fff712d79": "\\beta F\\left(K^{*}, L^{*}\\right) = \\beta F\\left(K,L\\right) + \\frac{1}{2}\\log\\left[\\sinh\\left(2K\\right)\\sinh\\left(2L\\right)\\right]", "8172130159f523136a1cd8df32e0db0c": "= x(p) = x \\,", "8172397d6278d98a2da932157614c8d7": "\\frac{u_{i}^{n + 1} - u_{i}^{n}}{\\Delta t} = \\frac{a}{2 (\\Delta x)^2}\\left(\n(u_{i + 1}^{n + 1} - 2 u_{i}^{n + 1} + u_{i - 1}^{n + 1}) + \n(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n})\n\\right)", "81723b24ab19f514f0d50ce42be00f2a": "\\zeta(s,q)=\\frac{1}{s-1}\n\\sum_{n=0}^\\infty \\frac{1}{n+1}\n\\sum_{k=0}^n (-1)^k {n \\choose k} (q+k)^{1-s}.", "81724fd8483fc1f8e5052f874208a773": "X^\\epsilon = \\{y\\mid |x\\cdot y - 1| \\le\\varepsilon \\text{ for all } x\\in X\\}.", "81728764c645b1cc56d7e244572effc8": "\\sum_{i=1}^{k} \\sum_{w \\in V} f_i(s_i,w)", "8172a603670b5d12c2752f31f9d0e7f3": " [B] = \\begin{bmatrix} 0 & -b_z & b_y \\\\ b_z & 0 & -b_x \\\\ -b_y & b_x & 0 \\end{bmatrix}.", "8172bcc0e6bed83d963218dc4c7e8602": "S_\\eta = \\frac{1}{\\beta}\\sum_{i\\omega} g(i\\omega)", "8172cd507ea4ef45b7eef006c8a17d55": "X+Y \\sim B(n+m, p).\\,", "8173345a98e56a46de572f836cf10802": " D = \\frac{ 2A }{ 2A + B + C } ", "81733494e0a964b506e21c93d67e2b1b": "2^{-n}", "817340b0d7ce8eb3742536c5221a941c": "1 \\over 20", "8173a11e21e36b78b3f0edf8827cfe36": "\\mathbf{f}(\\mathbf{v}) = \\mathbf{f}_1(\\mathbf{v}) + \\mathbf{f}_2(\\mathbf{v})", "8173cfc69c6da3a24c8256ecb45cb888": " \\Lambda = 4\\pi n \\lambda_D^3 ", "8173f74b80036ef2b66cc811879a99a3": "\\widehat{\\mathbb{Z}}", "81740e5577561d05b2d75e89769f8177": "\\phi_a \\left(\\omega \\right) = \\mbox{Arg} \\left(1 - a e^{-i \\omega} \\right)", "817427ff49debedbd26cf450b7224a41": "F(x) = \\Pr(X \\le x) = p.\\,", "8174428dda89ee80f000ce6677059e37": "v_{Al}=0", "8174f8705682dbba5f49cf1df0203935": "\\|\\mathbf{x}\\| := \\sqrt{\\mathbf{x} \\cdot \\mathbf{x}}.", "81752463763811c3e0c30c4f31c19e12": "(b_i)_{i\\in I}", "8175949c25384767c69c87d01900dd8c": "A_\\text{ellipse} = \\pi a b", "8175978b501fef748f65871aed378f6a": "E_{l} = \\frac{2l+5}{4(l+1)},", "8176180c4a8a09f2103bcf2af4f4ed13": "\\mathcal{D}_{\\mu} = \\partial_{\\mu}+\\Omega_{\\mu} \\times ,", "817636552c8ee2a8586d90b9cb938c9a": "\\operatorname{MSE}(S^2_{n-1})=\\operatorname{E}((S^2_{n-1}-\\sigma^2)^2)=\\frac{2}{n - 1}\\sigma^4", "817674f6aa2d6d6b5ba92e8ba2739b12": " 10^{-5} ", "81767d5da03e01e00037a74bc92df33f": "C_{V,m}=\\frac{3R}{2}+R=\\frac{5R}{2}=2.5R", "81768b8736700eb98ab4a0ad82df0a59": "\\left[x_1^{(1)} \\right] \\supset \\cdots \\supset \\left[x_1^{(k)} \\right], \\cdots ,\n\\left[x_n^{(1)} \\right] \\supset \\cdots \\supset \\left[x_n^{(k)} \\right]\n", "817717621c643979cb054f1142502d2f": "H=\\int d^3x \\left[{1\\over 2}\\pi^2+{1\\over 2}(\\nabla \\phi)^2+{m^2\\over 2}\\phi^2\\right].", "817742e03bf804871da411cb67008ef4": " f = -1 + 10x + 6x^2 + x^3 ", "817791caace3243150dbf1591cf0c5e9": "\\Psi_nx", "8177d4d50fbccad27f3aeae79eae6790": "P(\\mathbf{s}|spike)", "8177e0b3faff04c352afc42a57cc4b89": "\\sin \\theta \\simeq \\theta, \\quad \\theta \\ll 1 \\,", "8177ec13f49eebe642226e67a13e3648": "(x - k)", "81781acdf66cd055c22bdd772a409401": "s_i:\\,U_i\\to\\, PX.", "81781eb22ed84cd79bd7f732313af865": " (\\Omega, \\mathcal{F}, \\mathbb{P}) ", "81788f2392c3f304a59c0b6c645997a8": "\\begin{bmatrix} b_{11} & b_{12} \\\\ b_{21} & b_{22} \\end{bmatrix}", "8178b201469e2d8e3246bf9fe94d7e10": "U={\\bold 1}'\\otimes\\dots\\otimes{\\bold 1}'", "81791be2ef9b73caa79e6304ecb33ab0": "1 - 1 + 1 - \\cdots = \\frac{1}{2}\\ (\\Re)", "81791df1862f742d021e135469ee7171": "n \\times B", "817996ebb4d6bb2f02651bee3907acf8": "\\left(\\tfrac a\\cdot\\right)", "8179a96b614e1530627948b372e761eb": "\n (\\boldsymbol{\\nabla}\\cdot\\boldsymbol{S})\\cdot\\mathbf{a} = \\boldsymbol{\\nabla}\\cdot(\\boldsymbol{S}\\cdot\\mathbf{a})\n ", "8179e3859083b939c7f21d42e806d461": "\\scriptstyle\\pi(n)", "8179eef272595387f536a9bc62dc74ec": "ND(X_1,\\ldots,X_n) = \\frac{D(X_1,\\ldots,X_n)}{H(X_1,\\ldots,X_n)} .", "817a0fae40e540a1d995618655de50ee": "{WI}", "817a5d1e47e1fcae96f3ae7637624133": "(w\\in L' \\Leftrightarrow f(w)\\in L)", "817a73fef66ec9e0f75f35bd1b8477ea": "\\mathbb{Z}[\\pi_1(X)]", "817ac7dbffe6ffbecde38875760698a2": "(\\mathrm{id}_{\\alpha}, \\mathrm{id}_{\\beta})", "817aef5903881f4178a4dc84fc7a83e1": "\\scriptstyle {\\varphi \\choose 1}", "817b1f45e9ed66afd5456c4bcfa923cf": "e^- + H_2O \\longrightarrow O^+ + H_2 + 2e^-", "817b57c29211536a8bcd9092e7f691b7": "\\sigma_a(k)", "817b8f5ad334baaef89215b3f0510fa3": "g:\\mathbb R^{k+1}\\mapsto \\mathbb R", "817bc0e9907031ef451cb59ad3033b99": "\\alpha_{1}(a,\\, b)", "817bc96a1ed5aa04786ef4d75f9cae91": "w_i = \\frac{n'_i}{n_i}", "817bdd16b6549c599e9dabd9ec590a63": "e^-", "817c09b67a7c41ea6b6712c3e383e387": "\\mathbb{C}^+ = \\{z \\in \\mathbb{C} | {\\rm Im}(z) > 0\\}", "817c1ff4ae507f6e84e944276bfc02c0": " \\dot{z}(t) = \\frac{h(z(t))}{\\eta(\\varepsilon)} \\dot{u}(t) \\left\\{A(\\varepsilon) - \\nu(\\varepsilon)\\left[\\beta\\operatorname{sign}(\\dot{u}(t))|z(t)|^{n-1} z(t) + \\gamma |z(t)|^n \\right] \\right\\} ", "817c48a596502855578fa1fc4c67501c": "\\operatorname{Pic}(X)", "817c4e058301f315a442d99bc236a6c8": "R_L(x)=\\max_{y} \\max_{x^* \\in X^*} \\{ \\|y-x^*\\|_L : f(y)\\leq f(x) \\}", "817c760b7b61fa39ca83c8cecbcac296": "(\\dot x,\\,\\dot y)", "817c95134776c86fde9f4bb95269fa53": "\\frac{1}{z_2 - T} = \\frac{1}{z_1 - T} \\cdot \\frac{1}{1 - \\frac{z_1 - z_2}{z_1 -T}}", "817cbec64347527631931188be34fee0": "SubCipher_1=ENC_{f_1}(k_{f_1},P) ", "817cc862385e6cde2b10fd499a358782": "T^{\\mu\\nu} \\,", "817cebb78317abffbd5766e2ce8b9840": "\\hat{t}(\\tau) = \\max \\{ t \\in [0, T] | \\hat{\\tau}(t) \\leq \\tau \\}.", "817cf8d060e7cc440859045356a08530": "\\mathbf{A} = \\frac{\\mathbf{J} + i \\mathbf{K}}{2}\\,,\\quad \\mathbf{B} = \\frac{\\mathbf{J} - i \\mathbf{K}}{2}\\,.", "817d21e43cfb23e35dbadecdd8d28623": "U=\\sum^{i=1}_{i=np} w_i\\left(y_i^{observed} - y_i^{calculated}\\right)^2", "817d4c04a9fcc646c1440ec2bcae0f12": "a\\uparrow\\uparrow\\uparrow b=A(4,b)", "817d54e4e4f8f3c19097b390e09ecd40": "R = \\exp([\\omega]_\\times \\theta) = \\sum_{k=0}^\\infty\\frac{([\\omega]_\\times\\theta)^k}{k!} = I + [\\omega]_\\times \\theta + \\frac{1}{2!}([\\omega]_\\times\\theta)^2 + \\frac{1}{3!}([\\omega]_\\times\\theta)^3 + \\cdots", "817d67fdeae16d0996854a5c69bf0dd1": "-x \\ ", "817d7c09a84725b7e28efe9b96d03822": "\\tilde{H}_{ij}(y) = H_{ij}(y)/H_{rr}(y)", "817db2b0b610930d9732654cd3469a2b": " \\ \\theta ", "817db6f1a3402f9812bc980a26d4025b": " \\left (\\frac {\\partial L}{\\partial \\mu}\\right)_{T, V, Area} = \\, -N", "817dbf82639554952cb666ddfb3d86f3": "\\psi(\\alpha) = \\varepsilon_\\alpha = \\phi_1(\\alpha)", "817df544e8529aaf5076045772a4bb88": "\\phi(x_{\\pi(1)})\\cdots \\phi(x_{\\pi(n)})|\\Omega\\rangle", "817f34d7976039caba8ed6b7054773c2": "\\forall x \\, A(x) \\Rightarrow A(a/x),", "817f6abe288c5ff155263921bcc223f4": "(d_1,e_1,f_1) \\cdot (d_2,e_2,f_2) = (d_1^{e_2}d_2^{e_1}, e_1e_2, [d_1,d_2]^{1+e_1e_2}f_1^{e_2}f_2^{e_1}) \\ . ", "817f992c14ed5713d398b68f1d595522": "s\\approx 0.01L ", "817fb48d7ceac2731abaf96e1d017eaa": "\nR(16.26^\\circ) = \n\\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix} = \\begin{bmatrix}\n0.96 & -0.28 \\\\\n0.28 & \\;\\;\\,0.96 \\\\\n\\end{bmatrix} \\qquad (\\text{rotation by }16.26^\\circ )", "817fe73ec9b3348d8f52f3ba5e7bff85": "gS", "817ff2f46ed219943bd310baf268e42c": "F_{\\alpha\\beta}", "817ffe3af2bec8988754cb1ebfc268b1": "\\displaystyle \\Beta(\\alpha,\\beta)=\\int_0^1 t^{\\alpha-1}(1-t)^{\\beta-1} \\, dt ", "818015f5f1b02d3f15fdd2c2efee3513": "L(s,\\chi)=\\prod_p\\left(1-\\chi(p)p^{-s}\\right)^{-1}\\text{ for }\\text{Re}(s) > 1,", "81804799ce2f10b12bb26e6c289c7a50": " (1 - 2^n) (0:0:0:1) + 2^n (X:Y:Z:1) \\ ", "81806344483eb0e7e7374c014a39583c": "K_p", "818066815c2524206d8dd66f6ed5055e": "\\begin{matrix}\nc_{\\nu_j}|\\dots,1,\\dots \\rang= |\\dots,0,\\dots \\rang & c_{\\nu_j}|\\dots,0,\\dots \\rang=0 \\\\\n{c^{\\dagger}}_{\\nu_j}|\\dots,0,\\dots \\rang=|\\dots,1,\\dots\\rang& {c^{\\dagger}}_{\\nu_j}|\\dots,1,\\dots \\rang=0\n\n\\end{matrix}", "81807dc2a6108a8498899c162d2bc7a7": " \\frac{b-a}{6} (f_0 + 4 f_1 + f_2) ", "81807f57c0c275b3946274569b0c2321": " \\epsilon=\\frac{T}{\\delta}\\delta^{p+1}=T\\delta^p", "8180ab2d3eccbb375c9a13477d81f7dc": "{\\sigma_i}^2\\,", "818133d6381cd796b2a14afa97e38460": "\n\\langle M_{ap}^2 \\rangle (\\theta) = \\int_0^{2\\theta} \\frac{\\phi d\\phi}{\\theta^2} \n\\left[\\xi_{++}(\\phi)+\\xi_{\\times\\times}(\\phi)\\right] T_+\\left(\\frac{\\phi}{\\theta}\\right)\n= \\int_0^{2\\theta} \\frac{\\phi d\\phi}{\\theta^2} \n\\left[\\xi_{++}(\\phi)-\\xi_{\\times\\times}(\\phi)\\right] T_-\\left(\\frac{\\phi}{\\theta}\\right)\n", "8181ce8b88a343dfb755a2e849ebc19f": " G_1 = b_1 Z^{-1} \\, ", "8181df3e57e37b5dae21c8e451b0ca7f": "\\mu \\colon T^{2} \\to T", "81829eb72e92e089abefd16c1158d1d5": "F_1 P_1=S_1 F_2\\,", "8182a50f97d59500aa80f44d1d8f5182": "2a^2= \\frac {1}{R_c s_o}", "8182e620622f22e6c8f83009181271e2": " \\, P(x;\\;y_0,\\;y_1,\\ldots,\\;y_n)=\\sum_{(a_0,\\;a_1,\\ldots,\\;a_n)} A_{(a_0,\\;a_1,\\ldots,\\;a_n)}(x)\\cdot(y_0)^{a_0}\\cdot(y_1)^{a_1}\\cdot\\ldots\\cdot(y_n)^{a_n}\\!", "8182ef0f9307542e92ab8af9ecab6868": " a = a_1 + i a_2 ", "81830b5e58d23e1228152c8303623447": "x \\not\\in L \\Rightarrow \\mathrm{Pr}[A\\,\\mathrm{accepts}\\,x] \\le 1/2", "81831b066d03f59ab4d1ace8fbe2a9c5": "b(n)={n\\choose (n-3)/2}(2^{-(n+3)/2} - n^2 2^{-n-6} - n2^{-n+3}),", "81832d2a174b8a2b0c74ed30b89f28ec": "x = \\sin t + 2 \\sin 2t", "81838927250c1e3ceefc96ddc3797f63": "F(\\mathbf{k})= 1 / \\Omega", "8183bc36a8a814b1f04ce91b13221186": " y = Ak ", "8183d8535f60508e6f61b661b19536fa": "\\sum F_i=0", "81843524c3ea8951ef7bb15866ed0b5a": "\\rho_{AB}\\,", "818470bfa974408ba2095df99f2f58d8": "T, p, \\{N_i\\}", "8184949d2201622180fd241d3cb6137a": "f(V_j)=0\\,", "818538746fe64a1c8bfa83b6c585c4d5": "F(s)\n\n=\\int_{-\\infty}^\\infty f(r)\\,dx\n\n=2\\int_s^\\infty \\frac{f(r)r\\,dr}{\\sqrt{r^2-s^2}}", "81858c92121900780d40223a46e1d888": "\\hbox{d}A_1", "81859f69a68f29e87bcae8505d8ed06a": "\\begin{bmatrix}\na & b & c & d & e \\\\\nb & f & g & h & d \\\\\nc & g & i & g & c \\\\\nd & h & g & f & b \\\\\ne & d & c & b & a \\end{bmatrix}.", "8185b36f413e1d6dc139f1bb26bdc88b": "\\textstyle \\sin \\pi - \\theta = \\sin \\theta", "81861bf1fceaef3c67fc0daeff357a8f": "(\\Sigma \\cup N)^{*} N (\\Sigma \\cup N)^{*} \\rightarrow (\\Sigma \\cup N)^{*} ", "81863eb47db97899b4eb944435c40345": " \\frac{\\partial \\Pi}{\\partial \\Delta X_{kl}} = 0 ", "81864793bc171f0eec5fb6ce194a99ac": "MMOS_n(r_1(r),r_2(s)", "818648f4aeb83a295080234d9a3054aa": "\n\\frac{x^{2} + y^{2}}{a^{2} \\cos^{2} \\nu} - \n\\frac{z^{2}}{a^{2} \\sin^{2} \\nu} = \\cosh^{2} \\mu - \\sinh^{2} \\mu = 1\n", "818677f406a08f065756d3a86791a675": "n_{a1}(\\mathbf{k})", "81867c14e69e1050bcab49f228a0f6ae": "\n\\mathbf{\\hat{b}_{t:T}}(i) =\n\\frac{\\mathbf{b_{t:T}}(i)}{\\prod_{s=t+1}^T c_s}\n", "8186aa720978dd8f102321ee417e45f6": "P (x,y,z,\\theta,\\phi,\\lambda,t)", "8186af662a714453f7f0161cd36c0b8e": "L_B", "8186db112717e69dba3c6383b796a957": "\\delta \\colon [a, b] \\to (0, \\infty),\\,", "81872f486ebc25642719a8d156f307ec": "\\displaystyle\\overline{d(\\lambda)}\\tilde{F}(\\lambda)", "818734189b968c5ae2416b3d0c5e7afe": "y \\le 0.992 - x", "81877aac686b34cb35f36ef1ab115f70": "x \\times 0 = 0", "818787ceeeab9ddf4f492f77f52e1a37": "\\begin{align} \\bigl\\||f|^{1/p}\\bigr\\|_1 &= \\bigl\\||fg|^{1/p}\\,|g|^{-1/p}\\bigr\\|_1\\\\\n&\\le \\bigl\\||fg|^{1/p}\\bigr\\|_p\\,\\bigl\\||g|^{-1/p}\\bigr\\|_q\n=\\|fg\\|_1^{1/p}\\,\\bigl\\||g|^{-1/(p-1)}\\bigr\\|_1^{(p-1)/p}.\\end{align}", "8187a8327ecc973737828c127e2fdd02": "P({\\rm unknot})=1", "8187fd9ba4d9370ba7852792cf437d07": "c\\le s+2", "818817b6c8b5bf8a840f22687725fc3d": "i = 1,\\ldots,m", "81881a0f79d70413bbca47b3437644ab": "\\mathrm{d} s = \\left(\\frac{\\partial s}{\\partial v}\\right)_T \\mathrm{d} v.", "81889a64772dc7dcfc0c62095ebf0033": "P(\\overline{R_1}\\cos(\\overline{\\theta_1}),\\overline{R_1}\\sin(\\overline{\\theta_1}))\\overline{R_1}d\\overline{R_1}d\\overline{\\theta_1}", "81890b3361252971395e208419f3b7ec": "\\mathcal{E}_1", "818947d9b6eb107ab445c6000610d90d": "\\zeta _{max}=2^{-M} \\frac{\\pi \\mbox{GCD}(\\Delta F,2^W)}{\\sin \\left( \\pi \\cdot 2^{-P}\\mbox{GCD}(\\Delta F,2^W) \\right)}", "8189846f3341900ea731b17001e9bfe0": "q_T = \\sqrt{q_x^2 + q_y^2} \\,", "8189c0f5eb235eae399ce7c2f3a05997": "\\operatorname{pf}\\begin{bmatrix}\n0 & a_1\\\\ -a_1 & 0 & b_1\\\\ 0 & -b_1 &0 & a_2 \\\\ 0 & 0 & -a_2 &\\ddots&\\ddots\\\\\n&&&\\ddots&&b_{n-1}\\\\\n&&&&-b_{n-1}&0&a_n\\\\\n&&&&&-a_n&0\n\\end{bmatrix} = a_1 a_2\\cdots a_n.", "8189c5c14f7f679f11b62e14c7a5fe43": "(x_0, y_0),\\ldots,(x_{k}, y_{k})", "8189cc49f540649052ccce43e8760b24": " S_h(f)(x) = f(x+h) \\, ", "8189d4ee4ce432444bf8863a03571975": "T(n) = 2 T\\left(\\frac{n}{2}\\right) + O(\\log n)", "8189d5256b9718f1c8065744af692c7b": "x = \\frac{u}{u^2+v^2+w^2},\\quad y = \\frac{v}{u^2+v^2+w^2},\\quad z = \\frac{w}{u^2+v^2+w^2}.", "818a440d8b89a1d3d21be5b065f42a42": " \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)}) = \\mathrm{Re} (\\mathbf{E}_0^* e^{-i(k z - \\omega t)}),", "818a7a7822dd7b6eba0b2c297861fd45": "\\mathrm{Re}(\\gamma) = \\mathrm{Im}(\\tilde{k}) = \\alpha_{abs}/2", "818a8109cc8a546666103228f88a117f": "x_1 = \\alpha \\cos(\\chi/\\alpha),", "818a8d65e96bef7ba139c14065124ead": "a_1+a_2+a_3 \\leq b_1+b_2+b_3", "818acfe83db608012e1f5274d781d99f": "{\\tilde{A}}_{7}", "818ad98dea44f39f0f4b3942b7e1e304": "\n[d(\\rho, \\rho+d\\rho)]^2 = \n\\frac{1}{2} \\mbox{tr}\\left[ \\rho \\frac{L_{\\mu} L_{\\nu} + L_{\\nu} L_{\\mu}}{2} \\right] d \\theta^{\\mu} d\\theta^{\\nu}\n", "818b1a576b86f05e766979f9671becad": "m = 2790^{2753} \\; \\operatorname{mod}\\; 3233 = 65 ", "818b2fb93d096101190ed6787a323dc6": "a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = (ababababab^{-1})^6 = 1, \\,", "818b9e87aa76b49096d1d7e939d84051": "\\theta^i(v)=\\langle e_i,v\\rangle", "818b9ed491e79a3e8d245739a711d2b9": "\\scriptstyle A \\or B", "818be6cc5c5572d02c2d5ad9b2885025": "\n\\begin{align}\n& x + y = a,\\ x - y = b,\\ xy = c, x^2 + y^2 = d, \\\\[8pt]\n& x^2 - y^2 = e,\\ x^3 + y^3 = f,\\ x^3 - y^3 = g\n\\end{align}\n", "818bfbecf121fc5f8c0c75a5558fe702": "C^{cand} := \\emptyset", "818c4eb83eeb6f4a31e11f5bf613167c": "=\\frac{1}{N}\\sum_n \\langle n|H|n\\rangle+\\frac{1}{N}\\sum_n \\langle n-1|H|n\\rangle e^{+ika}+\\frac{1}{N}\\sum_n\\langle n+1|H|n\\rangle e^{-ika}", "818c54b1cc2cd9be2a067c79ec48ac82": "Y \\subseteq X^+", "818c5d05a6050b186de6b9c48aa9e13b": "\\langle B_0, \\langle B_1 \\ldots \\langle B_{n-1}, z\\rangle \\ldots \\rangle \\rangle ", "818c937fc5ce08be658f8a01bc788501": "T_c = \\sqrt{\\frac{9}{16 \\pi f_d^2}}", "818c9ef44caca0edec20c7c7f8ef3704": "\\textstyle \\leq \\lambda l", "818d0420a77094d062d5cebc19630fba": "a \\mathrm{Tr}[ F \\wedge F ]", "818d12fb631c8769a4df425954015187": "X_u \\perp\\!\\!\\!\\perp X_v \\mid X_{V \\setminus \\{u,v\\}} \\quad \\text{if } \\{u,v\\} \\notin E", "818d34df50b2009d0c3ebe3afcc6546f": "\\displaystyle \\partial_t u + \\partial_x^3 u + \\partial_x f(u) = 0", "818d44877c33cf491bc70c205a0c291d": "n \\over 2", "818dd0eed668d7d9b2d95aa0817b1bcd": "\\frac{d^{2}Y}{dy^{2}}+k_{y}^{2}Y=0 \\ \\ \\ \\ \\ \\ \\ \\ (18) ", "818de9bff9a3e862f7a9a92d8436c177": " \\bar{X_i} \\sim \\mathcal{N}_p \\left(\\mu_i, \\Sigma_i/n_i \\right), ", "818e7f446ccfdf5906b5315e00c5d7d7": "X \\underline{\\bowtie}_{\\mathbb{F}} Y", "818ea06df8f1c2596cc03a99cca95302": "\\pi^S_k(X)", "818eb0b637cfbf422b0a682e0f38df72": "k_B = R / N_A\\,", "818eb453c92ceb1e42c85029e1afbf97": " \\mathbb{P} (Y \\le y|X) = \\begin{cases}\n 0 &\\text{for } X^2 \\ge 1-y^2 \\text{ and } y<0,\\\\\n \\frac12 + \\frac1{\\pi} \\arcsin \\frac{ y }{ \\sqrt{1-X^2} } &\\text{for } X^2 < 1-y^2,\\\\\n 1 &\\text{for } X^2 \\ge 1-y^2 \\text{ and } y>0.\n\\end{cases} ", "818ee0dc2b98bb9827b724169be9fcb5": "u = 3 \\frac{c^2}{1-c^2} \\operatorname{sech}^2 \\frac12 \\left( cx - \\frac{ct}{1-c^2} + \\delta \\right),", "818ef1a570344cbfeb3b363d34efb01f": "{v_{air}}", "818f1cc78286b0c6be5e49cd83615c72": "1+r\\sum_{i=0}^{(g-3)/2}(r-1)^i", "818f292d9977cfb0e4f12fa7668b319d": "\\det(B)=\\pm1", "818f922a7c31c0cc47da6c341b7c9ee2": "\n0=0 +\n\\int \\mathrm{d}^4y \\delta^4(x-y)\n \\langle 0|j(y)|p\\rangle;\n\\quad\\Leftrightarrow\\quad\n\\langle 0|j(x)|p\\rangle=0\n", "818f9371e1803817d057bb9faa292a8b": "b_{\\nu, n}(0) = \\delta_{\\nu, 0}", "818fb0fbc8a58259f9dd6678eabeb91b": "B_i (1,2)", "819003fb6a41c203a918326a0acea3fb": "q_f^{}", "8190c68219097b06d14f3ba47f20331d": "\\scriptstyle \\left(u_i\\right)_{i \\in I} \\,\\in\\, \\prod_{i \\in I}k", "8190c9c96711f9398ec507a52e3d1514": "\\sup_{x \\isin X}", "81917f39b1c92e52631378bedd37692f": "\\mathfrak{p}_i = \\mathfrak{p}'_i \\cap A", "81918263ff65a25f9aab20b7ac89ee71": "S(\\vec{r},\\hat{s},t)", "8191b320a7bb150778eb4a605aa5041e": " = \\int\\limits_{-\\infty}^\\infty f(t)\\cos\\,{2\\pi \\nu t} \\,dt - i \\int\\limits_{-\\infty}^\\infty f(t)\\sin\\,{2\\pi \\nu t}\\,dt,", "8191ce075efb6e409b9ca02cf4f03288": "R_i F(M) = H_i F(E_*),", "8192080825303157aeda8df22dfb516e": "F(x)=\\frac{p(x)}{\\prod_{j=0;\\,j\\ne k}^n(x-z_j)}", "819226af0dacfa23174eaae82ba0cdef": " \\langle L,L\\rangle \\equiv 0", "81922ee7a1040b82a215f2ecc897ac1b": "B(y;s)\\subseteq B(x;r)", "81923664036cc86c664c3b7c2f1929a4": " \\frac{\\hbar^{2}}{2m\\psi} \\left( \\nabla \\psi \\right)^{2} - U\\psi = \\frac{\\hbar}{i} \\frac{\\partial \\psi}{\\partial t} ", "819238c42435812614a329d7e55cbe72": "F_\\beta = \\frac{(1 + \\beta^2) \\cdot (\\mathrm{precision} \\cdot \\mathrm{recall})}{(\\beta^2 \\cdot \\mathrm{precision} + \\mathrm{recall})}\\,", "81929bad6529d26438b6bf62f117743b": "M_{2,n}", "8192a977f9ab184a73f68dfb844b046f": "{v^2\\over r}= {g\\tan \\theta}", "8192f1bc9fb189c85c57154eb4643ad5": "{n+k-1 \\choose k-1}.", "819346a13583b907a8c93d72bc00fa44": "\\partial\\mathcal{L}/\\partial(\\partial_b A_a) = F^{ab}", "819363dad8b6aa3da624ea876d5a7956": "u < X \\leq v\\,.", "81936531a827b8bdb78554a170072ff8": "\\overline{\\Delta}= \\{z \\in \\mathbb{C}\\,|\\, |z|\\le 1\\} ", "8193ba7fb681615ab525286e5aa5ddba": "\\mathbb{Z}[\\pi_1 (X)]", "819484a2275eb9f438e3accaa77e0fd6": "\\,=(G'WG)^{-1}\\Big(G'W\\Omega WG - G'WG(G'\\Omega^{-1}G)^{-1}G'WG\\Big)(G'WG)^{-1}", "81949783913318a0bd40a77d9a2b9e80": "\\mathbf{x}_{k+1}\\leftarrow \\mathbf{x}_k+\\gamma(\\mathbf{s}_k-\\mathbf{x}_k)", "8194a2c35aa7d1a7784720b464344777": "\nP_\\mathrm{avg} = \\frac{\\Delta W}{\\Delta t}\\,.\n", "8194b20b3dc51a1a620a152b299b5080": "L^1(\\mathbb{T})", "8194f233185ebf061b769bcbfecdb0a7": "\\, S=\\gamma_0 A/4\\gamma.\\!", "81951bbb171baa20228e0e335d955a71": "CD = K \\times \\sqrt{CV_a^2 + CV_i^2}", "819543be2773e1df4901518ada74bd7a": "\\pi R S", "819568186936ff229ca39399e23fbde3": "V_x: \\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix} \\mapsto \\begin{pmatrix}-x\\\\y\\\\z\\end{pmatrix},", "8195905f4bb8112145259a2c4fd21315": "J_+ \\equiv J_x + i J_y, \\quad J_- \\equiv J_x - i J_y", "8195b32adec08631be9b99435c63f3a3": "\\left(\\prod_{i=1}^{d-1}\\alpha^{ck_i}\\right)\\det\\begin{pmatrix}\n1 & 1 & \\cdots & 1 \\\\\n\\alpha^{k_1} & \\alpha^{k_2} & \\cdots & \\alpha^{k_{d-1}} \\\\\n\\vdots & \\vdots && \\vdots \\\\\n\\alpha^{(d-2)k_1} & \\alpha^{(d-2)k_2} & \\cdots & \\alpha^{(d-2)k_{d-1}} \\\\\n\\end{pmatrix} = \\left(\\prod_{i=1}^{d-1}\\alpha^{ck_i}\\right) \\det(V).", "8195d33a4c83b64e1c3e72072e0e5f90": "\\ R\\ ", "8195eaac244b791fec53973f5f1e23a5": "z_{n+1} = \\mathcal M z_n = \\gamma z_n \\left(1 - z_n\\right),", "81960fd6e729f664278db90735dc3cd1": "= {1 \\over 2} ([F, G]^{IJ} + [\\mp i * F, G]^{IJ})", "81961596f2377bfde2c726a79ec8d662": "H_r = \\lbrace p \\ :\\ w = r \\rbrace ,", "81964ed9b48e033a3d694fd24c5c1657": "\\sqrt{n-1}\\,s/\\sigma", "8196dcea2831206e30c4dd5ca0cf2488": "C\\in\\mathrm{GL}_n (\\mathbb{C})", "819719fd9e2cf4cde391d973159d16e8": "q_{jk}x^k y^j", "81975fef056067f01e06402d94fb09d2": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{5}{\\sqrt{6}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm1\\right)", "8197976d90fe6282008cf1549ef6b148": " \\log(y) = f(X) + \\mathrm{er} ", "8197986614810d10d4648b24cc8bda26": "{A_i}'", "819835ee4b56483794ace833528bc248": " f(z) = \\frac{\\alpha - i\\beta}{2}z + \\frac{\\alpha + i\\beta}{2}\\bar{z} + \\eta(z) z\\, ", "8198aa496d9d1519479b430aaa1a1d85": " \\mathcal I_{\\Sigma} ", "8198b05c6534dbbf0e40a50389091fd5": "L = 2\\alpha_{0}l", "819923fd52f495fa78278ec95b2e8b7a": " G_F^2 T^5 \\sim \\sqrt{G T^4}", "819968744d78ebb4871cce765f21f51a": "\\lambda > 0, \\nu \\geq 0", "8199d981b7b2bb28be34b42de03211f9": "\\mathbf{y_1}+\\mathbf{y_2}=\\mathbf{x_1}+\\mathbf{x_2}+\\mathbf{e_3} \\in \\mathbf{C_2}", "819a3c99bd75227585ce55cc70ea7732": "\\mathcal{H}_{int} = - \\sum_{\\alpha} c_{\\alpha} Q_{\\alpha} u(x-x_{\\alpha})", "819a562cfed8c08cf2a8ec981bcf6a50": "T_{game}", "819a7b4f84f28b0e496526ffbfaaee47": "\n\\left[Q(\\mathbf{k}),Q^\\dagger(\\mathbf{l})\\right]\n= \\delta(\\mathbf{k}-\\mathbf{l})- \\Delta(\\mathbf{k},\\mathbf{l}). \\quad\\quad (8)\n", "819b273fd2c730b3682b81c8bb4ee053": " \\int_{\\gamma} g(\\zeta) d \\zeta", "819b538d88954e1618a6d68b0d7d7660": "\\Omega \\in C(\\alpha,\\rho)", "819b7551e087b2056e66b9c270766364": "\n\\lambda_1 = \\mathrm{E}X\n", "819b8ef2b9458961c7ed74639698d37b": "x_\\mathrm{a}(t) = A(t)e^{j\\phi(t)},\\,", "819ba09a5b6a755b2d738bbad15f9771": " du = \\frac{h}{k T} \\, d\\nu ", "819be69606321c07f116a1e07248cefe": " \\theta \\in (0,\\infty) ", "819c754503d59664e4b3390243274741": "f(k;s,N)=\\frac{1/k^s}{\\sum_{n=1}^N (1/n^s)}.", "819c788ca13bd51018a48b2e18de6e8a": "P(\\theta)", "819c9be9f35b51b6934264e6c79d55fa": "\\{ 2 t + 1 ~|~ t \\in \\mathbf Z\\}", "819d0d7f63b5fa04c6a7e4ac1a6e011b": " r = 0 ", "819d3aef443524c1c8b3a87cdcbeeda5": "Tr[\\Phi]=0", "819d9a3219abc0477c625ea8a0ab6c5b": "\\frac{dy}{dt} + \\alpha y = 0, y(1)=2, y(2)=1", "819db34e6369500afbc04e7fbc61def5": "\n\\Delta (x_{\\perp })=\\Delta _{\\mathcal{L}}+\\Delta _{\\mathcal{G}}.\n", "819e0171837dfa616443677967f07bed": "\\Gamma(3/2) = (1/2) \\cdot \\Gamma(1/2) = \\sqrt{\\pi}/2", "819e09717309975636cb6d32d0ba3dcc": "\\Omega_1(t) =\\int_0^t A(t_1)\\,dt_1,", "819e228e7ba35b38ff89c698eb7c0f2e": " \\lim_{x \\to c} f(x) = 0,\\ \\lim_{x \\to c} g(x) = \\infty \\! ", "819e5720969e862cd3dc97874da8e273": "\n W^{oo} = W \\,\n ", "819eabe67358eac0e0e8568a7a6ed8be": "f(a).", "819f599ac8d27df1cceabce478c2ca84": "\\; - \\sum_j p_j \\log p_j,", "819f728066f19094c77ce00715cc25f1": "{D}_{5}^{(1)}", "819f7adfafbf9ebec0feef853fa650d8": " \\sinh z = - i\\sin iz = z \\prod_{n=1}^{\\infty} \\left(1 + \\frac{z^2}{n^2\\pi^2}\\right). ", "819f92840b1a799973ab2d1e7d50a2ec": "a_{A} \\frac{(A - 2Z)^{2}}{A}", "819f97bac6dbe83d6f3580f44cf08c87": "\\tan\\left(\\frac{\\pi}{2} + n\\pi\\right) = \\infty\\text{ for }n \\in \\mathbb{Z},", "819fa9b5e3765e6db457fbfd406365ed": " \\mathbf{} \\rho ", "819fe88cf251cd4346765de1c32236dd": "pV=nRT=\\operatorname{constant}", "819ff266849bce5a9ab219ccc39da16d": "\n\\text{Tr}\\left\\{ \\Pi_{\\rho,\\delta}^{n}\\rho^{\\otimes n}\\right\\} \n\\geq1-\\epsilon,", "81a02c4eed2fff35d37d39166e35cf63": "\\sigma_\\mathrm{n} \\,\\!", "81a072b847a5aa1c72632209d6b7364c": "V = E_f d + E_e\\left(2\\lambda\\right)", "81a07d720f047f8b55dd365a6d8511ca": " L = 0.2 \\, f^{0.3} \\, d^{0.6}", "81a0818aa6756e750499cc745740533e": " X \\mapsto (X^{-1})^T ", "81a110aaf093c5a18e5aa40f26051131": "\\prod X_i", "81a235b44440997c13c4fd1307a135bc": "\n\\int\\limits_{0}^{1}dy(1-y^{\\beta })^{1/\\alpha }=\\frac{1}{\\beta }\n\\int\\limits_{0}^{1}dzz^{\\frac{1}{\\beta }-1}(1-z)^{\\frac{1}{\\alpha }}=\\frac{1\n}{\\beta }\\Beta \\left(\\frac{1}{\\beta },\\frac{1}{\\alpha }+1\\right). \n", "81a249c47b7d6efb33355ebf0c074fb5": "x'^2 + y'^2 + z'^2 = r'^2.", "81a2e2eeb358f1accb6e7f5e8ad9b6ff": "h < \\frac{2a}{\\epsilon u}", "81a2e79b4a6a4431852ca185dc1b381f": "X_{2\\pi}(\\omega) = \\pi \\sum_{k=-\\infty}^{\\infty} \\left[ \\delta (\\omega - a - 2\\pi k) + \\delta (\\omega + a + 2\\pi k) \\right]", "81a2e9972801f6d9901dfd5fa1383e3a": "\n\\hat{\\mathbf{k}} = P_\\mathbf{\\mathbf{k}} - \\bar{P}_{\\mathbf{k}},\n", "81a31a9f12fe29388299e0d0092707ef": "\\sum A_{exp}", "81a32404e36e0d57be05b62687f3c117": "\\widehat{\\mathbf{C}}", "81a3af531c8db5da7bc00a8a653810c6": "\\displaystyle{g_3^2=g_1(-1) ,\\,\\, g_3 g_1(a)g_3^{-1}=g_1(a^{-1}),\\,\\, g_1(a) g_2(b) g_1(a)^{-1}=g_2(a^{-2}b),\\,\\, g_1(a)=g_3 g_2(a^{-1}) g_3 g_2(a) g_3 g_2(a^{-1}).}", "81a419fe095ad9cd396af9a38ac4953c": "\\ F(aK,aL)b", "81b1cb05b40b69d117bac21785d21bb0": "\n g\\;(f\\;a\\;b\\;c)\\;:\\;E \\to F\n", "81b1da507ce698e0c2dc27b8aceee800": "m \\geq 4", "81b233698d7eb53e27393bee6e627b34": "\\hat\\nu_1", "81b234bd172fdb0fb93663e51421f68d": "V : \\mathbf{S} \\rightarrow \\mathbf{R}", "81b315bbbff97ee64161543129ecbb32": "A \\rightarrow ((B \\rightarrow A) \\rightarrow A)", "81b34465c86615163bd9077a61fa210f": "\\operatorname{Tr}_{L/K}(\\alpha)=\\sum_{j=1}^n\\sigma_j(\\alpha).", "81b35f5f94cdf77ebc634d5311508412": "y=-b/2", "81b38a2308b3fcc675df1ee6729e096b": "\\text{change open}_0 \\equiv (\\text{open}_0 \\not\\equiv \\text{open}_1)", "81b3920707ff0dd3842e6998d9b9e05b": "\\max_{x \\in \\bar \\Omega} u(x) = \\max_{x \\in \\partial \\Omega} u(x)", "81b397fe6c2471dd23d450811fb19186": "\\sigma_{yz}\n=-\\frac{\\partial^2A}{\\partial y \\partial z}", "81b3cea0ef9e1318d7fc199d5e25ad22": "\\textstyle \\big( (0,1), \\mathcal{F} \\big) ", "81b3f46a8e5c711de10cf9b77f8f0f09": "W=\\sum_{j=1}^Jn_jV_j \\, ,", "81b41be48a1971bdf671830f7669ded0": " \\mathcal{E}^0 \\subsetneq \\mathcal{E}^1 \\subsetneq \\mathcal{E}^2 \\subsetneq \\cdots ", "81b42b3ee5c4e310d6d3d71883764fea": "A_0(x) = \\pm \\sqrt{ 2m \\left( V(x) - E \\right) }", "81b45e70f0e8492c3fe64521037db210": "\n\\omega =\\frac{(1-\\alpha )E_{0}}{\\hbar }\\left[ \\frac{1}{n^{\\frac{\\alpha }{\n\\alpha -1}}}-\\frac{1}{m^{\\frac{\\alpha }{\\alpha -1}}}\\right] \n", "81b4a7e6b0cfcc2c695aee0892762770": "\\bar g:X\\times_Z Y \\rightarrow X", "81b4c8dd7cbec41cae5ef37da5644e99": "\\beta \\,", "81b532981dd96eea9c795e6fd2010aeb": "\\hat{\\pi}", "81b5391797775ef800408f76cf7468d0": "\\Delta{f_{echo}} = t_rk", "81b585c335f6293eb97ba8260a5a97d6": "\ny = \\beta_0 + \\beta_1 x + u, \\,\n", "81b5de37e8dec70027064bccd6daaff4": "\\int_{-\\infty}^{+\\infty} P_{2n-1}(\\phi,\\, \\partial_x \\phi,\\, \\partial_x^2 \\phi,\\, \\ldots)\\, \\text{d}x\\,", "81b602b95e1f8eb830ba433d4f180a6b": "k' - k \\pm q = \\begin{cases} 0 & \\text{ } \\\\ R & \\text{Umklapp-process} \\end{cases} ", "81b62c7af99837d95ab2ef875d576302": "\\rho(x) = \\sqrt{\\mathfrak{Re}^2[f(x)]+\\mathfrak{Im}^2[f(x)]} \\quad (5) ", "81b6c02758c5d9b2f2236e64a998eed4": "O( n^{34}k^{34}d^8 log^4(n)/ \\sigma^6 )", "81b6c4774f5a947ae103da90e38d727b": "Q_{n+1}(z) = P_n(z) - z^{2^n} Q_n(z) . ", "81b710d48898534987f3bcf5a2fda023": " f = x^{11} + 2 x^9 + 2x^8 + x^6 + x^5 + 2x^3 + 2x^2 +1 \\in \\mathbf{F}_3[x]", "81b764d83edf374bd3f4a40fb657229b": "(4)\\quad y_2=\\frac{0.24}{2}\\sqrt{1+8(224.65)}-1=4.97\\;ft", "81b790d55f7d27410309e1414b38857b": "t_{i=2}=\\Delta t_{i=2}+\\Delta t_{i=1}= 43.4\\text{ s}+44.9\\text{ s} = 88.3\\text{ s} ", "81b7958caeda4c9bb2cf621f2a003cd3": "T_c=\\left(\\frac{n}{\\zeta(3/2)}\\right)^{2/3}\\frac{2\\pi \\hbar^2}{ m k_B} \\approx 3.3125 \\ \\frac{\\hbar^2 n^{2/3}}{m k_B} ", "81b7b6acff0f0aad71c837c669c717af": " 5.307\\times 10^{-10}\\times 60\\times 60\\times 24\\times 10^9\\approx 45850 \\text{ ns} ", "81b8198c9cfcf71864477b1bdb02b158": "\\phi_a\\to\\phi", "81b8713e4d9d03990d939b7a1b34e97c": "u\\stackrel{*}{\\Rightarrow} v", "81b87782d8f5df854862b03b73faa349": "U{}^0_n", "81b89c1c800136c35bd6354e156b42df": "S[\\Lambda_{rot}] = \\left(\n\\begin{array}{cc}\ne^{+i\\phi\\cdot\\sigma / 2}&0\\\\\n0&e^{+i\\phi\\cdot\\sigma / 2}\n\\end{array}\n\\right)", "81b8b2294d0b88c40e6d3e08423cf13c": "(x_{\\bar{q}},y_{\\bar{q}})", "81b8bfbf1e815f1ef1bbdf424be4e8d4": "-\\infty<\\eta<\\infty", "81b8f57823eb5b32916850d1e61b7aea": " D_n(z)=d_0+d_1 z+d_2 z^2+ \\cdots + d_{n-1}z^{n-1} + d_n z^n ", "81b90600631cf830bc8a675b5c45eabb": "X_nY_n \\ \\xrightarrow{d}\\ cX ;", "81b930fcc059baa2254d43e56f005c70": "\\,x(-t)", "81b93288b41fb4d1e0d25a5e9ed6a018": "h_i(x) = 0 \\ ", "81b97ebdeaea34d1a1be5b403116b955": "x^2-y^2=N", "81b9ad4c51b91ae2c788fac33df26158": "f_1 \\Leftrightarrow f_2", "81ba6cd602b42361f2193fa2e054181e": "{3 \\choose 1}_q = \\frac{1-q^3}{1-q}=1+q+q^2", "81baab8c6d6bbe28dfb2102211c787c4": "\\frac{x^2}{a^2}\\pm\\frac{y^2}{b^2}=1", "81bae870a4d2218d738687248daca04e": "\\lim_{n\\to\\infty} (a_n \\pm b_n) = \\lim_{n\\to\\infty} a_n \\pm \\lim_{n\\to\\infty} b_n", "81bb6688a236177c8211c7bb72628300": " \\left [ \\hat{A}, \\hat{B} \\right ] \\psi = \\hat{A} \\hat{B} \\psi - \\hat{B} \\hat{A} \\psi . ", "81bbb7f44befc649686ae1e33fff662e": "t_{ff}=t_{orbit}/2 = \\frac{\\pi}{2} \\frac{R^{3/2}}{ \\sqrt{2 G(M+m)}}", "81bbe9fe615f86d38b662cfec2d53463": "= a(x, \\sigma(x), \\sigma'(x))dx + b(x, \\sigma(x), \\sigma'(x))\\sigma'(x)dx + c(x, \\sigma(x),\\sigma'(x))\\sigma''(x)dx \\,", "81bc51722bf4482fd1736002d137503c": " \\log \\left( \\frac{X_n}{n} \\right) ", "81bc66edcb952fcdfcbcf83d507ae4d4": "Y(a,z)1 \\in a + zV[[z]]", "81bccc7cb49f7a557dfa27059355a9c5": "X(t)=a\\,\\cos t", "81bcdd0884018e287a721c73ac1ce7dc": "\n\\begin{bmatrix}\nX\\\\Y\\\\Z\\end{bmatrix}=\n\\begin{bmatrix}\n0.4124&0.3576&0.1805\\\\\n0.2126&0.7152&0.0722\\\\\n0.0193&0.1192&0.9502\n\\end{bmatrix}\n\\begin{bmatrix}\nR_\\mathrm{linear}\\\\ \nG_\\mathrm{linear}\\\\ \nB_\\mathrm{linear}\\end{bmatrix}\n", "81bce0f0b479fd0bafa39bdd5daa92b8": "(NP\\backslash S)/NP", "81bd2fadafaf42f169e1c259499fe4e8": "i^{-1}\\mathcal{F}", "81bd71c7f05ad39c984e6641394e70aa": "A=B=2", "81bda322a8f8335c36f815f041c80037": "X_1, X_2, \\ldots, X_p ", "81bdc013e7be35a5dbf1e08f085c2ded": "\\begin{align}\\frac{M}{B} &= \\frac{C+D}{C+R} = \\frac{C+D}{C+R}. \\frac{D/(CR)}{D/(CR)}\\\\\nM & = B. \\frac{C+D}{C+R}. \\frac{D/(CR)}{D/(CR)}\\\\\n & = B . \\underbrace{\\frac{\\tfrac DR(1+\\tfrac DC)}{ \\tfrac DR + \\tfrac DC }}_{\\textrm{multiplier}}\\end{align} ", "81be15fa494861be6369fadf74947f1d": " \\mathbf{\\hat{r}}_i", "81be30d83c898417078b4158b189fa8e": "\n\\frac{m^{*}_{2D}(\\alpha)}{m_{2D}}=\\frac{m^{*}_{3D}(\\frac{3}{4}\\pi\\alpha)}{m_{3D}} ", "81bea0a87ae9de939b2267345fa0153b": "(B,\\mathfrak{m}_B)", "81bec2b5ad812e1cf43d344e80709cc1": "-\\oint\\frac{\\delta Q}{T}\\leq 0", "81bedbeb2355dc1a6f12497e936347c9": "Rx \\le c", "81bf10be6a9c1f1a4ad9a4b7bcec1cf7": "2 f", "81bf2a608a6b8a89482b1ba3a2d1a0f6": " Q(p, w, r).", "81bfaeaf0964c5b797119a23c0ce5892": "\\pi_1(X,w)", "81bfcba1f7fbdcd67b842729f1c6332b": "\\mathbf{q}_F", "81bffc4dbe52d4bd18d969ad51614d74": "\\frac{\\lambda}{\\rho c}", "81c00013e4d25a1016c335d27c7edddb": "\\tfrac{BO_2}{EO_2} = \\tfrac{5}{3 + 2} = 1", "81c099a05609fe7b02145f0d4dd656f3": " T(z)= \\sum_{n\\ge 1} T_n \\frac{z^n}{n!}=\\log\\left(\\frac{1}{1-z}\\right).", "81c154e289180f989c673aa8c111faad": "\\boldsymbol{\\Sigma\\Tau\\Upsilon\\Phi\\Chi\\Psi\\Omega} \\!", "81c16df08b6e7995c11b91bd3c6ed8c7": "m=2^M", "81c224aca7e4951ba6547faeb477abd1": "\\mathbf{1}_r", "81c239f0a65f37990a394ca2812ee7b7": "\\int_{c} \\mathbf{F}\\ dc\\ \n=\\int_{c\\circ{\\theta}_{[a,b]}}\\ \\mathbf{F}\\ d(c\\circ{\\theta}_{[a,b]})", "81c25223e0d7ab8ec7f1de6b3d7fdc30": "10^{10^{10^{10,000,000,000}}}", "81c25f28307afdfb9f0ea72b96614981": "R_h = \\frac{A}{P}", "81c285d3a5ef389923ec57b8819ec248": "0\\to A\\to T_1\\to \\dots\\to T_n\\to 0", "81c2dc66daf2a160fa2d1314a440e714": "D = y \\oplus x", "81c2f8ec27b6ad2b953cca94ccaf1bf8": "v(n,d), \\, ", "81c36e8400865b3f39f53e3a9fb9cfde": "\\tau = 0, 1, \\ldots", "81c38353307e45394f967c3718a0df58": "n=1; \\quad s+1", "81c3b62416b039907f6aaec72aafa29d": "E' \\rightarrow M", "81c3c41704ff8ac37388a8fd14e37e1f": "\\displaystyle M_{2}", "81c43aae1220f38677b7afa5a613a1cf": "k_{f}", "81c443d294208bd0a94993386f3a573b": "h_l = f_D \\left ( \\frac{L}{D} \\right ) \\left ( \\frac{V^2}{2g} \\right )", "81c4491695b5471e39489198438980bd": "\\mathrm{Ann}_R(M)", "81c48de58966ebbf4458fbbd867648dc": "\\displaystyle c = max_P I(P, \\zeta)", "81c4acf0a1dece5cb51ba2bdd5d7bb27": "H_n(X)=Z_n(X)/B_n(X). \\, ", "81c4ebe01ddcd4c67e8c97a46f4f6c10": "\\scriptstyle \\leq-2\\times10^{-16}", "81c55ca41f7ac922a845f8daf07e9aca": "(xy)^{\\lambda} = y^{\\lambda}x^{\\lambda}", "81c58df51bf5b543d9fabc85af627176": " (\\cdot,\\cdot,\\cdot) \\colon V\\times V \\times V\\to V.", "81c5a1f8fa4b89185ec92b6ed90b00ec": "\\epsilon_{abc}", "81c5bd53c8b348116e7f4b5dd1cd887a": "Expr \\rightarrow Int\\,ExprRest\\,|\\,String\\,ExprRest", "81c637535ea3cf64aa2cb882ac1cf324": " \\langle -\\nabla f(\\mathbf{x}_k), \\nabla f(\\mathbf{x}_k) \\rangle = -\\langle \\nabla f(\\mathbf{x}_k), \\nabla f(\\mathbf{x}_k) \\rangle < 0 ", "81c656aa10f8f759c18b372e9be3960d": "J=\\left[\n\\begin{array}{cc}\nF_x & F_{\\lambda}\\\\\n\\end{array}\n\\right]\\,\n", "81c66c124263bc93c45d6fc3cd6e077b": "\\displaystyle ||u||_{L^2}^2\\leq \\lambda_1^{-1}||\\nabla u||_{L^2}^2", "81c68bac39f78e65fdba701bce524907": "O(|E| \\log |V|)", "81c6aa812f651e83c7109d2468229a03": " \\psi=(\\nu U x)^{1/2} f(\\eta)", "81c721a1524fcfa33722499d00929e2e": "\n\n\\left[\n\\begin{array}{ccccccccccc}\n\n1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n\n1 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 0\\\\\n\n1 & 2 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\\\\n\n1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 0\\\\\n\n1 & 0 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1\n\n\\end{array}\n\\right].", "81c729f7be385d99a853915a484ef36e": "\\overline{Q}^{\\mathrm{day}}=0", "81c78c98b9d0a98b59020524e564d0f6": "S_2=\\frac{1}{4\\pi\\alpha'}\\int d^2z\\sqrt{\\gamma}\\left[\\gamma^{ab}G_{\\mu\\nu}(X)\\partial_aX^\\mu\\partial_bX^\\nu+\\alpha'\\ ^{(2)}R\\Phi(X)\\right],", "81c806ceab768dbf4bda8b390e0b3b68": "0.\\overline{46153}\\overline{8}", "81c80b2e53901cbd51384fd5da9f5533": " \\lambda \\mathbf{A} = \\lambda \\begin{pmatrix}\nA_{11} & A_{12} & \\cdots & A_{1m} \\\\\nA_{21} & A_{22} & \\cdots & A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n1} & A_{n2} & \\cdots & A_{nm} \\\\\n\\end{pmatrix} = \\begin{pmatrix}\n\\lambda A_{11} & \\lambda A_{12} & \\cdots & \\lambda A_{1m} \\\\\n\\lambda A_{21} & \\lambda A_{22} & \\cdots & \\lambda A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\lambda A_{n1} & \\lambda A_{n2} & \\cdots & \\lambda A_{nm} \\\\\n\\end{pmatrix}\\,.", "81c8151bb99d4d97d0aae1a8c3a61c19": "\\frac{\\partial \\mathbf{\\hat{s}}} {\\partial \\varphi} = -\\sin \\varphi\\mathbf{\\hat{x}} + \\cos \\varphi\\mathbf{\\hat{y}} = \\boldsymbol{\\hat \\varphi}", "81c82bb1c28e9ba2a78e64231ed2c34d": "S=0", "81c875502d90c7b5cdf24778f4f93331": "\\frac{1+2}{3+4}+5=\\frac37+5.", "81c8c9235cec6de52cfce32c4ae6460f": "4 km^2", "81c8e0cb5daf6c2772a34521c1f224b6": "\\quad N_C = 2\\left(\\frac{2\\pi m_e^* kT}{h^2}\\right)^\\frac{3}{2}", "81c9207b43c899b273483444e3bcf87c": "\\textstyle G_T", "81c92d35373e4e3c9fd26ba8ed7d23d0": " \\lim_{n \\rightarrow \\infty} \\sigma^2_P = \\bar{\\sigma}_{ij} ", "81c966fc61e792edf8fb34586d022ffa": "\\textstyle-\\frac{3}{4} + i\\epsilon", "81c9a005a35f08df77c50b9dbb00f8b5": "\\frac{\\partial C}{\\partial x} = \\frac{1}{2}\\left(\n\\frac{(C_{i + 1}^{j + 1} - C_{i - 1}^{j + 1})}{2 (\\Delta x)} + \n \\frac{(C_{i + 1}^{j} - C_{i - 1}^{j})}{2 (\\Delta x)}\n\\right)", "81c9b97a9d8f71bb0ffce859cc0a3e0f": "\\int u\\,\\mathrm{d}v = u v - \\int v\\,\\mathrm{d}u", "81ca21467485bd7a6ba785196b686653": "| \\mathrm{Bi} ( x + iy) | \\, ", "81ca64a5f6421ec749e8718c62832aa4": " \\text{DJIA} = {\\sum p \\over d}", "81cb05f9007dbd5af1d39048d7b84eb8": "\\bigstar \\bigstar | \\bigstar ||", "81cb7652368b406c02941691088ec254": "O( n^{\\log_{2}7}) \\approx O(n^{2.807})", "81cb8429e6537e76e60e09979b505555": "H_0(P,\\Z) \\cong \\Z, \\ H_1(P,\\Z) \\cong \\Z, \\ \\text{and} \\ H_2(P,\\Z) \\cong \\Z. ", "81cbbc4af23bfab82f088b7a930dc8dc": "x \\not\\in B", "81cc0da4d6c7d24bcfb9158f8f111038": "q = \\mathrm{ceil}(y) = \\left\\lceil y \\right\\rceil = -\\left\\lfloor -y \\right\\rfloor\\,", "81cc8c8da62ff777aad1f8bac5e87bff": "f_\\text{0}", "81ccb7648546216f26b068359eb4f7a5": "v = \\sum_i v^i[\\mathbf{f}]X_i,", "81ccda37e61bb1eed96684f5d39da8a6": "\\operatorname{P}(n) \\ge \\operatorname{P}(n+1) + \\operatorname{P}(n+2)\\, ", "81ccda92a79139a8be155adce88ecc1c": "restrict(F, x, 0)", "81cd073fd199e80ddd5948450fb1be2f": "\n\\underbrace{ \\frac{\\partial k}{\\partial t}}_{ \\begin{smallmatrix}\\text{Local}\\\\\\text{derivative}\\end{smallmatrix}}\n+\n\\underbrace{\\overline{u}_j \\frac{\\partial k}{\\partial x_j}}_{ \\begin{smallmatrix}\\text{Advection}\\end{smallmatrix}}\n= - \n\\underbrace{ \\frac{1}{\\rho_o} \\frac{\\partial \\overline{u'_i p'}}{\\partial x_i} \t} _{ \\begin{smallmatrix}\\text{Pressure}\\\\\\text{diffusion}\\end{smallmatrix}}\n- \n\\underbrace{ \\frac{\\partial \\overline{k u_i'}}{\\partial x_i} \t}_{ \\begin{smallmatrix}\t\t\t\t\t\t\t\t\t\t\\text{Turbulent}\\\\\t\t\t\t\t\t\t\t\t\t\t\\text{transport} \\\\\t\t\t\t\t\t\t\t\t\t\t\\mathcal{T}\t\t\t\t\t\t\t\t\t\t\t\\end{smallmatrix}}\n\t+ \\underbrace{ \\nu\\frac{\\partial^2 k}{\\partial x^2_j} \t\t\t\t\t\t}_{\\begin{smallmatrix}\t\t\t\t\t\t\t\t\t\t\\text{Molecular}\\\\\t\t\t\t\t\t\t\t\t\t\\text{viscous}\\\\\t\t\t\t\t\t\t\t\t\t\\text{transport}\t\t\t\t\t\t\t\t\t\t\t\\end{smallmatrix}}\n\t\\underbrace{ - \\overline{u'_i u'_j}\\frac{\\partial \\overline{u_i}}{\\partial x_j} \t\t}_{\\begin{smallmatrix}\t\t\t\t\t\t\t\t\t\t\\text{Production}\\\\\t\t\t\t\t\t\t\t\t\t\t\\mathcal{P}\t\t\t\t\t\t\t\t\t\t\t\t\\end{smallmatrix}}\n\t- \\underbrace{ \\nu \\overline{\\frac{\\partial u'_i}{\\partial x_j}\\frac{\\partial u'_i}{\\partial x_j}} \t\t\t\t\t\t\t\t\t\t\t}_{\\begin{smallmatrix}\t\t\t\t\t\t\t\t\t\t\t\t\\text{Dissipation}\\\\\t\t\t\t\t\t\t\t\t\t\t\t\t\\epsilon_k\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{smallmatrix}}\n\t- \\underbrace{ \\frac{g}{\\rho_o} \\overline{\\rho' u'_i}\\delta_{i3}\t\t\t\t}_{\\begin{smallmatrix}\t\t\t\t\t\t\t\t\t\t\t\t\t\\text{Buoyancy flux}\\\\\t\t\t\t\t\t\t\t\t\t\t\t\tb\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{smallmatrix}}\n", "81cd2b2a7c2d99da0f0d661ec74a8bdf": "SO(3)", "81cd31afaac47b665d10b19fe81e9bef": "c_j^i", "81cd8b2cc67fe3a72c1357ee41bb1514": "H^i(\\tau_{\\geq 0} A) = 0", "81cd978bf709b49b6e292ae50afd0030": "A_0(x) B_0(x) = 0 \\;.", "81cdf7461ae3cd18f09047d48bc295f5": "T U v = U T v", "81ce17b9bc744df94ff92013929b26fa": " A_{OL} = \\frac { \\beta i_B } {i_S} = g_m R_C \\left( \\frac { \\beta }{ \\beta +1} \\right) \n\\left( \n\\frac {R_1} {R_{22} + \n\\frac {r_{ \\pi 2} + R_C } {\\beta + 1 } } \\right) \\ . ", "81ce3dcb5d47b8fc317aea61aae995a3": " i \\ = \\ \\left({FV \\over PV}\\right)^{1 \\over n} - 1 \\ = \\ \\left({200 \\over 100}\\right)^{1 \\over 5} - 1 \\ = \\ 2^{0.20} - 1 \\ = \\ 0.15 \\ = \\ 15% ", "81ce85f374c0e1dd595526d365e45b50": "\\mathbf{\\hat{d}}_\\mathrm{i}", "81ce87bfe84460829ceb8fb081011176": " m( x ) = \\frac{ 1 } { \\pi | 1 - x^2 | } ", "81ceaeba7833f5ef6cfb201e4405909d": "H^m", "81ceb59d62c48f699ed0e2c70c80313c": "\\limsup_{n\\rightarrow\\infty}\\frac{\\sigma(n)}{n\\ \\log \\log n}=e^\\gamma.", "81cebdc66376ff7a6bb57aa8963359e1": "\\scriptstyle\\vec x", "81cf101b40caccde32439977dba76d66": "w(x) = g\\circ\\phi(x)", "81cf1b24ce6c1235ecc452116a1f1669": "\\chi_s~", "81cf1d703feef5346287092f6cc172e0": " \\mathfrak{p}= \\mathfrak{p}_{0} \\subsetneq \\mathfrak{p}_{1} \\cdots \\subsetneq \\mathfrak{p}_{h}=\\mathfrak{q} ", "81cf28176a74ef47e24980d3177f4960": "\\sum_{\\nu=1}^{n}\\nu(\\nu-1) b_{\\nu, n}(x) = n(n-1)x^2", "81cf92b76eef896a54652c86ca080000": "\n(a_2 x_1 + b_2 x_2 + c_2 x_3) b_1 - (b_1 x_1 + c_1 x_2) a_2 = d_2 b_1 - d_1 a_2\n\\,", "81cfb120c159ba8b0003a7d3b558ece2": "(E_r-E_y)", "81d05f6293bcabffe53ce1d0ce01a1ae": "{\\mathit{He}}_n(x)=\\frac{n!}{2\\pi i}\\oint\\frac{e^{tx-t^2/2}}{t^{n+1}}\\,dt", "81d091ca87c6aff3367183f71d3cf2af": "dT_{G-F}(F) = \\lim_{t\\rightarrow 0^+}\\frac{T(tG+(1-t)F) - T(F)}{t}", "81d0a9cc9b6e2c43d353f60d85fb3dba": "\\scriptstyle Y ", "81d0a9e61a135db6b351842de1dcae13": "S={1\\over 16\\pi G}\\int d^4 x\\sqrt{-g}[R+\\omega K_\\mu K^\\mu R+\\eta K^\\mu K^\\nu R_{\\mu\\nu}-\\epsilon F_{\\mu\\nu}F^{\\mu\\nu}+\\tau K_{\\mu;\\nu}K^{\\mu;\\nu}]+S_m\\;", "81d0aebc310d99d93310211058e51fa6": "\\textrm{lfp}", "81d10d98a89dad7390a4398520e912fb": "\\ \\mathrm{p}K_{\\mathrm a} = - \\log_{10}K_{\\mathrm a}", "81d16a77c8f5bf91d3e475f113738890": "[C_i,C_j]=0", "81d175997752b117c18ab33fbfcc34e3": "\\lambda \\sum_k (A_{ik}B_{kj}) = \\sum_k ( \\lambda A_{ik} ) B_{kj} = \\sum_k A_{ik} ( \\lambda B_{kj} ) ", "81d253756a7175bba40368a5509e2872": " f: \\R^n \\to \\R^k\\!", "81d26eaf48065c831715d19784d55621": "\\{v_n\\}", "81d33a4d0131498d509dbf806cdb655c": "P(x) = p_0 + p_1 x + p_2x^2 + \\cdots + p_n x^n", "81d33bb83a6dd202fa54ecb4ea8394b6": "\\frac{d}{dy}\\ln y = \\frac{1}{e^{\\ln y}} = \\frac{1}{y}.", "81d357e44c183195745cddb462b8cbf8": "t \\mapsto t_+^x", "81d3626486cccd47033c819b3fcbe7e7": "U-TS\\,", "81d36af15b45b87c089516cf3a867c81": "=\\frac{2n}{2n+1} \\cdot \\frac{2n-2}{2n-1} \\cdot \\frac{2n-4}{2n-3} \\cdot \\cdots \\cdot \\frac{6}{7} \\cdot \\frac{4}{5} \\cdot \\frac{2}{3} I(1)=2 \\prod_{k=1}^n \\frac{2k}{2k+1}", "81d3ab985ebfef43d3da8b7f0b91d4ae": "S = \\frac{U}{T}+\\frac{p V}{T} + \\sum_{i=1}^s (- \\frac{\\mu_i N}{T})", "81d3df9124df783d8dda6daa223fe7bf": " \\Delta K = K_{max}-K_{min}", "81d494930c1eb595e9bf37cb6b19cc0f": "\\big\\{0,1\\big\\}^{32}", "81d4e08a4e855522a686d63e3b38544d": "d = 2.375 - \\left ({\\text{INT} \\over \\text{ATT}} \\times 25 \\right )", "81d50b512a10a93f47f8ea4a4f8c4e64": "2.9330", "81d53b44f7c97c7a9187ba4cd923e805": " \\tau(x)=\\inf r(x,y,z),\\, ", "81d56b52cc511c5dc70b488864365ad8": "\\| u \\|_{L^{p} (T; X)} := \\left( \\int_{T} \\| u(t) \\|_{X}^{p} \\, \\mathrm{d} \\mu (t) \\right)^{1/p} < + \\infty \\mbox{ for } 1 \\leq p < \\infty,", "81d598f34fa5241c8a8415f842ee55a0": "(A, \\leq)", "81d5afa6ef4a9155cb6ffa1d7f9aa77b": "\\displaystyle \\exists x_1 \\exists x_2 \\phi(x_1, x_2) \\quad \\mapsto \\quad\n\\exists x_1 \\forall y_1 \\exists x_2 \\phi(x_1, x_2)", "81d5c2735edf2c44ccdd44e6b17f7df1": "d\\lambda = -c_2/(x^2T\n)dx", "81d5cfc95d2a5a69ba7946ade91ca6bf": "Weight=\\frac{Term\\ Frequency}{Document\\ Frequency}=\\frac{Frequency\\ of\\ the word\\ or\\ expression\\ in\\ the\\ Text\\ Sea}{Number\\ of\\ documents\\ containing\\ the\\ expression\\ or\\ word}", "81d5e6e1defc3b5d511689ee20cb52d6": "B(U),B(V)", "81d68f8feab90fbdd94aedc81d01c78e": "(x_0, y_0),\\ldots,(x_{n-1}, y_{n-1})", "81d6d22a1701e9230cafd8a80c6bcb5c": "\\mbox{Cov}(f_n, f_m)", "81d71057fff38503bc07bb54619fd029": "W^{(-1)}(t) := \\int_0^t W(s) ds", "81d735ba490754b08f63ba64054f1ee6": " T_{m-1}(z)=\\frac{\\delta_{m+1}(1+z) T_m(z) - T_{m+1}(z) }{z}", "81d737215dd240282f124e3d026e1d71": "\\log \\left (1 - x \\right ) \\approx -x ", "81d795aca0545950b584d710ca73a85b": " \\left.\\frac{\\delta S}{\\delta \\phi}\\right|_{\\phi = B} = 0 ", "81d7c474959d8e023a265cc9e92d38fd": " = \\left( \\sum_{i=1}^n \\left| x_i - y_i \\right|^p \\right)^{1/p}", "81d81a103293a23e335c123fe0ecefab": "M_i^2 = M_i,", "81d83cdda8aa8fe2018915c9eab54572": " \\nabla ({\\bold u} {\\bold u}) = [\\nabla ({\\bold u} u_x),\\nabla ({\\bold u} u_y),\\nabla ({\\bold u} u_z)]", "81d9009eea6c18ff36de8f46fa1203ff": "\\sigma_m^2", "81d902e1d482d9efe3fd941205abbba4": "\\hat{r}_{k+1} \\leftarrow \\hat{r}_k- \\alpha_k \\cdot \\hat{p}_k A^T ", "81d92aa28f5653b63f4e4c5203907ab7": "G:\\{0,1\\}^\\ell \\to \\{0,1\\}^n", "81d95826e8bdf4b4563d64d38b7cab59": "A_n(R)", "81d95bc38b3f5f09eb3deb7d33037f80": "Z_{\\mathrm{eff}}= Z - s.\\,", "81d9717a3c499c51ce4efb720c5155c7": "\\mathrm{tfidf}(t,d,D) = \\mathrm{tf}(t,d) \\times \\mathrm{idf}(t, D)", "81d999094b809b497339998780d07ab9": "\\gamma_1(x) = x", "81d99ea4ea9aa7a9e82c65a9f9d156f6": "\\beta(v_1, \\dots, v_m) = \\alpha(\\widetilde{w_1}, \\dots, \\widetilde{w_{k-m}}, v_1, \\dots, v_m), \\quad \\widetilde{w_i}\\text{ the lifts of } w_i.", "81d9ab88a21720d60b6f2077da766e3d": "k \\leq n-1", "81d9b8c0216665439b1d5c078d97b53a": "8 e", "81da1ea4fb66181aa573f95da0080b08": " \\gamma = 0.6745 \\times \\sigma .", "81db277f0beb1987bb2683f19686df6f": "(0,\\ \\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm1)", "81db3b8969e6cd819961f8a23c1bb13f": "\\operatorname{NW}(X,Y) = \\operatorname{NW}(X^l,Y^l) + \\operatorname{NW}(X^r,Y^r)", "81db4ce76416ff67a5c2527b9e1cedc2": "S = - k_\\mathrm{B}\\sum_i p_i \\, \\log \\, p_i", "81dc0959cb81f4e6762f1a87b820b423": " (1+\\sqrt{2})^4 = 17+12\\sqrt{2} \\approx 33.97056", "81dc643ec35bd454639d7c390d8d62f1": "\\frac{ \\sum_j^t \\alpha_j h_j (x)}{\\sum |\\alpha_j|}", "81dce678e675ee4a252f253845261e50": "F = dA + A \\wedge A \\, ", "81dcf523d708cfc91b1ac747f0716a04": "\\mathrm{supp} (\\mu) := \\overline{\\{ A \\in \\Sigma \\mid \\mu (A) > 0 \\}}.", "81dd9c497ad2322c70deb5e11fd2f78f": "Q^S=Q^D=Q>0", "81de21eeb53d730d2bb09177b0180e4e": "\\text{Upper fence} = Q_3 + 1.5(\\mathrm{IQR}), \\,", "81de7c6efdb35afd42e30ca2adb87e07": " P= \\{g\\in G| \\tau(g)=g^{-1}\\}.", "81df24206a1ca76146d291ad484af8c1": "\\mathrm{H}(X_b)", "81df93812085b928452e6a394becc49f": "\\mathrm{ind} Df_s\\left( x \\right) = n", "81dfa79ba7f2cdd745af0d539d9c80f1": " H_0 \\,", "81e0375a9d804f0aee6ccdcb662fcc8d": "Lc(z)=z\\log(2\\sin \\pi z)+\\frac{1}{2\\pi}\\, \\text{Cl}_2(2\\pi z)", "81e0aa83466d455c561a0dd8c4a6ef67": "x_{j1},...x_{jn}", "81e0b2261ab258a8e281be5fde75f37b": "\n \\mathrm{N}\\; \\frac{1}{\\alpha\\;\\sqrt\\pi}\\; e^{-\\frac{x^2}{\\alpha^2}}dx\n ", "81e1694b0f099f0a6171e4d51f1ded8c": "0=p_{r}-p_{fr}.", "81e1a68393d269d00833e22f5f3de71e": "T_{\\text{test}} = U_{\\text{test}}^{-1}", "81e21f27d117eb50f1029ad2df878a7d": "\\tau(X, X')", "81e2a3e3e64a71863042ff6851334fcd": "f(w)", "81e2d20df1c922071ca1e6f4f3fd6264": " {\\Gamma \\vdash M : A \\qquad \\qquad A =_\\beta B \\qquad \\qquad B : K \n\\over {\\Gamma \\vdash M : B}} ", "81e2f312e496686e81201ed0eb0bfe25": "P_0\\geq 0", "81e34b04c4be70569ef2aa43a2180326": "\\Delta T_{\\rm f} = T_{\\rm f}(solution) - T_{\\rm f}(solvent) = - i\\cdot K_f \\cdot m ", "81e375948186c1521145c7fcbc0d1d1c": "\\mathrm{coker}\\,f := W/f(V) = W/\\mathrm{im}(f).", "81e39e69b50261d7569d8919537f77cc": "\\beta = \\sqrt{k_{0}^{2}-\\left ( \\frac{\\pi }{a} \\right )^{2} }< k_{0}", "81e3e76a7b537c0d3c73919dba087531": "|\\mathbf{a}|^2|\\mathbf{b}|^2(1-\\cos^2\\theta) = |\\mathbf{a}|^2|\\mathbf{b}|^2\\sin^2\\theta", "81e40bc124ca55145c974d32c6f5698f": " h(g(\\zeta))=\\sum_{n\\ge 1} {\\lambda_n\\over n} \\zeta^n+\\beta +\\sum_{n\\ge 1} \\beta_n \\zeta^{-n},\\,\\,\\,\\beta_n=\\sum_m c_{nm}\\lambda_m.", "81e41b1d54e352a3ddf28fa43af51f23": "\\omega_{sp}=\\sqrt{3}\\omega_p", "81e437b0faf6408280d757e1e5bc3493": "\\frac{d}{dx} \\ln{( 1+x^\\alpha )} \\leq \\frac{d}{dx} ( \\alpha x ) ", "81e43840097b56894c6b84b26c3bb83f": "\\mathbf A_+=\\langle F,R,V\\rangle", "81e43d81ea36044c2c5f735c0638ab3c": "\\frac{1}{30} + \\frac{1}{60} = \\frac{1}{20}", "81e43e6fcb7003c15beced3dd7658e53": "y(t) = Y_0(t,t_1) + \\varepsilon Y_1(t,t_1) + \\cdots.", "81e46b0975b519ac331add7157fa26d0": "\\tfrac12 k x^2", "81e4d15b4acefb71fbd27128eb973882": "\\nu_i = \\frac{dN_i}{d\\xi} \\,", "81e4d64a78f5fd83b7eba018ec186b95": "1.8 + {LWL\\over1.8}", "81e50194f15e6adb9d96d9745693c109": " \\Phi \\subseteq H ", "81e5382b144a7899c134e7dbfca94c29": "f(k)", "81e58aa59b7c62fe76084ceed4e19945": "\\frac{1}{n^3-n}", "81e5dd51a006c233f7d2e0b92a88f2ff": "5) KDC \\rightarrow B : S_{KR_{KDC}}[ID_A||KU_A]||E_{KU_B}[S_{KR_{KDC}}[N_A||K||ID_B||ID_A]] ", "81e5e79fff7b51b7bd9e469e44844dc0": "\\rho=r^2/4", "81e5f5c52f0fae06697a1a2d8c0aebae": "t<\\tfrac{1}{2}p+n-1", "81f0cb60aa712011b1ac925e0a38066e": "\\textstyle G_o=2e^2/h", "81f10623059b24b2fb20acc14a5733fe": " \t\\scriptstyle p = p^{\\star}_{\\rm A} x_{\\rm A} + p^{\\star}_{\\rm B} x_{\\rm B} + \\cdots", "81f1354fe345a3a7bee121a602f0f910": "2 \\left (x - 75 \\right )(y+5)", "81f17b5a7fdcace2398454af004d3289": "F\\equiv U-TS", "81f17c110fa06d08ec460100aefb47f5": "= a C \\frac{\\sin\\frac{ka\\sin\\theta}{2}}{\\frac{ka\\sin\\theta}{2}}\\left(\\frac{1 - e^{iNkd\\sin\\theta}}{1 - e^{ikd\\sin\\theta}}\\right)", "81f1a84da74c67bccf48dbca26d6761f": "\nS= m \\int d\\tau (\\frac{1}{2} \\dot{x}^\\mu \\dot{x}^\\nu g_{\\mu\\nu}+(B_\\mu+\\tilde{B}_\\mu) \\dot{x}^\\mu )\\,.\n", "81f1dc858007099b9d57b97971e94af2": "H(4,q^2) :\\ s=q^2,t=q^3", "81f230e717d8a0a64bbf1c3f780a6edb": "\\hat{\\boldsymbol{x}}_{k|k-1} = f(\\hat{\\boldsymbol{x}}_{k-1|k-1}, \\boldsymbol{u}_{k-1})", "81f27e5e02cd2a77bf975c97d2cbd4e9": "\\sigma_\\mathrm{n} = \\frac{1}{2} ( \\sigma_1 + \\sigma_2 ) + \\frac{1}{2} ( \\sigma_1 - \\sigma_2 )\\cos 2\\theta\\,\\!", "81f2bf22992b562fd4eac74006a6b7d1": "I = \\emptyset", "81f2f938eda8f260ec77223dbcae166c": "~f(x'y')= c_1 x'+c_2 y'+c_3 x'^2+c_4 y'^2+c_5x'y'\\cdots", "81f3908f7b21f6d9357a4996e5db6bd5": "h'\\left(k\\right)=\\left\\lfloor\\frac{k}{11}\\right\\rfloor\\mod 11", "81f3cdf7679598d49995988a076cb3da": " M \\cup_f H^j", "81f3ef3a6ce910f891c816e5c3414114": "2Z_{sc}", "81f43d6888342193d9b264db59ef7ef4": " \\iint_{\\Sigma} \\nabla \\times \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{\\Sigma} = \\oint_{\\partial\\Sigma} \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{r}, ", "81f47e2f0a5d6a741b5f853c2389245a": "\\Delta t = \\Delta v L / c^2", "81f4944faad478f5708f7ac30e1ddbba": "\\delta=\\frac{\\lambda}{\\mathrm{2NA}}", "81f4bb262234dfdccd940171a7869d3b": "t'\\in [0, 1]", "81f4d2557b171134c72a3457c77d3ee4": "s_n = (Ta)_n = \\sum_{k=0}^\\infty T_{nk} a_k.", "81f4ff1357cd51b2ca6a70c2b88fff2e": "\\begin{align}\n S_{HSV} &=\n \\begin{cases}\n 0, &\\mbox{if } C = 0 \\\\\n \\frac{C}{V}, &\\mbox{otherwise}\n \\end{cases} \\\\\n S_{HSL} &=\n \\begin{cases}\n 0, &\\mbox{if } C = 0 \\\\\n \\frac{C}{1 - |2L - 1|}, &\\mbox{otherwise}\n \\end{cases}\n\\end{align}", "81f5031ae845276e0df209ac2bb4ca30": "C = \\frac{\\mu HP_c^{1/\\gamma}}{\\rho_c k}", "81f5580d33ad61ed1b5301c62c3917bf": "A_0=\\bigcup_{m\\in\\mathbb{N}}A_{1,m}", "81f5632fa21de74197b15db7e9e057f9": "Y = y l = j \\omega C l ", "81f57fdb204fc51b96e48a83856e6ea9": "\\psi = (\\mathbf E - i \\mathbf B) / \\sqrt 2", "81f602b092c21d6485365d87fe79a811": " \\epsilon(abc) = \\sum \\epsilon(ab_{(1)})\\epsilon(b_{(2)}c) = \\sum \\epsilon(ab_{(2)})\\epsilon(b_{(1)}c)", "81f617535ef4521b84a227b8fc41effc": "\\sigma^2 \\approx \\frac{N}{N-1} \\bigg/ \\left( \\frac{1}{\\mu}+ \\frac{1}{m_1-\\mu}+ \\frac{1}{n-\\mu}+ \\frac{1}{\\mu+m_2-n} \\right)", "81f62a2e94f60861094b114e986cdafe": " y(x) = c_{1}e^{3x} + e^{-x}(c_{2} \\cos x + c_{3} \\sin x) + xe^{-x}(c_{4} \\cos x + c_{5} \\sin x) \\, ", "81f6928b0db2925807e8a879985140ee": "\\omega=0.2", "81f6a2ea32a66399f5aa4ac4f6834f73": "\\textstyle\\frac{\\log10}{\\log2} \\approx 3.3219 \\approx \\frac{10}{3}", "81f6cdd6c0a229f7bc7cd5f5c2cd7276": " x_1, \\dots, x_k ", "81f6d07e9f1be0144dfb503975e7b0d6": "\\left|\\frac{a-1}{a+1}\\right| \\le 1.", "81f7060c3731a70ac497d9302d0438a0": "\\triangle x_j + \\lambda x_j = 0", "81f725b06b59823904b1c7f4fefbb32a": "p_B(b, \\lambda)", "81f7a1c01f67baaf6a583f0fe9793963": "L = L_1 + L_2 \\,", "81f8032713a29376b8ac2e94836e353a": "\nO_t\\left\\{\\int_{-\\infty}^{\\infty} c_{\\tau}\\ x_{\\tau}(u) \\, \\operatorname{d}\\tau ;\\ u\\right\\} = \\int_{-\\infty}^{\\infty} c_{\\tau}\\ \\underbrace{y_{\\tau}(t)}_{O_t\\{x_{\\tau}\\}} \\, \\operatorname{d}\\tau.\n\\,", "81f806999ed310a9903b6e035b0077fe": "1.8\\times10^{308}", "81f80c4e9b62c3d288c65388e3c6f70a": "\\scriptstyle \\sum_{k\\,\\ge\\,0} a_k(z\\, - \\,a)^k", "81f84b8a0219a6c0105c20872b8b7835": "\\nabla_{\\dot\\gamma(t)}\\sigma = 0", "81f86e579e290537155897cd61520f20": "r \\leq d \\sqrt{\\frac{n}{2(n+1)}},", "81f879f2bc75a678f2f6157094225561": "\\mathfrak{p} R_\\mathfrak{p}", "81f8ed97daf2dd5fb288d7422483d8b0": "1;24,51,10=1+\\frac{24}{60}+\\frac{51}{60^2}+\\frac{10}{60^3}=\\frac{30547}{21600}\\approx 1.414212\\ldots", "81f903a52f61e1f52abb8b55e0bc1f7b": "W = \\mu + \\sigma \\left(\\frac{U_1}{U_2}\\right)^\\gamma", "81f904817fbefc26d3777d1cacc62585": "\\vec \\mu_S= \\vec S g_S \\mu_B", "81f922342a437bacf39d96840f1d1f8c": "\\mathbf{k} = \\left ( 2\\pi/\\lambda \\right ) \\mathbf{\\hat{e}}_{\\angle} \\,\\!", "81f93f17ee3fd02716882142ca10ed1c": "K_2(k) = k^\\times\\otimes_{\\mathbf Z} k^\\times/\\langle a\\otimes(1-a)\\mid a\\not=0,1\\rangle.", "81f94dc96ca458a3c8da91a39c29604a": "\\frac{p^p-1}{p-1}", "81f9a2715f3493799b4152b9aca67923": "\\arccot (-x) = \\pi - \\arccot x \\!", "81f9c708984d2c005fadeff8ba058064": "\\scriptstyle\\mathbf{E}[ X] = \\frac{\\alpha}{\\beta}", "81fa05bfc1beaded5222afb0da11c360": "T : \\mathcal{D} \\to \\mathcal{D}", "81fa0964217090b1c074040d052aef60": " h_k = h_{d-k} \\quad\\textrm{for}\\quad 0\\leq k\\leq d. ", "81fa5073c4d2050bf1ede27e3592b666": "X*X*X*X", "81fa80a4f0c79197c486f369f75e49e9": "{\\Omega}", "81fa843ee67127b94fdad8956c99195e": "[0:0:0:1]", "81fa8da0e8ac7996ebfbaf69468a2969": "y\\in s_j", "81fb365975c291658b669314b1c50451": "(2s + r)^2 = 2(2s^2 + r^2) + 4s^2r", "81fb676bb4023aa0f5f8f3ef46152a07": " F(t) = -kx(t) = m \\frac {\\mathrm{d}^{2}}{\\mathrm{d}{t}^{2}} x \\left( t \\right) = ma. ", "81fb7cc7b5141dbe308b044cd3304d1e": "\\{6\\}", "81fb83cb4f75a06985e140a59bb81043": "\\theta(a^g) = \\theta(a)^g. \\, ", "81fbd6e29cda2a6709846076086829ef": "p_{e'}^{\\, 2}c^2 = (hf - hf' + m_{e}c^2)^2-m_{e}^2c^4. \\qquad\\qquad (1) \\!", "81fbffb934d45b6dfae15b2cb10ad031": " SO(m,\\mathbb C) ", "81fc795738f6412b8a2157e53f002759": "-\\rho u_{i}^\\prime u_{j}^\\prime", "81fcc37e56ddd3aff41c555d71ebfce5": "{P_0}\\,", "81fd940d63c32492f4f05bb1eccc4a9a": "2T(k,r)+T(n-k,r-1)", "81fe084aa3c9df338a52a1c3bc961d62": "\\begin{align} h &=\\sum_i dy_i \\; dy_i\n= \\sum_i d\\sqrt{p_i} \\; d\\sqrt{p_i} \\\\\n&= \\frac{1}{4}\\sum_i \\frac{dp_i \\; dp_i}{p_i} \n= \\frac{1}{4}\\sum_i p_i\\; d(\\log p_i) \\; d(\\log p_i)\n\\end{align}", "81fe3b59dc00af38a7e94f57210f9c09": "\\sup_{T \\in F} \\sup_{x \\in U} \\; \\|T(x)\\|_Y < \\infty.", "81fed06fdeda99f9636671569c0fef49": "0.998351", "81feedb4919bc3801314d80a0d9e359d": "Q_{cold}=\\frac{Q_{hot}T_{cold}}{T_{hot}}", "81ff04f661822e8be8c1bcca740ef1bd": "0<\\Delta<1", "81ff1818adebd840326d15877ed08223": "c\\cdot x'", "81ff1d6241a95337c55124820f904d2b": "E_2(l, m, t) = A \\left( l, m, t - \\frac{R_2}{c} \\right) \\frac{ e^{-i \\omega \\left( t - \\frac{R_2}{c} \\right) }}{R_2}", "81ff4bfb4fd17a2b20c884ebce2a10c6": "\\bar{\\eta}(E) = \\frac{1}{2\\pi}\\left [E\\sqrt{2N-E^2}+2N \\arcsin \\left( \\frac{E}{\\sqrt{2N}} \\right )+ \\pi N \\right ] ", "81ff5c2da7fb87ee5764bc2d11aa0fe8": "\\Sigma_1 \\in \\mathbb{F}^{m \\times r}", "81ff8838b50363f69dac66253381ef97": "F = k_\\theta / L", "81ff9b5122d093f6792db1f4c781f885": "\\sigma(A,B)", "81ffd989367ebaabd66903c948760523": " \\overline{U}^2 \\left({1- {R \\over 2C_p}}\\right) -\\overline{U}{F\\over \\dot m} +{HR \\over C_p}=0.", "82006fa4e0b55b47d6a419d4a3316140": "p_{\\mathbf{R},i}", "8200e51cee24fe595d26a88defbf0935": "E_u(x,y,z)=e^{j(k_x x + k_y y + k_z z)} ", "82014a806b51c6a4ee8b50681800a08b": "V(\\boldsymbol\\omega)=V_{\\mathrm{ac}}\\cos(\\boldsymbol\\omega t) \\;", "82014cdbaf1452b98e50b049cfda207c": "\\mathcal{L}=-T\\sqrt{-\\det\\left(\\eta+2\\pi\\alpha'F\\right)}", "82019f82463720bade30101e63eb7fb8": "0.\\overline{4}", "8201b199e53419c80a35b17848a9a273": "n=\\log{q}", "8201ba6397c892b8a51d48bb0e94f0eb": "\nTr = \\frac {\\int_{t=-\\infty}^{t_0} P_x \\, dt}{E_x}\n", "8202278a788d6c774c219dbd686f8823": "\\! R_{jm}", "82022882ce2ba334704f532161441d87": "b_1 = \\frac {n+1}{3(n-1)}", "82027e282a4cab1583529ef3e0a7b532": "R^*", "820284e9359bf9916aeb5da5d84c94d9": "\\frac{1}{|G|} \\sum_{g\\in G} |(Y^X)_{\\omega,g}|", "8202a719d25f8fb410107718f6fee1c1": "\\theta(t) = \\theta_{\\mathrm{eq}}", "8202a94b8e465eea1f8d34f4607702c5": "x\\neq x'", "8202df67ba5b1e1d58403eeb75003c5a": "c^{2} d\\tau^{2} = e^{\\nu(r)} c^2 dt^{2} - e^{\\lambda(r)} dr^{2} - r^2 d\\theta^{2} - r^2 \\sin^2 \\theta d\\phi^2 \\;", "820307cd6cbc8a437960d311bea3bc3c": "\\lim_{n \\rightarrow \\infty} \\frac{m}{n} = \\frac {6 \\ln 2 \\ln 10}{ \\pi^2} \\approx 0.97027014", "820341dc67dd887805f2620efabe6669": "4.7", "82034fdad76c78d41821134a0ca70cc0": "\\tan\\frac{\\theta}{4}", "820365eba7778c53cb5dc24c78ca9ae0": "z-\\alpha_1, \\,", "820371c6b4b7c53edc5a9330e9d201c6": "E_g(f; N) = {1 \\over N } \\sum_i^N { f(x_i)} / g(x_i) ", "82037ea5ecef06d8adf2ec4d05ccbec0": "\\!\\mathcal A \\models_Y^+ \\phi", "820459374eee0c216763c80b43bd3b33": "f,g: M\\to S^n", "8204887315614d84b5fdb048d7c6a7dc": "\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}", "820517ec01098f0fb2a2aeb3849bf93b": "f(z) = (p,-\\sin q)^{\\mathrm T}", "82052693f515b7ffa5acd5d69cd8fd54": "d_1(q^c)", "820566c18590c9630e4de7321c8aac3c": "\\mu\\!\\left(\\varnothing\\right) = \\mu\\!\\left(E\\right) - \\mu\\!\\left(E\\right) = 0", "8205af4a9e26b40370e349745b687223": "L(\\gamma)^2\\le 2(b-a)E(\\gamma)", "8205d1998f2172c941489a2317e5a046": "(y_1,\\alpha_1 y_1) = (y_2,\\alpha_2 y_2)", "8206229d8b91a1e68db3fc16398eee71": "v_1,\\ldots,v_m", "82063864f5c0e7dae64755e55b253ef3": "\\mathrm{Var}\\left(\\hat{Z}(x_0)-Z(x_0)\\right)=\\underbrace{c(x_0,x_0)}_{\\mathrm{Var}(Z(x_0))}-\n\\underbrace{\\begin{pmatrix}c(x_1,x_0) \\\\ \\vdots \\\\ c(x_n,x_0)\\end{pmatrix}'\n\\begin{pmatrix}\nc(x_1,x_1) & \\cdots & c(x_1,x_n) \\\\\n\\vdots & \\ddots & \\vdots \\\\\nc(x_n,x_1) & \\cdots & c(x_n,x_n) \n\\end{pmatrix}^{-1}\n\\begin{pmatrix}c(x_1,x_0) \\\\ \\vdots \\\\ c(x_n,x_0) \\end{pmatrix}}_{\\mathrm{Var}(\\hat{Z}(x_0))}\n", "820644e7a1d9e03c5fc1faeab1a22814": "\n \\log \\Gamma \\left(\\frac{1}{2}-z \\right) + B_1(z) \\log 2\\pi-\\frac{1}{2}\\log 2+\\pi \\int_0^z B_1(x) \\tan \\pi x \\,dx", "820682791767c143e816ed500bf27e6a": "A = UP;\\,", "8206f76f6fff9222a5b27455e13e09b2": "\\phi: T \\times T \\longrightarrow V", "820713fd8e83c0a05937e67ed6dac9ba": "(N \\cup \\Sigma)^*N(N \\cup \\Sigma)^* \\to (N \\cup \\Sigma)^*", "82072c2ae85333003afc6d780e8f34d1": "f(z) = uz + c \\ ", "820753a179510eef4eead527c7670027": "e^{i \\gamma z}= \\Gamma(s)\\cdot\\sum_{k=0}i^k C_k^{(s)}(\\gamma)(s+k)\\frac{J_{s+k}(z)}{\\left(\\frac z 2\\right)^s}.", "8207a89fb48f4f34e56eabf4e1aa689c": "(r(t))_{t\\in [0,T]}", "8207cad9b8699879d5d7ad797282cf22": "s_1=\\alpha^{-7},", "8207f551291e4e62edd0754585eb5db7": "\\mathbf{y}_i \\in \\mathbb{R}^d", "820816ffc64c88d2dbd804553427e93b": "\\left(\\frac{\\psi-i}{\\psi+i}\\right)", "820841f06d2b9587835287008c191d02": "\\tbinom{t}{k}", "8208515e566ed30fada2fc829a670802": " P^{MAP}_j ", "820855a3bfbb3dff4b2c7b2a2e5839c3": "B(Homeo(F))", "8208b63d1c1b6b0056f0e5b328a4cf5c": "\\alpha_1^v , \\alpha_2^v , ... , \\alpha_n^v \\in \\Phi^v", "8208e0dd4c4f89e8762c9eed1b37061c": " c_{(\\pm)} ", "8209314fa34bac7f31b5f6b4ba0660f8": "\\mathbf{J} =\\mathbf{J}_{\\text{f}} + \\mathbf{J}_{\\text{M}} + \\mathbf{J}_{\\text{P}} ", "8209326ef40297f59937de95c553d010": "y \\in C_{in}^{\\alpha_1}-\\{ 0 \\}", "82093bd70be3a5cf2a3098aff40349ac": "d(f^\\tau(x), f^\\tau(y)) > \\mathrm{e}^{a\\tau} \\, d(x,y).", "82095e9f7da27f5ee19fb1c82d1be1d3": "1/k^2", "8209a70961db3d047b7cb4ef49af4a3c": " r_{it} - r_f ", "820a236742d7f605bfd4ec0db13c2e81": "x_1 = 1-(a-3b)(a^2+3b^2), x_2 = (a+3b)(a^2+3b^2)-1", "820ab561ca42b8c22abf8c7aaffbdb18": " K_{n,n} ", "820aff0700f39504636305f5266bf9bd": "S_\\alpha(\\beta)\\in \\Phi", "820b675bf8b5f26ead22b62dbfc0b12a": " \n \\limsup_{t\\rightarrow\\infty} \\frac{1}{t}\\sum_{\\tau=0}^{t-1}\\sum_{n=1}^N\\sum_{c=1}^NE\\left[Q_n^{(c)}(\\tau)\\right] \\leq \\frac{B}{\\epsilon}\n", "820b70d43ec4be2d0e1c1b936f6aed18": "\\mathrm{Kn} = \\frac{\\mathrm{Ma}}{\\mathrm{Re}} \\; \\sqrt{ \\frac{\\gamma \\pi}{2}}.", "820ba8e0dc66e4874c9d5d981508fdad": "i\\frac{\\partial \\Phi}{\\partial \\xi} + \\frac{1}{2}\\frac{\\partial^2 \\Phi}{\\partial \\,s^2} = 0", "820bbfe1f01da7fb40c8da98b3e2fd6c": "\\sqrt[9]{0.6}\\sqrt[28]{4.9} = 0.99999999754\\ldots \\approx 1", "820be0fed8f61f5e77d501717297ce71": "t_{\\alpha\\beta} = |\\det (d\\phi_\\alpha\\circ d\\phi_\\beta^{-1})|^{-s}.", "820c01b0dd82204b0d96852def5b652d": "\\scriptstyle dB / dr\\ =\\ - D B ", "820c43a9bc83698803a9b1e06ab31f29": "\\binom{n+\\delta-1}{\\delta-1}", "820c6df93f287a8de35cd9be5b163fa9": "+a/2\\,", "820d368682e8fc1956e9c407256240d0": "\\mathrm{\\Lambda}(X)", "820d4936393cb302d19157f42cc5f630": "s=\\bigcup a", "820e33a6bd329e86b03040a30accdc10": " Z = \\sum_{s} \\mathrm{e}^{- \\beta E_s}", "820e4097700d9b7569b1d32a72982bd1": "\\,\\!min(-2 \\overrightarrow{v_r} \\overrightarrow{v_i^0},-2 \\overrightarrow{v_r} \\overrightarrow{v_i^1}) = max(\\overrightarrow{v_r} \\overrightarrow{v_i^0}, \\overrightarrow{v_r} \\overrightarrow{v_i^1})", "820e6215bfbb4f323d7b4f17029aaf5d": "\n\\begin{align}\nP_1(x) & {} =1 + x \\\\\nP_2(x) & {} =1 + x + x^2 - x^3 \\\\\nP_3(x) & {} =1 + x + x^2 - x^3 + x^4 + x^5 - x^6 + x^7 \\\\\n... \\\\\nQ_1(x) & {} =1 - x \\\\\nQ_2(x) & {} =1 + x - x^2 + x^3 \\\\\nQ_3(x) & {} =1 + x + x^2 - x^3 - x^4 - x^5 + x^6 - x^7 \\\\\n... \\\\\n\\end{align}\n", "820e70eb123a5ec6ef96d711a770c194": "\\mathbf{x}=\\mathbf{o} + d\\mathbf{l}", "820ea9960a8c4bf1466ee5fe26e6e6ed": "\\scriptstyle (1 + x)^n = \\sum_{k = 0}^n \\binom{n}{k} x^k", "820eb5b696ea2a657c0db1e258dc7d81": "cm", "820efc4b528fce64d3bb0a0fa34930cd": "3y^2=x^3+7x^2+x", "820f0ab519462adbd4e2505eaffb846d": "\n\\text{If }q \\equiv 3 \\pmod 4 \\text{ then}\n", "820f0ec00729a22d4a23766e861323d4": "H=H(S,P).", "820f4c5defacf9dbe524b9160315adf6": " L_z \\Phi_s = \\pm \\Lambda\\hbar \\Phi_s ", "820f730235ab9b3e6f898d545cb57e7e": " \\beta : B_X \\times B_X \\stackrel{*}{\\longrightarrow} B_X \\stackrel{\\ell}{\\longrightarrow} \\R; \\ \\ \\beta(g,h) = \\ell(g*h) , ", "820f80da1c9490bdeed9209d0d1b75db": "\n\\hat{s} = \\frac{\\vec{r}_1}{||\\vec{r}_1||}\n", "820f8afd811272f995aa5a0a3ab0ee6c": "M_{UT} = \\begin{bmatrix}2.00 & 0.0443 \\\\0.0443 & 0.0104\\end{bmatrix} ", "820fc29cdc7fd47972f574f0c600826f": "\\alpha\\rightarrow\\beta_1\\mid\\beta_2", "820ff4a82250d917d37ea9420060d56f": "x_1,\\dots,x_n\\,\\!", "82103b86d9780e2a9030b4f4de8597ed": "\\max P_X(x) \\langle x | \\mathcal{E}(\\rho_B^x)|x \\rangle~.", "82103e8c43a8542bf3e1647e9bf1674d": "\\cos \\theta_o \\,", "82103ee083578084a09a4406f633755c": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{2-n})(1+p^{-n/2})}", "82104ba64fa1542ab752e62d5904070f": "\\omega(B) \\leq \\frac{1}{2} \\int |df|^2,", "82106d5c89925c351d0866a6cd89576f": "k_\\text{cat}", "8210741e21b4a876782eb3ac38398a53": " a \\triangleright b = a b a^{-1}", "8210806613648cbd78d3cd1864957748": "r_1 = r_2 \\approx 0 \\,", "8210ae1369f0a4b026cf3b317014c8d5": "Y_{out} = y_{22} - \\frac{y_{12}y_{21}}{y_{11}+Y_S}", "8210d6de0525484c8c66f8999518255c": "\\bar{\\theta}", "8211032047fa0fdecb852edaf4643cde": "E \\subset \\mathbb{R}^n", "82117913520c534ac0d08070b9cefe8b": "t_{a}^{i}", "8211be577271b725b68c508237ad01b7": "\\displaystyle{ W(x,y)=e^{ix\\cdot y/2}U(x)V(y).}", "82124cd4a287dd4a839e3a6cd3a00490": "\\frac{1}{b-a} \\int_a^b G(t) \\, dt=\\ G(x).", "821298197ebd03e1d75e6c67af9e854c": "S = \\frac{h_f}{L} = \\frac{10.67\\ Q^{1.85}}{C^{1.85}\\ d^{4.87}}", "8212cd9206f66f04808ccedbc765c2d4": "c_{n-1}\\alpha^{n-1} + c_{n-2}\\alpha^{n-2} + \\cdots + c_1\\alpha + c_0", "8213695e07a19e166d376e34412ef43e": "\\hat{G}_{yy}(j\\omega)", "821386a9d7e7e9ad24921d169cbd3a25": "v(u(\\Gamma_s))={{D(\\Gamma_s)-1} \\over {k}}=G(\\Gamma_s)", "8213a43bd397553328b90865e11ae25f": "R_{\\quad \\mu \\nu }^{\\alpha \\beta }", "8213ad0a882928f8314f33a84ffec8cb": "\\alpha^6+7\\alpha^5+8\\alpha^4-15\\alpha^3+26\\alpha^2-8\\alpha+8=0\\,", "8213d2407604401ec0f79d48bfeed04d": "s'\\,", "8213f11e67660fc60ccdeb73c9f80707": "N_e^{(v)} = N_e^{(F)} = {4 N_m N_f \\over N_m + N_f}", "82141bfbef2fd34006df9ec2d976123a": "ei(X(mech,x_1)) = H[p(X_0(mech,x_1)) \\parallel p(X_0(maxH))]", "821556d74a42dfe0b0b49d366c792591": "S = \\{x, y, z\\}", "82156edc0784f0056d9f18260365105b": "\\exp \\left(-\\beta H(x_1,x_2,\\dots) \\right)", "8215aa330e0fafd1ba1d8779945b88b0": "q_{ab} (x)", "8215c811412e29425321dc8966f83cfe": "10^{3}", "8215c9f1c6c8024cd5d7cd478cd0995a": "t_r =\\left ( \\frac{1+k^1}{1+(t_M/t_c)k^1} \\right )t_M", "8215fd3e95442d37a081968cfeb23d2e": " \\left\\lfloor \\frac{\\lfloor x/m\\rfloor}{n} \\right\\rfloor = \\left\\lfloor \\frac{x}{mn} \\right\\rfloor ", "8216d23b88337edc6de00f652719c8ca": "a^2 + b^2 + c^2 + d^2 = 1.", "8216ddfff9f42ae511bf05a9cd8c22b5": " L [y(t)] = f(t)", "8216e48218aaae9a47bb11595d9cbb2a": "\\int_{a}^{b}i(t)dt=\\int_{a}^{b}C\\frac{dv} {dt}dt.", "8216f39faa05933caf5883bbeaf7c1fc": "\nJ: J^2_0(\\mathbb{R}^2,M) \\to J^2_0(\\mathbb{R}^2,M) \\quad / \\quad J([f])=[f \\circ \\alpha]\n", "82179022f430fdc4978e347076aadc21": "\\mathbf{G}(s)", "82179e7e54aa5ac6434b589366a90ae2": "\\gamma = \\frac{1}{\\sqrt{1-(\\frac{v}{c})^2}}", "82183d9ec6375e5b0d8267fec693c693": "\\lambda(y_i)=C", "82185401faa208014b3373cd006b3a1f": "C_{\\bar v} ", "821886db24809e6036897ba3aa9aa67b": "(a|b)^*a(a|b)(a|b)(a|b)", "8218beb5ef4c2f8449578ac5472508bc": " \\qquad \\qquad \\mathrm{H}_{ph-e} = -\\frac{e_c}{m_e}(a+a^\\dagger)\\mathbf{a}_e\\cdot\\mathbf{p}_e = -(\\frac{\\hbar\\omega_{ph,\\alpha}}{2\\epsilon_o V})^{1/2} (\\mathbf{s}_{ph,\\alpha}\\cdot e_c \\mathbf{x}_e)(a+a^\\dagger)(ce^{i\\mathrm{\\kappa}\\cdot\\mathrm{x}}+c^\\dagger e^{-i\\mathrm{\\kappa}\\cdot\\mathrm{x}}), \t\\ \\ \\ \\ ", "8218c4bc7a48804a95b6b654a43949f3": "\\ln \\rho_{nk} = \\operatorname{E}[\\ln \\pi_k] + \\frac{1}{2} \\operatorname{E}[\\ln |\\mathbf{\\Lambda}_k|] - \\frac{D}{2} \\ln(2\\pi) - \\frac{1}{2} \\operatorname{E}_{\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k} [(\\mathbf{x}_n - \\mathbf{\\mu}_k)^{\\rm T} \\mathbf{\\Lambda}_k (\\mathbf{x}_n - \\mathbf{\\mu}_k)]", "8219536c07bc2d279a6987376b737e71": "\\forall i \\ \\ [u_{i-1},u_i]\\subset U_{\\delta (t_i)}(t_i)", "8219732144b16ec5496bbd8778233348": "\n \\begin{matrix}\n \\underbrace{b_{}^{b^{{}^{.\\,^{.\\,^{.\\,^b}}}}}} \\bar a = {_c} \\bar a\n\\\\ \n c \\mbox{ copies of } b\n \\end{matrix}\n ", "8219ea565b2159e64c39e2dac6ade568": "X^a \\mapsto \\nabla_{(a}X_{b)}-\\frac{1}{n}\\nabla_c X^c g_{ab}", "8219eb3acb9f62e76cdbd679ee72f20f": "\\nu = \\frac {(\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2})^2} {\\frac{S^2_1}{n_1(n_1 - 1)} + \\frac{S^2_2}{n_2(n_2 - 1)}} ,", "821a20d7ebe1f3bf196053d76faf005f": "\\textstyle a(x) + x^bb(x) = d(x)(x^{2l-1}+1)", "821a2a69eac5217559527bc3eb7b2f6b": "\\left[ \\begin{matrix} n \\\\ k \\end{matrix} \\right] = s(n,k)", "821a755793394f96e8ac009e698be085": " P \\otimes Q", "821ad6a24fcab4f45a1c86b63e4a7e0f": "\nv_0 = 0\n\\,\\!", "821b8c852cbfcec919871e474fe11292": "\\bigcap_{i\\in J} A_i.", "821b8fe94536959a0577dd69e4acb01d": "2 \\Delta \\nu", "821baed134422ada05094c9288a8ca77": " M = ", "821bb6108520e7a0fdde8490173fe3f2": "(x',y') = (x, x s+y)\\,", "821bc03e8ce90a22f77a4712e760cbf5": "p = 1/3", "821c14bba64af186da305e2479c91649": "n=1,2,\\dots", "821c3a2d8594569a4ba31f9ce48e9aa2": "m_{inf} (R,T)", "821c5b10e1d1a219bae08d0ff949b5c0": "\n\\int_0^1 \\sqrt{1-x^2}\\; dx = \\int_0^\\frac{\\pi}{2} \\sqrt{1-\\sin^2(u)} \\cos(u)\\;du = \\int_0^\\frac{\\pi}{2} \\cos^2(u)\\;du=\\frac{\\pi}{4}\n", "821c8213f57e32a2f2dbe12b4b70220d": "a^{(N-1)/2} \\equiv -1\\pmod{N}\\!", "821cf607c26e45e0bed9cca077005388": " \\left( a - \\lambda \\right) \\left( b-\\lambda\\right) -c^2 = 0 ", "821d0c659b51b22e126e7de9a153340b": "1 - \\Chi^2_r(X^2)", "821d23dbbeb45b97a21974eb8b6a17a9": " \\begin{align}\n& \\phi \\to \\phi[1]=\\phi\\Lambda-\\sigma\\phi \\\\\n& U \\to U[1]=U+[J,\\sigma] \\\\\n& \\sigma = \\varphi\\Omega\\varphi^{-1} \n\\end{align} ", "821d61f851cbae4b2773973f87b60593": "n_{i1}=f_in_{I1}", "821e1fc93e22d055565bac94badaaf06": "\\mathbf B", "821e355c95f02bdef81e0642dd1d3f29": "V=\\frac16\\pi h\\left[3\\left(r_1^2+r_2^2\\right)+h^2\\right]", "821e7bf22cc7660f3f998f8450294165": "\n\\begin{alignat}{2}\n & \\mathrm{Glucose} + \\mathrm{ATP}\\xrightarrow[\\mathrm{Phosphorylation}] {\\mathrm{Hexokinase} + \\mathrm{Mg}^{++}} \\textrm{G-6PO}_4 + \\mathrm{ADP} \\\\\n & \\textrm{G-6PO}_4 + \\mathrm{NADP}\\xrightarrow[\\mathrm{Oxidation}] {\\textrm{G-6PD}} \\textrm{G-Phosphogluconate} + \\mathrm{NADPH} + \\mathrm{H}^{+} \\\\\n\\end{alignat}\n", "821ec559d84359254db9117508e543f1": " \\frac{i\\omega}{\\omega_c'} \\to Q \\left( \\frac {i\\omega}{\\omega_0}+\\frac {\\omega_0}{i\\omega} \\right)", "821edb77e878c351632784dd82453af7": " \\exp \\left( i t \\right) ", "821f4d26b2853509fcde885a715b761f": "nR = H(W) = H(W|Y^n) + I(W;Y^n)\\;", "821f5f456c678b0f36dcb00ef105349f": "y = Y(u, v).\\,", "821fc5071f6bdb70efa18c1ce40a95b6": "\\frac{1}{\\rho}\\frac{\\,d\\left(\\,P-P_e\\right)}{\\,d\\,X}+\\tfrac12\\,\\left[\\frac{\\,f}{\\,D}+\\frac{f_e}{D_e}\\left(\\frac{F}{F_e}\\right)^2\\right]\\,W^2\n+\\left[\\left(\\,2-\\beta\\right)\\,-\\left(\\,2-\\beta_e\\right)\\left(\\frac{\\,F}{F_e}\\right)^2\\right]\\,W\\tfrac{\\,dW}{\\,dX}\\,=\\,0", "821fe993c064e58d6110c5809c2ad971": " P_1 = \\left[ \\begin{matrix} \\exp(i \\theta/2) & 0 \\\\ 0 & \\exp(-i \\theta/2) \\end{matrix} \\right] ", "82201f99193092a3aab13d6a375bd0ba": "f(z) = \\sum_{n=-m}^\\infty a_n e^{2i\\pi nz}.", "8220b0b78def6d3a3b750e373626d7f9": " k_X ", "8220b3eb38848bf7933fbff5ff4b8e15": "x(0) = x_0, \\;\\;\\;\\; \\xi(x(0)) = \\nabla u(x(0)).", "8220b7e1ad412885509333ab34c3f36e": "a(t) = 2\\cdot 4^t-1", "8220d360d07643d6a09ec18b53f98517": "z(1-z)\\frac {d^2w}{dz^2} + \\left[c-(a+b+1)z \\right] \\frac {dw}{dz} - abw = 0.", "8220fee7e78d309096b420aec20ea4c1": "A = 250a^2 \\cot \\frac{\\pi}{1000} \\simeq 79577.2\\,a^2", "8221435bcce913b5c2dc22eaf6cb6590": "2.5", "82218f582e4ad996b71e2edc8cfefa88": "\n f_1(x) = f_2(x) = \\sum_{m=1}^\\infty E_m\\sin\\frac{m\\pi x}{a}\n", "8221a2f059ee4312a4f8419caf582f1f": "{\\Bbb C}P^{n-1}\\times {\\Bbb C}P^{m-1}", "822290d41217c4ef3ce3d0a32a44323a": "\\theta_\\text{crit} = \\arcsin\\left(\\frac{n_2}{n_1}\\sin\\theta_2\\right) = \\arcsin\\frac{n_2}{n_1} = 48.6^\\circ.", "8222b0da280944de2be767c411db9611": "{\\rm d} x/{\\rm d} x", "8223452c9289563679cc172e5531e768": "153=1^3+5^3+3^3", "822364b4d770d0e259abba81fdde3dc0": " \\delta Q_{rev} = 0\\,\\!", "82236931ec8b9fbb8635231f1e7f06ca": "*[F , *G]^{IJ} = {1 \\over 2} \\epsilon_{MN}^{\\;\\;\\;\\;\\;\\; IJ} (F^{MK} (*G)_K^{\\;\\;\\; N} -(*G)^{MK} F_K^{\\;\\;\\; N})", "8223e84d047a75c7e1873efcb0c6b09d": "{\\nabla}^2 \\Phi = 4\\pi G \\rho.", "8224013f5f5bea2ccf6474bf99a32442": "\\lambda = - \\beta(x)", "822418bf9592747019b5942bdf67fdb0": "es \\in E \\Longrightarrow s \\in E.", "822419440ad07b900c01269fc21e6e9a": "f_x(1,0) = p_x(1,0) = a_{10} + 2a_{20} + 3a_{30}", "82242da728a9e9c402aaccddc68ddf85": "1 + 2 + \\cdots + n = n(n+1)/2 = O(n^2)", "82243d17ec9725a0004269ddc898ce2d": " a = \\frac{v_e^2}{2l} ", "8224591d295ba6f0f6b3ab7859e44f23": "\n\\beta = 1- \\frac{ Expected~backorders~per~time~period}\n{Expected~period~demand}\n", "82246f5037571adfd93ec2e0a5898aaf": "x = -5 \\quad\\text{or}\\quad x = -1.", "8224e1f266c040ed4b0a81c21e5a76ff": "\\frac{\\pi^2\\sinh(s t)}{st(\\pi^2+s^2 t^2)}\\,e^{\\mu t}", "822516c7447766e9760f3e7f521bab35": "\\begin{align}A(z) &{} = \\prod_{\\beta \\in \\mathcal{B}}(1 + z^{|\\beta|}) \\\\\n &{} = \\prod_{n=1}^{\\infty}(1 + z^{n})^{B_{n}} \\\\\n &{} = \\exp \\left ( \\ln \\prod_{n=1}^{\\infty}(1 + z^{n})^{B_{n}} \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{n = 1}^{\\infty} B_{n} \\ln(1 + z^{n}) \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{n = 1}^{\\infty} B_{n} \\cdot \\sum_{k = 1}^{\\infty} \\frac{(-1)^{k-1}z^{nk}}{k} \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{k = 1}^{\\infty} \\frac{(-1)^{k-1}}{k} \\cdot \\sum_{n = 1}^{\\infty}B_{n}z^{nk} \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{k = 1}^{\\infty} \\frac{(-1)^{k-1} B(z^{k})}{k} \\right),\n\\end{align}", "822578a4f777a3c54aa32213abaede98": "MK=(MK_0,\\ MK_1,\\ MK_2,\\ MK_3)", "822582c6e86a5fd0433f842966538c0d": " \\eta_0 |\\Psi\\rangle = 0 ", "8225c7791410b47d397e81f3edef7b49": "[\\hat{A_{\\pm}},\\hat{H}_{\\text{JC}}]=\\Omega_{\\pm}A", "82266710c7e67796962cdf6a52f24185": "r_m = a\\, \\frac{\\varphi^2}{2}", "8226c96f990b2702ccdcd78b7f63ed59": "F_t= \\sin(u)\\ F\\,", "8226da50af0cf14ac91994ab1f29c62d": " \\cos E = \\frac{x}{a}\\quad \\mathrm {and} \\quad \\sin E = \\frac{y}{b} \\ .", "822772778747bbc64e7d5d73594dd73e": "Q_\\max:E(Q>Q_\\max)>Qm", "8227d2c6d0877307586fd47372c78ea1": " -\\frac{\\hbar^2}{2m}\\nabla^2\\psi(\\mathbf{r}) + V(\\mathbf{r})\\psi(\\mathbf{r}) = E\\psi(\\mathbf{r}) ", "82283d303cac9192622a9971031e0940": "y^* =\\frac{y}{L_c}", "8228920949bcec7bb2898bf1a1c3a035": "G(x)=\\frac{1}{2\\pi} \\int_0^1\\int_{-\\infty}^\\infty |f^\\prime(x+ty)|\\,dy", "8228f74b4df8f51c8af6f2d90cb3514c": "V_{0}=218", "8229243e32672c50a0b86a12cd894a14": "D = -4(\\sin\\theta)^2", "82292845a9a19570dc310b24517d20b8": "\\omega^2 = G \\, \\frac{M}{d^3}.", "822939e2e11975bcd58fe726f32600da": " \\{ \\, | e_i \\rangle \\, \\} ", "8229be4ea633624cf1d4923a16dd5818": "Z'_{St}", "8229e0f77212c01a7b601da0bf3d1ba8": " q_{k+1} =y_k + q_k - x_{k+1}. ", "8229f3471ab7f1b338f91f2a5bf1e659": "\\mathbb{S}^d=\\bigcup_{\\langle T_i: i \\in d \\rangle} S^d_{\\langle T_i: i \\in d \\rangle}.", "8229ffb8d729626772a9c411282a377a": "\\textstyle \\mathrm{Distance\\ in\\ parsecs}=\\frac{1000}{\\mathrm{parallax\\ in\\ milliarcseconds}}", "822a0457182f184f623f80c11d953124": "X = \\frac{1} { \\cos\\, z + 0.50572 \\,(6.07995^\\circ + 90 - z)^{-1.6364}} \\,,", "822a2295ce34ac36cee959ef20f2fe5d": "[X]\\in D", "822a2fba73008a6aa4fdd6fcb0f55011": "l_{eff} = V_0 / E_s \\, ", "822a955ef21c217a5d61657b59fa18ab": "w \\ne 1/4", "822b19ece2f860c95b987538a283a1a8": "[0 : 1]", "822b21762e4091540ee0d42d854044ca": "\\mu \\nabla^2 w", "822b81da835cb8d52dafc7423567357f": "(x,y)\\,\\!", "822b97dfab637271b2157518f80affd7": "P(E,\\Omega)\\leq P(E,\\Omega_1)", "822bcf440ea18b95bfa133fbc33e0342": "P\\left(Searched|Known\\wedge\\delta\\wedge\\pi\\right)", "822bdd7d593e4a01561490a3ba4ae28b": "R^r f_* \\to R^r f_* \\circ g'_* \\circ g'^*.", "822c308850944cac6dc7776523e56782": "\\Lambda(P)=-\\log P(x)+K(P) \\geq K(x)-O(1)", "822d3012a4a30b4dca5bd9756adeaa1c": "\\, b_L = x - 5 \\,", "822d31576e8a244b9036002cc1ddab49": "\\dot{q}^i = \\frac {\\partial H}{\\partial p_i}", "822d42d12339f95834e84778edc6d06f": "perceived-reduction % = 100 \\times \\sqrt{actual-reduction % \\over 100}", "822d464ec15fec0db3077e296012432e": "\\frac{(24V)^{1/8}}{\\sqrt{\\pi}}", "822d8c7446df95bb924b8dcddcf083c5": "A \\xrightarrow{f} B \\to C(f) \\to", "822db3846c145a42612ec585e8f82ffb": "\\nabla_{\\mathbf v} f{\\mathbf u}=f\\nabla_{\\mathbf v} {\\mathbf u}+{\\mathbf u}\\nabla_{\\mathbf v}f", "822dbfa271ed99bedbe60b86610fdd6a": "h(x) = x^2 + x - 1", "822dd1816f25b4deb1f4c9d5913a286a": "x^13", "822ded7cbe09583637121a4c7aabd06c": "\\mathbf{B}(\\mathbf{r},t)=-\\frac{\\mu_0q}{4\\pi}\\left[\\frac{c\\,\\hat{\\mathbf{n}}\\times\\vec{\\beta}}{\\gamma^2R^2(1-\\vec{\\beta}\\mathbf{\\cdot}\\hat{\\mathbf{n}})^3}+\\frac{\\hat{\\mathbf{n}}\\times[\\,\\dot{\\vec{\\beta}}+\\hat{\\mathbf{n}}\\times(\\vec{\\beta}\\times\\dot{\\vec{\\beta}})]}{R\\,(1-\\vec{\\beta}\\mathbf{\\cdot}\\hat{\\mathbf{n}})^3}\\right]_{\\mathrm{retarded}} \\qquad (1)", "822dfa4cb3c47bbc205b5316ff698e74": "\\frac{\\pi r^2}{h^2}\\left(\\frac{h^3}{3}\\right) = \\frac{1}{3}\\pi r^2 h.", "822e371ddb8f66bcfd5941f41c7d5133": "\\frac{d\\mathbf{u}_\\mathrm{n}(s)}{ds} = \\mathbf{u}_\\mathrm{t}(s)\\frac{d\\theta}{ds} = \\mathbf{u}_\\mathrm{t}(s)\\frac{1}{\\rho} \\ ; ", "822e88881057f2e5a45374a846751288": "-0.6038 .. 7.5913", "822ec6f0d9b1e0e01d509b1cbde146bf": "(\\lambda_1,..., \\lambda_n)", "822f2349b010b90867ac6d79f53602be": "[x, y]", "822f2f5606660daa461794ee5f55984b": " G(z) = \\sum_{n\\ge 1} \\left(\\frac{1}{|C_n|}\\right) g(z)^n = \n\\log \\frac{1}{1-g(z)}.", "822f5a24955a28ee0d6d9e65bba04445": "M=B\\frac{d^2 \\theta}{dt^2}", "822f893e5aee6193bbeae66315439fd2": "{n \\over p} - 1 = n'-1", "822f8a8e63932b68b0770a52a4e8a18e": "\n\\begin{align}\ne_0 & = 1 \\\\[6pt]\ne_1 & = \\sum_i x_i & & = \\sum_i \\tan\\theta_i \\\\[6pt]\ne_2 & = \\sum_{i < j} x_i x_j & & = \\sum_{i < j} \\tan\\theta_i \\tan\\theta_j \\\\[6pt]\ne_3 & = \\sum_{i < j < k} x_i x_j x_k & & = \\sum_{i < j < k} \\tan\\theta_i \\tan\\theta_j \\tan\\theta_k \\\\\n& {}\\ \\ \\vdots & & {}\\ \\ \\vdots\n\\end{align}\n", "822f8cf6515199917b500aa6e9784708": "34 n^3 + 51 n^2 + 27 n+ 5", "822f8e01473e6693a1640ed618641a77": "(10)\\quad \nds^2=-e^{2\\psi(r,\\theta)}\\,\\Big(1-\\frac{2M}{r} \\Big)\\,dt^2+e^{2\\gamma(r,\\theta)-2\\psi(r,\\theta)}\\Big\\{\\,\\Big(1-\\frac{2M}{r} \\Big)^{-1}dr^2+r^2d\\theta^2\\,\\Big\\}+e^{-2\\psi(r,\\theta)}r^2\\sin^2\\theta\\, d\\phi^2\\,.\n", "822fa5c38894d832f534cc157e921b5b": " \\displaystyle{[a_m,a_n]={m\\over 2}\\delta_{m+n,0},\\,\\,\\,\\, [b_m,b_n]={m\\over 2}\\delta_{m+n,0}},\\,\\,\\,\\, a_n^*=a_{-n},\\,\\,\\,\\, b_n^*=b_{-n}", "82300450496736f32867fae5fcc3ddc4": " f = \\sum c_k h_k,", "823005e0153c444ed2a2ad9481d647c5": "\\frac{\\partial \\phi}{\\partial x_i} + \\frac{\\partial \\phi}{\\partial y}\\frac{\\partial y}{\\partial x_i} = 0 ", "82300a2a5fc92f29de75d9e7d15d6885": "E(t,f)", "8230a85e35ffc590f8b0298cba133ad2": "D_i=\\left(C_0^*e^{-\\hat{\\alpha}'_i-\\beta^T{X}}\\right)^{1/d}", "8230f32a8811ceb1e986c2879d692b9e": "\\tau_{a}", "823144e0c7e3ca28401f6baed87f39a0": "H^2(M^{2n}; \\mathbb{Z})", "8231dab183397c1299e3a17d200c4fac": " \\mathbf{F} = \\frac{\\gamma(\\mathbf{v})^3 m_0}{c^2} \\left( \\mathbf{v} \\cdot \\mathbf{a} \\right) \\, \\mathbf{v} + \\gamma(\\mathbf{v}) m_0\\, \\mathbf{a}.", "8231fb50b22364bed871fb66087ca140": " \\gamma_{B,A} \\circ \\gamma_{A,B} = Id", "823200a71b73194f8a516cdd869d5552": "{d^2 x \\over dt^2} = - m \\omega^2 x.", "823219be29a13efc93fe881f102719f2": "Q=Q_A \\cup Q_N", "82325f1eb018a62698ea59d6893982f0": "\\hat{\\rho}(\\tilde{p}):= \\max \\ \\{\\rho\\ge 0: p\\in P(s),\\forall p\\in B(\\rho,\\tilde{p})\\}", "8232684c84302cd613f4025ae0a24b1c": "E = \\frac{F - 650}{8}", "823274b070946e76854e034891211cb7": "{{\\Delta}E_t(S_{t + k})}", "82329f3706cde007f75128c7a3ebfb42": "\\scriptstyle \\sqrt{1/2} \\ \\approx \\ 0.707", "8232a9132f270e9caa4b8f4839cf2c87": "C^N", "8232adee4cb2caaa237fb64e425fa25a": " V_1-V_2 = \\frac {V_B}{A_v} \\ .", "82334efe209a2d09c53233dc99106c81": "\\theta = \\lbrace a, \\mathbf{R}, \\mathbf{t}\\rbrace", "823377e13ea4857bb1db81eaedd8fc2e": "C^\\perp", "8234145dc197774224cade1049571f44": "\\psi_n(x,t) =\n\\begin{cases}\n\\sqrt{\\frac{2 }{L}} \\sin(k_n x - \\frac{n \\pi x_0}{L})\\mathrm{e}^{-\\mathrm{i}\\omega_n t}, & x_0 < x < x_0 + L,\\\\\n0, & \\text{otherwise,}\n\\end{cases}\n", "82342cf92532335c24ce300b11ea6a01": "\\lambda = n", "823445cac0d8d76bf05726a309a61a8f": "\n\\begin{align}\n\\Pr(Y_i = 1) &= \\Pr(Y_{i,1}^{\\ast} > Y_{i,k}^{\\ast}\\ \\forall\\ k=2,\\ldots,K) \\\\\n&= \\Pr(Y_{i,1}^{\\ast} - Y_{i,k}^{\\ast} > 0\\ \\forall\\ k=2,\\ldots,K) \\\\\n&= \\Pr(\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i + \\varepsilon_1 - (\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i + \\varepsilon_k) > 0\\ \\forall\\ k=2,\\ldots,K) \\\\\n&= \\Pr((\\boldsymbol\\beta_1 - \\boldsymbol\\beta_k) \\cdot \\mathbf{X}_i > \\varepsilon_k - \\varepsilon_1\\ \\forall\\ k=2,\\ldots,K)\n\\end{align}\n", "8234554031f3ea3fbd4e568914e8ae7a": " \\mathrm{Var}[X] = \\mu_2' = \\kappa_2 + \\kappa_1^2 \\, ,", "8234969ebc31a97f3c17f8b0e39317af": "L_{QY} = \\frac{\\mu_0}{\\lambda_0} = H/m^2 \\ ", "8234c8a9274213c17ac5efbdd048bff2": "d \\approx 4.12 \\cdot \\sqrt{1500} = 160 \\mbox { km.}", "8234da36925d523efb8231b0c68fc0fb": " W(x) ", "8234f25406fb510dc42d5e79c688deea": "\n\\left\\lfloor\\frac{n}{m}\\right\\rfloor \\le k_i \\le \\left\\lceil\\frac{n}{m}\\right\\rceil,\\ \\sum_{i=1}^m{k_i} = n,\n", "82353ed6e2c35b2a5154395e0e9f8d42": " f\\ x = x^2", "82353f2ace9f0134589972809dc46984": "2\\over\\sqrt{15}", "82355e287fd7158fcb6cd79102ee34d1": "\\frac{N_2}{N_1} = \\exp{\\frac{-(E_2-E_1)}{kT}},", "8235a46d1741b811e5241b7b3a359dce": "\\alpha t/d^2", "8235b7fa5d6dd5182fc0f994916f9655": "G(B) = \\operatorname{Hom}_{\\mathcal{L}} (B,I)", "8235ebd2c2f09892361b634cee62acaf": "\\int_{-\\infty}^{\\infty}dx\\,P(x,p)=|\\varphi(p)|^2", "8235f9c43ac1f83ebde98fae3b3b7a65": "h_\\mathrm{int}", "823612d0693bddf1070c7da50dc2c553": "19.6~", "823635ce4b6cb8611120147d5c63298a": " \\int_\\mathcal{V} \\partial_\\gamma T^{\\beta\\gamma} cdt dxdydz = \\oint_{\\partial \\mathcal{V}} T^{\\beta\\gamma} d^3 \\Sigma_\\gamma = 0 ", "823694a00c7e3658af935685f8fe7250": " {\\dot{m}}_D ", "82370d701c0522474447ef955ff7aa82": "\\,\\phi", "82371df2f284e3bc798a0cfebb01705b": "I_{AB} = \\frac{g_se}{h}\\int_{E_{F_B}}^{E_{F_A}}{M(E)f^'(E)T(E)dE}", "823725a8211b8144ffe7fd5a8b78af72": "\\Delta\\rho= \\rho_1-\\rho_2", "823735c1fb02411a553e43b1a320028c": "\\mathbf A (\\mathbf{r}, t) = \\frac{\\mu_0}{4 \\pi} \\int \\mathrm{d}^3 x^\\prime \\frac{\\mathbf{j}( \\mathbf{r}^\\prime, t_r)}{ \\left| \\mathbf{r} - \\mathbf{r}^\\prime \\right|},", "8237597da9ec2f86d168a84ff12c26a2": "q = a + bi + cj + dk \\!", "8237c459e42727c1e1792ef4d6b2f0f0": "\\frac{\\mathrm{d}^2}{\\mathrm{d} x^2}\\left(EI \\frac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\\right) = q.\\,", "82381ee77d44d5b5ac6401c4a3d1f244": "K_I = \\sigma \\sqrt{\\pi a}\\,", "8238d23d43813de09f877e758c97e198": "\\frac{128}{65}", "8238d29a3a36595e906d042b8d936c0d": " \\frac{0.0248*\\frac{1000 \\mbox{ g}}{18.053 \\mbox{ mol}^{-1}}}{1-0.0248}*151.17 \\mbox{ mol}^{-1} = 213.4", "8238d9abb6f9f4f4b4ceca4440a7b7d9": "I(x):=\\{t \\in T : (t,x) \\in U \\},", "8238dcbafeed6abe4c67e6c028ee0062": "w,b,d,", "82390d0e44f651b70171035aa42e64ee": "c=\\inf\\{\\,x\\in(a,b]\\mid |f(x)-f(a)|>\\varepsilon(x-a)\\,\\}.", "823921d4ad84e25204490f7b5777969b": "\\sigma_\\infty = f\\sigma_f + \\left(1-f\\right)\\sigma_m", "82398074e3f5d6186e0f855df146ffa7": "\\text{iii) Inv}(s_A,\\boldsymbol u_i) =s_A/\\max_{j=1,\\ldots,m}\\{u_{ij}\\}", "8239df00e29c53fceb2341336baec872": "\\mathcal{I}(\\theta) = \\mathrm{E}(\\mathcal{J}(\\theta))", "8239fd77ff55354ce83c68e38e16e7b5": "\\mathbf{R}_i=\\mathbf{R}+\\mathcal{R}\\mathbf{r}_{io}", "823a3f622bfef53624853808a11435cf": "\n2\\rho\\Omega\\times{\\mathbf u}=\\nabla \\Phi -\\nabla p,", "823a46425aa2c16377bad746dafcea94": "{\\phi_n(x)}", "823a4aaa818ae7b750266b04dca4a699": " \\xi < 0 \\,", "823a65a32e6402048b076b1dd3ebf5eb": "\\langle T\\varphi,\\psi\\rangle = \\langle\\varphi, T^*\\psi\\rangle", "823a75d54eb9b2228a91c6675e392bcf": "G = G^{\\alpha}\\,+ G^{\\beta}\\, + G^{\\mathrm{S}}\\,,", "823aacb6bc34912b203dd292b6ee9db3": " E = h\\nu ~.", "823ae357cfd8f732094c5a08b6ecf8b2": " \\star^{-1} = s\\star ", "823b01bc5a2f5ea1826714409185e467": "A - \\left(V^{M}_{N \\setminus \\{b_i\\}}-V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_i\\}}\\right).", "823b3fa6c0178066808120ab4b8eba6e": "E = X_1^4 + X_2^4", "823b57aa89886edcf6ea229c33dc7fa8": "=\\operatorname E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )^2\\right ]+\\operatorname E\\left [ \\left ( m(\\vartheta)-\\Pi\\right )^2\\right ]", "823bb032336a607fc48b85369e4d36c9": "\\scriptstyle (\\Omega)", "823bc7611ba0036d3550e8a648c6418f": "\\pi \\approx 3", "823c1329c8c59845e265785c992501b6": "\\, b_R = y - 5 \\,", "823c1deaf631edd12a4a53b6ecf1acd3": "\\int f\\,dg = f g - \\int g\\,df.", "823c6b9005dfc7b9f62d81b1535972ae": "B_{n}=\\frac{1}{\\pi}\\displaystyle\\int^{2 \\pi}_0\\! f(x) \\sin{nx}\\, dx\\qquad (n=1,2,3, \\dots)", "823c9ff3334a493bded8a11290465a20": " a_i \\ge g_i ", "823d26fb58bf4d18b40ec8af0ed92422": "\\mu<\\mu^{cf(\\mu)}\\le\\mu^{\\kappa}=(\\lambda^\\kappa)^\\kappa=\\lambda^{\\kappa\\cdot\\kappa}=\\lambda^\\kappa=\\mu", "823d27e1f9aab3646721076afe3cd761": "V(x;\\sigma,\\gamma)=\\int_{-\\infty}^\\infty G(x';\\sigma)L(x-x';\\gamma)\\, dx',", "823d445d0e5099115adacba066aaf0f3": "P=\\frac{e^2}{6\\pi \\varepsilon _0 c}\\gamma ^6\n\\left [ \\left | \\dot{\\vec{\\beta }} \\right |^2\n-\\left | \\vec{\\beta}\\times \\dot{\\vec{\\beta }}\\right |^2 \\right ]\\qquad (5)\n", "823dae0f1452276e801d270903951d2c": "G_2(\\bold{q},\\boldsymbol{\\alpha},t)=S(\\bold{q},t)+A, ", "823dc0c23977342ab11987251cb7d73b": "\\cos(\\bold{k}\\cdot\\bold{r}) = \\frac{1}{2} [ e^{i \\bold{k}\\cdot\\bold{r}} + e^{-i\\bold{k}\\cdot\\bold{r}}] ", "823dd172ee4bf576bebce1edd32a9a9a": "G\\colon (X\\times \\{0\\} \\cup A\\times I) \\rightarrow Y", "823e00e0d588aa080659ec090a6bdecb": "\\sum\\limits_{i_1,i_2, \\cdots ,i_k\\in\\left\\lbrace 1,2, \\cdots ,n\\right\\rbrace} a_{i_1,i_2, \\cdots ,i_k} X_{i_1} X_{i_2} \\cdots X_{i_k},", "823e015770155ccd2679cfc22a7d15e1": "\\mathit{(x, Y)}\\,", "823e1cdfb182ecf1b305e125d6bbe189": "\\bar{F}(x) := \\Pr[X>x] = \\int_x^{+\\infty} f(u)\\, du", "823e25f74d569104cd09ca0f1ca669a2": "q_0 \\ge 0", "823e34d39dcc12d1d8d0a2ff4ee748cb": "\\,l_{x+t}", "823e3a393b5cc9e1b4a0cf91ed6d7e3b": "{e}^{x}\\propto x\\text{ },\\text{ }\\left(x\\right)< 1", "823e5e69fba800c58bb06b3c7d4e365d": "\\mathbf{F}' = \\mathbf{F} - 2m \\mathbf{\\Omega} \\times \\mathbf{v}_{B} - m \\mathbf{\\Omega} \\times (\\mathbf{\\Omega} \\times \\mathbf{x}_B ) - m \\frac{d \\mathbf{\\Omega}}{dt} \\times \\mathbf{x}_B \\ , ", "823e7b721d2b88be99cab59da7519dfd": "\\scriptstyle 26", "823e7b85458466ddc2f1f720090d3026": "\\phi R\\,", "823f2301626a76bfc6793b841506dad2": " k_0^2 - \\omega_0^2\\, \\mu_0\\, \\varepsilon_0 = 0.\\, ", "823f23f27f18718fcd0e25eace285546": "\n\\begin{align}\n\\nu\\omega_1&=a\\omega_1+b\\omega_2\\\\\n\\nu\\omega_2&=c\\omega_1+d\\omega_2\n\\end{align}\n", "823f273c31924792e75f0c6f9cbdb673": "p=\\frac{x\\frac{\\partial f}{\\partial x}+y\\frac{\\partial f}{\\partial y}}{\\sqrt{(\\frac{\\partial f}{\\partial x})^2+(\\frac{\\partial f}{\\partial y})^2}}.", "823f28c70e8a271d4b88a14ccc761dbb": " \\mathbf{v}_i\\otimes \\mathbf{w}_j", "823f9f976de497a741f54e196eebb3ef": "\\textstyle{2x \\over x^2-1}", "823fc8ae83da975306f53142bd37b5d5": "R = I + \\sin\\theta [\\mathbf{k}]_\\times + (1 - \\cos\\theta) (\\mathbf{k} \\mathbf{k}^\\mathsf{T}-I)", "823fdcf6875935db40be587f64f346d4": "v_{n+1} = U_{2n + 1}(\\beta), \\ v_{n} = U_{2n - 1}(\\beta), \\,\\!", "823fee20ce2fa3985c7dc9ad97773085": "\n\\mathbf{F}_{21}(\\mathbf{x}_{1},\\mathbf{x}_{2}) = m_{2} \\ddot{\\mathbf{x}}_{2} \\quad \\quad \\quad (\\mathrm{Equation} \\ 2)\n", "824016ce94a31fa2ef77c41d9951cd14": "^\\prime", "8240592985be5e3e70b2463150a8a1e6": "{\\frac{(s-1)}{s}}", "8240b6e7d08f13ed4d206b31c22a01bd": "\\epsilon_{ox}", "824109116a23a5af178c2b4f351580ca": "\n\\sum_j {e^{ - \\lambda _i - \\lambda _j - \\beta C_{ij} } } = T_i;\n\\sum_i {e^{ - \\lambda _i - \\lambda _j - \\beta C_{ij} } } = T_j\n", "82412732f62567cec1a46ccf4e05ac3a": "\\mathcal{L}_{X}", "82413237c7835a85613fcccb4a1f882d": "e = \\sum_{k=1}^\\infty \\frac{k^2}{2(k!)}", "824133cf312b9764f2b786f1749cc82a": "{{Thames Tonnage}} = \\frac {({length}-{beam}) \\times {beam}^2} {188}", "824143654770eb5b2ac70a8ce050ba81": "S' \\cap S_{k-1}^{\\perp} \\ne {0}.", "824187a8ef3f9aade0b04ad970d94040": " \\left ( \\frac{Power_2}{Power_1} \\right ) ^ 2 ", "82419290c5e2eb163d2c93f69e987b00": "\\frac{\\pi}{\\sqrt{18}} \\simeq 0.74048.", "824264070801cbd92d2240e37bdff02b": "c_1=", "824267d3bf303e2669d9ee1554e5cabd": "\\phi^x_t", "824267ede99c5da22b924360a22426df": "\n\\overline{\\theta}=\\mathrm{Arg}(\\overline{\\mathbf{\\rho}}).\n", "82428735fb49ef2e268477b9fb788d5e": "f_i = \\Pr\\{N = i\\}", "82430baed7a80713691715a99efd0ce1": "\n\\left(\\frac{\\partial \\mathcal{E}}{\\partial T}\\right)_Z=\n-\\left(\\frac{\\partial S}{\\partial Z}\\right)_T\n", "8243357c23dce51f13ac4752ad7671a4": "d=(m^2-3n^2)/g, \\, ", "8243937d2dfb52ba8899a2717aefc7ef": "d:=\\deg(b); \\quad c:=\\text{lc}(b);", "8243a363ce72204d2ba5c5b433cea6de": "Q = 1 - \\frac{W}{n!} = \\sum_{p=0}^n \\frac{(-1)^p}{p!}, ", "8244a463fca7342ef08f1933b6e20e55": "f(k)=k", "8244c908cae70823a338ed2ebfffadc6": "M_X^c + \\overrightarrow{XG} \\times mg = \\dot {H}_G + \\overrightarrow{XG} \\times ma_G", "82458ba315830a07fea8f106e53669f7": "\\{ q_i , p_j \\} = \\delta_{ij}", "82462eea2d9005f015785c4bb68a42e9": "\n\\begin{align}\n\\left\\{\\begin{matrix} n \\\\ 2 \\end{matrix}\\right\\} & = \\frac{ \\frac11 (2^{n-1}-1^{n-1}) }{0!} \\\\[8pt]\n\\left\\{\\begin{matrix} n \\\\ 3 \\end{matrix}\\right\\} & = \\frac{ \\frac11 (3^{n-1}-2^{n-1})- \\frac12 (3^{n-1}-1^{n-1}) }{1!} \\\\[8pt]\n\\left\\{\\begin{matrix} n \\\\ 4 \\end{matrix}\\right\\} & = \\frac{ \\frac11 (4^{n-1}-3^{n-1})- \\frac22 (4^{n-1}-2^{n-1}) + \\frac13 (4^{n-1}-1^{n-1})}{2!} \\\\[8pt]\n\\left\\{\\begin{matrix} n \\\\ 5 \\end{matrix}\\right\\} & = \\frac{ \\frac11 (5^{n-1}-4^{n-1})- \\frac32 (5^{n-1}-3^{n-1}) + \\frac33 (5^{n-1}-2^{n-1}) - \\frac14 (5^{n-1}-1^{n-1}) }{3!} \\\\[8pt]\n& {}\\ \\ \\vdots\n\\end{align}\n", "82469207d80389ec2181f868ebb9f49b": "\\Sigma _i x_i^* \\in Y + \\{\\omega\\}", "8246b3a03f298ee0313ac521c32f9b37": " \\sec \\theta =\\!", "8247258c58843dc4adf1daa46de11e69": "\\vec a=\\vec b-\\vec c\\,,", "8247408ad6fa5bbc3eb4975662fe99f2": "R=\\left\\{-1, +2\\right\\}.\\,\\!", "82478b023fdaf0352440e31ad352c96e": "\\left\\{ \\begin{matrix}\n 1 & \\mbox{if } k = 0 \\\\\n \\frac{\\sin\\left(\\frac{\\pi W k}{N}\\right)}\n {W \\sin\\left(\\frac{\\pi k}{N}\\right)} & \\mbox{otherwise}\n \\end{matrix} \\right. ", "8247985985422e7270004c483d2cf589": "Qe^{rt} = \\int_0^t Q e^{r(t - a)}\\ell(a)b(a) \\, da ", "8247f4f8c596e43b0c5059d9bbdbc1df": " d \\sigma^1 = p_y \\exp(p) dy \\wedge dx = -\\left( p_y dx \\right) \\wedge \\sigma^2 = -{\\omega^1}_2 \\wedge \\sigma^2", "8247f65212a145b5c215e46425ea1953": "\\Phi_S", "8248818dfd480c88f57637677ceac92b": "[A,B] = AB - BA.\\,", "824894065bef7120ef7e458dcc31add0": "\\gamma_{r}", "8248b3c6a68c88d20a6c5f2f2aadf39f": "\\pi(r - r')", "8248b74ce0a094db048bd97af91166fa": "\\left(\\frac{\\partial p}{\\partial V}\\right)_T = \\left(\\frac{\\partial^2p}{\\partial V^2}\\right)_T = 0", "82491a87b35a3b354af3685cb0389eec": "(2n+2)\\pi", "8249270668da8ed8ffa97626abfc9cee": "\\boldsymbol{L}_{y}", "82493e8fa292c77426082269c20bede1": "M,w \\models S", "8249455993618da958ab67e148ae59b2": "\\omega^{A}_{D} = (\\omega^{A}_{B}\\otimes \\omega^{B}_{D}) \\oplus (\\omega^{A}_{C}\\otimes \\omega^{C}_{D})\\,\\!", "824966ca06788a01f5f1568572cf4f97": "y' = y^2", "824967b2fc4ef32b651aec9f6a455440": "max \\sum_{i=1}^{N} C_x(p_*)-C_x(p_i)", "8249be5dda3fefa94ab414683d4f7dcc": " x_i \\in \\mathbb{R}^n", "8249cb7e30221b3b3f30b0877b937127": "C=\\frac{p_x\\rho r^4}{\\eta \\dot{m}}", "8249cd6b9cb3e1f09df43264275a6e8c": "d_i: S_n(X)\\to S_{n-1}(X)", "8249f2bd66050505806c184b9ea4572f": " \\dim\\left\\{\\R^k[x_1,\\ldots,x_m]\\right\\} = \\sum_{i=1}^k \\frac{(m+i-1)!}{(m-1)! \\cdot i!} = \\left( \\frac{(m+k)!}{m!\\cdot k!} - 1 \\right) . ", "824a3253f048d926133d638b67b52852": "\\mathbf{e}_1, \\mathbf{e}_2", "824a56dd50bae11750797b4e64ba9deb": "=\\alpha \\frac{\\partial ^2 \\bar v}{\\partial s^2}+\\beta \\frac{\\partial ^4 \\bar v}{\\partial s^4}", "824a681d55f1243a6fd2d8c60d25a24c": "E_1 \\wedge E_2 = E_1\\cap E_2", "824a91ed8fe19911f8c0d7bb29b94eaf": "\\mu = \\mu ^\\circ + RT\\ln \\frac{f}\n{{f^\\circ }}", "824b5a2c7fe9184b5079961fff8e5f7a": "g_{1,2} = \\frac{a_1a_2^2}{r_LL_1(a_2^2L_1+a_1^2L_2)} ", "824b66fe8443bb8a025a8c48f1edcdf7": "\\mathbf{k}\\times \\mathbf{F}_l\\left(\\mathbf{k}\\right) = \\mathbf{0}.", "824b7d0ed8656dc6da1cde1f4e1c4532": "\\ \\gamma", "824ba54b0c7fef6024a8200659eccf8d": "\nA_\\bar p = \\frac{1}{\\pi} \\int_{\\Omega} V_{\\bar p,\\hat\\omega} (\\hat n \\cdot \\hat\\omega ) \\, \\operatorname{d}\\omega\n", "824bba204fff950650d28857e5c23f69": "x_3 = x_2 - f(x_2)\\frac{x_2-x_1}{f(x_2)-f(x_1)}", "824bdc6bbfe75f61c8c2344dda3462e5": "\\zeta-\\zeta_0", "824bf068d02e0f529c0d167b9107be89": " \\operatorname{tr}(A^* A) \\ge 0 ", "824c1d1e1fd266d0d3614a971ed62211": "U_q(G)", "824c9093666dbae95c7dcfb444c10e9f": " \\min \\left\\{ \\int_0^T C(t,X_t,u_t)\\,dt + D(X_T) \\right\\}", "824cbd88506ecb73eaf9c7f1e065c084": "\\scriptstyle B\\times C", "824d2ae539d1df558d6098338539e96d": "xF", "824d59baa99daf2734b7323e9e689bc2": "P^{\\prime }", "824dc5d83b49716274ac79091d34c1df": " |\\phi \\rangle ", "824dead5ee5f48815bad1af3237297f6": "\\int_a^b \\varphi(t) \\, \\mathrm{d} t := \\sum_{i = 0}^{k - 1} c_i | t_{i + 1} - t_i |.", "824df68a66a295c512fdd925a40a3eef": "\\operatorname{VAR}(S)=\\frac{2(n^3-\\sum t^3_i)+3(n^2-\\sum t^2_i)}{18}", "824e3cf293b15ea2bdcdfa1f5e4298c1": "3.140\\dots\\!", "824e45bc02cd4032df2a339e285ca937": "f(0^+)=\\lim_{s\\to \\infty}{sF(s)}.", "824e64de5e601995e8214af5a9bb7d89": " SM = 100\\% \\cdot \\frac{\\dot{m_w} - \\dot{m_s}}{\\dot{m_w}} ", "824eaee2e5c170683f0dbeab8d73e2c0": "d_H\\phi=dx^\\lambda\\wedge d_\\lambda\\phi, \\qquad \\phi\\in\nO^*_\\infty(Y)", "824ebc7028964b1759eb94b6aa58e047": " \\mathbb C_c [ G(K) / G(O) ]^{G(O)} \\cong K_0(G^L-Rep).", "824f78d13b2973243ffb26bc2fe23622": "h\\in L^1(\\mathbb{R}^n)", "824f93f1cae0a9689373cc4cde6f339a": "\\mathbf{x}'(t) = \\mathbf{Ax}(t) + \\mathbf{b}", "824fc1f6cbac18bcf596ecbfb4ad7b9a": "w(n) = \\cos\\left(\\frac{\\pi n}{N-1} - \\frac{\\pi}{2}\\right) = \\sin\\left(\\frac{\\pi n}{N-1}\\right)", "824fe1d23d9cf5189eac2606545fb68c": "x_{n1}", "82500b53093db0e150eba665452853ac": "f_\\textrm{p}", "8250112f61f370f93f3416c0d3342a50": "a \\ln \\frac{x_1}{x_2}", "8250356817c293d4f3eabc1a83cd97b1": "W = cL^b\\!\\,", "825055e10f6cf47e04ccca1aacd276db": "\\lambda_\\text{b:air} ", "825119635828ba11248c46981c63ca64": "t(i\\Delta \\alpha) = interpolation(\\{ 2\\pi k , T_k \\} , \\alpha _i)", "82514f801f5665b27d5d628fda85903e": "A_i M_j \\subseteq M_{i+j}", "82517fb9f02b7df283a8ccd22d9d5def": "~ A = N_1\\sigma_{\\rm pa} -N_2\\sigma_{\\rm pe} ~", "82518767c43a8c0099e33e4c90d4fdc2": "H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2", "8251be83cd9595fb3f93e23de9a58b3c": "L(x) = \\ell(x) \\sum_{j=0}^k \\frac{w_j}{x-x_j}y_j", "8251fbf0f4ffb85ed951c35f10762d56": "\\lambda = 500\\mbox{ nm},\\, L = 1\\mbox{ cm},\\, n = 1.5", "8252073b6029f11114a79961183f9661": "\\pi = \\tfrac{157}{50}", "82523b1d631101af8bf6bd39a14f0b51": " d(g \\cdot h) = gd(h)g^{-1} \\! ", "825289770e93a127830f899fa7f0ec33": "\\begin{align}\n\\frac{e^{itz}}{z^2+1} & =\\frac{e^{itz}}{2i}\\left(\\frac{1}{z-i}-\\frac{1}{z+i}\\right) \\\\\n& =\\frac{e^{itz}}{2i(z-i)} -\\frac{e^{itz}}{2i(z+i)} ,\n\\end{align}", "8252d11a47e33ca21bd07c0591f46245": "m_\\odot", "82531d3280c4eca8ac3f5ff6c02acc79": "r_\\mathrm{e} = \\frac{e^2}{m_e c^2} = 2.817 940 3267(27)\\times 10^{-13} \\mathrm{cm}", "8253310a0dcd7e8733ca0d5ad7f2217e": "u_t = k u_{xx},\\ ", "82534982241a06e9b01ab937a6ff03f4": "f''(x) + p_1(x) f'(x) + p_0(x) f(x) = 0.\\, ", "825384bd82d368f612867271a249a2bf": "\\boldsymbol H(\\boldsymbol r,\\ t) = \\frac{1}{\\mu_0} \\boldsymbol B(\\boldsymbol r,\\ t)\\, ", "82539dfea2b783567ee4f21b863bf219": "\\frac{\\partial}{\\partial t} \\rho = - \\{\\,\\rho ,\\mathcal{H}\\,\\}.", "82539f8b37e1856202f82373ac4b1b81": "M=\\begin{bmatrix}\n1 & 0 & 0 & -2\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 1\n\\end{bmatrix}", "8253f45facc5f04054793a2bb77f4bd2": " L^2 +W^2", "8254474bc8e5abe7651c5e81f1f3fda6": "f(x,y) = f(x,x) = x^2", "825456c19caceab181bfdf0105ecb9df": " ({L^{1}}(\\mathbb{R}),\\star) ", "82550771f6432153f27e0ebb1d8cbf42": "U_{\\mathrm{out}}= V_{sig}\\cos\\theta", "825520725ee9771d3d67994055d97766": "Y_2\\sim\\Gamma(r,\\lambda)\\!", "825597f144771d96513d86816830f0cd": "Z = \\sum_{n=0}^{\\infty} e^{-E_n/kT}", "8255b1a4591bbffcf9ffb6fd90f029ad": "\n\\exp(i\\eta(\\hat{a} + \\hat{a}^\\dagger)) = 1 + i \\eta(\\hat{a} + \\hat{a}^\\dagger) + O(\\eta^2)\n", "825626adcc39d9c11d94a16ec7a18e8d": "\n\\mathbf{G}_x = \\begin{bmatrix} \n+1 & 0 & -1 \\\\\n+2 & 0 & -2 \\\\\n+1 & 0 & -1 \n\\end{bmatrix} * \\mathbf{A}\n\\quad\n\\mbox{and}\n\\quad \n\\mathbf{G}_y = \\begin{bmatrix} \n+1 & +2 & +1 \\\\\n\\ \\ 0 & \\ \\ 0 & \\ \\ 0 \\\\\n-1 & -2 & -1 \n\\end{bmatrix} * \\mathbf{A}\n\n", "8256b75bc1df2f68398d26cf39198cc8": "= 2 \\gamma^\\mu \\eta_\\mu^\\nu - \\gamma^\\mu \\gamma_\\mu \\gamma^\\nu \\,", "8256d8677239bf332fb2a8c4dcf85d23": " t^2 - 2tx + y(x) = 0. \\ ", "8256d90b22f10799bc67c4d93003b7bf": "3 \\times 13^2 + 9 \\times 13^1 + 8 \\times 13^0", "825737a789d6572dd9172de3f7603e5c": "\\varepsilon=10^{-2^n}", "82573a8b69d9e852ea68fc342200b49f": "\ndp(x,t) = L[p(x,t)] dt + p(x,t) [c x- c \\mu(t)]^T \\eta^{-\\top}\\eta^{-1} [dz-c \\mu(t) dt],\n", "82573aa83a671826f31c14d8ec2b4970": "n_d + N_d \\cdot (n_{d-1} + N_{d-1} \\cdot (n_{d-2} + N_{d-2} \\cdot (\\cdots + N_2 n_1)\\cdots)))\n= \\sum_{k=1}^d \\left( \\prod_{\\ell=k+1}^d N_\\ell \\right) n_k\n", "82578d635836f237b0eba223f29d0b71": "ct\\sqrt{-1}", "82579eeffb38df026711cafe6698c61b": "BI {{=}} (AB)W_1 + (SN)W_2\\,\\!", "8257ad2af011c73b232bbd3945b2b716": "\\sup_{\\| u \\| = 1} | B(u, v) | \\geq k \\| v \\|", "82585cd60211a7dd677a2976d708eb61": "\n y = h \\ast x =\n \\begin{bmatrix}\n h_1 & 0 & \\ldots & 0 & 0 \\\\\n h_2 & h_1 & \\ldots & \\vdots & \\vdots \\\\\n h_3 & h_2 & \\ldots & 0 & 0 \\\\\n \\vdots & h_3 & \\ldots & h_1 & 0 \\\\\n h_{m-1} & \\vdots & \\ldots & h_2 & h_1 \\\\\n h_m & h_{m-1} & \\vdots & \\vdots & h_2 \\\\\n 0 & h_m & \\ldots & h_{m-2} & \\vdots \\\\\n 0 & 0 & \\ldots & h_{m-1} & h_{m-2} \\\\\n \\vdots & \\vdots & \\vdots & h_m & h_{m-1} \\\\\n 0 & 0 & 0 & \\ldots & h_m\n \\end{bmatrix}\n \\begin{bmatrix}\n x_1 \\\\\n x_2 \\\\\n x_3 \\\\\n \\vdots \\\\\n x_n\n \\end{bmatrix}\n", "82588bd18d9601c7b498fa3c1795e57a": "\n\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4)\n", "82590ca8a0e0b8e373cd73c020f0279e": "\nP_r=P_i - P_d\\,\n", "8259201394cafc63a2f833a779f82dd5": "(F \\downarrow b)", "82592f2faa6c5b2c8686856a5d31f66e": "H_n = 1 + \\frac12 + \\frac13 + \\cdots +\\frac{1}{n}", "82598c600a484b6c50bec8c026e5e209": "\\alpha(C\\!\\ell_n^\\pm(\\mathbf{C})) = C\\!\\ell_n^\\mp(\\mathbf{C})", "8259d48a3ee935e8ab1883621136bbf7": "\\operatorname{Pref}_L(s) = \\{t \\vert s=tu \\mbox { for } t,u\\in \\operatorname{Alph}(L)^*\\}", "8259e4a90e110709ea1e2f5ac3bf330b": "y \\in \\mathbb{R}", "825a094c65fb8dd2fb30877e9840565c": "i = 0, 1,..., 29", "825a6fde7a50d58dd240cba66db4a332": "\\begin{bmatrix} y'\\\\ y_2' \\\\ \\vdots \\\\ y_k' \\end{bmatrix} = G_t \\begin{bmatrix} w_1\\\\ w_2 \\\\ \\vdots \\\\ w_k \\end{bmatrix}", "825b0ee7b8dec16e903e1f63997d2254": "f=\\frac{g}{ t_a+(n-1)\\cdot t_0} ", "825b13a6e0903d2cd6d8c33684ae5780": "G = \\left ( \\frac{\\pi k}{\\theta} \\right )^2 \\ e_A ", "825b15186084a3f4e153021ac5396d54": "\\gamma _{1}\\cdot p_{\\perp }", "825b3dec69df6f00fde0fa67909cf19b": "\\overline{O_iO_j}^2=((R-R_i)-(R-R_j))^2+(R-R_i)(R-R_j)\\cdot \\frac{\\overline{K_iK_j}^2}{R^2}", "825b3fd5bafbc46b9a560ea9f16b21dd": "a_n", "825b82deecf5d426b426fe3517978eba": " e^{i \\pi/4}\n\\begin{pmatrix}\n1 & 0 \\\\ 0 & -i\n\\end{pmatrix} ", "825b9fc80e5d7ef9699a14784a732077": "H,K", "825be27531cdc17c9ad050c7a2fe3cf5": "H_{\\aleph_1} \\subset L(R) \\,.", "825bf628a3288897d9ca75ad9afebab8": " \\frac{\\partial u}{\\partial \\nu}(x_0) > 0", "825c1a10baf7e3d15b3208fa32ef478b": " A=\n \\begin{bmatrix}\n 16 & 3 \\\\\n 7 & -11 \\\\\n \\end{bmatrix}\n", "825c325129660e48091690d91f614266": "K = k_e + k_h.", "825c8d81aac287544cc430e539bf6e12": " Q_A = \\mathcal{M}.Q.", "825cff6de987b8a1c5f020ce4f0cb3eb": "\\boldsymbol\\psi(t,\\mathbf{r}_0)", "825d3ca8d658153ad6b962fbaeb81283": "\\theta \\geq 0", "825d41a70aa7a9cc3b075d0cc566bb77": "[X]_t=\\int_0^t\\sigma_s^2\\,ds.", "825d8a32b17e630134e48dd44f7909ec": " t = -\\tau\\,\\ln\\frac{N}{ N_0} \\approx 10360", "825d960fec1cf1308384d5c83261f526": "C^{\\infty}(M)", "825dafc906eee8c415bb6c7ea0c2faca": "P_{ij} = \\frac{\\exp(\\lambda EU_{ij}(P_{-i}))}{\\sum_k{\\exp(\\lambda EU_{ik}(P_{-i}))}}", "825dd686fc5eb7ab571a302c0d538622": "\\boldsymbol{\\mathcal{A}} = t_a \\boldsymbol{\\mathcal{A}}^a \\,.", "825e376f9069a6ecdf58c229b792b906": " b = \\frac{L}{4\\pi d^2} ", "825e663ca636d5f09d68a4857ee6e443": " Ax = b.", "825e9a5b1b2c78cefadc9ba1a573e39e": "\\lambda = \\frac{c}{f} = \\frac{\\text{speed of sound}}{\\text{frequency}}", "825ed83e42a05b56d0cdb27457935633": "\\frac{d\\sigma_\\mathrm{n}}{d\\theta}=0\\,\\!", "825edca74ccb83f1b026c368752d990e": " N= 2 \\epsilon g(E_f). ", "825f81e1cad368e74f8a6611f77f4680": " h_{0}= {T0_{1}V_{0} \\over 2cos(i_{c}) }", "825fc8b31b28faa370959911fde75019": "\\sin (\\alpha + \\beta) = PB = RB+PR = AQ+PR = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta\\,", "825fcb4b28bba6f925eac8e80117d8f8": "\nf(N_1,N_2,\\ldots,N_n)=\\ln(W)+\\alpha(N-\\sum N_i)+\\beta(E-\\sum N_i \\varepsilon_i)\n", "8260781f91b25839c344b7f0bf895ccd": " {P_1 \\over P_2} = { \\left ( {D_1 \\over D_2} \\right )^5 }", "8260bb9adfbaf667a9288d73314857a8": "\\ \\begin{array}{rrcl} & (\\dot{A}-wA+Aw)^* &=& Q(\\dot{A}-wA+Aw)Q^T \\\\\n\\Rightarrow & \\bar{A}^* &=& Q\\bar{A}Q^T. \\end{array}", "826128936d51d7dbf268bbf2922b03fa": "\\scriptstyle p_1, p_2", "82613f5a0c13db67daf65e9d616904d9": "f_\\theta(X)", "82614d71d30b1dcf49afba903fb0d890": "\\hat{\\mathbf e}_i = \\frac{\\mathbf e_i}{h_i} = h_i \\mathbf e^i = \\hat{\\mathbf e}^i", "82617a2943480df55794159da461c383": "\\scriptstyle v_1,\\dots,v_{n-1}", "8261ecce2790b41becc5937ba9f3f33b": "1 \\mathrm{\\ rad} = 180^{\\circ}/{\\pi} \\approx 57.3^{\\circ}.", "82620cf525c6ed14bfb7a91e88b4b868": "d = r \\, \\Delta\\sigma.", "82620eaa589a29c21643798c57c71e14": "G = \\frac{1}{n}\\left ( n+1 - 2 \\left ( \\frac{\\sum\\limits_{i=1}^n \\; (n+1-i)y_i}{\\sum\\limits_{i=1}^n y_i} \\right ) \\right ) ", "826215a477f87bc5a094c70adba43673": " r_k ", "82623aa5fa2034ee8d953ec1fd5390f1": "b_k = 1/\\vert k \\vert", "8262eab7915d2ca1728756b460511b94": "\\sigma(t)= E_\\text{inst,relax}\\epsilon(t)+ \\int_0^t F(t-t^\\prime) \\dot{\\epsilon}(t^\\prime) d t^\\prime", "82631bd98500270e9e9c15a2d6812ff7": "\\psi(\\omega) = \\omega^{\\omega^\\omega}", "82631e953bd6025a2b9c825ef9dfec3f": "\\mathbf{X^TM^{-1}X \\boldsymbol\\beta=X^T M^{-1} y}", "82637378fa49c3e2b851ac146f3870ac": "{\\mathbf E} = -\\nabla\\varphi - \\frac{\\partial{\\mathbf A}}{\\partial t} - \\nabla \\frac{\\partial{\\psi}}{\\partial t} = -\\nabla \\left( \\varphi + \\frac{\\partial{\\psi}}{\\partial t}\\right) - \\frac{\\partial{\\mathbf A}}{\\partial t}", "8263a8a82ce24c466acc0c157df1b00d": "E(x)=a^x\\,", "8263b237418eb6aeb3c63ac416fa8006": "\\text{A} \\mapsto 1, \\text{B} \\mapsto 2, \\text{C} \\mapsto 3.", "8263b7844454ac2eb60e9e1bbc2815e9": "\\ln(\\Omega)", "8264024cab15cb0f633f786bd0cb2c66": "Q = \\frac{ \\sqrt{R_3 R_4 C_2 C_5} }{ ( R_4 + R_3 + |K| R_3 ) C_5 } ", "826438c2aac9c7b423777b5995f8626f": "P^{-1}\\cdot Q", "82643b3e2ab0b060291440cf96629543": "E=\\hbar\\omega=h\\nu=\\frac{hc}{\\lambda}", "82643bfbbe225c0ae7247198d78cca38": "f(j) M(i,j) = f(i) M(j,i)", "826456462c537229c5f654d498bf1074": "\\mathcal{H}\n_{1}\\Psi =0", "826461e0e9ea7fce855c2c2569fd4387": "\\sum_{a=1}^{p-1} a^{p-1} \\equiv (p-1) \\cdot 1 \\equiv -1 \\pmod p.", "826493fbe319671d8dd2aa6711227414": "d^2", "8264c584a44bb23085570d595ac1fe49": "f'_+(t) \\triangleq \\limsup_{h \\to {0+}} \\frac{f(t + h) - f(t)}{h}", "82650a91246c2f132a95fa35007f2d27": "R_2=\\frac{L_2}{k_2A}", "82651fe6d484db0dd94c76ce6d4e7a81": "x^2 + \\frac{b}{a} x + \\frac{c}{a}=0.", "82652413c83152cdbe6945b3b35474e3": "\\dfrac{1}V\\! \\times ", "82654377f3858e625f68e508d2011a42": "\\scriptstyle Q ", "82655bc0c3b4a9921f156d06a929d679": "A^{0} = \\frac{-1}{\\sqrt{3}}\\ \\sigma^{z}", "8265ac4047930ebeb19593ecb2e98af2": "ie \\;\\;\\; ^\\or \\qquad oi \\;\\;\\; ^\\mathfrak{7} \\qquad ow \\;\\;\\; _\\and \\qquad ew \\;\\;\\; _\\cap", "8265ce0d288343f918c9f49c16830739": "\\zeta(s,a)=a^{-s}\\cdot{}_{s+1}F_s(1,a_1,a_2,\\ldots a_s;a_1+1,a_2+1,\\ldots a_s+1;1)", "82667087cb77f2ef64704931fc09db71": "P \\left\\vert\\left[ \\left\\{ a \\right\\} \\right]\\right\\vert Q", "8266739bba355265dc4da4824cf25787": "\\mathit{prob}_{\\mathit{after}}(\\psi \\rightarrow \\phi) = \\sum_{i,j} |\\psi_i |^2 |\\phi_j |^2 \\lang j | i \\rang \\lang i | j \\rang = \\sum_{i} |\\psi_i^* \\phi_i |^2 ", "826676a6a5ad24552f0d5af1593434cc": "E=mc^2", "8266b318562185f0bc65e7f6cd2a6274": "1 - \\lambda K", "8266bb21c655c9dc496209b9f8bac19a": "qp", "8266dad43f0ac4eb15280073aab95fd3": "a = \\frac{0.427\\,R^2\\,T_c^2}{P_c}", "82671999bd17b9ffff8ec39ad4f1de59": "T=p*q + k", "82674f1b6d487d21cfce19a7467b901d": "g = {Plowback\\ ratio}\\times {return\\ on\\ equity}", "826785105a1f9ae26d611d7d06238cbc": " \\Delta(x)p_{1^{(n)}} = \\sum_{\\lambda\\vdash n} \\Delta(x)s_{\\lambda}f^{\\lambda} ", "8267a632910f694f4032e441740f6c89": "C_{P}-C_{V}", "8267b9975cfc802a9b092742c21471b7": "P = (0,0)", "8267c8a93507030d2fa442f91e064dd9": "R=\\sum_{t=0}^\\infty \\gamma^t r_{t+1},", "8267d0fae2677f131c327d9da39e3686": "\\Pi_1^1", "8267fef7501539aa41073b47b6436aa1": " \\delta_p(a) = \\frac{a - a^p }{p} ", "826801dbc57799b4d71c777903c7d670": "\\frac{\\partial (\\mathbf{a}\\cdot\\mathbf{x})}{\\partial \\mathbf{x}} = \\frac{\\partial (\\mathbf{x}\\cdot\\mathbf{a})}{\\partial \\mathbf{x}} =", "82680977184cd26cda5bc36fdb887515": " \\vec{u'}=\\left (\\sum_{\\sigma}\\frac{\\rho^{\\sigma}\\vec{u^{\\sigma}}}{\\tau_f^{\\sigma}}\\right)/\\left(\\sum_{\\sigma}\\frac{\\rho^{\\sigma}}{\\tau_f^{\\sigma}}\\right) ", "82688acc06c5f6cfedaf8eb791ae80ca": " Z_N = \\int_{-\\infty}^{\\infty} \\ldots \\int_{-\\infty}^{\\infty} d\\sigma_1 \\ldots d\\sigma_N exp \\left[ K \\sum_{\\langle jl \\rangle} \\sigma_j \\sigma_l + h \\sum_j \\sigma_j \\right] \\delta \\left[N - \\sum_j \\sigma_j^2 \\right] ", "82689ec82c0bdc2b3b3af8f7ada4e15f": "\\frac{(3+5)}{2} = 4", "82694c670dac3244a9f95da8341e9384": "(2p)(2q)=-1,", "82697e5383524bfbed35eddde6b0d61f": " \\operatorname{Ei}(x)=-\\int_{-x}^{\\infty}\\frac{e^{-t}}t\\,dt.\\,", "8269862b69d8be220d5ec538277fa78d": " H(s) = \\frac{Y(s)} {X(s)} ", "826a05adc386c4ea499c9c8c32b30b59": "B_1=-\\frac{1}{2}.", "826a1931499ed27178c4cea1b98c1f91": "(\\hat{H})", "826a357e7b17cbb5dab831b43d13d9a8": " |S(f)(z)| \\le 2(1-|z|^2)^{-2},", "826a3bf628aaf73cd737d4a760419727": " v(n)", "826a58d881791ca139e56ec999824d61": " I=G^{-1}G = \\frac1d \\sum_\\alpha \\left[ (d+1)\\Pi_\\alpha \\odot \\Pi_\\alpha - I\\odot \\Pi_\\alpha \\right]", "826a6ed0ebf6dc20353c209f97dabd12": "{\\rm Inj}(M)", "826a9ab805a11c9f4410014f893f9825": "\\Sigma_f(\\mathbf{r},E^{\\prime},t)", "826aa9070e3c86bb59a448b417767d93": "\\lim_{n \\to \\infty}S(n)= s3^n \\, ", "826b46736b56710cd1cf18d33be129bd": "\\frac{d}{dt} \\langle Q \\rangle = 0 ", "826b5eb0fa32d6a8f2a7538419a51a99": "E[l]", "826b6daa8c7e808196d48e9d74d0d18c": "\\mu \\colon Y \\to X", "826b726c4f41f1fada9e066cca9fb4b5": "\\operatorname{Inn}(\\mathfrak{g})", "826ba7b6795141c41ccac59462973f9c": " r_t", "826bc41d7d914987a17788f18dc3c4c0": " \\binom nk = \\frac{n!}{k!\\,(n-k)!} \\quad \\text{for }\\ 0\\leq k\\leq n,", "826bcdacaf05b72b35c33420d1e81655": "f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\exp\\left(-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right)", "826c10d89bbfc05ade3a35700c4019c8": "\\neg P \\or Q", "826c13e412518867e45e7f5f43383e0e": "R \\leq \\frac{1}{2}\\log \\left(1+ \\frac{P}{N}\\right) + \\epsilon_n", "826c599d20960d8336fdd9a261f96f5b": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 1}{40 \\choose 1} \\end{matrix}", "826cd7f9b41515455433c3577c2ec435": "\\pi_n^S", "826cf4e77e85204033f188dd93968bf3": "x = 2t_1 + 3t_2,\\;\\;\\;\\;y = 5t_1 - 4t_2,\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;z = -t_1 + 2t_2", "826d133e4ffac866ee58d5d2e02d0f2c": "\n\\begin{bmatrix}\n1 & 1 & 1 \\\\\n1 & 1 & 1 \\\\\n1 & 1 & 1\n\\end{bmatrix}\n", "826d41a8daf44ec790e95f5e17aef297": "\\scriptstyle\\sigma_x^2", "826da6bf07af71a859cab5d687d7afa5": "\\rho_S(0) \\mapsto \\rho_S(t)", "826dca2f981b58e2a7fb9f2db90f4986": "A \\to \\widehat{a}bc", "826dccaa27ae147626c2c2e73eacf25c": "A \\to Z", "826e4615d01ec608ff2d802c4810d86c": "\\eta > 0\\,\\!", "826e6764af22b6b1e163676f7d608485": "r(x_1,x_2)=r(0,0)e^{-c(x_1-x_2)^2}", "826e688311f4b013086c9e3a66cfa0d7": "A \\rightarrow \\alpha", "826e796d50bdc7b39d5d8859f2731629": "s_i=\\pm 1\\,", "826eaa1b2480050e65bd9487b41c4f61": "\\mathbf{r} = (x,y)", "826ed2ba4952b2ae6e5699cb2e155cdc": "\\left(A_x\\right)_{m'n',mn} = \\delta_{n'n} \\left(J_x^{(m)}\\right)_{m'm}\\,\\quad \\left(B_x\\right)_{m'n',mn} = \\delta_{m'm} \\left(J_x^{(n)}\\right)_{n'n}", "826ef5f88ce58bc0cfe2b3a1923d3964": "\\{a_n\\}, \\{b_n\\}", "826f1338bbd6ec13eb1871b85632affe": "q_T = \\frac{V}{\\Lambda^3} = V\\frac{(2\\pi mkT)^{\\frac{3}{2}}}{h^3}", "826ffad288ab24abe82782cee280343d": "\\hat{A_{l}} = \\sqrt{a_{j}}\\langle{k}|\\hat{U}|{j}\\rangle.", "8270176eaa29fa10156c10a798ffb304": "F_{t_2,t_1} \\circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p)", "82703018acd27c51954eb3951824e78f": "d\\mathbf{E}", "82704c88e58c3bc15cbed4a3657188c2": "f([1,2],[5,7],x) = ([1,2] \\cdot x) + [5,7] = 0\\Leftrightarrow [1,2] \\cdot x = [-7, -5]\\Leftrightarrow x = [-7, -5]/[1,2],", "82709f89a390fb9f512fbccdb906e456": " {f_o}=a_1 \\cdot sin(\\omega t)+a_2 \\cdot sin(2 \\omega t)+a_3 \\cdot sin(3 \\omega t)+..", "8270b0ee16322239f9138410f9a38d86": "x_2*w_2", "8270e7b25a779c24435d613e3bf43053": "a_{8}=a_{9}", "82718653b970457f3420149b21af2138": "\\mathbb{C}^2", "8271a7d8c11eb797bd1d294e63152077": "(ij)j = kj = -i", "8271b10f8550acc8296f226e78ce5b43": "\\sigma_x = E\\epsilon_x\\,", "8272422567dbef5a550a56d8a8cc29f2": "A \\le B", "827249c173ed7cd4fbebdd5b820a8561": "[3,4] \\;\\supseteq\\; [3.1,3.2] \\;\\supseteq\\; [3.14,3.15] \\;\\supseteq\\; [3.141,3.142] \\;\\supseteq\\; \\cdots ", "8272556f4b60f2bc3f780caafdea3d78": "w\\neq 1", "8272ee905279ae86dc5af2d8a4d04dee": "\n\\tau \\ = \\ RC ", "82736ceefdf8196f65d698b44b5af894": "L(n , k )=0", "827451b1cb7196c02796aaa94af256ad": "\\begin{align} 2\\cdot R_*\n & = \\frac{(83\\cdot 3.24\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 58\\cdot R_{\\bigodot}\n\\end{align}", "8274521fcd564c3542da03ddaa298de4": "e^{\\frac{-iE_{+}t}{\\hbar}}", "82746bf0d65799016e843f1be31a528d": " \\operatorname{de-lambda}[F = \\lambda P.E] ", "82747f4315c1ca56056e2bc10566d609": "K(\\Pi) \\subset \\mathbb{R}^d", "8274a5cc5d54ed55199962f1752d9fd9": " T = \\{j | \\mathbf{x}_j = 1\\} ", "82754ad6212df18381245b6e9a404400": "~A \\leftrightarrow B", "82755a01c6f356d9ada2ee8078bad145": "(x,y)=(0,0)", "827573c71755f3c5e1533d931e77586c": "\\langle\\chi_{k'}|\\big(P_{A\\alpha}\\chi_k\\big)\\rangle_{(\\mathbf{r})}", "8275779f3f4e5757377018012fc1e299": "X = j s \\frac {10^p-1}{F}", "82757e5d73276169b6d12217bd98a022": "\\text{H. sapiens} \\subset \\text{Homo} \\subset \\text{Primates} \\subset \\text{Mammalia} \\subset \\text{Animalia}", "8275826554113e50d4d970759e29e396": "{\\mathbf B}", "827588f69e2e7d1860199f476e98ae1c": "\\mathbb{F}_q^n =\\cup_{c \\in C} B(c,d-1).\\, ", "8275b64b3baf744afbd9c86445ab03c8": "E(z|x_A+y)=0", "82760418f366ae4c27d5903ed782dd18": "X \\hookrightarrow Y,", "8276840e61270918784b905f89e27c46": "\\tilde{\\phi}(p)", "827712ad9a1c77874f8d6c2b01c98243": "\\log S_t", "8277199efa65d500879c91978686d2a7": " \\lambda\\neq 0,1 ", "827747ab35b3e6a5107aed47ef76aabd": "\\log^2 N", "8277e0910d750195b448797616e091ad": "d", "8277edda7618a3979a5d9f746dfc5a2b": "\\varepsilon \\sim t^{-2(p_1+p_2)}=t^{-2(1-p_3)},\\ u_{\\alpha} \\sim t^{\\frac{(1-p_3)}{2}}.", "82789e896ea830a3468ce5876d0a981a": "Int2\\,", "8278a4854c84dbb777ae1dd579bd2eb1": "Z(X,t)=\\prod_{i=0}^{2\\dim X}\\det\\big(1-t \\mbox{Frob}_q |H^i_c(\\overline{X},{\\Bbb Q}_\\ell)\\big)^{(-1)^{i+1}}.", "8279d231bc136e9505c545e2266fd304": " -x \\partial_z + z \\partial_x \\,\\!", "827a3fd56bea5a86f7d8dae532de6d27": "\\theta(t^{}_n,j)\\sim \\mathrm{Normal}(\\theta(t^{}_{n-1},k^{}_j), a^{m-1} \\Phi)", "827a537b6e1ed5bcd76d04902ccd97d2": "(t,\\epsilon)", "827a879fb6ce1779a07ecc0adf7a41b5": "\\scriptstyle I_y=R-\\frac{1}{2}I_x", "827a9f912c496082b4eec2ed79b66bec": "V(\\mathbf x)", "827aa0521848c45b5df3b1f8570cc389": "\\kappa = {{\\pi e^2} \\over {\\epsilon_0 m^2 \\omega^2}}\\sum_{k,k'} |e \\cdot p_{cv}|^2 \\delta[\\Epsilon _c (k') - \\Epsilon _v (k) - \\hbar \\omega]\\delta_{kk'} ", "827aca8baad038b2249b7e31a6f9a4d4": "g-2", "827ad61a67e52dc0956532093f7bacd6": "\\Sigma_{j \\in S} p_j \\leq \\Sigma_{a \\in T^*} W(a)", "827ad8d7bdb5ed427940b29a7c552062": "E_v = \\log_2 {\\frac {A^2} {T} } = \\log_2 {\\frac {B S_x} {K} } \\,.", "827aec8ed0e01c763de9542c30496d7b": "\\text{watt} = \\frac{\\text{joule}}{\\text{second}} = \\frac{\\text{newton}\\times\\text{meter}}{\\text{second}}", "827b409318043f6296c6e593d11a1e28": "F_1 F_2 = x^4 + bx^2 + cx + d", "827bf2badea034426932c43694b920ae": "y = \\sin(c t) - \\sin(d t)^k ", "827bfc20cd8de6d8603d6e14ecfa3bef": "\nD_B(\\rho_1,\\rho_2)^2 = 2(1-\\sqrt{F(\\rho_1,\\rho_2)}), \n", "827c20dc368ed480bb85e736f4e19ead": "p(\\mbox{height} | \\mbox{male}) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}}\\exp\\left(\\frac{-(6-\\mu)^2}{2\\sigma^2}\\right) \\approx 1.5789", "827c2541dacb40edf6689e3ba14484c6": "X\\colon \\Omega \\rightarrow \\mathbb{R}", "827c596c76b623a9710402dd087d6e1e": "\\langle \\psi|\\hat{H}|\\psi\\rangle = \\int \\psi^*(\\mathbf{r}) \\left[ - \\frac{\\hbar^2}{2m} \\nabla^2\\psi(\\mathbf{r}) + V(\\mathbf{r})\\psi(\\mathbf{r})\\right] d^3\\mathbf{r} = \\int \\left[ \\frac{\\hbar^2}{2m}|\\nabla\\psi|^2 + V(\\mathbf{r}) |\\psi|^2 \\right] d^3\\mathbf{r} = \\langle \\hat{H}\\rangle ", "827cc96d3f8dcdddc38bfd931e46cd8a": "\\scriptstyle \\hbar\\;=\\;2m_e\\;=\\;e^2/2\\;=\\;1", "827cf42e6b8e9709ab92e3f241ae17b3": "\\tan\\phi^\\prime = \\frac{Z_r}{\\sqrt{X_r^2 + Y_r^2}} = \\frac{ N(\\phi) (1 - f)^2 + h}{ N(\\phi) + h}\\tan\\phi", "827d053a473f3264d48838f2c39b724c": "z_{/\\cong_{\\mathcal{B}}}", "827d64ee8cb5a8986c9c07727374d3f8": "\\nabla \\cdot \\mathbf{E} = \\frac{\\rho}{\\mathcal{E}_0}", "827d8e19e48789422978ccb50f82f043": "\\ddot r =0 \\ ", "827db7e90607e25c88a4f5be042f534a": "= d\\left( \\gamma + (\\dfrac{\\lambda}{d}) \\left( 1-\\gamma\\right)\\right)", "827dde5fe46cb99acd9f41b0fa7d4c44": " | q \\rangle, q\\in \\mathbb{R} ", "827de5a5fed42a72e19cf482f241f75e": "ES_{\\alpha}^t(X) = \\operatorname*{ess\\sup}_{Q \\in \\mathcal{Q}_{\\alpha}^t} E^Q[-X\\mid\\mathcal{F}_t]", "827e08f8b40a46823e1322e713dd366c": "x = im + j", "827e24f0289ad1138a437210dfd802e3": " f\\mapsto \\int_E f \\, d\\mu ", "827ea82891049bb5e2b2ed4c877f4d7d": "\nt = -RC \\times \\ln\\left(\\frac{1}{2}\\right)\n", "827eb0f2cd5d3ae26b9e52037cc06b4c": "g_{|U \\cap X} = f_{|U \\cap X}", "827eb2cf6f362e7f02fc48f94a486f28": "\n\\begin{align}\n\\varepsilon_N^2(t)&=\\mathrm{E}\\left[\\sum_{i=N+1}^\\infty \\sum_{j=N+1}^\\infty A_i(\\omega) A_j(\\omega) f_i(t) f_j(t)\\right]\\\\\n&=\\sum_{i=N+1}^\\infty \\sum_{j=N+1}^\\infty \\mathrm{E}\\left[\\int_{[a, b]}\\int_{[a, b]} X_t X_s f_i(t)f_j(s) ds\\, dt\\right] f_i(t) f_j(t)\\\\\n&=\\sum_{i=N+1}^\\infty \\sum_{j=N+1}^\\infty f_i(t) f_j(t) \\int_{[a, b]}\\int_{[a, b]}K_X(s,t) f_i(t)f_j(s) ds\\, dt\n\\end{align}\n", "827ece5a4b939c2954347f5ea9b8199f": "{\\mathbf E}", "827f07c87acc5534fb4e3153af07aaff": "u'\\left(x_0\\right)", "827f08042ca392429c02630e78c3ba34": "f{(x)}", "827f77b5e30eb14963b7a1c4eabbeeec": "(\\tfrac{np}{p}) = 0", "827fbed07df52b6584ba23bc6c791bfb": " \\lambda = \\frac{c}{n f} \\ ", "82804257ee8ede1bc74e65ce1c519ec0": "\\tfrac{2^n - 1}{2^n}", "8280702bb2a3f1aa83467a90e346ff98": " -r^{-2}~\\cos\\theta \\,", "8280820b8a86c322c0bb3337b45560a6": "H_{x}=L\\frac{\\partial T }{\\partial y}^{TM}+\\frac{1}\n{j\\omega \\mu }\\frac{\\mathrm{dL} }{\\mathrm{d} z}\\frac{\\partial T}{\\partial x}^{TE}= L \\frac{\\partial T }{\\partial y}^{TM}-\\frac{k_{z}}{\\omega \\mu }L \\frac{\\partial T}{\\partial x}^{TE} \\ \\ \\ \\ \\ (30) ", "8280d7ea972a3959ba0cc6589f27f3d9": " e_1, \\ldots, e_n ", "8280dbc2ad91e01a1e1ccde0a427828f": "V_q\\,\\!", "8280e19e6a68dd5d5f4f3c8c92ce3df2": "k_\\mathrm{cat}", "82813181689b67105c028403869f5608": "\nG_{1} \\equiv \\mathbf{q} \\cdot \\mathbf{Q}\n", "8281a3207eda58ee363dc5b64947236f": "F=\\{f_1,f_2,\\dots,f_m\\}", "8281a7b07dbfa0951d5190070a8f9f56": "\\rho_c = \\frac{3 H^2}{8 \\pi G}", "8281c26f770cb449995290d48aee2792": "a + c = b + c", "8282078c10474f722df2fd2f9705878d": " W_0^{1,p}(\\Omega):= \\left \\{u\\in W^{1,p}(\\Omega): \\exists \\{u_m\\}_{m=1}^\\infty\\subset C_c^\\infty(\\Omega), \\ \\textrm{such} \\ \\textrm{that} \\ u_m\\to u \\ \\textrm{in} \\ W^{1,p}(\\Omega) \\right \\}.", "8282316efc4fcd794ac8c84d804343b9": " \\mathbf{m} = \\int \\rho \\left ( \\mathbf{r} \\right ) x_i \\mathrm{d} \\mathbf{r} \\,\\!", "82825e8558d095162fa481977355d75f": "f(x;\\alpha,\\beta,c,\\mu)=\\frac{1}{\\pi}\\Re\\left[ \\int_0^\\infty e^{it(x-\\mu)}e^{-(ct)^\\alpha(1-i\\beta\\Phi)}\\,dt\\right].", "828295e8e9f6bc2413319d0562c24a1a": " (p+qX)(a+qX)=pa+(p+a)qX+q^2X^2=q(b+(p+a)X+qX^2)\\,", "8282f070803784b10e1612692ccf8b34": "\nWN(\\theta;\\mu,\\sigma)=\\frac{1}{\\sigma \\sqrt{2\\pi}} \\sum^{\\infty}_{k=-\\infty} \\exp \\left[\\frac{-(\\theta - \\mu - 2\\pi k)^2}{2 \\sigma^2} \\right]=\\frac{1}{2\\pi}\\zeta\\left(\\frac{\\theta-\\mu}{2\\pi},\\frac{i\\sigma^2}{2\\pi}\\right)\n", "82831f366b49a25d96468d3bd2eeb59c": "\\left| G \\right|^k", "828386419eecf265e65f248eef7f890e": "\nf_i(Z)=0 \\quad \\forall i \\in \\{1,\\dots,k\\}.\n", "8283a5a6c6ffb5e54269efe61a17b817": "w^3 = ~-1", "828401ca7b21accc4c00bd85e8de3c4c": "M_p 2^{p-1} = M_p (M_p + 1)/2 = h_{(M_p+1)/2}=h_{2^{p-1}}", "8284413ca82635043f57c6538bdbfa41": "\\Theta_{n,m}(\\tau, z + a)=\\Theta_{n,m}(\\tau,z), \\qquad a \\in \\mathbf{Z}", "8284517dff0a976a334f8adf2bb404eb": "\\Omega_*^{\\text{SO}}.", "82846848a4ff07dcad5b632455404e52": "H = \\{ x : x \\equiv 1 \\mod p \\}", "8284c02e53552d2e33b6bfecad848ff6": "\\aleph_0.", "8284c2b182ee7baceea27821587e2477": "n_{j}", "82850452e5bbab3e9006f084041649cf": "\\textstyle{\\mathrm{Var}}([x=i]) = p_i (1-p_i)", "8285619ad46cf8d8f53fff49801bd455": "\\left[ \\left[ [ \\mbox{un-} ] [ \\mbox{event} ] \\right] [ \\mbox{-ful} ] \\right]", "82857e9da48d3d90d9fe7b057e24b444": "|\\phi_n\\rangle", "8285c6e60f5bb3a5a4a3498deae7beb0": " \\operatorname{de-let}[f\\ (q\\ q)] ", "82862853d861d7c2dd8cf39d837f7d2c": "\\psi_n(x) = (2^n n! \\sqrt{\\pi})^{-1/2} \\mathrm{e}^{-x^2/2} H_n(x) = (-1)^n(2^n n! \\sqrt{\\pi})^{-1/2} \\mathrm{e}^{x^2/2} \\frac{d^n}{dx^n} \\mathrm{e}^{-x^2}", "828691a46a9e29daadfed772fb75025b": "\\cdots \\leftarrow C_{n}^* \\stackrel{ \\delta^n}{\\leftarrow}\\ C_{n-1}^* \\leftarrow \\cdots ", "8286bf724718e1f8d132ae07c7d5571b": "\\displaystyle{2X(c(t))\\cdot (c(t)-z)= 2 X(z)\\cdot (c(t)-z) - 2(X(z)-X(c(t)))\\cdot (c(t)-z).}", "8286ffe6dc105f824e3d74d051301f2c": "(g\\cdot f)(x)=f(xg).", "828750e32c412bc9a62218593b3fe869": " \\begin{bmatrix} x_1; x_2; \\dots; x_m \\end{bmatrix} ", "8287717266760ef799ba3200972d588c": "\\mathrm{dn}\\,(x)", "8287e6b8b4e84b3c7745d8c102b9948c": "L = I - D^{-1/2}SD^{-1/2} \\, ", "8287edc111f7bb5b35522288c70d4428": "T_a>T_L", "828855d16d49ce9e414a315e126fcc01": "e=1/l", "828858df53cba8024f33700338dab1a6": "\\displaystyle g(y) = x .", "82896d530eca47fec2e370b0b0ccfbad": " \\|\\psi\\|^2 = \\langle \\psi | \\psi \\rangle = \\sum_{j=1}^n | c_j |^2 \\,,\\quad \\|\\psi\\|^2 = \\langle \\psi | \\psi \\rangle = \\int d \\varepsilon \\, | \\psi(\\varepsilon) |^2 ", "82898d7ee7190d8294f15b043884db90": "\nJ_{0} = A_{3} \\,\n", "8289a605a4c16090c24868cac439dfdd": "(\\gcd[114-80,1649])\\cdot(\\gcd[114+80,1649]) = (17)\\cdot(97) = 1649.", "8289e8a805d7fd75ade60e3ec9762705": "aL", "828a25aed9d261288a78d4ecb842b9c9": "\\operatorname{Sl}_{2m+1}(\\theta) = \\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m+1}}", "828a407a7113064f06a5647dda1c837c": "\\omega=dp_1\\wedge dq_1+\\cdots+dp_m\\wedge dq_m", "828a61ec6e35038601a86a87fda1d404": "T_B = \\frac{B_B}{R_B \\cdot 0.96}", "828a8110aeb4278f9ecfe98d5111b888": "\\text{I}^+~+~\\text{M}~~\\xrightarrow{k_{i}}~~\\text{M}^+", "828ac96ac6913986579e5c5746cc9838": "-L=m^2.", "828ad0fd5bce75da0b6009053d158eaf": "\\mathfrak{P}^{123}", "828afa80dae21da491cf42170d16e5f0": "\\begin{align}\nE_{8} &= E_4^2 \\\\\nE_{10} &= E_4\\cdot E_6 \\\\\n691 \\cdot E_{12} &= 441\\cdot E_4^3+ 250\\cdot E_6^2 \\\\\nE_{14} &= E_4^2\\cdot E_6 \\\\\n3617\\cdot E_{16} &= 1617\\cdot E_4^4+ 2000\\cdot E_4 \\cdot E_6^2 \\\\\n43867 \\cdot E_{18} &= 38367\\cdot E_4^3\\cdot E_6+5500\\cdot E_6^3 \\\\\n174611 \\cdot E_{20} &= 53361\\cdot E_4^5+ 121250\\cdot E_4^2\\cdot E_6^2 \\\\\n77683 \\cdot E_{22} &= 57183\\cdot E_4^4\\cdot E_6+20500\\cdot E_4\\cdot E_6^3 \\\\\n236364091 \\cdot E_{24} &= 49679091\\cdot E_4^6+ 176400000\\cdot E_4^3\\cdot E_6^2 + 10285000\\cdot E_6^4\n\\end{align}", "828b05e5a184deac7eefc6ba1f7b2cb7": "I_x = I_y = \\frac{I_z}{2}\\,", "828b662b78330f2b38b0124f3683b34a": "f(O) \\in \\mathcal{B}", "828b89891c52c813a60ad0ac53b294ca": "1/5=0.2", "828c080b10c7a6a6045952202bfff763": "\\rho_{out} = S_{22} + \\frac{S_{12}S_{21}\\rho_s}{1-S_{11}\\rho_s}\\,", "828c4e5b38e95e93e321219f7472e27a": "[\\ ,\\ ] \\!\\,", "828c5e3c1d61f603935b7ec051985ef1": "f_k ", "828c77fd2839a5b480a510578b28e725": "\\nu_k = \\frac{(k-1)^{k-1}}{(k^2-2k)^{\\frac{k}{2}-1}}.", "828ca42d98a96772dc294beffc36b4a8": "\\Omega_{dH}", "828d5bc8ec56d05f5a291b004d68d122": "Q_{\\varphi}\\,\\!", "828d76a6a651bbcb043b23544c8ff75a": "\\begin{align}\nf &= \\sum_{k=0}^\\infty A_kz^{k+r} \\\\\nf' &= \\sum_{k=0}^\\infty (k+r)A_kz^{k+r-1} \\\\\nf'' &= \\sum_{k=0}^\\infty (k+r)(k+r-1)A_kz^{k+r-2}\n\\end{align}", "828d8b6e890c38890553bf5e4c6c5580": " \\frac{15}{4\\pi^4}\\int_0^x \\frac{t^7e^{2t}}{(e^t-1)^3} \\, dt .", "828da3418e10990c188497779c4ceb66": "\\sum_k A_{ik}(B_{kj} + C_{kj}) = \\sum_k A_{ik}B_{kj} + \\sum_k A_{ik}C_{kj} ", "828dc3e1281751b337b3bdabbde52c79": " p_y = p_\\text{T} \\sin \\phi ", "828dcdff118aea3db09ff7f24cf01b06": "F_0 = \\frac{1}{2\\pi R_{f1}C_1}", "828e15845230b2fc89c1e02d52c16191": "\\sum_{i=0}^{\\infty} \\, f(i) = a", "828e66087319d48a8a37232832be60e8": " =\\sum_{a^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}}}\\Pr\\left\\{ E_{a^{n}}\\right\\}\n\\Pr_{\\mathcal{S}}\\left\\{ \\bigcup\\limits_{b^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}\n},\\ b^{n}\\neq a^{n}}E_{a^{n}}^{\\dagger}E_{b^{n}}\\in N\\left( \\mathcal{S}\\right)\n\\right\\} ", "828f48d098596440f797ea0865af11e3": "M(x,y) = B \\frac{x (1+y)}{x+y}", "828f9f24d2d8979f98c99ad8a5ddb780": "C(f) := \\{ \\varphi_i \\in \\mathbf{P}^{(1)} | \\forall x.\\ \\Phi_i(x) \\leq f(x) \\}", "828fa6b3f497ea0b806cab8244d8da50": "\\sum_{i,j = 0}^{n} a_{ij} X_{i} X_{j}^{\\theta} =0", "828fb5fc6123e01c0cad8b9042ef7e33": " K_a = \\tan ^2 \\left( 45 - \\frac{\\phi}{2} \\right) \\ ", "828fe41487a2b71fb7e40873bdfae1d4": "\\cos{\\frac{A}{2}}=\\sqrt{\\frac{ad}{ad+bc}}=\\sin{\\frac{C}{2}},", "828fefaa9be5104d77362b6eab0d1e92": "\n Y = X + \\frac{1}{s}\\left((p_1+p_2)Y - (z_1+z_2)X\n\t\t\t\t\t+ \\frac{1}{s}(z_1 z_2 X - p_1 p_2 Y)\\right).\n", "828ff6d9cccbaaf0946db0d5908d100b": "f^{-1}(B)\\,", "828ff71c55602b54a5a4ea9af6beb621": "\n\\mathcal{A}^B\\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B) = \\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B).\n", "82904e0849e683cb8abfc2079e828f20": "\\,\\langle a_i^+a_j^+\\rangle", "829063e3f9e4cec930fee5a42092b920": "\\tilde{n} = n + z^2", "8290e80b20ed99a84ba1a638695b74c6": "J\\vdash K", "829171020c6b33c7ae4d8975f58d937d": "\\displaystyle{(f,g)_\\sigma \\le \\|f\\|\\cdot \\|g\\|,}", "829174dec1ce189b9b6e6a8e5d6a23fa": "\\begin{align}\n\\ln\\, \\mathcal{L} (\\alpha, \\beta|X) &= \\sum_{i=1}^N \\ln \\left (\\mathcal{L}_i (\\alpha, \\beta|X_i) \\right )\\\\\n&= \\sum_{i=1}^N \\ln \\left (f(X_i;\\alpha,\\beta) \\right ) \\\\\n&= \\sum_{i=1}^N \\ln \\left (\\frac{X_i^{\\alpha-1}(1-X_i)^{\\beta-1}}{\\Beta(\\alpha,\\beta)} \\right ) \\\\\n&= (\\alpha - 1)\\sum_{i=1}^N \\ln (X_i) + (\\beta- 1)\\sum_{i=1}^N \\ln (1-X_i) - N \\ln \\Beta(\\alpha,\\beta) \n\\end{align}", "8291aede8041aafb209e40fb9d5296e8": "\\vec{I}=\\int_{t_1}^{t_2}{\\vec{F} \\mathrm{d}t}", "82924b52e96f40c0df9616439d8d52ac": "\\Phi (\\mathbf{r})=\\int_{\\mathbb R^3}\\,{\\rm d}^3\\mathbf r'\\;\\frac{\\operatorname{div}\\,\\mathbf{v}(\\mathbf{r}')}{4\\pi|\\mathbf{r}-\\mathbf{r}'|}\\,.", "829271e01b8794fa09f5aeae83361091": "\\operatorname{pred}(n)", "82927480b6597025587a8f89a24408fc": "\\scriptstyle q \\;\\in\\; \\Omega", "82927cf8dd4434cf976986e2292e0c8c": "x=c_1", "8292b76d6fbd66420c727075f8cb7e16": "\\vec w", "8292c535b50967f1ec41807facba6cec": "\\mathbf{H_1}\\mathbf{y_1} = \\mathbf{H_1}\\mathbf{y_2}", "82931708255c5328459fd2519f553c9e": "\\ln(1+r)\\,", "8293273d9499f5ed56600eb8a72f51ac": "P_-=\\frac{1}{2}\\left(I-\\frac{A}{\\alpha}\\right)", "829333a948166d3fba335f1eaaae8c3e": "\\frac {\\theta_c + 1} 2", "82934f00e66ddc7fe7dd1a3c6d9da526": "\\beta'=\\beta", "82938e2034dac035e17e77d207d253f0": " 4 ", "82939090f599d57b6b20371c751b7bb2": "(0...\\pi/6)", "829404c0d5866b8ad28dd0fd238d7e2b": "\\, k\\pm\\tfrac{2\\,\\pi}{a}", "8294c8de54f6e6c1cc712eedae87bc15": " \\Phi(\\Psi_\\varepsilon(\\tilde{x})) = \\Phi(\\Psi) + f(\\varepsilon) g(\\tilde{x}) + o(f(\\varepsilon)) ", "829503f10f1207ad5e4363ce50b06034": "f_2(z) = \\,_1F_1(a+1;b+2;z)", "82953ba40094d8b092bb2352884de679": "\\varphi_r \\geq 0", "82954bc40edb7b5d28464637e517aae4": "f\\colon V_0\\rightarrow V^\\prime", "8295a7e73111c73f06da6aa5ab2dc100": "r_e = \\dfrac{\\int\\limits_{r_1}^{r_2} \\pi \\cdot r^3 \\cdot n(r)\\,dr}{\\int\\limits_{r_1}^{r_2} \\pi \\cdot r^2 \\cdot n(r)\\,dr}", "8295c696eb227de2439bcec90c40d260": "x:\\alpha \\vdash \\lambda\\ y.x : \\forall\\beta.\\beta\\rightarrow\\alpha", "8295e5bb437c7983b5a481dfe3f3f0e4": ", \\textbf{Q}", "829621454667c86a7605681fa4d9ab22": "\\mathbb{Z}^*_p", "829626aabc2f58a865a00b905a95b59c": " \\begin{pmatrix}\n\\gamma \\\\\nZ^0 \\end{pmatrix} = \\begin{pmatrix}\n\\cos \\theta_W & \\sin \\theta_W \\\\\n-\\sin \\theta_W & \\cos \\theta_W \\end{pmatrix} \\begin{pmatrix}\nB^0 \\\\\nW^0 \\end{pmatrix} ", "8296810da8692daf2532bebe8d21183e": "x_4 s", "829689bd4cc39dc64f112f7a40ec9249": "|n_1 n_2 \\cdots n_N; A\\rang = \\frac{1}{\\sqrt{N!}} \\sum_p \\mathrm{sgn}(p) |n_{p(1)}\\rang |n_{p(2)}\\rang \\cdots |n_{p(N)}\\rang\\ ", "8296cdc0850d46cde2d901df530adde1": "y(x)=x\\frac{dy}{dx}+f\\left(\\frac{dy}{dx}\\right).", "8296f41e0f63155fbcc633935456ce71": " 2n-1", "829750527942f75a182776a0f1960028": "\n\\begin{array}{rcl}\n{\\rm Si}(x) &=& x \\cdot \\left( \n\\frac{\n\\begin{array}{l}\n1 -4.54393409816329991\\cdot 10^{-2} \\cdot x^2 + 1.15457225751016682\\cdot 10^{-3} \\cdot x^4 - 1.41018536821330254\\cdot 10^{-5} \\cdot x^6 \\\\\n~~~ + 9.43280809438713025 \\cdot 10^{-8} \\cdot x^8 - 3.53201978997168357 \\cdot 10^{-10} \\cdot x^{10} + 7.08240282274875911 \\cdot 10^{-13} \\cdot x^{12} \\\\\n~~~ - 6.05338212010422477 \\cdot 10^{-16} \\cdot x^{14}\n\\end{array}\n}\n{\n\\begin{array}{l}\n1 + 1.01162145739225565 \\cdot 10^{-2} \\cdot x^2 + 4.99175116169755106 \\cdot 10^{-5} \\cdot x^4 + 1.55654986308745614 \\cdot 10^{-7} \\cdot x^6 \\\\\n~~~ + 3.28067571055789734 \\cdot 10^{-10} \\cdot x^8 + 4.5049097575386581 \\cdot 10^{-13} \\cdot x^{10} + 3.21107051193712168 \\cdot 10^{-16} \\cdot x^{12}\n\\end{array}\n}\n\\right)\\\\\n&~&\\\\\n{\\rm Ci}(x) &=& \\gamma + \\ln(x) +\\\\\n&& x^2 \\cdot \\left(\n\\frac{\n\\begin{array}{l}\n-0.25 + 7.51851524438898291 \\cdot 10^{-3} \\cdot x^2 - 1.27528342240267686 \\cdot 10^{-4} \\cdot x^4 + 1.05297363846239184 \\cdot 10^{-6} \\cdot x^6 \\\\\n~~~ -4.68889508144848019 \\cdot 10^{-9} \\cdot x^8 + 1.06480802891189243 \\cdot 10^{-11} \\cdot x^{10} - 9.93728488857585407 \\cdot 10^{-15} \\cdot x^{12} \\\\\n\\end{array}\n}\n{\n\\begin{array}{l}\n1 + 1.1592605689110735 \\cdot 10^{-2} \\cdot x^2 + 6.72126800814254432 \\cdot 10^{-5} \\cdot x^4 + 2.55533277086129636 \\cdot 10^{-7} \\cdot x^6 \\\\\n~~~ + 6.97071295760958946 \\cdot 10^{-10} \\cdot x^8 + 1.38536352772778619 \\cdot 10^{-12} \\cdot x^{10} + 1.89106054713059759 \\cdot 10^{-15} \\cdot x^{12} \\\\\n~~~ + 1.39759616731376855 \\cdot 10^{-18} \\cdot x^{14} \\\\\n\\end{array}\n}\n\\right)\n\\end{array}\n", "829765045ca7f6dcf958e9b3e8238a15": "x \\in V", "82978ed70df8182bdfbbc7735826dd4c": " |\\Psi|^2 = \\Psi^*\\Psi = \\left( {a \\over \\sqrt{a^2+(\\hbar t/m)^2} }\\right)^3 ~ e^{-{\\bold{r}\\cdot\\bold{r} a \\over a^2 + (\\hbar t/m)^2}} ~.", "8298377a34f206a8e40f7174c50c2082": "N(E_F) \\Delta", "82987fab3a5e5ac48354292a4007ad21": "L^p(\\partial \\Omega),", "82988300184d1b9126845e57c3c2cd22": " E=\\frac{1}{\\overline{P_m(2\\eta)}} \\sum_{j=0}^m a_j e^{\\lambda_j\\eta x} \\mathcal{F}^{-1}_{\\xi}\\left(\\frac{\\overline{P(i\\xi+\\lambda_j\\eta)}}{P(i \\xi + \\lambda_j \\eta)}\\right)", "829888fd43966faae2ab850dc897cd46": "C_3 = \\langle u \\mid u^{3} = 1 \\rangle", "82988fef087fc74fc3073bd2a25cfa08": "\\binom nk = \\frac{n(n-1)(n-2)\\cdots(n-k+1)}{k!}.", "8298c94648a655857f1c8b603a622e98": "P\\cdot p_{\\perp }=P\\cdot x_{\\perp }=0\\,.", "8299b20e28ea569314b44e6e835ddd32": "X\\setminus A \\in U", "8299c438b1b47715b0bd274a77862af1": "x_1y_2-x_2y_1,\\,x_1z_2-x_1z_2,\\,y_1z_2-y_2z_1,\\,x_1w_2-x_2w_1,\\,y_1w_2-y_2w_1,\\,z_1w_2-z_2w_1.", "829a1b6a91fdaacfc33b5453284609cb": "\\operatorname{sign}(p_1(\\xi))= \\operatorname{sign}(p'(\\xi));", "829a388f3a9f2caa57bfadb9a3043daf": "\\frac{\\dot m_0}{\\dot m_{01}} = \\sqrt{\\frac{T_{01}}{T_0}} \\sqrt{\\frac {\\epsilon_0^2 - \\epsilon_2^2}{\\epsilon_{01}^2 - \\epsilon_{21}^2}}", "829a3d4f80106e513b60c048450200cf": "g^{x_3}", "829a7732407cb8f4239e2ab823168587": "d1=dimeter = 1.082532 * P", "829a8392a613b8f34342a80e96ef39f2": "\\sqrt{-1}(\\partial\\bar\\partial f-\\omega)", "829b397cf431be5e4024cb4b86942b20": "\\|x\\|= 0", "829b8a338367e6b0f69ad9bf5c583c30": "\n\\hat{m} = \\frac{\\vec{r}_1 \\times \\vec{r}_2 }{||\\vec{r}_1 \\times \\vec{r}_2 ||}\n", "829bb5ede0770ccbc9ad6e6b549dcd83": "\\mathfrak{P}^{28}", "829bbbb374b9fdbd2a7cf4aee6ee7c44": "\\Lambda \\gg 1 ~ (\\Gamma \\ll 1)", "829c1f0a892a61a065dd48c6caa3b563": "bp_2", "829c2a9d1fb4c807ba58c6319056cd98": "F^{\\alpha\\beta} = \\frac{\\partial A^\\beta}{\\partial x_\\alpha} - \\frac{\\partial A^\\alpha}{\\partial x_\\beta},", "829c40b8d6c2e6ef35d0e2509779bc3a": "\\sigma(T)-\\mathcal{B}(\\overline{\\mathbb{R}})", "829c4e3fe6d64d5b49ab36af0ce9767e": "I(T)\\ =\\ A \\exp \\left( \\frac{-Q}{kT} \\right) \\exp \\left( \\frac{-16 \\pi \\gamma_{sl}^3}{3 \\Delta H_s^2} \\cdot \\frac{1}{kT} \\cdot \\frac{T_m^2}{\\Delta T^2} \\cdot f(\\theta) \\right)", "829c6afa56849244fc7c257acdc01b0e": "e^{-\\frac{x^2}{2}}\\cdot H_n(x) = 2^{n/2-\\frac{3}{4}}\\sqrt{n!}(\\pi n)^{-1/4}(\\sinh \\phi)^{-1/2} \\cdot \\exp\\left(\\left(\\frac{n}{2}+\\frac{1}{4}\\right)\\left(2\\phi-\\sinh(2\\phi)\\right)\\right)\\left[1+O(n^{-1}) \\right],", "829c6edf9f4460897219c67b7ee13862": "r=\\frac{W^2}{8H}+\\frac{H}{2}.", "829c9ca8b85a604723f05bb9bc238ffc": "S_{23}, S_{32}", "829ccff65e37752be0684b94d2c9ad62": "X_{k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1} = \n \\sum_{n_1=0}^{N_1-1} \n \\left( \\sum_{n_2=0}^{N_2-1} x_{n_1 N_2 + n_2 N_1} \n e^{-\\frac{2\\pi i}{N_2} n_2 k_2 } \\right)\n e^{-\\frac{2\\pi i}{N_1} n_1 k_1 }.\n\n", "829ceeacf7ecd054bfa4fe4b68bc0803": " m_{\\mathrm{e}} \\ ", "829d2b0e365b7b8a49d88d217526c8ea": "n(\\vec r),", "829d2d8fc25698e3a880c0e8abf37e77": "\\tilde G", "829d49bf349eb94dbfc755c0a10e5a53": "R_t=w_{1t}r_{1t}+w_{2t}r_{2t}+\\cdots + w_{nt}r_{nt}", "829d5193b98673957246a86fc1363a27": "l = 303^\\circ - \\arctan\\left({\\sin(192^\\circ.25 - \\alpha) \\over \\cos(192^\\circ.25 - \\alpha) \\sin 27^\\circ.4 - \\tan\\delta \\cos 27^\\circ.4}\\right)", "829da909cf0072317906029cf6ca8c2a": "F(x;\\alpha,\\beta) = I_x(\\alpha,\\beta) = 1- F(1- x;\\beta,\\alpha) = 1 - I_{1-x}(\\beta,\\alpha)", "829dd2053c2c9e277f78ab1476811396": "L=\\frac{g}{\\omega_n^2} = \\frac{I_P}{mr}.", "829dec6847a80f1e78fc151ccabc4f67": "\\hat{b} = u \\hat{a} + v \\hat{a}^\\dagger", "829e0abb8d1bb4e9d9e613ad25925cc6": " \\text{markup} = \\frac{0.5}{1 - 0.5} = 1 = 100%", "829e4153fc98296c25e658578e4f6180": "\\{\\delta_a : a\\in E\\}", "829e43818e771448857b276853f83212": "f(V^{{1 \\over 3}}) {1 \\over 3} V^{-{2 \\over 3}}={1 \\over 3 V log({5 \\over 3})}", "829e62cb8cdf3d6eb60eb2e4d78d7258": "S={1\\over 16\\pi G}\\int d^4x \\, \\sqrt{-g}L_\\phi+S_m\\;", "829e83ddedcef08f6a6f43d3cbc8ede2": "{\\arg\\min}_S E(x, S, C, \\lambda)", "829e8d9fc4d0a40f89e3aa341b0737f0": "\\lambda = k^2 - k", "829ec65824c7bf8ec0e1b7f701f65bb1": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 1\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 1 & 0 & 1\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 1 & 0\n\\end{array}\n\\right] .\n", "829f4c03dbb49ef835ac815087dc450b": "(\\alpha|g\\rangle+\\beta|e\\rangle)|0\\rangle\\leftrightarrow|g\\rangle(\\alpha|0\\rangle+\\beta|1\\rangle)", "829f83a174e4541820f211d6e83255cb": "\\mathbf{r}=x_j\\mathbf{e}_j\\,\\!", "829fd644ceed787437a41be6533188f6": "L=min(P1,P2)", "829fd707c4bb961443d8e5b73155344d": "k\\ ", "829fe3e740c621ee416085ee540405bb": "\n\\mathbf{g}_k = \\lim_{T\\to\\infty}\\frac{1}{T}\\int_0^T\\sum_{i=1}^n \\mathbf{1}_{\\{r_t(i)=k\\}}\\,d\\log\\mu_i(t)\n", "82a005f145da5680a70ab9cb239d1283": "\\frac{\\partial \\Pi_1}{\\partial q_1} = \\frac{\\partial P(q_1+q_2)}{\\partial q_2} \\cdot \\frac{\\partial q_2(q_1)}{\\partial q_1} \\cdot q_1 + P(q_1+q_2(q_1)) - \\frac{\\partial C_1 (q_1)}{\\partial q_1}=0.", "82a01356445cddc74dc9a5b2fe4f5a8d": "E_\\gamma \\ll m_ec^2", "82a08b018e3104d71d0fdae36fed5746": "B A B^{-1} = \\begin{bmatrix} \\sqrt 2/2 & \\sqrt 2/2 & 0 & 0 \\\\ -\\sqrt 2/2 & \\sqrt 2/2 & 0 & 0 \\\\ 0 & 0 & -\\sqrt 2/2 & \\sqrt 2/2 \\\\ 0 & 0 & -\\sqrt 2/2 & -\\sqrt 2/2 \\end{bmatrix}.", "82a0eabc76755611461a2779c7045582": "\\coth x = {\\rm{i}} \\cot {\\rm{i}}x \\!", "82a106f1980b46dc955537e3378d96f4": "\\beta =5", "82a1209335eb80f57c231aad3b4a5649": "\\arctan z = \\frac{z}{1+z^2} \\sum_{n=0}^\\infty \\prod_{k=1}^n \\frac{2k z^2}{(2k+1)(1+z^2)}.", "82a1848a49fa6e09632de018d79c6432": "\\Box p \\rightarrow \\Box \\Box p", "82a19a183ea387e48e91dbd98d8c989b": "\\zeta(s)", "82a1e6aa9a019b070e5e4ece31258918": "E < 0:~~~~~~~~ R = \\frac{M}{2 E} (1 - \\cos\\eta)~,~~~~~~~~ (\\eta - \\sin\\eta) = \\frac{(-2 E)^{3/2} (t - t_B)}{M}~;", "82a1f31119fdff878c7a5afd349f3e03": "[A]_0-[C]\\approx \\;[A]_0", "82a29be83ceaf9fc37abd26da5946dd8": "\\mathbf{T}=\\mathbf{a}\\otimes\\mathbf{b}\\otimes\\mathbf{c}, \\quad T_{ijk}=a_ib_jc_k ", "82a308a258da78ef1d92791c54c5c272": " V = \\frac {c}{n} + v \\left(1 - \\frac{1}{n^2}\\right) ", "82a32937736dc2b6b0ff4af9de570607": "B(u, v) = (Q(u+v)-Q(u)-Q(v))/2.", "82a35d8c6a6d233dce462cdaaf26e11c": " q_\\mathrm{LH} \\ = \\ \\sqrt{4\\pi} \\ q_\\mathrm{G} ", "82a367abe2823996f0b03601fd866dfa": "2^{p-1} \\equiv 1 \\pmod{p^2}\\,\\!", "82a372c03873beec502206d86c01e192": "\\int \\frac{e^{2\\lambda x}}{ae^{\\lambda x} + b} \\; \\mathrm{d}x = \\frac{1}{a^2 \\lambda} \\left[a e^{\\lambda x} + b - b \\ln\\left(a e^{\\lambda x} + b \\right) \\right] \\,", "82a384b84ebb9d0b067f4b13fa13a922": "h(p, u^*) = x(p, e(p, u^*)). \\,", "82a39eaac5244121ce4a318c370fe73d": "\\bold{u}", "82a3bc56fbbf39accbe841f27b930c33": "\\textbf{k}_0=2\\pi/\\lambda_0", "82a3da57c4ff3ce25c7189f0027f1087": "\\text{Hom}_Y(Y', X) \\to \\text{Hom}_Y(\\text{Spec} K, X)", "82a3e268b3f1103596444bee5e436b0f": "\\sum_{n=0}^\\infty (-1)^n z^{2n}.", "82a3e5afe051d5950188e64e65c62f3a": "\\begin{align}\n\\iint_{\\mathbf{R}^2} e^{-(x^2+y^2)}\\,d(x,y)\n&= \\int_0^{2\\pi} \\int_0^{\\infin} e^{-r^2}r\\,dr\\,d\\theta\\\\\n&= 2\\pi \\int_0^\\infty re^{-r^2}\\,dr\\\\\n&= 2\\pi \\int_{-\\infty}^0 \\tfrac{1}{2} e^s\\,ds && s = -r^2\\\\\n&= \\pi \\int_{-\\infty}^0 e^s\\,ds \\\\\n&= \\pi (e^0 - e^{-\\infty}) \\\\\n& =\\pi,\n\\end{align}", "82a4045cc0fd6f4c2f48567c088c9753": " k^{-3/2} ", "82a4813cf95d3161c1cf748f1be2be40": "f_a", "82a4c7c474f7cb8db296b465fc3b1810": "(c_0,..,c_{n-1})", "82a5063c753a28a609ae5ba83f60cf0f": "\nH^{(0)} \\psi^0_k = E^{(0)}_k \\psi^0_k, \\quad k=0,1, \\ldots, \\quad E^{(0)}_0 < E^{(0)}_1 \\le E^{(0)}_2, \\dots \n", "82a52e1d203a00a77d4b222e4630b023": "ESR", "82a5e6b146c80d627bea9379c9d688a1": "\\left(\\begin{array}{c} E(z+L) \\\\ F(z+L) \\end{array} \\right) =\n M\\cdot \\left(\\begin{array}{c} E(z) \\\\ F(z) \\end{array} \\right)", "82a5fe9044068e2780f6a46a6421e186": "e^{(2-i)x} = e^{2x} e^{-ix} = e^{2x} (\\cos x - i \\sin x)", "82a685078a24c3f1c2ba972b5f80cf0b": " \\dot{\\mathbf{x}}(t) = A\\mathbf{x}(t) + B\\mathbf{u}(t). ", "82a6be207f64dd5247a91a3688b3d7a0": " E_n = \\frac {E_1}{n^2} = \\frac {-13.6\\text{ eV}}{n^2}, \\quad n=1,2,3,\\ldots ", "82a72c543cfd8b5e2da1344f7b7cd676": "\\textbf{Ra} = \\frac{\\Delta\\rho g L^3}{D\\mu}", "82a75a47cbe0771f334a28eadb1cc940": "=t+\\tfrac{1}{2}p q+t'+\\tfrac{1}{2}p' q'+p q'-\\tfrac{1}{2}(p+p')(q+q')", "82a75ec8c8b6c1c1d5a120d48e7bf9e1": "\\vec{\\alpha}", "82a828f1144c50d49214975c6c52b0c6": "p_0,p_1,\\dots,p_n", "82a8aa4fe9eaa5de655ee2e5c0407cee": "\\hat{U}_{nn} = {1 \\over 2} \\sum_i \\sum_{j \\ne i} \\frac{Z_i Z_j e^2}{4 \\pi \\epsilon_0 \\left | \\mathbf{R}_i - \\mathbf{R}_j \\right | } =\n\\sum_i \\sum_{j > i} \\frac{Z_i Z_j e^2}{4 \\pi \\epsilon_0 \\left | \\mathbf{R}_i - \\mathbf{R}_j \\right | }. ", "82a8ab35b69b78c0b324318e66bd748e": "M_s = M_N \\cdot \\ldots \\cdot M_2 \\cdot M_1.", "82a8c0da97fad6818b65effce01246ea": " + 3m_1m_2m_3m_E + 3 = 0.", "82a921e33b51b6bcfb499ded8fc48f65": " R=\\frac{1}{2}+\\frac{V_f}{2U}(\\tan{\\beta_3} - \\tan{\\alpha_2}) ", "82a9397c5a066899652d4b54b61c94ee": " \\mathbf{p}=q\\mathbf{d}, ", "82a9c16e25904fd5dc7aa5addaef616d": "M = -6 +12/ \\sqrt{2} -8/ \\sqrt{3} +6/2 - 24/ \\sqrt{5} + \\dotsb = -1.74756\\dots.", "82a9c4014d48e12a9d55985d1007dda9": "\\textbf{S}_k", "82a9f1d9018f83471027ced083706dd6": "\\frac{Q}{t} = Av\\,", "82a9f91ed08fb00ee32272f7a3e1898e": "K_p T_u/8", "82aa5b785dfe6fbd8e106192851727f9": "(a+bi)\\left(\\frac{a}{a^2+b^2}-\\frac{b}{a^2+b^2}i\\right)=\\frac{(a+bi)(a-bi)}{a^2+b^2}=1.", "82aa7a09f9d4133472a202a5097f708d": "\\mathbb{Q}(\\sqrt{-5})", "82aa97f3edada68db2db2c14351659b5": "y_4 = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4", "82aace6643c386adad02b8106ecb3a62": " m^{3/2} \\rho(m)=\\frac{\\gamma}{m}\\big[1+(q_o-1) \\beta _o m\\big]^{\\frac{1}{q_o -1}}=\\frac{\\gamma}{m}[1+(q'_o-1) m]^{\\frac{\\beta _o}{q'_o -1}}", "82ab02856feabf5fe5baafd55d0b18bb": "\\Delta \\bar{E}_{UVW}", "82ab68f63741781dd188070d5d00f885": "\\| \\varphi \\|_{\\alpha, K_i} = \\max_{x \\in K_i} \\left |\\delta^{\\alpha} \\varphi \\right | .", "82ab909af015813357772aad9596657f": " n = 21 ", "82abb7a2e51f7cf262902236803d4ccd": " B=U +P_RV -T_RS-\\sum_i\\mu_{i,R}N_i \\qquad \\mbox{(2)} ", "82abfa6a9ccc21503684457e5d91f027": "p_1^2+p_2^2+p_3^2-n^2\\left(x_1,x_2,x_3\\right)=0", "82ac6068216fd3cbe01ce2ce8c70fd9a": "\\overline{C}", "82ac6098fc0ca9f69ac8b4dd03eda350": "f^{-1}(D)p_n(x)=np_{n-1}(x). \\,", "82ac74e99cb21f30fb2d0aa19f6dc81a": "\\hat{\\alpha} = \\underset{\\alpha}{\\operatorname{arg\\,min}} \\, D_\\alpha ", "82acaf7ce354fd1fd782a2f755770f9a": "\\left\\{\\begin{array}{l}r\\\\p\\\\q\\end{array}\\right\\}", "82acc367f18f7092e172da08932804b2": "R \\left[ \\psi(\\mathbf{x},t) \\right], S \\left[ \\psi(\\mathbf{x},t) \\right]", "82acd65e0217c637823d95a3e50474ef": "p(z)=\\sin(z)", "82acd662c218b262dc43ed6a3ea6c451": " ( x^\\lambda,\\sigma^m, \\sigma^m_\\lambda) ", "82ad0da02e3d01b060d165abdb9aace2": "-\\infty < u,v,x,y < \\infty", "82ad10ddc4db4965ec0223b83b26154c": " EF = \\frac{N_e - N_n}{N_e} ", "82ad13129c5d89aa6101232877a0f278": "\\mu_j=2^{j/2}\\frac{\\Gamma((k+j)/2)}{\\Gamma(k/2)}", "82ae21db40908993a511f53609904145": "Gr(2,4)\\, ,", "82ae3db94a615911cc95c82099ba23f2": " f: W \\to \\mathbb R", "82aea758c7dca1cf99bc8f8733954ab7": "(u-iv)(u+iv)-w^2=0\\,", "82af00f728097e617efff11e69dcf669": "a_0 + \\sum_{n=0}^\\infty \\frac{(-1)^{n}}{k_{n+1}k_{n}}.", "82af42b1eb3bcf9562c46f15fd966f27": "\\; \\Phi(f) = f(T).", "82af7df472302cf38c9919c67e56abe1": " (0.000069\\ ,\\ 0.001285)\\,", "82b0054b6fc57e05c547b45e158b3014": "(L^*_1,C^*_1,h_1)", "82b0135832f60ab02d9b16c1c95b5260": " \\operatorname{E}[X] =\n \\operatorname{E} \\left [\\begin{pmatrix}\n x_{1,1} & x_{1,2} & \\cdots & x_{1,n} \\\\\n x_{2,1} & x_{2,2} & \\cdots & x_{2,n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n x_{m,1} & x_{m,2} & \\cdots & x_{m,n}\n \\end{pmatrix} \\right ] =\n \\begin{pmatrix}\n \\operatorname{E}[x_{1,1}] & \\operatorname{E}[x_{1,2}] & \\cdots & \\operatorname{E}[x_{1,n}] \\\\\n \\operatorname{E}[x_{2,1}] & \\operatorname{E}[x_{2,2}] & \\cdots & \\operatorname{E}[x_{2,n}] \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\operatorname{E}[x_{m,1}] & \\operatorname{E}[x_{m,2}] & \\cdots & \\operatorname{E}[x_{m,n}]\n \\end{pmatrix}.", "82b08a96248b53bcd1a799d3355fbc86": " P(X_1,\\ldots, X_n)=Q(e_1(X_1 , \\ldots ,X_n), \\ldots, e_n(X_1 , \\ldots ,X_n)) ", "82b0e002cbb46fe90661cde00c67115b": "t^2\\lll x", "82b16aef51b2e045acc30060f05699a5": " C_j = C_i g^{-1} ", "82b17fd67d4335dc49e1d2daa9478bc9": " \\Delta\\bold{u} = \\bold{u}_1 - \\bold{u}_2 ", "82b1836fde02919531556a7fdf7b4db8": "M_\\Delta f(x) = \\sup_{x \\in Q_x} \\frac{1}{|Q_x|} \\int_{Q_x} |f(y)| dy", "82b1c33367261db8ec96e5f3baeea23a": "M \\in \\{0,1\\}^*", "82b20d585c65a498143e1efda64eefa5": "p(x,y,z)=", "82b22be68a8d6fbf9f6faec5a379e2d7": "\\frac{\\part^2 u}{\\partial x^2} + \\frac{\\part^2 u}{\\partial y^2}=0,~", "82b26bf7f8d9566b9504a88c14a5fafd": "E\\subseteq\\mathbb{R}^2", "82b29275f7b24a3cdf2a14c35e1600bd": "(\\dfrac{1}{2}).\\alpha N \\delta \\left((\\dfrac{\\delta}{2})-(\\dfrac{\\lambda}{d})\\right) =\\left((\\dfrac{1}{4}).\\alpha N (\\delta^2- O(\\dfrac{\\lambda}{d})\\right)", "82b2a481ca660c7a0de98410d5b7be6f": "f_{*} : H_n\\left(X\\right) \\rightarrow H_n\\left(Y\\right)", "82b2ab0ba6218e5b654f36b5aebe74be": "D_$ = DV01 = V \\cdot ModD / 100 ", "82b2b82ab60ce9c518151ebae2ba5093": "\\gamma(r) = b_i\\cdot bj\\gamma_0(r)V", "82b326dfccd0dcc2645310359886ff53": "f_n=0,g_n=0", "82b3803ca578345270baa2b380219e8f": "P(x)=2x^4-3x^3+x^2-2x-8.\\,\\!", "82b3aba224075a9a6fff643ebbefe61a": " P = \\frac{I_\\perp - I_\\|}{I_\\perp + I_\\|}\\ ,", "82b3adee59574b294bed697327f8858f": "g(x),", "82b43341c3e38f30fe2e07a3e7f15648": "\n\\operatorname{dVar}^2_n(X) := \\operatorname{dCov}^2_n(X,X) = \\tfrac{1}{n^2}\\sum_{k,\\ell}A_{k,\\ell}^2,\n", "82b46a47bccf6c73b52a6faebfba3fb8": " \\prod_{p} \\Big(1 - \\frac{1}{p}\\Big)^7 \\Big(1 + \\frac{7p+1}{p^2}\\Big) = 0.0013176... ", "82b46c2faf20da792b2859d48c90799c": "\\mathfrak{s}_0", "82b5b0674b9cf64a3a45326c0336935e": "\\lambda >0 ", "82b5faf7b5a272a1496b3a95d5eab650": "P^TBP=D_B", "82b60f0e2ab335cdfe32e04274209786": "= f(t-T).\\,", "82b623e1675388bacb57cd06ccfb0123": "\\frac{k}{i} i= (-j)i = -(ji) = -(-k) = k", "82b64a76ddb49669453e0ae98392fc68": "\\int_L \\overline{f(z)}\\,dz = \\int_L \\bar{f}\\,dx + i\\int_L \\bar{f}\\,dy = \\int_L (u,v)\\cdot d\\mathbf{r} + i\\int_L (-v,u)\\cdot d\\mathbf{r},", "82b725180b42971508f46ef0a67b27a0": "\n s^2 = \\frac{n}{n-1}\\,\\hat\\sigma^2 = \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\overline{x})^2.\n ", "82b7eaed56317b8e118c197095d150a3": "s_b(z)=\\log_b(1+b^z)", "82b80106bda5159214c2d8f8fad0189b": "0\\leq s \\leq t", "82b818bec321a85a618a9762591e82a6": " \\rho_f ", "82b827160ae4452a2340ff2388049433": "\\operatorname{Var}[c\\chi^2(k')] = 2c^2k' .", "82b8dfcceb6bbf0f009868271da4609c": "2T_n(x)^2 - 1 = T_{2n}(x) \\, ", "82b91e775b9acf163f79aaf764a08791": "\n\\begin{pmatrix}\n\\alpha_1 & 0 & 0 & & \\cdots & & 0 \\\\\n0 & \\alpha_2 & 0 & & \\cdots & & 0 \\\\\n0 & 0 & \\ddots & & & & 0\\\\\n\\vdots & & & \\alpha_r & & & \\vdots \\\\\n & & & & 0 & & \\\\\n & & & & & \\ddots & \\\\\n0 & & & \\cdots & & & 0\n\\end{pmatrix}.\n", "82b93965a0963cd32c43e6bace953d2a": "[H]", "82b9ad7f058f71809361d53b038f350a": "\\scriptstyle\\mathcal{X}", "82b9adb086444d1cb0858ee83d7a46e7": "\n\\mathrm{Fr}=\\frac{v^2}{gl}=\\frac{(lf)^2}{gl}=\\frac{lf^2}{g}.\n", "82b9c60c302c2d4add8deaf34fff07c4": "\\mu \\perp \\nu.", "82b9e99a2093bb31f27a5bedb83204c6": "V_{normal}=0", "82ba3b3d3ab5254c20acd0eadc2a3e3a": "\\partial_t n(t,a) + \\partial_a n(t,a) = -\\mu(a) n(a,t) ", "82baa66d192984a5896a5ac1ae1ee11f": " 2^\\kappa > \\kappa^+ ", "82babf9266d8cb6d7f42fffe47f40df2": "K_b", "82bad142dc455ced5dec3d3a6c5d6efd": "\\Phi:f(x)=f(y)", "82bb57545433e232a063372cc4abc645": "TP_s = 2~ \\mbox{if}~g\\ >= 0.04 ", "82bb5fa2a004e048775818b7a517a898": "{\\scriptstyle{1 \\over n^2 \\sqrt 5}}", "82bb8c4ca511bc2792728385e22b4224": "g(x) = f(Ax+b)", "82bbbc8e1760d6f80275c379ec13e778": "\n (2) \\qquad \\frac{\\rho C_0^2 \\chi}{(1 - s\\chi)^2}\\left(1 - \\frac{\\Gamma_0\\chi}{2}\\right) + \\frac{d e_0}{d V} + \\frac{\\Gamma_0}{V_0} e_0 = 0\\,.\n ", "82bbf13640e29d967874b605dbfce006": "\\mathrm{P} = \\frac{w_s}{\\kappa u_*}", "82bc0803fbc53e8e45d4871df5afbb1b": "P_1 \\sqcap P_2 \\equiv P_1 \\lor P_2", "82bc202a339d093a125f38c015e58c9c": "\\mathbf{n} \\sim \\mathcal{CN}(\\mathbf{0},\\,\\mathbf{S})", "82bc6c4f50532ff38c30b110e6f3e12b": "\\frac{\\partial E_A}{\\partial \\theta} = 0 ", "82bc721b786eba03f113b90ea2bd8d54": "\\{p_n(z) \\}", "82bccf29c794575b6c30dbdb7c24e04a": "\\Delta L = 20\\log 10 = 20 \\ \\mathrm{dB/decade}", "82bcd6a74dcf07f9642819cf0e7df23d": "\\underline{\\ell}", "82bd0ec5bb3ee1ee08dc13a64292b760": "c_n = \\frac{\\Gamma[(n+1)/2]}{\\pi^{(n+1)/2}}.", "82bd48f6b64a29569e6b7de6b49cc8eb": " f_{M_t,W_t}(m,w) = \\frac{2(2m - w)}{t\\sqrt{2 \\pi t}} e^{-\\frac{(2m-w)^2}{2t}}, \\qquad m \\ge 0, w \\leq m.", "82bd85a75a36d0b889e1f7f33bb4cd97": "\\langle s,t \\mid (st)^2 = s^3 = t^3 \\rangle.", "82bdbe40d4dee1c1f94bea4e6a94a4a8": "|x - y| + |u - v| = 0 \\!", "82bdddc78514f20aa382bd082872fe7d": " \\lambda = \\lambda_1+\\cdots+\\lambda_n.", "82be017acc86fef7b4400be1f9d72978": " C (1 + |\\xi|)^{-M} ", "82be1a35e200ac6659ca2fbe53b5cf30": "H = \\frac{R T}{M g_0} ", "82be74380f2bb86b3104ecbe5311f076": " k \\leftarrow 0 ", "82be78d470b94d79261419ec5e59be02": "s(x^n)=s(x)^n", "82bee8bdab4b48422d60d0b2e8c68bf5": "A \\subseteq \\N", "82bf136c82cede883cf03b6293c8bbad": "\n Px = EI\\,\\frac{d\\varphi}{dx} \\qquad \\text{and} \\qquad -P = \\kappa AG\\left(-\\varphi + \\frac{dw}{dx}\\right) \\,.\n ", "82bf2618095905fb50f7fc604ad0cf76": "1 < s \\le n", "82bf7945ddb4e553f86d90e66880a750": "\\textstyle I(X_r \\land Y_r)", "82bf8388647147dd56e40d0209882911": "\\sqrt{\\alpha}", "82bfa7542a845b350800d44c4c18b705": "2.03694", "82bfa7f0ce22a3e81d3f4fea45328e6b": "F(t_0,t_1,\\ldots) \n= \\sum\\langle\\tau_0^{k_0}\\tau_1^{k_1}\\cdots\\rangle\\prod_{i\\ge 0} \\frac{t_i^{k_i}}{k_i!}\n=\\frac{t_0^3}{6}+ \\frac{t_1}{24} + \\frac{t_0t_2}{24} + \\frac{t_1^2}{24}+ \\frac{t_0^2t_3}{48} + \\cdots\n", "82c0a1c79a9866e28717fd5c7386f7ed": "\\operatorname{Dirichlet}_N(\\beta N \\boldsymbol\\eta)", "82c0b575c3d04b866cd5935fdac09658": "(X_0, X_1, \\dots)", "82c0f1c6c164350ca24958283ef4abb0": "\n\\tau(a) = \\frac{1}{\\varphi(a)} \\sum_{0 \\leq bT\\}}A(X_t-X_T).", "82e0c2d83aad3ab96f159fe9a0823483": "\\begin{align}\ns &= \\bigl( F(f)\\circ\\pi_{F(\\mathrm{dom}(f))}\\bigr)_{f\\in\\mathrm{Hom}(J)} \\\\\nt &= \\bigl( \\pi_{F(\\mathrm{cod}(f))}\\bigr)_{f\\in\\mathrm{Hom}(J)}.\n\\end{align}", "82e0da1846d2f88ee036d67abf56152b": "\nJ\\!D\\!N = \n\\text{day} + \n\\left\\lfloor\\frac{153m+2}{5}\\right\\rfloor +\n365y+\n\\left\\lfloor\\frac{y}{4}\\right\\rfloor -\n\\left\\lfloor\\frac{y}{100}\\right\\rfloor +\n\\left\\lfloor\\frac{y}{400}\\right\\rfloor -\n32045\n", "82e12eb0c94d0a4534495f89fc71dd27": " u{\\partial \\upsilon \\over \\partial x}+\\upsilon{\\partial \\upsilon \\over \\partial y}=-{1\\over \\rho} {\\partial p \\over \\partial y}+{\\nu}\\left({\\partial^2 \\upsilon\\over \\partial x^2}+{\\partial^2 \\upsilon\\over \\partial y^2}\\right) ", "82e158fc44e5ad64b531219059246d15": "\\left(\\mathcal B([0,\\infty))\\otimes \\mathcal E^*\\right)^{\\lambda\\otimes \\mu}", "82e1adf102486bccf2d6ab8e4390d404": "A_{\\mu} = \\frac{Qr^3}{r^4 + a^2z^2}k_{\\mu}", "82e1b9efdaa8e077783c2bc60db3964b": "x, y \\in \\mathbb{C}", "82e1c1b52b7cb7daee63ceafd048e5ee": "1, i_1, i_2, i_3, \\varepsilon{}_4 , \\varepsilon{}_5, \\varepsilon{}_6 , \\varepsilon{}_7 ", "82e1e4d6f7dcb7541c909461997c9032": "\n\\operatorname{var}(\\textbf{X})\n=\n\\operatorname{cov}(\\textbf{X})\n=\n\\mathrm{E}\n\\left[\n (\\textbf{X} - \\mathrm{E} [\\textbf{X}])\n (\\textbf{X} - \\mathrm{E} [\\textbf{X}])^{\\rm T}\n\\right].\n", "82e2418285836524eccafe979adb92e9": "\n\\hat{z}_\\mathtt{KED} (\\mathbf{s}_0 ) = \\sum\\limits_{i = 1}^n\nw_i^\\mathtt{KED} (\\mathbf{s}_0 ) \\cdot z(\\mathbf{s}_i )\n", "82e2823ce638a1a65240b965fa59056c": "f \\sim g_1", "82e2989421313470173a9c0735734de7": "54.8 \\times 10^6", "82e29ba50f8284651b2492f018e215c9": "\\ \\sigma_{tresca}=\\sigma_1-\\sigma_3 > \\sigma_{max} ", "82e2b7454a6aff85e528bd6180efa6db": "H_2 = h_{vel} + h_{ele}", "82e2c5631a14946556c5450fa21904dc": "\\sum_{n=2}^{\\infty} \\zeta(n,\\bar{a},b) = \\zeta(\\overline{a+1},b)", "82e2eaacdf498114fa15a555f7722395": "c^* = \\frac{p_1 \\times A_t}{\\dot{m}}", "82e30ec143b968eb6a68329d35bf6ee1": "A^2=AA", "82e343e026875571f64095e14f58dec9": "f = \\frac{8 \\tau_w}{\\rho V_{avg} ^ 2} ", "82e3496fa2fd614255224c82a45b888b": "\n\\nabla_\\mu\n(nu^\\mu)=0.", "82e3a758752a581fbe648e18501b703e": " u(x+h) = u(x) + h(3u(x)+2). \\, ", "82e3f07d2ddd51a1f22b41b8e0d9a0a1": "\\displaystyle \\left\\{ 1, 2, 3 \\right\\}", "82e43a90edcf8f787b96556eb1a22c00": "\\scriptstyle T' \\,=\\, \\frac{1}{\\Delta f}", "82e4491e43d84c3c6a10e5f02c6b2cc4": "ab+2", "82e524097f4dd357a8da5640c427714e": "\n m\\rightarrow 0\n", "82e568a2883e3b388c65c900ae64f5d7": "r\\in\\mathcal{R}\\,\\!", "82e578ed56edcd4b21b4fc9093bcff53": "\\cos\\left[\\frac{\\pi}{N} \\left(2N-n-1+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right)\\right] = -\\cos\\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right)\\right]", "82e582a682a083d9ef5a6181304bfdc6": "z^1", "82e5fd5a213b53b2a0e0289279ce1a81": " \\frac {[(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)]^2} {(C_C C_L +C_L C_i+C_i C_C)(R_A//R_i) (R_O//R_L) } \\,\\! ", "82e60c58ad3b6647657abbaba26bd40f": "\\Diamond \\phi", "82e61f36d1095abb62ea3f31934e1bf9": "\\frac{V_{ss}}{2}", "82e628862dd4030c46f476cf680e607f": " \\mathrm{grad\\,}\\,\\varphi = \\left( \\frac{\\partial \\varphi}{\\partial x}, \\frac{\\partial \\varphi}{\\partial y}, \\frac{\\partial \\varphi}{\\partial z} \\right) ", "82e65acd3b48c1481ab12101e499799e": "+1,-1", "82e67e20ad8c862fe6768dc162c0471b": "f W g", "82e6832b308f31ac4aa53322b62e7d02": "\\mathbb{Z}[\\sqrt{3}] ", "82e6c9c205f19e4b3fa15243abbade4f": "\\int_a^b\\lang\\psi|\\psi\\rang\\, dx=1", "82e709a352d4b1d77f8697a1982838b0": "\nE(\\mathbf X) = \\frac{\\mathbf\\Psi}{\\nu-p-1}.", "82e7146a5ee70d5e48e4007a7d521425": "p_x", "82e737f7a570e4546f7f9a26d853f84a": " u_n ", "82e74680a1e7048af3641ea03bfb467f": "D_x E_x = E_x D_x = 1", "82e753f55dd77480fe5c79ac8980a86f": "\\rho = \\rho(\\mathbf{r},t) = \\rho(x,y,z,t)", "82e77c40b53f436ef7ffc54c93efce85": "P= {q^2\\omega^4\\ell_\\circ^2 \\over 12\\pi\\varepsilon_\\circ c^3}", "82e79155b1afc9ade8fd153d345b125c": "A \\mathbf{f} +B \\mathbf{f}'=\\mathbf{0}. ", "82e8cc291c7e2bfac6ed7c03e61023f9": "n=\\left\\lfloor\\frac{n}{m}\\right\\rfloor + \\left\\lfloor\\frac{n+1}{m}\\right\\rfloor +\\dots+\\left\\lfloor\\frac{n+m-1}{m}\\right\\rfloor.\n", "82e906993369dab1a659429d85aeee9d": "\\cot A = {\\csc A \\over \\sec A} ", "82e90a03b1af19e18267aede76fcdce0": " C(a,b) = 2\\cdot A(a,b) - H(a,b) = a+b- {{2ab} \\over {a+b}} ", "82e91a2ad7711fce9d2e6e245d0ba058": "\nn(\\overline{\\mathbf x}-\\boldsymbol{\\mu})'{\\mathbf \\Sigma}^{-1}(\\overline{\\mathbf x}-\\boldsymbol{\\mathbf\\mu})\\sim\\chi^2_p ,\n", "82e96530bc34ea6f40fc9b48ff9797ad": "\\operatorname{Pr}(X=x_k)=c \\exp\\left(\\sum_{j=1}^n \\lambda_j f_j(x_k)\\right)\\quad \\mbox{ for } k=1,2,\\ldots", "82e9906023ba1d2242eec9af50ad937e": "\\textstyle \\left ( {\\frac h {2\\pi}}\\right)", "82e99ced607f42619274b387763df7b8": " I_\\mathbf{g} = \\left | \\psi_\\mathbf{g} \\right |^2 \\propto \\left | F_\\mathbf{g} \\right |^2. ", "82e9a0f2145f28d452c7bd403a619631": "\\Delta^+\\subset\\Delta", "82e9a6371d47fc4212271e607d78b919": "2^X", "82ea7a5181754535e84e25b497fc3deb": "\\textstyle\\frac{2}{2}", "82eaaf184eab3acfa9ade9ad920f675b": "f(\\vec x) = g(\\vec x) + C \\|\\vec x\\|_1", "82eac65712f9c072e035b9c1a0e689ee": " Pos = P_t + \\left (\\acute{P}_t - P_t \\right)\\hat{T} ", "82eae033c9ecfc0ece6e0e2b032ebe5b": "\\alpha(p-\\xi)^a(q-\\xi)^b=\\alpha'(p'+\\xi)^{a'}(q'+\\xi)^{b'}\\!", "82eb161703dd4bbc62d7cae88b18338b": "p(\\sigma^2|D,I) \\propto p(\\sigma^2|I) \\; p(D|\\sigma^2)", "82eb34b1cbcc4c49e719aaf1d4a43d89": "\\leq m", "82eb6b16e226e0f3fff92dbd5d669b78": "|\\psi_{2n}\\rangle := |n+1,g\\rangle", "82ebb3648ea3d8f05c9091508af81efc": "\\frac{1}{H_{N,s}}\\sum_{n=1}^N \\frac{e^{int}}{n^s}", "82ebba5ae07595072cb8ed1a3a86c5a6": "\\omega_R = \\Omega \\ ", "82ec17783f136365676f1dbc0997db2f": "\\begin{align}\n\\int_0^\\infty\\;\\frac{\\sin\\,x}{x}\\;\\mathrm{d}x &\\to \\int_0^\\infty\\;e^{-\\alpha\\,x}\\;\\frac{\\sin\\,x}{x}\\;\\mathrm{d}x,\\\\\n\\int_0^{\\frac{\\pi}{2}}\\;\\frac{x}{\\tan\\,x}\\;\\mathrm{d}x &\\to\\int_0^{\\frac{\\pi}{2}}\\;\\frac{\\tan^{-1}(\\alpha\\,\\tan\\,x)}{\\tan\\,x}\\;\\mathrm{d}x,\\\\\n\\int_0^{\\infty}\\;\\frac{\\ln\\,(1+x^2)}{1+x^2}\\;\\mathrm{d}x &\\to\\int_0^{\\infty}\\;\\frac{\\ln\\,(1+\\alpha^2\\,x^2)}{1+x^2}\\;\\mathrm{d}x \\\\\n\\int_0^1\\;\\frac{x-1}{\\ln\\,x}\\;\\mathrm{d}x &\\to \\int_0^1\\;\\frac{x^\\alpha-1}{\\ln\\,x}\\;\\mathrm{d}x.\n\\end{align}", "82ec22715a3f4901008635ee3e51ff46": "\\mathbf{Y}_{10}= \\sqrt{\\frac{3}{4\\pi}}\\cos\\theta\\,\\hat{\\mathbf{r}}", "82ec81d32149fda74515256a828bb0e6": "1-hmcr", "82ec9415ccb0ab4b3ae723bd8a91a460": "P(T) \\approx 10^{-21.2} ~~~~~~~~~~~~ (10^{4.9} < T < 10^{5.4} K) ", "82ecc1f6e0df90b9b40a69d3c81db934": "\n\\partial_{t_i} P(\\mathbf{x},\\mathbf{t}\\mid \\mathbf{x_0}) = D_i \\partial_{x_i}^2 \\int_0^{t_i} k_{\\alpha}(t_i-u_i) P(\\mathbf{x},\\mathbf{t}^{'(i)},u_i\\mid \\mathbf{x_0}) \\, du_i;\\qquad -M\\le i \\le M. \n", "82ed59729e66436c8f4238235a70d03d": "dV = dx\\,dy\\,dz.", "82ed683857c689b3b27eab06f752e1a1": "\\scriptstyle \\tilde x", "82ed881b4ab0755e252fc13173abba3c": "(k+1) O(\\log \\log n)", "82ed941ae5e3966e5e6084a8b6f494c5": "F(\\ast,\\lambda)", "82ed9efc5ffb085b70d3752ff4d685ec": "\\mathfrak g_P", "82edc071096723c09213fb2ae3f41308": "f = {n\\sqrt {T \\over \\rho} \\over 2 L} = {n\\sqrt {T \\over m / L} \\over 2 L}", "82ee4d4c57fc5fd61837bc3b21d38b78": "D=D^{(0)}+\\epsilon D^{(1)} ", "82eec2b6fe8e6ae5a5958a186fb11422": "(n-1)J_n = \\frac{\\sin{ax}}{a\\cos^{n-1}{ax}}+ (n-2)I_{n-2}\\,\\!", "82eee0c20aba0fa3ef0550e416c962f0": "\\sigma _a", "82ef6124286e8b40a33c220de4371a23": "\\{a,b,a^{-1},b^{-1}\\}", "82efe0fa9b27e1d8205509f30301b3c1": "\\|X - \\mu\\|_\\alpha", "82f0413fb9addc2259e5e25f37b0ad4a": "\\mu_0=k\\Pi_0^{n-1}+\\tau_0\\Pi_0^{-1}", "82f07ffd0f4a83a11717b285da24d211": "a = (a_0:a_1:a_2: \\dots),", "82f0a1b5d193e08f3ba94821a3f74fba": " \\lfloor js/r \\rfloor", "82f0a310208336e41933d0b0644af2a9": "\\dot {U}_m", "82f0b293ffbdca18ef5dfd3d5dcb6b1c": "a, 2a, 3a, \\ldots, (p-1)a \\quad\\quad (A) ", "82f12c0842efb02e32d1eccc2aff1993": "(\\hat {\\textbf{q}}_i)", "82f1d27f1e0e57bea7e456bd7bf27b2e": "\n S \\approx 0.35 a h\n ", "82f20aaa0130d9e8a438bddf24725473": "L=[(\\Sigma\\cup\\{\\epsilon\\}) \\times \\Gamma] \\cup [\\Sigma \\times (\\Gamma\\cup\\{\\epsilon\\})]", "82f2221b89d6a72c18f57f69dece87bd": " \\sum_{i=1}^N m_i(\\mathbf{R}_i-\\mathbf{R})=0,", "82f27b1bcff5bef88ab86a642cc328a9": "\\phi_{2,m}", "82f2c7812e922296f448d05aa03b7c09": "u(0,x)= 0", "82f2fdc092d3615aeb4c7ad8cfd856c3": "\\left( \\mathcal{P} (M), \\pi \\right)", "82f32896647dbc72f9a65c02664e932f": "k^{-m}<1", "82f345081f5cc03b8612137a8f42e159": "h_1(X_1) = X_1\\,.", "82f35266d8ed375757ebd4cc87ba374a": "\nb'(s') = \\eta \\Omega(o\\mid s',a) \\sum_{s\\in S} T(s'\\mid s,a)b(s)\n", "82f369a90333270f4218657e750f3bdc": "\\mathbf{F} = q \\left[\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B} \\right].", "82f3ad1dd32180b504c613078ae724f9": " 0\\rightarrow \\Omega^0(M) \\xrightarrow{d_0} \\Omega^1(M)\\xrightarrow{d_1} \\cdots\\xrightarrow{d_{n-1}} \\Omega^n(M)\\xrightarrow{d_n} 0 ", "82f3c0d75dfaf856cc0d3fc827149a0c": "N = mg \\cos(\\theta) ", "82f41b7efa6b30b03c29a2acb6690d60": "P(A)=1-\\left(\\frac{5}{6}\\right)^{6} = \\frac{31031}{46656} \\approx 0.6651\\, ,", "82f4240c35cb86be8a8b515809676570": "F_{\\mathrm{kf}} = \\mu_{\\mathrm{kf}} F_\\mathrm{N}", "82f42966dc1729c58c3bf794616f59ba": "\\epsilon_{f} = 2 \\nu \\bar{S_{ij}} \\bar{S_{ij}}", "82f49039bf2e2e97c37791efeea7ceaf": "{2\\pi}\\over 3", "82f4e6f37e76c44d0708b86cc7103e5b": "\\mu' ", "82f5036feb3abacd4b576fa1de43c1e1": "K[T]", "82f510fe4b3357ac4bfe40d65e159e46": "w(t) = \\frac{at + b}{ct + d}", "82f513879686c99a7b0af827d1b90e7b": "\\forall m [Sm=0 \\rightarrow \\bot].", "82f533ea8104f14566c62ab9a4946557": "\\mathcal{M}_0", "82f53887aab5000f53b4e3c38dccd749": "\\kappa > \\aleph_0;", "82f5a2f23dc86af0762435a73157d8fd": " {\\lambda_n}^{+} \\sim \\frac{n^2 \\pi^2}{\\left(\\int_a^b \\sqrt{(w/p)_{+}(x)}\\, dx\\right)^2},\\quad n \\to \\infty, ", "82f5d221068cc7ee26c5c5148d89fc71": "_2^1\\text{S}^\\gamma= ^{14}\\text{N}^{15}\\text{NO}", "82f6124760060324eabb2c189857737b": "\\Sigma _{YY} = \\operatorname{cov}(Y, Y)", "82f62832c9234088aba4a58320750873": " Target = \\mathrm{round} \\left( \\frac{\\min \\left( \\max \\left( ATEE - 1000, 1000 \\right), 2500 \\right) }{35} \\right) ", "82f685b89081f7a6f17c68e48afe1175": "\\cos a \\sin c = \\sin a \\,\\cos c \\, \\cos B + \\sin b \\,\\cos A", "82f69f608a2997064b4432ec39d1a27c": "\\omega=2 \\pi \\nu", "82f6a06f5066cb1d4a4ab2cf84108192": "Q=Q(\\pi/2)", "82f77d4eb9d82d2a79edfe82c62e9a3a": " {\\textstyle \\sum} a_kz^k = a(z) \\, (\\boldsymbol B), ", "82f77faac7556965d86f8495cc761819": "\\mathrm{TAS}=\\mathrm{EAS}\\sqrt{\\frac{\\rho_0}{\\rho}}", "82f7a4026868a73943dd87032a9ae314": "g(x)=\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}", "82f7af24a573168f81bd491d78fc23d1": "v\\in E^s", "82f7d7eb48426eef4781005bcf7f5dc8": " \\textbf{y}(t) = \\begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} \\end{bmatrix}\\textbf{x}(t). \\,", "82f7dc1b199c7bf33d61e4a43416b395": "V_Z", "82f7e04f723da5d60066dff663d98acf": " | 3 \\cdot 5^e | = \\lim_k |\\{e \\} (k) | ", "82f7e9e07a6493e3e58d19e025bd4ec2": "\\mathbf{f}_f \\in \\mathbb{R}^3", "82f8434287a100ec5dd2046f2878a302": "E_j^{rot}", "82f8a9df26ec45845c0fe907174c8817": "F\\left(x\\right)=x^2\\sin\\left(\\frac{1}{x}\\right)", "82f96556c26a145776c1884680275ae2": " 20 \\mathrm{m^2} ", "82f977a11ac71455bfdc1046f54b10eb": "\\textbf{diag}(Ax+b)", "82f985cd5004592221ce235596bf23ec": " i_2=i_t=i -i_r\\, ", "82f98a23a456ca2cafdc27ba7a2f33c7": "y_{j+1}-x_{i+1} < \\delta < \\frac{\\varepsilon}{2r(m-1)},", "82f9b66f98356b7c82758c9f1cbaa1b8": "\n (2.1)\\quad\n E (\\mathcal{A}f)(Y) = 0\\text{ for all } f \\quad \\iff \\quad Y \\text{ has distribution } Q.\n", "82fa32e605cd6a67acf525947fee94e1": " -[R]^2= -\\begin{bmatrix} 0 & -z & y \\\\ z & 0 & -x \\\\ -y & x & 0 \\end{bmatrix}^2 = \\begin{bmatrix}\n y^2+z^2 & -xy & -xz \\\\ -y x & x^2+z^2 & -yz \\\\ -zx & -zy & x^2+y^2 \\end{bmatrix}.", "82fa6fdcdc832c806246335f7c0b5f1b": "\nK_S=\\gamma\\, P\n", "82fa80a42381966098d99a99abfa1455": "U_A ", "82fa9eee68ffe5cc14c76c4b24c8b926": "\\exp(-E_a/RT)\\ \\ ", "82faaab339c768e55eb6d0c62b6533c3": "\\Delta x \\, \\Delta p\\gtrsim h\\qquad\\qquad\\qquad (1)", "82fabe2ccd57024569a9b8a5dc0801ad": "2\\times 2 = 4,", "82fb24d379ed2b43c3e7de09910a8014": "F_l", "82fbcdd1bc0cae54ae7bbb932207c524": " y' = p(x)y + q(x)y^n. ", "82fcb7283731a563af4e8cb2452f65a7": "T_C = \\frac{-a+b+c}{a} : \\frac{a-b+c}{b} : 0", "82fcef5e8f4416aeccdc93284c9c7790": "\\scriptstyle \\eta(x) ", "82fd9655a6e2a4a633ba85a42d7cb726": "S(T) = \\frac{C T^A}{\\exp\\left(\\frac{B}{T}\\right)-1}", "82fd9a54d05ea0407c7c6f05e1afd15f": "\\cos(B x)", "82fe45c18258e88c16f80fa1f0b3474d": "e=2+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{4+{}\\ddots}}}}}", "82fe70c7375dbc5aab428656d5000fd4": " I - A ", "82fede4bc9528623e965e8895e68622e": "B_2=ik_0(C_re^{iak_0}-C_le^{+ak_0})", "82fedfced28c7a3569dd7f28588c34c1": "e_q(x) = [1+(1-q)x]^{1 \\over 1-q} \\text{ for } 1+(1-q)x >0 \\text{, otherwise } e_q(x) = 0 \\text{ if } q<1 \\text{, } =+\\infty \\text{ if }q>1", "82fee6810731a148e949120af5e2fd40": "((x, y), z) = (x, (y, z))", "82ff13b8dc04ceebeb44930187bac3a3": "\\Omega = \\frac{\\Delta PE}{\\Delta t}", "82ff1be7873fb8f6dac54fa3b7203888": "n!^{(k)}", "82ff328aba184884d8f8da43d516679e": "2g\\sqrt n", "82ff39b25a9d305dd29824c0d69d4b82": "0<|q|<1", "82ff41714ca257a9e445569ff8635678": "H \\left|0\\right\\rangle = \\frac{\\hbar\\omega}{2} \\left|0\\right\\rangle", "82ff85ca39bff256e8570fb269d8f37d": "\\pi=\\frac{4\\sqrt{3}}{3Z} \\!", "82ff8a2e53aaf8a39a64c3a2be9488a6": "\\text{var}\\,(Y) = a\\mu^p", "82fface902d6532624795518f2aa2ffb": "\\left( H_x - i\\hbar \\frac{\\partial}{\\partial t} \\right) K(x,t;x',t') = -i\\hbar \\delta(x-x')\\delta(t-t')", "82ffc03634a5d9a76d1b613409db958d": "\\mathbb{S}^n", "830014a1cd6f3b7500a0185a2d118e71": "\\psi_0\\mathit(V)\\!", "830038dff7301fb6857405512cf92b3f": "I_B \\to 0 \\ ", "830086541799ffd45bfa3b37e3f0a319": "N(E_F)", "830095864c421255702b7d509fa62890": "\\xi\\Big.", "8300c80d2afaa679b93617f29cc64d3c": " \n \\int_0^{2 \\pi} {d\\varphi \\over 2 \\pi} \\exp\\left( i p \\cos\\left( \\varphi \\right) \\right)\n=\n\\mathcal J_0 \\left( p \\right)\n . ", "8300f01d3d4356b501861b27b4209f24": "x \\nabla y = x + y + yx - xyx - yxy ", "8300ff485b4b11cf85e1fcc130c6b3bb": " \\frac{2^{r_1}\\cdot(2\\pi)^{r_2}\\cdot h\\cdot \\operatorname{Reg}}{w \\cdot \\sqrt{|D|}}.", "830108e5b8a1bb8c4fbdf84c91e869b2": "a=2k \\Leftrightarrow b=2k'", "83011c23d1d11e7c8b6c4033a4a3a255": "N((ab)\\sigma)", "83011e179356d98b8bc49777d4c9a3be": "\\mathit{d_H^{RC}}(\\mathcal(O)) \\geq \\mathit{d_{min}}", "830128da085abd5b4c3b95553cbbc5ad": "\\scriptstyle{ \\{\\emptyset\\} }", "830132ff68bc93c6042115a9a6d7d06d": "\\phi_\\lambda: W(\\lambda)\\times W(\\lambda)\\to R", "8301361a6626ddf5e850c323fa494b92": "\\frac{(xyz - 1)^2}{(xy + y + 1)(yz + z + 1)(zx + x + 1)}.", "830151e757685ef34150c4f3386e7c6a": " A^+ = \\langle A \\rangle _0 + \\langle A \\rangle _2 + \\langle A \\rangle _4 + \\cdots ", "83016b3021dccec0dce492b8fb143057": "D(f)", "83018742ce5ddbfe98abb1cf659cb729": "R_{a,\\theta} \\ ", "8301c27e4a2e1d1f933732b6cf31deed": "\n\\Pr_{x \\in_R D_n}[t_A(x) \\geq t] \\leq \\frac{p(n)}{t^\\epsilon}\n", "830231f641350910ea79e6a48535f554": "|i-j|>1.", "830294f3578f344bf85b4ff851efb34e": "Q_{total} = {2 \\over 3} \\pi \\tau \\omega \\left (R_{shoulder}^3 - R_{pin}^3 \\right ) ", "8303650b391f8199979a679fd51c5de5": "z=7.0", "83037d23b1bd7faf4e6ed85cc0a1d584": "n_2 (\\lambda)", "8303adc95a7d04285c7c89da4342d1b5": "\\mathcal{N}B(\\phi_i)=B(\\phi_{i+1})", "8303ddd98125ad150b97d7cfaa36f2da": "G^1=G", "830453599e826d47c4297924187a387f": "H = H_1", "830472b802959435ce0fdd61233f518e": " c^\\star \\approx 0.4773 ", "830484d92a657612e7526a5210a4ca01": "N_\\uparrow", "830499d4f5ec8e8c816fb8745c23217c": "V^1 \\ \\stackrel{\\mathrm{def}}{=}\\ \\rho^{i}(u^{1})\\frac{\\partial}{\\partial x^{i}} + \\phi^{\\alpha}(u^{1})\\frac{\\partial}{\\partial u^{\\alpha}} + \\chi^{\\alpha}_{i}(u^{1})\\frac{\\partial}{\\partial u^{\\alpha}_{i}}.", "83049a98a97f000e0cff3765cfab6325": "B \\subseteq V", "8304c7d83bac92c274cc411ca26ebfd7": "N_P=\\min\\{N_U\\ ,\\ N_D\\}", "8304c7fa15cd730d6735c1c863bafca8": " \\Delta_r G =\\left(\\sigma \\mu_S^\\ominus+\\tau \\mu_T^\\ominus -\\alpha \\mu_A^\\ominus- \\beta \\mu_B^\\ominus \\right) + RT \\ln \\frac{a_S^\\sigma a_T^\\tau} {a_A^\\alpha a_B^\\beta} =0", "830511e65db0c4ff1c01be208835b31e": "\\mathbf{B}=\\mathbf{C}=\\begin{bmatrix} \\alpha^2 & 0 & 0 \\\\\n0 & \\alpha^{-1} & 0 \\\\ \n0 & 0 & \\alpha^{-1} \\end{bmatrix}\\,\\!", "83054bf551c8f66f3faca2d9d367a464": "\\textstyle r", "8305bf9ee4e2dae91a2a91ef898a31cb": "\n\\lambda_n \\leq R_A(x) \\leq \\lambda_1 \\quad\\forall x \\in \\mathbb{C}^n\n", "8305d496fa5c0138a04879726b73eb71": "\\sum_{1 \\le i < j \\le n} \\left(a_ib_j-a_jb_i \\right)^2 = | \\mathbf a |^2 | \\mathbf b |^2 - (\\mathbf {a \\cdot b } )^2\\ , ", "8305f47244d98c44eb508c8d0fd46bf1": "u_{max} = u_{static}DAF", "83060e547b387dd45a71c01729d1b28e": "1 \\times \\sqrt{5}", "83064d6b1c4121b38af7ca229a188833": "z^2 \\equiv \\prod_{p_i\\in P} p_i^{a_i} \\pmod{N}", "8306a6f87993791b74283aab0eca4d8a": " t \\rightarrow t + T_{1/2} \\,\\!", "8306bd117e59f87513a845c2cd939211": "disc(\\mathcal{H}) = O(\\sqrt t)", "8306e69bea678cf9971a2ba4f2befc3b": "\\le 200", "8306f696e78dceb28d4cc9f3bcf935d2": "\\Delta G^\\ominus=\\Delta G^\\ominus_{Fe}+\\Delta G^\\ominus_{Ce}", "8307e349a9872f6fd5ca99adecd37f00": "Z_0 = \\sqrt{\\frac{\\mu_0} {\\epsilon_0}}", "830802072fcb5f7c7b11eb18c0f3dd5c": "f\\in L^2(0,\\infty)", "8308076a654fefad56bc7dd84149bdd4": "R^{\\frac{1}{K}}", "8308702cdf4f0275984aeb31543567da": "f=h+\\sum_{g\\in G} q_g\\,g,", "83089788cf1e4d308b4fbcb7cdc1e27d": "M(a,c,z) = \\lim_{b\\to \\infty}{}_2F_1(a,b;c;b^{-1}z)", "8308abf1f79d80025403faef95bbf5e3": " (\\mathbf{\\mu_L})_z=-\\mu_B m_\\ell.\\,", "83093113e6550f3993eb4cb36a01df3a": "y'(\\phi)=a\\cos\\phi", "830942badb6c5c44b3f1d921ed6b9f58": "-4f[n+3] +2nf[n+2] + n(n-4)f[n+1] +2f[n] = 0.", "830976c0fd640f0bf0ff93b262d75019": "\\mathbf{D} \\cdot \\mathrm{d}\\mathbf{A} ", "8309881512b509e73234cad254f2a3f9": "\\frac{2}{x_1-x_0} k_0\\ +\\frac{1}{x_1-x_0}k_1 = 3\\ \\frac{y_1-y_0}{(x_1-x_0)^2},", "8309a30dee2a2999bc5afea0bebc628d": " f(n) = \\begin{cases} n/2 &\\text{if } n \\equiv 0 \\\\ (3n +1)/2 & \\text{if } n \\equiv 1. \\end{cases} \\pmod{2}", "8309f9f32b46b0ca9c7fa46e0d0942ae": "u=Aw", "830a057a77caf999db70a454269eb6f2": "{d\\tau}^2 = - g_{\\mu \\nu} dx^\\mu dx^\\nu \\,", "830ac1205cd46f4e916607a4da3c7486": "\\displaystyle E(P:\\psi:\\nu:x) = \\int_K\\psi(xk)\\tau(k^{-1})\\exp((i\\nu-\\rho_P)H_P(xk)) \\, dk", "830ac8c1c07cd594947c962b51369ac2": "\\beta \\geq 0", "830ad00556803dae6aaabaa17bd09459": "|a_{q_j}^{-}|", "830ae8174356a2263ccf3a7fc3f245dd": "\\left|B_{2n}\\right| \\sim \\frac{2(2n)!}{(2\\pi)^{2n}}", "830b545b0bbdb52c3adb4bdd9d913f79": "\\tfrac{s^2 \\pi^2}{3}", "830b8b766d030691b252719ab9e94a7d": "\\mathbf{F} = Q \\cdot (\\mathbf{v}\\times\\mathbf{B} )", "830bf76d9526a749c3867764c55e9ed8": "{q^{2}\\over 4}+{p^{3}\\over 27} < 0 ", "830c04689fd2b7f27eb83f621b0751bf": "C_{10}", "830c27c57a7ec5b2be6c1e2383406779": " q=q_v(T)^3 ", "830c4dc53be10cd1f353fcad91dddab7": "\\scriptstyle N/n_i", "830d266eb485aa96431d1801000037d2": "\\sigma^2=\\operatorname E\\left [ s^2(\\vartheta) \\right ]", "830d296a0e047513956a4e20d328e0b2": "\\|x \\times y\\| = \\|x\\| \\|y\\| \\sin \\theta.", "830d3ec71fedf8bbda933b8b1259961b": "\\Lambda_1\\cup\\Lambda_2=\\mathbb{L}", "830d41b235cc241f8cbacb0a8ecb1f16": "\n (\\or L)\n ", "830d43f51eda447ecbfde8992e0d9d89": "0 -\\frac{1}{n}}", "830f536be751ae81b1826cbb22712841": "(A^TC_Z^{-1}A + C_X^{-1}),", "830f7625af8fd0d7782248ec98a47a5c": "\\mathsf{NP} \\subseteq \\mathsf{P/poly} \\implies \\mathsf{AM} = \\mathsf{MA}", "8310446858b32af49bc5ff2c86a74d15": "L = 4\\pi R^2T_{eff}^4", "83107e579177155a2e57226fb5ca7aad": " \\left(a, y\\right) > \\left(b, x\\right)", "8310802bc4bf8bf9a617b80defc221a3": "\\scriptstyle\\mathbb{E}", "831094593d48d1c81c58aa031a6f4e82": "\n\\begin{array}{rrcl}\nn=0: \\quad & T^0 (x,y) &= & \\mbox{Input Image}(x,y) \\\\\nn=1: \\quad & T^1 (x,y) &= & T^0 \\left( \\bmod(2x+y, N), \\bmod(x+y, N) \\right) \\\\\n& &\\vdots \\\\\nn=k: \\quad & T^k (x,y) &= & T^{k-1} \\left( \\bmod(2x+y, N), \\bmod(x+y, N) \\right) \\\\\n& &\\vdots \\\\\nn=m: \\quad & \\mbox{Output Image}(x,y) &=& T^m (x,y)\n\\end{array}\n", "83114f454c39d52b1f0d88097d9ff643": "\\exists{x}{\\in}\\mathbf{X}\\, P(x)", "8311a36dd2d3af5d55da5094c0eb5efc": " {{R_L+r_O} \\over {(g_m+g_{mb}) r_O +1}} ", "8311b6b984f8412002f98c74f95ffda6": "P=\\int_0^\\infty d\\nu \\int_h d\\Omega\\,B_\\nu \\cos(\\theta)", "831220cbba25ee27930f60c272507019": "A_0={\\sqrt{3}\\over4}a^2\\,", "8312472ff5e591ae7fb460bc81d3bc02": "r=C_S^2 \\frac{K_2C_B}{K_1C_A}", "8312b980edb2f3080e73a834ddccc39c": "\\left | v - w_{q'} \\right \\vert \\le \\left | v - w_q \\right \\vert \\forall q \\in \\mathbb{N}", "8312c8f707968a5da956f07daa63fbf5": "\\scriptstyle(6.9\\pm4.5)\\times10^{-11}", "8312d9160afbfef0125be9059dd2f89f": "x = t ", "83130f036ca966b7a62aa646fce4ab98": "\\mathrm{SO}(4n,\\mathbb C)", "83136843b2ca34a16d456d848c2e5f5b": "i = \\sigma_1 \\sigma_2 \\sigma_3", "831437dc2a660f2f6577c12acc1778aa": "\\begin{align}\n{\\mathbf{A=LDL}^\\mathrm{T}} & =\n\\begin{pmatrix}\n \\mathbf I & 0 & 0 \\\\\n \\mathbf L_{21} & \\mathbf I & 0 \\\\\n \\mathbf L_{31} & \\mathbf L_{32} & \\mathbf I\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n \\mathbf D_1 & 0 & 0 \\\\\n 0 & \\mathbf D_2 & 0 \\\\\n 0 & 0 & \\mathbf D_3\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n \\mathbf I & \\mathbf L_{21}^\\mathrm T & \\mathbf L_{31}^\\mathrm T \\\\\n 0 & \\mathbf I & \\mathbf L_{32}^\\mathrm T \\\\\n 0 & 0 & \\mathbf I\\\\\n\\end{pmatrix} \\\\\n& = \\begin{pmatrix}\n \\mathbf D_1 & &(\\mathrm{symmetric}) \\\\\n \\mathbf L_{21} \\mathbf D_1 & \\mathbf L_{21} \\mathbf D_1 \\mathbf L_{21}^\\mathrm T + \\mathbf D_2& \\\\\n \\mathbf L_{31} \\mathbf D_1 & \\mathbf L_{31} \\mathbf D_{1} \\mathbf L_{21}^\\mathrm T + \\mathbf L_{32} \\mathbf D_2 & \\mathbf L_{31} \\mathbf D_1 \\mathbf L_{31}^\\mathrm T + \\mathbf L_{32} \\mathbf D_2 \\mathbf L_{32}^\\mathrm T + \\mathbf D_3\n\\end{pmatrix}\n\\end{align}\n", "83147567f0c73373542dcd6b2f12115e": "r = \\frac{\\gamma m_0 v}{q B}", "8314a1cca4a0e95600361b5351ba7855": " n~r^{-n-1}~\\sin(n\\theta) \\,", "8314a583791dc70baa8d0ed7450e83d1": "\\angle A = \\angle B = \\frac{180^\\circ-\\angle C}{2}= 90-\\frac{\\angle C}{2}\\!", "831501aabcfc4c679e13a76bb8823b2f": "X \\cup \\{A \\vee B\\}", "831556678730c86d360692fb72cd8bdb": "\nd_{iu} = {\\arg\\max_{d}} ~ U(d,E[x])\n", "83155b04022219485559a324ee8b8be2": "[0]p\\,\\!", "8315b42fc1fa8e66652f67b765c8258d": "L = 0.299R + 0.587G + 0.114B", "8316a20321b742e3f08a249c07e2c24b": "\\{(x_1, y_1), ..., (x_N,\\; y_N)\\}", "8316e19eb24db96ecc8b33930f712e4a": "\\{1, \\ldots, q-1\\}", "8316efbc8bce6b9b0d18ffabf75cfb41": "\\sum_{i=0}^n \\tbinom ni q^i(1-q)^{n-i} = (q + (1-q))^n = 1.", "831704c3eeb7b8392aec8ae944df6167": "\\sin(\\pi z)", "83171050c3a53071a90d50860a81d52f": "f(x)=b.\\,", "83171db9da9266a06620806ba5ef2cbc": "T_1, \\ldots, T_n", "831785603c6afd3d618a6ea11acc77c8": "\\cos(a n)", "8317882d0fdcb08d849a06351f5dac4a": "\nT_{i,j}(m) = \\begin{bmatrix} 1 & & & & & & & \\\\ & \\ddots & & & & & & \\\\ & & 1 & & & & & \\\\ & & & \\ddots & & & & \\\\ & & m & & 1 & & \\\\ & & & & & & \\ddots & \\\\ & & & & & & & 1\\end{bmatrix}\n", "83180c8555e5067408ab2b1f9d4b9eb9": "\\forall\\alpha.\\alpha\\rightarrow\\alpha \\sqsubseteq\\ int\\rightarrow int", "83181b8b6eade11c324cdbfaef3bf413": "\\frac1{x^m}=x\\Bigl(\\frac1{x}\\Bigr)^{m+1}\\le (m+1)!\\,x\\sum_{n=0}^\\infty\\frac1{n!}\\Bigl(\\frac1x\\Bigr)^n\n=(m+1)!\\,x\\exp\\Bigl(\\frac1x\\Bigr),\\qquad x>0,", "83183575e6e651b48972d8cb2c57f75c": "C:P \\times W", "83186f2e4a33cdb82ec37faf63533feb": "H_1(M;\\mathbb{Z})", "83189329ea88522cffac272904d35001": "g^{\\alpha\\beta\\gamma}_{a\\bar b}", "8318baf3eab039191c62f311a9712af7": " \\ddot{\\mathbf{r}} = n\\, \\mathrm{grad}\\, n ", "83192480281cdbc9a9436d9661f59ffa": "\\theta \\leftarrow \\theta + \\eta \\cdot \\mathbf{F}^{-1} \\nabla_\\theta J", "831941193b30c64e77a1e38baa21c747": " \\omega = \\frac {2}{ \\hbar } \\mathcal{E} f( \\mathbf{R} )\\langle \\mathbf{p \\cdot \\epsilon} \\rangle \\ . ", "8319535d4acf5210b9902fc80cc6d8ea": " {G^a}_b \\, {G^b}_c \\, {G^c}_d \\, {G^d}_a = -R^4", "83195e4635ccbe055e00791f05dbf85e": "g\\mapsto \\Psi_g", "8319c0a0680c91f78bcd519990219128": "GPA = 4-3\\times(100-x)^2/1600 \\ \\ (60\\leqslant x\\leqslant 100) ", "8319f01cae0e5752ba2e719544659959": "\\nabla", "831a2648174d23218dd8c9cffeaf821f": " \\left|c_g\\right|^2 + \\left|c_e\\right|^2 = 1 ", "831a30fd12ca76f550c425764070cdde": " E(h) = Ch^n ", "831a543e073fdb52d031b001306f8bef": "r = \\frac{a \\Delta t}{2 (\\Delta x)^2}", "831a6a9d22b24cdfd903b72ea5b3f066": "l \\simeq 1.05 r", "831a9f7e9c0a31b5da3e32ed7901d85b": "(H_{jk})^q=1 {\\quad \\rm for \\quad} j,k=1,2,\\dots,N. ", "831aba7f69385795a599ffda7bcbffa5": "[f]([\\mathbf{x}]) := f(\\mathbf{y}) + \\sum_{i=1}^k\\frac{1}{i!}\\mathrm{D}^i f(\\mathbf{y}) \\cdot ([\\mathbf{x}] - \\mathbf{y})^i + [r]([\\mathbf{x}], [\\mathbf{x}], \\mathbf{y})\n", "831ad59a2c2db4d582cb96430f4a9067": " \\operatorname{smoothstep}(t) = 3t^2 - 2t^3 ", "831ad96a7ae6239b2cea73698c269d3e": "\n\\begin{bmatrix}\n A_{11} & A_{12} & A_{13} \\\\\n A_{22} & A_{23} & A_{24} \\\\\n A_{33} & A_{34} & A_{35} \\\\\n A_{44} & A_{45} & A_{46} \\\\\n A_{55} & A_{56} & 0 \\\\\n A_{66} & 0 & 0\n\\end{bmatrix}.\n", "831b0e4ccc85d73029a13c4aaad45f3e": "\\overline i", "831b1f634e1a02dd0494bcec52c41b9b": "C_V=1.93..R\\left( \\frac{T}{T_B}\\right)^{3/2} ", "831b46aa3c7b6429bbfce34cb2fb5484": "m=\\infty", "831b53423e3e62e77244f0138699830a": "\\partial_\\hat{t} \\varphi + \\partial_\\hat{x} \\varphi + \\varphi\\, \\partial_\\hat{x} \\varphi - \\partial_\\hat{t}\\, \\partial_\\hat{x}^2 \\varphi = 0\\,", "831b56eb89143ad21fb385e33798420f": "\\exp[i(kx+mz-\\omega t)]", "831b8d68276888ad6578234ae10f522e": "U = -m/\\rho", "831b95720069e31094927bcce1fbcf3f": "\\phi_k(\\mathbf{R}),\\; k=1,\\ldots,K", "831ba1f7a848cb0fc30c2ba6623b9d75": "\\sqrt{i} = \\frac{1}{2}\\sqrt{2} + i\\frac{1}{2}\\sqrt{2} = \\frac{\\sqrt{2}}{2}(1+i).", "831ba656abce555547505ed293ee01c8": "P(k)-i", "831bde21cd18cf42a25a72edcc8474bc": "\n\\hat{\\phi^{\\prime}} = (1 - \\hat{G}) \\hat{\\phi}.\n", "831c24e7b174c823950fb83afeb72ec4": "Dw_1(M)=[G]", "831cc185f60bf45d45580a15387560e1": "g^i", "831cd5727f37371ebbaebf65e943ce18": "O_n+4T_{n-1}=T_{2n-1}.", "831cdac54123a3feb5a81b9abf015066": "\\operatorname{Specht}(\\lambda)", "831d444186a0ac6bccb3abd49857fc4d": "u=0 \\ \\forall (x,y) \\in r=a, 0\\leq \\theta \\leq \\pi \\ ", "831d448028853c3a62e974a7c4d84406": "\\Delta f = f(\\mathbf{x}+\\Delta\\mathbf{x}) - f(\\mathbf{x}).", "831d47656c1e047ef70d7d4e6d551678": "v(t) - \\ ", "831d628f09d7e3768cabba863399ede9": " h(X) = -\\int_X \\log \\frac{\\mathrm d \\mathbb P}{\\mathrm d\\mu} \\,d\\mathbb P, ", "831d65af1b81296d2de2a4d61e90431d": "^*\\mathbb{Z}\\setminus\\mathbb{Z}", "831d751626a68c074fd5f1480aa9dfed": "x_2 = \\frac{k_1}{k_2} x_1 .\\,", "831dc6a75d968630e1d67a35d5363e05": "average_{12}=\\frac{f_1 + f_2}{2}", "831dd154df3642bc1089bab04f92896a": " Z \\le \\frac{ ( k_2 - k_1 )^2 + 4 + \\sqrt{ 16 ( 1 - \\rho^2 ) + 8 ( 1 - \\rho )( k_2 - k_1 ) } }{ ( k_1 +k_2 )^2 }.", "831dd1887c125baa929fe20ff4bed84f": "[\\nabla^2 + E]\\psi(\\bold{r}) = V(\\bold{r})\\psi(\\bold{r})", "831e1564ad25fced5e5edd8ec1d8a3e2": "R_1+R_4=R_2+R_3", "831ec02b31d5e770dfe67f831d272043": "\\lnot(A \\land B) \\Leftrightarrow (\\lnot A \\lor \\lnot B)", "831ed80e1a2f87a29e209f644e187815": "K = -R_4/R_1\\,", "831f08c7340e913c5308ff2370da6827": "t'_{\\sigma}=\\sum_{{\\theta}=1}^k \\left(\\delta_{{\\sigma}{\\theta}}t_{\\theta}+\\frac {c^2} {v_{\\sigma} v_{\\theta}}\\beta^2({\\zeta}-1)t_{\\theta}\\right)-\\frac {1} {v_{\\sigma} }\\beta^2{\\zeta}x_{k+1},", "831f742c5f19ad8e2961b920017af4ca": "x \\in [0; +\\infty)\\! \\text{ for }q \\ge 1 ", "831fd7677c56a89b3a51b3b45902e0e0": "z\\frac{d^2w}{dz^2}+a\\frac{dw}{dz}-w = 0.", "831fddd78dc1db6bff2479e776b79a98": " \\frac{a}{\\sin A} \\,=\\, \\frac{b}{\\sin B} \\,=\\, \\frac{c}{\\sin C} \\,=\\, D \\!", "831fe9d142b0b153b1949352281f838b": "\\{\\mathbf{Z}_k\\}", "8320228c30792d4e53a050cedafd5822": "\\gamma_s(1)", "83204dff53aeca10bf583cb07be96103": "\\scriptstyle \\arg (Z)", "8320cf33094b295a6a258c245b5475f4": " \\lambda = \\lambda_{0} + \\frac{\\pi x}{2 \\sqrt{2} \\cos \\theta}, \\,", "8320d1005489da544f0244dd5ee9af63": "s_0", "8321125ca0d25b4a7519318ef7e582be": "\\frac{16}{9}\\div \\frac{704}{576}={\\color{blue}\\frac{16}{11}}", "832139474de09c32f5a359c32f6ddc25": "v_0,v_1,\\dots,v_n", "8321901b0fbdab80ddf1ab4997350424": "g:Y\\to Y'", "83220343f03f1ecffbfd844f6fe6cfd9": " \\boldsymbol{\\omega}_\\mathbf{N} = {1\\over 2}\\mathbf{N} \\times \\mathbf{N'} = {1\\over 2}(-\\kappa \\mathbf{N} \\times \\mathbf{T} + \\tau \\mathbf{N} \\times \\mathbf{B}) = {1\\over 2}(\\kappa \\mathbf{B} + \\tau \\mathbf{T}) ", "83222560c722f8ce7f1e6f7474758de9": " = {\\mathbf{T}(t) \\times \\mathbf{T'}(t) \\over 2}. ", "83224725ca1da6e734c75c990b545519": "e^{a + bi} = e^a (\\cos b + i \\sin b)", "8322ac6258781bcced7c517077c3a4c1": "\\Delta ( {\\hat \\Psi}^{\\otimes m}, \\Psi_{id}^{\\otimes n} )", "8322ceea5be3c1e253186d40529b5413": "\\begin{alignat}{2}\nT & = 2\\pi \\sqrt{L\\over g} \\left( 1+ \\frac{1}{16}\\theta_0^2 + \\frac{11}{3072}\\theta_0^4 + \\cdots \\right)\n\\end{alignat}", "83232ae0a30ec561e89ca8f5650ef767": "[0,2,-1]',", "8323c6e570fbcec0f917476c44b0b3c5": "\\nabla \\times (\\vec u \\times \\vec v) = \\vec u \\, (\\nabla \\cdot \\vec v) - \\vec v \\, (\\nabla \\cdot \\vec u) + (\\vec v \\cdot \\nabla) \\, \\vec u - (\\vec u \\cdot \\nabla) \\, \\vec v", "8323fed7f274ac07a9aa7734f66c3ff0": "\\sqrt{2} = {a\\over b}", "83241748391ee0bfd71ef372a7468db3": " \\operatorname{Hom}(P,-)\\colon\\mathcal{C}\\to\\mathbf{Ab}", "832446a1d659dfea6929e33c31d4eed0": "E= {mc^2 \\over \\sqrt{1-\\displaystyle{v^2\\over c^2}}}", "83245d81953439e96be45e6a2b37e56c": "\\mathbf{E} [F \\, \\delta u] = \\mathbf{E} [ \\langle \\mathrm{D}F, u \\rangle_{H} ].", "8324b1b2a869dfd9a971c306ae6c3e1a": "{\\nabla}^2 \\varphi = \\frac {1}{r^2} \\frac {\\partial }{\\partial r} \\left ( r^2 \\frac {\\partial \\varphi(r)}{\\partial r} \\right )= \\frac{\\part^2 \\varphi(r) }{\\partial r^2} + \\frac{2}{r} \\frac{\\part \\varphi(r) }{\\partial r} = \\kappa^2 \\varphi(r)", "8324ff26c3285ebf5e08f01528dbfea2": " \\begin{align}\n p_4'(x) = \\sum\\limits_{j=0}^4 f(x_j) \\ell'_j(x).\n\\end{align} ", "8325307973d16e123e21263c3b4b4f59": "\\Phi'(x) = A(x) + i B(x)", "83255d50e4fd8cd890a85c7e03a68ca2": "\\operatorname{Int} E_i", "83257a137987741980f01876f87955a0": "\\cosh x = \\prod_{n = 1}^\\infty\\left(1 + \\frac{x^2}{\\pi^2(n - \\frac{1}{2})^2}\\right)", "8325ae34b1be262d5a79af196ec629f7": "f(x)\\nabla^2f(x) \\preceq \\nabla f(x)\\nabla f(x)^T", "8325d935c68f36aed15d07cab68744f4": "-D \\nabla^2 \\Phi + \\Sigma_a \\Phi = \\frac{1}{k} \\nu \\Sigma_f \\Phi", "8326011d05a55b8403f2020edcce930f": "\\Lambda^k \\mathbb{C}^{n}", "8326037e408b9f63379af60fdf2f5b13": "\n\t\t\\hat{y_i} = \\left\\{\n\t\t\t\\begin{array}{ll}\n\t\t\t\ty_i 1 \\le i \\le m \\\\\n\t\t\t\tsign(g_{3-j}^{t-1}(\\boldsymbol{x_{3-j,i}})) m < i \\le n\n\t\t\t\\end{array}\n\t\t\\right.\n\t", "832687010a7c934cf6fef15139b2e8c6": "(n_{l-1}'\\dots n_0')_{\\text{NAF}}", "8326878542f3cc38c020c5cae2c878ac": "\\theta_e", "8326b3c4975c3f1d81c718c882bd8d1e": "\\mathcal{H}\n_{i}\\Psi =0", "8326beea41c46eace60a7e67e8230676": "\\nu(d)=\\max\\{w \\mid w\\geq\\mu(w-d-1)\\} \\, ", "8326fb702b47e4fbab2366e6370092ce": "c_p/c_v = 1.4", "83270dc0b0a51d15214b0adf9c53f39f": "f(t,x,V)", "83276da8ae0aa6f7041a25bdc55121de": " a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 \\qquad\\qquad(1')", "832802a5f3beeb4e3e3bdc0d36cd038b": "R_c = Z_{21} \\qquad R_a = Z_{11} - Z_{21} \\qquad R_b = Z_{22} - Z_{21} \\, ", "8328426e3a70f003c080fe3c1114b327": "D(X) < \\mathbb{E}[X] - \\operatorname{ess\\inf} X", "832879c1d1d078616e6ff3cded317915": " E_2 = y_2 + \\frac{q_2^2}{2gy_2^2}", "83287a3d0587e74a71644fdaacdd0b2d": "ac = bd", "8328836f682492e72b666090c1878d87": "\\mathrm{Hom}(Y, X) \\rightarrow \\mathrm{Hom}(V, X) {{{} \\atop \\longrightarrow}\\atop{\\longrightarrow \\atop {}}} \\mathrm{Hom}(S, X)", "8328b3ef25a1e102ee14d093e5580ead": "0.\\overline3_{10} = 0.3333333\\dots_{10}", "8328b6e0a6ccb1a98d55138a23a6f7c3": "\\mathcal{M}\\{\\xi\\in B\\}=\\begin{cases} \\underset{f(B_1,B_2,\\cdots,B_n)\\subset B}{\\operatorname{sup} }\\;\\underset{1\\le k\\le n}{\\operatorname{min} }\\mathcal{M}_k\\{\\xi_k\\in B_k\\}, & \\text{if } \\underset{f(B_1,B_2,\\cdots,B_n)\\subset B}{\\operatorname{sup} }\\;\\underset{1\\le k\\le n}{\\operatorname{min} }\\mathcal{M}_k\\{\\xi_k\\in B_k\\} > 0.5 \\\\ 1-\\underset{f(B_1,B_2,\\cdots,B_n)\\subset B^c}{\\operatorname{sup} }\\;\\underset{1\\le k\\le n}{\\operatorname{min} }\\mathcal{M}_k\\{\\xi_k\\in B_k\\}, & \\text{if } \\underset{f(B_1,B_2,\\cdots,B_n)\\subset B^c}{\\operatorname{sup} }\\;\\underset{1\\le k\\le n}{\\operatorname{min} }\\mathcal{M}_k\\{\\xi_k\\in B_k\\} > 0.5 \\\\ 0.5, & \\text{otherwise} \\end{cases}", "8328c60270f4eb6ceaf13eba871fe7cf": "\\alpha,\\alpha^2,\\ldots,\\alpha^{n-k}", "8328d79a28784b954c2c3780a9fd41cc": "27 (b^2 + c^2 - a^2)^2 (c^2 + a^2 - b^2)^2 (a^2 + b^2 - c^2)^2 \\leq (4 A)^6.", "8328f297fa3e2291e896989182f53187": "L \\varpropto M^3", "832936ad89a09c96a4ea55ce90d075fb": "\\Phi_d(x)\\,", "83299d3bb8a4d3e4a1ff43eb5bfaaebd": "\\psi=1\\,\\!", "8329cbab81a52fa1bf582f37b784470e": "Xg(Y,Z)", "832a277b086dcdb24d489356b5c3c31e": "h(\\mathbf{X}) \\leq \\frac{1}{2} \\log[(2\\pi e)^n \\det{K}]", "832a7f54661fc6eceb76d965d6bd3063": "\\scriptstyle{\\vec{E}_0 = \\hat{\\epsilon}E_0}", "832a9adb163dfff7a36c9f51297346d3": "\\mu\\!\\,", "832b017f3170fd8ca401c9a4bbaf9768": "\\tfrac{2}{n}", "832b37b098877cb18567ed7042b25609": "\\left(|\\Delta|\\ll\\Omega_{\\perp} \\right)", "832b5d53c37e62fe3797dc26d04e9ef8": "\nG_B(p,E) = { - i \\over - E - {i\\vec{p}^2\\over 2m} + i\\epsilon}\n", "832b83301aad07226e8ae81e3ad966cc": " Y(x) = R_n(x) - \\frac{R_n(x)}{W(x)} \\ W(x) = 0 ", "832c0e56d4da22183d8270390630d628": "\\mathbf{P} = (E/c, \\mathbf{p}) = \\hbar(\\omega /c ,\\mathbf{k}) = \\hbar \\mathbf{K}", "832c5c453a444410b097165c5e102cd1": "\\nu_a", "832c69bbd60edcd6b9c40d79a54ebbb9": "a_{31} x_1 + a_{32} x_2 \\le b_3", "832cab6245118c67b73c5ef0be7cf7e8": "M_x", "832d50f456e9ed96f9879a15e518ba72": "\\nabla_X g = 0", "832d89b6a7c5db8a2afb5227116a7d7a": "X_{2}^\\mathrm{opt}", "832dc0559aac42a0fe4c2f157d59e373": "\\boldsymbol{r_1 = x_1 - x_2} \\ , ", "832dcb5066083f9c20210d6faefa54bf": "Z[\\pi_1(X)]", "832dce12eb6cf8cb4bb68f41d589e25a": "\\Kappa", "832dec27d15f24afcf0918b445133ca4": "X(z)", "832eb65f5c05ca84dc0f93d75dfd69e5": "\n f(\\mathbf{x}_k) - l_k = O(1/k) .\n", "832ed09dcd28c059d72da7157c5f83d3": "0\\leq\\theta\\leq\\pi", "832f00b1380e58a068ea4c387a45a8b2": "\\phi(t) = 2\\pi \\left( \\left( f_0 \\,-\\, \\frac{\\Delta f}{2}\\right) t \\, + \\, \\frac{\\Delta f}{2T}t^2 \\, \\right) ", "832f2d5390fade1bf411a3117cabb51a": "\\operatorname{cov}(Y, Y)", "832f3ebb344145f41c3438537ca58b1e": "< T_{cut}", "832f556c3aedd0f95b405242f6d2f15e": " M = Kf_m - N", "832f59195f57d86e9e4cf81036019895": "\\pmod p", "832f5b3b4a8a5334a18bc6f6042266d0": "{d (\\rho m_{fu} ) \\over d t} + div(\\rho m_{fu} u) = div(R_{fu} .grad m_{fu}) + S_{fu} ", "832f97a79009644c2af2b9efb820283f": "\\{\\{a\\}, \\emptyset\\}", "832f9f0189156a7d7c89499850bf0ca1": " Q = F \\cdot \\left[\\sum_{i=1}^n a_{i}X_{i}^{\\frac{(s-1)}{s}}\\ \\right]^{\\frac{s}{(s-1)}}", "832fa9791d9397a6b4eb364466b90b5a": "J_q(\\delta) \\equiv_{def} (1-{1\\over q})(1-\\sqrt{1-{q \\delta \\over{q-1}}}) ", "832fb55ac46178a46a15b8d6c6b36859": "C_k \\cong \\ker (A_k\\to A_{k+1}) \\cong \\operatorname{im} (A_{k-1}\\to A_k)", "833000f41d4a0a9cb1ed603c69bc465f": " a(x,t) = \\left( \\frac{\\partial}{\\partial t} + \\frac12 \\frac{\\partial^2}{\\partial x^2} \\right) p(x,t). ", "83300447ab6d262a1a11159e53c17c99": " \\alpha \\ ", "8330095afe7267e71a7f4270df21a721": "X_1^n", "83300de0774d93c1404b9359b2546ea4": " \\sigma^2 = <(A-)^2> ", "83303c7922b0afa719530fb035d60aaf": "\\Gamma(\\gamma)_s^t\\dot\\gamma(s) = \\dot\\gamma(t).\\,", "8330491e4f7cf3d77e2ef3f9ee8fc8e3": "m_2,", "83309e4bc000dd81cb0bfd67f338a2a4": "B = {E^{2 \\over 3} \\over 3000}", "83311fb482f9322a9e14ec2c0ab4f9be": " h_{y} = e_{x} / \\eta ", "83313301f831e45ffecc5a042b9d0ab7": " |E(K)|+|E^d(K)| = 2 q+2 ", "83316685aa2a35b40a0287ed188d30d0": "w_1 \\gets 0", "83325272132d31d2f6248e9b1c782f36": "U_1=R_1 - {n_1(n_1+1) \\over 2} \\,\\!", "833261f1aad3efb93f207f21747f3ad0": "(3,0)", "8332740de1dcd06cdbc73a833efcf9bc": "\\mathfrak{A}=\\langle A, E\\rangle ", "8332af35a95e60b177b21798da2e7b10": "A_3 \\rightleftharpoons A_1", "833362234fb0077e0e8ea3122b6688b1": "T \\propto M^{-1}", "8333d795bd5394ce072ec64cf8af22e2": "\\mathbb{X}^{(-n)}", "833420f9ee8c4aab38a87c994a1ddca0": "G_{ijkl}=(\\gamma_{ik}\\gamma_{jl}+\\gamma_{il}\\gamma_{jk}-\\gamma_{ij}\\gamma_{kl})", "8334359816923a90e9d2bfdeed4c1ed7": "(x'(t),\\ y'(t))", "8334461c88b9bc14a0231ee43fc56c55": "Q=\\begin{cases}\n0 & \\text{for } P \\leq 0.05S \\\\\n\\frac{(P-0.05S_{0.05})^2}{P+0.95S_{0.05}} & \\text{for } P>0.05S \\end{cases}", "83347e6909d013b4caff1226f87bdb6c": "f_X(x|Y=y) = \\frac{f_{X,Y}(x,y)}{f_Y(y)} ", "8334bf25d637e67791ece5d21b972e2e": "\\ln \\left( \\frac{p_1}{p_2} \\right) = \\frac{g}{R \\cdot \\bar{T}} ( z_2 - z_1 ). ", "833509c887d9d7e53cc21c7360476e22": "\n\\lim_{n \\to +\\infty} \\left( \\frac{\\int_a^b e^{nf(x)} \\,dx}{\\left( e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}} \\right)} \\right)\n\\ge \\lim_{n \\to +\\infty} \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\delta\\sqrt{n (-f''(x_0) + \\varepsilon)} }^{\\delta \\sqrt{n (-f''(x_0) + \\varepsilon)} } e^{-\\frac{1}{2}y^2} \\, dy \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) + \\varepsilon}}\n= \\sqrt{\\frac{-f''(x_0)}{-f''(x_0) + \\varepsilon}}\n", "83351dc3ee21f84272960e22f7c4ac72": " \\frac{1}{4\\pi} \\left(\\rho,R,L\\right) ", "8335aa4c0ee6f743a3839f95c6af9f44": "\\operatorname{var}(p'\\mid p)", "83362d53318ce01f8cdfce45b2a97e8b": "m(\\varnothing) = 0 \\,\\,\\,\\,\\,\\,\\! ; \\,\\,\\,\\,\\,\\, \\sum_{A \\in P(X)} m(A) = 1. \\,\\!", "833641252d62187e2bae4e35e3aae9db": "\\,\\! t = \\tau t_c \\Rightarrow dt = t_c d\\tau \\Rightarrow \\frac{d\\tau}{dt} = \\frac{1}{t_c}.", "8336a8f1d0621cb1ec8a82f2a7133197": "h_i(\\mathbf{p},u) = x_i (\\mathbf{p}, e(\\mathbf{p},u))", "8336ae9fb44dc9ed130c82601291cd1b": "|q\\bar{q}\\rangle = -|\\bar{q}q\\rangle", "8336b8492120e029a3e7fc5a7249a663": "\n(\\alpha s + \\beta t + \\gamma u)^n \n= \\sum_{\\begin{smallmatrix} i+j+k=n \\\\ i,j,k \\ge 0\\end{smallmatrix}} {n \\choose i\\ j\\ k } s^i t^j u^k \\alpha^i \\beta^j \\gamma^k \n= \\sum_{\\begin{smallmatrix} i+j+k=n \\\\ i,j,k \\ge 0\\end{smallmatrix}} \\frac{n!}{i!j!k!} s^i t^j u^k \\alpha^i \\beta^j \\gamma^k \n", "8336cf4c9c0db4750df5a4d3cf553ac4": "h(G) = \\min_{0 < |S| \\le \\frac{n}{2} } \\frac{|\\partial(S)|}{|S|},", "8336e6c39d1a3139c04bc4029cb4103c": "\\operatorname{det}(\\mathbf{A}+\\mathbf{UV}^\\mathrm{T}) = \\operatorname{det}(\\mathbf{I} + \\mathbf{V}^\\mathrm{T}\\mathbf{A}^{-1}\\mathbf{U})\\operatorname{det}(\\mathbf{A}).", "8336f73cef914c82e5ce05efe1570f22": " K( x , t ) = \\log( \\operatorname{ E } ( e^{ t x } ) ). ", "833716f08fb56a0a9dfd61c768e3d304": "\nx |1\\rangle + y |2\\rangle + z |3\\rangle\n\\,", "8337d4350e5b14648b366e496eb93da4": "100(1-\\alpha)", "8337ddc4af46dad092ae28948b383050": "\\hat{t}", "83388c6c2d8fb0c597de92302d782f2b": "H(\\mathbf{x})=-\\frac{1}{N}\\sum_{t=1}^N\\ln p_\\mathbf{x}(\\mathbf{x}^t)", "8338f09df4c3c0ff0dec1af7525f72fa": "1.00U($1\\text{ M}) > 0.89U($1\\text{ M}) + 0.01U($0\\text{ M}) + 0.1U($5\\text{ M})\\,", "8339187914aa94b8b11cd57870747467": "\\mathbf{L}=\\sum_n \\mathbf{r}_n\\times m_n \\mathbf{v}_n", "833947c13cefd5654c641a68f63b99e8": "\\sum_i K_i(p_1,\\ldots,p_i) z^i = \\sum_j K_j(p'_1,\\ldots,p'_j) z^j \\cdot \\sum_k K_k(p''_1,\\ldots,p''_k) z^k ", "833968fc5dfe95117a130a5b779efcca": " \\exists x \\in a", "83397c1c964d06411156fbde4b6322fe": "\\tfrac{mile}{hr}", "833994326ce5fc459010bfd5aa160c17": "(f\\star g)", "8339bb4b183a3b66de87444e4531896e": "\\textstyle\\frac{K_{\\nu/2} \\left(\\sqrt{\\nu}|t|\\right)\n \\cdot \\left(\\sqrt{\\nu}|t| \\right)^{\\nu/2}}\n {\\Gamma(\\nu/2)2^{\\nu/2-1}}", "833a23525a4404d07a032aab80c2cf32": "A={a_1, a_2, \\ldots a_n}", "833a485fd6403c619573caa9010fb729": "\\mathrm{sinc}\\Bigl(\\frac{k}{m}\\Bigr)", "833b0178cfd1b33df1023cb8c09e7235": "\\scriptstyle A\\ =\\ {2}a^2\\sqrt{3}\\ \\simeq\\ 3.464102 a^2", "833b06ab086b314ff83d80c166cf8c92": " v_\\theta", "833b2df36f6a3338a410414e8dfb181d": "\\int\\limits_\\Omega f\\text{div}\\mathbf\\varphi \\leq \\int\\limits_\\Omega\\left|\\nabla f\\right| ", "833b4099c9263e63999a76fa645f8940": "A \\lor B \\in S", "833b51a853519ebd7aa5c650843e5fe3": "\\left[J_i, J^2 \\right] = 0", "833bb26a9a67840871a838a8b226031e": "t^d - c_1t^{d-1} - c_2t^{d-2}-\\cdots-c_{d} = 0 \\,", "833bb2da9cf22d8791c961de10bec1a2": "\\mathbb{A}_{\\mathbb{Z}}^n", "833bd5301ccbf2ba0e68006b947652e9": "\\log(1+x)", "833c08c8185697b8fa670b1503bc1df2": " = 3 N k T D_3(T_D/T)\\,,", "833c1a72a853ff9e50d90fbc658e96f3": "\\omega'", "833c83e66791318440175196e75a000a": "k^{ion}", "833c93d86f1786684c9995a2c49e7bbc": "\\int_0^1 \\ln\\frac{x^2-1}{x\\ln x}dx=-1+\\ln 2+\\gamma.", "833c971beae6ef4fa0a919d7b0017c96": "\\,x \\prec z", "833cad23c550c1d3972e5581290393be": "A_{T_p}=\\{g\\in A \\;|\\; \\exists n\\in \\mathbb{N}\\;, p^n g = 0\\}.\\;", "833d2aa38ff7371fead36cfdcfb193f2": "{\\lambda}<20{\\mu}m", "833d518247df0c5ea7de1d53a51cc77d": "S_1 \\cup \\dots \\cup S_B", "833d5623f4229b6ec09065409a14c8bb": " \\prod_{1\\le i 0 ,\n\\end{align}\n", "83408fda93648d0f33d3c96082342c28": "\\sin(\\pi/2 - x)", "8340a02889c6dc74a880a751545264fb": "\\text{apparent immersed weight} = \\text{weight} - \\text{weight of displaced fluid}\\,", "8340c39b25efbd646f0fffe4560b43ab": "\\sigma_X", "8340d4e4334f67ebe1537173c729777a": " d = \\frac {v^2 \\sin 2 \\theta}{g} ", "8340efb2226ab086c8b400dc9336fa90": "x^5+20x^3+20x^2+30x+10 ", "8340f997f2e048877e803b85c1d439a9": "\\kappa x. (h\\circ \\operatorname{lift}_\\tau(x)) = h", "83410993ce6595cb2c9de388a09b07e0": "\\frac{a_n}{a_0} ", "83412f6e062d29a6b377ce3bf45cb563": "\\Sigma_s", "83419a17b72c74e5ccd955106975eb5c": "\\mathrm{Mg_3Si_2O_5(OH)_4 + Fe_3O_4 + CH_4}", "8341b8409ece8f73ddabc5498f7482d1": "\\lambda \\in (-1,+\\infty)", "83424703ea9a5254fa76dcac43e4b02a": " \n\\begin{align}\n\\text{ker} \\; \\pi &= \\{ (0,q): (0,q) \\in X \\} \\\\\n& = \\{ (0,q): \\phi^\\prime(q) =0 \\} \\\\\n& \\cong \\; \\text{ker} \\; \\phi^\\prime \\cong K^\\prime.\n\\end{align} ", "83427bc2ee216a93612ea90d2bd9ad24": "J_2 = 1.08262668\\times10^{-3}", "834281736738d17bc602210a8765f960": " \\{na; n \\in \\mathbb{N} \\} ", "83429b9b12c5da86a2ecf68d99f0d18d": "\n\\Delta(x_1 \\wedge x_2) = 1 \\otimes (x_1 \\wedge x_2) + x_1 \\otimes x_2 - x_2 \\otimes x_1 + (x_1 \\wedge x_2) \\otimes 1.\n", "8342b6a597dd20d416e4689ef141ae3c": "M_{\\mathrm{right}}^{\\mathrm{fixed}} = \\int_{0}^{L} q_0 \\frac{x}{L} dx \\frac{ x^2 (L-x)}{L^2} = \\frac{q_0 L^2}{20}", "83433ce8c881e15aa955ed7291626364": "\\varphi_{stator}=[L_{ss} ] i_{stator}+[L_{sr}]i_{rotor}", "8343b509dc3aadda8c19f00b021825d9": " \\nabla\\cdot \\mathbf{v} = 0.", "8343b71a33d43de2c5d24ec91dfa2517": "\\left| \\sum_{i=1}^n x_i \\bar{y}_i \\right|^2 \\leq \\sum_{j=1}^n |x_j|^2 \\sum_{k=1}^n |y_k|^2 .", "83445c4c57933ce34cee0f4b1e08d977": "s(1-s)y'' + (1/2-s)y' + n^2(y-1/2) = 0.\\,", "8344c8bd77827bc15ff10f0711b039e2": "4+64+4=72", "8344f38d16771e68b1c2d7fb47e0f5bf": " X^{m,s} = \\{ v \\in D' | \\forall \\alpha \\in N^n, |\\alpha| = m, D^{\\alpha}v \\in H^s \\}", "83467c2d58f965aaeff46d8674ed3023": "\\Omega = -d\\Theta = g^*\\omega", "8346de5ec8994e099a1664ce9e201f55": "E = \\sqrt{ (mc^2)^2 + (pc)^2 } \\,\\!", "8346edbc6c963c3a332f7c857a2dc1a8": "(r \\cdot f + s \\cdot g)'(x) = r \\cdot f'(x) + s \\cdot g'(x).", "8347167d91bac14bb6a6a1c3ae0a8a7c": "p^a q^b ", "83472014ac15d0a517558616fab59b7a": " \\flat:TM \\to T^*M ", "834757bd65c8ad506ba6cf3568d9e680": "\\Phi_n(x)=\\frac{x^{n}-1}{\\prod_{\\stackrel{d|n}{{}_{d0", "8351d5f0e46e7e881960510d769ecb9c": "\\mathbf{F}(\\mathbf{X_0}) = T \\nabla_{\\mathbf X} S(\\mathbf{X})|_{\\mathbf X_0}", "8351f0bbcedccf281e74e1ebb0f3a60c": "\nT", "8351fd3b3cb11f7c7290cb9da3032f70": "\\mbox{Pearl-Index} = \\frac{\\mbox{Number of Pregnancies} \\cdot 12} {\\mbox{Number of Women} \\cdot \\mbox{Number of Months}} \\cdot 100", "83521ad36c200f3e7e603be1db7ce1cd": "\\,\\Sigma_{xx}", "83525001e888aec1c2f1860b1a75f637": "k^3", "8352862424846cd4a610cf7bf43fb96c": " \\Delta\\lambda \\approx \\Delta \\theta \\left(M {\\partial\\theta\\over\\partial\\lambda} + {\\partial\\phi_{2,m}\\over\\partial\\lambda} \\right)^{-1}", "83529c3d67a4d755117b247464a721cb": "\\mathrm{d}\\eta=\\left({\\partial C \\over \\partial y} - {\\partial B \\over \\partial z}\\right)\\mathrm{d}y\\wedge \\mathrm{d}z + \\left({\\partial C \\over \\partial x} - {\\partial A \\over \\partial z}\\right)\\mathrm{d}x\\wedge \\mathrm{d}z+\\left({\\partial B \\over \\partial x} - {\\partial A \\over \\partial y}\\right)\\mathrm{d}x\\wedge \\mathrm{d}y.", "8352eb852718abf2d5e4943ce0950b47": "1,\\;1,\\;3,\\;5,\\;9,\\;15,\\;25,\\;41,\\;67,\\;109,\\;177,\\;287,\\;465,\\;753,\\;1219,\\;1973,\\;3193,\\;5167,\\;8361, \\ldots", "8352fcf611a8624658f37d56a95b20b0": "\\pi^{-1}(p)", "8353833fd405a32aa39eb65222702be5": "\\mathrm{LE}(\\gamma)", "8353a9b18d01423af2fdab76a75160e2": "t_w", "8353b5a3b211c388ef0f82efed3b0e5d": "\\Omega \\ \\stackrel{\\mathrm{def}}{=}\\ 4 \\pi \\rho G T^2", "8353e4259703044b5c7098d7275dd550": "\n\\begin{array}{rrl}\nT_{K_X}: L^2([a,b]) &\\rightarrow & L^2([a,b])\\\\\nf(t) & \\mapsto & \\int_{[a,b]} K_X(s,t) f(s) ds\n\\end{array}\n", "8353f4256da83af623e78f98d7d907ce": "\\sum_{n=0}^\\infty \\frac{x^n}{n!}", "835401f002d01fcd494786ceb88938be": "\\left\\lbrace c_j \\right\\rbrace", "8354039eddf38fd989a6a04b71c3aa97": "\\frac{\\Gamma(x)^{x-1}}{\\operatorname{K}(x)}\\,", "8354cd3a80d09df3015197837703e99a": "\\scriptstyle\\sqrt{n}/2", "8354d5c28fda218b865c7e0252dce26f": "\\frac{1}{2}\\sqrt{\\frac{1}{30}(61421-23\\sqrt{5831385})} ", "8354d744b1b4b1995a7b25a22d9e65de": " V = \\sum(v_i) ", "8354e16faba95f09e9a0ba6bee0967d5": "\n\\phi(x) = \\int {dk\\over (2\\pi)^d} \\phi(k) e^{ik\\cdot x} = \\int_k \\phi(k) e^{ikx}\\,.\n", "8355681903e83ec9743732f8a30109ef": "\\frac{\\partial i_l}{\\partial x}=-i_m\\ ", "8355fd5de2dc56afbef3c961a67c091f": "\\mathbf{A}=\\mathbf{Q}\\mathbf{\\Lambda}\\mathbf{Q}^{T} ", "83564ffe583c0bf8d49dc7a210508088": " Q = \\lim\\limits_{\\Delta t \\rightarrow 0}\\frac{\\Delta V}{ \\Delta t}= \\frac{{\\rm d}V}{{\\rm d}t}", "8356e595be07370d3c571354a36c6a89": "\\frac{dx(t)}{dt} = 1/3*(-(a+1)*x(t)+a-c+z(t)*y(t))+((1-a)*(x(t)^2-y(t)^2)+(2*(a+c-z(t)))*x(t)*y(t))", "83577310c81fdae19255e04b739ed884": "\\Delta_{\nput}(K_p,\\sigma_0)=-1/4", "8357b8dfcac1913892c4c18daa30d042": "\\boldsymbol{p}=m\\boldsymbol{v}", "835800c68e6af989fc974875b8b07980": " t \\rightarrow 0, \\text{ let } G = \\int^T_0 S(t)x(t)dt", "8358d794d471015d7fddfbc5484a09b8": "0\\leq z<\\infty\\,", "8358da79689836e15fddd4aece082a7f": "W_1", "8358e31fc4b33e9d29370dc0cb64f9bb": "\n\\mu u_{i,jj}+(\\mu+\\lambda)u_{j,ij}+F_i=\\rho\\partial_{tt}u_i\n\\quad \\mathrm{or}\\quad\n\\mu\\nabla^2\\mathbf{u}+(\\mu+\\lambda)\\nabla(\\nabla\\cdot\\mathbf{u})+\\mathbf{F}=\\rho\\frac{\\partial^2\\mathbf{u}}{\\partial t^2}.\n\\,\\!", "8359446eb9578b954d7fb3d5532ce599": "4 \\times 10^{-3} \\, V \\cdot s^{-1} \\cdot m^{-1}", "8359ad1dbde3289520c403a72b7cfc5d": "J(\\phi) = A \\cos^2 \\phi + B \\cos\\,\\phi + C", "835a2251898f6b50a76f683a5ad9b5ef": " ModD = MacD ", "835a28d5d056242c2fa45f6968b41773": "\n\\mathrm{d}s^2 = \\left(1-\\frac{r_{s}}{r} \\right)^{-1} \\mathrm{d}r^2 + r^2\\mathrm{d}\\phi^2,\n", "835a84ecc27dbd5a79b4fda535e0ea97": "AB = G", "835a9bd5d8848dab09ced445414f81f3": "\\lim_{n\\to\\infty}\\left(\\sum_{p\\le n}\\frac1p -\\ln\\ln n-M\\right) =0,", "835ac75708fcefa1539da8c0cbc12912": "N=\\int d\\epsilon \\, g(\\epsilon) \\, n(\\epsilon)", "835adfb440fa404c7e83c82e5933178f": " 1 + \\sqrt{\\lambda_i \\lambda_j}\\cos{\\omega_{ij}} = 0", "835ae12a85fb67412969b68076180b3e": "\\displaystyle{RK=AR.}", "835afdf573c86d148208f330e8ebaf91": "E << E_a ", "835b4c212a127e7ed98063c301f74f6e": "\\mathbf{F}\\cdot\\delta(\\mathbf{r})", "835c05fe227acd7ad3c27941428b3f2e": "\\boldsymbol{\\alpha}e^{xS}\\boldsymbol{S}^{0}", "835c6381b8a8f19c565ba0ed227a155f": "0\\leq x_i\\leq 1", "835cbeb2365693561eae5e1a748ad520": "{\\bar{BH}}_3", "835d04e191758597e18e0657b307670a": "(Q,\\Sigma,T)", "835d7697e168f6c905131faa012ed41c": "(A.1.e)\\quad \\Phi_{,\\,\\rho\\rho}+\\frac{1}{\\rho}\\Phi_{,\\,\\rho}+\\Phi_{,\\,zz} =\\,2\\psi_{,\\,\\rho}\\Phi_{,\\,\\rho} +2\\psi_{,\\,z}\\Phi_{,\\,z} ", "835e13eb1c62d5a9f1af31e47ffdabfe": "\n \\begin{align}\n P_n & = \\frac{1}{2}\\int_0^a\\left[\\left(\\int_{-b/2}^{b/2}n_x(x,y)\\,\\text{d}y\\right)\\left(\\cfrac{d w_x}{d x}\\right)^2 +\n \\left(\\int_{-b/2}^{b/2}y n_x(x,y)\\,\\text{d}y\\right)\\cfrac{d w_x}{d x}\\,\\cfrac{d \\theta_x}{d x} \\right.\\\\\n & \\left. \\qquad\\qquad +\\left(\\int_{-b/2}^{b/2}y^2 n_x(x,y)\\,\\text{d}y\\right)\\left(\\cfrac{d \\theta_x}{d x}\\right)^2\\right]\\text{d}x\\,,\n \\end{align}\n", "835e3237dd755a654ec3b1a8f1126dbc": "r^{\\nu-2s}\\Gamma\\left(s,r^2 h\\right)\\,", "835e34c1c775b391fea9015efe48329b": "D_0=\\sum_{v=0}^{v_{max}} \\Delta G_{v+\\frac{1}{2}}", "835eeeb2ed61ebb066a574f7aee53c2e": "\\mathbf{q}_2 = \\sin(\\alpha/2)\\cos(\\beta_y)", "835f6ce2afc119d660a5c8b31df5d94a": "F = S_0 e^{(r - q)(T - t)}\\,", "835fa1e28e3b3914249fc556c5b8bdb1": "\\mathrm{not}~t \\equiv \\mathrm{false}", "835fac5395b9687cfc823ba497f746d8": "A = Attr_i(U)", "835fbb177686c7a200d5f324a4fd3473": "df(p)\\colon T_p M \\to T_{f(p)} N.", "836000fb4ebd3b6190d5e044be1cfcbb": "\\Phi [A] = \\hat{O} \\Psi [A]", "83603ed920fca4ef4d22328861c9b666": "p=g\\eta\\rho-\\sigma\\eta_{xx},\\qquad\\text{on }z=0.\\,", "8360407818043c7674053895515f752e": "(A+B+C)/4", "836058a681dfc8404852d2b4375586af": "\\pi : M^{2n} \\rightarrow P^n", "83606c752f4b2b3e450724d1d91614e9": "k_2(s) = u_0 + u_1 s^1 + l_2 s^2 + l_3 s^3 + u_4 s^4 + u_5 s^5 + \\cdots \\,", "836079f66bbbad8b637a0f7a68b6a61b": " \\|\\mathbf{a} \\times \\mathbf{b}\\|^2 + (\\mathbf{a} \\cdot \\mathbf{b})^2 = \\|\\mathbf{a}\\|^2 \\|\\mathbf{b}\\|^2\\,", "8360cc3e1d0afa55cfd4898073a9d519": " \\mathbf{X}\\mathbf{v}_j ", "8360d3fb079869decb368fc079f6ac31": "\\beta = \\frac{\\mu_1 - \\mu_2}{\\sqrt{\\sigma_1^2 + \\sigma_2^2 }}.", "8360f47bf2508ee4ebec05058c43d168": "\\displaystyle x \\in (-\\infty, +\\infty)\\!", "836118bc92899a0a78e1ae7a54aca64f": "\\eta_N", "83619633b7bd4a9a3a5df7db250faa58": "\\bigcap_{n=1}^{\\infty} A_n \\in \\mathcal{R}", "836197507a73502cb66a61b1b70f8849": "\\frac{t}{N} = -\\frac{2z}{(1-z)^2}.", "8361b33e5c8e3fe7b4991f4d6d4e3117": "mu_2 - mu_1 + mP_2 v_2 - mP_1 v_1 = 0", "8361e09c3973100b384630e7a8c2976a": "\\mbox{CAR} = \\cfrac{\\mbox{Tier 1 capital + Tier 2 capital}}{\\mbox{Risk weighted assets}}", "8362039b92c5b8fb87e2040608ea444a": "Spin(8)", "83622aa0b4314e7b9334a174bdfbb0b2": "(0)^2 - (cT)^2 = (vt)^2 - (ct)^2 \\,", "836249d25ff1e3569fb77f337d1ed248": "V_T", "83624c199a814c19343084c8e89c2808": "r1", "8366f972f88419fb2c0a8a3f7b15951e": " e^{i \\phi_{k}} = \\left( \\lambda_{1} / |\\lambda_{1}| \\right)^{k} ", "8367038f1fc0b1bcf48c9bd0c4fd1dc6": "\\sum_{j \\in S\\setminus \\{i\\}} \\pi_i p_{ij} = \\sum_{j \\in S\\setminus \\{i\\}} \\pi_j p_{ji}.", "8367130590b226c0ccd9e430da6178c1": "\n\\text{time-slots}\n\\begin{matrix}\n\\text{transmit antennas}\\\\\n\\left \\downarrow\n\\overrightarrow{\n\\begin{bmatrix}\ns_{11} & s_{12} & \\cdots & s_{1n_T} \\\\\ns_{21} & s_{22} & \\cdots & s_{2n_T} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\ns_{T1} & s_{T2} & \\cdots & s_{Tn_T}\n\\end{bmatrix}\n}\\right.\n\\end{matrix}\n", "83672543b3672b87540ed481552f4ef4": "e : X \\to \\prod_{U\\in \\mathcal{T}(X)}S = S^{\\mathcal{T}(X)}", "83678c388bbfac06beedebb4fdf8fb2a": "\\nabla^2 \\nabla^2 \\Phi = 0", "8367d15c4fba670729dd10c49cfa6e9c": "\\Delta p \\propto \\frac{L}{D} \\cdot \\frac{1}{2}\\rho V^2.", "836815f21dee5838ada1fd43b07c630d": "c^*", "836850ff11ea5ef1c50c8e30e5b64bf5": "x = \\frac{1}{2} \\left( -p \\pm \\sqrt{p^2 - 4q} \\right).", "8368933505aaada9da1406a5f1b24d8e": "\n\\begin{align}\nH(x,y) & = \n\\frac{1}{2}\\frac{\n\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial y^2} - \n2 \\frac{\\partial S}{\\partial x} \\frac{\\partial S}{\\partial y} \\frac{\\partial^2 S}{\\partial x \\partial y} + \n\\left(1 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right) \\frac{\\partial^2 S}{\\partial x^2}\n}{\\left(1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2\\right)^{3/2}}.\n\\end{align}\n", "8368cd5236ab4e739ae916b7494da52a": "x_{i},y_{i}", "83693f3887e7c39b8314f90fba633802": " \\in L ", "836958285e3958324cc972ab5a7dfe25": "K=\\sqrt{\\frac{I_C}{m}}.", "8369ae61e0a424c9211c4e5c7cb212cb": "X=\\{x\\mid f_1(x)=\\cdots=f_k(x)=0\\}", "836a06d2421844e8e0eb30454148acac": "\\displaystyle e^{-\\frac{1}{2} \\boldsymbol \\nu^{\\mathrm T} \\boldsymbol \\sigma \\boldsymbol \\sigma^{\\mathrm T} \\boldsymbol \\nu} ", "836b256e5e027ba8007c779245300640": " X^{571} + X^{10} + X^5 + X^2 + 1 ", "836b60f200be7bebe2d42d28215660cb": "M_4", "836b71274e2c99f435f321049a40fe37": " x = \\frac{2 \\pi r} {\\lambda}. ", "836b74f73a2fe95e236e7d4417e25636": " O = 1 - \\frac{ 1 }{ 2 }| p_i - \\frac{ 1 }{ K } | ", "836bb2056cf62d45588b424893041aeb": "C_n(\\varphi)=-\\frac{1}{n}\\sum_{k=0}^{n-1}\\frac{1}{\\binom{n-1}{k}}.", "836beab179255b8ab1207d235c559d59": " \\alpha \\approx \\frac{v}{c}", "836c75a188c05ef6cd4fa0ba6ee9fb0f": "u + 1 = 0", "836d5b90e2526229afc6ad0829fede97": "\\beta^{loss}", "836d8ce45811022e3e97c38bf9335963": "c_{2}(t)=H(t,1)=H(\\ominus{\\gamma}_{3}(t))=\\ominus{\\Gamma}_{3}(t)\n", "836dbaae6ea8b5f7499faea8f4221152": "\\frac{\\partial^2\\psi}{\\partial x^2} + \\frac{\\partial^2\\psi}{\\partial y^2} \\equiv \\psi_{xx} + \\psi_{yy} = 0.", "836de19d3eafca62136b54e8d8fba98e": " P(a,b) = \\int d \\lambda \\cdot \\rho(\\lambda) \\cdot \\cos^2(a - \\lambda) \\cdot \\cos^2(b - \\lambda) = \\frac{1}{8} + \\frac{\\cos^2 \\phi}{4} ", "836df74718242319625844afd89a1954": "f(\\vec{x})", "836e01fecf688afb264f3555261e486f": "\n\\Gamma(p_1,\\ldots,p_n)=\\int\n\\prod_{i=1}^{n}\n \\left\\{\n \\mathrm{d}^4x_i\\ \n e^{i p_i\\cdot x_i}\n \\right\\}\n\\langle 0|\\mathrm{T}\\ \\varphi(x_1)\\ldots\\varphi(x_n)|0\\rangle\n", "836e0c892999d0c869a913e13216d8bc": "f(x^\\alpha)", "836f75a617d04809a43c51e8dfb235f2": "\\exists c>0, F(x)=O(e^{cx}), x\\to-\\infty;", "836fc0d3b1b2e078e35df741872945b9": "\\scriptstyle| i \\rangle", "8370351606b1798a6951c1fef94df6a8": "p_4(x)", "83705c797d7d7dc52dfd48b5c64c5d26": "A_\\mathrm{v} = \\begin{matrix} \\frac {R_L}{R_S}\\end{matrix} \\ \\ \\mathrm{ or } \\ \\ A_\\mathrm{v} = g_m R_L ", "837079c5c56c381ce98be907b377c9f5": "c(\\nu)\\langle n, k \\rangle := \\nu(i)", "8370d4b0f38f46150fb96d03d0a3a23f": " \\theta^\\hat{1} = dz, \\; \\; \\theta^\\hat{2} = dr, \\; \\; \\theta^\\hat{3} = \\frac{r \\, d\\phi}{\\sqrt{1- \\omega^2 \\, r^2}}", "8370df1dcb67c3071d49547ca5132208": "J_0, J_1, \\cdots", "837112dd8105fc4c7593583795e0d1f2": "\\frac{24}{3}\\,\\bmod\\,5 = 8\\,\\bmod\\,5 = 3", "83719e4a8980d5afac5de5b7f7cd3d07": "\\mathcal{H=\\lambda }\\left[ \\left( p-A\\right)^2 + (m+S)^2 \\right] \\equiv \\lambda\n(p^{2}+m^{2}+\\Phi (x,p)). ", "83727810c0ab6af03109d26737663b17": " n/r+1", "83729fd5efbf129cf0dfab4cc1133373": "X=x_i", "83736f51fb7523e64ea35974ab87f43a": "\nf(P) = \\sqrt{\\operatorname{vec}^{\\top}(D)\\Big( \nW - W (I_n \\otimes P) \\big( (I_n \\otimes P)^{\\top} W (I_n \\otimes P) \\big)^{-1} (I_n \\otimes P)^{\\top} W\n\\Big) \\operatorname{vec}(D)}.\n", "83749a003418aa75d7e274eaaa14b202": " n(r)", "83749f5ad681a32c0e59754ce55437d0": "\\gamma([0,t])", "8374ca79ffedc4f2eaf31f4998e0a16c": "C_0 = E_0\\left[\\max\\left(S_Te^{-\\mu T}-X_Te^{-rT},0\\right)\\right]", "8374f7ebb1850f3d61d1b80ebd6b6c2c": "{\\mathcal{I}}_{i, j} = {\\mathcal{I}}_{j, i}", "83755b56fc7df85f124d54c7f3c6649a": "\\frac{d\\phi_d}{dl} = \\int_{r_i}^{r_{o1}} B(r) dr = \\frac{\\mu_d I}{2 \\pi} \\ln\\frac{r_{o1}}{r_i} ", "8375b0da7b8fed74549982c6295a6a8a": "g_{ij}(\\beta) = \\frac{\\partial^2}{\\partial \\beta^i\\partial \\beta^j} \\left(-\\log Z(\\beta)\\right) = \n\\langle \\left(H_i-\\langle H_i\\rangle\\right)\\left( H_j-\\langle H_j\\rangle\\right)\\rangle", "8375ea3a677833a12d2e102be15e376c": "\\textstyle \\mathbf{v}_1, \\ldots, \\mathbf{v}_m \\in \\mathbb{Z}^n", "8376416cc047f9e1144b3ed29632aecc": " = \\frac{v\\left(1-\\frac{1}{n^2}\\right)}{1+\\frac{v}{cn}}\\approx v\\left(1-\\frac{1}{n^2}\\right)", "8376672ce411a9c9d8f3fa56f877f5bf": "g_t=g(m_t,s_t)", "83767479266048b1e5eaba3a4a6d7fee": "\\frac{1}{2i} \\int_L \\overline{z} \\; dz", "837686aa2ef32eefdb7662b67bdcec5b": "u=\\frac{0.4661x+0.1593y}{y-0.15735x+0.2424}, \\quad v=\\frac{0.6581y}{y-0.15735x+0.2424}", "8376c3627a6f121baf62fa7e3abb03a2": "\\vec{B} = \\vec{\\nabla} \\times \\vec{A} \\,", "8376fa0562a4dcc20bafe2fcac0db467": "\\Omega(d\\log n) \\leq t(d,n)", "83771956fe9d0d2d9b1f9f236d3e1115": "{\\Delta h_f}", "83773fe0853cdcd4cb9c2609bba6df3d": "\\R^2 = \\R\\times\\R", "837767bec2697b047a385ab27ec8fb4e": "W(s,\\xi)=U(s)(1-e^{-s\\xi})/s", "837800261888eb52d78e5b91ccdd3244": " g\\in\\mathcal{H} ", "837803caeb6ce5800b080a7accf4a292": "w = g(z)\\,", "83785915a30f585982c8f1903be9620a": "8\\sqrt 3", "8378785e0e0a6caa4a2c3e41dbf8bf54": " Z(G) = \\frac{1}{|G|} \\sum_{g\\in G} \\prod_{k=1}^n a_k^{j_k(g)}.", "8378dda4c280c9289d1ba9f8f638cc42": "\\delta_{\\pi}\\,\\!", "8378fc5867c463bf3920d4d649969098": "a_8", "837941be90ab1f9309c97f821d784b5b": "t = 1/x", "83794d160f31179def39a1a9de455bfa": "\\vert{\\Psi_{\\mathbf{p}}^{(\\pm)}}\\rangle = \\vert{\\Psi_{\\mathbf{p}}^{1}}^{(\\pm)}\\rangle + G^1(E_p \\pm i0) V^{2} \\vert{\\Psi_{\\mathbf{p}}^{1}}^{(\\pm)}\\rangle", "83795f2d96efa5263ed1b175bcdbed1e": "\\ll 1\\,\\!", "83795fc103bcab3407be0f5201a1ea58": "\\frac{d y_p}{dt} = 2 A t + B", "8379b4d62270b52c21bf6dbb8e503656": "\\sqrt{\\mathbf{H}_{ii}} = \\left(\\frac{4}{d+2}\\right)^{\\frac{1}{d+4}} n^{\\frac{-1}{d+4}} \\sigma_i", "837a5aab7ee6ea0f1910867a5152f7a9": " \\langle x,w \\rangle \\in L ", "837a8a81557059b9d34491f67c6a4d13": "c_i \\in \\{0,1,\\ldots,a-1\\}", "837a8c0a28687c1ef3371fceda3c3c88": "R_\\mathrm{ab} = R_aR_b(\\frac 1 R_a+\\frac 1 R_b+\\frac 1 R_c) = \\frac{R_aR_b(R_aR_b+R_aR_c+R_bR_c)}{R_aR_bR_c}=\\frac{R_aR_b + R_bR_c + R_cR_a}{R_c}", "837a8d3b93a6210cc4158980661a942b": "\\sqrt{\\frac{10}{63}}\\!\\,", "837ae7d5f99c48c9f71fcfcd7c53688c": " p_{ xx } = \\frac{ \\sum_{ i = 1 }^K x_i p_i }{ N_x } ", "837afd43e3684880cedc007d4a64a1a9": "m \\ = \\ \\left({ I t\\over F }\\right)\\left({ M \\over z }\\right) ", "837b1fde06283f11452c680a7de4941b": " u \\mapsto u(x) ", "837b7cf1ae802a63c5e3f147c2fcbed7": " h(g) - h(f) \\geq 0 \\!", "837bc10ba02c950ef2f6090b19d64073": "\\mathbb{E}[2e' + s']", "837c415707129f7194026fef639d1800": "n=\\frac{V_{\\rm{universe}}}{V_{\\rm{particle}}}=\\frac{4 \\times 10^{80}{\\rm\\ m}^{3}}{3.38 \\times 10^{-18}{\\rm\\ m}^{3}}=1.18\\times 10^{98}", "837c7b39952293689a68a30eb051e955": "\n p(r) = p_0\\left(1-\\frac{r^2}{a^2}\\right)^{1/2}\n ", "837c9f1adf5d0e370c9a8972e0ccde69": "X_{n}", "837ca8fabbfb95e5d61341c51b0b6dbe": "\\mathcal O_Y", "837cabc5ab3fc444801425dafd0c2482": "X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w},\\ ", "837cc476b8b74538ac518a128045e27b": "\\sum_{i=0}^{\\infty} \\frac1{s_i} = \\frac12 + \\frac13 + \\frac17 + \\frac1{43} + \\frac1{1807} + \\cdots.", "837dc6906567bdfcc4845f78a62a1f92": "Q=\\begin{bmatrix}\nx_1 & q_{1\\,2} & q_{1\\,3} & \\cdots & q_{1\\,n} \\\\\nx_2 & q_{2\\,2} & q_{2\\,3} & \\cdots & q_{2\\,n} \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\nx_n & q_{n\\,2} & q_{n\\,3} & \\cdots & q_{n\\,n}\n\\end{bmatrix}", "837dc75077e095f812f22ce48e664f32": "\n \\hat{Q}_n(\\theta) = - \\Bigg(\\frac1n\\sum_{i=1}^n g(x_i,\\theta)\\Bigg)' \\hat{W}_n \\Bigg(\\frac1n\\sum_{i=1}^n g(x_i,\\theta)\\Bigg),\n ", "837dd32c5d859cbff5c780f4c88bad4b": "\\alpha_9\\le\\frac{35}{54}\\ ,\\quad \\alpha_{10}\\le\\frac{41}{60}\\ ,\\quad \\alpha_{11}\\le\\frac{7}{10}\\quad\\ ,", "837e3176ef622e33625c2b18288f0464": "\nW_{0} (x) = L_1-L_2+\\sum_{\\ell=0}^{\\infty}\\sum_{m=1}^{\\infty}\\frac{(-1)^{\\ell}\\left [\\begin{matrix} \\ell+m \\\\ \\ell + 1 \\end{matrix}\\right ]}{m!} L_1^{-\\ell-m} L_2^{m}\n", "837e812c3d2d0f4b45cb702912e85fa8": "(RQ^{r}y')' + [{\\lambda}_n-{\\lambda}_r]RQ^{r-1}\\,y = 0\\,", "837f5c81ecc70fac760de14ef84418c1": "(1+x)^{3}p(\\tfrac{4x}{1+x}) = 43x^3-35x^2-7x+7", "837f7f3017168f7416e368ec80f8fb38": "\\R \\cup \\{+\\infty\\}", "83800ce4d6fbbfd78cff5a5ceb6f1275": "W_{10}=Y_{10}+800(x_{n,10}-x_{10})+1700(y_{n,10}-y_{10})", "83804234a92d46074d0ec4d47614f3ab": " \\frac{1}{p} - 1 = \\frac{(1-p_1)(1-p_2)\\dots(1-p_n)}{p_1 p_2 \\dots p_n} ", "8380428f46002a10d08d2e16c9129760": "\nA(t,T) = \\left(\\frac{2h \\exp((a+h)(T-t)/2)}{2h + (a+h)(\\exp((T-t)h) -1)}\\right)^{2ab/\\sigma^2}", "8380d788a0eff29662f3867470645ad2": "\\{1,\\ldots,d\\}", "8380e87ed08c4884ecdd60fb4dc31a49": " (x + y)^* = x^* + y^* ", "83810f6e67cddc5fe315cc4c82191999": "q(x)=x^2+2x-3=(x+3)(x-1)", "83814090232ca459b1b1dc6d93bcc364": "\\tfrac{355}{113} ", "83815a30d8d430691b91b5b1da8a7410": "\\Phi(\\cdot)", "83819d9d5f4a7d556f1aa85d0a4f588d": " \\Delta y ", "8381fcc907ddd83ae6ebdad96f2d1f9e": "a,b,s,t", "83820e2049b513d8e2372ce6684214c6": "\\alpha=\\left[\\begin{smallmatrix}a_{00}&a_{10}&a_{20}&a_{30}&a_{01}&a_{11}&a_{21}&a_{31}&a_{02}&a_{12}&a_{22}&a_{32}&a_{03}&a_{13}&a_{23}&a_{33}\\end{smallmatrix}\\right]^T", "83822e62ded5807b3a30ea5e7d58603a": "\\mathbb{H}^{\\times}", "83823149e91f7a213f643b20f26e0354": "T_{\\alpha\\beta\\mu}", "838244bd8f3464c3bdfe9d83055700b7": "\n\\frac{1}{n} \\sum_{i=1}^{n} \\frac{1}{2}\\log\\left(1+\\frac{P_i}{N}\\right) \\leq\n\\frac{1}{2}\\log\\left(1+\\frac{1}{n}\\sum_{i=1}^{n}\\frac{P_i}{N}\\right)\n\\,\\!", "8382543dc3326e01ae81be622f944a78": "\\{-\\infty, 1,2,\\ldots,\\ell-1,\\ell\\}\\times\\{1,2,\\ldots,\\ell-1,\\ell,\\infty\\}", "83826c6b8583ba3b169b0d4e42de78d8": " \\cosh c=\\cosh a\\,\\cosh b", "83829f96aba92aa6e0428904f418e981": "\\displaystyle X(s)", "8382bd55b8e586b56060558b84cc6a56": "\\Pr[\\text{miss a specific min-cut}] = (1-P(n))^{O(\\ln ^2 n)} \\le \\left(1-\\frac{c}{\\ln n}\\right)^{\\frac{3\\ln ^2 n}{c}} \\le e^{-3\\ln n} = \\frac{1}{n^3}", "8382d283b65990971abb30185282d700": "\\begin{bmatrix} 1 & 0 \\\\ -G & 1 \\end{bmatrix} ", "8382ff50302067bad93ee3e108e385fb": "\\text{dist}(S(t)u_0,\\mathcal M)\\leq c_1\\exp(-c_2t)", "838314f35341c63fab04d3590fdf3cfb": "x\\left(\\frac{1}{y}\\right)", "8383ce229f8160beb2c9801be06484d0": "\\textstyle g(X)=\\sum_k g_k X^k", "8383eac5678548473c32b9c809d63f15": " t > 1,\\ \\ S_{t} = \\alpha \\cdot Y_{t-1} + (1-\\alpha) \\cdot S_{t-1}", "8383f171178f84e114511d9a726552cb": "x_1,\\ldots,x_j=a_1,\\ldots,a_j", "8383f2480de71dc03701f53b8988ca14": "\\nabla \\cdot \\mathbf{\\sigma}", "83840a345bdd7445e32466a576272600": "\\phi:\\mathbf A \\times \\mathbf B \\rightarrow \\mathbf C", "83842000220a080b2cd3301b9d30d5cf": "m_\\mathrm{p}\\frac{\\mathrm{d}\\mathbf v_\\mathrm{p}}{\\mathrm{d}t}=\\sum\\mathbf F + \\frac{\\rho_\\mathrm{c}V_\\mathrm{p}}{2}\\left(\\frac{\\mathrm{D}\\mathbf u}{\\mathrm{D}t}-\\frac{\\mathrm{d}\\mathbf v}{\\mathrm{d}t}\\right),", "83842153ff21ff730ac2be5c1c0c6a2b": " d\\sigma^0 = -x \\, dt,\\;\\; d\\sigma^1 = dx,\\;\\; d\\sigma^2 = dy,\\;\\; d\\sigma^3 = dz", "83847adcd4f62ba98ed1bcc39ca287a9": "\nx = \\cfrac{1}{1 + \\cfrac{a_2}{1 + \\cfrac{a_3}{1 + \\cfrac{a_4}{1 + \\ddots}}}}\\,\n", "8384c2896f15e07464b188afff911758": "\\vec{h} = (1/\\lambda_h)AA^T\\vec{h}", "8385d202308afc5ca61d407024c4c547": " \\frac{d(dP/dt)}{dP} = - \\lambda(-b+g).", "838604b99fb0df8b60bc5dd1f505d558": "\\phi:X\\to X'", "838610315a45b11b0cfdb105eb9052c4": " M_{\\Delta^{\\alpha}}f(x)=\\sup_{x\\in Q\\in \\Delta^{\\alpha}}\\frac{1}{|Q|}\\int_{Q}|f(x)|dx.", "8386ab0937dda15b083d151bab72c036": "\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} n^c_{\\mathbf k} =\n( \\Omega_{\\mathbf k}^\\star \\, p_{\\mathbf k} - \\Omega_{\\mathbf k} \\, p_{\\mathbf k}^\\star ) \n+ \\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} n^c_{\\mathbf k}|_{\\text{corr}} \\, ,\n", "8386c96f9f4d39e693d7a70208ce84ec": "\\mathcal{O}(D^{1/4+\\varepsilon})", "83874cbbaef0f4c0ffa3328d5d66fe04": "\n\\Delta^1_{\\rm LAT}=\n\\frac{\\pi a(1 - e^2)}{180(1 - e^2 \\sin^2 \\phi )^{3/2}} \\,\n", "838753dabead80c4acbdd83017e8c464": "F(k;k_0)=\\left\\{\\begin{matrix} 1, & \\mbox{if }k\\ge k_0 \\\\ 0, & \\mbox{if }k y_0, \\\\ \\pm B(y)A(x) & \\text{if } x_0 < y_0. \\end{cases} ", "83a33b86fa9a53a8cc191ae4984bc10f": " k", "83a348d7bc7ca64a9e8178da6ebfc927": "n(n+1)", "83a349d8619f89cb9dc54609bc85a11c": "O\\left(\\frac{\\log d}{d^2}\\right)", "83a351d2b139a88e29dd057ec75f5bf9": "\\frac{b^2}{p^2}=\\frac{2a}{r}-1", "83a399bd905f757ee59d5527ac788163": "\\nabla \\cdot \\mathbb{T}", "83a424f84b2953d67e81987d260a64b2": "E_{n_x,n_y,n_z} = \\frac{\\hbar^2\\pi^2}{2m} \\left[ \\left( \\frac{n_x}{L_x} \\right)^2 + \\left( \\frac{n_y}{L_y} \\right)^2 + \\left( \\frac{n_z}{L_z} \\right)^2 \\right]", "83a42fcf961461975373d01af28cc882": "D \\colon \\mathfrak{g} \\to \\mathfrak{g}", "83a490702d4130f00ebef7c30e5421f8": "\\,J^+(x)", "83a4a1a7a20e91b03761aef90145d5b2": "(a+b)^{n}", "83a4bf63c395f42199f774bad832d43d": "p_i \\in \\operatorname{Ker}(A-\\lambda_{i} I)", "83a500240a87c1d6d4686c677b028786": " \\gamma =-2 \\,", "83a507d8960861efcd18556be8cd38e7": "\\!(\\mathcal A, X, \\phi) \\in \\mathcal T", "83a51bc8b373be9d8a6819b1a4d6f9ac": "= a_1 \\mathbf{e_2 e_3} +a_2 \\mathbf{e_3 e_1} +a_3 \\mathbf{e_1 e_2} = (\\star \\mathbf a )\\ ;", "83a526db3e512d2fc3c5b34cbb9671fd": "N(t) = \\int n \\,dx \\,dy \\,dz. ", "83a54512f9ce15de469f49ad18ac652e": " =, \\ \\in , \\ \\le,\\ <, \\ \\sub, ", "83a590faecc9861487d610f688b9ba46": "q \\ = E C_{min} (T_{h,i} -T_{c,i})", "83a627d6323f111a01595dc5d3e45064": "v(p)", "83a6690269556800159b52dd8c5d36f2": "D(f) = \\frac{df}{dx}\\text{.}", "83a67f730bf2a2cdce2b2168c7aae21a": "[\\sqrt{LOA\\cdot 3.28}+(Beam\\cdot 3.28)\\cdot 1.58]", "83a6a6352e560874920ffc3a789a547e": "\\mathrm{tan(Z) = \\frac{sin(LHA)}{sin(lat) \\cdot cos(LHA) - cos(lat) \\cdot tan(dec)}}", "83a6ee01d56559bfaf991e8d4cbb674a": "\\Phi^{-1}(x)/\\sqrt{\\frac{\\pi}{8}}", "83a6faa52be9e1d3f5cdbce52825e436": "|a, b, c\\oplus ab\\rangle", "83a740c714bdf8a55d666668573c06f0": "\\frac{1}{T_s} \\sum_{k = -\\infty}^{+\\infty} H \\left( f - \\frac{k}{T_s} \\right) = 1 \\quad \\forall f", "83a78d72d358b72bd387c2f5db21df91": "\\left\\{ {a_1 , \\ldots ,a_n } \\right\\}", "83a7a2ff08e415dd7013c5e1830b05ce": "1393-985\\sqrt{2}=-0.00035\\ldots", "83a8629992113ad991acef826e7c36a6": " b + c ", "83a88ab12cf3296e031df84985733d33": "a=2", "83a89f4fe56324a581b3faa408b486f8": " G_x(t,f) = \\int_{-\\infty}^\\infty e^{-\\pi(\\tau-t)^2}e^{-j2\\pi f\\tau}x(\\tau)\\,d\\tau ", "83a8fe4ac30af2c3b36f5559dccab018": "\\sigma=.032", "83a90b5c67a80cee7a56f7397cbba4dc": "K(\\sqrt[p]\\epsilon)/K", "83a9c5737d7ce81e9ea5a58c9b993c56": "\\mathbf{e}", "83aab664ab9dc7c0f3aa77b9730948b7": "\\scriptstyle +\\colon\\, F \\,\\times\\, F \\;\\to\\; F,\\,", "83aabd894a9392feb949004dda1aeafa": "{n!\\over (k-1)!(n-k)!}u^{k-1}\\cdot du\\cdot(1-u-du)^{n-k}", "83aabf752f1bc1645bb0fcb0f869347d": "c(\\lambda)=c_0\\cdot\\prod_{\\alpha\\in \\Sigma_0^+} {2^{-i(\\lambda,\\alpha_0)}\n\\Gamma(i(\\lambda,\\alpha_0))\\over\n\\Gamma({1\\over 2}[{1\\over 2} m_\\alpha + 1 +i(\\lambda,\\alpha_0)])\n\\Gamma({1\\over 2}[{1\\over 2} m_{\\alpha} + m_{2\\alpha} +i(\\lambda,\\alpha_0)])}", "83ab6ca8d623501befb481c340651ad1": "\\mathbf{M}_{2} := (\\mathbf{A}_{2,1} + \\mathbf{A}_{2,2}) \\mathbf{B}_{1,1}", "83ab848c9c86e4452873dfa706608d4f": "V = \\nabla f = \\bigg(\\frac{\\partial f}{\\partial x_1}, \\frac{\\partial f}{\\partial x_2}, \\frac{\\partial f}{\\partial x_3}, \\dots ,\\frac{\\partial f}{\\partial x_n}\\bigg).", "83abaf300a81993ca1f6f6431cbce3a0": " C_P = A + 2B(T) - C(T^{-2}) + \\textstyle \\frac {1}{2} D(T^{-0.5}) + 3E(T^2) \\,", "83ac14d1b290d0897c308954264cac27": "\\rho\\,\\bar{\\boldsymbol{u}}_S", "83ac2b1a77ce20a15a1ab9a91d6b0c1d": "n=k=0", "83acb0db5171eb2aec2b2ff227d2647e": "\\theta \\vdash_{\\vec{s}} \\phi", "83acf1a2613bc131f2af412aeca09f32": "T^{\\mathrm{SW}}_p", "83ad00b0c7d84f9cd64b05d5129042ac": "\\mathbf r", "83ad2e71bf20b4367760669a53c34500": "\nu_i = \\frac{x_i-Q}{9\\cdot{\\rm MAD}}.\n", "83ada98cc1014ca868e35879580b39b5": "\\emptyset=\\mathcal S\\ne \\mathcal R", "83adb079b9814e1c5465f7e043ba3a87": " x_{k+1} = x_k - \\nabla f(x_k) ", "83adffc02349f869067f2a54e9134216": "\\Gamma = \\Gamma_{\\max} \\frac{C}{a+C}", "83ae018d9f2729fbb05a0d8540bf3594": "=\\sqrt{\\cosh^2(t)}", "83ae7cd0580f839e7d3b86cdea7c72ab": "M= \\langle Q, \\Gamma, b, \\Sigma, \\delta, q_0, F \\rangle", "83aec75f0dc5f44c7186a8c35da87aed": "V_M", "83aec89723c240ff265767f5d5bf0d91": "\nM_{pair} = \\sum_{x=1}^{N} \\sum_{y=1}^{p} \\sum_{a=x+1}^{N} \\sum_{b=1}^{p} \\left(P(r_{x}^{y})P(r_{a}^{b})E_{xy}(r_{x}^{y}, r_{a}^{b})\\right)\n", "83aeee2aba54e693fe614763063d71ff": "\\operatorname{div}( a\\mathbf{F} + b\\mathbf{G} ) \n= a\\;\\operatorname{div}( \\mathbf{F} ) \n+ b\\;\\operatorname{div}( \\mathbf{G} ) ", "83aef03d0bc5506b359037b4589c98f7": "\\begin{matrix} R\\ \\bowtie\\ S \\\\ a\\ \\theta\\ v\\end{matrix}", "83aef82c004ef46020ac8ff7e55c1fb3": "\n\\frac{\\mathrm{d}P}{\\mathrm{d}t} + P/\\alpha = \\frac{\\mathrm{d}(a*r(t))}{\\mathrm{d}t}\n", "83af189ff9188bbe94f443fa59338f2c": "\\alpha_i^\\vee\\ ", "83af21a11e0e4b5161fa6c9b4d39913a": "\\mathsf{\\Alpha \\Beta \\Gamma \\Delta \\Epsilon \\Zeta \\Eta \\Theta} \\!", "83af27004ca7af784774a38a13cb5179": "\\chi_1(n)", "83af6b80c3a1789e628015344ef1d161": "\\,1 + 3x + 2x^2 - 4x^3 + \\cdots", "83af9076bd21eeb790d68303c64bf31a": "\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "83b0627b8a49c58b17c78b4523f522ba": " \\partial_n = d_0 - d_1 + d_2 - \\cdots + (-1)^n d_n. ", "83b08453f4197d78025b7af0f4b71186": "x_k", "83b0c5c4b1db85a24700eeb28d6084c0": "\\{\\breve\\theta_1,\\ldots, \\breve\\theta_N\\}", "83b12a38cbebfa833d5ca01c2dc497dc": "\\theta=45^\\circ", "83b135e668aeb7db237e59b03d184f37": "\n \\varphi_f(t;\\sigma,\\gamma) = E(e^{ixt}) = e^{-\\sigma^2t^2/2 - \\gamma |t|}.\n", "83b1fed376a959cde6e62ae2ad3a019e": "f(x)=\\textbf{c}_Pe^{\\textbf{A}_Px}\\textbf{b}_P.", "83b2f511b137e2026830b735e47f2b4e": "\\operatorname{P}(A|\\mathcal{B}) = \\operatorname{E}(\\mathbf{1}_A|\\mathcal{B}) \\; ", "83b32df4b734004c5009fa6bb362b8a8": "\\mathbf{S} = \\frac{q^2}{4\\pi c}\\left|\\frac{\\mathbf{n}\\times(\\mathbf{n}\\times\\dot{\\boldsymbol{\\beta}})}{R}\\right|^2.", "83b337770e19b816aea3d8693d0f95e2": "\\int_{x_s(t)}^{x_2}w_t \\, dx\\rightarrow0", "83b3711bd507707de6c6c46c54b52839": "\\ Kc/\\Delta R(\\theta \\rightarrow 0, c \\rightarrow 0)=1/M_w", "83b3e7da52372938f3d4018c72f3573f": "-C = B", "83b4382c1745e9f7b2803d18d4e373f3": "B\\cong S_3", "83b4da1ab8cc20d2b2fa811e8674036f": "\\cap N = 1", "83b4eb096245ee56d031fe7be0bdb61e": "\\hat{\\mathcal{H}} \\hbar \\omega [\\hat{a}_1^2(t) + \\hat {a}_2^2(t)] = \\hbar \\omega [\\hat {a}\\hat {a}^\\dagger + 1/2] \\ ", "83b50a5b6fea0af32a7e0c19f07c2e52": "\\, e^{+j\\omega t}", "83b5281fa9825c50dab8af07679fe558": "L = \\mathbb{Q}[\\alpha] = \\mathbb{Q}[y]/q(y) = \\prod_{i=1}^n \\mathbb{Q}[y]/q_i(y)", "83b5484d0e431d5e91c4b0ab0a392c57": " i^*(TM)\\cong TX/\\nu \\oplus (TX)^\\omega/\\nu \\oplus(\\nu\\oplus \\nu^*), \\quad \\nu=TX\\cap (TX)^\\omega,", "83b58bfd4a6b7f356b30d9970050b72a": "{10}^{\\,\\! 4 \\cdot 2^{1000}}", "83b5a40f23dff2746187f07c8fc67b10": "e^{-2\\gamma}", "83b5cdf7c23bdb135ef63e679a3e6962": "s_1^3+s_2^3=A", "83b5dc271860eb1fe327cf7a6fe72fa2": "Q_c^{(c)}(t)=0", "83b5e1ea8fff29cb67cb060b0b51f8ce": "d=i/(i+1)", "83b610dd26176b630dfe06454c4fd021": "\\mathbf{P}^5", "83b61965ac6c08dd48ad8950757da2d3": "\\phi \\circ \\hat{f} = \\hat{f} \\circ [n] = [n] \\circ \\hat{f}.", "83b632c555875e81e94941a3d22c5ef0": "x\\sim y\\,", "83b657185e32a9d64bdf00892478f50f": "pn=p_B n_B\\, e^{-\\varphi_\\mathrm{B}/V_\\mathrm{th}}", "83b65fb51b3d66f0f6b029eb9a3485fe": "0011010100010100010100010000010100000100010100010000010000010100000100010100000100010000010000000100", "83b6c264c49c37f793646c3a1b5bce31": "\\lim\\limits_{R_{\\text{NIC}} \\to R_s+} R_s \\| (-R_{\\text{INIC}}) \\triangleq \\lim\\limits_{R_{\\text{INIC}} \\to R_s+} \\frac{-R_s R_{\\text{INIC}}}{R_s + -R_{\\text{INIC}}} = \\infty.", "83b6f62a77c50b99927a79e7025e8065": "\\Delta\\lambda = \\lambda^{\\mathrm{state 2}}_{\\mathrm{observed}} - \\lambda^{\\mathrm{state 1}}_{\\mathrm{observed}}", "83b6f9afb0995d92d6fe9d23ed120841": "\\omega_i = \\min(\\Delta(C_\\text{in}(y_i^\\prime), y_i), {d\\over2})", "83b7215c32b98ece653af54f31161ab5": " (x_2,y_2) ", "83b7691d1568446612cf223a2fdef368": "\\alpha = 1.202 \\exp\\left(-0.30288\\,T_r\\right).", "83b77c5af3242e1ee20a6ef5e21d8636": "S=\\begin{pmatrix} \np_m & 0 & \\cdots & 0 & q_n & 0 & \\cdots & 0 \\\\\np_{m-1} & p_m & \\cdots & 0 & q_{n-1} & q_n & \\cdots & 0 \\\\\np_{m-2} & p_{m-1} & \\ddots & 0 & q_{n-2} & q_{n-1} & \\ddots & 0 \\\\\n\\vdots &\\vdots & \\ddots & p_m & \\vdots &\\vdots & \\ddots & q_n \\\\\n\\vdots &\\vdots & \\cdots & p_{m-1} & \\vdots &\\vdots & \\cdots & q_{n-1}\\\\\np_0 & p_1 & \\cdots & \\vdots & q_0 & q_1 & \\cdots & \\vdots\\\\\n0 & p_0 & \\ddots & \\vdots & 0 & q_0 & \\ddots & \\vdots & \\\\\n\\vdots & \\vdots & \\ddots & p_1 & \\vdots & \\vdots & \\ddots & q_1 \\\\\n0 & 0 & \\cdots & p_0 & 0 & 0 & \\cdots & q_0 \n\\end{pmatrix}.", "83b77d3362f2cc240d381d459c67075c": "\\frac{{}_{(1)1}\\partial x^2}{\\partial x}=1\\,\\!", "83b78ba3b7a98d5567fd3fd9df164f37": "L \\subseteq f^{-1}(f(L))", "83b790929b4e15abe0febbb9fecb682a": " u(0)=0 ", "83b79eba76a1adc95552d7084d3eb0f5": "\\,T_C = 5 + N_{8}", "83b7a01ace114ca11f72c1c7eafe260a": " V_{Out} = \\frac {V_{i} \\cdot (C_{1}+C_{2}) - (d-1) \\cdot V_{r} \\cdot C_{2} + V_{os} \\cdot (C_{1}+C_{2}+C_{p})} {C_{1} + \\frac {(C_{1} + C_{2} + C_{p})} {A} } ", "83b82b6ebabaadd6b25e0a95ab519fc5": "k^m", "83b887728e6421a1e563f0d47abba1d1": "\\cup x\\in U", "83b8bfac22fe16d7f8ffd94c8bc038f0": "x' = -a\\lambda'\\,\\qquad\ny' = \\frac{a}{2}\n \\ln\\left[\\frac{1+\\sin\\phi'}{1-\\sin\\phi'}\\right].\n", "83b8d7d31bafb3f4987f16ab903de556": "B_n=\\sum_{k=0}^n \\left\\{{n\\atop k}\\right\\}.", "83b98b611b3f7001dcf6155a84b2ceb6": "\\beta^{-m}", "83ba22393adfe9d878eef301dd13afc9": "\\Pi (t,f) = W_h(t,f) ", "83ba36cfaa2c46060d66edd41eab9834": "\n\\text{Tr}\\left\\{ \\sigma\\right\\} -\\text{Tr}\\left\\{ \\Pi_{N}\\cdots\\Pi\n_{1}\\ \\sigma\\ \\Pi_{1}\\cdots\\Pi_{N}\\right\\} \\leq2\\sqrt{\\sum_{i=1}^{N}\n\\text{Tr}\\left\\{ \\left( I-\\Pi_{i}\\right) \\sigma\\right\\} },\n", "83ba422a50cb4f6cd83e012939ad3a0a": "(x+y)^n=\\sum_{k=0}^n{n \\choose k}x^{n-k}y^{k}", "83ba951729cb7af9ca8730b4b1fb8dee": "z^{\\text{nad}}", "83baa6cedcf6102b6e7ecc3d13caaf99": " \\frac{1}{p!} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} a^{\\lbrack \\nu_1 \\dots \\nu_p \\rbrack} = a^{\\lbrack \\mu_1 \\dots \\mu_p \\rbrack} ,", "83bac89388b4d70f82a0b5bd9d95813a": "\n \\cfrac{\\mathrm{d}}{\\mathrm{d}t}\\int_{a(t)}^{b(t)} f~\\text{dx} = \n \\int_{a(t)}^{b(t)} \\frac{\\partial f}{\\partial t}~\\text{dx} + \n\\frac{\\partial b(t)}{\\partial t} f(b(t),t)\n-\\frac{\\partial a(t)}{\\partial t} f(a(t),t) ~,\n", "83bb2943b2f0324b76b58fa3b5cbb575": "\\{ e_1, \\ldots, e_n \\} \\subset H_1", "83bbc6b6b400953cd7294fe9b45a7b9f": " \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} V] ", "83bbdd134a7dbb70519d97ce54a4adcb": "k = (dA/dt)/A", "83bc2854180720e15bb483972c95525e": "P(T_1>t)=P(N(t)=0)=P [(N(t) - N(0)) = 0] = \\frac{e^{-\\lambda t} (\\lambda t)^0}{0!} = e^{-\\lambda t}.", "83bc2e4094b6c20fe19695bb36c10587": " _2F_1(a,b,c;w)", "83bc31431dbf0c1d03c7e91c95bb30ff": "D(s)=a_ns^n+a_{n-1}s^{n-1}+\\cdots+a_1s+a_0", "83bc6a7d151075baa6a11316ab54e37e": " - 4b", "83bcb8fbeecca8b0c8aaecce9b907c2b": "\\{T\\mathbf{w}_1, \\ldots, T\\mathbf{w}_n \\}", "83bcd7a6d7df8413528958f55c92da24": "p_{ij} = \\Pr(X_{k+1}=j \\mid X_k=i). \\,", "83bd0226b97684514755ec18d6796d06": "\\sin(kx) = \\frac{e^{ikx} - e^{-ikx}}{2i} , ", "83bd63de1abc466325f6fadc39728dbe": "\\iota:G\\to G", "83bda1a31b8f9556c2f3f24b75a3a201": "A_{[\\alpha} \\left(B_{\\beta]\\gamma\\cdots} + C_{\\beta]\\gamma\\cdots} \\right) = A_{[\\alpha}B_{\\beta]\\gamma\\cdots} + A_{[\\alpha}C_{\\beta]\\gamma\\cdots}", "83bdcec79871925283dce188186c50e2": " \\Delta p_{\\rm b.a.}(t_{j+1})\\to\\infty", "83bdd207e2566523f6eb740d69096661": "B \\subseteq r T", "83be0ba5d1cebadd89f39d1867d692d5": "\\rho = \\frac{c}{\\epsilon_m}", "83be5cd4d1619d74670d1f44cfe2b5e6": "\\scriptstyle \\, C^j(t)", "83bec8decabbd2acb9ce7c9fa84daac8": " V = \\sum_{i=1}^N V(\\bold{r}_i,t) = V(\\bold{r}_1,t) + V(\\bold{r}_2,t) + \\cdots + V(\\bold{r}_N,t) ", "83bf3c1337756513cb48a0507b444ffb": "\\mathrm{E}_{\\mathrm{RP}}(g)=c/p", "83bf4764298c3667c6cc9719b4ddfb57": "E^R_k = E_k\\Omega_k", "83bf75f19534d065c2480a51340887aa": "Df = \\left( Dg \\right)^{-1}~,", "83c04107d0bb20e5968a11ae2b61eaf6": "10^{10^{10^{10^{13}}}}", "83c0530f2e84596e0445732a5bd5d273": "\n\\rho(\\mathbf{k},\\omega) = \\frac{1}{\\mathcal{Z}}\\sum_{\\alpha,\\alpha'} 2\\pi \\delta(E_\\alpha-E_{\\alpha'}-\\omega)\\;\n|\\langle\\alpha|\\psi_\\mathbf{k}^\\dagger|\\alpha'\\rangle|^2\\left(\\mathrm{e}^{-\\beta E_{\\alpha'}}-\\zeta\\mathrm{e}^{-\\beta E_{\\alpha}}\\right),\n", "83c0a68c0577f4d1f5a7c116c00d4406": "c_k=\\sup_{|n|\\ge k} |a_n|", "83c0c719acfcb65c65eeee515a6e477c": "V_\\max =\\frac{8}{3\\sqrt 3} abc, \\qquad V_\\min = 8abc.", "83c1d818167cb86540b05c744fc376b8": "\\nabla = \\nabla_0 + iA", "83c27221641f3efc8815fd868e788dcd": " \\mathbf{a}' = \\mathbf{a} ", "83c2932401a381e33d47df7d55ce162e": "\n\\delta(x)\\approx \n\\left(\n\\frac{\\nu x}{U}\n\\right)^{1/2}.\n", "83c2946fef4bf8002c81d6e3f7751e46": "[L_{rs}] = \\begin{bmatrix} L_{aA} & L_{aB} & L_{aC} \\\\ L_{bA} & L_{bB} & L_{bC} \\\\ L_{cA} &L_{cB} &L_{cC} \\end{bmatrix}", "83c2d541cb24aa7786002e1d548433c2": "\\left\\{\\begin{matrix} - & c_1 & \\cdots & c_{d-1} & c_d \\\\ a_0 & a_1 & \\cdots & a_{d-1} & a_d \\\\ b_0 & b_1 & \\cdots & b_{d-1} & - \\end{matrix}\\right\\}, ", "83c2f53fc0ff0aa1071a6781ef413286": "[Q]^{t(n)}", "83c31350fe4c287c8b92e5633b0b5de4": "x=[1,1]", "83c32f11ba2610426ac58deb9c7cf584": "\\sigma[1] \\ldots \\sigma[L]", "83c3607c6ce475dbb45d55f73399e2c6": "\n \\underset{n\\to\\infty}{\\operatorname{plim}}\\;T_n(X^{\\theta}) = g(\\theta),\\ \\ \\text{for all}\\ \\theta\\in\\Theta.\n ", "83c3ad3e872dcdecc033a6278bdc90a7": "\\chi = 3590{{Z_{\\rm eff}}\\over{r^2_{\\rm cov}}} + 0.744", "83c3e3321f8517a86bc418f7f1964ea4": " F = Se^{rT}", "83c40080ddd1a6d1dd6183388dcb8d5b": "\\mathrm{Ai}(z) = \\frac{1}{2\\pi i} \\int_{C} \\exp\\left(\\tfrac{t^3}{3} - zt\\right)\\, dt,", "83c406cff4bcbffeef2a74a3d7560484": " \\mathbf{A} \\oplus \\mathbf{B} = \\mathbf{A} \\otimes \\mathbf{I}_m + \\mathbf{I}_n \\otimes \\mathbf{B}. ", "83c40f4b4d0e32e153566b07e661ba7b": "\n\\left( X_{1}| X_{2} \\right) :=\nv_{1} w_{2} + v_{2} w_{1} + \\mathbf{c}_{1} \\cdot \\mathbf{c}_{2} - s_{1} s_{2} r_{1} r_{2}.\n", "83c4238c84741a5e700441aaa959c1b3": "\\displaystyle{E=\\oplus E_i.}", "83c42dc09ea47da30886fd063a2a52bf": "s_{\\mathrm{polar}}:(r, \\theta) \\mapsto 1, \\quad v_{\\mathrm{polar}}:(r, \\theta) \\mapsto (1, 0).", "83c44d203808bda54b465767c0898f91": "\\varepsilon_i=y_i - \\hat\\beta_1 x_{i1} - \\cdots - \\hat\\beta_p x_{ip}.", "83c46a10de9602b6d09a63de09eff7b6": "(x-7) (x-2)^{28} (x+3)^{21}", "83c492d703badc317e859ea319b48950": "X_{i(K+1)} = 1", "83c4a0f64b5c308332e71b11e874d4cc": "\\phi_i", "83c4c9e72f8dbf0f6e8a7ab46663059b": "x^i\\circ f(t)=tv^i", "83c54551e2ef2c882d22d0437035a5a6": "c_n(t) = c_n^{(0)} + c_n^{(1)} + c_n^{(2)} + \\cdots", "83c5805d537dd85ffcf57603be2d4e94": "M_{i,f}", "83c630f6a9f91186ea0e491e7e822095": "\\frac{E_{tot}}{c^{2}}=\\frac{E_{em}+E_{p}}{c^{2}}=\\frac{E_{em}+\\frac{E_{em}}{3}}{c^{2}}=\\frac{4}{3}\\frac{E_{em}}{c^{2}}=\\frac{4}{3}m_{es}=m_{em}", "83c6e44e44c9dc9ab2e936a83eab50ec": " \\frac{\\partial F}{\\partial t} = \\lambda \\kappa(t) - 1 \\ . ", "83c70b0f558df95d8faefeb2df9f6e19": "\\mathcal{F}^4(f) = f,", "83c7511d18b5626c53d29d33fe36a987": "(a, b, c, d) = (\\pm 1, 0, 0, 0)", "83c768367d27f84949a9409d15a54248": " y''+4y'+4y=\\cosh{x}.\\;\\!", "83c778b76fe5569adc1b6d9d669d3c1e": "1030003_{2i}", "83c7c46d68679c227b4f8f03e54c9231": "\\mathfrak {q}", "83c81b383ef6f85f26610167d8cde57b": "a(t) = a_1(t) + ja_2(t) \\ ", "83c88601d40e3fc9165264981c19b4ef": "-\\rho(X)", "83c88787747f4dd121594168f17b135a": "\\mathbf{y}(t) = \\mathbf C \\mathbf{x}(t) + \\mathbf D \\mathbf{u}(t) + \\mathbf{v}(t)", "83c8bbd198e9efad614f6bacf8dd8ed2": "\\sqrt{8/k}", "83c8e04613e8c29203e8b3d5967a6717": "\n\\delta(a, b) = \\begin{cases}\n1 & \\text{if } a=b \\\\\n0 & \\text{if } a", "83db0f12a0759052a500a385599a3f94": "v_i (\\empty) = 0", "83dbc92cf92c931d57a7d88631badc46": "x^2-ny^2=1", "83dc0a3e5ffa66c6a43ec668fdbe2c1e": "C_n = \\frac1{n\\ln^2 n} - \\mathcal{O}\\left(\\frac1{n\\ln^3 n}\\right),\\quad n\\to\\infty,", "83dc1b9a5a6b284f114c481e6129aab3": "a \\not\\!\\triangleleft S - \\lbrace a \\rbrace", "83dca5da491261def7310f2d949c026b": "f(x) = \\frac{1}{\\sigma x \\sqrt{2\\pi}} \\exp\\left(-\\frac{(\\ln x - \\mu)^2}{2\\sigma^2}\\right)", "83dcdd20cb06638c88055a2214851654": "[0,1.7]", "83dce4001cfa5f770fcedada6d4a77e0": "\\lambda=\\frac{A}{N_A \\sigma_{abs}}", "83dd3e321b8c1751646bec79056185d6": "{6\\choose 6}{43\\choose 0}\\over {49\\choose 6}", "83dd47300feec27711cbb8efe7072b9d": "U' = \\phi^{-1}(U)\\;", "83dd5c1d36f0942fadd0723aef5cad05": "\\vec n + p \\to d + \\gamma", "83dde7e9359a4cf40a46ec2aa94a875a": "\\Omega_{0}", "83de0220f9be6fb88bcf353dd6da2508": " f(\\infty):= \\lim_{z\\to\\infty}f(z) ", "83de08644e6fcf7d9fdad859f043eed7": "T_c \\approx \\frac{1}{D_s}", "83de29c2857895fcd13017c33930e7e1": "J_1\\frac{\\partial T_1}{\\partial x}=\\gamma(T_2-T_1)", "83de3524895ed02f7d1e095fcc73dda7": "\\frac{3-2\\ln2}{6}0", "83e36dd6f5e9a6a3e1e4f29add8bae2d": "u(c,t) = u_c(t).\\,", "83e37b7246fdfcb99b2754210ebeae27": "\\rightarrow ", "83e3f249506fcdcde7e4cc54f7e98cd6": "\\textstyle \n (1/f)'(x)=-\\frac{f'(x)}{f(x)^2}\\ \n", "83e4342493a6edd1fbc623a9e83d13e2": " \\Box p \\equiv \\neg \\Diamond \\neg p, ", "83e46bf9511ee036a25cd9bcff4c3888": "x \\ne 0", "83e494bd35e65a71064e5731844e85d0": "\n \\boldsymbol{\\sigma} = -p~\\boldsymbol{\\mathit{1}} + \\cfrac{\\mu J_m}{J_m - \\gamma^2}~\\boldsymbol{B} \n ", "83e4b5e67ed8292368d3ad778f6e5009": "\\scriptstyle 59/60^{n+1} + 59/60^{n+2} + \\dots = 1/60^n", "83e55de18ad579105399538de925cc74": "p V^\\gamma\\;", "83e5645175feb7a010b1acbf147b2324": "\\sum_{n=2}^{\\infty} \\zeta(n,a,\\bar{b}) = \\zeta(a+1,\\bar{b})", "83e5bc61e38375c0c7b6ef153f9c9600": "p_1=2,p_2=3,p_3=5, \\dots ,p_t", "83e6202dd6cd85d3213eceea87d94264": "Q_{xx}+Q_{yy}+Q_{zz}=0", "83e68ae6d8cc4c3c728d653419c50782": "\\hat{h}(\\xi)=a\\cdot \\hat{f}(\\xi) + b\\cdot\\hat{g}(\\xi).", "83e6fd13f93a1d2bb39354b74b127483": "\\displaystyle{\\Gamma_i=Z(G_i)/Z(G_i)\\cap K_i}", "83e7281b64ebaa18819cb7e6e4bc3ea6": "\\{S(\\omega)\\} = \\{{\\text{length}}^2\\cdot\\text{time}\\}", "83e73859fc3efc22b6e247011c436340": "l_a=(-\\frac{F}{2},1,0,0)\\,,\\quad n_a=(-1,0,0,0)\\,,\\quad m_a=\\frac{r}{\\sqrt{2}}(0,0,1,\\sin\\theta)\\,.", "83e7394af2fa46ff9f02c31879a920e2": "F_2=bS(b^{-1})\\cup S(b). \\, ", "83e7868c54d1f5b94ccf13a95d92154b": " \\lim_{h \\to 0} \\frac{\\psi_1(h)}{h} = \\lim_{h \\to 0} \\frac{\\psi_2(h)}{h} = 0 ", "83e7fb8f12468a44c5cad5296515c356": " Z = \\frac{Z_1 Z_2}{Z_1 + Z_2}", "83e82eadc8e024564aa41b5c5d69b505": "\\bigcup_{i \\in \\mathbb{N}} S_i", "83e83ba7982226b700e21f00650a5df0": "V_{\\rm w} = {b\\over{N_{\\rm A}}}", "83e8c9710450a2511bd20d9035431e11": "\\cos(\\alpha) \\cdot \\sin(\\beta) = -Z_2,", "83e8cd0dfad498c1f0fd3f18e2214836": "-14.1014+17.5597(x+\\tfrac{3}{2})-10.8784(x+\\tfrac{3}{2})(x+\\tfrac{3}{4}) +4.83484(x+\\tfrac{3}{2})(x+\\tfrac{3}{4})(x)+0(x+\\tfrac{3}{2})(x+\\tfrac{3}{4})(x)(x-\\tfrac{3}{4}) =", "83e8f16a383202af27e2b95c2fd0efca": "\n\\text{Critical band rate (bark)} = [(26.81 f) / (1960 + f )] - 0.53 \\,\n", "83e90ed46998859a86a1ff65c028d8da": "S_{800}", "83e93eb5c8644c173a61c5f815446345": "\\pi^{ij} = \\sqrt{^{(4)}g} \\left( {^{(4)}}\\Gamma^{0}_{pq} - g_{pq} {^{(4)}}\\Gamma^{0}_{rs}g^{rs} \\right) g^{ip}g^{jq}", "83e9405594cddffbd2ce32928f4d1fc5": "u(0)=u(1)=0,", "83e96d434fc2aa5e235f64de554fbca6": "\nH = V(x, y)\n", "83e9cf33edd22efd95ec9ce73e5aa81c": "a+b+2c=0", "83ea141465cc1da9526aa10cbe241ee8": "F:\\mathbf{Set}\\to\\mathbf{C}", "83ea273cc257e98314f110cb43a14b63": "\\lfloor x \\rfloor", "83eaba908782232257869f82c0baef96": "\\mathcal{L}\\left\\{f(t)\\right\\} = \\mathcal{B}\\left\\{f(t) u(t)\\right\\}.", "83eb7fde7a20d164b5b48cf70793148e": " \\bar r_1 ", "83ebbdf9e1c179d4dab6563e902c14b8": "\\scriptstyle T_\\text{B}", "83ebcfaabb3605046dc6680bbb6db3df": " \\scriptstyle\\operatorname{erf}\\,\\left(\\,\\frac{a}{\\sigma \\sqrt{2}}\\,\\right)", "83ebd3137f0a350b58a963ebffffb1a4": "TE_i > GT", "83ebe7e97eb1374e7b7535ab9d3f1ef1": "T^{\\hat{\\nu}}_{\\hat{\\mu}\\hat{\\underline{\\alpha}}} = 0", "83ebf1e52c0751dae18d910f74dc3818": "H_{n}", "83ebfbb4cc9944ff5522f1f15fa69854": "d_B = 4.1,\\mbox{ } 3.4,\\mbox{ } 2.0, \\mbox{ and } 1.8", "83ec10e4b60986e95b6339ecaaa27d05": "S_1, \\ldots, S_n", "83ec51b6bbee0f1d78dbd29eac292aa2": " \\Phi = \\iiint L_\\lambda \\mathrm{d} \\lambda \\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) \\mathrm{d} \\Omega", "83ec5c60763cc04d322737344f0c9cd6": "\\mu = B/H\\,", "83eca4fa396872771c0d9ea940bb85bf": "\n\\begin{align}\n M_\\mathrm{ss} & ~=~ \\tfrac{1}{2} \\cdot (1 + \\tanh [\\tfrac{V-V_1}{V_2} ]) \\\\\n N_\\mathrm{ss} & ~=~ \\tfrac{1}{2} \\cdot (1 + \\tanh [\\tfrac{V-V_3}{V_4} ]) \\\\\n \\tau_N & ~=~ 1 / ( \\phi \\cosh [\\tfrac{V-V_3}{2V_4} ] )\n\\end{align}\n", "83ecba5b98f3b86d9d415429f530cf00": "\\frac{y^T (D - W) y}{y^T D y}", "83ecc5e4839eeceb7752cb9d84aa1b22": " \\frac{d}{dx} \\left(\\sum_{1 \\le i \\le n} f_i(x)\\right) = \\sum_{1 \\le i \\le n} \\left(\\frac{d}{dx}f_i(x)\\right) .", "83ecf2618532dd30fba7328415f8856c": "\\mathbf{b}_{u,v}", "83ed092e1218c98748f7c93a4bfd4f15": "h:A \\to B", "83edc26f34cfedb8fb42a483ece09fab": "Q\\left(\\frac{\\part\\varphi}{\\partial x_1}, \\ldots,\\frac{\\part\\varphi}{\\partial x_n}\\right) =\\det\\left[\\sum_{\\nu=1}^nA_\\nu \\frac{\\partial \\varphi}{\\partial x_\\nu}\\right]=0.\\,", "83edfe007514df140293cfb116397ab0": "C_G(a)", "83eecae90698095e2a1da238f24ddf2e": "a_{ij}, b_j, c_i \\geq 0", "83eef093418cbc12bd23da517582ac63": "S(t|\\theta)=S(\\theta t)", "83ef36083c5960970278befea50347ee": "\\nabla\\times\\left(f(r)\\mathbf{\\Psi}_{lm}\\right) = \\left(\\frac{\\mathrm{d}f}{\\mathrm{d}r}+\\frac{1}{r}f\\right)\\mathbf{\\Phi}_{lm}", "83efa486bb94e9928d1ed40357f92f23": "{\\mathbf e}_0", "83f029a64da44e2ddf799ee8959539c5": "\\mu'_6=\\kappa_6+6\\kappa_5\\kappa_1+15\\kappa_4\\kappa_2+15\\kappa_4\\kappa_1^2\n+10\\kappa_3^2+60\\kappa_3\\kappa_2\\kappa_1+20\\kappa_3\\kappa_1^3+15\\kappa_2^3\n+45\\kappa_2^2\\kappa_1^2+15\\kappa_2\\kappa_1^4+\\kappa_1^6.\\,", "83f08115cb8d594fbed89a4d33e62e51": "a_2(z)f''(z)+a_1(z)f'(z)+a_0(z)f(z)=0.\\;\\!", "83f0dc036788c10de842df5c32c4c026": "\\star (\\mathbf u \\wedge \\mathbf v )=\\mathbf {u \\times v}\\,,\\quad\\star (\\mathbf u \\times \\mathbf v ) = \\mathbf u \\wedge \\mathbf v \\,,", "83f0e3dd7022ad20ea0ef4d849c56b5c": " w_f", "83f0fd86432bc3cd019caa82dab42843": "C_n={1 \\over n+1}{2n \\choose n},\\, ", "83f114a219b1dbdd5dc72c01ad3362ad": "\\Phi^5 - 2000\\varphi^4(\\tau)\\psi^{16}(\\tau)\\Phi - 1600\\sqrt{5}\\varphi^3(\\tau)\\psi^{16}(\\tau)\\left[1 + \\varphi^8(\\tau)\\right] = 0\\,", "83f164e03536f9851d5caf306ac9696d": "\\mu \\; ", "83f19002526471b692187f090d8fe8a2": "\\;\\;\\frac{1}{\\sqrt{y}} \\; f\\left(\\frac{1}{\\sqrt{y}} \\right) \\;\\;", "83f1afb0c7aa46e2ed59fa68be652c0d": " y_{n+1} = y_n + h f(t_{n+1}, y_{n+1}). ", "83f1cc449de9fdff1ac5cda7d03a55c7": "u(x, y) = \\log \\left( \\frac{\\cos (x)}{\\cos (y)} \\right).", "83f20d3a87a967f7ea73d9942821ea4d": "x^*=f(x^*)", "83f21599f43c228fbc1c2375accedab7": "A = (a_{v,t})", "83f260873f74825283702bdeb5769a80": "\\displaystyle{Cf=E\\star f,}", "83f2b158f9dc4ce5b83fe475e0615682": "f(x) = e^{-\\pi x^2}.", "83f2c5ae0cb0ccbd08d8440219fbe652": "11.29 = \\frac{9.8 \\times 3600 \\times 3600}{1852 \\times 6076.12}", "83f2e1edaf5720b7d7136b7f793745e1": " \\mathbf{z}_n ", "83f3288dcb916bb2fc265b289eae251d": "G = \\tfrac18\\pi \\log(2 + \\sqrt{3}) + \\tfrac38 \\sum_{n=0}^\\infty \\frac{(n!)^2}{(2n)!(2n+1)^2}.", "83f34249246febf6ac8d78577ad7d03a": "\\log \\, R_e", "83f369b8f54ce3e7efcf7573e0b61419": " \\det A = \\sum_l (-1)^{i+l} a_{il} \\cdot \\det A^{il}", "83f3b19ea9fb42b3e0b74e87c729f330": " v_2 = \\frac{a_{21}-\\alpha}{2r}", "83f3cbc085416cccf5a466b5fd959fe4": "\\,\\! \\lambda^*", "83f3d271ad118ce467a0274a00be5bbd": "f(x) = (Ax, x), \\; \\|x\\| = 1.", "83f3dc6c3a134c63520bf7157c9904e5": "Bu=u,~~~Cx=\\int_0^1 x(\\xi)\\,d\\xi,~~~D=0.", "83f41c476ae7b841dc4ba3408b1bcb84": "(B y + \\beta + 1)^n>B^n x + \\alpha\\,", "83f47d29d872f3f51ec500c5eeda9614": "\\pi_0(PSO) \\cong 1", "83f4bca735bf0b76abd678f56288ec1b": "\\bigoplus_{i+j=n} C_{i,j}", "83f56d8cd242c023cd40fcc0d372ee98": " \\mathbf{\\left(A\\times B\\right)\\cdot}\\left(\\mathbf{C}\\times\\mathbf{D}\\right)=\\left(\\mathbf{A}\\cdot\\mathbf{C}\\right)\\left(\\mathbf{B}\\cdot\\mathbf{D}\\right)-\\left(\\mathbf{B}\\cdot\\mathbf{C}\\right)\\left(\\mathbf{A}\\cdot\\mathbf{D}\\right) ", "83f56f37a245ccaf8c885814074777f6": "TE", "83f59a49e452a9279299e84eb83a01ac": "A_2B_2,\\ A_4B_4", "83f5aba50ae1ddfb52a23390b23b9ff0": "\\{w^{2^{k-1}}|w\\in\\{a,b\\}^*\\}", "83f5b12dcb8c1b86268a4b2567722e7e": "\\frac{M(x,y)}{N(x,y)} = \\frac{M(tx,ty)}{N(tx,ty)} = \\frac{M(1,y/x)}{N(1,y/x)}=f(y/x)\\,. ", "83f5b2fa0898635b8b49bd16eba5dd37": "Q=\\int j_0 d^3x = \\omega\\varphi_0^2\\frac{4}{3}\\pi R^3.", "83f5d149bb005168288c0cda8751554e": "y = X\\beta + \\varepsilon", "83f6094d3253e193158f1f258e79aafa": "\\overline{M R}", "83f60d6361c00047046a83ed8a1ed5fe": "\\textstyle(x\\pm1, y\\pm1, z)", "83f6c41a89082ba107850e4ea6f10b39": "(0, -\\gamma, \\gamma),", "83f711cdab7d8d1536e25e37a1bc5292": "g'(a)", "83f712a789064504d9cf7b016fd9986f": "\\mathrm{GL}_n(\\mathbb{C})", "83f7da5c6def90efd9b6fa4a83ded728": "\\pi=\\{V_1,\\dots,V_k\\}", "83f8030c11bca55795e8b37387f1c537": " y_{n+1} = y_n + h f \\Big( t_n + \\tfrac12 h, y_n + \\tfrac12 h f(t_n, y_n) \\Big). ", "83f80e24ff364ca5790b3382d54a97f7": " Y = \\frac{1}{iX} = - i\\frac{1}{X} =iB ", "83f834dbe394852044b5935c28ada95a": "x_i = r z_i \\qquad\\qquad\\qquad\\qquad\\qquad 2\\le i\\le n.", "83f835ae637c88c25ec955aba9eefa83": "(\\pi,\\lambda)", "83f858d79b902ec550edb634cf33c58c": " \\hat{f} \\, \\hat{f}^\\dagger \\,= 1 - \\hat{f}^\\dagger \\, \\hat{f} = 1 + :\\hat{f} \\,\\hat{f}^\\dagger :", "83f89867516254f1aa8be566faebb455": "\\left(\\frac{d}{dz}\\right)_q z^n = \\frac{1-q^n}{1-q} z^{n-1} = \n[n]_q z^{n-1}", "83f89af399283965f78b78d4461b54ca": "P=t^n+a_{n-1}t^{n-1}+\\cdots+a_2t^2+a_1t+a_0=(t-x_1)(t-x_2)\\cdots(t-x_n).", "83f8b12651fd3c3873150529bf6d3549": "c_{3,1}(\\widehat{a}, w(a\\widehat{b}cd, \\widehat{b}c), \\widehat{d})", "83f8bcbe1b323b64a49c62746faa57c0": "\\mbox{LOP2}=120+40+180", "83f8c65571caf629b7638a700b55e4f3": "\\exists! \\!\\,", "83f8dd612f66b0562964787c517fbd79": "a_n = n^{-1} e_n,\\ \\ \\text{where}\\ \\ \\{e_n\\}_{n=1}^{\\infty}", "83f95a22b47d30a5b413e8282fad9f24": "\\kappa(K_m,~Q_{m+1}) \\neq 0", "83f977bd7a334a741bdfe8c244c73305": "\\begin{align}\n \\Delta p_{\\text{B}}(0) &= D_2 \\\\\n \\Delta p_{\\text{B}}(W) &= D_1 W + \\Delta p_{\\text{B}}(0)\n\\end{align}", "83f9a2eea1512318951c703963281fa7": "C_{4,2} = (9 + 1) / 2", "83fa5721c93fb640f04bbf4e9c7ece44": "\n\\begin{align}\n& {} \\qquad \\Pr(|X-\\mu| \\geq k\\sigma) = \\operatorname{E}(I_{|X-\\mu| \\geq k\\sigma})\n= \\operatorname{E}(I_{[(X-\\mu)/(k\\sigma)]^2 \\geq 1}) \\\\[6pt]\n& \\leq \\operatorname{E}\\left(\\left({X-\\mu \\over k\\sigma} \\right)^2 \\right)\n= {1 \\over k^2} {\\operatorname{E}((X-\\mu)^2) \\over \\sigma^2} = {1 \\over k^2}.\n\\end{align}\n", "83faacf69f10d79a05cc07dd625a7797": "x_{ij}=0.", "83fab270b77e15554e2821cf1362c1d2": "is\\_ carrying(o,s)", "83faf38c149ce7da2348bde1501a11e0": "C_z=C_L-C_d*\\alpha", "83fb0f1d624e63b5d2a8047e9174288e": " \\left(\\begin{matrix}\\alpha & \\beta\\\\\n \\overline{\\beta} & \\overline{\\alpha}\\end{matrix}\\right)", "83fb894e74dbe07d7dd95709423afa53": "(n\\,\\bmod\\,r)", "83fbaf54cd8227c625734f070a9dc50e": "\\frac{z}{1-\\exp(-z)}", "83fbb847b21d39e1c2c34a3dab4b4143": "\\mathcal B([0,\\infty))\\otimes \\mathcal E^*", "83fbff809a284a4cd4e155dbb26436b9": "g(y) \\approx \\int_{\\mathbb R} e^{-2\\pi ixy} f(x)\\,dx\\text{ and }f(x) \\approx \\int_{\\mathbb R} e^{2\\pi ixy} g(y)\\,dy,", "83fc13421ff616fd6d958b18e43d84cf": "\n\\left[b(\\mathbf{k}),b^\\dagger(\\mathbf{l})\\right]\n= \\delta(\\mathbf{k}-\\mathbf{l}). \\quad\\quad (7)\n", "83fc4b8b37bdd58bb27d8c6b5e2b6da3": "\\displaystyle{\\partial_{z}(I-U\\mu)F= UG.}", "83fc6be8f4e6129971b16da57811f9fb": "\\mu_{T,e_1}=X^3+4X^2+X-1", "83fcbf31663a745145608820ec97bb0e": "\\mathbf{G} = -\\mathbf{G}", "83fcd101735c57b3666bc24bbd6dd81b": "f_1' = f_1' \\!.", "83fd6cd1bece8bd932aa052169308a0d": "T(y)", "83fd743b78173e119d0c88ec7094e402": "{\\mathrm{Nu}}_D \\ = 2+ 0.43 \\mathrm{Ra}_D^{1/4} ", "83fda358ed8ae58970cd06253f0187fa": "e \\ne e^{\\prime}\\,", "83feb1b265d68772d07baf5578f8e0f2": "M = \\sup_{x \\in I} \\frac 1 {2}\\left |{\\frac {f^{\\prime\\prime} (x)}{f^\\prime(x)}}\\right |. \\,", "83fec576a26531e353d1b1c9d1eabbc9": "\\mathbb {RP}^2 ", "83fed7c2468fde95c0ba3c1af1062c6d": "\\displaystyle{f(\\theta,t)=\\theta + f_1(\\theta)t + f_2(\\theta)t^2 + \\cdots=\\theta + g(\\theta,t)}", "83fee028818cee6ba5abd432e9901b94": "\\|g\\|_\\infty\\omega_1+\\|f\\|_\\infty \\omega_2", "83ff03ce12dba54c8333b7e77d07e433": "{{N_i}\\over{N}} = {{g_i e^{-E_i/k_BT}}\\over{Z(T)}}", "83ff21018dd23ab19a95b96a4e9737e6": "t\\otimes v\\in kG\\otimes_{kCent(g)}X=V", "83ff4679428925a9e34581f4e0792d20": "K_H(t,s)=\\frac{(t-s)^{H-\\frac{1}{2}}}{\\Gamma(H+\\frac{1}{2})}\\;_2F_1\\left (H-\\frac{1}{2};\\, \\frac{1}{2}-H;\\; H+\\frac{1}{2};\\, 1-\\frac{t}{s} \\right).", "83ff46c4264f6d27f5274f3b7415b91e": "\\mathrm{horizon}_\\mathrm{miles} \\approx \\sqrt{2 \\times \\mathrm{height}_\\mathrm{feet}}.", "83ff4fb0898f1ce5d2e042be6bf87108": "\\varphi = \\frac{1+\\sqrt{5}}{2} \\approx 1.61803", "83ff5384522b9208b07c57a2ff9a66ae": "4 \\left( \\left( {L \\over 2} \\right)^2 \\sqrt{3} \\right) = 4 { {L^2} \\over 4 } \\sqrt{3} = L^2 \\sqrt{3}.", "83ff815a20e7d6fa64e7093cc898c4bc": "\\text{(3)} \\qquad \n \\sigma_y(\\varepsilon_{\\rm{p}},\\dot{\\varepsilon_{\\rm{p}}},T) = \n \\sigma_a + B\\exp(-\\beta(\\dot{\\varepsilon_{\\rm{p}}}) T) + \n B_0\\sqrt{\\varepsilon_{\\rm{p}}}\\exp(-\\alpha(\\dot{\\varepsilon_{\\rm{p}}}) T) ~.\n", "83ff88535e2aefbee168a47e1c20fd2d": "p^2 + q^2 = 5\\left(\\frac{c}{3}\\right)^2.", "83ff970e2f4cd4cc2ee45a475256a8ec": "m \\ll M \\ ", "83fff98c69a1b615747c884449c14203": "z=f(\\zeta) = \\int^\\zeta \\frac {\\mbox{d}w}{\\sqrt{w(w^2-1)}}\n=\\sqrt{2} \\, F\\left(\\sqrt{\\zeta+1};\\sqrt{2}/2\\right),\n", "84000773f58e04a855b488cfed1690e8": "P_1^2,P_2^0,P_3^0", "84002fa21d68eebc2076853db2ccac27": "|\\omega(n)-\\log(\\log(n))|<{(\\log(\\log(n)))}^{\\frac12 +\\varepsilon}", "84007cf3dbfa79d20532830fa7f49b73": "\\int_1^\\infty \\frac{dx}{(1+x^2)(1+x)^2} = \\frac{1}{4}(1-\\ln 2).", "84010570e40f57a8fc228d3b9cd12cc3": "x = 0.8707 \\times l(\\varphi) \\times \\lambda", "840119b21561db8d3525f97e052db51c": "\\mathrm{in}(f) = 0", "84015e8a1aef3f6ddcd6ebdd2bcac51c": "\\text{bind}\\colon A^{*} \\to (A \\to B^{*}) \\to B^{*} = l \\mapsto f \\mapsto \\begin{cases} \\text{nil} & \\text{if} \\ l = \\text{nil}\\\\ \\text{append} \\, (f \\, a) \\, (\\text{bind} \\, l' \\, f) & \\text{if} \\ l = \\text{cons} \\, a \\, l' \\end{cases}", "84016b5f92efacb6e7c471db5b4db655": " C_2^\\perp", "840182077a412c368481f1eb9d6eb51b": "\n(y - y_1)(y - y_2)(y - y_3)(y - y_4)(y - y_5)\n", "8401a6ed412a50e95108b64d14be8cb1": " \n b=0. \\,", "8401cc7e9237d8b6dd1f0faea742f579": "\n\\begin{align}\n\\frac{dG}{dt} & = \\sum_{k=1}^N \\mathbf{p}_k \\cdot \\frac{d\\mathbf{r}_k}{dt} +\n\\sum_{k=1}^N \\frac{d\\mathbf{p}_k}{dt} \\cdot \\mathbf{r}_k \\\\\n& = \\sum_{k=1}^N m_k \\frac{d\\mathbf{r}_{k}}{dt} \\cdot \\frac{d\\mathbf{r}_k}{dt} + \\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k \\\\\n& = 2 T + \\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k\\,,\n\\end{align}\n", "8402ab46601d523ee3310f59c14b53ad": "w(z) = e^{-z^2}\\operatorname{erfc}(-iz) = \\operatorname{erfcx}(-iz).", "8402bb9d136c73d89dd9062f8c49d8cc": " \\frac{1}{\\pi} = \\frac{2\\sqrt{2}}{9801} \\sum^\\infty_{k=0} \\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\\!", "840311bdec1b184e5993f8dabdae0005": "\n\\begin{matrix}\nX_k & =\n& E_k + e^{-\\frac{2\\pi i}{N}k} O_k \\\\\nX_{k+\\frac{N}{2}} & =\n& E_k - e^{-\\frac{2\\pi i}{N}k} O_k\n\\end{matrix}\n", "84035c02bd1916c538ddf679fd5b863c": "[\\mathfrak{g}, \\mathfrak{g}]", "84037ac2ca001c0ec4ddbca41884f8a1": "[0,1)", "8403f16b7635bb6e2ff362251f183003": " \\pi_{k\\neq 2}(PU(\\mathcal H))=0", "84045f7b4797d7ef1a69290a8f35e293": " \\varepsilon = \\frac{1}{L} u ", "8404bccb9b5a9c314d5c25da1a27bfcc": "U_\\alpha S^\\alpha = P_\\alpha S^\\alpha = 0 ", "8404efc4230216bb3f6876529f83d756": "\\text{with}", "8404f35dd94de68dd1ace0c5884c5d3a": "C N^{1-c}", "840555d40a27558661e24d6a67ffc3d5": "[F , G]^{IJ}", "84062498a3a97aca8f9c0ea2cf39a816": "{}^{z}e", "84064ac604f560b88e722d2ce6448059": "\\epsilon_n", "84066f2a845e4cb29703cbb11641d018": "\\left\\|\\mathbf{L}-\\sum_{k=0}^N\\mathbf{x}_k\\right\\|\\to 0\\quad\\text{as }N\\to\\infty.", "84067dff010d41ca5a77e76a9941875a": "2mP", "840684c6a16beaedcbf3e3c5dee8c5ea": "g_* : H_1 S \\to H_1 S", "8406da38055079b35c0e5c66f47eb53c": " J(r)=\\frac{1-D_o(r)}{1-H_s(r)} ", "84072f40c2c9d8e15b15cd77b4609ff8": "\\operatorname{R}(x) - \\frac1{\\ln x} + \\frac1\\pi \\arctan \\frac\\pi{\\ln x}", "840782107f111e53f3f7073ade75b98f": "\n\\frac{\\partial f}{\\partial \\mathbf{p}} = \\mathbf{\\hat{e}}_x\\frac{\\partial f}{\\partial p_x} + \\mathbf{\\hat{e}}_y\\frac{\\partial f}{\\partial p_y}+\\mathbf{\\hat{e}}_z\\frac{\\partial f}{\\partial p_z}= \\nabla_\\mathbf{p}f\n", "8408546e7986f985ad7d0acca506c194": "H=\\left(2-\\frac{R^2}{r^2}\\right)\\frac{p^2}{2m} + \\frac{1}{2}\\left(1+\\frac{R^2}{r^2}\\right)mgz - \\frac{(\\vec{r}\\cdot\\vec{p})^2}{mr^2} + u_1 p_\\lambda", "8408be74e22c8b8e8685a001af329fea": " G^{\\hat{a}\\hat{b}} = -\\Lambda \\, \\left[ \\begin{matrix} -1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{matrix} \\right] ", "84092c68ffd2a1b7d7909cedd5d93bcb": "\n T = 1 - 0.17 \\cos ( \\bar{H}^\\prime - 30^\\circ )\n + 0.24 \\cos (2\\bar{H}^\\prime)\n + 0.32 \\cos (3\\bar{H}^\\prime + 6^\\circ )\n - 0.20 \\cos (4\\bar{H}^\\prime - 63^\\circ)\n", "84094724b29540123dc68df97c5ed813": "\nP_{\\ell}(\\cos \\gamma) = \\frac{4\\pi}{2\\ell + 1} \\sum_{m=-\\ell}^{\\ell} \n(-1)^m Y^{-m}_{\\ell}(\\theta, \\varphi) Y^m_{\\ell}(\\theta', \\varphi')\n", "8409d4115c4d427cd6136041621f5779": "\\begin{align}\nu'(x) &= \\lim_{h \\rightarrow 0} \\frac{\\int_{y_0}^{y_1}f(x + h, y)\\,dy - \\int_{y_0}^{y_1}f(x, y)\\,dy}{h} \\\\\n&= \\lim_{h \\rightarrow 0} \\frac{\\int_{y_0}^{y_1}\\left( f(x + h, y) - f(x,y) \\right)\\,dy}{h} \\\\\n&= \\lim_{h \\rightarrow 0} \\int_{y_0}^{y_1} \\frac{f(x + h, y) - f(x, y)}{h} \\,dy\n\\end{align}", "8409f01256ff1494daa9d5dc3ff8dcef": "\\exp(t X) g", "840a17f37426f307e55b5f1ac1c9358f": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=12,\\cdots ,17", "840a1efe67db57c10565c84db134dfb2": "\\eta \\left\\{\\eta_bE + \\frac{Px}{v}\\right\\} = \\left\\{W C_{rr1} + N C_{rr2} v + \\frac{1}{2}\\rho C_d A v^2\\right\\}x +Wh + \\frac{N_a W v^2}{2g}", "840a3d17f5b0929ad2b3168babf7ef45": "\\mathrm{d}H = C_p\\mathrm{d}T+V(1-\\alpha T)\\mathrm{d}p.", "840a5c8d25f15c0bce4f3a4576d4e02f": "\\overline{16}", "840a7a155b9b9102028d525798ad1aa8": "\\exp\\left[it\\mu_1+it\\mu_2 - |c_1 t|^\\alpha - |c_2 t|^\\alpha +i\\beta_1|c_1 t|^\\alpha\\textrm{sgn}(t)\\Phi +i\\beta_2|c_2 t|^\\alpha\\,\\textrm{sgn}(t)\\Phi \\right]", "840a86b9b8c9f99619d8f340c7f19688": "ACN = \\ell^2 + \\ell + m", "840ac5deea953a2f4138110173003a65": "\\epsilon = a / L_{sd}", "840ace4a6a4dc84bc867de925696b97d": "\\ddot{X}", "840b0eb643c3298361d93a66a77503c5": "O \\times O \\subset O\\subset U\\subset \\operatorname{Sp} \\subset\n\n\\operatorname{Sp} \\times \\operatorname{Sp} \\subset \\operatorname{Sp}\\subset U\\subset O \\subset O \\times O. \\, ", "840b76d9a109b737a49b95e071fc5567": "L \\otimes_k L", "840b8cf690bd7875a128cfd833bb6eaa": " k_1 \\ ", "840b90727b30918428a03ccc292be2d8": " \\sigma = g\\tau", "840b97b5edeb8dae74af04bd6b05c2df": "\\nabla\\cdot\\mathbf{D}= \\rho_{\\text{free}}", "840ba4800b7e1c3b784f3890753bf79c": "\\Delta u * w_{r,s} = u*\\Delta w_{r,s} = u*\\chi_r - u*\\chi_s=0\\;", "840c0ceb89b5f288c58214e9ad6aa7ce": "\\rho_{m}=-\\nabla\\cdot\\mathbf{M}", "840c0f0e4a0d183f4637caed8b5f0644": "f^{(n)}(x) = \\lambda^n\\exp(\\lambda x)", "840c13016e22c4e3c7012e82daca6fe4": " \\|x\\|_0 ", "840c51f15eb0786ca0bdfdc0c601049a": "\\displaystyle Q", "840c5f07ba089f85798dfe7c95ed45e7": "E=E(q_{k},p_{k})", "840cb221ff33e3172bcd603ef264dba2": "A \\to (abc, 0)", "840d11070f750636c73d2cde226d3950": "\\mathbf{a} \\times \\mathbf{b} = \\begin{bmatrix} a_x \\\\ a_y \\\\ a_z \\end{bmatrix} \\times \\begin{bmatrix} b_x \\\\ b_y \\\\ b_z \\end{bmatrix} = \\begin{bmatrix} 0 & -a_z & a_y \\\\ a_z & 0 & -a_x \\\\ -a_y & a_x & 0 \\end{bmatrix} \\begin{bmatrix} b_x \\\\ b_y \\\\ b_z \\end{bmatrix} = \\mathbf{\\hat{a}} \\mathbf{b} ", "840d1ddfeec85b3b472227b109937327": "n_{2}=\\frac{p_{2} V_{ref}}{z_{f2} R T_{ref}}", "840d2e2183d17a85f11280500e33b62f": " a_k, a_{k-1}, a_{k-2},\\ldots, a_1, n+1, b_1, b_2, b_3,\\ldots b_{n-k} \\, ", "840e8624605b9df47d8b45c2ae46e8c5": "\\,\\!\\rho", "840eca0e3384ff06c50874598a6cd8d9": "X_\\mathrm L = \\frac {V_\\mathrm P}{I_\\mathrm P} = \\frac {2 \\pi f L I_\\mathrm P}{I_\\mathrm P} ", "840ecc0df1357002e4077bbc4899ea1a": "Fred(\\mathcal H),", "840f107b4f2f6a81b2f8fb02a3077c2a": "\\int_0^\\infty {\\sqrt{x} \\over x^2+6x+8}\\,dx.", "840f44f78e53ed237e77f7703b780d6b": "f=\\frac{d(a-c+1)}{a-d}", "840f9ea54a284dddc201b25aeda332a4": "\\theta_1,\\theta_2,\\theta_3,\\theta_4\\,", "840f9fa1460af8398810d899684c5cb6": "\n{\\Delta}T_F =\\frac{{\\Delta}H^{fus}_{T_F}-2RT_{F}{\\cdot}\\ln(a_{liq})-\\sqrt{2{\\Delta}C^{fus}_{p}T^{2}_{F}R{\\cdot}\\ln(a_{liq})+({\\Delta}H^{fus}_{T_F})^2}}{2\\left(\\frac{{\\Delta}H^{fus}_{T_F}}{T_F}+\\frac{{\\Delta}C^{fus}_p}{2} - R{\\cdot}\\ln(a_{liq})\\right)}\n", "840fbb57757c64a7611577c60cda96a6": "\\mathbf{A}^* \\,\\!", "840fe6357110661caa50fddc528e7371": "\\left\\vert \\langle \\psi_{x+} \\vert \\psi_{x+} \\rangle \\right\\vert ^2 = 1 ", "84100959a67e9f5911bac45f30a8b935": "X[(p+m)\\Delta_{F}]", "84102aedc38bebaacad4aa4f3ead46ec": "r = r_+ := M + \\sqrt{M^2 - Q^2 - J^2/M^2}", "84104a80e0563d3f83000a9e0ce755d9": "\\mathbf{u}\\rightarrow\\mathbf{u}.", "84109b7dae87e29839f2fc1aecfa7da7": "dx^{0(2)} - dx^{0(1)} = \\frac{2}{g_{00}} \\sqrt{\\left ( g_{0\\alpha}g_{0\\beta} - g_{\\alpha \\beta}g_{00} \\right ) \\,dx^\\alpha \\,dx^\\beta}.", "84109c3cdc56b1b60edb22dfe7916754": " \\rho = 1/2 ", "8410b13da688fc4747f07bedd4e6d6af": "\\sgn(z) = e^{i\\arg z}\\,,", "8411159c639fe3b3f46f11f79da1688c": "p_A", "8411320ede02bb93c8e4380d4a77d967": " \\mathcal{L}(\\mu,\\Sigma)=(2\\pi)^{-np/2}\\, \\prod_{i=1}^n \\det(\\Sigma)^{-1/2} \\exp\\left(-{1 \\over 2} (x_i-\\mu)^\\mathrm{T} \\Sigma^{-1} (x_i-\\mu)\\right) ", "8411392d51c50497d5115202502fda5e": "\\left( \\frac{1}{2} \\cdot 3\\right) + \\left( \\frac{1}{2} \\cdot 5\\right) = 4", "8411acf2758b77085081dd6849c82c1c": "\\int_V \\left[ \\mathbf{J}_1 \\cdot \\mathbf{E}_2 - \\mathbf{E}_1 \\cdot \\mathbf{J}_2 \\right] dV = \\oint_S \\left[ \\mathbf{E}_1 \\times \\mathbf{H}_2 - \\mathbf{E}_2 \\times \\mathbf{H}_1 \\right] \\cdot \\mathbf{dA} .", "8412042799abbf80bd144fe3fa78e533": "\\rho = \\alpha V_p^{\\beta}", "84120c7531a916982187d3e86779cc64": "p=x^2+\\;\\,y^2\\text{ if and only if } p=2 \\text{ or } p\\equiv 1 \\pmod4,", "841227a3d2111497ae60a87ed10e9e98": "\\partial^2/\\partial t^2", "84122f75280472816f4230402a4e4704": "\\textstyle \\sin(a x) = ", "8412446b326f60c0dd4f4fca0375e008": "B_n = \\{Z, S, (p_i^m)_{i \\le m}, E_k : k < n\\}", "84124b5a3ecb687f5bafb68d71f1e529": "E[a\\xi+b\\eta]=aE[\\xi]+b[\\eta]", "84125566515325ab3983c3d7385f9021": "\\psi(\\bold{r}) \\approx e^{ikz}+f(\\theta)\\frac{e^{ikr}}{r}.", "841287c561744ff080700e0fa98896a3": "e^{-i\\int H(t) dt_{op}}\\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}", "84129563f1b8906922b3bfe8d4d1d2f8": "\\kappa_1=\\kappa_2=200", "8412aaba16f5252c1c629628a85a7a99": "-\\sqrt{g}(\\nabla^{i}\\nabla^{j}N -g^{ij}\\nabla^{n}\\nabla_{n}N) + \\nabla_{n}( \\pi^{ij}N^{n} ) - N^{i}{}_{;n}\\pi^{nj} - N^{j}{}_{;n}\\pi^{ni}", "8412e81168f007d18ac855f0192058ef": "\\tilde{F}(x) = \\sum_{n \\ge 0} |T(F_n)| x^n.", "8412f1f2cc68ba2347bca65ac78426bc": "\\mathcal{E}(k)", "8413219ccf6d69558eeaf7952f2d9947": "R_{ij}=\\overline{\\upsilon_i^\\prime \\upsilon_j^\\prime}", "841374e01f2af4c84aec44062da8db6c": "\\frac{1}{z}", "84138da054f191a2c5bf8470d1df2d3a": " \\sigma = F/A \\,\\!", "8413be34d17218bccd9b168a146d2552": " \\begin{bmatrix} V_1 \\\\ I_2 \\end{bmatrix} = \\begin{bmatrix} h_{11} & h_{12} \\\\ h_{21} & h_{22} \\end{bmatrix} \\begin{bmatrix} I_1 \\\\ V_2 \\end{bmatrix} ", "84146f52b7be7f905d114d4b5c3b59da": "\\operatorname{H}^*(X; G)", "8414fbc30810ef8018b9c50563e5cce6": "\\mathbf{\\mathit{\\rho}}", "8414fde712aa258cfce70bfd824f5053": "| B(u, u) | \\geq c \\| u \\|^{2}", "8415515a69c4e6bc05bc56ad1b2786bc": "2^{\\aleph_0+n}\\, = \\,2\\cdot\\,2^{\\aleph_0+n} ", "841562647dae41023b69f7f1a284d970": "0 = \\psi(1-x_n) = \\psi(x_n) + \\frac{\\pi}{\\tan(\\pi x_n)}", "8415b0c6fe8a0f246efc011e810b67f0": "t = \\tanh\\tfrac{1}{2}\\theta = \\frac{\\sinh\\theta}{\\cosh\\theta+1} = \\frac{\\cosh\\theta-1}{\\sinh\\theta}", "8415e4ad63a5ca2164d8e685c325b429": "m=\\frac{\\sqrt{3}}{2}\\frac{q}{m_e}\\hbar", "841669f772223f251cf646b0deb92c64": "(\\mathbb{Z}/n\\mathbb{Z})", "841672af1a09b9ec3a873534940cc75f": "\\mathbf{P}=(m_0 c,0,0,0)", "8416ff100e833938edd60ff437b900ac": "y^2=P(x)", "8417097be99b88bc31392cdac1c596ca": " x = x", "84171b5678a2cf0ea2a532e9c849b0d8": "S =\\frac{kA}{4}", "841734d907adcc2c0f60aa4c5cb2bc21": "\\oint \\mathbf{H} \\cdot d\\boldsymbol{\\ell} = \\oint (\\frac{\\mathbf{B}}{\\mu_0} - \\mathbf{M}) \\cdot d\\boldsymbol{\\ell} = I_{\\mathrm{tot}}- I_{\\mathrm{b}} = I_{\\mathrm{f}},", "84173e6e0106d58349f6639c27db901d": "e^{i\\pi} = -1", "84174c33e205c44a3c850084e932a4de": "\\left\\{\\frac1{16}, \\frac{33}{16}, \\frac{17}4, \\frac{105}{16}\\right\\}", "841758b9877b7cf85c4f36657c95915c": " N(t) = N_0 e^{-rt}", "8417aab5fc7cf2eb3109b98bc086648b": "\\ EX(p)", "8418999a177079ac36ec94cb90b532f2": "\n\\delta L = \\epsilon \\frac{d}{dt} G(\\mathbf{q}, t)\n", "8418ce4b9ab1a23f48c17c6727ee6d74": "(x)_{\\infty} = \\prod_{m=1}^{\\infty}(1-x^m).", "8418f516832d217e4734c9dee73dd242": "\\Delta_\\lambda", "84190ae5f46b79c8d911540beee8d11f": "2^2P[S_2=k]", "84191972ac078712cce6e99d4f1cf74e": "\\ln x = kt + \\ln a\\,", "84192a9b49ed1bf01da22dd03ad53ea9": "\\frac{2x^6-4x^5+5x^4-3x^3+x^2+3x}{(x-1)^3(x^2+1)^2}=\\frac{A}{x-1}+\\frac{B}{(x-1)^2}+\\frac{C}{(x-1)^3}+\\frac{Dx+E}{x^2+1}+\\frac{Fx+G}{(x^2+1)^2}", "8419740b53ea212da899da6fd8de7112": "p(\\bar{S}_{2t})/p(S_t)\\,\\!", "84197562c18a5815ba75ad51604cb9f5": " \\mathbf{MTF_{atmosphere}(\\xi,\\eta) \\cdot\nMTF_{lens}(\\xi,\\eta) \\cdot } ", "8419bc13f0b74f6c7643c0052e3846fe": "\\mathrm{H}", "841a4893c0ade83354b3ad0d34152c32": "\\scriptstyle{\\vec{r}(u(t),v(t))}", "841a532a46c35e01e199468f9e4f8c01": "\\tilde{H} = \\sqrt{det (q)} H = \\epsilon_{ijk} F_{ab}^k \\tilde{E}_i^a \\tilde{E}_j^b = 0", "841a8b50c43efa9907da5040adfa86c3": "\\hat{G}(\\boldsymbol{k},\\omega),", "841ab962e4ee5b331e38c5d2a6882cf5": " P_1,P_2,\\cdots ", "841abd62e7de99c00555115cc4913fde": "\\tau_{\\geq 0}, \\tau_{\\leq 0}", "841ad57ebc62a3b2764dbb54707cb9e6": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon} \\frac{1}{R} =\n\\frac{q}{4\\pi\\varepsilon} \\frac{1}{\\sqrt{r^{2} + a^{2} - 2 a r \\cos \\theta}}. \n", "841afe9e0a14fb5e02479072a1fcc70c": " \\operatorname{E}(\\widehat{\\theta}) - \\theta = \\operatorname{E}(\\widehat{\\theta} - \\theta ) ", "841b1c2fc05f210369e7c3620874b40b": "\\begin{align} A & = 12 \\tan\\left(\\frac{\\pi}{12}\\right) r^2 = \n 12 \\left(2-\\sqrt{3} \\right) r^2 \\\\\n & \\simeq 3.2153903\\,r^2.\n \\end{align}", "841bc86b0eeb26fcfb797dce9d481804": "\\mathcal{W}()", "841c2b773ad9571edeb4b6f0def72caa": "f(\\zeta) = \\int_{-\\infty}^\\infty F(x)e^{i x \\zeta}\\,dx", "841c3a6798912d8c0ff1b73d0b375958": " x^{1/n} = \\sqrt[n]{x} ", "841c586cf361ed795252c149c6918c76": " \\lambda_{2} = 1 ", "841c897850783d696744e80ae5e419b9": "f(S_i, S^i_1, \\dots, S^i_K)", "841c9535f58d13bd803023ca484222cd": "\\overline{P}(Cl^{\\ge}_t)", "841ccbe8dc96a695219b5d059b610664": "\\frac{\\theta \\vdash \\phi \\quad \\theta \\wedge \\phi \\vdash \\psi}{\\theta \\vdash \\psi}", "841d015501e167e5afa912ec057b5ddc": "(x_1, x_2, \\dots) \\mapsto (0, x_1, x_2, \\dots).", "841d1d61c3c0890229f1bfc61151e881": "a=1+2+3+\\cdots+n= \\frac{n(n+1)}{2}. ", "841d37bf115377ffff9ffd95c78c49dc": " \\cfrac{\\qquad }{ \\{p\\} \\vdash p}\n", "841d79f5740d5cf6d6e85b535ff3ae10": " \\mathcal{H} = \\sum_i \\dot{x}_i p_i - \\mathcal{L} = \\sum_i \\frac{ (p_i - e A_i)^2 } {2 m } + e \\phi. ", "841d9d2c567ee3676c7768f44ee7b9fc": "\n\\begin{align}\nf(x; 0, 1)\n&= { 1 \\over \\pi } {\\Gamma(5/3)} {}_2F_3(5/12,11/12;1/3,1/2,5/6;-4x^6/729) \\\\\n& {} \\quad{} - { x^2 \\over 3\\pi } {}_3F_4(3/4,1,5/4;2/3,5/6,7/6,4/3;-4x^6/729) \\\\\n& {} \\quad{} + { 7x^4 \\over 81\\pi } {\\Gamma(4/3)} {}_2F_3(13/12,19/12;7/6,3/2,5/3;-4x^6/729) ,\n\\end{align}\n", "841daec288cea111ec3eeed9166ed80f": "A = \\begin{bmatrix} 1 & 3 & 2 \\\\ 2 & 7 & 4 \\\\ 1 & 5 & 2\\end{bmatrix}.", "841dc4965af0f753c35e406221112472": "\\scriptstyle\\hat U/\\sqrt{2}", "841dd70659430496451d4360834c7dbf": "p_2 \\circ u=q_2", "841e485e1caeab15faaed0f8fd07adbc": "0 = (h-1)m + c .", "841e89932e77af82d500b27a29d865b3": "\\scriptstyle \\sqrt z", "841e8b3a86d9b6411dbb6ea1a26109cf": "\\tfrac{9}{16}", "841e9cc2e548f5d1e68d1835f3467916": "\\lambda x. sleep'(x)", "841f1be20ab2b9d2a1cf1ea463d39eb9": "F\\subseteq K", "841f2e4ed0bc7ae05341e0afe8591814": " Q^{n+1}_i = Q^n_i - \\frac{1}{\\Delta x } \\int_{ t^n }^{t^{n+1} } \\left( f( q( t, x_{i+1/2} ) ) - f( q( t, x_{i-1/2} ) ) \\right)\\, dt. ", "841f3709f2deb3e9c8cfaaa821217eab": "(\\exists x \\phi) \\land \\psi", "841f423cf96d2079814c4a881a94fd73": "x_{1}=a_{1}cos(\\theta)", "841f92628ac352664cad8dcd59767d5b": "I_0 := 1", "841fa77c17aaf7d610b718fe342be1cf": " L_i ", "841fdcfadcc9228a95abf00f4bf0c7a4": "I_{\\alpha\\beta\\gamma}", "8420106bd2db83dc9d888ca9263e1888": " \\nabla \\cdot \\mathbf{u} = 0 ", "842040e067e60d44bb5725394c74d1fc": "f''(x)=\\frac{g''(x)[h(x)]^2-2g'(x)h(x)h'(x)+g(x)[2[h'(x)]^2-h(x)h''(x)]}{[h(x)]^3}.", "842045c9b8252d88c0ee036ad1478876": "\n\\begin{align}\n\\mathcal{M}(n-1,n)&\\leq\\mathcal{M}(n,n+x)\\leq\\mathcal{M}(n,n+1)\\;\\;\\mathrm{when}\\;0< x\\leq 1\\\\\n\n\\frac{\\log\\left(\\Gamma(n)\\right)-\\log\\left(\\Gamma(n-1)\\right)}{n-(n-1)}&\\leq\n\\frac{\\log\\left(\\Gamma(n)\\right)-\\log\\left(\\Gamma(n+x)\\right)}{n-(n+x)}\\leq\n\\frac{\\log\\left(\\Gamma(n)\\right)-\\log\\left(\\Gamma(n+1)\\right)}{n-(n+1)}\\\\\n\\frac{\\log\\left((n-1)!\\right)-\\log\\left((n-2)!\\right)}{1}&\\leq\n\\frac{\\log\\left(\\Gamma(n+x)\\right)-\\log\\left((n-1)!\\right)}{x}\\leq\n\\frac{\\log\\left(n!\\right)-\\log\\left((n-1)!\\right)}{1}\\\\\n\\log\\left(\\frac{(n-1)!}{(n-2)!}\\right)&\\leq\n\\frac{\\log\\left(\\Gamma(n+x)\\right)-\\log\\left((n-1)!\\right)}{x}\\leq\n\\log\\left(\\frac{n!}{(n-1)!}\\right)\\\\\n\\log\\left(n-1\\right)&\\leq\n\\frac{\\log\\left(\\Gamma(n+x)\\right)-\\log\\left((n-1)!\\right)}{x}\\leq\n\\log\\left(n\\right)\\\\\nx\\cdot\\log\\left(n-1\\right)+\\log\\left((n-1)!\\right)&\\leq\n\\log\\left(\\Gamma(n+x)\\right)\\leq\nx\\cdot\\log\\left(n\\right)+\\log\\left((n-1)!\\right)\\\\\n\\log\\left((n-1)^x(n-1)!\\right)&\\leq\n\\log\\left(\\Gamma(n+x)\\right)\\leq\n\\log\\left(n^x(n-1)!\\right)\n\\end{align}\n", "842086c8d1602faa1181de2f3bb1bdda": "i,j,k\\in\\{-1,0,1,2\\}", "84209d1e1a7fa521ece791a309608032": "\\chi_1(n)=0", "8420ceb181caddec308f11482f104c37": "\\mathcal{F}, \\mathcal{M} ", "84213aa256479916eb942300a0b6f6bf": "y=\\pm\\sqrt{1-x^2}. \\, ", "842153c440b299876684cfd269dd87c9": " \\mathbf{\\bar y}, \\mathbf{\\bar y}' ", "84218e676e04a6b04e1bd976c9b7dee4": " \\varepsilon_2", "84219111c421e2715e0610b62997b0af": "\n \\underline{\\underline{\\mathbf{A}_1}} = \\begin{bmatrix}-1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\mathbf{A}_2}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\underline{\\underline{\\mathbf{A}_3}} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix}\n ", "842230cff157e288e768b32e1dadb951": "\\max(\\operatorname{sum}(S_1), \\operatorname{sum}(S_2)) \\le 4/3\\mathrm{OPT}", "8422a7d5d7ca11b31cb3e2bfa462f276": "\\frac{V}{\\sqrt{2E}} = \\left(2\\frac{M}{C}+\\frac{1}{3}\\right)^{-1/2}", "8422b99f7423637b923ce1da0d5533e1": "\\Gamma(X,\\mathcal F)", "8422c0561b3b63025da040954c44a0a8": "r=p(1) = 1+p_1+p_2+\\cdots+p_n", "8422d05f9870a2ea0a750604659832ae": "G_1/Z_1", "84230622774940e385ced780b1302689": "E = hf-\\phi", "842332c5fde62d78cd46e6aa132008e3": "\\mathbf{H}_\\alpha(x) - Y_\\alpha(x) \\rightarrow \n \\frac{1}{\\sqrt{\\pi}\\Gamma(\\alpha+\\frac{1}{2})} {\\left(\\frac{x}{2}\\right)}^{\\alpha-1}\n + O\\left({(x/2)}^{\\alpha-3}\\right) ", "84238b7c41c86dbb174b7d2f8aeb53a2": "f_{xyz} = N_3^c \\frac{xyz}{r^3} = \\frac{1}{i \\sqrt{2}}\\left(Y_3^2 - Y_3^{-2}\\right)", "8423fc4942186bc4961f03947a92d738": "\\begin{align}\n u &{}= f'(x) \\\\\n du &{}= f''(x)\\,dx \\\\\n dv &{}= P_1(x)\\,dx \\\\\n v &{}= \\frac{1}{2}P_2(x)\n\\end{align}", "84245a4ce2d0b38a2294faeab91b19c3": "L=D_x+D_y^2", "8424a5f5c0fad81ba419c44bbd43c991": "H(p) \\le L(p) \\le n H(p) ", "8425495c665c12b5e080b28a6fb9602a": "\\mathcal{J}_\\lambda", "842558e6923b58c80605b2d331b070de": "\\lambda b^*(f)=b^*(\\lambda f), \\,", "842562a32114418516af495318e61593": "\\displaystyle\\sum_{i=0}^n c_{ri}", "8425a8ccb2939e12390ec8f89f086a00": "S_{\\alpha \\beta \\gamma} = S_{\\alpha [\\beta \\gamma]}", "842615085a6202543fa3cd845490ddd0": "T^a = \\frac{\\lambda^a }{2}.\\,", "842647221504c643c9400798d273229c": " \\det \\mathbf{A}={{\\prod_{i=2}^n \\prod_{j=1}^{i-1} (x_i-x_j)(y_j-y_i)}\\over {\\prod_{i=1}^n \\prod_{j=1}^n (x_i-y_j)}}", "84270173cc033b219279e61f732d90c5": "u \\in K_M \\Rightarrow u1 \\in A", "842704e569f87e7f268e159121c841d3": "\\mathrm{Re}(\\tilde{\\epsilon}/\\epsilon_0) = \\frac{c^2}{(\\omega^2)(\\mu/\\mu_0)}(k^2-\\frac{\\alpha_{abs}^2}{4})", "842709873f1e53513df623724fad3489": "G_{\\frac{\\lambda}{2}}\\,\\!", "84278b601de6f2353dba336db2d7cbbc": "\\scriptstyle{AB_{i}=Rt_{i}-Rc_{i}}", "84278dea31c944fad5b16607c1b5d93f": "\\mathbf{E}_{inc}= \\sum_{n=1}^\\infty \\sum_{m=-n}^n a_{mn} \\mathbf{M}^1_{mn}+ b_{mn} \\mathbf{N}^1_{mn}.", "842798d888922e99b2f94eec4287641c": "\\delta\\phi\\,", "8428364ab53bd99a875b3ffb307a8b20": "r_\\mathrm{min}=\\frac{p}{1+\\varepsilon}.", "84284fa514562a74c2621c19be921571": "d^n : C^n (G,M) \\rightarrow C^{n+1}(G,M) ", "8428b668b2df3cc4289de7d60d6d1fe1": "\\text{Doppler Frequency} = \\left (\\frac { 2 \\times \\text{Transmit Frequency} \\times \\text{Range Velocity}}{C} \\right)", "8428b800920d49d1c3306a50f6dd48aa": "\\mu^{2}>0", "8428c642eb90d45202eabd6dc81ac023": "j a +b", "8428ccce453183ed111c77a60ab7b383": "f\\colon S^2 \\to M,", "8428e0ca5ef7b525360cf33cf72b8288": "\\omega = \\omega_k", "84290e7b9711c81414acfc548025dd28": "p_3 \\equiv 7 \\pmod{8}", "8429130b5521403bc932513d8206cef8": "r_2 = \\frac{r}{2(s-b)}(s+e-r-d-f),", "8429183498d83b61ace3a81dbd2c0977": "{z_1^2 z_2^2 \\cdots z_k^2 \\equiv \\prod_{p_i\\in P} p_i^{a_{i,1}+a_{i,2}+\\cdots+a_{i,k}}\\ \\pmod{N}\\quad (\\text{where } a_{i,1}+a_{i,2}+\\cdots+a_{i,k} \\equiv 0\\pmod{2}) }", "84294f99da9a4431aed69266dd3ed406": "|F(k)|^2=|F(-k)|^2 \\,", "842958412b0f9a50127d73086c1328ed": "\\bigcup\\nolimits_k \\operatorname{supp}(\\varphi_k)\\subset K.", "842977e25cdd251321f1cbafcfd0ac4f": "\\left(-\\frac{\\partial V}{\\partial t} - \\frac{1}{2}\\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2}\\right)\\Delta t = r\\left(-V + S\\frac{\\partial V}{\\partial S}\\right)\\Delta t", "84297b8adbacca9a5bc15f78ba2305bf": " n + 1 ", "8429905126e5fe992782821165514410": "\\displaystyle{ \\Phi(t)=\\Pi(t)v}", "842a44d5672e3d098d93afd1de452829": "\\theta_X=dx^\\mu\\otimes \\partial_\\mu", "842a45387333d0d80c3aa6e5ec4bb440": "\\sum_{n=2}^{\\infty}1/\\prod_{i=2}^{n}S(i)\\approx 0.719960700043\\ldots", "842a7b1f09711a563c2ff463f5359ec3": "2\\le i\\le F", "842a9eb91479ebef72e937647fe97431": "p_{1},\\ p_{2},\\ p_{n}", "842ac55f9df40f159f83b518ffcb59f0": "f(R)", "842acd4f277cbfbc9dd9853d4e70ca3d": "S^+_i", "842b2c82cb12c607bd90126df6d4698d": "\\text{Gal}(L/\\mathbf{Q}_p)=(R \\rtimes Q) \\rtimes P", "842b455731528e62580c23ac1262b3ef": "t = \\frac{\\mu - \\bar{x}}{s / \\sqrt{n}}", "842b672c15ff57e3dc0a0957560aa10e": "(x)=-1", "842b841ce24d7b0f9b0a5878fd6b482a": "x \\in I,", "842bc513a0ad2adfe6efc8fbd4c8ff98": " 0\\leq i \\leq j \\leq n", "842bf24d792f989745f58a3171c67bed": "\\frac{u_{i}^{n + 1} - u_{i}^{n}}{\\Delta t} = \nF_{i}^{n + 1}\\left(u,\\, x,\\, t,\\, \\frac{\\partial u}{\\partial x},\\, \\frac{\\partial^2 u}{\\partial x^2}\\right) \\qquad \\mbox{(backward Euler)}", "842c2d7137e3f693182124c3c150fd0f": "s\\approx{R_r^'/X}", "842c82f435340f319f423aa50380dcaf": "\\mathbb F_n", "842c8a485fce198e1a90e5bd940b77aa": "x \\odot a_j", "842ca08f875185c282527a31cdb857a4": "\nP^\\star_{\\mathbf{k}} = \\langle \\hat{a}^\\dagger_{c, \\mathbf{k}} \\hat{a}_{v, \\mathbf{k}} \\rangle \\,,\n\\qquad\nP_{\\mathbf{k}} = \\langle \\hat{a}^\\dagger_{v, \\mathbf{k}} \\hat{a}_{c, \\mathbf{k}} \\rangle \\,,\n", "842ce7d24ed12f74427282f256bd1db0": "\\left(2+\\sqrt{-6}\\right)^4 = \\left(-2+4\\sqrt{-6}\\right)^2 = -1-3\\sqrt{-6} .", "842d22513a5e066050d638d313c890a8": "6 \\sqrt{2} \\approx 8.48", "842d234001825d2a956d7d2dc2eddef8": "U(t) = e^{\\frac{-iHt}{\\hbar}} = e^{\\frac{-iat}{\\hbar}} (cos(|\\mathbf{r}|)\\sigma_0 + sin(|\\mathbf{r}|)\\hat{r}\\cdot\\mathbf{\\sigma});", "842d89e833343c70d28f120d59a10370": "F_{BH} \\; = \\; \\left( \\; \\frac{h_B}{100} \\; \\right)^2", "842df3fb83e3d8bf357208bcd1460079": "r \\approx i-p\\,\\!", "842e15e4a0a6a823c5f011b79c9b3aa3": " \\bar X \\xrightarrow{n \\to \\infty} N(k, 2\\cdot k /n ) ", "842e21403431fecdf5c15c63e505992b": "\\|P_n-P\\|_\\mathcal{F}=\\sup_{f\\in\\mathcal{F}}|P_nf-\\mathbb{E}f|\\to 0.", "842e4ba4c83a690d1ce988b13421aa26": " X_1,\\dots,X_n : \\Omega \\to \\mathbb{R}, \\,", "842e523cdb842a8e3c64f71d99a5d79f": "\\varphi(p)", "842e52fc5a072cc9ad515a71f7860e8a": "2\\|x\\|^2+2\\|y\\|^2=\\|x+y\\|^2+\\|x-y\\|^2. \\, ", "842eb0bb94f4a7635cc651df441fb839": "\\widehat{\\varepsilon}", "842ebaf1c8dfbb687c02be5259231a73": "d (1 + \\cos \\alpha)", "842f1625998382b242715412d7d554a2": "x_1,x_i", "842f1b544158dd3f6679bb88ab9060f9": "\\mathrm{abs}(a-b)", "842f2fe2586aad39888e6ea1eb655006": "\\delta _\\kappa^2=\\frac{2\\kappa V_m}{\\omega C_p}.", "842f3f9563fee9619ae14c7812fa03dd": " P = \\begin{bmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}, \\qquad Q = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix}, \\qquad \nz= \\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}, ", "842f5edd8d2a3fa49049997d6803bead": "Q_H = Q_C + W_{in} \\,", "842f7c36d1a22d5c2158e9a1cd161381": "K_w = [H^+][OH^-]\\,", "842fa03ab2bbf1c9b9162c30ebe8d636": "\\sum_{x\\in R}\\left\\vert F(x+h)-G(x) \\right\\vert", "842fa48178744c0e17ffc54ecfe0f0f8": "r_{\\rm PT}", "842fcc4f94bb7cc9c3ac9d636d9c8d40": " \\det \\left(-\\frac{d^2}{dx^2} + A\\right) \\qquad (x\\in[0,L]), ", "842fcfa876e86766dfba4c163c400a64": "\\tan \\frac{\\pi}{4} = \\tan 45^\\circ = { {\\sin \\frac{\\pi}{4} } \\over {\\cos \\frac{\\pi}{4} } }= {1 \\over \\sqrt2} \\cdot {\\sqrt2 \\over 1} = {\\sqrt2 \\over \\sqrt2} = 1. \\,", "842fd5e5775ee971d5175b1294c3736c": "\\mathcal{R}=(\\mathbb{R},0,1,+,\\cdot)", "843072b9ca118a7f10a0500cf41d3e19": "7\\times 2", "84308375b34be8f7148e71e7abaff0da": "\\bar{R} = \\frac{(1+\\alpha)^n}{(1+\\alpha)^n+L(1+c\\alpha)^n}", "8430a5fd5a67469a5e084e558b9ef5af": "k[t]_{(t-a)}/(t-a)k[t]_{(t-a)} \\cong k", "84319dd9f88e7f9727d8fd73a4de031b": "\\mathrm{d}\\,{\\star \\bold{F}} = {\\star \\bold{J}} ", "8431bb3a8e0a3e4046264c30b95cf7a8": "\\Beta(\\tfrac{1}{2}, \\tfrac{1}{2}) = \\frac{1}{\\pi \\sqrt{p(1-p)}}.", "8431ccea697303d512bf8322051aec92": "n! \\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.", "84326d47666dcf8139ed8b7e7a4e6241": "(f_\\bullet)_*:H_\\bullet(A_\\bullet, d_{A,\\bullet}) \\rightarrow H_\\bullet(B_\\bullet, d_{B,\\bullet})", "84327da4fe139f3b958f817acfe22837": "E(e) = L_e(1) + L_e(2) + \\cdots + L_e(x)", "8432b61c5ec3167cb873361b8dd7a410": " M^n \\mathbf{e}_1 = M^n \\mathbf{u} = a^n \\mathbf{e}_1,", "8432f4d349cf8b5a00f2c87db26eda7d": "L(G) = \\left \\{ a^{n}b^{n}c^{n} | n \\ge 1 \\right \\}", "843344c13012405081501ea91a1d48da": "\nM_{1} =m_{1} \\cosh \\mathcal{L} +m_{2}\\sinh \\mathcal{L}, ", "8433baf5e37a14901aefb55d363de765": "\nl = \\frac{1}{\\beta} \\left[(n+1)\\pi - \\arccot\\left(\\frac{\\omega L}{Z_0}\\right) \\right]\n", "8433c4f70adbde40ade5a1fe51fe95b8": "H(f)(x) = \\frac{1}{\\pi}\\lim_{\\varepsilon \\to 0} \\int_{|x-y|>\\varepsilon} \\frac{1}{x-y}f(y) \\, dy. ", "84341cba3b26468f76d4aeb011a458c9": "f\\!\\left(x\\right) \\leq f\\!\\left(y\\right)", "84346a3e650ed932272789c8ca9fe207": "x * (y + z) = (x * y) + (x * z)", "84349e71f0cddefab8f1cad3e45b7aac": "A_{R_A} = A", "84349f9141c13140e24eb69d71da59d9": "\\psi^* = \\sqrt{1-\\alpha^2} \\vert M_d \\rangle - \\alpha \\vert L_{np} \\rangle ", "843520667ed60aa6268b0b13c5562eae": " u(x,t) = X(x) T(t).", "843562e71bf2e8f1f09167037efe1c69": "\\lim_{n\\rightarrow\\infty}\\; \\frac{1}{n} \\sum_{k=0}^{n-1} f\\left(T^k x\\right) = \\frac 1{\\mu(X)}\\int f\\,d\\mu ", "8435665c2bcb5996d960d3dd7b35b13d": "\\displaystyle w_t=3u_xw+6uw_x-4w_{xxx}", "84359c8f33cf5d07e4f014528da82117": "M \\rarr T", "8435a3d65a613c6ca2b23659bac21a3a": " \\frac{d\\rho}{dr} = -\\frac{\\rho(r)g(r)}{\\Phi(r)}.", "8435ad1102edb514be4f46bddb51d744": "\\begin{matrix} \\frac{27}{25} \\end{matrix}", "8435e378d64e33b9b3ab9c9376e070e0": "\nNS_i = NSE_i + NSD_i\n", "8435fc4d57308d0699ff3485f24f5953": "\\operatorname{dim}[C(\\cdot)] = q \\times n", "8436110492d3b4ae4d27fe114fb2a677": "\\,E_{\\alpha}", "84363fdfb5b1f4bc2471f6d308e67faa": "\\frac{{}_1F_1(a+1;b+1;z)}{{}_1F_1(a;b;z)} = \\cfrac{1}{1 + \\cfrac{\\frac{a-b}{b(b+1)} z}{1 + \\cfrac{\\frac{a+1}{(b+1)(b+2)} z}{1 + \\cfrac{\\frac{a-b-1}{(b+2)(b+3)} z}{1 + \\cfrac{\\frac{a+2}{(b+3)(b+4)} z}{1 + {}\\ddots}}}}}", "84367e773dc121817468a7f672b7415a": "d(\\vec x_i,\\vec x_j)=(\\vec x_i-\\vec x_j)^\\top\\mathbf{M}(\\vec x_i-\\vec x_j)", "8436bbf4fde01ca35e9373eb2bb53798": "\\mathbf{x}^{(n+1)} = k_1\\dot{\\mathbf{x}} +\\cdots + k_{n-1}\\mathbf{x}^{(n-1)}.", "8437266c3c2c7e8e23e52a31b6f62113": " \\sum_{k=0}^{\\lfloor\\frac{n}{2}\\rfloor} \\tbinom {n-k} k = F(n+1).", "84372bdafd9a98bb3e3eb47dcd54d116": "\n\\begin{align}\n\\cos x\\cdot \\cos y & = \\frac{(e^{ix}+e^{-ix})}{2} \\cdot \\frac{(e^{iy}+e^{-iy})}{2} \\\\\n& = \\frac{1}{2}\\cdot \\frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\\\\n& = \\frac{1}{2} \\bigg[ \\underbrace{ \\frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\\cos(x+y)} + \\underbrace{ \\frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\\cos(x-y)} \\bigg] \\ \n\\end{align}\n", "8437936d4a9554ef2f1eb2deb6a99f46": "n/K", "8437f1ec1a6726a1570b5f7d678c6520": "\n \\sigma_{11} = 2~\\left(\\lambda^2 - \\cfrac{1}{\\lambda}\\right)~\\cfrac{\\partial W}{\\partial I_1}~.\n ", "8438597b950374f51930b80fb2815cbb": "\nS_{\\alpha \\beta \\gamma} = S_{\\beta \\alpha \\gamma} = - S_{\\beta \\gamma \\alpha} = - S_{\\gamma \\beta \\alpha} = S_{\\gamma \\alpha \\beta} = S_{\\alpha \\gamma \\beta} = - S_{\\alpha \\beta \\gamma}\n", "8438771a89a2b979004453cd1a63ef46": "\\frac{1}{L(\\chi,s)}=\\sum_{n=1}^{\\infty} \\frac{\\mu(n)\\chi(n)}{n^s}", "843882542054a2922a88f0334dd4f6db": "\\scriptstyle\\Delta t ", "8438a52bb5507401b812dcdd0b8cf4e1": "\\vec{M} = \\sum \\mu_i", "8438da4183559533acceb537c6652971": " \\log: V\\subset G\\rightarrow U\\subset g. \\, ", "843902c7faf89d6435368d727f9da1bd": "rp_n", "8439aaca27b5fcbf648ea400bd154585": "B\\cap S=U", "8439e216e134531f649d94db02b23830": "\\scriptstyle \\mathbf{M} = \\boldsymbol{\\mu}/V = M_s \\left(\\alpha,\\beta,\\gamma\\right)", "8439ec551d6e79e915921cb96d7ff6b2": "j \\in C_j", "843a332516aa62370920a51350522acd": "t(G)=\\frac{1}{n} \\lambda_1\\lambda_2\\cdots\\lambda_{n-1}\\,.", "843a46c5c7b4767c8ef8a69addde6e88": "n^*=0", "843a4efa3249edec5af2e189aa466c6d": "A(x) = - \\int {1\\over e^{-4x}} xe^{-2x} \\cosh{x}\\,dx = - \\int xe^{2x}\\cosh{x}\\,dx = -{1\\over 18}e^x(9(x-1)+e^{2x}(3x-1))+C_1", "843a718aa7f75284fccaadd9e2584998": "\nh(78) =\n \\mathrm{round}\n \\left(\n \\frac {46 - 1} {63}\n \\times 255\n \\right)\n=\n \\mathrm{round}\n \\left(\n 0.714286\n \\times 255\n \\right)\n=\n182\n", "843b3571641ac4b8593b1f1309a05b08": "2\\theta=2\\cos^{-1}(|t|)", "843b5b84307b8ebb4662b226d6e43fa1": "\\left.\\frac{\\partial f(\\boldsymbol{x})}{\\partial x_i}\\right|_{\\boldsymbol{x} = \\boldsymbol{a}} = A_i (\\boldsymbol{a}) ", "843b6448d16b5258b7f8b055c64a10df": " s^2 F(s) - s f(0) - f'(0) \\ ", "843b6fe08647502408e40c131151afca": "\\alpha_3=a+c+f;\\quad \\beta_3=b+c+e+f;", "843bb4eca98673340ddd094dfa443a86": "\\int\\frac{\\mathrm{d}x}{\\sin ax\\cos^n ax} = \\frac{1}{a(n-1)\\cos^{n-1} ax}+\\int\\frac{\\mathrm{d}x}{\\sin ax\\cos^{n-2} ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "843c26ab4ac73ac29a1bcb5a51937afb": "D_{ob}", "843cb6b27f4d7ef98183e4a536362a9f": "\\mathbf{p}_r\\in \\mathbb{R}^3", "843cc22bc5c6d890c06132b4ec561ca9": "\\nu \\in \\mathcal{O}_k ", "843d122cfd390b3bfdc7b1f30821e8a1": "\\omega^3", "843d26ccab464a0462e9f5ebee044cee": "\nr_{s} = \\frac{2GM}{c^{2}}\n", "843d2f6f30ba9f378e26518ee13f7613": "\\frac{\\partial}{\\partial x} \\left[ K_{xx} \\frac{\\partial \\phi}{\\partial x} \\right] + \\frac{\\partial}{\\partial y} \\left[ K_{yy} \\frac{\\partial \\phi}{\\partial y} \\right] + \\frac{\\partial}{\\partial z} \\left[ K_{zz} \\frac{\\partial \\phi}{\\partial z} \\right] = S_{S} \\frac{\\partial \\phi}{\\partial t} - q", "843d434ed66f06994110d93f8b7b4085": "s(x)=x+xs(x)+x^2s(x)", "843d4b82c314514dd7fd24b03f71a94c": "{\\mathcal K}_n(x) = x^{-n/2} K_n(2 \\sqrt{x}).", "843da2b9ef0a3c9d2d67f103654be122": "=4\\pi (n-l)! (2\\zeta)^n (ik/\\zeta)^l Y_l^m({\\mathbf{k}}) \\sum_{s=0}^{\\lfloor(n-l)/2\\rfloor} \\frac{\\omega_s^{nl}}{(k^2+\\zeta^2)^{n+1-s}}", "843dc8a1a0ad53c2687cc1b3aa7390b6": "\\left( \\hat{x},\\hat{y},\\hat{z},\\hat{b} \\right) = \\underset{\\left( x,y,z,b \\right)}{\\arg \\min} \\sum_i \\left( \\sqrt{(x-x_i)^2 + (y-y_i)^2 + (z-z_i)^2}- bc - p_i \\right)^2", "843e04e7d3e53ab7b14e911efb9fb48e": "\\left\\lceil\\frac{|S|}{r(S)}\\right\\rceil=\\left\\lceil\\frac{kr(S)+1}{r(S)}\\right\\rceil=k+1", "843e9a8e257ff01eb50f5a487343b4b3": "\\Omega_{1} \\subseteq \\Omega_{2} \\subseteq \\dots", "843ea07b20f96a476f3638064f207b96": "\\frac{1}{1 - z + {\\scriptstyle\\frac{1}{2}}z^2}", "843f287c93734d4d7fd266d22d810610": "~A \\leftrightarrow B \\leftrightarrow C~~\\Leftrightarrow", "843f3d2f92506e7ba303d6f4ab39f0e8": "\\eta\\left(-\\frac{5\\xi_1^2-\\xi_2^2}{2\\xi_1^2(\\xi_1^2-\\xi_2^2)}c_\\eta(0,\\xi_1) - \\frac{n_\\eta^\\prime(\\xi_1)}{2\\xi_1^2}\\right)+(1\\leftrightarrow 2)", "843f59a6d7d2cf4e0a21257a47550df9": "\\begin{bmatrix}\\,\\,\\,2 & 3 & 5 \\\\ -4 & 2 & 3\\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z\\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}.", "843f81dac04c803e20ffa0db891e0a89": "R_F(x,y,z)=2R_F(x+\\lambda,y+\\lambda,z+\\lambda)=\nR_F\\left(\\frac{x+\\lambda}{4},\\frac{y+\\lambda}{4},\\frac{z+\\lambda}{4}\\right),", "843fac37ceaf3ba689b513a5d7a65ab0": " \\Delta h_{meters} = 0.067 D_{km} ^2 ", "843fc014c78e475512d34d518af824c6": "\\flat", "843fd50f4b6c9905719bcc112cf59459": "\\delta_{ext}", "84403da30f36a8ad11a749aa39d5b69e": "u(A_1,\\dots,A_n) \\preceq u(B_1,\\dots,B_n)", "8440428384c1b05ef097efba7f8bbc98": "1\\le x_{3},x_{5} \\le 5", "8440572137c571fa34a5ec20de16253a": " {\\zeta_g} = {\\hat{k} \\cdot \\nabla \\times \\overrightarrow{V_g}}", "844058c7c0eab67606df84aa9e2351a2": " \\widehat{\\boldsymbol{\\beta}}_{ols}", "84406453c2f2958580b89d00edc2f0f2": "{AE}_{9}", "8440680fc9f6f51a342441f643c41566": "\n\\mathbf{r}^{\\prime} = \\mathbf{r} - \\mathbf{r}_{0} - \\mathbf{v} t\n", "84409b3ea5c4950fe507bf1cdf92f4ec": "1.2\\cdot10^{-7}", "8440bdee473bfb8725a9844d5304adee": "\\hat{\\phi}_j", "8440f994ee2c9b4b8e79b98e73ed505d": "\\sqrt{\\hat{V}(d)}", "8440fc1f073697f2ddb6a7d82d6ed094": "\\mathbf u_0(t)", "844117277297261d037db21c803a0fe1": " \\eta_C =\n\\begin{cases}\n \\frac{T_{cold}} {T_{warm} - T_{cold}}, & \\mbox{if } T_{cold} < T_{warm} - T_{cold} \\\\\n 1, & \\mbox{otherwise}\n\\end{cases}\n", "844152b9085a964da851f8c2dfabc026": "\\left\\{{n \\atop 1}\\right\\} < \\left\\{{n \\atop 2}\\right\\} < \\cdots < \\left\\{{n \\atop K_n}\\right\\},", "84417eefcbddcea82295b8f9a39d74ec": "\\Gamma_{k}", "844183583df6e8f4337776a0af576a10": "\n\\wp(z;\\omega_1,\\omega_2)=\\frac{1}{z^2}+\n\\sum_{n^2+m^2 \\ne 0}\n\\left\\{\n\\frac{1}{(z+m\\omega_1+n\\omega_2)^2}-\n\\frac{1}{\\left(m\\omega_1+n\\omega_2\\right)^2}\n\\right\\}.\n", "84420d54f487ea8eea1c78fed4110735": "X \\sim {\\rm Arcsine}(a,b) \\ \\text{then } kX+c \\sim {\\rm Arcsine}(ak+c,bk+c) ", "844220befc45742f2d55e6457f3c7200": "S_{xx}(\\omega)=\\int_{-\\infty}^\\infty \\,\\gamma(\\tau)\\,e^{-i\\omega\\tau}\\,d \\tau=\\hat \\gamma(\\omega). ", "844234c0f2a312ef4a38d0aef6b3d143": "SO(n,1)", "8442a5e0c14360520e7814bf3aeeca54": "A= (a_{ij})", "8442c6d81106f1c8c5a45b48a36eaa38": "\\tan\\lambda = \\frac{Y_r}{X_r}", "844310625ff642d50d920ced1f40641f": "x=\\pm\\sqrt{y}", "844318dd34053c3da3dbbe6af298e221": " \\hat p ", "84435aaf71303c67310a0fe82b863147": "I = C \\frac{dV}{dt}", "8443c7771db44e36a99a523f029a8fde": "w_{3}=.5714", "84441ee8034ea041e109f2b717b31d9c": "\n\\lambda = \\sqrt \\frac{r_m}{r_l}\n", "84445342d2c02ea08ef064eddcb31ca7": "s = \\|\\mathbf r \\|^2 = \\mathbf r^{\\rm T} \\mathbf r = \\mathbf r^{\\rm T} Q Q^{\\rm T} \\mathbf r = \\mathbf u^{\\rm T} \\mathbf u + \\mathbf v^{\\rm T} \\mathbf v ", "8444bbcacb14dd1ec67351341144c73c": "u \\ge c", "8444f7e712d17f771eca2f04932e9aa8": "\\bar{t}t", "844502847888c3c652e00c8dd51cebdf": "i^2 = (e_1 e_2 e_3)^2 \n= e_1 e_2 e_3 e_1 e_2 e_3\n= - e_1 e_2 e_1 e_3 e_2 e_3\n= e_1 e_1 e_2 e_3 e_2 e_3\n= - e_3 e_2 e_2 e_3\n= -1", "84454735735c9c2554df1d9fba5e35fb": " \\int_{-\\infty}^\\infty \\left| G_x(t,f) \\right|^2\\,df = \\int_{-\\infty}^\\infty e^{-2\\pi (\\tau-t)^2}\\left| x(\\tau) \\right|^2 d\\tau \\approx \\int_{u-1.9143}^{u+1.9143}e^{-2\\pi (\\tau-u)^2}\\left| x(\\tau) \\right|^2 d\\tau ", "8445630e7d884dce57c4e1ceb2b19637": "F_\\nu(H)=\\bigoplus_{n=0}^{\\infty}S_\\nu H^{\\otimes n} =\\mathbb{C} \\oplus H \\oplus \\left(S_\\nu \\left(H \\otimes H\\right)\\right) \\oplus \\left(S_\\nu \\left( H \\otimes H \\otimes H\\right)\\right) \\oplus \\ldots", "84459602d991e44e84359a5b1196ea8e": "\\frac{1}{z} = z^{-1} = z^{n-1} = \\bar z.", "8445c1b5faa10677cfc4d8c49d0770c9": "(2)^e = (1 + i)^2", "8446923779b6ab0d27a0c5f710272ed2": "x + \\dot{x} > 0", "8446b5faf592598a449ddf25ba724774": "s-s' = k^{-1}(z-z')", "8446c09fa182ab2890f4de17acbe7e73": "C = 2 \\pi r = \\pi d\\!", "84470ba814e3bd0693b88853a23f44e5": "\\displaystyle{\\partial_{\\overline{z}}F =0.}", "8447379bf3121e4cbbcd5690052611eb": "h(v) = sgn(v \\cdot r)", "8447c74fffdb3cadc8e91356180b67c3": " p = \\rho g h", "84481b59173771cfc6f1f77237ac51fd": "a_{n+2} = a_{n+1} + 1", "84484c86b6e34b851cc96dc5e4bd2b81": "A \\rightarrow aA", "8448c6f28276cddd5881dcb2a2e4e377": "\n\\begin{pmatrix}\n x' \\\\\n y'\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n \\cos \\theta & - \\sin \\theta \\\\\n \\sin \\theta & \\cos \\theta \n\\end{pmatrix}\n\n\\begin{pmatrix}\n x \\\\\n y\n\\end{pmatrix} \n", "8448c7015fe91f3f79819363ba9385b5": " \\frac{1}{2\\pi i} \\int_C\\frac{\\phi(\\zeta)d\\zeta}{\\zeta-z}, ", "8448e72dd7be1547ba8b1dbafe082234": "O(n^{\\rho}(kt+d))", "84496c155ac42ee2213b1ae5b94b689a": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot (26.73\\cdot 3.99)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 22.9\\cdot R_{\\bigodot}\n\\end{align}", "84497e54fd0da0bbc161550a6fc45ec4": "F_M : Mod(R) \\to Mod(R), \\quad N \\mapsto M \\otimes_R N", "84499fe8814ee8c61cb5e2bcc1697738": "y = \\frac{h}{b^n} x^n", "8449c390d6236e0f7f04e5b12368c024": "\\phi_k\\,", "8449e566754db8927626eba106db8ce9": "\\zeta_\\mathrm{r} = 8 \\pi \\eta r^3", "844a0a2fe24d8024dbef28556c019812": "J(k) = \\int h(x) e^{i k f(x)} \\, dx", "844a0ea38dca00c3c4ee4c8610f6e0d8": "\\epsilon_\\mathrm{reversible} = \\frac {\\sigma_0} E, ", "844a1055df4f08de5e146c93251b24be": "\n( \\mathbf{I} - \\mathbf{X} ( \\mathbf{X}^T \\mathbf{X} ) ^{-1} \\mathbf{X}^T ) \\mathbf{A},\n", "844a44ea46785d61302e14cc6060de73": "-552\\pm 80", "844a749fd669b8e4bf7ce10c579850ed": "even(s(s(0)))\\leftarrow \\hbox{not } even(s(0))", "844a8afe76185a47d6bbaf65664e6b30": "\\begin{matrix} {2 \\choose 2}{3 \\choose 2}^2 \\end{matrix}", "844aaca57cbd194e3777cd7a5bdaf004": "\\triangleleft", "844ac9c180615e9b377eaf80ab7199e6": "\\Sigma^0_{m+1}", "844ad3781c0df790371b38df6bd8e187": "\\tan\\theta_\\mathrm{c}=\\frac{|V_{us}|}{|V_{ud}|}=\\frac{0.22534}{0.97427} \\rarr \\theta_\\mathrm{c}= ~13.02^\\circ. ", "844b05331b81029e9e3fa1aab8ae91d0": "\\kappa^+ = |\\inf \\{ \\lambda \\in ON \\ |\\ \\kappa < |\\lambda| \\}|", "844b2af23a45ec59c1142df549123719": " (\\Delta \\otimes \\mathrm{id})\\Delta (h)=h_{(1)}\\otimes h_{(2)}\n\\otimes h_{(3)} \\in H\\otimes H\\otimes H", "844bd5cf4694b5e54162f85b9c808aab": "{\\rm blanc}(x)= {\\rm blanc}(2x)/2+s(x)", "844bde09fe3107c2e0560df694b96cc5": "\\sum_x q_x c(a, x) \\leq C", "844c29c862babf02bce889ed7da73407": " =e^{j(k_x x + k_y y)} e^{\\pm j z \\sqrt{k^2-k_x^2-k_y^2} }", "844c2fb563b7cb55409d58ec87397e48": "f(x) = \\rho(x)e^{j\\theta(x)} \\quad (4) \\,", "844c5f391cea5a2385ac4996462aeb9f": " \n\\Omega_{Z,[t_l,t_u]}=\\{(z,t)^*| z \\in Z \\cup \\{\\epsilon\\}, t \\in [t_l, t_u] \\}\n ", "844c6b701f84f4b3cbb10df0c2763eca": "| \\phi_i \\rang", "844c72ca5ed1c6172a9e4909302b4ff1": "f: A \\rarr B", "844c7fac61139d40311f054bfaea8f89": "\\phi_4", "844ca1b153565941d2befb572be7f671": " \\frac{dx(t)}{dt} \\ = Ax(t) + Bu(t)\\,\\!", "844cba6bb1bfa662383a5075683fbf6e": "S = \\sum_{i=1}^n(x_i-\\bar{x})^2.", "844ce3558a77d476c19dba5436ed9fc0": "\\sigma_{XX}", "844d438374cbfe9a807d6ae07baa972f": "a_W", "844daa64d775ca322d55545290c4eaec": "\\rho = \\rho_0 e^{-y / H} \\,,", "844e3207bc349228c76a6208d7c60dee": "\\mathrm{SL}(3,\\mathbb C)\\times SL(3,\\mathbb C)", "844e87b31b501e0931f8a11d473e4b88": "\\scriptstyle f(x) = a", "844e8cee2925d52573f69f1998e47b60": "4.2) \\ \\mbox{Adopters}\\ += \\mbox{New adopters }\\ ", "844ed6c378b01d3a33ea337ace170bb2": "\\mathbf{c} = c_1^1c_1^2\\cdots c_1^{n_T}c_2^1c_2^2\\cdots c_2^{n_T}\\cdots c_T^1c_T^2\\cdots c_T^{n_T}", "844f329b2a2fdbb64bc4ed004cacf094": " \\Re[S(x)] = 0, \\, \\forall x \\in \\mathbb{R}^n ", "844f4ae2a53abbeb45a72acc8c9e8416": "\\kappa \\sim 8\\cdot 10^{-7}", "844f508ca08a59b44d5cff4505770818": "\\kappa:= -m^al^b\\nabla_b l_a\\,\\hat{=}\\,0", "844f75522304f3ccd49e9c5f03b2ae8c": "010^{10}", "845bb3587e79a2610f38c1d79a82a6bb": "H^i(X, \\mathcal{F}) \\simeq H^{n-i}(X, \\mathcal{F}^\\vee \\otimes \\omega_X)'", "845bcdf16c1eff2eb69df2b717ce1e7c": "\\dot V=C(p_1-p_2)", "845be6203533f9666b553cd1375ffebc": " \\hat{t}", "845bee581ab780f721e2f2cf14aafcab": "D_{2N}", "845bf5e44d8e37db52c2f6d274adfb84": "\\scriptstyle D^2\\,", "845c2e3b70eeb96ae2a06c4eb1b32e57": "\\hat{T}_n = - \\sum_i \\frac{\\hbar^2}{2 M_i} \\nabla^2_{\\mathbf{R}_i} ", "845c66ad13aacb89f4fc723adf8dbb41": "Y \\sim \\Beta(\\alpha, \\beta, a, c).", "845ca4353384685b926fd7f804b4cc15": "\\tfrac12, \\tfrac13", "845cad42779f43f592267b3d811e4d8f": "N=92", "845cdefe767e3f7b1e904b5b5ea92746": "(y,z)\\in T", "845d5e5eb8d9eb4b9e26cd613fa0f333": " \\sum_{n=1}^\\infty a_n ", "845daa16e2c71979a4356b8371c65124": "s \\models_K f_1", "845e016fd270b45b75efa29fc4a4f643": "E = U \\frac{(c_{y2} + c_{y3})}{\\frac{1}{2} c_{2}} ", "845e0b4f8920c93470906956cbe434a0": "p = (x^3 - y^3) / (x - y)", "845e6ab111bca3c5d2a045ddd28783e3": " y < x ", "845e8d03b813362739598a56d915a109": "P(T|B)", "845ec4d27d94bb21a01613efea3108ba": "\\mathbf{\\left( J^TWJ +\\lambda I \\right)\\Delta \\boldsymbol \\beta=\\left( J^TW \\right) \\Delta y}", "845f4c816281a5f167afef1db8f239c8": " \\tfrac29 ", "845f873d460a05a9a2cc1ad2fac5fd9c": "+\\frac{V_{nn}V_{k_1k_2}V_{k_3 n}V_{k_2 k_3}}{E_{k_1 n}E_{nk_3}E_{k_2 n}}\\left(\\frac{1}{E_{nk_3}}+\\frac{1}{E_{k_2 n}}+\\frac{1}{E_{k_1 n}}\\right)+\\frac{|V_{k_2 n}|^2V_{k_1k_3}}{E_{nk_2}E_{k_1 n}}\\left(\\frac{V_{k_3 n}}{E_{nk_1}E_{nk_3}}-\\frac{V_{k_3k_1}}{E_{k_3k_1}^2}\\right)", "8460c72284692beb84f33bd1b7070ac3": "(x + 1)^{3}p(\\frac{1}{x+1}) = 7x^3-7x^2-35x+43", "8460fce7ce512a3662f24a6aca7fb035": "\\mathcal{M}\\left\\{\\prod_{i=1}^n\\Lambda_i\\right\\}=\\underset{1\\le i\\le n}{\\operatorname{min} }\\mathcal{M}_i\\{\\Lambda_i\\}", "846146039f65a51e7cb7b94031e1fb3e": "\\begin{matrix}\n & \\text{n} & \\rightarrow & \\text{p} & + & \\text{e}^- & + & \\bar{\\nu}_\\text{e} \\\\\nL: & 0 & = & 0 & + & 1 & - & 1 \\end{matrix}", "84617995478b792636dabb5e9f112532": "\\tau_{oct}=\\tfrac{\\sqrt 2}{3} \\sigma_y\\,\\!", "846188f02db3e2750524673d48c95511": "(x^2,\\sigma)(x,\\sigma)=(x,1)", "84618b346486431f95e2203810a02fd9": "\nJ:TTM\\to TTM; \\qquad J_\\xi X := \\operatorname{vl}_\\xi(\\pi_{TM})_*X, \\qquad X\\in T_\\xi TM\n", "8461920c88c8d9c85e7a68f8d4edf662": "m_F", "8461987e9339358f191279774c8c0879": "X=\\{\\} ", "84621eac13f46dfdcbf30adeb0e79891": "a=\\sqrt{2}(r_A+r_0)=2(r_B+r_0)", "84624905c73282e003c659d1ebba29a3": "f(x;\\mu)=\\mathrm e^{-\\mu x} \\mu^x / x! I_{\\{0,1,\\ldots\\}}(x)", "846253f935cc6f5090d537028edcc941": "\\mathcal D_x", "84627e7323a0f81be306ccdec85058c7": "0 \\leq \\lim_{(x,y) \\to (0, 0)} \\frac{x^2 y}{x^2+y^2} \\leq 0", "8462bdd69045f3243195e068a296f103": "C : \\mathcal{X} \\to [0, + \\infty]", "84632a43c5b2194c66d00b5edc0be2ef": "N =R/d", "84633a0147055d4ba24be9fd84d6607f": "\\mathbf{P} \\left [ X_t \\in S \\right ] = \\int_{S} \\rho(t, x) \\, \\mathrm{d} x.", "84634d775fd6fa25028172a9dd6755a6": "x_j = 2\\pi j/N", "8463b953cd55e363806955c1ee7ea1fd": "q(\\xi ,\\tau_1 + \\tau_2 ) = q(\\star q(\\xi ,\\tau_1 ),\\tau_2),", "8463bc430d945ee6b9e092a23f85fe16": "\\eta^{\\mu\\nu} =\\eta_{\\mu\\nu} = \\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n0 & -1 & 0 & 0\\\\\n0 & 0 & -1 & 0\\\\\n0 & 0 & 0 & -1\n\\end{pmatrix}", "84643337575e6385c278071e55ac2614": "m\\frac{{\\rm d}^2 \\mathbf{r}}{{\\rm d}t^2} = q\\left(\\mathbf{E} + \\frac{{\\rm d} \\mathbf{r}}{{\\rm d}t} \\times \\mathbf{B}\\right) \\,\\! ", "8464584072bfe4c021c66892d5d1bb84": "2^2 - 2 - 1 = 1", "84648de97c79b8f4bff32e028ddf91a1": "z=-1.", "84649448136ddc1e8589f980b6887711": "\\delta(v+w)=\\delta(v)+\\delta(w)", "84649553e9b57e87b8244a88811b3d99": "\\frac{\\partial^2 z'}{\\partial t^2} = \\frac{g}{\\rho_0} \\frac{\\partial \\rho (z)}{\\partial z} z' ", "8464ba2e6c306d559388b1fec8ebf61a": "p(e)", "8464f2134cdd53d3120f5cd8bc73363b": " \\mathbf{V} = \\{ V[j, k] \\} ", "8464ffbe7078da7f2b4ce192db8cccdf": " \\int_{t,t'} e^{-t(k^2+m^2) - t'((k+p)^2 +m^2) } dt dt'\\,. ", "8465f0fc30a74726a35639942aaee13a": "c(x,y)=\\mathrm{Cov}(Z(x),Z(y))", "8466097ba7ee8be60c9569d8d72c69bd": "\\displaystyle D^A\\phi=0, \\qquad F^+_A=\\sigma(\\phi)", "8466186c8dcb8e4dc743bf3f9cc8559d": "[\\mu,\\sigma]", "84661dfbf3690c953ee24c8fcb2565ca": "\\sum_{k=0}^{\\infty} k r^k = \\frac{r}{\\left(1-r\\right)^2} \\,;\\, \\sum_{k=0}^{\\infty} k^2 r^k = \\frac{r \\left( 1+r \\right)}{\\left(1-r\\right)^3} \\, ; \\, \\sum_{k=0}^{\\infty} k^3 r^k = \\frac{r \\left( 1+4 r + r^2\\right)}{\\left( 1-r\\right)^4}", "846642d8f87ee6b8e701e51b987e3fec": " E = q, V = q, Y = (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f)), X = \\{\\} ", "8466b336d609d3d2e0d89b62168cd39e": "\nH_\\mu (s,t)=E_\\mu (\\nu t^\\mu (e^{-s}-1)), \n", "8467622e97e7c7c154db5bf61cd1763b": "\\frac{d^2N(z)}{dz d\\Omega} = \\frac{r^2(z)}{H(z)}\n\\int_0^{\\infty}{f(\\mathcal{O},z)d\\mathcal{O}}\n\\int_0^{\\infty}{p(\\mathcal{O}\\vert M,z) \\frac{dn(z)}{dM} dM} ", "8467bf186541b6a2ccd4d7be49f62542": " \\frac{1}{1 - p z} = 1 + p z + p^2 z^2 + p^3 z^3 + \\cdots,", "84680273ced5055b2b1c0ce7fcd25ab9": "4 \\pi \\, \\left( T_{\\rm grav} \\right)_{ab} = \\phi_{,a} \\, \\phi_{,b} - 1/2 \\, \\eta_{ab} \\, \\phi_{,m} \\, \\phi^{,m} ", "84681bfdbdde548fe9f30d83e1e3d820": "xy^q-yx^q\\ne 0", "84682693a3a254a4a70b3560a6643601": "t \\rightarrow t+C", "84686cfb4e47e70b7ade1afb8a78f2c1": "1-RR.", "8468a884afacfee8c128b4af3fec6796": " WF_A(f) \\subseteq WF_A(Pf) \\cup {\\rm char}\\, P.", "8468e68aae6e574ca46d30900db79fc0": "(p_1,\\,p_2,\\,\\dots,\\,p_n)", "846904a6254fc2e6c3cfb553dcae4c92": "(x_0, y_0) \\in D", "84691c4795922f01dc76325e95edc0ac": "\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix} =\n\\begin{bmatrix} r\\cos\\theta \\\\ r\\sin\\theta \\\\ z \\end{bmatrix}.", "846921a5190541f00324c1ad82a638c3": "\\ker (j') = W^\\perp", "846a2c96ad4f2295761f98d2abd5a05e": "2\\gamma_\\mu^*(t) = \\sum_{i=1}^n \\mu_i(t) \\tau_{ii}^\\mu(t)", "846ac859f597991ab27dbc35a3011ddb": "2.1417", "846c52fbe32b124e232fd1c6f64a031f": "-125''\\sin(D)", "846c6d6065196573b39176232c397c2a": "1 \\leq Z(G) \\leq Z^2(G) \\leq \\cdots", "846c7c57f6bbb05666d67ffa24aa2b8a": "\\displaystyle \\frac{1}{r}=\\frac{1}{h_a}+\\frac{1}{h_b}+\\frac{1}{h_c}.", "846c8523844e4771541f8142061e8285": "A\\equiv r(mod n)", "846c8611bdbc1a5cbda5b347bd555ba6": "d=\\frac 1 2 \\,l", "846cb42ab1af34d8270104cd7cc59528": "5^n", "846e0ac477de1954643fdf2f5c2a897c": "k_d = \\left( \\frac 1 {k_a} + \\frac 1 {k_t} \\right) ^{-1} ", "846ea4ea85551f56f22ef51fd6fecae2": "\\sigma(0)=A_0\\,", "846f031153db29cb5e99cc81a5bc354e": " U(1)\\times U(1)", "846f53c9f23cb561a8abb3a71fea4eba": "m_{0}", "846fed1d963ed1e4a8c876d7a34b8fb2": "|c_n-c|\\le\\frac{|b-a|}{2^n}.", "84704b9e4e73980c94b452c363375a36": "\\frac{\\partial k_{i}}{\\partial t} = m\\Pi_{i} = m\\frac{\\eta_{i}k_{i}}{\\sum_{j}\\eta_{j}k_{j}}", "847076a5363e1024e078b2c5004ffbca": "\\gamma_{yz}=2\\epsilon_{yz}", "84707b9e3e0ae888b7514822c429f552": "1 \\le i \\le j \\le n.", "8470b873326f1a516b8561af9475900b": "d'(X,Y) = n + m - 2 \\cdot \\left|LCS(X,Y)\\right|.", "84717170808ca07571e8169f3381d798": "g(x,t)", "84719de54354ebac73b95207b4a86bd9": "D_{n}=\\langle r, s \\mid r^n = 1, s^2 = 1, s^{-1}rs = r^{-1} \\rangle", "8471c72a8e1bd1ee12df71435589e7ea": "{\\mathbb F}={\\mathbb C}", "8471fb44647e67cae2b9a76d56355c93": "I = \\frac{V}{R}.", "8472032cee73db54e353475e03d18d49": "\n S(z) = \\sum_{i,j=1}^n z_i z_j h_{ij}(z).\n", "8472cab8b05a31482b8546adfb271c4e": " y_{n+2} = y_{n+1} + \\tfrac32 hf(t_{n+1},y_{n+1}) - \\tfrac12 hf(t_n,y_n). ", "8472fb431421a4e877ccb15919ac2c73": "x^3=x+1\\, .", "84730845d29d83fed375973c12f76869": " \\max_b W(b,p) = \\sum_i p_i \\log_2 o_i - H(p) \\, ", "84731d5f674eb2202aa6ac29797a4fa7": "k=\\operatorname{tr}\\left[\\left(I-H\\right)^T\\left(I-H\\right)\\right]", "847324bd9321c9b43796c45e49f3b2bb": "z^5 - 5z - 4t = 0\\,", "847386229be999c565c19b3f41176101": "x' = x + t_x; y' = y + t_y", "8473a24a4b92460d1589b8122526b915": "s_N[n]\\ \\stackrel{\\text{def}}{=}\\ \\sum_{k=-\\infty}^{\\infty} s[n-kN],", "8473bc5e122ddd458a2452e2f82d16ba": "\\theta + \\mathrm{d}\\theta", "8473f25ef4042a8bcc816696eef36b0c": "P_0^h (S) = \\lim_{\\delta \\downarrow 0} \\sup \\left\\{ \\left. \\sum_{i \\in I} h \\big( \\mathrm{diam} (B_i) \\big) \\right| \\begin{matrix} \\{ B_{i} \\}_{i \\in I} \\text{ is a countable collection} \\\\ \\text{of pairwise disjoint balls with} \\\\ \\text{diameters } \\leq \\delta \\text{ and centres in } S \\end{matrix} \\right\\}", "8473fd88a5d2ac8d949707d6a19ad079": "\n f(x,y,\\eta,t) = \n \\left[ f \\right]_0\n + \\eta\\, \\left[ \\frac{\\partial f}{\\partial z} \\right]_0\n + \\frac12\\, \\eta^2\\, \\left[ \\frac{\\partial^2 f}{\\partial z^2} \\right]_0\n + \\cdots\n", "84740d57e989f81e2243c7a95d0d6b9c": " [h(T) \\psi](x) = [h \\circ f](x) \\psi(x). ", "8474155b8ac3df77225e88fe744668f9": "\\mathcal F_{n+1} ", "847415f02670ec51228db86cea753b93": " h_{FM} = \\Delta f/f_m \\,\\!", "84748a81cd322bc6970277551621cbc0": "\\lambda (x): \\ [\\operatorname{STATE} \\ x]", "84749245c2923da6ad1bbc94e16c2b69": "E_\\mu \\|h(x) - \\mu\\|^2 = E_\\mu \\left( d\\sigma^2 + \\|g(x)\\|^2 + 2\\sigma^2 \\sum_{i=1}^d \\frac{dg_i}{dx_i}\\right).", "847497a8a0bf4283a64443c2ffa18455": "\\Omega = \\dfrac{\\mbox{V}}{\\mbox{A}} = \\dfrac{\\mbox{m}^2 \\cdot \\mbox{kg}}{\\mbox{s} \\cdot \\mbox{C}^2} = \\dfrac{\\mbox{J}}{\\mbox{s} \\cdot \\mbox{A}^2}=\\dfrac{\\mbox{kg}\\cdot\\mbox{m}^2}{\\mbox{s}^3 \\cdot \\mbox{A}^2}= \\dfrac{\\mbox{J} \\cdot \\mbox{s}}{\\mbox{C}^2} = \\dfrac{\\mbox{1}}{\\mbox{S}} = \\dfrac{\\mbox{s}}{\\mbox{F}} = \\dfrac{\\mbox{W}}{\\mbox{A}^2}", "84749e8cb9d228d3e510ff5e2ddaea52": " f'\\left(x\\right),", "8474a83609b2ecdd17e8fe9c25437ebb": "B(f):=b^*(f)+b(f)", "8474c9ee1747f87993db30037c8972ca": " \\frac{\\partial \\rho \\mathbf{U}}{\\partial t} + \\nabla \\cdot\\phi\\mathbf{U} - \\nabla \\cdot\\mu\\nabla\\mathbf{U} = - \\nabla p ", "8474df661e27786d9e936ffc4409d7eb": "N_{(+)}", "8474ef408f50c31ca2381002b17e96a2": "k_C(f)=\\frac{\\dot{W}_{12C}(f)}{\\sqrt{[\\bar{W}_{11L}(f)+\\bar{W}_{11C}(f)][\\bar{W}_{22L}(f)+\\bar{W}_{22C}(f)]}}.", "8475537484f9fe476f940641214a84bd": "K =\\frac{n(A \\cap B)}{\\sqrt{n(A) \\times n(B)}}", "8475ae69a8d75869a895611460b5e766": "\\forall x \\in X,\\; \\langle x, (T^* - \\lambda) \\varphi \\rangle = \\langle (T - \\lambda) x, \\varphi \\rangle = 0.", "8475d11d4c36c413b86adbf1acd27105": "2^{p+1} = 2 \\pmod q", "8476216fcd6a136854cbedfd4115c374": "R_{P}(t)= \\frac{ \\displaystyle\\sum_h \\sum_{d_h\\neq 0} \\sum_{\\gamma_h} \\frac{d_h q}{^{d_h}M_{P_h}} \\ {^{d_h}_{c_h}}P^{\\gamma_h}_h (t) }{ \\displaystyle \\sum_h\n\\sum_{d_h\\neq c_h} \\sum_{\\gamma_h} \\frac{(c_h-d_h) p }{^{d_h}M_{P_h}} \\ {^{d_h}_{c_h}}P^{\\gamma_h}_h (t) }. ", "84767887246235a8b677855c2c8b1292": " c = f(k) - (n+d)k", "8476b5798c6e3f59660bd21db2a9cfba": "A_{\\text{i}} \\triangleq \\frac{i_{\\text{out}} }{ i_{\\text{in}} } \\,", "8476e13bd84667de7f0deab94c57a026": "C_{t-1} = 0.9 Y_{t-1}", "84774b9db04a0de84fb0c0d0ca96f413": "\\mathbf{R} = \\mathbf{Kr} + \\mathbf{R}^o \\qquad \\qquad \\qquad \\mathrm{(2)}", "8477ca3dbb285f4c9fa59b584f4136fe": "\\varphi_{\\overline{X}}(t)= \\varphi_X\\!\\left(\\tfrac{t}{n} \\right)^n", "8477f9c891dcefa68dc92811a3b685c6": " [S] \\gg [P] ", "84782ab2a06eef0e75b227a31e3c196f": "\\frac {v'} {f} = \\sin \\theta \\left [ \\frac {1} {\\tan \\left ( \\psi - \\theta \\right ) } + \\frac {1} {\\tan \\theta } \\right ] \\,;", "847859295d170409d649479bd286ad0c": "f = \\mathcal{F}^{-1}(\\mathcal{F}f) = \\mathcal{F}R\\mathcal{F}f = \\mathcal{F} (\\mathcal{F}^{-1}f).", "84787c182ef40abff3844b12334f7735": "a =(1,2), b = (1,2)(3,4)\\cdots, c = (2,3)(4,5)\\cdots", "84789ff70311c39be3dcc592903dd1fa": "k^{p^{-\\infty}}", "8478c02600f617190b8504043dadf38b": "\\limsup_{n \\to \\infty}a_n", "847920d83211bec9343138a974597355": "\\delta \\approx 0.01", "847934a3e014f88cf9608e40cd998401": "f\\,g:(x)\\mapsto f(x)\\,g(x)", "84794f66c2f166b6ee3f35a7de4e013d": "x_1-x_0", "847951766b21e136017f0193e3abf079": "n\\sqrt{2}", "847a40074a58d18b467693c3814778dd": "\\beta_0=1", "847abc5d8bebcb654e24ee48eb257612": "\\hat{t}=-\\sin u\\ \\hat{G}\\ +\\ \\cos u\\ \\hat{H}\\,", "847ace0ccb83364527750d38fd39a11c": "\\pi/2+\\varphi+\\theta_0", "847ae72700115e7cd125326f18104af7": " f(\\Omega) \\,", "847ba71de5397af6fadbce987db921a6": "\\scriptstyle y\\in f(x)", "847bb8167d4d52d6e41c778bbadb6722": "\\ C_d = \\frac{\\epsilon}{4 \\pi \\delta}", "847bc0b8ebc8a1f397e1d24ebfb7a374": "\n\\frac{1}{42}, \\qquad \\frac{5}{42}, \\qquad \\frac{11}{42}, \\qquad\n\\dots, \\qquad \\frac{31}{42}, \\qquad \\frac{37}{42}, \\qquad\n\\frac{41}{42}.\n", "847bf766cc9fd962dbfd166bcfaf9668": "(x_{i1}-\\dfrac{x_{i1}+x_{i2}}{2})=\\dfrac{x_{i1}-x_{i2}}{2} ", "847c82c6970ff18a079b393afd5e803a": "\\tfrac{a}{c} + \\tfrac{b}{c}", "847cb7e9711a49d272186186155b800f": "x^5-3x+1=0", "847ceeacfe3db4bf64a55bed94fe0f6e": "R \\ge \\tfrac{1}{5}", "847cf1a10d1e347324173a1ae87e5a0c": " \\widehat{\\Omega}_{OLS} ", "847d34ecba1959bb1cec72f226685795": "u\\times v\\in U", "847d5128661733e47e0736791031186e": "(n = 2)", "847d59b3bb3b72b6510412bb4709383b": "\\mathbb{Z}/4\\mathbb{Z}", "847dd76517d0ddd0c4d61ed49adc9094": "g_{00} \\ne 0", "847de0bac7f2b4d8c35134108fd4f60f": "x_\\epsilon\\in X_\\epsilon", "847e037e4c6240cbdd2ac41023f75db0": "G=P\\cdot K,", "847e43a55009d7ec59f7b2fbd7fdd439": "\\frac{x-1}{{{x}^{2}}+12}", "847ec794bd9d31db53ae9cd2cc8f49ee": "N_j\\log|u^j|", "847eca37af47a3c2e64e359dd2226dce": "N_1/(N_1\\cap N_2)\\cap N_2/(N_1\\cap N_2)=\\{0\\}.", "847ef73f406469e25dd1bfdf2056c8be": "V(S) = \\{x \\in K^n : f(x) = 0 \\mbox{ for all } f \\in S\\},", "847f2563cf8f3209d0dc63f8adfeb25c": " \\propto \\exp{-t/\\tau} ", "847f76d76dc052e787491869c6200c2a": " \\Box=\\bigtriangledown^2 ", "847fa281bb89dd86aa1feaed408f6a5c": "\\circ \\!\\,", "847fc975a5287af584b2e264f9d2d8d7": "T \\subset L\\,", "848045328558e710a8ee2d33b1158f62": "\\textstyle\\sum\\frac{n^m}{n!}", "8480a3b7200851e0ac9fccb2808eab03": "\\Lambda_{i i}", "8480dbdb9fba88b5045a8c28e09f8066": "P_n^{(\\alpha, \\beta)}", "8480e04d4f2afed71b1060cf51762365": "Q_\\text{phred} = -10 \\log_\\text{10} e", "84810839a67fe0ddb69165f98f97576e": "\\mathfrak c=\\aleph_2", "848199b69f7948aff6c79176f801ed6f": "p(I | \\theta_{fg}) ", "8481b9760137213830567de30535c14f": "bcab", "8481cb64461771bb10063792d87e24c8": "H\\ltimes N", "8481d7e5ce2825dd22c20c42ee261f0d": "\\beta\\left(\\pi\\left(x_0,m\\right),i\\right) = \\mathrm{rem}\\left(x_0, \\left(i+1\\right)\\cdot m+1\\right)", "8481ffafc593987e501ebb3f5e0c844e": "m^2>0", "84828eaeb152fac750ae2ce4c39d3567": "\\mathbf{R_0} = \\frac {\\mathbf{V_1}} {\\mathbf{I_w}} ", "8482c05f21adbea1873714b0ee15eb39": "Z^{(\\ell)}_{\\mathbf{x}}({\\mathbf{y}}) = C_\\ell^{((n-1)/2)}({\\mathbf{x}}\\cdot {\\mathbf{y}})", "8482c176e052cd21bb4656a92829b90a": " \\beta (s) ", "8482c3343f882888543ba4d66a2dd24a": " \\alpha = 0 ", "8483b93db0bd31f14d77e3550cf46cbb": "q \\leftarrow \\hbox{not } p", "8483ba697c3faf8b90cd2c58bbb21bad": " v=M \\sigma \\kappa ", "8483c463d848ecd9ce2764fc546149d8": "y=Ax+z", "8483f036585b2ed35449a811893841e6": "\\begin{align}\n(1-r) \\sum_{k=0}^{n} ar^k & = (1-r)(ar^0 + ar^1+ar^2+ar^3+\\cdots+ar^n) \\\\\n & = ar^0 + ar^1+ar^2+ar^3+\\cdots+ar^n \\\\\n & {\\color{White}{} = ar^0} - ar^1-ar^2-ar^3-\\cdots-ar^n - ar^{n+1} \\\\\n & = a - ar^{n+1}\n\\end{align}", "8483f4fe8fa69bf401b1ad1691fe3114": "(p_{1}^{2}-p_{2}^{2})\\Psi =-(m_{1}^{2}-m_{2}^{2})\\Psi", "8483f8a79cd91768b01d4b1638ff554e": "p_1 + p_2 = p_3 + p_4 \\,", "8484782a045c7781ddf24d3fb96d6198": "\n\\Phi_0 [\\mathbf{r}] = \\frac{3 k_B T}{2 N b^2} \\sum_{l=1}^n \\int_0^1 ds\n\\left| \\frac{d \\mathbf{r}_{l} (s)}{d s} \\right|^2,\n", "848493156d199c8b35106431b2781e8a": "\\lambda \\ge 0", "8484a716f439425db0d9938b3d6d33ff": "r + dr", "8484cdc1fa5bc790ce8184a9178abb7c": "\\neg ( (\\forall x')(\\exists y') (\\forall u)(\\exists v)(P)\\psi(x,y|x',y') \\wedge (\\forall x)(\\forall y) ( (\\forall u)(\\exists v)(P)\\psi \\rightarrow (\\forall u)(\\exists v)(P) \\psi ) )", "84851e7c0d01a09f646b8cc089a87ce8": "f(x;\\sigma,n) = \\int_0^{\\infty} \\frac{re^{-r^2/2\\sigma^2}}{\\sigma^2} \\tau(x,r;n) \\,\\mathrm{d}r,", "84852e59f6edfd1e0a0aad08336ce391": "\\boldsymbol{u}(\\boldsymbol{x})", "848540adabe1728ef200a64d151912d7": "=\\frac{\\rho}{(2 \\pi RT)^{D/2}}e^{-\\frac{(\\vec{e})^2}{2RT}}e^{\\frac{\\vec{e}\\vec{u}}{RT}-\\frac{\\vec{u}^2}{2RT}} ", "848567d83c7daa7c569b2e3050b8f772": "\\gamma_n", "8485be7e9b05cd6a3455088f1a3c05b3": "\\text{Humidex} = \\text{Air temperature}\\ +\\ 0.5555 \\times (6.11 \\times e^{5417.7530 \\times \\left(\\frac{1}{273.16} - \\frac{1}{\\text{dewpoint in kelvins}}\\right)} - 10)", "8485f5b6a6cc53c1053ba812bbf8b688": "\n (m\\mid n,k) =\n \\begin{cases}\n \\frac{\\binom{m - 1}{k - 1}}{\\binom{n}{k}} &\\text{if } k \\le m \\le n\\\\\n 0 &\\text{otherwise}\n \\end{cases}\n", "84860d87bc3fccbdf936f7faf46352a9": "\\left [\\begin{smallmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & -2 \\\\\n0 & 0 & 1 \\\\\n\\end{smallmatrix}\\right ]\n", "848613ed032279c0c1317f79e3ea6c87": "\n\\begin{align}\n{d\\over{dt}} {{\\partial T}\\over{\\partial \\vec \\omega}}\n& = {{\\partial T}\\over{\\partial \\vec \\omega}} \\times \\vec \\omega + {{\\partial\nT}\\over{\\partial \\vec v}} \\times \\vec v + \\vec Q_h + \\vec Q, \\\\[10pt]\n{d\\over{dt}} {{\\partial T}\\over{\\partial \\vec v}} \n& = {{\\partial T}\\over{\\partial \\vec v}} \\times \\vec \\omega + \\vec F_h + \\vec F, \\\\[10pt]\nT & = {1 \\over 2} \\left( \\vec \\omega^T \\tilde I \\vec \\omega + m v^2 \\right) \\\\[10pt]\n\\vec Q_h & =-\\int p \\vec x \\times \\hat n \\, d\\sigma, \\\\[10pt]\n\\vec F_h & =-\\int p \\hat n \\, d\\sigma\n\\end{align}\n", "84863a597a962a977746db2b45c5c559": "\\ \\frac {dU} {dH} = \\zeta \\frac {U(h)} {h}\n", "84863dd0cfa0ddf22dc387ea6dfedef8": "r^2+1\\equiv 2 \\text{ mod } 3", "84863fd8cc818b4156a5fa61a85601a9": "g_{44}", "84866049a2fd49104814e136f4504565": "I_w=\\frac{S_{spw}-S_{cw}}{1-S_{cw}-S_{or}}", "8486fca0daa0100da5e3aa97e515df3f": " \\mathbf{B} = \\frac{\\mu_0}{4\\pi}\\iiint_V \\ \\frac{(\\mathbf{J}\\, dV) \\times \\mathbf{\\hat r}}{r^2} ", "848706b51c9dfe443e2da4cc6cfea885": "\\mathbf Z/n", "8487e08e8733cf27fa25beb8a5e23601": "\np^{\\sum x_i}(1-p)^{n-\\sum x_i}=p^{T(x)}(1-p)^{n-T(x)} \\,\\!\n", "8487e68a3c2d340dbaf3e4aea7ffbfa9": " |Y \\rangle ", "848876a025fe223b689fd89e63df5b0f": "Q^\\mathbf{Z}=\\{ s=(\\ldots,s_{-1},s_0,s_1,\\ldots) : s_k \\in Q \\; \\forall k \\in \\mathbf{Z} \\}", "84889708a2b21605f7b9907e24507130": "V\\le_{RK}U", "8488ea4d8701e2dbf3ce3fb4ea09c893": "\n\\log \\left( \\frac{P_A }\n{1 - P_A } \\right) = \\beta _0 + \\beta _1 \\left( c_A - c_T \\right) + \\beta _2 \\left( t_A - t_T \\right) + \\beta _3 I + \\beta _4 N = v_A \n", "8488f92d7f692b6488e67c987b7a861d": "\\tilde C", "84890b645c9b033d56fe6f322b34a64b": "d\\geq 4\\,", "8489342d5e42281ae9a94dd3c98e833f": " f(\\gamma) \\in \\beta \\times \\{x\\}", "8489971ae8581908facf0c676851c109": "\\mathit{MPC}<\\mathit{APC}", "8489ac2cbd9efabb1e7339eab9a01488": "x^+ \\to x^+", "8489e3eae0382b73a4d40ae0be011dd3": "\\xi^3+\\xi^2+\\xi=1, \\,", "8489e53e7a86fdf4462dbc59adf5fb81": "\\widehat{H_{\\mathbf{R}} f} = \\left (i\\chi_{[0,\\infty)} -i\\chi_{(-\\infty,0]} \\right ) \\widehat{f},", "848a04757375998735065c926adf36a5": "\\frac{I}{I_0}=e^{\\alpha_n d}, \\, ", "848a3a2cfebd8211bb7f242bb08ba015": "\\hat{f}(\\xi)\\to 0\\text{ as }|\\xi|\\to \\infty.", "848a7007756fdf20c79a60055b8d7643": "({\\mathbf{s}}_0)", "848afd5c85c03ef4fc2bfae917996f0e": "v_\\mathrm{in}", "848b25c52f81d477f4f11375b5c0166f": "{\\left(P_r + \\frac{3}{V_r^2}\\right)\\left(3V_r-1\\right) = 8T_r}", "848b617fc32228e0c639e65721eefbc6": "l(f)=L(f^{-1})^{-1}", "848b7a30124d3a52020668cf10684ab4": "\\frac{d^nf(t)}{dt^n} = \\mathcal{F}^{-1}\\left\\{(i \\omega)^n\\mathcal{F}[f(t)]\\right\\}", "848b82b51a54e7d752ca0d036b16cb58": "O(log(B))", "848b8388ab49c3061a1e06ec72686d73": " \\vec{k} = \\frac{\\ |n| \\omega}{c}. \\,", "848b8ba5c56c0471d1b999556c524520": "a _0", "848bdfc27553660218af7aff91454dc3": "PL_0", "848bf27356a0385f9358b0c734e64e5c": "p(x|C_1) = (0.25, 0.75, 0.75)", "848c0f14037aa0eeea6b7e5258a73816": "\\;\\;\\quad \\rho=\\sqrt{r^2-2Mr}\\,\\sin\\theta\\,,\\quad l_+ l_-=(r-M)^2-M^2\\cos^2\\theta\\,,", "848c51a76bea27edfa7d97d5ce9a9f90": "\\{true, false\\}", "848c7244d368bd252857de6cd5b32127": "R_1 \\| R_2 \\| R_3", "848c9590b78636c0a9347b5ae24c215f": "A_{t}(x), B_{t}(x)", "848caa5c1a9cb63826cd2ba9a2c85c3c": " 1\\leq i\\leq r ", "848cb571e26e32f3acf0c6f293ef81a2": "\\mathbf{\\sigma} = \\mathbf{E}(\\mathbf{\\epsilon} - \\mathbf{\\epsilon}^o)+\\mathbf{\\sigma}^o = \\mathbf{E}(\\mathbf{Bq} - \\mathbf{\\epsilon}^o)+\\mathbf{\\sigma}^o\\qquad \\qquad \\qquad \\mathrm{(5)}", "848cc3e1dfdeef18395549d6aa5917db": "dW_t \\,", "848cf6e187687392772a9c358956db5d": " X/C_1 \\; + \\; X^2/C_2 \\; + \\; X^3/C_3 \\; + \\; X^4/C_4 \\; + \\cdots.", "848d13e9db82673ef488f09607855055": "\\Gamma\\simeq 175", "848d1a6080e8f7eef77c1a824a59f28a": "\\mathbf{x}=\\{x_i|1\\le i\\le k\\}", "848d2c4b278c34b9252e66c438af6948": "\\Psi(t)=\\frac{2}{\\sqrt{3}}\\pi^{-\\frac{1}{4}}\\left(\\sqrt{\\pi}(1-t^2)e^{-\\frac{1}{2}t^2}-\\left(\\sqrt{2}it+\\sqrt{\\pi}\\operatorname{erf}\\left[\\frac{i}{\\sqrt{2}}t\\right]\\left(1-t^2\\right)e^{-\\frac{1}{2}t^2}\\right)\\right).", "848d3becea0effb2d8e4c07260d4908d": " \\text{DF} = \\frac {i^2 \\text{ESR}} {i^2 |X_{c}|} = \\omega C \\cdot \\text{ESR} = \\frac {\\sigma} {\\varepsilon \\omega} = \\frac{1}{Q} ", "848d4b5448a62e420c1c3c0340e645ad": "\\tfrac{q-1}{q-2}", "848d5eb13e4dd76995962638c1729e05": "(* \\rightarrow *) \\rightarrow *", "848db3caf781d98091af6b802a2dbcb4": "x\\in \\mathcal{F}^{n}", "848dfcb1691da51c7b910ce5356be07f": "\\ p(x,y) = 2xy^2+x^2-y^2+3x+5y-8", "848e09abf1743a84bcd8002783493dea": " \\langle f/\\varphi \\rangle_\\rho = \\langle T_\\rho (f)/1 \\rangle_\\rho", "848e1ac7251cbd31a9d7d53d8e058128": "p[A,B] = 0", "848e73e90aa7b0b9db6938cf6660991d": "\\vert d \\rangle", "848ef468a86fbfd6bbaf2b6e0e39968c": "T(n) = S(n) + T(n(1-p)), \\, ", "848f0e398ab42485c69838cad4c4acb7": "x_\\beta", "848f5af34fc7b0ac2ebe1ece37a6dcfc": " F_7 = q, F_6 = x, A_3 = A_6, A_4 = A_7, S_3 = S_6, S_4 = S_7 ", "848fa2f5c0f3f007232b61555b7117d4": "\nS = \\frac{1}{2} \\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k}^{2}\n", "848fba304332140cad81995d67ab6e66": "m\\times 1", "848ff17a236019eb4c09c550a587e796": " t_0\\ ", "848ff9b26009f6022dfdd4048e33ca38": "\\displaystyle{T_\\varphi f(\\theta)= f(\\varphi^{-1}(\\theta)) -{1\\over 2\\pi}\\int_0^{2\\pi} f(\\varphi^{-1}(\\theta))\\, d\\theta ,}", "849079621d5f24f91a1201c703911faf": "\\sum_{k=1}^{n-1} \\sin\\frac{2\\pi k}{n}=0\\,\\!", "849157421b189a0ec2a82900b49a5554": "\n \\sigma_{11} - \\sigma_{33} = \\cfrac{2C_1}{J^{5/3}}(\\lambda_1^2-\\lambda_3^2) ~;~~\n \\sigma_{22} - \\sigma_{33} = \\cfrac{2C_1}{J^{5/3}}(\\lambda_2^2-\\lambda_3^2)\n ", "84919040ee8c29f4ba67c9bc69c20279": " \\,X ", "84925c05787f8cdf3b18e1fee8a29a77": "\\phi_c\\,", "849321906fd38351b74fe27d7a732a43": "\\sigma_{zz}<0", "84935f08ab0572600ec567e64263a768": "\\partial N", "8493712986961cf553503b7d855dac7d": " M(\\emptyset)=0", "8493aa00cec6dffb7fbc505b1926a314": "\n [(2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)]^\\frac{1}{2}\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n = \n \\langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\\rangle.\n", "8493b762f07b8b7c3686550adead362f": "\\{ (t, 2 t + 1) ~|~ t \\in \\mathbf Z\\}", "8493c3fba8388aaa263ac1aeecbf8b61": " h = \\frac{\\sigma_x^2 - m^2}{(\\sigma_x^2 - m^2)+\\sigma_w^2}, \\quad c = \\frac{\\sigma_w^2}{(\\sigma_x^2 - m^2)+\\sigma_w^2} m , ", "8493e5163eb443cf873542df07faf73b": "\\frac{\\partial \\mathbf{v}}{\\partial t} + ( \\mathbf{v}\\cdot\\nabla ) \\mathbf{v} = -\\nabla p + \\nu\\Delta \\mathbf{v} +\\mathbf{f}(\\boldsymbol{x},t)", "8494031ceda18b850f601c83deeaa6bc": "C_{p_u}", "8494560fbd848c4708dba3e1f5bd8a58": "v^2 = -\\langle v,v\\rangle", "8494799492f40acfe13089fd4de4a81f": "\\rho_{Out}\\,\\!", "8494cd151f595db26a1ba379041c3e0c": "\\Gamma=\\{\\mathbf{g}\\in C([0,1];H)\\,\\vert\\,\\mathbf{g}(0)=0,\\mathbf{g}(1)=v\\}", "84954c3b93f7543f5f59afc33f2df492": "\\geq 1\\,", "84959fd50b5098fe6e88bf17e376de73": "\\mathbf{Y}(s) = C \\mathbf{X}(s) + D \\mathbf{U}(s)", "8495dea579204872327a8c14b8bfd925": " \\sqrt{a (1-e^2)} \\cos i", "8495fe440d8a228c5998b53d5433900b": "\\mathcal{B}(p,q)", "849647484f18445de0b4d5fd37302529": "\\ell_0=2\\ell\\tan\\theta\\sin\\theta_0\\,", "8496661ffd0ca762b84d4bc3155c4968": " \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0", "8496d8506cf36d686ab04822b4310dcb": "\\sum^{\\infty}_{t=0} {\\gamma^t R_{a_t} (s_t, s_{t+1})} ", "8496eedef373c5bc3264eb663b3fcce8": "\\kappa_{\\lambda}", "84971f8b3dd64d130cd0aadae89d041d": "\\hat{\\theta}_i", "849735c442f49d85720b4c9ab72e9e5a": "F_1 = -k_1 x_1= F_2=-k_2 x_2 .\\,", "849743a674fe419134286ab0305f8e50": "C=\\frac{\\partial P}{\\partial \\rho}", "8497550217bfcf0ac55a391ff7040db4": " \\frac {V_O} {V_S} = \\frac {1} {1 + j \\omega \\left(C_2 (R_1+R_2) +C_1 R_1 \\right) +(j \\omega )^2 C_1 C_2 R_1 R_2 } ", "84977d7d75b1e0f8ce4f9a3eef30d057": "\nd_A= \\frac{x}{\\theta}\n", "8497acb46551b319b78d20d97b819e58": "\\pi_{30}(S^{16})\\neq \\pi_{30}(S^{31})\\oplus \\pi_{29}(S^{15}) . \\,\\!", "8497b1adfdfec0a34754bd3d9f963d4c": "P(x) = \\prod_{n\\ge 1}(1-x^{2n-1}) = Q(x)/Q(x^2)", "8497b1da60fb7dd308eefdabfd8125d1": "\\Delta(t)", "8497d763c4d6c06fbe0898e0e2c60a27": "x y x^{-1} y^{-1}", "84982f0d179b5f945e6ae098c19a2f87": "\\Lambda x\\in C. J(x)\\vdash K", "8498e1cea633404a14db0f7021299209": " b_j = \\prod_{f(i)=j} a_i, \\,\\, j=0,\\dots,n, ", "8499de7b86c30a5aae92667d2b38c97b": "A=xy", "849a14a4a06a130b52c0a9bf8334ed31": "\\mathfrak g", "849a8e13a339ebe22aee2b90ba1f218a": " h=\\tilde{h}\\circ \\varphi", "849a9e48d8e14ea9eceeff409ffe5b42": " V = \\{p, q, m\\} ", "849ab395c9f4e3898aab4de28d3ecc7f": "x_e=L_e", "849ab7a3011bc3d9c16cab9706ea0abf": "\\operatorname{E}(X)=\\mu\\,\\! ,\\quad \\operatorname{E}(\\Tau)= \\alpha \\beta^{-1}", "849b00c3033ac638b8e0731b8a43ac55": "\\overline{x_i}", "849b22bf62e5e550d61e0766b3579669": "W^{-1}", "849b75fb0a0019577abd8f7032d68731": "\\epsilon_{abcd}\\,", "849b86501b407e5956295563f7a782aa": "Gr_p", "849b919225ea865c6ce7ce9170e3d04c": "\\scriptstyle{\\left\\langle E\\right\\rangle}- \\frac{\\varepsilon}{2}", "849ba151491edffed1538bea6fb6e608": " p_\\mathcal{N}(x | C) ", "849be9f2bab07e9bc526b03d6ed4d8f7": "\\hat{\\beta}({r_{\\rm w}}),", "849c072b963d1b8c63ef866996cf33cb": "(v_{i-1},v_i)", "849c0a0bd8215c2fc76a76e7c3f43761": "k'_z = -\\sqrt{\\frac{\\omega^2}{c^2}-\\left(k_x^2 + k_y^2\\right)}", "849c22dd3a77908bed9716f1257a9179": "w_i \\in T", "849c39b36386236b10c98adf9f822a40": "\\lambda/40", "849c4eb77dfb2e2225e120c523e7b23c": "\\frac1{\\mu \\big( B_{r} (x) \\big)} \\int_{B_{r} (x)} f(y) \\, \\mathrm{d} \\mu(y) \\xrightarrow[r \\to 0]{\\gamma} f(x),", "849c83dfa3238ed35e5f6cc5a4b37bb3": "26*36*36", "849c96983e134159f2d7da012e2fef32": "\\lessgtr \\lesseqgtr \\lesseqqgtr \\gtrless \\gtreqless \\gtreqqless \\!", "849ca11a24454b542f6f42df3afc6cd4": "R = k(10g + s) \\, ", "849cd807f551242177a403c2b6797f00": "h(n,\\Delta)", "849cfd94b469abaa99488478253ef86d": "\\Lambda \\ll 1 ~ (\\Gamma \\gg 1)", "849d070985ab4e21a2e02329b2386bf9": "B^{64}\\,", "849daa6c4466e13c6b19c00da85dde85": " dt_- = \\frac{-\\omega \\, r^2 - \\sqrt{(1-\\omega^2 \\, r^2) \\; (dz^2+dr^2) + r^2 \\, d\\phi^2}}{1-\\omega^2 \\, r^2} ", "849df106622c2c2df21e91158749feb2": " \\sum_{k=0}^n f(k) = \\int_0^n f(x)\\,dx + {f(0) + f(n) \\over 2} + \\frac{B_2}{2}(f'(n) - f'(0)) - {1 \\over 2}\\int_0^n f''(x)P_2(x)\\,dx. ", "849df1393a04897a92083d747851be4a": "(x_2)", "849e11e592e8637a290ef5a671722b46": "(r, \\theta, \\phi, t) \\rightarrow (r, \\theta, \\phi, -t)", "849e2407888690ddf60ed5d2ac60699d": "Z={\\rm Tr} [e^{-HT / \\hbar}]", "849e469456f3a7357f60013c1fd2f714": "K = 1,\\quad K = x + y, \\quad K = xy,", "849e625f2645da70bfe23c6bd04b5bca": "\\,\n\\mathbf{s} = \\mathbf{v} + \\mathbf{u}\n", "849ef5217ff9ae4e49d76c4bc9c143bb": "\ne = [2;1,\\mathbf 2,1,1,\\mathbf 4,1,1,\\mathbf 6,1,1,...,\\mathbf {2n},1,1,...] = [1;\\mathbf 0,1,1,\\mathbf 2,1,1,\\mathbf 4,1,1,...,\\mathbf {2n},1,1,...],\n", "849ef85e22fafd64e1d83242e66327af": "N = R_{\\ast} \\cdot f_p \\cdot n_e \\cdot f_{\\ell} \\cdot f_i \\cdot f_c \\cdot L\\cdot f_d", "849f1d940d90debec9ac49d1033dd61c": "c_1 \\approx 2 / n^2", "849f395bfc987894d142dd5aad0603af": "\nu \\sim\\ f_u(\\alpha).\n", "849f3f3cc5b1bbbd2cb08fadd709a47f": "d_{BS}\\!\\,", "849f9df2d516521b9163ae28c1fbc16a": "C_{n}", "849fb31fb0bc020c3772985682abc6c8": " \\frac{-1+\\frac{2a e^{\\tfrac{a}{b}} {\\rm E}({1-\\tfrac{1}{b}},\\tfrac{a}{b}) }{b}}{a^2} ", "84a03cc5629db409b07564a77e7a5fe1": "\\displaystyle A^\\top Ax=A^\\top b", "84a0993f1e084f2edcfea829f79644d9": " MII = \\frac{ II - \\frac{ A }{ T } }{ 1 - \\frac{ A }{ T } } ", "84a0cf936f289c5aefd5fbb8045ceb02": " \\begin{align}\n\\lfloor x \\rfloor +\\lceil -x \\rceil &= 0 \\\\\n-\\lfloor x \\rfloor &= \\lceil -x \\rceil \\\\\n-\\lceil x \\rceil &= \\lfloor -x \\rfloor\n\\end{align}\n", "84a10f49cb0f2e126d14ebf91d59667a": "\\mathcal{P}_\\alpha", "84a134d9887822c2a82bb2ac220b343e": "p_k(g) = \\sum_{a=0}^k C(2k+1, 2a+1) \\frac{\\sqrt{2}}{\\pi} \\left(a - \\begin{matrix} \\frac{1}{2} \\end{matrix} \\right)! \n{\\left(a + g + \\begin{matrix} \\frac{1}{2} \\end{matrix} \\right)}^{- \\left( a + \\frac{1}{2} \\right) } e^{a + g + \\frac{1}{2} }", "84a1f001082d89f9b167a96f38de733e": "\\forall y (y \\in s \\iff (y \\in L_{\\omega+1} \\and (\\forall a (a \\in y \\iff a \\in L_5 \\and Ord (a)) \\or \\forall b (b \\in y \\iff b \\in L_{\\omega} \\and Ord (b)))))", "84a2495d7af242d2e3f56f56a66faaa2": " \\ a^2p_a+b^2p_b - {} \\ ", "84a2867c20e53d419d45a8b4bf5a3712": "c \\;", "84a2f6a917d40499af7c217eab2916b4": "\\begin{align}\nb_n &= b_{n-1} = 0,& f_n &= f_{n-1} = 0,\\\\ \nb_i &= a_{i+2}-ub_{i+1}-vb_{i+2}&f_i &= b_{i+2}-uf_{i+1}-vf_{i+2}\n \\qquad (i=n-2,\\ldots,0),\\\\\nc &= a_1-ub_0-vb_1,& g &= b_1-uf_0-vf_1,\\\\\nd & =a_0-vb_0,& h & =b_0-vf_0.\n\\end{align}", "84a31b2a9f761ee8d86d74bc3857e8a3": "\\leq_{\\rm m}^{\\rm P}", "84a32cef9df050bd679675dd4b5d7c04": " \\frac{1}{k^2 + i \\epsilon} - \\frac{1}{k^2 - \\Lambda^2+ i \\epsilon} ", "84a335a1470185f9757873ebae5ca4c8": "H\\leftarrow s(H)", "84a40dfe88e9816aacaf1f630890a53f": "\\int_0^{2\\pi} \\frac{dx}{(a+b\\sin x)^2}=\\int_0^{2\\pi} \\frac{dx}{(a+b\\cos x)^2}=\\frac{2\\pi a}{(a^2-b^2)^{3/2}}", "84a44a6dc6241f1d870bbddc2ee335c8": "x = \\mathbf{s} \\cdot \\mathbf{v}", "84a475c68953bb07936e7b52840f2cdf": "\\frac{1}{\\tau_U}=2\\gamma^2\\frac{k_B T}{\\mu V_0}\\frac{\\omega^2}{\\omega_D}", "84a4781cc8a90d198b648d4c77e4c43a": "\\lambda_A\\circ(\\varepsilon_A\\otimes A)\\circ\\alpha_{A,A^*,A}^{-1}\\circ(A\\otimes\\eta_A)\\circ\\rho_A^{-1}=\\mathrm{id}_A", "84a4f4b10e1b2ffd9a8c326d2a4ee6ff": " \nb=\n\\begin{cases}\n1 & \\text{if } \\exists i \\in D: (x, x_i) \\in C_{xx},\\delta_{ext}(s_i, t_{si}, t_{ei}, x_i)=(s_i',1)\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n", "84a4f5160b088c6f50f185ff4f431c71": " G \\approx 6.674 \\times 10^{-11} {\\rm \\ N}\\, {\\rm (m/kg)^2}.", "84a50306636628a6df4a0e818b95f1aa": "f(n) = \\Theta\\left(n^{c} \\log^{k} n\\right)", "84a54194db253590972154cac98cc4e6": "v^*_j \\in V^*", "84a6077fc36813024a089d65f2d72510": "Ax \\leq b", "84a67f79df352d3fb2960f95657bb330": "1\\le j < l\\le n", "84a6a9930c7d9693c8b1284a6974389e": "x^k \\equiv 1 \\pmod{n} ", "84a71b938098fb7feacd4f761b651c40": "Z = \\zeta^N.", "84a726257a8822975afeaf96dfb7cf72": "1 \\triangleleft G", "84a7302534ffded0ecb7743545e7e8a1": "t_{\\frac{1}{2}}=\\frac{1}{r}\\ln\\left(\\frac{1+e^{rT}}{2}\\right)", "84a7437b3a496cdb32625e60179ee02a": "\\mathcal{Z}:=\\mathbb{Z}[v,v^{-1}]", "84a80c8c596f900832cf689b243f3995": "a_\\text{f}", "84a83015b02edaa7cb965adbd12212fd": "\\scriptstyle \\hat z' \\;=\\; R_{\\beta'} \\hat z R_{\\beta'}^\\dagger", "84a890f4805084bacc7b59534ceff9bf": "x(yz) = (xy)z.", "84a8915577add0595a110fe83f41527e": "\\sum_{n \\geq 0} T^n \\cdot \\frac{1}{2 \\pi i} \\left ( \\int_{\\Gamma} \\frac{d \\zeta }{\\zeta^{n+1-k}} \\right) = \\sum_{n \\geq 0} T^n \\cdot \\delta_{nk} = T^k.", "84a9206b01bb159320cb59c9c1c4632e": "\\begin{align}\n\\sum_{j} \\delta_{ij} a_j &= a_i,\\\\\n\\sum_{i} a_i\\delta_{ij} &= a_j,\\\\\n\\sum_{k} \\delta_{ik}\\delta_{kj} &= \\delta_{ij}.\n\\end{align}", "84a92d1433e9866c02a096d30d5a0146": "\\nabla^2\\Psi=4\\pi G\\rho", "84a9bc23d4a7df269a7caa9ebb350cb3": "a\\ \\pmod{ \\Phi_n(q)}", "84a9e54c2d18958c5cf6b7fc12827187": "d(t) = (2/\\sqrt{\\pi}) a_{eff} T_1 \\sqrt{\\kappa t} + d_r", "84a9fd06e3139ce97a9bb057525fa6f6": " K_p = \\tan ^2 \\left( 45 + \\frac{\\phi}{2} \\right) = \\frac{ 1 + \\sin(\\phi) }{ 1 - \\sin(\\phi) } ", "84aa3b6c5acde665276be575f12057c9": "\\psi(z,q)=\\frac{\\zeta'(z+1,q)+(\\psi(-z)+\\gamma ) \\zeta (z+1,q)}{\\Gamma (-z)} \\, ", "84aa5950a328a24f46d3802fb8e03f4d": "(\\lambda^3,\\lambda)", "84aa70d3082944c09e6e5507feeb9f3f": "JMJ=M^\\prime", "84aaea514d02ed261a0004c2a7f4ea4f": "E_a^i = |det (e)| e_a^i", "84aaf1a2a6b006ef37a836c58b0f9d04": "\\Pr(R_{i}<\\underline{R})", "84ab0a4f42b2c0bdb278073283371dc3": "C(a,b,0)=1", "84ab8c06d20296cc83b858cd1f35838f": " D_{24576}=A_{24576}- A_{12288}=0.0000001021", "84abc49f808854c1a86d34d763385c10": "\\ S(f)", "84abce65ccacf7279b0100152255bf3b": "\\frac{\\partial^2 E_A}{\\partial \\theta^2} > 0.", "84abe2b6b86b2187508c0a9bd91060cf": "(-Y_a,X_a)", "84ac043e846216ae7f33fb3802f8162b": "\n\\frac{\\Omega}{4\\pi}=\\frac{\\Omega^{*}}{4\\pi}\\text{sign}\\left(\\left(r_{34}\\times r_{12}\\right)\\cdot r_{13}\\right).\n", "84ac334f0382762467317b0f8d2e2bc4": "dg/dh", "84ac3d9d2ce21c60cb2e98e2f2195508": "(4x_2 + 2x_1 + x_0) (4y_2 + 2y_1 + y_0)", "84ac3fc51f054f3a55769a541768048e": "\\mu_3=\\frac{k_0 p_3}{p_0} = \\frac{10\\times 0.2}{0.5}=4.0", "84ac754429c08eb678fc8a8fdaba6ae9": "U{\\partial v'\\over\\partial z_1} \\sim O\\left({a^2 \\over r^2}\\right).", "84accb0bfd37769db2da974fa2376356": " \n\\Phi(x_i)^{}_{} ", "84ad35a30959d9ed3e267cf61245a0f3": "a = \\sqrt{\\gamma RT}", "84ad63832eb813076a2a865d231f2da3": "\\mathfrak{P}^{118}", "84ad7b85aef2c4e3581cd7c25e3e21db": "= \\frac{1}{2}pq\\sin A + \\frac{1}{2}rs\\sin C.", "84ad9801a585a139d8a8bc56ad015a2a": "f(u) * f(v) = f(u * v)", "84ade6b00fefaef8833ed7b56e950df9": "n_1sin(\\theta_1)=n_2sin(\\theta_2)", "84ae22ba71f6c06c851c4f6d2b113240": "(t-1)^n+(-1)^n(t-1)", "84ae37ebe1094a2a7ef3e38beeac1278": " V(a,z) = \\sum V(a,n)z^{-n-\\alpha}.", "84ae3f625f5b752a646cacc764e7e0c6": " \\!\\ \\varphi^n = \\varphi^{n-1} + \\varphi^{n-3} + \\cdots + \\varphi^{n-1-2m} + \\varphi^{n-2-2m} ", "84ae77119a3a4a192f010ebf19217e7f": "\\mathcal{A} \\models \\Gamma", "84ae8d19409d20cdef0ebb373ee36253": "\\displaystyle{J=\\begin{pmatrix}0 & 1 \\\\ -1 & 0\\end{pmatrix}.}", "84ae9d72bb1f08e37859b2a1ecbbc6b0": " f(i)M^{k+1}(i,j) = f(i)\\sum^{N}_{n=0} M^k(i,n)M(n,j)", "84aeadbc23fc031d218afdb1b395d36e": " r+2s=b. \\,", "84af0e3b3a0c2d31d22b4eac9b969b27": "{m \\choose 0}_q ={m \\choose m}_q=1 \\, ,", "84af394c0a99391fd79285dea2162883": "\\int_a^b w(x) P_n(x) P_m(x) dx = \\delta_{mn}.", "84af5c002f714dcb76f361d86687df18": "\\frac{K_1}{k_2}(1+\\frac{k_3}{k_4}) + V_p", "84af83bab90cd1f6af678f4dbaee1c4c": "\\scriptstyle C'_0 \\;=\\; M_C \\,\\oplus\\, I \\,\\oplus\\, \\bigoplus_{i=1}^{k-1} C'_i", "84af8eab2b2cb10abaef2b968eaafa40": " \\Delta p \\equiv p - p_c ", "84afbc0908b8f3fb5fd345ca0ee90e3e": "Q\\ =\\ \\sqrt[3]{\\frac{\\Delta_1 + \\sqrt{\\Delta_1^2 - 4\\Delta_0^3}}{2}} \\quad\\qquad\\qquad {\\color{white}.}", "84b01a3789dfd2ffc9cbad132a80b45d": "\\langle B, E^+(A-T)| e^{-1}\\alpha_e(g)e=\\omega_e(g) \\text{ where }e\\in E^+(A-T), g\\in G_e \\rangle ,", "84b03b73df977b9ce82f034f1257e56e": "\\frac{256}{243} = \\frac{2^8}{3^5} \\approx 90.2 \\text{ cents}", "84b0c2eff3c4b69bf8c53baeba1da364": "\\mu \\sim \\nu \\iff \\mu \\ll \\nu \\ll \\mu.", "84b0cd0dba46382cf1736dac5785fe19": "\\int_V \\sigma_{ji,j}\\, dV + \\int_V F_i\\, dV = 0\\,\\!", "84b14cfdba3f9802da2d86e84694fd8a": "\\gamma v t' = \\gamma^2 v t - \\gamma^2 \\beta^2 x \\,", "84b1aba6b6a63be2d44d5d792482acd7": "\\pi=(\\pi_i)", "84b1b8759ee51108f861f2dee1813f83": " X \\to Z", "84b1bf11da407972e4fed9d724309e66": "(a_1,\\ldots,a_m)\\in A", "84b1c1d75c9bf7d6b9d58c8b52d3fadc": "(\\operatorname{\\mathbf{Proj}}\\, S)|_{p^{-1}(U)} = \\operatorname{Proj} (S(U)).", "84b212f2c5c33cbcd61257ff94ca8782": "\\bigstar || \\bigstar \\bigstar |", "84b237fa7629494a6933dd916a32c597": "{}^{n}2 = \\operatorname{A}(4, n - 3) + 3", "84b270f087151fd31b62597ddcb617b8": "\\scriptstyle{\\langle\\hat{A}\\rangle}", "84b31f2850be6cdeb5265bb609aefea6": "v_s=\\frac{1}{2\\pi r}\\kappa.", "84b32d4aea507572812771155659fd04": "\\scriptstyle \\sigma_{ij} = -pI", "84b354389165dba9ac0e61f8dc41b994": "A^{**}=\\alpha_{m-2}P^{**}A_{d}^{**}\\left(P^{**}\\right)^{T},", "84b372fa71eac7c126eeaed7b72f3e7b": "\\hat{r}\\ ,\\hat{t}\\,", "84b37c0f31ad25d2a3dbb3d1b1f96658": " b:\\mathbb{R}^k \\to \\mathbb{R}^k ", "84b39d38b7668c8582f8d55c9ba89b72": " \\mu\\ ", "84b3dfb3b2aecb26233bc8c9c618bedc": "\\le r", "84b42753cd107aa95197af020f51d7c1": "A,f \\in F_2[T],f \\ne 0\\,", "84b42d5f80e3053208a6e6b4b5d9b9ee": "b, b^{n_1}, (b^{n_1})^{n_2},...", "84b44f9c974adc1aaef94cdf1032c9c5": " \\text{(2)} \\qquad W = Q = Q_{in} - Q_{out} ", "84b48c8215604ff9f9b31a467b440cb2": "\\theta(\\xi)", "84b4c9d1f6b1e333539c8c936f431b8d": " \\int T(0,x) \\mathrm{d}x = x T(0,x) - \\frac{1}{4 \\pi} \\ln(1+x^2) + C ", "84b4d4f570b8175bc3dc27621830fbed": " \\ddot{\\varepsilon} = -\\frac{m^4}{L^8} \\, (m^2+L^2) \\, \\varepsilon + O(\\varepsilon^2) ", "84b5654fc2072dab3a3156e002a40897": "8\\pi G/c^4", "84b5775d2fd02c5570303d1507d611a0": "\n |j_1 m_1\\rangle |j_2 m_2\\rangle |j_4 m_4\\rangle |j_5 m_5\\rangle, \\;\\; \n m_1=-j_1,\\ldots,j_1;\\;\\; m_2=-j_2,\\ldots,j_2;\\;\\; m_4=-j_4,\\ldots,j_4;\\;\\;m_5=-j_5,\\ldots,j_5.\n", "84b59c687f73bf70ace546672366f75c": "p_k(E,\\mathbf{Q})\\in H^{4k}(M,\\mathbf{Q})", "84b6034213d7b78c31d7b5ccaa4db77e": "a^*(z)", "84b605f11d7045688acc1ab169290d9b": "\\{e^{inx} \\quad| \\quad n \\in \\mathbb{R}\\}", "84b61ee86163ad6d10b835f158ddd3c2": " \\mathfrak{g} > [\\mathfrak{g},\\mathfrak{g}] > [[\\mathfrak{g},\\mathfrak{g}],\\mathfrak{g}] > [[[\\mathfrak{g},\\mathfrak{g}],\\mathfrak{g}],\\mathfrak{g}] > \\cdots ", "84b666615eb5c8f39e77cb8956ee9385": " x^{**} = (x^*)^* = x ", "84b6a1a7e28b0c17938d5ac8d4565c74": "P = {(4x - 6) \\choose 5} \\div {50 \\choose 5},", "84b6ab29dd6097592849cd24d27bb3db": "\\frac{d \\sin\\theta}{ds} = \\cos \\theta \\frac {d\\theta}{ds} = \\frac{1}{\\rho} \\cos \\theta \\ ", "84b6ab60cbb6d1c05a6b378fd0d8bae9": "(x_1-a)x+(y_1-b)y = (x_1-a)x_1+(y_1-b)y_1\\,", "84b6b8ff30b1b4da7ddff213dbf4a75d": "f(c^+)", "84b7179235da7bab630005fcf9da1263": "\\mathbb{P}^n\\mathbb{C}", "84b726e98caf993f94bf1a24bbb2cbe6": " X \\sim D(a,b,p) \\iff \\frac{1}{X} \\sim SM(a,\\tfrac{1}{b},p)", "84b73fe6a45bf6a0e58d9bf1ea3d7f4e": "\\mathbf{u}=\\mathbf{z}\\times\\nabla\\psi'\\equiv(-\\psi'_y,\\psi'_x,0)", "84b741e5767df75e593e4d4ed9fb683d": " \\Gamma (s,x) ", "84b7562b9ad76a723a716be1ba144fc6": "a=0.9, b=-0.6013, c=2.0, d=0.50", "84b759cd32074059ac5f299173658cdd": "|{\\psi_{gr}}\\rangle=\\lim_{\\tau\\rightarrow\\infty}\\frac{e^{-\\tilde{H}\\tau}|{\\psi_P}\\rangle}{||e^{-\\tilde{H}\\tau}|{\\psi_P}\\rangle||},", "84b7c18462cfb0b891f775b670c1a29f": "t(\\phi)", "84b7d31110f196df1a715147462e908a": "f(x_0)=0", "84b7f965383b3bc567bba4f0f194607e": "A \\to \\alpha_1 \\And \\ldots \\And \\alpha_m", "84b83bb7bfec1c2b7d17265b6382b3a2": " \\mathbb{C}^{n}", "84b8fe1bf2bef5839eb54c8485a2e429": "\\Psi(x,t)", "84b907f4f45e33b7d4d191be44419ebb": "\\mathbf{Gr}(r, \\mathcal E)(T)", "84b9205000cb021cc39cbb1d0d6fe32a": " K = \\lim_{r\\to 0^+} 3\\frac{2\\pi r-C(r)}{\\pi r^3}.", "84b99c69ab330420f32476c40e8ef09d": "d_i=x_i-x_0 \\,", "84ba5705ddeef8df41eb898cb87ba575": " r \\in \\lbrace 0, 1 \\rbrace^{\\lfloor \\log^2n \\rfloor} ", "84baabbbc4de6c648983b76bc4d10ecf": "P_1>P_2", "84bad237cc3ed9d4cc67153179233846": "\\vec T_{b0}", "84bb051968893fd9cbb806a9313a3b66": "\\scriptstyle k \\;\\geq\\; 1\\,", "84bb4ff5772e3825832774d20173d184": "x=t a + (1-t) b", "84bb6b4f8ec9f3b1bcdc1d8364338cef": "\\mathcal{D}^{\\mu \\nu} \\, = \\, \\frac{1}{\\mu_{0}} \\, g^{\\mu \\alpha} \\, F_{\\alpha \\beta} \\, g^{\\beta \\nu} \\, \\sqrt{-g} \\,.", "84bbbfff1657267b86b5775fd93908a6": "\\Delta G_i^{stat} = \\sqrt{\\sum_x (ln P_i^x)^2}", "84bbe84c34b87c272350dadf5b80cc24": "\\alpha \\mbox{ and } u_{n}", "84bc37770ed1854821c108d923b4e4fd": "\\ \\mbox{Bond Order} = \\frac{(\\mbox{No. of electrons in bonding MOs}) - (\\mbox{No. of electrons in anti-bonding MOs})}{2} ", "84bc421e107d9461a0757b767c0fbbc1": "u_\\lambda(T) = {8\\pi h c\\over \\lambda^5}{1\\over e^{h c/\\lambda k_\\mathrm{B}T} - 1}.", "84bc5ee660c1aa66b8a915e8b33f4c1f": " \\mbox{lim}_{t \\ \\infty} x(t) = \\infty ", "84bc8cb168b955d01e013802afab6e63": "\\frac{\\eta_{sp}}{c}", "84bca81df69957153d3dc9dbfeec0ddc": "\\mathbb{F}_2^n", "84bdd9f56fdc5cf50e08e81928617f3e": "\n\\left( \\boldsymbol{\\nabla} \\cdot \\mathbf{E} - \\frac{\\rho}{\\epsilon_0} \\right)- c \\left( \\boldsymbol{\\nabla} \\times \\mathbf{B} - \\mu_0 \\epsilon_0 \\frac{\\partial {\\mathbf{E}}}{\\partial {t}} - \\mu_0 \\mathbf{J} \\right)+ I \\left( \\boldsymbol{\\nabla} \\times \\mathbf{E} + \\frac{\\partial {\\mathbf{B}}}{\\partial {t}} \\right)+ I c \\left( \\boldsymbol{\\nabla} \\cdot \\mathbf{B} \\right)= 0 \n", "84be27e9ce9650a9427053211b613549": "\\widetilde{f}", "84be34c29aafc5fd440ee5eeac8f3a0f": "D_M", "84be3c4dfd02876e2190dd6240410b4c": "\\mathbf x[k+1] = e^{\\mathbf AT}\\mathbf x[k] + \\left( \\int_0^T e^{\\mathbf Av} dv \\right) \\mathbf B\\mathbf u[k]=e^{\\mathbf AT}\\mathbf x[k] + \\mathbf A^{-1}\\left(e^{\\mathbf AT}-\\mathbf I \\right) \\mathbf B\\mathbf u[k]", "84be406cb43e8508311ab5baf8695169": " pc=E {v \\over c} ", "84be4b8d8e57f5bed3189e9c55d1c3f0": " \\frac{1}{\\sqrt{4\\pi\\varepsilon_0}}\\left(q, \\rho, I, \\mathbf{J},\\mathbf{P},\\mathbf{p}\\right) ", "84be6d11f03ee5270c398dfbb202bb62": " \\mathcal{D}_n ", "84bea3173fcf7c56e378c9a94aeed3f1": "\\scriptstyle E^*\\otimes E^*", "84bea6c5582e37286d87c03120cfcd6e": " \\lambda = \\frac{1}{MTBF}", "84bebb31a9eb58a7054a3b9a9f3f1643": "P(x,D){u(x)} = \\delta(x) + \\omega(x),\\,", "84bebfadbe6ec8ddd5596c64dfc2866c": "{E} = \\frac{V_1^2-U^2+2UV_1\\cos\\alpha_1}{2}", "84bee3e41585fa0d7d8916923050b8bf": "Z = Z_0 \\frac {1 + \\rho}{1 - \\rho}", "84befb894bc5f1d3706972966d793fd8": "\ne_2 =\n\\begin{pmatrix}\n0&-1\\\\\n1&0\n\\end{pmatrix}\n", "84bf26b658ddef145bc078cae6be2168": "\\mathrm{%O_2 \\ in \\ combustion \\ gas} = -0.00138 \\times (\\mathrm{% \\ excess \\ air})^2 + 0.210 \\times (\\mathrm{% \\ excess \\ air})", "84bf745ad2797d90ece229d27eadd284": "c_s:=c_{{\\rm eff}}", "84c0310d4af69570ebc9a905c947a53a": "P'(t) = P(t) Q", "84c035dacacd36b0a52ac8597f9f90e4": "Ax^2\\ +\\ Bxy\\ +\\ Cy^2\\ +\\ Dx\\ +\\ Ey\\ +\\ F\\ =\\ 0,\\ B \\ne\\ 0", "84c06e02509223a079fa6dcf8f0c5d4d": " P_B = \\sum_{i=0}^{u}\\sum_{j=0}^{u}f_ig_j\\beta_{ij} ", "84c0bb818d08ef30e6d23d1f04a846fe": "(2,5,6)", "84c14b11dec6f03e601631727b69be4b": "(a_1,...,a_d)", "84c1900ed75347d769bd6fdba8d5d691": "\\theta\\left(g_{n}\\right) = n\\pi.", "84c1a8e275576901b4ee96d783923b17": "V_\\mathrm{s} = RI_\\mathrm{s}\\,\\!", "84c237497d846a75df9cae7bf189b56a": "E M_+^2 \\approx 1.64493 \\ldots \\ , \\ \\ \n Var(M_+) \\approx 0.0741337 \\ldots.", "84c248cd29d8aa0db7148385c7468ef9": "U= \\textstyle{\\frac{2}{3}}X", "84c28738a736e3a24863a1f87f6bf9b9": "\\Sigma_{v \\in V} d(v)|A_{S_{(v)}}| ", "84c297d2dea799623f70f8d73dc7f1a3": "\\mathcal{P}=\\{\\mathbb{P}(x; \\mu, \\sigma) = \\frac{1}{\\sqrt{2 \\pi} \\sigma} \\exp\\left\\{ -\\frac{1}{2\\sigma^2}(x-\\mu)^2\\right\\} : \\mu \\in \\mathbb{R}, \\sigma > 0\\}", "84c31f7af2d038bf9f4a1cb98ce2715f": "f \\mid g", "84c3d00dda7377f382e999e15b582e23": "\\tau(p)\\equiv \\sigma_{11}(p)\\ \\bmod\\ 23^2\\text{ if } p\\text{ is of the form } a^2+23b^2", "84c3db4ce8c19d55cc867150edaa9ea7": "\\tilde{\\nu}=0", "84c40473414caf2ed4a7b1283e48bbf4": "(", "84c40b0949a08f29c7a4ace8048895f0": "\\alpha < \\min(1, e^{-\\beta \\Delta E} )", "84c48493bd6b9122a50caf4cda204107": "q_0=0", "84c544ef07dff3c84d54cc427751c018": "n\\geq \\sum_{i=0}^{k-1} \\left\\lceil\\frac{d}{2^i}\\right\\rceil.", "84c581a1b75196b21329142ef2916e7d": "\\lambda _1 + \\lambda _2\\,", "84c5985eb1a8f8ac63180881b460828d": "\\int_a^b f(x)d_hx", "84c5bfaa45e8929538a6a06d81610dd2": "[u][v]", "84c5d62acae4c373e4aeb96eb8c91a6c": "10^{-16}", "84c62bb6912552343de1b0c43ab64772": "C_p - C_V = T \\left(\\frac{\\partial p}{\\partial T}\\right)_{V} \\left(\\frac{\\partial V}{\\partial T}\\right)_{p} ", "84c650459bdc1c089b378247e65636cf": "A_n = n V_n = \\frac{n \\pi ^ {n/2}}{\\Gamma(1+n/2)} = \\frac{2 \\pi ^ {n/2}}{\\Gamma(n/2)}\\,,", "84c6bafebdbd302dc4e6b3d114143ba7": "\\mathrm{Ref}_a(v) = -\\frac{a v a}{a^2}", "84c6e0c89de6228d8c3a9324336c9982": "x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+sx+t", "84c6fbc3e2eed8320d8ddb33a042c2e0": "Pr(K=k) = \\sum\\limits_{A\\in {{F}_{k}}}{\\prod\\limits_{i\\in A}{{{p}_{i}}}\\prod\\limits_{j\\in {{A}^{c}}}{(1-{{p}_{j}})}}", "84c74a2e631f88cfe96b5dadc0032b64": "\n I[f] = \\int_{x_0}^{x_1} \\mathcal{L}(x, f, f', f'', \\dots, f^{(n)})~\\mathrm{d}x ~;~~ \n f' := \\cfrac{\\mathrm{d}f}{\\mathrm{d}x}, ~f'' := \\cfrac{\\mathrm{d}^2f}{\\mathrm{d}x^2}, ~\n f^{(n)} := \\cfrac{\\mathrm{d}^nf}{\\mathrm{d}x^n}\n ", "84c74f33372e4161de3740d238b7afd7": " \\sigma^2 / \\lambda", "84c754a6da18b9ee77636eb0d40d55b3": "\\alpha(f)", "84c75a906e1e7548f6df1e4e3aa3af52": "Pmf", "84c76f159153fbd5d44c9085c7c03e60": "g_{X+Y}=g_X+g_Y", "84c7922cd2e5b240578e6100f46e261e": " P_{\\mbox{lacunary}} ", "84c7cf3e3b826b30fb3c0ebe78794751": " F^{el}(t) = a k_i u(t) ", "84c7d542af6612dcf5653cd731d5f57a": "\\omega(\\mu) = \\omega(0) \\cdot {\\rm det} \\,( I - \\mu D^{-1}) . ", "84c82d14abe92eae15c528ad70091aed": " \\epsilon_i \\ ", "84c8412f2d31d3476d435171865c65a0": "Y_j = \\frac{1}{35} (-3 \\times y_{j - 2} + 12 \\times y_{j - 1} + 17 \\times y_j + 12 \\times y_{j + 1} -3 \\times y_{j + 2})", "84c860219011154e6a3ea1cf50b27eda": "= m \\Omega^2 r \\ . ", "84c885506d9efde5d5711291347b85e0": "\\partial_i", "84c89daa76dbccb4066d61c795f5eb5a": "\n\\frac{\\partial^m \\Gamma (s,x) }{\\partial s^m} = \\ln^m x \\Gamma (s,x) + m x\\,\\sum_{n=0}^{m-1} P_n^{m-1} \\ln^{m-n-1} x\\,T(3+n,s,x)\n", "84c8a846fb5003a6d1b03dd58225523e": "\\Pi_{H}\\,\\!", "84c956d7f37613235b9ee9154d572563": "2^4\\cdot 3\\cdot 5\\cdot 7", "84c95cf1575d35e6a705c6ef9741ab35": "\n f:= \\cfrac{2^{m-1}(1-R)+(R+2)}{(2+2^{m-1})(1+R)}(|\\sigma_1|^m -|\\sigma_2|^m) + \\cfrac{R}{(2+2^{m-1})(1+R)} (|2\\sigma_1 - \\sigma_2|^m + |2\\sigma_2-\\sigma_1|^m)- \\sigma_y^m \\le 0\n", "84c991c30f0e01fd65d5b459b757e00a": "\\mu \\frac{\\partial}{\\partial\\mu} y \\approx \\frac{y}{16\\pi^2}\\left(\\frac{9}{2}y^2 - 8 g_3^2\\right)", "84c9a6237e7c3af7ec83dba7cb978a54": "G_{a/b}=\\arctan\\frac{b}{a}", "84c9dba9c982d5fce02db596d67fa8c8": "\\rho = \\frac{\\Gamma}{K_{eq}}", "84c9e289ba65713f368fafd8997aeba4": " r=\\sqrt{(x-x')^2+(y-y')^2+z^2} ", "84c9f5bbc8be9cce9b7a6b138ac03f8b": " c^* ", "84c9f8654d6fa6e98c38c98aefdf0bb7": " \\{a^{2^{n}} | n \\geq 0 \\}", "84ca04b81144595440996c412d6ad329": " k_2 - k_1 \\ge 2 \\lambda ", "84ca2f61a02f52c06f812ddea3735d13": "\\mathcal{SHIQ}", "84ca4db17d814c0ee2e7a3040f8a4a93": "-\\cot\\delta' = \\frac{-\\cos\\delta}{\\gamma\\cdot(\\sin\\delta+\\beta)}", "84caba6983481de53a566973d1009ee1": "\n g_{\\vec z}(X)=(X-z_1)\\cdots(X-z_n).\n", "84cad83b6f4781c225ec11ff5214215d": "(x_1,y_1), (-b_2,a_2)", "84cb347ac876dc13245aad86572b2363": "g(T(\\mathbf{x})|\\lambda)", "84cb3ed282350380cca980000c0fa068": "e^{\\frac{2\\pi i}{8}} = \\sqrt{i}= \\frac{\\sqrt{2}}{2}+i\\frac{\\sqrt{2}}{2}.", "84cb64b6bacdb09907eeeeb66522f3ad": "\\rightsquigarrow x \\cdot (y \\cdot z)", "84cb668627372fb3dda863de06f86a54": "\\| P(T) \\| = \\sup_{\\lambda \\in \\sigma(T)} |P(\\lambda)|.", "84cb76df446b476be5f4bd9c3c674a13": "\\!V", "84cb78c3ba9962a1cb4f049283b1ea23": "Q_B = \\sqrt{2\\pi \\alpha^3m_0c^2C_B} = e. \\ ", "84cb78f7129cbb6c9b6e012b5ed35a97": "\\xi - \\ ", "84cb8708061306a11d72f6bcd9864207": "\n\\begin{array}{rcl}\nY' &=& Y' + 16\\\\\nU &=& U + 128\\\\\nV &=& V + 128\n\\end{array}\n", "84cbd5f5e18d254e600de00fc25e4f85": "0 \\leq j \\leq 31", "84cbfe9187bd9d304f877047d8ac5a00": "\\phi = \\omega t - kz", "84cc1c247eb7a5041c42eb04c7dee01e": "\\langle U \\rangle = \\frac{3NP}{2\\beta}- \\langle U_{\\mathrm {spring}} \\rangle + \\langle U_{\\mathrm{internal}} \\rangle ", "84cc54cb77bcefee789bfa2ad480af79": " V^\\pi(s)= R(s) + \\gamma \\sum_{s'} P(s'|s,\\pi(s)) V^\\pi(s').\\ ", "84cc93d3ae30cdcbdaef940103b48be4": "W(x)", "84cd5e378030157ee9415c339147ffab": "\\bar{8}", "84cd79da2205710f4044e610bbab0e53": "X^T1 = W", "84cdb4330a0705d8b1535dddbfdefd33": " \\rho(x,y)", "84cdd9748cf185355d289379a949e0ce": "f_i^{(0)}", "84ce223a3f1f61e673f70fa4461b75d0": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=\\frac{r_s c^{2}}{2 h^{2}}+\\frac{3 r_s}{2}u^{2}-\\frac{G Q^{2}}{4 \\pi \\varepsilon_0 c^{4}}\\left(\\frac{c^{2}}{h^{2}}u+2u^3\\right)", "84ce762ffe5e6ac12666823b9ff138a8": "\\mathbb{H}_{inv} \\, \\mathbb{H} = \\mathbb{I}", "84ce8a46dfa0228acb903858d3ba90a2": "P = P_1 \\cdot P_2", "84ceb2513893acb7c9ea001cc9053995": "x_{i+1}", "84cec31265f0a6144b63ad994adf7201": " \\frac{d}{d r} \\left( p +\\frac{B_\\theta^2}{2 \\mu_0 } \\right) +\\frac{B_\\theta^2}{\\mu_0 r}=0 ", "84cf10a24f32083e95d1a4c398632c90": "\\frac{1}{1 - \\eta} = \\frac{1}{\\sqrt{1 - \\frac{v_e^2}{c^2}}}", "84cf401e1c9daa1a28f1a124a54d0fbf": "f = S \\to AA", "84cf7d7bd4c6613492591b3d6ebe6e6e": "\\delta Q\\, ", "84d012085969edb80c21f2678a30e66b": "W_i = M_i \\oplus E(M_i,B_i)", "84d03af644caab43f60e21fdd27b4952": "P_1,\\ldots,P_8", "84d05bf7e942ef729c8ac3aac42597ff": "l(C) = \\sum_{x\\in\\mathcal{X}}l(C(x))\\mathbb{P}[X=x]", "84d0a79cfade9fb5fb8fe18529752f5f": "\\eta_{\\mu\\nu}\\,", "84d0b4b6e7035fb85e8280dd55a7d6d1": " C_{1\\epsilon} = 1.44 ", "84d12c2ae492c417cc80164000af76e1": "y\\in \\omega(x,f)", "84d148a8dbf441e543b5ea1aebd5fba2": "(!x)=\\frac{\\Gamma(x+1,-1)}{e}", "84d14d85dc52a4025234e9f9185b9df0": "B(x)>{{B}_{x}}(x)", "84d159c2b3ecaa97ff80e0f170edfd05": "\\mathrm {DOF} = \\frac\n{2 N c \\left ( m + 1 \\right )}\n{m^2 - m^2_{\\mathrm{h}} } \\,.\n", "84d1a2254fb46c3fe1a0325e13bbe273": "(\\mathbf{r}_i - \\mathbf{r}_j)^2 - L_{ij}^2=0, \\, ", "84d1cef595009658eef96c77d2b899a9": "64 \\times 60 = 3,840\\,", "84d1d75fa6c4976440e45746002e5c74": "\\boldsymbol{r'}", "84d26f88801f3e037c12effe5296bd7d": "C(D,G) \\cong L(G)/\\ker(\\alpha), ", "84d2942685b0b048e2e42649d4d58608": "\\beta= {v_1,v_2,\\ldots,v_n}", "84d2f3bf60636d49e0540dd1a342d883": "1/f", "84d31c2e5d2d4da85cad02066135d05c": "\\epsilon_1", "84d346d0167a7a101379480700f04478": "\\Theta^k = d\\theta^k + \\sum_{j=1}^n\\omega^k_j\\wedge\\theta^j = \\sum_{i,j}T_{ij}^k \\theta^i\\wedge\\theta^j.", "84d38695c038fd16540cfd7d4c07be2b": "\\bar{\\partial} f (z,\\bar{z}) = g(z)", "84d3a33fde552fc6148664c09686965b": "S=E", "84d3afdec8b047d157d0c81ac2625158": "|E_{-}\\rangle", "84d42d4792a26cf3735d969c7a44817f": "\\vec{v} \\cdot \\vec{w}", "84d47e9d7567156106d3641eda5af97f": "F(x;\\alpha)=1-\\frac{\\exp(-\\alpha x)}{(1+\\exp(-x))^\\alpha}, \\quad \\alpha > 0 .", "84d47f2670238a959529d2b8888b07fa": "E(x,z,t) = A_m a(x,z) e^{i(k_0 n z - \\omega t)}", "84d4931db88c19db8dfbdcd16c9d3e08": " \\sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}=\\sum_{k=0}^{d}h_{k}t^{d-k}. ", "84d4aa2046ead53bc51430c7a5c7bd73": "(A,B)\\,\\!", "84d546d84e15b7865393d8ad10981dfb": "\\frac{dx}{dy}=-\\frac{1-\\alpha}{\\alpha}\\left(\\frac{x}{y}\\right).", "84d5507b942c2aca1f2acd8474c8f18e": "H(V)=-\\sum p(v)\\log p(v)", "84d5793bb25e21d932b8f0f9d0b89605": "i\\hbar\\frac{\\partial\\Psi(\\mathbf{r},t)}{\\partial t} = \\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r}) + g\\vert\\Psi(\\mathbf{r},t)\\vert^2\\right)\\Psi(\\mathbf{r},t). ", "84d5c7c1a7c8c482edc55ba9ef3e15fa": "0.4 \\leq \\rho_{BC} \\leq 0.5", "84d5e20b47d13366c9cfd159b45d005d": "\\phi(\\alpha,\\beta,\\gamma)", "84d67bdb56b7dac1baa63052a9b66679": "x = \\psi(\\phi x)", "84d693311fbbed09bf1f1262e4e42a35": "a'+b=1", "84d6b12645e52c51a47987819aaa131e": "1+3=4 \\bmod 13", "84d71a3e1858e014492911f142703fa3": "\\displaystyle \\sin{\\frac{A}{2}}\\sin{\\frac{B}{2}}\\sin{\\frac{C}{2}}=\\frac{1}{8}.", "84d7262079dd534135405a03971ea575": "A \\xrightarrow[-f]{} \\alpha", "84d7cdd7acb77545bbde8c8933183a8d": "!n = n[!(n-1)] + (-1)^n", "84d7ce950234838668713c2053d2fa16": " \\begin{align} \n\\mathbf{A} & = (A^0, \\, A^1, \\, A^2, \\, A^3) \\\\\n& = A^0\\boldsymbol{\\sigma}_0 + A^1 \\boldsymbol{\\sigma}_1 + A^2 \\boldsymbol{\\sigma}_2 + A^3 \\boldsymbol{\\sigma}_3 \\\\\n& = A^0\\boldsymbol{\\sigma}_0 + A^i \\boldsymbol{\\sigma}_i \\\\\n& = A^\\alpha\\boldsymbol{\\sigma}_\\alpha\\\\\n\\end{align}", "84d7d756246cf37b4f8d52d27b90cb35": "\nA_{p}(\\kappa)=\\frac {I_{p/2}(\\kappa)} {I_{p/2-1}(\\kappa)} . \\,\n", "84d7f05f12665955cbc2013704384b23": "V = \\sum_{i=1}^3 V_i", "84d7fbd01871bbd2bf1efc47d2a98f0f": "0\\rightarrow \\mathbf{Z}/p\\mathbf{Z}\\rightarrow K \\xrightarrow{x\\mapsto x^p-x} K\\rightarrow 0", "84d85341cb5c941ca05e5f4db6ece32f": "a, b \\in \\mathbb{R}, a < b", "84d85cdd8ed2f8a282dd6818629d3e9e": "\\operatorname{Cl}_s(\\theta)", "84d91e6a1a4e6cf194b7d697fbfb44e6": "n(\\vec r)", "84d926876fdd9ea16a1bde2ff3f42ba2": " \\dot{\\mathbf{x}} = f \\left(\\mathbf x \\right) ", "84d94c1856d60912c31d7ca3db192212": " \\rho(r)", "84d9b9930b61e3bdb167599b61eaeccb": "{\\color{Blue}~2.34}", "84d9c985a9cec51c164bd40a9c1cb4ab": "\\mu_{1/n}^{* n} = \\mu.", "84da0c7bf81303c7df20b3a0d4ebc8c3": "v_{\\mathrm F}", "84da3f29d27525fa77fe4b87108bcdd4": "(g^b)^a", "84da4cd3bec565f132187a9bd8b0bfd9": "r=r_0", "84daa6b6d8a13a058dcb1bc78556f0c0": "\\delta(S)", "84dad8835577c18a8d8c63eac0e1f806": "k_\\mathrm{spec} = \\frac{1}{\\sqrt{(N\\cdot L)(N \\cdot R)}}\\frac{N\\cdot L}{4\\pi\\alpha_x\\alpha_y} \\exp \\left[ -2 \\frac{\\left(\\frac{H\\cdot X}{\\alpha_x}\\right)^2+\\left(\\frac{H\\cdot Y}{\\alpha_y}\\right)^2}{1+(H\\cdot N)} \\right]", "84daeadc81b0a449647af4b2dfa0a456": "\\overline x", "84daf62b2b5c4c6d52e9dc72999516ed": "f:D \\times D^{'} \\rightarrow D^{''} ", "84db06e18e21ab418cd89cb9805cf20b": "\\text{reduction ratio} = \\frac {\\text{flex spline teeth} - \\text{circular spline teeth}} {\\text{flex spline teeth}}", "84dbb3eae6e7e27b525f2d1f6cc3d907": "\\ -\\frac{d[A]}{dt} = 2k[B]^2", "84dcf82dbcd26df11a8b7187c3d95a57": "\\dot{r}<0 .", "850b0ee43f85b0f61296408c94861cb3": "(\\tau{\\to}\\tau){\\to}\\tau", "850b1fbc753f8b37f6be3f96b435470c": "f_y(x):= f(x,y)", "850b351e24c99fba992e1299fe9a30c2": "E[\\pi_H]=p A + (1-p) C", "850b56005b02388f50ffc9d6c6950fcc": "\n\\frac{\\partial^\\beta f(t)}{\\partial t^\\alpha}=\\lim_{t_1 \\rightarrow t}\\frac{f^\\beta (t_1)-f^\\beta (t)}{t_1^\\alpha-t^\\alpha}\\,, \\quad\\alpha>0, \\beta>0", "850b5bb5758cd37f720c943a0a9629c0": "|c_{i,j,k,...}|^2", "850bb083621d8334a8b891f4379ef852": " A_{24576} < \\pi < A_{24576} +D_{24576}", "850bd5cb65eb898a239f76b9febeae60": "\\tilde{P}; \\tilde{w} = Rec(w',\\tilde{s}).", "850c0fb51780d71f9f788f3eb263ed4e": " [E]_0 = [ES] \\left ( \\frac{K_m}{[S]} + 1 + \\frac{K_m[I]}{K_i[S]} \\right )= [ES] \\frac{K_m K_i + K_i[S] + K_m[I]}{K_i[S]}", "850c1dd887f47537b93f7559ea071ccb": "QH^*(X, \\Lambda) = H^*(X) \\otimes_\\mathbf{Z} \\Lambda.", "850c403baebd6c716fbcbb0d7d90e63e": "\nX = \\mathbf{fA}v[\\mathbf{fA}] = \\mathbf{f}v[\\mathbf{f}].\n", "850c7ed7d778feca8d747c349cb66bc9": "y_1=-\\cos \\Omega \\cdot \\sin \\omega - \\sin \\Omega \\cdot \\cos i \\cdot \\cos \\omega", "850c96c1bf1c8eab8cd36b58d644140b": "N+P=n \\quad (3) \\,", "850c9d44fb00dcbb45207a2d4278b243": "y ~=~ \\beta_0 + \\beta_1 T + \\beta_2 S + \\beta_3 (T \\cdot S) + \\varepsilon", "850cdde1393baf9cd6cc2af30b7371c1": "\\sqrt{-g} = t^{p_1 + p_2 + \\cdots + p_{D-1}} = t", "850d58bb66aa44d98adaa2ca41b04165": " \\phi(\\vec{r}) = -{\\frac{1}{A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{{(\\vec{r}-\\vec{r}') \\bullet{} \\vec{F}(\\vec{r}')}}{|\\vec{r}-\\vec{r}'|^n}d\\tau'}}", "850d6ac77221ca286a3fd7043cc073e1": "X \\subseteq \\operatorname{cl}(X)", "850d86956f9c01be30753e68dcfda55e": "w_j = W_j / \\sum_{j=1}^{n}W_j, j = 1, 2, . . ., n ", "850dbaa086e6ad99e4bf5b11462a9d86": "f^m(n)", "850dfde0cfa1c38368aecd24628ba820": "\\delta Q_0", "850e131acdc62e97541780ff71b4faec": "d(\\sigma)=\\frac12 (d(\\sigma 0)+d(\\sigma 1))", "850e9712f97e1f663b9e7b8ea2980e54": "K(u|\\eta) = A(\\eta+u) - A(\\eta),", "850eb2bc5f12cdac788f79b790726415": "c^T", "850f2d3ef51570833a8b23e9c83e0da1": "\\Pi^V_2", "850f81286d0f9f3505c5c917c0dac929": "2k+1", "850f8a3d10b81ae37a44c3edc41e4af4": " r^{6.5-lm} ", "850f8c9c781dd4290bf474b206e52201": "\\langle \\mathbf{z},\\mathbf{w}\\rangle = \\sum_{n=1}^\\infty z_n\\overline{w_n},", "850fa4f15475d4a9684c24457b5fd76a": "\n\\begin{align}\n\\mathbf{B}(\\mathbf{r}, t) &= \\boldsymbol{\\nabla}\\times \\mathbf{A}(\\mathbf{r}, t)\\\\\n\\mathbf{E}(\\mathbf{r}, t) &= - \\boldsymbol{\\nabla} \\phi (\\mathbf{r}, t) - \\frac{\\partial \\mathbf{A}(\\mathbf{r}, t)}{\\partial t}, \\\\\n\\end{align}\n", "850feb907a49c3ba08bfd18a8e285546": " f : (\\mathbb{C}^n,0) \\to (\\mathbb{C},0) \\ . ", "85100f83e9d70b652df930e1324042cd": "(x-x_0)^2+(y-y_0)^2=r^2.", "85101564316c920424d0b0d88a454b53": " \\sigma_{\\mathrm{ess},1}(T) \\subset \\sigma_{\\mathrm{ess},2}(T) \\subset \\sigma_{\\mathrm{ess},3}(T) \\subset \\sigma_{\\mathrm{ess},4}(T) \\subset \\sigma_{\\mathrm{ess},5}(T) \\subset \\sigma(T) \\subset \\mathbf{C}, ", "85105e006d62681f263a7e4011eaa925": "\\begin{pmatrix} a & c \\\\ b & d\\end{pmatrix} \\leftrightarrow q = \\frac{(a+d) + (c-b)i + (b+c)j + (a-d)k}{2},", "8510ffa81f2dc7934e9718dc94099e45": " \\mathbf{r}\\,", "85111f25624dcfec4d3a1c11a5eb650d": "d=\\frac{ \\lambda}{2 (n \\sin \\theta)}", "8511349c6cd82c7afdfaae492a387a64": "\\int_0^1 x^{4n}(1-x)^{4n}\\,dx\n=\\frac{1}{(8n+1)\\binom{8n}{4n}}.", "8511cc8f7be6d9a35547f02f113db147": "\\frac{1+3x+4x^2+12x^3+8x^4-8x^5}{(1-x^2)(1-16x^4)}", "85123917f52b986a6fef29641205546d": "bandwidth = frequency/Q", "8512835b0c432050a7f2c77d0831792e": "M_{PAW} = \\frac{f * T_i}{60} * (P_{IP} - PEEP) + PEEP", "85128f037f485a4a482a8b9f8b4cc20d": "\\begin{bmatrix} \\cos \\theta & \\sin \\theta & 0 \\\\ -\\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}", "8512ff4af67804fd2fcfe751e77838c1": "g_w", "85134143bfea7109921fe94ffb989ca7": " L = mvr = n \\hbar ", "85135469e9e1c45d2d0a9901da3292c9": "\\left(\\frac{a}{a-2it}\\right)^{\\frac{p}{2}}\\frac{K_p(\\sqrt{b(a-2it})}{K_p(\\sqrt{ab})}", "8513760391ebabf7f1f28110f46514cc": "Z_1^'\\cap Z_2^'", "85138261c57eaad61471dd1c75cf075d": "\\hat{M} = (\\hat{m}_{ij}) \\in \\mathbb{R}^{I\\times J}", "851389b8df543a9251ad9531877be13d": "2^m, 1, 2^m", "85139a2c28165fd88e725c3328ece8a4": "\\begin{array}{lll}z& = & r \\left (\\cos \\varphi + i \\sin \\varphi\\right) \\\\\n& = & r e^{i \\varphi}.\n\\end{array} \\,\n", "8513a0a58e56c2992bdced3ea701d25d": " {d^2 \\mathbf{h} \\over d\\tau^2} + R \\mathbf{h} = 0 ", "8513e13f662ef60c841c851fe548d770": "F(p_1, p_2, p_3,p_4) = \\displaystyle\\sum \\limits_{r_1,r_2,r_3,r_4} f(r_1,r_2,r_3,r_4) \\cdot(-1)^{p_1r_1 + p_2r_2 + p_3r_3 + p_4r_4}.", "8513fa76b89bb5317bb2129abd167898": "\\aleph_{\\beta+1}", "8514a8aa510784ed1da6a28bd6b41f1c": "\n\\bar{T}_\\text{fixed} = 4N_e\n", "8514dff80e6cc0618cb5a01134aca207": "\\psi(\\mathbf{r},t) \\rightarrow e^{ig\\bar{\\theta}(\\mathbf{r},t)}\\psi(\\mathbf{r},t) ", "8514ff67c3ecad588655aad14f64ee08": "\n {\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}~\\text{dV}\\right) = \n \\int_{\\Omega(t)}\\frac{\\partial \\mathbf{f}}{\\partial t}~\\text{dV} + \n \\int_{\\partial \\Omega(t)}(\\mathbf{f}\\otimes\\mathbf{v})\\cdot\\mathbf{n}~\\text{dA}\n = \\int_{\\Omega(t)}\\frac{\\partial \\mathbf{f}}{\\partial t}~\\text{dV} + \n \\int_{\\partial \\Omega(t)}(\\mathbf{v}\\cdot\\mathbf{n})\\mathbf{f}~\\text{dA} \\qquad \\square\n }\n", "8515044da4f5680795453375cfb63920": "\\ MCI_t = MCI_{0}exp[(r_t - r_0) + (a_{1}/a_{2})q_t].", "85152bda8c90cc760b256b6f77f1b212": "\\begin{align}\n\\eta' &= (\\tau\\ast\\sigma)\\circ\\eta \\\\\n\\varepsilon' &= \\varepsilon\\circ(\\sigma^{-1}\\ast\\tau^{-1}).\n\\end{align}", "85155489a39365c0209d36421eb5aebc": "\\int_0^1 p f(p)\\,dp = {s+1 \\over n+2}.", "8515c5d89288e50752ea47d7c818ddcd": "\\delta^{ij}", "8515eb41c698b3c52e1d07ea917bed60": "9+5 = ?", "8516a7749c1e4706bd89ac8010aae09e": " \\beta\\to\\alpha\\vee\\beta ", "8516ac9a3ff95af33a172bac5ee209bf": "M\\big(M(x, M(x, y)), M(y, M(x, y))\\big)=M(x, y)", "8516f5204537856de6e8345b4d358e88": "\\left \\| s_p(t) \\right \\|^2=\\left \\| s_{p+1}(t) \\right \\|^2 + \\left| B_p \\right|^2,", "85170af6ea51242e648c02780dbd44be": "\\prod U_i", "85170c0df91cff2c8646af78045c27f2": "P|n_1, n_2; A\\rang = - |n_1, n_2; A\\rang", "85171c33b7bd40653b41cf2e5da7abcf": "\\left\\|\\mathbf{a}\\times\\mathbf{b}\\right\\|=\\sqrt{\\mathbf{a}^2\\mathbf{b}^2-(\\mathbf{a} \\cdot \\mathbf{b})^2}.", "8517616b0758ea3b2bb7e3732036af7e": "\nm_1 \\ddot{x_1} + { (c_1+c_2) } \\dot{x_1} - { c_2 } \\dot{x_2}+ { (k_1+k_2) } x_1 -{ k_2 } x_2= f_1,\n", "85176d4e0d9d0026117cd431beae9bd0": "(x_0,t) \\sim (x_0,t')\\,.", "85177a2723b834f749bbc6dfa3f1f571": "w^2+x^2=1,", "85183a1bd949078e6c254b7463647d47": "\n\\begin{align}\n& {} \\operatorname{mfnchypg}\\left(\\mathbf{x};n,\\mathbf{m}, (\\omega_1,\\ldots,\\omega_{c-1},\\omega_{c-1})\\right) \\\\\n& {} = \\operatorname{mfnchypg}\\left((x_1,\\ldots,x_{c-1}+x_c); n,(m_1,\\ldots,m_{c-1}+m_c), (\\omega_1,\\ldots,\\omega_{c-1})\\right)\\, \\cdot \\\\\n& \\qquad \\operatorname{hypg}(x_c; x_{c-1}+x_c, m_c, m_{c-1}+m_c)\n\\end{align}\n", "85186cd3ba7a4e181fe9e718983dd1a6": "\\textstyle{\\int_M K\\,d\\mu}", "85188c47d562b5a818daf779b320a49f": "\n\\hat{b} = \\frac{2}{N} \\sum\\limits_{i=1}^N R_i \\sin{\\theta_i}\n", "8518bea326341101dc7ce435097e0821": "K(M)", "8518e1d7d2bad9dd23044f8acc0cd1d0": "Z(A) = \\{a\\in A : [a,x]=0 \\text{ for all } x\\in A\\}.", "85191924fc8149139862452c4a22b9b6": "(S,\\rho)", "851926f4c67eaa44c5666a61bcb54b0e": "V = L_1 \\oplus ... \\oplus L_n", "85192726fae3af3dfb31fc5919543722": "X'=\\cup_i L(A_i) ", "851994fc2e24d372e3d28ce1daa39461": " S = bc \\sin A = ac \\sin B = ab \\sin C \\,", "851acb9ef37cf9470e975aa963efdf89": "\\scriptstyle |\\phi\\rangle\\in\\mathcal{H}", "851ae4849c3e225959f88f2476ba5db5": "U(x,y)\\geq U(x',y')", "851afd1102e54c5f55d58ff1a1cb1315": "{\\mathrm{h}} \\ = \\frac{k 0.54 \\mathrm{Ra}_L^{1/4}} {L} \\, \\quad 10^5 \\le \\mathrm{Ra}_L \\le 2\\times 10^7", "851afe475e002949cd6d01453af6f4a9": "\\gamma_k(X)=X\\circ B_k", "851b3c9bc6aca7725992f99be62cf286": "w_{0}-w_{a}", "851b9cbf11aca7730f03d56b35e7148e": "\\tfrac{BO_2}{BE} = \\tfrac{1}{2}", "851be3f2f7835c584eb02853417106a1": "a^i = \\sum_j a_j g^{ij}\\,", "851be5fe52cb22160130e7a719b78538": " P = P_0 e^{-M g z/R^*T}\\,", "851be610b0bcdd3594f4404c34041cf8": "T\\,\\xi=\\mathrm{constant}", "851befb0aaebd05ba5e16c463b1b8a34": " \\frac{d^2u}{dx^2} + \\omega^2u = 0. ", "851c37d65f52d816b898f11b1f348231": "n \\leq N", "851c40b2bd5ce96685f9070871fb4077": " f_3\\equiv w_{yy}+4x^2w_x-8x^2z_y-8xw=0, f_4\\equiv z_{xx}+\\frac{1}{2x}z_x=0\\}", "851c9d2a29cc0e91053302cbaea8f8e7": "{{H}=J{{{\\otimes}}}H_C{\\otimes}H_D{\\otimes}K}.\\, ", "851c9e8e766bcb32c398ab747c26afcc": " \\sum_{j=1}^{N_s}(B_{ij}) = 1", "851cd7dacd9bd9afcf97ec66ab9f42e8": " f(x) = e^{a x^2 + b x + c}", "851cd93734612e0df4bec09144d4dc98": "U_M", "851ce771b393430444b9c894f5b830d1": "\\gamma \\subseteq \\delta", "851d36e2145d1a4b54b4293ded8819ca": "\nJ_F(x,y)=\n\\begin{bmatrix}\n {e^x \\cos y} & {-e^x \\sin y}\\\\\n {e^x \\sin y} & {e^x \\cos y}\\\\\n\\end{bmatrix}\n", "851de02ebd0004a30a2628a87f653b65": "\\varepsilon{}_n^2=+1", "851e0ad0a2aef922fcaff0d7c2d31908": "I_\\mathrm{rev} = \\left|g_\\mathrm{rev}\\right| = \\left|20\\log_{10}\\left|S_{12}\\right|\\right|\\,", "851e20da790d71c3ed2308458f392eaa": "S_x(t,f) \\approx S_y(t-af,f) \\, ", "851e6fada5b031742ed65af2f39fb03e": "G_{i}", "851f004ace45f60388695eae38dd6517": "z^m e^{a+bz+cz^2}\\prod_n \\left(1-z/z_n\\right)\\exp(z\\operatorname{Re}\\frac{1}{z_n})", "851f2342995a3cc40aee95ce750ee6bc": "\\widehat G(\\omega)\\,", "851f3531697faf91d71d4710c1a7d3d3": "\n\\sigma_i = 0.18\\,\\mathrm{nA} \\; .\n", "851f7439eea441da1503b755f3d0e147": "\\displaystyle \\Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2.", "851f752a53273ea9d10164bc0dbf3716": " \nE_{act} /{RT_m}^2 = v_1/\\beta* e^{-E_{act}/RT}\n", "851f989d4ffd1cf2e5d556d55634fb8b": "(x_2, y_2)", "851fe7322c1ca116854c51c79d5153ce": "\\{3, 7\\}", "851ffc531f462fd9c1fd2bcc1340c453": "\\equiv", "851fff8f7b0376ff4770908f93eda579": "G(\\omega)=\\frac{1}{\\sqrt{1+\\frac{1}{\\alpha^2 T^2_n(1/\\omega)}}}", "852036f2ea099b648ddfd499ef81254a": "m_0=m_1=m_2=.....=m_{p-1}=0", "85207ed39dbb816af7cdb6d85c203707": " \\sum\\limits_{a=k}^{b}f(k)= \\sum\\limits_{k=0}^m \\frac{B_k}{k!}\\left(f^{(k-1)}(b)-f^{(k-1)}(a)\\right)+R(f,m). \\ ", "8520884a2f33e440d8d1cc4bc07185e0": "\\frac{dS}{dt} = - \\beta SI + \\mu (N - S) + \\gamma I ", "85208d0b3ac174c1803b51c844cab453": "X(a,b,c;z,w)", "8520a08c8a8127448137354dbe218872": " V_{r2} ", "8520a25e47a6eea04db1a2973bc0416b": " (\\Omega,\\mathcal{F},P) \\,", "85212f26e7803e7ef8d5efb8d0a852fc": "\\|\\phi\\|_{X^*}=1", "85214a765af559390d21ff2345d58dd1": " \\det X < 0.", "85217f17897193479987c83e918fddf9": " \\delta \\approx 4.91 \\sqrt{ {\\nu x}\\over u_0} ", "8521c9965a95d8ff3cec1046c590a784": "r_0,r_1,r_2,\\ldots", "8521e1012a0f5494404c1995eaddd1ea": "u,v\\in V", "8521f734fde4ffb61cb3a65b5c4b16e0": "\\{l^a,n^a,m^a,\\bar{m}^a\\}", "852213e966cd91f807689824a5abfd03": "l^\\theta", "8522cde4492ced5002befa13d82ffba5": "\\operatorname{MUAE} (\\bold{H}) = \\operatorname{E}\\, \\operatorname{sup}_{\\bold{x}} |\\hat{f}_\\bold{H} (\\bold{x}) - f(\\bold{x})|.", "852330587070382d5a1320fef5543840": " x_2 = 1, \\, ", "85236106e26e0f5391f81f7f2d9bdb2b": "\\text{Income} + \\text{Capital Gain} = \\text{Total Return}", "85237b2b9620f46b9470763b747fe752": " \\vec{a} = \\frac{-1}{\\rho} \\vec\\nabla P", "85239b479e0a0f651d0ae2c921114629": " {d^2 x^\\nu \\over dT^2} {\\partial X^\\mu \\over \\partial x^\\nu} =- {d x^\\nu \\over dT} {d x^\\alpha \\over dT} {\\partial^2 X^\\mu \\over \\partial x^\\nu\\partial x^\\alpha}", "8523afef1a6f476d3c6c7a81cdf62fba": "\\hat{S}_{i}\\in\\mathcal{O}_{SB}(\\mathcal{H}_{SB})", "8523fedaf00f1c42bc763f9b007e654b": "R^i f_* \\mathcal{O}_Y = 0", "85240e05d315633280a7ac74058116f0": " a^2 + b^2 = (a+bi)(a-bi). \\,\\!", "8524917ab955fd7d70b80571dcbada46": "B^{II}", "85249339c204ccbc7f9d3d069ab1bcde": "b^{RC}", "8524d68572d9a4536652588fc46f8923": "= \\nabla \\times \\mathbf{A}", "8524eb1789cf2093cfccc4c297138c7f": "\\ell_1", "8524fc13565b867d117a08d40509720f": "\\varepsilon:FG\\rightarrow I", "85253d689ca7a0da557666ef4a8afa5b": "g_{ff}", "852554b3f4f5669053ffe0f66dc5d4ba": "RL(\\mathrm{dB}) = -20 \\log_{10} \\left| \\mathit \\Gamma \\right|", "85256bbfc4405a95f42d8daa1035fe96": "\\left(\\frac{k_x^2}{n_o^2}+\\frac{k_y^2}{n_o^2}+\\frac{k_z^2}{n_o^2} -\\frac{\\omega^2}{c^2}\\right)\\left(\\frac{k_x^2}{n_e^2}+\\frac{k_y^2}{n_e^2}+\\frac{k_z^2}{n_o^2} -\\frac{\\omega^2}{c^2}\\right)=0\\,.", "8526051c4251f8d145364da7f238b4b0": "\\sigma_{21}(\\nu) = A_{21} { \\lambda^2 \\over 8 \\pi n^2} g(\\nu)", "85261078343b9a0f479afe55768c84d8": "\\nabla_X\\psi=\\lambda X\\cdot\\psi", "85264ef5f7c1c8d08041f3125d7ede08": "2^{n}\\times 2^{n}", "8526dd808bb4b7dd112251c5bd258051": "P_N(N_R)=\\frac{N!}{2^NN_R!(N-N_R)!}", "8526f72b1b28df88fbac0c9855aefdc1": "L=-F\\dot{v}^2+2\\dot{v}\\dot{r}\\,,", "852716161007f17558d3191d1b987698": "\nI =\n\\begin{bmatrix}\n \\frac{3}{5} m h^2 + \\frac{3}{20} m r^2 & 0 & 0 \\\\\n 0 & \\frac{3}{5} m h^2 + \\frac{3}{20} m r^2 & 0 \\\\ \n 0 & 0 & \\frac{3}{10} m r^2\n\\end{bmatrix}\n", "85274ec94917c6a0a0739c18951ef002": "\\scriptstyle P_s", "85276bcb78a34ce3db39266f59b89565": "\\boldsymbol\\xi", "8527713f45c18d9f1cb91541aebe42be": "u(d)", "85278323a1bd3846040ae8491205c62e": "\n\\operatorname{tr_B} \\left( | \\psi \\rangle \\langle \\psi | \\right )= \n\\operatorname{tr_B} \\left( \\sum_{i, j} \\sqrt{p_ip_j} |i \\rangle \\langle j | \\otimes | i' \\rangle \\langle j'| \\right ) = \\sum_{i,j} \\delta_{i,j} \\sqrt{p_i p_j}| i \\rangle \\langle j | = \\rho.\n", "852787e093483da972b90a9d861caa74": " \\sigma^\\alpha \\rightarrow \\tilde{\\sigma}^\\alpha\\left(\\sigma,\\tau \\right) ", "8527b4db0711e9823ac9fdc3b09198c7": "N\\ge L+M-1,\\,", "85280e10389392f039dc868cae3f934e": "\n\\eta\\left(\\frac{i}{2}\\right)=\\frac{\\Gamma \\left(\\frac{1}{4}\\right)}{2^{7/8} \\pi ^{3/4}},\n", "852815c9e1645c5bdc8117afd5c5e15c": "p=1/2^n", "8528727d369a3045f6be17d24df98434": "\\lbrack t_0 .. t_1 \\rbrack", "852880f43a07bf304db295b25acd570a": "x = L/L^*", "852888e512d9033b591b5128bfc8290a": "\n= 2/3 \\cdot 5.625 \\mathrm{mA} = 3.75 \\mathrm{mA}.\n", "8528a614597a7ee70a21ece518b929d6": "|\\Psi(t)\\rangle", "8528b4209bc7d79bccd744c85e5329af": " \\int_{t,t'} e^{-(t+t')(k^2+m^2) - t' 2p\\cdot k -t' p^2}\\,, ", "8528bdb1b66daa971d4598253b5dd4b0": " 1 - \\sqrt{R} ", "8529134983297ca75ce71d843777b94c": "\\mu_1,\\dots,\\mu_p", "852940119a7e646a53e46fac91df15e6": "\\alpha_c \\,", "852965c3cccc8b90ee5391d29fde8d90": "G\\subseteq U(H)", "8529e2a1d289fe8a6caf3a9f04c1ec03": "\\Box ( \\Box p \\rightarrow p )", "852a3c25130b935f04bd311bf94141cc": " \\ \\zeta", "852ad9630eba058c5420003e3578d1f5": "f_a(y) = a^2 + ay + y^2. \\,", "852b16fd8e573db7a35375c22abc28ce": " G = G_0 = e^{\\gamma_0(\\nu) z}", "852b83360aa22e6d2f0ca5c02efedfc9": "N = b - n + s \\ ", "852b942d94a5b2d72931022976e9b5c5": "M \\subset \\mathbb{R}^n", "852c5dbd726cc2ef65d7658df35e8b77": "X \\vdash a\\mbox{ iff }a \\in X.", "852cc33ce3bf44184e3d98c349f2095f": "\\frac{3\\eta^2 + \\pi^2}{4} = 230\\times10^3~\\mathrm{MeV^2}/c^4", "852cce29c3e04de80deb88278b1bf3a8": "f(A,B)", "852cd2421177a277ee93b93658228fff": " \\Phi_E = \\,\\!", "852d03420a0c8212771c09c6c7154393": " f(x)", "852d06967a8c6c8c486b773aad8492e8": "\n\\int_0^{R_{max}} 4\\pi r^2 \\rho (r)^2 dr=\\frac{4\\pi}{3} R_s^3 \\rho_0^2\n\\left[1-\\frac{R_s^3}{(R_s+R_{max})^3}\\right]\n", "852d72da04f14eb11652cb8af70d8683": "f_*([Y]) = n [f(Y)]\\,\\!", "852d8319bf4e6813e99c243874fd0af3": "R(r) = c_1 J_0(\\lambda r)+ c_2 Y_0(\\lambda r).\\,", "852d85471eafda1eb1d49c5618d7e65e": "\\alpha = { \\omega_0 \\over 2 Q } = \\zeta \\omega_0 = {1 \\over \\tau}", "852dbc657b44a95942c6112ea56cc5d8": "t_1 \\rightarrow t_0 \\Rightarrow \\left | \\mathbf{f} \\right | \\rightarrow \\infty", "852dc7d91b79d714aeb12831b3020b5e": "\n\\begin{align}\nf=\\epsilon_5-2kT\\sum_{n=1}^\\infty \\frac{\\sinh^2((\\tau-\\lambda)n)(\\cosh(n\\lambda)-\\cosh(n\\alpha))}{n\\sinh(2n\\tau)\\cosh(n\\lambda)}\n\\end{align}\n", "852ddea7c647db4fb9224d7428a02d0f": "y^2 + \\frac{2ay}{\\sqrt{1+y'^2}}=b^2", "852e05252898af9a78be3abbf86fe227": "\\nu=3/5", "852e06922dbf110152591fe1e5a7f41f": "y^e", "852e104e442aaa58bde107682b5fa3d4": "Y \\prec Z", "852e247778fab068598e6677788de970": "\\displaystyle{L(a)b=ab}", "852e54d78bc2c96cb16d73d2ebf0f47c": "z\\in[0,\\infty)", "852e941cb1d871fa15a7bd8c0860f1bd": "P_N ", "852ea3ed724576f3ac302d2875bbf1b1": "E=\\int_{t_1}^{t_2}dt=-\\int_{W_1}^{W_2}\\frac{dW}{F}=\\int_{W_2}^{W_1}\\frac{dW}{F}", "852f2ff5735d1d217ad386210ef75982": " \\mathrm{Run time} = \\frac{\\mathrm{Instructions}}{\\mathrm{Program}} \\times \\frac{\\mathrm{Cycles}}{\\mathrm{Instruction}} \\times \\frac {\\mathrm{Time}}{\\mathrm{Cycle}}", "852f89a6778fb48023b4ea2a7c5fb067": "M_T \\leq M", "852fadbdb64df5efe4a21ffe633bc151": " x \\nearrow y ", "852fe9f438b52428e99d17759d6b1ee9": "h<", "852fef1bb10dac5f1b203a71b319c809": "\\operatorname{boxcar}(x)= (b-a)A\\,f(a,b;x) = H(x-a) - H(x-b),", "853063ab9ac62dfa12e889b8d01730a2": "\\left\\{\\begin{array}{ll}0 & n \\le 1\\\\ 1 & \\text{otherwise}\\end{array}\\right.", "8530835ed9cabfa270e1f660e71912fa": "\n\\begin{pmatrix}\\alpha^{-3}+\\alpha^{5}x+\\alpha^{7}x^2&\\alpha^{7}+\\alpha^{-3}x\n\\\\\n\\alpha^4+\\alpha^{-5}x&1\\end{pmatrix}\n\\begin{pmatrix}\\alpha^{-5}+\\alpha^{-4}x&1\\\\ 1&0\\end{pmatrix}\n\\begin{pmatrix}\n\\alpha^{-3}+\\alpha^{-2}x+\\alpha^{0}x^2+\n\\alpha^{-2}x^3+\\alpha^{-6}x^4\\\\\n(\\alpha^{7}+\\alpha^{-7})+\n(\\alpha^{-7}+\\alpha^{-7}+\\alpha^{4})x+\\\\\n(\\alpha^{-5}+\\alpha^{-6}+\\alpha^{-1})x^2+\\\\\n(\\alpha^{-7}+\\alpha^{-4}+\\alpha^{6})x^3+\\\\\n(\\alpha^{4}+\\alpha^{-6}+\\alpha^{-1})x^4+\n(\\alpha^{5}+\\alpha^{5})x^5\\end{pmatrix}=\n", "85316c6b7844d4ffca47008f1ca355dc": "\n\\beta=f_2(d_a)= \\left\\{ \\begin{array}{ll}\n\\beta_{min} & \\mbox{if } d_a \\leq d_2 \\\\\n\\kappa_3+\\kappa_4 d_a & \\mbox{if } d_2 < d_a < d_3 \\\\\n\\beta_{max} & \\mbox{otherwise.}\n\\end{array}\\right.\n", "8531bfc43c9b19e5585a94526e251c5c": "\\Box_i, i\\in \\{1,\\ldots, n\\}", "8531c0d3ff05bf667800c775066356ce": "\\mathbb{Z}(\\eta)", "8531db638b6d5a2377f501940b9a9fb8": "|p_{x,y},i+1_{x,y};\\uparrow\\rangle", "8531e074b5b5268579c151d8d769ddc8": "1,2, \\dots, N", "85328ac5c729fd873a5e1884aca3e25e": "\\mathcal{A}^a_\\mu(x) \\,", "85329348a37c5c788717a657ac4d8e0c": "(V_R)^j_i ", "853307b5dd787d640753313e2e6f19ce": "x_t^{[m]} = ", "85333635b56e9d6067590e96492722bf": "(Sv)(ds)", "8533367d239b3ea119caa3cc4541e99d": "D_L = \\sqrt{\\frac{L}{4\\pi F}} \\,", "853365458023f74ff09b2a260087996c": "C \\in \\mathcal C", "85336bdd303c8e18e45aaf78b501efde": "\\sigma\\lbrack\\text{LMA}\\rbrack = c \\sqrt{\\frac{P_{margin} (1 - P_{margin})}{r}}", "8533c06d36120b5390bf8766c639c9f8": " x_{j} = j\\,\\Delta x \\ ", "8533d0b0ddd2e25a0df11d4f0941e37a": "(\\mathbf x_k)_{k\\geq 0}", "8534ae4e43620220b8a22df9b3c10ac7": "P(X \\le 82) = P\\left(Z \\le \\frac{82 - 80}{5}\\right) = P(Z \\le 0.40) =0.15542 + 0.5000 = 0.65542", "8534ca36d579c83375e4086956013e79": "\\chi_-^y = {1 \\over \\sqrt{2}} \\begin{bmatrix}\n 1\\\\\n -i\\\\\n\\end{bmatrix} \n", "8534e54451a7920f9646ecb8f57642ae": "\\textstyle f^{-1}(B), ", "85352b5b8832903500d7f7d2873e5144": "\\scriptstyle M", "85355a31f889407da5ab6d02d8819f97": "\n \\boldsymbol{\\nabla} \\mathbf{u} = \\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{X}} ~;~~\n \\boldsymbol{\\nabla} \\mathbf{x} = \\frac{\\partial \\mathbf{x}}{\\partial \\mathbf{X}} \n", "853568325ce55dc6915e4e28e0e566d5": "\\text{left} = 2i + 1", "8535862490b6a021035dea2ce759887d": "R_0(f) \\leq Q_1(f)Q_2(f)^2 \\log N", "85358fa848b386b008c6f2fd9fb7a80e": "\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\vec{F}(\\vec{r}')d\\tau'} ", "853599929943d9d5bb97f3a4281d1ec2": "|\\Phi^+\\rangle", "8535caf93d533194d1ff2dbc2a8b9fb0": "\\int_{C_{0}} F(p) \\, \\mathrm{d} \\gamma(p) = \\mathbf{E}[F]", "853631595384a349e81c8c12246cef2e": "(a, b, c) = (18720, \\sqrt{211773121}, 7800)", "8536382215b3ea20ea6c90a22d3bae39": "{\\frac{x^TAx}{x^Tx}}={\\frac{x^TB^TBx}{x^Tx}}={\\frac{||Bx||^2}{||x||^2}}", "8536467b94badf92fdcf2202ecb8e381": "X - Y", "853688b058ea4495fcb3132c5d16927c": "a > 0,\\ b > 0,\\ a \\ne b", "8536bd76ef63c4cab332b5c4d4660867": "\\sqrt{S} = \\sqrt{a}\\times2^n", "8536c3f3fd0ee0fc8309860879cf82f2": " X^\\alpha \\rightarrow X^\\alpha + \\omega^\\alpha_{\\ \\beta} X^\\beta ", "8536dd63b03ef03b2e83d3426642267a": "\\alpha_b", "8536e66d79a962bad8809935be0c2dc9": "\n\\begin{align}\nS_{12}&=R_2^2 (\\alpha_2-\\alpha_1)\n+ b^2 \\int_{\\lambda_1}^{\\lambda_2} \\biggl(\n\\frac1{2(1 - e^2\\sin^2\\phi)}\\\\\n&\\qquad\\qquad{}+\n\\frac{\\tanh^{-1}(e \\sin\\phi)}{2e \\sin\\phi}\n- \\frac{R_2^2}{b^2}\\biggr)\\sin\\phi\n\\,d\\lambda,\n\\end{align}\n", "85372068e76b3983bad8927415bdca80": "V_e \\, ", "85377d006aa40eff309e0af204e0b5a0": "E_{x,zx} = \\sqrt{3} l^2 n V_{pd\\sigma} + n (1 - 2 l^2) V_{pd\\pi}", "85385b4341c7abc1d6fcf33a643f619e": " \\langle A_0 B_0 \\rangle + \\langle A_0 B_1 \\rangle + \\langle A_1 B_0 \\rangle - \\langle A_1 B_1 \\rangle \\le 2\\sqrt{2} ", "85388310e1bdcd9f0c131ca5fae0c714": "P^{(n)}_e", "85388c695a9d4df7ca792ba725bc5e35": "\\|x+y\\|^2=\\langle x+y, x+y\\rangle= \\langle x, x\\rangle + \\langle x, y\\rangle +\\langle y, x\\rangle +\\langle y, y\\rangle, \\, ", "8538fe6d021f7e3bc506b1ed892bfeba": "\n\\sigma^y = \n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix}\n", "85390d0db05224d889d41d625b670850": "\\delta\\ ", "853959b340037f9ca5d413d16fef4202": "1 - \\frac{2}{4} = 0.5 = 50%", "85395a87e90bc1cf72c1bfda0084cef0": "\n\\tau_{\\rm{ty}}=\\frac{\\Delta x^2}{6D}.\n", "8539b4c88d13ca7d75c68a5646b628b7": "\\widehat{\\mathbf{Z}} = \\textstyle\\varprojlim_n \\mathbf{Z} / n\\mathbf{Z}", "8539d2b816e2db7c878861212b5a9a31": " \\begin{matrix} \\frac{1}{2} \\end{matrix} ", "8539e4f03894fe7cc7e9e77c1c94ec53": "\\mathit{x}\\in\\mathbb{R}^{+}", "8539ef1fba74a70f5a77fcc3f25c1659": "BCD", "853a1e27896a309fe7f57068834bb138": " \\, c ", "853a90949ba1605422cb732583bebd41": "C(u)=\\sum_{i=1}^k R_{i,n}(u)\\bold{P}_i", "853aac55ada93755fb4be25eab15656f": "D_{(Q)} = \\frac{\\tau_{(Q)}}{Q-1}", "853ab60e02db1b6ed97270d89bceebd7": "T_{\\alpha\\beta\\mu}=T_{[\\alpha\\beta]\\mu}", "853addabd670f93ca73d507a1e9fc15e": "\\lambda_{0}=\\mathrm{Z}\\mathrm{o}\\mathrm{n}\\mathrm{e}\\times 6^\\circ - 183^\\circ\\,", "853ae90f0351324bd73ea615e6487517": ":", "853aefff5e4d89484df381c379b0e717": "\\tilde{f}", "853b273f2257928f709a8b8380e91179": "a^3 + 3a^2b + 3ab^2 + b^3", "853bb3c348828756032a05b47dcbd6f6": "1+{1 \\over 2}+{1 \\over 3}+...+{1 \\over p-1} \\equiv 0 \\pmod{p^2} \\mbox{, and}", "853bb7f8e8c03c0736b82888aaed7999": "\\sum_{m,\\boldsymbol{R_n}} b_m ( \\boldsymbol{R_n}) \\ \\varphi_m (\\boldsymbol{r-R_n+R_{\\ell}})=e^{i\\boldsymbol{k \\cdot R_{\\ell}}}\\sum_{m,\\boldsymbol{R_n}} b_m ( \\boldsymbol{R_n}) \\ \\varphi_m (\\boldsymbol{r-R_n})\\ .", "853bf4d5a67b471c1629ed626ee4b334": " [A_0,A_1] = 0 ", "853c179505d01d6180020e9ab07a4790": "4^4 - 52 = 204", "853c4a19c12658d2f370f9802ac969e5": "J_\\mu^{em} = \\sum_f q_f\\overline{f}\\gamma_\\mu f", "853c684541f29c485d48048f90e09508": "\\mathfrak{P}^{32}", "853c8a2d27c8624f0119ce5dba04f435": "r(l) = \\infty", "853cd1174954cc061b46ee1b4f4ff2b2": "P(m)", "853d6b07b26761382558b4b03c4a1f34": "\\rho_{1}=P\\sigma", "853dbfd5b977a456e8d3c88d4f530a7b": "h_1, \\ldots, h_{d},", "853e1ab64989261aa1065ae5719aa8c5": "a_{i_k}\\in{\\overline{a}}", "853e48fa3c5e6ab9e3b41764c4603e83": "\\Phi =\\frac {L\\ V'}{A\\ \\Delta\\!P}", "853e82b2fefd32927946034032b20151": "\\cot\\frac{\\pi}{10}=\\cot 18^\\circ=\\sqrt{5+2\\sqrt 5}\\,", "853f12f1f56c3c4a352abeac3dd695e0": "\\alpha \\models \\neg \\mu", "853f3a7f99a49c6a4140a98e1285375d": "\nc \\,F_1(a,b_1,b_2,c; x,y) - (c-a) F_1(a,b_1,b_2,c+1; x,y) - a \\,F_1(a+1,b_1,b_2,c+1; x,y) = 0 ~,\n", "853f79d49ec7d2109119c4d8d257851c": " \\lim_{z\\to 0} \\frac1{z}\\left\\{\\frac1{\\Psi(1-z)} - \\frac1{\\Psi(1+z)} \\right\\} = \\frac{\\pi^2}{3\\gamma^2}.", "853fe8f231ecf803a9ec1743733cbf78": "(q_1,q_2,q_3)=(\\lambda,\\mu,\\nu)", "853fee4c81ff3724447f1069c0dbba9f": "MU_{(p)}", "853ff298572018b713f1bee84a43caa2": "{{P}_{V}}{{\\phi }_{\\gamma (u,\\xi )}}({{u}^{'}},{{\\xi }^{'}})", "85400fc4811e90097ca710ffa11da4e7": "|f\\rangle=|(\\hat{A}-\\langle \\hat{A} \\rangle)\\Psi\\rangle ", "85403ec5c8e026f69b3a2efc024ed29a": " \\mathbf x ", "854045f4cfc30719aef65ed7e8a7774f": "\\operatorname{perm} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = ad + bc. ", "8540545d461d7ebb5cf011bee1cfa5d0": "V^{-1}", "8540897f0dd0572a44d362cdb7c1ad7b": "{m^+}_3 = [15.854, 0.683]", "854178254b3dc33caf2872c6b03abbea": "B_1 \\cong A_1.", "8541c4b525afdf662b08b7ee90131cba": "\\scriptstyle P_B(\\lambda_B)", "8541d44bee8ce6e9cc29b9d6f1f78185": "\\bold{j}_\\perp = (1/\\mu_0)\\nabla B_z \\times \\hat{\\bold{z}}", "854315fd0686b1f398aaf72528b99e42": "S(u,v) = 1 - D(u,v)", "8543fc489977b4db6ec945c6ccd184e0": "\\lim_{\\alpha = \\beta \\to 0} \\gamma_1 = \\lim_{\\alpha = \\beta \\to \\infty} \\gamma_1 =\\lim_{\\nu \\to 0} \\gamma_1=\\lim_{\\nu \\to \\infty} \\gamma_1=\\lim_{\\mu \\to \\frac{1}{2}} \\gamma_1 = 0", "85441931b346461c1fb587cdd8216ab8": "s(n) < c_2 4^n n^{-{5 \\over 2}},", "8544bbbf62a08870b6d45b0a7bf8fca9": " H_A \\otimes H_B .", "8544e329cc304149618656106b81114c": "ds^2 = \\sum_{i,j}^n g_{ij}\\, dx_i\\, dx_j ", "854564f117d125f684087efc9b41dde7": " {\\rm Homeo}(X)", "85458463111f093b3a1ddf3f0aa5ec7c": " D_{j} = A_{jj} - \\sum_{k=1}^{j-1} L_{jk}L_{jk}^* D_k ", "8545a124e4d01c4d6f9e2297194829bc": "\n\\sqrt{114} = \\cfrac{\\sqrt{1026}}{3} = \\cfrac{\\sqrt{32^2+2}}{3} = \\cfrac{32}{3}+\\cfrac{2/3} {64+\\cfrac{2} {64+\\cfrac{2} {64+\\cfrac{2} {64+\\ddots}}}} = \\cfrac{32}{3}+\\cfrac{2} {192+\\cfrac{18} {192+\\cfrac{18} {192+\\ddots}}},\n", "8545c2ec4845c6fb8cb2cc6d21f0835d": "s^2+1.4142s+1", "8545e919959a018d39f65a07e8e17bc5": "\\hat A_i", "854625eef134b89f08d9e7e78c042c96": "\\Psi(x)=e^x", "8546c435af307d99656831daef3758da": " \\langle\\dot{O}\\rangle = \\operatorname{tr}\\left(O\\rho\\right) ", "8546ccb1fc4d1678a5531b83998ae4e5": "\\tau_\\ast", "8546d73c2319c4fd5c422395ad7cd0fb": "\\lambda'", "854704af4c78c2d950b60bae098de9ec": "\\,\n{\\mathbf{U}}_{||} = \\boldsymbol{0} \\ , \\quad {\\mathbf{U}}_{\\perp} = \\mathbf{U} \\ , \\quad \\mathbf{V} \\cdot \\mathbf{U} = 0\n", "8547553f1859a297a665b25e88b07acc": "\n1 = \\frac{f \\sigma}{m \\sqrt{s}} \\rho^{2/3} M^{1/3}\n", "85475fa30b59d8b8f76fb930224a694d": "\\scriptstyle O(n\\log n)", "854774b3095663bec8929d1170fc3c6a": "\\forall a\\forall b\\forall c\\forall d\\forall e\\forall G\\forall H \\;aH\\and bH\\and eH\\and cG\\and dG\\and eG\\rightarrow\\exists f\\exists I\\exists J\\; aI\\and cI\\and fI\\and bJ\\and dJ\\and fJ", "85478ed913ab74df3b8d8ecf21e76f27": "u^{*}_{z}", "854791216471bbe479c38891638e4d02": "\\begin{bmatrix}\ne+i\\overline{e}+jv+k\\overline{v} \\\\\nu_r+i\\overline{u_r}+jd_r+k\\overline{d_r} \\\\\nu_g+i\\overline{u_g}+jd_g+k\\overline{d_g} \\\\\nu_b+i\\overline{u_b}+jd_b+k\\overline{d_b} \\\\\n\\end{bmatrix}_L\n", "8547fdf339c5ef662326d7e594a675a1": "u \\in H^n(K(G,n);G)", "85481482b82544e93cdc3fecf894b418": "0I_i ", "854ef23267f3af8430a55c8cbe4a3805": "d/(1-d)%.", "854f12f38efbb5980abddeb4c7031777": "\nV_C(s) = \\frac{1/Cs}{R + 1/Cs}V_{in}(s) = \\frac{1}{1 + RCs}V_{in}(s)\n", "854f22fc627951ab5e9a38f2d5f0d8d4": "{\\frac{1{,}000{,}000{,}000}{24 \\times 365.25}}", "854f48e2bcfda2d9501fee55af9e4982": "\\overline{I}(h) = \\overline{\\overline{I}}(h),", "854f64903966a1e12626ec0d9a7801a5": "y_\\text{SDM} = \\operatorname{Quantize}\\left( \\int \\left( u - y_\\text{SDM} \\right)\\right).\\,", "854f96966e4749d6f9a3e359c6c7c373": "P\\left( \\frac{L-\\mu}{\\sigma} < Z < \\frac{U-\\mu}{\\sigma} \\right) = \\gamma.", "854fb09168a25030ee1ca81baa24d9b1": "\\beta=\\arccos\\left(\\frac{a^2+c^2-b^2}{2ac}\\right)", "855003eb26fe5be594609bd5cf5d3073": " \\|C_h f\\|^2 = |a_0|^2 + \\|hC_hU^*f\\|^2 \\le |a_0^2|+ \\|C_h U^*f\\|^2.", "855052c479e3579f6b550f0b6bc9df2a": " \\delta(\\mathbf{x})\\sim\\mathcal{GP}\\big(\\mathbf{h}^\\delta(\\cdot)^T\\boldsymbol{\\beta}^\\delta,\\sigma_\\delta^2R^\\delta(\\cdot,\\cdot)\\big) ", "85505fdc5ae1c3ca97015118516a0892": "\\frac {\\zeta(2s)}{\\zeta(s)} = \\sum_{n=1}^\\infty \\frac{\\lambda(n)}{n^s}.", "8550826ce28c74d34bb2878718d528f5": "\\mathcal{O}(x^{1/3}\\log x).", "8550b7fcd5115cf65f4a46b0c16dda2f": "(N,V,T)", "8550f8fb9a781bc75082037c824aa493": "|\\mathbf{u}| \\approx v", "855127bd517cbe695f5b1498f391fb36": "\n\n\\mathbf{F} =\nm \\mathbf{g} \\left ( \\mathbf{r} \\right ),\n", "855186afdd4627d1a873e70661c51ecd": "T(1)", "85518c077729c400d870648431ef7132": "G_X (\\Beta(\\alpha, \\beta) )=G_{(1-X)}(\\Beta(\\beta, \\alpha) ). ", "8551c0154d99da8f7b818aae21fa1a52": "p_{s's}(a).", "8551e6e3d1b7eb0ec3ef755730a6079e": "x_i = i \\Delta x", "85521a0646a4699ac843071949d3d9d5": " \\hat{\\alpha}= (\\alpha,\\dot{\\alpha}) = 1,2,\\dot{1},\\dot{2}", "85525d9bdafe7249ed0a72e1748ffa4d": " PW_x(t,f) = \\int_{-\\infty}^\\infty ST_x(t, f+\\nu/2) ST_x^*(t, f-\\nu/2) e^{j2\\pi\\nu\\,t} \\, d\\nu", "8552617f087430c72036dfdeaa061921": "a_p = a(1-{ 2f\\over 3})", "85528c7b0f0576ba26330dd344c8fb82": "\n\\begin{align}\n A_0 &= \\quad a(1-e^2)\n \\left(1+\\frac{3}{4}e^2+\\frac{45}{64}e^4+\\frac{175}{256}e^6+\\frac{11025}{16384}e^8 \\right) \\\\\n A_2 &= -\\frac{a(1-e^2)}{2}\\left(\\frac{3}{4}e^2+\\frac{15}{16}e^4+\\frac{525}{512}e^6+\\frac{2205}{2048}e^8 \\right)\\\\\n A_4 &= \\quad\\frac{a(1-e^2)}{4}\\left(\\frac{15}{64}e^4+\\frac{105}{256}e^6+\\frac{2205}{4096}e^8\\right)\\\\\n A_6 &= -\\frac{a(1-e^2)}{6}\\left(\\frac{35}{512}e^6+\\frac{315}{2048}e^8\\right)\\\\\n A_8 &= \\quad\\frac{a(1-e^2)}{8}\\left(\\frac{315}{16384}e^8\\right)\n\\end{align}\n", "8552d5cc88d6f541335b1ded813e1233": " [2]P ", "8552d87a7ba898fbb1c122be18506450": " (\\hat{c}- \\hat{a}) = \\frac{\\sqrt{\\text{(sample variance)}}}{2}\\sqrt{(2+\\hat{\\nu})^2(\\text{sample skewness})^2+16(1+\\hat{\\nu})}", "85532a505c06bf6acb547bff862481de": "W=u_x - i u_y,", "8553413e83860e79f6b712259bebe263": "1 \\mapsto \\begin{pmatrix}\n 1 & 0 \\\\\n 0 & 1\n\\end{pmatrix}", "855354a2c1543f4e63f1ccf1d49b1c6e": "|c|=(|c_1|^\\alpha+|c_2|^\\alpha)^{1/\\alpha}\\,", "85535c81d3a9b3ca4f45c456b145d6db": "H = - \\frac{\\hbar^2}{2m} \\sum_i \\nabla_i^2 + \\sum_{i < j} U(|\\mathbf{r}_i - \\mathbf{r}_j|) ", "85537b353831cd7c6b7b8bf308ef8217": " w^2+x^3+y^5=0. ", "8553a4b8cd5e4fc21b3f1395ece4b11e": "(((\\exists x Rx) \\land \\lnot (\\exists x Sx)) \\to \\forall x Tx) \\Leftrightarrow ((\\exists x Rx) \\to ((\\lnot \\exists x Sx) \\to \\forall x Tx)).", "8553b02decd21e3979737b174d095915": "\\begin{matrix}\\text{If } x(t)=0\\text{ for }t>t_0\\text{ then } W_x(t,f)=0\\text{ for }t>t_0 \\\\ \n\\text{If } x(t)=0\\text{ for }t \\gamma_1, h(x) > \\alpha_1", "855c7c92155235df7debb1498bae86bf": "\\left\\{ Q, \\overline{S} \\right\\} = \\left\\{ \\overline{Q}, S \\right\\} = 0", "855c89d3445b71e2bcfa4435c734bd6c": " a_i \\leftarrow principal~eigen~vector(A^TA) ", "855ca40c4d9d3cb653b82347e1a0ecf9": "\\left(x,y\\right)", "855ce787beba6ce758e4ba0f31777927": "[0,b)", "855d2b0b45b972736563568be7702b67": " \\boldsymbol{\\omega} \\times \\mathbf{T} = \\mathbf{T'}, ", "855d2f83e2ca2a8933e67f4c9a6cac24": "L\\preceq N", "855d56d607dbd0b9f7ed18f23d32fede": "\\scriptstyle f\\colon \\Omega \\to {\\Bbb R}", "855d5e7259ceb2905ebe2bf3fdbba93a": "Hom_{\\mathfrak{C}}(-,Q)", "855d82703f385d72f22aa5aad49ee610": " C_{\\max} ", "855da9f65363fe0f05f5013916fbbc07": "m_{ij}\\geq 2", "855dca71fca759a9fccfea9d299d388a": "\\frac{\\binom{n}{2}! }{1^{n - 1} \\cdot 3^{n - 2} \\cdot 5^{n - 3} \\cdots (2n - 3)^1}", "855e7e46cd6dc8b474a1d4707c77f05b": "\\sum_{n=1}^\\infty \\left(\\frac{1}{2}\\right)^n = \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\cdots", "855e7e6eec871a8aedc27ace9b488e5c": "\nI(t) = \\frac{1}{L}\\int_{0}^{t} E(t-\\tau) e^{-\\alpha\\tau} \\left ( \\cosh \\omega_r\\tau - { \\alpha \\over \\omega_r } \\sinh \\omega_r\\tau \\right ) \\, d\\tau\n\\text{ in the overdamped case }(\\omega_0 < \\alpha)", "855e94c470f0f1150e5b1fbfccbbc174": "\\langle p' | J^0 (0) | p \\rangle ", "855ef10dff9665aab5ace45775ed7476": "\n\\begin{align}\n\\sin \\theta & = \\cos \\left(\\frac{\\pi}{2} - \\theta \\right) \\\\\n& = \\frac{1}{\\csc \\theta}\n\\end{align}\n", "855ef46c20dc01ba31ae4f1f671b0cd9": "m_i = \\{ p_1, p_2, ..., p_n \\}", "855f64d276d99174350a10e032c22fed": "g_n(G_{n-1}(z))", "855fd210feff630b6495523451551ef2": "\\begin{align}\nh'(0) &= \\lim_{t \\to 0} \\frac{h(t) - h(0)}{t - 0} \\\\\n&= \\lim_{t \\to 0} \\frac{g(y+tz) - g(y)}{t} \\\\\n&= \\lim_{t \\to 0} \\frac{1}{t} \\left (\\frac{\\langle T(y+tz), y+tz \\rangle}{\\|y+tz\\|^2} - \\frac{\\langle Ty, y \\rangle}{\\|y\\|^2} \\right ) \\\\\n&= \\lim_{t \\to 0} \\frac{1}{t} \\left (\\frac{\\langle T(y+tz), y+tz \\rangle - \\langle Ty, y \\rangle}{\\|y\\|^2} \\right ) \\\\\n&= \\frac{1}{\\|y\\|^2} \\lim_{t \\to 0} \\frac{\\langle T(y+tz), y+tz \\rangle - \\langle Ty, y \\rangle}{t} \\\\\n&= \\frac{1}{\\|y\\|^2} \\left (\\frac{d}{dt} \\frac{\\langle T (y + t z), y + tz \\rangle}{\\langle y + tz, y + tz \\rangle} \\right)(0) \\\\\n&= 0.\n\\end{align}", "855ffdc32084e7c39121accbd112caed": "Pr(H),", "85607f2862ecf5fd36ff19d5e67b6481": "\\hat{f} + \\alpha b", "8560a3b1a89eb576496afb18a98474a9": "P_n(r) \\propto r^{n-1}e^{-r^2/2}", "8560ba8d052d7976d002acc597238468": "\\sum_{i=1}^n \\Pr(x_i\\mid I) = 1.", "8560c09e6986caafdd4b12e1465b1c20": "\\chi=\\sum_{s\\in W} a_s \\lambda_s", "8560ca33eebf428e5bf50b9b03b4750e": "1 < 2^r+1 < 2^n+1", "8561a5ebe40dc60b6a99bb6286e82aef": "f(d,s)", "8561c0e811366f5c4035855402106c7f": "\\xi_k = \\frac{1}{\\|\\mathbf{X}_{k-row}\\| \\|\\mathbf{X}^{+}_{k-col}\\|}", "8561d434e29666a905a657d2c13837af": "\\lambda (\\partial A)", "8561e4bcd898a36efbd58b53d2a1e072": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\n \\end{Bmatrix}\n =\n \\begin{Bmatrix}\n j_4 & j_5 & j_3\\\\\n j_1 & j_2 & j_6\n \\end{Bmatrix}\n =\n \\begin{Bmatrix}\n j_1 & j_5 & j_6\\\\\n j_4 & j_2 & j_3\n \\end{Bmatrix}\n =\n \\begin{Bmatrix}\n j_4 & j_2 & j_6\\\\\n j_1 & j_5 & j_3\n \\end{Bmatrix}.\n", "8562142cd4d68a34ae2a8e9b3fd041d6": "\\mathrm{NPV} = -123400+\\frac{36200}{(1+r)^1} + \\frac{54800}{(1+r)^2} + \\frac{48100}{(1+r)^3} = 0.", "85623d2c8a4fb3d2d91d48d3d47a3d5f": "\n \\tfrac{1}{2}|\\sigma_2-\\sigma_3|^n + \\tfrac{1}{2}|\\sigma_3-\\sigma_1|^n + \\tfrac{1}{2}|\\sigma_1-\\sigma_2|^n = \\sigma_y^n \\,\n", "8562785536a147ee4751ac9ea3a636cb": "S^1 = \\{ ( \\cos{\\phi}, \\sin{\\phi} ) \\, | \\, 0 \\leq \\phi < 2\\pi \\}.", "85628a7ea66127e288dd82024e1826b3": "\\displaystyle K=ab", "8562bc4a4ca4defe44eb662a1df68ab6": "Y^2+a_{3,2}Y-a_{6,4}\\ ", "85637937ae7cb548095199d247812a62": "\\begin{align}\\dot x(t)&=f(x(t),y(t),t),\\\\0&=g(x(t),y(t),t).\\end{align}", "8563b13611b19b7931e025213ab2ef24": "\nE \\in [24V,26V]\n", "8563d153684ea747acee02600755c617": "\\frac{dP}{dr} = -\\rho(r)g(r),", "8563d6ea344d865019948ade70887429": " \\frac{\\partial F}{\\partial \\overline{z}} = \\mu(z) \\frac{\\partial F}{\\partial z},", "8563ff89e9ca4ef4f1da48f43476753b": "0 < N_0+N_1 < 2(b^k-1) \\, ", "85642c2488304fa77ad40dbf670eb9e7": "\\mathfrak{sl}_2(\\mathbb{R})", "8564baf009f4ed4f3b4fe5d1f18e258b": " A = -B ", "8564dc95485be27affb8bdc856c9cdaf": "A^\\mu = (\\vec{A},\\phi)", "85654da71dc3ab53ed87d567c66c7660": "U_j", "85655e1dcfbc35e972c1a0a278a06be0": "\\mathbf{Q}(\\mathbf{r}) = Q_i(\\mathbf{r})", "8565be970ec7aaee7ae8efa8cd94e0d6": "\\frac{d[A]}{dt} = \\frac{k_1k_3[ABCD]}{k_2[D]}", "8565c5c4dbee0113b024c7e0a99c9b7c": "\\psi = \\frac{p}{\\gamma} = \\frac{p}{\\rho \\, g}", "85660e289ef58e23917f1267358b81d2": "\\|Av\\| \\le c \\|v\\| \\quad \\mbox{ for all } v\\in V", "85661aab917b6b44ee22063f20442ae2": "\\mathbf{L}=\\frac{\\epsilon_0}{2i\\omega}\\sum_{i=x,y,z}\\int \\left({E^i}^{\\ast}\\left(\\mathbf{r}\\times\\mathbf{\\nabla}\\right)E^{i}\\right)d^{3}\\mathbf{r} .", "856643604e4dcc5b1670f373ae787c1f": "U_{(k)} \\sim B(k,n+1-k).", "85664d831dcd9b3a1ad45a9901691432": "h(X) = -\\int_\\mathbb{X} f(x)\\log f(x)\\,dx", "85668947a4a20213f9e57950f8a24601": "\\,\\Delta(x-y) = G_{adv} (x-y) - G_{ret}(x-y)", "8566c017716bc9da8136d44e7dcdb944": "R_{ab}=\\, 0", "856716b6dd4c7d4631e61638cd3cc59f": " \\lambda = ({\\gamma*H^3 \\over \\mu})^{1/4} ", "856755afb5503fe0c328d5055a13b92a": "\\{(s_1,\\ldots,s_n)\\in I_{s,t}^n\\mid s_i=s_j\\}", "856783aa6f5f18a94bfe71970eea7a1a": "\\sin^2\\theta+\\cos^2\\theta=1", "85678c5c91dea1a1f0a8aca1f4e1baf7": "(A_2)", "8567d1295570c3b9f1a861b735853179": "w_i^\\alpha(\\vec r) ", "85681f9995f00e839519992d64292235": "\nR(s,t) = \\frac{\\operatorname{E}[(X_t - \\mu_t)(X_s - \\mu_s)]}{\\sigma_t\\sigma_s}\\, ,\n", "8568ab5ba4094308e1ac94c773c370e5": "k=\\log n", "8568d050e69f0895f0a5ce5fa5c31752": " T/n ", "8568df00373df90282ef5bdfb8f1d509": "\\rho \\;", "8569109ae702466b58e17c84e9a4bb3f": "\\prod_{n=1}^{3}{\\mathbb{R}} = \\mathbb{R}\\times\\mathbb{R}\\times\\mathbb{R} = \\mathbb{R}^3", "8569213bcb5ecf175a1fc3f6190e5de1": "\\mathit{L}_{{\\omega_1},\\omega}", "856937aff3930e830d257eea60ad7d5b": "y(t) = 10 x(t)", "856956a1cbc3dab96bb45e5413335d69": "\\displaystyle{ |(z,\\zeta;w,1)| \\le 8|z-w|,\\,\\,\\, |(f(z),\\zeta; f(w),1)|\\ge |f(z)-f(w)|/8,}", "85698bd4f733213262b289011d9aee15": "\\mbox{EY} = \\frac{\\mbox{SM}\\times\\mbox{DM}}{\\mbox{SM-DM}}", "856a2ca6ceed5a9170ea5c1d4b26922e": " V_1 = \\frac{2}{R[u]} \\left( \\int_{x_1}^{x_2} \\left[ p(x) u'(x)v'(x) + q(x)u(x)v(x) -\\lambda u(x) v(x) \\right] \\, dx + a_1 u(x_1)v(x_1) + a_2 u(x_2)v(x_2) \\right) , \\,", "856aa5d1d6c2afcaa93eea54f9103915": "\\chi_2(\\omega) = -{2 \\over \\pi} \\mathcal{P}\\!\\!\\! \\int \\limits_{0}^{\\infty} {\\omega \\chi_1(\\omega') \\over \\omega'^2 - \\omega^2}\\,d\\omega' = -{2 \\omega \\over \\pi} \\mathcal{P}\\!\\!\\! \\int \\limits_{0}^{\\infty} {\\chi_1(\\omega') \\over \\omega'^2 - \\omega^2}\\,d\\omega'.", "856ace3debdbc98595fbb9a2dadf4be7": "OPSBI = \\frac{AB*(H+BB+HBP)+TB*(AB+BB+SF+HBP)}{AB*(AB+BB+SF+HBP)} * 1000 + RBI", "856ae12427ef87e602574ed8418928b0": " u_0(x)", "856ae95cdbd87bd33019b432f58f7ea0": "x_0 \\in X_m", "856b1e54beefb1711ca345d3a680f077": "\\ \\Delta H(T)=\\Delta H(T_d)+ \\Delta C_p[T-T_d] ", "856b2a49f56b5e0a4114b201b9a2500b": "\nv^{ij}{}_{;k}=\\nabla_k v^{ij}=\\frac{\\partial v^{ij}}{\\partial x^k} +\\Gamma^i{}_{k\\ell}v^{\\ell j}+\\Gamma^j{}_{k\\ell}v^{i\\ell}\n", "856b4bb5fb7f4cad6bdf2a73da33780b": "\\sum_{k=1}^{10} f(k,n),", "856bbf2fbd70018c416effd78cf43abc": "Pr(s'|s,a)", "856c3f35d4c09151b3fb3f61f711b271": "q\\geq 0, ", "856c5c820ca890897954073f0083b2e3": " (\\forall x) Eq(x, x) \\wedge (\\forall x,y,z) [Eq(x, y) \\rightarrow (Eq(x, z) \\rightarrow Eq(y, z))] ", "856c73932b8ecc3d80f854dfe8a46547": " {G^a}_a = t_1 = a_1", "856cae24b0363ab388fddf71da8a1130": "u_n = \\sum_{k=1}^N U_k e^{iknd}", "856cf7c3fb2f85b5279339ae90954008": "y^2(a^2-x^2)=(x^2+2ay-a^2)^2.", "856cf829315c093a2af0985cb836c07a": "X \\sim F(\\alpha,\\beta)\\,", "856d3ebd0ea0958e4fa4e86f6d91ee51": "E\\{.\\}", "856d55f6b336067ad5ba6a9b16603a04": "[\\ A] \\{\\Delta X_3\\} = \\{\\ B_3\\}", "856d91df9127944e48ecb104e55b7a9f": "G_{j}", "856df5a474db3c1aff7f44cd42a0a213": "\\chi (X, Y) = \\sum _{j=0} ^n (-1)^j \\; \\mbox{rank} \\; H_j (X, Y). ", "856e0351955d4b6876179c5ec7d11a6f": "\\alpha_2 = 0.4", "856e40a68d62f3344929abed145fd553": "\\sum_{n=1}^\\infty \\frac{(-1)^n}{(n+1)(n+2)} = 2\\ln 2 -1.", "856e7bc0332b534039b67ddcb0c29e00": "S[A[i+1]+H[i+1],n]", "856e91589e0afa26085f32a79cc0ebcb": "|S| \\leq d", "856ed1f17429c72198f917d6a4a39a66": " \\mathcal{G}_{\\alpha \\beta} \\equiv \\acute{R}_{\\alpha \\beta} - {1 \\over 2} \\acute{R} g_{\\alpha \\beta} ", "856ed429969df9e5e72fb15eef347040": "b^{-2}", "856edf7462e2e63b9624d125f8a35fd2": " \\bar{z}/z = 1", "856f3a9565d7232524c611e4aa84d8cf": " \\left(\\frac{{\\rm d}}{{\\rm d}x}\\,\\ln(\\Gamma(x+1))\\right)_{x=n} +\\, \\gamma.", "856f5915cfa31d9e4535a180303bbe58": " \\nabla\\times\\mathbf{F} = \\boldsymbol{0} \\,\\!", "856f6d3666a2486c11f4410f7643e59c": "\\scriptstyle (5.1\\pm2.9)\\times10^{-5}", "856f83fc489e613781b28242fedf6baa": "E=\\sqrt{k^2+m^2}", "856fa7bcfe9dc8c02bb84d26c9766c82": "\\mathrm{Ta_c}", "856fdfbb367bbdca287f2902ae053e3d": "W\\circ f^*=f_*\\circ V", "856ffc507426af67e220f5218e579a45": "y = mx + b \\,", "8570113b83c2b147c12a89ef4fa2069d": "(\\nabla\\cdot\\nabla) \\mathbf{A} = (\\nabla_i \\nabla_i) \\mathbf{A} ", "857083505825bd9e48896a363160ef00": "\\sin \\theta \\sin \\varphi = \\frac{1}{2}\\cos(\\theta - \\varphi) - \\frac{1}{2}\\cos(\\theta + \\varphi)", "8570c783591f075b579e502083c742c7": "U_{in}", "85715c2b56eae8ecb135223b835ef57a": "T \\sin \\varphi = \\lambda_0 gp,\\,", "85716782bfc75c42080ea71e9f92a98b": " \\pi [s,t] = e^{i s x_0}. \\quad ", "8571a4c0347d8e0795ecceecd1261f8f": "0.99 \\cdot 100", "857212be7dec9116d527ae525a2e9f3e": "I(v) = \\{ i \\in I \\mid a_i(v) \\ne 0 \\}", "85724338810f42bf26ee235e9dac8231": "=\\sum_\\alpha (m_\\alpha \\dot{\\vec{x}}_\\alpha t-m_\\alpha \\vec{x}_\\alpha)", "85732cd05b851026b3df6cfc93f74734": "\\int_0^1 \\lim_{n\\to\\infty} f_n(x)\\,dx = 0 \\neq 1 = \\lim_{n\\to\\infty}\\int_0^1 f_n(x)\\,dx,", "85732cef10f4db6722c7f44bd24927a2": "\\tbinom mr_q=\\tbinom m{m-r}_q", "8573713b12b97773545ac2e49ea17175": "H_i(V; \\bold Z)=0,\\text{ for }i>n. \\, ", "85737c4df20efb018a4eeb6c6ccd0cc8": "SL(2)", "85738880d3b2cb45ddbc5a49b391ba33": "N_{\\mu\\nu...}\\sqrt{-g}", "8573aba80ded9ec6f532d1b815b6b464": "\\mathit{k_p}", "8573c831956bcdf34966fd80c6f73878": "(g,g^a,g^b) \\, ", "8573f8cd973536b879d17a1b8de22268": "\n\\sigma_\\omega^2 =\\frac{1}{2\\pi E} \\int |\\omega-\\xi|^2|\\hat{\\psi}(\\omega)|^2d\\omega\n", "857452ce5062325107c4fcb156a1e548": "\\gamma_{\\mathrm{rad}}", "857468cf338d78e4385ed923eeaab6b0": "\\mu_W", "857481d130e0446923898f88383879fd": "R_{-}", "8574889b7d0cc431bc31f5593a5808c1": "\n\\begin{align}\n{}\\quad \\begin{vmatrix} t+ E_{11}+1 & E_{12} \\\\\nE_{21} & t+ E_{22} \n\\end{vmatrix}\n& = (t+ E_{11}+1)(t+ E_{22})-E_{21}E_{12} \\\\\n& = t(t+1)+t(E_{11}+E_{22})+E_{11}E_{22}-E_{21}E_{12}+E_{22}. \n\\end{align}\n", "8574f592b40ccc0aedf716113478aaf3": "\\frac {T_1}{T_2} = \\frac {\\rho_1}{\\rho_2} \\times \\frac {D_1^3}{D_2^3} = \\frac {\\rho_1}{\\rho_2} \\lambda^3", "857632894fc8ff40a5541a6e5a25d980": "\\Pi_0(x) = \\operatorname{li}(x) - \\sum_{\\rho}\\operatorname{li}(x^{\\rho}) - \\ln 2 + \\int_x^\\infty \\frac{dt}{t(t^2-1) \\ln t}.", "85765a784d4b93d3a499cbcb1a18e8b9": "\\frac{}{\\Gamma \\vdash S: (\\alpha\\!\\rightarrow\\!(\\beta\\!\\rightarrow\\!\\gamma))\\!\\rightarrow\\!((\\alpha\\!\\rightarrow\\!\\beta)\\!\\rightarrow\\!(\\alpha\\!\\rightarrow\\!\\gamma))}", "85768d38ceff5725e7a69bb0c916c8c4": "\n\\begin{align}\nb_{0,0}(x) & = 1, \\\\\nb_{0,1}(x) & = 1 - x, & b_{1,1}(x) & = x \\\\\nb_{0,2}(x) & = (1 - x)^2, & b_{1,2}(x) & = 2x(1 - x), & b_{2,2}(x) & = x^2 \\\\\nb_{0,3}(x) & = (1 - x)^3, & b_{1,3}(x) & = 3x(1 - x)^2, & b_{2,3}(x) & = 3x^2(1 - x), & b_{3,3}(x) & = x^3 \\\\\nb_{0,4}(x) & = (1 - x)^4, & b_{1,4}(x) & = 4x(1 - x)^3, & b_{2,4}(x) & = 6x^2(1 - x)^2, & b_{3,4}(x) & = 4x^3(1 - x), & b_{4,4}(x) & = x^4\n\\end{align}\n", "8576a3c704dbd2ab0ffed196359d8de5": "z\\,\\!", "8576be7d2b488c3612c5c443c6825829": "\\nabla p = -\\left[(1/\\mu_0) \\nabla^2 A\\right]\\nabla A-(1/\\mu_0)B_z\\nabla B_z. ", "8576d7fbe12632e735f9ad51b139167b": "dv_1 \\cdots dv_n = |\\det(\\operatorname{D}\\phi)(u_1, \\ldots, u_n)| \\, du_1 \\cdots du_n", "857726ce620c893ce556a258922188b0": "\\tau_{\\ast} = \\frac{\\tau}{(\\rho_s - \\rho) g D}", "8577653db692aea3b76789ef4ca4683f": "-\\frac{\\pi}{2}<\\theta<+\\frac{\\pi}{2}", "857792f8d6da285f6fdfd3bbcb476c9b": "N(d_1)", "8577af5a75a4130efa4ce04bfddcdf43": "\\zeta(\\bar{a},b)+\\zeta(b,\\bar{a})=\\zeta(b)\\phi(a)-\\phi(a+b)", "8577c4067c125c56e233ca0fa32a3d0b": "\\mathbf{H} = \\sum_k \\sum_{s = 1}^3 \\hbar \\, \\omega_{k,s}\n\\left( b_{k,s}^{\\dagger}b_{k,s} + 1/2 \\right).", "85782715cbbefa6c964e179393969d29": "P=p_1+p_2=(w,\\vec 0)", "857895d4b7fff381f10d13c80b863d41": "\n\\left( {z\\,\\, - \\,\\,{\\rm E}[z]} \\right)^2 \\approx \\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)^2 \\left( {x_1 - \\bar x_1 } \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right)^2 \\left( {x_2 - \\bar x_2 } \\right)^2 \\,\\, + \\,\\,\\,2\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right)\\left( {x_1 - \\bar x_1 } \\right)\\left( {x_2 - \\bar x_2 } \\right)", "85789ce8808aeea61757fb7626efb797": "\n A = \\left[ \\begin{array}{rrrr} a & b & b & a \\\\ b & -c & c & -b \\\\ b & c & -c & -b \\\\\n a & -b & -b & a \\end{array} \\right], ", "8578b474ebfb31d63c8abe2a2be7a80e": "\\mathrm{Im}~f(\\bold{\\hat{k}}', \\bold{\\hat{k}})=\\frac{k}{4\\pi}\\int f(\\bold{\\hat{k}}',\\bold{\\hat{k}}'')f(\\bold{\\hat{k}}'',\\bold{\\hat{k}})~d\\bold{\\hat{k}}''.", "8578c0e54cd490f6cf8af36e009fc594": "\n \\max_{\\pi_{i}, \\theta} \\sum_{i=1}^n \\ln \\pi_{i} \n ", "85790d3525f8102970301983a0ae7789": "H_1 \\cap \\cdots \\cap H_p", "85799588842ef378af1cd457bd44feb2": "\\{\\bullet\\}", "8579bb0c429039daf7323992edb5086c": "\\begin{align}\n\\mathcal{L} \\left\\{f(t)\\right\\} & = \\int_{0^-}^{\\infty} e^{-st} f(t)\\,dt \\\\[8pt]\n& = \\left[\\frac{f(t)e^{-st}}{-s} \\right]_{0^-}^{\\infty} -\n\\int_{0^-}^\\infty \\frac{e^{-st}}{-s} f'(t) \\, dt\\quad \\text{(by parts)} \\\\[8pt]\n& = \\left[-\\frac{f(0^-)}{-s}\\right] +\n\\frac{1}{s}\\mathcal{L}\\left\\{f'(t)\\right\\},\n\\end{align}", "857a3c917e386f8f7577c112fb4e894a": "p(n,x_1,\\ldots,x_k)=0.", "857ab7c6046c342c686136a5b95e8e5a": "\\mathbf{\\theta}", "857ab825ebd9a320e7cb3a18adfce403": "\\rho_{\\mathrm e}", "857af4754f567b3bb4b2965bcc344aa1": "\\frac{1}{2}\\hbar \\omega", "857b089b1de5cddbece6eb3484759e31": "\\Pr \\left( X > 0, Y > 0 \\right)\n = \\int_0^\\infty \\int_0^\\infty f_{X,Y}(x,y)\\,dx\\,dy.", "857b8c96af66f8629553482bc26d6db1": " k \\in \\{1,...,p\\}, \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{k}) \\succeq 0 ", "857b8e51a5c180d81a7d14ff3baa5723": "\\bar F(A) = \\frac 12 p(A) = \\frac 12 \\cdot \\frac 13 = 0.1666...", "857b8fc72a8464c180f3bcf3cfda3e86": "f^i(p)", "857bebfeff87e868022c8ee868c276ee": "\\mathcal{F}(H(u))(\\omega) = (-i\\,\\operatorname{sgn}(\\omega)) \\cdot \\mathcal{F}(u)(\\omega)", "857bf307b5231c41556aabd77c1dc750": "262144 \\cdot \\left(\\textstyle{\\frac 9 8}\\right)^6 = 531441", "857e05539a0637d13a7b3abb484c07ca": "{F^{jb}}_{;j} = 0", "857e3600ae1b49ef4635e67bb5fc980f": "R_{\\mu \\nu} - {1 \\over 2}g_{\\mu \\nu}\\,R + g_{\\mu \\nu} \\Lambda = {8 \\pi G \\over c^4} T_{\\mu \\nu} \\,.", "857e5732376821dc914f2a9711f33e2e": "e^{\\frac{2\\pi i}{N}n m}", "857f0468245f6cc9abfd96e2c569341e": "I = m (a^2+b^2) \\,\\!", "857fa624c887484fc166728cc072be0b": "P(N) = \\frac{k}{N}", "857fd1aa4315afca1e5b59df14fc8d1f": "\\operatorname{dim}S = \\operatorname{dim}R + \\operatorname{dim}S/\\mathfrak{m}_R S", "857fdba5f830c0503cfcac77bb138762": " \\{ x\\in V: \\|x\\| = 1 \\}.", "85801beceddad5be5946109673a15e63": "\nK(x-y) = \\int_0^{\\infty} e^{-{(x-y)^2\\over\\Tau} -\\alpha \\Tau} d\\Tau\n\\,", "8580873bf02e307139224dd3284c8a70": "\\mathbf{P}\\subsetneq \\mathbf{EXPTIME}", "8580e4bdc24cba33732a6f77d03b04da": "Z_{B \\otimes B^{\\text{op}}}(k \\otimes B^{\\text{op}}) = B \\otimes k", "8581252a4d8665656ee9f8d55d7ababf": "Q \\leftarrow (q+1)P", "8581554b4f76092e2fd3873dd53b2fae": " \n\\eta_{00} = \\langle \\mathbf{e}_0 \\bar{\\mathbf{e}}_0 \\rangle =\n \\langle 1 (1) \\rangle_S = 1, ", "858186dcbbf09a34408315fe77ff6744": "\\Pr(S|W)", "85818ff6a179e86e749f591dfabac12f": " S^2 P \\mapsto TM/P", "8581d9bfd6ca55ddbe7dfab0f8edbf7f": "\\kappa_3=0", "85820d65721541974ed85ec421e41fbb": "\\alpha x = (\\alpha x_1, \\alpha x_2, \\ldots, \\alpha x_n) \\,", "85825e55cdcdf0b3d863eb5234e3765a": "\\frac{v_{\\text{R}} \\left( t \\right)}{i_{\\text{R}} \\left( t \\right)} = \\frac{V_p \\sin(\\omega t)}{I_p \\sin \\left( \\omega t \\right)} = R", "85827f29b3f67b1530b71431a6034b33": "1\\ \\mathrm{J} = 1\\ \\mathrm{kg} \\left( \\frac{\\mathrm{m}}{\\mathrm{s}} \\right ) ^ 2 = 1\\ \\frac{\\mathrm{kg} \\cdot \\mathrm{m}^2}{\\mathrm{s}^2}", "8582d0b3df9c631b76ff9ed70e782380": "\\bar{X}(k)=\\frac{1}{k} \\sum_{i=1}^{m_k} \\bar{X}_i(k)", "8582f374083f8a4673fa5672cc2b6a55": "\n G = K_{\\rm I}^2\\left(\\frac{1-\\nu^2}{E}\\right) + K_{\\rm II}^2\\left(\\frac{1-\\nu^2}{E}\\right) + K_{\\rm III}^2\\left(\\frac{1}{2\\mu}\\right)\\,.\n ", "858332cc4066305c496a42ba8631443b": "z\\ne \\beta = \\lim_{n\\to \\infty} \\beta_n", "85838e524ce3808e62083e18d19809f4": " \\mathbf {g} = g(r)\\mathbf{ \\hat n }", "8583b66f235f8a5217e1fe5a2b513106": " V_0 = 1/\\sqrt{1-v^2} \\ , \\quad V_1 = v/\\sqrt{1-v^2} \\ .", "8583e9c95245931bd111bce9e7c0dd64": "\\hat{f}(\\xi)=i^{-k}f(\\xi)", "8583f8576879be19b4b41e38d89845d9": "+\\infty\\!\\,,", "858409a1f65c5165350030db5f3b4590": "\\int_V \\left[ \\phi(x') \\delta(x-x')-G(x,x') \\nabla^2\\phi(x')\\right]\\ d^3x' = \\int_S \\left[\\phi(x')\\nabla' G(x,x')-G(x,x')\\nabla'\\phi(x')\\right] \\cdot d\\hat\\sigma'.", "8584706958a9cda9dfb25f791728a487": " \\frac{S_n-n\\mu}{\\sqrt{n}} \\rightarrow \\xi ,", "8585388bcc0a44d57b975c178f7e759b": " \\{ \\hat{X_s}\\hat{Z_{sr}} | s \\in \\mathbb{F}_d \\} ", "85854f02bf5bba519ebb0ae72871bc2b": "(x_1,(x_2,\\ldots,x_n))", "85857fc41efba34012910220d2d19271": " \\varrho^{(m)}_{G,1...m} \\approx z^m \\exp(-\\beta U^{m}_{1...m}) ", "8585ba28c6a7856e21dab52059e8add8": "\\theta_N(x)", "8585c0669709eeca617591972165bc87": "\\delta x_{WP}", "8585cf254cb0d162d553d90fe810aa32": " f(f(x,y),z) = f(x,f(y,z))", "8585d0e1c16bb41575008e018925e72e": "|R|^2 = |C|^2 + \\left|\\frac{1}{n-2}\\left(\\mathrm{Ric} - \\frac{s}{n}g\\right) \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc g\\right|^2 + \\left|\\frac{s}{2n(n-1)}g \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc g\\right|^2.", "8585dc9d3a384022748125e6e3c7e25d": "f(x;\\alpha,n,\\bar x,\\sigma) = N \\cdot \\begin{cases} \\exp(- \\frac{(x - \\bar x)^2}{2 \\sigma^2}), & \\mbox{for }\\frac{x - \\bar x}{\\sigma} > -\\alpha \\\\\n A \\cdot (B - \\frac{x - \\bar x}{\\sigma})^{-n}, & \\mbox{for }\\frac{x - \\bar x}{\\sigma} \\leqslant -\\alpha \\end{cases}", "8585e1aad8ad325e7b7c7b75e971f4c0": " V_0^2 - V_1^2 = 1 \\ ,", "85864c8b0ec67a4f4b3a094d0f67c17a": "\\Delta f_{\\text{FWHM}} = \\sqrt{\\frac{8kT\\ln 2}{mc^2}}f_{0}", "8586755f456c0e22b9c399da4fb850b0": "\\mathrm{error}\\bigl(x(t_0 + 4\\Delta t)\\bigl) = 10\\,O(\\Delta t^4)", "85868e7c8b71c1cd521a06944ec951f6": " [U_g \\psi](x) = \\sqrt{s(g,g^{-1}x)}\\ \\Phi(g, g^{-1} x) \\ \\psi(g^{-1} x). ", "8586a2ace1ea6397923b1afa2b460f8a": "K > 1.", "8586cea3e9c43eac449c5a9c12bebacb": " (n{\\in}\\mathbb{N})\\, P(n) ", "8586f84176874086c3a3fbe7f0239534": "\n\\bar x =\\alpha \\ \\hat e_3 \\quad -\\infty <\\alpha < \\infty\n", "85870b5a8df62a80bc6a79cc8ce6878a": " \\binom n k E_k E_{n-k}. \\, ", "85870d3ffde14e51bc87a09dfeb6440c": " \\lim_{y \\to c} u'(y) = u'(c) ", "85871611cd240e7ba0f13562566bd2fa": " \\langle L_1,L_2,L_3\\rangle = L_1 \\cdot L_2 \\times L_3 = 0. ", "8587499cb44c406683a8093f8013a585": " \\mathbf{F} = m\\mathbf{a} ", "85878a09a78fea1dd20bf4168e1a1df0": "\n\\nabla^2 = \\frac{1}{h_\\theta h_\\varphi}\\left[ \n\\frac{\\partial}{\\partial \\theta} \\frac{h_\\varphi}{h_\\theta} \\frac{\\partial}{\\partial \\theta}\n+\\frac{\\partial}{\\partial \\varphi} \\frac{h_\\theta}{h_\\varphi} \\frac{\\partial}{\\partial \\varphi}\n\\right]=\n \\frac{1}{R^2}\\left[\\frac{1}{\\sin\\theta}\n\\frac{\\partial}{\\partial \\theta} \\sin\\theta \\frac{\\partial}{\\partial \\theta}\n+\\frac{1}{\\sin^2\\theta}\\frac{\\partial^2}{\\partial \\varphi^2} \n\\right].\n", "8587dc7dac003428af6ed99ae0837b52": " g(x)=\\frac{1}{cosh(\\pi x)}", "8587f57bc9551ab05a2fb20b941317e7": "\\sum_{i=1}^m b_{ij} x_i \\leq B_j,", "85880b70be08ef49c78c674a899640ca": "V^+=\\bigcup_{i \\in \\N \\setminus \\{0\\}} V_i = V_1 \\cup V_2 \\cup V_3 \\cup \\ldots.", "85885567b03a8157ca84fee6faf4236a": "d\\tau = \\sqrt{\\left [1 - \\left (\\frac{R \\omega}{c} \\right )^2 \\right] dt^2 - \\left (\\frac{R\\omega}{c} \\right ) ^2 \\,dt^2 + 2 \\left ( \\frac{R \\omega}{c} \\right ) ^2 \\,dt^2} = dt. ", "85885c3a71edcaf4a7dc31d849ff58aa": "I_x(q_2) V_x + I_y (q_2) V_y = -I_t(q_2)", "858871700993199c32c7478adaba445a": "x\\in Ex", "8589352e739c7d55c5f8a7138c846e99": "C_1=\\sqrt{\\frac{4 - c_1 + c_2}{12}}\\approx 0.337754", "85894ab0dbcdbff689f3975f57d70a3e": "\\alpha(\\cdot)\\overline{\\alpha}", "85894f64ce4a9a38cc56952406eff2e0": " \\operatorname{sink-tran}[\\lambda N.B, X] = \\lambda N.\\operatorname{sink-tran}[B, X] ", "858950f4decd1376c2294b8e3d73dd0b": " ~U^{m}_{1...m} ", "858990780a3807ef95a467f86d0a656f": "R(T) = R_0[1+\\alpha (T - T_0)]", "8589b97b376f493b4f3a56859c37cee5": "H_{n-1}(\\mathcal{A})", "8589bcd09ff6ccc5f2c7fc9b493980d3": "\n\\sin\\alpha_0 = \\sin\\alpha_1 \\cos\\phi_1.\n", "8589d225c62f9b366838073ac6b1838b": "\\widehat{V}(\\mathbf{r},t)\\psi = V(\\mathbf{r},t)\\psi ", "858a5405a2ed5ea90c69bf98a55d504c": "\\{\\mathbf{A}_k\\}", "858a6bac6beb9d56637395a79aa73a9c": " P(c) \\to\\ \\forall{x}{\\in}\\mathbf{X}\\, P(x).", "858b5afefffe84aa05b309dd4a806cd1": "\\psi = \\angle P_1KA", "858b7e9be497570de15cff760753ebe6": " W = - \\alpha n R T_1 \\left( \\left( \\frac{V_2}{V_1} \\right)^{1-\\gamma} - 1 \\right) ", "858b93cab3e6656cc2584f5b62278ca1": "S_2 = 4\\pi \\,", "858b991a396cf4fd78f7cdfdf6f8215d": "\np_y(y) = p_x(\\phi^{-1}(y)) ~ \\left|\\frac{d\\phi^{-1}}{dy}\\right|. ", "858bc3d7aff39c8ad097bb9948874086": "\n P \\left(\\text{test error} \\leq \\text{training error} + \\sqrt{h(\\log(2N/h)+1)-\\log(\\eta/4)\\over N} \\right) = 1 - \\eta\n", "858bfb675cca6ec4ebe662e04d90ddf9": "p(M_1)=p(M_2)", "858cb3ed4a64ca65a9c4f684fabd8090": "\\underline{\\underline{A}}", "858cb876637af74bb99603098311a1d2": "{d (\\rho u ) \\over d x} = 0.", "858d392a568229b1f0982b7256413c9e": "\\Lambda\\subset\\Complex", "858d625eeac2253cf19d9b5f28a1c6ed": "\\varphi_{h(x)} \\simeq \\varphi_{\\varphi_x(x)}", "858d72e97eef6ecf6286661771bd924c": "\\frac{d}{dx} (x) = \\frac{1}{x}x = 1", "858d8e93dc71efc4a040bcf82ba96366": "\\delta z", "858ddd91e0083dc9b365bd5caa802905": "{\\vec a}, y", "858de01cd4ca42e10f64a7f2b1e7040b": "Q_{s1} = q_{1} + q_{2} + q_{3} + q_{4}\\!", "858e046d233c0a9042bb222a2fe09186": "\\frac {dv/dt}{v} = g_v=\\frac {s(1-u)}{\\sigma} -(\\delta +\\alpha + \\beta).", "858e3bf2c82d4088d090f3b82b37c1a2": "\\pi_3\\left(\\frac{SU(N)_L\\times SU(N)_R}{SU(N)_\\text{diag}}\\cong SU(N)\\right)", "858e64f8097381743406293474646d8e": "-v_n(t+\\tau)^2/2b_n+v_n(t+\\tau)(\\tau/2+\\theta)-\\left[x_{n-1}(t)-s_{n-1}-x_n(t)\\right]+v_n(t)\\tau/2+v_{n-1}(t)^2/2\\hat{b} \\le 0", "858eadd3a6990a1b187eeaa5e07b552c": "\\ (r,\\ \\theta_\\text{el},\\ \\phi_\\text{az,right})", "858f030e1fbe8865f4634fabde6edee7": "s\\rightarrow \\infty", "858f4813db2dbc259f817b995a0702d2": "\\phi \\to \\left( \\psi \\to \\phi \\right) ", "858f53dbf2d125c8435858bab5fe8b3b": " T(n,k) = a(GCD(n,k))", "858fc5fd51b2a5a4bbbef97c4229434a": " F : \\mathrm{L(A)}^N \\to \\mathrm{L(A)} ", "858fd9e392e1fdb2a0928e3a2986c5af": " D_{\\mathrm{KL}}(P\\|Q) = \\int_X p \\ln \\frac{p}{q} \\,{\\rm d}\\mu.\n\\!", "858ff8a54890bfcf62c0158a259ff957": "\n \\Psi \\rightarrow \\Psi^\\prime = \\Psi R_0\n", "85902d0e8736cbeab7d31b5c18243d9c": "\\frac{\\pi}{C\\, \\sqrt{2}} = \\frac{\\sqrt{\\pi}}{2}", "85904d33d3d93ef4ee16325f8563e6e3": "l_{\\mathrm{tot~core~run}}", "85908079e0840ff602ca3f6762ef2f62": "B\\exp(-t)", "8590e18e83f7dcc1b316b31ad04d7216": "a_L = n", "8591947ab747b35b6c49905385d9d116": "10^3", "8591ba4b7005d1b145504ff95f2b7021": " \\frac {2 \\pi Q}{N[k]} ", "8591eaeae9b31f754a01f598a8e10ee2": "\\rho_{\\mathrm{primary}}", "85922f031feb7b8dd3beacacf54abbe7": "t + \\Delta t", "85927882530552b9e1636b8f0c26380c": "r-3", "8592e862eb4678736ccc6603b5f9a807": "wRAA = \\tfrac{wOBA - .320}{1.25} * (AB + BB +HBP + SF + SH)", "8592ec56d7c1df614e9ff09788b805eb": "\\mathrm{C}_R(S)=\\{r\\in R \\mid rs=sr \\text{ for all } s\\in S\\}.\\,", "8592fafad55cab551af3480f1d317844": "f(z)=\\sum_{n=1}^\\infty (e\\rho\\sigma/n)^{n/\\rho} z^n", "85937ad4d457dd81b546716ddfa2c9f0": " K^0(X) ", "85940fd80463f80a252a85a3c1e712c2": "\\nabla _A", "85941fe4167dad3d6ff134de95945a30": "\\Delta_0 = \\frac{\\partial^2}{\\partial x^2} +\\frac{\\partial^2}{\\partial y^2} \n= 4 \\frac{\\partial}{\\partial z} \\frac{\\partial}{\\partial \\bar z}", "8594219888c36b1a2db081e09b4c05b2": "\n\\int_{T} f(\\mathbf{r}) \\ d\\mathbf{r} = 2A \\int_{0}^{1} \\int_{0}^{1 - \\lambda_{2}} f(\\lambda_{1} \\mathbf{r}_{1} + \\lambda_{2} \\mathbf{r}_{2} +\n(1 - \\lambda_{1} - \\lambda_{2}) \\mathbf{r}_{3}) \\ d\\lambda_{1} \\ d\\lambda_{2}\n\\,", "8594562d9ae9cf9f539a5c5fe410fb28": "f(x,y) = g (r) h (\\phi)", "8594b103f22fa98bf343ed1fe947d3b0": "\\|x\\|^2 + 2\\|x\\|\\|y\\| + \\|y\\|^2 = \\left(\\|x\\| + \\|y\\|\\right)^2 \\ .", "8594b43708a7374df578e95c2f132671": "2+5+6=13=0 \\bmod 13", "8594d09bcb0986a6f5d709d33aa8f4a6": "n=478", "85950bc28cc6c359edddf5933aa2aa1a": "t^{1/3}", "859548224a5c49251b7a5f18ba4f6a63": "\nB_n = \\sum_m u_m \\frac{\\partial \\phi_m}{\\partial p_n},\n", "8595501c52818fe30de29c386bce4c89": "\\ln \\frac{\\hat{\\alpha} - \\frac{1}{2}}{\\hat{\\alpha} + \\hat{\\beta} - \\frac{1}{2}} \\approx \\ln \\hat{G}_X ", "8595553e5a95fa43cf37c55e2bd61c56": "b = (b_1, \\ldots , b_r) \\in \\mathbb{F}_p^r", "85959a888b48dadc27c254283722e51c": "\\sin x = 2^n \\sin\\frac{x}{2^n}\\left(\\prod_{i=1}^n \\cos\\frac{x}{2^i}\\right).", "8595e733c72c0a1a0961461658e710e8": "\\sum_{k=1}^m k^3 =\\left[\\frac{m(m+1)}{2}\\right]^2=\\frac{m^4}{4}+\\frac{m^3}{2}+\\frac{m^2}{4}\\,\\!", "8595f152f486c75a3821e0b5c33c7d66": " \\langle x\\vert \\psi\\rangle = \\psi(x;\\theta)", "85962ad5fee90ecdaa0af5ec9c4fb99a": "\\scriptstyle QPC", "85966d1fa08888ec5d1ba66affed1677": "\\sqrt{abcd}", "8596be9c2876960cc248a6f765122989": "j=1,\\dots,N", "8596dd82c1c439c13b2798a750141009": "x\\in Y", "8597739c63195002475d8bedb87fba07": "\\,\\!h", "859782c1c372fca0ce8dfd294648e142": "\\mathbb{E}\\|\\theta-\\hat{\\theta}\\|^2 = \\text{var}(\\hat{\\theta}) =\n\\frac{\\sigma^2}{N}", "8597dfeb5cae909ac7bf58fc40ce54ab": "[D,P_\\rho]=-P_\\rho", "8597f42761b8d114296f1917c77c11a2": "\\scriptstyle\\boldsymbol{a}(\\boldsymbol{x})=\\left(a_{ikjh}(\\boldsymbol{x})\\right)", "8597f90f537b798002cecc74dd20a208": "x_n \\rightharpoonup x.", "85987aa798f18b0bb12ebb8c93a924fb": " \n |\\alpha\\rangle \\equiv |x,p\\rangle \\qquad \\qquad \n x \\equiv \\langle \\hat{x} \\rangle \\qquad\\qquad p \\equiv \\langle \\hat{p} \\rangle\n", "859882c254ca0cbf281b33c64caab9b7": " \\omega = \\frac {2 \\pi}{T} \\ ", "8598b26cf52f1ec5792b0c0edf202e4d": "\n \\dot{\\xi} = \\frac{1}{2}\\left(\\cfrac{4\\pi\\rho}{3M}\\right)^{1/3}\n \\left(\\cfrac{\\mu(p,T)}{\\rho}\\right)^{1/2}\n", "8598c6d9bff820a478e4eb05963ab57d": " C^T_{(-)}=-C_{(-)};~~~C^2_{(-)}=-1 ", "85991473abfa8df90f113d19d7c32c90": "\\arccos (-x) = \\pi - \\arccos x \\!", "85993e504524b1f07cad1986bc8c5809": "\\mathbf{F} = 2 x\\mathbf{i}+y^2\\mathbf{j}+z^2\\mathbf{k}.", "8599791a9e1b5efaf844b9a8761b2516": "A=(6+5\\sqrt{3})a^2 \\approx 14.6603...a^2", "8599845daa0214eb1f7537edcbda947e": " \\mathrm{S} \\subseteq \\mathrm{N} ", "859987976ea63d644b1ac26d9b026db5": "x, ", "85998de46151e16ba13039b510108fce": " i_1,\\cdots,i_k >1,", "859a25409f35f6fb73e4c87e08daecb4": "m+\\Delta m\\,", "859a47e4d6284f6aab5b8a52c4c275be": "(a - aba)(a^{-1} + (b^{-1} - a)^{-1}) = ab(b^{-1} - a)(a^{-1} + (b^{-1} - a)^{-1}) = 1.", "859a7784036202363bde8a42bb3e1396": "\\phi (-a)=\\phi (a)=0", "859b9e6dafe5cd8d255dd02c254d6848": "\\{R_1,R_2,R_3\\}", "859ba136ada3ccd9a703fa40cfa62084": "s(b,c)+s(c,b) =\\frac{1}{12}\\left(\\frac{b}{c}+\\frac{1}{bc}+\\frac{c}{b}\\right)-\\frac{1}{4}.", "859bb4f4fded094896ba405edef1cf60": "L_3=\\ln\\left(R_3\\right)", "859be21fa2f2f02c213791bf014a2c88": "1/\\sqrt{2}(\\pm 1,\\pm 1,0)", "859c203a55d34091609f52d1fbd7cc9b": " e^{\\lambda x}", "859c49636e38d7da118cf2a92476b93b": "\\|f\\|_p = \\Bigl(\\int |f(x)|^p\\,dx \\Bigr)^{1/p}", "859c636fb5c54450dda94da72772adc2": " GL_n \\times GL_m ", "859c8dcc3acd91eaef593284d0def5b9": "T'\\in\\mathcal{F}", "859ca96a1d1079e534eb6d681f9cefcd": "|s-b_k| < \\begin{matrix} \\frac12 \\end{matrix} |b_{k-1} - b_{k-2}|", "859cc697c535dcaf791959eae42c588f": "\n\\left[\n\\begin{array}{rrrrrrrr}\n-10 & -10 & 4 & 6 & -2 & -2 & 4 & -9 \\\\\n6 & 4 & -1 & 8 & 1 & -2 & 7 & 1 \\\\\n4 & 9 & 8 & 2 & -4 & -10 & -1 & 8 \\\\\n-2 & 3 & 5 & 2 & -1 & -8 & 2 & -1 \\\\\n-3 & -2 & 1 & 3 & 4 & 0 & 8 & -8 \\\\\n8 & -6 & -4 & -0 & -3 & 6 & 2 & -6 \\\\\n10 & -11 & -3 & 5 & -8 & -4 & -1 & -0 \\\\\n6 & -15 & -6 & 14 & -3 & -5 & -3 & 7\n\\end{array}\n\\right]\n", "859cdfb20c1a6efa2b006c1dc18c2840": "GWP \\left(x\\right) = \\frac{\\int_0^{TH} a_x \\cdot \\left[x(t)\\right] dt} {\\int_0^{TH} a_r \\cdot \\left[r(t)\\right] dt}", "859d025c9944ba2e1c55cdef0b449e4d": "\n\\begin{array}{rcl}\nE\\{e^2[n]\\} &=& E\\{(x[n]-s[n])^2\\}\\\\\n&=& E\\{x^2[n]\\} + E\\{s^2[n]\\} - 2E\\{x[n]s[n]\\}\\\\\n&=& E\\{\\big( \\sum_{i=0}^N a_i w[n-i] \\big)^2\\} + E\\{s^2[n]\\} - 2E\\{\\sum_{i=0}^N a_i w[n-i]s[n]\\} .\n\\end{array}\n", "859d7f5b825765e8c659b7d20776e87a": "\\|x+y\\|_{\\infty}=\\|(1,2)\\|_{\\infty}=2", "859d995c6d52342d0964c52a9cdb3636": "b(T)\\,", "859d9c27bf88530d890ea0bc8f755a2c": " p(r) ", "859e1085aad43c5e9c9c416ed6017ad4": "M_{\\theta}", "859e20a6dd7e0b7e0f25ff6a4d9a4563": "\\frac{1}{\\mu} = \\frac{1}{\\mu_{\\rm impurities}} + \\frac{1}{\\mu_{\\rm lattice}} + \\frac{1}{\\mu_{\\rm defects}} + \\cdots", "859e5ddb0f4ae2b9c9fea659e2aeb118": "\\sin(\\alpha + \\beta) = \\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta", "859e722b46129e660c5e4da710892728": "h \\in C", "859e87fdd2890ef6c9dca2df5a27b6ef": "(N-b)/2", "859ec69ed276def02fe6ddb2d737cabc": "(C, x_1, \\cdots, x_n, f)", "859f164597f4d6b962c31c0a6de81c4a": "(*)", "859f69c43dc292aca8a23931f45a27cf": "Y_{5}^{-4}(\\theta,\\varphi)={3\\over 16}\\sqrt{385\\over 2\\pi}\\cdot e^{-4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot\\cos\\theta", "859f82ffdb1b404acc43f4b18e56a43e": "X_{AC} = X_{ref} + c{dC_m\\over dC_z}", "859f8c99bf2cae24688c6bebd9e7fe48": "w_1 = a(1 - z)", "859fabdf7a51c4459dcd56dfdba09c70": "\\scriptstyle\\min(N_t, N_r)", "859fd8360cecd7b851bc6cd54c243b79": "X,A\\not\\vdash B", "859fe9433a6b9ab8ac0d999db75ae6ac": " \\mathbf{S}(\\mathbf{p}(t))=\\sum_{r=1}^R \\mathbf{S}_r w_r(\\mathbf{p}(t)), ", "85a00a47e511ea9fd992bdd3914c1896": "X(\\omega) = \\frac{2\\pi}{M}\\sum_{k = -M/2+1}^{M/2} \\delta \\left(\\omega - \\frac{2\\pi k}{M} \\right) \\,", "85a04a2915829d81cace3525c2f7e8d1": "\\dot{x} = f(t,x)+G(t,x)u", "85a069e5f8d023b2db482c39b3f361f6": " \\simeq 1/2 ", "85a076a535995fac8002224659f593fe": "(z,z_2;z,z_4) = (z_1,z;z_3,z) = 0\\,", "85a0dc1aca78c02c104f8e08fe9c6c8f": " \\frac{\\Gamma(\\tfrac{1}{5})\\Gamma(\\tfrac{4}{15})}{\\Gamma(\\tfrac{1}{3})\\Gamma(\\tfrac{2}{15})} = \\frac{\\sqrt{2}\\sqrt[20]{3}}{\\sqrt[6]{5} \\sqrt[4]{5-\\frac{7}{\\sqrt{5}}+\\sqrt{6-\\frac{6}{\\sqrt{5}}}}}", "85a16c717f5b2d8ea19319a6448e66ca": "0.\\dot{6}", "85a1c5dba0cf04aa33c4b353b5dbd67c": "c_c^{-1}\\approx n/4", "85a1e85d432ebf01e832e7a1b5139846": "f(\\langle x_0, 1 \\rangle)", "85a2903d44b2c56fad57d7b2e00d267a": "VT^n = C", "85a3074b9045f71dacbb335dfc2be7b5": "D_x=\\{z\\in\\mathbb{C}: \\left|z\\right|\\leq \\|x\\|\\},", "85a352687a4bf6170510f5f1eda4ffdd": "c=12", "85a36ec2a032e732b68ed33ac2ca94f7": "Z_{t}", "85a373d0e8770c42c019031401bb2b76": "\\left|\\sum_{k=1}^m\\|x_k\\|-\\sum_{k=1}^n\\|x_k\\|\\right| = \\sum_{k=n+1}^m\\|x_k\\|< \\varepsilon.", "85a37431af8cb3f51f723d2386ac6e91": "\\vec{x}= \\begin{pmatrix} X_{vega} \\\\\nX_{vanna} \\\\ X_{volga}\n\\end{pmatrix}\n", "85a38dade184e8136ebae26998a86fb7": "R_2 = \\frac{V_{Z}}{I_{R2}}", "85a398a20b40c8ad372ed906d5e85f0d": "\\ln r = \\ln a -\\ln b", "85a3a1cf1f2b8b5801dfa16bd51d41b5": "H(\\sigma) = - J\\sum_{}\\sigma_i \\sigma_j.", "85a3b1e67b86e761a00002a37a52a978": "f(x)=\\frac{g(x)}{h(x)}\\,\\!", "85a43adb43c65361ac65caf12442a2e6": "b \\approx \\log \\frac{1-\\beta}{\\alpha}", "85a446c8cbd5a865dca2cc25471df41b": "\\frac{\\mathrm{d} \\Phi(\\mathrm{d}A,\\theta,\\mathrm{d}\\Omega,\\mathrm{d}\\nu)}{\\mathrm{d}\\Omega} = L^0(\\mathrm{d}A,\\mathrm{d}\\nu)\\,\\mathrm{d}A\\,\\mathrm{d}\\nu\\,\\cos \\theta", "85a45381564f5a73db520e4ef1757b6b": "X=L^2(a, b), ", "85a4c76bab3fe7256306e5a8975bea95": " \\xi = \\sum_{i} s_i ", "85a504ae40e19b28479adf382b90ab0e": " \\zeta_i = \\frac{x_{max}}{2 \\pi} \\sin(\\theta_i) ", "85a51d41cffacdd03d55b88cffa6fa9a": "M=(m_{ij})", "85a57b75c3a82271bec4bc3eac332017": " F = \\tfrac{1}{4} + \\left(\\tfrac{1}{4}\\right)^2 + \\left(\\tfrac{1}{4}\\right)^3 + \\left(\\tfrac{1}{4}\\right)^4 + \\cdots = \\tfrac{1}{3}.", "85a5e3752844672d55d3c767c1963ab1": "\\bold{w}", "85a738e973bfca6895fd54fabb277f85": "n=\\lfloor x\\rfloor ", "85a7b5cd312cbc1c141d62253fcfa4f8": "g_0 = G \\, m_\\mathrm{Earth} / r_\\mathrm{Earth}^2 = 9.8331\\,\\frac{\\mathrm{m}}{\\mathrm{s}^2}", "85a7e7fae6827e4b2c2e8287578fefa0": "\\forall (a,b)\\ S (a,b) = S (b,a)", "85a82c561f00ffe0668e1f0fb639da49": "m\\overline\\psi\\psi", "85a8414905c71ffe2e99a2c750a14788": "\\scriptstyle F(\\boldsymbol{v}) ", "85a87e0649ab3287b87363fbdb5801ef": "g^{(1)}(0)=1", "85a88f0c92e0e98520ea8d7f35431aef": " \\sum_{n=1}^\\infty\\frac{H_n^{(r)}}{n^m}<+\\infty ", "85a923e49619bb24111ea3723dd602df": "c=DH(a,b)", "85a94aec9d9435691648cea62d499e85": "\\mathbf{D(t)}", "85a94f3a9ad3c2ae5c5a142d843332e7": "\n\\frac{\\partial{\\mathbf u}}{\\partial z}=0,", "85a9570be7a43d1eec28dd15b857cf03": "U_n(x) = xU_{n-1}(x) + T_n(x)\\,", "85a957523794480a36e7f5ab3080f0fa": "\\bar c_i = \\sum_{j=1}^n \\mbox{no. of defects for } x_{ij}", "85a95c49865b4c4d7fe1fc53596d4392": "MA = \\frac{F_B}{F_A} = \\frac{a}{b}.", "85a98952ed59acfe72c0c1fc4487f7a7": "\\int \\cos{x}\\, dx = \\sin{x} + C", "85a9ae19262e9c47f868f33773e7481c": "t = \\sqrt{\\frac{30 * 180}{2.0 * .308^3 * 3.83(1+3.83^2)}} = 39.2511937", "85a9d35a08d07eba53e774eb8c8647e4": "\\tfrac{1}{2}Q_n", "85aa3b7179adc3c37d33dc5f510d557e": "e^{x} = \\sum_{0}^{\\infty}x^{i}/i!", "85aa73083c9b0a9b666edf2099adcd18": "\\mathrm F(E) = \\coprod_{x\\in X}F_x.", "85aa8b6f190f74ecc49fd71ba290ad5c": "\\Theta(|V|^2)", "85aac9f2cd87ef6bb79ea4691d4ecbaa": "u(t,x,y) \\mapsto [Lu](\\omega,x,y)", "85aaf09709c2bdf8e0ccb111b4b94598": "L\\equiv\\partial_{xy}+\\frac{2}{x-y}\\partial_x-\\frac{2}{x-y}\\partial_y-\\frac{4}{(x-y)^2}", "85ab089f815aa37c9ceecb969f933ba5": "\\sum_{n = 1}^\\infty \\Pr(E_n) = \\infty", "85ab1a31cddfe0b261c6f9d1dbcf4be6": "(c+d)", "85ab58610ea4c5dcc6d5ecbff9b86559": "s \\in S \\cup \\{e\\}", "85ab5d5c4a4aa505e1aa2aac91ee1efa": "y^2 = x^3 +cx^2 +d", "85ab6c90ba2e83ad4fe9285eeb8ef755": "u(x,0) = \\left \\{\n\\begin{array}{cl}\n1 & 0.4 \\leq x \\leq 0.6 \\\\\n0 & \\text{ otherwise }\n\\end{array}\n\\right\n.\n", "85ab8fb11883534e2113343d90fced79": "\\lambda = 1", "85ac5319cad22a0ef8b244397b615b5e": "\nR=g^{ij}R_{ij}=g^{ij}g^{\\ell m}R_{i\\ell jm}\n", "85aca07a0c5d0682bebaaf5add0197a0": "\n \\hat{H}_{\\mathrm{lm}}=-\\sum\\mathcal{F}\\,\\hat{B}\\hat{X}^{\\dagger}+\\mathrm{h.c.}\\,,\n", "85ad2ed323a4f1356e6a57e4796c57e5": "1\\le q < 3 ", "85ad75f90a80c58258e699f4036b575b": "{\\text{Metabolisable Energy}} = \\left(\\text{Gross Energy in Food}\\right) - \\left(\\text{Energy lost in Faeces, Urine, Secretions and Gases}\\right).", "85ad7b5359665631e10474b017fab034": "\\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}} = \\frac{1}{1-2^{-s}}\\cdot\\frac{1}{1-3^{-s}}\\cdot\\frac{1}{1-5^{-s}}\\cdot\\frac{1}{1-7^{-s}}\\cdot\\frac{1}{1-11^{-s}} \\cdots \\frac{1}{1-p^{-s}} \\cdots.", "85adf4fb9944665150850b9f03544a12": " Ax = \\lambda x", "85adff1556ef27bd2558043e841cf2e9": "r_1\\,r_2,\\,\\ldots,\\,r_k", "85ae1b6d7da3fb0e005325502fc3d79d": "d_o", "85ae5a8d2b7e6f7d685858b0b53616b4": "\\begin{align}\n(x-a)^2 - x^2 & = r^2 - R^2 \\\\\n a^2 - 2ax & = r^2 - R^2 \\\\\n x & = \\frac{a^2 + R^2 - r^2}{2a}.\n\\end{align}", "85ae9a84327342feb1fa97449c6e1902": "B_1=x^2/2 \\qquad 0 \\le x \\le 1", "85aeba9ee388181344457154a3b8fd9a": " r = n(m-n) ", "85af90f3d77942cd0a123f4ec97a22e7": "\\exp\\left (\\frac{{\\rm i}}{\\hbar}\\epsilon\\, \\,\\sum_{j=1}^{n} L \\left (\\tilde x_{j},\\frac{x_j-x_{j-1}}{\\epsilon},j \\right )\\right )", "85afa51dc0084a8898edf0c969865f3d": "c-v", "85afb9f97ef807ac7809c4869bf86df5": "a(v)", "85afbbe2c51dd24ce08bf774a8c87353": "\\begin{smallmatrix}10^{-0.13} = 0.74\\end{smallmatrix}", "85afc95e83abc5d755c5604d1f077be3": "p_b=\\tfrac{2bT}{a^2+b^2-c^2},", "85afcb8e8ccfb841bcb233c5d284a78b": "h_{l}", "85afde7778eb4cee7093a10e758d47af": "d^2G_\\Sigma =n^2 d\\Sigma \\cos{\\theta_\\Sigma} d\\Omega_\\Sigma = n^2 d\\Sigma \\cos{\\theta_\\Sigma} \\frac{dS \\cos{\\theta_S}}{d^2}", "85b01d9487f24a2496ee38f3a1f303c4": "\\{W_j\\}", "85b03ae5ea3a62e67851c1d308895654": "\\sum_{d=0}^{\\infty} \\frac{k}{2^d}=2k", "85b0653ab319b36c62f88f9394846959": "\\gamma_i = {\\phi_i}^2", "85b078da772ee9e0773bcb4f0041b586": "\\epsilon_k", "85b08aacbd9bae8eb5b6a88fd7397295": "x_i\\not\\ge y", "85b0b47dca221bc47554ab86b2ae8751": "\nK_H(x') = x' - x_{step}.\n", "85b13d66020728a3888823765acd55a7": "\\frac{d^2 \\chi}{d \\tau^2} + \\frac{b}{\\sqrt{ac}} \\frac{d \\chi}{d\\tau} + \\chi = F(\\tau). ", "85b1d3b76b1702a1039e04ae85815b84": "\\sigma : G/H \\to G", "85b22183eb12e8e6b2ec84820a5953b2": "\\sqrt{\\frac{Z_{In}}{Z_{Im}}}", "85b23dc92df286a08778a880349eebd1": "\\mathbf{v}_{n+1/2} = \\mathbf{v}_{n-1/2} + \\mathbf{a}_{n}\\Delta t_n", "85b27cf749a9d8694cbc034f3c394625": "\\kappa=1", "85b2a98e86e23d40369469281cfa01e4": "\\frac{I_\\mathrm{max}-I_\\mathrm{min}}{I_\\mathrm{max}+I_\\mathrm{min}}.", "85b35383e7d598707b1ce8c81bd57610": "a_{n}=Aa_{n-1}+Ba_{n-2}.", "85b36877fe074683d03163c22dabc0f3": "\\alpha_{\\text{o}}", "85b368bfc581d2df0b872b835cc8a372": "n = p \\cdot q", "85b3be62c1012d18335fca546ca79627": "\\ K=4\\pi^2 n_0^2 (dn/dc)^2/N_A\\lambda^4", "85b3cd7ecdee642c27914c5e0213d334": "\\sum_{i=1}^N E_i = \\operatorname{I}_H, \\quad E_i E_j = \\delta_{i j} E_i,{\\quad}E_i=\\left|\\phi_{i}\\right\\rangle \\left\\langle \\phi_{i}\\right|.", "85b3d56f9cdabc0b2bd29a278160391e": " Q_1^{-1}XQ_1", "85b3e1737a0547b1add5bf917e5903f6": "j:V\\rightarrow N", "85b3f1ea8afeb048f9673671765bea4c": " \\tilde{s}_N ", "85b408a76c8d92fc4e4eede6b8ae2511": "\\varepsilon_{\\rm{p}}i", "85b426cc846020f6bf42f74c50f8f899": "J(a) = \\frac{\\Gamma(a/2)}{\\sqrt{a/2 \\,}\\,\\Gamma((a-1)/2)}.", "85b44635cfd7dd765a00c5f3798c4016": "s_{AB}: A \\otimes B \\simeq B \\otimes A", "85b4ca8f8ad213fe3752d63819318ee0": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4) \\, ", "85b4f9b3de5a31c4a8b0ae3b9e81ca3f": " \\left[\\dfrac{n}{m}, \\dfrac{k}{m},\\dfrac{d}{m}\\right]_{q^m}", "85b52f022b0af118684b34c4bb906059": "\nx = a \\left. \\sigma \\right. \\tau\n", "85b577788ee8f0e87fb1ff749d273da4": "\n \\mathrm{PL}(\\omega) = \\mathrm{Im}\\left[ \\sum_\\lambda \\frac{ F_\\lambda S_\\lambda }{E_\\lambda - \\hbar \\omega - \\mathrm{i} \\gamma_\\lambda(\\omega)}\\right]\\, ,\n", "85b5ab806f79ebc0e6dcfaa1293fc31b": "f(t) = f_0 + k t", "85b5db1c78ce459d6d95955a63da78c4": "\\Delta^\\ast", "85b5fa7f482b88b2f6df658ba8644ceb": "\\scriptstyle B (r)\\ =\\ \\sum_{i=1}^{n} c_i e^{-r t_i} ", "85b65a32308bbe650745cd0744190647": "(x, 0) \\sim (x',0)\\,", "85b675e3dafeb77f3f279c327025f701": "\\frac{\\partial y}{\\partial v}.", "85b6835ef3f368658e121270fed07f27": "S_{M_a} \\subseteq [t]", "85b6e32c09131fe2db1d9d11d1793a4d": "\\theta^\\prime(0)= -\\frac{\\ln \\pi + \\gamma + \\pi/2 + 3 \\ln 2}{2} = -2.6860917\\ldots", "85b6fb257a35869df68ff189250f99b1": "\\scriptstyle T_{3} =L/\\sqrt{c^{2}-v^{2}}", "85b70ccd8a68430dbf628ed5a98d854a": "\\dot{z} = Az", "85b74879711179913f2ee7c17e804b1d": "Pr(c_{A} = c_{B} = 0)\\leq \\frac{1}{2} + \\epsilon ", "85b7d690923bbdf11de89816350dc064": "(f + g)' = f' + g'\\,", "85b7da7a8d54534a51f8230713b4c6df": "Z = z l = (R + j \\omega L)l ", "85b7db17bc5c60214f6b2eecf365b105": "\\ln\\begin{vmatrix}\\cfrac{K-P}{P}\\end{vmatrix}=-kt-C", "85b838fc219d579b8384b33d60fad418": "\\Delta\\subset{\\mathbb{C}}", "85b864ec6f09a6cf8b4ade144283eff2": "W = \\sum_{i=1}^n Y_i", "85b8705470d227bb2ce966348983cd24": "g^{x_2}, g^{x_4} \\neq 1", "85b8b45b9c19ac74aafb8f2a7d933e15": "\\phi_{intf}", "85b8d3c766fe447edb77a534aca9973f": " {} + b\\cdot d", "85b8dabb1ed1144e32f86bfec6b8da6a": "\\ C_i ^T ((X)) ", "85b8db02f50f1d4f46b304e4916da8b4": " 8658\\sqrt{10} \\, ", "85b8dd6197c0646d6b160ba55d24bdae": "\\Lambda_k^n", "85b8e0cecf0e9c88a73dc5630d31cad7": "~\\sigma~", "85b900886e2619d7d50c7301d991b6df": "f(T_2,T_3) = \\frac{g(T_3)}{g(T_2)}.", "85b905895f30f3a8bb068736215b6ac0": "x,y\\in\\Bbb F_p", "85b95f6fc6494a70d8d7355c8f3fb81f": "g(y^p) = a_{m} y^{p^m} + a_{m - 1} y^{p^{m-1}} + \\cdots + a_{1} y^p + a_0", "85b967c6b32cfedb78e5c2e168288cd3": "\\mathcal{M}_{g,n}", "85b99fd4a969be1bd351b4938045e3fd": "\\alpha + 2 y \\not=0,", "85b9a3ecf9504b611f6927c20edc4417": "\\nabla^2 B_x = {\\partial \\over \\partial x}{\\partial B_x \\over \\partial x} + {\\partial \\over \\partial y}{\\partial B_y \\over \\partial x} + {\\partial \\over \\partial z}{\\partial B_z \\over \\partial x}.", "85b9f78e6089d4d61c9bc4fdac4bcac7": "\\begin{pmatrix} \\frac{1}{c}\\frac{\\partial \\phi}{\\partial t'} & \\frac{\\partial \\phi}{\\partial x'} & \\frac{\\partial \\phi}{\\partial y'} & \\frac{\\partial \\phi}{\\partial z'}\\end{pmatrix} = \\begin{pmatrix} \\frac{1}{c}\\frac{\\partial \\phi}{\\partial t} & \\frac{\\partial \\phi}{\\partial x} & \\frac{\\partial \\phi}{\\partial y} & \\frac{\\partial \\phi}{\\partial z}\\end{pmatrix}\\begin{pmatrix}\n\\gamma & -\\beta\\gamma & 0 & 0\\\\\n-\\beta\\gamma & \\gamma & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\n\\end{pmatrix} \\,.", "85ba0cdcf172dd274a9bb9029657bb01": "H_c = J_c\\sum_{(i,j)} \\cos \\left( \\theta_{s_i} - \\theta_{s_j} \\right)", "85ba1a707dec379f7f6c51546ec9f198": "c_1=c_2=0", "85ba3aa878275407a820d31b82b58f42": "\\prod_p^\\infty \\left(1-\\frac{1}{p^s}\\right) = \\left( \\prod_p^\\infty \\frac{1}{1-p^{-s}} \\right)^{-1} = \\frac{1}{\\zeta(s)}. ", "85ba78196f89b376134096910b3a2db1": "\\chi:\\mathbb{Z} \\to \\{0,1\\}", "85bab7940a5a36ebad2cbfa9edc28634": "x \\mapsto R^2 \\frac {x} {|x|^2} = y \\mapsto T^2 \\frac {y} {|y|^2} = \\left( \\frac {T} {R} \\right)^2 \\ x. ", "85bac5db0503eb4eb9e9912789ae25a7": "d S(t)", "85bacdc2ebb16f9a69625439ce333d22": " \\Gamma(z+1)=z \\, \\Gamma(z) ", "85bb4692e8dae6f1fe12dc9123ae9dd0": "= \\frac{31!}{2} \\cdot 60^{31} \\cdot \\frac{160!}{2} \\cdot \\frac{12^{160}}{3} \\cdot \\frac{320!}{24^{80}} \\cdot \\frac{6^{320}}{2} \\cdot \\frac{320!}{8!^{40}} \\cdot \\frac{2^{320}}{2} \\cdot \\frac{160!}{16!^{10}}", "85bb4e62e39cccbfb0b79168ad2b4133": " r_1 \\ ", "85bb74d6ec363fe1cd55a1ec7de024ad": "f(z)=\\sum_{n=0}^\\infty f_n z^n", "85bbd2e2af6e3d5d458d1b9e64b0cdcc": "\\frac{\\partial^p}{\\partial x_1^{p_1}\\partial x_2^{p_2}\\ldots\\partial x_n^{p_n}} f(x_1, x_2, \\ldots, x_n) \\equiv \\frac{\\partial^{p_1}}{\\partial x_1^{p_1}} \\frac{\\partial^{p_2}}{\\partial x_2^{p_2}} \\cdots \\frac{\\partial^{p_n}}{\\partial x_n^{p_n}} f(x_1, x_2, \\ldots, x_n)", "85bbfaf498bd0defbd497ea53a716f4b": "\\overline{x} =\\left[ \\begin{array} [c]{c}\\bar{x}_{1}\\\\ \\vdots\\\\ \\bar{x}_{p}\\end{array} \\right] = {1 \\over {n}}\\sum_{i=1}^n x_i", "85bc566f93a4f8290a16e5ce6302a8fd": "\\frac{1}{C_p} \\left ( \\frac{p_\\circ}{p} \\right )^\\kappa \\left [ \\frac{\\partial}{\\partial x} \\left (\\frac{dQ}{dt} \\right ) \\right ]", "85bc681639c80d3638ce6ef6b3518e2c": "\\delta_{max}", "85bcb37dee46f0ac47a376a4a31ff826": "\\left(x, y\\right) \\in F", "85bcb45bfa882b34e9ea5d0865a4f404": "{g}_{\\kappa\\lambda}", "85bccb2c20160c0d7fa185fb512f126e": "f \\in \\mathcal O(X)", "85bcefb1b4996fcd824263b1e4afb502": "\\overline{\\mathbf{GP}}", "85bd10334617e6715403aeffc1cb18a9": "\\sim p", "85bd53bbcb428989ade02e573f54ae0f": "R_{vs}", "85bd57f1ab335bb6a2bcde2d0bfd2b67": "\\textstyle \\alpha_i : K \\to \\mathbb{R}", "85bd6d4d9a7efa8ad3d228e01ac11c3a": "E_k = \\begin{matrix} \\frac{1}{2} \\end{matrix} mv^2 \\times\\left(\\frac{1\\mbox{ ft}\\cdot\\mbox{lbf}}{7000\\mbox{ gr}\\times 32.163\\mbox{ ft}\\mbox{/s}^2}\\right)", "85bd8648e34c5c05244699379bdae252": "\\min \\{g'\\} = \\sum_{p_i \\in P} r(p_i) + \\sum_{q_j \\in Q} c(q_j).", "85bdedc8491e4c2e208fdd2c2f2c868d": "C^{2/n+1}\\theta(C\\xi)", "85be0a0378fd0f6924b169218baefc7f": "t_n = t_{n-1}t_{n-2}t_{n-3}", "85be200b5333946736b65c126719723f": "\\alpha'_1,\\dots, \\alpha'_n", "85be5981fa48852e3da098cbdc55176b": " \\phi(\\vec{r}) = -\\frac{1}{4\\pi}\\iiint_{\\vec{r}'} \\frac{\\vec{E}(\\vec{r}') \\bullet (\\vec{r} - \\vec{r}')}{\\|\\vec{r} - \\vec{r}'\\|^3}d\\tau' ", "85be608bb00ab137eec4818d4df1ef04": "\\operatorname{Trans}_{z_{n - 1}}(d_n)\n = \n\\left[\n\\begin{array}{ccc|c}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & d_n \\\\\n \\hline\n 0 & 0 & 0 & 1\n \\end{array}\n\\right]\n", "85be7dcd684f8cb25b6c07a3a1411fba": "{\\partial \\Phi \\over \\partial t}", "85bf33f46a5fbef77e3740478fba172c": "\n\\text{N} = TN + TP + FN + FP\n", "85bf4fa692b0424d79ed0850da3a96ed": "\nS = \\int_{xt} \\psi^\\dagger \\left(i{\\partial \\over \\partial t} + {\\nabla^2 \\over 2m}\\right)\\psi - \\int_{xy} \\psi^\\dagger(x) \\psi(x)V(x,y) \\psi^\\dagger(y)\\psi(y).\n", "85bfa261bc877a62fd17894cacceb7f2": "\\textstyle x^2 \\equiv \\prod_{i=0}^k p_i^{e_i}", "85bfca3a63f55e28f9b3c6b5809fe588": "Z_{TS} ", "85bfdeab0f4057ea3dcd64a97542c3fa": "I_D = \\frac{\\mu C_i}{2}\\frac{W}{L}(V_{GS}-V_{th})^2.", "85bfe1052d6026f70ecd4fd630453514": "r_j\\!", "85c01860f88dc2655d52c3022fb00d4e": "Q(\\mathbf{Z}) = \\prod_{i=1}^M q_i(\\mathbf{Z}_i\\mid \\mathbf{X})", "85c03a9f0d908989e381383b67a34a2c": " x' ", "85c0427e242a84f96783c762dfe80771": "\\ p(x) = 0 ", "85c0690cda616a18bc86361f7089d1fe": "R = \\frac{u'(c_{t})}{\\beta u'(c_{t+1})}", "85c080f3d47235e4260238d58eb50d08": "R[t] \\otimes_R R[t] \\simeq R[t_1, t_2]", "85c08488998b289d7eab144a47c79ef5": " c=g^m \\cdot r^{n^s} \\mod n^{s+1} ", "85c0a764614a4b915c35c8a466ab5688": "\\frac{pV_m}{RT} = 1 + \\frac{B}{V_m} + \\frac{C}{V_m^2} + \\frac{D}{V_m^3} + \\dots", "85c0dde13174077cef32d65f6c1e9c2b": " T=\\{T_{ij}\\} ", "85c0f97764e704e17abcb74d8881687a": " \\gamma_0(\\nu) = \\sigma_{21}(\\nu) \\cdot \\Delta N_{21} ", "85c0fff7c2814085d668957573f1d1f2": "wp(S,\\ x=0 \\vee x=1)", "85c10a05d652be5d55d52d28536ba04e": "c = 0.585", "85c14bceecce905fa2c0661b03bca239": "w : t", "85c174d89ed27d0741744ae7a1161b3d": "\\lambda^\\Lambda(\\mathrm{d}\\omega)", "85c1b0e6242022b7e23a0923a811531d": "t_r\\cong\\frac{0.35}{BW}\\quad\\Longleftrightarrow\\quad BW\\cdot t_r\\cong 0.35", "85c1c59da5f904e20aefce2149ad9da7": "\\textbf{x}=V\\textbf{y}", "85c21315eb5322d9b7c63b5876f04377": "\\alpha \\in \\mathbb{R}^+", "85c2362cb0f3fa63c8807bc9e0a28298": "\\frac{d^2 V}{dz_kdu_q}=0", "85c285b08a6c481b14bd6acaa08bd5bd": "n\\leq 28", "85c295110d5a0c9a0772b59362f32a87": "D_A*F_A+[\\Phi,D_A\\Phi]=0,", "85c2b3a9d9cd9130ea3ad3859f915348": "| 0\\rangle", "85c2fde3febcf17d844f937ce7c0ea70": "\\hat{e}", "85c338efe77e88aa49230b900881c4aa": "P \\rightarrow Q", "85c3d5e5d81664bbfab79b3f3355159d": "\\sum_{n\\ge 0} \\Phi_n(g(z))\\zeta^{-n} =1+\\sum_{n\\ge 1} \\left(z^n +\\sum_{m\\ge 1} c_{nm}z^{-m}\\right)\\zeta^{-n},", "85c3e4a7884ac9688b7860e3041e1d55": "\\delta_\\epsilon\\Phi=(\\epsilon^* Q+\\epsilon Q^\\dagger)\\Phi.", "85c421d5574964d3d1fdce5f0e911f59": "\\alpha=1\\quad\\quad\\quad\\quad\\sum_{i=1}^n\\frac{x_i^n}{\\Pi_i(x_1,\\ldots,x_n)}=\\sum_{i=1}^n x_i", "85c44eaefc6ad02a1b13439e24d53558": "u'' -Ru' +Su=0.\\!", "85c46b73df57837b6b732b0c2d0a024a": "i \\in \\left\\{ {1,2,\\dots,k } \\right\\}", "85c4bcbf965c578041c5c5d285e6dce1": "\\alpha = \\arctan \\left({R \\over L}\\right) - \\arccos \\left({\\sqrt{L^2+R^2} \\over 2\\rho}\\right)", "85c4d8422bbd624a51409e48349364a3": "x y z^2", "85c54d383710d3b394c7851615a2d57e": "\\left(\\frac{1}{2},\\frac{\\sqrt{3}}{2}\\right)", "85c55d6f8d7f1b182be26732fc776714": " \\boldsymbol{x}|_t = \\Phi_{\\{H(t)\\} \\left(t,{\\boldsymbol{x}_0} \\right)}. \\, ", "85c56a8dd7e79654610a252f93f0cb62": "y|\\textbf{x}", "85c56e2ff22b4b281fd34abc2142425f": "\\scriptstyle \\frac{1}{k}", "85c571c4e725d43a9cb3d4b92c53c4f7": " w_j \\leftarrow w_j - \\alpha_j v_j - \\beta_j v_{j-1} \\, ", "85c5a1e56607dcaf679c60a035181a40": " s = u t+ \\frac{1}{2} at^2 = \\frac{1}{2} (u+v)t ", "85c5b61c94be0285f82fe3f928f611e5": "\\begin{align}\n & \\hat\\beta = 61.6746 \\\\\n & \\hat\\alpha = -39.7468 \\\\\n \\end{align}", "85c5bda786ea69bbddb4be78be20c88f": "W_\\text{pressure}=F_{1,\\text{pressure}}\\; s_{1}\\, -\\, F_{2,\\text{pressure}}\\; s_2 =p_1 A_1 s_1 - p_2 A_2 s_2 = \\Delta m\\, \\frac{p_1}{\\rho} - \\Delta m\\, \\frac{p_2}{\\rho}. \\;", "85c5cb7c4e2ae43f3450a97e3e6a9a39": "S_k", "85c5ebc994277dfb78d5176bfc75faa1": "f \\equiv -\\infty", "85c64b919939830bffd4c53381be8511": "\\frac{c}{n}", "85c65fdac284d7a4060530c928c91e81": "f(x):= \\int_{-\\infty}^x F(u)\\,du = \\operatorname{E}\\left[\\max(0,x-X)\\right] = x-\\operatorname{E} \\left[\\min(x,X)\\right]", "85c687cf57e045d46aa9fb9830baae62": "Z_{\\cdot\\cdot} = \\frac{1}{N} \\sum_{i=1}^{k} \\sum_{j=1}^{N_i} Z_{ij}", "85c6c2fb3556d5bcb663709e6c563f54": "(\\tfrac{11}{5}) = +1: \\qquad \\tfrac{1}{2}\\left (5(\\tfrac{11}{5})+3 \\right )=4, \\quad \\tfrac{1}{2} \\left (5(\\tfrac{11}{5})- 3 \\right )=1.", "85c6f2ded561655d6753c890bee0df13": "\\csc\\varphi = \\frac{1 + t^2}{2t},", "85c727fe1b76d72cec263984bef8ae9e": "a_a = (a,0,0)", "85c743e868e9e6fd6c3b3801b224ac18": "P(x) = x\\frac{1}{1-P(x)}", "85c779a1aebf892f3ab320c3bed460bc": "B\\setminus A:=\\{ r \\in R \\mid Br \\subseteq A \\}\\,", "85c79f64ff04f088f757408869f00058": "p_2=m_1q_3(1+m_1)\\ ,", "85c79fed4f834d45d2f1c23c2b1b30b3": "\\forall i < n \\; \\left(a_i < m_i \\right)", "85c7c83284989a23237c6fd10e7baed4": "\n\\delta(C) = \\frac{1}{l} \\int_0^l h(C - C^\\prime(\\vec{r})) \\, ds\n", "85c7d15973e44808cc4f30d66fe22552": "\\frac{\\partial u}{\\partial \\mathbf{X}} + \\frac{\\partial v}{\\partial \\mathbf{X}} ", "85c84320c7f347c47f3b469ec4afa3c8": "X \\sim \\mbox{Scale-inv-}\\chi^2(\\nu, \\tau^2)", "85c86886c6b8b9ae918b7e01ff073b01": "F_Y(y) = 0\\qquad\\hbox{if}\\quad y < 0.", "85c8750588dcefad1f4c26b2adf00e43": "Y=F_X(X)", "85c882d70d5fee04f52c63fba4762e20": "\\displaystyle{(Tf,g)=(u,(I+L)v)=(u,v)_{(1)}=((I+L)u,v)= (f,Tg).}", "85c8a39cdd6f77be4f8f8b452468ee11": "\nG = G[\\tilde{S}(\\omega)] = \\left[\\int_{-\\infty}^\\infty \\mu(\\omega) \\tilde{S}(\\omega)^{-1} \\, d\\omega\\right]^{-1} \n", "85c92bc4f0b57854196995f2ba9eaf1c": "a\\sin x+b\\sin(x+\\alpha)= c \\sin(x+\\beta)\\,", "85c93b4c93febbec35d34d451f3c0ead": "\\langle N,E,< \\rangle", "85c95211d7ee3c595b4fef19af01d353": "\\lim_{n\\to\\infty}nu_{n}=0,", "85c97db8e6d50ee02eb81165e59cae4e": "2 c \\frac{d\\tau}{dq}", "85c9add5e7d8ac3bc6f50d8d5fdf448b": "i(V)=\\frac{enS_z}{4}\\int\\limits_\\sqrt{2eV/m}^\\infty f(v)\\left ( 1 - \\frac{2eV}{mv^2}\\right ) vdv", "85ca3a88cf7436a2e514f3f0fe3ab1fd": "\\prod_{n=0}^\\infty {p_{n+1}}^{a_n}, a_n \\in \\lbrace 0, 1 \\rbrace,\\text{ and }p_n\\text{ is the }n\\text{th prime}. ", "85ca76c9f37466472017719f3fcad5e5": " \\Phi_M ", "85cadf8077b173acc7ef1d65b78dcaed": "{}^{n}a = \\exp_a^n(1)", "85cae66ac25d7660967bbdaebc934e32": "V_c \\, = \\, 25.2 + 2.80 * MW + \\sum_{j=1}^{35} n_j \\Delta_j", "85cb1a90aae8581e5e35a651ef843bca": "X^\\text{tr}AX \\equiv A\\ \\bmod\\ p^r", "85cb7d8fe1be79de45b6b59d8cb6673b": "V(t)\\ ", "85cbb164bb6818e758f54eb63c47990f": "\\mu(x) > \\epsilon", "85cbcadda03ab29a86ea30ae30c43778": "\\mathbf{(J^T J + \\lambda\\, diag(J^T J))\\boldsymbol \\delta = J^T [y - f(\\boldsymbol \\beta)]}\\!", "85cbfcfe77a11f40109a794111a05ca3": "e(t) = s(t + \\alpha) - \\hat{s}(t),", "85cc03254a4e25de9fb102a94bcb378b": "(c')^+", "85cc2fbee939d7a713afcfde872efa2a": "\\frac{d^3\\mathbf r} {ds^3} + \\alpha\\frac{d\\mathbf r} {ds} = \\mathbf{0}", "85cc3c0e33c5f32ce79d6a2fc2b43ff9": "p_{k,i}^C", "85cc878672fc4440d900b27437b2ce74": "\\lambda\\geq0", "85cc93dc1614adbca55601de9ac60fcb": "\\mathbf{ x}(4) = [u(4)\\, u(5)]=[85\\, 80]", "85cd1091ae2bf1231f9dc9b44dda338b": " {\\tau_G}^{-1} = \\frac{2 \\nu}{\\pi d_G} \\left[ 1-exp\\left(-\\frac{\\pi^2}{4}\\nu_G \\right) \\right] ", "85cd375a581fc5f9cd45dd3cd327d20b": "\\Delta = I - M", "85cdea7485096cbb75166697187f50cd": "\\frac{v_m}{v_r}=\\frac{p_m}{p_r}.", "85ce11e60d960dc0527bf0890e4d4358": "A^i_a", "85ce28618ed8e0bea583b855f60d5086": "\n\\gamma(\\vec{s}) = \\; \\gamma(s, \\theta) = \\; -\\frac{\\pi}{2} \\, C_s \\, \\lambda^3 \\, s^4 \\; + \\; \\pi \\lambda \\, z(\\theta) \\, s^2 \n", "85ce3dcb852f908ead794fe69c01ef72": "\\Box p", "85ce4a9de491d4de18e4c07b74ddff93": "U_\\nu~d\\nu = \\left(\\frac{N\\,h\\nu}{V}\\right) P_\\nu~d\\nu = \\frac{4\\pi f h\\nu^3 }{c^3}~\\frac{1}{e^{(h\\nu-\\mu)/kT}-1}~d\\nu.", "85ce5bc6dff5736876b098157a40a852": " U_n = \\beta s_n + \\varepsilon_n ", "85ce7c1c45dba49dbaa57a047fc2b1f1": " E \\ ", "85ce8e09fe868ec8d79fdbcf5fdb3aa2": "\\in T", "85cec63e0c1a0ea84f6151f2a20fc9a7": "B\\subseteq|\\mathcal A|", "85cf44aedc564d4ee0e5a1f2449d2458": "\\epsilon_m = \\frac{[E]_0}{[S]_0 + K_m} \\ll 1", "85cf6bfa5fd3a7d19c22ec8d113f6698": "A + (\\lnot A \\cdot B) = A + B", "85cf7fbbbaa9e3eaadbdaaab085c857b": "\\displaystyle A^{(2)} = P^{1}AP^{1}", "85cfe0d4d0a8235932ee53e5d0d02494": "\\mathbb{Z}.", "85d0088f7ce1543ff00d88650b74c0d9": " U_\\omega |x\\rang = |x\\rang \\qquad \\mbox{for all}\\ x \\ne \\omega", "85d0575e5feb5ba52f83b2fe4654641d": "S=\\{s_1,s_2,s_3,\\dots\\}", "85d05d6209771cc5534373e7388b688b": "\\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} ", "85d09b948359e3b64946862a2559f5ca": "\\,o", "85d0a69c23ad5529c873d782dfbcea9e": "\n s_{(d_1, d_2, \\dots , d_n)} (x_1, x_2, \\dots , x_n) =\n\\frac{ a_{(d_1+n-1, d_2+n-2, \\dots , d_n+0)} (x_1, x_2, \\dots , x_n)}\n{a_{(n-1, n-2, \\dots , 0)} (x_1, x_2, \\dots , x_n) }. ", "85d0cf6958594f53c890ddfa1603c1ee": "d\\theta = \\frac{1}{x^2+y^2}\\left(-y\\,dx + x\\,dy\\right),", "85d0e9e69590b04c55daee7f0c81d186": "\n\\begin{align}\ny[n] = \\sum_{m=1}^{M} h[m] \\cdot x_k[n-kL-m]\n&= x_k[n-kL] * h[n] \\\\\n&\\stackrel{\\mathrm{def}}{=} \\ y_k[n-kL].\n\\end{align}\n", "85d0f722ee04a66750cd2b81285d006c": "p_{s,\\ell}(z)", "85d0f941d86d65040d8f80b31a856809": "f+g:(x)\\mapsto f(x)+g(x)", "85d0fd06679d66d1a7bb661a9027c5d7": "\na_{i}\\bar{a}_{i}b_{j}\\bar{b}_{j}+a_{j}\\bar{a}_{j}b_{i}\\bar{b}_{i}-a_{i}b_{i}\\bar{a}_{j}\\bar{b}_{j}-\\bar{a}_{i}\\bar{b}_{i}a_{j}b_{j}=\\left(a_{i}\\bar{b}_{j}-a_{j}\\bar{b}_{i}\\right)\\left(\\bar{a}_{i}b_{j}-\\bar{a}_{j}b_{i}\\right),\n", "85d10764e67417b6a71a7fa770967931": "T_{\\mathrm V}", "85d127f226cdd2c3bf43ca61f51c66dd": "\\ A = 2R(c\\tau/2)(\\tan\\theta/2)\\sec\\psi", "85d1756078dd0a1697e4567bac183127": " x^3yxyxy ", "85d1818fc679199fe09cdd57be0e9f4e": "\\textit{dau}(e,t) \\leftarrow \\textit{par}(h,m) \\land \\textit{par}(h,t) \\land \\textit{par}(g,m) \\land \\textit{par}(t,e) \\land \\textit{par}(n,e) \\land \\textit{fem}(h) \\land \\textit{fem}(m) \\land \\textit{fem}(n) \\land \\textit{fem}(e)", "85d18509eea9705fd06b91fc8c4a4a3e": "\\sin(\\alpha + 180^\\circ) = -\\sin(\\alpha)", "85d18f6356bacafed472ae76203e6934": " Y= \\frac{P+B}{1-\\alpha} \\,\\ ", "85d1b18179d30abcc712f7ca7bc068ff": "f = \\sum_{k=k_0}^{+\\infty} c_k T^{k/n}", "85d1c30b2cd9f0d2c2d98ffbf96158c8": "S_{xy}(\\omega) = \\lim_{T\\rightarrow\\infty} \\mathbf{E}\\left\\{\\left[F_x^T(\\omega)\\right]^*F_y^T(\\omega)\\right\\}.", "85d1c762627d9edf16df234ea2709bfe": "j^{\\mu}(x)=f^\\mu(x)-\\frac{\\partial}{\\partial (\\partial_\\mu \\phi)}\\mathcal{L}(x) Q[\\phi] \\,", "85d20e57328b4f3c548c7a67f5bc814c": " \\textstyle C_P=A+q \\cdot P\\,\\ ", "85d241c2624de4c022330da5e9dd32d0": "0x \\neq 0\\ ", "85d25f095e24e8b107f8c240d700030b": "b_n\\!", "85d26b0d5c75bdbe9280838ff6dc51ac": "1+\\frac14+\\frac{1}{4^2}+\\frac{1}{4^3}+\\cdots = \\frac43.", "85d29dfe72838006d8adf98ae49056dc": "\\hat F = \\frac{i-0.3}{n+0.4}", "85d2ef3e7b3251594111f6fde632fa24": "\n \\operatorname{E}[s^2] = \\sigma^2, \\quad\n \\operatorname{Var}[s^2] = \\sigma^4 \\left (\\frac{2}{n-1} + \\frac{\\kappa}{n} \\right) = \\frac{1}{n} \\left(\\mu_4 - \\frac{n-3}{n-1}\\sigma^4\\right),\n ", "85d2f13015b9c15944289385600092df": " a\\!\\!\\!/ := \\gamma^\\mu a_\\mu ", "85d31d4c11c8346cb86d216dad316b34": "\\mathbb{A}_4", "85d358d388f9c92a5d9e6cd104d98711": "P^{2} \\psi = e^{i \\phi} \\psi", "85d3d0e9a0ec9afe39fc73381949cead": "\\left({c\\over p}\\right)", "85d3ded587ff8fb2de8e8efa5cfbab41": "R \\in S", "85d4a2d4677c489b03614e27730cb32f": " \\operatorname{de-let}[M_1\\ N_1] ", "85d4d81d547697cb8c327d82a23ccfe4": "\\eta_i = \\frac{K_c + K_i}{K_c+K_i + K_i1 + K_i2},\\quad i = 1,2", "85d5520f37adac55c9dacec50b765087": "\\beth_0(\\kappa)=\\kappa,", "85d58ae074808634dbbf85123e1097be": " R_q ", "85d58fa9d6222ffa6b765d15bb346871": "\\scriptstyle x[n]\\cdot e^{-i 2\\pi f T n}", "85d59395ca8274a2e0b195554b8ba8d2": " E(\\hat x,u,y),W(\\hat x),I(\\hat x,u)", "85d5a8fa78f0ca683194fbf75315b753": "\nr(t) = -\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t} = v_n \\sigma^n * e^{-E_{act}/RT}\n", "85d6085f059537a372f8834946d34225": "1/(z-w)", "85d6601fdec9148dba950c7b32e4819a": "G^2(\\omega)=\\left |H(j\\omega)\\right|^2 = \\frac {{G_0}^2}{1+\\left(\\frac{\\omega}{\\omega_c}\\right)^{2n}}", "85d663bb18685baf4490cf9029ea264d": " \\int_{-\\infty}^\\infty \\frac{\\sin^2(\\theta)}{\\theta^2}\\,d\\theta = \\pi \\,\\! \\rightarrow \\int_{-\\infty}^\\infty \\mathrm{sinc}^2(x)\\,dx = 1. ", "85d6c77aacaa25ff5a2bd1b7c1c4dc3f": "\\varphi_f", "85d71fdad8d50cd7d19088927c44b967": "x/n.", "85d749a624698378c29d2554191ad0c4": "x_{k+1}, \\ldots, x_m", "85d76cb321214a9a9d3017ae56149292": " \n\\text{Minimize: } \\lim_{t\\rightarrow\\infty} \\frac{1}{t}\\sum_{\\tau=0}^{t-1}E[p(\\tau)]\n", "85d79092121144cd523d1d9b90059a82": " dS = \\mu S \\,dt + \\sigma S \\,dW_t. ", "85d7a165ae559ca85f9cdc2418373712": "\\,g(X,X) > 0", "85d7a9f7252e07ef6e21bfefffe4ab85": "\\begin{align}\n \\color{BrickRed} P &\\color{BrickRed}{= 2x + 3y + 5} \\\\\n \\color{RoyalBlue} Q &\\color{RoyalBlue}{= 2x + 5y + xy + 1}\n\\end{align}", "85d7e164d2c11fded28ad9c4f9fcad46": "\\frac{(7+2+1+7+6)^2}{5} + \\frac{(11+6+10+7+3)^2}{5} + \\frac{(5+3+4+11+4)^2}{5}", "85d7e27dd3e441ded828880bde7730d5": "~g~", "85d8b274957517f6b3da78354085a0f5": "W=\\sqrt{K/h_0}", "85d9130aeb7f675d7a3210e0f77aa1ed": "\\scriptstyle w[(N-1)/2] = 1,", "85d91a37860aaa55da52ddd6f1c16088": "SH_k(\\text{point}) = 0", "85d92b7617c95979421ea7f74712671f": "q^{ab} E^i_a E^j_b", "85d9baf9c814edcd44a6792aab4f4bc9": "79 \\times {4 \\choose 1} +\n 283 \\times {4 \\choose 1}{9 \\choose 1}{3 \\choose 1} +\n 222 \\times {4 \\choose 1}{10 \\choose 2}{3 \\choose 1}^2 = 390,520\n", "85d9bc8fea7a3d6f2b9a09890c81e1bf": "y=\\dot{M}_O - \\hat{\\dot{M}}_O", "85da171f47421d8637a5e3ccd721ec7e": "\n \\left( 1 + \\frac{n - x}{\\left[x + 1\\right]F\\left[\\frac{\\alpha}{2}; 2(x + 1), 2(n - x)\\right]} \\right)^{-1}<\n \\theta <\n \\left( 1 + \\frac{n - x + 1}{xF\\left[1 - \\frac{1}{2}\\alpha; 2x, 2(n - x + 1)\\right]} \\right)^{-1}\n", "85da4128e908652bbbd7fd6055fec06f": "\n \\hat{w}_n = A_1\\cosh(\\beta_n x) + A_2\\sinh(\\beta_n x) + A_3\\cos(\\beta_n x) + A_4\\sin(\\beta_n x) \\quad \\text{with} \\quad \\beta_n := \\left(\\frac{\\mu\\omega_n^2}{EI}\\right)^{1/4}\\,.\n ", "85da529ad2f8631bd9efebe85d01c6a9": "Y(0) = \\ln I(0)", "85da7ddd8b12edcefff877c70a9a3bed": "\\lambda_2=\\frac{(y_3-y_1)(x-x_3)+(x_1-x_3)(y-y_3)}{\\det(T)}=\\frac{(y_3-y_1)(x-x_3)+(x_1-x_3)(y-y_3)}{(y_2-y_3)(x_1-x_3)+(x_3-x_2)(y_1-y_3)}\\, ,", "85dac695e4ffd72b60e37a4683df76d6": "\nN(E)\\, dE = N_0 |E|^{-9/4} dE,\n", "85db218fd80dd6dce1149a8684e96d62": "\\ Z_{\\text{inductor}} = j \\omega L", "85dbae3e7d3a2942598beff1ab0e1bda": "x_{n+1} > x_n", "85dbc695027f556eb39c5ea8dd015083": "P= -4\\partial_{x}^3+3(u\\partial_{x}+\\partial_{x} u)\\,", "85dc11cf6892ca4d50c04af6d79ffe33": "p^{(h+1)}=\\frac{-n(1-p^{(h+1)})} { \\ln( p^{(h+1)}) \\sum_{i=1}^n\n\\{1-(1-p^{(h)})e^{-\\beta^{(h)} x_i}\\}^{-1}}.", "85dc416a77b69520fd2b27a3100a12a5": "\\begin{matrix}{5 \\choose 5} = 1\\end{matrix},", "85dc4599e21f4bf6dcf1b64862e33806": "g_p(u)=1", "85dc8ac7b6db1500646b12bf209cdcbe": "P_n^{(\\alpha, \\beta)} (1) = {n+\\alpha\\choose n}.", "85dcbb5b0aaa4071e20d9c1a8c2e585d": "1 \\over 22", "85dcd33a9293dfe1c02682fa5a2ae54f": "\\langle X(t),X(s)\\rangle_s = \\exp(-(\\lambda+\\mu)|t-s|) \\operatorname{var} \\{ X \\}_s.", "85dcf12f2ef2ff51040e4e7bd60a00a6": " f\\in", "85dd21c1772c1a3f7c1aa9e3e1d3c485": "\\left\\{x, y\\right\\}[z]+\\left\\{y, z\\right\\}[x]+\\left\\{z, x\\right\\}[y]=0.", "85dd26811e5fec5ca79480cd0eed6399": "v_{t} = \\frac {v_{a}}{M} = \\frac {52}{81.25} = 0.64", "85dd6281cd26f1f82c23dff33399fb6b": " a^{(p-1)/q}\\not\\equiv 1\\pmod p ", "85dd825027058b5e2e73bf849e6d0230": " k_c ", "85de0ed9bb6c0a0cfe0917a77d60c091": "p \\supset q", "85de158ef9c3ce81d11fc6692163a8c2": " \\frac{2M+T}{3}. ", "85de510fafbd374adf77f82f95e15bea": "f_{x x} = \\frac{d^2f}{dx^2}. ", "85debaebadfb8d43f2ffc0c26ef7c6aa": "\\sin^2 \\theta + \\cos^2 \\theta = 1 \\, ", "85dec51e9c4b6f8fc49223220f84fa95": "{{F}_{n}}={{P}_{0}}+{{P}_{1}}*{{g}_{1}}\\left( {{r}_{1}} \\right)+\\ldots \\ldots +{{P}_{2}}*{{g}_{n}}({{r}_{n}})", "85df41d237c155d7b4fb27ebddf392b8": "U_t \\,", "85df442a2a1c35458b89f184e8eeeaef": "\n \\mathbf{u}^* = \\mathbf{u}^{n+1} + \\frac {\\Delta t}{\\rho} \\, \\nabla p ^{n+1}\n", "85df4d008e8520fe5c6c9fede9e96d96": "x^n=\\left( x_1 \\ldots x_n \\right)", "85df70b12ed8dad002b992e22080e509": "d(p,q)=\\frac{1}{2} \\log \\frac{|q-a||b-p|}{|p-a||b-q|}", "85dfc0dc439aca568a3764294de2f54e": "U-V=1 ~ ", "85dfda607884266b379fc707bab36840": "V = (n-1)\\frac{S_n^2}{\\sigma^2} ", "85e05c94fbc1b4b880ac4038b68584ae": "(S,T,W)\\!", "85e0677e2fb6bb495716994d4bb8ef9c": " u - u^* - r|d-d_1| - P \\ge 0 \\,", "85e0c7bd9063d9596a9f11ca19657fb0": "F(x_1,x_2,\\dots,x_n)", "85e0d87c4b7982124e65b6ba895e5894": "f(x)=\\Omega_\\pm(g(x))", "85e0e7d31d416a3516f198e3d32bf5ac": " |\\phi\\rang ", "85e0ecf8b1c4440c21d2c6bda5d22a48": " A_w = \\frac{\\langle\\phi_1|\\hat{\\mathbf{A}}|\\phi_2\\rangle}{\\langle\\phi_1|\\phi_2\\rangle}.", "85e12d940d5750986a4711bb5f314eb2": "f_i, \\pi_i, i=1,2", "85e139a7d5fa4dd5aad7ce83c55a5331": "\\int \\operatorname{sech}\\,x \\, dx = \\arctan\\,(\\sinh x) + C", "85e17085a970dbc7e5d229858731aefb": "r^{n}=Ar^{n-1}+Br^{n-2}", "85e1904c92b405e445d2c5d12368c2fa": "|P| = 2N^3", "85e1e6fa594f3cd5a94504cb7dedf6e2": "\\frac{{dY}}{{dx}} = \\frac{1}{h}\\left( {a_1 + 2a_2 z + 3a_3 z^3 } \\right) = \\frac{1}{h}a_1 \\text{ at } z = 0, x=\\bar x", "85e1f4acd4e75b6d6ef2cffd53e40d64": "\n\\cos(2 \\vartheta)=2\\cos\\vartheta \\cos\\vartheta - \\cos(0 \\vartheta) = 2\\cos^{2}\\,\\vartheta - 1 \\,\\!\n", "85e222aee4d765d86951384c92b2a815": "\\ (0)(x^2+1)=0", "85e22683116ee1466ea1a9ec5036fc20": "Y_0", "85e22c55cb9375b85efba3508390ca0e": " \\nu_i ", "85e23ff534b2db110ec5dea4278df36e": "\\mathrm{tri}(x) \\ ", "85e2406e94c253d97cf6a1f72f1e1371": " d_{B,n} \\circ f_n = f_{n-1} \\circ d_{A,n}", "85e247da69ad1cacf520d73b8ff4ef26": "R \\colon \\mathbf{B} \\to \\mathbf{C}", "85e2712e6be46fb22733644550b0547b": "K_{M}", "85e2a3180cf3259f3a5c85226f6c4f57": "c^d = (b^{\\log_b (c) })^d = b^{d \\log_b (c)} \\,", "85e2a37ce4fdfd3e760ab778f0066fc4": "\\scriptstyle\\lesssim10^{-14}", "85e2eada2ac36d790f63cff10872b253": "\\sum_{i=0}^{n}a_i\\mathcal{L}\\{f^{(i)}(t)\\}=\\mathcal{L}\\{\\phi(t)\\}", "85e2f9ee118bddcbee90d6ee81d2e4fb": "\\deg(Q)=\\deg(\\varphi(Q)),", "85e335df4dd3f8d027b97c93c69afa66": "\ng^{(2)}(\\tau)={{\\langle a^{\\dagger}a\\rangle_C}\\over{\\langle a^{\\dagger}a\\rangle}}\n", "85e373e256b755c1a553762ed9130ff4": "A_o(T_o)=\\mathrm{max}_{S_o}(U_o(S_o)-T_oS_o)", "85e399e85258e5befbe3b55f4c1541a8": "b=\\arccos\\left(\\frac{\\cos\\beta+\\cos\\gamma\\cos\\alpha}{\\sin\\gamma\\sin\\alpha}\\right),", "85e3be99016a3afb64bc5f746a6da5f3": "\\frac{G^k e^{-G}}{k!}", "85e3bf20a107178f7876d89c872b6214": "R_{ab[cd;e]}^{}=0", "85e3e077fc1c6bb08bdf34e9d9702320": "-2 \\log \\big( p(y|\\hat \\theta_0)\\big)", "85e3e16a83a85f49e3b4d88379bdcd2d": "f(x,y,z),", "85e40a9085558c36abe8de4ba10ce961": "x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \\cdots + a_0 y(x) = 0.", "85e421432c6ba12f52aa17c6f1c880d6": "G_I = \\pi n_I S \\sin^2 \\alpha=G_R = \\pi n_R \\Sigma \\sin^2 \\beta", "85e4c27e3c18313b1ad6a969ecbea53c": "B' ^3\\Sigma_u^-", "85e4c80f0648fab2684b9b4445104c65": "\\displaystyle \\delta(x)\\,", "85e4efd97319c7ec30522039c0a1341a": " Q_{H}=\\int_{H}T_{H}ds \\,", "85e547e799e1be0e3f9b6861f227de74": "P(X)=X^3-2", "85e57b614265c598dcbb9e54a55b48bc": "(\\lambda 1 1) (\\lambda \\lambda \\lambda 1 (\\lambda 1 (3 (\\lambda \\lambda 1)) (4 4 (\\lambda 1 (\\lambda \\lambda \\lambda 1 (\\lambda 4 (\\lambda \\lambda 5 2 (5 2 (3 1 (2 1)))))) 4 (\\lambda 1))))) (\\lambda \\lambda \\lambda 1 (3 ((\\lambda 1 1)", "85e59525d03020560aabcb57fee9d79a": "\\Gamma_{2}(s)= {\\Gamma}_{4}(1-s)=\\ominus{\\Gamma}_{4}(s)", "85e618827ba2c73dd19e5a6e8a31db5e": "\\alpha_3 \\alpha_2 \\alpha_3 \\alpha_1 = bba + ab + bba + a = bbaabbbaa = bb + aa + bb + baa = \\beta_{3} \\beta_{2} \\beta_{3} \\beta_{1}.", "85e658223c114a27bca16e3075a41556": "\\begin{align}\nx_1&=\\frac{f_1(t)}{g_1(t)}\\\\\n\\vdots\\\\\nx_n&=\\frac{f_n(t)}{g_n(t)},\n\\end{align}\n", "85e6a46c5c31875903da42985969b364": "\\begin{align}\n\\hat{\\boldsymbol\\sigma} &= \\frac{\\tau \\hat{\\mathbf x} - \\sigma \\hat{\\mathbf y}}{\\sqrt{\\tau^2+\\sigma^2}} \\\\\n\\hat{\\boldsymbol\\tau} &= \\frac{\\sigma \\hat{\\mathbf x} + \\tau \\hat{\\mathbf y}}{\\sqrt{\\tau^2+\\sigma^2}} \\\\\n\\hat{\\mathbf z} &= \\hat{\\mathbf z}\n\\end{align}", "85e6c302572475bd9a4c436a823ace86": "\n\\lim_{\\Big((x,y)\\to(0,0)\\,:\\,y=x\\Big)} \\frac{xy}{x^2+y^2} = \\lim_{x\\to0} \\frac{x^2}{x^2+x^2} = \\frac12.\n", "85e6c495be4fd4063c10b1d41d9a25ca": "\\left|\\sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i) - s\\right| < \\varepsilon.", "85e6ec1f9e2053ec41d7cdf96377f1b4": "\\int^T_0 Q(t,s)R_X(s,t)ds + \\frac{N_0}{2} Q(t, \\lambda) = R_X(t, \\lambda), 0 < \\lambda < T, 0 y", "85ef194ee62f0575e1ad7cab7d001a32": "y_{n-1}", "85f007fcd1c4f1e3b94ca7384b94c937": "\nF(\\rho, \\sigma) = \\operatorname{Tr} \\left[\\sqrt{ | \\phi \\rangle \\langle \\phi | \\sigma | \\phi \\rangle \\langle \\phi |} \\right]\n= \\sqrt{\\langle \\phi | \\sigma | \\phi \\rangle} \\operatorname{Tr} \\left[\\sqrt{ | \\phi \\rangle \\langle \\phi |} \\right]\n= \\sqrt{\\langle \\phi | \\sigma | \\phi \\rangle}.\n", "85f06cc816f93b8a6dbb1a494f813358": "near", "85f0931848ec5dc9c4c108158cd8739f": "R=R(\\varphi)=\\sqrt{\\frac{(a^2\\cos\\varphi)^2+(b^2\\sin\\varphi)^2}{(a\\cos\\varphi)^2+(b\\sin\\varphi)^2}}\\,\\!", "85f0d83cf701bece2827923633599fb3": " v(z)= z p(z),\\,\\,\\, \\Re p(z) \\le 0, \\,\\,\\, p^\\prime(0) < 0.", "85f0f341b1ec4bf112a70ed895945eab": "\\,\\!K", "85f16b44f5e5a4acde09c0ecbcb00ebb": " x_1 + x_2 + x_3 + x_4 = 10", "85f17d85569a93e708544ee993de7327": "\n \\frac{\\partial \\Phi}{\\partial t}\n + \\tfrac12\\, \\left| \\mathbf{u} \\right|^2\n + g\\, \\eta\n = 0\n \\qquad \\text{ at } z=\\eta(x,y,t),\n", "85f2024e082ddbc706d8256b4a7fc06e": "(c_{12}-a_{12})+(c_{12}-b_{12})", "85f2a50ed9bb22d0d8d95298b9f61fbc": " 1<|z| \\varepsilon \\right ) \\leq 2\\exp \\left (-\\frac{n\\varepsilon^2}{ 2 (1 + \\frac{\\varepsilon}{3})} \\right).", "860e7cac5d357563b62d4226fa9e9fc2": "E = \\beta_1 - \\beta_2", "860e92c990a39013b43c6848496a6795": "P(x) = D(x)Q(x) + cx + d = ((x - a)^2 - b)Q(x) + cx + d, \\,\\!", "860e936ee3b97dc3f46eeb734ff86019": "\\phi \\in \\operatorname{Gal}(L/\\mathbf Q)", "860f04adc74ae0a07831b9bd2ce40bde": "N_\\epsilon(x)", "860f0ce4df01044c8d18814178b4959c": "a=(x+2)(y+2)", "860f2b9c95ac7253838cb4d1075045e7": "A = Q \\begin{pmatrix}R\\\\O\\end{pmatrix}, \\qquad Q^*Q = I,", "860f3452b6fbc77ee8683a01ded56f11": "\\scriptstyle{{r\\over c}}", "860f4548ae35d0c8fcbe4c8588684974": "\nt^n = (1 - x_0\\cdot t) \\dots \\cdot (1 - x_n\\cdot t) \\cdot\n(p_0[x_0,\\dots,x_n] + p_1[x_0,\\dots,x_n]\\cdot t + p_2[x_0,\\dots,x_n]\\cdot t^2 + \\dots) .\n", "860f4e8acef037293d89280cdc837098": "10\\,\\mathrm{g/cm^3} \\,\\lesssim\\, \\rho_c \\,\\lesssim\\, 10^3\\,\\mathrm{{g}/{cm^{3}}} ", "860f6587711fcbada1e5f654f8435b7d": "\\int_\\Omega \\varphi\\circ g\\, d\\mu \n\\geq \\int_\\Omega (ag + b)\\, d\\mu \n= a\\int_\\Omega g\\, d\\mu + \\int_\\Omega b\\, d\\mu \n= ax_0 +b\\cdot1\n=\\varphi (x_0) \n= \\varphi (\\int_\\Omega g\\, d\\mu),", "860f95235ef58c29440b153370613ce0": " \\scriptstyle{\\Phi(\\frac{b - \\mu}{\\sigma}) =1}", "860fa33c6ac17efcac1055eeb8524637": "S_{i,t}", "860fb22ec2067b121f2faf8f51e82487": "\\delta_{a}^{b}", "86103b8ffd3d7881774974d64698cc66": "\\displaystyle{T(f)=Pm(f)P}", "86105916104fe0904df6191daa84583c": "f_{x}=f'_{x},\\ f_{y}=f'_{y}\\cdot\\sqrt{1-\\frac{v^{2}}{c^{2}}}", "86107739545a1ec69ce1cc92d685ea50": " 1 + (f(x))^2. \\, ", "861084e698e49e30c109baa7ab2bba66": " \\frac {\\mu_0 l}{2\\pi} \\left( \\ln\\left(\\frac {2d}{a}\\right) + \\frac {Y} {2} \\right)", "8610a86cf9e27ef1da44a304abcfa4b7": "a+ar+ar^2+ar^3+ar^4+\\cdots = \\sum_{k=0}^\\infty ar^k = \\frac{a}{1-r} \\Leftrightarrow |r|<1 ", "8610b43dffcdda36ad54e1f1dd3f4371": "c_m=\\frac{1}{2\\pi}\\int_\\Gamma \\ln\\left(\\frac{\\sinh\\gamma}{2\\pi(\\cosh\\gamma-\\cos\\theta)}\\right)\\cos(m \\theta)\\,d\\theta", "8610c62bdf6cb53e1083e731c09750af": "\\cos\\delta'=\\frac{x_1'}{-c\\,t_1'}\n=\\frac{\\gamma \\cdot(x_1-\\beta \\cdot c\\,t_1)}{-\\gamma \\cdot (c\\,t_1-\\beta \\cdot x_1)}\n=\\frac{-c\\,t_1 \\cdot \\gamma \\cdot \\left ( \\frac{x_1}{-c\\,t_1}+\\beta \\right)}{-c\\,t_1 \\cdot \\gamma \\cdot \\left( 1 + \\beta \\cdot \\frac{x_1}{-c\\, t_1} \\right) }\n=\\frac{\\frac{x_1}{-c\\,t_1}+\\beta}{1 + \\beta \\cdot \\frac{x_1}{-c\\, t_1}}\n=\\frac{\\cos \\delta +\\beta}{1 + \\beta \\cdot \\cos \\delta}", "8610e4136d2f102316728f9fc2779088": "a(t,x)", "8610f8a20a7c2cbb929e1354543f3a37": "\na_n \\equiv \\frac{f^{(n)}(0)}{n!}\n", "8611754cc24fd6e2e132795cdce16e4a": " \\delta \\mathbf{r}_j = \\frac{\\partial \\mathbf{r}_j}{\\partial q_1} \\delta{q}_1 + \\ldots + \\frac{\\partial \\mathbf{r}_j}{\\partial q_n} \\delta{q}_n,", "8611b9b95677adc50ec3034e1eec8805": "T_{02} = T_{01} + \\frac{q}{C_p} ", "8611e3b4bb391ab65f58b4a068a7ca7b": " \\cfrac{\\Gamma \\vdash B, \\Delta}{\\Gamma \\vdash A \\or B, \\Delta} \\quad ({\\or}R_2)\n ", "8611ed01316a65edfb9392ce1eb0159b": "\n \\begin{align}\n P_t & = \\left(\\int_{-b/2}^{b/2}q_x(y)\\,\\text{d}y\\right)w_x -\n \\left(\\int_{-b/2}^{b/2}m_x(y)\\,\\text{d}y\\right)\\cfrac{d w_x}{d x} +\n \\left[\\int_{-b/2}^{b/2}\\left(y q_x(y) + m_{xy}(y)\\right)\\,\\text{d}y\\right]\\theta_x \\\\\n & \\qquad \\qquad -\\left(\\int_{-b/2}^{b/2}y m_x(y)\\,\\text{d}y\\right)\\cfrac{d \\theta_x}{d x} \\,.\n \\end{align}\n", "8611fddf6402334121ae20043f646a83": "A=\\{n^2; n\\in\\mathbb{N}\\}", "8612c4f345315d593ceb6e341ec39ad7": " f(\\alpha \\mathbf{v}) = \\alpha^k f(\\mathbf{v}) ", "8613537f903b2fc8caeb94347c02f866": "a\\equiv b \\pmod{ n} \\implies z^a = z^b.", "86136ea50698b93560334445eb764cd5": "\\begin{matrix} {11 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "8613985ec49eb8f757ae6439e879bb2a": "90", "86139fbc22c684849757209d2cd408a4": "\\lambda=F/EI", "8613ce7d35a968204ab0de93b15b5fb7": "\\scriptstyle{I\\geq 1}", "861413b04410e02dac2084bc2cf69375": "\\bar B(\\mathbf x_1,\\|\\mathbf h_0\\|)", "86145d8210be613fa61cdd01f2f423a1": "\\textstyle\\pi = 176 \\arctan\\frac{1}{57} + 28 \\arctan\\frac{1}{239} - 48 \\arctan\\frac{1}{682} + 96 \\arctan\\frac{1}{12943}", "8614656e4f19b55592aee61f0643b3d4": "\\Gamma(\\theta)= \\frac{sin \\theta - X}{sin \\theta + X }", "8614c59b1ebe7e290ec89b0bcc854c10": "m_{\\text{fil}}", "8614ca71fd8e5efb250b84ea7f45e022": "\n\\mbox{div} X = \\frac{1}{\\sqrt{|g|}} \\partial_i \\left(\\sqrt {|g|} X^i\\right)\n", "8614d4b1194eb1ba49e7df2a1ee30e75": "x_1x_2 + y_1y_2(n^2-a) = 1", "8614dc8013253c8494b61f979a42aa05": "\\scriptstyle f:\\mathbb{R}^p\\rightarrow\\mathbb{R}^s", "8615111a9f6a75a6ff8fd337e9986199": "F \\subset P", "86153b59a32d5cabf7564d7230136333": "Y = X(t)", "861553616465c5438b7d7377b0c95926": "|z_n| \\ge 1", "86157d732d86fa20aafcc6243b14314a": "F(n + 1) = f(F(n))", "8615a7162f8528ed7559fb8d34d15670": "\\prod _x a^{\\frac{1}{x}} = C a^{\\frac{\\Gamma'(x)}{\\Gamma(x)}} \\,", "8615c16df00cc859d730e28b963f71ae": "\\omega_{jk} e_l = e_l \\omega_{jk} \\,", "8615d357d957a94455eba454f5448fb5": "Q(X,Y)", "86166b174fa3a598d39e201e09d4dc02": "x^a \\mapsto \\frac{v^a(x)}{\\sqrt{\\sum_b(v^b(x))^2}}", "86166f7a4ea6aeb31a3b7685b0b943f7": "\n{\\sigma_r^2\\over\\sigma_t^2} = 1 + {r^2\\over r_a^2}.\n", "8616e4c1e7157bdc0f1d59a6dfd7a8ab": "\\Pi_{\\kappa}", "8616ef9dc8c9872b110e6799521c4208": "a_{\\mathrm t1}", "861713992589271808c9b47156215ccc": " (x,y,z)\\mapsto\n\\begin{cases}\n(x+2z, z, y-x-z),\\quad \\textrm{if}\\,\\,\\, x < y-z \\\\\n(2y-x, y, x-y+z),\\quad \\textrm{if}\\,\\,\\, y-z < x < 2y\\\\\n(x-2y, x-y+z, y),\\quad \\textrm{if}\\,\\,\\, x > 2y\n\\end{cases}\n", "861714de260c8ce312cca3a4fc843b06": "M\\otimes_R-:R\\mbox{--}\\mathrm{Mod}\\rightarrow \\mathrm{Ab}", "86176ffef4b71bf1b1ff81d066535ba3": "\\beta_k (E_{ij}) = E_{ij} \\otimes I_2", "8617c005357629f2c808cc236c592bc6": "(+)(x^3)(\\sin x) - (3x^2)(-\\cos x) + (6x)(-\\sin x) - (6)(\\cos x) + C \\,. ", "8617e170be64de9e8f751c89c9e6fc67": "Z = \\sqrt {j \\omega \\mu \\over \\sigma + j \\omega \\varepsilon} ", "8617e4d715fb890690806bca047af3f9": "dU = nC_vdT\\,\\!", "86180356a48eba8614e117aaf730430c": "\\hbar=1", "861818012ebd4029588d9d6b143466dc": "SR = D / (D+B)", "861844a92fbe2eab86037cf3fe758cfa": "\\ X = |Z| \\sin{\\theta} \\quad", "861860fe54d0d11fdc3ee9b9818386c3": "B \\in S", "8618c23398e57c490bc3a2cab3737c89": "A_2=B_2", "8618d9672e59dc8f88c89e1423b31dee": "\\sqrt{\\tfrac{11(t-4)}{2(20t-37)}} \\scriptstyle{\\approx 7.4474}", "8618df8834260ddd81f2ee89ce164c53": "F_{ab} = a\\,\\mathrm{inf}_a \\,u_b - a\\,\\mathrm{inf}_b \\,u_a \\,", "861934479d3ecbb5f8b7519ea60d7d8a": "\\int_{-1}^{+1} \\sqrt{1 - x^2} g(x)\\,dx.", "86194e949d6af153146c2ab3856f86ff": "\\|Tf\\|_{q,w}\\le C\\|f\\|_p", "86197583975211aae1d1778819b16805": "\\{\\mathrm{onions, potatoes}\\} \\Rightarrow \\{\\mathrm{burger}\\}", "8619bccb090916abedcfcfaf8ce84e02": "\\|0\\| = 0.", "861a2598209857741f02abe422cf8d64": "D=D_1+D_\\infty", "861a4bab07b54085009b4a589e2622ad": "f_i(\\vec{x},t)", "861a5d7fdd7cc07aa1d294951676162b": " \\tilde{\\chi}(t)=\\tilde{\\chi}(0) -i\\int^t_0 dt' [\\tilde{H}_{BS}(t'),\\tilde{\\chi}(t')] ", "861ac4ad4dc954a1a2b230b1f826734f": "1+1 = 2", "861ae9f21d5744fa568db66e2a500dac": "P[f] \\propto \\exp \\left( { - \\beta \\left\\| {\\hat Pf} \\right\\|^2 } \\right)", "861afa62680b435210b9b6ae0455a081": " Xgroup + Y \\xrightarrow[transferase]{} X + Ygroup ", "861b0c598d5be2c12b25a5daaa6335e5": "\\varphi_\\gamma(\\beta)", "861b0fde2616aebcf2a3502ca8522c95": "\\mathrm{d}S_r=r^2\\sin\\theta\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\varphi.", "861b53f07a7bf3864d46e304fb2d96ad": "\\prod_\\varnothing{} = \\{ f_\\varnothing: \\varnothing \\to \\varnothing \\} = \\{ \\varnothing\\}.", "861b5561ebe69c485fbe3047b993e908": "\\sigma_{xz}\\sigma_{xy} - \\sigma_{yz}\\sigma_{xy} - \\sigma_{yz}\\sigma_{xz}", "861b856d1ca396d8c647eaf5a008fe67": " t= \\frac{T-T_{m}}{T_{m}} ", "861ba3090130ff00cccf6b4fbc931673": " \\!\\ R - \\lfloor R \\rfloor = c/R ", "861bc970d13a35e363d1653262e5be62": "t_D = L^2/D", "861bd0795e80b572ed5cbf0fc5db99c1": "|\\textbf{k}_0|", "861c3348889b8a79b9e57ca7bc7e0502": "\\mathbf{U}(t)", "861c45068151296c6b5eabe61d2ebf89": "\\frac{1}{k} \\sum_{d | k} \\mu(k/d) \\gcd(N^d - 1, MN - 1) ,", "861c48d2c94252de6063282dd457b8f5": "a(Z)=\\frac14(1+Z)^2\\,((1+Z)+c(1-Z))", "861c76157873437d9fff4f2a6f0952ca": "O[n + A(S)]", "861c946eec5b32e81d77ceffc0560f07": "\\mathbf{v}_p = \\hat{\\mathbf{k}} \\frac{\\omega}{|\\mathbf{k}|}, \\quad \\mathbf{v}_g = \\vec{\\nabla}_{\\mathbf{k}} \\, \\omega \\,", "861ca6cc79d75890b7a7df5506d80dd9": "y \\in W.", "861d3a40b663987e7ff66bf5b4e01bea": "(b_n + a_n i)", "861d4eb53d26dc8b0d1d68b2f649f855": "\\forall x \\Big( g(x) \\vee \\exists y R(x,y) \\Big) \\iff \\forall x \\Big( g(x) \\vee R(x,f(x)) \\Big)", "861d9c8092af1c462e0adae5221a187f": "\\beta \\in (V\\cup\\Sigma)^{*}", "861df74596abb976c25bcec0d09e08c9": "\\mathcal A", "861e29bff8f5100c17b05289a7853fff": "\\mathbf S1:\n\\begin{cases}\nx^2+y^2=1\\\\\nkx^2+z^2=1\n\\end{cases}", "861e478eaa0325e1114601f8a39e5090": " \\psi(x, t+dt) \\approx e^{idt\\hat D}e^{idt\\hat N}\\psi(x, t)", "861e512e99d0071d54e48e8ac0d3d8f1": "k_B T B", "861ec38344f65d866980b33b52611a80": "N_P", "861f26a5d91e17258eef562c116dfaa2": "q_2=\\frac{5000-q_1-c_2}{2}", "861f4537852b74257dc14a479168133a": "\\sqrt{163}", "861f506d12f0a12cfcc0b514798dddfb": "2 = 1/(1-1/2)", "861f5f1a257257fe2a392983a72ee78c": "\\displaystyle E_T=E_A+E_H ", "861f97dba464a7e5e07bbe73f6b428e8": "\\theta(n)", "861fadc85d84ba3da62cd25047f3ef61": "t = 10^{-33}", "861ffd911d49b21dd119b7db3b274023": "\\lbrack\\mathbf z\\rbrack_1 = \\begin{bmatrix} R_1 + R_2 & R_2 \\\\ R_2 & R_2 \\end{bmatrix}", "86201ed88e14902b216037f7e1176ebb": "\\hat{p} = -i\\hbar\\frac{\\partial }{\\partial x}", "8620273e55c302c19d7adf62bc2171f9": "\\bar{T} (\\bar{3},1)_{\\frac{1}{3}}", "8620437b918ba6ada551f47fae96dfd1": "\\int_S 1 \\,dS. ", "8620a62f2f59c8ca57948284f3215eb3": "\\begin{align}\n&m_{AB} \\cdot m_{BC}\\\\\n&=\\frac{\\sin \\theta}{\\cos \\theta + 1} \\cdot \\frac{\\sin \\theta}{\\cos \\theta - 1}\\\\\n&=\\frac{\\sin ^2 \\theta}{\\cos ^2 \\theta -1}\\\\\n&=\\frac{\\sin ^2 \\theta}{-\\sin ^2 \\theta}\\\\\n&=-1\n\\end{align}", "8620be6fa3963dca9bc1eae3006ae27a": "u+H(x,\\nabla u) = 0", "8620bee90b8c4bd1ede09afef8255373": " A_0 \\,", "86219db36e515ede7d2963f249994cc7": "L_1 = p_1/3", "8621b75809fd3502fb6b202e5ec245b4": "\\dot{x} = -x \\qquad \\text{(i.e., } \\sigma(x,\\dot{x}) = x + \\dot{x} = 0 \\text{)}", "8621bf65c21551dd27219cba5a9c8d49": "\n\\begin{array}{rcll}\n\\min &~& f(\\mathbf{x}) & \\\\\n\\mathrm{subject~to} &~& g_i(\\mathbf{x}) = c_i &\\text{for } i=1,\\ldots,n \\quad \\text{Equality constraints} \\\\\n &~& h_j(\\mathbf{x}) \\geqq d_j &\\text{for } j=1,\\ldots,m \\quad \\text{Inequality constraints} \n\\end{array}\n", "8622181f2de8614666fd24d86f7ce250": "m(x)\\leq \\liminf_{y\\in S_x} \\frac{f(y)}{g(y)} \\leq \\limsup_{y \\in S_x} \\frac{f(y)}{g(y)}\\leq M(x)", "86221e6b7ddea89a4e056c7d662aed60": "fs^2 \\approx 447", "8622268a8ee64566220b57e901418448": "\\mathrm{E}\\left[\\frac{a(n,k,X)}{j(n,X)}\\right]=a(n, k,X(\\Omega))", "86223fabb6b5473e51f0465ecab2aff4": " y_{ni} = \\begin{cases} \n1, & if \\, U_{ni} > U_{nj} , \\quad j \\not= i,\\\\\n0, & otherwise\\end{cases}", "862250337b431e257369ba6344685aad": "E_K = \\frac{p^2}{2m}", "86225672e13c229816de4ddbd8f7968f": "V_0\\subset V_1", "86226bfeb36bd05327f03fa068244de9": "u+u'+p q'-\\tfrac{1}{2}(p+p')(q+q')", "8622a0ed4fe3da3def6743b1dc317012": "V_{\\mathrm{o}} = A_\\mathrm{d} (V_+ - V_-) + \\tfrac{1}{2} A_\\mathrm{cm} (V_+ + V_-),", "8622b3282a43eb32cbc025740156f5de": "f(\\mathbf{a} + \\mathbf{v}) - f(\\mathbf{a}) \\approx f'(\\mathbf{a})\\mathbf{v}.", "8623275ec9eb4311474d10ec0b7f4160": "\\hat{X(t|T)} = \\int^T_0 Q(t,s)Y(s)ds", "86236aceab233b3610464302734be39e": "\\scriptstyle<5\\times10^{-30}", "8623bdb04d482f96842e720b37b13210": "-1.8107", "8623c71c6b122b553fafa87c0101f4a0": "Q^\\dagger", "8623fbe27a0ab4e803ba151062625588": "(G,q,g_1,g_2,c,d,h)", "862420f683e44d5fd7ad8546c610e79e": " \\rho = \\frac{ N }{ \\pi R^2 } ", "86242bdfd0ce4e13f00f9808a617f90a": "n_1^2, n_2^2, \\dots, n_j^2, n_1^3, n_2^3, \\dots", "8624807462a074297588a23fa7b20035": "ai + b", "862489a56dbf818d7e05516ececae1d8": "L = \\{ l_e(f_e) = a \\cdot f_e + b \\; | \\; e \\in E, \\; a \\geq 0, \\; b \\geq 0\\}", "86249c42968466273e6ceb3de5798a7d": "\n\\begin{array}{lcl}\nK,N &=& \\text{as above} \\\\\n\\phi_{i=1 \\dots K}, \\boldsymbol\\phi &=& \\text{as above} \\\\\nz_{i=1 \\dots N}, x_{i=1 \\dots N} &=& \\text{as above} \\\\\n\\mu_{i=1 \\dots K} &=& \\text{mean of component } i \\\\\n\\sigma^2_{i=1 \\dots K} &=& \\text{variance of component } i \\\\\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& \\mathcal{N}(\\mu_{z_i}, \\sigma^2_{z_i})\n\\end{array}\n", "86249e228d94bd3dcdc58355df8d30fc": "I-1", "8624a9c11b023b2723940e76bae6ea8a": "\\Phi \\in \\mathcal{A \\otimes A \\otimes A}", "8624c3694bf8d786aba4c979661866b2": "w'", "862529c50f6289c84efb90d314682ded": "y_{2c} = \\frac{2}{3} \\times E_2c = \\frac{2}{3} \\times 6.00 = 4.00 \\text{ ft}", "86255a97413ee33a074089522d4a7a11": "0\\le d_i\\le |B_i|", "8625b4c272eb3b053ef6fb9668023a95": "\\theta_X=dx^\\mu\\otimes\\partial_\\mu", "8625fa9658e767c9bb9294c77a2f39a5": "\\zeta = {R \\over 2} \\sqrt{C\\over L}", "86261d40dc2b8831b2a402ba0dd98b5c": "\\tilde{\\mathbf{q}}^{i}=(X^{\\dagger})^{i}_{j}\\mathbf{a}^{j}", "86263dddb1cdf828ffc3ab41e5baddef": "\\mathbf{\\sigma}", "862694d26282a2c9a75fe872bb4c2446": "= \\sgn( \\sin (\\theta+ \\frac{\\pi}{2})) \\sqrt{1 - \\sin^2\\theta}", "8626af7dc1d191c67ed8cf654863541c": "\\pi(\\theta)", "8626b0c14a48867bf18389d77c900294": "IF(x; T; F):=\\lim_{t\\rightarrow 0^+}\\frac{T(t\\Delta_x+(1-t)F) - T(F)}{t}.", "8626b8e0ed86d7b22d5e1fc4f025b666": " 1/(U \\cdot A) = 1/(h_1 \\cdot A_1) + dx_w /(k \\cdot A) + 1/(h_2 \\cdot A_2) ", "8626d5353c6eefb801a0d310e508c502": "f_T(x)", "862728ec009f8126282b35d04e398429": "\\langle W,R,\\Vdash\\rangle", "86272f78c16e46781ab4e35f7fb32caa": "d \\ne 1, N", "8627700e3bb46e50b824c56c53dd5466": "\\prod_{j=1}^{n-1}(N-R-j)\\approx (N-R)^{n-1}", "8627d2988e3f441eec236fe5ce062e82": "\\text{extend}: (\\mathrm{W} \\, A \\rarr B) \\rarr \\mathrm{W} \\, A \\rarr \\mathrm{W} \\, B = f \\mapsto (\\text{fmap} \\, f) \\circ \\text{duplicate}", "8627d7bd5cdf9e4ae68acf06bb0e6498": "t=t_0, \\, r=r_0", "86280ae451627e529c7a17538b62f0a8": "\\lim_{p\\rightarrow\\infty}\\|f\\|_p=\\|f\\|_\\infty,", "862893ea5c68d89eab788815eb93c86b": "\\{\\{x^3+2x^2-x-1,\\frac{x+3}{x+2}\\},\\{x^3+6x^2+5x+1,x+2\\}\\}", "86289cad71eb66db1f483da31067e840": "\\frac{2^{nH(q)}}{n+1} \\leq \\tbinom nk \\leq 2^{nH(q)},", "8628a02f0678750bb764e8b7f247f0ba": " \\hat h_{V,\\phi,L}(P) = 0 ~~ \\Longleftrightarrow ~~ P~{\\rm is~preperiodic~for~}\\phi.", "8628a067f04e03a8d50891099ad64dab": " Z\\gamma ", "86298243f6959f7d3720e6fef10f650f": "\\wp_\\tau'(z)", "862999fa80921d352d285c919476f1c5": " J^\\mu ", "8629c69eb3c9bad6342adb1996e832a0": "I_n=\\int \\frac{(px+q)^n}{\\sqrt{ax+b}} dx\\,\\!", "8629d10f84c71a81f7fc7d98c8976790": "\n\\operatorname{Li}_2 (z) = -\\int_0^z{\\log (1-t) \\over t} \\,\\mathrm{d}t.\n", "862a7ad6f6fcf76f5af65000bb21310e": "R^{\\frac{3-n}{n}} M^{\\frac{n-1}{n}} = \\frac{K}{GN_n}", "862a9479b98ef49fb1efbc1bda833564": " \\pi_1 = \\frac{\\mu(c_p)}{k} = Pr", "862ab9aa54b66f5f380423c8b2286635": "student = bad \\lor medium", "862b08adb4dfb4ce7d3a4691919cd346": "a,b \\in X", "862b75a2bdef64740d3aaa94e7335af2": "\nc \\,F_1(a,b_1,b_2,c; x,y) + c(x-1) F_1(a,b_1+1,b_2,c; x,y) - (c-a)x \\,F_1(a,b_1+1,b_2,c+1; x,y) = 0 ~,\n", "862c5834426249416575a50d2953096b": "\\mathbf{C}=\\mathbf{X}^{-1}", "862c688f0795aae9e60ae0ebc538f69d": " (\\{y \\in C | x \\preceq y \\preceq z\\}) < \\infty ", "862c6dfb65d04b44626214bcee175412": "g(T) = T", "862cab15b24279cb443431c3944fadff": " r\\theta={\\frac {r\\sin \\theta }{\\cos \\theta\n }}-(1/3)\\,r\\,{\\frac { \\left(\\sin \\theta \\right) ^\n{3}}{ \\left(\\cos \\theta \\right) ^{3}}}+(1/5)\\,r\\,{\\frac {\n \\left(\\sin \\theta \\right) ^{5}}{ \\left(\\cos\n\\theta \\right) ^{5}}}-(1/7)\\,r\\,{\\frac { \\left(\\sin \\theta\n \\right) ^{7}}{ \\left(\\cos \\theta \\right) ^{\n7}}} + \\cdots", "862d5168765656f718afeac6f2297db1": "\nP_{i} :=\nE\n\\left[\n\t\\left(\n\t\t\\textbf{x}_{t-i} - \\hat{\\textbf{x}}_{t-i\\mid t}\n\t\\right)^{*}\n\t\\left(\n\t\t\\textbf{x}_{t-i} - \\hat{\\textbf{x}}_{t-i\\mid t}\n\t\\right)\n\t\\mid \n\tz_{1} \\ldots z_{t}\n\\right],\n", "862db590fec853832d7305a215fde235": " f(x_1+x_2)=f(x_1)+f(x_2) ", "862dbe28c636ec76549776d0e5597dd7": "\\varphi_1 > 0", "862ddd79ad06fe6c1b2b44b23a46f71c": "\\arctan\\frac{b}{a}", "862e0dfb75c7700a2f0a1e3f527a7c54": "|\\psi_2 \\rangle", "862e839854c21dccbaa8d6970c637585": "J(x)\\in X''", "862e99b83c09f1fcbe587c21bed20bf1": "\\Delta\\,\\!", "862ef4c18cc412276f264c9759dbb194": "\\mathbf{r} = \\mathbf{x}+\\mathbf{e}_5 = \\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 1 \\\\ 1 \\\\ 1 \\end{pmatrix}", "862f2f9a609da35e33dfcb35521451d0": "P_x(\\omega) = \\frac{1}{2} \n\\left(\n \\frac { \\left[ \\sum_j X_j \\cos \\omega ( t_j - \\tau ) \\right] ^ 2}\n { \\sum_j \\cos^2 \\omega ( t_j - \\tau ) }\n+\n \\frac {\\left[ \\sum_j X_j \\sin \\omega ( t_j - \\tau ) \\right] ^ 2}\n { \\sum_j \\sin^2 \\omega ( t_j - \\tau ) }\n\\right) ", "862f9447a863bc042efe56d877ec52f7": "C = f(T)", "8630089e1b8ff849775e9c1bda148e58": "\\frac{\\delta F^j_k}{\\delta t} =\\frac{\\partial F^j_k}{\\partial t}+CN^i \\nabla _i F^j_k", "8630342c4adb15cad07c54f5baa66eb5": "\\sigma_{A}", "8630cdf4c84dd629d5afd3427153e56f": "((\\mathbb{Z}/9\\mathbb{Z})^{\\times}, \\{1, \\circ\\})", "8630cedbec314ca7060f0da7e3bbffef": "y= \\frac{y'}{x'^3} ", "8630e2bb5350601a0ab96567fb9a6442": "(f+g)(x) = f(x) + g(x), \\forall x\\in E", "863124b0a8bff7f14ba7e46ab1c4ba8a": "\n\\alpha^3 = \\alpha^2 + 1", "86313bb22a897f054e379b31655d4864": "x\\le y\\land y\\nleq x", "8631640acd193ef75de0fbfb3f0a9402": "C_x(\\tau) \\,\\! \\mbox{ where } \\tau = t_1 - t_2.", "8631709e0ce501057018ab9af5018fcd": "\n\\boldsymbol{x}\n\\sim\nN_d\n\\left(\n \\boldsymbol{\\mu} \\left( \\boldsymbol{\\theta} \\right)\n ,\n {\\boldsymbol C} \\left( \\boldsymbol{\\theta} \\right)\n\\right)\n", "86318a40523dd1f3235d8d2a2a6b8bbd": " J_D = \\frac{I_D}{ S}= -\\frac{I}{ S}= \\varepsilon_0 \\frac {\\partial E}{\\partial t} = \\frac {\\partial D}{\\partial t} \\ , ", "8631d190fd7413031a3885f2363373d7": "\nK \\approx \\int_{0}^{1} \\frac{dy}{\\sqrt{1 - y^{2} }} \\left( 1 + \\frac{1}{2} k^{2} y^{2} \\right) = \\frac{\\pi}{2} \\left( 1 + \\frac{k^{2}}{4} \\right)\n", "86321f9de05c223edfcee2132e8b13c0": "\\{A,B,C,a,b,c,P,Q,R,O\\}", "86323023c352683929e5d6de0a546aad": "\\displaystyle \\frac{\\partial^2 u}{\\partial x^2}-\\alpha^2 \\frac{\\partial^2 u}{\\partial y^2}=-2\\alpha^2 \\mathbf{S}\\cdot\\left(\\frac{\\partial \\mathbf{S}}{\\partial x}\\wedge \\frac{\\partial \\mathbf{S}}{\\partial y}\\right)", "863241d161c675b325112b9365dd2ccb": "(\\Delta t)^{1/2}", "863256ea3a6f9e63574bf975951699e7": " \\scriptstyle \\beta, ", "8632e5ff9b5cde7fb42aa9891d6a5bfc": "P_m = \\frac{1}{2\\mu_0}B^2\\,", "8632fee98a46257cbb9a9329940c87ca": "\\int x^m\\arcsin(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arcsin(a\\,x)}{m+1}\\,-\\,\n \\frac{a}{m+1}\\int \\frac{x^{m+1}}{\\sqrt{1-a^2\\,x^2}}\\,dx\\quad(m\\ne-1)", "863368759bf3f752e2fc5da9d6338da4": "\n\\int_0^\\infty J_\\nu(kr)J_\\nu(k'r)r\\operatorname{d}\\!r = \\frac{\\delta (k-k')}{k}\n", "86339244834e580d4a531cd42790bd46": "\n\\begin{align}\nVIS & == [IS | P]\\\\\nTCL_{T_{1}}^{1} & == [VIS | P_{T_{1}}^{1}]\\\\\n&\\dots\\\\\nTCL_{T_{1}}^{n} & == [VIS | P_{T_{1}}^{n}]\\\\\nTCL_{T_{2}}^{1} & == [TCL_{T_{1}}^{i} | P_{T_{2}}^{1}]\\\\\n&\\dots\\\\\nTCL_{T_{2}}^{m} & == [TCL_{T_{1}}^{i} | P_{T_{2}}^{m}]\\\\\n&\\dots\\\\\nTCL_{T_{3}}^{1} & == [TCL_{T_{2}}^{j} | P_{T_{3}}^{1}]\\\\\n&\\dots\\\\\nTCL_{T_{3}}^{k} & == [TCL_{T_{2}}^{j} | P_{T_{3}}^{k}]\\\\\n&\\dots\\\\\n&\\dots\\\\\n&\\dots\n\\end{align}\n", "8633be0b31cecccc2e0c2ece0cab2d29": "J'' = J'-1 ", "86346811838b4d6d4c40af75d8330f3b": "G\\not\\rightarrow H", "86347b5a8e309da7165eddde10673de0": " \\nabla \\cdot \\mathbf{B} = 0 \\,", "8634ff038c046583e8254a4204b5c05f": "{\\Delta \\left( {{{\\partial v} \\over {\\partial T}}} \\right)_P = - \\Delta \\left( {\\left( {{{\\partial v} \\over {\\partial P}}} \\right)_T } \\right) \\cdot {{dP} \\over {dT}}}", "86353736bb1de3d0354a74100b402590": "{}E[X_t|\\{X_{\\tau} : \\tau \\le s\\}] \\ge X_s \\quad \\forall s \\le t.", "86355e1622461adac0031526b4c5dd5a": "w\\ ", "86356ff3e752ade3a0c85c65f63ff3b8": "\\frac{ \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n}{ \\sqrt{2\\pi \\left(n-k\\right)}\\left(\\frac{n-k}{e}\\right)^{n-k}k!} p^k (1-p)^{n-k}\\rightarrow \\frac{ \\sqrt{n}n^np^k (1-p)^{n-k}}{ \\sqrt{n-k}\\left(n-k\\right)^{n-k}e^kk!}\\rightarrow \\frac{n^np^k (1-p)^{n-k}}{\\left(n-k\\right)^{n-k}e^kk!}.", "86357ba0a9417fc8a440dd06877dd484": "\\sum_{j=0}^\\infty \\hat{\\psi}(2^j\\gamma)\\overline{\\hat{\\psi}(2^j(\\gamma+q))}=0", "86358b5f60c7c82d53fe415f0ef574a7": "\n\\begin{align}\n E&=E0+x(\\lambda,\\phi),\\\\[1ex]\n N&=N0+y(\\lambda,\\phi)-k_0 m(\\phi_0).\n\\end{align}\n", "8635982e2f226ad35a8c34ef7c686585": " \\frac{VK}{Y} . k ", "8635a7ae9bf7419b059bd831c412392b": "\\lambda = 3,\\, \\beta = 2.5", "863613c0dc1dc46d183b2cf1e174358e": "Z^m_n", "8636be7822566be261d1a8a4dbd783a9": "p(x) \\propto \\exp\\left( -\\!\\int\\!\\!\\frac{x-a}{b_2 x^2 + b_1 x + b_0} \\,\\mathrm{d}x \\right).", "8636cf49763aeb941a7818a734135716": " \\mu(E) = \\sup \\{\\mu(K): K \\subseteq E, K \\text{ compact}\\}.", "8636deb25fd64de9583774c84091d299": "\\sqrt2 \\,", "8636f27984e591ad9d6f58c00c103b2e": "h^{(i)}(x)=f^{(i)}(x), \\qquad i=0,1,2.", "863738c0ec5c6eb0fc23a31d46dc4894": "N \\in \\left( \\left[ 0, l - 1 \\right] - M \\right)", "86374cdc43b1dc18cfffa5d875022b23": "\\mathcal{Q} = \\{ \\mathit{A}, \\mathit{T}, \\mathit{C}, \\mathit{G} \\}", "863776cf00761299d7cb7a4cc4d7e715": "D^{\\prime}", "8637978935911550aad7420328272d85": "\\frac {p}{2}", "863798bd1208b3edda37b5fb73b6d0b3": "\n\\begin{pmatrix}\na & c \\\\\nb & d \\\\\n\\end{pmatrix}\\begin{pmatrix}\n\\psi_1 \\\\\n\\psi_2 \\\\\n\\end{pmatrix}=\\begin{pmatrix}\n\\chi_1 \\\\\n\\chi_2 \\\\\n\\end{pmatrix}\n", "8637eb86a3522896934392ffae75a20b": "\\alpha\\in {\\Bbb C}\\backslash {\\Bbb R}", "8637fd652dbeeffca21afe99bc0047eb": "P=(x,y)", "86380e85e9a8bf04e2cd94c9fc0aa10b": "A \\rightarrow A", "86381f55dfe8fc80da7896a922e47d79": "I_{x}", "863826aeaf528c83d933e39eae78c014": "like\\langle Mary, Sue\\rangle", "863843fa7bb6b1636767959daaabf58e": " \\int_0^2 \\! \\int_{0}^{\\pi/2} \\! \\int_0^2 \\! \\bar{f}(r,t,h) \\, dh \\, dt \\, dr = 12 + 10 \\pi", "8638492bac5f83b8eb96c3ce33e63568": " R=\\sqrt{\\frac{l^2+m^2+n^2}{8}} ", "8638a2f02a0f01696d284346e169e792": "\n \\delta_t = \\frac{1}{2} \\Rightarrow \\operatorname{E}[A_t - B_t] = \\frac{\\alpha_t \\mu}{\\alpha_t \\mu + 2\\epsilon}\\left(S_G-S_B\\right) \\;.\n ", "863911f8b9ce9fc690f44315c7a18c41": "(a+b\\sqrt c)+(j+k\\sqrt c)\n=(a+j)+(b+k)\\sqrt c", "86394216d89f67d96fa1f3c1d80bde42": "y/a = \\sin(t)\\,", "86397949359662783836aeaa9345f3cd": "B \\to C(f), C(f) \\to A[1]", "8639a53ab7b579d4b24372b1972d908a": "\\sum_{n = 1}^m (a_n + |a_n|)", "8639ad9def457d50a53dda1786424535": "\\sum_{n=0}^{\\infty}M_n", "8639c16149f016f0fb660c17407de704": "\\delta \\mathbf{r}", "863a1a313a683885261a0fa85ddc07aa": " p \\approx 2^{-20} ", "863a58f603a80e07131feec17be73cac": "M \\le n < M + R.", "863ac58e068d4077f1bf6215cc4e58c7": "(1-p) \\,", "863ad2152876a07e8f1ca42367dea28b": "\\mathbf{x}^\\prime = [x[0], x[1], \\ldots x[N - 1]]^T", "863b09658b1bc6efb9a7d26625367cbf": "\n\\hbox{Monthly inflation } = 100 \\times \\left(\\left(1+\\frac{\\hbox{inflation}}{100}\\right)^{\\frac{1}{12}} -1\\right)\n", "863b15c99c3db71b69906b6510e620cb": " B=C^\\mathrm{T}AC. ", "863b49f3364e1635c586b4b957b228af": "\\Gamma = \\frac{2 \\pi\\, \\Delta n\\, L}{\\lambda_0},", "863bd12573fa23779b6e8a478221f8af": "\\mathbf{e}_r = \\begin{pmatrix}\n \\cos\\theta \\cos\\zeta \\\\\n \\cos\\theta \\sin\\zeta \\\\\n \\sin\\theta\n\\end{pmatrix} \\quad\n\\mathbf{e}_\\theta = \\begin{pmatrix}\n -\\sin\\theta \\cos\\zeta \\\\\n -\\sin\\theta \\sin\\zeta \\\\\n \\cos\\theta\n\\end{pmatrix} \\quad\n\\mathbf{e}_\\zeta = \\begin{pmatrix}\n -\\sin\\zeta \\\\\n \\cos\\zeta \\\\\n 0\n\\end{pmatrix}", "863be301d44035aeaa060bd757b0d674": "f^\\prime(x) = p", "863c52514755891569fe7a2b4e222840": "S_n(s)", "863c6aecb564978f0adc5e5cc766e18a": " j_1 ", "863ca86ea0dab5970aa2dcbf6ff608c4": " P_{nj} = \\Pr( U_{nj} > U_{ni} ) ", "863ce1b832f752206df09e01a5a91617": "_{q.(p+p')\\,}\\!", "863d7f230d8a79d783140fa74d800276": "\\chi_{ZXX}=\\frac {1}{2}N_s[\\langle\\cos \\theta \\sin^2 \\theta\\rangle\\beta_{Z'Z'X'}+\\langle\\cos \\theta\\rangle\\beta_{Z'X'X'} - \\langle\\cos \\theta \\sin^2 \\theta \\sin^2 \\Psi\\rangle(\\beta_{Z'X'X'} + \\beta_{X'X'Z'})]", "863dc0f02e805453f3d05ae4510dd96d": "\\sum _x \\Gamma(x)=(-1)^{x+1}\\Gamma(x)\\frac{\\Gamma(1-x,-1)}e+C", "863e48a750e924a107438d43abb46f97": " 1x_n \\geq x^* \\Rightarrow x_n \\succ x_m", "865e3973583acb7a5009e93e2135f185": "\\scriptstyle K>0,", "865e9435edd3cd575422a846c4cfcf98": "a_1 + a_2 + a_3 + \\cdots + a_n = \\sum_{i=1}^n a_i", "865ee1c0a17e027ff2a5e859a59d0d33": "\\beta=0,", "865eeafdd2cd5e049a5dacd278f65d3b": "j \\in m", "865ef8f7e85adaf929d52f220e4e0ee7": "\\mathcal{L}_\\mathrm{Dirac}(\\psi) = \n\\bar{\\psi}(i\\partial\\!\\!\\!/-m)\\psi ", "865f63631b684c7ab8881b63f5af1371": " P = K \\rho_c^{1+\\frac{1}{n}} \\theta_n^{n+1}", "8660a7491251bc2a25a511fddf5e8017": "\\mu=w'\\cdot\\tilde\\lambda", "8660ac55ba6d51649204454103422052": "C_{P} - C_{V} = VT\\alpha\\left(\\frac{\\partial P}{\\partial T}\\right)_{V}\\,", "86617769da51abb21a087d2de0efd71f": "\n z^2\\, \\frac{\\partial^2 \\theta}{\\partial z^2} \n + z\\, \\frac{\\partial \\theta}{\\partial z}\n - z^2\\, \\theta \n =0.\n", "8661b89ac820f421a91aaf586bfd6aa8": "\\Omega^p(M,E) = \\Gamma(E\\otimes\\Lambda^pT^*M).", "8661e0b5ff1f1fd790ab9cbd07982ee1": " u_w = \\phi(x,y,u,A,w(A)) \\,", "8661e692232628e8cb6ab9b616db180d": "r_i \\equiv U(0,1)", "86625a55fe27937098ef4b3f42e1cbdf": " \\rho_\\perp ", "866298b74c59e6152235fb79b16676ac": "\\left(i\\omega V_c + \\frac{1}{RC}V_c \\right) \\cdot e^{i\\omega t} = \\left( \\frac{1}{RC}V_s\\right) \\cdot e^{i\\omega t}", "86630556c1e37004e2adef8bfffaf5dd": "\\frac1{\\det(A)}M=A^{-1},", "8663138c6b85761dff4af3e0b05fc36c": "\\Psi(x)=\\sin{(x)};\\ -\\pi \\le x \\le\\pi ", "86635b5bc4ac08dafe45934e1c288c69": "s_i = 0.5 + r_i", "8663aef10b438439f8f98e7558efabbc": "x (\\tau), \\;\\;\\;\\; t (\\tau)", "8663d2b6520b7faef9f2f137bab0496c": " (1\n-\\lambda_{1}L)(1 - \\lambda_{2}L)y_{t} = \\varepsilon_{t} ", "8663d5791308cb4a4c927457c63a643a": "(f, \\, g, \\, f^x, \\, g^y)", "8664018c7dc379dfd258853d7961dbe8": "O\\left(n\\right)", "866411ade50714e8d3adf7e53eb89719": "\\rho_\\mathrm{out}\\,", "866413da87c00cb18c4b2d73a1a817b7": "\\dot \\Phi", "86642b8e0d651f139f85823459142c66": "A(\\rho)=\\int_A |z|^{-2}\\,dx\\,dy= \\int_{0}^{2\\pi}\\int_{r_1}^{r_2} r^{-2}\\,r\\,dr\\,d\\theta = 2\\,\\pi \\,\\log(r_2/r_1).", "86642fc817940a0ef37573faa30d72e4": "P = 2.5C.n^6-7", "866490b3ebfb5745325c5ca8133a1018": "A\\leftrightarrow\\neg\\neg A", "8664d711f5b077f0872dc256244f7096": "U=m g l (1-\\cos(\\phi)) ", "8664dd66b88d25c06f985cac720c6b9d": "\\mathbb{C}S_nc_\\lambda", "8664e7e42ef8e6ea8d6802414af28a68": "G_{S_n}(z) = \\operatorname{E}(z^{S_n}) = \\operatorname{E}(z^{\\sum_{i=1}^n a_i X_i,}) = G_{X_1}(z^{a_1})G_{X_2}(z^{a_2})\\cdots G_{X_n}(z^{a_n}).", "86650c012b44542acea5c892e010ca0f": "h = \\mathrm{ana}\\ f", "86654a069b9c92f30509570e2ee3e0ba": "H^3(X,\\mathbb{Z})", "86655bdb4386d4a21007051ea0e3e3ca": "A = \\frac{1}{2}\\sum_{i=0}^{n-1} (x_i\\ y_{i+1} - x_{i+1}\\ y_i)\\;", "86657f001dcecee03955e8cc5fadbf23": "KA=z(i-i*)+k", "866590d57315ef0ad56029a88d5318c5": "A=(X,Y,S,ta, \\delta_{ext}, \\delta_{int}, \\lambda) ", "866595677739af2255dc29067e65f29c": "\n\\big( D \\partial_{x_j} P(\\mathbf{x},t\\mid \\mathbf{x_0}) \\big)_{x_j=x_{j+1}} = \\big( D\\partial_{x_{j+1}} P(\\mathbf{x},t\\mid \\mathbf{x_0})\\big)_{x_{j+1}=x_j}; \\qquad j=-M,\\ldots,M-1, \n", "8665adcf83766dc3f6a464c1b891245b": "\\frac{p_1+p_2}{q_1+q_2}", "8665c5352b3db2e4fe93c3bc66acbff6": "\\,\\omega a", "8665fe2cdab2e96f9770aa40cf616e92": "(G,n):(\\mathcal C,\\otimes,I)\\to(\\mathcal D,\\bullet, J)", "866669a1db08700ff83b99663f58f466": "\\mathbf{X}^{\\rm T}(\\mathbf{A}+\\mathbf{A}^{\\rm T})", "8666756b9458fe068d3d3769b68bcdef": "(3,4)", "86669f058952a75e8aa2336f06cbb908": "g_{0k} \\ne 0", "8666b3e2c02f8b4370281ccbda952618": "\nY^{\\prime\\prime}={255/1.402}\\cdot Y^{\\prime}\n", "8666c3328df499fe91e7bbff52e7bef6": "\\mathbb{P}^x\\{\\tau<\\infty\\}", "8666c599af3194d0564b6426ba3edc57": "\\mathrm{NPV}(i, N) = \\sum_{t=0}^{N} \\frac{R_t}{(1+i)^{t}}", "8666e099625c8625538737b4942b5047": "i,\\,j,\\,\\ldots", "8666e98f0213c6e405b17510a5a90e46": "\\bar{A}=\\frac{\\int{Ae^{-\\beta H(q_1, q_2, ... q_M, p_1, p_2, ... p_N)}d\\tau}}{\\int{e^{-\\beta H(q_1, q_2, ... q_M, p_1, p_2, ... p_N)}d\\tau}}", "8667702bc0382f58a99eff62b4e1ce15": "\\Delta\\,P+\\tfrac{\\rho f}{2\\,D}\\,W^2\\Delta\\,X+\\tfrac{\\rho}2\\Delta\\,W^2\\,=\\,0", "86677689fbc98db2f6d04026c9bdd52e": "P^* = \\begin{pmatrix}\n h_{\\mbox{e}}^* & g_{\\mbox{e}}^* \\\\\n h_{\\mbox{o}}^* \\leftarrow 1 & g_{\\mbox{o}}^* \\leftarrow 1\n \\end{pmatrix}\n", "8668dd090624fc0083726bf5af631a03": "\\vee", "86691c960d00f6c7f70c72dc0b33aa6b": "\\mathbb{E}(y_i)", "8669235af8511e9a3487d321750c3f93": "\\mathbf{x}(n) = \\left[x(n), x(n-1), \\dots, x(n-p+1)\\right]^T", "86692962f9f90b7424f33f3fc7d59129": "\\left\\lceil \\frac{9^k-4^k}{5\\cdot4^{k-1}} \\right\\rceil", "866975e9d1566d7a18586ff1e2e0c466": "F(R)R_{\\mu\\nu}-\\frac{1}{2}f(R)g_{\\mu\\nu}+\\left[g_{\\mu\\nu} \\Box-\\nabla_\\mu\n\\nabla_\\nu \\right]F(R) = \\kappa T_{\\mu\\nu},", "86698a91dff848e72bd35f470840b39a": "\n\\iint_D u\\, dx dy = \\sum_{j=1}^k c_j u(z_j),\n", "866a5c96d4348add7f183a0a5674b5ae": " \\ddot{x}(t) = \\frac{qE_0}{m}~\\sin(\\omega t) ", "866a71c4896e81588d4f98111322e107": "\\mathbf A \\neq 0", "866a9dbae9e495e4489dd2bbd1d1b457": " \\mathbf{R}^o ", "866b3d0601d6ee2f6cf7a054c08a81ec": "y = \\pm e^{\\left(-\\int f(t)\\,dt\\right) + C} = \\pm e^{C} e^{-\\int f(t)\\,dt}", "866b420f9e47986003835085e27f013d": "DPW = \\left\\lfloor\\frac{\\pi d^2}{4S}\\right\\rfloor", "866b90762d73de49d32b615b99c1b7b0": "\\begin{cases}\\mu + \\sigma \\frac{(\\ln2)^{-\\xi}-1}{\\xi} & \\text{if}\\ \\xi\\neq0,\\\\ \\mu - \\sigma \\ln\\ln2 & \\text{if}\\ \\xi=0.\\end{cases}", "866bbf4c32d30ea6d3e622251dd5b455": "\\scriptstyle K \\;=\\; \\prod_{i \\in I}\\mathrm{Ker} F_i \\;=\\; \\bigcap_{i \\in I}\\mathrm{Ker} F_i", "866bd6149f3deff3981f3227642e0ea7": "\\frac{\\nu}{2}\n\\!+\\!\\ln\\left(\\frac{\\tau^2\\nu}{2}\\Gamma\\left(\\frac{\\nu}{2}\\right)\\right)", "866c29ee32431ec76249f96b1b2355d8": "m_{biomass,\\, X} = m_{resource,\\, X} \\frac{c_{resource,\\, X}}{c_{biomass,\\, X}}", "866c30f1d0faa1813a0fe27a29ba66b8": "\\cosh^2 (x) - \\sinh^2 (x) = 1", "866c42015fe05e7b78d8a713b0f7f709": " S(t) + I(t) + R(t) = \\textrm{Constant} = N ", "866c8a1468e7d349db1e0af4d22f698b": "C_0 = 0", "866c96d9782f04dc7aaa404711cabef9": "f(\\theta,\\phi)", "866cbd9a56d26495c7e7903df71fe30b": "\\left [ \\mathbf S \\right ] = \\begin{bmatrix} S_{11} & S_{12} \\\\ S_{12} & -S_{11} \\end{bmatrix} ", "866cf4856c14b95b3edd156f6984189d": "\\scriptstyle S^2\\tilde{\\times}S^1", "866da551dcbd128aaef971564aeba5e2": "\\det(V) = \\prod_{\\mathbf{c}} \\left( c_1\\alpha_1 + \\cdots + c_n\\alpha_n \\right), ", "866dba7ccb017e70356141aeac420a69": "b^{2}(w^{2},m_{1}^{2},m_{2}^{2})", "866dd94c993fab596f67ada889320107": "g^k \\mod q= (-1)^{e_0}2^{e_1}3^{e_2}\\cdots p_r^{e_r}", "866def8bb3c555756f0587c6ac35bd25": "\\scriptstyle O(nh\\sqrt{\\log h})", "866dff9877c86f34a9d3001b5b28e7ce": " \\frac{K\\Sigma_{ii} - n }{n(K-1)} ", "866e31bf17be4da975bbc8b2fc69227d": " \\mathbf e_i \\cdot \\mathbf e^j = \\delta^j_i", "866e6ec0e47e5a42ad9606fd7c597103": "e_3=\\overline{e_1}", "866ef8dc8df3551e907dfd92570ff388": " \\mathbf{E} = \\mathbf{U} \\, \\mathbf{\\Sigma} \\, \\mathbf{V}^{T} ", "866f3c82486a22dafde3f9ec0dbfa5a4": "\n\\psi=P_j \\sim 2^{-2j}\\frac{(2j/e)^{2j}\\sqrt{4\\pi j}}{[(j/e)^j \\sqrt{2\\pi\nj}]^{2}}\\ =\\ \\frac{1}{\\sqrt{\\pi j}} \\sim \\sqrt{\\frac{2}{\\pi}} \\frac{1}{\\sqrt{N}}.\n", "866fa012bb7fa0731fa8e35f89c5eb08": "e^{i m \\phi}", "86700e6e0407ed6339c3632b3dff48e2": "\n\\chi(n) =\n\\left(\\frac{-4}{n}\\right)= \n\\begin{cases}\n\\;\\;\\,0 & \\mbox{if } n \\mbox{ is even}, \\\\\n\\;\\;\\, 1 & \\mbox{if } n \\equiv 1 \\mod 4, \\\\\n -1 & \\mbox{if } n \\equiv 3 \\mod 4.\n\\end{cases}\n", "867068688691d165865009c2b043b55a": "\\delta=0,w(x_1,x_2)=1", "8670aaa72cb428b6c5f8b2bbce7042b9": "g(y) = h(2) = y \\mapsto f(2,y)", "8670b6d979158426be1bed616855cf72": "4 \\le N \\le 7", "8670d7d454d31475504ea680166040ca": "-2 \\log \\big( p(y|\\hat \\theta_0)\\big) ", "8671645b35844237075a4e6f4f3c2d8b": "\\mathrm{Pr}_\\mathrm{m} = \\frac{\\mathrm{Re_m}}{\\mathrm{Re}} = \\frac{\\nu}{\\eta} = \\frac{\\mbox{viscous diffusion rate}}{\\mbox{magnetic diffusion rate}}", "8671a11389deec320afd2442e8db719e": "XX^\\top", "8671f2dfdcd1a49f7060d7bdc34a77c0": "\\bar{U}", "8672c2943da760e0ab905d6cb33878de": "\\int a^2 dm", "8672f49d0bd16ca787800f55d3888262": "\\underbrace{ a+b+\\cdots+z }_{26}", "867357e623c86e3a055a945ccd02a785": "\\varphi=\\frac{1+\\sqrt{5}}{2}", "86736dcf4dae8cd3164ff04e45441ed3": "[L_{ij},H]=0 \\,\\!", "8673b7a2a772a0deddb5d7feba562794": "\np(1)= \\frac{1}{N}, ", "86740579fc9fa5630615051f4f25cbe8": " J^{i}={\\frac{1}{2}}\\,\\sum_{jk}\\,\\epsilon^{ijk}\\,J^{jk}, K^{i}=J^{0i}", "867419cb2575f32f9d32db48a20e8c79": "M_n(R[X])", "867443557c16ff2f6f2a627d3a9ac989": "Wins = 63.83 + 0.68*fWAR", "867453a82914ddf48a3822638a971bf4": "v(\\alpha) = \\alpha^i", "8674a114c0d0a842d26260862cde8735": " (\\pi \\circ \\varphi)(x,v) = x ", "8674b8275be7fdb91c2aca8274938b0b": "\n\\begin{align}\n\\mathbf{w}^{\\text{T}}\\mathbf{S}_W^{\\phi}\\mathbf{w} & = \n\\left(\\sum_{i=1}^l\\alpha_i\\phi^{\\text{T}}(\\mathbf{x}_i)\\right)\\left(\\sum_{j=1,2}\\sum_{n =1}^{l_j}(\\phi(\\mathbf{x}_n^j)-\\mathbf{m}_j^{\\phi})(\\phi(\\mathbf{x}_n^j)-\\mathbf{m}_j^{\\phi})^{\\text{T}}\\right) \n\\left(\\sum_{k=1}^l\\alpha_k\\phi(\\mathbf{x}_k)\\right)\\\\\n& = \\sum_{j=1,2}\\sum_{i=1}^l\\sum_{n =1}^{l_j}\\sum_{k=1}^l\\alpha_i\\phi^{\\text{T}}(\\mathbf{x}_i)(\\phi(\\mathbf{x}_n^j)-\\mathbf{m}_j^{\\phi})(\\phi(\\mathbf{x}_n^j)-\\mathbf{m}_j^{\\phi})^{\\text{T}} \n\\alpha_k\\phi(\\mathbf{x}_k) \\\\\n& = \\sum_{j=1,2}\\sum_{i=1}^l\\sum_{n =1}^{l_j}\\sum_{k=1}^l \\left(\\alpha_ik(\\mathbf{x}_i,\\mathbf{x}_n^j)-\\frac{1}{l_j}\\sum_{p=1}^{l_j}\\alpha_ik(\\mathbf{x}_i,\\mathbf{x}_p^j)\\right)\n\\left(\\alpha_kk(\\mathbf{x}_k,\\mathbf{x}_n^j)-\\frac{1}{l_j}\\sum_{q=1}^{l_j}\\alpha_kk(\\mathbf{x}_k,\\mathbf{x}_q^j)\\right) \\\\ \n& = \\sum_{j=1,2}\\left( \\sum_{i=1}^l\\sum_{n =1}^{l_j}\\sum_{k=1}^l\\Bigg( \\alpha_i\\alpha_kk(\\mathbf{x}_i,\\mathbf{x}_n^j)k(\\mathbf{x}_k,\\mathbf{x}_n^j)\\right.\\\\\n& \\left.{} - \\frac{2\\alpha_i\\alpha_k}{l_j}\\sum_{p=1}^{l_j}k(\\mathbf{x}_i,\\mathbf{x}_n^j)k(\\mathbf{x}_k,\\mathbf{x}_p^j)\n\\left. + \\frac{\\alpha_i\\alpha_k}{l_j^2}\\sum_{p=1}^{l_j}\\sum_{q=1}^{l_j}k(\\mathbf{x}_i,\\mathbf{x}_p^j)k(\\mathbf{x}_k,\\mathbf{x}_q^j) \\right)\\right) \\\\\n& = \\sum_{j=1,2}\\left( \\sum_{i=1}^l\\sum_{n =1}^{l_j}\\sum_{k=1}^l\\left( \\alpha_i\\alpha_kk(\\mathbf{x}_i,\\mathbf{x}_n^j)k(\\mathbf{x}_k,\\mathbf{x}_n^j) \n - \\frac{\\alpha_i\\alpha_k}{l_j}\\sum_{p=1}^{l_j}k(\\mathbf{x}_i,\\mathbf{x}_n^j)k(\\mathbf{x}_k,\\mathbf{x}_p^j) \\right)\\right) \\\\\n& = \\sum_{j=1,2} \\mathbf{\\alpha}^{\\text{T}} \\mathbf{K}_j\\mathbf{K}_j^{\\text{T}}\\mathbf{\\alpha} - \\mathbf{\\alpha}^{\\text{T}} \\mathbf{K}_j\\mathbf{1}_{l_j}\\mathbf{K}_j^{\\text{T}}\\mathbf{\\alpha} \\\\\n& = \\mathbf{\\alpha}^{\\text{T}}\\mathbf{N}\\mathbf{\\alpha}.\n\\end{align}\n", "8674c70f7fa2b8fc64cd7106dc5308d8": " \\mathbf{K} = \\begin{bmatrix} K_{xx} & 0 & 0 \\\\ 0 & K_{yy} & 0 \\\\ 0 & 0 & K_{zz}\\end{bmatrix} \\ ", "8674d0edb565e5da10a2cab0c8bd353d": "\\lim_{n \\to \\infty} \\frac{\\text{N}(n,S)}{n} = 1.", "86751ed798ae042e0732cdfe6ce30201": "\\mathrm{Var}(f) = \\frac{\\sigma_a^2(f)}{4 N_a} + \\frac{\\sigma_b^2(f)}{4 N_b}", "8675823ac613cfbaeb608ce5dd02200c": "\\omega_0.", "86758b9c53eefb5a51702dac445bfd03": "\\nabla \\times \\mathbf{E} = -\\frac{1}{c}\\frac{\\partial \\mathbf{B}}{\\partial t}", "867596b9b09d7122c59697b391bcabb6": " E(1+\\alpha E) = \\frac{\\hbar^2 k^2}{2m^*} ", "867597ff13d48db7f78d22360d688e16": " \\nu = n - p", "8675d59a33d2f170a2c1c572a3cce4d4": "A \\cup B \\in \\mathcal{R}.", "8675e959605516f2ea946d3f331a259f": "M^\\beta_{ii}", "8675f23ac90f90c1b8a0d0ef9b339950": "\\ln{T_{eff}} = A\\ln{L} + B\\ln{M} + const", "8676422a85806013561aed2127f1161b": "\\textstyle p(x) \\mid (x^r - 1)", "8676544c98fffad7e5bd67a50fb5cfab": "A < B < \\sqrt{M_p}\\text{, as }B/A\\text{ and }M_p \\rightarrow \\infty\\text{, the number of prime divisors of }M", "86766eed7f3446bfb2a652d3157a1d24": "\\exp((\\log \\log \\log k)^c)/\\log k", "8676e83110e033b0a8f5cbd74abd2d0b": "\\partial (u,x_1) = x_2", "8676fc4f6f2d75799a559944124874b6": " Q_3 \\rightarrow O\\,", "86771371622486afe21eed9c886ae8fb": "\\scriptstyle \\phi (t) \\;=\\; \\phi_0 \\,+\\, n \\omega t \\,+\\, a \\sin( \\omega t)", "86774cbc3b99ab9b339a752d3ed21b71": "r = n =", "86775eb9c6ba62f74795694fc32b456e": " \\mathrm{DIBL} = - \\frac{V_{Th}^{DD} - V_{Th}^{\\mathrm{low}}}{V_{DD} - V_{D}^{\\mathrm{low}}}, ", "86777d1e53b958fc3719bcd3909660c4": "\\pi_{n-k} V_k(\\Bbb R^n) \\simeq \\Bbb Z", "8677c758324366e4844aa485696538a9": "\\mbox{Apply}:([Y\\to Z]\\times Y) \\to Z", "86782eebfa3c1404ddf1506ad966d23f": "\\frac{d}{dt}\\langle \\sigma \\rangle = i\\left(-\\Delta_a \\langle \\sigma \\rangle - ig\\langle a \\sigma_z \\rangle\\right) -\\gamma \\langle \\sigma \\rangle ", "867851c6276044d05e71db266639adad": "\\langle x,y+z\\rangle=\\langle x,y\\rangle+\\langle x,z\\rangle.", "86791e35533c04bc3be259f7888e1dda": "P^\\perp:=\\{\\langle\\vec x\\rangle\\in {\\mathcal P} \\mid f(\\vec p,\\vec x)=0\\}", "8679d819a40a2d865be0dd77b9a0b4d1": "\\frac{s+1}{n+2}", "8679e0109a3d6977393018e4aa6a6ced": "\\scriptstyle \\{a_i\\}", "8679e4c92863f6419be05c02ba89fad8": "T = T_{eff}", "867a0dddd7d16614b0668a87e89579ed": "U_n(P,Q)=U_n(2S,S^2) = nS^{n-1}\\,", "867a1af55f352571051edd4efdf71f72": "(g^{bc})^a = g^{bca} = g^{abc}", "867a2e90a3faca6141dcbf3255776a19": "\\Theta_k", "867a55552678a70e72007d1a2f349f40": "D\\,\\! ", "867a8481c270b4c717e4de127a2b0d83": " |z-c| = \\varrho", "867a8964e398f065ce99abfe7ba22c2c": "\\mathrm{% \\ excess \\ air} = 1.2804 \\times (\\mathrm{%O_2 \\ in \\ combustion \\ gas})^2 + 4.49 \\times (\\mathrm{%O_2 \\ in \\ combustion \\ gas})", "867b607097df03839661c780d9acbf0d": " (a^2 - 1)y^2 + 1 - x^2 ", "867ba96cfd2e52ea71cc70b1094c110c": "n\\sin\\theta=n_0\\sin\\theta_0\\,", "867c8e9b139e685e518e8650a3667908": "\\max_i\\left[w_{i}+\\mathrm{length}\\left(c_{i}\\right)\\right]", "867ca32c4a66e7ec264f2d0d054df645": "\\int_{\\mathrm{Vol}} \\nabla \\cdot \\mathbf{F} \\ \\mathrm{d}_\\mathrm{Vol} = \\oint_{\\partial \\mathrm{Vol}} \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{\\Sigma}", "867ca491df718d7feef6ed7ce97ffbe2": "\\frac{1}{2}\\sigma^2", "867cdcffab24450e548c925d608460c0": "[C_i,E]=i\\hbar P_i", "867d406cc5b9cac95d243f3cf850e6ec": " \\left ( \\frac{d\\,\\ln\\,P_A}{d\\,\\ln\\,x_A} \\right )_{T,P} = \\left ( \\frac{d\\,\\ln\\,P_B}{d\\,\\ln\\,x_B} \\right )_{T,P} ", "867d6481297534e23299b1f472c50915": "f'(r) = 2r \\not\\equiv 0 \\,\\bmod{p}", "867d66eaa14f430405628f23c07cd997": "G_{ik}\\,\\!", "867d95a3fba4f4138e78631887b4b126": "P_i\\,", "867db6bea4d3c1127cb18f6b75954933": "\\displaystyle \\Phi(0)=I, \\,\\,\\, \\Phi(n)=T^n,\\,\\,\\, \\Phi(-n)=(T^*)^n, ", "867e0388fa1720ab1b7941b7c3dbeb44": "\\displaystyle \\frac{J_1\\left(\\sqrt{\\omega_x^2+\\omega_y^2}\\right)}{\\sqrt{\\omega_x^2+\\omega_y^2}}", "867e8997be02cae7b6884cd6b9ab5483": "\\alpha_1=2", "867eaea63afa40814793c4b2be697672": "p \\rightarrow p^\\star", "867ec8817bd89ccb0ab060289788d570": "\\pm\\phi/e", "867efa68230810cbd803629e49b693ad": "W_j^\\mu", "867f444ea0e16efd450bcaa11dd044ad": "F(x; -\\ln \\sigma, 1/\\alpha, 0)", "86800b825f9039def1f930ba0628ebf8": "s_{zj}=0", "8680a643cddd22d29e2834e0958fdf7d": "b_Y\\, ", "8680de367116f5a1476ffd5cdf642328": "g^{\\rho\\mu}R_{\\rho\\sigma\\mu\\nu} = R_{\\sigma\\nu}", "8680ed2f5407ed9424d77caf5d0cbcb6": " e= \\lim_{n \\to \\infty} n\\cdot\\left ( \\frac{\\sqrt{2 \\pi n}}{n!} \\right )^{1/n} ", "8680f722a3c5c4c68aed0843febe262d": "R^{n}", "86810b8db33158e5b65b11e180999d4e": " \\phi = \\pi ", "8681881280862b27c78ecdbf5a7937b7": "\\mathbf{u} ", "8681bd9fa380faf7f8eac34f7af53c1d": " h_k = h_{d-k} \\, ", "8681c0d804ec4c916ff8f7e633af2d00": "T(a,0,b)=T(0,a,b)=b\\quad \\forall a,b \\in R", "8681fbf488f9d92d194c9fa3ec05648f": "f(x)=x^x", "86822c94d44b11b4f849dc173a1d76e9": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.38971 & 0.68898 & -0.07868\\\\\n-0.22981 & 1.18340 & 0.04641\\\\\n0.00000 & 0.00000 & 1.00000\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "86824946aac005c24b5344624f3dd12d": "h \\colon [0, \\infty) \\to \\R", "86826cf2e829deff86e7e6f8795f2c73": "\n\\operatorname{probit}(p) = \\sqrt{2}\\,\\operatorname{erf}^{-1}(2p-1).\n", "8682718d6389ed4e1c72609905379dd3": "\\Sigma_\\text{int}", "8682e0803fa65bf875931b6e7bd1f7f9": " \\phi\\colon M\\to N", "8682e9f5a5522251f4be4b92c7e1b29b": "f^{-1}(V_i) = U_i", "86831c61f9e35f7bd04f097c009b6e5e": "\\begin{pmatrix}\n0 &1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & ... \\\\\n0 &7 & 2 & 9 & 4 & 11 & 6 & 1 & 8 & ... \\end{pmatrix} ", "8683830a678c729bc410dd37302be58c": "\n\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|} = \\sum_{\\ell=0}^\\infty \n\\sum_{m=-\\ell}^{\\ell}\n(-1)^m I^{-m}_\\ell(\\mathbf{r}) R^{m}_\\ell(\\mathbf{r}')\\quad\\hbox{with}\\quad |\\mathbf{r}| > |\\mathbf{r}'|.\n", "8683ca001066b97d40aab83ca58bf54a": "\\displaystyle{k(t,t)=-{\\kappa(t)\\over 4\\pi}.}", "8683d98533a1372e6cd0414742edfc8c": "\\varphi\\circ f:{\\mathbb R}\\rightarrow {\\mathbb R}", "868437bf2eaf3db69cac976e6be3f720": "\\widehat{P1O1Q}=\\widehat{P2O2Q}", "86844217091264e975fb6017ce421db0": "T_{ab}", "868489581dc7f1d8beff8228c056d30e": "\\mbox{grad}\\,c = {\\partial c \\over \\partial x} \\mathbf{\\hat{x}} + {\\partial c \\over \\partial y} \\mathbf{\\hat{y}} + {\\partial c \\over \\partial z} \\mathbf{\\hat{z}} = \\nabla c.", "86848b3a3289077a785c485ecb00b8d2": "\\Pr_R[|p'_R(x,y) - p(x,y)| \\geq 0.1] \\leq 2 \\exp(-2(0.1)^2 \\cdot 100n) < 2^{-2n}", "868499a6a360fdc9e2d6f77f8775a8bc": "\\sum_{k=0}^\\infty {2k + \\alpha \\choose k} z^k = \\frac{1}{\\sqrt{1-4z}}\\left(\\frac{1-\\sqrt{1-4z}}{2z}\\right)^\\alpha, |z|<\\frac{1}{4}", "8684dc6ef009238d30e37ef970111b78": "T_n(x) =\n\\begin{cases}\n\\cos(n\\arccos(x)), & \\ |x| \\le 1 \\\\\n\\cosh(n \\, \\mathrm{arccosh}(x)), & \\ x \\ge 1 \\\\\n(-1)^n \\cosh(n \\, \\mathrm{arccosh}(-x)), & \\ x \\le -1 \\\\\n\\end{cases} \\,\\!\n", "86851aab14ffb778e3c5573ca3cfd94f": "\\begin{align}\ng_{\\kappa\\lambda ; \\alpha} & = 0 \\\\\n(\\sqrt{-g}\\;^W)_{; \\alpha} & = (\\sqrt{-g}\\;^W)_{, \\alpha} - W \\Gamma^{\\delta}_{\\delta \\alpha} \\sqrt{-g}\\;^W = \\frac W2 g^{\\kappa\\lambda} g_{\\kappa\\lambda,\\alpha} \\sqrt{-g}\\;^W - W \\Gamma^{\\delta}_{\\delta \\alpha} \\sqrt{-g}\\;^W = 0 \\,.\n\\end{align}", "86851cbcc21ad30873cf4c09458db1af": " P= \\dot{m}(h_2 - h_1 +[\\frac{V_2^2}{2} - \\frac{V_1^2}{2}])\\,", "868529fa6cf567ab193f1e3e1e96874e": " \\epsilon = 1 - \\frac{\\omega_s}{V_s \\rho_c} ", "86853233ca7f14ab8c5822d7932360f6": "\\sin\\frac{\\pi}{10}=\\sin 18^\\circ=\\tfrac{1}{4}\\left(\\sqrt5-1\\right)\\,", "868592f8569573326f3aa25e845db8c3": "c_1({\\mathbf C\\mathbf P}^1\\times {\\mathbf C})=0.", "86860a0f02dedc50f3a14ab1ab546eb8": " \\delta (T_0)_{isentropic} = U \\frac{(V_{f2}\\tan\\alpha_2 - V_{f1}\\tan\\alpha_1)}{c_p}\\,", "86864970a22a48b97c0324ccf00817b6": " B_1(x) = \\frac{\\mu_0 I R^2}{2(R^2+x^2)^{3/2}}", "86865c7c490c1fdd76c2a54e0c99af94": "HJD = JD + \\frac{\\vec{r} \\cdot \\hat{n}}{c} ", "8686cbd6e291c5dbe8e53ba0f1456ed5": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\n \\end{Bmatrix}\n = \\sum_{m_i} (-1)^S\n \\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & -m_3\n \\end{pmatrix}\n \\begin{pmatrix}\n j_1 & j_5 & j_6\\\\\n -m_1 & m_5 & m_6\n \\end{pmatrix}\n \\begin{pmatrix}\n j_4 & j_5 & j_3\\\\\n m_4 & -m_5 & m_3\n \\end{pmatrix}\n \\begin{pmatrix}\n j_4 & j_2 & j_6\\\\\n -m_4 & -m_2 & -m_6\n \\end{pmatrix}\n.\n", "8686ce4920910ff51295fc635e3e2954": "\\max S\\leq s", "8686ee1e7717262551d47ee2bd7112f6": " {\\hat H} = \\hat T + \\hat V ={\\hat p^2 \\over 2m} + V( \\hat x ) ", "868711fbc6c22046bd7729704a55b1fb": "A_m(4,3) = 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, \\ldots ", "86871a142654d7fa7d41809b35316de1": "(\\Omega,U,X,S,p,q)", "868763112a09dabbcc3efc3e6f450b89": "\\displaystyle r=\\sqrt{eg}=\\sqrt{fh}.", "8687adf3fe8bf201d94f8a76653b3bf8": "D_-(x)", "8687e547fbe8ccef70eb9317b2b1aa82": "\n \\frac\n {\\exp\n \\left(\n -\\frac{1}{2\\sigma^2} \\lVert s_j - T(m_i, \\theta)\\rVert^2\n \\right)}\n {\\sum_{k=1}^{M} \\exp\n \\left(\n -\\frac{1}{2\\sigma^2} \\lVert s_j - T(m_k, \\theta)\\rVert^2\n \\right) + (2\\pi \\sigma^2)^\\frac{D}{2} \\frac{w}{1-w} \\frac{M}{N}}", "868836207ac7794c25b3273d89cfe61e": "A_3", "868838d8a9da4d36dabf0fbcab6c6885": " p_{S_{i}}", "868850f5d53916b4f37ada8bf6908711": "\\Gamma \\tau=\\tau\\rfloor\\Gamma=\\tau^\\lambda(\\partial_\\lambda +\\Gamma^i_\\lambda\\partial_i)\\subset HY", "86886d84f08c9329fef4f8e67b307b75": "\\Delta W_m", "8688956a50ddee60f25c4ddaf975461d": "E=\\begin{matrix}\\frac{1}{2}\\end{matrix} m v^2 - \\frac{G M m}{r} = \\frac{-G M m}{2 a} .\\,", "86889d217d19bfa20a3579f1786fc350": "m=\\lambda^{-1} \\ln 2 .", "8688c419802f5f0f0ff6dfdcbcef0a0d": "S\\to X", "8688efa0f2ad31b35d57808956071451": "\\mbox{affinity}=k[A]^{\\alpha}[B]^{\\beta}\\dots\\!", "868913d0e12c8c84036b8e224b2314f9": " \\Pr [a \\le X \\le b] = \\int_a^b f_X(x) \\, dx .", "868926b41c068a7fea45b0822fb0b6f3": " m \\geq n ", "868976d60b9392ef8e59cf8e8099caad": "\\overline{A_t}", "8689ea2adb6c6cea8da67f251d360645": "\\frac{n + 1}{n} = 1 + \\frac{1}{n}. ", "8689fef37a90072102d9685f52ff398e": "~~~\\or~~~", "868a4722c64ebbb40c0e8eeb127a25fd": "0 = (x+y)\\wedge (x+y) = x\\wedge x + x\\wedge y + y\\wedge x + y\\wedge y = x\\wedge y + y\\wedge x", "868a6a60cae3b5656d81d0914c3a4dc6": "R(a)=Jn(m,k*a)=0", "868b49a424ab64ac56f7e5d41d05fcab": "p_{c} = \\frac {\\gamma} {L_{c}} = \\sqrt{ \\gamma \\rho g}.", "868b4d4d99591de470ade1a628458c16": "M _{BA} ^f = \\frac{Pa^2b}{L^2} = \\frac{10 \\times 3^2 \\times 7}{10^2} = + 6.300 \\ kN\\cdot m", "868b89f8bfabda795e77c570b39b209e": "\\mathbf{W}\\mathbf{H} = \\mathbf{V}", "868c09cd0a278f64e65a55e684bce431": "x^6 + y^6 = x^2", "868c27022d578ad702bdf1b56b52db8c": "\\displaystyle u_{xxx}-\\frac{1}{8}u_x^3 + u_x\\left(Ae^u+Be^{-u}\\right)=0.", "868c44567ae2cc7def91cdc1dec5fb72": "c = 13", "868c670940c7913e3afc9675453733e4": " \\mu(t) = \\text{E}(X(t)) ", "868ca864c08e3a1808bfbac2d43e528d": " \\phi(z_{a^n_1}...z_{a^n_k}, z_{(n-1)m+2}, z_{(n-1)m+3}...z_{nm+1}) ", "868cbc0541904f85fac31a2ec63e4e64": "I = (\\mathbb{U},\\mathbb{A})", "868cd228661357fa7a9a9a13f1bc60d8": "x = 2^0 + 2^2 + 2^3 + 2^4", "868d03b3c15306f9678f9275dedd7ef1": "\\gamma0 \\\\ \n 0 & \\textrm{if} \\; y_i^* \\leq 0\n\\end{cases}", "86ad2309cdf5c94226494fe2bb893a47": "\n i\\bar{\\partial} \\Psi \\mathbf{e}_3 \\rightarrow \n i\\bar{\\partial} \\Psi R_0 \\mathbf{e}_3 R_0^\\dagger R_0 =\n ( i\\bar{\\partial} \\Psi \\mathbf{e}_3^\\prime ) R_0,\n", "86ad601ce410b64186c7f0d5439d2ff4": "m_{rel} = E / c^2 \\!", "86ada6c75990658a0afe9f93da90440a": "y=m x+c", "86adc52f87435bd6783db578753f2b19": "R(c) = cA(c)=\\frac{-cu''(c)}{u'(c)}", "86ade08b2125dc2f3f826a12682157eb": "i=0,1,\\dots", "86ae9400e7e718663116db037e004e96": " \\langle , \\rangle ", "86aeaebb0eeafcfe52ee94cb726810c9": "K^\\mathrm{T}K", "86aeb0c4b0fad9ec991dff5854fb2d36": "R \\rightarrow X \\times X", "86aeb1ac915102e4105ba4cf8dd0f6c1": "\\scriptstyle\\mathcal{F}(W)", "86aebcf375c79b1b3ee86fab144d11f9": "\\Pr[Y(t) \\leq x] = \\left[1-e^{-\\lambda(t+x)}\\right] - \\int_0^t \\left[1 - 1+e^{-\\lambda(t+x-y)}\\right]d(ay) = 1 - e^{-\\lambda t}.", "86af397e81b979500bb6068794198202": "(x^{q^{2}}, y^{q^{2}}) + \\bar{q}(x, y)", "86af6586e756a45e20c720d0d9686173": " R_{ab}-\\frac{1}{2}R g_{ab}=\\kappa P_{ab}", "86af8f7d30121c5d0469a432b95902f6": "\\zeta(0) = -\\frac{1}{2};\\!", "86af90977c9762e9c85b3e9ab09dea14": "\\mathrm{argmin}_m\\,\\mathrm{E}((X - m)^2) = \\mathrm{E}(X)\\,", "86afb98fee7e80d496c55a43f6fb423e": "E_{ABC}=E_A-E_B-E_C-F(BC:x)+R_{xin}+R_{xex}", "86afd85facca9e107b8416a670e54b05": "\\int\\frac{dy}{y(1-y)}=\\int dx,", "86aff842154cac469bc7718fe8613afc": "S(\\beta_1, \\beta_2)", "86b00edb3553d501bd887f1de8813281": " \\nu (x) := \\begin{cases}\n0 & \\text{if } x < 0, \\\\\nx & \\text{if } 0< x < 1, \\\\\n1 & \\text{if } x > 1. \\end{cases}", "86b113a3b0f2e6bea2620f2a2e88b3f0": " \\angle BAC", "86b11dd484fd4691a1c814ffc76f5ef2": "\\operatorname{exp}\\ m\\ n = m^n = n\\ m ", "86b145a6bd5aaef6e12f1db4c64a00fc": "H \\ ", "86b14ef44cac35971e08fc4751b90ae5": "F_M(\\phi) : M \\otimes_R K \\to M \\otimes_R L", "86b14fc73fded60ed428726a5cff46ea": "\\xi=1/\\lambda_1T_G, ", "86b194fabdf2d6275a0726a34872614e": "\\vec x \\to r\\left( \\vec x; q \\right)", "86b1d4b39aa195de081782c2c188d19c": "k=\\sin \\alpha\\,\\!", "86b1fb44bb5cfab0a7c8ccfa91eb9cf3": "S_s \\frac{\\partial h}{\\partial t} = -\\nabla \\cdot (-k\\nabla h) - G. ", "86b1fb88003107097b2ae34c3b11e811": "f_Y(y) = f_X(g^{-1}(y)) \\left| \\frac{d g^{-1}(y)}{d y} \\right|.", "86b20c30952bd33a4fd2f4ffadfd3499": "\\frac{dq} {dt} =- \\iiint\\limits_V\\left(\\nabla\\cdot\\mathbf{J}\\right)dV.", "86b27388466e62e40c0d2d35b2754321": "\\langle\\phi^0\\rangle = v", "86b2af4388dc05408909fe43e69cb3d9": "\\sum_{i=1}^n\\left(\\operatorname{E}\\left[\\widehat{g}(x_i)\\right]-g(x_i)\\right)^2\n=\\operatorname{E}\\left[\\sum_{i=1}^n\\left(y_i-\\widehat{g}(x_i)\\right)^2\\right]-\\sigma^2\\operatorname{tr}\\left[\\left(I-L\\right)'\\left(I-L\\right)\\right].", "86b2ceae90386ef95e596b7f172cf049": "\\begin{align} y(N)\\quad & = s(N) - e^{-2 \\pi i \\frac{K}{N}} s(N-1)\\quad \\\\\n & = (2 \\cos(2 \\pi \\omega) s(N-1) - s(N-2)) - e^{-2 \\pi i \\frac{K}{N}} s(N-1) \\\\\n & = e^{2 \\pi i \\frac{K}{N}} s(N-1) - s(N-2) \n\\end{align}", "86b2f4040021beaa9a2eabbc8c1bf50d": "c_n = \\sum_{i=0}^n a_i b_{n-i}\\;", "86b32015594c436e5ce767f1c90b011b": "\\scriptstyle\\vec{A}", "86b323986aeec13777b07328959bbdb4": "\\mathbf{x}'(t) = \\mathbf{F}(\\mathbf{x}(t)).\\!\\,", "86b37d4024cee325f4c6699383a4e4c0": "{{v}_{MIRROR\\_OUT}}", "86b3e58880e38d656592d2b0d45d28a0": " \\begin{align}\n\\frac{d z}{d t} &= r[s(x-x_R)-z],\\\\\n\\end{align} ", "86b461c679e90cc24dc854910f5709b7": "AA^T", "86b4c94c84d928041674a54c0d57c5f4": "u_0,\\dots, u_k \\in \\mathbb{R}^k ", "86b50519fe68c5545889bbe6b5e601e2": "\\mathcal P:=(\\R\\cup \\{\\infty\\})^2=\n\\R^2 \\cup (\\{\\infty\\} \\times\\R) \\cup (\\R\\times\\{\\infty\\}) \\ \n \\cup \\{(\\infty,\\infty)\\} \\ ,\n \\ \\infty \\notin \\R", "86b50a1f112774d59170a764a4da96c2": " x = r\\cos A + \\sqrt{l^2 - r^2\\sin^2 A} ", "86b58b333113110e52b535154e6d9df0": "\\forall v \\in W \\ (w, v) \\not \\in R", "86b5ad601f2f0c4fed4a0eec1d7898de": " u_{80}(\\mathbf{r}) = \\bar{u}_{hh}(\\mathbf{r}) = \\left | \\frac{3}{2},-\\frac{3}{2} \\right \\rangle = -\\frac{1}{\\sqrt 2}|(X-iY)\\downarrow\\rangle ", "86b5fdad319ec3a3ad00a781ce18dbda": "f(x+y)^2 = f(x)^2 + f(y)^2\\,", "86b62bcba8e7cc05e6ed44fa35483e7d": "dm/2.", "86b691775359d8df26f28161f5fa1c51": "\\delta_s=-k \\cdot r_0", "86b6cbda960ff29d4b97e188bd5db580": "\\psi,\\; A,\\;m", "86b6eed190ae34e57be36de162f85eac": "h = \\frac{d}{2\\sqrt{2}\\tan{\\frac{\\theta}{2}}} \\approx \\frac{d}{7.0006},", "86b72a763e0bae71d58b5142611417c6": "[L_{ij},E]=0", "86b733676e1d99819f194101da7f8302": "\\mathcal{O}_{\\mathbf{Q}(\\sqrt{-3})} = \\mathbf{Z}\\left[{{1 + \\sqrt{-3}} \\over 2}\\right]", "86b744dd78c768a79972a5a45542a9b7": "Y \\subseteq R^m.", "86b76e060947af6ea3ed8869833dde59": "H(z) \\, H_{inv}(z) = 1", "86b7867370f2e29c2bd657789b809d36": " \\log\\rho-\\log\\sigma=T^*\\Bigl(\\log T(\\rho)-\\log T(\\sigma) \\Bigr).", "86b7bcf3334adaa0b3015e40753894ea": "\\tfrac{abc}{2(a+b+c)}.", "86b80f2d1f97f002111bf891f7906718": "B_k = A_{I \\cup \\{ k \\}}", "86b810942a681ef657d76295708bc287": "a(u,v) = f(v)", "86b8168dad0bc3cae3a23995c93d7c32": " V \\simeq \\oplus_{\\lambda \\in P(\\mathfrak{g})} ( \\oplus_{i=1}^{d_\\lambda} M_{\\lambda}) ", "86b826dee5c983a904b57c761b840364": "E_1=x_0+x_1+x_2", "86b82776ca82c2d1a98b3d6cfddaa634": "\n\\operatorname{E}\\left[ \\left. \\left( \\hat\\theta\\left(X\\right) - \\theta \\right)^2 \\right| \\theta \\right] = \\int \\left(\\hat\\theta - \\theta\\right)^2 f \\, dx.\n", "86b8556c4d76baed73d85d0a674239ba": "A(x) = \\sum_n a(n) x^n = \\prod_n (1-x^n)^{-p(n)} \\ ", "86b872c015c67929be9313f066c60de4": "P( A_1 / A_3) = 1.067 > 1,\\text{ and } P( A_2 / A_3) = 1.059 > 1. \\, ", "86b893e5c3fb28a65a72884bb63d489e": "x^T Q x", "86b8a486e5bb8b4dfbb81b76223b766b": "u_2(\\mathbf{x},z_1,z_2)=-\\frac{\\partial V_1}{\\partial \\mathbf{x}_1 } g_1(\\mathbf{x}_1)-k_2(z_2-u_1(\\mathbf{x}_1)) + \\frac{\\partial u_1}{\\partial \\mathbf{x}_1}(f_1(\\mathbf{x}_1)+g_1(\\mathbf{x}_1)z_2)", "86b972652838802a48f3e2c5a248db67": "\\Psi(x) \\approx \\frac{ C_{+} e^{+\\int dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}} + C_{-} e^{-\\int dx \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}}}{\\sqrt[4]{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}}", "86b98ce5b7f50e2dc1a728402056853c": "\\int f = \\int_0^\\infty f^*(t)\\,dt", "86b9b7d0baf98c267a81881cf1b47de7": "w \\equiv z", "86b9bdf8568688848783a196006698ea": " y''-xy=0\\,", "86ba337f7370d3f7fc3e4fed064defef": " \\tan \\theta = \\frac{v^2\\pm\\sqrt{v^4-g(gx^2+2yv^2)}}{gx} ", "86ba5dc319494f2b5d432a59e1d66b4b": "\\log \\left( \\mu \\right)=c^{\\mu }+a_{j}-d_{i}", "86ba91d4d636d11c45a57a67f1ea8deb": " 2^{\\aleph_0}", "86ba97decd88f583e40525acab2b26b2": "c\\;", "86ba995e4fb38392b48925e4a8a55372": "\\omega_f := f^* \\omega_G,", "86ba99cbdaffb676966a25b3d7cc0635": "a_{\\alpha} u^{\\alpha} = 0", "86babc02dbf097b031f0c9e00515b02f": "\\alpha \\|u-u_h\\|^2 \\le a(u-u_h,u-u_h) = \\|u-u_h\\|_a^2 \\le \\|u - v\\|_a^2 \\le \\gamma \\|u-v\\|^2", "86bad0041ce5653abf8f2cd21e10aac4": "V \\ \\stackrel{\\mathrm{def}}{=}\\ \\rho^{i}(x,u)\\frac{\\partial}{\\partial x^{i}} + \\phi^{\\alpha}(x,u)\\frac{\\partial}{\\partial u^{\\alpha}}.\\,", "86bae0e592f0356c0ce7dde132c38f22": "\\textstyle K_{pub} = sP", "86bb429ae4f89fc350209a76049bdbf4": "\\lambda \\mathbf{x}_i^\\top \\mathbf{v}=\\mathbf{x}_i^\\top C\\mathbf{v} \\quad\\forall i\\in [1,N]", "86bb4ffd93de4d80864ee4731d63eb45": "g_1^q\\equiv1(mod P)", "86bc31a86267fafd48cdcd8c00d787bc": "H_X^P \\to H_X^P", "86bc942f27d966f5613e29a3d78f66e4": "|\\lambda|^k\\leq \\|A^k\\|", "86bca201f064be5c5988308462913a50": "\n\\sum_{n=1}^N \\frac{1}{n^2} < 1 + \\sum_{n=2}^N \\frac{1}{n(n-1)}\n= 1 + \\sum_{n=2}^N \\left( \\frac{1}{n-1} - \\frac{1}{n} \\right)\n= 1 + 1 - \\frac{1}{N} \\; \\stackrel{N \\to \\infty}{\\longrightarrow} \\; 2.\n", "86bcf67b8a7d747a7eb30defd28b1378": "\\mathcal{S}_{I}", "86bdcbb3f4ef129329263bdb869091a4": " \\scriptstyle\\mathbb{R}^n ", "86bdf20df900f441cdccf85cb6c161b3": "\\tilde\\triangle f = e^{-2\\varphi}\\left(\\triangle f -(n-2)\\nabla^k\\varphi\\nabla_kf\\right)", "86bdf7d3501cfd2d5ad0fa92b0a81624": "\\rho_c = \\frac{3 H^2}{8 \\pi G};", "86be00f002409478813ebeba14672704": "C = E(P)", "86be5ee6414a1c3d58c946ae4314264d": "\\{x_1r_1+\\dots+x_nr_n \\mid n\\in\\mathbb{N}, r_i\\in R, x_i\\in X\\}\\,", "86be836669817e2e1980ed10fb25572e": "J_z = \\int_A \\rho^2\\,\\mathrm dA = \\int_A (x^2 + y^2)\\,\\mathrm dA = \\int_A x^2\\,\\mathrm dA + \\int_A y^2\\,\\mathrm dA = I_x + I_y", "86bea1b990616f5e5eaef6322ea58775": "\\chi'' = \\left( 1 + \\frac{2\\chi}{\\mathfrak{M}^2} \\right)^{-1/2} - e^{-\\chi}", "86beda289cf75c2acaed4e709edec745": "\\log_{10} 10000 = 4", "86bee43ccbdda3b8bfd0b1fbef16e926": "(J^k_{x_0}f)(z)=f(x_0)+(Df(x_0))\\cdot z+\\frac{1}{2}(D^2f(x_0))\\cdot z^{\\otimes 2}+\\cdots+\\frac{D^kf(x_0)}{k!}\\cdot z^{\\otimes k}", "86bf03762981b3680d314ce9e08c69d1": "int(A)", "86bf126cdbf78e4b50c1ff1fe2400cfe": "a\\ne b = c \\ne d, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ, \\zeta = 120 ^\\circ", "86bfd76cd736dd6bd9b0dc5c92dbc7be": "G_{}^{}", "86c00578f42354f5423ec2bc71738967": "\\operatorname{\\Gamma L}(V) = \\operatorname{GL}(V) \\rtimes \\operatorname{Gal}(K/k).", "86c00e59689d6da165005d2dd66a6e8e": "\\frac{d}{dz}\\log(f(z))=\\frac{f'(z)}{f(z)}", "86c0767b6d29fc731cf396fd73ec31a1": "\\psi_n(y) = \\sqrt{\\frac{2}{L}} \\sin{\\left(\\frac{n \\pi y}{L} \\right)}, \\,", "86c09b900e208a876d42fe749aa38f64": "d_v V - E + d_f F = 2 D.", "86c0cd1791d92e85c9b20d0cd0150f00": "\\overline{u'v'}", "86c0cfa660a397133a43d1a3632cd9aa": "K(a,b;t)=\\langle x=a|e^{-\\frac{i\\mathbb{H}t}{\\hbar}}|x=b\\rangle =\\int d[x(t)]e^{\\frac{iS[x(t)]}{\\hbar}}.", "86c0d9fbae0b97bcce386c44cc21bc1b": "\\displaystyle \\eta_t + \\alpha \\eta \\eta_x + \\int_{-\\infty}^{+\\infty} K(x-\\xi)\\, \\eta_\\xi(\\xi,t)\\, \\text{d}\\xi = 0", "86c12e05bbb44538b67acc34800ed244": "{c \\textrm{~is~a~constant~of~type~} T}\\over{\\Gamma\\vdash c \\Rightarrow T}", "86c15531669ce0ff2a81bf9810a700ba": "D = \\{ x^2 + y^2 \\le 4 \\}", "86c159674b48281e7c211ab76512ed12": "\n\\left(-\\frac{\\hbar^2}{2m}{\\partial^2\\over\\partial\\mathbf{r}^2} + V(\\mathbf{r}) + {4\\pi\\hbar^2a_s\\over m}\\vert\\psi(\\mathbf{r})\\vert^2\\right)\\psi(\\mathbf{r})=\\mu\\psi(\\mathbf{r}),\n", "86c19035d5e23ab295dac44825deda9a": "\\mathbf{P}=\\mathbf{T}{v}", "86c196c2a753533f3b4d8b98d3ccbdb1": "K \\approx n_1 + n_2 - 3.48 ", "86c1980eea8958179b9ec01461ca8b0b": "V_{ACDA} \\approx \\sqrt{529.8+ 20.9 d_{ACDA} } - 23.0", "86c1c573dc38193f71673f3943f88639": "\\displaystyle{\\dot{\\mathbf{v}}\\cdot \\dot{\\mathbf{v}} =1,\\,\\,\\, \\ddot{\\mathbf{v}}\\cdot \\dot{\\mathbf{v}} = 0.}", "86c1fa0450f3e739df6e1821dd93a11b": "\nm(\\varphi)=6367449.146\\varphi\n -16038.509\\sin 2\\varphi\n +16.833\\sin4\\varphi\n -0.022\\sin6\\varphi\n +0.00003\\sin8\\varphi\n", "86c28ba7f9ebd958610d1f5389d95faf": "S_5 \\ge A_5 \\ge \\{e\\}", "86c2a3796d200e63e0de45a1b3818b31": "(R_i)", "86c2ad554166012ec407608b3a372515": "\\Pi_{(A:\\mathcal{U})} A\\to C", "86c2c88a08c838a3da9c145afb826e26": "\n \\mathbf{b}^1\\cdot\\mathbf{b}_1 = A~\\mathbf{b}_1\\cdot(\\mathbf{b}_2\\times\\mathbf{b}_3) = AJ = 1 \\quad \\Rightarrow \\quad A = \\cfrac{1}{J}\n", "86c30b4f44bdea34d27474547a9aaa56": "E_c=33w_c^{1.5}\\sqrt{f'_c}", "86c315fc0c5b9c89fcacc75b9de758fb": "\\mathrm{C_6H_6} + \\,n\\ \\mathrm{CH_3Cl} \\rightarrow \\mathrm{C_6H}_{6-n}(\\mathrm{CH_3})_n + n\\,\\mathrm{HCl}\\,", "86c32e61103dd2e080a17ce28b66b58d": "E = h\\nu.\\,", "86c365d14908c7cce4ba99e9fb4425c6": "\\gamma_{SG}\\ =\\gamma_{SL}+\\gamma_{LG}\\cos{\\theta}", "86c3a3ac1efb0ca10d426aace9803e6c": "B = \\alpha_1 \\alpha_2 + \\alpha_1 \\beta_1 + \\alpha_1 \\beta_2 + \\alpha_2 \\beta_1 + \\alpha_2 \\beta_2 + \\beta_1 \\beta_2\\,", "86c487ce36c60d750f7b9a5aed49581a": "E = \\int_\\text{magnet 1} \\mathbf{M}_1\\cdot\\mathbf{H}_\\text{d}^{(2)} dV.", "86c4a923615626e5a126a4dc43514457": "w_t = (1- \\eta x_t)/x_t", "86c4acb8b43a7f2ee4e32cf997d98c60": "\\min_{\\lambda\\in\\sigma(A)}\\left|\\frac{\\mu}{\\lambda}-1\\right|=\\min_{\\lambda\\in\\sigma(A)}\\frac{|\\lambda-\\mu|}{|\\lambda|}\\leq\\kappa_p (V)\\|A^{-1}\\delta A\\|_p", "86c5aeb1ec8e0c2ac8ee28913211223f": "p(\\tilde{x}=i) = \\frac{{\\alpha_i}'}{\\sum_i {\\alpha_i}'}", "86c63adad2ebb44724cbde0790fac63c": "\\hat{H} (x)", "86c653d8d5ec3e66a30a21a68485db23": "f(X\\mod P)=a", "86c6703b3dd53dd1a6616c04caea34c1": "\\neg \\text{open}_0", "86c6a61f62fe5c19471d511feaaf2d85": "A \\succ 0", "86c6c0afb7f1d874618a9009fdd8b953": "\\det(H)=a_n\\, n^{-1/4}(2\\pi)^n \\,4^{-n^2}", "86c70ffea114d8e243e7dee3bf347c72": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{0}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "86c7217946527f14259e165db4c56e08": " Fr_1 = \\frac{15}{3.75 \\times \\sqrt{(32.2)(3.75)}} = 0.36", "86c7604c411ae27dec3540d4c22fa7ef": "\\Psi_c\\,", "86c78be48242b65273a609f980a79e3e": "\\mathfrak{h}^*_\\tau", "86c7d0f26f467c3022501dc936936f96": "W = \\int_{t_1}^{t_2}\\mathbf{T}\\cdot\\vec{\\omega}dt.", "86c7d5236ec1d33175c41d4cc3bd5b66": "\\psi_1(-L/2) = \\psi_2(-L/2) \\,\\!", "86c7fdfea7150e43394b0c3557223653": "\n\\Xi_1(a_1,a_2,b,c;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a_1)_m (a_2)_n (b)_m} {(c)_{m+n} \\,m! \\,n!} \\,x^m y^n ~,\n", "86c81661580e0dc7652f61a85c659ed6": "w_{t+1} - w_t", "86c83bfd9cd7a1ff615c03fc104c0d2d": "\n\\frac{d}{d E_1} \\Omega = \\Omega_2 (E_2) \\frac{d}{d E_1} \\Omega_1 (E_1) + \\Omega_1 (E_1) \\frac{d}{d E_2} \\Omega_2 (E_2) \\cdot \\frac{d E_2}{d E_1} = 0.\n", "86c845094c70fffbc33e397638b8c7fc": "\\scriptstyle a \\,-\\, b", "86c8c5817e0af321ce614845c7e9a8b2": "\\operatorname{Hom}(A, \\text{--}), \\operatorname{Hom}(\\text{--}, A),", "86c8c8e825b137074ceeab983e006089": "\\;\\;\\quad \\rho=\\sqrt{r^2-2Mr+Q^2}\\,\\sin\\theta\\,,\\quad l_+ l_-=(r-M)^2-(M^2-Q^2)\\cos^2\\theta\\,,", "86c9102398d3c531a0271d89857f90d5": "\\subset B", "86c9adfccde4e57b3778a5381a5eba1d": "\\scriptstyle X_t ", "86c9c16bc74ab54bdd78dd0f9c375354": "\\scriptstyle (a\\, \\mid\\, b)^*a", "86ca1cf9114a71b7d79db423356bfe69": "\\tau_{ij}=\\lambda\\delta_{ij}e_{kk}+2\\mu e_{ij}", "86ca6d056f115ab757e7ea69b7d952d7": "\\Omega = \\bigcap_{j=1}^{n} \\Omega_j", "86caa23f8c0514dc35067b60ffaf61b2": "{{u(X)}^{*}} = \\sum\\limits_{i=1}^N \\alpha_i\\phi \\left( r_i \\right),\\qquad(4)", "86cb7ab5367893e360337e4af5e0bd2f": "F_{A0}", "86cb83e9b0a2d5fe6937a7965a4fe936": "\n \\boldsymbol{S} = \\phi^{*}[\\boldsymbol{\\tau}] ~;~~ \\boldsymbol{\\tau} = \\phi_{*}[\\boldsymbol{S}]\n", "86cb9a54525dce8af30297d62ecb3192": "\\delta_k>0", "86cba6967dacc169104f2f51c133712a": " \\alpha>=d-1 ", "86cbaf1d9b5bda3f33e594389f6df308": " x^{m+1} - x^m - x^{m-1} - \\dots - x - 1 = 0 ", "86cbc72590fede3f001bc3752e771caa": "\n\\begin{matrix} X_k & = & \\left\\{\n\\begin{matrix}\nE_k + e^{-\\frac{2\\pi i}{N}k} O_k & \\mbox{for } 0 \\leq k < N/2 \\\\ \\\\\nE_{k-N/2} + e^{-\\frac{2\\pi i}{N}k} O_{k-N/2} & \\mbox{for } N/2 \\leq k < N . \\\\\n\\end{matrix}\n\\right. \\end{matrix}\n", "86cbff00ea34242607652f75292e27aa": "v_i(x) := s_i\\, u_i(x) + r_i", "86cc096ecb16cd9cbd5f50794576a380": "W_x(t, f)= \\int_{-\\infty}^{\\infty}R_x(t, \\tau)e^{-j2\\pi f\\tau}\\, d\\tau,", "86cc132b0c4c21a6d9499b7d21596b54": "\\forall i, j \\in \\mathbb{N} \\backslash \\{0\\}: i,j \\leq n,", "86cc19515a215faa8f2787bb56fb872c": "s(m,n)=(n,n)", "86cc235e1ee8d2a8a1e1a8997665eca6": " -p(x_1)u'(x_1) + a_1 u(x_1)=0, \\quad \\hbox{and} \\quad p(x_2) u'(x_2) + a_2 u(x_2)=0.\\,", "86cc327477edbef7964cb2139ddcbcf2": "Y_{p} = (C_{0} + I_{0})/(1-c)", "86cc7552c1b3c5bf24cfcc5e63359bd2": "B(aq) + H_2O(l) \\leftrightarrow HB^+(aq) + OH^-(aq)", "86cc9c04d7099cc1a4fed4d635d051d7": "\\hat h_L", "86ccc531047c210db84bd3b29abbf8db": "\n\\bar{x}^j = \\bar{x}^j(x^i)\n", "86cccf1d0cb2a971ce9fd4b580960499": " U=L^T,\\,", "86ccd68fd8bbbf5d0b53a52702446d48": "x=r(t-\\sin(t))\\,", "86cd30d8747e6ffd96710f2c6ff75221": "\\eta_Y \\circ F(f) = G(f) \\circ \\eta_X", "86cd58135697b5de24b08c8f0749c109": "\\ddot{\\vec x}(t)=A(\\vec x(t))", "86cd58e33f957aa87096dac92c48d83f": " \\mathbf{A\\times}\\left(\\mathbf{B}\\times\\mathbf{C}\\right)=\\left(\\mathbf{A}\\cdot\\mathbf{C}\\right)\\mathbf{B}-\\left(\\mathbf{A}\\cdot\\mathbf{B}\\right)\\mathbf{C} ", "86cdacd46642595da9ceb5f2ab695a6f": "\\rho = \\frac {M P}{R\\,T}", "86ce23cd8b50b281caac02eea9c9c337": "\n\\begin{matrix}\n\\Phi^2&\\Phi^A_B \\Phi^B_A\\\\\n\\Phi^3&\\Phi^A_B \\Phi^B_C \\Phi^C_A\\\\\nH_d H_u&{H_d}_A H_u^A\\\\\nH_d \\Phi H_u&{H_d}_A \\Phi^A_B H_u^B\\\\\nH_u \\mathbf{10}_i\\;\\mathbf{10}_j&\\epsilon_{ABCDE} H_u^A \\mathbf{10}^{BC}_i \\mathbf{10}^{DE}_j\\\\\nH_d \\mathbf{\\bar{5}}_i\\;\\mathbf{10}_j&{H_d}_A \\mathbf{\\bar{5}}_{Bi} \\mathbf{10}^{AB}_{j}\\\\\nH_u \\mathbf{\\bar{5}}_i N^c_j&H_u^A \\mathbf{\\bar{5}}_{Ai} N^c_j\\\\\nN^c_i N^c_j&N^c_i N^c_j\\\\\n\\end{matrix}\n", "86ce28521b25bb9bc9cd8cb37d036465": "e, e'\\in E", "86ce2db62cf3e16261aa3dfe241221a9": "u_8 = \\tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+ax_8^2)x_{16} - 2x_8(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +bx_8 x_{16})}{c}", "86ceaf088122369a25c36c8a653eaf65": "1-(1-a)^k (1+ a k) ", "86cef5ac9f6c862c70bf8e27aa9f0a83": "\\displaystyle P(w,x|d) = \\sum_{z=1}^Z \\sum_{c=1}^C P(w,x|c,z)P(c)P(z|d)", "86cf2228d841fbd9d395e362c4a0c18c": "ce^{i\\psi}=i b\\sin\\alpha e^{i\\theta}+a\\cos\\theta\\ e^{i\\theta} - i a\\sin\\theta\\ e^{i\\theta}=(a\\cos\\theta\\ +i(b\\sin\\alpha-a\\sin\\theta)e^{i\\theta},", "86cf32d084c2b5431fc69905511eebf3": "\nS_0[p] =\n\\begin{bmatrix}\n(I_x[p])^2 & I_x[p]I_y[p] \\\\[10pt]\nI_x[p]I_y[p] & (I_y[p])^2\n\\end{bmatrix}\n", "86d01fbd4cfb85b47e9390fae432a593": "r d\\theta\\,", "86d02316311a132408ec20e9bfe88a93": " \\|f\\|_{A^p}^p = {1\\over \\pi} \\iint_D |f(z)|^p\\, dx\\,dy", "86d024412ef8ca2ef2fb1a6fdcb0c6c0": "k(\\mathbf{x_i},\\mathbf{x_j})=\\tanh(\\kappa \\mathbf{x_i} \\cdot \\mathbf{x_j}+c)", "86d02bee7ed9f0d7de514151ecc726fe": "T^{\\mathrm{H}}_p", "86d0378be94db93cd1247aa16b54b01c": "{d\\tau}^2 = - \\eta_{\\mu \\nu} dx^\\mu dx^\\nu \\,", "86d089c4cd0c12f98c26c289d52f2b49": "\\ p", "86d203ca589b8fcea5b07d9a5c0adb30": "f^{(n)}(x) = \\begin{cases}\\displaystyle\\frac{p_n(x)}{x^{2n}}\\,f(x) & \\text{if }x>0, \\\\ 0 &\\text{if }x \\le 0,\\end{cases}", "86d2183915679fd26215e7ea90679c8e": "{B_i = \\hbar / 4 e l_i^2}", "86d232ab7bcf9684f6ef01fd688466eb": "n > d_1^i + \\cdots + d_m^i. \\, ", "86d2e178469d244832bb26b72b0b7524": "\\tilde{Q}(\\omega)", "86d40a645abd8963f984ceee253a65ed": "T = \\mbox{either }\\sum_{y=0}^\\infty \\ell_y v_y\\mbox{ or } \\int_0^\\infty \\ell_x v_x\\,dx.", "86d4fba3d993670071835a9f9b564429": "\\boldsymbol{P}", "86d52135c98c0c598c585abb81eeb63f": "\n\\nabla (\\varepsilon (r)\\nabla \\phi (r,t))\n= -(\\rho_\\text{ions}(r,t) + \\rho_\\text{perm}(r))\n", "86d52b043455fe50603244d84398c679": " (\\alpha f)(x)=\\alpha f(x).\\,", "86d52b18dab6c67c5923690dfa4fe11b": "n=1+C_D/C_{OX}, \\, ", "86d5521eb779bde723874af390971dd7": "\\begin{bmatrix}.7&0\\\\.3&1\\end{bmatrix} ", "86d567a3fa958e239e942a2d448636f0": " M \\oplus F = G . \\, ", "86d5722eabafe008fcccd370e7c97da4": "\\theta \\approx \\pi", "86d5d628fa27106c7b9b55632ab1881d": "a_2= k_1(x_2-x_1)-(y_2 - y_1)=-3.3750", "86d5feac679b1fc432ace5f1f45c0c96": "\\frac{1}{(R-x)^2}+\\frac{1}{(R+x)^2}=\\frac{1}{r^2}", "86d60596e469eed9c46c4249fea311d8": "C_\\epsilon (\\Delta)", "86d61b5eec1c67debb1fb12728633c37": "\\mathbf{a}\\cdot\\mathbf{\\Delta k}=2\\pi h", "86d69efdb9ff8fec8da7fb4c95f7fb90": "\\|A\\|^2_{HS}={\\rm Tr} |(A^{{}^*}A)|:= \\sum_{i \\in I} \\|Ae_i\\|^2 ", "86d6ad21f716361920843e9892b4915b": "I =I_0{{\\left(\\frac{\\sin \\left(\\frac{\\pi a}{\\lambda } \\sin\\theta \\right)}{\\frac{\\pi a}{\\lambda }\n\\sin\\theta }\\right)}^2}{{\\left(\\frac{\\sin \\left(\\frac{\\pi }{4} N \\sin\\theta+\\frac{N}{2} \\phi \\right)}{\\sin \\left(\\frac{\\pi }{4}\n\\sin\\theta+ \\phi \\right)}\\right)}^2}", "86d6ad8d45ee497e0b8b2758fe999f99": "L_1 - L_4", "86d6c9cfed7b6fe18e846bfd38377f86": "f_n:\\R \\to \\R", "86d6eb27b28741011dc4b73315415c44": "m^2\\phi^*\\phi", "86d73cd90fc95a0f1ac2c87fdde9655d": "= -8/4=-2 \\,", "86d78499e0f4b14065e0ce0f15484160": " E' = \\left( 1 + \\frac{\\epsilon_0 }{2m_0 c^2 } \\right ) \\epsilon_0 \\quad \\mbox{and} \\quad V' = \\left( 1 + \\frac{\\epsilon_0 }{2m_0 c^2 } \\right ) V ", "86d7b29c01a33e609dd22dafc45063a4": " \\dot{G}_i = 0, \\qquad \\dot{\\varphi}_i = F(G), ", "86d7c8e98f8c5a321cc87109ace6c55f": " H^1(k,G)\\rightarrow\\prod_s H^1(k_s,G)", "86d7cee9f76b4a3a36aa12c1e4bb7a27": "\\varphi(1),\\ldots, \\varphi(r)", "86d7ffd50e192299cb9d1b463d3e0feb": "\\smile \\frown \\wr \\triangleleft \\triangleright\\!", "86d87095974bc7b71ed8857c885ed20f": "G_X (\\Beta(\\alpha, \\beta) )=G_{(1-X)}(\\Beta(\\beta, \\alpha) ) ", "86d8a9929f5dde51d0723b94fe198fdf": " \\left [ \\nabla^{(2)} \\right ]_{\\alpha \\beta} = \\frac {\\partial^{\\,2}} {\\partial r_{\\alpha} \\, \\partial r_{\\beta}} \\qquad \\qquad \\text{where} \\quad \\alpha, \\beta = 1, 2, 3 \\, . ", "86d8ca9b3f33b62a7b54432509dc0cc8": "\\frac{\\partial |\\mathbf{X}|}{\\partial \\mathbf{X}} =", "86d8d92aba9ecf9bbf89f69cb3e49588": "GG", "86da303d0bae4c2ee7421a60fd625c89": "x^r", "86da5860ffff29fcbb48213c54a1c1b4": "C^3", "86da58805da22ab24a587f240cc74dbf": "((\\sigma_{ij} - \\bar{\\sigma}_{ij}) \\mathbf{\\hat{n}}) \\cdot \\mathbf{\\hat{n}} = -\\gamma \\nabla_{\\!S} \\cdot \\mathbf{\\hat{n}}", "86da6de1331f2b5d34d59eb9aa5073f6": "I = {V \\over R}", "86da7a73c5c1354e604458d6ddd0924c": "T_0 = T_0(\\varepsilon) > 0", "86dad6aca9d21c3bb25882cb06c16b25": " \\left(g_{ij}\\right) = \\begin{pmatrix}g_{11} & g_{12} \\\\g_{21} & g_{22}\\end{pmatrix} =\\begin{pmatrix}E & F \\\\F & G\\end{pmatrix}", "86db3c3f7f7220f7801f110cf0a45b78": "\\frac{|v^{+}\\rangle - i |w^{+}\\rangle}{\\sqrt{2}} \\frac{|v^{-}\\rangle - i |w^{-}\\rangle}{\\sqrt{2}}=\n\\frac 1 2\\left(|v^{+}\\rangle|v^{-}\\rangle - i |v^{+}\\rangle|w^{-}\\rangle-i|w^{+}\\rangle|v^{-}\\rangle-|w^{+}\\rangle|w^{-}\\rangle\\right).", "86db3d24cc3328c8fef94531e7028554": "1/{\\eta}<1", "86db40df86598737fc47047fe56f6518": "\\boldsymbol\\tau = \\mathbf{r}\\times \\mathbf{F} = \\mathbf{r}\\times (m \\boldsymbol\\alpha \\times \\bold r) = (mr^2)\\boldsymbol\\alpha = I\\alpha \\bold e,", "86db5659dfa8dbe49aa772353f57a9d6": "\\wp(z) =\\frac{1}{z^2} + z^2 \\sum_{k=0}^\\infty \\frac {d_k z^{2k}}{k!} =\\frac{1}{z^2} + \\sum_{k=1}^\\infty (2k+1) G_{2k+2} z^{2k}.", "86db62bad22e3c996f5409964cdb00ec": "\\oint_{C} {f'(z) \\over f(z)}\\, dz=2\\pi i (4-5)", "86db6a093fa01480673a49c130e35d48": "\\dot{y}(t) = ay(t-1)", "86dbc28a2eef3c7e69f185cdf269c6e1": "d(f(0), f(z)) = \\rho (0, z)", "86dc1242b55b20ba2541a5b997517e43": " \\mathbf{n} = \\frac{\\mathbf{b} \\times \\mathbf{d}}{|\\mathbf{b} \\times \\mathbf{d}|} ", "86dca1d6d1cd42d3a37b3f1c9681dfb8": "\nE(V) = E_0\n + \\frac{ K_0 V }{ K_0' } \\left( \\frac{ (V_0/V)^{K_0'} }{ K_0' - 1 } + 1 \\right)\n - \\frac{ K_0 V_0 }{ K_0' - 1 }. \\qquad (8)\n", "86dd0e4d349b92d9a4a9a1e96bf942d5": " = \\sum_{n=0}^\\infty \\left(\\sum_{i=0}^n a_i b_{n-i}\\right) (x-c)^n.", "86dd7d69cce267ed2471a57f85857124": "\\mathbf{\\tilde{H}}=\\mathbf{B}^{-1}\\mathbf{H}", "86de3684776c74417d6df57db1131f32": "k_B T = {m\\overline{v^2}\\over 3} ;", "86de5efe031e12e851c7dd0c8c2186bf": "f(A) = \\frac{1}{2\\pi i} \\oint_C {f(z)(zI-A)^{-1}}\\, \\mathrm{d}z. ", "86de7509a9c4e12228466cc5117b08d9": "a_0 ", "86dea90e788dff8b44449fd9e6d49563": "0 \\to \\mathcal F(U) \\to \\prod_{i \\in I} \\mathcal F(U_i) {{{} \\atop \\longrightarrow}\\atop{\\longrightarrow \\atop {}}} \\prod_{i,j \\in I} \\mathcal F(U_{ij})", "86deba63540688127fa237bcbe324b77": "\\rho \\bar{u_j}\\frac{\\partial \\bar{u_i} }{\\partial x_j}\n= \\rho \\bar{f_i}\n+ \\frac{\\partial}{\\partial x_j} \n\\left[ - \\bar{p}\\delta_{ij} \n+ 2\\mu \\bar{S_{ij}}\n- \\rho \\overline{u_i^\\prime u_j^\\prime} \\right ].\n", "86ded479a049228f7e75a0c3c6377fff": "q(t) = \\int_0^t i(\\tau)d\\tau = {1 \\over \\mathrm{R}}\\int_0^t {f_{em}}(\\tau)d\\tau = {1 \\over \\mathrm{R}} [\\Phi(0) - \\Phi(t)]", "86dee47081b949516f090a81ab663945": "\n \\frac{(k+r-1)\\cdots(r)}{k!} = (-1)^k \\frac{(-r)(-r-1)(-r-2)\\cdots(-r-k+1)}{k!} = (-1)^k{-r \\choose k}.\n \\qquad (*)\n ", "86def9ea2833e4211b4025d8025e3a49": "00g_1, \\cdots, 00g_{k_2}, 01g_{k_2}, \\cdots, 01g_1, 11g_1, \\cdots, 11g_{k_2}, ", "86deffc02c3988a1c42630da950cfa90": "\\beta_F = \\frac{I_{\\text{C}}}{I_{\\text{B}}}", "86df16fda134e9ea6ee27ca7d29f14cb": " p_k ", "86df5dc871f2ec639b93bde2a33c93b2": "Current Account = (Private Savings - Investment) + (Tax - Government Expenditure)", "86e0167741ca345120427ad12ec10994": "R_O = \\frac {V_x} {I_x} = r_O \\left( 1+ \\frac { \\beta R_2} {( R_1 \\parallel r_E ) + r_{\\pi} +R_2} \\right) ", "86e039f133a5678bbd54912949e84749": "H=H_\\mathrm{sys}+H_B+H_\\mathrm{int}", "86e0851909d5598cc67d68aa8fe771a0": "P(\\emptyset)=0", "86e0a253a29d4682b563e4311ebcae2e": "\\epsilon^T\\Lambda^T\\epsilon=\\epsilon^T\\Lambda\\epsilon", "86e0c1602bd3b88805c58b840f7ad3d2": "\\ H\\Psi = E\\Psi", "86e18059b298591bd2fcaf2746b5faa4": "u(x,t)=\\int_{0}^{t}\\int_{0}^{\\infty} \\frac{1}{\\sqrt{4\\pi k(t-s)}} \\left(\\exp\\left(-\\frac{(x-y)^2}{4k(t-s)}\\right)-\\exp\\left(-\\frac{(x+y)^2}{4k(t-s)}\\right)\\right)\nf(y,s)\\,dy\\,ds ", "86e1c67b2bdc8f14c04ee1d5fd962840": "G_k \\subset G", "86e21c673f090180c932e803b70c66d7": "g(v, w) = g' \\left( f_{*} v, f_{*} w \\right). \\, ", "86e2231cfe5e41fd3fb60aff412b43cf": "\\mathbf{P}^\\infty(\\mathbf{R}) = \\mathbf{R}^\\infty/\\mathbf{R}^*", "86e22d51f7f42668d02071be841e63e2": "h_{i+1}(x)", "86e272c79a618a1ce719a9ac0cc2d02d": "y'' = -y.\\,", "86e2892286a632c8ba5101580dfc5f54": "H_\\mu \\, ", "86e34b74b09f8debabc0d4da6afe05f6": "\\overline{\\rho}", "86e351e09e1527ad9e8d5fa5c4ab7b4a": "\\mbox{E}", "86e359cace00d93fac42fca77e4b25df": "\\mathbb{H}/\\Gamma,", "86e381f0f0789730f0120e8116117def": "\\frac{73,550\\ \\mathrm{N}}{(445\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=16.85", "86e3b3d90d6c00a8b244c4320ffe37e9": " \\mbox{SWAP} = \\begin{bmatrix} 1&0&0&0\\\\0&0&1&0\\\\0&1&0&0\\\\0&0&0&1\\end{bmatrix} ", "86e3c5976e518ace481a1cf93ce4879b": "w^\\ast", "86e3f833f8273d89ccfb4c046369a1b1": "\n\\begin{matrix}\n\\mbox{independent}\\qquad\\\\\n\\underbrace{\n \\overbrace{\n 3,\\quad\n \\sqrt{8}\\quad\n },\n 1+\\sqrt{2}\n}\\\\\n\\mbox{dependent}\\\\\n\\end{matrix}\n", "86e406c2ba13322cf8d4faadc939632b": "R_{i-1} < u R_N \\le R_i", "86e42e967cdf841b06570689e5d7d920": "\\sigma'_{ij}=a_{im}a_{jn}\\sigma_{mn} \\quad \\text{or} \\quad \\boldsymbol{\\sigma}' = \\mathbf A \\boldsymbol{\\sigma} \\mathbf A^{T},", "86e437df3161cc87a67f94f2e9176c71": "Y = (Y_1,\\dots,Y_n)", "86e46b3576b5cdec095a1088c56f5eff": "\\left(a, r, w\\right)\\succsim \\left(c, s, v\\right)", "86e476bd5935dad20983a9ea3c024933": " {{{K \\choose k} {{N-K} \\choose {n-k}}}\\over {N \\choose n}} = {{{n \\choose k} {{N-n} \\choose {K-k}}}\\over {N \\choose K}}.", "86e58d2fdcadbe27460fd1db5afc56ea": "\nL_A(\\Omega)=\\left\\{ u\\in M_f(\\Omega):\\|u\\|_{A,\\Omega}=\\inf\\{ k>0:\\int_\\Omega A\\left( \\frac{|u(x)|}{k} \\right)~dx\\leq 1 \\}<\\infty \\right\\}.\n", "86e5a969f4777467f2ff615620f48a57": "M_n=\\max\\{X_1,\\ldots,X_n\\}", "86e5e9bde8be1bc2f2791877a91818cc": " H_{\\frac{1}{6}} = 6-\\tfrac{\\pi}{2} \\sqrt{3} -2\\ln{2} -\\tfrac{3}{2} \\ln{3}", "86e63dc6c79490c90bf653cdefd59891": "a_m + b_m(Y - T)", "86e66312d234b4e2a7008b17a7412da4": "\\mathbf{D}", "86e67cfba6f380ba43210a2aa07cbf76": "\\frac{\\mathrm{d}U}{\\mathrm{d}t} = \\dot Q - \\frac{\\mathrm{d}(pV)}{\\mathrm{d}t}.", "86e6e31ecba92abde13c4cf3e1571b7a": "zx\\leq zy", "86e6eccefea977ae70bf9ccc33e81130": "e^{-2\\alpha t}", "86e6f0a333933a602c7c4c962aac0ac4": " \\overrightarrow{Q_o} ", "86e709e4a7b5a93f2146b3ea7a6ab390": "Fluid \\ required \\ in \\ first \\ 24 \\ hours = (4 \\cdot Persons \\ weight \\ (kg)) \\cdot Percent \\ body \\ surface \\ area \\ burned", "86e749abc597e906820ee6883590b38a": "\\begin{align} e & = \\sum_{n = 0}^\\infty \\frac{1}{n!} \\\\ \n& = \\frac{1}{0!} + \\frac{1}{1!} + \\frac{1}{2!} + \\frac{1}{3!} + \\cdots \\\\ \\end{align}", "86e77f69e89200fb6178862204951a90": "\\mu_{HB}", "86e84c33aecc920a2d0df48431f37c04": "\\frac{1}{2^{2m}(2m-1)!} \\left[\\psi_{2m-1}\\left(\\tfrac{j}{2p}\\right)+(-1)^q\\psi_{2m-1}\\left(\\tfrac{j+p}{2p}\\right)\\right] ", "86e874d9cc74544ece12f21d4b701eed": "\n C\\geq\\frac{\\sqrt{10}+3}{6\\sqrt{2\\pi}} \\approx 0.40973 \\approx \\frac{1}{\\sqrt{2\\pi}} + 0.01079 .\n ", "86e89592b6e10690a882b0b5e9394e37": "CA=(X-M)+NY+NCT", "86e8a1d43f386a6f8c46fb473cc643da": "X_n \\subset L^1(\\mu)", "86e8b371e8e97721503be8c7eba99530": "K/n", "86e8cf55ae1e4c62c86097eefec0cb71": "n_k", "86e938e482d54915f795eb158e1c6d93": "\\cap_k U_k = \\pi(p)", "86e96e7534306b58d4036c453a7290f5": "A = \\begin{pmatrix}\n12 & -51 & 4 \\\\\n6 & 167 & -68 \\\\\n-4 & 24 & -41 \\end{pmatrix}.", "86e976885c5ee43acc43157a32ac5e30": "X = BS", "86ea3c5f68036f893b8bc0bcbc3a7dad": "\\Pr \\left[ X > x \\right]=x^{-3}\\mbox{ for }x>1,\\ \\Pr[X<1]=0", "86ea4c386034b34f0ca0d9e602131569": "\\Pr(z \\mid d)", "86eaba9291523c7ae9d1415161e481f4": "A \\ang \\theta.", "86ead65e78489a688682c6eb0a866fb7": "(\\alpha\\, , \\, \\beta)", "86eb01bf23bc5a12c668226244870469": "p^\\mu=\\int d^3x\\, T^{\\mu 0}(\\vec{x},0)", "86ebf3b65140cad269853c3ebceb3350": "{\\frac{1}{k!}}c_{0}\\int^{\\infty}_{0} f_k e^{-x}\\,dx = c_{0}[(-1)^{n}(n!)]^{k+1} \\qquad \\mod (k+1).", "86ec10a9641650c37941396723f35c97": "\n\\Sigma (E) = \\int_{H < E} d\\Gamma.\\,\n", "86ec7b1c0792ab5b9d2ed7cd29460684": "v_s^2", "86ec8c09899653201a4c4b8a4d28d0ca": "(\\exists x.\\ [x=x]) \\rightarrow \\exists x.\\ [D(x) \\rightarrow \\forall y.\\ D(y)]", "86ed34d0d82a24b59330851d4734c723": "f(u,v) \\ge 0", "86ed3caf41866add512e88d10dc8d7ab": "\\vec k", "86ed907117fafdd328a551652dffa199": "A\\mathbf{x}=\\mathbf{b}", "86ede7086cda6eda1e2fafb1c7b40723": "f_M (v) = \\frac{4}{\\sqrt{\\pi}}\\frac{v^2}{v_p^3}\\exp \\left (-v^2/v_p^2\\right )", "86ee18bcd1ae4be237ca25c4f130f423": " n=dim({\\mathbf{x}}(t)) ", "86ee4036514fe3bb345c497b48e4e824": "U = e_U T", "86ee51cbfc8c5bb35a8f206196ef805b": "\\frac{[W + D-d] T^Q}{M N_a}", "86eee4e52ae5bf03bf0f9960a3e57ce8": "\\ V_m=\\pi R \\tan(\\theta/2)R \\tan(\\phi/2)(c\\tau/2)", "86eef2817d971ea33bef858711f0e0c9": "D(A) = \\big\\{ u \\in H^{2} ((0, 1); \\mathbf{R}) \\big| u(0) = u(1) = 0 \\big\\}.", "86ef34eee50a117ed3f2e44ff18a9db5": "i_D = \\frac{Q_D}{\\tau_T} \\ , ", "86ef8fe5773fb794a758ec7930855059": "r_2 = r_1 + tp^1 = 3+1 \\cdot 7 = 10 =13_7.", "86ef9bab4f9041d43493918729ff3ec5": " \\frac{dV}{V} = \\frac{2 \\omega_e R_e (\\cos \\varphi_i - \\cos \\varphi_o)}{c}", "86efb30d807d1d48270b014861770cb8": "\\nu_p:\\textbf{Q} \\to \\textbf{Z}", "86efc02ba1e99582d86e0d74cc6a610b": " R=\\frac{1}{1+k^1}", "86efd87ca0a0f66e30cc47836b3c875e": "R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\\,", "86f049a2b640bb581829a1e1619e63f6": "\\begin{matrix}{r \\choose 1}\\end{matrix}", "86f050861b59b4eb258bdc58be2b16ef": "|B|^{|A|}", "86f05952a9fea2a9703cdd8311c439d2": "\\varphi_m(x)", "86f06f8c7f52b6882ea498ec85ac1d63": "\\Box\\Diamond p \\rightarrow \\Diamond \\Box p", "86f07e149891702b24e6c04bcb41b5e7": "\\hat{\\sigma}", "86f0adfdbe9c50fc1f8373344b824786": " \\ K = -1+X^2-X^{10} ", "86f0fcea1428a7db952f25b84df4f78e": "\\begin{align}\n\\pi_0 (KO) &= \\mathbf Z\\\\\n\\pi_1 (KO) &= \\mathbf Z/2\\\\\n\\pi_2 (KO) &= \\mathbf Z/2\\\\\n\\pi_3 (KO) &= 0\\\\\n\\pi_4 (KO) &= \\mathbf Z\\\\\n\\pi_5 (KO) &= 0\\\\\n\\pi_6 (KO) &= 0\\\\\n\\pi_7 (KO) &= 0\n\\end{align}", "86f117f484064f4063477c9977d326cb": "\\|\\sigma\\| = \\| T\\| = 1", "86f1a74b96ca2d94ab9ade99f474abcd": "M_{x} = \\int\\rho c_{x2}dy", "86f1e0c7e578039552785fe46f740f3a": "x_0:=\\int_\\Omega g\\, d\\mu,", "86f2251cd083533f4a432e87f3a145a1": " N = \\frac{V }{8\\pi \\hbar \\omega}\\mid \\mathbf{E} \\mid^2 . ", "86f238960e3fce1182feb520bdc84da9": "{{V}_{CC}}-{{v}_{MIRROR\\_OUT}}", "86f2731a5fd77df21f376feef68e35ef": "\\eta_{sp} = [\\eta] c + k [\\eta]^2 c^2 + \\cdots\\,", "86f28d2da6025e06e07221dc875c2216": "\\begin{smallmatrix}L \\end{smallmatrix}", "86f2d5e0e8c975ed4e27dd0a75b5f342": "R=2\\times T", "86f2fa23a7f1816ad54f2a68a1d7e3af": " x_{i+1} = x_{i} -f(x_i)/f^{\\prime}(x_i) ", "86f3485d37903ed1852a1ab173c5c0a6": " M = \\mathop{Med} F ", "86f353f4b8d7d2d50e24c6589afa54e8": " S_{\\Gamma_1}= S_{S_{\\Gamma_0}} \\, ", "86f3e1147c85bda337ea9e2107aac88e": "\\vec{v}\\left(t + \\tfrac12\\,\\Delta t\\right) = \\vec{v}(t) + \\tfrac12\\,\\vec{a}(t)\\,\\Delta t\\,", "86f4035141132acdac7c4bd7db296b5c": "H(\\mathbf{R}_{ij}) = \\frac{\\mathbf{I}_i \\cdot \\mathbf{I}_j}{4} \\frac{\\left | \\Delta k_m k_m \\right |^2 m^*}{(2 \\pi )^3 R_{ij}^4 \\hbar^2} \\left [ 2 k_m R_{ij} \\cos( 2 k_m R_{ij} ) - \\sin( 2 k_m R_{ij} ) \\right ] ", "86f476e3405b2d87a5c6ced12fbdca4d": "\\mathrm{(Fe,Mg)_2SiO_4 + n H_2O + CO_2}", "86f49f10428aa114e8cc192fbc29ef1a": "\\text{charge} = \\text{capacitance} \\times \\text{voltage}", "86f4a10149f6cadef2f6e6e3cef399fd": "{13 \\choose 2}{4 \\choose 2}^2{11 \\choose 1}{4 \\choose 1} = 123,552", "86f4bb2d9ed983e782a08def82bd87f4": " \\omega \\in \\Bbb{R}^{3}", "86f4cf01aa4ef9a8d6f35c24aa53f1f6": "\\frac{P_{\\rm min}}{\\dot m} = \\int_1^2v\\mathrm{d}p.", "86f4d2e53f1937900040952b1b42dbb6": "f'(x_0) = K > 0", "86f51a5275d32972288fbea78bc82ba3": "\\color{red}\\land", "86f51b3c44764536d8b2ba5e6a01ceae": "\\omega_{c} = \\frac{1}{RC}", "86f51d84d1d2131f5d8332134d627ad2": "\\mathbf{x}_0, \\mathbf{x}_1, \\mathbf{x}_2, \\dots", "86f5261f410a20264d2421af9ab29d22": "(\\alpha \\to \\alpha) \\to \\alpha", "86f5a630e41a47c19cead3bcd8b27187": "\\eta_m", "86f5a7c758b366f84ed15a2472f6dd5e": "u\\left(z_1\\right)", "86f5bc24703239ac094ed4848424188f": "\\begin{matrix} {49 \\choose 2} = 1,176 \\end{matrix}", "86f63c6f84e39ad9afcb7c090cba1473": "a_j := [y_0,\\ldots,y_j]", "86f649d998e4e2dd6deaa88189cffa29": "X=X_1^2", "86f678cfd4387c0f615abf611b2e2341": "A \\ne B", "86f68ed54c474f9a7a6c95eaeac5cc9c": "\nh^2 f_n y_n = 2y_n-y_{n-1} - y_{n+1} - \\frac{h^4}{12} \\frac{f_{n-1} y_{n-1} -2 f_{n} y_{n} + f_{n+1} y_{n+1}}{h^2} + \\mathcal{O} (h^6)\n", "86f694d28225e9aabbe3d352e19c61a4": " \\alpha_1 = \\alpha_3 \\,", "86f6b341449f3f66a84edab9d8c89333": "\\operatorname{plus}(m, n)= m+n", "86f6c19bdb0aaea20e7cb1b3e26dd7a5": " {\\Delta H^o} = {RT^2}.\\frac{d \\ln K}{dT} ", "86f6cf9b97a8311f51188de1c2297595": "\\partial{T} ", "86f7015cb3691ad2408f6af87ddffd58": " (\\partial G)_H=-(\\partial H)_G=-V(C_P+S)+TS\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "86f78b2bca701cc3d0dd865907d330b7": " e^+e^- \\to \\omega \\pi,~~ \\eta \\pi \\pi,~~ \\phi \\pi ", "86f79306895b63d5963776bd76ee71dc": "-2y+x+z=0", "86f798ca600c4d2566d6b0d0358b4ba9": "BM=2y\\sin{\\theta}.", "86f79c944643bcc7087903a7bcafe3eb": "codegree", "86f83df0f82fa67982898507797395c4": " \\theta_{n+1}\\sim N(\\widehat{\\mu}_\\pi,\\widehat{\\sigma}_\\pi^{2}) ", "86f8b04b4a8368b0bd0ecd41b0242085": " 2\\,\\mathrm{Im}[\\hat\\chi(\\omega)] = \\omega\\beta \\hat A(\\omega).", "86f8cf6915fe9f1615828624afe9e034": "0\\le s(x) \\le 1/2", "86f909e643ef888402c0c1a583325f2e": "\n\\mathbf{a} \\cdot (\\mathbf{a} \\times \\mathbf{b}) =\n\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{a}) =\n\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{b}) = \n\\mathbf{a} \\cdot (\\mathbf{a} \\times \\mathbf{a}) = 0\n", "86f92ad4ed85a1a9c7c215cc88760d62": "\\textrm{Gal}(F_n/F)\\cong \\mathbb{Z}/p^n\\mathbb{Z}", "86f9976b409d72bf669860eb6f80c19c": "\\dot{\\varkappa}_a^b + \\xi^{-1} \\varkappa_a^b - {\\lambda_a^b}^\\prime = 0,", "86f9a3a2cc344b78b146c202b2226bae": "[-\\pi, \\pi]", "86f9bb7d7eed192ab7b1914d78fed07b": "Q_A = e\\gamma + Q_{CA} \\,", "86f9dcc6623fcd4e01411ca0c9a8cc2d": "w_i^+=w_i-\\sigma_ix_i\\Theta(\\sigma_i\\tau)\\Theta(\\tau^A\\tau^B)", "86fa5139304f744597644137496698ac": "\\partial \\Omega_D \\cap \\partial \\Omega_N=\\varnothing ", "86fa63e468c32a09eef79dd037818d10": "q_{on1}", "86fa8c04db1774d292a8839badd1667d": "MD(\\varphi \\wedge \\psi) = max(MD(\\varphi), MD(\\psi))", "86fb2854fff4e8fe481edef4991a4f10": "1 = 2A + B \\,\\!", "86fb3104bb7f6107dc179680db982367": "\\scriptstyle\\hat\\theta", "86fb52a8762288041f5e41e93796a62e": "\\{ x \\mapsto f(u), y \\mapsto f(f(u)) \\}", "86fb61579f373761934adee5111ff52f": "A=\\tilde{A}^T\\tilde{A}", "86fb7f13fe828d3e1a0e27f8b830fae0": " l=b_i+1 ", "86fb849b1897294bddfc426964662c8c": "\\begin{align}\n & D[\\partial_i\\parallel\\cdot] = D[\\cdot\\parallel\\partial_i] = 0, \\\\\n & D[\\partial_i\\partial_j\\parallel\\cdot] = D[\\cdot\\parallel\\partial_i\\partial_j] = -D[\\partial_i\\parallel\\partial_j] \\ \\equiv\\ g_{ij}^{(D)},\n \\end{align}", "86fb91471d8dc26d8af95c5d89485464": "H(s) = \\frac{\\theta_n(0)}{\\theta_n(s/\\omega_0)}\\,", "86fba0df1fb8d2928df8c55f81b09701": "L_{ij}=-\\frac{V}{R}\\sum_r w^{\\rm eq}_r \\gamma_{ri}\\gamma_{rj}", "86fba22a33e8474df9804104dc7d9569": "[v]_B = [ \\alpha_1, \\dots, \\alpha_n ].", "86fbb04c12b8429bc33ba79baafe14f3": "Z_{22} = {((1 - S_{11}) (1 + S_{22}) + S_{12} S_{21}) \\over \\Delta_S} Z_0 \\,", "86fbcbbb98e1416931a11fe46362f4e4": "\\mathrm{DK}=\\mathrm{KDF}(\\mathrm{Key}, \\mathrm{Salt}, \\mathrm{Iterations})", "86fc196a05325bbd6082f58fd078d3f0": "=-\\operatorname{tr}(\\gamma^\\nu)", "86fc83385a00ecdf612090b7a1b093cc": "\\begin{align}\nn_1^2 &= \\frac{\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_2)(\\sigma_\\mathrm{n} - \\sigma_3)}{(\\sigma_1 - \\sigma_2)(\\sigma_1 - \\sigma_3)} \\ge 0\\\\\nn_2^2 &= \\frac{\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_3)(\\sigma_\\mathrm{n} - \\sigma_1)}{(\\sigma_2 - \\sigma_3)(\\sigma_2 - \\sigma_1)} \\ge 0\\\\\nn_3^2 &= \\frac{\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_1)(\\sigma_\\mathrm{n} - \\sigma_2)}{(\\sigma_3 - \\sigma_1)(\\sigma_3 - \\sigma_2)} \\ge 0.\n\\end{align}", "86fcec825af79d6b137e524fa0e34a3a": "2\\psi", "86fcefa27a6329ebd030e7f480177b50": " F(1+r) ", "86fd118b8394ff0671999fc3daf1c884": "c_1 (1,0,2) + c_2 (0,1,0) = (c_1,c_2,2c_1).\\,", "86fd2507eed78ca83f00aaace8ed07ac": "r\\colon \\Sigma \\rightarrow 2^\\Sigma ", "86fd68dd4e2474f784b5c999b1e77a4f": " Z_{\\mathrm{F}} \\; [=z_{\\mathrm{S}} {d_{\\mathrm{F}}}^2] ", "86fd72437840e42b60f95d9fdce0dd22": "[f, g]", "86fd9523a49fd885177dcd7476de1c09": "C_{v1}", "86fd9a8645216855d688d4be71df78af": " r = c - e.\\,", "86fe5db167e90ce73a25d21772ba63da": "n\\sim N(0,\\operatorname{diag}(\\Sigma))", "86fe82bb1b86977218b8f104911d9481": "a\\to b", "86fe9fa731835d24f479932f8ca6b5f8": "\\scriptstyle G \\ll \\omega C", "86fefb04c64786e90825136bb7684887": "X_n = \\{-1,1\\}^n\\,", "86ff076b182d42a4e4088591ab8a3adc": "F L < k_\\theta", "86ff8cfa5f3daf94033dc8d3002800f5": "\n \\delta U = \\delta W \\implies\n \\int_L \\left[\\left(\\frac{\\partial M_{xx}}{\\partial x} - Q_x\\right)~\\delta\\varphi - \\left(\\frac{\\partial Q_{x}}{\\partial x} + q\\right)~\\delta w\\right]~\\mathrm{d}L = 0\n", "86ff9837a166793c3916e8f6509d3b7e": "N(v)\\setminus\\{u\\}", "86fff32d1281388762cb287cd6f60de4": "10^{19}-10^{22}", "870007fcea5ae6715093205ab71f648c": "r=\\frac{(k_0^2-k_1^2)\\sin(ak_1)}{2 i k_0k_1 \\cos(ak_1)+(k_0^2+k_1^2)\\sin(ak_1)}.", "87004eb3e5d957b480fbc1fa8052c0bc": "MDD=Max_{te(start,end)}(DD_t)\\text{ where }DD_t=\\begin{cases} \\displaystyle 1-\\frac{P_t}{P_{t-1}/DD_{t-1} }&\\text{if }P_t-P_{t-1}<0\\\\ 0&\\text{otherwise}\\end{cases}", "8700881d2114d35cbb523b14404449b2": "\\tilde{u}(t)", "8701300a9286d4fb7e03ee3f202d09a6": "p \\in S", "870190713b62bb9e9e1aa6784802f30e": "\\Phi=f(t)\\exp{(i\\theta (t))}/\\sqrt 2", "8701a60daa7ac7cd63bb6ca873991920": "X\\times Y ", "8701ab034de280cbc63c2f5c1290f548": "\\underline{y} = \\mathbf{C}\\underline{x}+\\mathbf{D}\\underline{u}", "8701da62f1cc14c6e74208df7d0fe4db": "P(t,T) = E_{Q_*}\\left[\\frac{B(t)}{B(T)}|\\mathcal{F}(t)\\right] = E_{Q_*}\\left[\\frac{D(T)}{D(t)}|\\mathcal{F}(t)\\right] ", "87026b8378a6af2f3a0f0a43094b590b": "(n,1)", "87028ef2f4e997be22b3516e1df2d2a9": "V(| \\mathbf{r}_1 - \\mathbf{r}_2 | ) = - \\frac{G m_1 m_2}{| \\mathbf{r}_1 - \\mathbf{r}_2 |} \\, ,", "87036ac5393d3db49f56eca88a3592d2": "f(x) = a^{b^x}=a^{(b^x)}", "87037e463ce9dbf35dc554d504b80905": "\\angle PCP' = 2(\\varphi+\\theta_0)", "870414acae4381536d45d1df19122d69": "s_1s_2=-3p", "87043d54f67af0265c6389660cdd5051": "n=2; \\quad s^2+3s+3", "8704aaa42038c9e2e1d325aba05f5659": "\\delta Q\\ =C^{(V)}_T(V,T)\\, \\delta V\\,+\\,C^{(T)}_V(V,T)\\,\\delta T", "8704ad072c51029fc0f006ae0682f24e": "\\kappa=\\frac{r^2+2r'^2-rr''}{(r^2+r'^2)^{3/2}}", "87055b431c53308a61053283fedfe257": "\\mathfrak{g}_\\alpha=\\{X\\in \\mathfrak{g}: [H,X]=\\alpha(H)X\\,\\,(H\\in \\mathfrak{a})\\}.", "87055ee4222f03aa91a9e9bbad04f8a6": "H_{O}", "8705601e61823c2be8f924b53e48512a": "\\tilde{x} = \\frac{x}{h}", "87057f1695959627d6e4a13529a75671": "\\mathrm{R_2SO_4\\ +\\ H_2O_2\\longrightarrow\\ R{-}O{-}O{-}R\\ +\\ H_2SO_4}", "870585caddb5be4d5067477c750019ff": "h_B={P_\\mathrm{atm} \\over \\rho g} - {v_B^2 \\over 2g}.", "870590b4ae3d39ca26b9523b597c79c4": " u^{\\alpha} T_{\\alpha \\beta} u^{\\beta} \\rightarrow T_{00} ", "870593c0dd268a0cd706a3333d4e654c": " \\langle I(\\mathbf{q}) \\rangle \\sim \\langle \\left | \\phi(\\mathbf{q}) \\right |^2 \\rangle = N \\left | f(\\mathbf{q}) \\right |^2 S(\\mathbf{q})", "8705cc453ea6526b8b67c6e4da2b4d8f": "f(u,v) = (0,0)", "87060eea2892ea817f1b2910d8f11bf7": "\\begin{align}\n\\cos 2\\theta &= \\cos^2 \\theta - \\sin^2 \\theta \\\\ &= 2 \\cos^2 \\theta - 1 \\\\ \n&= 1 - 2 \\sin^2 \\theta \\\\ &= \\frac{1 - \\tan^2 \\theta} {1 + \\tan^2 \\theta}\n\\end{align}", "870642ebac7465e235405dfad81ca06e": "\\sum_{a=1}^N\\mu_{an}^{(c)}(t)", "87067255a183e51cd29b926d1bbe777d": "b^{2}(w^{2},m_{1}^{2},m_{2}^{2})=\\frac{1}{4w^{2}}\\left\\{\nw^{4}-2w^{2}(m_{1}^{2}+m_{2}^{2})+(m_{1}^{2}-m_{2}^{2})^{2}\\right\\}\\,.", "870692d172ae7e6a783dab50537bcf42": "a_{-1} = a + \\sqrt{a^2 - b^2} \\, ", "87069d0b9cbfd259ce8e4a8198f527fe": "4/3", "8706a4ec5f3dccc638cc012416b71e4d": "\\, b_k^\\dagger", "8706bb24dc564cd6038103860bb81e19": " \\frac {N_2} {N_1} \\approx \\frac {c_1} {c_2} \\left ( \\frac {m_2} {m_1} \\right )^2 = \\frac {l_2} {l_1} \\,.", "8706c85d3c709f40ea4b35808b273684": "h_{0j}\\,", "8706d147514dc3dd1930e5da76631182": "{}_sY_{\\ell m} = \\sqrt{\\frac{(\\ell-s)!}{(\\ell+s)!}}\\ \\eth^s Y_{\\ell m},\\ \\ 0\\leq s \\leq \\ell;", "87077b135cad06a81c3c331e2fb09aff": "\\overline{u} = \\frac{u_*}{k}\\ln\\frac{z}{z_o}", "8707aea7e89da53763c1b0cd804085f4": "f(y)=\\frac{1}{y^2}-x", "8707c6b8e8d61f484e7b4b99dd39a0bb": "\\scriptstyle \\mathbf{F}\\times \\mathbf{G}", "870875c8ea104b946bbf0cc828ea168c": "(J, g)", "87088d4df28506e7f320ad0bb75dd07c": "\n \\hat{y}|_{x=\\xi} \\in \\Bigg[\n \\hat\\alpha + \\hat\\beta \\xi \\pm\n t^*_{n-2} \\sqrt{ \\textstyle\\frac{1}{n-2} \\sum\\hat{\\varepsilon}_i^{\\,2} \\cdot \n \\Big(\\frac{1}{n} + \\frac{(\\xi-\\bar{x})^2}{\\sum(x_i-\\bar{x})^2}\\Big)\n }\n \\Bigg] .", "8708cc407de91b48d0a7f223d585bfb5": "\\ y_n= \\sum_{k=0}^{n-1} h_{k} x_{n-k}", "870915327ed57f65bade690f1757927b": "\\overline{O_R P}", "87093e13f35ec113eecfabac8311be17": " \\ f_L = \\ {64 \\over Re}\\left[1 + {He\\over 6 Re} - {64\\over3}\\left({He^4\\over {f}^3 Re^7}\\right)\\right]", "8709793311c34f92e984529a2187b598": " \\int_{Y^{-1}(B)} X(\\omega) \\ d \\operatorname{P}(\\omega) = \\int_{B} g(u) \\ d \\operatorname{Q} (u). ", "8709e366a8659294e83aa0b1aab588df": " F_i = M_i ^\\dagger M_i ", "870a5bbed9d585c5cfd5426dcbe7e871": "Notional_\\text{fixed leg}(t) = N \\times \\prod_{i=1}^{n-1} \\left( 1 + \\kappa \\right)^ {(\\frac{1}{252})} ", "870a649c3a66bdc4b712e34daae3878e": " \\cos{\\omega_{ij}} = -\\frac{1}{\\sqrt{\\lambda_i \\lambda_j}} ", "870a6dba627d750037dade3d027e3247": "L^\\dagger_\\Phi", "870a8f573a2bcf84f37fb6d3638b9a8b": "\\scriptstyle\\theta", "870b06c06e70431d289f4751aaf9a67e": " { n \\choose k } + { n \\choose k-1 } = { n+1 \\choose k }.", "870b0b527f965f52b2729d94c77d5cfe": "k_B T_e>Z_i^2 E_h", "870b8bd87266ccfc24e983eb233b4cac": "\\nu=2\\tan^{-1}\\left[\\sqrt{\\frac{1+e}{1-e}}\\tan\\frac{E}{2} \\right]", "870bde97dbb184ce5eb701c0765e1d5f": "\\left.\\begin{matrix}\n\\xi_1^2-\\xi_2^2&=x_1\\\\\ni(\\xi_1^2+\\xi_2^2)&=x_2\\\\\n-2\\xi_1\\xi_2&=x_3\n\\end{matrix}\\right\\}\n", "870c0d371a23ebf34430c5f881696eee": "s = \\sqrt{\\frac{1}{N-1} \\sum_{i=1}^N (x_i - \\overline{x})^2}.", "870c2eb1e5c3edebf42a09c0cb166607": "\\mbox{eGFR} = \\mbox{exp}{(1.911+ 5.249/{Serum\\ Creatinine} - 2.114/{Serum\\ Creatinine}^2 - 0.00686 \\ \\times \\ \\mbox{Age} - {[0.205\\ if\\ Female]})}", "870c4808e87a7353fc1748940b4f9576": "\\Delta_1(f_i) = k_i^{-1} \\otimes f_i + f_i \\otimes 1", "870c6fd2d1813d16d9326ee26fbf9ba3": "\\bigwedge P = \\bigwedge \\{ x \\in L \\mid x \\ge f(x) \\}", "870c83e288e8b9ebae8175b42fbefdfb": "\\mathbb{R}^n \\backslash \\lbrace 0 \\rbrace", "870cb422618df6880a286c9fe571505a": "x(t)=(x_1(t),\\dots,x_n(t))", "870cc8c04f5d1830b43676d0f9a28e66": "\\scriptstyle\\mathbb{E}\\{X\\}", "870ce572cffac6cafe7480a0bbe201dc": " \\int | K(x, y) |^2 dy < \\infty ", "870cf01e2d39103083a06175b54bbae4": "q = 11", "870d61583e8fcdf427a68d9cc95b108c": "S_n = \\sum_{k=1}^n a_k.", "870d977ca2e85029611e206520632c33": "\\int_0^\\infty\\frac{\\sin(x)}{x}\\,\\mathrm{d}x=\\lim_{b\\rightarrow\\infty}\\int_0^b\\frac{\\sin(x)}{x}\\,\\mathrm{d}x=\\frac{\\pi}{2}.", "870dde8ba4758f81ac44e10b20b00aab": "\\nabla^2\\Phi=4\\pi G{\\rho},\\,", "870df669efb7b2a2f7ee4df7348f5800": " \\frac{P(x)}{(1-x)(1-x^2)\\cdots(1-x)^n} = \\prod_{(i,j)\\in \\lambda} (1-x^{h_{(i,j)}})^{-1} ", "870df7c1f0742798b6c67ceccf655a53": "c=1+3.535\\omega+0.533\\omega^2", "870e5a1b3219129925be3b38f09521d6": " N = a^\\dagger a = 1 - a a^\\dagger ", "870eb40a378510216042d7a0172b2280": "\\sum_{1\\le k\\le n \\atop (k,n)=1}\\!\\!k = \\frac{1}{2}n\\varphi(n)\\text{ for }n>1", "870eb9e1bc984be3de645f9d51f6d725": "c_{ijkl}", "870ee77007779e42f941a7a283e2ac7c": " P_{ij}(t)\\ ", "870f2a92cba06beeab78aafc636c087d": "(A, \\delta_Z, \\varepsilon_Z)", "870f62b62c4a49813491131fee943398": "P_{2r} = A_{2} \\frac{\\mathrm{G_{1}}(\\theta,\\Phi)}{4 \\pi r^{2}} P_{1t}", "870f63995df37de977280ad341fe5227": " p \\times \\left(p-k\\right)", "870ff0330fe8555339bcc8a4c7db4de8": "1 x = x = x 1 ", "8710133df18941dd8bdbad4774bc1cc0": "\\int (px+q)^n\\sqrt{ax+b} dx = \\frac{2(px+q)^{n+1}\\sqrt{ax+b}}{p(2n+3)}+\\frac{bp-aq}{p(2n+3)}I_n\\,\\!", "8710572e7ca8bdea007b38b72ddfa92d": "f(x_i,\\boldsymbol \\beta)\\approx f(x_i,\\boldsymbol \\beta^k) +\\sum_j \\frac{\\partial f(x_i,\\boldsymbol \\beta^k)}{\\partial \\beta_j} \\left(\\beta_j -\\beta^{k}_j \\right) \\approx f(x_i,\\boldsymbol \\beta^k) +\\sum_j J_{ij} \\,\\Delta\\beta_j. ", "871074ebb3f64a8e09cc0d328b6ff3ae": "\\displaystyle \\langle i,j \\rangle ", "87108f0d3c74c88a4d867c9852df6e8a": "F_{\\mu\\nu}=T^aF^a_{\\mu\\nu}", "8710c36adfe8c8af3ef4007ed4bfbbc2": "\\Chi_H=\\left( \\frac{\\partial H}{\\partial p_i}, \n- \\frac{\\partial H}{\\partial q^i} \\right) = \\Omega\\,\\mathrm{d}H,", "8710c7f8c6d80f53e045a3695e5eea45": "T = 4 t\\left(\\theta_0\\rightarrow0\\right),", "8710fadd84735627e49369db085c29c4": "\\; (A - 4 I) p_4 = p_3. ", "87112cfb755068b6d84a7538b5145ca7": "I_n= -\\frac{\\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+\\frac{a(2n-3)}{2(n-1)(aq-bp)}I_{n-1}\\,\\!", "87113266eef7b991a0ccdb2528cd68ce": "A,B,C,D \\in \\mathcal{A}", "8711414994efd2511eda6f2a4a76e891": "\\,\\alpha,\\beta>0", "87114687931e35cf73899349da656c31": "\n\\mathbf{R}^\\mathrm{T}\\mathbf{R} = \\mathbf{R}\\mathbf{R}^\\mathrm{T} = \\mathbf{I},\n", "871168c78a15388179a816ec74f7abbb": "\\chi_\\mathrm{w}", "8711743d45eaf741b8b6046d7ee3a307": "\\rho_S = \\sum_i F_i \\rho_S (0) F_i^\\dagger ", "87117ec68a453ab7669543470176f441": "I^{n} M \\cap N = I^{n - k} ((I^{k} M) \\cap N).", "87118e98457ba350f0eebafcef5cf6b5": "\\theta = \\tan^{-1} \\frac{B}{B_H}\\,", "8711e5e938526e9a4fbbc91aaa02d711": "C_{s}", "8711e6e617a9e3c8a835ae224ee11987": "\\mathrm{Spf}(R[[T_1,...,T_n]])", "8711f57bc355dc7ed64970569c81a246": "\\operatorname{Pr}(Y_i=y_i\\mid \\mathbf{X}_i) = {n_i \\choose y_i} p_i^{y_i}(1-p_i)^{n_i-y_i} ={n_i \\choose y_i} \\left(\\frac{1}{1+e^{-\\boldsymbol\\beta \\cdot \\mathbf{X}_i}}\\right)^{y_i} \\left(1-\\frac{1}{1+e^{-\\boldsymbol\\beta \\cdot \\mathbf{X}_i}}\\right)^{n_i-y_i}", "8712e5ec56139a4618080c3e2dc5c2b7": " u(x,t) = \\sum_{n} a_n (t) X_n(x)", "87130abf37bbfee8af61ae903455c064": "\\Lambda^q(E)", "871312ed6a1d1d69eddbeb1238d76f53": "\\Delta u_n = 2^n", "871336fd08c9e517633c07f98f292236": "W_0:C\\rightarrow C", "87134b99cbfe695c7c72f0978e440c0c": "\\mathbb{R}^d.", "87136851e86bbcb9d17ac5c911ca4829": "\\operatorname{Ass}_R(M)", "87136e6781bde7ffdc654be94af31be3": " X^\\mu(\\tau, 0) = b^\\mu, X^\\mu(\\tau, \\pi) = b'^\\mu \\ ", "871398f0ba83dde261f8b2ebe1b32d38": "W^t= -W", "8713d972ed6bd30f9f2eeec80cc1b286": "\\mu/\\sigma", "8713f06f9a5141dbf946cbfba13e6ab6": " m(r,f)=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}\\log^+ \\left| f(re^{i\\theta})\\right| d\\theta. \\,", "8713f9bd1f94a0e12fd05065849a6366": "\\tan(e)=\\frac{a}{d}", "87143adcdcd8e7d456d24d8896a2c629": "1\\leq k \\leq l-1", "87144f039104de5a4fe50f24ab1b91a1": " F_{diff} = exp (-bD_{int}+ K(bD_{int})^2/6) \\,", "8714d99880eda1b0590df16bf69d1a13": "420, 3360, 30240, 403200, 4019400, 80166240, 965284320, 12173441280, 162850287600,\\ldots", "871558fa026b7a681d9434663c0ae7c4": "PV = \\sum_{k=1}^{n} C(1+i)^{-k} = \\frac{C}{i}\\left[1-\\frac{1}{\\left(1+i\\right)^n}\\right] = C\\left[\\frac{1-(1+i)^{-n}}{i}\\right], \\qquad (1) ", "8715804445bd1862bba49051bc3d783a": "Dy(x) + f(x) y(x) = g(x).", "87159deb6b4ea1bc8162e8d40f1d5fe5": "\n\\begin{align}\n\\frac{\\Delta^\\acute{n}F(P_0)}{\\Delta_1P^\\acute{n}} & =\\frac{\\sum_{I=0}^{\\acute{N}}{-1\\choose\\acute{N}-I}{\\acute{N}\\choose I}F(P_0+I\\Delta_1P)}{\\Delta_1P^\\acute{n}}; \\\\[10pt]\n& \\frac{\\nabla^\\acute{n}F(P_\\acute{n})}{\\Delta_1P^\\acute{n}} \\\\[10pt]\n& =\\frac{\\sum_{I=0}^{\\acute{N}}{-1\\choose I}{\\acute{N}\\choose I}F(P_\\acute{n}-I\\Delta_1P)}{\\Delta_1P^\\acute{n}};\n\\end{align}\n", "8715b7cc38583bba52e171db4d9dea37": "^{2S+1} \\Lambda (v)", "8715ce9d0498759366bc1134fa286843": "\\scriptstyle H \\ = \\ 2 \\sqrt{\\frac {\\gamma} {g \\rho}}", "87160308f579cf998d96b2558f238e4e": "f(x,y,z) \\longrightarrow f(\\rho \\cos \\theta \\sin \\phi, \\rho \\sin \\theta \\sin \\phi, \\rho \\cos \\phi)", "87164a9d2e357f99823e23e2ddfdddf9": "K_n=K(1/L_n)", "87166e3be5e2cebeaf27d49bb704310d": "{\\mathcal N}({\\mathbf 0},{\\boldsymbol\\Sigma})", "8716d4abcf2df3d86aef1d5e327c631f": "x_{mn}=\\sum_{k=-p}^p \\sum_{l=-q}^{q} X_{kl} e^{\\frac{-i 2\\pi k \\xi_m}{L_\\xi}}e^{\\frac{-i2\\pi l \\nu_n}{L_\\nu}}+\\epsilon_{mn}, \\quad m=1,\\dots,M;\\ n=1,\\dots,N.\\,", "87170afcc74f7bd000a3f717a7375d2d": "\\left( {z - \\mu } \\right)'\\left( {Az - A\\mu } \\right)", "87174578f3ad6f8c630ad1a82d596a8e": "\\hat{\\pi}_{\\psi^T}", "8717a741dc6bf80ead01ee93aec01633": "a_i = b_j", "8717ad61271309384c9a7c0e8a79807f": "p^2=N(g)=N(h)N(k)", "8717bd5dc484a9ea02a271339052d497": "\\phi_3=0^\\circ", "8717d7ef1ff89acc4e0a3242559909d0": "0\\leq i \\leq 2", "87180a1727a6f866d5cc7476d4cbba08": "c_\\lambda = a_\\lambda \\cdot b_\\lambda", "8718230ad462a0eb407f8de9f0e777d2": "\\Eta ", "871848bb8d2011bbcf6872ba16ad1c84": " f(x) = \\int_{c}^{x} R \\left(t, \\sqrt{P(t)} \\right) \\, dt, ", "87185032a523d4e9069399befaddef18": " F(x) = 1 - \\exp\\{ - \\alpha(\\log x - \\log \\sigma) - \\beta(\\log x - \\log \\sigma)^2, x \\geq \\sigma,\n", "87185e56aa4e6f1d3c95f38cf1448655": " \\nu \\approx \\mu \\frac{3 k - 0.8}{3 k + 0.2} ,", "8718a2c1690c17e101221e5e9a7d8b2d": "3^3+4^3+5^3=6^3.\\,", "8718c1d769a5ed5d0b19799526238042": "F(\\mathit{z}, \\mathbf{x}) = \\operatorname{Prob} \\lbrace Z(\\mathbf{x}) \\leqslant \\mathit{z} \\mid \\text{information} \\rbrace . ", "871987813eed7c23d9b326d0979da0aa": "s_i^2 ", "8719bb254b974f6582837fd921e5d3b6": "x(t) = t - \\lfloor t \\rfloor = t - \\operatorname{floor}(t)", "8719d7a8e88f42c281c4e38be561fb51": " x = \\frac{z-z_{\\text{min}}}{z_{\\text{max}}-z_{\\text{min}}} , \\qquad z_{\\text{min}} \\le z \\le z_{\\text{max}}. \\,\\!", "8719e21ae7933cac26b52fe62d37da99": "u_1,...,u_n", "871a75225f937914208c7f9fb0d7b820": " \\boldsymbol{e}_k\\,", "871a994baa59c996bef6e6c106513208": " (1-x)^t \\ge 1-xt. ", "871aa33b156bc37c4303cf37f012addb": "\\tilde{Z}[\\tilde{J}]=\\int \\mathcal{D}\\tilde\\phi e^{-\\int d^4p \\left({1\\over 2}(p^2+m^2)\\tilde\\phi^2-\\tilde{J}\\tilde\\phi+{\\lambda\\over 4!}{\\int d^4p_1d^4p_2d^4p_3\\delta(p-p_1-p_2-p_3)\\tilde\\phi(p)\\tilde\\phi(p_1)\\tilde\\phi(p_2)\\tilde\\phi(p_3)}\\right)}.", "871acb1553714b6164afa039793cb6bf": "\\, 4x^5 + x^3 + 2x^2", "871ade49cd07ecb682da475b0a658310": "\n\\begin{align}\n&\\dot{\\hat{\\mathbf{x}}}(t) = \\mathbf{F}(t) \\hat{\\mathbf{x}}(t) + \\mathbf{B}(t) \\mathbf{u}(t)\n\\text{, with }\n\\hat{\\mathbf{x}}(t_{k-1}) = \\hat{\\mathbf{x}}_{k-1\\mid k-1} \\\\\n\\Rightarrow\n&\\hat{\\mathbf{x}}_{k\\mid k-1} = \\hat{\\mathbf{x}}(t_k)\\\\\n&\\dot{\\mathbf{P}}(t) = \\mathbf{F}(t)\\mathbf{P}(t)+\\mathbf{P}(t)\\mathbf{F}(t)^T+\\mathbf{Q}(t)\n\\text{, with }\n\\mathbf{P}(t_{k-1}) = \\mathbf{P}_{k-1\\mid k-1}\\\\\n\\Rightarrow\n&\\mathbf{P}_{k\\mid k-1} = \\mathbf{P}(t_k)\n\\end{align}\n", "871b354d265716a852eea9970e53ff5e": "n_t", "871bf229abd1688a3440f09d9a071fcf": "\\displaystyle (ab)^2= A_2A_0-2A_1A_1+A_0A_2 = 2\\Delta.", "871c1fcc584f07e9602a4de8e0b1e6b2": "|\\gamma(s, b) - \\gamma(s, a)| \\le \\int_a^b |t^{s-1}| e^{-t}\\,{\\rm d}t = \\int_a^b t^{\\Re s-1} e^{-t}\\,{\\rm d}t \\le \\int_a^b t e^{-t}\\,{\\rm d}t", "871c6319effb13bdefce1fade7ae89b7": "1/\\binom{6}{3} = 1/20 = 0.05,", "871c6f5296abddfad7eba180f79646b3": "\\begin{align}\n \\sin \\theta &\\approx \\theta \\approx \\arcsin \\theta \\\\\n \\tan \\theta &\\approx \\theta \\approx \\arctan \\theta\n\\end{align}", "871c7219c9830dbd0bc4838d8b97d55f": " \\Gamma_{a_1 a_2} ", "871cac36c226d429123d04acd7ce849c": "\\langle\\partial_i,\\partial^j\\rangle", "871d25806a2c075c16b5d2120bfca1c4": "\\sigma _0", "871d750c307c95da42b13a8e9e3a1626": "v_{i,j}", "871d8eb206c3be107041f25ef9afb4fb": "p: A \\rightarrow B\\,", "871dd39407437a188b2c44454ae675e2": "m\\times r", "871e1e29bce5e75aad1461da101f94c6": "\\left({c\\over p}\\right)=1", "871ec7c92d61e4ac62978ccd1d30e7fc": "R_\\min", "871ef1f171b09796d51bc6ea306791f9": "~ \nG = \\sigma_{\\rm e}N_2 - \\sigma_{\\rm a}N_1\n~", "871f14c82572ca4708150d04e231ef8f": "\\binom{n+k-1}k\\frac{\\Gamma(\\alpha+n)\\Gamma(\\beta+k)\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha+\\beta+n+k)\\Gamma(\\alpha)\\Gamma(\\beta)}", "871f3d7ec80220f6103f51ba92c34b5b": "\\forall i \\in \\left[ 0, l - 1 \\right], {n_M} \\ge {n_i}", "871fdfe74cf3aa76153282f044cf5577": "H(\\xi )=H(\\star q(\\xi ,\\tau ))", "871fe468100384b8dd6baf829f69c2dc": "R=\\sum_{t=0}^{N-1} r_{t+1},", "871ff20ae08c6581f4b0ef34673de0d6": "P =(a,b) = \\overline{P}=(a, -b-h(a))", "871fff53e8980b8ac4029077c70efdf9": "n=1, d=2,", "872006363ef0483a5a55656d78457dbd": "\\Delta J = 0, \\pm 1", "8720248cdd01481ab4c8387030ae6c3e": "X\\to \\mathbb A^d_k", "872086ef28badfaf3f5666ff2378cec9": "p_{n,k}'(x_{n+k})", "8721121a115be37c86b366e88c09e17a": "\\Delta w''=0^*", "87211cf660b43fa68c7707988032209c": "ds^2 = \\sum_{i,j=1}^n g_{ij}(p)dx^i dx^j", "87218460f9e8fb2815baa09e81c2a3ec": "(I-A)S=S-AS=I-A^{n+1}", "8721f2fdb5599c960d0f5145e80b971a": "X(s)=(sI-A)^{-1} x(0) + (sI-A)^{-1}[BU(s)+EW(s)] ", "8721ff01e36260e1ca62f22967b016c5": "N_B", "87220ddf966df6ba4ee2ceba3851d904": "-\\frac{\\hbar^2}{2m_b^*} \\frac{\\mathrm{d}^2 \\psi(z)}{\\mathrm{d}z^2} + V \\psi(z) = E \\psi(z) \\quad \\quad \\text{ for } z < - \\frac {l_w}{2} \\quad \\quad (1)", "8722359e0a559c2290919d863cc907db": "(\\{a_{ij}\\},k)", "8722439958298b3dfbd60cded8990f4e": "\\triangle", "87229bdea416f669306b306abdc51436": "L_k, k=1\\ldots q", "8722dd8b2278fd28705dfae2f9a95a5c": "E_m(x)=\n\\sum_{n=0}^m \\frac{1}{2^n}\n\\sum_{k=0}^n (-1)^k {n \\choose k} (x+k)^m\\,.", "8722fc1eabf959c86f93a4babe4bcd58": "\\, \\tilde{t}_\\text{r}", "87230df1799a00ae2e99f65e28ceaf05": "\\scriptstyle \\omega", "8723233ba2955e1cafe8bcb3c8b0187c": "\\nabla_{a}\\omega_{b}=\\hat{\\nabla}_{a}\\omega_{b}-Q_{ab}{}^{c}\\omega_{c}", "87236c166e1402455b164c0e91e9c204": " f[x_0,\\dots,x_n] = \\frac{f^{(n)}(\\xi)}{n!}.", "8723ad1a02d6e41ab541918c8fb499cd": "x+ \\infty = \\infty \\quad \\text{if}\\quad x\\not= \\infty", "8723c3655b22d58a7c2bbb1a62cb5b3a": "i=1,2,\\ldots\\}", "87241bfd20804f098fe65b1dba3c2acb": "\\omega_A=1-\\omega_X=1-\\frac{W_X}{W_X+W_A}", "8724746b18aeb66a14ce793dc99f94e8": "\\Psi_1(q) = -1 + \\sum_{n \\ge 0} { q^{5n^2}\\over(1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^{5n+2}) }", "8724a55d8b0c50efadad900e528e40c6": "T \\colon A \\to A", "872511e44df4811c1e228bac20439800": "\\frac{\\partial C}{\\partial x}_{x=z} = \n\\frac{(C_{i + 1} - C_{i - 1})}{2 \\Delta x} = 0.", "87255f615abf3ca1b9124662afd19c08": "\\textstyle t + 1", "87257366cb857817a7ddc0086c118ce5": "P_{\\mathrm{out}}", "87259641a3b9c16fe34ecb3ab53cee9f": "K^{\\ominus} =\\mathrm{\\frac{\\{A^-\\} \\{H_3O^+\\}} {\\{HA\\} \\{H_2O\\}}}", "8725966f6b40bbd899b1166e72a9e5b8": "\\, e^{i t^\\mathrm{T} \\mu - \\frac{1}{2} t^\\mathrm{T} \\Sigma t}", "8725cf2d2435069ad55df9a28abf1fc5": "A = l\\cdot (\\epsilon_{X} c_{X} + \\epsilon_{Y} c_{Y} )=l\\cdot\\epsilon \\cdot (c_{X} + c_{Y} )=l\\cdot\\epsilon\\cdot c", "87269347dcf7d876166fca649ac23b29": "\\left(1, \\frac{q_k}{p_k}\\right)", "8726a3f9489b358a8331ce85a13cc8d0": "(Z,\\ X,\\ Y)", "872731aa3142c4eb374c2133a458754a": "a_0 + a_1 p^1 + a_2 p^2 + ...", "8727772e5eca1e70260c7e90bb30fbd5": "H(s)H(-s) = \\frac {{G_0}^2}{1+\\left (\\frac{-s^2}{\\omega_c^2}\\right)^n}.", "8727a81f5ee26fc39ffaab765c8e9c8f": "\\delta: Q \\times \\Gamma^k \\rightarrow Q \\times (\\Gamma \\times \\{L,R,S\\})^k", "8727db319318efcf04bb5ac9f1512a2b": "\\sqrt{\\frac{1}{12}}\\!\\,", "8728c4e90f8fc683b4814bc133411c8f": "\\sigma: V \\to V", "872912aaafc75caee2018c4043d6b275": "\\int{\\frac{dx}{a^2-x^2}}=\\frac{1}{a}\\tanh^{-1}{\\frac{x}{a}}", "872921f0d4e9ee51be534f72c2fcfdd8": "\n\\begin{align}\n3 \\times 11 & = 3 \\times (1\\times 2^0 + 1\\times 2^1 + 0\\times 2^2 + 1\\times 2^3) \\\\\n& = 3 \\times (1 + 2 + 8) \\\\\n& = 3 + 6 + 24 \\\\\n& = 33.\n\\end{align}\n", "8729457d8af22ca062e5d5d7e15eaa0a": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = \n-\\nabla p + \\nabla \\cdot \\left(\\mu (\\nabla \\mathbf{v} + (\\nabla \\mathbf{v})^T)\\right) + \\nabla \\left( - \\frac{2\\mu}{3}\\nabla \\cdot \\mathbf{v}\\right)\n+ \\rho \\mathbf{g}", "87296c59c6a26b61e4d82bba97318fff": " c = e = \\hbar = 1 \\ ", "87297fc76cc4e2ff2dbb991707db0bc7": " \\mathrm{distance} = \\frac{\\mathrm{width}}{2\\tan\\left(\\frac{1}{2}\\mathrm{fov}\\right)} ", "872999cadb70805c8ca7361415a01e6a": "\\ a_0 + \\cfrac{1}{a_1}", "8729ce31f15359e76bfe708904676643": "\\mathrm{Ber}(X) = \\det(X_{00} - X_{01}X_{11}^{-1}X_{10})\\det(X_{11})^{-1}.", "8729f7540c1b4c28661713172de096b2": "2, 3, \\ldots, m", "872a9563384648175945de98e6fdc845": "d_J = {M_{01} + M_{10} \\over M_{01} + M_{10} + M_{11}}.", "872a96bb3731d3e5c618bd9164585c8e": "H \\in \\mathbf{H}", "872acdc54ab304fa874336d04bc466c6": "e^{i \\varphi}", "872ad8255cd13ec84c142850a9cf8b16": " [\\operatorname{p.\\!v.} (K)](f) = \\lim_{\\varepsilon \\to 0} \\int_{\\mathbb{R}^{n} \\setminus B_{\\varepsilon(0)}} f(x) K(x) \\, \\mathrm{d} x. ", "872aed6f32775fc764972c0b40daa7ed": "\\Bbb Z[t^\\pm]", "872af04659c74f05750e603288395007": "\\hat{u}_{n,\\alpha}", "872af1cb3f6e1a4e2d7681843b27caa3": "R, R[\\epsilon]", "872b023b194bda9b7838893f419ec277": "\n d = \\cfrac{\\pi a}{2 E^*}(p_0 + 2p_0') = \\cfrac{a^2}{R}\n ", "872b5251b43698fb4b5fc75d23dfd0ed": "\\tan 3\\theta = \\frac{3 \\tan\\theta - \\tan^3\\theta}{1 - 3 \\tan^2\\theta}\\!", "872b6693f2a0af3d7774a5acf17b1191": "H_\\epsilon", "872c060a215c9c109f236ffe4a97662f": "i_1, i_2, \\ldots", "872c7791fc705e6a86d2f74b5055156d": "| \\phi_{\\beta/\\alpha} \\rangle", "872c8197dd81e96a162c16760d5dfd4c": "\\hat \\lambda^k = \\frac{1}{N} \\sum_{i=1}^N (x_i^k - x_N^k)", "872c87433f1c75c7d8d3625ea643572b": "\\rho_c(\\mathbf r, t) = \\Phi_L(\\mathbf r, t) \\tilde{\\Phi}_L(\\mathbf r, t)", "872c884050a935434b4e0ed132ee8392": "(x,\\dot{x})=(0,0)", "872cdbfc3f22375ce0232baaad5caee6": "\\varepsilon \\to 0", "872cdc939b6e8109572f5e42e17382c9": " = (\\boldsymbol\\beta-\\hat{\\boldsymbol\\beta})(\\boldsymbol\\Sigma_{\\epsilon}^{-1} \\otimes \\mathbf{X}^{\\rm T}\\mathbf{X} )(\\boldsymbol\\beta-\\hat{\\boldsymbol\\beta})", "872d3c2eb2b49a73ee5b4edd816e1beb": " \\mathbf{v}_\\mathrm{p} = \\mathbf{\\hat{e}}_{\\parallel} \\left ( \\Delta r /\\Delta t \\right ) \\,\\!", "872d558969eb57bb8ea6b1f7d9c4a6fa": "\\frac{d P}{d\\Omega} = \\frac{q^2}{16\\pi^2 \\varepsilon_0 c} \\frac{|\\hat{n} \\times ((\\hat{n} - \\vec{\\beta})\\times \\dot{\\vec{\\beta}})|^2}{(1-\\hat{n}\\cdot\\vec{\\beta})^5}", "872e0322d76878e3896dd8de5a345efb": "\\left \\{ \\sqrt{\\pi} \\right \\}", "872e2e164531e240794bb31a81698935": "C_{t} = C_{0} + cY_{t-1}", "872eabb88940f5801766a60482b987e1": "\n\\langle z \\rangle=\\frac{1}{1-i/\\lambda} .\n", "872ee879a2a9501357f503f54b76531e": "(p,l) \\in I,", "872f2a5c89c4569bcbfc6dacbb1f25fe": " E \\to \\pm \\infty ", "872f483a6bf878f69bb5a3897dcccefb": "\\displaystyle{([L,\\psi]f,g)=([\\Delta,\\psi]f,g),}", "872fc1ab30c37efeb3f183fcd62ff1fc": " t,", "873023401988a44be208b38c7fe09e20": "e_1 , \\ldots , e_n", "873025972271b12541ef622107450e28": " \\tau_{dynamical} = \\sqrt{\\frac{2R^3}{GM}} \\sim 1/\\sqrt{G\\rho} ", "87302a94b4e46028cfa548fbaf76af88": "a = b = 0", "87302ae5a8ac68c67c697a1e9ea2a7b1": "\\varphi\\circ g (x) \\geq ag(x)+ b", "87308a12e95adf3aa9f1158161d346fd": " \\mathrm{Gr}_L = \\frac{g \\beta (T_s - T_\\infty ) L^3}{\\nu ^2}\\, ", "8730e872e66d09da17671583e60c1e30": "R(t,T)", "87313a19b0bec44220d041b1dc0e692c": "t(v, w^*) := w^*(v) \\in F", "87315d5056e8971955a8f0f55aee164f": "\\Box P\\rightarrow P.", "873166617aa2a89e1fed347e9b34a358": "f(0) = f(L) = 0", "8731a2b6414532594c347f1a0b8282c3": "\\mathcal{H}^{(0)}", "8731a64e248f2e826de81dda44dde457": "\\lambda_J = \\frac{c_s}{\\sqrt{G \\rho}} \\simeq (0.4 \\mbox{ pc})\\left(\\frac{c_s}{0.2 \\mbox{ km s}^{-1}}\\right)\\left(\\frac{n}{10^3 \\mbox{ cm}^{-3}}\\right)^{-1/2}.", "8731cff43f60d6afe6cb7c31f918ba43": "\\frac{2\\pi}{T}=\\sqrt{\\frac{2g^2}{U^2}}", "8732099f74d777a67257cb2f04ead3d8": "x_2", "873240566d9ff05654579da61a39eb3f": "\\{O_{1},O_{2},O_{3},O_{7},O_{8},O_{10}\\}", "8732bbce1e74039df303739162cde469": "\n\\vec x_1\n=\n\\vec{x}_0 + \\vec{v}_0\\Delta t + \\tfrac12 \\vec a_0\\Delta t^2\n\\approx\n\\vec{x}(\\Delta t) + \\mathcal{O}(\\Delta t^3).\\,\n", "8732d68fa1839333f8877affc6139dfb": "pa=r_\\max r_\\min=b^2\\,.", "8732ddec41c4e5614616d0dd29f95fd3": " f(x) = (1 + x)^{\\alpha}.", "87335cbb192a4f679bf61d1f2e59280f": "\\frac{1}{(4\\pi it)^{n/2}} e^{i|x|^2/4t}", "8733bdab9c83218ad4158a1a4ee0a025": "f(n) = \\Theta\\left( n^{c} \\log^{k} n \\right)", "8733efb379d727f072c02f1f22fa8adf": " \\Phi_0 (z)", "8733eff5b24f9be2b5c0e6df484d6cc4": "e_1, e_2, \\ldots , e_n", "873417751fc018ab8f370e1af0d2732c": "\ns'^2 = \\begin{bmatrix}c \\Delta t' & \\Delta x' & \\Delta y' & \\Delta z' \\end{bmatrix}\n\\begin{bmatrix} -1&0&0&0\\\\ 0&1&0&0 \\\\ 0&0&1&0 \\\\ 0&0&0&1 \\end{bmatrix}\n\\begin{bmatrix} c \\Delta t' \\\\ \\Delta x' \\\\ \\Delta y' \\\\ \\Delta z' \\end{bmatrix}\n", "8735994a72ec87c66028dede0c2d9b17": "\\epsilon_0 (k)", "8736094da7a785780cc3c9e04f67cb30": "\\frac{\\partial M_r}{\\partial c} = \\frac{c(c + \\alpha)_r (c + \\beta )_r}{(c + 1)_r (c + \\gamma)_r} \\left\\{\\frac{1}{c} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta + k} - \\frac{1}{c + 1 + k} -\\frac{1}{c + \\gamma + k} \\right) \\right\\}. ", "87361ca989e6e1bfa830aae853d48586": "\\aleph_{\\omega_1}", "873666000c98470b805cd27bf3233c22": " \\langle f_i |", "87366991b6ca15ca50fab9981050f03e": "r_{\\mathit l \\mathit l^{\\prime}}", "873674f9d437d4fc1cdc1ced08d21833": "p_i = h_i(v_{-i}) - \\sum_{j \\neq i} v_j(a)", "87367e18af54a5bffb9c8bcf5ec0587c": "\\mathbf{d_H}(X, Y) \\leq t", "873729a10b3e299bda70c97a7bd09818": "\\pi_*\\colon H^*(E)\\longrightarrow H^*(M),", "8737329fb785ee86e1cf5d15990645d9": "abs(\\lambda) < 1 \\,", "87374966239e809d9248e29049e0388c": "q + 1 - t \\equiv t \\pmod 2", "87379a762d6bc1bc0a45a5ae178cb6db": "\\{\\varsigma:= e^{i\\phi}\\cot\\frac{\\theta}{2}, \\bar{\\varsigma}=e^{-i\\phi}\\cot\\frac{\\theta}{2}\\}", "87379b338e3a3fe541006398928c0080": "a = d \\ne b = c, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ, \\zeta = 120 ^\\circ", "8737df269f7d10c59afb4e9a3878b06c": "A(\\boldsymbol\\eta) = -\\ln g(\\boldsymbol\\eta)", "8737e605b5a47a8cfd423c3d4ee879e2": "\\lambda_1, \\cdots, \\lambda_n", "87382de7f827d28a813e674a85b46735": "\\;_2F_1", "87383a63013d8948014cbbad862e474d": "j^k=\\sum_{m=0}^{k}{j^{\\underline{m}}}S(k,m)\\!", "873857ae9cbb4515c7a58876faf6003d": "\\mathbb{P}\\biggl(\\bigcup_{i=1}^n A_i\\biggr) =\\sum_{k=1}^n (-1)^{k-1}\\binom nk a_k", "8738811dfa6f20550a872c445478444e": "\\exp[\\tau (D_T + D_V)]", "87393cc9931aa8e72d7b06cc5f1b622f": "\\hat{\\beta}_1 ~=~ (y ~|~ T=1,~ S=0) - (y ~|~ T=0,~ S=0)", "87395c68ec3cf6db9d0ee78f11cfb0dd": "{dx_1 \\over dt} = r_1x_1\\left(1-\\left({x_1+\\alpha_{12}x_2 \\over K_1}\\right) \\right)", "8739fe962292e24040da413c100a0d49": " \\displaystyle{(\\pi(S)\\pi(R))^3 =\\mu \\pi(Z),}", "873a242de8c595471c0097fb791a6319": "b = 2 \\times 3 = 6 \\, ,", "873a3ae745aeb179e32ffebbfc58b391": "\\textstyle 0.5%=\\frac{1}{200}", "873a673930cea8791545725ba73c0fdc": "\\frac{1}{r}\\frac{d}{dr}\\left (r \\frac{d\\phi}{dr} \\right )= \\kappa^2 \\phi \\ ", "873b0acd4cefd9823e43a7429130c5e2": "\\sum_{R\\in G}^{|G|} \\chi^{(\\lambda)}(R)^* \\, \\chi^{(\\mu)}(R)= \\delta_{\\lambda\\mu} |G|,", "873b53f88807dccd6965ee6e71d0d374": " -\\infty \\text{ to } +\\infty ", "873bc2db7acdc6b8dfa4b88e4018e4e2": "\\begin{bmatrix}x \\\\y \\\\ z\\end{bmatrix}=\nx(0)\\begin{bmatrix}e^{2t}(1+e^{2t}-2t) \\\\-e^{2t}(-1+e^{2t}-2t)\\\\e^{2t}(-1+e^{2t}+2t)\\end{bmatrix}\n+y(0)\\begin{bmatrix}-2te^{2t}\\\\2(t+1)e^{2t}\\\\2te^{2t}\\end{bmatrix}\n+z(0)\\begin{bmatrix}e^{2t}(-1+e^{2t})\\\\-e^{2t}(-1+e^{2t})\\\\e^{2t}(1+e^{2t})\\end{bmatrix} ~,", "873bc3d205ae5055c1e0796281e89be1": " \\mathbf{B} = \\nabla \\times \\mathbf{A} ", "873bc57b7d76e04e165416e8bc897a5a": "k(\\mathbf{x},\\mathbf{x'}) = \\sum_{j=1}^N \\frac{1}{\\alpha_j} \\varphi(\\mathbf{x},\\mathbf{x}_j)\\varphi(\\mathbf{x}',\\mathbf{x}_j) ", "873bd0acfa5b32d680a30c1217eb9a69": "2 a", "873bd4d18cf07ed41645669660dd1618": "\\{1,2,...,n\\}", "873bf5c40f21274458df987f25a8aaf5": "b, ", "873c09cd711d141c520783e1e1a8034d": "\\phi: \\ell_\\infty \\to \\mathbb{R}", "873c179db3da1df43a7a65278349d364": "\\vec x_n", "873c31026f9575ba7e58e0aee68ccb36": "\\varphi:W_{m_1}\\longrightarrow W_{m_2}(M)", "873c469c472db68480379cbe86e518dc": "A = \\cup _i A_i", "873d05f768e53cbaba997432cd7e8b83": "\\{ \\mathcal{H} f \\} = \\Re \\{ \\mathcal{F}f \\} - \\Im \\{ \\mathcal{F}f \\} = \\Re \\{ \\mathcal{F}f \\cdot (1+i) \\}", "873dad9afe54d7e094a348ae1210097a": "\\gcd(a,b) = \\gcd(a - b,b)\\quad,", "873e07e28f1554a6ab2023cb1a3d18ad": "\\nabla \\cdot \\bold{D} = \\rho_f \\ , ", "873e58b774ec8ed03b68adb3444f87aa": "\\chi'=0", "873e90618ce943747fffc83a43ba8233": "Q(a,x)", "873eb88bd384968f91c34022de57b95d": "\\displaystyle \\operatorname{tri} (a x)", "873f128e032f348eb2d14aeca904b849": "\\textstyle {n \\choose n, 0}, {n \\choose n - 1, 1}, \\cdots, {n \\choose 1, n - 1}, {n \\choose 0, n} ", "873f323e18017528de3ae4a8a0af04aa": " y_{n+1} = y_n + hf(t_{n+1},y_{n+1}). \\qquad\\qquad (6)", "874030b68bf9fa134d9c2434cfd05d57": "\\frac{8L^3}{E^*wt^3}", "87403d70e847f3177e5448ff0ca18501": "q = \\frac{p}{c}", "874047c69eab5e7d5ec8d9c600642669": "C_{p0}\\;", "874055a767049ab44f426299a50e8726": "\\{f(x_i)\\}", "8741009d5184ca17e8b11bea63ad2c64": "G\\left(\\frac{a+b}{2},\\frac{e}{2}\\right)", "8741178b9ddd2261f67558f4724463ed": "V(j)\\approx 0", "874145deb545dc2aec39c9474235ebc2": "\\alpha^{\\mathrm{N} \\mathfrak{p} -1}\\equiv 1 \\pmod{\\mathfrak{p} }, \n", "87414a9a2f02eb661a5ea4f22918dd24": "\\bigwedge (\\neg x_i) = \\neg\\bigvee x_i", "8741bafc49586b9845f36852132f75a4": " E_2-E_1", "87424079238ee3bbdef326eb500ba525": "\\mathbb{X}", "87425f379d2c7436ff00afe71a22665a": "\\sin \\frac{n\\pi x}{L}", "87427aa54ac407d65efa1a0ebeadb866": "\\|y(t)-x(t)\\| \\rightarrow 0 ", "8742810730994954c7a66058b4936231": "\np_{\\varphi} = - ac\n", "8742c5ed9207abb313307ee414bf6680": "E_\\uparrow(k), E_\\downarrow(k)", "87432ad949467633d71c3cdfd98f5d86": " F_2 = o, S_2 = \\_, A_2 = p ", "8743c66a7105aab5205b68913d3eb3e5": "{\\bar{\\mathbf{x}}}_k", "8743e62051441e087e43f00902e55431": "C_P \\, = \\, \\sum a_i - 37.93 + \\left[ \\sum b_i + 0.210 \\right] T + \\left[ \\sum c_i - 3.91 \\cdot 10^{-4} \\right] T^2 + \\left[\\sum d_i + 2.06 \\cdot 10^{-7}\\right] T^3", "8744117c0af678e4fd4d0d86957366b1": " w(x) \\propto \\exp\\left(S(x)\\right). ", "874496a4b27dea3c56809294ea7bb9c4": "\\min_x \\|x\\|_1 \\quad \\mbox{subject to} \\quad y = Ax.", "8744a4cbfac05ddc48692e7f3a44873c": "X=(z+a)^\\mathrm T A(z+a)+c^\\mathrm T z= (x+b)^\\mathrm T D(x+b)+d^\\mathrm T x+e ,", "8744cfc4c24ea193939946bfcaa9488e": " \\lim_{x \\to c} f(x) = L \\, ", "8744f6c290c999076457bbb620ee1dff": "\\gamma(1),\\dots,\\gamma(n)", "874523c9cd58c8df01d67f282114b1e5": "\\bar I_{\\text{L}}=\\frac{-V_o}{(1-D)R}", "8745b04b49bbb692b8ad96ed20dbcbee": "\\left( x_k,y_k \\right)|_{k=1}^{m_1}", "8745f1c21059ef580894cdd1c254d11e": "\\frac{P}{Y}= \\frac{g_n}{s_c}k", "87463b24c2c5085259a570e66d21a863": "N_{NE}\\approx H_{NE},", "8746be72ea8c63ad355e11d1662caf3c": "\\mathcal{H} = \\mathcal{H}(2,q^2)", "8746db19329be1c8864464f539d30142": "\n\\begin{align}\n{\\rm i} \\frac{\\partial }{\\partial t} {\\mathbf F}^{\\pm} \\left({\\mathbf r} , t \\right) \n& =\n\\pm v {\\mathbf \\nabla} \\times {\\mathbf F}^{\\pm} \\left({\\mathbf r} , t \\right)\n- \\frac{1}{\\sqrt{2 \\epsilon}} ({\\rm i} {\\mathbf J}) \\\\\n{\\mathbf \\nabla} \\cdot {\\mathbf F}^{\\pm} \\left({\\mathbf r} , t \\right) \n& = \n\\frac{1}{\\sqrt{2 \\epsilon}} (\\rho)\\,. \n\\end{align}\n", "87472fc6a64b891cc98a20140c6a92b3": "\\mathrm{so}(1,7) \\,", "87475f4fc9f9bd0cb60cf5a59e70ca07": "If \\ Z_1,Z_2 \\in \\mathcal{L}\\text{ and }\\lambda \\in [0,1] \\text{ then }\\varrho(\\lambda Z_1 + (1-\\lambda) Z_2) \\leq \\lambda \\varrho(Z_1) + (1-\\lambda) \\varrho(Z_2)", "874783ca01e7afe573686ee551dbbe3f": "(x_i\\ ,\\ y_i)", "874785379b98fba4ee9226c1296af7e9": "r \\in \\mathbb{R}", "8747b77b2c4e9ae8d67bae5fece2f5d7": "\\tfrac{1}{2}\\scriptstyle{\\sqrt{t+1} \\approx 0.842509}", "8747d7c2f4d80fd5ab9f6fbec01d8f62": "\\max_{a \\leq x \\leq b}|f(x)-p(x)|.", "874850e73b595ada768e9fc35bfc7e8d": "p_s(t) = \\exp \\left(-\\int_0^t h(u) \\, du \\right).", "87486963cd94ec148f129559fb62c96d": "\\textstyle V(z)", "87486d12317a1dd6711cef7883898ad4": " \\mathbf{V} = -\\int_{S_t} \\mathbf{u}^T \\mathbf{T} dS - \\int_{V} \\mathbf{u}^T \\mathbf{f} dV ", "8748a74574502d2206e95901e16ebf26": "X\\times \\mathbb{Z}", "8748df79bb8c0a0ced0ba5fd05917382": " p_d ", "8748f86644f3ac94b866bcfc2e4fa3cc": "k_{ni}\\left(k_{ij}\\right)^{-1} u_j= \\delta_{nj} u_j = u_n = -\\frac{k_{ni}}{\\phi\\mu}\\left(\\partial_i P-\\rho g_i\\right)", "87490bedab9026d55ad8b17ff5777f09": "P_2(x)=0 \\,", "874910dfe35174fef6795a79aff40152": "\\Pi_{1}", "8749b78c028011b49fce65e879f7afa1": "v \\in R\\,", "874a30112cf821161aaa0f81d6162dd6": "\\left(Y,Y^*\\right)", "874a841c233d9ae5010097c875915db2": "y-xz/w", "874a899ac71c7abb204de549f5f48279": "a > b > c", "874ae6058e46d7f0a6dc1ac55a908263": "T x = \\left( x_{1}, \\frac{x_{2}}{2}, \\frac{x_{3}}{3}, \\dots \\right)", "874b5032d13c163ea9ac077344687ea9": "R_f~", "874b65c6070349a299f753cdf54105d8": "\\varphi=1", "874b8c0f29246032c2735d328033685a": "\\vec J = \\sigma \\vec E\\,", "874c47ac3e9766e430be4393dca36f21": " {\\partial u\\over\\partial x}+{\\partial \\upsilon\\over\\partial y}=0 ", "874c9da3839f79c9955507545f8cd82e": "\\Delta G_S^\\circ = \\Delta G_A^\\circ + \\Delta G_B^\\circ - \\Delta G_{AB}^\\circ ", "874cd64b44e46b771abeb6b69f82a07d": "\n\\begin{align}\nK & {} = \\frac{1}{4}\\sqrt{(P)(a-b+c)(b-c+a)(c-a+b)} \\\\\n& {} = \\sqrt{s(s-a)(s-b)(s-c)}\n\\end{align}\n", "874d27f3ebd4970a240a772d181f0b91": "\\begin{align}\n \\vec{\\nabla} \\times \\vec{B} &= \\alpha\\vec{B} \\\\\n \\vec{v} &= \\pm\\beta\\vec{B}\n\\end{align}", "874d5d2b78a5f24238239557e35d6a31": "\na_0 = 1, \\quad b_0 = \\cos\\alpha,\n", "874d64fee8e3276bc0e3e5e7d5bf91ac": "\\bar{L} = \\frac{L^*_1 + L^*_2}{2} \\quad \\bar{C} = \\frac{C^*_1 + C^*_2}{2}", "874d895a43b60cb82133c45fc4029fc6": "k_3 \\gg k_2", "874d8e1d1b1554a9a80ff2c898df24da": " \\widehat{\\mu}_X = \\frac{1}{n} \\sum_{i=1}^n \\phi(x_i) ", "874d98cdf4bbcfbccbe24e54e859b67b": "K_{BS}", "874daf6cdce5a2d32d60a7f8c72f5f36": "(x*y)*z = x*(y*z)", "874dbadeddf7e8ba1a749d18478d7b55": "A_{circle}=\\pi\\left(\\frac{1}{2}\\sqrt{r-r^2}\\right)^2", "874dcf9837a4fb73795d5b17fa3b741e": "\\left \\langle \\alpha, \\beta\\right \\rangle := \\frac{1}{ \\left | G \\right | }\\sum_{g \\in G} \\alpha(g) \\overline{\\beta(g)}", "874de890e0137e264476d8c183b3c340": " r = {aW^{-0.25}}", "874df84dd3b185a99374a9710505250a": "\\mu_1=0\\,", "874dfd68289a229975454120dcf84c16": "A \\bigcap B", "874e080502787f148888ca8b609114a6": "-u", "874e23ce2f2f95d30c06fe2681c3cb76": "\\mathbf{\\hat{e}}_1", "874f1b8e47f64d17933f0b6e1e593b97": "X_i^p", "874f295c1dba91e46caab41759ec11a9": "\\times\\zeta(2)^2", "874f3fe168f1cd56dcdf7a20c3907332": "\n\\mathbb{C}^{k\\times k}\\otimes\\mathbb{C}^{m\\times m}\\cong\\mathbb{C}^{km\\times km}.\n", "874fbefa63add452c721fa19f42b085d": " a^{(n-1)/2} \\not\\equiv \\left(\\frac{a}{n}\\right) \\pmod n", "87506a7d0f961bd786944a2dad4d684f": "\\forall i\\neq k, p_i, p_k", "875099f3576ccefbf41b1a716176339b": "\n(h_\\text{eff})_{ab}=\\frac{1}{E}[(a_L)^\\alpha p_\\alpha-(c_L)^{\\alpha\\beta} p_\\alpha p_\\beta]_{ab}.\n", "8751155223686d83a9827fbbd94abd16": "\\nu_1, \\nu_2 , n, k", "87511c45505ed61f6ae2d7742ad4d8fc": "\\psi = \\rho^2\\xi^2 \\,", "8751875cb7dfc62e1ae80404fd52e883": "a \\leftrightarrow b", "8751d43bdfd9d6e26271ac3cb403ad74": "\\mathbb{R}P^1", "8751db55a2dab3b2249bce3775b5b52a": " SPW_x(t,f) = [ q\\,\\ast\\, PW_x (.,f)] (t) = \\int_{-\\infty}^\\infty q(t-u) \\int_{-\\infty}^\\infty w(\\tau/2) w^*(-\\tau/2) x(u+\\tau/2) x^*(u-\\tau/2) e^{-j2\\pi\\tau\\,f} \\, d\\tau\\, du", "8751e0dc7ad825898df78dc929178b7b": "\n\\int (A+B\\,x) (a+b\\,x)^m (c+d\\,x)^n (e+f\\,x)^p dx=\n -\\frac{(A\\,b-a\\,B)(a+b\\,x)^{m+1} (c+d\\,x)^n(e+f\\,x)^{p+1}}{b (m+1) (a\\,f-b\\,e)}\\,+\\,\n \\frac{1}{b (m+1) (a\\,f-b\\,e)}\\,\\cdot\n", "8752048598e779dfcce390e2f362dcb8": "00100111_2", "875208da04f8c687a77f2d9f1af48bb9": "\n\\mathbf{A} = \\frac{\\mu_0}{4 \\pi} \\frac{q\\mathbf{v}_q(t_{ret})}{\\left| \\mathbf{r} - \\mathbf{r}_q(t_{ret}) \\right|-\\frac{\\mathbf{v}_q(t_{ret})}{c} \\cdot (\\mathbf{r} - \\mathbf{r}_q(t_{ret}))}.\n", "87520ba6765abbc60497fcc4b53bab5a": "\\alpha|R\\rangle+\\beta|L\\rangle", "8752407d15919dddd63e99db01b8070f": "e^2=(a^2-b^2)/a^2", "87527f8c0720e295df3541c8625fe2e7": " S - I = \\sum_{k=2}^p {\\frac{B_k}{k!}\\left(f^{(k - 1)}(n) - f^{(k - 1)}(m)\\right)} + R ", "8752819fbe9e78fa6d568fa3cc4f731c": "{{h}^{{}^{4}\\!\\!\\diagup\\!\\!{}_{3}\\;}}={{h}_{conv}}^{{}^{4}\\!\\!\\diagup\\!\\!{}_{3}\\;}+{{h}_{rad}}{{h}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{3}\\;}}", "87529ec5889f076ca6b66f70cebb767e": "(2,\\sqrt{-3})", "8752d385383a686ab95912bc3bba723e": "\\tfrac{2\\pi}{e}", "8752e4bf733ab303e21a6741c1c393c7": "\\psi_n({\\mathbf{x}})=e^{i{\\mathbf{k} {\\mathbf{\\cdot x}}}}u_{n{\\mathbf{k}}}({\\mathbf{x}}), \\qquad \nu_{n{\\mathbf{k}}}({\\mathbf{x}}+{\\mathbf{a}})=u_{n{\\mathbf{k}}}({\\mathbf{x}})", "8753291c434442e6fed9ee34370ab515": "\\rho _{\\alpha -} ^{i_0 } ", "8753ccb08c52393d2ef1f20c542aa933": "M(t,p_1,\\dots,p_n)", "875565713131bf7eae031a86e745b4c5": "y'+a(x)y = b(x)", "87559a7923be8b4b850b9e133d24b52c": "R(p,t)(r)", "8755c380d9911460cadf0d5ad49aaea3": "1/32 = 0.03125 \\approx 0.03", "8755dc0d1b7f079416fa5f7ae5072b58": "\\sqrt{\\frac{\\nu}{\\nu+1}}\\mu.", "87567e37a1fe699fe1c5d3a79325da6f": "\\varphi", "8756994c051d367c4f0f171c95251218": "\nK(x)=\n\\begin{cases}\n-\\rho(x,X_{in}) &x\\in{X_{in}} \\\\\n\\frac{\\rho(x,X_{in})}{\\rho(x,X_{out})-\\rho(x,X_{in})} &x\\not \\in{X_{in}}\n\\end{cases}\n", "8756d20764102ad06b2f862ad81b95ed": "(\\mathbf F^X)_0 = \\bigoplus_{x\\in X}\\mathbf F.", "875766fc4267eb31a2f6efdd3cf8092f": "c_\\mathrm{A}(f(x))", "8757a2d5d7ae37ef8e2094a93b887267": "P^N=P\\times \\cdots \\times P", "8757ae251e1d1aadd1554a86357f452e": "\\operatorname{E}(X) = \\mu-\\frac{\\sigma}{\\xi}+\\frac{\\sigma}{\\xi}g_1 ,", "8757bec6e17bee5e2a225846461e8633": "R^\\gamma{}_{\\beta\\gamma\\delta;\\varepsilon} \\, - R^\\gamma{}_{\\beta\\gamma\\varepsilon;\\delta} \\, + R^\\gamma{}_{\\beta\\delta\\varepsilon;\\gamma} \\, = 0", "8757dbad1a4532abe7706f08b2f558b9": "\\delta t = \\left( 2 - {B\\lambda \\over 2} \\right)b t", "8758304e279e32591f186ab98adb6e9d": "J_\\nu(z) = \\frac{(\\tfrac{1}{2}z)^\\nu}{\\Gamma(\\nu+1)}\\,_0F_1(;\\nu+1;-\\frac{z^2}{4}),", "87585fa29b3bec0d3ad85e382bfff29c": " u_1 = \\begin{pmatrix}\n0 & i\\\\\ni & 0\n\\end{pmatrix}\n\\qquad\nu_2 = \\begin{pmatrix}\n0 & -1\\\\\n1 & 0\n\\end{pmatrix}\n\\qquad\nu_3 = \\begin{pmatrix}\ni & 0\\\\\n0 & -i\n\\end{pmatrix} ~,", "87591ff49a216ca80121d625bcdeade2": "l={\\varphi}_{{\\lambda}}\\circ\\delta_{[1,{j},{c}]}", "8759af407781fbaa92e92f1332eb62c2": " l^{*}=n^{*}-1 ", "875a0d1061160d0a9da3a14c387020ff": "W : [0, T] \\times \\Omega \\to \\mathbb{R}", "875a1f1571214bd804f675cf8a674fa6": "f_1(t)\\,", "875a419bb0507ba4a52426c52163635b": "x_{ij}", "875a7852586c067216bc27b780050362": "R_P", "875aa1668bac4f24ca58054bc659cc36": "k=2 \\pi /\\lambda", "875ad271cfd7f2760dadd8d5f81ee74d": "|\\psi_\\text{NOON} \\rangle = \\frac{|N \\rangle_a |0\\rangle_b + |{0}\\rangle_a |{N}\\rangle_b}{\\sqrt{2}}, \\, ", "875ae2c04451769cb994bd566ffcdf91": " (P - O)+O = P", "875b0b252c0f9e8e83ad898fd37f35c8": " g(z) = z + b_1 z^{-1} + b_2 z^{-2} + \\cdots ", "875b13b2c4df21aab9bca640bf2c7176": " Z[J] = \\int \\mathcal D \\phi e^{i \\int d^d x (\\mathcal L [\\phi(x)] + J(x) \\phi(x))} ", "875b19bee4919653fe320883bfa64f99": " \\mathbf{x^T A x} + \\mathbf{b^T x} + c ", "875b6a81670f5847f434c2c1cb3b4c94": "g=h", "875b874c68ced8710b17449031ab6ffe": "\\lim_{n\\rightarrow \\infty}f_n", "875bca51f6aa6fe0a50b15dc27de01c8": " (\\lambda x.(\\lambda q.q)\\ \\lambda f.f\\ (x\\ f)\\ (x\\ f))\\ \\lambda f.\\lambda y.f\\ (y\\ y) ", "875c883873b1cbf78cb5552edd7e4e72": "1+\\lfloor\\log_2 n\\rfloor", "875ceb8b4a6f27727ca99bea14b37512": " Ex. 1.", "875d1609e9cf09385344058f434925ae": "V\\otimes V^*\\otimes V^* \\otimes V\\otimes V^*.", "875d3d4abe510d205efc62556715b0ca": "Q_{in:k} = C_k \\rho |\\mathbf{u}|(k^*_s-k)", "875d888cab95ae6df556d56c15587d79": "X \\times \\hat X,", "875d88e431430328782c29b08d9baae9": "P_2 \\uparrow S(X,J)", "875d9c2c01873e2be197237541529000": "\n\\frac{\\mathrm{D} \\boldsymbol{v}}{\\mathrm{D}t} = -\\frac{1}{\\rho}\\nabla p,\n", "875dccc42e906cdf9949c5eda3b91a30": "\nr_2(n) = 4\\sum_{d|n}\\chi(d),\\;", "875de13cb3b5d825d55f1114ffb560a7": " ((\\lambda x_1 \\ldots x_{A_1}.\\lambda c_1 \\ldots c_N.c_2\\ x_1 \\ldots x_{A_2})\\ v_1 \\ldots v_{A_2}) ", "875df220c654afd434c49fa333c62527": "h_F^{(1)}(z)=\\frac{\\beta}{1+e^{-\\beta z}}=\\beta n_F(-z)=\\beta(1-n_F(z))", "875e84f4f1c5d3e8bdb32a9b64a0c41e": "\\{ n+m, n^{2}, n \\bmod m, 2^{n} \\}", "875eb5db317a477d07681278a7446cd7": "O(mdr) = O(n)\\,", "875ec5234cb7f063914f9ce7cc44ceda": "E-N+P", "875eccfff316c745fb104355a4c62a9e": "a_n = k_1 r^n + k_2 n r^n + k_3 n^2 r^n.", "875ed3c165e7458bec983de2912d60c1": "x = c", "875edd6519add4cf1b6b07496cec5e69": "\\vec u(t) = (x(t), y(t)) ", "875ee94e9a829492706a103cca05f7b0": " \\begin{align}\nK &= K_+ + K_- \\quad\\text{with}\\quad K_- = -K_+^T, \\\\\nL &= L_+ + L_- \\quad\\text{with}\\quad L_- = -L_+^T. \n\\end{align} ", "875f2c9dda2bb296919c518286b6e347": "\\langle X_i Y_j \\rangle = 0", "875f6ca9cf15e998055e5efdd7f01a73": "S_r", "875fe536d28fb2cb29dbf3b35f4129a2": "\\vert{\\Psi_{\\mathbf{p}}^{1}}^{(\\pm)}\\rangle = \\vert{\\Psi_{\\mathbf{p}}^{\\circ}}\\rangle + G^\\circ(E_p \\pm i0) V^{1} \\vert{\\Psi_{\\mathbf{p}}^{1}}^{(\\pm)}\\rangle", "876075df9ff085442618de58df5abb52": "C_{SP}(t,\\omega) = \\iint g_{SP}(t^'-t,\\omega^'-\\omega)C(t,\\omega^')\\,dt^'\\,d\\omega^'", "87608e5f44f63c0d566e21935e5c7ce0": "s=m_1^2 + m_2^2 + 2p_1 \\cdot p_2 \\,", "8760f74d862ec09ecda8d0d3ebe82be0": "\\omega_z", "876110131e3df0985815ea50833df7e1": " a_r = 2 \\Omega u \\cos \\phi + \\frac{u^2 + v^2}{R}. ", "8761190a51b68c31116fcac14ffd285b": "([1,\\frac{\\beta-t}{\\beta}],[\\frac{2\\beta-t}{\\beta}],1-p)", "876150706598f78ba74859c074d66dcc": "b_1=", "876156dc75923284d045b173992de71a": "\\mathfrak{P}^{52}", "87615f14bd93f606ec3ad79698fd0ea3": "\\sigma(f)=\\int_G f(g)\\sigma(g)\\, dg", "876206432c25cdef5722d9a86bc7326f": "\\begin{align} \\mathcal{Z}\\{nx(n)\\} &= \\sum_{n=-\\infty}^{\\infty} nx(n)z^{-n}\\\\\n&= z \\sum_{n=-\\infty}^{\\infty} nx(n)z^{-n-1}\\\\\n&= -z \\sum_{n=-\\infty}^{\\infty} x(n)(-nz^{-n-1})\\\\\n&= -z \\sum_{n=-\\infty}^{\\infty} x(n)\\frac{d}{dz}(z^{-n}) \\\\\n&= -z \\frac{dX(z)}{dz}\n\\end{align} ", "87629eaf762c4f4a92da3825d82721b7": "\\begin{bmatrix}y_{t} \\\\ y_{t-1}\\end{bmatrix} = \\begin{bmatrix}c \\\\ 0\\end{bmatrix} + \\begin{bmatrix}A_{1}&A_{2} \\\\ I&0\\end{bmatrix}\\begin{bmatrix}y_{t-1} \\\\ y_{t-2}\\end{bmatrix} + \\begin{bmatrix}e_{t} \\\\ 0\\end{bmatrix},", "8762b229e034256880690f604ec77095": "x = a\\lambda\\,,\\qquad\ny = a\\ln \\bigg[\\tan \\bigg(\\frac{\\pi}{4} + \\frac{\\phi}{2} \\bigg)\\bigg]\n = \\frac{a}{2}\\ln\\left[\\frac{1+\\sin\\phi}{1-\\sin\\phi}\\right].\n", "8762bf8352f15460502eb545a16ecf25": "(x_n),", "8762c7f7b9c6a4f99af218df8aaeedb7": " \\int_{\\phi(U)} f(v)\\, dv \\;=\\; \\int_U f(\\phi(u)) \\; \\left|\\det \\phi'(u)\\right| \\,du", "8762e0d730267d41473d27dfc7fae2fb": "x = [a_0; a_1, a_2, a_3] \\;", "8762e20226e466347b44ce6a962b25da": "\\lambda=20", "8763072adfa5a9e839d7263813a0d731": "P_1 \\times P_2 = \\{ (x,y,z,w) | (x,y)\\in P_1, (z,w)\\in P_2 \\}", "876342a57a37cbd28e5d10bac6946573": "\\mathrm{SNR} = \\frac{h^\\mathrm{H} s s^\\mathrm{H} h}{ h^\\mathrm{H} R_v h }.", "8763a0bd8538b07f8f09fd8ef609cb8d": "\\tau\\rightarrow\\tau", "8763a6d11248687f13aee261b4558365": "T:=(\\Sigma, R)", "8763f6af736904af7224b4e6ac4697ce": " \\begin{align}\nA_1 &= c_1 r_1 = \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix} \\begin{bmatrix} 1/7 & 1/7 \\end{bmatrix} = \\begin{bmatrix} 3/7 & 3/7 \\\\ 4/7 & 4/7 \\end{bmatrix} = \\frac{A+2I}{5-(-2)}\\\\\nA_2 &= c_2 r_2 = \\begin{bmatrix} 1/7 \\\\ -1/7 \\end{bmatrix} \\begin{bmatrix} 4 & -3 \\end{bmatrix} = \\begin{bmatrix} 4/7 & -3/7 \\\\ -4/7 & 3/7 \\end{bmatrix}=\\frac{A-5I}{-2-5}.\n\\end{align} ", "87640d2f88f38a0cf7005f5dceea4a98": "U_o", "8764248a7ea0d7180775abf7311c8d70": "\\frac{d m(t,V)}{d t} = \\frac{m_\\infty(V)-m(t,V)}{\\tau_\\mathrm{m} (V)} = \\alpha_\\mathrm{m} (V)\\cdot(1-m) - \\beta_\\mathrm{m} (V)\\cdot m", "87645e955842e7572b41455eef000095": "r_a+r_b+r_c+r=AH+BH+CH+2R,", "8764646ddaf6f92c370a89502d70b234": "\\Phi(\\xi,\\,s) = \\mathrm{Ai}(\\,s - ( \\xi/2)^2 ) \\exp(i(\\,s\\xi/2) - i(\\xi^3/12))", "87648121f423bcb9c5c5e31ed86fe5dd": "R (\\rho) = Jn (m, k\\rho)", "8764ac16f9d32e59c06b4b8a85d71efd": "\\bigg(\\prod_{i=1}^n x_i \\bigg)^{\\frac{1}{n}} = \\sqrt[n]{x_1 \\cdot x_2 \\dotsb x_n}", "8764b5aa109f029351ebca4cd302e1cf": "\\frac{\\Gamma' \\vdash b : Y \\qquad \\Gamma \\vdash a : X\\backslash Y}{\\Gamma'; \\Gamma \\vdash ba : X}[\\backslash E]", "8764c8ee4822c2f1511e87cb819fcbe2": "\n= \\delta(\\mathbf{p}^{\\prime}-\\mathbf{p})\n\\sum_\\mathbf{k} \\left[ \\left| F(\\mathbf{p}/2-\\mathbf{k}) \\right|^2\n n_a(\\mathbf{k}) + \\left| F(\\mathbf{k}- \\mathbf{p}/2) \\right|^2\nn_c(\\mathbf{k}) \\right] ", "876524dbc471c3024eda9d302e0fb222": "H(X_2),", "87652d5671b37f077d5dcf8fba1e47fe": "\\angle ADC=\\angle CDB", "876546e4d29fad0ef803f3580566ddd1": "On(\\text{coin},\\text{white area})", "876556da1e225660011e51e112c2534d": "\n \\begin{cases}\n a+\\frac{\\sqrt{(b-a)(c-a)}}{\\sqrt{2}} & \\mathrm{for\\ } c \\ge \\frac{a+b}{2}, \\\\[6pt]\n b-\\frac{\\sqrt{(b-a)(b-c)}}{\\sqrt{2}} & \\mathrm{for\\ } c \\le \\frac{a+b}{2}.\n \\end{cases}\n ", "876577cdac3a2350f4a3fb9d4de2f1ce": "Q_{T_j}", "8765b10e3626c1cc6dd95e9c9ed7f572": "H + 1^2 , H + 2^2 , H + 3^2 , H + 4^2 , ... , H + k^2", "8765bf00e00dbef15af48df1e19c5727": " y= \\begin{cases}\n0~~ \\text{if}~~y^* \\le 0, \\\\ \n1~~ \\text{if}~~00).", "87702ded532360e9aed916d11c998886": "c \\approx 3\\times 10^8~\\mathrm{m/s}\\,\\!", "877060b3d2878568b851013adbf2388f": "\n\\begin{align}\n\\left[ R_{pq}(\\theta_2)\\, R_{pq}(\\theta_1) \\right]_{m,n} =\n\\begin{cases}\n\\ \\ \\ \\ \\delta_{m,n} & m,n \\ne p,q, \\\\[8pt]\n-i e^{-i\\theta_1}\\, \\sin{\\theta_2} & m = p \\text{ and } n = p, \\\\[8pt]\n-i e^{+i\\theta_1}\\, \\cos{\\theta_2} & m = p \\text{ and } n = q, \\\\[8pt]\n\\ \\ \\ \\ e^{-i\\theta_1}\\, \\cos{\\theta_2} & m = q \\text{ and } n = p, \\\\[8pt]\n+i e^{+i\\theta_1}\\, \\sin{\\theta_2} & m = q \\text{ and } n = q. \n\\end{cases}\n\\end{align}\n", "8770fc731459329625d4f1b9bf467450": "\\nabla^2 U = -{\\partial^2 (M_x B_x + M_y B_y + M_z B_z) \\over {\\partial x}^2} - {\\partial^2 (M_x B_x + M_y B_y + M_z B_z) \\over {\\partial y}^2} - {\\partial^2 (M_x B_x + M_y B_y + M_z B_z) \\over {\\partial z}^2}", "877138dbb8c7ee495d73947506ceb6ee": "\\langle \\Psi_m , \\Psi_n \\rangle = \\int\\limits_{-\\infty}^\\infty d x \\, \\Psi_m^*(x, t)\\Psi_n(x, t) = \\delta_{mn} \\,,", "877164887c8dfb4a6a12b9b44ea992e9": "\\frac {\\Delta G_{mix}} {RT}= \\frac {\\phi_1} m_1 ln\\phi_1+ \\frac {\\phi_2} m_2 ln\\phi_2+\\chi \\phi_1\\phi_2", "87716fe9f2f822a2fb3f1c2cf2063d1d": "\n\\begin{array}{lll}\nv(\\varnothing) = 0 & & \\scriptstyle{\\text{Strictness property}}\\\\\nv(U)\\leq v(V) & \\mbox{if}~U\\subseteq V\\quad U,V\\in\\mathcal{T} & \\scriptstyle{\\text{Monotonicity property}}\\\\\nv(U\\cup V)+ v(U\\cap V) = v(U)+v(V) & \\forall U,V\\in\\mathcal{T} & \\scriptstyle{\\text{Modularity property}}\\,\n\\end{array}\n", "877190fe1ca8d2218f7cc6f181f2fe20": "\n (\\mathcal{R} \\circ \\mathcal{E})(\\rho) = \\rho \\quad \\forall \\rho = P_{\\mathcal{C}}\\rho P_{\\mathcal{C}},\n", "87720a564198ff34dbb9cd3e0d98069d": "g(x) = \\sum_{n=0}^{\\infty} g_n x^n", "87728d86939b28e9f4c986e4fbeed8cd": " n = a m^b / D^2 \\, ", "8772d7328f264b3980fe923e74af6f6d": "\\dfrac{ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h}{ix^4-jx^3-kx^2-lx-m}=nx^3+ox^2+px+q+\\dfrac{rx^3+sx^2+tx+u}{ix^4-jx^3-kx^2-lx-m}", "8772fdbf584118397e1c582fdc160423": "|{\\Phi^{[k+1..N]}_{\\alpha_k}}\\rangle=\\sum_{\\alpha_{k+1},\\alpha_{k+2}..\\alpha_{N}}\\Gamma^{[k+1]i_{k+1}}_{\\alpha_k\\alpha_{k+1}}\\lambda^{[k+1]}_{\\alpha_{k+1}}\\cdot\\cdot\\lambda^{N-1}_{\\alpha_{N-1}}\\Gamma^{[N]i_N}_{\\alpha_{N-1}}|{i_{k+1}i_{k+2}..i_N}\\rangle", "87730b7b6c20f4e8d8d906470de5be86": " P(x>1) + (Cb>>2) + (Cb>>6)", "878a4dd0bff132c7f62ff89de9f802a8": "u = \\exp(aj) = \\cosh a + \\jmath \\sinh a", "878b41222bb6f474cd46c310045c2830": "\\left( \\dot{X}^a \\, X_b + {X^a}_{;b} \\right) \\, X^b = {X^a}_{;b} \\, X^b - \\dot{X}^a = 0", "878b57a3c31454f927ac3f147db5e407": "\\eta = 1 - \\sqrt{\\frac{T_c}{T_h}}", "878b724c4bb84e60e1363927961a85d3": "\n| \\langle \\psi _{\\rho}| \\psi _{\\sigma} \\rangle | \n= | \\langle \\Omega | ( \\rho^{\\frac{1}{2}} \\otimes I) ( \\sigma^{\\frac{1}{2}} V_1 \\otimes V_2 ) | \\Omega \\rangle |\n= | \\operatorname{Tr} ( \\rho^{\\frac{1}{2}} \\sigma^{\\frac{1}{2}} V_1 V_2^T )|.\n", "878baf93d368ed1e175539e492811b10": "dS ", "878bd532f1718635c637124be801e4d9": "1/n", "878c3a834b9495c23c9b7173ee07b7bd": "M_{12}", "878c8680d8c6de1a2ab422973286ff09": "f^{\\mathrm{e}}", "878c8b54e5ae9d35278ec58de9c39574": "\\displaystyle{G(c,p)= \\left({c\\over p}\\right)G(1,p)}", "878cc95bc041e4da21843cbdf952db46": "P_{r}=\\frac{\\exp\\left[-\\beta H\\left(r\\right)\\right]}{Z}\\,", "878d6a4618625a9faf54aaa70cb9ff1a": " a<0", "878dac18936558f6ee940cbb1b7f1c8f": " k=-5,-4, \\cdots ,5", "878db0c1ef8e26f5993b1a72b892e4b3": "\\lim_{n\\to\\infty} {D_{n,k} \\over n!} = {e^{-1} \\over k!}. ", "878db7b6d818ac5ed300a67162267fe4": "h_\\ell^{(1)}", "878e040d330775179be2df13e7c96c3f": "e(p_{1},p_{2},u)", "878e0f331adc6fa4511367db6c8adcf0": "\\eta(\\bar{X})\\cdot v", "878e1b2b63f2b6c0a4bf9dc606cf6523": "CAT(0)", "878ec6d2a9d345f2d3fb04a4eb0f4656": "\\sigma_k(n)", "878ee7675064370b5f3ad3326de91308": "\nG_\\mu (s,t)=E_\\mu (\\nu t^\\mu (s-1)), \n", "878fa07bb1e29840f4e7716aed345d96": "\\int \\phi^2\\, \\text{d}x,", "878fa94cb9cd1226056cf0ebfa0fd3ad": " \\tau_p = \\frac{\\rho_p d_p^2}{18 \\mu} (1 + 0.15 Re_p^{0.687})^{-1}, ", "878fbeda6c916e0f22c74a019adf5a33": " L/D = \\frac{\\mu_0}{2 \\pi} \\ln \\left( \\frac {b}{a} \\right) \\, ", "878fe65f552edc434403dd1e52a038ed": " T : (x, y, z) \\rightarrow (y z, z x, x y), ", "879029b4b7bdc7d93cf0d0bcd9e8e49b": "\\mu_r = (1 - \\frac{\\phi}{A})^{-2},", "87902a1be44473b5bb0266290a5e7940": "F(t) = S(t)\\times (1+r)^{(T-t)}\\,", "87906a238200610c80577da5fe3b9cf6": " \\delta (x-y), ", "87909ba5c05c3d7d460d85e64b617a2c": "\\displaystyle W_{\\zeta}(y|x) = \\sum_{s \\in S} W(y|x, s)P_{S_r}(s)", "8790bc42a6090ac9c8055ab539d9d9f2": " {M} \\,\\overset{\\underset{\\mathrm{def}}{}}{=}\\, ({A} {Q}^{-1} {A}^{T})\\,", "8790c2ce161360ee9ac34fcb29926d01": "\\hat{a}_{j}", "8790d00ed7495623d730bc4cf171cc50": "\\hat f(n) := \\frac{1}{2\\pi} \\int_0^{2\\pi} f(t) e^{-int} dt ", "8790d8b08ba50c7c52c8edfe98894d30": "Y_n(x)", "87910b2194f9e9fe7f7168977ad32d68": "\\nabla_\\mathbf{Y} f = \\operatorname{tr} \\left(\\frac{\\partial f}{\\partial \\mathbf{X}} \\mathbf{Y}\\right).", "87914146b3fefaea90285b1da9a48df2": "(bc)^{|n|}+(ac)^{|n|}=(ab)^{|n|}", "879160cdef87fa8d1e6e56f411076cbf": "U(\\vec{r},t)\\!", "879191d6cfffa873f6a3c16dafa66f1a": "\n \\sum F = -10 + R_a - (1)(x-10) - V_2 = 0\n ", "8792228e189861daa853c1190803a771": "\\Delta^* \\colon H^\\bullet(X \\times X) \\to H^\\bullet(X)", "8792bdaa5c8350ac83d609c84e55f381": "\\ t \\in \\mathbb{R}", "879307ea63561dc59d7cc12803eeeb8d": "T_{In}\\,\\!", "87931d5352fe34102a3e4864b07faea2": "\\displaystyle \\frac{\\sqrt{2\\pi}}{\\Gamma\\left(\\alpha\\right)}u\\left(\\pm\\omega\\right)\\left(\\pm\\omega\\right)^{\\alpha-1} ", "8793d72eb8f8bb27b265908483a10bdb": "I_1,\\dots,I_n", "87942eb4393e452835ae577ab3f7559b": "f(-ia,-ze^{(1/4)\\pi i}), f(ia,-ze^{-(1/4)\\pi i})\\text{ and }f(ia,ze^{-(1/4)\\pi i})\\,", "87947175571104af966b572886145d0a": "X\\, .", "87948aefc3858202789965c9eb123d96": " x,p ", "8794cd493ad2cfca222c68f13acff182": "dt=abc\\ d\\tau.", "8794e24b0d65e4c75fd51b5c662728cb": "\\tau(s,a,t)", "87957040518f8eead09db67c0964e134": "\\hat{\\rho}_{XY\\cdot\\mathbf{Z}}", "87958516000b9c30a1caec066359bca4": "\\frac{\\partial \\mathbf{f(g(u))}}{\\partial \\mathbf{x}} =", "8795da71d54b243e90e8f51beff7afbc": "x < 10^4", "8796625fa69a2a2140aba319e8e62f16": "R \\ge 1/\\limsup_{n \\rightarrow \\infty}{\\sqrt[n]{|c_n|}}.", "87968a164a7e241e16d180aff88d1cde": "|{\\Phi^{'[{{JC}}]}}\\rangle=\\sum_{i,\\alpha}\\Gamma^{'[{{C}}]i}_{\\alpha\\beta}\\lambda_{\\alpha}|{{\\alpha}i}\\rangle.", "87968e0de388c94fcb2400bcbe1ec55a": "\\displaystyle \\Delta", "8796ce4f0c4a635985e613b715d5ff56": "p_6(x)", "8796ff3d6f9fabd4adf29a790a5dd798": "\\begin{align}\n G(x,y,z,t)=\\ & (z(x^2+y^2+z^2+t^2)+x (6x^2-2y^2-2z^2-2t^2), \\\\\n & \\ t x \\sqrt{2}+y (6x^2-2y^2-2z^2-2t^2)).\n \\end{align}", "8797004c193f1cd4cde66a9118acc9f1": "|H:H\\cap K| \\le |G:K|,", "87977c69a2ac79f23be1510da8a1e196": "\\textstyle \\beta>0", "87978fec8a454ed7de534c1424f11095": " D_\\alpha = \\sum_n n \\frac{|W_n^\\alpha|}{|W|} ", "87986c11517af361197847527d18dead": "G-SPAB = \\frac{GS}{AB} ", "879895bf47e9ae839207c80a30f9a37c": "\\mathbf{a} \\times \\mathbf{b} = \\mathbf{\\hat{a}} \\mathbf{b} ", "87990405515871c78cd40e599fbe0ba8": " \\!\\ \\delta_S^n = K_n\\delta_S + K_{n-1} ", "8799210524bd0a8dec4a63f6e0d7dd04": "k = \\tan \\frac{\\alpha}{4}, \\tan \\frac{\\alpha+2\\pi}{4}, \\tan \\frac{\\pi - \\alpha}{4}, \\tan \\frac{3\\pi - \\alpha}{4} \\,", "879933de168130f9ce50e8fafd45ee45": "F_{\\mathbf P_1}", "879940bb2335b29950238729f1ad04b3": "\\frac{\\omega}{c}a << 1", "8799876deecc372b9c0563ef0e0bc846": "y'+3y=6t+5,\\qquad y(0)=3", "87999e9a847d373d062c24646697f85d": "V = {2 \\pi a \\over \\lambda} \\sqrt{{n_1}^2 - {n_2}^2}\\quad = {2 \\pi a \\over \\lambda} \\mathrm{NA},", "8799cc8ad9c89ed383698006bdf415a3": "G_0=0", "879a0990a7986f2ac360fc76b9272f45": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 1}{4 \\choose 2}{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "879a524acd44fc58fe69afd3588d359f": "\\left(\\frac{\\partial^2 \\Delta S_{mix}}{\\partial x^2}\\right)_{T,P} = -nR\\left( \\frac{1}{x} +\\frac{1}{1-x} \\right)", "879ac510dc6f2f6c1477d151ade90fc1": "m_e(f) \\in \\mathbb{F}_p", "879acf7ee60cf7d5d8b8f1fd6da758c3": "a(x)=\\int{f(x)\\,dx}.", "879ad69b8dd14eb1e1218d535b0bd641": "K \\subset \\cup_{k \\in K} V_k", "879b8dbdbfc356dbf11835054a12ac8a": "\\textstyle U_{N} (\\mathbf{r}_{1}\\, \\ldots, \\, \\mathbf{r}_{N})", "879bf220854e2a4387ebe82417ffdf77": "\\nabla p^{n+1}\\cdot \\mathbf{n} = 0", "879c121284cc636e7d41407ad57eb2dd": "\\sigma_{0,\\,0.25}", "879c99a63c6ec03ec4968a6da3384194": "\\ln(\\sqrt{2 \\pi e} \\sigma Z) + \\frac{\\alpha\\phi(\\alpha)-\\beta\\phi(\\beta)}{2Z}", "879cbc7ae9d4b00589d778fb47dbf145": " C_n\\times X^n\\to C_n", "879cc548fd94a8aeb5aea339c94c9f36": " x_i \\mapsto (\\ell_i, r_i) ", "879d6a1cf63000e400efc02232dfb258": "E_1=-b/a", "879d709f9da3f3bda0fb92eb5db1508a": "\\dot{p} = \\gamma\\left( -(1+i\\Delta)p + xD\\right) ", "879da9fe542e17a64e96d7f0db3c530b": "a^{4n}-b^{4n} = \\left(a^{2n}+b^{2n}\\right)\\left(a^{2n}-b^{2n}\\right).", "879dea70ceeeafbfd14cb91df8564ecd": "\\mathrm{v}_\\perp=|\\mathrm{\\mathbf{v}}|\\,\\sin(\\theta).", "879e16f42a4f30a662df8e4b925890b7": "\\boldsymbol\\omega \\times \\boldsymbol{v} = \\dot{\\mathbf{Q}} \\mathbf{Q}^{-1}\\boldsymbol{v}", "879e5fd8be04c5de7dfa018eea7421d9": "\\begin{align}\n &\\Pr(N=n\\mid M=m,K=2) \\\\\n = {} &(n\\mid m) \\\\\n = {} &\\frac{\\mathcal{L}(n)}{\\sum_n \\mathcal{L}(n)} \\\\\n = {} &[n \\ge m]\\frac{m - 1}{n(n - 1)}\n\\end{align}", "879e882936c437d70e0452c98a8f6056": "\\mathrm{erf}(x)= \\frac{2}{\\sqrt{\\pi}}\\int_0^x e^{-t^2} dt=\n\\frac{2x}{\\sqrt{\\pi}}\\,_1F_1\\left(\\frac{1}{2},\\frac{3}{2},-x^2\\right).", "879ea62496a24018fbd52814a67a47a9": "\\phi^2\\chi^2", "879edfae60246d6621578b0dc168cd14": "f^\\text{inc}(x):= \\arg \\sup_t \\,t\\cdot x-\\int_0^1 \\max\\{t-f(u),0\\} \\, \\mathrm d u,", "879f0c46c96da246d1492ac006cbedf4": "\\mathbf{T}q = \\frac{\\mathbf{T}\\alpha}{\\mathbf{T}\\beta}.", "879f456572784fce80803aabf8e6d023": "3J' = \\ln(\\cosh(4J)).", "879f51a150194a4e8d5e93770c857849": "\\displaystyle{\\mathfrak{H}=\\mathfrak{H}(\\overline{\\Omega})\\oplus \\mathfrak{H}(\\overline{\\Omega^c}),}", "879f8907e67a3892f8a54685e4dc6a99": "M_0, \\dots, M_k", "879f95507b0aac992b8dfde94717d6fd": "W_{ad} = \\gamma_m + \\gamma_o - \\gamma_{mo} ", "87a0321dbbed4dfa251561127212c918": "\\frac{A^*-B^*}{2}", "87a04fc2e6c43ee338d34dfc889a2115": "\\scriptstyle (X,\\tau,\\tau^*)", "87a0a4898515c330e37c5f76b655e465": " r_{0ij} = 2^{1/6}(\\sigma)", "87a0a86b49def7c0c76edb96b33a8728": "\\omega = \\angle R_N(1) = \\tan^{-1}\\frac{im\\{ R_N(1) \\}}{re\\{ R_N(1) \\}}. ", "87a0f9cf3a6ee1260039ef4f7a9abfbd": " a_i\\wedge \\alpha_j", "87a133253bbadbe2c49b530f52f3143c": " \n \\eta_{\\mu \\alpha} \\left ( \\partial^2 + m^2\\right ) D^{\\alpha \\nu}\\left ( x-y \\right ) = \\delta_{\\mu }^{ \\nu} \\delta^4\\left ( x-y \\right ) \n", "87a13ae194e86391102161492b9192f8": "|\\mathcal Z|=6", "87a1a2b14330de0429b77ef36da021a8": "K(\\vec{x},\\vec{x}';t)=\\prod_{q=1}^N K(x_q,x_q';t)", "87a1c94a79093a85567cc42f65f72d57": "D_{P}", "87a287b635f80ca5bddba4c0f4dbc89b": " \\boldsymbol{ \\nabla \\times B} = \\mu_0 \\left(\\boldsymbol J +\\varepsilon_0 \\frac {\\partial \\boldsymbol E}{\\partial t}\\right) = \\mu_0 \\left( \\boldsymbol J_f +\\frac {\\partial \\boldsymbol D}{\\partial t}\\right) \\ , ", "87a290ce4f977cf40360c135bd875914": "\\partial/\\partial c^a", "87a31de0f7ea37a7a8cb668d1ee3f89f": "\\iota_\\epsilon", "87a329ed3f2b88cc682d4315318d8a77": "v_{(G; c)}(\\emptyset)=0", "87a369a753c5391dcf5b302ad50f6e16": "C := \\{ \\{ x_n : n_0 \\leq n \\} : n_0 \\in \\mathbb{N} \\}.\\,", "87a371930137c6cbd998b0a2a3e03638": " \\mathbf{u}= (U_1/U_0, U_2/U_0, 0) ", "87a399778843e541d9a750d6facadf80": "S''_{zz}(0)", "87a3a15bff4f75f9412744a7a56d3770": "a \\times 2a \\times 3a \\times \\cdots \\times (p-1)a \\equiv 1 \\times 2 \\times 3 \\times \\cdots \\times (p-1) \\pmod p.", "87a44fc64dcf33973dfa983ca8447198": "\\vec \\omega", "87a49a5e3aed248a16ada8f84e64c033": "f : A \\rightarrow B", "87a4ce884b481d5498e01274db1da75f": "^{3}He + ^{3}He = ^{4}He + 2 p ", "87a52b84607ef352db570f8c5d7a7814": "1 - w^{n}", "87a55440b5d361befa0c2bb499e12a23": "\\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3 = 0.", "87a5e2d0da29cc48c032ace8d192a327": "R_\\lambda(A)", "87a6144055f761ec005b9bc61b350a5c": "((P \\and Q) \\to R) \\Leftrightarrow (P \\to (Q \\to R))", "87a66df997f57679807fe298f800c252": "\\omega(x,\\overline{\\theta},\\theta)^*=\\omega(x,\\theta,\\overline{\\theta})", "87a6e0c69133b0e6870c2b60c2973b38": "A^T P + P A = -q q^T\\,", "87a726a924d60490a9cef184899bd7d6": "n(q)", "87a7910a648846b53079c10741c3f0c0": "\\textstyle{\\varepsilon '(\\mathbf{s})}", "87a7ffa3295fbfdb2cba16f60fd2d96b": "\\mathbf{y}=\\mathbf{Xr}+\\mathbf{e}", "87a8d3a72104ff376ca6b4e471e69c20": "(\\mathbf{W}^i -\\mathbf{G}^i)\\cdot(\\mathbf{W}^i -\\mathbf{G}^i) - (\\mathbf{W}^1 -\\mathbf{G}^1)\\cdot(\\mathbf{W}^1 -\\mathbf{G}^1) =0,\\quad i=2,\\ldots, 5.", "87a938e106d505cd5670d2b7e7d73aee": " \\Delta=2(\\frac{K}{N}-p) \\! ", "87a94f75f031e0b8342c47ed4cf7d285": "\\vec a_g", "87a96624668573b77ba60ad258c4b224": "\\tbinom n{k_1,\\cdots,k_n}", "87a9768799dffdf95f9085ec2b9e2794": "\\Sigma^1_k", "87a9d74c43e82ac1fa508a53c404e000": "(u_j,h_j)\\in U_j \\times Homeo(F)", "87a9f08cde9be4c4decfed19d506b5bc": "y \\ge 0.440", "87aa92524c4e0b3a254db865233483db": "\\langle \\cdot,\\cdot\\rangle", "87ab2274cf71643f8857998def03df81": " h^2 \\lambda + 2 = (4 \\beta^2 - 2). \\,\\!", "87ab27a467854ec218746aac3150f18b": "Y_j=y_j", "87ab7fc9a353734c6a0889678131d9a9": "\\beta _p(T,p)\\ ", "87aba1cd267eb9e7533bde6712ff6685": "\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n}", "87ac0e057905a66e902b92bf59a61979": " \\sum_\\text{cyclic} a^2S_A = a^2S_A + b^2S_B + c^2 S_C = 2S^2 \\quad\\quad \\sum_\\text{cyclic} a^4 = 2(S_\\omega^2-S^2) \\, ", "87ac8824b9df5e203d67bf479d0a8972": "z_{b}", "87ac991f0862dbdae073faf4c37f677d": "\\gamma \\vec{u} = d\\vec{x}/d\\tau", "87ac996ae6a1671e8fa980a664f0ac59": "\\phi _n", "87acb6b9c332a73b5d315f87de20613d": "Ef(x) = \\frac{d \\log f(x)}{d \\log x}.", "87acb9fa4c895d0d2b9a6c79df8bf278": "\\mathrm{{}^{90}_{38}Sr}\\rightarrow\\mathrm{{}^{90}_{39}Y} + e^- + \\bar{\\nu}_e", "87acd4e60d9b001a9befccee0ed486b7": "s=\\left(x_1\\left(x_3\\right),x_2\\left(x_3\\right),x_3\\right) \\ ", "87acf1fd02bd06f70d7388e1c056f39e": "\\nabla \\times \\mathbf{A} = \\displaystyle{1 \\over r\\sin\\theta}\\left({\\partial \\over \\partial \\theta} \\left( A_\\varphi\\sin\\theta \\right)\n - {\\partial A_\\theta \\over \\partial \\varphi}\\right) \\boldsymbol{\\hat r} +\n \\displaystyle{1 \\over r}\\left({1 \\over \\sin\\theta}{\\partial A_r \\over \\partial \\varphi}\n - {\\partial \\over \\partial r} \\left( r A_\\varphi \\right) \\right) \\boldsymbol{\\hat \\theta} +\n \\displaystyle{1 \\over r}\\left({\\partial \\over \\partial r} \\left( r A_\\theta \\right)\n - {\\partial A_r \\over \\partial \\theta}\\right) \\boldsymbol{\\hat \\varphi},", "87acff52eaeae61c8efeb23f7bd5f593": "\\sin (\\arccot x) = \\frac{1}{\\sqrt{1+x^2}}", "87ad0644f6e33b39b1b43c29ac9f604d": "\n\\hat{w}^{(L)}_k = w^{(L)}_{k-1}\n\\frac{p(y_k|x^{(L)}_k) p(x^{(L)}_k|x^{(L)}_{k-1})}\n{\\pi(x_k^{(L)}|x^{(L)}_{0:k-1},y_{0:k})}.\n", "87ad368081633fe8011f8124f3c1e457": "f_*([Y] \\cdot f^*([Y'])) = f_*([Y]) \\cdot [Y'].", "87ad5e6b91a03e0a301ee93f1a71d782": "2 N + 4", "87aea817e1d23ee4ef5a873cea262daa": "b_1 \\geq b_2 \\geq \\cdots \\geq b_n,", "87aec0bd9a7d876f0d1e78ea0afa798c": "\\nabla \\cdot \\mathbf{u} = 0", "87af111e1e31583398a99a3ba695649e": "H(x-a)\\,", "87af4af8eef1b4abe25971bb0bd96eee": "Z_{ab}g^{ab}=\\, 0.", "87af5e5d453571fe3676b519831f78d0": "= 2\\ \\gamma^\\rho \\gamma^\\nu - 2\\ \\gamma^\\nu \\gamma^\\rho + 4\\ \\gamma^\\nu \\gamma^\\rho \\,", "87af7e46230471f9d60f06e5ddc1849d": "\n G = \\cfrac{1}{2}\\left[\\cfrac{1}{(\\sigma_3^y)^2} + \\cfrac{1}{(\\sigma_1^y)^2} - \\cfrac{1}{(\\sigma_2^y)^2}\\right]\n ", "87af98d147f5c08f7c549c8db4f7136c": "y^5 - y + a = 0\\,", "87b002cf2c8323fbafb1ffd71381877b": "\\frac{D\\rho}{Dt} + \\rho (\\nabla \\cdot \\mathbf{v}) = 0.", "87b03e99633ed3fa85e2903f9f8ba5aa": "B^s_{p,q}(\\R)", "87b04471815d8a84672f9a4838e915b5": "H_n(x)=(-1)^n e^{x^2}\\frac{d^n}{dx^n}\\left(e^{-x^2}\\right)", "87b05f185bf5b513ea8ba21457ac0619": "x\\cot(x)", "87b0633b3ed5e2dbbfc410ad82359a8e": "\\prod_{i \\in I} S_i ", "87b0f6d1f8b2a37dbb289dc958b7a644": "V_M(\\lambda)", "87b0fe9b18c25b742d38ef74d23cfea6": "T^{\\mu\\nu} = \\frac{2}{\\sqrt{-g}}\\frac{\\delta (\\mathcal{L}_{\\mathrm{matter}} \\sqrt{-g}) }{\\delta g_{\\mu\\nu}} = 2 \\frac{\\delta \\mathcal{L}_\\mathrm{matter}}{\\delta g_{\\mu\\nu}} + g^{\\mu\\nu} \\mathcal{L}_\\mathrm{matter}.", "87b11523daac3ccbd32f4356bfb3bda0": "FK=(FK_0,\\ FK_1,\\ FK_2,\\ FK_3)", "87b1acf6ef617d34cede362f8df3a4ef": " \\Sigma E_n \\to E_{n+1} ", "87b1dac74314df781539d6bb72e6d787": " \\chi_w(k, n)= 2^{n/2}\\int_{-\\infty}^{\\infty}x(t)\\phi(2^nt-k)\\, dt", "87b214659d1d3f240f500152f2f1b04b": "\\Phi=M\\Theta", "87b251077e0e58d3fa194875ec6570d8": "{\\tilde{A}}_2", "87b27108fa5f54f7ae05d54763d5a837": "\nP(r,t)=A\\frac{\\lambda+1}{\\lambda}\\frac{(e^{t/\\tau})^{b/2D}}{1-e^{-t/\\tau}}\\left(\\frac{\\sinh(\\frac{t}{2\\tau})}{\\lambda}\\right)^{\\frac{b}{D}+1}\\left(\\frac{4\\lambda^2}{(\\lambda+1)^2e^{t/\\tau}-4\\lambda}\\right)^{\\frac{b}{D}+\\frac{1}{2}}.\n", "87b2ba6fdf0e8c06c9d4ea3d07ada2fc": "q_{i, i-1} = i\\, \\lambda", "87b2bcbea3d16c33b6b0b67ac2a52945": "\n\\begin{matrix}\n R = Q^{T}A =\n\\end{matrix}\n\\begin{pmatrix}\n 14 & 21 & -14 \\\\\n 0 & 175 & -70 \\\\\n 0 & 0 & 35\n\\end{pmatrix}.\n", "87b2ec464e3dcb4c5522c4d90cd2c762": "\\displaystyle\\pi(D)f =\\lambda_D f,", "87b345d10a266c75fa0d2ca959b14387": "\\sum_{i = 0}^\\infty 2^{-i-1} = 1.", "87b393ea6778445576e100273b94f532": "m\\in\\mathbb N", "87b3cc050f3c9c3f2125825734a06be6": "\nE=0.337 + \\frac{RT}{2F} \\ln a_{\\rm Cu^{2+}}\n", "87b3ee1e6e87d42f82c6127e814d8fec": "3{n+1\\choose0}_2-{n+2\\choose0}_2=f_n(f_n+1)", "87b43165ac7ffd86b94d089357b215b9": "\\ y[n] = b_0x[n] + b_1x[n-1] + b_2x[n-2] - a_1y[n-1] - a_2y[n-2] ", "87b434db2b37129414511cd7222ecef5": "\\left(r,\\theta\\right)\\in \\left[0,1\\right]\\times S^{n-1}", "87b435d84f895dcedcb026f49a6ea738": "\\mathbf{p}=n\\frac{\\mathbf{ds}}{ds}=\\left(p_1,p_2,p_3\\right)", "87b43d93a68691751b1e83bf8adf060b": "\\mu_1 = 2", "87b4572fce89712d6d4197de1d69afb7": "\\Theta_{j}", "87b466255ba4621cfc70b7a0ff6729b3": " g_{\\mu\\nu} = \\begin{pmatrix}\n-1 & \\gamma \\cos( \\theta ) \\cos ( \\phi ) \\frac{v}{c} & \\gamma \\cos( \\theta ) \\sin ( \\phi ) \\frac{v}{c} & -\\gamma \\sin ( \\theta ) \\frac{v}{c} \\\\\n\\gamma \\cos( \\theta ) \\cos ( \\phi ) {\\frac{v}{c}} & 1 & 0 & 0\\\\\n\\gamma \\cos( \\theta ) \\sin ( \\phi ) {\\frac{v}{c}} & 0 & 1 & 0\\\\\n-\\gamma \\sin ( \\theta ) \\frac{v}{c} & 0 & 0 & 1\n\\end{pmatrix} ", "87b48f8299d933cf23c7c63e04eadaa6": "\\Gamma, \\Pi \\vdash \\Delta,\\Lambda", "87b525508b06b907d586dc927ac6a6b1": "{\\mathbf{}}\\tau(t)\\hat{P}(t)=\\hat{P}(t)\\tau'(t)=\\hat{P}(t), \\tau'(t)\\hat{S}(t)=\\hat{S}(t)\\tau(t)=\\hat{S}(t)", "87b527bc759bb86fdc721fc7ae9ad86a": "\\int\\sinh^n ax\\,dx = \\frac{1}{a(n+1)}\\sinh^{n+1} ax\\cosh ax - \\frac{n+2}{n+1}\\int\\sinh^{n+2}ax\\,dx \\qquad\\mbox{(for }n<0\\mbox{, }n\\neq -1\\mbox{)}\\,", "87b58b38fd1f3b29053f59bd8fbf58d2": "\n\\begin{bmatrix}\n0 & 1 & 0 \\\\\n1 & -4 & 1 \\\\\n0 & 1 & 0\n\\end{bmatrix}\n", "87b5adc519d005ff9247e238ae80221b": "B, C, M", "87b613f390f02af17b46b4808485c8d6": " E = - \\frac{1}{n^2} \\frac{Z^2e^2}{2a_0} = - \\frac{Z^213.6eV}{n^2}", "87b64381aa9032516a346326d24f8227": "\nY = C / X^n, \\,\n", "87b7175e926acf51587d762950be6feb": "X(t) = X_1(t) + \\cdots + X_n(t)", "87b78f725543be41e1e4cd57793d071f": " a_S = (\\pi_{10}\\pi_{21} - \\pi_{11}\\pi_{20}) / \\pi_{21} ", "87b81832f5f4ec84a60991c29292740e": "\n |((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\\rangle.\n", "87b8460d4225495c56fe32dda8a32f4e": "\\phi(a_i)=\\displaystyle\\sum_{k=1}^q\\phi_{k}(a_i).w_{k}", "87b868560e92cf929755e314e112d926": "P_X", "87b886ca249c653afdca5b8a213f03a4": "\nH=\\sum_{i=1}^N \\left(-{\\hbar^2\\over 2m}{\\partial^2\\over\\partial\\mathbf{r}_i^2}+V(\\mathbf{r}_i)\\right)\n+\\sum_{i=\\sum\\limits_{n=0}^\\infty nP_\\mu\n(n)=(\\mu |\\varsigma |^{2\\mu })/\\Gamma (\\mu +1)\n", "87c0025f17df7070e127ea8a6d0b9ce3": "Q = {{125-(70)} \\over {1.5+0.1+4}} = 9.8 \\ \\mathrm{W} ", "87c018d9fe7ebd337aaba0e220f06ec0": "\\mathcal{S} = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-h} h^{ab} g_{\\mu \\nu} (X) \\partial_a X^\\mu (\\sigma) \\partial_b X^\\nu(\\sigma)", "87c02c7422fb97efac82be73dee23fb5": " BS=\\frac{1}{N}\\sum\\limits _{k=1}^{K}{n_{k}(\\mathbf{f_{k}}-\\mathbf{\\bar{o}}_{\\mathbf{k}})}^{2}+\\frac{1}{N}\\sum\\limits _{k=1}^{K}{ n_{k}(\\mathbf{\\bar{o}_{k}} (1 - \\mathbf{\\bar{o}_{k}} } ) )", "87c052cff7a729e3c72b20d39ef5c11a": "\\begin{align}\n\\text{Pr}\\left( \\max_{1 \\leq i \\leq n} S_i \\geq \\lambda\\right) &=\n\\text{Pr}[Z_n \\geq \\lambda] \\\\\n&\\leq \\frac{1}{\\lambda^2} \\text{E}[Z_n^2]\n=\\frac{1}{\\lambda^2} \\sum_{i=1}^n \\text{E}[(Z_i - Z_{i-1})^2] \\\\\n&\\leq \\frac{1}{\\lambda^2} \\sum_{i=1}^n \\text{E}[(S_i - S_{i-1})^2]\n=\\frac{1}{\\lambda^2} \\text{E}[S_n^2] = \\frac{1}{\\lambda^2} \\text{Var}[S_n]\n\\end{align}\n", "87c0696b2e2699cef0eb94f852b87ccf": "(M',\\mu',\\eta')", "87c0ae9a537b91559d46ea7613789003": "\n\\frac{r_1}{A} = 0.38+0.2\\log\\frac{M_1}{M_2}\n", "87c0d859c061b2f2fce4ca8c2df4f9e0": "\\left\\langle U \\right \\rangle = \\frac{n^2 \\epsilon_0}{2} |E|^2 ", "87c115089f9522d5cda7c975ebe3146b": "| \\triangle CDA|=| \\triangle CBA|", "87c1897bfc893459a207d57fbec4fee6": "[P_i,D]=i P_i, [K_i,D]=-iK_i,\\,\\!", "87c1adefee44d4d19aa511a25d971f7c": "y^*_i = \\hat{y}_i + \\hat{\\epsilon}_j", "87c1ae7022485e6fa73b73c53a1b7424": "y^2 = x^3\\ ", "87c1f57891cd554900d263cb51c22817": "w \\in L(G)", "87c23d84448086829ac6eeebee232774": "\\operatorname{Ind}X\\le n", "87c24f43e826b0dcf4993f653890b0e7": "\\vec{D}\\cdot\\vec{\\pi}_A-\\rho'(\\pi_\\sigma,\\sigma)=0", "87c251c0ea1da08c36544e808cc38751": " {\\mathcal{B}}_n \\hookrightarrow {\\mathcal{C}}_n ", "87c25cfc935c1f4071aea966fca07279": " S(1 \\lor 3) \\to 2 \\lor 4 \\qquad S(2 \\lor 4) \\to 1 \\lor 3", "87c26cf6a5842923c31ec9cf3a6f1209": "\n f(k;\\mu_1,\\mu_2)= e^{-(\\mu_1+\\mu_2)}\n \\left({\\mu_1\\over\\mu_2}\\right)^{k/2}I_{|k|}(2\\sqrt{\\mu_1\\mu_2})\n ", "87c29f8e4f23cce961ec81f339cc8f8e": "C_{12} = G_{8} + P_{8} \\cdot C_{8}", "87c2a6b2fc57fc6a2822496e6f3b18ac": "\\left[\\begin{array}{c} L_R \\\\ L_G \\\\ L_B \\end{array}\\right]=\\mathbf{P^{-1}}\\left[\\begin{array}{ccc}X_w/X'_w & 0 & 0 \\\\ 0 & Y_w/Y'_w & 0 \\\\ 0 & 0 & Z_w/Z'_w\\end{array}\\right]\\mathbf{P}\\left[\\begin{array}{c}L_{R'} \\\\ L_{G'} \\\\ L_{B'} \\end{array}\\right]", "87c2b6b7710decd0871551ed05af9af4": "f_{+} (t) = \\lim_{s \\downarrow t} f(s);", "87c2eaf142d2ffa19f6a8d785a1c3b2f": "10^{-19}\\,\\text{GeV}", "87c39820c60da822a5153fd88e91d43b": "\\nu^2", "87c3c229aad336c66d1c6b66226d709b": "\\phi=\\sum_{i=1}^n f_i(x_i)", "87c3f44b9523b05f60a4c7d2a81b4c59": " (F \\otimes 1) \\circ (\\Delta \\otimes id) F = (1 \\otimes F) \\circ (id \\otimes \\Delta) F ", "87c4330e20d5d88a3db3a2bc049540ab": "C = 4 a E(e)", "87c459fc65fa240dab8992656ff44490": "\\dot m=\\dot m_{in} +\\dot m_{gen.}", "87c4c18009225e998bee271fab61ae5b": " \\wedge (\\forall x,y,z) [Eq(x, y) \\rightarrow (Eq(z, x) \\rightarrow Eq(z, y))] ", "87c5135a3688d8277316501c03edef31": "Y_{10}^{5}(\\theta,\\varphi)={-3\\over 256}\\sqrt{1001\\over \\pi}\\cdot e^{5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(323\\cos^{5}\\theta-170\\cos^{3}\\theta+15\\cos\\theta)", "87c551e4db96d4aa2f874ad61e9d4e94": "\\lambda_1 x_1 + \\lambda_2 x_2", "87c564cac73a7f8d54b05fbd7f66c52f": "\\chi_T", "87c5e0a8342cfec19e8a3530436fff43": " D\\left (y(x)e^{\\int f(x)\\,dx} \\right )=g(x)e^{\\int f(x)\\,dx}", "87c635f6725b8d812e5c7e5eaef92143": "\\left \\{ a_1,\\ a_2,\\ a_3,\\dots \\right \\}", "87c670b9d6326543e56cc161277f9c13": "A_\\Sigma = \\int_\\Sigma dx^1 dx^2 \\sqrt{\\tilde{E}^3_i \\tilde{E}^{3i}}", "87c6cc738cbf4dcf0b807f077b22f818": "\n\\sum\\left(X_i-\\overline{X}\\right)^2 \\sim \\sigma^2 \\chi^2_{n-1}.\n", "87c773cf09625e3c6150fbd187ec3710": "A=\n\\begin{pmatrix}\nap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\\\\nbp+aq-dr+cs&-bq+ap+ds+cr&-br+as-dp-cq&-bs-ar-dq+cp\\\\\ncp+dq+ar-bs&-cq+dp-as-br&-cr+ds+ap+bq&-cs-dr+aq-bp\\\\\ndp-cq+br+as&-dq-cp-bs+ar&-dr-cs+bp-aq&-ds+cr+bq+ap\\end{pmatrix}\n", "87c7752635ced5c1a47d1180ae588925": "\\sum_i \\Big(\\sum_\\alpha a_{i\\alpha} X^\\alpha\\Big) \\otimes b_i \\mapsto \\sum_i \\sum_\\alpha a_{i\\alpha} b_i^p X^\\alpha.", "87c7be4aafe7cf18c46f28c6ea7c8887": "\n \\sigma_{xx} = \\frac{3z}{2h^3}\\,M_{xx} = \\frac{12 z}{H^3}\\,M_{xx} \\quad \\text{and} \\quad\n \\sigma_{yy} = \\frac{3z}{2h^3}\\,M_{yy} = \\frac{12 z}{H^3}\\,M_{yy} \\,.\n ", "87c7e18d7642a9cf86ca2ae36d07d3a4": "(c + 1)_{\\gamma - 1} = (c + 1)(c + 2)\\cdots(c + \\gamma - 1).", "87c7eba2bc038e07fab3859ea793d77b": "\\hat\\mu_n=\\int_0^{2\\pi}{\\rm e}^{-in\\theta}\\frac{d\\mu(\\theta)}{2\\pi}=0,\\ ", "87c808c77e4b22a4f834c99b8c2f4bd8": "\\lambda_M", "87c8109fdda234c541c9b407772f8be6": "P(T)= 0.5\\times0.4 + 1\\times0.6 = 0.8", "87c83ed46009c45b931560102643a031": "S = \\sqrt{ax+b}", "87c8577c8777e549346f35e9b455b690": " u_{10}(\\mathbf{r}) = u_{el}(\\mathbf{r}) = \\left | S\\frac{1}{2},\\frac{1}{2} \\right \\rangle = \\left|S\\uparrow\\right\\rangle ", "87c869932b22f41294a519a61ff13f86": "x-x = 0x^2\\ ", "87c89d8715e117cfd2dc1dc30eaa90d9": "\nj_{\\text{t}}=j_0\\left(\\exp(\\alpha_{\\text{o}}\\,f\\, \\eta)-\\exp(-\\alpha_{\\text{r}}\\,f\\,\\eta)\\right)\n", "87c8a8d7754b573f74595fab2f3f53db": "[n)", "87c8b5920153c7db5c787ee8b6bd086d": "P=\\sum_{i=0}^{e-d}\\binom{e}{d+i}\\left(\\frac{a}{b}\\right)^{a+v}\\left(\\frac{b-a}{b}\\right)^{e-d-i}.", "87c8c0e7fbfd0fc0179865b3e77e030c": "\nQ=\\sum_{i=0} Q^{(i)}r^i=Q^{(0)}+Q^{(1)}r+\\cdots +Q^{(n)}r^n+\\ldots\n", "87c8cd87fd0e69c38380536ecf64e311": "M \\leftarrow \\frac{M}{p_i}", "87c93da7b7325256a361de7a1ba32b67": "M1 = 1 - \\sum_{ i = 1 }^K p_i^2 ", "87c976a3462ad2a36ceeba7d865a1d54": "\\begin{align}\n3x+5y&=2\\\\\n5x+8y&=3\n\\end{align}\n", "87c9a52bac1b51045eb4adb72a112efd": " Com.1", "87c9c97a99915f72eecb2b9264fc990e": "y = \\varphi(\\mathrm{net})", "87c9ec3c195aad26a460b07c51ba8eab": "Q_1(z, v)", "87c9ff6c6379ed972cd1392d902ebbe5": " \\Box (p \\rightarrow q)", "87ca0e059020c9a0536b60caef7a50ef": "\\mathbf{Y}_i", "87ca7352aeb6d5576db95825c6357b69": "\\scriptstyle \\frac{\\lambda^k x^{k-1} e^{-\\lambda x}}{(k-1)!\\,}", "87ca9591949e89c22462882bf5ebc4b5": "\\sqrt{\\operatorname{ann}_R(M)} = \\bigcap_{\\mathfrak{p} \\in \\operatorname{supp}M} \\mathfrak{p} = \\bigcap_{\\mathfrak{p} \\in \\operatorname{ass}M} \\mathfrak{p}", "87cb16343e58f7df11e382ba5350ee6a": "K_a = \\frac{[HG]_{eq}}{[G]_{eq}([H]_o - [HG]_{eq})}", "87cb20375c378a8003b07c711ff6d56d": "Y_i \\in \\mathbb{R}^n", "87cbd8698395cd38c6fa8a53ea6a1c9f": "P(X_1, \\ldots,X_{n-1},0) ", "87cc14b01d5df1dd09ae6e21f1f65e7b": "f=\\sum qf_q", "87cc1d837e8b4d8b8b18f22eb122d98b": "\\csc \\left(\\pi z\\right)", "87cc4c7c3350c1c9c89ac87dfd3e4bfe": " CPP = MAP - JVP ", "87cc525964d3694ba04cdaf35e075299": "kT/q", "87ccaa47c3b2c36b7a12cf05775aae83": "P(t) = \\frac{M_a}{r}(1 - e^{-r(T-t)})", "87ccc24ddbad4b2b24ab9b7b683c6625": "{\\rm add}({\\mathcal L})", "87ccf599f5ac7dabeb37582543a4aec6": " \\begin{align}\n& \\begin{vmatrix} t+ E_{11}+1 & E_{12} \\\\\nE_{21} & t+ E_{22} \n\\end{vmatrix}\n=\\begin{vmatrix} t+ x_1 \\partial_1+1 & x_1 \\partial_2 \\\\\nx_2 \\partial_1 & t+ x_2 \\partial_2\n\\end{vmatrix} \\\\[8pt]\n& = (t+ x_1 \\partial_1+1 ) ( t+ x_2 \\partial_2)- x_2 \\partial_1 x_1 \\partial_2 \\\\[6pt]\n& = t(t+1)+ t( x_1 \\partial_1 + x_2 \\partial_2) \n+x_1 \\partial_1 x_2 \\partial_2+x_2 \\partial_2 -\n x_2 \\partial_1 x_1 \\partial_2\n\\end{align}\n", "87ccfa2706d2e25ed282fd050890b1ef": " \\sqrt {\\frac{BA}{k * n}} ", "87ccfd82d48ed08aa457da5c4ca1d165": " \\frac{u_{i+1}- 2 u_i + u_{i-1}}{h^2} = u''(x_i) + \\mathcal{O}(h^2). ", "87cd5071d947b9a47443b297f2124017": "S_t\\!", "87cd6ebef7db0b6a78511529ea3f6800": " C_{i,j} ", "87cd7f4f0ae24fd2ffb7336ac97abf35": "\\textstyle{\\frac {\\log(50)} {\\log(10)}}", "87cd955e46f71bdd7f8030216bddd44c": "f(x) = \\frac{\\lambda^k}{(k-1)!} x^{k-1} \\exp(-\\lambda x)", "87cdbd46ee759441f7541d9fd95fa984": " y=\\frac{Y}{Z} ", "87cdd2e0237caa44ba1390f070d2fb54": "U = 0,\\,", "87ce0eb50fc41f4b697ea542834c48de": "\\neg \\neg p", "87ce21639019c3c4d95fbbd0b8a79609": "w''\\,\\!", "87ce3cfe398469bae9dff96037817955": "i = I/12", "87ce774152cf96ec9f0227e5d2da057e": "\\sigma = \\frac{-1+i\\sqrt{7}}{2}.", "87ce801918f8056c0e77e0aea0cb319c": "\\left|\\sum_{n=M+1}^{M+N}\\chi(n)\\right| =O\\left( \\sqrt q \\log q\\right),\n", "87cece37c2f1334b21b7e46ef01ddff1": "\\ S_{2,t} = F(t_2) (m_2 + F(t_1) m_1 ) (1-F(t_3)) ", "87cef9cbbbebca5a00ef9a32ca06811b": "\\Delta \\xi=\\frac{\\Delta n_i}{\\nu_i}", "87cf4e58ed500c4b672460938ced552d": "d= \\sqrt{A}", "87cf60716634e608cc5bee62deaece23": "K(u|\\eta) = A(\\eta+u) - A(\\eta)", "87cf83faf7a3bb253ac7216b1da6cac6": "\\neg \\neg p \\vdash p", "87cfb190e0c4dab87265dc5b53a2a010": "{\\tilde \\Psi}(A)= \\sum_i \\Psi_i (A) = \\sum _i M_i A M_i", "87cfb7b9871aff4b4a813b867924dfd4": "\\{ 0 \\} \\times A \\subseteq \\{ 0 \\} \\times \\mathbb{R},", "87d08e940c7f06a0fdf6323f0a838ed7": "c_n \\in GF(p^2) \\text{ for } n \\in \\mathbb{Z}", "87d0e46a0f74ce1eff2ecc3db5f5ea53": "C \\to \\mathbf{P}^1", "87d140e412e0965b46abc1b53a7da635": "\\mathbf{A} = A \\mathbf{\\hat{e}}_{\\parallel} \\,\\!", "87d145866a9493551e2e3ed8af16843a": "\\begin{bmatrix}\n1 & \\lambda_3 & \\lambda_3^2 & \\lambda_3^3 & \\cdots & \\lambda_3^{n-1} \\\\\n0 & 1 & 2\\lambda_3 & 3\\lambda_3^2 & \\cdots & (n-1)\\lambda_3^{n-2} \\\\\n0 & 0 & 1 & 3\\lambda_3 & \\cdots & \\tfrac{1}{2}(n-1)(n-2)\\lambda_3^{n-3} \\\\\n1 & \\lambda_4 & \\lambda_4^2 & \\lambda_4^3 & \\dots & \\lambda_4^{n-1} \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & & \\vdots \\\\\n1 & \\lambda_n & \\lambda_n^2 & \\lambda_n^3 & \\cdots & \\lambda_n^{n-1}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\lambda_3^{n+m} \\\\\n(n+m)\\lambda_3^{n+m-1} \\\\\n\\tfrac{1}{2}(n+m)(n+m-1)\\lambda_3^{n+m-2} \\\\\n\\lambda_4^{n+m} \\\\\n\\vdots \\\\\n\\lambda_n^{n+m}\n\\end{bmatrix}", "87d1787ce5bad9c8e9bdd2fde094a55b": "a(p) \\ne 0", "87d1ca73e0773a94467deee2b935b153": "xy>0", "87d21ca42d1d250ee7a5c696aa768981": "{\\bar{Q}}_9", "87d2204532412253dc414627d4b0f3ed": " OS = \\frac{x_p - x_f}{x_f}, ", "87d2c3307e3e88c1a9fe645654d8e066": "A(0)", "87d2f35f0e9e115af56d6657c83b1d51": "\\Sigma^0_k", "87d317b6eb53cc7d2de7f80eeffae801": "V_b\\ ", "87d34236a21e8167bfa9f19970e73648": "NF_4", "87d3501009f917f8b6cfa6a4f000e363": "Tr(g^{xy})", "87d364165667e7d8216d5f9c9f9e7a34": "T_{{\\mu}v}= \\rho\n\\begin{pmatrix}\n1 & v_x / c & v_y / c & v_z / c\\\\\nv_x / c & v^2_x / c^2 & v_x v_y / c^2 & v_x v_z / c^2\\\\\nv_y / c & v_y v_x / c^2 & v^2_y / c^2 & v_y v_z / c^2\\\\\nv_z / c & v_z v_x / c^2 & v_z v_y / c^2 & v^2_z / c^2\n\\end{pmatrix}", "87d3679fb7f04a907d3affdad85eab4b": "T_0^*", "87d3705ff5f2682c2c66405e2c488e1d": "\\epsilon={v^2\\over{2}}-{\\mu\\over{r}}", "87d3cbc1bcbf81c7ac16697e1e553eb8": " \\frac{1}{2}+\\frac{1}{2} \\prod_{p} \\Big(1 - \\frac{2}{p^2}\\Big) = 0.661317... ", "87d3e1ba5346f5a6b757e86cc6fd7e71": "c_1\\neq0", "87d44a8453daea7fd608fd7a8233e577": "\nq_L \\rightarrow e^{i\\theta} q_L \\qquad\nq_R \\rightarrow e^{i\\theta} q_R ~,\n", "87d473516cb23919124acc2ebab730e2": "\\kappa = H(K)", "87d491690abc2d7fbec610e8b34a40da": "\\mathsf{plus}\\ x\\ y \\Leftarrow \\underline{\\mathrm{rec}}\\ x\\ \\{", "87d520d8b215465a14c8c8273daae70e": "\\left( \\lim_{x\\to 8^+} \\frac{1}{x-8} = \\infty \\right) \\Rightarrow \\left( \\lim_{x\\to 3^+} \\frac{1}{x-3} = \\omega \\right)", "87d5246bbf8e8c517c7f47285555e819": "N_k = \\sum_{n=1}^N r_{nk} \\, ", "87d56dd64f1986ad96831598486a2c7d": "X(\\omega(Y))=Y(\\omega(X))=0,", "87d5a27722e7aae148d0747f7477ea0d": " \\max_{(\\Delta ,{{x}}) \\in {V}} \\left\\{ { - \\left\\| {{x}} \\right\\|^2 + \\operatorname{Tr}(\\Delta )} \\right\\} ", "87d5a6b4d50aeb116f5941b6b0538da8": "b_{\\nu_j}|\\dots,n_{\\nu_{j-1}}, n_{\\nu_j}, n_{\\nu_{j+1}},\\dots \\rang=\\sqrt{n_{\\nu_j}}|\\dots,n_{\\nu_{j-1}}, n_{\\nu_j}-1, n_{\\nu_{j+1}},\\dots \\rang", "87d5d3f5e89b571484d1edd047922898": "X \\xrightarrow{id} X \\to 0 \\to", "87d5d59d44986e9e72d3c8a171d8c161": " A = 1 + \\frac{u^2}{16384} \\left\\{ 4096 + u^2 \\left[ -768 +u^2 (320 - 175u^2) \\right] \\right\\} ", "87d6a92af3785c1ab1863844add28dfe": "\\gamma(h)=(s-n)\\left(\\left(\\frac{3h}{2r}-\\frac{h^3}{2r^3}\\right)1_{(0,r)}(h)+1_{[r,\\infty)}(h)\\right)+n1_{(0,\\infty)}(h)", "87d6b4faf16294f34b3a5fd9d13dee1c": " h^{-1}\\bigg(T(h) x - x\\bigg) ", "87d6b85206f1db31f2d0244460df71d0": "2f_\\mathrm{IF}\\!", "87d75df23390e6e4a6e6c79d224f6ffd": "\\begin{align}\n\\operatorname{E}[\\ln X] \n&= \\int_0^1 \\ln x\\, f(x;\\alpha,\\beta)\\,dx \\\\\n&= \\int_0^1 \\ln x \\,\\frac{ x^{\\alpha-1}(1-x)^{\\beta-1}}{\\Beta(\\alpha,\\beta)}\\,dx \\\\\n&= \\frac{1}{\\Beta(\\alpha,\\beta)} \\, \\int_0^1 \\frac{\\part x^{\\alpha-1}(1-x)^{\\beta-1}}{\\part \\alpha}\\,dx \\\\\n&= \\frac{1}{\\Beta(\\alpha,\\beta)} \\frac{\\part}{\\part \\alpha} \\int_0^1 x^{\\alpha-1}(1-x)^{\\beta-1}\\,dx \\\\\n&= \\frac{1}{\\Beta(\\alpha,\\beta)} \\frac{\\part \\Beta(\\alpha,\\beta)}{\\part \\alpha} \\\\\n&= \\frac{\\part \\ln \\Beta(\\alpha,\\beta)}{\\part \\alpha} \\\\\n&= \\frac{\\part \\ln \\Gamma(\\alpha)}{\\part \\alpha} - \\frac{\\part \\ln \\Gamma(\\alpha + \\beta)}{\\part \\alpha} \\\\\n&= \\psi(\\alpha) - \\psi(\\alpha + \\beta)\n\\end{align}", "87d7630eafe834d4e0b534dc4eecc8c6": "\\textstyle x_{k+1} = T x_k", "87d79d7f28f8b8028a33d69b61127c6b": "f_w(\\theta)", "87d7a0f6479b8c10789afa4980ad952b": "\\epsilon_\\textrm{max}\\approx 0{.}86", "87d7c55e80865bbf365b0203a8326240": " \\sum_{i=1}^k p_i = 1. ", "87d7dd482830d4703cb311770f618a0a": "t \\sigma_2 \\equiv t_2", "87d821536757657b8bf85044aebfbd94": "\\frac{Z/z_n}{Y/y_n}-1=\\frac{Z/z_n-Y/y_n}{Y/y_n}", "87d83b56cd04978f548f13bdfeeb1045": "\\rho(b,a+\\pi)=-\\rho(b,a)", "87d83d9b24321dbab3f073dcadd5b671": "\\mathbb{Z}_q^n", "87d853f67fadfff693b19fd48d3a8c9f": "\\cos^2\\theta=\\frac{1}{\\tan^2\\theta+1}", "87d85c23f4ae3d770189ba18dc281e75": "\\ddot{r} = \\frac {d^2r} {d\\theta^2} \\cdot {\\dot {\\theta}}^2 + \\frac {dr} {d\\theta} \\cdot \\ddot {\\theta}", "87d8a81e8172d09f4427d9c58846e8fd": "F(c)", "87d8ad50b2a8a0ce483daa05faab52c7": " g_{ij} = \\begin{bmatrix} \\sigma^2+\\tau^2 & 0 & 0\\\\0 & \\sigma^2+\\tau^2 & 0\\\\0 & 0 & \\sigma^2\\tau^2 \\end{bmatrix} ", "87d8b63ba1aa7773bdb4dbd7198b8f55": "r_i = v_{Ti}/\\omega_{ci} = 1.02\\times10^2\\,\\mu^{1/2}Z^{-1}T_i^{1/2}B^{-1}\\,\\mbox{cm}", "87d8ecc8351557aff0f0cd45e1fa5aea": "\\left\\{A_n\\right\\}_{n\\in\\mathbb{N}}", "87d900d51696edeb3fc4653d7ca9d131": " US . \\frac{Y}{K} = \\frac{1}{10} \\times \\frac{1}{10} \\times \\frac{1}{2} = \\frac{1}{2} ", "87d95185407b5eccb9395431f0bad43d": " h_{g;k} ", "87d9febb376208f3f1e629dfa177cb9c": "q''_n(x_n)\\ =-2\\ \\frac {3(y_n - y_{n-1})-(2k_n+k_{n-1})(x_n-x_{n-1})}{{(x_n-x_{n-1})}^2}=0,", "87da07b2fcc1af7143e5c7fefd859c69": "\\mathrm{soc}(M)\\,", "87da0b57ad0acc8c34ac1d973a28db59": "G-S", "87da12f898709d4902e31b28a56b0d18": "\\mathrm{PH}_{d}(\\boldsymbol{\\tau},{T})", "87da2507523066be8126a8de33d847c4": "\n\\begin{array}{lll}\n\\displaystyle\\frac{\\part f}{\\part x}(0, r_{o}) = 0 , &\n\\displaystyle\\frac{\\part^2 f}{\\part x^2}(0, r_{o}) = 0, &\n\\displaystyle\\frac{\\part^3 f}{\\part x^3}(0, r_{o}) \\neq 0,\n\\\\[12pt]\n\\displaystyle\\frac{\\part f}{\\part r}(0, r_{o}) = 0, &\n\\displaystyle\\frac{\\part^2 f}{\\part r \\part x}(0, r_{o}) \\neq 0.\n\\end{array}\n", "87da96744a038106fecf655780737b49": "\\scriptstyle \\left(1 \\,-\\, \\frac{i\\,t}{\\beta}\\right)^{-\\alpha}", "87daa7220d2dec8fb6b29f512154ad58": " \\sigma\\ _i ", "87dabbdd9eac50bc70930e6da0f55f40": "\\forall n \\in \\mathbb{N} : \\lim_{k \\to \\infty} a_n^{(k)} = x_n", "87dafd750c89a4bc7644610686da3ad5": "\\tau(a) = \\frac{12}{\\pi^{2}}\\ln 2 \\ln a + C + O(a^{-1/6-\\epsilon})", "87db10a676a4b7e517f5709a3475199b": "\\mu_\\mathrm{B}", "87db32177427b77598d1f95bc982c6ce": "\n W \\approx C_0 + C_{ij}E_{ij} + \\frac{1}{2!}C_{ijkl}E_{ij}E_{kl} + \\frac{1}{3!}C_{ijklmn}E_{ij}E_{kl}E_{mn}+\\cdots\n ", "87db81b813fc7391baa2ce8807a6d84b": "\\vec{F}(\\vec{x})", "87db8de83c49ac64a337c342c0a95efa": "|V(x,y,t)-V(x,y,t+1)|>\\mathrm{Th} \\, ", "87db969d91274d6df73e0c0bbe7945b8": "\\mu(x\\otimes y)\\cdot z = x\\cdot\\mu(y\\otimes z)", "87dbeee0594e57f44c41bf80e0148512": "^{15}\\text{NO}_3^- \\rightarrow^{14}\\text{N}^{15}\\text{NO}", "87dd6cf20a3658a0f16d452545c92c95": "MG(a_1,\\dots,a_n) \\le \\frac{e}{n(n+1)}\\, \\sum_{1\\le k \\le n} k a_k \\, ,", "87dd6deddcdb6dfa686dece1a0a7ee37": "f(x)=\\prod_{j\\in Q}(x-\\zeta^j)", "87dd87ab075134d2db31733121934496": "I(X_1;\\ldots;X_{n+1}) = I(X_1;\\ldots;X_n) - I(X_1;\\ldots;X_n|X_{n+1}),", "87dd9e833d58394690e907aed4c2e01c": "d=2", "87ddb24422ef140d054c92452e01767b": "r,\\theta,\\phi", "87ddd56f99495a82a92886810be350e8": "\\sum_{i=m}^n a_i = a_m + a_{m+1} + a_{m+2} +\\cdots+ a_{n-1} + a_n. ", "87ddeb36bb3fa70d5582510ba540e38e": "f_\\text{zf} = {1 \\over 2 \\pi R_\\text{f} {(C_\\text{i} + C_\\text{f})}}", "87de2f18c4f22cd1fae4b19657af76aa": "r = 1 + i", "87deda7d13915f61a525e16cbfbf5e66": " \\widehat{\\mathbf{J}} = \\widehat{\\mathbf{L}} + \\widehat{\\mathbf{S}} ", "87dedfb119ea680e204b79a34361208d": "\\mathbf{e}_{1}(s) = \\mathbf{\\gamma}'(s).", "87df0dd6b6af0aba5a81ec17477f3057": " \\omega ^2(\\mathbf{q}) ", "87df48143ec1d8556ae7cf01a759ad13": "|10 \\rangle ", "87df69bca200f7b3371c597ac54937b4": "\\beta_w", "87df9b49fd224a5adae628cce4762c4a": "\\sup \\{ \\|T\\| : T \\in \\Gamma \\} < \\infty", "87dfb58691debfa603f5b0f54bd5c987": " (a(x_0 - m) + b(y_0 - n))^2 = (ax_0 + by_0 - am -bn )^2 = (ax_0 + by_0 + c)^2", "87dff2fe448bc9891df511b508c30cd7": "i = \\frac {V_\\text{in}} {R_g}", "87e028d52e8ca1ce8418dee7d0cab470": "G_F ", "87e0a31eb5902cb094c55a876068fe81": " \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix} = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\begin{bmatrix} V_2 \\\\ -I_2 \\end{bmatrix} ", "87e0bf6ffb688f0d85c39c9523b7ab7b": "W = (A^TC_Z^{-1}A + C_X^{-1})^{-1} A^TC_Z^{-1},", "87e142af5a33fc53eac636f07d29a173": "A_{f}(\\infty) = F_\\infty ", "87e166c445ddb749f133b5efebf1dc8e": "\\omega_1 + \\omega_2 = \\omega_2 + \\omega_1 ", "87e1e7fa848025e205e724f3c96266f9": "\nC_p = \\frac {V}{L_{pp} \\cdot A_m}\n", "87e20b5b5efeb9c03b03ab3c69cccde5": "{\\bar{X}}_5", "87e2305b3005d83e7f8f808754569917": "dx \\wedge dy.", "87e2380322f2cb6542418807ba343d39": "F_{B \\rarr A}", "87e2aa1c49a3a8d89db40c8e96358735": "[f]_{0,\\alpha;\\Omega} = \\sup_{x,y\\in \\Omega} \\frac{|f(x)-f(y)|}{|x-y|^\\alpha}.", "87e2b980a91974ad5f108aa1b84fdd7e": "w \\in V", "87e2f297587c83a1cdf44b117322ced1": "\\sigma_y^2(\\tau) = \\frac{1}{2}\\langle(\\bar{y}_{i+1}-\\bar{y}_i)^2\\rangle", "87e31276bfdd0842602e60c1eb0c225c": "A\\frac{d^2\\phi}{dt^2}+H(\\frac{d\\theta}{dt}+\\Omega \\phi)=Wh \\phi ", "87e34bf44433148a651b5e9e14808720": " (\\text{trunc}_v(x_i), \\, \\text{trunc}_v(x_{i+1}), \\, ..., \\, \\text{trunc}_v(x_{i+k-1})) \\quad (0\\leq i< P) ", "87e3a6fccb370fa902aac7951617fc3a": "\\sum_i \\sum_j \\mathrm{P}(X=x_i\\ \\mathrm{and}\\ Y=y_j) = 1,\\,", "87e3a7a237e50d6d439787b192749eb6": "\\sqrt{\\mu/\\epsilon}", "87e3b2ccf00b6b5338f1d4121545bc83": "\\overset{\\sim}{\\rightarrow}", "87e3b2ea1a5c0bfd2e6cca013baafceb": "R_H", "87e3d516005275432144a232b9528508": "\\displaystyle{T_K\\varphi + T^*_K\\varphi= v_\\pm +\\partial_{n\\pm}u=\\mp\\varphi/2 +T_K\\varphi +\\partial_{n\\pm}u,}", "87e3dbf165b2e3b5fbea8de41909b4ac": "\\hat{T} = \\frac{\\bold{\\hat{p}}\\cdot\\bold{\\hat{p}}}{2m} = \\frac{\\hat{p}^2}{2m} = -\\frac{\\hbar^2}{2m}\\nabla^2", "87e47a19db942bbfab7c1a5e9e3a6cf7": "P\\Gamma L", "87e48f5c671346ea946443e7b4a966b0": " \\xi = f'^{-1}\\left(\\frac{f(x)-f(y)}{x-y}\\right) ", "87e4bc6b87c6db994045fe45c4b2198c": "x_n = -n +\\frac{1}{\\ln n} + o\\left(\\frac{1}{\\ln^2 n}\\right)", "87e4e22272277d9dde6dad74b54c5cca": "-|e|", "87e527e4bf47aebc1e146cbf4c642d2b": "P(X_t|X_{t-1})", "87e63cb78708deba8ebca18816d9c0d9": "(a_n)_{n\\in\\mathbb{N}}", "87e6653543e337e63ae99e6036143c14": "E_{so}=(\\hbar^2e^2)/(4m^2c^2)[j(j+1)-l(l+1)-3/4]/((a_0)^3n^3(l(l+1/2)(l+1))]", "87e671507ee8c7b46aa1d9c521016d45": "\\mathbf{6}\\otimes\\mathbf{6}\\otimes\\mathbf{6}=\\mathbf{56}_S\\oplus\\mathbf{70}_M\\oplus\\mathbf{70}_M\\oplus\\mathbf{20}_A", "87e69ea540fc6d58f815988a182c94f5": " \\sum_{k=0}^\\infty |B_k(a_k - a_{k+1})|", "87e6b4698aeeb82a451258c876275555": "D^{+}(\\mathcal{S})\\cup \\mathcal{S}\\cup D^{-}(\\mathcal{S}) = \\mathcal{M}", "87e6b705a49a209d2c5733322d85b306": "X \\sim \\chi_1(x) \\,", "87e6d528041e2e18fd3d68bb54913945": "c z^{p^e} \\in I^{[p^e]}", "87e71d78f26b20775b81b1d5b35b21cd": "\\liminf_n d(X|n)=\\infty", "87e7499dc811a39a72d961ce458d9383": "|\\alpha|^2c=1", "87e7543c967c985f5f83d263f485ffbc": "\n\\left(\\frac{13}{9907}\\right) \n=\\left(\\frac{9907}{13}\\right) \n=\\left(\\frac{1}{13}\\right)\n=1 \n", "87e79d7fe42db97539a6bbccafb847be": " X_\\alpha ", "87e7f0085595d6e28dc4dbfa716529ac": "\\lambda=\\pm 1", "87e7f1a5fc8d2b3c2af50bee86596803": "T(X) = m^T R^{-1} \\sum_{n=0}^{M-1}X_n.", "87e8666378a37b9421582a6c15627566": "u^2 + v^2\\,", "87e88f98d21a73b154c2233266e71706": "v_e = c \\ \\sqrt{2 \\eta - \\eta^2}", "87e8d4e1c84bcfcc752c7197dc562ab9": " \\mathbf{E}[X;P]", "87e8da32b1f0c7d69dd66255223cedc1": "\\begin{align}\n r &{}= \\sqrt{(-2.4327)^2 + 4^2} = 4.6817 \\\\\n c &{}= -2.4327 / r = -0.5196 \\\\\n s &{}= -4 / r = -0.8544\n\\end{align}\n", "87e8ebbc4d2e8396b057f199c94121be": "N(1 + \\sqrt{5}) = 6", "87e8fbf3e39e628681ec0317a91ecb84": "(2\\pi)^{z/2}\\text{exp}\\left( -\\frac{z+(1+\\gamma)z^2}{2} \\right) \\exp \\left[\\sum_{k=2}^{\\infty}(-1)^k\\frac{\\zeta(k)}{k+1}z^{k+1} \\right]", "87e8ff35dc697204ad40af18b1665b1d": "\\Psi(x,t) = \\left(\\frac{1}{x_0 \\sqrt{\\pi}} \\right)^{1/2} \\cdot \\frac{e^{-x_0^2 p_0^2 /2\\hbar^2}}{\\sqrt{1+i\\omega_0 t}} \\cdot \\exp{\\left(-\\frac{(x-ix_0^2 p_0/\\hbar)^2}{2x_0^2 (1+i\\omega_0 t)}\\right)}.", "87e92ed4126a011f7cbe1dd5ffeb773e": "\\scriptstyle P(\\eta_t(x)=1)=\\rho", "87e9554c86996a40b17ce9ab054c62bc": "U_s = \\frac{2}{9}\\frac{\\left(\\rho_p - \\rho_f\\right)}{\\mu} g\\, a^2", "87e99c7f3ff80f7db92f84bdcc28b60c": "f*(2\\pi \\delta)=f \\,", "87e9afaa0abf1c4c5b04f960b44505ca": "S = \\begin{bmatrix}\nT1 & T2 \\\\\nW(A) & \\\\\n & W(B) \\\\\nW(B) & \\\\\nCom. & \\\\\n & W(A)\\\\\n & Com. \\end{bmatrix}", "87ea7ecde1334ac716b5dd7d384cdaf2": "N^* = \\sum_i {\\bar \\lambda_i} P_i.", "87ea8c4ebc55b663fdd8ae9254eaea72": "D_1 = (1-c)\\;D_0", "87ea9c8d03488ba4a0183d14cd271cec": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "87eab8bab1baf8d2496319afda6cc066": " \\int f_n \\geq \\int \\limits_{E_n} f_n \\geq \\alpha \\int \\limits_{E_n} \\phi ", "87eae75a713fac496055e5a11bc8dd74": "\\tfrac{n}{m}", "87eaecfa39a86dd5733dde2f737d57df": "\\left.-\\frac{1}{\\hbar^2}\\int_{t_0}^t dt_1\\int_{t_0}^{t_1} dt_2e^{\\frac{i}{\\hbar}\\lambda V(t_1-t_0)}H_0e^{-\\frac{i}{\\hbar}\\lambda V(t_1-t_0)} e^{\\frac{i}{\\hbar}\\lambda V(t_2-t_0)}H_0e^{-\\frac{i}{\\hbar}\\lambda V(t_2-t_0)}+\\ldots\\right]|\\psi(t_0)\\rangle .", "87eafbc4885976e36e5eab959497e407": "A\\cap B^{-}=\\emptyset=B\\cap A^{-}", "87eb35b25f6b71a1e94a2485f7e485ea": "C_{\\beta I}^{\\;\\;\\; K} e^\\beta_K e^\\alpha_J,", "87eb670cad058f3cb20a95607477c03f": "V_x = (V_{1,x},\\dots,V_{n,x})", "87eb956b11f91719dd7af275f79576a3": "v^\\mathrm{H}", "87ebef53114e47473bcc9e696c4f319e": "\\text{duplicate}: \\mathrm{W} \\, A \\rarr \\mathrm{W} \\, \\mathrm{W} \\, A = \\text{extend} \\, \\text{id}", "87ebf06e39f6ebc635ae38159f1df0ae": "\n\\Big|\\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix}' M_i \\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix} -\\lambda\n\\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix}' M \\begin{bmatrix}y_i&Y_{-i}\\end{bmatrix} \\Big|=0\n ", "87ebf49a838d251cb3bc9efa7a677093": "q=3/2", "87ecb1addd5d837558069a6786037273": "f_{a}(k+1) = g^{a_{1}a_{2}^{x_{2}} \\dots a_{n}^{x_{n}}}", "87ecc855190170cdc6b4cf26aa641d90": "(x - \\alpha)^{u}", "87ed372fa3443f86ed4b8f369a15df30": "\\hbar \\frac{dk}{dt}=-eE", "87ed4ed17f4467afdde00cac39504e39": "s = {1 \\over 0.031 \\times 1.016^t + 1}", "87edfe888aac5a538b197f2c2a4cfcd6": "y(x) = \\frac{\\int b(x) M(x)\\, dx + C}{M(x)}.\\,", "87ee4f0cd2ad51c410d54a4ec2391cda": "\\Omega_{-1,1/2}\\propto\\binom 1 0", "87eea4a19a98e79c5734054ad2156911": " \\displaystyle T = \\frac {2} {3} \\frac {K} {N k_B}.", "87efb0cd6cb29398f69aae7f67825e27": "C\\sim N^{-0.75}. \\, ", "87efb7b8826cbb4fffcfd5b5af0a4892": "\\sigma_{r\\theta}\\, ", "87efcb67fa618275d415333685835af5": "\\mbox{ex}(n; K_r(t)) = \\left( \\frac{r-2}{r-1} + o(1) \\right){n\\choose2}.", "87efdd9621b85350c611795a22d3b19e": "\\cos(\\pi/4)", "87f033b6a67e9e22670826ab9f5d85e8": "\\Delta S_{fus} = \\frac{\\Delta H_{fus}}{T_m}", "87f04fab244783895fc5318b5b200964": " I = \\int_S J {\\mathrm{d} A} . \\,\\!", "87f0666420d98c658e6483b7620da550": "y \\mapsto y-\\frac{(2y+1)(x^4+x^3-3x^2+3x-1)}{x^3(x-1)^3}", "87f0cec1dfd0aa1eb6120ea64248fa90": "(\\epsilon_e^*,\\sigma_e^*)", "87f0f2a8d2f8dc802c00e3f30739ea5b": "f(u) = 2^{-n_u},\\quad u \\in U", "87f0f6449e69caecf87c2785e2e095ac": "S = - m\\circ (S\\otimes P)\\Delta,", "87f101c6ba60a61197e0f1e942b24ea9": "\nx = b_0+\n\\frac{a_1}{b_1+}\\,\n\\frac{a_2}{b_2+}\\,\n\\frac{a_3}{b_3+}\\cdots\n", "87f10394d952870a253e1e99094c43c8": "D = P^2 - 4Q", "87f18216aa67c6e9391d60fdb06236a8": "\\color{red}\\text{red}", "87f1a33ba953007d110a14901356d7b7": "a^{'}=b^{'}=c^{'}=8R\\sin(A/3)\\sin(B/3)\\sin(C/3), \\, ", "87f1c9ab8451f8011f2482c8d9480166": "\\, m", "87f2212e11896753f69dfda8cce64064": "F[t_1, t_2, \\dots, t_n]", "87f23ca501b363a12c17b6ef395526dc": "\\textstyle a a = a \\, , \\quad b b = 0 \\, , \\quad a b = b a = 0 ", "87f27ffae41f6a66c553e8815e2f067d": " \\operatorname{lift-choice}[\\lambda x.f\\ (x\\ x)] ", "87f292a9203f16abaa7d57b1265ad4a1": "v_{th}=\\sqrt{\\frac{2k_BT}{m}}", "87f296642f2c85b1db580b1507529cdd": "\\operatorname{MH} (\\bold{H}) = \\operatorname{E} \\int (\\hat{f}_\\bold{H} (\\bold{x})^{1/2} - f(\\bold{x})^{1/2})^2 \\, d\\bold{x} .", "87f2a9961edc0b9e260deef06e597411": "\\begin{bmatrix}\nc\\,t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix} =\n\n\\begin{bmatrix}\n \\Lambda_{00} & \\Lambda_{01} & \\Lambda_{02} & \\Lambda_{03} \\\\\n \\Lambda_{10} & \\Lambda_{11} & \\Lambda_{12} & \\Lambda_{13} \\\\\n \\Lambda_{20} & \\Lambda_{21} & \\Lambda_{22} & \\Lambda_{23} \\\\\n \\Lambda_{30} & \\Lambda_{31} & \\Lambda_{32} & \\Lambda_{33} \\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\nc\\,t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix}.\n", "87f2ad2153cd6aa0d26feb0e056a78a7": " V_{BE1} = V_T \\ln \\left(\\frac{I_{C1}} {I_S} \\right) = V_A\\ . ", "87f3331465886112a04aaad821fa6d11": "\\nabla F", "87f33751d6d50a91053ec83f52596ba1": "r=R_0\\ .", "87f3488f8c20ab4fc5b9784445b6901f": "\\Phi^+(\\gamma) = \\{\\alpha \\in \\Phi | (\\alpha, \\gamma) > 0\\}", "87f380f1e1a16c03331e43119e8c9147": " \\mathbb{Z} \\times BU ", "87f392413dca9fd1fc53eeae25ec9e38": "\\!\\,\\log_{10} N = a - b M", "87f3a60d70ab900865a9edfe2750bab1": "\\mid \\psi_{y+} \\rangle", "87f4038362ab36843474c15e530f708f": "d([L]\\mathbf{v}, [L]\\mathbf{w})^2 = (\\mathbf{v}-\\mathbf{w})^T[L]^T[L](\\mathbf{v}-\\mathbf{w}).", "87f4a92a08779d287234c1826eba23c8": "\\nabla G = \\mathbf{F},", "87f4e04e2584b2deccd709dc942ee1d1": "\n \\begin{align}\n w_3 = \\frac{1}{300EI}\\Bigl[&30 R_a (-50 + x)^3 - 2 M_c (-50 + x)^2 (25 + x) - \\\\\n & 625 (-141875 + x (8400 + (-162 + x) x))\\Bigr] \\,.\n \\end{align}\n ", "87f4ee798d0f9831a044f064dfde3e7b": "x^{\\prime}=\\gamma(x-vt),\\quad y^{\\prime}=y,\\quad z^{\\prime}=z,\\quad t^{\\prime}=\\gamma\\left(t-x\\frac{v}{c^{2}}\\right)", "87f505928f904646c6fedd17820bb754": "B \\rightarrow A: \\{T_A + 1\\}K_{AB} ", "87f51a9ed63f60c2e03c522bb86d304f": " \\mathbf{\\hat{\\mu}}=\\mathbf{\\mu}+K\\left( \\mathbf{d}-H\\mathbf{\\mu}\\right) ,\\quad\\hat{Q}=\\left( I-KH\\right) Q, ", "87f54b3b993f9eaf43c0b27600b977df": "Q = K H^n", "87f55dbf7e2fed27f974418420f52bed": " \\mathbf{q} = 2 \\mathbf{k}_F ", "87f5aab272cdfffaa5a3a3c367b09f97": "{\\mathbb E}_\\theta", "87f5ebd73fb9b737a172b2c291aaf7b8": "\\overline{P}_-:=\\{Q\\in \\mathcal P \\ | \\ Q\\parallel_- P\\}", "87f653aa731756c7aae0091953f64561": "R=\\left(\\frac{1}{2}\\left(1+\\sqrt{5}\\right)\\right)a\\approx1.61803...a", "87f659ef14381de7c12f87588f36248d": "p_{01} \\leftarrow x^3-x^2-2x+1, M_{01}\\leftarrow \\frac{x+2}{x+1}", "87f6bfcfea626e870b2c63a71fb45836": "\\int_0^T \\sum_{d=1}^D |\\theta_d(t)|^2 dt < \\infty", "87f6c3fa3a353820fc88937449a3a49e": "\n \\omega=\\frac{j\\pi v}{l}\\ ,\n ", "87f75caf772792d844e5424ed88066d1": "\\scriptstyle -4,8(3,7)\\times10^{-8}", "87f77702bbf62354e09622b462c0788e": "T_{[abc]} = \\frac{1}{3!} \\, \\delta_{abc}^{def} T_{def} .", "87f786d428ebf282f4dbaa78f92b528f": "u_{23}", "87f7d10ba1aec6522bb920108b8d3592": "V = 0.877(R'-Y)", "87f801f4891aa730587375218e31474c": "(S ; \\vee, \\wedge, 0)", "87f80f3d6c425e15bc4b9a9f01605118": "\\deg(v)<\\deg(a)-\\deg(g)", "87f81bd95d9e88d63015a81bc7f164e8": "\n\\frac{1}{R_{\\rm hyd}} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{N^{2}} \\left\\langle \\sum_{i \\neq j} \\frac{1}{r_{ij}} \\right\\rangle\n", "87f82dde4bb378c81314c9192e942c0c": "\\langle k | x_j(t) x_{\\ell}(0) | k \\rangle = \\frac{\\hbar}{Nm\\omega_k} \\cos \\left[ k(j-\\ell)a - \\omega_k t \\right] + \\langle 0 | x_j(t) x_\\ell(0) |0 \\rangle ", "87f845abd484f6d440c1d8972d52aa31": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 1}{4 \\choose 1} +\n {13 \\choose 2}{4 \\choose 2}^2 +\n \\left [ {13 \\choose 2}{4 \\choose 2}{12 \\choose 2}{4 \\choose 1}^2 - 2,304 \\right ] = 85,368\n", "87f881c95863db961e19154547c199a4": "\\tilde Y(s) = \\{\\mathcal{L}^*F_Y\\}(s) = \\int_0^\\infty e^{-st} \\lambda e^{-\\lambda t} dt = \\frac{\\lambda}{\\lambda+s}", "87f90159a40bc7091ea99c3bd5a310d8": "\\gamma \\in G", "87f9537b7fd9a368bfb4da7c50754b6c": "X^\\pi = \\begin{bmatrix}D & C \\\\ B & A\\end{bmatrix}.", "87f95cbed3e5ac14a836fb721893b2d2": "m_{t+1} = m_t - {\\Delta v}_t + {\\Delta x}_t \\,.", "87f97da9b8e997898a5a7d2e61d2ec09": "L \\left( x, y, \\sigma \\right)", "87fa250e2162219f16dbead680c1a52f": "\\bigwedge_{i=1}^n p^\\epsilon_i", "87fa37da33e5dcd0707755005c87b4a9": "k^{}_1,\\dots,k^{}_J", "87fa5f03f2a70aa276e230bf3b7f96ba": "\n\\Delta(\\mathbf{p}^{\\prime},\\mathbf{p}) =\n\\sum_\\mathbf{k} F^\\dagger(\\mathbf{k}) \\left[\nF(\\mathbf{p}^{\\prime}/2-\\mathbf{p}/2+\\mathbf{k})\na^\\dagger(\\mathbf{p}-\\mathbf{p}^{\\prime}/2-\\mathbf{k})\na(\\mathbf{p}^{\\prime}/2-\\mathbf{k}) \\right. ", "87fa90e8e05508228e4fbe34f8938000": "R_n(\\lambda(x);\\gamma,\\delta,N)= {}_3F_2(-n,-x,x+\\gamma+\\delta+1;\\gamma+1,-N;1).\\ ", "87fabbb9fe7f5e73670351774ae166d6": "D \\subset L^2(\\mathbb{R}^n)", "87fb02664f0f4bb90821a3b74ea1a75c": "\\lambda^d - c_1 \\lambda^{d-1} - c_2 \\lambda^{d-2} - \\dots - c_d \\lambda^0 =0. \\, ", "87fb49d7128dc823718b329e84140347": "z_1 = X + Y\\,\\!", "87fb7e40623688935792995843ac053d": "\n-min\\ \\{ I(X;Y), I(Y;Z), I(X;Z) \\} \\leq I(X;Y;Z) \\leq min\\ \\{ I(X;Y|Z), I(Y;Z|X), I(X;Z|Y) \\}\n", "87fba9282d2d9cfaaae1627bf2e054ec": "L^2/T", "87fc2bb9f655c313e27f70d67040e2c3": "\\left(\\frac{2}{p}\\right)", "87fc582b5b6f286dea5c5eba273d4023": "P,", "87fc5feb0f5bcbb63d474388e361ab4f": "2P[S_1=k]", "87fcb01f585169ddd48301870e1696cf": "\\omega^{\\omega^{\\varepsilon_0+\\omega}}", "87fcd08f66a9d5bd46f079ccfe856a57": " \\left\\{ | \\phi_k \\rangle \\in \\mathbb{S}^d \\right\\}_{k=1}^n ", "87fd5b9a78e4ad4e209bfd748c599bac": "G_{ik}=\\frac{1}{4\\pi\\mu r}\\begin{bmatrix}\n\n1-\\frac{1}{2b}+\\frac{1}{2b}\\frac{x^2}{r^2} &\n \\frac{1}{2b}\\frac{xy} {r^2} &\n \\frac{1}{2b}\\frac{xz} {r^2} \\\\\n\n \\frac{1}{2b}\\frac{yx} {r^2} &\n1-\\frac{1}{2b}+\\frac{1}{2b}\\frac{y^2}{r^2} &\n \\frac{1}{2b}\\frac{yz} {r^2} \\\\\n\n \\frac{1}{2b}\\frac{zx} {r^2} &\n \\frac{1}{2b}\\frac{zy} {r^2} &\n1-\\frac{1}{2b}+\\frac{1}{2b}\\frac{z^2}{r^2} \n\\end{bmatrix}\n\\,\\!", "87fe69d7f8d11958eecf32c1993f3503": "\\|x\\| := \\sqrt{\\langle x,x\\rangle}.", "87fe7a61f86936cc4f076f31c28e5664": "P(x)=R(x) \\cdot S(x).\\,\\!", "87fed4a234e4ba565182f48ab34ab0de": "(\\Sigma,\\alpha)", "87fefcd6b5d4c334246c7bc2a5b14855": "\\frac{1}{T} \\int_T^{2T} Z(t)^2 dt \\sim \\log T", "87feffcae42ff3e508af15b88240634d": "(x_1-\\overline{x},\\,\\dots,\\,x_n-\\overline{x}),", "87ff354c0e336d16079dd37eb164f13e": "\\alpha=\\frac{1}{d}\\left\\vert\\ln\\left(\\frac{1}{2R^{2}}\\left(\\sqrt{4R^{2} + \\left(\\frac{I_0}{I}(R^{2}-1)\\right)^2}+\\frac{I_0}{I}(R^2-1)\\right)\\right)\\right\\vert", "87ff36da0df5aed9a13e40cebfd073a4": "H = x\\frac{d}{dx}-y\\frac{d}{dy}", "87fff9c8cf2bdb8e2ca6edc67b7b5c99": "A(z) \\cdot B(z).\\,", "8800171bb0fc3d3e5b8f54c0ebda5095": "x \\in [-1,1]", "8800267caeb4252ce528ee3c646e6722": "\nQ_H = k_H C_p \\rho v (T_a - T_s)\n", "88005b6c0da9d403a94fb14f622f6755": "1+(1/n)", "8800648cbe02c884f7f9fad384db32fb": "\nT_{ij} = a\\frac{{P_i P_j }}\n{{C_{ij}^b }}\n", "880067e68e665dbee7df511d716603a0": "\\vec{r}_u := \\frac{\\partial\\vec{r}}{\\partial u}", "880148ba5f5f0129569171bb7c27e93c": "A \\cup U = U\\,\\!", "880166f4f7fcf95aa2008ae429b566cc": "\\{\\neg\\tau, \\rho_2, \\rho_3, \\rho_4, \\dots, \\rho_n\\}", "8801a9473d7c8078bfbfbb28a1368215": "S(\\beta)=\\sum_{j=1}^n |r_{(j)}(\\beta)|^2,", "8801e23c0c24af9432d1e4f6cee62c5a": "x(2,3) = 2, \\text{ and } y(2,3)=3.", "8801f2cb86d2b50fc891acf4b1b72007": "p(y,x_1,x_2,\\dots,x_m)=0.\\,", "8802296c9e6c50001eabceb46e2c08a7": "\\mathbf{A}\\mathbf{x} = \\lambda \\mathbf{x}", "8802b53c2b5401ca2104c3bfe106f800": "\\varphi_{i,j}(a) = b'", "8802c0457eb662275afd7a1bd60fcdd8": "\\sim ~ 1 ~ S/m ", "8802f1a3c5a03fee1362cc911847976a": "\\displaystyle E/V = K_1 \\sin^2\\theta + K_2 \\sin^4\\theta + K_3\\sin^4\\theta\\cos 6\\phi ", "88032e1e3666b9d17a2b8dd5fc534bec": "\\widehat K/\\widehat H", "880355818b23f54d41e5949b2f7002bf": "\\tilde{r}_E", "880380895eec73ca7c93dcbd10f99387": "\n\\begin{align}\nA &= \\frac{Z^2\\alpha_{fine}^3}{(2\\pi)^2}\\frac{|\\mathbf{p}_f|}{|\\mathbf{p}_i|}\n\\frac{\\hbar^2}{\\omega} \\\\\n\\Delta_1&= -\\mathbf{p}_i^2-\\mathbf{p}_f^2-\\left(\\frac{\\hbar}{c}\\omega\\right)^2+2\\frac{\\hbar}{c}\\omega|\\mathbf{p}_i|\\cos\\Theta_i, \\\\\n\\Delta_2&= -2\\frac{\\hbar}{c}\\omega|\\mathbf{p}_f|+2|\\mathbf{p}_i||\\mathbf{p}_f|\\cos\\Theta_i.\n\\end{align}\n", "88038ce8929a69b46eebf9d2ed696649": " R_2 ", "8803d790f6ea723a8a9c17a9c07b119c": " R_\\text{in} = {v \\over i} = -R ", "8803ede77943c11bd407db04746ba351": "U=D+(\\frac{wvkj}{w+v})", "8803f018d3760669867ab3c56481f9a0": "w_i = \\frac {\\pi} {n}.", "8803f3f305eeb516c563487b57617f6a": "n \\le 30", "88046917b785a8cb02ca9b2e3939024c": "H(M)\\subset T(M)", "88048d503bd5cbc7cf5532e0fbb94a87": " \\frac{1}{|B|} \\int_B |f(y) - g(y)| \\, \\mathrm{d}y \\leq \\sup_{r>0} \\frac{1}{|B_r(x)|}\\int_{B_r(x)} |f(y)-g(y)| \\, \\mathrm{d}y = (f-g)^*(x).", "8804d09b7ae5e0fe26117ed0758513fc": "\\mathcal{I}/\\mathcal{I}^2", "8804fe140069a3c90350a9c756f26152": "M_{earth} = \\frac{gR_{earth}^2}{G}\\,", "8805abfa3077c9197dea6abb476d833a": "\\nabla J(\\mathbf{x}) = 2\\mathbf{B}^{-1}(\\mathbf{x}-\\mathbf{x}_{b}) - 2\\mathit{H}^T\\mathbf{R}^{-1}(\\mathbf{y}-\\mathit{H}[\\mathbf{x}])", "8805cbdf83c2f6af3321c04961e7e127": "\\mathbf{r}_{k}", "8805e51637cfa60b4c404b3ae5e40913": "3^\\frac{12}{13}", "880617d69481088041405394a2a6c5a3": "(ap+bq)^2", "880648eff6f49d500bb0aaecdb8d3499": " \\phi(s,t) ", "88064db78059f85b7812c060ee603670": "x^x\\,", "88065bca6b92b91b744f6296196f0f78": " c^{2^{S-1}} \\equiv z^\\frac{p-1}{2}\\equiv -1 \\pmod p", "88066c4b9af77b9232e3c8a81e287067": "\\prod_{i=1}^{p-1} i^{p-1} \\equiv +1 \\pmod p.", "88070680c00b108ac78f303aa226e54d": "\\forall a, b \\in X,\\ a R b \\or b R a\\or a=b.", "88071c7132604a43ddb6fa584e53f6e3": "K_{5}", "88074706f4f6ed52a68258c6152e2969": "\\overline{T}_1=\\frac{1}{L}\\int_0^LT_1(x)dx", "88077663fb84c20738a14af3b348dc38": "\n\\Big|\\int_a^b e^{i\\lambda\\phi(x)}\\Big|\\le c_k\\lambda^{-1/k},\n", "8807e2bb0db03e5520141388761da956": "\\displaystyle s^2=3\\sqrt{3}T.", "88081c7d5ad14647def95977e55773b1": "\\delta_{ij}", "88084057916888a53f9094ca5847197d": "f(z) = 1/(z^k-1)^2, g(z) = z^{k-1}\\,\\!", "8808a3e562439b83156a29c11a15b596": "M-p", "8808acaabe36a3781c085eb3487a1ba6": " (1+x^2) \\frac{dy}{dx} = n (1+y^2)", "8808c94ff49f2590c249e12a789a9af9": "\n\\mathcal{A} = \\mathbf{A} + \\frac{mq}{2} \\left[ \\left( \\mathbf{r} \\times \\mathbf{E} \\right) \\times \\mathbf{r} \\right] ,\n", "88093558bf10fa1492e3a6df51f04fec": "q_\\textrm{test}", "8809430d90f2a6f1167f56183f3082fb": "\\|R(hA)\\|\\leq 1", "8809a225691d67fb0a3203ccd9993152": "\\prod\\limits_{j=1}^n (1-{p_j}+{p_j}{e^{it}})", "8809a637eb546b8567402e117b222ab9": "\n\n\\sigma_{ji,j}+ F_i = 0\n\\,\\!", "8809b8368880e66fa0a4182db5c9e28c": "L(x,u) = f(x) + \\sum_{j=1}^m u_j g_j(x)", "8809c3d191df8e2591cab129efadfefb": "\\Delta\\, ", "8809e7d34f7e1ddeeb1214881170b60e": "\\frac{\\partial \\mathcal{H}}{\\partial q_i} = - \\frac{\\partial \\mathcal{L}}{\\partial q_i} \\,, \\quad \\frac{\\partial \\mathcal{H}}{\\partial p_i} = \\dot{q}_i \\,, \\quad \\frac{\\partial \\mathcal{H}}{\\partial t } = - {\\partial \\mathcal{L} \\over \\partial t} \\,.", "880a4f7960d5bc52fc127b420b92bac5": "\\mathrm{^{239}_{\\ 94}Pu\\ \\xrightarrow {4(n,\\gamma)} \\ ^{243}_{\\ 94}Pu\\ \\xrightarrow [4.956 \\ h]{\\beta^-} \\ ^{243}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{244}_{\\ 95}Am\\ \\xrightarrow [10.1 \\ h]{\\beta^-} \\ ^{244}_{\\ 96}Cm} \\quad; \\quad \\mathrm{^{244}_{\\ 96}Cm\\ \\xrightarrow {5(n,\\gamma)} \\ ^{249}_{\\ 96}Cm}", "880aaa315a0c8fee5faff347cffeffa3": " P_{ij} = \\tfrac{1}{2}(\\vec{\\sigma}_i\\cdot\\vec{\\sigma}_j+1)\\,.", "880acfbe42459c31e04a4b3b5203f65e": "1\\times 3^{-3\\,\\,} + 1\\times 3^{-4\\,\\,\\,} + 2\\times 3^{-5\\,\\,\\,} + {}", "880b3c59665daa34607c3ac58a2731a4": "S \\approx L + \\frac{x^2}{2L}+\\frac{x a}{2L}", "880b802794dacf26bc1dbe215a059d5d": "G(z_{t}, \\zeta, c)x= 1-exp(-\\zeta(z_{t}-c)^{2}) \\zeta>0 ", "880ba8a2026e5b7d4cef92423c374e3a": " A \\otimes B \\cong B \\otimes A ", "880bad92d02e4cdbc873fd274b1c393b": "{d \\over dt}\\left\\{ Y \\right\\} = k_2 \\left\\{ X \\right\\} \\left\\{Y \\right\\} - k_{3} \\left\\{Y \\right\\} \\,", "880c2649824028985f9d597e224072e4": "K(m)", "880c58fbe6079406350bb579b98c6640": "|f'(t)| \\,", "880c868dcd32f0e20e98a006d23aa1d1": "\\ \\sigma_{rc}", "880d2672ade233f67f9286456b65ec90": "\n(p \\rightarrow q) \\rightarrow p \\Rightarrow\n\\overline{p \\rightarrow q} \\or p \\Rightarrow\n\\overline{\\overline p \\or q} \\or p \\Rightarrow\n(p \\and \\overline q) \\or p \\Rightarrow\n(p \\and \\overline q) \\or (p \\and 1) \\Rightarrow\np \\and (\\overline q \\or 1) \\Rightarrow\np \\and 1 \\Rightarrow\np.\n", "880d5395c1f72ae7d57769f36cca3e2c": "\\left(\\sum_\\alpha c_\\alpha X^\\alpha\\right)\\times\\left(\\sum_\\alpha d_\\alpha X^\\alpha\\right)=\\sum_{\\alpha,\\beta} c_\\alpha d_\\beta X^{\\alpha+\\beta}", "880d7d66a7046ceb1c5fef69efa6533e": "\\mathcal{Y}=\\ker(G')", "880da8676e73e275de25df4dcdd8e7de": "\\frac{ n_\\text{upper} }{ n_\\text{lower} } = \\exp{ \\left( -\\frac{ E_\\text{upper}-E_\\text{lower} }{ kT } \\right) } = \\exp{ \\left( -\\frac{ \\Delta E }{ kT } \\right) } = \\exp{ \\left( -\\frac{ \\epsilon }{ kT } \\right) } = \\exp{ \\left( -\\frac{ h\\nu }{ kT }\\right) } (Eq.1) ", "880dcc186d5421ae461570a8a9f5d346": "\\scriptstyle{t+{r\\over c}}", "880def1400d3ee5849cc6aafb1febfae": "z = \\mathrm{round}(x, m) = \\mathrm{round}(x / m) \\cdot m\\,", "880e42643940cd75f6ce0c19a0a3b177": "I_n = \\int x^n e^{ax} dx . \\,\\!", "880e4bbadf02a392db81378cae27a2fc": "\\Gamma_8", "880e5323098cf5da29adfcb36642bf5e": "\\ u_y = W + u_1 - u_2,", "880eb5629c99088c42c0cec35b7a0632": "A \\widehat{=} 0", "880f115ef12f6a851e62772582e7882e": "{dY}/{dK}=MPK=r", "880f1674e213ab6d5b801bd4f37720ed": "dW_{stored} = i ~ d\\lambda \\;", "880f4236562b3d06d68be8c891dc6162": " \\psi(Y) = \\psi(y_{16} \\mathcal{k} y_{15} \\mathcal{k} ... \\mathcal{k} y_2 \\mathcal{k} y_1) = (y_1 \\oplus y_2 \\oplus y_3 \\oplus y_4 \\oplus y_{13} \\oplus y_{16}) \\mathcal{k} y_{16} \\mathcal{k} y_{15} \\mathcal{k} ... \\mathcal{k} y_3 \\mathcal{k} y_2", "880f93a4040c1db14ec6a881d0a4e5cc": "\n\\sigma_{i0} = \\mathrm{amf}_i (\\theta) \\sigma_{i \\theta}\n", "880f9d1022e9c34d154b091c7f80e564": "V(\\underline{t})", "880faf90284e145e0dc6e0ee0ba9716c": "\\sum_{i = 1}^{m} c_{i} n_{i} = n.", "88101ef0318daec377be6359459c60b8": " \\eta(\\hat x,x)", "88104192230d10e9d1106ee91e1e7aac": "\\displaystyle \\gamma_\\mu = \\eta_{\\mu \\nu} \\gamma^\\nu = \\left\\{\\gamma^0, -\\gamma^1, -\\gamma^2, -\\gamma^3 \\right\\},", "8810cee35146d6b13381dd1413bef17c": "\\mathcal L = (f(t)-1)^2 \\,,", "88111d669c26fd35e7f30641687a03e8": "\\star \\mathrm{d}x=\\mathrm{d}t\\wedge \\mathrm{d}y \\wedge\\mathrm{d}z", "881154b2ca0c1d8535acbcaf1c089de0": "\\mathcal{P}^{\\prime\\prime}", "8811aea12b514bd85fb50cf8bd7bd90d": " \\|f\\|_{C(K)} = \\max \\{ |f(x)| : x \\in K \\}, \\quad f \\in C(K). ", "8811afe17f27386f08e8c9bee49e1868": "F_q", "8811baa576bc5cb0c30b43ddf864ba57": "f(t') = |\\mathbf{r} - \\mathbf{r}_s(t')| - c(t - t) = |\\mathbf{r} - \\mathbf{r}_s(t')| \\geq 0", "8811ff8681915291c361608589576efc": "E_T,\\, H_T", "88123e72979532650aa6be2f1c06f906": "f^{-1}(y) = ab^{-1}y\\mbox{.}", "8812442960a88922103457dfacaf5d4e": "as^2 + bs + c = 0\\,", "8812907ddc6eec3ad09010e77d012ef7": "1 \\in \\mathbb{N}", "8812c27b2f5b398936460273a08c2054": "x(n) = \\sum_{i=1}^p A_i e^{j n \\omega_i} + w(n)", "8812cffdabc6335e704be334649a490d": "Name = 2^{(\\text{number of flags on note}+2)}", "8813110c60da24c34affa9b5079b6bce": "B \\in K\\backslash\\{0\\}", "8813e5e118b674053f7c05febdf3c59d": "m_{i_1}, \\ldots, m_{i_k}", "8815098188bcfe0a0341b09169186fdf": "\\frac{\\pi}{4} = 4\\arctan\\frac{1}{5} - \\arctan\\frac{1}{239}", "88151448a34d578123896b1c9c5aff69": "\\displaystyle{z=C^{-1}(\\overline{w}).}", "88152dbdfee59601464874486da3cc95": "\\mathrm{BkF_3\\ +\\ 3\\ Li\\ \\longrightarrow \\ Bk\\ +\\ 3\\ LiF}", "8815306411ca2c5be0c960424c214644": "O(n^{\\lceil \\frac{d}{2} \\rceil})", "881533372318eca35d021f3e9633b097": "\\pi_0(\\operatorname{GL}(n,\\mathbf{R}))=\\mathbf{Z}/2", "88153feb6e85c75e8a3f6cdd62f8c374": " u \\in \\mathbb{R}^n", "88154f2be18c154af36fde9d5c6b4c76": "F_{12}=\\sin{(\\alpha)} d\\alpha\\wedge d\\theta", "88156b7e25398ea7303b9308687c0fbf": "\\nabla \\left(\\frac{n_1}{n}\\right)", "8815718fd1b36f7d46470c22311a49be": "\\alpha(0):=0\\ ", "8815c4b3967872a007f910c2adb0eb08": "\\omega = -d\\theta = \\sum_i dq^i \\wedge dp_i", "8815c6a664124db23051298d2e652af4": "N(0,\\sigma_Z^2I)", "8815ca6390116f4952afe2e5f64ba34f": "= a\\left(1 - 3 \\tan^2 \\frac{\\theta}{3}\\right)= a(1 - 3t^2) ", "8815d1b91f2731adf96dd017b38c7cca": "\\vdash\\varphi\\rightarrow\\psi", "88164354652817ffee36d596a683a2d4": "[a]", "881678827b9d30ea3020d5866f61ff38": " \\ v_{2} ", "8816eaf74ab6c24a051e336b6c5aa8ad": "H < G", "88172a0cac7b5e1940352fad39f9dce3": " RTF = \\frac{P}{I} ", "88174a4271c155db867607dbafe33737": "\\left[\\frac{n}{n + p}\\right] = 0.21", "88177aea5154918e5e70dd08819abf6a": "\\Sigma^f", "88179ab77f79b8b8f1001ea73c342652": " \\frac{g_m R_{\\text{S}}}{g_m R_{\\text{S}} + 1} ", "8817b727a01500e56582e363d1a53bf7": "O_{10}", "8817c7784bc42ff3aa033c9d079ae364": " | \\overline{j , m} \\rangle = \\sum_{mm'} {D(R)}^{(j)}_{m'm} | j , m \\rangle ", "8817defd9641d6e2b07420ae4bb0a89c": "\\hat{\\Theta}_{m}", "88186b707564581b3c0fbd50e1edfd38": " \\mathbb{C} ", "8818939d9616d93f6a38e83dcfe91c4e": "B^+", "8818b1ff17b3ec8757dd0651c3bf311f": "x^4 - Ax^3 + Bx^2 - Cx + D = 0", "8818f0ce1404a8b4ac7429a58f8e1f2e": "\\alpha < \\frac{\\pi}{6}", "8818fdd72ad4270f78f20e29117285c7": "H(t) \\equiv \\frac{\\dot a}{a}\\!", "881901e8181d8e7083154a702fed48e1": "5_H", "88194a297d1cc17acb8295c80fe191f7": "H(|f|^2) + H(|g|^2) \\ge \\log(\\pi e)\\quad\\textrm{for}\\quad g(y) \\approx \\frac 1{\\sqrt{2\\pi}}\\int_{\\mathbb R} e^{-ixy}f(x)\\,dx.", "88195b14f22caebf2ad99346be0f5088": "p(x):=\\sum_{i=0}^n p_i(x)", "8819683a72852fa92e5b0312071d4eb0": "f(\\theta)", "88199ee2b55f047f1b2868d265e7b37c": " \\sum_{i=0}^{\\infty} \\frac{1}{2^i} = \\lim_{x\\to\\infty}\\frac{2^x-1}{2^{x-1}} = 2,", "8819d00b41a5a8ddfc1d5467d962479f": "B\\cap\\left(\\bigcup_\\alpha A_\\alpha\\right)=\\bigcup_\\alpha\\left(B\\cap A_\\alpha\\right)", "8819ee9be8c673461da790fef94d9850": "\\textbf{G}(s)=-1=e^{j(\\pi+2k\\pi)}", "8819f7b564b3fb7784c8a315949d6874": "\\int f(x)\\, dx.", "881a4938d25fa6cf3f70c825c33bafde": " \\subset ", "881a72fd6c8353e116c043a9d38f88c5": " A_{mn\\mu\\gamma} = \\frac {m + 1}\\pi \\sum_{i=2}^{images} \\sum_{x=1} \\sum_{y=1} U(i,\\mu,\\gamma) [V_{mn}(r,\\theta)]^* P_{i_{xy}}", "881a97e352032b1711a4941cfb920bb6": " \\psi\\left(\\frac{1}{3}\\right) = -\\frac{\\pi}{2\\sqrt{3}} -\\frac{3}{2}\\ln{3} - \\gamma", "881aa7b22b0aff7abc1fa0b159b6f114": "\\displaystyle g^{ij}(x,\\xi)", "881adb99d2bacce0cc45155e25302ab7": "\\lambda ^{6/5}", "881adf68a56aa362e2fa5f7a420f9a3f": "H_p(X, \\mathbb{Z}) \\simeq H_p(Y, \\mathbb{Z})", "881ae90bc026569867b8b9f7b0f587ed": "j_s = {\\mu}_s j_r", "881b44d393c94049df905077c4622bdc": "\\boldsymbol{\\omega} \\times \\boldsymbol{r}", "881b847d6494b31408e4ad52b40c02e3": "M\\Rightarrow_{mem} N", "881bce6e24a65345c4f5ad3c37b841ee": "L_\\mathrm {L1} = \\sqrt{Z_\\mathrm {i \\Pi} Y_\\mathrm {i T}} \\ e^{\\gamma_\\mathrm L}", "881c1e778c47245631450be5c124eac0": "\\scriptstyle I ", "881c276ce458b90024691e2a4678b80d": "\\mathcal{I} \\subset \\mathcal{O}_X", "881c652a8be8a9d05a036921fa49c4bf": "\\eta_c", "881c777ff925b3598e07524a5d094fe2": "\\begin{matrix} \\frac {1} {g_m} \\end{matrix}", "881c7d434244c23479e83747b1502fb7": "A_1\\, v_1\\, =\\, A_2\\, v_2\\, =\\, A_3\\, v_3,", "881cd330ff95847fa51fd68712afb739": "\\Delta_{\\mathrm{LB}} = \\frac1{\\sqrt{\\det(g)}} \\sum_{i = 1}^{m} \\frac{\\partial}{\\partial x_{i}} \\left( \\sqrt{\\det(g)} \\sum_{j = 1}^{m} g^{ij} \\frac{\\partial}{\\partial x_{j}} \\right),", "881ce592b7f13173ddd1425da6415a6a": "\\frac{\\delta^3}{\\delta J(x)^3}", "881d00cd20f6a970f204b12c737456f1": "T(n) = 2^{O(r\\cdot \\sqrt{n})}", "881d08b0fbe46d5c90b9e2f38c0b9776": "\\Delta U_t= U(t)-U(t-)", "881d166c360d37b821b25405903fb86e": "\\textstyle\\mathcal{R} = 2\\alpha R+W/kT", "881d1f40f57941a28a2814abd048192a": " e^\\nu_\\mu = \\delta^\\nu_\\mu \\,", "881d2d22cddd007253407316d5b8da5f": "x_{n+1} = x_n + d\\; \\frac { \\left(1/f\\right)^{(d-1)} (x_n) } { \\left(1/f\\right)^{(d)} (x_n) } ", "881d4c81dd75bed0f01b370f792404c6": "e^{-\\frac{1}{4}x^2(1-2\\ln x)}\\,", "881d639f37530c1d9b879fed4d5b2660": "\n \\boldsymbol{F} = \\boldsymbol{1} + \\gamma~\\mathbf{e}_1\\otimes\\mathbf{e}_2\n ", "881d6ba5afab7da9b98aaf5d2f91a924": "\\scriptstyle \\Gamma", "881da4037a464c1582355fe883fcf0eb": "\\begin{align} 2\\cdot R_*\n & = \\frac{(170\\cdot 3.67\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 134\\cdot R_{\\bigodot}\n\\end{align}", "881dc83118f7a7ec1c42d5bdbd9add4e": "| g^{(1)}(\\tau)|", "881dca2b04505f8d7b966b27aacf2b59": "l_x(y) = xy, x \\in \\mathfrak{g}, y \\in U(\\mathfrak{g})", "881e0200b376a57cb0650bb4d8b62999": "{\\mathrm{length}}_\\gamma(t)", "881e562eb9e997e03452d288ab9d3cfa": "\\frac{\\partial}{\\partial J_B}\\langle \\sigma_A\\rangle=\n\\langle \\sigma_A\\sigma_B\\rangle-\n\\langle \\sigma_A\\rangle \\langle \\sigma_B\\rangle\\geq 0\n", "881e572ceb643d5b79b06c19cf245038": "x\\in [{-2}, {-0.5}]", "881e8f7c35d3dcff572ef4b857fe8619": "S = \\frac {1}{T} U + \\frac {P}{T} V - \\sum_{i=1}^s \\frac {\\mu_i}{T} N_i \\,", "881ed780a5ed47ef1ac8762d298916c0": "\\left|\\frac{p(z)-q(z)}{(z-z_0)^{k+1}}\\right|", "881ee7219971d71e314adbc83843a43c": "V = \\frac{Mp}{Mb}0.2018 D", "881f28c289b3bc2c2ad11471bfa65119": "\\mathcal{C}_{\\varepsilon} (\\mu) (z) = \\int \\frac{1}{\\xi - z} \\, \\mathrm{d} \\mu (\\xi),", "881f4489c6b12eaac338b21a2283e7c6": "-\\Phi^+", "881f652d523e35481be202ed0205e73d": "\\delta \\subset (Q \\times \\{ \\mathit{root}, \\mathit{left}, \\mathit{right},\\mathit{leaf} \\} \\times \\Sigma \\times \\{ \\mathit{up}, \\mathit{left}, \\mathit{right} \\} \\times Q)", "881fb8f89225f5f9d86fee07218c4560": "_{s.2.right\\,}\\!", "881fdaac6c2ef4883d2c05c73ea08fb4": "\\left|s\\textbf{I}-\\textbf{A}\\right|=0.", "8820181f95b2d44ff4c5b089c2420647": "\\tfrac{2}{3} \\pi r^3 \\rho_\\text{water}.", "88207b6e67b2d788d5ab8b06ca6e6e24": "\\, S=A/4\\!", "88207bc086c9d879c22807479d29a9e2": "\\sigma'", "8820efc21f6b18be8c50ac06f9d19c12": "\\mathbf{M} = \\mathbf{A}^{T}\\mathbf{B}", "88211202fb8a8af146fb190752cecd94": " \\omega_0 = \\lim_{t\\downarrow0} \\frac1t \\log \\| T(t) \\|. ", "8821170a4e233e33c5fbb919c082433e": "\\scriptstyle\\sqrt2", "882125e8d22cfbf1333df819726c7cfe": "\\alpha_{avg}", "882181de7da7261839ecd2f41e35edf6": " \\left\\{ x \\in S^n | \\mathrm{dist}(x, x_0) \\leq R \\right\\} ", "8821aeb070c3f9e9e1e8f690d78ada6b": " q_{ult} = 0.867 c' N '_c + \\sigma '_{zD} N '_q + 0.3 \\gamma ' B N '_\\gamma \\ ", "88221a2af8d2ab8ae359232a44a93a94": "F = F_{DC} + F_{\\omega} + F_{2 \\omega}", "88224f703497b568e67e2f69b4e86492": "F_a \\times Ptot = PaCO_2", "882259b97f15a89fd6780df8f35e7535": "(\\mathbf{q}, \\mathbf{p}) \\rightarrow (\\mathbf{Q}, \\mathbf{P})", "88227278f591826eab949983a5f7e2e7": "E \\Psi(\\mathbf{r}) = \\left[ \\frac{-\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r}) \\right] \\Psi(\\mathbf{r})", "88229ff4f1b9d893dec307f39aad434d": "\\ S,", "8822adb2d7fc6494336892463a7895ac": "\\mathcal{L}:L^2(0,\\infty)\\to \nH^2\\left(\\mathbb{C}^+\\right)", "8822caa62e45ee7e7d44990c87869674": "re^{-r^2/2}. \\, ", "8822dc272660866ec8f83de25b6dc8a4": " \\alpha_k = \\frac{\\mathbf{r}_k^\\mathrm{T} \\mathbf{r}_k}{\\mathbf{r}_k^\\mathrm{T} \\mathbf{A p}_k} = \\frac{\\mathbf{r}_k^\\mathrm{T} \\mathbf{r}_k}{\\mathbf{p}_k^\\mathrm{T} \\mathbf{A p}_k} ", "8822e548c3dc5e3ec3e5c83d55f3a27f": "L = mr^{2}\\omega", "88238090b1883da4b190cfbff9801dcf": "ax^2 + bx = c", "8823ac8be1c6605550bea71e92e4c86f": "65^2", "8823bce6b916cd178dc3318ecb08d2a7": " {}+63030812099294896\n x^8-311333643161390640\n x^7 \\,\\!", "8823c7d72e1b365d21bdcd284199d9a7": " x^5-10x^3-20x^2-1505x-7412", "8823d0b72ab8d2c9496f72c410a33762": "=\\frac{1}{3}\\cdot\\left(\\frac{1}{2^{1-1}}+\\frac{1}{2^{2-1}}+\\frac{1}{2^{3-1}}\\right)", "8823fa511ce56c0e2a1bef6f35b31d01": "\n\\frac {d^2 f} {dx^2} - 2 x \\frac {df} {dx} + \\lambda f = 0.\n", "882499d14378fdbe63e85f87fab957c5": "\\log_2 3", "88249a3470e7108ebc9baec1b737b40a": "\\left(1-\\frac{2}{n}\\right)\\times 180", "8824b92768012da89047c0b96d5d75aa": "\\dot{m}\\;v_{e-act}\\,", "8824c1cf7cad86199d6385360c2c152f": " \\widehat{U}(\\Delta t) = I - \\frac{i}{\\hbar}\\Delta t \\widehat{E} ", "882513e073c4a74f75da924a07e10163": "\\scriptstyle d\\sigma^2 \\;=\\; dx^2 \\,+\\, dy^2 \\,+\\, dz^2,\\; \\forall x\\,>\\,0,\\; \\forall y,\\, z", "88255b98a3b9431dd0159aa3249ae27d": "C \\propto (-\\tau)^{-\\alpha^\\prime}", "88257bb43d5fa8c6963f948edbda8b91": "P = \\rho v L/t.", "882592ee33db0d35ba2850cde43a0aa7": "\\textrm{throughput} = \\textrm{sales\\,revenue} - \\textrm{direct\\,material\\,costs}", "88266a6facfce469ca37987426dbb6d2": "PV_{1} = \\frac{$100}{(1.05)^{1}} = $95.24 \\, ", "88272ceb888fbac8955aacb5b254c1ac": "\n{\\hat \\theta_w} = { K \\over a_n },\n", "8827462dbbf8b3f4d3262cfc660ed531": "\\displaystyle{(Hf)^2= f^2 +2H(fH(f)).}", "882771dabf671ddd9de0c63c84355e2a": "f(a_q)=a_0", "8827846d75e837d92b744175f3def0a3": "\\mathfrak{p} \\mapsto D_\\mathfrak{p}, \\operatorname{Spec}(D) \\to", "8827d1e0ac4eeba926ac377bbb720c42": "g\\in A_e", "8827f9879db9ca0e30668dff8ef3678a": "\\prod_{r=0}^{k-1} \\frac{\\alpha+r}{\\alpha+\\beta+r} ", "88282c9831a2e97eb7afd927e7b9f15a": "m_s \\in \\{-s,-s+1\\cdots s-1,s\\}\\,\\!", "88285219e8c64e783ae1fa95c0b3195c": "\\phi(z)=\\,:\\,\\psi^\\dagger(z)\\psi(z)\\,:", "882a2071b0be6baa2a486ac911ce0fee": " d", "882a6ad8f65e1ef2dd77addf8c7363d1": " l_1", "882a81af40900a32b1eb0007cc635b57": "(e^{\\sigma^2}\\!\\!-1) e^{2\\mu+\\sigma^2}", "882aa58b9be34b3d5f458685e98c4b64": "0 = \\nabla_\\ell g_{ik}=\n\\frac{\\partial g_{ik}}{\\partial x^\\ell}- g_{mk}\\Gamma^m{}_{i\\ell} - g_{im}\\Gamma^m{}_{k\\ell}\n=\n\\frac{\\partial g_{ik}}{\\partial x^\\ell}- 2g_{m(k}\\Gamma^m{}_{i)\\ell}.\n\\ \n", "882aae311f2bfe330e285621e57ca7f8": " \\frac {(m_s \\psi g-m_s g)^2}{2k}=\\frac {1}{2} m_s v_r^2", "882aea75f5a3986c6a06f21cf9570bb9": "y_p = A t^2 + B t + C", "882b29ed7d24c3042e05f77722a17ade": "\\triangleright \\!\\,", "882b8de2d62383c4178fc74e45091c2c": "\\sigma(X_t)", "882be76636cf5e9e7d5845ed95d1f2d1": "\\frac {d M_z(t)} {d t} = i \\frac{\\gamma}{2} \\left ( M'_{xy} (t) \\overline{B'_{xy}} - \n\\overline {M'_{xy}} (t) B'_{xy} \\right )\n", "882cb2afa0831580ff3a764bb5246cfb": "\\mathrm{SCl_4 \\ \\xrightarrow{-15^oC}\\ SCl_2 + Cl_2 }", "882cccffc1bcb351a65838fd62911622": "N(r)=\\sum_{n=0}^{r^2} r_2(n).", "882d0965e0c9bc26d9345589a8e1365e": "\\omega_a \\ll 2/T", "882d22b7b951deb5e509f639019c7a77": "F = E A \\alpha_L \\Delta T", "882d6424aae639248eb3ea7da1bdd824": "k_{b_{n+1}}", "882da1fa841b7a070ad854fd7ce50871": " u_i^{n+1} = \\frac{1}{2}(u_{i+1}^n + u_{i-1}^n) - \\frac{\\Delta t}{2\\,\\Delta x}( f( u_{i+1}^n ) - f( u_{i-1}^n ) ).", "882dbdfab902b4ef2355038f40cd64c0": " G_{\\sigma} ", "882ddb93020b103b54fd5a7f6441816d": "\\mbox{blade element efficiency} = \\frac{V_a}{2\\pi Nr}\\cdot\\frac{1}{\\tan(\\varphi+\\beta)}", "882e3a8bc7524d75e38983ee57766657": "12\\# = 2 \\times 3 \\times 5 \\times 7 \\times 11= 2310.", "882e74a5300730dc096ca5723201a686": "\\int_{0}^{T} \\cdots\\, \\mathrm{d} \\sigma_{t}", "882e8cbbd6490418fb852bea1f650fde": "\\mathit{d_H}(\\mathcal{E}_w \\geq \\mathit{d_{min}})", "882e8e65c1a749fcf25a6b02a89e9b45": "J(4,2)", "882ec1c100a7a0b29ec4f52bc9efab9e": " h_{\\mu\\nu}(t,z;\\omega)=A^{+}(\\omega)(t-z)e^{+}_{\\mu\\nu}+A^{\\times}(\\omega)(t-z)e^{\\times}_{\\mu\\nu} +h_f(v_\\mathrm{g} t-z;\\omega) \\eta_{\\mu\\nu} ", "882f61ab11bfd900c3abe5f5e3f2df0b": "N \\approx \\frac { v_{\\mathrm N} - v_{\\mathrm F} } { 2 c }\\,.", "882fa53e26c7800d6491943e6a42fead": "w \\cdot a_{21}", "8830072519d94404f2791e29f34f79d3": "H^*(X) = H^*(X, \\mathbf{Z}) / \\mathrm{torsion}", "883016344e756bb80fdf1a301cc15a3b": "P_d", "8830410bda5d01dcf1d644f69f027488": "\\delta={\\min_{r,s}}{}_{+}\\|x_{r}-x_{s}\\|, \\quad \\tau=\\min_{r,s}{}_{+}\\|\\lambda_r-\\lambda_s\\|, \\, ", "88305207e7c2719827e660cd569b2e66": "\nI(\\omega)= \\left| \\int| E(\\omega+\\Omega)| |E(\\omega-\\Omega)| \\times \\text{exp} \\{i[\\phi(\\omega+\\Omega)+\\phi(\\omega -\\Omega)]\\} \\mathrm{d}\\Omega \\right|^2\n", "883059d20a8b1e71acca74bec76c973e": "e^{\\lambda I + N} = e^{\\lambda}e^N. \\,", "8830c82771fa4c004017388c3f71fc6b": "n\\,\\,\\mathsf{nat}", "8830e70abbf6e72657da8167c8397b70": " A_{FB} = \\frac {A_{OL}} {1 + \\beta A_{OL}} \\ , ", "88310bddf1b58deea6dd4f878a6a38f2": "\\int_{0}^{\\Delta t_i} r_i(t') dt' = \\ln(1/u_i) ", "88314eb607aa9c67664fcf2bc4731d17": "a^{\\dagger}(k)", "8831534a0c444e6e1f6aa0ad59cd961d": "\\begin{matrix} {2 \\choose 1}{10 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "8831603960ff540d2b1ce9ca85c2166f": "f(x) \\ ", "88317dd63297f33c1f6bd17592bd2a11": "2^{2^i}", "88318d2a41a053d280ffac0013a074f7": "\\operatorname{Tr}(\\cdot)", "8831ad7a19a53a24b791172516918266": "c_L(s', x) \\geq c_L(s, x)", "8831ae867de1d61b35cf6508d5366b5a": "U^{\\otimes r}\\otimes (U^{*})^{\\otimes s}dU", "8831b71b9af7559e1dbcf843f182b69a": " \\sigma_k ", "8831eeb4c1cfceffe5a78840cb2c4a24": "2^m,m+1,2^{m-1}", "883241a09f793dabec6df71da16dcd63": "g(x, y) = C", "883259e325f9c29de3bf3910345e8e8b": " \\operatorname{drop-params}[g\\ m, D, V, [F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "88330c33f7fed83ebea7cffb78440d19": "L(f^{-1})", "8833a343d980569d3b386d1f1e5f0c39": "\\frac {d M_x(t)} {d t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _x", "8833a3633facfe9b39a87413e0c157f2": "C_{xy} = \\frac{|G_{xy}|^2}{G_{xx} G_{yy}}", "8833e3006fd2257345c3e30261a8330f": "\n\\mathbf{p} \\cdot \\dot{\\mathbf{q}} - H(\\mathbf{q}, \\mathbf{p}, t) = \n\\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t) + \\frac{\\partial G_{1}}{\\partial t} + \\frac{\\partial G_{1}}{\\partial \\mathbf{q}} \\cdot \\dot{\\mathbf{q}} + \\frac{\\partial G_{1}}{\\partial \\mathbf{Q}} \\cdot \\dot{\\mathbf{Q}} \n", "8833fe42841151af3f09db3c0870870b": "|j\\rangle", "88340166bd285ce8d28350ad1682fb9e": "2/\\mu", "883456018e0e31c6cd52a52deba2c678": " L := F(t,(x,y)) - F(t+\\varepsilon,(x,y)) = 2\\varepsilon^3+6\\varepsilon t^2+6\\varepsilon^2t-(3\\varepsilon^2+6\\varepsilon t)x = 0. ", "88345c0cf7ac552c514d7e1a28121acd": "X_3 + a X_4", "883525668e4bd8fe7908be27903a8c1d": "(X): \\mathfrak{g} \\to \\mathfrak{g}, X \\in \\mathfrak{g}", "883537f8db8cd0cd580a88bf11e49513": "a^{\\frac{b^n - 1}{b - 1}}x^{b^n}", "88357c599da34160d93a43f906d0c5f1": "SHS^{-1}=H", "88359d9b8e9679bf510d56c55bf5feec": "\\dot{y} = \\varepsilon \\overline{h}(x_1, y, \\varepsilon) \\;\\;\\;\\;\\;\\; y \\in \\mathbb{R}^q", "8835aabf566573ef7da4343754c44ec1": " \\quad H^A \\Phi_n^A = E_n^A\\Phi_n^A\\quad", "8835bc2f89d00f13b6ca83606995a356": " Y(x_1,\\dots,x_n) = a_0+\\sum\\limits_{i = 1}^n {a_i} x_i+\\sum\\limits_{i = 1}^n \n{\\sum\\limits_{j = i}^n {a_{i j} } } x_i x_j+\\sum\\limits_{i = 1}^n \n{\\sum\\limits_{j = i}^n{\\sum\\limits_{k = j}^n {a_{i j k} } } }x_i x_j x_k+\\cdots ", "8835cf3b7714de66eca05012eef62caa": "e^{j \\omega t},", "8835ee2dcf514971d1726d83ce6b48d5": "\\ell_i(x)", "8835fcc3980bf934c2e35e8bc808dbc9": "\ni\\frac{\\partial u}{\\partial t}\n=-\\frac{\\partial^2 u}{\\partial x^2}-|u|^{2k} u\n", "883624ec718e1daa5bebee6ccee4de5f": " c_3 = 4.777114035, \\,\\!", "8836720debbc5677cb66c7af0238a522": "a,b \\in Z\\cup\\{-\\infty,+\\infty\\}", "8836ad4e2337f981c3e2ccae114ca284": "(1-\\beta)", "8836cca63bdb282b5aee63b53bdfa13d": "\nv_1 = v_0(x) - z \\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x} ~;~~v_2 = 0 ~;~~ v_3 = w_0(x) \n", "8836d66d9c258e3ae4ab38d74c933461": "f(rm + sn) = rf(m) + sf(n)", "88372ac2465c6ba135f018e140498938": "\\mathbf{s}_j", "8837337ddfce9d292c05e580fbd265d9": "T_M(n)\\in O(n^{k})", "8837389479bfab276ca3a6cacd4a6514": "\\big. F_0=-k_BT\\ln Z_0 ", "8837620c33299433062ad0e5b59746b4": "C=\\prod_p\\left(1-\\frac{1}{p}\\right)^7\\left(1+\\frac{7p+1}{p^2}\\right) = 0.001317641... ", "8837d8e4b9d2c6a067ccbf0745aeae0e": "(a, b, c)", "8837df45e6cc70b20f1c232847a27d71": "\\rightharpoonup ", "8837f6990431697bb98cb05ee9336bad": "\\sigma(t)=t", "88382a0f6c309b87f46662b4ef26e551": "\\frac{dx}{dt}=-\\lambda x+f(Wx)", "88384a3fcb24e8eb18d19cde30850efb": "\nS(t) = \\begin{cases}\nt^3 & t \\ge 0\\\\\n-t^3 & t < 0\n\\end{cases}\n", "883854b8761a316599d13cbacf6c6550": "1 \\mbox{ and }-1 \\,", "883866274e7ca606be07ffb4b1bc28b8": "T_P(X)", "88391696013c0c0c2659bcece4311aae": "(A - 2Z)", "883a49cae894d520ed138875379c6d98": " \\ln(a^{b}) = b \\ln(a)", "883a6a8d71c28cf41585e2034078900c": "\\left\\langle\\nabla^2\\left(\\frac{-e^2}{4\\pi\\epsilon_0r}\\right)\\right\\rangle_{at}=\\frac{-e^2}{4\\pi\\epsilon_0}\\int d\\vec{r}\\psi^*(\\vec{r})\\nabla^2\\left(\\frac{1}{r}\\right)\\psi(\\vec{r})=\\frac{e^2}{\\epsilon_0}|\\psi(0)|^2", "883aa6e4139ce241560bf2bf3b00e11f": "\\psi(\\bold{r}_1,...,\\bold{r}_j,...,\\bold{r}_k,...,\\bold{r_N})=+\\psi(\\bold{r}_1,...,\\bold{r}_k,...,\\bold{r}_j,...,\\bold{r}_N)", "883ad077b5fb3824c5d3242f30ffbb69": " \\frac{h}{\\sqrt{p\\left( 1+m \\right)}} = \\sqrt{GM}= k ", "883ae9b4e859989648890e5732e46c91": "\\arctan\\big(\\tfrac{1/300}{12}\\big)", "883c6429ab4d475cd324513ed1cbc6cd": "\\mathbf{j}_\\mu = {1\\over 2}\\bar{\\psi}_L \\gamma_\\mu\\boldsymbol{\\tau}\\psi_L", "883cc0c11b75d487fa4bde27479381e0": "\\delta I = I_c \\cos(\\phi_0) \\delta\\phi\\,", "883cc364e0f7e0dcbe1fe264731556a3": "\\frac{\\partial y}{\\partial \\mathbf{x}'}", "883cdcbaed02167f0fd34b9894b188df": " \\int f \\, \\mathrm{d} \\mu \\geq \\lim_k \\int f_k \\, \\mathrm{d} \\mu. ", "883cffdb1e98c304167ebd488641851e": "\\,g", "883d7c8259e9c68264c2261c9a402810": "\\mathcal{H} \\lbrace x(t) \\rbrace \\ \\stackrel{\\mathrm{def}}{=}\\ \\widehat{x}(t) = \\frac{1}{\\pi}\\int_{-\\infty}^{\\infty}\\frac{x(\\tau)}{t-\\tau}\\, d\\tau \\ ", "883d8d468ec83cd32194d0c925e905f6": "f_k = \\sum_{j=0}^{n-1} v_j\\alpha^{jk}.\\qquad (2)", "883d9499efaa81d8d4d9460dfa2f6761": "\\mu^2\\frac{d\\alpha_s}{d\\mu^2}=\\beta(\\alpha_s).", "883db3cbda3a2d86faf495c50c92f511": "\\zeta(7)=\\frac{19}{56700}\\pi^7 -2 \\sum_{n=1}^\\infty \\frac{1}{n^7 (e^{2\\pi n} -1)}\\!", "883e06fa4cda55c5dbc22d6b158cfdf0": "D_{L_{CO}}", "883e91c63dcb8fbfd25163d6e545fc72": "H(x) = \n\\begin{cases}\n1 & \\text{if } x\\ge 0\\\\\n0 & \\text{if } x < 0.\n\\end{cases}", "883f204668e2f4b61ed20942291cd559": "\\lim_{r\\to +\\infty}z_\\mathrm{approx}(r)=\\frac{1}{2}\\frac{r_s}{R^*} = \\frac{GM}{c^2R^*}", "883fa0de4f715cc077b3f54bd5f6f1ff": "U_p = ", "8840643738d19d63691712ecebbcc384": "3 \\times 3 = 9", "8840a07cb6e9429b23415e99de9f773c": "\\scriptstyle \\mu S(t) ", "8840eed9f074cd3daf2d912bb964d6ac": "{dt \\over t} = 2 - \\lambda B", "884108bfea1ea81a942f2924b7932fd0": "\\bar{x} = \\frac{1}{n}\\sum_{i=1}^n x_i.", "8841499cc8e0b9e6048fc2a2bb0cd4eb": "N = N_1 \\cdot N_2 \\cdot \\ldots \\cdot N_d", "8841a25c72fec1605cdfa9aeb8d1fab6": "\\textstyle X=x ", "8841b445e590037198e7a1919e406db8": " y_i=\\sum_{j=1}^{K}\\beta_j X_{ij}+\\varepsilon_i \\quad \\forall i=1,2,\\ldots,n", "8841b5f8eb2c1f91781b1e5a78928a24": " W\\left ( J \\right ) =\n - \\iint d^4x \\; d^4y \\; J_1\\left ( x \\right ) {1\\over 2} \\left [ D\\left ( x-y \\right ) + D\\left ( y-x \\right )\\right ] J_2\\left ( y \\right )\n", "8841d5a3ce3f7ddb21e4d412f0206d55": "G(\\chi)=\\mu(N/N_0)\\chi_0(N/N_0)G(\\chi_0)~", "884213980adda53a7a671e8476ecd601": "d_3d_2d_1d_0", "88422ef163384d3f813d3e6ca057ee00": "d\\mu=\\frac{dx\\,dy}{y^2}.", "88426552aab4441c220d62a94dde24a6": "\\mu=\\mu_x+i \\mu_y", "88427dd9316b8f397a7700e01770175a": "\n \\begin{align}\n \\mu(t,r)&=\\alpha(t)r+\\beta(t)\\\\\n \\sigma(t,r)&=\\sqrt{\\gamma(t)r+\\delta(t)}\n \\end{align}\n", "884286394c2010570be2d5bcd67ed5de": "\n\\frac{\\theta_2}{\\theta_1} \\Big|_{\\theta_1 \\ne 0} = \\frac{k_\\theta}{F L} - 1 \\approx \\left\\{\\begin{matrix} 1.618 & \\text{for } F L/k_\\theta \\approx 0.382\\\\ -0.618 & \\text{for } F L/k_\\theta \\approx 2.618 \\end{matrix}\\right.\n", "88429512077b3355baf040c1a7a0b027": "X_\\epsilon", "8842acfc5b974d5a232f8fb0171eee43": "\\Delta x = R_W \\cos(\\theta)(T_1+T_2) \\frac{\\pi} {T_R}", "8843505ee0d4759e373e306c2d94d388": "(\\phi ')", "8843a1a618fdf87abfb82405fa8d49c7": "\\overline {OD}", "8843ca79f7df986df12dd7a03fb074d0": "\\{g_\\alpha(k)\\}", "884402eea6fc0e346f75c60110643b1e": "\\frac{dA(\\beta)}{dt} = - \\frac{M}{N_\\nu} [ f'' + 2 \\eta^2Y + 2Y\\beta^2 ] \\beta^2 A(\\beta) ", "8844590194972cd77fed945c0ebcfa65": "\\textstyle \\lambda^{-1/2}", "884496c13d57018cf0283763a1cf3296": "(n-1)!\\ \\equiv\\ 0 \\pmod n.", "88449ca5365fc85a4051a4f39cf72e31": "n_y", "8844a5666eaf39d0a36bc1b3c19d54fd": "\\operatorname{predicate}\\ x\\ \\operatorname{then-clause}\\ \\operatorname{else-clause} ", "8844bc453766d1d08d431b4431f69b1d": "z=0.\\,", "8845399f29959593ffef9e11f9c43d5e": "\nF_n=\\begin{bmatrix}\n0_{1\\times 2^{n-1}} & 1_{1\\times 2^{n-1}} \\\\\nF_{n-1} & F_{n-1} \\end{bmatrix}.\n", "88455aa7feacd597b5e3dd17f65bd605": "(\\lambda, \\lambda +2\\rho)", "884567511360863c845871b18de26c62": " C_R^{\\text{Gauss}}(u) = \\Phi_R\\left(\\Phi^{-1}(u_1),\\dots, \\Phi^{-1}(u_d) \\right), ", "88459cf6c4fbc408569642dbc39363f4": "x = (x_1,\\ldots,x_k), x_i, m_i \\in \\!F_p", "8845b817849dc15584ba93219a72966b": "\\overline{P}(Cl_t^{\\geq})", "884696058c95fe7db66cb40d207f7183": "(\\mathbf{a}\\times \\mathbf{b})\\times \\mathbf{c} = -\\mathbf{c}\\times(\\mathbf{a}\\times \\mathbf{b}) = -(\\mathbf{c}\\cdot\\mathbf{b})\\mathbf{a} + (\\mathbf{c}\\cdot\\mathbf{a})\\mathbf{b}", "8846dfb3fb2c4e298467a0d76d54ff1b": "r\\dot\\theta^2", "8846f3a166725cfb91cceb18baf08346": "s_t=(s^i_t)_i", "8847023b3b3bf9d0673538aca11d9e38": "\\mathbf{L} = \\hbar\\sqrt{\\ell(\\ell+1)}", "884727456e9c4a36e0223032bf6062ee": "{\\bar{VP}}_3", "8847594c66a9f28f70a9b78d2f48f585": "r\\left(\\varphi\\right) =\n\\left[\n \\left|\n \\frac{\\cos\\left(\\frac{m\\varphi}{4}\\right)}{a}\n \\right| ^{n_{2}}\n+\n \\left|\n \\frac{\\sin\\left(\\frac{m\\varphi}{4}\\right)}{b}\n \\right| ^{n_{3}}\n\\right] ^{-\\frac{1}{n_{1}}}\n", "88476ac02f7e70920c94b358748dc668": "\\mathbf{i.o.-SUBEXP} = \\bigcap\\nolimits_{\\varepsilon>0} \\mathbf{i.o.-DTIME} \\left (2^{n^\\varepsilon} \\right).", "88479f585c380fa199bff4f827996f98": " \\lfloor x \\rfloor = \\sum_{n=-\\infty}^{\\infty}n[n \\le x < n+1],", "8847e24220b1de65831cbb47e9d1036e": " W_c(t) = \\int_0^t e^{A\\tau} B B^T e^{A^T \\tau} d\\tau = \\int_0^t e^{A(t-\\tau)} B B^T e^{A^T(t-\\tau)} d\\tau", "884851ef88ece159b10c7c80f760f215": "x(r,\\theta) = r\\cos(\\theta) - (1/2)r^2 \\cos(2\\theta)", "884864a480f40937c7b831fcf3c0742c": "\\Sigma = \\mathrm{E}(\\epsilon_t \\epsilon_t') = \\begin{bmatrix}\\sigma_{1}^2&0 \\\\ 0&\\sigma_{2}^2\\end{bmatrix};", "8848cfeded7478a0506ac6ddb08678cd": "\\phi:\\bigoplus_{i\\in F}R\\to M\\,", "8848d70e4dcc1f25247ab14f6fb8411e": "1.8050", "8848fb66a1276b58d4b7727fca8b6da1": "\\!\\delta V \\approx \\pi y^2 \\cdot \\delta x.", "8849366c07eb3b86a0af9e788ac3e06a": "\\Delta G^* = \\frac{16 \\pi \\sigma ^3 T_m^2}{3\\Delta H_v^2} \\frac{1}{(\\Delta T)^2}", "88493a5191faae9fe54e041a99a185c1": "{1\\over{a}}", "884950956b3b227ed29fd1b74ff7c3af": "\\frac{\\sum_{i=1}^K \\lambda_i}{\\sum_{i=1}^N \\lambda_i}\\geq \\alpha", "88495b37373c3c7b82b5c8d6e330dc8f": "VAS(x^3+2x^2-x-1,\\frac{x+3}{x+2}) ", "88496309a7caa69a9afadf9d8b17b905": "(g,g^a,g^b), \\, ", "8849ec194c0ab05b5f70c4b3a4bcdcf9": "\\sum_{r=1}^g d_rc_{rs}=b_nd_s, s=1,2,...g", "8849f7b5137a7873c902ec60420208a9": "I(Mary : N) = m : E", "884a0002e1a3d367e1b186e615d24b1e": "R_i = \\tilde{x}_i / p_i", "884aabd8a73415b79879618fece29fa7": "v_m^2 dx^2=v^2 ds^2=v^2 (dx^2+dy^2)", "884ac88766ceb0b3dadd2eb4452eacf9": "S(\\beta)", "884b2843c9525c50af4d5166dc912b15": "\\frac{d\\mathbf{L}}{dt} = \\mathbf{r} \\times \\frac{d\\boldsymbol{p}}{dt} + \\frac{d\\mathbf{r}}{dt} \\times \\boldsymbol{p}.", "884b6541be9ce4a9be0512416ea5c521": "BI {{=}} (W_1)AB[(b) + (e)] + (W_2)SN[(n) +(m)] + (W_3)PBC[(c) + (p)]\\,\\!", "884bbbd1af8d1fc417c9429fd89e5dcb": "\\begin{alignat}{7}\n x &&\\; + \\;&& 3y &&\\; - \\;&& 2z &&\\; = \\;&& 5 & \\\\\n3x &&\\; + \\;&& 5y &&\\; + \\;&& 6z &&\\; = \\;&& 7 &\n\\end{alignat}", "884bc0ab7d18e9517fa44b1b264e486c": "C = 0.8", "884c364f5417e0c7609db9b15f0c05bb": "\\displaystyle{\\mathrm{ind} \\, T(f)= \\dim \\ker T(f) - \\dim \\ker T(f)^*.}", "884c3f7d725ffd6dbdff2052f9af9ee0": " \\langle Mod(T), \\prec_{\\mathcal{F}} \\rangle ", "884c758dda09f9cd59f187a53ab0b398": " \\mathbb C^8 ", "884cb0a7205bd4ec53454feea0e0eb77": "80\\times0.8=64", "884d0824b7b1007c4840928c991853b3": "H_{7}\\approx H_{f}\\,\\ at\\ T_{7} \\,", "884d0c7fdb74f413f05e8224ac9b369d": " (\\kappa-2)~r^2~\\sin\\theta \\,", "884d12585d2a1b9ca68003a3ff5100bd": "y \\in \\mathcal{X}", "884d12f4be6a1c2d7ee24df0f2876adb": "f(\\det((a_{i,j}))) = \\det ((f(a_{i,j})))\\,", "884d174949920c6acd2a01255a52e909": " (\\vdash p)\\rightarrow(x \\pmod 2 \\equiv 0)", "884d2ec0b0ad0ae0b1a76807e55c579e": "\\frac{a_n \\alpha^n}{2^{n-1}}", "884d71f9e321034a2504099454253fb3": " s\\sim\\mathcal{N}(\\mu,\\sigma^2),x=\\textrm{max}(0,s), ", "884dd9cc4f1479df9872e55a001d614f": "\\mathbb Z_5^3", "884e31f472840e257fac94a56a27e050": "\\frac{ \\partial Y}{ \\partial K} = {\\alpha}[K(t)]^{\\alpha-1}\\cdot [A(t) L(t)]^{1-\\alpha} = \\frac{ {\\alpha}Y }{[K(t)]} ", "884e4c19e1b873253a8be5085517864d": "X_1 Y_2 X_3 = \\begin{bmatrix}\n c_2 & s_2 s_3 & c_3 s_2 \\\\\n s_1 s_2 & c_1 c_3 - c_2 s_1 s_3 & - c_1 s_3 - c_2 c_3 s_1 \\\\\n - c_1 s_2 & c_3 s_1 + c_1 c_2 s_3 & c_1 c_2 c_3 - s_1 s_3 \n\\end{bmatrix}", "884e59ffe4ced8c80a692a559aec54a8": "\\boldsymbol\\Sigma_{12} \\boldsymbol\\Sigma_{22}^{-1} \\left(\\mathbf{a} - \\boldsymbol\\mu_2 \\right)", "884ead2dfe02dd86707db99c861c1d34": "\\ln P = 23{.}7836 - \\frac{3782{.}89}{351{.}47 - 42{.}85} = 11{.}52616367 = \\ln(101332\\,\\mathrm{Pa}).", "884eb3f41fac09df284ebed81479a5dc": "\\sum _x \\log_b x = \\log_b \\Gamma (x) + C \\,", "884ede43386afdf3d544ec7cf25696c0": "Q = \\int_{t_{\\mathrm{i}}}^{t_{\\mathrm{f}}} I\\, \\mathrm{d}t ", "884f1963733d3c9968c8aa4508f5fb14": "\\frac{A}{\\sin a} = \\frac{B}{\\sin b} \n= \\frac{C}{\\sin c}", "884f2cd7b4b6a889dd6ec4fc26c33bae": "A ^3\\Sigma_u^+", "884f322e5705d85c7e7713e982eb1cb9": "\\text{Var}(\\hat \\theta)", "884f3c8b59f6ca501a41f6b1f293cad5": "\\theta=(\\theta_p)_{p\\in\\mathbb{P}}", "884f4b3416042b64a71860e4fa27e7ab": "\n\\tau = \\ln \\frac{d_1}{d_2}\n", "884f8494b8dd8a24cd52541c29aecc09": "P(\\zeta) = e^{i\\omega_{k}\\zeta}", "884fb4cd081686db41a0769885ac7992": "4! = 1\\times2\\times3\\times4 = 24", "884fd041b6989c41de5ac9e3fba5974e": " \\frac{P_B(t) - P_\\infty(t)}{\\rho_L} = R\\frac{d^2R}{dt^2} + \\frac{3}{2}\\left(\\frac{dR}{dt}\\right)^2 + \\frac{4\\nu_L}{R}\\frac{dR}{dt} + \\frac{2S}{\\rho_LR} ", "884fe1db3faaac3a6990938995e538d3": "C_xH_y + zO_2 + 3.71zN_2 \\to xCO_2 + \\frac{y}{2} H_2O + 3.71zN_2", "88504b74baa794da95709fbdfb3cb8af": " B_k+\\frac {y_k-B_k \\Delta x_k}{\\Delta x_k^T \\, \\Delta x_k} \\, \\Delta x_k^T ", "88505ff158346b6d8e9d9b7bbfef3acc": "\\mu (K_{\\varepsilon}) > 1 - \\varepsilon. \\,", "885076528cc14b970007803c47a3c501": "\\int {dx \\over x}", "8850a6af3d933ea8815aa572ba1b75a6": "\\displaystyle J_{2}, J_{3}, J_{1}", "8850a84aad986f96d1d356edb612d7f6": "\\hat{\\mathbf{H}}", "88510f05bb8a29bc09885ade703297fe": " \\frac{d}{dx}\\ln(x) = \\frac{1}{x},\\qquad x > 0.", "8851224c2597dccd0b4276b5c2ee0b2d": " \\sec \\theta = \\frac {1}{\\cos \\theta}", "88512ab12706879fec83c0c3aa79931f": "n \\times 1", "8851308fee5ca6e580001b7b45c2368f": "c \\sqrt{X}/ \\log X\\ ", "8851459c6facf738d11d0ba8b1198831": "\\lim_{t \\to + \\infty} \\mathrm{dist} \\left( \\varphi (t, \\vartheta_{-t} \\omega) (B), \\mathcal{A} (\\omega) \\right) = 0", "8851817882d505582689635b136c2c3e": "\\scriptstyle R/(I_1 I_2\\cdots I_k) \\,\\simeq\\, R/I_1 \\,\\times\\, \\cdots \\,\\times\\, R/I_k", "88520b6a52d29605ea110a16b4839529": "\n\\left(\\frac{\\partial S}{\\partial V}\\right)_T =\n+\\left(\\frac{\\partial p}{\\partial T}\\right)_V\\qquad= -\n\\frac{\\partial^2 A }{\\partial T \\partial V}\n", "88526d911c5c7180ddd862c62b7817e8": "\\frac{x+2}{x-2}=0", "88528b99d6f552994df31c30392f8483": "\n\\nabla \\mathbf{F} = \\frac{1}{a(\\zeta^2+\\xi^2)}\n\\left\\{\n\\frac{\\partial}{\\partial \\zeta} \\left(\\sqrt{1+\\zeta^2}\\sqrt{\\zeta^2+\\xi^2}F_\\zeta\\right) +\n\\frac{\\partial} {\\partial \\xi} \\left(\\sqrt{1-\\xi^2}\\sqrt{\\zeta^2+\\xi^2}F_\\xi\\right)\n\\right\\}\n+\\frac{1}{\\sqrt{1+\\zeta^2}\\sqrt{1-\\xi^2}} \\frac{\\partial F_\\phi}{\\partial \\phi}\n", "88529179b875d343cb271be3de0d8ccb": "L(\\mathbf{u},U)L(\\mathbf{v},V)=L(\\mathbf{u}\\oplus U\\mathbf{v}, \\mathrm{gyr}[\\mathbf{u},U\\mathbf{v}]UV)", "8852df397511e1dbb1f35fb7921860dd": "\\boldsymbol\\mu_2", "8853432b9487e3e413ab6666136aa5af": "\\tfrac{1}{n}\\mathbf{1}'\\mathbf{v}", "885389bf5a71ff54297c575eb00e70ae": "f(z)=\\sum_{k=0}^\\infty a_k z^{n_k}", "8853b0232c4727670ea8136c41bf196c": "R_1, R_2, ..., R_n", "8853bb2af59516dd8ae2c874aa21f832": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}~ \\cfrac{\\partial W}{\\partial \\boldsymbol{F}}\\cdot\\boldsymbol{F}^T ~;~~ J := \\det\\boldsymbol{F} \\qquad \\text{or} \\qquad\n \\sigma_{ij} = \\cfrac{1}{J}~ \\cfrac{\\partial W}{\\partial F_{iK}}~F_{jK} ~.\n ", "88543cea9cb78f00186e38ec00d41343": "H^i(X, \\mathcal{L}\\otimes \\omega_X) = 0", "885475d1c001c43a68372c299665d3e1": "B = \\mu_0 \\mu_{\\mathrm{r}} \\frac{N I}{l}.", "8854d76ad82c1809f8ec3fa38a70a060": "\\alpha_2\\;", "8854d9f0ac2830338cd5410f44d6ebff": "n = m", "8855ba5398600fb2511788513b66ff6e": "f(x,y) = \\frac{x^2y}{x^4+y^2}.", "8856f191870e42a3ed223d9ee2cc66d7": "B(\\frac{N_1}{N_{f1}} + \\frac{N_2}{N_{f2}} + ... + \\frac{N_k}{N_{fk}}) = 1", "885701681976e3d9d5661eef4e5a4804": " \\, m = 1, \\ldots, M ", "88572542da66ea4f1200610a492b9ac5": "\\textstyle\\mathcal{F}f := \\lim_{k\\to\\infty}g_k", "885756675a108c25426a478aa8e9101d": "\\boldsymbol{\\chi}", "8857905d8d4a67fc00fafacf8d5ff4a0": "\\mathbf{B}(\\mathbf{m}, \\mathbf{r}) = \\frac {\\mu_0} {4\\pi r^3} \\left(3(\\mathbf{m}\\cdot\\hat{\\mathbf{r}})\\hat{\\mathbf{r}}-\\mathbf{m}\\right) + \\frac{2\\mu_0}{3}\\mathbf{m}\\delta^3(\\mathbf{r})", "8857c3c67ca03aabb98d0391a825e68a": "C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2", "8857d89f213e630556ead407cea65baa": "\n \\boldsymbol{u}=\\boldsymbol{u}^{(0)} + \\boldsymbol{u}^{(1)}=\\boldsymbol{x}-\\boldsymbol{X},\n ", "8857e53dffc76657316efbe796282c16": "\\displaystyle 9R^2=a^2+b^2+c^2.", "885841afe916be57e45d699e9acaa305": "SG_{\\rm H_2O} = 0.998203/0.999840 = 0.998363 ", "8858437b58d904947bead17529696386": "{\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y = \\frac{1}{{\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y}.", "88588bacf0dad76870d7ede67a0ddbdf": "IL", "8858b7ed25a1f7f484f9c8bdbbed1ad9": "\\,\\mu(x)S(x) = f_X(x)", "885913bed9613a67eb5bb53d78abb3b8": "\n\\|f_{\\mathbf{w}} \\|_k = \\|\\mathbf{w}\\|.\n", "885948dd73a772069230d3a3c8e8bd88": " \\Pi_2 := \\prod_{p \\geq 3} \\left(1 - \\frac{1}{(p-1)^2}\\right) = 0.6601618158\\ldots.", "885968c0444cb499c1d8723cc53a1c60": "H_{D}(x,s)", "88598385e04c7766ee4c7c83323deb6c": "d^nf(x,\\Delta x) = \\left.\\frac{d^n}{dt^n} f(x+t\\Delta x)\\right|_{t=0}", "8859c17837b4484fc3c8deabc459a317": "(\\epsilon_r)\\,", "885a4aa4fd43feee4e230c4727aef1e0": "S(G)", "885a65611e1d797ed882669399498a7d": "\\mathbb{E}(W_i) = \\frac{1-\\rho_i}{2} \\mathbb{E}(C) + \\frac{(1-\\rho_i) \\text{Var}(C_{i+1})}{2 \\mathbb{E}(C)}", "885a9a21ec24728e8494568f95345fa8": "\\begin{align}\n\\text{PL} &= \\text{TR} - \\text{TC}\\\\\n &= \\left(\\text{C}+\\text{V}\\right)\\times \\text{X}\n - \\left(\\text{TFC} + \\text{V} \\times \\text{X}\\right)\\\\ \n &= \\text{C} \\times \\text{X} - \\text{TFC}\\\\\n &= \\text{TCM} - \\text{TFC}\n\\end{align}", "885ad03ce6f4328cc46027fcebb99101": "\\tfrac{|AB|}{|BC|}=\\tfrac{|AC|}{|AB|}=\\phi", "885b19c39d4f51482d6fd229ece9008a": "dS+dS_h+dS_w\\ge 0\\,", "885b2ccc601aa233100ffd19c2130553": "x_0^*", "885b747b269b7c9942e3f8bcd58274d7": "\\int\\frac{\\cosh^n ax}{\\sinh^m ax} dx = -\\frac{\\cosh^{n+1} ax}{a(m-1)\\sinh^{m-1} ax} + \\frac{n-m+2}{m-1}\\int\\frac{\\cosh^n ax}{\\sinh^{m-2} ax} dx \\qquad\\mbox{(for }m\\neq 1\\mbox{)}\\,", "885ba6aa825c2f39069115e9e0a80cb9": "\\chi_{0} (\\mathbf{q} | \\Gamma) = \\xi_{0}(\\mathbf{q}) exp [-i \\int_{\\Gamma} d\\mathbf{q} \\mathbf{'} \\cdot \\mathbf{\\tau} (\\mathbf{q} \\mathbf{'}|\\Gamma)]", "885be7f1b3eda075714571a5a535036d": " L^p ", "885bf4b65bb145c479b0a76179554f5b": "\\scriptstyle f_\\mathrm{blue},\\,", "885c015432f067da6b387d8c9010225a": " A = \\int_a^{b} f(x) \\, dx", "885c67cc290354286d5f8340cee63398": "(n, l, m, s)", "885cb6ab6588b777f446ea88fa85f606": "\\gamma = \\tfrac{1}{2}+\\int_0^{\\infty} \\frac{\\overline {(x+1)\\cos(\\pi x)}}{x+1} dx", "885d57fb33872c277cfab7fcb0bc94c1": "v \\vee a \\vee v : v \\in B", "885d6614d2d89928f903346705cd4437": "\\mathbf E_{1s} = <\\psi_{1s}|\\mathbf - \\frac{\\nabla^2}{2} - \\frac{\\mathbf Z}{r}|\\psi_{1s}>", "885d8603191e25a2f98e7292edc5d90b": "\\bar{\\mathbf{F}}_p", "885d8a2182843ae79b6f5c50edc9c5fa": "\n\\bar{\\boldsymbol\\mu}\n=\n\\boldsymbol\\mu_1 + \\boldsymbol\\Sigma_{12} \\boldsymbol\\Sigma_{22}^{-1}\n\\left(\n \\mathbf{a} - \\boldsymbol\\mu_2\n\\right)\n", "885e0c8e700d2de92d8031c6cf1b5f99": "V_{loop}*dF_{O_2loop}=(Q_{feed}*F_{O_2feed}-V_{O_2}-(Q_{feed}-V_{O_2})*F_{O_2loop})dt", "885e38cb70b3009cd34778b874d410a7": "w^{k+1} = S_{\\gamma}\\left(w^k - \\gamma \\nabla F\\left(w^k\\right)\\right).", "885ea8e5a869bd36193f6911577b7a4d": "\\kappa = \\frac{d\\varphi}{ds},", "885f3679424686f6f7cd2c75348d70d3": "q>(N^{1/4}+1)^2", "885f58a02f8565564cf27bc5a93a6319": "y<10^{-5}", "885f724cb849abde102322860791f661": "\\mathrm{NS}(A)", "885f7a5a257cd35d130a64878d4b61de": "c_{\\{j,k\\}}=\\begin{vmatrix} a_{3j} & a_{3k} \\\\ a_{4j} & a_{4k} \\end{vmatrix} ", "885f816264d7971af75d18e605df0172": "+\\omega_c", "886012481a9967ce8eebb16d942b8e31": "a \\cdot \\mathcal{D} = a \\cdot \\bar{\\mathsf{h}}(\\nabla)+\\mathsf{\\Omega}(\\mathsf{h}(a))", "8860370af76c01de5337d4626c2678f4": "Q1", "88603fb431044cfd8556bf1a0192552d": "\\alpha,\\,\\beta>-1", "8860768aa737a8750c6a10c8b8e16144": "\\bold{\\hat{p}} = -i\\hbar\\nabla - q\\bold{A} ", "8860bb07f7295f36dc06c203f37e0404": "\\boldsymbol{\\nabla}\\mathbf{u}'", "8860d8433aff21a1a11a3e734b658b7c": "1-z^{-1}", "88610cad7f79baed19c8fa23e826a6cb": "\\mu(x,y)=\\begin{cases}\n1 & \\text{if }y-x=0, \\\\\n-1 & \\text{if }y-x=1, \\\\\n0 & \\text{if }y-x>1,\n\\end{cases}", "8861d9d54ff238ea8151591f64fbccb0": "\nS_{\\text{Grav}} = \\frac{1}{16 \\pi G} \\int d^4 x \\, \\sqrt{-g} R \n", "8862286c02dc807b7793982e7a6dc421": "E(z)= \\exp \\left[-4 \\left( \\frac{\\left(z-z_0 \\right)}{I_c} \\right)^2 \\right]", "88624b1e344b1b23d6f2c6a7a4a701ea": "\\mathbf{ x}(4) = [u(4)\\, u(5)\\, u(6)]=[85\\, 80\\, 89]", "88627872c151c4e01ac1888bcdb8a14f": "a\\uparrow^n b", "886381267ad6d7ba6add96b6bd03a3e4": "\\tfrac{1}{2}\\left(\\tfrac{a}{c} + \\tfrac{b}{d} \\right ) \\pm \\tfrac{i}{2\\sqrt{3}} \\left (\\tfrac{b}{d} - \\tfrac{a}{c} \\right )", "8863815cf9f7b14a198adabf496feb4e": " \\text{Undefined for }2 \\le q <3", "88638fe87552e4f1159618a4c072c929": "\\hat H =- \\frac{\\hbar^2}{2I} \\left [ {1 \\over \\sin \\theta} {\\partial \\over \\partial \\theta} \\left ( \\sin \\theta {\\partial \\over \\partial \\theta} \\right ) + {1 \\over {\\sin^2 \\theta}} {\\partial^2 \\over \\partial \\varphi^2} \\right]", "88639321822658555b6eca1d8ec1aa48": "x =(A^{\\mathrm{T}}A)^{-1}A^{\\mathrm{T}}b,", "8863ffb80b155dac614595e01ea26a44": "B_n = \\frac{1}{e}\\sum_{k=0}^\\infty \\frac{k^n}{k!},", "88643a35135f962e7e20348987bbb440": "-4-4\\gamma", "8864475d0a480f752620773e836dfee8": "\nf(x) + f(-x) = 0. \\,\n", "886458e5bb8047a08b6c7017439c1722": "U = X", "88646360d74353d4fd6a2ccf23643fe5": " P(a) \\to\\ \\exists{x}{\\in}\\mathbf{X}\\, P(x)", "886501725eacff0fdeacba52fc9ac4ac": "\\textstyle\\binom{2m + 1}{m}", "886505e4ad104e427b5f8c00c3600ea7": "p\\times k", "886535cf383a670d24e77b967415d883": "\\gamma = 0.787", "88658890bdfb1bf820932486095c0107": "\n a X^k \\; b X^l = ab X^{k+l}", "8865a6b0e2d9c5a0935fa9102a8a4515": "\\mathcal{O}*\\,", "8865d5ac8628bb8826634e1e65804d08": "\\mathbf{W} = {}^\\star(\\mathbf{J}\\wedge\\mathbf{p}).", "8865e4171c60801aaf8b1e6e7289685c": "\\tfrac {mg}{kg}", "8865e4b5e1dfb0fdaba033a24cb24315": "\n\\int_0^{+\\infty} q(t)\\log \\Bigl(1+\\frac1{t^{2\\alpha}}\\Bigr)\\,dt\\leq\n\\pi \\alpha \\prod_{k=1}^{n-1} \\Bigl(1+\\frac{\\alpha}{k}\\Bigr)=\n \\frac{\\pi}{\\mathrm B (\\alpha, n)}.\\,\n", "8865f4471d1e34863573afc903116cc1": "p,q,r,...", "88661f4d42f50c3f20d081ddf582155d": "\\sum_{n=0}^{\\infty}u_{n}=\\sum_{n=0}^{\\infty}\\sum_{k=1}^{m}\\frac{a_{k}}{n+b_{k}}=\\sum_{n=0}^{\\infty}\\sum_{k=1}^{m}a_{k}\\left(\\frac{1}{n+b_{k}}-\\frac{1}{n+1}\\right)=", "88666db69b5bc096f8d4775970880319": "d = \\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta},", "8866e41d7a86896a1500e4ca04a4b515": "\\Pr(S|W) = \\frac{\\Pr(W|S) \\cdot \\Pr(S)}{\\Pr(W|S) \\cdot \\Pr(S) + \\Pr(W|H) \\cdot \\Pr(H)}", "8866e980e452669a159afd1db69504a2": "z = \\theta", "886736c4ea85297f6bce325cd647ae2e": "\n \\nabla^2\\nabla^2 w = 0 \\,.\n ", "88677469c20f035df0c3b6e08c1dfe21": " = -\\frac{T_0}{c} \\int d\\mathcal{A} ", "88679185f3ce9d4567b49c9b763700ad": " CHI = \\left ( \\frac{V}{V_0} \\right )^3 + \\frac{3}{2}\\left ( \\frac{R}{R_0} \\right ) \\left ( \\frac{V}{V_0} \\right )^2 ", "88679f7cc0495e6d5d068f908b086c75": "r0", "887fc412492d9381297373ca34bceee2": "\\iota_i", "887fed71ae1ecb0176a64f20bffa8c85": "xI-B", "88800c800331abca162fe6d0ece853a6": "T = 25 C", "88802ce0b10ff908e221cb6f5396d82f": "-c_3\\,", "8880730ed474a7591dd2d08212c2e6da": "(n-k)^2", "8880c4458566fd386b12603052bdd22a": "\\operatorname{Minority}(x,y,z) = x \\oplus y \\oplus z.", "8881335ad5d5bcdf1d31162c3c67b52c": "V'", "88819584d28c7219259ad58ac875c29e": "\\delta_X\\colon x\\mapsto d_x", "888197794b2d9f23cfccd1e3fe9aee68": "c^2 \\approx a^2 + b^2 - 2ab\\cos(C) . \\,\\!", "8882108740611b87da23a6c9d586cb78": "\\neg D(f(x))", "8882d4fd6101021ae960e0ef89f71b06": " \\frac{1.32\\times 10^{16} }{1600[year] \\times 226} \\simeq {3.7} \\times 10^{10} [\\text {Bq/g}] ", "88837efec693ace008fd8e0f145c3482": "i_K\\,\\omega(X_1,\\dots,X_{k+\\ell-1})=\\frac{1}{k!(\\ell-1)!}\\sum_{\\sigma\\in{S}_{k+\\ell-1}}\\textrm{sign}\\,\\sigma \\cdot\n\\omega(K(X_{\\sigma(1)},\\dots,X_{\\sigma(k)}),X_{\\sigma(k+1)},\\dots,X_{\\sigma(k+\\ell-1)})\n", "8883845a415ba20c1c66cfc46565b54a": "\\scriptstyle y\\in I", "8883b51d7a24bcc3857f8fca2d743bbb": "{\\rm Fm}", "8883b5ccd982cb67cdac49fc1b5ee6d9": "VTX=TX\\times TX", "8883f6f12d74032da69d263f9d5f7805": "(a \\cdot c)(b \\cdot d) = (a \\cdot d)(b \\cdot c) + (a \\wedge b) \\cdot (c \\wedge d)\\,", "88841fe536b183f2587d1ffbf2ac7127": "p^1 = \\mathbf{b}_1\\cdot\\cfrac{\\mathbf{e}_1}{|\\mathbf{e}_1|} = |\\mathbf{b}_1|\\cfrac{|\\mathbf{e}_1|}{|\\mathbf{e}_1|}\\cos\\alpha = |\\mathbf{b}_1|\\cfrac{dx}{dq^1} \\quad \\Rightarrow \\quad \\cfrac{p^1}{|\\mathbf{b}_1|} = \\cfrac{dx}{dq^1}", "888456ff4aaf569491be6acc005e204e": "t=s\\,", "88845a4dfd839b03407873af6e071dcc": "= - \\left ( t_j-y_j \\right ) g'(h_j) \\frac{ \\partial \\left ( \\sum_{k} x_k w_{jk} \\right ) }{ \\partial w_{ji} } \\,", "8884879c723c6b8f3efe343cee9c46d1": "f(s_0)=g(s_0)=x_0", "8884ff8a29ba8c7c32e08c279ecc22b7": "\\begin{align}\n &\\exp(\\pm t) \\, \\left( \\frac{y}{x} \\, \\partial_t \\pm \\left[ y \\, \\partial_x - x \\, \\partial_y \\right] \\right)\\\\\n &\\exp(\\pm t) \\, \\left( \\frac{z}{x} \\, \\partial_t \\pm \\left[ z \\, \\partial_x - x \\, \\partial_z \\right] \\right)\\\\\n &\\exp(\\pm t) \\, \\left( \\frac{1}{x} \\, \\partial_t \\pm \\partial_x \\right)\n\\end{align}", "88856985a472c48f139f3e2ad06c973f": "(a,b)\\!\\in \\text{E}", "88859e38eb8a738777eff20625d90d03": "||\\alpha|| = \\min(\\{\\alpha\\},1- \\{\\alpha\\}),", "8885b81a82f208c2f7a1d27e5e01e082": "p \\in \\operatorname{cl}(A) = \\overline A", "8885e73f5aab92c65616cb13c0b89be1": "U(\\mathfrak{g})", "888646df6e46bf9080812cbe2e074e10": "v_e", "888652521aef86b68342c02e38a0ebe5": "\\delta_3", "888660ca99e0d0fee54d4282545ac7b1": "m(t)", "8886d1c31486dac4f46256d6f18317f4": "A =\n\\int_0^\\theta\\int_0^r dS=\\int_0^\\theta\\int_0^r \\tilde{r} d\\tilde{r} d\\tilde{\\theta} = \\int_0^\\theta \\frac{1}{2} r^2 d\\tilde{\\theta} = \\frac{r^2 \\theta}{2}\n", "88872ac9116c251c79a8fbfc3d5a4d58": "~(((x_1 \\leftrightarrow x_2) \\leftrightarrow x_3) \\leftrightarrow ...) \\leftrightarrow x_n", "8887a8512ac4dd9ee803a526cc9f6ad8": " b \\rightarrow 0 \\ ; \\ V_0 \\rightarrow \\infty \\ ; \\ V_0 b = \\mathrm{constant} \\,\\! ", "8888173b73dc1d032e94c8977179ae6d": " y' = f(t,y), \\qquad y(t_0) = y_0, \\qquad t \\geq t_0 ", "88881ef29a2650196b29792b94349c6f": "\\mathbb{S}_4\\times\\mathbb{Z}_2", "88881f8c279c24525da5d5532d81abd4": "\\mathbf e_i \\cdot \\mathbf e_j = 0 \\quad \\text{if} \\quad i \\neq j", "8888363a8eb37dcb38a86294f06cd72d": "\\delta=\\frac{2 \\varepsilon \\rho}{3 D_0}", "88890c6d227cac2b85498e4406288330": "\\begin{array}{rcl}\nF(x) & = &\\displaystyle x-x^2+x^4-x^8+\\cdots \\\\[1em]\n & = & \\displaystyle x - \\left[(x^2)-(x^2)^2+(x^2)^4-\\cdots\\right] \\\\[1em]\n & = & \\displaystyle x-F(x^2).\n\\end{array}", "88891da5501f68bcb08bfb7404485706": " \\frac{120}{90}=\\frac{4}{3} \\,.", "88891ded3b2ea18471f6b259f6b860dd": " {\\rm div\\,\\,}\\vec B \\equiv 0,", "8889942d9bb3526fbd431f4e4ed88ac4": "\n\\begin{align}\n\\tan\\left(\\frac{\\eta}{2} \\pm \\frac{\\theta}{2}\\right) & = \\frac{\\sin\\eta \\pm \\sin\\theta}{\\cos\\eta + \\cos\\theta} = -\\frac{\\cos\\eta - \\cos\\theta}{\\sin\\eta \\mp \\sin\\theta}, \\\\[10pt]\n\\tan\\left(\\pm\\frac{\\theta}{2}\\right) & = \\frac{\\pm\\sin\\theta}{1 + \\cos\\theta} = \\frac{\\pm\\tan\\theta}{\\sec\\theta + 1} = \\frac{\\pm 1}{\\csc\\theta + \\cot\\theta}, ~~~~(\\eta = 0) \\\\[10pt]\n\\tan\\left(\\pm\\frac{\\theta}{2}\\right) & = \\frac{1-\\cos\\theta}{\\pm\\sin\\theta} = \\frac{\\sec\\theta-1}{\\pm\\tan\\theta} = \\pm(\\csc\\theta-\\cot\\theta), ~~~~(\\eta=0) \\\\[10pt]\n\\tan\\left(\\frac{\\pi}{4} \\pm \\frac{\\theta}{2} \\right) & = \\frac{1 \\pm \\sin\\theta}{\\cos\\theta} = \\sec\\theta \\pm \\tan\\theta = \\frac{\\csc\\theta \\pm 1}{\\cot\\theta}, ~~~~(\\eta=\\frac{\\pi}{2}) \\\\[10pt]\n\\tan\\left(\\frac{\\pi}{4} \\pm \\frac{\\theta}{2} \\right) & = \\frac{\\cos\\theta}{1 \\mp \\sin\\theta} = \\frac{1}{\\sec\\theta \\mp \\tan\\theta} = \\frac{\\cot\\theta}{\\csc\\theta \\mp 1}, ~~~~(\\eta=\\frac{\\pi}{2}) \\\\[10pt]\n\\frac{1 - \\tan(\\theta/2)}{1 + \\tan(\\theta/2)} & = \\sqrt{\\frac{1 - \\sin\\theta}{1 + \\sin\\theta}}.\n\\end{align}\n", "8889ac3266fd23652d80d8413f90e7c3": "f(i) \\in A \\setminus W_i.", "888a0b8d358e1b73889c7a5559b734be": " E_i E_{i\\pm1} E_i=E_i, ", "888a1f2a2895d2ae858147d0dd71383b": "v_0 \\approx \\frac{k_{cat}}{K_M} [E] [S] \\qquad \\qquad \\text{if } [S] \\ll K_M", "888a7ede64252745bca712953dcfa5d3": "\n\\sum_{\\delta\\mid n}d(\\delta^2)=\nd^2(n).\n", "888a9840e27d06e666c32845f130de31": " G(n, \\tbinom{n}{2} p) ", "888af71bd33f7c834785ffb99af77467": "r ", "888b53a37e85b2f1fca5698fe5c8e785": "\\mathbf{M} = \\chi_\\text{m} \\mathbf{H}", "888b6f9be761cac04b11566afd4d469a": "\\sup_{\\theta\\in\\Theta} R(\\theta,\\delta')\\leq c \\inf_\\delta \\sup_{\\theta \\in \\Theta} R(\\theta,\\delta).", "888b744ff0fef0a706e5c43f1ecd9d16": "\n \\underline{\\underline{\\mathsf{A}_\\sigma}} = \\begin{bmatrix} \n A_{11}^2 & A_{12}^2 & A_{13}^2 & 2A_{12}A_{13} & 2A_{11}A_{13} & 2A_{11}A_{12} \\\\\n A_{21}^2 & A_{22}^2 & A_{23}^2 & 2A_{22}A_{23} & 2A_{21}A_{23} & 2A_{21}A_{22} \\\\\n A_{31}^2 & A_{32}^2 & A_{33}^2 & 2A_{32}A_{33} & 2A_{31}A_{33} & 2A_{31}A_{32} \\\\\n A_{21}A_{31} & A_{22}A_{32} & A_{23}A_{33} & A_{22}A_{33}+A_{23}A_{32} & A_{21}A_{33}+A_{23}A_{31} & A_{21}A_{32}+A_{22}A_{31} \\\\\n A_{11}A_{31} & A_{12}A_{32} & A_{13}A_{33} & A_{12}A_{33}+A_{13}A_{32} & A_{11}A_{33}+A_{13}A_{31} & A_{11}A_{32}+A_{12}A_{31} \\\\\n A_{11}A_{21} & A_{12}A_{22} & A_{13}A_{23} & A_{12}A_{23}+A_{13}A_{22} & A_{11}A_{23}+A_{13}A_{21} & A_{11}A_{22}+A_{12}A_{21} \\end{bmatrix}\n ", "888b7ee6b19ba8f1c399ca0c13e3bea9": "\\overrightarrow{cd}=\\overrightarrow{cs}+\\overrightarrow{sd}", "888bc6abfcc9fd1b8884009c9e873c2f": "\nH=VD=\\begin{pmatrix}\n1 & 1 & 1 & \\cdots & 1\\\\\nL_0^1 & L_1^1 & L_2^1 & \\cdots & L_{n-1}^1\\\\\nL_0^2 & L_1^2 & L_2^2 & \\cdots & L_{n-1}^2\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nL_0^t & L_1^t & L_2^t & \\cdots & L_{n-1}^t\n\\end{pmatrix}\n\\begin{pmatrix}\n\\frac{1}{g(L_0)} & & & & \\\\\n & \\frac{1}{g(L_1)} & & & \\\\\n & & \\frac{1}{g(L_2)} & & \\\\\n & & & \\ddots & \\\\\n & & & & \\frac{1}{g(L_{n-1})}\n\\end{pmatrix}\n", "888c1ebbd5e150b2de3cd17fe6eb5b06": "R^* = 8.314\\,32\\times 10^3 \\frac{\\mathrm{N\\,m}}{\\mathrm{kmol\\,K}}. ", "888c6fbefe3d611ac508ca1941252f41": "x \\in \\mathbb{F}_q", "888cb2a3ffbbe9d745126e3c5f3ce131": "\\scriptstyle \\hat{\\mathbf{e}} \\;=\\; [e_1\\ e_2\\ e_3]^\\mathrm{T}", "888cde090b230197d02e27114f594632": "N_0 - N_1 + N_2 - N_3 = 0\\,", "888d1c35fc98aef372be7e30f4f483b1": "\\begin{align}\np(\\mu|\\sigma^2; \\mu_0, n_0) &\\sim \\mathcal{N}(\\mu_0,\\sigma^2/n_0) \\\\\np(\\sigma^2; \\nu_0,\\sigma_0^2) &\\sim I\\chi^2(\\nu_0,\\sigma_0^2) = IG(\\nu_0/2, \\nu_0\\sigma_0^2/2)\n\\end{align}", "888d9e41886ab50b70a7236ef15768d4": " D_{\\text{max}} = \\frac{\\beta}{4} + \\sqrt{2} + \\frac{1}{\\beta} ", "888e1c96fff4db338ef058c3b23e65d6": " = ( 2 + 2009 + 502 - 20 + 5) \\mod 7 = (2 + 0 + 5 - 6 + 5) \\mod 7 = 6 ", "888e6402c2ed353897434be7d0db5656": "\\langle s,t \\mid (st)^2 = s^3 = t^4 \\rangle.", "888ea1e557846d408e728748633c9484": "X \\to w", "888ecf7b586f352a0c757c4adec3f206": "I_{n,m}= \\int \\frac{dx}{x^m(a^2-x^2)^n}\\,\\!", "888f7d3397a758214587377189a9e288": "(D_L)", "888fa252e9cde41ef196d5154cd7219c": "\\partial V", "888fe41cdc313b8277894719298fa88e": "\\mu_{i} = 0", "88901a458bb84203fe2688785bd529f3": "[x]^\\omega", "8890225a71650fa756d97f34d1036519": "\nc_0+c_1 p+c_2 p^2 \\equiv a_0+a_1 p+a_2 p^2+b_0+b_1 p+b_2 p^2 \\mod p^3\n", "88902490493b8e9814811bbd72858ef2": "\\{T_z\\}_{0 \\leq \\mathrm{Re} z \\leq 1}", "889053ba71f907a4fb87fe818c36e064": "P(A_2|B) = \\frac{1}{P(B)} \\cdot P(B|A_2) \\cdot P(A_2).", "8890541da1d1e4feca986c45120f3a88": "\\sigma^m", "88907823cd713caad4e5be7e5c915a44": "\\bar{g}(w,c)>0.", "8890c10c29ab0d42972d049110a7d5a2": " c/nv > 1 ", "88912e4b398ea6d6cec76c0f54c44c56": "\\mathrm{\\tfrac{u\\bar{u} + d\\bar{d} - 2s\\bar{s}}{\\sqrt{6}}}", "88914be16a8757bc3fc7ac75b373f999": "c_i(a)=L_{a_i}(a),", "889180679a74a9efdb6be8d77bb19f23": "Y_{6}^{-3}(\\theta,\\varphi)={1\\over 32}\\sqrt{1365\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(11\\cos^{3}\\theta-3\\cos\\theta)", "8891cc52f6f629b97ca41427c0f8e834": "^{14}\\text{NO}_3^- +", "88922d9f85de41cc667f6b704f8d7e27": "a'^{(g+1)}", "8892525f81d72057b10b7a565dcc4825": "\\displaystyle k=k+1", "88925d0abbfc5d9184354a928279e608": "\n\tE_2 = \\begin{pmatrix}-0.179186290535454826\\\\ 0.741917790628453435\\\\ -0.100228136947192199\\\\ -0.638282528193614892\\end{pmatrix}\n", "88927fe8f46f4768724fbe8f4160d64b": "x \\ ", "8892a99bc14ebeb6ddbb2fcde8f5da3b": "y = {xR_1 \\over L_1}", "8892c5741ccd5be1f55fff8aabf7197d": "\\lang n^{(0)}|n \\rang ", "88935976360801cb33297c0c2478991a": "\n\\ln \\gamma_2^{\\infty } =\n\\frac{\\nu_2}{RT}\n\\left[\n\\left( \\lambda_1 - \\lambda_2 \\right)^2 +\n\\frac{q_1^2 q_2^2 \\left( \\tau_1^T - \\tau_2^T \\right)^2}{\\psi_1} +\n\\frac{\\left( \\alpha_1^T - \\alpha_2^T \\right) \\left( \\beta_1^T - \\beta_2^T \\right)}{\\xi_1}\n\\right] + d_{12}\n", "8893e7760ec0e5f850a8bae9bf4201ae": "p_i=x_ip_i^*", "8893f13667ef5f6e1b3cf167c1ab0414": "f(S_i) = f(S_i, S^i_1, \\dots, S^i_K), \\, ", "8893f25a404eee24248280293c961765": "S_L \\,", "88940c46d5828b1c79ea3efa5e6e7f73": "H(\\pi) = \\sum_i C\\left (p_i,q_{\\pi (i)} \\right )", "889416bc9630b10072fd7cc713ea8c0a": "B_i", "8894629e9a743529be6e10599188e7c8": "K_{83}", "889478d9ec7798baeca745b0c4b8e920": "\\displaystyle{c((ad)b) + d((ac)b) + a((dc)b)= (cd)(ab)+(da)(cb)+(ac)(db).}", "889483c1b689444358950611f3fb29c8": "\\mathbf{E(r)} = \\frac{1}{4 \\pi \\varepsilon_0 } \\sum_{i=1}^{n} \\frac{q_i \\left( \\mathbf{r} - \\mathbf{r}_i \\right)} {\\left| \\mathbf{r} - \\mathbf{r}_i \\right|^3}", "8894d54c98ee5bc71848b938897b8f8b": "a, b \\in \\mathrm{Q}", "8894fe9cc11390c7f2cc3dad93a57c0e": "v = \\prod_{k/2+1}^k p_i", "8895211147cfec0103c49eb567532310": "\\frac{(\\mathrm{plyr\\ PTS}-\\mathrm{plyr\\ FGA}+\\mathrm{plyr\\ REB}+\\mathrm{ plyr\\ AST}+\\mathrm{plyr\\ STL}+\\mathrm{plyr\\ BLK}-\\mathrm{plyr\\ PF}-\\mathrm{plyr\\ TO}+(\\mathrm{team\\ wins}\\times10))\\times250}{\\mathrm{team\\ PTS}-\\mathrm{team\\ FGA}+\\mathrm{team\\ REB}+\\mathrm{team\\ AST}+\\mathrm{team\\ STL}+\\mathrm{team\\ BLK}-\\mathrm{team\\ PF}-\\mathrm{team\\ TO}}", "88953ab8c147b92bf751b249ef7b15b5": "d = ((h + 5) \\mod 7) + 1", "88954281cc0cd3a6dc2ef12b4f7954c5": " \\mathbf{A} = \\sum_i\\mathbf{a}_i\\mathbf{b}_i = \\mathbf{a}_1\\mathbf{b}_1+\\mathbf{a}_2\\mathbf{b}_2+\\mathbf{a}_3\\mathbf{b}_3+\\cdots ", "8896273c0c1b7d53b3bbb5d8e0d76c17": "(\\mathbf{x},\\mathbf{x'})", "88969856b24032db521f2fe1a1eee463": "E[\\textbf{v}_k\\textbf{v}_k^T] = \\textbf{R}_{k}", "88970871e2f9f78bb9caa1fb8b043b14": "G(s) H(s)=\\frac{K_d}{s}", "88970c2ecec33615557e22968ec4ee37": "p=n, k=1", "8897118b6a7da1f7f0113a25abe858bc": "\\mathbf A (\\mathbf r , t) = \\frac{\\mu_0}{4\\pi}\\int \\frac{\\mathbf J (\\mathbf r' , t_r)}{|\\mathbf r - \\mathbf r'|}\\, \\mathrm{d}^3\\mathbf r'\\,.", "8897328afcd09075d1f6e5b8ef92b575": "{K_w} = [H_3O^+] [OH^-]", "88973c7a1aa439907cdb3b6457e3e5f0": "T_i f(x) = \\frac{\\partial}{\\partial x_i} f(x) + \\sum_{v\\in R_+} k_v \\frac{f(x) - f(x \\sigma_v)}{\\left\\langle x, v\\right\\rangle} v_i", "8897c279b856a6ccc2e76cf4b00ab702": "\\ddot{x} + x + \\varepsilon\\, x^3 = 0\\,", "889818d181b8ba20ca2c878fedf5329b": "\\triangle\\delta = -\\frac{v}{c} \\cdot \\sin \\delta \\cdot \\frac{180^\\circ}{\\pi}", "88981ffee3d44d9be6c902d44deee8a9": "\\Delta x \\Delta p = \\frac{\\hbar}{2} \\sqrt{\\frac{n^2\\pi^2}{3}-2}", "889836ac326faa8824db43678472e82f": "P: U \\to S", "88989f7911ac394e8ba43bb2a39cc680": "[f(\\theta_{i-1}), f(\\theta_i)]", "8898f225e0b6bc9144fea8cf744c9bca": "VAG(x^3 -7x + 7,(\\frac{3}{2},2)) ", "88994061f7bbc7494200bdfb1238690c": "B_1 = \\{x : |x|<1\\}", "88997d172fb58d11740a0a2bbb3923a1": "\nG(\\mathbf{r}) = \\frac{e^{- \\lambda r}}{4\\pi r}.\n", "889a497808a1101369a71d5c7c5bd743": " \\sum_{i}{ k}_{i } ", "889a536fd05058611aa9122080e9b07b": "\\scriptstyle\\lim\\limits_{m\\rightarrow\\infty}\\,\\sum\\limits_{n=0}^m\\,a_n", "889a975d5e5807818efe590628740d4c": " \\cos \\left(\\frac{c}{R}\\right)=\\cos \\left(\\frac{a}{R}\\right)\\cos \\left(\\frac{b}{R}\\right) +\\sin\\left(\\frac{a}{R}\\right) \\sin\\left(\\frac{b}{R}\\right) \\cos \\gamma \\ .", "889b3b1d0f3a624069dfe09661377f01": "\\operatorname{Cov}(z', z'A')=\\operatorname{t}(AV).", "889b55fa437542ab417ea7bcd8eaa792": "g(k) = \\int_k^\\infty \\!xf(x)\\, dx\n = e^{\\mu+\\tfrac{1}{2}\\sigma^2}\\, \\Phi\\!\\left(\\frac{\\mu+\\sigma^2-\\ln k}{\\sigma}\\right).", "889b6c0279f9f5e7df0bcd87a7e110f9": "T = (\\delta_{ij}t_i)_{m\\times m}", "889b81671d1cb843aeb41c8cbcf3ec66": "x^2-m^2y^2=bx+cy", "889bd311968993d4460f76e7b838c489": " F_{pp}=\\frac {\\Delta \\epsilon_{pp}} {\\Delta \\epsilon_{inelastic}}, F_{cc}=\\frac {\\Delta \\epsilon_{cc}} {\\Delta \\epsilon_{inelastic}}, F_{pc}=\\frac {\\Delta \\epsilon_{pc}} {\\Delta \\epsilon_{inelastic}}, F_{cp}=\\frac {\\Delta \\epsilon_{cp}} {\\Delta \\epsilon_{inelastic}} ", "889bf24f1f2ca96cd1f4b7f5a74a43ca": "\n\\hat{H}(s) = \\lim_{N\\to\\infty}\\int^N_{-N} \\mathrm{e}^{-2\\pi i x s} H(x)\\,\\mathrm{d}x = \\frac{1}{2} \\left( \\delta(s) - \\frac{i}{\\pi}\\mathrm{p.v.}\\frac{1}{s} \\right).\n", "889c055b64430bec1f3c6bed34c7ad73": "\\mathfrak{h},", "889c0618e6377230c97dedeb830ff286": "\n\\begin{matrix}\nz_0 = -1 & f[z_0] = 2 & & & & & & & & \\\\\n & & \\frac{f'(z_0)}{1} = -8 & & & & & & & \\\\\nz_1 = -1 & f[z_1] = 2 & & \\frac{f''(z_1)}{2} = 28 & & & & & & \\\\\n & & \\frac{f'(z_1)}{1} = -8 & & f[z_3,z_2,z_1,z_0] = -21 & & & & & \\\\\nz_2 = -1 & f[z_2] = 2 & & f[z_3,z_2,z_1] = 7 & & 15 & & & & \\\\\n & & f[z_3,z_2] = -1 & & f[z_4,z_3,z_2,z_1] = -6 & & -10 & & & \\\\\nz_3 = 0 & f[z_3] = 1 & & f[z_4,z_3,z_2] = 1 & & 5 & & 4 & & \\\\\n & & \\frac{f'(z_3)}{1} = 0 & & f[z_5,z_4,z_3,z_2] = -1 & & -2 & & -1 & \\\\\nz_4 = 0 & f[z_4] = 1 & & \\frac{f''(z_4)}{2} = 0 & & 1 & & 2 & & 1 \\\\\n & & \\frac{f'(z_4)}{1} = 0 & & f[z_6,z_5,z_4,z_3] = 1 & & 2 & & 1 & \\\\\nz_5 = 0 & f[z_5] = 1 & & f[z_6,z_5,z_4] = 1 & & 5 & & 4 & & \\\\\n & & f[z_6,z_5] = 1 & & f[z_7,z_6,z_5,z_4] = 6 & & 10 & & & \\\\\nz_6 = 1 & f[z_6] = 2 & & f[z_7,z_6,z_5] = 7 & & 15 & & & & \\\\\n & & \\frac{f'(z_7)}{1} = 8 & & f[z_8,z_7,z_6,z_5] = 21 & & & & & \\\\\nz_7 = 1 & f[z_7] = 2 & & \\frac{f''(z_7)}{2} = 28 & & & & & & \\\\\n & & \\frac{f'(z_8)}{1} = 8 & & & & & & & \\\\\nz_8 = 1 & f[z_8] = 2 & & & & & & & & \\\\\n\\end{matrix}\n", "889c0b1ee8296d0b92141ac2170df6f8": "\\dfrac{5}{\\sqrt{3} + 4}\\,\\!", "889c19c882a9192a551c76d3aa353133": "\\Delta t * \\Delta \\omega \\geqq \\frac{1}{2}", "889c37b76abfca75c97c1b9eae13e7e6": "\\mathcal{Z}_0", "889c58edac3e17cf134a4388d6831c05": "\n\\mbox{SAIDI} = \\frac{\\sum{U_i N_i}}{\\sum{N_i}}\n", "889c61ff76d4104ceab852ecf6fb6a5b": "\\Delta S=\\int_i^f\\mathrm{d}S=\\int_{V_{0}}^{2V_{0}} \\frac{P\\,\\mathrm{d}V}{T}=\\int_{V_0}^{2V_0} \\frac{n R\\,\\mathrm{d}V}{V}=n R\\ln 2.", "889c92a4f06dde296bd4e3d82f14b898": " x_1+y_1+D ", "889cbb1528d0eb3133fca781f7786949": "\\prod_{\\sigma\\in S_N}(b(1) e^{\\gamma(\\sigma(1))}+\\dots+b(N) e^{\\gamma(\\sigma(N))})", "889cc9f41e5a60ce9e426dc97d39a4df": "m=\\frac{8}{3} \\cdot \\frac{h \\, \\varepsilon_0}{c^2}", "889cccfc69b0b048ce70d8b7a35f23c8": "\\tau_\\mathrm{min} = -\\frac{1}{2}(\\sigma_1 - \\sigma_2 )\\,\\!", "889cfa6548f232787c647b3765a54978": "\\pi_j(.)\\ ", "889d1d37d295e8460f3d68c23bf4da77": "g(A)=\\bigg|\\bigcap_{i \\in \\underline{m} \\backslash A} A_i\\bigg|,~~ g(\\underline{m}) = \\bigg|\\bigcup_{i \\in \\underline{m}} A_i \\bigg| \\qquad\\text{and}\\qquad g(A)=\\mathbb{P}\\bigg(\\bigcap_{i \\in \\underline{m} \\backslash A} A_i\\bigg),~~ g(\\underline{m}) = \\mathbb{P}\\bigg(\\bigcup_{i \\in \\underline{m}} A_i\\bigg)", "889d558178c37b8eff642f89b9fdc781": "\n y_{ir} = x_{ir}^\\mathsf{T}\\;\\!\\beta_i + \\varepsilon_{ir}, \\quad i=1,\\ldots,m.\n ", "889d7d8f31a019b3bdc0d1c55ee4f7f2": "\\rarr ", "889da662087dcd9a0a442ec99cd01166": "\\delta \\left(Q_i,[x_1,x_2...x_n]\\right)=(Q_j,[y_1,y_2...y_n],d)", "889e0b0e1954831e0cbf251ac21ad027": " \\langle \\psi |\\hat{S} |\\psi\\rangle = \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 ", "889e6ed2247c343e4a225d1f46b2dc27": "(\\omega_c - \\omega_a)", "889edd37d8f11aa8d036b8f3641847e5": "9x+1", "889ef0dc88083456f9cd81307324d0cb": " T_{ij}=(\\mathcal L_U g)_{ij} = U_{i;j}+U_{j;i}\n", "889f25fd0c9460a1615e10ce44a0a11e": " \\frac {m} {k^{1+\\operatorname{Re}\\,\\alpha}}\\le \\left|{\\alpha \\choose k}\\right| \\le \\frac {M} {k^{1+\\operatorname{Re}\\,\\alpha}}, \\qquad\\qquad(5) ", "889f373402fdfc9ab624dc9abd25a15a": " \\sigma_t ", "889f3d26b42e00c76dda508f8f3c05ed": "\\phi_{bh}", "889f5935c0f35be63da1c104341c3ba3": "\\ R_2 = r", "889f9f036674a0ce39d1619e66d9cb63": " \nF \\sim \\mu_0^2 \\sigma v m^2 L^{-3}\n", "88a0d43cd3d4a0d8c93dae53fff92714": "f_s:=\\delta(s)+\\inf_{y\\in X}\\{f(y)+sd(x,y)\\}, \\quad \\mathrm{for} \\ s\\in\\mathrm{dom}(\\delta):", "88a14dba83b56d0a6a7e393a5755d43c": "\\delta:Q\\times\\Sigma \\to P(Q)", "88a17ea48d0bb0edeff71e9ffea3517b": "O(n^{3.5} L^2 \\cdot \\log L \\cdot \\log \\log L)", "88a19f0ad007596b24b36aab0da6796a": "\\mathcal{L}_X \\psi := X^{a}\\nabla_{a}\\psi\n-\\frac14\\nabla_{a}X_{b}\n\\gamma^{a}\\,\\gamma^{b}\\psi\\, ,", "88a216c79902b312a68d26fa6c1316ae": "v \\in V^*", "88a21e6a3e2ebbd7deb5212b0baa4058": "n - m", "88a23d4ea1cb24ceb10c30cfa939b4db": "\n\\lambda_r = r^{-1} \\sum_{k=0}^{r-1} {(-1)^k \\binom{r-1}{k} \\mathrm{E}X_{r-k:r}},\n", "88a28e44c19988a5e3651b2b3a9c4df2": "A = S\\,\\mbox{diag}(e^{i\\theta_1},\\dots,e^{i\\theta_n})\\,S^{-1}.", "88a2e776574c00c628e669e238e8baa5": "\\tau_b=\\rho g h S", "88a352b3f78b26b4e7defc2f3d31253a": "\\scriptstyle a = \\lambda H/T_H", "88a37ffebabfe900ea029b6f1d8ad4d4": "\\lang S \\rang = [\\lang S_x \\rang, \\lang S_y \\rang, \\lang S_z \\rang]", "88a408ac5d04d19f9ed7b5fbd5d7ef8a": "{{g}_{m}}=\\frac{{{V}_{T}}}{{{I}_{C}}}", "88a4157b1f3ad2625f5dfc9e9e8cc42f": " r_O \\gg R_C\\|R_L \\ \\ \\left( \\beta \\gg 1 \\right)", "88a479f39389f35054169833a4f69f28": " v\\notin C", "88a49ba50a641d9f31475e248fd9516f": " S_{G}^m", "88a4eef9206a71579fedd4b46274f4ad": "\\Psi(\\mathbf{R},\\mathbf{r})", "88a5085827228ced131503646f014d61": "\\mu_{nb}(t)", "88a55bcaaf78ed6c174f054e1273907a": "\\tfrac{m}{n}=\\sqrt{3}", "88a5738c1c4e6856758644f3920aa452": "\\nabla \\times ( \\varphi \\mathbf{F}) = \\nabla \\varphi \\times \\mathbf{F} + \\varphi \\nabla \\times \\mathbf{F} ", "88a5940a241e754686edd50b6e40a497": "\\mathcal O(X)", "88a5ab96042d5ac17ddecbebf56fd1b6": "n^2 = (n-a)(n+a) + a^2", "88a5afa415a867613ae53557900ada2f": "{N}", "88a5dfc6eabd6282ea264d7ff3077011": "P\\in \\mathcal Q", "88a6345ef62d4e211f666eed279dcd60": "\\frac{\\hat{dx_i}'}{dt}=\\hat{v_i}'\\equiv i[\\hat{H}'_0,x_i] = \\beta \\frac{p_i}{p^0} = \\beta v_i ", "88a6471deb1dafb0eb2d6db2f7c91129": "\\nabla f(\\rho, \\phi, z) = \n\\frac{\\partial f}{\\partial \\rho}\\mathbf{e}_\\rho+\n\\frac{1}{\\rho}\\frac{\\partial f}{\\partial \\phi}\\mathbf{e}_\\phi+\n\\frac{\\partial f}{\\partial z}\\mathbf{e}_z\n", "88a71f98a44c56da1d838a259ad90ada": "V_g f (x) = \\langle f, \\pi(x)g \\rangle", "88a73f4296e91fbff1497eac4afc408f": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx = \n \\frac{x^{m+1} \\left(a+b\\,x^n\\right)^p}{m+n\\,p+1}\\,+\\,\n \\frac{a\\,n\\,p}{m+n\\,p+1}\\int x^m \\left(a+b\\,x^n\\right)^{p-1}dx\n", "88a75ba350e7897c55ff37511f1a98f9": "1 (1 + \\delta)\\mu) & \\le \\inf_{t > 0} \\frac{\\mathbf{E}\\left[\\prod_{i=1}^n\\exp(tX_i)\\right]}{\\exp(t(1+\\delta)\\mu)} ) \\\\\n& = \\inf_{t > 0} \\frac{\\prod_{i=1}^n\\mathbf{E}[\\exp(tX_i)]}{\\exp(t(1+\\delta)\\mu)} \\\\\n& = \\inf_{t > 0} \\frac{\\prod_{i=1}^n\\left[p_i\\exp(t) + (1-p_i)\\right]}{\\exp(t(1+\\delta)\\mu)}\n\\end{align}", "88b8c657eca1ed752531b977859697dd": "\\displaystyle{\\mathfrak{g}_{\\mathbf{C}} = \\mathfrak{n}_- \\oplus \\mathfrak{t}_{\\mathbf{C}} \\oplus \\mathfrak{n}_+,}", "88b8feee9ac6d70a42f71c85422b41ac": "\\scriptstyle \\tan \\delta'=y_1'/x_1'=3/7,50555\\;\\rightarrow\\;\\delta' = 21,79^\\circ", "88b9331da3b9c9532f0e4331b6aaef38": "*\\frac{1}{3*\\sqrt{x(t)^2+y(t)^2}}", "88b9aceaab9ca9ab39bb81a30bc99868": "W_z", "88b9b8abbdafcd5f7eda4d7848f64289": "\\operatorname{GL}(2,\\mathbb{Z}/2\\mathbb{Z})", "88b9cf8ce728bb6fece679320637844f": "u(t)\\leq \\alpha+ \\int_0^t f(s)\\,w(u(s))\\,ds,\\qquad t\\in[0,\\infty),", "88ba110765e76371ba52f1560d906deb": "\\sum_{i=1}^{k}d_i \\leq k(k-1) + \\sum_{i=k+1}^n \\min(d_i,k) \\quad \\text{for } k \\in \\{1,\\dots,n\\} \\, .", "88ba3b1852a48e2b186c59665120d6e3": "x<\\min \\varphi", "88ba6d919c726529c78eae5ac42cb351": "v_i (a) - p_i \\geq 0", "88baee54e129d26b313a5bae78842506": "\\mathbf{P} = AB^T - BA^T", "88bb09d671a519ed93657113c6596232": " \\alpha_i ", "88bb1a770ce0cbbd6277997615a070b8": "x' = Ax,", "88bb68557ee5ab2ef9a9f211942d9637": " n = 2^{m_1} + 2^{m_2} + \\cdots + 2^{m_k} ", "88bb7f870845f7aae0e8fe85d2c3a457": "\\ SF^{n+1}", "88bbbf762d5498b2c74be8ae496e385b": " \\gamma=\\sqrt{M^2\\left(M^2+\\Gamma^2\\right)}", "88bbca77eaa7b9e40501a43ebd561bf4": "\\ln f_2 - \\ln f_1\\over\\ln 10", "88bbecc139ce0e4539726fb72544e50d": "\\cos(A - B) = \\cos(A)\\cos(B) + \\sin(A) \\sin(B)\\,", "88bbf08e06ad32663c957f7f0823dfff": "\\kappa=10", "88bc10a0e7a633caa488534b642b2fb9": "a,d\\in K\\backslash\\{0\\}", "88bc3b8045b62c9631daeff6677d17a2": "\\theta_{(\\ell)}", "88bca17bccd61a446234ae5aa7efba33": "W= \\frac{q_1q_2}{4\\pi\\varepsilon_0}\\frac{1}{r}", "88bcd1dcc0d13c32c151dbd707a25361": "\\tilde{h}", "88bcd4a56f69a0832d058239317fd23f": "\\Delta(c)=\\sum_{(c)} c_{(1)}\\otimes c_{(2)}.", "88bd14395f6eb79b775b773648c843a1": "\n\\begin{align}\n\\delta &= \\int \\frac\n{\\sqrt{b^2\\sin^2\\beta + c^2\\cos^2\\beta}\\,d\\beta}\n{\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}\n \\sqrt{(b^2-c^2)\\cos^2\\beta - \\gamma}}\\\\\n&\\quad -\n\\int \\frac\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega}\\,d\\omega}\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}\n \\sqrt{(a^2-b^2)\\sin^2\\omega + \\gamma}}.\n\\end{align}\n", "88bd338e5b904fd57b5636302b1c73bb": "a_S + b_S P = a_D + b_D P", "88bd3b7246577ab33f8903b66711c096": "a_{j} \\ne 0 ", "88bd44f37a478dd6ed322cdb9386ffa8": "\\mathbf{k_i}", "88bd9412ce55e79396e16be77f436fbf": " {k = 0.01720209895 \\ A^{\\frac{3}{2}} \\ D^{-1} \\ S^{-\\frac{1}{2}} } \\ ", "88bd9fdfd10e5cb00fbf98328e3ed69d": "\\Sigma_{2}^P \\cap \\Pi_{2}^P", "88bda60d6c34b5da515258860c941d85": "\\hat{H}(x)", "88bdbd98c1c0ecb7824fb3f2690061f7": "\\langle a,b \\,\\vert\\; aba=baa, bba=bab\\rangle", "88bde220dbcba7610bd6b20106437754": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1=[A_{12}+2(A_{21}-A_{12})x_1]x^2_2\n\\\\ \\ln\\ \\gamma_2=[A_{21}+2(A_{12}-A_{21})x_2]x^2_1\n\\end{matrix}\\right.", "88be1787662ae90af4b7b744f911628a": "/\\beta", "88be75a12898c5fd78ddb472026d312a": "M_{2413} = \\begin{bmatrix} &1&& \\\\ &&&1 \\\\ 1&&& \\\\ &&1& \\end{bmatrix}", "88be9a1a716b8d3a06d39b0c5b0cdab0": "F_X = F_Y", "88bec7b36385107d34ae6d974a1fc3b3": "|\\beta A| = 1\\,", "88becc81d9aedf439edd3e3ff08c9361": "\\left [ \\mathbf z \\right ] = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{12} & -z_{11} \\end{bmatrix} ", "88befe513e1085d2e2d62e97eff74e54": "p = \\frac{\\partial L}{\\partial \\dot{q}}", "88bf504f2520913fe802b5a4950de1cb": " g_{ab} = \\eta_{\\mu \\nu} \\frac{\\partial X^\\mu}{\\partial y^a} \\frac{\\partial X^\\nu}{\\partial y^b} \\ ", "88bf8b476e41d3f05504a11ec454e5e0": "\\mathrm{adj}(\\mathbf{AB}) = \\mathrm{adj}(\\mathbf{B})\\,\\mathrm{adj}(\\mathbf{A}),", "88bfaf41ad647c59256f4013f97e4a83": "(3) \\,", "88bfdbdda304ed86056d573752f03e3d": " H_k^2 = U \\tan^2 \\theta_k + V \\tan \\theta_k + W ", "88c03cf5bc17122cad028110eacd8ab7": "T \\subset S \\rightarrow S/D", "88c0ad0e724d4a5855a98bad95db70ca": " A \\rightarrow A + \\alpha. \\,", "88c136e69fb4ac7017ef375e8829c02c": "(n-1)\\times (n-1)", "88c14463c49d918450dc5584c08683fa": "f = \\frac{1}{T}\n\n= \\frac{1}{\\ln(2) \\cdot (R_2 C_1 + R_3 C_2)}\n\n\\approx \\frac{1}{0.693 \\cdot (R_2 C_1 + R_3 C_2)}", "88c16ed13db38d6cc8a019a3a3a458ac": "|U_1||U_2|", "88c23c986b2e1114d7fd951a8a677ca4": " i^i_{cap} + i^i_{ion} = i^i_{long} + i^i_{electrode}", "88c23e796e235e0c01149e8739b747d3": "\\begin{align} \nM_X(\\alpha; \\beta; t) \n&= \\operatorname{E}\\left[e^{tX}\\right] \\\\\n&= \\int_0^1 e^{tx} f(x;\\alpha,\\beta)\\,dx \\\\\n&= {}_1F_1(\\alpha; \\alpha+\\beta; t) \\\\\n&= \\sum_{n=0}^\\infty \\frac {\\alpha^{(n)}} {(\\alpha+\\beta)^{(n)}}\\frac {t^n}{n!}\\\\\n&= 1 +\\sum_{k=1}^{\\infty} \\left( \\prod_{r=0}^{k-1} \\frac{\\alpha+r}{\\alpha+\\beta+r} \\right) \\frac{t^k}{k!}\n\\end{align}", "88c2f032564b707cf720e2337bf519e4": "\\displaystyle{\\widehat{Eg}(m,n)=\\lambda_{n}^{-1} \\widehat{g}(n)\\cdot {(1+n^2)^{k-1/2}\\over (1+n^2+m^2)^k},}", "88c30572bd52395cfd1fd446028e36b2": "k \\ll \\pi / a ", "88c310082155a4dadd293b5b62d42bf7": " \\wedge (\\forall x_1...x_k, y_1...x_k) [(Eq(x_1, y_1) \\wedge ... \\wedge Eq(x_k, y_k)) \\rightarrow (A(x_1...x_k) \\equiv A(y_1...y_k))] ", "88c34386d8dc2af3f26da0bbd38613c7": "(\\mathfrak{g}, s)", "88c3811289323ec6f23db1f785717f24": "\\mathrm{[OH^-]} = \\frac{K_{\\mathrm w}}{\\mathrm{[H^+]}}", "88c38e0b27335660eaed3bfad999a972": "D_0(f)=\\int_{-\\infty}^\\infty x^2|f(x)|^2\\,dx.", "88c457558eb7658e80ba7bb59a98c8d8": "AE = AF = CD = 2", "88c45d40efeac66cb761001ec3ea6440": "t_{LL}^{\\mu \\nu}\\,", "88c472c8a3d825cfbb5bf2d3496d7580": " \\bar{\\alpha}", "88c4ad260e959c343c2caf43b40a77d1": "\\mathsf{G}", "88c4c77bccfde2de110ad963c2b1ff97": "L = - \\int d^4 x \\sqrt{- det (g)} (- g^{\\mu \\nu} \\partial_\\mu \\varphi \\partial_\\nu \\varphi - V (\\varphi))", "88c51cd24f68294a2c1ad02daea1cbcf": "\\widehat{\\delta}", "88c5474f62d008f88cd9c40b6bb75f78": "\n2x_{1} = D + \\frac{r_{1}^{2} - r_{2}^{2}}{D}\n", "88c5926833291387a41cc5a5c39ec1a0": "\\sqrt{(x'-x)^2 + (y'-y)^2 + \\cdots}", "88c5c990bd5685f9a65f63d7da858a44": "\\theta=90^\\circ", "88c5e357df6b6f78253d12b77b10733e": "GDP = C + I + G + \\left ( X - M \\right )", "88c60468122fcc43d60cbec8934c255e": " \\sum_{i,j} de_i de_j' = 0 ", "88c6167eeec85e790e291d073522c5ab": "A + e^- \\overset{M}{\\to} A^-", "88c673d58ea8219907ddfd09411cdea9": " H(.,j)= (I-M_{-j})^{-1}e", "88c6c298b1f04464ad14ded6949423cf": "\n \\tau_m = \\tfrac{1}{8}\\left[-A \\pm \\sqrt{A^2 + 4(A\\sigma_m + B^2)}\\right]\n ", "88c7162c0fc9e1cec686e653ba6c0f05": "a = \\frac{v^2}{r} \\, ,", "88c72a22a80c32f21ae391a751a8622f": "k(t) = \\frac{6\\cos(t)(8\\cos(t)^4-10\\cos(t)^2+5)}{(232\\cos(t)^4-97\\cos(t)^2+13-144\\cos(t)^6)^{3/2}}\\,, ", "88c7414d4b4d182de0a1e07ca5d6eba6": " \\omega = \\mu B/\\hbar", "88c756b76f72855e55149c0383635bf0": " \\sin(k t) = \\frac{\\exp(i k t) - \\exp(- i k t)}{2 i} = \\frac{z^k - z^{-k}}{2i}", "88c76da6f1765be3b493b1a553401488": "Ih", "88c775f9b48c17a894b6e0c33e8ee8f6": "F(x)=\\sin x", "88c78a9e94f71991a20dd9bcd72d526d": " K_a = \\frac{\\cos \\beta - \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}{\\cos \\beta + \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}", "88c7a46b0cdb7391e024d30c8a9b92bf": " y = \\frac{C}{e^{\\int_{s_0}^{x} P(s) ds}}", "88c7c21866d78713c954825fe01e6be2": "\\operatorname{succ}: 2^P \\times O \\rightarrow 2^P,", "88c7cabaea133dbf461e8052fe0c58de": "\\begin{align}\\mathrm{Obj}({\\mathcal{C}_T}) &= \\mathrm{Obj}({\\mathcal{C}}), \\\\\n\\mathrm{Hom}_{\\mathcal{C}_T}(X,Y) &= \\mathrm{Hom}_{\\mathcal{C}}(X,TY).\\end{align}", "88c831ac80f3dfc62dbd7b792215d1d6": " 2\\cdot6-4\\cdot5+5\\cdot4-3\\cdot3+2+3 = A\\cdot(0+0) + B\\cdot( 2+ 0) + 8 + D\\cdot0 ", "88c837575410ea7ca706b8d012ceeba1": "\\displaystyle B \\cdot \\cos(\\theta)", "88c8829f4becfdc7951c4800ac2450a1": "n{\\rm C} + (n+1){\\rm H}_2 \\rarr {\\rm C}_n{\\rm H}_{2n+2}", "88c8a2afffe9393cd812b18638e2d50a": " \\gamma = \\lim_{n \\to \\infty} \\frac{1}{n}\\, \\sum_{k=1}^n \\left ( \\left \\lceil \\frac{n}{k} \\right \\rceil - \\frac{n}{k} \\right ),\n", "88c91f1ead345e0ecaf4b993a0b48fcd": " 10^{16}", "88c97c56ea23504e14bca8c69ec0f742": "f \\colon V \\to W", "88c9eab34058481bad0b1617a375350e": "1 / \\sqrt{det (q)}", "88ca0465230c5fc292d5511c0aa9585a": "\\frac{\\Delta G^\\ominus}{T} = -R \\ln K ", "88ca0482e722ee35c08f02332182401f": "r_n = a", "88ca27adea9eaaa8a839df4ffc369d00": " f_Z (z) \\sim \\frac{1}{2} \\frac{4^{4/3} |z|}{\\operatorname{Ai}' (\\tilde{a}_1)} \\exp \\left( - \\frac{2}{3} |z|^3 + 2^{1/3} \\tilde{a}_1 |z| \\right)\n\\text{ as }z \\rightarrow \\infty\n", "88ca2849b6983bdef7879ade5527dd3d": "k = 0", "88ca2936e8b9a0f830af8f37a6d254e7": "{{ q-2 } \\over { (2q-3)^2 (3q-4) \\lambda^2}} \\text{ for }q < {4 \\over 3}", "88ca42da623c24b2b708ccaac507b9b3": "\\{X_1,\\ldots,X_n\\}", "88ca4498f4bd28a9f1821ffd46883c71": "h(x_2,\\ldots,x_n) = f(0,x_2,\\ldots,x_n) \\oplus f(1,x_2,\\ldots,x_n)", "88ca67743d1198878f96cd0a714badd1": "T = \\left ( \\frac{(1 - A) L_0}{\\epsilon \\sigma 16 \\pi a^2} \\right )^{1/4}\\,\\!", "88caa612ca87ad19c1de541aadacfb1f": "Lu = -4\\frac{n-1}{n-2} \\Delta u + Ru,", "88cb7dfe15c89d5842e38fe28fb26408": "\\scriptstyle \\frac{2}{\\sqrt{k}}", "88cb9c89a63fe4944734ed1c8f2b30fa": "\n\\mathbf{v} = v_{r} \\mathbf{\\hat{r}} + v_{\\varphi} \\hat{\\boldsymbol\\varphi} = \\dot{r} \\mathbf{\\hat{r}} + r\\dot{\\varphi} \\hat{\\boldsymbol\\varphi}\n", "88cc14332e22f8c5148eecff25ba4db3": "D^{\\frac{3}{2}}f(x)=D^{\\frac{1}{2}}D^{1}f(x)=D^{\\frac{1}{2}}\\frac{d}{dx}f(x)", "88cc77d326be3e5e11e2895792bbbc82": " \\,a", "88cc9f55cc96b1e3337baf46de26d195": "x + ax = b", "88cccbed2f0901e3abe9bb9e2fc9b1fe": "\\mathbf{p}\\cdot\\mathbf{J}\\left|\\mathbf{p},\\lambda\\right\\rangle=\\lambda |\\mathbf{p}|\\left|\\mathbf{p},\\lambda\\right\\rangle", "88ccfd211d889023eaa8902890a0f068": "{\\tilde{B}}_{3+}", "88cd56cf8a8cf1a90da9109569ca0c9b": "(k(n-k))", "88cd718762e76f91f59662db5229adc7": "\\left| \\int_a^b f(x)\\,dx - (b - a) f(a) \\right|\n = \\left| \\int_a^b (x - a) f'(v_x)\\, dx \\right|", "88cda6d98af681cc80c03beb4e259395": "= M M^* N N^* = M^* M N^*N. \\,", "88cdb53ff9ad5ba3315dbb03c5b8fa1f": "Z_3 = 4X_1(XX_1+aX_1+1) \\, ", "88cdbb23ad7ce75ca40d9f4193c0a0bd": "v_i^*P_{i'}\\left(AM^{-1}\\right)r_k=0", "88ce03d0c0d91b6e56cf5f4a98cb755b": "T_P(Y)", "88ce06e89a1af00871e8c0c65456724c": " \\Phi = \\frac{\\pi}{2 \\eta} \\frac{|\\Delta P|}{\\Delta x} \\int_{0}^{R} (rR^2 - r^3)\\, dr = \\frac{|\\Delta P| \\pi R^4}{8 \\eta \\Delta x} ", "88ce471b25c0a1a88b4eb834c2ca8e43": "R_{xy} (\\omega) = X(e^{i \\omega})^* \\cdot Y(e^{i \\omega}) \\!", "88ce7f60b3af63059527ebece6661dea": "N = IN", "88ce88951f968efe07f8f7723010aa35": "U_{i+1} \\subseteq U_i", "88ce8abad96f288fdb7d25c1b44e9b17": "140^2", "88ceb47062ef3a67abe89463e200f98f": "\\forall i \\ \\ t_i-\\delta(t_i)< u_{i-1} \\leq t_i \\leq u_i < t_i + \\delta (t_i). ", "88cf160ac8f2705a3645976cd676a50c": "\\tfrac{W}{L}", "88cf422df3402ec0f6c6f9f24dede0c8": "\n \\dot{\\boldsymbol{\\varepsilon}} = \\mathsf{E}^{-1}~\\dot{\\boldsymbol{\\sigma}} + f(\\boldsymbol{\\sigma}, \\sigma_y)~\\boldsymbol{\\sigma} \\quad \\mathrm{for}~||\\boldsymbol{\\sigma}|| \\ge \\sigma_y\n ", "88cf95a695fa16946a39f064773c3cd3": "c = \\frac{\\lambda}{N_0}.", "88d07124cffe7c7f22f7d20d323adf37": "b_{t+1}", "88d0e3747718d7bcf3df9498ff3785dc": "\\beta = \\frac{\\mu_D}{\\sigma_D}.", "88d1149a617cf27d818ed96ac0bf70c7": "\\operatorname{U}(n,K/k,\\Phi)(R) := \\left\\{ A\\in \\operatorname{GL}(n,K\\otimes_k R) : A^*\\Phi A=\\Phi\\right\\}.", "88d16ec3779a6895702da5f92e2ac572": "\n\\begin{align}\nA z \\bar z + B z + C \\bar z + D & = 0 \\\\[6pt]\nA \\frac{1}{w} \\frac{1}{\\bar w} + B \\frac{1}{w} + C \\frac{1}{\\bar w} + D & = 0 \\\\[6pt]\nA + B \\bar w + C w + D w \\bar w & = 0 \\\\[6pt]\nD \\bar w w + C w + B \\bar w + A & = 0.\n\\end{align}\n", "88d1aaed0f4f5f16f6e6596a2a255805": "2\\le \\nu(W)\\le \\# N", "88d1c2d945a448f132494fd3f1f8762b": "N \\times 1", "88d1f88aceef56cad0c852f75c024528": " (\\mathrm{OPT} - c) \\leq f(x) \\leq (\\mathrm{OPT} + c).", "88d1fdbf7648d9f55af039440847756c": "A(D)", "88d200a3509342f0a29ccf5d1f9b7c2f": "\\mathbf{I} = \\mathbf{a}\\hat{\\mathbf{a}} + \\mathbf{b}\\hat{\\mathbf{b}} + \\mathbf{c}\\hat{\\mathbf{c}}", "88d21cda345b71d7d88168cb5335a292": "G_{k,n}(z) = f_n\\circ f_{n-1} \\circ \\cdots \\circ f_{k+1} \\circ f_k(z).", "88d2288380207f901e6454ac1b73e651": "d_I", "88d2440de21de5c75e80fac34a491c35": "\\displaystyle{(F,G)_\\sigma^\\prime=\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty F(x)\\overline{G(y)} |x-y|^{2\\sigma-1}\\,dx\\, dy.}", "88d245bc9a5640a494d5d366f117c7bc": "{T}_{4}/{T}_{1}={T}_{3}/{T}_{2}", "88d250a2fb45faa8468ab10c5309fbb2": "L = \\{ \\mathbf{u}+t\\mathbf{v} \\mid t\\in[0,1]\\}", "88d251c1e3c37b857b31686d29078f32": "\\! F^*(x)", "88d25326300b23a8079153fb12f166d1": " X = \\alpha_p", "88d2742941ac08f381c929ace9df9452": "(x-i0)^\\alpha = \\lim_{\\epsilon\\downarrow 0} (x-i\\epsilon)^\\alpha", "88d283b76393ef17b1a7eca57f6b14f4": " M(x)=\\sup_y |f(x+iy)|, \\,", "88d2901cd951c2d09f008e882fa38c3e": "\\overline E", "88d2d4742f340c0b8f3b2284e3bd9028": "\\Sigma_*", "88d31d40fa1cb7ea145ce8c4aa089faa": " y \\in Y^\\phi", "88d34524fb622645046b2ea5e438f5df": "\\textstyle P_2(f(\\Omega_1))=1 ", "88d35dc936bcb64bf983a0105fdb7fb3": "\\Lambda_E = 4\\pi G\\rho/c^2", "88d3b7f404ec5a820f5cda992f16c71e": " p = h/\\lambda.\\,", "88d3d676ef89d6dd2b58f6dfce85f5f4": "N \\pi_{0} ", "88d412f0e13275e3d58c4264eb28388c": "\n\\mathcal{L}_{2}=\\frac{F^2}{4}{\\rm tr}(\\partial_{\\mu}U \\partial^{\\mu}U^{\\dagger})+\\frac{\\lambda F^3}{4}{\\rm tr}(m_q U+m_q^{\\dagger}U^{\\dagger})\n", "88d43f64c5dfa5d788bae52827388b41": "N_r = mg\\left(\\frac{L-b}{L} - \\mu \\frac{h}{L}\\right)", "88d4489b33bc859f3a4abced0ad10d44": "\\!\\, N ", "88d4505eb15f9f61bf0b1823a83de06d": " \\mathbf{w}_n(p_n(t)) ", "88d45f3e6e6288e727feab78a4cf9355": "\\gamma_{\\lambda,\\nu}(\\omega)", "88d4754ec48ba9bc3c842f0f417c1a8b": "K'\\,", "88d488dceac353b2349effb212c90fab": "\n\\begin{pmatrix}\n ct' \\\\\n x'\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n \\cosh \\phi & - \\sinh \\phi \\\\\n - \\sinh \\phi & \\cosh \\phi \n\\end{pmatrix}\n\n\\begin{pmatrix}\n ct \\\\\n x\n\\end{pmatrix} \\text{ where } \\phi = \\operatorname{artanh}\\,\\frac{v}{c} \\text{,}\n", "88d4a185bf82d897dc0db4ed972150c9": "\\text{CBW} = \\phi_t \\cdot \\text{SF} \\cdot \\text{Qv}", "88d4d9721a9f3cdbc2e0f6d501d6dc05": " \\widehat{a}(\\widehat{U}|\\alpha\\rangle) = \\widehat{U} \\alpha e^{-i\\theta}|\\alpha\\rangle = \\alpha e^{-i\\theta}(\\widehat{U}|\\alpha\\rangle) ", "88d523b27260a325bef211d17cc8a6f3": "t\\colon V \\otimes V^* \\to F", "88d5570116c449d8cfb8099eb2e50e2d": "a\\to (b\\wedge c)= (a\\to b)\\wedge (a\\to c)", "88d56a2c3aaa3368c2241c088cfafebd": "a+d=b+c.", "88d56f874eda4acdd8fa95867d9a16f3": "0.487 < p^{*}/p_0 < 0.587", "88d57e3376686b7405a2f03d411964e7": " \n Z_\\mathrm{OC} = -j Z_0 \\cot (\\beta l) \\,\\!\n", "88d5f19d9a922d80220cd0f6fdb6dfde": "\n\\text{The congruence }x^2 \\equiv 2 \\pmod p \\text{ is solvable if and only if }p\\equiv \\pm 1 \\pmod 8.\n", "88d5f4aaa97ceb23cde5dd5e112f1265": " \\alpha_1 ", "88d604fc8bbba724c1cc08b5b8502677": "T(n) \\cdot \\frac{1}{P(n)} = O(n^2\\log ^3 n)", "88d644fe503e37642beb5d55b1229223": "\\operatorname{E}[|T_N|]\\le\\sum_{n=1}^\\infty\\operatorname{E}[|X_n|1_{\\{N\\ge n\\}}],", "88d70e9ec2627df2f217d986f015b1d1": " f(g)= \\int_{\\mathfrak{a}^*_+} \\tilde{f}(\\lambda) \\varphi_\\lambda(g) \\,\\, 2^{{\\rm dim}\\, A}\\cdot |b(\\lambda)|\\cdot |d(2\\lambda)|^2 \\,d\\lambda, ", "88d773e1e51b906a2ef695033507c3bb": "\\text{strong bond}\\rightarrow\\;\\dashv\\!\\overset{\\textstyle \\top}{\\underset{\\textstyle\\bot}{0}}\\!\\dashv\n\\qquad\\text{and}\\qquad\n\\text{strong bond}\\rightarrow\\;\\vdash\\!\\overset{\\textstyle \\bot}{\\underset{\\textstyle\\top}{1}}\\!\\vdash", "88d7b872b58bdf0b0c142a8753133b2f": "|D| = \\mathbf{P}(\\Gamma(X, L))", "88d7cfdb08668380e64d906c2f34c0c7": " B_X := A_{n,0} / I_X \\, . ", "88d7d11781b5d63e15e73a06801f3009": " u(0) = u(\\pi) = 0.", "88d8525c54acc9206622e5e7807d6f00": "\nA_{T}^x:=\\sum_{\\gamma \\in S_{T}:a\\to \\alpha\\setminus\\{a\\}} x^{\\ell(\\gamma)},\\quad\n B_{T}^x:=\\sum_{\\gamma \\in S_{T}:a\\to \\beta} x^{\\ell(\\gamma)}, \\quad\nE_{T}^x:=\\sum_{\\gamma \\in S_{T}:a\\to \\epsilon \\cup \\bar{\\epsilon}} x^{\\ell(\\gamma)}.\n", "88d85c9353085b5b9499b69cc8dde907": "\\theta(L)=\\theta_L", "88d860101b90be6893df46a44e8c5d0b": "\\scriptstyle{R_{J}(x,y,z,q)}", "88d89a2a32470eeb8ba435de9802f097": "Z_L = \\frac{V_L}{I_L}=\\frac{V_S}{I_S}", "88d9239c39e10efb2f441ab3347fb55c": "R^a{}_{bcd}", "88d976228e458f8a674f0e381f307926": "\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{7} + \\frac{1}{11} + \\cdots = \\ln \\ln (+ \\infty)", "88d9adca934d40c80ece0895b2a63b88": "\\cos^2 \\alpha = 1 - \\sin^2 \\alpha \\,", "88d9b62eac4746f5efb033fd1afd9f24": "u\\,dv+v\\,du+du\\,dv=u\\,dv+v\\,du,", "88da29d6ecd8683ae919bf4fc689d662": "g(x) = (x-3)(x-3^2)(x-3^3)(x-3^4) = x^4+809 x^3+723 x^2+568 x+522", "88da4fbb6fe155955848dc1712d3f51d": "\\mathbf x = \\begin{bmatrix} x_1 & x_2 & \\dots & x_n \\end{bmatrix}.", "88da50b90442ed4c2418f87603c1f2d5": " W_{DM}(t) = \\frac{q}{2m} (L_x + 2S_x) B_0 \\cos \\omega t \\, ", "88da9daf6626abac9014f7bda26499a0": "E_{\\rm dn}=E_{\\rm ext}\\,e^{-cm},", "88da9fa8422d9c8f013efe82bfb3401b": " \\lambda_v ", "88dab9c0a09af2bf61a158ce91d87479": " \\Delta(x) = \\sum_{w\\in S_n} \\sgn(w) x_1^{w(1)-1}x_2^{w(2)-1}\\cdots x_k^{w(k)-1} ", "88dad17c519e9bb127941a4dc22a3f3a": "\\Delta^2 = \\frac43\\, \\frac{1}{\\eta_1 - \\eta_3}", "88db01e38e98ff8df13e5bbb3a6adc05": "\\scriptstyle A=\\{x,y,z\\}", "88db1deaed2111351cf38356f4eecd1f": "T(n) = T\\left (\\frac{n}{2}\\right )+n(2-\\cos n)", "88db27111442b0639ca7043dec452638": "K_\\alpha(x) = \\frac{\\pi}{2} \\frac{I_{-\\alpha} (x) - I_\\alpha (x)}{\\sin (\\alpha \\pi)}.", "88db34725361e46f940c9ea22fa00483": "\\mathbb{P}_n", "88db5ec5de084b1dff906623cde52486": "\\int_{-1}^1 f(x)\\, dx \\approx d^T c = d^T D y = (D^T d)^T y = w^T y", "88dbc6b8b1ee34b9a058eca40dae6a5b": "r_k=\\frac{s_k}{s_\\lambda^m}", "88dbd2d10d3ae7e8fded6c1fb2c6de7b": "u\\left(A\\right)\\geqslant u(B)", "88dbed8f70ec15435a6f8a402110c835": " {f_o} {u_g} = - {\\partial \\Phi \\over \\partial y}", "88dc0ae098ea0d398769a81d82b18414": "H = E + pV,\\,", "88dc119fcd0c94210bd01c69ea47d830": "q\\;", "88dc1f7c204ec8686fd31a572b9cf999": "\\zeta(\\tfrac{1}{2} +it) = Z(t)e^{-i\\theta(t)}", "88dcd14db5cb3f2ae31cb92b911db62b": " \\tan \\alpha = \\frac {\\sin L}{(\\Lambda - \\cos L) \\sin \\phi_1 }", "88dcf8c5ed5e92ed3cd5e8d578119166": "(\\mathbf{x}, z_1)", "88dd124add3eeeaaef883266af063718": " \\hat{a}\\mathsf{S} = (a, b)(\\mathbf{S}, \\mathbf{V}) = (a \\mathbf{S}, a \\mathbf{V} +b \\mathbf{S}).\\!", "88dd13fdfbc3cff02bad391cae75e4d1": "\n \\limsup_{n\\to\\infty} \\operatorname{Pr}\\big(g(X_n)\\in F\\big) \\leq \\operatorname{Pr}\\big(g(X) \\in F\\big),\n ", "88dd6c80e4af219bb249ff37db514d48": "\\Delta p \\ge \\frac{\\hbar}{2\\Delta x}.", "88dd9048ac307230e44fb2f8c5e72cc4": "\\lambda_{100}", "88dda0b0fa3686dc6ec222dc00d2940b": "\n\\Delta w_{k} \\equiv \\oint \\frac{\\partial w_{k}}{\\partial q_{k}} dq_{k} = \n\\oint \\frac{\\partial^{2} W}{\\partial J_{k} \\partial q_{k}} dq_{k} = \n\\frac{d}{dJ_{k}} \\oint \\frac{\\partial W}{\\partial q_{k}} dq_{k} = \n\\frac{d}{dJ_{k}} \\oint p_{k} dq_{k} = \\frac{dJ_{k}}{dJ_{k}} = 1\n", "88de0fb87f66be89d37f7db0c2725c24": "(T,\\eta,\\mu,m)", "88de19c993b2eb812d41917ad936b25f": " \\theta_\\mathrm{right} ", "88de31ab0023761e8c282e313b1949c9": "\\begin{array}{lcl}\nS(x, y) &=& \\left( \\frac{x}{x+1}, \\frac{y}{2} \\right) \\\\\nR(x, y) &=& \\left( 1-x, 1-y \\right)\\,.\n\\end{array}", "88de5641e8c9d130351042a3b91d6549": "\\pi,", "88de6e2f3b69b7d77f7e99f371af7e35": " \\left(\\frac{P_1}{P_2}\\right) = ", "88df0b48eff7018fef0c91fa2a9408c6": "\\langle \\phi_n \\rangle ", "88df4118feebaa8afaf55d0e6170ab36": "\\blacksquare(O\\underline{A} \\to \\underline{A})", "88df9e8d6ff2d92d88155c123a4c2899": "\\{-1,0,1,2,3\\}", "88dfa53ab3a45a151c7e3a674870dfe5": "\\textstyle-\\frac{3}{4}", "88dfa8951894a4a71741014ff8868679": "C_m[\\xi_0, \\ldots, \\xi_k]= \\{s \\in Q^\\mathbf{Z} : s_m = \\xi_0, \\ldots ,s_{m+k} = \\xi_k \\}", "88dfac10bf07afb2ba47390a04ea48ee": "\n\\begin{align}\n& N^2 + (2N - 4)^2 = 16\\\\\n\\Rightarrow\\ \\ & 5N^2+16-16N = 16\\\\\n\\Rightarrow\\ \\ & 5N^2 = 16N\\\\\n\\Rightarrow\\ \\ & N = \\frac{16}{5}\\\\\n\\end{align}\n", "88dfc47b391fb241c60c090d94cd4205": "\\int\\arccos(x)\\,dx=\n x\\arccos(x)-\n {\\sqrt{1-x^2}}+C", "88e007c8c899be7f34399c14388035ba": "\\phi_n x = \\phi_n y", "88e017f449d38cbbb1fec9aa80787330": "(11)\\;\\;\\quad ds^2=-\\Big(1-\\frac{M}{r}\\Big)^2 dt^2+\\Big(1-\\frac{M}{r}\\Big)^2 dr^2+r^2 \\Big(d\\theta^2+\\sin^2\\theta\\,d\\phi^2\\Big)\\;.", "88e06e9edf6d28d3c62e0137970b4278": "\\begin{alignat}{2}\nA - B & = A + (-B) \\\\\n& = A + \\bar{B} + 1 \\\\\n\\end{alignat}", "88e0ae24c4fc2edabb84c5ca46457d3d": "= \\frac{\\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta}{\\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta}\\,", "88e0d199f4ae4c09c2ff95eab49858e5": "PR(E).", "88e1049bf7031ddb6355dcbb93d3f4f0": "\\Omega f (x) = \\limsup_{r \\to 0} f_r(x) - \\liminf_{r \\to 0} f_r(x)", "88e184f6602814ceb66d7226ec3ec36e": "\n\\Gamma_{xy}(f)= \\Lambda_{xy}(f) + i \\Psi_{xy}(f) ,\n", "88e1b15ff9f9d1abf0b581a05c8dddc6": "b^2 \\nmid f", "88e1c31d6bc6c9c4540b341a72530266": "\\Delta G_m", "88e1dbb330f94109cadd0effeba60292": " \\delta = \\frac{ | \\mu_1 - \\mu_2 |}{ \\sigma } ", "88e1f997d682cd7ac407ee0f7bff639f": "X = \\lbrace x_i | i \\in I_v \\rbrace,", "88e215ae36fca99682bd5733b82f42a8": " s_{ij} = \\exp{\\left[ \\frac{d_{1} - d_{ij}}{b} \\right]} ", "88e27c7ebcede65d54d00e02bfd6f1e5": "\\begin{matrix} {12 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "88e2c8d5b69c5791ca2a15605373e906": " \\partial_x (\\alpha\n\\partial_y) = \\partial_x (\\alpha) \\partial_y +\n\\alpha \\partial_{xy}.", "88e2ce7bf41433a4e96eed0f418195c8": "H_i M \\simeq H^{n-i} M", "88e2fea7ebf58caaeffc58650f93a45f": "E(q)", "88e31bf4a909121e5b81ce7f23df3d65": "in_b", "88e382e5d488926355e625eee6d415c1": "T_G(0,2)", "88e38a7bdfc2190c6562ed012bc2a407": "z = A^+ b", "88e3c88b4e043ad44761397790bc3211": "a_k = r_k \\cos(\\phi_k) = 2 f_0 \\int_{0}^P y(t) \\cos(2 \\pi k f_0 t)\\, dt, \\quad k \\ge 0\\,", "88e3d5ffba4f5427829882ae8b167af2": "2\\times 10^0 + 3\\times 10^{-1} + 5\\times 10^{-2}", "88e3d641af4cf7206539dc2e939b19d7": "W_a", "88e4068ab8fb33de3b47f5f4bfabb0b1": "\\mathbf{x},\\dot{\\mathbf{x}},\\dots,\\mathbf{x}^{(n)}", "88e41d808273fe4d9b51051e95088050": "d_{ik}", "88e44a9c823b794979be741ac81036f6": "\\sum_{ij}{e_{ij} = 1},\\quad\\sum_{j}{e_{ij} = a_{i}},\\quad\\sum_{i}{e_{ij} = b_{j}}", "88e4c68ac31fbb0aaf56209aca27bbaf": "U P P = P U P.\\,", "88e51ca6ac515457ffed1b32265e0614": "s_1^2", "88e59605e4371a2e01222445afe9a007": "K^M_2(\\mathbb{Q}_p)", "88e5fd8a515775cc6b2b545d96f1e84a": "\ng = exp(-2Dk_{1}t+\\phi_{0})\n", "88e5fe49fc565e8e952452c071eeef27": "E_1 = E_2 > E_3", "88e60b7bb1c12e9c26f3007958144ad7": "\\scriptstyle T_2", "88e61e26ab83803bbadefc3cb740e28b": "T_{cr} = (g\\tau)^{3/2}\\hbar^{2}/(k m_{\\mathit{eff}}) = (g\\tau)^{3/2} \\times 10^{-11}", "88e627e225051ab8c2332012918f1d28": "[d_0, d_1, \\ldots d_{N - 1}]^T", "88e63ef1fb825da45ee1949fa36f28f6": "\\displaystyle \\frac{\\displaystyle \\delta\\left(\\xi - \\frac{a}{2\\pi}\\right)+\\delta\\left(\\xi+\\frac{a}{2\\pi}\\right)}{2}", "88e6431e34c7dd057ee2d4d9dd82a4db": "\\sum_{n\\ge1}MG(a_1,\\dots,a_n) \\le\\, e\\, \\sum_{k\\ge1} \\bigg( \\sum_{n\\ge k} \\frac{1}{n(n+1)}\\bigg) \\, k a_k =\\, e\\, \\sum_{k\\ge1}\\, a_k \\, ,", "88e64dbe5e1a959110c2cd5839ea1b2d": "\\frac{1+p}{\\sqrt{pr}}", "88e65b8f3a44966082b7f4675e9a2701": "\\int_0^1 \\frac{\\ln (1+x)}{x}\\, dx= \\frac{\\pi^2}{12}", "88e67695a55b0ceabc30a24cd7c615be": "\\phi_{\\beta/\\alpha}", "88e68cc0d2cb604493bff285f5df67bf": " VT^{f/2} = \\operatorname{constant} ", "88e696dffccb39ba9a645cb55fe811fe": "J_k/J_{k-1}", "88e69bb56008356eda91e03ab94f43d7": " q \\equiv 1 \\bmod 2n ", "88e6ae8c4b34436651c5687adaaf2093": "u_n (x) = e^{- n x} \\, ", "88e71548da0461b5cfff1b80fbfe3d0b": "\\mathbf{L} = -i\\hbar\\mathbf{x}\\times \\nabla = L_x\\mathbf{i} + L_y\\mathbf{j}+L_z\\mathbf{k}.", "88e71b0edde4e84efb9ea4b6d82b60f3": "f(T) = \\int_{\\sigma(T)} f(\\lambda)\\,dE_\\lambda.", "88e71f120ae31dd9237ea205a9904059": "\np_5(x) = x^5 + 11x^4 + 5x^3 - 179x^2 - 126x + 720 \\,\n", "88e72ed416104efcc797b4473d3d95d8": "\\text{im}(C)\\subseteq\\overline{\\text{im}(B^*)}", "88e72fa27e5da9d399d15db5e5a387fa": "\n\\textrm{efficiency} = 1 - \\frac{q_C}{q_H} = 1 - \\frac{T_C}{T_H}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(7)\n", "88e790f53ffcea8d2804a0a33c3a5e05": "N_R = {\\text{Convective Inertial Force}\\over \\text{Shear Force}}\\,\\!", "88e7a11c68eaafbc15c5685417e7d2f2": " \\mathbb C^{16} ", "88e830dc70d5c2cd89a31ba9298d0ecc": " d\\alpha =\\beta\\wedge \\gamma,\\,\\, d\\beta= \\gamma\\wedge \\alpha, \\,\\, d\\gamma=\\alpha\\wedge\\beta.", "88e83d38029b9641301fb23a30c1258c": "4 \\pi G \\mu [\\vec{x},t] = \\nabla^2 \\zeta [\\vec{x},t] ", "88e872b39bd18a1a95fddaaea5ae4be0": " y_s(x) = -(1/4) \\cdot x^2 \\,\\!", "88e88b4a44b7c8e57ff5459606a7944e": "\\scriptstyle \\inf\\varnothing=\\infty", "88e894bf7bce4a476b4343c72884a53b": "b_{k+1}=1", "88e8a76bdcc25d6e188fb786b45c708b": "\\scriptstyle A \\and B", "88e8cf48c878ede1d1441976c87794df": "+\\lnot F^n \\leftrightarrow \\lnot +F^n", "88e8d0e3854f5381f363f75783fdf697": "f^+(x,u) = \\limsup_{h\\to 0^+} \\frac{f(x+hu) - f(x)}{h}", "88e8d360fb22d4f3cec787938bf0d5ca": "E^2(\\mathbb{C}\\mathbf{P}^\\infty)", "88e8de99d5ccdd542e6ab1743c78a3da": "\\scriptstyle \\tilde{t}_i \\;=\\; t_i \\,+\\, \\delta t_{\\text{clock},i} (t_i)", "88e9905a8bb5b04895236e17a539f3e4": "y=r(\\varphi)\\sin\\varphi \\,", "88e99f0b764d313c50a5f4fdd8a7947e": "S_n", "88e9aa1a574229f5cd66a2bd0724e75d": "f: M \\times [0, 1] \\to N", "88e9f18cd24f22133b1f2b406ba14c07": "I = \\int_a^b f(\\mathbf{r}(t)) |\\mathbf{r}'(t)|\\, dt.", "88ea67f78f66ea40dfa3e4f1887d4729": "\\begin{align} g &=\\frac{GM}{R^2}\\\\\n\\frac{dg}{dR} &= -\\frac{2GM}{R^3}= -\\frac{2g}{R} \\end{align}", "88ea86af5d50da984292d575ee64cf20": "\\scriptstyle E(T)", "88eabda918e8e0b3a94a6999bf0a166d": " 2 p_3 \\cdot p_4 \\,", "88ead7b855856c90599a581b3f42d2f6": "uv=-\\frac13p\\,.", "88eaf1c5a4cc5bfcb36b7b851106b837": " A=\\iint_D\\sqrt{\\left(\\frac{\\partial f}{\\partial x}\\right)^2+\\left(\\frac{\\partial f}{\\partial y}\\right)^2+1}\\,dx\\,dy. ", "88eb2bd6e3c999f47d967600965aa047": "\n\\begin{align}\nm_2 & {} = 123456 \\\\\nm_1 & {} = 78901234 \\\\\nm_0 & {} = 56789012 \\\\\nn_2 & {} = 98765 \\\\\nn_1 & {} = 43219876 \\\\\nn_0 & {} = 54321098 \\\\\n\\end{align}\n", "88eb371eac52312c6e7faf51967c4675": "\\sum_{k=0}^m k^{n-1}=\\frac{B_n(m+1)-B_n}{n}\\,\\!", "88eb5df6ed6e9e3ae976b4b4d793e2f0": " f(p) \\neq f(q) ", "88ebb8b497bda89de28f4608ceb6b93a": "( r^{\\prime}, \\theta^{\\prime}, \\phi^{\\prime})", "88ecc5865c2e3af64b7d5011c25729f8": "\\mathcal{S}(X)", "88ecd7f0f0bfb4bbeb52f516d4590384": "{\\eta} = \\frac{\\tanh h_T}{h_T}", "88ed34acd6e724154711ec9755a7b2f3": "Price_{begin}", "88ed65dbfbc8771e4c5d9443a313a62b": "\\sigma_3=0", "88ed801d0a5cd2189a80ac1697ff9b6a": "=\\mu \\cos \\theta\\ ", "88ed8734ec058fa55a08782bd158c459": "\\begin{matrix} {4 \\choose 3} \\end{matrix}", "88ede1bae14c566ecbfdb9b599404d13": "d(\\gamma_0(\\pi),\\gamma_\\tau(\\pi))=0 \\,", "88edea3bde0c86e0b687a169fc4a6ad2": "\\rho = {{nM}\\over{V}}.\\ ", "88edf48499e8ac95a0e9a068bb186a05": " \\forall x\\,Px \\Rightarrow \\exists x\\,Px", "88ee2dd858a6e98714512587e0a10e32": "\\lambda=W_k(a)", "88eeb0cec5e46b35ca129a1cd4a8853b": "\\left(1+\\sqrt{-5}\\right)\\left(1-\\sqrt{-5}\\right)", "88eec553d7c5bff51ed871f3c7188b3c": "\\psi (z,t) = \\sqrt{P_0} \\left[ 1-\\frac{4 \\left( 1 + 2 i \\dfrac{z}{L_{NL}} \\right) }{1+4 \\left( \\dfrac{t}{T_0} \\right) ^2 + 4 \\left( \\dfrac{z}{L_{NL}} \\right) ^2} \\right] e^{ \\dfrac{i z}{L_{NL}}} ", "88ef9c3ea09d80a924def5a07379f532": "H := 2h", "88efbad9623bacf6c8567390882508c9": "\\Delta E'^{\\mathrm{sys}}=Q'_{in}+ W'_{in} - Q'_{out} - W'_{out}\\,", "88efe4c2b714223a60dc083aa3faaac5": "v(\\sigma)", "88f04459f5845f1cf4113b12fc57cea8": "\\sqrt[3]{\\frac{\\rho_{\\mathrm{secondary}}}{3 \\rho_{\\mathrm{primary}}}}", "88f06ba426d055439d1ecad6a2b4fd55": "\\phi_{-k} = \\phi_k^\\dagger, \\pi_{-k} = \\pi_k^\\dagger,", "88f07580d84e07d962d8a7e8fed9c024": "\\phi:\\mathbb{R}\\to\\mathbb{R}", "88f0950f1396a79be8e466c11f477626": "\\psi_1(\\alpha)", "88f10c69929fc858238ab88bfc4b6aee": "i\\prec\\prec J", "88f1841f966dc539665bcb1e9353a442": "Q(x + \\alpha,y + \\beta) = \\sum_{u,v} x^u y^v \\Bigg ( \\sum_{i,j} \\begin{pmatrix}i\\\\u\\end{pmatrix} \\begin{pmatrix}i\\\\v \\end{pmatrix} a_{i,j} \\alpha^{i-u} \\beta^{j-v} \\Bigg )", "88f1bb79cb45084b4424ebf71ee60180": "\\hat{E} = i\\hbar\\frac{\\partial}{\\partial t} \\,\\!", "88f1c0a4f66eaaea8efa7b3a0438d285": "\\Rightarrow_{g} BAAA \\Rightarrow_{g} BBAA \\Rightarrow_{g} BBBA \\Rightarrow_{g} BBBB", "88f25f8d658cf8a1b78902fd9be0558d": "\\mathrm{NPV}(i)", "88f2ad8a6881fd939e75c4794a9b6d0f": "\\frac{\\partial \\rho uu }{\\partial x} + \\frac{\\partial \\rho vu }{\\partial y} = \\frac{\\partial \\frac{ \\nu \\partial u} { \\partial x}}{\\partial x} + \\frac{\\partial \\frac{ \\nu \\partial u} { \\partial y}}{\\partial y} - \\frac{\\partial p}{\\partial x} + S_u ", "88f2b2d3d36051bc8dbb6c9e92f32477": "\\tau = \\mu \\frac{U}{h} = \\frac{2 \\pi r \\mu N}{c}", "88f32b29854da75288ac810bac667bb0": "b = y_0^2 - x_0^3 - ax_0\\pmod n", "88f35a123eb40fa8b33b9c9af7ac64f3": "i\\hbar\\frac{\\partial}{\\partial t}\\hat{U}(t,t_0) = \\hat{H}(t)\\hat{U}(t,t_0)", "88f385cf447fd84120960a5d820c4679": "D_{10}", "88f3d6a208872375b8bb0afadaa07301": " \\text{MSE} (\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{MSE} (\\widehat{\\boldsymbol{\\beta}}_{k}) \\succeq 0 ", "88f44ffe398f97ade177af8f760d3bd9": "T_n(x) = \\tfrac{1}{2} (U_n(x) - \\, U_{n-2}(x)). ", "88f469274496152b34c34b0d7a618850": "{\\bar{Z}}_3", "88f46bd07d5143f6e1db15b8294b2238": "\\log(p(X_i))", "88f4c8475be0b013a9c066bd67d358c5": "\\alpha\\|x(0)-x_e\\|e^{-\\beta t}", "88f4fd454f873b3f2aae17fe408800b1": "\\sigma^{n}(A) := \\frac{1}{\\alpha(n + 1)} \\lambda^{n + 1} ( \\{ t x \\mid x \\in A, t \\in [0, 1] \\} ),", "88f519a95a13a0cab6efd62797b34005": "\n \\begin{align}\n \\eta(x,t) =& a \\left\\{ \n \\cos \\theta \n + \\tfrac12 (k a)\\, \\cos 2\\theta \n + \\tfrac38 (k a)^2\\, \\cos 3\\theta \n \\right\\}\n \\\\ &\n + \\mathcal{O}\\left( (ka)^4 \\right),\n \\\\\n \\Phi(x,z,t) =& a\\frac{\\omega}{k}\\, \\text{e}^{kz}\\, \\sin \\theta\n + \\mathcal{O}\\left( (ka)^4 \\right),\n \\\\\n c =& \\frac{\\omega}{k} = \\left( 1 + \\tfrac12 (ka)^2 \\right)\\, \\sqrt{\\frac{g}{k}}\n + \\mathcal{O}\\left( (ka)^4 \\right), \\text{ and}\n \\\\\n \\theta(x,t) =& kx - \\omega t,\n \\end{align}\n", "88f51ea1d4bebc64b3044de65a82f98c": "\\log p(\\mathbf{X}|\\boldsymbol\\theta)", "88f5213eddd8fa7f71c83a8afb969f5a": " \\sin\\phi = \\frac{1}{\\cosh a} ", "88f5587a3d3b53c7807708d0a5cb62cc": "y_{p_1}", "88f5b270f370a945f8442400a7b48824": "(\\ \\bar{r}\\ ,\\bar{v}\\ )", "88f5efb04a8882647ad9ee0df37af182": "\\mathrm{S}_{\\mathrm{O},\\mathrm{P}}\\,", "88f60a384dc99d6e6cdd43c4015caa73": "\\tilde{S_{n}^m} = -\\frac{S_{n}^m}{\\mu\\ R^n}", "88f65241f1c90f5f6ba6e8931dfa7af1": "r\\in{\\mathbb Q}(x)", "88f727447c7c7d845e6a0d9019ecc167": "\\forall X \\, \\forall Y \\, ( \\forall z \\, (z \\in X \\leftrightarrow z \\in Y) \\rightarrow X = Y).", "88f756b6ff53fe90476b3d3891a7765c": " f(n_1,n_2,\\ldots,n_m) = \\frac{n! \\,\\Delta(n_m,n_{m-1}+1,\\ldots,n_1 + m - 1)}{n_m ! (n_{m-1} + 1)! \\cdots (n_1 + m - 1)!}", "88f7b9fe0421829dbd17dbf598452e13": "\n\\frac{\\partial \\mu}{\\partial \\theta_m}\n=\n\\begin{bmatrix}\n \\frac{\\partial \\mu_1}{\\partial \\theta_m} &\n \\frac{\\partial \\mu_2}{\\partial \\theta_m} &\n \\cdots &\n \\frac{\\partial \\mu_N}{\\partial \\theta_m}\n\\end{bmatrix}^\\mathrm{T};\n", "88f7e9771b21bc526cfc51220b72d6a1": "Q(U)", "88f8020b19138e04d1b37887fd498721": "\\Box p \\rightarrow p", "88f8316f06c3333342d02fd3ab2a2da7": "\\zeta (s,q)=\\Phi(1, s, q).\\,", "88f84aae3148dc18ab4f9dc8a56964a6": "S^2=\\frac{ O(3) }{ O(2) }", "88f89f5de870d611dd98c06c2ba58c4b": "\\int_0^1 x^2 j_\\alpha(x u_{\\alpha,m}) j_\\alpha(x u_{\\alpha,n}) \\,dx = \\frac{\\delta_{m,n}}{2} [j_{\\alpha+1}(u_{\\alpha,m})]^2.\\!", "88f8d12dcb21bd175230e4ea775411d5": " P_{emission} = \\frac {E_f}{c} = \\frac {\\varepsilon\\sigma}{ c } T^{4}", "88f9727dd019e049cc7c1b623e9780c2": " f\\ v_1 \\ldots v_n ", "88fa3714a406aa2021778c03216008b6": "f^*(\\theta)", "88fa40882af9318212decd3e2ceacf31": "\\gamma_{F}^{-}(\\overrightarrow{x},a,s)", "88fac524196151e5b98177dfd484a3c4": "E \\neq 0", "88fb2b9f2c32aefe475be5c43e52ae3d": "a_{t,j_t}\\neq0", "88fb585179a375192a11c46000279cca": " b = x^\\frac{1}{\\log_b(x)}.", "88fbe163f796216577702e7a409428ca": " A = 6 \\pi a^2.", "88fc1bac34c9cf37b12caeb3192b61f1": "P_\\lambda=\\sum_{\\mu\\le \\lambda}u_{\\lambda\\mu}m_\\mu", "88fc3805c097f8722bac7b77020f91b0": "\n\\frac{dP}{d\\zeta} + \\frac{1}{2} P = 0\n", "88fcffd5715759963b223c999d45c58c": "\\tilde\\lambda+\\delta", "88fd1133277e4caae86810611f904805": "{2 \\pi\\over 3}", "88fd1920cb75f43d4acb78b3666f5557": "\\scriptstyle \\Vert x(t)\\Vert\\,\\to\\,\\infty", "88fd3f92578691a3681ec66a143819d1": "\\Delta G = \\frac{4}{3} \\pi r^3 \\Delta G_v + 4 \\pi r^2 \\sigma", "88fd5a7af3d403de91cc9d02dd049c90": " \\nabla^2 \\varphi =-\\dfrac{\\rho}{\\epsilon_0}\\,,\\quad \\nabla^2 \\mathbf{A} =- \\mu_0 \\mathbf{J}\\,,", "88fd6a5df1b388cfd1a62043f7acd847": "\\scriptstyle s(t + \\alpha)", "88fe26eda76ddf5f8cf63e123d6ff4be": "b_0 = 0", "88fe58381a697502dd19785241a81b3b": "\\textstyle {(4+0+0)!\\over 4!\\times 0!\\times 0!} \\ {(3+0+1)!\\over 3!\\times 0!\\times 1!} \\ {(2+0+2)!\\over 2!\\times 0!\\times 2!} \\ {(1+0+3)!\\over 1!\\times 0!\\times 3!} \\ {(0+0+4)!\\over 0!\\times 0!\\times 4!}", "88fe8b66ceeba57ea0bd8046c25579b0": "P(H|I)+P(T|I)=1 \\rightarrow 2 P(H|I)=1 \\rightarrow P(H|I)=0.5 ", "88feab4003d370c7e3a2b33c992f2fb9": "x \\leq M \\land y \\leq M \\land x = M", "88febac849e1017ad4640c92352f23bd": "X_{\\mu\\nu}=\\partial_\\mu X_\\nu - \\partial_\\nu X_\\mu + g f^{abc}X^{b}_{\\mu}X^{c}_{\\nu}", "88fee43ecaa79c160c2b84b354154dbf": "\\mathbf{F} = \\frac{q_1 q_2}{4\\pi\\epsilon_0 \\left|\\mathbf{r}\\right|^2}\\hat{\\mathbf{r}}", "88ff4a6f25e9a1675fc702f8ee41a28a": "\\arctan", "88ffa346a8f26217f133075474a943d7": "x < y \\Rightarrow x + z < y + z", "88ffbc34c27d7ea216e4c0702223a4a5": " z(x,y,\\alpha,a,q) = \\frac{q}{2\\pi}\\left[\\ln(x+iy- ae^{i\\alpha}) - \\ln(x+iy + ae^{-i\\alpha})\\right]", "890012af1ebb1d08320245be0794b7e2": " \\mathbf{C}_{1}, \\mathbf{C}_{2} ", "8900b2cf2f948a6bb56b73ef93e405a8": " Y=BZ +U \\, ", "8900b5b4de2a17837be925401369967f": "\\begin{array}{lll} |g'|^2 &=& \\rho^2 (h')^2 \\\\ g' \\cdot g'' &=& \\rho^2 h' h'' \\\\ |g''|^2 &=& \\rho^2 \\left( (h')^4 + (h'')^2 \\right) \\end{array}", "8900cd8e53fa3edcc243e3b8601c220a": "\\left (T_\\varepsilon(D)-T^\\prime_\\varepsilon(D) \\right )f(w)= \\frac{1}{\\pi}\\iint_{U_\\varepsilon} {\\partial_zf(z)\\over z-w}dxdy-{1\\over \\pi}\\iint_{V_\\varepsilon} {\\partial_zf(z)\\over z-w}dxdy+{1\\over 2\\pi i}\\int_{\\partial U_\\varepsilon} \\frac{f(z)}{z-w}d\\overline{z}-\\frac{1}{2\\pi i}\\int_{\\partial V_\\varepsilon} {f(z)\\over z-w}\\, d\\overline{z}.", "8900e4d21276ed2cba80e2a0c32d031c": "\\hat n\\cdot \\hat n=1", "89014ec1b187c9493f399d4827d43983": "K_n = \\sqrt{1+2^{-2n}}, \\tan(\\phi) = 2^{-n}.", "89017605ed6697dfc1a10870e2009ef2": "N = N_\\mathrm {TOT} 10^{-bM} \\ ", "8901763e1d4cf7325c1196c1cff4f5cd": "x^3-6x^2+11x-6\\,", "8901cc600d048b033fc49d12eaad5f13": "F: \\mathbf{Set} \\longrightarrow \\mathbf{Set}", "8901ce42e022d34d4cbdf0a2aa305b81": "\\| \\mathbf a \\|^2 \\ \\| \\mathbf b \\|^2 - (\\mathbf {a \\cdot b } )^2 = \\sum_{1 \\le i < j \\le n} \\left(a_ib_j-a_jb_i \\right)^2 \\ , ", "890227711b79c1f420225f5518ab10e6": " -i\\hbar \\partial_{t_1} U_\\epsilon(t_2,t_1) = U_\\epsilon(t_2,t_1) H_\\epsilon(t_1) ", "890270dcf2ba7a3e1b9970771be56654": "L = \\{a^nb^nc^n|n \\ge 1\\}.", "89027c7019902e10e785a5b9e60818e0": " \\frac{8! \\times 3^7 \\times 12! \\times 2^{10} \\times 24!^3}{4!^{12}} \\approx 2.83 \\times 10^{74}", "8902b7f6ff057c3e843198fc03426c2e": "\\mathbf{p} = \\left[ x,y,z \\right]", "8902e9b5f68626d77399ca5846d709b5": ":\\sigma(\\alpha)\\mapsto \\{\\sigma(\\beta):\\beta\\preceq\\alpha\\}\\,", "8902ee7c1c8c1e01a9a785e08712bc23": "\\xi = 0", "8903464aa7bcd9233fbd24a985f40641": " X_{01} ", "890356a10835df60a2c6a99875aed74f": "G(\\mathbf{x},\\mathbf{x'})", "890384bbfd697b22cb9fbba15c9520be": "C = f(\\sigma, \\cdot) \\,", "8903ed161a6e9f936edcdba1e381052b": "\\alpha_{mk}", "89046f93e91f08da2f7e5c62dd22834d": " \\nu^* \\approx 2.7 ", "89047247841d093fc980cd9355ec3ef7": "\n\\begin{align}\n\\beta f &= \\dfrac{\\beta^2 J^2 q^r}{4} - \\dfrac{r\\beta^2 J^2 q^r}{2} - \\dfrac{\\beta^2 J^2}{4} + \\dfrac{\\beta J_0 r m^r}{2} + \\dfrac{r\\beta^2 J^2 q^{r-1}}{4\\sqrt{2\\pi}} \\\\\n&\\qquad + \\int \\exp\\left(-\\frac{z^2}{2}\\right)\\log \\left(2\\cosh\\left(\\beta Jz\\sqrt{\\dfrac{rq^{r-1}}{2}} + \\dfrac{\\beta J_0 r m^{r-1}}{2}\\right)\\right) \\, \\mathrm{d}z\n\\end{align}\n", "89048b7287297ebfac4e0067047889f1": "\\hat n=(n_1,n_2,n_3)", "8904d6ee7cadac1be37358b3b97721f6": "D(n)=\\Pi_{k=1}^{\\infty}\\left\\{1-[1-\\Pi_{j=1}^n(1-p_k^{-j})]^2\\right\\},", "890573c49df8317c5bb156c8a02abf28": "\n\\begin{align}L & = \\mathrm{Kinetic~Energy} - \\mathrm{Potential~Energy} \\\\\n & = \\frac{1}{2} m \\left ( v_1^2 + v_2^2 \\right ) + \\frac{1}{2} I \\left ( {\\dot \\theta_1}^2 + {\\dot \\theta_2}^2 \\right ) - m g \\left ( y_1 + y_2 \\right ) \\\\\n & = \\frac{1}{2} m \\left ( {\\dot x_1}^2 + {\\dot y_1}^2 + {\\dot x_2}^2 + {\\dot y_2}^2 \\right ) + \\frac{1}{2} I \\left ( {\\dot \\theta_1}^2 + {\\dot \\theta_2}^2 \\right ) - m g \\left ( y_1 + y_2 \\right ) \\end{align}\n", "89058e9abef5a98ad99ff7bc8137d2b3": "\\int_{[a,b]} K_X(s,t) e_k(s)\\,ds=\\lambda_k e_k(t)", "8905c716cffb2363c70fcb51d094652a": "P_{\\rm emt\\,bb} = 4 \\pi R_{\\rm E}^2 \\sigma T_{\\rm E}^4 \\qquad \\qquad (4)", "8905e956ab870e8cded8139514e70372": "B=QTZ^*", "890623a436ed651ccf38f956a419a999": "d_\\lambda=e_\\lambda", "89064de8554b8c3fd1a6caf413820a48": "\\operatorname{ch}(M_w)=\\sum_{y\\le w}P_{w_0w,w_0y}(1)\\operatorname{ch}(L_y)", "890651e8fc123c97fe4647cc83fed02e": "(x,\\ x)_K = \\{\\{x\\},\\{x, \\ x\\}\\} = \\{\\{x\\},\\ \\{x\\}\\} = \\{\\{x\\}\\}", "89069a028a54b1aa79341e898d49a47b": "\\Psi^k \\chi (g) = \\chi(g^k) \\ . ", "89069fa17df77a92cd2e215339bc3519": "S(w)", "8906cb7116922be72ae88163ef309c81": "A_{\\alpha\\beta}=-\\frac{i\\lambda}{\\hbar}\\int_{t_0}^t dt_1\\langle\\beta|V(t_1)|\\alpha\\rangle e^{-\\frac{i}{\\hbar}(E_\\alpha-E_\\beta)(t_1-t_0)}", "89070437a0d20890e374782173af6213": "||\\psi \\rangle|^2 = 1", "89077649879b3e849f4184bfb6e695c9": "I_{max} = \\frac{|A_m|^2}{2 \\eta_0 / n} = \\frac{1}{X_0^2 k_0^2 n |n_2|}", "8907781a23df2301346bfe4523b7f137": "X \\to ( X \\to ( \\cdots (X \\to ( X ) \\to q)\\cdots ) \\to q ) \\to q", "890797977843e7dc4a9fb6503d8ede7f": "B(\\beta(t))", "8907f723827bad6091b1c3c7072f0a7a": " i_{electrode}^I = \\frac {I_{electrode}^i}{A_i}", "8908147c2601381d2fffbfeb10d8aeee": "x[T]y", "89082ae9d74d936ce85a0e8481a7367b": "x = \\rho \\cos \\varphi", "8908898ec45fd93098ae88569bfd7379": "{\\tau \\rightarrow \\infty}", "89089ea98f01a70c1b4ec2f2cb9f7a63": " \\mathbf{F} = \\frac{\\gamma(\\mathbf{v})^3 m_0}{c^2} \\left( \\mathbf{v} \\cdot \\mathbf{a}_\\parallel \\right) \\, \\mathbf{v} + \\gamma(\\mathbf{v}) m_0\\, (\\mathbf{a}_\\perp + \\mathbf{a}_\\parallel ) \\, . ", "89090b1f870f110066910561d955ef46": "S[\\rho] = \\int_{\\mathbf{R}^{n}} \\rho(x) \\log \\rho(x) \\, \\mathrm{d} x", "8909348ee2310d25e100d7ecca5e52f2": "V_{wind} = \\frac {\\mu'} {k} \\ ln ( \\frac {z + z0} {z0}) ", "89098b2bc8d123a455f5c944aa623b89": "F(x) = 1 - e^{-x^2/2\\sigma^2}", "8909bd54bdf39653cddfff96fdb23679": "\\nabla \\times \\textbf{A} = \\textbf{H}\\,", "8909c4e2c70a2fa59268da357b326644": "\\left(\\frac{1}{3},-\\frac{2}{3},\\frac{1}{3};-\\frac{2}{3},\\frac{1}{3},\\frac{1}{3};-\\frac{2}{3},\\frac{1}{3},\\frac{1}{3}\\right)", "8909c8ac8cbe4c802f48c0d8639675d4": "\\lim_{k\\rightarrow \\infty} \\oint_{\\Gamma_k} \\left|\\frac{f(z)}{z^{p+1}}\\right| |dz| < \\infty", "8909c9679e9c498d6e1250cb95b1f7b7": "\\ldots f(x,q_p) \\leq r_p\\,\\!", "8909dae53b3b4cdf8d571dfb6f02da24": " \\vec{j}\\left(\\vec{r}\\right) ", "890a08636ec40d6b1d8dab258bcd56ba": "\\vert\\phi\\rangle", "890ab7a232273772fddd16dfd491d349": "\\ X(z) = \\frac{1}{1 - 1.5z^{-1}}\\ ", "890abfcbdf815b66d10b8af1f286779a": "\\scriptstyle\\langle", "890b1ee2f1e06dfad358613a9619995e": "y \\approx Dx", "890b55b3c264a864caff16046f004afe": "\\sum_{i=a}^b f(i)", "890b8eb348681c678e86c56056130dc1": "\\textstyle\\sum_{i=1}^n (x_i - \\bar x_n)^2", "890b97d10253d446dafcf4e68ce56ae2": "T = \\left \\{ ( x,y, z) \\in \\mathbf{R}^3 \\ : \\ x^2+y^2+z^2 \\le 4 \\right \\}.", "890bdfd68990b0e32be5087a1457ec51": "\\displaystyle{ W_{{\\mathcal F}_n}(z)f(w)=e^{-|z|^2} e^{w\\overline{z}} f(w-z).}", "890c59dbd71cec158b7c7498f0a57f89": " F_A ", "890c7a880b20609ffe2f6f6d3996c3ba": "k_f [E] [S] = k_r [ES]", "890ca0bd50ab57a8a11d41d6a6f1d33a": "U_i = AU_i", "890ca1db4c2cec60c55037d48ce66965": " \\hat{f}(k,y)=C(k)F(k,y), ", "890caa99f6b8616e501a285b730a40e7": "g(k) = 2^k + l - 2", "890cf73a70ba476badbb7916a4c0658f": "\\gamma=-1\\,", "890dc8b93b4d63626fc21d9329ddfc6d": "\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} p_{\\mathbf k} = \\Delta \\varepsilon_{\\mathbf k} \\,\np_{\\mathbf k} + \\Omega_{\\mathbf k} \\, (n^c_{\\mathbf k} - n^v_{\\mathbf k})\n+ \\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} p_{\\mathbf k}|_{\\text{corr}} \\, ,\n", "890e2f1e7d175e90c5c06196cc31aeb2": "(I,Q)", "890e9b98672cea6dde52603cf7510506": "\\vert\\psi_{n}\\rangle (\\pi_{n}=1)", "890ebf93c73feb41952cc37d7517eeee": " \\operatorname{colog}_b\\ x = \\log_b \\left(\\frac{1}{x} \\right) = \\log_b 1-\\log_b x = -\\log_b x = \\log_{1/b} x.\\, ", "890eec29adb8ef56da632430b882de1b": "a, c \\in A", "890f3bcab5f545be20ab11a2d1a1b3d5": " (R,\\Delta ,\\varepsilon )", "890f560df0cf9432ce726b1632f6996c": " F(x,y,u,p,q)=0, \\,", "890f5b409e4aeb0fab7eda73ff653552": "k=1,\\dots,n-1", "890f768d03a781dae87fdd90df390f3f": "f_{i_1, \\dots, i_n}", "890f845d753a298d73d2ed410c98c304": "\\scriptstyle \\psi_k", "890fa34bba4b6c232178450c63402562": "\\rho_{L}\\,", "890fafc3367fe26d08ca0e94bf0d52a9": "F : \\mathcal{C}\\longrightarrow \\mathcal{C}", "891021d81f2aaa8c147712171f224410": " p_m = 2 ", "891038991013f862a1f1da406594cb58": "c_p = \\sqrt{\\frac{g\\lambda}{2\\pi}} = \\frac{g}{2\\pi} T \\qquad \\scriptstyle \\text{(deep water),}", "89108c167f469aa3253b25e09df69086": "f = pq", "89109320b491e63740e248929a83880f": "U{}^2_6", "89109e68dec98b22ceabb237a640b0f9": "\\begin{align}\nq \\colon [a, b] \\subset \\mathbb{R} & \\to X \\\\\n t & \\mapsto x = q(t)\n\\end{align}", "8910afe7d0e7cc4544645982efb5bdea": "\\alpha d_{1} (x, y) \\leq d_{2} (x, y) \\leq \\beta d_{1} (x, y).", "8911accb901a91cef823c04a89366626": "\\tan(\\eta/2)=\\tanh(t/2\\alpha)", "8911f8f22b9a41f3b6f3369bd0233fdd": "\\operatorname{E}(I) = n\\sigma^2 + n\\mu^2", "8912cec4f8064c4b57c14750a4e7bd86": "N = Q-1", "8912d5247d4fd28a229f12dbde5659e1": "\\mathcal{L} = \\mathcal{L}\\big(\\varphi(x),\\partial_\\mu\\varphi(x)\\big).", "8912f684c5f116306bc5b03d9fc678f1": "\\begin{array} {l}\n\\frac{\\partial ^2 f}{\\partial x^2}=\n\\frac{f\\left(x + \\Delta x,y\\right) + f\\left(x - \\Delta x,y\\right) - 2f(x,y)}{\\Delta x^2} - 2\\frac{f^{(4)}(x,y)}{4!}\\Delta x^2 + \\cdots\n\\end{array}", "891306333ec2f34be482d00bbba31690": "\n\\begin{align}\n\\sum_{n=1}^N \\sin n\\theta & = \\frac{1}{2}\\cot\\frac{\\theta}{2}-\\frac{\\cos(N+\\frac{1}{2})\\theta}{2\\sin\\frac{1}{2}\\theta}\\\\\n\\sum_{n=1}^N \\cos n\\theta & = -\\frac{1}{2}+\\frac{\\sin(N+\\frac{1}{2})\\theta}{2\\sin\\frac{1}{2}\\theta}\n\\end{align}\n", "89135ae0eb5bcc1af22e8a6d910c39d8": "\\mathcal V", "89136f4b35f3611b31e7b9d45227a605": " \\mathbf{L}^\\mathrm{T} \\mathbf{F} \\mathbf{L} =\\boldsymbol{\\Phi}\n\\quad \\mathrm{and}\\quad \\mathbf{L}^\\mathrm{T} \\mathbf{G}^{-1} \\mathbf{L} = \\mathbf{E},\n", "891375b994c155bd003a60668930b526": "U_s(|s\\rang-\\frac{2}{\\sqrt{N}}|\\omega\\rangle) = (2 |s\\rang \\lang s| - I)(|s\\rang-\\frac{2}{\\sqrt{N}}|\\omega\\rangle)=2 |s\\rang \\lang s|s\\rang-|s\\rang-\\frac{4}{\\sqrt{N}}|s\\rang \\langle s|\\omega\\rang+\\frac{2}{\\sqrt{N}}|\\omega\\rang=", "891378f35e6a68544be4fabd772240aa": "g^r ~\\bmod~ p", "89138102d1d02d5917dad65a5bd651fd": "N_i\\,", "89138364bc32f400f6214c5f85aaa29c": "\\lim_{n\\to\\infty} diam(C_n)\\rightarrow d(x,x')>0", "8913d79406e5169dbaf3ab934827d957": "(i-1,j-1,y_3)", "8913f221fff811756ac0bd11e02ad0cd": "r_\\mathit{inner} = \\frac{r_{s} + \\sqrt{r_{s}^{2} - 4\\alpha^{2}}}{2}", "891442075431ccbfe5c98def08b7b5fc": "s[k]", "89150405e766d197943f85e0e849778e": "\\mathit{succ} : \\mathrm{N} \\rightarrow \\mathrm{N}", "8915386a85e286d45be6dcc7906fd936": "(\\gamma,\\delta)", "891552dd4ff360d93c2322c855bbc412": "[J_i,J_j]=i \\epsilon_{ijk} J_k,\\,\\!", "891569e11e82012327be2a88e44bdda5": "\n\\text{volume}(\\Omega) = \\int_\\Omega \\nabla\\cdot\\vec F d\\Omega = \\oint_S \\vec F \\cdot \\hat n dS.\n", "89159ff035ed4006b64e0e5079d983d5": " X + A X = B ", "8915dd6df38a39fb3c7eb0fb9e519848": "\\int_R d^3 \\mathbf{r} f(\\mathbf{r})\\delta(\\mathbf{r} - \\mathbf{r}_0) = f(\\mathbf{r}_0)", "8915e519cca691d5fc78a6ca84dbd97c": "0.61\\lambda/NA,", "89160cb3116c69036b2e4d315e2b2d63": " {\\rm ad} ", "89168332d539c69187b5c12ccb92e629": "x\\rightarrow t,\\quad y\\rightarrow q,\\quad y'\\rightarrow \\dot{q},\\quad F\\rightarrow L,\\quad J\\rightarrow\\mathcal{S}", "8916a630398017091ae4e4c683e4f97a": "p(m) = 1- e^{m/V},", "8916dc0aa34b1137af273064ed586520": " g_i \\in K_i ", "8917291714057e4cd3894dfa9a254710": "\\Theta^{ij}_{\\alpha\\gamma}=\\sum\\limits_{\\beta=1}^{\\chi}\\sum\\limits_{m,n=1}^{M}V^{ij}_{mn}\\Gamma^{[C]m}_{\\alpha\\beta}\\lambda_{\\beta}\\Gamma^{[D]n}_{\\beta\\gamma}.", "891747eb449e894237fb7f5dab103fc9": "k_r^+=k_r", "89176c08d0f775b764aedd58f3980ff6": "\\arcsin (1/x) \\,= \\arccsc x \\,", "8917792739b053524c9062c773f5762b": "n_1\\times n_1", "8917be4b19cfd6b7bc8104ae739baa2a": " \\overline{7}=7", "8917ec82f77abe32398d31d06ae3943e": "\n x( t ) = \\sum_{n=1}^{ \\infty }\n\\left[\n\\lim_{ r \\to 0^+ } \\left(\n {\\frac{ t^{ \\frac{ 2 }{ 3 } n }}{ n! }}\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{ d } r ^{\\,n-1}} \\! \\left(\n r^n \\left( \\frac{ 3 }{ 2 } \\Big( \\sin^{-1}( \\sqrt{ r } ) - \\sqrt{ r - r^2 } \\Big)\n \\right)^{ \\! -\\frac{2}{3} n }\n \\right) \\right)\n \\right] ", "891816094fce6d6aa0022380bae89535": "\\deg(P \\circ Q) = \\deg(P)\\deg(Q)", "891817e0db4d51c9a9759bc6cbeed780": "-\\frac{\\varepsilon_0}{q}\\nabla^2", "891869b8fac15d3c00e6e8509d49e8ab": "\\tilde{A} = ( 1 + \\tilde{C} ) ( 1 - \\tilde{C} )^{-1} ", "8918b6c52186ea8efec8e3a4531488c3": " l_3= a_{00} - \\mathcal{L}(p_9)+p_3p_9, ", "8919964edd166a997db18e3691fea037": "x_i = \\alpha \\cosh(t/\\alpha) z_i, \\qquad 1 \\leq i \\leq n", "89199c8b06dba140f27743b143f3742c": "\\scriptstyle \\hbar^2 l(l+1)/ 2m r^2", "89199d5e16212eaa7988b3fab3ea5cc1": "m_i = ", "8919a4aa0f593fdf763b31e87bb8ec1a": "\nU=\n\\begin{bmatrix}\n\\cos\\theta & \\sin\\theta & 0 \\\\\n-\\sin\\theta/\\sqrt{2} & \\cos\\theta/\\sqrt{2} & \\frac{1}{\\sqrt{2}} \\\\ \n\\sin\\theta/\\sqrt{2} & -\\cos\\theta/\\sqrt{2} & \\frac{1}{\\sqrt{2}} \n\\end{bmatrix}.\n", "8919b1442aaf51f5bb63776efca10599": "\\arccos\\left(\\frac{2-n}{n}\\right)", "8919bb8a74eb3385aaa640a8e6f6bc09": "\\langle G',S' \\rangle", "8919c25e97bb0816c48578f9c16b565b": "0\\rightarrow B\\rightarrow E\\rightarrow A\\rightarrow 0", "8919eab97bb944073bffcd23d58285f5": "\\begin{align}\n \\mathopen{}\\left(\\Delta A_\\rho - \\frac{A_\\rho}{\\rho^2} - \\frac{2}{\\rho^2} \\frac{\\partial A_\\phi}{\\partial \\phi}\\right)\\mathclose{} &\\hat{\\boldsymbol\\rho} \\\\\n+ \\mathopen{}\\left(\\Delta A_\\phi - \\frac{A_\\phi}{\\rho^2} + \\frac{2}{\\rho^2} \\frac{\\partial A_\\rho}{\\partial \\phi}\\right)\\mathclose{} &\\hat{\\boldsymbol\\phi} \\\\\n+ \\Delta A_z &\\hat{\\mathbf z}\n\\end{align}", "891a117427fc732aa2a4ea36ba6b3a5b": "p_3(n)", "891a8d8b447c159257d5274105d5fc2a": "\\nabla\\cdot(\\nabla\\times\\mathbf{B})=0", "891a98cbb6b6e0573c3cf39767087554": "C\\ell_{p+8,q}(\\mathbf{R}) = C\\ell_{p+4,q+4}(\\mathbf{R}) = M_{2^4}(C\\ell_{p,q}(\\mathbf{R})) .", "891ac2f3cd54a3938ad90790f4c0d280": "\\varphi(x) \\in L^p[0,1]", "891b32e3038a3e275403e5181c2fbacb": "\n\\frac{\\partial \\vec{u}}{\\partial t}\n+ (\\vec{u} \\cdot \\vec{\\nabla}) \\vec{u}\n= - \\frac{1}{\\rho} \\vec{\\nabla}P\n+ \\nu \\nabla^2 \\vec{u}\n- 2\\vec{\\Omega} \\times \\vec{u}.\n", "891b5915df5a4a3443dc59d821d1ffc6": "X \\mathbf{\\operatorname{f}} Y", "891b8ebb7e0b873363571058e32f85dd": "\\scriptstyle y \\in R", "891b99b9f32a9e06b584c129d4f8fa1f": "Q^\\mathrm{T} Q = I \\,\\!", "891bf84814c84ad36d6f3febdd0fd62c": "\\rho(e)", "891c2782b18f3e6a92f1b2e6cac0e019": "\\langle\\gamma\\rangle", "891c96862489b681c4fc743d19c82865": "X=TP^{'}+E \\,", "891d43bf481a4235451c0e9e21d113ef": "H = - \\sum f_i \\log f_i ", "891d4ab97025df1db701b45e0bb1fc5d": "\\lim_{R \\rightarrow \\infty} \\int_{|\\xi| \\leq R} \\hat{f}(\\xi) e^{2 \\pi i x \\xi} \\, d\\xi = f(x)", "891d8dd4eaa6a5019dd41b75d04820fb": "C_{in} = [q] \\rightarrow \\{0,1\\}^q", "891dde35e444dad7ddefb2a71bae7794": "\\int |f(x)|^2 {\\rm d}x ", "891e0183c31b50f3f466f6a47f1d7fbd": "\\mu(w)= 2^{-2|w|-1}", "891e2315db30ac4fbaad84b04c60db30": "x \\ge 1", "891e3423923199af218e0a43b08c4a16": " I_{xy} =\\int\\int xy \\, dm ", "891e4179e926b61339e0c25f9925525a": " |S(\\rho_A)\\,-\\,S(\\rho_B)|\\,\\leq \\, S(\\rho_{AB}) \\, \\leq \\, S(\\rho_A)\\,+\\,S(\\rho_B) ~. ", "891e58c9a600b8a68c735ddc7df20d4f": "\\langle d,\\varphi \\rangle", "891e60cdd17c8050fa9b534942f54fc3": "\\beta = \\min(\\theta_i, \\theta_r)", "891e6b6c45b14a6f5ff73a47d20a4c4b": "\\{ sI_n | s \\in F\\setminus\\{0\\} \\}", "891e93c1af73e03611450dfef5531eba": "\\frac{1}{1-iZ_o / \\Omega}", "891ee2991250f72210d74df6447b0c14": " D_\\text{wave} = \\frac{64 V^2}{\\pi L^4} \\rho U^2 ", "891f0621871d55f2ec52215a4755f5f7": " U(\\infty)={\\mathbb C}.", "891f3004c08b06587307a6ceec1edb22": "\\{p_{n}\\}", "891f51a8ece6e48c1e6903f4dc937f14": " \\frac{ 2}{ a+b }", "891f6d25ef5148cf0757e2169dea5277": "\\Psi(0)=0", "891f82dcae80ca1860c97de3e0b4465d": "\\delta(\\beta) < \\delta(a_{t,j_t})", "891ff39ab2d8269c65e4b27b1ea32b6f": " (1-\\epsilon)(k\\ln k)^{2-\\epsilon}<\\ln A(k)<(1+\\epsilon)(k\\ln k)^{2+\\epsilon} ", "8920046e696ab1d537bd0a193a3b454d": " \\begin{align} y_1 + y_2 & = A \\sin \\left ( k_1 x - \\omega_1 t \\right ) \\\\\n& + A \\sin \\left ( k_2 x + \\omega_2 t \\right ) \n\\end{align}\\,\\!", "89200b27d0cf2633d7b442cdbdebeae1": "\\bold{P} = \\frac{\\partial L}{\\partial \\dot{\\mathbf{r}} }", "892051a82ed27b93ebd1973313e88551": "\\prod R_i", "8920ccb1ca45fb3f88ea11ad7937bd0e": "If: {\\delta} = \\frac{d_B}{D_B} \\qquad Then:d_B = \\delta \\cdot D_B \\quad And: R_B ={\\left ( {\\frac {\\delta \\cdot D_B}{2}} \\right )}", "8920d47ef92cadd24543efcb4a6713e5": "\nF(Z,T) = \\frac{2 (1+S)}{\\Gamma(1+2S)^2} (2 p \\rho)^{2S-2} e^{\\pi \\eta} |\\Gamma(S+i \\eta)|^2,\n", "8920f58601066c2d0aa117e140f88daf": "E^{2}=p^{2} c^{2} + m^{2} c^{4}.", "892111585e336e590cc0a138af6a4752": "\\mathrm{P}(A|BC) = \\frac{\\frac{4}{40}}{\\frac{4}{40} + \\frac{1}{40}} = \\tfrac{4}{5} \\ne \\mathrm{P}(A)", "89211c9ad523ad807064b403acfb30ab": "\nWg(d,\\sigma) = \\frac{1}{q!^2}\\sum_{\\lambda}\\frac{\\chi^\\lambda(1)^2\\chi^\\lambda(\\sigma)}{s_{\\lambda,d}(1)}\n", "89212a2a0097c2a7465121193e68ed98": "0\\le C_{xy}\\le 1", "89215d06e399e113483455ca6cac12ff": "e-\\operatorname{cr}(G) \\leq 3n-6", "89216e885659c5c38efd1207b637903c": "{\\mathrm S\\mathrm O}(n)", "89217c3227b257a13ee29c50109f1480": "\\tan A = \\frac{a}{b}", "89225e7ef8621bc05f01c00e928ccfcc": "\\mathbf{E}^{-1}\\mathbf{A}(\\mathbf{E}^{-1})^\\mathrm{T}\\mathbf{\\hat{x}}=\\mathbf{E}^{-1}\\mathbf{b}", "89227b5d81750d37004d4d952f1c9bc5": "\\frac{\\partial u}{\\partial n}=h\\left( x,y \\right),\\ \\ h\\left( x,y \\right)\\in \\partial \\Omega_N,", "8922c564c6599bbe9d626fec6cc271e7": "T_{\\mathrm {1e}}", "8922cd9005a88e624b0d61c20a81f730": "\\begin{cases}\nT &= \\gamma \\left( t - \\frac{v x}{c^{2}} \\right) \\\\ \nX &= \\gamma \\left( x - v t \\right)\n\\end{cases}", "89233d68f998fef36ebe4093e243a0d0": "u(0)=u_0", "89235bb69fb1aea476f149e34b1eb935": " 4pq ", "8923a54c98fc1d0e617d9949889fa54b": "\\kappa_p", "8923fdc376f23341a67e3e178da95693": "\\Gamma_r", "89240b42c11d575e0ec425d2375416e0": "\\mathbf{A = L D L}^{*}", "892487e228342916b7a1a40f779b3d8e": " p^v_m = \\frac{1}{m! \\Upsilon_v} \\int \\left [ \\prod_{i = 1}^m \\psi^v_i \\right ] \\varrho^{(m)}_{G,1...m}(\\chi^v) d\\boldsymbol{r}_1...d\\boldsymbol{r}_m, ", "892488393350d24a4525a39232560820": "\\bar{x} \\pm 2 s", "89251f2aebe24ccede48caefc43db390": " \\lambda_n(t) = {n \\choose 2} \\frac{1}{N_0 e^{-rt}}", "892528c7ec26331d1ae50660f7faabf2": "\\left\\langle \\eta_{i}\\left( t\\right)\\eta_{j}\\left( t^{\\prime}\\right) \\right\\rangle =2\\lambda k_{B}T\\delta _{i,j}\\delta \\left(t-t^{\\prime }\\right) ,", "89257b3d963879fc4c918bef2dd0a849": "v_e = \\frac{2 \\cdot \\pi \\cdot 1\\,AE}{365,25\\,d} = 29,78\\,km/s ", "892592c4ed6faf3c2ca5411365a9b916": " A_n = \\left | A_n \\right | e^{i \\left ( \\mathbf{k}_\\mathrm{n}\\cdot\\mathbf{r} - \\omega_n t + \\phi_n \\right )} \\,\\!", "89259ce260881c0acd730b578a290675": "\\scriptstyle{R_D}", "8925b66a8b84a1bb9c16f594f5e159e6": "{\\partial \\operatorname{Li}_s(e^\\mu) \\over \\partial \\mu} = \\operatorname{Li}_{s-1}(e^\\mu) \\,.", "8925b9d2023bbd540ded18c0141db7d5": "Rj_!j^! \\to 1 \\to Ri_*i^* \\to Rj_!j^![1],", "8925c5825ef10dfd10cf20ef141e3ede": "|\\phi\\rang+e^{i\\theta}|\\psi\\rang", "8925cc353ffa2603d5fb6207bb5309f0": "{\\tilde{D}}_4", "89263760c4349b38d5d03437b8747ed8": "D_{4d}", "89264b18dbfbf45e3eae084550107f68": "\\sin(\\theta) \\approx \\pi - \\theta", "8926674f6327d6c379af795ce5e50221": "A_0[x_0, \\dots, x_n], \\operatorname{deg}x_i = d_i, A_0", "8926a01289703763caf8c6afd2289b89": "\\textstyle A^{-1}B", "8926d6c6d90e5e876135d7d366215475": "\nP_{ij} = {2 \\over 3} (E_i -E_j)^4 |X_{ij}|^2\n\\, .", "8926fa70a1c6be81014f1e996b340c0b": "\\overset{\\textstyle v_1}{\\underset{\\textstyle i_1}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoondown}} \n\\stackrel{\\textstyle\\stackrel{\\textstyle R}{\\upharpoonright}}{1}\n\\overset{\\textstyle v_2}{\\underset{\\textstyle i_2}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightharpoondown}}\n", "89274c6d14b6fb9bc6c72b2302432165": "+ 7 \\cdot 9^7 + 7 \\cdot 9^6 + 7 \\cdot 9^5 + 7 \\cdot 9^4 ", "89275d80ff36806a9c8f5220b18219e4": "m = s", "89278a291736c22807ad51df17d22970": "\\lim_{t\\rightarrow \\infty}P(M_t > x)=\\exp(2\\mu x/\\sigma^2 ), x \\geq 0; ", "89279718b9d1a0afcc70d2660c342037": "\\ln C_{\\mathrm{Artin}} = - \\sum_{n=2}^{\\infty} \\frac{(L_n-1)P(n)}{n}", "8927cf60fdc761fe74aeff635ff36a08": "\\begin{alignat}{2}\nij & = k, & ji & = -k, \\\\\njk & = i, & kj & = -i, \\\\\nki & = j, & ik & = -j. \n\\end{alignat}", "89280933ea8c2a152b791f19780a55b6": "B\\cap C=\\emptyset", "8928115c6d52d8e78ac50b5acb97779b": "A:=e^*T^tG=\\bigcup_{p\\in M} T_pM=TM", "89281bf42bc124884905e1d2995d4a68": "c \\in C \\implies wt(c) \\geq 2(1-\\varepsilon)\\gamma n\\,", "892821cbaa1123b20fc52709d8fd72eb": " \\Pr(X=1) = 1 - \\Pr(X=0) = 1 - q = p.\\!", "892897ee69281cb2c09e427f3a3e051d": "W_h=N_h/N", "8928e225ae711b4b6df05963c66ecae2": "S=-\\sum_{i\\in I} p_i\\ln p_i", "8928ec373d717568accc8f61cbfa0fa8": "ln\\left ( \\frac{A_{final}}{A_{initial}} \\right )=\\left ( \\frac{-{\\pi}t}{Q} \\right )f +ln(RG)", "89296a663d3fedd800e4cfac292be03e": "\\beta'_k \\sim \\mathrm{Beta}\\left(1,\\alpha\\right)", "8929b16980a2d49747e3183f9be55940": "(\\forall x)", "8929e8c602243c270d89df2ffe6c704f": "\\sqrt{\\Omega}\\!", "892a211611186d4c892b8043fb0b6066": "\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}=\n\\begin{bmatrix}\n0.7990&0.4194&-0.1648\\\\\n-0.4493&1.3265&0.0927\\\\\n-0.1149&0.3394&0.7170\n\\end{bmatrix}\n\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}\n", "892a598dc9412b2bdde984a9ad96f617": " 0 \\rightarrow H_i(X, \\mathbf{Z})\\otimes A\\rightarrow H_i(X,A)\\rightarrow\\mbox{Tor}(H_{i-1}(X, \\mathbf{Z}),A)\\rightarrow 0.", "892a7bbd4484b83bd725759e6ae06a03": "2 \\otimes_M \\mathbf{v}_m", "892a89115e898fbdb56568db2078f0ab": "\\exp[-(1+D/(D^2-4F)^{1/2})z/2]w(z)", "892a8ad6d8cc50cf99c75fd947d3f2bc": "\\omega_1=e^{i\\frac{2\\pi}{3}}=-\\tfrac{1}{2} + \\tfrac{\\sqrt{3}}{2}i.", "892aeb555729fdf23fc9ffedae9f5990": " \\mathrm{Rot}(\\theta) \\, \\mathrm{Rot}(\\phi) = \\mathrm{Rot}(\\theta + \\phi), \\ ", "892af04fd5092791613de3f23258ce64": "f_x(a,b)=e^x\\log(1+y)\\bigg|_{(x,y)=(0,0)}=0\\,,", "892b0bc06511a15bed3007fda2e443b9": "\\mathbf{\\hat{n}}=n_x\\mathbf{\\hat{x}}+n_y\\mathbf{\\hat{y}}", "892bbdceb61d03b844d259d19e3f78f5": "E_h \\approx 27.2", "892bc6e6d37ee4df0ff5898e453f19a5": "Q(x,y) = (y - 4x^2) (y + 6x^2)", "892bdb93ef16ba5833ae3a03130c0b6c": "(e,\\omega)", "892c07306884b38802a78c6904322522": " \\varphi(a_m X^m + a_{m - 1} X^{m - 1} + \\cdots + a_1 X + a_0) = \na_m \\theta^m + a_{m - 1} \\theta^{m - 1} + \\cdots + a_1 \\theta + a_0.", "892cbd318609092f3a00aea7725f0722": "f(y; \\theta)", "892cfd0f9ce3c50f5861851fecd801de": "X_n = 0,", "892d3bf15bf98237e9042dfec2141dff": " x^{\\frac{\\log(\\log(x))}{\\log(x)}} = \\log(x) ", "892d818ead8190abe10c9cb6c601614a": "t_i : U_i \\to G\\,", "892dad4243bce60db094740ad83fd9e7": "\\Bbb Z[\\Bbb Z] \\equiv \\Bbb Z[t^\\pm]", "892e1bbc21fd76461950e7f28c7414f1": "\\delta(q,a,q^\\prime)=0", "892e2e7aa307af499c05c7dd4d448343": "\nRi = \\frac{g\\Delta \\theta _{v}/\\theta _{v}}{[(\\Delta U)^2 + (\\Delta V)^{2}]/\\Delta Z}\n", "892e76c845d0a7e377ddcd87adfb20ad": "K = \\tfrac{1}{4}(a^2-b^2-c^2+d^2)\\tan{A}.", "892ee4378b998408a185ff34f86208e0": "(\\frac {dx}{dt})^2 + (\\frac {dy}{dt})^2 + (\\frac {dz}{dt})^2 = 1", "892f2c0b01bf6bec5a06521558c87b19": "F = \\emptyset", "892f30aacae940361207e1b5496f6786": "\\dot a", "892f56385188dbc21d9124bf16a66ba5": "\\bar{y}\\,\\!", "892f81319cb2d50161946183f9d46e3d": " J = (1 + u^2 + v^2)^4/81,\\ ", "892fbde897d2777477ecde9f89426cf2": "df(x) = f'(x) \\, dx", "892ff32806b08fffc06479703b8183bb": "\\Omega BQC", "89303e92b590b803ef5be97f99f25bbb": "\\sigma'_j=\\sigma_j\\sigma_{j-1} \\qquad j\\ge 2.", "89308e2aac71316db19a96912165a259": "f(\\boldsymbol\\chi,\\nu)", "8930fdded019f16e645d9472004312e9": "\\boldsymbol{\\hat \\varphi} = - \\sin \\varphi\\mathbf{\\hat{x}} + \\cos \\varphi\\mathbf{\\hat{y}}", "89310f161555fd5549757095e2050702": "A \\sim B", "89313ad9c0fae172380265c54a9442cd": "(X_{ij})_{i \\in n_j, j \\in m}", "8931680404a96bdf9def7bc1b14a4bf1": "\\sqrt[p]{\\sum_{i=1}^nw_ix_i^p}\\leq\\sqrt[q]{\\sum_{i=1}^nw_ix_i^q}", "8931aad31dded12e00c5f548452f564a": "m_{\\mathrm{r}}", "8931b800c6b0f70003008fe5f6f4784c": "[x, y^{-1}, z]^y\\cdot[y, z^{-1}, x]^z\\cdot[z, x^{-1}, y]^x = 1", "8931e102c8152326e4458362ccda5813": "S = k_B \\ln\\Omega = k_B\\oint\\limits_{H(\\vec p, \\vec q)=E} (d\\vec p\\,)^n d\\vec p\\,)^n", "893258f1150df41c4b3138eca598705b": "L=\\lambda W", "89328905d9881cc85e44787c8b40b921": "2x^4+3x^3-4x^2+5x-6", "8932c64c98386e3c643c5e6b7e299744": "x_i^q", "893327eece6ed5f2ab90486a8a804c40": "f({{v}_{1}},{{v}_{2}})", "89333213285dd2ef0a3c9e3796c992dd": "\\eta = {E_x \\over H_y}", "893395189cda420221f5db69bf8e89c1": " \\mathbf{k}_0 = k \\left( \\cos{\\theta_0} \\mathbf{\\hat{x}} + \\sin{\\theta_0} \\mathbf{\\hat{z}} \\right) ", "8933f124185e0eb611073b1c802d849a": "0 \\leq r_1 < m_1", "8933fce11a2d04e1400e749b44dce205": "X_A", "89345f9930d2eca0fd286fac05de6d39": "\n\\begin{array}{cccc}\nX_{11} \\\\\nX_{21} & X_{22} \\\\\nX_{31} & X_{32} & X_{33} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{array}\n", "893477f31c03e63902c03e170cfdb9a1": "f_k (g_i)", "8934a2409e454c18d9854a92e43e9c37": "[a]p \\equiv \\neg\\langle a \\rangle \\neg p\\,\\!", "8934af4cb1db5fe421cca1ac78016cf9": "\n -10 - (1)(15) + R_a + R_b + R_c = 0 \n ", "8934da601be45e970847ee1e3aa84d76": "\\frac{\\partial^2}{\\partial x^2}", "893512864fb2fc3ecdb9401ec5b292e0": "\n\\Sigma_{i=1}^n k_i \\sqrt[r_i]{x_i},\n", "89351b87b91b3f0ea32250e9d62f759c": "A\\big\\| f \\big \\|^2 \\leq \\sum_{i \\in \\mathcal{I}} v_i^2 \\big\\| \\pi_{w_i}f \\big \\|^2 \\leq B\\big\\| f \\big \\|^2", "893548bb9473f5505e76560f542dbe6a": "\\arccos\\left(\\tfrac{-1}{n}\\right)", "893575e617cf2f9641ba10a46d84ee07": "((((A\\to B)\\to(C\\to\\bot))\\to D)\\to E)\\to((E\\to A)\\to(C\\to A))", "89359dedd2737c9352c958ab6bd7ba2e": "Z=1", "8935cf624b74cd0589cf468f10cd1d46": "f(\\theta) = \\sum_{k=1}^{\\infty}\\beta_k\\cdot\\delta_{\\theta_k}(\\theta)", "8936199275e3687edbda4e07f37f040f": " u_j^{n+1} ", "8936e5a463e47c3b68bfbb7381b7316c": "\\Vert f\\Vert^2_{L^2}= \\int_{-\\infty}^\\infty \\vert f(x)\\vert^2 dx", "8936e9bc2066fb61a58410b77f8c4b30": " \\widehat{y_i}", "8936ecf44b963bdf97246fe4ba52c731": "X^*(s) = \\sum_{k=0}^\\infty x(kT) e^{-kTs}", "893726ebece1e8f60e652132f38d75ff": "\\log_2 (1+|h|^2 SNR)", "8937473204d5d2d6fe9a7871bc86388b": "w_c=", "8937f7c7be72f04c5bda04f543b06753": "KP(x) \\leq \\ell(\\overline{x})+338", "89384b40c8d8c3d7efb13a4d054b8d92": "T\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}", "8938a9bdf85243a76957e3822bc6d2a8": " \\frac{dP}{ds} = n \\nabla n, \\,", "8938cf8aa4150437e0842dc5912a452e": "\\dfrac{\\left(k!\\right)^{k^{n-1}}}{k^n}", "89390d45eb59641588cc2ff54adb0779": "\\! 0.5 \\leq f \\leq 0.8 ", "89392c50c8f2723be1fa3feef89a48ee": "5! = 5 \\times 4 \\times 3 \\times 2 \\times 1 = 120. \\ ", "893950886604f6e51c819b405bb53620": "{}^\\mathrm N \\mathbf v^\\mathrm R = {}^\\mathrm E \\mathbf v^\\mathrm R + {}^\\mathrm N \\mathbf \\omega^\\mathrm E \\times \\mathbf r^\\mathrm R", "89396af0a003a9ab486fc6819555ea11": " C < N ", "89397b3e6314760e16165e302ea5fa72": "p(N) \\ne M", "893988d479ce1dce24ff1a9ef728ec9d": "\\frac {\\partial g}{\\partial y}(x,y)=0.", "89399236a4a33cf5a08d732ccdae3bd4": "\\Delta S^\\ddagger", "8939a32fb36ea398e4fd890223ea2e3c": "{\\mathcal{I}}_{a, a},{\\mathcal{I}}_{c, c},{\\mathcal{I}}_{\\alpha, a},{\\mathcal{I}}_{\\beta, c}", "893a05aaf679a04cc6e8df8b3cb89ba8": "A_{n} = H_{n}^{T}B_{n}H_{n}", "893a7423fd11e1b0f638f2b12d4ee2aa": "\\Delta<< P", "893a78bdedf27d85ba87f6d4618e4bb8": "\\scriptstyle Z_{\\mathrm {iT}m}", "893ab9a73996b51f2771c7633db539b0": "r = \\Sigma_i r_i s_i", "893ad9b4fae1b67ad101150d7854d212": "\\overline{u_i}", "893af46d3ae18ed1e7e0cec403fdbaba": "\\cdots \\gamma\\right] \\leq e^{-\\Omega (\\gamma^2 (1-\\lambda) k)}.", "893e6e720765628949b3ee26d5474379": "\\sup(C\\cap \\alpha)=\\alpha\\ne0", "893ebfcdce357afa7e4ec741f3c80ae2": "\\scriptstyle a_{\\rm C}", "893ed5d3b43a378c807ecf9b82ffc277": "\\bigcup X \\subseteq X", "893ed9944517245d192678719b665f60": "s=(s_1,\\ldots,s_n)^T.", "893f2c69f7cd9415f5f835c0f82f37c0": " e^{j k f \\cos \\theta} \\,", "893f327cbd7eb56a085c27e16c858fa5": "p\\times p", "893f77dd71baef5870230d4d07ba8f5d": "\n\\frac{\n\\int{ e^{i \\int{ -\\frac{1}{2}f(x) \\cdot K(x,y) \\cdot f(y) dxdy} + \\int{ J(x) \\cdot f(x) dx} }[Df] }\n}\n{\n\\int{ e^{i \\int{ -\\frac{1}{2}f(x) \\cdot K(x,y) \\cdot f(y) dxdy} } [Df] }\n} \n=\ne^{i \\frac{1}{2}\\int{ J(x) \\cdot K^{-1}(x,y) \\cdot J(y) dxdy } }\n", "893fa48bcd268d48141ef25b0000ec90": "I_{\\mathrm{in}}=I_{\\mathrm{out}}=I", "893fed4b7a89d4dd86ac0ce9c7521813": "\\operatorname{traces}(Clock) = \\{\\langle\\rangle, \\langle tick \\rangle, \\langle tick,tick \\rangle, \\cdots \\} = \\{ tick \\}^*", "89400c1a055aac87f87c9363807458da": "\nF(k)=\\sqrt{2\\pi}\\sum_m i^m e^{i m\\theta_k}\n\\sum_t f_{mt} \\int_0^R r^m [1-(r/R)^2]^t J_m(kr)r\\operatorname{d}\\!r\n", "894034a729c1e62871c30e0102d7eb32": "\\theta \\in [0,\\pi] ", "8940561c918e03151e10455ae63d4eb3": "R=e(g,g)^{1/x}", "89406954b70a9e1a01697eb64e067c92": "\n\\nabla^2 \\mathbf{E} - \\frac{n^2}{c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{E}\n= 0.", "89406f5b2631c4f27c57a11434309de0": "\\operatorname{Li}_{-3}(z)=\\sum_{k=1}^\\infty k^3 z^k =\\frac{z(1+4z+z^2)}{(1-z)^4}\\,\\!", "8940c478b2763c94fd8883102c5355c5": "(H_t)_{t=0}^T", "894104e7c23c16c30e6897c3cbb60eb7": "s=0^n", "89414924522b8e76775748fe9c185ce4": "h(x_1,\\dots,x_{n+m-1})=f(x_1,\\dots,x_{n-1},g(x_n,\\dots,x_{n+m-1})),", "8941820f3c9f7c9d6455b1b389414ca8": " \n \\textbf{(4)} \\quad \\hat{\\textbf{x}}_{k} \\leftarrow \\hat{\\textbf{x}}_{k} + (\\alpha)\\ \\hat{\\textbf{r}}_{k} ", "8941a0b12f98f15444d295abee163e31": " {f (x^{*}) \\leq f (x) + \\epsilon, \\qquad \\forall{x}\\, \\in X} ", "8941bbc6d83a11779743a7e530817e12": " (j^{2}_{p}\\sigma)^{*} \\theta \\,", "8941d760980ac6bfca982ccb32f43676": "Q\\, ", "8941f9f95d81255aef7f851dfdd452d5": "a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}=\\frac{mv^2}{\\sigma^2+mv^2}\\bar{X_i}+\\left ( 1-\\frac{mv^2}{\\sigma^2+mv^2} \\right )\\mu=Z\\bar{X_i}+(1-Z)\\mu", "89428df936f7c4ec40085d3a5cd4b08e": "r=r(\\theta)", "8942957c1071b833be0a3cd7dbe362aa": "\\nu(W)=3", "89429ef7db78592bd557c81729ff3e09": "X = \\exp(Y)", "8942b7080fab496aa54e35b186111fde": "\\operatorname{tr}", "894358c2b7a851e051b103ec8262beb9": "W \\times V^* \\to \\operatorname{Hom}(V,W)", "89437fc4c82fa5ecee2df31db9b0e953": "7 \\over 10", "894380926fc806d1bf68ba2dad4efd97": "\\textstyle x^k - 1 = (x-1)(1 + x + \\ldots + x^{k-1})", "89439e24053dd5b1916b6fa6c6b86014": "\n\\Delta \\bar{e}\\ =\\ -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ \\left(-e_h \\hat{g}\\ +\\ e_g \\hat{h}\\right)\\ =\\ -2\\pi\\ \\frac {J_2}{\\mu\\ p^2} \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ \\hat{z}\\ \\times \\ \\bar{e}\n", "8943e5cb9e452234b947e8d33d185b77": "1.5 {\\rm\\ {\\mu}m}", "8943ee29aef0750fb15ed99b251db2a8": " R = \\sum_{k=1}^{M} -p_k \\cdot \\log_2\\left(p_k\\right) ", "89446ecb22acf6c07af190c4855fa8c9": "<2n", "8944f0705620f9bfa8acfd729708a144": "\\sigma_{ij} = -p\\delta_{ij}+\\epsilon_{ij}", "894517a3901d82c7a644d8ba6b9684f1": " (s,t_s,t_e) ", "89452c98e64a2875d37db07c4e3a5236": "B^{*} = B_{0} \\bigg(\\frac{1}{r_{n}} + 1 \\bigg)", "8945306c54ab328de76f792efc735a81": "239+169\\sqrt{2}=478.00209\\ldots", "8945555710d1b03717d6e84beae1873c": "\n\\left[ \\frac{\\partial}{\\partial t}, G \\star \\right] = \\left( \\frac{\\partial G}{\\partial \\Delta} \\star \\phi \\right) \\frac{\\partial \\Delta}{\\partial t}.\n", "89460371f1804dc295be5c4cf2d83367": "u_{k-2}", "89462c9480d2a23d2d64f329e9badfac": "y:B", "894679ae0f78b99bb2e4da220b48051c": " j=\\pm i q Z_{DP}\\cdot \\bigtriangledown u(r,t)\\sqrt{N_{q}+\\frac{1}{2}\\pm \\frac{1}{2}}\\delta _{k', k \\pm q} \\; \\; (14)", "8951b8aac3f397360db9d528cb341ff6": "\\bar{u}", "8951bc53be014c8de89a393eb04d553a": " E \\subset \\Sigma^* ", "8951d185360e9bda92eaad7abe5a09bc": " \\stackrel{\\triangledown}{\\mathbf A} = \\frac{D}{Dt} \\mathbf{A}-\\frac {\\dot \\epsilon} 2 \\begin{pmatrix} 4A_{11} & A_{12} & A_{13} \\\\ A_{12} & -2A_{22} & -2A_{23} \\\\ A_{13} & -2A_{23} & -2A_{33} \\end{pmatrix} ", "8951f82113514435ccc43c92ce278852": "= H_a \\left(j \\frac{2}{T} \\cdot \\tan \\left( \\omega T/2 \\right) \\right) \\ ", "8952a375bd06b0ec29334f69e39daeed": "E_{ii} x^k_1= k \\delta_{i1}x^k_1", "8952ddaf45e6980a97304ac2f57a64be": "a(1-e)", "89530e1f6e38ed1e404f94eb0cf9831f": "(x+3)^2 - 4 = 0.\\,\\!", "89539fa189d040a54aa99ce27cf2415b": " |\\delta| < |b_k - b_{k-1}| ", "8953acdd10a39ed51b0199a96ea90ba4": "A=B_0 \\cup B_1 \\wedge B_0 \\cap B_1 = \\varnothing \\wedge A \\not\\le_\\alpha B_i (i<2).", "8953e64efba9de4918cda22f4c037b8b": "\n \\varepsilon_{11} = \\cfrac{\\partial u_1}{\\partial x_1} ~;~~\n \\varepsilon_{12} = \\cfrac{1}{2}\\left[\\cfrac{\\partial u_{1}}{\\partial x_2} + \\cfrac{\\partial u_{2}}{\\partial x_1}\\right]~;~~\n \\varepsilon_{22} = \\cfrac{\\partial u_{2}}{\\partial x_2} \n ", "8953ecc0c9e498c179a5b81b6663026a": " \\mathbf{F} = q\\mathbf{E} ", "8954217b397e94acb9346aa1ca9583c1": "\\sum_{n=1}^{p} a_{n}^{+} = a_{\\sigma(1)} + \\cdots + a_{\\sigma(m_1)}, \\quad a_{\\sigma(j)} > 0, \\ \\ \\sigma(1) < \\ldots < \\sigma(m_1) = p.", "8954239a69629960b4c50b8f2f1cd199": "b_n=-\\sum_{d|n}\\frac{1}{d}=", "89543c9f1f8113555a21092b8a30bdb6": "\\text{Dividend Yield} + \\text{Growth} = \\text{Cost Of Equity}", "89546733381c2c93e009d4e339061828": "\\mathcal H=\\{e\\in\\mathcal E^\\bullet\\mid\\Delta e=0\\}.", "89546ec9b7dfbaf51076600ae98049bf": "\\ x^3 + d = bx^2", "8955446db6255c6fdc9b55a48e36376a": " F_\\mathrm{f} ", "89557f3e7c522cfae9280c4b9c440430": "\\Delta\\varphi_a^*=\\Delta\\varphi^*_b+2\\pi\\frac{\\Phi}{\\Phi_0}+2\\pi n.", "8955c28e3c823c8896d3c83efccec471": "\\gamma_{n,\\mathbf{H}}:=(E(\\gamma_{n,\\mathbf{H}})\\to\\mathbf{P}^n(\\mathbf{H})),", "8955cac974f3037e3d2232b361a8ac44": "c=m^2-n^2 \\, ", "895620879605ce2533c70ebbab617482": "a=b^l.", "89570ac2f3c8fb49d8774067c32b6556": "P(r) = \\frac{4 \\pi r^2}{(2/3\\; \\pi \\langle r^2\\rangle)^{3/2}} \\;e^{-\\,\\frac{3r^2}{2\\langle r^2\\rangle}}", "89570c0037f309c5dbf7cdc2a248edae": "\\theta(\\cdot \\cdot)", "89579227cb95ab3c74088cd81cff9294": "n_{0, t+ 1} = f_0n_{0, t} + \\cdots + f_{\\omega- 1}n_{\\omega - 1, t} = \\lambda n_{0, t}.", "8957a28394bfaadbb4cbd523033977fd": "\nQ(x) =\\frac{1}{2} - \\frac{1}{2} \\operatorname{erf} \\left( \\frac{x}{\\sqrt{2}} \\right)=\\frac{1}{2}\\operatorname{erfc}\\left(\\frac{x}{\\sqrt{2}}\\right).\n", "8957c6ca758f1d6adef60a5c5f0f1e40": "\\Bbb{R}((G))", "8958030e74e5bbe4f531d933f4777f4c": "f(z)=c_1z+c_0 + c_{-1}z^{-1} + \\dots, \\qquad c_1\\in\\mathbf{R}_+.", "895816861c543506ee03e47a93eddb6e": "\\ \\delta I_R", "895819b9e55703adb01bd57426ff42d6": "\\frac{X/x_n}{Y/y_n}-1=\\frac{X/x_n-Y/y_n}{Y/y_n}", "89586142ad2eda4802c526ee8df7912d": "\\Pr(\\overline X - \\mathrm{E}[\\overline X] \\geq t) \\leq \\exp \\left( - \\frac{2n^2t^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right),\\!", "8958d824b4574098c5694ce44b8f8c96": "\\zeta _{max} \\approx -6.02 \\cdot P\\;\\mbox{dBc}.", "8958ed3416881012d029ffd8aed2cad5": " y_s'(-2 \\cdot c) = -(1/2) \\cdot x |_{x = -2 \\cdot c} = c. \\,\\!", "89599b0b5580fcc1171559e1ab722c41": " X_k = \\sum_{n=0}^{N-1} x_n z^{nk}\n\\qquad\nk = 0,\\dots,M-1, ", "8959a9416103c38248192d53a6a74bab": "R_{i-1}(\\Delta t) < u R_N( \\Delta t ) \\leq R_i(\\Delta t)", "895a11d91bb31b664b271392c4842d29": "A_1,A_2,\\ldots A_m", "895a20d4b78183229cc578e8f5c8762f": "\\rho' = \\mathrm{Ad}_\\psi (\\rho)", "895a628fbb9b3d8237732ff72b284f5f": "n_1, n_2 \\rightarrow \\infty", "895ad02a8267c02ce2403e0a45e13f98": "\\iota^6-2\\iota^5+13\\iota^4-15\\iota^3+16\\iota^2+28\\iota+8 = 0", "895b306fcdfd2e21ad0d2d0696da1187": "\\mathcal{E}= - B \\ell v,", "895b45d286bd64d46fde1c7b129b7c0f": "Q(x) = \\sgn(x) \\cdot \\Delta \\cdot \\left\\lfloor \\frac{\\left| x \\right|}{\\Delta}+\\frac1{2}\\right\\rfloor", "895b9bb0b0a63e30e88437e60c1f7c13": "\\mbox{RowSize} = \\left\\lfloor\\frac { \\mbox{BitsPerPixel} \\cdot \\mbox{ImageWidth}+31\n }{32}\\right\\rfloor \\cdot 4,", "895bc523216231aa0f1395a6cdb8edae": "E_\\text{k} = \\int \\mathbf{v} \\cdot d \\mathbf{p}= \\int \\mathbf{v} \\cdot d (m \\gamma \\mathbf{v}) = m \\gamma \\mathbf{v} \\cdot \\mathbf{v} - \\int m \\gamma \\mathbf{v} \\cdot d \\mathbf{v} = m \\gamma v^2 - \\frac{m}{2} \\int \\gamma d (v^2)", "895bc6b6f418043190fe7ee4705a4514": "F_i=-k x_i \\,", "895be88126213511dd498dcbfc21a529": " \\int f \\, d \\mu = \\int g \\, d \\mu. ", "895c07624bfa5fbdfaf52cc1733538f7": " \\frac{d(m\\vec{V})}{dt} = \\sum{\\vec{\\mathrm{F}}_i} = 0 ", "895c0b17185c09ee4519ce01650ecc69": "F^{-1}(Y)", "895c1ca5625e6b920964be7096acd7d7": "r_\\mathrm{per}=(1-e)a\\!\\,", "895c770b977642fe0a02bd81c20b0a80": "\\tan\\frac E2 = \\frac\n{\\tan\\frac12a\\tan\\frac12b\\sin C}{1 + \\tan\\frac12a\\tan\\frac12b\\cos C}.", "895cf893f5e42e9ee8dead370854f01c": "\\forall j, 2\\leq j \\leq T ", "895d21d92537173ef86a6c9a9a9928be": " 1 + \\xi(x-\\mu)/\\sigma \\geqslant 0", "895d9a0dbd66bca504d7ff0de0688af2": "j^{\\alpha}_i", "895d9a81bd0b4417612f5743b178ce4b": " S:=\\operatorname{CompleteSystem}(S)", "895e0eda39095d36bb1322732f1881c1": "\\,w_i (n+1) ~ = ~ \\frac{w_i}{\\left( \\sum_j w_j^p \\right)^{1/p}} ~ + ~ \\eta \\left( \\frac{y x_i}{\\left(\\sum_j w_j^p \\right)^{1/p}} - \\frac{w_i \\sum_j y x_j w_j}{\\left(\\sum_j w_j^p \\right)^{(1 + 1/p)}} \\right) ~ + ~ O(\\eta^2)", "895e2f7d5a3e8fbb36e7d0e06a3c5b40": "f(x)=\\cos(2\\arccos(x))", "895eb82beb5dd5830d1e5fdd599086d6": "u=x^2", "895f5c4832056fdc000ba4acb7bb9a50": "j \\neq k", "895fe1f65376b6a1535edadf4de04bb8": "l \\prec \\partial\\{({I}^{2},{\\varphi}_{\\lambda},{S}_{\\lambda})\\}_{\\lambda\\in\\Lambda}", "89607134575748e32b6ecd13cc8b6759": "\n{\\mbox{KILL}}[s] ", "8960b785835b98c814c2b592167c6f19": "L_k(t)", "8960cd60ab84f038726ae4f92db8fb65": "P_c = N_0 R P_S P_t P_R P_g", "89613382e53b3329df4f0cd6802d5a7a": "\\mbox{Capital Employed} = \\mbox{Total Assets} - \\mbox{Current Liabilities}", "896157cbb4ea63a0329a298b62b6d836": "F_4", "89617aedd97c34364bbf253ecfa6dd10": " c_{0,j} ", "8961f693859700259bd1ba1d1bbabf2b": "\\operatorname{pos}(U \\cap V) \\leq \\min \\left( \\operatorname{pos}(U), \\operatorname{pos}(V) \\right)", "89620d2be91d35ce5f3885fe88368824": "R_{xx}=(2,9,14,9,2)", "896210fa842a15b1d4725adc66328160": "(\\forall x \\forall y \\, \\mathop{\\leq}(\\mathop{+}(x, y), z) \\to \\forall x\\, \\forall y\\, \\mathop{+}(x, y) = 0)", "89623505ea5df0398376005f88196916": "\n \\boldsymbol{R}^T~d\\mathbf{f} = (\\boldsymbol{P}^T\\cdot\\boldsymbol{R})^T\\cdot\\mathbf{n}_0~d\\Gamma_0\n ", "8962719d247170245a5fc83d9bd48e7e": " \\qquad z_{n+1} = z_n^2 ", "89636050d846d1c50fe2f5e86808cfc7": " \\int_0^1 fg \\geq \\int_0^1 f \\int_0^1 g,\\, ", "8963d3c40a758f932c5c26a585eb7eb6": " \\alpha = \\frac{e^2}{4 \\pi}. ", "89640b8ddf7c8434cb30fce04a7ff394": "m \\ ", "89643973bd6f1e58ba188c8b932000f4": "Z=X+Y", "8964fa9076451d488062837b12eea970": "\\Bbb{Z}/2^{n-1}\\Bbb{Z}", "89657c8d56d2b94fca2d99616e9e6761": "{\\operatorname{d} \\over \\operatorname{d}x} e^x = e^x", "8965c798ca6441b435cccafdd10129a8": " \\hat{\\mathbf{h}}(n+1) = \\hat{\\mathbf{h}}(n)+\\mu\\,e^{*}(n)\\mathbf{x}(n)", "8965dc0569d247e21f9534017d729f79": "\\overline{P}(Cl_2^{\\geq}) = \\{x_2,x_3,x_4,x_6,x_7,x_8,x_9,x_{10}\\}", "89662e6c795f6dc29c4e9ad031568da3": "a_x\\,dx + a_y\\,dy + a_z\\,dz", "896640fcac2e5645446a68739b21c2b4": "(x+az, y+bz, 0)", "89667c3fb656c656fc4ae677ea0fc669": "A \\xrightarrow{f} B \\to C(f) \\to A[1]", "89668481b70629b71e5f5f81fe9202da": "p(x_t, x_{t+\\tau_{1}}, x_{t+\\tau_{2}} .. x_{t+\\tau_{k}}) = p(x_{t'}, x_{t'-\\tau_{1}}, \nx_{t'-\\tau_{2}} .. x_{t'-\\tau_{k}})", "896685d02c858e0534bca7be6390af59": "\\tfrac{5}{12}", "8966b1266ec73bad952ee84d75755201": "\\partial_x\\, \\partial_y + a\\,\\partial_x + b\\,\\partial_y + c = \\left\\{\\begin{array}{c}\n(\\partial_x + b)(\\partial_y + a) - ab - a_x + c ,\\\\\n(\\partial_y + a)(\\partial_x + b) - ab - b_y + c .\n\\end{array}\\right.", "8966b1a8b6e8b5a3d87ad060d00ec0d9": "R = \\rho x / A \\,\\!", "8966b81105fe258980b15eab2c6a1aef": "D_X(f^!N) \\cong f^*(D_Y(N)),", "8966ee144c60856ca6800d81c30616ec": "G = f^{64}(4),\\text{ where }f(n) = 3 \\uparrow^n 3,", "8966fb3fd33765f5f51fc3c375a2ecd2": "\n P(d|q) = P(q|d)\n", "896718ce35578f2b28422c6006f8d963": "\n\\sum_{j=1}^2 \\epsilon_{\\mu}^j(\\mathbf{p}) \\epsilon_{\\nu}^{j*}(\\mathbf{p})\n== \\sum_{j==1}^2 \\epsilon_{\\mu}^{j*}(\\mathbf{p}) \\epsilon_{\\nu}^j(\\mathbf{p})\n= \\delta_{\\mu \\nu} - {p_{\\mu} p_{\\nu} \\over E^2}.\n", "8967473f0927d5a53c9738453629e6cf": "\\hat{I},\\hat{J}", "8967582f4fa0023e3970669220d7cbea": "p_\\sigma", "8967e0e0532eb2ccb8fdbd6612357db6": "(\\mathbf{e}_1 + 3\\mathbf{e}_2 +2\\mathbf{e}_3)\\wedge(4\\mathbf{e}_1-\\mathbf{e}_2+\\mathbf{e}_3)", "896832ebad69dff2cbfc99cc36f40338": "F \\to dFd (60.4%)| LS (39.6%)", "8968539b9a7a87f16ab6463830ff9f93": "\\exist x \\exist y Lyx", "8968789d994de66aa960caa018cc2ce8": "V_{L2}=V_o", "89687c4f28b56d99910aff2486da22f5": "L \\left( x, y \\right)", "89688de843173a843f18e10f38938fbb": " S = \\frac {\\Delta f_N} {f_N} ", "8968e5fe6025908f6319ecc3cf6f27ab": "-20 \\log {\\omega \\over {\\omega_\\mathrm{c}}}", "8968e6e8d7fb0af794fb2e4c620dadcf": "P_N = P_0(1+r)^N - c {{(1+r)^N - 1}\\over r}", "8969038a007726d2f555459756e9172c": "\\operatorname{Pr}(a_1,\\dots,a_n; \\theta)={n! \\over \\theta(\\theta+1)\\cdots(\\theta+n-1)}\\prod_{j=1}^n{\\theta^{a_j} \\over j^{a_j} a_j!},", "896948fa8f5e10be0e07cbe185314d23": "D_{\\infty h}", "896988541cc49716df87b702f23396d5": "Ps", "8969955b8c851f651c7c78b7533bc9ae": "I_{-\\frac{1}{2}} (z)= \\sqrt{\\frac{2}{\\pi z}}\\cosh(z) ;", "8969a23dbe9348cb4f5de520f5946fb6": "\n\\frac{1}{\\langle I_x^2 \\rangle \\langle I_y^2 \\rangle - \\langle I_x I_y \\rangle^2}\n\\begin{bmatrix}\n\\langle I_y^2 \\rangle & -\\langle I_x I_y \\rangle\\\\\n-\\langle I_x I_y \\rangle & \\langle I_x^2 \\rangle\n\\end{bmatrix}.\n", "8969acaa2d6cf6dd1329705b4e18b210": "\\lVert A_i x + b_i \\rVert_2 \\leq c_i^T x + d_i,\\quad i = 1,\\dots,m", "8969aee0d90a08a16239eaeae32e44d1": "B(S)", "8969b33cc66a89cef868b9bce8605dca": "1/{\\sqrt{N}}", "8969f711a3a09186f05f74e768279400": "\\beta = \\frac {v}{c}=\\sqrt{1-1/\\gamma^2},", "896ae13fd399536ed31a8ede338fa59a": "\\underline{\\int_{a}^{b}} f(x) \\, dx \\leq \\overline{\\int_{a}^{b}} f(x) \\, dx ", "896b03c3bea4ba452b5e9e452917c1c8": "T\\times T\\le G\\le T_0 {\\rm wr} Z/2Z", "896b4d2a01d6405d244d258abe2a6670": "\\, _2F_1(a,b;1+a+b-c;1-z)", "896b55cd7728c9985011b543a443c67f": "\\coprod_{i,j}U_i\\cap U_j{{{} \\atop \\longrightarrow}\\atop{\\longrightarrow \\atop {}}}\\coprod_iU_i\\rightarrow U", "896b6803cc982d23628e43578533322f": "\\rightarrow - - \\leftarrow", "896bac14fc269ccdffa69094ad9b8ca7": " \\ \\displaystyle \\varphi(q,\\alpha,u) \\ ", "896bc3f1ed6395d72f5c3f3ad80e3123": "\nG = \\frac{\\displaystyle\\max_{i=1,\\ldots, N}\\left \\vert Y_i - \\bar{Y}\\right\\vert}{s}\n", "896bc8adbfb943271931422d7a69dbc9": "\\sum_{i = 1}^\\omega \\frac{\\ell_ib_i}{\\lambda^i} = 1, ", "896bfac1134795e4f847fc637f5d788f": "G * H = \\langle x, y \\mid x^4 = y^5 = 1 \\rangle.", "896c046e5e3e8fb1c4718bd65c57eb7d": "\\phi^+", "896c646e176435a4a9680abcadc9b618": "\\overline{x}\\langle y \\rangle", "896cd9a61bc004fdabec55b11fa5dc48": "U = \\frac{3GM^2}{5r}", "896d06a4eda4aa8a02ca6415cd07ba96": " \\mathbf{r}_i = (\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{R}, \\quad \\mathbf{v}_i = \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{v}.", "896d1a09590af48c9adc5088b513ad63": "\\textstyle v^2 = (v_i + V)^2 = (\\mathbf{v}_i + \\mathbf{V}) \\cdot (\\mathbf{v}_i + \\mathbf{V}) = \\mathbf{v}_i \\cdot \\mathbf{v}_i + 2 \\mathbf{v}_i \\cdot \\mathbf{V} + \\mathbf{V} \\cdot \\mathbf{V} = v_i^2 + 2 \\mathbf{v}_i \\cdot \\mathbf{V} + V^2", "896d6fb4b971c4a17adb8e0617267cff": "\\dot{J}_{ab} = J_{an;b} \\, X^n - E[\\vec{X}]_{ab}", "896d7bdbccc1550372c6a9119ed8f605": "\\ddot{a}", "896d83b598ea7c3c987cc81709cf5fe4": "q^N < \\frac{\\varepsilon(1-q)}{d(x_1, x_0)}.", "896e9ce05f9ec41ccfd0f518df6f3a71": "O_n={n(2n^2 + 1) \\over 3}.", "896f4b15dfff2dfa39d8e16c740103e7": " u = i \\pi z", "896faa99602e5a7c0fad72b3ad997db0": "I_t = I_{t-1} \\times \\prod_{j = 1}^{N(t)} \\left( \\frac{e_{j,t}}{e_{j,t-1}} \\right)^{w_{j,t}}", "8970146fef700449bf2d04a4be22d8a8": "m_1 \\ ", "89702d79061a905c0c74c91d13d6d572": " \\frac{\\partial u}{\\partial t} = 6u\\frac{\\partial u}{\\partial x} - \\frac{\\partial^3 u}{\\partial x^3}. ", "897036f4f85391ff81b6091a3f3f45f3": "m_{\\tau} \\approx m_b", "89706a25536a5ab255d61e286701077c": "\\frac{\\partial y}{\\partial \\mathbf{n}}(\\mathbf{x}) = f(\\mathbf{x}) \\quad \\forall \\mathbf{x} \\in \\partial \\Omega.", "8970785fee77e4ab2e4da90f69673624": "K = \\sqrt{\\left(s-a\\right)\\left(s-b\\right)\\left(s-c\\right)\\left(s-d\\right)}.", "8970b51c5d59d0ebff7bb6a9034d15b2": "r^n \\equiv 1 \\pmod{B}", "89715b34cd02e8dfce5c88fdb9322ec5": "\\exists x G(x,y_1\\dots, y_n)", "89716486e49db9ff1a98ca633b8c13fd": "\\displaystyle{H=KAK,\\,\\,\\,H = K\\cdot \\exp \\mathfrak{m}}", "8971966feb770bcf0b8576795b59d99a": "\nf(x_1,x_2,...,x_{n-1},x_n) = \\sum_{i = 1}^{m} \\tfrac{1}{c_{i} + \\sum\\limits_{j = 1}^{n} (x_{j} - a_{ij})^2 }\n", "8971a25a2210661d7416c17d8c970832": "F(s) = \\mathcal{L}\\left\\{ f(t) \\right\\}", "897277295652d67d727445e03705366c": "c = \\frac{K_d^R}{K_d^T}", "897279981bb156bbac0a8b21aa844337": "g(x,u) \\le b + \\beta \\cdot dist(u,\\mathcal{N}) \\ , \\ \\forall u\\in U", "89727a69e29932ae6f6ca695235f4f8d": "\\displaystyle \\partial_t u + \\partial_x^3 u - 6\\, f(t)\\, u\\, \\partial_x u = 0", "8972a20eb0beda399e7da80adde40031": " \\chi(D) = \\chi(0) + \\frac{1}{2}(D.D - D.K) ", "8972b61090a69fb209c9128004d0348d": "(\\nabla u)(x(s)) = \\xi(x(s)).", "8972bb38d68680df176e26c682cdafa6": "\\,\\! \\left(\\frac{I_z^+}{I_u}-\\frac{I_z^-}{I_u}\\right)=\\left(\\frac{I_x^+}{I_u}-\\frac{I_x^-}{I_u}\\right)\\left(\\frac{I_y}{I_u}\\right)", "8972e6d34d81712dffe73b5d22934789": "{10}^{\\,\\! 4\\cdot 2^{30}}", "8973127b7c09dedca1351d3167b96980": "ds^2 = -dt^2 + \\alpha^2 \\cosh^2(t/\\alpha) d\\Omega_{n-1}^2.", "89735c3af695b5be10cfce04ccf9b2df": " H = T(t,v) + V(t,x). \\, ", "8973881c4ccfab003c68a622584f3aa8": "\\mathfrak{P} = \\{U_p(b)\\, |\\, p,b \\in \\mathbf{Z}^+, p \\text{ is prime}\\}", "897389829455eb52bd5e215149b52ece": "\\{F_{\\alpha} \\}", "897430fd2beabc8f6593eb8aff7f691b": "\\scriptstyle\\mathbb{Z}\\times\\mathbb{Z}", "89743e10a43054ae74e3efd5ace78138": "a_{33}=\\frac{2}{x_2-x_1}", "89744da6df5653ff44bd8aa45c243d85": "n^{g(n)}", "89745e042a2395252a4e70edbadd5bec": "\n\\varphi(x)=\\sqrt Z \\varphi_{\\mathrm{out}}(x) +\\int \\mathrm{d}^4y \\Delta_{\\mathrm{adv}}(x-y)j(y)\n", "897491922b39aa26859c54097412c58a": " p_i = 2^{i} a - 1 ", "8974f47c326450af47f4ea8fe9145a93": " \\lceil x \\rceil", "89758b5ac237e2b5279f72b7d99eec29": "a:A", "897595b543106b3794025a2da410cffc": "D = R C_h", "8975bbcad400f86d30e1e160ed537968": "w_{ij}\\ge 0", "8975d454d4065da06b724fccf14a5328": "v_1 ' + v_c", "8975f05c55cf0a084640b8eb84557d16": "R_0 = 1-\\log_2(1+2\\sqrt{p(1-p)})", "897619eea21496aee3d496725ac062dc": "F_{in} = \\sin \\gamma_{in} \\left[\\begin{array}{c}0 \\\\1 + r^{TE}_{10}\\textit{e}^{ i\\Phi_{in}} \\\\0\\end{array}\\right] + \\cos \\gamma _{in} \\left[\\begin{array}{c}\\cos \\theta ^{in}_{1}(1-r^{TM}_{10}\\textit{e}^{i\\Phi_{in}}) \\\\0 \\\\ \\sin \\theta ^{in}_{1}(1+r^{TM}_{10}\\textit{e}^{i\\Phi_{in}})\\end{array}\\right]\n", "89763adb26d25d06d7e9392b07110af8": "\\psi(\\Omega+2)", "89763f65afc6104563d7c450d19493fc": "\\chi_{v}", "89768062986e04b2a3d9dd4616b68ffe": "A(\\mathbf x) = \\sum_{i=1}^n x_i \\mathbf a_i.", "89770d586aff0eed559fb627280a7141": "(E,\\vec pc)", "8977739f8fda64344419640d9f4c60fd": "f_1+f_2", "8977d79a4da93b1547e2cb51d0f02f8a": "t \\equiv 1 \\pmod{13}", "89782097016c73398f730313c603ef28": "k'd=0, \\pi, 2\\pi, \\cdots\\,", "897836175d886fe2d043b19ea9fbc76b": "H^k(BO(k))", "8978a121dfdfd51e23524f689031beec": "\\operatorname{tr}(ABCD) = \\operatorname{tr}(BCDA) = \\operatorname{tr}(CDAB) = \\operatorname{tr}(DABC)", "89792914a9bf17f1383c349e6fb8cab0": "S_{kn}", "89792fdd0cbb6caf22f0cf83e9ef9421": "\\int_0^\\pi \\sin mx \\sin nx\\, dx=\\begin{cases}\n0 & \\text{if } m\\neq n \\\\ \n\\pi/2 & \\text{if } m=n \n\\end{cases}\n\\ \\ m,n \\text{ integers}", "8979740d5547f247b6fbe3c3f256f287": " f s^2 \\approx 447", "897977520a7d2180bf08e9c61befd1d8": " i \\hbar \\frac{d}{dt} | \\psi_{I} (t) \\rang = H_{1, I}(t) | \\psi_{I} (t) \\rang. ", "8979853083dfd2e0553c28a280b952d7": "\\,\\!GenerateCandidateSubspaces(S_k)", "8979acf69006c48c3286405908b848b1": "A=\\{0,1,2,3,\\ldots,\\aleph_0\\}", "8979b25b13a747d95f15d20cf3ae7d90": "f(t) = e^{i2\\pi\\xi_1 t}", "8979c395289d319f4d23434f47f2443d": " \\sigma \\le 10 | X | ", "8979d6362b5ce923b27158197d40479d": "\n\\begin{bmatrix}\n\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}&\n\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\\\\n\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}&\n\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 1 \\\\\n\\end{bmatrix} .\n", "897a79aca9d4d926da9ff2dc4a95d15a": " F_1 L = L ", "897a80006b937c5c46daa9504558ab96": "f\\chi_{\\{|f|>1\\}} \\in L^{p_0}", "897a9c2dc49c83435b705eb85a50b686": "y(1) = 1", "897ad29c67e86fe4381fe0d9fce58793": " \\Delta H_{ad} ", "897b097a0ff9ee6fe579c1d9129978ef": "f(AA)", "897b4164a5823d40b2d469e8db5bfe5c": " d= \\frac{1}{2} (f_A + f_B) ", "897b4b11aefe32d0e7597f39c7e50e39": "\\rho= z\\frac{d\\log(\\kappa)}{dz} =z-7z^2+58z^3-519z^4+4856z^5+\\cdots.", "897b6c7fed29221fb69093d77499b8d0": "{z}^{\\mathrm{T}}({Mz}+{q}) = 0\\,", "897b989fc5a515d2604fb6e3fc02edc8": "V(\\Gamma)", "897c37268a4a8a859b822c2418e6a86a": "\\left\\| Eu\\right\\|_{W^{1,p}(\\mathbb{R}^n)}\\leq C\\left\\|u\\right\\|_{W^{1,p}(\\Omega)}.", "897c4cb98431e82c4261caaff1283add": " f = f \\left ( t \\right ) ", "897c733da5e63b96c432978eab1583db": "\\omega^2+1", "897c960c2e7da8d43f43581cebd0a3b0": "\\theta(\\boldsymbol{x},t)\\,", "897cbcc209eeeb69f53d19345d39f79f": "\\displaystyle{Z=\\begin{pmatrix}A & B \\\\ B & D\\end{pmatrix}}", "897cd95a21511fd8b561dd8ab9cc13b6": "\\begin{align}\n\\boldsymbol{r}_i^\\mathrm{T}\\boldsymbol{r}_j&=0\\text{,}\\\\\n\\boldsymbol{p}_i^\\mathrm{T}\\boldsymbol{Ap}_j&=0\\text{.}\n\\end{align}", "897d110ebde33adf93886b5f20c0509a": " \\prod_{n=1}^{\\infty} \\frac{4n^2}{4n^2-1} = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\frac{8}{7} \\cdot \\frac{8}{9} \\cdots = \\frac{4}{3} \\cdot \\frac{16}{15} \\cdot \\frac{36}{35} \\cdot \\frac{64}{63} \\cdots = \\frac{\\pi}{2} \\!", "897ddfdf92de0687c7a19e244e4ee5bf": "a_{\\overline{n}|i} = \\frac{1-\\left(1+i\\right)^{-n}}{i},", "897dfff5922fc40ee50c83dd92f8817f": " \\begin{pmatrix} y_1 \\\\ y_2 \\\\ 1 \\end{pmatrix} = \\frac{f}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ \\frac{x_3}{f} \\end{pmatrix} \\sim \\begin{pmatrix} x_1 \\\\ x_2 \\\\ \\frac{x_3}{f} \\end{pmatrix} ", "897e1e556b154ca1dd5dbd7c4696727a": "(n \\times n)", "897e8f2aa290a7ea40beee39794c757f": "x_n=x(n\\,\\Delta t)", "897ec18af4b4369c4f359b373da8fbc1": "\nq_{xy} = \\frac{\\sum (x-\\bar{x})(y-\\bar{y}) w(x-\\bar{x},y-\\bar{y}) I(x,y)}{\\sum w(x-\\bar{x},y-\\bar{y}) I(x,y)}\n", "897f770701f4028f063e106af4ba2829": " V_S \\, ", "897f82f3c308672155e6e52c3abbebe9": "\\psi(x)=\\phi(x)\\qquad\\forall x\\in U,", "897f84127b2c2935652daa6669f65718": "P_{k_1,k_2,\\dots, k_m}=p_{k_1}\\smile p_{k_2}\\smile \\cdots\\smile p_{k_m}([M])", "897fdab24d23224b84fc36f39ee448e5": "|n_0,n_1,\\cdots,n_k\\rangle_\\nu, = |\\psi_0\\rangle^{n_0}|\\psi_1\\rangle^{n_1} \\cdots |\\psi_k\\rangle^{n_k}", "897ffa6dea45d6bc2353b3c790dacc0c": "x = \\left[ 2, 5 \\right]^\\mathsf{T} ", "898014e593f626aa90e3ef8be1351e64": "\\sigma_3 - \\sigma_2 < 0.", "89802182fbb3f3f9ab72fcfe967f8350": "\\partial_-f(a):=\\lim_{{\\scriptstyle x\\to a-\\atop\\scriptstyle x\\in I}}\\frac{f(x)-f(a)}{x-a}", "898024f075e93a8a725496802f27baba": " \\int_\\mathbb{R}^\\oplus H_x d \\mu(x). ", "8980297739dc3468c6379fea65d202b4": "\\,\\gcd(a,b)^{-1} = \\operatorname{lcm}(a^{-1},b^{-1})", "898029cbe499fa8957ba009c24588976": " 2< \\alpha < 4 ", "898046bbf3918bb4b0c3caad848abd05": "\\mathrm{H}\\left(X_b\\right)=\\mathrm{H}\\left(Y_b\\right)=\\mathrm{H}\\left(X_b,Y_b\\right)", "8980803e9e9f86411b1a731a9365690e": "h_{k-2}, h_{k-1}, v_i", "8980ddee59834e5dd92c90f1effb4692": "\\Psi_0(\\vec r_1,\\dots,\\vec r_N)", "89811361c00e2501a1feb29185eff3b0": " y(t) = Ye^{j\\omega t} = |Y|e^{j(\\omega t + \\arg(Y))} ", "8981304cd2cd95628201462de1386a03": "\\mathrm{Nu}_x\\ = 0.332\\, \\mathrm{Re}_x^{1/2}\\, \\mathrm{Pr}^{1/3}, (\\mathrm{Pr} > 0.6) ", "89813b09e9f0beb4e0e2206fbcda0a49": " v(1), v(2),\\ldots,v(d)", "8981458fbfbe9415b074571fec60f0cd": " x_1 ", "8981d7236c5d1c11a55ee81c0c55044f": " \\textbf{g} = -1 + X^2 +X^3 + X^5 -X^8 - X^{10} ", "8981ffc79ed83a9810f08d253c2862e5": "s, h \\models P \\Rightarrow Q -\\!\\!\\ast\\, R", "89820f339c237b52cbec2ed5eb734a7e": "m < n\\,", "8982215c8e33a19335d97f210483a7c1": "d=\\frac{v^{2}}{2\\mu g}", "89823f936af385029c49b030148fa7e6": "h_{10}", "898246cc4dee90a91c6326f5d50c21de": "\\frac{\\text{data written to the flash memory}}{\\text{data written by the host}} = \\text{write amplification}", "89824e7af4645cd9e047b288de8bf8aa": "\\displaystyle x^n f(x)\\,", "89825f9da6e9fc83d662ad39fd276cd9": "E_{obs|SCE}=497 mV - 241 mV=256 mV", "898311afc891c36519a8c61dbcff0cb5": "f(z)={{1 \\over (z+i)^2} \\over (z-i)^2},", "89832cccf46b1868bce8e676d1692207": "\\Pi^E_k=\\mathrm{coNE}^{\\Sigma^P_{k-1}}", "89834a4725e2cff329c6e2f0bc699564": "p=\\left\\{1, 1.5, 2\\right\\}", "89845d8485778a1bc6d450987bb18093": "\\sigma_{\\mathrm{tr}}", "8984602087e341e0632de6c9c47b1215": "\\ (c\\tau/2)\\sec\\psi", "898476fae36a2c7187aaa6d776cd630f": "\\int_\\gamma f(z)\\,dz=F(b)-F(a).", "89847ae47ccb8b03ec2e46d33b4f9a6f": "{BE}_{n}", "89849d87cbb8e530b4f2f6187f1af051": "\\Psi_j", "8984cb69969611cbc60e3a67eb128a90": "d^2=(R+h)^{2}-R^2= 2\\cdot R \\cdot h +h^2", "898561ad08184181c751f1262f9e69ef": "\\begin{align}\\Phi_{Y,X}(f) = G(f)\\circ \\eta_Y\\\\\n\\Phi_{Y,X}^{-1}(g) = \\varepsilon_X\\circ F(g)\\end{align}", "8985765e8af5ee28129a4abca2a967c0": "R[\\sigma_1, \\ldots, \\sigma_n]", "89860ce14664bde8ca532947273466cf": "\\ell = \\sqrt{2rs-s^2}", "89867c6294abce1664b580d3ec3b9599": " \\frac{\\Delta H}{\\Delta P}[X] = X P^2 + PXP + P^2X", "89869b0a5cf90ff07d720af087e33211": " \\part \\over \\part x ", "8986c81a29ee1212f7f3087c36447eea": "\\begin{align}\n\\left|\\int_0^\\delta g^{(N+1)}(t^*)\\, t^{\\lambda+N+1} e^{-xt}\\,dt\\right| &\\leq \\sup_{t \\in [0,\\delta]} \\left|g^{(N+1)}(t)\\right| \\int_0^\\delta t^{\\lambda+N+1} e^{-xt}\\,dt \\\\\n&< \\sup_{t \\in [0,\\delta]} \\left|g^{(N+1)}(t)\\right| \\int_0^\\infty t^{\\lambda+N+1} e^{-xt}\\,dt \\\\\n&= \\sup_{t \\in [0,\\delta]} \\left|g^{(N+1)}(t)\\right| \\,\\frac{\\Gamma(\\lambda + N + 2)}{x^{\\lambda+N+2}}.\n\\end{align}", "898707eb1b6d33cc432555071df00736": "\\scriptstyle px \\;\\equiv\\; p_\\mu x^\\mu", "89876bcec04b9db975e41c2e2039c4aa": "T = {E \\over 18}", "8987abcbbd1d68c9bbedc148adbcff1d": "L^s_{\\mathbf\\xi}(\\mathbf{x})=w_{\\mathbf\\xi}(K_n(\\mathbf\\xi,\\mathbf{x})-T_n(\\xi_i)T_n(x_i)),\\quad s=1,2,3,4,\\quad i=2-(s\\mod 2).\n", "8987d1fe1628ccf47543c4c50e969615": "I=I_0 \\, e^{-\\mu \\ell}", "8988023a41157ba4d899260585cb192b": "[HG]_{eq} = \\frac{K_a[G]_o[H]_o}{1}", "89889080045ab04b86be96a0b0e22333": " \\left\\{ \\frac{a_1 + a_2 + \\cdots + a_n}{n} \\right\\}_{n=1}^\\infty \\in \\ell_{1} ", "8988aebd2f702bf6589795f08a2aae66": "P_{ex}\\propto \\mid F_{in}\\cdot e_{ex}\\mid^{2} ", "8988bcb4ab493fc90375e3e90e34dd45": "\\displaystyle{B ={1\\over 2\\pi} \\int_0^{2\\pi} \\varphi(\\theta) U_\\theta A^{(1)} U_\\theta^* \\, d\\theta.}", "8989049f76fcc431aae79d9fff0713e0": "4 \\times (2 + \\overline{2}) \\,", "898920f00c958bb269464274f90eceaf": "\\scriptstyle{\\lambda_a^a}", "898929f718525948776ea7e6019290a4": "\n\\mathbf{u}= \\nabla \\times \\boldsymbol{\\psi}\n", "898930e8d81b289a2f7fa0975c33fded": "e_i\\cdot e_j=[\\mathbf{D}]_{ij}", "898a299aac280734d64b5f27decf28d0": " A^j_l", "898a420abeb2c9c41ab88170320d5895": "|\\psi_0\\rangle", "898a593594d19f3ff388c13ce6f1ab3e": "a_{V}", "898a6b907dff13227fa91259fef0b205": "\\text{tell} \\colon W \\rarr (W \\times 1) = w \\mapsto (w, ())", "898a7bb38acdee643637d2f0168c8703": "uv=qvu", "898aa4d663307f958b853a6132baa3f3": " Y = \\alpha X + \\beta ", "898aae5c6cbafa37360c0d0ba46cf0f6": "\\prod _x ax = C a^x \\Gamma (x) \\,", "898abbe64f2c4a6a75ce262416b77ca5": "R(P_i)= R_i", "898abd4f23e5b65c57beb35a0e13deac": "\\langle E_1(l, m, t) E_2^*(l, m, t) \\rangle = I(l, m) \\frac{e^{i \\omega \\left( \\frac{R_1 - R_2}{c} \\right) }}{R_1 R_2}", "898abda84fb14558478d2c02986164a7": "a \\in \\mathbb R", "898b24316c2def5e965d48450997f50e": "\\int \\frac{ R}{x^{2}}\\,dx=- \\frac{ R}{x}+a \\int \\frac{dx}{R^2}+ \\frac{b}{2} \\int \\frac{dx}{ R}", "898b4bfb912e60e9faf606c039f4437c": "=\\frac{{\\pi}^2}{10}-\\operatorname{arcsch}^2 2", "898b7d7e64ad5fa32f4cb496df60648e": "\\begin{align}n! = \\Pi(n) &= \\prod_{k = 1}^\\infty \\left(\\frac{k+1}{k}\\right)^n\\!\\!\\frac{k}{n+k} \\\\ &= \\left[ \\left(\\frac{2}{1}\\right)^n\\frac{1}{n+1}\\right]\\left[ \\left(\\frac{3}{2}\\right)^n\\frac{2}{n+2}\\right]\\left[ \\left(\\frac{4}{3}\\right)^n\\frac{3}{n+3}\\right]\\cdots. \\end{align}", "898bed473023de58dcbc1bca3bf9e167": "{\\rm cof}({\\mathcal K})=\\max\\{{\\rm non}({\\mathcal K}),{\\mathfrak d}\\}", "898c481146256858bd17deb6da90bfd1": "y = e^x", "898c5b0b8ca4654bc74c029138df8276": "\\left | \\{ \\Omega g > \\varepsilon \\} \\right | \\le \\frac{2A}{\\varepsilon} \\|g\\|_1", "898c6a339810ea3ea5722880221dbff1": "\\int\\frac{\\mathrm{d}x}{1+\\cos ax} = \\frac{1}{a}\\tan\\frac{ax}{2}+C\\,\\!", "898c760aca818723e58ae4b2ec5fb9cb": "\\alpha_p = \\frac{\\mathrm{d}L / L} {\\mathrm{d}p / p} = \\frac{p}{L} \\frac{\\mathrm{d}L}{\\mathrm{d}p} =\\frac{1}{L} \\oint \\frac{D_{x}(s)}{\\rho(s)}\\mathrm{d}s", "898e142fa176398d8fb89ecf66b7b3f1": "\\sqrt{25} = 5.\\!\\,", "898e2b16fee5e957710eec6c39399024": "q_{2}", "898ee2ba7900df1c60c980bfc02ba017": "d_i s_j = d_{j-1} s_i\\,", "898f68b7d5641890d71d039045cf3d8b": "\\langle T_v\\exp_p(v), T_v\\exp_p(w_N)\\rangle = \\langle v, w_N\\rangle = 0.", "898f7e1bd94bd7ef7e47a0d5b63655bf": "\\textstyle C_i(S) = \\sum_{e \\in P_i} \\frac{c_e}{n_e}", "898fbcca8b5c207afb59a88955deaba3": "\n K_{\\rm IA} = \\sigma \\sqrt{\\pi a}\\left[1 + \\sum_{n=2}^{M} C_n\\left(\\frac{a}{b}\\right)^n\\right]\n ", "898ff6455b671f99636bdfb1eebb158d": "P_3=(x_3,y_3)=2P_1", "89900132874642b09ead4d3e0e0f7d0a": "E_1:\\mathrm{ player\\ 1\\ wins}", "89902b3d4d87620b4883c97769fa6856": "12 + 13 + 21 + 23 + 31 + 32 = 132", "899037bf9e138502ee9b7225644fbfbf": "out\\;m.P", "89905b7a0c2bfdd9d4dac3e27a60232b": "\\frac{1}{j}", "89909f36d5ac33791ec02f1209c1ead9": "\\mathsf{f}(1)=1", "8990c8aadb51774ef3bb3236e0927a68": "\\dim_{\\operatorname{Haus}} \\leq \\dim_{\\operatorname{lower box}} \\leq \\dim_{\\operatorname{upper box}}.", "8990ee3cdaefcc337a785bf1efea5766": "f: \\mathbb{R} \\rightarrow \\mathbb{R}", "8991d2f9b2e365237a6dc416e7bc20a7": " =\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\}\n\\right\\} +\\text{Tr}\\left\\{ \\hat{\\Pi}_{\\rho,\\delta}^{n}\\rho^{\\otimes\nn}\\right\\} ", "8991d9d74068de5c7b2c5ab50a85159c": "\\langle g_1,g_2,g_3,\\ldots | R_1, R_2, R_3,\\ldots\\rangle_P", "89921d0a114b42d641cd4ba1c6f8a98e": "\\mathbb{N}.", "89921ff6bcea24cac953ff3f318ec6f8": "2P^2", "89922ea19d5d0db35872977554e9e322": "\\omega_k", "899257be812b9aba820d038be4e58e27": "E = \\frac{1}{2} C V^2", "8992a57d69979b2c600e75b9c5c379b0": "r=\\frac{\\sin^2\\theta}{\\cos\\theta} = \\sin\\theta \\tan\\theta", "8992f5cffdd561a52339d557859a2bbc": "90\\frac{\\textrm{lb}}{\\textrm{ft}^3}\\leq w_c\\leq155\\frac{\\textrm{lb}}{\\textrm{ft}^3}", "8993059a7bff0beef5efc4a7e3c63056": "\\forall X_2 - X_1 \\in L_d^p(K) \\Rightarrow R(X_2) \\supseteq R(X_1)", "89931cfd88db85d48559045bea965d47": "\\textstyle [\\sigma]", "89933109f1e9d1855449e140c56a21ff": "r = \\frac{a \\left( 1-e^2\\right)}{1+e\\cos \\theta} \\ . ", "89940d2cccd3c1733e7652abd014fd1f": "\nE=1+\\sum_{n=1}^{\\infty} \\frac{1}{2^n(2^n-1)}\n", "8994435099bea0b9096ab607824bef9f": "\n\\overline{\\boldsymbol\\Sigma}\n=\n\\boldsymbol\\Sigma_{11} - \\boldsymbol\\Sigma_{12} \\boldsymbol\\Sigma_{22}^{-1} \\boldsymbol\\Sigma_{21}. \n", "8994576a156b956729c2ed0f9869b861": "n = 2^m - 1", "89945f041471a78a8d54e7f4c6f139b9": "\n\\left\\{ D_{i}, L_{j}\\right\\} = \\sum_{s=1}^{3} \\epsilon_{ijs} D_{s} ~.\n", "89949690b2568e754fa21ace5e021b28": "\n\\sigma_D^{(k)} = \\underset{\\sigma_D = s\\sigma_I^{(k)},\\; s \\in [0.5, \\dots, 0.75]}{\\operatorname{argmax}} \\, \\frac{\\lambda_\\min(\\mu(\\mathbf{x}_w^{(k)}, \\sigma_I^{k}, \\sigma_D))}{\\lambda_\\max(\\mu(\\mathbf{x}_w^{(k)}, \\sigma_I^{k}, \\sigma_D))}\n", "8994a9b1007fe5fc2e1c915319c1a168": "z \\mapsto z^n", "8994b9074efb74b4bd852744ab8fc4f1": "\\exists p\\,\\phi", "8995137e71e79b1bebff0d49d55577c8": "X\\left( t\\right) ", "899522c6f4469e510354c41bd0ef121b": "\\begin{matrix} {2 \\choose 1}{3 \\choose 3}{45 \\choose 2} \\end{matrix}", "8995272dc7107053026a3e139f24ceeb": "X(a,b,c;z,w) \\in V[[z,w]][z^{-1}, w^{-1}, (z-w)^{-1}]", "899543529f788aa0acb3b882bff7a876": "f, g: X \\rightarrow Y", "899552fb221caa1980e8abaa547863aa": " {{E(b)} \\over {E(e)}} ", "8995549fedd7431652316528e96504f0": "n=0,1,\\ldots", "8995594e456fc4e232d78299dce9e1c5": "(\\mathbb{Z}/q\\mathbb{Z})^*", "89957dd8e7967f0aeb39821866534843": "\\left( 3,4,5 \\right)", "8996584981f888698b87d0e765b95852": "\\Gamma_0", "8996d84c9ea40f70167303e18a710fe1": "q^m q^n = q^{m+n}", "8996e55101526f85132e725141e2d59c": "f = \\mathrm{id}", "8997093cf9a83e7bee0ab77d75e9754a": "f(x_1,\\dots,x_i,\\dots,x_j,\\dots,x_n) = f(x_1,\\dots,x_j,\\dots,x_i,\\dots,x_n) = -f(x_1,\\dots,x_i,\\dots,x_j,\\dots,x_n),", "89973cede940a078beb789a7983e9856": "T\\{\\}", "899770c4f1cb98c9532947e8826f606b": "F(x)=\\frac{1}{\\sqrt{4\\pi}}\\int_{-\\infty}^\\infty f(y) \\; e^{-\\frac{(x-y)^2}{4}} \\; dy = \\frac{1}{\\sqrt{4\\pi}}\\int_{-\\infty}^\\infty f(x-y) \\; e^{-\\frac{y^2}{4}} \\; dy,", "89979615bd01d3cf0ae593bfbe0ecc3b": "i = \\{i\\}", "8997b66fd127810e9cb896c229ba0b2c": "A \\rightarrow S: \\left . A,B,N_A \\right .", "8997dad65c75fa8a49899891eb3f573e": "\\lang S \\rang ", "8998259c61938d1747afc9ac4709150a": "\\mathcal{E}^4", "8998b455e81f05c9414d66cfe145346d": "\\mathfrak c ", "8998fa04764ed8e6513f1b7e584fae06": " G_{ \\infty } = \\frac { \\beta i_B } {i_S} = \\left( \\frac {\\beta} {\\beta +1} \\right) \\left( 1 + \\frac {R_f} {R_2} \\right) \\ . ", "89995282ad2669f68f0c7283c051734b": " c(E^*) = 1 - \\lambda_1 + \\lambda_2 - \\cdots+ (-1)^g \\lambda_g. ", "8999910211b384f18aa1189f65eccbda": "[ \\mu (A + B) ]^{1/n} \\geq [\\mu (A)]^{1/n} + [\\mu (B)]^{1/n},", "89999889d032531576c2fff3e67dcf3b": "\\scriptstyle \\partial_t", "899a49e7bad315c3bcc8e17cdc719935": " y=l\\cdot x+m ", "899a55554a02765d28ace09100fb46ff": "\\begin{array}{cc}\n \\begin{array}{rrr} \\\\ &1& \\\\ 2&& \\\\ \\\\&&/3 \\\\ \\end{array}\n \\begin{array}{|rrrr} \n 6 & 5 & 0 & \\text{-}7 \\\\\n & & 2 & 3 \\\\\n & 4 & 6 & \\\\\n \\hline\n 6 & 9 & 8 & \\text{-}4 \\\\ \n 2 & 3 & & \\\\ \n \\end{array}\n\\end{array}", "899a7bbe1db5c4ca68d002b33b928691": " \\mathbf{r}_{k+1}^\\mathrm{T} A \\mathbf{p}_k = \\frac{1}{\\alpha_k} \\mathbf{r}_{k+1}^\\mathrm{T} (\\mathbf{r}_k - \\mathbf{r}_{k+1}) = - \\frac{1}{\\alpha_k} \\mathbf{r}_{k+1}^\\mathrm{T} \\mathbf{r}_{k+1} ", "899a8a0ef68638974f1a09daa154595d": "\\frac{d}{dt}\\hat{\\boldsymbol{\\imath}}(t) = \\Omega (-\\sin \\Omega t, \\ \\cos \\Omega t)= \\Omega \\hat{\\boldsymbol{\\jmath}} \\ ; ", "899a983afe0418412d4c36f273a3c4dc": "\\mathbf{u\\times v}=(u_2v_3\\mathbf{i}+u_3v_1\\mathbf{j}+u_1v_2\\mathbf{k})\n-(u_3v_2\\mathbf{i}+u_1v_3\\mathbf{j}+u_2v_1\\mathbf{k}).\n", "899af4f382524319eefd764349b567ac": " m\\geq 3 ", "899b67d9fcc74c6aff25f71579a09838": "\\mathbf{S} \\cdot {\\rm d}\\mathbf{A}", "899b740df427612505dbf07ba7d29889": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 9.608804 \\log_e(T+273.15) - \\frac {6339.983} {T+273.15} + 78.35769 + 8.183476 \\times 10^{-06} (T+273.15)^2", "899bb6ef776677016444995ed3a65c49": "\\mathbf{S}_{1i}", "899bde96a29aa27e68247bd20306d297": "\\frac{1}{1-2^{k+1}}\\sum_{i=0}^k \\frac{1}{2^{i+1}} \\sum_{j=0}^i {i \\choose j} (-1)^j (j+1)^k ", "899bef40c9cc7eea3ac0a7896fa7e7cf": "f \\in \\mathfrak{m}", "899c25a26d7ccf4176a6f2d73ceb0f6e": "\\{1, 1 + x, 1 + x + x^2, 1 + x + x^2 + x^3 \\cdots \\}", "899c4b748896440dcc89fdef2148d733": "\\Phi(t) = \\omega_nt + \\phi(t) = 2\\pi v_nt + \\phi(t) \\, ", "899d33555fca61340b69472f69fea993": "c/\\omega_{pe} = 5.31\\times10^5\\,n_e^{-1/2}\\,\\mbox{cm}", "899d76c16dd4824a47510a53a48a00d0": "\\lambda(g):h\\mapsto gh,\\text{ for all }h\\in G.", "899d8ec93ae53e81b6586303cc697745": " \\|x\\|_H = \\sqrt{\\langle x, x \\rangle}, \\ \\ \\text{where} \\ \\ \\langle \\cdot, \\cdot \\rangle \\colon H \\times H \\to \\mathbb K", "899d9ad5fceb74d2d3979f46fadbefb1": " \\kappa = - \\xi \\,", "899dbaa664bd6202839f9bdad69c290b": "L\\setminus p", "899e9bdcc1aebf82fdc045051367d4f0": "\\boldsymbol\\Theta ", "899f417692724313a629b5cd9e4dddf7": "f : V_1\\times V_2\\times\\cdots\\times V_N \\to \\mathbf{R}", "899f4b411e6a72c0196be2176541ac7a": "T_{first} = \\left \\lceil \\dfrac {V_{max} C R_{s1} f_{clk}} {V_{ref}} \\right \\rceil", "899f99079424ff25b2ea31b9af616c08": "\\mathbf{\\lambda}_i=\\frac{||\\mathbf{p}_i^T\\mathbf{X}_1||^2}{||\\mathbf{p}_i^T\\mathbf{X}_2||^2}", "89a00209258e5cb1a2b4c7b242c7a9c7": " f(g(z))=\\sum_{-\\infty}^\\infty c_n z^n", "89a08025f6e1a7b132c68f0f0150386b": "B_I N", "89a084a36c9344fec4055335ad141b1f": "K_m(S1, A), K_x (S0, response)", "89a0a2bbf4c162d7186f1f464cc603ff": "\\Theta(\\sqrt n)", "89a0d21bc068b28dccebd04223481a29": "1 < h < p", "89a130d0aedcf047e17f42f494b85d80": "M_C(B)=M_C(\\beta)", "89a14448694bce6c3354b3f68f53e3c2": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "89a1ac7d00d60bcefdafb25f813f07e8": "f_Y(y \\mid X=x)f_X(x) = f_{X,Y}(x, y) = f_X(x \\mid Y=y)f_Y(y). ", "89a1bd1c62a106d46368614970820cdd": " q_j= - \\lambda_{ij}\\frac{\\partial T}{\\partial x_i} \\,", "89a1cdcb353133be28ec2ba43a6da39e": " \\bold{B} =\\nabla A \\times \\hat{\\bold{z}} + B_z \\hat{\\bold{z}}", "89a1d62607aba550b47344dded01dca3": " -2\\sqrt{a} \\int \\frac{vdv}{\\sqrt{1+bv^2}}", "89a1dbdc4f87ed055b71171c496a1ea4": "A^2 = B^2 = \\ldots = 1.\\,", "89a20e7ba47d32112b58d7c4e38c8bfc": "\\mbox{diff}(M_x)", "89a2261aa8a1d7ff7452e9c71a09de34": " P= \\dot{m}c_p(T_2 + \\frac{V_2^2}{2c_p}-[T_1 + \\frac{V_1^2}{2c_p}])\\,", "89a27265904c1cc608d4757c0122f5c5": "x_1, \\dots, x_k \\in U", "89a2a76456473e2742437b0ad50b675d": "\\Pi(k_i)=\\frac{k_i + C \\sum_{(i,j)} k_j}{\\sum_j k_j + C \\sum_j k_j^2}", "89a2e7b06fe66843230d68fd1eab747c": "\\operatorname{Ad}_g = (d\\Psi_g)_e : T_eG \\rightarrow T_eG", "89a36db7ea636dabc4ea67d1cd3f3b8b": "deg_X(Q)", "89a3cde35cfbbc7d6d59c73cf4b97f2e": "H\\left(A,C\\right) = \\left\\{00,1,01\\right\\}", "89a3e69e262ec0ffc332cb8cd91598ff": "\\Delta = K^\\times \\cap (L^\\times)^n.\\,\\!", "89a423434a56feb94f2d6995c1ed7b87": "M_{\\mathfrak p}", "89a431be2e34bec43c4118c456922344": "\\sqrt{\\lambda_i}", "89a451a69c88dad425339c4b48b861c7": "\\left[(H_2O\\right)_nH]^{+} + S \\to SH^+ + nH_2O", "89a4581ad7dd6d109b04c7bfd858dc63": "(U_1^i,U_2^i,\\dots,U_d^i)=\\left(F_1(X_1^i),F_2(X_2^i),\\dots,F_d(X_d^i)\\right), \\, i=1,\\dots,n.", "89a485ff267cb44746b8b0949545807d": "E_i = \\{(X_1^n(i), Y_1^n) \\in A_\\varepsilon^{(n)}\\}, i = 1, 2, \\dots, 2^{nR}", "89a4cffdce011bd998395cddc17490d2": "|\\Psi\\rangle_-", "89a524f66adf331893429b06247d3eca": "\\left(\\frac{\\partial U}{\\partial V}\\right)_{S,\\{N_i\\}}=-p", "89a525c523d9b9e6caed7d30a0cb5ea0": "\n\\{\\phi_2, \\phi_3\\} = 2 r^2 \\neq 0.\n", "89a543dde2f4c130e95742a704777906": "\nQ=\n\\begin{pmatrix}\n-\\frac{1}{2}S+T & 0 &0 \\\\\n0 &-\\frac{1}{2}S-T & 0 \\\\\n0 & 0& S\\\\\n\\end{pmatrix} \n", "89a54564a280f08af932c95f2d69a1f9": "\\scriptstyle \\varphi\\,", "89a5456dcdc7f99d5962466534d9d9a5": " F({\\bold x}) := (f_1({\\bold x}), \\ldots, f_n({\\bold x})) , ", "89a594ad278082709683a46eb53bdd14": "\\zeta_{m\\cdot n}(\\xi,\\epsilon)=\n\\zeta_m\\left(\\xi,\\sqrt{\\frac{1}{\\zeta_n^2(L_m,\\epsilon)}-1}\\right)", "89a59b98b79415a1b0fdb929e30bcc8a": "\n\\alpha_V = \\frac{1}{V}\\,\\left(\\frac{\\partial V}{\\partial T}\\right)_p\n", "89a601a1f4b49927e05892761cd845b8": " \\eta :k\\to R ", "89a6807c126e62636837f0e42f9b2ed9": "\n\\| {d\\over dt} Q_A |0\\rangle \\| \\approx {1\\over A} \\| Q_A|0\\rangle \\|. ", "89a6b3b2c514cbd5747218e8b36d19da": "\n\\begin{bmatrix}\n 0 & 1& -1\\\\\n 1 & 0 & 0\\\\\n 0 & 1 & 2\n\\end{bmatrix}\n", "89a6ebc66d1be204c40733d4de2d217c": "\\int_{-1}^{+1} \\frac {f(x)} {\\sqrt{1-x^2} }\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)", "89a700b27acdeed3b45e4301678debb8": "e\\begin{Bmatrix} p, q , r \\end{Bmatrix}", "89a70832f9a0da2f6120d1a84193c9e8": "\nK = K_0 e^{\\delta t}, \\quad L = L_0 e^{\\nu t}, \\quad P=P_0 e^{\\eta t}. \n", "89a717e0eb9581ce979409297c9d1b31": "c (x, y, z) = \\frac1{R}.", "89a7186c2435be093d8567c5cd102ecf": "\\log X_i", "89a7b0a9c3292602b2507f507b63bf03": "\\boldsymbol{U}_p=\\text{d} \\boldsymbol{X}_p / \\text{d}t", "89a7c189d6ba70e8dd2445cc6d14c606": "\\begin{align}\nr &= \\sqrt{x^2+y^2+z^2} \\\\\n\\theta &= \\arccos(z/r)\\\\\n\\phi &= \\arctan(y/x) \\end{align}", "89a7d597023a9941536171cfcb562845": "\\operatorname{E}\\bigl[(X)_r\\bigr] = \\frac{n!}{(n-r)!} p^r, ", "89a7f20378d5c4b029e70f59eb7bfd05": "S(\\boldsymbol{\\beta}) \n= \\bigl\\|\\mathbf y - \\mathbf X \\boldsymbol \\beta \\bigr\\|^2 \n= (\\mathbf y-\\mathbf X \\boldsymbol \\beta)^{\\rm T}(\\mathbf y-\\mathbf X \\boldsymbol \\beta) \n= \\mathbf y ^{\\rm T} \\mathbf y - \\boldsymbol \\beta ^{\\rm T} \\mathbf X ^{\\rm T} \\mathbf y - \\mathbf y ^{\\rm T} \\mathbf X \\boldsymbol \\beta + \\boldsymbol \\beta ^{\\rm T} \\mathbf X ^{\\rm T} \\mathbf X \\boldsymbol \\beta .", "89a7fff1b3ea6470c6683a77173516a8": " P^2 = \\begin{bmatrix} 0 & 0 \\\\ \\alpha & 1 \\end{bmatrix} \\begin{bmatrix} 0 & 0 \\\\ \\alpha & 1 \\end{bmatrix}\n= \\begin{bmatrix} 0 & 0 \\\\ \\alpha & 1 \\end{bmatrix} = P. ", "89a83d43c605d2a02342cd0eb9078684": "S(\\rho_{ABC}) + S(\\rho_B) \\leq S(\\rho_{AB}) + S(\\rho_{BC})", "89a8b2b106e29cbe8fec0fc10c644faa": "\nS_M(n) = \\frac{\\theta}{n}\\frac{\\Gamma(J_M+1)\\Gamma(J_M+\\theta-n)}{\\Gamma(J_M+1-n)\\Gamma(J_M+\\theta)}\n", "89a8d38a37385b04e3040299b7014ebe": " P \\, ", "89a9146b795203ccccf802dd52f3168c": "P(A|B) = \\frac{P(B | A)\\, P(A)}{P(B)}\\cdot \\,", "89a917bbcfa5711d56fabdd2a64b3c18": "f_* \\mathcal{O}_X = \\mathcal{O}_S", "89a94b0d1280fd08bac26e6800a0985a": "x_{n+1}, x_{n+2}, \\dots, x_n.", "89a964c5e0862dca6f0133bbc0c2a322": "m g", "89a9934d6a4b66b811ecb54dbec4a910": "\nH_t(p,x) = {p^2\\over 2m} + {m \\omega(t)^2 x^2\\over 2}\n\\,", "89a99f88cd1c07b2fe6597c27ffa3b3e": "\\theta\\hat{\\mathbf{n}} = \\theta(n_1, n_2, n_3)", "89a9ff7cede6f3c87e2ac0ee32f6e489": "e(k,i+1) = e(k,i) - v_i(k)e_b(k,i)\\,\\!", "89aa04b49528f7e01509b3833b0b5b29": "t_{LL}^{\\mu \\nu} = - \\frac{c^4}{8\\pi G}G^{\\mu \\nu} + \\frac{c^4}{16\\pi G (-g)}((-g)(g^{\\mu \\nu}g^{\\alpha \\beta} - g^{\\mu \\alpha}g^{\\nu \\beta}))_{,\\alpha \\beta}", "89aa15c75573acd8ad422a1cfc8e4ff8": "\\textrm{Hom}_R( - , I)", "89aa40c2021aa4c9902cfefd8780dec1": "j^2 = +1", "89aa6574d581a82fc65cb0d508124b9a": " X_{t}=(1,y_{t-1},y_{t-2},...,y_{t-p})\\,", "89aa9aacfd333d1ac4689aaf1a47ff81": "0 \\leq k \\leq \\dim{X}", "89aaa5b6b5c766b47974cbb98cf4dd9b": "(2r)^d", "89aac7a8d750e2ab0e907904f96a1ae5": "y_{it} = \\alpha + \\beta' X_{it} + u_{it}, ", "89ab3dd2b9f63147c8ce4369aa0857a1": "u(t),v(t)", "89ab7e841984f48e5f6113aee51a22bf": "\\boldsymbol{\\lambda \\Gamma} =0 \\;\\; \\left(\\mbox{i.e.}\\;\\; \\sum_r \\lambda_r \\gamma_{ri}=0\\;\\; \\mbox{for all} \\;\\; i\\right)", "89abbe398e8b5d4ca4a2a22b19767b1d": "3.\\mu_{5,2}(p_{3}) = \\alpha_{5}(p_{3}) ", "89abe5b4661a3a67d265d3cfcc9feeb8": "\\lim_{x \\to \\infty} \\frac{L(ax)}{L(x)}=1.", "89ac15cadd38140601888999ece2db96": "y_d=\\alpha+\\beta x_d +\\epsilon_d. \\,", "89ac4f0a3caf9c770b97fa677fdf15f9": "\\epsilon_b \\,", "89ac715350a45ef7869aede1703aee5e": "\\tau_a^* (R_jf) = R_j(\\tau_a^*f).", "89acdf83d9cd692048909c353ca07459": " \\sec \\theta = \\frac {\\mathrm{hypotenuse}}{\\mathrm{adjacent}} = \\frac {h}{b}", "89ace20d8f8cfe29eea0cdb81af9cfe9": "\n\\begin{align}\ns_t& = \\alpha x_{t-1} + (1-\\alpha)s_{t-1}\\\\[3pt]\n& = \\alpha x_{t-1} + \\alpha (1-\\alpha)x_{t-2} + (1 - \\alpha)^2 s_{t-2}\\\\[3pt]\n& = \\alpha \\left[x_{t-1} + (1-\\alpha)x_{t-2} + (1-\\alpha)^2 x_{t-3} + (1-\\alpha)^3 x_{t-4} + \\cdots \\right]\n+ (1-\\alpha)^{t} s_0.\n\\end{align}\n", "89ad7199810bd1e2eef354b6c53dd1df": "L_d(i)\\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\frac{d(N\\Phi)}{di} = \\frac{d\\Lambda}{di}", "89ad74bce36706b2c2918ce6536c6ce7": " R(A\\|B):= {\\rm Tr}(A\\log A) - {\\rm Tr}(A\\log B).", "89ad75a6e2460015184363bf45d97b5e": "\\begin{align}\nd f & = \\frac{\\partial f}{\\partial t}dt \n+\\left(\\frac{\\partial f}{\\partial x}dx\n+\\frac{\\partial f}{\\partial y}dy\n+\\frac{\\partial f}{\\partial z}dz\n\\right)\n+\\left(\\frac{\\partial f}{\\partial p_x}dp_x\n+\\frac{\\partial f}{\\partial p_y}dp_y\n+\\frac{\\partial f}{\\partial p_z}dp_z\n\\right)\\\\\n& = \\frac{\\partial f}{\\partial t}dt +\\nabla f \\cdot d\\mathbf{r} + \\frac{\\partial f}{\\partial \\mathbf{p}}\\cdot d\\mathbf{p} \\\\\n& = \\frac{\\partial f}{\\partial t}dt +\\nabla f \\cdot \\frac{\\mathbf{p}dt}{m} + \\frac{\\partial f}{\\partial \\mathbf{p}}\\cdot \\mathbf{F}dt\n\\end{align}", "89ad763106bd54218e740c5c16bc395c": "\nU = \\prod_{k} \\mu_i^{(k)} \\cdot U^{(0)} = \\prod_{k} (\\mu^{-\\tfrac{1}{2}})^{(k)} \\cdot U^{(0)}\n", "89ad95c71df72c03bc5ee0a8a72e33c9": "\\frac{1 + z + {\\scriptstyle\\frac{1}{2}}z^2 + {\\scriptstyle\\frac{1}{6}}z^3+ {\\scriptstyle\\frac{1}{24}}z^4}{1}", "89ad9a225e31c3711e63b56c2fb42601": "x+y=y+x", "89adac91c6be6a123fa465e529f72880": "p = a(1-e^2)\\,", "89adde97e2edbc3e5780bec4f663db0d": " v = \\sum_{k=0}^\\infty \\alpha_k b_k \\ \\ \\overset{\\textstyle P_n}{\\longrightarrow} \\ \\ P_n(v) = \\sum_{k = 0}^n \\alpha_k b_k", "89ade284187c7382ac4098915173603a": " e^-, \\mu^-, \\tau^- ", "89adf0be55fa4d3025d50d091fc8e510": " \\scriptstyle \\left\\langle {n \\atop m} \\right\\rangle ", "89ae2361ec452fa620474f8b460cd651": "2 \\notin A \\Rightarrow \\sigma A \\le \\frac{1}{2}.", "89ae6b48fe1cdad1c32f080a777c2593": "y_k = \\varphi \\left( \\sum_{j=0}^m w_{kj} x_j \\right)", "89ae78be880a004aa5404ac874a01bff": "u_2", "89ae8cfb6cfeedec21cf98e99a38f8c1": "X = x\\frac{d}{dy}", "89ae9fd5755e69b34184c0e3bcdd2b75": "J(S)=\\{0\\}\\,", "89aef2f655db11b031d3f272a2fd2ad7": "F = \\frac{dp }{d t}. ", "89aef89dd2377edfc724bfa7aec39c5d": " -\\dot{S}(t) = A'(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B'(t)S(t)+Q(t)", "89af2073841fe093579f09d5f7e54143": "\\,\\!K_X = \\bigwedge^n\\Omega_X,", "89af23f15a82c465a4d8cb68d9637708": "\\scriptstyle I_n", "89af7747c117d75155f0adbda8383587": "\\frac{350}{429}", "89af8e07422396e37e0bbbc7497b238e": " \\mathbf{p} = \\frac{h}{\\lambda}\\mathbf{\\hat{k}} = \\hbar \\mathbf{k}", "89afb8b3950a22f39dc0daa8377212b6": "R = \\frac{1}{2\\sum is_i - 1}", "89afd9723b0ae02df682dfe69a4f9022": "U_t + A(U)_x = 0", "89afffd4dee73cc2640cf747172cc550": "\\tfrac{\\varepsilon}{n}", "89b020dc5f65e2949c174f88c6bcdef8": "\n \\Phi = \\beta x - \\gamma t + \\sum_{n=1}^\\infty B_n\\, \\biggl[ \\cosh\\, \\left( nk\\, (z+h) \\right) \\biggr]\\, \\sin\\, (n\\theta),\n", "89b04233469f46c3306c3686c87a9b98": " g(\\gamma)=\\frac{3}{2\\gamma} (1+\\frac{1}{2\\gamma^{2}}sinh^{-1}(\\gamma)-\\frac{\\sqrt{1+\\gamma^{2}}}{2\\gamma})", "89b06291f26bc18935524705d18f9d02": "G_1' \\cup G_2' \\subseteq G'", "89b065fa30e6c9d692a5c990ca965511": "x_{n+1} = x_1^2+\\cdots+x_n^2", "89b0910f6cdd84c902d6498c91af49ea": "Q(D_{2^{j+1}}\\varphi) = 2\\cdot (h * Q(D_{2^j}\\varphi))", "89b0d17236e05012d25efd7990be938f": "\\boldsymbol\\omega", "89b114d4c6d5247730bb60db847ff547": "\\gamma_1 > 0, \\alpha_1 > 0", "89b11e229b384e8e4da89c3540842068": " dx= a\\,dt + b\\,dB.\\!", "89b1787135c95d42db26e556c3527db1": "\\psi(\\psi(\\psi(0)))", "89b1a5b0e53334a08880159ae790828f": "\\frac{ \\partial V}{\\partial h} = \\frac{\\pi r^2}{3},", "89b1cf675c22d6a69465887dbe4883f4": "\\tfrac{\\mathrm{u\\bar{u}} - \\mathrm{d\\bar{d}}}{\\sqrt 2}", "89b2136e9a4f6561ffda88a70e7c54f2": "y=-1/(2\\lambda)", "89b21b5a80e80a859ba6386d8a9be2b3": "i \\in \\{1, ..., n\\}", "89b2604f146d592c9472ba7dad07d36b": "R^{n+1}", "89b3623cb7420749728972205e543458": "\\overline{\\mathop{\\rm span} (x_n)} = H", "89b37ccc87ecf921cd13f6ceea9fc5a0": "\\frac{\\partial I}{\\partial t} = I_J \\cos \\phi \\cdot \\frac{\\partial \\phi}{\\partial t}. \\ ", "89b3b1b528b40fd7676b00f2f58212c1": " \\wp, \\wp' ", "89b3e3836124c5d76cfbf4c6cafc722c": "\\frac{dx}{dt} = r(1 - \\cos t)", "89b40010328e9b87467c2676ac74f9d5": "x_0 + 1", "89b4148cf9f1cb2f21c29afaa90ed4e7": "N_k = \\left(6\\sqrt{N_{k-1}} - \\sqrt{N_{k-2}}\\right)^2,\\text{ with }N_0 = 1\\text{ and }N_1 = 36.", "89b47d7587d606127b3c06a851e68035": "A_\\varepsilon = \\{ x \\in X \\, | \\, d(x, A) \\leq \\varepsilon \\}", "89b4b08896b7110ddb06bf486b6791ec": "\\Vvdash \\nvdash \\nVdash \\nvDash \\nVDash \\!", "89b4c76780eb1c1ded9d4a02b05f5668": " \\mathrm{A} + \\mathrm{BX} \\rightleftharpoons \\mathrm{B} + \\mathrm{AX} ", "89b4e5234d1b273c0613d749d3427a20": "\\frac{d[P]}{dt} = k^{\\ddagger\\ominus}[\\mathrm{AB}]^{\\Dagger} = k^{\\ddagger}K^{\\Dagger }[A][B] = k[A][B]", "89b55f37905e688bf9e3bcd69e0a7b08": "g, h \\in \\mathfrak g", "89b5bd9754ea75add5da9bbee3fedafc": "1 \\over 4", "89b5da5e3fa7ac0fa948a603263d6fce": "V=\\mathbb{E} \\left[ \\left(X - \\mu \\right) \\left( X - \\mu \\right)^T \\right]. \\, ", "89b5ef61d7b692f22cd10ebcf5fc2613": "\\oint_C {1 \\over z^5}\\left(1+z+{z^2 \\over 2!} + {z^3\\over 3!} + {z^4 \\over 4!} + {z^5 \\over 5!} + {z^6 \\over 6!} + \\cdots\\right)\\,dz.", "89b60c1cb6b2fb46251d8d0f0b89ee5f": "L^n,\\ ", "89b6480918e205d56a47dee862130c3a": "\\operatorname{El}(a)", "89b6bdd7d914fe67d241e97aa07919ce": "r=s", "89b6dfbba94a012a6b81aad4c4585e2e": "E(x) = \\sum_i (x-p_i)^\\top (\\hat n_i \\hat n_i^\\top) (x-p_i).", "89b6f7191b8663c50ffc4320b3a232bd": "ds^2 = -\\exp(2 \\, \\phi) + \\exp(-2 \\phi) (dx^2+dy^2+dz^2", "89b78cbfc85aa24f8c8e250aaff86272": " x =[1,2] ", "89b7d1e62a1fe43295f4f54e9efa5338": "\n\\begin{align}\n\\mathrm{P}\\{d=1\\}& = \\frac{1}{n}\\\\[2pt]\n\\mathrm{P}\\{d=k\\}& = \\frac{1}{k(k-1)} \\qquad (k=2,3,\\dots,n). \\,\n\\end{align}\n", "89b7f27b87ef0391e97355e143fc1951": "\\mathbf{A}_{\\text{Electric dipole}}(\\mathbf{x},t) = \\frac{\\mu_0}{4 \\pi} \\frac{e^{i k r - i \\omega t}}{r} \\int d^3\\mathbf{x'}\\mathbf{J}(\\mathbf{x'})", "89b802af1111c77ee90e6bc1ad3d9058": "\nJ = \\int_0^{2\\pi} p {\\partial x \\over \\partial \\theta} d\\theta\n\\,", "89b86e1626a6690b12315eda1f43859b": "\\|f(x)\\| < \\epsilon", "89b878a6f3eb610fae7df849a7a4eda6": "(-1)^r (\\chi(\\mathcal{O}_X) - 1),", "89b89caf08022b34cff01740dfa09e64": "E_\\varepsilon(\\lambda)", "89b8c987b4908f1a6734c2d4d1f8d9cc": "K = \\{(x, -x) \\mid x \\in \\mathbb{Q}\\}", "89b8e56db915b585c730fe3602bcbb0e": "a = \\frac{G M}{r^2 + e}", "89b8fe4da2a4ee32e3e2a93ac006273c": "\\ {\\mu}_a= \\frac \\mu{1+N\\mu} ", "89b9406fe795d87ce9ca3d3483d8d64a": "x_0+N=\\{x_0+n: n\\in N \\}", "89b9815f48d899ea55b8f297e1e1cd36": "t \\rightarrow\\infty.", "89b99daf320cb1543cb82038c18c0e4b": "\n\\varphi\\left(e^{-6\\pi}\\right) = \\frac{\\sqrt[3]{3\\sqrt{2}+3\\sqrt[4]{3}+2\\sqrt{3}-\\sqrt[4]{27}+\\sqrt[4]{1728}-4}\\cdot \\sqrt[8]{243{\\pi}^2}}{6\\sqrt[6]{1+\\sqrt6-\\sqrt2-\\sqrt3}{\\Gamma(\\frac{3}{4})}} \n", "89b9c9589ce7573337dcaf597a762b28": "\\begin{matrix} {10 \\choose 1}{4 \\choose 3}{44 \\choose 2} \\end{matrix}", "89b9f180bfe162c43d474d59260af28f": "\\langle X, \\mathcal{T} \\rangle", "89ba2218d1b8903a2202b95affd89e88": "\\sum_{k=1}^\\infty \\frac{\\cos(k\\theta)}{k}=-\\frac{1}{2}\\ln(2-2\\cos\\theta), \\theta\\in\\mathbb{R}\\,\\!", "89ba28697d2c4a21a4922192a2e97c0f": "\\mathbb Z^n", "89ba470162d14e9a51d3c86411fd1b47": "x = b!\\,\\biggl(\\sum_{n = 0}^{\\infty} \\frac{1}{n!} - \\sum_{n = 0}^{b} \\frac{1}{n!}\\biggr) = \\sum_{n = b+1}^{\\infty} \\frac{b!}{n!}>0\\,,\\!", "89ba4912fc9391077c27ca77b0613b94": "Au^+ (s) + 2CN^- (aq) \\rightarrow Au(CN)_2^- (aq)", "89ba8acad3c7bb6c9c87458c359b2a6b": "\\frac{dY^{S}(L^{D})}{dL^{D}} = \\omega", "89bb70013d08989c779f0e2880753671": "H(n)", "89bb839a6f85c9b267973f3fb094129a": "[0, \\infty) \\times \\Omega", "89bbbbf8f0473ee16134479fb812ccb4": "\\langle f(\\theta)\\rangle=\\int_{-\\infty}^\\infty p(\\phi)f(\\phi+2\\pi a)d\\phi.", "89bc0e09763fa93b9668f01d9ee43b0b": " f(x) = a x (1-x) ", "89bc313780d760fe28e986b4c55ded94": "f(x) = \\begin{cases}x^2\\sin{(\\tfrac{1}{x})} & \\mbox{if }x \\neq 0, \\\\ 0 &\\mbox{if }x = 0\\end{cases}", "89bc58b1ff7d7a367d7984bf8d0c45e8": "\\rho_c = \\frac{1}{\\Delta \\nu V}", "89bc6265fdce9407dbbb9e44b3898b7a": "\\mu^'_n = 2^{n-1}(n-1)!(k+n\\lambda)+\\sum_{j=1}^{n-1} \\frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\\lambda )\\mu^'_{n-j}. ", "89bcb89db32d03d7280aa4fa47ab6f1a": "D = e^{-r\\tau}", "89bcd2171c22144cb3bb8bc5b0371151": "\\mathbb E[F(x,\\xi)]", "89bcf2d9607725613f4b0d9b5242ad7c": "{\\mathit{He}}_5(x)=x^5-10x^3+15x\\,", "89bd08f13b4dc73089d232d4704af538": "\n\\begin{align}\n\\hat{A} \\hat{B} \\hat{C} \\hat{D} \\hat{E} \\hat{F}\\ldots &= \\mathopen{:} \\hat{A} \\hat{B} \\hat{C} \\hat{D} \\hat{E} \\hat{F}\\ldots \\mathclose{:} \\\\\n&\\quad + \\sum_\\text{singles} \\mathopen{:} \\hat{A}^\\bullet \\hat{B}^\\bullet \\hat{C} \\hat{D} \\hat{E} \\hat{F} \\ldots \\mathclose{:} \\\\ \n&\\quad + \\sum_\\text{doubles} \\mathopen{:} \\hat{A}^\\bullet \\hat{B}^{\\bullet\\bullet} \\hat{C}^{\\bullet\\bullet} \\hat{D}^\\bullet \\hat{E} \\hat{F} \\ldots \\mathclose{:} \\\\\n&\\quad + \\ldots\n\\end{align}\n", "89bd1c5efd6a9f185f2646a1338efa39": "t \\in T.", "89bd3761e72f4222748a623589f1e1bb": "\nM = \\sup\\{f(x) \\mid x \\geq 1\\}", "89bd5a203cca89c288aaa072f1462af3": "\\pi_i (B^+C)", "89bd72767d7de958ac6134ba770ce68b": "v(w, P)", "89bd805f6f752730e30a97b484111ace": "\n \\nabla^2\\nabla^2 w = \\cfrac{q}{D} ~;~~ D := \\cfrac{2h^3E}{3(1-\\nu^2)}\n ", "89bdcf4f84466004b5712f33f3a774ce": "\\,\\!m", "89bddeeb83ffad6e4a3c95df5d45031c": "\\nabla_m R^n {}_{ik\\ell} + \\nabla_\\ell R^n {}_{imk} + \\nabla_k R^n {}_{i\\ell m}=0,\\ ", "89bdeceab49000b9411aa2ff29ef08af": "(u_1,u_2,\\dots,u_m)", "89be13c11a2fcfc88d0a387151cdf6c2": "u^2 + v^2 = |w|^2\\,.", "89be42cbcfa8e0365fd70a56344d1a8d": "\\alpha A +\\beta B ... +ne^- \\rightleftharpoons \\sigma S+\\tau T ...", "89be52d683ee46eb5f696e7a9f3ae983": "X\\in\\mathfrak{X}(M)", "89be79dc5f6ff79f6560b3a8f033a787": "r = r_1 {\\left( 1 + \\frac{r_s}{4 r_1} \\right)}^{2}\\, ,\\quad r_1 =\\frac{r}{2}-\\frac{r_s}{4}+\\sqrt{\\frac{r}{4}\\left(r-r_s\\right)} \\,.", "89bea0624a5754dc737594163a1d6ee1": "\\operatorname{Cl}_{2m+1}\\left(0\\right)=\\operatorname{Cl}_{2m+1}\\left(2\\pi\\right)=\\zeta(2m+1)", "89becb2e924c6e33e1d5769366fbd6ca": "f'_+(t),\\,", "89befcb114bdfe7e7f629b5bcfaa48d1": "\\ H = \\frac{h}{h_0} = \\frac{c_pT}{c_pT_0} = \\frac{T}{T_0} ", "89bf928b3b2e6c95db544b8aa666eefa": " M \\mapsto S + \\lambda \\overrightarrow{SM}, ", "89bf9672283d852a58f8acb4d5886f66": "y_i = \\pm 1", "89bfaa90d7f4cab8af9bb92698098aa4": "k'=p_0/k", "89bfed480e5c89eff5dd6efdb9e847f9": "\\frac{d}{dy} = \\frac{dx}{dy}\\frac{d}{dx} = 2 y \\frac{d}{dx} = 2 \\sqrt{x} \\frac{d}{dx}, \\text{ and }", "89c026e77153189a6797961a4455897b": "\\sum_{n,k} \\frac{1}{(n+k)!}{n+k\\choose k} x^k y^n = e^{x+y}.", "89c04e97a92d803b8b91b4a4f3ffa544": "a=\\sqrt{Rd}\n", "89c0615290cc07f957f5d644076ad25b": "\\dot\\sigma", "89c0c6c79ef739a2da324649e86b9eb3": " L_b ", "89c0d4eb32fe9bacfc15e4443a74c6d3": "N_{\\text{var}}=\\frac{N_{\\text{vol}}}{2\\sigma_{\\text{strike}}}", "89c0e0e8dc86411bc9b5c2e0ba5602b8": "X_{\\mathcal{T}}", "89c0e55ce392a05e412ca3debe81a320": "\\boldsymbol{E}(r,\\theta,\\phi)=\\frac{e^{-jkr}}{4\\pi r}{\\boldsymbol{\\hat{u}}}(\\theta,\\phi)", "89c10da6253ba172d32f1a6209df5521": "X^\\flat (Y) = \\langle X, Y \\rangle", "89c121f65e31d2f2b12833475dac6873": "F3 = \\begin{bmatrix}\nT1 & T2 \\\\\n & R(A) \\\\\nR(A) & \\\\\nW(A) & \\\\\n & W(A) \\\\\nAbort & \\\\\n& Commit \\\\\n &\\end{bmatrix}", "89c12b8132ca2d7ce4fd8dc5ab950bf0": "v(x,t)=\\sum{u_n(x,t)}", "89c1938ed7ca9dc7be37461c071ca3c6": " \\mathcal{A}\\left\\{x[n]\\right\\}\\ \\stackrel{\\text{def}}{=}\\ \\sum_{k=n-a}^{n+a} x[k]", "89c1cce125c2996056b06517195492cc": " GB2(y;a,b,p,q) = \\frac{|a|y^{ap-1}}{b^{ap}B(p,q)(1+(y/b)^a)^{p+q}} ", "89c26d24b19ad8d828571b706ed2a61f": "(X-r)|P", "89c2c9e4dde059304e71933a8ba57d24": " P(0)= E \\left({\\mathbf{x}}(0){\\mathbf{x}}^\\mathrm T(0) \\right).", "89c2e2e59a9025afd4e9137f2603a96b": "F_t = E_t(S_{t + 1}) + P_t", "89c2e5f6bec9467c14dd8e9ff6022522": "\\Delta f\\,\\!", "89c35808c7f96f4d7b4dcbc14d991dd0": "E(p,t)", "89c35f42524b9b737cd437dac8bf413f": "\\alpha \\geq 0", "89c36955d4fde406e986bb410b5c255e": "\\frac{dq_1}{d\\sigma} =\\frac{\\partial P}{\\partial u_1} \\quad \\quad \\frac{du_1}{d\\sigma} =-\\frac{\\partial P}{\\partial q_1}", "89c38b3980595ba802ac306e5c90582a": "r= \\frac{M_2 -M_1}{M_1}", "89c39ab73952440d9e7f4e42663ee3ea": "u\\cdot v", "89c3beedfb635fca92a585e6ca4f5ef7": "e = \\frac{V_V}{V_S} = \\frac{V_V}{V_T - V_V} = \\frac{n}{1 - n}", "89c3f26161d0ee91b482a757c3d17efa": "\\mathbf{E}(\\mathbf{r}, t) = \\frac{1}{4 \\pi \\epsilon_0} \\int \\left[ \\left(\\frac{\\rho(\\mathbf{r}', t_r)}{|\\mathbf{r}-\\mathbf{r}'|^3} + \\frac{1}{|\\mathbf{r}-\\mathbf{r}'|^2 c}\\frac{\\partial \\rho(\\mathbf{r}', t_r)}{\\partial t}\\right)(\\mathbf{r}-\\mathbf{r}') - \\frac{1}{|\\mathbf{r}-\\mathbf{r}'| c^2}\\frac{\\partial \\mathbf{J}(\\mathbf{r}', t_r)}{\\partial t} \\right] \\mathrm{d}^3 \\mathbf{r}'", "89c452dc3a032894f09b820576158ef0": " z_{cr} = 0\\,", "89c4d562c3414bf185474cc079b4bce8": "\\bold {n}_1 \\times \\bold {n}_2", "89c51b090bacb5170f688159beb303be": "|p_i\\rangle", "89c55cd1b934839237a1b3be0dfbe2db": "( < \\alpha_i , \\alpha_j > )_{i,j=1}^n", "89c5a977565cdd0a6b2912ecb5be5e21": "\\textstyle (P_1,L_1)", "89c5c26389d49a337fc0a2d44624cfae": "\\tilde A", "89c5e36062b7cd01f5d186f8b23f441d": "\ng_{0}(\\varepsilon) = \\frac{\\nu |\\mu(\\varepsilon)|^2}{4 \\varepsilon_0 \\pi n} \\left( \\frac{2 m_{\\mathrm{r}}}{\\hbar^2} \\right)^{3/2} ~,\n", "89c5f56b9af8638cb80b4ee28416a952": "\\ddot{\\bold{r}} \\times \\bold{H} = -\\frac{\\mu}{r^2}\\bold{u} \\times (r^2\\bold{u} \\times \\dot{\\bold{u}}) = -\\mu\\bold{u} \\times (\\bold{u} \\times \\dot{\\bold{u}}) = -\\mu[(\\bold{u}\\cdot\\dot{\\bold{u}})\\bold{u}-(\\bold{u}\\cdot\\bold{u})\\dot{\\bold{u}}]", "89c603adec264a14e4dfaca563552419": "\\langle \\bar{q}^a_R q^b_L \\rangle = v \\delta^{ab} ~,", "89c6255af70329224e1ced9bf695ff14": " e^{-\\alpha|t|} \\ ", "89c6294f708790e08769d18f25c0ed3c": "GL(6,\\mathbb{R})", "89c6523eca079d432535dac23a612444": "L=L'_{0}/\\gamma", "89c6608dd06bef00dd507e743946b61e": "P = \\lbrace x, y \\rbrace", "89c6c4fd1185e6eaa471b97265fd9ee4": "\\scriptstyle\\partial_x", "89c6d5b781a07b1a3ec958b42a416c6c": "\\begin{matrix}\\mathrm{Cabtaxi}(9)&=&424910390480793000&=&645210^3 + 538680^3 \\\\&&&=&649565^3 + 532315^3 \\\\&&&=&752409^3 - 101409^3 \\\\&&&=&759780^3 - 239190^3 \\\\&&&=&773850^3 - 337680^3 \\\\&&&=&834820^3 - 539350^3 \\\\&&&=&1417050^3 - 1342680^3 \\\\&&&=&3179820^3 - 3165750^3 \\\\&&&=&5960010^3 - 5956020^3\\end{matrix}", "89c6e8782f5b08e057458333b3beb8c6": "\\langle x(t) \\rangle = \\langle x \\rangle_0 - \\beta f_0 A(t),", "89c6f48ee0ff6b04f7aebc48283eb9ae": "\\mathbb{W}", "89c6f5d1b8ddd0a9532a010dfb6c534f": "{\\mathbb B}", "89c6fbb7b9dfe48d98bf1a6c3fa7124d": "r\\times X:I\\times X\\to PX\\times X", "89c73badd0167f92b2848d0694c53d18": "\\begin{align}\n\\sin(x)&=\\sin(y)\\\\\n\\cos(x)&=-\\cos(y).\\\\ \n\\end{align}", "89c779619fa4782790a2539d20af0a03": "\\operatorname{Var}(X) = \\operatorname{Cov}(X, X).", "89c783b29fd77eda3028d4f4fdc2dc77": "z_{n-1}", "89c7c20211366e678e9a9b9b81c94cbe": " V = p, W = f, E = (p\\ f)\\ (p\\ f), Y = (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "89c7d2b40e5ac0a1330d36e1d4b1c054": "V_\\max1 - (V_\\max1 - V_\\max2 ) \\cfrac{[X]}{[X]+K_x} ", "89c83a8dfdf751487852c4d243dfb9ae": "\n \\begin{align}\n F &= U_0 & \\qquad & \\text{ for } z = 0,\n \\\\ \n F &\\to 0 & \\qquad & \\text{ for } z \\to +\\infty.\n \\end{align}\n", "89c841bc50c8d3d46da87bd92210818d": "G=\n\\begin{array}{c}\nu \\\\\n\\longrightarrow \\\\\n\\left[\n\\begin{array}{rrrrrrrr}\n-415.38 & -30.19 & -61.20 & 27.24 & 56.12 & -20.10 & -2.39 & 0.46 \\\\\n4.47 & -21.86 & -60.76 & 10.25 & 13.15 & -7.09 & -8.54 & 4.88 \\\\\n-46.83 & 7.37 & 77.13 & -24.56 & -28.91 & 9.93 & 5.42 & -5.65 \\\\\n-48.53 & 12.07 & 34.10 & -14.76 & -10.24 & 6.30 & 1.83 & 1.95 \\\\\n12.12 & -6.55 & -13.20 & -3.95 & -1.87 & 1.75 & -2.79 & 3.14 \\\\\n-7.73 & 2.91 & 2.38 & -5.94 & -2.38 & 0.94 & 4.30 & 1.85 \\\\\n-1.03 & 0.18 & 0.42 & -2.42 & -0.88 & -3.02 & 4.12 & -0.66 \\\\\n-0.17 & 0.14 & -1.07 & -4.19 & -1.17 & -0.10 & 0.50 & 1.68\n\\end{array}\n\\right]\n\\end{array}\n\\Bigg\\downarrow v.\n", "89c85ba52c965c216b01c462a9cf3d00": "\\beta=L_a^{-1}", "89c90f8e6146adcad8ba043d22f6b739": "W = \\Delta KE.", "89c9119c1647bebf9b7762d819d9dbbb": "\\mathbb F_n^n", "89c921306a7646f99f65b0afa404616c": "T - t_0", "89c94049331c03f4cf43b363b6806c37": "o(n^{3+\\epsilon})", "89c948262c4478dd8de630fa031c3cb3": "\\tilde{\\phi}_{\\mathbf{R},i}", "89c98e0182505b23d02e7f204f7e9e90": "T = a + b \\log_2 \\Bigg(1+\\frac{D}{W}\\Bigg)", "89c9c75d895a09d1a895a999035bf5f1": "ab=\\sum_{i=0}^m \\sum_{j=0}^m 2^{w(i+j)}a_i b_j = \\sum_{k=0}^{2m} 2^{wk} \\sum_{i=0}^k a_i b_{k-i} = \\sum_{k=0}^{2m} 2^{wk} c_k ", "89ca340aa4ad23cad307e41ff40182a7": " (\\nabla f)_x\\cdot v = \\mathrm d f_x(v)", "89ca5934761f5bf30ab1a75711dc3760": "1 \\,-\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,-\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,-\\, \\cdots \\;=\\; \\ln 2.", "89ca59ec83a71c7d447ec05d033bd41b": " {R^\\rho}_{\\sigma\\mu\\nu} = \\partial_\\mu\\Gamma^\\rho_{\\nu\\sigma}\n - \\partial_\\nu\\Gamma^\\rho_{\\mu\\sigma}\n + \\Gamma^\\rho_{\\mu\\lambda}\\Gamma^\\lambda_{\\nu\\sigma}\n - \\Gamma^\\rho_{\\nu\\lambda}\\Gamma^\\lambda_{\\mu\\sigma}.", "89ca9cca2dd74e7157afba5844373469": "k(G)", "89cb4802bd22373d4cf9553e5f21a227": "\\begin{align}\n\\dot c_1&=\\frac i2\\Omega_p c_3\\\\\n\\dot c_2&=\\frac i2\\Omega_c c_3\\\\\n\\dot c_3&=\\frac i2(\\Omega_p c_1+\\Omega_c c_2).\\end{align}", "89cb7739ba9372700c1c3c48e4882f21": " \\rho_j = \\int\\limits_{-\\infty}^{\\infty} R_{z}(\\phi)|\\psi_j\\rangle \\langle\\psi_{j}|R^{\\dagger}_{z}(\\phi)p(\\phi)\\, d{\\phi}", "89cbd58a97bb6ce8e5676c6f24bec457": "q\\in H_0", "89cc0e4a525f91c61ff7741130123854": "p_\\alpha = \\operatorname{sup}\\{p_\\beta \\vert \\beta<\\alpha \\}", "89ccf4fca0e9c69279beb331d3e687d9": "\\tilde{\\theta_i} = E(\\theta|k_i) = \\frac{k_i + \\widehat{M}\\hat{\\mu}}{n_i+\\widehat{M}} = \\frac{\\widehat{M}}{n_i+\\widehat{M}}\\hat{\\mu} + \\frac{n_i}{n_i+\\widehat{M}}\\frac{k_i}{n_i}.", "89cd450cbb693d6135aa2d9b442ee49c": "=\\mu N\\,", "89cd5efa704c5c34b62549c005b3becd": "\\mathbb{E} \\left[ \\left( \\int_{0}^{T} X_{t} \\, \\mathrm{d} W_{t} \\right)^{2} \\right] = \\mathbb{E} \\left[ \\int_{0}^{T} X_{t}^{2} \\, \\mathrm{d} t \\right],", "89cd699a347fd99542cd5b98b9b02ed6": " { mv^2 \\over 2 } = h \\nu - 13.6 eV", "89cd8a06c81cab237b635eeee6f24e21": "R_\\infty=\\frac{\\alpha^2}{2\\lambda_e}", "89cd8ec5cbb327fc7ed6ad735c1dd2d5": "f_1(\\cdot)", "89cd9730e5fd585c4a0235727c752d46": " R. ", "89cdaade8ab2be96ff644dad9846ffe7": "T^*", "89cdacec933c51ba49e5f169b3f8c1d1": "p(X,A,\\textbf{h}, \\omega | \\theta) ", "89cdc7a303e599b1bba0201fac0fa300": "\\frac{1}{2}\\omega^2 Y_1^2 m_1", "89ce01a43c02c327eecb10bc039e91c3": "\\sin\\theta = \\left (\\frac{ m\\lambda}{2\\Lambda} \\right)", "89ce40f0499448ff66cd7e785b66de4e": "(B_s)^{-1}(A)", "89ce5ee6333d5036f1b2dc2af0faf578": "{49\\choose 6} = 13,983,816", "89cec393cf370c9e1aeb992e58edaa99": "m = Z m_{p} + N m_{n} - \\frac{E_{B}}{c^{2}}", "89ceea89c813bb1605374af7323c1cfe": "R(x)=B(x)e^x-A(x)\\,", "89ceef28ea4df70449d0d2f81e2adccf": " \\omega_c = \\frac{R_0}{L}", "89cf67e25398af83cf15e462c6f3e88b": "k_1< k_2 < \\cdots 0} \\alpha}.", "89d57488de49f3358761afd539a56421": " C_2 \\cong \\{\\pm1\\} \\cong \\operatorname{O}(1) \\cong \\operatorname{Spin}(1) \\cong \\mathbb Z^*", "89d5b562dd4aba10b0e2b7099e74d0de": "\\pi:\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}.", "89d5d7be7e7be8377d17c8475e7e2b0a": "\\nabla\\times(\\nabla\\times\\mathbf{B})=\\nabla\\times(\\alpha\\mathbf{B}) ", "89d621266ffa715918ddef902b0aec0a": " T_n (x) ", "89d623e324bb186bcbf18a3777c0ade6": " \\|x\\| =\\sqrt{\\langle x, x\\rangle}.", "89d63950993cee5e13721214836336ba": "y \\vee x \\vee y = y", "89d6873b82791f111f6a38824803438b": " U-B = M_U - M_B\\!\\,", "89d768052c3cd1b859b26b8d2c407303": "H_1(\\mathrm{A}_3,\\mathbf{Z})=\\mathrm{A}_3^{\\text{ab}} = \\mathrm{A}_3 = \\mathbf{Z}/3", "89d774eafe9722fd16140981ff3e0a6d": "w = z^{1-a}u,", "89d792075ab01b69f196fec6975aea0e": "\\preceq", "89d7a57f44479b2056f57917afe768aa": "A\\equiv((B\\equiv(A\\equiv C))\\equiv(C\\equiv B))", "89d7b0c3bb496bd3940c10302f28121c": "\\mod 8", "89d7b73007774eb3de166a86b7686ac6": "\\mathbb{Z}[i]", "89d7d4bccc2d45250802cef0e964023b": "\\lim_{t \\to 0} K(t,x,y) = \\delta_x(y)\\rm{\\ \\ for\\ all\\ } x,y\\in\\Omega", "89d7d4cd0eadc4cb87f2e73fb669bc20": " (1 - \\delta_X(t))^{j}, \\ j = 1, \\ldots, n.", "89d7d778eae0db18eb3bd728cfd4f1f4": "q(\\alpha^i)=nv_{n-i}", "89d809eecbcc8a8884ce1e7dc23a3d6a": "\n\\chi_s(z) = \\tfrac {1}{2} \\left[ \\operatorname{Li}_s(z) - \\operatorname{Li}_s(-z) \\right] .\n", "89d83bd6b3db24f577ba386bd384239b": "q^{i}(\\xi,\\tau)\\wedge q^{j}(\\xi,\\tau)=\\xi ^{i}\\wedge \\xi ^{j} ", "89d86ab46448cf8496faa2aca07ee168": "\\mathrm{Var}(z, [a, b]) = \\sup \\left\\{ \\left. \\sum_{i = 1}^{k} \\| z(t_{i}) - z(t_{i - 1}) \\| \\right| a = t_{0} < t_{1} < \\cdots < t_{k} = b, k \\in \\mathbb{N} \\right\\}.", "89d88048f20ad7fe221d0d963e94824f": "\\hat{H}_{0}", "89d89a867bbce10193e3d7f15e4288fd": "\\hat R_{\\mu}", "89d89ab2c842901ce529947dce4a2063": "\\mathbf{a}_{\\mathrm{r}} \\ \\stackrel{\\mathrm{def}}{=}\\ \\left( \\frac{d^{2}\\mathbf{r}}{dt^{2}} \\right)_{\\mathrm{r}}", "89d8db322ef2ec0cac8b54f35b2f4c49": "\\{q_j\\}_{j\\in \\N}", "89d8ff183667e13a9f50034508724970": "\\theta = \\omega t", "89d9053b65087ff8952e625aa4e287b1": " \\pi_F\\circ\\varphi = \\pi_E ", "89d921d9e024ea96f898cde8e6d67a2b": "\n \\boldsymbol{P} = 2~\\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{C}} \\qquad \\text{or} \\qquad P_{iK} = 2~F_{iL}~\\frac{\\partial W}{\\partial C_{LK}} ~.\n ", "89d93af0b1f14d0b4eb09e83f6059fa1": " M_{\\mathrm{Pl}}^2 = M_{\\mathrm{Pl}_{3+1+\\delta}}^{2+\\delta} n^{\\delta}. ", "89da153e127575762a086544c49ba3e3": "-\\frac{1}{f}", "89da23bdfa2bf27acae25bebbbc937e5": "P_X^*", "89daf2dacfa1b90d25b2a3715d3e5ebe": "\\pi^{ab}", "89db2baedf2f0119661edd513dc4e477": " \\frac {(m_s \\psi g-m_s g)^2}{2k}+\\eta \\mu S m_s g (\\psi -1)=\\frac {1}{2} m_s v_r^2", "89db5bfed25b3e5130597bd8bc314fe9": "\\lang B,(--),()\\rang", "89db66f64d8fb0ba94b8382ecc145391": "s_n(t) = \\sqrt{\\frac{2E_b}{T_b}} \\cos(2 \\pi f_c t + \\pi(1-n )), n = 0,1. ", "89db6a5333eb8b0230f239935b2af96a": " \\mu(g) = \\lim \\mu(f_n),", "89dbe7566f5b328cd8fb42c824edbaf5": "DTS = \\exp ((9.5\\times10^3).(\\frac{T - 288.2}{288.2T}))", "89dc43c667f806b263d73451d81c5283": "f^{0}(0) = x = 6", "89dc6c78fa3a1ea7503555b187691df8": "\\Iota \\Kappa \\Lambda \\Mu \\Nu \\Xi \\Pi \\Rho \\!", "89dcadd3e96c8b1beb57728a2877cd91": "\\left |\\beta\\right |=\\tfrac{1}{3}", "89dcb196c1179bb542c9526d20be3857": "r = \\left\\lfloor \\sqrt{2k+\\sqrt{2k}} \\right\\rfloor,", "89dcd0d2e9c8c729b0475c5691ec117e": "k=1,2,\\ldots", "89dce6fc7acc592d224ea5bb377cc5eb": "{\\alpha}<-1.6", "89dd25de577cf4840ba3149ee936de80": "\\mathbf{\\mathit{S}}", "89dd46dfd5ac422d57ddaa6237c56f8c": "[0,255]", "89dd7d0ac068cb80afed396e003f98e7": "3n^3/2", "89dd933e7f73db006496399c985dda96": "\n\\phi = \\arctan\\left(\\frac{y}{x}\\right)\n", "89dde79fcfb8b82265fd9c777965fb05": "\\gamma=\\epsilon^9/108-", "89ddf20ec45395aeafae2c2d7a746db3": "x^*[n]", "89de0038ed5b279499a601c086b2f4ac": "(M \\cup_f H^i) \\cup_g H^j", "89de41ebeb24ce484486e6526fb809e3": "\\int_c S(c, c) \\to \\prod_{c \\in C} S(c, c) \\rightrightarrows \\prod_{c \\to c'} S(c, c'), ", "89de4bed697ba94a8fac3ff76728b629": "E_{out}=4\\pi R^2_\\oplus \\sigma T^4_{eq}", "89defcf158bedc3501b551aa7e384a2f": "G_k \\cup \\{ E_{k+1} \\}", "89df126d24ddaf2d0472285dcff4bc91": "A \\cup B = B \\cup A\\,\\!", "89df16d1caf129bc30f7b41348d24d68": "V_{\\mathbf{k}}", "89df688d499e9766560be4a8027fb9da": "v_{evac} = C_f\\, c^* \\,", "89df7401551d525a04db9dc12e4684ba": "x\\times y", "89df8a127b1699369774696fabd10ed2": " b^*_2 ", "89dfc997619e6cfeb74e9067e6e04c87": " \\bar{z} = x - iy = |z| (\\cos \\phi - i\\sin \\phi ) = r e^{-i \\phi} \\ ", "89e0056bd7562cfc01d250e6c9125ef6": "(p_3(t) x_3^\\prime)^\\prime + q_3(t) x_3 = 0, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x_3(a)=1,\\,\\, x_3^\\prime(a)=R_3\\,", "89e01e28c051c8e5cc40c787a7f132f3": "\\lambda = \\arcsin\\left(\\frac{z}{\\sqrt{z^2+\\rho^2}}\\right)", "89e0366771babb033c0f286bdce34f3a": "v_i (X) = 1", "89e05e2e05804405beb7646eb4a1016a": "g_1\\in G", "89e0a1d3f5531f05bff57f0574bed384": "\\mbox{vec} (\\mathbf{H}) \\sim \\mathcal{CN}(\\mathbf{0},\\,\\mathbf{R})", "89e0a2f36ab480ba2b9a1196c0aed89b": "x = \\sum _{i=1} ^n \\alpha _i v_i", "89e0a5ff1a3f303e6aa047b22febfb8f": " s : G \\times X \\rightarrow [0, \\infty) ", "89e0afca6f9dcae5f6ec3b4c1c37c2e4": "\\left( \\Phi_{X} (\\omega) \\right) (t) := X_{t} (\\omega).", "89e0e879e9a2ed9f9340f069b7dbe655": "B_{2n}(x)=\\frac{2(-1)^{n-1}(2n)!}{(2\\pi)^{2n}} \\, \\sum_{k=1}^{\\infty}\\frac{\\cos 2\\pi kx}{k^{2n}}", "89e0ffe9dfc20029b3d33bd14509bf29": "{\\mu}m", "89e156ad1117e8b0651176fd15c4a9f5": "\\varphi(B)=\\frac{-1}{\\varphi(A)}, \\quad \\varphi(C)=\\frac{1}{\\varphi(A)}, \\quad \\varphi(D)=-\\varphi(A). ", "89e1d00255b5cf0aa0580ef022f182a0": " \\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial}{\\partial \\theta}[\\rho v] = 0, ", "89e2452cfbaa487ea4a2c07b979094bb": "y(x)\\approx 2x\\left\\lceil\\sqrt{n}\\right\\rceil", "89e250fae737192968ae6899112ff87e": "\\begin{align}\nt' &= \\gamma \\left( t - \\frac{vz}{c^2} \\right) \\\\ \nx' &= x \\\\ \ny' &= y \\\\\nz' &= \\gamma \\left( z - v t \\right)\\\\\n\\end{align}", "89e264b032b4cf8498af5197fa525273": "\\sin(\\alpha)^2=1-\\cos(\\alpha)^2", "89e2887a958b9b7bad8968d6a3d1c0d1": "\\lambda(L)/2", "89e2924b508410fe00c8046508466f43": " \\hat{S}^2 ", "89e2c0ab12759d684eb063495d9a74ba": "4^{4-2}=16", "89e2ca0edf96c962d122abb4f55107a1": "a_E = \\sqrt{2 \\lambda L_E} \\qquad a_H = \\sqrt{3 \\lambda L_H}", "89e32b59117814b777699f710349bb86": "\\gamma = \\sqrt{1+\\left ( \\frac{p}{m_0 c} \\right )^2 } ", "89e3549d2a9422cc00b8c7ee3fce0497": "\\Gamma^{*}", "89e3b8ddc238b5951f3934a4ad597c0c": "\\frac {\\partial} {\\partial q} \\zeta (s,q) = -s\\zeta(s+1,q).", "89e3c825aa3f1068fb170ce0ff652d45": "P(u)=u_{tt} - c^2\\Delta u\\,", "89e420e2527d78483f8ab4f0570ed373": "\\overline{x}\\langle z \\rangle.P | x(y).Q \\rightarrow P | Q[z/y] ", "89e536c7232f46a85584d5863598030a": "\n^*\n\\begin{Bmatrix} P \\\\ \\oslash \\end{Bmatrix}\n\\begin{Bmatrix} w \\\\ m \\\\ \\oslash \\end{Bmatrix}\n\\begin{Bmatrix} l \\\\ r \\\\ y \\\\ n \\\\ \\oslash \\end{Bmatrix}\ne\n\\begin{Bmatrix} l \\\\ r \\\\ y \\\\ n \\\\ \\oslash \\end{Bmatrix}\n\\begin{Bmatrix} w \\\\ m \\\\ \\oslash \\end{Bmatrix}\n\\begin{Bmatrix} P \\\\ \\oslash \\end{Bmatrix}-\n", "89e5863c8fa817457dad0375c72e0dc6": "{\\mathbf{K}}_\\mathrm{T}=\\frac{[cis-enol]}{[diketo]}", "89e5912491880ac8f861a334e0726d4b": " \\begin{align}\nx_i = \\frac{i}{N}.\n\\end{align}", "89e59c7d3cdc72af74537c3a15fe2223": "A\\in\\mathcal{V}\\,", "89e5aad04241b63284c21469db829193": "K^k_r", "89e5bdc961f959d3545fd178658f836b": "\\phi_{;\\mu}\\;", "89e5dda1d8bf5a8566ecfd708df46d1b": "\\mathbf{T}(\\mathbf{X}) = \\sum_{i=1}^N \\mathbf{T}(x_i) .", "89e663ce1d81742296aaf3ef4bce9db4": "T \\vdash S", "89e67654b4e8d0c87c73b21401ade785": "c(\\overline{root1})=11,\\ \\ c(\\overline{root2})=5", "89e6e14eef6a361bd8d9cbcad5bb98f5": " \\varphi(x) = f(x)+ \\lambda \\int \\limits_a^b K(x,t)\\,\\varphi(t)\\,dt. ", "89e751961172ea0b6b6c7968aa1acb41": "\\zeta_X(s) = \\prod_p \\zeta_{X_p}(s).", "89e766e9f05f5a3ff8b68db8c900d4ed": "\\mathcal{B} \\left\\{f(t)\\right\\} = F(s) = \n\\int_{-\\infty}^\\infty e^{-st} f(t) \\,dt.", "89e7ddea25d21557b523042e632eb1cf": "\\int \\frac{1}{\\sqrt{a^2+x^2}}\\,dx", "89e850747ce99b04579f5a37cf21f870": " x^{\\log_b(y)} = b^{\\log_b(x) \\log_b(y)} = b^{\\log_b(y) \\log_b(x)} = y^{\\log_b(x)} \\!\\, ", "89e888fcfe0a59ec049dc9ae509f768f": "MinVol(M):=\\inf_{g}\\{Vol(M,g) : |K_{g}|\\leq 1\\}", "89e96b4e4bb95f0073d0e8a766c7180d": " \\mathbf{E}_{\\rm est} = \\mathbf{U} \\, \\mathbf{S} \\, \\mathbf{V}^{T} ", "89e99198bad45a01df7b2634bf368eb6": "\\rho = \\frac{1}{4}\\begin{pmatrix}\n1-p & 0 & 0 & 0\\\\\n0 & p+1 & -2p & 0\\\\\n0 & -2p & p+1 & 0 \\\\\n0 & 0 & 0 & 1-p\\end{pmatrix}", "89e991c70f089f676e2d1d4fcac765e7": "g_{\\mu\\nu}", "89e9c902fed032a267e19bcbbe710b75": "\\overrightarrow{a}", "89eaff44dacb455aca540625cbe4cd98": "\\neg p", "89eb0448913d4b93f07148f3f9196b80": "\nX_{6}=[3,7].\n", "89eb4d5abd8caec6046112ca865c9976": "dE_\\theta(t+\\textstyle{r\\over c})=\\displaystyle{-d\\ell \\sin\\theta \\over 4\\pi\\varepsilon_\\circ c^2 r}{dI\\over dt}\\,", "89ec2868cc2e7d7d349f2e47ec28a3aa": " h_0 = f(iv) \\oplus iv ", "89ec78b553386e1bbad277b62e146dd2": "a^{n-1} \\not\\equiv 1 \\pmod{n}", "89ec7dcfab68fe9fa5a75a4ff1952a4f": "n = m^k", "89ecdf8f38b39f9b8131aa2fd31f25d2": "h_a(x) = (a\\cdot x\\,\\, \\bmod\\, 2^w)\\,\\, \\mathrm{div}\\,\\, 2^{w-M}", "89ece24911e79d018934c8ce57c45969": "(q_1^*,q_2^*)", "89ed394d0ad743a350cd59d2c5165e41": "\\scriptstyle{\\left(\\frac{dB}{dt}\\rightarrow\\infty\\right)}", "89eda00eda7f7823a6b3b85b0aa9d4bd": "F: C \\to \\mathbf{Set}", "89edeec4daf75c58aa28b7af1ba34365": "\\frac{dy}{dt} = iPe^{it}", "89ee3ba55f4db78bdcaeb20f1f40c6eb": "X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow {w} ", "89ee89b4fe44d8d915f0b7f9669bc474": " F = f( Y ) ", "89ef2cea7af73c738f3b09ab908fab71": "V_t \\times F_e = V_t \\times F_a - V_d \\times F_a ", "89ef5ab7989dbb4badee8713f968b09e": "\n\\frac{A\\hbox{ prop} \\qquad B\\hbox{ prop} \\qquad A\\hbox{ true} \\qquad B\\hbox{ true}}{(A \\wedge B) \\hbox{ true}}\\ \\wedge_I\n", "89ef646e1721fff27c097cff44503319": "P_2(x)", "89ef8fd88c0761109d299f2b469830d3": "\\frac{5\\pi}6\\!", "89efbaa5cb1d023c6a646b66730e8cd5": " \\tau^{-1}(\\beta)(\\alpha) = \\sum_{i\\in I} \\beta(i)(\\alpha(i))", "89efc832a6bd4a5fdeefd909ffcf05f2": "\\rho^{T_B} = I \\otimes T (\\rho) = \\sum p_i \\rho^A_i \\otimes (\\rho^B_i)^T ", "89efd398b3202dcde47e7143ef18cf97": "\\exp(j\\theta) = \\cos \\theta + j \\sin \\theta\\,", "89eff17bb11349e3a5675b8e7ef97dfb": "(x + 1)^{3}p(\\tfrac{1}{x+1}) = 56x^3+56x^2-56x+8", "89eff515efb5c60bcbafd65e2bdaf585": " \\operatorname{de-lambda}[x = \\lambda x.f\\ (x\\ x)]", "89f00436b046ce71b4bfe5f2adbf8214": "\\scriptstyle f|_{S\\cap V} = g|_{T\\cap V}", "89f0488ca62ff1041262ffbb20be067a": "U_p=\\frac{F^2}{4\\omega^2}", "89f08099c6d6fd7dcea4b6fb4f017a58": "1,208 \\times {4 \\choose 1} + 432 \\times {4 \\choose 1}{39 \\choose 1} + 64 \\times {4 \\choose 1}{39 \\choose 2} = 261,920", "89f0bd5419d047a36bcd771571882fb6": "2g+2", "89f0eb104fa5bd292b636e3c02ea6b4a": "(\\sin(t), \\cos(t))", "89f0eb2c88cdaf936a9b341f1035c964": "f\\left(\\frac{n}{m}\\right)=f\\left(\\frac{1}{m}+\\cdots+\\frac{1}{m}\\right)=f\\left(\\frac{1}{m}\\right)^{n}", "89f0f325b1b253437c2379817f05044a": "f(x) = \\sum_{i=1}^n p_i \\delta(x-x_i).", "89f0fd5c927d466d6ec9a21b9ac34ffa": "325", "89f1181d3c4adf5ffa6d5d3e43674a19": "\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}} = \n \\cfrac{\\partial W}{\\partial I_1}~\\cfrac{\\partial I_1}{\\partial \\boldsymbol{C}} = \\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{\\mathit{1}} \n ", "89f13bf4e07088f5deab1570535a7894": "\\Delta I = Nk[\\Theta(V/V_o)+\\frac{3}{2}\\Theta(T/T_o)]", "89f15d06a6319df0fbcaa1ed785eb256": "(M_0,M_1)", "89f16fc859e5c18fe9f24f63c9ac624f": "\\scriptstyle \\mathbb{P} ", "89f17fc46c0ff8e92fe31d89c6ba8b94": "S = \\frac {u_G} {u_L} = \\frac {U_G(1-\\epsilon_G)} {U_L \\epsilon_G} = \\frac {\\rho_L x (1-\\epsilon_G)} {\\rho_G(1-x) \\epsilon_G}", "89f1b85bc0188ca2cc60deced25efe8e": " \\min |x_i-x_0|\\le \\gamma\\le \\max |x_i-x_0|. ", "89f2596c184e136f628e7a71b8d5e0f0": "C_i'", "89f2866c2d89238961234f4f9a418e39": "1 \\times \\sqrt{6}", "89f28af2ae7ea496f45922d89a4ad51d": "\\{x[m];\\ m\\}.", "89f2a99389dcd2ad818db98be26d6f8d": "\\pi_A(\\{ \\langle A=a, B=b \\rangle \\}) \\setminus \\pi_A(\\{ \\langle A=a, B=b' \\rangle \\}) = \\emptyset\\,,", "89f2eb8b6676972253f00bef7361a9a6": " \\sum_{i=1}^N a_i [G/G_i],", "89f329f179297592800f526ee0477154": "v,A(v),A^2(v),\\ldots,A^{d-1}(v), ~\n P(A)(v), A(P(A)(v)),\\ldots,A^{d-1}(P(A)(v)), ~\n P^2(A)(v),\\ldots, ~\n P^{k-1}(A)(v),\\ldots,A^{d-1}(P^{k-1}(A)(v))", "89f3a6e81a2f91cd1b06025ffe4ff2f8": "S_0 < \\cdots < S_k", "89f3aed9c2e61efcaf4947fce08ee9e2": "\\Delta_2^0", "89f3fa3c4cf5c866dfd9588db141bb4b": " a_0 E_\\mathrm{h} / \\hbar = \\alpha c", "89f40d4e19863323097d8c692e42d0cb": "(a\\otimes b)(c\\otimes d)", "89f4a69e7d79d3656dba5d267d1fe414": " \\therefore J_n = -\\frac{\\cos{ax}}{(n-1)x^{n-1}}-\\frac{a}{(n-1)^2}\\left (-\\frac{\\sin{ax}}{x^{n-1}}+aJ_{n-2} \\right )\\,\\!", "89f4b522ac6aee10b3bb4e39ababbe7c": "\\left(\\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+2\\sqrt{2})\\right)", "89f4b70ca740f11b18ec035ac8c92621": "\n\\sigma_{\\bar{x}}^2 = \\frac{ 1 }{\\sum_{i=1}^n w_i},\n", "89f4bf650e0d73dc941c7042b98c0298": "2^{-\\delta n + 1}", "89f4c0e5df471299420da891cec9444b": "c_1=h_{1k}.\\,", "89f4f580f30df34ec52c08b7388ef0b7": " \\hat{H} = \\hat{T} + \\hat{V} ", "89f58f51b71f7c2e54003117d0b0dfdd": "\\scriptstyle \\pi = \\{ 3, \\frac{25}{8},\\frac{201}{64}, ... | 4, \\frac{7}{2}, \\frac{13}{4}, \\frac{51}{16},... \\}", "89f59dc4f60d391856468c228cb94498": "C_{qs}^0", "89f5a52791bd413686b44b38a25f8616": "0 \\leq \\frac{h(x)}{h(x) + 1} = h(y) < 1 ", "89f6206d902ad88ffd26a7bdf7123d06": "\n\\frac{1}{r} = A \\cosh\\left( k\\theta + \\varepsilon \\right)\n", "89f620a0fc39488e8b8a46e3276d7928": "(t,x) \\longmapsto X(t,x)=X_t(x) \\in T_xM", "89f626356e42d1053fdf1ca065e9968a": "\\mathbf{p}=\\left(p_{I},p_{X},p_{Y},p_{Z}\\right)", "89f674be19205a04baa65110228ec7aa": "\\rho^\\nu_\\mu", "89f6afe826b2c6ae118a48a1ee4bbc7a": "\\lnot (\\exists x P(x))", "89f6d2b1f9a789c678bc7ae73b6542d8": "x^2+px+q,", "89f703a98c5e711686615d79a6b753d0": "\\neg Q \\to \\neg P", "89f72005f28746f134c2a45e0cd68bd3": " \nL(t) = \\frac{1}{2}\\sum_{n=1}^N\\sum_{c=1}^N Q_n^{(c)}(t)^2\n", "89f72d5898742339620aded91f804eb5": "\\dot{\\textbf{x}} = X(\\textbf{x})", "89f74fbf30ed711232022aad15f229bc": "f:G\\rightarrow H", "89f7656a6027815bf039561871d18a48": " {a_b}_{c_b}= {a_c}_{b^c} ", "89f78a1a3061f469ce98abb8348d488d": "1\\le p < \\infty", "89f79dece87fd660171af79dd9d21a97": "\\mathbf{T}^1", "89f7a4f1708398f75eaa39d9f53483cf": "L_{p,\\mathrm{loc}}(\\Omega)=\\left\\{f:\\Omega\\to\\mathbb{C}\\text{ measurable }\\left|\\ f\\in L_p(K),\\ \\forall\\, K \\subset \\Omega, K \\text{ compact}\\right.\\right\\}.", "89f7d89deb67e6e070dc09dccb3f6cea": "(-1)^{i-1}.", "89f8030e9b691d4072fb4fff33731963": " k=1,2,... ", "89f8383964c81af91b2ca62ccee7300e": "a * b := \\sum_{A \\in H_2(X)} (a * b)_A \\otimes e^A.", "89f84364daa93b056b86d138a645617b": "A[n] \\times A^\\vee[n] \\longrightarrow \\mu_n", "89f880b59e6cb06e2c6ed45501a88c95": "\\delta=\\zeta=0", "89f8a46bf0d0f1ce74071ac33d30cf72": "\\bar{\\eta}", "89f8da3be458ec6d777f98800393a7be": " \\vec{e}_0 = \\frac{1}{f(r)} \\, \\partial_t ", "89f95351031143684a14bdf2bfcc957c": "\n= \\prod_{p=1}^P \\left ( \\frac{G(A(h_p)|c_p,V_p)}{G(A(h_p)|c_{bg},V_{bg})} \\right )^{b_p}\n", "89f968654a221f1220e9bb5656c7b66c": "\\lambda_0, \\lambda_1, \\dots, \\lambda_{k-1}", "89f96d5642fb016610ea6c2490cb97fa": " F^r(x, u, \\frac{\\partial^{|I|}u }{\\partial x^I})=0, \\quad 1\\le |I|\\le k ", "89f9b5c8122708bdb9a1b178c06fd7d7": "m_N \\geq \\beta_N", "89f9d0e3e950e91fd88ae6356c718593": "n=2^{(k+1)}-2", "89fa1c82a4adaa0252cdcf65373ba25e": " (1-z)^w = \\sum_{n\\ge 0} {w \\choose n} (-1)^n z^n =\n\\sum_{n\\ge 0}\\frac{(-1)^n}{n!} s_n(w) z^n.", "89fa395eae7e295a6f7cb47322307ea6": "\\mathbf \\nabla \\cdot \\mathbf A' = 0", "89fa5da362e8232f584e231e5b4670da": "ds^2=\\phi^2dt^2-\\psi^2[dx^2+dy^2+dz^2]\\,", "89fa74fdd08c32eb69254dbb24572e28": "\\alpha_{y} = f \\cdot m_{y}", "89fa92de7203940a68aab87b06aea7f7": "a_{i,j} ", "89fa970e572053ba1bdb47e347c5a9fb": "q = w + xi + yj + zk\\,", "89faebcea43223abb4635fda3435a31f": "\\; = -(\\sum_{x,y} p(x,y) (\\log \\sum_{y'} p(x,y') + \\log \\sum_{x'} p(x',y)))", "89fafc3ff9b5497056fcdfd3f0b95fb0": "a_4= \\lfloor 36^\\frac{1}{2} \\rfloor = \\lfloor 6 \\rfloor = 6, ", "89fb027751a4486369273dd32e41cdd2": "\\int^{2\\pi}_0 |f(re^{i\\vartheta}) |^2 \\, \\mathrm{d}\\vartheta = \\int^{2\\pi}_0 {f(re^{i\\vartheta})}\\overline{f(re^{i\\vartheta})} \\, \\mathrm{d}\\vartheta", "89fb2c3b3dd6ce207b0ed1ee8125095a": "Z_n^m \\rightarrow Z_j", "89fb3b2563937a6c187d541a4adee5d2": "D_2 = [R] + [S] - 2 [O]", "89fb566a9f16de48c9498d4c586604fd": "p\\in R^{2}", "89fbb6b7354423428a726783136ac5b2": "\\frac{1}{2}+\\ln\\left(\\frac{b-a}{2}\\right)", "89fc29a1a5246e6f2b1e69ce75776d80": "A x^2+ B x y + C y^2 = 1 ", "89fcd47eaa0f4db5b3926fcb2219907f": "E\\supseteq K(X)", "89fcd74bf58a8ad72f1c212f06df1977": "X = \\cup_i X_i", "89fd370c30fbc6401fb4ef5d24fab87b": "\\phi(z)=\\int_z^\\infty \\frac{\\partial n(z,\\rho)}{\\partial z}\\,\\epsilon(\\rho)\\,{\\rm d}\\rho", "89fd5cb8dac3bfccdba18831b37c754c": "p^{q}", "89fd90eb28467c5efc1ffd6fca83e896": "\\mathbf{z}(t)=e^{tA}\\mathbf{z}_0.", "89fdaad09a1edc1c92200fb7848c1730": "\\limsup_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{P}_{\\varepsilon} \\big[ d(Y_{\\varepsilon}, Z_{\\varepsilon}) > \\delta \\big] = - \\infty.", "89fdb7bd563b2bd6820e0b46fc51188c": "\\mathbb{T}^n = \\underbrace{S^1 \\times S^1 \\times \\cdots \\times S^1}_n", "89fe018732b264fdd60d394d129a54d7": "F^{(loc)}_1(X)", "89fe022964f73dba8499f2b54afeff91": "u^k", "89fef3359f78403b2a73fe06fd8732c4": "\\varphi_n(t)\\to\\varphi(t) \\quad \\forall t\\in\\mathbb{R},", "89ff084f3f0c7f75fcd5d2ad878f68ac": "\\langle 0 |\\hat{H}|0\\rangle = 0", "89ff93b87df954d126e06e205bf46e34": "3x^2-1 = 3(x-1/\\sqrt{3})(x+1/\\sqrt{3})", "89ffa52112566e00e24ee8162507a07a": " \\mathbf B = \\frac{\\mu_0}{4\\pi} I \\int_C \\frac{d\\mathbf l \\times \\mathbf{\\hat r}}{r^2}", "89ffabd5ddeba5beb9c4b967abe60c4f": "y_{i,j}=y_i+t_j.", "89ffb5be48467abd13d93c294a0d6687": "\\lambda_p = \\left[282.93807+1.7195\\left(\\frac{D}{36525}\\right)+3.025\\cdot 10^{-4}\\left(\\frac{D}{36525}\\right)^2\\right]\\mbox{ deg}", "89ffbdbacfab011b9a10c34e594cc889": "(A-\\lambda I)^k\\mathbf{v} = \\mathbf{0}.", "89ffc74061dcb810c6339a3a5bd668a5": "L^2=-1", "8a002ee8a85c9751951b8c7db3762f95": "\\hat{H} = E_0 + \\frac{1}{2} k \\left(\\hat{x} - x_0\\right)^2 + \\frac{1}{2m} \\hat{p}^2", "8a01550771b7bfe48dff5e00ea5f780f": "z = 0, x^2 + y^2 = r^2", "8a019d08cd775425d00c6572a1c45518": "L_{AB} = L_{BA} = L_{AC} = L_{CA} = L_{BC} = L_{CB} = - \\frac{1}{2} L_{ms}", "8a01d6456cf7a66b1494213fdba7a2cd": "\\mbox{For } a > 1: \\,", "8a01d6fa52598810e07dc01ee3264d19": "V_\\text{out} = V_\\text{in} + i \\times R_f = V_\\text{in} + \\left(\\frac {V_\\text{in}} {R_g} \\times R_f\\right) = V_\\text{in} + \\frac{V_\\text{in} \\times R_f} {R_g} = V_\\text{in} \\left(1 + \\frac{R_f} {R_g}\\right)", "8a020a0f94e5d8b395e1685f0e37a141": "S_2 = E \\setminus S_1", "8a0214452d24bf1200ab9159ffd30d31": "\\dot{p}_i = \\{ p_i , H \\}", "8a027173e92e2b7236f74ae55b9d196b": "Q_{\\nu}=1-e^{-\\tau_{\\nu}}=1-e^{-\\tau_0 (\\nu / \\nu_{0})^{\\beta}}", "8a02f4cee127e01e001131d119963996": "\\mathcal Y", "8a0311813eaa828dff0b4b15af2926e8": "(K-3)(K^3-K-2)(K-1)^2K^2,", "8a031f3057dc9f1da39bffd60d7cca8d": "{1-d\\over1+d}\\|Y-D\\|_1\\leq \\|R-D\\|_1\\leq \\|Y-D\\|_1,", "8a0365eca6c740d7d871d93b4469bbb7": "F_A, F_B, F_C, F_D", "8a0369ebc2f595c38a66d400a6d9c49e": "p_x(n+x)", "8a03b72e75d4560696fe470dcf77f821": "Q^* = Q", "8a0414ffc9def867a268274a31a2d932": "\\left[J_z,J_\\pm\\right] = \\pm\\hbar J_\\pm.\\quad", "8a042c5d2b3935f1c1eb573be806b55c": "\\lambda_f(t) \\le \\frac{C^p}{t^p}", "8a0435a99c1ad1320127b30df866e417": " d_{ib} = |x_{ib} - \\bar{x}_{b} | ", "8a043d550a229d15b7e5e82d1acf6f8a": "\\textbf{P}_{k\\mid n} = \\textbf{P}_{k\\mid k} - \\textbf{P}_{k\\mid k}\\hat{\\Lambda}_k\\textbf{P}_{k\\mid k}", "8a0448e6a79f718410af219357a01087": "i=1, \\dots, 7.", "8a0469b38c786d17411bf93bb6b1a972": "u_P=t_P=0", "8a04811b218239f73c31ce63e7c59be3": "\\dot{x}(t)=2t", "8a04886c9c54c90f8aa45ace837f964f": "A = \\bigcup A_i", "8a04a4df932b64f25a67d7a5227418e6": "P_0,P_j", "8a04cd219ceeffacbaefbf3c795516ff": "\\scriptstyle \\wedge", "8a05791ab51622f4ad5ab310bcdfc16c": "w_1,w_2", "8a05efbd7da76cd6f7856adfa4ba8166": "\\Pi_{i=1}^l h_i^{m_i}", "8a05f3ddbf67682651cf56efe2a2a49a": "(abc \\dots xyz)", "8a060e32239b1978570735b901ae4849": "\\ell_P=\\lambda_e\\,\\frac{\\sqrt{\\alpha_G}}{2\\pi}", "8a063395ded15845176d0b1e07824cbf": "g_{i}", "8a06616709e5ec9447302b63eae7124c": " \\dot{x}_k=h_k+g_{kl} \\xi_l,", "8a06e45f6b5a146463065aebae485017": "\\phi_{\\overline{X}}(t) = \\mathrm{E}\\left[e^{i\\overline{X}t}\\right]", "8a076ea478fc25f84c3f87b1f99662bb": "r>2GM", "8a07aa96d3d09fe6067b746839bf8382": "|\\mathbf{P}| = \\sqrt{x_P^{\\ 2} + y_P^{\\ 2} + z_P^{\\ 2}}.", "8a07ae38a876fe25a10bcccb2ee53046": "p_a\\left(\\Sigma^*\\right)= \\left(\\Sigma-a\\right)^*", "8a07da23fb36d55f47fab8bb6a18c478": " \\lim_{N\\rightarrow \\infty} {\\rm det} P_N m(e^f) P_N = \\exp \\sum_{ n>0} na_n a_{-n},", "8a0820d8cedfaa8d1bcc7f6d67a80535": "\\phi_i + \\omega \\left( \\frac{b_i - \\sigma}{a_{ii}} - \\phi_i\\right)", "8a084d5626ade1b74585299ad4d5f3ef": "\\Delta \\epsilon = \\int v\\, d (\\Delta v)", "8a085760f2da209966aaaac1292f8641": "x_k = 0", "8a085b75a91d27ecb14a13395c18609a": "\\begin{cases}x^* \\log(x^*) - (1 + x^*) \\log(1 + x^*) & \\text{if }x^* > 0\\\\ 0 & \\text{if }x^* = 0\\end{cases}", "8a085d045d42057436657f94cac47718": " F(T;H) = \\max_{|t-T|\\le H}\\bigl|\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)\\bigr|,\\quad G(s_{0};\\Delta) = \\max_{|s-s_{0}|\\le\\Delta}|\\zeta(s)|. ", "8a088253d6860a0456eb2f965dbfd6d0": "V_{od} = V_{out+} - V_{out-} = V_{id} \\times Gain", "8a089d74c0ecc5181d659f35a0440ac5": "p_{(A,B)}", "8a08b9077e6913e71f4f3d3f2d4460bd": "\\frac{\\mathrm{d}}{\\mathrm{d}t} f(p,q,t) = \\frac {\\partial f}{\\partial q} \\frac {\\mathrm{d}q}{\\mathrm{d}t}+ \\frac {\\partial f}{\\partial p} \\frac {\\mathrm{d}p}{\\mathrm{d}t} + \\frac{\\partial f}{\\partial t}.", "8a08bb342996f0cd6c009072877320ef": "Y_j", "8a09544f2c08daae9614da2817522b67": "c_8 = 8.5282 \\times 10^{-4}, \\,\\!", "8a097dcf960756387e4b21484d00b04d": "-3.0479", "8a0993f2668c17caae8f131ea3ea49c3": " q_j = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0_n & \\operatorname{e}_j \\\\ 0 & 0 & 0 \\end{bmatrix}, ", "8a09ce068c0ce029a0523ffc56c26f0f": "\nD(a\\Vert u) = a \\left( \\log a - \\log u \\right) + (1-a)\\left( \\log(1-a)-\\log(1-u) \\right)\n", "8a0a143aadc5b733f73dc741a7a44dbd": "\\frac{dX}{dt}=\\frac{ce^{kU}}{a(t)}\\sqrt{1-\\frac{dU^2}{c^2dt^2}}", "8a0a83bed549da70fb23b60984daaa29": "x\\to \\rho x", "8a0a97a1810aca87f609c82442093e63": " f(\\lambda_1, \\ldots, \\lambda_r)\n = \\sum_{j_1, \\ldots, j_n = 1}^r V(K_{j_1}, \\ldots, K_{j_n}) \n \\lambda_{j_1} \\cdots \\lambda_{j_n}, ", "8a0b15ee2908e984fe49d04ecdc97cce": "P=\\overline{n}^\\nwarrow.\\overline{n}^\\nwarrow (n^\\nwarrow.n^\\searrow(~))", "8a0b30a5e33aed7a0f40a05382dc9f79": "\\scriptstyle \\nabla\\cdot\\vec{M},", "8a0b40530da4d7d6df05d51f4e797e88": "\\mathbf{a} \\times \\mathbf{b} + \\mathbf{c} \\times \\mathbf{d} = (\\mathbf{a} - \\mathbf{c}) \\times (\\mathbf{b} - \\mathbf{d}) + \\mathbf{a} \\times \\mathbf{d} + \\mathbf{c} \\times \\mathbf{b}.", "8a0baf58b570c3ff4c445ec95dd9814d": "\\mathcal{O}_{\\mathbf{R}^n}", "8a0bc037c7e435b8803153aded96b7ea": "c_0(x) = \\cos {\\sqrt x},\\text{ for }x > 0", "8a0c3547b770ca1570cfca7edaacdfa9": "f \\equiv +\\infty", "8a0c544c461c76f70ddfd27d8f4755ab": "\\psi(x+1) = -\\gamma + \\sum_{k=1}^\\infty \n\\left( \\frac{1}{k}-\\frac{1}{x+k} \\right).", "8a0c8cc4ff765f9462bde58d578e6255": "\\mbox{Copper Loss} \\propto I^2 \\cdot R", "8a0cd056909cf1bc4a52c3f71ffa209b": "C_f = N_v - U_e - \\frac {B_n}{2}", "8a0ce2794a57928fe4678f2181af2faa": "\\sum_{j=1}^J p_{0j}=1", "8a0d7f67ba3a4ec8378df0c642fb9c60": "G_0, G_1, \\ldots,\nG_{k-1}", "8a0d8cd2a637b18fd4fdcc72a5ae7d66": "\n\\begin{align}\n\\dot y &= D Y \\\\ \n\\ddot y & = D^2 Y \\\\\n&{} \\ \\vdots \\\\\ny^{(\\beta)} &= D^\\beta Y\n\\end{align}", "8a0db85e22eca8c2b37212e56e0edebc": "\\frac{(k-1)\\gamma_E}{k} + \\ln \\frac{\\lambda}{k} + 1", "8a0dc526c84ee0ebdea43791f310c7fd": "\\Phi(x)=M\\{\\xi\\leq x\\}", "8a0dcc7bb4376c9997c9160881686a1c": " t_n = \\frac{1}{n}\\sum_{k=1}^n s_k ", "8a0e11d5cc7a3bb65f1566ae341ecb02": "d_v=m d_h", "8a0e15754d57459f53a0fd6c0d269089": "\\operatorname{Cl}_2(\\pi) = 0", "8a0e511cf9000c96dd656cd47a34fa5e": "x/b", "8a0f75684f78e6e7bc8820d3980886b3": " \\begin{bmatrix}AA\\end{bmatrix}\\begin{bmatrix}C^{j+1}\\end{bmatrix}=[BB][C^{j}]+[d]", "8a0f765656037aa95b39b39f614af01f": "g_m = \\begin{matrix} \\frac {2I_D} {V_{\\mathrm{eff}}} \\end{matrix}\\,", "8a0fb1fd69c78ea9cccfa0025aa8a02d": "\\kappa^+\\to(\\alpha)^2", "8a100213cf2949d109b0d5e64c5c285e": "-r^2 f(r)\\,", "8a108a28c6fa2a4ae740e500c321c288": "\\ker\\, f", "8a10de66691cfd029919c639cf6cd5ca": "X_N(t)", "8a10ed638541f25b9d65edebe33df195": "a = \\aleph_\\omega", "8a110aedd4589636f6f09e5717752612": "\\scriptstyle \\ VDOP = \\sqrt{\\sigma_{z}^2} ", "8a1120db2abc01871f32f67f4478d0e7": "C = \\frac{PV_\\text{float}}{\\sum_{i=1}^n ( N \\times \\tilde\\delta_i \\times P^D(\\tilde t_i))}", "8a1134c5b58c66b7c7a6d9eb6b3b8ea9": "c_2=1.4388 \\times 10^{-2} \\text{m·K}", "8a113b334b6fcfd2c148920b000fc620": "\\int_0^{\\infty}x^{\\alpha+1} e^{-x} \\left[L_n^{(\\alpha)} (x)\\right]^2 dx=\n\n\\frac{(n+\\alpha)!}{n!}(2n+\\alpha+1).", "8a118fd4e2b6360a5e4b17f5086391d2": "ax^2+by^2=z^2", "8a11a1150fb8fb4a88bff4c53a5290cf": "\\mathrm{NapLog}(x) = \\frac{\\log \\frac{10^7}{x}}{\\log \\frac{10^7}{10^7 - 1}}.", "8a11b8d826ce8fce54f9ac4e74b88349": "\\ A0 = A", "8a11ccaee3696c14ef72e14f062a7dae": "\\cong_{\\mathcal{B}}", "8a11dee042c1e943e8bde6916ef7a86c": "y_{1,t} = c_{1} + A_{1,1}y_{1,t-1} + A_{1,2}y_{2,t-1} + e_{1,t}\\,", "8a11e377ad7ed1264bd7edb45974ce71": "\\begin{align}\n& \\mathbf{A^{-1}} = \\frac{1}{47} \\begin{pmatrix}\n18 & 13 & 16 \\\\\n11 & 21 & 15 \\\\\n13 & 12 & 22\n\\end{pmatrix}, \\quad \\mathbf{B^{-1}} = \\begin{pmatrix}\n\\frac{1}{6} & 0 & 0 \\\\[4pt]\n0 & \\frac{1}{4} & 0 \\\\[4pt]\n0 & 0 & \\frac{1}{5}\n\\end{pmatrix},\n\\end{align}", "8a1211661c7eca6d3589a9925bfcfca6": "\\quad C>0", "8a1211ccd771e155b6c7588b9e24d9b6": "a_2 b_3", "8a1261d2ae0123144e6089528be5998b": "\\|x\\|^2=\\langle x,x\\rangle=\\sum_{v\\in B}\\left|\\langle x,v\\rangle\\right|^2.", "8a126e2e03e52066c1a290e95aafe156": "x_6", "8a133558c4bd8cbfdc8cd24cfbc65aea": "d_{1,0}^{1} = \\frac{-\\sin \\theta}{\\sqrt{2}}", "8a135225f8eb14ece0e6737467754a20": "F_1\\cap F_2,\\cap F_3", "8a13a1aac025dd32595a101a92d2ab64": "\\frac{n}{cm^2 * s}", "8a13c5b6f93eda9dda3dd3c656ba5a20": "\\mathbf{rank}_1(x)", "8a140337171d690f8dd0eebd94448bf0": "P\\,", "8a146bcb7883de2b2b6568b9eeb7413c": " s' \\equiv (m')^d\\ (\\mathrm{mod}\\ N). ", "8a14a8ed127ec6b68d17ea78c45ee509": "\\lim_{q \\to 1\\pm} \\Gamma_q(x) = \\Gamma(x).", "8a14e9013aba78ff51250c109efd84a8": "\\varphi_u :\n\\left\\{\n\\begin{array}{l}\n\\Sigma^n \\rightarrow \\mathbb{R}^{\\Sigma^n} \\\\\n s \\mapsto \\sum_{\\mathbf{i} : u=s_{\\mathbf{i}}} \\lambda^{l(\\mathbf{i})}\n\\end{array}\n\\right.\n", "8a1555e0955047d01ac345c11f44ef4c": " \\mathbf{x'} = \\begin{pmatrix} \\frac{d x_{1}}{d t} \\\\[2mm] \\frac{d x_{2}}{d t} \\\\[2mm] \\frac{d x_{3}}{d t} \\end{pmatrix} ", "8a15b06559f285dd169705a8ee878330": "c^2=a^2+(a+1)^2=2a^2+2a+1", "8a15ba4199597bde57229dc9e6fd093b": "\n\\frac{\\partial\\operatorname{atan2}(y,x)}{\\partial y} =\\frac{\\partial\\arctan(y/x)}{\\partial y} = \\ \\frac{x}{x^2 + y^2}\n", "8a15d86725f957ef7fa8903164f51028": "H = \\{ h_{a,b} \\}", "8a15e8c3568dc8f591d1262794921b9e": "O(\\varepsilon \\sqrt{\\log n})", "8a162dcdbb0096732672082576377982": " \\frac{{d^2 x(t)}}{{dt^2 }} + \\bar{c}\\frac{{dx(t)}}{{dt}} + \\bar{k}x(t) = \\bar{p(t)}", "8a163d8cc9e9324a568c4f28767dd83e": "\\mathbb{S}_4\\;", "8a16667dfce07b7a79a16077d770346c": "M'\\models_{\\Sigma'}\\sigma(\\varphi)", "8a16913936ca42948160ed4ca2259c1e": "K_1 = 90", "8a16bf7e0f188b5ac694a9f8f83c8b01": "c_{i+1}", "8a16d7f7b02c377643be2bbe1840a586": "\\beta_1 \\vee \\beta_2", "8a173467fc6403665c9a266890066ba7": "O(k*n).", "8a1757fcf73d8eddcad29dd50588489e": "- \\lambda \\mathbf{v} = m \\mathbf{a} = m {\\mathrm{d}\\mathbf{v} \\over \\mathrm{d}t} \\, .", "8a178cef95ee5b765434f32857b3802c": "\nf_X(\\vec{x}) =", "8a17929730159dd1440a93e485de0a45": "f(a)", "8a17ac9fd3027d98dd0c9081aad1118b": "=\\cdots=(a'c+(b'c+ac'))+bc'=(a'c+(ac'+b'c))+bc'=\\cdots=(a'c+ac')+(b'c+bc')=", "8a17bf52c4412b4390513cb4d500280a": "F_{X/S} = (F_X, 1_S).", "8a17db051cc73a9567f9f293c5bdc33e": "|\\mathcal{F}|", "8a184688d5730153f204c1889b666df2": "\\alpha^4", "8a18af18f5496224ad28545bbf423727": "\\ E[M] =\\ \\sum_tF_t", "8a18c4673fc8c8e067eaace2c4db8d06": "x[n] \\ \\stackrel{\\text{def}}{=}\\ x(nT) \\qquad \\forall \\, n \\in \\mathbb{Z},", "8a190a51aa1215bb77bd6ea3a7408118": "u(r_0)=0", "8a191c9e087c3f115aeea755ec8eac2e": "\\Delta {\\mathcal A}=\\bar{\\mathbb D}", "8a195c666efdac536d0e20b7f1c6aae8": " \\mu_0 = \\frac{Z_0}{c} \\,", "8a1a0959ad7719a49573f3078c794de6": "\n D\\left( k \\right) \\mid_{k_0=k_B=0}\n=\n{1 \\over \n k^2 + k_B^2 }\n\n", "8a1a212a9b2af11e1ab10ec6bd361628": "\\pi^+", "8a1a27fb57d0e0a7e0b079889a834166": "(\\pi_0^\\text{pr} F)(X) = \\pi_0 (F(X))", "8a1a347cd47ecd5bc441fce7afc05d0f": "\n\n{{documentation}}\n", "8a1a7596cc55c5b3aed413c825fe952f": " y = \\gamma x' + \\delta y'", "8a1b27fc45957031334b89f8d6253b79": "\\gamma_s:[0, 1]\\to \\mathbb C", "8a1b2f3763da933fc12f43df3fcba6ed": "\n \\begin{bmatrix}\n \\lambda & 0 & 0 \\\\\n 0 & \\lambda & 0 \\\\\n 0 & 0 & \\lambda\n \\end{bmatrix} \\equiv \\lambda \\boldsymbol{I_3}\n", "8a1b733944ac6e15ce2a6678c1e47434": "\n\\begin{align}\n\\sin 0 = 0 & \\text{ and } \\sin{2x} = 2 \\sin x \\cos x \\\\\n\\cos^2 x + \\sin^2 x = 1 & \\text{ and } \\cos{2x} = \\cos^2 x - \\sin^2 x \\\\\n\\end{align}\n", "8a1c4b924703f4bf441d204c25ad5641": "\\begin{pmatrix}\nu_1(x) & u_2(x) \\\\\nu_1'(x) & u_2'(x) \\end{pmatrix}\n\\begin{pmatrix}\nA'(x) \\\\\nB'(x)\\end{pmatrix} =\n\\begin{pmatrix}\n0\\\\\nf\\end{pmatrix}.", "8a1c4e736a16649844d69c2ef1c9c562": " G_2 = b_2 Z^{-2} \\, ", "8a1c88533d1a3577cf2c800b94442ecf": "\\operatorname{dim}[B(\\cdot)] = n \\times p", "8a1d07e75eac37428e6201df4efa21c3": "v_{\\text{L}}(t) = L \\frac{\\operatorname{d}i_{\\text{L}}(t)}{\\operatorname{d}t}", "8a1d2e6b3decb99343d3f7cc8d374a6b": "= \\alpha", "8a1d66d54140e1270cf2e7d90b478acb": "\\Omega=d\\omega +{1\\over 2}[\\omega,\\omega]=D\\omega.", "8a1dda02ac557e40b660f907be61f18d": "\\Delta = \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} ", "8a1e5010ddc3df3c0a141a15084d8c5a": "\\frac{ \\sum_{i=1}^n w_i x_i}{\\sum_{i=1}^n w_i} = \\frac{w_1 x_1 + w_2 x_2 + \\cdots + w_n x_n}{w_1 + w_2 + \\cdots + w_n}", "8a1e664da1f54f0245c648c0c609ae13": " X_n = \\left \\{x \\in X \\ : \\ \\sup\\nolimits_{T \\in F} \\|T (x)\\|_Y \\le n \\right \\}. ", "8a1e960928bf72d0207253bd99904703": "g=a_0e_1a_1\\dots e_na_n,", "8a1f4f2a0d7060cc1eede8029c0092e0": "\n\\forall \\sigma\\ _i \\in\\ \\Sigma\\ ^i \\quad \\quad\n\\pi\\ (\\sigma\\ _i ,\\sigma\\ _{-i} ) \\le \\pi\\ (\\tau\\ _i ,\\sigma\\ _{-i} )\n", "8a1f89e88f9089107c98648a6a9b3efe": "s'\\in X", "8a1fa39c7c93aa76a139d8c0bd9df0ab": " \\displaystyle\\frac{d}{dt}(af+bg) = a \\frac{df}{dt} + b \\frac{dg}{dt},", "8a1fb006869b61b6db96e03dc017fda5": " \\Pr(X_1 + \\cdots + X_n > k) \\leq \\Pr(Y_1 + \\cdots + Y_n > k).", "8a2026e4ef5245b116252f2b303c3392": "F_2=\\frac{F_{load}}{\\left [\\frac{Cos(\\alpha )Sin(\\beta )+Sin(\\alpha)Cos(\\beta )}{Sin(\\alpha )}\\right ]} \\,", "8a20a29a20c0c913156816282695388a": "\\varphi=\\sigma\\circ\\beta\\circ\\pi", "8a20c08f8d298cb0c7fdeb820eaca46f": "\\operatorname{arcosh}\\frac{|\\langle x, y\\rangle|}{\\sqrt{q(x)q(y)}}\\,.", "8a20d6529ab5f08704499a894d3cacda": "h = (b-a)/N", "8a2107ac843561d9025a46dfde377755": "\\mathbb{P}^1", "8a215d93ee702698e83d42e8e0035ae7": "{p+1}", "8a21a2219881d1f00924962c4b29bd59": "(j_1j_2j_3)", "8a21fa13b66872f876ec92ecd1a36967": "F,G\\in \\mathcal{P}", "8a2237a4aa4df556195f09d1144b369b": "M\\sqrt{v}", "8a2253c47c606c7db5785ee424a59689": "\\frac{\\partial{G}}{\\partial{\\theta_{i}}} = -\\frac{1}{R}[p_{i}^{+}-p_{i}^{-}]", "8a227cabcf6115e39ac046c59f78665f": "\\sin a, \\cos b, \\tan c, \\cot d, \\sec e, \\csc f\\!", "8a22846521ea37bb5d3824da6bbc2b8e": "\\mathrm{sys}", "8a22d7ffbe9425d5a666342838ad4956": "c_{12}-a_{12}", "8a22e1ec83a7af30ceb65aa946e51b6d": "{CE}_{7}", "8a22ee71c5584be06f08160da15ebaf4": "\\mathcal{U}", "8a2375c3a4f067652078d57d0fb7db0b": "u(c) = \\frac{c^{1-\\rho}}{1-\\rho}", "8a239c1b81dfe3bebf8c07e0d2cc6c8b": "\\begin{matrix}{r \\choose 1}{r - 1 \\choose 1}\\end{matrix}", "8a23b4f00e76149fe04b9f5f85a7621e": "p_{0 \\tfrac{1}{2}} \\leftarrow 64x^3-112x+56", "8a2472f04e2a8788a81fef13bb07ae7d": "\\|x+y\\|^2 + \\|x-y\\|^2 = 2(\\|x\\|^2 + \\|y\\|^2),", "8a24768719c4e0e9853740bee0dbd8d0": "F_i \\,", "8a24ac7f93250fff68757c2ab4b578b3": "y(x)=c_1 x^{\\lambda_1} + c_2 \\ln(x) x^{\\lambda_1}.", "8a253bcee532b083ebbc74ced3ee4b79": "\\frac{T }{\\sqrt{\\mathrm{Var}(T)}} \\sim \\mathrm{N}(0,1).", "8a254711c9b2a1b5f4fff3ceb029d96f": "\\tilde{h_1} \\leftarrow h_1^r rem P", "8a25679f93015a011d40009dd8e2ee57": "\\color{Red}\\text{Red}", "8a258bd40c6fbab48ceb3d7d7c68dccc": "(p-1)!\\ \\equiv\\ -1 \\pmod{p^2}", "8a26196589926ebcca0d1b89be8afa37": "\\psi\\, =\\, \\eta\\, \\sqrt{c_p\\, c_g},", "8a264154b143c5c3c35727812fc55e19": " \\sigma_Y^2 = \\alpha^2 \\sigma_X^2. ", "8a26b0702886de75731533bb341d1fcc": "K=\\mathbb Q", "8a26be0e1fc450185f611f5fe83b893a": "\\left\\lfloor \\frac{5N}{11} \\right\\rfloor, \\left\\lfloor \\frac{5}{11}\\left\\lfloor \\frac{5N}{11} \\right\\rfloor\\right\\rfloor, \\ldots, 1", "8a26de32eba0908205cad3c9f4f04acb": "ax + by = \\gcd(a, b)\\,", "8a274a46d3cb822a9e0c41b68b11d1fe": "c=m(m^{2}-2n^{2}), \\, ", "8a2781278eff147527c0a21013216c39": "\n \\dot{\\mathbf{F}} = \\frac{\\partial \\mathbf{F}}{\\partial t} = \\frac{\\partial}{\\partial t}\\left[\\frac{\\partial \\mathbf{x}(\\mathbf{X}, t)}{\\partial \\mathbf{X}}\\right] = \\frac{\\partial}{\\partial \\mathbf{X}}\\left[\\frac{\\partial \\mathbf{x}(\\mathbf{X}, t)}{\\partial t}\\right] = \\frac{\\partial}{\\partial \\mathbf{X}}\\left[\\mathbf{V}(\\mathbf{X}, t)\\right]\n ", "8a287bdfcf4c2770d67f73dde9d90f20": "\\! e^{it^T\\mu - \\sqrt{t^T\\Sigma t}}", "8a287fa9dfc262287ce056e307a268c6": "\\begin{align}\n\\mathbf{T}^k = \\begin{pmatrix}\n(\\frac{1}{4})^k & \\frac{k}{2^{2k - 1}} \\\\[4pt]\n0 & (\\frac{1}{4})^k\n\\end{pmatrix}.\n\\end{align}", "8a292c1e734bf87b539e4a02b848b6e1": "\\xi(M,g,\\theta)", "8a29465cb518df809db660e2e7f69daa": " v(0)=v(L)=0", "8a294db4565c3bfe6975b5050e5e8965": "x^a y^b z^c", "8a299a0fca7df0c71ca290aa0b694f64": "x(p_1,p_2,w) = \\left(\\frac{aw}{(a+b)p_1}, \\frac{bw}{(a+b)p_2}\\right).", "8a2a2048882d2a71413bb5db9d9e4bf0": "d(\\mathbf{x}, \\mathbf{y}) = \\|\\mathbf{x} - \\mathbf{y}\\| = \\sqrt{\\sum_{i=1}^n (x_i - y_i)^2}", "8a2a26af29d4a9bd3a2df0782ff593e9": " P_{F} (x) ", "8a2a486c98811fd451d568f5f7cdb929": " C\\ell_{p,q+2}(\\mathbf{R}) = \\mathbf{H}\\otimes C\\ell_{q,p}(\\mathbf{R}). ", "8a2a5a97c5e763fed9c56dc1d67f5911": "\n\\int_X f(x) \\, d\\mu \\geq \\int_X s(x) \\, d \\mu = \\varepsilon \\mu( \\{ x\\in X : \\, f(x) \\geq \\varepsilon \\} )\n", "8a2a71547511c702dcc00a26bd26dbd0": "\\beta = \\chi^{-1}\\alpha^{-1}\\chi,", "8a2ad1b49308b6039f7f469c9bf03dd1": " \\int x^{\\ast }(s)\\cdot ds=0. ", "8a2af29894a8b1db889fa307035d9335": "z_0 = 0 ", "8a2af6d3f81497bd953ce11ff582bc86": " N(0,1) ", "8a2b0c026932ac25ad59bcde10bce202": "abc=\\Lambda t,\\ \\tau=\\Lambda^{-1}\\ln t+\\mathrm{const}", "8a2b27f66ca28c9af54c45edf6cfb4f6": "\\partial_t v=-\\delta_u H(u,v)", "8a2b561d143354e0325fa5842f7a81f8": " 0.75 > \\beta > 0.5", "8a2b9656057f2d224d46719dcfc23414": "p(x_j|x_1,\\dots,x_{j-1},x_{j+1},\\dots,x_n) = \\frac{p(x_1,\\dots,x_n)}{p(x_1,\\dots,x_{j-1},x_{j+1},\\dots,x_n)} \\propto p(x_1,\\dots,x_n)", "8a2c15667e9ef41b025ec8e7ea3768cb": " \\mathbf{d^2F} = -\\frac{k I I'} {r^3} \\left[ \\mathbf{r} (\\mathbf{dsds'}) - \\mathbf{ds (r ds')} -\\mathbf{ds'(r ds)} \\right] ", "8a2c2d9af667b95624a3e2a3bad5fc77": "\nP= \\frac{Li}{1-e^{-n\\ln(1+i)}}\n", "8a2c37c996841d13c510acce5b16e907": "L_{1} \\pitchfork L_{2} \\iff \\forall p \\in L_{1} \\cap L_{2}, \\mathrm{T}_{p} M = \\mathrm{T}_{p} L_{1} + \\mathrm{T}_{p} L_{2}.", "8a2c81b9ae28e402c1ab6483d2ce82da": "\n\\mathbf A = \\begin{bmatrix} a & c\\\\b & d \\end{bmatrix}\\,\n", "8a2cc34bd09f05110e3918b5194ec892": "e_q(x) = [1+(1-q)x]^{1 \\over 1-q}", "8a2d549a03005573dcd2134e3ed6253d": "\\sum_j \\; u(y_j) \\; P(y_j).", "8a2d564608826c12ddfac08d96dcad63": " \\Delta K = k_i (sin \\theta_i ) - k_s (sin \\theta_s) = ", "8a2dbdc815e7b911018a97d5f2b6afc0": " UPI = { Return - RiskFreeReturn \\over ulcer \\, index } ", "8a2e048e16d5a0bd35fe463fd5d049bb": "\n\\left(\\frac{d(QM)}{ds}\\right)(QM)^T \n= \\left(\\frac{dQ}{ds}\\right)MM^TQ^T\n= \\left(\\frac{dQ}{ds}\\right)Q^T\n", "8a2e4eee19a04a2d38966d9ed04bd8ed": "x^{n-1}.", "8a2f0f0e398f2231dd58656f257b6fbf": "f^h_{\\mathbf{k}}", "8a2f1bce558b015146148776d055a8b5": "\\partial f(x) = \\mathrm{conv} \\left\\{ \\frac{\\partial \\phi(x,z)}{\\partial x} : z \\in Z_0(x) \\right\\}", "8a2f399eda42a68fc69721cc7ab449bc": "(l+m+n)\\times (l+m+n)", "8a2fa5492a7209f74f208be971d0bf82": "P + Q = 3x^2 - 2x + 5xy - 2 - 3x^2 + 3x + 4y^2 + 8 ", "8a2fa64b8a1e7cfe783e3403c22e1f4a": " g(z)=0 ", "8a2fb1d8442fc3b9f8e482a46c952145": "u(x,t) = A(x,t)\\sin (kx - \\omega t + \\phi) \\ , ", "8a2fd0ce0f324c6f674e56e06e69d02e": "\n\\begin{align}\nC_0^\\alpha(x) & = 1 \\\\\nC_1^\\alpha(x) & = 2 \\alpha x \\\\\nC_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\n\\end{align}\n", "8a300ca5386a42c008cb68e47bfb432e": "\\lambda(s)= e^{\\mu s}", "8a300cab185752a43307e42f161455e2": "\n \\begin{align}\n M_{xx} & = -D\\left(\\frac{\\partial^2 w}{\\partial x^2}+\\nu\\,\\frac{\\partial^2 w}{\\partial y^2}\\right) \\\\\n & = q_{x1}\\left(\\frac{x-a}{b}\\right) - \\left[\\frac{3yq_{x2}}{b^3\\nu_b\\cosh^3[\\nu_b(x-a)]}\\right]\n \\times \\\\\n & \\quad \\left[6\\sinh(\\nu_b a) - \\sinh[\\nu_b(2x-a)] + \n \\sinh[\\nu_b(2x-3a)] + 8\\sinh[\\nu_b(x-a)]\\right] \\\\\n M_{xy} & = (1-\\nu)D\\frac{\\partial^2 w}{\\partial x \\partial y} \\\\\n & = \\frac{q_{x2}}{2b}\\left[1 - \n \\frac{2+\\cosh[\\nu_b(x-2a)] - \\cosh[\\nu_b x]}{2\\cosh^2[\\nu_b(x-a)]}\\right] \\\\\n Q_{zx} & = \\frac{\\partial M_{xx}}{\\partial x}-\\frac{\\partial M_{xy}}{\\partial y} \\\\\n & = \\frac{q_{x1}}{b} - \\left(\\frac{3yq_{x2}}{2b^3\\cosh^4[\\nu_b(x-a)]}\\right)\\times \n \\left[32 + \\cosh[\\nu_b(3x-2a)] - \\cosh[\\nu_b(3x-4a)]\\right. \\\\\n & \\qquad \\left. - 16\\cosh[2\\nu_b(x-a)] +\n 23\\cosh[\\nu_b(x-2a)] - 23\\cosh(\\nu_b x)\\right]\\,.\n \\end{align}\n", "8a30528c3d0034fda250f35a5f9ffb18": " \\frac{d^2 \\psi}{dz^2} = \\kappa^2 \\psi ", "8a3063881c650c1ec09a45651109b0ad": "\n\\left[ \\hat{S} ~ \\vdots ~ \\hat{M} ~\\vdots~ \\hat{S} \\times \\hat{M} \\right] = A \\left[ \\hat{s} ~\\vdots~ \\hat{m} ~\\vdots~ \\hat{s} \\times \\hat{m} \\right]\n", "8a30a23d2ef460c32a7b5e6c0f8c73f6": "y\\in\\mathfrak{Y}", "8a30e0ca16c768742e3581d321c5849a": "\\sqrt{p_1^2+p_2^2+p_3^2}< \\lambda^{-1} \\, ", "8a31359cc015cc4049439ba2793c56a3": "\\left(\\tfrac an\\right)=\\left(\\tfrac am\\right)", "8a31a1569c7214ca049959ff3b4e5c15": "\\Bbb{Q}_3", "8a31c8aaf2f5934c82fe173f0c716f07": "\\int \\frac{x^{2}\\,dx}{R^3}= \\frac{(2b^{2}-4ac)x+2bc}{a(4ac-b^{2}) R}+ \\frac{1}{a} \\int \\frac{dx}{ R}", "8a31d8c8739b51390ad22257542ea993": " KR_n^G(\\coprod_{j\\in I_i} G/H_j\\times S^{i-1})\\rightarrow KR_n^G(\\coprod_{j\\in I_i} G/H_j\\times D^i)\\oplus KR_n^G(X^{i-1})\\rightarrow KR_n^G(X^i) ", "8a31f1fed7f7bed7dc56036aaff03364": "-\\scriptstyle\\mathrm{Im}\\,f(x+iy).", "8a3201bce018d5846a878f8235a91323": " H( Y ) = \\sum \\frac{ x_{ kj } }{ Y } log \\frac{ Y }{ x_{ kj } }", "8a3340ccc6bb6c49035f77a127e9a6ea": "\\mathbf{1_N}", "8a335e2b424f76963f64efcad9f34382": "\\alpha \\in (0, 1]", "8a33b2009601f68187e84b2c11951212": "C_{i0}, C_{k1}", "8a33c04011fb51196c4d8b9b9202caf7": " N : W \\to 2^{2^W} ", "8a344e07f66a61ea3a0add4c97a5fcab": "[\\bar p,\\bar\\xi, N(\\bar\\xi)]", "8a3482fea8c54377ee2cd64fd2c4569e": "\n\\begin{align}\nU(\\theta) \n&= a e^{\\frac { i\\pi S \\sin \\theta }{\\lambda}} + a e^{- \\frac { i \\pi S \\sin \\theta} {\\lambda}}\\\\\n&=a (\\cos {\\frac { \\pi S \\sin \\theta }{\\lambda}} +i \\sin {\\frac { \\pi S \\sin \\theta }{\\lambda}} )+a (\\cos {\\frac { \\pi S \\sin \\theta }{\\lambda}} -i \\sin {\\frac { \\pi S \\sin \\theta }{\\lambda}} )\\\\\n&=2a \\cos {\\frac { \\pi S \\sin \\theta }{\\lambda}}\n\\end{align}\n", "8a348f3213ecedfe60e8b25c8bfc6a89": " J_+ = \\{ j = 1,2,...,n | j ", "8a34b1bc9cba8927b0bdd5df0e9fba3e": " \\langle x | s \\rangle = \\sum_{j=1}^\\N\\, \\langle x | j \\rangle \\langle j | s \\rangle", "8a34c11755e8b6db314718c569931478": "l(s)", "8a34ede05fdce77abe3c153887c9e562": "\\sum_{i=1}^k \\left(\\frac{X_i-\\mu_i}{\\sigma_i}\\right)^2", "8a350aa8407dae9c6be7a6e86d819c4e": "p + q = r_0 + r_1 X + r_2 X^2 + \\cdots + r_k X^k,", "8a3518d0c322cbfa3221abe289bb82a1": "-u''=f", "8a351f4da00414b9428451acbc7f1f7b": "N=-{{1}\\over{2\\pi i}} \\oint_{\\Gamma_s} {D'(s) \\over D(s)}\\, ds=-{{1}\\over{2\\pi i}} \\oint_{u(\\Gamma_s)} {1 \\over u}\\, du", "8a35701976fa92cceedcdfd375eb1652": "\n \\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon} = w_{k,l}~\\mathbf{e}_k\\otimes\\mathbf{e}_l = \\boldsymbol{\\nabla}\\mathbf{w}\n ", "8a357cbcad31c5ab039d1f2ffce0fd47": "\\mu,\\nu \\ ", "8a358d34a8dff7127017ba6f8a8f4673": "1/[y_1, y_2] = [1/y_2, 1/y_1]", "8a35b59c36d9b8e3ea06d4077e1bc958": "\n\\mu _z \\,\\, \\approx \\,\\,\\mu ^2 \\,\\, + \\,\\,\\,{1 \\over 2}\\,\\,\\sigma ^2 \\,\\,{{\\partial ^2 z} \\over {\\partial x^2 }}\\,\\,\\, = \\,\\,\\,\\mu ^2 + \\,\\,\\,{1 \\over 2}\\,\\,\\sigma ^2 \\,\\,\\left[ 2 \\right]\\,\\,\\,\\, = \\,\\,\\,\\mu ^2 + \\,\\sigma ^2", "8a3677a381dced255d8b67ea5c9ee70c": " \\mathrm{id}_A=\\mathrm{id}_A^\\dagger\\colon A\\rightarrow A", "8a3692ef966ec7d03a02c73291943a0b": "p=", "8a369bbc32e1e94db63325e435369164": "M(y) = \\int_{\\mathbb{R}^{m}} H(x,y) dx.", "8a369ce9e9f64d7bf2d03cabc1d945c8": "R = r_0 A^{1/3}", "8a36e2eea17703131dba02f25a2a181f": "A = (C_{a} - C_{i})g/1.6P", "8a36e3fa878e4ab4580d72f50ed79b54": "\\varphi(t) = \\left\\{\\begin{array}{cc}e^{-\\left(\\frac{1}{1+t}+\\frac{1}{1-t}\\right)},&-1x^{\\sqrt{2}-1-o(1)}", "8a5b44b170c4ed8e8164570bab00973a": "h : \\mathbb{R} \\rightarrow \\mathbb{R}", "8a5b6ac67d4a1a0737608ea7959e17ac": "P_{global}(k_{i})=\\frac{k_{i}}{\\sum_{j}k_{j}}", "8a5b9b4f3b389978af7eb35fe7a4e8f0": "\n[L_z,X-iY] = -(X-iY) \n\\,", "8a5be0e4de4a47feb76283bd9001dbb3": "a^ib^j(ba)^k", "8a5c029570aec7cb4cc1d1de00d1bd89": "\\langle\\psi|\\phi\\rangle", "8a5c7a509d3e39bed6282d07e7da61f3": "\\frac{7}{2}", "8a5c7eb60ceb18a28dc1e230fdc07b31": "\\frac{dF}{dL}", "8a5c897f63ec5aa833b336f1ce3af4d5": "c_k = \\frac{f^{(k)}(0)}{k!}.", "8a5cd781f9d71bbc2e609d312d96b95c": "\\lbrace e^{ar}e^{br} :\\ 0 \\le a,b < \\pi \\rbrace.", "8a5d0baf59e2d39dcacefb2339e4fdcf": "n_{1}\\hbar\\omega_{1} + n_{2}\\hbar\\omega_{2}+ ...", "8a5d30b61588142358446c24b6c6e2a9": "\\gamma_0 x", "8a5d619b570ebe91be420f2d1a2fda3a": "E_{2k}(\\tau)=\\frac{G_{2k}(\\tau)}{2\\zeta (2k)}= 1+\\frac {2}{\\zeta(1-2k)}\\sum_{n=1}^{\\infty} \\frac{n^{2k-1} q^n}{1-q^n} = 1 - \\frac{4k}{B_{2k}} \\sum_{d,n \\geq 1} n^{2k-1} q^{n d} ", "8a5d794f2058b30fe1731a8c1a470729": " \\hat{E} \\,\\!", "8a5d9663f4a9b8d94cd18fe76cd7f112": "\\left(u^{+i}\\right)^* = u^-_i", "8a5da260384c5966ab45f3f2295e6dfa": "\\sum_{i \\in S_j} x_i = 0", "8a5e1d44957211fb289f20b831bc1df8": "\\mathrm{QSym}_n = \\mathrm{span}_{\\mathbb{Q}} \\{ M_\\alpha | \\alpha \\vDash n \\} = \\mathrm{span}_{\\mathbb{Q}} \\{ F_{\\alpha} | \\alpha \\vDash n \\}. \\, ", "8a5e59c920f11580bc0bcd841ae240d7": "|\\mathbf{X}|{\\rm tr}(\\mathbf{X}^{-1}d\\mathbf{X})", "8a5ee487c1212ca3e60e40f5f8c5e9bd": "e \\in E", "8a5f3a3164940f5f5883e06eea2218e9": "\\mathcal{P}(\\mu_{X^{(i)}Y^{(i)}}) = 1/N \\text{ for } i=1,\\dots, N", "8a5fc676298e17c49e2ec73adf33dffc": "M_x = \\lim_{m,n \\to \\infty}\\,\\sum_{i=1}^{m}\\,\\sum_{j=1}^{n}\\,y{_{ij}}^{*}\\,\\rho\\ (x{_{ij}}^{*},y{_{ij}}^{*})\\,\\Delta\\Alpha = \\iint_{}{} y\\, \\rho\\ (x,y)\\,dx\\,dy", "8a5fd3fba3b3a4b6e3abd00a4f491ecc": "f(z) = P(z)/Q(z) \\,", "8a6051d846046bf1387818410208ace4": "\n\\begin{align}\nR'_c & = \\frac{R_c}{\\sqrt{2 R_c L_s}} \\\\\n & = \\sqrt{\\frac{R_c}{2L_s}} \\\\\n\\end{align}\n", "8a6092426bc99a5def0b346d28e8afae": "\\gamma^\\mu \\gamma_\\mu \\,= \\gamma^\\mu \\eta_{\\mu \\nu} \\gamma^\\nu = \\eta_{\\mu \\nu} \\gamma^\\mu \\gamma^\\nu", "8a60ea9802cabdcceac3bb0893b5fb8c": "\\mathfrak{so}_{10}", "8a613d2f04476c0f4801c52c5676e37b": "{\\pi\\over 5}\\ {\\pi\\over 5}\\ {2\\pi\\over 3}", "8a6169ff8fed51e5038280f37407b7b1": " x_{n+1} - x_n = b_n(\\alpha - N(x_n)), \\qquad \\bar{x}_n = \\frac{1}{n} \\sum^{n-1}_{i=0} x_i ", "8a617cad7239c722414ae49d321a2fba": "\\mathrm{RH}_\\mathrm{2} + \\mathrm{O}_\\mathrm{2} \\rightarrow \\mathrm{R }+ \\mathrm{H}_2\\mathrm{O}_2", "8a61fdf08cd2c873cce0d388e6360ec9": " = \\mathbf{u}_{\\rho} \\left[ \\frac {\\mathrm{d}v_{\\rho}}{\\mathrm{d}t}-\\frac{v_{\\theta}^2}{\\rho}\\right] + \\mathbf{u}_{\\theta}\\left[ \\frac{2}{\\rho}v_{\\rho} v_{\\theta} + \\rho\\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{v_{\\theta}}{\\rho}\\right] \\ .", "8a622fc6266436f275b4225368673c1f": " d = r \\left( 2\\;\\frac{M}{m} \\right)^{\\frac{1}{3}} ", "8a624a4c2d94ecbc398b284fb5b76d14": "\\mathcal{A} (\\omega) \\subseteq X", "8a6276bfef387d856b1ef70ead5996ab": "f_! M = M \\otimes_R S", "8a62849baebc7f2d4a8e10d9942c129c": "R_{5,3} = 55 r^5-90 r^4+36 r^3", "8a62d05764a1a17bef582ff0d46ebfaf": "\\sigma :\\{\\;1, \\cdots ,n\\;\\} \\to \\{\\;1, \\cdots ,n\\;\\}", "8a63031a3bb111142fe89581dd549011": " x_k(t) \\,\\!", "8a6351ee2b303b94f6cf8bbe47a5b365": "10 \\equiv 1 \\pmod{3}", "8a6393df19e6e49ac790e4cbbf03bd5a": "X \\to Y", "8a63a5aa40c4030eaa5340141a7ed275": "(x + y)(x - y)", "8a6435fb3ba551282a275016ca1e39f8": " \\textbf{P}(t)=[T(0,\\textbf{d}(t))]\\textbf{p} = \\textbf{d}(t) + \\textbf{p}.", "8a644a75bf2b33188805ad5b677bd766": "Q(x) = x^T L x + \\gamma \\left((1-x)^T F (1-x) + x^T B x\\right) = \\sum_{e_{ij}} w_{ij} \\left(x_i - x_j\\right)^2 + \\gamma \\left(\\sum_{v_i} f_i (1-x_i)^2 + \\sum_{v_i} b_i x_i^2 \\right),", "8a649cc933dcbae878580ebe157c0ea2": "arctan", "8a64be25a690052ff5b20ad198b690cb": "M_G = (EG \\times M)/G", "8a64d905132e7bfa04b57ab55ffe2785": "(45, 13); ", "8a653de12596e0096326d2647fd36ccf": "4*x*b_{7}=4*x*(x+(b_{7}-a_{7})+(b_{8}-a_{8}))=4*x*( x+ 161)=4*x^2+644*x", "8a65c1778271a3cfa35988cdb12efb9f": "\\displaystyle u_t=(u^4)_{xx}+(u^3)_x", "8a65dea7392d357769d1f275ba27f5fc": "(X_1,\\dots,X_n)\\sim\\mathcal{N}(0,1)", "8a660a60f3da1f0a70ff852cd7ce4e21": "G(\\alpha_i)", "8a662133eec52d0fec13984154cd32e7": "\\alpha(+1/2)=\\beta(-1/2)=1", "8a66254b3a4cd0edb6d7acf2f9bd66f4": "\\tfrac{dI}{dT} = \\varepsilon E - (\\gamma + \\mu)I ", "8a66298547e4f43d65c245b5296f1902": "|w| = const", "8a6657ce46a5775e5c59dec59c60a29a": "1 + p_1z + p_2z^2 + \\dots = (1 + q_1z + q_2z^2 + \\cdots) (1 + r_1z + r_2z^2 + \\cdots)", "8a67006477ef68fb2984aa99238e545a": "t=t_2\\,", "8a67229a1c78a5935d3ad090afe950c9": "\np_{i}\\left(k\\right)=E_{i}\\left(k\\right),\n", "8a67b4c0a233df8cc4a1237f0521e800": "\\Pr\\left[\\frac{1}{k} \\sum_{i=0}^k f(Y_i) - \\mu < -\\gamma\\right] \\leq e^{-\\Omega (\\gamma^2 (1-\\lambda) k)}.", "8a67bd502d5cbbd03e97d30b2dfa46e7": "x \\equiv a_i \\pmod{m_i}", "8a67fb7f02e4e3c621a6d6374e3e1e89": " \\frac{3\\eta + \\pi}{4} = 445~\\mathrm{MeV}/c^2", "8a68134ca05d7fea2320fe5b56cb8570": "F=- \\star F", "8a68251ed75e6a72d6b97e9cc146db4b": " \\varphi_1, \\dots, \\varphi_n ", "8a683f5c64fa801e7ec21651dad47f8d": "M = \\frac{1}{1 - \\int_0^L\\alpha(x)\\, dx}", "8a6846285aa94e3b2e5ff467d26df74b": "\\Delta S = n R \\ln \\frac{V}{V_0} = - n R \\ln \\frac{P}{P_0} .", "8a68990f93e17897d5564e35c3b7678a": "\\sum_{(x,y) \\in C^2, x\\neq y} d(x,y)", "8a68ab2a8a20b6ce846083487b9416d0": "\\mathbf{E_T}=\\mathbf{E_0}e^{i(\\mathbf{k_T}\\cdot\\mathbf{r}-\\omega t)}", "8a69261e569142edd9e0c06644a762e2": "P^{-1}=I.", "8a6935adec9377d8cb12a7c76fb79e0d": "\\left(\\gamma^5 \\right)^2 = I_4, \\quad \\mathrm{and} \\quad \\gamma^\\mu \\gamma^5 = - \\gamma^5 \\gamma^\\mu \\,", "8a699c38e65cf03a543e4743c5ecf894": "E_M", "8a69f530ddddea73cd899f78175d27ae": "(2)\\qquad m_{q,\\ell}(M_{ETC}) \\cong \\frac{g_{ETC}^2 \\langle \\bar T T\\rangle_{ETC}}{M_{ETC}^2} \\cong \\frac{4 \\pi F _{EW}^3}{\\Lambda_{ETC}^2}\\,.", "8a6a6b4a1387217c9f33d0bd68b5a1fe": "\\mathbb{Z}[t, t^{-1}]", "8a6ab282b2dbdac1e0ed3d6f00a56af3": "\\mathbf{D_{1}} = \\mathbf{D_{2}} ", "8a6b0bc97c4bb5b3eac79b15f69a084c": "B_{\\lambda^j}(x_0) \\cap \\Omega^c", "8a6b499412296b7b0f8aae10f5ba0d34": "2k_BT", "8a6b52f71059a0dac96974018d59c343": "\nz = e^{i\\theta} \\quad\\Rightarrow\\quad z^{\\frac{1}{2}} - z^{-\\frac{1}{2}} = \n2i\\sin{\\textstyle \\frac{\\theta}{2}} \\ne 0\n", "8a6b5ab46e06fa60418f7c34e624b076": "A[i]", "8a6b7fc6582d57234f8216c178580a25": "239-169\\sqrt{2}=-0.00209\\ldots", "8a6bcf7abe9700966d1ec6be3c24d917": "\\mathfrak{g} \\cong T_eG", "8a6c1f511dd4a491df65c40f10b5de0d": " \\vec{\\ell} = \\partial_u - H/2 \\, \\partial_v", "8a6c54dbcf09ece5f65e81996369cfb6": "\\varepsilon_{ij}\\,\\!", "8a6c661f8b5fecfa85d8904bcdf9e543": "K/k", "8a6c73042510714b9fd44ea97407ef8a": "A^{\\ast}\\,", "8a6c77f69880c9df492512a7b559fc6f": " \\begin{align}\n R_k & = \\begin{pmatrix}\n \\cos \\frac{2\\pi k}{n} & -\\sin \\frac{2\\pi k}{n} \\\\\n \\sin \\frac{2\\pi k}{n} & \\cos \\frac{2\\pi k}{n} \\end{pmatrix}\n \\ \\ \\text{and} \\\\\n S_k & = \\begin{pmatrix}\n \\cos \\frac{2\\pi k}{n} & \\sin \\frac{2\\pi k}{n} \\\\\n \\sin \\frac{2\\pi k}{n} & -\\cos \\frac{2\\pi k}{n} \\end{pmatrix}\n .\n \\end{align}\n", "8a6c88e15f46d53a303d11ad7c86f6a9": "A/Hz^{1/2}", "8a6ca9eefd96ad82b51c97f672d90253": " B_1 \\in \\Sigma_1,\\ B_2 \\in \\Sigma_2. ", "8a6cc376456019cf9d8469576c11055c": "\\mathsf{\\Iota \\Kappa \\Lambda \\Mu \\Nu \\Xi \\Pi \\Rho} \\!", "8a6cf22ff80e402b3c20b99b1f1f579f": "H_3O^+ + R \\longrightarrow RH^+ + H_2O", "8a6cf65d4ad8ebf76ac018c865582a06": "B\\to A", "8a6d22471212158472ca4cee7b8e45e9": "\\scriptstyle \\sqrt{10}", "8a6d31bf0206039e73ad1f392fd017d9": " \\mu_P(x,y)=(-1)^{|y|-|x|} \\text{ for all } x\\leq y.", "8a6d3d61d7496c6985f384ea135e76d7": "p(w)=(|w|_{a_1}, |w|_{a_2}, \\ldots, |w|_{a_k})", "8a6e6108e33d43b26f9f8f975903fd20": "m u^2 /2", "8a6e8d7169841ac4f4392e59d6a2ad61": "q(t)=\\int d^3\\mathbf{x'}\\rho(\\mathbf{x'},t)=\\int d^3\\mathbf{x'}\\rho(\\mathbf{x'})e^{-i \\omega t}=q e^{-i \\omega t}", "8a6ebd886d1c76bcd43b412d27897d74": "k_n=\\sqrt{{k_z}^2-4\\pi({\\rho}_n-{\\rho}_0)}", "8a6ec528cd5d005ba7cb091d60b08fe7": "u=\\alpha L/2 ", "8a6ec5fd44986f40062e762c86fb0004": "\\mathrm{erfc}(x) = \\frac{e^{-x^2}}{x\\sqrt{\\pi}}\\left [1+\\sum_{n=1}^\\infty (-1)^n \\frac{1\\cdot3\\cdot5\\cdots(2n-1)}{(2x^2)^n}\\right ]=\\frac{e^{-x^2}}{x\\sqrt{\\pi}}\\sum_{n=0}^\\infty (-1)^n \\frac{(2n-1)!!}{(2x^2)^n},\\,", "8a6f45a0d4be399888b21e576cc7e537": "\\{ \\mathbf{e}_\\mu \\}, ", "8a6f645ae10e07477fc799f9096c6825": "\\operatorname{succ} \\mathbin{:} \\mathbb{N} \\to \\mathbb{N} ", "8a6f8e76c57282a4b4adc9be89e475ff": "|\\mathcal A|", "8a6f9e02717de7dd588968d510fcbab9": "\\forall x_1, \\ldots , x_{k(i)} \\in X^1", "8a6ff4eba93b55942001c5b2e929e744": "\\tau_g \\ ", "8a704f7303189da3093cc8570e235093": "r_m = a\\frac{1}{4} \\left(3 +\\sqrt{5}\\right) \\approx 1.309016994 \\cdot a", "8a70684b0e428f91b75757e77efae265": " \\sigma(n) \\le H_n + \\ln(H_n)e^{H_n},", "8a709dfdfac20be3b3762834364941f0": "\\Omega G \\,", "8a711e2312dcaad7d67d22208f84579f": "a_1 \\ge 0", "8a71382fb23f4d3098054f44ebba92ea": "\n\\Pr \\left\\{ f(X_1, X_2, \\dots, X_n) - E[f(X_1, X_2, \\dots, X_n)] \\ge \\varepsilon \\right\\} \n\\le \n\\exp \\left( - \\frac{2 \\varepsilon^2}{\\sum_{i=1}^n c_i^2} \\right) \n", "8a71400c30e5935fe859ce3c8d825495": "b_s = 0", "8a714900d563d60f4193f450d29df26e": "\\;\\lceil x \\rceil = n\\;", "8a715320eae9a5097bb85093adf8e5b2": "\\oint_{C} \\frac{f'(z)}{f(z)}\\, dz = 2\\pi i \\left(\\sum_a n(C,a) - \\sum_b n(C,b)\\right)", "8a71bf2f67620fff55e178b564809f9d": " a = 4^2-3^2 =7 \\,,\\ b = 24 \\,,\\ c = (4^2 + 3^2)=25", "8a71cd4716278dfa326549e6bdd0eace": "\\mbox{LOP}=\\mbox{TH}+\\mbox{RB}+180", "8a71f76b511450bb097dd0783c72c41f": "i, j = 1, 2, a, b", "8a72a0ff3a473f7c73e2d151f2b7925e": "\\omega = (b,d,u,a)\\,\\!", "8a72a8d8d8472700e31d7154ad2087dd": "4s^2 + 4sr + r^2 = 4s^2 + 2r^2 + 4s^2r", "8a72b92c18a408783ee6d06c644a66ab": "Nc\\left(\\frac{2^p+1}{3}\\right) = 2^{p-1}(2^p-1),", "8a733480538164feec9465da79f90c2c": "D_X(\\alpha Y + \\beta Z) = \\alpha D_XY + \\beta D_XZ, \\qquad \\alpha,\\beta\\in\\mathbb R", "8a7373ec67f4bbdd10985c1a9bf10658": "\\sqrt{k}\\,\\theta", "8a73de7227d1c248c6d0caf044868275": "\\frac{f}{m} = \\frac{1}{\\rho} \\frac{dp}{dx}.", "8a73eae7a6c65adf6cb1e390c3fe43be": " v-u,\\, x, \\, y", "8a73eb1afadefecba05800ec90d56c97": "\\Omega^p_X(\\log D)", "8a742ba2a308bc86ad300d3a23607023": "p_K (x) = \\inf \\left\\{r > 0: x \\in r K \\right\\},", "8a744155897ca01f656804f208505e5d": "T \\leq S", "8a74778c2d8a78ffdada33e3c453b837": " {m_\\mathrm{e} v^2\\over r} = {Zk_\\mathrm{e} e^2 \\over r^2} ", "8a74884fe242164cd8d20c32f34ee64b": "V^{(2)} = \\frac{\\pi}{12d}(r_1+r_2-d)^2[d^2+2d(r_1+r_2)-3(r_1-r_2)^2].", "8a756bb2c8a9298f273f175d83d5c50e": "T \\vdash \\neg\\varphi", "8a7581fd49ef0bba2af26dad2d081d4e": "\\aleph_{\\alpha},", "8a75ece0666a9d55df7f02e48d4b0734": "U_t=\\exp(it\\sqrt{\\Delta})", "8a75fb89515184704df36950498d1adf": "\\mathbf{l_0}", "8a760ff332dbd776b84e051d2274613e": "W =\\frac{L}{\\lambda} = \\frac{2}{10} = 0.2 ", "8a766d0e8505acb076a6f3e506b37560": "p(\\lambda |D,\\mathbb{M}) \\propto p(D|\\lambda ,\\mathbb{M})p(\\lambda |\\mathbb{M})", "8a76aea7dad60dc08eaa4c04ddfda823": "E_i=h\\nu_i=\\frac{h}{2e}\\frac{2e}{T_i}=\\frac{h}{2e}I_i ", "8a770742fe895434e02c7f71d16b02bb": "F(\\nu)", "8a776681e295055a6b76c1febee56d39": "1 = \\frac1{2\\cdot 3 \\cdot 1/2} + \\frac1{3 \\cdot 4 \\cdot 1/2} + \\dots + \\frac1{(2n-1) \\cdot 2n \\cdot 1/2} + \\frac1{2n \\cdot 1/2} ", "8a77cd795e9e2a401d29e9db7feae1e8": "\\varphi_\\gamma(\\alpha)=\\alpha", "8a7856b018c362530b773ffad5345a82": "N_r\\times 1", "8a78612fa3e56e785025f07aab16e927": "\\alpha_k \\le 3", "8a7866153ffe7f94eb666ca4da92b99f": "{f_i}_i \\subseteq [D \\rightarrow D^{'}]", "8a78e712b209d3b4ce70275e26324771": "{\\mathbf{R}} = \\mathbf{r} - \\frac{\\mathbf{r}_+ + \\mathbf{r}_-}{2} , \\quad \\hat{\\mathbf{R}} = \\frac {\\mathbf{R}}{R} \\ , ", "8a790aa55c01d1070d899d07c7c02673": "(5)\\qquad \\dot{m} = C\\;A_2\\;\\sqrt{2\\;\\rho_1\\;P_1\\;\\bigg (\\frac{k}{k-1}\\bigg)\\bigg[(P_2/P_1)^{2/k}-(P_2/P_1)^{(k+1)/k}\\bigg]}", "8a79325ce4929f93841601baa74553a4": "\\forall{x}{\\in}\\mathbf{X}\\, \\lnot P(x)", "8a79974a43f4faae3de6d0433c053731": "\\frac{1}{\\lambda} = \\frac{4}{B}\\left(\\frac{1}{2^2} - \\frac{1}{n^2}\\right) = R_\\mathrm{H}\\left(\\frac{1}{2^2} - \\frac{1}{n^2}\\right) \\quad \\mathrm{for~} n=3,4,5,...", "8a79a152170c6b51eb8c5430d7ded494": "\n\\oint z^{m+n-1}\\,dz = 2\\pi i \\delta_{m+n}\n", "8a79ad3c3fea277280cc518a0fd4ed0b": "z(r) = -4 + 2 \\sqrt{r-1}, \\; 5 < r < \\infty", "8a7a3139031aa8a5d06bcae1d4896037": " \\frac{dG(t)}{dt} = 2H(t)-G^2(t). ", "8a7a88186951ac932ebfe3b3e7beb903": "b_{k+1}", "8a7ab20ec0ab3262ce329c7dcb399a4e": "***", "8a7acdc3f8efefcbd438a66614d24dc1": "\\textstyle\\frac{\\pi}{2}", "8a7af201bcbdad44061eafdd3d39d8ea": "S_i = \\{s_i\\colon T_i \\rightarrow A_i \\mid (s_i(t_i),t_i) \\in C_i, \\forall t_i\\}.", "8a7b3bd3795d417e71fae23e47cde39c": "X_{ii} \\geq 0", "8a7b766d71b55914d5e8752931071481": "abc'", "8a7b91508c09b0f0a8d32143c9fd81c3": "f: \\mathbb{T} \\rightarrow \\mathbb{R}", "8a7bfad7955c51e761ac94b50bb4acbe": "p_2=\\textstyle \\frac{1}{3}\\ ,", "8a7bfe117b85e39d17bf9d016c751c42": "t_1,\\ldots,t_k.", "8a7c20a9004267c7fe2fb34f97a4eccc": " A^\\mu ", "8a7c43f0a71ad839463051e30f91e0df": " \\left( -{1 \\over 2} a x^2 + Jx\\right ) = -{1 \\over 2} a \\left ( x^2 - { 2 Jx \\over a } + { J^2 \\over a^2 } - { J^2 \\over a^2 } \\right ) = -{1 \\over 2} a \\left ( x - { J \\over a } \\right )^2 + { J^2 \\over 2a } ", "8a7c6983fa461cd4cda8ec8b0c2c5827": "O(nD^{-1/(k-1)})", "8a7c6fe733403672d578901b08d6a377": "\\mathrm{sat}(T)=(T):h^\\infty", "8a7d79cdbac3ce7017656845240086e5": "(\\mathbb{Z}/16\\mathbb{Z})^\\times", "8a7da7cfd562f642487025a7e25ceaaa": "\\mathbf{\\Rho}=\\frac{\\mathrm{d}\\mathbf{\\tau}}{\\mathrm{d}t}=\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\frac{\\mathrm{d}\\mathbf{L}}{\\mathrm{d}t}\\right)=\\frac{\\mathrm{d}^2\\mathbf{L}}{\\mathrm{d}t^2}=\\frac{\\mathrm{d}^2(I\\cdot\\mathbf{\\omega})}{\\mathrm{d}t^2}", "8a7e511e065cfb072f464c67a8f1e91f": "\\frac{1-e}{p}", "8a7e6b71c563cb1edfd67dce249db1af": "\\lceil X \\rceil", "8a7e8e64049154d26241d4824c1e676d": " t_d = {v_s \\eta_d \\over Q_f \\Phi_f} ", "8a7f3aa3d97a65e6725dda0d9c1d040c": "\\Delta \\,.", "8a7f6512f47442b0d2404979a7f363bf": "=a^n-b^n.", "8a7f73c0dccd1514af2cc724f9c8d0bc": "D_e = - \\frac{{P\\left( {y + \\delta y} \\right) - P\\left( {y - \\delta y} \\right)}}{{2 \\cdot P\\left( y \\right) \\cdot \\delta y}}", "8a7fcffbb624d96453c67b2b7ca2d714": "G_{T_2}^\\ominus = G_{T_1}^\\ominus + (C_p^\\ominus - S_{T_1}^\\ominus)(T_2-T_1) - T_2 \\ln{({T_2}/{T_1})}C_p^\\ominus", "8a7ff020e85bd951c5b653e213d7144c": " \\frac{1}{1}\\frac{y \\cdot r}{x} + \\frac{1}{5}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}+ \\quad \\cdots", "8a7ff241a411d1e1db9c833ad19e70a5": "D_n ", "8a7ff738b7d86dc071271b1b7100a5ae": "E = \\hbar \\omega.", "8a8049894f120bf496e88afd98572833": "\\alpha = \\tfrac{30}{10}", "8a80956dcc7cc0d586d930b758b56f3b": "E_{z,3z^2-r^2} = n [n^2 - (l^2 + m^2) / 2] V_{pd\\sigma} +\n\\sqrt{3} n (l^2 + m^2) V_{pd\\pi}", "8a80a045968e3abd5585227ff4d58915": "\\mathbf{C}^2", "8a80bd0e735dc75ed89c3d4b05423ce4": "\\mathbf{\\hat{r}}\\cdot\\mathbf{\\hat{e}}_3", "8a8136f1073fabc069d7bd1e4eece9e9": "(a,b,k) = (8, 1, 3)", "8a81495e22a8e7b1d59f297944a2c830": "\\Omega_{\\rm matter}=0.266", "8a81792a79250443aec1a924480ddadc": "f'\\,", "8a817b90ba2e939411620710d311a4ca": "EE(i)=\\sum_{k=0}^{\\infty} \\frac{(A^k)_{ii}} {k!}", "8a819830f9052988dd9881e9e89d92ff": "\\Psi_k(X)", "8a81b3ea71604896f182eb63c8ad998a": "a(t) = 2\\cdot 2^t-1", "8a82083fd8f1f9c8894dafca92895f43": "S^{-1} = \\{ x^{-1} : x \\in S \\}", "8a82099d2b5b85e05b49ea0d3bcb68cc": "H \\otimes D", "8a824447127923cae024b9ca0f1b8470": "\\operatorname{median} (\\Beta(\\alpha, \\beta) )= 1 - \\operatorname{median} (\\Beta(\\beta, \\alpha))", "8a826f776103126e610a820a56d5e102": "R_{0}", "8a829739a00037135aa717c8d5608994": "\\frac{\\mu}{m_1} + \\left(1-\\frac{n-\\mu}{m_2}\\right)^{\\omega} = 1", "8a82ad7c3e749b57e48ce83864a5558b": "\nx = -b-\\cfrac{c} {-b-\\cfrac{c} {-b-\\cfrac{c} {-b-\\cfrac{c} {-b-\\ddots\\,}}}}\n", "8a8340aa59052ebdb7297a58f18aceb5": "f=z^p-1", "8a83b688a26286ad77c929437b05bb7a": "e^{\\mu t}\\,\\mathrm{B}(1-st, 1+st)", "8a8403df308e8059926ff0a6dc29a8df": " dA = |X_u \\times X_v| \\ du\\, dv", "8a8459cf6b8808354031079c4d1b58fc": "\\sqrt[5]{100} \\approx 2.512", "8a84aaedf06ccdd27ec1e544eedcf108": " \\frac{ \\Gamma (-1+\\frac12(k_1+k_2)^2) \\Gamma (-1+\\frac12(k_2+k_3)^2) } { \\Gamma (-2+\\frac12((k_1+k_2)^2+(k_2+k_3)^2)) } ", "8a84c02fbf7acc3f6c94d9a4af02b79f": "O\\left(\\frac{N}{C} \\lg(N)\\right)", "8a85956421d30cd8917e2fea051da4a5": "\\int_S K \\; dA = 2 \\pi \\chi(S).", "8a86644c61a48450ca1e608ae08e7fb6": "ax^2 + bx + c\\,\\!", "8a867bc7518422686cf059cb031ca5e5": "U \\mapsto K_q(U)", "8a86b18083f82a1e5175df43ba5fbd97": "\\zeta(i_1,i_2,\\cdots,i_k)=\\sum_{n_1> n_2>\\cdots n_k\\geq1}\\frac{1}{n_1^{i_1} n_2^{i_2}\\cdots n_k^{i_k}}", "8a86d022c53f39f484a75ba4118ba818": " \\hat{H}_0 = c \\vec{\\alpha} \\cdot \\vec{p} + \\beta m_o c^2 + \\frac{1}{2} ( I + \\beta ) V ", "8a86d695f75d80357e3d40304206c7de": "\\scriptstyle n \\times m", "8a86f2c2a63c05b904d3c5befb28bc4d": "\\theta(G)", "8a8710d922065a674b812afeec3ba87b": " f^{0.5} \\, ", "8a871d885100d9007164b44778fc472f": "u_{n-1} \\qquad\\quad u_n \\qquad\\quad u_{n+1}", "8a872c49921699308046b9021e02d1e9": "L_{a}", "8a875d17fda5b074240cc155a790164d": " g(x) = (x-K)^+ ", "8a88137a6858b71aec8fa9a77c62174d": "\nv^\\text{of}=1 - \\frac{|m_l|^2}{2|\\vec p|^2}\n+ \\sum_{djm} (d-3) |\\vec p|^{d-4} \\, Y_{jm}(\\hat p) \\big[(a_\\text{of}^{(d)})_{jm}-(c_\\text{of}^{(d)})_{jm}\\big]\n,\n", "8a8821c536ba4781ab549bae0a5b3e0f": "r\\geq 1, ", "8a8836d85e7deb2c046d515c22c12bb7": "F_{LNA}", "8a888b7b17e23f678ad19d2d442c3085": "\\pi_1(U,w) = \\langle u_1,...,u_k | \\alpha_1,...,\\alpha_l\\rangle", "8a888ee50d535187a0222dc1afa1c0b6": "\\Lambda(V\\oplus W)= \\Lambda(V)\\otimes\\Lambda(W).", "8a88931c06b07a29ed1a6674961392ae": "\\sin\\theta = \\pm \\sqrt{1-\\cos^2\\theta} \\quad \\text{and} \\quad \\cos\\theta = \\pm \\sqrt{1 - \\sin^2\\theta}. \\, ", "8a88f202a8cabdae0e7f2872e918ed04": "\\omega_X = p^!(k) , \\,\\!", "8a88f9199c15a4db6326291238f6d185": "k^2-k=(v-1)\\lambda", "8a89181da7fa7b112f6a868c2991a894": "\n \\hat\\beta = \\frac{\\tfrac{1}{T}\\sum_{t=1}^T(x_t-\\bar{x})(y_t-\\bar{y})}\n {\\tfrac{1}{T}\\sum_{t=1}^T(x_t-\\bar{x})^2}\\,,\n ", "8a895a776595246c822eb40ba24b7575": "\\operatorname{var} \\left [\\frac{1}{1-X} \\right ] =\\operatorname{E} \\left [\\left(\\frac{1}{1-X} - \\operatorname{E} \\left [\\frac{1}{1-X} \\right ] \\right)^2 \\right ]=\\operatorname{var} \\left [\\frac{X}{1-X} \\right ] =\\operatorname{E} \\left [\\left (\\frac{X}{1-X} - \\operatorname{E} \\left [\\frac{X}{1-X} \\right ] \\right )^2 \\right ]= \\frac{\\alpha(\\alpha+\\beta-1)}{(\\beta-2)(\\beta-1)^2 } \\text{ if }\\beta > 2", "8a897f69175143c6b0d3b189095e9cab": " Ax = b", "8a899e6747c737e812d892342717d1bc": "\n\\operatorname{var}(z)\n=\n\\operatorname{E}\n\\left[\n (z-\\mu)(z-\\mu)^{*}\n\\right]\n", "8a89e806b51d20cd65decbc2406fa50d": "\\!\\epsilon_{\\mu}(k)", "8a8a32a5f2328d5e55f1a69f656925fd": "\\vec{E}^i", "8a8a579aca195538da1c62f158494d5c": "\\sum_{n=0}^\\infty \\mathbf{x}_n", "8a8a70d136de22a81fd98bc33029c8d6": "G^k", "8a8aaafd54e62d8f5290f546f3d379fe": "-\\frac{a^2b^2}{p^2}+r^2=a^2-b^2", "8a8aff585305b95b51439fdeb94f7856": " A_v = 132.715(OG - FG) = (OG - FG)/0.00753\\, ", "8a8b4e979e6a0c57ef2282f0a8d3c3d8": "1/t", "8a8b8f6b4150238884522bf5d95fbd5d": "m = \\frac{\\nu+1}{2}, \\!", "8a8b9618d8dcbfce22b7b20ca23c227c": "p \\rightsquigarrow p'", "8a8bb54d62da730f2cbc19b476904324": "[(x + \\Delta x) - x]/x = (\\Delta x)/x,", "8a8be32410a3afbae88a76a801ed8a84": "\\,[\\mbox{R}(z,t),H]=0", "8a8c38e6d17e4df0227449f89d45d172": "{{i}_{test}}={{i}_{e3}}+{{i}_{c1}}=2{{i}_{c1}}\\text{ or }{{i}_{c1}}=\\tfrac{1}{2}{{i}_{test}}", "8a8c4f62b42e0dcc81394c5ab6b01c4f": " A_t ", "8a8c6cc36014c471cec3906dd55684cb": "\\vartheta^a", "8a8c8e3332093d6dec060ab809968c09": "S_1(t)", "8a8cc4ce5b46c2fad4826afa247bc10b": " E\\Psi = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\frac{\\partial^2}{\\partial x_n^2}\\Psi + V\\Psi \\, .", "8a8cd6edd422cb5fcf360d1350a9582b": "S_n(X)", "8a8ce6b68cb520020fad091f0187197e": "p' = p*", "8a8d1e66f4cd7c1355dfbe04c00d2de1": "\\sigma = \\mathrm{Diag}(1,0,\\dots,0) \\,", "8a8d289be29551a5b391873fb5c2200a": "\\Lambda_{LH} = \\sum _{i=1...p}(\\lambda_{i})", "8a8d39c78ad177f07539a3251cc551d7": "\n \\displaystyle \n \\sum_{k=0}^{q}\n p\n =\n q p\n", "8a8d50cc0272ae92f0d80ef6de343f49": "\\hat\\beta = (X^\\mathrm{T}X)^{-1} X^\\mathrm{T}y", "8a8da60f8265e71b8c6e47c987c0fa4e": "\\scriptstyle 2\\sqrt{q}", "8a8e302603074d90171acf7daa58a6d7": "\nN_k\\ge \\underbrace{e^{e^{\\cdot^{\\cdot^{e}}}}}_{k\\ e'\\text{s}}=e \\uparrow\\uparrow k\n", "8a8e75593fb7cd647f1ddb9c126b89bb": "K=\\frac{a+b}{|b-a|}\\sqrt{ab(a-c)(c-b)}.", "8a8f07debdfc22b3f3aea64f34fe2819": "V_\\mathrm T = V_\\mathrm {iL}[(1+\\mathit \\Gamma)\\cos(\\beta x) + i(1-\\mathit \\Gamma)\\sin(\\beta x)]\\,\\!", "8a8f1e8e0a73d8e44a17653f830f7947": "i=0", "8a8f205871423e498a164603b1e2aac0": "n_T - I", "8a8fbd637f52be90029d54d1f11cba7e": "\\boldsymbol{\\Sigma}^0_2", "8a901b596b023239d4cbf312e8f247ac": " (e_s)_{s \\in S}", "8a9078797a8136556b369da02c10410b": "\\theta = \\frac{\\pi}{3}+2m\\pi \\quad[m\\in\\mathbb{Z}]", "8a907f251d04f68cf23f1a7e1dc428d5": "\\lambda = \\sqrt{x y} + \\sqrt{y z} + \\sqrt{z x}", "8a90ce20efd292c84d042a778a689eb7": "\\tbinom{n}{2}", "8a90daee7687e0caef620c54c288347c": "\\Delta=L/32", "8a91cf2d8dd13230211725e45feff4c0": "\\omega_r(\\phi) d\\phi", "8a92396f93a5a7dfc13f7a9bc7a9a4ca": "x_n , \\dots , x_{n+k}", "8a9271de96919edfc0ee84fbb9762296": "A_0 = 0", "8a9298103b0fa023b56ad94b60c8273d": "{\\tilde K}_1(\\mathbf{Z}[\\pi_1(Y)])", "8a92d4c76615e6082b6eb6ee5cec2c28": "\\scriptstyle\\hat{Q}_n", "8a92e962d07c4003db64ab33f76643bc": "(G, q, g, h)", "8a931d409ccdc175230455169e3095d2": "g_S = |g_e| = -g_e.", "8a9336e17306bcadf0039084bceee619": "f = f(E-J^2/2r_a^2)", "8a93518d80d399843fb9b7316c55df44": "\n\\{x, y\\}_{DB} = -\\tfrac{c}{q B}\n", "8a935c4878b308ff225d5d88cb60acee": "\\forall n, h(n) \\leq C(n)", "8a9371d58f2a28b7d8afee688ef934fe": "CFM = \\frac{3.16 \\times W}{\\text{allowed temperature rise in} ^\\circ F}", "8a93f7f0a348a1fb745dc06f8e2dc350": "O = I^2/16", "8a94068254ef51366054f4cd462657d9": " E^\\ominus\\left( \\mathrm{A}^{+} \\vert \\mathrm{A} \\right) ", "8a940786de48dfe7e8673007b4dd531a": "D_{4+k}", "8a942bf64837b0a7b0c77a42ab97c13b": "\\sqrt{a} + \\sqrt{c} \\le 1 \\ ", "8a943a760e171c277ac975221f6304e3": "\\int_0^\\infty f(t) e^{-tz}\\,dt \\to 0", "8a943f66606a8e91427ce1b6f6c82e6f": "x\\in F_n", "8a9486e865825f3a56d0b03b060f8ea8": "q_1,\\ldots,q_K", "8a94c6a10a04e24d83590ed186d48681": " \\lambda_i = - \\frac{2n+\\alpha+\\beta+2}{n+\\alpha+\\beta+1} \\frac{\\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)}{\\Gamma(n+\\alpha+\\beta+1)(n+1)!} \\frac{2^{\\alpha+\\beta}}{P'_n(x_i)P_{n+1}(x_i)}, ", "8a94f2637986d0caf5ad6a9b9ce68889": "{\\bar R}\\mathsf{G}:=R\\mathsf{G}(b,c)", "8a94f6e5b4290eaee4dcc14026ac325d": "\\sum_{\\ell m} {}_s\\bar Y_{\\ell m}(\\theta',\\phi') {}_s Y_{\\ell m}(\\theta,\\phi) = \\delta(\\phi'-\\phi)\\delta(\\cos\\theta'-\\cos\\theta)", "8a9519479d0bb121ac7e052d5472ee71": "h(x) \\oplus h(y) ~\\bmod~ m", "8a95860761da29164b89123a8d1c7339": " x_k = \\cos\\left(\\tfrac{k}{n}\\pi\\right),\\quad k=0,\\ldots,n.", "8a95b265d046660a2b84fa330aa9596e": "a<0\\,\\!", "8a95f0f7e330cb96f092d50a9cd1e831": "n^{O(1/\\varepsilon)}", "8a95f146c65fedbdfe65ab9a9da9377d": "\\ c", "8a95fc6b4f1e2a4421a53f12b80099b2": "\\mathcal{B_A}", "8a965d680da08cff1edd1cb38f8439ba": "\\overline{Q}", "8a968ac3bc44c7bb262fad787e995d77": "= c \\mapsto f \\mapsto k \\mapsto c \\, \\left( a \\mapsto f \\, a \\, k \\right)", "8a96cd7abd43104620a73c40edeba872": "\n \\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x} = -2fh~\\left(\\cfrac{C_{11}^{\\mathrm{face}}}{C_{55}^{\\mathrm{core}}}\\right)~\\cfrac{\\mathrm{d}^3 w_b}{\\mathrm{d} x^3}\n ", "8a96dd184b92dc1ea5d425f54341296f": "dx_i(\\Delta x_1,\\dots,\\Delta x_n) = \\Delta x_i,", "8a96f52c5f1360ec85c9fb02a8c1cdf3": " \\Delta P", "8a96fb10b2065e2f0282954fd10e0e87": "{q^2}={gy_c^3}", "8a971687040726c10d052f492f3e5c27": " \\psi (0) = \\frac{1}{\\sqrt{4\\pi}}\\,2 \\left( \\frac {Z}{n a_0} \\right)^\\frac {3}{2} \\text{ for } l = 0 ", "8a9727d94575bb1c7e8ffbe4bba19494": " U = U_{thermal}.\\;", "8a9775222cd86a1d4381c66fa636b813": "a | b \\| c", "8a977580b4c120c9e392dbe8b003ac68": "\\mathbf{r}=[1.625,\\ 0.75,\\ -0.875,\\ -1.5]^{T}.", "8a97b8c61eee5f0ccf6a32368fd91e5f": "(y,w)", "8a97f04e155499c3d2d0e75459f5c671": " \\frac{1}{i\\omega L}", "8a9829cc715ebd890d416d3e158daefe": "1-\\beta", "8a982bf5f84091612ec1af5f1242596b": " | \\Omega^{-1} - 1 | ", "8a986510fc42c2f25bdd9236ce148150": " [\\mathrm{M}/\\mathrm{H}] = \\log_{10}{\\left(\\frac{N_{\\mathrm{M}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{star}} - \\log_{10}{\\left(\\frac{N_{\\mathrm{M}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{sun}} .", "8a98732c91396fe9a7d2702827aba681": "C_D(\\{x\\}) = \\{d | x \\epsilon A_X^{(\\beta)}(\\{d\\}) \\}", "8a988a02ace7ed2d41497914b6a416d5": "\\,pL+(1-p)N \\prec pM+(1-p)N.\\,", "8a98cb13fc02936b21882ddc4e6f92ef": "\\{ x \\in \\mathbb{R} | x \\geq a \\}", "8a9908236e5ae8928740261866c178cc": " + \\ln\\Gamma_p\\left(-\\Big(\\eta_2 + \\frac{p + 1}{2}\\Big)\\right)", "8a99d7eb3353257b649bcac332382831": "f(x) ", "8a9a1de881763a79f7d00c789eb9462f": "\\Pi = \\sum_{j=1}^n D_j R_j^2 + \\sum_{i=1}^3 \\left(\\sum_{k=1}^m C_{ik} \\Delta X_{ik}^2 - \\sum_{k=1}^m P_{ik} \\Delta X_{ik}\\right)", "8a9af19df666972c5e59929990df4051": "\\begin{align}\n\\boldsymbol{\\sigma}' &= \\mathbf A \\boldsymbol{\\sigma} \\mathbf A^T \\\\\n\\left[{\\begin{matrix}\n\\sigma_{x'} & \\tau_{x'y'} \\\\\n\\tau_{y'x'} & \\sigma_{y'} \\\\\n\\end{matrix}}\\right]\n\n&=\\left[{\\begin{matrix}\na_{x} & a_{xy} \\\\\na_{yx} & a_{y} \\\\\n\\end{matrix}}\\right]\n\n\\left[{\\begin{matrix}\n\\sigma_{x} & \\tau_{xy} \\\\\n\\tau_{yx} & \\sigma_{y} \\\\\n\\end{matrix}}\\right]\\left[{\\begin{matrix}\na_{x} & a_{yx} \\\\\na_{xy} & a_{y} \\\\\n\\end{matrix}}\\right] \\\\\n\n&= \\left[{\\begin{matrix}\n\\cos\\theta & \\sin\\theta \\\\\n-\\sin\\theta & \\cos\\theta \\\\\n\\end{matrix}}\\right]\\left[{\\begin{matrix}\n\\sigma_{x} & \\tau_{xy} \\\\\n\\tau_{yx} & \\sigma_{y} \\\\\n\\end{matrix}}\\right]\n\n\\left[{\\begin{matrix}\n\\cos\\theta & -\\sin\\theta \\\\\n\\sin\\theta & \\cos\\theta \\\\\n\\end{matrix}}\\right]\n\\end{align}\n", "8a9b20a2bd5f595c4de56fb02812b2b7": "x^2 = y^2 \\mod n", "8a9b4dd0dc07cff08a3ea7470af8fdc3": "\\frac{dL}{dt}", "8a9b747e59646a42870037123ce786e6": "C_i^- = max \\lbrack 0, \\left ( T - K \\right ) - x_i + C_{i - 1}^-\\rbrack", "8a9be6d0ef8e2474fcdc05587474e7e7": "\\sin (a)=\\sin (b+c)=\\sin (b)\\cos (c) + \\sin (c)\\cos (b)", "8a9bf17f6f3f4bd6b6a9d7b5d0344b27": " V_1 = Z_{11} I_1 + Z_{12} I_2 ", "8a9c11d5c99ca61576ab6d457f18a936": "\\int\\arcsec(x)\\,dx=\n x\\arcsec(x)-\\operatorname{arctan}\\,\\sqrt{1-\\frac{1}{x^2}}+C", "8a9c849d0b3cadc53351dfedb8207177": "\\Gamma^0=(\\Gamma^0_{ij})", "8a9c85dcbb7070c36b4f906f0d065ecf": "{Q_u}={\\eta_0}.{Ic}.{Ac}-{U}.{Ar}.({Tr}-{Ta})", "8a9ca22a3f3724f31b65b91f658a213a": "t^{-n}\\langle S, \\varphi\\circ\\mu_t\\rangle = t^k\\langle S,\\varphi\\rangle", "8a9d0ede2235e5af729069ac12361b3a": "1\\leq\\lambda<\\infty", "8a9d17bc0f3725f192f6f5030b09c27b": " \\; 0(x) ", "8a9d58d2805cd6395ea13e46982c8b26": " g = g_{11}g_{22}g_{33} = h_1^2h_2^2h_3^2 \\quad \\Rightarrow \\quad \\sqrt{g} = h_1h_2h_3 = J ", "8a9d5903c6b4715c8fc99e49203923bb": "b(x) + 1", "8a9db3d3b1f97175e175f0c53bf386ba": "\\begin{align}q_0\\,\\,\\,\\,\\, & = \\,\\,\\,n,\\quad \\\\\n q_{i+1} & = \\left \\lfloor \\frac{q_i}{5} \\right \\rfloor.\\,\\end{align}", "8a9df28aab0800a6169caf4050ab1f91": "\\hat{M} = \\lim_{\\eta\\rightarrow\\infty} M^{(\\eta)}.", "8a9e1d82e7e2fcc6c48bdb296426cd8e": "\n3 \\cos \\theta_1 + \\cos \\theta_2 > 2, \\,\n", "8a9e247ce498d0db8fa41dab6cd000bd": "y=1 + rx", "8a9e5038df7d072fcb9956c1b04fd462": "\\Psi_0=\\Psi_1=0", "8a9eb0a32d828e2c19653a3ec85a582e": " \\mathbf{M} = \\begin{pmatrix} \\frac{d m_{1}}{d x_{1}} & \\frac{d m_{1}}{d x_{2}} & \\frac{d m_{1}}{d x_{3}} \\\\[2mm] \\frac{d m_{2}}{d x_{1}} & \\frac{d m_{2}}{d x_{2}} & \\frac{d m_{2}}{d x_{3}} \\end{pmatrix} ", "8a9eb5ebf18b3a33a0f2cdaae7b2b007": "\\pi_i=\\begin{cases}\n0 & \\text{ when } i=0\\\\\n(1-\\delta)\\delta^{i-1} &\\text{ when } i>0\n\\end{cases}", "8a9ece426b2be5a8fa06e087ab710e54": "\nJ \\frac{\\partial^{2}\\varphi}{\\partial t^{2}} = N\\frac{\\partial w}{\\partial x} + \\frac{\\partial}{\\partial x}\\left(EI\\frac{\\partial \\varphi}{\\partial x}\\right)+\\kappa AG\\left(\\frac{\\partial w}{\\partial x}-\\varphi\\right)\n", "8a9edf98a5b049bc1a5c1fb0b272cf63": " \\dot{V}(\\mathbf x) \\le 0 ", "8a9f71d9cfcc3b83b98cd70cfd526298": "P = ( v_1, v_2, \\ldots, v_n ) \\in V \\times V \\times \\ldots \\times V", "8a9fac398a32c739c3617d4e48538cbc": "y\\in\\mathcal{F}", "8a9ff4d4685908b68744acc539fda629": "X_{(n)}", "8aa01db479c30452eae3596971d3dbe8": "\n\\begin{array}{ll}\n & P\\left(Spam\\wedge W_{0}\\wedge\\cdots\\wedge W_{N-1}\\right)\\\\\n= & P\\left(Spam\\right)\\times P\\left(W_{0}|Spam\\right)\\times P\\left(W_{1}|Spam\\wedge W_{0}\\right)\\\\\n & \\times\\cdots\\\\\n & \\times P\\left(W_{N-1}|Spam\\wedge W_{0}\\wedge\\cdots\\wedge W_{N-2}\\right)\\end{array}\n", "8aa04a3fa3b8e67fdd5b6019350e835d": "\\gamma_n/(2\\pi)", "8aa075c9aa4b47600354a1177734d0a4": " | \\psi_0 \\rangle = \\frac{1}{\\sqrt{2}} \\bigg(|0 \\rangle |f_k \\rangle |f_k' \\rangle + |1 \\rangle |f_k \\rangle |f_k' \\rangle \\bigg)", "8aa10fc3118c6a5ac516a87b1c84a731": "\\log\\sigma_{t}^2=\\omega+\\sum_{k=1}^{q}\\beta_{k}g(Z_{t-k})+\\sum_{k=1}^{p}\\alpha_{k}\\log\\sigma_{t-k}^{2}", "8aa119b065fdc28c69575f520bfbdc8f": "u_2 = \\left[\\begin{array}{c}1/2\\\\ -\\sqrt{3}/2\\end{array}\\right]", "8aa144b46d6ffa8088cab3054747456d": "\\frac{\\partial U}{\\partial g}\\approx\\left.\\frac{\\Delta U}{\\Delta g}\\right|_{c.p.}", "8aa1560a0e0a4f9f5a4610dcba87206b": "\\frac{1}{2}\\pi\\left(\\left(2x+y\\right)^2-2x^2+y^2\\right)=\\frac{1}{2}\\pi\\left(2x^2+4xy+2y^2\\right)=\\pi\\left(x^2+2xy+y^2\\right)=\\pi\\left(x+y\\right)^2=\\pi\\left(r_1+r_2\\right)^2", "8aa15913ce98b9872279b3a9e3155e8b": "\\Phi \\subseteq H \\subseteq \\Phi^*. ", "8aa183b04203da4ee5d9910ac0517b73": "add(x,1) \\in T", "8aa19b86bbafdfce934a20cffe1b83d8": "Q_{s4} = q_{1} - q_{2} - q_{3} + q_{4}\\!", "8aa1a1d6765abb2a6dcce7bf8d8e530f": "\\exists~ a\\in\\mathbb{R}", "8aa1e4dbdf5239b873f51ebde3d73d25": "\\left|v-\\bar{v}\\right|<9\\times10^{-5}", "8aa20671c8f801ab2f2e793441d78de5": "O((Nmlogq)^2)", "8aa21382fc0a3b7b49e0c0019467a4b0": "\\frac{1}{|\\ln(x)|}", "8aa2484fe28619ade4baa900db1527fb": " -\\infty=r_{0} ", "8ac777d34ce3066610c4e10536ca66b8": " a_n(t) \\ ", "8ac77a273fb4fdad811dc556c0857e7a": "V=\\frac{\\sum_i \\rho_i C_i}{\\rho}\\, .", "8ac7bb2751607975f3cc8fc045c3e6e9": "1!+2!+3!+4!+5!", "8ac7ea376a4a33a6f866388665dc4b50": " \\begin{align}\n&\\lim_{\\alpha = \\beta \\to 0} \\text{excess kurtosis} = - 2 \\\\\n&\\lim_{\\alpha = \\beta \\to \\infty} \\text{excess kurtosis} = 0 \\\\\n&\\lim_{\\mu \\to \\frac{1}{2}} \\text{excess kurtosis} = - \\frac{6}{3 + \\nu}\n\\end{align}", "8ac845bea56928f52907b93cf24e749a": "\\mathit \\Gamma = \\frac{V_\\mathrm r}{V_\\mathrm i} = \\frac{I_\\mathrm r}{I_\\mathrm i} = \\frac {Z_\\mathrm L - Z_\\mathrm 0}{Z_\\mathrm L+Z_\\mathrm 0}", "8ac85c5d59183c726d174aad37bff944": "C_i^+ = max \\lbrack 0, x_i - \\left ( T + K \\right ) + C_{i - 1}^+\\rbrack", "8ac90c40f88bc3b44660b7e88636fc62": " \\mathbf{a} = - {|\\mathbf{\\Omega|}}^2 \\mathbf{r}(t) \\ .", "8ac95b33783f7c5780280a2717267d41": " G_1=\\alpha^2p_ip_i", "8ac9728907c4d2d2f1ecf6008c6642e9": "\\begin{matrix}\n\\end{matrix}", "8ac9d01c78da2376cb8ffdc248d27d4d": "K = \\left | \\frac{Q_L}{W} \\right | \\,\\!", "8ac9fa2dd4cf0a61196c4b15489f6e72": "\\frac {q_H}{T_H} - \\frac{q_C}{T_C} = 0,", "8aca33844937161e4657eb56f1d3b061": "\\text{Hom}_S(_SM,N)", "8aca39bb0ee7e4940cacda8cfd93e0e6": "\\tfrac13 \\left( \\eta' \\right)^2 + 2\\, \\left( 1 - c \\right)\\, \\eta^2 + \\eta^3 = r\\, \\eta + s, \\,", "8aca4c723b625bdc3bb3393016beef70": "\\frac{d^3}{dx^3}\\bigl(f(x)\\bigr)\\ \\mbox{or}\\ \\frac{d^3y}{dx^3}\\,.", "8aca6ce94f25719e1488e299d2f7a9b2": "\\beta_S", "8acaedbc680a69ce6c5772f3c260b10e": "D\\beta-\\delta\\varepsilon=(\\alpha+\\pi)\\sigma+(\\bar{\\rho}-\\bar{\\varepsilon})\\beta-(\\mu+\\gamma)\\kappa-(\\bar{\\alpha}-\\bar{\\pi})\\varepsilon+\\Psi_1\\,,", "8acb628bd4312af950cdd7ed69dee932": " f_X(x) = \\sum_{i=1}^n f_{Y_i}(x)\\;p_i,", "8acb80f719178531620deaf8a4c5797f": "a^{1/m}", "8acbd8c1ef5572fa4fb2dd782a4f5fa6": "\\begin{align}\n & E_{J}=\\frac{J(J+1)\\hbar ^{2}}{2I};\\text{ }g_{J}=2J+1 \\\\ \n\\end{align}", "8acbdf491d62f6871a10f1c6f6b891bd": "\\mathbf {\\beta}^{\\rm T} = [\\mathbf {\\beta_1}^{\\rm T} \\mathbf {\\beta_2}^{\\rm T}]", "8acc30c164bcf9267bc648cd7a7e6033": "\\rho:G\\rightarrow \\mbox{Aut}(V),", "8acc3946e7682b6c6dd56257c5fd3418": "\\exist\\, w\\in (V\\cup\\Sigma)^{*}: (S,w)\\in R", "8acc444f433dab0808fd49b21064ae87": "\\frac{C_{P}}{C_{V}}=\\frac{\\left(\\frac{\\partial P}{\\partial T}\\right)_{S}}{\\left(\\frac{\\partial P}{\\partial S}\\right)_{T}}\n\\frac{\\left(\\frac{\\partial V}{\\partial S}\\right)_{T}}{\\left(\\frac{\\partial V}{\\partial T}\\right)_{S}}\\,", "8acc7ef8fe779a2cfefdf995ff49f35a": "\n \\operatorname{atan2} (y, x)=2 \\arctan \\frac{\\sqrt{x^2+y^2}-x}{y}.\n", "8acc9c7c7f22d06f6a3bf7eb3704bf79": "\\iota^* \\circ s = \\mathrm {Id}", "8accbae5ab34d6382367b17a60cdcbd9": "P(\\text{ill}\\cap\\text{negative})=P(\\text{ill})\\times P(\\text{negative}|\\text{ill}) = 1%\\times1% = 0.01%.", "8acd03774eb5a76e7596c2ac4718ed95": "D(t) \\geq (A \\otimes (S_1 \\otimes S_2))(t)", "8acd4d38a21f0ad95bbc187b8dcd220a": " \\alpha = \\arccos \\frac{b^2 + c^2 - a^2} {2 b c}", "8acd8b1142aa1f24ebcb93c3b63cd4ca": "\nf(x,t) = \\int F(s,t) \\delta(x - Z(s,t)) \\, ds\n", "8acd9a2b5037c665c11302180dcbd285": "^{+6}", "8acda4dbfc0907ffb9b531c7359748ac": "\\scriptstyle |\\text{Pad}_n^s|=N=\\frac{(n+1)(n+2)}{2}", "8ace96b92f9eeade329c673296fedb04": "\\mathbf{v}_k=\\left[v_0, v_1, \\dots, v_{k-1} \\right].", "8acf2a6e30bad02d919e15d3b3df9bbe": "J_{reacting} = k_i C_i", "8acf42e9b1ece81ddb05e79e253bbe0f": " \\beta = \\frac{1}{k \\, T}", "8acf4b131f2d93048faf4c20aeea52b3": "\\Pi^{(i)}", "8acf55e00839bc03d88ec0da0a241161": "\\nabla _\\mu T_{em\\ \\nu }^{\\mu }=4\\pi f_\\nu \\left( \\phi \\right)\nT_{em}=4\\pi f_\\nu \\left( \\phi \\right) \\left( 3p_{em}-\\rho _{em}\\right)= 0", "8acf6d5028ae467b725031bafbeb2e2c": "[M\\rightarrow Fred(\\mathcal H)]", "8acf7b03c36f3487cd2c3b05f117d352": " T(\\sigma, \\ldots ,\\rho, {\\mathbf u}, \\ldots, {\\mathbf v}) \n\\;\\mbox{ or as }\\; { T^{\\sigma \\ldots \\rho} }_{ {\\mathbf u} \\ldots {\\mathbf v}}\n", "8acf7dfa57181c340b34fe88f3705d21": "IV_{\\mathrm{out-of-the-money}}= 0 ", "8acfa532c5598c6e3281e24504a3b99c": " a = ", "8acfab0fd032c03f800fb0be3a6d58b5": "C' \\to C= \\frac{\\omega_c' Q}{\\omega_0}C' \\, \\lVert \\,L= \\frac{1}{\\omega_0 \\omega_c' Q}\\frac{1}{C'}", "8ad0bd8f9fb047209df924e3ff1206b4": " \\gamma_{ws}^0 \\,", "8ad0decbf6eb623b1d6b7f978fd06a31": "x_{1}:T_{1} \\dots x_{n}:T_{n}", "8ad0e7423e4831a38d98be1180654b9d": "+ \\frac{\\partial}{\\partial y} E_{ext} (\\bar v_i) \\Bigg\\} ", "8ad0f2a813baad058961a776e923644a": "L=\\lim_{x \\to a} g(x)\\leq \\liminf_{x\\to a}f(x)\\leq\\limsup_{x\\to a}f(x)\\leq \\lim_{x \\to a}h(x)=L,", "8ad0fa12c97de29516a03a099561b8cd": "\\tilde{\\phi}(k,t) = \\exp(R(k)t)\\;.", "8ad116caf2b9f6cee0f252d150938c90": "M(\\vec X) = \\left[ {\\begin{array}{*{20}c}\n 0 \\\\\n 0 \\\\\n\\end{array}} \\right]\n", "8ad15354b1e9ed2f4255c7335c02afe6": "\\oint\\frac{dz}{z}=2\\pi i\\!", "8ad19ac330f15086336b324062cf957c": " n + \\cdots + 3 + 2 + 1 ", "8ad1b9a1ddd8b2b30efa23532cb5ca64": "\\mathcal{C}[x_1,x_2,\\dots,x_D]=\\left[\\frac{x_1}{\\sum_{i=1}^D x_i},\\frac{x_2}{\\sum_{i=1}^D x_i}, \\dots,\\frac{x_D}{\\sum_{i=1}^D x_i}\\right],\\ ", "8ad1eaf1a41fb5bc27a5cbb57697de3d": "\\varphi=G", "8ad288f2d1ee237b2bbac0e2620834f5": "\\Delta{V}\\,", "8ad29c96e4c3b289a28ec9c71eae0566": "f(x) = \\prod_{s \\in \\mathbb{F}_q} \\gcd(f(x),g(x)-s).", "8ad2b56f99eefed49c54de3f374f8105": "\nD\\simeq 6250 M_\\odot^{1/2}\\mathrm{kpc}^{-1},\n", "8ad2f68b254d85437ebef096548977f5": "\\scriptstyle g\\in \\mathcal{F}(V)", "8ad32d5f98ec06f5e152c9386aa94813": "O(n^2(\\log(q)+n))", "8ad36aae72698efe317d423c62750334": "\\textstyle e", "8ad37cd713d6b26801ea6ee273b1035b": "\\overset{\\cdot}{x}_{1}(t) =x_{2}(t) ", "8ad38db7c9f5c4d3e21f82b3dcad8761": "\\ \\left[\\nu \\right]", "8ad39a9d1fef702f13546fe04ddbb0a1": "Z_t= \\sum_{i=1}^{t} Y_{i} \\text{ for } t=1,2, \\dots ,n \\, ", "8ad3a9f624a0cf8f2b88533086ab8b65": "0 = P_0(1+r)^N - c {{(1+r)^N - 1}\\over r}", "8ad40ed8e6987f75f6517b3342757f17": "C_h", "8ad4190e57e19fb06cd73f52f00eb4cf": "\n\\frac {\\partial} {\\partial x} F_3(a_1,a_2,b_1,b_2,c; x,y) = \\frac {a_1 b_1} {c} F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) ~,\n", "8ad46042763dc4d859236c541b6f28a4": "(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))^2-(\\mathbf{o}-\\mathbf{c})^2+r^2", "8ad468b0c5ca1f830ed05ef147186fa2": "\\mathcal{H}=\\frac{1}{2\\sqrt{\\gamma}}G_{ijkl}\\pi^{ij}\\pi^{kl}-\\sqrt{\\gamma}\\,{}^{(3)}\\!R=0", "8ad4709b4f325709d0e6fda654edf0e5": " f_{a}\\;(6)", "8ad498d05410912442ec612e54d2837b": "u(x,t) = \\frac{1}{\\gamma_n}\\left [\\partial_t \\left (\\frac{1}{t} \\partial_t \\right )^{\\frac{n-3}{2}} \\left (t^{n-2} \\int^{\\text{average}}_{\\partial B_t(x)} g dS \\right ) + \\left (\\frac{1}{t}\\partial_t \\right )^{\\frac{n-3}{2}} \\left (t^{n-2} \\int^{\\text{average}}_{\\partial B_t(x)} h dS \\right ) \\right]", "8ad4d69cfc616ad628c396f31e20bcae": "c_g = \\frac{\\partial\\Omega}{\\partial k}", "8ad4db09913072eb8f56b9c0aaada732": " x= (x_1,\\ldots,x_k )", "8ad4ed3ca88b5b6412a448fb28de1b8e": "f(1) - f(0) = \\int_{f(0)}^{f(1)}\\;\\mathrm{d}f = \\int_{0}^1 0\\;\\mathrm{d}\\varphi = 0", "8ad50e31cfbcfbff85ba67f23343cc48": " \\lambda ", "8ad53cfb1dd8aad7c215404142ef3f25": "x \\mapsto f(a) + f'(a)(x-a)", "8ad540e66ae9a2040a707aedad2ad68c": "\n{\\left(\\mathcal{I} \\left(\\theta \\right) \\right)}_{i, j}\n=\n\\operatorname{E}\n\\left[\\left.\n \\left(\\frac{\\partial}{\\partial\\theta_i} \\log f(X;\\theta)\\right)\n \\left(\\frac{\\partial}{\\partial\\theta_j} \\log f(X;\\theta)\\right)\n\\right|\\theta\\right].\n", "8ad5b03e41c0e36feccaf9e86f8f8448": "A_1, \\ldots, A_n \\vdash B_1, \\ldots, B_k,", "8ad5f203212aab1878f7eebded21fc60": "p_c", "8ad659f84f36165383cbc5a22b48e96a": "0.\\overline{0011} = 11.\\overline{1100} = \\tfrac13+\\tfrac13\\mathrm i\\sqrt 2", "8ad672392d0c4ccac09b0d7e4b5042d8": "\\,P \\approx H", "8ad6fe77334f5da119802ae19b2dfa37": "(1)\\quad y_2=\\frac{y_1}{2}\\left(\\sqrt{1+8 F r_1^2}-1\\right)", "8ad708a272ed69abfff20e4047c34738": "\\textstyle \\zeta_G(\\alpha)=4\\zeta(\\alpha-1)", "8ad723acfadacbe456b7a8a24b9e9315": "S := S-Y+X", "8ad738c6b7fb2cb52ea48acc02725bbc": "A_d \\cong Sym^d k^n", "8ad73c78214f78972b6b6b97976c4339": "{D}={E h^3 \\over 12 ( 1 - \\nu^2 ) }", "8ad7a434823506a1be2f16068b002eae": "\\bold{r} = 0", "8ad836c7b7b8504f22982549e64209c5": "\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty \\mathrm{rect}(t)\\cdot e^{-i \\omega t} \\, dt\n=\\frac{1}{\\sqrt{2\\pi}}\\cdot \\frac{\\mathrm{sin}\\left(\\omega/2 \\right)}{\\omega/2}\n=\\frac{1}{\\sqrt{2\\pi}} \\mathrm{sinc}\\left(\\omega/2\\pi\\right),\\,\n", "8ad83c8fe86f15b03cbb3aef550c8312": "|f(x)|\\le M", "8ad84d41d6a0d61c31c3f1ecb5e28f1f": "X_0=x", "8ad8be7ee6eb78354736a81b3d9cb501": "\\operatorname{sgn}(\\sigma)", "8ad8d024cf859b688b0e23f9290e75b0": "\nF_{i,j}(n) =\n\\begin{cases}\n1 , \\quad n=1,2, \\\\\nF_{i,j}(n-i-F_{i,j}(n-1))+F_{i,j}(n-j-F_{i,j}(n-2)), \\quad n > 2.\n\\end{cases}\n", "8ad90327a83d3e9ecfb9bd23c6e085bf": "\n\\sum_{m=1}^\\infty \\left(\\left\\lfloor\\frac{n}{m}\\right\\rfloor-\\left\\lfloor\\frac{n-1}{m}\\right\\rfloor\\right) = 2.\n", "8ada0ee074ab9831db51655b2a8eb6ec": "\\color{blue}\\mathcal{M}", "8ada205f2599ae0d30d189e9cd698b0b": "\\frac{i}{k}=j", "8ada2a636f981bb600d6eae0f9e22ee9": "\\ f ", "8ada4780c626ddf3d226fa2070b50bb0": "t'=t-ra", "8ada6ffd158d1b3c88b53654db9b763d": "Q(x) \\approx x\\prod_{p\\ \\text{prime}} \\frac{1}{1+\\frac{1}{p^2}+\\frac{1}{p^4}+\\cdots} = \\frac{x}{\\sum_{k=1}^\\infty \\frac{1}{k^2}} = \\frac{x}{\\zeta(2)} ", "8ada959157671212adbc086f516dde39": " x_2 = \\exp {\\left(- \\frac {28100 \\mbox{ J mol}^{-1}} {8.314 \\mbox{ J K}^{-1} \\mbox{ mol}^{-1}}\\left(\\frac{1}{298}- \\frac{1}{442}\\right)\\right)}= 0.0248 ", "8adac3dea5173f82cc6a0870722ffe38": "X_1,X_2,\\dots", "8adb74e76e3778c3433da7056dee77eb": "\nV(\\mathbf{r})=-\\frac{Ze^{2}}{|\\mathbf{r}|},\n", "8adb8c2136aaefc54a8ca862fc62bad7": "f':\\bar{V}\\otimes V\\rightarrow \\overline{\\mathbf{C}[G]}", "8adbab94bb7fdd426db361b6e6f37559": "\\frac{\\partial u}{\\partial x}(x,y) = 0.~", "8adc333a339c83bd2ab6fb723156a177": "\\sigma_L=\\sigma_L^D + \\sigma_L^\\pm ", "8adc3c7bdee2223977382e41246b20b2": "v\\approx\\hat{v}=\\sum_{i=1}^n\\theta_n\\phi_{n}", "8adc52f24c659d62686679676263bed2": "\n\\begin{align}\n x\\left(\\theta\\right) &= {|\\cos \\theta|}^{\\frac{2}{n}} \\cdot a \\sgn(\\cos \\theta) \\\\\n y\\left(\\theta\\right) &= {|\\sin \\theta|}^{\\frac{2}{n}} \\cdot b \\sgn(\\sin \\theta).\n\\end{align}\n", "8add05a0932ac749755f24ee3f53dbfb": " \\begin{bmatrix}c & -s \\\\ s & c\\end{bmatrix} ", "8add0bb3dae1cbdc7a95910dc031985c": " g: X \\rightarrow \\mathbb{R} ", "8add1849f240d8853ccc8d24ec0d4b36": "R_1 = R/fR", "8add46a2c7936d154d321f0fcec1e96b": "\\sum\\{v\\colon v > 0,v\\in S\\} - \\sum\\{v\\colon v < 0, v\\not\\in S\\}", "8add80577ced8a058b79806735ac636c": "|\\langle x,y\\rangle|\\le\\|x\\|\\cdot\\|y\\|", "8addf23495c6251fcca726a6a48c2791": "W(T_i)", "8ade3a9605a444640eddcacd5da1e9e3": " \\Pr[c \\in B(y, pn)] = \\Pr[y \\in B(c, pn)] \\, ", "8ade642d4c7e8e3c9042965d615f2c63": "\\mathrm D(M) := \\mathcal R \\mathrm{Hom} (M, D_X) \\otimes \\Omega^{-1}_X [\\operatorname{ dim} X].", "8ade767aad5b310e65edf1a6494d8a38": "F_0 = F", "8ade85f510a71be51c743a2c02e198f1": " h_\\alpha(n) = h_{\\alpha[n]}(n) \\,\\!", "8adeb276dddb3aac74c428a94c226f65": "\\nabla\\cdot (\\varepsilon_0\\mathbf{E} + \\mathbf{P}) = \\rho_\\text{f} ", "8adf41be35b9bb7c379c05f130837467": "\\Phi = S - \\frac {U} {T} = - \\frac{A}{T}", "8adf8e2b51b6627e2c109a5ec9ecfec5": "H^q(U, \\mathcal{F})", "8adfc7381ef47254104a3ddbd53cbaeb": " \\mathbf{r}_0 ", "8adfd7b37f426601fe95107cb6fd49f8": " E_{r}(R) = \\max_Q \\max_{\\rho \\varepsilon [0,1]} E_{o}(\\rho,Q) - \\rho R. \\;", "8adfe8e95f145b310982a719c8f18b21": "\\Lambda_n[v_n]", "8ae01824c5f1620e38af8e688a5daa98": "H_{\\mathbf{x}_0}(f)", "8ae064d6bad2e4e560285ee0f0b5959e": "dA=-SdT-PdV+\\sum_i \\mu_i dn_i,\\,", "8ae07caa93f64c34e4f4fcb2e61aea27": "(X_2, Y_2)", "8ae0d045798f2a208c8330d3b70ea4af": "\\theta_1,\\dots,\\theta_n", "8ae16d4d3d4b3c0bda52a45ef6c47af0": "L, R", "8ae188b1a2dddc2cb6e0c3ea93835faf": " A = \\bigcup A_i", "8ae1c14417e172d7347652a763a7e3ed": "S\\otimes_R A", "8ae1c4b4fa94a2564d893b4e6837721b": " p = \\frac{v^2\\pm\\sqrt{v^4-g(gx^2+2yv^2)}}{gx} ", "8ae1dacba0eb8b0c04d0620a17cc780b": " R^3", "8ae1e1a23b663fd3d4d6087f85ea663d": "|\\lambda_1| + |\\lambda_2| \\leq 1 ", "8ae1eaf14dc1ae7683f8dcee2b8f4639": "\n\\hat{P} \\Psi\\big(1,2,\\ldots, N\\big) \\equiv \\Psi\\big(\\pi(1),\\pi(2),\\ldots, \\pi(N)\\big) = (-1)^\\pi \\Psi(1,2,\\ldots, N),\n", "8ae258890375bbf8b83c2b018b781a6c": "x^\\alpha = (ct, x, y, z) \\,.", "8ae2adf88462dc2608c35519b9f39cac": " x(\\lambda) = \\frac{x_0 - (x_0^2+y_0^2) \\, \\lambda}{1 - 2 \\, x_0 \\, \\lambda + (x_0^2 + y_0^2) \\, \\lambda^2} ", "8ae2fcea4b87a2d821b5a607534114fa": " \\log_b(x y) = \\log_b (x) + \\log_b (y) \\,", "8ae31570261ba282f3e692fae83bc000": "P\\left(L_{k}|R_{k}\\wedge\\delta\\wedge\\pi\\right) = P\\left(L|R\\wedge\\widehat{\\delta}\\wedge\\widehat{\\pi}\\right)", "8ae31b4a2522d86869df197ba6b477e4": "V(A,1)\\,", "8ae3226fa454df957b5fe1389072fd6e": "\\textstyle \\bar{M}_{\\mathrm f} R \\bar{M}_{\\mathrm f}^{-1} = T^{-1}", "8ae33a35c11c8e5786d7833b9826908b": "\n\\mathbf{\\hat{r}} = (\\cos \\varphi ,\\ \\sin \\varphi)\n", "8ae37586bc1e1ab2671eb1d3981ca908": " L_2 = -R_L y_{22} \\, ", "8ae3cbeee6dbaa5344e7de5da35bb176": "D_b", "8ae3f9f9398d771e5634657e0d5f27a8": "A^\\mu = \\frac{d U^\\mu}{d\\tau} \\,.", "8ae41786daffee5c74baa5a4bec8c657": "\\displaystyle{f_z=g_z.}", "8ae4297660f789749c55451074a6d0d3": "I_X = \\frac{Y_X} {Y_{Total}}I_T", "8ae454b786f7bc19d9577eca55d5730a": "\\{ e_i \\}", "8ae49c3b8ffdec2584ca36be96b7604e": "(Z-Z')^{\\otimes n}", "8ae4c9b6dfe595f59b388ecf5ad29312": "\\bar{x} \\!\\,", "8ae4cf306cd34ca0395f55a523cd0ed1": "\nc = \\Sigma _{XX} ^{1/2} a,\n", "8ae4d84398c1a78d40c9228b1e34feed": "f(x) = C \\cdot g(x)", "8ae554519293983f5e82398030e923ea": "\\{\\psi(\\rho,z)=0, \\gamma(\\rho,z)=0\\}", "8ae5627ba2f6bbc1578ecce9478fd35c": "\\begin{align}\n \\boldsymbol{\\nabla}\\phi & = \\cfrac{\\partial\\phi}{\\partial \\xi^i}~\\mathbf{g}^i \\\\\n \\boldsymbol{\\nabla}\\mathbf{v} & = \\cfrac{\\partial (v^j \\mathbf{g}_j)}{\\partial \\xi^i}\\otimes\\mathbf{g}^i \n = \\left(\\cfrac{\\partial v^j}{\\partial \\xi^i} + v^k~\\Gamma_{ik}^j\\right)~\\mathbf{g}_j\\otimes\\mathbf{g}^i\n = \\left(\\cfrac{\\partial v_j}{\\partial \\xi^i} - v_k~\\Gamma_{ij}^k\\right)~\\mathbf{g}^j\\otimes\\mathbf{g}^i\\\\\n \\boldsymbol{\\nabla}\\boldsymbol{S} & = \\cfrac{\\partial (S_{jk}~\\mathbf{g}^j\\otimes\\mathbf{g}^k)}{\\partial \\xi^i}\\otimes\\mathbf{g}^i \n = \\left(\\cfrac{\\partial S_{jk}}{\\partial \\xi_i}- S_{lk}~\\Gamma_{ij}^l - S_{jl}~\\Gamma_{ik}^l\\right)~\\mathbf{g}^j\\otimes\\mathbf{g}^k\\otimes\\mathbf{g}^i\n \\end{align} ", "8ae5d15f449e9680605f034529f089f8": "f(\\cos \\theta) = \\frac{a_0}{2} + \\sum_{k=1}^\\infty a_k \\cos (k\\theta)", "8ae5d987231d6490e01b0c63a7b9be8b": "\\hat M", "8ae5e9ab19ef0c3b51a9f95ed567cc4c": "\n0=\\partial_t \\theta(x)=-\\partial_u H(\\theta,0)=\\frac{1}{2}E'(\\theta),\n", "8ae64844b68c89d074671a370cab15ea": "{n!}_F := \\prod_{i=1}^{n} F_i,\\quad n \\ge 1, \\text{ and } 0!_F := 1, ", "8ae65d3093a1aba7215ca32f1f2d990d": "\\alpha_{\\mathrm n}", "8ae6c4c53fd44765242e959c4a1c1bd2": "Y_\\mathrm{srgb}=\\begin{cases}\n12.92\\ Y, & Y \\le 0.0031308\\\\\n1.055\\ Y^{1/2.4}-0.055, & Y > 0.0031308.\n\\end{cases}\n", "8ae8a46275e690a60b8127f1a25c4e86": " T_{\\alpha \\beta} {}^\\gamma ", "8ae8a679a42589ac3ec00f5a0039ca87": " \\gamma_{1-2} ", "8ae8d37085761955a83024267418989d": "\\hat{\\textbf{y}}_{k\\mid k} = \\textbf{z}_{k} - \\textbf{R}_{k} \\textbf{S}_k^{-1/2} \\alpha_{k} ", "8ae8d5a27df04a45968f249d9b6961cd": "a \\equiv \\frac1x \\pmod y.", "8ae8f33efc78267d2525ef55873272cd": "\\{0\\} = J_0 \\subset \\cdots \\subset J_n \\subset A", "8ae8fa3793a2d85cd4a489137d50f664": "A^{*}=F_{3}\\cdots F_{m+1},A^{**}=F_{4}\\cdots F_{m+1}", "8ae90aee058155fd85b370e060b0be9d": "C_n=\\int_0^4x^n\\rho(x)dx", "8ae922088ca1c8a199b5d1f3ddf29ca6": "L=r \\times p \\,\\!", "8ae946a20dc964aa206eb38853d8e547": "f_s\\,", "8ae95c70db0d160c6492c3a0854fbc43": "x \\times x \\times x", "8ae99a0a5e96759aec0f0f7149627fbf": "I ", "8ae9cc02cfa4fa71c0051c061cbf40c7": "I_X = \\frac{Z_T} {Z_X+Z_T}I_T \\ ,", "8ae9ceb262a41968c544d2379b11ce9e": "\\tbinom L d ", "8ae9e62c038b87698d67b75d5dca6765": "{1\\over T}=p(X\\ge{x_T})", "8aea2356402017c262ff63a7b24d9b04": "M^T\\eta M = \\eta.\\,", "8aea4477fbbad74af6dc774e0f872f52": "{v^2 \\over 2}+\\Psi+{p\\over\\rho}=\\text{constant}", "8aea9fef7c7fc318573a50e6fa92b347": "\\alpha \\in \\mathfrak{h}^*", "8aeaa9809e3055895dfdcf09b5a87db0": "t_1 \\mathbf{v}_1 + \\cdots + t_k \\mathbf{v}_k \\;\\ne\\; u_1 \\mathbf{v}_1 + \\cdots + u_k \\mathbf{v}_k", "8aeab5c29edccb0d1f3003d2150ea6a8": "\\left\\{ Q,Q^\\dagger\\,\\right\\}=2i\\frac{\\partial}{\\partial t}", "8aeb39e207e0a9c1c1fb052d4c1dffcb": "\\approx 0.98/\\sqrt{n}\\,", "8aeba121403f34c0d6e3ce9b6894f12d": "{\\mathcal L}_{xy}^8", "8aebb1df8eb917afc657944c349c146c": "=\\frac{[g(x+y+\\delta)-g(x+y)]-[g(x+\\delta)-g(x)]}{g(y)}", "8aebd6565e2ae27c5a296af22b3993d5": "\\mathbb{I}_{ \\{ x \\} } = y_i \\wedge \\ldots \\wedge y_d ", "8aec73023233ebed28a2b104cd284fdd": "\\pi_{n-k} V_k(\\Bbb R^n) \\simeq \\Bbb Z_2", "8aed91f32efa174b09e62fc4ba6e235d": " u(r) = r R(r)", "8aee3a5ce78cd673abe9acd11c13ede0": "\\|\\theta\\| \\rightarrow \\infty\\,\\!", "8aee4344dc3614940b902bb99a3dc43d": "\\; E(B) = V F(B) V^*.", "8aee83f36caee95ee48fae38efeb9c86": "\\dot{\\theta} = - \\frac{\\theta^2}{3} - 2 \\sigma^2 - {E[\\vec{X}]^a}_a", "8aeeecf609bbadc262fb98387cf13396": "(a_0,a_1,\\dots)\\,", "8aef13874b335a2c5261d387cb033241": "N =2\\times\\frac{1}{8}\\times\\frac{4}{3} \\pi n_F^3 \\,", "8aef18dcf2ed10953440ec639de45cec": "\n \\displaystyle \n \\frac\n {K}\n {3 N}\n =\n \\frac\n {k_B T}\n {2}\n", "8aef3bb4a0c2147a7c75bfe0b4bcaf6d": "\\mu \\neq \\mu_0.", "8aef45e0586f2a3a9f8fd1566b7f6273": "\\rho\\in\\Omega^3(M,\\mathbb{R})", "8aef7ce7c0afd8c66a3886be27ed91fa": "\\sim\\!\\!\\phi", "8aef8b2d037af610399cad7282032184": " \\varepsilon_t = X_t - \\sum_{i=1}^p \\varphi_i X_{t-i} = \\left(1 - \\sum_{i=1}^p \\varphi_i L^i\\right) X_t\\,", "8aef91f6d552beff09d3217df1c98317": " \\Delta(q)=q\\left(\\prod_{n=1}^{\\infty}(1-q^n)\\right)^{24}=\n\\sum_{n=1}^{\\infty} \\tau(n)q^n, \\quad q=e^{2\\pi i\\tau}, ", "8aefd1ba90c33e297008429916443637": "\\mu \\theta", "8af065b78bbd1ad73bec0d00d6f99da2": "|\\psi(t)\\rangle = e^{-\\frac{i}{\\hbar}H_0(t-t_0)}|\\psi_I(t)\\rangle", "8af06d10c736797ff41350289c414370": "{\\text{Delay}}_{E}", "8af095205ac21e0765bb0708e630393b": "\\vdots\\,", "8af09dd3f41c52a41780757f237d9816": "t_{A,B}=m_{A,B}\\circ(\\eta_A\\otimes 1_{TB}):A\\otimes TB\\to T(A\\otimes B)", "8af0c50bf7789e949eb99ff1a5f9c5ba": "E_r=(-1/2)mc^2\\alpha^4[-3/(4n^4)+1/{n^3(l+1/2)}]", "8af12e5fc8585814315a840bb1cafd80": "(q^2 - 13q + 24)\\left(\\tfrac{1}{24} + o(1)\\right),", "8af148e30d29251fbc9c5694ae7f2e21": "h\\not= 0", "8af1a532147ea84ea18f046b4525ffee": "d\\sigma > 0", "8af1deaa7b8bb72f08b4edc3cd0b445a": "|s_1| = 5", "8af2441f2dacc18aa354062cdb5262f6": "k_\\text{B}T = \\frac{\\hbar a}{2\\pi c}.", "8af33b83c205fcc5b1d45cdc2a2692af": "\\frac{2187}{2048} = \\frac{3^7}{2^{11}} \\approx 113.7\\text{ cents}", "8af33e0b787d43cf297e8979c6a652d6": "R(f) = P(f(X) \\neq Y)", "8af389b9556d747e62180020812389df": "\n\\operatorname{Li}_s(e^\\mu) = {\\Gamma(1 \\!-\\! s) \\over (2\\pi)^{1-s}} \\left[i^{1-s} ~\\zeta \\!\\left(1 \\!-\\! s, ~{\\mu \\over {2\\pi i}} \\right) + i^{s-1} ~\\zeta \\!\\left(1 \\!-\\! s, ~1 - {\\mu \\over {2\\pi i}} \\right) \\right] \\qquad (0 < \\textrm{Im}(\\mu) \\leq 2\\pi) \\,.\n", "8af391660d6eead48df96e945b797ac9": "P=t*A +(1-t)*R ", "8af3d0a66cf728f095af073004f61343": "\\Omega=\\{v_1,v_2,\\dots,v_n\\}", "8af40c12492e07647f2d1d2df2b6bd68": "k = 223", "8af4164f3c0c642e450a3f46c5fe928f": "A\\subseteq A\\bullet B", "8af4222a2a01fb6bdde7939596999470": "\\varepsilon =\\frac{\\Delta L}{L_0}=\\frac{L-L_0}{L_0}", "8af42e190d12fa325624e3ea3d0f6297": "\\langle z,z\\rangle", "8af4435cad4ad54182389bc20e9be0a4": "M_2^* + M_1 \\xrightarrow{k_{21}} M_2M_1^* \\,", "8af44414bec905a90e2283bb5c240886": "\\tau,", "8af457b06a16823cb0443d745f3864c9": " F_3 = x, S_3 = \\operatorname{false}, A_3 = \\_ ", "8af4b550540ea10b66ec7d96b9b3c8e5": "\nH(X,Y) = H(X) + H(Y|X)\n", "8af4c2439033182f37bbff2f3352adfd": "u=\\int u' dx", "8af4ffcba5ce0eaa61c95ef8e12e385b": "X \\in L", "8af53ae54b9efb6a27174865f0ac4088": " E = p, G = (p\\ f), H = (p\\ f), Y = (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "8af543ecac943edabb10af185770bd2d": "\\rho(\\mathcal M) = \\rho(A_1 \\dots A_t)^{1/t}.", "8af54d02057e92a90195f4128ec44cda": " p_i \\propto x_i^{ - ( 1 + 1/ \\beta) } ", "8af5a85d1e6c938f8beb0a683debc669": "\\mathrm{RAT}(N)", "8af5cafba3756ac691c29bb8c10211c1": "x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)}. \\,\\!", "8af5dcc0ac829e33ff41185e51e535e1": "D^{\\alpha}", "8af5edc4440ea049517b353ddb9792d2": "\\phi = -GM/r \\,.", "8af60267e19c780debba17b7f77cd22c": "M^2 \\equiv S \\times \\sigma_B + \\overline{R_F}", "8af6048eb05f3162cd823542db05671f": "\\boldsymbol{u}", "8af631d1f408bc5ca938eec4936bc17e": "y=\\sin^{3}(2\\pi t)+ \\sin(2\\pi nt)", "8af693730b1de1a266811d4e3752b7f4": "yri[lb]", "8af69a093ac685e6c5c89767eadfee0f": "{\\Gamma\\vdash e \\Leftarrow \\tau}\\over{\\Gamma\\vdash (e\\!:\\!\\tau)\\Rightarrow \\tau}", "8af779661aff877a13aa135ce59283f5": "P = \\frac{R\\,T}{V_m-b} - \\frac{a\\,\\alpha}{V_m\\left(V_m+b\\right)}", "8af7be993604d3c86eee12ae095810b4": "c(t,s)=-1", "8af812aa9813162675fd9126053f0a2f": "\\tilde{S}_F(p) = {1 \\over \\gamma^\\mu p_\\mu - m + i\\epsilon} = {1 \\over p\\!\\!\\!/ - m + i\\epsilon} ", "8af81559ffe31d5d5fe5cdc64ae8b617": "l_1x+m_1y+1=0\\,", "8af81cb6067153fec90eb28c6132860a": "|\\mathcal{F}|>\\sum_{i=0}^{k-1} {\\tbinom{n}{i}}", "8af838b2b277e05fe91569557de06350": "x^2 - Ny^2 = k_1 k_2", "8af83f470484bc5570ee466ffa3a0ca2": "v_s(t)=\\frac{dx_s(t)}{dt},", "8af86136ca4870a22e366b0659859024": " \\left(\\frac{M}{p}\\right) = (-1)^{(p-1)(M-1)/4} \\Bigg(\\frac{p}{M}\\Bigg) ,", "8af8a50da866700010cddad72b32bd4c": " \\left|\\Psi\\right\\rang = {1 \\over \\sqrt{2}} \\left|1,n\\right\\rang \\left|2,n\\right\\rang + {1 \\over \\sqrt{2}} \\left|1,n\\bot\\right\\rang \\left|2,n\\bot\\right\\rang ", "8af8d09fcc503a33827b103b0e5933d8": "\\mathbf H= \\mathbf F^{-1}\\,\\!", "8af8df37f02b60fbff45b7573c2d2bc1": "L^2 \\equiv L_x^2 +L_y^2 + L_z^2", "8af92f4ea9fa5fed5374b254c98b8b76": "\\textstyle P_{X_{r}S_{r}Y_{r}}(x,s,y) = P_{X_r}(x)P_{S_r}(s)W(y|x,s)", "8af930dd06d3cc4a9fd35aa8536af56f": " s F(s) - f(0) \\ ", "8af930ec9f84d46348ece59da46592d8": "\\alpha \\subseteq C", "8af94e904a68b29289a31ae784216f82": "(a;q)_\\infty = \\prod_{k=0}^{\\infty} (1-aq^k).", "8af99b88e2533248c8d0437cfb205f40": "\\tilde{C}", "8af9a05768c6261b04e1829f0d0ca759": "4^a(8b + 7)", "8af9aa06b3f0ad286e71be61df2949f7": "\nT_{grav} = rmg\\cos(\\Omega_0t)\n", "8afa0d12e4aa0ad86668277500134989": "\\Delta s = \\frac{\\lambda}{2 \\sin \\varphi} ", "8afa12de90024548406a275e543a7c6a": " (0.00, 1.25);", "8afa25c47e5083db09de77256ab7318f": "0 \\leq j \\leq r-2", "8afa3558ae7db2ee06d5d55b55b51797": "\\Psi_0\\,\\hat{=}\\,0\\,,\\quad \\Psi_1\\,\\hat{=}\\,0", "8afa3803e11d3765c25c45f1598340fc": " \\frac{dH_p}{dt}=H_p(z)H(z)+c", "8afa3b48d9715e3784ff5d9c4257d7b5": "\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\,,\\quad\n\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\,,\\quad\n\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}\\,,\\quad\n\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}\\,,", "8afa768cd9009384b07bd0f41d5a3222": "\\lambda=(I-P')^{-1}a=\\begin{bmatrix}\n1 & 0 & 0\\\\\n-0.5 & 1 & 0 \\\\\n-0.5 & 0 & 1\\end{bmatrix}^{-1}\\begin{bmatrix}\n0.5\\times5\\\\\n0.5\\times5\\\\\n0\\end{bmatrix}=\\begin{bmatrix}\n1&0&0\\\\\n0.5&1&0\\\\\n0.5&0&1\\end{bmatrix}\\begin{bmatrix}\n2.5\\\\\n2.5\\\\\n0\\end{bmatrix}=\\begin{bmatrix}\n2.5\\\\\n3.75\\\\\n1.25\\end{bmatrix}\n", "8afaa82e08666fe6dce1e524444455d1": "\\frac{\\sigma\\sqrt{2}}{\\sqrt{\\pi}}", "8afabdf6a943419f62aeb79c1afb0658": "\\omega_0 > 0\\,", "8afad07af4d4daa0ebc1a8a61efe2531": "\n\\begin{align}\nW_x(t,f) & {} = \\int_{-\\infty}^\\infty e^{i2\\pi k(t+\\tau/2)^2}e^{-i2\\pi k(t-\\tau/2)^2}e^{-i2\\pi\\tau\\,f} \\, d\\tau \\\\\n& {} = \\int_{-\\infty}^\\infty e^{i4\\pi kt\\tau}e^{-i2\\pi\\tau f}\\,d\\tau \\\\\n& {} = \\int_{-\\infty}^\\infty e^{-i2\\pi\\tau(f-2kt)}\\,d\\tau\\\\\n& {} = \\delta(f-2kt) ~.\n \\end{align} ", "8afad5b9a0f7069d12e74032f18253d2": "\\dot{z}_1 = f_1(\\mathbf{x},z_1) + g_1(\\mathbf{x},z_1) u_1(\\mathbf{x},z_1)", "8afb3eb5de2ea2494f5ee9aa7ca28ca2": "\\approx 0.167 \\,", "8afb5ace80092489ff82bb839fbd11be": "F'(y) = 2 y \\int_y^\\infty \\frac{f'(r)}{\\sqrt{r^2-y^2}} \\, dr.", "8afba0c622afb8fb5da8faf4493f61f2": " g(z) = g(a) + g'(a)(z-a) + {g''(a)(z-a)^2 \\over 2!} + {g'''(a)(z-a)^3 \\over 3!}+ \\cdots", "8afbe518001344929dad32111f8d0772": "h\\colon Y\\to X", "8afbf7e084fd949bef5be12bc1b1bcae": "\\gamma\\in R^{k}", "8afc466d1d3d23267cd9f34d4991b94d": "a - b = a + (-b),", "8afc62fad3a509343ad435e30d833ae5": "p_{X^{n}}\\left( x^{n}\\right) ", "8afcacce631d1a6e94a49cc0d876569e": "x=2y, dx=2\\, dy", "8afccb4be4986488b4de7622e4709768": "K(U_1, \\dots , U_d)", "8afd0525881e80450feb55f0a3cf3414": "\\left | d(u) - d(w) \\right | \\notin T", "8afd457c2c6e90953d14b4b568e7265b": "\\mathbf{F}\\cdot{\\rm d}\\mathbf{r}", "8afd45f6a2d4bc70bb4a3baf1c6d120b": "GBWP = {A_1}(\\omega )\\cdot\\omega = \\frac{{{H_0}}}{{\\sqrt {1 + {{\\left( {\\frac{\\omega }{{{\\omega_c}}}} \\right)}^2}} }}\\cdot\\omega \\simeq \\frac{{{H_0}}}{{\\sqrt {{{\\left( {\\frac{\\omega }{{{\\omega_c}}}} \\right)}^2}} }}\\cdot\\omega = {H_0}\\cdot{\\omega_c} = const.", "8afd8b5e712e57b9071176011ef3ff74": "\\mathbf{v}_x(\\mathbf{u}_y\\mathbf{w}_y+\\mathbf{u}_z\\mathbf{w}_z)-\\mathbf{w}_x(\\mathbf{u}_y\\mathbf{v}_y+\\mathbf{u}_z\\mathbf{v}_z)", "8afd9e0399e33aec3104b0273f798ff4": "b_j=\\int f v_j dx", "8afdc2e4c5af61debc91547b8e5e24c9": "\\scriptstyle |0\\rangle_A |1\\rangle_B", "8afe048e7286e4dd3651d9560bbf9227": "\nh_2 = 2 \\frac{dA_2}{dt} = r^2 \\frac{d\\theta_2}{dt} =\nk r^2 \\frac{d\\theta_1}{dt} = 2 k \\frac{dA_1}{dt} = k h_1\n", "8afe184fb10cffceb28fbb15eb7f1756": "\\psi \\equiv \\phi", "8afe18ba426078802af16274f2b6b9fd": "\\mathbf{x}^{k+1}", "8afe505057d8fbbf7c78316976023114": " f(y) \\ge f(x) + \\nabla f(x)^T (y-x) + \\frac{m}{2} \\|y-x\\|_2^2 ", "8afe560e4a38969ae54226d983859045": "\\Gamma \\left[{1 \\over 2} + \\left( \\frac{n+l}{2} + 1 \\right) \\right]\n= \\frac{\\sqrt{\\pi}(n+l+1)!!}{2^{\\frac{n+l}{2}+1}} = \\frac{\\sqrt{\\pi}(n+l+1)!}{2^{n+l+1}[\\frac{1}{2}(n+l)]!},", "8afe58686184911d3702f88176b12ed5": "X^u", "8afe8a5ab8aa5763d831a8e7d4e35387": "2\\pi \\sqrt{1\\ \\mathrm{m}\\over g} \\approx 2.0064\\ \\mathrm{s}", "8afec14025a23ae539912b3ca2039389": "d_i=\\frac{ (v_T-v_i)-m_i (u_T-u_i) }{\\sqrt {1+m_i^2}}", "8afec1817e6e39ef27bf12f881ff14ae": "\\mathrm{O}(n) \\to \\mathrm{O}(n+1) \\to S^n,", "8afec71f3cebff8a6340c67f15aab4cc": "m[W]", "8afed4d4061bcea9485dda8a07c09ad6": "N_j = \\sum_{i=1}^e \\lambda_i \\frac{\\partial F_i}{\\partial r_j} ", "8afed845302735d9d779fa7476e2f314": "\\displaystyle{U_{\\alpha+\\beta}=\\varepsilon(\\alpha,\\beta) U_\\alpha U_\\beta}", "8afefdc101cd08e7ab9cffecb06131ee": "X_\\mathrm{horiz} \\approx 37.20 \\,.", "8aff04756c4d9c330356dc9cd0ff7552": "\\frac{d(n)}{n^\\varepsilon}\\geq\\frac{d(k)}{k^\\varepsilon}", "8aff0aeaa5507c72201b566aed5050c2": "100\\uparrow\\uparrow\\uparrow n", "8aff0ca8cc3572a132cbbac92a4c725c": "\\begin{align} \\ln 2 &\n= \\frac{1}{2} + \\frac{1}{2 \\cdot 2^2} + \\frac{1}{3 \\cdot 2^3} + \\frac{1}{4 \\cdot 2^4} + \\frac{1}{5 \\cdot 2^5} + \\cdots \\\\ &\n= \\sum_{k=1}^{\\infty}\\frac{1}{2^k \\cdot k} = \\frac{1}{2} \\sum_{k=0}^{\\infty}\\left[ \\frac{1}{2^k} \\left( \\frac{1}{k + 1} \\right) \\right] \\\\ &\n= \\frac{1}{2} P\\left( 1, 2, 1, (1) \\right).\n\\end{align}", "8affcd8f89ac75983e4b3cd21fb93c37": "c_n (t) = 0\\,", "8b000dc50ef8b51e36062482cf93a004": "\\sqrt{SS_\\text{tot}}", "8b00ab835539873b6bf2a9e2acb727e2": "\\mathcal{H}^m", "8b00af2c2457f025c9362b8ba3ef3f19": "\\begin{bmatrix}3 & 1\\\\7 & 5\\end{bmatrix}\n\\rightarrow\n\\begin{bmatrix}1.8125 & 0.0625\\\\3.4375 & 2.6875\\end{bmatrix}\n\\rightarrow \\cdots \\rightarrow\n\\begin{bmatrix}0.8 & -0.6\\\\0.6 & 0.8\\end{bmatrix}", "8b011fc1aee9820df43cf8f88bba4a25": "\\alpha : I\\rightarrow T_pM", "8b015624a166690e5b4e2aaa66413057": " m(x,\\omega) ", "8b015fe10b27fd42c2cd884d1cc4099b": " x_1^n + x_2^n +\\cdots +x_n^n = -n\\lambda x_1x_2\\cdots x_n \\, ", "8b019f146942c226124358d7fa4aef19": "(z\\omega)[X]=\\bar{z}\\omega[X]", "8b01a6852354a86342d33f0b194fbb90": " {\\mathit l = \\mathit l^{\\prime}} . ", "8b01c7738992277b626c94298796870e": " \\forall \\ x,y,z\\in X \\ \\ \\neg (xRz) \\and \\neg (zRy) \\Rightarrow \\neg (xRy),", "8b01cd63da12e97e532de8b0bdc9fcd1": "p = 0.5 \\left ( \\frac{c}{u_{m}} \\right )^3", "8b01df6045fa87323e3ff465dadbf8a9": " x_n=n^{-1/2}\\lambda_n=\\overline{y_n}", "8b020457f3534b6fe836ffadd7cc29ef": "\\begin{align}\n\\sum \\limits_{n=1}^{\\infty} a_nx^n \n= x &+ {m \\choose 1}\\sum \\limits_{a=2}^{\\infty} x^{a} \n+ {m \\choose 2}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} x^{ab} \\\\\n&+ {m \\choose 3}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} \\sum \\limits_{c=2}^{\\infty} x^{abc} + {m \\choose 4}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} \\sum \\limits_{c=2}^{\\infty} \\sum \\limits_{d=2}^{\\infty} x^{abcd} +...\n\\end{align}", "8b023088802f528f7293da638b27babc": "A = (1/2F) (2\\sqrt{2}/\\pi)|v| = 2.1\\times10^{-6} \\text{m}", "8b02324aa3acd37a6f3af9e6d120d55e": "b^*=b", "8b023abce00516fbba79ca43f3758e06": " v(z) = \\left( 1 + C \\mathrm{exp}\\left(\\pm{z}/{\\sqrt6}\\right) \\right)^{-2} ", "8b02615644200444c0c5337b50f469f4": "\\begin{matrix}{4 \\choose 2}{4 \\choose 1}{32 \\choose 1}\\end{matrix}", "8b026c6588e88be70123c39bd1a3aa45": " f(S_{x,i}, g) ", "8b029b6b75548f6067e1dd9f7a320f5f": "p_{k} = \\frac{k^{-\\tau}\\mathrm{e}^{-k/\\kappa}}{\\mathrm{Li}_{\\tau}(\\mathrm{e}^{-1/\\kappa})}\\ \\mathrm{for}\\ k\\geq 1", "8b02c2e847bdb083f9b6dd2eae8a8f3f": "\\operatorname{grad}(f_t)", "8b02cf3a30a3b860e01173806ec8cfb0": "x=1,y=2", "8b0315031e8f3aa736b8bd3d1bf8b196": "k\\leq1000", "8b031630d7d834af3caab1a8b86919ca": " A = E + L ", "8b032d184a23535239e347f91af1bf1c": "D_0(X)", "8b033a10d9fc843fd1eac54523a2d2f2": "c_i^1 \\ne c_i^2", "8b037aeb57c85fa1b2312c47b2ed029e": "\\alpha \\approx 1/137", "8b038e2bbbae5743c4b9abe459562220": "Z_n = Z_0 + \\frac{800}{E+n} (W - W_e)", "8b03bea88df4eb87d57bf362bb2a5604": "\n J_1 := \\int_{-h}^h \\rho~dx_3 = 2~\\rho~h ~;~~ \n J_3 := \\int_{-h}^h x_3^2~\\rho~dx_3 = \\frac{2}{3}~\\rho~h^3\n", "8b0406cb3204074212bb16d0ffa85b3d": "|E| = \\aleph_0", "8b041db19e7fd52be1fd4fa55b0b6eeb": "\n\\Delta F\\left(r\\right)=-kT\\ln\\frac{P\\left(r\\right)}{Q_{R}\\left(r\\right)}-kT\\ln\\frac{Z}{Z_{R}}\n", "8b0472d6a37eec5c86e6bf59de6d8703": " X_{f}", "8b049283a1be6683688aa065cd3e91ad": "I^+", "8b04a321c8b246e4d6a26d674edf7cd9": " b_0 ", "8b04d5e3775d298e78455efc5ca404d5": "first", "8b054b9fc3d623df54c51574631889fc": "\\psi^R = \\frac{1}{2}(1+\\gamma_5)\\psi", "8b0550119fbebf2299cceb9de4f48f0d": "q \\,\\in\\, J", "8b059528c4f5fb4748ffc3640d458474": "Q_\\text{accept}", "8b05de68bf1f4b51d3d5e247260cb3fc": "\\ Z = |Z| e^{j\\theta} \\quad", "8b05e91bd5ccde2e1f7ec32782283dcc": "\\scriptstyle \\Delta_T \\;\\approx\\; \\alpha RC", "8b0659ac7e195d4fe885662d795c89e3": "f_s = 100e6", "8b068026bb9b466ce141d63d7f4d6138": "L_2=0", "8b06c3ec2ceb1eca38c1ab62424537ec": "\\int f(x)\\,dx = -\\cos x + C ", "8b06c83d779521954fbf4df01bad9acf": "\\mathrm{Var}(\\varepsilon_i)=\\psi_i", "8b06d0f3ff29743fde9de5eeba19c727": "A=(a_1,\\ldots,a_n)", "8b06d43cceeed417abf5c7c9359bddd5": "36^3", "8b06e6abaea7b3b4f51f0012efb305a3": "S(d)=\\{1,2,\\dots,k\\}", "8b070962b15edf6809824847bc6ab368": "c = \\frac{1}{1 + \\tfrac16\\, \\left( \\kappa h \\right)^2}\\, \\sqrt{g\\,h},", "8b07137b0ba2a0699d10dd3a2918c69b": "\n{52 \\choose 7} = \\frac{52!}{7!(52-7)!} = \\frac{52!}{7!45!} = \\frac {52 \\times 51 \\times 50 \\times 49 \\times 48 \\times 47 \\times 46} { 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1} = 133{,}784{,}560\n", "8b072dc4ba86c6008887fd8004865c61": "n'", "8b072e80530d69c85ff00b6dd5672fae": "\\chi=\\alpha-\\beta=k\\ln \\xi+\\mathrm{const},\\,", "8b07590fb68cb65ef73bb6feec6ba5b7": "V=B_1^Y(\\underline{0})", "8b0768e04282850d552af9e1d60fce5c": "f \\mapsto D(f) = \\sum_{i=0}^n a_i \\frac{d^i f}{d x^i}", "8b07a5231497eea78d3100f2287c062e": "\\Theta = \\sum_{k=1}^n x_k \\frac{\\partial}{\\partial x_k}.", "8b07e7af78cb2e0dc9158427ae695161": "G \\cap D_r.", "8b08998266540ce5776ab59361f9df1c": "r(v):V \\to \\mathbb{Z}", "8b08dd54fb73745f66fdcdcb07c3e4d9": "b_1, b_2", "8b0912dec4a1b37d2829a9f1a895ff97": "V^*Y\\to Y", "8b0949f1187bf26ef34c58fa019e816b": "\\mathbb{N}^n", "8b09672db89f62437f49c717617c1ee8": "\n=\\frac{\n (0.99 \\times 0.01 \\times 0.2)_{TTT} + (0.8 \\times 0.99 \\times 0.2)_{TFT}\n}\n{\n (0.99 \\times 0.01 \\times 0.2)_{TTT} + (0.9 \\times 0.4 \\times 0.8)_{TTF} + (0.8 \\times 0.99 \\times 0.2)_{TFT} + (0.0 \\times 0.6 \\times 0.8)_{TFF}\n}\n", "8b0975a0bc26073d23252dd4e3d13cda": "m\\geq 1", "8b097ed01fc1311cadbb22980b44c600": "\\tilde E_6, \\tilde E_7, \\tilde E_8", "8b099531153674f7c5947f718e61a800": "\n t = constant + {D_d D_s \\over D_{ds} c} \\tau, ~ \\tau \\equiv \\left[ { (\\vec{\\theta}-\\vec{\\beta})^2 \\over 2} - \\psi \\right] \n", "8b099cab0f3f6fd80d14890c219903d9": "\\mathbf{B} = B_x \\hat{x} + B_y \\hat{y} + B_z \\hat{z}", "8b099f533c96cba6a9eca100c5274ad3": "2(\\eta+1) ", "8b09c16f1cec9e5c96186b33a314a2eb": " \\lambda f.(\\lambda x.x\\ x)\\ (\\lambda x.f\\ (x\\ x)) ", "8b09ec32a5f743a547aafd598875a507": "B \\models R(h(a_1),\\ldots,h(a_n)).", "8b0a4074cca231c97a741f1c9e084a33": "\\alpha^{-1}=\\bar\\alpha\\mathrm N(\\alpha)^{-1}", "8b0b24dafbf5e28aca353ec2f748d050": "r_s = \\frac{2GM}{c^2} \\;", "8b0b74d3531b711dbbd4c38aff33c056": "\\omega(x)\\,\\!", "8b0b83a233e30bec482a4e6af564e1de": "w_2l^*", "8b0bdb227ed0817766aeec5cf3ea422c": "\\omega_p = -7.44\\times10^{-7}\\ rad/s", "8b0c109c7525a89d35c8ae14fdf10a09": "\n\\cdots\\left\\vert\n\\begin{array}\n[c]{c}\nIII\\\\\nIII\\\\\nIII\\\\\nIII\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{c}\nXXX\\\\\nZZZ\\\\\nIII\\\\\nIII\n\\end{array}\n\\left\\vert\n\\begin{array}\n[c]{c}\nXZY\\\\\nZYX\\\\\nXXX\\\\\nZZZ\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{c}\nIII\\\\\nIII\\\\\nXZY\\\\\nZYX\n\\end{array}\n\\left\\vert\n\\begin{array}\n[c]{c}\nIII\\\\\nIII\\\\\nIII\\\\\nIII\n\\end{array}\n\\right\\vert \\cdots\n", "8b0c185c11b153602a609a8b950deadc": "\\delta\\tau-\\Delta\\sigma=(\\mu\\sigma+\\bar{\\lambda}\\rho)+(\\tau+\\beta-\\bar{\\alpha})\\tau-(3\\gamma-\\bar{\\gamma})\\sigma-\\kappa\\bar{\\nu}+\\Phi_{02}\\,,", "8b0c86632a659287b8abe314b9c65dbe": "X ^1\\Sigma_g^+", "8b0c885970cd624b9e3c0951bf0230f1": "\\frac{1}{\\log n} < n^\\epsilon", "8b0ca80408df4bb5888116f5f9aa00e2": " f(z)=\\frac{e^{iz}}{z} ", "8b0ced1c2cf39ba2dd1fbe1ea79315a0": "(\\phi \\to \\lnot \\phi ) \\to \\lnot \\phi ", "8b0cfda9d1167a4387b52d38fb4c3f43": "\\scriptstyle Z^d ", "8b0d8ec773099a0eefe2f440b2cdc8b6": "S\\cong e\\mathbb{M}_{n}(R)e", "8b0db9c863801161a91a41ef7b855ebf": "\\Gamma_w = \\frac{\\Gamma_P + \\Gamma_E}{2}", "8b0e2960ae330b6b52c4a0742c289757": "L(\\psi) \\ge 3l+16", "8b0ea7878edf6fa5f6b5155d9b92ac14": "\\dim(R/P) + \\dim(R/Q) \\le \\dim(R)", "8b0eac60a6b9d8e7a0c8d708b2ac04ca": "\\Delta K = m\\int_{t_1}^{t_2}\\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}dt = \\frac{m}{2}\\int_{t_1}^{t_2}\\frac{d}{dt}(\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}}) dt = \\frac{m}{2}\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}}(t_2) - \\frac{m}{2}\\dot{\\mathbf{X}}\\cdot \\dot{\\mathbf{X}} (t_1) = \\frac{1}{2}m \\Delta \\mathbf{v^{2}} , ", "8b0eda81994ce4199b3572c9ce955bc4": "Y_{2}^{-1}(\\theta,\\varphi)\n={1\\over 2}\\sqrt{15\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot\\cos\\theta\\quad\n={1\\over 2}\\sqrt{15\\over 2\\pi}\\cdot{(x - iy)z \\over r^{2}}", "8b0f167432bd531a9eee57b9b0aed09c": "dU = T \\, dS - \\sum_i X_i \\, dx_{i} + \\sum_j \\mu_j \\, dN_j\\,", "8b0f55b15c05710c97a6c3df6649d3a7": "T_k", "8b0f7830403c81074e2464b51d5cf16e": "\\frac{|ax(t)+by(t)+c|}{\\sqrt{a^2+b^2}}", "8b0fc38ea759443f8c985231060d6ea4": "\\int_\\mathbb{R}\\left(\\int_\\mathbb{R}|f(x,y)|\\,dx\\right)\\, dy", "8b0fc7899ad2d91aaaaeebaf4d0d268d": " h_n = \\sum_{i=0}^{n-1}{(4i+1)} ", "8b1008a23d760e2587006aabb1016108": "A(t).\\,", "8b106b811877550aff2a5c7c6f54ac68": "O(kn(\\log n)^2)", "8b1096599dfb6d30fc8d51e8b112cf21": "\\sum_{n=0} \\sqrt{\\frac 2 \\pi} x^{n+\\frac 1 2} e^x K_{n-\\frac 1 2}(x) \\frac {t^n}{n!}= e^{x(1-\\sqrt{1-2t})}.", "8b10b8698fc330801ced047c90aaea73": "n'=N(k,d)-d", "8b10cd41b4a3e38cf9a85968b7113204": "\\text{Var}\\left(y_d - \\left[\\hat{\\alpha} + \\hat{\\beta}x_d\\right]\\right) = \\sigma^2 + \\sigma^2\\left(\\frac{1}{m} + \\frac{\\left(x_d - \\bar{x}\\right)^2}{\\sum (x_i - \\bar{x})^2}\\right) = \\sigma^2\\left(1+\\frac{1}{m} + \\frac{\\left(x_d - \\bar{x}\\right)^2}{\\sum (x_i - \\bar{x})^2}\\right) .\n", "8b115b4dbd6e5effe3f4e8f9cabd97d6": "A_{n,\\sigma}(s,t)=\\{(s_1,\\ldots,s_n)\\in I_{s,t}^n\\mid s_{\\sigma(1)}0\\}\\in\\Delta", "8b127e9a573ebc6f6b24253284185ee2": "\\kappa = \\frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}},", "8b12973f5f467863fcb264ab422c70dc": "f^{-1} : Y \\to X", "8b12aae67620f8874be0058b2e0b55b5": "\n\\begin{align}\n\\Sigma & \\ll p^{1/2} \\log p , \\\\[6pt]\n\\Sigma & \\ll 2 R^{1/2} p^{3/16} \\log p , \\\\[6pt]\n\\Sigma & \\ll r R^{1-1/r} p^{(r+1)/4r^2} (\\log p)^{1/2r}\n\\end{align}\n", "8b1376f6f7340002f6dd1807fb005a9b": "Q(S_i,r)", "8b139df36d0148d4b14b68fe239fc233": "\\sigma_{\\rm long} = \\frac{p(r - 0.4t)}{2tE}", "8b13e6a6e0eacf26a41e3c4427995567": "d_{FG}=d_F+d_G.", "8b13ffc02a7c8e4b4b0d09208258262e": "B=\\begin{bmatrix}\n1 & 1 & 1 & 1\\\\\n1 & 1 & 0 & 0\\\\\n1 & 0 & 1 & 0\\\\\n1 & 0 & 0 & 1\\\\\n\\end{bmatrix}", "8b141f94d4371ad99206ca92a896986d": "\\forall", "8b143b50e038dda88b1b0255bd849b57": "f(x,y,z)=C(t)\\text{ whenever }x - y + z = t.\\, ", "8b1477b0a37ddbe845626b224cf919da": "Y_{3,0} = \\omega_ey_e", "8b14be560334f9815d90e0d93ba94787": "\\left(N=\\textstyle\\sum N_i\\right)", "8b14d57bb0b9179bc27f87738a411da5": "\\textstyle R_{x}w_{k}=\\mu a(\\theta_{k}),\\ or\\ W_{k} = \\mu R_{x}^{-1}a(\\theta_{k})", "8b14dee6fb8c36e633402dcd86c326fe": "P-P^*", "8b14f598112b843a401e9db5c8425118": "f_i = \\sum_{j=0}^m \\sum_{i=1}^n a_j x_i^j", "8b1527cfba5002a6eb750ec737057b7f": "f'(x)=\\overbrace{\\prod_i(f_i(x))^{\\alpha_i(x)}}^{f(x)}\\times\\overbrace{\\sum_i\\left\\{\\alpha_i'(x)\\cdot \\ln(f_i(x))+\\alpha_i(x)\\cdot \\frac{f_i'(x)}{f_i(x)}\\right\\}}^{[\\ln (f(x))]'}", "8b155deef970a727b11772af065feeda": " \\mathbf{x} = \\mathbf{a} + \\lambda \\mathbf{b}", "8b15c3419fb5266a19afbdf6bf27e55d": "m(A\\cap B) = 0.", "8b15e9f104dff202eeeb5e751e1731ea": "E\\{Z(x)\\}", "8b160d4db5db0f2f7c0733b2c7e37de1": "\\rho = \\sum_s p_s | \\psi_s \\rangle \\langle \\psi_s |", "8b16425fe8f246f5363206e75bb7f4a5": " R_{i,j} = R_{j,i} ", "8b164340557ebca271f03c8edee2161c": "\\|A\\|_{op} \\ge 0 \\mbox{ and } \\|A\\|_{op} = 0 \\mbox{ if and only if } A = 0 ,", "8b1643ce1edf0b62e053357b72a8f76d": "e = 2^{16} + 1 ", "8b165ecabff286937ccfde9355a8481f": " \\nabla \\times \\mathbf{B}_\\text{g} = 4 \\left( -\\frac{4 \\pi G}{c^2} \\mathbf{J}_\\text{g} + \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}_\\text{g}} {\\partial t} \\right) ", "8b1663a14d0f683d7bc533eb02ca8b05": "\\omega=\\sum_{i 0 ", "8b3096d0e824f9b3076d14e0129adb43": "1-e^{-9.7/4.5}", "8b310e65ac1e5bce66b94328d70c64a1": "T_{p^k},\\ T_{q^\\ell},\\ T_{r^m}", "8b31167bd9dbcc856436015746b5003d": "(e^{\\frac{1}{x}}y')'+\\frac{2 e^{\\frac{1}{x}}}{x^3} y =0.", "8b31722adc96317f6ef0114e15d5eba2": "\n\\begin{bmatrix}\n 2 & 1 \\\\\n-3 & 1 \\\\\n-1 & 1 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nX_1 \\\\\nX_2 \\\\\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n-1 \\\\\n-2 \\\\\n 1 \\\\\n\\end{bmatrix}\n", "8b3174652465c7d1d01ac2d518893ac4": "\\left(3\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ 0\\right) \\pm \\left(0,\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "8b317cb5fdfda09266079f487a484fd1": "i=1,2,3\\ldots,N", "8b32067fe833508edc39a996fc9c7e59": " x \\to (y \\to (x \\and y)) = 1 ,", "8b320ffd8d8d4b0da249ff1074fa9554": "\\scriptstyle U_1", "8b321a6f900186e2c6dd4f186488efc8": "_{q \\rightarrow p}\\!", "8b327e2a620ae3220310c098f779021f": "P(x) = 0", "8b32914dd956ac43cd32a51095bff4a4": "\\tilde{L}(\\mathbf{\\alpha})=\\sum_{i=1}^n \\alpha_i - \\frac{1}{2}\\sum_{i, j} \\alpha_i \\alpha_j y_i y_j k(\\mathbf{x}_i, \\mathbf{x}_j)", "8b32c1bec6057020f51cc2441b1db841": "\n\\det\\left(\\frac{\\partial}{\\partial_z} - L(z) \\right) =\\sum_{i=0}^n H_i(z) \\left(\\frac{\\partial}{\\partial_z}\\right)^i. \n", "8b3350569437ef4493c24b4a901df77d": "\\theta(x)", "8b336e58826d052df5a15ccc15fb82be": "sP", "8b339125383031d76f2d1126c1fcef9d": "r = ae^{b\\theta}\\,", "8b339cee960ff9434baf287fb8d81314": "( z_2 - z_1 ) = \\frac{R \\cdot \\bar{T}}{g} \\ln \\left( \\frac{p_1}{p_2} \\right) ", "8b33a26a54921f41aa5669b0fd500151": "\\tan \\theta = \\frac{e^{i\\theta} - e^{-i\\theta}}{i(e^{i\\theta} + e^{-i\\theta})} \\,", "8b34092c4e0609d04c0c0ae576865150": "P(G, t) = P(H, t)", "8b340a5c07fff1ebce704fd289f0855c": "n \\times nr", "8b3426885bb3ba7979716f329bc735de": "O(Nmlog^2(Nm)loglog(Nm))", "8b346d6aa2564842ee74c0a3987bb4e6": "\\Delta a = \\sqrt{\\langle\\hat a^2\\rangle-\\langle\\hat a\\rangle^2}", "8b348d46be8c601218fbdf9a26135cf3": "x^6-2x^5+x^4-x^2+x-1,", "8b34bec5f997a8c0d2011b502ad8641e": "\\int_0^\\infty \\cdots \\int_0^\\infty G'(u(s_1, \\ldots, s_l)) \\times h(s_1 + \\cdots + s_l)ds_1 \\ldots ds_l < k_2", "8b34d3a7fb774e57bc23c6b8e7e8ccb6": "\n\\nabla_i \\vec V := \\frac{\\partial\\vec V}{\\partial x^i} - \\vec n = \\left( \\frac{\\partial v^k}{\\partial x^i} + v^j \\Gamma^k{}_{ij} \\right) \\frac{\\partial\\vec\\Psi}{\\partial x^k}.\n", "8b351e69f1566b19d20150db1ec12b21": "\\Delta G_{ad} = -RTln(K_{ad})", "8b356b5dc2fe3243aec0d8a732c25068": "AB^+ + M \\to A + B^+ + M", "8b3581824c6329d96235e285e2f6a92e": "\\displaystyle r_a+r_b+r_c+r=a+b+c", "8b35c451d7a327953f4ceabc89b18063": "\\Psi_0 := C_{\\alpha\\beta\\gamma\\delta} l^\\alpha m^\\beta l^\\gamma m^\\delta\\ , ", "8b35e6e5c5d2dd49c763e9ee5b264980": " exp(-d/dz)(Id + E/z) ", "8b362706cc2570bb2fd4282498c8a226": " g( y ) = \\frac{ 1 }{ \\sqrt{ k \\pi } } \\frac{ \\Gamma\\left( \\frac{ k + 1 }{ 2 } \\right) }{ \\Gamma\\left( \\frac{ k }{ 2 } \\right) } \\frac{ 1 }{ y^2 \\left( 1 + \\frac{ 1 }{ y^2 k } \\right)^{ \\frac{ 1 + k }{ 2 } } } .", "8b369186c80831c0d259862ed9defe9c": "\nH_{p}x(n) = \\sum_{\\tau_1=0}^{M}\\sum_{\\tau_2=\\tau_1}^{M}\\cdots\\sum_{\\tau_p=\\tau_{p-1}}^{M}\n {h_{p}(\\tau_{1},.\\,.\\,,\\tau_{p})\\prod^{p}_{j=1}{x(n - \\tau_{j})}} .\n", "8b36ad3708722b79bf05f36a8cb98a11": "\\lambda=NBA", "8b37142d3f58b2cf707f3836b001cdab": "(\\Sigma E)_n = E_{n+1}", "8b379104a98954f284579e11d074b6c5": "v_x\\,\\mathbf{e}_x + v_y\\,\\mathbf{e}_y + v_z\\,\\mathbf{e}_z,", "8b382e2c227bafbe8c7caec60c4c6c04": " \\mathbf{\\mathfrak{T}} ", "8b38372f7f11f861c9b979cb7933a1eb": "b_c\\;", "8b3858120613024c333b0fc303903ca5": "N_\\mu^\\perp", "8b39037ee9a908cb423a6aeb7e0b9037": "\\scriptstyle 2k+1", "8b3919c162338465a60bb09fcf20789e": "\\psi_1(\\alpha) = \\frac{\\part^2\\ln\\Gamma(\\alpha)}{\\partial \\alpha^2}=\\, \\frac{\\part\\, \\psi(\\alpha)}{\\partial \\alpha}", "8b391b067a1aedb0b1587017ef5053bd": " K_t(x) = {1\\over \\sqrt{2\\pi (i t + \\epsilon)}} e^{ - x^2 \\over 2it+\\epsilon }\\,", "8b3a23064008578303aac8a772c74d57": "\\sigma,", "8b3a248130663140376a3136e01c01ff": "\\langle\\langle\\;\\cdot\\;\\rangle\\rangle", "8b3a2ce6ffa9151517897b6e95f4baed": "\\kappa(X_1,\\dots,X_n)=\\sum_\\pi \\kappa(\\kappa(X_i : i\\in B \\mid Y) : B \\in \\pi),", "8b3a3841d7adc8bcc62d16f0a5c62ce3": "z=8.6", "8b3a57f0dc42c62cf4d7740b235b823f": "T = \\{w_1, \\ldots, w_m\\}", "8b3afbd0ffffea5e21670460844cdac1": "\\sum_{-(m-1)/2}^{(m-1)/2} z^2= {m(m^2-1) \\over 12}", "8b3afcf7c31fbfbe1b9872b9c03ebb84": "[x,[x,x]] = 0", "8b3b0e4f26e206600f8abb2abe15449b": "L_{g}L_{f}h(x) = \\frac{\\operatorname{d}(L_{f}h(x))}{\\operatorname{d}x}g(x).", "8b3b2531431962cd0ef6acd734c09fb0": "\n\\begin{align}\n\\Delta^\\acute{n}F(P_0) & =F^{(\\acute{n}-1)}(P_1)-F^{(\\acute{n}-1)}(P_0), \\\\[10pt]\n& =\\frac{F^{(\\acute{n}-2)}(P_2)-F^{(\\acute{n}-2)}(P_1)}{\\Delta_1P}-\\frac{F^{(\\acute{n}-2)}(P_1)-F^{(\\acute{n}-2)}(P_0)}{\\Delta_1P}, \\\\[10pt]\n& =\\frac{\\frac{F^{(\\acute{n}-3)}(P_3)-F^{(\\acute{n}-3)}(P_2)}{\\Delta_1P}-\\frac{F^{(\\acute{n}-3)}(P_2)-F^{(\\acute{n}-3)}(P_1)}{\\Delta_1P}}{\\Delta_1P} \\\\[10pt]\n& {\\color{white}.}\\qquad -\\frac{\\frac{F^{(\\acute{n}-3)}(P_2)-F^{(\\acute{n}-3)}(P_1)}{\\Delta_1P}-\\frac{F^{(\\acute{n}-3)}(P_1)-F^{(\\acute{n}-3)}(P_0)}{\\Delta_1P}}{\\Delta_1P}, \\\\[10pt]\n& = \\cdots\n\\end{align}\n", "8b3b91a9fbe9d58cc963e08a3a457d1a": "\\left|\\mu_1 - \\mu_2\\right| > 2\\sigma,", "8b3bbc76d1acc50519ff73a9d50d7c45": "DM(x1^{2^{|x|^c}})", "8b3bdaab5699b2c9fc1a82d016466126": "\\scriptstyle{c}", "8b3bf7afa789ca80574c234116d53c49": "{d \\over dx} \\left[ (1-x^2) {d \\over dx} P_n(x) \\right] + n(n+1)P_n(x) = 0.", "8b3c5890efc083e802e3b89392a8b4a4": "\\delta_c(A)", "8b3c6d962ce31d71d71eed432b88ad94": "w_2l^*h>p/2", "8b8977980e0eb1ab74d992d615d79d53": "{\\tilde{B}}_n", "8b89921c8065d94bfca965c2a1da5d1c": "-i(r\\bar{g}-g\\bar{r})/\\sqrt{2}", "8b89a43d93ebcb54fa54d24afb9e51b8": "\\nabla f", "8b89d0af76dd4579a240f0b32a2ba6e8": "k(t)", "8b89d577860dbfad983570452719c549": "\\mathbf{B}_{r}(x) := \\{ z \\in X | d(x, z) < r \\}.", "8b89e3afb508a34c99d5d81ed9783f50": "\\nabla\\cdot\\mathbf{B} = \\rho_m", "8b8a08b3fa6c938ada6dfd602e046cfb": "f : X \\to Y", "8b8a62e2cdc95f9c948c2ff74c46f9b5": "\\tfrac{26}{11} \\simeq 2 \\tfrac {36}{100}.", "8b8ac44386c613bf63d7b64a29e3b38f": "M_T(u) \\equiv E[e^{u^{\\rm T} T(x)}|\\eta] = \\int_x h(x) e^{(\\eta+u)^{\\rm T} T(x)-A(\\eta)} dx = e^{A(\\eta + u)-A(\\eta)}", "8b8adc0991f2a532d81bc83adb1b53a8": "a_N = \\frac{\\sum_{i=1}^N \\mu_i(T)}{\\log(N)}", "8b8b1a3d3603c496c5bbadbda2ea0cbd": "\\Sigma^1_n \\subset \\Sigma^1_{n+1}", "8b8b5ae9583c4d3b4e06e8b6b8050d07": "G/G_n", "8b8b6c755d5adc52fd2833d56b8dfbdf": "R[\\Delta^\\bullet]", "8b8b84c1917067fc3c968fbfe91e32e0": " \\mathbf{F} = \\iiint \\! ( \\rho \\mathbf{E} + \\mathbf{J} \\times \\mathbf{B} )\\,\\mathrm{d}V. \\,\\!", "8b8b901ec30379587f416b8041f6779a": "\\begin{array}{cc} \\begin{array}{rrrr} \\\\ \\\\ j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} & & oj & ok & ol & & & \\\\ & nj & nk & nl & nm & om & & \\\\ a & b & c & d & e & f & g & h \\\\ \\hline a & o_0 & p_0 & & & & & \\\\ n & o & & & & & & \\\\ \\end{array} \\end{array}", "8b8bcb72b9257b80aa84679da7b598ae": "(1,\\sin\\theta,\\cos\\theta)", "8b8c047c9988a3537c885cc884f17910": " \\exp_p", "8b8c21347dfd84493de12939fea3f3ef": "g_i = g(\\left\\lbrace x_i \\right\\rbrace)", "8b8c275c826a486a3862522824b8c754": " F=0.9", "8b8c2bd50d0a241b9ba2aec9e513b289": "\\phi_k = \\pi", "8b8c3795702979840464f2f0620baaef": "C_\\text{out}(m) = ((C_\\text{out} (m)_1, \\ldots, (m)_N ))", "8b8cdac77d1e77528fede886130222b4": "R(\\hat{n},360^\\circ) = 1", "8b8d1025c8740c08fa387feef6ae4bd6": " Peak \\ Criteria \\begin{cases} \\mathrm{\\left (\\frac {\\Delta Amplitude}{\\Delta Frequency} \\right)Cell(n-1) < 0} \\\\ \\mathrm{\\left (\\frac {\\Delta Amplitude}{\\Delta Frequency} \\right)Cell(n+1) > 0} \\end{cases}", "8b8d45611c9bf0c258628adc38b69edc": "(d^\\nabla)^2\\sigma = F^\\nabla\\wedge\\sigma.", "8b8dcb83e9deb96b0bd3ac1b00f267a0": "\\mathrm{d}H = T\\mathrm{d}S+V\\mathrm{d}p + \\sum_i \\mu_i \\mathrm{d}N_i", "8b8e1ecc2c7c96971eafac98c495551d": " z=i\\omega", "8b8e4f71ad79595f1dc69d4e11ec781d": "\\delta r = y \\cdot \\delta t - MD \\cdot \\delta y + \\frac{1}{2}C \\cdot \\delta y^2 + O\\left( {\\delta t^2 ,\\delta y^3 } \\right)", "8b8e5ecd89aab5723de62b796bab3571": "h^2=pq \\Leftrightarrow h=\\sqrt{pq}", "8b8e608ada8ed89d1e6d37c9b73d38b5": "T_C = \\frac{C \\lambda }{\\mu_0}", "8b8e66829f7e11fb233fea99826e3ac7": "[H]_{eq}", "8b8ea82628946e01a679d42c36d99f4a": "\n\\begin{matrix}\nI_1\\dot{\\omega}_{1}+(I_3-I_2)\\omega_2\\omega_3 &=& N_{1}\\\\\nI_2\\dot{\\omega}_{2}+(I_1-I_3)\\omega_3\\omega_1 &=& N_{2}\\\\\nI_3\\dot{\\omega}_{3}+(I_2-I_1)\\omega_1\\omega_2 &=& N_{3}\n\\end{matrix}\n", "8b8ec663032a67e4cf582f7c6f97ce00": "X^2-Y^2-Z^2 = 0", "8b8edc69532dbd6ff2156100697e579b": " S = -k \\langle \\log P \\rangle = - \\frac{\\partial \\Omega} {\\partial T}, ", "8b8f27396fa20a38140b4ba285dcd2d0": " \\mathrm{Proj}_{{\\hat{u}}}\\,{v} = \\hat{u} ( \\hat{u} \\cdot v)", "8b8f625a0d9114ce5d823e249b573f8d": "g\\cdot(aH) = (ga)H.", "8b8f71f617f8bc0f5e66e5139d612879": "\\frac{\\partial U/\\partial x_i}{p_i}=\\lambda~~\\forall i", "8b8f7a21c2068c8ba5b9f10123b88f99": "V^{\\mathbb C}= V^{+}\\oplus V^{-}", "8b8fb7ab3eb6beb7f08c838a21136d8a": "f(x) = \\begin{cases}e^{-\\frac{1}{1-x^2}} & \\mbox{ if } |x| < 1, \\\\ 0 &\\mbox{ otherwise }\\end{cases}", "8b8ff499a062fd067df84f9501cf6094": "G = (\\{S, A\\}, \\{a,b\\}, S, \\{f,g,h,k\\})", "8b8ffc06b77fc4ce7ac39bfb09cc89f0": "h_e(X)", "8b901f2bf3658c8de2e935b62a2389a6": " k_{act} = A\\cdot e^{-\\frac{\\Delta G^{\\ddagger}}{RT}} ", "8b90996c2ab0210c7737945db97da036": "a_{p,t}=\\frac{(p)_t}{(t+1)_t}a_{p-t,0}=\\binom{p}{t}\\frac{a_{p-t,0}}{t+1}.", "8b90c81c379edcad07751b8ce2e16ca1": "\\oint_C f(z){g'(z) \\over g(z)}\\, dz", "8b91513e5d3902785f67a652ed6ad245": " \n\\int_o^{\\infty} {k\\; dk \\over k^2 +m^2} J_1^2 \\left( kr \\right)\n\\rightarrow\n{1\\over 2}\\;\\left( {1\\over mr}\\right)\n . ", "8b916dd07e1a8b01dc6ffc39720c7ad7": "\\sqrt{\\frac{g}{k}\\, \\tanh\\, (k\\, h)\\,}", "8b91fa8a186973cd55df66eaa33bb36e": "\\textstyle \\zeta_{G}(\\alpha)=2^{d}\\zeta(\\alpha-d+1)/\\Gamma(d)", "8b9259bf1044ff51b16c2ce37b736bb9": "\\beta_{n} = i k_{n}d_{n}", "8b9268bc7fac167a0ea55dae0e26a8e7": " x^2 = y^2 \\pmod{b} ", "8b92756d7bb38e2ca80476299172679e": "\n\\sum_{n\\leq x} s_n(G) \\sim x^\\alpha\\log^k x\n", "8b92913c57cb9d8bc4ca6d670527d0e9": " P_x = \\varepsilon_0 \\chi_{xx} E_x", "8b930003da6fb282dcfe8854976472e9": "class_i", "8b93086f6d43233efe11a4453aed7cf4": "\\hat{S}(z) = \\exp \\left ( {1 \\over 2} (z^* \\hat{a}^2 - z \\hat{a}^{\\dagger 2}) \\right ) , \\qquad z = r e^{i\\theta}", "8b936dd63f7d07014f155691bc0c5220": "= -i \\lim_{\\varepsilon\\rightarrow 0^+} \\int_{-\\infty}^\\infty \\frac{f(E)}{E-i\\varepsilon}\\,dE = \\pi f(0)-i \\mathcal{P}\\int_{-\\infty}^{\\infty}\\frac{f(E)}{E}\\,dE,", "8b93795b2aa680d6285bf52fee066aba": "c+L=W", "8b93ba0ded62139b1b9bbc60ddf8317c": "F^mx_1...x_mG^nx_1...x_n \\leftrightarrow (F^m \\times G^n)x_1...x_mx_1...x_n.", "8b93e9bacfa19f574f9441ae6833cd29": "\\sigma_H(\\omega)\\ \\stackrel{\\mathrm{def}}{=}\\ \n\\begin{cases}\n e^{+i\\pi/2}, & -\\pi < \\omega < 0 \\\\\n e^{-i\\pi/2}, & 0 < \\omega < \\pi\\\\\n 0, & \\omega = -\\pi, 0, \\pi\n\\end{cases}", "8b9416f15934b1624488f5f2e92178ca": "\\mu_{\\operatorname{\\inf}}", "8b9469d5d27e9d7c8706841dae942432": "a \\neq 0", "8b947dbbfb65f1f7fab9c264f01f975d": " \\Gamma^r_{tt}=\\frac{BB^{\\prime}}{2}, \\; \\Gamma^r_{rr}, \\; \\Gamma^r_{\\theta\\theta}, \\; \\Gamma^r_{\\phi\\phi}=-Br\\sin^2\\theta ", "8b94c05254db39f6a04cc37292a051e9": " \\nu d = 2 - \\alpha = 2\\beta + \\gamma = \\beta(\\delta + 1) = \\gamma \\frac{\\delta + 1}{\\delta - 1}\\,", "8b955d37f640caa89a39879a4b36125d": "\\textstyle E = (0\\textbf{1000011}0)", "8b958c492e4176ee946c8ffaeeabdc93": " 6 ", "8b95e591c8720e79247b2347e25ccde9": "D_{ii} = \\sum_{j=1}^{l+u} W_{ij}", "8b95f77257288bb4b64c65398359a661": "(G,n):(\\mathcal D,\\bullet,J)\\to(\\mathcal C,\\otimes,I)", "8b95f8beb78a88d38c235091aefb408d": "\\tilde{y} = \\mathbb{H} \\, \\tilde{x}", "8b96e93669c7b555e25fe101cb7ec725": "\\operatorname{Tr}(\\rho^2)=(\\operatorname{Tr} \\rho)^2", "8b97a133557539e2cdd91c3291a17a95": "\\mathbf{F}_4", "8b97c2b02f18f2ed6eddc9dcf07e5e85": " \\chi^2_{hs} = \\frac{(2i+2j-h)^2}{h}. ", "8b9824664578e1597eb2dd7b67b7301c": " Y(s) = { I(s) \\over V(s) } = \\frac{s}{ L \\left ( s^2 + 2 \\alpha s + {\\omega_0}^2 \\right ) } ", "8b982ab626c2f402e425a97ccb66abc0": "\\tfrac{1}{r_1}<|z|<\\tfrac{1}{r_2}", "8b982cf22d8533964ec3d7cd1664fbf9": "Z_j=X_{1j}-X_{2j} \\quad\\mbox{for } j=1,2,\\ldots, n,", "8b987fe8da1e7c176d00f2b6771739f3": " x,y,z ", "8b98e296e8f1d9b81d56663150afd0dc": " \\mathbf{L} = (-\\sum_{i=1}^n m_i [\\Delta r_i]^2)\\boldsymbol\\omega = [I_C]\\boldsymbol\\omega,", "8b98f0070ee15cdd5388348ec50c224f": "\\partial^{2} f(x)/\\partial x_{i}^{2}<0", "8b98f170f39f25b9577d2cf9d4c9113a": "|T(f+g)(x)| \\le C(|Tf(x)|+|Tg(x)|)", "8b995f6d8ef2c95650f52b4eebee00c0": "\\xi_{[t]}=(\\xi_{1},\\dots,\\xi_{t})", "8b9995e8e22768f85b282bb12724f1de": "\\ U_a (z) = AU_0(z/b)^2 + \\Delta U_0(z/b)+(1-A) U_0 + U_p \\qquad(11)", "8b99dcb08e5fb6b9ffeee6f8e238d607": "\n\\begin{align}\n\\cot x & {} = \\sum_{n=0}^\\infty \\frac{(-1)^n 2^{2n} B_{2n} x^{2n-1}}{(2n)!} \\\\\n& {} = x^{-1} - \\frac{1}{3}x - \\frac{1}{45}x^3 - \\frac{2}{945}x^5 - \\cdots, \\qquad \\text{for } 0 < |x| < \\pi.\n\\end{align}\n", "8b99e2132498cf9449349e53fa1bef1a": "l_{1}=\\sin \\Theta \\cos \\varphi\\, ", "8b9a125adde871eae8fd51da442da96a": "A \\vec e_j = a_{1,j} \\vec e_1 + a_{2,j} \\vec e_2 + \\ldots + a_{n,j} \\vec e_n = \\sum a_{i,j} \\vec e_i", "8b9a5ce24b15b5a794b99853305007cd": "G|_{S}", "8b9a7115b949a2a73e2534c452167779": "f: \\mathbb N\\to [0,\\infty) ", "8b9a8f292e1e8fc0cda93584edccab47": "\\chi=V-E+F=2\\ ", "8b9ae3c1c18e24a68195bd4c94ced7ec": " \\lim_{(a,b) \\to (0,0)} {a \\over b} ", "8b9ae5d72c992086da6b4ef8d6583cb0": "g_0 \\approx 1", "8b9aee345fab5cab6798e02cfddccd5b": "x \\to +\\infty", "8b9b5a240f565284f7523151cd81e026": "R^i f_* F", "8b9b6580a387d8f0067b389a2ed22603": "\\frac{\\mu\\left( T^n A \\cap B\\right)}{\\mu\\left( B\\right)}.", "8b9bb2d826a02c8b19189693abd14c54": "\\|f\\|_\\infty = \\operatorname{ess\\ sup}_{x\\in S}|f(x)|.", "8b9bc555641ae700e2847bd0c1c57b81": "B = K(\\theta,x^*,x^*) - K(\\theta,x^*,x) K(\\theta,x,x')^{-1} K(\\theta,x^*,x)^T ", "8b9bd25b3ee7d2ce83aaa6c8143517b7": "M^{4m+2}", "8b9c6caddbf4b3d224c9a63630952b2b": "\\textstyle n(t)", "8b9c96e37e3d6baa45beeb03a278cb81": "W_{out} = l F_{out} \\,", "8b9cc9f082682eb4b840e7dcbd94248b": "S_{i,j}", "8b9cd6f0801bdd96074a621389d1e507": "C = \\mathcal{F}\\cdot\\Sigma\\cdot\\mathcal{F}^*\\,\\!", "8b9cec63e1c8f14263d6fd073f7ba7c6": " \\int_0^\\infty { e^{-x^2} \\ln x }\\,dx = -\\tfrac14(\\gamma+2 \\ln 2) \\sqrt{\\pi} ", "8b9ced9375a3e84e947100f0aaa20467": "\\left \\vert \\frac{e}{N}- \\frac{k}{d} \\right \\vert \\le \\frac{1}{dN^{ \\frac{1}{4}}}", "8b9d73b5ea3587b5c890171dc59366c2": "q = 0", "8b9dba8ed315b954cff5db33774a39f2": "y= 0", "8b9e39a7cdabf4c8ee887a384c9c63b3": "2e^2/h", "8b9e8a031bac0f763f78e5bb5e941d45": " 6 \\div 2 = 3.", "8b9eb28cece82d3fa017de85e38149e9": " F(x_0,y_0,u_0,p,q) =0.\\,", "8b9ef83f1fcc507df1fe0c7e0086c77a": "(P^{(\\pm)} [F , G])^{IJ} = [P^{(\\pm)} F, G]^{IJ} = [F , P^{(\\pm)} G]^{IJ} = [P^{(\\pm)} F , P^{(\\pm)} G]^{IJ}", "8b9f05d6a91e30fc6134d384fcf198dc": " \\mathrm{Re} = { \\rho {\\ V} D \\over {\\mu}} ", "8b9f3a1796a8e229ba0cbef54e8e4e5d": "(\\{a_{ij}\\},k,D,\\overrightarrow{T})", "8b9f5f0c9a051596d253abd12cb55573": "\\mathrm{CAL} : E(r_{C}) = r_F + \\sigma_C \\frac{E(r_P) - r_F}{\\sigma_P}", "8b9f881047640b76932c34a73764aacb": " U_e(\\mathbf{r}) := -\\int_{\\gamma[\\mathbf{a},\\mathbf{r}]} \\mathbf{F}_e(\\mathbf{u}) \\cdot d\\mathbf{u} ", "8b9fa2e47e64829b064ef770cae4e9f1": " = \\frac{1}{n} \\left[{(n-1)x_k +\\frac{A}{x_k^{n-1}}}\\right]", "8b9fd6aacd85f0b12e32e4bddd6c386c": " \\Delta \\colon X \\to X \\times X", "8b9fde3d09d877114dc2e76942d93643": " T(V) \\subseteq C \\subseteq \\hat T(V) \\text{ and } \\hat\\Delta(C) \\subseteq C\\otimes C \\subseteq \\hat T(V) \\hat\\otimes \\hat T(V),", "8b9ff94a4c494d2721134fa3c063d9d3": "2\\pi/a", "8ba035424cf524c97d8df817e0f72574": "M_{xy}(t) = M_{xy} (0) e^{-i \\gamma B_{z0} t} = M_{xy} (0) \\left [ \\cos (\\omega _0 t) - i \\sin (\\omega_0 t) \\right ]", "8ba0366adb02aebfeeb198ea5990c3a3": "\\text{dim}(\\text{im} (T)) + \\text{dim} (\\text{ker} (T)) = \\text{dim} (V).", "8ba156da9e01c305a2f7a4e2a2560b87": "BO \\to BG \\to B(G/O).", "8ba159f05283b2ef6cef74f37d10ad30": " b_0 \\Psi = 0 \\left.\\right. \\ .", "8ba17b9129b7066f338e09acf2cc18c5": "\\tau_{max},\\tau_{min}= \\pm \\sqrt{\\left (\\frac{\\sigma_{x} - \\sigma_{y}}{2}\\right)^2 + \\tau_{xy}^2}\\,\\!", "8ba1ef17789cfbe9bed7944b4b12354c": "F_{n+1}^{(r+1)}=F_n^{(r+1)}+F_{n-1}^{(r+1)}+F_n^{(r)}", "8ba225b17d72bd6b246122aab3a29621": " \\left( \\begin{smallmatrix}\nB_1 & * & * & \\cdots & * \\\\\n0 & B_2 & * & \\cdots & * \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n0 & 0 & 0 & \\cdots & * \\\\\n0 & 0 & 0 & \\cdots & B_h\n\\end{smallmatrix}\n\\right)", "8ba2ab1f012eb9088bcbec207148f9c2": "\\Delta (x)=W^{A}(x,x)-W^{B}(x,x),", "8ba2db1e86156516ec85bb68eb7b64d3": "a.0", "8ba2ea095132777568cefdf046d5a020": "\\lim_{t\\to 0} \\sum_n \\sgn(\\omega_n) e^{-t|\\omega_n|}", "8ba2fb24343233ce7abe6afcca4cd076": "V = 2 \\arctan\\left(\\frac{S}{2D}\\right)", "8ba3a09dd3507b8c14486efe48d44dc6": "M=E-e\\sin E", "8ba3a1cf8b377f607f8208c6b38027f7": "L^p,\\ 1\\leq p<\\infty", "8ba3af81a467345813481c8314d98ece": "\\frac{5}{2n-1}", "8ba43b6a12f00a0896a67a7774814068": "e^\\mu_a , \\mu = -1, \\dots, -n, a = 1, \\dots, n", "8ba43bef0f55f360e64ed701a5981ce5": "\nf^\\star\\left(x^{*} \\right)\n= \\begin{cases} 0, & \\left|x^{*} \\right| \\le 1\n \\\\ \\infty, & \\left|x^{*} \\right| > 1.\n \\end{cases}\n", "8ba46d20c74df2b2999c300f0290244e": "\nJ = \\Psi {\\Psi}^\\dagger = -\\frac{p}{m},\n", "8ba536bdac30f5c811f680842d90a8c4": " \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 = \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i^2 b_j^2 + \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_j^2 b_i^2 - 2 \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i b_j a_j b_i .", "8ba5773754d721084042b8ce7f3b2073": "dA = A_f . A_0^{-1}", "8ba5a34dc6e8b0ab968686cf8680b162": "\\langle \\cdots \\rangle", "8ba5a43aa60431272c104c5651d2ac91": "-(I_2-I_3)/3", "8ba5b5846513b3b5440070ddbde26d8b": "\\delta Q\\ ", "8ba5bb7b3a77905c8b68ec5df9a0454c": "X(s) = L\\{x(t)\\}\\ \\stackrel{\\mathrm{def}}{=}\\ \\int_0^{\\infty}{x(t)e^{-st}dt},", "8ba5e05fd3a470ea27bd38c5bd6e4350": "\\gamma_2 = 12\\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}", "8ba60f0734540724ecf097946c4470d3": "\\upsilon_D \\,", "8ba611f108f1143ed8ebe7571ef147d3": " 0; \\gamma > 0\n", "8badec3309a15d24c646801838770482": "R_1 = \\frac{R_bR_c}{R_T}.", "8bae03cb614eb53e00df0918fc85f80b": "r_i \\neq 0", "8bae093df568a9e942d9c9e4c4d8aa42": "\\mathbf{R}=\\left(\\cos{2\\omega t},-\\sin{2\\omega t},0\\right)", "8bae1b353970853b05703aa1b0560aee": "A=S,...,J", "8bae7450644ab3a522c009b5d5fa538b": "\\left\\| \\mathbf{d_2} \\right\\|", "8bae7486d77007d031bc6da25e72434c": "\\frac {m_2} {m_1} = \\frac {c_2} {c_1} = \\frac {l_2} {l_1}.", "8bae78828a115c9fdccc6ba86abccd81": "\\mathrm{add}_c(x)=c +x, ~~~ ~ \\forall x \\in \\mathbb{C}", "8baeb8833e1cf1eb73f5be90e9e347e1": "{\\mathfrak A}_P:= ({\\mathcal P}\\setminus\\{P\\},\\{z\\setminus\\{P\\}|P\\in z\\in{\\mathcal Z}\\}, \\in)", "8baedab173b6351f04a8937c3d8bb268": " t_n\\in [0,\\infty]", "8baedfd33248e04ea39f8871c82e382f": "\\Delta G=R*T*\\Phi_{v,fresh}*(C_{feed}-C_{fresh})\\left[\\frac{ln \\alpha}{1-\\alpha}-\\frac{ln \\beta}{1-\\alpha}\\right]", "8baf17612eb56c66a73750b53f26709b": "\\Phi(x_s^{i_p},x_b^{j_q}) = \\exp ( -E(x_s^{i_p},x_b^{j_q})/K_BT)", "8baf1d17f409e9fcafa1bc8cd3b5174b": "c^d = (1+n)^{m} \\;mod\\; n^{s+1}", "8baf31773effc3771f2af38a4560da91": "\n\\begin{array}{lcl}\n\\boldsymbol\\phi &\\sim& \\operatorname{Symmetric-Dirichlet}_K(\\beta) \\\\\n\\boldsymbol\\theta_{i=1 \\dots K} &\\sim& \\text{Symmetric-Dirichlet}_V(\\alpha) \\\\\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& \\text{Categorical}(\\boldsymbol\\theta_{z_i})\n\\end{array}\n", "8baf3a30a96aa0df935ff603124b723f": "\n\\operatorname{Li}_2(z) = -\\int_0^z{\\ln(1-u) \\over u}\\, \\mathrm{d}u \\text{, }z \\in\\mathbb{C} \\setminus [1,\\infty)\n", "8bafd9f942f5feb526b5ce03a2f634a9": "\\sum_{i=1}^d (d+1-i)b^i", "8bafe70b88cc1838ccba4ee3e82da436": " S = \\int dx^4 \\frac{1}{4}F^2 = \\int dx^4 \\frac{1}{8}(F\\pm\\tilde{F})^2 \\mp \\int dx^4 \\frac{1}{4}F\\tilde{F}", "8bb0051b78934fbde4613fb76b83c74c": "p(x) = x^3-x^2-2x+1", "8bb04fc14d3394397c30c9ea86343fc1": "\\frac{\\omega - \\omega_0}{\\omega_0}\\thickapprox -\\frac{1}{W}\\iiint_{V}(\\frac{\\Delta\\epsilon}{\\epsilon}\\cdot\\bar{w_e}+\\frac{\\Delta\\mu}{\\mu}\\cdot\\bar{w_m})dv\\,", "8bb0dbe8eef27dea312b8bc39c9265d6": "\\log \\mathcal{L}(\\alpha,\\beta \\,|\\, x) = \\alpha \\log \\beta - \\log \\Gamma(\\alpha) + (\\alpha-1) \\log x - \\beta x. \\, ", "8bb1231c5a318540cbb355a7f3d2387b": "-\\frac{d[A]}{dt} = k[A]^2", "8bb13081b2a88eeffd89f4ce9eccda3c": "\\zeta^{\\prime}(-6) = -\\frac{45}{8\\pi^6} \\zeta(7)", "8bb1646992bda47f11afd6b07dd4a407": " L^1(\\mathbb{R}^d) \\cap C_0(\\mathbb{R}^d) ", "8bb1a370104515e14872843159b49519": "\\scriptstyle f_s = 1/T", "8bb1ba865ea6b8e2a1ceb6031bd40cf9": "g^\\prime \\equiv g \\frac{\\rho-\\rho_0}{\\rho_{00}}", "8bb29b5360ff88fc2270dd85fffbe1ed": " \\delta_\\bullet^i= f^i, \\,\\,\\, \\delta^i_{[t_1,\\dots,t_n]} = \\sum_{j_1,\\dots,j_n=1}^N (\\delta^{j_1}_{t_1} \\cdots \\delta^{j_n}_{t_n})\\partial_{j_1} \\cdots \\partial_{j_n} f^i.", "8bb2adbd60e348da502afbf9a8f93c86": " (\\partial_{c v} f)(p) = c (\\partial_v f)(p)", "8bb2fa4d51b576e0d2bc17fd647ad8ae": "H_{11} = H_{22} = \\alpha \\,", "8bb3297ef982c87df3a3240ad90b0d8a": "L(x_1, \\cdots, x_N, t) = \\int_{u_1=-\\infty}^{\\infty} \\cdots \\int_{u_N=-\\infty}^{\\infty} f_C(x_1-u_1, \\cdots, x_N-u_N, t) G(u_1, t) \\, du_1 \\cdots G(u_N, t) \\, du_N,", "8bb3a21702d34e04ea0b6f7467911a8a": "\nc \\ \\stackrel{\\mathrm{def}}{=}\\ \\lambda_{x}^{2} - \\lambda_{y}^{2} \n", "8bb3cea26e55e007fa3fbe513ea6cf1c": " <\\Omega ", "8bb4370405a4347fdf7a5ef1fd7d59a5": "f_{g^{-1}}", "8bb4afe4f1f6d06d50dfdf9530c243a1": "\\max(|t|,|\\mu|) \\ll N \\ll \\frac{2}{\\sigma^2}\\ln\\frac{2}{\\sigma^2} ", "8bb4d918f9d2e53fe99e88bb6f935a47": "{\\rm d}N {\\rm d}x \\simeq \\sum {\\rm d \\ell}", "8bb4e356faf61d0bd07991fa4922266d": "(u_g,v_g)", "8bb50182544fd3fad1411f6e021bddf9": " n\\log k", "8bb50605ff63759107f02187b2ee1a8d": "\\vec x", "8bb5bd97e8bfa244f150fbd9d588eef9": "Q = 1 - \\sum_{i=1}^{N} \\left(\\frac{\\tfrac{1}{2} - s_i}{\\tfrac{1}{2}}\\right)^2 \\times s_i,", "8bb6119fe935caec7f59a9cbbfc4e792": " x(t) = A\\cos\\left( \\omega t+\\phi\\right), ", "8bb635441563240c623c3b2cd62fdd4f": "e = 1", "8bb64aa6c58a24251ccb8c9d3f3e8b2a": "f(x) \\in A[x]", "8bb73bd1477fbff7141400deac9e0214": "x \\cdot y = (RxR^{\\dagger}) \\cdot (RyR^{\\dagger})", "8bb7e95a6b9f56f6b3ba02fc0653df68": "D = \\sum_i n_i p_i\\ ", "8bb812b0dcbb64a760a07746fa026a0d": "\nf(\\gamma z) = \\prod_{i=1}^m j(\\sigma_i(\\gamma), z_i)^{k_i} f(z).\n", "8bb8493318ce7a043639a662c0240380": "\\tfrac{\\log n}{\\log \\gamma (n)}", "8bb87c6790872889703b6282e1e40f52": "d(x,(p,n))=\\|(x-p)\\cdot \\hat n\\| = \\|(x-p)^\\top \\hat n\\| = \\sqrt{(x-p)^\\top \\hat n \\hat n^\\top (x-p)}.", "8bb89f6e73094bb224ddd7de8944609f": "g^{\\beta\\alpha} \\left[00,\\alpha\\right] = \\frac{\\varepsilon}{2} \\gamma_{00|\\beta}", "8bb8d1cd72b5d410b3aaf68e75aa207d": "\\bar a = \\exp\\left(\\frac{4\\sqrt{\\log N}}{\\sqrt{2e}\\log\\log N}\\bigl(1+o(1)\\bigr)\\right),", "8bb8e5f002975648112e4b5348dd57c8": "\\prod f_i : X \\to \\prod Y_i.", "8bb9225dac9232ab16258200a154f52d": "R_\\mathrm{E} = R_\\mathrm{L} \\parallel r_\\mathrm{O}", "8bb93e1d750627f73f6a0541453c8c07": "\\pi_1(A)=\\pi_1(D^2)={1}", "8bb94e5c4fa401c27be7182e574f1d02": "B(f)", "8bb9bb8a7e16a5817e21a576c065f318": "d\\mathbf{S} = \\mathbf{\\hat{n}}dS", "8bba1fac5fcc8a9ed04fe802c859a896": "V=(2\\pi)^d \\lim_{R\\to\\infty}\\frac{N(R)}{R^{d/2}}\\,", "8bba38e7ee4174361c1eb34ccaf1fbcd": "\\displaystyle{\\int_{-\\infty}^\\infty {|\\lambda|\\, dt\\over t^2 + \\lambda^2} = \\int_{-\\infty}^\\infty {dt\\over t^2 + 1}=\\pi<\\infty.}", "8bba3f4b79440b82e29685e31b4c9c57": "OPT(\\sigma)", "8bbb042b288398daf9c47ba296c2a497": "-vMv^{-1}", "8bbb1f4c5412f04833b582f07299320d": "G_1\\setminus G_0", "8bbb1ff26acf08ef4051ba7048025aa1": "G_{i_0 + 1} = \\cdots = G_{i_0 + p i_1} = G(1) = G^{i_0 + 1} = \\cdots = G^{i_0 + i_1}", "8bbb5a8d6cb2a8a68e859fd3f5ec1546": "\\int\\frac{dx}{xr} = -\\frac{1}{a}\\,\\operatorname{arsinh}\\frac{a}{x} = -\\frac{1}{a}\\ln\\left|\\frac{a+r}{x}\\right|", "8bbb9939c5f7449df1e1ae45c6f89b71": "x_{ni}", "8bbb9b80f013c1cfe89dc53bb9c9afcd": "\\neg\\exists", "8bbbb4a5954cc4cf2ece4b22ddc58134": " -\\frac{d}{dx} \\left[\\frac{ n(x,f_0) f_0'}{\\sqrt{1 + f_0'^2}} \\right] + n_y (x,f_0) \\sqrt{1 + f_0'(x)^2} =0. \\,", "8bbbd844def35bb267706a966175b5ce": "a=\\bar a", "8bbc189a1f810308a8d997447e3dcfa6": "\\chi = \\frac{F_D}{3 \\pi \\eta V d_e}", "8bbc431c416c822d47a714a5cb2205be": "\\rho>2\\,", "8bbc89f304c89ec23c57bed95720ada4": "\\scriptstyle a^{p-1} \\equiv 1 \\pmod p.", "8bbcc1c8a07eb746ebade74776e80344": " (\\neg A)^*", "8bbce29a474190af6e48843e4a5ad0bd": "r = 2b\\cos{\\theta \\over 3}", "8bbcfaa5666cd2e677bf96b4b14fd07f": "\\ C_1^3 (3)=\\frac{17}{49}", "8bbcfcc6b7f97b62f26031e130870154": "\\hat f(k) = \\int_{\\Omega} f(x) u_k(x) \\, dx.", "8bbd5901e2ea71eb396b7d4eb957db0e": "x= \\frac{-b \\pm \\sqrt{b^2-4ac} }{2a}. ", "8bbdbcbf3c9a751047b58f3bc538c75c": "\\{ C \\}", "8bbdcee50405253995800b3bc9f66095": "i\\mathbf{B} = (\\mathcal{F}\\wedge\\gamma_0)\\gamma_0", "8bbdd12fbd9c6b4de39db0aa5a613ae2": "x = 28.\\,", "8bbf27840e4247fc0f60a34f43a0ba54": "[0,X]", "8bbf43ab867b0e8f602bf63cd47a0e4d": "\n\\sum_{\\stackrel{1\\le k\\le m}{ \\gcd(k,m)=1}} \\gcd(k^2-1,m_1)\\gcd(k^2-1,m_2)\n=\\varphi(n)\\sum_{\\stackrel{d_1\\mid m_1} {d_2\\mid m_2}} \\varphi(\\gcd(d_1, d_2))2^{\\omega(\\operatorname{lcm}(d_1, d_2))},\n", "8bbf44a9657233167b188a36b7313bab": "a\\times10^{2n}", "8bbf8c729aa92d223c5d2f351066ed1a": "\\lambda_m", "8bbfa248148535243c5fed4de471d825": "\\mathbf{D_{xx}},\\,\\mathbf{D_{yy}}", "8bbfe07a07beea91ac8b0d5c7a60a813": "A=\\sum_k \\alpha_k\\text{ and }N=\\sum_k n_k\\text{, and where }n_k=\\text{number of }z_n\\text{'s with the value }k\\text{.}", "8bc0283fa273cd5d5e45aff01cba878b": "180-\\tfrac{360}{n}", "8bc0481ba60e3233b5b8d21ebddd27d5": "\\left|(1-z)\\sum_{k=n}^\\infty s_kz^k \\right| \\le \\epsilon |1-z|\\sum_{k=n}^\\infty |z|^k = \\epsilon|1-z|\\frac{|z|^n}{1-|z|} < \\epsilon M \\!", "8bc15390c98f88cd3f0c402497e709cb": "\\Delta_{SO}", "8bc1c182589eafc9e1363a2865adedc6": "\\frac b d-\\frac{a+b}{c+d}={{bc-ad}\\over{d(c+d)}} ={c\\over{c+d}}\\left( \\frac{b}{d}-\\frac a c \\right). ", "8bc1c1cdd12a79c6ccb33db28ed8af00": "\\mathbf{S}_k", "8bc1ff4a2bc80e4bdcf2e8e708783477": "\\exists j", "8bc23bf1213ed5e943a8132c90ad4461": "G(y)=z", "8bc26d93a20e340f5ed2397372b1fb9e": "W_{-1}", "8bc2c91dcb709b3f6740405f091a4497": "\n W = \\sum_{i,j=0}^n C_{ij} (\\bar{I}_1 - 3)^i (\\bar{I}_2 - 3)^j + \\sum_{k=1}^m D_{k}(J-1)^{2k}\n", "8bc2c988ea632b93a71a1301055df742": "N/S", "8bc30b54e294d8da321dd0d16aa7ed5e": " \\begin{bmatrix} x_0 & y_0 & z_0 \\\\ x_1 & y_1 & z_1 \\\\ x_2 & y_2 & z_2 \\\\ x_3 & y_3 & z_3 \\end{bmatrix} ", "8bc32f5a84fe6eb21e14b143b28b7d64": " A^T A ", "8bc331b256d2170ef22da1cbf7e00c8a": "z \\mapsto z^{k}.", "8bc38c803fca17df5b84a851122adcf0": "(f \\circ g \\circ h \\circ j)(x)", "8bc3967cf96c0baf05bf902fdce5f68f": "k[X]\\to k[X,Y]/ \\langle Y^2-X^3-X \\rangle.", "8bc3a738ff9fe5681adb5900ea8a7235": "x(i,j)", "8bc3a7e80988236e8f017205f413461c": "f(g(x))", "8bc3dc11d416c68b70df1fadcce3bbe6": "(X,Y)\\ ", "8bc3e5f30efb8870ce1b9a0dbb45b43a": "\\mbox{Diameter Ratio} = 1-(1-0.2)^{\\frac{1}{2}} \\approx 10.6%", "8bc44b2163d62b7ba1e2ce50662d8559": "1_{n = 0}", "8bc44eaec5cd0254de244fbced8accc8": "G = H - TS = U + PV - TS", "8bc45c8bb4c49e08bac40a6eb240deaa": "(10^{-20} \\text{ erg/s/cm}^2\\text{/Hz})", "8bc4c44b0e0b7ec5455595a57049320f": "Y_{r,b}=X_{r,b}.a_{r,b}+\\varepsilon_{r,b}", "8bc4c8d98d85d472a3988a3f3d892509": "\\displaystyle{T_{Z_1}T_{Z_2}=(D_1+A_2)^{-1/2}T_{Z_3}}", "8bc5009d2905e7154a1d8d2e3122278c": "\nK(x-y,\\Tau) = \\int_{x(0)=x}^{x(\\Tau)=y} e^{i \\int_0^\\Tau \\sqrt{{\\dot x}^2} - \\alpha d\\tau}\n\\,", "8bc52e5c9a9fc458205fea482243c75c": "S=\\sum_{i=1}^n \\left[ y_i - \\sum_{j=1}^r f_j (\\beta_j 'x_i) \\right]^2 ,", "8bc54edd3e10926c3d985f17883a79d5": "\\mathbf{j}_f=\\left \\{ \n\\begin{matrix}\n-(m \\mathbf{v}_r \\cdot \\mathbf{\\hat{t}})\\mathbf{\\hat{t}}\n& \n\\mathbf{v}_r \\cdot \\mathbf{\\hat{t}} = 0 & m \\mathbf{v}_r \\cdot \\mathbf{\\hat{t}} \\le j_s\\\\ \n-j_d \\mathbf{\\hat{t}}\n& \n\\text{(otherwise)} \\\\\n \n\\end{matrix}\\right.", "8bc550e1413f46c6332dad62ebe71bd3": "\\mathbf A=\\langle A,\\wedge,\\vee,-,\\Box\\rangle", "8bc555b48f219bc6878bb76562c7b336": "\\rho_{AB},\\rho_{BC},\\rho_{B}", "8bc56d09fc9349df6b89d0ab26e96589": "N = q^k - 1", "8bc5b3da09f5bf37a7cb35265c1b7229": "h(v) = h(v_0) + \\Omega (|v_0|) \\ ", "8bc5e92acf434210d18231d6fb27aa36": "\nDG(f;s) DG(g;s) = DG(f*g;s)\\,\n", "8bc5f7af56bc80a152f7a9caba1e9d6e": "Z_\\Gamma(s)=\\prod_p\\prod^\\infty_{n=0}(1-N(p)^{-s-n}),", "8bc6ac748fade2b79ffcd8cf6e8e21ea": "\\ \\mathbb{R} ", "8bc6d88aa4aefb54c24b89e121fa52ff": "\\cot\\frac{11\\pi}{60}=\\cot 33^\\circ=\\tfrac{1}{4}\\left[2-(2+\\sqrt3)(3+\\sqrt5)\\right]\\left[2-\\sqrt{2(5-\\sqrt5)}\\right]\\,", "8bc6de01b82c4f6e701e62b4b661a110": "\n\\frac{d V}{V} = -\\frac{d P}{K_0 + K'_0 P}. \\qquad (5)\n", "8bc6e089d20a994c35048243f8154cdc": "\n-\\Bigl\\langle \\frac{dL_{z}}{dt} \\Bigr\\rangle = \n\\frac{32G^{7/2}m_{1}^{2}m_{2}^{2}\\sqrt{m_{1} + m_{2}}}{5c^{5} a^{7/2} \\left( 1 - e^{2} \\right)^{2}} \n\\left( 1 + \\frac{7}{8} e^{2} \\right)\n", "8bc7259e33f5c4fbc5fdc634b6e93ec4": "\\varrho_{A_1\\ldots A_m}", "8bc75333dac14e2e11e1364291cb8e8e": "Re_p", "8bc75eef33c45cc777f50e0b1befaa4b": "\\sigma _{ph}=e\\left ( N_{d}-N_{d}^{+} \\right )\\frac{sI\\mu \\tau }{h\\nu } ", "8bc7627774da35811c5ff4c9f7b64e5c": "\\varphi(r)=P_\\rho(\\cosh r) = {1\\over 2\\pi} \\int_0^{2\\pi} (\\cosh r + \\sinh r \\, \\cos \\theta)^\\rho \\, d\\theta,", "8bc78467e594e0fdc6fb3d9303044c99": "C=c_p\\frac{dm}{dt}", "8bc7be7beeea6a42ccdf059f3f4e3f93": "4\\log_2 n - 4", "8bc7dc8db05321da7b37e9c40a74d164": " \\left( {P \\over P_0} \\right) = \\left( {V_0 \\over V} \\right)^{\\gamma}. ", "8bc7e825440a7b52da529253d01ae615": "{d \\over dt}\\left\\{ Y_2 \\right\\} = \\left\\{B \\right\\} \\left\\{X_2 \\right\\} - \\left\\{ X_2 \\right\\}^2 \\left\\{Y_2 \\right\\} + D_y\\left( Y_1 - Y_2\\right) \\,", "8bc818b0d950a035417e4cba75081f3c": "f_Y(y) = f_X(x)/|g'(x)| \\, ", "8bc82a0025250ec72970578771adbdc4": "A_{21}=\\frac{8\\nu^2 \\pi^2 e^2}{m_e c^3}~\\frac{g_1}{g_2}~f_{12}", "8bc85ee2a87b70702476e5c28e8086d4": " \\sigma(\\pi) = \\prod_{c\\in\\pi} (-1)^{|c|-1},", "8bc877c7162696628e3b3548baee4b8f": "X[\\sigma] \\to \\alpha Y[\\sigma] \\beta", "8bc88cf71b9329c24e2394c77e90f2de": "d(x,y) = \\# \\{i : x_i \\not = y_i \\}", "8bc88d93901a31605c6b36134c963ee5": "\n -1 < \\Re(t) < 1\n", "8bc8e918bb711f7938a031ce579cef6c": "|\\text{RVB}\\rangle=\\sum_C|C\\rangle", "8bc8f94d777ffaa530c0f2403c9eef5b": "\\, \\Delta h", "8bc917697850c3766cb0ee001d88d6c9": "m \\geq 2", "8bc925ea8c2a857e87b382d003acd2cf": "O(M) \\le \\ln S \\le O(M)+ \\ln M. \\ ", "8bc93520eab9bc0d70ef1b9d9199716b": "\\frac{e^{-c}c^k}{k!}", "8bc954e206148f5f0525ac05874176f6": "x<^*y\\iff x\\in P\\land[y\\notin P\\lor\\{x\\leq y\\land y\\not\\leq x\\}]", "8bc959c4fbd19514edb216d677cf72f8": "III_0", "8bc98749209053c031f213a21e173520": "I_i^k", "8bc9997fb6e4ee86fbca03aab79a5726": "L\\otimes_KL", "8bc9c598e22f84ce9182092e09b86f13": "\\tan (\\arcsec x) = \\sqrt{x^2-1}", "8bca6d8c0ae0997a806a07be80b26a54": "\\begin{align}\nf = \\Phi_{Y,X}^{-1}(g) &= \\varepsilon_X\\circ F(g) & \\in & \\, \\, \\mathrm{hom}_C(F(Y),X)\\\\\ng = \\Phi_{Y,X}(f) &= G(f)\\circ \\eta_Y & \\in & \\, \\, \\mathrm{hom}_D(Y,G(X))\\\\\n\\Phi_{GX,X}^{-1}(1_{GX}) &= \\varepsilon_X & \\in & \\, \\, \\mathrm{hom}_C(FG(X),X)\\\\\n\\Phi_{Y,FY}(1_{FY}) &= \\eta_Y & \\in & \\, \\, \\mathrm{hom}_D(Y,GF(Y))\\\\\n\\end{align}\n", "8bca727b7916ffb40d920cedd3ace541": "\\mathbb{Q}(\\sqrt[3]{2})", "8bca867243548a103bac5641bf1187c0": " S(x+t) = (1-t) S(x) + t S(x+1), \\qquad 0 0, \\forall p \\in \\partial \\Omega, \\mu \\left( \\Omega \\cap \\mathbb{B}_{r} (p) \\right) \\leq C \\sigma \\left( \\partial \\Omega \\cap \\mathbb{B}_{r} (p) \\right)", "8bd1494025f22a27d4985071e11196c6": "f = aA\\,", "8bd14d6e3d0e6994e0d9f59a613caa67": "D^{-1}(b - Rx^{(k)}) = Tx^{(k)} + C", "8bd1cf9050467869cb72fa6e11c1d8a7": "g(Z_{t})", "8bd1ec24d1ca9a58bf0c545d3f858a74": " {q}_{i}^e ", "8bd2863a6ce21f62a836cc7c335f021a": "\\mathrm{SNR} = \\frac{ | {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{-1/2}s) |^2 }\n { {(R_v^{1/2}h)}^\\mathrm{H} (R_v^{1/2}h) }\n = \\frac{ \\alpha^2 | {(R_v^{-1/2}s)}^\\mathrm{H} (R_v^{-1/2}s) |^2 }\n { \\alpha^2 {(R_v^{-1/2}s)}^\\mathrm{H} (R_v^{-1/2}s) }\n = \\frac{ | s^\\mathrm{H} R_v^{-1} s |^2 }\n { s^\\mathrm{H} R_v^{-1} s }\n = s^\\mathrm{H} R_v^{-1} s.\n ", "8bd2974bd88cc21885a1f70f4e313cd2": "\\theta^*\\in \\Theta", "8bd29763101fc39d09fcf9dee9eaa42d": "\nE = \\frac{1}{2m} \\left| \\mathbf{p} \\right|^2 + V(\\mathbf{r})\n", "8bd2a25bc88400e33d22fd6c91001508": "\\mathbf{p}_k=(x_k,y_k,z_k)", "8bd2bfa86521d825feccebb546a9811e": "\\mathsf{Q^D} = a - bP", "8bd2d92d0c18447616117a1ae2c20752": "\\frac{bc}{b+ c - a} : \\frac{ca}{c + a-b} : \\frac{ab}{a+b-c}", "8bd2e5b6344897fae00f5d354971fc5c": "m\\geq N(\\sigma)", "8bd2ebe969e3e920db6910708da3acad": "z=\\frac{\\lambda_o-\\lambda_e}{\\lambda_e}", "8bd34fd3f17bca07aece81a2900da32f": "-k x = \\omega^2 m x", "8bd3738451ad5a031e61fc886c4d77ce": " \\int_{x_1}^{x_5} f(x)\\,dx = \\frac{2 h}{45}\\left( 7f(x_1) + 32 f(x_2) + 12 f(x_3) + 32 f(x_4) + 7f(x_5) \\right) + \\text{error term}, ", "8bd385e5a9eec9c189057f96b3ae0fa0": "f(\\mathbf{x} + \\mathbf{y}) = f(\\mathbf{x}) + f(\\mathbf{y}) ", "8bd3d31034dc2cebca5ab58c326b369e": "X_1, X_2, \\dots", "8bd419655dc95e5da1968f6ddf6ce5a5": "X_{i,\\text{inst}}=\\frac{\\dot{n}_{i,\\text{react}}}{\\dot{n}_{i,\\text{in}}}", "8bd419b38f2171ac6c55d0272ecab029": "\\partial \\mathcal{L}_1/\\partial\\psi=0", "8bd44b999c486c52ec410496e1e9d2e5": "i\\partial_t\\psi=-{1\\over 2}\\partial^2_x\\psi+\\kappa|\\psi|^2 \\psi", "8bd484245a9b0debed1bc10abb5144a6": "-\\frac{p_{n-1}}{n\\cdot p_n}", "8bd4a75a89a8c08667e1d078339e304d": "z=z+ln(\\lambda)", "8bd4f3d82089a3a329d89ebaaf656ab8": "\\Phi: \\mathbb{R} \\times X \\to X", "8bd55327bc1bdf1c09530bb537c3cceb": " x \\le 2 ", "8bd574a60b7c8727a670b023c50e70a5": "\\beta(A\\cdot\\varphi,\\psi) = \\varepsilon\\varepsilon_k \\beta(A\\cdot\\psi,\\varphi)", "8bd5de1b5779a0b6a866424269befe9f": " \\frac{dx}{dt}=rx-x^3. ", "8bd5e6b0a7b0197426633e3176982f91": " a_{ij} > 0 ", "8bd656398703f8ef3449a88897c6ec1c": "A\\Rightarrow_{amb} B", "8bd68d69ebfc6f2ebb668047661c3389": "\\mathbf{W} (C_{0} \\setminus \\mathbf{B}_{c} (0; \\| \\cdot \\|_{\\infty})) \\equiv \\mathbf{P} \\big[ \\| B \\|_{\\infty} > c \\big],", "8bd6cdac6d8f06ed94c201a6b9804b8b": "q:=0; \\quad r:=a; \\quad d:=\\deg(b);", "8bd6e5991c40001f950c0312f3b68f1f": "T_f + T_g = T_{f+g}.\\,", "8bd70657b70e17eb8c0055ac4441c5a9": "S_{2k}", "8bd78d6b271a5a92b3311ebea80eef4b": "R_N", "8bd7aa079d5f4a5f71cf7c1f59df3a46": "\\pi_{\\mathbf{f}}|_{U(N-1)}= \\bigoplus_{f_1\\ge g_1 \\ge f_2\\ge g_2\\ge \\cdots \\ge f_{N-1}\\ge g_{N-1}\\ge f_N} \\pi_{\\mathbf{g}}", "8bd7e321a9d9f3066b762b9526c5e571": "\\nabla_k", "8bd7ed80f38833d7018bf8104cf73ef8": " \\begin{pmatrix}\n 0 & -1 \\\\\n 1 & 0\n\\end{pmatrix}", "8bd7fb88b060e1f5579ea2c197568d28": "Z=\\textstyle{\\frac{3}{2}}U-3V+2W", "8bd8044a61295138e9e0daa644c88f92": "\\mathbf{A} \\rightarrow \\mathbf{A} + \\nabla \\psi\\,.", "8bd804c9e487a4579e6f6350eccf97cc": "-(E - e\\phi) \\psi_{-} + c\\boldsymbol{\\sigma}\\cdot \\left( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right) \\psi_+ = mc^2 \\psi_{-} ", "8bd8798f1005a45fb66c561f9abcc459": "\\mathcal{M} = \\{ \\mathbf{p}\\in\\mathbb{R}^N \\}\\,.", "8bd8a24d0f765e323180903d4238ea22": "\\left( \\begin{smallmatrix} 4 & 0 \\\\ 0 & 4 \\\\ \\end{smallmatrix} \\right)", "8bd8cd2c80b14a2a0c763717e653d764": "\\operatorname{P}(A|\\mathcal{B})=1_A.", "8bd8cfb48bc4b4be1a8ccd41e875d432": "\\Delta\\sigma=\\arccos\\bigl(\\sin\\phi_1\\sin\\phi_2+\\cos\\phi_1\\cos\\phi_2\\cos\\Delta\\lambda\\bigr).", "8bd8faa06fc8fe5f052370778c4c1355": "e=\\lim_{n \\to \\infty} \\left [ \\frac{(n+1)^{n+1}}{n^n}- \\frac{n^n}{(n-1)^{n-1}} \\right ]", "8bd904c1aea03a05dc0ae7196c721bd5": "u_1 = u_2, \\qquad \\partial_nu_1 = \\partial_nu_2", "8bd99eed536abab2403e51e7d04326e8": " \\quad ( ^{(1)}, \\ ^{(2)}, \\ ^{(3)}, \\ ^{(4)}, \\ \\dots )\\,\\!", "8bda46f2a1c71f301c6901eee614e0da": "L>f", "8bda50aa04e8b0e3418807f46d5dfa9e": "\\cup_i \\, U_i ", "8bda78cd6f97e621ece12fd2fa42a12d": "[X\\; Y]", "8bda9f09fc5b8330f5241f20976b248d": " \\frac{1}{T} \\int_{0}^{T}\\frac{\\beta(t)}{\\mu+\\nu}dt < 1 \\Rightarrow \\lim_{t \\rightarrow +\\infty} \\left(S(t),I(t)\\right) = DFE = \\left(N,0\\right), ", "8bdac46f92bdaede7a67f799d58cc7dd": "t_n =\n\\begin{cases} \nf_j & \\text{if } m = 1 \\mod 4 \\\\\n1-f_j & \\text{if } m = 3 \\mod 4\n\\end{cases}", "8bdaeb9cd41a636c9b417edd7eb9cd72": "\\Gamma_1\\subset\\Gamma_2", "8bdb05c5f7e16e2d0de27b22abfbfaad": "\\displaystyle{{1\\over \\pi} \\int_{\\Gamma} {F(z)\\over z-x} \\, dz.}", "8bdb1ea46685195d8f6cb68b7275c786": "z_c", "8bdb65b59417aff5cfc08dc6c0b1ed4f": "\n00 \\to I, \\,\\,\n01 \\to X, \\,\\,\n11 \\to Y, \\,\\,\n10 \\to Z\n", "8bdbcd5bd460dc8ada2d7a2655853a04": "\\sigma_2 = \\sigma_\\min = \\sigma_\\text{avg}-R", "8bdbfb38d4d5d23ea8c5728ee09116b3": "(\\bigcup_{C\\in \\mathcal{C}} C)\\cap \\Delta_{X} = \\bigcup_{C\\in \\mathcal{C}} (C \\cap\\Delta_{X})=\\empty", "8bdc6d11dded72e8b5f336f5c1612c24": "p(b) = 1/2", "8bdc992f15e25a5f42704d0996c8ef82": "\nN_i = \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}} \n", "8bdcdf17819676b0712cad5e7bcdb64e": "p = \\frac{mv}{\\sqrt{1-\\frac{v^2}{c^2}}}", "8bdcefc0ce280ef12a166a1b30a23dd9": "x \\setminus \\N", "8bdd0d9eaa297ac0ff158d7377d53164": "\\arccsc z = \\arcsin \\frac{1}{z} \\quad z \\neq -1, 0, +1 \\,", "8bdd1dd1e7608eb8709b169f0461a669": "S = \\frac{V_w}{V_v} ", "8bdd68a694047f2b91e4777462dbbe4b": "\\mathcal{P}_n", "8bdd9e7b41905788cf1f0a3e6622685a": " n+6 ", "8bddf35da3ead4be87db55e9527c762d": "\\nabla\\cdot\\mathbf{E}(\\mathbf{r}) = \\frac{\\rho(\\mathbf{r})}{\\mathcal{E}_0},", "8bde45b59aab63ce696ccac425309190": "\\Omega \\,", "8bdea5e8b906675766fe6ace3ca72e53": "\nf( \\hat{\\mathbf{k}}) P_{\\mathbf{k}} = f(1) P_{\\mathbf{k}},\n", "8bdef79baabd620fc7195db6a0087e4b": "(m\\times n)", "8bdf2f51a0cd68805210d329bd0c3644": "\\overline{\\mathcal{M}}_{g, n}", "8bdf399a4523479320ccfed982a4ffa8": "\\mathbf{[a]}", "8be04c8f0693fcf4349db7045096fb86": " a^{b^c} = a^{(b^c)} ", "8be0b1d1faed24319e1c568a8c381872": " \\mathbb{A}_F =F\\otimes_\\mathbb Z \\mathbb{A}_\\mathbb{Z}", "8be10e69bac07102f876088cd33fe562": "\\tau\\,\\!", "8be1bf97890dcf08171d9e8cf4d838ba": " [X(m),\\Phi(a,n)]= \\Phi(Xa,m+n).\\,", "8be20c6c4fceb3edbaaeb87abdf42bd9": "(-1,1)", "8be23ca5aeab1aa645ede1fdba3d0e13": "\\begin{bmatrix}M\\end{bmatrix}\\begin{bmatrix} \\Psi \\end{bmatrix}\\begin{Bmatrix}\\ddot{q}\\end{Bmatrix}+\\begin{bmatrix}K\\end{bmatrix} \\begin{bmatrix} \\Psi \\end{bmatrix} \\begin{Bmatrix} q\\end{Bmatrix}=0.", "8be257ca3407260d128637b6bfaa7fec": "\\mbox{i.e. }\\ddot{u} = - \\nabla_{u} V(t, u).", "8be2d004b61d707ff83e111305665e7e": "\\mathsf{S} = [\\hat{A}]\\mathsf{s}, \\quad (\\mathbf{S}, \\mathbf{V}) = ([A], [DA])(\\mathbf{s}, \\mathbf{v}) = ([A]\\mathbf{s}, [A]\\mathbf{v}+[DA]\\mathbf{s}).", "8be34491ed0b8a52d9c01f2048230a01": "\\mathcal{E}(cg) = \\mathcal{E}(g)", "8be3dcbf1b749b593468a278b40be932": "T_{MC}(s)", "8be3e8b1fde46141a892b458d5c55c72": "\\left ( \\sum_{i=m}^{m+n-1} x^i \\right ) \\bmod G(x)", "8be424466d3869ab49703d84751ead6e": "\\scriptstyle{\\bar{l}^2}", "8be43a17de8fef29bc871d12a38ed3b5": "\\displaystyle M=N^{-1}", "8be4510d60f279229a58e0ce6d46989a": "\\sum_{\\{i\\}} \\vert \\lambda_i \\vert < \\infty. \\, ", "8be473c9b59cfb77e4d78ab67b75ac05": "x=(x_1,\\ldots, x_n)", "8be4f3444ee4088c4bdc9f5a96002a30": "\\tan \\theta =\n 2\\cot\\beta\\frac{M_1^2\\sin^2\\beta-1}{M_1^2(\\gamma+\\cos2\\beta)+2}", "8be5527ff583d98e9b7ec905d24741bc": "\nt^{\\prime} = t - t_{0}\n", "8be5761d29d87048eddfa0cd105503ca": "1\\to \\mathrm{Pic}^0(V)\\to\\mathrm{Pic}(V)\\to \\mathrm{NS}(V)\\to 0.\\,", "8be57dce4dee97f0c2ab6194b4d449bb": "y^2 = (20-a)(5+a)\\,", "8be5930458e4dff10425b744a687654a": "00.", "8bf38dff20ac1c4f55572a90298059bd": " N = 2\\left[ \\begin{matrix} u^2+v^2 & u+iv \\\\ u-iv & 1 \\end{matrix} \\right]. ", "8bf3cda9f7f33f346df58c475750f021": "\\exp : \\mathcal O_M \\to \\mathcal O_M^*,", "8bf40c11c2de8de985b3146a19968778": "\\frac{D \\mathbf{v}}{D t} = -\\nabla p + \\frac{1}{\\mathrm{Re}} \\nabla^2 \\mathbf{v} + \\mathbf{f}. ", "8bf4175b3a7561fc812a96fa95f1d746": "\\arg\\max_W \\mathbb{E} \\left[\\sum_{v \\in V} \\log P(v)\\right]", "8bf42048b87d1911d5d79d577cccad1c": "|\\lang\\psi|\\phi_i\\rang|^2", "8bf45f64a12e6532f92c3c3a69167372": "\\vec{J}(\\vec{r},t)", "8bf47f4869ef9ddfa3660e56e5ec48ae": "r=\\frac{(1+i)}{(1+\\pi)}-1 \\simeq i - \\pi ", "8bf4b3605ce90bc300422db91ca670a7": "\\sqrt[3]{\\ \\sqrt[3]{2}\\ - 1}= \\sqrt[3]{\\frac{1}{9}} - \\sqrt[3]{\\frac{2}{9}} + \\sqrt[3]{\\frac{4}{9}}. ", "8bf4c46c9d80dd30d568a774fed0cf25": " \\Delta_h[f](x) = f(x + h) - f(x). \\ ", "8bf4f4b74a363ad5494b6554d251b55e": "\\xi\\omega_0=1", "8bf58ee47d6055866f0548bf9387359b": "D = \\frac{2}{3 n\\sigma}\\sqrt{\\frac{kT}{\\pi M}}", "8bf5b8a12078695dea2c35266c0f4053": "(7 \\times 7) - 7 - 5", "8bf5e4cf78f7611d9ec72dce844cabdc": "\\frac{\\frac{L_p}{A}\\frac{N_r}{C}-\\frac{N_p}{C}\\frac{L_r}{A}}\n{\\frac{N_p}{C}\\frac{E}{A}-\\frac{L_p}{A}}", "8bf5e92e786941939680c88a2a9b00e0": " 0< N < A ", "8bf5fc82894b33cc8a25a1e8870224f0": "-0.022\\pm0.020", "8bf632b29a8805645233b531d448c07a": "M_{2,n}\\! = M_{2,n-1} + (x_n - \\bar x_{n-1})(x_n - \\bar x_n)", "8bf749a045c82bb6f654108b4f80b927": "\\{ w \\in \\mathbf{C}^n \\mid \\lVert z - w \\rVert < r \\}.", "8bf7586e24ef4d6b04abc8842e382ca7": " E(F(X + \\varepsilon\\varphi))= E \\left [F(X) \\exp \\left ( \\varepsilon\\int_0^1 h_s\\, d X_s -\n\\frac{1}{2}\\varepsilon^2 \\int_0^1 h_s^2\\, ds \\right ) \\right ].", "8bf7a485dfaca28e6ea331840bd4d389": "\\delta_4", "8bf7aa5aa2ff48a01c0df9c981d1fe12": "\\ MSD= 6 D_a t^\\alpha \\, ", "8bf7b6372c681ea49ef305ad78fedc3d": "\\psi_0=\\psi_0\\big(\\vec{\\sigma},\\vec{\\rho}\\big)=\\Big({\\textstyle\\sum\\limits_{i=1}^n\\sigma_i^2}\\Big)^{-3/2}\\cdot\\sum\\limits_{i=1}^n\\rho_i.", "8bf7bd84abdd432df1eb560cc4043c26": "r \\approx 1.45", "8bf7c539a34d94c96f38a37f3087c7bb": "\\ \\mathbf U(\\mathbf x,t) = \\mathbf x - \\mathbf X(\\mathbf x,t) \\qquad \\text{or}\\qquad U_J = \\delta_{Ji}x_i - X_J =x_J - X_J", "8bf80f56f09854d5d192ca3cb511f0fe": "J \\subseteq P.", "8bf8eed73b2511779aa8ba0ea68c406e": "\\ v_w(h)", "8bf9584c0559006723fd6c907b5e7669": "\\boldsymbol{\\psi}\\left(\\boldsymbol{\\theta}\\right) = \\boldsymbol{\\theta}", "8bf97bda87145f1d55b2a52aba43b3c7": " L_t = (L^x_t)_{x \\in E}", "8bf9e97cea9c00453832b861c5cbef7c": "n = \\frac{\\lambda}{\\Delta x}\\,\\!", "8bfa0fc60d3fd2d94eecbf46ef6926a2": "\\frac{p}{q}=\\varphi\\!", "8bfab2e334dfc8f98b8c246af0875f9d": "K_\\textrm{a} = \\frac{[\\textrm{H}^+][\\textrm{A}^-]} {[\\textrm{HA}]}", "8bfaf7be049bd4fa5e254020733fe354": "(B)", "8bfb0e53fa331d545b15ab28507df416": "y \\in S ", "8bfb277eedd86b05dc6358a876b1f9b0": "\\scriptstyle\\vec{H}", "8bfb37c59b2f0ef53897f03322bc1339": "(2+\\sqrt{2})^2 \\approx 11.66 \\approx {36.6\\over \\pi}", "8bfb3e073497e706fcafa398ba78029b": " c \\equiv m^e \\pmod{n}.", "8bfb43d10e04807cacf770d04a575949": "{\\sqrt{x_{2p-2}^2+x_{2p-1}^2+x_{2q-2}^2+x_{2q-1}^2}} = {\\sqrt{1/2+1/2}} = 1", "8bfb56611b332f24040cce7ed4f59278": "\\sum_i a_i \\sin(x+\\delta_i)= a \\sin(x+\\delta),", "8bfb9d262b5f39f29e49cd2cfb52a912": "\\ell \\leq \\ell_n", "8bfba2cbc63d212e22a76ce54df4ef09": "\\neg AF\\phi \\equiv EG\\neg\\phi", "8bfbc76ef63523347de50008a4a32657": "\\|\\mathcal{F}f\\|_{L^{p'}(\\mathbb{R}^d)} \\leq \\|f\\|_{L^p(\\mathbb{R}^d)}", "8bfbd014b485302475649a2b60d10284": "\n \\boldsymbol{\\sigma} = \\mathsf{C}:\\boldsymbol{\\varepsilon}\n ", "8bfc01d4943d7e9aca42cee74fac9fd4": "c^2-a^2=b^2", "8bfc07a00db9e4ac60cbf7ca3df05e5d": "~(X + iP)~", "8bfc22c16843f04d28b65f8d940881d7": "E = \\hbar v_F\\sqrt{k_x^2+k_y^2}", "8bfc243b6671062f1c1d6b80d96e239a": "\\int (\\ln x)^2\\; dx = x(\\ln x)^2 - 2x\\ln x + 2x", "8bfc70368eb6dcff679e6132c61e0c57": "\\langle Ux, Uy \\rangle = \\langle x, y \\rangle", "8bfc7c5c551d49aafca49191c33343d2": " \\bar{1} = 1 ", "8bfc7ca1db355202086941da2899d097": "D(f) = O(R_2(f)^3)", "8bfc879207f97a51e0274a5c9d98870c": "{C}_{8}^{(1)}", "8bfcbe499837df4621a4101413941399": "W_\\gamma [A] W_\\eta [A] = W_{\\gamma \\circ \\eta} [A] + W_{\\gamma \\circ \\eta^{-1}} [A]", "8bfcc910200487166d7a04de3c41866b": "E_{0}\\sum_{t=0}^{\\infty }\\beta^{t}\\left[u(c_{t})\\right]", "8bfd18fd616785b14c07df6e56acaeea": "O(\\sqrt{n\\log n})", "8bfd4097cd5ca332ba1bab83be196656": "\\epsilon_{**}", "8bfd4eaa594e037da7298e517b0a4287": "F(x,y) = f(x) + g(Tx - y)", "8bfd670aea05de00aef50c2a8327ed42": "\\mathbf{R} = (\\mathbf{I}-d \\mathcal{M})^{-1} \\frac{1-d}{N} \\mathbf{1}", "8bfd7d245f39bd40d37676427873c481": "H_1(\\mathrm{A}_n,\\mathbf{Z})=0", "8bfe340de5ae0fab4e2380560e442384": "Y^TA_iY\\sim\\sigma^2\\chi^2_{r_i}", "8bfe4faf4debbc851ac17e429939f188": " h^{\\mu}=P^{\\mu}/Mc ", "8bfe79b323657f98ed9cecf63b5e1651": "d = N", "8bfeecc7942595a3a7d3636ff5cab5bf": "y_0 = 1", "8bff440c66f453c6d9d0f9faecd4e887": "I_{\\pm}", "8bff5ba22a29fb652678c10f6efac9b4": "\\theta = \\arctan \\left( \\frac{\\text{opposite side}}{\\text{adjacent side}} \\right)", "8bff9cfcee5723738e347f2df494030f": "\\Pi_{k=2}^{\\infty}{\\zeta(k) ^{-a_k}},", "8bffb9d90c671bc767d7c92cb95cc268": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon r} \\sum_{k=0}^{\\infty}\n\\left( \\frac{a}{r} \\right)^{k} P_{k}(\\cos \\theta ) \\equiv\n\\frac{1}{4\\pi\\varepsilon} \\sum_{k=0}^{\\infty} M_{k}\n\\left( \\frac{1}{r^{k+1}} \\right) P_{k}(\\cos \\theta )\n", "8bfff2c57a23f1a4abca477577bb82b9": " \\hat x-x ", "8c00217c793824c048392d72318b349c": "\\,\\xi", "8c0040de03e9fffda365a108bd76350d": "K = \\tfrac{1}{2}(ac+bd)\\sin{\\theta}", "8c0098cbdd7b2483c5a695e594d6402b": "e''=\\sqrt m", "8c00a8f2e3e7d5b93a6f793d2f0e07a6": " R^n {}_{ik\\ell;m} + R^n {}_{imk;\\ell} + R^n {}_{i\\ell m;k}=0 \\ ", "8c00e5daedea8ea6c4a23b81dc151258": "\\displaystyle 8pq\\le (a+b+c+d)^2", "8c012ab41dd57e88892a1196a9f86e4e": " Acoplanarity \\equiv \\phi_2 - \\phi_1 - \\pi ", "8c013ab8449fd181ad6beae159a52fe4": "G(\\mathbf{r,r'})", "8c016353aff42cec344d79a8b308abd3": " \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\left(\\frac{\\partial^2 \\mathbf{S}}{\\partial x^{2}} + \\frac{\\partial^2 \\mathbf{S}}{\\partial y^{2}}\\right)+ \\mathbf{S}\\wedge J\\mathbf{S}\\qquad (4)", "8c016d12892d1e15708eaf10e17d487a": "p_{\\text{i}}", "8c017714ddf8b7109225173d6c1f9fbc": "B_1 = {\\alpha_0, \\alpha_1, \\ldots, \\alpha_{m-1}}", "8c017f719aeebd6da05388d98d334606": "\\begin{align}\ne^{\\pi \\sqrt{19}} &\\approx 96^3+744-0.22\\\\\ne^{\\pi \\sqrt{43}} &\\approx 960^3+744-0.00022\\\\\ne^{\\pi \\sqrt{67}} &\\approx 5280^3+744-0.0000013\\\\\ne^{\\pi \\sqrt{163}} &\\approx 640320^3+744-0.00000000000075\n\\end{align}\n", "8c01c87a9ffe0bad8e8b4de1a0e669d0": "(\\nabla \\times \\mathbf{F} )^k = \\epsilon^{k\\ell m} \\partial_\\ell F_m", "8c022c3276303db6a0ee5b461dc15cb6": "\\nu: [x:y:z] \\mapsto [x^2:y^2:z^2:yz:xz:xy]", "8c0237526daf9c843272cba002b1a6b7": "\\begin{align}\nI(a)^2 & = \\left ( \\int_{-a}^a e^{-x^2}\\, dx \\right ) \\left ( \\int_{-a}^a e^{-y^2}\\, dy \\right ) \\\\\n& = \\int_{-a}^a \\left ( \\int_{-a}^a e^{-y^2}\\, dy \\right )\\,e^{-x^2}\\, dx \\\\\n& = \\int_{-a}^a \\int_{-a}^a e^{-(x^2+y^2)}\\,dx\\,dy.\n\\end{align}", "8c025358593e2ef2786e66175d97e6fd": "h_{15}", "8c027a5c7a221c5c2f990088f3652616": "O(VE)", "8c029b4a1a84e001d76c3c8795cea26c": "\n \\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = Df(\\boldsymbol{S})[\\boldsymbol{T}] \n = \\left[\\frac{d }{d \\alpha}~f(\\boldsymbol{S} + \\alpha\\boldsymbol{T})\\right]_{\\alpha = 0}\n", "8c03b8fe36946a55cd4253def6e1b78c": "\\psi(\\boldsymbol{r+R_{\\ell}}) = e^{i\\boldsymbol{k \\cdot R_{\\ell}}}\\psi(\\boldsymbol{r}) \\ , ", "8c03e9cad086ba0ffcbf618ed6f8fe3f": "q(x) = q_0 x^m + \\cdots\\,", "8c04997a3301a444b9a1d7e94112c95d": " \\mathrm{atomic \\ percent} \\ (\\mathrm{i}) = \\frac{N_\\mathrm{i}}{N_\\mathrm{tot}} \\times 100 \\ ", "8c04aefa31e421e8a4c59e5448025ad1": "a_1 \\rho \\cos(\\theta)", "8c056fb0614289c5227fb26d940b3973": "W = \\left\\{w_{1},w_{2},\\cdots,w_{n}\\right\\}", "8c057b3a647fc79e83f8be77af391160": "(q^0,q^i,q^i_0)", "8c058275d1434ed4d2d0ab5a2b57b09f": "V_\\mathrm{M}=U_\\mathrm{CKM} \\cdot U_\\mathrm{PMNS} \\ ,", "8c05eb07f629e83d206bbe4649fd5e73": "\\tilde{\\Phi}_{\\ell r}", "8c05ed6c700503438b7f4b81d478e96a": "\\psi _{2m + 1}", "8c061a2981279a9e424f3427bb264630": "a_1,..,a_k", "8c062280fe153f2d6c315963d2211ebb": "X, Y, W", "8c0632548f38c609f769ad956d29c32f": "\\tan(\\alpha)=\\frac{v}{c}=\\beta", "8c0670a110369879a028be48ce2c0f10": "2t>1", "8c06817a1b46461646f4b6f2f1beea3a": "H_\\infty(A) = - \\log(\\max_{\\mathrm{a}}P[A = a])", "8c069cb20e8f7bf65c2cd3d5b617b6e5": "\n\\begin{pmatrix}\n A & B \\\\\n C & D \n\\end{pmatrix}\n=\n\\begin{pmatrix}\nI \\\\\nC A^{-1}\n\\end{pmatrix}\n\\,A\\,\n\\begin{pmatrix}\nI & A^{-1}B\n\\end{pmatrix}\n+\n\\begin{pmatrix}\n0 & 0 \\\\\n0 & D-C A^{-1} B\n\\end{pmatrix},\n", "8c06a4ebb8ecaf2254e7e2bf1c7167c5": "\n\\sum_{mn} \\psi_m^* A_{mn} \\psi_n\n", "8c06f4f0bd1d4e562a6503d7319706a2": "T_\\delta", "8c06f77a048c7032b6258f4ceddcac77": "s_{y}", "8c070b724e9fa7ca0ebf18f77bc5781b": "\\sum_{k=0}^\\infty ar^k = \\frac{a}{1-r}", "8c076dd5a1ebae2e5ad16ad3c79251aa": "t_{il}", "8c077465b0893f1067b9399b7d0f8e96": " \\begin{matrix} v_{21}+v_{22}-v_{21} = 1 \\\\ v_{22}- v_{22} = 0. \\end{matrix}", "8c0785f0458068dfa9c5bf1fb0900b16": "L_1\\colon U_1 \\to \\mathbb{C}", "8c07b3a166adeea63fbf4e8c8b0aa38e": "\\cos (\\arctan x) = \\frac{1}{\\sqrt{1+x^2}}", "8c07b73c9bb4bf14cbc69a1c413a3904": "H(a) = \\{ P \\in X_F : a \\in P \\}", "8c07e81a9320c6a95f433f0784f2bb8c": "f: Y \\to X", "8c07ed41fad6130ae2f953353e7e4d0d": "\\text{Area}=\\tfrac{q^{2}cuv(v^{2}-u^{2})}{2} \\,", "8c082f4d8a61a9f5fa9655dc1538ab72": "p(g(Y),Y)", "8c0870064d63c05343fe9292003c27bb": " [ A | I ] = \n\\left[ \\begin{array}{rrr|rrr}\n2 & -1 & 0 & 1 & 0 & 0\\\\\n-1 & 2 & -1 & 0 & 1 & 0\\\\\n0 & -1 & 2 & 0 & 0 & 1\n\\end{array} \\right].\n", "8c08e95edac7d4b2c91b3bf73cbb1153": "\n\\begin{bmatrix}\nc t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\gamma&0&0&-\\beta \\gamma\\\\\n0&1&0&0\\\\\n0&0&1&0\\\\\n-\\beta \\gamma&0&0&\\gamma\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nc\\,t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix} ,\n", "8c09170797e58a7b1299cb10569ee55f": "p(\\chi)\\!\\,", "8c0926374bfbd6a90c49dd50f43daa9b": "\\Lambda \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{h^2}{2\\pi m_ek_BT}}", "8c097d13b3acc1d4785d4fde2b88b01e": "\\pi_* (f)=0", "8c097dbbd1c79a63dec6692e2a54740a": "\\cos (\\theta) = 0", "8c098cc4a42925afafda499af6e5fc64": "\\begin{align}\n a_1 &= h_{\\mbox{e}}\\cdot a_{0,\\mbox{e}} +\n h_{\\mbox{o}}\\cdot a_{0,\\mbox{o}} \\rightarrow 1 \\\\\n d_1 &= g_{\\mbox{e}}\\cdot a_{0,\\mbox{e}} +\n g_{\\mbox{o}}\\cdot a_{0,\\mbox{o}} \\rightarrow 1\n\\end{align}", "8c0a2fea2322263de6bb41ab8ac4a5f1": "x < w_1 < w_2 < y", "8c0a56dc546fe3ce06100559a4eb567f": "N_1 \\cup N_2", "8c0a6878f8d17d0a65c512c7e0ca46c7": "E_{d/s} = (1.5)\\biggl(\\frac{q_3^2}{g}\\biggr)^\\frac{1}{3} = (1.5)\\biggl(\\frac{20^2}{32.2}\\biggr)^\\frac{1}{3} = 3.47\\text{ ft}", "8c0adf9f1c1f183e4be98ce60c85c34b": "\\displaystyle G_0", "8c0b63664610269cec9a28ff6c9ae5d8": "2 \\pi \\left( \\frac{m}{k} \\right)^{1/2}", "8c0bba3801bf7a0b60c9d6ff8c8fd795": "SU_{\\mu}(2)", "8c0be15a7f13979a9fe15fc503398db1": " K = -\\frac{1}{2} e^{-\\varphi} \\left(\\frac{\\partial^2 \\varphi}{\\partial u^2} + \\frac{\\partial^2 \\varphi}{\\partial v^2}\\right),", "8c0c31adbbb2f18417b9476617043833": "\n \\begin{cases}\n v_i-\\max_{j\\neq i} b_j & \\text{if } b_i > \\max_{j\\neq i} b_j \\\\\n0 & \\text{otherwise}\n \\end{cases}\n ", "8c0d3703df8ef5ac61fc6791333fdebe": "V_o = -A_v V_i", "8c0d4ca2fef4c72d739ffef525815010": "D \\cdot r", "8c0e058063edae6ec22947fc9093e3ef": "E = A s f(\\ell/A), \\!", "8c0e1ef36f07d58523ac92c938930e7b": " \\, A_{(0,\\,0,\\;1)}(x)=x^2\\,", "8c0e2421515d8318eaebca84a4537067": " f_{Y|X=0.5}(y) = \\frac{ f_{X,Y}(0.5,y) }{ f_X(0.5) } = \\begin{cases}\n \\frac{1}{ \\pi \\sqrt{0.75-y^2} } &\\text{for } -\\sqrt{0.75}1}S(i_1,\\cdots,i_k)={n-1\\choose k-1}\\zeta(n)", "8c1125120f72459fefb27ee362d26739": "R(t) = 1 - \\sum_{k=1}^K \\frac{\\alpha_K}{\\sum_{l=1}^K \\alpha_l} (1-R_k(t)) \\prod_{j=1,j\\neq k}^{K} \\alpha_j \\int_t^\\infty (1-R_j(u))\\text{d}u", "8c11368cb3b0e7d794cd818d7fddf9e9": "(A - \\lambda I)^k", "8c1168997a1cb93e022cf457b407d867": "r \\geq 2R", "8c11afb0e0bbc5bf3c4bcc80f268eec1": "p-1 = 2r", "8c121cdd8eec4c21f64c0d666fdae9d6": "(1 + n)^D = 8", "8c129a4987ca69dc355cf0921621339c": "\n\\vec{I} = \\begin{pmatrix}\n0 \\\\\n{RR}^{mkt} - {RR}^{BS}\\\\\n{BF}^{mkt} - {BF}^{BS}\n\\end{pmatrix}\n", "8c12e6899e77b735a808f67fdeece5f1": "z(\\beta,\\mu)= e^{\\beta \\mu}", "8c131a9f3a4668341639b8a50286d24f": "\\textstyle a\\in(0,1)", "8c1338e19da5b07c693de56a60cebd89": "\n\\varphi\\left(e^{-3\\pi}\\right) = \\frac{\\sqrt[4]{27\\pi+18\\sqrt3\\pi}}{3\\Gamma(\\frac{3}{4})}\n", "8c1385bd14d59a54e197579b19becafa": "-\\sqrt{\\frac{1}{10}}\\!\\,", "8c1392ce871a020d0ef9e95323c5cc26": "\\Pi = \\gamma_{0} - \\gamma", "8c13dfdb19d3fa85f158f3650d1b7ae9": " \\sim 10^{561} \\,\\!", "8c140f57dd756d13355e03e15d907331": "x \\leq M \\land y \\leq M \\land y = M", "8c14130dbe36e22ea489d21357b72749": " B(\\mu_f)=\\{x\\in M:\\frac{1}{n}\\sum_{k=0}^{n-1}\\delta_{f^kx}\\to\\mu_f\\}. ", "8c142408731c53a5be513201d0b139ab": "\\begin{align}(ax^2&+bx+c)(dx^3+ex^2+fx+g)\\\\\n&= adx^5 + (ae+bd)x^4 + (af+be+cd)x^3 + (ag+bf+ce)x^2+(bg+cf)x+cg.\\end{align}", "8c1438e167c636f9ec1c914e20b12c67": " \\int_0^T \\|A(s)\\| ds < \\pi", "8c14478d6108003b34d47ff141503c17": "B_{\\alpha\\beta}=\\partial_\\alpha u_\\beta-\\partial_\\beta u_\\alpha", "8c148a1df486659f2d258aeed5cdd7e9": "\\tau(i_1,\\cdots,i_k)", "8c14f67185eed4b0614e22cb54cd5747": "b + d = b + a + c ", "8c14ffc373d86766cb4e095149841355": "1\\ \\mbox{Pa}\\cdot\\mbox{s} = 1\\ \\mbox{kg}\\cdot \\mbox{m}^{-1}\\cdot\\mbox{s}^{-1} = 10\\ \\mbox{P}", "8c1533a46cac6263cce8177c6ca81ee7": " \\int_a^b \\omega(x)\\,f(x)\\,dx - \\sum_{i=1}^n w_i\\,f(x_i)\n = \\frac{f^{(2n)}(\\xi)}{(2n)!} \\, (p_n,p_n) ", "8c154f576b6c8a6db8eddf12a8ff12f8": "R_{k}", "8c155111d805d796686f8759d71fc07c": "D_i>1", "8c1558729c5de05d835408c51f27bdfe": "v = Ki\\,", "8c156f155ffcd951e8ed32e7cd094a9f": " \\alpha_x ", "8c15a09b857181293515caee43c288c5": "m_e \\,", "8c15d71bb25605b6e926771672464755": "z=0^-", "8c1603e2b47a08ed38e0ef7543fa0b5a": "\\log_{10}\\left(\\frac{Z/X}{Z_\\mathrm{sun}/X_\\mathrm{sun}}\\right) = [\\mathrm{M}/\\mathrm{H}]", "8c164617b8b6ab24d0e8c3ee429f9848": "Z,F_2[T],b^{\\ast},B^{\\ast},W^{\\ast}", "8c166871434fef489ce3cb742f871568": "Tds = dh - (\\frac{P}{\\rho})dp", "8c1670b0ddedec5a48c7209144328ff0": "{\\delta Q}=T\\mathrm{d}S", "8c167bec558e89f16b0efedf777efd1d": "\\phi^{\\#}: \\mathcal{O}_{\\phi(x)} \\overset{\\sim}\\to \\mathcal{O}_x, \\, s \\mapsto s \\circ \\phi", "8c16e3a8944d40f9e8e6205c910acdfe": "\\omega=2\\pi f", "8c16e68bbe71fbe14299932d869009ab": "{\\lambda}", "8c17143f5d6a1381f98852ed69b0755a": "H = -\\int_\\Gamma f_{WC}(\\theta;\\mu,\\gamma)\\,\\ln(f_{WC}(\\theta;\\mu,\\gamma))\\,d\\theta", "8c17170d0e49e824c27ef1199da00179": "E_0 = |E(0,0)|", "8c1731eb2f1591e7229fe57dea8cb1c3": "A_n.", "8c176762e341fb7aa995345ffafe4f06": "x \\in \\mathfrak{h}", "8c17939d83c63520005ad0e355e8927e": "H = pu(t) - \\frac{u(t)^2}{x(t)} - \\lambda(t) u(t) ", "8c17b903cf1389e4829e279b282593ca": "X_{\\overline{s}} = X \\times_S {\\overline{s}}", "8c17fd9c38e286109e8ecfcb6f17c453": "a_{0} \\neq 0\\, ", "8c18311959e80a065fe4f10545cffb38": "[ \\cdot ]", "8c186dc67d021cf635b46118f8258a29": "\\mathfrak {pq}", "8c1929ac884dee2943bc7242dd1dbc98": "N=rol(\\bigoplus_{j=1}^d S_j,u)", "8c19dc52cbf5635018fd3c09b338043c": "\n\\frac{\\partial \\mathbf{y}}{\\partial x} = \\left[\n\\frac{\\partial y_1}{\\partial x}\n\\frac{\\partial y_2}{\\partial x}\n\\cdots\n\\frac{\\partial y_m}{\\partial x}\n\\right].\n", "8c19e1deac3c60060d35cf5bd12b3616": "s=2n", "8c19f1a18a9524141c06cfde128fb851": "\n\\tau_{xy} \\leftarrow\n(1-\\rho)\\tau_{xy} + \\sum_{k}\\Delta \\tau^{k}_{xy}\n", "8c1a58a841fab198fe7924562e9b2721": "1+\\sqrt{-19}", "8c1a910f16c24262c27e11fdfd13b5e2": " SD( H ) = \\frac{ 1 }{ N } [ \\sum p_i [ log_e( p_i ) ]^2 - H^2 ] ", "8c1ab2b5a001c44b7edfba3ad20caa6d": "a \\in \\{ 1,\\cdots,9 \\}", "8c1b16356456b6797ef823c10beb2d4a": "E_{rest}=(m_{rest})c^2\\!", "8c1b2d91f0ad006356fbc6706b4b74d2": "\\partial : D^k \\to D^{k-1} ", "8c1b4fe153db843c83ecfeab432839e7": "h\\theta(L)=-k\\left.\\frac{d\\theta}{dx}\\right\\vert_{x=L}.", "8c1b5dacc4067a1ba751e2e8c77f2ace": "\n\\begin{align}\n \\mathbf{b}_m\\times\\mathbf{b}_n & = \\frac{\\partial \\mathbf{x}}{\\partial q^m}\\times\\frac{\\partial \\mathbf{x}}{\\partial q^n}\n = \\frac{\\partial (x_p~\\mathbf{e}_p)}{\\partial q^m}\\times\\frac{\\partial (x_q~\\mathbf{e}_q)}{\\partial q^n} \\\\[8pt]\n& = \\frac{\\partial x_p}{\\partial q^m}~\\frac{\\partial x_q}{\\partial q^n}~\\mathbf{e}_p\\times\\mathbf{e}_q\n = \\varepsilon_{ipq}~\\frac{\\partial x_p}{\\partial q^m}~\\frac{\\partial x_q}{\\partial q^n}~\\mathbf{e}_i\n\\end{align}\n", "8c1b78fd09b095a3f60a1bad05b9789e": "L(5) \\propto \\sum_{i=1}^{4}i+\\sum_{i=0}^{4}i=20,", "8c1ba9f501655b75f5fe40df9ea82b41": " x \\in G ", "8c1bb3e30f748e9bc3967f87b61c13b9": "S(z) + i C(z) = \\sqrt{\\frac{\\pi}{2}}\\frac{1+i}{2} \\operatorname{erf}\\left(\\frac{1+i}{\\sqrt{2}}z\\right)", "8c1bef159f3175de7024dab96a8d80e6": "y = 2 \\times (x + 10) - 10", "8c1c2a5ce20369ff7b9de3bee5cda260": "f, g: M \\to \\R", "8c1c3b7f8ce2411c82b583fce6f68f20": "\\frac{\\partial}{\\partial\\tau}\\bigg|_{\\tau=0}d(\\gamma_0(t),\\gamma_\\tau(t))=|J(t)|=\\sin t.", "8c1c758dc065abd24f5d2d2216c171ff": "\\det(\\mathsf{A}-\\lambda \\mathsf{I}) = 0.\\ ", "8c1d149786d32017ac59aeed6f102d94": " M_{i,j} = \\left|\\left|A_i-A_j\\right|\\right|_p = \\sqrt[p]{\\displaystyle\\sum_{k=1}^{n}\\left|a_{k,i}-a_{k,j}\\right|^p } ", "8c1d19e9c9ad7682630e20bcb0e3af54": " f(x) = c_0 + c_1 x + \\dots + c_{n-1} x^{n-1} ", "8c1d234b93b4acfc56cbb26e088e9d4b": "-\\tfrac {1}{12} \\pi^2 \\,", "8c1d410d3fa3e06c0386d3493f865d31": "\n \\frac {\\partial p^{n+1}} {\\partial n} = 0 \\qquad \\text{on} \\quad \\partial \\Omega\n", "8c1d965ffb0a863c56ba306a6e95cc47": "\nR_n^m(r)=(-1)^{(n-m)/2}\\int_0^\\infty J_{n+1}(k)J_m(kr)\\operatorname{d}\\!k\n", "8c1d9b44e9788fa26a3c06448950ca2a": " M_1 (X,Y,\\vec Z)", "8c1daf366885f70b882aed28a88371fb": "\\varphi_{i,j}(a') = b", "8c1dd3a68fb9f7edd1ed7384b8f9bacc": "\\frac{\\partial}{\\partial t} \\rho(x, p) = \\left[- \\frac{\\partial H(x, p)}{\\partial p} \\frac{\\partial}{\\partial x} + \\frac{\\partial H(x, p)}{\\partial x} \\frac{\\partial}{\\partial p} \\right] \\rho(x, p)", "8c1ddc1b71aa933543636a88b3bb73bb": "\n\\sigma(x,y) = \\operatorname{E}{\\big[(x - \\operatorname{E}[x])(y - \\operatorname{E}[y])\\big]},\n", "8c1de77b058a292516db9d4d77ed4121": "V_{in} = -V_{ref}\\dfrac{t_{d}}{t_{u}}", "8c1e648cf77ebf67bb27d4b1d35ac8c9": "1/v^2", "8c1e702915bbe6ff769edf20506134be": "L_{q}\\left[1/3,\\sqrt[3]{32/9}\\right]", "8c1ecde0107d83903f5e8814f711f72b": " y(t) = y(t+P) \\ ", "8c1ed4190886184afcef09f7220d9c36": "\\begin{align}\n\\alpha \\colon \\mathrm{G}_{1,3} & \\rightarrow \\mathbf{P}^5 \\\\\nL & \\mapsto L^{\\alpha},\n\\end{align}", "8c1ef927fae600308d201265aa43e5e2": " E_a = \\sqrt{2 E_t E_k + (m_t c^2)^2 + (m_k c^2)^2} ", "8c1f278591248b05e1661d6c2092659f": "\n \\begin{align}\n \\eta(x,t) =& \n a\\, \\left\\{ \n \\cos\\, \\theta\n + ka\\, \\frac{3 - \\sigma^2}{4\\, \\sigma^3}\\, \\cos\\, 2\\theta\n \\right\\}\n \\\\ &\n + \\mathcal{O} \\left( (ka)^3 \\right),\n \\\\\n \\Phi(x,z,t) =&\n a\\, \\frac{\\omega}{k}\\, \\frac{\\cosh\\, k(z+h)}{\\sinh\\, kh}\n \\\\ & \\times\n \\left\\{\n \\sin\\, \\theta\n + ka\\, \\frac{3 \\cosh\\, 2k(z+h)}{8\\, \\sinh^3\\, kh}\\, \\sin\\, 2\\theta\n \\right\\}\n \\\\ &\n - (ka)^2\\, \\frac{1}{2\\, \\sinh\\, 2kh}\\, \\frac{g\\, t}{k}\n + \\mathcal{O} \\left( (ka)^3 \\right),\n \\\\\n c =& \\frac{\\omega}{k} = \\sqrt{\\frac{g}{k}\\, \\sigma}\n + \\mathcal{O} \\left( (ka)^2 \\right),\n \\\\\n \\sigma =& \\tanh\\, kh\n \\quad \\text{and} \\quad \n \\theta(x,t) = k x - \\omega t.\n \\end{align}\n", "8c1f291d11c4e12f2e02a2a2354ea48c": " \\int_0^t | H_s | | K_s | d \\langle M,N \\rangle_s \\leq \\sqrt{\\int_0^t H_s^2 d \\langle M \\rangle_s} \\sqrt{\\int_0^t K_s^2 d \\langle N \\rangle_s} ", "8c1fe5286c23f1a750dd369a60cb760a": "e^{-j\\omega t}", "8c1fea5e415a43ce5d6279b627bed6dd": "\nE(V)\n= \\frac{n\\theta}{\\theta} - \\frac{n(1-\\theta)}{1-\\theta}\n= n - n \n= 0.\n", "8c1ff08e73bb8162b9dc562d92e33eae": "\n\\begin{matrix}\n\\frac{\\partial z}{\\partial t} & = & \\frac{\\dot{x} y - x \\dot{y}}{y^2} \\\\\n&& \\\\\n& = & \\frac{a(1-Q)xy - x (aQx + by)}{y^2} \\\\\n&& \\\\\n& = & a(1-Q)z - (aQz^2 +bz) \\\\\n&& \\\\\n& = & z(a(1-Q) -aQz -b) \\\\\n\\end{matrix}\n", "8c204102674fc88e9448e9b88cd2b863": "\\left[v, R\\right] = 0", "8c209b82e94ebd4339cca50901ff62d4": "(1+x)^{0.5} = \\textstyle 1 + \\frac{1}{2}x - \\frac{1}{8}x^2 + \\frac{1}{16}x^3 - \\frac{5}{128}x^4 + \\frac{7}{256}x^5 - \\cdots", "8c20be8e9e251d7ff766a16cbbf3aa54": " M(\\vec X,Y)", "8c2160e32cd740b1273f0e3d2dae32b3": " \\omega=\\nabla\\times\\mathbf{u}.", "8c217af53b53a4379d280a1f8e29028c": "H(\\vec{x},\\vec{x}')>0\\,\\!", "8c218c7e122b42eee9d1decc114c44c0": "Q(x_1,\\dots,x_n) = x_1^2 + \\cdots + x_p^2 - x_{p+1}^2 - \\cdots - x_{p+q}^2.", "8c21db5983ee0e751f436aaee6d9589f": "wv\\in E", "8c21f735232a12c299ecda7f749e9903": "\\scriptstyle \\tilde{r}_i \\;=\\; -c (\\tilde{t}_i \\,-\\, \\tilde{t}_{\\text{rec}})", "8c22487246db9f7d40d2ca48211ede52": "\\frac{\\partial C}{\\partial \\sigma}", "8c2265f041030256984c2f011e7d342f": "X_w(a,b)=\\frac{1}{\\sqrt{|a|}} \\int_{-\\infty}^{\\infty} x(t)\\psi^{\\ast}\\left(\\frac{t-b}{a}\\right)\\, dt", "8c22ac08c2fc550efe78cc57ef9132e7": " 0 < | x - 5 | < \\delta \\ \\Rightarrow \\ | x - 5 | < \\varepsilon / 3 ", "8c22d546d36875cc264e3108be14e506": "\\frac{d^2\\beta}{dt^2}+(\\frac{2k}{MV}+\\frac{2kb^2}{VI})\\frac{d\\beta}{dt}+(\\frac{2kb}{I})\\beta=0", "8c2305f0a6b23bba2394bbf0f6cc0125": " Q_x=\\int_0^b q_x dz = -k b\\frac{\\partial h}{\\partial x}", "8c23258103904c29a8558bea963826d4": "w_{i} = \\frac{a_{n}}{a_{n-1}}\\frac{\\int_{a}^{b}\\omega(x)p_{n-1}(x)^{2}dx}{p'_{n}(x_{i})p_{n-1}(x_{i})}", "8c239bfc14be6825e0497fa304795dd0": "- 1.3816 \\times 10^{-7} \\left( 10^{11.344 \\left( 1-\\frac{T}{373.16} \\right)} -1 \\right) ", "8c23a7527afcd5ad9d2fe0e0b08ff543": "\\left(\\frac{p}{5}\\right) = \\begin{cases} 1 &\\text{if }p \\equiv \\pm1 \\pmod 5\\\\ -1 &\\text{if }p \\equiv \\pm2 \\pmod 5 \\end{cases}", "8c23ba2a8d48c14bf5005d01cc230aac": "\\mathcal{S}", "8c23d468403b7fbc0e53abb1b2f5de61": "\\frac{A}{A^*} = \\frac{1}{M}\\left[\\frac{1+\\left(\\frac{\\gamma-1}{2}\\right)M^{2}}{\\left(\\frac{\\gamma+1}{2}\\right)}\\right]^{\\frac{\\gamma+1}{2(\\gamma-1)}}", "8c242379dcba59472881c79634cc68e2": "P = \\text{work done per unit time} = \\frac {QV}{t} = IV \\,", "8c2446c5233e582d4f0cd6eabc9c3b2e": "\\overline{Y}_j", "8c24c022a5f04499b007ce840a149c7f": "\\ \\Epsilon=[\\frac {\\partial(slope)} {\\partial(L/K)} \\frac {L/K} {slope}]^{-1}", "8c24d8d8b4b53e14916f3c99d16adf2a": "P_{\\nu_a\\rightarrow\\nu_b}\\neq P_{\\bar\\nu_b\\rightarrow\\bar\\nu_a}", "8c24e90800f2f4d45755cc8fcb7f359f": "\n\\begin{matrix}\n\\_\\_\\_\\\\\n???\n\\end{matrix}\n", "8c252ebcd1d714d9e6c80ede7f151a9f": " f=f_0(v)+f_1(x,v,t)+f_2(x,v,t)", "8c25782b52fe8978b020fa69148b9583": "N-n_i", "8c258bbfdca7a858e02f57602b059fb4": " ds^2 = du^2 + 2\\cos\\varphi \\,du\\, dv + dv^2,\\, ", "8c25b067541149926fd70a85ad72d8bc": "4p^3 q", "8c25da528a837e32f08869b22b7a6cb7": "\\Psi_i", "8c25ebe13f9de3c9612dff0e9830c199": " \\partial_{ [ \\alpha } F_{ \\beta \\gamma ] } = 0 ", "8c267bdac530ee35282bd5587b0cc3f9": "\\mathrm{div}(G)=\\sum_{P\\in C} {\\mathrm{ord}}_P(G)P", "8c26d9467429fcff5dd6e87ab3f80bcd": "\n\\text{volume} = \\frac{\\sqrt {\\,( - a + b + c + d)\\,(a - b + c + d)\\,(a + b - c + d)\\,(a + b + c - d)}}{192\\,u\\,v\\,w}", "8c26fd833f0c1a31c2d02aeda04c186e": "aR=\\{ar \\mid r\\in R\\}\\,", "8c271c8a6537151514042f87a9e10592": " \\mathbb E ", "8c2742aa2dc490d2abf77c2c135a8773": "[HG] = \\frac{[H]_0K_a[G]}{1+K_a[G]}", "8c275bce6b81552c20ba9a7d502b5baa": "\\psi=\\begin{pmatrix} \\psi_L \\\\\\psi_R \\end{pmatrix},", "8c2783f54c7f6fa3014873d6b403ee36": "2d-\\tfrac{4d^2}{3l}", "8c27ddc6b014da25afb78f8b768dd9be": "y = \\frac 5 4 \\ln\\left[\\tan\\left(\\frac 1 4 \\pi + \\frac 2 5 \\varphi\\right)\\right]", "8c27e7e4307671273358e80a56d0b006": "{11628 \\choose 1} = {153 \\choose 2} = {19 \\choose 5}", "8c281ab07da0b995c27b2aee8d09961e": "g^+=1_{\\Omega'\\backslash U}g_0^+", "8c2836c0c5c0e7e07f6f117d828c1cf6": " \\dot E ", "8c287aebff24e6aaf58524a507c73476": "k[U]_{\\mathfrak{m}_x}", "8c28902af04dbf3cd7dbb1172f83867d": "\\Delta_+(x-y) = \\langle 0 | \\Phi(x) \\Phi(y) |0 \\rangle ", "8c2897476032900f016ebe108bfd5e1d": "y_{t}= y_{t-1}+\\varepsilon_t.", "8c28b564561e9169fc8c55f6e0a678a4": "\\lambda_{\\hat{x}}\\odot \\pi_z", "8c28f1d2564981665a8d69c9271b5910": "\\alpha_1 = 0", "8c293cbcf1b0d6cda56b8b39b5aa027a": "T_i \\uparrow", "8c29d1c7a48d5471c37ceaa9b6f01682": "B-V", "8c29ef666f1008e8791bc9a06d6f92a9": "x\\mapsto {\\rm Tr}\\begin{pmatrix}{1\\over \\lambda+1}&0\\\\ 0&{\\lambda\\over \\lambda+1}\\\\ \\end{pmatrix} x.", "8c2afa1652163feeaefbe27ec4a58316": "\\mu^'_1=k+\\lambda", "8c2b23b10678f8459ab5f6904ba06972": "\\{U=\\infty, V>0\\}\\cup\\{U>0, V=\\infty\\}", "8c2b50ebecbdfb3c7c2ca407174b6fa8": "f\\mapsto f(x_0)", "8c2bca9ca4892ba63e5adceaded0e640": "\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c}) = \\mathbf{b} \\cdot (\\mathbf{c} \\times \\mathbf{a}) = \\mathbf{c} \\cdot (\\mathbf{a} \\times \\mathbf{b}), ", "8c2bd234b384287beeab1f422f6d8831": "\n\\| x-P_K(x) \\| \\leq \\| x-y \\|\n", "8c2c64b011baa7efd255c4800db7a6e5": " \\xi^2\\frac{d\\theta}{d\\xi} = C_1-\\frac{1}{3}\\xi^3 ", "8c2d7feb85c2aae1c5e34ecdf4f2e60a": "T_{i_1 k_1 \\dots k_N}^{i'_1 \\ell_1 \\dots l_N} = \\sum_{r_1,\\dots,r_{N-1}} R_{i_1 k_1}^{r_1 \\ell_1} R_{r_1 k_2}^{r_2 \\ell_2} \\cdots R_{r_{N-1} k_N}^{i'_1 \\ell_N}\n", "8c2d9e551c3be511ac02d451e037ee9e": "\\arg\\zeta(s)", "8c2da521c681dbca6984520ec593611d": "z^{4n}-1 = (z^{2n}-1)(z^{2n}+1)", "8c2dd9d660690412489da7702758378d": " x_3' = {\\lambda'}^2 - x_1(x_1-x_2)^2 - x_2(x_1-x_2)^2", "8c2e0b46b4a319da24a0aef26c8b8ccf": "\\Delta_rG<0", "8c2e15e48287b95deca19c14d0ebdd2e": "(\\mu - \\nu)/\\sigma,", "8c2e54678a11ba2783af1aec9867c68e": "f \\lambda_{\\epsilon,g}", "8c2e57268ed1fe6c3fb09339a7d300a0": "\\left(x^r\\right)' = rx^{r-1},", "8c2f16909ee6e594176d66899f1c2961": " 0.08 x_0.", "8c36b710c7f25189b39c14e2c15894dd": "f = \\frac{\\cosh\\, \\bigl( k\\, (z+h) \\bigr)}{\\cosh\\, (k h)},", "8c36b79586b80d54794bb71f0fb6c907": "[\\mathbf x], [\\mathbf y]", "8c36eefd9a171e15f07265ce6fa2ee1c": " \\frac{d}{d t}N_1=-c_2 N_2", "8c371111428996391b238a2983abc860": "\\epsilon \\in V\\mathfrak{E}", "8c37223a6d126d5bbdc97bfb490cbb9d": "\\langle r^2(\\tau)\\rangle", "8c37343e3e927c81d1d34579added6ce": "Q'_{total} = Q'_{out} + Q'_{lid}", "8c37344dc11cbf1e4785be23244cbca7": "A=2.", "8c380361593608570ffd1e2013557443": "\\,\\!U[n]", "8c38116c320f53a411c14e98e7ca0786": "H_n(D^n,S^{n-1})\\cong H_{n-1}(S^{n-1})", "8c381d05b91181a07b9f8c8b9ae9330a": "\\displaystyle w(n,1)=1", "8c383ea4fd2c45548fec1f7648b431dd": "\\tan \\alpha \\approx \\alpha \\quad , \\qquad \\tan \\beta \\approx \\beta\\,\\!", "8c3859e7e4cd76bb99cd71846782a650": " \n {v'}^i = \\frac{d{x'}^i}{d\\lambda\\;\\;} =\n \\frac{\\partial {x'}^i}{\\partial x^j}\n \\frac{dx^j}{d\\lambda} =\n \\frac{\\partial {x'}^i}{\\partial x^j} {v}^j\n", "8c3885b4d4510cfdb141de05881cf571": "k_N", "8c38e569fd26e2bb86be342d6818cd82": "(H,D)\\,", "8c393e2e42df39939507a4391172e517": "X_i(t=0)\\frac{}{}", "8c3943d47972d7b0a1d5979bba1bd8ae": "\\beta = \\mathop{\\rm Im}(m)\\,", "8c399ad267b259a02909192824e45374": "f^*(x^*) - \\langle b,x^* \\rangle", "8c39bd10340015746aa767a582734aeb": "\\Phi\\left(\\frac{x-\\xi}{\\omega}\\right)-2T\\left(\\frac{x-\\xi}{\\omega},\\alpha\\right)", "8c39ceeec9257b1f9d1e9c87df29fb65": " -\\log P(x)>0 ", "8c3a2bc72515c9c6053de6ddf3264b5e": "\n L_{n+1} = 2L_n - 3(-1)^n. \\,\n", "8c3a4412cd65299be235afc20f00b332": "R'=R^\\frac{1}{2.19921875},", "8c3a4722cf7be61a9167f51449baaa70": "\\mathbf{p}_A", "8c3a961dd5a008fc9d523b6d32306f15": "\\frac{51}{50}", "8c3aad9c66a226115d49fdc1e5fcc2d8": "\\{1,e_1,e_2,e_3,e_1e_2,e_1e_3,e_2e_3,e_1e_2e_3\\}\\,", "8c3ac9b8a0cba59ee2581900cb713890": "P(x) \\gets (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\gets Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "8c3ad15b1369570a17f7e7ea6e561497": "H'", "8c3ae12f5d5d1f6833c35e9efeddbc8a": "n=1632, k=1269, t=34", "8c3b16cf744125003b59f482f1542c2e": "G_{ab} = 8 \\pi T_{ab}.", "8c3b34e9c0461e18c328df1056bb0b6a": "z_N=\\sum_{j=0}^N a_j e^{i k_j t}", "8c3b504aa26285a532e2b546614af0c5": " \\sigma^{2}_{P} = \\mathbb{E}\\left[\\sum^{n}_{i=1}x_i R_i - \\sum^{n}_{i=1}x_i\\mathbb{E}[R_i]\\right]^2 ", "8c3bae49c8fc8b0b04d60de7da1ee35a": "h[k]", "8c3c255781053b53379847fbd77d9e1a": "{d \\over dt}\\left\\{ B \\right\\} = k_+ \\left\\{ A \\right\\} \\left\\{B \\right\\} - k_{-} \\left\\{B \\right\\}^2 \\,", "8c3c3b2ec22c175e09e06d2a1b67689e": "1/\\sqrt{g_{pav}(E)}", "8c3c59987b1b49d36541d1d0e97dcc1b": "(D^+)_{ij} = (D_{ij})^+", "8c3c73eb7df9724fee4ffb798656ff8a": "T_A^2", "8c3ccae07924785a466e80b021016edc": "\\{1,\\dots,K\\} ", "8c3cd6b52bc70ff670ff85fccdf4df8c": "\\sqrt{3}\\sin(\\phi)", "8c3d06ed9d402b6f0309f6bda1e49488": " U(\\mathfrak{g},e)", "8c3d0db0dc4b147c70084981e36045d5": "F(s)=p_\\mathrm{mid} + s\\cdot\\mathbf{v}^{\\mathrm{perp}}.", "8c3d95a4a0be207bdcb9610b624c88ce": "E(\\mathbb{F}_{q})", "8c3daba3264eef892973defd61b733c8": "S=\\sum_s s^2 n_s/p_c \\sim |p-p_c|^{-\\gamma}\\,\\!", "8c3e799630d7d054284c0dcde00c6b25": "\\pi^1", "8c3eabba1c7a602758c760f40682904d": "(S_1, S_2, S_3)", "8c3eccb99fe008f5e431d6bd0526ae74": "\\left|f\\left(\\frac{p}{q}\\right)\\right|\\geq\\frac{1}{q^d},", "8c3f03546b1147c30d9a242f0e510096": "\\begin{matrix}\\operatorname{Ta}(1)&=&2 &=& 1^3 + 1^3\\end{matrix}", "8c3f0a9af79ba7cad5eb4b526a793fc0": "\n\\mathcal{U}(0,{\\tilde{u}}) = \\{ {\\tilde{u}} \\}\n", "8c3f1fc8ae1bdc177621fbe35046ae63": " P_e = \\sum_{k=\\frac{n+1}{2}}^{n} \n{n \\choose k}\n\\epsilon^{k} (1-\\epsilon)^{(n-k)}", "8c3f3ea16b0158d5a1d8fc5d4e9ce762": "G_{\\mu \\nu} + \\Lambda g_{\\mu \\nu}= {8\\pi G\\over c^4} T_{\\mu \\nu}", "8c3f628e5f439f09eeb1ea6a0e8e107d": " \\left| \\lambda_{n+1} / \\lambda_1\\right|, ", "8c3f7deeaa41901324ab8f53f0357a05": " \\Delta N \\Delta \\phi \\geq 1 ~.", "8c3f80c93c8ca4b2bbc0ebe5bb7f2395": "x\\mapsto (x+1) \\cos(\\pi x)", "8c3fa079f6f01d074bb9af01d78432ee": " DK_j ", "8c401d24190952d806d8c81082fb0f40": "\\mathbf{x}=\\mathbf{D^{-1}}\\mathbf{X}", "8c402fe4aa1bcb4bf48c57eaab21eb47": " F_i ", "8c4054f1038b8ff0fc17b70f155b87cc": "?\\times 0=0.", "8c407a41fe4f69cb9b7496f25b9d2e29": "k= \\frac{\\gamma_2 - z_\\infty}{\\gamma_1 - z_\\infty}\n= \\frac{Z_\\infty - \\gamma_1}{Z_\\infty - \\gamma_2}\n= \\frac {a - c \\gamma_1}{a - c \\gamma_2},", "8c407a81b2043b4fb9bff83db0087198": "f_g", "8c40811974ea845df34cf10c45144c14": "\\overrightarrow{Y}=Y_{o} \\ \\ \\ \\ \\ \\ \\ \\ \\overleftarrow{Y}=Y_{\\varepsilon }\\frac{Y_{o}+jY_{\\varepsilon }tan(k_{x\\varepsilon }b)}{Y_{\\varepsilon }+jY_{o }tan(k_{x\\varepsilon }b)}", "8c40b9086945cb7189ca52f8e807e1e3": " \\sqrt{2/5} ", "8c40e8dbc457a11d26187c56dd8a2aac": "||f'||_{\\infty}", "8c4118ef13d92b5d5116267c7b87eaac": "x^{\\ast }(t)\\in X^{\\ast }(t)", "8c415f599f1bfc82db379bae2c5d6b21": "\\mathfrak{P}^{3}", "8c418795143ceb67a325bcb2fa4005ac": " x \\mapsto 2x ,", "8c41e1e9e677619f801453adf869e6d8": "\\lim_{h \\to 0}{ \\left({f(x(1+h))\\over{f(x)}}\\right)^{1\\over{h}} }=\\exp\\left(\\frac{x f'(x)}{f(x)}\\right)", "8c4292fcc5c9fc4bc52d460651ef4c64": " \\varphi(\\lambda) := \n\\begin{cases}\n\\lambda' : n - g_{i-1} = (\\ell + 3i -1) + (\\lambda_1 - 1) + \\dotsb + (\\lambda_\\ell - 1) &\\mbox{ if } \\ell+3i \\geq \\lambda_1\\\\\n\\\\\n\\lambda' : n - g_{i+1} = (\\lambda_2 + 1) + \\dotsb + (\\lambda_\\ell + 1) + \\underbrace{1+\\dotsb+1}_{\\lambda_1 - \\ell - 3i - 1} &\\mbox{ if } \\ell+3i < \\lambda_1\n\\end{cases}\n", "8c42985c7837ab0a30281510c2e2741a": "\n \\eta(\\xi) = \n \\eta_2 \n + \\left( \\eta_1 - \\eta_2 \\right)\\, \\operatorname{cn}^2 \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m \\end{array} \\right).\n", "8c42b0d40c5dd8a8bc8b3fce494c8188": "\\begin{array}{rcl}formula\\ mass & = & mass_{flour} \\times formula\\ percentage\\\\ \\frac{formula\\ mass}{formula\\ percentage} & = & mass_{flour}\\\\\\end{array}", "8c42c6b0a41ef7c375904c4a34b4d7e7": "\\sum_{k=0}^{m-1}f(x+k\\omega)", "8c42dbf5c794acb7358f47f67e58c92a": "\\phi_{P}\\,= \\phi_{e}", "8c42dea309670a01454949f7d4d4381f": "U(t+\\bigtriangleup t,w)=U(t,w)\\exp \\bigg(|\\frac{w}{w_h}|^{-\\gamma} \\frac{|w|\\bigtriangleup t}{2Q(w)}\\bigg) \\exp \\bigg( i|\\frac{w}{w_h}|^{-\\gamma} w\\bigtriangleup t\\bigg) \\quad (1.8.a)", "8c4303b5a0c612bdf7343a1893eb89ef": "\\lim_{k\\to+\\infty}f_k(x)=1.", "8c431a3d328ca6c5122e9e13dbb06421": "x_n y_m^{-1} \\in U", "8c4405c38ef9bd722977ae69124290ce": "\\int_{\\mathbf{R}^n} f(x)x_1^{i_1}\\ldots x_n^{i_n}\\, \\mathrm{d}x, ", "8c440ebc5516f7126345bd1abae75330": "c_j^{'}=0", "8c44266fbff16429056dd87ae229c7fd": "\nc = \\frac{Z}{(2 \\pi mkT)^{3/2}}", "8c4431fb726ba23ff4d826afe661754e": "F^{-1}=F^{\\dagger}", "8c4448a594db856394e75d75ad459fa0": "\\mathcal{L} = \\frac{-1}{4\\mu_0}F^{ab}F_{ab} + j^aA_a.", "8c4476cc096cc3015f5ada1b73397c05": "\\sigma = \\pi r_I^2", "8c44820572cf735fdc3595be220d72a0": "\n\\rho \\left( 1 \\right) =\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{bmatrix}\n\\qquad\n\\rho \\left( u \\right) =\n\\begin{bmatrix}\nu & 0 \\\\\n0 & 1 \\\\\n\\end{bmatrix}\n\\qquad\n\\rho \\left( u^2 \\right) =\n\\begin{bmatrix}\nu^2 & 0 \\\\\n0 & 1 \\\\\n\\end{bmatrix}.\n", "8c44dad8ecfc7537e9aace7241fa7e65": "\\Psi_{mn}", "8c44df5dd0e5a00cde2c5ef389296a4a": "f,g : M \\to N", "8c44e5d4b690d80b2e95f1ac5f9d1cd9": "\\sum_{i=1}^n (X_i-\\bar X)=0", "8c45235d61b79efa22861b28c13ad149": "I_1, I_2", "8c452f3e9afccc872e2000a4f800f63d": "g: M\\to [0,1]", "8c45db3dae62f4d0d2a00a5df7a43631": "\\lfloor n/k \\rfloor^{k-(n\\bmod k)}\\lfloor n/k+1 \\rfloor^{n\\bmod k}.", "8c45e518eaf08637b3d39f95a5ee0d1f": "\\Delta\\Delta G^\\ddagger", "8c45e5f0b094e2830c980adf93412348": "\\text{CR} \\in [0,1]", "8c45e65a873116f008bce083f0ad1dd4": "(x_1,...x_n)", "8c45f27208f64d116da22b4d3083939a": "\\otimes_{n=1}^{\\infty} \\psi_n", "8c46493fd1ae053d0ac3548729726dea": "\\bar{X} \\pm t_{n-1,0.975} S / \\sqrt{n}", "8c464cdccc038b1ea7010147b234295b": "\\Delta P = P_m - P_e", "8c4687f3342a96b8a5eefa4ce9bc703d": "2^{N-1}", "8c46d668c0dbc745a6fc5a9276046875": "B \\rightarrow c \\mid cd", "8c46e65a30bfa84404c070db32120d0f": "{\\rm for}\\quad \\left|x-1\\right| \\leq 1\\quad {\\rm unless}\\quad x = 0 \\,.", "8c470c1c45ea2cb0fa247944946a601b": "\\rho_c a^2 - \\rho a^2 = - \\frac{3kc^2}{8 \\pi G}", "8c472f9e1940cd7f91a06865b5a030e9": " 2d \\ ", "8c474c6051a8976ce9ebcd131800b09d": "K_q = L_d!/(tf_{t1,d}!tf_{t2,d}!...tf_{tM,d}!)", "8c475e252ea0a2c65200e1b1bbed7b7f": "d_{min}, d_{mean}", "8c47a8c08a8b4130b5b36a3d84470fb8": "\\lambda(\\rho^{\\prime}, \\theta^{\\prime})", "8c47b1c9970e0891ac533a6a8180af4b": "f(a,b,c) = g(h(c,a),h(a,b)) \\!", "8c47b872ca37e94374a3f1801101393c": "\\left |\\psi(t)\\right\\rangle = \\sum_{n=0}^{\\infty}c_{n}\\exp\\left(-i E_{n} t\\right)\\left |\\phi_{n}\\right\\rangle", "8c47be74bc6dae2c950a8658a6f0b803": "j + \\lfloor js/r \\rfloor = j + \\lfloor j(s - 1) \\rfloor = \\lfloor js \\rfloor.", "8c47c9a1896b9c0dd8190f20bfffb459": "\\rho(E)", "8c47cfb4c7c3de5135e825bfca7e8a21": "2 \\cdot 3 \\cdot 23", "8c4842f34221854bb4e1a3273f4d2680": "X_1 \\text{ and } X_2", "8c4867f5b9c4f2ec8e38c156a9e838c8": "(A\\equiv(B\\equiv C))\\equiv(C\\equiv(A\\equiv B))", "8c489d0946f66d17d73f26366a4bf620": "Weight", "8c48a8195c243bb06c932ab3b0dde8c7": "\\,f \\colon X \\times Y \\rightarrow Z", "8c48a984d813438277d99eeac41f8f57": "\\sum_{j=1}^{N(t)} w_{j,t} = 1", "8c48ca49cdf2e47e5bbcebc8b41f5b2a": "\n x_1 \\begin{bmatrix}a_{11}\\\\a_{21}\\\\ \\vdots \\\\a_{m1}\\end{bmatrix} +\n x_2 \\begin{bmatrix}a_{12}\\\\a_{22}\\\\ \\vdots \\\\a_{m2}\\end{bmatrix} +\n \\cdots +\n x_n \\begin{bmatrix}a_{1n}\\\\a_{2n}\\\\ \\vdots \\\\a_{mn}\\end{bmatrix}\n =\n \\begin{bmatrix}b_1\\\\b_2\\\\ \\vdots \\\\b_m\\end{bmatrix}\n", "8c48d8dd1205f016edc12c265987872c": "R/P^i", "8c48e93d4a15683e3f2818e854565e6d": "M^{N+1}{\\cdot}(N-2)", "8c48f0040b7b3e3019e67b158f3a7cb3": "+ \\frac{10,000}{510,260} log_2\\left(\\frac{10,000/510,260}{510,000/510,260 * 10,060/510,260}\\right)", "8c49145544fca24efb8de07eb1275c09": "w_{i} = \\frac{1}{p'_{n}(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx", "8c496a7539733ba959ffe798d7d1eee7": "\\displaystyle{[X,X^2]=aI,}", "8c49c79a7fb6fc4b12a982eb5e406346": "\\omega^2\\phi+\\vec{\\partial^2}\\phi-g\\sigma^2\\phi=0 ", "8c49e5864eabec30e707db225542398d": "\\|\\ \\| _{2} ", "8c4a60d473be7bba3a3fd4f442baf705": "|z|=a", "8c4abf4c548dc6393abfc932251cdf7e": "\\Delta(a_n)", "8c4adcd8d425afe79d9b9feab198ebb9": " U_0(x) = 1 \\,", "8c4b273c6508c821146d86cca24f11cd": "\\mathbf{v}_2=(v_2,\\underbrace{0,\\cdots,0}_{n-1})\\,\\!,", "8c4b464f991a18f45b06c67fa4039f74": "f(t_{i+1},\\tilde{y}_{i+1})", "8c4cda170a645203fd226e17746edda6": "f(x)=\\begin{cases} g(x)&\\text{if }g(x) < \\infty\\\\0&\\text{otherwise,}\\end{cases}", "8c4d3fd558b862cc9926705adb841f09": "I_{abs} + k \\left( \\frac{\\partial T}{\\partial z} + r \\frac{\\partial T}{\\partial r} \\right) + \\rho_l \\nu_i L_v - \\rho_v \\nu_v (c_p T_i + E_v) = 0", "8c4e3d3ec57c3a103e2ba6aaa6dd3fac": "\\sqrt{ - 1}\\,^{\\sqrt{ - 1}}=e^{ - \\frac{1}{2} \\pi}", "8c4e473aba41d931123f271cf0a2de3d": "O(\\log^3 n\\log\\log n \\log\\log\\log n)", "8c4e5a89ef7be5b52cc85f235b2912a3": " \\vec a_1 ", "8c4e90874802bd5fbddd657fefe0d59b": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "8c4eca512051289ee7937ce29c225876": "\\mathbf{v} (t) =v(t) \\frac {\\mathbf{v}(t)}{v(t)} = v(t) \\mathbf{u}_\\mathrm{t}(t) , ", "8c4ee139c1468c5b3955de79728d5385": "t_\\lambda", "8c4f1b0b47817c6a5a45bb641bcd97c8": "p_1/P_2", "8c4f1e830be7a88068671f94a67151c5": " l_z = \\hbar \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right ). ", "8c4f3a67d4ee53d52ff17d050a6b3c8b": "d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\\!", "8c4f4a9d81d022188476d9010ea6e6ea": "p(\\mathbf{x}, t)", "8c4f6240b7dc948a535afd091a0f28e0": "\\delta\\psi = \\left(\\frac{\\delta p}{2p} + i \\delta \\alpha\\right) \\psi", "8c4f71d4ccdae36d75c3539bd05ef5ef": "U_{2n} = U_n V_n \\,", "8c4f82ab4777f195dd82d3fab88ec389": "a(t) = (1-d\\cdot t)", "8c4f8a4749d8db347b1b23c70948f729": "m=(m_0, \\dots, m_{L-1})", "8c4fbf6039679ea7dd7eb06c2b8a90bf": "\\sum_{n=0}^{\\infty}u_{n}=\\sum_{n=0}^{\\infty}\\sum_{k=1}^{m}\\frac{a_{k}}{(n+b_{k})^{r_{k}}}=\\sum_{k=1}^{m}\\frac{(-1)^{r_{k}}}{(r_{k}-1)!}a_{k}\\psi^{(r_{k}-1)}(b_{k}),", "8c4ff2a4abf42f02ea7a193e462eeb51": "C= \\frac{\\varepsilon A}{d}", "8c4ffa72f5b179879f1e2be56af17041": " (1 - \\delta_X(t))^{n-1} < t / 2, ", "8c508a0140c9e0a5ffe517cc6e19589d": "p V = n R T\\,\\!", "8c50c55e90e2c73c982604fff26d75ba": "H_q^{-1}(\\frac{1}{2}-\\varepsilon)", "8c51798399ab31fbf37464a366eccda3": "\\sqrt{1- v^2 / c^2}", "8c51ac7b364c0075984ba73080f9177d": "\\|\\mathbf{v}\\|^2 = \\sum_{k=1}^3 \\|\\mathbf{v}_k\\|^2.", "8c5260c272213750c6063d23392002a7": "\\frac{U_e} {A_0 L_0} = \\frac {E {\\Delta L}^2} {2 L_0^2} = \\frac {1} {2} E {\\varepsilon}^2", "8c531cc06711a4251138f6360dcc2418": "\\Delta>>P", "8c53352b3311298a453454b666cba4af": " \\sum_{j=1}^{n_{P}} W_{j}\\frac{\\operatorname{d}Z_{j}}{\\operatorname{d}t} = \\sum_{k=1}^{n_{f}} W_{k} f_{k} + \\sum_{j=1}^{n_{P}} w_{j} p_{j} + \\sum_{j=1}^{n_{t}} w_{j} t_{j},\\quad j=1,\\dots,n_{p}+n_{t} ", "8c537977076365c9314c45d704cbb4e9": "E[X_t] = \\mu_t", "8c5383408844dc5de8215bcd220a4699": "\\scriptstyle \\left(1 \\,-\\, \\frac{1}{e}\\right)", "8c540f283c3b1aadf25a1632e04e910b": "A^\\nu = (\\phi, \\mathbf{A})", "8c546a3c39fc720a0c55c33e04f7dc3f": "{\\rm tr}(d\\mathbf{X})", "8c546b319dbdcd6a8901adaa9162605c": "P_c^n(0)", "8c5549b55433ae024e24cbca0a9eb5da": " P_a + P_m + P_d=1 ", "8c55b1dc2932f41c103ee55968f37ba6": "AIC = \\chi^2 + 2k", "8c55c465bc1a74219658a87dd9a38112": " | \\psi \\rangle \\to | \\phi \\rangle \\,", "8c55d713dd80a18e5c5f81c223e4bbd2": "= \\frac{- g'(x)}{(g(x))^2}.", "8c55e53074cfd8b0616d84ba525aa6b2": "\nx + iy = a \\ \\cosh(\\mu + i\\nu)\n", "8c55ef5d86124afa4ec730fd8e4aea52": "\\lnot \\exists x Sx", "8c55f8f6bf8017731c9ab4c0350bd5f2": " E_b: L^2(R)\\rightarrow L^2(R), (E_bf)(x)=e^{2\\pi ibx}f(x)", "8c56212cfc8cae690e15878208f7627a": "{}- e^4 \\left( \\frac{ (\\bar{v}_{k} \\gamma^\\mu v_{k'} )( \\bar{u}_{p'} \\gamma_\\mu u_p)}{(k-k')^2} \\right) \\left( \\frac{ (\\bar{v}_{k} \\gamma^\\nu u_p )( \\bar{u}_{p'} \\gamma_\\nu v_{k'}) }{(k+p)^2} \\right)^* \\,", "8c5674882f18bbc4e13c1ebdabaff9a6": "A\\ =\\ C", "8c56754710091f68d65de51095b8491a": "ax^3+bx+c=0", "8c56a78bff16d9b47d2d58aa5b2eac8f": "A^{+} = \\sqrt{\\frac{2}{3}}\\ \\sigma^{+} ", "8c5704cd837c7d149777b4d7c1acb0b5": "\\boldsymbol{\\lambda}=(\\lambda_r)", "8c5735d1f8b43afc6f6cb39d871af219": "\\hat\\ell(\\!x_0,\\gamma|\\,x_1,\\dotsc,x_n) = n \\log (\\gamma) - \\sum_{i=1}^n (\\log [(\\gamma)^2 + (x_i - \\!x_0)^2]) - n \\log (\\pi)", "8c575343df6e788478b986275639ea66": "y(x) = a\\times\\csc\\left(\\frac{2\\pi}{p}x\\right)\\left\\vert\\sin\\left(\\frac{2\\pi}{p}x\\right)\\right\\vert", "8c57717c0078edc5002b55a5d1e13a5f": "\\textit{on}(0) \\leftrightarrow \\textit{on}(1)", "8c5784b27794103ee67ee4abde1db7d4": "\\mathbf p = \\{p_{1}, \\ldots, p_{n}\\}", "8c581b3df27c47863ab39f3144c233a4": " \\phi(t+T)=\\phi(t) \\phi^{-1}(0) \\phi (T).\\ ", "8c58480c9529634bdad5cf85f4181085": "W = Q \\int_{a}^{b} \\mathbf{E} \\cdot \\, d \\mathbf{r} = Q \\int_{a}^{b} \\frac{\\mathbf{F_E}}{q} \\cdot \\, d \\mathbf{r}= \\int_{a}^{b} \\mathbf{F_E} \\cdot \\, d \\mathbf{r}", "8c58563fe2e21fdde50fab1de418f777": "T(A,B) = {A \\cdot B \\over \\|A\\|^2 +\\|B\\|^2 - A \\cdot B}", "8c58cb0e160eb2ab418b328649dc2540": " \\mathbf v_i ", "8c58d343d0084c192109be2e42ca3a14": " \\operatorname{tr} \\mathbf{M}_{\\mathbf{X}_1 + \\mathbf{X}_2}(\\theta) \\leq \\operatorname{tr} \\left [ \\left ( \\operatorname{E} e^{\\theta \\mathbf{X}_1} \\right )\n\\left ( \\operatorname{E} e^{\\theta \\mathbf{X}_2} \\right ) \\right ] = \n\\operatorname{tr} \\mathbf{M}_{\\mathbf{X}_1} (\\theta) \\mathbf{M}_{\\mathbf{X}_2}(\\theta) ", "8c59271aacb66e1c5de4b021c18f8f81": "y'=-\\infty", "8c593b245dc3de96357975b5e136eddc": "\\mathbf{1}_{(-\\infty, \\lambda]}", "8c593c3d9bb03cae903183c0dacac0b6": " \\|f^{(k)}\\|_{L_\\infty(T)} \\le C(n, k, T) {\\|f\\|_{L_\\infty(T)}}^{1-k/n} {\\|f^{(n)}\\|_{L_\\infty(T)}}^{k/n} \\text{ for } 1\\le k < n.", "8c59a76e9da0ea31498450c9a6d522d0": "{43\\choose 6-n}", "8c59b5eebb345a16ca3ccc2570eec0a7": "\\scriptstyle{U(\\vec{r})}", "8c59c882165c93f6873272b4b196df39": "\n\\frac{V_{out}}{I_{in}} = \\frac{R}{1+sRC}\n", "8c59e1f34407876525ea0789d5bf15b0": " 50-4Q=0", "8c5a225e66b74fe4b9b70cf6904ee85d": " \\sum_{j=0}^n (-1)^j\\tbinom n j P(m+(n-j)d) = d^n n! a_n", "8c5a2aa1aeb768e11ffa797064ad1267": "\\pi\\;", "8c5a341a67b803cb46ed1e32796530c7": "G(1;x)=\\sum_{n=0}^{\\infty} x^n = \\frac{1}{1-x}", "8c5a9923deb65fa9bbe875f52f3463ba": "V/Hz^{1/2}", "8c5b0ee670b0eb64c7bba4e2d1f913af": "\\mbox{Debt ratio} = \\frac {\\mbox{Total Liability}} {\\mbox{Total Assets}}", "8c5b3b1841f76cf29d602524bc32b16f": "{10 \\choose 1}{4 \\choose 1}^5 - {10 \\choose 1}{4 \\choose 1}", "8c5baa2f38039458bd0d09984032057e": "\\mathbf{\\nabla} \\cdot \\mathbf{D} = \\rho_\\text{free}", "8c5bb2376806adb3d5f8a245343b96b4": "\\C\\cup \\{\\infty\\}", "8c5bba142fe47a939330053a200521a1": "\\langle\\phi(k) \\phi(k')\\rangle = \\delta(k-k') {1\\over 2(d - \\cos(k_1) + \\cos(k_2) ... + \\cos(k_d)) }", "8c5bdf8650569353574739498c9b1621": "{y_1^2 \\over 2} - {y_2^2 \\over 2} - {F_d \\over \\gamma} = {\\rho \\times q \\times (v_2 - v_1) \\over \\gamma}", "8c5be8754c1a91acbb4bb14c67ace60f": "\n \\mathbf{v} \\times \\mathbf{c} = e_{ijk}~v_j~c_k~\\mathbf{e}_i\n ", "8c5c1b7958250e00dcfc3ef4b814a686": "\\textstyle\\frac12", "8c5c2338543e949a41c98887a3a050b1": " P(X) \\in F_q[X] ", "8c5c797a0d006e78519881f04a93391e": "(x_0,y_0)\\,", "8c5ca5890399cb8b9011a2400caea7b0": "\\scriptstyle \\lambda_1,\\, \\ldots,\\, \\lambda_r", "8c5caf9b396e5d92e38cf95c791ce26e": "\\varphi : P \\to G", "8c5ceccea81fcc0a4f3c3a7636b37238": "h^{-1}\\,", "8c5d146c539aad52e7bb9ba8f6fdf140": "V \\,=\\,\n\\left(\\mathbf{e}_1;\\mathbf{e}_2;\\mathbf{e}_3\\right)\\,=\\,\n\\mathbf{e}_1\\cdot(\\mathbf{e}_2\\times\\mathbf{e}_3) \\,=\\,\n\\mathbf{e}_2\\cdot(\\mathbf{e}_3\\times\\mathbf{e}_1) \\,=\\,\n\\mathbf{e}_3\\cdot(\\mathbf{e}_1\\times\\mathbf{e}_2)\n", "8c5d5026473b710e2ade5b17ca35ca6c": "r_3 = (S \\to a, \\{r_3\\}, \\emptyset)", "8c5d629de7354c4bd4ec2f0a0a446a89": "P(i,j)=\\frac{|U_i \\cap V_j|}{N}", "8c5dd0adcbf8a16b54c8b634c52d8d5d": "P^{2m}(R)\\geq\\frac{P^{m}(V)}{2}\\,\\!", "8c5e4ffe093a0f3a72e037cd70843883": "\\scriptstyle \\hat F_n(t)", "8c5e7bf4805a0b077a7ec163639ba588": "x'=-{1 \\over x}.", "8c5e9fa099443ecfb48738d59aeec553": "H_q \\cong C_2 * C_q,", "8c5ea68f4ef0de8793799c24eccb2eab": "\\omega_g", "8c5ecac14baf8e2ff09c1d341e37ecfa": "2\\otimes\\overline{2}=3\\oplus 1.\\ ", "8c5edbaf7beac7904bbf1c7405fd6110": "Q_{\\alpha}", "8c5eecbd51895d1ec08c784d22cf4f17": " \\frac{dy}{dx}\\,\\cdot\\,\\frac{dx}{dy} = 2x \\cdot\\frac{1}{2x} = 1. ", "8c5f07909c6bdb0e0dc4a3d079d70533": "\\Delta G = \\Delta G^0 + \\sum_{p}^{}{(T_i-T^0)(-\\Delta S^0_i)} + \\sum_{n}^{}{(T_i-T^0)(-\\Delta S^0_i)} ", "8c5f44b90082219bd13ee1dc049bdd91": "N > 3", "8c5f608952fc1d735af9d3a293212519": "\\,j", "8c5f7292cd4847de9730795b92a34e57": "a\\in{\\mathrm S\\mathrm O}(n)\\,.", "8c5fe65e70ee9dd8d3947376e1e5851e": "n^{\\star} = J_{date} - 2451545.0009 - \\dfrac{l_w}{360^\\circ}", "8c603b9465ce914723458a536316fea6": "path_{max}", "8c6056572a557dfe8e78639f56ca0b81": "F_{net} = m a = m g - {1 \\over 2} \\rho v^2 A C_\\mathrm{d}", "8c606e49c1c76be6f5c6db6ba19aa0ca": "pH = -log(x)", "8c608e44d539ced63c1c2570fa30f5b6": "(Z, g_{\\boldsymbol\\theta})", "8c60d2e74f149123ecf64e3d19188abf": "\\bigoplus_i T_i", "8c60dd390cb36cfa612684da1228f47f": "\\mathrm{Ar{-}COOOH\\ +\\ Ar{-}CHO\\longrightarrow\\ 2\\ Ar{-}COOH}", "8c6126d2020ae15a5b4aa2e0f01a811b": "i^* \\theta_1(\\gamma^1) = \\theta_1(i^* \\gamma^1) = \\theta_1(\\gamma_1^1) = w_1(\\gamma_1^1) = w_1(i^* \\gamma^1) = i^* w_1(\\gamma^1)", "8c6183462a52072964c878046d02d90a": "M_{\\ell m}", "8c6246842b723b17ce47d80983496ec5": "V_\\mathrm {dc}=V_\\mathrm {av}=\\frac{3{\\sqrt 3}V_\\mathrm {peak}}{\\pi} \\cos \\alpha", "8c62bf8c45bd7ad7e9bac7420f998dbc": "~ \\left ( {\\partial T\\over \\partial V} \\right )_{p,N} \n= -\\left ( {\\partial p\\over \\partial S} \\right )_{T,N} ~", "8c637d343f82d6af3d9356baa6364792": "\\varphi \\leftrightarrow \\psi\\,\\!", "8c63f5802564ab8758ebbdd6d5d1212c": "\n\\begin{align}\nU(\\rho,z)&=2 \\pi a \\int_0^{W/2} J_0(2 \\pi \\rho' \\rho/\\lambda z) \\rho' d \\rho'\n\n\\end{align}\n", "8c64058e86fb4bc38994e23db664e398": "\\left[\\mathcal{A},\\mu\\right]", "8c64430c782cd21f961d75f57db82d76": "c_g = \\frac{\\Lambda_g}{\\tau_g}.", "8c645b9da4e4ad85defa9f6c7dd11b8c": "\\text{Cost at A} = \\text{Cost at B}\\cdot \\frac{\\text{index at A }}{\\text{index at B }}", "8c64bb39f01d8493e7fd984a22af353d": "k^*:T_P(Y) \\rarr T_{f^{-1}P}(X)", "8c64dbe3ae3e3684c2d04a9c073d5b72": "\\rho=|\\vec S|/Mc", "8c64f24aefd655fcff8ff35dbecbc939": "R(\\theta, \\delta) = \\mathbb{E}_\\theta L\\big( \\theta, \\delta(X) \\big) = \\int_X L\\big( \\theta, \\delta(x) \\big) \\, \\operatorname{d} P_\\theta (x) .", "8c64f88b0a3d78b101f9e5fea1889ed2": "nCF", "8c652394f4a63e0a8c5fe780b99f98d7": "\\frac{|v-c|}{c}<4\\times10^{-5}", "8c65c63f7e0718506624455663bda664": " \\vec \\sigma (\\phi, \\theta, t) = (t s \\cos \\theta \\cos \\phi, t s \\cos \\theta \\sin \\phi, t s \\sin \\theta, t) ", "8c66680402f2af0d982592ed69752f29": "={\\Delta C \\over{\\Delta F}} \\,", "8c6669940878da5a08601689020afc47": "q(x_1)q(x_2) \\cdots q(x_n)", "8c66b56ecb5ac2209a7a128638955942": "n > 0 \\Rightarrow q = \\left\\lfloor \\frac{a}{n} \\right\\rfloor", "8c66e70b3dec533324da2f92ec5b09a0": "\\lim_{n\\rightarrow \\infty} \\frac{c_n}{c_{n+1}} = r.", "8c66f90ddf179d1aea0cca3ef9404500": "\\nu^2 = \\frac{K}{1+K}\\Omega", "8c673bcf1a3a30fb1b3b0c419c7f4f7a": "\\gamma^5=i \\gamma^1 \\gamma^2 \\gamma^3 \\gamma^4 = \\begin{pmatrix} I_2 & 0 \\\\ 0 & -I_2 \\end{pmatrix}. ", "8c676c87c2cf5ce4829c54e29b91c60e": "s \\leftarrow p.", "8c679e9670ff02ee570a67c334e37ac7": "t^\\ast", "8c67caa575ee563cc207da50f58be9d8": " 3 \\times 4 ", "8c67e5a86dc7b7ec9cddb166b47b9ca8": "M_\\text{s}", "8c6822f59c0ff1778c22d415ccaab88e": "\\scriptstyle{\\underline{\\underline{Q}}}", "8c686a66b78fe310d024130badfd3f28": " 2 \\psi = \\theta, \\,", "8c68a973fcc5e34d737707e66ef0a88f": "\\rho = \\Omega^\\Omega 2 + \\Omega^{\\psi(\\Omega^\\Omega 3)}", "8c68d606c7cf77d6fc0debc539c72c35": " h_{ii}=h_{ii}^2+\\sum_{i\\neq j}h_{ij}^2 \\geq 0 ", "8c690f509ed862767fff5d7f973c8c35": "g^{\\prime}(x):=\\frac{1}{3}\\sum_{j=0}^\\infty \\frac{a_j}{\\sqrt[3]{(x-q_j)^2}}>0,", "8c69710badf4d59edcfab46d9c407e19": "\\{j_1,j_2,j,m\\}", "8c6a23a60783d354a48fa059a29c9b5b": "cY(t)", "8c6a3e5efd2625b17e56fb07b426169a": " \\acute{F}^{\\mu \\nu} = {\\Lambda^{\\mu}}_{\\alpha} {\\Lambda^{\\nu}}_{\\beta} F^{\\alpha \\beta} \\, ,", "8c6a5deb6fa4ecdd0ec0a29fa7a5fa34": "\n\\frac{\\partial \\Gamma (s,x) }{\\partial s} = \\ln x \\Gamma (s,x) + x\\,T(3,s,x)\n", "8c6ad1c9cdb5b5b32d87eb95354e7ff0": "\\alpha = \\alpha' + \\alpha''", "8c6b0a2d53ae74453a07e2759b545bc1": "< 0.08M_\\odot", "8c6b2d78ef8c61c594753c930b8c5f22": "T\\approx A^{-1}", "8c6b9f195a6255a25fb6ffd82322e663": "\n\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1\n", "8c6bb67f936472ce7b354b14df94d4d4": " T(r,f) = N(r,a,f)+m(r,a,f) + O(1),\\,", "8c6c82bb123666ef0156f8004671f581": "\n\\langle \\mathfrak{p}\\rangle = 0.\n", "8c6ca45ce6b4fa539286b5a9d0c98408": "\n C_{QP}=q_0\\Delta{t}-k_0\\Delta{x}=q_0(t_P-t_Q)-k_0(x_P-x_Q) \\qquad (5)\n", "8c6cab1bc397becc1dfea0c668202944": "f:\\mathbb{Z}_m^n \\to \\mathbb{Z}_m", "8c6ce0381218cce10d9592ae7068a5dd": "k_{cat}/K_{M}", "8c6cfd47448ab526ae5c431733df20f8": " E_n = -{1\\over 2} {{e_M}^2\\over r} = -{1\\over 2} \\left ( {m{e_M}^4\\over \\hbar^2} \\right ) {1\\over n^2} ", "8c6d64f70133b965b3e7f23c6b53c60b": "\n\\begin{array}{l}\n\\displaystyle\nF_Z^{\\text{GNIG}} (z|r_1,\\ldots,r_p,r;\\,\\lambda_1,\\ldots,\\lambda_p,\\lambda) = \\frac{\\lambda ^r \\,{z^r }}{{\\Gamma (r+1)}}{}_1F_1 (r,r+1, - \\lambda z)\\\\[12pt]\n\\quad\\quad \\displaystyle - K\\lambda ^r \\sum\\limits_{j = 1}^p {e^{ - \\lambda _j z} } \\sum\\limits_{k = 1}^{r_j } {c_{j,k}^* } \\sum\\limits_{i = 0}^{k - 1} {\\frac{{z^{r + i} \\lambda _j^i }}{{\\Gamma (r+1+i)}}} {}_1F_1 (r,r+1+i, - (\\lambda - \\lambda _j )z) ~~~~ (z>0)\n\\end{array}\n", "8c6e33c7cc097e0b9ad1322fb4c00e96": "\\pi^{-1} \\mathcal{I} \\cdot \\mathcal{O}_{\\tilde{X}}", "8c6e36e355f6e7b1130d8bf18f2eb742": " \\vee ", "8c6ea4bbfb84735376a86c227ffec122": "\nP(\\mathbf{Y}|\\mathbf{X},f) = \\mathcal{N}(f(\\mathbf{X}),\\sigma^2 \\mathbf{I}) \\propto \\exp\\left(-\\frac{1}{\\sigma^2} \\| f_{\\mathbf{w}}(\\mathbf{X}) - \\mathbf{Y} \\|^2\\right),\n", "8c6edd760ee8c0f01cc5945bff56bcfc": " O\\left(\\frac{(\\log N)^s}{N}\\right) ", "8c6eeb9d794937840bcf6a27fc52bb87": "A_m", "8c6ef8324c56e9c26afc8a5617dda227": "J_\\mu^{em}", "8c6f7a44c8da3281096a74a8ee105fb9": "J=\\tfrac{1}{2} \\textbf{x}^{\\text{T}}(t_f)\\textbf{S}_f\\textbf{x}(t_f) + \\tfrac{1}{2} \\int_{t_0}^{t_f} [\\,\\textbf{x}^{\\text{T}}(t)\\textbf{Q}(t)\\textbf{x}(t) + \\textbf{u}^{\\text{T}}(t)\\textbf{R}(t)\\textbf{u}(t)\\,]\\, \\operatorname{d}t", "8c6fded627df68b06775acecde3fd702": "\n\\boldsymbol{\\sigma}^{dev} = \\boldsymbol{\\sigma} - \\frac{1}{3} \\left(\\mbox{tr} \\ \\boldsymbol{\\sigma} \\right) \\mathbf{I}\n\\,\\!", "8c7021a4abd14613ede1c99e95f22c91": " \\frac{1}{2} + i \\frac{ 2\\pi n}{\\log n}. ", "8c70756b4a94c24c165bac9aaa76a243": "a_nx^n + a_{n-1}x^{n-1} +\\cdots + a_1 x+ a_0 = a_n(x-x_1)(x-x_2)\\cdots (x-x_n)", "8c70877ef748a6634a541d2731473543": "\\mathbb{E}[|T_j|] = \\frac{t}{d}", "8c70ce348ebcc7ba9644748513711d8e": "\\frac{b+ c - a}{a} : \\frac{c + a-b}{b} : \\frac{a+b-c}{c}", "8c70fefe8c715ee049de76d703e4586d": " \\!\\ \\delta_S^5 = 29\\delta_S + 12 = [82;82,82,82,\\dots] \\approx 82.01219 ", "8c7218f4789552016e2c6563ea440fe3": "l \\equiv 0 \\pmod{4}", "8c721bb776047975617f979369fc1f17": " = \\int\\!\\frac{dt}{t} = \\ln|t|+C = \\ln|x+\\sqrt{x^2+c}|+C", "8c724a34910a14b0fdef5595da4160ca": " x^{(1)}= Tx^{(0)}+C ", "8c72712bb38082f32f797def2f867148": "\\delta_\\xi", "8c7378153a7d1cd36e0e5da7700ea9a9": "\\operatorname{ch}(V) = \\frac{\\sum_{w\\in W} \\varepsilon(w) \\xi_{w(\\lambda+\\rho)-\\rho}}{\\prod_{\\alpha \\in \\Delta^{+}}(1-\\xi_{-\\alpha})}", "8c737c44446e6045876750ccb0b40c61": "{\\textit{VAR}_\\text{err} = SS_\\text{err}/n}", "8c739d234693673243908b7db70feb6f": " k_+ \\left\\{ A \\right\\}^\\alpha \\left\\{B \\right\\}^\\beta = k_{-} \\left\\{S \\right\\}^\\sigma\\left\\{T \\right\\}^\\tau \\,", "8c73eb0e3525c5cb4ab7fca21ac3a67d": "b(0)\\geq 0", "8c73f25e90ba16e6f19d68dea3007106": " T = {2h_{0}cos(i_{c_{0}}) \\over V_{0}} + {X \\over V_{1}} = T0_{1} + {X \\over V_{1}}", "8c743befc76f1eabb7e03e6ac1657ad7": "S_n = B", "8c7441ee2a48b3cdf7689ecb8c449e08": "(\\neg p \\land p) \\to q", "8c746d4641ebf2027519656c1c888698": "\n \\frac{1}{r}\\cfrac{d }{d r}\\left[r \\cfrac{d }{d r}\\left\\{\\frac{1}{r}\\cfrac{d }{d r}\\left(r \\cfrac{d w}{d r}\\right)\\right\\}\\right] = -\\frac{q}{D}\\,.\n", "8c74badb2b8173911cf0e764f7242f3d": "\\lim_{r \\to 0} \\frac{\\gamma \\big( M \\cap B_{r} (x) \\big)}{\\gamma \\big( B_{r} (x) \\big)} = 1.", "8c74c1d644f09b87ef2f69cc52dd528a": "\n\\begin{align}\na^{(\\mu)}_\\mathbf{k}(t)\\, &\\rightarrow\\, \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}}\\, a^{(\\mu)}(\\mathbf{k}) \\\\\n\\bar{a}^{(\\mu)}_\\mathbf{k}(t)\\, &\\rightarrow\\, \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}}\\, {a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\\\\n\\end{align}\n", "8c74c8206666b5fd522ae85c19836ecc": "\\frac{d}{dx}g(x)=\\frac{d}{dx} \\left(\\sum_{i=1}^{k-1} f_i(x)\\right)=\\sum_{i=1}^{k-1} \\frac{d}{dx}f_i(x)", "8c750145c45dc4a267f2f112577f939e": "\\alpha_n < 0", "8c75181c22c57ed7b35dd9710a01aa75": "-V_{\\rm peak}", "8c757bbcc6c31abee0d88314e9ab8e3f": "\\forall x:~P(x) \\to Q(x)", "8c757dd749aed9b7f4ccafd2ee2cba31": "\\frac{a_0 + a_1 x^1 + \\cdots + a_{d-1}x^{d-1}}{1-x^d} = \\left(a_0 + a_1 x^1 + \\cdots + a_{d-1}x^{d-1}\\right) + \\left(a_0 + a_1 x^1 + \\cdots + a_{d-1}x^{d-1}\\right)x^d + \\left(a_0 + a_1 x^1 + \\cdots + a_{d-1}x^{d-1}\\right)x^{2d} + \\cdots.", "8c76039635d1288254f6446a68b50095": " R_P(t) = \\frac{ 15 \\ {_2^1}P(t) }{ 14 \\ {_2^1}P(t) + 29 \\ {_2^0}P(t) }", "8c7620ddb9fb2966cfe3a90ae2295371": " \\left ( \\frac{P(X_1^n(i'))}{P(X_1^n(i))} \\right ) ^s \\geq 0 \\, ", "8c762ccd1e7cf8d1737d809061711474": "\\frac {u'} {f} = \\frac {u} {f} \\frac {1} {\\cos \\theta}\n = \\frac {m + 1} {m} \\frac {1} {\\cos \\theta} \\,,", "8c7632a12d695bd2146633117113cf5d": "\ng^v\n= c_0 c_1^i c_2^{i^2} \\cdots c_t^{i^t}\n= \\prod_{j=0}^t c_j^{i^j}\n= \\prod_{j=0}^t g^{a_j i^j}\n= g^{\\sum_{j=0}^t a_j i^j}\n= g^{p(i)}\n", "8c7642cca7f987ac5b0aed1de8b80990": "\\left \\lfloor \\frac{i-1}{k} \\right \\rfloor", "8c765ca1a59cc0b6eee93a5a0ce96c70": "{E} = {V_1^2-\\frac{V_1^2}{2}-\\frac{U^2}{2}+\\frac{2UV_1\\cos\\alpha_1}{2}}", "8c7690ab3c4084efd52fede8e26152cf": "S\\cdot~A_{x+t:\\begin{smallmatrix}\\hline~n-t|\\end{smallmatrix}} - NP_{x:\\begin{smallmatrix}\\hline~n|\\end{smallmatrix}} a_{x+t:\\begin{smallmatrix}\\hline~n-t|\\end{smallmatrix}}", "8c76e20444331b02e19be15ca9495cbd": " \\underbrace{\\forall x \\in \\mathbb{R} \\, \\, \\forall \\epsilon >0} \\, \\exists \\delta > 0 \\, \\forall h \\in \\mathbb{R} \\, \\left( \\, |h| < \\delta \\, \\to \\, |f(x) - f(x+h)| < \\epsilon \\, \\right) ", "8c770666b8fdac44d8dcfbd41628b84a": "\\dot{Q}''", "8c77c30b1b27cdcd43fa62e8764c31ab": "N_b", "8c780aec90579a1188eb5310090c946d": "\\widehat{a}_j\\rho \\rightarrow \\alpha_j P(\\mathbf{\\alpha},\\mathbf{\\alpha}^*).", "8c78444ef850b9377f55243dc392e1a8": "0> \\sigma ", "8c78d7ca127c44638411d1c6e14fa137": "\\Gamma^\\infty (E)", "8c79256a202f095c128472f5840214d4": "2^{n-2}", "8c7933f713c8667464a701a7b6879ed4": "c \\phi(x)", "8c795c0bc4b184c48edbf5a50b3fb784": "\n \\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}}{\\mathrm{d}t} = \\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}_{\\mathrm{e}}}{\\mathrm{d}t} + \\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}_{\\mathrm{vp}}}{\\mathrm{d}t} ~.\n ", "8c7980b58b068bee856fecb0d03b359f": "\\mathcal{L}(X,Y)", "8c7980d4b33899bb369c33326f3d4acc": " F=\\sum_{i,j=1,2}\\frac{1}{2m} |(\\nabla - ie A) \\psi_i|^2 + \\alpha_i |\\psi_i|^2 + \\beta_i|\\psi_i|^4 - \\eta( \\psi_1\\psi_2^* + \\psi_1^*\\psi_2)\n+ \\gamma [(\\nabla - ie A) \\psi_1 \\cdot (\\nabla + ie A) \\psi_2^* + (\\nabla + ie A) \\psi_1^* \\cdot (\\nabla - ie A) \\psi_2] + \\nu |\\psi_1|^2|\\psi_2|^2 +\\frac{1}{2}(\\nabla \\times A)^2 ", "8c79a89bf719c9556473c9c786ef21fd": " \\mathbf{Gr}(r, \\mathcal E)(k)", "8c79ac82787da02453245af254bf277c": " \\R^n", "8c79e059f4d2712c770f44f25f84d429": " \\cot \\theta = \\frac {\\csc \\theta}{\\sec \\theta}.", "8c7a213453d7e4e8268152fb35e101cd": "a_1+a_2+a_3+\\cdots", "8c7a2a75a22c3ccff9fddcef89fb8019": "{-L, \\dots ... ,L}", "8c7a3007b523e0d80a81c661e5f1edb2": "q_p(ab)\\equiv q_p(a)+q_p(b) \\pmod{p}", "8c7a407164765e1094f85c802a8866b5": "\\mu_k(s)", "8c7aab243266b42aed273c6e3f675026": "e_A", "8c7ab8b528a74c2133a30773c5b8d156": "\\tilde R_{ij} = R_{ij} - (n-2)\\left[ \\nabla_i\\partial_j \\varphi - (\\partial_i \\varphi)(\\partial_j \\varphi) \\right] + \\left( \\triangle \\varphi - (n-2)\\|\\nabla \\varphi\\|^2 \\right)g_{ij} ", "8c7b04cc2beb8e2bf385dd4e5409a2a5": "U_0 = U_0(\\boldsymbol{\\epsilon})", "8c7b3e5ac38c6e359042ce97931bbe55": "\\operatorname{Var}[X] = E\\!\\left[X^2\\right] - E\\!\\left[X\\right]^2 = \\sum_{i=1}^n \\frac{2}{\\lambda_i^2}p_i - \\left[\\sum_{i=1}^n \\frac{p_i}{\\lambda_i}\\right]^2 \n = \\left[\\sum_{i=1}^n \\frac{p_i}{\\lambda_i}\\right]^2 + \\sum_{i=1}^n \\sum_{j=1}^n p_i p_j \\left(\\frac{1}{\\lambda_i} - \\frac{1}{\\lambda_j} \\right)^2.\n", "8c7bac361b66549eb300397d4e68d8bc": "A \\in \\mathcal{S}", "8c7c3e0bec01a4a2bc7236c0313cd5a6": "\\frac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}", "8c7c915929972f9cf30996479573ef2e": "W=\\int_0^t\\mathbf{F}\\cdot\\mathbf{v}dt =\\int_0^tkx v_x dt = \\frac{1}{2}kx^2. ", "8c7ce314ce0bc8e6a66f06122119d0cf": "p'_x(x,y)=p'_y(x,y)=p'_\\infty(x,y)=0,", "8c7ce6d73b084c7ea8c8ce34b50c1f53": "C_\\gamma = C_\\beta \\cap \\gamma", "8c7d288d3a0f4ecfc47f00b74f328464": " H_2", "8c7d5c2534ec5a16aff516e7d1b02600": "v(a+b) \\le v(a) + v(b) ", "8c7da21d9f227ea5b8351b51db74c049": "3x - 3", "8c7dbf20cc6eee769eb156111bf46135": " = \\frac{1}{2} \\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu + \\gamma^\\nu\\gamma^\\mu) = \\frac{1}{2} \\operatorname{tr} \\left( \\{\\gamma^\\mu, \\gamma^\\nu\\} \\right) \\,", "8c7dd8454a89cc7a2ace7f6ffa2afc65": "\\omega\\omega'", "8c7e1bbe567dea9aa0b2d88a3db538ec": "\\alpha_{i}: A_{S_{i}} \\rightarrow R", "8c7e1c3d2e2ea2963b1ae5b1fc6505e8": "g(y) \\approx \\int_{-\\infty}^\\infty \\exp (-2\\pi ixy) f(x)\\, dx,\\qquad f(x) \\approx \\int_{-\\infty}^\\infty \\exp (2\\pi ixy) g(y)\\, dy,", "8c7e6965b4169689a88b313bbe7450f9": "kn", "8c7e76d0b0023c86adcc58d700b9bf6c": "b_n = E[\\hat{g}_n|u_n] -\\nabla J(u_n) ", "8c7f4f4160ea4805c71d73c3e566ff77": "\nP-49s_2^2 = x_1^2 + 2x_1x_2 + 14x_1^2x_2 + x_2^2 + 14x_1x_2^2.\n", "8c7fbc0e527c77fce61b5661667ead83": "F(\\sqrt{-1})", "8c7fdf9bbf24caa50a96b3ac98e428d0": " C_p={SSE_p \\over S^2} - N + 2P, ", "8c8055ce2bd23c9160a84a6c28dda51d": "\\psi=\\frac{p}{\\rho g}", "8c80fc50570cfae0f24f0b4708ad6ea9": "a < 2^w", "8c81656355dafa701a726a9f5b83feb1": " \\int_{f^* > \\lambda} f \\,d\\mu \\ge \\lambda \\cdot \\mu\\{ f^* > \\lambda\\} ", "8c817ac74f4c7d72d61d5b9ab7a3a72f": " {C_{c}^{\\infty}}(\\mathbb{R}) ", "8c820a235ef26b3829bf7bd2ff7b2ffc": "n(t) = \\frac {K} {c+t}", "8c8236a53995349cc6a8f6fd08e88e6b": " c \\,=\\, \\frac{1 - \\left(e^{it}-1\\right)^2}{4}.\\,", "8c82a0b75f2df50890b7a875763bcc81": "\\hat{S}(r)a\\hat{S}^{\\dagger}(r)=\\mu a + \\nu a^{\\dagger}", "8c82a9c7044e16db64231f986b539278": " k_\\mathrm{cat}[ES]", "8c82c91e8340345b369e2974327911e1": "\\Re s > -1", "8c834d30bbe046c99321eb71949382e5": "[t]", "8c835043c53c80b3e5c2a234758ec783": "|z|\\to\\infty", "8c837b189144aeb71b9f29ec07d9c553": "c(x)", "8c840aa7de447438e70897dc449bf797": "\\Lambda^0 = \\operatorname{Obj} ( \\Lambda )", "8c841c5bdbd7028c7992411a28127c7a": " \\displaystyle{\\sigma(g)= M \\overline{g} M^{-1},}", "8c844b8781a536c41f8e7f0de0b7f1df": "\\eta_e = \\frac{\\sigma_n}{\\dot{\\varepsilon}}\\,\\!", "8c848db7066abc6ee527cda00a4f1fdf": "\\pi_{ij}", "8c849c30a01f25df09a8c3afb9c4814b": "\\sum_{n=0}^\\infty \\left|a_n\\right|", "8c84a44ce5f92ff8a6404eabe8c5cd9c": "\\mu_6^{'}=48\\sigma^6+72\\sigma^4\\nu^2+18\\sigma^2\\nu^4+\\nu^6\\,", "8c851519c74badfaea652a333a082262": "\\phi_k(x)", "8c85179e1f60e1a56a817fc0f6c590dd": "\\nabla^\\Gamma_\\mu g_{\\nu\\alpha}=0 ", "8c85747a639d1f14091b2c874ecffb26": "w=3", "8c857e7eb92d0b3e8c95ab2380ce6212": "\\operatorname{dim}", "8c85cea5d4592376a15ff1bf37da1fcf": "\\pi^3\\approx 31,", "8c85fb80593fe369f591498ca0550e74": "e(T) = 1", "8c86a94e7cd3f0f0dc558959d6076bba": "D = \\int_0^{q_n} \\frac{[2\\lambda + (1 - 2\\lambda)q]}{(1 - q)[\\lambda + (1 - 2\\lambda)q]}dq = \\frac{1}{1 - \\lambda} \\int_0^{q_n} \\left[\\frac{1}{1 - q} + \\frac{[\\lambda(1 - 2\\lambda)}{(\\lambda + (1 - 2\\lambda)q}\\right]dq.", "8c86db3d49d606682af22e3014f75625": " \\int_a^z \\, \\int_a^x \\, h(y) \\, dy \\, dx \\ ", "8c86de824e31438743617147322c67a9": "\\langle f_n, x_n \\rangle = \\int_{-\\pi}^\\pi 1\\, dx = 2\\pi.", "8c874976945367e2e90f834a78016429": "\\Pr(M=m\\mid N=n,K=1) = (m|n) = [m \\le n]\\frac{1}{n}", "8c8762fe819481ec09d9bda641aa6690": "\\varphi_x \\in A^*", "8c87a21cbb89d070b57639601651f26c": "p_i = b_{i+1}", "8c87c0817813a29ee6eaf214a7b4f46a": "v_e=v_0 [H^+]_0/[OH^-]_{0^{ }}", "8c87c6db3bb391e4fe33c1c6f7531994": "\\sqrt{2}^{ \\sqrt{2}^{\\sqrt{2}^{\\cdots}} }", "8c881b25f7fd5dd38ca425a300ef13df": "h^*- \\Delta p^*= \\left( \\frac{1}{{R_1}^{*}} + \\frac{1}{{R_2}^{*}}\\right).", "8c8839ceea0771cdbf71329dd19aab3f": "\\begin{align}\n h_1(X_1,X_2,X_3) &= X_1 + X_2 + X_3\\\\\n h_2(X_1,X_2,X_3) &= X_1^2 + X_2^2 + X_3^2 + X_1X_2 + X_1X_3 + X_2X_3\\\\\n h_3(X_1,X_2,X_3) &= X_1^3+X_2^3+X_3^3 + X_1^2X_2+X_1^2X_3+X_2^2X_1+X_2^2X_3+X_3^2X_1+X_3^2X_2 + X_1X_2X_3.\n\\end{align}", "8c883dbb8bfdd5e82ac9d5842fa82f58": "\n(-\\partial \\bar{\\partial} + \nA \\bar{A}) \\Psi - i( 2e\\left\\langle A \\bar{\\partial} \\right\\rangle_S + eF) \\Psi \\mathbf{e}_3 = m^2 \\Psi\n", "8c8858ca3f0770490d04cfde3a986043": "\\sigma_X = \\sqrt{\\sum_i{\\sigma_{X_i}^2} + \\sum_{i,j}\\operatorname{cov}(X_i,X_j)}", "8c887e3b1ddc65d82dff8d6e4ec95275": "I \\subset R", "8c888c79474fb12c8899f84653a8e82b": "\\mathbf{\\Delta}^0_\\alpha", "8c88a56d9eea511621179a8dd3a67c8a": "\nV_b = 10^{-3}S\\left(-\\frac{49.185}{T}+0.532\\right)\n", "8c896043a60629f6b204900554566980": "s_{int}", "8c899ff9390564eb72c9d395e651f5e7": "\\sum_i \\Big(\\sum_\\alpha a_{i\\alpha} X^\\alpha\\Big) \\otimes b_i \\mapsto \\sum_i \\sum_\\alpha a_{i\\alpha} b_i^{1/p} X^\\alpha.", "8c89bada032807134515e3fdd722667d": "{\\rm ATIME}(C,j)=\\Sigma_j {\\rm TIME}(C)", "8c8a1de22ba9a3579d78310e67260372": "\\tilde{G}_0 \\cong \\mathbb T^m \\times K.", "8c8a6b4ef38e7e05bfa8a1fa90639a56": " arg_c(z) = arg(\\Phi_c(z)) \\,", "8c8ad7bb0591176aff8002c93632aa97": "\n\\frac{\\partial Er}{\\partial f_i(t)}=\\int_{[a, b]} \\left(\\int_{[a, b]} K_X(s,t) f_i(s) ds -\\beta_i f_i(t)\\right)dt=0\n", "8c8af1137d31811a8a7d3832ba1d9c18": "\\left(\\rho=1000 \\frac{kg}{m^3} \\right)", "8c8b2f742d8d7512e7b8b5be740af3ef": "\\scriptstyle\\ [\\![1,k]\\!],\\ ", "8c8b38d7f3e0fbec51df0d47e4f04539": "\\mathbb{R} \\times \\mathbb{R} \\times \\mathbb{R} \\times \\mathbb{R}", "8c8b4080743cc4b3cc1caeddfea1aaef": "i\\omega V_c + \\frac{1}{RC}V_c = \\frac{1}{RC}V_s \\quad\\quad(\\mathrm{QED})", "8c8c83ab8aa7c04cdc4171c5479935d0": "X_{\\sigma(X, X^*)}", "8c8c83aef447d941e9d888bc148dc4f9": "\\det(I_\\mathit{m} + cr) = 1 + rc", "8c8c9d0de8f46d1f24c03860accb870e": "\\mathcal{S}(\\mathfrak{a}^*)^W", "8c8cda7ac5213c4e43fe5c93a3279c19": "u_c", "8c8cfc7608c3e8146548247fe116e52d": "\n\\gamma_P(\\textbf{Cl}) = \\frac{\\left|U - \\left( \\left( \\bigcup_{t \\in T} Bn_P(Cl_t^{\\geq}) \\right) \\cup \\left( \\bigcup_{t \\in T} Bn_P(Cl_t^{\\leq}) \\right) \\right)\\right|}{|U|}\n", "8c8d10efb438c0c7c7bad8412ebb2d73": "\\frac{27}{16}", "8c8d1b99145134bd081a55b885b62653": "\\lim_{n\\rightarrow \\infty, n\\in \\mathbb(Z)} \\frac{1}{(2t)^{2n}+1} = \\frac{1}{0+1} = 1, |t|<\\frac{1}{2}", "8c8d310c47a3241d2b59bea12bd87442": "\\omega_f=\\log^{-2}(\\delta^{-1})", "8c8d5cc98a115cd4128ec452cf850320": " \\eta, \\zeta \\mapsto \\int_M \\eta\\wedge\\zeta", "8c8d67175c9080f48833cca814bc6011": "a \\lor(a \\land b) = a", "8c8d7e784fe825e6698b12cff168bc38": "\\tau = \\frac{1}{\\lambda}.", "8c8dc40f302bdb49703ea1e9445c45dc": " -\\frac 2 h \\leq P(x,p) \\leq \\frac 2 h.", "8c8def31522184e959cbe01dddc35314": "(p-1)! \\equiv -1 \\pmod p", "8c8df46f6ed94e3e8d8adf737a49a9b0": "P \\psi = \\pm e^{i \\phi /2} \\psi", "8c8e03f08673beb2134d55c8c478e2df": "l: S \\to a", "8c8e315365aa5ada0ab43448e40657ef": "x \\leq_K \\lambda_x e", "8c8e8346bb004ccd37caf9c5fbc69c2f": " \\frac{V}{T} = \\frac{nR}{p}", "8c8eabc557ceeec3a9e8a61702741985": " \\Phi \\left ( ka \\right ) = \\frac{N \\left ( ka \\right )}{D \\left ( ka \\right )} ", "8c8eedea4021c3e96de5ba251f2b3b2c": "\\sum_{\\Lambda}{1\\over|\\operatorname{Aut}(\\Lambda)|} = {|B_{n/2}|\\over n}\\prod_{1\\le j< n/2}{|B_{2j}|\\over 4j}", "8c8f1fda26b8e3a7a4ca5d2e218b0500": " p_j", "8c8f2f3d8af17aa52efdd9b4be19e5af": "j_3(x)=\\left(\\frac{15}{x^3} - \\frac{6}{x} \\right)\\frac{\\sin(x)}{x} -\\left(\\frac{15}{x^2} - 1\\right) \\frac{\\cos(x)} {x},", "8c8f4a73474d7f455eade4a626d3bac1": "x_3 = \\frac{x_1 y_2 + y_1 x_2}{1 + dx_1 x_2 y_1 y_2} = 1", "8c8f57782b08ff34f5e348190508021f": "\\Delta\\theta_{i}=\\sum_{j=1}^{N}R_{TH_{ij}}(t)P_{j}(t)", "8c8f6aee34345cb8eb5e83697d0e418a": " \nL(t) = \\frac{1}{2}\\sum_{i=1}^KQ_i(t)^2\n", "8c8f751ddc0ef83de800bd3fe73be3bf": "\\{ v_i \\}_{i \\in \\mathcal{I}}", "8c8f9fe57de871478818a107b4f6c98b": "2^{2^{\\aleph_0}} > 2^{\\aleph_0}", "8c8fe04084a85d4bffccc77ded61d8d3": "v_p = 422 ", "8c908b865fa629cc3a8650293d25c9cf": "y_i \\in \\{ - 1, + 1\\}", "8c9142c6c96bd7c10d9fff4ac6a47070": "R>2.", "8c919978e234d39c24c0b7db6e3793dc": "\nf_3^2=b^2-c^2=(b^2+\\lambda)-(c^2+\\lambda).\n", "8c91a035c4aa7d0529afe78f3f0b9144": "S_{i,0}=50", "8c91f32eb417712184272ee37cd638a9": "A\\cup\\{a\\}", "8c922f214b189df93db65289f0dc3f5d": "-\\sqrt{1 - (x^*)^2}", "8c923f161004e700ae3ed75c15c09b53": "\n\\left(\\nabla^2 v(\\mathbf{r}- \\mathbf{R})\\right)_{\\mathbf{r}=\\mathbf0} = \\sum_{\\alpha=x,y,z} v_{\\alpha\\alpha}(\\mathbf{R}) = 0\n", "8c92534da0021e2458e4bd93063fadfb": "\n\\begin{bmatrix} x' \\\\ y' \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 1 & 0 & t_x \\\\ 0 & 1 & t_y \\\\ 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\\\ 1 \\end{bmatrix}.\n", "8c92d476d32f00ed1d570a1f839b089c": "|a-b| \\ge |(|a| - |b|)| ", "8c937755f0ea33282d935569e22579b6": "\\Psi \\left[ \\psi(\\mathbf{x},t) \\right] = R \\left[ \\psi(\\mathbf{x},t) \\right] e^{S \\left[ \\psi(\\mathbf{x},t) \\right]}", "8c93a0cbc1896c8f3cdcc65dc981d407": "S=I(r)", "8c93b3a1e2ad2df6dfb5740b3692b2df": "f_r(\\mathbf x,\\, \\omega_{\\text{i}},\\, \\omega_{\\text{o}},\\, \\lambda,\\, t)", "8c93e5c039a42549575c6fd5e3e39f68": "|a\\rang", "8c94134f25efe7680f9743b2774e8e1f": "2D", "8c9422e618a9f0fc9d2438fb166ec0e5": "\n{\\varphi}_{{\\lambda}_{1}}\\circ\\delta_{[1,{j}_{1},{c}_{1}]}[I]=\n{\\varphi}_{{\\lambda}_{2}}\\circ\\delta_{[1,{j}_{2},{c}_{2}]}[I]\n", "8c944b98368f6c5c99d00b17b54bd676": "L_2 = x_3 p_1 - x_1 p_3\\,.", "8c94747cd1089fd054427c29552fd5ed": "{\\rm GF}(q)", "8c9478cbc5ab66e2d9f5d3c50a04d2fa": "\\omega_f(x_0) = \\lim_{\\epsilon\\to 0} \\omega_f(B_\\epsilon(x_0)).", "8c949faf124eabaf5c2ac933d14f4c51": "A = UP\\,", "8c94ace6266ba81eb794dbdbbdfb2505": " \\mathit{L}_D = \\sqrt{\\frac{\\varepsilon_{\\mathrm{Si}} k_B T}{q^2N_d}}", "8c94c9f2749ea7f043cd41bbd61dde77": "\\begin{matrix} {4 \\choose 2}{3 \\choose 1}^2{9 \\choose 1}{4 \\choose 3} \\end{matrix}", "8c9529577b857a585b35c5a34abf1160": "f(x)=\\sqrt{x}", "8c95bcf95098a1cd74165cf3734addb9": " Hom(i,j)", "8c95bdfe35bf5f2bd530eb6d755ff345": "s_{-i}", "8c967e6a7490914e5d38a353809ee377": "\n\n\\begin{align}\ng_1 (\\mu) = \\eta_1= X_1 \\beta_1 + \\sum_{j=1}^{J_1} {h}_{j1}(x_{j1}) \\\\\ng_2(\\sigma) = \\eta_2= X_2 \\beta_2 + \\sum_{j=1}^{J_2}{h}_{j2}(x_{j2}) \\\\\ng_3(\\nu) = \\eta_3 = X_3 \\beta_3 + \\sum_{j=1}^{J_3}{h}_{j3}(x_{j3}) \\\\\ng_4(\\tau)=\\eta_4=X_4 \\beta_4 + \\sum_{j=1}^{J_4}{h}_{j4}(x_{j4})\n\\end{align}\n", "8c97374323da39b98e2a36526c3d94bc": "1\\leq m\\leq n,", "8c973b9f3908e58ce9562f7bc9707158": "\\phi^B_e", "8c97b5485af020c996d37625ba2299f5": "w = {\\rm min} ( \\frac{1}{\\epsilon}, \\log_2N)", "8c97d6d842702d9e3ee7dc4c77f896d7": "\\pi d^2 / 4", "8c98011ad396607f89f506dc2d146bd3": "\\begin{align}\nx,y & \\sim N(10,\\sqrt{10}) \\\\\nz & \\sim N(30,\\sqrt{10}) \\\\\n\\end{align}", "8c982c741c15bcc1a0acf7991268cf95": "-\\log\\!\\left(\\frac{\\exp(-\\theta t)-1}{\\exp(-\\theta)-1}\\right)", "8c99608ae0c09527bf7e7d3df449950a": " f_1\\ge f_2 \\ge \\cdots \\ge f_{n-1}\\ge|f_n|", "8c999d6455b724069f4f1ce4a933e92b": "u_{21}", "8c99da2a428fcfff0e5132a6e0d2ca9e": "\\prod B_i", "8c9a04c24e7e1c767e39d0ffe4a83e00": "U \\subseteq Y", "8c9a2a11516dc81480318811dc87cf31": "G_X =e^{\\operatorname{E}[\\ln X]}= e^{\\psi(\\alpha) - \\psi(\\alpha + \\beta)}", "8c9a4493e575c43a5e70126519a6250c": " \\frac{1}{E^\\prime} - \\frac{1}{E} = \\frac{1}{m_{\\text{e}} c^2}\\left(1-\\cos \\theta \\right) ", "8c9a59b8a6783857c90736d17fc2de68": "\\bar{x} - \\bar{y}", "8c9a5d7ceb8ea3ef1224fff91fce656e": "s \\notin \\alpha, t \\in \\gamma", "8c9ac4471eccef40a4256d9367940c63": "J_{transit} = J^{\\star} + 0.0053 \\sin M - 0.0069 \\sin \\left( 2 \\lambda \\right) ", "8c9adeffb75bc3fb5bcdbd5f23586996": "\\mathfrak{F}=\\left\\{\\left(-\\infty,t\\right] : t\\in\\mathbb{R}\\right\\}", "8c9b832b262e41504e92620c5fd9de18": "\\mathrm{d} X_t = \\mu X \\mathrm{d} t + \\sigma X d W_t", "8c9b9419feada144d60ac3758300f6f1": "g_1\\,=\\,h_1.", "8c9bb2ea24b298b45f5065f0c0f1f426": " G(y) = \\Pr(Y \\leq y) = \\Pr\\left(X \\geq \\frac{1}{y}\\right) = 1-\\Pr\\left(X<\\frac{1}{y}\\right) = 1 - F\\left( \\frac{ 1 }{ y } \\right).", "8c9bf754de7b80bb33b15c33822f1a46": " E = m c^2 \\,", "8c9bffdaebe1a6a9f9bd55a2b5da936c": "C_*(M, (f, g))", "8c9c0134a6f003af7327242884a302ba": "f = f_0\\left(1+\\frac{v}{c}\\right)", "8c9c11f5142005e581dab628bfb3a2eb": "(\\bar{x},\\;\\bar{y})", "8c9c404b6cc06dcb1b883835980c03b9": "\\Delta H (T) / m", "8c9c6d77fe1c55e541e97bc25738185f": "\\delta_{ij} = \\begin{cases} 1 & i=j\\\\ 0 &i\\not=j \\end{cases} ", "8c9c8cf38948f2de4dbf6123e66399c3": "d_0=1\\,\\!", "8c9cb1997e86f9299e21a7926dbe4fff": "f'(x)\\cdot\\frac{1}{g(x)} + f(x)\\cdot\\left(-\\frac{1}{g(x)^2}\\cdot g'(x)\\right)\n= \\frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2},", "8c9cb254a5e388f2bcaf294e52d745a6": "\\textstyle W", "8c9cb757a71315393c753065d0489493": "\n\\begin{align}\n& {} \\quad \\sin\\angle OAB\\cdot\\sin\\angle OBC\\cdot\\sin\\angle OCA \\\\\n& = \\sin\\angle OAC\\cdot\\sin\\angle OCB\\cdot\\sin\\angle OBA.\n\\end{align}\n", "8c9d29b5a3025bf0fc33fbb766bd59f4": "x(t) = pt^2 + h; \\ \\ y(t) = 2pt + k \\, ", "8c9d71e34691b6375ed2197ba52a32d5": "n = \\sum_{c=1,k=1}^v o_{ck}", "8c9dcafdbcae2652f07fa63359bf356b": "V_\\mathrm{RMS}", "8c9e96ba6faa9e70807700113d45ab4f": "\\,_tp_x\\!", "8c9ec32248ac71768c8b30b5dbce4827": " \\mathbf{v}_{\\parallel} = (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k} ", "8c9ee3c3f00971407f153890e197c4bc": "f_{PAD}=-f_{0} \\frac{ v}{ c_{0}} cos \\alpha +f_{0} \\frac{v}{c_{a}} cos \\theta", "8c9f02c9aadca7b8d1adcdc845b764e9": "\\omega^l = \\{ (e,f)\\, :\\, ef = e \\} ", "8c9f0b126b05d44ab0a1681589547486": "c_P \\in\\Z", "8c9f0b71c7c12efb79a5d0d889180d7f": "0\\times 1 = 0\\times 2.\\,", "8c9f481d3575a833558b34648f4e021f": " R^n ", "8c9f4a2fb42d2cc7c0ef150a9c40a468": " \\to\\{\\mbox{scalar fields on }U\\} \\;", "8c9f7faec8bfb766668839fe25a028b7": "h={1\\over d}*a", "8c9f9f4c1c0a25f6b1b152d737d218d5": " \\int_{-\\infty}^{\\infty}R(x) e^{-2\\pi ifx}dx ", "8ca042e8ff30aba99a78e069db08b58a": "[x,y]", "8ca06e8cb0f462f5bf35ae6b2b58eae5": "\\exists \\phi", "8ca0744962ce1829ac8155242b989ed9": " \\mathbf{j} = {\\hbar \\over m} {1 \\over {2 i}} \\left( \\psi ^{*} \\nabla \\psi - \\psi \\nabla \\psi^{*} \\right) = {\\hbar \\over m} \\operatorname{Im} \\left( \\psi ^{*} \\nabla \\psi \\right),", "8ca0911abf9aea2f4c361ff11a3eab57": "\\nrightarrow", "8ca0b1e0e65a9b5955d422753b3fc9bb": "\\frac{1}{4 \\, \\pi} \\int_\\Omega |f(\\Omega)|^2\\, d\\Omega = \\sum_{\\ell=0}^\\infty S_{f\\!f}(\\ell),", "8ca0f010c78965a81bc1c4238f04917d": "\n\\overline{n_\\mu ^2}=\\sum\\limits_{n=0}^\\infty n^2P_\\mu (n,t)=\\overline{n}_\\mu\n+\\overline{n}_\\mu ^2\\frac{\\sqrt{\\pi }\\Gamma (\\mu +1)}{2^{2\\mu -1}\\Gamma (\\mu\n+\\frac 12)}. \n", "8ca104782ec1b84ec107d49e318401f3": "b = 2mn", "8ca10e2af6fdf91c5dddf191b3be742a": "\\frac{\\partial \\Phi}{\\partial t} = \\frac{1}{i\\hbar}H\\Phi", "8ca14c31a4216b7933638256d658d8c2": " \\left(x+\\frac{1}{2}\\right)^2+\\frac{3}{4}=x^2+x+1.", "8ca1511f378f19e0dbea61794d6978ba": "N_{samples}", "8ca17c28856f625dc857edfa049a29c9": "F_{\\gamma} \\sin \\alpha", "8ca19816e3cf45305e52e1922138d0b5": "x^2 + ux + v", "8ca1b66cd7ed3bee930d10cfe1cae2c6": "R/A", "8ca1c17a00f76a7c6fe8033608702373": " p_{3,4}(x) \\, ", "8ca20f5f3c338610774786957f4e9c1c": "\\cos(x) = \\sin(x + \\pi/2),", "8ca2521faf860da4e38b669113d9bd54": "h\\nu \\gg kT", "8ca26236cf190eb1fb83805916881a5b": "y (1 \\times 1)", "8ca29da756003fc6f3b658cea66e838a": "z(v) \\rightarrow z(v) - \\mathrm{deg}(v)", "8ca2bb7c9ff46fc8c9161251167416f3": "dI", "8ca2e86e52104d76da54d4c84a570bef": "F_m", "8ca2ed590cf2ea2404f2e67641bcdf50": "a-b", "8ca31232e12aad1b3c507d40109ff38d": "\\operatorname{var}[\\hat\\alpha] \\geq \\frac{1}{\\mathcal{I}(\\alpha)}.", "8ca33b2d1ca00385f5dce296366797e4": "d_M", "8ca3407caaba5b559bab6799fbb3cba6": "35 = 2^{2^2+1}+2+1, ", "8ca3ad7eec2f644e40fcb94088717491": "\\bar{g}\\in \\bar{G}", "8ca421e877ba1a9fe9bd3e529e01db02": " \\sum_{i < j} \\frac{1}{|\\mathbf{r}_i-\\mathbf{r}_j|} ", "8ca4527377681894e33387d839c47a40": "\\mathcal{P}(\\kappa)\\,", "8ca4f0d76cd0c72a982a5e345d5bdc13": "\\exp \\begin{pmatrix}0 & a \\\\ a & 0 \\end{pmatrix} =\n\\begin{pmatrix}r/q & p/q \\\\ p/q & r/q \\end{pmatrix}", "8ca537c751164f29865c6b398b551582": " U_{\\alpha_1 \\ldots \\alpha_n} = U_{\\alpha_1 \\ldots \\alpha_{n-1}} \\cap U_{\\alpha_n}", "8ca5c2a25e8488b1954e2a862e34cd22": "{f'(z)\\over f(z)}={-m \\over z-z_P}+{h'(z)\\over h(z)}", "8ca5e095b1efebed40498cc19a16b12d": "R \\to \\hat{R}", "8ca5f706bcbe976ec684f76a050ea1e6": "f(k,0)=(k-1)(f(k-1,0)+f(k-2,0))", "8ca6104be08a7e0bfa0d510a1c0f0731": "U_g = 0", "8ca61a25463d67e634a0201a6731f5fc": "G(\\omega)=1 - \\frac{1}{2}\\omega^{2n}+\\frac{3}{8}\\omega^{4n}+\\ldots", "8ca62aaae03f85e325c08938f6589643": "X^*_{b(X^*, X)}", "8ca65626a28d04f4738bca5e3472d06d": "\\nu(\\gamma,z)", "8ca66f208af2896bbe1a2fa790bbfecd": "F L \\sin \\theta = k_\\theta \\theta", "8ca6f795934aa91d72c0d7891e8a27aa": "-\\phi \\,", "8ca706cf03b835883732b3c3702c3a95": "\n\\frac{d^{2}q}{dt^{2}} + \\omega_{n}^{2} q = -\\omega_{n}^{2} f(t) q\n", "8ca72d938732936bbb3cce1f9ce47a01": "\\frac{\\mathrm{D} \\boldsymbol{u}}{\\mathrm{D} t} = - \\frac{1}{\\rho}\\boldsymbol{\\nabla}p + \\boldsymbol{\\nabla} \\Phi", "8ca786515f8613e4846bab4e76ba7aab": " ~\\epsilon_{t-1} < 0 ", "8ca79fe1d7dafc83b56b609c4b5e2359": "y_i=y_{i-1} \\oplus x_i, \\qquad (1)", "8ca7c066bb3d7010288f4048095422f6": "\n A = (2 L^\\prime_a + M^\\prime_a + \\textstyle{\\frac{1}{20}} S^\\prime_a - 0.305) N_{bb}\n", "8ca7ed8d6a126e1c71792452d01198eb": "dl^2 = \\frac{dr^2}{1-\\kappa\\frac{r^2}{R^2}} + r^2d\\theta^2 + r^2\\sin^2\\theta d\\phi^2", "8ca83bdba8c3686a6c9fa2977abe603f": "2\\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y) + (E(Z(x))-E(Z(y)))^2", "8ca881466cdc8b2c284b16db0d8de706": "\\int\\frac{dx}{xR}=-\\frac{1}{\\sqrt{c}}\\operatorname{arsinh}\\left(\\frac{bx+2c}{|x|\\sqrt{4ac-b^2}}\\right), ~ c < 0", "8ca8b0db084034347a6e4ea4b76f321d": "\\ MU_y ", "8ca8c48a6db2ba417d6de45a86362b1e": "\\left\\{\\beta'_k\\right\\}_{k=1}^\\infty ", "8ca8deb7f181328001d8668fe9533a89": "t=\\pm\\tfrac{1}{1,3}", "8ca8e04ae11575b7034153a733dbfcfa": " W_\\text{charging} = \\int_0^Q \\frac{q}{C} \\, \\mathrm{d}q = \\frac{1}{2}\\frac{Q^2}{C} = \\frac{1}{2}QV = \\frac{1}{2}CV^2 = W_\\text{stored}.", "8ca8e9b3ffcd0d637824fb097b08e4d6": "P(x) \\sim \\frac{C}{D} \\int_2^x \\frac{dt}{(\\log t)^m},\\,", "8ca908284d9ed16c4176114c3ad4bef1": "n\\ge N", "8ca953e896b4427db035a173c2f3e432": "\\Omega_{\\Lambda} \\simeq 2/3", "8ca970610bc8766a0b00f2f1aa78e108": " I_S = \\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{S})\\cdot (\\mathbf{r}-\\mathbf{S}) \\, dV,", "8ca9c3d2036989fd7d65959f66f1457e": "id_\\tau:\\tau{\\to}\\tau", "8ca9e4e82afeeb81085bb43fc7298b4a": "\\hat{\\boldsymbol {\\beta_1}}", "8ca9f062053d8154668e7c798af529c2": " | \\psi \\rangle = \\frac{1}{\\sqrt{2}} \\bigg( | 00\\ldots0 \\rangle + |11\\ldots1 \\rangle \\bigg). ", "8caa1d1b075151e8ee42ac557a348446": "\n \\quad (1) \\qquad \\frac{u_i^{n+1} - u_i^n}{\\Delta t} + a \\frac{u_i^n - u_{i-1}^n}{\\Delta x} = 0 \\quad \\text{for} \\quad a > 0\n", "8caa96af1f76005183aac66ed32d28cd": "n\\geq 3,", "8caacecc74851f032752033d6968c3a3": "0 < \\int_0^1 \\frac{x^{2n+2}}{1+x^2}\\,dx \\;<\\; \\int_0^1 x^{2n+2}\\,dx \\;=\\; \\frac{1}{2n+3} \\;\\rightarrow\\; 0 \\text{ as } n \\rightarrow \\infty.\\!", "8cab15923b8b83e9c4eb7d155cde6914": "\\begin{matrix} 64 \\times {4 \\choose 1}{3 \\choose 1}{3 \\choose 1}{10 \\choose 1}{3 \\choose 1} = 69,120 \\end{matrix}", "8cab2f372325bec87894c9c61301447e": "\\sqrt \\lambda", "8cabfaf2c833d7596e0a18192ff34a0a": "\\scriptstyle{\\mathrm{R}^-}", "8cac7b5021c0fa3c54eb947c094eda6f": "I_{s,t}^n\\subset\\bigcup_{\\sigma\\in S_n}A_{n,\\sigma}(s,t) \\cup \\bigcup_{1\\le i 0,", "8cb355099616a7c6a3665b20c713ed0e": "G_i=G/V_i", "8cb3629f8f43cc9728b728002a269a29": "x_n>b-\\varepsilon", "8cb3759048428d0bc8b0d4a31572434b": "\n\\frac{dV}{dt}=\\int_A \\mathbf{v}\\cdot\\mathbf{n}dA=\\int_V \\nabla \\cdot \\mathbf{v}dV\n", "8cb39a00361c0ef0fd6a3b92359cdcc1": " S = \\{ (x_1,x_2) | \\dot{V}(x_1,x_2) = 0 \\} ", "8cb3eebfd52ac0f270df4acec054b244": "f(x;P_{\\rm{80}},m) = \\begin{cases}\n1-e^{ln\\left(0.2\\right)\\left(\\frac{x}{P_{\\rm{80}}}\\right)^m} & x\\geq0 ,\\\\\n0 & x<0 ,\\end{cases}", "8cb4246e398c00f33cde01ac3784140c": "X_N", "8cb4759fa54c4fc9f49980f56f9e2a45": "\\epsilon_{Qmin}=\\frac{1}{\\sqrt{L_n(\\xi)}}", "8cb49836d7197a25d2e030f18fe7c738": " P(E, \\Omega) = \\int_{\\Omega} |D\\chi_E| ", "8cb5162ce40724e40c4182ae083c3529": "\\widehat{\\boldsymbol\\Sigma} = {1 \\over n}\\sum_{i=1}^n ({\\mathbf x}_i-\\overline{\\mathbf x})({\\mathbf x}_i-\\overline{\\mathbf x})^T", "8cb525da11370e70b46da839155a60aa": "\\mu =\\frac{2.25}{TW^{0.15}}", "8cb53369899e52fe9f8aadf2898a89e5": "p_0 = \\frac{\\rho . V^2}{2} + p", "8cb58c41a783dab56e43ad8efae2ed7f": "\\cos x\\not=0", "8cb62a3e0ffc188b80d9af1e59699098": "sl(n)", "8cb633ec1d8e9a7db3e22e41595a6c90": "\\mathbf{j}_5", "8cb67671e70ac0c27273ade531a6d370": "\nP-P_{i} =\n\\sum_{j = 0}^{i}\n\\left[\n\tP^{(j)} H^{T}\n\t\\left[\n\tH P H^{T} + R\n\t\\right]^{-1}\n\tH \\left( P^{(i)} \\right)^{T}\n\\right]\n", "8cb67c09d9573b96af0cf3520d300306": "\\{O_{1},O_{2}\\}", "8cb680ca1b518eb26c9508547c4f4821": " f(x,y,z)=0 \\iff f(\\lambda x, \\lambda y, \\lambda z) = \\lambda^k f(x,y,z)=0.", "8cb6a087b5b2285f3730ff551f2bec7a": "g(h) = c^2/(H+h)", "8cb6b08b7d289a363ba0667819ac9961": "\\begin{align}\n\\tau_\\mathrm{n}^2&=(\\sigma_2-\\sigma_3)^2n_2^2n_3^2 \\\\\n\\tau_\\mathrm{n}&=\\frac{\\sigma_2-\\sigma_3}{2}\\end{align}\\,\\!", "8cb6fec6280b8bc5b25e561303595c14": "E_K(E_K(P)) = P", "8cb75a61d1a97d99f4c5b8b24271fba6": " \\bigg| { \\partial^2 A \\over \\partial z^2 } \\bigg| \\ll | k^2 A |. ", "8cb7b5bbea43f25e5257651a65c7270f": "\\sum_{k=1}^n |x_k\\,y_k| \\le \\biggl( \\sum_{k=1}^n |x_k|^p \\biggr)^{\\!1/p\\;} \\biggl( \\sum_{k=1}^n |y_k|^q \\biggr)^{\\!1/q}\n\\text{ for all }(x_1,\\ldots,x_n),(y_1,\\ldots,y_n)\\in\\mathbb{R}^n\\text{ or }\\mathbb{C}^n.", "8cb7c5dcff006a28e9ff9ae21d59289a": "ZFC\\vdash\\forall T(\\operatorname{Fin}(T)\\land T\\subset ZFC\\rightarrow(ZFC\\vdash \\operatorname{Con}(T+H)))", "8cb7cefe4779aff744dd41731a0ed227": "a=-\\infty", "8cb900817aef25443fed2fb8c593b5c1": " V = \\sqrt{ -\\frac{R}{\\rho} \\frac{\\partial p}{\\partial n}}", "8cb92466efc26ccf673578de97e74998": "\\begin{cases}\nx' = \\gamma x - bct \\\\\nct' = \\gamma ct - bx . \\,\n\\end{cases}", "8cb933c553ba0e6fe2e119ab24ff8654": "\\lambda = -\\alpha\\frac{\\gamma}{M_\\mathrm{s}},", "8cb941e92d9dde895202d69df90b4222": "\\begin{align}\nN(t) & = \\mathcal{L}^{-1} \\{\\tilde{N}(s)\\} = \\mathcal{L}^{-1} \\left\\{ \\frac{N_o}{s + \\lambda} \\right\\} \\\\\n& = \\ N_o e^{-\\lambda t},\n\\end{align}", "8cb9592cfc5733573464a030c3d72584": "T_{\\rm a}\\,\\!", "8cb97f28799cc8697880dc8b946337d5": "H_k(M,\\partial M;\\mathbb{Z}_2) \\to H_{k-1}(\\partial M;\\mathbb{Z}_2)", "8cb9af2b68d0b6d54f16ff9f854bae4f": "\\nabla \\times \\nabla \\times \\textbf{A} - k^{2}\\textbf{A} = \\textbf{J} - j \\omega \\epsilon \\nabla \\Phi \\,", "8cb9fa1df6d2dd12581eee9756ba14d2": "A_0X_0", "8cba59f94797fe379ef64444150046db": " \\partial (a \\otimes b) = \\partial a \\otimes b + (-1)^{|a|} a \\otimes \\partial b ", "8cba9a8d83763cf6e133e976267e4778": "f = (f_1, \\dots, f_m)", "8cbbe52f85c619f9c22ac827b1ac9503": " I_{corr} = I_{obs} \\exp \\left (-G\\alpha^2 \\right ) ", "8cbc0031ea03e5f02b8ce25e2aa0ecc1": " \\frac{1}{\\sqrt{2\\pi}} ", "8cbc0538420a2f2d5685f3ca22f96638": "\n\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ 4 & 5 & 7 & 6 & 8 & 2 & 1 & 3 \\end{pmatrix} =\n\\begin{pmatrix} 1 & 4 & 6 & 2 & 5 & 8 & 3 & 7 \\\\ 4 & 6 & 2 & 5 & 8 & 3 & 7 & 1 \\end{pmatrix} =\n(14625837)", "8cbc5e0f14b0d812f906ecb111d78035": " \\forall x\\, (\\text{FARMER} (x) \\and \\exists y \\,( \\text{DONKEY}(y) \\and \\text{OWNS}(x,y)) \\rightarrow \\text{BEAT}(x,y)) ", "8cbca42b15fc674374150332e2fc24d6": "K^a = \\tilde{E}_i^a \\tilde{E}^{bi} (N \\partial_b M - M \\partial_b N) / (det (q))", "8cbccf951e934328746fc0de48301c37": "48 = 110000 \\rightarrow 110001", "8cbd76868b5c624329b1673ba2ad1043": "g(y_1,\\dots,y_n;\\theta)", "8cbe2017e4a526d3c770fbba66cdd1e2": "\\scriptstyle\\infty", "8cbe5bda2a3fddf8527e0af38e382307": "\\mathsf{P^{\\sharp P}} = \\mathsf{MA}", "8cbebbdbadc817c79b98407eaef94488": "-S = \\{ -s | s \\in S \\}.", "8cbec0bad953ab38975f14f6ff833ecb": "\\mathcal{C}(\\mathbf{X})", "8cbec7d0b373a878d0d744a7b241ebbf": "f(z) = u(x,y) + iv(x,y),", "8cbee05e8e743302ad9d76d0f7f1091f": " u_1 ", "8cbf06547b740df4cc5d96d4aade556b": "F\\left(0\\right)=0", "8cbf06ac3f3945d7354fd5c1f95220d3": " y'(t) = f(t,y(t)), \\qquad y(a)=y_a ", "8cbf308f71cfdc615ed1db69da55f6fc": "\\mathbf{D}_i = \\begin{cases} 1, & \\mbox{if }L(p_i)=0\\ \\\\ 0, & \\mbox{otherwise} \\end{cases}", "8cbfb353bcabbd809d523057f39beffe": " V_p = 0.5 \\ S \\ d \\ \\ln(D/d)", "8cbfbe7238959a7134734efe5e984976": "v_i^*=\\bar{v_i}", "8cbfca86bcf03b0efa1b757b623bca80": "f_\\text{o}(x) = \\tfrac12[f(x)-f(-x)]", "8cbff723f812d6bc57cbb072fffe15f7": "\\displaystyle{\\sum_{n\\ne 0} |n||a_n|^2 < \\infty.}", "8cc019ba3f72497332aba0478d6b8330": " Ke^{-rT} ", "8cc0659c9f136888361630b4131349e5": "f(Y)=Y^p-X=Y^p-\\alpha^{p}=(Y-\\alpha)^p", "8cc09310d3705da8e774617cee5b7c85": "\\left( \\pi _{1}^{2}+M_{1}^{2}\\right) \\psi =0,", "8cc0da2e3780955eb4b63be16343c0aa": "{\\mathcal A}^*", "8cc1612d2a67f078867cfce5989bd88e": " {x} = {Q}^{-1}({A}^{T} {\\lambda} - {c})\\,", "8cc1792b557257bc6d2bb8c54247c6ea": "\n\\begin{align}\n\\Phi_5(z) \n&=z^4+z^3+z^2+z+1 \\\\\n&= (z^2+3z+1)^2 - 5z(z+1)^2\n\\end{align}\n", "8cc17fc5e1a6167e7a75a2bca7c37f8d": "E_{x,t}^c", "8cc1ea620a6b99bb863cf9e91b177608": "|P(x)|", "8cc201bd05e7cf1f9e578bb5d3022663": "{\\rm diam}(M,g_i)\\le 1/n", "8cc20fdb980724243d62c0183d240fe1": "W_E = Y \\sigma^2 ", "8cc23cac3b6220e5abf6d1c36f18b435": "\\sigma_0, \\sigma_1,\\,\\dots\\,,\\sigma_n", "8cc27b872d3661c577b84e09aafac74c": " \\bar f =\\frac 1{\\mu(X)} \\int f\\,d\\mu.\\quad\\text{ (For a probability space, } \\mu(X)=1.) ", "8cc2d9c5eb452cd0fcb074cad56aa1ff": "\\begin{bmatrix} N \\\\ M \\\\ \\end{bmatrix}^{-1}.", "8cc2e0c2bc3f8ad050fc68f851178654": "\\pi_n(X)", "8cc2e7240164328fdc3f0e5e21032c56": "NN", "8cc2eb2414ea672b80576cd7146e8b98": "\\mathrm{H_2O(l) + C_5H_5N(aq) \\leftrightarrow C_5H_5NH^+ (aq) + OH^- (aq)}", "8cc3d26e3f473282104fbdb0d77b3b42": "\\varepsilon_{\\color{Violet}{3}\\color{Orange}{\\color{Orange}{2}}\\color{RedViolet}{4}\\color{Violet}{3}} = -\\varepsilon_{\\color{Violet}{3}\\color{Orange}{\\color{Orange}{2}}\\color{RedViolet}{4}\\color{Violet}{3}} = 0", "8cc3dea934428758924c10952adeeaba": "kN + 1", "8cc3e4890b6d88ae7dc419a186468bfd": "\\scriptstyle x^2/2", "8cc3ffba98d27e868b1907e003ad5e72": "{R \\over R_0} = {\\frac {[A_2]/[A_1]}{[A_2]^0/[A_1]^0}} = {\\frac {[A_2]/[A_2]^0}{[A_1]/[A_1]^0}} = \\frac{1-F_2}{1-F_1}=(1-F_1)^{(k_2/k_1)-1}", "8cc44482ab588f401aed55db44b69978": "\n\\nu(q) = \\sum_{n\\ge 0} {q^{n(n+1)}\\over (-q;q^2)_n} \n", "8cc45e3f3297008f1f5185668049af55": "{\\tilde{C}}_{2n+1}", "8cc468ae7af219c64ca5f43a228baf4c": "\\textstyle \\text{Slope}_{\\text{ideal}} = (\\Delta y /h) ", "8cc4b097696204720f36847255a95428": "\ndV = a^{3} \\cosh\\mu \\ \\cos\\nu \\ \n\\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right) d\\mu d\\nu d\\phi\n", "8cc4bd88d091943256fd90957f1dbb3c": "\\tilde{H}_{n+1} = \\begin{bmatrix} \\tilde{H}_n & h_{n+1} \\\\ 0 & h_{n+2,n+1} \\end{bmatrix}, ", "8cc4fcb6aa09a54d4cc48fe0a6fdd8ee": "\\mathcal{E}^{(0)} \\subset \\mathbb{R}^n", "8cc5a8aea74fb9938dbf5f6279ed612c": "\\textstyle a(x) + x^bb(x) = 0", "8cc5d42f5f10d8f6ec742d7e2a6ff414": "x=1+\\epsilon x^5.", "8cc5f577693898dc911ff3fb06e8fa56": "(A+B)_{ij}=A_{ij}+B_{ij}.\\,", "8cc5f579e135049dd82aa8b64d91a1b2": " F = \\frac{m_1 m_2}{r^2} \\ .", "8cc625b5c3f7ed80ae5a86c8d2d926df": "(p \\to q) \\vdash (\\neg p \\lor q)", "8cc62a2dbb2d14c934bb784f030e6909": "A=\\bigcup_{i=1}^n A_i", "8cc65e00daf16a941469e75059eeb583": " A_{\\mathcal{B},\\varepsilon}\\ \\mbox{is a preclass} \\iff \\forall x,y\\in A, \\parallel \\boldsymbol{\\phi}_{\\mathcal{B}}(x) - \\boldsymbol{\\phi}_{\\mathcal{B}}(y)\\parallel_{_2} \\leq \\varepsilon.", "8cc6a338a636c628a1a54b89f9028b88": "\\mathfrak h ", "8cc71c0948c0b537ad065d434c4d6aa5": "v, w \\in C_0", "8cc729116cb2ad26f0774d7c4c7867f0": "lk(f(c_1),f(c_2))=lk(c_1,c_2)", "8cc74312a41d3204e59ea26bb5cf81b5": "S=\\cup_{i=1}^l S_i", "8cc7658b7dee04f0a34f2f1f4023956d": "\\max_{a\\in A}C_{a,f(a)}", "8cc7940966cd9ef472de351d72e85226": "\\sigma_{\\varepsilon} ", "8cc7bdf7be6edfaa86adab989823f5a7": "f (c) = \\int f(e) d w(e).", "8cc80aea177f3e624f4e88d34086f427": "f^{-1}(\\mathfrak{b})", "8cc80b759897e87215b47b2d768157cc": "F(x,y)=P(X \\le x, Y \\le y).", "8cc8f93f929a6918c7b5e5581147c687": "x', y'', f', f''", "8cc941905019abee77e9e5695b42c013": "\\frac{d\\ \\operatorname{Im} \\{V_c \\cdot e^{i\\omega t}\\}}{dt} + \\frac{1}{RC}\\operatorname{Im} \\{V_c \\cdot e^{i\\omega t}\\} = \\frac{1}{RC}\\operatorname{Im} \\{V_s \\cdot e^{i\\omega t}\\}", "8cc964650ecb718a0591fa3bca3be734": " \\eta_{NCA}=1-\\sqrt(\\frac{T_{cold}}{T_{hot}})", "8cc97fd85e1bd7153d2d700746c6a5f2": " [((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2 - ", "8cc9a28521ff9b5102369980047b046f": "N_I=\\alpha_0 n_h J", "8cca0cb8dfab6d328889e15d506dfd6f": "\\begin{bmatrix}\nA & B/2 & D/2 \\\\\nB/2 & C & E/2 \\\\\nD/2 & E/2 & F\n\\end{bmatrix}.", "8ccaa08e5b35fd534a70c53c3dcff8e4": "D_i J=J D_i = v_i J. \\qquad (2)", "8ccb233fce9bc841d0e9baca7549dd3d": " p_\\text{TDC} = 1 \\text{ bar} \\times 10^{1.4} = 25.1 \\text{ bar}", "8ccb43cc961af50a15e6a2a98d79b68c": "\n \\boldsymbol{U} = S^{ij}T^m_{.n}g_{jm}\\mathbf{b}_i\\otimes\\mathbf{b}^n = S^i_{.m}T^m_{.n}\\mathbf{b}_i\\otimes\\mathbf{b}^n\n = S^{ij}T_{jn}\\mathbf{b}_i\\otimes\\mathbf{b}^n\n ", "8ccb6e0a1c2e5ce01c6d0c16369ea262": "b a_1, \\ldots, b a_n", "8ccb86e6ce22cc1707e9cd8ace9de672": "\\frac{x}{576}\\times \\frac{59}{54}=\\frac{4}{3}\\Rightarrow x=\\frac{576\\times 54\\times 4}{59\\times 3}\\approx 702.915254", "8ccc11b1a9ee6c16f8fed0bb76594058": "\\frac{{6 \\choose 4}{43 \\choose 2}}{{49 \\choose 6}}\\approx\\frac{1}{1,032.4}", "8ccc619c7bbfa9d4e550cad335c38177": "\\left(\\tfrac{2at}{t^2+1}\\right)^2", "8ccc80cfb7356a1ffe3dff9212271951": " \\underline u_i = min\\{u_i(p_k^*):K(p_k^*) u(p_k^*) =Q(p_k^*), p_k^*\\in L\\} ", "8cccd578ee01ba7ee4c9ab6a7a23769d": "H_{inv}(z)", "8ccd0bc42f6d6fc98d5544f399565a2b": "F^{-1}(y) \\leq x", "8ccd29403ed466cf4ccbb40393d15a28": "\\scriptstyle 0 \\;<\\; x_2 \\;<\\; b \\;<\\; a", "8ccd3d6d02408c5c8cf0fd75dcb8596a": "\n2 T = \\mathbf{p} \\cdot \\dot{\\mathbf{q}}\n", "8cce6ec098649c60897841e47400dd05": " A \\,\\!", "8cce98f301d50d657d3f2fed74803fde": "\\scriptstyle\\frac13\\,2", "8ccf251e8b3fdb4a0d950faa40031aea": " G_r^\\pm", "8ccf36d35fa291a4fd95a7443b133443": "U(x) = \\sum_{i\\in \\mathbb{Z}} L(x-i)", "8ccf6bd6caf81b1a8396e99755edd61c": "\\Phi_{ij}(\\mathbf{q}_i, \\mathbf{q}_j)", "8ccf93672e4e1eefbf57e60a826bebb4": "M \\to N ", "8ccfc151d0f2f1185e62b41c5e54ba6a": " m_a c^2 + K_a + m_X c^2 + K_X = m_Y c^2 + K_Y + m_b c^2 + K_b", "8ccfdf921a369089fdc09fddc1a1a697": "i = 1, \\dots, n.", "8ccfefa52dd9ce9ea09d6c0cd275ff93": "f_{\\varphi} \\,.", "8cd00ab1e3162db5b301798e31a061fb": "|z| = 2", "8cd02366bb27999098fbe2fbb1744719": "T_f = \\frac{T_s + T_\\infin}{2}", "8cd02bccdf381d23801230ed32547fe7": "\\mbox{Pf}(\\Omega) \\neq 0", "8cd045d398804afd5a5f23acf11e8734": "Q(p) = \\tan (\\pi(p-1/2)) \\!", "8cd09f966daf813a4a2092069b86dffe": "\\left(\\frac{dn_2}{dt}\\right)_{\\mathrm{spontaneous}}=-A_{21}n_2\\,.", "8cd0a3e465088f08bd051eb3164659b5": "FO+\\text{pos}\\,TC", "8cd0d00c7150d700eeef58a413e736b8": "M_L\\rightarrow e^{i\\beta}M_L\\text{ and }(\\mu_R)^c\\rightarrow e^{i\\beta}(\\mu_R)^c", "8cd12d24b899a312dceb9487fe25c81f": "\\delta={2k\\ell\\over\\cos\\theta} - k_0 \\ell_0\\,", "8cd1cb7341ed9d3094804e70cd572cac": "a \\in (\\mathbb{Z}/n\\mathbb{Z})^* ", "8cd1d7809c3e02a05ffc12b57c036f33": "E\\left[\\delta_i\\right] = 0 ", "8cd2771d2a113b4a871b77572932dbe3": "\\mathbf{F}^\\alpha\\ ", "8cd328d531acf0b5fb0f44739c0caff7": "\\psi^{\\mathrm I}", "8cd34f3cf60d9dbeda62a857c45348cd": "\\scriptstyle\\mathbf{a} \\,=\\, \\mathbf{T}^{-1}\\mathbf{v}", "8cd41a7a129bdcced1c91356dffbd542": "\\{X_n\\} = \\{ \\{0\\},\\{1\\},\\{0\\},\\{1\\},\\{0\\},\\{1\\},\\dots \\}.", "8cd49b07f891f8687135280617652c28": "\\ \\ R_S \\gg r_E ", "8cd4ae0c03604a96170f7228135912d6": "\\mathcal{L}(\\varphi,\\partial\\varphi,\\partial\\partial\\varphi, ...,x)", "8cd4bf994641986cc41c823586b02adc": "1414^3+2213459^2=65^7\\;", "8cd4f0dea16bad9884700f66cc91cf88": "BB=A^*A", "8cd4f45077c7557f1c1cb24ed390edb9": "p_i | ({n \\over p_i} - 1)", "8cd50ea587a576d0cc31676ce7e87155": " - \\sum_{i=1}^n p_i \\ln q_i \\geq - \\sum_{i=1}^n p_i \\ln p_i ", "8cd557a3facbbbb9613d21ff00e8c0a8": "(J^3_0g)(x)=x-\\frac{x^3}{6}", "8cd55ffc976f0adcf3f64d8223e8097c": " g_{xx} = g_{yy} = \\exp (2 p)", "8cd56aaf9f0f3db658427738721b62a3": "\\displaystyle\\varphi(3^k) = \\varphi(2\\times 3^k) = 2\\times 3^{k-1}.", "8cd580ea999caef9fc8734bce18dfd63": " c_j^\\dagger | N_1, N_2, \\dots, N_j = 1, \\dots \\rangle = 0. ", "8cd591771d19039b2b877ed9916b91a6": "S \\!", "8cd5bc620bd98b07f28ae60c6433aea5": "\\mathbf{h}(n)", "8cd5fb063d5c7fa01807566795245a04": "Z_\\infty = \\frac{a}{c}", "8cd6007e073d10b6a5c4488b23051663": "d_{\\bar k}", "8cd637054e5f6d798609df3204295638": "\\operatorname{E}(w_i\\,\\Delta z_i) = \\operatorname{E}(w_i z'_i) - \\operatorname{E}(w_i z_i)", "8cd691ff8621840fc87b883bf3b0237d": "q = V_1y_1 = V_2y_2", "8cd69a07f9e0562d82776c0ea18317bd": "\\begin{align}\n&\\left(\\frac{1}{4},\\frac{1}{4},\\frac{1}{4},\\frac{1}{4}\\right)\\\\\n&\\left(\\frac{1}{2},\\frac{1}{2},0,0\\right)\\qquad\\text{2-term truncation}\\\\\n&\\left(1,0,0,0\\right)\n\\end{align}", "8cd7593fc718a080cb2191e5936dc2d1": "E[(y_i - g_i)^2] = E[\\epsilon^2] + E[(f_i - g_i)^2]", "8cd774d50d285e0684fb75e5e75fe6ca": "L_{-n}", "8cd78578b8ca428937fbe1c3e02a93f8": "\\textstyle c_n \\neq 0", "8cd7a194825a990728e82c02ad2a8d60": "(p+q)^{8.3}=\\sum_{k=0}^\\infty {8.3 \\choose k} p^k q^{8.3 - k}.", "8cd7cc1f44c1a51a1077d08456fc2816": "y_2 = \\frac{2y_1}{-1 + \\sqrt{1+8\\frac{gy_1^3}{q^2}}} =0.45 ft", "8cd7e91168fa2646a2d1e12e8c72544c": "S = k_B \\ln \\Omega, \\,", "8cd80a896a5f48a59f0893566900a5e1": "\\left(\\frac{f}{g}\\right)' = \\frac{f'g - g'f}{g^2}\\quad", "8cd833fd2c73a6b4289f1527aa64d475": "\\frac{\\partial \\; \\textbf{a}^{\\rm T}\\textbf{x}\\textbf{x}^{\\rm T}\\textbf{b}}{\\partial \\; \\textbf{x}} = ", "8cd87d233968c189b98d6fa606841b8a": "f(x)=\\frac{ \\sgn{(x-a)}-\\sgn{(x-b)}} {2(b-a)}.", "8cd8ad69fe1467baf353d9de2d9977bd": "\\mathbf{j} = \\frac{d\\mathbf{a}}{dt} = \\frac{d^2\\mathbf{v}}{dt^2} = \\frac{d^3\\mathbf{r}}{dt^3} ", "8cd8ba8605f5b2a4050f33e8e5990c11": "k = \\frac{(a + d) + \\sqrt {(a - d)^2 + 4 b c}}{(a + d) - \\sqrt {(a - d)^2 + 4 b c}}.", "8cd903f1b32f90674a4c6b4d5cd13896": "a^{p-1} \\equiv 1 \\pmod p \\quad \\quad (X)", "8cd9102219adf5ae3b5b803b632e4957": "\\rho(\\tau)", "8cd91d29865f83ee50c6b4cf8ac667e4": "M(Q,\\Sigma,T)=\\{T_w \\vert w\\in\\Sigma^* \\}.", "8cd91f8f4df6cdd6d5351166671b3e58": "D = Eh^3/[12(1-\\nu^2)]", "8cd93370e42e093a23effd4af583cc55": "V_{1} (K, L) \\geq V(K)^{(n - 1) / n} V(L)^{1 / n},", "8cd961be9efd42c258fe0d706aeefe3a": " C(r,z) = 2\\pi S(r)\\int_{0}^{\\infty} G(r'',z)\\exp\\left [-2\\left (\\frac{r''}{R} \\right )^2 \\right ]I_0\\left (\\frac{4rr''}{R^2} \\right ) r'' \\, dr'', \\qquad(7)", "8cd9b503666dc099d8825a05b82120a1": "\\mathbf{C}^\\beta\\ ", "8cda888dc9e250370da8df8afe57750a": "Blind \\ Velocity = \\left (\\frac {C \\times PRF}{2 \\times Transmit \\ Frequency} \\right)", "8cdabb9ce193d77a85f9548a241170c2": " P_B ", "8cdaed0a883ca95d9ed07fda5f221101": "\\epsilon_\\mu^1(n)", "8cdb32422b78c66cb0f314e343499730": " = \\lim_{h\\rightarrow 0} \\frac 1 {h^2} \\int_{-\\infty}^\\infty \\left[\n \\left( 1 - \\frac{\\mathrm dX_\\theta}{\\mathrm dX_{\\theta+h}} \\right)\n +\\frac 1 2 \\left( 1 - \\frac{\\mathrm dX_\\theta}{\\mathrm dX_{\\theta+h}} \\right) ^ 2\n + o \\left( \\left( 1 - \\frac{\\mathrm dX_\\theta}{\\mathrm dX_{\\theta+h}} \\right) ^ 2 \\right)\n \\right]\\mathrm dX_{\\theta+h},\n", "8cdb523c1b7078fe9dc7e2d5da04ccec": "\\sin(f(x,y) \\cdot \\omega_c)", "8cdb839f32270f2c3253c4a0c973c418": "\\mathrm{A}_s", "8cdbb388c7ab04eaf74a372769a732de": "S() \\to x\\text{:}A()\\ B(x) \\to x\\text{:}(a\\ A())\\ B(x) \\to x\\text{:}(aa\\ A())\\ B(x) \\to x\\text{:}aa\\ B(x) \\to aa\\ B(aa)", "8cdbcf5ab89f40ef3199aa5560d179b2": "c_{k}=\\frac{y_{k}}{n}", "8cdc4edbb4c40111448a2afb21225be7": "P = U^2/R", "8cdc625058c08dc9dc4358534eec819e": "(\\mbox{fuel-to-oxidizer ratio based on mass})_{st} = \\left(\\frac{m_{\\rm C_2H_6}}{m_{\\rm O_2}}\\right)_{st} = \\frac{1 \\cdot (2 \\cdot 12 + 6 \\cdot 1)}{3.5 \\cdot (2 \\cdot 16)} = \\frac{30}{112} = 0.268 ", "8cdc7fd369947cce01f8c7516b7968e6": "1/n \\to 0", "8cdd9b3d5d75d8fccd3ca15194e5b7c8": "\n\\begin{align}\ny_1 & = {f_1}^*(x_1, x_2, \\ldots, x_n) \\\\\ny_2 & = {f_2}^*(x_1, x_2, \\ldots, x_n) \\\\\n& {}\\ \\vdots \\\\\ny_m & = {f_m}^*(x_1, x_2, \\ldots,x _n )\n\\end{align}\n", "8cddca2013cdb13588374d935439d26b": "{\\mathcal{C}_{CE}}", "8cddf0511e17848cbce12a4fcc0fd012": "\\forall w_1,\\ldots,w_n \\, \\forall A\\, \\exist B\\,\\forall x \\in A\\, [ \\exists y \\phi(x, y, w_1, \\ldots, w_n) \\Rightarrow \\exist y \\in B\\,\\phi(x, y, w_1, \\ldots, w_n)]", "8cddf277cded534b0c5b83155582e42b": "\\begin{bmatrix}1&2&1\\\\-2&-3&1\\\\3&5&0\\end{bmatrix}", "8cde4b08cfcf6dddde3ea9a89ae995d0": " ab = a \\cdot b + a \\wedge b ", "8cde56810328751f5895d4d4c2388f35": " (R,\\delta) ", "8cde5cdff011b472025ec1a674fd1423": "L(x) = \\frac{x-1}{p}", "8cdefd39af9e2732c19ef1095de5b405": "k_{\\rm A}=k_2=k_{\\rm E}/c^2", "8cdf541912c372a5f6a78cb129a04ad0": "\nV_C = \\frac{1}{C}\\int_{0}^{t}Idt\n", "8cdf6095e6bcb6cbb3b30116ccea2313": "X_{1:S},...,X_{S:S}", "8cdf70c70c7fa3cf984af868855af135": "\n\\frac{1}{r} \\frac{\\partial}{\\partial r} \\left(r \\frac{\\partial}{\\partial r} \\left(\\frac{1}{r} \\frac{\\partial}{\\partial r} \\left(r \\frac{\\partial \\varphi}{\\partial r}\\right)\\right)\\right)\n + \\frac{2}{r^2} \\frac{\\partial^4 \\varphi}{\\partial \\theta^2 \\partial r^2}\n + \\frac{1}{r^4} \\frac{\\partial^4 \\varphi}{\\partial \\theta^4}\n - \\frac{2}{r^3} \\frac{\\partial^3 \\varphi}{\\partial \\theta^2 \\partial r}\n + \\frac{4}{r^4} \\frac{\\partial^2 \\varphi}{\\partial \\theta^2} = 0\n", "8cdfa7bc29deaf4589b4c1e9ec766ddf": "-\\otimes_R-:\\mathrm{Mod}\\mbox{--}R\\times R\\mbox{--}\\mathrm{Mod}\\rightarrow \\mathrm{Ab}", "8ce03af545a8c84b63cadee89d348a62": "\\operatorname{max}\\{K-S_{T},0\\}", "8ce060ee8cd588a4261cf78954b1be06": "\\Sigma(2k) > 3 \\uparrow^{k-2} 3 > A(k-2,k-2) \\quad (k \\ge 2),", "8ce08700c981c75c938c07b4ab48702b": "\\cos \\omega_\\circ = \\dfrac{\\sin(-0.83^\\circ) - \\sin \\phi \\times \\sin \\delta}{\\cos \\phi \\times \\cos \\delta}", "8ce0ac02b21849a645fd60ac09b366cd": " V_{hex}=\\begin{bmatrix} T1 & T1 \\\\ T2 & -T2 \\end{bmatrix}", "8ce0bbe1babe33d56c4474bee0040e77": "\\tilde c_1 = c_1\\,", "8ce0fedb29fbaeef97d4c0863d7d1661": "Z_0=\\sqrt{\\mu_0/\\epsilon_0}", "8ce1026fe4afe2726d8f203b47bf9744": "U_{22} - U_{12}", "8ce1545e9a62bc3c7bedc8ef7e870219": "U(1)_R \\approx S^1", "8ce15c26b09aaa6a1e0925f7ede13294": "|\\mathbf{Q}| = K - L", "8ce17edd92db2532c634c3e6cb059968": " \\lim_{\\alpha\\rightarrow 0}\\text{EVaR}_{1-\\alpha}(X)=\\text{esssup}(X) ", "8ce18d743f33fe1731623259fd728dbd": "\\mathrm{Re}_p=\\frac{U_p D}{\\nu}", "8ce1b06171af73309f8b79065e501f44": "X\\;R\\;Y", "8ce1e3f506cca7bc14e009c4b19ff9e9": "\\nu = 0", "8ce1fd9b87443c94b4efc4090381b9d9": "\\sum_{k=0}^\\infty \\frac{z^{2k+1}}{(2k+1)!}=\\sinh z\\,\\!", "8ce21161380ae32cd005596022de48af": "\n\\begin{align}\n1 + \\sum_{i} \\left\\lfloor \\frac{x}{p_i} \\right\\rfloor - \\sum_{i < j} \\left\\lfloor \\frac{x}{p_i p_j} \\right\\rfloor & + \\sum_{i < j < k} \\left\\lfloor \\frac{x}{p_i p_j p_k} \\right\\rfloor - \\cdots \\\\\n& \\cdots \\pm (-1)^{N+1} \\left\\lfloor \\frac{x}{p_1 \\cdots p_N} \\right\\rfloor. \\qquad (1)\n\\end{align}\n", "8ce2e3bdb1ea167b8fc5c0590a919942": "\n\\mathcal{A}", "8ce3116361ee41d0d2f76a04492bc58f": "\\lim_{t \\rightarrow \\infty} P_{ij}(t) = \\pi_{j}\\,,", "8ce3253a21c5e62331c1829677288f84": "E\\subset\\mathbb{R}", "8ce329daf837f3049551ab4b71fa5da8": "\\omega^{(k)}=\\sum_{i_1<\\overline{16}_{-1H}>16_1 16_1", "8ce4b16b22b58894aa86c421e8759df3": "k", "8ce5265cb0a80aef7f2245f6d51a3fc7": "a \\leq b \\Longleftrightarrow a=eb,", "8ce5b45465c2c202cb52cee4e0da080a": "\\beta \\gamma", "8ce5c1e19e207f3d4f9cb4ef80d5034f": " \\int_0^t\\frac{v_{\\text{in}}}{R_{\\text{1}}} \\ dt\\ = - \\int_0^t C_{\\text{F}} \\frac{dv_{\\text{o}}}{dt} \\, dt", "8ce5c565280c57bdd136cccbe00dfcac": "\\,|u|^2 - |v|^2 = 1", "8ce5e79573be17e92a452f1ffe38a7bf": "d\\mathbf{r} = (dx, dy)", "8ce60a6795b2f9972d66a159a1804da2": "xa = \\sigma(a) x \\;\\forall a \\in R", "8ce60c9a14b47708f3768bc8557c5888": " \\mathbf p_1 =\n\\left( 1 + {1\\over 2} { v_1^2\\over c^2 } \\right)m_1\\mathbf v_1\n+{q_1\\over c}\\mathbf A\\left( \\mathbf r_1 \\right)", "8ce63fb2c19f9f90ad3e2933fa1b64f9": "\\mathfrak F\\subset TX", "8ce685e8f036fb00a82a46d106a9f179": " d(\\mathbf{L}, \\mathbf{x}) = ", "8ce69cd6d0bf66fb3fb85bcc9de064ae": "-4-4\\lambda", "8ce6c553ce6fd87df9bb382dc236bb9c": "\\begin{align}\n\\begin{cases} \\gamma_{1}:[0, 1]\\to D \\\\ \\gamma_{1}(t) := (t,0) \\end{cases}, \\qquad &\\begin{cases}\\gamma_{2}:[0,1] \\to D \\\\ \\gamma_{2}(s) := (1, s) \\end{cases} \\\\\n\\begin{cases} \\gamma_{3}:[0, 1] \\to D \\\\ \\gamma_{3}(t) := (-t+0+1, 1)\\end{cases}, \\qquad &\\begin{cases}\\gamma_{4}:[0,1] \\to D \\\\ \\gamma_{4}(s) := (0, 1-s)\\end{cases}\n\\end{align}", "8ce70fdda4094a1e9bc5c3c8dbd169ac": "w(x)\\oplus w(y)=w(x\\oplus y)", "8ce72b178f6edf61d63c51fd7881bc4c": "P_3\\in\\{0,1\\}=\\{\\mbox{1st},\\mbox{2nd}\\}\\,", "8ce76c4db4b21da7fde0c96a8484aa91": "\\phi(\\eta)={max}_\\theta\\{\\theta^i\\eta_i-\\psi(\\theta)\\}", "8ce78765b25c951d77ea5af68de94335": "\\Psi_0=\\Psi_1=\\Psi_3=\\Psi_4=0", "8ce796de98b74705b10b3549a076d2ae": "V \\setminus \\{0\\}", "8ce79e8ec4d5be61a170d92997fb6f61": "\\zeta_4=\\beta_4-\\gamma", "8ce7bc37965c24723357cb8ef1b8b691": "\\displaystyle \\mathbb{E} X_t = \\begin{cases}\n 0 &\\text{for } 0 \\le t < 1,\\\\\n -1 &\\text{for } 1 \\le t < \\infty.\n \\end{cases} ", "8ce7d9a7c3f3703655506fa2d7a56f1c": "g(\\mu)=g_0", "8ce7f2ced5b55654edb86bb9cefb944e": "|A|", "8ce84208b1897557de7c6601e5c0fc20": "\\tfrac{n(\\sum c_j \\bar X_j)^2 }{\\sum c_j^2} ", "8ce91474779c9115367408f3651ba4e6": "j=0", "8ce934ec6d37bf9bcf2bbe52e1014260": "\\varphi(.)", "8ce9528fdad174201cfb5f7426e0c0c4": "V=\\sum_{n=0}^\\infty\\sum_{m=0}^\\infty\\,Z_{mn}(\\zeta)\\,\\Xi_{mn}(\\xi)\\,\\Phi_m(\\phi)", "8ce9900e4041d7f52c065a04b7b11008": "T(B)", "8ce9e1f4996f725250b60c61d809a088": "F(\\gamma(t),\\dot\\gamma(t))=\\lambda", "8ce9ea7993b3024e0b20fdee6582203f": "\\left|\\rho_\\mathrm{in}\\right| < 1\\,", "8cea238f3e01052304c145379723ea33": "\\displaystyle{\\|K_r\\|_1 ={1\\over 2\\pi} \\int_0^{2\\pi} K_r(e^{i\\theta})\\, d\\theta =1.}", "8cea58a19292583bb4a6c106abf275cb": "\n\\cfrac{\n \\begin{matrix}\n \\cfrac{}{A \\ true} u \\\\\n \\vdots \\\\\n B \\ true\n \\end{matrix}\n}{A \\supset B \\ true} \\supset_{I^u}\n\\qquad \\cfrac{A \\supset B \\ true \\quad A \\ true}{B \\ true} \\supset_E\n", "8cea67ed1339b441a7965283c3ad37a0": "\\lim_{n\\rightarrow \\infty} (A_n)^{1/n} = \\lambda", "8ceb0620651253e0eb048bdec4e19cfe": "\\beta_{rel} = { \\beta + \\beta \\over 1 + \\beta ^2 } = { 2\\beta \\over 1 + \\beta^2 } \\leq 1.", "8ceb63016049cef8d148a240eca8e5f1": "Q = I - (1+w)\\mathbf{v}\\mathbf{v}^H", "8cebdb026772e5bc12f415ddb85d66d8": "\\mathrm{-C(=O)-CH_{3}}", "8cebff20ceb493bf7cf9f4aed0949ec3": "DF(u) : \\psi\\mapsto dF(u;\\psi)", "8cec0930ca9420bb8bf79befa2232769": "\\frac{n+\\alpha}{n+1}\\,", "8cec3694136d2053cac65bdf624356c0": " \\delta_{ij} = \\begin{cases} 1, & \\mbox{if }i = j \\\\ 0, & \\mbox{if }i \\ne j \\end{cases} ", "8cec3ea8aba5b9146749cad8b8d40135": "b_j=\\sum_{i=1}^Rn_{ij}", "8cec692b2137a322a4becf5d66be45c6": "\\mathbf{p}_\\mathrm{k} = m\\mathbf{v} \\,\\!", "8cec9aaa6d457e5c7ad738bcc98d88f7": " \\operatorname{E}[{}_{t}L|K(x)>t] = \\operatorname{E}[v^{K(x)+1-t}|K(x)>t] - P_{x}\\operatorname{E}[\\ddot{a}_{\\overline{K(x)+1-t|}}|K(x)>t]", "8cecb49d54303a92388e704545aa51d2": "\\Sigma_\\mathrm{left} = \n\\begin{pmatrix}\n\\psi_{\\alpha}\\\\\n0\n\\end{pmatrix}\n", "8cececfd32e721a94ce6cb3dd1ec1d1d": " d_S = {|E_S|\\over|V_S|} ", "8ced236b594a6672ec60a36a8964dec2": "\\omega\\cdot2+\\omega=\\omega\\cdot3", "8ced3d96509ce391473d81c059c6a44b": "(1 + i)\\ ", "8ced6341f12e11df28a06649dc1b1244": "4\\sqrt{0.25/n} = W", "8ceda38005104e91c32642a2c41a4e2f": "\\mathbf{\\otimes}", "8cee26b52be73cec859d14482cd8ffd2": "I_O = C_O L\\;", "8cee2df418ae3457f9db9e9e1025b9df": "g\\le (n-1)m(m-1)/2+m\\epsilon", "8cee35c13ad00d8f840b7eb5c9b5e8f4": "\\scriptstyle b_{i}", "8cee3aecea1dbdc4b3ac915d6bab02c8": "\\scriptstyle \\frac{d\\mathbf{q}}{dt}", "8cee5296a62f4a6662ebfed674c084d2": "z=e^{i\\theta}", "8ceeabef2f940ba6288ec5ea30da21b5": "a_k=\n\\left( \\begin{matrix} \n \\cos k\\alpha & -\\sin k\\alpha \\\\ \\sin k\\alpha & \\cos k\\alpha \n\\end{matrix} \\right)\n\\left( \\begin{matrix} e^p & 0 \\\\ 0 & e^{-p} \\end{matrix} \\right)\n\\left( \\begin{matrix} \n \\cos k\\alpha & \\sin k\\alpha \\\\ -\\sin k\\alpha & \\cos k\\alpha \n\\end{matrix} \\right)\n", "8cef18a81a1a0ca55fdc9bb51b737913": "B(f)=0.17+20\\log_{10}\\left(R_B(f)\\right)", "8cef6407fbe58fc5abf11cf5390516bf": "\n f = \\cfrac{1}{2\\pi a}\\left(\\cfrac{3\\gamma~p_A}{\\rho}\\right)^{1/2}\n ", "8cefb3f6f38022ba9e0c0cd8f2d26230": "1-1/e - o(1)", "8ceff9908a40becf959054eeb6551401": "6 = 2(3) = (1 + \\sqrt{-5}) (1 - \\sqrt{-5}).", "8ceffbbcbbfc1a19d3fd30d02310c4f1": "\n(\\Delta e_g,\\Delta e_h)\\ =\\ -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ 3 \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)\\ \\left(-\\left(e_h+\\frac{J_3\\ \\sin i}{J_2\\ 2\\ p}\\right) \\ ,\\ e_g \\right) \n", "8ceffd70888c282c4a38b6b5185330b8": "N_{i,n}(u)", "8cf05fa63837703e9038b69c72afe5f3": "^{3}He + ^{4}He = ^{7}Be + \\gamma\\ ", "8cf066ca6753972371f55ae5a0021f5a": "\\scriptstyle Ar(\\Delta ^n)\\, \\to\\, \\mathcal{C}", "8cf09bb3af8784764b202bfee6cd14e0": "L_{\\kappa} \\models ZFC", "8cf0f4f4fe18f159858236c76ca64000": "z=L", "8cf134334f2edf605754dbc64f88ddde": "\\textstyle x^{k-p}-1 = (x-1)(1 + x + \\ldots + x^{p-k-1})", "8cf16369bccb501b0be71576c391b9a7": "\\psi \\mapsto e^{i \\epsilon^{IJ} (x) \\sigma_{IJ}} \\psi", "8cf1c4964dd8b9746ca5b6feb65103d3": "V(1 - 1/e)", "8cf2ecdf47ac4c715c573b372cf5515b": " W(z) = \\prod_{p \\in P(z)} \\left(1 - \\frac{w(p)}{p} \\right) . ", "8cf309aa1bcccb4a2db4ae55f9c55f24": "\n\\begin{align}\n\\langle\\Psi|\\hat{F}|\\Psi\\rangle &= \\sum_{i=1}^{N}\\ \\langle\\phi_{i}|\\hat{f}|\\phi_{i}\\rangle, \\\\\n\\langle\\Psi|\\hat{F}|\\Psi_{m}^{p}\\rangle &= \\langle\\phi_{m}|\\hat{f}|\\phi_{p}\\rangle, \\\\\n\\langle\\Psi|\\hat{F}|\\Psi_{mn}^{pq}\\rangle &= 0.\n\\end{align}\n", "8cf38560d8d817bdf22aa4c650d7d429": "P_0=|000\\rangle\\langle000|+|111\\rangle\\langle111|", "8cf392011c0e819cc6f85019e9791188": "\\Lambda = \\Lambda _1 + \\Lambda _2 \\circ T, ", "8cf3b38642458c34a3b2d45692b18dec": "P_{cr} = \\frac{2Eh^2}{\\sqrt{3(1-\\mu^2)}}\\frac{1}{R^2}", "8cf416e2da2049e19ef989cf99f3ffa7": "\\sum f(\\Lambda')", "8cf44bc58730e8d7a18c8bc696dcc554": "\\max(\\kappa,\\aleph_0);", "8cf45bf0777b696eb77dbad34cda6ad4": " A,B ", "8cf46a8391747b6cbd3c2416bf88ec97": "\\bot\\in\\Gamma", "8cf47ea0efa9f8ec197eb176d561b289": "\\scriptstyle f(e)", "8cf485fc9f92a332ac05819e50009193": " \\frac{du}{dx} = u^2 + 1. ", "8cf4b81a6533a77ff128eaea3128d366": "k=\\frac{\\sin \\gamma \\sin \\alpha_2 \\sin \\beta_1}{\\sin \\delta \\sin \\alpha_1 \\sin \\beta_2}", "8cf5069ec7e39346875199da5ec7a0b6": "\\Pi_{1}=\\left\\vert +\\right\\rangle\n\\left\\langle +\\right\\vert ", "8cf51265a2064f8f042ef75080fb82e9": " \\int_{t_1}^{t_2}W v_z dt = \\frac{m}{2}V^2(t_2) - \\frac{m}{2}V^2(t_1). ", "8cf5c166f17e9882a7cf88d7e4eb2cc1": " \\operatorname{cov}(\\mathbf{A X} + \\mathbf{a}) = \\mathbf{A}\\, \\operatorname{cov}(\\mathbf{X})\\, \\mathbf{A^{\\rm T}} ", "8cf5f78ae831ae21de3ced83271652ec": "0<\\mu\\leq 1", "8cf62e973eecf42c2a5ae7ed88db8adb": "m_{L}=\\frac{m_{em}\\left(1-\\frac{1}{3}\\frac{v^{2}}{c^{2}}\\right)}{\\left(\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)^{8/3}},\\quad m_{T}=\\frac{m_{em}}{\\left(\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)^{2/3}}", "8cf65fa5acf02b635f6d848237084529": "\n \\rho~\\dot{e} - \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} + \\boldsymbol{\\nabla} \\cdot \\mathbf{q} - \\rho~s = 0 \n \\qquad \\implies \\qquad\n \\rho~\\dot{e} - \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} = - (\\boldsymbol{\\nabla} \\cdot \\mathbf{q} - \\rho~s).\n ", "8cf69d7ace2a994a5c15e31513deabb4": "{m}^3", "8cf6d4ae2157b81be4a432edc583b8b6": " \\mathbf{e}_2 \\mathbf{e}_3 = -\\mathbf{e}_3 \\mathbf{e}_2, \\,\\,\\, \\mathbf{e}_3 \\mathbf{e}_1 = -\\mathbf{e}_1 \\mathbf{e}_3,\\,\\,\\, \\mathbf{e}_1 \\mathbf{e}_2 = -\\mathbf{e}_2 \\mathbf{e}_1,\\!", "8cf6ef5e80d850924b93528ee4046d66": "v_r = \\frac{v}{RT_c/p_c}\\,", "8cf6facb7c924ccf6617f0f16a1a5450": "\\bold{B} \\cdot {\\rm d}\\bold{S}", "8cf782be116b0ecfed3f13c17189ba4d": "x^\\gamma", "8cf791f0269a30540c37dd7501ae67c2": "-\\delta X_{1:S}", "8cf7ab9b67fc2a82a9068d7e053c4a63": "R = N\\int\\Phi(E)\\sigma(E)dE", "8cf7d75c055489ec609e8e5b11658e2f": "K = \\sum_{j} \\pi_j m_{ij}", "8cf7df15badbd96d371a1059709aa01b": "\\xi_1=\\pm \\sqrt{\\frac{x_1-ix_2}{2}},\\quad \\xi_2=\\pm \\sqrt{\\frac{-x_1-ix_2}{2}}.", "8cf811f726e5e0c0e86a9b2ff644346a": "0.999\\ldots = \\lim_{n\\to\\infty}0.\\underbrace{ 99\\ldots9 }_{n} = \\lim_{n\\to\\infty}\\sum_{k = 1}^n\\frac{9}{10^k} = \\lim_{n\\to\\infty}\\left(1-\\frac{1}{10^n}\\right) = 1-\\lim_{n\\to\\infty}\\frac{1}{10^n} = 1.\\,", "8cf82874d50f721823520bae5b917527": "\\tan A = \\frac {\\textrm{opposite}} {\\textrm{adjacent}} = \\frac {a} {b}.", "8cf82e4f685bbac82dc7253f33f55199": "\\ g \\mapsto g^{\\omega}", "8cf853b9883aa1fb522d45d8cd82019e": "\\frac{ \\partial }{\\partial x_j} \\left( \\overline{\\sigma}_{ij} - \\tilde{\\sigma}_{ij} \\right)", "8cf88f5c9a1139bed12eb73f31de3a54": "\n \\Pr[a\\le X\\le b] = \\int_a^b f(x) \\, dx\n ", "8cf9111ae933940c2e74d9592abba36e": "\\Omega=\\{e_1,e_2,\\dots,e_n\\}", "8cf943f35f95da0c266ec28738154362": "\\vec{C}", "8cf964f4989b19b0ecd03688eb5ebe81": "|X-O|", "8cf96ef47b8c5118ca716f08727a3471": "2^{2^n}+1", "8cf98f68cddfd1125240152b1c9bfb67": "x \\in \\mathbb{R}", "8cfa7676d4c5bdb75344e449b9ceae1b": "P_2 - P_1 = \\frac{1}{2} \\rho (C_s^2 - C_u^2) ", "8cfa93bf683195ddf39236119358a44a": "M_B = \\begin{pmatrix} 0 & -B_{12} & B_{31} \\\\ B_{12} & 0 & -B_{23}\\\\ -B_{31} & B_{23} & 0 \\end{pmatrix}.", "8cfab6508f3c13e17b297f99c089f3f7": "|r-k|^2", "8cfae69bfa35114e325f5946fcbdc729": "E \\exp(i u^T X)=\\exp\\{-(u^T\\Sigma u)^{\\alpha/2}+i u^T \\delta)\\}", "8cfb69a870b7dfbd6e944a2c9c001b4d": "V_r", "8cfb733d1700d75b8f936928bfa654df": "\\mathbf \\nabla \\cdot \\mathbf A' = - \\mu_0 \\varepsilon_0 \\frac{\\partial \\varphi'}{\\partial t}", "8cfb84fc9ec19923dcdecd4673110a60": "\\bar c^a f^{abc}\\partial_\\mu A^{b\\mu}c^c", "8cfbabffc7ee98d1cd617c7853cf31d2": "Out = \\overline{AB}", "8cfbd12c0edd72e0a13fb9046f1e75eb": "\n\\mathrm{Hg} = -\\frac{1}{\\rho}\\frac{\\mathrm{d} p}{\\mathrm{d} x}\\frac{L^3}{\\delta^2}\n", "8cfc11e989ea2eb708fd0a1062f8d344": "(x,g) \\mapsto (x,xg)", "8cfc5b85d0c2b070f15be0ce089e3762": "E = \\frac{R T}{z F} \\ln\\frac{[\\text{ion outside cell}]}{[\\text{ion inside cell}]} = 2.303\\frac{R T}{z F} \\log_{10}\\frac{[\\text{ion outside cell}]}{[\\text{ion inside cell}]}.", "8cfc667f992544d1c841ace247169e82": "f_n: X_n \\to S_n", "8cfc725d718709390fde9bd679681241": "A (u(t)-u(t-\\tau)) \\, ", "8cfcbc707cbc787b66e5e09baa80a97b": "(x_1,y_1,0)", "8cfd04ac02acef40dfb12dcf45f8afa9": "\\frac{\\partial u}{\\partial \\mathbf{x}} + \\frac{\\partial v}{\\partial \\mathbf{x}} ", "8cfd0fd7d9f0e84fdc566bb53f745df9": "\\frac{1}{2}\\left(1 + i \\sqrt{3}\\right)", "8cfd3861e82a5a2020cba5a9876e9d8a": "P(t,x,y) = \\frac{1}{2\\pi} \\, \\int_{(x-x')^2 + (y-y')^2 < t^2} \\frac{p(x',y') \\, dx' dy'}{ \\left[ t^2-(x-x')^2-(y-y')^2 \\right]^{1/2}} ", "8cfd5c51693ec3609052b805276ddfe4": "\\cosh x = \\sum^{\\infty}_{n=0} \\frac{x^{2n}}{(2n)!} = 1 + \\frac{x^2}{2!} + \\frac{x^4}{4!} + \\cdots\\quad\\text{ for all } x\\!", "8cfd71b79dfd8b2cb65aa3ef3ed86160": "\\rho\\frac{D\\mathbf{v}}{D t} = -\\nabla \\pi + \\nabla \\cdot\\mathbb{T} + \\mathbf{f}", "8cfd741f1a78d84f30dbcf4d5a4d83bf": "\\rho w_{tt} + \\mu \\Delta\\Delta w = 0.\\,", "8cfdcc2929cbb0128d25f5c24972f266": "t,x,y,z", "8cfdd715c3a77d5609aab79dca8042c4": "\\left( \\frac{2}{3} \\right) ^4 \\times 2^3", "8cfde9cc9a0911c5609598fb2ebca20e": "16.2\\pm0.3", "8cfe6d9c1cfb098735cf64234dd2bcd4": "\nM = \\sum_{i=1}^{C} \\frac{k_i(k_i-1)}{2}\n", "8cfeabcc82bc703f7628f566d5cd2254": "|B^A|", "8cff6bd27b8a858970b85fed34e12796": "P_{bc} \\approx \\frac{P_{sc}}{\\frac{1}{2}k} = \\frac{4}{k}\\left(1 - \\frac{1}{\\sqrt M}\\right)Q\\left(\\sqrt{\\frac{3k}{M-1}\\frac{E_b}{N_0}}\\right)", "8cffb91a880104cd99f2663662ea3713": "\\mathbf{P}_{k|k} = (\\mathbf{I} - \\mathbf{K}_{k}\\mathbf{H}_{k})\\mathbf{P}_{k|k-1} ", "8d004a49afd91b936c15e0a91c5366b0": "\\liminf_{n\\to\\infty}d(n) = 2\n", "8d007681974ff15d6fdefa112368ddec": "\\begin{align}\n a_{n+1} &= \\frac{1}{2}(a_n + g_n)\\\\\n g_{n+1} &= \\sqrt{a_n g_n}\n\\end{align}", "8d0091cc890bd74aaa5c725505ab6972": " \\lambda=f^{-1}\\left[(f-a)(f-b)-(1+f-d)(1+f-e)\\right].", "8d009e9f09ebdf12ccd3263574090f05": "(\\mathcal{L}_{\\!X} f)(p) \\triangleq X_p(f) \\triangleq (Xf)(p) ", "8d016db94a170e5fa65099603f0e0614": "y = \\bar{y}", "8d019c7f9268aacf08f126966da175da": "\\{K_j\\}", "8d01ce7653944ce71f0728337c11ad42": " \\qquad \\qquad \\mathbf{q}= \\alpha_{te}\\alpha_{ee}^{-1}\\mathbf{j}_e-\\frac{\\alpha_{tt}-\\alpha_{te}\\alpha_{ee}^{-1}\\alpha_{et}}{T^2}\\nabla T.", "8d01fd88af3de1b4bc96b98671778404": "\n \\begin{bmatrix}\n 1 & -\\tfrac{2}{3} & -\\tfrac{11}{3} & 0 & 0 & -\\tfrac{4}{3} & -20 \\\\ \n 0 & \\tfrac{7}{3} & \\tfrac{1}{3} & 0 & 1 & -\\tfrac{1}{3} & 5 \\\\\n 0 & \\tfrac{2}{3} & \\tfrac{5}{3} & 1 & 0 & \\tfrac{1}{3} & 5\n \\end{bmatrix}\n", "8d02729e022fa1fd8c34741bab93b66d": "\\ell = 1/\\sqrt{2} n \\sigma \\,\\!", "8d02bc6103faf2491e9f30a619b439f6": "w(f) \\leq (4/3) \\cdot \\min_{f^{*}} \\{ w(f^{*}) \\}", "8d0398538f85d228e043c62752cd64ad": "R\\bowtie S=\n \\begin{matrix}\n \\texttt{name} & \\texttt{age} & \\texttt{income} \\\\\n \\texttt{A} & \\texttt{34} & \\texttt{20'000} \\\\\n \\texttt{B} & \\texttt{47} & \\texttt{32'000} \\\\\n \\end{matrix}\\,", "8d0419e8eb2186e80c974118f38e51d8": "d\\in S_B", "8d041f0b6ce851b39e2963f737e89a30": " {1,2,...,m} ", "8d0427bb3bdf9f771e8ed74a3ebb1b24": "{rK}/{Y}", "8d04575ab8a9719f320edb85eba5c45e": "\\omega_e\\chi_e", "8d04a6d889e99605226fd81340b546bb": "\\cot (\\alpha + \\beta) = \\frac{\\cos (\\alpha + \\beta)}{\\sin (\\alpha + \\beta)}\\,", "8d04bb219bbc5c52a32c6f7df9dc6f51": "\\frac{\\partial f}{\\partial x} = f_x = \\partial_x f = \\partial^x f, ", "8d050e8544f55fd8e6abf54accc541f5": "C_{P,m} - C_{V,m} = R", "8d05251c81a4b758a5d0b11c3b09fa56": "\\sigma^2 = \\langle (\\phi - \\bar\\phi)^2 \\rangle", "8d052b5ed6ce9473485cec9ef463c343": "d(x,y)=\\sum_{i=1}^\\infty \\frac1{2^i}\\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.", "8d057cac3fad3732d5544c0ebb0f2da4": "\\int x R\\,dx = \\frac{R^3}{3a}-\\frac{b(2ax+b)}{8a^{2}} R - \\frac{b(4ac-b^{2})}{16a^{2}} \\int \\frac{dx}{ R}", "8d0592628cb891479e0b0edfc338de5a": "z = r e^{i \\varphi}.\\,", "8d059325f1360a1ea795c23ea898fcb7": "S_{+}", "8d05d6fe8e98ec4ceeb8afb18c3f9d9b": "A \\Psi(\\mathbf{r}) \\ \\stackrel{\\text{def}}{=}\\ \\lang \\mathbf{r}|A|\\Psi\\rang ", "8d05f4944b89477dfbcc290e6c39ec61": " y_{1}(x) = e^{ax} \\cos bx \\, ", "8d062e2e5170c49232b14b2ac2b0888e": "u(x,y)=u(x,-y), \\quad v(x,y) = -v(x,-y) \\quad", "8d0650536bd60032f8e420a3e99b5930": " \n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{deposit}\\\\\n\\text{accumulation}\n\\end{array} \\right]=\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{deposition}\n\\end{array} \\right] -\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{re-entrainment of}\\\\\n\\text{unconsolidated deposit}\n\\end{array} \\right]\n", "8d066fbd6196c14dab475a2c8d6401a6": "A+D=0", "8d06860ce1832883b71e5b79f682da61": " \\liminf_{N \\to \\infty} S_N f(x_N) \\geq f(x_0^-) - a\\cdot (0.089490\\dots).", "8d06b410a041e4ae4dfbe2b207370498": "q = \\iiint\\limits_V \\rho dV.", "8d06f199caacb59bedd8b0c9134d186e": " \\hat{H} = \\frac{\\hat{\\mathbf{p}}\\cdot\\hat{\\mathbf{p}}}{2m} + V(\\mathbf{r},t) \\,,\\quad \\hat{\\mathbf{p}} = -i\\hbar \\nabla ", "8d070eac0725af35e550046dbc5f0a1e": "A_i\\subseteq\\Bbb{R}^m", "8d072ceaa802beec4f0040d4179f7138": "\\Phi_b", "8d073b86b3b538fc4feea1a482f691e9": "\\{e_1,\\ldots,e_n\\}", "8d078e0584137d22b38e0e3f0d4daee9": "g_{\\mu 3}=\\, 0", "8d08313aafbea367a529110638907f98": " N = N_1 + N_2 + \\dots + N_n; \\Psi(x) = \\sum_{i=1}^n \\|x^{(i)}\\|_2 ", "8d08d3cc6ed0ed3aaf169f9c1e7659eb": "\\, \\frac{e^{it\\mu}}{1 + b^2t^2}", "8d0917a8f643e6c87ddb3d491703662e": "\\int e^{cx}\\sin bx\\; \\mathrm{d}x = \\frac{e^{cx}}{c^2+b^2}(c\\sin bx - b\\cos bx)", "8d09d923b20d30f92380328038cc7cfe": "N(t) = N_0 e^{-\\lambda t}", "8d09db7791333fa402fbacb5c1b7e1a9": "= \\pm\\frac{\\sin \\theta}{\\sqrt{1 - \\sin^2 \\theta}}", "8d0ad8ec612a698b8d82f8dd0961feb8": "\\mu(X) = \\sigma \\sqrt{\\frac{\\pi}{2}}\\ \\approx 1.253 \\sigma", "8d0b02088ab5399d641ddda333ec2373": "\n\\begin{pmatrix}\n A & B \\\\\n C & D \n\\end{pmatrix}\n=\n\\begin{pmatrix}\nI & 0 \\\\\nC A^{-1} & I\n\\end{pmatrix}\n\\begin{pmatrix}\nA & 0 \\\\\n0 & D-C A^{-1} B\n\\end{pmatrix}\n\\begin{pmatrix}\nI & A^{-1} B \\\\\n0 & I\n\\end{pmatrix}\n", "8d0b2f954750e03551d540368d50cf8c": "\\|f-f_s\\|_{\\infty,X}=\\delta(s);", "8d0b39b21dca326c32b1a2f86bb09ba4": "F_Y(y)=\\frac{F(y)-F(a)}{F(b)-F(a)} \\,", "8d0b777509c1b30ad43aee6c741f3741": "\\overline\\psi\\to \\exp[-iQ\\phi(x)]\\overline\\psi", "8d0b9572406ec8402e37a364182e1188": " e_s ", "8d0bd4ff9aefa337b161ad7c1a32cf00": "\\sum_{r \\in R} s(r)r", "8d0c2d60fe470425581471ab5829b891": " \\begin{align} \n\\langle \\alpha',j'm'|[J_{\\pm}, T_q^{(k)}]|\\alpha,jm\\rangle\n& = \\sqrt{(j'\\pm m')(j'\\mp m'+1)}\\langle \\alpha',j'm'\\mp1 |T_{q}^{(k)}|\\alpha,jm\\rangle\\\\\n& \\qquad -\\sqrt{(j\\mp m)(j\\pm m+1)}\\langle \\alpha',j'm' |T_{q}^{(k)}|\\alpha,jm\\pm 1\\rangle \n\\end{align} ", "8d0d5492c9dd188583b9b4422b1d1a46": " \\mathbb{Q}^d ", "8d0d8ead02d9e3f2500ec8850dd38d7f": "T \\ll T_c\\,", "8d0dab171ca7323f99aa2d4deecc4abf": "f(r,\\frac{r^2}{p})=0", "8d0de7b2bc4d4c8de130d6714af68a43": " \\alpha =\\frac{k}{\\rho C_p} ", "8d0e6652c144fe1641ba9c0aa192a018": "\\tau_\\lambda", "8d0e68b6ca706085a636eaf86e749999": "\n \\Gamma_{ij}^k = \\cfrac{\\partial \\mathbf{b}_i}{\\partial q^j}\\cdot\\mathbf{b}^k = -\\mathbf{b}_i\\cdot\\cfrac{\\partial \\mathbf{b}^k}{\\partial q^j}\n ", "8d0e8eb2998e2cbb0f2f8ce7d36a2c38": "\\log_2(M)", "8d0f29924ffd9459a3aa414f5042dd2d": "\\theta_{cp}\\;", "8d0f51a5257ecf580cf91a4d5e4d3cce": " \\|S_1 S_2\\| < 1 ", "8d0f586781acf15fb01affaedef2261e": "s_P=\\sum_{i=1}^m x_i", "8d0fd48c64bdd225dc2c4cfcf67c8236": " \\Phi = \\frac {\\textrm{Number of photons emitted}} {\\textrm{Number of photons absorbed}} ", "8d0ffac658d8c61ff35273cff2d48d8d": " \\omega _1 = \\omega _0 + \\alpha t \\,", "8d1006f3cb8a7a38ae720fb699460404": "A_{g} + S \\rightleftharpoons AS", "8d1012777b87466643a11b2aadbc7f92": "C(X)", "8d101cefe17684bbc0e74590e4f3d9e7": "\\Psi(x) = C_A Ai\\left( \\sqrt[3]{v_1} (x - x_1) \\right) + C_B Bi\\left( \\sqrt[3]{v_1} (x - x_1) \\right)", "8d101eb4335def0ceaa76076c793debc": "\nM M' = \\begin{pmatrix}\n\\det(M) & m_{1,2} & m_{1,3} & \\ldots & m_{1,k-1} & 0 \\\\\n0 & m_{2,2} & m_{2,3} & \\ldots & m_{2,k-1} & 0 \\\\\n0 & m_{3,2} & m_{3,3} & \\ldots & m_{3,k-1} & 0 \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots & \\vdots & \\vdots \\\\\n0 & m_{k-1,2} & m_{k-1,3} & \\ldots & m_{k-1,k-1} & 0 \\\\\n0 & m_{k,2} & m_{k,3} & \\ldots & m_{k,k-1} & \\det(M)\n\\end{pmatrix}\n", "8d10bcefda4ca9796ae80f0f0abc645c": " (q^{i},p_{j}) = \\delta^{i}_{j} . ", "8d10e4f747726913cf3969aca22065a7": "\\mathrm{NF} = 10 \\log(F) = 10 \\log\\left(\\frac{\\mathrm{SNR}_\\mathrm{in}}{\\mathrm{SNR}_\\mathrm{out}}\\right) = \\mathrm{SNR}_\\mathrm{in, dB} - \\mathrm{SNR}_\\mathrm{out, dB}", "8d111bbb28dce7282cc0bfe306fd112a": "MA = \\frac{W}{T} = n.", "8d11721ffa09efb6c6f33be26982dc58": "\\sqrt{\\frac{5}{8}}\\!\\,", "8d118dbefdf6d52262ffd7129ed7f0ab": " \\cos ^2 \\theta - \\sin^2\\theta=\\cos 2\\theta \\qquad \\text{and} \\qquad \\sin 2\\theta= 2\\sin\\theta\\cos\\theta", "8d11e0a2f86e7b211786ed44d679bf09": "V^{\\otimes r} = V \\otimes \\cdots \\otimes V \\quad (r\\text{ factors}). \\, ", "8d12044a9eade18b0b6c8fce02ab0959": "\\operatorname{tr}\\ \\operatorname{id}_{\\mathbf{R}^2} = \\operatorname{tr} \\left(\\begin{smallmatrix} 1 & 0 \\\\ 0 & 1 \\end{smallmatrix}\\right) = 1 + 1 = 2.", "8d123c6fae3baeec43c6d0a91e0ea1c2": "|c_n|\\leq(t+\\epsilon)^n", "8d1241c72008906cd9c94348bafd5090": "\\scriptstyle \\frac{1}{\\lambda}=\\frac{E}{hc}", "8d125cbd7cce5096b466575b920c4c67": "(a\\uparrow ^m)^b", "8d125d67c6e8c5bdb917bd23ba5d869c": "d \\leq n-k+1", "8d127ec5c16a13aa9a7189f16b9cece0": "(Q_Hx_1,x_2,y_1,y_2)\\phi(x_1,x_2,y_1,y_2)\\equiv\\begin{pmatrix}\\forall x_1 \\exists y_1\\\\ \\forall x_2 \\exists y_2\\end{pmatrix}\\phi(x_1,x_2,y_1,y_2)", "8d1286ca500c379b81714334afd2ae1c": "\\begin{align}\n&Z_h^2 \\left( \\langle fg\\rangle_h - \\langle f \\rangle_h \\langle g \\rangle_h \\right)\\\\\n &\\qquad= \\iint d\\mu(x) \\, d\\mu(y) f(x) (g(x) - g(y)) e^{-h(x)-h(y)} \\\\\n &\\qquad= \\sum_{k=0}^\\infty\n \\iint d\\mu(x) \\, d\\mu(y) f(x) (g(x) - g(y)) \\frac{(-h(x)-h(y))^k}{k!}.\n\\end{align} ", "8d12a327637dfaa3cd5cd84667b0894a": "p = \\frac{Nm \\langle v^2 \\rangle}{3V} = \\frac{nM_m \\langle v^2 \\rangle}{3V} = \\frac{1}{3}\\rho \\langle v^2 \\rangle \\,\\!", "8d12b8849fcf184f52144a1be05cf9cd": "Cz", "8d12ccbc5e9360c738846d8e85e4153a": " E_K", "8d1307f5fde582a869d9be758b06fc78": "\\scriptstyle \\sum_x |x|^2p(0,x)\\le\\infty ", "8d13334dc52a040bbbb8e2ff93e55ce3": " \\min_x \\| f - D x \\|_2^2 \\ \\text{ subject to } \\ \\|x\\|_0 \\le N, ", "8d134380780efb0896bfda0d6c9cfde8": "\\displaystyle pq = ac + bd.", "8d1363fd9781efc603213390e8a7aaec": " F_{hullcourse} = b \\times V_b^2 = - F_{forward}", "8d1384918c66ab41e64d915c586a338a": "{\\bar{S}}_4", "8d13a5fe9c74f2a13e93d90fffc1c117": "\\overline{\\mathbf x}=\\frac{\\mathbf{x}_1+\\cdots+\\mathbf{x}_n}{n}", "8d13c8a7736c7d3bbab40fba6d43a464": "C=\\frac{8L^3}{E^*wt^3} \\left(1+3c \\left(\\frac{t}{L}\\right)^{2-n}+\\frac{\\alpha_s E^*}{4 \\mu} \\left(\\frac{t}{L}\\right)^2 \\right)", "8d14503365c19b6e24fb996d336d1f93": "l_\\alpha", "8d15f986a6cc87c9d366fc6032a4a2af": "B(t,t_0)=\\int_{t_0}^tb_\\mathrm{in}(t^\\prime)\\mathrm{d}t^\\prime\\,.", "8d1626e162e880dcbeed3b06e1c67d9e": "c_{v} \\frac{dT}{dt} + p \\frac{d\\alpha}{dt} = q + f", "8d16932cd2d7d0d5aa264f213956f8d3": "\\,c", "8d1695fb4855b8bc4d54ca9a206a1847": "\\frac{7\\cdot\\pi}{6}", "8d16be724b6dd63430946b24aca29779": "bx-a", "8d1710aef2dce718fa584d13e54a0829": "S = k \\log \\Omega_{0}", "8d1722023d661b2e48fb01ef60c3dc77": "X \\sim N(\\mu,\\sigma^2) \\Leftrightarrow \\frac{1}{\\sigma} (X - \\mu) \\sim N(0,1).", "8d17701ce8a10fd24b7238ebbe409354": " \\frac{d T(t)}{d t} = \\frac{d\\Delta T(t)}{d t} = - \\frac{1}{t_0} \\Delta T(t)\\quad ", "8d177e9ddd406b820e2bb19062e3545c": "x_{ik}", "8d177ebe141c9a90993f295ed8725c06": "| \\Psi \\rangle=\\sum\\limits_{i_1,..,i_N=1}^{M}\\sum\\limits_{\\alpha_1,..,\\alpha_{N-1}=0}^{\\chi}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}\\Gamma^{[2]i_2}_{\\alpha_1\\alpha_2}\\lambda^{[2]}_{\\alpha_2}\\Gamma^{[3]i_3}_{\\alpha_2\\alpha_3}\\lambda^{[3]}_{\\alpha_3}\\cdot..\\cdot\\Gamma^{[{N-1}]i_{N-1}}_{\\alpha_{N-2}\\alpha_{N-1}}\\lambda^{[N-1]}_{\\alpha_{N-1}}\\Gamma^{[N]i_N}_{\\alpha_{N-1}} | {i_1,i_2,..,i_{N-1},i_N} \\rangle", "8d17c9f84e5bcb0557698337f6c81302": "\\nu(\\emptyset)=0", "8d17d5f2d04fc5eb75d23302f5649039": "p_1,\\, \\dots,\\, p_k", "8d17f3db8ab10beabb0c6256acdd46e1": "u''", "8d1824c1c4fa075b9d25166adc7e325d": " \\{x\\,\\mid\\,f(x) > t\\} \\in X\\quad \\text{for all}\\ t\\in\\mathbf{R}. ", "8d18563b1fe045d03aab7f6a19cfbece": "\nf(\\theta\\circ(\\theta_1,\\ldots,\\theta_n))\n=\nf(\\theta)\\circ(f(\\theta_1),\\ldots,f(\\theta_n))\n", "8d1892ba821a3d2a2cc20d708cce1b82": "A(w)= \\sum_{n=0}^\\infty a_n w^n \\quad", "8d18a5571009da46300ce40fdcf12031": "u(ci, x, y) = m_1 + m_2 + m_4 + m_7", "8d18a8b55192dfad3f8844303a35759b": "\\displaystyle i\\partial_t\\psi=-{1\\over 2}\\partial^2_x\\psi+\\partial_x(i\\kappa|\\psi|^2 \\psi)", "8d18f1060b971f924176be294ab1c7b7": "\\Delta_{\\mathrm{ret}}", "8d18f593c73579dc3d5895fd018e4d8a": "Y_{t} = Y_{t-1} = Y_{t-2} = Y_{p}", "8d18f729a39095f579348d52e46fc585": "e=5", "8d198e70b5fb0215e96092a8636b04d5": " Q^{n+1}_i = Q^n_i - \\lambda \\left( \\hat{f}^n_{i+1/2} - \\hat{f}^n_{i-1/2} \\right), \\quad \\lambda = \\frac{\\Delta t}{\\Delta x}, \\quad \\hat{f}^n_{i-1/2} = f^\\downarrow\\left( Q^n_{i-1}, Q^n_i \\right) ", "8d199669c948a5719ec870c4098bdee6": "T_{i_{\\sigma 1}i_{\\sigma 2}\\dots i_{\\sigma k}} = T_{i_1i_2\\dots i_k}", "8d19cc1bff6517cb8ad9b021687fa007": "C\\ell_{p+1,q+1}(\\mathbf{R}) = \\mathrm{M}_2(C\\ell_{p,q}(\\mathbf{R}))", "8d19d2f8772f41e9827fbb79c98764ce": "\n\\begin{matrix}\ns &=& \\sqrt{x^2 + y^2}\\\\\n\\alpha &=& \\arctan{(y/x)}.\n\\end{matrix}\n", "8d19f59af6754dfe688ce7f9b3484544": "\\frac{e^{st/u}}{2\\pi i}", "8d1a3299c1db097002179c2fb189576f": "(\\lnot \\alpha)", "8d1a7ba6120269b2ea1939aab5740658": "\\mathbf{e} = e_x\\hat{x} + e_y\\hat{y} + e_z\\hat{z} ", "8d1a7c3bb249fa1003b92564881c16d9": "T \\hat{\\mathbf{S}} T^\\dagger = - \\hat{\\mathbf{S}} ", "8d1a7ceb19dabd614baccdb9d5ce22ba": "IdB", "8d1a8438c1f792843b1de141c66fc2b7": "M_+(z) = u(z) + i v(z)\\!", "8d1af58d997b6ccce42547c8cd201a97": "S(v,w,u) = (1 - p)^{-1}.", "8d1b0901d02f13f009929efefdb96bce": " \\left ( {\\partial S\\over \\partial N} \\right )_{V,U} = - { \\mu \\over T } ", "8d1b2fbbf800a9b960e05da481f0e7f6": "\\scriptstyle \\gamma_\\mathrm{ls}", "8d1baa7f04ce92880fb24998cc61ef4f": "\\left(\\frac{{}^{187}\\mathrm{Os}}{{}^{188}\\mathrm{Os}}\\right)_{\\mathrm{present}} = \\left(\\frac{{}^{187}\\mathrm{Os}}{{}^{188}\\mathrm{Os}}\\right)_{\\mathrm{initial}} + \\left(\\frac{{}^{187}\\mathrm{Re}}{{}^{188}\\mathrm{Os}}\\right) \\cdot (e^{\\lambda t}-1),", "8d1bf8a3cc74288545760d1d1709cdfa": "{P}^{5}-5{P}^{3}Q+5P{Q}^{2}\\, ", "8d1bfe250d88c25f88f87558a510f951": "\\left( \\overline{D}^2 - 8R \\right) f", "8d1c1a4ce7a02791a00e488972969c9e": "d z = {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y \\left [ {\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z d y + {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y dz \\right ] + {\\left ( \\frac{\\partial z}{\\partial y} \\right )}_x dy,", "8d1cbda01e8ab4b6f660abcec9033cf9": "d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}", "8d1cca13aa769db0a12364122dc75e56": "\\gamma \\!\\,", "8d1d121a54d1801208f71fa3dff54310": "x \\bullet y = \\mbox {round} (x \\circ y)", "8d1d2c9be75f61d5f0949290fa5504f4": "x\\rightarrow x + \\xi_1x^2+\\xi_2x^4+\\xi_3x^8+\\cdots", "8d1d538d30d9e8651b678480c46b5ba3": "\\int \\sec{x} \\, \\tan{x} \\, dx = \\sec{x} + C", "8d1d78d2946c1827553aa64ad4e93a74": " (\\tfrac{4}{3}+5)\\times 3 ", "8d1d848c3d2b986c510efee50dc5d196": "\\textstyle b + l_2 - 1 = 2l - 1 + \\delta", "8d1db44883a9f180f0c0222760b25194": "(p^a)' = ap^{a-1}\\textrm{,}\\!", "8d1dc3c660433f289282debc4dba780a": "expr", "8d1dceac0e18739e11f81a80abe89c1f": "\\Phi(\\alpha) = \\langle \\phi | \\alpha\\rangle", "8d1e6a5889429a024d8dc1c54c0de56d": "C_D\\;", "8d1e79003375a2080a5300c021e5da28": "\\{b\\in A : b E a\\}=\\{b\\in B : b F a\\}", "8d1eaec3562c27cfd1c546d6870dfa42": " (A \\circ B)_{i,j} = (A)_{i,j} \\cdot (B)_{i,j}", "8d1f1409f0b2121a18891c263be5cd73": "L_\\mathrm{total} = L_1 + L_2 + \\cdots + L_n", "8d1f243ba6da01a7aadc1e872ca62c97": " = e ^ {V / V_e} = M", "8d1f4ab04e2ed5358838fca29ac958a9": "\\delta (u, v) = 2 \\frac{\\lVert u-v \\rVert^2}{(1-\\lVert u \\rVert^2)(1-\\lVert v \\rVert^2)},\\,", "8d1f746a72ef899eddedc196b77d32cd": "a_{T+1}", "8d1f7629d801509d5de78eceb9a18eeb": "\\frac{dp_\\alpha}{d\\tau} = \\frac{1}{c}\\left[ q_{\\mathrm e} F_{\\alpha\\beta}v^\\beta + q_{\\mathrm m} {\\star F_{\\alpha\\beta}}v^\\beta \\right]", "8d1facc6535cb97311f2c03515a51146": "M^2 \\,", "8d1fcd83952721fdbea924d28c7d20c4": "\\frac{\\pi}{4} = \\sum_{n=0}^\\infty \\frac{(-1)^n}{2n+1} \\approx 0.785398", "8d1fde9a1d6f275f54d1922108f37cd8": "E_m(x)=\n\\sum_{k=0}^m {m \\choose k} \\frac{E_k}{2^k}\n\\left(x-\\frac{1}{2}\\right)^{m-k} \\,.", "8d200dec42034d0593c0379d67472f79": "\\begin{bmatrix}\n0 & 1 & 0 & 1 \\\\\n1 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 1 \\\\\n1 & 0 & 1 & 0\n\\end{bmatrix} \\text{ and } \\begin{bmatrix}\n0 & 0 & 1 & 1 \\\\\n0 & 0 & 1 & 1 \\\\\n1 & 1 & 0 & 0 \\\\\n1 & 1 & 0 & 0\n\\end{bmatrix} \\text{ and } \\begin{bmatrix}\n1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1 \\\\\n1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1\n\\end{bmatrix} \\text{ and } \\begin{bmatrix}\n1 & 0 & 0 & 1 \\\\\n0 & 1 & 1 & 0 \\\\\n0 & 1 & 1 & 0 \\\\\n1 & 0 & 0 & 1\n\\end{bmatrix}.", "8d203cd17e44003b650118bb8e82904c": " \\lim_{y \\to a} E(u(X)|X>y) = E(u(X)) ", "8d206989438793f52d216dc7d113ee56": "x^{ 11 }+x^{ 9 }+1", "8d21ab8b6449a74ba14d6c15a9de540f": " {1 \\over {|\\mathcal{A}|\\cdot|\\mathcal{B}|}}\\sum_{x \\in \\mathcal{A}}\\sum_{ y \\in \\mathcal{B}} d(x,y)", "8d21e9a1ffccf68c19edbc7000a41def": " 0 \\to H^0(\\mathcal{O}_X) \\to H^0(K) \\to H^0( K|_D) \\to H^1(\\mathcal{O}_X) \\to ", "8d21ebece7d053dcda36d458a0d7dbd1": " (g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu) (g_{\\lambda \\nu ,\\mu} \\dot x^\\nu \\dot x^\\mu + g_{\\mu \\lambda ,\\nu} \\dot x^\\mu \\dot x^\\nu + g_{\\lambda \\nu} \\ddot x^\\nu + g_{\\lambda \\mu} \\ddot x^\\mu) ", "8d2208df41b812f5487cbcff46926458": "L_z/\\hbar", "8d221e210c40e97040c93f8be923d88e": "S(\\tau_B)-S(\\tau_A)= \\int_A^B dS=\n\\int_{\\tau_A}^{\\tau_B} \\frac{dS}{d\\tau}d\\tau=\n\\int_{\\tau_A}^{\\tau_B} \\frac{S(\\tau + d \\tau)-S(\\tau)}{(\\tau + d\\tau)-\\tau}d\\tau\n", "8d2228d96d563a5c6f46de3f4b1d12c2": " c_c=2m\\omega_n ", "8d2229a09798cb43c994c8cc9fbfbcf2": "\\vec{E} \\times \\vec{B}", "8d2254c70db7bb3409203cfbfb9cce3a": "\n\\begin{align}\n & \\left[\n \\frac{\\partial^2 \\Phi}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi}{\\partial z}\n \\right]_0\n + \\eta \\left[ \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi}{\\partial z}\n \\right) \n \\right]_0\n + \\left[\n \\frac{\\partial}{\\partial t} \\left( |\\mathbf{u}|^2 \\right)\n \\right]_0\n \\\\ & \\quad\n + \\tfrac12\\, \\eta^2 \n \\left[ \\frac{\\partial^2}{\\partial z^2} \n \\left(\n \\frac{\\partial^2 \\Phi}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi}{\\partial z}\n \\right) \n \\right]_0\n + \\eta \\left[\n \\frac{\\partial^2}{\\partial t\\, \\partial z} \\left( |\\mathbf{u}|^2 \\right)\n \\right]_0\n + \\biggl[\n \\tfrac12\\, \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\left( |\\mathbf{u}|^2 \\right)\n \\biggr]_0\n \\\\ & \\quad\n + \\cdots\n = 0,\n\\end{align}\n", "8d2271c24df06480d57eb4b03bd2e2e0": "D\\Delta u_\\varepsilon (y) + \\frac{1}{\\gamma}F(y)\\cdot\\nabla u_{\\varepsilon}(y) = -1 ", "8d2289fe1b3e910267db6f7ff6aa670b": "m\\ \\arg(z) - n\\ \\arg(z-a) = \\arg(z^m (z-a)^{-n} ) = const", "8d22e27089904bc8266daf7c2459d7fd": "V = \\frac{1}{4} (15+7\\sqrt{5}) a^3 \\approx 7.6631189606a^3", "8d230354d61bbeb5f89db5780353713d": "S^2_{n+1} = \\frac{1}{n+1}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2", "8d238bc729e7dcec29b87f0ba6af5d8d": "t_1\\ ", "8d246a37d276a73fc0bde676f4ca289d": "y=\\phi", "8d24798fe50b190d57cb59e86b253770": "|c|<1", "8d249cbc8ca62392c07ea84620ff788d": "U_{2k}=U_k\\cdot V_k", "8d24b99f2418700f0af0b974c0f60d30": "\n\\frac{d^2x}{ds^2}+\\Gamma^{i}_{jk}\\frac{dx^{j}}{ds}\\frac{dx^{k}}{ds}+\\Delta^{i}_{jk}\\frac{dx^{j}}{ds}\\frac{dx^{k}}{ds}=0~~~~~~~~~~~~~~(2)\n", "8d24d31a08abee359d8f29ea7e25691c": "1,\\;\\omega,\\;\\omega^2,\\;\\ldots,\\;\\omega^{n-1},", "8d24db9d1f6fddf1c038a2c51a94b2f7": " \\vartheta_4(z,\\tau) = 0 \\quad \\Longleftrightarrow \\quad z = m + n \\tau + \\frac{\\tau}{2} ", "8d2507bc15214c32b72f97b8459061fa": "\n\\operatorname{log-odds} \\{X_{n1}=1 \\mid \\ r_n=1\\} = \\delta_2-\\delta_1,\\,\n", "8d2518ec5bbf47635b45f10336fcb340": " [X_i]=\\begin{bmatrix} 1 & 0 & 0 & r_{i,i+1} \\\\ 0 & \\cos\\alpha_{i,i+1} & -\\sin\\alpha_{i,i+1} & 0 \\\\ 0& \\sin\\alpha_{i,i+1} & \\cos\\alpha_{i,i+1} & 0 \\\\ 0 & 0 & 0 & 1\\end{bmatrix},", "8d252cdd158fdcb5e3eaf8fcfc21d2cf": "end\\, for", "8d25314197f5f038dc2e6c501f80c752": "C_A (A+1)x = (A-1)x. \\quad \\mbox{i.e.} \\quad C_A = (A-1)(A+1)^{-1}.\\,", "8d2545d69bda68c9ad86c6ed560acfb1": "\\textstyle\\sinh x = \\sum_{n=0}^\\infty\\frac{x^{2n+1}}{(2n+1)!}", "8d254aa8e4178545d9340c92b63a7418": "|M| = |A|-1", "8d254aaf40db8d903275cb3d54c06f8e": "\\mathcal{L}_H = [(\\partial_\\mu -ig W_\\mu^a t^a -ig'Y_{\\phi} B_\\mu)\\phi]^2 + \\mu^2 \\phi^\\dagger\\phi-\\lambda (\\phi^\\dagger\\phi)^2,", "8d25e3a37823ae053210bf496f5d2878": "\\kappa = \\frac{0.60-0.54}{1-0.54} = 0.1304", "8d2645535798ffcd98eb5b3f547ffc09": " \\dot{\\hat{x}} = \\left [ \\frac{\\partial H(\\hat{x})}{\\partial x}\n\n\\right]^{-1} M(\\hat{x}) \\, \\operatorname{sgn}( V(t) - H(\\hat{x}) )", "8d2685cdfc3a8147792d088a6565ac57": " \\langle\\mathbf{\\hat X}\\rangle \\rightarrow \\langle\\mathbf{\\hat X}\\rangle + \\varepsilon ", "8d2688eacfaaa425776f91da1fb46ebe": "p(X_1,X_2,...,X_n) = \\int \\prod_{i=1}^n p(X_i|\\theta)\\,dP(\\theta).", "8d269c52cf6cbfe5b8bb547e0a8819b9": "m:T_qQ \\to \\mathbb{R}", "8d26ddf98c9e166b971daf1a6c6ad41d": "\\ y_2 = y'y_c = (0.93)(2.3 ft) = 1.0 ft ", "8d26eeff63f15689f5d6161d00e56df6": "= f(z) + f'(z)\\cdot(\\xi-z) + \\dots + \\frac{f^{(n)}(z)}{n!}\\cdot(\\xi-z)^n", "8d26fa1a66411dff141d3ac8f612dbb7": "\\mu-\\beta\\log\\left(\\tfrac{X}{Y}\\right) \\sim \\mathrm{Logistic}(\\mu,\\beta) ", "8d2709d4ad96eb586be41e363d12fe2e": "X(f)\\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty} x(t) \\ e^{- i 2 \\pi f t} \\ {\\rm d}t,", "8d27546c84451d3d2a47c67ace97cfa9": " P( | \\mu - \\sigma | ) \\le 0.5 \\, ", "8d278d07c2a5fd9efa325613ed22fef8": "\n( C_1 \\circ C_2) ( [x]) =C_1 ( C_2 ([x]) ) \n", "8d288bbcf455d6784ef1da4e8150a35a": "\\left|\\sigma_{ij}- \\lambda\\delta_{ij} \\right| = -\\lambda^3 + I_1\\lambda^2 - I_2\\lambda + I_3=0\\,\\!", "8d28bff79008eb975a172a9e8971081a": "v = \\frac{c}{\\sqrt{1 + \\frac{e + P}{2 P_m}}}", "8d2924e9f1c55a0ee15987f6412d7887": "\\pi_6 = \\frac{r}{a} = R(r).", "8d292bece156356f1b52bcc7cfdd0e54": "Sym^2 C", "8d292d99561cb8fe842ac6bdc1ce6afd": "V_{2n}", "8d2934ca2702991b3eb0f483fe62c7fe": "f(x) = \\|x\\|_X, \\ \\ \\| f \\|_{X'} \\le 1.", "8d2997f062c7c2ab6d0293669203fc66": "(\\lambda x : x + 1) 3 _\\beta\\rightarrow 4", "8d29a1cf48b1498370f629e8c79324c1": "\\tfrac{0.5}{T}", "8d29ad944b23986a7bee93bc052177a5": "\\Delta\\omega = \\frac{1}{C R_0}", "8d2a6f9bc9e3c290a8d3a0782e96d61d": "\\textstyle P(A\\mid[x]) < \\alpha", "8d2ab54bf21eded68153eed2c458690d": "J^+(x) \\cap J^-(y)", "8d2ac5d27e304f21f76beb997c306748": "\\partial_{t} \\pi^{ij} = -N\\sqrt{g} ( R^{ij} - \\frac{1}{2} R g^{ij} ) + \\frac{1}{2} Ng^{-1/2}g^{ij} ( \\pi^{mn}\\pi_{mn} - \\frac{1}{2} \\pi^{2} ) - 2Ng^{-1/2} ( \\pi^{in}\\pi_{n}{}^{j} - \\frac{1}{2}\\pi\\pi^{ij} )", "8d2ad848e34329035a6de5be273a5d1e": " \\langle \\mathbf{x},\\mathbf{y} \\rangle = g_{\\alpha\\beta} \\, x^{\\alpha} \\, y^{\\beta}.\\, ", "8d2ae2f36f3e0fa76b05902bc2e82c66": "(E, \\mathcal E).", "8d2ae80d1bb3b0658560488a761d17cc": "x^2(x^2-x-1)+(x^2-1)", "8d2af285de3fa7563b0da5312b3db5ec": "Z(G) = 2\\left(e^{-\\alpha}\\right)^{|E| - r(E)} \\left(4 \\sinh \\alpha \\right )^{r(E)} T_G \\left (\\coth \\alpha, e^{2 \\alpha} \\right).", "8d2b23555c4a68f9747ad76d73e9b754": "\\hat{I}_{S}, \\hat{I}_{B}", "8d2b3d106f413fa0ddb8a7c8c031a0b2": "\\Bigg[\\frac{\\alpha}{\\pi}\\Bigg]=\\Bigg[\\frac{\\alpha}{\\theta}\\Bigg]", "8d2b41f0110944302cc5932cd394d096": " pE \\equiv m B \\equiv IA \\,\\!", "8d2b5015572a18d29be94a150a5c2ffe": " \\mathcal{F}", "8d2b729a82bca24181b21bf40d3b6504": "P_\\mathrm{ion}", "8d2bf18dffee8005942fdde98a69f3a1": "\\beta\\,\\!", "8d2c548dfed9b727f0620a6b4b1c5814": "a < -1", "8d2c825eba8a879106afe2174bf99c45": "E^\\frac n 2 ", "8d2c83f59f0db31488f41141e381d0f3": "(\\boldsymbol\\mu,\\boldsymbol\\Lambda) ", "8d2c8e0e92871afbf2cffbaa7add792a": " \\prod_{p} (x-p^{-s})\\approx \\frac{1}{\\operatorname{Li}_{s} (x)} ", "8d2cc2bc96cc4da988c1618318c019d1": " \\langle A, B \\rangle = \\operatorname{Tr}(A^* B) ", "8d2ce2e7b43a2d26eab9026559960a14": "r_\\theta = {d r \\over d \\theta} = {-a \\over \\theta^2} ", "8d2ce778643ad1eb81182d19cc419ab5": "\n \\gamma_{n,q}= m_{n,q} \\gamma_{0,q} \\qquad \\quad \\textrm{for} \\quad n=1,2,3,4 \\quad \\text{ and } \\quad q = 1,2, \\dots ,Q\n", "8d2cfefff5c74893dfc57c24364bfa0e": " \\dot{y} + hk_3 ", "8d2d80ef00296b9ed491d989b30cebd1": "g_j", "8d2d912cab6f008aba1bd45963ab6e43": "\\ \\displaystyle \\varphi(q,\\alpha,u)\\ ", "8d2d9c23c2bde29ce3f1a7714b826f6b": "\\{e| e\\in E^+(A-T)\\}", "8d2dc58b763978691a808f980a60df2a": "\\begin{matrix}\n\\xi = \\frac{a}{l}\\sin\\varphi_0 ~\\cos\\nu t\\;.\n\\end{matrix}", "8d2dfc94afc0642d5b5e36e4c4a8d06c": "t=\\left\\{ \\begin{matrix}\n \\frac{\\ln \\left( \\frac{k_{1}}{k_{2}} \\right)}{k_{1}-k_{2}} & \\, k_{1}\\ne k_{2} \\\\\n \\frac{1}{k_{1}} & \\, otherwise \\\\\n\\end{matrix} \\right.", "8d2dff96ebd830b86ec27e773f4dadc4": "\nm \\, \\frac{d}{dt} \\, \\vec{v} = - \\nabla( V + Q ) \\; ,\n", "8d2e53b68ad5b1e1486edb412ea0db2a": "A = C ", "8d2e5fbc7e1d1de4dd4f81445f70dd07": "p(x) = \\varepsilon x^3-x^2+1", "8d2e6e4abdd3f29084a3dad32c0277a8": "f(\\cdot, y) : A \\to R^n", "8d2e83f5cf174712ab89305770659543": " A_i \\to A_i' = A_i + a_i ,", "8d2eaeee156d1db121cc794cb6097aca": "\\mathcal{O}\\,", "8d2ef9888cc712e7ef518f1734221670": " \\part u / \\part n", "8d2f2cc73a500d63bc3c086b918bdab3": "(\\phi \\wedge \\neg \\phi) \\to \\psi", "8d2f343f524588b534571d1620d85ebf": "Y\\to X", "8d2fac6a1d684b019bfe993e24e51de0": "S=s_1,s_2, ..., s_n", "8d2fc970d8276ec40ffa6eac1c847661": "H_TB = y \\sum A_{exp}", "8d2fd2a78aed7a76ca5c1a0d7219c7fe": "\\frac{I_{3}}{\\sigma_{xx} \\sigma_{yy} - \\sigma^2_{xy}}", "8d2fd3fd9d77db0a7a591df8325f448c": "Sales = {A}*{Y^B}", "8d300e6ca9b2f5d7c7130c8788c637d5": "ab=a", "8d3011c91a4779e6e10266fc45ab34b9": "\n\\ R_q = R_0(1-q)\n", "8d3063c05c6e40331cdbf893a067658a": "\\pi_t = 1/(PL_t)", "8d30ab2fba522b8ff9871c9183f55007": " (\\alpha e^{-i\\theta} = 2^{-1/2}[q_{\\alpha} cos(\\theta) + p_{\\alpha} sin(\\theta)] + i2^{-1/2}[-q_{\\alpha} csin(\\theta) + p_{\\alpha} cos(\\theta)], \\widehat{U}|\\alpha\\rangle = |\\alpha e^{-i\\theta}\\rangle)", "8d30b33f1e30dbafd73f7f25d727e19d": "H > 0, \\qquad 1 \\ll U \\ll V, \\qquad 0 < b-a \\leq V", "8d30ca4e510dbcfae8d3de793452a897": "\n\\left(\\frac{\\gamma}{\\mathfrak{p} }\\right)_n = \\zeta_n^{b(1)+b(2)+\\dots+b(m)}.\n", "8d312a571ea10ce3796d18e248b88686": "\\frac{n(a_1 + a_n)}{2}", "8d314be5d24f76801a8f0df8700ccd41": "|x_1,x_2,x_3\\rangle", "8d3151268cd3b93956a3b67bc6338ba6": "\\zeta_K(s)", "8d315b2d6b9aa02db5b3d5f3a7b691cd": " a_k ", "8d31703831eac1098527eaa7f62d21ef": "\\tau_{cap} \\left(\\omega \\rightarrow 1 \\right) = 0", "8d31a5fd68de5033dacd40a078c01fc1": "\\mathbf{H} = \\begin{pmatrix}\n\\mu B(t)-\\hbar\\omega_0/2 & a \\\\\na^* & \\hbar\\omega_0/2-\\mu B(t) \\end{pmatrix}", "8d31f6479a264706464d2c7bbeec8446": "\nG(a,c) = G(a,0,c) = \\begin{cases} 0 & c\\equiv 2\\mod 4 \\\\ \\varepsilon_c \\sqrt{c} \\left(\\frac{a}{c}\\right) & c\\ \\text{odd} \\\\\n(1+i) \\varepsilon_a^{-1} \\sqrt{c} \\left(\\frac{c}{a}\\right) & a\\ \\text{odd}, 4\\mid c.\\end{cases}\n", "8d32e5068cf11358abd1f8b3f85acc3b": "E = X D^t", "8d331bac28506fe0823230c359dc4dac": "k^a \\nabla_a k^b = \\kappa k^b ", "8d335b2a391bff5784b3051e99176ea0": "\\begin{cases}\nt' &= \\gamma(t - vx/c^2) \\text{ where } \\gamma = 1/\\sqrt{1-v^2/c^2} \\\\ \nx' &= \\gamma(x - vt)\\\\\ny' &= y \\\\ \nz' &= z\n\\end{cases}", "8d336a90c02f18129ad09f6815528531": "\\operatorname{dim}(\\mathcal{U})=k\\leq \\operatorname{dim}(\\mathcal{W}):=l", "8d33af3ddf74e4628221e5f6fa5527b1": "(x)_n = x(x-1)(x-2)\\cdots(x-n+1).\\,", "8d33e94774fcd57782bf01c208fd4847": "\\tbinom{k}{2}", "8d33f40723c9b4a0ece112676269e15f": "\\mathbf E_{1s} = <\\psi_{1s}|\\mathbf - \\frac{1}{2r^2}\\frac{\\partial}{\\partial r}\\left (r^2 \\frac{\\partial}{\\partial r}\\right )|\\psi_{1s}>+<\\psi_{1s}| - \\frac{\\mathbf Z}{r}|\\psi_{1s}>", "8d3401f3ba00b0980cebe82fe9073cf4": "\\mathbb{P}(N\\ge k)=\\sum_{n=k}^m(-1)^{n-k}\\binom{n-1}{k-1}S_n,\\qquad k\\in\\{1,\\ldots,m\\},", "8d341c1417cc9a65797174565888053c": " f(z) = \\frac{g(z)}{(z-a)^n} ", "8d345bf95c2fbc3875b23023bd188ecc": "a \\in [b]", "8d346d1850beaa8eb7bfd2859de0358a": "a = \\iota x (\\phi (x) \\land x=a)", "8d346de3111b450210dd10790970511e": " x_{n+k+1} = x_{n+k} - \\frac{f(x_{n+k})}{p_{n,k}'(x_{n+k})}", "8d347fd2fca7ae129310fba161266bfe": "f \\, (x)", "8d34aed815a988d3e058e2669798522e": "\\mathfrak{P}^{16}", "8d34b08e8c8d4b64772cc99a980d4b2c": "\\frac{1 - \\cos x}{x}", "8d34b1cf94df0f481a49a757e0f68ca0": "(A,\\rightarrow)", "8d34fb6bca1403ca067cad1f8e787c53": " \\lambda \\approx 550", "8d3531802748d28cc053362c478beb47": " \\phi_{kn}(r) = \\max \\left[ 0, \\min \\left(2 r, \\left(1 + 2r \\right)/3, 2 \\right) \\right]; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{kn}(r) = 2", "8d3532ed62d25085fd896b1381c55e91": "\\int_0^1 B_n(t) B_m(t)\\,dt =\n(-1)^{n-1} \\frac{m! n!}{(m+n)!} B_{n+m}\n\\quad \\mbox { for } m,n \\ge 1 ", "8d3568f70c3fcc5f0c4faff09c90ab35": "\\sin k\\pi=0", "8d356eccbaedcb9a3f58a5e2fcb62fed": "{2n\\choose n}", "8d35baff37ddc092b563de3cd4d4732d": "\\ \\hat{X}(f) = G(f)Y(f)", "8d35ccf383120a398914d24a627f9a4f": "\\rho_{A^*}\\circ(A^*\\otimes\\varepsilon_A)\\circ\\alpha_{A^*,A,A^*}\\circ(\\eta_A\\otimes A^*)\\circ\\lambda_{A^*}^{-1}=\\mathrm{id}_{A^*},", "8d35de48b780c9212cc4d75dfc6b09d7": "\\mathcal{E}^5", "8d365921911cc8fb978eaa03779cdab0": "\\tan(\\theta - \\phi) = v/c", "8d368f8f0da570ce1d67a0034579cb72": "\\partial_{\\text{out}}(S)", "8d36b2676f1451a017458b681f872f3d": "p_y = N_1^c \\frac{y}{r} = i\\frac{1}{\\sqrt{2}} \\left(Y_1^1 + Y_1^{-1}\\right)", "8d36d09cf96ae3f973f2d644d94a765b": "r=k C_S^2 \\frac{K_1K_2C_AC_B}{(1+K_1C_A+K_2C_B)^2}", "8d370f870d09c517fb1095d3ad94a2c4": "\\theta^{*}", "8d379b6c35d47235ccf4269bf5b1692d": "\\mathbb{R},\\,\\mathbb{C}", "8d383b0b030eb2208a491abd3aec2fe4": "(x_p, y_p) + (x_q, y_q) = (x_r, y_r)", "8d384a0c4581625172b38ffda54b856b": "O(2^K n^2)", "8d387a30d9048b5530d6475380b61321": "\\sigma_0(n)=d(n)", "8d38884509774b0b27ada79a373bc577": "w_{ij} > 0 ", "8d38fd7b589ba245909b3ae6fed19989": "\\sin(1/x)", "8d390ec8ec31c06a961e4c9881a42740": "U_{ij}(r_{ij}) = \\sum \\frac {z_i z_j}{4 \\pi \\epsilon_0} \\frac {1}{r_{ij}} + \\sum A_l \\exp \\frac {-r_{ij}}{p_l} + \\sum C_l r_{ij}^{-n_l} + \\cdots\n", "8d391c7ec4800c59945386ae65970822": "\\textrm{Kendrick~mass~(F)} = \\textrm{(observed~mass)} \\times \\frac{\\textrm{nominal~mass~F}}{\\textrm{exact~mass~F}}", "8d3939bbad63a9ec8c4dc4933ca44538": "\\mathcal V_1, \\mathcal V_2,\\dots,\\mathcal V_{n+1}", "8d39cf4f851c2529676d4b09e6b7837a": "f\\colon U \\to \\mathbb{C}", "8d3a0e5ed346fc8afa0d58a8d4d8865e": "\\sum_ \\mathrm{sym} x^3 y^0 z^0 \\ge \\sum_\\mathrm{sym} x^1 y^1 z^1 ", "8d3aa5ed4362dfaff67caedc6255663c": "H^{p,q}=\\overline{H^{q,p}}", "8d3aba5ebc8dc8899b8b2f46629dea4b": "\n\\begin{align}\n\\propto & \\frac{\\prod_{i\\neq k}\n\\Gamma(n_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i)}{\\Gamma\\bigl((\\sum_{i=1}^K\nn_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i) +1\\bigr)} \\prod_{i\\neq k} \\frac{\n\\Gamma(n_{(\\cdot),v}^{i,-(m,n)}+\\beta_v)}{\\Gamma\\bigl(\\sum_{r=1}^V\nn_{(\\cdot),r}^{i,-(m,n)}+\\beta_r \\bigr)}\\\\\n\\times & \\Gamma(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k + 1) \\frac{\n\\Gamma(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v +\n1)}{\\Gamma\\bigl((\\sum_{r=1}^V n_{(\\cdot),r}^{k,-(m,n)}+\\beta_r)+1\n\\bigr)} \\\\\n\\propto & \\frac{\\Gamma(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k +\n1)}{\\Gamma\\bigl((\\sum_{i=1}^K n_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i)+1\n\\bigr)} \\frac{ \\Gamma(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v +\n1)}{\\Gamma\\bigl((\\sum_{r=1}^V n_{(\\cdot),r}^{k,-(m,n)}+\\beta_r)+1\n\\bigr)}\\\\\n= &\n\\frac{\\Gamma(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k)\\bigl(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k\\bigr)}\n{\\Gamma\\bigl(\\sum_{i=1}^K\nn_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i\\bigr)\\bigl(\\sum_{i=1}^K\nn_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i\\bigr)} \\frac{\n\\Gamma\\bigl(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v\\bigr)\\bigl(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v\\bigr)}\n{\\Gamma\\bigl(\\sum_{r=1}^V n_{(\\cdot),r}^{k,-(m,n)}+\\beta_r\\bigr)\n\\bigl(\\sum_{r=1}^V n_{(\\cdot),r}^{k,-(m,n)}+\\beta_r)} \\\\\n\\propto & \\frac{\\bigl(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k\\bigr)}\n{\\bigl(\\sum_{i=1}^K n_{m,(\\cdot)}^{i,-(m,n)}+\\alpha_i\\bigr)} \\frac{\n\\bigl(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v\\bigr)} {\\bigl(\\sum_{r=1}^V\nn_{(\\cdot),r}^{k,-(m,n)}+\\beta_r)}\\\\\n\\propto & \\bigl(n_{m,(\\cdot)}^{k,-(m,n)}+\\alpha_k\\bigr)\\frac{\n\\bigl(n_{(\\cdot),v}^{k,-(m,n)}+\\beta_v\\bigr)} {\\bigl(\\sum_{r=1}^V\nn_{(\\cdot),r}^{k,-(m,n)}+\\beta_r)}.\n\\end{align}\n", "8d3ae0b6c887c43cfb2af2175d3a796a": "z_\\mathrm{f}", "8d3ae12705140b302de5896dd313ec16": "Y_{2}^{2}(\\theta,\\varphi)={1\\over 4}\\sqrt{15\\over 2\\pi}\\, \\sin^{2}\\theta \\, e^{2i\\varphi}", "8d3b197eb2a3517fd15804602dec9359": " Q = \\begin{bmatrix} 0.36 & 0.48 & -0.8 \\\\ -0.8 & 0.60 & 0 \\\\ 0.48 & 0.64 & 0.60 \\end{bmatrix} . ", "8d3b286cde325b4281b183adc08b09ca": "[t^n, t^{n+1}]", "8d3b627869e6cb7c3e5c5c9132f88088": " \\mu=\\mu_d + \\mu_a = \\mu_d + \\sum_{n=1}^N \\kappa_n \\delta_{X_n}, ", "8d3b7749051e018d9dadaa769495bc00": "\\omega=\\sqrt{gk}.", "8d3c1bb6bf86d4292107123c4b6a986e": "R_b = \\frac{R_\\mathrm{ab}R_\\mathrm{bc}}{R_\\mathrm{ac} + R_\\mathrm{ab} + R_\\mathrm{bc}} ", "8d3c39499f08c2bdeb4aadc0882cfc89": "\\bar{\\delta}m^a-\\delta\\bar{m}^a=(\\bar{\\mu}-\\mu)l^a+(\\bar{\\rho}-\\rho)n^a+(\\alpha-\\bar{\\beta})m^a-(\\bar{\\alpha}-\\beta)\\bar{m}^a\\,.", "8d3c578b32fbdca4a38c9f8f063b79a0": "\\Delta S = nC_v \\ln \\frac{T}{T_0} + nR \\ln \\frac{V}{V_0}", "8d3c7010892fc6fa61b90989e8b190c4": "\\begin{align}\n \\frac{\\delta \\rho }{\\delta t}+\\nabla _{\\alpha }\\left( \\rho V^{\\alpha } \\right) &= \\rho CB^{\\alpha }_{\\alpha } \\\\ \n \\\\ \n \\rho \\left( \\frac{\\delta C}{\\delta t} + 2V^\\alpha \\nabla_\\alpha C+B_{\\alpha \\beta }V^\\alpha V^\\beta \\right) &= -\\rho^2 e_\\rho B^\\alpha_\\alpha \\\\ \n \\\\ \n \\rho \\left( \\frac{\\delta V^\\alpha}{\\delta t} + V^\\beta \\nabla_\\beta V^\\alpha - C\\nabla^\\alpha C - 2CV^\\beta B^\\alpha_\\beta \\right) &= -\\nabla^\\alpha \\left( \\rho^2 e_\\rho \\right)\n\\end{align}", "8d3c8a722820030ed377de43116109b1": "\n\\begin{matrix}\nL & \\rightarrow & e^{i2\\theta'}L, \\\\\nQ & \\rightarrow & \\mbox{Re}\\left(e^{i2\\theta'}L\\right), \\\\\nU & \\rightarrow & \\mbox{Im}\\left(e^{i2\\theta'}L\\right).\\\\\n\\end{matrix}\n", "8d3c8b6e2bfe05876935777cc425664e": "\nr_{Q}^{2} = \\frac{Q^2 G}{4\\pi\\varepsilon_{0} c^4}.\n", "8d3ca48c9fd9a3445df6456c4edcc4d1": " \\operatorname{Inv}(N,F):=\\{x\\in N: F_t(x)\\in N{\\ }\\text{for all }t\\} \\subseteq \\operatorname{Int}\\, N, ", "8d3d7af9a1d612c5c1e9bf69289b6617": "\\Gamma(s,x)", "8d3e0fc1be9d3b283864722251a0cc9f": "\\sum_{n=0}^{\\infin} x^n =\\frac{1}{1-x}\\forall x<1", "8d3e12283e2cf362f76925d8eadf9e52": "\\beta = {1\\over kT}.", "8d3e422f5b9d4a42cefbfcff13b91179": "r'^2 \\equiv nt' \\pmod p", "8d3e8aaa4a3480639cbc03958a4d4a3c": "f\\left(r,\\theta,t\\right):=\\left(r^2-1\\right)e^t,", "8d3e91335fc398662f2f1aba470eb0d9": "\\sum_{n=0}^\\infty |a_{n+1} - a_n| < \\infty", "8d3ece542a783c585cb461b088e17ee3": "D\\frac{\\partial}{\\partial n}p_\\varepsilon (x,t) - \\frac{p_\\varepsilon (x,t)}{\\gamma} F(x)\\cdot n(x)=0 \\text{ for }x \\in \\partial \\Omega - \\partial\\Omega_a", "8d3f321ab14cbc213a9118159e72021b": "\\, a_{\\| m} = (a\\cdot m)m^{-1} ", "8d3f5e8c7ecd07500ece1ed9e0c0966c": " \\begin{align}\n \\sum\\limits_{i=1}^{m} y_i &= x_m &1\\\\\n y_k &= x_1 \\cdot \\prod\\limits_{l=1}^{k-1}\\gamma_l &2\\\\\n \\Rightarrow \\sum\\limits_{m=1}^{i}y_m &= x_1 + x_1 \\sum\\limits_{j=1}^{i-1}\\prod\\limits_{k=1}^{j}\\gamma_k = x_i &3\n\\end{align}", "8d3f699a71f4340a2f340235cfe04bb9": "2^{81}", "8d3f86d3dd6fc6c80cafacf36bf2dff0": "Q(\\sqrt{q^*})", "8d3fbde66fcd3c48c3cb3361797a9845": "\\inf_{\\lambda\\in\\Lambda}f_\\lambda", "8d3fc4aae41343d06b38574e283bc58c": "\\hat{\\tau} \\colon [0, T] \\to [0, + \\infty);", "8d3fe58e25e4ee71e957697b60557eeb": "(a_1, \\ldots, a_k),", "8d40057231405abb8b0e7c99d3465b42": "\\mathbb{A} (\\textbf{n}) \\textbf{g}=\\mathbb{C}\\left(\\textbf{g}\\otimes\\textbf{n}\\right) \\textbf{n}", "8d40cb682fdbcf87c224091a57bf0164": "\nk=\\frac{m\\pi }a,\\quad \\quad m=1,2,3,... \n", "8d411726e0b720da9ec8fbce7e55359c": "x_k \\,", "8d412243792fc16836f0c132cd78455b": "K_i(e) = \\{ s \\in S | P_i(s) \\subset e\\}", "8d417e770d570f64b37f6e608113c06b": "\\beta_j^+", "8d41812c874957837851e822700160d9": " \\frac{dA}{dt} = \\{A,H\\} + \\frac{\\partial A}{\\partial t}\\,. ", "8d41b24bde23d2ba1d439f0989855a54": "\\nabla(z) = \\begin{cases}\n\\frac{n+1}{2}z^2 + 1 & \\text{if }n\\text{ is odd} \\\\\n1 - \\frac{n}{2}z^2 & \\text{if }n\\text{ is even.} \\\\\n\\end{cases}", "8d41beb2b04e859f35026e2b89f569ce": "x + N := \\{x + n \\mid n \\in N \\}", "8d41dc057a65b3c78de8b70a194a4322": " \\nabla \\cdot \\nabla u= 0 \\, ", "8d41ff80571bb7ad14cfffee513a4fba": "\n \\Delta x_{\\mathrm{b.a.}}(t_j) = \\frac{\\Delta {p}_{\\mathrm{b.a.}}(t_j)\\vartheta}{M} \\,.\n", "8d4267b4b78fddc15425fba6e7772ecc": "\\scriptstyle \\{0,\\, \\pi \\}", "8d42867a78f49f31770902aa35e2f578": "*\\!", "8d429e51a25790b94a557f7b7756f910": " x_{i} \\geq 0 \\; \\; \\; \\forall i \\in \\{1, 2, \\ldots, n \\} ", "8d42d17d07675f82509419c78bfb91d2": "n\\ m\\ f = m^n\\ f ", "8d430d384d402a84d282c983c5ad6e25": "M'\\prec_K M", "8d43605911a82b3ede3eff8ac31ca646": " s:\\, \\, a+bi\\mapsto a-bi\\ . ", "8d436b8dcd8c58742f2dbb7a10f89c4c": "\\displaystyle c_p=\\frac{\\lambda}{T}=\\frac{\\omega}{k}", "8d437a8467e057303b1087a06514ba3a": "\\left(\\scriptstyle L^1_{\\text{loc}}\\right)", "8d4385fd5da9b9fbca8a8f179f751c92": "P[a/y]", "8d43f1bfec73a9905b937f6e46dc9411": "v = \\frac{1}{\\sqrt{LC}}", "8d44113f345f832d1dc65c381aba04bc": "At(x)", "8d442c6ef1cc7bbc31982f5947f6bb60": "Q(x,y) = y - 4x^2", "8d4442b6140ef3347239d91e54e7d3c7": "V = \\begin{pmatrix}1 & -1 \\\\ 0 & 1\\end{pmatrix}.", "8d44c26bb644f6d3631edf35e2667317": "L=L(a-\\theta)", "8d45258fca01e2519870c96628be54e1": " A = \\sqrt{AA^{*}} = \\sqrt{\\sum_{n=1}^N \\sum_{m=1}^N \\left | A_n \\right | \\left | A_m \\right | \\cos \\left [ \\left ( \\mathbf{k}_n - \\mathbf{k}_m \\right ) \\cdot \\mathbf{r} + \\left ( \\omega_n - \\omega_m \\right ) t + \\left ( \\phi_n - \\phi_m \\right ) \\right ]} \\,\\!", "8d45794d798ff04d1ef0b58bed9377a1": "x =\\frac{(y - k)^2}{4p} + h;\\ \\,", "8d45c151756b996cc3ca57699df19870": " \\{\\,D_i : i \\geq 1\\,\\}", "8d45f391b9eb2cfba5ca1c719fd469e9": "\\varphi(n) = (p-1)(q-1)", "8d46188d9377680959b332e4f78a5af0": "f=f^*_\\phi\\circ\\phi", "8d4637b398efb938658ca3bab87cdbbc": "\\Delta f \\equiv \\nabla^2 f", "8d468b2676211144c4c843bd2ddb345b": "\\Delta = \\frac{\\partial V}{\\partial S}", "8d46a509cb7956aac55c48431679f7eb": "\\mathbb E[\\bar v_N] = \\frac{1}{N} \\sum_{n=1}^N \\mathbb E[v_n]", "8d4748713c0aa02679d990bdb21247b7": "(b, a') = ga", "8d478730ab4492bc21ca09b56c6d2aa7": "\\left( \\begin{matrix} n \\\\ 3 \\end{matrix} \\right)", "8d47a710595b868121206c0d73aca773": "\\rho=\\psi^\\dagger \\psi, \\quad \\mathbf{j} = \\psi^\\dagger \\gamma^0 \\boldsymbol{\\gamma} \\psi \\quad \\rightleftharpoons \\quad J^\\mu = \\psi^\\dagger \\gamma^0 \\gamma^\\mu \\psi ", "8d47ab5d3e12e65985e987297b5f0036": " \\forall x \\in M: \\quad \\eta_x = \\sum_{i,j = 1}^{m} {\\eta_{i,j}}(x) \\cdot \\left( \\frac{\\partial}{\\partial x_{i}} \\right)_{x} \\otimes \\left( \\frac{\\partial}{\\partial x_{j}} \\right)_{x} ", "8d47c072b192611bc3d72b2df1a1c1f3": "(x + 2y,y)", "8d47c9aebce9bfaf3d919fe9fc5fe711": "(G, G^+) = \\varinjlim (\\mathbb{Z}^{n_k}, \\mathbb{Z}^{n_k}_+) ,", "8d4844b5e0c24639b8360aa322e06209": "f(z) = \\frac{az+b}{cz+d}\\;,\\quad \\mbox{where } a,b,c,d\\in\\Bbb{C} \\mbox{ and } ad-bc \\ne 0.", "8d488a964e19b75a71c54834a1657749": "=m_1^2 + m_2^2 + m_3^2 + m_4^2 + 2 \\left( m_1^2 + p_1 \\cdot \\left( p_2 - p_3 - p_4 \\right) \\right) \\,", "8d489ddc8a0b6c213c274b6083655adb": "\\sigma' = \\sigma^t - u\\,", "8d48a39bd6e2c9db6e4c212bf24d7f69": "Q< 0", "8d48d1eacb15bece557995a26af38574": "x^k_T", "8d490458eba1eb30a05a2ff0c7baf137": "k > 2,", "8d490ba824ced469bbd0b5dab71db48b": "B = A^C\\,\\!", "8d4912aaa9ded3cb89599f1b54833095": "\\mathbf{\\nabla}\\varphi", "8d4962d170413c3265a493b3ba8520e0": "\\widehat{c_v} = \\frac{s}{\\bar{x}}", "8d4984ed88a42ed08f15b6a83f7f1300": "\\begin{align}\n g_{11} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_1}{V_1} \\right|_{I_2 = 0} \\qquad g_{12} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{I_1}{I_2} \\right|_{V_1 = 0} \\\\\n g_{21} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{V_1} \\right|_{I_2 = 0} \\qquad g_{22} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{I_2} \\right|_{V_1 = 0}\n\\end{align}", "8d499ae4abe2e0b6880f5a947563a5cc": "\n\nF(t) = (K + iC) x (t) = K (1 cm) sin (wt) + C (1 cm) cos (wt) \\!\n\n", "8d49a96010a0b53b4064b6089ddd9e7a": "x^2-2x+1=0", "8d49c662cb34a4c5efd3090689a98446": "e^{i\\mathbf{K_{2}}\\cdot\\mathbf{(R)}}=1", "8d4a494db63b617d941d434ff1c7652f": " p = \\frac{2\\pi r_A}{N_A} = \\frac{2\\pi r_B}{N_B}.", "8d4a79af1fd8eae160c5cbe4262c9ca7": "f(x,\\cdot)", "8d4aa7b7bed8077f7d974de33b12e70d": "\\mathbf{p} = 2q\\mathbf{a}\\,\\!", "8d4ad7aaeeb5ad53e31dda2154918b51": "A=\\begin{bmatrix}\n2 & -1 & 1 \\\\\n0 & 3 & -1 \\\\\n2 & 1 & 3 \\end{bmatrix} ~.", "8d4b6af8f2d947433dc0466b3dfe7b4d": " \\operatorname{de-lambda}[(\\lambda F.E) L] ", "8d4ba275e32405de642d057ea40122cf": "\\frac{\\text{individual's outcomes}}{\\text{individual's own inputs}} = \\frac{\\text{relational partner's outcomes}}{\\text{relational partner's inputs}}", "8d4bb41302958eb4e8158eea59a7f7c8": " w_{t+1} = w_t + f(x, y) - f(x,\\hat y).", "8d4bc8f0d46d8f866400c3d480167402": "\n a_{mn} = \\begin{cases}\n 0 & m~\\text{or}~n~\\text{even}, \\\\\n \\cfrac{16q_0}{mn\\pi^2} & m~\\text{and}~n~\\text{odd}\\,.\n \\end{cases} \n", "8d4bcf8da8d95ed1029fc1376c0d6a04": "A = f^{-1}(0).\\,", "8d4c250deef4869307763379c7990b10": " {1 \\over 0.3546} ", "8d4c4dae054c77936fa763f842e1f93f": "T_{(i_1i_2\\dots i_k)} = \\frac{1}{k!}\\sum_{\\sigma\\in \\mathfrak{S}_k} T_{i_{\\sigma 1}i_{\\sigma 2}\\dots i_{\\sigma k}}", "8d4cb7aceb772a5f3c6ccd10928b8415": "L_n = \\varphi^n + (1-\\varphi)^{n} = \\varphi^n + (- \\varphi)^{- n}=\\left({ 1+ \\sqrt{5} \\over 2}\\right)^n + \\left({ 1- \\sqrt{5} \\over 2}\\right)^n\\, ,", "8d4cc0a434aa73e4e1fc12427bccbea4": "\\begin{align}\n D &{}= T - C \\\\\n &{}> G_n \\\\\n P_n &{}= C + G_n \\\\\n &{}< C + D \\\\\n P_n &{}< T\n\\end{align}", "8d4da15447e98a71b5b1a3f17ff1922f": " (a*b)*(c*b)=(a*c)*(b**c) ", "8d4dff327201b7b52cde35b46a36cdc8": "\\rho(\\Omega) = \\sum_{i=1}^n \\rho(X_i)\\rho(X^i).", "8d4dfff01839096395bb224633cac917": "\\forall a \\forall b \\;a \\wedge (a \\vee b) = a ", "8d4e31c029497c9c3fe4d2ff39492a61": "D=s(s\\otimes s)^T+s(s\\otimes n)^T+s(n\\otimes s)^T+n(n\\otimes n)^T,", "8d4ed0050808814af0e3d77a653ff6e4": "\\sum_{n=0}^\\infty B_n(x) \\tilde{B}_n(y) = \\delta (x - y)", "8d4f2496abe6ff8bd71230bb988b5da1": "\\mathbb{C} [ S_n] ", "8d4f55d9cc7d144e1d3e7ce31d2fb78a": " (X\\cup X^{-1})^* ", "8d4f6421dd3ce40dd948ff1b496548bb": "\\pi_3(S^2)", "8d4f93aaf4e63f6f1100296e43ba55e6": "x\\in s_i", "8d4f9831ab13277d8bce8fbbf1344b11": "\\,\\ \\tan x", "8d4fa0c61fceaec17477883262099f49": "* \\rightarrow *", "8d502cb0afcfb6c56e5212245a18cb62": "\\sum_{k = 0}^{\\infty} \\frac{1}{(16^k)(8k+1)} = \\sum_{k = 0}^{n} \\frac{1}{(16^k)(8k+1)} + \\sum_{k = n + 1}^{\\infty} \\frac{1}{(16^k)(8k+1)}. \\!", "8d503eeadb9a9eb730fb8057b85ce855": "\\tfrac{2\\lambda}{E+\\lambda+R}", "8d5093799ad539e36cf1ccb3eef2d4ed": " \\frac{1}{\\cos \\theta} \\cong \\frac{1}{1 - \\frac{\\theta^2}{2}} \\cong 1 + \\frac{\\theta^2}{2} ", "8d512f3e0a08a57a662ebc1a725d2f99": " \\exp\\left( \\left. \\frac{ d f^t(x) }{dx} \\right|_{x_0} t\\right) ", "8d51a2b675b3fc1900d3f74fb5ac20c3": "\\tau^* = \\tau e ^{-i \\phi}", "8d51a9b23c6ae0598ed3296ba0df8a06": "a = e", "8d52014c846b30e331da52ee6c958cfe": "\\frac{dy}{dx} = mx^{m-1} \\,", "8d5238eb835828522fe22e4ebc17a0ba": " x^\\mu \\to x'^\\mu ", "8d52869fe67ba1752567049efeac5a2b": "R_J(x,y,z,p) = \\tfrac{3}{2}\\int_0^\\infty \\frac{dt}{(t+p)\\sqrt{(t+x)(t+y)(t+z)}}", "8d536b56e6c416c984159c0217063d89": "\\ \\tau_{D,GR}=\\frac{\\omega_{xy,G}^2+\\omega_{xy,R}^2}{8D_{GR}}", "8d539517dab97963d3487f1da89af7b2": "\\pm1/2", "8d539b6635bf46127b558754fb3cb56e": " J_x \\equiv J_1 = i\\left.\\frac{\\partial \\widehat{R}(\\theta,\\hat{\\mathbf{e}}_x)}{\\partial \\theta}\\right|_{\\theta = 0} = i\\begin{pmatrix}\n0 & 0 & 0 \\\\\n0 & 0 & -1 \\\\\n0 & 1 & 0 \\\\\n\\end{pmatrix} \\,, ", "8d53bb2da462a964785b7b25e24c27e4": "\\mathrm{slog}_b(z) \\approx \\begin{cases}\n\\mathrm{slog}_b(b^z) - 1 & \\text{if } z \\le 0 \\\\\n-1 + \\frac{2\\log(b)}{1+\\log(b)}z + \n\\frac{1-\\log(b)}{1+\\log(b)}z^2 & \\text{if } 0 < z \\le 1 \\\\\n\\mathrm{slog}_b(\\log_b(z)) + 1 & \\text{if } 1 < z\n\\end{cases}", "8d53e41afe5da5e111f59d7aa2814154": "\n\\left\\langle \\frac{dG}{dt} \\right\\rangle_\\tau = \\frac{1}\\tau \\int_{0}^\\tau \\frac{dG}{dt}\\,dt = \\frac{1}{\\tau} \\int_{G(0)}^{G(\\tau)} \\, dG = \\frac{G(\\tau) - G(0)}{\\tau},\n", "8d540f1d3b37f642e91e7596ac7c4c4f": "vt_{p-r}=\\frac{v^2}{2 \\mu g}", "8d544a7faaa807fc0f47f7d394e8f9da": " \\| \\mathbf{E}' - \\mathbf{E}_{\\rm est} \\| ", "8d54a7dd99068c55060be5370be0015d": "2^{(3^4)}", "8d54b80748c917c083d53a84dd684e92": " \\Delta^2=-\\frac{691}{1728^2\\cdot250}\\det \\begin{vmatrix}E_4&E_6&E_8\\\\ E_6&E_8&E_{10}\\\\ E_8&E_{10}&E_{12}\\end{vmatrix}", "8d55321f485d68c666d704a8df57087b": "\\vec{R}", "8d55975ca14202065509ccf1d9c4e633": "{\\eta}_{anode}", "8d55c35a1731a9c5ee97e5d59e7b74ac": "\\theta_{11}", "8d55dc1f6f32fde7fc18bd27a8f73d63": "(x_1-\\bar{x}) + \\dotsb + (x_n-\\bar{x}) = 0", "8d5606315d6461ed404b4eb5afe4ceff": "\n W \\approx \\frac{1}{2!}C_{ijkl}E_{ij}E_{kl} + \\frac{1}{3!}C_{ijklmn}E_{ij}E_{kl}E_{mn}+\\cdots,\n ", "8d560f41974462d055f4d145f1ddc7ad": "M_1(\\phi,\\tau) = \\dfrac{\\phi_1(\\theta,\\tau)}{\\phi_2(\\theta,\\tau)}M_2(\\phi,\\tau)", "8d5614686cfa44d565bd5c95714e5ea7": "\\sqrt{x+1}\\sqrt{x-1}=\\sqrt{x^2-1}", "8d56d3a088d2b6b97c1b027c4c024d24": "\\mu = \\frac{M}{m}", "8d56dcb4281d4eca6eba6200fc2be343": "\\langle \\Psi | = a^{*} \\langle \\psi | + b^{*} \\langle \\phi | ", "8d5732883f8b81729035c3443dbd3307": " V_0 = -I_0 L \\alpha \\beta \\left(\\frac{1}{\\beta}-\\frac{1}{\\alpha}\\right) ", "8d576d612543db39cac697dba3f4e6e4": "\\mathbf{c} = \\mathbf{a} * \\mathbf{b}", "8d57ea205872339cd9428bfcfba545b8": "F(X) = E(F(X)) + \\int_0^1 H_t \\,d X_t ,", "8d5819cad3f7edfca91a84becf010394": "p(x) = g(x/|x|) |x| \\,", "8d5841442716990879f4f8a1b303416f": " f_{\\theta}(x) = x^2 + e^{2 \\pi \\theta i} x \\,", "8d5860e775a907c9abc0aeb3ec5b6fea": "x_d\\,\\!", "8d58c93ca7bca9dbb7438d35ff30cea1": "\\Lambda^{m\\mid n}.", "8d5959e377313b5d0aaa1b0a060fe6a2": "\\sin \\beta = 0.5", "8d59c5b8c00c95b7a7b8ff1f7c522944": "\\mathbf{k_I}", "8d59cb6e950ace4c4c9f53c8895e84c5": " u\\in V", "8d5a8115faadd6dd6759242cc4bf1919": "D=\\Pi_{x\\in X} D_x .", "8d5a953bb1517326888ebe7d6f9fa8f2": "\n\\bigl( \\begin{smallmatrix}\na&b\\\\ c&d\n\\end{smallmatrix} \\bigr)\n", "8d5a95b4afa34ff24eae5c8b1125059c": "\n\\begin{align}\n\\operatorname{tri}(t/a) &= \\int_{-\\infty}^\\infty \\mathrm{rect}(\\tau) \\cdot \\mathrm{rect}(\\tau - t/a)\\ d\\tau \\\\\n&= \n\\begin{cases}\n1 - |t/a|, & |t| < |a| \\\\\n0, & \\mbox{otherwise} .\n\\end{cases}\n\\end{align}\n", "8d5ab10893248af6c23ed75b9aa9792d": " R = \\left \\{ x \\in X \\ : \\ \\sup\\nolimits_{T \\in F} \\|Tx\\|_Y = \\infty \\right \\} \\neq \\varnothing", "8d5ae814783076d8ec3f46a774135565": "f(x) \\mapsto \\int_I \\frac{f(t)-f(x)}{t-x}\\rho (t)dt", "8d5aefeed6916d97d77e351aa50c2e5f": "\\Delta \\phi < \\frac{\\pi}{2} ", "8d5b0f4dcbc2f0b6f23d0e301dd12d88": "\\begin{align}\nt' &= \\gamma \\ (t - vx/c^2) \\\\\nx' &= \\gamma \\ (x - v t) \\\\\ny' &= y \\\\\nz' &= z ,\n\\end{align}", "8d5ba2e5d13697e2668bbafd0f83b3ae": "D_{i,j}^{KL}=D^{KL} \\Big[ p(y|x_j) \\,|| \\, p(y| c_i)\\Big ] \\Big)", "8d5bb76c1090510aff09b20af4337f5c": "1~\\mathrm{V}", "8d5bdbf51b4696586e78a9e5890ea80e": "I(\\hat x,u)", "8d5c0e0394b5f0318a6d412fa9f73649": "x=\\begin{pmatrix}\n 1 & 1 & 0\\\\\n 0 & 1 & 0\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix},\\ \\ y=\\begin{pmatrix}\n 1 & 0 & 0\\\\\n 0 & 1 & 1\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}", "8d5c10863c31ce200ae04d0fd7cb15a1": "\\int^\\infty_0 y_\\lambda J_\\lambda d\\lambda", "8d5cc580e86a3012a45fd6a97b78ff54": "E_\\infty^{p,q} = \\frac{Z_\\infty^{p,q}}{B_\\infty^{p,q}+Z_\\infty^{p+1,q-1}}.", "8d5ceebe4d8f74f995b1547ed090b142": "C_n(\\varphi)=(-1)^{\\frac{n+1}{2}}\\frac{\\left(\\frac{n-1}{2}\\right)!}{\\sqrt{n!}}", "8d5d3cae234d363ba85f729b41ce378b": "z_T\\, ", "8d5e031fdb255e5fa630371acfaae25c": "U(\\phi)", "8d5e68657092ea8df501270db48a4282": " m_H=\\sigma_0\\sqrt{2\\lambda}", "8d5e749b2fb932597bcc047b5c03517d": "\\max_{w(\\cdot)} E\\left[ y(\\hat{e}) - w(y(\\hat{e}))\\right]", "8d5e935320a9887332463b9c4247ae35": "\\lim_{x\\to 0} \\frac{1}{|x|} = \\infty", "8d5f07f7ed91a506fec47ed2a16ec528": "\\frac{a}{b} < \\frac{ad + bc}{2bd} < \\frac{c}{d}.", "8d5f39fc26bd4dad5beee59f0d098660": "u(\\eta)/U(x)", "8d5f63cc4dfd2d13c128c4187adc3c13": "\\mathbf{s}^K", "8d5fb1d840eca7480b8187c0a7639ece": "q\\in E(T(p,x))", "8d5fdde8e385b624c21152dc2d06d722": "\\{\\infty\\}", "8d5fed746122e478eb5694a918bf29fe": "E = \\mathbb{Z}^+", "8d5ff3794434d15dfff1e2495500c935": "( \\Delta \\boldsymbol{x})", "8d607fb3c042dc10d2b411a118ca04aa": "v_\\mathrm F", "8d60d0cf33090a87249cc45c17f87467": "\\Phi \\left(\\eta,\\tau \\right) = \\exp \\left[-\\alpha \\left(\\eta \\tau \\right)^2 \\right], \\, ", "8d610c5b97c1151bfd6effc9c34b97eb": "T_{comm}", "8d61177c69e50d6a94f5866d4785bace": "\\displaystyle i\\partial_t\\psi = (-\\tfrac12\\Delta^2 + V(x) + g|\\psi|^2) \\psi ", "8d622cd52360983935f337e5d2879fc5": " \\textstyle 1 = \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{16} + \\cdots", "8d62740b43df1a90b88a0ae3ec684197": "t_n=t_0+n\\,\\Delta t", "8d62ccc81990a1af34b54befe484ea77": "E^*(\\mathbf{CP}^\\infty)\\times E^*(\\mathbf{CP}^\\infty) = E^*(\\text{point})[[x\\otimes1, 1\\otimes x]]", "8d62e469fb30ed435a668eb5c035b1f6": "y_i", "8d630a1273d90db9ad73c439274270b0": "y=Hx+w", "8d632033658de1c292e2aaeb55db58da": " \\dfrac{\\partial g_t(z)}{\\partial t} = g_t(z)\\dfrac{\\zeta(t)+g_t(z)}{\\zeta(t)-g_t(z)}.", "8d63a4125ae317d963dd7e3ee0e0ccf3": " x \\preceq x ", "8d63bfa0aab9617490509a8a6cb1f613": "\nU(P_1,r) \\propto J_0^2(\\frac{\\pi r d}{\\lambda b})\n", "8d63f9070922cc094ce627a5eb9eb8fa": "\n\\sqrt{11} = 3 + \\cfrac{1}{3 + \\cfrac{1}{6 + \\cfrac{1}{3 + \\cfrac{1}{6 + \\cfrac{1}{3 + \\ddots}}}}}\n", "8d641b31b92a2bcc42475b2616d131bd": "L (1 - \\cos\\theta)", "8d6440f80112b8d74670070a56130e25": "2^{283}", "8d6449451ded23c1631fae8e11c7346b": "\\alpha_1+\\alpha_2+\\cdots+\\alpha_n=1.", "8d6462b62299238d39b5991140dc8cfc": "_n\\!\\!\\diagdown\\!\\!^k", "8d646bb9ab260119b0364a15d941dc87": "\\mathcal{H}^A\\otimes\\mathcal{H}^{B^L_j}", "8d647776d883f381488980a1bdbf0a99": "1 \\leq p \\leq \\infty, \\ 0 < \\theta < 1, \\ s= (1-\\theta)k + \\theta (k+1)= k+\\theta. ", "8d64795eed0c3233828de78512aa559b": " |U_{ai}|=1/\\sqrt{3}", "8d64922e085d003f7477bfbb840c4883": "\\begin{pmatrix}\n 1 & a & c\\\\\n 0 & 1 & b\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}=y^bz^cx^a\\, .", "8d649776fcdb18e38d2e61c6fbbb957e": "\\frac{d P/P}{d t}=\\frac{d M/M}{d t}", "8d64e2dc8a98915400dd620254073360": " {q} \\,\\overset{\\underset{\\mathrm{def}}{}}{=}\\, (- {A} {Q}^{-1} {c} - {b})\\,", "8d65252213a246ed816a5100174e22d0": "\\Theta+d\\Theta", "8d65531ad08305169738c89385ae7974": "aba^{-1}b^{-1}=1", "8d657460dab94a830e3b15aba0e5699b": "dI = \\beta I dz", "8d6589cf28eb47a12c9a299e28e06540": "\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix} =\n\\begin{bmatrix}\n\\gamma(-v) & \\delta(-v) \\\\\nv\\gamma(-v) & \\gamma(-v)\n\\end{bmatrix}\n\\begin{bmatrix}\nt' \\\\ x'\n\\end{bmatrix},\n", "8d65be83d88b5ff6ff4f8c65d4923b79": "(s)_+ = max(s,0)", "8d65cce3d9ad05b4ad19faf801575cb7": " A(a,b) = {{a+b} \\over 2}\\text{ and} ", "8d6613fc4fd8c7c814eb253a6e8d51d6": "\\left | \\ln \\left (\\frac{S(n, t)}{n!} \\right) \\right |, ", "8d66521edcf212006563dae917e273d3": "\\mathit{3}^{k-1}", "8d667ef7c67de22fe53ff7b07ca82c68": "\\tfrac{1}{2}(d - \\lambda_2) \\le h(G) \\le \\sqrt{2d(d - \\lambda_2)}.", "8d669312095159ee4fa9d26c32de5878": "\\boldsymbol{\\mathbf{U}} = \\frac{d \\boldsymbol{\\mathbf{X}}}{d \\tau} = \\left(\\frac{c dt}{d\\tau} , \\frac{d\\mathbf{x}}{d\\tau} \\right)", "8d672bbfdd2c9a7a483744825c16bf67": " \\psi(z) ", "8d677e3ac74a44d2969d0b3f2a4a2c77": "\\gamma_c(A)", "8d679919b333e26bf40eedd58c8e514d": "\\mathcal{D}_T", "8d67aff17645c18c5bc0fc7990578e75": "U_i\\hookrightarrow X", "8d6823c85badfc57adf2758b2ffedfa0": "(\\vec{x},\\vec{y},z)", "8d682fbae5992f3749ad879b4705428d": "(\\hat{B}^\\dagger_\\omega)", "8d68536702b7310aa12e105914e6fa80": "\\Phi_{u_g}(\\Omega)=\\sigma_u^2\\frac{2 L_u}{\\pi} \\frac{1}{1+ (L_u \\Omega)^2}", "8d68d3118f1c122dc12065d87e8248a8": "D(a,b;c)=\\sum_{n \\bmod c} \\left( \\Bigg( \\frac{an}{c} \\Bigg) \\right) \\left( \\left( \\frac{bn}{c} \\right) \\right),", "8d6915735f56c027509c56470881306d": "u_r(\\mathbf{p})", "8d6974f9bc7c554eb402287c61c478c2": "w_2 = (w_2 \\sqrt{T_3}/P_3) * (P_2/\\sqrt{T_2})* (P_3/P_2) / \\sqrt{T_3/T_2} \\,", "8d6998e67f48f45765b52d74f701306c": "S[\\sigma] \\to aS[f \\sigma]c", "8d69b2b4bc1d83feba9047b9dbdca459": "\\text{If }S_{i+1} = R, \\quad f_i(a_1,\\dots,a_i,a) = (1-a)f_{i+1}(a_1,\\dots,a_i,0) + a f_{i+1}(a_1,\\dots,a_i,1)", "8d69eb0252a700bacbad0b7319138685": "\\pm f_d", "8d6a1736298535ae46b8b8bb6e5e20d7": "J_\\mu^B", "8d6acef64db49362f2ae19f7ef5a7b82": "U=B_1^X(\\underline{0})", "8d6b2ea3f4bdc37c7c43858999db9622": "F(\\mu)=\\textstyle\\frac34{\\mu^2(\\mu-2)^2\\over 1-\\mu}\\;", "8d6b68bad0cf58fc5338b716491b4f89": "Q_{ijkl}", "8d6b71da47181982182260c41f2114b5": " h_x'(x) = h'(x); ", "8d6bbcd679e64b12787522488a342163": "t=\\tfrac{x-x_{i-1} }{ x_{i}-x_{i-1} }", "8d6c10ea0325d40d89a3433814535919": "\\Omega_c ", "8d6c2482f954ee1f80cd53ccbc606750": "\\sqrt{\\frac{5}{14}}\\!\\,", "8d6c327c3f00305245c4cb08159c7ea7": "\\mathrm{[H]^2 + 6.46\\times 10^{-5}[H] - 6.46\\times 10^{-7} = 0} ", "8d6cdd761cd3ce7e8cbf2ecf9e1929bd": "{0 \\choose 0}_q = {1 \\choose 0}_q = 1", "8d6d036bc419d1f5401eaee5514bd690": "q^{1-s}", "8d6d18680e9e2021f078fa420e49fac4": "\\,\\frac{e^{hk}p^k(1-p)^{n-k}}{1-p+pe^h}", "8d6d56e4f315fbc4db341e12cde974a6": " H = H^1_0(\\Omega) \\times L^2(\\Omega)", "8d6d7185f96069ffb240ef28e75f0d5f": " x^2 + y^2 = (1 - x)^2, ", "8d6d779a83aa7cc54476ae0447645d67": "t=\\hat\\beta_j/\\hat\\sigma_j", "8d6d9daafda44d4cc64e42e89c6b86af": "x(t) = A_1 \\sin (tf_1 + p_1) e^{-d_1t} + A_2 \\sin (tf_2 + p_2) e^{-d_2t}, \\,\\!", "8d6dabe461a1590a1029cde352d1dc18": "f(n) = 3 \\rightarrow 3 \\rightarrow n \\!", "8d6db2c14230b5b6099b0aa87b44988d": "p^{\\deg(Q)}\\prod_{P(x)=0} Q(x),", "8d6dc813b48ee9984f524713630f3c08": " M_{i,j} = M_{i,k} ", "8d6dcb36e8b7e6c9a69ce2cac4d70d1e": "K_R = 0.2126", "8d6dd57aea417ab4a59dd5e5bb064ac7": " \\mathbf{y} \\sim \\mathbf{C}_{0} \\, \\mathbf{x} = \\left ( \\begin{array}{c|c} \\mathbf{I} & \\mathbf{0} \\end{array} \\right ) \\, \\left ( \\begin{array}{c|c} \\mathbf{R} & \\mathbf{t} \\\\ \\hline \\mathbf{0} & 1 \\end{array} \\right ) \\mathbf{x}' = \\left ( \\begin{array}{c|c} \\mathbf{R} & \\mathbf{t} \\end{array} \\right ) \\, \\mathbf{x}' ", "8d6e1ee04508461ffccedf0342e7ab7a": "(\\ldots)_S", "8d6e631d68bbbb7767772263129d8a57": "W(T,G) := N(T)/Z(T).\\ ", "8d6e7167f92d06851d8ae9ff012eb65c": "\\frac{\\part}{\\part t}=K\\frac{\\part}{\\part t_1}+K^2\\frac{\\part}{\\part t_2} \\qquad s.t.\\ t_2(\\text{diffusive time-scale}) \\ll t_1(\\text{convective time-scale}) ", "8d6f93cc6a9cea2dfe0d13badaf12e1e": "A,B\\in \\mathbf{H}_n", "8d6fc562786a632404c50efdcf423f2d": "C_\\mathrm{even} := \\oplus_{n \\text{ even}} \\, C_n", "8d6fdbc6cb50baeb46a4a7b8032654c6": "\nK_0 = - V \\left( \\frac{\\partial P}{\\partial V} \\right)_T.\n", "8d707f825c6ef08fc1246483eb4acdb0": "u_2=\\mbox{Im}(y_1)=\\tfrac{1}{2i} (y_1-y_2) =e^{2x}\\sin(x),", "8d70ce6e92b40334d8bbebff2f05ee1e": "\\{3\\}, \\{4\\}, \\{5\\}, \\{\\frac{5}{2}\\}", "8d70dd79525b7d872740e8d723cd379b": "\\{ www : w \\in \\{a,b\\}^{*} \\}", "8d7107b5647acfdc72356e8aaf7a2c99": "1.62400\\pm 0.00005", "8d7121a7c771356f6a62c34a1d60c97d": "C_{V}", "8d7172282c767a47f917796dd7b0b249": "0.000975482 \\times W^{0.46} \\times H^{1.08} ", "8d7190301b71daaed4821645cc747e57": "\\displaystyle{\\sum_{i,j} |(R_i v,R_jv)| \\le \\left(\\max_i \\sum_j \\|R_i^*R_j\\|^{1\\over 2}\\right)\\left(\\max_i \\sum_j \\|R_iR_j^*\\|^{1\\over 2}\\right)\\|v\\|^2.}", "8d71b586e556d277654cfaf171118147": "d \\approx 2{.}455 \\cdot R \\cdot \\sqrt[3]{ \\frac {\\rho_M} {\\rho_m} } \\,. ", "8d71bae048f57bc030c91e4fadaabfd6": "|q|_\\ast=1", "8d71f22e401d5ebe32c5c8b725910f13": " 54 \\times 1 = 27 \\times 2 = 18 \\times 3 = 9 \\times 6. \\, ", "8d722d388fb0b000484c422251e6a594": "\\sum_{n=0}^{\\infty}U_n(x) t^n = \\frac{1}{1-2 t x+t^2}; \\,\\!", "8d724c81ced9ce9a64c21a6abcffe687": "d\\bold{A}+\\bold{A}\\wedge\\bold{A}", "8d731c9bdbca6e21fb6c1b6d2eb7f219": "\\Pi_C\\,", "8d7320c7f13740a787e5c2f8f38f04ed": "b_{1}-d=(3/5)b_{1}", "8d735f49bd89fa7190ca459be7308fa6": "\\scriptstyle H_n(z) \\,=\\, \\left[1 - z^{-1}\\right]^2", "8d736840273feb4615296bbef06ee1c0": "p_H(\\mathbf{x}|\\boldsymbol\\alpha) = {\\displaystyle \\int\\limits_\\boldsymbol\\theta p_F(\\mathbf{x}|\\boldsymbol\\theta)\\,p_G(\\boldsymbol\\theta|\\boldsymbol\\alpha) \\operatorname{d}\\!\\boldsymbol\\theta}", "8d7399fb56ac81bedcdc15e97006b1e6": "\\mathbf{H}~", "8d73bbd27ef15d8baf03e93d3e001f0b": "\\operatorname{Ind}(f)(s) = \\frac{1}{|H|} \\sum_{t \\in G,\\ t^{-1} st \\in H} f(t^{-1} st).", "8d73d6d1a6a2f11828d70b3e8fa364f1": " \\phi^i ", "8d7400587203f9a930826f10fcad1eaa": " P O_L = Q_1^{e_1} \\cdots Q_n^{e_n}, ", "8d740fd1a8ce1b225c872ea60b1bb959": "x^{\\mu} \\rightarrow x^\\mu + \\delta x^\\mu \\!", "8d74a6c9ee54f79178e74ed91a68c6c1": "\\gamma_s x_s", "8d74a7c2c6f78518b9125b7cd8b9c337": "\\partial_\\mu\\partial^\\mu A^\\nu = e \\overline{\\psi} \\gamma^\\nu \\psi ", "8d74fa87f9c8138f6e391af05fdef606": " b_j\\ge nb_{j-1} -\\sum_{i=2}^{j} b_{j-i} r_i.", "8d750ced645e9259f6f4d3606009ec8d": "Z \\approx {377 \\over \\sqrt {\\varepsilon_r} }\\,\\Omega", "8d751f9b8b67ade962e0501cc251255e": "L_n", "8d7531405552d5abbef3eef8af1bef8d": "I_{n+\\frac{1}{2}} = I_{\\frac{2n+1}{2}} =\\int \\frac{1}{(ax^2+bx+c)^{\\frac{2n+1}{2}}}dx = \\int \\frac{1}{\\sqrt{(ax^2+bx+c)^{2n+1}}}dx\\,\\!", "8d756c28b59daf1549f2a852ef921def": "x \\cup \\{x\\}.", "8d767337169af4079868f429baba7c15": "a_0 \\, ", "8d76799e330fcf98af81cb1ae64a55f1": "0 \\to L \\to L' \\to P \\to 0", "8d76a0d8ce45c1a051c399417dcffd60": " \\mathbf{S} = {{E_t^2 \\over \\mu_0 c}}\\mathbf{\\hat{r}} = {{e^2 a^2 \\sin^2(\\theta)} \\over {16 \\pi^2 \\varepsilon_0 c^3 R^2}} \\mathbf{\\hat{r}} ", "8d76b15b14b7b8d3f614a07190437667": "\\sum_{s=1}^{m+n}{x'_{r,s}}=1", "8d76c8245781c14b26312cc1639db5cc": "\\left\\{\\mathcal{F} f\\right\\}= F(s=i\\omega) = \\frac{1}{\\sqrt{2\\pi}}\\left\\{\\mathcal{B} f\\right\\}(s)", "8d76ff3e2b19a93eca08a17732a691bd": "\\sigma_\\infty = E_c\\epsilon_c", "8d7741be1fe522bd03b45fc887503042": "(x_2,0,0)", "8d777f385d3dfec8815d20f7496026dc": "data", "8d77f870ea61a240658edc1ef22ca750": "\\boldsymbol\\Sigma_{XY} = \\boldsymbol\\Sigma^T_{\\mathit{YX}} = \\mbox{cov}(\\boldsymbol{X}, \\boldsymbol{Y})", "8d77fc463f04bdd35d9379cfd30bef8e": "r_{yield} = y \\cdot \\delta t", "8d7849cc8b48fa064e62426a29994817": "360/n", "8d78aa0c16eca6e94ce7abe9d6fb7cd5": "\\text{ask} \\colon E \\rarr E = \\text{id}_E", "8d790af5ac9b24d6733772af1e2d585f": "V \\not\\in Fin", "8d791cfaef93312072ae12046af8971f": "{\\dot{\\theta}}", "8d795e92dfbec18871f90e9c1a1ecee3": "z_{\\alpha/2}:=\\Phi^{-1}(1-\\alpha/2)", "8d7992f9a82c506d7fa3ebb847a78804": " {\\eta_{\\alpha \\beta}}{d X^\\alpha \\over ds}{d X^\\beta \\over ds}=-1.", "8d79bff6f8b08e931c86b66dead5912b": "\n\nRi = \\frac{Ei}{Ee} \\!\n\n", "8d79f387a83a215dadd421f66aece309": "\\scriptstyle{I>0}", "8d7a08c4bfc0bbe6811512fa75dccdd0": "||x-y||=2\\sin\\left({{|\\alpha-\\beta|}\\over 2}\\right).", "8d7a4b78eac0be848209a8bd1c46a9ee": "= 2 \\pi \\int_{-\\infty}^{0} f(t)\\, dt + 2 \\pi \\int_{0}^{t} f(\\tau)\\, d \\tau", "8d7a84b86268ee1e1d4b79a6d3a92cd1": "\\int_{H^3} F \\,dV =4\\pi \\int_{-\\infty}^\\infty F(t) \\sinh^2 t \\, dt,\\,\\,\\, \\int_{H^2} f \\,dV =2\\pi\\int_{-\\infty}^\\infty f(t) \\sinh t \\, dt.", "8d7a96a1d90c926cdb8a2a4b35059afc": "\\!\\,\\gamma(a) = \\gamma(b)", "8d7ab21bd24ea04d5eb5874f18915cae": " f: \\mathbb{C} \\setminus \\{a_k\\} \\rightarrow \\mathbb{C}", "8d7af37936ba55ec18e6cc7620bbc669": "R_\\mathrm{fb}=29 \\cdot R", "8d7af8b2e284d93f39f1e77e3b5e9c7d": "F(x_1 + \\Delta x) - F(x_1) = \\int_{x_1}^{x_1 + \\Delta x} f(t) \\,dt. \\qquad (2)", "8d7b42176def90793c4d64a1ecf4f892": "{\\vec {\\tilde x}} = - {\\frac{{\\vec K}}{\\sqrt{M^2c^2 - {\\vec P}^2}}} +{\\frac{{\\vec J \\times \\vec P}}{{\\sqrt{M^2c^2 - {\\vec P}^2} (M c + \\sqrt{M^2c^2 - {\\vec P}^2})}}} + {\\frac{{\\vec K \\cdot \\vec P\\, \\vec P}}{{Mc\\, \\sqrt{M^2c^2 - {\\vec P}^2}\n(Mc + \\sqrt{M^2c^2 - {\\vec P}^2}) }}}", "8d7b53f88da8abe84f3a7ea21088ceb2": " y - l = Uk - (1 - Q)l + r ", "8d7b5bb5c0fd9b6847a7622b7302c0e0": "z^n", "8d7b743bcca084eff26e439b4baf3376": "\\left(\\!\\!{n \\choose 0}\\!\\!\\right) = 1,\\quad n\\in\\N, \\quad\\mbox{and}\\quad \\left(\\!\\!{0 \\choose k}\\!\\!\\right) = 0,\\quad k>0.", "8d7b766e270ba2d720d2e5b030582b39": "Erosion Number = \\frac{t_{diffusion}}{t_{erosion}} = \\frac{^2\\pi*k}{4D*(ln-ln\\sqrt[3]{M \\over N_a(N-1)*p})}", "8d7bc5bd72dd6be730b78bdc6814d30a": "(c_{\\alpha_{l-1}\\alpha_{l}})^2 = \\sum\\limits_{i_l=1}^{d}(\\Gamma^{[l]i_l}_{\\alpha_{l-1}\\alpha_{l}})^{*}\\Gamma^{[l]i_l}_{\\alpha_{l-1}\\alpha_{l}}", "8d7c7383b35f31e7aa719f7ae09c72ed": "\\Psi'_\\theta(0) = \\mu_\\theta,", "8d7c823e762db2ed5248cf7e4e9a9f0a": "g'(x)=\\bigl((\\det\\Phi(x))'-\\det\\Phi(x)\\,\\mathrm{tr}\\,A(x)\\bigr)\\exp\\biggl(-\\int_{x_0}^x \\mathrm{tr}\\,A(\\xi) \\,\\textrm{d}\\xi\\biggr)=0,\\qquad x\\in I,", "8d7cb8e9292fa3489dd44ec2263ab19d": "E_\\mathrm{nonbonded}", "8d7d5b31f699645602284cf6b24bafaa": "\\scriptstyle \\mu_{\\rho} ", "8d7dafebd9a76f7c1bc6dcfa1553f763": "\\circ L \\circ ", "8d7e63405fe2accfda480952dd54801b": "\\displaystyle{\\left\\|\\sum_{m,n\\ge 0} {1\\over m! n!}z^m w^n P^m Q^n v\\right\\| \\le C \\sum_{k\\ge 0} {(|z|+|w|)^k \\over k!} \\|D^{k\\over 2} v\\| <\\infty.}", "8d7e99c73cd5a10adaaf4c9f9a520368": "CS", "8d7f70a0d3197d50fe9e6b557387116e": "\\beta_\\gamma(\\{g_\\alpha^*\\})=0", "8d7f821884bf4cbe8dccf311f05ce16b": "\\begin{align} \\frac{d P(\\mathbf{X}, t)}{dt} & = \\Omega \\left( \\mathbb{E}^{-S_{11}} \\mathbb{E}^{-S_{21}} - 1 \\right) f_1 \\left( \\frac{\\mathbf{X}}{\\Omega} \\right) P(\\mathbf{X}, t) \\\\\n& = \\Omega \\left( f_1 \\left( \\frac{\\mathbf{X} + \\mathbf{\\Delta X}}{\\Omega} \\right) P \\left( \\mathbf{X} + \\mathbf{\\Delta X}, t \\right) - f_1 \\left( \\frac{\\mathbf{X}}{\\Omega} \\right) P \\left( \\mathbf{X}, t \\right) \\right),\\end{align}", "8d7fa5f5a64da2c4b8ca99148b8cbd26": "M(b,b,z)=\\exp(z)", "8d7fb16fbf525e7eaada75441f0fc006": "\\partial S:=\n\\partial\\{({I}^{2},{\\varphi}_{\\lambda},{S}_{\\lambda})\\}_{\\lambda\\in\\Lambda}", "8d8015eea5d42984a6129f5105a70c07": "\\lceil k/16 \\rceil", "8d80545fecc947134718f00a7f28941c": "G_{\\delta}", "8d806ecc76fe8d418768b781dfd542bd": " \\mathbf{a} = \\mathbf{a}_\\parallel + \\mathbf{a}_\\perp\\,,\\quad \\mathbf{v}\\cdot\\mathbf{a}_\\perp = 0 \\,,\\quad \\mathbf{v}\\cdot\\mathbf{a} = \\mathbf{v}\\cdot\\mathbf{a}_\\parallel \\,, ", "8d80758ef67e5dacda523ddd521d81b5": "m c^2 / \\sqrt{1 - v^2/c^2}", "8d80cb19cb03bb9ec3290bcdd969ecdb": " C_D", "8d80cb54e731d56e368888604cf22b04": "\\displaystyle{2Q(Q(a)b,b)-2Q(Q(b)a,a)= L(a)Q(b)L(a) - L(b)Q(a)L(b).}", "8d810cbec8bfc1d8185cd6458fd6c6ff": "\\begin{align}\n \\Psi({\\mathbf{r}}) = \\exp(i{\\mathbf{k\\cdot r}}) - \\frac{m}{2\\pi\\hbar^2}\\int\\frac{\\exp(ik\\cdot {\\mathbf{|r-r'|}})}{{\\mathbf{|r-r'|}}}\n V({\\mathbf{r'}})\\Psi({\\mathbf{r'}})dr'\n \\end{align}", "8d812f87fe929afd0c46d23fc9d1ec1f": "\\Delta \\tau = \\int_0^{\\Delta t} \\sqrt{ 1 - \\left(\\frac{v(t)}{c}\\right)^2 } \\ dt, \\ ", "8d8168598a01716b9d9e0ff961f99994": " \\phi : (x, y) \\mapsto (x^{q}, y^{q})", "8d8195a14996e330ea60352cb1c4880e": "\nV(\\mathbf{r}) = \\frac{-\\mu_1}{r_1} - \\frac{\\mu_2}{r_2}\n", "8d81abe458ab40efa312ef51f6c289a1": "\\frac {dm} {dt} = k_t (C_b-C_i)", "8d81d1ffc6bacc5037e35eb410260c6a": "=\\operatorname{st}\\left(\\frac{2x \\cdot dx + dx^2}{dx}\\right)", "8d81f17e4407d7e1f457537cfca5086f": "E = E_0", "8d8203473e40e5ad94fb4606ee48b8cd": "\n ", "8d82234835ea099241a413a1dc5296ee": "G(F, H) := \\, G_{abcd}F^{ab}H^{cd}\n", "8d8245a55a39c4966cbe4cb1d5e87739": "v_1 = \\begin{bmatrix} -1 \\\\ -1 \\\\ 2 \\end{bmatrix}, \\quad v_2 = \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix}, \\quad v_3 = \\begin{bmatrix} -1 \\\\ 0 \\\\ 2 \\end{bmatrix}.", "8d829c81b723e90fee51affcf9b21ff8": "\\phi(x) \\geq 1+\\delta", "8d82ec3f7a81728ab7d0fdbd1dfb39ad": "\\{x_i, y_i\\}_{i=1}^n", "8d831210716b988c3c1a57f4549befd5": "\n y_i = \\begin{pmatrix}Y_{-i} & X_i\\end{pmatrix}\\begin{pmatrix}\\gamma_i\\\\\\beta_i\\end{pmatrix} + u_i\n \\equiv Z_i \\delta_i + u_i,\n ", "8d8327bd37bb63ff7930832078193401": "\\scriptstyle \\eta(.) ", "8d833ee92e187c0859f55337b00caf18": "\n\\hat{f}(x) = n^{-1} h^{-1} \\sum_{i=1}^{n} K\\left(\\frac{x-x_i}{h}\\right)\n", "8d837f3d9bf5a7ea8deb2b925d1720bf": "{\\rho}\\frac{\\partial \\mathbf{v}}{\\partial t}+{\\rho(\\mathbf{v}\\cdot\\nabla)\\mathbf{v}} = -\\nabla p+\\nabla\\cdot\\sigma,", "8d839d3912de67da69c2b5a0bbf83c28": "J_z^2", "8d840ca64b401121e07c645015037a17": "0 0 \\text{ and } c,a \\neq 0 ", "8d856b6b7c759817b59bdc4d3e99b7b4": "\\beta=\\|f\\|_p^p,", "8d857d4efddd7d9060e01c546fe7360b": " X_n \\,", "8d85c7b9818946111d8fd556eb3d6c83": "N_{\\rm A} = \\frac{A_{\\rm r}({\\rm e})M_{\\rm u}}{m_{\\rm e}}.", "8d8601e24edc92fa5a770fa670b8a08a": "\\{v_1,v_2\\},\\{v_2,v_3\\}", "8d869d0471fa44aa310f7c743eedcf16": "H=(V,X)", "8d86a06bba8a7ceee8fbc922ac446d72": "\\textstyle \\theta \\approx 0.5", "8d86bd9b2d24d382d80ac9f96e1b51d6": "\\nabla \\cdot \\mathbf{B} = 0 ", "8d8797cc429a63964cdf560c1a43032e": "\\sqrt[3]{\\frac{1}{2}-\\frac{1}{6}\\sqrt{\\frac{23}{3}}}", "8d87fa744e7800bba11f3bb80312307b": "T_z", "8d88400cec93580aff31c1c804cbfd72": "\\Lambda^{\\mu'}{}_{\\nu}", "8d887a172449333bbd6148c82641cf18": "u=t-r^*", "8d88a43872d6db6535d8672a15f09ce2": "n_1\\,\\!", "8d8953f3b87c38a53988d22aa526a3b4": "\\varphi_0(0)", "8d89766e19b99ec07b37b90c4a91978b": "\\left \\langle G,S,M\\right \\rangle", "8d899424f5ba3291cc82e7a460ec2dfb": "\\Omega\\left(\\min\\left\\{\\frac{n}{\\min(p,1-p)},\\frac{n^2}{\\log n}\\right\\}\\right)", "8d89ef7ebc0b04bb5ef3b118d138eb96": "f:G\\rightarrow \\mathbb{C}\\backslash\\{0\\}", "8d8a1bc7c9f79887e6aabb11643b8c86": "p \\star (q \\star r) := (p \\star q) \\star (p \\star r)", "8d8a1da9dec7bdbbd61fa96dbcac78aa": "q_0 = \\sqrt{\\frac{2\\hbar}{\\rho_0}} = \\frac{e}{\\sqrt{2\\pi \\alpha}} \\ ", "8d8a5488971d64e9cb57bf4e4eecf87b": "s_1 = 1", "8d8a6417ec7daa115049367b77f325d0": "n_i=0\\,\\!", "8d8a7714183dc3edc96a5fb58ecf27f4": "M(0,b,0) \\to T_{b/2\\pi}", "8d8a99a84e2f5f4f8076c1cbf5b7178c": "t_{2g}", "8d8af88f75c40a038b80b334168692f8": "\\lim_{k\\to\\infty} \\int_S f_{n_k}\\,d\\mu=\\liminf_{n\\to\\infty} \\int_S f_n\\,d\\mu.\\ ", "8d8b0b1528fa0167557a80ff45b57d4d": "\\frac{d^{2}y}{d\\theta^2} + \\cot \\theta \\frac{dy}{d\\theta} + \\left[\\lambda - \\frac{m^2}{\\sin^2\\theta}\\right]\\,y = 0\\,", "8d8b0f883809289ed59769517b6f1e02": "(p(n)/2)/(p(n)/2 +p(n-1)/2).", "8d8b0fba52f8f19ea1736d810106ed77": "U^i = \\frac{dx^i}{d\\tau} = \n\\frac{dx^i}{dx^0} \\frac{dx^0}{d\\tau} = \n\\frac{dx^i}{dx^0} c\\gamma = \\frac{dx^i}{d(ct)} c\\gamma = \n{1 \\over c} \\frac{dx^i}{dt} c\\gamma = \\gamma \\frac{dx^i}{dt} = \\gamma u^i ", "8d8b10e9acd1b6d3d244136a4b6b98d5": "1 = cov (\\tilde{m}, \\tilde{R}) + E(\\tilde{m}) E(\\tilde{R}).", "8d8b2759d53723524ad03fbdde926aac": "F_0 = S_0 e^{rT}", "8d8bba1752d3fd6a536e5de8b983f3e1": "1-x^2\\,", "8d8bc7c7cef41f5439d2243c4217fa24": "\\tau_{c},", "8d8c9e4cfd9e14b04db1e11eaa63f162": "E(0) = \\tfrac \\pi 2 ", "8d8ca1e5f188ce129524171a26292df9": "l=0,1,\\ldots ,k-1\\,\\!", "8d8cd7ab23d54fd3f787b214de97989d": "12 \\sum^\\infty_{k=0} \\frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}=\\frac{1}{\\pi}\\!", "8d8d192742842df91e9067d42b7ed336": "\\mathbb{F}L:TQ \\rightarrow T^* Q", "8d8d25d0ff0a0e42f4aea2cace567765": "a_{x} = \\frac{dv_{x}}{dt}", "8d8d383bab760e5f0a7bfafcdba751fe": "\\gamma(X^*, X) = c(X^*, X)", "8d8d3be9312a170e7359062d07f636a2": "a_i = min\\{a_1,a_2,\\ldots,a_n\\}", "8d8d414e2bf4e62d6d6874e7b79e12f5": "h_{XY}\\ ", "8d8d8621f790dad17cb9596719fb6a69": "t \\mapsto \\left( t - \\tanh{t}, \\operatorname{sech}\\,{t} \\right), \\quad \\quad 0 \\le t < \\infty.", "8d8de34e30ec539ffa29f0c97abcae57": "\\mathbf{\\Phi}_{lm} = \\vec{\\mathbf{r}}\\times\\nabla Y_{lm}", "8d8dfdb1a79df178116abbbf838495d2": "S^{op} \\to \\Delta^{op} Sets", "8d8e041c408a26de2811615e3478fec1": "\\mathbb{A}^2_k", "8d8e11ea42b1b074137583dfb78fbe62": "\n \\displaystyle \n S(n,g) \n =\n \\bigcup_{k=0}^{n}\n S(n-k,g-1)\n", "8d8e9242f2ec99ce65c83711c0a2d765": "2\\pi\\,\\!", "8d8e95a352d3e8f0d4e881f6333140b0": "T_e+R_e=1", "8d8fceed624880d3750bae2fe43f66d1": "V_H = \\frac{V_{13} + V_{24} + V_{31} + V_{42}}{8}", "8d8fd8a558392aa442c77932fd649b22": " dV= r^{-3}\\, dx\\,dy\\,dr", "8d901fdcbbcdc2dc5876c70713378450": "r^{\\otimes}(X) = \\max\\{|z|:z\\in\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( X \\right)\\}.\n", "8d905825e5602f25538d75a7e05d1d26": "(1+z)^\\alpha = \\sum_{k=0}^\\infty {\\alpha \\choose k} z^k, |z|<1", "8d906e707d0e7f64ce940e31eda2c195": "W_\\mu\\;", "8d90bf7127721e93dab9087adfce9a5c": " a = 0.1388 \\cdot \\frac{d^2}{t_{1/2}} ", "8d912e35fe39a34f2bb96afbe624992b": " y_i = \\beta_0 + \\beta_1 x_i +\\epsilon_i, \\,", "8d91a1fd2d2be6009982b643dbc4186d": "\\bold{F}(\\bold{x})", "8d91a895e833b39b89f3081bfaeb3cbf": "S_{Basin}", "8d91ba544d17a73bd1a67df596392a6e": "\n\\begin{align}\nx_s(t) & {} = x(t) \\ T \\sum_{n=-\\infty}^{\\infty} \\delta(t - nT) \\\\\n& {} = T \\sum_{n=-\\infty}^{\\infty} x(nT) \\delta(t - nT)\n\\end{align}\n", "8d91c26166bacd25e725bd7a698425e2": "\\Delta \\theta = \\Delta \\theta_2 - \\Delta \\theta_1 ", "8d921ef026a02ca1e78160fa5d632b43": " i\\hbar\\frac{\\partial}{\\partial t}\\psi=-\\frac{\\hbar^2}{2m}\\nabla^2\\psi -\\frac{1}{4 \\pi \\epsilon_0} \\frac{Ze^2}{r} \\psi", "8d92372daf8683c51451a9455747430d": "= - \\frac{1}{2 T},", "8d923ba946730b51131c88b9e6d26ebd": "C_n \\ltimes C_{q^n-1}", "8d924fc26c53ebea27c03ba3ad340eb8": "\\frac{1}{28} + \\frac{1}{49} + \\frac{1}{196} = \\frac{1}{13}", "8d9265bc619582f7dcc06de0d9817359": "{}^{2}x", "8d93367a0426713cbf13b5b1a527d8d5": "\\mathbf{A}_{\\text{Magnetic dipole}}(\\mathbf{x},t) = \\frac{-i k \\mu_0}{4 \\pi} \\frac{e^{i k r - i \\omega t}}{r}\\mathbf{m}\\times\\mathbf{n}", "8d934537be02716f056eb002df172676": "s_0 = \\omega^{2^0} + \\bar{\\omega}^{2^0} = (2 + \\sqrt{3}) + (2 - \\sqrt{3}) = 4.", "8d9362e33643325db87da3ccd35cd139": "C_l=\\int\\limits_{LE}^{TE}\\left(C_{p_l}(x)-C_{p_u}(x)\\right)\\,d \\frac{x}{c}", "8d937ad9255a3dbac13421863cba306c": "\\sum_{i=1}^r N^2(0,1) \\sim \\chi^2_r \\qquad r=1,2,\\dots", "8d93f06c229ac455e9997d3943c98c95": "gC_H(P)\\in O_p(H/C_H(P))", "8d93fd2446c9d8248f4411ed5024fd82": "\\hat{\\boldsymbol {\\beta_2}}", "8d94072be529bca57814c8408d060b6c": " H(X|Y) = \\mathbb E_Y [H(X|y)] = -\\sum_{y \\in Y} p(y) \\sum_{x \\in X} p(x|y) \\log p(x|y) = -\\sum_{x,y} p(x,y) \\log \\frac{p(x,y)}{p(y)}.", "8d9436a9c31b3b5cd28345e32a9ba893": "\\psi_1(x)= \\frac{1}{\\sqrt{k_1}} \\left(A_\\rightarrow e^{i k_1 x} + A_\\leftarrow e^{-ik_1x}\\right)\\quad x<0 ", "8d9460224f42e34c6fc44e49f39b1584": " x ~ (y) ", "8d946770ab1a4941400bd512a27dd684": "\\int_{-\\infty}^\\infty v_n^2(x) \\, dF(x)=\\int_0^1 u_n^2(t)\\,dt.", "8d947cc454ce58e3de00709f46ad57f2": "\\lambda \\lambda \\lambda. 1 3 2", "8d94c207bccdf817e2fa355c62c0651e": "V:\\mathbb{R}^2\\to\\mathbb{R}^2", "8d95687ed618095ec7c69a6501a55014": "X^{\\prime}=r\\left(1 - \\frac{X}{K}\\right)X", "8d95beb2656ffeaae442a52dec392b33": "\n\\begin{matrix} \n & X & & & U & & \\Sigma & & V^T \\\\\n & (\\textbf{d}_j) & & & & & & & (\\hat{\\textbf{d}}_j) \\\\\n & \\downarrow & & & & & & & \\downarrow \\\\\n(\\textbf{t}_i^T) \\rightarrow \n&\n\\begin{bmatrix} \nx_{1,1} & \\dots & x_{1,n} \\\\\n\\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\\\\nx_{m,1} & \\dots & x_{m,n} \\\\\n\\end{bmatrix}\n&\n=\n&\n(\\hat{\\textbf{t}}_i^T) \\rightarrow\n&\n\\begin{bmatrix} \n\\begin{bmatrix} \\, \\\\ \\, \\\\ \\textbf{u}_1 \\\\ \\, \\\\ \\,\\end{bmatrix} \n\\dots\n\\begin{bmatrix} \\, \\\\ \\, \\\\ \\textbf{u}_l \\\\ \\, \\\\ \\, \\end{bmatrix}\n\\end{bmatrix}\n&\n\\cdot\n&\n\\begin{bmatrix} \n\\sigma_1 & \\dots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\dots & \\sigma_l \\\\\n\\end{bmatrix}\n&\n\\cdot\n&\n\\begin{bmatrix} \n\\begin{bmatrix} & & \\textbf{v}_1 & & \\end{bmatrix} \\\\\n\\vdots \\\\\n\\begin{bmatrix} & & \\textbf{v}_l & & \\end{bmatrix}\n\\end{bmatrix}\n\\end{matrix}\n", "8d95cd2c13c0c3f9acef586fe0a5f2aa": "F_N\\;", "8d96302cb7e9235b7ff6119b7a8dbbc0": "K_{\\lambda\\mu}= K_{\\lambda\\mu}(1)=K_{\\lambda\\mu}(0,1). ", "8d9677dfbd5a5f8e782b3f61e897a0bf": "g(s,t)=(\\vec x_s(s,t)\\times \\vec x_t(s,t))\\cdot \\vec v=0", "8d9684df01e31deb5e4cfc5351a736ca": " z = {1\\over 2} (x^2 + y^2) \\left({1\\over a^2} - {1\\over b^2}\\right) + x y \\left({1\\over a^2}+{1\\over b^2}\\right) ", "8d96bf3f81e05e6ba007247843cbe591": "z = \\rho e^{a j} \\!", "8d96d1646c8caa06c8376d0a5d983b3e": " x_2 = r\\, \\sin\\theta\\, \\sin\\phi \\,", "8d96f5f9eb13d21d493263536644df77": "r,s", "8d96f601cecba807e71c1954ab65eec2": " M(s) = \\int_{-\\infty}^{\\infty} e^{sx} n(x) dx ", "8d971b3f3d6eba00a1b75d68bab5d2c9": "(3,2)_{-\\frac{5}{6}}", "8d9727529eacceb736ad575113fb6dca": "\\Omega = 2\\pi \\left (1 - \\cos {\\theta} \\right) ", "8d973e0ae4ced7f21ea078dfa65da86e": "|a_{n+1}+a_{n+2}+\\ldots+a_{n+p}|<\\varepsilon", "8d97a22dfb457154f4a3a4ff40bc9312": "S>F,\\ S=F,\\ SP(Y|\\theta_{true})", "8da49c8abed96ee65d574a96c22e8a91": "\\scriptstyle R_\\oplus", "8da49d659366e7a25569690f7d53b93e": "\n \\hat{w} = A_1\\cosh(\\beta x) + A_2\\sinh(\\beta x) + A_3\\cos(\\beta x) + A_4\\sin(\\beta x) \\quad \\text{with} \\quad \\beta := \\left(\\frac{\\mu\\omega^2}{EI}\\right)^{1/4}\n ", "8da4bf284f091f4b39a6503d548aed75": "\\quad h\\;=\\;\\dfrac{\\delta y}{a\\delta\\phi\\,}=\\,\\cos\\phi", "8da598356b4ffea7eb247501c9976546": "V(\\theta)", "8da627a5f2a2b3b0b191b8dadeb9ffc6": " \\sigma( \\mathcal{I}) \\subset D ", "8da6329800d8d0a92646e46911460342": "K*P \\subseteq K+P", "8da64bce99b4ced3ac9fc344aa38c425": " \\left|\\sum_{m=1}^N \\sum_{n=1}^N \\alpha_m \\alpha_n\\left[ {f^\\prime(z_m)f^\\prime(z_n) \\over(f(z_m)-f(z_n))^2} - {1\\over (z_m-z_n)^{2}}\\right] \\right| \\le \\sum_{m=1}^N \\sum_{n=1}^N \\alpha_m\\overline{\\alpha_n} {1\\over (1-z_m\\overline{z_n})^2}.", "8da70dfa53b876b5b8e42727649a2f1c": "\\frac{\\partial^2W}{\\partial x^2}+\\frac{\\partial^2W}{\\partial y^2} = \\frac{1}{c^2}\\frac{\\partial^2W}{\\partial t^2}.", "8da7520bb190e69171b81cc4bb14ab64": "\\mbox{where }\\lambda=\\frac{1}{2}-\\frac{i}{2\\sqrt{3}}\n\\text{ and }\\lambda^*=\\frac{1}{2}+\\frac{i}{2\\sqrt{3}}.", "8da7c47dfd5565b9a19fb10b1de62da4": "\\frac{\\mu_1-\\mu_2}{(\\mu_1+\\mu_2)^{3/2}}", "8da7f7f816391596a0a656f2732ce17f": "F(y,z_1, \\dots, z_n)", "8da806e204a9820d9aca145b2d83c545": "\\displaystyle \\sqrt{2 \\pi}\\cdot\\frac{\\delta(\\omega-a)-\\delta(\\omega+a)}{2i}", "8da819eb05e7715a20b4d4093792e5db": "T \\equiv", "8da8213d93e9f2096220402ae8acdce9": "\\begin{pmatrix} p_{k-1} & p_{k} \\\\ q_{k-1} & q_{k} \\end{pmatrix}", "8da82444dc99ba78858d4b0832ee1f26": "\\eta^ST\\cdot\\eta^T", "8da82ad38cd90332282d4002340ca1ae": "\\scriptstyle\\aleph_\\alpha", "8da8800336c0528b20b45095decb8ec9": "\\theta = b(x, \\sigma(x), \\sigma'(x))\\theta_{0} + c(x, \\sigma(x), \\sigma'(x))\\theta_{1}\\,", "8da8bc48477d4ccfc06070d3c83c672d": "\\mathbf\\Sigma_e = \\lambda\\mathbf\\Sigma_i", "8da8d2c58f78c0f323d67bea65257e4b": " f(x;\\lambda) = \\begin{cases}\n\\lambda e^{-\\lambda x} & x \\ge 0, \\\\\n0 & x < 0.\n\\end{cases}", "8da8d5e0f2d5ba9c6e1c05fa1ebeb31d": "h/R = \\rho_a/(3 \\rho_s)", "8da8fb022ce3aa8bdc033571e519c9c9": "P_d(x,y,z)+P_{d-2}(x,y,z) + \\cdots P_0=0,", "8da92771107ec3b99347dc51caed1895": " S(r) = kdr^{d-1}. ", "8da92c894551bc540cee367a165eb71f": "\n\\begin{align}\n\\omega^1 & = \\cos\\theta \\, \\mathrm{d}s,\\quad \\omega^2 = -\\sin\\theta \\, \\mathrm{d}s\\\\\n\\omega_i^j & = -\\omega_j^i\\\\\n\\omega_1^2 & = \\kappa_g \\, \\mathrm{d}s + \\mathrm{d}\\theta\\\\\n\\omega_1^3 & = (\\kappa_n\\cos\\theta + \\tau_r\\sin\\theta) \\, \\mathrm{d}s\\\\\n\\omega_2^3 & = -(\\kappa_n\\sin\\theta + \\tau_r\\cos\\theta) \\, \\mathrm{d}s\n\\end{align}\n", "8da9436bdafcdc76644ddb0684f47b68": "\\phi_{sl}=\\frac{\\rho_{s}(\\rho_{sl} - 1)}{\\rho_{sl}(\\rho_{s} - 1)}", "8da9864f259189d015e1a2fa860adaea": " \\text{EU} {{=}} 50.356 + (0.4 \\times \\text{Wright}) + (0.0008814 \\times \\text{Wright}^2) - (0.0000001116 \\times \\text{Wright}^3)", "8da98b90fbb3dd3c448663550ec35a7a": "f^{*}(\\cdot)", "8daa3f990d609a43d4eff79f5365629d": "\n\\left\\{b+1,\\prod_{i=a}^b g(i)\\right\\} \\equiv \\left( \\{i,x\\} \\rightarrow \\{ i+1 ,x g(i) \\}\\right)^{b-a+1} \\{a,1\\}\n", "8daa6df9b331fdb83ab0b6582e446e3e": "\nD_P^-(x) = \\{y \\in U \\colon x D_p y \\}\n", "8daac66c802afb3e7b60f38b91e81232": "\\begin{align}\n&-\\frac{4}{3}i \\left [ \\oint_{C_1} \\frac{\\frac{z}{(z+\\sqrt{3}i)(z-\\sqrt{3}i)\\left(z+\\frac{i}{\\sqrt{3}} \\right)}}{z-\\frac{i}{\\sqrt{3}}}\\,dz +\\oint_{C_2} \\frac{\\frac{z}{(z+\\sqrt{3}i)(z-\\sqrt{3}i)\\left(z-\\frac{i}{\\sqrt{3}}\\right)}}{z+\\frac{i}{\\sqrt{3}}} \\right ] \\\\\n&= -\\frac{4}{3}i \\left[ 2\\pi i \\left(\\frac{z}{(z+\\sqrt{3}i)(z-\\sqrt{3}i)(z+\\frac{i}{\\sqrt{3}})}\\right)\\Bigg|_{z=\\frac{i}{\\sqrt{3}}} + 2\\pi i \\left(\\frac{z}{(z+\\sqrt{3}i)(z-\\sqrt{3}i)(z-\\frac{i}{\\sqrt{3}})} \\right)\\Bigg|_{z=-\\frac{i}{\\sqrt{3}}}\\right] \\\\\n&= \\frac{8\\pi}{3} \\left[\\frac{\\frac{i}{\\sqrt{3}}}{(\\frac{i}{\\sqrt{3}}+\\sqrt{3}i)(\\frac{i}{\\sqrt{3}}-\\sqrt{3}i)(\\frac{i}{\\sqrt{3}}+\\frac{i}{\\sqrt{3}})} + \\frac{-\\frac{i}{\\sqrt{3}}}{(-\\frac{i}{\\sqrt{3}}+\\sqrt{3}i)(-\\frac{i}{\\sqrt{3}}-\\sqrt{3}i)(-\\frac{i}{\\sqrt{3}}-\\frac{i}{\\sqrt{3}})} \\right] \\\\\n&= \\frac{8\\pi}{3} \\left[\\frac{\\frac{i}{\\sqrt{3}}}{(\\frac{4}{\\sqrt{3}}i)(-\\frac{2}{i\\sqrt{3}})(\\frac{2}{\\sqrt{3}i})}+\\frac{-\\frac{i}{\\sqrt{3}}}{(\\frac{2}{\\sqrt{3}}i)(-\\frac{4}{\\sqrt{3}}i)(-\\frac{2}{\\sqrt{3}}i)}\\right] \\\\\n&= \\frac{8\\pi}{3}\\left[\\frac{\\frac{i}{\\sqrt{3}}}{i(\\frac{4}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})}+\\frac{-\\frac{i}{\\sqrt{3}}}{-i(\\frac{2}{\\sqrt{3}})(\\frac{4}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})}\\right] \\\\\n&= \\frac{8\\pi}{3}\\left[\\frac{\\frac{1}{\\sqrt{3}}}{(\\frac{4}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})}+\\frac{\\frac{1}{\\sqrt{3}}}{(\\frac{2}{\\sqrt{3}})(\\frac{4}{\\sqrt{3}})(\\frac{2}{\\sqrt{3}})}\\right] \\\\\n&= \\frac{8\\pi}{3}\\left[\\frac{\\frac{1}{\\sqrt{3}}}{\\frac{16}{3\\sqrt{3}}}+\\frac{\\frac{1}{\\sqrt{3}}}{\\frac{16}{3\\sqrt{3}}} \\right] \\\\\n&= \\frac{8\\pi}{3}\\left[\\frac{3}{16} + \\frac{3}{16} \\right] = \\pi.\n\\end{align}", "8daacdd63add1a6010d90d50f6b31287": "[\\chi(a)]_m=[\\eta(m)]_a", "8daadc755544d004ab47d6de94aac603": " e^{-x^2} ", "8daaec29229349c86fac499b70a2f9a1": " \\hat \\Sigma = \\frac{1}{T-kp-1} (Y-\\hat{B}Z)(Y-\\hat{B}Z)^'.", "8dab2b7a89017e53773e9cd229024f69": "S^{\\prime \\prime \\prime} = (S^{\\prime \\prime})^{\\prime} \\subseteq S^{\\prime}", "8dab79fb803f1fd7c0d7e51f53af91fc": "P f(x) = \\sum_{i, j = 1}^{n} a_{ij} (x) \\frac{\\partial^{2} f}{\\partial x_{i} \\, \\partial x_{j}}(x) + \\sum_{i = 1}^{n} b_{i} (x) \\frac{\\partial f}{\\partial x_{i}} (x) + c(x) f(x),", "8dab854f9a6557a4a0929ef4aa66ff83": "M_r\\approx\\frac{a+b}{2}\\,\\!", "8dabb3547e797588cdf3d53ed0fe3840": "(mx+p)(nx+q),\\,\\!", "8dabbebd6ee290c4146185e9594a3d5e": "w_i' = \\frac{w_i}{\\sum_{i=1}^n{w_i}}", "8dabcc43d3a23771e0bf9b083acf0276": "\\textstyle{\\frac {\\ln 2} {\\ln \\sqrt{2}} = 2}", "8dabcf474eacd6552e93bdcb128e2112": "f(x)=ax+b,", "8dabd14e6d0c54783df977686fd133c5": "\\Lambda_{\\chi} = 4\\pi F", "8dabd754114991e7c8613e4a71de6a7f": "g_N = 0 ", "8dac106412bbbf5ff2b51c7e890b8da2": "\\begin{array}{ll} \n p_x(x) = 0 &\n |x| \\ge a \\\\\n p_x(x) = \\frac{\\mu p_0}{a} \\left( \\sqrt{a^2-x^2} - \\sqrt{a'^2-x'^2} \\right) &\n a - 2a' \\le x \\le a \\\\\n p_x(x) = \\mu p_n(x) & \n x \\le a - 2a'\n \\end{array}\n", "8dac66d1fbbf1f63074f06b7a5fa34af": "\n \\begin{align}\n W_{2n+1}W_1^3 &= W_{n+2}W_n^3 - W_{n+1}^3W_{n-1},\\qquad n \\ge 2, \\\\\n W_{2n}W_2W_1^2 &= W_{n+2}W_n W_{n-1}^2 - W_n W_{n-2}W_{n+1}^2,\\qquad n\\ge 3,\\\\\n \\end{align}\n", "8dac687113037c47c0fd05bab531f110": "Q = Ze \\ ", "8dac72375ca10cfbaf88706677ea84fc": "\\begin{align}\n Y &= Y_n f^{-1}\\left(\\tfrac{1}{116}\\left(L^*+16\\right)\\right)\\\\\n X &= X_n f^{-1}\\left(\\tfrac{1}{116}\\left(L^*+16\\right) + \\tfrac{1}{500}a^*\\right)\\\\\n Z &= Z_n f^{-1}\\left(\\tfrac{1}{116}\\left(L^*+16\\right) - \\tfrac{1}{200}b^*\\right)\\\\\n\\end{align}", "8dacafd83e3a092334138cf6b146a45a": "V=V_0 e^{-5} \\approx 0.0067V_0 ", "8dacbe88e234817064df13b95dc0b03f": "\\beta(X, X')", "8daccf8fcd60f22c12fa871279aed6e9": "m\\colon I\\to Z", "8dad07c7edc7553e99073b33d3eefd27": "\n|R_j - R_i| > t_{1-\\alpha/2,bk-b-t+1}\\sqrt{\\frac{2\\left(A-C\\right)r}{bk-k-t+1}\\left(1-\\frac{T_1}{b\\left(k-1\\right)}\\right)}\n", "8dad1f18d2a4c5374b51337b7a7089a2": "\\begin{matrix}\np \\oplus q & = & (p \\land \\lnot q) & \\lor & (\\lnot p \\land q) & = & p\\overline{q} + \\overline{p}q \\\\\n\\\\\n & = & (p \\lor q) & \\land & (\\lnot p \\lor \\lnot q) & = & (p+q)(\\overline{p}+\\overline{q}) \\\\\n\\\\\n & = & (p \\lor q) & \\land & \\lnot (p \\land q) & = & (p+q)(\\overline{pq})\n\\end{matrix}", "8daddc6510f7c78dd1bdb182275e257d": " \\mathbf{A}_P = [\\dot{\\Omega}]\\mathbf{P} + [\\Omega][\\Omega]\\mathbf{P}, ", "8daddd3becf6dd3d5870fd2ffd26e926": "(\\lambda y.x)[x := y]", "8dae15391990e3023539618879d23d2e": " \\mathrm{res}(P(-z),Q(z))=\\mathrm{res}(Q(-z),P(z))", "8dae3ae1047e3b8c583a67b30d58b070": "x:1{\\to}\\tau_1", "8dae3b2372709f33cb76a0610e27c912": "\\lambda_{a;c}\\equiv \\partial_c \\lambda_a-\\Gamma^b{}_{c a}\\lambda_b", "8dae42e042a664161a81fb5721cfddcc": "(1 - T2) * G1 + G2", "8daed7fc81d8dadcf42f6e5aac3bfa95": "\\Lambda^n = d^{-1} (n)", "8daef4f31d7e81111d4b47aac8351056": "{\\mathbf r}", "8daf19537c33bdbb6bf970c457744856": " \\begin{align}\nZ &= \\sqrt{z} (1_{\\!N} + S) (1_{\\!N} - S)^{-1} \\sqrt{z} \\\\\n &= \\sqrt{z} (1_{\\!N} - S)^{-1} (1_{\\!N} + S) \\sqrt{z} \\\\\n\\end{align} ", "8daf46c1aa6ed7080487e0da4ddaf2c4": "a^2 + b^4", "8dafc814fac4e4dcbe25ff18e2e12102": "t(x) = \\begin{cases}\\big(1+(\\tfrac{x-\\mu}{\\sigma})\\xi\\big)^{-1/\\xi} & \\textrm{if}\\ \\xi\\neq0 \\\\ e^{-(x-\\mu)/\\sigma} & \\textrm{if}\\ \\xi=0\\end{cases}", "8dafd75f459ccbdd51bb778a9ea84696": "\\frac{\\text{P}}{120}", "8db075d8727efd952ae9aa9ddc72e215": "g_i(x_1,x_2,\\dots,x_n) = \\gamma(h_1(x_1,x_2,\\dots,x_n), h_2(x_1,x_2,\\dots,x_n), \\dots h_p(x_1,x_2,\\dots,x_n))", "8db085dd3c69bdc77640e70ffc270859": "\\displaystyle{R=\\sum a_{ij}R_i^*R_j}", "8db0b662fa9251025056db2a559edf22": " L \n\\approx \\frac{S}{\\left(\\frac{\\varphi}{\\theta}+1\\right) S \\sin \\theta - 1} \n= 1 \\left/ \\left( \\left( \\frac{\\varphi}{\\theta} + 1 \\right) \\sin \\theta - \\frac{1}{S} \\right) \\right.\n", "8db0cfdc3e8012e6f3305647c1a8e75a": "\\frac{1}{\\sqrt{z-\\cos\\psi}}=\\frac{\\sqrt{2}}{\\pi}\\sum_{m=-\\infty}^\\infty Q_{m-\\frac12}(z) e^{im\\psi}", "8db0d01d654d359fba2bd143fc245ab2": "\\phi(c_d)", "8db0eb87d6997c572eb934bd08df4f99": "R_r(N) = \\frac{\\varepsilon_m - \\varepsilon_p(N)}{\\varepsilon_m - \\varepsilon_p(N-1)}", "8db0ed0d2dea4d5440e351e5cb8a52cb": "V=L^3", "8db119c7ab7fbc0e67d237d4d45786b5": "f(x)=2x^3-6x^2+2x-1\\,", "8db196fb7418a660abc61087d1bd3fb4": "\\Omega(c)", "8db1bc7c79d58c3dcaa6733279465afa": "\\scriptstyle{M_r}\\,\\!", "8db1c171b8d09b10c93a5b7cc686c267": "\\scriptstyle{\\lambda_i}", "8db1c60583b352c814bfe7c173887f9a": " \\langle\\mathbf{\\hat P}\\rangle \\rightarrow \\langle\\mathbf{\\hat P}\\rangle ", "8db259fbdb4b468d582358b08d521c31": " [x_1,x_2]= 2 i \\lambda x_3,\\ [x_2,x_3]= 2 i \\lambda x_1,\\ [x_3,x_1]= 2 i \\lambda x_2", "8db29312062e5ca07bcd9e71a959320f": "\n\\{\\phi_2, H\\}_{PB}+\\sum_j u_j\\{\\phi_2, \\phi_j\\}_{PB} = -\\frac{\\partial V}{\\partial y} - u_1 \\frac{q B}{c} \\approx 0.\n", "8db2bfbbe96bb6ef59bacb47409a9bfb": "I (\\mathcal K_X - D) = \\mathrm {dim} H^0 (X, \\omega_X \\otimes \\mathcal L(D)^\\vee) ", "8db2f483a9dfe249c76584933e92b13f": "o(X^{\\epsilon})", "8db361a862292f1f9986b834a12f9a49": "\\forall y \\in Y \\setminus \\{0\\} \\quad \\exists x \\in X : \\langle x,y \\rangle \\neq 0", "8db3e2b577e8ee934a6077810415dc5a": "r = a \\left ( 1 - e \\cdot \\cos{E} \\right ) \\ .", "8db435dcf1497fdf586dc092ad7b156c": "1000_b", "8db47f00f7c9985b1cc9bbf323742ca5": "\\langle x_i, x_j \\rangle^3 = x_ix_jx_i", "8db4ce44577c33d51fdfc897847fb0f3": "\\vec{u_0} = (0, \\ldots ,0)", "8db50b5761fe08090afea5a7e95e15e2": "\\langle 0 | \\hat{O} | 0 \\rangle \\neq 0", "8db59c81b362ae06f366fa6540848a0c": "\n\\|\\mathbf{b}\\|\\ = \\textstyle\\frac{a}{2}\\sqrt{h^2+k^2+l^2}\n", "8db5d02d8b4212de262b7705c3e1050b": "p=\\frac{l^2(4m^2+a^2)-m^2}{4}, \\qquad \tb=l(4m^2+a^2)-5p-2m^2, \\qquad c=\\frac{b(a+4m)-p(a-4m)-a^2m}{2}", "8db634cc7594395c137f9552a55d739b": "E_{p}/E_{s}=\\pm j", "8db64a3d8a926ec76ce237e18604df82": "H(f)", "8db6ac0d782f73c855f98e6da215b791": "v_\\theta", "8db6b6ceca095e847272349cd6c7815d": "\ndE(u,\\psi)=\\langle f(u),\\psi \\rangle\\,.\n", "8db77c1db1b64f77c15e545ce6d0f841": "B =\n\\begin{pmatrix}1&1\\\\3&1\\\\0&2\\end{pmatrix}", "8db7a880b9180fb935a03a24c1791a61": "\\sigma_{i j} \\equiv \\epsilon_0 \\left(E_i E_j - \\frac{1}{2} \\delta_{ij} E^2\\right) + \\frac{1}{\\mu_0} \\left(B_i B_j - \\frac{1}{2} \\delta_{ij} B^2\\right)\\,", "8db8071d3701476abd2ae54cece957d2": "\\begin{matrix} \\frac{15}{7} \\end{matrix}", "8db821790df0757ea94e33854a2be09e": "\\boldsymbol\\theta'", "8db88902425395a53274a536f9acafc2": "\nE(Y^2) = E(X+Z)^2 = E(X^2) + 2E(X)E(Z)+E(Z^2) = P + n\n\\,\\!", "8db8b9755b30434a775c85a9ba042135": "\\bar x=\\frac {1}{25} \\sum_{i=1}^{25} x_i = 250.2\\,\\text{grams}.", "8db94b89b3d73b48dc2e6d2a69f8640e": "\n\\frac{dx(t)}{dt} = -F(x(t))-P_{N_K(x(t))}(-F(x(t))).\n", "8db96b7f19b662240f82fcad942cf7fc": " \\mathcal{G}^1", "8db995838da5a101f0dd2ff081258119": "\\delta \\mathbf{r}_i = \\sum_{j=1}^m \\frac {\\partial \\mathbf {V}_i} {\\partial \\dot{q}_j} \\delta q_j,\\quad i=1,\\ldots, n.", "8db99695ed6964b6c9c259458a50712f": "\\sigma(f(x)) = f(\\sigma(x)).", "8db9f9980d085b9184a30924aa6c6853": "d_2", "8dba0675ae4cf6f83c1d60c639199227": "\\kappa_z", "8dba1a8777c9ebb27989e974b7774d3c": " T^{\\mu\\nu} ", "8dba833ad1c6ee6e906b37920e36bad0": "\\Gamma^\\lambda_{\\mu\\nu}", "8dba845d40b18c328bdc4091423d00a5": "\\begin{align}\n& (-i\\hbar \\gamma^\\mu \\partial_\\mu + mc)_{\\alpha_1 \\alpha_1'}\\psi_{\\alpha'_1 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2s}} = 0 \\\\\n& (-i\\hbar \\gamma^\\mu \\partial_\\mu + mc)_{\\alpha_2 \\alpha_2'}\\psi_{\\alpha_1 \\alpha'_2 \\alpha_3 \\cdots \\alpha_{2s}} = 0 \\\\\n& \\qquad \\vdots \\\\\n& (-i\\hbar \\gamma^\\mu \\partial_\\mu + mc)_{\\alpha_{2s} \\alpha'_{2s}}\\psi_{\\alpha_1 \\alpha_2 \\alpha_3 \\cdots \\alpha'_{2s}} = 0 \\\\\n\\end{align}", "8dba9dea241789707792fe15c4a08bfb": "q \\leftarrow \\mathrm{not}~p", "8dbaf8ac915fe6c0f72a731104ee183c": "\\frac{\\pi}{\\sqrt{m}\\, K(m)}\\, \\operatorname{sech}\\, \\left( \\frac{3\\, \\pi\\, K'(m)}{2\\, K(m)} \\right) = \\tfrac{1}{16}\\, m + \\tfrac{1}{32}\\, m^2 + \\cdots,", "8dbb6da1bd20c91187bddc2a1254ad33": "\\overline{MR}", "8dbb754171c2c5302031a8964e1cd49a": "F(x; d_1,d_2)=I_{\\frac{d_1 x}{d_1 x + d_2}}\\left (\\tfrac{d_1}{2}, \\tfrac{d_2}{2} \\right) ,", "8dbb7b8d44398137544e0ba39945cefb": "q \\equiv e^{2\\pi \\rm{i} \\tau}\\,", "8dbbb590a9ff816e7f62c6ce8f3eebb0": "\\alpha _{ij}^{ar} ", "8dbbcbb0e878bab88516ad2b7d27f8e0": "C([0,1])", "8dbbd203e43194e55141c9e1509cdf24": " \\zeta (\\rho) ", "8dbbead80afc516b544f8627fd8ef15f": " \\gamma(t)\\,= \\, \\begin{pmatrix} \\cos(3t) \\\\\n\\sin(2t) \\end{pmatrix}\\,. ", "8dbc1d4bb63b754bf24725958a84fd15": "P(A|B) = \\frac{0.99\\times 0.001}{0.99 \\times 0.001 + 0.001\\times 0.999} \\approx 0.5, ", "8dbc7f788e77e5484f4dad18b1f4a9e2": "\\,pL + (1-p)N\\, = \\,M\\,", "8dbc9f1c6a0bd7e50c33200bbb5a4de6": "{{v}_{GS2}}+{{v}_{GS4}}-{{V}_{TH4}}", "8dbce60e60a368022e21b741ac1eac39": "163=4\\cdot 41-1", "8dbd1e08e9db45f7eef2f103cd95537d": "X_1,X_2,X_3,\\dotsc,X_n", "8dbd2f1f878b62b60fb918c5331baddb": "\\begin{array} {l}\n2f'(x_0)=\n\\frac{f\\left(x_0 + h\\right) - f(x_0)}{h}\n-\\frac{f\\left(x_0 - h\\right) - f(x_0)}{h}\n-2\\frac{f^{(3)}(x_0)}{3!}h^2 + \\cdots\n\\end{array}", "8dbd375defae922bfd12055e7ccea794": "S' = \\frac{S\\,L}{2\\,Q}.", "8dbd626b6da2b094bb140d57b3d393b7": "\\beta =3", "8dbdb022beccd0794ac387b46d6a850b": " n = k \\cdot \\frac{1}{d} ", "8dbdedd164a9fab7c4df9606974aef7e": "p(\\Delta X ) = \\frac{1}{\\sqrt{(2\\pi)^N \\frac{k_B T}{\\gamma} |\\Gamma^{-1}|}} exp\\left\\{ -\\frac{1}{2} \\left(\\Delta X^T\\left( \\frac{k_B T}{\\gamma} \\Gamma^{-1} \\right)^{-1} \\Delta X \\right) \\right\\}", "8dbe15af3641f004ae18917b399a8daf": "e_C(x) := \\sum_{j = 0}^\\infty \\frac{x^{q^j}}{D_i}.", "8dbf1794d34855b43dc8459d5f724fc6": " H_1 \\ge H_2 ", "8dbf82d6b16273a32e8d4d77e91f7184": " f: \\mathbb{R} \\to [-1,1] ", "8dbf9fc1ba9a9d8308be2d385891a478": "x=(\\pi-y)\\, ", "8dbfb71aa8d919616cc438b8da40ac07": "X= \\int_{380}^{780} I(\\lambda)\\,\\overline{x}(\\lambda)\\,d\\lambda", "8dbfc54de011ca6ab2193349957f921c": "~\\Phi_{15}(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1", "8dbfdc5f55876dc225778e70b6e52716": "H^1(X, \\mathcal{O}_X^\\times) \\stackrel{c_1}{\\to} H^2(X, \\mathbf{Z}) \\stackrel{i_*}{\\to} H^2(X, \\mathcal{O}_X).", "8dbfe80e98096b722bc16a2e94bcd3a2": " \\sqrt{5} ", "8dbfffc5d0fc6f119ef9094ca48e455f": "r_i n_i + e_i = 1", "8dc016b10f90d25f5ef0235c61a2e92d": "b x^3+(2 c-a)x^2 y+(d-2 b)x y^2-c y^3", "8dc047f1ab141d872b848bea3704fb16": "\\rho^0", "8dc0726e7939728d78a6915a4b8f5f31": "g : T_pM \\times T_pM \\to \\mathbb{R}", "8dc07276d2b9f3530a254a9d1578d1ff": "1_{T}", "8dc077183f64071dfe8a5fdccd2233ce": "\\gamma=\\beta=0", "8dc09ae53af7861aceaed2f83c56266e": "\\varepsilon = \\varepsilon_1 - \\varepsilon_0 \\sim \\operatorname{Logistic}(0,1) .", "8dc0c9763d6840d72a08a52baa700d71": "\\|x\\|_{\\theta,q; K} = \\left( \\int_0^\\infty \\bigl( t^{-\\theta} K(x, t; X_0, X_1) \\bigr)^q \\, {dt \\over t} \\right)^{1/q}, \\ 0 < \\theta < 1, \\ 1 \\leq q < \\infty,", "8dc1108fa267f5557c5c513715b6936f": "\\displaystyle P=P_w+P_c", "8dc16c2e5e63bf6fd53a2d9bb255e2c2": "\\tan{\\nu \\over 2} = \\sqrt{{{1+e} \\over {1-e}}} \\tan{E \\over 2}.", "8dc17d07563915639be70694fa05d10f": "\\sum_{k=0}^\\infty \\frac{1}{1+F_{2k+1}} = \\frac{\\sqrt{5}}{2},", "8dc1afcbc29a7682ef6ded6db4118e3f": "\\mathbf{y=x+e}", "8dc1b08bfa49a25fa48d92039d2aa57b": "\\cos (\\alpha - \\beta) = \\cos \\alpha \\cos \\beta + \\sin \\alpha \\sin \\beta\\,", "8dc1d17f4cfe1f24e19ea2e5c811b0f1": "L(s) = \\left(\\frac{s}{\\omega_{gc}}\\right)^\\alpha", "8dc1d56baed5ad78856bde14f0890829": "\n\\left( X_{1} - X_{2}| X_{1} - X_{2} \\right) = \\left( X_{1}| X_{1} \\right) - 2 \\left( X_{1}| X_{2} \\right) + \\left( X_{2}| X_{2} \\right).\n", "8dc20d26fe44f088fe1a6fbe9b83ee68": "{q(t+\\Delta t)} \\approx e^{\\Delta t A} q(t) ", "8dc21659c38149d1e560f3ece15dfa61": "\\scriptstyle \\gamma \\,\\in\\, \\Gamma", "8dc227bf45a769411fba0e02211c50b7": "v=xbay", "8dc28a0cc76b526b1f9fe2f8e0365afd": "\\Sigma^+", "8dc29e910ba13bd638d859ad04e3a728": "\n W = \\cfrac{\\mu}{2x}\\ln\\left[1 - (I_1-3)x\\right] ~;~~ x := \\cfrac{1}{J_m}\n ", "8dc2b551c671a4cbd1a47da6aabd4290": " K(x, y) = \\frac{\\sin \\pi(x-y)}{\\pi(x-y)} ", "8dc3165127c0a051eab59388a8bd85b6": "|a|<1;\\Re(s)<0 ;z\\notin (0,\\infty). ", "8dc32cceb6194fa5f575b00194d43e3a": "\\mathfrak{m}_B", "8dc33800831c3c255949e21994644e06": "V_2(\\mathbf{x},z_1,z_2) = V_1(\\mathbf{x},z_1) + \\frac{1}{2}( z_2 - u_1(\\mathbf{x},z_1) )^2", "8dc3bc49839e8a78ee2fff49cbb6a671": "\\pm \\mathbf{j}_r = \\pm j_r \\mathbf{\\hat{n}}", "8dc3c80382d36770ce1d55fd9bc857d7": "(0, \\, \\omega-x)", "8dc450b19be83d6c0d0e9e8cd025f6db": "a_{15}", "8dc4a8ffacef3624aa0dbfd18b7a2645": "\\mathbf{a}\\cdot \\mathbf{b} = \\|\\mathbf{a}\\|\\|\\mathbf{b}\\|\\cos\\theta = \\|\\mathbf{b}\\|\\|\\mathbf{a}\\|\\cos\\theta = \\mathbf{b}\\cdot\\mathbf{a} ", "8dc4d6a95bddf0efeb53f4bcbff85c78": "f(x+\\delta)-f(x) = 2x\\delta + \\delta^2 = \\delta(2x+\\delta)\\ ,", "8dc4e9b51e0163abb12682fed644bec8": " AB + C \\rightarrow A + BC", "8dc4eef060814e559aff4c5fac3f51fe": "S_{1}", "8dc608627da05cbf462058c8aa0f0159": "T=a\\dot\\gamma(t)+bt\\dot\\gamma(t)", "8dc639c3fa687ef0a816d77d61d1d4b3": "\n (\\nabla^2 + \\frac{\\omega^2}{c^2}) \\psi(x,y,z) = 0.\n", "8dc6746416453d83e83e57fcc84f0a6b": "H=v^2 sin^2(\\theta) /(2g)", "8dc69e13fa9b650f0a3fbaa04219b38b": "\\Phi_S^R", "8dc715ebb19d974d3aab41139a452f06": " \\alpha < 0 ", "8dc7ccad4a5d11df2425a555762dbdd5": "r(v,u) = - f (u,v)", "8dc7dffd8fa13fb12d64fe738500ddc8": "{\\mathbf x} \\; \\leftrightarrow \\; {\\mathbf x}^{'} \\; \\leftrightarrow \\;{\\mathbf x}^{''}", "8dc86ef3277d43e796954a007d4749fe": "\\Omega(n^{r/2})", "8dc8a30964b1ef46d62f2c481cc25d41": "y_j(t+\\Delta t)=\\frac{y_j(t)}{|FF_j|}\\sum_{k\\in \nFF_j}s_k", "8dc8b40bea27052c6ab892eb6e2d9549": "P_{\\mathrm{out}}=K_p\\,{e(t)}", "8dc8baabffb768dc5b1ef5003cc1f0aa": "\\,|b+\\rangle = D^{-1}(y, t) |c+\\rangle", "8dc97254265e69c44bbac8dc8021fd0d": "w=\\frac{az+b}{cz+d},", "8dc97ea29b82f742bc615e355a595eb8": " {}^\\infty {i} ", "8dc986d213c376c8018a319090cfb4df": "f_1,f_2 \\in C^\\infty(G)", "8dc9ac661c13469f9ca4a12f3bc5778e": "N_q, N_r, N_t", "8dc9e1d04e75c7cd5005c4374e735dba": "M_Q", "8dca64f35915796a1f58040735955f98": "~u=GL~", "8dca86ba06603d4452a1f300f3f1a858": " \\exp\\left\\{ { r_1^2 \\ln r_1 + r_2^2 \\ln r_2 + 2r_1r_2 \\ln S \\over (r_1+r_2)^2} \\right\\} ", "8dcaca3cf711b0eb1a297b04cc0bcc5a": "\\hat{f} = f(\\hat{\\xi}) \\equiv \\sum_{s=0}^{\\infty } \\frac{1}{s!}\n\\frac{\\partial ^{s}f(0)}{\\partial \\xi^{i_{1}}...\\partial \\xi ^{i_{s}}} \\hat{\\xi}^{i_{1}}...\\hat{\\xi}^{i_{s}}.", "8dcb2809520d96c7fd43785b89b6a419": "a b \\le \\frac{a^p}p + \\frac{b^q}q", "8dcb4054cddf3f5c00a73ed4cd4bb3e8": "\\frac{L(x^2,y^2)}{L(x,y)} = \\frac{x+y}{2}", "8dcb89a6a3b5aed209b49fb275cafb1d": "H\\ ", "8dcbbd8a26042d465d2a3168d0f1a6e3": "\\frac{78}{{a^*}^2} + \\frac{52}{{a^*}} + 33", "8dcbe52c54b60c635047b5695ff5e18a": " \\langle A,k,B,x \\rangle \\in L ", "8dcbe65b7a89721690d87fd8ae73a691": " \\theta ", "8dcbec897fcf90e1e3d4f1c661e7d201": " S = \\int { dQ \\over \\tau } ", "8dcc3352698dbb02fc68eba16025702f": "I=I(i-E(\\pi), Y_{-1}) \\, ", "8dcca16a6f01d121265982b9d7a420c5": "i_s = g_m \\left ( v_1 - v_2 \\right )", "8dccb0fb4f7e3b60ce26f3aa4fcfdde2": "DR_{T/D}^{V/S}", "8dccbb627dcd1005a38a66476a64e9b9": "\\frac{\\partial }{\\partial t}f\\left( x^{\\prime },t\\right) =\\int_{-\\infty}^\\infty dx\\left( \\left[ D_{1}\\left( x,t\\right) \\frac{\\partial }{\\partial x}+D_2 \\left( x,t\\right) \\frac{\\partial^2}{\\partial x^2}\\right] \\delta\\left( x^{\\prime }-x\\right) \\right) f\\left( x,t\\right).", "8dccc478ab55fd98af0c19912871a664": "e^{i\\theta} = e^{i\\arg z} = \\sqrt {\\dfrac{z}{\\overline z}}", "8dccf045be97b120cff5a14070254e2c": "[B]_e", "8dcd00954bed70e12f41f3fd4dc76199": "\\operatorname{ht}(\\mathfrak{q}) = \\operatorname{ht}(\\mathfrak{p}) + 1", "8dcd90e7fc898f4f53eaa7f23e1fc19e": " E(\\hat x,u,y),W(\\hat x),I(\\hat x,u),\\eta(\\hat x,x)", "8dcdfb53339153bd7d7a01b5f34f464f": "\\Delta(C_{in}(C_{out}(m^1)), C_{in}(C_{out}(m^2))) \\ge dD.", "8dce1ddb308e8dbb24737db9e7e8fb80": "\\{U, V\\}\\,", "8dce227794927774ffe1c44e8ff3d2ce": "\\digamma :\\pi_1 (X, t)\\rightarrow \\mathrm{Sym}\\,f^{-1}(t)", "8dce6d345cc4da29a0bc0a07763e7a54": "d_h(f(x)) = f(x + h) - f(x) \\,", "8dceed86424a014b260384020316ee6f": "\\lim_{n\\to\\infty}\\ln n\\prod_{p\\le n}\\left(1-\\frac1p\\right)=e^{-\\gamma},", "8dcf170ca6d57c8367b4b676d9dfc6ce": "\\chi(M) = \\frac{3}{2}|\\tau(M)|,", "8dcf7bea90e8b747e052f4043fd0282d": "LMP = T \\cdot (C + log(t))", "8dcfeeac00f3c2d43aa2a8ae720195a8": "(1+|\\mu|)^2\\textstyle{\\left|\\frac{\\partial f}{\\partial z}\\right|^2},\\qquad (1-|\\mu|)^2\\textstyle{\\left|\\frac{\\partial f}{\\partial z}\\right|^2}.", "8dd06ff66a264cf511e4761ed7d22698": "\\textstyle (1 + x + \\ldots + x^{p-1})", "8dd0c538a75adcdd0d028bfd00da104a": "\\begin{align}\n& \\mathbf{D} = \\begin{pmatrix}\n6 & 0 & 0 \\\\\n0 & 4 & 0 \\\\\n0 & 0 & 5\n\\end{pmatrix}, \\quad \\mathbf{U} = \\begin{pmatrix}\n0 & 2 & 3 \\\\\n0 & 0 & 2 \\\\\n0 & 0 & 0\n\\end{pmatrix}, \\quad \\mathbf{L} = \\begin{pmatrix}\n0 & 0 & 0 \\\\\n1 & 0 & 0 \\\\\n3 & 1 & 0\n\\end{pmatrix}. \\quad (14)\n\\end{align}", "8dd0c7198846ada898576223d690781c": "\\phi_i(v)= \\frac{1}{|N|!}\\sum_R\\left [ v(P_i^R \\cup \\left \\{ i \\right \\}) - v(P_i^R) \\right ]\\,\\!", "8dd0d0dcdaf6c95881085c61f3f2b804": "[x_0:\\ldots:x_n]", "8dd1098acbda801c4a919fdf1b31b49a": " \\frac{3}{8}\\sqrt{5}", "8dd14b1e21dc7a58cd3cb3129cf1e714": "C^\\infty_0(\\mathbf{R}^n)", "8dd18da2d4c291bdb09498be7410a8a8": "\\binom{n}{j}-\\binom{n}{j-1}", "8dd1c8d755ab5276193c42ddca01714b": "x\\neq x^{\\prime}\n", "8dd1d16b8a48594efea4627112daf138": "\\lambda_\\max = {hc\\over x }{1\\over kT} = {2.89776829\\ldots \\times 10^6 \\ \\mathrm{nm} \\cdot K \\over T}.", "8dd1dfc9085a8ee591e71061d859f1f8": "1 \\leq i,j \\leq n-1", "8dd27b5721e69b09a1ed26de568ea4cb": "\\frac{2d_1^{d_1/2}d_2^{d_2/2}}{B(d_1/2,d_2/2)}\\frac{e^{d_1z}}{\\left(d_1e^{2z}+d_2\\right)^{\\left(d_1+d_2\\right)/2}}\\!", "8dd2905600f176656a57f7a590c8d10a": "z=-L", "8dd2c24f125069bc335ac0f9c342240f": "\\mu_1(X+Y)=\\mu_1(X)+\\mu_1(Y)\\,", "8dd2ce87deb5e592ac1f0b4774fdc57e": "\\left(\\tfrac{1}{2},\\tfrac{1}{2}\\right)\\otimes \\left(\\left(\\tfrac{1}{2},0\\right)\\oplus \\left(0,\\tfrac{1}{2}\\right)\\right)", "8dd2fc3655f6cac953cdb9346c82d79c": " \\frac{1}{\\sqrt{4\\pi}} \\left(\\mathbf{E}, \\varphi\\right) ", "8dd375014d5d80d7ebc559503389fc75": " =\\frac {1}{4 \\pi \\varepsilon_0}\\int \\bold{\\nabla_{\\bold {r_0}}\\cdot} \\left( \\bold{p} ( \\bold{ r}_0 ) \\frac {1}{|\\bold r - \\bold{r}_0|} \\right) d^3 \\bold{ r}_0 -\\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac {\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 )}{|\\bold r - \\bold{r}_0|} d^3 \\bold{ r}_0 , ", "8dd3b8524bbf60a6454a0d4060523edd": "k_{\\rm on}", "8dd3f7aace93c0eb35885f03089bf9f4": "\\scriptstyle\\Omega=\\mathbb{R}^n", "8dd48ad82c400267b34e543a43898975": "H_1 X", "8dd4d4c8c8252a05c2f5bcee9853c13c": "\\sqrt{2} = 1 + \\frac{1}{3} + \\frac{1}{3\\cdot4} - \\frac{1}{3\\cdot 4\\cdot 34} \\approx 1.4142156 \\ldots", "8dd4eadf7cbeae9d2a9508c20ab745d8": "A_s A_r \\subseteq A_{s + r}.", "8dd52d12291f7fb29eb805220085da1f": "\\mathrm{d}U = \\delta Q\\ - \\delta W\\,", "8dd54ac98862fe5dca2a88b6608a2efe": "\\frac{ \\hat y-\\bar{y}}{s_y} = r_{xy} \\frac{ x-\\bar{x}}{s_x} ", "8dd561ce07003f472c79edf351db79d2": "Ratio.of.indv.space/rentspace=unit.space(num.users.individual.consumed/num.max.rent.units.stocked)", "8dd5b7a1b2b2225dc947f89f95e725a7": "\\mathfrak{g}_0=\\mathfrak{k}_0\\oplus\\mathfrak{p}_0. ", "8dd5d5be4ea058afe96f9a0746720bfc": "\\frac{\\partial n_1}{\\partial t}=D_n \\frac{\\partial^2 n_1}{\\partial x^2}+\\mu_n n \\frac{\\partial E}{\\partial x}+\n\\mu_n E \\frac{\\partial n_1}{\\partial x}-\\frac{n_1}{\\tau_n}", "8dd65bab593f474cdd7d9e36ff6e6ef5": "P_{11}>0", "8dd67b271b421f274f0da11f81dd00c8": " \\varphi(x) = f(x) + \\lambda \\int \\limits_a^x K(x,t)\\,F(x, t, \\varphi(t))\\,dt. ", "8dd6b9e44bc9a0f10940966120dcc543": "=-2D", "8dd70810fd630d70859c68476e26a8af": "A = \\frac{1}{2} \\times \\frac{5t^2\\tan(54^\\circ)}{2}", "8dd76fd827ac308ae127f270d588ea88": "\\{\\psi_{jk}: j, k \\in \\Z\\}", "8dd782163a0139cdd39d884686f341ab": "\\frac{x}{t}+\\frac{y}{1-t}=1", "8dd786a0c2f6c4ed55b7018c316b758f": " ^y x = y + 1", "8dd7ae1d8473fdd6f6fff2797502ce52": "\\alpha=\\rho", "8dd8065c52a9ca2a1c1d675be055ac4f": " = \\frac{d}{dt} \\langle \\psi | Q | \\psi \\rangle \\,", "8dd8103970eb2ac566b3b643485bc3a7": "Z_S", "8dd887d65eb091f41eef673dcad2bcd6": " \\exp_p : T_{p}M \\supset V \\rightarrow M ", "8dd8ba1c4dcbad0b4ef12b710183031b": "p^3", "8dd90a3f4478dbd1a560b935510a634c": "W_{AB}", "8dd92d746a8f51760c7eb613c081d833": "A=2 D_{ox} (\\frac{1}{k_i} + \\frac{1}{h_g})", "8dd93436c52651449b4b3b8e857b2815": "||u|| < \\epsilon", "8dd9353c524814435c57b5842cd77704": "P_n(A) = {1 \\over n} \\sum_{i=1}^n I_A(X_i)=\\frac{1}{n}\\sum_{i=1}^n \\delta_{X_i}(A)", "8dd9af42e91892208498d8911bf80cee": "MUAMA = \\frac{MUAMC^2}{4 \\pi}", "8dd9e89ce5578bc23c645e258a4984a1": " p \\longleftarrow (\\phi n)^3 ", "8dd9f0722c7d932cc0458487687bbdfb": " \\sum_{r\\neq s}u_r\\overline u_s\\csc\\pi(x_r-x_s) ", "8dda2cadce26713fe93df8cffc69a68c": "\\alpha^e = -1", "8dda748668c66fb07b4740428e4dcb8a": "w \\gg h", "8ddacefefa0108166062415f26a30865": "\n\\varphi(p^k) = p^{k-1}(p-1).\\;\n", "8ddadb440399f5f12dd876a68980cc48": "f: S \\to AA", "8ddafc7927053d5097738367da5198d4": " = { \\left( (x,y) - (x_0,y_0) \\right) \\over u_x v_y - u_y v_x} \\begin{pmatrix} v_y & -u_y \\\\ -v_x & u_x \\end{pmatrix}", "8ddb4710a0adbac8b2053abfeeaafce9": "\\gcd(a, \\gcd(b, c)) = \\gcd(\\gcd(a,b), c).\\;", "8ddbdb17baaf0d21127307a38263f7f3": " | \\psi_0 \\rangle = \\frac{1}{\\sqrt{2}} \\bigg(|0 \\rangle + |1 \\rangle \\bigg) |f_k \\rangle |f_k' \\rangle ", "8ddbf0f47383b7ab8ab3abeb314f0751": "3 \\times 5 \\times 5 = 75", "8ddc12e22ad21af10bcf6805805c2643": "\\widehat{q}", "8ddc1c9b69bd3a22f5dbb0c9b4691191": "\\left(2\\sqrt{\\frac{2}{5}},\\ 0,\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "8ddc27874210ff78dfafafeeae2e1b36": "s = X_{00} = Y_0^0 = \\frac{1}{\\sqrt{4\\pi}}", "8ddc866f49a0b504ef2a0c135bac046b": "k_1/k_2=-dn_2/dn_1", "8ddca693d0aaa671fc0383a33b2d1881": "\n\\begin{align}\\mathrm{Pr}\\left( \\sqrt{\\left( X-\\mu\\right)^T \\, V^{-1} \\, \\left( X-\\mu\\right) } > t\\right) &= \\mathrm{Pr}\\left( \\sqrt{y} > t\\right)\\\\\n&=\\mathrm{Pr}\\left( y > t^2 \\right) \\\\\n&\\le \\frac{\\mathbb{E}[y]}{t^2} .\\end{align}\n", "8ddcab4efac2c57b502f852729cff556": "\\frac{\\partial}{\\partial t} \\{W|P|Q\\}(\\alpha,\\alpha^*,t) = \\left[(\\gamma+i\\omega_0)\\frac{\\partial}{\\partial \\alpha}\\alpha + (\\gamma-i\\omega_0)\\frac{\\partial}{\\partial \\alpha^*}\\alpha^* + \\frac{\\gamma}{2}(\\langle n \\rangle + \\kappa)\\frac{\\partial^2}{\\partial\\alpha\\partial\\alpha^*}\\right]\\{W|P|Q\\}(\\alpha,\\alpha^*,t)", "8ddcea788e2bfbd337ccf743cc362473": "1\\leq j\\theta_1", "8ddd1c91e37edfcbf5b3648c037f6dc9": " d_{i} = \\sum_{j} p_{ij} u_{j}\\,", "8ddd47d032a2b30e0c07079d70eca21d": "[(f(U) = 0) \\vee P] \\wedge [(f(V) = 1) \\vee P]\\,", "8ddd9db9e3bb41bdada9b72a395eb232": "\\overline{g}(x)= \\sum_{n=0}^\\infty s_n \\frac{x^n}{n!}", "8dddd65ed4b1df54a9a1c10e3eda97a5": "|N(x)|\\leq \\left ( \\frac{2}{\\pi}\\right ) ^ {r_2} \\sqrt{|\\Delta_L|}N(\\alpha)", "8dde731ce72d35c4c97320afaa210f5b": "(m,l)", "8dded0e396cfffe91fe61c0f5638918d": "\\scriptstyle(-0.14\\pm0.78)\\times10^{-13}", "8dded57ff95fbb7edb569f29172e8f15": "x:\\text{Spec}(k)\\to X", "8ddf2bab8ea2cf8bc23ad76b8f97a760": "Rx=Q^Tb", "8ddf3ea325351149da370d3ef1431f1f": "\\{1\\} = A_0 \\triangleleft A_1 \\triangleleft \\dots \\triangleleft A_n = G", "8ddf428754264f7920990178accd0ed5": " \\mathbf{V} \\cdot \\mathbf{W} \\le \\| \\mathbf{V} \\|\\| \\mathbf{W} \\| . ", "8ddf5a5a55242845c207a08a66734e17": "\n \\begin{bmatrix}\n 1 & 2 & 3 \\\\\n a & b & c \\\\\n x & y & z\n \\end{bmatrix}\n", "8ddf64593833e2eb5819c8edc7bf8ccc": "\\Delta S_2 = R * interception_2;", "8ddff807a5f3c26a1261ab2c1a8d70ae": "K*P=(K-\\neg P)+P", "8de00e764fb9da17d2756f327529fbfc": "\\binom nr \\times \\binom nr", "8de044b05649df54fcf6f5d948e9a052": "f(\\tau) = g[\\log(\\gamma \\tau)]", "8de08651551b0ea4d02e06b4a810500a": "\n h = \\int_{-\\infty}^\\infty f(x;\\,k)\\ln f(x;\\,k) \\, dx\n = \\frac{k}{2} + \\ln\\!\\left[2\\,\\Gamma\\!\\left(\\frac{k}{2}\\right)\\right] + \\left(1-\\frac{k}{2}\\right)\\, \\psi\\!\\left[\\frac{k}{2}\\right],\n ", "8de10033d25460cb5d3aa426604b8e7a": "{\\partial u \\over \\partial x} + {\\partial v \\over \\partial y} + {\\partial w \\over \\partial z} = 0.", "8de1b711fe458ef34dbb7b3ca65fb430": "\\boldsymbol{J}_{a}=-D_{a}(\\nabla n_{a}+\\frac{z_{a}en_{a}}{kT}\\nabla\\phi)", "8de1d35d1aa4d26173e05d8afdc47f2b": "\\lambda_{i} = \\lambda", "8de24af05f52dd8011d7d10c9655b678": "\\mbox{percent yield} = \\frac{\\mbox{actual yield}}{\\mbox{theoretical yield}} \\times \\!\\, 100", "8de2b72de4c6ab86f466473d8f0170f6": "\\ j = 1,2,3", "8de30666a7ad413a6fe2dcff3e2745d9": " -\\ln \\left ( \\frac{S_{T}}{S^{*}}\\ \\right ) = -\\frac{S_{T}-S^{*}}{S^{*}}\\ + \\int\\limits_{K \\le S^{*} } (K-S_T)^{+} \\frac{dK}{K^2}\\ + \\int\\limits_{K \\ge S^{*} } (S_T-K)^{+} \\frac{dK}{K^2}\\ ", "8de349b7671c746c957af00e216e2981": "K=\\sup_{t\\in[0,1]}|B(t)|", "8de3a4c56ec52d94ec92f3af31bffa24": "{{\\left\\| f \\right\\|}_{\\Beta ,q}}<+\\infty ", "8de412771f818bdeb8d48036ef559d39": "-\\frac{\\partial \\rho}{\\partial t} = \\nabla \\cdot (\\rho v^{\\psi})", "8de429f3361069c3daf7f7fad3c99899": "C\\subset \\mathbb{Z}", "8de4575bd6ad6673bfe010c0f03502f0": "p_M(\\lambda) := \\sum_A \\mu(\\emptyset,A) \\lambda^{r(M)-r(A)} \\ ,", "8de49e4dc03a96dbef8b4af5737a1140": "\\forall \\mathcal{F} \\,\\exists A \\, \\forall Y\\, \\forall x [(x \\in Y \\land Y \\in \\mathcal{F}) \\Rightarrow x \\in A].", "8de5526743bde347bc84a522aa3c613d": "\nIME_i = h_i^t \\times \\left( G_i - G \\right) \n", "8de65ef2a60da4eae42555a67ad79f90": "\n (23) \\qquad u_s = c_0 + s\\,u_p = c_0 + s \\,u_2\n ", "8de676e41ff8b05a83846f1847567ea1": "\\lambda \\geq \\kappa ", "8de69f7c7f63b7f09f8c8a26f8cefe9b": "\n\\frac{d^2x^{\\mu}}{d q^2} + \\Gamma^{\\mu}_{\\nu\\lambda} \\frac{dx^{\\nu}}{d q} \\frac{dx^{\\lambda}}{dq} = 0\n", "8de6aec4a891c6996da14077026c305d": "\\frac{\\partial\\rho}{\\partial t}=-\\{\\,\\rho,H\\,\\}", "8de70fac02678a44fb6bf615620a5509": "t = \\left \\lceil{\\sqrt{kn (1 - \\frac{1}{r})}} \\right \\rceil", "8de711161135a66a25816d55863db2c0": "\\frac{d^{2}X}{dx^{2}}+k_{x}^{2}X=0 \\ \\ \\ \\ \\ \\ \\ (17)", "8de75f395a38169896030b75701e832b": "\\alpha_k\\geq0,\\qquad\\sum_{k=1}^\\infty \\alpha_k^2 < \\infty,\\qquad \\sum_{k=1}^\\infty \\alpha_k = \\infty.", "8de7628f5ccb57abafaea9fe30ae44ac": "|\\psi (\\mathbf r_1,m_1;\\dots ;\\mathbf r_N,m_N)\\rangle.", "8de772a7e9fd492078dda19d389492b8": "\\hat{y} = Hy,\\,", "8de7bef571c58c96eb34589c9ca4764f": "\\int \\left(e^x + \\cos{x}\\right) \\,dx = \\int e^x \\,dx + \\int \\cos{x}\\ \\,dx = e^x + \\sin{x} + C", "8de7c50c2b3b3e50d0d0d63e6270a2af": "A = \\frac{a\\alpha p}{ R^2\\,T^2}", "8de83db7aec2eb7498a207f67135e8a3": " a_i (t+1) = a_i(t) + \\nu \\big [ y(t) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ] \\rho \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big ) ", "8de85f2a0b2bf965fdb004f4e1af3fdb": "\\Pr(z_{dn}=k\\mid\\mathbb{Z}^{(-dn)},\\boldsymbol\\alpha)\\ \\propto\\ n_{k,d}^{(-n)} + \\alpha_k", "8de8d9022c800498f590dcab03eaf0b0": "{P_\\mathrm{atm} \\over {\\rho}}={v_B^2 \\over 2}+gh_B", "8de940378846f9a8dddde71196f04882": "\\frac{f(x)-f(y)}{x-y}", "8de9c6bb8e514d327cc88e3a4389ac77": "\\sin \\theta ", "8dea6d3772dda581bcd36f9064d72b62": "{R=\\tfrac{1}{n}}", "8deaa661b07ddd12a15beaf388efa7df": " \\text{Variance} = \\frac{1}{T}\\ \\int\\limits_{0}^{T}\\sigma^2 dt\\ = \\frac{2}{T}\\ \\left ( \\int\\limits_{0}^{T} \\frac{dS_{t}}{S_{t}}\\ \\ - \\ln \\left ( \\frac{S_{T}}{S_{0}}\\ \\right ) \\right ) ", "8deb056e23f0aca29e9f69600fdc29a9": " -(\\Delta\\mathbf{r}_i)\\times\\mathbf{S}\\cdot(\\mathbf{S}\\times(\\Delta\\mathbf{r}_i)=-\\mathbf{S}\\cdot[\\Delta r_i][\\Delta r_i]\\mathbf{S},", "8deb3ec2dc2a7cf05458b45e9781c005": "(D + w w^{T})e_i = De_i = d_i e_i", "8deb4ee0b08cccb2e11c78287c49cbc4": " = \\frac {A_0} {1+ jf/f_C + \\beta A_0} ", "8deb798a937c8ff1e068a5018da56e71": "R(\\alpha_i) = y_iE_1(\\alpha_i)E_2(\\alpha_i) - y_iE_2(\\alpha_i)E_1(\\alpha_i) = 0", "8deb834d914f84c39e1642c8fe64a184": "\\neg\\neg A \\rightarrow A.", "8deb9cc78bed58444f4cea15497007c8": "g_j =\\prod_i g_j^i =g_j^{trans}g_j^{rot}g_j^{vib}g_j^{e},", "8dec559e201a7b6a0f99baeaa1731051": "e_i", "8ded2d88a382bfbc5016daa2c8380544": "|r'(s),\\ r\\theta'(s)| = 1", "8ded386f390a9788bfbdabfc86fd8378": "Z=[N,N]", "8ded7c4c6c63a31a38f7138bca5c768f": "\n\\sum_{\\delta\\mid n}|\\mu(\\delta)|=\n2^{\\omega(n)}.\n", "8dedae6b5608e46c42b2ea125aabd0ee": "\\hat{\\Omega}", "8dedaeff081ef6d489d8dcb968ff83e3": "\\theta_B = \\arcsin{\\frac{m \\lambda}{2d}} \\ .", "8dedea28c0db57cb470d5b8d141ec473": " y_3 = 4x_2^2 - 2x_3 \\, ", "8dee39b4b77e098883ea1263a3e41afc": "\\gamma_{i+1}", "8dee834b9c6bc52b92982e3f4edc6a22": "T(n, t) = I_n(\\alpha t) ", "8deee96988dd671e968e0974ccde2964": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{(m+n(2 p-1)+1) x^{m+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{(m+1)(m+n+1)}\\,+\\,\n \\frac{n\\,p\\,x^{m+1} \\left(2 a+b\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}}{(m+1)(m+n+1)}\\,+\\,\n \\frac{2 c\\,p\\,n^2(2 p-1)}{(m+1)(m+n+1)} \\int x^{m+2n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}dx\n", "8def12a7f11ba3468a7ab912da32c5c8": "\\Omega(1/n^8)", "8def3e26c70c3b142668164527bf9276": "z=2\\pi", "8def7711f2f9d804663e72850d795c3a": "a\\times2^{2n}", "8def87a948c05bdf8d2174b50a88643b": "\\phi_{xx} + c_3 \\phi_{yy} = ( |u|^2 )_x.\\,", "8defec645462870c00549545da97d32d": "\\begin{align}\ny_{n+1} &= y_n + \\tfrac{1}{6} h\\left[k_1 + \\left(4-\\lambda\\right)k_2 + \\lambda k_3 + k_4 \\right]\\\\\nt_{n+1} &= t_n + h \\\\\n\\end{align}", "8deff46bfe6d9cc591f2d29aeaf021dd": "N_t\\times 1", "8df0070301d2c858f745526ae74f58df": "C=0", "8df00afd462b7d0f24a745315e021a3f": "z=e^{i\\theta}=e^{i\\phi}.", "8df00e15cc5f5c966ce2d08c0adfd4a1": " A^{[k]}_{ij} = \\int_{T_k}\\nabla\\varphi_i\\cdot\\nabla\\varphi_j\\, dx.", "8df021e4ae08ad7fadec2b3208136320": "\\mathcal{K}(X)=C_C(X)", "8df0bc84de29da888a0e7df6682b4021": "\\{S_n(X)\\}", "8df0ea82f3745cb087a6db79c7e40bcf": "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx.", "8df0fb39b203443c6223cb8bd4de5b80": "k \\geq 0", "8df1268e5bdbfce2b449ab34227f678a": " f_0", "8df153825848c0a87b663695050bea9a": " h = \\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}, \\quad \ne = \\begin{bmatrix}\n0 & 1\\\\\n0 & 0\n\\end{bmatrix}, \\quad \nf = \\begin{bmatrix}\n0 & 0\\\\\n1 & 0\n\\end{bmatrix} ", "8df1758f4aff31cf9827dacce0c0819a": "p_{ij}", "8df1a404b7d7de246b2d2da03f1c6ec1": "n \\leq 15", "8df24edc6178c7ae89056232917de0d5": "\\mu, \\nu \\in \\mathcal{P}_p(\\mathbf{R})", "8df254fd0506ccb0c4451533f4f9c4df": "\\operatorname{or} \\operatorname{true} \\operatorname{false} = (\\lambda p.\\lambda q.p\\ p\\ q)\\ (\\lambda a.\\lambda b.a)\\ (\\lambda a.\\lambda b.b) = (\\lambda a.\\lambda b.a)\\ (\\lambda a.\\lambda b.a)\\ (\\lambda a.\\lambda b.b) = (\\lambda a.\\lambda b.a) = \\operatorname{true} ", "8df2eb5b10fe6528f9b8c67654c8c4bd": "\\sqrt{n} + \\lfloor\\sqrt{n}\\rfloor", "8df2fb82432fdb2db2c0f3ef7601ac22": "n < 0 \\Rightarrow q = \\left\\lceil \\frac{a}{n} \\right\\rceil,", "8df32754611c53c88e00377523bef2ed": "p(y_i | x')= \\sum_j p(y_i | c_j) p(c_j | x') )= \\sum_j p(y_i | c_j) \\tilde p(c_j) \\,", "8df3a9b52d75ec0d3cc4e60114a31ed5": "Q^j_i", "8df3cb32263ab87c373d47b470cee5bd": "i=0,\\dotsc,n", "8df3e60243071586fefe0cd4989056ea": "\\mathbf{c}(U) = \\aleph_0 ", "8df3f0b563bf3b2675139588fd453c15": " \\psi = R (\\rho e^{i \\beta})^\\frac{1}{2} ", "8df419fccbabda562a248e36a2120df4": "x^{\\lambda}(xy) = y", "8df43b61adcc04ad33a426d063e13c2f": "d_f(x,y)= | f(x)-f(y)|", "8df4444d54a187a6e86521ac35a78412": "(v_1, J v_1, \\ldots, v_n, J v_n)", "8df464f6d96772d3beff2e66353bb00f": "\\frac{1}{R_a}+\\frac{1}{R_c}=\\frac{1}{R_b}+\\frac{1}{R_d}.", "8df4bbaf8162d21137505ea089d20859": "\\frac{79,470,000\\ \\mbox{MW·h}}{(365\\ \\mbox{days}) \\times (24\\ \\mbox{hours/day}) \\times (18,300\\ \\mbox{MW})}=0.4957 \\approx{50%}", "8df4e1b3a8beab4df9c7b1b8c93762a7": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 9.200978\\log_e(T+273.15) - \\frac {6354.898} {T+273.15} + 75.65058 + 7.374814 \\times 10^{-06} (T+273.15)^2", "8df4efc5a5d47cf3fcea5beda1438cfe": "F \\sim \\frac{4\\pi\\mu\\ell u}{\\ln(\\ell/a)}", "8df5261aef7c9f6606ee440efafdc45d": " ds^2 = -\\frac{1}{\\kappa^2} dt^2 + \\kappa^2 \\, ( dx^2 + dy^2 + dz^2) ", "8df5441a379c3f2ed18b5f97b676a087": "=\\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac {\\bold{p} ( \\bold{ r}_0 )\\bold{\\cdot } d \\bold {A_0 } } {|\\bold r - \\bold{r}_0|} \\ ,", "8df5584a1445ce6018cd0d4cf6856d1c": " \\mathrm{Im}(\\tilde{k}) = -\\alpha_{abs}/2.", "8df57dc42aaadb4c26613b2ddfd2386d": "\\mathbf{H}(\\mathbf{x_1}-\\mathbf{x_2})=0", "8df5f78ddfc61293b7fd07b4c855ab71": " M_{pq}=\\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{-\\infty}^{\\infty} x^py^qf(x,y) \\,dx\\, dy", "8df60ab71dfb8c4320a09a3ec27604ea": "\n \\sigma_{11} = \\frac{3x_3}{2h^3}\\,M_{11} = \\frac{12 x_3}{H^3}\\,M_{11} \\quad \\text{and} \\quad\n \\sigma_{22} = \\frac{3x_3}{2h^3}\\,M_{22} = \\frac{12 x_3}{H^3}\\,M_{22} \\,.\n ", "8df6415930d76ed85a16f4cb928d8ad2": "d(f(x),f(y))=rd(x,y)", "8df6460aea00dc45708d89f5791024a2": "d_{X[n]} = (-1)^n d_X.", "8df66efff341458c7d463e34cd491492": "(x_1 - x_2)^2", "8df68f09908a728c9f67a36ec0cb9ea9": " x \\sim \\mathcal{N}(\\mu, \\tau^{-1}) ", "8df6b4a2d27419f9be18806fdcf7cc68": "B_e = EP_b\\,", "8df6fa15a164d58a5467b726edde7270": "FF=\\alpha\\frac{UU_m}{1-UU_M}+1", "8df70e3ba42d586ede1ffade0aa3119f": "m g - {1 \\over 2} \\rho v^2 A C_\\mathrm{d} = 0", "8df73a4b8701f0477135ffa58131a335": "\\neg \\Box B", "8df76d6c418a7b3626bd4b7c4c53dce5": "\\tau \n= \\int \\frac{dt}{\\gamma} \n= \\int \\sqrt {1 - \\frac{v(t)^2}{c^2}} \\, dt \n= \\int \\sqrt {1 - \\frac{1}{c^2} \\left [ \\left (\\frac{dx}{dt}\\right)^2 + \\left (\\frac{dy}{dt}\\right)^2 + \\left ( \\frac{dz}{dt}\\right)^2 \\right] } \\,dt,", "8df7782f49c2c3631ffd31cc3f21ec5b": "\\scriptstyle A\\,", "8df817eef518fede6126e007cad5ba99": "\\quad \\frac{\\partial T}{\\partial \\dot{q}_j} = \\sum_{i=1}^n m_i \\mathbf{\\dot{r}}_i \\cdot \\frac{\\partial \\mathbf{r}_i}{\\partial q_j} \\ .", "8df8a30223333aeb3890c95a56d77151": "(I) - (S)^H (S)\\,", "8df8df683e00791bba81aa22bf19c054": "(r-1, r-2)", "8df8eba3d4d596daad0b53bf455537fb": "\\mathcal{H}^{(1)} = \\left(\\frac {e^2}{R_{ab}} + \\frac {e^2}{r_{12}} - \\frac {e^2}{r_{a1}} - \\frac {e^2}{r_{b2}}\\right)", "8df9038795246fe004bb61d31627a3d4": "x_i, y_i \\in X", "8df933991a0bddf2fdd560955ee5f47d": "\n \\cfrac{\\partial \\sigma_{xx}}{\\partial x} + \\cfrac{\\partial \\sigma_{zx}}{\\partial z} = 0 ~;~~\n \\cfrac{\\partial \\sigma_{zx}}{\\partial x} + \\cfrac{\\partial \\sigma_{zz}}{\\partial z} = 0\n ", "8df942e95c68f027df14675903d319f6": "y_{n+s}", "8df959661c4d10f6151a4d2ccef44b69": "\\scriptstyle \\vec{L}", "8df9618bf981dfa12df66c007dbc630e": "K[X]/(X^2+X+d)", "8df99493576c3dc24feffb6818840791": "\\frac{dR}{dt}=\\frac{R(t)}{2},", "8dfa4827475b0bc19815dc9c5c7ef22e": "\\frac{1}{N_\\mathbf{P} D}(\\operatorname{tr}(\\mathbf{\\hat{\\mathbf{S}} \\operatorname{diag}(\\mathbf{P}^T\\mathbf{1})\\hat{\\mathbf{S}}}))-a\\operatorname{tr}(\\mathbf{A}^T\\mathbf{R})", "8dfa92312f784a2ecabed90c223271ac": "\\chi (X) = \\sum _0 ^n (-1)^j c_j.", "8dfa9694754c292ea7cc50ccafaed4c3": "\\delta(P,Q) \\le \\sqrt{\\frac{1}{2} D_{\\mathrm{KL}}(P\\|Q)}", "8dfaf8281769c217b7e78b27a4747285": "x^2+y^2", "8dfb4d52e556af007f6980032c6a9c24": "\\begin{smallmatrix}10^{-0.32} = 0.48\\end{smallmatrix}", "8dfb90237f0b1e4825147cbf9729ae0d": "\\frac{p}{q} = \\frac{a + c}{b + d}.", "8dfbaca5d13b716cf273b1f56fcb7c29": "\\pi D/2", "8dfbcff4b9be68eb1803de4f6a6885f2": "H \\subset L", "8dfbfcffaab2e565eea3c38dc2d9507d": "{\\rm Cin}(x)=\\gamma+\\ln x-{\\rm Ci}(x)\\,", "8dfc413f82773880d082c49fec649c4a": "a\\pm \\frac{|n-a^2|}{2a\\pm \\frac{|n-a^2|}{2a\\pm \\frac{|n-a^2|}{2a\\pm \\cdots }}}", "8dfc831293fd2a616d52a6268b08c26c": " \\qquad \\left(1+ \\frac{1}{n-1}\\right)W_n=W_{n-2}", "8dfc894b3a626e8a7368e1c654be66a4": "f_\\mathrm{image}(N) = |f - Nf_s|,\\,", "8dfc958dc7ed1e33802974b717a3aa91": "\\approx 158.82", "8dfc95cbf5777e94d04d64fe6694fba3": "p_1e_{k-1}=ke_k+r(2)\\,", "8dfce2923c7f8c52c11dc4f8759d3f29": "n(\\vec{r})=\\frac{8\\pi}{3h^3}p_f^3(\\vec{r}).\\ ", "8dfd6ade75f64749ec481491800726f0": "\\scriptstyle xy \\,+\\, 1", "8dfda9d0b912c41c7e61769529d0a096": "\\lim_{k\\rightarrow \\infty} d(\\Gamma_k) = \\infty", "8dfdbdc36ef7f50102ae8532053cd186": "\nf( \\hat{\\mathbf{k}}) = f(1) P_{\\mathbf{k}}+f(-1) \\bar{P}_{\\mathbf{k}}.\n", "8dfdc816876e25b20c2da19a2bc63866": "\\mu = 0,1", "8dfdcbdab97dd2dfd240e1ad5d03ed8f": "(5x+8)", "8dfdcf62d7ba09f5b98dbd2c1ef84f4b": "H(\\xi )=Tr[\\hat{B}(\\xi )\\hat{H}]", "8dfde068780a2051f4f3f40a8b88fdce": "\\dot{\\hat{x}} = A \\hat{x}+ B u + L \\left(y - C \\hat{x}\\right) ", "8dfdea5b69f0aec4eefcc9fe4c2ad8dd": "\\omega_{\\Psi}\\simeq\\sigma", "8dfe15ab43c27fbc763fba838a2f8d44": "|x| = \\begin{cases} x & \\mbox{if } 1 \\leq x \\leq \\frac{p-1}2, \\\\ p-x & \\mbox{if } \\frac{p+1}2 \\leq x \\leq p-1. \\end{cases}", "8dfe89ba46b28a0561cf1962dd4328d4": "u_{e}(x)= U_{0} \\left( x/L \\right) ^{m}", "8dfed39c1f390efcc53b2dd02d0ab073": "\\exists x\\,\\neg P(x)", "8dff02c55c965b032a518413ad55f7e1": "\\phi(t)\\,", "8dff36682d9f8b8f6f5bec6bb39244b3": "(0.70-1)^2 = 0.09", "8dffb568032894e18636914fb959bac5": "3 = \\lambda f.\\lambda x.f\\ (f\\ (f\\ x)) ", "8e007dbb5a41ffbdeac1e527543ba0af": "P_a^b", "8e00edd13aa02bec9292cdb1d5034db4": "D_{\\mathrm N} = \\frac {f^2 / ( N c ) + f} {2} = \\frac {H}{2}\\,.", "8e00f8f0817b216fc2d9f094dcd15429": " \\langle{\\varphi_2} |\\big( P_{A\\alpha}\\varphi_1\\big) \\rangle_{(\\mathbf{r})} = \\big(P_{A\\alpha}\\gamma(\\mathbf{R}) \\big) + \\langle\\chi_2| \\big(P_{A\\alpha} \\chi_1\\big)\\rangle_{(\\mathbf{r})}.\n", "8e0127b53c428eaa3f699535964b2b6e": " de/dt = P_0 \\exp{[-k h]}\\ ", "8e0145b7e1552df9070d8bfb1763ee51": "(x)_v =\\left(x_{e1}, x_{e2}, x_{e3}, x_{e4}\\right)\\in C_o", "8e018fc1acb493303fed222a798f75eb": " p_{i,i}(x) = y_i, \\, ", "8e01a82bb324e2aa95e082ce39ad2312": "\\tan\\frac \\theta 2 = \\sqrt{\\frac{1+\\varepsilon}{1-\\varepsilon}}\\cdot\\tan\\frac E 2", "8e01ad20c11042b41772b76c484bcabe": "\\operatorname{E}\\left[\\frac{X}{Y}\\right]\\approx\\frac{\\operatorname{E}\\left[X\\right]}{\\operatorname{E}\\left[Y\\right]} -\\frac{\\operatorname{cov}\\left[X,Y\\right]}{\\operatorname{E}\\left[Y\\right]^2}+\\frac{\\operatorname{E}\\left[X\\right]}{\\operatorname{E}\\left[Y\\right]^3}\\operatorname{var}\\left[Y\\right]", "8e01db094c1ff951feed52effdc8990d": "x_1, x_2, x_3, ...", "8e01f30222ae1da6c74c30b426ece0d0": "G_3=\\langle L^2,R^2,F^2,B^2,U^2,D^2\\rangle", "8e021a799321394b60926cf684a1b4f1": "\\mbox{At x=2}, B_2=B_3=0.5; \\frac{dB_2}{dx}=\\frac{dB_3}{dx}=-1", "8e0258045d3dd54576160c5ead32ec6d": "\\mathbf{r} = r \\mathbf{\\hat r} ", "8e028f69ec12ceac3cd738d264ed51aa": "\\vec{r'}", "8e02aeddab8a9f24517e3235bfd26606": "\\displaystyle 2^{-\\delta}\\Gamma(\\delta+1)\\left|\\boldsymbol \\omega\\right|^{-n/2-\\delta}", "8e02ba92c7f64e8593438b4f42a38721": "\\scriptstyle m \\equiv n\\pmod{p^{b-1}(p-1)}", "8e02bee4b3fa6993715ee80215bf732c": "F(w) ", "8e02eaca8e560f63c66747df9fcfb3a7": "y_i^{\\prime\\prime} = y_i'", "8e02f476aac410f3e1b60e593b818cf4": "a\\cdot c", "8e0317897ec82b9ff1057a26995d2c1f": "- \\otimes_k F", "8e03dfd4973e1b7248c454f577a79133": "\\scriptstyle \\dot m_0 \\,", "8e049a95dd0cb42be9a8df7af13c9d93": " u \\mapsto \\ln|u-1| ", "8e04ea2fbed21e24f37b273140d25ad4": "E\\!", "8e04f6c038363a34d8fb3e04c1388d14": "[UT]_\\alpha^\\gamma=[U]_\\beta^\\gamma[T]_\\alpha^\\beta", "8e05248adb54363f474695154e1708ea": "[C] \\in F", "8e0552a516084be5003d5d42920f0d03": "\\mathcal{S} = \\int \\left( -\\begin{matrix} \\frac{1}{4 \\mu_0} \\end{matrix} F_{\\mu\\nu} F^{\\mu\\nu} \\right) \\mathrm{d}^4 x \\,", "8e056cae5e5ce629f9958abc6b8b7e52": "\\nabla = \\mathbf{\\hat{x}}{\\partial \\over \\partial x} + \\mathbf{\\hat{y}}{\\partial \\over \\partial y} + \\mathbf{\\hat{z}}{\\partial \\over \\partial z}", "8e05825cfe22e960ae94037787e51128": "I_i(v)", "8e05b343e6f032a95b21c3e3eb614fed": " \\Psi(\\mathbf{r},t) = A e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} \\,\\!", "8e05c340e7d97ec32762fa1677e7de48": "\\begin{matrix}{4 \\choose 1}^2{3 \\choose 1}\\end{matrix}", "8e05ca56efcd2c52365f004b6017de4b": " u(w)= w-be^{-aw}", "8e05fbf03e691a3eeae66b6190c875b1": "\nm_b = m_\\mathrm{o} \\cdot \\left( 1 - \\frac{\\rho_\\mathrm{f}}{\\rho_\\mathrm{o}} \\right)\\,\n", "8e06439b249594a585e433951964485c": "Y_1 X_2 Z_3 = \\begin{bmatrix}\n c_1 c_3 + s_1 s_2 s_3 & c_3 s_1 s_2 - c_1 s_3 & c_2 s_1 \\\\\n c_2 s_3 & c_2 c_3 & - s_2 \\\\\n c_1 s_2 s_3 - c_3 s_1 & s_1 s_3 + c_1 c_3 s_2 & c_1 c_2 \n\\end{bmatrix}", "8e068955c0fdbb6a6167fe9b3c4243d4": "x(T)", "8e070e76428226d12fb35cdf33ac3185": "(C_k, \\partial_k)", "8e07c4aa0347c64f2b2814e5bfcb0d56": "Ja=\\frac{C_{p,f}(T_{sat} - T_w)}{h_{f.g}}", "8e07fca5f0a3d59cc23b9cfdadbe049b": "F = I_{sp}(vac)\\,g_o\\,\\dot{m} - A_e p_o", "8e08250cf19472b6310647927edabe25": "r = a \\cdot (1-e \\cdot \\cos E)", "8e083e765e0519c662a56356d1b7cc19": "\\scriptstyle U{}^2_4", "8e0863d5e0ff46557dfd2d193cb3889c": "u_2 \\ge P", "8e086971f8b90da68d68d90cabb74a47": "f_\\ast\\mathcal{O}_Z\\cong \\mathcal{O}_X/\\mathcal{I}", "8e08a910e13a87c41b98fcf36b6cac45": "A =\\frac{1}{2}A_{n} \\, r^{n-1} I_{(2rh-h^2)/r^2} \\left(\\frac{n-1}{2}, \\frac{1}{2} \\right)", "8e0961e759c0e22b8b2b00d698c448bf": "\\left \\{ S_1,\\ S_2,\\ S_3,\\dots \\right \\}", "8e09748f6f6195e66aefaf110b892262": "\\varnothing \\subseteq A \\subseteq S\\,\\!", "8e0976110876ff72c442c7ae9ad5bdbb": " yM(xy) + xN(xy)\\,\\frac{dy}{dx} = 0 \\,\\!", "8e097e73384a09fa6a2b625e6857b80b": "\\zeta_n\\in\\mathcal{O}_k.", "8e09df07c1e94717f145e6d687455723": " \\sigma_d = 0 ", "8e0a271ba421cbca5e424c0f1034e1ad": "\\varnothing \\notin P", "8e0ac238fc0f9cb5ea868b444ccaf25e": "I_L^\\mathbf{c}", "8e0ac9c6fed0e5ca4e205817fa3484ca": "k_\\mathrm{PE}=\\bar{\\gamma} n\\Delta G_\\mathrm{PE} \\tau", "8e0aeaecc1ad909a9580174823a12e31": "\\sum F_i = I", "8e0b03e883068dc26e0a6ed0243037e9": "{\\mathbf{p}}_{\\text{crystal}} \\equiv \\hbar {\\mathbf{k}}", "8e0b23c07a8863bc21a30c7228cbc247": "\\scriptstyle d\\,=\\,a(1\\,+\\,2cos{30^\\circ}\\,+\\,2cos{60^\\circ})", "8e0b73172758324a062e502690b45440": " \\operatorname{Pr}(Y \\leq a) \\leq \\operatorname{Pr}(X\\leq a+\\varepsilon) + \\operatorname{Pr}(|Y - X| > \\varepsilon).", "8e0b8f9b7d18c87a29b1a4abd9766c0b": "S_N[k] =\\frac{1}{NT} \\underbrace{\\sum_N s_N[n]\\cdot e^{-i 2\\pi \\frac{k}{N} n}}_{S_k}\\,", "8e0bdbc00c4c94eb99d9809e9d6f3db4": "P_8(x)=3x^2 \\,", "8e0bdd7ea476cb3211f75631152ab244": "\\mathbb Q^r", "8e0be5604811af47884173d610fb5a49": "(AB+BC) - (AC'). \\,", "8e0c150b88ff7526aef71b94594991cd": " < \\alpha", "8e0c9214ae4b6de7eeb9e94cc56c41bf": "\\psi' = Q_1 x_1 R x_1 Q_2 R x_1 R x_2\\dots Q_m R x_1 \\dots R x_m [\\varphi]", "8e0c96de2bd7ab84510b416944e70d6b": "x^{(n)} = {(-1)}^n {(-x)}_{{n}} .", "8e0ca41b4b9b378054cf2dece4c53734": "S N'(d_1) \\sqrt{T-t}\\,", "8e0cb4f5f73dec020432b4bf1b35484c": "X_{C}\\approx 10^9 ", "8e0ced822cb907e9b55ee64ec2e8c1ff": "\\sum_{h \\in G}\\Delta_{C[G]}((g,hkh^{-1}))", "8e0d39b1cc44481da88c859023c0d907": " [ I | B ] = \n\\left[ \\begin{array}{rrr|rrr}\n1 & 0 & 0 & \\frac{3}{4} & \\frac{1}{2} & \\frac{1}{4}\\\\[3pt]\n0 & 1 & 0 & \\frac{1}{2} & 1 & \\frac{1}{2}\\\\[3pt]\n0 & 0 & 1 & \\frac{1}{4} & \\frac{1}{2} & \\frac{3}{4}\n\\end{array} \\right].\n", "8e0d8fe8821da06e49ae81673a089b80": "s_2 = (\\triangle^2 a)_0 = -(-a_2+a_1)+(-a_1+a_0) = a_2-2a_1+a_0", "8e0da893b8d83c0442a5997d9657e3ee": " \\forall x \\in A: \\ \\ \\sum_{i=1}^n E(xa_i)b_i = x = \\sum_{i=1}^n a_i E(b_i x). ", "8e0dac65269a88903db9f1821a5a090d": "Y^{\\mu\\nu}\\sigma_{\\mu\\nu}", "8e0db3b9353ada462eb22b359360eb80": "(\\Omega\\wedge\\bar\\Omega - \\omega^n/n!)", "8e0e6375ddb842bff97cdf5ad4bcb63a": "A(T) = {1 \\over 15} \\arccos \\left( {\\sin(\\arccot(t+\\tan(L-D)))-\\sin(L)*\\sin(D) \\over \\cos(L)*\\cos(D)} \\right)", "8e0eb14157052775882d5236c8646be0": " v_2 = \\sqrt {2g\\bigl(E_1-y_2\\bigr)} ", "8e0f2644640dfc6d7d334f836b92bb07": "\n\\operatorname{P}( Z \\ge \\theta \\operatorname{E}[Z] )\n\\ge \\frac{(1-\\theta)^2 \\, \\operatorname{E}[Z]^2}{\\operatorname{var} Z + \\operatorname{E}[Z]^2}.\n", "8e0f46af5230b35072ee3448af844018": "\\hbar=h/2\\pi", "8e0fac3f93ad72489f6a84b52a16016c": "\\Omega_c^m(M)", "8e0facea4c2deac97182bf328a41e929": "f(x,y,z) = x^2 + y^2", "8e10173e9462b4835e0cea29bc3ced73": "\\ldots + d_3\\cdot (2i)^3+d_2\\cdot (2i)^2+d_1\\cdot (2i)+d_0+d_{-1}\\cdot (2i)^{-1}+d_{-2}\\cdot (2i)^{-2}+d_{-3}\\cdot (2i)^{-3}\\ldots", "8e10229df0e2821c167ce4745b6d7b51": "\\pi=\\epsilon+\\pi^W-bY+bY_{-1}+\\gamma \\Delta Y^W+\\delta \\Delta G-f(\\Delta i^W+\\Delta \\epsilon^e)", "8e1051771a00db4e26742cde057266dd": "N_w", "8e1070c8ff32fa3160275965f5cce68e": "S = K \\ln I", "8e1098c9fefd065ee3895f16dde55857": "\\varphi\\in sen(\\Sigma)", "8e11240d05d1818608534c9fac121f64": " I=\\frac{1}{2}mR^2 ", "8e11393fbaad264615d4ffe6c006cbc6": "\\mathbf{G} := \\bigcup_{n=0}^\\infty \\mathbf{G}_n", "8e1169693359b3b6db67b8a36ae4cb8c": "X_k^{(m)}", "8e11bab34b37dcd0a097ceaef9ea4827": "CEncode\\,", "8e11ed1330d15329770b92f9bbf88a72": "q_{jk}", "8e120947ef4135506aca82baf7811006": " L = D - A ", "8e12c91677b3b3df266a770b22c82f2f": "= ", "8e12f78121981a48eb7d34e4f0b05aac": "\\omega_f^2 = \\omega_i^2 + 2 \\alpha\\theta", "8e13058db9b4e28a1b26a93703d72e1f": "EC=\\begin{pmatrix} I_r \\\\ 0 \\end{pmatrix}", "8e1343a051697c06206ca4b91ad8565c": "\\mathcal{L}(p_\\text{H}=0.5 | \\text{HH}) = P(\\text{HH} | p_\\text{H}=0.5) = 0.25.", "8e13565fb7ba7f174b3377dd5a7e680f": "f(x,x)=0", "8e1381689e204c60c1275a06d1984699": "\nc \\,F_3(a_1,a_2,b_1,b_2,c; x,y) + (a_1+a_2-c) F_3(a_1,a_2,b_1,b_2,c+1; x,y) - a_1 F_3(a_1+1,a_2,b_1,b_2,c+1; x,y) - a_2 F_3(a_1,a_2+1,b_1,b_2,c+1; x,y) = 0 ~,\n", "8e13c090a8807b90252bb8d98e87fada": "s(t) = A\\cdot \\sin(\\omega t) = A\\cdot \\cos\\left(\\omega t -\\begin{matrix} \\frac{\\pi}{2}\\end{matrix}\\right)\\,", "8e13dfd45542a4f90f85e43f1c356ec6": "\\mathrm{SO}_4^{-\\bullet}+\\mathrm H_2\\mathrm O\\longrightarrow \\mathrm{SO}_4^{2-}+\\mathrm{OH}^\\bullet+\\mathrm H^+", "8e1426a5609d29f3914601aedbc66f78": "(\\mathbf{S})\\,\\mathrm{d}a=-\\frac{i}{\\hbar}[a,H_\\mathrm{sys}]\\mathrm{d}t-\\frac{\\gamma}{2}\\left([a,c^\\dagger]c-c^\\dagger[a,c]\\right)\\mathrm{d}t-\\sqrt{\\gamma}\\left([a,c^\\dagger]\\mathrm{d}B(t)-\\mathrm{d}B^\\dagger(t)[a,c]\\right)\\,.", "8e1439aa0a3872ffe49528595ff1cafe": "F|i-i_F|(\\frac{1-(1+i)^{-N}}{i})", "8e145b8dafdaa8d052c4ffccd57d13d3": "S=\\left\\{ z: |z|0 \\right\\}", "8e145f74f8325308f41c934b8a48f1ab": " {1\\over AB}= \\int_0^1 {1\\over( vA+ (1-v)B)^2} dv ", "8e146ebb6585829b5a3f0e3bb1c509cf": "u=\\frac{4 \\times 0.4402}{-2 \\times 0.4402 + 12 \\times 0.4031 + 3}=0.2531", "8e146ff496e1fd2d8abfe430e512daf0": "\\frac{3x + 5}{(1-2x)^2} = \\frac{A}{(1-2x)^2} + \\frac{B}{(1-2x)}.", "8e147cfd6b752162445f61d0e5c5596a": "\\phi =90{}^\\circ -\\psi \\,.", "8e14a217fb2c0b7165c2dda84e8fd776": "h_{\\mathrm{FOH}}(t)\\,= \\frac{1}{T} \\mathrm{tri} \\left(\\frac{t-T}{T} \\right)\n = \\begin{cases}\n\\frac{1}{T} \\left( 1 - \\frac{|t-T|}{T} \\right) & \\mbox{if } |t-T| < T \\\\\n0 & \\mbox{otherwise}\n\\end{cases} \\ ", "8e14fe062aeb0d89e32483757bc0a4b0": " (x^\\lambda, \\sigma^m, y^i, \\widehat y^i_\\lambda, y^i_m), ", "8e15aeaa556b543c5cc34acea1058432": "\\sqrt[n]{|a_n|} = \\sqrt[n]{|c_n(z - p)^n|} = 1,", "8e15ba7e2a48f8fd81a5d031ede19290": "\n \\begin{align}\n E_{11} & = -x_3\\,\\frac{\\partial^2 w}{\\partial x_1^2} \n + \\frac{1}{2}\\left(\\frac{\\partial w}{\\partial x_1}\\right)^2 \\\\ \n E_{22} & = -x_3\\,\\frac{\\partial^2 w}{\\partial x_2^2} \n + \\frac{1}{2}\\left(\\frac{\\partial w}{\\partial x_2}\\right)^2 \\\\ \n E_{12} & = -x_3\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2} \n + \\frac{1}{2}\\,\\frac{\\partial w}{\\partial x_1}\\,\\frac{\\partial w}{\\partial x_2}\\\\\n E_{33} & = 0 ~,~~ E_{23} = 0 ~,~~ E_{31} = 0 \\,.\n \\end{align}\n ", "8e15ce82c787a1fd44c80e6766111b5d": " t'=t-t_o ", "8e160f37486696302325611eb218cee1": " \\alpha = K(x)+c", "8e1662982f54688660a160f0be74020c": "c' = \\sum_{i = 1}^n \\alpha_i w_i.", "8e166317e7b8a8ddb8f237427a60a2c2": "n=(k+1)/2", "8e167fedf2943798cd3142cd281da732": "\\textstyle \\frac{0}{0}\\times 1 = \\frac{0}{0}\\times 2.", "8e16950b51d915988796a0bcced59d99": "\\frac{|AE|}{|AB|}=\\frac{2}{\\pi}", "8e16b311916b2b678ecab4dea4b5cc2f": "{10}^{\\,\\! 4 \\cdot 2^{70}}", "8e16d937f723990863a9d79e33a333d0": "E(\\omega)/\\Delta\\omega", "8e170ff7e8ad40adaf6f6e566154c719": " R_{in}(fb) = \\frac {V_x} {I_x} = \\frac { R_{in} } { \\left( 1 + \\beta A_i \\right ) } \\ . ", "8e1786f6933622e4a931409dba0071e4": "x_{.,j} = \\sum_{i=0}^{2}{x_{i,j}}", "8e17aa63cb56edf45c64c476424b7ba6": "M(\\lambda x, \\lambda y) = \\lambda^n M(x,y) ", "8e17b31bab86327c82728bab6a294d67": "\\boldsymbol{\\beta},", "8e17b9250224d3f9c335b03f4e06a973": "(m^2 + n^2 + p^2 + q^2)^2 = (2mq + 2np)^2 + (2nq - 2mp)^2 + (m^2 + n^2 - p^2 - q^2)^2.", "8e17d34c105ea086e9a42a45509c3639": "\\Delta_K(t) = f(t)f(t^{-1})", "8e17e54e7acdf48067dddbdc430831d5": "\\hat\\theta_n", "8e1859c2283feef98096fe8fbb604409": "{{D}_{KA}}=4850{{d}_{pore}}\\sqrt{\\frac{T}{{{M}_{A}}}}", "8e1875bb7fc438efa61e9fb1050a6e6a": " \\Delta(f) :=\\frac{1}{\\sqrt{\\rho}}\\Delta_{\\pi}(\\sqrt{\\rho}f)", "8e187ee9fc651daed7722fa720613b99": " \\delta \\phi = \\frac{2 \\, m}{R}", "8e189aa0780d4629caa68b655290c42f": "\\mathbf{DTIME}(f(n)) \\subsetneq \\mathbf{DTIME}\\left (f(n)^2 \\right).", "8e18c09fbc6becd5196dc8bd68ee9bb5": "\\mathbf{r}' = \\mathbf{r} + \\left(\\gamma - 1 \\right)\\mathbf{r}_\\parallel - \\gamma\\mathbf{v}t \\,. ", "8e18ce823239056615a4d3ce9e924175": "\\langle x,y \\mid xy = yx \\rangle\\,\\!", "8e18d3f61276d5b10a55fee9df3cb540": "\\psi(\\alpha)<\\delta", "8e198941d0d04d2520ae04a290a14d61": "Q_{xx} = X'X.", "8e199b073265161013eba149b5833cc0": "\\ell(p_i,p_j)", "8e19b6dcd64bdbaae254ef16df5d7a72": "\\mathrm{SQNR} = 20 \\log_{10}(2^Q) \\approx 6.02 \\cdot Q\\ \\mathrm{dB} \\,\\!", "8e1a35a1e74defecde4dcdd6570f8394": "P(E|H_1) = 30/40 = 0.75", "8e1a6865b9bdc2626ee1916d22946235": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}{10 \\choose 1}{4 \\choose 2}{9 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "8e1a7d87e3da11dd5a973fe98d6004ec": "F_{12}=F_1F_2", "8e1ac35acc5e0e03f34f6eecffd5411a": "e=E_1||C_1||...||E_t||C_t\\,", "8e1ad78c6c0f912dced4bd5a4d64a619": "\\scriptstyle{E_b}", "8e1b70a1b02f6bfb1a1f3eee3616a719": "Y_3 = D(H-X_3)-2BG = 296\\sqrt{3}-144\\sqrt{13}", "8e1b8f4e2ad36e76a1bc5fff7d8fe8fa": "b_i = \\frac{n_i}{m_{\\rm solv}} ", "8e1c031fd729a7fe9fa703295e832ebb": "AdjustedIndex = \\frac{Index - ExpectedIndex}{MaxIndex - ExpectedIndex}", "8e1c056c7dbb42a7c135c99f9f55b3b3": " \\operatorname{Maps}_*\\left(\\Sigma X,Y\\right)\\cong \\operatorname{Maps}_*\\left(X,\\Omega Y\\right)", "8e1c0dd3b3757afeee5e8bb63bdc7b00": "\\mathbf{G_2}", "8e1c22691c7b005730676d3557a1ccff": "D^{+}", "8e1c47960df7b217162442d5e1c0e3a2": "\nw(n) = \\left\\{ \\begin{matrix}\n \\frac{1}{\\exp(Z_+)+1} & 0 \\leqslant n < \\epsilon(N - 1) \\\\\n 1 & \\epsilon(N - 1) < n < (1 - \\epsilon)(N - 1) \\\\\n \\frac{1}{\\exp(Z_-)+1} & (1 - \\epsilon)(N - 1) < n \\leqslant (N - 1) \\\\\n 0 & \\mbox{otherwise} \\\\\n\\end{matrix} \\right.\n", "8e1c887daab43f7c158ca90aae342857": " c_n = \\sum_{k=0}^n (-1)^{k} {n\\choose k} a_k, \\qquad (n \\geq 0).", "8e1d31848b99dbcf59d0039f62227f78": " -\\det X = x^2 -|w|^2.", "8e1d5951b37686035b537394df3dc510": "\\pi_{4k+2}^S \\to \\mathbb{Z}_2", "8e1d61ced78141a872963fe07047636b": "\\phi(0)=0", "8e1d788e99e6c05c85bbda37d7a6456a": "\n\\phi^\\dagger \\gamma^0 \\psi = \\langle \\bar{\\Phi}\\Psi + \n (\\bar{\\Psi}\\Phi)^\\dagger \\rangle_S \n", "8e1d90b8a6d168a962a80092a11c04b2": "R = I + [\\omega]_\\times \\sin(\\theta) + [\\omega]_\\times^2 (1-\\cos(\\theta))", "8e1daba3210fb19033d1cf5a639a1045": "\\displaystyle V_{\\rm peak}", "8e1e1e65e87580fd18c3f04b58e7aef4": "T_k(\\omega) \\to T(\\omega),\\qquad \\forall \\omega.\\,", "8e1e5916dcc686cd711434059b2960cc": "e^{-k\\cdot r^2}", "8e1eb92d8506cd514c0c01ad6a778dfd": "\\lambda(C_1 \\cup C_2) = \\lambda(C_1) + \\lambda(C_2) ", "8e1ec05b97f59a59cdd7aa416a1d3038": " \\bold{T} = \\sum_i N_i \\bold{a}_i. ", "8e1f293accbacfe2a9c32de6ca5c9946": "\\begin{bmatrix}\n 1 & x_1 & x_2 & \\dots & x_n \\\\\n 1 & y_1 & y_2 & \\dots & y_n \\\\\n 1 & z_1 & z_2 & \\dots & z_n\n\\end{bmatrix}\n", "8e1f61aca2cad7912853a4e41402ca0a": "a_n = \\int_0^1f(\\theta)e^{-2\\pi in\\theta}\\,d\\theta.", "8e1f7b45270063bfe3df6c62e972a00b": " F^k_{ji}=\\partial_j A^k_i - \\partial_i A^k_j", "8e1f87ee027659538507e54a65f2b116": "\\varepsilon_x ", "8e1f98ea6d1695a9db9a5da853646e34": "\\mathbf{p} = m \\mathbf{v}.", "8e1fbdcd94b4dff3753d63d54aa34251": "P'_z(a,b,0) =p_{d-1}(a,b).", "8e1fbdd40c38805287868e6182c56e70": "\\bar{x}_i=\\bar{\\mathbf{e}}_i\\cdot\\mathbf{x}=\\bar{\\mathbf{e}}_i\\cdot x_j\\mathbf{e}_j=x_i \\mathsf{L}_{ij} \\,, ", "8e1ffc16a397359870f5e25b844d69e5": "u'_i", "8e202ac8a8a8173c17582473c560127c": "\\int_0^{\\pi/3} \\tan x \\, dx=2\\int_0^{\\pi/4} \\tan x \\, dx=\\ln 2.", "8e20843ffc4650ff08bafbfce7a79cb0": "c=13", "8e20a06dee28f42d1733c26788d0d183": " \\frac{\\pi}{4} = 44 \\arctan\\frac{1}{57} + 7 \\arctan\\frac{1}{239} - 12 \\arctan\\frac{1}{682} + 24 \\arctan\\frac{1}{12943}", "8e20a8bf62ca86c6f6a4212122b9f480": "A_w^2 = \\left(\\frac{\\pi D^2}{4}\\right)^2 = \\frac{\\pi^2 D^4}{16}", "8e20bad21ceab084b41746d0ed9f5e56": "P(B\\mid C) > P(B\\mid C,A)", "8e20c79043d2d0991b5f64cae295fa68": "\n\\mathbf{M} \\cdot \\frac{d^{2}\\mathbf{q}}{dt^{2}} = \\mathbf{f} = -\\frac{\\partial V}{\\partial \\mathbf{q}}\n", "8e2139255104b186f0a77b001af36cf9": "\\forall n \\geq 1, \\forall x \\in A \\ : \\ |f_n(x)|\\leq M_n,", "8e215a8a670d2942268cc506a89bf500": "[H_{0}, S] = 0", "8e217751aba984e0af05c044e2c5e8fc": "U_\\lambda", "8e221cadad38f0fe1477a354aad371dc": "\\mathrm{d}U={\\delta Q}-{\\delta W}.", "8e2242e9cbcbac8fafa939b0874b4702": "P = \\left\\langle -\\frac{\\partial \\varphi }{\\partial V}\\right\\rangle _{t},", "8e224eb855a6a93b13e464bb3398d726": "\n\\sigma^2 = \\max \\left\\{ \\bigg\\Vert \\sum_k \\mathbf{B}_k\\mathbf{B}_k^* \\bigg\\Vert, \\bigg\\Vert \\sum_k \\mathbf{B}_k^*\\mathbf{B}_k \\bigg\\Vert \\right\\}.\n", "8e225141b841de5891a6d52029b3bd5f": "\\partial = \\nabla - \\mathbf{e}_4\\frac{1}{c}\\frac{\\partial}{\\partial t}.", "8e227fc1120bd30c933148c5c83f3db0": "A = 4\\pi r^2.", "8e229a50a28aded3bdc05308471ea16f": "\\mathbf{P}(u,v) = (P_1(u,v), P_2(u,v))", "8e22a77feac45c854ba56a90996a3c7e": "\\alpha>0\\,\\!", "8e22b2f52f978bb80f0568738493d0f6": "e(v)=\\frac{C(v)}{F(v)}=\\frac{\\int\\limits_{0}^{v}{{}}xf(x)dx}{F(v)}", "8e22c8f346798539ed46ee086d6755bf": "\\chi_E(x)=0", "8e22de3ee8b05e3cf7bcd5eec8245365": " (2)\\qquad \\int_X p(x)K(x,y)\\,dx\\le\\beta q(y)", "8e2329a0b6d346d285a1faa309c3b666": "\\pi_1,\\pi_2,\\dots,\\pi_{k-1}\\,\\!", "8e234e268278cb421771dfbaabca20ce": "{R^0}_{101} = \\frac{-f'' \\, g + f' \\, g'}{f \\, g^3} ", "8e2371b989a02cccfa6e9098d6a7e38c": "\\scriptstyle {\\aleph_1}", "8e243eb5a975c2659f58e75896adfbf2": "K_f", "8e24531faff241ff459554681ccb35cd": "-\\pi<\\phi\\leq+\\pi", "8e24843448395fd29492dbcbe75a6180": " (a_1 + ib_1) \\cdot (a_2 + ib_2) = (a_1a_2 - b_1b_2) + i(a_1b_2 + a_2b_1). ", "8e24f00f29d8a430d38a563d2c23ed21": "|Q| 0", "8e401614571a89a60fb36167e7d417ef": "\\Gamma\\vdash A\\to B.", "8e40f2dd6a2d1643cc97ebe880570166": "\n\\frac {1} {R_0} = \\frac {A} {L} \\Rightarrow R_0 = \\frac {L} {A}.\n", "8e411f40ca9306427471e4952de2d7a5": "g^x ~\\bmod~ p = y", "8e41a7d50af608bf1ad01d10ca599352": "\\scriptstyle IV", "8e4228fa4fced9b5a5fc1eaef2a609a8": "p(a,n,x_1,\\ldots,x_k)", "8e4231b7b76eb8cf17afb96217e0cb5f": "\\textstyle\\frac{\\Delta{\\rm Principal}}{{\\rm Principal}} \\approx -\\frac{\\Delta r}{r}", "8e4266a4cc879489c6e2f6416af20dcc": " T_\\text{m} = {1 \\over \\beta T_\\text{c}} + (1- {1\\over \\beta}) T_\\text{m}^\\circ ", "8e4306e0393894dafc784ed1543e21ed": "\\mathrm{C^{\\alpha}_{i+2}}", "8e43842130ee347060e9e3d5513c8ef2": "x \\ll z", "8e43bf466a61085092d5db671573345d": "\\omega=1/2", "8e43c834c20494e5bcd0776595563670": "\nY = \\cosh^{2} \\xi - \\cos^{2} \\eta\n", "8e43d8cb1e557720e76598a02ee74386": "W_{t_1}-W_{s_1}", "8e43dc0679a87ff18e39075403176e1e": " -\\omega^2 \\, [FLu] + \\omega \\, [Fp] + [Fq] - (m^2+n^2) \\, [FLu]", "8e4466a2b9386a49c151ca11212e3756": "\\omega = \\frac{df}{f} =\\left(\\frac{m}{z} + \\frac{g'(z)}{g(z)}\\right)dz", "8e4488d6d4650f3ee409e07061362629": "W_q^{(n)}=\\max_{n-1\\geq k \\geq 0}P^{(k)};", "8e44b8db770e4e590692718af8f527f0": "L_s(i)\\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\frac{N\\Phi}{i} = \\frac{\\Lambda}{i}", "8e4544e06bdc8bd4d538beaf719caabd": "\\frac{1}{\\gamma} = \\sqrt{1-v^2/c^2\\,}.", "8e458df2124803a17245f60ff74fde89": "s(t) = \\int_{-\\infty}^{\\infty} S(f)\\ e^{ i 2 \\pi f t} df\\,", "8e46494aede5b41808fb2849bfe05992": "a_n = a_{n-d}, n\\geq d", "8e46523dc1c40ae58011a9afdfaeae43": "\\varphi^n = \\varphi^{n-1} + \\varphi^{n-2} = \\varphi \\cdot \\operatorname{F}_n + \\operatorname{F}_{n-1}.", "8e4670de03258f99e4edcfd91ba156b1": "\n \\psi_w(u) = \\operatorname{E}[\\,w_te^{iux^*}\\,] \n = \\frac{\\operatorname{E}[w_te^{iux_{1t}}]}{\\operatorname{E}[e^{iux_{1t}}]}\n \\exp \\int_0^u i\\frac{\\operatorname{E}[x_{2t}e^{ivx_{1t}}]}{\\operatorname{E}[e^{ivx_{1t}}]}dv\n ", "8e4689c503349e3bbeda543da60167d4": "\\hat{\\mathrm{Td}}(E)(1-\\epsilon(R(E)))", "8e46944bed1bcbcbbb8324aedb2d699c": "\n\\begin{pmatrix}\nu'\\\\x'\\\\y'\\\\z'\n\\end{pmatrix}\n=\n\\begin{pmatrix}\na&-b&-c&-d\\\\\nb&\\;\\,\\, a&-d&\\;\\,\\, c\\\\\nc&\\;\\,\\, d&\\;\\,\\, a&-b\\\\\nd&-c&\\;\\,\\, b&\\;\\,\\, a\n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\nu\\\\x\\\\y\\\\z\n\\end{pmatrix}\n", "8e47107d7188607a1415802f0365c689": "2^{\\aleph_0}", "8e478ee0825b8ed0d63bcca83274f5f9": "z=\\pm \\sqrt{r_1^2-x^2-y^2}.", "8e479aeaa80a243e8bae74a73c1e6bd5": "\\hat{g}_N(x_N)", "8e47e27bdca31a416aa0cbfc59a91143": "x_N * y,", "8e480d3de6947aa3eb266a768a40116c": "+ 7 \\cdot 9^{(9+2)} + 7 \\cdot 9^{(9+1)}+ 7 \\cdot 9^9 ", "8e4815d718e637f3959fc0d23b11f004": " \n\\begin{bmatrix}\n \\mathbf{e}_1'(t) \\\\\n \\mathbf{e}_2'(t) \\\\\n \\mathbf{e}_3'(t) \\\\\n\\end{bmatrix} \n\n=\n\n\\left\\Vert \\gamma'\\left(t\\right) \\right\\Vert\n\n\\begin{bmatrix}\n 0 & \\kappa(t) & 0 \\\\\n -\\kappa(t) & 0 & \\tau(t) \\\\\n 0 & -\\tau(t) & 0 \\\\\n\\end{bmatrix} \n\n\\begin{bmatrix}\n \\mathbf{e}_1(t) \\\\\n \\mathbf{e}_2(t) \\\\\n \\mathbf{e}_3(t) \\\\\n\\end{bmatrix} \n", "8e488cc6f6e8ffae6d57db7899cf93c1": "x^{y/2}", "8e489d2696b686495457155a014d87b7": "\\textstyle x_1(t) = u(t)", "8e48c749de41b46133f247511ad0ecbf": " = \\sqrt{4 \\pi \\left (\\frac{m}{2 \\pi k T} \\right )^\\frac{3}{2} \\frac{3}{8} \\pi^{\\frac{1}{2}} \\left( \\frac{2kT}{m}\\right)^{\\frac{5}{2}}}\\,\\!", "8e49535746310e1412c974687bc8e59f": " d = \\frac{c_t}{gr_t \\times gr_d}", "8e496c74f3de8919422ce27f26e7aac3": "\\omega^{A}_{x}", "8e49786ad1c811799aaab4e26fe0a0db": "\\int_0^{\\theta} \\operatorname{Cl}_{2m+1}(x)\\,dx=\\operatorname{Cl}_{2m+2}(\\theta)", "8e49ab29dc8566ba65ad8bf271cae8c9": "00\n", "8e5035b8959bf5f71b8109c13d41ea01": "\\mathbf{A}^{\\mathrm{T}} = -\\mathbf{A}^{*} .", "8e504752bc08f8d8a018e3a37702715b": "\\overline{A}=\\{x\\in\\mathbb{N}|x\\not\\in A\\}", "8e505d27cc02ca8378e9d5af6df49785": "P(X_1,X_2)=X_1X_2-X_2X_1=0~", "8e508209b1d6b0f4324398b02bbf45d2": "F(u) = (u \\cdot d)^2 - (d \\cdot d) (u \\cdot u) (\\cos \\theta)^2", "8e508719427fcb814b07e839614ab1eb": "Adv(T_d)=-\\mathbf{V}\\cdot\\nabla T_d \\!", "8e50a4f404509b3b494771517c70788b": "c_\\Lambda(\\eta,d\\xi)", "8e50ac36cedf60b6dd84e6d6ecb4c985": "K_n(A) = K_0(S^n(A))", "8e512f24bf125fc49bfab90abe98a23d": "\\frac{dp_{\\mu}}{d\\tau}\\frac{dp^{\\mu}}{d\\tau} = \\beta^2\\left(\\frac{dp}{d\\tau}\\right)^2 - \\left(\\frac{d\\mathbf{p}}{d\\tau}\\right)^2,", "8e51d27a914d7cd2aab8c18076f4b7d5": "D_n(0)=1.", "8e51ee10798ad0b8de1de5addc03b956": "\n\\begin{bmatrix}\n -415 & -30 & -61 & 27 & 56 & -20 & -2 & 0 \\\\\n 4 & -22 & -61 & 10 & 13 & -7 & -9 & 5 \\\\\n -47 & 7 & 77 & -25 & -29 & 10 & 5 & -6 \\\\\n -49 & 12 & 34 & -15 & -10 & 6 & 2 & 2 \\\\\n 12 & -7 & -13 & -4 & -2 & 2 & -3 & 3 \\\\\n -8 & 3 & 2 & -6 & -2 & 1 & 4 & 2 \\\\\n -1 & 0 & 0 & -2 & -1 & -3 & 4 & -1 \\\\\n 0 & 0 & -1 & -4 & -1 & 0 & 1 & 2\n\\end{bmatrix}\n", "8e52172a4df1037215b124de72363d18": "s = {v+u \\over 1+(v/c)(u/c)}. ", "8e5227d5f5d0a122e3c9586a94e1cd8d": "R_{f1} = R_{f2}", "8e528ef0b47225464a546e11f47fd6d7": " A_{ik} = \\sum_{j = 1}^R S_{ij} \\frac{\\partial f'_j}{\\partial \\phi_k} = \\frac{\\partial (\\mathbf{S}_i \\cdot \\mathbf{f})}{\\partial \\phi_k}, ", "8e52ac01caea8eac5934b73d1af498e8": " \\rho_b ", "8e53600f8fca182cb24c704a01953ac3": "L(x, t) = \\sum_{n=-M}^{M} f(x-n) \\, G(n, t)", "8e537ed2fbb011baccbe695655e299f2": "\\vec{c}_3(u)", "8e53839cfc6d6068f8125b1440a1b076": " \\beta = \\tfrac{v}{c}", "8e53bc368a06fbb7fe1be0a904b594aa": "\\frac{\\partial(F_1,\\dotsc,F_m)}{\\partial(x_1,\\dotsc,x_n)}", "8e53c561630c51fc59cb816b69e1096b": "\\mathrm{Mg_3Si_2O_5(OH)_4 + Fe_3O_4 + MgCO_3 + SiO_2}", "8e53c80bb82d9f280e9a3ae0ada597d2": " \\ge T(\\Sigma^1_1) ", "8e53ee8873f272d57107a9027570ff8a": "\\mathrm {rect} (x'/W)* \\sum_{n=0}^N \\delta (x'-nS)", "8e53fb1832e00708a62b70b569071d61": "(x+5)(4x+3y)\\,", "8e53fd42e341f5e424cab5e6af079d4e": "y*(i,a)", "8e5457e826f044784328105cb8d17008": " E=\\iint \\left[(I_xu + I_yv + I_t)^2 + \\alpha^2(\\lVert\\nabla u\\rVert^2+\\lVert\\nabla v\\rVert^2)\\right]{{\\rm d}x{\\rm d}y} ", "8e54d0572d922c15f85cc43581a3d003": "Sq_p^{2k(p-1)+1} = \\beta P^k", "8e5517e65e1db6d4e931793f41283a6e": "\\det\\begin{bmatrix}\\frac{d\\beta}{ds} & \\frac{d^2\\beta}{ds^2}\\end{bmatrix} = 1", "8e55643690091aa2c0cf7fc022ee371b": " \\theta=2 \\ ", "8e55a2a41128647586c33de13e02b439": " \\ k(x,\\cdot)", "8e55ef2fcdb53dfe7c446fc16de291a3": "\\text{M}^+~+~\\text{M}~~\\xrightarrow{k_{p}}~~\\text{M}^+", "8e5602a13cc3a7172602c72d3fe0f56a": "A_{{xx}}", "8e5620ad6cdd23a6c4ff4411afbfe9f8": "\\begin{align}b(e_i,e_j) &= b(Q(e_i),Q(e_j))\\\\ &= b(\\sum_{k} Q^k_i e_k, \\sum_{n}Q^n_j e_n)\\\\ &= \\sum_{n,k} Q^k_i Q^n_j b(e_k,e_n)\\\\ &= \\sum_{k} Q^k_i Q^k_j \\end{align} ", "8e5628ad8ac62d674dc503d029a196e5": "\\operatorname{Pr}_\\mbox{acc} (\\sigma) = \\Vert P_\\mbox{acc} |\\psi^\\prime\\rangle \\Vert^2", "8e562a891e6908f5f345aa80a5b3ff4c": "\\mathbf{V}=-\\mathbf{T}-\\mathbf{\\omega}\\times\\mathbf{P}", "8e5655e6c19a41a704a5866782183378": "T_\\pi (x_0, x_1, \\dots, x_n) := \\left( \\frac{x_{\\pi (0)}}{x_{\\pi (n)}} , \\frac{x_{\\pi (1)} - x_{\\pi (0)}}{x_{\\pi (n)}}, \\dots, \\frac{x_{\\pi (n)} - x_{\\pi (n - 1)}}{x_{\\pi (n)}} \\right).", "8e56aa4c3beed496faf6dd2f944286af": "H^2 f(x) = D f(x) = \\dfrac{d}{dx} f(x) = f'(x) ", "8e56f55bd63720720f6b9e8a69b83be6": "\\text{Card} \\, P \\cap \\{x_1, ..., x_{b^m}\\} = b^t", "8e5737b07692c4912a1d50d6c3d6af5e": "\\Delta = g_2^3 - 27g_3^2", "8e5788a4b656e8ef94065a8beed0e539": "c_2 = \\mp c_1\\,", "8e578a92efd21b44d2aabfc13fd0455d": " -e^{-r \\tau} \\Phi(d_2) \\, ", "8e57d4bf4536035bb611f1810c0ba0b2": "-\\frac{\\partial\\mathcal{E}(n)}{\\partial v_j(n)} = e_j(n)\\phi^\\prime (v_j(n))", "8e57e5fb32edec7f7da506ae3ed398ce": "m \\times m", "8e591bfb39be4deac3e58ccd8433badf": "P\\cap K\\subset Q\\cap K", "8e594697928e72a281df16159149dcd1": "y_{\\text{min}} \\le y_0 + u \\Delta y \\le y_{\\text{max}}\\,\\!", "8e595835e53f4e791842f61f45ec9c9a": "\\mathrm{Ass}(M)", "8e59d5c22fda7f76839113cb357b2de3": "T = \\cos \\theta", "8e59facec7b0b64217e51d101cbc45d8": "f(z):=f(z)+g(z)", "8e5a4791beb376561031d80df4423fca": "\\mathcal{F}(f_1+f_2) = \\mathcal{F}_{L^1}(f_1) + \\mathcal{F}_{L^2}(f_2)", "8e5abee3899ac16282f07bd7c036c2c9": " \\ NTU", "8e5ac31fa44cd603edacbd7d58edbcae": "\\int|x|dx=\\frac{x|x|}{2}+C,", "8e5b13059a44734b93d65a8b83003ea7": "\\mathbf{K}=\\mathbf{k}-\\mathbf{k^\\prime}", "8e5b20866c94883447ec20fbb5e73e1e": "\nx_{n,i}=x_{n-1,i} \\oplus v_{k,i}. \\,\n", "8e5b236dd13e7d9abb93fd30dda7b904": "E_{kinetic} = \\tfrac 1 2 m v^2 \\,\\!", "8e5b91560042c377a92ab77856550406": "\n a_1^\\prime = a_1^* + \\frac{a_1^*}{2} \\left( 1 - \\sqrt{\\frac{\\bar{C}^7}{\\bar{C}^7 + 25^7}} \\right) \\quad\n a_2^\\prime = a_2^* + \\frac{a_2^*}{2} \\left( 1 - \\sqrt{\\frac{\\bar{C}^7}{\\bar{C}^7 + 25^7}} \\right)\n", "8e5bdfd3e77417df5203a0c85a469e27": "\\Gamma^\\lambda {}_{\\mu \\nu}", "8e5bf56bb29efcb86846423eab501fb5": " T^m_n(V) = \\underbrace{ V\\otimes \\dots \\otimes V}_{m} \\otimes \\underbrace{ V^*\\otimes \\dots \\otimes V^*}_{n} .", "8e5c0e43fa4ce2a72404345275d89d85": "(a_1,a_2,\\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \\ldots (a_m;q)_n", "8e5c3b2549c72d02c4d9c893e6df7ccf": "\\begin{matrix} {4 \\choose 2}{3 \\choose 2}^2{2 \\choose 1}{3 \\choose 1} \\end{matrix}", "8e5c424526d86755c6094d2f3de6157e": "f'(x_0) \\le 0", "8e5ccb70c2b84160728cdcff87f582d5": "s\\mapsto T_s", "8e5ce6d0c4a043e794dedc5db0ac3b9d": "R_{\\rm l}(r).", "8e5d38a9ed847f03fb50e9aff08c81b8": "(xx)(x(xx))", "8e5d5c3af687913fb3bec893027d3b83": "I_s(\\phi) = I_{c}\\sin(\\phi-\\varphi_0)", "8e5dad26b3ce522c4a0b96ae1e519cf1": "S = (N_{s} - N_{r})/N_{s}", "8e5dda60286b1b5b1937a32c4e6fbbbc": "I=I_{in}\\frac{(1-R)^{2}(1-L)}{1-R^{2}(1-L)^2}", "8e5e0471ca3689a1ef5db91dcbe97c6c": "p=\\operatorname{tr}\\left[L\\right]", "8e5e57528b7c7a74c7f1d84278480413": "\\Gamma_2 = \\frac{Z_2 - R_1}{Z_2 + R_1}", "8e5e7b32d7d8ed0db67f40ee4adbb096": "a=-\\mu g", "8e5e830322b7acb84cb2ac09486b3f6e": "\\textstyle f : \\Omega \\to \\mathbb{R}, ", "8e5e9c002fc6fbe0a7a3425e5b2465a8": "\n H^{(\\lambda+1)}(X)\n =\\frac1{X-s_\\lambda}\\cdot\n \\left(\n H^{(\\lambda)}(X)-\\frac{H^{(\\lambda)}(s_\\lambda)}{P(s_\\lambda)}P(X) \n \\right)\\,,\n", "8e5ea1816b983bec3b19cd1438434a80": " \\nu_1 {\\rm X}_1 + \\nu_2 {\\rm X}_2 + \\cdots + \\nu_r {\\rm X}_r \\rightarrow \\eta {\\rm Y} \\,, ", "8e5eccc6517ca0305f06d326335c0a1f": "W(C)", "8e5ee73e2e7e3a8b869e8b243bbe33f3": "e(x_1,x_2)", "8e5ef4d032a632b79d522dabd0b1e6b4": "p_{4}\\}", "8e5ef85c3d933b8185f4981ed531b4ad": " m\\frac {d^2x}{dt^2} = - \\frac {2e}{r_0^2} \\big( U + V\\cos \\Omega t \\big) x . \\qquad\\qquad (9) \\!", "8e5fa721c586d4a8e2a87011126a2944": "SG_A = {gV(\\rho_s - \\rho_a) \\over gV( \\rho_w - \\rho_a)} = {( \\rho_s - \\rho_a) \\over ( \\rho_w - \\rho_a)}. ", "8e5fc2c8783bbdd9e5e1d72837030e16": "\\mathcal{H}_{1} =\\pi _{1}^{2}+M_{1}^{2}, ", "8e5fd739998cba7dad1671506ae9209c": "(\\operatorname{arsech}\\,x)' = -{1 \\over x\\sqrt{1 - x^2}}", "8e6053e6cedfeb2a29f3a569f7546221": "\\Delta\\phi_1-\\delta\\phi_2=\\nu\\phi_0-2\\mu\\phi_1+(2\\beta-\\tau)\\phi_2\\,, ", "8e608d3c6423110b836869ee3d809349": " \\vec{p}_3 = \\frac{1}{\\sqrt{1-\\omega^2 \\, R^2}} \\; \\frac{1}{R} \\, \\partial_\\Phi + \\frac{\\omega \\, R}{\\sqrt{1-\\omega^2 \\, R^2}} \\, \\partial_T", "8e609cea448ca9fd758208be86ddca8a": "P = \\{P_{1},P_{2},P_{3},P_{4},P_{5}\\}", "8e60d24b1875d039ea949bc50545b32c": "N = \\sum_{i=1}^k n_i", "8e610601934a6f5be1bcf186b174068d": " \\prod M_{n_i}(D_i) ", "8e6177f7dbdca246cee440f33c983c48": "f(x_0)=-14.1014", "8e617cef93046bcc5ceeb0752a45cb17": "\\forall x\\in X: 0\\ast x=0 ", "8e619b7d9ab291d2060bad0c33739575": "\\delta = C_c\\sqrt{4\\left(\\frac{\\Delta I_\\text{obj}}{I_\\text{obj}}\\right)^2 + \\left(\\frac{\\Delta E}{V_\\text{acc}}\\right)^2 + \\left(\\frac{\\Delta V_\\text{acc}}{V_\\text{acc}}\\right)^2},", "8e61d28e0006aa7f9910ddd10973f7d9": "D^k\\!f(p)\\colon \\mathbb{R}^m\\times\\cdots\\times\\mathbb{R}^m \\to \\mathbb{R}^n", "8e61e8b49ac39638cfff699331e0d692": "y = R_1 + (x - L_1) \\tan(\\phi_2)\\;", "8e61f6ffa256fbfdab53e0c808590572": "\n E_0=\n\\left( { a^2 \\over 2 \\pi L_B}\\right) v^2\n", "8e622d6decb07e930a6610600691bd80": "U = \\begin{pmatrix}\na & b \\\\\n-b^\\star & a^\\star \\\\\n\\end{pmatrix}", "8e62329cd6f6d84d5e5c65206e54dcf2": "G(n)=n \\oplus \\lfloor n/2 \\rfloor", "8e6259072264c851a265d4b33a4ddfc1": "P_C = \\frac {1}{n}\\cdot\\sum\\left(\\frac {p_{t}}{p_0}\\right)", "8e6280579f6f6fcbc357d50733e2cd01": "x^2 - 2\\operatorname{Re}z\\,x + |z|^2", "8e62ab23d4d8366a139880ba47d961ad": "p \\in (1, \\infty)", "8e63329290d337fe7edf2ba5ae98a687": "{I}_{n}=\\underset{-\\infty }{\\overset{\\infty }{\\int }}\\frac{{e}^{x}}{{\\left({e}^{x}+1\\right)}^{2}}s\\left(x\\right){x}^{n}dx", "8e6347e8299c221a9dc50ac23f1d1d5b": " \\left(1/2,\\frac{+ \\sqrt{3}}{2}\\right) \\;\\; \\mathrm{and} \\;\\; \\left(1/2,\\frac{-\\sqrt{3}}{2}\\right) ", "8e635f243c959ef29b3d06e5eb215a7f": " \\beta_{0} ", "8e639a52689cab07e69af610ee93e3ee": "P = \\frac{2}{3}\\frac{q^2 a^2}{c^3},", "8e6476a2d4cd4a967e638587b5e17abd": "u|_{\\partial\\Omega}", "8e6483e2e1d687079cf0be05cbbf8fa9": "Z = Y^{-1} \\,", "8e64889d5e801f003fa560bc31a7bc90": "\\epsilon_2(n) * \\psi_k(n) = \\sigma_k(n)", "8e64a5a17361f29522320246b9003738": " I(x,y) ", "8e64c40667c4128bf3541f6898b4c921": "\\ddot{r} = g \\frac{1-\\mu}{1+\\mu}=g\\frac{m-M}{m+M}", "8e64cb6355a79281ddc913d9f3da03da": "L^2 (R)", "8e650d3192be94fb132e0dd063be846e": "\\mathrm{End}_\\mathbb{C}(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\cdots\\otimes\\mathbb{C}^n).", "8e65aa695b8354bca56075d04873f99a": "\\frac{1}{2}\\left[\\hat{p}^2+\\hat{q}^2\\right]=\\hat{T}\\hat{I}\\hat{T}^{\\dagger}.", "8e663b7365df57044436a40397c1d2b1": "~ {\\rm BesselJ}[n,x]~", "8e6661904b08729e755bd05177960a53": "U = YZ", "8e666b888031968c3c3cc973a9306fd1": " 0 = {1 \\over 2} {d \\over d\\tau} \\left ( v_{\\mu}v^{\\mu} \\right ) = {d v_{\\mu} \\over d\\tau} v^{\\mu} = a_{\\mu} v^{\\mu} ", "8e669605a625382746a88470d19ee811": "\\frac{d\\ell/dt}{\\ell} =g_\\ell=g_q-\\alpha", "8e66bc29c959c7aacb1d5b411617ef9f": "\n \\int (d+e\\,x)^{m+1} (B (b\\,d+2 a\\,e+2 a\\,e\\,m+2 b\\,d\\,p)-A\\,b\\,e (m+2 p+2)+(B (2 c\\,d+b\\,e+b\\,e m+4 c\\,d\\,p)-2 A\\,c\\,e (m+2 p+2))x)\\left(a+b\\,x+c\\,x^2\\right)^{p-1}dx\n", "8e66e65a321cb8408378c4fa18aa839d": "c \\in C \\setminus [1/4,+\\inf ]", "8e6706c621f7558d3a25239746b73b8b": "\nA(z) \\propto s^{l(t)}\n", "8e672bfffd7ff8f0bae846585d749f65": "\n\\begin{pmatrix}\n1 & \\,\\, 0 & \\,\\, 0 & \\,\\, 0 \\\\\n0 & a_{11} & a_{12} & a_{13} \\\\\n0 & a_{21} & a_{22} & a_{23} \\\\\n0 & a_{31} & a_{32} & a_{33}\n\\end{pmatrix}.\n", "8e676e037c8faeb7f3cd4b096e22ba7a": "\\mathcal{T},", "8e6781d211967686c44e549d7ff743a6": " \\omega_p = \\frac{1}{\\sqrt{L C}} = \\sqrt{\\frac{2 e I_0 \\cos \\delta_0}{\\hbar C}}", "8e6796f8257eceac1887e845dc6f6312": "\n\\mathbf{b}_1 = p^1\\mathbf{e}_1 + p^2\\mathbf{e}_2 + p^3\\mathbf{e}_3 = \\cfrac{\\partial x_1}{\\partial q^1} \\mathbf{e}_1 + \\cfrac{\\partial x_2}{\\partial q^1} \\mathbf{e}_2 + \\cfrac{\\partial x_3}{\\partial q^1} \\mathbf{e}_3\n", "8e681b85e2ea0786a5bcdc939df91b34": "x \\notin L", "8e68294c12730e1df68c5d06cc37604a": "\\scriptstyle x_2^2 - kBx_2 + B^2 - k > x_2^2 + k + B^2 - k", "8e68409e65c7dccfeacea5ff0fbaea3e": "\\scriptstyle D_F(3\\rightarrow 1)= 4(0)-1+3-2=0", "8e688b2bfafa94225c77e6526a09fada": "\\mathrm{Pr}(Q_{|x|}(x)=0)\\geq \\tfrac{2}{3}", "8e68c4ba317f46cd46a0a2b2018e44e0": "E_2\\,", "8e691e78e3ed07ccaa57f7532e7b5a91": "B_n(x)=\\sum_{j=0}^n\\binom{n}{j} B_jx^{n-j}", "8e6978b0c51e826d2ddaeac21a7340d5": "A(\\theta)", "8e69c7351bf783f2281e81bd0205d652": "I_0=\\frac{4e}{h}kT=\\frac{kT}{eR_0}", "8e69ecfb41ce4583469a7adebfe61101": "\\Phi,", "8e6a8c8a766f7b57c6502254e2751b98": "\\tfrac{1}{2}.", "8e6b0c7b6c0cb31ab01c209d0d24acf1": "1 = \\frac {E_t(S_{t + k})} {F_t}", "8e6b22400c682f3067129e17af13afd2": "\\int_0^{\\lambda_2} \\frac{c_1}{\\lambda^5[\\exp(\\frac{c_2}{\\lambda T})-1]}d\\lambda\n=c_1(\\frac{T}{c_2})^4\\int_{c_2/(\\lambda_2 T)}^{\\infty}\n\\frac{x^3}{\\exp(x)-1}dx\n", "8e6b34aac3f91a6e4621b7a78fcb8add": " \\begin{bmatrix}\n 0\\\\ 1\n\\end{bmatrix}.\n ", "8e6b38d0ffe9923fbf0199f294d3d497": "2^\\kappa=\\kappa^{++}\\,", "8e6b3cd3b7a2b19ac5fbad9fe53d9e0d": "\\begin{align}\n\\int_0^\\pi f(x)\\sin(x)\\,dx\n&=\\sum_{j=0}^n (-1)^j \\bigl(f^{(2j)}(\\pi)+f^{(2j)}(0)\\bigr)\\\\\n&\\qquad+(-1)^{n+1}\\int_0^\\pi f^{(2n+2)}(x)\\sin(x)\\,dx,\n\\end{align}\n", "8e6b3d63b3460345e1cfd96431443e48": "\n \\omega_{ij} = -\\epsilon_{ijk}~w_k ~;~~ w_i = -\\tfrac{1}{2}~\\epsilon_{ijk}~\\omega_{jk}\n ", "8e6b7c8896866e41efc801978594ccb5": "U \\in \\R^n", "8e6ba967645c302e1f2a60ec9c341e5c": "a_{1}", "8e6c1ac2d0e6dd80f34273d0c293d45e": "\\scriptstyle\\vec{j}", "8e6c4332bfdcdc4a47ad56c48b6b0ac3": "\\langle t,R \\rangle", "8e6c61b840f70f990def294ca0fc8112": " O=\\{o_1,o_2,\\dots,o_N\\}", "8e6ca55bc099a08b4b7d41c8c7470f40": "\n u(x) - u_\\epsilon(x) = O(\\epsilon), \\quad 0 < x < 1\n", "8e6cbbe17309914004f27eb22f610624": "\\mathrm{Rb_2S_2O_7 + H_2O \\longrightarrow 2\\ RbHSO_4}", "8e6d7573dbadb40e9df686d3e861bf1a": "\\frac{3x^2 + 12x + 11}{(x+1)(x+2)(x+3)} = \\frac{1}{x+1} + \\frac{1}{x+2} + \\frac{1}{x+3}", "8e6d9f53fda40667e9733e9d80901f40": "{D}_{10}^{(2)}", "8e6dc65f15c001b997b5c5a025ab5bba": "\ny_{n+1} = \\frac {\\left( 2-\\frac{5 h^2}{6} f_n \\right) y_n - \\left( 1+\\frac{h^2}{12}f_{n-1} \\right)y_{n-1}}{1+\\frac{h^2}{12}f_{n+1}} + \\mathcal{O} (h^6).\n", "8e6e8c6a152ca6cd464cee10ca978a2f": "d\\mathbf{l} + \\mathbf{l_0}", "8e6eaaa00752547e5c898d60ab4f2daa": "h_{0i}=w_i", "8e6eb8fe734a56a5f810d957e1deef8d": " \\sigma(E) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{k=1}^N \\operatorname{Meas}(E, X_k) ", "8e6ec9a09053daf05001b57d68ca3412": "\\frac{{\\rm d}}{{\\rm d}t}", "8e6f144b496d4e38a813dd287cc76887": "M \\Rightarrow_{mem} N", "8e6f1bac9c5d5167a14f61c8ff547133": "\\mathbf{S}(\\mathbf{p}(t))=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(p_n(t)) ", "8e6f22511e88fa43d21a89839b16742c": "\\ z=(d/2) \\sqrt{(\\xi^2-1)(1-\\eta^2)} \\sin \\phi, ", "8e6f456b073b99b6aa2e7a11c5d49b64": "\\hat{O}^\\dagger", "8e6f9bedf8b9b33da8787fac152b9e83": "\\rightsquigarrow x", "8e6fd50493b85c1e84cef0df0b7d0d1a": "x^2+ux+v=(x-s)^2-t^2", "8e703e626bd2e243455498e6f4106469": "\\begin{pmatrix} A_{11} & A_{21} & b_{1} \\\\ A_{12} & A_{22} & b_{2} \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} x' \\\\ y' \\\\ 1 \\end{pmatrix}.", "8e7043d17ca9443da24c491967c87972": "[j\\varepsilon,(j+1)\\varepsilon]", "8e70e5b349c4c4b4962f08fd6bc7834e": "S''+S'^2+\\left(\\frac{c}{x}-1\\right)S'-\\frac{a}{x}=0\\,", "8e70f71396027e9eccfc75e0fa99235e": "\\lim_{x \\to a} f(x) = 0", "8e71584994fd218d8652681a0cd08f5c": "\\nu=\\frac{\\mu}{\\rho}", "8e71e14d5e8712df56560c43afa7d496": "L^\\infty and H^\\infty", "8e723832dbddfe5fbbf014c40dc04e0e": "\\Sigma\\left|a_n\\right|^2 > \\Sigma\\left|b_n\\right|^2\\,", "8e7244d64b757750fd1ced2423197d79": "\\Rightarrow \\Leftarrow", "8e7257d3912189391ec7d9ff34a4442c": "\\tfrac{n}{2}\\sin{\\tfrac{2\\pi}{n}}", "8e72ae767363f1e90f5bfb146812815a": " L \\le \\frac2\\pi \\log(n+1) + 1.\\quad ", "8e72b6dbdf5711ed1df27ee1ba89e6be": "\\mathfrak M (K,\\rho)", "8e72b9f0b009da5d13c109ff8f9758ac": "\\frac{\\partial}{\\partial x}(\\rho u \\phi)\\,= \\frac{\\partial}{\\partial x}\\Gamma\\frac{\\partial \\phi}{\\partial x} ", "8e72cc738a4d9c4e76f430353096cfb2": "\\mathbf{f}=\\mathbf{Ax}\\,", "8e72f6be4e9fc95e365d887fb6de7005": "\\begin{align}\n\\int\\frac{dx}{\\sqrt{a^2-x^2}} & = \\int\\frac{a\\cos(\\theta)\\,d\\theta}{\\sqrt{a^2-a^2\\sin^2(\\theta)}} \\\\\n&= \\int\\frac{a\\cos(\\theta)\\,d\\theta}{\\sqrt{a^2(1-\\sin^2(\\theta))}} \\\\\n&= \\int\\frac{a\\cos(\\theta)\\,d\\theta}{\\sqrt{a^2\\cos^2(\\theta)}} \\\\\n&= \\int d\\theta \\\\\n&= \\theta+C \\\\\n&= \\arcsin \\left(\\tfrac{x}{a}\\right)+C\n\\end{align}", "8e7305bed41e2cf6c5026878e938e559": "(\\nu)", "8e7355e2edaef0a07b956dd03d46d78a": "I_{\\lambda}=\\frac{2 hc^2}{\\lambda^5}\\frac{1}{ e^{\\frac{hc}{kT \\lambda}} - 1}", "8e736b32572736dcd88b98193aa1133c": "\nV_{n}(c)=\\sum_{t=c}^{n-1}\\left[\\prod_{s=c}^{t-1}\\left(\\frac{s-1}{s}\\right)\\right]\\left(\\frac{1}{t+1}\\right)\n+\\left[\\prod_{s=c}^{n-1}\\left(\\frac{s-1}{s}\\right)\\right]\\frac{1}{2}={\\frac {2cn-{c}^{2}+c-n}{2cn}}.\n", "8e7375e18fabf0a1090292570e43f9b0": "a(x) \\equiv b(x)\\cdot v(x) \\mod g(x)", "8e73b0cd02309ccf5c94fa2e974bcbc8": "\\!\\psi(R_1 \\ldots R_n)", "8e74172fcceb0b2ad65f03992e69a36f": " \\sum_{e\\in E} w(e) = 1.\\ ", "8e74377a0d184d8f3f64585a6acd26a7": "T(n) = n + T \\left(\\left\\lfloor \\frac{1}{2} n \\right\\rfloor \\right)", "8e74acd14572c111580021021e8c53c5": "\\widehat{D}_{i} = \\frac{\\partial}{\\partial x^{i}} + u^{k}_{i}\\frac{\\partial}{\\partial u^{k}} ", "8e74c0bbdb9a67f046d88de93f84e2b6": "\\eta\\!", "8e74cf618900659ca85cceb0a14e142f": "\\sin \\delta = \\sin \\varepsilon~\\sin(\\theta - \\varpi )\\, ", "8e74d8145382219c2ed707e489dd7529": "\ng_{jk}(\\theta)\n=\n\\int_X\n \\frac{\\partial^2 i(x,\\theta)}{\\partial \\theta_j \\partial \\theta_k}\n p(x,\\theta) \\, dx\n=\n\\mathrm{E}\n\\left[\n \\frac{\\partial^2 i(x,\\theta)}{\\partial \\theta_j \\partial \\theta_k}\n\\right].\n", "8e7544e7bdeac11e6c1826e13a903e39": "p_{\\mathbf{s}}", "8e75661d8196ae7baea44c9e88f9a5ef": "\\frac{\\gamma(d/p, (x/a)^p)}{\\Gamma(d/p)}", "8e7592875c681d37c3c97286a6ec6fcf": "\\mathbf{R} = 2 \\mathbf{\\hat{d}}_\\mathrm{n} \\mathbf{\\hat{d}}_\\mathrm{n}^\\mathrm{T} - \\mathbf{I};", "8e75ed63ecc7eba0dcdae9eaf0ba6e56": "G(a,c)=G(a,0,c)", "8e7606b2c2bb510dbe64cd497b831b05": "\\ominus", "8e7790ce94acb332db762f235269357f": " {\\rm det}(I+A) \\cdot {\\rm det}(I+B) = {\\rm det}(I+A)(I+B). ", "8e77998f219e41e349ea6a3c9fe668c2": "\\pi_{n+2}(S^n)=Z/2Z", "8e779cb91d1f5a2a56dcc22b533211ab": " F = \\gamma F_{\\mathrm{M}}.", "8e77f017aa06831be4eed5bb2c2c1d0c": "V_{\\alpha}\\setminus\\bigcup_{\\xi<\\alpha}V_{\\xi}\\,", "8e77f738ee6c91ce207906fbfa9259dd": "\\left\\langle f^*(x) \\cup L_i(M),[M] \\right\\rangle \\in \\mathbb{Q}", "8e782692b80004fbab99ae7aa114f54d": "(\\lor_{\\mu < \\gamma}{(\\and_{\\delta < \\gamma}{A_{\\mu , \\delta}})})", "8e78572406d48704d0bf18ae272541b0": "\\eta^{\\mu \\nu}=\\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 \\end{pmatrix}\\,", "8e7870956a833bf8ff7198cf5952f891": " \\begin{align}\n\\text{specificity} & = \\frac{\\text{number of true negatives}}{\\text{number of true negatives} + \\text{number of false positives}} \\\\ \\\\ & = \\frac{\\text{number of true negatives}}{\\text{total number of well individuals in population}} \\\\ \\\\\n& = \\text{probability of a negative test given that the patient is well}\n\\end{align}\n", "8e788d79c930447977ab5168fea2bff2": "\\begin{align} \\langle\\psi_{\\varepsilon}|\\mathbf{\\hat P}|\\psi_\\varepsilon\\rangle & = \\int_{-\\infty}^{\\infty} \\, \\psi_{\\varepsilon}^{*} (x) \\, \\left(-i\\hbar\\frac{d}{dx}\\right) \\, \\psi_{\\varepsilon}(x) \\, dx \\\\ \n& = \\int_{-\\infty}^{\\infty} \\, \\psi^{*}(x-\\varepsilon) \\, \\left(-i\\hbar\\frac{d}{dx}\\right) \\, \\psi(x-\\varepsilon) \\, dx \\end{align} ", "8e78a2595344d6251efe6a49d76abadf": " \\vec{\\mathbf{F}}=k\\vec{\\mathbf{x}} \\ ", "8e78ae4303a802feeca16369aae1c95f": "\\begin{Bmatrix} p \\ \\ \\\\ q , r \\end{Bmatrix}", "8e78d7f0778b9b9d27e4bb9642464398": "\n\\begin{array}{rcl}\nmass_{ingredient} & = & \\frac{formula\\ mass}{formula\\ percentage} \\times baker's\\ percentage_{ingredient}\\\\\n& = & mass_{flour} \\times baker's\\ percentage_{ingredient}\\end{array}", "8e791d6af506c0c6b9da7f754b30b53f": "\\mathbb{R}_{p,q}\\,", "8e799f28a7f096a3fc2932c154e30122": "E_{GR} =- \\frac{3}{5}\\frac{GM^2}{R}", "8e79b78c669df244f30514ec27e79cb5": "\\begin{align}\n ax &= ay &\\quad \\rArr & \\quad ax-ay = 0 \\\\\n & &\\quad \\rArr &\\quad a(x-y) = 0 \\\\\n & &\\quad \\rArr &\\quad x-y = 0 \\\\\n & &\\quad \\rArr &\\quad x = y.\n\\end{align}", "8e7a52541498fe1bc2f335fbe55ac659": "p = r_p(1 + e),\\,", "8e7a6dbad11ff9da6dcb7921ccf8e11c": "\\boldsymbol{\\sigma}(\\alpha u + \\beta v) = \\alpha\\boldsymbol{\\sigma}(u) + \\beta\\boldsymbol{\\sigma}(v)", "8e7a6f6b0821d4505ccbc996eaa57d38": "2(1-\\varepsilon)\\gamma n > |T| > \\gamma n\\,", "8e7a7445c1704384e2bb93660f3fe27f": "wp(S, R)", "8e7adb2ee596e3a1dac8e465ed6a0158": "J-1", "8e7ae064584eb5d9a9154ecdae6df0bb": "g = h^r\\text{ mod }p", "8e7b34511b0355306ac8be40ce8a9804": "\\displaystyle{f_r(z))=\\sum_{n\\ge 0} r^n a_n z^n}", "8e7b4035b1515ac62c397b8a0c4d3add": "a^2 - b^2", "8e7b78fd8561d5351b4213b5bae07543": "X_L = \\sqrt{\\Big(R_{source}+jX_{source}\\Big)\\Big((R_{source}+jX_{source})-(R_{load}+jX_{load})\\Big)}", "8e7b86528dbcc4b8275444e111958eef": "\\,\\mathrm{slog}_b(1) = 0.", "8e7bad81a848fff2ce43b98f29f8c723": "|+\\rang", "8e7bb743e2aac8157af5e42cfd41c1d6": " 5^{1/4} \\approx 1.495349", "8e7bd30a8bd8320331a56bd7892de602": "C(X_1,X_2,\\ldots,X_n)", "8e7bd992f4dc6520f7da5aa086519211": "p(drunk|D) = \\frac{p(D | drunk)\\, p(drunk)}{p(D)}", "8e7c357822326d3acd7a71380cdcdd8e": " z(t,x) ", "8e7c971b1127de3d664f4bab36ffcb3c": "1/c^2", "8e7ccbb11a385ea52f71b61fa391167a": "\\omega_B = ae|E|/\\hbar", "8e7cce5d11bf1cf57bd866d200586d6d": "(X,A)\\,\\!", "8e7d0dda2db8bd269ad7f51997853876": " \\le d ", "8e7d44821db6bf5fe3897fd2b8fb1e8f": " V\\otimes W", "8e7dadc10359bf64f7eaaa61854c086b": "\\textstyle P(A\\mid[x]) \\leq \\beta", "8e7dd2b91738f5be47dae9abc2b483e4": "\\lnot \\psi\\,\\!", "8e7dd58d77c392f109ec8df64aa48906": "\\Bbb{Z}\\left[(1+\\sqrt{-19})/2\\right].", "8e7dd5d3e76aa952e21999a5537dcffb": "dy", "8e7e046b407acb4e75e7c5894aa52bfe": "u\\in \\mathcal{D}'_{L^p}", "8e7e9ab8734f01b2036fbef9dc90d5e9": "\\frac{X_i - \\bar{X}}{s_X},\\,\\bar{X}=\\frac{1}{n}\\sum_{i=1}^n X_i, \\text{ and } s_X=\\sqrt{\\frac{1}{n-1}\\sum_{i=1}^n(X_i-\\bar{X})^2}", "8e7ea6536c30ad7f336269eb1dd13302": "I_{t}", "8e7eacf7cdf0d8751f01b36ea6d3b538": "\\ln p(x|\\theta) = \\ln p(x|m_k,\\sigma_k^2 C_k) = -\\frac{1}{2}(x-m_k)^T \\sigma_k^{-2} C_k^{-1} (x-m_k) \n \\,-\\, \\frac{1}{2}\\ln\\det(2\\pi\\sigma_k^2 C_k)", "8e7f44d120f1585832dc08f2fa00debe": " \\frac{d\\theta_{T}}{dt}=\\frac{s_{Tx}+s_{Ox}}{R_T} \\,\\!", "8e7f61c239e6d307ab9b56cbddc4f04b": "= \\frac{32!}{2}\\cdot 60^{32}\\cdot \\frac{80!}{2}\\cdot \\frac{24^{80}}{2}\\cdot \\frac{160!}{2} \\cdot \\frac{12^{160}}{3} \\cdot \\frac{40!\\cdot 80!}{2} \\cdot \\frac{6^{80}}{2}\\cdot \\frac{2^{40}}{2} \\cdot \\frac{320!}{24^{80}} \\cdot \\frac{6^{320}}{2} \\cdot \\frac{320!}{24^{80}} \\cdot \\frac{6^{320}}{2} \\cdot \\frac{240!}{(6!)^{40}} \\cdot \\frac{2^{240}}{2} \\cdot \\frac{320!}{(8!)^{40}} \\cdot \\frac{2^{320}}{2} \\cdot \\frac{480!}{(12!)^{40}} \\cdot \\frac{2^{480}}{2} \\cdot \\frac{80!}{(8!)^{10}} \\cdot \\frac{160!}{(16)^{10}} \\cdot \\frac{240!}{(24)^{10}} \\cdot \\frac{320!}{(32)^{10}}", "8e7f95c710b558537e4477d77b69ae2a": "\\phi^{18}=2889+1292\\sqrt5 \\approx 5777.999827\\,", "8e7fd163c8f4c7c4b485595ab194b250": "Q_{feed}", "8e7feb8d1ad2882571aba22be306d553": "\\scriptstyle u_i", "8e80061f051e94a086167a05fa669746": "p_1 \\left( \\frac {1}{u} \\right )=p_1(u),\\ p_2 \\left( \\frac {1}{u} \\right )=p_3(u),\\ p_3 \\left( \\frac {1}{u} \\right )=p_2(u)", "8e8038aec51e0ffa0b311ee01f81f069": "a_{k+1}= \\begin{cases}\n \\left \\lfloor a_k^{\\frac{1}{2}} \\right \\rfloor, & \\mbox{if } a_k \\mbox{ is even} \\\\\n \\\\\n \\left \\lfloor a_k^{\\frac{3}{2}} \\right \\rfloor, & \\mbox{if } a_k \\mbox{ is odd}.\n\\end{cases}", "8e805a44331ef70e16b29a68708b841c": "\\Lambda(A_1:A_2|B) = \\frac{P(B|A_1)}{P(B|A_2)}.", "8e80654b273deb3d1c2a04a547580e15": "\\theta_3 (X) = \\begin{pmatrix} 0 & I_m \\\\ -I_m & 0 \\end{pmatrix} X^T \\begin{pmatrix} 0 & I_m \\\\ -I_m & 0 \\end{pmatrix}.", "8e807ad45b0d2059682de53d24938a49": " x-y ", "8e80a6d0cc5c23ae418b5167bf9a7b51": "\\alpha = v_d/v_p", "8e80aeaccc834b3a65d1a5f0bbb9d1e8": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V\\left[\\mathbf{F}\\cdot \\left(\\nabla g\\right) + g \\left(\\nabla\\cdot \\mathbf{F}\\right)\\right] dV\n =", "8e80ee205a3ed5f79bbdb3c9a90ea06c": "|\\psi(0)\\rangle", "8e8108c6cbd9b58053dd597df524ea5e": "\\ E(r)=Re(\\Psi(r))", "8e8158421496aceb007fd02dc7c473b1": "\\textstyle\\sum_{i=0}^\\infty a_i X^i", "8e8190b1650f5d8b5763bf4c0c48d750": "J_-J_+ = (J_x - iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 + i[J_x,J_y] = J^2 - J_z^2 - \\hbar J_z,", "8e81f73db2c6cebb8c3c69f1a3b23af2": "\\text{Hypothesis }H_1:\\text{ Leak}", "8e81fa839f0cc133f5ea98f2c82564ac": "H^{(n)}", "8e82438bf45fe87819b9d07dc0b80a94": "\n{\\Vert \\mathbf{u} \\Vert}^2 {\\Vert \\mathbf{v} \\Vert}^2\n=\n({\\mathbf{u} \\cdot \\mathbf{v}})^2 + {\\Vert \\mathbf{u} \\times \\mathbf{v} \\Vert}^2\n", "8e826a46b0202ee256dbbe8233c6d674": "\\alpha_1(k)", "8e82852bf4d53afc90b5e051ded3ac70": "E_G (\\varphi \\wedge C_G \\varphi)", "8e829f20db7382138200ffa1e34bd937": "S^j", "8e82bc3fe57d70ea8cec904b0413663b": "p_{ij}^{+}", "8e82fd96c605f6ead3e5cba5c2b146f6": "\n\\psi=-\\frac{A}{r}\\sin\\theta.\n", "8e8313d85afcaaedb9a24b2070e4b482": "F r_1^2=\\frac{x^2}{2}+\\frac{x}{2} \\Rightarrow 0=\\frac{x^2}{2}+\\frac{x}{2}-F r_1^2.", "8e8370c58f0fa0cf032ebf947834a4e4": "a^*(z)=\\sum_{n=-N}^N a_{-n}^*z^{-n}", "8e83826ba1618e33c2a43ed631ffa2fd": "-L_3 \\rightarrow L_3 ", "8e8391d2432e555db66dbfc098d0e8f9": "K=\\sqrt{(s-a)(s-b)(s-c)(s-d)} \\,", "8e839f91c679340d381e75ec9f01d213": "(D - \\lambda I)q + w(w^{T}q) = 0", "8e8402a639fd4e1c43d3b56f22afa232": "|I\\cap B_i|\\le d_i", "8e84ad9c0ceb8019451768188c99d702": "pq=ac+bd", "8e84d58206b65aa48dd1f9761823cf5e": " \\hat{f}_1^{(i)} ", "8e84eb458653f695e3ed0ade17b38f64": "=\\binom{n}{r} \\times p^r \\times (1-p)^{n-r}", "8e8502fd26c7f38100b65cf9402cc6b1": "\\left.\\right. |z|<1 ", "8e855525bf640a2e251d62a854575958": "t _1", "8e859dfddcc4cdd1baf7fcc2c6c441c7": "\n\\Delta_{\\psi} S_x \\, \\Delta_{\\psi} S_y \\ge \\frac{1}{2} \\left|\\left\\langle\\left[{S_x},{S_y}\\right]\\right\\rangle_\\psi\\right|\n=\n\\frac{1}{2} (i \\hbar S_z)\n=\n\\frac{ \\hbar}{2} S_z\n", "8e85fd61abc27b0ceb7c8bbe389cd6e7": "\n \\hat{f}_\\bold{H}(\\bold{x})= \\frac1n \\sum_{i=1}^n K_\\bold{H} (\\bold{x} - \\bold{x}_i)\n ", "8e86509343dc5953aaefda481e680e0a": "\\operatorname{cov}(x_i, y_j)", "8e8652e14dd8f03ae504ee89a1c842e3": "\\sqrt{-1 \\times -1} = 1", "8e86647a004d77a3ce739ff13cec08b9": "[1,0,0]^\\mathsf{T}", "8e8667fd3fd1d14a44e0a2efbce62a86": "\\begin{align}\n\\int f(\\sin(x), \\cos(x))\\,dx &=\\int\\frac1{\\pm\\sqrt{1-u^2}} f\\left(u,\\pm\\sqrt{1-u^2}\\right)\\,du && u=\\sin (x) \\\\\n\\int f(\\sin(x), \\cos(x))\\,dx &=\\int\\frac{1}{\\mp\\sqrt{1-u^2}} f\\left(\\pm\\sqrt{1-u^2},u\\right)\\,du && u=\\cos (x) \\\\\n\\int f(\\sin(x), \\cos(x))\\,dx &=\\int\\frac2{1+u^2} f \\left(\\frac{2u}{1+u^2},\\frac{1-u^2}{1+u^2}\\right)\\,du && u=\\tan\\left (\\tfrac{x}{2} \\right ) \\\\\n\\int\\frac{\\cos x}{(1+\\cos x)^3}\\,dx &= \\int\\frac2{1+u^2}\\frac{\\frac{1-u^2}{1+u^2}}{\\left(1+\\frac{1-u^2}{1+u^2}\\right)^3}\\,du = \\int \\frac{1-u^2}{1+u^2}\\,du\n\\end{align}", "8e86e0b3e286652601e2f795d42877bf": "{\\mathbf e}'_i", "8e870e24ee13826e4bb6bd44dbff7b82": "\\textstyle\\sigma ", "8e870eb4e2d2d5e00b30661618008ff0": " \\sigma = \\begin{pmatrix} \n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\\n3 & 8 & 1 & 2 & 4 & 7 & 5 & 6\n \\end{pmatrix}.\n", "8e8722145a62da53e872567d97db8075": "q = - \\frac{\\partial F_3}{\\partial p} = \\frac{-1}{Q}", "8e872817d4e79efbde100f06f25b82b5": "\\sum_\\pi (x)_{|\\pi|},\\,", "8e87c30645f962bfb79ea06abd914522": "\\textbf{max}\\left( 0,\\text{log}\\tfrac{N-df_{t}}{df_{t}} \\right) ", "8e87e00d4f9f42d5bcbcbda602c69937": "x / (\\log x)^k", "8e87e10cd4ef7b89a8a3370eefeefb78": "P(n)= 1-\\left(1-\\frac{1}{2} P\\left(\\Bigl\\lceil 1 + \\frac{n}{\\sqrt{2}}\\Bigr\\rceil \\right)\\right)^2", "8e884eb24163fe18bdfaa389368a3018": "MB(X,Y,W)", "8e8858d36f31a9b2bdb31fbeb060300e": "I = {Q \\over t} \\, ,", "8e88591d005803018b832784216d0f9d": " p \\times p", "8e88e942e13a0b02da285618f24b1a8c": "-(z\\log z-z)-\\frac{z^2 \\gamma}{2}- \\sum_{k=1}^{\\infty} \\Bigg\\{ (k+z)\\log \\left(1+\\frac{z}{k}\\right)-\\frac{z^2}{2k}-z \\Bigg\\}", "8e88ec2c3d0ea7725eaac06c63f59e53": " {(rad/s)} ", "8e89dee6744a4d0a02322a61fb3aa6cf": "L^q(R^n)", "8e8a0d6382ad153ab14d8af421916583": "\\nabla^2 \\mathbf{B} = \\frac{1}{\\lambda^2}\\mathbf{B}, \\qquad \\lambda \\equiv \\sqrt{\\frac{m c^2}{4 \\pi n_s e^2}}. ", "8e8a4e4eb54918f170c9591e880a289e": "\\scriptstyle R[X]/(X^2 \\,+\\, 1)", "8e8a66abe241a54a69d6f229548ec31a": " M= \\Gamma\\backslash G /K.", "8e8a8556bde695befa51c4e6d4e24fc2": "= \\int u \\,dx - \\int v \\,dx", "8e8a912dd4799bff7b509f9ec2fa850b": "0.05 < {p}/{p_0} < 0.35", "8e8b01af699602c8ba172dfeeccb72ac": "\n {D\\vec{\\omega} \\over Dt} = (\\vec{\\omega} \\cdot \\vec{\\nabla}) \\vec{v},\n", "8e8b83081ba5a00102476934aad15d66": "[v,w] = X_{\\omega(w,v)}", "8e8bba7e6fe43bf920b3ce8cbd2f8f6f": " f(t) = a_1 \\cdot g_1(t) + a_2 \\cdot g_2(t) + a_3 \\cdot g_3(t) + \\dots + a_n \\cdot g_n(t) ", "8e8bc12de34546a18af0d993eda83079": "\\ \\bar{E}= \\frac{\\sum_{i=1}^N {E_i}}{N} ", "8e8c22746186c4389b34181fc35917b1": "\\Omega M", "8e8c318e91d48e251e31c13c78e1e969": "\\lambda_i=\\frac{(r-1)^i}{2i}", "8e8c3699ba2b3a9977fe4cbcb9f99442": "\\text{fmap} \\colon (A \\to B) \\to \\left( \\left( A + E \\right) \\to \\left( B + E \\right) \\right) = f \\mapsto a \\mapsto \\begin{cases} \\text{err} \\, e & \\text{if} \\ a = \\text{err} \\, e \\\\ \\text{value} \\, f \\, a' & \\text{if} \\ a = \\text{value} \\, a' \\end{cases}", "8e8c37cb7075cc6abc9b15e4e05d655f": " \\frac{\\partial \\phi}{\\partial t} = \\frac{\\partial }{\\partial z}[(1-\\phi) u], ", "8e8c3fbcf8669ad9e7719b7af696b408": "Np", "8e8c434f641003d9b7403a35bfc610fd": "x_{\\sigma (1)}y_1 + \\cdots + x_{\\sigma (n)}y_n", "8e8c449002949c668b3faac7e60efb78": "\\hbar k_z", "8e8ca12be6c7582f0baae3bb1825529c": " \\left ( \\frac{d}{dx} - r_{1} \\right )^{k}y = 0 ", "8e8ca4aada60ba8295c34c5a379fa00d": " (y_1',...,y_n') ", "8e8cf68e2c5cd170fb3220d75cbed58d": "\\lambda^*=(-1)^\\frac{N\\lambda-1}{4}\\lambda.", "8e8d0deeef03dfeda78e9918ef12d326": "(G, i)", "8e8d0e218828b718639ff30a0893949a": "v_{n+1}(x)=\\int\\! \\int\\ \\cdots \\int v \\ (dx)^{n+1}.\\!", "8e8d589a46efb600d3fb7f7f045a28e8": "\\sigma_x = \\sigma_y = 1", "8e8d86f32ed919bf309672ecc9935f23": " \\int_0^r 4\\pi u^2 \\,du", "8e8e24d1f7e1c9417071005b6bb9087c": "x_{B}=x_{A}+L", "8e8e83cc54c1882b5c21b7545f38975e": "\\mathcal{I}(\\boldsymbol\\beta^{(t)})", "8e8e8c8725460d574f454e51aa2a05a2": " P_j = AB ", "8e8e9274d32b14514dcc280af50195bf": "a_f", "8e8eccc4a67e01cee0791117e1aa318d": "\\sum_{j}\\gamma_j \\delta O_j = 0\\,\\!", "8e8f14a2f3f38466a85c9bd34257ab9c": "\n\\sigma(t)= \\sum_{mn}^{} { A_{mn} [\\ln(1+t)]^m (\\epsilon'_0)^n} ", "8e8f96cd9087ce92eaaee13d8af96ea6": "\\mathbf{1}^T\\mathbf{P}\\mathbf{1}", "8e8f9c80c109063f529ebbff7f390c24": "5 \\le D \\le 13", "8e8fde035d49773cd2c139a84a9a3e22": "b_{FB}", "8e90153a86c9b5bda506671f20f7cdbd": "n(\\vec{r})=\\frac{8\\pi}{3h^3}p_{f}^3(\\vec{r}) .", "8e902ebe43fbd5c8c4a7dec705723044": "\\overline{X_j}", "8e90536734f8ba8155e6c0e78565b6e4": "\\partial f(x)(h)=-x^{-1}hx^{-1}", "8e90b109464dfa62b23b058145ea89f5": "f(T)=(T-2I)^{-1}(T-5I)^{-1}.\\,", "8e90c15fce220935d878237d6a5a02cc": "\\Delta E < 0", "8e910d54b6cb951f17dc73377c1ba977": "\\hat c = C(\\omega A^{-1} \\hat a + (1-\\omega)B^{-1} \\hat b) \\, .", "8e911b2ef0faa550a8a5f19517c50db1": "g<1", "8e9120f5ea4ed2b61da43cd88e218269": "A_{ad}", "8e9131c0d73961f6a7fbd207207e0da6": " \\tau \\in [0,2\\pi] ", "8e915d78db7914dcd59383d23d1ca3f5": " \\delta y=y^2\\delta p/Ec ", "8e919a854707b0a0ed1f14abc606ff71": "S[j,n]", "8e91b498d88365f06334d7bd0da3ce30": "\\Delta_K(t)=\\pm 1", "8e920d832888ec8e9857947c1f5ad512": "\\Xi = \\Xi_k + \\Xi_e", "8e920f4878766d9298bbf003098e2532": "c_k = \\frac{(-1)^{k-1}}{(k-1)!} (-k+a)^{k-1/2} e^{-k+a} \\quad k\\in\\{1,2,\\dots, a-1\\}.", "8e921a48bdaa45bf6f9ad168f00c26a7": "\n\\alpha_p = Prob\\{Period~demand \n \\le \\; {Inventory~on~hand~at~the~beginning~of~a~period}\\}\n\n", "8e922f4c413c49b2d7929e35aeea6c04": "G_L = (V, E_L, c_f|_{E_L}, s,t)", "8e928191664d5c25cd30d6a3a194c8b7": "[{w} \\mapsto 1, {x} \\mapsto 1, {y} \\mapsto 2, {z} \\mapsto 2]", "8e9293be93c858482c386e5d598314c7": "\\Rightarrow \\Pi_1 = \\bigg(\\frac{a - b.q_1 + \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2})\\bigg) \\cdot q_1 - C_1(q_1).", "8e92bab9ab264909928116570178451d": "v_1, ..., v_n", "8e92ddb263024975bb36f2580f336ce7": "|n|_2 = 2^{-\\nu_2(n)},", "8e92f8b8309159f3b9b71ad34b848fde": "U = \\text{close}_\\text{now} - \\text{close}_\\text{previous}", "8e92faf59878c9413e45f77b07081f39": " ~{\\mathcal B}^{(m,t)}_{1...m+t} ", "8e92fb76109678ad7bfdd89020278063": "E[2n+1]", "8e9325fb86f889207cf77fbfa6b18a04": "n = 0,1,5", "8e9332fa4c5aa355dd6b9a36f2e666de": "\n\\bar{a} = -2d_1 \\left( \\bar{ \\omega} \\times \\bar v \\right) - d_2 \\left[ \\bar{ \\omega} \\times \\left( \\bar{ \\omega} \\times \\bar{r} \\right) + 2\\left( \\bar{ \\omega}\\bar{r} \\right) \\bar{ \\omega} \\right]\n", "8e93e161336e005f309aa0acff5833c5": "\\ \\phi_i = \\sum_{r} c_{ri} \\chi_r ", "8e940d60ad33b880e9106d301019a064": "p(n) = a + bn\\,", "8e9410c50b3f09ffd1b7b0313cf57d36": "P_k:\\R\\rightarrow[0,1]", "8e94877ad746448754c7000c75bdb9a9": "\n\\mathbf{c}_\\mathbf{0}^\\mathtt{KED} = \\left\\{ C(\\mathbf{s}_0, \\mathbf{s}_1\n), \\ldots , C(\\mathbf{s}_0, \\mathbf{s}_n ), q_0 (\\mathbf{s}_0 ), q_1 (\\mathbf{s}_0 ), \\ldots ,q_p (\\mathbf{s}_0 )\n\\right\\}^\\mathbf{T}; q_0 (\\mathbf{s}_0 ) = 1\n", "8e952fe99618de7bcde81e24b409dd7f": "|A\\rangle_C", "8e95f3f741e89eec348aacaa4859f4b0": "\np_{\\mathbf{Y}}(Y)=\\frac{p_{\\mathbf{y}}(\\mathbf{y})}{|\\frac{\\partial\\mathbf{Y}}{\\partial \\mathbf{y}}|}=\\frac{p_\\mathbf{y}(\\mathbf{y})}{p_\\mathbf{s}(\\mathbf{y})}\n", "8e9603fb1e1b17d85ca8497883c81047": "d_i\\times (\\prod _{j \\neq i} d_j)", "8e96107f8acd5a803c538aa3b7cd9dc0": " y_{3i} = \\begin{cases} \n y_{3i}^* & \\textrm{if} \\; y_{1i}^* >0 \\\\ \n 0 & \\textrm{if} \\; y_{1i}^* \\leq 0.\n\\end{cases}", "8e96750c4d1a645b6ef94691a46bf7d4": "\\frac { N } {\\sqrt {T_{01}}}\\ ", "8e968360acff7df190b0f0c2b63f139b": "F_0 = XZ - Y^2", "8e9687007adb1f809c123bbccb7fe107": "L \\circ L^{-1} \\neq \\mathrm{id}", "8e96a641544c0371ec58952c15f33bc1": "\\vec{H_{in}}=\\eta \\vec{H_{0}}", "8e96fd937037432a585586a791bff9d5": "\\forall a,b \\in A, \\;\\; f(a)=f(b) \\Rightarrow a=b", "8e970908d6e93233b5a664e19c42c242": "\\ell(A)", "8e97237f880b3afaaa74c58eaf796cb6": "\n\\begin{array}{ccccccccccccccc}\n0&&1&& 4 && 9 && 16 && 25 &\\ldots & (n-1)^2 && n^2 \\\\ \n&1&&3&& 5 &&7 &&9 && \\ldots && 2n-1 & \n\\end{array} \n", "8e976c997bfd53f2121ce1e268a0aa0e": "(\\rho \\mathbf {u})_r -(\\rho \\mathbf {u})_l\\,=0; (OR) F_r-F_l=0; (OR) F_r=F_l=F;", "8e97b01b469c102f162e32d36ed9c9b0": "m=p-n\\, ;", "8e97d75e125e7dd9f9e8f8edfd1c2004": "O(k^{2/3} \\log k)", "8e97ef2420307bd4c04395f59b73cd82": "L^{\\,p}", "8e984ddd55e6257748f93f6e5a47257e": "{\\mathbb Z}[x]", "8e98c6b14aff15f91f372271b3ff8e26": " f(x) = \\tan\\left( \\frac x 2 + \\frac\\pi4 \\right). ", "8e99011ff4f339139f7d69c1afc5ce82": "\n\\begin{align}\n\\ln \\frac{\\Pr(Y_i=1)}{\\Pr(Y_i=K)} &= \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i \\\\\n\\ln \\frac{\\Pr(Y_i=2)}{\\Pr(Y_i=K)} &= \\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i \\\\\n\\cdots & \\cdots \\\\\n\\ln \\frac{\\Pr(Y_i=K-1)}{\\Pr(Y_i=K)} &= \\boldsymbol\\beta_{K-1} \\cdot \\mathbf{X}_i \\\\\n\\end{align}\n", "8e9921e08ed68c791ba4b40cb5b1c7a2": "p_0,p_1,p_2,\\ldots,p_m . \\,", "8e9945b1a85c5144eba99c1b5c924ee2": "{\\gamma}=\\left(\\frac{\\mathit{c}_{p}}{{c}_{v}}\\right)", "8e9945dbff7d0d0935e46723acd2b32b": " m' \\equiv m r^e\\ (\\mathrm{mod}\\ N) ", "8e9976d6c17b3b72705459185b490afd": " A = A(h) + a_0h^{k_0} + a_1h^{k_1} + a_2h^{k_2} + \\cdots ", "8e9991110e6772b4cb97985c49b4d2ee": "\\tfrac{-(1-p)\\log_2 (1-p) - p \\log_2 p}{p}\\!", "8e99cdbb8603772b9a150c9c7d7f8bb7": "b=V_{\\infty} sin(\\alpha)", "8e9a37225f442526c307d5693ef99538": "\\|f\\|=\\sup_{x\\in [0, 1]}|f(x)|.", "8e9a684b7314e8ab2a3e6cba26c7d02e": "e^{i(\\mathbf{k}-\\mathbf{k^\\prime})\\cdot\\mathbf{R}}=1", "8e9a6ff64ddb8349f8ad863700c88d81": "F(\\mathbf{x}) = F(\\mathbf{p}) + J_F(\\mathbf{p})(\\mathbf{x}-\\mathbf{p}) + o(\\|\\mathbf{x}-\\mathbf{p}\\|)", "8e9abe82ed72ad3820a9369741defdaa": "\\forall (u, v) \\in E \\ f(u,v) \\le c(u,v)", "8e9ac70f37aab9b1f6c57b4bb5937dd4": "\\mathbf n\\,\\!", "8e9ad32285b368d0dde94564a6154124": "2m_e \\vec{v}_s=\\frac{h}{2\\pi}\\vec{\\nabla}\\varphi+2e\\vec{A}.", "8e9b1473221e38b276b3b2c0b07b59ca": " V(x(t), t) = \\min_u \\left\\{ C(x(t), u(t)) \\, dt + V(x(t+dt), t+dt) \\right\\}. ", "8e9b2af538728869703dc63ad2141e75": "\\sum_{p \\in P} 2^{-|p|} \n\\overset {p: \\text{ Halted program}}\n \\underset{ P:\\text{ Domain of all programs that stop.}}\n{\\scriptstyle {|p|}:\\text{Size in bits of program }p}", "8e9b431f2aa3cf6d54542656823a44bf": " DL_i = \\lbrace \\hat{y} \\in R^m ", "8e9bde2cf27f081095ad741f39f190c6": "D_1 = \\sum_{P \\in C}{\\mathrm{ord}_{P}(f)[P]}, D_2 = \\sum_{P \\in C}{\\mathrm{ord}_{P}(g)[P]} \\in \\mathrm{Princ}(C)", "8e9c274eb99affd5d6a0e480acbcbc8b": "a \\to b \\to (n-2) ", "8e9c5fee65a4f32abccd0e83ff203e39": "O(N^2)", "8e9c72086ec03f0cde5d0240db16ea67": "\\textstyle \\lim_{n \\to \\infty} \\left(1 + \\frac{x}{n} \\right)^n", "8e9cdb8d925d4a23dbbacd86b9d2a8fd": "\\varphi=\\exists x_0\\cdots\\exists x_{n-1}(x_01", "8eb24476b54094b7122be750df40b99b": "\\varphi\\, ", "8eb25e99f2d275b9c25c72c682877767": "[\\rho^a(\\vec{x}),\\rho^b(\\vec{y})]=if^{ab}_c\\delta(\\vec{x}-\\vec{y})\\rho^c(\\vec{x})", "8eb284ea3ed2d8ca9e601f341048d87a": "G_{n}", "8eb2b4a37a4be6e29ea25e9dfe1b90ad": "d + p = ^{3}He + \\gamma\\ ", "8eb2ff5b8244738cbbe7b3fa07a79597": "\\int_E g_k\\,d\\mu\\le\\int_E f_n\\,d\\mu,", "8eb39d7b947d53c0f8c3d8eda26e03b8": " r = 0.5+0.2\\,\\sin\\theta + 0.2\\,\\cos3\\theta", "8eb39ef1b1e8e49753ea7d48b0f6add7": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{33} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix}\n = \\cfrac{E}{(1+\\nu)(1-2\\nu)}\n \\begin{bmatrix} 1-\\nu & \\nu & \\nu & 0 & 0 & 0 \\\\\n \\nu & 1-\\nu & \\nu & 0 & 0 & 0 \\\\\n \\nu & \\nu & 1-\\nu & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & (1-2\\nu)/2 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & (1-2\\nu)/2 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & (1-2\\nu)/2 \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{33} \\\\ 2\\varepsilon_{23} \\\\ 2\\varepsilon_{31} \\\\ 2\\varepsilon_{12} \\end{bmatrix}\n ", "8eb3c83503dbc915bf30811d467901c7": " \\beta E_{t}R_{t+1}\\frac{u^{\\prime }(c_{t+1})}{u^{\\prime }(c_{t})}=1 ", "8eb3cc3b6c9309735288de69d5cdbc04": "\\hat{H}_v(t)|\\Psi(t)\\rangle=i\\frac{\\partial}{\\partial t}|\\Psi(t)\\rangle.", "8eb43af77f8ab9e75699106bc3bf16a9": " \\ N(0,n) = \\{ (x,y) \\in {\\bold R}^2 : x^2 + y^2 < 1/n^2, \\ y > 0\\} \\cup \\{0\\} . ", "8eb44813cbdb346254544ce45e0915ee": "\\lim_{x \\to a} g(x) = b \\Rightarrow \\lim_{x \\to a} f(g(x)) = c", "8eb4937338fb14129de35ac10e86e4e0": "Y_t=X_{t}-m \\quad \\text{ for } t=1,2, \\dots ,n \\,. ", "8eb4d5d0885eaae4c0bd11435f5d2f01": "\\begin{align}\n& \\mathbf{B} = \\begin{pmatrix}\n6 & 0 & 0 \\\\\n0 & 4 & 0 \\\\\n0 & 0 & 5\n\\end{pmatrix}, \\quad \\mathbf{C} = \\begin{pmatrix}\n0 & 2 & 3 \\\\\n1 & 0 & 2 \\\\\n3 & 1 & 0\n\\end{pmatrix}, \\quad (11)\n\\end{align}", "8eb4f4043494d40ca031f09dde558b9a": "I_\\mbox{nonlinear}", "8eb5543b5dea4343bb812e938d4e4687": "S(T) = \\frac{C}{\\exp\\left(\\frac{A}{T^2} + \\frac{B}{2T}\\right)-1}", "8eb56b43e0c00adbe1f0b9584ac0a635": "X^\\mu", "8eb5751ae5f92ad98d3da72ec73cbe06": "u_x\\,\\!", "8eb59ef9fd71089cb5093d96aa756a1b": "\\varepsilon=W/l", "8eb5b69cbce3b6d19a90991a4975b668": " R[\\varphi] =\\int_{x_1}^{x_2} r(x)\\varphi(x)^2 \\, dx.\\,", "8eb5e358fba5adc2e1b99421cb671e8e": "\n \\mathbf{b}_i = \\boldsymbol{F}\\cdot\\mathbf{e}_i\n", "8eb5fd4609492ff750102572c36e7da6": "mS(\\psi) = \\langle mS, \\psi\\rangle = \\langle S, m\\psi\\rangle = S(m\\psi).", "8eb61e1bde49855814c959924e6aeb17": "W_{2\\,p}", "8eb658bed4dad2d7ffc4495dd4eef8dd": "((\\operatorname{trace}_{V}(T))^2)_{j_1 \\dots j_N }^{\\ell_1 \\dots \\ell_N} ", "8eb66e265940cf0990838f6da50e610b": "id_{\\Pi(G)}", "8eb6b33bff14318e491d98bf0ce58b33": "|\\psi_t\\rangle = U(t)|\\psi_0\\rangle", "8eb6e21c4d3bc3fc32c1d6efd6b132f1": "\\widehat{G},", "8eb70086b61a480ecf0a5f52029c9c73": "\\textrm{haversin}(\\theta) := \\frac {\\textrm{versin}(\\theta)} {2} = \\frac{1 - \\cos (\\theta)}{2} \\,", "8eb7030d855bf88f0bbb5730c8b6e673": " a^7 - b^7 = (a - b)(a^6 + a^5 b + a^4 b^2 + a^3 b^3 + a^2 b^4 + a b^5 + b^6).\\,\\!", "8eb70f08602416570f19119898a30934": " I_1=I_2= 2 I_3", "8eb75adbe302e31ca9993d027fe873b8": "\n\\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 1 & 0 \\\\\n\\end{bmatrix}\n", "8eb76bee747597d3b4a69c83ffa5c11b": "Y = (y_1, \\dots, y_m)'", "8eb77df72429735c199f2eb18105ecd5": " \\text{diffusion vector angle between }B_X\\text{ and }B_Z = \\arctan \\frac{B_Z}{B_X} ", "8eb84ae2d6f24bf6c6e73d3d032b0815": "\\phi \\to \\phi ", "8eb8610b9aa699c98a1527c075ea4766": "(S\\downarrow T) \\rightarrow \\mathcal{B}", "8eb8a6a47120b048a26ad2dd69e2427b": " \n \\nabla^2 \\varphi + {{\\partial } \\over \\partial t} \\left ( \\nabla \\cdot \\mathbf{A} \\right ) = - {\\rho \\over \\varepsilon_0} ", "8eb8a819a3027287edf8e30df366149e": "\\operatorname{cl}(X)\\subseteq \\operatorname{cl}(Y)", "8eb8db7ad47cfef3e38a2a63167c9df2": " n \\, / \\, \\bar{n} ", "8eb91baa9a2e573b6c373d4a392496bd": "\n\\begin{align}\n\\mathbf{x}_{0}&=\\underset{\\mathbf{x}\\in \\mathbb{R}^{2\\times 2}}{\\operatorname{argmin}} \\int_{\\mathbf{x'}\\in N}(\\nabla I(\\mathbf{x'})^{\\top}(\\mathbf{x}-\\mathbf{x'}))^{2}d\\mathbf{x'}\\\\\n&=\\underset{\\mathbf{x}\\in \\mathbb{R}^{2\\times 2}}{\\operatorname{argmin}}\\int_{\\mathbf{x'}\\in N}(\\mathbf{x}-\\mathbf{x'})^{\\top}\\nabla I(\\mathbf{x'})\\nabla I(\\mathbf{x'})^{\\top}(\\mathbf{x}-\\mathbf{x'})d\\mathbf{x'}\\\\\n&=\\underset{\\mathbf{x}\\in \\mathbb{R}^{2\\times 2}}{\\operatorname{argmin}}\\, (\\mathbf{x}^{\\top}A\\mathbf{x}-2\\mathbf{x}^{\\top}\\mathbf{b}+c)\n\\end{align}\n", "8eb92f18d7157413d3dda2f836e7e9f2": "\\{...\\}", "8eb941c129cddeaf354b050557092ab8": "dU=T\\,dS-P\\,dV+\\mu\\,dM", "8eb941f3e9d4fc02c268c44a08adfdee": " \\left| r \\right| = -r, \\text{ if } r < 0 .", "8eb96650f5a4ae6593e175d866e8c0c7": "p = {RT\\over{\\underline{V}}}", "8eb9b78c8a31f40c794895c31fb0850e": " \\mathcal{H}f(x) = \\frac{1}{\\pi} \\, \\mathrm{p.v.} \\int_{-\\infty}^\\infty \\frac{f(x-t)}{t} \\, dt = \\left(\\frac{1}{\\pi} \\, \\mathrm{p.v.} \\frac{1}{t} \\ast f\\right)(x)", "8eba224d514ad2ca02232b579a26b301": " \\mathbf{A}(\\mathbf{r},t) = \\frac{1}{4\\pi \\varepsilon_0} \\, \\nabla\\times\\int d^3r' \\int_0^{R/c} d\\tau \\tau { \\mathbf{J}(\\mathbf{r'}, t-\\tau)}\\times { \\mathbf{R}}/R^3 ", "8eba7b11931a37dad456f0136c0c4ee8": "F_n = F_{n-1} + F_{n-2},\\!\\,", "8ebafc63b67764064ed09d724cc916d5": " K = \\alpha \\ln( 1 + \\frac{ N }{ \\alpha } ) ", "8ebb01cf1b3c65617c51fc4f8f318fc8": "n^2 / 4 ", "8ebb5e11150bc32c82d41106e41f7edd": "(U_n)_{n = 0}^N", "8ebc1cd8c804fd533f359b0d9bc7c0d5": "\\hat{f}(\\omega) = \\int_{\\mathbf R^n} f(x) e^{-i\\omega\\cdot x}\\,dx.", "8ebd108acd32beb7c6abf12307c09360": "\\chi_1(z',z'')= z' \\times z'' \\, ;", "8ebd394610085651d370e7a7dcbc7b0c": "\\mathbf{A}^3", "8ebdd99077865ce639a71e8245c213f8": "z_T\\,", "8ebe35b28912fb354421e45dae528758": "s + t \\ge \\lfloor w/2 \\rfloor - 1", "8ebe89dce0bc2058776b3fc1a2906010": "\\{1,2,\\ldots,n\\}", "8ebe9ea1bf176343d6ce3232cab5e97d": "E_{k} \\, = \\, - \\, p_{\\beta} \\, u_{\\text{obs}}^{\\beta} \\, - \\, m \\, c^2 \\, .", "8ebee6c17581b0ca3b8b9563959ae55b": "\\scriptstyle{T^2(q)}", "8ebf11612677e6412771036ad8412be2": " R^2_6(\\rho) = 15\\rho^6 - 20\\rho^4 + 6\\rho^2 \\,", "8ebf2acaeaab9052e37a557b72c07ab3": "P \\vert Q", "8ebf3236eb30049316fbb275e468a1da": "\\gamma_d = \\frac{G_s\\gamma_w}{1+e} = \\frac{\\gamma}{1+w}", "8ebf5e63a05e9f7d1fd59959e1a84e13": "\\log(\\exp X\\exp Y) = X + \\frac{\\text{ad} _X ~ e^{\\operatorname{ad} _X}}{e^{\\operatorname{ad} _X}-1} ~ Y + O(Y^2),", "8ebf8bc97b732dacbfabbcc28d5f9a0a": "\\textstyle \\sin A'", "8ebfb638380811789ff573ddf6aef7ef": " \\theta \\le \\nu \\le \\mu ,", "8ebfbeb0e281133a94c9e2696f188388": "h_a=\\frac{2\\sqrt{s(s-a)(s-b)(s-c)}}{a}.", "8ebfe0a2745b3d35b033c6206fa18ddd": "h f/c", "8ec02b481f6f77af23dc9a59343aba0a": " \\frac{dN(t)}{dt}=rN(t) ", "8ec07fd0ceb9e53d4f1ea6a5c3903bae": "\\displaystyle{\\varphi_{cb}(a)=a^{b-c}}", "8ec0c0ba9d3797e9c444eaf951e9de93": " k_{f_1} ", "8ec0e94c89431481155e565b7304d611": "(1/4) \\cdot w(f) = (1/4) \\cdot \\sum_{e \\in E}f_e(a_{e}f_{e}+b_{e})", "8ec18767ba3c47bcc61280d08cc38e01": "\\scriptstyle\\Delta uv=\\pm 0.05", "8ec1eb711bbeb60cb89fadd393c96aa4": "p_{x_k|y_{1:k}}(x|y_{1:k})", "8ec2c63d125a9f77868cfa23ca03dfb6": "(L + \\lambda) \\textbf{P}_{k-1\\mid k-1}^{a}", "8ec2edaf687516e3e470df2f468ba0cc": "q = \\frac{-1}{2},", "8ec2fe8d83035ceae24b4a6a8cbda97a": "Ff(\\mathbf{x}) = \\int_{\\mathbf{u}\\in C(\\mathbf{x})} f(\\mathbf{u})\\,ds(\\mathbf{u})", "8ec34ce3786c2f1d6ca4a74290d25545": "\\tfrac{1}{X} \\sim \\mbox{Inv-Gamma}(k, \\theta^{-1})\\,", "8ec387847e49796ca83e044d6b7cadfc": "\\psi(\\theta)", "8ec3b6ace7db6bebd3e359cdf7289233": "G(s) = K_c \\frac{(\\tau_i{s}+1)}{\\tau_i{s}} (\\tau_d{s}+1)", "8ec3cb6d97d6a42e9f179b4478e6815c": "\\mu :G \\times G\\rightarrow G", "8ec4020e63e96d9ad5a275ecb6a4f8b9": " \\sigma_i ", "8ec4a106e96b6b1f5c95ea8147100875": "Q = Q_0 + \\Delta Q", "8ec4c57a234147e015054c5f31feefe5": " \\overline u_i \\approx u_i(p_0)+\\left|\\sum_j\\frac{\\partial u(p_0)}{\\partial p_j}\\right|\\Delta p_j ", "8ec4e5b22c9150fbc55799767d78e962": "\\tfrac{{{f}_{T}}}{3}", "8ec5155a71e792f9f264a9254ff7592d": "\\left(\\frac{n+1}{n} \\left(\\frac{1}{2}\\right)^n\\right)/\\left(\\frac{n}{n-1} \\left(\\frac{1}{2}\\right)^{n-1}\\right) = \\frac{n^2-1}{2n^2} = \\frac{1}{2} - \\frac{1}{2n^2}.", "8ec54b57b9d0c9de38b84639fb47f000": "\\vec{X}=\\vec{e}_0", "8ec55e17eaa46ecd4c27c1e683a74f9b": "\n \\boldsymbol{\\varepsilon} = \\frac{1}{2} [\\boldsymbol{\\nabla}\\mathbf{u} + (\\boldsymbol{\\nabla}\\mathbf{u})^T]\n", "8ec5fdb1898c18601fa443bec11b60d7": " D=\\left[ \\mathbf{d}_{1},\\ldots,\\mathbf{d}_{N}\\right] =\\left[ \\mathbf{d}_{i}\\right], \\quad \\mathbf{d}_{i}=\\mathbf{d}+\\mathbf{\\epsilon_{i}}, \\quad \\mathbf{\\epsilon_{i}} =N(0,R), ", "8ec60cb4d7b345ae663ce36262bc31b7": "\\vec R_{cm} = h \\mathbf{\\hat e}^1", "8ec60ee0d87d52376e9dea78533fa4ab": "\\mathcal{F}_\\alpha f (u) = \\int K_\\alpha (u, x) f(x)\\, \\mathrm{d}x", "8ec668fe8744b8836d601d5964f8bc81": " Y\\to \\Sigma \\to X", "8ec66e03586a26c1a7406fd9c3dff76b": "e^{i \\omega}.", "8ec6a2dbfb68e24e6f0edbfc40dd361c": "(K^{\\cdot,\\cdot}, d, \\delta)", "8ec77c94698e73cb14f0935a3dd7d1c7": "\\mathrm{var}(\\hat{\\theta})\n\\geq\n\\frac{1}{I(\\theta)}\n", "8ec7c678c70a14419a05e0b4e773cf98": "(\\sqrt{2}, 1, -1)", "8ec80ee0fd4fd5c756be85e2702f978b": " \\mathbb{R}^8", "8ec827ffec391a090cf888f9e613b2a7": " f_0 = E(Y) ", "8ec83ac801c0fca98176db317a45fe5d": "P(A \\leftarrow B)", "8ec861c1f5c4af9d94ecdda2c127edd0": "D=\\frac{1}{2}\\sum_{j=1}^k \\sum_{s \\in K_j}\\|s-W_j\\|^2", "8ec88045875253c74051c1ec5095e604": "\\scriptstyle \\dot q_i", "8ec896976d3d48b4ac0b6813ade55c3b": "0\\leqslant x \\leqslant 40", "8ec8c20cf23187623b9adeed9721e7c6": "F(x,y,\\ldots) = \\varphi(x,y,\\ldots) + i\\psi(x,y,\\ldots) ", "8ec8d7c8e9362d093cd1e3fce058b06d": "\\begin{matrix} \\frac{25}{21} \\end{matrix}", "8ec8f1234e57f139a068e89eb3b2e5fa": "10^7", "8ec90aa045d37005980236fb53ac38ed": "H = \\plusmn 2n \\kappa_1 \\kappa_2", "8ec9108ebc2946a516e00934af7f7f52": "\\operatorname{cl}(A) \\leq D", "8ec93a4689743b5f63e9149d9913eef0": "x' \\in X^*", "8ec97bd7fd0448abb1a6cf9374670bc8": "(\\log n)(\\log n + 1)/2", "8ec9a2ac045e22174768888c10c7b033": "w^k(0)+y^j\\frac{\\partial w^k}{\\partial y^j}(0)=\\left(v^i(0)+x^j\\frac{\\partial v^i}{\\partial x^j}\\right)\\frac{\\partial y^k}{\\partial x^i}(x)", "8ec9d7b7ca60d9a70c1bd70b6943857f": "h = \\ell\\left(\\cos\\theta-\\cos\\theta_0\\right)", "8ec9ed3c7543e2c6a4d060376450e92a": "\\Xi", "8ec9faa93e4c5af5edb115570ac404a9": "I(\\theta) = I_0 \\,\\operatorname{sinc}^2 \\left( \\frac{d \\pi}{\\lambda} \\sin\\theta \\right)", "8eca064ba37438a6472737504ebedc87": "10 \\times \\log_{10} 1000000 = 10\\times 6 = 60", "8eca614c696a49d85eef2bc8a8ab1feb": "10^{-\\frac{30}{10}} = 10^{-3} = 0.001", "8ecaa905c2dce891988427da29b76aa2": "\\quad\\quad\\quad T(t) = \\sin(\\omega t + \\psi),\\ ", "8ecadfb8140e81f89d145082bee6e2b9": "\\begin{align}\nx&=a\\,\\cos u\\cos v,\\\\\ny&=b\\,\\cos u\\sin v,\\\\\nz&=c\\,\\sin u;\\end{align}\\,\\!", "8ecb34cfec6ca7cce4d49f72b7a5f693": "\\frac{\\mathrm{d}^2 x^\\mu}{\\mathrm{d}s^2} = - \\Gamma^\\mu{}_{\\alpha\\beta}\\frac{\\mathrm{d} x^\\alpha}{\\mathrm{d}s}\\frac{\\mathrm{d} x^\\beta}{\\mathrm{d}s}", "8ecb50d6d9510ad7d8d1724dc2781b2e": "H^\\dagger B^{\\mu\\nu}B_{\\mu\\nu}H/\\Lambda^2", "8ecb5374e0935421089238b648f1c953": " \\frac{d^2 \\sigma}{d\\Omega_{k^\\prime}d(\\hbar \\omega_k^\\prime)}=\\frac{\\omega_k^\\prime}{\\omega_k}\\sum_{|f\\rangle}\\left | \\sum_{|n\\rangle} \\frac{\\langle f | T^\\dagger | n \\rangle \\langle n | T | i \\rangle}{E_i - E_n + \\hbar \\omega_k + i \\frac{\\Gamma_n}{2}}\\right |^2 \\delta (E_i - E_f + \\hbar \\omega_k - \\hbar \\omega_k^\\prime)", "8ecb766875c222ce803d948354dfb39f": "3 \\cdot 2^n", "8ecbc22f175442b4384443418de0be58": "b_2=\\frac{r_1,c_3 - b_1\\times a_2}{a_1}\\mbox{ with remainder }r_2", "8ecbec5a87f05f85a17042193ae9bb3b": "\n F_{11}\\mathbf{e}_1 + F_{21}\\mathbf{e}_2 = \\mathbf{e}_1 \\quad \\implies \\quad F_{11} = 1 ~;~~ F_{21} = 0\n ", "8ecbffcefa04b40bf94fde462ed990f1": "\\begin{align} \\left[ HA\\right] _{i} & =H\\mathbf{x}_{i}-H\\frac{1}{N}\\sum_{j=1}^{N}\\mathbf{x}_{j}\\ & =h\\left( \\mathbf{x}_{i}\\right) -\\frac{1}{N}\\sum_{j=1}^{N}h\\left( \\mathbf{x}_{j}\\right) . \\end{align}", "8ecc110fa98ac3a3f9680089aed3d80b": "\\mathrm{SNR\\ (in \\ dB)} = 10\\log_{10}{S \\over N}. ", "8ecc369d49ae3f6bb02503b3ba1b50f8": "\\pi R^2 P", "8ecc62bfd31a91cb9eee51fd6c407b1b": "K_\\nu=\\frac{1}{2}\\int^1_{-1}\\mu^2 I_\\nu d\\mu = \\frac{a}{3}", "8ecc7519b26ee94612e10488b18ed8cc": "U^C = \\varnothing", "8ecc751ffba15b1a09287b83fffb202b": "Q(1,0)=Q(0,1)=0", "8ecc8c5fb98d07fc3368eb54a8990fe2": "\\mathrm{SU}(6)\\cdot\\mathrm{SU}(2)", "8ecc9af69023dfc2714f5bca377d4cf8": " \\theta, \\sigma, \\nu ", "8ecd56594e8a9f4313717d5915817d2f": "\\aleph_{\\omega}", "8ecd641659f2313688d074cbb41efdef": "\\sum_{n=s}^t f(n) = \\sum_{n=s+p}^{t+p} f(n-p)", "8ecd78ec886121667adb8ed76a7dc90c": "f |_{P \\cap I}", "8ecdc5fa7fc952ccaac32140fd5eba73": "\\Box(\\Box p \\rightarrow p) \\rightarrow \\Box p", "8ecdddcc901c5a86b6cf02d61bb9882e": "\\mathbf{Sv}_i = \\mathbf{T}\\mathbf{T}^T\\mathbf{v}_i = \\lambda_i \\mathbf{v}_i", "8ece284b5c1f66a5f577f67b2dc429c3": "R(u) \\le R(t)", "8ece332105df48109f0a311fcde3dfb0": "e^{\\log\\frac{1}{|x|}}=\\frac{1}{|x|}", "8ece85cb33958049eef81d08bafdd324": "{{\\exp(2z)-1} \\over {\\exp(2z)+1}},", "8ecec0bf17f96bde8a2f5db8502154ed": "x \\rightarrow x", "8ecef9500c4ea9751f3b65bc34a420c9": "(A-B)^6 = A^6-6A^5B+15A^4B^2-20A^3B^3+15A^2B^4-6AB^5+B^6.\\,", "8ecf193af98a4bff67e617f93d2cb2a1": "\\psi_{4,3,1}", "8ecf4669950e06af32145ac71b624f23": "\\left(\\frac{\\partial g(T,P)}{\\partial P}\\right)_{T}=RT\\left(\\frac{\\partial \\ln f}{\\partial P}\\right)_{T}", "8ecfa2fae68e21ebd5ea8cac75fc5896": "(D_1 + D_2)(f) = D_1(f) + D_2(f)", "8ed040d50aa9c92080b8e2f70948a27b": "\\frac{8 \\pi G}{c^4} T^{\\alpha}_{\\beta} = G^{\\alpha}_{\\beta}", "8ed07fb490e26db71cd982715dd7631c": "t= \\sup_{\\theta\\in [0,2\\pi)} h(\\theta; f)\\,", "8ed14f797ca3633ddae915b436127cdd": "T_X \\to T_Y", "8ed1611d3b1fe1d9cae7b7de8b997813": "\\left(\\mu_y\\right)", "8ed18ef6966c4fc1e041707c8589727f": "\n\\partial^\\gamma(uv)=\\sum \\binom\\gamma\\alpha \\partial^\\alpha u,\\partial^{\\gamma-\\alpha}v\n", "8ed205fb203cb1b15a4acec78bbdea56": " S ", "8ed23e7522e76defbb539960b365f9c7": "C_V + nR", "8ed26401e888563af0bec5c6b1d8fb0e": "\\mu_i=\\partial F(T,V,N)/ \\partial N_i", "8ed27d584d6dca333fb9d48db0d716b5": "\\Psi = c_a \\psi_a + c_b \\psi_b", "8ed29fc06ba3656c8d61f98fcb18d812": "\n\\frac{1}{r} = A \\theta + \\varepsilon\n", "8ed2e683fdac8ff413e9e6196d1d3d08": "\nL_n = 3 T_{n-1}= 3{n \\choose 2};~~~L_n = L_{n-1} + 3(n-1), ~L_1 = 0.\n", "8ed2f6e30f091a1e19b60dc0e2644ed9": "Nil_*(R)\\,", "8ed32f051e455da2aab458b3b6ac87b6": "\\Gamma(z+1) = \\sqrt{2\\pi} {\\left( z + g + \\frac{1}{2} \\right)}^{z + \\frac{1}{2} } e^{-\\left(z+g+\\frac{1}{2}\\right)} A_g(z)", "8ed371a4f91be8ed7b9047f0b626a78b": "|y|\\in O(|x|^{k})", "8ed3ce166ebdd3530a047ae98b9a92ec": "b_1=3\\ \\frac{y_1-y_0}{(x_1-x_0)^2}", "8ed3cf4501ba15640a9aad92e2fd84cd": "x \\leftarrow w \\stackrel{*}{\\rightarrow} y", "8ed46e6f10541561bb27d9be1ab3c94a": "x_0\\in X", "8ed4de2045f3435ec3af8b9e7306929d": " k - 1 \\, ", "8ed4e9202729fee1ea98e86a0f41ecf1": "\\mathcal{E}(n)=\\frac{1}{2}\\sum_j e_j^2(n)", "8ed4f26924b51bfc8a3a07bd3d18c755": " \\mathbf{y} = (y_1,y_2,1) ", "8ed4f69e249a5bd635167df0974c9e68": "(x_k, y_k) = d_B Q_A", "8ed585389ea36ed2f1e89440b47da7f8": "3 \\cdot m\\ ", "8ed5fd41e6356c4c5ba83d4c815fc90a": "Q(x, y) = \\frac{1}{\\pi}\\frac{x}{x^2 + y^2}", "8ed607918102315e1d2d83ee666ae120": "v = \\phi_{L/F}(u)", "8ed6201ac5f0c0b640c2e3038cbdcfc4": "(4,5,6)", "8ed6cafa4f361569deec7944616dfea1": "R_\\text{series}={\\pi\\over6}Z_0 \\left({L\\over\\lambda}\\right)^2 \\qquad \\text{ for } L \\ll \\lambda,", "8ed751cc012b34e325638c86b6c611fb": "a\\neq 0, \\frac{9}{4}", "8ed8015c5560102677b0d46d22c2e729": "\\sigma_{A \\lor B}(R)=\\sigma_{A}(R)\\cup\\sigma_{B}(R)", "8ed80c35b90a8b0d433d955207fb6b42": "d^2y/dt^2 = b(t)", "8ed80d7ecae9440b317c408a6f3624f0": "\n|n_1, n_2, ... n_k \\rangle\n\\,", "8ed8328687157e225aff29197caf692a": "\\hat\\mathcal{O}", "8ed83287ad1db7dde4a48f306564342e": "K[[T^{\\mathbb{Q}}]]", "8ed845ee63e24aa29665fe0d1d961adc": "\\scriptstyle x_t", "8ed8a804e7cede127befbe0c34f2cec6": " -\\frac{3617}{510} ", "8ed8d369811eccf8e3db3f828a6836d7": "\\displaystyle\\gamma^\\mu\\gamma^\\nu\\gamma^\\rho\\gamma_\\mu=4\\eta^{\\nu\\rho} I_4", "8ed8ddc1ad7d516bbb188fab9a3afda9": "x \\equiv y", "8ed940ee94cdd331c9ae9a469e5b3a0d": "x\\approx y", "8ed99c851b24c820d950124d3fb0614f": "\n\\left. + \\left[Q_R^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{1*}(\\mathbf{p})\n+ Q_L^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{2*}(\\mathbf{p})\n\\right]e^{-i p x} \\right\\}. \\quad\\quad\\quad (5)\n", "8ed9aaa9b2f88afce09ef050dcce02f1": " \\rho(z)", "8ed9c8886a088fba71d921a510ba6d4e": "\\{l^a,n^a\\}", "8eda4f665705300c5d4b0753b67ce856": "f_1=\\sqrt{\\frac{{(\\frac{{T}}{{m}})}}{{2L}}}", "8eda649f742e6e02bf8f4692d0d1a98c": "7-5\\sqrt{2}=-0.07106\\ldots", "8eda66e640792ec2bb7e52a4431cd8e2": "f = \\mathbb{E}[f] + \\int_{0}^{T} a_{t}^{f} \\, \\mathrm{d} B_{t}", "8eda738a458a8435fc745f6bf95d82b1": " \\diamondsuit ", "8eda7d97d62f8aec2ca0811c4a5ecb0e": "\\begin{align}\n\\rho_A(\\hat{a},\\hat{a}^{\\dagger})&=\\frac{1}{\\pi}\\sum_{j,k} \\int c_{j,k}\\cdot\\hat{a}^j|{\\alpha}\\rangle \\langle {\\alpha}|\\hat{a}^{\\dagger k} \\, d^{2}\\alpha \\\\\n&= \\frac{1}{\\pi} \\sum_{j,k} \\int c_{j,k} \\cdot \\alpha^j|{\\alpha}\\rangle \\langle {\\alpha}|\\alpha^{*k} \\, d^{2}\\alpha \\\\\n&= \\frac{1}{\\pi} \\int \\sum_{j,k} c_{j,k} \\cdot \\alpha^j\\alpha^{*k}|{\\alpha}\\rangle \\langle {\\alpha}| \\, d^{2}\\alpha \\\\\n&= \\frac{1}{\\pi} \\int \\rho_A(\\alpha,\\alpha^*)|{\\alpha}\\rangle \\langle {\\alpha}| \\, d^{2}\\alpha,\\end{align}", "8edb5d4f5b2eecb77785690c57b55611": "\n\\begin{array}{rcl}\nL &=& +\\frac{1}{2\\kappa^2}eR-\\frac12e\\overline{\\psi}_M\\Gamma^{MNP}D_N[\\frac12(\\omega-\\overline{\\omega})]\\psi_P\\\\\n&&+\\frac{1}{48}eF^2_{MNPQ}+\\frac{\\sqrt{2}\\kappa}{384}e(\\overline{\\psi}_M\\Gamma^{MNPQRS}\\psi_S\\\\\n&&+12\\overline{\\psi}^N\\Gamma^{PQ}\\psi^R)(F+\\overline{F})_{NPQR}+\\frac{\\sqrt{2}\\kappa}{3456}\\varepsilon^{M_1\\dots M_{11}}F_{M_1\\dots M_4}F_{M_5\\dots M_8}A_{M_9 M_{10} M_{11}}\n\\end{array}\n", "8edb91fa5a6cbc9a8c459aadbf6afb12": "c>0 \\,", "8edb977b35ed5b01ae026b8207258837": " P_B,1=p_0+g\\rho_1z\\!", "8edbb2b3cc3e13748159ffe5f048cc13": "\\frac{\\partial L}{\\partial q} = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{\\partial L}{\\partial \\dot{q}}", "8edbc6da0d1ed8e3730d3b59b612a021": "C_{1k}", "8edbf9e535d0191fad1673393d5952ae": "Y=1.2219V_J-0.23111V_J^2+0.23951V_J^3-0.021009V_J^4+0.0008404V_J^5", "8edc169da7777b54027cdccfce20471b": "N = a^2 - b^2.", "8edc257c7ad7a7d9da67c1bf8d37bbac": "\\Delta^2 (x_i) = \\Delta(\\Delta x_i) = x_{i+2}-2x_{i+1}+x_{i}", "8edc77721f53636d4935f6c040e2070d": "N_f, N_V", "8edc7bdf1363d6ddffcb512089b2f2f0": "\\mathcal M_X=(Z,g_{\\boldsymbol\\theta})", "8edc7ef656a857874414d9a355fbde35": "\\pi:E\\to X", "8edc8997eb3b42448650c8fa098b5f15": "\n \\eta_{\\mu\\nu} = \\langle \\mathbf{e}_\\mu \\bar{\\mathbf{e}}_\\nu \\rangle_S\n", "8edcb91fa2e8a6276ec96ea97f1aa130": "\\mathbf{x} = \\{x^1,x^2,x^3\\}", "8edce063487ead6feb2c9602b0f1f23c": "\\phi^-(x)", "8edcff96235991ccf9f3b801d08ceead": "S(\\nu) = \\frac{1}{\\pi f_d \\sqrt{1 - \\left(\\frac{\\nu}{f_d}\\right)^2}},", "8edd4db561acdfcd8964a666b4eea509": "\\langle a \\mid a^n = 1\\rangle.\\,\\!", "8edd6a0606a5f1e17c4b1538ad77f746": "{\\mathbb Z}/n{\\mathbb Z}", "8eddc1cc420dd5d2d90032d0409497f3": "\\sum_{i=1}^n \\mathrm{Poisson}(\\lambda_i) \\sim \\mathrm{Poisson}\\left(\\sum_{i=1}^n \\lambda_i\\right) \\qquad \\lambda_i>0 \\,\\!", "8eddd69d2feb2fefceddfa170c72f70d": "\\Re s > 0.", "8eddec690a1655b06f90f001ba9e302a": "x\\in L", "8ede46a02958abc5faf2d7c6e428d691": "s =\\frac {1} {2} (\\sqrt 5 - 1)", "8ede4e8e2689c93ed4ea8b38fd867db4": "{0^2 \\over 2}+g(0)+{P_\\mathrm{atm} \\over \\rho}={v_B^2 \\over 2}+gh_B+{P_B \\over \\rho} ", "8ede5e56c36b853f9f53d49a971df868": "\\begin{align}\nT & = \\left ( \\frac{(1 - \\alpha) L_0}{\\epsilon \\sigma 16 \\pi a^2} \\right )^{\\frac{1}{4}} \\\\\n & = \\left ( \\frac{(1 - 0.0436) (3.827 \\times 10^{26}\\ \\mbox{W})} {0.9 (5.670 \\times 10^{-8}\\ \\mbox{W/m}^2\\mbox{K}^4) 16 \\cdot 3.142 (3.959 \\times 10^{11}\\ \\mbox{m})^2} \\right )^{\\frac{1}{4}} \\\\\n & = 173.7\\ \\mbox{K}\n\\end{align}", "8edeac15bf95818a9934bb0c1409f40d": "\\textstyle\\mathbf{w} = \\sum_i \\alpha_i y_i \\varphi(\\mathbf{x}_i).", "8eded5e61d0cd950e3ef67b58231559c": "\\bar{\\partial}:\\Gamma(\\Omega^{p,q})\\rightarrow\\Gamma(\\Omega^{p,q+1})", "8edee640b790265fce0333788e301e51": "P\\cdot f\\left( q\\right) =\\sum_{n}\\left( f,\\Phi _{n}\\right) \\Phi _{n}\\left(A\\left( q\\right) \\right).", "8edf1e0fed5d25c2a6dd769a37062524": "j = 1,2,\\cdots,m-1\\, ", "8ee0799a3a62ee34160e0a2eebc65715": " \\boldsymbol \\beta^{(s+1)} = \\boldsymbol \\beta^{(s)} - \\left( \\mathbf{J_r} \\right)^{-1} \\mathbf{r}(\\boldsymbol \\beta^{(s)}) ", "8ee07d8ba4244c9ae999bad38933a153": "\ndA = i [G,A] ds\n\\, .", "8ee0c40ca0fb824a7dfd02130ff8066c": "\n(\\nabla^2 + k^2) u = -f \\mbox{ in } \\mathbb R^n \n", "8ee0ef37e91e9abdcde90c4771b5ee25": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n0 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 1\\\\\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 1\n\\end{array}\n\\right] .\n", "8ee14a521fe231d30fdccd12b4ebe085": "\\sigma_\\text{ts}", "8ee21701a9eb5ce7656286eca7fc5e46": "\\Delta:=\\omega_L-\\omega_0", "8ee2c72e48af980f9259263310205c7d": "f=f_\\mathrm{c}\\frac{m_0}{m_0+T/c^2} \\, ,", "8ee328fe5af3931569ee40ae42b9cf52": " \\frac{1} {1-RL} = \\frac{-(RL)^{-1}} {1-(RL)^{-1}} = -\\frac{1} {R} L^{-1}- \\left(\\frac{1} {R} \\right)^2 L^{-2} - \\left(\\frac{1} {R} \\right) ^3 L^{-3} - . . . ", "8ee34f862018634ec7d565553907ae18": " \\Chi_w(k, n)= 2^{n/2}\\int_{-\\infty}^{\\infty}x(t)\\psi(2^nt-k)\\, dt", "8ee375dc19a80f55d023f486a911dcb7": "\\rm H_2O+CO_2 \\leftrightarrow H_2CO_3 \\leftrightarrow H^++HCO_3^-", "8ee3b4e1e08740e694f21c1fe988a105": " dE = c \\times MAXQ ", "8ee3d0c45d0a7dbc4d9a081db8bb7d1b": "\\mathrm{T}_1^\\infty=\\bigcap_{k=1}^\\infty\\mathrm{T}_1^k", "8ee3de171d3ccc7da3a622ab344e7f6d": "\\operatorname{cn}^2(u,k) + \\operatorname{sn}^2(u,k) = 1,\\,", "8ee47f338083083b6bd25bfbefca3799": " z = \\int X(x) e^{ax}\\, dx \\quad\\text{ and }\\quad z = \\int X(x) x^A \\, dx", "8ee48619c38d84f1c18f34c494ba99df": "A = \\sum \\alpha_k", "8ee493de6bc615f645c3dcbae117e8b8": "M(x) = \\int V(x)\\, dx", "8ee4a66eaaf52dd76844805a78cb805a": " \\mathbf{C} ", "8ee4bfa783d369fbb7bd14e26da3ab57": "0=\\det \\begin{bmatrix}\\dot{\\mathbf{x}}, &\\ddot{\\mathbf{x}}, &\\dots, &{\\mathbf{x}}^{(n)} \\end{bmatrix}\\dot{}\\, = \\det \\begin{bmatrix}\\dot{\\mathbf{x}}, &\\ddot{\\mathbf{x}}, &\\dots, &{\\mathbf{x}}^{(n+1)} \\end{bmatrix}", "8ee4f4714930df493afa4e4025c654ee": "\\frac {E(S_{t+k})} {S_t} - 1 = \\frac {(i_$ - i_c)} {(1 + i_c)} = E(e)", "8ee50721a2ff104f373f39c341d4b727": "\\displaystyle \\frac{1}{|ab|} e^{\\frac{-\\left(\\nu_x^2/a^2 + \\nu_y^2/b^2\\right)}{4\\pi}}", "8ee5965eb7738926e967edfa8b8df41a": " \\mathbb C^2", "8ee5e2a0a1376326f8e95b5238749423": "K=r(r+\\sqrt{4R^2+r^2}).", "8ee5f79716c172a3eb4c71a5dd442cfd": "\\frac{\\frac{L_\\beta}{A}\\frac{N_p}{C}-\\frac{L_p}{A}\\frac{N_\\beta}{C}}\n{\\frac{N_p}{C}\\frac{E}{A}-\\frac{L_p}{A}}", "8ee668c1012dc569526903b2f183772d": "\\int\\frac{x\\;\\mathrm{d}x}{1+\\cos ax} = \\frac{x}{a}\\tan\\frac{ax}{2} + \\frac{2}{a^2}\\ln\\left|\\cos\\frac{ax}{2}\\right|+C", "8ee68c6da0f72202b45bbd99694095b4": " \\{A,H\\} = {d\\over dt}A~, ", "8ee70e0e3afd6586fdbfb47a2ab7cc4b": "g(\\vec r)\\sim |\\vec r|^{-2(d-d_\\text{f})}\\,\\!", "8ee7183c598ac81ebc315528d1440b35": "F(r) =k \\frac {m_1 m_2}{r^2} exp(-\\alpha \\cdot r)", "8ee7a90320316c24485f579366ad6bc0": "\\frac{dy}{dt} = - y(\\gamma - \\delta x)", "8ee7acddfca34b30f5d2995a91678920": "\\displaystyle{Q(a)=2Q(a,1)^2 - Q(a^2,1)= 2L(a)^2 - L(a^2).}", "8ee7ad58db6793c9b2739f35c2c6ce95": "H(s) = \\frac{1}{(s+a)(s+b)} = \\frac{1}{s+a} \\cdot \\frac{1}{s+b} ", "8ee7d4ccf35e5c5ffcac38fd8fae9ac8": "[-\\psi]", "8ee7f0bb10b084cf82456a6416daf344": " \\phi_i \\leftarrow \\frac 1 {a_{ii}} (b_i - \\sigma)", "8ee8589ee4e774c56002d4e847ec21fe": "f(x)=\\sum_{k\\in\\mathbb{Z}^n} e^{2\\pi ix\\cdot k} \\, \\hat f(k).", "8ee8b1324ee2e8f5d964850522707852": "a_{n+1} = \\frac{a_{n} + \\left(\\sqrt{k}(l+\\frac{1}{2}+n)-E_{l}\\right)a_{n-1}}{(n+1)(2l+2+n)}. ", "8ee8dbd4a27091b93b2d73b76ffe6d35": "F(M') \\rightarrowtail F(M) \\twoheadrightarrow F(M'')", "8ee8f8816e3ee6a4fa3997239a1ac087": "{{n+1} \\choose {k+1}}", "8ee92cf7a4a4bc52415f11b015b35613": " \\frac{\\partial \\bar{u_i}}{\\partial t} + \\overline{\\frac{\\partial u_iu_j}{\\partial x_j}}\n= - \\frac{1}{\\rho} \\frac{\\partial \\bar{p}}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 \\bar{u_i}}{\\partial x_j \\partial x_j}.\n", "8ee93e08328479e3098873f1e27a10bd": "Z_0 = 276 \\ln \\left(D/d + \\sqrt{(D/d)^2-1} \\right) = (120/{\\sqrt{\\epsilon_r}}) \\cosh^{-1} (D/d)", "8ee940c3ef9e5e4593a15df266afd6fc": "\n\\ell = \\ln L^* = \\sum_{i = 1}^n \\left[ Y_i v(x_\\text{auto} ) - \\ln \\left( 1 + e^{v(x_\\text{auto} )} \\right) \\right]\n", "8ee945bd8f15ffb1434b2a28e8177510": "(A\\oplus B)\\oplus C = A\\oplus (B\\oplus C)", "8ee952fbd38c74cf80f2a6f8c9c22743": "\\alpha,\\beta\\in E", "8ee9ce1b5d1fd879e2eac7eb254f5dd9": "\n\\cfrac{\n A \\vee B \\hbox{ true}\n \\quad\n \\begin{matrix}\n \\cfrac{}{A \\ true} u \\\\\n \\vdots \\\\\n C \\ true\n \\end{matrix}\n \\quad\n \\begin{matrix}\n \\cfrac{}{B \\ true} w \\\\\n \\vdots \\\\\n C \\ true\n \\end{matrix}\n}{C \\ true} \\vee_{E^{u,w}}\n", "8eea135e03d9ff9037bdb5661459a238": "\\alpha<\\omega_1,n<\\omega", "8eea2812b383f2906531d7a653975b3b": "E^0_q", "8eea98dca6d52e82a1c7badcea372ff0": "EP = \\frac{YA + p - 1}{2p - 1}", "8eeaaf920758db4970f768e1a84736f6": "g(\\xi) = \\frac{1-\\xi^2}{(1-\\xi)^3} = 1 + 2 \\xi + 5 \\xi^2 + 7 \\xi^3 + \\dots ", "8eeb16dc70032c664139d11ee820a6ab": "b_i : A \\longrightarrow R_+", "8eeb42704e5933b36d90865c12d713ed": "\\begin{Bmatrix} q , p \\\\ r , s \\end{Bmatrix}", "8eeb60cd07f4186495fd56f5ac16a07d": "\\psi\\phi", "8eeb6eb5dbd597c1b6f80bee36bca698": "-4/3", "8eebbbd25bf42000e5e2f0aebdccc444": "\\Delta G^{TS-D}_{W}", "8eebf3416b84954ed718f93ea2997bf4": "C_V = 3Nk\\left({\\varepsilon\\over 2 k T}\\right)^2 {1\\over \\sinh^2\\left({\\varepsilon\\over 2kT}\\right)}.", "8eec01f2a9930f83b22b72f3af8c2e46": " f = \\frac{sm_{fu} - m_{ox} + m_{ox, 0}}{sm_{fu, 1} + m_{ox ,0}} ", "8eec3a50bf8216c8fcebfcbd8cdfc815": "f = 1", "8eec78781b8a5a432c5c6c511ab6d2a1": "- \\log_b a = \\log_b \\left({1 \\over a}\\right) = \\log_{1 \\over b} a", "8eed0934fca41a9e157fadf1d7acad70": "w(x)dx", "8eed75403a5de81547da5c1fe5cebdff": "P \\left( {\\frac{V}{R}} > q \\right) ", "8eedee906d2e14c11fb53cc8937cafd6": " \\frown \\colon H_\\bullet(X) \\times H^\\bullet(X) \\to H_\\bullet(X)", "8eee5f2f535d03d6127c1316a16525f5": "F_0,F_1,F_2,F_3", "8eeecc43706ee992b46dde239851cbe5": "e^{\\hat{M}}_N", "8eef3575364608f088fd6c3caea2a20c": " F \\equiv |\\vec F| = |\\vec J + \\vec I|", "8eefaad78070091b39e8011ee520593b": "\\mathrm{col}_n(L),", "8eefab561ad54b4f56011054c893f021": "H(z) = \\mathcal{Z}\\{h[n]\\} = \\sum_{n=-\\infty}^\\infty h[n] z^{-n}", "8eefba2a60cb53cfe4f32e6c76eb2959": "\\beta_{2}=\\frac{3(25n^4-13n^3-73n^2+37n+72)}{25n(n+1)^2(n-1)}", "8eefc3fc0c64a659575cbcffa0438bfc": "\\omega^i", "8eeffd5e839b3c003bfab3b615b493e0": " \\,Q_0 = \\{(s_0,ta(s_0),0)\\}", "8ef01d869e6a8d8bf1677f9151e2a06a": "\n\\int \\exp\\left[ i \\int d^4x \\left ( \\frac 1 2 \\varphi \\hat A \\varphi + J \\varphi \\right) \\right ] D\\varphi \\; \\propto \\;\n\\exp \\left( { i\\over 2} \\int d^4x \\; d^4y J\\left ( x \\right ) D\\left ( x - y \\right ) J\\left( y \\right ) \\right)\n", "8ef081e877ca0ef57f7461c0759713de": "\\mathrm{ATIME}(g(n))\\subseteq {\\rm DSPACE}(g(n))", "8ef08f1ea7b97bf084039583fd2cd073": "1+\\sum_{n=1}^\\infty m_n t^n/n!=\\exp\\left(\\sum_{n=1}^\\infty\\kappa_n t^n/n!\\right) ,", "8ef0e6f735615727a0fdfc489dce23dd": "r = \\left( 1 + { i \\over n } \\right)^n - 1 ", "8ef0e798b206cf67d73b333c918d82be": "x^2 - 20 x + 100 = 81 x", "8ef168a7d7a45b891141796928318f76": "\\hat\\theta(\\theta)", "8ef170e4e7539cd1a7e9231f92443a30": "\\bar{I_L}", "8ef195d539dd809a69509c6fe04fb528": "i^*TM", "8ef1c0251cfbcae831b1b153998df789": "\\begin{matrix}{4 \\choose 1}^3{52 - 4r \\choose 1}\\end{matrix}", "8ef1cf523127b6067a5e40ab681e3a6e": "-GM/R^2,", "8ef1f10f2b06b84caf03af900dfb7ee9": "m_v", "8ef2134f8abe3d7d100cd052cef2c36b": "x_{i+1} = (x_1 + \\cdots + x_{i+1}) - (x_1 + \\cdots + x_i) = (i+1)CA_{i+1} - iCA_i\\,.", "8ef213d32f667845ba7775d7be5c8045": "\\frac{3x + 5}{(1-2x)^2} = \\frac{A}{(1-2x)^2} + \\frac{B}{1-2x}", "8ef219bbe6bbb9991b5172d9258b97ad": "a,b \\in V", "8ef2ab8e4a6b95c21e47d374834b5b78": "2^{22}", "8ef32051e8a1d303894c926cd03f7875": "s_{X_1}^2", "8ef3429b699688bebb8ec3d404d844ad": "f_{transit} =0.45\\frac{v_h}{L}", "8ef349fb889c8182b70e67e697a6d713": "\\bigcup_\\theta G_\\theta", "8ef39bc7e767e8bd9c947e4eaefb85aa": "\\begin{align} 2\\cdot R_*\n & = \\frac{(147\\cdot 2.44\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 77.2\\cdot R_{\\bigodot}\n\\end{align}", "8ef4735e815b609f83f3fd83e732b4d6": "\\Lambda=\\Lambda_0", "8ef48e01b2b71804ebbaa15302eca60c": "\\scriptstyle\\mathcal{H}", "8ef493de4f1ffa145f0cef3fa690a851": "\n {{n+1} \\choose {k+1}}_q = \\prod_{i=0}^k \\frac{q^{n+1-i}-1}{q^{i+1}-1},\n", "8ef530bec0f1c17552689c98f034cc90": "\n _{(x)}\\Gamma_{ij}^k = 0\n", "8ef53a9639cc4b661dda6dc58a5ed2f8": "\\langle \\psi^*\\psi\\rangle=\\eta\\langle \\mathcal{T}_\\tau \\psi(\\tau=0^-) \\psi^*(0)\\rangle\n=-\\eta G_\\eta(\\tau=0^-)=-\\frac{\\eta}{\\beta}\\sum_{i\\omega}G(i\\omega)e^{i\\omega 0^+}", "8ef5a0c8a583d4ea0f4f0ff52fc60723": "h_j : \\,\\!\\mathbb{R}^n \\rightarrow \\mathbb{R}", "8ef5f32848cec5b3b08132613ae3515f": "\\dim(U+W) = \\dim(U) + \\dim(W) - \\dim(U \\cap W).", "8ef60214b88a434b2b5dc4d6fb6924d5": "V = \\frac{n}{12}hs^2 \\cot\\frac{\\pi}{n}.", "8ef60bb2409fbef18f9b827b4ebf1d1d": "\\delta_1=(Tp)(0)=|a_0|^2-|a_n|^2.", "8ef626e74eaa1578aef2b2858e670bda": "\\omega= \\operatorname{arg}(\\ y_3\\ ,\\ x_3\\ )", "8ef637c675bfb1885564f5ccd1b7acdd": "\\frac{d[A]}{dt} = k_1[ABCD]", "8ef6441ebc1571e7477e2f51e85c3192": " \\textstyle \\prod_{i=1}^r (\\frac {p_{i} - 3}{p_{i} - 1})^{1/2} ", "8ef655bc6440112be198603cf2a020b6": "\\nabla \\gamma_{12} = -F_{12} - \n\\tan{\\gamma}_{23} (- F_{13} \\cos\\gamma_{12} + F_{23}\\sin \\gamma_{12})", "8ef668b1b24a1d087b554f909decbc13": "\\frac{\\mathrm{e}^{tb}-\\mathrm{e}^{ta}}{t(b-a)}", "8ef6f0eb753ffd6611e7bee8d715e06a": "f(\\varepsilon_{\\rm{p}})", "8ef6fb69c259498b6bae1820e54f2da9": "\\omega\\colon TP\\to \\mathfrak h", "8ef70083c4780d0e9eee34e51024aae9": "X(u,v)", "8ef741c79f71e3fa00f0df60e2595495": "\n \\bold A_x=\\frac{\\partial \\bold f_x(\\bold s)}{\\partial \\bold s}, \\qquad\n \\bold A_y=\\frac{\\partial \\bold f_y(\\bold s)}{\\partial \\bold s} \\qquad \\text{and} \\qquad\n \\bold A_z=\\frac{\\partial \\bold f_z(\\bold s)}{\\partial \\bold s}.\n", "8ef7be2ad7c731d817583fb7e1d55e84": "\\sigma_y = 0", "8ef7c2ccde0e2f42933cbf729da5cdfe": "\\sin(i)=(1/j)(2^{-1}\\bmod{p})\\cdot(\\epsilon^{i} - \\epsilon^{-i}),", "8ef8e63e2cb8a209af7bcdfe11f47b6b": " A' = \\iint \\limits_\\mathrm{top} d\\mathbf{A} \\cdot \\mathbf{\\hat{r}}, ", "8ef911d0c35f0a739c0c4bac6041cd78": "Z=(z_{ij});\\quad z_{ij}\\leq 0, \\quad i\\neq j.", "8ef93f402e460b995448549ac962834d": "S -A = R (1 + \\mathrm{APR}/100)^{-t_N} + \\sum_{k=1}^N A_k (1 + \\mathrm{APR}/100)^{-t_k}", "8ef942e2a2d510393c4df07c2a58fe45": "\\vartheta_{11} (z; \\tau) = e^{iz + i \\pi \\tau / 4} \n\\int_{i - \\infty}^{i + \\infty} {e^{i \\pi \\tau u^2} \n\\cos (2 u z + \\pi \\tau u) \\over \\sin (\\pi u)} du", "8ef9b5c99c58572ee1896f7b4a9e9d86": "\\mathbf{A} = \\mathbf{Q}_{kl} \\mathbf{Q}_{mn} \\mathbf{Q}_{pq}", "8ef9b83cb2186bc1e2765be53b299f47": "\nX^{\\{6\\}}=]-\\infty ,\\infty [.\n", "8efa1744d47ff8cb2706843fc00d34ab": "y_{x}^{e}=\\,", "8efa53faac1d4364cff3846ddd709f87": "c^2 = (b+d)^2 + h^2,\\,", "8efa58ddf0d24c1d11e4a6f1bde546b6": "\\geq(u^*) = \\{x \\in \\textbf R^L_+ : u(x) \\geq u^*\\}", "8efa7f89c88c838ad44e9bde5a906fb7": "Q_j = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial T}{\\partial \\dot{q}_j} \\right ) - \\frac {\\partial T}{\\partial q_j}", "8efadf73291afef2d42a5925d6756dfb": "\\zeta:", "8efb4532fa97616c6cee8138bedc955c": "D^2+E^2>4(A+C)F", "8efb8b8ecb97b586c935e2e1a6b5bd02": "\\frac{p_t}{p} = \\left(\\frac{T_t}{T}\\right)^{\\frac{\\gamma}{\\gamma-1}}\\,", "8efb8c5cb742fe75c5ad035c86617d64": "s^*=\\frac{s}{L_c}", "8efb93386fedb181fa420dcd8a2195ef": "\\{English(Fred) \\vee Irish(Fred), \\neg English(Fred), \\neg Irish(Fred)\\}", "8efbc35bbfe88264973589aa237c9dd1": "P(x)\\in\\mathbb{Z}[x]", "8efbcac7194f6e30ad186032e2b3847c": "T(n) = n + 10 n\\log_2 n", "8efbf86c65155d1f6ebba5031ae54e88": "\\sqrt[3]{x} = \\sqrt[3]{r}\\exp ( \\tfrac13 i\\theta ).", "8efc0452ac3bea3ee8a8a71a35b4580e": "\\mathcal{H}_g", "8efc172f2827250ce315088cd5e1bf41": " Cxy \\rightarrow \\exist z[SCz \\and Ozx \\and (Pwz \\rightarrow (Owx \\or Owy)).", "8efc42a0342b978ca714e9efa9334170": "e_{\\mu}^a", "8efc8b1060436a5ddf225dc8ceb6a09b": "\\alpha_e", "8efc8dfbceb9d7ced385143e138dcedd": "\\frac{ \\partial Y}{ \\partial L} = \\frac{ (1 - {\\alpha})Y }{[L(t)]} \\text{ and } \\frac{ \\partial Y}{ \\partial A} = \\frac{ (1 - {\\alpha})Y }{[A(t)]} ", "8efca960b209402104b448a5ad9486a8": "\\lnot ", "8efcb298dd1d2cea6ad19b8eabee1191": "x \\equiv \\frac{y-1}{n} \\pmod{n}", "8efcc33b187482bcd54992ada4bd74c5": "\\mathbb{O}", "8efcddc849dc24ac030b70ae523a7f17": "X_{hG} = X\\times_G EG.", "8efd21c4fef4431baf69cb0da26e5548": "\\{ v, T(v), T^2(v), \\ldots, T^r(v), \\ldots\\}", "8efd6c427e52df779baf4f5ace0d4554": " \\bar V_t = r_{t} + \\gamma \\sum_{i=0}^{\\infty} \\gamma^{i} r_{t+i+1} ", "8efd9badebcbf0a92176b62a2c4ce602": "v_F", "8efdd82ecb9fc5198ce56bb951f1cae7": " \\frac{\\partial V}{\\partial t} + \\frac{1}{2}\\sigma^2 S^2\\frac{\\partial^2 V}{\\partial S^2} + rS\\frac{\\partial V}{\\partial S} - rV = 0 ", "8efdeae602b098799bead78603f7fabd": " \\sum_k \\xi_k H_2 \\left(\\frac{1 + \\sqrt{(1- 2 \\,\\eta\\,p_k)^2+4 \\,\\eta\\, |\\gamma_k|^2}}{2} \\right) \\geqslant H_2 \\left(\\frac{1 + \\sqrt{1- 4 \\,\\eta\\,(1-\\eta) (\\sum_k \\xi_k p_k)^2}}{2} \\right) ", "8efe51d7882c158612d466601b904e62": "t = -\\frac{2z}{(1-z)^2},", "8efe5df8e33523aeb34109dda47e0f3c": "P_{i}(x)", "8efe889996b08868dd95d062ebd16b02": "a=bq+r", "8efed306dee65458b7e192fd67166878": "a \\equiv a^{\\lambda(n)+1} \\pmod n ", "8efed31f853ba565b69ed2594895dad3": "y(t) = \\sin(t),\\,", "8efee8cf21a4c8445da300c990cd6cf1": " E_{0}=|e|V_{0}", "8efefbdd90e2f9ed31c755ee56ddcc32": "\\begin{align}\n\\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial \\bar{x}^{\\nu} \\partial x^{\\beta}} \\, + \\, \\frac{\\partial \\bar{x}^{\\rho}}{\\partial x^{\\sigma}} \\frac{\\partial^2 x^{\\sigma}}{\\partial x^{\\beta} \\partial \\bar{x}^{\\rho}} & = \\, \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\nu}} \n\\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial x^{\\sigma} \\partial x^{\\beta}} \\, + \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\sigma}} \\frac{\\partial^2 x^{\\sigma}}{\\partial x^{\\beta} \\partial \\bar{x}^{\\nu}} \\\\\n\n& = \\, \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\nu}} \n\\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial x^{\\beta} \\partial x^{\\sigma}} \\, + \\, \\frac{\\partial^2 x^{\\sigma}}{\\partial x^{\\beta} \\partial \\bar{x}^{\\nu}} \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\sigma}} \\, = \\, \\frac{\\partial}{\\partial x^{\\beta}} \\left( \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\nu}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\sigma}} \\right) \\\\\n\n& = \\, \\frac{\\partial}{\\partial x^{\\beta}} \\left( \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial \\bar{x}^{\\nu}} \\right) \\, = \\, \\frac{\\partial}{\\partial x^{\\beta}} \\left( \\mathbf{4} \\right) \\, = \\, 0 \\,.\n\\end{align}", "8efefec3778f793cc117f028bdfc4805": "\\exists x(x^2=y)", "8eff65013a62e583ce248fb92ae74ccb": "P(A \\cap B) \\ = \\ P(A) P(B)", "8eff6a08a02003f9d698073c45256a34": "v,u", "8eff71426c5c8adeefdf734b8ab7cb0a": "Df = - f^{\\prime\\prime} + qf.", "8effff999de692c242b9f7a539c63e58": "\\cos", "8f0002d95c87352ef447cd17247a612e": "\\begin{array}{rl}\nVar(\\epsilon(x_0)) &= Var\\left(\\begin{bmatrix}W^T&-1\\end{bmatrix} \\cdot\n\\begin{bmatrix}Z(x_i)&\\cdots&Z(x_N)&Z(x_0)\\end{bmatrix}^T\\right) =\\\\\n&\\overset{*}{=} \\begin{bmatrix}W^T&-1\\end{bmatrix} \\cdot\nVar\\left(\\begin{bmatrix}Z(x_i)&\\cdots&Z(x_N)&Z(x_0)\\end{bmatrix}^T\\right) \\cdot\n\\begin{bmatrix}W\\\\-1\\end{bmatrix}\n\\end{array}", "8f001beda345e3925abe3e0606065900": "1 \\leq i \\leq n, 1 \\leq j \\leq s", "8f005cd858b7467134f298a2186d1df9": "m \\left \\{x:\\, x\\notin \\cup J_m^*, |Tb(x)| \\ge \\lambda \\right \\} \\le \\lambda^{-1} A \\|b\\|_1 \\le 2 A\\lambda^{-1}\\|f\\|_1.", "8f006a6ac8de764c4211f694daff6a85": "Q = \\frac{X_C}{R_C}=\\frac{1}{\\omega C R_C}", "8f007f58cd0106730e59a7e9b02d0336": "\\mathrm{d}F = 0, \\quad \\mathrm{d}{*F} = 0.", "8f00a4a954aab309dc6dfe630b030fbf": "\n\n\\sum_{i = 0} ^n (-1)^i \\; \\mbox{rank} \\; H_i (X_n, \\empty)\n\n= \\sum_{j = 0} ^n \\; \\sum_{i = 0} ^j (-1)^i \\; \\mbox{rank} \\; H_i (X_j, X_{j-1})\n\n= \\sum_{j = 0} ^n (-1)^j c_j.", "8f012428cdea8581be8ef8247c594d1f": "F_\\nu \\propto \\begin{cases} {\\nu^{2}}, & \\nu<\\nu_a \\\\\n {\\nu^{1/3}}, & \\nu_a<\\nu<\\nu_c \\\\\n {\\nu^{-1/2}}, & \\nu_c<\\nu<\\nu_m \\\\\n {\\nu^{-p/2}}, & \\nu_m<\\nu\n\\end{cases}", "8f0155e456c92b38ffa9bdb7c36afef8": "(\\nabla \\mathbf{A})_{ij} \\equiv (\\nabla \\otimes \\mathbf{A})_{ij} = \\nabla_i A_j ", "8f0174a52dfd3a248c4237ba6744e49a": "\\pi_1(X,A)", "8f017a125286a5a328f8a67e0ce3b2e5": "\\binom nk=\\frac{n(n-1)\\cdots(n-k+1)}{k(k-1)\\cdots1}.", "8f01c0c8ba93dbc4d5a0ff59a9916fc0": "h_i(x) \\leq 0", "8f01d8005d259b4d0e2b5ae8a660dc25": "\\boldsymbol\\beta^k.", "8f034b447804c8ee49ccf9cde3bc38bd": "\\frac {dx(t)}{dt}=Ax(t)+Bu(t)+Ew(t) ", "8f037617a320d235b67ae1d79a6a519b": "x^2y'' + xy' - \\left (x^2 + \\tfrac{1}{9} \\right )y = 0.", "8f038d1c38aaf3491a7cf18724e8a215": " h(x)=a_1x+a_3 ", "8f0390378410b68fef9d5b0b2953e726": "y_{4}=3-1", "8f03ac2b1ac9b377d8603026d5312d76": "1x^2", "8f03ad02c6cad600d2c00a972a9bcce6": "020", "8f03c5810ca4c9b1e92866f28b7e285e": "\\left( c - c_o \\right) = A\\, \\cos\\, \\beta x", "8f04032bbe0e71f8f6fdb2e1ae9e16b0": " F_p\\, dp + F_q \\,dq =0, \\,", "8f04d70944bff5f3afffe01f210d0eb7": "c_0^p\\equiv (a_0+b_0)^p \\mod p^2", "8f05399c1f599e71094dcbcb6b35fce2": "1:\\sqrt{2} \\approx 0.707", "8f055f99e9cda4c3c3a0088f0ac12451": "h \\sim \\frac{R}{Q \\beta}", "8f05e9bd2879428e9a6818799823dec5": "w(\\hat{t}) = w(V.\\hat{n})", "8f06093855e99d195be477cc200cf3ed": "R_\\mathrm{E}/hc", "8f06114e773e9ba9598ce05c40b0ffa4": " B(r)=\\frac{D_\\odot}{r^3} ", "8f0687266beb629ac28a073285ef8819": "E=E^0 - \\frac{2.303RT}{F} \\text{pH}", "8f06fe2d9207849ed48858bdecec4d95": "\\bigvee_{j=1}^m \\bigwedge_{i=1}^n L_{ij},", "8f0724932fe1ab1d08d64f0ce43d42a5": "a<\\infty", "8f07540db6b6d334682fd832912148e6": "\\langle\\mathbf{p}\\rangle = q \\tau \\mathbf{E}.", "8f075611fca774c450a5cb29414a1457": "\\sigma \\in \\Delta, a \\in A_i", "8f086116cc85e3be5ea96ebc471ffdc7": " \\text{(4)} \\qquad W_{net} = W_{1\\to 2} + W_{3\\to 4} ", "8f08e83b5f964a921990a8dd5c4257ee": " \\hom(A\\times B, C) \\cong \\hom(A, C^B) .", "8f0950b77f165ea6692c3067e5115ede": "f_t=f_c+(f_0 - f_c)e^{-kt} ", "8f09a1f122b7dbc41e81754de514c41d": " P_n(x) = k_n x^n + k'_n x^{n-1} + \\cdots + k^{(n)} ", "8f09baae0fe9725b4a9c84ca1a923b06": "\\varepsilon_3", "8f09df9a0460ded841016faa062e7c5c": "V_m||W_m", "8f0a11093c886da8ba7a1e0c28266b98": "P \\subset \\mathbb{R}^n", "8f0a1cbe13b623938086dbcbe1cf3bf5": "Q = \\begin{pmatrix} \\begin{pmatrix} \nq_{1,1,1} \\\\ \\vdots \\\\ q_{1,1,i} \n\\end{pmatrix} & \\cdots & \\begin{pmatrix}\nq_{1,n,1} \\\\ \\vdots \\\\ q_{1,n,i} \n\\end{pmatrix} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\begin{pmatrix} \nq_{j,1,1} \\\\ \\vdots \\\\ q_{j,1,i} \n\\end{pmatrix} & \\cdots & \\begin{pmatrix}\nq_{j,n,1} \\\\ \\vdots \\\\ q_{j,n,i} \n\\end{pmatrix} \\end{pmatrix}\n", "8f0a5014fb776dd05fd360ae38d0223f": "I = \\tfrac{1}{2}m r^2", "8f0addfd624d3eb5eed243de4fc68223": "\\mathbf{H} = \\mathbf{B} / \\mu_0 - \\lambda^\\prime \\mathbf{M}", "8f0b4ac42b8c9109c0fbfa5ce6a60550": "m(m-1) = l(l+1)\\,, ", "8f0b65e035fba8e9a9a23d5567ce2c22": "|0\\rangle\\pm e^{i\\theta}|1\\rangle", "8f0b91de02d6ab1c1063c0b959e2c5ef": "O(\\log w(x))", "8f0ba5a34f069ff9343941998c28336b": "\\scriptstyle2\\hat U", "8f0be7cc0312d0e7f2a2af5b0e523033": "{}_RQ_P", "8f0c3f0f2657d5bbc3d88c7ccc14f38e": " \\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p ", "8f0c4460458214848a55745c9f3fdd87": "h(x) = y \\mapsto f(x,y)", "8f0c48c2bf4079ce6afd6ddb6d6f98b6": "a \\equiv i\\cdot m \\cdot (x-y)^{-1} \\pmod{p}", "8f0c4fa26a69778fe30b6ba854d85e2c": "M_f = (([0,1]\\times X) \\amalg Y)\\,/\\,\\sim", "8f0c5ed644d78fb7ff7b4aac460027bb": "\\mathbf{S}=1/\\mu_0 \\mathbf{E}\\times\\mathbf{B}\\,,", "8f0c75feac1b08c3318178512e1424b7": "\\pi_{f_\\Lambda}^{-1} = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix},", "8f0c7f2808fec1bb0e8b803cf5c8fca4": "A^{k+1} A^D=A^k, \\quad A^D A A^D=A^D,\\quad A A^D= A^D A.", "8f0c947fe56f971d4d3063432b636dcf": "K_{1,n-1}", "8f0cec1f5e4d26b84dd0a989c6e46dca": " 0.007241 \\times W^{0.425} \\times H^{0.725} ", "8f0d2ff15463fd630d2f8ab094673b47": " \\frac{1 + 0.02}{1 + 0.10} - 1 = -7.27\\%", "8f0d49531cb1d1a07813b725bc96b4e1": "K_e(2,p)", "8f0d5352a737b40da61974130980fbc8": "[H]^2 + K_a[H] - K_a C_a = 0", "8f0e448671d407993f96e3c816373e2f": "(N + 1/3)p + 1/3\\,", "8f0e55ddbd577b7da0c270d9720c69b0": "d\\epsilon_{i,j}^e", "8f0e943baf6d610282d144039eed3f2b": "A_{430} = A_{530} \\times 10^{H_{L}/10}", "8f0ee24a9fa1e60c02aed796ec34686f": "\\mathcal{D}_a", "8f0eedcd53ad5d7be57b43f50621bba6": "\\underline{x}(t)=x_c(t)+j x_s(t).", "8f0efa587debfe9e071cd200212a5335": " P = diag(A).", "8f0f0f6b6f9b19adf178848829ff0f33": " {z g^\\prime(z)\\over g(z) -w}=\\sum_{n\\ge 0} \\Phi_n(w) z^{-n}.", "8f0f808d5b8cb4bc42f4fb1cf2637831": "y = r \\cos b \\sin l", "8f0f87f397ebc6aebd3124603896b869": " S_{CHSH} = E \\left ( {A_0, B_0} \\right ) + E \\left ( {A_0, B_1} \\right ) + E \\left ( {A_1, B_0} \\right ) - E \\left ( {A_1, B_1} \\right ) ", "8f0f88b21e54b0db780969b82b638d26": "\\mathfrak g = [\\mathfrak g, \\mathfrak g]", "8f0f9cec91338bff07336293eb62831f": " E_n-E_m \\approx h(n-m)/T", "8f0facc0abf558df38102a94e086d9c0": " z = z_0 + ct \\,", "8f0fe8a791fc92853460e1bc918ddf52": "\\tfrac{Q - 1}{Q + 1}", "8f0ffd51ad5cd2b9aac23b632e4cc33a": "B, B'", "8f104e480b4d175bd35d4502cbd6468a": "(M, V) = (M, V) \\# (P, V_\\infty).", "8f10555ce14787fa4191ee722ff88466": "\\mathrm{corr}(X,Y)=\\rho_{X,Y} \\,", "8f10c20cc3ee16ec54081576699542a3": "\\mathbf{u}\\cdot\\mathbf{v} = \\tfrac{1}{2}\\left(\\|\\mathbf{u}+\\mathbf{v}\\|^2 - \\|\\mathbf{u}\\|^2 - \\|\\mathbf{v}\\|^2\\right).", "8f10ee219b79fecd37ef192c664f35f7": "T_3 = (T_1Z_1)^2 + (T_1Y_1)^2 - (Z_1Y_1)^2", "8f110ebcc40c6b28786ab2de4d882e73": "\\gamma(x,y)", "8f11153a8992df98f896a8b176a21ee6": "\\displaystyle | I( i \\omega ) | = | Y(i \\omega) | | V(i \\omega) |.\\,", "8f116956d08b7019c114240f63279725": " \\hat{\\textbf{x}}_{t\\mid t-1} ", "8f11aa9d48f371b445c6eb2a7536b5a7": "d(x, y)", "8f11ec2f6ca5745136a358f86e667a35": "\\Theta_m", "8f120c2afdf8e5e66933aa02670d2770": "\\{\\pi_1, \\pi_2, \\ldots, \\pi_j\\}", "8f1212190c87c364c6d85c608e1e553c": "dU^2", "8f121462c8d4a1c025676b1ca7f88a61": "K(\\tau_1,\\tau_2) = \\begin{matrix} \\sum_{\\{i,j\\}\\in P} \\bar{K}_{i,j}(\\tau_1,\\tau_2) \\end{matrix}", "8f126cd7854b87c3384976559dcbc5b2": " \\and (S_3 \\implies (\\operatorname{equate}[A_3, f] \\and V[F_3] = A_3)) \\and D[F_3] = D[f] ", "8f127eda876ed40580d1205cca440386": " \\sum_{i} y_{ij} \\leq 1 ", "8f128922c9a74eae61d437729142ebd9": " \\displaystyle{H^\\varepsilon f \\rightarrow -i f}", "8f13a2377dc880cc10fcfa721641f93f": "\\vee,", "8f13b0f34530c2e71a1a3ab63be961eb": "1959 = [39, 9]_{50}", "8f13c8ecb51c8869d95f567683d4c600": "\\varphi_1 ", "8f14122ef336f60768e9dcd11dd1cfa0": "{C(x)}={\\int_\\Omega e^{-{{(G_a* \\left \\vert v(x+.)-v(y+.)\\right \\vert ^2)(0)}\\over h^2}}}dz", "8f14b548c48d655b97ab1a173acb2427": " Y = C+I ", "8f14e45fceea167a5a36dedd4bea2543": "7", "8f14e9172c1d2b7ad7f95870ddd30acd": "(1 \\otimes \\Delta)(R) = R_{13} \\ R_{12}", "8f14edd72626c3df2e3c67192fadbb66": "PI = \\frac{v_{systole} - v_{diastole}}{v_{mean}}", "8f14f1f01ff64fa3a0977645b5827232": "a = a. + j.t + \\frac{1}{2}st^2 ", "8f152bb64a444ba6cbde4f098dee959f": "a \\cdot b = a + (a \\cdot (b - 1)),\\,\\!", "8f15309d6be4a244f5b7ebadb07cd0fb": "(x-a)^2H(x-a)", "8f154e28926a79eecd36115dbb5d1f07": "\\big((n-1)^2+n^2+(n+1)^2\\big)^2-2\\big((n-1)^4+n^4+(n+1)^4\\big) = (6n y)^2 = (4A)^2", "8f156f2555eb8fead45b0ec0c46bb29f": " \\rho (i) ", "8f15979a338dd00c62c56c0dbcc87784": "{\\Omega^\\hat{m}}_\\hat{n} = d{\\omega^\\hat{m}}_\\hat{n} - {\\omega^\\hat{m}}_\\hat{\\ell} \\wedge {\\omega^\\hat{\\ell}}_\\hat{n}", "8f159839d9d128176fb793be8285b584": "\n\\frac{1}{\\exp\\left(iy\\right)-1} = \\frac{\\exp\\left(-i\\frac{y}{2}\\right)}{\\exp \\left(i \\frac{y}{2}\\right) - \\exp\\left(-i\\frac{y}{2}\\right)} = \\frac{1}{2i} \\frac{\\exp\\left(-i\\frac{y}{2}\\right)}{\\sin\\left(\\frac{y}{2}\\right)}\n", "8f15b4e84bf31ed6cf79d240dfb28764": "\\frac ab = c", "8f161bc809666d9423430aace5d74912": "P_i / \\epsilon_0 = \\sum_j \\chi^{(1)}_{ij} E_j + \\sum_{jk} \\chi_{ijk}^{(2)} E_j E_k + \\sum_{jk\\ell} \\chi_{ijk\\ell}^{(3)} E_j E_k E_\\ell + \\cdots \\!", "8f1668f14f3cd06cf3ca77f1c771f5b5": "\\Psi_B(N_A+1,N_A+2,\\dots,N_A+N_B)", "8f17064ee62e42bfd3854b288b37ad32": "\\kappa = \\aleph_{\\alpha^+} \\,.", "8f173dcc17f18779058eebd347b479c4": "P_{min} = mva_{min}", "8f1765950e619b391c32897b70370a6d": "-S[i]-S[j]", "8f17b9b1284ad31a153c18f245294f5c": " X = \\{ x \\in \\mathbb{R} \\,:\\, 0 \\leq x\\} ", "8f180900b16a8c1c68b4822a3af76c91": "T^{\\mu\\nu} = \\frac{\\hbar^2}{m} (g^{\\mu \\alpha} g^{\\nu \\beta} + g^{\\mu \\beta} g^{\\nu \\alpha} - g^{\\mu\\nu} g^{\\alpha \\beta}) \\partial_{\\alpha}\\bar\\phi \\partial_{\\beta}\\phi - g^{\\mu\\nu} m c^2 \\bar\\phi \\phi .", "8f18f0b2df3be3cf53cc0f95298fca6a": "\\rho d=\\alpha -\\frac{\\boldsymbol\\Gamma}{2}\\, ", "8f1904e3ee7425ab46d6b09cad9ba43b": "\\pi(x) = [x]", "8f1957cb486cfa88f043a9eb896ec9b6": "M_{sup} (R,T)", "8f196546fef48de07c38b3169a0064fb": " B = \\frac{1}{3}\\left[\\left(n_\\text{u} - n_\\bar{\\text{u}}\\right) + \\left(n_\\text{c} - n_\\bar{\\text{c}}\\right) + \\left(n_\\text{t} - n_\\bar{\\text{t}}\\right) + \\left(n_\\text{d} - n_\\bar{\\text{d}}\\right) + \\left(n_\\text{s} - n_\\bar{\\text{s}}\\right) + \\left(n_\\text{b} - n_\\bar{\\text{b}}\\right)\\right]", "8f19776281103ea0b507a97ffc6afecc": "\\frac{F}{C}", "8f197fdf44a614895f9f5f47b3aacf99": "q=1-\\omega_{\\mathrm{s}}/\\omega_{\\mathrm{p}}", "8f198053f426e135790c5a9af72755e9": "\n \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) , \\left( A \\rightarrow \\left( B \\or C \\right) \\right) \\vdash \\lnot A , \\lnot A\n ", "8f19b54fd133d970d38ff6657edf0aec": "\\phi_n^{(k+1)} = -n\\phi_{n+c}^{(k)}", "8f19c671facbc4fa007f0f78ec965268": "f(\\operatorname{cl}(A)) \\subset \\operatorname{cl}'(f(A))", "8f19d8b2ab049bd030e9daff2ad5ac68": "\\frac{1}{N} \\sum_{n=0}^{N-1}U^n", "8f19f53c15602fee7a98600214e4e6ef": "f(x) > Y", "8f1ac927faad85a148851858c90e2e34": "x\\mapsto a\\uparrow x", "8f1b493bc855c19af859c944042c1f66": "\\frac{1}{R} =\\frac{2ky}{rd}", "8f1b4f93758af281283898998ef6ace9": "f(x)=2x\\sin\\left(\\frac{1}{x}\\right)-\\cos\\left(\\frac{1}{x}\\right)", "8f1b5282eca1336b9d6e6707a98bf669": "E[\\mathbf{x}]=0", "8f1b6234c66aad322b8dd1b086190e27": "\\displaystyle \\sin( ax)", "8f1b8782ccc0de1865baebdd4d6bc7da": "\\rho_{m,t}\\;", "8f1baf2045173c7cad01a5615c4a4dd1": "p(v)", "8f1bbb5f5c8a76d0d83f0cdab5eb3bf8": "O(n\\log(n))", "8f1bcbf994d0bbf7d4c1ac9df0c14d5d": "Z_0=\\frac{9.8\\times 3\\sqrt{\\pi/2\\times PSD\\times f_n\\times Q} }{f_n^2}\\quad Z_c=\\frac{0.00022B}{chr\\sqrt{L} }", "8f1c5b0dbdf35352dcca2a46112219a9": "var(Z(x))", "8f1c8052dc89a0df26386f19cf18c23e": " \\Theta=A=\\Bbb{R}^1 ", "8f1c9903bb7dd2b0685222f53b767b92": "\\mathfrak{P}^{15}", "8f1cfdce44c48bbe2b1e67cbb7677e4e": "B_a(x)=-a\\zeta(-a+1,x)\\,", "8f1d4cc9d1da3c9c9d17609119d1f513": "\nz_{ij}=\\left\\vert y_{ij} - \\tilde{y}_j \\right\\vert\n", "8f1d4ec6a420712b8de694c54d3a8e5b": "g_2(\\tau)=\\frac{\\langle I(t)I(t+\\tau)\\rangle_p}{\\langle I(t)\\rangle_p^2}", "8f1d8ae21b48324cf0defb0186fe5544": "1.6507", "8f1deb681123483951c1e1baa2642bfd": "\\lim_{r \\rightarrow \\infty} h_{ab,pq} = O(1/r^3)", "8f1e261d9a1f5aa7e38f42a120b40f57": "d^{-1}", "8f1e28e87d791583fa7efffad2e48ccf": "H = \n\n\\left[\n\\begin{array}{cccc}\n 0&0&1&1\\\\\n 1&1&0&0\n\\end{array}\n\\right]\n", "8f1e57f8ab9ea69b519649229352d250": "G_{th} = \\frac{\\pi^2 {k_B}^2 T}{3h} ", "8f1f2bc339a16249ed0edd6c0d67f6ec": "\\scriptstyle V \\;=\\; -\\mu/e", "8f1f2cb72889ab870ec9a8a77acd0802": "\\times 10^{-6}", "8f1f56c08f93a3a0589a20b64ef42462": "(a, b)", "8f1fa3d983d7a27137e33a738d3db65e": "\\langle\\phi(x_1)\\cdots\\phi(x_n)\\rangle_{con}=(-i)^{n+1}\\left.\\frac{\\delta^n E}{\\delta J(x_1)\\cdots \\delta J(x_n)}\\right|_{J=0}", "8f1fbb61f423bd807b16a3e1965e2f39": "1-(1-x)(1-y)", "8f20509eae491d103bdb761ae03a09d5": "GR_L, GR_T", "8f206d19714550b54c6d732eceb54d32": " (x - 3)(x - 2 - 5i)(x - 2 + 5i).\\,", "8f20ae609f2173dc7ca8e6f2dfc0f28d": "dx^2 + dy^2 \\neq dl^2", "8f20fb7778268e9693aefc5bcf9dafa3": "C_p\\ ", "8f210c89c6b709cefaa5dde5c1818f96": " \\!\\ \\left({1 \\over R}\\right)c = R - \\lfloor \\operatorname{Re}(R) \\rfloor ", "8f22363c22767ac5eae795c103f38256": "\\sin\\delta\\neq0", "8f2276474fd384a12edcd6ec51feade1": " |\\mathit{after}\\rang = \\sum_i |\\epsilon_i\\rang \\lang i|\\psi \\rang", "8f22fd2ceb52c7349cde7b6d60cce743": "\\sigma(V)\\in\\partial X", "8f23357c62daa6223e8cd03d37780002": "R_i\\ ", "8f2338cbe24d8396bf43c59bd3921d68": " \\alpha=K_G^{-1}\\left(\\Phi_M^G \\right)^T \\hat{V}^\\pi. ", "8f23857e58d4068f1f8416a5d1f36c9f": "\\left(\\begin{array}{rrrrrr}\n 2 & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 3 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 2 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 3 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 3 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 1\\\\\n\\end{array}\\right)", "8f23983f0fabe647d7d9b6ae7349cde1": "f(x,y) = A \\exp\\left(- \\left(a(x - x_o)^2 + 2b(x-x_o)(y-y_o) + c(y-y_o)^2 \\right)\\right)", "8f23ce97d50fcbea6faa65818d88024c": "\\mathbf{x}(i)=[x(i), x(i-1), \\dots , x(i-p) ]^{T}", "8f23f2dd588a62ffae5178774d0a69bd": "\\iiint_T \\rho^2 \\sin^2 \\theta \\rho^2 \\sin \\theta \\, d\\rho\\, d\\theta\\, d\\phi = \\iiint_T \\rho^4 \\sin^3 \\theta \\, d\\rho\\, d\\theta\\, d\\phi", "8f240b44f12b06fe92e0e32f19d79ef4": "R \\cdot U(\\$100) + (1-R) \\cdot U(\\$0) > B\\cdot U(\\$100) + (1-B) \\cdot U(\\$0) ", "8f24199f66430c6bd54da2ed21005ef2": "W_E = W_E(1) - W_E(2) ", "8f243f648d80991933e8ac1ab1df5ab1": "|T_k| = 2^{2^k}", "8f246cb4a9779b3e52d4a03e1e76e93a": "\\frac{1}{2a} + \\sqrt{a(1-e^2)} \\cos i \\approx {\\rm const}", "8f2535baedd520f9494cd05d618030eb": "A_k \\in M", "8f25443a2408fe2804b18205be2aa729": "C^\\mathrm{op}", "8f260b1296ea1f6203fc3083113744fe": "\\displaystyle \\hat{r} = \\lim_{s\\to\\infty} \\hat{r}(x,s)", "8f26429ab05931f5d205b6d759a4cb42": "f:\\; M \\to {\\Bbb R}", "8f2662691f67ccc4f87d0d72a40df5d8": " +\\frac 1\\phi \\left( \\nabla _\\mu \\nabla _\\nu \\phi -g_{\\mu \\nu }\\Box\n\\phi \\right), ", "8f266bd2f12618b54a51644e00b9b7d3": " \\nabla^2\\phi=0.", "8f268efb8ba6b9fb684d73b2d949dbe2": "C_{8} = G_{4} + P_{4} \\cdot C_{4}", "8f26e1dc4de785627eb95ea7abed8172": "\\underline{v}\\in V", "8f271b8b9fa827430d469171ce436289": "Vol(B(y,(p+\\epsilon)n)/2^n", "8f272c4d13125780d39dd75ddea5b87b": "\\exp(n^{0.504})", "8f2736b098b8441f3432641407b61061": "f*f = \\tau \\cdot f", "8f276ae5f85ac09adba3665b1089ddce": "M\\left[\\begin{matrix}a_1 \\\\ \\vdots \\\\ a_d\\end{matrix}\\right] \\geq \\left[\\begin{matrix}0 \\\\ \\vdots \\\\ 0\\end{matrix}\\right]", "8f27a1ea7cbd8f188388ea07908f6838": "\\scriptstyle{\\hat{H}(t_1)}", "8f27a24db75337ca74bb444a8d5a0a02": "E_{em}/c", "8f27c1dce22dcda137302eca83992fd5": "f(\\mathbf{x}) = \\frac{1}{2\\pi}\\left\\{\\frac{d}{du}\\int_0^u F^*(Ff)(\\cos^{-1}v,\\mathbf{x})v(u^2-v^2)^{-1/2}\\,dv\\right\\}_{u=1}.", "8f2869e13e46d02c1153659d97ad51a1": "\\operatorname{Var}(X+a)=\\operatorname{Var}(X).", "8f28991ed07cf703ec90e5811363e412": "\\displaystyle S(n,g)", "8f28afa6b99cf13e23fd4b7fcb3132c7": " \\mathbb Q ", "8f28ed0221541ff916e12ffe4bde13b4": "q\\le(\\log x)^N.", "8f292d2efedeff3620294715b9d698c8": " \\hat{F}_{i} ", "8f29749e53a93d1091fe68a1a20a3e8f": "(p \\leftrightarrow q) \\vdash (q \\leftrightarrow p)", "8f29796bec5ceb3b594faadea199f8b5": " \\Phi_N = \\int_S \\mathbf{N} \\cdot \\mathrm{d}\\mathbf{S} \\,\\!", "8f2996040e14a8148b5070abd949156d": "(2)\\; L=220*0.5*\\tanh\\left(\\frac{4.5-1}{22}\\right)", "8f29be7dc7f9ad49cf0cd0f2d5603644": " 1 \\le x_1 \\le 10 ", "8f29cc33d14129c1d219737e7c300c9a": " \\langle f, \\psi \\rangle = \\int e^{i \\phi(x,\\xi)} L \\left ( a(x,\\xi) \\, \\psi(x) \\right ) \\, \\mathrm{d} x \\, \\mathrm{d} \\xi ", "8f2a2a0d2e927f60be5001d320b5e0ee": "\n\\begin{align}\nf({aa}) & = { 9 \\over 49 + 42 + 9 } = { 9 \\over 100 } = 0.09 = (9%) \\\\ \n\\end{align}\n", "8f2a2c0cd2a425e6f65944de3144b621": "\\Delta\\Theta = \\frac{4\\pi v \\Delta t}{\\lambda}", "8f2a57326195c042b6c09e6be1699fbf": "\\mu_{k+1}", "8f2a7418d973aae1af3603b57f18f0c5": "\\sigma (42) = 96 = 3 \\times 4 \\times 8 = \\sigma (2) \\times \\sigma (3) \\times \\sigma (7) = 1+2+3+6+7+14+21+42", "8f2a775b226d042167b2a3e0077c1948": "n_\\mathrm{A} = n_\\mathrm{B} \\frac{R_\\mathrm{B}-R_\\mathrm{AB}}{R_\\mathrm{AB}-R_\\mathrm{A}} \\times \\frac {1+R_\\mathrm{A}}{1+R_\\mathrm{B}}", "8f2a7ec9229f5f644269e0ec1222590d": "\\lim_{\\beta \\rightarrow 0}H(f) = \\mathrm{rect}(fT)", "8f2a8a7b756831915e450aed39f1e72d": " y_{i+1} = y_i + \\tfrac12 h \\bigl( f(t_i, y_i) + f(t_{i+1},\\tilde{y}_{i+1}) \\bigr). ", "8f2aa8f67021996ea2801126fa5cc2d0": "\\mathcal{E}_*", "8f2ac19688698b6050386d010d062dd5": "\\operatorname E(g(X)) = \\int_{-\\infty}^\\infty g(x) f_X(x)\\,dx.", "8f2af0be1fb89cc26826c2ea9bdd7ab2": "Q_0=\\mathbb{I}~", "8f2b2296725fac33a4e04c54024b1347": "\\begin{align}\n \\mu &= \\sum_{m=k}^N m\\frac{\\tbinom{m - 1}{k - 1}}{\\tbinom Nk} = \\frac{k(N + 1)}{k + 1} \\\\\n \\Rightarrow N &= \\mu\\left(1 + k^{-1}\\right) - 1\n\\end{align}", "8f2b2aa3cf643f36c7fa97d0a9c1aa15": "\n {\n \\begin{align}\n \\dot{\\rho} + \\rho~\\boldsymbol{\\nabla} \\cdot \\mathbf{v} & = 0 \n & & \\qquad\\text{Balance of Mass} \\\\\n \\rho~\\dot{\\mathbf{v}} - \\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\sigma} - \\rho~\\mathbf{b} & = 0 \n & & \\qquad\\text{Balance of Linear Momentum} \\\\\n \\boldsymbol{\\sigma} & = \\boldsymbol{\\sigma}^T\n & & \\qquad\\text{Balance of Angular Momentum} \\\\\n \\rho~\\dot{e} - \\boldsymbol{\\sigma}:(\\boldsymbol{\\nabla}\\mathbf{v}) + \\boldsymbol{\\nabla} \\cdot \\mathbf{q} - \\rho~s & = 0\n & & \\qquad\\text{Balance of Energy.}\n \\end{align}\n }\n ", "8f2b2da882eb0308f586ba991c506d9f": "\\scriptstyle \\{U_{k+1}-V_k,k\\geq0\\} ", "8f2b3dd96eba2932f998ed0a366cdfd9": "\\mathbf{B}'(\\mathbf{x}',t) = \\mathbf{B}(\\mathbf{x}'+\\mathbf{v}t).", "8f2b4efaacce793523ef7118bec97770": " \\alpha_c + \\beta_c = -\\frac{B}{A} = 1 ", "8f2b6006cfa9a654263a95994df2cbdd": "d_i \\colon N(C)_k\\to N(C)_{k-1}", "8f2b9d9ccb540b458b69df8c3a2a5d21": "(z \\subseteq x) \\Leftrightarrow ( \\forall q (q \\in z \\Rightarrow q \\in x)).", "8f2bdd0e4c2a794e56003c2ae03b13fe": "m_h^*", "8f2c40ebdd6537d40d682d7ee3bc71d9": "(s,u_1,u_2,v,e)\\,", "8f2c64c68da8053b1c928238819113a8": "f = \\frac{A}{B}\\,", "8f2c95ded8f24e5de3d776e4ac334c8b": "\\sigma(A+\\mathfrak{G}^2)>\\sigma(A)\\text{ for }0<\\sigma(A)<1.\\,", "8f2cc7df89442e15f178d1205d58fa1b": " C_{ijkl}", "8f2cd779c93cc08449a872b543a0ee11": "\n|\\mathbf{A}|_+ = \\lim_{\\alpha\\to 0} \\frac{|\\mathbf{A} + \\alpha \\mathbf{I}|}{\\alpha^{n-\\operatorname{rank}(\\mathbf{A})}}\n", "8f2ceeaa3726b74bd3527a342291f56e": " \\mid \\mathrm{T_{High}}(f) \\mid = \\frac { f/f_1 } { \\sqrt{ 1 + (f/f_1)^2 }}, \\ ", "8f2d093be7f26ab140eea5f4b75ebca2": "| x \\rangle", "8f2d18adf83592a82f33001198b98e7a": "y_j \\in F(p)", "8f2d228635271967c9379fb92e829c74": "v=6 m+1", "8f2d25fdde0b39230fb0fe08ad51b30c": " s>0 \\,. ", "8f2d51ac6b2733a9f0657019bd96eec0": "i\\frac{d\\Psi(x,t)}{dt}=-\\frac{1}{2}\\frac{d^2 \\Psi(x,t)}{dx^2} + V(x)\\Psi(x,t).", "8f2d7c9a2f3c93115a17ffc0ef46ff79": "\\hat{\\mathbf{r}}_{ij} ", "8f2d7d10c33468d0891d220fd436042d": "\\begin{bmatrix} 1 & a \\\\ 0 & 0 \\end{bmatrix}.", "8f2dd9adc1139b33c05bb76e29238d84": " E \\rightarrow (E+(n-i)\\delta_{ij}) ", "8f2e647778aee85792d738026de1dc1e": "_{y_\\vee x}\\!", "8f2edd182a8bfa89fee6a3a268d14337": "\\begin{align}\n\\frac{d\\mathbf p}{dt} &= \\mathbf F \\\\\n\\frac{d}{dt}\\int_V \\rho\\mathbf v\\,dV&=\\int_S \\mathbf t dS + \\int_V \\mathbf b\\rho \\,dV. \\\\\n\\end{align}", "8f2edf27e65deae22346a3504bcd2c0d": "(A^+)^+=A\\,\\!", "8f2f1410ddb1e9687eb0140d1c25dc0a": "{\\mathbf{}}\\Psi^1_i=\\left(A_i-B_i(B'_iS_{i+1}B_i+R_i)^{-1}B'_iS_{i+1}A_i)\\right)\\hat{P}_i\n\\left(A_i-B_i(B'_iS_{i+1}B_i+R_i)^{-1}B'_iS_{i+1}A_i)\\right)'", "8f2f2cba44912c291582efc64d5d304a": "\\delta x ", "8f2f5b066bf5e0f1a5394febe26a6cd7": "{ {\\ln\\left(2\\right)} + {\\pi\\over6} }", "8f2fad1676f338861e61349863939f37": "\\scriptstyle g\\approx9.81", "8f2fdebdc49aeebd4681e54fa612f47e": "f'(x) = f(x)\\times \\Bigg\\{\\frac{g'(x)}{g(x)}-\\frac{h'(x)}{h(x)}\\Bigg\\}=\n\\frac{g(x)}{h(x)}\\times \\Bigg\\{\\frac{g'(x)}{g(x)}-\\frac{h'(x)}{h(x)}\\Bigg\\}", "8f306dcda6cf31d045522f2c25039de5": "\\tau \\gg \\frac{1}{2\\pi f_H}.", "8f309618b3e0ff73c44dabb93a099f9c": "H_k=y^{\\rm T}_k s_k/y^{\\rm T}_k y_k", "8f309d453344552aae28d6ca26c1df3f": " P = \\pi_2 \\, ", "8f30b2208fe593dd16427c1e2886f021": " f^*(g^*(z))\\neq (g\\circ f)^*(z).", "8f30b86535786d0a446080bc25ec6b86": "8 \\times 4", "8f30f8bb8a098abbf40212634b0b7b91": "g_i \\leq n", "8f317152a906169f4f1d603f2670c821": " p_n'(0)=\\kappa_n= \\,", "8f317cc045a053417d2da6b1ae093e45": "\\lim_{x\\to 0}-x^2 = \\lim_{x\\to 0}x^2 = 0", "8f3197181b3ceacb84fa13c17a98decb": " n\\leq D^{\\beta-1}+\\beta", "8f32594d5c13fe2b29bae0152b312687": " m = 2n + p/4", "8f32831a29dfc813cbd05278709706f0": "\\mathrm{Tr} (\\rho (ghg^{-1}))=\\mathrm{Tr} (\\rho(g)\\rho(h)\\rho(g)^{-1})=\\mathrm{Tr}(\\rho(h))", "8f32ee7eab8968c6467e39e9cabca9d1": "C = -1.5 \\times 10^{-2}", "8f33063579bcfcef334309f49fb4bf61": "C(x)=\\int_0^x \\cos(t^2)\\,\\mathrm{d}t=\\sum_{n=0}^{\\infin}(-1)^n\\frac{x^{4n+1}}{(2n)!(4n+1)}", "8f337023369d8d3e39a5ca2190586a51": "\\hat{H} = \\frac{\\hat{p}_x^2}{2m} + \\frac{1}{2} m \\omega_c^2 \\left( \\hat{x} - \\frac{\\hbar k_y}{m\\omega_c} \\right)^2.", "8f33a1f2a96440427e39064c01edacaa": "Y''_j = \\frac{1}{7h^2} (2 \\times y_{j - 2} - 1 \\times y_{j - 1} - 2 \\times y_j - 1 \\times y_{j + 1} + 2 \\times y_{j + 2})", "8f33b1eec06435699bdf911a427e86a5": " i = k^*+1,\\dots,n", "8f340df561c5efea1048d76e198e8435": "\\displaystyle{K_f=V(f)PV(f)^{-1} - P}", "8f342169eb53704c14d62d2eab14aa76": "\n\\begin{align}\n \\sqrt{ax^2+bx+c} \\;=\\; xt\\pm\\sqrt{c}.\n\\end{align}\n", "8f343c5f86d429aafb82d0b60c4e9daa": "\\operatorname{Lan}_FX", "8f345af7dd4d5a2622c6eeefb0eca738": "y_i = z_i + y_1^{r^{i-1}}", "8f34a3e1dbc06a0adece16d5e904f56d": "\\alpha_i = \\alpha_j", "8f34ac868e8957d6ba7fdc0ca429bc04": "\\sigma_{\\alpha\\beta}", "8f34cbac89200a7d78e120b46904bf1c": " x(t) = f(t) + \\sum_{i=m}^n k(t, s, x(s))", "8f352e440943a2d7698c8bc1ffa36237": "\\bar{I}_1, \\bar{I}_2, J", "8f3590aabdc1f6d41824587cb1a5d386": "x_0 = i,\\quad x_{n+1} = \\frac{x_n + |x_n|}2,\\qquad\\lim_{n\\to\\infty} x_n = \\frac2\\pi.", "8f359697a05695440b0c97c640b7c082": "(\\forall) \\frac{\\forall x . \\gamma(x)}{\\gamma(x')}", "8f35a2f12fc3ff12b1119d7e825bff3f": "(hg,hgs)", "8f35a9bcbcb718f514769ac92c938c64": "1 \\leq j < k \\leq n", "8f35cb28b69df238516307697cf77af4": "s 1 + \\delta \\,\n", "8f38ee5ea6a4afababefebe3fe7ecc54": "\\displaystyle \\partial_x(\\partial_t u+u \\partial_x u+\\epsilon^2\\partial_{xxx}u)+\\lambda\\partial_{yy}u=0", "8f38f7b1865f3fc84b9e880d0d27b696": " = \\left(\\frac{\\mathrm{d}^2s}{\\mathrm{d}t^2}\\right)\\mathbf{u}_\\mathrm{t}(s) - \\left(\\frac{\\mathrm{d}s}{\\mathrm{d}t}\\right) ^2 \\frac{1}{\\rho} \\mathbf{u}_\\mathrm{n}(s) \\ , ", "8f38fc0e8bbfd742a18f5efb48a9ef0f": "\\tfrac {PL_t}{PL_{t-1}} - 1", "8f39c108505932931d2eaf6d6a9d0b8c": "|Y^X/G| = Z_G(t,t,\\ldots,t).", "8f39c7f161b4f165a939cba81af3333f": " -\\frac{ \\frac{\\beta}{\\beta + 1} r_O + r_E }{ r_O + r_E } ", "8f39c85eb5aa0077b531513ec8cae9aa": " {q}_{i}^e, {q}_{j}^e ", "8f39e28fd88432e7af24ea3da39618b9": "D_i=B_i-A_i", "8f39efc18471630a1fbb4d4f5c383064": "P' = QPQ^{-1}", "8f3a07602f5e1ba202c5eddd00780e24": "\\mathcal{A}_\\mu", "8f3ac7701095f3bb9f01b9fc28d1562a": "\\delta = \\omega - \\omega_a", "8f3b1c8fcec8eeefe17a9dd51427cdde": "C^{(1,1)}", "8f3b3f91cb29a669c4ce371e48f996fe": "\\varphi = h f_0.", "8f3b916e5d779e30ce719086f9b4d57f": " (x_0,x_1,x_2,x_3,x_4)\\cdot (y_0,y_1,y_2,y_3,y_4) = - x_0 y_0 + x_1 y_1 + x_2 y_2 + x_3 y_4 + x_4 y_3.", "8f3b9c722a1c3db0d7a201b243c162c9": "\\mu_{z} ([t_{1}, t_{2})) = \\lambda ([\\hat{\\tau}(t_{1}), \\hat{\\tau}(t_{2})) = \\hat{\\tau} (t_{2}) - \\hat{\\tau}(t_{1}) = t_{2} - t_{1} + \\int_{[t_{1}, t_{2}]} \\| \\mathrm{d} z \\|.", "8f3ba282a0516dd573fe5759d957d7b4": "\\{ x \\in X | I(x) \\leq c \\} \\mbox{ for } c \\geq 0", "8f3ba5f59aab4564735bb63513d1e853": "\\begin{matrix}\\text{If } y(t)=\\sqrt{a} x(a t) \\text{ for some } a > 0 \\text{ then }\n\\\\ W_y(t,f)=W_x(at, \\frac{f}{a}) \\end{matrix}", "8f3c00e8fb2991a522c9729eabbf49c4": "(x_m-\\varepsilon)^{2^k}=x_m^{2^k}-\\varepsilon\\,{2^k}\\,x_m^{2^k-1}=y_m+\\varepsilon\\,\\dot y_m.", "8f3c76ecb16fc01645508991f1ddf5e4": "\\mathbf{STFT}\\{x[n]\\}(m,\\omega)\\equiv X(m,\\omega) = \\sum_{n=-\\infty}^{\\infty} x[n]w[n-m]e^{-j \\omega n} ", "8f3cde4de7d31bd882fcea9814682b8b": "\\textstyle{\\frac{1}{n}}", "8f3cf6eaad36ef49cf27166cb61ecc2d": " A(x,t) = \\frac{3v_{\\parallel}}{\\alpha} \\mathrm{sech}^2 \\left [ \\frac{\\sqrt{v_{\\parallel}}}{2} \\left ( x-v_{\\parallel} t \\right ) \\right ] \\,\\!", "8f3d4b0711af615bdfe08c9ea8e347d2": "A_\\mu", "8f3d88afb19e64ecfbd38a573578b48d": "\nP(Z 4000", "8f508dc66ad89a4f412eb2c73c2fdab7": "\n\\tau = \\frac{\\rho c_p V}{hA_s}.\n", "8f50f6434ec9e681d02fe2cb7f9c155b": "[J_i,P_j]=i \\epsilon_{ijk} P_k,\\,\\!", "8f510ce6b0e57eb30e8bd38303e1b02b": "\\left(\\frac{\\dot{a}}{a}\\right)^2 = \\frac{8 \\pi G}{3} \\rho - \\frac{kc^2}{a^2} \\,", "8f51180404d21194b71a97e3e102081d": "=\\frac{e^4}{(k-k')^4}\\left(4 \\left( {k'}^\\mu k^\\nu - (k' \\cdot k)\\eta^{\\mu\\nu} + k'^\\nu k^\\mu \\right) + 4 m^2 \\eta^{\\mu\\nu} \\right) \\left( 4 \\left( {p'}_\\mu p_\\nu - (p' \\cdot p)\\eta_{\\mu\\nu} + p'_\\nu p_\\mu \\right) + 4 m^2 \\eta_{\\mu\\nu} \\right) \\,", "8f51a24ae3a13546159a7c237160e125": "v \\ge P", "8f51e4570a9c512a859d2a83b4054bb5": "\n w^0_{,1111} + 2\\,w^0_{,1212} + w^0_{,2222} = -\\frac{q}{D}\n ", "8f51ef3b9040a527832bebba66e286ac": "fw", "8f522a22c27ca2fb997d2c56e8d0ebea": "\\phi_{sl,m}=\\frac{M_{s}}{M_{s}+M_{l}}=1-\\frac{M_{l}}{M_{s}+M_{l}}", "8f5250ddb7e90bc8e858c962f0583441": "z(v)\\geq \\mathrm{deg}(v)", "8f527275baed1901853fcdc5115d94e6": "\n\\begin{align}\n& 1^2 + 0^2 + 0^2 + 0^2 \\\\\n& 0^2 + 1^2 + 0^2 + 0^2 \\\\\n& (-1)^2 + 0^2 + 0^2 + 0^2.\n\\end{align}\n", "8f52a8a2a2451d198b499ce0b59c9661": "y.\\,\\!", "8f52d9c4a3bb52072e6d18aec578951a": " \\mathbf{H} = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix} ", "8f53055051d1991ed052d4580fac9571": "\\hookrightarrow", "8f5351f4bcfa3a94aefe5ab6dc82f19d": " r\\left(\\sqrt{5}-1\\right) = \\frac{a}{10}\\sqrt{50+10\\sqrt{5}} \\!\\, ", "8f53b8edfcc9ff6bd13ae6d846aa0b77": "\nk = \\frac{\\log(1 - p)}{\\log(1 - w^n)}\n", "8f53d0e59eac5a523894e812e27bdaac": "I = \\Sigma I_{a,i} + \\Sigma I_{c,i}", "8f53d4f81e3f790f9927dfb9a17ea32b": " ( \\cdots ( A_N \\otimes A_{N-1} ) \\otimes \\cdot ) \\otimes A_2 ) \\otimes A_1 ) ", "8f5413d189296ec2f0030ad79fdac45d": "\n{\\rm Pr}\\Big(\\hat{f}(x)-w(x) \\le f(x) \\le \\hat{f}(x)+w(x) \\;\\;\\;\\; \\text{ for all } x\\Big) = 1-\\alpha.\n", "8f542dfa4bb8bf64038e426ce1897e62": "C_{PL}", "8f5433db273738af73898c9730f4ced5": " \\Delta \\ell \\approx 1.220 \\frac{f \\lambda}{D} = 1.22 \\lambda \\cdot (f/\\#)", "8f54354928faf540e0195ca149a63750": "\\lambda= \\mu_1 + \\frac{b_0}{b_1} - (m+1) b_1\\!", "8f5491100c8046c87803861dcf98cd3c": " y - f(b_k) = \\frac{f(b_k)-f(a_k)}{b_k-a_k} (x-b_k). ", "8f552b0a95b9c45838d258a6e6b38236": "L \\leftarrow 1", "8f5561b1a587725a452c1ef5f145c2ef": "\\kappa = \\frac{\\alpha[1-(1-\\alpha)\\beta]\\phi}{1-\\alpha}", "8f5562858f12dfd21ef250ce492ea095": "{\\rho} = \\frac{U}{V_1}", "8f5573fdcd20a84ee5f3e65e85c6ce01": "H(\\mathbf x)", "8f55ccc41c53d6b85f68e20b57a670c5": "h\\in H,\\exists i\\,\\!", "8f55e79d3441ce6d12182511dfb1fae2": "V =\\,", "8f564d4187b320641a5176a8a1690373": " g_B = g_{obs} - g_\\lambda + \\delta g_F - \\delta g_B", "8f5709d1953dfc84bb1765b6d4d9f454": "F = [S - PV(Div)] \\cdot e^{r \\cdot (T-t)} \\ ", "8f5780972bc4d4773853d9ff9cb4823e": "X_1,X_2,\\ldots", "8f57c216ae0e05c3deb4877e3fd5ff39": "x = x' + 1, \\ y=y'", "8f57c9bbb52672fcb42e99bf43376842": "\\left(\\prod_{i=1}^n a_i \\right)^{1/n} = \\sqrt[n]{a_1 a_2 \\cdots a_n}.", "8f57ed0d0ccdc4d72c45f440dab6857d": "\\sum_{(I,m)=1} \\chi(I) N(I)^{-s} = L(s, \\chi)\\, ", "8f581bf7c2ac7a3a34d4dffd0d78e1e3": "\n\\begin{align}\nA_{m_j} &\\approx \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f\\bigl(X_1\\left(s\\right),X_2\\left(s\\right),\\dots,X_n\\left(s\\right)\\bigr)\\cos\\left(m_j\\omega_j s\\right)ds := \\hat{A}_{m_j}\\\\\nB_{m_j} &\\approx \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f\\bigl(X_1\\left(s\\right),X_2\\left(s\\right),\\dots,X_n\\left(s\\right)\\bigr)\\sin\\left(m_j\\omega_j s\\right)ds := \\hat{B}_{m_j}\n\\end{align}\n", "8f581f275eed6d10a097719f2d640643": "\n\\dot{x} = \\{x, H\\}_{PB} + u_1\\{x, \\phi_1\\}_{PB} + u_2 \\{x, \\phi_2\\} = -\\frac{c}{q B} \\frac{\\partial V}{\\partial y}\n", "8f582366eda38461d60604583853bb7f": " x \\in [u_{\\ell},u_{\\ell+1}] ", "8f582c1d40e4b0303d9373c406827ab6": "SG_\\text{true} = SG_\\text{apparent} - {\\rho_\\text{air} \\over \\rho_\\text{water} }(SG_\\text{apparent}-1)", "8f5865edf736e279674e3bae3ca3a90a": "\\lambda_2,", "8f58874de7be7c1d264d8b974db4e3d0": "(p_2(x) y^\\prime)^\\prime + q_2(x) y = 0 \\,", "8f593902bcfc312fd59d3186abcf1b1e": "D=\\{z\\,\\colon |z|<1\\}", "8f5950ed650a59ffa837ae8993d11fe0": "F_n(a, 0)", "8f596685a622f3491be8a3ad3877e871": "L_{RX}", "8f5991a48e150916605dc3d50ab6ab84": " \\iiint_V \\nabla \\cdot \\nabla u \\, dV =-1.", "8f5a1d5bdf54f21cab2e03931e1d3039": "D_+f(t)\\,", "8f5a71e992445e2d05a6c54bdd5d596e": "\n(\\mathbf{\\gamma }\\cdot \\mathbf{p-}\\beta (\\varepsilon -A)+m+S)\\psi =0.\n", "8f5a90879f1bfce76a0ef93ca22ba364": "\ni\\hbar \\frac{\\partial}{\\partial t} \\psi(\\{\\mathbf{r}_i\\},t) = H(t)\\psi(\\{\\mathbf{r}_i\\},t)\n", "8f5ac1de5877458bc3d600961c2d34ae": "\n Q_{\\rm TE_{mnp}} = \n \\frac{Z_0 lwh}{4R_s} \\frac{k_{xy}^2 k_r^3}\n {\\zeta l h \\left(k_{xy}^4+k_x^2k_z^2 \\right) +\n \\xi w h \\left(k_{xy}^4+k_y^2k_z^2 \\right) +\n lw k_{xy}^2 k_z^2}\n ", "8f5ad865232f986ebbbecc476e3520a3": "R_{\\alpha}", "8f5b05c4ac9d0ced1fbd7fd09d09f3c0": "l(I)=b - a", "8f5b62619703e06463b6622ee56eb9be": "e^{-1/t}\\; \\operatorname{Ei}\\left(\\frac{1}{t}\\right) = \\sum _{n=0}^\\infty n! \\; t^{n+1}. ", "8f5b73033bc858c76daac05464990a32": " \\begin{align}\n\\nabla \\cdot \\mathbf{D} &= \\rho / \\beta, \\\\\n \\quad \\nabla \\cdot \\mathbf{B} &= 0, \\\\\n \\quad \\kappa \\nabla \\times \\mathbf{E} &= -\\frac{\\partial \\mathbf{B}}{\\partial t}, \\\\\n \\quad \\kappa \\nabla \\times \\mathbf{H} &= \\frac{\\partial \\mathbf{D}}{\\partial t} + \\mathbf{J} / \\beta,\n\\end{align}", "8f5b7fab7df30b218f7d96da0ecd37be": "D=b^2-4ac.", "8f5bf59d1f2f82c4970edc6b8a6fcd35": "\n\\begin{align}\n\\boldsymbol{\\Tau}_{\\boldsymbol{1}}(z)& = \\tau_0\\circ\\tau_1(z)& =&\\quad b_0 + \\cfrac{a_1}{b_1 + z}\\\\\n\\boldsymbol{\\Tau}_{\\boldsymbol{2}}(z)& = \\tau_0\\circ\\tau_1\\circ\\tau_2(z)& =&\\quad b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + z}}\\,\n\\end{align}\n", "8f5c5c8abda019d8f4b1d06beb5e2d7b": "\\mathrm{Res_0}\\big(u(1/V(z))\\big)=\\mathrm{Res_0}\\Big(\\sum_{k\\geq 1}u_k V(z)^{-k}\\Big)=\\sum_{k\\geq 1} u_k \\mathrm{Res_0}\\big(V(z)^{-k}\\big)\n", "8f5c7f033273d04e5e5e9351df41b390": "(A.1.b)\\quad \\psi_{,\\,\\rho\\rho}+\\frac{1}{\\rho}\\psi_{,\\,\\rho}+\\psi_{,\\,zz}=e^{-2\\psi}\\big(\\Phi^2_{,\\,\\rho}+\\Phi^2_{,\\,z}\\big)", "8f5c8072fb9af60c0d37061769863924": " E_{\\pm}=\\frac{1}{2}(E_{1}+E_{2}+W_{1}+W_{2}) \\pm \\frac{1}{2}\\sqrt{(E_{1}-E_{2}+W_{1}-W_{2})^{2}+4W^{2}} ", "8f5cae1be0add93c6e82a0f9177097f1": "u=\\frac {w\\ell} {q} = \\frac {w} {a}.", "8f5cd6f027aa6a596ed181bd2bed5a7d": "\\varphi : \\mathbb{R} \\times \\Omega \\times \\mathbb{R}^{d} \\to \\mathbb{R}^{d}", "8f5cdaae1040b50ee2e55776e88b288b": "\\tilde{A}(\\omega, z+h)", "8f5cf4760102e6ad965bae867fdb7b3e": "\\frac{\\tfrac DR(1+\\tfrac DC)}{ \\tfrac DR + \\tfrac DC }", "8f5d0c8c16df7a21c15d973228d1efae": "b = \\arcsin \\left( \\frac{\\sin a\\,\\sin \\beta}{\\sin \\alpha} \\right),", "8f5d4ae915acfba1104073be2c1993df": "\\frac{2m}{\\hbar^2}\\left(V(x)-E\\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \\cdots", "8f5de2391de2a65aa990921671ecfe82": "\\Delta^{-1} \\,", "8f5de29b020b777d97cd2e1314ee0fc9": "\\otimes \\colon \\mathbf C\\times\\mathbf C\\to\\mathbf C", "8f5e3c04558fcfb29ab50986f009a28f": "(L + R) - (L - R) = 2R", "8f5e676da1b42d721a95da00905ad249": "p>p_0", "8f5e6f271f28bc908528fea9ab13cb34": "u_1(t)=x(t)", "8f5e8b2feae26ff02365392a9d94f669": "q_{i,t}", "8f5f2d3eacdbcd44cf5797e3bf05b6b5": "\\mathbf{p}_2-\\mathbf{p}_0", "8f5f54da087170a6200215f3b657b4a8": " g = 357.528^\\circ + 0.9856003^\\circ n ", "8f5f75124e4599293d74f04c74e0d0c3": "O_{p^\\prime}(G)", "8f5fa9f8812709ee715ea55b4b46deca": "a S^1 = \\{as \\mid s \\in S^1\\}", "8f5fb729cd33331e6865c5205010b524": "f(n-1)", "8f5ff1384c2f0874858e0c1fb13eea00": "\\theta\\mapsto z = e^{i\\theta} = \\cos\\theta + i\\sin\\theta.", "8f6053fc256c8bea397cdbb3624ac5bf": "\\begin{alignat}{7}\n 2x &&\\; + \\;&& 3y &&\\; + \\;&& 5z &&\\; = \\;&& 0, \\\\\n-4x &&\\; + \\;&& 2y &&\\; + \\;&& 3z &&\\; = \\;&& 0.\\\\\n\\end{alignat}", "8f6086ab06063fece931df0b3854ce96": "\\lim_{b\\to\\infty}\\int_0^b\\frac{\\mathrm{d}x}{1+x^2}=\\lim_{b\\to\\infty}\\arctan{b}=\\frac{\\pi}{2},", "8f609e42ae0db5351b63a175d10a7d89": "\\hat u (\\xi) := \\int e^{- i y \\xi} u(y) \\, dy", "8f60a7dbb1f3002dbcc224d07285aac3": " \\mathbf{c} \\mathbf{d}^{\\mathrm T} = \\begin{bmatrix}\nc_1 d_1 & c_1 d_2 & c_1 d_3 \\\\\nc_2 d_1 & c_2 d_2 & c_2 d_3 \\\\\nc_3 d_1 & c_3 d_2 & c_3 d_3 \\end{bmatrix}\n", "8f60c7562c88c83d832b1d4a2c165fec": "\\sigma_{-i}", "8f6109a74e9c60df51b6ae0a82ecc849": "(x_1,y_1),(x_2,y_2)", "8f6126f893f1a3af18a8fefcf7a76980": "\\,I^-(y)", "8f61669622522582a2d9b0a6a6319ccd": "\\textstyle m = \\left(\\frac{t_2}{n}\\right)", "8f616fafae07a4767781a816dc22497a": "\\Lambda(V) = T(V)/I\\ ", "8f6179074821873224d1f69737e7d7d3": "{a\\pi\\over 4}\\ {b\\pi\\over 4}\\ {c\\pi\\over 3}", "8f6191bfb723ef3bad87f780cf709ea6": "l_{k}\\, ", "8f6197af8b2dd6e04951a66f40a9f9cd": "\\textstyle\\vec{M}_G", "8f619e71b198bb19f98da6c6b54dd411": "\\mathrm D_{\\mathsf C}=\\sup_{{\\boldsymbol S},c}\\#{\\boldsymbol S}(c)", "8f61e3768b7bb99a0cab3a82e3424e97": "g=\\lambda^T\\lambda", "8f62114105e7eeba743abcd97db6275c": "R = {D}\\cdot {P} = {D_{max}\\cdot (P_{max} - \\frac{P_{max}} {2})}\\cdot \\frac{P_{max}} {2} = {D_{max}\\cdot \\frac{P_{max}^2} {4}}", "8f6254807e7c3955255ff33aa6dac0ad": "\\varphi_{xx} + \\varphi_{yy} = 0.", "8f6258a791536ed2c76cf0d3c477cf36": "N_{rot}=\\binom{D}{2}=D(D-1)/2", "8f626212d0044363759c799221be2c7c": "\\Delta\\lambda_0", "8f6263b454e3fd451d8b4255d8559334": "(\\forall z\\in\\mathbb{C}):|g(z)|=\\frac1{|f(z)-w|}<\\frac1r\\cdot", "8f626d011aaaca0b27c9921b5128613f": "Z=\\sum_{n=0}^{\\infty } \\frac{((2n)!)^3(42n+5)} {(n!)^6{16}^{3n+1}}\\!", "8f629145bab7a7dce0cbd471429b5a02": "\\begin{matrix} {3 \\choose 3} \\end{matrix}", "8f63b2ed60d7e79d0c25bc898477e6da": "82^2", "8f63dbf2b8879bdc7c8a05121e0e81b0": " r \\in \\overline I", "8f64096df1cfc729fdd0902e23a8b854": "T(\\varphi)(x) = f(x) \\varphi (x) \\quad ", "8f64196446a823ca818929bdf02ba9be": "m_1-m_{ref}=-2.5\\log_{10} \\left ( \\frac{I_1}{I_{ref}} \\right )", "8f644136c89241e31897b9c944fc037f": ",", "8f6487f8e7a683dcf41f402834bd80c1": "E_6", "8f649b2fadbfd8fd036c41c51e2198e0": "C_k/C_d", "8f64a71bc90f3f382812dbf083286645": "\\succeq", "8f64e188d18c82ebd3cbb708e5ce37cc": "U_{\\alpha \\beta} n_{\\alpha}n_{\\beta}", "8f64f48e5947a640dae3e6f8476ee3c9": "H_{Y'} = -\\sum_{y'\\in Y'} \\nu(y') \\log \\nu(y') .", "8f64fbcb5b0d4ad9230bdb78ba0b8b12": "(\\mathbb{Z}/(2), A_3, \\mathbb{Z}/(2))", "8f657643e970ef57a767b420cf05a5eb": "\\begin{align}\n\n\\mathbf B &= \\mathbf {\\nabla \\times A}, \\\\ \n\\mathbf E &= -\\frac{\\partial}{\\partial t} \\mathbf{A} - \\mathbf{\\nabla}V \\ . \n\\end{align}", "8f658dbff07b8295f90355ee6c0bceb4": "C_p = \\frac{5}{2}nR\\;", "8f65a1076b5b5e2d8476c0660cd965d1": "{4 \\choose 1}^7 - 844 = 15,540", "8f65af76055a522881260760b262d955": "\\sum_{k=1}^n {-\\ln U_k} \\sim \\Gamma(n, 1)", "8f65c0f0178db5d2e3528d78934e328d": "\\frac{1 + {\\scriptstyle\\frac{4}{7}}z + {\\scriptstyle\\frac{1}{7}}z^2 + {\\scriptstyle\\frac{2}{105}}z^3+ {\\scriptstyle\\frac{1}{840}}z^4}\n{1 - {\\scriptstyle\\frac{3}{7}}z + {\\scriptstyle\\frac{1}{14}}z^2 - {\\scriptstyle\\frac{1}{210}}z^3}", "8f65fcdd871b2baa2d73e31d04bf4b7a": "\\sup_{\\theta} R(\\theta,\\delta)=\\lim_{n \\rightarrow \\infty} r_{\\pi_n} \\,\\!", "8f6663e2846c55ccd94771455734335b": " n\\,\\leftarrow\\,n + 1", "8f6670491ec22ae77850c8130e3589eb": "k/2^m", "8f6693bcb14bf33779c4ce9d93bbaa52": "f(x) = \\sum_{k=1}^{\\infty}f_k x^k", "8f66980163f5ecc3508b03bffec54c0f": "\\overrightarrow{a b}", "8f66b792cefd666f267829fcdfb9b3b4": " FV \\ = \\ PV (1+i)^n \\ = \\$ 10,594 \\times (1+.07)^{20} \\ \\approx \\$ 10,594 \\times 3.87 \\ = \\$40,995 ", "8f66ccf5c2e980d64988e656e6ad1020": "\\tilde{\\xi}_1(\\alpha_1,\\beta_1,\\alpha_2,\\beta_2,\\eta,\\gamma,\\nu),\\tilde{\\xi}_2(\\alpha_1,\\beta_1,\\alpha_2,\\beta_2,\\eta,\\gamma,\\nu)", "8f66cd9dad30056bf08d36e7aa9cc6bf": "(x-\\alpha^2)", "8f66ea9f9e17fcc5456adac283704500": " F_4 = q ", "8f66f299eb778eaff3ff7f0a77933a87": "\\mathcal{L} = K + \\frac{1}{N_0} \\int^T_0 Y(t) \\hat{X(t|T)}dt", "8f671a5be2b6c2440d41a702adab99d9": "\nk = \\frac{1}{2}\\rho \\frac{\\lambda}{4},\n", "8f6725f7fa006cd170c2948405fe3287": "a_{Y}", "8f6738c1387399d371ede6328a418b35": " \\pi^2 (R-r)^2", "8f674382bf903f10a8d91bafa6557af3": "\\sigma_c=\\inf\\{\\sigma\\in\\mathbb{R}:\\sum_{n=1}^{\\infty}a_n e^{-\\lambda_n s} \\text{ converges absolutely for any } s \\text{ where Re}(s)>\\sigma\\}", "8f67483edd2f5253b8c6d1c0e68daa19": "\n\\Delta^{(2)}(\\sigma)=((x_{n+2}^2-x_{n+1}^2)-(x_{n+1}^2-x_n^2))_{n=0,\\dots,M-3}=(x_{n+2}^2-2x_{n+1}^2+x_n^2)_{n=0,\\dots,M-3}.\n", "8f67f864f534b81a98d27e778301da5f": "x*y := y\\cdot x", "8f6827000305f156bad7a069f670fa1b": "(\\alpha f + \\beta g)' = \\alpha f' + \\beta g' \\,", "8f683fd648317ba2fb889e1e3125174f": "C_{qr}", "8f6890eff9849b72c3790b5b8d22894a": "\nG \\equiv -\\mathbf{Q} \\cdot \\mathbf{P} + G_{2}(\\mathbf{q}, \\mathbf{P}, t)\n", "8f68d0f8798568e5b5a4050a6dfe7160": "y(m,n) = y^{+}(m-n) + y^{-}(m+n)", "8f6928cc96fce360a99d2c8ef457cddc": "(n \\theta_\\mu) \\mod 2\\pi", "8f693b249353ba73e9295a12aa4d3240": "b_q = e^{-\\frac{2\\pi i}{N} g^{-q} }.", "8f695c511c7ef4f59f984e4a411c0113": "\\Sigma=(\\mathbb{T},\\mathbb{W},\\mathcal{B})", "8f69ae943aab8ac5037c53d8eb60b15c": "m_{i,j}", "8f69eab2a59ba279702b69d5b0ae443b": "F_3(x)=x^2+1 \\,", "8f6a04b09784c663d7667d636882f957": "H_d(z) = H_a(s) \\bigg|_{s = \\frac{2}{T} \\frac{z - 1}{z + 1}}= H_a \\left( \\frac{2}{T} \\frac{z-1}{z+1} \\right). \\ ", "8f6a69abcedcbbfe29bdd419b8cb744e": "y - y_1 = m(x - x_1).\\!", "8f6a7090577d5781db6d68dd1a2fc73f": "X({3\\pi/2},\\phi)=(2\\cos\\phi,2\\sin\\phi,-1)\\,", "8f6a96a13ad674b04d2640bbfd000072": " E_a = m_k c^2 + m_t c^2 ", "8f6ae64a8b0ee6462728cfd4f821cd4b": "\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty} |\\hat{\\psi (\\omega)}|^2 \\, d\\omega", "8f6b5657a31a689cb0e952f452b75898": "E = \\frac{V_1^2}{2} + \\frac{V_{r2}^2 - V_{r1}^2}{2}", "8f6b61510eea6c3ecae3502ae6c5650d": "\\mathbf{(J^{T}J)\\boldsymbol \\delta = J^{T} [y - f(\\boldsymbol \\beta)]} \\!", "8f6b674511311c3ba76c7d977aec2cbc": "x^2 + y^2 = 1.", "8f6b81b191e6275e976fbd9b12d22114": "\\nabla \\sigma = \\nabla \\; \\vec{u}^T \\mathbf{M} \\vec{v} - \\lambda_1 \\cdot \\nabla \\; \\vec{u}^T \\vec{u} - \\lambda_2 \\cdot \\nabla \\; \\vec{v}^T \\vec{v}", "8f6ba8e4592ad5ded4770064f5b4987a": " A = \\frac {\\theta}{2\\pi} \\pi R^2 \\ , ", "8f6bf1fd8ce804dc4292b715603d9b80": " f(x_0), \\dots, f(x_n) ", "8f6c30b44a553499d1a3efa4c630be40": " \\begin{align}\n\\dot{a}(n,t) &= a(n,t) \\Big(b(n+1,t)-b(n,t)\\Big), \\\\\n\\dot{b}(n,t) &= 2 \\Big(a(n,t)^2-a(n-1,t)^2\\Big).\n\\end{align}", "8f6ca2876cbf2ef35cb10dcb8c0b3ade": "\na_B=2\\pi G\\rho H,\\,\n", "8f6cbe47c1a5bda2e880b5f4c3458ef6": "x = a \\frac {\\sin [m p + \\theta_0] \\cos n p}{\\sin [(m - n) p + \\theta_0]},\n y = a \\frac {\\sin [m p + \\theta_0] \\sin n p}{\\sin [(m - n) p + \\theta_0]}\\!", "8f6ce6c57b51f659d079bf933332200d": "e< \\varphi (N) ", "8f6d3e29c9d5872efec12c05c3dd9b38": " c: 2^N \\to \\mathbb{R} ", "8f6dd691ba2500b743516f8b80a240a0": "A x = b\\,", "8f6de95c7feebecae165f2e5ae674d60": "u(\\mathbf{r},t)=A (\\mathbf{r}) T(t).", "8f6dfc04ef51d33f8f6552b8d3cb2a26": "(3^k)", "8f6e173a1af877f1426eef9d3b8c2f88": "(X_i)_{i\\geq 1}", "8f6e395588af8809153ab6d42de38d77": "L^{q_0}(\\mu_2)", "8f6ee529467cac6deebbb9d757f40ae8": "m: \\mathbb{R} \\rightarrow \\mathbb{R}", "8f6f02ca02232d87e79409752841afee": "(m_i,M/m_i) = 1", "8f6f2c06a00e910f4d65a3ac37a39386": "4) \\ \\left(x+1 \\right)^2=3", "8f6f471d1ebacce6dc11c64525fabb93": "(x_0 + \\alpha t,y_0+\\beta t)", "8f6f5284a718dafbbd34c4559872ffb1": "\\delta A=\\{\\delta x:x\\in A\\}", "8f6f56cb7a2838baafd3fc5bdb32c171": "\n \\begin{align}\n \\varepsilon_{\\alpha\\beta} & = \\tfrac{1}{2}(u^0_{\\alpha,\\beta}+u^0_{\\beta,\\alpha})\n - x_3~w^0_{,\\alpha\\beta} \\\\\n \\varepsilon_{\\alpha 3} & = - w^0_{,\\alpha} + w^0_{,\\alpha} = 0 \\\\\n \\varepsilon_{33} & = 0\n \\end{align}\n", "8f6f7ef436ac2bffe142aef7999d27a7": "\n \\frac{\\partial p(x,t \\mid x_{0})}{\\partial t}=D\\frac{\\partial^2p(x,t \\mid\nx_{0})}{\\partial x^2},\n", "8f6f9a66deeffd466151242a19ef3ede": "\\textbf{D}_{I}", "8f6f9d5bf06a7f7e51bd389be8cba202": "t+dt\\;", "8f6fce330b125a03afbff4b2d5a3e05d": "g = g_{\\mu\\nu}dx^\\mu dx^\\nu.\\,", "8f700d45c089fb988b79584468d9c0c5": "a u + b \\frac{\\partial u}{\\partial n} =g \\qquad \\text{on} ~ \\partial \\Omega\\,", "8f7026132864b459c67f63e3f152d843": "f(z) = 1/z", "8f70346de692b9e9d55bc9c6d74c1f27": "v=-ux/y", "8f7061f69423ef9bbb4668baf5c3d66d": "g \\in \\langle F \\rangle", "8f70792a963e98509b3050fa1d14afbe": "\\int\\frac{dx}{R} = \\frac{1}{\\sqrt{a}}\\,\\operatorname{arsinh}\\frac{2ax+b}{\\sqrt{4ac-b^2}} \\qquad \\mbox{(for }a>0\\mbox{, }4ac-b^2>0\\mbox{)}", "8f7124e96f05e4952d4fb076488cd84f": "(\\{z\\in \\mathbb{C} : |z| \\geq 1\\}) = \\{z\\in \\mathbb{C} : |z| > 1\\}.", "8f718d7667d47795fdb2bf72c63c2bc9": "x\\equiv\\pm \\frac{L'}{3M'}\\pmod{q}", "8f719c58c7904363bf4ae7f1d1f86117": " \\text{skewness}(Y) =\\text{skewness}(X) = \\frac{2 (\\beta - \\alpha) \\sqrt{\\alpha + \\beta + 1} }{(\\alpha + \\beta + 2) \\sqrt{\\alpha \\beta}}.", "8f71b58651e4fd8298b89a9b86494352": "U(0)=0", "8f71d0e40a1f271a7688fa54d7a3622f": "a \\# h", "8f72357517dcf9bbdcda5b08ee004f5f": "Y_{\\ell}^m", "8f72870068af466e436a961418d1883a": "p(Y)", "8f72c261baa3a4301be396bcbfa1b28b": "\\mathit{K}_1", "8f72c454dbbaab19d683ce58b2813c7f": " I = \\iint \\mathbf{J}_\\mathrm{m} \\cdot \\mathrm{d} \\mathbf{A} ", "8f72cd2b8e5b4a130fb5601e7fd820e3": "\\left( {j - \\sigma } \\right) > 0 \\ ", "8f72e39c441441e34dd7d973e4428539": "\\mathrm{\\Delta G = \\Delta H - T \\cdot \\Delta S}.", "8f72eb76d418261bf6c5fb5d9e962ac6": "\\! L(y, F(x)),", "8f734da1da5af7e14c6d0e66c59ef33e": "s_x^2 = \\frac{1}{N} \\sum_{n=1}^N (x_n - \\bar{x})^2", "8f735e44ea6a28e55bfc23e1494227fc": "H(0)", "8f7376bdf8fcdaaa774529a779c9eed9": "A \\quad \\text{and} \\quad \\tilde A", "8f737aa13505fda2bfcb86fe6dbb3ee3": "T_L = \\frac{1}{ \\frac{1}{T_d - 56} + \\frac{\\log_e(T/T_d)}{800} } + 56", "8f737be0546a82ccdde164db0cd18942": " u_g u_h = u_{gh} c(g,h) . ", "8f7392f96315195301b945fe3c2a1260": "ln(Q)", "8f73ed1f4819417bc02dbc75319d2acd": "z = x/(\\sqrt{2}\\sigma)", "8f7422c65ec2257a0536b6bfe58df4ae": " {64 \\choose 8} = \\frac{64!}{8!(64-8)!} = 4,426,165,368.", "8f74747c0fce5773b427107f4584f3a3": "Re_w=\\frac{u_\\tau k_s}{\\nu}.\\ ", "8f74966c40d6a016989ef9b3a8722f60": "x = \\frac{\\alpha - E}{\\beta}\\,", "8f749f378cd4de59f8a9451f5524457c": "\n\\mathcal{H}=\n{\\displaystyle\\bigotimes\\limits_{i=0}^{\\infty}}\n\\ \\mathcal{H}_{i}.\n", "8f74e55622ba47b6fd36dee459035002": " V_R^\\circ = \\frac{j}{m} F(t_R - t_o) \\frac{T}{273.15} ", "8f750efd1b15bee713a7d8f5a11ba3ea": " \\frac{1}\n{{n^{i_1}}_{\\sigma(1)}{n^{i_2}}_{\\sigma(2)} \\cdots {n^{i_k}}_{\\sigma(k)}}", "8f7510a5b9be39c2aa0526cc8bac119e": "y (\\theta) = (R + r)\\sin\\theta - d\\sin\\left({R + r \\over r}\\theta\\right).\\,", "8f751ab6d0b4e37857fcb9318b7b7313": " K^{+}(k)\\hat{g}(k)=G^{+}(k)+G^{-}(k). ", "8f75716baf042b5749a4503886446699": "y_i=\\varepsilon_i(x);", "8f759465691e8fa5d2b01483e0095de7": "x_1^2,\\,x_2^2,\\dots,\\, x_n^2", "8f759fe0ce2888d23c40a919e72ebfbe": "\\mathbf{e}_i(t_0) = \\mathbf{e}_i \\mbox{, } 1 \\leq i \\leq n-1", "8f75b85789473ed25cc5d7347680c263": "C^\\infty_c(\\mathbb{R}^n)", "8f75ce9945430bac1ecab3892f93a462": "M_1 \\otimes_R N \\to M_2 \\otimes_R N", "8f75e9a2a29673292eb893f7b9ee3a18": "\\star \\mathbf e_{\\ell} = \\mathbf e_{\\ell} \\mathit i =\\mathbf e_{\\ell} \\mathbf{e_1e_2e_3} = \\mathbf e_m \\mathbf e_n \\,, ", "8f75f7e90a828a7b7e933777cb7dbe19": "(x_1,x_2,x_3,...,x_n)", "8f75fe29ca6f55dca4a029d578beecc7": "\\mathbf{h} = \\mathbf{r}\\times \\mathbf{v} = { \\mathbf{L} \\over \\mu } ", "8f761b951892d90deedb7a2eed8a7052": "~\\nu~", "8f762e130948b94e60bf0155e43b9552": "\n\\frac{\nP\\left( |X_t - \\phi_1(t)| \\leq \\varepsilon \\text{ for every }t\\in[0,T] \\right)\n}{\nP\\left( |X_t - \\phi_2(t)| \\leq \\varepsilon \\text{ for every }t\\in[0,T] \\right)\n}\n", "8f7658a234b39cd03712853b1640c639": "ed - 1 = h(p - 1)(q - 1)", "8f76651a02dea8337c0cf88f6e680eb5": "\\cos \\phi + i\\sin \\phi", "8f7685af494bc8e07941a34bcfd82c6e": "\\displaystyle K = \\tfrac{1}{2}\\sqrt{p^2q^2-(ac-bd)^2}", "8f76dadc706f67241a63c266d5246dd2": "I_y", "8f77066a081c7470b6281197a9e54ce0": "s=(s_1,\\ldots,s_n)^T", "8f770fe16fb098b03c5931f80bd1aee7": "\\omega \\in X", "8f771bc1b32729794ded2f43f30110d3": "P_n^m", "8f77807cdc451b2212af439ca34199d2": "\\forall n \\in \\mathbb{Z} (\\exists m \\in \\mathbb{Z} [m > n \\wedge P(m)] )", "8f77d120b1d39ad182ad3302bd292214": "\\delta(\\mathrm{state}_1, \\mathrm{read}, \\mathrm{pop}) = (\\mathrm{state}_2, \\mathrm{push})", "8f780130e0530af1814fa72308e3c91a": "\\,G(x,y)=u(x)-v(x)y", "8f7813b795b6eb4274fde8f34404a70c": " -\\,\\frac{8}{945} h^7 f^{(6)}(c) ", "8f781780c541cb43cac23d8578cacb5b": "u\\otimes_z v \\equiv Y\\left(\\begin{bmatrix} p \\\\ q \\end{bmatrix},z\\right)\\begin{bmatrix} r \\\\ s \\end{bmatrix} = \\begin{bmatrix} pr-qs - qrz \\\\ qr+ps -qsz \\end{bmatrix}", "8f783d0a9a92c65bd397de868151144c": "A(i\\omega)=e^{-\\gamma}\\left[ \\frac{4Z_IR}{(R+Z_I)^2-e^{-2\\gamma}(R-Z_I)^2} \\right]", "8f7914e8e81d01f3b18d84b9ffaf580d": "O(MN)", "8f793293023e307aa484c3c0554b2b05": "z_1= \\sin i \\cdot \\sin \\Omega", "8f7992eb689aaaa7a2d9a1c9b5e71161": "J:X\\to X''", "8f79af2fe9ce06b40f858cbfe9fcb30b": "G[\\mathbf{f}'] = \\left((Dy)^{-1}\\right)^\\mathrm{T} G[\\mathbf{f}](Dy)^{-1}\\,", "8f7a7546c84833f753903d27a48e6609": " \\lambda^2 = C/(GJ)", "8f7b02b055a53b745c1cb2bcd25e8789": " P(T) = \\text{max}\\left( S(T) - k A(0,T), 0 \\right),", "8f7b0bffd24dee92a528f56059ed343e": "(a_1, a_2, \\dots, a_n) \\equiv (X,Y,F)", "8f7b3baaca38dc2b8679ada34c3d51fb": "x^4(x^2-x-1)+1", "8f7b6a374265bae21a2a8a20329c66a7": "g\\!-\\!\\beta~", "8f7b9a37c0f10457f819969ac540187b": "= 3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow (\\cdots (3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3)))\\cdots ))\\, ", "8f7baa724af0e9092afdb05531c4496d": "f_v := \\inf\\{ f \\ge 0 \\mid v - f \\in d(X) \\}", "8f7bb509411af42189739664ca27416c": "C = V \\times S_{HSV}\\,\\!", "8f7bbf8a4680f5c1a3d6dfb9da1af2a6": "\\mu_1 \\ge \\cdots \\ge \\mu_n\\, ", "8f7c08b70faea7a38c51d41ac3316870": "k \\in m", "8f7cd9b9775c24f19bb6a1d98a83b326": "p_{i}\\left(k\\right)", "8f7cdd901a777b5b32a7f9ecfb531bb5": "H_M", "8f7d5bf8a2fa4fbd83d1084354127934": "\\, R_\\mathrm{E} = {1\\over 2} (m_\\mathrm{e} c^2) \\alpha^2", "8f7dfd99b8c9a8a579562b0a81f5cedc": "R_{N}(t) = \\frac{N_0}{2} \\delta (t)", "8f7e22e036e4be9a4d091a1e0cb29bb5": "e = \\frac{F}{N_{\\mathrm{A}}} ", "8f7e2ba3e29dedf41d0b83cdc2475420": "b_{H_2O}=(M_{H_2O})^{-1}=1/0.018,", "8f7e603742a8135ad8b8468f0a31f1d0": "\\textrm{dim}(\\mathcal{H}_A \\otimes \\mathcal{H}_B) \\le 6 ", "8f7e92bd3573b345ba05c2c413cec6b5": "\n\\mathcal{G}(\\mathbf{x},\\tau|\\mathbf{0},0) = \\frac{1}{\\mathcal{Z}}\\sum_{\\alpha'} \\mathrm{e}^{-\\beta E_{\\alpha'}}\n\\langle\\alpha' | \\psi(\\mathbf{x},\\tau)\\bar\\psi(\\mathbf{0},0) |\\alpha' \\rangle.\n", "8f7eba67313cec69ec67268e1bc8f08d": "F = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\nW(A) & \\\\\n & R(A) \\\\\n & W(A) \\\\\nCom. & \\\\\n & Com.\\\\\n &\\end{bmatrix} \nF2 = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\nW(A) & \\\\\n & R(A) \\\\\n & W(A) \\\\\nAbort & \\\\\n& Abort \\\\\n &\\end{bmatrix}", "8f7edd8f92d411a16beb30c72f36312c": "\\csc", "8f7efb2d5ea8a256a2c273db48f3a85d": " \\alpha \\in F ", "8f7efdb0f793b0f37e30254615137a11": "u'(x)=\\cos (x) \\sin^{n-2}(x)", "8f7f27a5a80c8523abf6b05b851e167b": "H(p, q) = \\frac {p^2}{2m} + V(q)", "8f7f2f4f0bd93b6cbddf026d7cf1f250": "\\mathbf{A}=\\mathbf{I}", "8f7f752b57a9c0ceff6fdbd7b19360dd": "\\Psi \\in \\phi ", "8f7f98f3ae4523b5c6905e5e1c881d7a": "\\partial_j f", "8f7ff182cb8c9115c3753d6a776cda8d": "(R^{\\gamma\\delta} \\, - \\frac{1}{2}g^{\\gamma\\delta}R)_{;\\gamma} \\, = 0", "8f801db293654b4fca9ab48b335e3625": "\\langle x-a\\rangle^1", "8f806388a14cc8a7faf46bace2f7329a": "f(i)= \\boldsymbol\\beta \\cdot \\mathbf{x}_i", "8f80af7b2f7657259d439b9bb0553005": "\n\\mathrm{SNR_{dB}} = 6.02 \\cdot (n-m)\n", "8f80e2e482d0df0f6420c5246cf18361": "q_t = a + \\alpha n_t + \\beta k_t + u_t", "8f8184092822d4790e7e391f992edd3d": "\\gamma_1 > \\gamma_2", "8f81d033b2293fc135c9ac65ad733de2": "Mg(x)", "8f82471f2bcc012469be243a355027b0": "\\scriptstyle \\log_{10} P_{mmHg} = 7.02447 - \\frac {1161.0} {224+T}", "8f82724f7e0dedbb2c3c34b5b80e91b5": "C_{SP}(t,\\omega) = \\iint C_s(t^',\\omega^')C_h(t^'-t,\\omega^'-\\omega)\\,dt^'\\,d\\omega^'", "8f828a18d2e5ceb503836c2c6cba8d86": "\nx_3 \\in [ 6,\\infty]\n", "8f82ad9c23528979015ac3896894aadc": "D=-i\\sigma_x\\partial_x-i\\sigma_y\\partial_y,\\,", "8f82f402c9c10d6fcbb521beecaa7375": "g(i\\omega)", "8f833185d915b1ae51a123a98da43bfa": "x^2(\\alpha(\\alpha-1)x^{\\alpha-2})-3x(\\alpha x^{\\alpha-1})+3x^\\alpha=\\alpha(\\alpha-1)x^\\alpha-3\\alpha x^\\alpha+3x^\\alpha = (\\alpha^2-4\\alpha+3)x^\\alpha = 0\\,.", "8f833489d5d0124c91679cac43277893": "\\mathbf{a}\\cdot \\mathbf{b} = \\sum{a_i \\overline{b_i}} ", "8f8340fea5be7d3bb75157c79fbbfcec": "B\\to F\\vdash((B\\to C)\\to F)\\to F.", "8f834c6d53f34488c8b06bd1b5450856": "\\bar{m}^a", "8f83527b21931fea49b4605420751a56": "A_o = A_{o}^{D} \\times A_{o}^{FI} \\times A_{o}^{L} \\times A_{o}^{C}", "8f839461f3014883904c3a8d7dcd8263": " \\lambda\\, ", "8f83e7d3ec56daf4036dba2ba226baf0": "P^0_0=1, P^1_1=P^2_2=P^3_3=-1", "8f8472f00932c36158677d9cab61505e": "f(x)=O(g(x))\\text{ as }x \\to a\\,", "8f8497f0255a19af2418256afca0ddbd": "\\{|x\\rangle\\}", "8f84ccf0af4280300b4487cd8508f56f": "\\sqrt(63342)=255\\frac{371}{511}", "8f84ce3b52dce79a339cdd6f818fb8d0": "\\rho = \\sqrt{\\frac{L}{C}} = \\frac{q(0)}{p(0)}", "8f852acb62e8c8cc61828801ba4c1ad4": "\\Phi(a)^*\\Phi(a) \\leq \\Vert\\Phi(1)\\Vert\\Phi(a^*a)", "8f8571c7907e3622325f4693a4e981bd": " \\left(p + \\frac{n^2 a}{V^2}\\right)(V-nb) = nRT.\n", "8f8593f758abf9bf7d16cafcb3d005ef": "h_{n,m}", "8f85b9c8cbd094d37e4164f4622829b4": "\\beta(X',X)", "8f861bac73b384bfc6e1df42b2d9ab7a": "\\left [\\begin{smallmatrix}2&-1\\\\-7&2\\end{smallmatrix}\\right ]", "8f8630f1b63df51d9b78d8601f84150d": "I(John : N) = j : E", "8f86646ff3a19914dd61da2aba9e928e": " T_{\\perp} \\equiv 0 ", "8f8689a30cf8327fbdf631273f597210": "\\! {\\sin x}/{x}", "8f86bb909c347a1c9ceb05aed359eda9": "{\\mathbb C}={\\mathbb R}\\oplus i{\\mathbb R}\\,", "8f873337e5693cda3ccab758caf93ef5": "\\scriptstyle\\gamma^\\sigma D_\\sigma\\!", "8f876015c928e579e00e66d87a403bba": "f_\\mu \\, = \\, F_{\\mu\\nu} \\, J^\\nu \\,", "8f8771a3cee8660657f9f01c798bb197": "\\,[H,\\mbox{T}(a)]=0", "8f8792a385dd1df54d8ebb4daa7296df": " O(1) ", "8f87b2c3a8fe76420041dd5334c2647b": "\\begin{array}{lcl}\n x' = x_0+r_2\\cos(\\omega_2 t+\\phi_2),\\ y' = y_0+r_2\\sin(\\omega_2 t+\\phi_2),\\ r_2\\ge 0\\\\\n\\therefore x = x_0+r_1\\cos(\\omega_1 t+\\phi_1)+r_2\\cos(\\omega_2 t+\\phi_2),\\ y = y_0+r_1 \\sin(\\omega_1 t+\\phi_1)+r_2\\sin(\\omega_2 t+\\phi_2),\\\\\n\\end{array}", "8f8852d6826c4bd8c82e54fb31a63efd": "- T", "8f88ee0cc317c2054b2484f0ead935c6": "[P_n,P_m]=0", "8f89096bb0127d277405b673142e4bce": "\\frac{s\\sin\\phi+\\omega \\cos\\phi}{s^2+\\omega^2} \\ ", "8f89185e9f1b3ed4e9f98042b332c9fb": "f(x)=\\textbf{c}_Ne^{\\textbf{A}_Nx}\\textbf{b}_N ,", "8f896ef6850affbc6607642b3b3bf5d5": "a_i=1 ", "8f8973a1aaba26e248bb4253b96db972": "\\scriptstyle \\hat{\\mathbf{e}}", "8f897ca65d3dd0c408c1cca602e9ca6b": "\\tilde{\\kappa}_{e-}=\\scriptstyle(-0.31\\pm0.73)\\times10^{-17}", "8f89a673f4bf71dd03729a6a9f3708b4": "\\alpha_{S}", "8f89f6d5e61a1e98355bca8128f68020": " GRLEX", "8f89fa98cd548002ac6195a9986d2469": "T_i = \\mathrm F(L_{i+1}' - R_{i+1}',K_i)", "8f8a147efb00cc8b9c6f9faf1feb1e00": "\\sum\\tau_j=0", "8f8a26de0202ddc8509e9dab125163ac": "\\frac{1}{3} + \\frac{1}{3} = \\frac{2}{3}", "8f8a9d05172aaec23d3d6d8b032e81f2": "d=a+b", "8f8aa6d4ad513c4acceb55a29ba3e487": "f:X\\otimes U\\otimes V\\to Y\\otimes U\\otimes V", "8f8b4b0b12bf04936692c7ab64c64579": "w_{ij} = \\exp{\\left(-\\beta (g_i - g_j)^2\\right)}.\n", "8f8b6a3fe752cef877315ae281d0a127": "\\tilde{S}_i", "8f8b901ba6e411bcac89fad54d3a2edd": "\n\\upsilon = \\frac{1}{\\sqrt{K\\rho}}\n", "8f8bbe75e1ba4618b900b6c8557b398e": "w_{d=1 \\dots M,w=1 \\dots N_d}", "8f8bd82dfab796c1823fb52f94048e33": "r'= {\\rm ROL}(l \\wedge KL_{i,1},1) \\oplus r", "8f8c0988f521e3707f242ae975a58a45": "S(a,b) = \\sum_{n=a}^b \\frac{p_n}{q_n}", "8f8c4567a01be68b1a46bf707f67a53a": " \\mathbf{v}_j \\leftarrow \\mathbf{v}_j - \\mathrm{proj}_{\\mathbf{v}_{i}} \\, (\\mathbf{v}_j) ", "8f8c58a675577741be31f8b3bf560298": "\\{a_1,...,a_r\\}", "8f8c5e5508b441f0787182638e117ecb": "\nH_\\kappa=\\prod_{(i,j)\\in\\kappa} h_\\kappa(i,j)=\n\\prod_{(i,j)\\in\\kappa} (\\kappa_i+\\kappa_j'-i-j+1)\n", "8f8c732b9e9007fdcc4cc6c446bf7a42": " \\varphi \\,", "8f8cd4f02fdb074fae2c361512bf1813": "\\gamma = \\alpha + i\\beta = 0 + i\\frac{1}{2} \\cos^{-1} \\left(1-\\frac{2m^2} {\\left(\\frac{\\omega_c}{\\omega}\\right)^2 - \\left(\\frac{\\omega_c}{\\omega_{\\infin}} \\right)^2} \\right)", "8f8cd53d44b78f122f5b2aa6a9642cef": " \\operatorname{P}[E_1\\cup E_2]=\\operatorname{P}[E_1]+\\operatorname{P}[E_2]", "8f8ce9267ba147fc193bea2693ecd267": " l^2 - r^2 = x^2 - 2\\cdot r\\cdot x\\cdot\\cos A + r^2[(\\cos^2 A + \\sin^2 A) - 1]", "8f8d1148c0d086ca4631970486d7cc3b": "\\sigma' \\neq \\sigma,", "8f8d28d7034acc8befa521948a338661": "\\delta_a(x) = \\frac{1}{a \\sqrt{\\pi}} \\mathrm{e}^{-x^2/a^2}", "8f8d2f400f39fd84b085e27223a3d90e": "E[Y|Z=z, X=1]-E[Y|Z=z,X=0]", "8f8dc6c2d7cf860327407e323e6ff3ed": " f \\not \\in FV(E) \\to (\\operatorname{let} f : f = E \\operatorname{in} L \\equiv (\\lambda f.L)\\ E) ", "8f8dd622225f7db3a1037c995f1366d9": "\\mathbb{E}[M_n-M_{n-1}\\,|\\,\\mathcal{F}_{n-1}]=0", "8f8df1f2d33e670b2afc883b0e281bff": "24\\Lambda\\,=0=\\,R_{ab}g^{ab}\\,=\\,R_{ab}\\Big(-2l^a n^b+2m^a\\bar{m}^b \\Big)\\; \\Rightarrow \\; R_{ab}l^a n^b\\,=\\,R_{ab}m^a\\bar{m}^b\\,,", "8f8f0a537f85014f796530aa624295c6": "k(C)/k(x)", "8f8f244059dea9571e4dc6efa8b992b3": "\n[\\partial_t(\\rho u_j)+\\partial_i(\\rho u_i u_j)+\\partial_j p] - u_j[\\partial_t \\rho+\\partial_i(\\rho u_i)]=\n\\rho \\partial_t u_j+\\rho u_i \\partial_i u_j+\\partial_j p=0\\,\n", "8f8f58ae35baad98a46f0522655d4dca": "\\tilde{\\Lambda}", "8f906ad9692aeba2aca5e86b43711064": "I_\\mathrm{GABA_A}(t,V) = \\bar{g}_\\mathrm{GABA_A} \\cdot ([O_1]+[O_2]) \\cdot (V(t)-E_\\mathrm{Cl})", "8f9097d9050fb619422b41ab3f3be304": "I_{C1} =\\frac{\\beta_1}{\\beta_1+1} \\left( \\frac {V_{CC}-V_{BE1}}{R_1}-\\frac{I_{C2}}{\\beta_2} \\right) ", "8f90b6581a0f7efbc18b466f9097c17f": "X_k = \\sum_{n=0}^{N-1} x_n \\cdot \\omega^{k n}.", "8f91073f02846c8115f817fb3500f8a5": "\n\\begin{align}\n\\dot{\\mathbf{x}}(t) &= \\mathbf{F}(t)\\mathbf{x}(t)+\\mathbf{B}(t)\\mathbf{u}(t)+\\mathbf{w}(t), &\\mathbf{w}(t) &\\sim N\\bigl(\\mathbf{0},\\mathbf{Q}(t)\\bigr) \\\\\n\\mathbf{z}_k &= \\mathbf{H}_k\\mathbf{x}_k+\\mathbf{v}_k, &\\mathbf{v}_k &\\sim N(\\mathbf{0},\\mathbf{R}_k)\n\\end{align}\n", "8f915870299d847e9dea6bf4337546d2": "\\kappa_{\\max} =\\frac{P_{\\max} - P_{\\exp}}{1-P_{\\exp}}", "8f9262352ed53ead5dda0e01cdbec11b": "\\frac{2}{\\sqrt{3}}", "8f92643a97baf956472c38fbaa76e18b": "\\textstyle P_{X_{r}S_{r}Y_{r}}", "8f9296994dbcd2b9eab8912e5363f4c4": "y^n = k(a - x)^px^m ", "8f931947978fcb668c93620a7a8ce3b9": "\n\\max_{\\alpha \\ge 0} \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} \\vartheta(q,\\alpha,u)\n", "8f9337b298b2c969f54f2af24014f761": "\\mathbf{F}_{ext} = m_{rocket}(t) \\frac{\\mathrm{d}\\mathbf{V}}{\\mathrm{d}t} + \\mathbf{V}(t) \\frac{\\mathrm{d}m_{rocket}}{\\mathrm{d}t} = - \\left(\\mathbf{v_e}(t) \\frac{\\mathrm{d}m_{gas}}{\\mathrm{d}t} + F_{other}(t)\\right).", "8f934f6c3aebe63b11310c118c320fae": "\\frac{N^2_{nl}}{2\\gamma^{l+{3 \\over 2}}}\n\\int^\\infty_0 q^{l + {1 \\over 2}} e^{-q} \\left [ L^{(l+\\frac{1}{2})}_{\\frac{1}{2}(n-l)}(q) \\right ]^2 \\, dq = 1.", "8f936ece9eeb340d1c7aa37f0c153757": "\\forall w\\,\\exists u\\,(w\\,R\\,u\\land\\forall v\\,(u\\,R\\,v\\Rightarrow u=v))", "8f937611808fe0e510f1184fa1306a8a": "\\mathbf{F}_2", "8f9376b2c202ffcb9e6188ba0ff3d764": "y_0(x) = -\\cos(x)/x", "8f940b6aae0ac2dda66083d39f232a43": "\\mathcal{O}_{\\mathbf{C}_p}", "8f9421bad588dbf663ecf93f26f7f5bc": "x + x + x", "8f94683cd42b1f173fa99329bbf0c052": "\\Delta \\beta", "8f94bce9e42f4c48263e243cc415ad4e": "m-1\\,", "8f95161ea583a3639617e12969e38882": "\\begin{align}\n(\\tfrac{2}{5}) &= -1, &F_3 &= 2, &F_2&=1, \\\\\n(\\tfrac{3}{5}) &= -1, &F_4 &= 3,&F_3&=2, \\\\\n(\\tfrac{5}{5}) &= 0, &F_5 &= 5, \\\\\n(\\tfrac{7}{5}) &= -1, &F_8 &= 21,&F_7&=13, \\\\\n(\\tfrac{11}{5})& = +1, &F_{10}& = 55, &F_{11}&=89.\n\\end{align}", "8f95bcb758f8d2b471f986b019e0d505": "CBR=\\frac {p}{p_s} \\cdot 100 \\quad ", "8f95c3e8951c5103887294c988c8cf6e": " {P_{avg} = {1 \\over 2} R_a \\, I_\\circ^2 } ", "8f9620c53bf613b02f467b42a3bc9215": " \\mathbf y ", "8f969a288c11561077c4314e14f28265": "p_1=1+m_1\\ ,", "8f969d3a92c2e1122d3abae66467b24f": "\\begin{bmatrix} 1 & 3 & 1 & 4 \\\\ 2 & 7 & 3 & 9 \\\\ 1 & 5 & 3 & 1 \\\\ 1 & 2 & 0 & 8 \\end{bmatrix}\n\\sim \\begin{bmatrix} 1 & 3 & 1 & 4 \\\\ 0 & 1 & 1 & 1 \\\\ 0 & 2 & 2 & -3 \\\\ 0 & -1 & -1 & 4 \\end{bmatrix}\n\\sim \\begin{bmatrix} 1 & 0 & -2 & 1 \\\\ 0 & 1 & 1 & 1 \\\\ 0 & 0 & 0 & -5 \\\\ 0 & 0 & 0 & 5 \\end{bmatrix}\n\\sim \\begin{bmatrix} 1 & 0 & -2 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}\\text{.}", "8f96b8156465f7b64818d041c2fe631d": "(x,y)\\mapsto (v(\\operatorname{arcosh} y)\\cos x, v(\\operatorname{arcosh} y) \\sin x, u(\\operatorname{arcosh} y))", "8f9705dea25f6523ac66dede7bc289fd": "\\frac{q_{rev}}{T} = \\frac{6008 J}{273 K}", "8f972ea1c2ec7d9e4f335f7424135b5e": "\\int_{-\\infty}^{\\infty}d\\epsilon_xd\\epsilon_yd\\epsilon_z", "8f974f2ab449e023be40cec79a3e22fe": "\\tilde{U}=\\{(s,t) \\in G/H \\times G/H: \\ \\ t \\in U \\cdot s \\}", "8f9774ed20d502280a69f599b7939eff": "\\ \\sigma_R", "8f97f642ed30fecbc12d75168b618ac6": "s(\\cdot,\\cdot)", "8f97fca178fbaf5bea2ecd0ffcfdd732": " F(R_1,R_2, \\ldots, R_N)", "8f980ab80b79f276df22bff678bb880f": " \\mathbf L ", "8f98330f9f69eb0cb2fb1d0e6d21e0d7": "TR = \\frac{I_C}{I_E}", "8f9839d635bf68407161826c4953d1fd": "\\sqrt{|V|}", "8f983fed9877557215a8765e471e4124": "f \\circ \\phi = \\psi \\circ g", "8f9840aac215a2c0c968ea0d85d65994": "g\\left(f_j(e_{ij})_{i \\in n_j}\\right)_{j \\in m} = g(f_i)_{i \\in m}(e_{ij})_{i \\in n_j, j \\in m}", "8f9843c72f9cb875a5f681f5fcd70641": "\\,\\theta", "8f98a181db4d1865b779b2c8b68e803f": " I_2 \\frac{1}{c}\\frac{\\partial \\psi}{\\partial t} + \\sigma_x\\frac{\\partial \\psi}{\\partial x} + \\sigma_y\\frac{\\partial \\psi}{\\partial y} + \\sigma_z\\frac{\\partial \\psi}{\\partial z}=0", "8f98cbf063b440cf055995496a4e6e6d": "\n \\begin{align}\n \\delta U & = \\int_{\\Omega^0} \\left[-N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta}\n + \\left(M_{\\alpha\\beta,\\beta}-Q_\\alpha\\right)~\\delta \\varphi_{\\alpha} - Q_{\\alpha,\\alpha}~\\delta w^0\\right]~d\\Omega \\\\\n & + \\int_{\\Gamma^0} \\left[n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta}\n- n_\\beta~M_{\\alpha\\beta}~\\delta \\varphi_{\\alpha} + n_\\alpha~Q_\\alpha~\\delta w^0\\right]~d\\Gamma\n \\end{align}\n", "8f98d6cd247fcb2817e5800ec2763af2": "\\frac{\\omega_{\\mathrm{obs}}}{\\omega_s} = \\frac{\\sqrt{1+\\beta}}{\\sqrt{1-\\beta}} \\,", "8f992c953c94273a204c624d90b6719f": " L_{t_0} := \\{ {\\bold x} \\in \\R^2 : F(t_0,{\\bold x}) = 0 \\} ", "8f9958b80048e7d48a1ae72704a21fe5": "2^E = \\{0,1\\}^E.", "8f99742affd58c1934b8b35973ba537f": " 0 \\leq p \\leq 1", "8f99806b6d760bab74fef0ce760c2db0": " \\textbf{f} = -1 + X + X^2 - X^4 + X^6 +X^9 - X^{10} ", "8f9989c4657f07027bbf691c55086c05": "\\theta = k\\lambda / d \\,", "8f99a219101ea1d76bc574be48fc9e6e": "p(x) = \\|x\\|", "8f99d3487ff273ccd1d0d1e925e8b61b": "\nZ(\\mathbf{s}) = m(\\mathbf{s}) + \\varepsilon '(\\mathbf{s}) + \\varepsilon ''\n", "8f99e932c2209a08b6a5d42fa4518391": "\\tau(\\varphi)\\ \\stackrel{\\text{def}}{=}\\ \\operatorname{trace}_g\\nabla d\\varphi = 0", "8f99e9ca48efdaf6f578fd6cac97595c": "X_1 , \\ldots , X_t = \\{V (x_0 , y_0 , i) : 1 \\leqslant i \\leqslant t\\} \\, ", "8f9a9a267f60f6bf4406351e6447cfc2": "N\\setminus S \\in W", "8f9a9e8c6d13c0b786f532baaaf0eba7": " \\beta = 0 ", "8f9aa35b1249a79395f077bcd40ee867": "v_{\\mathrm{Euclidean}}:(x, y) \\mapsto (\\cos \\theta, \\sin \\theta) = \\left(\\frac{x}{\\sqrt{{x^2 + y^2}}}, \\frac{y}{\\sqrt{x^2 + y^2}}\\right).", "8f9aac5179a527a42c3494fccd13f746": "f_a: [0,\\infty) \\rightarrow \\mathbf{R}", "8f9acfe197cde4dd89b24151f1584703": "s = \\sigma + i \\omega, \\, ", "8f9b0877ad9f3ef82bbe26f520ff7c4a": "X_i^2", "8f9b23be27574a264d2a22c58de1c4c1": " \\operatorname{Var}_{X_i}(E_{\\mathbf{X}_{\\sim i}}(Y|X_i)) \\approx\n { \\frac {1}{N} \\sum_{j=1}^{N} f \\left ( \\mathbf{B} \\right )_{j} \\left ( f \\left ( \\mathbf{A}^i_B \\right )_{j} - f \\left ( \\mathbf{A} \\right )_{j} \\right ) } ", "8f9b3c00f349825176bf5a8a7b77a0b3": " \\rho _{cv}^{{\\mathop{\\rm int}} } (k,t) = \\int {\\frac{{d\\omega }}{{2\\pi }}\\frac{{d_{cv} \\varepsilon (\\omega )e^{i(\\varepsilon _{c,k} - \\varepsilon _{v,k} - \\omega )t} }}{{\\hbar (\\varepsilon _{c,k} - \\varepsilon _{v,k} - \\omega - i\\gamma )}}(f_{v,k} - f_{c,k} )}", "8f9bf96916ef8f03a7a77559b2d9b6c2": "w \\cdot \\log(n/w) > r", "8f9c3e917acdb41d81132d976b619123": "\\tilde{H}_n\\left(S^k\\right)\\cong\\delta_{kn}\\,\\mathbb{Z}=\\left\\{\\begin{matrix} \n\\mathbb{Z} & \\mbox{if } n=k \\\\ \n0 & \\mbox{if } n \\ne k \\end{matrix}\\right.", "8f9c515f7fdd5fe9424cb404874ed200": "y = 3", "8f9c74f6cbe7aaba6b93ea130efa9eeb": "\n \\operatorname{Cov}[\\, \\hat\\beta,\\hat\\varepsilon \\,|X\\,] = 0.\n ", "8f9cc006015696b8c58ceb71f8747720": "y^2 \\equiv 4a", "8f9ddc4f3c87597ba27c7ae8a95bf3e5": "[x=y]P", "8f9e449ca9d2e84079f6e0e57430338f": "{\\mathbf{}}S_{i} = A'_i \\left( S_{i+1} - S_{i+1}B_i \\left( B'_iS_{i+1}B_i+R_i \\right)^{-1} B'_i S_{i+1} \\right) A_i+Q_i+\\tau'_{\\perp i}\\Psi^2_{i+1} \\tau_{\\perp i}, S_N=F", "8f9e45fd86bc2ca71deaa4ca903aecb7": "H_0|n\\rangle=E_n|n\\rangle", "8f9e67469611e464fcee1783ffeb2376": "V_\\mathrm{pp} = \\frac{I}{2fC}", "8f9ead2a0e512b17ea874dade5b447d3": "\\mu_{l}", "8f9ebf920afd9021e14775956ce2283f": "\\left( X;\\ast ,0\\right)", "8f9f19ebce441b4a2076b92466a2eeb1": "\\mathit{alg}_A", "8f9f6f153bed23b02cd8f13ca682be71": "x_1=\\frac{1+\\sqrt{-31}}{4}\\,\\!", "8f9f78842e48d9343a7e4c30d83b27e9": "\\varphi = {1 + \\sqrt{5} \\over 2}", "8f9f8db97abfb33c891a931a46ace0fe": " \\begin{align}\n E(x) & {} = \\frac{a + 4m + b}{6} \\\\\n \\operatorname{Var} (x) & {} = \\frac{(b-a)^2}{36}\n\\end{align}\n", "8f9fa226387cb2c8cc4052bd13a22062": "m_2L_2\\ddot{\\theta}^2+m_2L_1L_2\\ddot{\\theta}_1\\cos(\\theta_2-\\theta_1) - m_2L_1L_2\\sin(\\theta_2-\\theta_1)=-m_2gL_2\\sin\\theta_2.", "8f9fdb8a695aea926d9e48f91cdb08d7": "\n\\begin{align}\nx + {1 \\over x} &{} = \\left(x - 2 + {1 \\over x}\\right) + 2\\\\\n &{}= \\left(\\sqrt{x} - {1 \\over \\sqrt{x}}\\right)^2 + 2\n\\end{align}", "8f9ffd40661d514a9e6ec34c47669b0e": "\\frac{1}{2}\\left( \\frac{d\\tilde{a}}{d\\tilde{t}}\\right)^2 + U_{\\rm eff}(\\tilde{a})=\\frac{1}{2}\\Omega_c", "8fa0024997bb65c40743e4ddf4e1046a": "\\overline{A(x_1, ..., x_n)}", "8fa040578d15e1ca7462abe7c25c3251": "\\frac{6(V_c)^{2/3}}{A_c}", "8fa07e699ae2532c33f4762999214738": "S[\\phi,\\psi]=\\int d^dx \\;\\left[\n\\mathcal{L}_\\mathrm{meson}(\\phi) +\n\\mathcal{L}_\\mathrm{Dirac}(\\psi) +\n\\mathcal{L}_\\mathrm{Yukawa}(\\phi,\\psi) \\right]\n", "8fa08ac93be176c0470a467da93c0d2b": " \\nu = \\frac{V}{m} = \\frac{1}{\\rho} ", "8fa0aa627d5e95111dcdde7539d6a6b4": " \\mathbf{P} \\left( \\bigg\\Vert \\frac{1}{t} \\sum_{i=1}^t M_i - \\mathbf{E}[M] \\bigg\\Vert_2 > \\varepsilon \\right) \\leq d \\exp \\left( -C \\frac{\\varepsilon^2 t}{\\gamma^2} \\right).", "8fa0b02873e89f157b309f2097cee78b": "N=\\sqrt{g \\over L_\\rho}", "8fa0b7c5cd597516607b87807111f36e": "C_{\\mu}", "8fa0c6c9f11a1fff01f9403fd108b5cb": "\n \\langle p |q_j\\rangle = \\frac{\\exp\\left( {i\\over \\hbar} p q_{j} \\right)}{\\sqrt{\\hbar}}", "8fa1207f5782a894de7120c073ada7b2": "\\hat{G}_X = \\prod_{i=1}^{N} \\left (\\frac{Y_i - \\hat{a}}{\\hat{c}-\\hat{a}} \\right )^{\\frac{1}{N}}", "8fa14cdd754f91cc6554c9e71929cce7": "f", "8fa163f22e987e2f693d8e3fdedf4c6b": "\n\\partial_t u=-\\frac{\\delta\\mathfrak L}{\\delta u}\n", "8fa178c5d0d8a34454b288bd539e94c4": " C^{k,\\alpha}(\\overline{\\Omega}) ", "8fa1d0317e982f4d576e285b15f32b5c": "\\alpha = P(T\\in C|H_0). \\, ", "8fa1dc5dd1991046ac4c8519e24bf097": "(\\xi \\wedge \\eta)_{sup}(\\alpha)=\\xi_{sup}(\\alpha)\\wedge\\eta_{sup}{\\alpha}", "8fa203fbb8824c47cae18b7afe32754c": "P(s) = e^{-s}.\\ ", "8fa2536ce6c002c4895d29d47b811c4a": "\\cos(5\\pi/7)", "8fa255025c6030d3b1e10e519004be5f": "\n\\frac{x^{2}}{a^{2} \\cosh^{2} \\mu} + \\frac{y^{2}}{a^{2} \\sinh^{2} \\mu} = \\cos^{2} \\nu + \\sin^{2} \\nu = 1\n", "8fa2693f56ea23f419b6f19e78085517": "\\neg Pacifist(Nixon)", "8fa2af052807388be5b70dd4370a316b": " (\\dot{\\tau}e^{-[\\mathit{k}/{\\mathit{k}_\\mathit{D}(\\eta)}]^2})", "8fa2fe1814508a395dcd4d60f40938c2": "(\\ast_{v\\in V} A_v) \\ast F(E)", "8fa3076dcff6531d06bed6ee91ef0159": "I(V+k\\sin(\\omega t))", "8fa30e06b77bfd8e33509578484833fe": "\\Delta\\chi\\equiv\\chi_\\parallel-\\chi_\\perp=N", "8fa334e3560752745b0bffc940a6ee41": "c_\\mathrm d = \\dfrac{2 F_\\mathrm d}{\\rho v^2 A}\\, ", "8fa3390a4b5e9bbddcab6788f07c19bf": "\\mathbf{F}=m\\mathbf{a} = m \\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t}", "8fa3a76f9b931cc2877e0be935943d1f": " \\mathbf X'\\tilde{\\mathbf W}_\\delta\\mathbf X\\hat{\\boldsymbol\\theta} = \\mathbf X'\\tilde{\\mathbf W}_\\delta\\tilde{\\mathbf z} ,", "8fa3aa401f8f4283840a95a442953cf0": "\\frac{\\partial T}{\\partial x'} = \\frac{1}{v}\\cdot\\frac{\\partial T}{\\partial t}", "8fa3c650ea0d8a7c709173362f654b54": "\\Delta^{*}\\psi = -\\mu_{0}R^{2}\\frac{dp}{d\\psi}-\\frac{1}{2}\\frac{dF^2}{d\\psi}", "8fa3f1e3ec38ad5d496d1be192168699": "f \\colon U\\to U , \\,\\!", "8fa44828f253b241da1ef219bf65c050": "F_{2 \\omega} = - \\frac{1}{4} \\frac{dC}{dz} V^2_{AC} \\cos(2 \\omega t)", "8fa4e50041e09dcafc6bb880f768b75c": " \\mu_\\mathrm{B} ", "8fa4f4772971e14176c78ac0067c6e11": "n\\geq N", "8fa52da699395ddeeb32f908efafc052": "\\gamma_i = P_{i,i-1} / P_{i,i+1}", "8fa542beada5bacfc30f2e88964c0905": "\\frac{b t_c}{a} = 1 \\Rightarrow t_c = \\frac{a}{b}.", "8fa55211271fb15225a7fcc11aab0aa1": "C\\sqcup D", "8fa5a59eae43c3a9dcd034a7363bd14f": "\\left(-2\\sqrt{\\frac{2}{5}},\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{4}{\\sqrt{3}},\\ \\pm2\\right)", "8fa5e4966564a27bd079dc065b6cfb16": "f(z) = \\frac{a z + b}{c z + d}.", "8fa600d263015b6c03749c037b5a51b7": " X' \\widehat \\otimes_\\pi X \\ \\longrightarrow X' \\widehat \\otimes_\\varepsilon X ", "8fa65cdded9de74747d6c5f7b416539b": "\\sigma[1] \\sigma[2] \\ldots \\sigma[L]", "8fa68114756fe33f544e00d963286007": "\\mu\\left([x_0, x_1,\\ldots,x_n]\\right)=\n\\prod_{i=0}^n p_{x_i}", "8fa68c04fc4d4f9cbe396358c6a538e8": "\\sigma^2_{\\eta}", "8fa711e56e5b52fb93536960d1dac670": " y_{ }^{ } = O^T x ", "8fa716e721573dfd119422e8c1e8bbe9": "\\epsilon_r = \\epsilon_r'+j\\epsilon_r''", "8fa772765556f3fdced49c2683a04fe0": "\\ q=Q(x)", "8fa7e6bd1106c0e60c52279114786f95": "y_s", "8fa816f6e6031ff48acda6b660525712": " F(x)=1-\\sum_{i=1}^n \\alpha_i e^{-\\lambda_i x}=\\sum_{i=1}^n\\alpha_iF_{X_i}(x). ", "8fa889721e58a1f42232c0a3608524c0": "\\hat{f}(x)", "8fa898ddc7ffcf01f3a4cfa9199870ec": "\\frac{dW}{dt} = \\frac{DA(C_{s}-C)}{L}", "8fa8a2fde60cacfdec7544095045b011": "-1.01446", "8fa8d7a9307b762f57a0b5fe5ced56a5": "C_1=M_1^e\\, \\bmod\\, N", "8fa92a708c695c49d91d3317b260e503": "\\lnot Pxy \\rightarrow \\exists z[Pzx \\and \\lnot Ozy \\and \\lnot \\exists v [PPvz]].", "8fa98fba75c9f794fdf04ed07b750e8d": " \\mathcal{B} = \\mathfrak{P}(\\mathfrak{P}_{\\ge 1}(\\mathcal{Z})),", "8fa9b8e0e79a8a9f508ec83c62cf1410": "x_1^3+x_2^3=x_3^3+x_4^3", "8fa9c98a7bf54643dc9e616c2d55b7c5": "\\displaystyle{\\nabla S(\\varphi) = -D(\\varphi \\mathbf{n}) + S(\\partial_t (\\varphi \\mathbf{t})),\\,\\,\\, \n\\nabla D(\\varphi)=\\widetilde{\\nabla} S(\\dot{\\varphi}).}", "8fa9cee6d53cf4e77f6b5b9c66712d47": "\\Gamma(\\alpha,x)=x^\\alpha e^{-x} \\sum_{i=0}^\\infty \\frac{L_i^{(\\alpha)}(x)}{1+i} \\qquad \\left(\\Re(\\alpha)>-1 , x > 0\\right).", "8fa9f4f02dd35a4f76228959db4ff49e": "\n\\frac{d^{2}\\xi}{d\\tau^{2}} = 0\n", "8faa29e612f0bee303831a4adaefab2b": "\\left[{0\\atop k}\\right] = 0", "8faa7f0b21ca9cb9550eb152b9fe260b": "\\| C y_n - C y_m \\| = \\| (C-I) y_n + y_n - (C-I) y_m - y_m \\|", "8faad6f6de05b4ad85f18457dbf71daa": "P_2=(1,2,1,\\sqrt{5})", "8faaf579264ca00a564f4a1b6b01fd10": "\\beta_0 \\mbox{ and } \\beta_1 ", "8fab1a6abb269e29eb6e92ce4cf07749": "\\varepsilon(\\omega)=1-\\frac{\\omega_{P}^2}{\\omega^2},", "8fab417c8d2906fce0da963db640916b": " x_1,\\dots,x_n", "8fab5f00febf5adcb8ba4f0ed02853e8": "\n\\cfrac{1}{1 + \\cfrac{e^{-2\\pi\\sqrt{5}}}{1 + \\cfrac{e^{-4\\pi\\sqrt{5}}}{1 + \\cfrac{e^{-6\\pi\\sqrt{5}}}{1 + \\ddots}}}}\n= \\left( {\\sqrt{5} \\over 1 + \\left[5^{3/4}(\\varphi - 1)^{5/2} - 1\\right]^{1/5}} - \\varphi \\right)e^{2\\pi/\\sqrt{5}}.\n", "8fab878249d74fd3064e8834b09df016": "\n\\overline{ \\hat{ \\phi } } = \\hat{G} \\hat{\\phi} .\n", "8fab9310b295890ca95f817177b4bbe4": "\\lambda g(x)(y) = g(x,y).\\,", "8fac458e1ed10b8a67b468b9f637ee9e": "{n\\choose k}={49\\choose 6}={49\\over 6} * {48\\over 5} * {47\\over 4} * {46\\over 3} * {45\\over 2} * {44\\over 1} ", "8fac62a24bde65a2d759f4ab9df2bb19": "\\phi: V \\rightarrow V\\,", "8fac80756b28c132d6316662f620583d": "\\phi_a(z) =\\frac{z-a}{1 - \\bar{a}z},", "8fad0619dbca3515c06ca03c6c70c7e3": " A_{ij} = \\sum_{k = 1}^N P_{ki} Q_{kj}, ", "8fadf69fe62d58d0bb827675609c04fe": "(S, \\rho)", "8faeda1bcd27558be31790a82d46de47": "x_m \\not\\succ x_1", "8faf6e36ff53551183003d8521947012": "\\sum_{n=0}^\\infty a_n = \\lim_{N\\to\\infty} S_N = \\lim_{N\\to\\infty} \\sum_{n=0}^N a_n.", "8faf92ae5e9f4f293d8d2e3fb871f780": "\\beta < \\delta \\,", "8fafd420272f076cfb05b0d45b535442": "\\aleph_0", "8fb0025f3a3e8c25018f7e9f854eb17e": "\n\\begin{align}\nE(x) & = \\sum_i x^\\top \\hat n_i \\hat n_i^\\top x - x^\\top \\hat n_i \\hat n_i^\\top p_i - p_i \\hat n_i \\hat n_i^\\top x + p_i^\\top \\hat n_i \\hat n_i^\\top p \\\\\n& = x^\\top \\left(\\sum_i \\hat n_i \\hat n_i^\\top\\right) x - 2 x^\\top \\left(\\sum_i \\hat n_i \\hat n_i^\\top p_i\\right) + \\sum_i p_i^\\top \\hat n_i \\hat n_i^\\top p_i.\n\\end{align}\n", "8fb01c18e9278c6aee69e145204fd4f8": " V^{2} = x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + \\rho(x,u,u_{1},u_{2})\\frac{\\partial}{\\partial u_{1}} + \\phi(x,u,u_{1},u_{2})\\frac{\\partial}{\\partial u_{2}} \\,", "8fb076a295368bf5a22c57ae53714e55": "\nQ\\left(p;\\lambda\\right) = \n\\begin{cases}\n\\frac{ 1 }{ \\lambda } \\left[p^\\lambda - (1 - p)^\\lambda\\right], & \\mbox{if } \\lambda \\ne 0 \\\\\n\\log(\\frac{p}{1-p}), & \\mbox{if } \\lambda = 0,\n\\end{cases}", "8fb07af00d667fd86004f9ba621c8136": "du/dt+Au+f(u)=0", "8fb0c6fe7cf7bbae01399b4ecbe382a4": "E[Y] = \\mu = \\frac{\\sigma\\sqrt{2}}{\\sqrt{\\pi}}", "8fb118bd3f627b51996b89c7115a8a9e": "q=\\frac{pn}{n-p}", "8fb1696d661c3ec1b052a458f4bed227": " L(p, w, r),", "8fb17f33ecfea96d828bc790d6d40e0a": "k=4", "8fb21612b175e7abd6a11b92105132c8": "x\\mapsto x^2+c", "8fb2af3c4c811eb7f04c7166c2d4fa5f": "P: \\mathbb{Z} \\times U \\to U.", "8fb2b43781fcba81ff069c2b96933316": " n_i p_{jh}^{i} = n_j p_{ih}^{j} ", "8fb2c1ebb6cef03852d5499ad9db252a": "r \\cdot m = f(r) \\cdot m", "8fb3297ad1d41538c22f86675e79b268": "\\displaystyle{ B(a,b)B(a^b,c)=B(a,b+c)}", "8fb39fce548a77fc44f6da7586f9d0fb": "g_n(z)= \\lambda^{-n} f^n(z)", "8fb3b707015e315eb29070a3e4b57bd1": "\\int_0^1 x^{n-1}(1 - x)^{n-1} \\, dx", "8fb3bc1cd401e47b2d8f4b3f07d217f2": "\\mathcal{T}_{\\mathcal{A}}(x)\\,\\!", "8fb3ebc36a45cb5b22673d0cc0160e4b": "\\mbox{H} \n= \\dfrac{\\mbox{m}^2 \\cdot \\mbox{kg}}{\\mbox{C}^2}\n= \\dfrac{\\mbox{m}^2 \\cdot \\mbox{kg}}{\\mbox{s}^{2} \\cdot \\mbox{A}^2}\n= \\dfrac{\\mbox{J}}{\\mbox{A}^2} \n= \\dfrac{\\mbox{Wb}}{\\mbox{A}}\n= \\dfrac{\\mbox{V} \\cdot \\mbox{s}}{\\mbox{A}} \n= \\dfrac{\\mbox{s}^2}{\\mbox{F}} \n= \\Omega \\cdot \\mbox{s}\n", "8fb3f69f8d7ac9b8b230ca3093a7c468": "u=g\\left( x,y \\right),\\ \\ \\left( x,y \\right)\\in \\partial \\Omega_D", "8fb40314c5256d1e407199bb1d452c0e": " \\mathbf{\\hat{p}} = -i \\hbar \\nabla \\,\\!", "8fb43d239993a6daa031bc425050aa52": " \\sqrt[5]{34} = 2.024397458 \\ldots, ", "8fb4f8a7e41ca1a943fe1d436b0317d7": "A_m(0,3) = 1,3,3,1", "8fb4fda47d53a86b6aa60639b01be223": "x - 1 - \\frac{1}{2} - \\frac{1}{4} - \\frac{1}{5} - \\frac{1}{6} - \\frac{1}{9} - \\cdots = 1", "8fb5119ca19d99c8c070ce8df67d6b2a": "B({{v}_{2}})0.\n\\end{align}\n", "8fc086039255c431e511c67c88b79323": "D_0,D_1,D_2,\\dots\\ ", "8fc105528643ec9734489a2f112bbf17": "O(n\\log n)\\,", "8fc1351768bd6be21db516a18281368d": "{\\varphi^\\prime}_v = -\\varphi_v + \\frac{2}{\\beta} \\sin\\left(\\frac{\\varphi^\\prime - \\varphi}{2}\\right)\\text{ with }\\varphi = \\varphi_0 = 0", "8fc266502194c5b7296ac7721e91adde": " \\text{If } \\frac{a}{b} = \\frac{c}{d} \\text{ and } a \\neq b \\text{, then } \\frac{a+b}{a-b} = \\frac{\\frac{a}{b} + 1}{\\frac{a}{b} - 1} = \\frac{\\frac{c}{d} + 1}{\\frac{c}{d} - 1} = \\frac{c+d}{c-d}. ", "8fc27b209ba0923aa52897d4cb0a3656": "\\pm \\!\\,", "8fc37c8823d85e8df1f6ce2b76530866": " \\psi_P(x)\\,\\!", "8fc393bfd72e3f62f168b46d8d10983e": " Y=X\\beta+\\epsilon ", "8fc3bf94cbb3c8eb9016c0af20cb9672": "q = 0.85", "8fc3d82279dc960e0f8ff43918096cfb": "\\,Q\\times (\\ddagger\\Gamma^+)^* \\times \\Gamma^* \\times (\\ddagger\\Gamma^+)^*", "8fc422d218f22670a02fdfc3f8610bc8": " E = m c^2 + \\tfrac12 m v^2 ", "8fc44d24aac07d975b1fff7858d96fab": "\nh_{k}(\\mathbf{q})\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{g_{kk}(\\mathbf{q})} = |\\mathbf e_k|\n", "8fc4cc2976a44a2eef2ebbaa1c0f4bc8": "f(N_i) \\subseteq M_i", "8fc4e12b414fe3a306a5c1af7fe8bc26": " \\epsilon_{i}^\\mu \\epsilon_{j}^\\mu ", "8fc519c5fe8f80db3ddc080d43f5e5ea": "\\tilde{\\rho}(1)\\cdot v = v", "8fc52df077b0c5aafe71fa3c4c31fa54": " \\text{MSE}(\\widehat{\\boldsymbol{\\beta}}_{ols}) - \\text{MSE}(\\widehat{\\boldsymbol{\\beta}}_{k}) \\succeq 0 ", "8fc59a5f71f450d27f2561b46d478083": "\\alpha_s", "8fc5b3ee062b3c3c08728ecdd5a598f5": "A:B = \\left( \\tfrac{1+H}{2} \\right): \\left( \\tfrac{1-H}{2} \\right)", "8fc5fe5172e0d7aba534a8ba6ab622da": "f:[0,1] \\to \\mathbf{R}", "8fc6677e2a98a84d29b9976db28b8f15": "2^{i}*n_{i}", "8fc67dea7ce91a054e5ab0977f665654": "\\text{Trail} = \\frac{(R_w \\cos(A_h) - O_f)}{\\sin(A_h)}", "8fc68db27c8ade7bdaf2f8e898c1426f": "\\textstyle l(s) \\leq \\Lambda", "8fc6ab0547471909921b09701854a6c2": "\\mathbf{A} \\, \\mathrm{adj}(\\mathbf{A}) = \\det(\\mathbf{A}) \\, \\mathbf{I} \\,", "8fc6b452ff17766968f92a15e2d2f8ad": "c_m=\\frac{1}{2\\pi}\\int_\\Gamma \\ln(f_w(\\theta))\\cos(m\\theta)\\,d\\theta", "8fc6be8e1544aec08de4e32d6b11f130": "\\mathbf{A} ", "8fc720fbd3ee3f1b91ba438db82f2d4b": "\\alpha\\in C_\\beta", "8fc793c8987e1718fee264ba1e7e446e": " \\begin{matrix}\\frac{4\\pi}{a\\sqrt{3}}\\end{matrix} ", "8fc7af6756c1f2dcf547884acfb89ce3": "Z_c=\\frac{p_c}{n_c k_B T_c}", "8fc7af7dabbd98372012708abdb90587": "J^\\mu = \\partial_\\nu \\mathcal{D}^{\\mu \\nu}", "8fc7d0c8c4738f3ccdd80d889a859784": "v=Br=\\frac{1}{\\alpha_0}\\sum_{i=0}^{m-1}\\alpha_{i+1}(I-\\gamma P)^ir=\\sum_{i=0}^{m-1}\\alpha_{i+1}\\beta_i y_i.", "8fc82c5a48933e36fce5d3c23d6c236a": " \\textstyle \\mu = \\arcsin \\left( \\frac{c}{u} \\right) = \\arcsin \\left( \\frac{1}{M} \\right)", "8fc858b949e2228fc2003954572d2cec": "P'(s)", "8fc8867c907d7824f188b91778bd59db": "c=2qk(p-qh),", "8fc8870bf42e67d14cae0b3b90bd10df": "\\,\\gamma=0\\,", "8fc888e7635d021ab99ff44e3086d7c8": "u_1,\\ldots,u_m", "8fc89246bc0b9a4ff7cac0df1180c044": "\\Delta x\\Delta p \\geq \\frac{\\hbar}{2}", "8fc89c3e0d6422cc41a0b21c85cd29b2": "\\mu \\left( \\varphi_{t}^{-1} (A) \\right) = \\mu (A) \\qquad \\forall t \\in T, A \\in \\Sigma.", "8fc8f52c3455d96df5a323d9b8d67a7c": " v^2 = GM \\left( \\frac{2}{r} - \\frac{1}{a} \\right) ", "8fc93d1c774790cc7fe22c1857ef1dc1": "\\mathrm{Ext}^1_{kG}(Y,X)", "8fc94c578800998502e9778e737b1c6d": "\\mathbf X = ( X_1, \\ldots, X_n)", "8fc966f57282363d1365c9fb85bd4f07": "v(k)=\\frac{1}{\\hbar}\\frac{d\\mathcal{E}}{dk}", "8fc96c834ac5bee81cd5fdcb0ff6c1aa": "A< \\min \\left(1, \\frac{1}{M}, \\left| \\alpha - \\alpha_1 \\right|, \\left| \\alpha - \\alpha_2 \\right|, \\ldots , \\left| \\alpha-\\alpha_m \\right| \\right) ", "8fc9ee8c8a2407dd1701653624681166": " \\left(1+ z \\right)^{s}\\geq 1+sz", "8fca4f5cd0fa66838a75e06336774742": "(x_1, y_1), ..., (x_n, y_n) \\in \\mathcal{X} \\times \\R", "8fca588655df833ad1766f0f7ab3f9c6": "2n\\alpha(n)-O(n)\\le\\lambda_3(n)\\le2n\\alpha(n)+O(n\\sqrt{\\alpha(n)})", "8fcad2e5c6ee74b03daa89acf71f9082": "\n{4\\over 3} \\sqrt{2m}{ E^{3/2}\\over F } = n h \n", "8fcad89aeaf70e274b43d0ebd15a0d0f": "\\,g(X,X) < 0", "8fcb14465c19f966e1f0b83610071b03": "p_A(y) := \\sum_B (-1)^{|B|}y^{\\dim f(B)},", "8fcb3cf32399625de5d56bb3ad67b73d": "\n\\max_{f(X_1,X_2)} I(X_1;Y_2|X_2) = \\frac{1}{2} \\log(1 + (1 - \\beta) c^2_{21} P_1 ).\n", "8fcb429f10294afa963cad640cf04e7c": "\\textstyle\\binom{n}{k} \\binom{n - k}{k} / 2", "8fcb98b5b28da7127709ff139227fdb2": "F^{-1}(p) = -\\frac{2}{\\pi}\\, \\operatorname{arsinh}\\!\\left[\\cot(\\pi\\,p)\\right] \\! ,", "8fcbb1f8053b902621bb7fc1a65b6d5d": "\\{a,\\,b\\}^*", "8fcc1b80f5128b315de8cbc0154ae7c1": "\\scriptstyle x=l", "8fcc2aa1ca3a8ff74cebc80afabafae3": "\\textstyle \\lambda k", "8fcc32957fcd2e70f540d3f29f33466f": "\\pi_{a_1, \\ldots ,a_n}(\\sigma_A( R )) = \\sigma_A(\\pi_{a_1, \\ldots,a_n}( R ))\\text{ where fields in }A \\subseteq \\{a_1,\\ldots,a_n\\}", "8fcc5021dd1c6afb527dd7a18fc3b209": "1_{\\mathbf Q}", "8fcc648a777765ea6e3142751c9ec5ea": "\\nu_i", "8fcc754604f38c5c9446167160ccdd7a": "\\mu(B) \n\\le\\sum_{k\\in\\mathbb{N}}\\mu(E_{n_k,k})\n< \\sum_{k\\in\\mathbb{N}}\\frac\\varepsilon{2^k}\n=\\varepsilon.", "8fcc93f20dde90462b2aecc0b8451ea9": "(\\theta ) = \\tan^{-1}\\ (\\frac{X_C}{R_L}) = 11.31^\\circ ", "8fccbe9dfa9d4ff90403e822782b4913": "P(Y_k | X)", "8fccf7eeeec364322bb15cec2a1fc436": "a_i=[A_i]\\gamma_i", "8fcd01a17ad602c542f98b916cba57f4": "z=0", "8fcd4373e8818651285f3f58b7c8194f": "\\Delta w'=0^*", "8fcd4e32292f62f0d730c8db06a73176": "_{s.1.right\\,}\\!", "8fcd6a5cdeae796763508007b27dbc1a": "t' = \\gamma \\left(1 - av\\right) t", "8fcdc884a8a20a8e8515a8c28dbc67e7": "x^{(n)}=x(x+1)(x+2)\\cdots(x+n-1). ", "8fcdcd40adf2b0ae126a4bbc53a18733": "\\Gamma_g(N)=\\left\\{ \\gamma \\in GL_{2g}(\\mathbb{Z}) \\ \\big| \\ \\gamma^{\\top} \\begin{pmatrix} 0 & I_g \\\\ -I_g & 0 \\end{pmatrix} \\gamma= \\begin{pmatrix} 0 & I_g \\\\ -I_g & 0 \\end{pmatrix} , \\ \\gamma \\equiv I_{2g}\\mod N\\right\\},", "8fcddcda9ba1c146e59b178ffa160962": "\\frac{20}{11}", "8fce17e36729ecb2db46ad1b4325b28c": "E_q(t)", "8fce754bbbf8f451f454f5329bb48c06": "s_d) = 1 ", "8fce859ab2c41da7bdb6bd8811c27df8": "gate7", "8fce957c7ca0e43ca65885aae2b0e616": "\\delta=1", "8fcea5588df302b1bcaa0618dfd6d099": "H^{0}\\vert\\psi^{0}\\rangle=E_{n}\\vert\\psi^{0}\\rangle ", "8fcf2e1613b213553d607c5bcefeb806": "\\begin{array}{ccc}\\mathbb{C}&\\longrightarrow&\\mathbb{C}\\\\ z&\\mapsto&a+\\omega\\overline z\\mbox{,}\\end{array}", "8fcf539ba5f15bab1a64acaea080973f": "\\tfrac{(c-a)}{b}", "8fcf8547a4e918c00a7024464c980618": " \\left| {\\epsilon_{n+1}}\\right| = \\frac {\\left| f^{\\prime\\prime} (\\xi_n) \\right| }{2 \\left| f^\\prime(x_n) \\right|} \\, {\\epsilon_n}^2 \\, ", "8fcfa23a612473aedd09c1db34ee133f": "\\gamma^\\mu", "8fcfa794fdaa0836642a9a2517160d75": " (L - z)^{-1}, \\qquad z \\in\\mathbb{C},", "8fd0031e88468b55047fbedd451d8d4e": "\\cos A = \\frac {\\textrm{adjacent}} {\\textrm{hypotenuse}} = \\frac {b} {h}.", "8fd00fd821a3e89aae6f49d16266d8e2": "\\mathbf{\\chi} = \\sum_{i=1}^{c}n_i f_i", "8fd02c06d73e3c1b5cd91bbab9c09e18": "{\\eta_{stage}} = \\frac{Work~done~on~blade}{Energy~supplied~per~stage} = \\frac{U\\Delta V_w}{\\Delta h}", "8fd082536a0a420385519d1473c9d27e": "\\vec{a}", "8fd09184c310d3fc8801f3960a1a6676": " \\frac{5}{66} ", "8fd0e1c62ea736ca1cac40a56460e938": "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\sqrt{\\pi}", "8fd0e6876e4b72a1399b6170903e64d9": "\\displaystyle{gKg^*=K,}", "8fd0f2d8f3ac61a0a5c9b65c6c90d86b": " W_{2\\to 3} = \\int_{V_2}^{V_3} P \\, dV, \\, \\, \\text{zero work if V2 equal V3} ", "8fd12de2c173606520b9272f126bf9a1": "1 - O\\left(2^{-N\\frac{1}{\\log \\log n}}\\right)", "8fd1f595c29f710b915fa00d60704d83": "x_{3} = \\frac{ (\\mathbf{r}_{1} - y'_{1} \\, \\mathbf{r}_{3}) \\cdot \\mathbf{t} }{ (\\mathbf{r}_{1} - y'_{1} \\, \\mathbf{r}_{3}) \\cdot \\mathbf{y} } ", "8fd20a79673a41ee353cd97db2f37043": "c_{accept}", "8fd2735935235de75b4efd97d16649e0": "(\\Sigma \\cup N)^*", "8fd305da11c5d8ddcf436bad8681f087": "\n\\begin{align}\n\\frac{\\pi}{4} =& 183\\arctan\\frac{1}{239} + 32\\arctan\\frac{1}{1023} - 68\\arctan\\frac{1}{5832}\\\\\n& + 12\\arctan\\frac{1}{113021} - 100\\arctan\\frac{1}{6826318}\\\\\n& - 12\\arctan\\frac{1}{33366019650} + 12\\arctan\\frac{1}{43599522992503626068}\\\\\n\\end{align}\n", "8fd30d03185e6ca54248091bbc572fe4": " t' \\equiv c't \\pmod p", "8fd357b5fcc053610bf886355184d540": "N = 1,2,3,\\dots", "8fd38722d7708ebdb163ede506f98bcc": "\nV(t) = V_\\textrm{max} e^{-t /\\tau}\n", "8fd3a5e580be4ecb12c88b3cc6105bf3": "(r,0,0)", "8fd3b92a365e17579b86b8a878a2f707": "D + A \\rightleftharpoons DA", "8fd3fc3bea2f0034825ccc20691ae170": "K\\subseteq R", "8fd3ffd3c6c77c9087148e2926edd866": "b=ny", "8fd44ff3daccd838fb402f35c2b8f919": " F : M_n (\\mathbb K) \\rightarrow \\mathbb K ", "8fd45fa70764715e01ea64a5a7120143": "T^tG=\\bigcup_{p\\in M}p\\times TM \\subset TM\\times TM", "8fd479e3531c7b5d20ed0d1d9bc49ee2": " b=S+W_d-\\Delta_{SL}-Y", "8fd4988e0e5a866eabc31a64d0138dd2": "q_\\text{opt} = F^{-1}\\left( \\frac{p-c_v}{p+h}\\right)", "8fd4bd591a9a6985c0f9d442c7823a56": "g(x)^q \\equiv g(x) \\pmod{f(x)}.\\,", "8fd5358303969076271fb21c159ca199": "\\scriptstyle S_{eff_{\\ast}}", "8fd6557288c5abdd6749ac5a9246fd6d": "\n \\begin{align}\n m_1 & = \\int_{-b/2}^{b/2}m_x(y)\\,\\text{d}y ~,~~ m_2 = \\int_{-b/2}^{b/2}y\\,m_x(y)\\,\\text{d}y ~,~~\n q_{x1} = \\int_{-b/2}^{b/2}q_x(y)\\,\\text{d}y \\\\\n t & = q_{x2} + m_3 = \\int_{-b/2}^{b/2}y\\,q_x(y)\\,\\text{d}y + \\int_{-b/2}^{b/2}m_{xy}(y)\\,\\text{d}y.\n \\end{align}\n", "8fd6f11a95844063d58e45a63498ca86": "a = \\frac{4\\sigma}{c} = 7.5657 \\times 10^{-15} \\textrm{erg}\\,\\textrm{cm}^{-3}\\,\\textrm{K}^{-4} = 7.5657 \\times 10^{-16} \\textrm{J}\\,\\textrm{m}^{-3}\\,\\textrm{K}^{-4}.", "8fd703fa5a2758a26cae2e9e8882f369": "F(k,m)=km(k-1+m)+\\frac{k(k-1)(2k-1)}{6}", "8fd70b1abcf715b73efb9133e12928e4": "\\xi=x-ct,\\tau=(\\alpha h)ct, y(\\xi,\\tau)=u(x,t)", "8fd727bf12772e0fbfdc856c9870117f": "\n\\mathbf{N}_{\\epsilon} = \\mathbf{N}+\\epsilon\\mathbf{I}.\n", "8fd789bc1b4717e2c065196408a57075": "(X,X')", "8fd837edc7595f9dacc74018553120fa": "\\begin{alignat}{2}\nx &\\,=\\,& 1 \\\\\ny &\\,=\\,& -2 \\\\\nz &\\,=\\,& -2\n\\end{alignat}", "8fd8452ad1d0d9477ad0c0f469b264bf": "\\rho \\otimes \\omega", "8fd86c6025ca6ea393f0bb945546b331": "(f \\cdot g)'(x) = f'(x) \\cdot g(x) + f(x) \\cdot g'(x).", "8fd8f2b220b4251f70d1333a1acb7784": "\\{ A_1, \\dots,A_m \\}", "8fd9407cae804f7958e3b178748d077a": "g(\\mathcal{Q}-I)=0", "8fd9546e0290a12102e24174765ac7b7": " \\phi = f/P \\,", "8fd95f2c37f7fdf7e76738c53fbc4ea0": "\\ (u,v).", "8fd96792183a40310099f087931879bb": "T_{i+1}", "8fd97ce8a0e9f41c326cf653f89e10d0": "\\begin{align}\n\\mathcal{A}\\left\\{c_1 x_1[n] + c_2 x_2[n] \\right\\}\n&= \\sum_{k=n-a}^{n+a} \\left( c_1 x_1[k] + c_2 x_2[k] \\right)\\\\\n&= c_1 \\sum_{k=n-a}^{n+a} x_1[k] + c_2 \\sum_{k=n-a}^{n+a} x_2[k]\\\\\n&= c_1 \\mathcal{A}\\left\\{x_1[n] \\right\\} + c_2 \\mathcal{A}\\left\\{x_2[n] \\right\\},\n\\end{align}", "8fd9c326bc89bd4d2d305eebf388cf1e": " L_n^{(\\alpha)} (x) = \\sum_{i=0}^n (-1)^i {n+\\alpha \\choose n-i} \\frac{x^i}{i!} ", "8fda187038a32088953816a65a327c39": " \n\\operatorname{tr} \\, A = \\operatorname{tr} \\, B, \\quad \n\\operatorname{tr} \\, A^2 = \\operatorname{tr} \\, B^2, \\quad\\text{and}\\quad\n\\operatorname{tr} \\, AA^* = \\operatorname{tr} \\, BB^*.\n", "8fda3247a05af0532fbee9ccfbf36928": "j = i+k", "8fda397e344f9d930d01877a3a3253f8": "E^2 = m^2 c^4 + p^2 c^2", "8fda53dff9ff08ffa4580e00b66138d6": "{\\zeta_g (x,y)}", "8fdae14151df5c5ae8457a63691dca26": "fc/(r+f)>c", "8fdb1a97182f76cad203b7365ac44391": "M_{20}", "8fdb7d86e0aa405c95ae733fb127e08b": " 2 \\alpha = 2 \\zeta \\omega_0 = \\frac{\\omega_0}{Q} = \\frac{1}{C_1} \\left( \\frac{1}{R_1} + \\frac{1}{R_2} \\right) \n= { 1 \\over C_1 } \\left( { R_1 + R_2 \\over R_1 R_2 } \\right) .\\,", "8fdb811cc73f53e24377a9e5e9b5e2ec": "G(s_{0};\\Delta)", "8fdbd4177ecfb7b4cab0d6848cb0613a": "U_{emf}", "8fdbe7333c7fccda1485dc66600f2d6e": "U=\\{ s\\in \\mathbb{C} : |s-3/4|0}(1-e^{-\\alpha})}.", "8ff936854ecf408d33c4f62528417b6a": "\\operatorname{Cov}(X,Y) = \\displaystyle\\frac {\\sum_{i=1}^n x_i y_i - (\\sum_{i=1}^n x_i)(\\sum_{i=1}^n y_i)/n}{n}. \\!", "8ff9b4718a50311cbd2c284635ff0939": "\\sigma_{12} = k=\\frac{\\sigma_y}{\\sqrt{3}}\\,\\!", "8ff9c1b69b4201fec1b23780372d5cdf": "\\lambda_k", "8ff9cd4bbda04106f17bff1aa116c2f5": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} b(k) \\frac{z^k}{k} =\n\\frac{1}{1-z} \\sum_{k\\ge 1} \\frac{z^k}{k} =\n\\frac{1}{1-z} \\log \\frac{1}{1-z}.", "8ff9dbc2f535a9abf088ebc6bd60b6d2": "I_i(s)", "8ffa211ca0e25bd7dc602d9df7dd7aa0": "\\pi_{\\mathbf S}\\colon{\\mathbf S}\\to M\\,", "8ffa8ec056f7305863e674906c7b6c32": "P=\\begin{pmatrix}\n 3& 5& 1&-1& 0& 0& -4& 0\\\\\n 4& 4& 0&-1&-1&-2& -3&-5\\\\\n 8& 5& 0&-2&-5&-2&-11&-6\\\\\n 0& 9& 0&-1& 3&-2& 0& 0\\\\\n -1&-1& 0& 0& 0& 1& -1& 4\\\\\n 0& 1& 0& 0& 0& 0& -1& 1\\\\\n 2& 1& 0& 1&-1& 0& 2&-6\\\\\n -1&-2& 0& 0& 1&-1& 4&-2 \\end{pmatrix}\n\\quad\\text{one has}\\quad A=PCP^{-1}.", "8ffab5e789164e715fec152ab23149f9": "S_{*} (\\nu)", "8ffae617b9142e3ddff7427447e13ca9": "\n \\sigma_{th} = \\cfrac{16\\Delta\\gamma}{9\\sqrt{3}z_0} \n", "8ffafe5b11ca234a49a308a4064c08fd": "U \\rightarrow N", "8ffb08c5993b21c6f9905ebda4d0ff71": "\\varepsilon _{\\alpha \\beta }", "8ffb399d59d831010920cb2e70f1915d": "\\theta \\circ (\\theta,1)", "8ffb59ed9f28fedc0e0896bc03b5eb8e": "V^\\omega", "8ffb65e24ff6b5ef5d0395d0c161624a": "\\pm\\pi", "8ffb68465d3ba970f94f4fe65659ebc1": "\\beta(g) = \\lambda\\,\\frac{\\partial g}{\\partial \\lambda}", "8ffb83f102664f87aa82678e5b0121e9": " \\rho\\, _{vapor} = \\frac {m_{vapor}} {V} = \\frac {MW \\cdot n} {V} = \\frac {MW \\cdot P} {RT} = \\frac {MW \\cdot P} {N_0 k_B T} ", "8ffb878cc2f5cc2e636a7cf683195941": "D(z)", "8ffb9dd29cc13e37cd96fd3c19704f8d": "{{D_g u_g \\over Dt} - {f_o v_a} - {\\beta y f_o v_g} = 0}", "8ffbe82f0c7552974ff28f141eadf189": "\\mathcal{T}=\\ker(F')", "8ffc27b0649502217ec2f0baa8abae6a": "l =", "8ffc7387c56f4092c0965f3fedfeafaa": "1.61", "8ffc8bdf963a8cb26d91e0678b50d08a": "\\ B ={1 \\over n}(a_{ij}^{-1})^T.", "8ffcdb73b73c1ba6a2ec8e696c76bb9d": "\\Omega = \\Omega_{1} \\times \\Omega_{2}", "8ffd0b06db36671b87b5e0b96b18a94a": "a \\frac{\\partial u}{\\partial x} + \\frac{\\partial u}{\\partial t} = 0\\,", "8ffd224dca44b1bf5a10a697ae5f200a": "px = p_\\mu x^\\mu", "8ffd4f3130a6696933619db2f5a6192e": " (x =_{f} y)\\; :\\Longleftrightarrow\\; (f(x) = f(y)). \\! ", "8ffda26c9fd61622e1a7d376217bb7a1": "\\Phi(S_{\\mathrm{final}})\\ge 0", "8ffdb42706f6b027bb4cd07b304a935a": " \\sigma = c \\Delta v ", "8ffeae8970b94f7386b9378bfd53a870": "\n\\beta_{cr} = 1.935 f + 0.065 + s\n~", "8ffec7f6faae75cc89de5b73b99dcb29": "S_T", "8ffed68913fa9d7bd52f8acb3a20d6c9": "h(y,z,x_1,\\ldots,x_k)\\,", "8ffef72493100c13944cc44ae7efe17a": "e^{\\pi\\sqrt{163}}\\approx 262537412640768743.99999999999925007\\,", "8fff2aece191dfc7723fb30be57101dd": "f_z", "8fff439bd4dec6d9c42fdb796f9263b6": "h_1, ..., h_k", "8fffc74da9ad58c710b55fa326362566": " L_{\\left(p-k\\right)} ", "8fffcc1c4ed7ccfa5b3ad0f90d8e2ec6": "\\mu_0 \\vec{J} = \\frac{1}{r}\\frac{d}{dr}(r B_{\\theta}) \\hat{z} - \\frac{d}{dr}B_{z} \\hat{\\theta}", "9000111b73f30aca6cd3b96b10f62f28": "\\langle A_\\xi X, Y\\rangle = \\langle \\alpha(X,Y), \\xi\\rangle", "900064df81070a62cf7217a20548afe7": "\\frac{-d[COOH]}{dt}=k[COOH][OH]", "9000970b861d915e771d20248d76afc4": "\\scriptstyle{|z|\\le\\|T\\|.}", "9000ada0fcc97078eadd15f046973b3f": "E[(f_i - g_i)^2] = E[(f_i - E[g_i])^2] + E[(E[g_i]-g_i)^2] + 0", "9000c04ad03460cf771bb09079777ffc": "H(i,j) = \\max \\begin{Bmatrix}\n0 \\\\\nH(i-1,j-1) + \\ w(a_i,b_j) & \\text{Match/Mismatch} \\\\\nH(i-1,j) + \\ w(a_i,-) & \\text{Deletion} \\\\\nH(i,j-1) + \\ w(-,b_j) & \\text{Insertion}\n\\end{Bmatrix}\n,\\; 1\\le i\\le m, 1\\le j\\le n\n", "9000d041997b1b46476e21508abe0793": "\\psi_0(x) = x - \\sum_\\rho \\frac{x^\\rho}{\\rho} - \\ln 2\\pi - \\frac12 \\ln(1-x^{-2})", "9001075e89f245f1e23e84f0d8cec731": "K = n (n + 2)", "900150983cd24fb0d6963f7d28e17f72": "abc", "9001527e8620d8176dbe83bc2d46e92c": "\\delta_\\xi(t) = \\begin{cases}1 &: t=\\xi , \\\\0 &: \\mbox{else}.\\end{cases}", "90018758866bcf1de107fba1ea62ed5d": "A_\\delta ", "900202f91a60cbb19517cdd745037510": "\\scriptstyle A_{11},\\, A_{12}", "90020b0dee2d93817985d0e8eb2377db": "2^{2^{2}} + 1 = 2^{4} +1 = 17,", "90020b1d96a41998b103468f530a78ff": "\\Phi(x)\\; =\\;0.5+\\frac{1}{\\sqrt{2\\pi}}\\cdot e^{-x^2/2}\\left[x+\\frac{x^3}{3}+\\frac{x^5}{3\\cdot 5}+...+\\frac{x^{2n+1}}{3\\cdot 5\\cdot7\\cdot ...\\cdot (2n+1)}\\right]", "90020b524703178aa0dae6eec799f68f": "\nP\\approx P_0 \\left(1 + X + \\frac{X^2}{3}\\right)\n", "90022060f1e077a7f839e8e29e134a8e": " A-I ", "90024a2ee0b3caa84617deceeca658eb": "\\displaystyle{|(f(z_1),f(z_2);f(z_3),f(z_4))| \\le c |(z_1,z_2;z_3,z_4)|^d,}", "90029288dafe9e2b1a986e4b9bb333af": "b>c>a", "9002bed441233d77921b4bdda00d1198": "V_\\phi", "90032824e6456563f711270e1615e585": "r(S) + r(E-S) = r(M)", "90032d7914c55a794276afee3692334d": "\\hat S(n)", "9003350a937e07820748a5b7bd3d83b9": "\\displaystyle{\\mathfrak{g}=\\mathfrak{g}_{-1}\\oplus\\mathfrak{g}_0\\oplus\\mathfrak{g}_1.}", "9004535d6a80981fc8ed7d856a1b4729": "\\mathcal{N}_q \\left(\\boldsymbol\\mu_1, \\boldsymbol\\Sigma_{11} \\right)", "900489b47f4721662ca3fb57fdc185bd": "Y_{7}^{-7}(\\theta,\\varphi)={3\\over 64}\\sqrt{715\\over 2\\pi}\\cdot e^{-7i\\varphi}\\cdot\\sin^{7}\\theta", "90048a9d6273f14eca70c45d7bbb3c59": "f_1=18x^7", "9004c6f883a47d810ed0feb73ff48fb3": "\\scriptstyle f''(g(x)) \\,\\!", "9005326d5d4e2a78ec0d3cbb73a218fa": "a^6", "9005a4d5459fddfbd52a415d3a4be9c7": "\\breve{S}", "9005ad3b2b0c87afdd3c7a830ba547de": "1 / \\sqrt K", "9005d68b2e84ffb2c5d321068b151014": "s_\\lambda s_\\mu =\\sum c_{\\lambda\\mu}^\\nu s_\\nu", "9006193add5c1a47b8905c3b9d1ca535": "\\nu_\\mu\\leftrightarrow\\nu_\\tau", "9006a72dd0aad41e14c041570c7f5e92": "\\mathbb{J}(\\mathbf{r}) = {1 \\over 8 \\pi \\mu} \\left( \\frac{\\mathbb{I}}{|\\mathbf{r}|} + \\frac{\\mathbf{r}\\mathbf{r}}{|\\mathbf{r}|^3} \\right)", "90072359c50edd37bd2316f862fa8219": "(x_n)_{n\\in\\mathbf{N}},\\quad x_n\\in\\mathbf{K}.", "90072ad93433c37162bc51b8044c6a35": "\\begin{alignat}{7}\n2x &&\\; + && y \\;&& - &&\\; z \\;&& = \\;&& 8 & \\\\\n&& && \\frac{1}{2}y \\;&& + &&\\; \\frac{1}{2}z \\;&& = \\;&& 1 & \\\\\n&& && && &&\\; -z \\;&&\\; = \\;&& 1 & \n\\end{alignat}", "90075b5d93d85345af3b31eaa95b4188": "\\hat{f}_{-}", "90077e1173bd01c8a4b10e21e2d7a52e": "M=X^{nr}", "90079d06ae80da20a89992e400a9a2a6": "F = \\mathbf {Q}\\left(\\sqrt{(-1)^\\frac{p-1}{2}p}\\right)", "9007b03f9386412b5a023d176fec9ecb": "V(f,z) = V(f(x),y)", "9007f6fd93f6f82cac2c00c0edc4ae65": " v_d = v (1 - \\frac {\\rho_e}{\\rho_g}) ", "90080114b7438427ef221cc4ba755acb": "\\scriptstyle n > 3", "900882eed2630c5a738f06f852b391dd": "\\frac{x}{y}=\\frac{1}{\\lceil y/x\\rceil}+\\frac{(-y)\\,\\bmod\\, x}{y\\lceil y/x\\rceil},", "9009386d652afff047f035ecfcfddb06": "\\sigma =\\{(\\ldots,s_{-1},s_0,s_1,\\ldots) : s_k \\in S \\; \\forall k \\in \\mathbb{Z} \\}", "900958f5044582e29f26640163fc6d74": "\\kappa_\\nu^{-1}", "90095fb111f3fbaa558df91c2b9e9f4b": "\\langle p' | J^3 (0) | p \\rangle ", "9009a8557cca7662ff69cd6293ec5525": "\nJ=\n\\begin{pmatrix}\nx_0 & 1 & 0 & 0 & \\cdots & 0 \\\\\n0 & x_1 & 1 & 0 & \\cdots & 0 \\\\\n0 & 0 & x_2 & 1 & & 0 \\\\\n\\vdots & \\vdots & & \\ddots & \\ddots & \\\\\n0 & 0 & 0 & 0 & & x_n\n\\end{pmatrix}\n", "9009a92ad6853516a7ce9bcbfe01d464": "f_{x_0}(x) = f(x - x_0).", "900a765d95b3cfdfd74942def8da0cbd": "\\scriptstyle k_{op}", "900a9241fa99d41335d830d32758c92c": "\\mathbf{x}_{0i}^\\top[K_0] \\epsilon_{ii} \\mathbf{x}_{0i} + \\mathbf{x}_{0i}^\\top[\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} \\mathbf{x}_{0i}^\\top[M_0] \\epsilon_{ii} \\mathbf{x}_{0i} + \\lambda_{0i}\\mathbf{x}_{0i}^\\top [\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i\\mathbf{x}_{0i}^\\top [M_0] \\mathbf{x}_{0i}. ~~(6) ", "900b1e484c70e54a9e379744418c7210": "d = d^*(d^*-1)-2\\delta^*-3\\kappa^*,\\,", "900b4614f3aba542db462b1be96b9782": " \\lim_{\\alpha = \\beta \\to 0} \\operatorname{var}(X) = \\tfrac{1}{4} ", "900b71dba2a36cd5bffd16b415e746a8": "\\mathbf{\\hat p}'", "900b92b2cd00cd21f2aabd10693ad132": " X(f) = x(-t) \\, ", "900ba2a521d826519f8c80ce11d592cf": "\n{\\rm E}[z]\\,\\,\\, = \\,\\,\\,\\mu _z \\,\\, \\approx \\,\\,\\,\\,a\\mu _1^\\alpha \\mu _2^\\beta \\,\\, + \\,\\,\\,\\frac{a}{2n}\\left[ \\begin{array}{l}\n \\left( {\\alpha \\left( {\\alpha - 1} \\right)\\mu _1^{\\alpha - 2} \\mu _2^\\beta } \\right)\\sigma _1^2 + \\\\\n \\left( {\\beta \\left( {\\beta - 1} \\right)\\mu _1^\\alpha \\mu _2^{\\beta - 2} } \\right)\\sigma _2^2 + \\\\\n \\left( {2\\,\\alpha \\,\\beta \\,\\mu _1^{\\alpha - 1} \\,\\mu _2^{\\beta - 1} } \\right)\\sigma _{1,2} \\\\\n \\end{array} \\right]", "900bf720b77d243b83a1495d2f61429a": "\\hat{A}, \\hat{B}, \\hat{C}", "900c41bbebbbc02c42b14e3b17e091fd": " -\\bar r_1 \\times \\bar r_2\\ ", "900c6692a58793c7360df2a929a64ff2": "\n \\frac{d^4 W}{dr^4} + \\frac{2}{r}\\frac{d^3 W}{dr^3} - \\frac{1}{r^2}\\frac{d^2W}{dr^2} \n + \\frac{1}{r^3} \\cfrac{d W}{d r} = \\lambda^4 W\n", "900c86fbbdb4f61448bbc6a15d41edef": "x_1^2+x_2^2+x_3^2=1", "900cabbe50537066d66951f2fee6d0ec": "\\nabla \\times \\mathbf{B} = \\mu_0 \\mathbf{J} + \\mu_0\\varepsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t}", "900cdf4d8f8f2c18e2bd60da2b64cbb6": "\\{1,3, 8, 120\\}", "900cff3840a7901a30d4847c868ea669": " y_k ", "900d1bdf12049579943bc35fc0c19d05": "M(n) = (1+o(1)) 2^{n\\choose \\lfloor n/2\\rfloor}\\exp a(n)", "900d22542b31d41aeb60201dc2f7189c": "\\sum_x \\cot ax =-i x-\\frac{i \\psi _{e^{2 i a}}(x)}{a} + C \\,,\\,\\,a\\ne \\frac{n\\pi}2", "900d41f3b3ff7250b7ef0aed6ab4bd90": "\nz = a \\ \\sinh \\mu \\ \\sin \\nu \n", "900d4db8bc7fab8f2a910645e7e9537a": "P_1\\parallel_- P_2", "900d5492230e22d70f3d05b004ef342d": "\\rho_n", "900d6c013b97ea6d0f5e88d401894f17": "T_\\mu S", "900e0f689aa9687625d8aeef91185ad1": " p = k_{\\rm H}\\, c", "900e425c339b19183807d1b83a9969f7": "s_1(t) = \\sqrt{\\frac{2E_b}{T_b}} \\cos(2 \\pi f_c t) ", "900e84add68304fe10f2d920bb1ecfe1": "U_{L, i}/U_{L, i+1} \\approx l^+", "900f1561922da15a24c837a90efedad1": "13.798\\pm0.037\\,\\mathrm{Gyr}", "900f798695d444d7acbc1e1c95064520": "\\delta\\preceq\\theta\\delta", "900faf32d4e172b4ba2ebca338aded4c": " \\delta w \\le - dU + T_R dS + \\sum \\mu_{iR} dN_i \\,", "900fb001f74146c4f85b18a4dd884436": "\\mu = \\lambda\\mu'\\lambda.", "900fd2cc88207e78937d525ef5eea064": "\\Delta\\text{H}_{\\text{f}} = \\text{V} + \\frac{1}{2}\\text{B} + \\text{IE}_{\\text{M}} - \\text{EA}_\\text{X} + \\text{U}_\\text{L}", "900fdc55b28d7dace12a79720ac5ed46": "T_\\infty", "9010349c052cb96f95e8651de387f4e3": "B_{ijkl} N_j N_l m_i m_k > 0", "90105c487fff15c5088fae0887559a29": "\n\\Gamma_p(a)=\n\\int_{S>0} \\exp\\left(\n-{\\rm trace}(S)\\right)\n\\left|S\\right|^{a-(p+1)/2}\ndS ,\n", "901086de221d6d67ebf981e98f726932": "\n\\begin{align}\n\\Big\\lfloor \\lceil x \\rceil \\Big\\rfloor &= \\lceil x \\rceil, \\\\\n\\Big\\lceil \\lfloor x \\rfloor \\Big\\rceil &= \\lfloor x \\rfloor. \\\\\n\\end{align}\n", "9010966badc421b67a9c91daabd03f00": "(Df, f) \\ge M (f, f)", "9010f15e3b47943ea1db690a4da7c17e": "\\ \\frac{dY}{dK}=c \\Rightarrow \\frac{dY}{dK}=\\frac{Y}{K}", "901101d87d1770f85014cc7562544931": "\nJ_{k} \\equiv \\oint p_{k} dq_{k}\n", "90110bfc1eada2e47265dab0720a529d": "\\sigma^2 \\sim \\Gamma^{-1}(\\alpha,\\beta) \\!", "90110fe63b2f0fd15ff47560a7ce5bdb": "f(n) = a_n", "90111bbf29b5c4d4041630869dad3075": "\n\nx = Ax + d\n\n", "90111ffc89d6d88ecafa2ab6d1025908": "{{C}_{\\mu 3}}", "90114c263256e4c7522f5e52e71e0280": " F_{max} = \\frac{E d^4 (L-n d)}{16 (1+\\nu) (D-d)^3 n} \\ ", "9011a21308a8a6e81335c3e9455851c3": " \\lim_{x \\to 0} \\frac{x^{2}}{x} = 0, \\! ~~ (2)", "90123210fa7198ec63f9848b941b7aaf": "c_{13}", "901269c0e45aab9eb5379445b65cea58": "(y,Y)", "9012a5ab48ab12b0242a36ae321cb5df": "\\frac{U}{L^3} = \\int_0^\\infty u_\\nu(T)\\, d\\nu,", "9012daecc98e3bb15f302f3fbe2c3f94": "K_{i\\alpha j\\beta}", "9012fb7ef89d6d98284094346f931aa2": "W_{\\mu}=\\frac{1}{2}\\varepsilon_{\\mu \\nu \\rho \\sigma} J^{\\nu \\rho} P^\\sigma ,", "90132acebdd5ffa2ef30546fe53b5198": " \\vec{E} = q \\, \\sin(\\omega u) \\, \\vec{e}_2 ", "901340c482df89b944e0fc7dd35c7778": "\\begin{align}\n\\Delta\\varphi &=\\varphi(\\alpha+\\Delta \\alpha)-\\varphi(\\alpha) \\\\\n&=\\int_a^b f(x,\\alpha+\\Delta\\alpha)\\;\\mathrm{d}x - \\int_a^b f(x,\\alpha)\\; \\mathrm{d}x \\\\\n&=\\int_a^b \\left (f(x,\\alpha+\\Delta\\alpha)-f(x,\\alpha) \\right )\\;\\mathrm{d}x \\\\\n&=\\int_a^b \\left |f(x,\\alpha+\\Delta\\alpha)-f(x,\\alpha) \\right |\\;\\mathrm{d}x \\\\\n&\\leq \\varepsilon (b-a)\n\\end{align}", "90138b8506d64a00fafffeeeef45ad9e": "k_\\text{cat}/K_m", "901405e6b119cd4392e48632a1125d8e": "\\beta = \\omega \\sqrt {LC} ", "90143d8e603e30faa874a3ce830aa653": "\\mathbb{R}^{8}", "9014559ea75e9aa8d1c8d8a0d9f53ae1": "J^\\mu = \\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)}\\delta\\phi ", "90145b0b9c26606a727249c0b678dece": " Q(x_1,x_2,\\ldots,x_n)=\\sum_{i=1}^n a_i x_i^2 ", "901467ae4df920da11205f2689836880": "S_n=\\sum_{k=1}^n w_k,", "9014897c8fba3c3f608b4a7bb26d345f": "M_z =\n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & - 1\n\\end{bmatrix}\n= \\beta \\,.\n", "9014a6c846cdbb5cbd2c5cabdc27bd8a": "V_\\mathrm{in}", "9014cf2612daf4ec44324be43dcadb31": "x^8 + x^5 + x^4 + 1", "9014e1b00f6b7abe2e1f6c3b4ad761fd": "\n\\sigma_3 = \\sigma_z =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "901569b5af0b0fc532c6198c010c325b": " P = \\lim_{T\\rightarrow \\infty} \\frac 1 {2T} \\int_{-T}^T x(t)^2\\,dt.", "90157bc9d1edf546b823bd23d74928d0": "\\lim_{s \\to t} \\frac{|w(s)-w(t)|}{|s-t|} \\to \\infty.", "9015842c0f0e5d40419826903abfd504": "\\psi(1;\\theta)=0", "90158bb159600baee9f0a5006868dc65": "p^{-1} \\ne 1/p", "9015affebbdce179eb225aff5f16d910": "k^2 \\times k^2 \\to k", "9015b8624936b731334c8fbd4fa72302": "w_\\alpha(n) = \\begin{cases} \\lambda & g(\\pi(n)) \\le g(\\tilde{n}) \\\\ \\Lambda & \\text{otherwise} \\end{cases}", "9015cdbfdf3efef91a125e464c13b857": " g(x) = \\sum_{j=1}^d \\beta_j x_j", "90161174ebbee0401b25fbaa4a1bff49": "\\frac{d^n F}{d \\eta^n} \\Bigg|_{\\eta = 1} = n! \\ c_n", "9016371499a2d49d92800062ed2fdd4f": "\n\\pi_{4,1}(x) - \\pi_{4,3}(x), \\, ", "901722318dd9d4298a1b419a95fa412e": "o(v_1,\\ldots,v_n)|\\omega|(v_1,\\ldots,v_n) = \\omega(v_1,\\ldots,v_n),", "9017256ddcd054b727222aceae3c8cb6": "P(X_2 = 0)=p_2", "90174c9c7fc8e880de6c42979624ae63": "x^7+x^6+1", "90174e908facdf12b6573cc5d3201ae5": "m_1 = c^{d_P}\\text{ (mod }p\\text{)}", "90175b3c3d6370a3df874483caf5bc2c": "CAS", "901763fc089aa1fb9ab0ab44ca5fa5a3": "\\textstyle f(A) \\in \\mathcal{F}_2 ", "9017c2826b52e926e1d7d7ca8a1ad435": "\\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k+1}}{(2k+1)!}=\\sin z\\,\\!", "9017daee1ff68d87dab349b4064519a4": "\\delta =\\frac 14g_{Sun}\\left( \\frac lc\\right)^2 \\approx 2\\times10^{-12}", "9017e629195e253386a08be168dffce8": "E[l]=\\{P\\in E(\\bar{\\mathbb{F}_{q}}) \\mid lP=O \\}", "9017ed92ffe8790406bb8b6e1a1649a8": "\\chi_\\epsilon (x,y)", "90181018ede80448f66dd9ba4cb380ee": "{}^sE_{x,t}^c", "901820513b0f6b6197264238ac9b22e1": "\\frac{3 \\cdot23^{3/2}\\zeta_k(2)}{4\\pi^4},", "9018533f7e8d6771b17f3b6b626823f0": "y_i = \\mu", "90185e6e56d822899ff678e85501a242": "2347)", "90189346b5456253ed0e64d5e0665399": "\\Psi(\\vec r_1,\\dots,\\vec r_N)", "9018c1d0a5d711e0e5fa576f8a1d392e": "\\sigma^0", "9018cbaac3af68b31b8c7d8080b332c9": " R = Ext(w;x) ", "9018ce3b9b27d485c53d9622bb6673c7": "\n \\tilde{\\boldsymbol{U}} \\otimes \\tilde{\\boldsymbol{U}}\\, =\\,\n \\begin{pmatrix} \n \\tilde{U}_x\\, \\tilde{U}_x & \\tilde{U}_x\\, \\tilde{U}_y \n \\\\[2ex]\n \\tilde{U}_y\\, \\tilde{U}_x & \\tilde{U}_y\\, \\tilde{U}_y \n \\end{pmatrix}.\n", "9018f2abec853f42c5d1126c4fa4de13": "\\tilde{C_{n}^m} = -\\frac{C_{n}^m}{\\mu\\ R^n}", "9019073d5f36be40b8e8d4c82f799f5b": "\\Lambda(A_1:A_2|B) = \\frac{P(B|A_1)}{P(B|A_2)} ,", "901976e5e21136d48f43774418049bf8": " \\widehat{\\mu}(x) = \\int_{\\widehat{G}} \\overline{X(x)} \\, d \\mu(X), \\quad x \\in G ", "9019dd06f7d7370b3b530c8d273c7f74": "\\operatorname{Li}_{-1}(z) = {z \\over (1-z)^2}", "901a240bdbbdbf30ff44e22b7a569e0e": " \\mu = \\int{ x f( x ) dx }", "901a40e86c5ac2f7d589e39f1267b72a": "\\frac{c(r+z)}2\\geq \\frac{ax+by+cz}2", "901a786f3564d1268fd6e0022ffbc91b": "\\ x_{ik}", "901a83e6e3955c27506389b046dc61ec": " U^{\\dagger} ", "901a92e3888d93ac1a25c09e8f6300d1": "10^{10^{100}} = (10\\uparrow)^2 100 = (10\\uparrow)^3 2", "901b23eb9fe75109fb27161d2945cd31": " y(t_0) = y_0, \\, ", "901b3481ba78e052da2d621cba679714": "H^1(V)", "901b945964236677256a7a4249d86e3b": "s_i > 0", "901b9b3e6b899ce6984630a2196ded1d": "\\frac{4}{3}\\pi r^3", "901c2ebd9d497a635d882d296c490745": "\\begin{align} (A \\oplus B) \\oplus B =& A \\oplus (B \\oplus B) \\\\=& A \\oplus 0 \\\\=& A \\end{align}", "901c520f5edf0f008e70afc192a856ef": "x_a", "901c8dce98e621e62574489e2f0422dd": "c_{C_1, C_2}: C_1 \\otimes C_2 \\stackrel \\cong \\rightarrow C_2 \\otimes C_1.", "901c95e39a5642d093e7b79cb9d19590": " Z_0 = \\left \\{ \\begin{matrix} Z_\\mathrm f \\dfrac {\\lambda_\\mathrm g}{\\lambda} & \\text{(TE mode)} \\\\ \\\\ Z_\\mathrm f \\dfrac {\\lambda}{\\lambda_\\mathrm g} & \\text{(TM mode)} \\end{matrix} \\right .", "901cb4998c6d88e049c1b74a114d57c1": "\\delta = -1/8, \\epsilon = 0, f(z) = 2\\sinh(z/2)", "901d1927d480d993f8a9ae40994cf7e7": "|g\\cap \\mathcal Q|=1", "901d2c44341a3d7b2d195233e7cf2e4d": "=1/\\left(bs\\right),\\quad b,s>0,\\beta=1", "901d86e2f762a496cb6b6fdbc58b559f": "\\pi_{ij} = \\frac{1}{\\pi_{ji}}", "901d894eb545b2542e3fc268222f54c0": "\\_ \\in \\Gamma - \\Sigma", "901d9c3bd540dc7e5e54379be673ac1d": "Q(\\mathbf{x},\\mathbf{y})>0", "901dd0cd2bde000bf93a815799d98bfb": "\\frac{1}{\\sqrt{2^{n+1}}}\\sum_{x=0}^{2^n-1} |x\\rangle (|0\\rangle - |1\\rangle )", "901dd323ad83927e5f44eb8ddd42b917": "\\tau_n=\\tau_n(t)", "901ddbee8a78fb5dff393aa4a99348cd": "\n\\begin{array}{l}\n a = 2qp \\\\ \n b = q'p' \\\\ \n c = pp' - qq' = qp' + q'p \\\\ \n \\\\ \n \\text{radii} \\to (r_1 = qq',r_2 = qp',r_3 = q'p,r_4 = pp') \\\\ \n A = qq'pp' \\\\ \n P = r_1 + r_2 + r_3 + r_4\n \\end{array}\n", "901e059a8ab0a70979bf1510172095f5": "\\mbox{A}", "901e5b909bf90f2923bd13c40f9595b8": "(-1)^{\\left|a_{\\pi}\\right|}", "901e7bfd3ace204de13978c57eff347e": "\\arctan(x)+\\arctan(1/x)=\\left\\{\\begin{matrix} \\pi/2, & \\mbox{if }x > 0 \\\\ -\\pi/2, & \\mbox{if }x < 0 \\end{matrix}\\right.", "901e8f107f8f26dd7a63fd44da03bba1": "v = 0.99 c", "901e954945bd6cbf53e5a0507509890f": " I_m = \\frac { n IMC } { ( nm - 1 ) } ", "901e9b88605c5ee26c3725c08b0e22df": "b_i \\neq 0", "901ebab736a2e30b8f72f4d322f9de67": "b=a-1", "901edaa2f3452a0fc504677a4a543da4": "\\ \\ h 2", "9033db547982814c3a54ecd706199d72": "W_1 = K N_1 \\sum_{p=0}^n \\rho^p ", "9033de42fbdca351296810dd4cbab0db": "K={abc\\over u^2 v^2} ,\\,\\,K_m=-(u+v)\\sqrt{abc\\over u^3v^3}.", "9033e0e305f247c0c3c80d0c7848c8b3": "!", "903462c87af5cebda2a3d81649f8d2ea": "x'_{k+1}=-c^2\\beta^2\\zeta\\sum_{{\\theta}=1}^k \\frac {t_{\\theta}} {v_{\\theta}}+{\\zeta}x_{k+1},", "903468d4d6126419abb264461f1a54c0": "0 f_{xy}^2", "903f49dc3c42d7bf47e1daf09da8d7b9": "\\frac{x_1 + x_2 + \\cdots + x_{2^k}}{2^k} > \\sqrt[2^k]{x_1 x_2 \\cdots x_{2^k}}", "903faf99a14b55b7ad3d1020786c49a8": "S^3", "903fe49a8273c966ba885226fe0e8e08": "t^{[k]}=t(t+1) \\cdots (t+k-1)", "9040e2904d293a6bcce1a337e645c485": " 2r \\left(\\sqrt{2}-1\\right) = R\\sqrt{2-\\sqrt{2}} \\!\\, ", "90411ea87b4ea59fd06af7efee4a51af": "[\\lambda,\\mu]", "90416b0ebc37046bdd3e7d5c5dc292ed": "R = \\frac{h \\cos{\\theta} }{1 - \\cos{\\theta} }", "90419638577702ed2504c40009a64b01": "1\\ \\mbox{P} = 0.100\\ \\mbox{kg}\\cdot\\mbox{m}^{-1}\\cdot\\mbox{s}^{-1} = 1\\ \\mbox{g}\\cdot\\mbox{cm}^{-1}\\cdot\\mbox{s}^{-1}", "90419aedea48e7452b31e5b2e5b75fdc": "\\frac{(1,0,0)\\cdot (0,1,1) }{ \\Vert (1,0,0)\\Vert \\Vert (0,1,1)\\Vert }= 0", "90426c90979898cfb964f12af21a81c6": " \\phi_j(x_2) ", "9042d8a5ffdf35a3b7e7d6a22aea5c5a": " \\therefore b + a + c = 180^\\circ.", "9042e7ad041b6f69e6bc67416b67826f": " \n\\begin{align}\nA_+ & = \\left\\langle \\mathbf{e}_+, \\mathbf{A} \\right\\rangle = -\\frac{A_x}{\\sqrt{2}} + \\frac{iA_y}{\\sqrt{2}} \\\\\nA_{-} & = \\left\\langle \\mathbf{e}_{-}, \\mathbf{A} \\right\\rangle = +\\frac{A_x}{\\sqrt{2}} + \\frac{iA_y}{\\sqrt{2}} \\\\\n\\end{align} \\quad \\rightleftharpoons \\quad A_\\pm = \\left\\langle \\mathbf{e}_\\pm, \\mathbf{A} \\right\\rangle = \\frac{1}{\\sqrt{2}} \\left( \\mp A_x + iA_y \\right) ", "9042edf0aea62d0d0766240e6a0e78dc": "\n \\Beta(x,y)\\cdot(t \\mapsto t_+^{x+y-1}) = (t \\to t_+^{x-1}) * (t \\to t_+^{y-1}) \\qquad x\\ge 1, y\\ge 1,\n\\!", "9043046bd62fcccc17bf8094339690e6": "\\tilde{x} = x(t) \\, \\forall \\, t \\in \\mathbb{R}", "90430693c76b194832878759e88d8488": "|\\psi\\rangle \\to \\psi (x,y,z)", "9043282ee01d68c2aee7564a20b0b0d5": "|11\\rangle", "90433ecd70fe6a472af576863b50bdad": "\\mathcal{L}_\\mathrm{G}=\\frac{1}{2\\kappa}R \\sqrt{|g|} ", "90439d43caf05af858d0f4bff56dfed2": "\\min(1,x^*_s)", "9043f81087a77a1c505c8441c90bbbea": "k_{ET}", "9043fc373f6b45009dc261504e95d07c": "\\phi(v_i)=U(v_i)", "9043fc69c59c0f34cb95026e485ea840": "w_i^t", "90440e169a2372e00d75bb9cf05fee18": "Y_m=Y_{m+1}=\\cdots.", "90441e312e8c6bbaee48d1190cf7c76c": " \\operatorname{charpoly}(\\rho(\\mathbf{Frob}(\\mathfrak{P})))^{-1}\n = \\operatorname{det} \\left [ I - t \\rho( \\mathbf{Frob}( \\mathfrak{P})) \\right ]^{-1}, ", "90443f5e7e55de14c60483d3537a02d5": "\\scriptstyle j \\;=\\; \\sqrt{-1}", "904451465f46ce8a9cf97ba13515d525": "L_p(\\Omega)", "904472e2de5a79c28caa61334cfd71f8": "\nI_{L,M}(\\mathbf{R}_{AB}) \\equiv \\left[\\frac{4\\pi}{2L+1}\\right]^{1/2}\\;\n\\frac{Y_{L,M}(\\widehat{\\mathbf{R}}_{AB})}{R_{AB}^{L+1}}.\n", "9044839340b80f35c9ea6d43342b884b": "\\Delta(l,m,n)", "9044fd17f52cd7a52ed42fe7e8d1105c": "\\frac{1}{p}+\\frac{1}{q}+\\frac{1}{r}<1.", "90451690a1a92a29abd1fdf43fc7d6a5": "\n\\frac{t}{t_0} = \\left(\\frac{r_0}{r}\\right)^2\n", "90453afcc42674d1b28da3bb2f84fa39": " (x^2-1)(x^2+x+1)", "9045651ad78bbbc8798af4ede77e601e": " E_{(v,N)}[n] = F[n]+\\int{v(\\vec r)n(\\vec r)d^3r} ", "9045ac6d418307b4645d5b2b761eaf35": "\\nu(\\rho) = \\frac{\\sum_{m=1}^{M-1}-\\kappa(K_m,Q_{m+1})}{\\sum_{m=1}^{M}\\kappa(K_m,Q_m)}", "9045aec7cfe4e2ec55b1708c66006e95": " L= \\sqrt{c^2 + P^2}. \\,", "9045b8d81d4ddf4729f9eec4d8386679": "A \\leftarrow \\alpha \\mathbf{x} \\mathbf{y}^T + A \\! ", "9045e107335fb853ba5722eb4c360071": " \\mathbf{c} ", "9045ea00c3dc3ea6e1e40fea591999cf": "\n E = \\frac{1}{V_0} \\int C_v dT \\approx \\frac{C_v (T-T_0)}{V_0} = \\rho_0 c_v (T-T_0)\n ", "904608bd4389cd8e3c35c62dd9b7d168": "p\\ f\\ x = f\\ (x\\ x) ", "90462737abb528aa8aaeb80caf5f35b0": "P_n^*=\\max \\left (\\left(\\frac{1}{\\lambda}-\\frac{N_0}{|\\bar{h}_n|^2} \\right),0 \\right)", "90463137165183e3ac5f245572bcc789": " r=k/h^2. ", "9046d0843eb3f711163d2bee86116b85": "x_d(t) = \\,\\!x(t + \\delta)", "90472bd7d8a2c93f241c8f11a5b392cb": "Z^{\\rm{rep}} = \\frac{4 c \\eta}{1-1.9 \\eta}", "9047316aa1f3a4bd922c9b9807b2102c": "n!/2\\,", "90473c4c9f161247d04cfc146f83ddad": "\\ L_1=a^{\\dagger}", "9047c163e0190278ef02df5bdd008a0c": "x_i, x_j \\in X", "9047f61ab987d3b20ce865f4e581a0bc": " \\vec{W}|_{\\vec{P}=0} =Mc\\vec{S}, ", "904800a125d13aa21b3cf4f8f7b37623": "\\frac{d L(F)}{d F} = \\frac{x(F)}{\\mu}", "904839cc2eddb04d174c384ae9b651d1": "O(n^{b+3})", "9048486e78f0b88cb6e31dc1b67fa51d": "\\mathfrak{P}^{81}", "904853b7f3082170a49c1d4aa7bf07d6": " p=\\pm 3", "904885b24b34bab1c4677cc95d4c2b6e": "\\mathrm{d}F= -S\\,\\mathrm{d}T - P\\,\\mathrm{d}V\\,", "904939c792a3fe063f6a2c39566020d9": "\\frac{1}{ \\gamma} = \\sqrt{1 - { (v/c)^2}}", "9049a840cf30ea38b0a0b5ceec5a861e": "\\scriptstyle S_{\\rm Fermi~gas} \\approx \\tfrac{\\pi^2 k}{-3e} T/T_{\\rm F}", "9049a848dae63b78d444bb938a7de466": "f,g", "9049e8bfe3aef21979073e5005cdf786": "p_1=P(Xx)=1-F(x).", "904a824ede3b5b72a0b7cc57cf498d3a": "k_2 / k_1 ", "904aad0f64a2b7fb7c7a38cfb8889a28": "\\lambda ( \\{ 0 \\} ) = 0", "904ab2310a0c5bf0af625ddacc666cab": "-\\infty < x<\\infty,\\,\\, t>0", "904ab26e5374bc0244192d7609d7189e": "l_{2}=\\sin \\Theta \\sin \\varphi, l_{3}=\\cos \\Theta", "904ab3c7d0fa918bed3991b481f90f7b": "H(mn) = H(m)+H(n)\\,", "904b2523f63ec8f8ed72352ef211b377": "Td_4 = (-c_1^4 + 4 c_2c_1^2+3c_2^2+c_3c_1-c_4)/720.", "904b68ac28513e48e0225433d96999fd": " \\mathrm{ind}\\,T := \\dim \\ker T - \\mathrm{dim}\\,\\mathrm{coker}\\,T;", "904b86294c6eb6aaa71e5b9e7719c4c7": "\\mathbf{p}_{e'} = \\mathbf{p}_\\gamma - \\mathbf{p}_{\\gamma'}.", "904ba0754eb5133eb5fb9d006b71c4b5": "\\psi_{III}(x) = C_I e^{i \\rho y} + C_{II} e^{-i \\rho y}\\quad y>1", "904bb29e82bb60cd6c21ad9d0c6d5de3": "e^{tA}=e^{s t}\\left( (\\cosh (q t) - s \\frac{\\sinh (q t)}{q})~I~+\\frac{\\sinh(q t)}{q} A\\right) ~.", "904bdc3c00d17b8d6d35b66a1d7ad519": " \\lim_{n\\to\\infty}\\; \\frac{1}{n}\\sum_{i=1}^n f(x_i) \\; =\\; \\int_{I^s} f ", "904cba74e2280872b61f2531b8fd69df": "d_K(f,g) = \\sup\\nolimits_{x\\in K} d(f(x),g(x)) + \\sum\\nolimits_{1\\le p\\le r} \\sup\\nolimits_{x\\in K} \\left \\|D^pf(x) - D^pg(x) \\right \\|", "904ceef58808fa8424afe382542fc78e": "(P \\and Q) \\vdash (Q \\and P)", "904d42e8e778f1ab920659e3f43c29b4": "\n\\begin{matrix}\n & & \\mathcal{R} & & \\\\\n & a & \\longleftrightarrow & b & \\\\\n \\omega^l & \\Big \\uparrow & & \\Big \\updownarrow & \\mathcal{L} \\\\\n & d & \\longleftrightarrow & c & \\\\\n & & \\mathcal{R} & &\n\\end{matrix}\n", "904d598186cbb907b0562321f29010ae": " n = 2.448 \\sqrt{ c } \\approx 2.5 \\sqrt{ c } ", "904dbe7d83b40f0c7990781c1373b9c0": "a>0,", "904e1c24f445a1f6b19a140bf8a44497": "|V_\\parallel|", "904e4d8d1b96fc9db1dd4861a7437b53": "(1-r)^2+h^2=y^2", "904e737471bf515c1dd4d01a988660ec": "\\nabla\\cdot\\left(f(r)\\mathbf{\\Psi}_{lm}\\right) = -\\frac{l(l+1)}{r}fY_{lm}", "904ec0caf3c558cd783cabf14728c296": "7^4 = P(7,5)K(4,1) + P(6,5)K(4,2) + P(5,5)K(4,3) + P(4,5)K(4,4) = 210*1+126*11+70*11+35*1 = 2401", "904ec24d2847ebffdb1395ad6e002d8b": "0 = i_0 < i_1 < i_2 < \\ldots", "904f486069a6e3719ff6d464a31cb6bd": "\\mathcal{D}_{L^p} = \\underset{n\\to\\infty}{\\underset{\\longleftarrow}{\\lim}} W^{n,p}(\\mathbb{R})", "904f6311600a8079d835d153023238c9": "\\mathbb{C}^n.", "904fba96172e5cb452b8371c4abe2906": "D_i>4/n", "904fc79952b02b6999117a0e8e4248ad": "\\frac{\\partial \\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y) }{\\partial \\alpha}= \\sum_{i=1}^N \\ln (Y_i - a) - N(-\\psi(\\alpha + \\beta) + \\psi(\\alpha))- N \\ln (c - a)= 0", "904fee6c837721be0b347e040de47a20": "E^2 = p^2c^2 + m^2c^4 \\;", "90506331733c5e1b55ce3d2242445a6c": "\n\\mathbf{P}(o_1, o_2, \\dots, o_t|\\mathbf{\\pi}) = \\prod_{s=1}^t c_s\n", "9050abc9679817551ac05430b79511cd": "A^{ij}", "9050eb7f2d3e8adca0889ef43261ba01": "\\alpha = 2", "905112aad4a616a7a462c4f52edd5c3e": " \\phi_x^*\\phi_x + \\phi_y^*\\phi_y = \\langle \\phi | \\phi\\rangle = 1 ", "90512669466b9064f07bb897be3f6cdf": "N = ", "9051450d4479a5def794e49c3315bbec": "\\scriptstyle \\left( \\lambda(t) = \\frac{f(t)}{\\bar{F}(t)}\\right)", "905146c7c2842de74e1f81c06f8fe440": "\\delta x=a\\delta\\lambda", "905156ae3ce9c085efed037ef3915c46": "(v,k,\\lambda,\\mu) = (16, 5, 0, 2)", "90529401638ee77edd7677c6e35ebd0f": " r_{ij} = x_{ij} / pmax(v_j), i = 1, 2, . . ., m, j = 1, 2, . . ., n, ", "90535d6f18fa9048dfc11d205bb852d8": "1x=x", "9053a20f078b81650a2ee3200ee3e6ea": "Q_2 = \\{O_{3},O_{5},O_{7},O_{9},O_{10}\\}", "9053c5e70e4c047976a65e9a04510257": " \\{ (1,D), (2,C), (3,C) \\} ", "9053c769a3534eab5de643fec1f663f8": "L\\ddot q + R\\dot q + q/C = e\\,", "905444cf7d96a0c134fd96b3f64ea008": "Z \\overline{Z}", "90545a0107b2a39447f5036d7701b77d": " \n\\mathbf{A}^{-1}=\\frac{1}{\\det (\\mathbf{A})}\\left[ \\mathrm{tr}\\mathbf{A}- \\mathbf{A}\\right].\n", "905499833ed98a61a49cac7e00c7c792": "\\phi(s)", "9054ce02633416a2fedfc1e01a0e4880": "\\scriptstyle{\\dot{\\boldsymbol{x}}}", "9054cfcb7c06b355a355f99eca9bb0dd": " M_{a} ", "9054f11437c733418a5c21fc878433f6": "O(2^{2L/3})", "90550e42f9b7a9bce5d34b61e23392ca": "\\Phi_L(\\mathbf r, t)", "9055535c2c267801ce17684b676653bc": "\\frac {\\partial M_x(t)} {\\partial t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _x - \\frac {M_x(t)} {T_2}", "905568386fb240e4707f885e67454b53": "f(x+y) \\geq f(x)+f(y)\\,", "9055d54f5c1e6baeb68aa93b87f688f0": " Q = S \\times \\mathbb{T}^\\infty \\times \\mathbb{T} ", "90560c2c4f4cb3096100e120c8f24ef8": "h(f)=\\frac{(1037918.48-f^2)^2+1080768.16\\,f^2}{(9837328-f^2)^2+11723776\\,f^2}\\ .", "90561a3042808443f2814ea0dde1cac1": " q=2 ", "90561a969a045218b6caca3b453af58c": " \\bold{B} = (\\partial A/\\partial y,-\\partial A /\\partial x,B_z(x,y))", "9056685ad8a014dfb58a74278a1dca29": "\\beta_{13}=K\\beta_{12}: K=\\frac{\\beta_{13}}{\\beta_{12}} \\,", "90568b879ada06c4c98ef45357c3ca49": " MC = \\frac{w}{MPL}", "9056bf4193f17cabdad45532c44369da": "P(G, t)", "90570442155501c310b0d248590bf421": "=\\sum_{g\\in G}\\langle X,g^{-1}\\rangle gY=\\sum_{g\\in G}\\langle X,g^{-1}\\rangle (g,e)[Y]", "9057c6b099c8bee255e776b8a14a2ed3": "f_m(r)= r^m\\sum_{t\\ge 0} f_{mt} [1-(r/R)^2]^t,\\quad 0\\le r\\le R.", "90581d96b500fd2d3fd701a583409cb8": "NC", "9058addad8635e4d8540865e2e4c9f50": "\\widehat x_\\mathrm{m} = \\min_i {x_i}.", "90593a8291d61acd38046056f90b1428": "28 = 2^2(2^3-1) = 1^3+3^3", "90596b90a87f27e6f1af8d3b59019efa": "U_1 = v_1-\\left(V^{M}_{N \\setminus \\{b_1\\}}-V^{M \\setminus \\{t_1\\}}_{N \\setminus \\{b_1\\}}\\right)", "9059eb60917daf68d34f993159e4c965": "E_{s,x^2-y^2} = \\frac{\\sqrt{3}}{2} (l^2 - m^2) V_{sd\\sigma}", "9059f83fb3b036c1112851888955c767": "\\forall D \\forall A \\, ( \\exists E \\, [ A \\in E ] \\implies \\exists B \\, [ \\exists E \\, ( B \\in E ) \\and \\forall C \\, ( C \\in B \\iff [ C \\in A \\and C \\in D ] ) ] ) \\,,", "905a6e0f6dcf687b6377738dfb5811c3": "10^{30}", "905aab7fee12bd4fd04c67dae7c0832c": "f=(f_1(x), \\dotsc, f_n(x))", "905ab2bcb26b66ee3d5cd8884a3544ec": "Z^0_2", "905ad3b31f0fd1c5f1f0225e0763696f": "\\mathrm{ad}\\colon \\mathfrak g \\to \\mathrm{Der}(\\mathfrak g)", "905b0b88585d439523dda9a1aa541d23": "p:\\Sigma^*\\to\\mathbb{N}^k", "905b3421a54d6ab10e69397362ad2072": "C_{\\infty v}", "905b566b5fb4359eb506b353ea3775f2": "p=2", "905bf5cacd7a7fa3a91c32a5b5da1f29": "\\gamma_3(t)=[\\cos((n+1)t),\\cos(nt)],\\quad t\\in [0,\\pi],", "905c64554320e3fb191e77f53ecc2940": "\\left\\{ \n\\begin{pmatrix}\na & b & 0\\\\ 0 & a & 0\\\\ 0 & 0 & a\n\\end{pmatrix}\n\\ :\\ a,b\\in\\mathbb{C}\\right\\}", "905c99290785f681430864195e8572a7": "S=\\lim_{t \\rightarrow \\infty} {e_t}/{t}", "905cb6037d800efe74b24b622ae57ccf": "(q,\\alpha) \\in \\delta(p,a,A)", "905cb7a62e7a17c18a3915115e103f87": " W_i = \\frac{CD-D_i}{CD}", "905cec3fc28d52da9d680143d2d8d59b": " A \t= \t\\sum_{i=1}^b\\sum_{j=1}^j R(X_{ij})^2 ", "905cfbfd71e72459aa4dd9d85ecc4696": "\\begin{align}\nT_1&=a+bz_1\t\t\\\\\nT_2&=a+bz_2\t\\\\\n\\vdots\t\\\\\nT_{n-1}&=a+bz_{n-1}\t\t\\\\\nT_n&=a+bz_n\t\\\\\n\\end{align}\\!", "905d9bee62cbc76be372e03683abc11d": "(1,0): Q = \\{(x,y) | (x-1)^2+y^2=1\\}", "905e752fa6697f2e1d99e5fc12c8d9dc": " \\det(\\rho_f(\\operatorname{Frob}_p))=p^{k-1} \\chi(p).\\ ", "905ea412d08bc675f746b2e62da3b0dc": " \\langle f , g \\rangle := \\int_a^b f(t) \\overline{g(t)} \\, dt. ", "905eff18cd861b8bb5b56b663ba2cd35": "Oscillator = (\\text{19 day EMA of Advances minus Declines}) - (\\text{39 day EMA of Advances minus Declines})", "905f33e7f4414b7d9ff559addd5c3fb3": "2g + 1", "905f612af7b4e0f763309a059430faeb": "\\gamma = V \\left(\\frac{dP}{dE}\\right)_V = \\frac{\\alpha K_S}{C_P \\rho} = \\frac{\\alpha K_T}{C_V \\rho}", "906027f81fde7c4445ab213f09638211": "(T f)(x) = \\int_{\\Omega} k(x, y) f(y) \\, \\mathrm{d} y", "90602dd5ce6e0eb5b18b1b1bcd0e3551": " |f(z)|\\le \\delta(r) |z|,", "90607cd5c860b480cee23767d821dd02": " = 1 + \\frac{1}{2(1)^2} ", "9060b35626acb8f9da2e23f95eac1f3c": " y_0 = A_0 e^{-\\omega x + \\sqrt{\\omega^2 - \\omega_0^2} x} = A_0 e^{-\\omega x} e^{\\sqrt{\\omega^2 - \\omega_0^2} x} ", "9060b8e250cf778c40b2a9a937d8aea0": "\\rho (a, b) = \\tanh^{-1} \\frac{| a - b |}{|1 - \\bar{a} b |}", "9060cd6548fe22de4bba833e80d2bbea": "\n(1)\\cfrac{\n (2)\\cfrac{\n {\\color{red}(1)}\\cfrac{C_1 (1,3)\\qquad C_2 (-1,2,5)}{C_3 (2,3,5)}\n \\qquad\n C_4 (1,-2)\n }\n {C_7 (1,3,5)}\n \\qquad\n (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{C_8 (-1)}\n}\n{\nC_9 (3,5)\n}\n", "9060e59147ca791d8f5cfce1cae096d4": "P< 4 \\rho^2 A\\ ", "9060f8815e5e12d68cdd28474d32b806": "\n\\frac{z}{c} = \\frac{x^2}{a^2} + \\frac{y^2}{b^2}. \n", "906100afb76665f48bfe6977df111bd9": "[J_f](\\mathbf{[x]})", "906109ee1a38fba435694e088efa4a98": " c_{2} \\, ", "906127a69c27bbde6ccaa1c4578ff822": " C_A(x) = C_{A0} e^{-k \\tau} \\, ", "906136fa7e6af1ab8243ff049ef2f021": "\n \\Delta h' = \\begin{cases}\n h_2^\\prime - h_1^\\prime & \\left| h_1^\\prime - h_2^\\prime \\right| \\leq 180^\\circ \\\\\n h_2^\\prime - h_1^\\prime + 360^\\circ & \\left| h_1^\\prime - h_2^\\prime \\right| > 180^\\circ, h_2^\\prime \\leq h_1^\\prime \\\\\n h_2^\\prime - h_1^\\prime - 360^\\circ & \\left| h_1^\\prime - h_2^\\prime \\right| > 180^\\circ, h_2^\\prime > h_1^\\prime\n \\end{cases}\n", "90614533bfacc5de2a71bc1c6a603f82": "x,y \\in [-1,1]", "906199136486441cd6c04aebd0b67aa1": "\\operatorname{dim}(S^k(V)) = \\binom{n+k-1}{k}", "9061a4e67a87f6e961f65ebd649aa136": " \\bold A = \\bold B - \\bold C, \\quad (2) ", "9061f722ba6e9c70f7a3988e26b59dc6": "\\pm \\sqrt{1-R^2}", "906283ee4dcc0dd6785f4e65f7c2699d": " \\langle k| H|k\\rangle =\\frac{1}{N}\\sum_{n,\\ m} e^{i(n-m)ka} \\langle m|H|n\\rangle ", "9062b9bfaff60e2e816bd17918103207": "\\phi: M \\rightarrow N ", "9062f37ae9aa3d3e104799e332ccda9e": "\\frac{dx}{ds}d^2x=d^2s=v\\ d^2t", "906343b027f88aacc86781f27804ab2a": "f_{uc}(\\langle \\text{Straße} \\rangle) = \\{\\langle S \\rangle\\} \\cdot \\{\\langle T \\rangle\\} \\cdot \\{\\langle R \\rangle\\} \\cdot \\{\\langle A \\rangle\\} \\cdot \\{\\langle SS \\rangle\\} \\cdot \\{\\langle E \\rangle\\} = \\{ \\langle STRASSE \\rangle \\}", "906376c531076f916fe0faa0e71e9adf": " \\begin{bmatrix}a & b\\\\ c & d\\end{bmatrix} ", "90638d83aef4dc3a47d9d920a49dc6b7": "\nx_N = \\sum_{m=1}^{N-1} \\frac{t}{(N-1)^2}\\sqrt{\\frac{N}{2\\pi(N-1)}}\\sum^{N-2}_{q=0}\\psi_m(q)_{(N+1)}F_N\n\\begin{bmatrix}\n\\frac{qN+N-1}{N(N-1)}, \\ldots, \\frac{q+N-1}{N-1}, 1; \\\\[8pt]\n\n\\frac{q+2}{N-1}, \\ldots, \\frac{q+N}{N-1}, \\frac{q+N-1}{N-1}; \\\\[8pt]\n\n\\left(\\frac{te^{\\frac{2m\\pi {\\rm{i}}}{N-1}} }{N-1}\\right)^{N-1}N^N\n\\end{bmatrix}", "9063b09460b3caf116e7ad8407f71295": "c_1 = \\frac{3}{8}", "9063cbfb8a4d47a513d3e22c39a71a0e": "\\scriptstyle x_n \\in U_n", "90641a7b107868aa2f7aee50ce75a6e4": " \\begin{matrix}\nI & \\equiv & \\langle E_x^{2} \\rangle + \\langle E_y^{2} \\rangle \\\\\n~ & = & \\langle E_a^{2} \\rangle + \\langle E_b^{2} \\rangle \\\\\n~ & = & \\langle E_l^{2} \\rangle + \\langle E_r^{2} \\rangle, \\\\\nQ & \\equiv & \\langle E_x^{2} \\rangle - \\langle E_y^{2} \\rangle, \\\\\nU & \\equiv & \\langle E_a^{2} \\rangle - \\langle E_b^{2} \\rangle, \\\\\nV & \\equiv & \\langle E_l^{2} \\rangle - \\langle E_r^{2} \\rangle.\n\\end{matrix} ", "90644f0884299a5174a9b1a0c5f6eb04": "0 = \\underbrace{1 + e_1^2 + \\cdots + e_{n-1}^2 }_{=: a} + \\underbrace{e_n^2 + \\cdots + e_s^2}_{=: b}\\;. ", "9064920ec290f0ed012507a90ff9a0be": "P(\\vec x|{\\rm class})", "906498f46d87a44405a827f4c14d4081": "\\liminf_{\\delta \\downarrow 0} \\delta \\log \\mu_{\\delta} (G) \\geq - \\inf_{x \\in G} I(x). \\quad \\mbox{(L)}", "9064b209224802bd3175fd8023014de5": "\n\\mathrm{Fr} = \\frac{v}{c}\n", "906546a490388e6e37748ce4c7261b36": "2^{n-k}{n \\choose k}", "90659996b459f730265a26d4fdd88edf": "B \\in \\mathcal{B}", "9065bf110d44e454c3a4cddd6bdac196": " T_\\mathrm{n} = \\sum_{A} \\sum_{\\alpha=x,y,z} \\frac{P_{A\\alpha} P_{A\\alpha}}{2M_A} \n\\quad\\mathrm{with}\\quad \nP_{A\\alpha} = -i {\\partial \\over \\partial R_{A\\alpha}}. ", "9066817120942ef3e98fc8e366289620": "\\lambda + \\alpha\\beta(1+\\beta)", "906688688b8b23ce8a20972fde98a4ad": "\\pi R^2", "9066e6274bef48c5421c179f37b92c5d": "\nP( \\gamma ) = \\int_{\\gamma} \\bar{j}(y) dS_y\n", "90674e89a1665b267ffda006f4b421b9": "I_1=I_0 \\sin^2\\left(\\frac {\\pi d \\sin \\alpha}{\\lambda}\\right) / \\left(\\frac {\\pi d \\sin \\alpha }{\\lambda}\\right)^2 \\ , ", "906753801edd310e30ce21bbc7de9f12": "|x| \\beta_2 > \\ldots > \\beta_k \\geq 0", "906dec662fd1856bedd8f726150d072f": "\\mathrm{0.\\overline{923076}}", "906e7b4cdde0ad448da967c8e42988f0": " \\tilde{u}(x):=\\{u(x):G(x,u,p)=0,p\\in {\\mathbf p} \\} ", "906e83369da7d98c01919f601b8a5b3a": "x_{N-0}=x_0", "906e8a3ec69819189da3e148c7ce57a5": "\tx^5-5p(2x^3 + ax^2 + bx)-pc = 0\\,", "906e98bc05ee3dc4070664521f13dc97": " N \\log N", "906ec66da6bbf2a2af053a1b4e1450ff": "\n H = U \\sum_i n_{i,\\uparrow} \\langle n_{i,\\downarrow}\\rangle\n +n_{i,\\downarrow} \\langle n_{i,\\uparrow}\\rangle\n - \\langle n_{i,\\uparrow}\\rangle \\langle n_{i,\\downarrow}\\rangle +\n \\sum_{i,\\sigma} \\epsilon_i n_{i,\\sigma}.\n", "906f5f05bb183a7f0f5ed386413b4c86": "\n\\tau = 1 | \\tau\\times\\tau | \\ldots\n", "906f8ddcf28d3637d1e002d9e5a30825": " c(x) = f \\cdot \\nabla V + \\nabla \\cdot \\Gamma \\cdot V", "906fa470cb18c3329f3a602eb05eb0d8": " D'(x)=\\left|\\begin{array}{ccc}f'(x) & g'(x)& h'(x)\\\\ f(a) & g(a) & h(a)\\\\ f(b) & g(b)& h(b)\\end{array}\\right|", "90706ba3ceca077e4ac0dfc6116adc7b": "\\Sigma_{n} x_n", "90707c60a65ced4ad7cc0b0559522f11": "\\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = Df(\\boldsymbol{S})[\\boldsymbol{T}] = \\left[\\frac{d }{d \\alpha}~f(\\boldsymbol{S} + \\alpha~\\boldsymbol{T})\\right]_{\\alpha = 0}", "9070a490545f737462900bf2bd1c44a0": "I_{t}=\\mu v\\sum_{i=1}^{\\infty}\\left(1-\\mu\\right)^{i}\\left(Y_{t-i}-Y_{t-i-1}\\right)", "9070bad48a1d306468e80a024773d1a5": "R_a \\le {R_n \\over \\Omega} ", "9070c47838e1a6806dcc7d421f9d36ce": "K(A) \\rightarrow D(A)", "9070caaa0c869f10b00b5083eaf0fa44": "\\frac {dA(t)} {A} = W \\cdot {dt}", "9070fade823f8f8f78018d43d0fa8123": " a\\otimes b - b \\otimes a - [a,b], \\quad a,b \\in L.", "907151898b62e32ddd6590f5f1a823da": "\n\\begin{matrix}\n\\begin{cases}\nX = a + \\sqrt{U(b-a)(c-a)} & \\text{ for } 0 < U < F(c) \\\\ & \\\\\nX = b - \\sqrt{(1-U)(b-a)(b-c)} & \\text{ for } F(c) \\le U < 1\n\n\\end{cases}\n\\end{matrix}\n", "907172befc1d646de74f52aa87ec9462": "\nx \\rightarrow \\begin{pmatrix} x^0 + x^3 && x^1 - ix^2 \\\\ x^1 + ix^2 && x^0-x^3\n\\end{pmatrix}\n", "9071954f12746732def725e43f91bb18": "M / {pV}", "9071ff01c781db029a632040a7fc72c3": " a(4q+2)+b(4p+2)=(2k+1)(4q+2)+(2k'+1)(4p+2)=4(2kq+2k'p+q+p+k+k'+1)", "90720bacc5ab0cc27abc30695600ceb6": "rank[\\Delta X'\\Delta X]=k", "907212a6a7e8df68df7581bf8481c606": "(\\mathbf u(s),\\lambda(s))", "907247b6cc2262a13b15c4f24c750c7a": "s=\\frac{\\log[1-c(1-ee'')]}{\\log[1-c(1+ee'')]}", "9072cbc1525aaf5f3a744151c13dac61": " (x_0,x_1,\\ldots x_n,x_{n+1},x_{n+2})\\cdot (y_0,y_1,\\ldots y_n,y_{n+1},y_{n+2})", "907343ddaafa5edd6187168607a7834c": "\nw = a + b(e + x + gy) \\,\n", "90735f88e08f855032cf3dc52cdca198": "\\left(g \\circ f\\right)\\!\\left(x\\right) = g\\!\\left(f\\!\\left(x\\right)\\right)", "907390eb04734bcb75cd2ebe65790c73": "\\hat{x} = W(y-\\bar{y}) + \\bar{x},", "9073a0af2bf83ec1965cdc0d44c3a6e3": "f(x+\\alpha)-f(x)", "9073a5da9cc1c54984e7d459557d0770": "w_j ( k + 1 ) = \\begin{cases} w_j ( k ) & \\mbox{if}j \\notin \\Nu_{k+1} \\\\ w_j ( k ) + LR ( k ) \\times (x_k - w_j ( k ) ) & \\mbox{if}j \\in \\Nu_{k+1} \\end{cases}", "9073c2c4580bc334a6e19bdf2cafd15a": "\\text{Utility} = \\ln W_T,", "9073e7f76b7d895cf6b65931d7a00427": "\\beta(\\alpha_s)=-\\frac{11N}{12\\pi}\\alpha_s^2-\\frac{17N^2}{24\\pi^2}\\alpha_s^3+O\\left(\\alpha_s^4\\right).", "9074405a1aeb2b91c003db60608401ef": "O(N^3)", "9074509ef480dfbf47cda1c907f07d9a": " - {A^{\\alpha ; \\beta}}_{ \\beta} + {R^{\\alpha}}_{\\beta} A^{\\beta} = \\mu_0 J^{\\alpha} ", "907458730b1cd65439c7f79bcf6a962d": "f(x)=\\exp^{[x]}_a (a^{x})=\\exp^{[x+1]}_a(x) \\quad \\text{for all} \\; \\; x > -2,", "90747c6bfb9ca0d9640885fed28a2b5f": "\\text{If }x + y + z = \\tfrac{\\pi}{2} = \\text{quarter circle,}\\, ", "907482354b593cd2e4db0a8cfddea06d": "66.1 \\times 10^3", "9074ccf7c9d171874a4738f0eb35d8c2": "\\{r^2-R^2,\\, H\\}_{PB} = 0", "9074d4c116aec5c3b0be1d0a2e92ec4d": "\\textstyle \\sin \\alpha = \\frac{\\text{opposite}} {\\text{hypotenuse}}", "9074e410bf923bc4c21947c940673438": "K_m^\\text{app} > K_m", "90751adb4fb9307e58738566678a3b42": " F^{-1}(u) = \\sigma \\exp \\sqrt{-\\frac{1}{\\beta} \\log(1-u)}, \\quad 0 < u < 1. \n", "907542ece7b6fa925ae4810951f15cc2": "F(N,V,T,\\lambda)=-k_{B}T \\ln Q(N,V,T,\\lambda)", "90756e7a50111e4756044befd50b2c8e": "\\frac{P}{K}(1-t_p)= \\frac{g_n}{s'_c}", "9075b7d94bffc740d22d12579308b6c4": " O(\\sqrt{n}))", "907601c8553601355ed01e6c7f81c245": "\\widehat{\\rho} = \\int \\varphi(\\alpha) |{\\alpha}\\rangle \\langle {\\alpha}| \\rm{d}^2 \\alpha,", "9076c631d96527dfe4d2319985a1382b": "C \\subseteq A \\subseteq X", "9076dd15b1985c7bdd981f1bc378e3c8": "c = \\frac{L}{2}\\;\\frac{{t_{up} + t_{down} }}{{t_{up} \\;t_{down} }}", "90771e5732e3001121173577ddbd11c6": "\\scriptstyle{\\hat{H}(t_0)}", "90772ad23d6bc97ecd95e74fa1a4a2a6": "\\nu \\le_1 c(\\nu)", "9077393a2ffa66fa4cb1cc79f7278757": "f_2(z)=1-\\frac{(1-i)z}{2}", "907748a470f31f7cecdd0b7f94eebad1": "\\sum_{i=0}^m|\\Delta F_i|^2\\!", "9077603708c9628a08008a71a88c12ab": "(2n+1)\\times(2n+1)", "90779d7015ca24997bbe15489de7b0df": "x^2+y^2+z^2+t^2=1", "9077d881fe24e9787b0162fb1b9b610e": "\\boldsymbol{z}_{k} = h(\\boldsymbol{x}_{k}, \\boldsymbol{v}_{k})", "9077f3dfe0239689cbf0b849fc53ae2c": "H^0(X, \\mathcal{O}(D))", "90785104a33d2443dc5bf25674c0570c": "r^{-2}", "90786e2b6fc430f7a8f38680ec489be7": "\\rho(x,u,u_{1}) = 1 + u_{1}u_{1}\\,", "907875ae6b85403d4b4fb4e3dfb11fac": " \\Lambda_{\\left(p-k\\right)}^{1/2} = \\text{diag} \\left(\\lambda_{k+1}^{1/2},...,\\lambda_p^{1/2}\\right)", "9078844a1c3c8be33e3c2005e07e5119": "\\langle f_i , \\, f_j\\rangle = \\, \\delta_{i,j}", "9078df815fa9576ed82e202c1432e954": "(\\mathbf u + \\mathbf v) \\times \\mathbf w = \\mathbf u \\times \\mathbf w + \\mathbf v \\times \\mathbf w", "90796e464c9d075efee8d75e458b986e": "(x+y)^4 \\;=\\; x^4 \\,+\\, 4 x^3y \\,+\\, 6 x^2 y^2 \\,+\\, 4 x y^3 \\,+\\, y^4.", "90797f1ead2078362894a464822ec477": "s_\\Phi(x) \\equiv x^q \\mod \\Phi.", "9079a8b0624d274e903e963e983344eb": "f^{-1} \\mathcal{I} \\cdot \\mathcal{O}_Y", "9079af1802512e74346d135ccdd4dabf": "k<=n", "9079c53b5049f6cc6592065b43131bcb": "\n2ik\\frac{\\partial E}{\\partial z}=\n\\Delta_{\\perp}E + 2 \\nu E + i G E", "907a47a1447ec1cd1feead66f29318e1": "K(S) \\leq \\alpha", "907a512182a8df70b5faf25adf1eefa2": "Cl_t^{\\geq}", "907a6be974a593f0ea64d2c002b8a2cf": "g = dx_1^2 + \\cdots + dx_p^2 - dx_{p+1}^2 - \\cdots - dx_{p+q}^2", "907a7556c5e08ec2c6e02e8d106bdc6f": "\\,V_{th}", "907aa97328927b5314e7ff74de973776": "\\nu >2\\!", "907ae0b7583c5917ed6f30b9769df561": "\\hat{H} \\equiv \\hat{H}_0 + \\lambda \\hat{H}_1 ", "907b039857894721fbcb2f11df31779e": "\\ \\displaystyle g=g(d,s)", "907b39aafb392566884e8af1964caa3d": "H(X|Y) = H(Y,X) - H(Y)\\,", "907b5bff41cacf29dc4b6f78a0eeb2d1": "p_r^{\\rm{sat}}{\\rm \\ at \\ } T_r = 0.7", "907b6a10630caee2f2e6a38885c98b23": " r_\\mathrm{corr} = nr - \\frac{ n - 1 }{ n } \\sum_{ i \\ne j 1 }^n r_i ", "907b7fcef54935e95678e23896997af5": "\\!t_1 \\ldots t_n", "907bc321c6314d9c0bce6b8571043b45": "F: C \\leftarrow D", "907be0408ea933c4391cc43325ac8a29": "[h,f]=-f", "907c467b4fd51055978aa71da9bc2c67": "z = x + y \\ \\jmath", "907cd1f7471c1acec0fa739cfffa2c6c": "u^2+v^2+w^2-w=0", "907d22176e59894579329b6c77232a82": "\\,\\sgn\\phi(c,\\bar{c}) = \\sgn\\left(c-\\left(\\frac{\\bar{c}}{c_0}\\right)^p\\bar{c}\\right) ~~ \\textrm{for} ~ c>0, ~ \\textrm{and}", "907d4daaf9ea08fc04bfcb6fbfcc4f26": "E = \\sum_{n=-\\infty}^{\\infty}(x[n]^2 - 2x[n]\\hat{x}[n] + \\hat{x}[n]^2)", "907dc4295b388ff9307e3cd8ad559d4a": " C\\,\\!", "907dce39480dd6cdb11385eb167b5dee": "\n\\mathbf{C'}\n = \\Big\\{\\;\n \\big( s_1, s_2,\\dots, s_{n} \\big)\n \\;\\Big|\\;\n s(a)=\\sum_{i=1}^n s_i a^{i-1} \\text{ is a polynomial that has at least the roots } \\alpha^1,\\alpha^2, \\dots, \\alpha^{n-k}\n \\;\\Big\\}\\,.\n", "907df0cbee248f4d9d73b0f5cee3f418": "\\frac{k^2}{l^2} = \\frac{bd}{ac}.", "907e08994342d26b1f69c97f90f09757": " R \\propto \\frac{\\int{p(I | \\theta, O_{fg}) p(\\theta | I_t, O_{fg})} d\\theta}{\\int{p(I | \\theta_{bg}, O_{bg}) p(\\theta_{bg} | I_t, O_{bg})} d\\theta_{bg}} = \\frac{\\int{p(I | \\theta) p(\\theta | I_t, O_{fg})} d\\theta}{\\int{p(I | \\theta_{bg}) p(\\theta_{bg} | I_t, O_{bg})} d\\theta_{bg}}", "907e3702939750f8692cd72e1d69144c": "{\\mathbf{}}S_i", "907e3e33a725b3eb00c608bb34a79e16": "\\mathbf{n} \\cdot \\mathbf{r}_0 = \\mathbf{r}_0 \\cdot \\mathbf{n} = -a_0", "907ef78148d80647ad166387d5e2a15d": " \\scriptstyle \\zeta '(2) \\,\\text{= Derivative of }\\zeta(2)= \n- \\sum \\limits_{n = 2}^{\\infty} \\frac{\\ln n}{n^2} = -0.9375482543\\ldots ", "907f1701c2c3c1f77ad6636bdde5bc99": "L(5;2)", "90800d1d375c481f927347191e35c459": "c \\mid b", "9080bdbdd9fcc89cd522a1b7210a1c24": "\\{0\\} = V_0 \\sub V_1 \\sub V_2 \\sub \\cdots \\sub V_k = V.", "9080c586dffb32468e188e2bfee7cd7b": "q\\,=\\,\\sqrt{3}/{\\pi}\\,=\\,0.551328895+", "9080d08b0ea3eb2fdb03c9c875d5ee36": "\nR_{U} = 1 - \\sqrt{\\frac{6(n+1)}{n(2n+1)}\\sum_{i=1}^{n}{\\left(R_{Fi}-\\frac{i}{n+1}\\right)^2}} \n", "9080e67c8345879f880cf6f734d1a0b1": "v_c = \\frac{p_T c^2}{E}", "908199fb130e5a2e2a094018ae941dc7": "a_k=\\mathbb{P}(A_I)\\quad\\text{for every}\\quad I\\subset\\{1,\\ldots,n\\}\\quad\\text{with}\\quad |I|=k,", "9081e4b218bc6d80197b419aed0e5dcb": "\n \\boldsymbol{\\sigma} \n = \\cfrac{2}{J}~\\left[\\left(\\cfrac{\\partial W}{\\partial I_1}+ \n J^{2/3}~\\bar{I}_1~\\cfrac{\\partial W}{\\partial I_2}\\right)~\\boldsymbol{B} - \n \\cfrac{\\partial W}{\\partial I_2}~\\boldsymbol{B}\\cdot\\boldsymbol{B}\\right] + \n 2~J~\\cfrac{\\partial W}{\\partial I_3}~\\boldsymbol{\\mathit{1}}~.\n ", "9082156ae7bfc7e6dfea1b7553d26e11": "\\left(\\frac{q}{p}\\right).", "90822f52297fa110f76997e6ca8e9b4d": "\\hat{\\lambda}_p", "9082a4be2747dad95e788a6c9d5b99bb": "\\bar{M}P=O", "9082b83c12d691ecf771c5cb362420ca": "\\phi_0 = \\begin{bmatrix}1 \\\\ 0\\end{bmatrix}", "9082e1fd1a1a2fdc1e2299ecb2b01a39": "R_4", "90830b4214fd8b76dc90d76de841a305": "\\sigma_I^{(0)}", "908313dff1fdad57580b4b14f3fa9e38": "\\boldsymbol\\theta^{(t)}", "90836a67fc84c239dcf8a2ccd910520d": "\\Delta X_n^1 = X^1(p_1, m) - X^1(p_1,m').", "9083b407dc2583aa764390314e34bc2a": "\n\\frac{d\\rho}{d\\zeta} = 0\n", "9083cf5f3a27186cb93538e50b373720": "\\psi(\\omega^3)", "9084717daacf3ae14f7a81483a6cd29b": "\\tan x = \\frac{\\sin x}{\\cos x}", "90848925a45605fd78812952e34b4eba": "r_s = \\left| \\frac{S_{12}S_{21}}{\\left|S_{11}\\right|^2-\\left|\\Delta\\right|^2} \\right|\\,", "90848ee09b4974dab65a50a2a7b223fb": "\\mathfrak{q}_1 \\supseteq \\mathfrak{q}_2 \\supseteq \\cdots \\supseteq \\mathfrak{q}_m", "9084c4345f0ad137220cd6568c902e17": "0 = \\nabla_\\nu (\\xi^{\\mu} T_{\\mu}^{\\nu}) = \\frac{1}{\\sqrt{-g}} \\partial_\\nu( \\sqrt{-g} \\ \\xi^{\\mu} T_{\\mu}^{\\nu}) ", "908512a09edc9c9390c58e8a45725225": " \\mathcal{S} = \\int \\, \\mathrm{d}^4 x \\sum_{i=1}^n \\left[ \\frac{1}{2} \\partial_\\mu \\varphi_i \\partial^\\mu \\varphi_i - \\frac{1}{2}m^2 \\varphi_i^2 \\right]", "90860f76f707d819d63c152506cf01d0": "\\mathcal{L}_Y", "908613df98fd3c7fe92ab9846fbb43ee": "D = \\frac{2E}{N(N-1)}.", "90861d468752bc5545f6ae68dca45b0a": " = \\mathcal{S} + {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-h} h^{ab} \\left( \\omega_{\\mu \\delta} \\partial_a X^\\mu \\partial_b X^\\delta + \\omega_{\\nu \\delta} \\partial_a X^\\delta \\partial_b X^\\nu \\right) + O(\\omega^2) \\, ", "9086293b996e894b5831ec37d4a52a34": "w(x;F_0)=[F_0(x)(1-F_0(x))]^{-1}", "9086497c478ec282f20ad7bc89c67145": "\\scriptstyle H(s)", "908662fc8a7ce37d4ab784fcac3950a9": "i \\wedge j \\rightarrow k", "90868a518ca69a4c360f855d7f0f2eef": "C_{i}^{RWB} = \\sum_{j \\neq i \\neq k} r_{jk} ", "908694b18bd50e7a7e85984001321d3f": "\\hat P_{PHD}(e^{j \\omega}) = \\frac{1}{|\\mathbf{e}^{H}\\mathbf{v}_{min}|^2}\n", "9086ad9bbaf1d6a457f5253fe0b5e6f7": "f_1(\\omega)\\,", "90871b35f768de5a4558a38f3f91762a": "\\theta(r)", "90873bdd76c89fda7238fff417d85e12": "\\|\\mu\\|_{TV} = \\mu_+(X) + \\mu_-(X)~,", "90875a81bf32875ce1b87a3fb0cb2544": "\\begin{pmatrix} \n1 & 0 & \\ldots & 0 \\\\\n0 & & & \\\\\n\\vdots & & A & \\\\\n0 & & & \\\\\n\\end{pmatrix}", "90877d0bd3ab089b49cad34e48f26a48": "\\scriptstyle \\cos\\theta \\pm i\\sin\\theta", "9087d45c14595d14556196654b0254ae": "\n \\frac{1}{n}\\sum_{i=1}^n (X_i-\\overline{X})^2 < \\frac{1}{n}\\sum_{i=1}^n (X_i-\\mu)^2,\n ", "9087e51a3a8de36945f1f8b5e98dcf6e": "\\mathfrak{sl}(n,F), \\mathfrak{o}(2l,F), \\mathfrak{t}(n,F)", "9087e73d011e822089521828d190e9b6": "n \\in \\mathbb{N}.", "90882dd576ddcfd0b333093c6b0e5715": "\\partial_i=(\\partial_i\\eta_j)\\partial^j=g_{ij}\\partial^j", "908917655dff28de43b7a82dec3dfcb1": "x=a\\sinh{u}", "90893323430dd57a7e480614cdf5bb30": "\nQ = \\frac{\\prod_j a_j^{\\nu_j}}{\\prod_i a_i^{\\nu_i}} \\approx \\frac{[Z]^z [Y]^y}{[A]^a [B]^b}.\n", "9089813d45eb2653be15711572f28842": "(gate3\\vee x2)\\wedge (\\overline{gate3}\\vee \\overline{x2})", "9089e7c715d5270d654f6f03e2727bc2": "c_{3,1}(\\widehat{a}, w(c_{3,1}(\\widehat{a}, \\widehat{b}c, \\widehat{d}), \\widehat{b}c), \\widehat{d})", "9089eba32fc734943d5350cf64bb8816": " \\frac{1}{1-x} = 1 + x + x^2 + \\cdots ", "908a248656f55a9fab174d5da327751d": "\\mathbb{Q}(A) = g(\\mathbb{P}(A))", "908a33d74d9057c251d4ac1d0a9e94c0": "\n\\mathcal{H}= \\begin{pmatrix}\n \\begin{matrix}\n 0&0\\\\\n 0&0\n \\end{matrix}\n & \\begin{pmatrix}\n\t\t \\cdots 0\\cdots\\\\\n\t\t\t \\leftarrow v^t\\rightarrow\n \\end{pmatrix}\\\\\n \\begin{pmatrix}\t \n \\vdots & \\uparrow\\\\\n 0 & v \\\\\n \\vdots & \\downarrow\n \\end{pmatrix} & B\n \\end{pmatrix}\n", "908a5096c3e76dbcfa6602e64420ee18": "\\sum_{s=1,2}{u^{(s)}_p \\bar{u}^{(s)}_p} = p\\!\\!\\!/ + m \\,", "908a82a85e40bcaa527b0d8745b9fe7a": "-7\\eta^2-14\\eta-3", "908aa6d31230f93b0898780c78b08d39": "\\frac{\\partial T}{\\partial t} = a \\frac{\\partial^2 T}{\\partial x^2} - \\epsilon u \\frac{\\partial T}{\\partial x} + \\frac{Q}{c \\rho}", "908ab10b5bf5011741d02035d965e41d": "a_i = \\arg \\min ~E\\{e^2[n]\\} ,", "908ac83197a8697bdca132b8fe698013": "\\frac {F_{out}}{F_{in}} = \\eta \\frac {2 \\pi r}{l} \\qquad \\, ", "908b693975c02f247af1e7cd57d0170c": "\\left( \\frac{3}{\\sqrt{10}},\\ \\sqrt{3 \\over 2},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "908bebd27d236c6e579b895c29d4effb": "\\operatorname{MSPE}(L)=\\sum_{i=1}^n\\left(\\operatorname{E}\\left[\\widehat{g}(x_i)\\right]-g(x_i)\\right)^2+\\sum_{i=1}^n\\operatorname{var}\\left[\\widehat{g}(x_i)\\right].", "908c040a554bbcedfb63a9f411f75490": " \\log_p ", "908c07fc271135c232e51fb3059e0cbd": "\\text{rate}=\\frac{V_\\text{max}[S]}{K_M+[S]}", "908c9f5658d91b83c82e6c582d3cb37d": "\\chi^2=wy^2\\;\\left [=193>\\chi^2(p=0.001,\\; df=1)=10.8 \\right ]", "908d0357d660e5123f17fd583266db01": "\\frac{\\partial N}{\\partial t} = -A_{21} N,", "908d1da202a72d564bf73aad0cc73724": "\\mathbf{v}'_1 = \\mathbf{v}_1 - \\mathbf{v}_0", "908d299d111cdbcefc1e3cb5b366d969": "\n\\begin{align}\nC_1(A) & = A \\\\[6pt]\nC_n(A) & = \\det(A)\\text{ if }A\\text{ is }n\\times n \\\\[6pt]\nC_k(AB) & = C_k(A)C_k(B) \\\\[6pt]\nC_k(aX) & = a^kC_k(X) \\\\[6pt]\n\\text{For } n\\times n \\text{ identity } I, C_k(I) & = I\\,, \\text{ the }\\textstyle{\\binom n k\\times \\binom n k} \\text{ identity }\\\\[6pt]\nC_k(A^T) & = C_k(A)^T\\,, \\text{ over any field} \\\\[6pt]\nC_k(A^*) & = C_k(A)^*\\,, \\text{ over } \\mathbb{C} \\\\[6pt]\nC_k(A^{-1}) & = C_k(A)^{-1}\\,, \\text{ for } n\\times n, \\text{ invertible } A\n\\end{align}\n", "908d55ee7e4c8d82917e0b179a9a40e3": " X = \\coprod_{j\\in J}X_j", "908d98e81a036cf12953f200ed05dad5": "m \\in \\mathbb{R}", "908dca448506d8856c37f507faa65d9b": " {ALE} = {ARO} \\times {SLE}", "908e3ea7038ef0b893234a2a3a8ff998": "\\hat{z} + \\Delta \\hat{z}", "908ee7f60df53648236c85006ce33081": "\\mathbf{P} \\left( \\bigg\\Vert \\frac{1}{t} \\sum_{i=1}^t M_i - \\mathbf{E}[M] \\bigg\\Vert_2 > \\varepsilon \\right) \\leq \\frac{1}{\\mathbf{poly}(t)}", "908f063388ab1108dbe6422206251681": "f(\\tau)\\,", "908f44261610c722071ce079edc6bdd6": "\\forall m\\colon f(n,m) = O(n^m) \\text{ as } n\\to\\infty", "908f466a189145a7c5d195635f37e808": "\\forall x\\,Fx \\rightarrow \\exists x\\,Fx", "90907612b007d4bcb57e200f0f81de14": " \\left ( \\frac{1}{8} + \\frac{2}{5} + \\frac{4}{10} \\right ) = \\left ( \\frac{37}{40} \\right ) = 0.925 = {\\mathbf{92.5%}} ", "909088f926c36821907b1d6e61949c42": "x_{2}(t)", "9090ffb2098bba21f9c2d50cc381df85": "\n\\int_{-\\pi}^{\\pi} \\cos((2m-n)x)\\sin^n x\\ dx = \\left \\{\n\\begin{array}{cc}\n(-1)^{m+(n+1)/2} \\frac{\\pi}{2^{n-1}} \\binom{n}{m} & n \\text{ even} \\\\\n0 & \\text{otherwise} \\\\\n\\end{array} \\right.", "90913f497f2be502aee4016699eedcfe": " H(t) \\simeq M|M|^{\\delta-1} f(t/|M|^{1/\\beta})", "909146061d4d51b5389a67cf31b2ba7d": "\\prod_{i \\in I} X_i = \\left\\{ f : I \\to \\bigcup_{i \\in I} X_i\\ \\Big|\\ (\\forall i)(f(i) \\in X_i)\\right\\},", "909155e20377d3091649c630739a3813": "\\mathbf{a}\\otimes \\mathbf{b} = \\mathbf{a}\\mathbf{b}^\\mathrm{T} = \\begin{pmatrix}a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n\\end{pmatrix}\n\\begin{pmatrix}b_1 & b_2 & \\cdots & b_n\\end{pmatrix}\n= \\begin{pmatrix}\na_1 b_1 & a_1 b_2 & \\cdots & a_1 b_n \\\\\na_2 b_1 & a_2 b_2 & \\cdots & a_2 b_n \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_n b_1 & a_n b_2 & \\cdots & a_n b_n \\\\\n\\end{pmatrix}.\n", "9091ad76d2abb779816c28594118f6a7": "\\mathcal{G}: \\mathbb{N} \\to \\mathbb{N} \\,\\!", "9091ccfadb92a813a7ce5ed4f92f73ae": "\\overrightarrow{q} = - k {\\nabla} T", "9091d313adac4cb9dd5fed7872a89f00": "H(X) = - \\sum_{k\\ge 1}p_k\\log p_k .", "909203145589513f65c8badbe341d0c9": "\\frac{f(x)-f(y)}{g(x)-g(y)}=\\frac{f'(\\xi)}{g'(\\xi)}", "90927dbdd8d26315a8023c5b7dc0d318": "\\frac{dx}{dt} = f(x, y) = x\\left(1 - \\frac{x}{K}\\right) - y \\frac{x}{1 + x}", "9092823ee991253a19b420e00411ca08": "|1\\rangle", "9092b8ad93ccbcfc1c918eccdf6d6931": "\\hat{G}_X < 1", "90933f3f934d2e195387f60c24b9a744": "\\lim_{n \\to \\infty} \\left(1 + \\frac{1}{n}\\right)^n", "9093c3b97994bb7f400ea44ccfa01616": "C_p = [U^2, D^2, F, B, L^2, R^2, R^2 U^\\prime F B^\\prime R^2 F^\\prime B U^\\prime R^2].\\,\\!", "9093cc72d2d40694b9361424cb0a6803": "(u,v)", "90940a24485f89992b16f04390a821d1": "\\underline{A_\\mu(x)A_\\nu(x')}=\\int{d^4k\\over(2\\pi)^4}{-ig_{\\mu\\nu}\\over k^2+i0}e^{-ik(x-x')}", "909466419bb946814a9156d98672129f": "\\mathbf{P}(\\mathcal G)(k)", "909493a8c4e1f88fc27093fc3411fd2a": "S = \\frac{R_{F_1}-R_{F_0}}{\\lambda_1 - \\lambda_0}", "9094a17e42ae30d9786c6d6b67089178": "\\boldsymbol{\\alpha} = \\frac{{\\rm d} \\boldsymbol{\\omega}}{{\\rm d} t} = \\bold{\\hat{n}}\\frac{{\\rm d}^2 \\theta}{{\\rm d} t^2} \\,\\!", "9094a4d6eda3a8d20f27fb6e35c3c02f": "\\psi \\to \\psi^\\prime = (1+i\\alpha\\gamma_{d+1})\\psi = \\sum\\limits_i \\psi_ia^{\\prime i},", "9094be63bd0064fb4d92936eb23684c5": "c(x, y, z) = \\frac{1}{R(x, y, z)},", "9094caf175ca9e68a4fd0af34bea1b63": "x_i=x_i(q_1,\\ q_2,\\ \\dots,\\ q_m,\\ t)\\ ,\\qquad\\qquad\\qquad i=1,\\ 2,\\ \\dots N. \\, ", "9094f33836b80e9c45d111bff87865ec": "A_i = \\frac {z_i q}{4 \\pi \\varepsilon_r \\varepsilon_0} \\frac{e^{\\kappa a_i}}{1 + \\kappa a_i}", "909503944634c607e4dfa99b1baaeff4": " \\phi(-x) R(\\pi)\\phi(x) \\ ", "909514155d171da3523f923a0f8f4360": " m = \\underset{x \\in M}{\\operatorname{arg\\,min}} \\sum_{i=1}^n w_i d(x,x_i) ", "90951ed862852a22fa0cd414038a4789": "\\pi : e_{ij} \\mapsto E_{ij} ", "909534300a5e9ee3fa26100fdc982d43": "= \\mathcal{L}_{V^{1}}(du - u_{1}dx) \\,", "909549a277df6c8d9a22cb2849476d9c": "\\phi_{ex}", "909556f1b29ae7330703f3de0ba32b89": "\\ddot y + 2n\\dot x = \\frac{\\delta U}{\\delta y}", "909556fa857ee01c3922112d897b72fe": "\\ln \\Gamma", "909578b7f0430ab1474e19971aaa4cb9": "m_\\mathrm{p}", "90959b10996d0ea331f2072c6f57fc7c": "\\mathfrak{sl}_n(\\mathbb{R})", "9095e331349280c34a6320f2ab6cd414": "\\sum_{i=1}^n \\mathrm{Gamma}(\\alpha_i,\\beta) \\sim \\mathrm{Gamma}\\left(\\sum_{i=1}^n \\alpha_i,\\beta\\right) \\qquad \\alpha_i>0 \\quad \\beta>0 ", "9095e39ca8fd5d4c5a780cb52d62310c": "\nF(x)=1-\\boldsymbol{\\alpha}e^{x\\Theta}\\boldsymbol{1}\n", "90960019195594a49df96b81e0331fa5": "\\mathbb{Z}/2\\mathbb{Z} \\xrightarrow{(1,0,0)} (\\mathbb{Z}/2\\mathbb{Z})^2 \\oplus \\mathbb{Z} \\xrightarrow{(0,1,0)} \\mathbb{Z}/2\\mathbb{Z}", "909648441e18854d927f0c62e7265113": "\\chi = 10^{-3}\\left [ 187 \\left ( E_{I} + E_{EA} \\right ) + 170 \\right ] \\,", "90965189b257c503b881572871b022ea": "\\lambda A + (1-\\lambda)B ", "9096bf7f114e4617ae05d20d1a9fa924": "x^{2^{j}s} \\not\\equiv -1 \\bmod\\,\\big(n,x^2-bx-c)", "9096c65fec82a640f6b61fabe82ec1b8": "\\langle 0,0\\rangle", "9097d080ddb6009f963ce583da2427d3": "\\mathcal{S}_\\mathrm{EH} + \\mathcal{S}_\\mathrm{GHY} = \\frac{1}{16 \\pi} \\int_\\mathcal{M} \\mathrm{d}^4 x \\, \\sqrt{g} R + \\frac{1}{8 \\pi} \\int_{\\partial \\mathcal{M}} \\mathrm{d}^3 x \\, \\sqrt{h}K,", "9097d2a74926f18f53a2f11131ea26a5": "c_{14}", "90981588430f3f2f04d4e3bd2e3a6a3a": "S_M - S_N = a_M B_M - a_N B_N + \\sum_{n=N}^{M-1} B_n (a_{n+1} - a_n) ", "90985d6b79437d5253e914388a556cf9": "\\phi \\in X^*", "909907eb90c57ed1f31f7c4fb4921496": "\\operatorname{E}[(X)_n]=M_X^{(n)}(1)=\\left.\\frac{\\mathrm{d}^n}{\\mathrm{d}t^n}\\right|_{t=1} M_X(t),", "90990f9d56dc363f6391544b8ade708c": "\\alpha = 1 - R^2", "909925b88861ff9b8db75fca3ddf86eb": "{\\vec{\\Omega }}'=\\frac{G\\cdot I}{c^{2}\\cdot R^{3}_{E}}\\left[ 3\\frac{\\vec{\\Omega }\\bullet \\vec{R}_{E}}{R^{2}_{E}}\\cdot \\vec{R}_{E}-\\vec{\\Omega } \\right]", "909941cb63538889423171ceb4913ac2": "T_a f(x) = \\left( I - aD + {a^2D^2\\over 2!} - {a^3D^3\\over 3!} + \\cdots \\right) f(x).", "9099653bd80495c2f100e430f9dc1cda": "\nC_{\\alpha IJ} e_\\beta^I e_\\gamma^J = 0 ,\n", "909997bcdeb0f844897ff48539c27f9b": "\n {A'\\,}^i{}_j= \\frac {\\partial {x'}^i} {\\partial x^l}\n \\frac {\\partial x^m} {\\partial {x'}^j} A^l{}_m\n", "9099f945c1c06d91b921bb57169dfeed": "f(1) = 6", "909a6b881143498359c9dc530b791f71": "K_d = k_\\mathrm{off}/k_\\mathrm{on}", "909a7af7904bb5dfb44ec1579372daf3": "M(a,b,1) = \\binom{a+b}{a}.", "909a7ee4fced78a9db918c3961555c27": " \\left[ \\begin{matrix} u^2+v^2+1 \\\\ 2u \\\\ -2v \\\\ u^2+v^2-1 \\end{matrix} \\right] ", "909a874065682f5f6de7ec9b75028d44": " R_s = \\frac{I_r}{I_0} ", "909ab595560e53adc888858398548685": "\\rho^2 = (x-x')^2+(y-y')^2 \\,", "909ac30294b05114c8c74ce831a574fc": "\\ge -1", "909b28fc991be60a5647cc726372e4e2": "G_0=1", "909b4eb1590d807142f670516bddc475": "0 < \\nu < 1", "909b7c7e1714eeece42fbbe0113136ee": "\\mathbf{K} = \\frac{2\\pi}{c}\\mathbf{N} = \\frac{2\\pi}{c} \\nu(1,\\hat{\\mathbf{n}}) = \\frac{\\omega}{c}\\left( 1 , \\hat{\\mathbf{n}} \\right) \\,. ", "909b98841ea85fe131df6bb4c474cc33": "a\\psi_n", "909b9bc774959e8c7c19c729dd92692e": "\n\\langle\\zeta_{i}(t)\\rangle=0\n", "909baadea19ad475196e67289e6a56df": "\\Delta{H}(a,Q^{(g)}", "909bc140ed36fa7077a5312981c88914": "M_1 = 2^m M + r_1", "909c332b32acc1bb68efd5bc0c167332": "\\sigma_-=\\sqrt{\\sigma_x^2+\\sigma_y^2-2\\rho\\sigma_x \\sigma_y}.", "909c47a7f7137ae76c3122f43d676de5": "\n\\mathrm{THD} = \\frac{ \\sqrt{V_2^2 + V_3^2 + V_4^2 + \\cdots + V_n^2} }{V_1}\n", "909c681745436ce26f947a5c048e9386": "\\varphi[x_1,\\ldots,x_m,y_1,\\ldots,y_n]", "909d0157e61e1ae696864563798dcf39": "dr(t) = (\\theta(t) - \\alpha(t) r(t))\\,dt + \\sigma\\, dW(t)\\,\\!", "909d538e2fbbebc490e52aee01fb3c45": "(A\\to B)\\to(\\neg B\\to(A\\to C))", "909d721ac02418af6ec8203d39334edb": "\\psi _{a,b} (t) = {1 \\over {\\sqrt a }}\\psi \\left( {{{t - b} \\over a}} \\right).", "909dae2a11fbfe97a75bebda3391ad6f": "\n{\\star \\bold{F}} = - B_x dx \\wedge dt - B_y dy \\wedge dt - B_z dz \\wedge dt + E_x dy \\wedge dz + E_y dz \\wedge dx + E_z dx \\wedge dy\n", "909dbc821224e3fd4468f6756171f169": "dg(x)=dg_1(x)-dg_2(x)", "909ddb1984011514f833819cb2e74541": "[E\\; F] = -[U_X\\; U_Y] \\begin{bmatrix}0_{n\\times n} &0 \\\\ 0 & \\Sigma_Y\\end{bmatrix}\\begin{bmatrix}V_{XX} & V_{XY} \\\\ V_{YX} & V_{YY}\\end{bmatrix}^* ", "909df05dd8a316aa1c6aa43b056505f1": "\\, =[10(x+y) - 100] + [100 - 10(x+y) + xy]", "909e673d2fdb80280442bfb1bf40a093": " F(x,\\theta)=\\int_0^\\theta {e^{-x\\sec(\\varphi)}}\\,d{\\varphi}.\\,", "909e9967ab5d55bc5e77d363e28626da": " \\frac{d u_j}{d \\eta}=T'\\frac{\\partial u_j}{\\partial t}+X'\\frac{\\partial u_j}{\\partial x}", "909f207e461329b32dcf8a6bd470610e": "x \\equiv n \\pmod{m_0}", "909f79e25aa2fea58add8c6897148886": "\\mathbb{D}_4\\;", "909fea713fdffabba8b3d5c145ee8221": " k = \\frac32 ( U I )^2, ", "909ff7654ab56ed5878744b9f7a7d7d7": "(i - 2)", "90a01b12bad50a8ba040805b7da851de": "S\\left|0\\right\\rangle = \\left|0\\right\\rangle \\Longrightarrow \\left\\langle 0|S|0\\right\\rangle = \\left\\langle 0|0\\right\\rangle =1", "90a04ba83db7fd08bee026fb0393ed52": "\n\\begin{align}\n\\lim_{x \\to 1} \\left( \\frac{x}{x-1} - \\frac{1}{\\ln x} \\right)\n& = \\lim_{x \\to 1} \\frac{x \\ln x - x + 1}{(x-1) \\ln x} \\quad (1) \\\\\n& = \\lim_{x \\to 1} \\frac{\\ln x}{\\frac{x-1}{x} + \\ln x} \\quad (2) \\\\\n& = \\lim_{x \\to 1} \\frac{x \\ln x}{x - 1 + x \\ln x} \\quad (3) \\\\\n& = \\lim_{x \\to 1} \\frac{1 + \\ln x}{1 + 1 + \\ln x} \\quad (4) \\\\\n& = \\lim_{x \\to 1} \\frac{1 + \\ln x}{2 + \\ln x} \\\\\n& = \\frac{1}{2},\n\\end{align}\n", "90a06463cdade7f4477e23240ee835b3": "P = C \\rho \\,", "90a0930c17ce190ea26438fa3de799bf": "\\gamma^{\\prime}_{\\rm p} = \\frac{\\mu^{\\prime}_{\\rm p}}{\\mu_{\\rm e}} \\frac{g_{\\rm e} \\mu_{\\rm B}}{\\hbar}.", "90a0935e7539efe5f1166c645c52dce5": "H(\\nu) = F(\\nu) \\cdot G(\\nu),", "90a0b10f34964af49ede20d7d6759513": "\\,\\omega_k", "90a11b9a3a40771c1539f7b2f18c3ea4": "\\frac{\\partial}{\\partial t}(u+\\tfrac{1}{2}\\rho V_i V_i)\n+ \\frac{\\partial}{\\partial x_j}(uV_j+\\tfrac{1}{2}\\rho V_i V_i V_j + J_{qj}+P_{ij}V_i)-nF_iV_i\n=0", "90a12167e92af36fa8d7cd6629f85962": "\\mathcal{K}_3(x; n) = -\\frac{4}{3}x^3 + 2nx^2 - (n^2 - n + \\frac{2}{3})x + {n \\choose 3}.", "90a130c0d721ee6ee4b1866bdbd313d6": "((a+b)+(a+c)+(b+c))\\left(\\frac{1}{a+b}+\\frac{1}{a+c}+\\frac{1}{b+c}\\right)\\geq 9.", "90a1cb3b224ccf7428bb7efd4d79bc2a": " \\hom(A \\times B, C) \\equiv \\hom(A, \\hom(B, C))", "90a1e7a0865bda2d0042873f3c08db91": "\\frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\\Gamma(k/2)}", "90a1fe25e6b731c566f5eaf437767305": "M_y= \\frac{80}{3}", "90a2373ee16786fcbb9faf2508f078af": "\\chi = \\frac{C}{T \\mu_0-C \\lambda}", "90a2971298d432cda9cf491c74862d70": "\\mathbb{G}_a", "90a31c308c95e905dbe45e3bf2ce28c6": "F > F(\\alpha/2,n_1-1,n_2-1)", "90a327c8dc22f709445475137ae6d1f0": "\\langle u,v\\rangle=1", "90a340b860c5ad38755145defdfb035c": "\\text{supp}(V)", "90a34c5feb01ce266340f81f2b97bf6b": "M_L", "90a3f81b9297dff4dee66b6d74546365": "\\langle\\langle e,t\\rangle,\\langle \\langle e, t\\rangle, t\\rangle\\rangle", "90a41b7da7cc7a7bc640dca013b4d123": " |\\lang\\phi|\\psi\\rang| = \\operatorname{cos}(\\theta), ", "90a43bc443e40eb118177b544af5840d": "\\Omega^{-1}", "90a43d88ecd4584707513063271775ba": "\\phi^+(x)", "90a47ce604934f5d9c61703e9cbc836b": " \\alpha(x,t)= \\int_{-\\infty}^\\infty dk \\, A(k) e^{i(kx-\\omega t)},", "90a52689b4cfe3f71b8d3064cc61b7b2": "\\begin{bmatrix}\n x \\\\\n y\n\\end{bmatrix}", "90a53d0bb7edcd999c0760b6b3e9c414": "J = \\int^b_aF(t, y(t), y'(t))\\,\\mathrm{d}t", "90a54317af1477684074c61143463637": " N = 3 \\,", "90a547fd570865d59518f572d72d50e4": "f(S_i) = f(S_i, S_{i+1})", "90a5497e166a6ccf8dd84f1dd866e604": "m_{(3,2,1)}(X_1,X_2,X_3)=X_1^3X_2^2X_3+X_1^3X_2X_3^2+X_1^2X_2^3X_3+X_1^2X_2X_3^3+X_1X_2^3X_3^2+X_1X_2^2X_3^3.", "90a58192b0b0c7d745e6a534146cd92b": "L\\subseteq\\Sigma^*", "90a623a47f710fe860ba05f526ad87ad": "\\mathbf{M}_{t,d}", "90a63d5e71a4dc6fb0297583717f9a05": "M_1=N\\left ( \\mathrm{d} \\Phi_2/\\mathrm{d} I_1 \\right )\\,\\!", "90a64979c7438a61cc3776b7cd89a4de": "\n \\sum_{q\\in\\mathbb{Q}} a_q\\varepsilon^q ,\n", "90a650a6433d76909a4f24003fa2cc5f": "\\operatorname{Var}_Y(Y) = \n \\operatorname{E}_N(N)\\operatorname{Var}_X(X) + \\left(\\operatorname{E}_X(X)\\right)^2\\operatorname{Var}_N(N) .", "90a673949e4141c4634f90e417b0174b": "\\int_a^b f(x)\\,dx \\approx (b-a) \\, \\frac{f(a) + f(b)}{2}.", "90a69b06f8ed51a380790efa5a9e719e": "\\frac1{k!}", "90a6b927cea6879a8d54b424a06ee59d": "!n = n! \\sum_{i=0}^n \\frac{(-1)^i}{i!},", "90a6ca957277b6380c4da165535a2a71": "J(5,2)", "90a6d67f0fe8d9dcb97e12003cb50d6c": "\\psi _{i}", "90a7a9e6e167c5c2f90e399b8ba29a2e": "\\,\\Delta w(t) ~ = ~ -\\mathrm{lower} [w(t) w(t)^{\\mathrm{T}}] w(t)", "90a7c45eaffbd575ca6fb361e6d170a4": "LR", "90a7ddfd1ced1f5dccabf14974850144": "\\Phi_N", "90a815c900df6091998138ff6a67483e": "\\iota", "90a8488df780629945f826a2166750ff": "\\text{SF}", "90a88188002eb2d08b73ded49a607651": "\\int_{-\\infty}^\\infty w(2^j t - k) \\cdot \\varphi(t - k') \\, dt = 0\\text{ for }j \\geq 0.", "90a89469fef2c95a69e1f67f2eb9de63": "L_{y}(\\mathbf{x}, \\sigma_{D})", "90a8aa6213678be57ddf8f97bdbf6cdc": "\\frac{nb+md}{db}", "90a8caca3e38820942d54c833c736964": "\\frac{X_b}{X_b^0-X_b}=\\frac{X_c}{1-X_c}\\exp ", "90a8cf530ae59623960ce4245b81c109": " (Y_i, X_{i1}, \\ldots, X_{ip}), \\, i = 1, \\ldots, n ", "90a96c21992f6717bb1002755ebcbe0e": "C \\to C", "90a96dc031e167e36f6108db9b9e0126": "\\bigcap", "90aa38df0b774e133e18786dd1244e2d": "((A \\lor B) \\land (A \\to C) \\land (B \\to C)) \\to C", "90aa5ab2356b006a7065f5128fe3221b": " U = (A^T A)^{1/2}A^{-1} ", "90aa654b86597648a6cbd045a5dbfc21": "X(t_i) = X(t_{i-1}) + \\Gamma^+_i(t) - \\Gamma^-_i(t).", "90aa7cf5785c2a8bc90d813ba46f1484": "\\ \\xi \\in C_i ^T((X)) , ", "90aad8fc1e7640f91acf948205fdd6d5": "U^{-1} C U = B", "90aad9f7e6ccda260def75d70d6031db": "^2_0", "90ab060f348dae7d50d691ffa2400ae7": "\\frac{1}{\\sqrt{r}}", "90ab1e9ef3f073e26bd482e94dc6cd16": "x^3=(3\\sqrt[3]{a^2-b})x+2a", "90ab3ee4f4815ca6bd6b5ba6f0c6852d": "\\alpha\\prime = 4\\,\\alpha", "90ab5072bdbe8b8ccd6fba1aa94dfdf3": "s = \\sqrt{x^2+y^2+z^2} \\,.", "90ab6d2400ee4cb6f9bd5136f088f44b": "\\lambda(\\phi)\\;", "90abec738afa2f16adfe0d9c7d2d94b2": " e(X) ", "90ac0b96d06431a8f0314bf530906493": "\\sigma_{max}", "90ac5bcee4651b5e2735c757e400a149": "\\theta_{\\mathrm{f}} - \\theta_{\\mathrm{i}} = \\omega_{\\mathrm{i}} t + \\tfrac{1}{2} \\alpha t^2", "90ac675182a355e53f9d4524ed1f0814": "\\delta(x,y)=\\Delta(x,y)/n \\text{ and } \\delta(w,C)=\\Delta(w,C)/n", "90ac95d86a9a0cc4a28f7b16986868f2": "g\\in H^p\\left(\\mathbf{T}\\right)\\text{ if and only if } g\\in L^p\\left(\\mathbf{T}\\right)\\text{ and } \\hat{g}(n)=0 \\text{ for all } n < 0,", "90acb8bca17072908ad1b846cf28d59c": "\\mathbf{Z}/2", "90acc741fee66a37a2ee0484d7d6a158": "J_h", "90acc7a13ea81bb6a7397653339be0b1": "d_5 b^5 + d_4 b^4 + d_3 b^3 + d_2 b^2 + d_1 b + d_0 + d_{-1} b^{-1} + d_{-2} b^{-2} + d_{-3} b^{-3}", "90acfc9ddc5b6d38c38ad12c9777b116": "\\textstyle E_{-}", "90ad25f7bea0e086483c209ae003faf2": " R= (\\frac{V_f}{2U})(\\tan {\\beta_3} - \\tan{\\beta_2}) ", "90ad3bbcf9aab22ae4ec631cb565f2ea": " \\lambda = \\lambda_B + \\lambda_C . ", "90ad5890545d04c7627da16412f93e07": "\\sigma_h \\cdot I_3 =\n\\left[ \\begin{array}{ccc}\n\\sigma_h & 0 & 0 \\\\\n0 & \\sigma_h & 0 \\\\\n0 & 0 & \\sigma_h \\end{array} \\right]\n", "90ad5d5f62b55704663579330a953dd3": "\n\\hat \\sigma _i \\,\\,\\, = \\,\\,\\,\\sqrt {{{\\,\\,\\sum\\limits_{k = 1}^n {\\left( {x_k - \\bar x_i } \\right)^2 } } \\over {n - 1}}} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hat \\sigma _{i,j} \\,\\,\\, = \\,\\,\\,\\sqrt {{{\\,\\,\\sum\\limits_{k = 1}^n {\\left( {x_k - \\bar x_i } \\right)\\left( {x_k - \\bar x_j } \\right)} } \\over {n - 1}}}", "90ad63c1e4db906cf3982b36c94112a5": "\\gamma_n(x) \\in I", "90adc0f011dd42dcd4b4b9ad063a9bf6": "b^0", "90ade9801cce811a5ff880423e2509f5": "T = T_a^a", "90ae2f27151a25f5bb218185c16f471e": "f \\in C^1(\\mathbb{T})", "90aec21b3eb3708bc6a14f9701ac7030": "P_X = X\\left(X^{\\mathrm T} X\\right)^{-1}X^{\\mathrm T}", "90aec922245805fa7902abd8166f2785": "J_n(\\mathbb R)", "90aef3e42aab4278264c72a09c7c65a3": "p = \\frac{e}{290} + \\frac{f}{4}", "90aeff328faab9855cd527a816f4e664": "n \\wr = (n-1) \\wr \\wr n \\wr \\wr", "90af01f58bae148853f7dfcbd1dfb1f1": "\\mathbf{P}_\\phi", "90af16bd39b635bb2d0bb58a4807c2a6": "R_2(x)=\\frac{x^2-4x+1}{(x+1)^2}\\,", "90af1ec3ab3f2ac47b2c7cb90a2e90aa": "\\sin\\theta = \\sin\\left(\\theta + 2\\pi k \\right),\\,", "90af23baa0b2148c00457233382c0d2c": "\\mathbb{F}_l[X]", "90af39c5a257ca405edf19d45abb854e": "X\\times (Y + Z)\\simeq (X\\times Y)+ (X \\times Z).", "90af56f0433e3b9b54e683d738aa20e5": " 6^{6^6} + 1 ", "90af7db6afc379ce51c61c5436f54d90": "x_{i_k}", "90af90d2baa3b8179b4f64f467effa51": "P(\\overline{C_1},\\overline{S_1})d\\overline{C_1}d\\overline{S_1}", "90aff6e002d24f07bafa3ce4563a36e7": "-100/60 = -1.66", "90b00b9f8c7d226138cf6157d3597a71": "\\sum_{k=1}^n k^{2m+1} = c_1 a^2 + c_2 a^3 + \\cdots + c_m a^{m+1}", "90b0546dc1544cdc2e257a1892900324": "d_2 = d_1 - \\sigma\\sqrt{T - t}", "90b08bdcd575905af962c23e63c5b3e8": " \nE\\left[\\sum_{n=1}^N\\sum_{c=1}^NQ_n^{(c)}(t)\\left[ \\sum_{b=1}^N\\mu_{nb}^{(c)}(t) - \\sum_{a=1}^N\\mu_{an}^{(c)}(t) \\right] |\\boldsymbol{Q}(t)\\right] \n", "90b09d74140b446a4799a600166aaed5": "\\mathbf{\\hat{\\Omega}}^\\prime", "90b15826b7e7387ca0075f4e12d9ecb2": "\\begin{smallmatrix}\\sigma \\end{smallmatrix}", "90b18caa08f64b39fd846544fde0fae9": "\\mathbf{X} = \\mathcal{F} \\left \\{ \\mathbf{x} \\right \\} ", "90b20c7b6739bf529b4aef0e0a57f7c6": "\\omega_\\oplus\\sim2\\pi/23\\,\\mbox{h}\\, 56 \\,\\mbox{min}", "90b2302fb60b72707d89762369131632": "x = x^\\mu \\gamma_\\mu", "90b2354355d0285322513335cf703ab6": "i_m^2 = -1", "90b2428b8b908f2b5e51e59a94c9b2c7": "\\frac{\\sin\\theta_\\mathrm{i}}{\\sin\\theta_\\mathrm{t}} = \\frac{n_2}{n_1}", "90b28e9d13e9ff5a6dc90d3efca22d86": " \\operatorname{Vec}(\\hat B) = ((ZZ^{'})^{-1} Z \\otimes I_{k})\\ \\operatorname{Vec}(Y) ", "90b313ecfb247b02ef1801b1dbb4489a": "\\,\\!n=0", "90b3277d3ee77e28d5b2fc6b1e87cba6": "d Y_t = (d_t-e Y_t)\\,dt + \\sqrt{Y_t}\\,f_t\\, dW_{2t}", "90b32971154e2b0000031f400aa038ff": "t_1 \\mathbf{v}_1 + \\cdots + t_k \\mathbf{v}_k.", "90b33e69f3b4a5e93ee49770d9799628": "\\textstyle\\frac{\\nu}{\\nu-2}", "90b3b0726400625551a782e59c4654e4": "\\textstyle z_{12} = z_{21}", "90b3bb7b2b7110ef900ec88344703650": "\\ ^2\\mathrm{D} + ^3\\! \\mathrm{T} \\longrightarrow ^4\\! \\!\\mathrm{He} + n + 17.6\\ \\mathrm{MeV} ", "90b48fea6a3bc604e702c0f91b16e19a": " \\langle \\chi | \\psi \\rangle = \\left( \\int d\\varepsilon' {\\chi(\\varepsilon')}^{*} \\langle \\varepsilon' | \\right)\\left(\\int d\\varepsilon \\psi(\\varepsilon) |\\varepsilon \\rangle \\right) = \\int d\\varepsilon' \\int d\\varepsilon {\\chi(\\varepsilon')}^{*} \\psi(\\varepsilon) \\delta (\\varepsilon' - \\varepsilon ) = \\int d\\varepsilon {\\chi(\\varepsilon)}^{*} \\psi(\\varepsilon) \\,.", "90b4fd9a5b6533bedc4b725f6e10d833": "dn_2", "90b5600c1d140c7792a13c4abfcb549c": "m = -l", "90b5aa43d62f1bbd7278ba55de76d9b7": "P_0 = \\frac {k_BT}{\\zeta_L} \\left ( \\frac {2 \\pi mk_BT}{h^2} \\right)^{3/2} ", "90b5bae1af75380a59d65f26c6c0d750": " n = \\frac{P}{100} \\times N + \\frac{1}{2}", "90b5d6dc475054404c57a1ec21265914": "\\mathrm{C_2H_5OH + 3\\ O_2 \\to 3\\ H_2O + 2\\ CO_2}", "90b5ff1c3c0b22a28389017d53477ca1": "\\begin{align}\\mathbf{e}_x & = - \\frac{1}{\\sqrt{2}} \\mathbf{e}_+ + \\frac{1}{\\sqrt{2}}\\mathbf{e}_{-} \\\\\n\\mathbf{e}_y & = + \\frac{i}{\\sqrt{2}} \\mathbf{e}_+ + \\frac{i}{\\sqrt{2}}\\mathbf{e}_{-} \\\\\n\\mathbf{e}_z & = \\mathbf{e}_0\n\\end{align}", "90b669748c8afb392b17b5d5779070c5": "= 448 + 40 + 6", "90b6dd941e62ee1e0ebb5479691da801": "\\lim_N a_N=+\\infty", "90b6de189e873171284129924e63323d": "S_\\mathrm {v}", "90b70da32f88d5326f4967b486b68273": "\\ell=2a", "90b78ff7a127c2d6a26bc90a7a1cc220": " Wf(\\lambda,z)= \\int_{G/K} f(g) \\pi_\\lambda(g)\\xi_0(z) \\, dg", "90b79547df7dce34480e6e7198e3d443": " l = n, n-2, \\ldots, l_\\min\\quad\n\\hbox{with}\\quad l_\\min =\n\\begin{cases}\n1 & \\mathrm{if}\\; n\\; \\mathrm{odd} \\\\\n0 & \\mathrm{if}\\; n\\; \\mathrm{even}\n\\end{cases}\n", "90b7a0c2338844a996d64a5213005c13": "R_v=+29", "90b7a1270d841245f718b51300f0facf": "X_{\\tau + t}", "90b7a1c16507f3836dec0963bd6e6181": "E[ f_j(X_{j}) ] = 0", "90b7acf7ae27dee235ba4dc018970924": "a_m - a_1", "90b821239df4be5666a30031c42a31bf": "x^{(k)} \\in \\Omega_x^{(k)},\\quad", "90b83633ba7a1f579ff7de2a7a997c6e": " \\rho (\\mathbf{r}) = \\sum_{i=1}^N \\, q_i \\, \\delta (\\mathbf{r} - \\mathbf{r}_i ),", "90b8396d7584044eeb1bbba0b4ec8531": "1.9423", "90b83f102ec3148a3319ea44fdedd394": "\\displaystyle\\sum \\lambda^m(x)t^m=\\prod_i (1+tx_i)", "90b88cdcd13e8c55540b72d5f41fd788": "\\epsilon>0\\,\\!", "90b8cde68c58225dad61988d39bf474d": "\\mathbf{E}(\\mathbf{x}) = \\sum_{i}\\mathbf{E}_i S(\\mathbf{x}_i-\\mathbf{x}),", "90b8dd05e0d0beb2e19f8c161cffae95": "P [C \\ge x]> P [B \\ge x]", "90b8f8489f86814947abdb5fe224dc33": "\\scriptstyle{-q}", "90b93f0c12878a5275d414b3513c5381": "E_\\text{P} = {\\frac{\\hbar} {t_\\text{P}}},", "90b940ed27556bc9ba4f388d9f0b368e": " \n\\begin{bmatrix}\n\\mathbf A^T \\\\ \\mathbf B^T \n\\end{bmatrix}\n^{+} = [\\mathbf A, \\mathbf B] ([\\mathbf A, \\mathbf B]^T [\\mathbf A, \\mathbf B])^{-1}. \n", "90b941b4eb5d302ed8be8260dcb29904": "\\ \\Delta S = \\frac{\\Delta s}{c_p} = ln\\left[M^2\\left(\\frac{\\gamma + 1}{1 + \\gamma M^2}\\right)^\\frac{\\gamma + 1}{\\gamma}\\right] ", "90b97d2798364749df4ec2d5ab8b566c": "E(s\\otimes s)=E(n\\otimes n)=s", "90ba1c60fcce40291c68ad5310e5a66f": "\\langle A(a) B(b') \\rangle = -\\frac{1}{\\sqrt{2}}", "90ba3e913fb14ec8ea57b03086e552c5": " G = \\frac{1}{\\mathrm{agm}(1, \\sqrt{2})} = 0.8346268\\dots.", "90ba4a79155ff5d092126e7cb15be424": "N_1 = N_2 = 8", "90ba6ff634f5bb7830a6040234614a1d": "H_j=\\sum_{i=1}^m\\alpha_j(E_i,E_i)", "90bb1b600088231fb7e381e6c092fe89": "\\frac{1}{KM}\\; {\\mathrm{tr}} A \\ \\in \\ \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left({A}\\right).\n", "90bb7b95892ee48b57338792a5ee573a": "x_0=2/3", "90bb82a4e9c6a39b076f13ca7e93b5b4": " \\lambda \\ = \\ \\sqrt {\\frac {r_{m}}{r_{i}}}", "90bb9d2b52b04275f7bd6499e3772c58": "\\eta_t = \\frac {h_3-h_4}{h_3-h_{4s}} ", "90bc1d60d7e53373c1f054d8ded89201": "{\\mathbf{p}}(t)\\in\\Omega", "90bc8d237f607c9ed249cc31ae285822": "\n\\begin{align}\n& \\Pr(z_n=k\\mid\\mathbb{Z}^{(-n)},\\boldsymbol{\\alpha}) \\\\\n\\propto\\ & \\Pr(z_n=k,\\mathbb{Z}^{(-n)}\\mid\\boldsymbol{\\alpha}) \\\\\n=\\ &\\ \\frac{\\Gamma\\left(A\\right)}{\\Gamma\\left(N+A\\right)}\\prod_{j=1}^K\\frac{\\Gamma(n_{j}+\\alpha_{j})}{\\Gamma(\\alpha_{j})} \\\\\n\\propto\\ & \\prod_{j=1}^K\\Gamma(n_{j}+\\alpha_{j}) \\\\\n=\\ & \\Gamma(n_{k}+\\alpha_{k})\\prod_{j\\not=k}\\Gamma(n_{j}+\\alpha_{j}) \\\\\n=\\ & \\Gamma(n_k^{(-n)}+1+\\alpha_{k})\\prod_{j\\not=k}\\Gamma(n_j^{(-n)}+\\alpha_{j}) \\\\\n=\\ & (n_k^{(-n)}+\\alpha_{k}) \\Gamma(n_k^{(-n)}+\\alpha_{k})\\prod_{j\\not=k}\\Gamma(n_j^{(-n)}+\\alpha_{j}) \\\\\n=\\ & (n_k^{(-n)}+\\alpha_{k}) \\prod_{j}\\Gamma(n_j^{(-n)}+\\alpha_{j}) \\\\\n\\propto\\ & n_k^{(-n)}+\\alpha_{k}\n\\end{align}\n", "90bcc750d4b3ce5b08b8782b8072ecc9": "[A]_t", "90bd728ea602b02a9ba4a80e36d89961": "x\\land (y\\lor z) \\Leftrightarrow (x \\land y)\\lor (x\\land z)\\,\\!", "90bd89f2e7a2ca33e3e056440d0fc560": " A(x_n)=x_n-\\frac{(\\Delta x_n)^2}{\\Delta^2 x_n},", "90bdcb146f1fdda0a90554d5d1a876ff": "\\scriptstyle\\hat\\varphi(t)", "90bde4cb6b6e88106b5ccbad5ca5b5c7": "\\int \\cosh (ax+b)\\cos (cx+d)\\,dx = \\frac{a}{a^2+c^2}\\sinh(ax+b)\\cos(cx+d)+\\frac{c}{a^2+c^2}\\cosh(ax+b)\\sin(cx+d)+C\\,", "90be2c5f8592ec15c968eb012be1bda1": "(\\star \\eta)_{i_1,i_2,\\ldots,i_{n-k}} = \\frac{1}{(k)!} \\eta^{j_1,\\ldots,j_k}\\,\\sqrt {|\\det g|} \\,\\epsilon_{j_1,\\ldots,j_k,i_1,\\ldots,i_{n-k}}", "90be6de7e2aa814ba32cc54025815fac": "H=\\{a,b\\}", "90bea3d59fe4c84d149508446fd1f607": " A' = V^{-1} A V.\\, ", "90bef8ff03115d31b134244bded37979": "f \\cdot v = g\\frac{\\partial P / \\partial x}{\\partial P / \\partial z} = g{\\partial Z \\over \\partial x}", "90beff13b9a92ab4fea935cd2ad62aa0": "KR_*^G(p):KR_*^G(E_{VCYC}(G))\\rightarrow KR_*^G(\\{\\cdot\\})\\cong K_*(R[G]).", "90bf23db98270d10b6cf6e9e329e1d7b": "\nA = \\frac{3}{2}\\sqrt{\\frac{\\mu}{2r_{p}^{3}}}(t-T)\n", "90bf59b2ea2832a3c07d93e7ca71f264": "\\overline\\psi \\gamma_5 \\gamma_\\mu\\psi", "90bf5c6768cf2f05604795bd729f9057": "(B, C)", "90bf6107ba30d423d29dc70ec3bf69c8": "\\beta = \\frac{\\mu_1 - \\mu_2}{\\sqrt{2}\\sigma}.", "90bf70448d7a85f2821cf028be78cb6e": "|\\beta \\rangle", "90bfb19dee8f457b72346072473c49f3": "x^0\\rightarrow +\\infty", "90bfe0ebd3d236726cdfb6d3e92fe0e8": "D_T = (I - T^* T)^{\\frac{1}{2}}", "90c014d1b9c4d3c51417046c51e4782e": "n+dn", "90c022c888edb97e541d0df29e2942ab": "\\nu_o\\,\\!", "90c0d2bb75999fd71ce123ecb1ae1d47": "\\beth_{d-2}(|\\alpha+\\omega|^{|\\alpha+1|})", "90c16ac5e4245f391dea23f62ec1714d": "N(g) = g\\bar{g} \\mid p_{i}^{2}", "90c1c4032eade76457b99a6debb6deb6": "N(E) = \\frac {V}{2\\pi^2} \\left(\\frac {2m}{\\hbar^2}\\right)^{3/2}\\sqrt{E}", "90c1eccd02d36d575692aff85ae48394": "H^{k+1}", "90c1fdb94961208c5a9440d97f182457": "x:\\tau", "90c22b9e991824602525c4d91d9e659a": "\\forall i0", "90ef145ec81e430e83f877b69866ee1d": "s_0 = \\Omega", "90ef1700ea00edb1c0e3a5af3639fc79": "\\sigma(z;\\Lambda)=z\\prod_{w\\in\\Lambda^{*}}\n\\left(1-\\frac{z}{w}\\right) e^{z/w+\\frac{1}{2}(z/w)^2}", "90ef31a7c7d1c29b8e572188c2cf12f5": " |i\\rang", "90ef4b6a2f7a432439efadcd074e4759": "\\mathit{d_H}(a \\bar{b}, d^{RC}c^R) = \\mathit{d_H}(a,d^{RC}) + \\mathit{d_H}(\\bar{b}, c^R) = \\mathit{d_H}(a, d^{RC}) + \\mathit{d_H}(c, b^{RC})", "90ef5ef17736293c4a09ddc81855076b": "(A + B)\\cdot(\\lnot A + B)=B", "90ef8327a68ca9b357adf8f7c5f0b8c3": "\\int s\\;dx = \\frac{1}{2}\\left( xs-a^{2}\\ln(x+s)\\right)", "90efaa373a8147a0f336c4154b5b1c9d": "RMS_{Total} =\n\\sqrt {{{RMS_1}^2 + {RMS_2}^2 + \\cdots + {RMS_n}^2} }\n", "90efac03cb3b0d537a460e02240a0912": "S_R", "90f0b969af677ba9a07b8c4a58b35726": " i\\leq j ", "90f0cb06e30060f64d1c7f941a1eb21a": "\\sum_\\ell\\sum_{\\ell'}.", "90f0f4c0bc869eec3de4c19d95fb867a": "\n\\begin{align}\n& {} \\quad \\int d{\\mathbf{\\hat n}}\\,{}_{s_1} Y_{j_1 m_1}({\\mathbf{\\hat n}})\n\\,{}_{s_2} Y_{j_2m_2}({\\mathbf{\\hat n}})\\, {}_{s_3} Y_{j_3m_3}({\\mathbf{\\hat\nn}}) \\\\[8pt]\n& = \\sqrt{\\frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4\\pi}}\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n -s_1 & -s_2 & -s_3\n\\end{pmatrix}\n\\end{align}\n", "90f139fc960e2a36df74173eb89f724f": "(U, m)", "90f143b5383e6ff378e26a6c2bcb368c": "V_{out,2nd order} = k_{2}\\{\\frac{A_{1}^{2} + A_{2}^{2}}{2} + \\frac{A_{1}^{2}\\cos(2w_{1}t)}{2} + \\frac{A_{2}^{2}\\cos(2w_{2}t)}{2} + A_{1}A_{2}\\cos[(w_{1}+w_{2})t] + A_{1}A_{2}\\cos[(w_{1}-w_{2})t] \\}", "90f14d8fd473cc2906c076b651cad6d2": "\\mathrm{ker}(\\partial_0),\\mathrm{Im}(\\partial_1)", "90f19ba85ee95bc7bcc233a6c28416c3": "\\scriptstyle \\gamma_{\\mathrm{la}} ", "90f207648f20851281b205ac3dba0aa2": "(S^3)=", "90f22b54b620528790bb56d1f9888bd8": "\\mathbf{F}(t)\\;", "90f24c093a426531b2432c43345bb491": "2-\\frac{1}{q}\n0", "91036a7b811888a69ad8d65bc8c36b2f": " E = i H ", "91036b4f3e6addfb9bba784d3bdda1dc": "K_i = k_{-3}/k_3", "9104040302aa2563b5289842cac8b739": "17^{14}\\ \\equiv\\ 25\\ \\not\\equiv\\ 1 \\pmod {71}", "9104971c5885161282d8826e21beabf8": " {G^a}_b \\, {G^b}_c \\, {G^c}_a = -R^3", "9105458ccd3ceda448a992dccd05985a": "(\\exists x \\ \\phi(x)) \\to \\neg (\\forall x \\ \\neg \\phi(x))", "910576390bb0ec37724f19ed329efd63": " n - d ", "9105b199b0441041a61195fedc101378": " dis(w,w') \\leq t", "9105e2713159f76b59ae41e92072710f": " \\bold{Ax} = \\bold{b} \\quad (2) ", "9106b2eb30b70da4bdf45a0ba2552539": "K^* = \\frac{\\sum\\limits_{x\\in PL} e^{-E(x)/RT}}{\\sum\\limits_{x\\in P} e^{-E(x)/RT}\\sum\\limits_{x\\in L} e^{-E(x)/RT}}", "9106e70c7d4878d411b5e41c23b8231b": "\\sum_{i=1}^n [{\\rm rank}(H\\cap a_iKa_{i}^{-1})-1] \\le ({\\rm rank}(H)-1)({\\rm rank}(K)-1).", "9106f4e167e620977368e254cc2a8506": "P'(q)<0", "910744070969abb2b0ac591dcd27116d": "\\frac{n^2-3n}{2}\\, ", "9107be8260762ba1af8d1f31cb2adb7b": "Pmf = \\cfrac{1 Pwo + 3 Pwf}{1 Pwo + 3 Pwf + 2 Pwo + 0 Pwf} = \\cfrac{1 Pwo + 3 Pwf}{3 Pwo + 3 Pwf}", "9108189e47a518c1c2b9b0c84b9e4b9e": "A=U - TS", "91085cb0f0eb3b6bd74a92b9a569dfd0": "X(x) = B e^{\\sqrt{-\\lambda} \\, x} + C e^{-\\sqrt{-\\lambda} \\, x}.", "910864d54904120dd20a20c01d4bc158": " = \\frac{c^4}{16 \\pi G} \\int d^4x \\sqrt{-g} R[g]", "91088a4503ee181707f392e64249bc55": "\\max \\left\\{ \\|x+y\\|, \\|y\\|\\right\\}", "91088f77332356b0dc2168cf3bf0a772": " \\mathrm{LiCoO_2} \\rightarrow \\mathrm{Li^+} + \\mathrm{CoO_2} +\\mathrm{e^-} ", "9108e32c880e360c430a8c61664eae1a": "A \\subseteq F(A)", "91091b0a0ce0eb482b6d3b287cba56b8": "\\sum_{i=0}^{63} 2^i.\\, ", "91094f3d5af32ee81a6c092e35ac292b": " \\begin{align}\n y_{i+1}^{j} &= y_i^{j} -\\frac{\\delta_{ij}}{q_i} \\\\\n \\sum\\limits_{i=1}^m y_i^{j} &= (k_{1}^j- k_{0}^j) + (k_{2}^j- k_{1}^j) + \\cdots + (k_{m-1}^j- k_{m-2}^j) + (k_{m}^j- k_{m-1}^j) \\\\\n &= k_{m}^j - k_{0}^j \\\\\n \\sum\\limits_{i=1}^m y_i^{j} &= k_{m}^j \\\\ \\\\\n y_1^{j} &= (k_{1}^j- k_{0}^j) = k_{1}^j \\\\\n y_2^{j} &= y_1^{j} -\\frac{\\delta_{1j}}{q_1} = k_1^{j} -\\frac{\\delta_{1j}}{ q_1 } \\\\\n y_3^{j} &= k_1^{j} -\\frac{\\delta_{1j}}{q_1} -\\frac{\\delta_{2j}}{ q_2 } \\\\\n & \\; \\vdots \\\\\n y_i^{j} &= k_1^{j} -\\sum\\limits_{r=1}^{i-1} \\frac{\\delta_{rj}}{ q_r}\n\\quad = \\quad\n \\left\\{\n \\begin{array}{lcr}\n k_1^j & \\text{for} & j \\geq i\\\\\n k_1^j - \\frac{1}{q_j} & \\text{for} & j \\leq i\n \\end{array} \\right. \\\\\n k_i^j &= \\quad \\quad \\; \\sum\\limits_{m=1}^i y_m^{j} \\quad = \\quad\n \\left\\{\n \\begin{array}{lcr}\n i \\cdot k_1^j & \\text{for} & j \\geq i\\\\\n i \\cdot k_1^j - \\frac{i-j}{q_j} & \\text{for} & j \\leq i\n \\end{array} \\right.\n\\end{align}", "9109569ee563000c627ef59fee7ee87a": "e\\rightarrow 0", "910958feb312c49c03399a01f7ae09fc": " \\forall x \\, (P(x) \\rightarrow Q(x)), \\forall x \\, P(x) \\vdash \\forall x \\, Q(x) ", "9109ab8f0a0b8bc43a6ef4d305f13064": " q (q_1,q_2,\\ldots q_n) = (q q_1,q q_2,\\ldots q q_n) ", "910a2117f54f821de3377d8fada73635": "8_{20}", "910a33357abfd00bdf0610a68864f5ff": "\\dot m_f^{k + 1} = \\dot m_f^{*} + \\dot m_f^{'} ", "910b2193e34677c49e34b1c39501ee70": "\\eta=-\\frac{1}{r}y_{min}", "910b64629230db0875381ee88d5123b5": "\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2} \\ge 2, \\, \\, \\, |\\cos{(4 \\omega t)}| \\le 1,", "910b68d6c4ace66b588bcd48eece1bb9": " W_x(t,f) = \\int_{-\\infty}^\\infty x(t+\\tau/2) x^*(t-\\tau/2) e^{-j2\\pi\\tau\\,f} \\, d\\tau", "910b9018af1ab4d2c1f05deadb280aed": "F_n^{(1)}=(nL_n-F_n)/5", "910bc31c496443eb82f7c9f0d9ae5b43": "Z\\bar{X}_i+(1-Z)\\mu", "910bd2d850e22eac2e7c588657f43fe2": "-i \\epsilon^{\\sigma 0 2 3} \\gamma_\\sigma \\gamma^5 = -i\\epsilon^{1 0 2 3}\n(-\\gamma^1) (i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3) = -\\epsilon^{1 0 2 3} \\gamma^0 \\gamma^2 \\gamma^3 = \\epsilon^{0 1 2 3} \\gamma^0 \\gamma^2 \\gamma^3", "910c026938a5d66a9ef30760ba6f2004": "\\mathbb{E} \\left[ | X_{t} - X_{s} |^{\\alpha} \\right] \\leq K | t - s |^{1 + \\beta}", "910c5697e4086f751246eed11bf19a50": "\\vartheta", "910c936ce1f0f65eb138cfa003072978": " R = A/\\mathfrak{m}^2 ", "910ce02d8a177413c32305835d2384ce": " 00 ", "91116e7f1660af407804df8b86bc834e": "\\Pr (K=k)=\\left\\{ \\begin{align}\n & \\prod\\limits_{i=1}^{n}{(1-{{p}_{i}})} \\qquad k=0 \\\\ \n & \\frac{1}{k}\\sum\\limits_{i=1}^{k}{{{(-1)}^{i-1}}\\Pr (K=k-i)T(i)} \\qquad k>0 \\\\ \n\\end{align} \\right.", "91119bd509ebc499d15aa5b15070742e": "11 = 2^2 - 2^0", "9111cefe7eb7895b2eaebdfc26af0730": "\\frac{1}{c(q)}\\int_{-\\nu}^\\nu E_{q^{2}}^{-q^2 x^2/[2]} \\, x^{2n+1} \\, d_qx=0 ,", "9111fefff8350b59ba1f271c934d2b0e": " \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix} = x_3 \\begin{pmatrix} y_1 \\\\ y_2 \\end{pmatrix} ", "911212db93da71a9727f50513eafdc04": " \\nabla^2 U_\\text{in} = \\nabla\\cdot\\mathbf{M}.", "9112267f5114dcf85723eedd3eb61a8e": "V_n(r_1,\\dotsc,r_n)=\\frac{\\pi^{\\frac{n}{2}}}{\\Pi\\left(\\frac{n}{2}\\right)}\\prod_{k=1}^{n}r_k", "9112726a3c9c83cdc9bd0fec33929271": "\n \\lim_{a \\to \\infty} \\limsup_{n \\to \\infty} \\mu_{n}\\big( \\{ f \\in D \\;|\\; \\| f \\| \\geq a \\} \\big) = 0,\n ", "9112ebd74d159e9002e537a7afaf0a07": " y_\\ell ( k r )", "911311a713b6bf7761e428943fc2726c": "t_\\text{P} = \\frac{l_\\text{P}}{c} = \\frac{\\hbar}{m_\\text{P}c^2} = \\sqrt{\\frac{\\hbar G}{c^5}} ", "9113682dd4d2b926c3c2d9c29bf82fea": "\\begin{matrix} {1 \\choose 1}{11 \\choose 1}{4 \\choose 2}{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "91139712a7d00f9afeffaee95eb8fc02": "{12\\over2}+{9\\over2}={21\\over2}=10{1\\over2}", "9113b1f4955dfdaea3f195ad68d73f3a": " \n \\begin{align}\n d\\mathbf{x} &=\\frac {\\partial \\mathbf{x}} {\\partial \\mathbf {X}}\\,d\\mathbf{X} \n \\qquad &\\text{or}& \\qquad \n dx_j =\\frac{\\partial x_j}{\\partial X_K}\\,dX_K \\\\\n &= \\nabla \\chi(\\mathbf X,t) \\,d\\mathbf{X} = \\mathbf F(\\mathbf X,t) \\,d\\mathbf{X} \n \\qquad &\\text{or}& \\qquad \n dx_j =F_{jK}\\,dX_K \\,.\n \\end{align}\n\\,\\!", "911457c33512a5391430560b00d609fd": "-(-1)^n\\det(A)I_n = A(A^{n-1}+c_{n-1}A^{n-2}+\\cdots+c_{1}I_n),", "91145860c1c765b42e51155a0c656c3b": "\\mathcal{T}.", "9114f305f323f34938e79dec611e4ac9": "a(t)=f^\\prime_t(0)", "9114fca5ae595aa1ecdbfb602bf59638": "\nf_{WE}(\\theta;\\lambda)=\\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty \\frac{e^{in\\theta}}{1-in/\\lambda} .\n", "91156e205bf7f678d40e86f9e58040b8": " S_{i,j}^{t+1}=0.", "911585e0a144f1817b0d18fdc40b81ca": "\\mathrm{rad}(\\varphi(z),D') = |\\varphi'(z)|\\, \\mathrm{rad}(z,D)", "9115b291e39d6c88425a0a4f8f884012": "V(A) := \\int_{\\Gamma_{M,A}}\\!\\!\\!\\!\\!\\!\\!\\theta(x) \\mathrm{d} \\mathcal{H}^m(x)", "9115f5bb2e53db27e7082ef5191a8c40": "\\int_{A\\times B} f(x,y)\\,d(x,y)=\\int_A\\left(\\int_B f(x,y)\\,dy\\right)\\,dx=\\int_B\\left(\\int_A f(x,y)\\,dx\\right)\\,dy.", "91160a5515592cd30e5b4f9e3d7b013a": " \\int_{S} \\mathbf{u}^{*T} \\mathbf{T} dS + \\int_{V} \\mathbf{u}^{*T} \\mathbf{f} dV = \\int_{V} \\boldsymbol{\\epsilon}^{*T} \\boldsymbol{\\sigma} dV \\qquad \\mathrm{(e)} ", "91163fe50686247507e15222e6368c1d": "\\delta_{x} (A) = 1_A(x)= \\begin{cases} 0, & x \\not \\in A; \\\\ 1, & x \\in A. \\end{cases}", "91163ff2ff8785ac77cd5f76812f7dfe": "\\mathrm{not}~r", "91164018f3f4ef39f676bbb2c22effe4": "S_{x_r}", "9116469d3bb2897be3454e4dcf70dc2c": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n \\frac{(d+e\\,x)^{m+1} (A\\,e (m+2 p+2)-B\\,d (2 p+1)+e\\,B (m+1) x) \\left(a+b\\,x+c\\,x^2\\right)^p}{e^2(m+1) (m+2 p+2)}\\,+\\,\n \\frac{1}{e^2(m+1) (m+2 p+2)}p\\,\\cdot\n", "91167468e522ab6e1b7f6ee1b0fbd004": " \\exists{x}\\, P \\qquad \\forall{x}\\, P ", "9116953fdaac1e88be37d03cb594393a": "\\mathbf{u}_\\mathrm{t}(s) = \\left[-\\sin\\frac{s}{\\alpha} \\ , \\ \\cos\\frac{s}{\\alpha} \\right] \\ ; \\ \\mathbf{u}_\\mathrm{n}(s) = \\left[\\cos\\frac{s}{\\alpha} \\ , \\ \\sin\\frac{s}{\\alpha} \\right] \\ , ", "91169acf4e8978291cbbae0ff6df7d47": "t_i= \\max\\{p_i+\\Delta\\, ,0\\},", "91169b210387f565c93d1a8370775a19": "\\{x, f, p\\}", "9116b864148c2178529a788eb487d91c": " m < -n + 1 ", "9116f673cf041196290afe098f0c9bcd": "(T-\\alpha)(T'x)=(T'(T~x)-\\alpha T'x)=0", "911729e489b9ef065b557be12e61933d": " k\\times k", "91172af0d2a724e6138f8b14d36d3780": "m(L\\cap (-m,\\, m))\\leq\\sum\\limits_{q=2}^\\infty\\sum_{p=-mq}^{mq}\\frac{2}{q^n}=\\sum\\limits_{q=2}^\\infty\\frac{2(2mq+1)}{q^n}\\leq (4m+1)\\sum\\limits_{q=2}^\\infty\\frac{1}{q^{n-1}}\\leq (4m+1)\\int^\\infty_1\\frac{dq}{q^{n-1}}\\leq\\frac{4m+1}{n-2}.", "91172ee8646a8a05fc67885e149739c7": "H(s)=\\frac{1}{1+\\underbrace{C_2(R_1+R_2)}_{\\frac{2 \\zeta}{\\omega_0} = \\frac{1}{\\omega_0 Q} }s+\\underbrace{C_1C_2R_1R_2}_{\\frac{1}{{\\omega_0}^2}}s^2},", "9117485fc6b7fc6104efc447a40fbcaa": "\\widehat{\\theta}(x)", "9117b590e1d54905cf4c72a77d4689a0": "H(z)\\ \\stackrel{\\text{def}}{=}\\ \\sum_{m=-\\infty}^\\infty h[m] z^{-m}", "9118042c965b9fe2edb5f79e8f3edcf3": "U^{-1}(x) = e^{-\\int_{x_0}^x A(x)\\,dx},", "911843de01b82bff69432caeea7ed39e": "\\rho_t(X)[\\omega]", "9118b7d2742b40c0e0dea7f03b75e836": "\\nu_f", "9118bfe8586981a6316af0d6265dc078": "\\varepsilon = \\varepsilon_{\\text{r}} \\varepsilon_0 = (1+\\chi)\\varepsilon_0,", "9118c36408e3aa0735caf600d90c3a12": "C_i\\le d_i", "9118d1797b3988fa31947ded9ca855a5": "\\frac{4\\cdot\\pi}{3\\sqrt{3}}", "9118d56a4ff2d4a559f39a5474b9762d": "p \\ ", "9118e50c0793c031986d97e53e2cebd5": "V(x)=W(x) \\exp\\left(\\int_{x_0}^x p(\\xi) \\,\\textrm{d}\\xi\\right), \\qquad x\\in I,", "91190bc1cc5f9cf0f50e82955184c55e": "1+\\tfrac1{24}\\omega^2\\Delta t^2+O(\\Delta t^4)", "91192bd10c633c5dcfa6c6dc8eaad6ea": "1_{GX} = G(\\varepsilon_Y)\\circ\\eta_{GX}", "9119629418465cc122f209d041b1e5c6": "p 0 ~|~ \\mu(A) \\leq \\nu (A^{\\varepsilon}) + \\varepsilon \\ \\text{and} \\ \\nu (A) \\leq \\mu (A^{\\varepsilon}) + \\varepsilon \\ \\text{for all} \\ A \\in \\mathcal{B}(M) \\right\\}.", "912489e4a15b86c91e35f3ba384ca994": "\\Gamma \\approx", "9124b145786d4608bb2c22ce763cf7e3": "\\textstyle \\{a_n, b_n\\}_{n=1}^N", "912553776c54e369a8c82c13184d2b16": " f\\in W^{1,p} ", "912571b225dcda5b233afd424f9b3b9f": "P_i+1=P'", "9125a35e2c36b7d40e8c9b1070199782": "F_i(x)=H_i(M_x);\\ ", "9125acdd641e0d9255b5439c0e9308d2": " \\lambda=(I-P')^{-1}a", "9126228761e89d4b01bf553fbf135def": "{\\vec r_s}", "9126454b1978f645430584b474e22baf": "\\|\\cdot\\|_F ", "91270417d2edf92fc5961199b099598a": "k_{av} = k_ak_f", "91270927c1c2074e10aa825c50100460": "E_o = \\frac{Q_{ms}(o)\\;[1-M_u(o)]}{Q_{ms}(f)\\;[1-M_u(f)]}", "912728cc34e0875071170b975b033668": "FWER=Pr\\left\\{ \\bigcup_{I_{o}}\\left(p_{i}\\leq\\frac{\\alpha}{m}\\right)\\right\\} \\leq\\sum_{I_{o}}\\left\\{Pr\\left(p_{i}\\leq\\frac{\\alpha}{m}\\right)\\right\\}\\leq m_{0}\\frac{\\alpha}{m}\\leq m\\frac{\\alpha}{m}=\\alpha", "91273d677d46a6f976aa149808957b96": " \\textstyle r_{ij} ", "91275eded2b1b4c400e06dd15fc47da2": "t_1 = 0, 7 \\, t_3^2 - t_2 \\, t_4 = 0, \\; 12 \\, t_4 - 7 \\, t_2^2 = 0", "91278fd77f6f5fbb1747efb6087b34d8": "\\tfrac{3}{3}", "9127d3e84ef8d548ebe078839600f764": "N^d", "9127eefdd0c719abe2a2963dabdfe470": " \\chi(gh) = \\chi(g)\\chi(h)", "91283b24648f7fcb9c82333be5a42c4b": "\\Phi' \\equiv \\Psi", "9128748d16121c87cd4238d870272a68": "\n\\beta \\, = \\, \\frac {\\Omega_e}{\\nu} \\, = \\, \\frac {e\\ B}{m_e\\ \\nu}\n", "912878d77751967c5ee937c757d16944": "n \\leq 3", "91287b48ea2038c5b3d9a609d0fad0fe": "A(\\eta_1,\\eta_2) = \\ln \\Gamma(\\eta_1+1)-(\\eta_1+1)\\ln(-\\eta_2).", "9128954d043168c46bd89abbe3479087": "{D\\overrightarrow{V} \\over Dt} + f \\hat{k} \\times \\overrightarrow{V} = - \\nabla \\Phi", "91289f128110c7f6a28979cf65acc30e": "\nf_t =\\ -J_2\\ \\frac{1}{r^4}\\ 3\\ \\sin^2 i\\ \\sin u\\ \\cos u\n", "9128fa7eb2a5d47e341a02bfdbf70547": "\\scriptstyle \\{i,j\\}", "91290649c41e1bf3fd6980ebfc07ee4c": "\\Theta_{n,m} (\\tau,z) = \\sum\\nolimits_{k\\in \\frac{n}{2m} + \\mathbf{Z}} q^{mk^2} u^{2mk}.", "912922a8b62d378ce7b16187f3a0f022": "W=\\sum_{i=1}^n X_i", "91294734534ea5896829c1d19b2550d2": "b=1/2", "912984caafb5d610d5474091bfb1a1c0": " \\mu_{-1}(S g) = \\mu((S g)^{-1}) = \\mu(g^{-1} S^{-1}) = \\mu(S^{-1}) = \\mu_{-1}(S). \\quad ", "9129873fed61ba9a815a8d5ef46a71d3": "t=\\sum_{i=1}^{N}t_{i}", "9129ae781be6c651c9d74015a6f22233": "\\displaystyle\\frac{\\delta F[f(\\rho)] }{\\delta\\rho(x)} = \\frac{\\delta F[f(\\rho)]}{\\delta f(\\rho(x) )} \\ \\frac {df(\\rho(x))} {d\\rho(x)} \\ , ", "9129c7cac5247aa89fc7eac358b98183": "1_{FY} = \\varepsilon_{FY}\\circ F(\\eta_Y)", "912a058af0c698f252ed903abf26833a": "V_2^{(2)}=V", "912a0951d8cbda359befcf0c2e8a2cbf": "2^{10} = 1024 ", "912a5e4885b976ca9c26b0a6be6a2402": "m \\in \\mathcal{B}_r", "912b06b87de8adb05daea007911e5e07": "f_1 : Y \\to X_1, f_2 : Y \\to X_2", "912b0768e3580b3c6a1d5b16e07d2478": "\\text{reach}(X) := \n \\sup \\{r \\in \\mathbb{R}: \n \\forall x \\in \\mathbb{R}^n\\setminus X\\text{ with }{\\rm dist}(x,X) < r \\text{ exists a unique closest point }y \\in X\\text{ such that }{\\rm dist}(x,y)= {\\rm dist}(x,X)\\}.\n", "912b8450d21794070d2331b6465b4456": "\\scriptstyle v_0 = 1-\\sum_{i=1}^n v_i", "912ba58daec777c483f9b09b7c940f2d": "\\dot{\\mathbf{x}}(t) = \\mathbf{A}\\mathbf{x}(t) + \\mathbf{B}\\mathbf{u}(t), \\quad \\mathbf{x}(0) = \\mathbf{x}_0", "912bcba3e44c48a57b4e09e3aaa76134": " |S| = \\sum_{k=0}^n s_k \\le {n \\choose \\lfloor{n/2}\\rfloor}.", "912cb520e19f5158849de0389aa6077d": "k \\rightarrow 0", "912ccb9c21c159d2af7af4eb5bb718df": "\\mu = \\mu_0 \\mu_r", "912d02ab761cc528e2d2f694f8607a3a": " \n\\beta^*=\\arg\\min_{\\beta\\in B} L[D(X),D^'(X,f_{\\beta})]\n", "912d236b5df12d25fccb2a3b345257db": "\\left(a, p, u\\right)\\succsim \\left(c, r, v\\right)", "912d430bafb59a4533bf902b9a00cbf9": " \\omega_j, j = 1,\\dots,n ", "912d8404c5c4c6900f9faec76c17022a": "\n\\begin{matrix}\n1 5 7 9 9 \\rightarrow &\n4 2 2 0 8 \\rightarrow &\n2 0 2 8 4 \\rightarrow &\n2 2 6 4 2 \\rightarrow &\n0 4 2 2 0 \\rightarrow &\n4 2 0 2 0 \\rightarrow \\\\\n2 2 2 2 4 \\rightarrow &\n0 0 0 2 2 \\rightarrow &\n0 0 2 0 2 \\rightarrow &\n0 2 2 2 2 \\rightarrow &\n2 0 0 0 2 \\rightarrow &\n2 0 0 2 0 \\rightarrow \\\\\n2 0 2 2 2 \\rightarrow &\n2 2 0 0 0 \\rightarrow &\n0 2 0 0 2 \\rightarrow &\n2 2 0 2 2 \\rightarrow &\n0 2 2 0 0 \\rightarrow &\n2 0 2 0 0 \\rightarrow \\\\\n2 2 2 0 2 \\rightarrow &\n0 0 2 2 0 \\rightarrow &\n0 2 0 2 0 \\rightarrow &\n2 2 2 2 0 \\rightarrow &\n0 0 0 2 2 \\rightarrow &\n\\cdots \\quad \\quad \\\\\n\\end{matrix}\n", "912d928e268a339bdaeae51d52d7c903": "\\frac{v_1^2}{2} = b(v_1)", "912db54513afbfbca6084cb130f33e9b": "\\pi_0(GL(V)) = (GL(V)/GL^+(V) = \\{\\pm 1\\}", "912dbbf00e92a10913bee7b21865b27a": "\\{\\lambda_i\\}_{i\\in\\mathbb{N}}", "912df964ca0333e1c4867d31f10012c5": "H^s(E)=0", "912e1b37f292e1d17c5b03568e67d012": "\\mathcal{O}(D)", "912e5c0903d34af49ea5488c8dfb2476": "(P,V)", "912e7d1dd79f9e9e70196b3432365ef5": "\\lambda_0 \\le \\lambda_1 \\le \\cdots \\le \\lambda_{n-1}", "912e978675593e46b9361c5e476b7807": "\\lim_{n\\rightarrow\\infty} \\sigma^2(n) ~ = ~ \\lambda_1", "912eb2daf2f3c192832981182c143098": "\\alpha^{-1} = 137.035\\,999\\,174(35).", "912ecad795bebcda7430f9ae2b5267b5": "\nf(n)\\le f(x)\\quad\\text{for all }x\\in[N,n].\n", "912ecd66e8df72cfdbeb18a3c23b14c5": "\\int_{-1}^{+1} \\sqrt{1-x^2} g(x)\\,dx \\approx \\sum_{i=1}^n w_i g(x_i)", "912ecf023e9238cbf517645e638385ff": "(x^\\mu, y^i)", "912edc4d6c449fb42bb227d85620c326": "\\Delta \\vec v = \\nabla^2 \\vec v", "912ee60b8622fd2c6f741e3c9f5f4a6d": "\\frac{25}{16}", "912f85fab3246d5956da2fc5bf9d643e": " {\\Delta x} \\,", "912fc162533356ef0528b3c98bb5ce68": "{h S}={0.1(D)}", "913007b9e05a7c90a9efe08b972306ba": "\\Delta H_\\text{max}", "91300931614a8c0e04d1403f09c0310a": "\n\\forall k \\ge 0: -(-1)^k k \\sigma_k = \\sum_{i=0...k-1} \\sigma_i Tr(M^{k-i})\n", "91300e3c12485d947330518a2e9fbd5a": "\\textbf{P}_{k|k} = \\textrm{cov}((I - \\textbf{K}_k \\textbf{H}_{k})(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k-1}) - \\textbf{K}_k \\textbf{v}_k )", "91301539fe644870cf35a3bd61cfccf4": "\\frac{dU_2}{dt} = \\int_0^L \\frac{du_2}{dt}\\,dx = J_2(T_{2L}-T_{20})=\\gamma L(\\overline{T}_1-\\overline{T}_2).", "9130289753db7a4482a074fedbcbfe53": "\\sum(r_j-r_i)(s_j-s_i) = \\frac16n^2(n^2-1) - nS ", "9130e636ef7d859895d8969a3830fd23": "Z = D \\,", "91313082c9fa5770d27dfd782713baca": "\\Phi_{,i} \\approx \\Gamma^i_{0 0} = {1 \\over 2} g^{i \\alpha} (g_{\\alpha 0 , 0} + g_{0 \\alpha , 0} - g_{0 0 , \\alpha}) \\,.", "91314631532a919732bd7a25a095fd96": "\\alpha A\\beta \\rightarrow \\alpha\\gamma\\beta", "9131489e9762f662a77bf2205a681bd2": "f(J_{\\lambda,n})=\\left(\\begin{matrix}\nf(\\lambda) & f^\\prime (\\lambda) & \\frac{f^{\\prime\\prime}(\\lambda)}{2} & \\cdots & \\frac{f^{(n-2)}(\\lambda)}{(n-2)!} & \\frac{f^{(n-1)}(\\lambda)}{(n-1)!} \\\\\n0 & f(\\lambda) & f^\\prime (\\lambda) & \\cdots & \\frac{f^{(n-3)}(\\lambda)}{(n-3)!} & \\frac{f^{(n-2)}(\\lambda)}{(n-2)!} \\\\\n0 & 0 & f(\\lambda) & \\cdots & \\frac{f^{(n-4)}(\\lambda)}{(n-4)!} & \\frac{f^{(n-3)}(\\lambda)}{(n-3)!} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & f(\\lambda) & f^\\prime (\\lambda) \\\\\n0 & 0 & 0 & \\cdots & 0 & f(\\lambda) \\\\\n\\end{matrix}\\right)=\\left(\\begin{matrix}\na_0 & a_1 & a_2 & \\cdots & a_{n-1} \\\\\n0 & a_0 & a_1 & \\cdots & a_{n-2} \\\\\n0 & 0 & a_0 & \\cdots & a_{n-3} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & a_1 \\\\\n0 & 0 & 0 & \\cdots & a_0\n\\end{matrix}\\right).", "9131554c44e74abe8cfa6ae6ec9b5529": "A(\\vec{k},\\omega)=Z\\delta(\\omega-v_Fk_{\\|})", "9131599093839bff064389af416d7f1c": "\\Lambda^{\\chi'}{}_{\\psi}", "9131c4c0da1b30d3ef2705cb1be9d34e": " \\Psi(x,t)= \\sum_n c_n\\Psi_n(x,t) = \\sum_n c_n\\psi_n(x) e^{-iE_nt/\\hbar}", "9131ce2d483e6955ace7fedb39da0c60": "\\mathbf{Z_{01}} = \\frac {\\mathbf{V_1}} {\\mathbf{I_1}}", "9131f86a753d5269216865e45ec13cc5": "r_m^+=r_m", "9132039d4c315608d84e2487bbfb4319": "\n W = C_{01}~(\\bar{I}_2 - 3) + C_{10}~(\\bar{I}_1 - 3) + D_1~(J-1)^2\n ", "91321a629d3a9a26a94d8214c8fc76b8": " U[\\varphi] = \\iint_D \\sqrt{1 +\\nabla \\varphi \\cdot \\nabla \\varphi} dx\\,dy.\\,", "91322eb4349347eaa2dbe404e9032209": " x_{m},\\, k\\!", "913233975f35c0ddba692fe71631b880": "\\zeta_0", "913238661c606ab288fa4745d92965c3": "D_7", "913247e0956d5ab69cf16e815c5ccaf0": "g(u) = A \\cos(h(u)) + B \\sin(h(u)) + C \\,\\!", "91325296d0bd8f0a8c28e954f99207bc": "\\scriptstyle\\Delta_M", "91329dc887b9e890829b209229cc25f7": "\\mathbf{\\Rho}=I\\frac{\\mathrm{d}^2\\omega}{\\mathrm{d}t^2}", "9132a2b076cd881bbfbec04656aa7a36": "C_3 \\times C_2", "9132b43b1c136e15fcb7ba2aeee25fab": "\n v_{i,jk} = v_{i,kj}\n", "9132cc5651688bbc49ddd7e68804e2a4": "\\textstyle \\frac{c}{\\sin C}", "9132da73d6f5b7c2ec93b506ed6c217f": "\\forall x", "9133067928eb30375404e514a5baf1f1": "\\iint", "9133652630435fff7b27c791f5115b8d": "_{q=p\\,}\\!", "913390594d7afc35809dc7b7d8d0e19c": " s *_l r(t) = r *_l s(t) = \\int_0^\\infty s\\left(\\frac{t}{a}\\right)r(a) \\, \\frac{da}{a}", "9133a7cfc5532aaa85b4d5dda1cffe21": "0<\\vert s\\vert< t < \\delta", "9134623bab339f26c1a17647383d152e": " \\prod_i A_i\\,, ", "913469c6306f69b5f56d723668235a79": " \\sum_{i=0}^{N-1} r_i = \\kappa ", "9134b9bf88c6e854b0405365e5cd8453": " \\frac{f(z)}{z} =\\left(\\frac{\\partial f}{\\partial z} \\right)(0) + \\left(\\frac{\\partial f}{\\partial\\bar{z}}\\right)(0) \\cdot \\frac{\\bar{z}}{z} + \\eta(z), \\;\\;\\;\\;(z \\neq 0). ", "9134f33537f58ff5cbca39f5312fc0b1": " z \\rightarrow \\displaystyle \\frac{1}{z},\\ z\\ne 0,\\ \\ 0 \\rightarrow \\infty,\\ \\ \\infty \\rightarrow 0 \\quad, ", "91350dffc21604436fecfa64ccd2b6ef": "\\scriptstyle p_A(t+\\frac{R}{c})", "91351eb34ed6209b284f4c1e5c03c306": "\\left\\{x_1,x_2,\\frac{1+x_2}{x_3}\\right\\},", "9135363c20ecb6b976f6036a1ca5df48": "u\\in B", "913555e444ef02353e16ac6d9fecb58d": "p\\in P", "91357cb2dc5dd24f224b0126a69406fc": "\\frac{\\delta S}{\\delta \\phi(x)}=-\\partial_\\mu \\partial^\\mu \\phi(x) -m^2 \\phi(x) - \\frac{\\lambda}{3!}\\phi(x)^3", "913583a44da2a4fe552663fe5742bf5c": "\nJ_x = \\frac\\hbar2\n\\begin{pmatrix}\n0&\\sqrt{3}&0&0\\\\\n\\sqrt{3}&0&2&0\\\\\n0&2&0&\\sqrt{3}\\\\\n0&0&\\sqrt{3}&0\n\\end{pmatrix}\n", "9135904e39d6cdeeaf8b6753088c9338": "\\frac{\\mathrm{d}}{\\mathrm{d} t} \\frac{\\partial L}{\\partial\\dot \\varphi_i} = \\frac{\\partial L}{\\partial\\varphi_i} \\,.", "9135a4e1934728a6b8f4e5d64d60ba8e": "E = \\hbar \\omega_0 ka ", "91361bcc504cd2870be2230d08f2e5b5": "\\mathrm{C=A\\ s}", "91361c532f59e96555e4d4548bf73080": "\\Gamma_d = -\\frac{dT}{dz}= \\frac{g}{c_p} = 9.8 \\ ^{\\circ}\\mathrm{C}/\\mathrm{km}", "91368860d0d7ef8c70516187587b4fa3": "(17)\\quad \\psi_{RN}=\\frac{1}{2}\\ln\\frac{L^2-(M^2-Q^2)}{(L+M)^2} \\,, \\quad \\gamma_{RN}=\\frac{1}{2}\\ln\\frac{L^2-(M^2-Q^2)}{l_+ l_-}\\,,\n", "9136fea5486d332179e7643a08ee0936": "\\operatorname{spectrogram}\\{x(t)\\}(\\tau, \\omega) \\equiv |X(\\tau, \\omega)|^2 ", "9137211969e5f70198a7714dab2ad18d": "b^2-4ac", "913724d8c06a847eb890e33f937a8642": "(1 - \\alpha_a)\\alpha_b = \\alpha_o - \\alpha_a", "91372a6f0817f1fea305bb3546096342": "M[a] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\na&0&0& \\cdots \\\\\na^2&0&0& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "91373d7450986870a28ed80f4a003e3f": "\\frac{\\partial\\theta}{\\partial t}~=~\\alpha~\\square^2\\theta~=~\\frac{-\\alpha}{C^2}~\\frac{\\partial^2\\theta}{\\partial t^2}~+~\\alpha~\\nabla^2\\theta ,", "913751bb42e6aa769c852a070e4e553d": "\nx^2 - 1 = 1.\\,\n", "913757897af81e67182bd62bed538500": " \\int_{[0,1]} 1_{\\mathbf Q} \\, d \\mu = \\mu(\\mathbf{Q} \\cap [0,1]) = 0,", "91376f90101ebab6ed67f926eaa3904b": "q=\\frac{1}{2}\\rho v^2", "9137a65038db1775685fbca8e0aa68ac": "M = PL/4", "9137deaf80edbace17962010a543798c": "\n\\bold{B}=(\\bold{v}\\times\\bold{E})/c^2\n", "9137fc9ce4827fa2f9d2bcdce09eebf7": "\\mathbf{v}_{\\mathrm{average}} = {\\Delta \\mathbf{r} \\over \\Delta t}", "913828304eba84a4e245a988e10af52b": "\\Gamma(t) = \\frac{1}{(1-t)^n} \\frac{1}{(1-t^2)^{n(n-1)/2}} \\ ", "9138b9933092c8a5c9d53f66f7227904": "\\omega \\approx \\omega_a", "913a00284a65bd38fb6a2a8c2b908b24": "\\sum_{n=1}^\\infty 1,", "913a051addb68d048cdfabc2c0f16850": "\\delta E(\\gamma)(\\varphi) = \\left.\\frac{\\partial}{\\partial t}\\right|_{t=0} E(\\gamma + t\\varphi).", "913a4348dd8be229af0fe2b32468a5a4": "G'=G \\backslash \\{C\\}", "913a54d479f688947c1cc6a66f7f1854": "\\langle u , v \\rangle = \\int_a^b u(x)v(x)dx ", "913a656d141f7014b0146ce4ac6961af": "u^TX \\sim s(\\alpha,\\beta(\\cdot),\\gamma(\\cdot),\\delta(\\cdot))", "913a74dd83f41ea50ab65bb649126666": "\\operatorname{gr}_n^{W} H = W_n\\otimes\\mathbf{C}/W_{n-1}\\otimes\\mathbf{C} ", "913aad03b6c3891fc78471e3bac98bab": "\n\\mathcal{I}(\\theta)=\\operatorname{E} \\left[\\left. \\left(\\frac{\\partial}{\\partial\\theta} \\log f(X;\\theta)\\right)^2\\right|\\theta \\right] = \\int \\left(\\frac{\\partial}{\\partial\\theta} \\log f(x;\\theta)\\right)^2 f(x; \\theta)\\; dx\\,,\n", "913ab86dbe26ec5e36606bbcab75fd7b": "a = \\sum_i {\\sum_j {x_i\\, x_j\\, a_{i\\, j}}}", "913baca5f4baf5c866da9cf66d01db7a": "\n\\begin{align}\n \\phi &= \\arctan2((q_1q_3 + q_2q_4), -(q_2q_3 - q_1q_4))\\\\\n \\theta &= \\arccos(-q_1^2 - q_2^2 + q_3^2+q_4^2)\\\\\n \\psi &= \\arctan2((q_1q_3 - q_2q_4), (q_2q_3 + q_1q_4))\n\\end{align}\n", "913be2f59174b62430e2b97b0874ec74": "b_a", "913c14cdf914ef509f8725cbf32ce018": "C_\\bullet:\\bold{Top}\\to\\bold{Comp}", "913c64d689d72d4f18d1b73514d0e191": "\nA^2= m^2 k^2 + 2 m E L^2 \\, ,\n", "913c8337f7e26e0948298fb283f479bc": " { \\partial T \\over T_m } = { \\gamma - 1 \\over \\ \\gamma } {\\partial p \\over p_m } ", "913ca8541d013009599a3a9a5d7816fc": "k_\\mathrm{spec} = \\frac{D F G}{4(E \\cdot N)(N\\cdot L)}", "913cb850395908859ae52e02beb643e5": " P \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix}\nx \\\\ y \\\\ 0 \\end{pmatrix}.", "913db6844be53079181ade7685a49acc": "\nP[T \\ge \\ t]=e^{\\frac{-t}{h}},\n", "913dd939f905e521376852cadeae5832": "^{n}a", "913e1be6b484227fd329eca0b40ab785": "Q(1)=0", "913e46c88bd5be6e2a53412d471a6a29": "I(\\nu,T) =\\frac{ 2 h\\nu^{3}}{c^2}\\frac{1}{ e^{\\frac{h\\nu}{kT}}-1}", "913e701bfd41194790d051576075b329": "F_1, F_2,\\ldots", "913e751d717c2ff7c5e16f8c1d6758bb": "D()", "913ee290c96f9f1a6cf26f1c1a4d5503": "F(X) == X \\rightarrow P", "913effbed323eeced5396a9bf7ab547b": "\n\\theta = L \\int^{u} \\frac{du}{\\sqrt{m^{2} c^{2} \\left(\\gamma^{2} - 1 \\right) - L^{2} u^{2}}}\n", "913f1dd551b1a644ca083f34f21ef32a": "\\hat{\\beta}_{[-i]}", "913f5c113f2a91f7ec7989bc062ea8dd": "\\displaystyle \\hat{f}(\\xi)=", "913f8fc3a74a327805a584946b520921": "f_{W}\\,", "91400f8bd5b910572721af0ff38260bd": "\\ v_{tune}(t) ", "914071b8934b1d6035495f1ed02aa5fd": "-h", "9140b785606eeea1edba2016201109dd": "\n\\Delta(t) = L(t+1) - L(t)\n", "914166a4b3643e572b86136fb9db3156": "p \\in F(k)", "9141cd3c38e4f3b9198e07b0e2e63073": "b + a i", "9141eaf0649478b7656e36f88d8f3244": "\\left(\n\\frac{9}{4 \\pi}\n\\right)^{\\frac{1}{3}} \\approx 0.894", "9141ee2d932aecccbd4759ba0eea585a": "F_p^k", "9142d580a456d197c215101c3471d2f1": "\\zeta_1=\\zeta", "914345475ae5c7b97d92d525b739739b": "[t,\\ t + \\Delta t]", "9143dfc4a440b237a7db2ef87f74e81d": "\n\\mu_i = \\mathrm{E}(X_i)\\,\n", "91442bfd69b03813902a4beba3d7ab17": "\\Zeta_2 = \n\\begin{bmatrix}\n0 & z \\\\\n1 & 0 \n\\end{bmatrix} \\in M_2(C( \\mathbb{T})).\n", "91446590a4bc66b68123f4cb47f21ce7": "\\sup_{t>0} \\frac{f(t)}{e^{bt}} < \\infty", "9144765516068bb9c6547254190c9a4b": " u(z):=\\sum_{k\\geq 1}u_k z^k", "9144d339329aafbdf68cf28f601d0d60": " \\frac{ \\sigma }{\\alpha-1}", "914506afe329be08d476d8dc57fb75c8": "{D}_{6}^{(1)}", "914551c6c1b6bb1d6c7072de34c4975e": " (1+x) ", "914577bfadb4714dd5ea10ecf0ae59ec": "f(\\mathbf{x}) = f_0+\n\\sum_{i=1}^nf_i(x_i)+\n\\sum_{i,j=1 \\atop i0", "9159e6132b4b09c20911f069c5174f6d": "V_D = n \\cdot V_T \\ln\\left(1+\\frac{I}{I_S}\\right) \\approx n \\cdot V_T \\ln \\left( \\frac{I}{I_S} \\right) \\Leftrightarrow", "915a0b85fd4876479499df02fe43638b": "\\approx 154.2857", "915a189636c6b6fe142b630da0105774": "\n \\frac{\\partial P_{ij}}{\\partial X_j} \\approx \\frac{\\partial P_{ij}}{\\partial x_j} + u_{p.j}^{(0)}\\frac{\\partial P_{ij}}{\\partial x_p}\n ", "915a246fe3bfd514b5630eb451f8224b": "Pr(W = w) = 2^{-nR}, w = 1, 2, \\dots, 2^{nR}", "915a601b7c51623f1565edae650f7b06": " \\hat{M} = \\left[\\begin{array}{c} M \\\\ A \\end{array}\\right] ", "915a61cf8c0d0c92ece136a1c54cef43": "|L\\rangle", "915a664dadf0528ce2a7189742ac1d1f": " X_\\leq(a) \\subseteq X_\\leq(b) \\mbox{ precisely when } a \\leq b . ", "915aa1ecea1fb104106760f7d8deec40": " g_x (m) ", "915aa7b05e586daa3caceafa68c25d35": "q_i\\!", "915b79b6127c97d8078fcc4970176027": "P(A)\\leq P(B)", "915ba1a990dd46a6b2e628d890886864": "R_o, X_o", "915bb2100e0660466a7467a4049b68ba": "v = (D_1-D_2)\\frac{dN_2}{dx},", "915be65ff2787ef3569685d60477d9ab": "\\frac{d}{dx}\\psi_L(0) = \\frac{d}{dx}\\psi_C(0)", "915c244f5c4c436638b8bf056c6d27d3": "0<\\varepsilon <\\!\\!< 1", "915c3553f18a0c1c68ed3f5a20f521d4": "\n{{\\sigma _z^2 \\,} \\over {z^2 }}\\,\\, \\approx \\,\\,\\,{{\\left( {2xy} \\right)^2 } \\over {\\left( {x^2 y} \\right)^2 }}\\sigma _x^2 \\,\\,\\, + \\,\\,\\,{{\\left( {x^2 } \\right)^2 } \\over {\\left( {x^2 y} \\right)^2 }}\\sigma _y^2 \\,\\,\\, + \\,\\,\\,{{2\\left( {2xy} \\right)\\left( {x^2 } \\right)} \\over {\\left( {x^2 y} \\right)^2 }}\\sigma _{x,y}", "915c52ffc4227886178e1149f9e76ad9": "\\scriptstyle\\hat{a}=x_{(1)}", "915c930ca93ffda0a1fcdf1be69adc07": "\\sqrt{\\frac{5}{12}}\\!\\,", "915cb16adc6851219c8f47525aaa3fcc": "\\frac{dq}{dt}=\\frac{M_q}{B}q+\\frac{M_\\alpha}{B}\\alpha+\\frac{M_\\dot\\alpha}{B}\\dot\\alpha", "915cf4a31bfe185c0db0c6baf6f8b42e": "A_{i1}", "915d395033f03c7e8b877bf645c5d9c9": "\\Delta S\n=\\int_{T_0}^{T} \\frac{C_v}{T}\\,dT+\\int_{V_0}^{V}\\left(\\frac{\\partial P}{\\partial T}\\right)_VdV.\n", "915d95a961d1728330b0dfef26ecf916": "f^{\\Delta} = f'", "915daff10a79f70f0e2e35281f85fd79": "\\operatorname{Sym}\\, T = \\frac{1}{k!}\\sum_{\\sigma\\in \\mathfrak{S}_k} T_{i_{\\sigma 1}i_{\\sigma 2}\\dots i_{\\sigma k}} e^{i_1}\\otimes e^{i_2}\\otimes\\cdots \\otimes e^{i_k}.", "915df51293ae0b3d1313243f4634b6cf": "s_a", "915e33529af66ef0354b6c807311d15c": "\\{ generate, hate, great, green, ideas, linguists \\}", "915e8bdcdee5e81dcae3c31083cf0b48": "E\\to P/G", "915ebbf18cc2cbccfa034d83d4d05551": "\\dot{g}\\ =\\frac{\\partial g }{\\partial v_1}\\ h_1\\ + \\ \\frac{\\partial g }{\\partial v_2}\\ h_2\\ + \\ \\frac{\\partial g }{\\partial v_3}\\ h_3", "915fa56d8fae021be3cf5de33120f7ef": "B \\land h \\land F^-", "915fe36de2a98938ba7202743927fbc8": " {\\rm NP} = \\Sigma_1^{\\rm P} ", "91600588d64566c71d05d4f1fb1c054f": "a, b, c, d \\in \\mathcal{B}_w", "916008f3aea0847651be44abceb0aae3": "R/M\\,\\!", "916030795a3a4b735846f950ca9f129e": "\\prod_{k=0}^{m-1}\\Gamma\\left(z + \\frac{k}{m}\\right) = (2 \\pi)^{\\frac{m-1}{2}} \\; m^{\\frac{1}{2} - mz} \\; \\Gamma(mz).", "916072694c1ccac907e9089ad400b958": "s = \\beta_m", "916075479e9f69248bc19852795789a5": "\nf_{\\mathrm{rms}} = \\sqrt {{1 \\over {T_2-T_1}} {\\int_{T_1}^{T_2} {[f(t)]}^2\\, dt}},\n", "9160b5bed6cb4fee7a872ad983009e66": "\n\\forall \\alpha \\in \\Lambda. \\, \\forall q' \\in S. \\,\nq \\overset{\\alpha}{\\rightarrow} q' \\, \\Rightarrow \\, \n\\exists p' \\in S. \\, p \\overset{\\alpha}{\\rightarrow} p' \\,\\textrm{ and }\\, (p',q') \\in R\n", "91610a1af7c4fc2c09fcc6423127c297": "O( n^2 / C)", "91627918a56cf8a2a0a215dae2459d28": "\\hat{X}^n", "9162794069c84c3962f6d62e8f9db944": "\\neg U", "9162afc2744f37162342fbe50303df22": "p_i=\\hat{p}_i-\\hat{p}_{i-1}.", "9162b53954ecf8f53da7b925cccc44d6": " Q ", "9162c92a384382c68616e694a3c72dbb": " d\\tau^2= \\left(1-\\frac{2M}{r} \\right)\\, dt_r^2 -2 \\left(1-\\frac{2M}{r} \\right)\\,f' dtdr - \\left( {\\frac{1}{1-\\frac{2M}{r} }}-\\left(1-\\frac{2M}{r} \\right)\\,{f'}^2\\right) dr^2 -r^2 \\, d\\theta^2-r^2\\sin^2\\theta \\, d\\phi^2\\,", "91635661e9e3a66456835bc132d82edd": "\\alpha_1, \\ldots, \\alpha_n \\vdash \\beta ", "9163878e45ace808e827b8ca5b6243f1": "\\left( 0,\\ \\pm1/2,\\ \\pm1/2,\\ -\\sqrt{1/8},\\ \\sqrt{3/8}\\right)", "9163b318cc0831cc0900c47c93f9b8b9": "\n\\begin{align}\n G(f) & = \\frac{1}{H(f)} \\left[ \\frac{ |H(f)|^2 }{ |H(f)|^2 + \\frac{N(f)}{S(f)} } \\right] \\\\\n & = \\frac{1}{H(f)} \\left[ \\frac{ |H(f)|^2 }{ |H(f)|^2 + \\frac{1}{\\mathrm{SNR}(f)}} \\right]\n\\end{align}\n", "9164252ef9ad9c84d1f2264703b1e001": "\\mathbf{k}_0 \\cdot \\nabla E_0 + \\omega_0\\, \\mu_0\\, \\varepsilon_0\\, \\frac{\\partial E_0}{\\partial t} = 0.", "9164d2842b51cf79c15e310b7491b352": " f(A)+f(B) \\geq f(A \\cap B) + f(A \\cup B) ", "9164d61f9c7cceb4f4baeb76b87e20ce": "\\Delta E_\\text{pot,gravity} = \\Delta m\\, g z_2 - \\Delta m\\, g z_1. \\;", "916525130414e6cc6261e8a92e2ed9ff": "b = \\sum_i {x_i\\, b_i}", "9165625c4f8173d114281cf3efbf55cc": " \\displaystyle wxw^{-1} = ||w||x", "9165b402786bf3408cda9ea2c9b3e65f": "\n\\begin{align}\n&F_\\theta = -\\frac{1}{r}\\ \\frac{\\partial u }{\\partial \\theta} = -J_2\\ \\frac{1}{r^4} 3\\ \\cos\\theta\\ \\sin\\theta \\\\\n&F_r = -\\frac{\\partial u }{\\partial r} = J_2\\ \\frac{1}{r^4} \\frac{3}{2}\\ \\left(3\\sin^2\\theta\\ -\\ 1\\right)\n\\end{align}\n", "9166684e56eebde0783fda25d916d377": "\\nu\\in\\mathrm{Tan}(\\mu,a)", "9166934c57a5c3e7dd9727b16e01a605": "m_{L}=\\frac{3}{4}\\cdot m_{em}\\cdot\\frac{1}{\\beta^{2}}\\left[-\\frac{1}{\\beta^{2}}\\ln\\left(\\frac{1+\\beta}{1-\\beta}\\right)+\\frac{2}{1-\\beta^{2}}\\right]", "916697644199b803b7b178e6516e84b9": "f(\\mathbf{Aa}) = 2pq", "9166c511be91998f37b5d00eb82aa9ec": " \\succeq 0 \\; \\forall \\; 1 \\leq j \\leq p", "9166f7aaecbb58f55d41b87a5d8404aa": "r_p < a", "916713c7ea68f03fff1726760fa3db26": "T=", "91673b4572fcac0a0ceac4a88e03df4a": "r={\\ell\\over{1+e}}\\,\\!", "9167433582c85ba2d932739c125875e7": "\\int_D f(x)\\,dx ,", "916769b14323ae408f692ba1d3a6b19f": "\\sin(x + 2\\pi) = \\sin x \\,\\!", "916774142dbb86bab80ec2b65c3ee45d": "p_{out}", "9168fae682716b87dbe2266eee315dcf": "\\forall u\\neg(uEu)", "9168fe6b1dba8ee37a53c7b2dd4a1733": " K(x) \\neq 0 ", "91690fcc12cedd88ff5bb184d8741edd": "m \\oplus t", "91691f43da3fae742d90c1509e8140da": " 0 \\,", "9169226ce344960791f07900a150bad1": "\nf_{sphere} = 6 \\pi \\eta R_{eff} = 6\\pi \\eta \\left(\\frac{3V}{4\\pi}\\right)^{(1/3)} \n", "91696a35969441c7f4c6dbb5fcfb39a4": " \\left|\\frac{a}{b} - \\frac{p}{q}\\right| = \\left|\\frac{aq-bp}{bq}\\right| \\ge \\frac{1}{bq},", "91698dbfb21b716436789a331c4f48de": "\\mathcal{H}=0", "9169b2a81b392707849e14665e3c9199": "\\omega_\\text {max}=\\frac{254}{\\ell/2}", "9169c4f60e7edb6a90dcf0aa9b76ce90": "f(\\textbf{x}_{r}) \\geq f(\\textbf{x}_{n}) ", "9169ce2e360277c15e5adde4145b4717": " \\boldsymbol {\\Phi} = \\frac{\\mathit MktCap - gross assets + total liabilities}{\\mathit MktCap}\n", "9169fcb0f51a86ba23c1c488f4124713": "y_1(t)\\,", "916a39cccf139a14ad637aa17b5ccae9": "\n\\phi_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\log \\left| R(t=0) - R_{+} \\right| - \\log \\left| R(t=0) - R_{-} \\right|\n", "916a4bf50e921bd0ac7d12f6e7acf9c6": "c: X \\rightarrow X^*", "916aa8e9810d0e9e6b5404a54850544a": "[\\eta]", "916b1c024b4c95f89a26b25b2d084c75": "y =\\mathbf{B'}x", "916b782e64bf5106a04e4778872a8818": "\\omega_m = \\frac{1}{\\sqrt{LC}}", "916b8907a98d90e71738debaf18554df": "\\sigma = \\sqrt[6]{\\frac{A}{B}}", "916bf0ad44e834e1edfa891fc43daf90": "g(A)", "916c3a7c22d04367c2c8a1873dc0b1f0": "\\tilde{n} = \\frac{c\\tilde{k}}{\\omega}", "916cacf2a97d5c455e55f228fc45a4af": " \\nabla f(\\mathbf{x}_k) \\rightarrow 0 ", "916ce9a34bcb74f264aa4d2deedfedc2": "\\mathrm{nopqrstuvwxyz} \\!", "916e048fd79d05fc531c867c5c2da1a4": "W(x_1)=-u'\\left(x_1\\right)v\\left(x_1\\right)", "916e4dda100f38e2c0fc3c85d2bf2f3a": " a^{(b^c)} = a^{b^c}", "916ece8ffacbc4979b80c5255920e476": "S = \\sum x ^ 2 - \\frac{\\left(\\sum x\\right)^2}{n}", "916edcfd2aa346c1b46df93dc1661dfd": "a>0\\,", "916f0331ef06174a540581d3703bfb43": "{\\left ( \\frac{\\partial y}{\\partial x} \\right )}_z = - \\frac { {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y }{ {\\left ( \\frac{\\partial z}{\\partial y} \\right )}_x }.", "916f22c64aca3b17244ea6e92f1eeef9": "F\\subset Q", "916f30788686a64d8cdc6d7a66fe2384": "F=Gm_{1}m_{2}/a^2", "916f612ff5015f6d7b3f9c6a045b1912": "S_N f(x) = \\sum_{n=-N}^{N} F_n \\, \\mathrm{e}^{inx},", "91701beadc22c5f2bc1c43e71161c81d": "d q_b = \\frac{d\\bold{d}}{|\\bold{s}|}\\cdot\\mathbf{\\hat{n}} ", "91701ea7904776a4097e9c211fe3681f": "\\langle E \\rangle = (3/2) \\langle k_BT \\rangle", "9170f7faf569077d4eaab612804aee77": "\\,m(b^2 + c^2) = a(d^2 + m^2)", "9170fe72ca2fed8d9e03b302fd117d2b": " f_x =f_{x,g} + f_{x,f} ", "9171da2c96624d2ba7397864e855c6c8": " = G_{\\infty} \\left( \\frac{T}{T + 1} \\right) + G_0 \\left( \\frac{1}{T + 1} \\right) \\ .", "9171e5b572f29f2cc3a3a8315e7a5c46": "\\mathbb{P} (X_{t} = \\tilde{X}_{t}) = 1.", "917202d405de9fd0a35bbb435ecfc8fb": "\\alpha\\|\\beta\\|_1", "91724710feb845330649eb1ce0207242": "\\Beta_m", "917289cce7d9d76f54576f82f2b10536": "C\\,\\vec e_i = \\lambda_i \\vec e_i = C \\begin{bmatrix}e_{i,n} \\\\e_{i,n-1} \\\\ \\vdots \\\\ e_{i,1}\\end{bmatrix} = \\begin{bmatrix}\\lambda_i\\,e_{i,n} \\\\ \\lambda_i\\,e_{i,n-1} \\\\ \\vdots \\\\ \\lambda_i\\,e_{i,1}\\end{bmatrix}", "917289db266e9e9ffd88ef7995cbfd8e": "\\ddot{u}_a \\ll \\omega^2 u_a", "91728c4007540c027192da916fc0a1a7": "l^2(\\mathbb{Z})", "9172a77e59f8828ee9d9626ca53ddd99": "d(x,z)=d(y,z)", "9172d2a69c1d7917f69f9d1cb7dc92a1": "M^\\beta=(X^{\\rm T} W X)^{-1}.", "917302a25d6e0f14bd4a53f7b0d2239b": "f(x)=\\lceil x \\rceil", "9173202cc6853ed43d8f8b0623022b1a": "\\Delta x_{l,o}=\\frac{x_{l,s} x'_{s,o}}{k_2}", "91732265e7b7aaf527ec3bc8b9641e58": "\\int{(e^{\\frac{k}{m}t}v_y)^\\prime \\,dt} = e^{\\frac{k}{m}t}v_y = \\int{ e^{\\frac{k}{m}t}(-g) \\, dt} ", "917324734bdc735ebdc93e958519dbe8": "\n\\int \\psi_0(x) \\int_{u(0)=x} \\left( \\int {\\partial S \\over \\partial u } \\epsilon + { \\partial S \\over \\partial \\dot{u} } \\dot{\\epsilon} dt \\right) e^{iS} Du\n\\,", "917405c089da743bfbe7e72857eefeea": " ds^2 = d\\bold{q}\\cdot d\\bold{q} = g(d\\bold{q},d\\bold{q}) ", "9174078e606cc36810fe21f221a812a6": "r_{x'y'} = \\frac{r_{xy}}{\\sqrt{r_{xx}r_{yy}}}", "91745c2d03c89e1a0d350e7703e486e2": "R_2 =45 \\times 0.0519 ", "91750124630c02867aec2a69ea1522ec": " \\psi_0 = \\angle DVC, ", "917528169da321ef8579fdc658df9ccd": "\\left ( \\frac{1}{\\sqrt 2}, \\sin(t), \\cos(t), \\sin(2t), \\cos(2t), \\ldots \\right )", "91759373e3a29f323b939d24e1fb8cb7": "\\mathbb{N}\\setminus A", "91763d5f43aa56eb91ae8ffea6816b00": " \\begin{align} \n\\left[x_l , p_m \\right]&=i\\hbar\\delta_{l,m} \\\\ \n\\left[ Q_k , \\Pi_{k'} \\right] &={1\\over N} \\sum_{l,m} e^{ikal} e^{-ik'am} [x_l , p_m ] \\\\\n &= {i \\hbar\\over N} \\sum_{m} e^{iam\\left(k-k'\\right)} = i\\hbar\\delta_{k,k'} \\\\\n\\left[ Q_k , Q_{k'} \\right] &= \\left[ \\Pi_k , \\Pi_{k'} \\right] = 0\n\\end{align}", "917643e7ec5030fbb601b238dab68571": "\\mathbb{R}^d_+", "91767400ed89d2f7daec7bdb2c8d1e17": "\\varphi(x)\\to\\varphi(S(x))", "9176e54f2ae4a0cb35bb944c57bd7f8d": "a \\cdot (1 + \\varepsilon_{11}) \\times a \\cdot (1 + \\varepsilon_{22}) \\times a \\cdot (1 + \\varepsilon_{33})\\,\\!", "9177000f38464d0f24b802b496914ee6": "\\; \\varrho_{A_1\\ldots A_m}\\to \n\\frac{\\sum_i\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n\\varrho_{A_1\\ldots A_m}\n(\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n)^\\dagger}{Tr[\\sum_i\n\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n\\varrho_{A_1\\ldots A_m}\n(\\Omega_i^1\\otimes\\ldots\\otimes\\Omega_i^n)^\\dagger]} . ", "9177579bd4fd26126b3485472d107a72": "(-\\infty,c)", "9177cd0407e429b1cd8283453a5ff3f5": "a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}", "9177e249858ce8bbd3d7c3b5a0a6250c": "\\overline{\\Lambda}", "917910553458d66d5a3020dcdc6aa8a2": "dI/dV=I/V", "917951536c51f3c08b33eb7992b5a8df": "c_1(\\mathcal{L})", "917958dd90dfcbc5b3379bb0329a0c67": "(I,J)", "91796befef6fc032c00efe0c29f3c83d": " \\begin{pmatrix} A & B\\\\ & -S \\end{pmatrix} \\begin{pmatrix} x_1\\\\ x_2 \\end{pmatrix}\n = \\begin{pmatrix} b_1\\\\ b_2 - B^* A^{-1} b_1 \\end{pmatrix},", "9179b66b36d9a89c84bf61e4262c5301": "X[x,y]=x+y'\\frac{x'^2+y'^2}{x''y'-y''x'}", "9179c0a90c0f4e9a711881a7e3760033": "\\rho\\left(\\mathcal{T}\\left\\{\\frac{\\delta}{\\delta\\phi}F[\\phi]\\right\\}\\right) = -i\\rho\\left(\\mathcal{T}\\left\\{F[\\phi]\\frac{\\delta}{\\delta\\phi}S[\\phi]\\right\\}\\right).", "9179f54e22ea5372f8da84f9c15378a0": "f(x_1, \\ldots, x_k) = b + a_1 x_1 + \\ldots + a_k x_k", "917a151c27524a3e44b9cf550227060c": "Y = AK\\,", "917a571d94aec364f3472812324071fb": "\\log((n/w)^w)= w \\log(n/w) = s", "917a5faf6f4f60fe552868105fb776dc": " L = \\sum_{i=1}^N T_i - \\frac{1}{2} \\sum_{i \\ne j}^N (V_R)^i_j ", "917a6f41d7e85e3c06bc176bbe352c95": " J=J_0 \\int_{-\\infty}^{\\infty} \\frac{\\mathrm{exp}(\\epsilon / d_{\\mathrm{F}})}{[1 + \\mathrm{exp} [(\\epsilon/d_{\\mathrm{F}})(d_{\\mathrm{F}}/k_{\\mathrm{B}} T)]]} \\mathrm{d}(\\epsilon/ d_{\\mathrm{F}}) = \\lambda_T J_0 ,..........(26)", "917a86c411d12ae0d0a815a061eb89ab": "e^{i \\pi} +1 = 0.\\,\\!", "917ab850cbb0685c8cd0d143d99f33c8": " \\frac{(1-\\chi _B) \\psi}{ \\|(1-\\chi _B) \\psi \\|} ", "917b15ded9591741a1c52cd1910e18d0": "\\Delta+\\sqrt{\\Delta/2}", "917b248e5e6e5bbb428abb9a358d0b53": "ds^2 = g_{\\mu\\nu}dx^\\mu dx^\\nu.\\,", "917bd9d4b4a1a3c4701cee45090707f8": "e^{i(kx - \\omega t)} \\to e^{i(kx - \\omega t) - \\frac{k}{\\omega} \\int^x \\sigma(x') dx'} ,", "917bf36e6e1b540cb7a87fb22e1d6757": "\n\\frac{30}{20 + 30} = 0.6\\,\n", "917c31e176067737a649bc34fc483fe2": " y_{n+1} = y_n + hf\\left(t_n+\\frac{1}{2}h,y_n+\\frac{1}{2}hf(t_n, y_n)\\right). ", "917c5388d53010bbf6fd83b4a35215a9": "\nF \\left(\\mathbf{x}^{(1)}\\right) = 0.5((-2.48)^2 + (-1.00)^2 + (6.28)^2) = 23.306", "917c6ded8d7cce627f486f8b815c6a8a": "\\Phi(a,b)", "917c7148303526bc82fddd96d9e3f1ea": "g\\cdot f(x)=f(g^{-1}x)\\,", "917c83caec0ddf71778589742f016232": "f : \\mathcal{X} \\mapsto \\mathcal{Y}", "917c84852f22d513b30227a110c7b6cb": "\\lim_{t\\to\\infty} \\int_a^t f(x)\\, \\mathrm{d}x", "917cc955210114e420505f357c3b284d": "\\textstyle{\\frac {\\log(5)}{\\log(10)}}", "917cc98c0f6615d0916799bf61c1b870": "\\left(D^2-\\alpha^2\\right)\\Psi_i=0,\\,", "917cea94cbf86868deee4adf5012eabf": "(Margin/COGS) * Annual Inventory Turns", "917d280438877b729751985a01fc228b": "+20", "917d371431fd21e6a83f68c78322ccbf": "H(X,Y) = H(X|Y) + H(Y|X) + I(X;Y)", "917d78f81b0dfda603ba80432da8ea19": "\\mathcal A_P", "917d7cc9bd225e396c27d24438bf06b5": "isopt(\\theta+\\pi)", "917dd5135ce58352b392e6bfc2c5ac72": "\\Re[ S_{xx}''(x^0) ]", "917e74b7ed483eb69a26acfb36460deb": "\\mathbf E \\approx \\mathbf e \\approx \\boldsymbol \\varepsilon = \\frac{1}{2}\\left((\\nabla\\mathbf u)^T + \\nabla\\mathbf u\\right) \\qquad ", "917e940dcef48b2c3ed4ebf4f8694e46": " E = \\hbar \\omega = {p^2 \\over 2m } = {\\hbar^2 k^2 \\over 2m } ", "917e94bc5ff4e60937fc73f402fa6ad2": " \\textstyle Y=k \\cdot (W+M)+W \\,\\ ", "917ea3e20f89c481d3fcc3b36dc873b3": "\\mathcal{O}(p^n)", "917f49f4265fb93cae53af309da98508": "P^{2m}(R)\\geq P^{2m}\\{\\exists h\\in H,|Q_{P}(h)-\\widehat{Q_{r}}(h)|\\geq\\epsilon\\,\\!", "917f7e15f868ae229071921f2fffb6eb": "\\omega_t=\\sqrt{g_{tt}}dt\\,,\\;\\;\\omega_r=\\sqrt{g_{rr}}dr\\,,\\;\\;\\omega_\\theta=\\sqrt{g_{\\theta\\theta}}d\\theta\\,,\\;\\;\\omega_\\phi=\\sqrt{g_{\\phi\\phi}}d\\phi\\,,", "917f9213bebba9e0cf50343b81c47821": "\\cdots \\le \\lambda_k \\le \\cdots \\le \\lambda_1,", "917fc0597218d87e66ba4dd8422560f5": "\nS_{AB} = S_B-S_A = {\\Delta V_B \\over \\Delta T} - {\\Delta V_A \\over \\Delta T}\n", "91801d960b23759d63e0e1d0bbc41afa": "Q^\\epsilon(M) := \\mbox{ker}\\,(1-\\epsilon T)", "9180a8d9f0216230180bfe9d9bfa804a": "(y + \\lfloor \\frac{y}{4} \\rfloor) \\bmod 7.", "9180c340134eab85f9ecaeaa9266796a": "(Y^X)_{\\omega,g}", "9180da590f65034233c760f1509f5e94": "(X,U,Y)=(\\varphi_g(x),\\psi_g(u),\\rho_g(y))", "9180e5bd143d8256a0e05d8f7d3b80b4": "\\neg EF\\phi \\equiv AG\\neg\\phi", "9180f87ee84c22b9df929a0833278c2f": "\\rho_1,\\,\\rho_2", "91815d056fa8d13827870bf971e5dd29": "{\\Psi}^{\\ast} = 1 - A{\\phi}^{\\ast} ", "91817e5be2de775db031a5ba8829756a": "u_x, u_y, u_z", "9181873f157fd4b005e08f22845e3834": "J(8,2)", "91818f6837e9502818ea442725abbd84": "G_y(x,y)", "9181dc328bb228121fcd4f9e6c19375c": "dS = \\left(\\frac{\\partial S}{\\partial E}\\right)_{x}dE+\\left(\\frac{\\partial S}{\\partial x}\\right)_{E}dx = \\frac{dE}{T} + \\frac{X}{T} dx\\,", "918238b4569ff2df00657ac4e3b9fd1e": "= I + uv^T A^{-1} - {uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \\over 1 + v^T A^{-1}u}", "91826557439f6cb27ca4f31f3a4ac7f7": "W_{H,p} ", "91831c7dcee18f70c1be0a75ba446874": "\\overline{R}", "918416b28aa8caa0931e767feff4afd8": "B^{ij} = \\frac{1}{I_{{ad}}} {c^{im}}_{n} {c^{jn}}_{m}", "91844b033dfa03c51f268cd074b01d23": "(L,\\in,U)", "91850840d0b2b4659c59dff22c65cd2e": "\\exists X \\forall n(n\\in X \\leftrightarrow n < SSSSSS0\\cdot SSSSSSS0)", "91852bf6d5e6dc2841e188839b68472c": "p_\\text{rep} = \\left[ 1 + \\left( \\frac{p}{1-p} \\right)^{\\frac{2}{3}} \\right]^{-1}.", "9185670db01058f556dc91018862aea1": "\\lim_{|a|\\rightarrow \\infty}\\langle\\Omega|A'_1\\cdots A'_n|\\Omega\\rangle-\\langle\\Omega|\\prod_{\\mbox{unshifted i}}A_i|\\Omega\\rangle \\langle\\Omega|\\prod_{\\mbox{shifted i}}U(a)A_i U^{-1}(a)|\\Omega\\rangle=0", "918581706d8fbf5962dedb3d8bd27100": "\\displaystyle e^wI_0(zw) = \\sum_np_n(z)w^n", "918598b484d027cc0e31473f66cb0860": "\\begin{align}\n f_{mnl} &= \\frac{1}{2\\pi\\sqrt{L_{mnl}C_{mnl}}}\\\\\n &= \\frac{1}{2\\pi\\sqrt{\\frac{1}{k_{mnl}^2} \\mu\\epsilon}}\n\\end{align}", "9185ad4909cb83be48dc4c4c2d9a4556": "\\theta(c\\cdot n)", "91865922ea7d641fbcedf405bf41e3d1": "t\\dot \\gamma(t)", "9186800cc893e257e496c0b46103efd5": "U_n(1)=n+1\\,", "9186a151892284f3a2ac3a3c7e7ae929": "f:X\\to\\mathbb{R}", "9186c844c56639166391a2c34831cac1": "\\hat{\\mathbf{k}}\\,\\!", "9186cd9880d8d53a5c7278f41bd36e88": "t=-\\infty", "9186d0c56bd4680fddd30b0e7ea63467": "\n\\begin{align}\nx^2 + bx& = -c\\\\\nx + b& = \\frac{-c}{x}\\\\\nx& = -b - \\frac{c}{x}\\,\n\\end{align}\n", "9186d9dfa3b6f229db401564b473aff8": "f(x)=5", "9186f9b27fc7241e8b2cb914b6862a4a": "Y|_I", "91872f1ce3e03618773585171661d85c": " \\tan \\theta_2 \\; = \\; \\frac {2sc \\sin^2 (\\omega/2)}{s^2 + c^2 \\cos \\omega} ", "91875902482076a029ce454d8cc9f651": "\\mathbf{Q_n} = {1 \\over n}\\sum_{i=1}^n (x_i-\\overline{x})(x_i-\\overline{x})^\\mathrm{T}.", "918771ca59cb8bd60f3285314872bef9": "P^{2m}(R)=P^{2m}\\{x:\\sigma(x)\\in R\\}\\,\\!", "9187daaa86bbb08223f5ba9353bb2f49": "s_2 = c_1e_2 = - \\frac{3}{16}", "918838c8c6871b15d39e101abb0137ae": "\\phi \\!", "91884f1452a21e581ba3f64c10f966fb": "b_k(x) = a_k + 2xb_{k+1}(x) - b_{k+2}(x)", "9188ccb26ab8e3875e3d7fbf61347c66": "\\Bigg[\\frac{i}{\\pi}\\Bigg]=i^{-\\frac{a-1}{2}},\\;\\;\\; \\Bigg[\\frac{1+i}{\\pi}\\Bigg]=i^\\frac{a-b-1-b^2}{4},", "9188ce5fa11d8509faad461916a7a599": "b=a\\cdot v_0/F_0", "91891152988b84449a986d8e3e7efaf9": "\\begin{align}\n y_1 &= 5x_2 \\\\\n y_2 &= 4x_1^2 - 2 \\sin (x_2x_3) \\\\\n y_3 &= x_2 x_3\n\\end{align}", "91892dfcdaa38292b9ed574d8ecda001": " a \\in I ", "91893869132693ec733523da01af1241": "\\Omega(\\log n)", "9189550e326d911861a83beddd31bef4": "T_c = 8a/27Rb,", "9189af813dd550a1ec53d9dff4806442": "\\Psi = (\\forall x')(\\forall x)(\\forall y) (\\forall u)(\\exists y')(\\exists v)(P)Q(x',y') \\wedge (Q(x,y) \\rightarrow \\psi )", "918a1098fcb3f96eca7fafa58b68db4a": "\\dot\\sigma^+ \\textbf{n}=\\dot\\sigma^- \\textbf{n},\\qquad {(2)} ", "918a161b4e5fc6b12a4945636a888491": " L_{\\rm ISCO}", "918a523cdc4f11d5f6b28c12a5f55d5d": "\\alpha\\simeq\\tfrac{1}{137}", "918a539345ef1f8262b689348b1838c5": " \\zeta = e^{2 \\pi i / m} \\ ", "918a7fed6d2de1f40a14963281340d4e": "\\alpha(X,Y) = \\sum_{j=1}^k\\alpha_j(X,Y)e_j", "918abd1ad7a687543f89b128e668ba26": "F_S", "918bbeb44938adfcde773a3cbe4a0b11": "k^{k^n}", "918c03645e4d4c560730c363ffaecaa8": "e^{ix}= \\cos x + i \\sin x,", "918c142dbe80528d1917a486202e274c": "u=\\frac{0.4661x+0.1593y}{y-0.15735x+0.2424}", "918c33163f2246a5467dab9215c9c7bc": "\n \\lambda = -0.924 \\ln(1-1.02\\beta)\n ", "918c4367a35683215a240397877a7730": "P_{(a, b, c)}\\;P_{-Q}", "918c6445115b3016a0ef50f668b513ef": "2\\uparrow\\uparrow\\uparrow\\uparrow n", "918c9434032647bb2afa8b6fbf55af85": "S_{i}\\cap S_{j}", "918ccce8785ae1330786e1dfc066d9ec": "{n\\over m}", "918d13cdfe9c03b550e27aae34120339": "\\scriptstyle e_2 \\times \\cdots \\times e_n = e_1,", "918d410cb663565dda0e36145186f345": "H^k(X)", "918d6ba8991f29ccd589429d532b6705": " \\langle q|\\mathbf{\\hat P}|\\psi\\rangle = \\frac{\\hbar}{i}\\lim_{\\varepsilon\\rightarrow 0} \\frac{\\psi(q+\\varepsilon)- \\psi(q)}{\\varepsilon} = \\frac{\\hbar}{i}\\frac{d}{dq}\\psi(q) ", "918de260f11e003a48b369c5576e9559": "s(x) = p(x) \\, x^t - s_r(x) = 3 x^6 + 2 x^5 + 1 x^4 + 382 x^3 + 191 x^2 + 487 x + 474", "918e7db8f10526a53e48c0152ff0d8db": "C = C_{0} e^{-Kt} \\,", "918e881425fe5570318ad1ea06abe25c": " + \\; C_{\\alpha I}^{\\;\\;\\; K} \\nabla_\\beta V_K - C_{\\beta I}^{\\;\\;\\; K} \\nabla_\\alpha V_K + C_{\\alpha I}^{\\;\\;\\; K} C_{\\beta K}^{\\;\\;\\; J} V_J - C_{\\beta I}^{\\;\\;\\; K} C_{\\alpha K}^{\\;\\;\\; J} V_J", "918ea6ff1c599f9895de4ce789e359e1": "B_k\\,\\!", "918fd9595f7695f45abc0deefe6e81bc": "\\frac1{2} \\sigma (x) \\sigma(x)^{\\top} = a(x) \\mbox{ for all } x \\in \\mathbf{R}^{n}.", "919008568394f1dc7d446db5a2b10912": " 1 = \\int\\limits_R d^3\\mathbf{r} \\, | \\mathbf{r} \\rangle \\langle \\mathbf{r} | ", "919032c6797d23e22c475f65e6c0b5c7": "(1+\\sqrt{2})*(1-\\sqrt{2}) = -1.", "91906796bac512c1b4d6d24be171f284": "\\int\\frac{\\tan ax\\;\\mathrm{d}x}{\\tan ax + 1} = \\frac{x}{2} - \\frac{1}{2a}\\ln|\\sin ax + \\cos ax|+C\\,\\!", "9190a90488b4a14a6c7c54d5f5cf2948": "a, a \\sqrt{\\varphi}, a \\varphi,", "919135d16af61df51f575738ac5b47d8": "R \\in \\mathcal{A \\otimes A} ", "919143defbe91e0723637c6a7ba7f080": "21\\times 21", "91915ddcbcb438146120be25e20bf18e": "\\oint_C {e^z \\over z^5}\\,dz", "9191b162aa91b08bf179e9f016e6b2d3": "\\mathbb{Z}/n\\mathbb{Z} = \\left\\{ \\overline{0}_n, \\overline{1}_n, \\overline{2}_n,\\ldots, \\overline{n-1}_n \\right\\}.", "9191b4aa7026ea4fd002cb7e3b9c0c09": "a_{9}*b_{7}", "9191ba6bdc968bd2645178ec6c1545bf": "v_1=v_2", "9191c8ca59d6ca6345d5ed9d0f54cfc0": "I(t) = \\frac{1}{\\tau}\\left(1-\\int_0^t E(t)\\ dt\\right) \\qquad E(t) = -\\tau \\frac{dI(t)}{dt}", "9192089bdbc0eb55a02b2a14a414a4f9": "\\scriptstyle \\mathcal{P}", "91920e2ef2283ef89b3c2c302614b76a": "(2ac)^2+(2bd)^2 = (a^2-b^2+c^2-d^2)^2", "91923b083a33516b2ae173d6a93c2573": "h_0", "91929e94d41e498956d5c9605f04238b": "\\omega (x) =\\limsup_{n\\to\\infty}\\omega(x,n).", "9192a370a965cb78a139d6d6e3615a4f": "|a_n|^2 = \\frac{1}{n^2}", "9192a79d0670b3a22039d78f0c7444b7": "m_e \\rightarrow \\frac{m_em_p}{m_e+m_p} ", "9192b3a53af34b845badf978cda00e22": "\\sum_{k=0}^{n} ar^{2k} = \\frac{a(1-r^{2n+2})}{1-r^2}.", "9192da97232327b4f762b89d82241c39": "\\int\\mathrm{e}^{-sx}\\,dg(x)", "91930dfaf782ee18216e9bd41ce2351e": "b^n/2", "919316e9b62d08bbed2ae068f305eead": "\\sup_{S_k} \\min_{x \\in S_k, \\|x\\| = 1}(Ax,x) \\le \\lambda_k.", "9193721df6bbc6eb38ef771c6ce12c17": "\\left(\\frac{a}{b}\\right)_3 = 1.", "919389bd535d319bf430c44c302f1c2e": " C_{12}=C_{23}=-1 ", "9193a5e4b5d01c8dc0dbaa2c7ef15cde": "\nG_0^{\\mathrm{R}}(\\mathbf{k},\\omega) = \\frac{1}{-(\\omega+\\mathrm{i} \\eta) + \\xi_\\mathbf{k}}.\n", "9193b64c7b432069dc85c5ffa9730d15": "\nM = \\left( \\begin{array}{cc} A & B \\\\ C & D \\end{array}\\right)\n", "91944834ddc23a4b8526244de2033bf9": "\\Delta = \\Delta(\\gamma)", "919484652a080afdfbc4ea2d1c4c7cff": "V+V^\\perp", "91948a5c506f5492db9eb93bf6104734": "(1-x^2)\\,y'' - x\\,y' + n^2\\,y = 0 \\,\\!", "9194a1c06bfc0458716a3177f426a162": " 4a^2x^2 + 4abx + 4ac = 0", "9194a76eecf3b4a5d108f25e1027612f": "\\langle \\phi | \\psi \\rangle = \\langle \\phi | \\psi \\rangle ^2 ", "9194e2b10090d9b43bd8b93507b3923e": "\\mathcal{L}_H = \\int_{t_0}^{t_1} \\iint \\left\\{ \\varphi(\\boldsymbol{x},t)\\, \\frac{\\partial\\eta(\\boldsymbol{x},t)}{\\partial t}\\, -\\, H(\\varphi,\\eta;\\boldsymbol{x},t) \\right\\}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t,", "91956cb00bf0f8701ebb0a20e039ecfa": "\\frac{\\mathrm{d}P}{\\mathrm{d}\\mathit{\\Omega}} = R(t')^2\\,[\\mathbf{S}(t')\\mathbf{\\cdot}\\hat{\\mathbf{n}}(t')]\\,\\frac{\\mathrm{d}t}{\\mathrm{d}t'} = R(t')^2\\,\\mathbf{S}(t')\\mathbf{\\cdot}\\hat{\\mathbf{n}}(t')\\,[1-\\vec{\\beta}(t')\\mathbf{\\cdot}\\hat{\\mathbf{n}}(t')]", "91957e5860f8d4eef786a9d3258d5753": "\\frac{1}{2} \\int d^3\\mathbf{x}\\left((\\mathbf{n}\\cdot\\mathbf{x'})\\mathbf{J}(\\mathbf{x'})+(\\mathbf{n}\\cdot\\mathbf{J}(\\mathbf{x'}))\\mathbf{x'}\\right)=\\frac{-i \\omega}{2} \\int d^3\\mathbf{x'} \\mathbf{x'} (\\mathbf{n}\\cdot\\mathbf{x'})\\rho(\\mathbf{x'})", "9195c45c09b79069a0d857d0989df247": "R_v", "9195ce9724ef277a3a45fe2ebba81e91": " m\\,", "9196500133e05a6855575adcdf2b4447": "\\scriptstyle(0,\\infty)", "91967f930d3ddeda9cc26a32e4fa1f4b": "v_{n+1}\\,\\!", "91968e7db22676a7302002e6bc8b7590": " l_{(+)} = {l\\over\\sqrt{1-v^2/c^2}}{\\sqrt{1-(2v/c)^2}} ", "9196e84b07340823ce4d80427d56bc3c": " ||\\mu_X - \\widehat{\\mu}_X ||_\\mathcal{H} = \\sup_{f \\in \\mathcal{B}(0,1)} \\left| \\mathbb{E}_X[f(X)] - \\frac{1}{n} \\sum_{i=1}^n f(x_i) \\right| \\le \\frac{2}{n} \\mathbb{E}_X \\left[ \\sqrt{\\text{tr} K} \\right] + \\sqrt{\\frac{\\log (2/\\delta)}{2n}} ", "91975079b063119fd007aa041e93c106": "\\varepsilon_\\omega", "9197f243e1ceebd9334927da660d97e0": "L_n^{(k)}(x)\\equiv\\frac{d^kL_n(x)}{dx^k}", "9197fccbe98c4b309eb86fb709d1da32": "g(x,y,z)=\\nabla f(x,y,z)\\cdot \\vec v=0", "919818353fe77de6ef6ea0f15337012d": "I_h: \\langle s,t\\mid s^3(st)^{-2}, t^5(st)^{-2}\\rangle.\\ ", "91985a768c96dc383e4571f3b19008a6": "f(\\mathbf{r}) =\n\\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{q}) \\right|\\mathrm{e}^{\\mathrm{i}\\left(\\phi+\\mathbf{q}\\cdot\\mathbf{r}\\right)} =\n\\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{q}) \\right|\n\\cos\\left(\\phi+\\mathbf{q}\\cdot\\mathbf{r}\\right) +\ni \\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{q}) \\right|\n\\sin\\left(\\phi+\\mathbf{q}\\cdot\\mathbf{r}\\right) = I_{\\mathrm{cos}} + iI_{\\mathrm{sin}}", "919860b52317a584e5de6f3257631d16": "x=y", "919866e8b7357c1472cd47ce710097fd": "[h,e]=e", "919868f63afcb06439fe9fc93b1abdc0": "V(f(\\vec{x}),y) = |y - f(\\vec{x})|", "919893ae308f6b81ca132aeb99f29631": "\\mathfrak{g}=\\mathfrak{k}+\\mathfrak{p},\\,\\, G=\\exp \\mathfrak{p}\\cdot K.", "919894a85a533d02bd4980ec03d17e27": "B\\in\\mathcal S_n(\\mathcal B_{1\\cdots n},\\mathcal D_{1\\cdots n})", "9198a0200d9ea12838db39eac07516b7": "U_k(\\omega)=-\\omega^2\\,\\!", "9199123a7cc0614374eedf0bccca6374": "R = V / I.", "91993583882187f825742d938aad0006": " \\varepsilon_t = \\left(1 - \\sum_{i=1}^p \\varphi_i L^i\\right) X_t = \\varphi (L) X_t\\,", "91994648180f6c289dc3d11e10626094": "\\sigma \\propto \\dfrac{Gb}{r}, ", "91997b84877686c77d48610d2e81601a": "\\operatorname{lcm}(a, b) = \\operatorname{lcm}(b, a),\\;", "9199cffd295fc187b4fea27660fd2f2d": "F:X\\times S \\rightarrow Y", "919a1a21ab85b237e8edd974dafa9fd0": "\\textstyle (\\mathbf{c} = \\mathbf{e}_1 - \\mathbf{e}_2)", "919a5d996d5692b035bce7672df28065": "M0\\,", "91b3fdd2e71b376e3399491d03187178": "f_s > R_N \\, ", "91b403ae250587e92517d72f648b8210": "\\mathcal{H}_{ji}", "91b41534a56ac3fa9a13c5aae894b9f8": "\\mathrm{FillRad}(X)=\\mathrm{FillRad} \\left( X\\subset\nL^{\\infty}(X) \\right).", "91b452e5867b943a6df79d99073b8a24": " (\\mathbf{AB})^\\mathrm{T} = \\mathbf{B}^\\mathrm{T}\\mathbf{A}^\\mathrm{T} ", "91b45336ca98ae773388653109fa6028": "\\mathrm{^{238}_{\\ 92}U + \\,^{1}_{0}n \\;\\rightarrow\\; ^{239}_{\\ 92}U \\;\\rightarrow\\; ^{239}_{\\ 93}Np + \\beta \\;\\rightarrow\\; ^{239}_{\\ 94}Pu + \\beta}", "91b4595aa57557d17ccd676a975fd4a4": "P \\cdot P^* = I", "91b4c5796971ec4ad5e41bf46442ef5c": "\\hat F(1)\\phi_i(1)=\\epsilon_i \\phi_i(1)", "91b4c6701d2f2d7fc243c347203a839c": "\\nabla = \\sum_i \\mathbf e^i \\frac{\\partial}{\\partial q^i}", "91b4f4820da8b44b333f2f62cfb8ad4f": "\\tau=-it(0\\leq\\tau\\leq\\beta)", "91b4f90769076aac0118d1a9a0484fe9": "\\Delta(fh) = f \\, \\Delta h + 2 \\partial_i f \\, \\partial^i h + h \\, \\Delta f.", "91b5093243237fe6276a3b93ba14d9db": "a+b\\sqrt{p^*}.", "91b5321f51901ffa549806a5caa2c83e": "\\begin{align}\nC(K) \\widehat{\\otimes}_\\varepsilon Y &\\simeq C(K, Y), \\\\\nL^1([0, 1]) \\widehat{\\otimes}_\\pi Y &\\simeq L^1([0, 1], Y),\n\\end{align}", "91b569f0d762da74f940570893b54613": " \\langle\\Sigma | \\Psi_1 *\\Psi_2 \\rangle = \\langle\\Sigma, \\Psi_1,\\Psi_2\\rangle \\ .", "91b5820112bf6732defec4d36c793960": "\\dot{u} = \\frac{1}{\\cos^2(\\theta/2)} \\frac{1}{2} \\dot{\\theta} = u^2 + \\Delta I", "91b5b27082d75a030ba56f3f5dfa35c1": "\n \\delta U = \\int_L \\left[-M_{xx}\\frac{\\partial (\\delta\\varphi)}{\\partial x} + Q_{x}\\left(-\\delta\\varphi + \\frac{\\partial (\\delta w)}{\\partial x}\\right)\\right]~\\mathrm{d}L \n", "91b65872687e2d8f7fef9ce194b36425": "c_P = \\frac{5 R}{2}", "91b6587baed5313982db8f3f69fe639d": " {\\Delta t} \\,", "91b66c3d45b774b556ca8156d147dc0d": "~n_1=N_1/N~", "91b6ae64ed4d93f771ffa1e3a6d05695": "3i_1+1i_2=12", "91b6b92f2e7b5e9c40a15a2ef7260f3d": "\\sigma = \\frac{F_n}{A}", "91b6e0bd8fcbd1dfd791978cddb0b816": "\nT_{\\delta}^{Y^{n}|x^{n}}\\equiv\\text{span}\\left\\{ \\left\\vert y_{x^{n}}\n^{n}\\right\\rangle :\\left\\vert \\overline{H}\\left( y^{n}|x^{n}\\right)\n-H\\left( Y|X\\right) \\right\\vert \\leq\\delta\\right\\} ,\n", "91b7b51b6d2095bc353d17c00db9b3e0": "P(x)=1+\\sum_{k=1}^\\infty p_k x^k=1+2x+3x^2+5x^3+7x^4+\\cdots", "91b7c20d69ba4b5419e6ab3d3d4a14e3": "d=\\sqrt{r^2+h^2}", "91b852dba5323b3379ffad3a96cfa8bf": " C(u)= ", "91b8e9ccb7bfbec1eafda0c0187adfcc": " f(x; \\mu,\\sigma,\\xi) = \\frac{\\left(1+\\frac{\\xi(x-\\mu)}{\\sigma}\\right)^{-(1/\\xi +1)}}\n{\\sigma\\left[1 + \\left(1+\\frac{\\xi(x-\\mu)}{\\sigma}\\right)^{-1/\\xi}\\right]^2}, ", "91b8fd4a81bebe9387ad8b813a8f3551": "\\rho(\\pi^*,a) = \\int_\\Theta L(\\theta, a) \\, \\operatorname{d} \\pi^* (\\theta)", "91b8fdada065fe7bf37be5ed90b96a33": "\n\\tau \\dot{w} = v+a-b w.\n", "91b9116a95bae791e51abfcac0a157e1": "g: X \\rightarrow \\mathbb{R}", "91b9a89421fc041af1545df8084a2113": "D_{F^*}^{q^*}(p^*, q^*) = D_F^p(q, p)", "91b9af42643eecbd26c5e1a5b1704b6f": "\\gamma(E) = \\int_E g\\, d\\mu ", "91ba2c3949b36a029c9faf81cc039f16": "\\langle E_i \\rangle = \\int dX_i\\,\\,\\alpha_i X_i^2\\,\\, p_i(X_i) = \\frac{\\int dX_i\\,\\,\\alpha_i X_i^2\\,\\, e^{-\\frac{\\alpha_i X_i^2}{k_B T}}}{\\int dX_i\\,\\, e^{-\\frac{\\alpha_i X_i^2}{k_B T}}} ", "91ba4e46c96bd25de6f5a149aec1c435": "\\varphi_\\gamma(\\alpha)", "91ba612456c58283b138b5b0821b445a": "\\left [ -\\infin,\\infin \\right ] ", "91baade61df54d9f60ec95665acffc88": "x\\in H_k(M)", "91bac35e4744013d4bc29aeddfd31f95": "L = k V^2 A C_l \\,", "91bb1c67c0e09df1a1f8fffc4719f5d0": " \\mathrm{II} = L \\, \\text{d}u^2 + 2M \\, \\text{d}u \\, \\text{d}v + N \\, \\text{d}v^2 ", "91bb86b48e6ed09c7483189f3bcc148d": "\n\\int_S f\\,d\\mu \\le \\liminf_{n\\to\\infty} \\int_S f_n\\,d\\mu\\,.\n", "91bb951815925e2d4d0bd6e323afac90": "[f_k[f_{k-1}[\\cdots[f_0,\\Delta]\\cdots]]=0", "91bb99e57f54bd85b8d55bf7c86479bd": " i\\hbar\\partial_t\\psi=-\\mu\\vec{\\sigma}\\cdot\\vec{B}\\psi ", "91bb9dc7950da13347b7b8302eae4435": "\\mathrm{succ}(u,v)=\\begin{cases} \n \\mathrm{next}(v,u) & \\mathrm{next}(v,u)\\neq \\mathrm{nil} \\\\\n \\mathrm{first}(v)&\\text{otherwise}.\n \\end{cases}\n", "91bc40def295221889e459444b3e8558": " f(x)= \\begin{cases} 5, & \\text{if } x=1 \\\\ \n -4, & \\text{if } x = -1 \\\\\n 2, & \\text{ otherwise. }\n \\end{cases} ", "91bc4af75ef682d2852db3b466b2fd2e": "R_C", "91bc73fd22440c56599d1ef95355b7a0": "\\hat{N} = \\eta_1 \\log_2 \\eta_1 + \\eta_2 \\log_2 \\eta_2 ", "91bc7f6042cdc86053ed52baabecc651": "T < 10^{-n}", "91bcdb768a3bd95812d66c186df821fe": "G=\\left\\{0,2,4,6,1,3,5,7\\right\\}", "91bcde8fb23621deba8c7ad90f4b209c": "\n\\sup_{t\\ge 0}\\inf_{s\\in\\R}\\Vert v(t)-e^{is}\\phi\\Vert_X<\\epsilon.\n", "91bd2c821d797babc30056faddc7119a": " v'(x) = - \\sin(x)", "91bd33b0d19d85281fd24184f846d4be": "\\tilde{g}_{lk}", "91bd566bedb52aee0b2d328c4db650a0": "\\sigma_{11} = \\sigma_{22}", "91bdb9d9fdd592e510b8ae250d53f574": "\\lambda^a\\,", "91bdcaf50d9fe33f77dcd42c873ba5a2": "{z\\choose n} = 0 \\quad\\text{for}\\quad n < 0.", "91bddeb2e69526e35237d5863169a480": "t = \\frac {I_{sp}}{a} \\ln \\frac {m_0}{m_1}", "91bddf338484ab934db7dd15febb8d99": "\\zeta_K(s) = \\zeta(s) L(s, \\chi)", "91bdea1830046357726007a359c4c472": "\\frac{\\gamma^2 \\theta^2}{1+\\gamma^2 \\theta^2}K_{1/3}^2(\\xi)", "91bdf24eb24ecd811fcafe18bc420620": "d_P(P_1,P_2)= (x_1-x_2)^2-(y_1-y_2)^2", "91bdfbe849449a19e39f2048e92a93a8": "gd_{\\mathbb{Q}}S_4=2", "91be118afa8d530d9653945e9b78e014": "var(x_j) = \\mathbf{\\sum_{k=1}^{P}}\\lambda_{k}\\alpha_{kj}^2", "91be21e536a353bbd419e9fce244b945": "\\mu\\mapsto\\hat{\\mu}", "91be6e178a8083bdac11d764158f6db6": "{p\\over 1-p} = e^{2\\beta JH}", "91be866487953bdafbe4046eef386d82": "B(r)", "91beb1d15ce28afaf7b10d0149a33ee3": "\\begin{array}{cc} \\begin{array}{rrrr} \\\\ \\\\ \\\\ \\\\ j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} & & & & qj & & & \\\\ & & & pj & pk & qk & & \\\\ & & oj & ok & ol & pl & ql & \\\\ & nj & nk & nl & nm & om & pm & qm \\\\ a & b & c & d & e & f & g & h \\\\ \\hline a & o_0 & p_0 & q_0 & r & s & t & u \\\\ n & o & p & q & & & & \\\\ \\end{array} \\end{array}", "91beb50ca46db51bbbfbedde95994438": "\\frac{1}{k T}\\equiv\\beta\\equiv\\frac{d\\log\\left[\\Omega\\left(E\\right)\\right]}{dE}\\,", "91beeb94100b869e7722e78c03307aab": "\\exists_{x\\in X} P(x)", "91bf3edf05e876384394f102c04f1ba5": "\\mu^+\\to e^+ + \\nu_e + \\bar\\nu_\\mu.", "91c01b4a8ead03f04ae43ec7ff76d3d2": "\\sum_{b\\in B} l_b b \\mapsto \\sum_{b \\in B} l_b^\\sigma b", "91c03612a7192c1e031d8573fe2a1279": "\\bar{x}(y)", "91c04a925f73103c195ad00e96453e4b": "f(x)=f_\\text{e}(x) + f_\\text{o}(x)\\, ,", "91c0b8e51fbdfbca3f56116f58bb1ecb": "X \\in C, Y^{*} \\in F", "91c0c5265380cca3ee61ec1896903fff": "\\scriptstyle \\phi(t)=t^2\\,\\log\\log\\frac 1t", "91c10f13a05bebb689739af60d540255": "L(\\hat{y}, y) = I(\\hat{y} \\ne y), \\, ", "91c12c8c7eb8a322d3df0556def18569": "d(H)", "91c18b9c49a2a6726d309bf1a04f09f0": "[\\theta_1,\\theta_2](t)=(\\theta_1 \\otimes \\theta_2 -\\theta_2\\otimes\\theta_1)\\Delta(t).", "91c1dfae1b1d3793389fb085f8719695": " (T_{i(x)}X)^\\omega/(T_{i(x)}X\\cap (T_{i(x)}X)^\\omega), \\quad x\\in X,", "91c20efd34d8559eac97586ba9b13bdc": "1^3 + 3^3 + 6^3 = 244", "91c21c5b0792a0358a7909c53ffe6ce1": "\\scriptstyle {\\mathbf V}", "91c2917ee97cb59dcdfd3211bed30034": "2^kd", "91c2967d7b16729e1201d7810153d9ac": "j\\in\\mathbb{N}", "91c2bb0581ab594f3923e7e332eae7ff": "\\!\\psi", "91c31c40e28d76d28faee97c1ccca4bf": " \\alpha_i=g_{ij}\\xi^i, \\quad F^2=g_{ij}\\xi^i\\xi^j, \\quad E = \\alpha_i\\xi^i - L = \\tfrac{1}{2}F^2. ", "91c32e5c3073095002f65435536fd949": "Q = ((\\text{Rest mass})_{\\text{before}} \\times c^2) - ((\\text{Rest mass})_{\\text{after}} \\times c^2)\\,\\!", "91c35ef56c6f915364dbb58a44ecab9c": "p_y = \\frac{1}{i\\sqrt{2}} \\left( p_1 - p_{-1} \\right) ", "91c3684b273b260cb920a4df985d1e0e": "d(A,B)=\\sum_{x\\in A, y\\in B} \\frac{d^2(x,y)}{|A|+|B|} -\n\\sum_{x,y\\in A} \\frac{d^2(x,y)}{|A|} -\n\\sum_{x,y\\in B} \\frac{d^2(x,y)}{|B|}.", "91c3720176f762c82557a965a77953ec": "T_{w,n}", "91c3737498d158b8557e57ca94c6b444": "P_c=\\frac{M}{\\left(0.34+\\sum G_i\\right)^2}", "91c3aff9ae5ac8f92c335f86df791a6d": "aaa = a,\taab = b,\taac = c,\taba = c,\tabb = a,\tabc = b,\taca = b,\tacb = c,\tacc = a,", "91c3f890826fadbd8d4ba3760c4f62f4": "1 \\; \\mathrm{C}/\\sqrt{4 \\pi \\epsilon_0} = 2997924580 \\; \\mathrm{statC}", "91c44b1897f9166cc2835f4b98cf006c": " \\xi_{ijl}\\geq 0", "91c457701481efccdc9012cf639dab9d": "D=\\frac{1}{2} ct = \\frac{1}{2} \\frac{c \\varphi}{\\omega} = \\frac{c}{4 \\pi f} (N \\pi + \\Delta \\varphi) = \\frac{\\lambda}{4}(N+ \\Delta N)", "91c474acddeedae9e86913ee8bb17ce0": "2^i", "91c4912dc992c103cc005e26042985ea": " E = p c\\,", "91c49d647bbd71bda092eb8e886c131f": "\\sqrt{x} = 1+\\frac{x-1}{1+\\sqrt{x}}", "91c4ad1fa594a502b8ce333d386eb800": "\\mathcal{S} \\subseteq \\mathbb{R}^n", "91c4fd4aaae86b5df6629905ea5ba894": " g(x) \\leq 0", "91c59edfe29427e7e249d4c63e88990d": "2v_1 + 3v_2 -5v_3 + 0v_4 + \\cdots.", "91c5b2bf8e9e132c4946c11e90ccafab": "0 \\le p_{ij} \\le 1", "91c5dee115afff9723bc39acab777cfd": "H_2(\\mathrm{A}_n,\\mathbf{Z})=\\mathbf{Z}/6", "91c5e39fb19e6869457f866d1d3159f7": "|i\\rang", "91c5f31dd29e7586439386a74a0c6fe3": "\\phi = \\frac{\\rho_\\text{matrix} - \\rho_\\text{bulk}}{\\rho_\\text{matrix}-\\rho_\\text{fluid}}", "91c6094e127a5e5896bbbc9857724df0": "L = \\{ww | w \\in \\{a,b\\}^+\\}", "91c6179e26f4350e5072cb0504faea2e": "\\mathbf u \\times (\\mathbf v + \\mathbf w)= \\mathbf u \\times \\mathbf v + \\mathbf u \\times \\mathbf w", "91c61c2662ddb2be0626a488ef4ea8dc": "\\int_0^{\\theta}\\log(1-\\sin x)\\,dx=-2G+2\\text{Cl}_2\\left(\\frac{\\pi}{2}-\\theta\\right)-\\theta\\log 2", "91c67c1e0f5b4f6a37bb4d81525a2a18": "|\\psi\\rangle_B= \\alpha |0\\rangle_B + \\beta|1\\rangle_B", "91c67ef152ab8d1d380323eaa5a0fc05": "G = \\{(k,Tk) : k \\in K)\\}", "91c6dea2e93feef2466550eca4c7d195": "b_k | k > 0", "91c711957fd2aca08e1bf568af11ccf5": "\\sum \\mbox{deg}\\,c_{i_j} = n", "91c749e91e8ffce2ecb234012558980f": "X_0 = X", "91c7ea350ac039e2cca992a45c64c313": "(X_1:Z_1)", "91c7f0380e8b5636fed300307413ef70": "{{1}\\over{2}}nMv^2 = E_\\mathrm{k}", "91c7f81f965d0d27c2bf422d583c3e7e": "b\\cdot x_1,\\dots, b\\cdot x_n", "91c828b5c73ccea3291cea92425c1922": " \\Phi = I \\mathrm{d} \\Omega \\,\\!", "91c833f73b006a253ea09fa860e702f6": "T = T_0", "91c8ac43c31cafd1596ca163ea604a3e": "\\ h ", "91c8d9c965894403f88017273628329e": "x = \\arcsin(\\cos(\\phi)\\sin(\\lambda))\\,", "91c99e8123a3f4c0c5f8ed2ce5447958": "\\textrm{cr}(K_n)=\\frac14\\left\\lfloor\\frac{n}{2}\\right\\rfloor\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor\\left\\lfloor\\frac{n-2}{2}\\right\\rfloor\\left\\lfloor\\frac{n-3}{2}\\right\\rfloor.", "91c99f3050730c9b9af68582086b6832": "\\gamma^0 \\gamma^0 = I_4 \\,", "91ca3514791745208615c47eaf86f712": "\\mathbf{e}_{i} \\mathbf{e}_{j} = \\sum_{k=1}^n c_{i,j,k} \\mathbf{e}_{k}", "91ca46f05dd618b555284eb8f228ca62": " (U_{t})_{t \\in \\mathbb{R}} ", "91ca543e3ed120890e8a792a0952d45a": " P_M = \\text{conv}\\{\\{1,1,0,0\\}, \\{1,0,1,0\\}, \\{1,0,0,1\\}, \\{0,1,1,0\\}, \\{0,1,0,1\\}\\}, ", "91ca9dd450f4782196a6d3c80b78c572": "A_{\\text{v}} \\triangleq \\frac{ v_{\\text{out}} }{ v_{\\text{in}} } = \\frac{ -g_m R_{\\text{C}} }{ g_m R_{\\text{E}}+1 } \\approx -\\frac{ R_{\\text{C}} }{ R_{\\text{E}} } \\qquad (\\text{where} \\quad g_m R_{\\text{E}} \\gg 1). \\,", "91caca39ea86e8d58e681f20375c736b": "\n\\{F,B,U,D,L,R\\}", "91caf6dbf9103f21714efa9ff589e125": "f(-x)", "91cb03b7327f9279fa79de394055b00c": "d^{2^{c' n}}", "91cb9fc7d0709721ddeae2504d29521a": "a^2+b^2-2ab\\cos{\\theta}=c^2, \\,", "91cbb77e4c5e5d0b51dc9e0b7e9738c9": "{\\Delta}^{*}", "91cbff84cf643028e2d71db79c8ad84f": "Q(-v)=Q(v)", "91cc11be00bee41b907ccfd5bf726d5c": "\\mathcal{BPP}\\subsetneq\\mathcal{P}/\\text{poly}", "91cc2bd2744d7f9a8dc2b6055ad01e7b": " Q_\\text{cmb}=Q_\\text{sec,core}+Q_\\text{L}+Q_\\text{G} ", "91cca8bb1c7cdd328a41586d515689b9": "\\sin \\widehat C = \\sqrt{1-\\cos^2 \\widehat C} = \\frac{\\sqrt{4a^2 b^2 -(a^2 +b^2 -c^2)^2 }}{2ab}.", "91ccb3952c2cd95eafbf5f003b48899c": "x = \\frac{\\alpha -1 -\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "91cccf79ba6673ca01961bb05636427f": "\\sum_{n=1}^k \\psi \\left(\\frac{n}{k}\\right)\n =-k(\\gamma+\\log k),", "91ccdbd48bfde3cfbc2c734748fa5e65": "t(1,n)", "91ccdfdde4745c7c724c10bea193b12a": "z\\le 0", "91ccf5388e1d7f12f03c67a3ad47f318": "X_0 = 0", "91cd167bed1daf440ac2544d3e67ed72": "\n\\beta \\,\\, \\approx \\,\\,\\frac{1}{2}\\,\\,a\\,b^2 e^{b\\,\\mu } \\,\\,\\frac{{\\sigma ^2 }}{n}", "91cd2c111b386b8fd8c0263b153b0bf3": "\\{x : A \\ |\\ P(x) \\bullet F(x)\\}", "91cd36b9830cce3f7d3cf41cbc41c1bc": "N_{0}=10000", "91cd5560b9ee2cebe0f690988f0e2ddf": "\\partial_{+}C", "91ce3df64ec3268ddb1a3ff2e0e332ea": "y = y_0 - \\frac{v_{\\infty}^2}{g} \\ln \\cosh\\left(\\frac{gt}{v_\\infty}\\right).", "91ce51740afbd2c52b45ebe335d8fb74": "\\Sigma = iGW - GWGWG + \\cdots", "91ce6000ec2e3f35c0727fa5b2b9bdbe": " \\sqrt{S} \\approx 6 \\cdot 10^2 = 600", "91cf3d4c6f9d8f31c42b442ca97e4668": "\\sqrt[12]{2}", "91cf60626088ed214b505ba324144854": "f_\\omega(f_1(3)) - 2", "91cf74c13e0c2efc538d706057149303": "\\tilde{J_3} = 2.532 \\times 10^{-6}", "91cf7ed251f6ce7ac7f3f61b49e1c034": " fu = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y}", "91cf80298e5929f6c9bae678f1f8eb3d": "_{lex}b", "91cf888ecf74c97c054c208afda5c70d": "\\frac{dI}{dt}", "91cfc006148702555f6c31bff1f24356": "\\textstyle y_\\mathrm {mn}", "91d0250aed342c89d69cbca20911752d": "\\Delta G_v=\\Delta H_v - T \\Delta S_v", "91d033e9e18a793505aa97f1b218d047": "\\aleph_1 \\leq \\kappa \\leq \\mathfrak{c}", "91d0824d493604d662c09684ebed867c": "\\int_a^c f(x)\\,\\mathrm{d}x\\,", "91d0a4df5cd30997a87c1adbab58fae4": "\\nabla\\cdot\\left(\\mathbf{E}\\times\\mathbf{H}\\right) = \\mathbf{H}\\cdot\\nabla\\times\\mathbf{E}- \\mathbf{E}\\cdot\\nabla\\times\\mathbf{H} ", "91d0b11974ae6847208653b6027fc824": "X_0,X_1,\\dots", "91d0d84bb60337411f328b7c97b36b44": "\\delta_z[f] = f(z) = \\frac{1}{2\\pi i} \\oint_{\\partial D} \\frac{f(\\zeta)\\,d\\zeta}{\\zeta-z}.", "91d17c663319ce91140d76d8a9c6bc01": "b_{ij} = s_j-s_i ", "91d21a2b671fae05350b8cbb65facaa4": "T = \\frac{K}{2r}", "91d223027b3aef1a65b47b88335fd4e6": "K=\\frac{\\sqrt{(a^2+b^2+c^2+d^2)^2+8abcd-2(a^4+b^4+c^4+d^4)}}{4}\\cdot", "91d28d4c74553b122a19b37332e63809": "\\sqrt{1.05}*2^{-0.25} m \\times \\sqrt{1.05}*2^{0.25} m \\approx 861.7 mm \\times 1218.6 mm", "91d2a799b96713862b4ac9a86191d3b0": "\\alpha = P_{impact}P_{fission} n_{avg} - P_{absorb} - P_{escape}", "91d2d51d27b19f69639fe6f0855ed890": "J_i", "91d2f274d877291fa7dd63a1650a2442": "\\scriptstyle E(\\exp([\\boldsymbol v]_{\\times}) \\boldsymbol R) \\;\\approx\\; E(\\boldsymbol R) \\,+\\, (\\boldsymbol \\omega(\\boldsymbol R) \\,\\times\\, \\boldsymbol L) \\cdot \\boldsymbol v", "91d31d0c7fbc30a039176da305a4d643": "\\frac{d\\sigma}{ds} = \\left|\\frac{dR}{ds}\\right|.", "91d3468fe138d95fb82aa9869f1795cf": " \\vec{B}=\\nabla\\wedge\\vec{A} ", "91d3483979b629f3ac95468e8265ade2": "S^*L(2,Z)", "91d3aee60458257cceed5ac1ab327b8a": "\\mathfrak{sl}(n)", "91d3c1ee000efeac5bcbbb60370f894c": " R_N =r_O = \\begin{matrix} \\frac {V_A + V_{CB}} {I_C} \\end{matrix} ", "91d460d33d9287692c9f134695e4c762": "\\lambda^2 + (a-1)\\lambda +b = 0.", "91d4837c3e849f464875ab1a05af4db6": "(l^k,r^k)", "91d4bff8dbfe2272108a652873de84e2": "(6)\\quad R_{ab}=8\\pi T_{ab}\\,.", "91d5300934e5499d35d5154bf7c17ebf": "\\overline {V^{\\mathbb C}}", "91d552814b8499310fd3f0c65fbcd176": "{{P}_{V}}[h,g](u,\\xi )=\\int_{-\\infty }^{\\infty }{h(u+\\frac{\\tau }{2}).}{{g}^{*}}(u-\\frac{\\tau }{2}).{{e}^{-i\\tau \\xi }}d\\tau ", "91d5d5bc17f6bb14ad0657c5e77f139a": "\\Omega+1", "91d5e7c83a6b72676d963f98bdf94dfa": "a_k |0\\rangle=0", "91d631778ee8d7806ffdc260c3b0f846": " \\frac{\\partial^2f}{\\partial x_1^2} + \\frac{\\partial^2f}{\\partial x_2^2} + \\cdots + \\frac{\\partial^2f}{\\partial x_n^2} = 0", "91d6d21393a6cf1e1a36c81d9c70f3f7": "A \\cap A^C = \\varnothing\\,\\!", "91d7240bb5b41e7bd44056098e6ee79b": "0 \\to M' \\xrightarrow{f} M \\to M'' \\to 0,\\ ", "91d7538efbe9bd56d35ca7dd700e45c2": " x = y = z = 1", "91d76d5fd074ebb99969f6a52c93bcca": " (1-\\kappa(A)^{-2}). ", "91d7b012f41165ed2e2219dda4b97d2c": "\\scriptstyle\\mathcal{I}_\\theta", "91d842a4548a30f515459444a0a8364d": " = \\frac{1}{2} (\\eta_{\\mu \\nu} + \\eta_{\\nu \\mu}) \\gamma^\\mu \\gamma^\\nu", "91d8548880de1c2d233db7c5bca751d4": "\\sum_{n=0}^\\infty \\, \\frac{(-1)^n}{2n+1} \\;=\\; \\frac{\\pi}{4}.\\!", "91d872e51c12ee4432bdadf14466598f": "X_0=\\{-2, -1, 1, 2\\}", "91d8890545f07d02ec1e15bafb1114b0": "\n\\begin{cases}\n\\dot{x}_1 = x_2\\\\\n\\dot{x}_2 = a(t,x_1,x_2) + u\n\\end{cases}", "91d8947f9447751708c1f4107beb7ab3": "g_i", "91d8ba462c301ccef330b74f772245bd": " H_n(x) = n! \\sum_{m=0}^{\\lfloor n/2 \\rfloor} \\frac{(-1)^m}{m!(n - 2m)!} (2x)^{n - 2m}. ", "91d930dc2577f123246cd686ff70bafd": "a.P + \\Omega", "91d9dda44d8f299e467b6ac158aafc2d": "\\Delta - I.", "91da1ce33a1b54051d0d1848cde4683d": "u+\\sum \\vec{b}(x_i) = 1\\,\\!", "91da3d9c78723e83dbd098875dde7ef2": "\\sum_i \\alpha_{i} A_i \\to \\sum_j \\beta_{j} A_j", "91daa36c2982b6f25a4c50eb9a64811e": "\\displaystyle{A^*A=I,}", "91dabca15320ef86cdd2137430c09770": "\\rho R / \\tau^2", "91daf0bec80a15d8698785b4071f27fb": " s_i = \\sum_{j=1}^n T_{ji} s_j ", "91db60af61604c0dba9c05b56c3cc5fa": "|N(T \\setminus S)| \\leq d(1-2\\varepsilon)\\gamma n\\,", "91db7011428129744764463b3f9e585f": "\\theta_1 = \\theta_{\\rm S} + \\frac{\\theta_E^2}{\\theta_1}", "91dba8fa589fb568139dbb5d90cb066d": "\\text{dV}", "91dbf79ec4a27e35ccbbb2cc7896a9d9": "\\Sigma=S^{1/2} B^{-1} S^{1/2}=S^{1/2}((1/n)I_p)S^{1/2}=S/n,\\,", "91dc16cf9d5367edf43292d616282530": "[M][N]=[M \\times N]", "91dc36020b012a6895f3125580ec1232": "j=1,\\ldots ,l", "91dc7cc1232c331e7c8a41274a5d2958": "s_a(t)\\ \\stackrel{\\mathrm{def}}{=}\\ s(t)+j\\cdot \\widehat s(t),\\,", "91dcd1eb2fc092f982e5e87b8685d280": "a^2 + b^2", "91dcde43b935a48f25ff7cf9eb93a04e": " \\hat{U}_{\\omega} = U - \\frac{1}{2} \\omega^{2} \\phi^{2} ", "91dd21e71ea5f827e9993fb93f536cbc": "\\frac{s\\cos\\phi - \\omega \\sin\\phi}{s^2+\\omega^2} \\ ", "91dd2aa82cda8dc66eb4a8c9c9e140e9": "V \\sub X", "91ddd6e603d26ad5c55c04c4347f8bf3": "X{_i^e} = 1", "91de0203ed5644ac13bb2a06b1379461": "-i \\epsilon^{\\sigma 0 1 3} \\gamma_\\sigma \\gamma^5 = -i\\epsilon^{2 0 1 3}\n(-\\gamma^2) (i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3) = \\epsilon^{2 0 1 3} \\gamma^0 \\gamma^1 \\gamma^3 = \\epsilon^{0 1 2 3} \\gamma^0 \\gamma^1 \\gamma^3", "91de16075db14a8bf2ac713bacb0636a": "\\,\\tau", "91de6f05b82688bb811ba2b6715a32df": "\\omega t < {\\pi -\\alpha} : I(\\omega t) = I_{tcr-max} \\sqrt{2} [-cos(\\alpha)-cos(\\omega t)]", "91de7d148960a5646c1af9928c4f44ac": "\\left ( C \\right ) \\left ( j \\right ) V \\left ( \\begin{Bmatrix}\n C \\\\\n :\n\\end{Bmatrix} \\right ) + Tone", "91de8f5d4918bb61f6ba794b1908d0c6": "e^2 = 0.00669\\,43800\\,22903\\,41574\\,95749\\,48586\\,28930\\,62124\\,43890\\,\\ldots", "91debf8abe21acaf2593b922641431b2": "\\vec{H}_{0}=H_{0}\\hat{z}=H_{0}\\cos\\theta \\hat{r}", "91def2c22f23c5e2a192b09ed4b0994e": "\\vert \\psi \\rangle = \\sum_{k=0}^n Z_k \\vert e_k \\rangle = [Z_0:Z_1:\\ldots:Z_n]", "91def435051253d7e29ecead5f5647a4": " s_i x= x- 2{(x,\\alpha_i)\\over (\\alpha_i,\\alpha_i)}\\alpha_i,", "91df1961cc8f0cb9149b6038d265b538": "Z(K[G]) = \\left\\{ \\sum_{g \\in G} a_g g \\ : \\ \\forall g,h \\in G, a_g = a_{h^{-1}gh}\\right\\}.", "91df288101878ae7453deee926dfef19": " 1_{\\mathbb{Q}} ", "91df428317330f731a9dfa1dd6422f5c": "\\tau_{\\mathrm{sen}}", "91dfb0fb3570b06a7f1506f1c51186c8": "(ab)c=a(bc)", "91dfc4eef97f475ebd8f7ef707521752": "N((a+bi)(c+di)) = N((ac-bd)+i(ad+bc)) = (ac-bd)^2 + (ad+bc)^2. \\,", "91dff0ce336f6cc08f854602607d0f80": "L_{pp, \\gamma-norm} = \\frac{t^{\\gamma}}{2} \\left( L_{xx}+L_{yy} - \\sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2} \\right)", "91e003c4ca22892a2e42059fc2ede573": "[1..r] = bcba", "91e016c2c0d41824b5dc241aa8b60c74": "\\{\\mathcal{L}^*F\\}(s) = \\mathrm{E}\\left[\\mathrm{e}^{-sX}\\right].", "91e039ce63894808a51213b2b8c35d22": "\\zeta_n^r\\equiv\\zeta_n^s\\pmod{\\mathfrak{p}}, \\;\\;0 0", "91e4697ca27ce46023270e40871f6828": " x^3 - 7x^2 + 41x - 87\\, ", "91e49a54b042baa57903e2431f9e4604": "f(3.0, 2.0) = 0.0, \\quad", "91e5355e87222101a43d133780d55c95": "a_{7}+b_{7}+c_{7}=b_{1}", "91e558ab89249793bab54bfe0f9843ae": "\\Lambda \\subset \\Re^n", "91e5b2ce5bdea7b4eaab324c494451e0": "\\mu = np", "91e5c243de9693c9ec0ed657bf60df3e": "MTF_{jitter}(k) =e^{-\\frac {1} {2} k^2\\sigma^2}", "91e5c8f4b3499a2d22e636b74ac3bacb": "*:\\mathbf{B}(b,c)\\times\\mathbf{B}(a,b)\\to\\mathbf{B}(a,c)", "91e5d3d63b474c2a109981f5d12418f0": "u_{i,j} = \\delta_{i}(j)", "91e5e0004cd99449aaf6647b02d4b7de": "(A,a)\\to (B,b)", "91e5ecbc076dbb82cc904756535dbfd7": "\\quad (A \\cdot B) + (\\lnot A \\cdot B)=B", "91e64183b03af5b6884af4896b92c4f7": " \\prod_{j=1}^{\\lceil\\frac{m}{2}\\rceil} \\prod_{k=1}^{\\lceil\\frac{n}{2}\\rceil} \\left ( 4\\cos^2 \\frac{\\pi j}{m + 1} + 4\\cos^2 \\frac{\\pi k}{n + 1} \\right ).", "91e6594c5746ebfde1e9873cb184513e": "Fr > 1", "91e65ffefe21c10bc12da392ddefd3d3": "v_A/c = 7.28\\,\\mu^{-1/2}n_i^{-1/2}B", "91e69693bdc955ca806c65af608522ca": " D = 1 + \\frac{ ( n - 1 ) \\theta }{ 1 + \\theta } ", "91e6b38dc9d1e3fbe5f5590104bc3ba2": "=\\frac{19\\cdot20\\cdot21}{1\\cdot2\\cdot3}.", "91e74b1c053e40dd2800df0c0d20bfa1": "M_{M_3} = M_7 = 127 ", "91e81983e2718323ec6f43847abe4d44": "b=\\lim_{n \\to \\infty} 2\\cdot\\frac{\\mid{P_c^n(c)\\mid\\cdot\\ln\\mid{P_c^n(c)}}\\mid}{\\mid\\frac{\\partial}{\\partial{c}} P_c^n(c)\\mid}", "91e858273e0d6f0eaa9e022f7e392025": "\\lambda(G)", "91e886525b7a21aa4dc5f47c9983d100": "L(F) = 1-(1-F)^{1-\\frac{1}{\\alpha}},", "91e8b5d65b4f00c1fbf9cdbd52de1380": "L_{\\alpha}=\\varepsilon_{\\alpha\\beta\\gamma}n_{\\beta}p_{\\gamma}", "91e8b887676b0dc8bb42925b354d0873": "PV_i", "91e8f8d13be2f5f82e5812c1d5116844": " \\|(x, y)\\|_1 = \\|x\\| + \\|y\\|, \\ \\ \\ \\|(x, y)\\|_\\infty = \\max (\\|x\\|, \\|y\\|) ", "91e9047731eb8a445be2fbcadcfe50c4": "f'(x) = \\frac{1}{2\\sqrt x}.", "91e91a7593ed1f621fa36f3859555a68": "k_B T_c/J = \\frac{2}{\\ln(1+\\sqrt{2})} \\approx 2.26918531421", "91e943789905fa8f4f3d298b4f37a311": "F=ma", "91e94599b5490db1e0d37c071d1fd38d": "\\frac{8}{9} \\sqrt[4]{2}", "91e998ed28a033bc9279555964260189": "\\text{Equivalent stiffness, }k_{eq} = \\int{EI \\bigg(\\frac{{d^2\\bar{u}}}{{dx^2}} \\bigg)^2}dx", "91e9c670ffe7816bc33282da60a877be": "z_1,\\ldots,z_k", "91e9cef2da5ceb8625e93ae7f118dbaf": "B_H (at) \\sim |a|^{H}B_H (t).", "91ea27d34be74f28971e21b1a44cdd84": "G_C \\to 1", "91ea6da09de15ec28c000e3d61f35ef0": "\\scriptstyle x^5 + (11/2) x^3 - 7 x^2 + 9 = 0\n", "91ea8bffa53ac8d50cadb327a67360bf": "(u,w)", "91ead9166094ef6e1adb643d8a571e13": "\\mathbf{} r ", "91eae262962c8a0dbe191a5822a98f4b": "\n \\underset{n\\to\\infty}{\\operatorname{plim}}\\;T_n = \\theta.\n ", "91eae83cadf672584e6ac521391c1570": "\\int_M \\mbox{Pf}(\\Omega)=(2\\pi)^n\\chi(M)\\ ", "91eb1eb8905df9ee178840cd5db4a203": " 100 \\times \\pi", "91eb5a411eb1b23564ed2acac23c960f": "\nr_{c} \\le R(q,u) \\longleftrightarrow \\alpha \\le \\varphi(q,\\alpha,u)\n", "91eba4f6a2a101dff663d21b93844038": "\\int\\frac {dy}{g(y)} = \\int f(x)dx", "91ebb9e74ab4db6c030d1e8f979e68bd": "\\alpha_L", "91ebcc092c7bcee634b4a55d6e9ae1be": "\nv = \\sum_{\\mathrm{cations\\ C}} P_{\\mathrm{C}} \\left[ \\mathrm{C}^{+} \\right]_{\\mathrm{in}} + \n\\sum_{\\mathrm{anions\\ A}} P_{\\mathrm{A}} \\left[ \\mathrm{A}^{-} \\right]_{\\mathrm{out}}\n", "91ebf84f0c642c925cf64731cc83b1e3": "x_n = 1/n", "91ec301a24185e55fb2fc1fbdad5f03a": "R=\n\\begin{pmatrix}\n\\cos \\theta & -\\sin \\theta\\\\\n\\sin \\theta & \\cos \\theta\n\\end{pmatrix},\n", "91ec3dbfba7410071f9949471ebfa5da": "n\\times{}n", "91ec5cfee6bbae3273678d4cc439d499": "d W = (I d \\mathbf{l} \\mathbf{ \\times} \\mathbf{B}) \\cdot d \\mathbf{r}", "91ec9744da07739c38d5f916acf598f6": "\\Phi_M\\,", "91ece3eaba98415348ada05a971860c0": "P(Fa|E) \\ = \\ \\frac{n_F+\\lambda P(Fa)}{n+\\lambda}", "91ecf47f123599ebf8d780a013d8d705": "\\forall^\\infty n\\in{\\mathbb N}", "91ed56d053163be781129fb7bb36e70f": "\\, =10(x+y-10) + (100-10x-10y+xy)", "91ed710455590145a5243a609d4561c8": " S = \\lbrace f_x:F \\to R \\; | \\; f_x(y)=\\delta_{xy} \\rbrace ", "91ed876841114a641d708ce17b3500ff": "R = k[x,y,z]", "91edac932cb9287f191edd474b5b2aa5": "\\eta_\\varepsilon = \\frac{1}{(2\\pi\\varepsilon)^{n/2}}\\mathrm{e}^{-\\frac{x\\cdot x}{2\\varepsilon}},", "91edae137f613877440994428128baae": "\n\\tilde{P}(q,\\omega) \\ \\ = \\ \\ \\frac{2Dq^2}{\\omega^2+(Dq^2)^2}\n", "91ee9efd115334bbd1e2c3d7535dfaa6": "\\sum_{n=s}^t \\ln f(n) = \\ln \\prod_{n=s}^t f!(n)", "91eec77d0d9a798d280dbdb74082e05a": "{\\color{white}.}\\qquad\n\\begin{align}\n\\lambda - \\lambda_0 &= (1-f) \\sin\\alpha_0\n\\int_0^\\sigma\\frac\n{\\sqrt{1 + k^2\\sin^2\\sigma'}}\n{1 - \\cos^2\\alpha_0\\sin^2\\sigma'}\\,d\\sigma'\\\\\n&= \\omega - \\sin\\alpha_0\n\\int_0^\\sigma\\frac\n{e^2}{1 + \\sqrt{1 - e^2\\cos^2\\beta(\\sigma';\\alpha_0)}}\\,d\\sigma'\\\\\n&= \\omega - f\\sin\\alpha_0\n\\int_0^\\sigma\\frac\n{2-f}{1 + (1-f)\\sqrt{1 + k^2\\sin^2\\sigma'}}\n\\,d\\sigma',\n\\end{align}", "91eece1af84a5ef5acb48c2673587c57": "\\psi(\\mathbf{x}) = \\psi_1(x_1) \\cdot \\psi_2(x_2) \\cdot \\psi_3(x_3)", "91ef23703b4b70119e54846b1a0e96de": "v = \\mathbf{f'}\\, \\mathbf{v}[\\mathbf{f'}].", "91ef7e0b8745e7ab759e1824f9aab144": "\n\\zeta_k = a_k + \\frac{1}{\\zeta_{k+1}} = [a_k; \\zeta_{k+1}], \\,\n", "91ef7e88d7047531301f520e34e6b442": "|R_f(\\tau)| \\leq R_f(0)", "91ef9e3360cea320a69620b00427beda": "V = \\frac{1}{3}a^2h = \\frac{\\sqrt{2}}{6}a^3", "91efd56602b5af3c5019164d48e65567": " 0< \\operatorname{P} \\{w: F(v_1, \\ldots, v_{n-1}, w) = \\mathbf{T}\\} \\leq a ", "91efe0ea57cd58c285a7e8eab14a2039": " \\phi :I_1 \\to I_2", "91f00cc068d29b8a52b876c3a504b8ed": "\\hat{v}^i = (R^{-1})^i_j v^j,", "91f03b987cd080a2e698acc779473675": "c_{12}-b_{12}", "91f06856be9d194aaa94e1fb42d199be": "\\ \\mathrm{Lie}_q\\,\\mathcal{F}=T_q M", "91f1175fadad8e7e0cab294c9595c663": "y_1, y_2, \\dots, y_n", "91f12de8c1cfa58ba3a146883e576478": "\\rho_{A}=\\mathrm{Tr}_B(\\rho_{AB})", "91f14e3ee99828eab5e85d86ca1a7ada": "d_i : \\mathrm{sing}_{i+1}(X) \\rightarrow \\mathrm{sing}_i(X) ", "91f14ff972a259861e2510dfe306600c": "n_k = 0", "91f1df5c7fa7a13a18f68191d8f7ca42": "\\left(\\frac{\\sigma_f}{f}\\right)^2 \\approx \\left(\\frac{\\sigma_A}{A}\\right)^2 + \\left(\\frac{\\sigma_B}{B}\\right)^2 - 2\\frac{\\sigma_A\\sigma_B}{AB}\\rho_{AB}", "91f2284ed967ceb50dbd57f46216d688": "\\scriptstyle B(b')", "91f23444af1a658b4417e333938c3377": "\\Theta(n\\log n)", "91f2508fe20658544c5c13dd16d1a171": "f(t) = \\frac{1}{2\\pi}\\left[\\frac{d\\phi}{dt}\\right ]_t = f_0-\\frac{\\Delta f}{2}+\\frac{\\Delta f}{T}t", "91f27410c138ff77d0dffa317c88cdb0": "E(x,y,z,t) = \\sum_{k_x,k_y} A(k_x,k_y) e^{i\\left(k_z z + k_y y + k_x x - \\omega t\\right)}", "91f2c09e883f39df462a50a0a4cb1a8d": "\\operatorname{lcp}(v,w)", "91f2d553b1e332e65f722f835ebf34c1": "\\Delta K = \\frac{1}{2}\\mu v^2_{\\rm rel}(e^2-1)", "91f2d6b55bdfc1daa514b9e79507af59": "(T + mN) \\equiv 0 \\pmod{R}", "91f2f0e08ec66e00b38ae52f36c96036": " \\frac{1}{2}[(\\kappa-1) \\theta~\\sin\\theta - \\{1 - (\\kappa+1) \\ln r\\} ~\\cos\\theta]\\, ", "91f32597fc5dd90846b891babe5f41e4": " \\det \\Bigl(\\begin{array}{cc} a+ib & id+c \\\\ id-c & a-ib \\end{array}\\Bigr) = a^2 + b^2 + c^2 + d^2,", "91f38c539c44b30e39ac859fed7b613d": "6^5 \\times 3 + 6^4 \\times 0 + 6^3 \\times 1 + 6^2 \\times 3 + 6^1 \\times 3 + 6^0 \\times 1 = 23671", "91f3aedde4ff0612447b5a6effb938e4": "P(\\gamma x) = P(x)", "91f3bd70ffb6160be7a465d5f67aefa2": "g_{00} \\,=\\, 1", "91f4f5fde06448561332b0f36c5a0973": "\\neg A", "91f5382d69a771cb0bd8bd9c8120a5b2": "\\{a\\rightarrow b, b\\rightarrow a, a\\rightarrow x, b\\rightarrow y\\}", "91f598a36ff4cccc6cb89d3ba665c6e0": "f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(k)}(x_0)}{k!}(x-x_0)^{k}+\\frac{R_{k+1}(x)}{(k+1)!}(x-x_0)^{k+1}", "91f59a5f97f9311665ffa50079082321": " M(s) = \\frac{G(s)}{(1+G(s))} ", "91f5dc5df91bac14c91575c2f0b0d67f": "{{N}\\over{N_0}} = {{e^{- \\nu /0.6952T}}}", "91f686813df40aeec7ddab4c52869ecf": "10_{42}", "91f69d18af75c25cee011288c31543d5": " k = 1,2,3", "91f6c0f03bc29e7533e62dffa41128bc": "(\\mathbf Z/p\\mathbf Z)^\\times", "91f6f11b30aae41f0af9b65919ad22e9": "R_M = (M - Q_2) - 2Q_1", "91f732a9d87b8663e5f7a185d185cd07": "\\scriptstyle c_i \\;=\\; x_i", "91f7341281e78ff5f08d6355c90df4fb": "r = \\frac{1}{R},\\ \\theta=\\Theta.", "91f762cb6c9882e5c05ddfe8a9f7b0d1": "\\epsilon_\\mathrm{thermal}", "91f77275355956ed265c2a0e49dbc2f7": "f(t)dt=S(t)-S(t+dt)", "91f77adc9cf2e24530adc632113ffb98": "\\sum_{\\Pi\\succeq\\Lambda}\\tilde{c}(\\Pi)=\\begin{cases} 1,\\text{ if } \\left| \\Lambda \\right|=k \\\\ 0, \\text{ otherwise }. \\end{cases}", "91f7c378b6ea9796a7e553cd572d4be0": "\\alpha=\\alpha_{0}", "91f89a09a02fbf73ee19fe04fccb5a98": "|n|_{\\ast}<1", "91f8eb4c4adb9a8003fb6917f21f8fe4": "\\mathbf r_{12}", "91f970ee58f0b812fb3af2003ce31f07": "\\frac{1}{x^2+1}, y=0", "91f9cd77b4ebfee73db1f02153c85be6": "\\frac{1}{h^2}=\\frac{1}{a^2}+\\frac{1}{b^2}+\\frac{1}{c^2}.", "91fa6a7a64f77bdbb10c103cebcaf382": "\\{m\\{nx\\}\\} = \\{mnx\\}.\\,\\!", "91faa98ea2ea28945d73e5de1a94f7cb": "\\frac{PV\\beta}{N}=\n-\\frac{\\Omega}{N}\\,", "91fb267daa601a8ed1f8093d754c743a": "X_0,Y_0,Z_0", "91fbaf18abf678ec8913c195e795e5f4": "\\frac{V_{1}^2}{2}+{h_{1}} = \\frac{V_{2}^2}{2}+{h_{2}}", "91fc30ea7e8e5d952fad7b354c49f238": "C= \\sum\\ V = 2 \\cdot\\ V_0", "91fc7fd528403a7fe76827285787b90c": "\\frac{X+1}{2}", "91fd05174d22dd8d6ec9f27a95db296d": "\\textstyle \\sigma_0, \\sigma_1, \\ldots", "91fd16f1ed14ae5db246e91528c810fc": "\\text{b. For unity power factor synchronous motors}", "91fd5cedd5ab6c125782993ec0f800d0": " \\Psi = e^{-i{E t/\\hbar}}\\prod_{n=1}^N\\psi(x_n) \\, , \\quad V(x_1,x_2,\\cdots x_N) = \\sum_{n=1}^N V(x_n) \\, .", "91fda0d0191bda14080324f9e5729311": "s^\\mathfrak{s}(t)", "91fdc0bcfa16bab9ba76b1c44257aac3": "f(x|\\mu,b) = \\frac{1}{2b} \\exp \\left( -\\frac{|x-\\mu|}{b} \\right) \\,\\!", "91fde58cb20e1bd996fbde38e913094e": "\\operatorname{wnchypg}(x-1;n-1,m_1,m_2,\\omega) \\frac{(m_1-x+1)\\omega}{(m_1-x+1)\\omega+m_2+x-n} + ", "91fdf52e66da376b27aeaa755824a514": "x, x'", "91fe33144595b06902c79906fda4a9a1": "\\frac{4!\\times 3^6}{2}", "91fe5528e533403641189597b9abab13": "(2^n)!", "91fe6f79ef05de458f95254d8f728adb": "\\tilde{E}^a_i", "91fef6f199ecb15ddda668e287062c1b": "\\mathbf{C}_a=\\frac{\\mathbf{C}_t\\times\\mathbf{V}_t\\times\\mathbf{M}}{\\mathbf{V}_a}", "91ff07bad408ce7f05f463d62f998b92": "m = \\sqrt{\\frac{A}{16\\pi}},", "91ff4a2a0e176650737cc73ae8f3efc3": "\\{S_n\\}_{n=0}^{\\infty}", "91ff5c402b308cdde5af57675410817b": "\\gamma_k \\gamma_0", "91ff841b26760d7b2e78178aa6bb9fe7": "\\epsilon_1 = -1", "91ffab57c15d362772dc410f1a373489": "\\theta = \\pm \\pi / 2", "92001b996cd10d8c09c30522bd5ea1df": " E_{-}\n", "9200258057474e1f58c0727dc9bfb1f3": " \\begin{align}\n E [X(t) | X(t-1) = i] &= (i-1)P_{i,i-1} + iP_{i,i} + (i+1)P_{i,i+1}\\\\\n&= 2ip(1-p) + i(p^2 + (1-p)^2) \\\\\n&= i.\n\\end{align}", "92006c5af4a85c0d50a005ec09d493f3": "\n\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|} = \\sum_{\\ell=0}^\\infty \\frac{4\\pi}{2\\ell+1} \n\\sum_{m=-\\ell}^{\\ell}\n(-1)^m \\frac{r_{{\\scriptscriptstyle<}}^\\ell }{r_{{\\scriptscriptstyle>}}^{\\ell+1} } Y^{-m}_\\ell(\\theta, \\varphi) Y^{m}_\\ell(\\theta', \\varphi').\n", "9200ce6ed8e00f9985440d5da34fff6d": " I(z) \\propto |\\mathbf{E}_0 e^{i(k z - \\omega t)}|^2 = |\\mathbf{E}_0|^2 ", "9200df5a1aa7b5c289904989026b1040": "S_{t}= \\sqrt{\\frac{1}{t} \\sum_{i=1}^{t}\\left ( X_{i} - u \\right )^{2}} \\text{ for } t=1,2, \\dots ,n \\, ", "9201423e86d9bd4959994d7121d9ab3b": "\\int_{\\sigma(T)} g \\, d \\mu_h = \\langle h, g(T) h \\rangle.", "9201da33a1413286f41d596c4075cbdd": "\\frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = \\frac{\\frac{f(x+h) - f(x)}{h} - \\frac{f(x) - f(x-h)}{h}}{h}.", "92021db91eea37b0c63ba5897a75d4df": " S = (1/2) \\int (H^2-E^2) dV + \\int \\rho(\\phi - \\vec{A}\\vec{u}) dV ", "92024f394e9cc3acdc6297a374658023": "\\chi_g= \\frac{C L (R_s-R_0)}{m}", "920286b81e60bc78200721b25ae308b4": "\\mu_S", "920296429e63cb89db8fcef9a629d046": "\n \\big|\\ln f(x\\,|\\,\\theta)\\big| < D(x) \\quad \\text{for all}\\ \\theta\\in\\Theta.\n ", "92029ae5206b33ca9b86111820280d85": "\\{(z,w)\\in \\mathbf{C}^2;~|z|^2+|w|^2<1\\}", "92030be7dfd954e1f256de1864a6cb5e": "\\operatorname{Tr}_B : T(H_A \\otimes H_B) \\rightarrow T(H_A)", "920319a0c2d48a7caad657efcaa68ad0": "W \\to \\mathbb{R}^1, (a,b) \\mapsto (a):", "92038619ad2fd6f8da686d6fb0a9c659": "F_{ax} = F_{eq} = 1", "9203867673c7b51497828c42c5b6c1ed": "\\begin{align}I & = \\int _0^\\infty \\frac{1}{\\sqrt{(x^2 + a^2) (x^2 + b^2)}} \\, dx \\\\\n & = \\int _{- \\infty}^\\infty \\frac{1}{2 \\sqrt{\\left( t^2 + \\left( \\frac{a + b}{2}\\right)^2 \\right) (t^2 + a b)}} \\, dt \\\\\n & = \\int _0^\\infty\\frac{1}{\\sqrt{\\left( t^2 + \\left( \\frac{a + b}{2}\\right)^2\\right) \\left(t^2 + \\left(\\sqrt{a b}\\right)^2\\right)}} \\, dt \\end{align}", "9203cae8da68777c16106f58db9e3043": "f(x):= \\begin{cases} -(x+1)/2, & \\mbox{if } x \\mbox{ is odd} \\\\ x/2, & \\mbox{if } x \\mbox{ is even}. \\end{cases} ", "9203cb8d154bbb369a952ffc2516a1dd": "ud | a,b,c", "9203cce3facd8a9e1e243a3b979df3bc": "= \\frac{\\sin \\alpha \\cos \\beta + \\cos \\alpha \\sin \\beta}{\\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta}\\,", "9203f409a9d6d67ad37a6ff5857fc352": "\\int_\\mathbb{R} g(x)\\,dF_n(x) \\quad\\xrightarrow[n\\to\\infty]{}\\quad \\int_\\mathbb{R} g(x)\\,dF(x)", "9203ff2cfc74d091671d7e612b6fd9ed": "(y_i ,\\alpha_i)^N _{i=1}", "9204170ed9098705f335c52d85c5427e": " F_fr\\ ", "920424640b818459e73dbb6507d76ad1": "\\exp(-4\\pi^2 t |\\xi|^2)", "92047045ab8945a18c4be097e12bc180": " n = ( t / D )^2 ( m + k ) / ( mk ) ", "920494a379b462dfb4bb4b9fbda91192": "~\\gamma=5~", "9204d76b8087f25ac75a1fdc3d6d9df2": "H_\\mathrm{D}=D_{IS}(\\theta)[2I_zS_z-(I_xS_x+I_yS_y)]\\!", "920515a6ebe506173a2cc10aef444aa1": "|c_i|u(x_i)", "920546341e79ca0dbcab781e7e34c89e": "16 \\tfrac 2 3 \\,", "9205512c444c2269187d2c984c072501": "\\mathbf{a} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\\\ a_4 \\end{pmatrix}.", "92055253384f096bec9d2ccf89f35245": "\\Delta \\phi = \\frac{4\\pi h \\cos i}{\\lambda} \\;", "9205a82a7881ffaae704a07e548b2fc6": "\\mathbf{a_{1}}", "9205d5a570aa20f14339552a9fba7199": " tan(\\phi) = \\frac {\\Delta D}{\\Delta L} = \\frac {C_D}{C_L}", "9205ebf2ad9f1b2d02059883fec3fb0a": "\\tau_2=L_D/(-w)", "92060adecaff777bc312c2dc46f1fd43": "P=(X_1 : Y_1 : Z_1)", "92062fa9e9454523fda48ee55b99bc7b": "\\textstyle P(x)", "9206363a64122755a600861341db6c4b": "\n{n \\choose k} + {n \\choose k-1} = {n + 1 \\choose k} \\quad\\text{for } 1 \\le k \\le n + 1\n", "920638a159bc90228adbebf8d6d3959d": "\\left \\langle N,e \\right \\rangle ", "92063a58fea67f52667ebe3e6c318aa4": " x = (\\mu_i - \\mu_i^{0})RT ", "92063fadc6ac5a85f8ed71d177258313": "\\mathbf{r}\\times\\mathbf{v}", "9206548e03a27ad54ea5814e4dc0a6d6": "{\\widehat{VV}}_3", "9206ba5bd8e0d4c73556e112f6cb07d0": "\\dot\\theta_c(t) = \\omega_c(t) =\n\\omega_c + g_v g(t)\\,", "9206fff2bc5a0a0d4c1401a8d784ec6a": "0 = J_\\mathrm{drift} + J_\\mathrm{diffusion} = -\\rho(x) \\mu \\frac{dU}{dx} + \\frac{D}{k_B T} \\frac{dU}{dx}\\rho(x) = -\\rho(x)\\frac{dU}{dx} \\left(\\mu - \\frac{D}{k_B T} \\right)", "920700bfc50dcb9278be7594df2662cb": "\\frac{I_{t}}{K_{t}^{*}}", "9207a3e08544787b5b87295612cd37bb": " \\nabla \\cdot (\\psi\\mathbf{A}) = \\mathbf{A} \\cdot\\nabla\\psi + \\psi\\nabla \\cdot \\mathbf{A} ", "9208ca819e9e3983cc1148fcf3d814f3": "\n\\begin{align}\n& {} \\qquad 152 q^{22} + 3,472 q^{21} + 38,791 q^{20} + 293,021 q^{19} + 1,370,892 q^{18} \\\\\n& {} + 4,067,059 q^{17} + 7,964,012 q^{16} + 11,159,003 q^{15} + 11,808,808 q^{14} \\\\\n& {} + 9,859,915 q^{13} + 6,778,956 q^{12} + 3,964,369 q^{11} + 2,015,441 q^{10} \\\\\n& {} + 906,567 q^{9} + 363,611 q^{8} + 129,820 q^{7} + 41,239 q^{6} + 11,426 q^5 \\\\\n& {} + 2,677 q^4 + 492 q^3 + 61 q^2 + 3 q\n\\end{align}\n", "92094ce5b3e737fb4dbb964e32f49052": "n = -\\sqrt4\\frac{\\ln\\left[\\left(\\ln\\underbrace{\\sqrt{\\sqrt{\\cdots\\sqrt4}}}_{n}\\right) / \\ln4\\right]}{\\ln{4}}", "920955921832b0a1fd833b87abbde606": "\\textbf{S}", "920968b2f67c5e8a2844e4b3fae0fd33": " \\pi r^2", "920a7a89198383e9ceb551c9865b30df": "R_\\text{merge}", "920a807f46eabfeefdc37dfd44442add": "y=pq.", "920a9fc90c14a07de79cd6511947d1ba": "m_0.m_{n-k+2}...m_n < m_1...m_k", "920aabc3964c950096004a9322bbe25c": "\n\\frac{M_1}{M_2}<0.8\n", "920aac84d5d2d6e7b13b6104bb646fdc": "\\hat{n} = \\frac\n {\\left( P_2 - P_1 \\right) \\times \\left(P_3-P_1\\right)}\n {\\left| \\left( P_2 - P_1 \\right) \\times \\left(P_3-P_1\\right) \\right|}.", "920ac981df80a6d9096cc9e480ef789c": "P_\\mbox{non}", "920ae461230742c1372b012181d40fc0": "\n\\bar \\sigma_N = \\sigma_0 \\left[ \\left( \\frac{l_0} D \\right)^{r n_d/m}\n +\\ \\frac{r l_0}D\\ \\right]^{1/r}\n", "920b2827051be17b32f9923f11aa92e4": "P \\to \\neg \\neg P", "920b58f0ad657f41f45bd833617e9a7f": " h = t_n - t_{n-1}, \\qquad n=1,2,\\ldots,N. ", "920b5d845eae396aee7bb4fe1cbe8dd0": "T = m\\dot x^2 + m\\dot y^2 + mR^2\\dot\\alpha^2 + 2R d \\cos\\alpha \\dot x \\dot \\alpha + 2R d \\sin\\alpha \\dot y \\dot \\alpha", "920bb7adbc40596bb43d17f7885af77f": "\\beta : \\mathfrak{g}\\times\\mathfrak{g}\\to k", "920bd849a814377b8162cb3409eb660b": "\\sin(\\tfrac{2\\pi nx}{P}+\\phi_n) \\equiv \\text{Re}\\left\\{\\frac{1}{i}\\cdot e^{i \\left(\\tfrac{2\\pi nx}{P}+\\phi_n\\right)}\\right\\} = \\frac{1}{2i}\\cdot e^{i \\left(\\tfrac{2\\pi nx}{P}+\\phi_n\\right)} +\\left(\\frac{1}{2i}\\cdot e^{i \\left(\\tfrac{2\\pi nx}{P}+\\phi_n\\right)}\\right)^*,", "920be10f708fc541ea0f2cf45e241fbf": "\\mathrm{H}(p, q) = \\mathrm{E}_p[-\\log q] = \\mathrm{H}(p) + D_{\\mathrm{KL}}(p \\| q),\\!", "920c46a47abc55e99ad04da27d0dae7d": "\\Gamma_{\\beta+1} [n+1] = \\varphi_{\\Gamma_{\\beta+1} [n]} (0) \\,.", "920c4ba5eeb8c8432756f7496f2b478a": "\n\\{D\\} = [d^t]\\{T\\}+\\left [ \\varepsilon^T \\right ] \\{E\\}\n", "920c6da0654a957948a04c8afa40402c": " D_L = \\sqrt{\\frac{L}{4\\pi F}}", "920c7e037cfcb4b39e643f6028d75286": " B(f)B(g)-B(g)B(f)=2i\\mathrm{Im}\\langle f,g\\rangle. \\,", "920ca6c78aff6736e5b88b605ae61402": "V_S", "920cc8c14eb6f9037109f622b04a0d63": "= \\frac {4}{3} \\sqrt{\\pi}\\,", "920cd9f97ac894f97d58277b2164fd55": "x_{\\mathrm{i}}", "920d03a69a67c7ce2f845a1b1df8fc47": "\\text{(5)} \\qquad \n \\frac{d\\sigma_e}{d\\varepsilon_{\\rm{p}}} = \\theta(\\sigma_e)\n", "920d25bcf1fb87881955e4ab6a619226": "(6)\\; % \\text{ efficiency}=0.55*100=55%\\text{ efficient}\\,", "920dcbf79560b98d8f675c8f47bd320e": "{x_1, ... ,x_n}", "920dd8c80a973069463809a0000a1260": "\n\\begin{align}\n \\Phi(x,y,z,t) & = + \\frac{1}{|k|} \\text{e}^{+|k|z}\\, \n \\omega a\\, \\sin\\, \\theta,\n \\\\\n \\Phi'(x,y,z,t)& = - \\frac{1}{|k|} \\text{e}^{-|k|z}\\, \n \\omega a\\, \\sin\\, \\theta.\n\\end{align}\n", "920ddc80b726adb00a52adb8197d69ce": " \\sigma(X + c) = \\sigma(X), \\, ", "920e1d58503d4282b56070db10ec70a5": "\\tau_{xy} = F/A \\,", "920e2d2b7ad07b72f29ab9689077f748": "B_{p,r}(z):=\\sum_{m=0}^\\infty A_m(p,r)z^m", "920e4891bcf6b0d4eb8b952bf1c0ab6f": "f\\in \\mathcal{L}^1(\\mu)", "920e98d7a99b2b04f41958805bf04552": "D_tV=\\nabla_{\\dot\\gamma(t)}V.", "920f2a43aa92cc957a08229cd1b840c1": "F_z=F_o \\cos(\\omega t), \\; z=z_o \\cos(\\omega t + \\theta)\\,\\!", "920f544a642ee6658853ef16357c90ff": "X_1, X_2, X_3, \\dots", "920f85c0f6bccfa9333b853b8882708e": "A = f_{i_1,i_2\\cdots i_k}e^{i_1}\\wedge e^{i_2}\\wedge\\cdots\\wedge e^{i_k}", "921012c40277760a97df7c77c927910b": "\nP = \\overline{\\begin{bmatrix} \\frac{di_1}{dt} \\frac{di_2}{dt} \\end{bmatrix}\n\\mathbf{D}\n\\begin{bmatrix} \\frac{di_1}{dt} \\\\ \\frac{di_2}{dt} \\end{bmatrix}}\n", "92101b594b14df776f777c8b8c4d54c6": "\\begin{bmatrix} X_{1(1)} \\\\ \\vdots \\\\ X_{1(k)} \\end{bmatrix}+\\begin{bmatrix} X_{2(1)} \\\\ \\vdots \\\\ X_{2(k)} \\end{bmatrix}+\\cdots+\\begin{bmatrix} X_{n(1)} \\\\ \\vdots \\\\ X_{n(k)} \\end{bmatrix} = \\begin{bmatrix} \\sum_{i=1}^{n} \\left [ X_{i(1)} \\right ] \\\\ \\vdots \\\\ \\sum_{i=1}^{n} \\left [ X_{i(k)} \\right ] \\end{bmatrix} = \\sum_{i=1}^{n} \\left [ \\mathbf{X_i} \\right ]", "92105dc62235484f4234e8d0b0036e8b": "\n\\left( \\frac{\\partial P_{m}}{\\partial q_{n}}\\right)_{\\mathbf{q}, \\mathbf{p}} = -\\left( \\frac{\\partial p_{n}}{\\partial Q_{m}}\\right)_{\\mathbf{Q}, \\mathbf{P}}\n", "92109cb167b0002bab041f2ddf818b69": "m=p", "9210a874a75bb6eb4e75614339c183d9": "\\boldsymbol\\eta_1 = -\\frac12\\mathbf{V}^{-1},", "9210c6cf023c0074416926761fe98649": " u_7 =0.90645 ", "9210d714abb1b597909712e648e5d291": "dU = \\delta Q_{rev} - \\delta W_{rev} ,", "92112ae170171462431d263a4e22894d": "(\\bullet)^T", "92113ff80960be866ef41486ece541fd": " \n\\mathbf{A} =\n\\begin{bmatrix}\n\\mathbf{A}_{1,1} & \\mathbf{A}_{1,2} \\\\\n\\mathbf{A}_{2,1} & \\mathbf{A}_{2,2}\n\\end{bmatrix}\n\\mbox { , }\n\\mathbf{B} =\n\\begin{bmatrix}\n\\mathbf{B}_{1,1} & \\mathbf{B}_{1,2} \\\\\n\\mathbf{B}_{2,1} & \\mathbf{B}_{2,2}\n\\end{bmatrix}\n\\mbox { , }\n\\mathbf{C} =\n\\begin{bmatrix}\n\\mathbf{C}_{1,1} & \\mathbf{C}_{1,2} \\\\\n\\mathbf{C}_{2,1} & \\mathbf{C}_{2,2}\n\\end{bmatrix}\n", "921162a76c9a79eb99998cca0a681a0a": " t-t_0 = \\log(x/x_0) ", "9211abe837d06c5ceba559d446b65545": "\\{\\alpha\\}", "92124ffb893ebee5d3affa33f7b713b1": " \\sum_n | \\langle e_n, x \\rangle |^2 \\leq \\| x \\|^2", "921290e998635459070db5189876a3a2": " \\mathbf{j} = {1 \\over 2m} \\left( \\Psi^*\\hat{\\mathbf{p}}\\Psi - \\Psi\\hat{\\mathbf{p}}\\Psi^* \\right)\\,\\!", "9212921365049ac23dfd9887500a2f19": "\\theta(n)\\geqslant\\left(\\frac{\\alpha}{c+v}\\right)\\left(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}\\right)\\,\\!", "9212c70ef1343e0cee105f4dd24bcc78": "\\vec{e}\\,^T \\vec{e}\\!", "92138bff89923f4e122e9739cf1b5116": "(r^2-a^2+b^2+c^2)^2 = 4b^2(r^2\\cos^2\\theta+c^2) \\, ", "9213a8951ae8e2e9510925aed79e7b27": "{dg_1\\over dt}={dg_2\\over dt}=0.", "92141569edada02a4b546d45cca2647a": "f(x;q)=D_xp(x;q)", "921498e54021ec6b2240fc8c02ff8aab": "\\mathbf{J} = \\int_{t_1}^{t_2} \\mathbf{F}\\, dt", "9214a9cd1861eb12c5cb5e1556ebb949": " \\tau = F^{-1}( 0.95 ) ", "92150603c5757f5f452eb7e75f9f51d9": "nD", "92151e71770d882ac62013ee6303c008": " V(t)=V(0)+\\int_{0}^{t}f_{t}(x^{\\ast }(s),s)ds. ", "9215abcbff2fcda9550c07a8703003a1": "x(i-S)-x(i-R)-cy(i-1)<0", "92162d49a43c634308535460dcee80f5": "\\omega = \\sqrt{\\frac{k}{\\mu}} ", "92171ef8b73169b51c85f87fa8a71a80": "X \\equiv a \\pmod{p}", "92179bb6d30ab2653261a97104b12fb7": "\n \\boldsymbol{S} = \\boldsymbol{F}^{-1}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{F}} \\qquad \\text{or} \\qquad S_{IJ} = F^{-1}_{Ik}\\frac{\\partial W}{\\partial F_{kJ}} ~.\n ", "9217bc1b88219d460c99139b2599bccc": "\n\\begin{array}{rcl}\n\\frac{dx}{dt}&=&a(x,y,z)\\\\\n\\frac{dy}{dt}&=&b(x,y,z)\\\\\n\\frac{dz}{dt}&=&c(x,y,z).\n\\end{array}\n", "921812a8581d0b9eff3c2abf0882fabe": "3x + 5y = 8.", "92186afd333d7ecdce6cfc0da7c895dd": "i_1 = 1\\,", "9218781ca0b157ae0baa08ef096d5ab7": "M(t_0,t_1)", "9218c35b045052abde762370df57c1c0": "\n{\\text{the}\\atop {NP/N,}}\n{\\text{bad}\\atop {N/N,}}\n{\\text{boy}\\atop {N,}}\n{\\text{made}\\atop {(NP\\backslash S)/NP,}}\n{\\text{that}\\atop {NP/N,}}\n{\\text{mess}\\atop {N}}\n", "9218fb299f81efaa0cf9d06ae5b4638d": "\\left|\\tfrac{\\omega(p)}p\\right|\\leq 10^{-14}", "92195eb9e29778fe8fa5bf0986198d46": "\\eta = \\text{d} \\ln T / \\text{d} \\ln n", "9219fcedf05de913d69365797d9c40f9": "{\\bar{\\Omega}}", "921a374df7273b457413d41733208bcd": "\\textstyle (1-p^2)^{N-2}", "921aab46606c08c7b2ef1316648bdc9a": "\n \\displaystyle \n V_{LJ} ( r_c )\n =\n V_{LJ} ( 2.5 \\sigma )\n =\n 4 \\varepsilon\n \\left[\n \\left(\n \\frac\n\t {\\sigma}\n\t {2.5 \\sigma}\n \\right)^{12}\n -\n \\left(\n \\frac\n\t {\\sigma}\n\t {2.5 \\sigma}\n \\right)^6\n \\right]\n \\approx\n -0.0163 \\varepsilon\n", "921ad3370387241d9e41e6b1c00c6ece": "\\delta \\alpha", "921af366838a151849f594be3afe2c89": "H_S", "921b4a09e53c9c3ba3b66c264a9e692e": " \\boldsymbol{\\omega} = (\\omega_1, \\ldots, \\omega_n)", "921b5c6b449ea0f5aa4ce9e68faa8452": "\n\nS_1 \\approx GM_1^2/c.\n\n", "921b734ab5175813c1256e31fd5a8df8": "K_a = \\mathrm{\\frac{[A][B]}{[AB]}}", "921b99acc3d1267654be1c41054e38a0": "y + 1", "921ba210fa3349fce7985027339611ef": "x = \\sum_{i=1}^n a_i \\otimes b_i", "921bef47bf29afc88b48db0f3763bdf9": "R_1 = 1\\ \\mathrm{k \\Omega}", "921c1302600d5283ae495795aa2ef060": "(n\\,n-1)(n-1\\,n-2)\\cdots(2\\,1)(n-1\\,n-2)(n-2\\,n-3)\\cdots,", "921c18b5d22a03b211e7fc1491794bfe": "\\lambda \\ne 0", "921c4974643f8cc9756a9fbc736da371": "G_{2} =\n \\begin{bmatrix} 1 & 0 & 0 \\\\\n 0 & c & -s \\\\\n 0 & s & c \\\\\n \\end{bmatrix}", "921c5811ba40f46c2fbcf4a16b70e90b": "f(x)=\\int_I\\frac{g(t)-g(x)}{t-x}\\rho(t)dt \\Leftrightarrow g(x)=(x-c_1)f(x)-T_{\\mu}(f(x))=\\frac{\\varphi(x)\\mu(x)}{\\rho(x)}f(x)-T_{\\rho} \\left(\\frac{\\mu(x)}{\\rho(x)}f(x)\\right)", "921c8a5c2912feaf962ba749248c78fe": "(\\exists x \\phi ) \\rightarrow \\psi", "921ccce6c11ae2ec51bab73a97e6038b": "a=\\det \\begin{pmatrix} z_1w_1 & w_1 & 1 \\\\ z_2w_2 & w_2 & 1 \\\\ z_3w_3 & w_3 & 1 \\end{pmatrix}\\, ", "921d077d4e50e3d899b57c41ed3ee98b": "\\left\\{ \\textbf{p}+\\textbf{v} : \\textbf{v}\\text{ is any solution to }A\\textbf{x}=\\textbf{0} \\right\\}.", "921d2a739442f212e144f4075accb29c": " C(i,p,q) = (x-\\overline{x_i})^p (y-\\overline{y_i})^q ", "921df2f71dbcdfe4db97b8706f42b89f": "\\sum_{i=1}^n c_i'(x) y_i^{(n-1)}(x) = b(x).\\quad\\quad {\\rm (vii)}", "921e01c76c65f14c7cce48b49fc54428": "M^d=P*L(R,Y) \\,", "921e039474339a8dd87c08503129f8a2": "\\rho_t\\,", "921e16645fee6f32e71d102a13c50831": "\nF = m \\cdot a\n", "921e1e5596f70a10e46847d703f81956": " P_1^{-1}A_1P_1=\\begin{bmatrix}0 & B_2 \\\\ 0 & A_2 \\end{bmatrix}", "921e68b91652fc9aacd948843e76ffa0": "\\frac{2}{10}=\\frac{1}{5},", "921ece9a5b19dd78abfd21550c62548d": " F(x,y,z,a)=0,\\,\\,{F(x,y,z,a^\\prime)-F(x,y,z,a)\\over a^\\prime -a}=0.", "921ed0068afbc6283a860fa20d39450b": "\nA \\bar\\psi F^{\\mu \\nu} \\sigma_{\\mu \\nu} \\psi\n\\,", "921ed41c2c5cf504f2a92b4000bc0494": " t_2", "921f2741c946f46ac2245795696fb304": "q(\\xi)=\\xi^2+a_0\\xi+b_0", "92200067546b58660134e58b893e33df": "G_\\mathbb{Q}", "92201224e59243e1fd595774d977cccd": "|v|mp+x \\right ) \\leq \\exp \\left (-\\frac{x^2}{2mp(1-p)} \\right ) .", "92277ad981346c79386ac027d9b50724": " \\scriptstyle\\text{partial}(f) \\colon (Y \\times Z) \\to N ", "922812621b52722670f90acfd00c36be": "b_0 \\centerdot 0 + b_1 \\centerdot 1 + b_2 \\centerdot 2 + b_3 \\centerdot 3 = 0", "92283f28847cdc088ee9ad6396e5cc4c": "d(x_n, x) < \\epsilon", "9228916cf34d3e235376b612b6d4ade3": "\\zeta\\left(\\frac{1}{2}+ig_n\\right) = \\cos(\\theta(g_n))Z(g_n) = (-1)^n Z(g_n),", "92289b92018873a30c96cd0fc46840e3": "r(u,v) = c(u,v) - f(u,v)", "9228a976bd673f0d848300d7c821f344": "EG \\times M", "9228aedc18d5537ca0e943f54d166c4a": " x_{3} ", "92298d214507ac97e0385400cf5c0a00": "y_2=\\left.\\frac{\\partial y_{b}}{\\partial c}\\right|_{c=\\alpha}.", "922a30bdf3ebdc9135c0398f57e5dd1c": "A_n\\left(\\frac\\pi2\\right)=P_n\\left(\\frac{\\pi^2}4\\right).", "922a36fd0a4541d7933b654260906a6c": "\\Sigma X_i = n\\!", "922bdde6102fb341f5955799f3fe0b8d": "v = \\frac{\\partial \\Psi(t, y)}{\\partial t} + \\frac{\\partial\\Psi(t, y)}{\\partial y} \\cdot w", "922c6ae87de662e2525d7193440370b5": "Num(S)\\times Num(S)\\mapsto\\mathbb{Z}=(\\bar{D},\\bar{E})\\mapsto D\\cdot E", "922c92687b1a39806f2bd3bffc1c384e": "\\theta^*_k", "922ca88b3d0cade0b68d2efb29413c00": "H = c\\sqrt{\\left(p - \\frac{q}{c}A\\right)^2 + m^2c^2} + qA^0.", "922d0df7004c0f70354a840ff7fd787c": "\\forall r > R", "922d3a5282957f7667aa3a4c3c7a1c71": " \\mathbf{F}(x,y)=2 y\\mathbf{i} + 5x \\mathbf{j} ,\\ ", "922db67d02b9fafbc022966ad2fa4069": " h(k) + y^2 \\pmod{b} ", "922dcc318b1e23e41a28c776997ac6bd": "J = A_{\\mathrm{G}} T^2 \\mathrm{e}^{-W \\over k T}", "922e448857f8c91e701ebc7ac76bc9d7": "H^2 \\equiv \\left(\\frac{\\dot{a}}{a}\\right)^2 = \\frac{8 \\pi G}{3}\\rho - \\frac{kc^2}{a^2}+ \\frac{\\Lambda c^2}{3},", "922e578330a8440ffbb9fb9d91117879": " 0 \\longrightarrow A \\longrightarrow B \\longrightarrow C \\longrightarrow 0", "922e8cf0f964449e0e694029b50ca40d": "\\psi(\\bold{r}) = \\phi_x(x)\\phi_y(y)\\phi_z(z)", "922e9a0da1163b39e877cc7a5d7ea029": "f(s')", "922ef43610231fffc05a174606c5fb8c": "\\,2\\omega", "922ef8d0381c9fb27c55e7caee9d51d3": "A = 2 + \\sqrt{3},", "922f2c1db61d9fd7ebc66b979f6e81e8": "Pr[A \\cdot B \\cdot r=C \\cdot r] = Pr[(A \\cdot B-C) \\cdot r=0] = Pr[D \\cdot r=0]", "922f4a1b1863b8479128a983782c06eb": "\\frac{F}{A} = \\frac {B_{sat}^2}{2 \\mu_0} \\approx 1000\\ \\mathrm{kPa} = 10^6 \\mathrm{N/m^2} = 145\\ \\mathrm{lbf} \\cdot \\mathrm{in}^{-2}\\,", "922f8e5e2f9cf03c1aa9e9555f2ef3b0": "\n\\Xi_2(a,b,c;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a)_m (b)_m} {(c)_{m+n} \\,m! \\,n!} \\,x^m y^n ~.\n", "922f9e2a2e59db2e4b526991ecc9cc3e": "I(X_1;X_1) = H(X_1)", "9230bdf99a2006a0fc05c197a2dbeed8": "f = \\left ( \\frac {c+v_\\text{r}}{c + v_\\text{s}} \\right ) f_0", "9230d065a31b31cf318e530ba438c5fc": "\\Sigma\\ :=\\ 0", "9230dca5433c68e2931a0c5cb058b113": "E_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\varepsilon _{o}\\varepsilon _{r}}(A \\ e^{-jk_{x\\varepsilon }w}+B \\ e^{jk_{x\\varepsilon }w})e^{-jk_{xo}(x-w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ (41) ", "9230e2c4766378eb4ed9cb00ac285e30": "\\gamma=\\delta", "9231391ac67dbb3744c06f699048d21d": "\\frac{ax+b}{cx+d}=\\frac{-ax-b}{-cx-d}", "92314f1fe4ff4416bc094f157a90e6e3": "\\varepsilon_i + \\varepsilon_j", "9231df0bcbdc3e8769bf2b66c0e98d27": "M_0=1,M_i=\\prod_{j=1}^i m_j", "9231f839b6731be8c2a3d3ffc73807f7": "\\scriptstyle \\sqrt{1/N}", "923210c23d68cb8fcefded3d723189ab": "f_{\\text{A}} = \\frac {\\sum_{i}(f_{\\text{Ai}}^2\\, \\cdot N_{\\text{Ai}})}{\\sum_{i}(f_{\\text{Ai}} \\cdot N_{\\text{Ai}})}", "92325ebcaf38c70794855a950d0332c2": "S_T=\\frac{D_T}{D}", "9232680c2d7dd2491a24fdb86fc01851": "M[f]_{jk} = \\frac{1}{k!}\\left[\\frac{d^k}{dx^k} (f(x))^j \\right]_{x=0} ~,", "9232726bcede7b8804592e2e69bf7e5a": "D\\subseteq C", "92329c3eed8439b71596675ae996a0b1": "EBADC", "9232c44da4cc8c4738e6d8864ac1b4d9": "N\\subseteq_d M\\,", "9232e1f4104156182b12baed414571ed": "\\wedge^m", "92330d07975a03529f8706e19591e401": "f(x)= k \\cdot g((x-\\mu)'\\Sigma^{-1}(x-\\mu))", "923333e496dc0c9c1707876cac52a92f": "f(x) = x^3-x-1", "92333d7e2d4b8ed93cc1b7c9297082c0": "\\blacklozenge", "92340e947efbed0f70dbcb7fde0159e4": "\\text{LHA}_{\\text{object}} = {\\text{GST}} - \\lambda_{\\text{observer}} - \\alpha_{\\text{object}}", "9234ab2b03d5b6dd55f118288c75c101": "\\langle g\\rangle", "9234cd7e59f60ba729ec4e989f568199": "\\displaystyle{\\widehat{a} =\\chi \\widehat{a} + (1-\\chi)\\widehat{a}=T+S,}", "9234d6feb01cc7448569a4469190e2e4": "\\left\\langle X\\right\\rangle_{0}", "9234f13356d31ad73aa0099e80d21486": "EL(\\Gamma_1)\\ge EL(\\Gamma_2)", "923512fc704f7c8f2d67e949f4e2d26d": "UCL_r = 3.267\\overline{MR}", "92351ae762b043ae85bc8a2a8a7297b4": "T_j = \\prod_{i=k_j}^1 (1-S^{(j)}_i)^{-1} w^{(j)}_i,", "92370e9baaeefe640080b2d4f389d041": "\\mathrm{Rot}_G", "9237454d12cd79c18b9d3e5eb05a817c": " q ", "92376159e217dc9aa0201bbd2f9dd89e": "inflation_t", "9237c658acf145c55cb513b2f4c1b0a7": "(1/x^2) - S = 0", "9238c57ffa5048b71d116cf566c21d44": "\\mathbf{\\delta}(\\mathbf{r})", "923908d46423c95eed9f1d3ab614854f": "dP = |\\psi(x, y, z)|^2 dV. \\, ", "923916a5e43b390071979d45105fb6be": "\\left\\lfloor \\frac{n}{m} \\right\\rfloor = \\left\\lceil \\frac{n-m+1}{m} \\right\\rceil = \\left\\lceil \\frac{n + 1}{m} \\right\\rceil - 1, ", "923916ac38177509d9180b27225532b1": "p_{k,S}^C={\\mathcal P}[C_k(s)=1 | s\\in S_k\\text{ with probability }\\mu_k(s)]", "92392a4baef1bf83cc2e6b30fa417d8c": " (\\arcsin x)' = { 1 \\over \\sqrt{1 - x^2}} \\,", "92393a02c43db648087a2adac848ca7f": "dX_t = \\left[\\mu(X_t,t) - \\frac{1}{2} \\frac{\\partial}{\\partial X_t}D(X_t,t)\\right]dt + \\sqrt{2 D(X_t,t)} \\circ dW_t.", "923989a3672b00025db632b9a9d1987d": "(X,\\sigma,\\tau)", "923996d516f42a83f68967e87c4bbc5c": "\\ell^2/A(\\rho)\\le w/h", "9239a96bac1a9cb97bd7df99c7aae44f": "\\frac{d^2 T_n}{d x^2} \\Bigg|_{x = -1} \\!\\! = (-1)^n \\frac{n^4 - n^2}{3}.", "9239cd718ec06fab6012aa903dd44dcd": "\\,{(1-d)}^{n}", "9239e23c2ff19a536c203be888509afe": " H = \\frac{ 1 }{ N - 1 } \\frac{ s^2 }{ m^2 } ", "9239fa5f89ea85da48c933aef37ece0f": "\\|f \\star g\\|_{L^{s}} \\leq \\|f\\|_{L^{r}} \\|g\\|_{L^{p}},", "923a25ac83109abc9694def341287c3d": "A \\Rightarrow B\\alpha \\Rightarrow A\\beta\\alpha \\Rightarrow \\ldots ", "923a64f6248cd1449e8fee64889cc1ed": "f(x)\\cdot\\left(1-\\frac{1}{1+g(\\vert x\\vert)}\\right)", "923a975a0b58506453ffa944182afd29": "\\sum_{k=0}^\\infty (-1)^k/(2k+1)\\!", "923aad8b7cc0ab4fe799df81526b6667": "\\frac{\\Delta P_p}{\\Delta P_i} = \n 1 - 0.24 \\beta - 0.52 \\beta ^2 - 0.16 \\beta ^3\n", "923ac33309f94ac0b89b3c7c9e799df4": " \\Delta\\lambda \\approx {\\lambda^2 \\over\\Delta x}", "923ad2d80578b6ea4ad09dada14dc94b": "j^\\text{th}", "923ae3b7cc5c46c2fc60aafb3571bac9": "z(\\hat{\\rho}_{XY\\cdot\\mathbf{Z}}) = \\frac{1}{2} \\ln\\left(\\frac{1+\\hat{\\rho}_{XY\\cdot\\mathbf{Z}}}{1-\\hat{\\rho}_{XY\\cdot\\mathbf{Z}}}\\right).", "923c30ca8aba238c9b3be24330107822": "\\sum M_B=0=\\sqrt{3}*F_{CD} \\Rightarrow F_{CD}=0", "923c8266fa483c22f5375049d392b718": "\\zeta(s) = \\pi^{s/2} \\frac{\\prod_\\rho \\left(1 - \\frac{s}{\\rho} \\right)}{2(s-1)\\Gamma(1+s/2)}.\\!", "923d11e1b50e55009da2019f11095eb6": "\\epsilon(v) = r", "923d171c7bdc2ef125b71cd2d8f31257": "\\ v", "923d2671c7a3b05c38a08f10bbd8f88d": "X \\in C", "923d2fdc8404316cfd88dff8d54a3d63": " \\mathbf{D}=\\begin{bmatrix}\n\\mathbf{A} & \\mathbf{B}\n\\end{bmatrix} ", "923d94a3c049142459200ac8de80562c": "\\textbf{P}_{k\\mid k} = \\textrm{cov}(\\textbf{x}_{k} - (\\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k(\\textbf{z}_k - \\textbf{H}_k\\hat{\\textbf{x}}_{k\\mid k-1})))", "923da662c588237ad6b20bc10e8606da": "\\,\\mbox{R}(z, dt) = \\exp\\left[- \\frac{i}{h}\\ (x p_y - y p_x) dt\\right]", "923db922542fbe09e7ff87bb31b2f310": "YX", "923dc1fd9afc1f53b8c1cb0e27826826": "\\begin{cases}\n \\ m+s\\Gamma\\left(1-\\frac{1}{\\alpha}\\right) & \\text{for } \\alpha>1 \\\\\n \\ \\infty & \\text{otherwise}\n \\end{cases}", "923e5377bb6261528e0990eab4b7fd0d": "V_L = V_{R_1} + V_D", "923e56917c04773b5f2aca2ee7a4e23a": "y = \\frac{x^3+2x^2+3x+4}{x}", "923e830cc8e98ac394fdf0d9594531eb": "\\mathbf{J} = \\mathbf{L} + \\mathbf{S}\\,\\!", "923ea142b803d568b4b2b00374aa1fdd": "\\Gamma_\\mathbf{R}(s)=\\pi^{-s/2}\\Gamma(s/2)", "923eb8bacbd914401f7d2e7cfc97bafc": "f,g\\in H", "923f2e9bc517deadebee548bc5eb9dfb": "\\beta = \\omega^{\\gamma_1}\\cdot j_1 + \\cdots +\\omega^{\\gamma_n}\\cdot j_n", "923f710277d1aec656ae7f1e7a3a3176": "B =\n\\begin{pmatrix}\n1&0&0&\\cdots&0\\\\\n0&\\omega&0&\\cdots&0\\\\\n0&0&\\omega^2&\\cdots&0\\\\\n\\cdots&\\cdots&\\cdots&\\cdots&\\cdots\\\\\n0&0&0&\\cdots&\\omega^{(n-1)}\n\\end{pmatrix}\n", "923f8fe7667e69ba748c009e3667b41b": "\\scriptstyle G_{f}", "923f971b1f74c600dd0aae274f5d868c": "(-i),", "923f9cefae019b8c7c74d882ea7d99f9": "G_{3}", "923fbd1d16d411a23b10809fb8fb7b87": "f:\\mathbb{R} \\to \\mathbb{C}", "923fd5c3fef65bb972daec4ee0f0571b": "\\pi_{k+1,k}^*: C^\\infty(J^{k}(\\pi))\\to C^\\infty(J^{k+1}(\\pi))", "924032799351f625d8437570ac9ca5ca": "\\begin{array}{rcl}\n\\mathrm{d}W(t)^2 & = & \\mathrm{d}t \\\\\n\\operatorname{E}[\\mathrm{d}W(t)] & = & 0\\,.\n\\end{array}\n", "924034a82f02a5a78e8f216b3f9d4285": "\n\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{r}} \\right) = \\frac{\\partial L}{\\partial r}\n", "92404e01b56a8d7701cd81beda631cb6": "(0,1,2,3,\\dots,22,23,24)", "924058f376015d4f96d4a59e9ab04848": " \\begin{align} \n\\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} &= \\sqrt{\\psi_1(\\alpha)\\psi_1(\\beta)-(\\psi_1(\\alpha)+\\psi_1(\\beta))\\psi_1(\\alpha + \\beta)} \\\\\n\\lim_{\\alpha\\to 0} \\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} &=\\lim_{\\beta \\to 0} \\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} = \\infty\\\\\n\\lim_{\\alpha\\to \\infty} \\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} &=\\lim_{\\beta \\to \\infty} \\sqrt{\\det(\\mathcal{I}(\\alpha, \\beta))} = 0\n\\end{align}", "92406b55028962a3a4e708cd158b6350": "(P_i)_{i\\in N}", "92406c12d31575d5db7224d5319f53d1": "\n\\begin{bmatrix}\n2 & 2+i & 4 \\\\\n2-i & 3 & i \\\\\n4 & -i & 1 \\\\\n\\end{bmatrix}\n", "9240ee028667013460e276f73db51c37": "y = \\frac{x^2}{4a}", "924103ab42a1d4d128400114311a81ca": "{L}", "924110d361b897b9e8b6ea1c95301dd4": "\\operatorname{E}[X] = \\frac{\\alpha}{\\alpha+\\beta}\\!", "92411463896619846248b3af0af05b60": "[v_x,v_y,v_z]", "92412a799c043f1846f361fd866e1e9d": "x=\\sqrt{a+\\sqrt{b}}+\\sqrt{a-\\sqrt{b}}", "92415aac440f0daee88c2c6b55d7f94e": "|z|^2 = 0", "924161fef7da20eaf7741cadec2f6b33": "1/|\\omega|", "9241c4d8a3bd1132810c3407ed914ff5": "01\\ \\text{is odd}\\right)", "92508cfd01459b0c2abd9923164b9266": "x^{(m)}_2", "92509a1670c35f69dd48d462d9d8bef0": "H_{3x,2}=\\frac{1}{9}\\left(6\\zeta(2)+H_{x,2}+H_{x-\\frac{1}{3},2}+H_{x-\\frac{2}{3},2}\\right),", "9250b6d2f8aa2b4987880ddfef543fd0": " M = \\begin{bmatrix} 3 & -4 & 1 \\\\ 5 & 3 & -7 \\\\ -9 & 2 & 6 \\end{bmatrix} ", "9250c9313d12a1d47f933d90fbaf3051": "\\operatorname{I}_{\\mathbb{P}^n}(S)", "9250f2a487d081142c58c3f73c99dd0d": "F_{gamma} \\approx 10^8\\left(\\frac{r}{d}\\right)^2", "925128f106e62ad0b11c920ecdcba961": "e^{ik_z z}", "92514bcf8527a45b50bd897403dc6370": "h\\ :=\\ f(h,\\ \\Sigma)", "925181c2fcddda3bc4ba7cfeaee2d516": " \\begin{align}\n y(t) & = H(i \\omega) \\ e^{i \\omega t} \\ \\\\\n & = \\left( |H(i \\omega)| e^{i \\phi(\\omega)} \\right) \\ e^{i \\omega t} \\ \\\\\n & = |H(i \\omega)| \\ e^{i \\left(\\omega t + \\phi(\\omega) \\right)} \\ \\\\\n\\end{align} \\ ", "925181c508645a6fb0778c9c2a93d163": " \\exists A \\in \\mathcal{A}", "9251bf137253b8be30838298fb55cc85": "|\\beta_1\\lambda_1+\\beta_2\\lambda_2|", "92526302e07bb6d749ec97c041f190b9": " \\operatorname{build-param-lists}[p, D, V, T_4] \\and \\operatorname{build-param-lists}[p, D, V, K_4] ", "925265a1060a04b6fda53f58f18b949e": "\\lambda(n)<(\\ln A)^{c\\ln\\ln\\ln A}", "92528fd64e6b2540e7df4b19842a87a4": "p_i(\\xi)=0", "925291d29ca9d635974077050a90e736": "{\\bar{DP}}_4", "92529acce7c4e2bbdc578351a42821f9": " E_x^0 = \\mid \\mathbf{E} \\mid \\cos \\theta ", "92529df53f97a54e931a913fbe7a29d7": " \\log_{10} (4) = \\log_{10} (2 \\times 2) = \\log_{10}(2) + \\log_{10}(2) = 0.602 ", "9252b0213999abdaebeb49563d5c92f9": "g (A, B) =-g (B, A) ", "9252b81abeaae88b6bf74da3b7bcb4ef": "\\psi(x,t) = \\langle x|\\psi(t)\\rangle", "925323172766a19040f83911e4784f95": "\\textstyle \n \\begin{bmatrix} A & B & x\\\\ C & D & y \\end{bmatrix}\n ", "925335f81021de4d22fde55ae7f0e86a": "Prerequisite", "9253eccd9f63b7267b68b987d1d80d5c": "\\mathbf x[k+1] = e^{\\mathbf A(k+1)T}\\mathbf x(0) + \\int_0^{(k+1)T} e^{\\mathbf A((k+1)T-\\tau)} \\mathbf B\\mathbf u(\\tau) d \\tau", "9254023d7a786f3c017f4083713ab14c": "\\mathbf \\gamma(t):(-\\delta,\\delta)\\to \\Phi_c", "92540a8c6c92c81accdf4d1a1beabca4": "\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}} = \n \\cfrac{\\partial W}{\\partial I_1}~\\cfrac{\\partial I_1}{\\partial \\boldsymbol{C}} + \n \\cfrac{\\partial W}{\\partial I_2}~\\cfrac{\\partial I_2}{\\partial \\boldsymbol{C}} + \n \\cfrac{\\partial W}{\\partial I_3}~\\cfrac{\\partial I_3}{\\partial \\boldsymbol{C}} ~.\n ", "925419f84c0e7047013f73014c9d9c7d": "M_{1x} + M_{2x} = F_{P1x} + F_{P2x}", "92545577b06ab8ce0a48da279227ef49": "{\\left ( {a^{ac}}^\\prime a_{bc} \\right )}^\\prime = 0, ", "925496c22b2e15052f56b6d8a0ad6a17": "SS(w')=s'=syn(x_w')", "9254bc6d234ad71754f0153371323934": "F^\\times", "9254c40babd5b16a26a3fa309ed3c866": "k := 1\\!\\,", "9254d01c818e7133823062df2a9a58ed": "XY=0", "9254d7923b3a31f215732ddf1aeabc75": "u_t = u^3u_{xxx}.\\,", "9255766032fe4728742b5f2808ace11a": " w = a u + b v + x", "925582fe4d7ca5221d21b605cb7a57fb": "\\{f_i : X \\to X | 1\\leq i \\leq N\\}", "9255b7be2c255272754cc776e2e1ee6b": "V= \\sqrt{2} (\\frac{5}{3}+\\sqrt{1+\\sqrt{3}}) a^3 \\approx 4.69456...a^3", "9255c55fa6d40d140be013909486fe24": "\\log (T^*T) \\ge \\log (TT^*).", "92563aadc1ad82f58d17ef7f2bf18666": " \\Omega_Q(B) \\psi = \\chi _B \\psi ,", "925664732027e5eb81507818a20551ed": "D_M(x) = \\sqrt{(x - \\mu)^T S^{-1} (x-\\mu)}.\\, ", "92567ee7bc7afe1b73a20d0a10b4ed17": "T = \\bigoplus\\nolimits_{i\\in I} (T_i, a_i, b_i),", "925694878a502a84706cdc1b74b15293": "1-2/2^s", "9256bdc31da4dfb19450c56554c90d5e": "\n\\eta(s)= \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n^s} = (1-{2^{1-s}})\\zeta(s).\n", "9256f3f75afa3feec2844c23572acae9": "D = c \\beta_1 \\beta_2 = c^2 + c", "9256f66530f811eb9725fc26afeabc4c": "t\\sigma_I^2/I", "92572c1c33762753bb73363a363120a2": "ax^2-2bx+c=0", "925757306cd373e91fccf9ac22dff356": "\\forall a, b, c\\in X\\,(a\\,R\\, b \\land a \\,R\\, c \\to b \\,R\\, c).", "9257740687390242606fcd8582d6ff2e": "O(''n'')", "925790ca497055487127cc92971e74fb": "F(w) = \\frac{1}{n}\\sum_{i=1}^n (y_i- \\langle w,x_i\\rangle)^2", "9257b08fcce5ad918e0573645430803f": "H(f) = \\lim_{y \\to 0}Q(-, y) \\star f", "9257b2ff53ce8b5099e67d513f477916": "E_{k0}={(k \\pi)^2\\hbar^2\\over2m_0r_0^2}={k^2h^2\\over8m_0r_0^2}", "9257ce093265ab77c4eeaca1ba459687": "a_\\text{Peano}=\\frac{(1+i)}{2}.", "925842e17022b3a5447076ed80526f40": "m^2-3n^2,", "9258a848f34b1beade053af97f315744": "n_{A^-}/n_{HA}", "9258dbb3f23ce268486287cd50b9b976": "\\mu_1(t)=\\left[\\Sigma_0^t p(i)\\,x(i)\\right]/\\omega_1", "92591f980670ebb89293ad0ce791031d": " \\phi = \\frac{1}{2} \\arctan \\frac{2 \\langle xy \\rangle}{\\langle x^2 \\rangle - \\langle y^2 \\rangle }.", "9259d70315bdd64ea96f75148315a7ff": " y_2 = 5x_3 \\, ", "925a3c03ba6fb34f810a5d92d43031b7": "\\{B_n(t),0\\leq t\\leq 0\\}", "925a3c9e9f82e0a12902ff3991b89bf9": "\\tfrac{730}{232}", "925a9035c2edbc2b98e182ba49111060": "P_r = \\frac { A_t A_r }{ r^2 \\lambda^2 } P_t \\,", "925ac617a149897c64d91a31f4c2ed49": "(i,m)", "925b2a54a84f3a591a4187e27eeaa919": "\nX^{\\{2\\}}=[3,4], \n", "925b51871d6ca9580e8ddfec1b8ad68e": " f(\\theta) ", "925b7342ffb1a569862c60d6828cf43e": "\n\\alpha = {K \\over K-1 } \\left(1 - {\\sum_{i=1}^K \\sigma^2_{Y_i}\\over \\sigma^2_X}\\right)\n", "925bdb11e4f93fc2464767caf6b3dd4a": "l_adx^a=\\frac{1}{\\sqrt{2}}(\\sqrt{g_{tt}}dt+\\sqrt{g_{rr}}dr)\\,,", "925c476624dec6a55cf96e8524b7783d": "\\int_{E}\\phi\\, d\\mu \\leq M\\mu(A),", "925d009d48eaab2b249175681b25071e": " K = \\langle a,b : a^4 = b^4 = 1, ba=ab^3 \\rangle,", "925d072570cffd60a48c48059e3a23e1": "{}^qH = \\ln\\left ( { 1 \\over \\sqrt[ q - 1 ]{{ \\sum_{ i = 1 }^K p_i p_i^{ q - 1 } } } } \\right ) = \\ln( {}^q\\!D )", "925d1fa1f634072cf4948abb7afb6236": "\\vec{F} = \\vec{F_x} + \\vec{F_y} + \\vec{F_z}", "925d72216158f5965bdf245043bfde44": "x \\in (a, b)\\!", "925d96f5cd607150887120c175838c3f": "\nu (y) = u_0\\frac{y}{h} + \\frac{1}{2\\mu} \\left(\\frac{dp}{dx}\\right) \\left(y^2 - hy\\right).\n", "925de87d3c7c8eb06739266bf3696ee5": "\\mathfrak{H}(k; \\gamma_1, \\gamma_2) = \\mathfrak{H}(1/k; \\gamma_2, \\gamma_1).", "925e5e97a5a5ce83be97004b4f84d05f": "\\theta_E", "925e9d0315ec1056a5f5195a8d3b927d": "\n\\langle I \\rangle_e \\sim L^{-3/2}\n", "925ebd68072da431fe24753f392da877": "\\log \\left (\\frac {k_s}{k_{\\text{CH3}}}\\right ) = \\delta E_s", "925ebe8f4b22d248c8d05ab081065a3d": "\\sin^2 (k_m \\Delta x/2)", "925ee28ad791df2abc6de6647179eca2": "\\tau_D=w^2/(4D)", "925f084d1b997aa5d91f066536814144": "N \\otimes_{R} M \\cong S", "925f94e45e34e12bd6594685b4bef284": "V_\\lambda(x,y,z) = x^3+y^3+z^3 - \\lambda xyz=0 \\;", "925fa816485cb32c4c0a793cd6fe6556": "\nh_{\\mu} = r \\sqrt{\\frac{\\mu^{2} - \\nu^{2}}{\\left( b^{2} - \\mu^{2} \\right) \\left( \\mu^{2} - c^{2} \\right)}}\n", "925fd5ff936d7564e4ed1e5ed3cf6df2": " A_0 =0", "92600b3335a10b681534717697fc4b2a": " \\gamma_V(t) = (tV^1, ... , tV^n) ", "926023dbf6c75019c4b477abd692d1e6": "(E_n)^\\infty_{n = 1}", "926039e793d6809d6515b561aa88f456": " \\hat N =\\gamma | \\psi|^2 ", "9260c1ec03b012623544ca863e316a65": "\\arctan x = \\arcsin \\frac{x}{\\sqrt{x^2+1}} ", "9260e3ae5ed1cacbc4b4769ea855e786": "\\gamma = \\alpha + j\\beta\\,", "9260ffd96fb821b5221eaf94f7984583": "S(\\rho \\| \\sigma) = \\infty", "92622e9dabaaa74d011abfee04bfd289": "m \\ddot{x} + { c } \\dot{x} + {k } x = 0.", "92626908a3803aa0ba24a22b6c47f4da": "(2\\pi |\\xi|)^{-\\alpha}", "92626c37facb2aa5a2982331b66d3a69": "\n{\\partial(\\rho u_j)\\over\\partial t}+\n\\sum_{i=1}^3\n{\\partial(\\rho u_i u_j)\\over\\partial x_i}+\n{\\partial p\\over\\partial x_j}\n=0,\n", "92629e155ca571e285e358f4b340ce0f": "\\theta(\\psi)", "9262e2a767afbff5aebc32e11e0d49c2": "\\lambda=e^{2\\pi}=1^{-i}", "9262e8633e00572e76bd1a8831f04574": "\n\\delta^{\\mu_1 \\dots \\mu_s}_{\\nu_1 \\dots \\nu_s} = {1 \\over (n-s)!}\\,\n\\varepsilon^{\\mu_1 \\dots \\mu_s \\, \\rho_{s+1} \\dots \\rho_n}\\varepsilon_{\\nu_1 \\dots \\nu_s \\, \\rho_{s+1} \\dots \\rho_n}.\n", "926355e4ecf03199b2a81975d3263298": "(v|0|0)", "926377ac8fb41c1fd3506abbaaf6a0d7": "p = max(a, b, c, \\ldots )", "92638bfa140b97edb7a208f01f1e78b4": "v_y", "926435b4224ee910519100e761c58ba3": "slt", "92643af3193e22ae6cb88e093e0e6359": "A= \\sum_{k=1}^{K} h(x_k)\n\\,\\Delta x_k", "9264515ba5feee7f31769c004cbe6226": "R_{\\text{3}}", "92645fe64e02a486f537a25eaaad44cc": "\\mathrm{d}k = -2x\\,\\mathrm{d}x", "9264adab397b9bd06a4fad450f9ad9d6": "\\mathit{F \\to dFd | LS}", "9264c9e2fdd41cb5505d9dd1269be996": "f: U \\to G", "9264ce47f48cc975356a7844042a5746": "(6,1,1)\\rightarrow (3,1)_{-\\frac{1}{3}}\\oplus (\\bar{3},1)_{\\frac{1}{3}}", "9264d287a04e895449a28a46e680592e": "N_2 = N (2 N_{\\operatorname{dg}} - I_t) - N_{\\operatorname{sq}},", "92653888dacd2f890523a5f4552c1122": " {{2^3 \\over x^5} = {8 \\over { 5^{5/4} }}}. ", "92655975fa16bec03a7bb29adc0cb506": "\\Delta \\left( \\frac {a\\tau+b} {c\\tau+d}\\right) =\n\\left(c\\tau+d\\right)^{12} \\Delta(\\tau) ", "92655e0b02a94a1651c6f0aedf38ae6a": "v_t - v_{xxt} = v\\, v_x.", "9265612ceee5d2d5dc0ccc169bd24672": " \\mathbf{H}_{1} = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & -1 \\\\ 0 & 1 & 0 \\end{pmatrix} ", "926567a3dd7523eab64b9ab8fab1256f": "a+b = b+a", "9265a082cead5b3bc0a294390309ab26": "(a+bi) - (c+di) = (a-c) + (b-d)i.\\ ", "9265aa305d7f990110d83b7e2129ca43": "\n\\frac{\\partial}{\\partial t} u_m(\\mathbf{r},t) + \\nabla\\cdot \\mathbf{S}_m (\\mathbf{r},t) = \\mathbf{J}(\\mathbf{r},t)\\cdot\\mathbf{E}(\\mathbf{r},t),\n", "9265b6e5e393a4bf5cc6451473438896": "r\\approx \\sqrt[6]{\\frac{2 B_0^2}{\\mu_0 \\rho v^2}}", "92660ea95a36cd68ea2fe1851e1f9865": "V^\\pi(s)", "9266803a0fe1e5607ce4dcdf19b50581": "\\scriptstyle|\\alpha\\rangle ", "926688a9967cc2ce1edab70c2b99da22": " x^0 = 1 ", "92669404f432b4f692d0042ec58055b9": "n = 10^5", "9266c6ca23313aa26062c605f268c8dc": "\\ F_{3dB} \\approx K/T_r ", "9266d15b630f77c02b55e29009d5bf31": "\\text{then }\\tan(\\psi) + \\tan(\\theta) + \\tan(\\phi) = \\tan(\\psi)\\tan(\\theta)\\tan(\\phi).\\,", "9267598edab3ace54d0ebf6f542be27d": "_{dual(q\\tilde{\\leftarrow}p)=q\\rightarrow p\\,}\\!", "9267f0f3be74526d727646585b70c07b": "C_{t+dt} = C_t - C_t * K_e*dt", "9268109b2ac8f54b79f485fb7563cf5d": "C_0 = 1 \\quad \\mbox{and} \\quad C_{n+1}=\\sum_{i=0}^{n}C_i\\,C_{n-i}\\quad\\text{for }n\\ge 0;", "926817c0c1abc6fe40a97aa3366dc1e1": "\\hat {\\textbf{Q}} = \\hat {w} \\hat {\\textbf{q}}", "926821975b7334a5780ecada0880e497": "\\sigma_{x}\\sigma_{p} \\geq \\frac{\\hbar}{2} ", "926831d9125d2f2903e6c414dacfb68e": "256\\uparrow\\uparrow 257", "92683561427d5899f457b06c7e6f63d7": "(q-1)N(q,n)", "9268f4888d73659d49a01207e414fbb5": " \\mu_\\mathrm{z} = \\gamma S_\\mathrm{z} = \\gamma m\\hbar.", "92690bfde4c90735a1e82222845fa8f4": "(3,2,1)_{1\\over 3}\\oplus(\\bar{3},1,2)_{-{1\\over 3}}.", "92691ae6468291414e3168af93c1f948": "p_*\\colon H_*(E) \\to H_*(B)", "92692ae49c27151c759063cc60239668": "p^*=p", "92694c2239bddcdd73c475dedd8a77e9": "uLv-vLu = \\frac {d}{dx} \\left( p(x)u \\frac {dv}{dx} \\right)-\\frac {d}{dx} \\left( v p(x) \\frac {du}{dx} \\right), ", "92698c4ffe9e4958fa393f4f894db30f": "\\frac{\\alpha^3}{4 \\pi} \\,", "9269c64bbea76e808d51a35b04319a90": "\\boldsymbol{u}_f(\\boldsymbol{x},t)", "926a34b19d146f7d24da8c2c1bb1f4e8": "v_\\mathrm{rms} = \\sqrt{\\int_{0}^{\\infty} v^2\\ p(v)dv}\\,\\! ", "926af7de19f2be7bfc19516b56234329": "s \\otimes n \\mapsto sn", "926b56f910d775fcae5cc7ad72908cc2": "D = B(0)", "926b5eb084727592786438a5989da93b": "a = (\\delta/(2g))^2 = 40", "926c203d4102f827e83fb9fea1a6e1ea": "g=\\frac{2}{g^R(\\sigma,\\sigma)}\\sigma\\otimes \\sigma -g^R", "926c39dc0f1943bdc1c029a0d5939dcf": "\\frac{\\varepsilon}{\\varepsilon_y}=\\frac{\\sigma}{\\sigma_y}+\\alpha\\left(\\frac{\\sigma}{\\sigma_y}\\right)^n", "926c428d82bd2c7edaae28784b17b671": "x'=A(t)x\\,", "926cb24ce667b52c7d09bf7fc6e73cf1": "C_1 = C_3 = 0", "926cf2e5769231208ff549dcd376b1fd": "E_\\mathrm{p,e} = \\frac{Q^2}{2C} = \\frac{1}{2}CV^2 = \\frac{1}{2}VQ \\,\\!", "926d99b3f400bc9aa9be49d0aa6a3c76": "A\\cdot (CVS+color) =A\\cdot (CVS) + A\\cdot (color)", "926dcfe7c896ef570eb6bc37e908cefd": "J(x) = D_x f", "926df57d1c8ee479a8acc268d26cd7ca": "\n\\begin{align}\n\\Psi^{(A)}_{n_1 n_2 \\cdots n_N} (x_1, x_2, \\cdots x_N) & \\equiv \\lang x_1 x_2 \\cdots x_N; A | n_1 n_2 \\cdots n_N; A \\rang \\\\[10pt]\n& = \\frac{1}{\\sqrt{N!}} \\sum_p \\mathrm{sgn}(p) \\psi_{p(1)}(x_1) \\psi_{p(2)}(x_2) \\cdots \\psi_{p(N)}(x_N)\n\\end{align}\n", "926e36b799e97d768e53d37993207d52": "T_{\\rm S} = 5778 \\ \\mathrm{K},", "926e511c31b77927c290127d5ef7ad5d": "\n\\mbox{If } \\left(\\frac{\\alpha}{\\mathfrak{a} }\\right)_n \\neq1\n\\mbox{ then }\\alpha \\mbox{ is not an }n\\mbox{-th power}\\pmod{\\mathfrak{a}}.\n", "926e6d2941dbeeb56ec8c94460c4dbe8": "$ \\lessdot Head^+(S)", "926e7add16cf1495d09dce147dea400d": "9^n-1=\\left(3^n+1\\right)\\left(3^n-1\\right)", "926e920b80a46b6c35cc1165ed4997dc": "1+o(1)", "926e9785a3b3ab25707188393ecdaa07": "\\begin{align}\n x &= \\frac{9u^\\prime}{6u^\\prime - 16v^\\prime + 12}\\\\\n y &= \\frac{4v^\\prime}{6u^\\prime - 16v^\\prime + 12}\n\\end{align}", "926e9feffd47b8a4b2e2f7f50cdf4a50": "\\sum_{n=0}^\\infty \\frac{2^n}{3^{n+1}} = \\frac{1}{3} + \\frac{2}{9} + \\frac{4}{27} + \\frac{8}{81} + \\cdots = \\frac{1}{3}\\left(\\frac{1}{1-\\frac{2}{3}}\\right) = 1.", "926eb96bdab76934503423545985a481": "\\mathbf{g} = - \\frac{GM}{\\left | \\mathbf{r} \\right |^2} \\mathbf{\\hat{r}} - (\\left | \\boldsymbol{\\omega} \\right |^2\\left | \\mathbf{r} \\right | \\sin \\phi )\\mathbf{\\hat{a}} \\,\\!", "926ee0fbde891fc760c386436896ba91": " T_i \\equiv 0 \\bmod\\ N_j ", "926f67c49f018bbfb9adf4e7f9d53570": " v_r ", "926f8be14ba352eef903f043addbfbc0": "\\begin{pmatrix} 73 \\\\ 5 \\end{pmatrix} = \n\\frac{73!/(73-5)!}{5!}\n=\\frac{69\\cdot 70\\cdot 71\\cdot 72\\cdot 73}{1\\cdot2\\cdot3\\cdot4\\cdot5}=15020334", "926f8c4fbd16c638fea3ea5a7301fd49": "{\\rm Con_c}\\,L\\cong S", "926fb3c27d38f7042d7e1279da57427d": "\\varphi(x,t)", "926fe2c84ab23ad25a5fd5de58f2e5a4": "\n q(x,y) = q_0 \\sin\\frac{m \\pi x}{a}\\sin\\frac{n \\pi y}{b} \n", "92703e5b05f18e1ea97ffad2b8a95070": "\\hat{m}_{ij}^{(2\\eta - 1)} = \\frac{\\hat{m}_{ij}^{(2\\eta-2)}x_{i+}}{\\sum_{k=1}^J \\hat{m}_{ik}^{(2\\eta-2)}}", "92708268cc968b9b1be6cf464b5222b5": " \n2^{-n(H(X)+\\epsilon)} \\leq p(x_1, x_2, ..., x_n) \\leq 2^{-n(H(X)-\\epsilon)}\n", "92719b7c8993efcdc5bbee3eb62cf042": "dU =C_{V}dT.\\,", "9271a120faecc5c0237d903da6f5e9e0": "(\\hat{m}_{ij}) := \\lim_{\\eta\\rightarrow\\infty} (\\hat{m}^{(\\eta)}_{ij})", "9271ed84eaf4c5b72865b6b231e3427a": "V_n(R) = \\int_0^R \\frac{2\\pi^{n/2}}{\\Gamma(\\frac{n}{2})} \\,r^{n-1}\\,dr = \\frac{2\\pi^{n/2}}{n\\Gamma(\\frac{n}{2})}R^n = \\frac{\\pi^{n/2}}{\\Gamma(\\frac{n}{2} + 1)}R^n.", "92724a89bc471094ac325cc9ac7fb878": " S_{ (a_{1}, ... a_{r}| b_{1}, ... b_{r})} = \\det ( S_{(a_{i} | b_{j})}) ", "9272840834acada16236440bf9c94630": "\n\\frac{dt}{d\\tau} = \\frac{\\partial H}{\\partial p_{t}} = \\frac{p_{t}}{c^{2} \\left( 1 - \\frac{r_{s}}{r} \\right)} = \\frac{a}{b \\left( 1 - \\frac{r_{s}}{r} \\right)}\n", "9272d933ce06c34dab290c082d91c789": " x = \\sum_{n=0}^{\\infty} x_n e_n, \\ \\ \\textit{i.e.,} \\ \\ x = \\lim_n P_n(x), \\ \\ P_n(x) := \\sum_{k=0}^n x_k e_k.", "9272e0fe7a7ac665f73928158a9a88a2": "\\varepsilon_p", "92730234139e985c6804ba7def1d8590": " \\exp \\left [ i\\left ( k x - \\omega t \\right ) \\right ] ", "9273094d9b9c766af772cc798a1d9efc": "(x-3)(x-1)^2(x+1)(x^2+2x-1)^2", "92731c5198f70090cb19cfdd7192ba79": " -\\frac{\\hbar^2}{2 m} \\int_{-\\epsilon}^{+\\epsilon} \\psi''(x) \\,dx + \\int_{-\\epsilon}^{+\\epsilon} V(x)\\psi(x) \\,dx = E \\int_{-\\epsilon}^{+\\epsilon} \\psi(x) \\,dx.", "92735317db670d6871e8e41456868af4": "t_{1}>t_{0}", "927355bf1f96976f85878ed61ba744eb": "p(T) = \\sum_{i=0}^m a_i T^i", "9273c8a73351bb6c7b245fa351e5e054": "\n\\frac{d}{dt} \\left(Y^{2} \\dot{\\varphi}_{r}^{2} \\right) = \n2\\frac{d}{dt} \\left( E \\chi_{r} - \\omega_{r} \\right),\n", "9273dee478eddf03dda12bcf038696dc": "I=\\int \\arctan (x) \\cdot 1 \\,dx. ", "9273e7b71c724e39da07d3edade74ae2": "R_a : x \\mapsto xa", "9273ee68401c1cb8a3bbba4439cea0be": "\\mathrm d \\colon S \\to M", "92742246429f40ae22dea71cfca76990": "\\partial / \\partial t", "927489eebea8cd18c6ef4781e6111b3a": "M_\\mu = \\bigoplus_{\\lambda} K_{\\lambda \\mu} V_\\lambda.", "92750ba4c3d4ce0d3a1d3b63b533f829": " \\frac{dN}{dt} = rN \\left(1 - \\frac{N}{K}\\right) ", "9275254830daa6170099f1882a078cf6": "\\bold{H}_{\\operatorname{AMISE}} = \\operatorname{argmin}_{\\bold{H} \\in F} \\, \\operatorname{AMISE} (\\bold{H}).", "92752ceb0ea262eb05627238ca4021c4": "\np \\overset{\\alpha}{\\rightarrow} p'\n ", "927531422c9bcb7239f42eb3298de2b3": "{\\Lambda^\\mu}_\\nu = \\left( \\begin{array}{cc} \\cosh \\tau & \\sinh \\tau \\\\ \\sinh \\tau & \\cosh \\tau \\end{array} \\right) \\,", "927546920f1447e9a6459843d5156b02": "(x_0, y_0),\\ldots,(x_k, y_k)", "927586a8c975ccc3dfafbb8bea5f2be2": "\\scriptstyle 1 \\,\\leq\\, j \\,\\leq\\, r ", "927594a8670b8aa2cbc42c6d46f62bd2": "F \\left[ \\hat{\\rho} \\right]", "9275be2aee1c8af3399ba1205766be96": "y=x^2,", "9275cd4cf0ef3cc29231a1dc7a5fa4ca": "\\frac{\\alpha - 1}{\\alpha + \\beta - 2},", "9275cf6751130291c0939c590ff7bd4b": "\n{dA_{mn}\\over dt} = i(E_m - E_n ) A_{mn}\n", "9275d7cb7da73d34250b700d23b62d2b": "V_s\\! = \\frac{4 \\pi}{3} r_\\mathrm{s}^3", "9275e8c8e967f7000d8e43f9c17095e1": "\\phi = (\\forall x)(\\exists y)(\\forall u)(\\exist v) (P) \\psi", "9276045de13b49323df13884fbab39e2": "k+2", "927605e833c5b5dada6783c06daccdcb": " f(Ty) = f(y). \\, ", "927615e88e3232e7454806d062c29c83": "T_{2} = R_{2B}(y)", "92763cb52be9a14ae1badffb99f7f85d": "\\Delta^{n-1} \\rightarrow \\Delta^n", "92763f0b83b0afa3c4a8e4cb372b2159": "\\lim_{n \\to \\infty} \\|Tx_n - \\lambda x_n\\| = 0", "92766e8ee96e980656adc20a65909a51": "R \\ge B > G", "92769b1d3f5f05b426f70c2c05c33a93": "45 \\cdot b_n", "9276c1dd710f8d64810dd4f223762b05": "\\scriptstyle a/\\infty = 0", "9276c298440d6eae4866608ccbcd7003": "P_n^{(\\alpha,\\beta)} (z) = \\frac{(-1)^n}{2^n n!} (1-z)^{-\\alpha} (1+z)^{-\\beta} \\frac{d^n}{dz^n} \\left\\{ (1-z)^\\alpha (1+z)^\\beta (1 - z^2)^n \\right\\}~. ", "92774ad67d0dd57a567eb14724cb14d0": "\\mathbf{\\Omega} ", "9277740d8e9a6795e75da5c8ac16010b": " \\lim_{x \\to 0} \\frac{\\sin(x)}{x} = 1, \\! ~~ (3)", "9277978b92f6e51a600fc8154aad6d7a": "[\\mathrm{OH}^{-}] = \\frac{K_W}{[\\mathrm{H}^{+}]}", "9277d04582e132606b9e6c36951a2238": " \n\\int_0^{\\infty} {k\\; dk \\over k^2 +m^2} J_{\\nu}^2 \\left( kr \\right)\n=\nI_{\\nu} \\left( mr \\right)K_{\\nu} \\left( mr \\right)\n\\;\\;\\; \\Re \\;{\\nu} > -1.\n", "9277f52b522a329deebd0c2de239b695": "\\mathbf{v'}_1 = \\mathbf{v}_1 - \\frac {j_r} {m_1} \\mathbf{\\hat{n}}", "9278168741de0ae7f99500070775a743": "\\lim_{\\Delta x\\to 0}{ \\Delta h \\over \\Delta x}", "9278fa8bf74b06e62a30ff96f6bd98fc": "\\det g", "927925e59efd71b053cade0b7b145b67": "|\\alpha|", "92793aecb674978b482b231809b0ea16": "0 \\rightarrow C_k \\rightarrow A_k \\rightarrow C_{k+1} \\rightarrow 0", "9279474ee46fa561adc0651576e401a8": "\\Omega(m) ", "927993bc2b246c5f475b3b70e36875bd": "\\varphi(r,\\theta) = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\frac{1-r^2}{1 +r^2 -2r\\cos (\\theta -\\theta')} u(\\theta')d\\theta'.\\,", "9279c11ba62338e0308058252e9f4727": " \\psi_0(x) = \\delta(x - y) \\, ,", "9279ed94110f38f2ca6db440094a38e2": "\\sum_{m=1}^\\infty\\sum_{n=1}^\\infty\\frac{m^2\\,n}\n{3^m\\left(m\\,3^n+n\\,3^m\\right)}", "927a58af8363ba949e7ea886048129b3": "\n F = \\cfrac{R_0}{(1+R_0)~R_{90}}~\\cfrac{1}{(\\sigma_1^y)^2} ~;~~\n G = \\cfrac{1}{1+R_0}~\\cfrac{1}{(\\sigma_1^y)^2}\n ", "927ab2abcb711e7e4460b9b0ceab8b8b": " \\lim_{n \\to \\infty} \\left(H_n - \\ln n\\right) = \\gamma,", "927b3e6cd6780ff74834374239bfcac0": "0\\le j\\le k", "927ba74c84b692cc725542c2d3e3d909": "\\lim_{n \\to \\infty}\\ _{ordinal}\\alpha = \\rho", "927bdb3f234e7b5ec7647811fa6e6087": "(\\Omega',\\mathcal{A}')", "927bf96f2eb988c5ee11c5cb4bc459e1": "A)", "927c08ae7834f77e4f87f1db0930dc6a": "f = \\frac {a-b}{a} \\ , ", "927c43e3c63d1922b68424dd30ec5e3b": " \\frac{dD}{dt} = -k_2 D + \\frac{S}{H} ", "927c4b0764a95d41c778b449c0433ec3": "d(\\boldsymbol{x},\\boldsymbol{y})=d(x_1, \\ldots, x_n, y_1, \\ldots, y_n)=\\sqrt{(x_1-y_1)^2+\\cdots +(x_n-y_n)^2}", "927c79634afdf913ea1abab44780f22f": "O(1/r^4)", "927ca166fdd3a7f02570bd71e321a0a0": " q_s = \\frac{Y_A - Y_B}{SE}, ", "927d41df3d2158fed548c8fd7bb6f449": "\\operatorname{lift}", "927d9589adddf402aabfc7e7cd325cda": "x=w\\,t", "927db217acb4bba44b732ddb546aad09": " \\mathit{CM} \\in \\Sigma_2^P (= \\exists^{\\rm P} \\forall^{\\rm P} {\\rm P}) ", "927dfc85626843398e90215294c80b0f": "\\beta\\equiv\\frac{1}{k T}\\,", "927e066b8829c1e8fb5328d5bd65778b": "x_1, x_2,\\ldots x_n\\in R^\\nu", "927effd4ec8517d09428a13817f7ad8a": "\nf_{P} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{2p^{2/3}}{S}\n", "927f2df15c5576d38ef36fbe56f7fcd3": "e_1, \\ldots, e_n", "927f644f9fe7dec3907b422ccc3ca65f": "V(j,t)", "927fbf540ae0b7108236aaa33881794e": "x_{i_1 \\cdots i_N} = \\sum_{r=1}^{R} a^{(r)}_{i_1} \\cdots z^{(r)}_{i_N}", "927ff2780fd25f283c15a583b5991a58": "S_y(f) = h_{\\alpha}f^{\\alpha}", "928037e7a74c457edc20c7f6cd1023e4": "AZ+BZ \\to AB+2 Z ", "92803bef035c966f08c2116cff67818c": "-e^{s-x} \\sum_{j=0}^{np-1} f_i^{(j)}(x)", "9280a897876419e8f70795b0e9b076bc": "\\Pi_2 = \\Sigma_2 \\,", "9280d0b498646555dd8bf984c059c215": "\n\\frac{\\mathrm d}{\\mathrm d x} A^+(x) =\n -A^+ \\left( \\frac{\\mathrm d}{\\mathrm d x} A \\right) A^+\n+A^+ A{^+}^T \\left( \\frac{\\mathrm d}{\\mathrm d x} A^T \\right) (1-A A^+)\n+ (1-A^+ A) \\left( \\frac{\\mathrm d}{\\mathrm d x} A^T \\right) A{^+}^T A^+\n", "9280d6fb2c71dd9893e7cb407ca5393f": " \\mathbf{V} ", "9280dc6dee815e269cd5237902c67153": "M_{1x} = \\dot{m}V_{1x} = - \\rho QV_1 \\quad and \\quad F_{P1x} = \\overline{P}_1A_1", "9280e860199a54a43f4237ec96334d06": " \\mathrm{Tr}(S^2_2) = \\displaystyle \\sum_{j,k} |\\langle \\phi_j |\\phi_k \\rangle |^4 = \\frac{2d^3}{d+1} ", "9280f443f65f84153e814861b94653dd": "\\ \\nabla_{\\mathbf x}\\mathbf U", "9280ff79f1dc6a0a37a302af15f5406e": "\n\nw[n] =\n\n\\left\\{\n\\begin{matrix}\n\n\\frac{I_0\\left(\\pi \\alpha \\sqrt{1 - \\left(\\frac{2n}{N-1}-1\\right)^2}\\right)} {I_0(\\pi \\alpha)},\n & 0 \\leq n \\leq N-1 \\\\ \\\\\n\n0 & \\mbox{otherwise}, \\\\\n\n\\end{matrix}\n\\right.\n", "92812be027b92553c6024b48208c2c0e": "\\mathbf x, \\mathbf y", "92813de18f8abebde93fcd90f0bb3ae5": "A^+ = (A^*A)^+A^*\\,\\!", "928184b2a78f18319cc8548954f7cdca": "(\\alpha f)(x) = \\alpha (f(x)), \\forall x\\in E", "9281a418ca7721c18512f5797d2063f1": "\\chi^2_{k-1,\\alpha}", "9281e5a265f4fd910f3b466ef92b1b5a": "\\theta = \\theta_0 /2", "9281fb4e0f68349264db2de9a7bd7071": "-i \\hbar \\frac{\\partial}{\\partial x}", "9282381947fca267b088630c5e126a1b": "A = \\begin{bmatrix} 1 & 3 & 1 & 4 \\\\ 2 & 7 & 3 & 9 \\\\ 1 & 5 & 3 & 1 \\\\ 1 & 2 & 0 & 8 \\end{bmatrix}\\sim\n\\begin{bmatrix} 1 & 0 & -2 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix}=B\\text{.}", "928246f9298ed889f8f805669b119ce7": " \\frac{\\mathit P/E}{g} ", "928247afdc42ed40ac66929b3383c6c4": "X_k =\n \\sum_{n=0}^{N-1} x_n \\cos \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) k \\right] \\quad \\quad k = 0, \\dots, N-1.", "92825c62c826df0ef2877cf51c0f8da3": " \\mathbf{A}\\cdot\\left(\\mathbf{B}\\times\\mathbf{C}\\right)=\\mathbf{B}\\cdot\\left(\\mathbf{C}\\times\\mathbf{A}\\right)=\\mathbf{C}\\cdot\\left(\\mathbf{A}\\times\\mathbf{B}\\right)", "9282cd2d3e26446a90b14656ada51202": "\\{e^*\\in E^*\\mid e^* \\text{ attains its supremum on } B\\}", "9282e4a57f4d33689a0e95207c6eba80": "\\mu=\\langle t_{i}, t_{j}\\rangle", "9282f72fda9a1c4b0bb508b492b0c6ea": "{|\\psi_\\alpha\\rangle}", "928380eed768b3aaafc8cbcdd15de103": "dx = xa~", "9283d3bcf6325a423b1dc0aab555c682": "\\sqrt{2}\\sqrt{3}=\\sqrt{6}", "928417fe5a8643d24b892eaaac95a1bf": "H(X_1)", "9284220ad3b340f2b5cc5c20d995c906": "i=1..n", "92846ebc522f0f03375441275cb5dbbb": "y_{2J}", "9284cf19a2ae0582adcd8418c108bc49": "\\mathcal{S}\n", "92850381b65dd3e147009b423892311e": "g_\\text{threshold}", "928518d8f3e628b501ff92cb118b46a4": "(r,a,b,s) \\in \\delta", "92852bfbd77b2e6c21065459bb74d11c": "\\lim_{n\\to\\infty}S(\\varphi_n)= S\\left(\\lim_{n\\to\\infty}\\varphi_n\\right)", "92854af921745d630775056d8fb55783": "d^2G_S =n^2 dS \\cos{\\theta_S} d\\Omega_S = n^2 dS \\cos{\\theta_S} \\frac{d\\Sigma \\cos{\\theta_\\Sigma}}{d^2}", "9285560454655cddad98af03392dfee3": "e_m(P + R,Q) = e_m(P,Q)e_m(R,Q)\\text{ and }e_m(P,Q + R) = e_m(P,Q)e_m(P, R)", "92856d90817947a65863b257986dc125": "L(\\theta|X=3)=\\begin{pmatrix}12\\\\3\\end{pmatrix}\\;\\theta^3(1-\\theta)^9=220\\;\\theta^3(1-\\theta)^9", "928570bd9f6bdda37df9a3007ec71ac2": " m^2 =\\left(\\frac{E}{c^2}\\right)^2-\\left(\\frac{p}{c}\\right)^2 \\,\\!", "9285c8229082407edd173579a0222f57": "\\int_E |f_n|d\\mu<1", "928699c495da67f13ae7d9d57dec4862": "S_{nn}\\,", "92869a9a5440efa4c0819946f501bed7": "\\overline{P}", "928739ee36278dac2acd7a0450142161": "F(x) = \\int_a^x f(t)\\, dt.", "92879924ba63fa4b8ad9baa08dcc1fdc": "\\biggl|\\int_\\Gamma \\frac{1}{(z^2+1)^2} \\, dz\\biggr|,", "92879f2ef31e3f5b3eeee370bf5544b5": "\\ \\sigma_{num} ", "928819bc87d01bfb17735516f5647c7b": "\\beta\\approx1", "92884356be7833cca5876b053cf761c6": " \\Pi(n ; \\varphi \\,|\\,m) = \\int_{0}^{\\sin \\varphi} \\frac{1}{1-nt^2}\n\\frac{dt}{\\sqrt{(1-m t^2)(1-t^2) }}.", "92885bad1f746bba67bc7faf1d52478a": "p_{1},...,p_{m}", "9288905560bdf0997f4f7d115631c05f": " t_d = {2v_0 \\sin(\\theta) \\over g} ", "9288bd47baae8acb85eebfae104c6449": "g_{\\mu\\nu}\\,", "9288c12ea8d8b31c7870889000cf7c29": "\n\\begin{align}\n\\mathcal{I}(\\theta)\n& =\n-\\operatorname{E}\n\\left[ \\left.\n \\frac{\\partial^2}{\\partial\\theta^2} \\log(f(A;\\theta))\n\\right| \\theta \\right] \\qquad (1) \\\\\n& =\n-\\operatorname{E}\n\\left[ \\left.\n \\frac{\\partial^2}{\\partial\\theta^2} \\log\n \\left(\n \\theta^A(1-\\theta)^B\\frac{(A+B)!}{A!B!}\n \\right)\n\\right| \\theta \\right] \\qquad (2) \\\\\n& =\n-\\operatorname{E}\n\\left[ \\left.\n \\frac{\\partial^2}{\\partial\\theta^2} \n \\left(\n A \\log (\\theta) + B \\log(1-\\theta)\n \\right)\n\\right| \\theta \\right] \\qquad (3) \\\\\n& =\n-\\operatorname{E}\n\\left[ \\left.\n \\frac{\\partial}{\\partial\\theta}\n \\left(\n \\frac{A}{\\theta} - \\frac{B}{1-\\theta}\n \\right)\n\\right| \\theta \\right] \\qquad (4) \\\\\n& =\n+\\operatorname{E}\n\\left[ \\left.\n \\frac{A}{\\theta^2} + \\frac{B}{(1-\\theta)^2}\n\\right| \\theta \\right] \\qquad (5) \\\\\n& =\n\\frac{n\\theta}{\\theta^2} + \\frac{n(1-\\theta)}{(1-\\theta)^2} \\qquad (6) \\\\\n& \\text{since the expected value of }A\\text{ given }\\theta\\text{ is }n\\theta,\\text{ etc.} \\\\\n& = \\frac{n}{\\theta(1-\\theta)} \\qquad (7)\n\\end{align}\n", "9288daebc7050bfb1895ac872320336c": "\\frac{6}{3}=2", "92891ac6a9ea8ca235b1f738b1ccd237": "\\left(\\sqrt{\\frac{2}{5}},\\ -\\sqrt{\\frac{2}{3}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm3\\right)", "92895eb9b9da9bc8e102dd1a38c9a0b6": "\\begin{align}\\frac{dx}{dt}&=f\\left(x,y,t\\right),\\\\0&=g\\left(x,y,t\\right).\\end{align}", "92895fff82a1c3299abebef9788a1a54": "\\tfrac{EO_1}{BO_1} = \\tfrac{3}{10}", "9289757f5d3f35ef90a3335487f354c5": "\\operatorname{Ad}", "928a017b1498c0fa81271c62160073c3": " (W(t): t \\in [0,1]) ", "928a093bd113324156aed473983ebe5f": "x^4+px^2+q=0", "928a1542e43058a426416b4d5b084429": "\\lambda (n) = (-1)^{\\Omega(n)}.\\;", "928a24f2cb6ded435555b33a5d403a17": "g \\circ f \\circ g = g", "928a2ab11dd474ac8cb744f21c9c5fed": "N(t) = C 2^{t/d}", "928a49501d71853d72dbe406d1f6c0dd": "a^2(y_0 - n)^2 + b^2(x_0 - m)^2 = 2ab(y_0 - n)(x_0 - m).", "928a7b40843d0391cff721435656b369": " \\mathbf{a \\cdot b} = | \\mathbf a | | \\mathbf b | \\cos \\theta , ", "928a81e79919b660776b3f22453a5176": "\n\\max_{f(X_1,X_2)} I(X_1;Y_2,Y_3|X_2) = \\frac{1}{2} \\log(1 + (1 - \\beta) (c^2_{21} + c^2_{31})P_1 )\n", "928b01d70dae0f1e1c8a1de982cf451b": "k \\in \\{0, 1, 2, \\ldots\\}", "928b151f210622f0b6c2a6036e8c892d": "A(A^*A) = (A^*A) A.\\,", "928b56d3627404bbc0e692f6dcea8eef": "(A*F)", "928bb2f4ffef4722c68fbcaeba986294": " \\quad (\\delta_q \\omega)(\\sigma) := \\sum_{j=0}^{q+1} (-1)^j \\mathrm{res}^{|\\partial_j \\sigma|}_{|\\sigma|} \\omega (\\partial_j \\sigma)", "928c8764533d486f9104d057c764b1c4": "|T_i|\\le\\sum_{n=1}^i\\bigl|\\!\\operatorname{E}[X_n]\\bigr|,\\quad i\\in{\\mathbb N}.", "928cad6b7fb2977897620fb55ead987d": "\\textstyle \\mathbb{Z}/p\\mathbb{Z}", "928caf364a32c972f5949b685b18d42b": "P\\vdash_L A", "928cbfc3914133b5710fc7bd149bd661": "\\beta \\mathbf{N}\\to\\beta \\mathbf{N}", "928cf4c1e8197e7a3d7204dad61193fa": "= \\left(f_u T^{\\otimes (\\deg u)}\\right)^{\\otimes |U|} \\left(f_v (T^{-1})^{\\otimes (\\deg v)}\\right)^{\\otimes |V|}.", "928d1b1774cb6e65abee7797912ddd41": "\\text{EMA}_{\\text{today}} = \\text{EMA}_{\\text{yesterday}} + \\alpha \\times\n\n(\\text{price}_{\\text{today}} - \\text{EMA}_\\text{yesterday})", "928d3a30982d0f7f349c2b3b63687137": "\n\\zeta^{\\alpha}_{st} = \\sum_{i=1}^N \\sum_{\\beta,\\gamma=1}^3 \\epsilon_{\\alpha\\beta\\gamma} \nQ_{s, i\\beta}\\,Q_{t,i\\gamma} \\;\\; \\mathrm{and}\\quad\\alpha=1,2,3. \n", "928d4fa7de82598e2a6b221cda908bf6": " H_{D}(x,R) = -\\sum_{i \\in (1\\dots r)} P_{D}(d_i,x,R) \\log P_{D}(d_i,x,R).", "928d5619a1ab67a2a603616d0779165d": "\\begin{align}\n\\nabla\\times\\mathbf{F} &= d\\omega_{\\mathbf{F}} \\\\\n\\psi^{*}\\omega_{\\mathbf{F}} &= P_1du +P_2dv \\\\\n\\psi^{*}(d \\omega_{\\mathbf{F}}) &= \\left (\\frac{\\partial P_2}{\\partial u} - \\frac{\\partial P_1}{\\partial v} \\right ) du\\wedge dv\n\\end{align}", "928d5971c23596dbe9945c8c58efe6ba": "\\int y \\, dx = \\int y \\frac{dx}{du} \\, du.", "928d6f1d3d19b1f6565433a99d3e0194": "\n\\operatorname{Li}_{s+1}(z) = \\int_0^z \\frac {\\operatorname{Li}_s(t)}{t}\\,\\mathrm{d}t \\,;\n", "928dbd6e431daf93a95c9ceaac594a19": "\\!\\mathcal A \\models_X^+ \\phi \\vee \\psi", "928dd222fe7179e366428cf7a562aafe": "f_\\omega(n) = f_n(n)", "928df6bc8ec9f1ffbabec2d7ed2a4aa7": "B_\\phi", "928e161ba9f90128bd0a3db52408d6d7": "\n\\Sigma_{ij}\n= \\mathrm{cov}(X_i, X_j) = \\mathrm{E}\\begin{bmatrix}\n(X_i - \\mu_i)(X_j - \\mu_j)\n\\end{bmatrix}\n", "928e5c872773abc6395f7ab90302a167": "R_2 = \\frac{V_{Z} + V_{D} - V_{BE}}{I_{R2}}", "928e5e2fc477f931f37905b7516c5b67": "\\langle\\phi^{i_1}\\cdots\\phi^{i_n}\\rangle_\\text{con}=(-i)^{n+1}E^{,i_1\\dots i_n}|_{J=0}", "928e80a7483623b461e2f273b70329ec": "+ c(x, \\sigma(x),\\sigma'(x),\\sigma'(x))d(\\sigma'(x)) + e(x, \\sigma(x), \\sigma'(x),\\sigma''(x))d(\\sigma''(x)) \\,", "928ecdca82b00ac3a7c6bd1e2ffc18e9": "v(k)", "928f0dc52fc7dd52448c3dc1fe062994": "i: Y \\to Z", "928f89d379a125d8891b8d74ca261a0e": " \\delta(t) \\ ", "928fb09adf59f0ec765c6e84b29b587e": "z_i=\\sum_{j=1}^2 c_{ij} y_j,", "92900f6f7d230101f1434e413e8616bb": " F = C \\times E ", "9290198303e1b8dea9b4f3fb955d1485": "g(z, u) = \\left(\\frac{1}{1-z}\\right)^u", "929019f88b77a0334043ea04bc15f72d": "X_1=0", "92904f3397138cd6c11a1bb28a04a7ef": " \\alpha = \\arccos \\left( \\frac{A\\cos \\beta-V}{W} \\right) = \\arccos \\left( \\frac{A\\cos \\beta-V}{\\sqrt{A^2 + V^2 -2AV\\cos{\\beta}}} \\right)", "92906bde9524c0f814dc14bd1f7a6772": "\\frac{\\Gamma, \\alpha \\vdash \\beta}{\\Gamma \\vdash \\alpha \\rightarrow \\beta} \\rightarrow I", "92906ee0aab61630651f09f0bd055ec3": "T \\cup E", "929093ee16ba2204b9d73e7145852f70": " m 2^{-k} \\approx (n 2^{-k})^{-1} ", "92909973e694992f987fc3aee3d8b6f3": "w^{-1}\\sum_{i=1}^n w_iY_i \\sim ED(\\mu,\\sigma^2/w)", "9290d8e85f304c4d307a2f32b7239f87": "\\tfrac {1}{4} \\pi^2 - \\pi i \\ln 2 \\,", "9290ddb6750fe4e1b3bc6c427fc1d8cb": "\\sqrt{n}(\\hat\\beta-\\beta) = \\bigg(\\frac{1}{n}\\sum_{i=1}^n x_ix'_i\\bigg)^{\\!\\!-1} \\bigg(\\frac{1}{\\sqrt{n}}\\sum_{i=1}^n x_i\\varepsilon_i\\bigg)\\ \\xrightarrow{d}\\ M_{xx}^{-1}\\cdot\\mathcal{N}\\big(0,\\sigma^2M_{xx}\\big) = \\mathcal{N}\\big(0,\\sigma^2M_{xx}^{-1}\\big)", "9291034375a69981432e475cbfecd183": "= g ((\\lambda x.g\\ (x\\ x))\\ (\\lambda x.g\\ (x\\ x)))", "92910a5004dc05fbf1f3dc97790e7c0e": "(\\boldsymbol{x}_n,y_n) \\in \\mathcal{X}\\times\\mathcal{Y}", "9291441d38dc38f08a56b6dedf318b41": "\\lim_{n \\to \\infty}{ B_n(f)(x) } = f(x) \\,", "92915e23273059c58ce915699aa91745": " S = \\begin{pmatrix}1 & -1 \\\\ 1 & 1\\end{pmatrix} .", "929186a79052d4b80fdcf1ecdbd909e9": " G_i - {G_i}^{-1}=m(1-E_i), ", "929199ae13617fd7b5743894d2254125": "\\chi_-(1)\\chi_-(2)", "9291f10d610ac3a35bcc98667319bfc8": "(n-2)\\times \\frac{180}{n}", "929209d8ea5785d7df49ce41df86ecff": "A^T A v=A^T b", "92921e8c0df043785dbd62875e7af3a1": "\n\\min_{\\alpha \\in \\mathbb{R}^p} \\frac{1}{2} \\|x - D\\alpha \\|_2^2 + \\lambda \\|\\alpha\\|_1,\n", "9292760d4354583594244e9e9e415294": " q \\geq 2", "92929419abc60a1a257536c509fd5db7": "Ra = -r,\\,", "9292a5e0881b60546378f67ad443463c": "T.", "9292c54e46b7e92713a0e2ad1c40fc87": "\\langle \\sigma_{1}, \\sigma_{2} \\rangle_{L_{0}^{2,1}} := \\int_{0}^{T} \\langle \\dot{\\sigma}_{1} (t), \\dot{\\sigma}_{2} (t) \\rangle_{\\mathbb{R}^{n}} \\, \\mathrm{d} t, ", "9292d15fb3207f1c3ed927bf7615b6d8": "p(y_k | c_j) \\,", "9292dd0dd69614684741230facd99ea1": "250 \\sin \\frac{1}{60} = 0.07272\\ \\mathrm{mm} = 72.72 \\mu\\ \\mathrm{m}.", "92931cc072769590750c8011ce42c786": "Q(1)=I. \\,", "92932f7603b29d79908c339bc470c4b5": "[.,.]_A", "9293368eced79a576126ca81fbee582d": "t' = \\frac{c}{a} \\sinh \\left(\\frac{a \\cdot t}{c} \\right)", "929347b46c0df88d035e587d02e41e89": "\\vec{p}_2, \\; \\vec{p}_3", "929361b2043f282f6b1314305e447370": "\\Sigma_{y=0},\\Sigma_{y=1} ", "929363de78db80eb04c257a7a69cb186": "5 * 10^{13} M_{sun}", "9293996d05880b645f51266a76a9dbdc": "V_L = V_i - V_o", "9293db15f8ca73d05ec0c194274ec7fb": "\\phi:A\\rightarrow \\operatorname{Aut}(G)", "9294324df0fb0af53245aafa73ee147c": "\\frac{\\det M}{\\det V}", "92946f7eb6369e344650aa555c85c795": "\\boldsymbol{J}_2", "92947d800786328f67c0f6666894c9ec": "x^\\alpha = x_1^{\\alpha_1} x_2^{\\alpha_2} \\ldots x_n^{\\alpha_n}", "9294d70590f16d4520b5d654b7359b4c": "v(L \\; p) = \\text{True}", "9294fb3493c5342b324e9e36ba894dd1": "Q = \\frac { V\\Delta H_0 K_a[H_0][G]}{1+K_a[G]}", "929514916dbe25e50fbc73083edc977c": " \\textrm{a} = {{v^2} \\over {r}}", "929515039f839b214ce3aee32e8a977a": "D = \\prod_{i=1}^n D_i.", "929519131c2475d6fe2014e7b9dc0713": "\\left(\\int_{-\\infty}^\\infty (x-x_0)^2|f(x)|^2\\,dx\\right)\\left(\\int_{-\\infty}^\\infty(\\xi-\\xi_0)^2|\\hat{f}(\\xi)|^2\\,d\\xi\\right)\\geq \\frac{1}{16\\pi^2}", "929565595de1223776e0371f03f71d73": "X \\to \\mathbb P \\Gamma (X, \\mathcal L^{\\otimes k})", "929589db71a66d7f8eb8ae28d7364a6c": " \\Lambda_\\gamma^2 + \\frac{\\partial^2\\ }{\\partial\\theta^2} = 0 ", "9295f2e14826f2a17ced410e9474a243": " B_3 = V^2 \\left[ \\frac{2Q_2}{Q_1^2}\\Big( \\frac{2Q_2}{Q_1^2}-1\\Big) -\\frac{1}{3}\\Big(\\frac{6Q_3}{Q_1^3}-1\\Big)\n\\right] ", "929649b7ec045ea9c7226dc99f69795e": "\\Delta_{\\theta} = \\{ 0 \\} \\cup \\{ t \\in \\mathbb{C} : | \\mathrm{arg}(t) | < \\theta \\},", "929658bc8606194fd4a138f2d1cc9646": "=({10^{1.0}})^{(3/2)}", "92965e23345ef8de5a895408384a1108": "x^{(k+1)} \\approx x^{(k)}", "92969ed84d096255098144c7735e4b51": "K_i (\\tilde M) : = \\mathrm{ker} \\{f_* \\colon H_i (\\tilde M) \\rightarrow H_i (\\tilde X)\\}", "9296a618dbbc4aeae582df82ddcd9762": "2 t", "9296a7459b2ffa8222f69a4a8e7b11f0": "\\mathbb{F}_{2^8}", "9296bc8a4c77eaa023707bba822d90f7": "\\frac{(k+1)p_k}{\\sum_k (k+1)p_k}=\\frac{(k+1)p_k}{m+1}", "9296d619e278b70fe5dc38e433c36217": "A^{n-1} v", "9296de6e37508a55d1589730dcd0258f": "x(\\tau)", "929730cb2888015b8f552664c033dbad": "|{\\mathbf P} (t)|^2", "929774ea9e552ca4126eeae0f450b061": "P_{(1)} \\ldots P_{(m)}", "9297cc7eda0ff003143c516784a665e3": "f(x) = {1 \\over \\pi (1 + x^2)}", "9297ddff2476bbb79d5f0d629af6a4dd": "0=\\vec\\nabla\\times\\vec v_s + (q/m)\\vec\\nabla\\times\\vec A = \\vec\\nabla\\times\\vec v_s + (q/m)\\vec B", "92983b2a15d54ef56c8346a88106d530": "\\ S_e ", "9298431e589d100a2b9da1a8c5f61031": "(d*3*3*d)\\times(d*3*3*d)", "929a5a8a0945ffd920727b070de26ee7": " S\\bigg(\\sum_{i=1}^k \\lambda_i \\, \\rho_i \\bigg) \\,\\geq\\, \\sum_{i=1}^k \\lambda_i \\, S(\\rho_i). ", "929ab4d3154d9fb4ebe7aa51d9666e39": "\\|x\\|_2\\le\\|x\\|_1\\le\\sqrt{n}\\|x\\|_2", "929ab923ab708be4118a336fbcf37c47": "\\binom{d+1}{2}", "929ada1bab7cd780973272c0713193db": "\\sum\\nolimits_n d(T^n(x),T^n(y))<\\infty.", "929b27d339fb26c686b4bb4e1e7437ba": "(x)_k=\\frac{\\Gamma(x+1)}{\\Gamma(x-k+1)}", "929b63f14cccfa8ecd50c1491b3845a1": "\nK_t = \\Phi \\exp(-t\\Lambda) \\Phi^T.\n", "929b7e91eb0d9b0a5d91e9a716464f5d": "\\delta n_k \\epsilon_k", "929b9a57bbe7c25d5a7b9be9cc404e5b": "\\Pr(Y_i=c) = \\operatorname{softmax}(c, \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i, \\ldots, \\boldsymbol\\beta_K \\cdot \\mathbf{X}_i)", "929b9c6c70ba9e9be32045a58453f6bf": "\\big\\{ x \\mapsto a x + b : a\\in\\R\\setminus\\{0\\}, b\\in\\R \\big\\}=\\big\\{\\begin{bmatrix}\n a & b \\\\ \\\\\n0 & 1 \\end{bmatrix}\\big\\}", "929ba7ea0731542ebe65ed53b92c80b0": "\\begin{matrix} {1 \\choose 1}{3 \\choose 1}{11 \\choose 1}{4 \\choose 2} \\end{matrix}", "929c31388ae39333c1544a5356784a08": "y \\,", "929c62a321a2dee23c7a2203cf1dd905": "\\beta^{T}_k = (\\beta_{1k},\\beta_{2k},\\ldots,\\beta_{J'_{k}\nk})", "929cf3eaf4e59383f4d01e4d1618e595": " L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f)~.", "929d1b0bdeb03d42236e2e7e95d2d81c": "2\\ln(2|\\mathcal U|) c\\cdot x^*", "929d1b66ac0d666f98ba684d787c0b68": "q=p^i,", "929d299f8bc72422ec121049565dde75": "\\operatorname{pred}", "929d36c5c18d5418b71280bef32ad591": "2ac = 0,", "929d4ca0748fe0c54d52d45995a1a75f": "\\frac{g(y)}{g(x)}", "929d5fe05a2ea29f2f975557386d0167": "O(e^M)", "929d948e14179f0042d2401778692a32": "h(\\bar{x}) = \\left( \\sum_{i=0}^{k-1} h_i(x_i) \\right)\\,\\bmod~m", "929d9e9a64c4606e02b38943683780ea": "\\begin{align}\n \\Delta n_{\\text{E}} (x'') &\\rightarrow 0 \\\\\n \\Delta n_{\\text{c}} (x' ) &\\rightarrow 0\n\\end{align}", "929de53fec37d7e720f48a70a97ce39a": "\\mu .", "929e4eb954b712d7787faabc91fbaa99": "\\pi_\\lambda(f)\\xi_0 =\\tilde{f}(\\lambda) \\xi_0", "929e669c7576d2e6995bda89a0ae883f": "\\sin\\theta \\;d\\theta = \\frac{s}{rR} ds.", "929e75803334ac82ab5a2237319cc93c": " \\int_k^{k+1} f'(x)P_1(x)\\,dx = \\int u\\,dv ", "929e84987897168296b0f7737dedbac6": "\\sigma_1, \\sigma_2, \\dots, \\sigma_m", "929e8fe79c92395adf1ae2ac3f2a97b0": "x \\equiv \\alpha y ", "929ea8af1682e18ff4d9ad07d75f8633": "c(t) + s\\cdot u(t)", "929f229e0f19fe46d509d8e2c246b1ca": " IFx_1x_2...x_n \\leftrightarrow Fx_1x_1...x_n \\leftrightarrow Fx_2x_2...x_n.", "929ff8399eb837158e989e50a20dfea3": "\nP(X,Y) = P(X) P(Y|X) \\,\n", "92a00eaab69fa3574bcf91e2ce163652": " E=[0,2) ", "92a087f94bc643e21f911cd69e3d142c": "(\\hat{\\epsilon}_1,\\hat{\\epsilon}_2)", "92a0a126dc256a026b8c51f9d6e70111": "\\begin{pmatrix}a & b\\\\c & d\\end{pmatrix}", "92a0ac2cec54d57eacbbd711d484ad16": "(\\alpha, *, f)", "92a0ad6a34e2e872f87c5d3c2bd519a2": " \\frac{1}{1+JF^2} ", "92a0c9bac70d0636034a88a1260e6dce": "\\rho_q(\\bold{r})= q n(\\mathbf{r})\\,.", "92a120c4b17b65ad9552701e141d5039": " \\operatorname{Kurt}[Y] = \\frac{\\kappa_4(Y)}{\\kappa_2(Y)^2} = \\frac{n \\kappa_4(X)}{(n \\kappa_2(X))^2} = \\frac{1}{n} \\frac{\\kappa_4(X)}{\\kappa_2(X)^2} = \\frac{1}{n}\\operatorname{Kurt}[X] .", "92a15e828174d8ed3d0f1a956de4868b": "E_\\mathrm{surf}=2\\pi l^2 \\sigma", "92a1806107c163bcadb3f726b55c0a22": "{1 \\choose 1}_q = \\frac{1-q}{1-q}=1", "92a1bf2bf0bb57d03a755a9420614a4e": "(X_1, X_2,\\cdots,X_K)", "92a1f4c1c6db467f58b3ebdb5bc5e5bc": "(\\mathbf{x},z_1)", "92a218e823142f8dc2fd06663cbfb493": "\\Delta^2(a_n)", "92a299effc060a80fbf2009842ac96c9": "\\exp(X) = 1 + X + \\frac{X^2}{2!} + \\frac{X^3}{3!} + \\frac{X^4}{4!} + \\cdots,", "92a2ac3bedd85dc75cfab6125db69037": "U_\\mathrm{E} = \\frac{1}{2} ( Q_1 \\Phi(\\mathbf{r}_1) + Q_2 \\Phi(\\mathbf{r}_2) + Q_3 \\Phi(\\mathbf{r}_3) )", "92a3075610444d5ce3ed2ebcd15afa56": "(G_0 : G_t)", "92a351ef1d8fdcd47ac30837174d5c64": "\\mathbf{\\dot{p}} = \\boldsymbol{\\mathcal{Q}}\\,.", "92a3a74f1feee8b1b94b04f88f86cc20": "a^0=1", "92a4070e1f194405c814dec806bbce5f": "AB=\\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix}", "92a4d79db86f5db78d6e0e3a945a0332": "\\beta_1 ", "92a51e051e46a9de1617c6a0523a704f": "n>j", "92a54b358b4cf53cca4095e4697e1004": "MC", "92a565b72b55111ab45a3c1119b75d50": "\\frac{\\text{d}C_i}{\\text{d}t} = \\text{k}_{1(i)} E\n\\overline{S}_i - [\\text{k}_{2(i)} + \\text{k}_{3(i)}] C_i \\qquad \\qquad (3b) ", "92a56ac00f89826f8f0e1b970f990adb": "U(N_{i,j})", "92a626be006fbdda47430738bdd4e4a8": " \\operatorname{F}_{t, s}: X \\rightarrow X \\quad \\forall t,s \\in \\mathbb{R} ", "92a643043717c84e9086eb800a0f010f": "\\theta\\in[1,\\infty)", "92a64e198141565d955e2b17dfc55a03": " \\mathbf{p} = q\\mathbf{d} \\ . ", "92a6935de81a189d5ce2dcca2c801676": "\\scriptstyle \\mathbb{P} =\\frac{1}{2}(\\boldsymbol{\\nabla}\\mathbf{u} + (\\boldsymbol{\\nabla}\\mathbf{u})^T) - p\\mathbb{I}", "92a6a13bcf723c7225bc6465573fbd59": " I = \\frac{1}{4} \\int_0^{2\\pi} \\frac{1}{1 + \\sin(t)^2} \\,dt.", "92a70090d3c99d4becf1cfef75c90778": "[x_1,x_2]\\in\\mathfrak{g}_{\\lambda_1+\\lambda_2}", "92a814fdd63206ae5f0ff7671d1126d9": "x=f(s|\\theta)+n", "92a826aaf8518d0458602bbbdbd97b6e": "k_i = k_{\\alpha_i}, q_i = q^{\\frac{1}{2}(\\alpha_i,\\alpha_i)}, [0]_{q_i}! = 1, [n]_{q_i}! = \\prod_{m=1}^n [m]_{q_i}", "92a854748dcca495af0675e81c5a4c84": "\\dot \\lambda", "92a862793fc1360f8d4c93aff1280bb4": "x\\cdot Du+f(Du)=u", "92a8d3d1faf04d5eab3ffced11482dd2": "W_Z= \\left(\\frac{z/z_n}{y/y_n}-1\\right)V_J", "92a908dd476e2d3b7b9ee90868c270be": "\\frac{15 \\mu}{\\lambda} ", "92a92b6815352d7d8c88854dbbaad6a9": " \\left(\\frac{\\ell}{m}\\right)", "92a97d7e62eb00f213d19c3c5df72556": "\\|x\\|_1 = \\sum_{i=1}^\\infty |x_i|", "92a9b7d68a372f3d0273030a8a524b9e": "SO(1,3) ", "92a9d94a3d5e91e8a5c61870abe3b24b": "(w,\\lambda)", "92aa2041f3fc833c69a8ac808875036d": "u_t =1", "92aa5366943075491e2980af3f4134bd": "\n\\frac{1}{3}\n\\begin{bmatrix} \n 1 \\\\ 1 \\\\ 1 \n\\end{bmatrix} \n*\n\\frac{1}{3}\n\\begin{bmatrix} \n 1 & 1 & 1\n\\end{bmatrix}\n\n=\n\n\\frac{1}{9}\n\\begin{bmatrix} \n 1 & 1 & 1 \\\\ \n 1 & 1 & 1 \\\\\n 1 & 1 & 1\n\\end{bmatrix} \n", "92aa810d64f16b106ebeff50378416e9": "\\frac{dG}{dt}", "92aad38c8960be14615eab0108f6095d": "\\begin{smallmatrix}\\frac{1836}{1837} \\approx 0.99946\\end{smallmatrix}", "92ab645ab7898177b26b4aad8fba5f5d": " \\cos \\theta ", "92ab6d6e686ba612d24217643dd33361": "a^{K(p-1)} \\equiv 1\\pmod{p}", "92aba0dcee76ff2ab8705af9bff872d9": "\\sin k\\theta = 2 \\cos\\theta \\sin (k-1)\\theta - \\sin (k-2)\\theta", "92abc29d99477f6b2834ac6dcb86551c": "\\sigma^*_i", "92ac31c17b4f0184061c3688dec8a242": "g \\equiv (f^{\\prime})^{-1}(p)", "92ac5056bb5054205ed9f1d894721118": "\\mathrm{FillRad}(M)=\\mathrm{FillRad} \\left( M\\subset L^{\\infty}(M) \\right).", "92ac55d2004126d87077d6d950f3819d": "\\sigma_z\n= -2\\frac{\\partial^2 C}{\\partial x \\partial y}", "92ac6fab7f7780376633d06fc0b98b86": "\\dotsb\\overset{\\partial_{n+1}}{\\longrightarrow\\,}C_n\n\\overset{\\partial_n}{\\longrightarrow\\,}C_{n-1}\n\\overset{\\partial_{n-1}}{\\longrightarrow\\,}\n\\dotsb\n\\overset{\\partial_2}{\\longrightarrow\\,}\nC_1\n\\overset{\\partial_1}{\\longrightarrow\\,}\nC_0\\overset{\\epsilon}{\\longrightarrow\\,} \\mathbb{Z} \\to 0\n", "92acc57261f673f4a112b6120c7322f9": "\\scriptstyle v_i = u_i/u ", "92ace52df817359b9357d1c6f725325a": "\nr_1 r_2 r_3 (p_1 p_2 \n+ p_1 p_3 + p_2 p_3) - 2 p_1 p_2 p_3 r_1 r_2 r_3 = 0 \n", "92acf810c260e34bacf40d6099721e09": "Z_R = R \\iff Y_G = G ", "92ad097037d52c0e97438f694729296a": "I = \\frac{V_{applied}-V_{cemf}}{R_{armature}}", "92ad0c99805eda8caf406dca6947eca5": "\\alpha_i\\in\\mathbb{R}^+", "92ad88798bafe5eac242a4053da1057e": "1 = \\sum_k u_{1k} \\kappa(u_{k1}) = \\sum_k \\kappa(u_{1k}) u_{k1}.", "92ae5b4819dc0030378894fc7b6f0ad8": " cz \\approx D H(t_0) = v_r \\ . ", "92ae7340113a8a68bf7d67069a17c106": "E_{i}(s_{i})", "92ae821610ad6a36f2911d1e196d2848": "\\beta\\left(\\pi\\left(x_0,m\\right),i\\right) = \\mathrm{rem}\\left(x_0,m_i\\right)", "92ae90313dc55e908469c75e490b95bb": "\\hat{I}", "92ae90451bd209fa5a233b4908a26fe9": "\n \\cfrac{\\partial \\mathbf{b}^i}{\\partial q^j} = -\\Gamma^i_{jk}~\\mathbf{b}^k ~;~~\n \\boldsymbol{\\nabla}\\mathbf{b}_i = \\Gamma_{ij}^k~\\mathbf{b}_k\\otimes\\mathbf{b}^j ~;~~\n \\boldsymbol{\\nabla}\\mathbf{b}^i = -\\Gamma_{jk}^i~\\mathbf{b}^k\\otimes\\mathbf{b}^j\n ", "92ae964a5054d973770606deb2c620ca": "Q(x_{\\ast})=0", "92af84b1bbc25d05426fd490858bb0fc": "B= (2 \\pi^2 \\Lambda^3) {1\\over (2\\pi)^4} { b \\Lambda} {1 \\over b\\Lambda^4} = {1\\over 8\\pi^2} ", "92af9f5f063d64eb16c4e0427bb31542": "\\forall x \\in I, \\forall r \\in R : \\quad r \\cdot x \\in I.", "92afc138f46436355c3a9bc0432b4032": "\nZ^T Q Z \\mathbf{y} = -Z^T \\mathbf{c}\n", "92afcd0228e69d9305356547915daac4": "\\sigma_+ = \\sqrt{\\sigma_x^2+\\sigma_y^2+2\\rho\\sigma_x \\sigma_y}.", "92affdccf7ec2bb7f50c98e8dffb7c25": " \n\\begin{align}\n\\frac{\\mathrm{d}}{\\mathrm{d}x} \\frac{f'(x)}{\\sqrt{1 + (f'(x))^2}} &= 0 \\\\ \n\\frac{f'(x)}{\\sqrt{1 + (f'(x))^2}} &= C = \\text{constant} \\\\\n\\Rightarrow f'(x)&= \\frac{C}{\\sqrt{1-C^2}} := A \\\\\n\\Rightarrow f(x) &= Ax + B\n\\end{align}\n", "92b0030e5528b13ea1636b24d1eef8f2": "(r,t,h)", "92b084669aff4abc181b1211f9b7456f": "\\mathbb{PT}", "92b09fa1636821ebf88af81e5f98221c": "\\qquad\\vdots\\qquad\\vdots\\qquad\\vdots\\,", "92b0dec60cc73b9cf30dd25beb026cfb": " u_1 - u_0,\\dots, u_k-u_0 ", "92b13b2f2f1efb10c30522f32f72e8e7": "\\varphi = \\frac{1}{4\\pi\\epsilon_0}", "92b153d4883517e244d6bd3f4ef1f9c7": "gN", "92b18e9313d1f348079be098b8309c17": "m \\not= p", "92b1930862d637cf4ca0f328392a0c68": "\\mathbf{Q}(\\sqrt{k})", "92b1a58edac48524e75171e3140451d2": "10\\log_{10}(4) \\approx 6.0206", "92b1bc26e6d95563701162c9854df08d": " 1\\le i,j\\le n ", "92b1c25b459aadf2880609b47c2af1b7": " \\dfrac{d^{\\frac{1}{2}}}{dx^{\\frac{1}{2}}}x=\\dfrac{\\Gamma(1+1)}{\\Gamma(1-\\frac{1}{2}+1)}x^{1-\\frac{1}{2}}=\\dfrac{1!}{\\Gamma(\\frac{3}{2})}x^{\\frac{1}{2}} =\n\\dfrac{2x^{\\frac{1}{2}}}{\\sqrt{\\pi}}.", "92b236cff9e259de9c0955621bae6042": "A \\in N", "92b252e55c375d14f59c78847470a784": "\\begin{matrix} \\frac{n}{2}\\end{matrix}", "92b261f5cfffb57b5c80eaf1a948ec71": "S^{0j} = 0\\,, \\quad S^{jk} = \\epsilon^{jk\\ell}S^\\ell \\,,\\quad S^\\ell = \\int_{\\partial \\mathcal{V}} \\epsilon^{\\ell mn}(x^m - x^m_\\text{com}) T^{0n} dxdydz ", "92b2bac6f804bd6849d5ff1114adf003": "\\partial_t g_{ij}=-2 R_{ij}", "92b307a45abd97a85faea9de65b668c2": "\\inf\\mathop{\\rm supp}\\,\\phi\\ast \\psi\n=\\inf\\mathop{\\rm supp}\\,\\phi+\\inf\\mathop{\\rm supp}\\,\\psi", "92b330ac82ca7320f514f0de8aa11b64": "L_+^kY = 0.", "92b388ddc77db39e88b2ff25d35d2a79": "\n {\\mathbf v} = \\sum_i v^i {\\mathbf e}_i = \n \\sum_i {v'}^i {\\mathbf e}'_i\n", "92b3c059cf78b3a971edf24d5665f940": "\\ q = 4\\pi n_0 \\sin(\\theta/2)/\\lambda", "92b3f28a63310f6dd94d3643cc14f02a": "r_k^*", "92b3fc603d18a58e2f7cf49b49dd9e13": "\\cos \\theta \\cos \\varphi = {{\\cos(\\theta - \\varphi) + \\cos(\\theta + \\varphi)} \\over 2}", "92b3fd2854a1c7b1dbfd56a074e72070": "C_{4k+2} \\cong C_{2k+1} \\times C_2,", "92b45cf2088cce076995dac41b93861d": "\\mu,\\nu,\\rho,\\dots", "92b4e78fd6b168954d439aeceee63760": "(b^{kn}+1) = (b^n+1) \\sum _{r=0}^{k-1} (-1)^r \\cdot b^{rn}", "92b51b6807482c733f540e89a7b86f88": "\\phi(x)\\phi(y)=-\\phi(y)\\phi(x)\\,", "92b54a4f11a8ad3d3ab78270a999e1ce": "m_p(f) = {p^{(rn(n-1)+s(n+1))/2}\\over N(p^r)}", "92b567e88c5728f8f85b14acb6abdfb5": "\\rho(x)\\le1/x!.", "92b5e082bcfb6d861f49e76ec8b7f19f": " G\\circ F", "92b612795a539bef535c8dc525d3308b": "f_2(z) = \\,_2F_1(a+1,b+1;c+2;z)", "92b68727874952b96de54cfb2c27839c": "1\\leq i\\leq k", "92b6892227b295ff39af326698178c56": "F=Kx", "92b6be422848bdee5434673883a7c9f4": "i\\vec{n}", "92b6c486e32c12205b00ed80cb368ff9": "b = 0.2", "92b6c6af6d32cf01dd927079ba6dff12": "r = 1\\ ", "92b7013cd5e342f4b9a5da186a136e67": "x_m = T x_0,", "92b70295d773248a6364f897ff75339e": "m+n\\ge l \\ge m-n", "92b76fbf77ebf0b86b15e0881b0a1a49": "u_t", "92b7c5c419959806593a02dd5ed0772a": "\\mathbf{c} = \\mathbf{a}\\times\\mathbf{b} = \\mathbf{T}\\cdot\\mathbf{b} ", "92b7ff8848004da9779f3a5ac0bcd3b9": "\nv_n = \\sqrt { 4 k_B T R \\Delta f },\n", "92b807e71fb25999416994c848e0d0a5": "\n\\bar{F} = -\\frac{GM}{R^2}\\ \\hat{r}\n", "92b85c5eb8a8db5b4e5154808159788b": "\n \\beta := \\sin\\left(\\frac{\\pi\\alpha_B}{\\alpha_A+\\alpha_B}\\right) ~,~~ \\alpha_A := \\frac{\\pi a}{2 d} \n ~,~~ \\alpha_B := \\frac{\\pi a}{4b - 2d} ~;~~ \\alpha_{AB} := \\frac{4}{7}\\,\\alpha_A + \\frac{3}{7}\\,\\alpha_B \\,.\n ", "92b86574bc2635ad4d43133b90ff919d": "R = |x-x'|, \\, ", "92b866dfc6e8763cd0e3b7724a633fcf": " \\,1/e = 0.368\\dots", "92b86836f07943d209981604ec73602e": "M_C(\\alpha)+M_C(\\beta) = 0. ", "92b88f40cf67fd12667c540509463f33": "y \\succ^p_W x", "92b8c82425fdd3286fc8f35ffc45f37e": "(v/m + s)n", "92b92be60eb67a95816fc92f13da0dc9": "\\rho(t-r,\\vec{x}') = M_1 \\delta^3(\\vec{x}'-\\vec{x}_1(t-r)) + M_2 \\delta^3(\\vec{x}'-\\vec{x}_2(t-r))", "92b949d1b28bedbbed0e69c084012eb7": " G(z) = \\exp\\left\\{-\\exp\\left(-\\left(\\frac{z-b}{a}\\right)\\right)\\right\\}\\text{ for }z\\in\\mathbb R.", "92b985abb478dafd4267369b3782d30d": "\\scriptstyle (f_n)", "92b9b40c17dc4094af789da3a883f411": " \\textstyle z_{ni}=z(x_{ni}, \\, s_n) ", "92ba08c2768cfe063528c64b4d7a2bf5": "\n\\begin{align}\n\\mathbf{c}\\cdot\\mathbf{c} & = (\\mathbf{a}-\\mathbf{b})\\cdot(\\mathbf{a}-\\mathbf{b}) \\\\\n & =\\mathbf{a}\\cdot\\mathbf{a} - \\mathbf{a}\\cdot\\mathbf{b} - \\mathbf{b}\\cdot\\mathbf{a} + \\mathbf{b}\\cdot\\mathbf{b}\\\\\n & = a^2 - \\mathbf{a}\\cdot\\mathbf{b} - \\mathbf{a}\\cdot\\mathbf{b} + b^2\\\\\n & = a^2 - 2\\mathbf{a}\\cdot\\mathbf{b} + b^2\\\\\n c^2 & = a^2 + b^2 - 2ab\\cos \\theta\\\\\n\\end{align}\n", "92ba2a21d079fe58aa082a4788abbae2": "-\\sqrt x.", "92ba69c54d4596f1cddc45b3f28f0593": "h(\\phi)", "92ba979ec8abc19bc5a55e16c9b92b02": "\\begin{align}4x + 2(-2x + 6) = 12 \\\\\n4x - 4x + 12 = 12 \\\\\n12 = 12 \\end{align}", "92baad2a553720d97c5e7ff9c9461265": "\\tfrac{tonne}{ha}", "92babca88326807400e7132d0697b692": " \\int_0^z \\log \\Gamma(x)\\,dx=\\frac{z(1-z)}{2}+\\frac{z}{2}\\log 2\\pi +z\\log\\Gamma(z) -\\log G(1+z) ", "92babdd490c6e2937392ddd2fcb374ac": "r_1 = (G \\cdot M/\\omega^2)^{1/3}", "92bacd0c5061d15b6fd214c100edea09": "2 p_1 \\cdot p_2 \\approx\\,", "92bb1ae64c2ecc7d4b64362deef7e6bd": "\\pi / 2", "92bb492b1ca16fd54607e735c0f6a31e": "r_g\\,", "92bb87751e808d2e52b4f89e289d1d82": "\\scriptstyle{\\mathrm {Diff}}(S^{3})\\simeq {\\mathrm O}(4)", "92bbde703193217f1110b0ab9c60810b": "\\Delta S", "92bbef0ef0aea4de9bb4bb1095b2b58f": "L_{AA} = L_{BB} = L_{CC} = L_{ls} + L_{ms}", "92bc3b857fc991dcf24fedf74cd840b7": "\\tfrac{d}{n} \\cdot |S| \\cdot |T|", "92bc637c9891d39ca1abf6e47479650f": "U = \\frac {\\mu'} {k} \\ ln ( \\frac {z + z0} {z0}) ", "92bc781e173288f63f3d680e68858027": "\\Gamma(t; 1, \\nu)", "92bd65d3543439cedc1048ffdb010e0c": "F_0 = 0,\\; F_1 = 1.", "92bdd6723a10f91df89ddc8d0565e02d": " \\frac{\\text{d } {_2^0}P}{\\text{d}t} = - \\frac{\\text{d}[{^0_2}S]}{\\text{d}t} ", "92bddb5e3909585ff72a92f02008bae3": "G = \\widehat{M}", "92be4adb08d0ab04c0f67bc8272bf2f2": " z_2(x,y)={\\mathcal E}_1(x,y){\\displaystyle\\int}\\frac{{\\mathcal E}_2(x,y)}{{\\mathcal E}_1(x,y)}F_2\n \\big(\\varphi_2(x,y)\\big)\\big|_{y=\\psi_1(x,\\bar{y})}dx\n \\Big|_{\\bar{y}=\\varphi_1(x,y)};", "92be686dd83457a98039980df694fedb": "= (-1) (+1) (+1) (+1) \\epsilon_{0123} = - \\epsilon_{0123} .", "92be6ce9a7737ab48b1e7b760eafa6d8": "\\Delta{z} = \\frac{1}{w}\\operatorname{cov}(w_i, z_i) + \\frac{1}{w}\\operatorname{E}(w_i\\,\\Delta z_i)", "92be91709fe763bcc2e15f7f2328ee87": "\\delta\\psi", "92bed30086c8060883c627ecd2d331bd": "T = 39.2511937 * .308 = 12.0893677", "92bf3b8dc8e24854acb8340d9b59b13b": "u_{\\bold{k}}(\\bold{r})", "92bf826f4c70ddaa8f9844f2b2c4e0ca": "q_p(p+a)\\equiv q_p(a) - \\frac{1}{a} \\pmod{p}", "92bfb1dabd5446c8fdbfe04a05c485b5": "\\varepsilon_i = \\varepsilon_i' + \\varepsilon_i'' +\\varepsilon_i'''", "92bfdffd2a20837bdf8f8195c958485c": " \\frac{n}{\\frac{2^k}{e}-k} ", "92bff11f9233b44572c76796135316e7": "\n \\begin{align}\n \\delta U & = \\int_{\\Omega^0} \\left[-\\frac{1}{2}~(N_{\\alpha\\beta,\\beta}~\\delta u^0_{\\alpha}+N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta}) \n + M_{\\alpha\\beta,\\beta}~\\delta w^0_{,\\alpha}\\right]~d\\Omega \\\\\n & + \\int_{\\Gamma^0} \\left[\\frac{1}{2}~(n_\\beta~N_{\\alpha\\beta}~\\delta u^0_\\alpha+n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta}) \n- n_\\beta~M_{\\alpha\\beta}~\\delta w^0_{,\\alpha}\\right]~d\\Gamma \n \\end{align}\n", "92bffb1d3440a62f67f64ba8ad6624b4": "~\\frac{2 \\log 2p }{p\\log 2}", "92c09ad47c092a001f38d476d466bd41": "\\sum_{k=1}^n a_n,", "92c0b69c556b2048cac58da7f079cd4c": "\\lbrace x i + y j + z k \\in H : x^2 + y^2 +z^2 = 1 \\rbrace ", "92c0d031b38ec49a35553664c024b7c8": "\\nabla^2 U = -M_x 0 - M_y 0 - M_z 0 = 0,", "92c0eab3e41f0517d39fc530a8369228": "P_D = 2\\rho \\bar{J}^2", "92c0f1d259807c5f68ff89cdeffdbd93": "t <_{KB} s\\,", "92c10739572240e7222bba3650c4862e": " Z_L\\,", "92c126256beabc0e72f549ebbd555ea5": "y\\in L", "92c1a9c289c1243ca3b19ee500a4044c": "f \\in L^2 (X, \\mu)", "92c1ed558554d87dcbd99c3ed567a25f": " COP_{cooling}=\\frac{| Q_{C}|}{ W}", "92c2a00bbfb46000e66c9eb9ef41661a": "F^{\\pm} := \\sum \\mathcal{Z}^{\\pm} t_j", "92c306bcbf4e323771562db3bc4f2723": " P( k_{ 11 } \\le X_1 \\le k_{ 12 }, k_{ 21 } \\le X_2 \\le k_{ 22 }) \\ge 1 - \\sum T_i", "92c46e393af0d4165c41939e4f5db276": "\\psi : M \\to {\\mathbf S}\\,", "92c49092b3080151acfa9fb0be1b1d7c": "\\sigma^{\\phi_t}(x)=\\Delta^{it}x\\Delta^{-it}", "92c4a8513d861b3bf00ac5596c65f2de": "\n(1) \\qquad H(x^*(t),u^*(t),\\lambda^*(t),t)\\leq H(x^*(t),u,\\lambda^*(t),t) \\,\n", "92c4d1c8f2a2d4b920b3301ff5211d67": "\\mathcal I", "92c514824f0bfa3bce63ee2e89c2f463": "b(z) < v", "92c5338f68bdd6fc7c17bb566cf1d672": "ax^2 + c = bx", "92c577418f7c6f5534961a1ed41b8824": "\\ V_l", "92c592c4c42f726168ef576a44f1a86f": " \\Phi(w,x) =", "92c5a4ed2a239f7ff836901741fcb7bc": "(x-4)(x-1)^2(x^2-3)^2(x^2+x-5)", "92c5c54dcfa554b6e3860e5756a3b7dc": "\\scriptstyle(0,\\, \\infty)", "92c5cd5847bbbcb6159c986d7b0d6d67": "\\tan\\frac{\\delta}{2} = \\frac{r_{ex}}{r_{in}}.", "92c6144a4b6f1252450050dd5485e09d": "e^{i k \\|\\mathbf{x}-\\mathbf{x'}\\|_2}=e^{i k (r - \\mathbf{n}\\cdot\\mathbf{x'} + O(1/r))}=e^{i k r - i k \\mathbf{n}\\cdot\\mathbf{x'}}(1 + O(1/r))", "92c66891d416a1dffcf4e3cd3c0dc54a": "L_{-1} = \\frac{\\partial^2}{\\partial z^2} + \\frac{\\partial^2}{\\partial \\rho^2} - \\frac{1}{\\rho}\\frac{\\partial}{\\partial\\rho}", "92c66fd783644bdc10eb27f2ff10e850": "\\tan \\xi_a = \\sin \\xi,", "92c6b935c277e19028ea7d9d306f2801": "\\displaystyle{g_t=\\psi_0 \\cdot g +\\sum_{i=1}^N \\psi_n \\lambda_{tn_i} g}", "92c6d73a721e00c70114baf7de58cf4a": "\n\\lambda =2\\pi/k > \\lambda_J = \\sigma_u^2/G\\Sigma\n", "92c73d143e442ce370acb3656308e773": "\\mbox{(2) }\\bigvee_{n=0}^\\infty T^n \\mathcal{K}=\\mathcal{B}", "92c79736816898bd1bb6323fc51828b3": "a, b \\in F.", "92c868714c6b9c70832f0557cbaa6e98": "\\mathrm{Ga} = \\frac{g\\, L^3}{\\nu^2}", "92c8892b20f60b475443c1205dcfd6b2": " M_s = \\log_{10}\\left(\\frac{A_{max}}{T}\\right) + 1.73\\cdot \\log_{10}(\\Delta) + 3.27 ", "92c8b00aaefa9fcf3863987e87bf2cf3": "\\textstyle\\Delta_\\alpha", "92c9134527161fd7453fe848b821d8c7": "\\theta _{1}", "92c931b043b154d951286e8d4fadecf1": "\\mathcal G(1,3)", "92c9a9b87dff2cf66b426ed16a9ee949": "\\ \\displaystyle \\psi(q,\\alpha,u) \\ ", "92ca2e13d9ef1d92298dfaaaf30616c1": "\\tfrac{M(1+\\nu)(1-2\\nu)}{1-\\nu}", "92ca2e1f967f6972538fb88338042d71": "\\chi : V^{\\mathbb C} \\to \\overline{V^{\\mathbb C}}", "92cb36bc6c36f238c391d2b3b2317fdd": "e_{ijk}", "92cb6ccbaacbbd9d89e47fd7912eac25": "\\scriptstyle G=\\langle H, t| t^{-1}C_1t=C_2\\rangle", "92cb8f56288c03164398847a289b4ecb": "W = \\dot{m}U({\\Delta }V_w)", "92cbd9a4028fd92c2c9b50402193f990": "F(s) = p_1 + s\\cdot\\mathbf{v}.", "92cc09ee390a53b27de385fca9838c93": "T_+~", "92cc367ca5b1fcf5b05bc0105b125163": "\\Chi^2 = ", "92ccf827568c5b025e48e1a707b4f7e7": "\\mathbf{x}_j \\,", "92cd0f4ac17c8c6a2a1d9e144fe54be6": "T \\subseteq R", "92cd4261814ad486213290d49d7f63a2": "\n\\Gamma_{\\sigma_1,\\sigma_2}:\\{\\mathbb{X}\\subseteq\\mathbb{R}^n\\}\\rightarrow\\{\\mathbb{Z}\\subseteq\\mathbb{R}\\}\n", "92cde66b45dd41eb4494a9ce50b878b6": " \nC = ET, \n", "92ceb3c8ef0b6a1770c5660a32448e7f": "\nD^*_N(x(1),\\dots,x(N))\\leq C'\\frac{(\\log N)^s}{N}.\n", "92cebf98a1ec13d81fb884fcff79428f": "-1+j0", "92cec1833b0109381b9730732fe130ee": " = G_0 + (G_0 - G_{\\infty} ) \\frac {-T} {1 +T} ", "92cf32ec1dc9db79c32d0f6c059b1172": "m(n) \\approx 0.570591", "92cfb3fe9caaa980e69483be5a512be7": " \\pi/4 ", "92d017cc0c668336d8f0ae2dbeb12241": "\n\\begin{align}\n\\sum_{k=-\\infty}^{\\infty} \\hat s(\\nu + k/T) \n&= \\sum_{k=-\\infty}^{\\infty} \\mathcal{F}\\left \\{ s(t)\\cdot e^{-i 2\\pi\\frac{k}{T}t}\\right \\}\\\\\n&= \\mathcal{F} \\bigg \\{s(t)\\underbrace{\\sum_{k=-\\infty}^{\\infty} e^{-i 2\\pi\\frac{k}{T}t}}_{T \\sum_{n=-\\infty}^{\\infty} \\delta(t-nT)}\\bigg \\}\n= \\mathcal{F}\\left \\{\\sum_{n=-\\infty}^{\\infty} T\\cdot s(nT) \\cdot \\delta(t-nT)\\right \\}\\\\\n&= \\sum_{n=-\\infty}^{\\infty} T\\cdot s(nT) \\cdot \\mathcal{F}\\left \\{\\delta(t-nT)\\right \\}\n= \\sum_{n=-\\infty}^{\\infty} T\\cdot s(nT) \\cdot e^{-i 2\\pi nT \\nu}.\n\\end{align}\n", "92d02e686382b7afb49d3549416ec5ba": "\\overline{O_iO_j}^2=(R-R_i)^2+(R-R_j)^2-2(R-R_i)(R-R_j)+(R-R_i)(R-R_j)\\cdot \\frac{\\overline{K_iK_j}^2}{R^2}", "92d03c82af42b7504a943b356a1feacc": "P_{n} = \\frac{C_{n} - \\frac{\\part A}{\\part n}(P_{a} - \\frac{\\part C}{\\part a})}{1 - \\frac{\\delta}{\\epsilon_{n}}}", "92d05188fe4bfdf5d001b0204736216a": "\\textstyle{\\sum_{n=1}^\\infty f_n(x,\\phi,\\psi)}", "92d0aa71e6ab8295182ee244cd19ada1": "\\vec{b} = \\overrightarrow{O'B}", "92d0dd129e34a8389525ca455f342fa8": " \\varepsilon_{ij} =\n\\begin{cases}\n+1 & \\text{if } (i,j) \\text{ is } (1,2) \\\\\n-1 & \\text{if } (i,j) \\text{ is } (2,1) \\\\\n\\;\\;\\,0 & \\text{if }i=j\n\\end{cases} ", "92d17d3f1eda732b242c9c36469136ca": "\n\\oint_{\\partial \\Omega} \\hat{n}(\\varepsilon \\nabla \\varphi)\n= \\frac{\\varphi_{i+1,j} - \\varphi_{i,j}}{h_i^x}\n\\left ( \\frac{h^y_j}{2} \\epsilon^x_{i,j} + \\frac{h^y_{j-1}}{2} \\varepsilon^x_{i,j-1} \\right )\n", "92d1c0c1b1f9be7074ff06e62a769224": "ES_q", "92d216bd9f4d446b49225e53716e7904": "\\ x_n", "92d217e72c40539bf890d5c41ac3964f": "-M+\\ln\\sqrt{2\\pi M} ", "92d248ab903a483730c2e726dd094968": "\\begin{align}\n \\int \\sinh (ax)\\,dx &= a^{-1} \\cosh (ax) + C \\\\\n \\int \\cosh (ax)\\,dx &= a^{-1} \\sinh (ax) + C \\\\\n \\int \\tanh (ax)\\,dx &= a^{-1} \\ln (\\cosh (ax)) + C \\\\\n \\int \\coth (ax)\\,dx &= a^{-1} \\ln (\\sinh (ax)) + C \\\\\n \\int \\operatorname{sech} (ax)\\,dx &= a^{-1} \\arctan (\\sinh (ax)) + C \\\\\n \\int \\operatorname{csch} (ax)\\,dx &= a^{-1} \\ln \\left( \\tanh \\left( \\frac{ax}{2} \\right) \\right) + C\n\\end{align}", "92d26be870fa41bb30fbdc39746703fd": "a_{i+1,i}", "92d272691f9d5c03d299ed6f4ea0abc8": "\\bold{u} = (u_1,\\ldots,u_n)", "92d28678c144bdc3047f06ecd78a9963": "f(c^-) = \\lim_{x \\to c}f(x)", "92d2de923212de14cb1c1da361491442": "\\chi^\\prime_l(K_{n, n}) = n", "92d32204103074e1fe74632a776f7cb6": "Cent(g)\\;", "92d3714882d3188ea9cf33caafb9c7a5": " |S \\rangle ", "92d3b5af14dca9f86eadfcc7ed93f322": "\\mathrm{RA} = 9 \\cdot \\frac{\\mathrm{R}}{\\mathrm{IP}}", "92d4158479c409ab18b51aa9f3de5cd3": "\\psi: N_{M_1} V \\to N_{M_2} V", "92d435395db55cffe080eac9334fa1e8": "a = b = 2", "92d4a39355a09709b52c8587df2bb839": "R^q f_* \\mathcal{F}", "92d4c91c8510ad699b5b0e1c02d6c0bb": "r\\cdot(xy) = (r\\cdot x)y = (-1)^{|r||x|}x(r\\cdot y)", "92d4db910fcee85c178d0f8dfaf612b5": "x=\\frac{1}{a_1}+\\frac{1}{a_1a_2}+\\frac{1}{a_1a_2a_3}+\\cdots.", "92d4fc28b9bfd738287204e427e560e0": "P_c^n(c)", "92d513ddc8635740a7aa9f391c20f44c": "N_e^{(F)} = \\frac{1}{2\\left(1-\\frac{P_{t+1}}{P_t}\\right)} ", "92d52258260c5b49b66dbc4df19fbfdf": "r_{o1}", "92d52a763d2d9f30df85d8bae519e672": "Q = \\chi x_c, \\ t = \\tau t_c, \\ \\ x_c = C V_0, \\ t_c = \\sqrt{LC}, \\ 2 \\zeta = R \\sqrt{\\frac{C}{L}}, \\ \\Omega = t_c \\omega.", "92d56ce0e925dfa83d186c1a831c780f": "\\dot{z} = -\\mu", "92d587a3c4ed8d47f15d939456cedb0f": "A = P_d/P_o", "92d5c4842e29904752aec8fbb0cefe45": "I(P)", "92d6001bd519f1761ff2bca6cbabd5a3": "\\boldsymbol{x}", "92d603d8c4d7871dbbca8b3499cac4f7": "\\gcd(N,R)=1", "92d611e5634e1cc577b3c32655a3b62a": "\\le 2^{-n(I(X;Y) - 3\\varepsilon)}", "92d633364afe1d58d2d2576a94f041bd": "x_{21}=\\xi\\sqrt{1-t}\\,", "92d69cd018c3bdbf042724265e905e6f": "T[f \\sigma] \\to T[\\sigma]b", "92d724e5f691e5e24a3fecd8e3b6e0cd": "S(Z,X,Y)=-\\frac12\\langle X,J(\\nabla_{Y}J)Z\\rangle-\\frac12\\langle Y,J(\\nabla_{Z}J)X\\rangle.", "92d7be834ae0fbe268e4c8f7e298e99c": "M=\\sum_{i=1}^e m_i", "92d89352e801ca70a62ed1d6600a68cc": "\\kappa(\\beta^*) = - \\mu \\beta^*", "92d899fa6d6be6438a0b2de46dc84f24": " u_{40}(\\mathbf{r}) = u_{hh}(\\mathbf{r}) = \\left | S\\frac{3}{2},\\frac{3}{2} \\right \\rangle = -\\frac{1}{\\sqrt 2}|(X+iY)\\uparrow\\rangle ", "92d8c35669b056b40d192750ad8239b4": "e\\;", "92d92a0e61376a0f6c0daaa4bfd52425": "\\mathrm{N}(\\alpha) = \\alpha\\bar\\alpha = a_0^2+a_1^2+a_2^2+a_3^2", "92d94d1314bff33f305461a293c4efd0": " \\frac{F_1}{F_0} = \\left(\\frac{x_1}{x_0}\\right)^m ", "92d9739682741c320fa4e6c58b94e6fb": "\\overline {AB}= dx\\,\\!", "92d9b18b3c3142cb700f7de84e08d0ed": "K_1(A)", "92d9e508150d57410afaa4b1e6be27cd": "Tr[(-1)^F e^{-\\beta H}]", "92da37221f795ded0c27bca8ec1fc979": "\\mathbf{x} = \\begin{pmatrix}e_1&e_2&i_{V_S}\\end{pmatrix}^T", "92da786c0a047263cb6c44cd2c516799": "\\frac {d^2 \\mathbf{u}_j (t)}{dt^2}= \\frac{d\\boldsymbol{\\Omega}}{dt} \\times \\mathbf{u}_j +\\boldsymbol{\\Omega} \\times \\frac{d \\mathbf{u}_j (t)}{dt} = \\frac{d\\boldsymbol{\\Omega}}{dt} \\times \\mathbf{u}_j+ \\boldsymbol{\\Omega} \\times \\left[ \\boldsymbol{\\Omega} \\times \\mathbf{u}_j (t) \\right], ", "92da91b24bd9b5bc520ebf65e34a06bd": "s, h \\cup h' \\models P \\ast Q \\Rightarrow R", "92dad96caef013cf0f1ddce25a72befd": "W(\\boldsymbol{F})=\\hat{W}(I_1,I_2)", "92db1554bbb5e6919a2acdd2a1a1710d": "a:=\\sqrt{a_0+\\frac{a_1^2}{4}}", "92db2c04a5c96f8a501f290991f835b2": "\\deg r_{i+1}<\\deg r_i.", "92db446407d5ef4855909610c21a53c8": "\\Delta x=-B^{-1}\\nabla f(x_k), \\, ", "92db4e0a1957d485e47067dd85ab9ee8": "(1-f)", "92db58c42d504cf3cb0547ad72bbba45": "z=b-\\sum_{i=1}^n c_i \\, ,", "92db9c89602a91a53ed21af9c8c45e11": " \\{\\phi_i\\} ", "92dc5b2cd29ccb1660d3f55d80f6fc1d": "\\mathrm{PH} = \\Sigma_2. \\,", "92dc8479f0b468bea555daf5e02865f3": "\\ll", "92dc9f40ff3c807078bc4bcbb81ffd10": "\\begin{array}{rcrcl}\nx&=&x_c+\\hat{x}&=&(R-r)\\cos t+\\rho\\cos \\frac{R-r}{r}t,\\\\[4pt]\ny&=&y_c+\\hat{y}&=&(R-r)\\sin t-\\rho\\sin \\frac{R-r}{r}t.\\\\\n\\end{array}", "92ddb189c1661d673e4357e5a1724487": " -K \\tau e^{-r \\tau}\\Phi(-d_2) \\, ", "92dddd7013ebbf04ae5ab2a0774e96ea": "BW = \\frac{1}{2}R_S(\\beta+1)", "92ddf899b6ef3e7941620a2b05360016": "i g", "9316c2a49797108946c5f88bf1cf9166": "v_\\pi(0)=\\infty", "9316d2cafde68de76d521cba78b8856f": " I_s = I ", "9316f039a1ddaf61aab806afdb67c30a": "\\alpha - 1", "9316f740abb3f3baaff016f1dc0c52a5": "BE = \\frac{885.975\\,M_x}{R - 738.313\\,M_x}", "93176f3c74437d11f719cf3f22113642": "\\sigma\\ge0", "9317ccb3848163f3c77ce01ed83d33b7": "(\\forall a,b\\in M)\\ a + b = 0 \\implies a = 0 = b \\!", "93182e067e58bc756deaae5547928f03": "\\mathrm{soc}(R_R)\\cong R/J", "9318415d2c78b9e4f4e02c8397f89426": "\\frac{\\sin(x)}{x} = \\prod_{n=1}^\\infty \\cos\\left(\\frac{x}{2^n}\\right).", "93184d704e544153075ad66ee50a11e6": "\\hat f", "9318a3eb53680b65c684e504703a9627": "t\\mapsto f(t)+(p-r(t)){f'(t)\\over r'(t)}.", "9318b6a117a38bc261ec5ad3e0570337": "g_{\\mu \\nu} \\!", "9318ba4bf5e9b1e99c85dbf104973e56": "\\mathrm{Hydrogen:} \\ 2c + 2d = 8", "93190a144098dd1d1c45007ab7169b5b": "\\omega = \\frac{1}{\\sqrt{LC}} \\ ", "9319368dd871b5eacaf9c39b1fed2305": "\\sum _x x = \\frac{x^2}{2}-\\frac{x}{2} + C", "9319c71ac4fa44cc1b99a4c7825eee78": " \\lambda_1, \\lambda_2, \\dots, \\lambda_p ", "9319e7f5be68492ea95f140e36007264": "\\begin{align}\\mathbf{B} &= \\mathbf{a} \\wedge \\mathbf{b}\\\\ \\mathbf{C} &= \\mathbf{a} \\wedge \\mathbf{c}\\end{align}", "931a2cc8055dce95961a93b0f7a01092": "3x + 2y = 12", "931a344e251d5be8840ca6b9348029ca": "T = rs", "931a7d9fdfa8a9623bb264e0a5ad8baa": "\\xi_\\lambda", "931aadfde85e223e007ce79489f121f5": "\\left\\{\\theta_k\\right\\}_{k=1}^{\\infty}", "931abca03d83cb4fb4353d5706b661d1": "k = k_+ + k_-", "931ad99e74e267ab1a0fd37f80ee972b": "Edim\\,G \\le 2\\,\\Delta(G)+1", "931add29bbcc0721e1cce79d48c9ffb5": "P[\\{t_i\\}]", "931b3f5c889a989e849219996a6a7e17": "m_n(x_n)=x_n", "931b40877c095d97d4724f018e97afd0": "2[\\sqrt{\\sigma_L^D\\sigma_S^D} + \\sqrt{\\sigma_L^-\\sigma_S^+} + \\sqrt{\\sigma_L^+\\sigma_S^-}]", "931c04701f775163c3444ed47cda34c9": "\\Lambda(x)=\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2.", "931c3633a18ee4ec047f8a25fffee47a": " fbsp^{(\\operatorname{m-fb-fc}) }(t) := {\\sqrt {fb}} .\\operatorname{sinc}^m \\left( \\frac {t} {fb^m} \\right). e^{j2 \\pi fc t} ", "931c6c07a84703d58f3520f85f9afaa0": "\n f_X(x) \\sim x^{ - (1 + \\alpha)} \\text{ as }x \\to \\infty, \\qquad \\alpha > 0.\\,\n", "931c7eb75c710279c0a0e103fa6170e0": "\n\\begin{align}\n\\langle H_{\\mathrm{kin}} \\rangle\n&= \\biggl\\langle c \\frac{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}{\\sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}} \\biggr\\rangle\\\\\n&= \\Bigl\\langle p_{x} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{x}} \\Bigr\\rangle + \n\\Bigl\\langle p_{y} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{y}} \\Bigr\\rangle + \n\\Bigl\\langle p_{z} \\frac{\\partial H^{\\mathrm{kin}}}{\\partial p_{z}} \\Bigr\\rangle\\\\\n&= 3 k_{B} T\n\\end{align}\n", "931cb7594e5370f756019269280aa278": "L[u]=0,\\quad \\text{on}\\ \\Omega", "931cece7766661a8fc579ae467efdb4c": " p_n(1)=\\mu_n'= \\,", "931d211a1318f4c9b0bf0be0f40ac25e": "n\\leq 72", "931e10d2a84321b01753e0876c10db1a": " x = \\log_{10} P_i ", "931f4192d367b69d3699c2312fea9611": "x_1, x_2, \\dots, x_n", "931f75382de3574673ad943082bf2900": " \\frac{\\partial \\bar{c}}{\\partial t} + \\bar{w} \\frac{\\partial \\bar{c}}{\\partial z} = D \\left( 1 + \\frac{a^2 \\bar{w}^2}{48 D^2} \\right) \\frac{\\partial^2 \\bar{c}}{\\partial z ^2}", "9320082fb297ce98a4d87240edbde2a8": "\\textit{state}", "93200cd51ced9e97f0117f3cede37977": "b(\\sigma) = \\sum_{c\\in\\sigma} b(c)", "9320dd79bd1bed770b260ba889e36e3b": "\\mathbb{K}=\\mathbb{C}\\,\\!", "9320fc0933e5dc5703e5774a13e51826": "\\textstyle \\text{Slope}_{\\text{ideal}} = (1/2) (\\text{Slope}_{\\text{left}} + \\text{Slope}_{\\text{right}})", "9320feea32db9888f9c271aa56e08ec2": "\\begin{align}\n\\vec{\\omega}&=\\left(\\begin{array}{ccc}\n1 & 0 & 0\\\\\n0 & \\cos\\psi & \\sin\\psi\\\\\n0 & -\\sin\\psi & \\cos\\psi\n\\end{array}\\right)\\left(\\begin{array}{c}\n\\dot{\\psi}\\\\\n0\\\\\n0\n\\end{array}\\right)+\\left(\\begin{array}{ccc}\n1 & 0 & 0\\\\\n0 & \\cos\\psi & \\sin\\psi\\\\\n0 & -\\sin\\psi & \\cos\\psi\n\\end{array}\\right)\\left(\\begin{array}{ccc}\n\\cos\\alpha & \\sin\\alpha & 0\\\\\n-\\sin\\alpha & \\cos\\alpha & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{c}\n0\\\\\n0\\\\\n\\dot{\\alpha}\n\\end{array}\\right)\\\\\n&{}+\\left(\\begin{array}{ccc}\n1 & 0 & 0\\\\\n0 & \\cos\\psi & \\sin\\psi\\\\\n0 & -\\sin\\psi & \\cos\\psi\n\\end{array}\\right)\\left(\\begin{array}{ccc}\n\\cos\\alpha & \\sin\\alpha & 0\\\\\n-\\sin\\alpha & \\cos\\alpha & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{ccc}\n\\cos\\delta & 0 & -\\sin\\delta\\\\\n0 & 1 & 0\\\\\n\\sin\\delta & 0 & \\cos\\delta\n\\end{array}\\right)\\left(\\begin{array}{ccc}\n\\cos\\Phi & \\sin\\Phi & 0\\\\\n-\\sin\\Phi & \\cos\\Phi & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\\\\n&{}\\times\\left(\\begin{array}{ccc}\n\\cos\\Omega t & \\sin\\Omega t & 0\\\\\n-\\sin\\Omega t & \\cos\\Omega t & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{c}\n0\\\\\n0\\\\\n\\Omega\n\\end{array}\\right)\\\\\n&=\t\\left(\\begin{array}{c}\n\\dot{\\psi}\\\\\n0\\\\\n0\\\\\n\\end{array}\\right)+\\left(\\begin{array}{c}\n0\\\\\n\\dot{\\alpha}\\sin\\psi\\\\\n\\dot{\\alpha}\\cos\\psi\n\\end{array}\\right)+\\left(\\begin{array}{c}\n-\\Omega\\sin\\delta\\cos\\alpha\\\\\n\\Omega(\\sin\\delta\\sin\\alpha\\cos\\psi+\\cos\\delta\\sin\\psi)\\\\\n\\Omega(-\\sin\\delta\\sin\\alpha\\sin\\psi+\\cos\\delta\\cos\\psi)\n\\end{array}\\right).\\end{align}", "93210b4544cbaea5c587ad1e1b8aabcc": "\\{a,\\ldots,b\\}\\subset Z_n", "932134fd6ec07590e235a66eacf761a5": "\\frac{-x^p}{p}", "932140b0ce77b37061c17a9191935824": " n \\geq4 ", "9321870df1112a2cdf8d3107526fb1fe": " C\\ell_{p+2,q}(\\mathbf{R}) = M_2(\\mathbf{R})\\otimes C\\ell_{q,p}(\\mathbf{R}) ", "9321e0af4c3c05fb14e71269a9b19b2c": "X \\sim \\beta^{'}(\\alpha,\\beta)\\,", "93223800edf622fafdf7f27b9b9ba936": "c_n(V)", "932279b7275fc392c1cb9d3a002354a1": "f(z),g(z)", "9322c1abd6070e9446837128ba4b4bea": "\\gamma\\in \n\\mathbf{F}", "9322e737060550aedf186d8e2d641f26": "\\displaystyle \\delta_t(f) = \\sum_s \\delta_t(P_s) a_s.", "93230b9bfd861a00ef96b1b9e9575684": "\\sum_{i=1}^k X_i=1.", "93230bbaa9e5ae645af4ebcf4360f876": "E/\\rho", "93232ff755d0aab9041d8d22f2fdbf96": "- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N\\partial \\beta^2} = \\operatorname{var}[\\ln (1-X)] = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta) ={\\mathcal{I}}_{\\beta, \\beta}= \\operatorname{E}\\left [- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N\\partial \\beta^2} \\right]= \\ln \\,\\operatorname{var_{G(1-X)}} ", "9323332a6bf2f822daf098545afecb77": "\\text{Span} \\{ \\mathbf{v}_1, \\ldots, \\mathbf{v}_k \\}\n= \\left\\{ t_1 \\mathbf{v}_1 + \\cdots + t_k \\mathbf{v}_k : t_1,\\ldots,t_k\\in K \\right\\} .", "932385c967e5ad27725f283551dc2900": "[a] = \\{b\\in X | a\\sim b\\}", "9323c2ea23f9f75d28a8103c26168310": "\\Gamma = \\cup_{i =1}^s \\gamma_{k_i} ", "9323e192b9beabe46de0807897385c71": "\\begin{align}R_{J}(x,y,z,p) & = 2 R_{J}(x + \\lambda,y + \\lambda,z + \\lambda,p + \\lambda) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \\\\\n & = \\frac{1}{4} R_{J}\\left( \\frac{x + \\lambda}{4},\\frac{y + \\lambda}{4},\\frac{z + \\lambda}{4},\\frac{p + \\lambda}{4}\\right) + 6 R_{C}(d^{2},d^{2} + (p - x) (p - y) (p - z)) \\end{align}", "9324a817a897607d2561dee1fe218157": "\\delta_P\\in\\mathbb N", "9324c4761b1de86c9456ff5e0b5d21a8": "\\ \\displaystyle \\alpha \\ ", "9324c9ed174a46866ccd8249e4f2d438": "\n\\begin{align}\n q_1 &= \\cos\\left(\\frac{\\phi - \\psi}{2}\\right)\\sin\\left(\\frac{\\theta}{2}\\right)\\\\\n q_2 &= \\sin\\left(\\frac{\\phi - \\psi}{2}\\right)\\sin\\left(\\frac{\\theta}{2}\\right)\\\\\n q_3 &= \\sin\\left(\\frac{\\phi + \\psi}{2}\\right)\\cos\\left(\\frac{\\theta}{2}\\right)\\\\\n q_4 &= \\cos\\left(\\frac{\\phi + \\psi}{2}\\right)\\cos\\left(\\frac{\\theta}{2}\\right)\n\\end{align}\n", "9324d8147e5191ff7c673526f42b2170": "d H", "9324fd6eaf3c7fc392b03d930614e964": "cF'(x)-cG'(x)=h(x)\\,", "93254018d509df65dbb6375742d3c24f": "g^{(n-1)}", "9325483bcde4c9c96afb5b9534942bef": "\\Theta_n/bP_{n+1}\\to \\pi_n^S/J,\\, ", "9325575847bc3a578a76f86fd992254f": "\\widetilde{H}_i(X;\\mathbb{Z})", "9325af2644fcd2572ad12a9170c4587f": " -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\frac{\\partial^2}{\\partial x_n^2}\\psi(x_1,x_2,\\cdots x_N) + V(x_1,x_2,\\cdots x_N)\\psi(x_1,x_2,\\cdots x_N) = E\\psi(x_1,x_2,\\cdots x_N) \\, .", "9325c2427c496fec2479b6296ccf5aad": "H=\\Delta_c a^\\dagger a + \\Delta_a \\sigma^\\dagger\\sigma+ig(a^\\dagger\\sigma-a\\sigma^\\dagger)+iJ( a^\\dagger-a)", "9325e2d32b100cbbe49b9afbc1479016": "\\tfrac{a}{c} + \\tfrac{b}{d} \\cdot \\tfrac{a}{c} = \\tfrac{a(d+b)}{cd}", "9326300a5fcc7987260c92cd252aaa32": " U_g W = \\{ U_{g} w: w \\in W \\}. ", "93264e149d46679c4c3016c7af91d79f": "\\neg c \\vee d", "9326bca8293b724be533078a1200cfde": "\\overrightarrow{DE}", "9326da492e20c6cda7f7f8a8f3e11f9b": "b_2\\,", "9326fa3e633cf170e8e850837b235fd4": " \\mathrm{S} = \\left( \\frac{r}{c} \\right)^2 \\frac {\\mu \\omega}{P} =\\left( \\frac{r}{c} \\right)^2 \\frac {\\mu \\omega L D}{W} ", "93275ead70bcdcfd6540c0cb9eea4433": "\\exists D. \\forall x. \\phi(x, D(x, z), z)", "93279aaa6b3eaf37063917ef5c1b751d": "\\mathbf{\\tau} = \\mathbf{r} \\times \\mathbf{F} = {{d\\mathbf{ L}} \\over {dt}}", "9327b5bea37e1bdd47cfb8600296f520": "\\delta t^\\prime = t_2^\\prime - t_1^\\prime = t_2 - t_1 - \\frac{v\\delta t \\cos\\theta}{c} = \\delta t - \\frac{v\\delta t \\cos\\theta}{c} = \\delta t (1-\\beta \\cos\\theta)", "932841ad7894482a55bee25ecbee2e5d": "\\tau_X", "932845f019edcb19064f6912458c6e86": "\\ldots + d_3\\cdot b^3+d_2\\cdot b^2+d_1\\cdot b+d_0+d_{-1}\\cdot b^{-1}+d_{-2}\\cdot b^{-2}+d_{-3}\\cdot b^{-3}\\ldots", "93285dce0ecdf8185b05dedbf0c10c89": "P( | X - m | \\ge ks ) \\le \\frac{ N - 1 }{ N } \\frac{ 1 }{ k^2 } \\frac{ s^2 }{ m^2 } + \\frac{ 1 }{ N }.", "932871cb99ded4ac9ca7fab3d7439073": "\\ M_{heel}", "93287d16268e8a73b389bd53b539577e": "\\binom{n}{k}_F = \\binom{n}{n-k}_F.", "93288a523bf1d8b08c69d65b6215129f": "\\begin{align}\\delta(t' - t_r')\n= \\frac{\\delta(t' - t_r)}{\\frac{\\partial}{\\partial t'}(t' - t_r')|_{t' = t_r}}\n=& \\frac{\\delta(t' - t_r)}{\\frac{\\partial}{\\partial t'}(t' - (t - \\frac{1}{c}|\\mathbf{r} - \\mathbf{r}_s(t')|))|_{t' = t_r}}\n= \\frac{\\delta(t' - t_r)}{1 +\\frac{1}{c}(\\mathbf{r} - \\mathbf{r}_s(t'))/|\\mathbf{r} - \\mathbf{r}_s(t')|\\cdot (-\\mathbf{v}_s(t')) |_{t' = t_r}}=\\\\\n&= \\frac{\\delta(t' - t_r)}{1 - \\boldsymbol{\\beta}_s \\cdot (\\mathbf{r}-\\mathbf{r}_s)/|\\mathbf{r}-\\mathbf{r}_s|}\\end{align}", "9328b851ef4d803c90cddb8fff917e8a": "\\deg P-\\delta+1", "932902ed1131c9d63465baa31173cb40": " \n \\begin{align}\n \\epsilon & = \\frac{C_{11} - C_{33}}{ 2C_{33} } \\\\\n \\delta & = \\frac{(C_{13} + C_{44})^2-(C_{33} - C_{44})^2}{ 2C_{33}(C_{33} - C_{44}) } \\\\\n \\gamma & = \\frac{C_{66} - C_{44}}{ 2C_{44} }\n \\end{align}\n", "93292e6811874f5275f68b5dd812a883": "M_N\\Delta t=x", "93294e0b79015fa8093e2c72de6d688c": "f = 2 \\Omega \\sin \\phi,", "9329b280195bbacf46077e4ee1abeb9f": "H = \\frac{G^2}{ A} .", "9329c3fc33949a4516db87a5a01c1d5b": "f_0.", "9329d854747248d68900ec6f23fff921": "\\scriptstyle \\Vert u \\Vert_{BV} < +\\infty ", "932a037f2e138c4a20198b8f3abd65cd": "\n\\begin{align}\n& {} \\quad x^2 + y^2 + \\frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x \\\\[8pt]\n& {} + \\frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0.\n\\end{align}\n", "932a09d93e412e3fc932a0dffa440856": "P = \\{n \\mid n \\mbox{ is prime}\\}.", "932a33d02a37d9ee26f0724ec58dc7c8": "449275\\cdots97101\\,449275\\cdots9710144\\cdots", "932a631a30622f9dac60591c6bc92f9e": "n_\\text{mean}", "932ac36e8ac39c8bea95601e8821d1c1": "~\\Phi_5(x) = x^4 + x^3 + x^2 + x +1", "932acb501991e0aad4037a9f75fe779e": "\\{c,c/2,c/3\\}", "932b38fdacf5999e5eb779034bd2b206": "C_{YX}=\\left[\\begin{array}{c}\nE[x_{4},x_{1}]\\\\\nE[x_{4},x_{2}]\\\\\nE[x_{4},x_{3}]\\end{array}\\right]=\\left[\\begin{array}{c}\n4\\\\\n9\\\\\n10\\end{array}\\right].", "932b391627a9caa90fcd6745ebd86986": "r=\\frac{b\\cos\\theta+c\\sin\\theta}{\\cos^2\\theta-m^2\\sin^2\\theta}", "932ba4b35ef7bdc0c3733ad101e7f149": "\\Delta X\\,\\!", "932bb987004477eb439ca137f8c03562": " \\frac{7}{12}\\,", "932c19babbd5f830dd6180a7550a7f0b": "\\ PV = NkT", "932c2dc018fd0b04ec4ee55092f8221b": "v(a) = 0 \\iff a = \\mathbf{0} ", "932d0ec79260e01afd1dd960c7bc69bb": "C_{2}", "932d3a4aa299afd474f6297096fc0e6f": "{}_a \\mathbb{D}^q_t \\left( f(x) \\right)", "932d4fc708253da976cde1242b371b6f": "u=\\hat{u}\\sin\\left( kx - \\omega t \\right),", "932db9787164493c342d18789f1da990": "\\displaystyle{(Cf)_z= Tf.}", "932dc5e1e9f4ad61ac6eefa2ab6c2acb": "\\delta_\\ell", "932dcc760f6aebef31448632c7248203": "{ESR}", "932dd4888d7d2f8906d0f2cf272bf4b8": "y^2\\,", "932de723494e838332961384ea0549e5": "\\dot x(s) = \\nabla_\\xi H(x(s),\\xi(s)), \\;\\;\\;\\; \\dot \\xi(s) = -\\nabla_x H(x(s),\\xi(s)).", "932df93d55ebeb052b25cffbef25c903": "\n\\begin{align}\nd\\log(S) & = f^\\prime(S)\\,dS + \\frac{1}{2}f^{\\prime\\prime} (S)S^2\\sigma^2 \\,dt \\\\\n& = \\frac{1}{S} \\left( \\sigma S\\,dB + \\mu S\\,dt\\right) - \\frac{1}{2}\\sigma^2\\,dt \\\\\n&= \\sigma\\,dB +(\\mu-\\sigma^2/2)\\,dt.\n\\end{align}\n", "932e71d3a17aef8c069efaa4f5076979": "(T_n)_{n\\in\\mathbb{N}}", "932e77f6d36eb9de34ee971a228ec8e4": "\\sum_{k=1}^{n(y)} \\left| \\frac{d}{dy} g^{-1}_{k}(y) \\right| \\cdot f_X(g^{-1}_{k}(y))", "932ec6ffb8982c0c24ef9da46f6198f9": "\n\\begin{align}\nr & = \\frac{g}{4k^2} \\\\\nk & = \\frac{1}{2} \\sqrt{\\frac{g}{r}} \\\\\nT & = \\pi \\sqrt{\\frac{r}{g}}\n\\end{align}\n", "932f19ee7d5e7d6715d20bd2f276775b": "\\lambda =\\frac{\\omega R}{U}", "932f36ed5d2ceff3a41f5b2abc068286": " H_D ", "932f617f8102e1f7c46c5ce29e3962f0": "(\\Omega,\\mathcal{F},P) ", "932f8bcec626a764fedde0eccaa063bd": "\\left(A + \\overline{D}\\right)", "932fcde0da39b88e602ca84cdc46c5f6": " M_P\\cong R_P ", "93300db106a0cd4a25a49d4402306a1b": "v_r \\neq 0", "93304c45f88b77247343b001e455643f": "{(\\eta_b)_{reaction}} = \\frac{2\\cos^2\\alpha_1}{1+\\cos^2\\alpha_1}", "9330af16419bad807b004109a842b12a": "\\operatorname{Var}[\\mathbf{X}]=\\operatorname{E}[(\\mathbf{X}-\\operatorname{E}[\\mathbf{X}])(\\mathbf{X}-\\operatorname{E}[\\mathbf{X}])^{T}]. ", "9330b442fd633b9d6d493aba2c620349": "f(z) = e^{R} \\left[ \\cos \\left( R\\tan \\theta \\right) - i \\sin \\left( R\\tan \\theta \\right) \\right] ", "93313d3c8bda1010be3b17f519dbef27": "\n\\hat g\\, = \\,{{4\\,\\pi ^2 L} \\over {T^2 }}\\,\\,\\left[ {\\,1\\,\\,\\, + \\,\\,\\,{1 \\over 4}\\sin ^2 \\left( {{\\theta \\over 2}} \\right)\\,} \\right]^2{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(2)}}", "933192686c58128d3892907f418d9628": " \\phi_2 ", "9331ae3503c59215308646a1a95ab5cc": "V=2U", "9331e965c90a1926eef4f8120024b18b": " \\sin \\theta = 1.22 \\frac{\\lambda}{D}", "9331eb8a7e5c7de05324df1aa4158b2f": "1 - Q_{\\frac{k}{2}} \\left( \\sqrt{\\lambda}, \\sqrt{x} \\right)", "933228d2cdec398844dbcfb17a2d7406": "{}^{(l)}G(t) = \\left( \\frac{d}{dt} \\right)^l G(t)\\ .", "9332d9546c8466a6b3084fb1e6e3a9e1": "\\pi:\\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}", "9332e38fccebc54d3632af06d480c896": " t_1,t_2,t_3,t_4 ", "933321b90e300e9bca47f9370e2b2f22": "E[\\varphi]=\\int_M |d\\varphi|^2\\,dV.", "933363138f3ffb2994b09b9e942f9c1b": "a_n = c_1a_{n-1} + c_2a_{n-2}+\\dots+c_da_{n-d}, \\, ", "93337d9c4e8cf04a2f98f48cc547e3ea": " LC_t = S\\Phi(a_1(S,m)) - me^{-r\\tau}\\Phi(a_2(S,m)) - \\frac{S\\sigma^2}{2r} ( \\Phi(-a_1(S,m)) - e^{-r\\tau}(m/S)^{\\frac{2r}{\\sigma^{2}}}\\Phi(-a_3(S,m))),", "93337f3c39c5fdc012a8ecd711e1f177": "\\langle\\sigma, {\\mathbf u}\\rangle", "933380096f1ab01363e647b5028afa5d": "G({f|}_A) = \\{ (x,y)\\in G(f) \\mid x\\in A \\}", "9333b343eb2d4c6c6d4249bf0b5e02ba": "2^n(9 \\cdot 2^{2n - 1} - 1).", "9333bf2a38f637453cb9103d822be8c2": "\\begin{array}{rcl}L_i & = & F_i(KL_i,KO_i,KI_i,L_{i-1})\\oplus R_{i-1} \\\\ R_i & = & L_{i-1}\\end{array}", "9334656b40645fda672002a7ebfa5bc2": "\\{y^*_k\\}_{k=1}^{M}", "93346f5bfabf776605b42912c8caf3b2": "\\mathbb{N} \\to \\mathbb{N}", "933474532b3fe7d35454f48d3bf3479b": "\\{1,\\sqrt{2}\\}", "93347647b5ac35b5a3aebf2bc9bcb785": "x_5=-0.9511", "9334a3564b01fb8b4178a3aafcb754ef": " \\sum ", "9334c76bde1941f63f7841d5f71b05e3": "\\mu=\\tfrac{1}{n}X\\mathbf{1}", "9334d095681a403788d94de4f56abfcf": "W_T=W_0R_1R_2 \\cdots R_T", "93350a7c765ed1840624d8d876ddb368": "0 \\cdot \\infty", "9335613eb6eef7a07e93fa016019da8a": "0\\le y \\le 3", "93359163583f454472f9f7f2de131400": " \\delta Q = T\\,\\mathrm{d}S_{\\mathrm e}\\,\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,\\,\\mathrm{d}S_{\\mathrm i}\\equiv\\mathrm{d}S_{\\mathrm {uncompensated}}.", "9336274d010f23f2aa591c595c43e3d1": "\\mathbf{E} = \\frac{1}{4\\pi\\epsilon_0}\\frac{q}{r^2}\\hat{\\mathbf{r}}.", "933651a35a1c0ba4beaff5a963588751": "g'N'=N' \\Rightarrow gN=N", "93368b3e592bf7a7286154fadc3d7ccf": "\\sqrt{\\frac{1}{14}}\\!\\,", "9336c4fe550cb58b3e3a560b1e5c4fcd": " I(\\omega,T) = \\frac{\\omega^3 }{4 \\pi^3}~\\frac{1}{e^{\\omega/T}-1} ", "9336ebf25087d91c818ee6e9ec29f8c1": "xx", "9336efc08105bd1ce2d16408b215cfbc": "H_i M", "933771ba26f54ab76df550353cb9a167": "s_1 = R(\\alpha^1) = 1011,", "933789f1dd1330a742480394598b33a5": "C'\\subseteq \nC", "9337d8fe8ca81dbdae55bd4cd00d6bae": " |\\nabla (\\mathbf{u})| ", "933823f46c05da0e933e3327844b539b": "Q=1", "93382b78af6939e0b263472ea278ce88": "s_\\Phi", "93383c212e2b016a5e66f30d1fc0dd83": "S^{\\prime}", "93385d094edd07b3d81aac2cefd815a2": "m_k(t)", "933861a2b9a1a38d85be67307ba60940": " \\Delta G_{micelle} = - \\frac{RT}{N} ln [micelle] + RT* ln (CMC) ", "9338706027c8ce9ebec40d226900ff83": "c_3=", "93387f6b461c533aa8e9f7fa2ac89119": "\\mathrm{Princ}(C)", "93388da1150f9b324ab03fced1fdf41e": "F'(x) = f(x)\\,", "93389c8a77d80ac929142250a634e357": "x_1,x_2,\\dots,x_L", "9338a713a21313e97a42eb4572ea193f": "\\omega_2(t):=\\frac{1}{t}\\int_t^{2t}\\omega_1(s)ds", "9338a85593b4331b7fead25fe91deb9c": "\\scriptstyle p \\, \\in \\, \\mathbb C ", "9338b671cd0cf59b3d4ea33177bf9e8d": "P_6(x)=2x \\,", "93391002825547ae18fe82b913b0a5da": "\\int\\frac{x}{R^3}\\;dx = -\\frac{2bx+4c}{(4ac-b^2)R}", "9339141d945d12ee253dae3f10a43f1a": " \\chi_U(g) = \\operatorname{tr}(U_g). ", "93394249f59dfde1dece617805f55512": "\\alpha<\\gamma", "9339600102454509251a124755d20e0a": "np=\\lambda", "9339d6ed8b28e8e3f9dc748e449d4ade": "\\scriptstyle X_{1/T}(\\frac{k}{NT})", "933a12704e5e269d8bc6acc837e0b0d4": "=\\operatorname{st}\\left(\\frac{(u + \\mathrm du)(v + \\mathrm dv) - uv}{\\mathrm dx}\\right)", "933a14cedca4ec456b45084988668eb0": "F_{\\mathbf P_2}\\circ f=F_{\\mathbf P_1}", "933ab94dc26e3dc94bae58dc741a979a": "SH_k^G(X)\\cong H_{k-\\dim(G)}^G(X)", "933b2266db87cdb03b715081e6891eeb": "\\, h_{ab}", "933b249003599f9208215c36e280266a": "\\operatorname{CAT}(-1)", "933b61297368c1af4504d3856df0f990": "h = \\left(q + \\left\\lfloor\\frac{13(m+1)}{5}\\right\\rfloor + K + \\left\\lfloor\\frac{K}{4}\\right\\rfloor + 5 + 6J\\right) \\mod 7,", "933b7db8fa10ba61e15669e5b6987805": "MSE^o", "933bac969694d1a2a4b20ea260a5c701": " \\ell_k(x) = \\prod_{i \\in \\{0,\\ldots,20\\} \\setminus \\{k\\}} \\frac{(x - i)}{(k - i)}, \\qquad\\mbox{for}\\quad k=0,\\ldots,20. ", "933cbe54ed4225f157fede4a478f95fa": "\\scriptstyle \\oplus", "933d37e50d24e4ff8f52b0282d820ee3": "V^\\natural", "933d629919fef2ea487eb6386547cf32": "(v,i)", "933d8cb1cad147ccd53b696d9b33b2a4": "T=\\frac{2\\pi U}{\\sqrt{2}g}", "933dc640e55323c06ad23d16beb5ae75": "x_{j,i}", "933e1a9ed0d6fca29e0d98c8515bdcf3": "(x,-y)", "933eafc892545c9c90ea997a9db1585a": "LFL_{mix}=\\frac{1}{\\sum \\frac{x_{i}}{LFL_{i}}}", "933eb474e0a59320807c10c585865a77": "A \\oplus B \\equiv \\mathrm{\\sim}(\\mathrm{\\sim}A \\mathbin{\\And} \\mathrm{\\sim}B).", "933f7bb687144369f8939772bbb6255f": "S = \\{\\{1, 2, 3\\}, \\{2, 4\\}, \\{3, 4\\}, \\{4, 5\\}\\}", "933fadfffa281cdc0e1a741c52e6f35f": "C_{4,1} = (1 + 1) / 2", "933fcb2a06997583ec5f7826f676ae1c": "x_{1},x_{2},\\ldots,x_{m} \\in \\mathcal{C}", "933fde7df309f1888f12100841d6eec7": "\\textstyle{3\\frac{\\log(\\varphi)}{\\log \\left(\\frac{3+\\sqrt{13}}{2}\\right)}}", "933ffa43ba9f17c270849697509fa49e": "\n\\sum_{k=j}^\\infty {k \\choose j} \\left( {-z \\over 1-z} \\right)^{k+1} = \\left[ \\left( {-z \\over 1-z} \\right)^{-1} -1 \\right]^{-j-1} = (-z)^{j+1} \\,.\n", "9341839df5a573a14e1892693a204b57": " \\frac {\\gamma}{\\hbar} B_0 = \\omega ", "9341cf25dc1da30970725e121ac68b81": "f_u(a)=a^u", "9341d897cfd7821e7a803a149f1ef897": "M_+^1(A)", "93420d2ba4924c12deca85fbc81ed12b": "V_{outsensmax}", "9342332adbfedea04bdc93298b54b941": "\n f(\\mathbf{x}^*) \\geq \\min_{\\mathbf{y} \\in D} f(\\mathbf{x}) + (\\mathbf{y} - \\mathbf{x})^T \\nabla f(\\mathbf{x}) = f(\\mathbf{x}) - \\mathbf{x}^T \\nabla f(\\mathbf{x}) + \\min_{\\mathbf{y} \\in D} \\mathbf{y}^T \\nabla f(\\mathbf{x})\n", "93423c3cd1477e5be082abe6b5bf1e89": "\\,R(X,Y)=\\Omega(X,Y),", "93424d4e053e3f0ed0965a6b3ee7f156": "f(x;\\mu',c,\\alpha,\\beta)dx=f(y;0,1,\\alpha,\\beta)dy\\,", "93428b544efd53cec8642c3eab85f571": "\n \\begin{align}\n \\nabla\\cdot\\boldsymbol{s} \\equiv \\cfrac{\\partial s_{ij}}{\\partial x_i} & = \n \\mu \\left[\\cfrac{\\partial}{\\partial x_i}\\left(\\cfrac{\\partial v_i}{\\partial x_j}+\\cfrac{\\partial v_j}{\\partial x_i}\\right)\\right] + \\lambda~\\left[\\cfrac{\\partial}{\\partial x_i}\\left(\\cfrac{\\partial v_k}{\\partial x_k}\\right)\\right]\\delta_{ij} \\\\\n & = \\mu~\\cfrac{\\partial^2 v_i}{\\partial x_i \\partial x_j} + \\mu~\\cfrac{\\partial^2 v_j}{\\partial x_i\\partial x_i} + \\lambda~\\cfrac{\\partial^2 v_k}{\\partial x_k\\partial x_j} \\\\\n & = (\\mu + \\lambda)~\\cfrac{\\partial^2 v_i}{\\partial x_i \\partial x_j} + \\mu~\\cfrac{\\partial^2 v_j}{\\partial x_i^2} \\\\\n & \\equiv (\\mu + \\lambda)~\\nabla(\\nabla\\cdot\\mathbf{v}) + \\mu~\\nabla^2\\mathbf{v} ~.\n \\end{align}\n ", "93428fd2db1b2adcb125d1d2aa3272d7": "\\frac {\\partial Y} {\\partial L}.", "9342a4b15c292b759fc8cd936e119448": "i(t) = Ae ^{+j \\omega_0 t} + Be ^{-j \\omega_0 t}\\,", "9342e83d1a61b7e2378a69a9495c019a": "n_0 = N_{\\mathrm{A}} / V_{\\mathrm{m}} \\,", "93441c76fb6288fd4db20552ee117ecd": "\\bigcup", "93446802ba54318c3c91b6032dcdf115": "\\epsilon_e", "93448d3178b83864ffd1008b46b79d89": "\\lim_{\\theta \\to 0}{\\frac{\\sin \\theta}{\\theta}} = 1", "9344a78b8eadea337b06e53f6c81330d": "G(k,\\chi) = \\sum_{m=1}^n \\chi(m) \\exp\\left(\\frac{2\\pi imk}{n}\\right). ", "9344c396cee7c7aad8b22ecc0281b681": "F_r = \\frac{GMm}{r^2},", "9344d01a570fd2d7d091d86dd4417325": "e^{x+iy}", "9344d9c19a0f828878209cfe6c69700c": "\n\\begin{align}\nI_1&=\\frac{2\\pi A}{\\sqrt{(\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\\Theta_+}} \\\\\n&\\times\n\\ln\\left(\\frac{(\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\\Theta_+-\\sqrt{(\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\n\\Theta_+}(\\Delta^{(p)}_1+\\Delta^{(p)}_2)+\\Delta^{(p)}_1\\Delta^{(p)}_2}{-(\\Delta^{(p)}_2)\n^2-4p_+^2p_-^2\\sin^2\\Theta_+\n-\\sqrt{(\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2 \\Theta_+}(\\Delta^{(p)}_1-\\Delta^{(p)}_2)+\\Delta^{(p)}_1\\Delta^{(p)}_2\n}\\right) \\\\\n&\\times\\left[-1-\\frac{c\\Delta^{(p)}_2}{p_-(E_+-cp_+\\cos\\Theta_+)}+\\frac{p_+^2c^2\\sin^2\\Theta_+}\n{(E_+-cp_+\\cos\\Theta_+)^2}-\\frac{2\\hbar^2\\omega^2p_-\\Delta^{(p)}_2}{c(E_+-cp_+\\cos\n\\Theta_+)((\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\\Theta_+)}\\right], \\\\\nI_2&=\\frac{2\\pi Ac}{p_-(E_+-cp_+\\cos\\Theta_+)}\\ln\\left(\n\\frac{E_-+p_-c}{E_--p_-c}\\right), \\\\\nI_3&=\\frac{2\\pi A}{\\sqrt{(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+\n}} \\\\\n&\\times\\ln\\Bigg(\\Big((E_-+p_-c)(4p_+^2p_-^2\\sin^2\\Theta_+(E_--p_-c)+(\\Delta^{(p)}_1+\\Delta^{(p)}_2)\n((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c) \\\\\n&-\\sqrt{(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+}))\\Big)\\Big((E_--p_-c)\n(4p_+^2p_-^2\\sin^2\\Theta_+(-E_--p_-c) \\\\\n&+(\\Delta^{(p)}_1-\\Delta^{(p)}_2)\n((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)-\\sqrt{(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+}))\\Big)^{-1}\\Bigg) \\\\\n&\\times\\left[\\frac{c(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)}{p_-(E_+-cp_+\\cos\\Theta_+)}\\right.\\\\\n&+\\Big[((\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\\Theta_+)(E_-^3+E_-p_-c)+p_-c(2\n((\\Delta^{(p)}_1)^2-4p_+^2p_-^2\\sin^2\\Theta_+)E_-p_-c \\\\\n&+\\Delta^{(p)}_1\\Delta^{(p)}_2(3E_-^2+p_-^2c^2))\\Big]\\Big[(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+\\Big]^{-1} \\\\\n&+\\Big[-8p_+^2p_-^2m^2c^4\\sin^2\\Theta_+(E_+^2+E_-^2)-2\\hbar^2\\omega^2p_+^2\\sin^2\\Theta_+p_-c(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c) \\\\\n&+2\\hbar^2\\omega^2p_- m^2c^3(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)\\Big]\n\\Big[(E_+-cp_+\\cos\\Theta_+)((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+)\\Big]^{-1} \\\\\n&+\\left.\\frac{4E_+^2p_-^2(2(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2-4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+)(\\Delta^{(p)}_1E_-+\\Delta^{(p)}_2p_-c)}{((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+)^2}\\right], \\\\\nI_4&=\\frac{4\\pi Ap_-c(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)}{(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+}+\\frac{16\\pi E_+^2p_-^2\nA(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2}{((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+)^2}, \\\\\nI_5&=\\frac{4\\pi A}{(-(\\Delta^{(p)}_2)^2+(\\Delta^{(p)}_1)^2-4p_+^2p_-^2\\sin^2\\Theta_+)\n((\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+)} \\\\\n&\\times\\left[\\frac{\\hbar^2\\omega^2p_-^2}{E_+cp_+\\cos\\Theta_+}\n\\Big[E_-[2(\\Delta^{(p)}_2)^2((\\Delta^{(p)}_2)^2-(\\Delta^{(p)}_1)^2)+8p_+^2p_-^2\\sin^2\\Theta_+((\\Delta^{(p)}_2)^2+(\\Delta^{(p)}_1)^2)]\n\\right.\\\\\n&+p_-c[2\\Delta^{(p)}_1\\Delta^{(p)}_2((\\Delta^{(p)}_2)^2-(\\Delta^{(p)}_1)^2)+16\\Delta^{(p)}_1\\Delta^{(p)}_2p_+^2p_-^2\\sin^2\\Theta_+]\\Big]\\Big[(\\Delta^{(p)}_2)^2+4p_+^2p_-^2\\sin^2\\Theta_+\\Big]^{-1}\\\\\n&+ \\frac{2\\hbar^2\\omega^2 p_{+}^2 \\sin^2\\Theta_+(2\\Delta^{(p)}_1\\Delta^{(p)}_2\np_-c+2(\\Delta^{(p)}_2)^2E_-+8p_+^2p_-^2\\sin^2\\Theta_+ E_-)}{E_+-cp_+\\cos\\Theta_+}\\\\\n&-\\Big[2E_+^2p_-^2\\{2((\\Delta^{(p)}_2)^2-(\\Delta^{(p)}_1)^2)(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2\n+8p_+^2p_-^2\\sin^2\\Theta_+[((\\Delta^{(p)}_1)^2+(\\Delta^{(p)}_2)^2)(E_-^2+p_-^2c^2)\\\\\n&+4\\Delta^{(p)}_1\\Delta^{(p)}_2E_-p_-c]\\}\\Big]\\Big[(\\Delta^{(p)}_2E_-+\\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\\sin^2\\Theta_+\\Big]^{-1}\\\\\n&-\\left.\\frac{8p_+^2p_-^2\\sin^2\\Theta_+(E_+^2+E_-^2)(\\Delta^{(p)}_2p_-c +\\Delta^{(p)}_1\nE_-)}{E_+-cp_+\\cos\\Theta_+}\\right], \\\\\nI_6&=-\\frac{16\\pi E_-^2p_+^2\\sin^2\\Theta_+ A}{(E_+-cp_+\\cos\\Theta_+)^2\n(-(\\Delta^{(p)}_2)^2+(\\Delta^{(p)}_1)^2-4p_+^2p_-^2\\sin^2\\Theta_+)} \n\\end{align}\n", "9344e109927e2dd3f8db68ebe9113455": "(P \\to (Q \\to R)) \\leftrightarrow ((P \\to Q) \\to (P \\to R))", "9344e23dc5a6cb82885a3ec9e1890852": "x = ja\\,", "934504bf330ca58e340bc29ef7069b8b": "\\Delta CS = \\frac{1}{2} \\left( {Q_1 + Q_0 } \\right)\\left( {P_1 - P_0} \\right)", "934554196a0530e4de0a9389b7639e55": "BT = \\frac {(CP-RP)*CS} {BR*AP}", "9345d55433a5d397fda79f63635306e5": "\\alpha_6 = {{2\\alpha_0 + 5\\alpha_1} \\over 7}", "9345e4bb73460898f153c15633ef22e4": " \\mathrm{cov}\\left (T_i,T_j \\right) = \\frac{ \\partial^2 A(\\eta) }{ \\partial \\eta_{i} \\partial \\eta_{j} }. ", "9345e523fd1b534775b2777111ffe03a": " E = h\\nu = \\hbar \\omega ", "9345f95c09684b002fe1e072a130ee5e": " p^4 \\operatorname{cr}(G) \\geq p^2 e - 3 p n.\\,", "934676cb2f1a6e24d30d10ac1dcdf1bf": "\n C_{ijkl}=C_{klij},\n ", "93468d6a5ea7225231b9a5ac71f089d6": "\\{ r \\in M \\mid \\mbox{for some } x \\in M, \\Delta = x r \\},", "9346eb116322d70f487721218d3bb346": "\\,W\\,", "934732c0ae5eee6508e3628d922fbaef": "\\mathbf{F} = (M, -L)", "934735a1d6a6105b1a4b8a711884b680": " \\frac {\\Delta \\epsilon_m} {2} = C(2N_f^{fatigue})^d ", "93478f60dcd002e98de8221004de8d9b": "\\delta Q=0 ", "9347b5b170f47866d0530fe61358011c": "f(\\mathbb{N} \\setminus A) \\subseteq \\mathbb{N} \\setminus B", "9347cfd83ffeb82cddfd1b41f7b28994": "m_{[i,\\epsilon]}", "9347d54bb7e263e5c3b9920e3e72c3cc": "\\{(gN,g'N') | (g,g') \\in H \\}", "934897e871c931d545a5b09a4eda7a33": "\\mathfrak{T}^{\\mu \\dots}_{\\nu \\dots} = \\sqrt{-g}\\;^W T^{\\mu \\dots}_{\\nu \\dots} \\,,", "9348de30894e78bce2d052d388950839": "\\frac{d\\nu(r)}{dr}=- \\left(\\frac{2}{P(r)+\\rho(r)c^2} \\right) \\frac{dP(r)}{dr} \\;", "9349a3c619a630062daa61ef95b643f5": "\\begin{smallmatrix}{h}\\end{smallmatrix}", "9349e5184e57e352884f754bf3139401": "JMJ\\subseteq M^\\prime.", "9349ecfff42b415f8e7dbabd25a7d266": "\\Delta t>0", "934a790703b2b9db5ee0596d2f03fd4c": "E = E_\\text{bonds} + E_\\text{angle} + E_\\text{dihedral} + E_\\text{non-bonded} \\, ", "934a800e026b2290d12b34b6fdf0e434": "I_{\\text{b}}", "934a8803865c260d2f122bc76d7e522d": "\n w \\; = \\; [\\Gamma(1+2m^{-1})\\Gamma^{-2}(1+m^{-1})-1]^{1/2}\n", "934aa2846e2985a09dd093fef01dc2e0": "\\text{Ultima}= \\frac{\\partial \\text{vomma}}{\\partial \\sigma} = \\frac{\\partial^3 V}{\\partial \\sigma^3}", "934adba659d63c4a8aad8566aee81032": "U = \\begin{bmatrix} M \\\\ N \\end{bmatrix} ", "934b1171c8cfbdc92cb63560e8b0c703": "CR=\\frac{V_1}{V_2}", "934b303c3d6403bc485b91a5c8caa3c8": "\\R^{1,n}", "934b4c0b6b512b09b01117152ebedd47": "\\mathcal{U},\\mathcal{S}", "934b5ae8ab12ee072c8d6d5b479ff28e": "\n(\\mathbf{u} \\mathbf{v})(\\mathbf{v} \\mathbf{u}) \n= ({\\mathbf{u} \\cdot \\mathbf{v}} + {\\mathbf{u} \\wedge \\mathbf{v}}) ({\\mathbf{u} \\cdot \\mathbf{v}} - {\\mathbf{u} \\wedge \\mathbf{v}})\n", "934b74c8fe33bd438023cc67ddc08870": "(\\Omega,\\mathcal{F},P)", "934be729f3af802d1e4d524b41527ce7": "P_i^x = \\frac{N!}{n_x!(N - n_x)!}p_x^{n_x}(1 - p_x)^{N - n_x}", "934bece99a02aee60d1a0b1d3f292da2": "f(3) ", "934bf8c7e7850f1b02704a1fc1fc385e": "BG = CE = 3", "934bffb41b7e703be57be15b0774e8f3": "i_{1}<\\cdots/", "935413540e3790944fdbde5422028705": "Q_{0} := \\{ (x_{1}, \\dots, x_{n}) \\in Q | x_{n} = 0 \\};", "935450220e06696f787f9fec6244b74e": "\\bar {x}_i", "93546ca590bad41506f21457d2580953": "(a,b),", "9354c0d9deda49f55426f87733043a53": "F_1,F_2", "9354e120e8f3d6f4b88c468d1dfa9dcd": " C \\approx 0.332 \\cdot B \\cdot \\mathrm{SNR\\ (in\\ dB)} ", "93553a506d0ba576d151ad988a9a732d": "T=\\left \\{ ( x,y) \\in \\mathbf{R}^2 \\ : \\ x^2+y^2\\le 1 \\right \\},", "93555a0a5c1de2938099be4e53713d61": "\\ ln(Y)=ln(A)+aL*ln(L)+aK*ln(K)+bLL*ln(L)*ln(L)+", "9355b47480832e00ec580c856948dca1": "\n[d(\\rho, \\rho+d\\rho)]^2 =\n \\frac{1}{4}\\mbox{tr}\\left[ d \\rho d \\rho + \\frac{3}{1-\\mbox{tr} \\rho^3} (\\mathbf{1}-\\rho)d\\rho (\\mathbf{1}-\\rho)d\\rho \n+ \\frac{3 \\det{\\rho} }{1-\\mbox{tr} \\rho^3} (\\mathbf{1}-\\rho^{-1})d\\rho (\\mathbf{1}-\\rho^{-1})d\\rho \n\\right]\n", "9355e9b1d69030dbf50d0f3d121fc122": "\n \\color{Gray}\n \\Rightarrow \\quad\n \\color{Black}\n \\frac{\\partial^2 \\Phi}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi}{\\partial z}\n + \\frac{\\partial}{\\partial t} \\left( |\\mathbf{u}|^2 \\right)\n + \\tfrac12\\, \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\left( |\\mathbf{u}|^2 \\right)\n = 0\n \\qquad \\text{ at } z=\\eta(x,y,t).\n", "93560487f615a94118f01c61156b2a81": "N = (Q,\\ \\Sigma ,\\ T,\\ q_{0},\\ A_{1}\\cup A_{2})", "935609ad3eb8978d0e5ad5fdbd9f1a28": "\\zeta\\,", "93564d09ce652d0b78e337c353b8eaf7": " |\\mathbf{F}_\\mathrm{h}| = m |\\mathbf{g}| \\frac { \\mathrm{sin}\\ \\theta}{ \\mathrm {cos}\\ \\theta} = m|\\mathbf{g}| \\mathrm{tan}\\ \\theta \\ . ", "93565b9da134d6bcd002a481f8e709ed": "X = \\frac{\\pi X^u + (1- \\pi)X^d}{1+R}.", "9356a6641c5d79f86dcf3c508fde5987": "(2^3/5!!)\\pi^2 = (8/15)\\pi^2 ", "9357647d2a288a323eb5830389a3b97e": "f(t') = |\\mathbf{r} - \\mathbf{r}_s(t')| - c(t - t')", "935780195fa75b02a6a9cf3a685195b5": "\\cap_{n=1}^\\infty A_n = A_1 \\smallsetminus \\cup_{n=1}^{\\infty}(A_1 \\smallsetminus A_n)", "9357b989d4a8861e78d500181aa5cb9a": "\\Theta_{1d}", "9357cbadb3645e96d6e8d2f577eb11e5": "M(10,1,5)", "935807b473faeae903f951754ea09f32": "2|A|-1\\,", "935813a8dbb9c24d465cf980c278b8b1": "\\textstyle J/kg", "935819d711fcad142df13795b1879a35": "\\displaystyle\\varphi^{\\prime\\prime} + \\coth r\\, \\varphi^\\prime = \\alpha \\, \\varphi", "93583f6f07002c794dd65c0dd298023e": "(C) \\frac{L_1 \\vee \\cdots \\vee L_n}{L_1'|\\cdots|L_n'}", "93588291f2aa4f616d5ca36d16cd326c": "\\psi= \\mathrm{Id} * |\\mu| ", "9358867f0a867ab3ad4777ef7987035a": "\\pi _{1} =p_{1}-A_{1}=[\\hat{P}(\\varepsilon _{1}-\\mathcal{A}_{1})+p],", "9358c1a9dacaa95a12d39374ac939e42": " \ny_i(t) = Y_i(x_1(t), ..., x_N(t)) \\text{ } \\forall i \\in \\{1, \\ldots, K\\}\n", "9358ead141c349dda77df052ab4ec2d9": " {n_{OH^- added}}={n_{HA initial}-n_{OH^- added}}", "935944d28ee1c69cc4ed28c6d72bc730": "\\alpha / n \\le 1 - (1 - \\alpha)^{1/n}", "93595b0b280fd37dd8c01fce43f75618": "R_\\text{k}", "93598619d32f04be6b46e9aca8a5b8a2": "K=\\mathbb{F}_q", "9359b909209550f55e2b90dae90ed9f4": "\n\\lim_{|z|\\rightarrow\\infty}U(a,z)/e^{-z^2/4}z^{-a-1/2}=1\\,\\,\\,\\,(\\text{for}\\,|\\arg(z)|<\\pi/2)\n", "9359cbdad9afd70d597506fb582eca49": "110.7\\pm 2.3%", "9359d02b110cfc0344781ae36fbe5e31": "E_{offset} = \\sum_{i=1}^{config} c_i \\sum_{j=1}^{hype} \\frac{EN_{ija} + EN_{ijb}}{2}", "935a689a190eb33be83cf9fb16e3baa0": " f_{0} = f_{1} + f_{1} - f_{2} ", "935a81a61449c044bd22d8ee9403d7bc": "\\operatorname{ad}:\\mathfrak{g} \\to \\operatorname{End}(\\mathfrak{g})", "935a9224d5792ee10870d45003a6d0df": "\\sigma\\;\\mid\\;0 \\equiv_{b} \\sigma", "935ac9be8652184ac3a15397155a0521": "\n \\boldsymbol{\\sigma} = 2\\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{B} - p~\\boldsymbol{\\mathit{1}}~.\n ", "935ae9165217b5282357f6c9e892ba0f": "W_{q,j}", "935b1c989305cb75a816b514e1952222": "A_n:=\\frac1n\\sum_{i=1}^n x_i,\\qquad G_n=\\biggl(\\prod_{i=1}^n x_i\\biggr)^{1/n}", "935b5b2e7aa799c0b8a101508aaa20dc": " \\underline{u}_D\\star = \\underline{u}-j\\omega_1L_1\\underline{1}_D", "935b672ec96e3d85088a0f3a1f3b6f0b": "sin(\\frac{m\\pi }{a}y) \\ \\ \\ \\ \\ (m = 1, 2, 3, ...)", "935b69e51b061a8956498203456b5056": "\\nu' = \\gamma \\nu \\left ( 1 - \\beta \\cos \\theta \\right )", "935c2a167bcd7dc79065928bbf1b8172": "c^d = (g^m r^{n^s})^d = ((1+n)^{jm}x^m r^{n^s})^d = (1+n)^{jmd \\;mod\\; n^s} (x^m r^{n^s})^{d \\;mod\\; \\lambda} = (1+n)^{jmd \\;mod\\; n^s}", "935c2ba0f1bccb980d70f8415ec055bc": " \\frac{dy}{dx} = g(x) ", "935c53e22c11df2767dbe5bf46fc22be": "\\overline{126}_H", "935cd709f0d0ca32e8e308f4a99e8f76": "f^{0}(x) = 6 + 2 \\times x", "935d08cbc3c5319314ba442ba2309c64": "\\vec{z}", "935d210fad4c3ab91ff0a65cb7d20a52": "\\frac{e_\\mu}{e_\\lambda}", "935d5db82eba1d4cca3c5a53a1552da1": "E\\{y_1\\} = E\\{y_2\\} = \\bar{x} = 1/2", "935d92837d96fa94c709ef9145d78b86": "\\int_{0}^{\\infty} x^n e^{-ax}\\,\\mathrm{d}x = \n\\begin{cases}\n \\frac{\\Gamma(n+1)}{a^{n+1}} & (n>-1,a>0) \\\\\n \\frac{n!}{a^{n+1}} & (n=0,1,2,\\ldots,a>0) \\\\\n\\end{cases}", "935d931a954decd13f2af8ffb180a168": "\\delta_{\\epsilon} S=\\bar{\\epsilon} \\psi", "935dd147e096334511ff8a74574e863b": "\\frac{-k}{i}=j", "935ddb09ace61dfc938bf3876d420f66": " E_2 = 4.00 + 2.00 = 6.00 \\text{ ft}", "935dfa04d520815f7c4a85cb62225d32": "U{}^4_6", "935e117772c62b0e12baf37141f4e03c": "{\\mu_t}_\\text{inner}", "935e24b6a08cc5e12cae47c75138c65b": "K_2 = 110", "935e7336814bf3bb73902e699599ea14": "P(x)=x(x-1)\\cdots(x-k+1)", "935e79bf7d6b5117a8f29644f15169d5": "\\mathbf{D} = C_D q A_{ref}", "935ed25b8812a6a64bd77f2c68db6cb0": "\\mu/\\rho", "935ed2b567c3ea9e3662c6d0ea585c78": "\\{\\sigma\\mathbf F;\\,\\mathbf F\\in C\\}", "935f37f1f3559662ad65cd876972a948": "~E_\\frac{M^{4+}}{AnO^{2+}_{2}}", "935f41dc8590845d5655f8aced9c389c": "H(\\theta|\\alpha)", "935fb95741436bbc791e59b3c6852ab1": "\\frac{1}{N_\\mathbf{P}}\\mathbf{M}^T\\mathbf{P}\\mathbf{1}", "936072f8f7685ccfbc216bab7e73c2c3": " m_1 = \\frac m2 , m = 2^k , k \\geq 1 .", "936097ade2d578eab9cdbd22e9a3f480": "(a;q)_0 = 1", "9360c7faa1d0790f3ab2acd2c89b3b88": "\\sigma_n(A)", "9360d2c79de73e141e391d96ae0770ba": "z=1", "93613b98b2578895780f87d42cdb5135": "\\mathbf{{\\Sigma}}_y = \\mathbf{B'}\\mathbf{{\\Sigma}}\\mathbf{B}", "9361988021552820f7701f6aec2bbf47": "f(x) = 0\\,", "9361cc4dff93cb94b674c69a8f9367da": "E = X_u \\cdot X_u = \\sin^2 v", "9361cf039818cd7b3c9beec611e5e379": " a\\div b=c", "9361ddac22e3048664604a1496834f3c": "\\displaystyle \\|u\\|_{L^\\infty(\\Omega)}\\leq C \\|u\\|_{H^1(\\Omega)}\\left(1+\\log\\frac{\\|\\Delta u\\|}{\\lambda_1\\|u\\|_{H^1(\\Omega)}}\\right)^{1/2},", "9361f5052b33eb241e8f90936289ffee": "K(x,x';t)=\\left(\\frac{m\\omega}{2\\pi i\\hbar \\sin \\omega t}\\right)^{1/2}\\exp\\left(-\\frac{m\\omega((x^2+x'^2)\\cos\\omega t-2xx')}{2i\\hbar \\sin\\omega t}\\right)", "9362a07eec4078744a51c829e1c4ba23": "\\varphi: H \\rightarrow S'", "9362b61dd45d297ead87fd8ccac0d554": "-177\\pm 10", "9362fbeb880fc3591f0357860142e1f1": "\n \\sum_{i\\in I}A_i\n ", "93632ecb0f2ceb95b1fe5fb733ab4d7b": " Sp(6,\\mathbb C)", "93633156e797acc0824080726d1230ea": "E = X_1^4 + X_1^3 X_2 + X_2^4", "936340f9440662d29d181335b634d015": "\\Phi_E =\\,\\!", "936342b61028f7b0765bf968459f9a34": "\\frac{dz(t)}{dt}=-\\beta*y(t)", "936348df1d7bf2d2ea3a1920d038550a": " |\\mathcal{F}| > \\sum_{i=0}^{k-1} {\\binom{n}{i}} ", "936354e19f12f3cd47881094836de5dd": "\\tau = 2/3 ", "9363c60b6c4dfda007e6f67f326eff2a": "R_i= p(L_i)\\,\\!", "9363eff9bd46a83a5344e602f0e8244d": "H^{2i}(X, \\mathbb{Z})", "9363f81453b9d33cac402f3f12b48d00": " \\gamma(\\chi) = \\left \\lfloor \\frac{7 + \\sqrt{49 - 24\\chi}}2 \\right \\rfloor.", "936412a2436bff37f1887462c364df52": "\\bold{r}\\rightarrow -\\bold{r}", "93647ad6db6f49bfd0de8ae220e34dcb": "\\theta = \\arcsin x = -i \\ln \\left(ix + \\sqrt{1-x^2}\\right) \\, ", "93648b3fb3b792f3a075a4c2fdd7977e": "\\ a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3}}}", "9364d4a75891c0bd26ad42a5d759c62d": "\n-\\mathbf{q} \\cdot \\dot{\\mathbf{p}} - H(\\mathbf{q}, \\mathbf{p}, t) = \n-\\mathbf{Q} \\cdot \\dot{\\mathbf{P}} - K(\\mathbf{Q}, \\mathbf{P}, t) + \\frac{\\partial G_{4}}{\\partial t} + \\frac{\\partial G_{4}}{\\partial \\mathbf{p}} \\cdot \\dot{\\mathbf{p}} + \\frac{\\partial G_{4}}{\\partial \\mathbf{P}} \\cdot \\dot{\\mathbf{P}} \n", "9364e29526aaae8f3b4d5ec23c233032": "\n\\sum\\limits_{s = - \\infty }^{s = + \\infty } {e^{ - {{i(mB - sV)\\pi } \\over 2}} J_{mB - sV} } (mBM)\n", "9364f2040d04554db61beb319bd14ddd": "m_\\text{P}=\\sqrt{\\frac{\\hbar c}{G}}", "9365089adf72414c0725ef63f6c99df5": "o(g(x))", "9365d005d5cd7e977a7cc1f3bab5b6d2": "\nc \\,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \\,F_3(a_1,a_2,b_1+1,b_2,c; x,y) + a_1 x \\,F_3(a_1+1,a_2,b_1+1,b_2,c+1; x,y) = 0 ~,\n", "9366004779128b15976b814c0ab13d59": "\\mathbf x = \\sum_{i=1}^n x_i \\mathbf{e}_i.", "936647334bd1602528177fc2bb33eeee": "B_4", "9366a78250ecf3819189e30ae6937f14": "F=-mh^{2}u^{2}\\left(\\frac{- \\varepsilon \\cos \\theta}{l}+\\frac{1 + \\varepsilon \\cos \\theta}{l}\\right)=-\\frac{m h^2 u^2}{l}=-\\frac{m h^2}{l r^2},", "9366f207da9c03605f0b71b1a8bb9ce4": "\\nabla(z)", "9366fe38c4f419908592fb0ce8084556": " E_t = m_t c^2 ", "93673e8431ca8122ee9d4c7e61b09229": "f:R\\rightarrow R/I", "93676b68ee33ed68c6b07321fbb5fddc": "x(1 - x)y'' + (\\gamma - (1 + \\alpha + \\beta)x)y' - \\alpha\\beta y = 0.", "9367704d5c2f14ba2c993c314220be04": "u:=\\int_a^b v(t)\\,dt.", "9367b119ad58383e7bfb871247df420a": "S \\in \\mathcal{S}", "936811a79a212c1134403ba079dabcb7": " f_2(X_1,X_2)f_3(X_1,X_2) ", "93686a205678a02400883dc8cd2b7f27": "y(t)\\,", "9368993c4d67ed3334bc516a22623067": "a = \\sigma_1 \\sigma_2 \\sigma_1", "9368fc40309b334da274db79c75cfcca": "M_{\\mbox{Effective}} = M \\times \\left( \\frac{\\mbox{Players}}{10} \\right)", "936931d0fdcba924a5b2768e0364dd27": "q = \\frac{\\nu+3}{\\nu+1}\\text{ with }\\beta = \\frac{1}{3-q}", "9369a97f6c7ce24db3a3402b3b6b6fd7": "\nQ^m_\\ell \\equiv \\sum_{i=1}^N e Z_i \\; R^m_{\\ell}(\\mathbf{r}_i),\n", "9369edbf7c07bf6f309b494b2a37db29": "var_{01}(p)=0", "9369f2243a4fa2cee66112a4f4914518": "\\overline{(z/w)} = \\bar{z}/\\bar{w}. \\,", "936a195dc019aaa077af0ba7a1f2378b": "r=f_2(\\theta)-f_1(\\theta),\\ r=f_2(\\theta)+f_1(\\theta+\\pi),\\ r=f_2(\\theta)+f_1(\\theta-\\pi),\\ ", "936a23ce09bef7f67579d60b1909c828": " i_{w,t} = \\frac{2hc^2}{w^5 (\\exp(hc/wkt) - 1)} ", "936a8960357a5ffd61167d52fa41ba32": "\\mathbf v_S \\times \\mathbf s", "936a8a915b268cb636b27b6ad832d3dd": " M \\times \\,^{\\prime\\prime} 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\mbox{-1} \\; 0 \\,^{\\prime\\prime} = M \\times (2^6 - 2^1) = M \\times 62. ", "936a993de4e6e8d4485f0711067cdf4b": "-2\\xi \\frac{\\mathrm{d} c}{\\mathrm{d} \\xi} = \\frac{\\mathrm{d}}{\\mathrm{d} \\xi} \\left[ D(c) \\frac{\\mathrm{d} c}{\\mathrm{d} \\xi} \\right]", "936ad7d3a72658f91324be22322db1ca": "\\boldsymbol\\omega_p = \\frac{(\\boldsymbol I_s - \\boldsymbol I_p) \\boldsymbol\\omega_s} {\\boldsymbol I_p}", "936b772090954dbb7dde61e016cfb737": "(x_0, x_1, ..., x_n)", "936bb9bfce16d88871eed6f64853a5cd": "f(s)=\\int_0^\\infty e^{-st} \\, d\\alpha(t)", "936bdebabb839c450a4214b465c8e40a": "\\mathbf{a}\\in\\mathbb{R}^d", "936bf7138cbabee0d4ac94b53caae8c9": "\\boldsymbol{K}_{k} = \\boldsymbol{P}_{k|k-1}{\\color{Red}\\boldsymbol{H}_{k}^\\top}\\boldsymbol{S}_{k}^{-1} ", "936c184526d637416b7dce5443a18156": " \\left [z \\right] ", "936c433b1ef836dd919cf05a3de3174e": " \\frac {1}{c^2} \\frac{\\partial^2}{(\\partial t)^2} \\psi - \\mathbf{\\nabla}^2 \\psi + \\frac {m^2 c^2}{\\hbar^2} \\psi = 0. ", "936c839c72b33bb74ee60557ada4033a": "x^2+y^2 = 1", "936ca8a69dd751bf10c774a7802252a1": "T\\left(\\int_\\gamma f(z)\\,dz\\right)=\\int_\\gamma (T\\circ f)(z)\\,dz.", "936cb4fe5b721184f1c53964f91c541b": "c^2 \\left(\\frac{t'}{2}\\right)^2 = h^2 + v^2 \\left(\\frac{t'}{2}\\right)^2", "936ce22c90a617fb1ca762548f8431ea": "\\frac{}{\\Gamma \\vdash \\alpha \\rightarrow (\\beta \\rightarrow \\alpha)} \\qquad\\text{Ax}_K", "936d51fd9e89047394b11530e799c253": "X_i \\cap X_j = \\varnothing,\\quad \\forall ij\\\\\n0\\;\\;\\;\\;\\mbox{otherwise}. \\end{cases}", "937ddc677ad9c01f9f16f0f5c038e539": "\\mathcal{U}(\\hat{\\alpha}(q,r_{c}),\\tilde{u})\\ ", "937decd2112fe27f47d4043a181bebb0": "m=0,1,\\dots,n", "937e0a76c42430121033f51b87139f7d": "{Z_0}^2 = \\frac {Z}{Y}", "937e15464886e12c911834a44868f9c9": " \\rho(x_1,x_2)=\\frac{\\mu^{(2)}(x_1,x_2)}{\\mu^{(1)}(x_1) \\mu^{(1)}(x_2) }, ", "937e16ad1c414be72be53e5cf5003fd4": "\\,^{238}_{92}\\mathrm{U} + \\,^{64}_{28}\\mathrm{Ni} \\to \\,^{302}_{120}\\mathrm{Ubn} ^{*} \\to \\ \\mathit{fission\\ only}", "937e559218ec2a425580dce6e48da962": "52^2", "937ebf888a4dcb44de9ba11526e6f8eb": "\n1_{1}1_{2}; \\gamma _{51}\\gamma _{52}; \\gamma _{1}^{\\mu }\\gamma _{2\\mu\n}; \\gamma _{51}\\gamma _{1}^{\\mu }\\gamma _{52}\\gamma _{2\\mu }; \\sigma _{1\\mu\n\\nu }\\sigma _{2}^{\\mu \\nu },\n", "937ed379d6c15cbb5629c5ba952fced6": "\\gamma'(t) = V(\\gamma(t))\\,.", "937ef0e34fb6d97ad3a5f12494a8b1fa": "Capex_t", "937f57b70422051dcb758bf5e645bc41": "P(c_j|I)", "937f651e5d05d10f798640b123ca47e5": "\\frac{D}{dx}\\frac{D}{dy}V-\\frac{D}{dy}\\frac{D}{dx}V=R\\left(\\frac{\\partial\\sigma}{\\partial x},\\frac{\\partial\\sigma}{\\partial y}\\right)V", "937f6ad48d5d4a328cec1a04591d6cd4": "(1.21688\\ +\\ 0.6)\\ A_x", "937fba453cfd08e6f5c4caa040ce0c4e": "\\displaystyle{a_n={1\\over 2\\pi}\\int_0^{2\\pi} f(e^{i\\theta}) e^{-in\\theta}\\, d\\theta.}", "937fbfb256a52852dc3ccfc08246307b": "\\Delta V_{ph} = V_f \\ \\Delta V = \\frac{4}{3}\\pi p_{f}^3(\\vec{r}) \\ \\Delta V .", "938019ca3cb7b70ff86a0d5b1ed16166": "\\textstyle p(x)\\leq 1", "938058c40837e75bcaec43e17241be8d": "\\scriptstyle q", "93805cf66baf3da6a57492a07797b1ca": "\\frac{x}{m} = Kp^{1/n}", "93806c6127ff770b0f8be512c8c2a40a": "\n\\begin{align}\nm_\\textrm{photon} &= 0 \\\\\nH \\,|\\,\\mathbf{k},\\mu\\,\\rangle &= h\\nu\\, |\\,\\mathbf{k},\\mu\\,\\rangle \\quad \\hbox{with}\\quad \\nu = c |\\mathbf{k}| \\\\\nP_{\\textrm{EM}} \\,|\\,\\mathbf{k},\\mu\\,\\rangle &= \\hbar\\mathbf{k} |\\,\\mathbf{k},\\mu\\,\\rangle \\\\\nS_z |\\,\\mathbf{k},\\mu\\,\\rangle &= \\mu |\\,\\mathbf{k},\\mu\\,\\rangle,\\quad \\mu=1,-1 .\\\\\n\\end{align}\n", "938092d4f74dd0ac63a040164a2cd944": "E(m) = x^\\sigma g^m \\mod n", "938096ff77c7f8e895a37209d801497c": " \\prod_{k=1}^{m} \\tan\\left(\\frac{k\\pi}{2m+1}\\right) = \\sqrt{2m+1}", "9380e1eb78bb91090f892d153a0e4443": "\\check{f}", "938108af898071fbcfab7859a7694866": "x^{ 13 }+x^{ 12 }+x^{ 11 }+x^{ 8 }+1", "938163e1ede834294d7564bc5f40a7f5": "\\forall u \\in V: u \\neq s \\text{ and } u \\neq t \\Rightarrow \\sum_{w \\in V} f(u,w) = 0", "9381ceb231a138f6954a05cc6c6c5211": "\\frac{\\partial}{\\partial t} \\pi_i = \\frac{\\partial {\\mathcal L}}{\\partial x_i}.", "9381e4365d3da6194f4c48beae1cabfb": "\\{ t \\in [0, 1] \\mapsto \\psi_{n,k}(t) \\; ; \\; n \\in \\N \\cup \\{0\\}, \\; 0 \\leq k < 2^n\\},", "9381ee36127118657ccc261e0f65723b": "\\mbox{Earnings Per Share}=\\frac{\\mbox{Profit- Preferred Dividends}}{\\mbox{Weighted Average Common Shares}} ", "9381f177dd8ba3a4bed401b2e8774bf0": " \\tilde{\\ell} ", "93822a0d4be57367481c3adc429e94b0": "f_{out}(x) = f_{in}(x) + \\beta f_{out}(x+1)", "938236a2a1feb342af33a4a107dd526f": "\\rho(p,q)\\,d^nq\\,d^n p", "938253d7be7db25970368a510b57aca9": "U= \n\\begin{bmatrix}\ne_{p} & e_{p+1} & \\cdots & e_{T}\n\\end{bmatrix}=\n\\begin{bmatrix}\ne_{1,p} & e_{1,p+1} & \\cdots & e_{1,T} \\\\\ne_{2,p} & e_{2,p+1} & \\cdots & e_{2,T} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\ne_{k,p} & e_{k,p+1} & \\cdots & e_{k,T}\n\\end{bmatrix}.\n", "93826b9de66fbfb38149bccf68230baf": "\\mathbf{R}_{x}(n-1)", "93827e7d7b52b0416f2aee2fd352495d": "(\\mathbf{n}\\cdot\\mathbf{x'})\\mathbf{J}(\\mathbf{x'})=\\frac{1}{2}\\left((\\mathbf{n}\\cdot\\mathbf{x'})\\mathbf{J}(\\mathbf{x'})+(\\mathbf{n}\\cdot\\mathbf{J}(\\mathbf{x'}))\\mathbf{x'}\\right)+\\frac{1}{2}(\\mathbf{x'}\\times\\mathbf{J}(\\mathbf{x'}))\\times\\mathbf{n}", "9382851724b14053d6adfb11ca3c3238": "(I / N C)", "9382938f659edd15d66522c194b5a4ac": "s_1, s_2, ..., s_k", "9382c3568a5eb674d70ad9e9e8b35d05": "\\hbar = \\frac{h}{2 \\pi}.", "9382d86d92e1a01c6eeed8a144a0fae8": "\\frac{\\partial \\delta}{\\partial x_j}(x)", "9382e9b402a348b8531a4db2c65204b5": "\\frac{\\eta_i}{c}", "9383237205563ac6e58313533a6d3ab1": "\\log_2\\!\\left (n \\right )", "93834bde205bcdda4323a27788056ceb": "\\scriptstyle r=\\sqrt{(m-M/2)^2+(n-N/2)^2}", "9383d103fc3a65bba569a99e98181ef0": "\\displaystyle{W(z)=\\prod_{j\\ge 1} (1-z/2^j),}", "93840b0a9a850da9d66f5734357bef68": "\\widehat{mn}=\\hat{m}\\hat{n}", "93842c414927d0b6f53983b9cf3051a7": "\\displaystyle \\times{}_3F_2(-n+2j,n-2j+\\alpha+1,(\\alpha+1)/2;(\\alpha+2)/2,\\alpha+1;t)", "9384c8180f2c036ceb1fc05f138d4c00": "\\bar{a}_n", "9384f3e00b4cfa98e04db5dffca63a0b": " \\psi_t(x) = {1\\over \\sqrt{2\\pi i t}} e^{ -i (x-y) ^2 /2t} \\, .", "93851f07417d616ec20be2ab2a94c70e": "\\overline{\\Gamma(T)}", "93854b7ef0a9b75bd9b2deb75337747d": "a=0.8", "938559aa7212d5a13f7eeedc54af8161": "\\mathit{C}_G \\varphi", "9386685fa477983c5cabf820914af5a1": "\\rho_\\text{matrix}", "938689ac37713685c35119ca774d88fd": "\\sup\\mathop{\\rm supp}\\,\\phi\\ast\\psi=\\sup\\mathop{\\rm supp}\\,\\phi+\\sup\\mathop{\\rm supp}\\,\\psi", "9386a89efee19ee0401165cb889e9d89": "\\mathrm{Hom}_\\mathcal{S}(X,Y)=\\mathrm{Hom}_\\mathcal{C}(X,Y).", "9386f10606d2285fd303fa107dde819a": "\\bar{Y}-\\bar{M}", "938729b9e140745b159f0f0257d5da9b": " \\mathbf{E}^\\dagger = \\mathbf{E}", "93873403f84896e4f9a76f900847ea33": "X_n\\xrightarrow{d}X", "93873b9c40e25258911a20d15314be8d": "\\deg(a_i \\otimes \\lambda_i) = \\deg(a_i) + \\deg(\\lambda_i).", "938790cd9aa53a94ea398ae436f5a9ef": "\\delta(t) \\, ", "9387a6b380234960a14d096a1eaa6535": "N_{\\epsilon}", "9387aa7bc8a5c4368789a85a5d86256c": "B : [0, T] \\times \\Omega \\to \\mathbb{R}", "9387b37b57ff41cc7fb17a29b6f541db": " \\psi\\wedge \\theta_1 + \\chi\\wedge \\theta_2=0 ", "938805aeda1fd53b470c89643223f3e2": "\\delta_a : a \\rightarrow a \\times a", "93889aa9ad31d7cb4f9ef64c5eb3fc9a": "\\displaystyle B = \\mu_0\\mu_r Ni/l", "9388b235e7f1312affad72fca3a11dd3": "V_{t,k}", "9388bafdcb5776dfb7ee970e04452a51": " F_\\varepsilon = F(x, \\, g_\\varepsilon (x), \\, g_\\varepsilon' (x) ) ", "9388bc3fff665887b334f192029bd53d": "h_{average}", "9388c89bdd1c831a83a7a6cbcd66eb11": "d\\psi=0 \\,", "9388fb86d72cf7e6eed79df5c82c3f7b": "ZFC+\\lnot \\operatorname{Con}(ZFC+H)\\vdash\\exists T(\\operatorname{Fin}(T)\\land T\\subset ZFC\\land(ZFC\\vdash(T\\vdash\\lnot H))\\land(ZFC\\vdash \\operatorname{Con}(T+H)))", "9389102b9d7b80d42174bc8821ffb4c3": "n_F(a+b)-n_F(a-b)=2n_F^\\prime(a)b+\\cdots", "93891b8532950353ca2314573419fcaa": "f_{t}", "93898e205d7cf5e3b0a0e107092602b1": "0_k", "938a62ab58d73b6dc31e017330cc2b76": "[n,k,2t+1]_{\\mathcal{F}}", "938a7fbcc414afd7b5c6378153ef4636": "y=\\cos x\\,", "938ab2244a4f8e9cfe5ede255f4bbfcc": "q'=W_1^TX^Ts", "938b1778bc9a266f5fa412dd263ff104": "\\mathrm{var}(\\hat{\\beta}) \\geq\\frac{1}{\\operatorname{var}[\\ln (1-X)]}\\geq\\frac{1}{\\psi_1(\\hat{\\beta}) - \\psi_1(\\hat{\\alpha} + \\hat{\\beta})}", "938b8bf405a00d00344ee9da2d434568": "E_n = \\varepsilon\\left(n+{1\\over2}\\right)", "938bdb86e6e89e17e91c67de5542d83a": "X_{\\mu > m - \\epsilon}", "938c4aa81e00947b8f83c35cfb837f6f": "U=C_{exp}-C_{obs}", "938d0ec22627efcbbc69dfe57e962e16": "dx / dt", "938d10e519a9519c66709b9567a84db8": "\\gamma(0) = p \\,", "938d4a21f629faee7717de12bf043f20": " R_{\\rm s}", "938d6a4ed2da6662fddcd96b01070658": "d_{\\rm f} \\gg D,", "938d8c9a95768b854494d636e8c53d27": "U^2 V^2 + V^2 W^2 + W^2 U^2 - U V W = 0.\\,", "938dc348d1907d391b04144ff2baf81d": " w^T \\Sigma w ", "938deea9db7b53d1fbb1b6be1479e50e": "I = \\mathbb{N}", "938e10f0e33d1e6e5091f82c1a5cc5b8": "XX_1 = X_1^2 \\, ", "938e29cbfb4175a4605fabe9b3defaaa": " e\\,(i)", "938e471583331a9741ecece9b5865e2d": "c_a = \\frac{1}{n}\\sum_{i=1}^n \\cos \\theta_i.", "938e7777e1ffba73d05748ba76edc2bd": "\\sqrt{S} = \\sqrt{a}\\times10^n", "938ecc3a6b649dd5016bf5b148619f66": " \\nabla \\cdot \\mathbf{H} = 0", "938f1320490eb82e5df8483d86170097": "\\{x,x^2,x^3,x^4,\\dots \\}", "938f32263fcd7fe0535c2bfbd5aa1c9b": "3987^{12} + 4365^{12} = 4472^{12}", "938fb45c49620cfbec3ccee64367dd5c": " \\mathcal{H}_\\mathrm{0} = \\mathcal{H}_\\mathrm{EZ} + \\mathcal{H}_\\mathrm{NZ} + \\mathcal{H}_\\mathrm{HFS} + \\mathcal{H}_\\mathrm{Q}", "938fd27710ba139815875cdb5731f4cc": "f(z)=z+2\\pi\\sin(z)", "938fd40263f4a9b63f5519f89355d7d2": "QH^*(X, \\Lambda) \\otimes QH^*(X, \\Lambda) \\to R", "938ffc9bd8128453e87bdf8ff072ac1c": "z = -(x+\\mu)/\\sqrt{2}\\sigma", "93904c08e4936d95e14827b64fd27af4": "a_0 \\approx 0.529\\;177\\;210\\;92 \\times 10^{-10} \\mbox{ m}", "93904e8d15ddf3b0208ff7684b4ff48c": "S_{k+1} := S_{k+1} \\cup cand", "93905bb8101dfe417e6139f6818f92bb": "| \\psi' \\rangle", "93906f7f334bfa5c41899e2274eada3e": "N_{B/A}(\\mathfrak a B) = \\mathfrak a^n", "939072b20b5e82f8d1576b9c8eee3981": " \\theta\\ ", "939074339b0eafb1bcb88e06689ff1f0": " \\partial \\ln X_2 = \\frac {\\Delta H^\\circ_{fus}} {RT^2}*\\delta T", "93907f2d5e3a0cff8507073e0971804d": " L^2 - 4\\pi A \\geq \\pi^2(R-r)^2. \\, ", "9390997f4fef43e427caa7cce03bf5d2": "\\Delta^{1,Y}_n", "9390dab5b6da0796e8e97953ae19f66e": "P = {C \\over C + F}", "9390e8bd065ca7e2b064525cba56e98f": "\\displaystyle{A\\sim S + SR +SR^2 + \\cdots}", "93917df63b744ceac64af1ec618a685c": "\\mathcal E", "9391c819e4054a02869b4d3434fe5f9b": "\\mathrm{pH} = \\mathrm{p}K_{\\mathrm a} + \\log\\mathrm{\\frac{[A^-]}{[HA]}}", "939200e17ffe84a3fa5c6994d3d01d83": "\\mathbf{J}", "939202e559f11825a6269cabf90b0479": " \\langle a\\rangle ", "9392303336ac515fc1a556696b66793b": "R_k(p)\\gtrsim k^2", "939231251580cd882ab2bc59dfc334b1": "T - t", "93924b61ba1e107c695c50ea989ef27d": "\\mathrm{UCL}_{1-a} = \\overline{X}_n + t_{a,n-1}\\frac{S_n}{\\sqrt{n}}.", "939290495395f72ae792d6190346a219": "\\sigma^2_n=\\frac{Q_n}{n}", "9392a9459f04d823bb623d4fe3f42e3b": "\np(\\boldsymbol{\\theta | x}) = \\sum_{i=1}^K\\tilde{\\phi_i} \\mathcal{N}(\\boldsymbol{\\tilde{\\mu_i},\\tilde{\\Sigma_i}})\n", "9392b187ccaa5d70939a14e253ecfe2e": " \\mathcal L_X\\omega = \\mathrm d (\\iota_X \\omega) + \\iota_X \\mathrm d\\omega. ", "9392d61c79945ddae1a8253e26ee886c": "\\mathrm{Exponent} = \\left(\\frac{R+RA}G\\right)^{.287} ", "93930eaefdaa1e7f5ceaeeb55c2b32d1": "W = \\frac {dA(t)} {dt} \\cdot A^{-1}(t) ", "9393482ab4a013bf9020de2a7ebf64f4": "K_{w^{ }}", "9393cb701efd9874f38be79bca77cfb9": "NU", "9393ceb6cafbb375556d5a7b3d6a790e": "F(n) = n(n + 1)\\,", "939427c809852815db32b86692d5e2f3": "u = 1 \\quad \\text{and} \\quad u = 8.", "93942a7bec11f1c7b1fafcf4bb0f2eff": "v_1=0;", "93942ac2c46a63e6564fe759012abcc8": "x_p^{(p-1)/2} \\equiv 1\\pmod{p}", "93942fdc894b2324c84a3d5b52a8f783": " 2d > n ", "9394a72975f4a843f917e4d88b1bdde6": "\n\\begin{align}\n\\varepsilon_{11} & = \\cfrac{\\mathrm{d}u_0}{dx_1} - x_3\\cfrac{\\mathrm{d}^2w_0}{\\mathrm{d}x_1^2} + \n\\frac{1}{2}\\left[\n\\left(\\cfrac{\\mathrm{d}u_0}{\\mathrm{d}x_1}-x_3\\cfrac{\\mathrm{d}^2w_0}{\\mathrm{d}x_1^2}\\right)^2 +\n \\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x_1}\\right)^2\\right] \\\\\n\\varepsilon_{22} & = 0 \\\\\n\\varepsilon_{33} & = \\frac{1}{2}\\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x_1}\\right)^2 \\\\\n\\varepsilon_{23} & = 0 \\\\\n\\varepsilon_{31} & = \n\\frac{1}{2}\\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x_1}-\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x_1}\\right) -\n \\frac{1}{2}\\left[\\left(\\cfrac{\\mathrm{d}u_0}{\\mathrm{d}x_1}-x_3\\cfrac{\\mathrm{d}^2w_0}{\\mathrm{d}x_1^2}\\right)\n \\left(\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x_1}\\right)\\right] \\\\\n\\varepsilon_{12} & = 0 \n\\end{align}\n", "9394ba7d281c50c6c6f2e741906c5842": "P(\\mathbf{Z}\\mid \\mathbf{X}) \\approx Q(\\mathbf{Z}).", "93959ff579bdd390cff249cb628a40e9": "G=(N,A,u)", "9395c831e8d0f0ca9dee38cab9869e82": "\\mathsf{G}(b,c){\\longrightarrow}R", "9395e27cbd6ff319d114b57941caa6fe": "\\varphi (i + 1 + 4(k - 1))= 8i + k,\\ i = 0, \\dots, 3,\\ k = 1, \\dots, 8", "9395e50060bcdceb73afb0edd89a63aa": "c_i > 1", "9396882d2831e70f587cc3b42527160c": "\\bar{5}_H", "9396ce3e8b8d4639d1f967b9577f3530": "x^2 + y^2 + z^2 +w^2 = \\textrm{constant} \\,", "9396cf73a3827fa4a0ce52c661c2adab": " k_Dn_An_B", "9396ffcda51235535a1ec12142bb2036": "\\tilde{\\theta}_i = k_i/n_i", "93972680a65c42a1533190e8d5842221": "A_{\\beta,s}x=\\sum_{m=0}^{\\min(s,k-1)}\\binom{s}{m}(\\lambda-\\beta)^{s-m}A_{\\lambda,m}x", "939727f6532edc07a554df67a8e8a345": "\\frac {m_1 (u_2 - u_1)}{m_1 + m_2}", "9397aa0de88c84033e2b86b5ae8c62a3": "\\models", "93981a8846576af3f0ddf623e9819835": "A_n = n A_{n-1} + \\frac{n}{n-2} A_{n-2} + \\frac{4(-1)^{n-1}}{n-2}", "93988ffa52f75373b4c76f86666460e8": "\\frac{1}{\\infty}.", "9398f6891e56698d6443135e33bd78a3": "\\rho_{01}(p)=0", "9399329e0791ef9196be4a68e6eef605": "a_0, a_1, a_2, a_3, a_4", "939974a71dda1b83cce5ab82a2d2cec1": "\\mu \\,", "9399d4db6d1dc1508686c78a2e781994": "\\mathbf \\nu (x) = \\int \\theta (x) = -\\frac{5}{12} x^4 + \\frac{75}{6} x^3 + C_3 x + C_4 (m)", "939acb60db19931b6b6c7582b24e22df": " Qf(x_1+x_2(x_1,\\lambda),\\lambda)=0 \\, ", "939b8b14d2e09bfc6bdabe7042282944": "\n \\sigma_{zx}^{\\mathrm{core}} = \\tfrac{2h + f}{4h}~C_{55}^{\\mathrm{core}}~\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x}\n ", "939b94f1cc358d1ce6db833d4940eb56": "\\zeta \\widehat{z} \\eta", "939bbbb451f7f7670193ccbec36b5be3": "(1/1!)\\pi^1 = \\pi ", "939beff9513804e38e62631e4bda7808": "V_h.\\ ", "939bf2cb6e49d0b0f3cd62fe874adb2d": "P(x,y,z)= {}^hp(x,y,z)", "939c240599ea44becc065ea2bcb8779e": "\\textstyle{\\int_W d\\omega = \\int_W 0 = 0}.", "939c3b21597bc21bbcb5f39bc087f2e5": "\\mathbf{a},b", "939c42c33f34ae9a83c7d52314f8f07d": "f(x)=\\begin{cases}\n\\frac{k}{\\lambda}\\left(\\frac{x}{\\lambda}\\right)^{k-1}e^{-(x/\\lambda)^{k}} & x\\geq0\\\\\n0 & x<0\\end{cases}", "939d3d5d25552b65c276d1f39193c1e3": "\n\\ P(x,t) = |\\psi (x,t)|^2\n", "939d4a09780ea15892bbff13e6d9f7a2": "\\frac{d^2 \\chi}{d \\tau^2} + 2 \\zeta \\frac{d \\chi}{d\\tau} + \\chi = F(\\tau) .", "939db3210551997b27f3704c092d97fe": "(mk)", "939e0cc80d80ab5d60348dbd9efe6a04": "\\sin {\\theta_0 \\over 2}=\\frac{1}{2}\\theta_0 - \\frac{1}{48}\\theta_0^3 + \\frac{1}{3840}\\theta_0^5 - \\frac{1}{645120}\\theta_0^7 + \\cdots.", "939e269e95d2712eda654f6e7b8dcd73": "\n dV = d\\mathbf{A}^{T}\\cdot d\\mathbf{L} ~;~~ dv = d\\mathbf{a}^{T} \\cdot d\\mathbf{l}\n\\,\\!", "939e60641fd39566edcff6bc62b7ca46": "x_{41}=\\xi\\sqrt{\\left(1-\\sqrt{t}\\right)\\left(1+t-\\sqrt{t(t+1)}\\right)}\\,", "939e657178605900f3a3796d7f784f2b": "w(S) = \\sum_{x \\in S} w(x)\\,.", "939ec67a43486b0f7c7bb829faefdcd8": "S:=\\{(p,q) \\in \\mathbb Q\\times \\mathbb Q: q>0\\}", "939ed9eb6b5f569d8b6d2d3be768cb14": "\\mathrm d \\varphi_x:\\mathbf R^m\\to\\mathbf R^n.", "939ef295589f955e7f97b7a62691e906": "r\\equiv 1\\text{ mod }4", "939f0cda77c2b3417276028c26c66906": "P(A_n)=0", "939f5d94f71d9c2ad16ab8d2c8e22fa8": "(U_{\\beta}, \\varphi_{\\beta})", "939f6a371f1d99ff1514c56bd34d6770": "\\langle Tu,v \\rangle = \\langle u, T^*v \\rangle", "939f71d713e093f950fbb054c71e3038": " -\\ln\\lambda", "93a0041faaf42183692582c582bacb34": "\ng(z, u) = \\left( \\frac{1}{1-uz} \\right)^{1/u}", "93a02f5b22c6d1e1dbab50fb922315d2": " \\hat{X} = \\sum_{k=0}^{d-1} |k+1 \\rangle \\langle k| ", "93a158ad4012109e0378773266bb4b93": "-1.5\\le x \\le 4", "93a220f8aed62b6159e5e5f67c83b91e": "(x, y) = \\{\\{x\\},\\{x, y\\}\\}", "93a258b38270b5a548e07a1165c7f56f": "\\arctan z = \\int_0^z \\frac{d x}{1 + x^2} \\quad z \\neq -i, +i \\,", "93a25db34e284b2cda695d44d8aa7628": "\n \\| E_k\\Omega_k - d_k x^k_T\\Omega_k\\|^2_F = \\| E^R_k - d_k x^k_R\\|^2_F\n", "93a299a1f92e2301706d22582015f12d": "\\mathcal{D}(A)=C^{1}[a, b]", "93a29b0afd81a554974f5b47588393ab": "x\\propto y", "93a2a669d3525870d5ef5c1809384d66": "X_1 (i,j)", "93a2c0c4bd771964e2843d0f45d83acb": " f(k;m,r,p)=0\\qquad \\text{ for }k\\in\\{0,1,\\ldots,m-1\\}", "93a2d5eaf2ec5a56ba312f8346478f19": "\\hat{F}_j = \\frac{1}{2}\\lambda_j ", "93a2d9e8588f44078a12ba11610e4ce5": "C_{A}", "93a34acfb788ebf10120003ad7e79363": "w_{k-\\ell+1}=\\cdots=w_k=0", "93a353536e38cf273cb9b0c629f8d5e7": " z_0 ", "93a37d9f1e44ce0e774a1f409eff5462": "Z_0=\\frac{1}{2\\pi}\\sqrt{\\frac{\\mu}{\\epsilon}}\\ln\\frac{D}{d}\\approx\\frac {138 \\Omega}{\\sqrt{\\epsilon_r}}\\log_{10}\\frac {D} {d}", "93a41918faf9da61dc4245ac0b4ea4a7": "\\leftrightarrow \\cup =", "93a41957792efab84ffe259af9cb4970": "=\\frac{\n1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots16\\cdot17\\cdot18\\;\\mathbf{\\cdot\\;19\\cdot20\\cdot21}}\n{\\,1\\cdot2\\cdot3\\cdot4\\cdot5\\cdots\n16\\cdot17\\cdot18\\,\\times\\,\\mathbf{1\\cdot2\\cdot3\\quad}},", "93a459e10c3c29e443715c697a51d788": "T_\\theta:[0,1)\\to[0,1)", "93a4a942bcd5cf595f8892f794b9b2f5": " \\varphi(L) X_t = \\theta(L) \\varepsilon_t \\, ", "93a4b920bbed41a41d23982eb1f42c8d": "\\psi(\\nu_1 , \\nu_2) = \\frac{1}{n-1} y^T(\\nu_1) . N . y(\\nu_2)", "93a4bb8a5700bd027e5d6adef9eb441a": "\\nabla\\times\\vec{B} = 0 ", "93a4d1e721c8dcc78c71771509656d1a": "\n F|\\sigma_2-\\sigma_3|^n + G|\\sigma_3-\\sigma_1|^n + H|\\sigma_1-\\sigma_2|^n = 1 \\,\n", "93a4d3e2443c7e649d1fd9627795a36c": "\\cos \\theta ", "93a5078f3a597ff0112fee0a9c0846f4": "\\left(H^*\\right)^* = H.", "93a60babee54d0c1ab1f6862015f49fa": "(A \\vee B)^*", "93a62db2203d39df882b86bc20c7ac4f": "\\textstyle \\delta_A>0", "93a6cb696bc7193dcdf96057d28b08f4": "k(\\phi)\\,", "93a6d3f9a88437d7bb29bbf0346f6d6f": " \\left|+x\\right\\rang, \\quad \\left|-x\\right\\rang. ", "93a6d859dcf61b30613c324e768569f9": "[ADO]=[BDO], [AFO]=[CFO], [BEO]=[CEO],", "93a6e4cc8a5e05d28a9a81ac956141d1": "p_{el}", "93a70d67c348f7913a2a01fd0e83f122": "z_i=Z(x_i)", "93a7375e38b23acfa68efe76ab4d570a": "F_1 = \\forall x_1 \\dots \\forall x_n \\exists y R(x_1,\\dots,x_n,y)", "93a76c9f4889ed84b7e361651e76f1be": " \\vert{\\Psi}\\rangle = e^{T} \\vert{\\Phi_0}\\rangle ", "93a77cf9df9f22c3f74ee685abfdfca9": " \\rho^{\\Gamma_A} ", "93a79904d394b6a52fe4ce2d6e89405c": " F_T = GMu\\frac{2dr}{d^4}", "93a7e1ecfdb6a4726b8bc2d6710a9219": "-------All officials are listed in the order provided by the official DoD order of precedence-------!>\n", "93a7ff62d9958e4df26a2d7f38ebe07b": "\\operatorname{E}[a X + b Y + c] = a \\operatorname{E}[X] + b \\operatorname{E}[Y] + c\\,", "93a8378b1bd5213c327f47c70bcf2f4e": "d\\sigma_t=\\alpha\\sigma^{}_t\\, dZ_t,", "93a8f2993744330c8b7ab77de0f96cad": " B_{ij} = \\frac{ Q_{ij}} {\\epsilon_{i} \\cdot A_{i} \\cdot \\sigma \\cdot T_{i}^{4}}", "93a94c0505843d0df51dd750bfae3987": "\nk = \\frac{1}{2}\\rho f,\n", "93a950e92caa613c42ee800f5665601a": "b_{\\nu,n}(x) = {n \\choose \\nu} x^{\\nu} \\left( 1 - x \\right)^{n - \\nu}, \\quad \\nu = 0, \\ldots, n.", "93a97ee7d529be2490c7ef132a63452f": "2^m+1|2\\lambda", "93a9b64f40a58184fbb1587915cb65b8": "\\mathbf{W}_t", "93aa72a3935feec537c911de6f80ec90": "\\ ^6\\mathrm{Li} + n \\longrightarrow ^4\\!\\!\\mathrm{He} + ^3\\!\\mathrm{T} + 5\\ \\mathrm{MeV} ", "93aa94204dda185da98c38f458360e81": "\\mathrm{arg} \\left[ \\mathrm{Bi} ( x + iy) \\right] \\, ", "93aae07b39067322444aca4631e7c264": "b_1 \\approx \n\\left[\\begin{matrix}\n -0.57927 \\\\\n -0.57348 \\\\\n -0.57927 \\\\\n\\end{matrix}\\right], ~\\mu_1 \\approx 5.3355\n", "93aaf561fb91e6f052c2dd7d538276d5": "f(x) = \\begin{cases} 0, & \\mbox{if }x =0 \\\\ \\sin(1/x), & \\mbox{if } x \\neq 0 \\end{cases} ", "93aafb0338d8a9c605a5f1a044f06ea2": " f(x,y)=0\\,", "93ab62f98200634244ba16bb96a39cb6": "Y \\equiv \\frac{1}{Z} \\,", "93ab82da0171a2c9938b269c422d7152": "\\dot S_{i} ", "93ab8875ebd23cb998d2fd3bc15b83d3": "\\scriptstyle{dr \\wedge \\omega = dx_1 \\wedge \\cdots \\wedge dx_{n+1}}.", "93ab8e8bd892e8fd10713751328cd4a2": "\\sigma(G_k)\\geq\\sigma(G)", "93abe79092eaff03bf25081bfc0e0545": "\n\\begin{align}\nf(x) & = (a+b)^2 \\\\\n& = a^2+2ab+b^2 \\\\\n\\end{align}\n", "93accabe78ba8801237570f817bb86ab": "U = - \\int\\limits_{x_1}^{x_2} \\vec{F}\\cdot d\\vec{x}", "93acce121ce044f0f88466909d053fea": "\\frac{\\partial\\mathbf{U}}{\\partial t} + \\frac{\\partial\\mathbf{F}}{\\partial x} +\n\\frac{\\partial\\mathbf{G}}{\\partial y} = \\mathbf{S}", "93acd156e94b2507dc131f2e6baa9060": "(X, \\tau)' \\simeq Y", "93acd9c3f0febe6f45e66354c2f0f8df": "4{ n \\choose 4 }", "93ad0a23c1b28f82fc2ee3890fe3fbc0": "\\phi(3)=2", "93ad1361eece534d79f47f99dcd63153": " B_k(j)=k\\sum_{m=0}^{k-1}{j\\choose m+1}S(k-1,m)m!+B_k \\!", "93adbec01d02fa66c595cc0ec7132c01": "(C\\times D)^{op} \\cong C^{op}\\times D^{op}", "93adc1fe1ce52d06d867e235a1230ae8": " h'(x) = a f'(x) + b g'(x).\\, ", "93adde6387e6d0245f591c16998023a9": "{x}=\\rho \\, \\cos\\phi \\, \\sin\\theta \\quad ", "93ae002ed06dd95c81a1634122dfe0ff": "\\,| \\mathbf{w} | ~ = ~ \\left( \\sum_{j=1}^m w_j^p \\right)^{1/p} ~ = ~ 1", "93ae26e608f0a4bc39516ed2f83e0125": "p(r) \\approx P + \\frac {2 \\gamma\\, \\rho\\, _{vapor} } {\\rho\\,_{liquid} \\cdot r}", "93ae5397e4436715c3a46e9bbef6d75d": " \\boldsymbol{\\tau} = \\int_{V_n} \\mathrm{d} \\mathbf{m} \\times \\mathbf{g} \\,\\!", "93ae6104acda6a1c8cd213979f7b39a1": "\\sum_{n \\le x} | \\omega(n) - \\log\\log n|^2 \\ll x \\log\\log x \\ . ", "93ae8a69d0d6287d6e89aea45081ea6e": "M_{t}", "93ae999954eab4b660de61f6ab2992a7": " r_{\\mathrm{A}} = a ( r_{-1} - r_1 ) = \\frac{dC_{\\mathrm{A}}}{dt} =0", "93aea4460d7750d7cec3159569bab188": "\n\\Pr(\\mid X - m \\mid > k) \\leq \\begin{cases}\n\\left( \\frac{2\\tau}{3k} \\right)^2 & \\text{if } k \\geq \\frac{2\\tau}{\\sqrt{3}} \\\\[6pt]\n1 - \\frac{k}{\\tau\\sqrt{3}} & \\text{if } 0 \\leq k \\leq \\frac{2\\tau}{\\sqrt{3}}.\n\\end{cases}", "93aeb864be67d29eb412ac490528e6c2": "\\mathbb{C}S_n \\rightarrow \\text{End} (V^{\\otimes n})", "93aeeffde09e5a62c0287a2891f9d7c9": "\\{q \\in \\mathbb{Q} \\mid q \\le x\\}", "93aef3e505156353d29923d27cb02630": "\\scriptstyle P \\left ( { {b}{|}{B,\\lambda} } \\right )", "93af0661766d162ae17dccbab1d73474": "{\\Gamma'}^i_\\lambda = \\frac{\\partial x^\\mu}{\\partial {x'}^\\lambda}(\\partial_\\mu {y'}^i\n+\\Gamma^j_\\mu\\partial_j{y'}^i). ", "93af0792b38ebbbdc5f1bb80bff0a636": "\n\\mathbf{Q} = ~~\\frac{\\partial G_{4}}{\\partial \\mathbf{P}}\n", "93af0aa721d86affa940e07ed2ebc6bc": " M_1 = p, N_1 = f, M_2 = x, N_2 = x, M_3 = f, N_3 = x\\ x ", "93af0fd78646a293b2fa725138e929ca": "p(x) = v_0 + v_1x + v_2x^2 + \\cdots + v_{n-1}x^{n-1},", "93af26a1a1915b407a9d22711cfaed5a": "\ny^2 +\n\\left( x - a \\coth \\tau \\right)^2 = \\frac{a^2}{\\sinh^2 \\tau}\n", "93af678f989cb30b21e49e8a4aabd9b1": "h=h_{\\overrightarrow{x}}", "93af9373c631de337b714c7b79d4010a": "\\scriptstyle \\binom{n}{\\lfloor n/2 \\rfloor}", "93afae77433c7ed2fe9e31671346c96e": " {v_x - v_S \\over v_\\infty - v_S} = {v_x \\over v_\\infty} = 1", "93aff2bf589a5e863d695a436a2dd708": "= \\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta\\,", "93b05c90d14a117ba52da1d743a43ab1": "\\frac{1}{2}", "93b081eec6885308511ff7acb75e5ae9": "= b \\lim_{N \\to \\infty} \\frac{\\frac{1}{\\left( \\frac{A}{N W} + 1 \\right)}\\frac{A}{W}(-1/N^2)}{-1/N^2}", "93b0892ab354348a95c95d94aebbd239": "\\sum_{k}{kp_k}=m", "93b0aacd658cc79e0dec113c18e97e87": "CLOSEST(M_o,N.k+ M_e)", "93b13aee6bf91f245ab0f5693bebfe86": "\n\\Omega = \\frac{\\sqrt{2}}{4} e^\\Phi [k^{ik}k^{jl}(\\omega_{i,j}-\\omega_{j,i})(\\omega_{k,l} - \\omega_{l,k})]^{1/2} = \n\\frac{ \\sqrt{\\beta} \\omega (r -3 m) }{ r- 2 m - \\beta \\omega^2 r^3 } \n=\\sqrt{\\beta}\\omega.\n", "93b1435011cb9b9a613a21be848917a1": "\\scriptstyle F(\\omega),", "93b17f8b01fb6d5376603c29b982e61f": "\\gamma X_1 \\theta_1 \\dots X_k\\theta_k \\delta", "93b1aaa477350609da64621bd55cd010": " \\begin{pmatrix} y'_1 \\\\ y'_2 \\end{pmatrix} = \\frac{1}{x'_3} \\begin{pmatrix} x'_1 \\\\ x'_2 \\end{pmatrix} ", "93b1d7002e6e1489da62deafca57e667": "\\int_a^b \\overline{\\psi_n(x)} f(x) \\,dx=0", "93b24bb339c0a2a36fbda8faa61b5cae": "\\mathfrak g \\times V\\to V", "93b2f95400e4555f033218d420a38dc5": "\\begin{vmatrix}\nx - x_1 & y - y_1 & z - z_1 \\\\\nx_2 - x_1 & y_2 - y_1& z_2 - z_1 \\\\\nx_3 - x_1 & y_3 - y_1 & z_3 - z_1\n\\end{vmatrix} =\\begin{vmatrix}\nx - x_1 & y - y_1 & z - z_1 \\\\\nx - x_2 & y - y_2 & z - z_2 \\\\\nx - x_3 & y - y_3 & z - z_3\n\\end{vmatrix} = 0. ", "93b32591a4d0491facd0a71f2d6cde6c": "\n\\frac{x^{2}}{\\mu - A} + \\frac{y^{2}}{\\mu - B} = 2z + \\mu \n", "93b3d51b3f940cbb5266e076e1a7e20c": "L' = L +\\frac{d}{dt}F(\\mathbf{q},t) \\,,", "93b43fbab46fe1068c2db199612e4ff9": "ds^2=-(1+2\\sigma)dv^2+2dv\\,dr+r(r-2\\sigma v)\\left(d\\theta^2 + \\sin^2 \\theta \\,d\\phi^2\\right),\\quad\n\\phi = \\frac{1}{2} \\ln\\left(1 - \\frac{2\\sigma v}{r}\\right),", "93b4520df7136a9198f248955fcfce99": "\\operatorname{E}(X^r)= a^r \\frac{\\Gamma (\\frac{d+r}{p})}{\\Gamma( \\frac{d}{p})} . ", "93b4780d3f3644961d6d02bfbab4570f": "\n\\mathrm{area}(D_r) = -\\frac 12\\, \\Re\\int_0^{2\\pi}f(r\\,e^{-i\\theta})\\,\\overline{r\\,e^{-i\\theta}\\,f'(r\\,e^{-i\\theta})}\\,d\\theta,\n", "93b47baf59ea142b485dc22577a56dac": "2m", "93b4a6a9705e808cdf661fd157e16f2d": "E[Y]_{11} = -2m/r^3, \\, E[Y]_{22} = E[Y]_{33} = m/r^3", "93b52cb15a659bf8fc6c1b93a02bc8a7": "r^i := Ax^i - \\rho(x^i) Bx^i,", "93b539ccaddace97b3ea8f8d7eb182c6": "\\lim_{n\\to \\infty }\\, 2^{n} \\underbrace{\\sqrt{2-\\sqrt{2+\\sqrt{2+\\text{...} +\\sqrt{2}}}}}_n= \\pi", "93b59be9fbec4e6b0ebe7a0d92a8350d": " Years \\; Until \\; FI = \\frac{\\frac{Yearly \\; Expenses}{Withdrawal \\; Rate} - Net \\; Worth}{Yearly \\; Earnings \\; After \\; Tax \\cdot Savings \\; Rate} ", "93b633cc395155fa7df4e841284698ca": "\\,w_i (n+1) ~ = ~ w_i + \\eta\\, y(x_i - w_i y)", "93b689f19b780a6d6aff419de2e757b9": "\nP-49s_2^2-14s_1s_2-s_1^2 = 0.\n", "93b6bbc912d3f4ed53f37180b14bccfd": "N>1", "93b6c739b5f7929b541426879d91e78c": "G_0^{-1} = h / 2 e^2 \\,", "93b7189719fb6319096475742ce849b4": "\\chi\\colon G \\to K,", "93b73f3f2f953f0847c39dd045fa4d6f": " \n\\begin{align}\ni_a(t)=&\\sqrt{2}I\\cos\\theta(t),\\\\\ni_b(t)=&\\sqrt{2}I\\cos\\left(\\theta(t)-\\frac23\\pi\\right),\\\\\ni_c(t)=&\\sqrt{2}I\\cos\\left(\\theta(t)+\\frac23\\pi\\right),\n\\end{align}\n", "93b7642f7578154cf2b822056155b50f": "Z_i", "93b797099da26772b9e53ab80ca59433": "q_1''(q_2)", "93b7a35460b0d704e8a2c7c7a533bc68": "\\varphi=\\neg\\exists y(y d \\ge 2", "93eabe2b54542004366c6658a6804638": "\\frac{\\partial L}{\\partial q_j }=0\\,\\rightarrow \\,\\frac{dp_j}{dt} = \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{q}_j}=0 ", "93ead807c00c4fd3a8d5fa19012eebf5": "R = I_0 \\supset I_1 \\supset I_2 \\supset \\dotsb", "93eb3c6096b5c50783b2227027dc963a": "\\nabla_{H} F : E \\to H", "93eb6d10ffd725826d73a5e4e92a40fb": "\\varphi_i\\varphi_j^{-1}\\colon (U_i \\cap U_j) \\times F \\to (U_i \\cap U_j) \\times F", "93eb7906ae1ddf2e21382626467fac5b": "Q = f(P)", "93ebe8636e1f8d60004fe33d1321674e": "\\exists ", "93ebf06daef0140f972258083da726fb": "C_{\\infty} = \\{x \\in X: \\exists (t_i)_{i \\in I} \\subset (0,\\infty), \\exists (x_i)_{i \\in I} \\subset C: t_i \\to 0, t_i x_i \\to x\\}.", "93ec1712d7f25ffefc5c8a3995260fa1": "w(z) = w_0 \\, \\sqrt{ 1+ {\\left( \\frac{z}{z_\\mathrm{R}} \\right)}^2 } . ", "93ec458c91998faa3abe12864ebebe4b": "N := (M-\\operatorname{int~im}\\varphi)\\cup_{\\varphi|_{\\mathbf{S}^p\\times \\mathbf{S}^{q-1}}} (\\mathbf{D}^{p+1}\\times \\mathbf{S}^{q-1}).", "93ec8b551c0dc99a355d3da2e102b452": "\nz = \n\\begin{bmatrix}\n\\lambda x \\\\\nx\n\\end{bmatrix}.\n", "93ec9940096cf2863b2a905a5efc7f49": "\\begin{cases} x = 2 \\\\ y = 3. \\end{cases}\\,", "93eccad93b432d2701a0273148f55e85": "N_f = \\frac{1}{2}(\\frac{\\sigma_a}{\\sigma'_f}\\frac{1}{1 - \\frac{\\sigma_m}{\\sigma_u}})^{\\frac{1}{b}}", "93ed07de9d4f5aadc2a408298b474090": " Speed \\ Criteria \\begin{cases} \\mathrm{\\left (\\frac {C \\times Doppler \\ Frequency}{2 \\times Transmit \\ Frequency} \\right) > Rejection} \\end{cases}", "93ed2b01a9067442b54e29b3e2f11fb6": "U \\cap V_n ", "93ed6624b9569e0f3bf7a5843300d74e": " y_2 = 5.55 \\text{ ft}", "93eda0e89b6de74dbb34abdcdbfbc37b": "\n\\Pr[X_1+ \\cdots +X_n>x] \\sim \\Pr[\\max(X_1, \\ldots,X_n)>x] \\quad \\text{as } x \\to \\infty. \n", "93edc0f37203f56c9fd79c14291ec1c1": " \\varrho(X)=\\int_0^{+\\infty}g\\left(\\bar{F}_X(x)\\right) dx", "93ede6319b352af3feb528d7205766cf": "2 \\pi i \\xi \\hat f(\\xi)", "93ee1f2dbf2b658ad0656e2130b4f7e3": "i\\rightarrow j", "93ee34c4001196a15f9266ff919efaf4": "\\mathrm{Nu}_D = \\frac{ \\left( f/8 \\right) \\left( \\mathrm{Re}_D - 1000 \\right) \\mathrm{Pr} } {1 + 12.7(f/8)^{1/2} \\left( \\mathrm{Pr}^{2/3} - 1 \\right)}", "93ee8c0a79958480a3b1bc8c3575dcbd": "B-A", "93ee9c6a217ceca129508d919c4a39f7": "\\textstyle D_{1}(i) = \\frac{1}{m},", "93eea9c8cc3a1e70204d9a898e716f36": "{\\mathbf a}\\times{\\mathbf b} = (a_2 b_3 - a_3 b_2) {\\mathbf e}_1 + (a_3 b_1 - a_1 b_3) {\\mathbf e}_2 + (a_1 b_2 - a_2 b_1) {\\mathbf e}_3.", "93ef2453d62038317c174a5e06039037": "\\begin{align}g_{n+2}&=\\frac{c^{n+2}}{x^2k(k+1)\\cdots(k+n-1)}\\cdot\\frac{x^2}{(k+n)(k+n+1)}f_{k+n+2}(x)\\\\\n&=\\frac{c^{n+2}}{x^2k(k+1)\\cdots(k+n-1)}f_{k+n+1}(x)-\\frac{c^{n+2}}{x^2k(k+1)\\cdots(k+n-1)}f_{k+n}(x)\\\\\n&=\\frac{c(k+n)}{x^2}g_{n+1}-\\frac{c^2}{x^2}g_n\\\\\n&=\\left(\\frac{ck}{x^2}+\\frac c{x^2}n\\right)g_{n+1}-\\frac{c^2}{x^2}g_n,\\end{align}", "93ef472ad1dc1ceaab3e1eb44ccca441": " f(p,k) < \\frac{k}{p} + \\frac{k}{p^2} + \\cdots = \\frac{k}{p-1} \\le k. ", "93ef6733d034e83972ed04dc2e2824f6": "\\mathcal{B}(\\Omega_1 \\times \\Omega_2)", "93ef773b61f9b59037f3098d78f7eea4": " P_d = \\mathbf{F}_d \\cdot \\mathbf{v} = \\tfrac12 \\rho v^3 A C_d", "93ef85c9fce7c7118aa5d9a28abdee9b": "Q=K^\\alpha N^{1-\\alpha}.\\,", "93efa4ff04769f6b9002f6fc4c9845f1": "x^{n+1} \\not\\equiv -c", "93f00499bfdac6f3c6385d07fc0f4e9c": " G_M = \\frac{3}{2}G_N", "93f0d9ca1a7fcf4bac925e7b420a1b23": "( 2 + 2 ) = ( 2 + ( 1 + 1 ) )", "93f16210ce204c3c67118bf9e4de2a91": " \\and S_4 \\implies A_4 = p ", "93f164e1a7d8ce706d7c48035efb80da": "B, B_1, B_2, \\ldots, B_m", "93f19b2f1166f34a271bbe9671e6fb42": "(a_{11} x_1 + K a_{12} x_2) (1+g) = x_1", "93f19f618e88330011dfce6032a381d3": "{\\rm rank}(M)", "93f1c012b01ea4efd1117de7c89ef46d": "E(y|x_d).", "93f1f1f47d017670ef586251ffbef790": "y=a\\sin \\theta \\sec^3 \\frac{\\theta}{3} = a \\left(3 \\cos^2 \\frac{\\theta}{3}\\sin \\frac{\\theta}{3} - \\sin^3 \\frac{\\theta}{3} \\right) \\sec^3 \\frac{\\theta}{3}", "93f21f9ac2d71448757c97fab527fb97": "K_{c/p}", "93f22e0dace0f28e5c4f86ccd3123575": "F(s) = (1 + \\theta s)^{-k} = \\frac{\\beta^\\alpha}{(s + \\beta)^\\alpha} .", "93f30fd332c4e8fea09e07b515abcd4a": "x \\in [0, 1]", "93f32e85ab1bfc29f8dfb62029bf01e9": "\\mathbf{f}=m_{rel}\\mathbf{a}", "93f38cbfe81b484bc7dd0438578bdac7": "y(t) |_{t=t_2} = h(t_2,t_1) \\,", "93f397371dc3f31d25872b2fd679771d": "\\omega = df", "93f399dd7a2b898c79f9a9d9ac108406": "R_{5,5} = r^5", "93f3f32819c51e18870e75c6cadbd519": "\\sqrt{(1 + x)}", "93f42f00d0c34b27cd6acf6daea1101b": "\\gamma_2 \\to 0", "93f43443d72f8979581b1a01ab49adbf": " \\vdash A \\or \\lnot A ", "93f48f594fb20fc14d93dae122aec0e0": " = -T \\int \\mathrm{d}^2 \\sigma \\eta^{ab} \\partial_a \\partial_b X^\\mu \\delta X_\\mu + \\left( T \\int d \\tau X' \\delta X \\right)_{\\sigma=\\pi} - \\left( T \\int d \\tau X' \\delta X \\right)_{\\sigma=0} = 0", "93f493c7520c4d90a537548578d0154c": "\\mathbf\\R_n", "93f4cde710d99ee01bf7bd93537fa2bb": "\\frac {F_w}{F_i} = \\frac {\\cos \\phi} {\\sin \\theta \\cos \\phi - \\cos \\theta \\sin \\phi } \\,", "93f52bf6531adab7fe076ff1fa0b1751": "\\hat\\sigma_i", "93f5a57860291fefd36fabcd3d62a239": "\\,_np_x = l_{x+n} / l_x", "93f5df74f5a41db8feb37ad52c5e4c43": "1\\rightarrow (1,1)_0", "93f5f6217ff0735c374ac66b44beb4cf": "(A \\to B) \\Leftrightarrow (\\lnot B \\to \\lnot A)", "93f60efcf1d5b436ced90c1f5f8502f3": "m(\\theta_0)=0", "93f62e936de567521f95dc48916b0381": "\\text{Drilling Time(T)} = \\frac{L}{f \\times N}", "93f63c8c9105396d44d12d46550f61ec": "C = c_0 + c_1 Y_d", "93f65060a4ff737ee695eb7fe21b14bc": "e^{-sT}\\dot{=}\\frac{(sT)^{4}-20(sT)^{3}+180(sT)^{2}-840sT+1680}{(sT)^{4}+20(sT)^{3}+180(sT)^{2}+840sT+1680}.", "93f6a2e9966045a46cd74dc195b1b032": "\\sigma^{\\psi_t}(x)=u_t\\sigma^{\\phi_t}(x)u_t^{-1} ", "93f6cedb4304e17c85b15550db455c60": "\\begin{matrix} {r \\choose 2}{4 \\choose 2}^2{r - 2 \\choose 1}{4 \\choose 1} \\end{matrix}", "93f796cfb4ac379e210559261615e1c3": "2^{-13}", "93f7c00a5cb008359697bc00b9bbbceb": "E_R = \\hbar \\omega_R", "93f7e0cb2971c52217aa59c7623de0ab": "\\begin{align}\nH_0 \\left (\\mathbf{P}^2(\\mathbf{R}) \\times \\mathbf{P}^2(\\mathbf{R});\\mathbf{Z} \\right )\\; &\\cong \\;h_0 \\otimes h_0 \\;\\cong \\;\\mathbf{Z} \\\\\nH_1 \\left (\\mathbf{P}^2(\\mathbf{R}) \\times \\mathbf{P}^2(\\mathbf{R});\\mathbf{Z} \\right )\\; &\\cong \\; h_0 \\otimes h_1 \\; \\oplus \\; h_1 \\otimes h_0 \\;\\cong \\;\\mathbf{Z}/(2)\\oplus \\mathbf{Z}/(2) \\\\\nH_2 \\left (\\mathbf{P}^2(\\mathbf{R}) \\times \\mathbf{P}^2(\\mathbf{R});\\mathbf{Z} \\right )\\; &\\cong \\;h_1 \\otimes h_1 \\;\\cong \\;\\mathbf{Z}/(2) \\\\\nH_3 \\left (\\mathbf{P}^2(\\mathbf{R}) \\times \\mathbf{P}^2(\\mathbf{R});\\mathbf{Z} \\right )\\; &\\cong \\;\\mathrm{Tor}^{\\mathbf{Z}}_1(h_1,h_1) \\;\\cong \\;\\mathbf{Z}/(2) \\\\\n\\end{align} ", "93f82be689cac157381c63fc643694eb": "\n\\mathbf{v}=\\nabla\\varphi ", "93f82be8ea0b85358dbe918a71a846af": "{\\mathbf \\Sigma}", "93f84ded6a40abc48a2017f09f336cdd": "\\lim_{k\\to\\infty}\\mathbf x_k", "93f86fa05dc99969bf8e83ae98817706": "\\Gamma(\\frac{1}{2})^2", "93f8f9b1c556f0591b7d9968477bc90c": "{\\rm C_2H_6} + \\tfrac{7}{2}{\\rm O_2} \\rightarrow 2{\\rm CO_2} + 3{\\rm H_2O}", "93f96db2c65139aedc3d445fef037cf0": " A_o = 0.9999 \\approx 52 \\ minutes \\ down \\ time \\ per \\ year", "93f9dc4db1b82845990c5124e8b1c45a": "A,B,C,D", "93fa107c108ffb26b5d6ea707b6c26aa": "|a\\rangle.", "93faa864c816378332d42ee21650a69e": "u = \\frac {x L}{\\lambda R} \\ , ", "93fb02bdfc021c54854dae5e1d41eaa9": "\\begin{align}\n & \\mu \\in \\left[\\, \\hat\\mu + t_{n-1,\\alpha/2}\\, \\frac{1}{\\sqrt{n}}s,\\ \\ \n \\hat\\mu + t_{n-1,1-\\alpha/2}\\,\\frac{1}{\\sqrt{n}}s \\,\\right] \\approx\n \\left[\\, \\hat\\mu - |z_{\\alpha/2}|\\frac{1}{\\sqrt n}s,\\ \\ \n \\hat\\mu + |z_{\\alpha/2}|\\frac{1}{\\sqrt n}s \\,\\right], \\\\ \n & \\sigma^2 \\in \\left[\\, \\frac{(n-1)s^2}{\\chi^2_{n-1,1-\\alpha/2}},\\ \\ \n \\frac{(n-1)s^2}{\\chi^2_{n-1,\\alpha/2}} \\,\\right] \\approx\n \\left[\\, s^2 - |z_{\\alpha/2}|\\frac{\\sqrt{2}}{\\sqrt{n}}s^2,\\ \\ \n s^2 + |z_{\\alpha/2}|\\frac{\\sqrt{2}}{\\sqrt{n}}s^2 \\,\\right],\n \\end{align}", "93fb1a6a4a2fd4390d92b867185410e2": " \\hat T_{2,-2} = + \\frac{1}{2} \\hat a_{-} \\hat b_{-} ", "93fb282f070f0ffcfeb45180b61f6bc0": " n = \\frac {1}{\\sqrt{f}}.", "93fb3ecaefd55f61177c0f882db2c1e3": " x = x_0 sin(\\omega t - kx)", "93fb4dc8b88058045539136ddb354875": "0 \\le \\theta < 2\\pi", "93fb50342df0d65edbdd5993a61d8f29": "w_i = \\sqrt{(\\beta (1- \\lambda_i )/\\lambda_i r_i}", "93fb7576034749f3d3fb32e173e3f398": "2^{n-1} + 1", "93fb8a57a527b9b3c8dd2025c8a7772d": "\\{x^n : \\hat{p}_{x^n} \\in A\\}", "93fbe15efb63dc3ab2b9da77ac9ee9c2": " \\omega=\\omega_i \\, dx^i ", "93fbfe4c1216c6f4323c1c24e2f7250e": "x_{-i}", "93fc030b5f340824391d797788695753": "P_\\tilde{a}:=LB < P < UB=P_0 < P < P_\\acute{n} \\,\\!", "93fc258979bec53c1d3b425e616bf954": "\\frac{\\delta W}{\\partial t} = u \\frac{\\partial W}{\\partial x} + v \\frac{\\partial W}{\\partial y} + w \\frac{\\partial W}{\\partial z}", "93fc4bc7f74dd39611c6d9713b243761": "ad-bc", "93fc9ab3b20eb5043473ea2a5fe30f6c": "I(v) = \\Omega (|v|) \\ ", "93fc9e139a949b668f88d22af558c3c0": "T(\\tau) = T_{0} e^{-\\beta \\tau}", "93fcee17a6e6cdb6743a2f3110c70f3d": "A(x,y)\\hat{\\bold{z}}", "93fd220e64c9c702ea918003f6eb0615": "\\scriptstyle x_i = g_i(r,s)", "93fd51ff9fb8677ac45bff7570989f9f": "\\frac{1}{4 \\, \\pi} \\int_\\Omega f(\\Omega) \\, g^\\ast(\\Omega) \\, d\\Omega = \\sum_{\\ell=0}^\\infty S_{fg}(\\ell),", "93fd9491c9fb5a45e8a2f596fed5aede": "N = \\frac{f}{(\\hbar\\omega\\beta)^3}~\\textrm{Li}_3(z)", "93fd96ae7b73b0009d94f161408e8b18": "\n(I_k \\otimes \\Phi)\n(\\begin{bmatrix} A_{11} & \\cdots & A_{1k} \\\\ \\vdots & \\ddots & \\vdots \\\\A_{k1} & \\cdots & A_{kk} \\end{bmatrix})\n=\n\\begin{bmatrix}\n\\Phi (A_{11}) & \\cdots & \\Phi( A_{1k} ) \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\Phi (A_{k1}) & \\cdots & \\Phi( A_{kk} )\n\\end{bmatrix}.\n", "93fe190431e7a4a684d6b39211c67d4b": " \\delta_s ", "93fe7726f71a4b84b09d74eec298408c": "V_c T^n=C", "93fed4519f916c1f4ad3ca0667b0651d": "\\deg(f_ig_i) \\le \\max(d_s,3)\\prod_{j=1}^{\\min(n,s)-1}\\max(d_j,3).", "93ff3fc93360b77326dea0fa7d25c21d": "{\\rm AP}=\\bigcup_{k>0}{\\rm ATIME}(n^k)", "93ff513d94b4469ff89dce262f51c959": "\n\\left[ T_\\mathrm{n} + E_\\mathrm{e}(\\mathbf{R})\\right] \\phi(\\mathbf{R}) = E \\phi(\\mathbf{R}) ", "93ff99aa85b95038b1d2748d250caf4d": "M \\times M", "93ffb5352d7860cb9aa5ac753877c1ac": " \\Omega = -\\frac{g_{t\\phi}}{g_{\\phi\\phi}} = \\frac{r_{s} r \\alpha c}{\\rho^{2} \\left( r^{2} + \\alpha^{2} \\right) + r_{s} r \\alpha^{2} \\sin^{2}\\theta}. ", "93ffb84364f0e58956fd0f3d8ca24e2a": " E_{2ss} = E_c ", "93fff95fa1c476a0165cbf7773ef158b": "\\lambda(\\mathbf{x},\\mathbf{y})=\\frac{1}{Q(\\mathbf{x},\\mathbf{y})}", "93fffa5aa0921905a126e83e44549ba8": "({v_0+v_i})10^{b_1E_i - b_0}", "94001d56702371ca76531f0343f9ee29": "\\mathcal{L}_\\mathrm{Yukawa}(\\phi,\\psi) = -g\\bar\\psi i\\gamma^5 \\phi \\psi", "94006b2a5a7808d87aeae326cdebe770": "\\sigma_j", "940079c704490aacb6e5e979817bdea8": "T_{ab} \\, = \\rho u_a u_b", "9400919888e865f3821acd60d5cda6f8": " \\lambda = \\frac{\\mathrm{tr}(A) \\pm \\sqrt{\\mathrm{tr}(A)^2 - 4}}{2}. ", "940091b8897577f3cc9b9d241a747e94": "\\Theta\\ ", "9400cd16cd4716c986c709a9978ba713": "E = \\sum_T W_T \\sum_R W_R \\cdot \\frac{\\int_{T}D_R (x,y,z)\\rho(x,y,z)dV}{\\int_{T}\\rho(x,y,z)dV}", "9400d2321e52671b144c59acfcc1416b": " \\sum_{n\\ge 0} \\frac{n!}{n!} z^n = \\frac{1}{1-z}.", "9400e115c3cc072ddd52a30e83349daa": "\\scriptstyle \\frac{1}{2}\\hbar", "9400e2c64d37cbed95e05948efab4e40": "10\\uparrow\\uparrow\\uparrow 3=(10 \\uparrow \\uparrow)^3 1", "9400ebf223f50ff8fccb32ed13ea819e": "c ", "940138a99a90e8d2789316facb152336": "B_k(x)=\\frac{1}{k!}(1+x)^k", "94014a4cb6c99935baeec9387b4a2f0d": " \\mathbf{a}(\\mathbf{b} + \\mathbf{c}) = \\mathbf{ab} + \\mathbf{ac} ", "940186be31e7eed9dec832b2ac64dc75": "\\rho = \\Omega + \\psi(\\Omega 2) = \\Omega + \\zeta_1", "9401878c39e1bf83c54063169bc84413": "f:X\\to X,", "94019bce65019a3c2af8eb2e023efc8d": "W(x)\\,\\!", "9401c9807195731f4e8538b4ee631f21": "\\varepsilon_\\cdot", "9401e97a5e06457151abe5486ea19843": "(5) \\,", "940220569cb60a2132b7fc65c707205c": "= H_a \\left( \\frac{2}{T} \\cdot \\frac{\\left(e^{j \\omega T/2} - e^{-j \\omega T/2}\\right)}{\\left(e^{j \\omega T/2} + e^{-j \\omega T/2 }\\right)}\\right) \\ ", "940245f7d148b0f7b7550ebae3aeb90e": "0 \\rightarrow K(H) \\rightarrow B(H) \\rightarrow B(H)/K(H) \\rightarrow 0", "9402c62f06e6ded63a9a949d698c10e6": "CBR = 2 (\\Delta f + f_m)", "9402ecfc7c7f1395de9e2257fae33bb1": "\\left| \\frac{z^n}{n!}\\right|\\le M_n , \\forall z\\in D_R.", "94033c04c68b44c9898fc781e3b30198": "\\eta, \\tau", "94039111ff7840e5133463b97c88b8ff": "\\Phi_j (z)= \\Phi_0 (z-jd) ", "9403a2b9b45529bd7025705a7526ac70": " \\nu_\\xi:E \\mapsto \\langle \\pi_T(\\mathbf{1}_E) \\xi, \\xi \\rangle ", "9403a5507bd833d7e3d9393245a607d8": "U,\\ \\Delta V,\\ \\Delta\\phi,\\ \\Epsilon", "9403c3da7d746ad8f91b63dbfbce5250": "\\tau =\\delta t/\\ln ({{D}_{0}}/{{D}_{1}})", "9403d35ca6c73417294413b3c6f83607": " \\frac{1}{(2\\pi)^n}\\int_{\\mathbb{R}^n} e^{i \\xi \\cdot (x - y)} \\sum \\limits_{|\\alpha| \\leq m} p_\\alpha(x) \\, \\xi^\\alpha \\, \\mathrm{d} \\xi. ", "9403dbdcf8a2793663ea77d8169a4f2e": "S_n = |a_1| + |a_2| + \\ldots + |a_n|,\\ T_n = |b_1| + |b_2| + \\ldots + |b_n|. ", "94040c122615c9bd0040c2ee85c7cb4f": "\\ \\Psi", "940426ecbefb5777baea356ad10b8bda": "a_1x_1+a_2x_2+\\cdots+a_nx_n\\geq b", "940454e5f5f7d82bab0c7624ec100d07": "\n \\cfrac{\\Gamma, A \\vdash \\Delta}{\\Gamma \\vdash \\lnot A, \\Delta} \\quad ({\\lnot}R)\n ", "94049acaffadc96c30025ac282de9c74": "y=y_c+y_p", "9404cc7defff8c46b2b9033618dc744f": "\\scriptstyle\\boldsymbol{\\sigma}=\\left(\\sigma_{ik}\\right)", "9404e36e2f16acacf0096e3ab7740d2d": "(Y/\\sigma)^2", "94053ed9b5f60d7d300ca5d7f4d8ef40": "\n \\boldsymbol{d} = \\tfrac{1}{2}\\left(\\boldsymbol{l} + \\boldsymbol{l}^T\\right) \\,,~~\n \\boldsymbol{w} = \\tfrac{1}{2}\\left(\\boldsymbol{l} - \\boldsymbol{l}^T\\right) \\,.\n ", "94056cc9c93999ef69c56e6e88066b53": "\n {\n \\begin{align}\n \\rho~\\det(\\boldsymbol{F}) - \\rho_0 &= 0 & & \\qquad \\text{Balance of Mass} \\\\\n \\rho_0~\\ddot{\\mathbf{x}} - \\boldsymbol{\\nabla}_{\\circ}\\cdot\\boldsymbol{N} -\\rho_0~\\mathbf{b} & = 0 & & \n \\qquad \\text{Balance of Linear Momentum} \\\\\n \\boldsymbol{F}\\cdot\\boldsymbol{N} & = \\boldsymbol{N}^T\\cdot\\boldsymbol{F}^T & & \n \\qquad \\text{Balance of Angular Momentum} \\\\ \n \\rho_0~\\dot{e} - \\boldsymbol{N}:\\dot{\\boldsymbol{F}} + \\boldsymbol{\\nabla}_{\\circ}\\cdot\\mathbf{q} - \\rho_0~s & = 0\n & & \\qquad\\text{Balance of Energy.} \n \\end{align}\n }\n ", "94057feaec1157509d3db8ed4487b000": " \\langle -x,x \\rangle= -1\\langle x,x\\rangle = \\overline{-1}\\langle x,x\\rangle = \\langle x,-x\\rangle.", "940596b4257b9fc3ebb8102c73b32577": "A(\\{i\\})", "94059bb88074d280b6902283edb526c7": "Tr", "9405a6cd8b2fdd00d39165162aa6cf7a": "(a \\times b) = (b \\times a)", "9405c8b3e1061e92bbd4331dfb940d2f": "\\times \\ \\sum_{t=p}^q (-1)^t \\frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n-w-t)!}", "9405d84e09bd64e6d4a1ab99f8d17e31": "(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2.\\,", "940617d12437c90dd14b10cd210d5d97": "\\theta_1,...,\\theta_N", "940663e236a465e38a8cf6ad3abb959a": "\\frac{\\partial h}{\\partial x_i} = \\frac{1}{|\\mathbf{x}|^2} - \\frac{2 x_i^2}{|\\mathbf{x}|^4} ", "94068396b58460cf9a08e50ed2247341": "a \\cdot x=b \\cdot x+c", "940690ce415863ede8c2d85f74de5739": "p_k \\circ i_l", "940698bb82dcc64ae763ab4c9a72e1c6": "s=k^{-1}(H(m)+xr)\\bmod\\,q", "94069c467df8eb1a68bcc51a1aa07205": "e\\in\\mathcal U", "94077356f860b830f8bcb8fb94c7a695": "e^{iar}/r\\,", "94077fc58f5f9e53870004c2fc2afd1d": "\\eta_p = \\frac{2}{1 + \\frac{c}{v}}", "9407855cb8f80ca3c8a720931b5f4d31": "T_{\\gamma(0)}M", "9407dff2d252c61d2803bdc407f425ce": "B \\in \\Sigma_Y", "9407e046b7ee3fcc10895ad2cf7a6da4": "\\operatorname{G}(-) \\cong \\operatorname{Hom}(_{S}P,-).", "9407e10ab424cb836a4d4f26d06eb2dd": "{SU(3)_C\\times SU(2)_L \\times SU(2)_R \\times U(1)_{B-L}\\over \\mathbb{Z}_6}", "9407ee708255beb1b7291b6e3b950f3c": "D\\Omega = 0.\\,", "94088e3b3a63fedd1b46fd77e8b8759f": "f(X)=g(X^p)=\\sum b_i^{p}X^{pi}=(\\sum b_iX^i)^p", "9408c23ce3d3e3da2c12ff0f451453c7": "I[w]", "9408e0c8a595ccae9b5e8ae0c40316ae": "\\phi\\colon G \\to H", "940900184d33959036960d31c36cc9f6": "Y \\sim \\operatorname{EV}_1(a,b)", "94090a61b16a0d3ba1208f63265ae761": "\\; \\Phi_x", "9409257cc162cef5b8bd14652f770561": " f:\\mathbb C\\cup\\{\\infty\\} \\to \\mathbb C\\cup\\{\\infty\\}; \\quad f(z) = \\frac{1}{1+z^2}", "94096294c4e5fc019683632a4b070263": " \\begin{align}\n& [\\hat{L}_a, \\hat{L}_b ] = i \\hbar \\varepsilon_{abc} \\hat{L}_c \\\\\n& [\\hat{S}_a, \\hat{S}_b ] = i \\hbar \\varepsilon_{abc} \\hat{S}_c \\\\\n\\end{align} ", "9409713ca653cd54c7663d9763928047": " A = tan \\beta_2 + tan \\alpha_3 ", "940998fa3bed8837891247bd5da11693": "d_9", "9409ba7944bf6ca0ada90a2aee0e481a": "w(x) = \\tfrac{qx^2}{48EI}(3L^2-5Lx+2x^2)", "9409bc4bf52acc7e2d8108f81c09c55d": "q\\cdot Id", "940a50b2f70c69cc86b2bb262d660bbc": " \\hat{x_{4}} = H_{4}^{T}y_{4} = \n\\frac{1}{2}\\begin{bmatrix} 1&1&\\sqrt{2}&0 \\\\ 1&1&-\\sqrt{2}&0 \\\\ 1&-1&0&\\sqrt{2} \\\\ 1&-1&0&-\\sqrt{2}\\end{bmatrix} \\begin{bmatrix} 5 \\\\ -2 \\\\ -1/\\sqrt{2} \\\\ -1/\\sqrt{2}\\end{bmatrix}\n= \\begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\\\ 4 \\end{bmatrix}\n", "940a5614f5ac47974b5d157d3efffa24": "V_t \\ = \\ a_0 \\cdot M \\sqrt{\\frac{T}{T_0}}", "940a6e19c60c1a7fc9585c3f8505e853": "f_0 = \\exp(-(a_1+a_2))", "940acf6195d419d8e039119c6329b30a": "\\pi_i(f)\\colon \\pi_i(X) \\overset{\\sim}{\\to} \\pi_i(Y)", "940ae9f684cb6b82ac32b00b81ba3a6f": "\\sqrt{-1}\\partial\\bar\\partial f", "940b2a1d28dee037af9a6842c2a9a47f": "D = \\sqrt{X^2} = \\sqrt{(R - \\mu_1)^2/S_1} = \\sqrt{(R - \\mu_1) S_1^{-1} (R - \\mu_1) }.", "940ba7bfd2a448389feb450683c884ee": "x^{i+1} := \\arg\\max_{y\\in span\\{x^i,w^i\\}} \\rho(y)", "940bef7af4aa960dca4fe31caffd01d4": "HT = G", "940c6f7b78582b22968b0fa373706e11": "\n\\ln (1+x)=\\frac{x^1}{1}-\\frac{x^2}{2}+\\frac{x^3}{3}-\\frac{x^4}{4}+\\frac{x^5}{5}-\\cdots=\n\\cfrac{x}{1-0x+\\cfrac{1^2x}{2-1x+\\cfrac{2^2x}{3-2x+\\cfrac{3^2x}{4-3x+\\cfrac{4^2x}{5-4x+\\ddots}}}}}\n", "940c9f1caba6bc559df6b8040cc639dd": "\\scriptstyle x=l-x'", "940c9fb9efe803a29121fe5f555d5ba0": "\\Sigma s_i \\otimes t_i \\mapsto \\Sigma s_i.t_i\\otimes 1", "940cddb64377363d309d71da61516f7c": "\np_{min} \\leq P(\\cdot) \\leq p_{max} \n", "940d1b9cbef9dcf863ae4df4d62786f5": " \\sigma_1 = -\\sigma_2 ", "940d3b4c18caded846d70c0c22690c3b": "\\frac{1}{p} + \\frac{1}{q} = \\frac{1}{r} +1", "940d6748ef869ab4c373721ae0be26c6": "x_{t+1}", "940d74ceba27c0085c97a2cefb885424": "C - Nx^2 = y^2", "940d8e57e3c6427fb8fb304d1e27973f": " P(X=x|Y=y) = \\frac{f_Y(y|X=x)\\,P(X=x)}{f_Y(y)}.", "940d97d88bf0b550b2a8f49678ac65df": "\\alpha=\\frac{c}{|c|}", "940e05486ca6a354c6b6c17b1cd198d5": "\\mathcal A.", "940e82813c68fd631326ee0ba0f92419": "\\{X_{l}(t): l=1,\\dots, L; t=1,\\dots, N\\}", "940e82c9cfb95c0ff7d4a0556ddc5cff": "\\operatorname{erfc}^{-1}(1-z) = \\operatorname{erf}^{-1}(z).", "940e9f76a7e6ad9116c37d2ca0dfe096": "\nW_0=10^{-12}\\ \\mathrm{W}=1\\ \\mathrm{pW}\\,\n", "940ec754905f30e782df99c67bf545f4": "AA^T=A^TA", "940ee4da7286e6caf288f3670705138b": "e^{(\\frac{x}{2})(t-1/t)} = \\sum_{n=-\\infty}^\\infty J_n(x) t^n,\\!", "940ef8d4e798117c29ebddff82b152a7": " S = \\{ n \\in \\mathbf{N} | n \\ge k \\} ", "940fe2ae7793fffc20c08223594bcee6": "2k_F", "9410267d8037712da0e4b6e87bd08220": "F_e \\times Ptot = PeCO_2", "94102eb534622dbf2e4a0792f525e345": "\\mathbf{X^TWX\\boldsymbol\\Delta \\boldsymbol\\beta=X^T W \\boldsymbol\\Delta y}", "94107bd4f8780d9de397b6bb18113783": " [H^{\\lambda,\\alpha}_\\omega u](n) = u(n+1) + u(n-1) + 2 \\lambda \\cos(2\\pi (\\omega + n\\alpha)) u(n), \\, ", "941179e3f935502809f3d6e88b0e64bf": "\\Gamma(n+1) = n\\Gamma(n)", "9411a2b38dcaadc15e0aa58d423c5d38": "W_\\alpha W_\\beta", "9411a8f395b311d614de8bc0675f1f8f": "\\lambda = \\int_0^\\infty e^{-t - E_1(t)} dt ", "9411c33e55b89639b604842fa24a27ab": "\\Pr (X=a) = q", "9411fb617bf2ab293703124277040d64": " \\begin{align}\n(a,b) = \\mathopen{]}a,b\\mathclose{[} &= \\{x\\in\\R\\,|\\,a", "941446c6f861a48e71edbb3574aeb76f": "p_0\\,", "94145eeafee9b12772570edf2b034890": "\\pi_{\\neg x}", "9414d9893825f6b4df083eb376812dec": "\\textstyle (2 r)^3 - \\frac{8}{3} r^3 = \\frac{16}{3} r^3", "9414eb27b83bea9b92c60508d0674f74": "d=4,6,8,\\ldots", "94153d74dffaf6cba4aa4a10c515d172": "\\phi(\\theta)=\\theta^i\\eta_i(\\theta)-\\psi(\\theta)=\\theta^i\\sum p F_i-\\psi=E[\\log;p-C]=-H(p)-E[C]", "9415495122df7ea49ad34934affdfb8b": "\\{^{i}_{jk}\\}", "94158a29925df7fa78c1b75cf8abdbd2": "Q[\\varphi] = \\int_{x_1}^{x_2} \\left[ p(x) \\varphi'(x)^2 + q(x) \\varphi(x)^2 \\right] \\, dx, \\,", "9415b51d078917bfbfd3ef51b78b2dd4": "\\text{Precision}=\\frac{tp}{tp+fp} \\, ", "941670eb147d32bae860f6c25dd0e8c8": "\\phi(a, 0, n) = a", "9416c436faba3bedd82dd93a10b9b657": "1-2^{-40}=1-10^{-12}", "94172846c22525ecc8d8aff36404f5a7": "\\mathbf{E}^{(r)}", "941765003ffa55ffad6b0fdfcc934f5b": " \\, F_h = (4\\pi \\epsilon_0/e^3) h^2 = (0.6944617 \\; \\mathrm{V}\\; {\\mathrm{nm}}^{-1})(h/{\\rm{eV}})^2. \\qquad\\qquad (10) ", "9417da0e2ed57ef0251d051aaa134f5b": "\\frac{S_o}{N_o}", "9417ee10a6708ce32daf5f069526e409": " S^{\\prime} \\times_{S} \\textrm{Spec}(\\overline{K}) ", "94180efd4c7f53047249c3e05e15f391": "s_{\\bar{Y}}=s/\\sqrt{N}", "9418599d032af4816498ba9492b2ed18": "d\\mathbf x_2\\,\\!", "9418cc4dec774a46ee0ec23ff27de5e6": "\n \\langle q_{j+1} | \\exp\\left( {- {i \\over \\hbar } \\hat H \\delta t} \\right) |q_j\\rangle =\n\\left( {-i m \\over 2\\pi \\delta t \\hbar } \\right)^{1\\over 2} \n\\exp\\left[ {i\\over \\hbar} \\delta t \\left( {1\\over 2} m \\left( {q_{j+1}-q_j \\over \\delta t } \\right)^2 - \n V \\left( q_j \\right) \\right) \\right]\n ", "9418d4e7632481ba23b3a23b1c7444fb": "\\mathrm{sinc}", "9418e403c4596b7e1fe4f033af00156d": "I'", "9418eaf60ee6dfecc37bfbbd266e0562": "\\scriptstyle \\lfloor n \\rfloor", "941921988a468cbe60b150b2fa9731c1": "\\mathrm{inv}_{x,y} :", "94195cb1f65ca5cd9d4484d9d919d64c": "\\scriptstyle \\varphi^{0}_n", "9419e3fa4b82d7c27b779e2af3575017": "\\varphi\\!", "9419ef5ce983605285f5d5e92fb94847": "\\begin{align}\n\\mathbb{E}[x] &= \\frac{ \\partial A(\\eta_1,\\eta_2) }{ \\partial \\eta_2 } = \\frac{ \\partial }{ \\partial \\eta_2 } \\left(\\ln \\Gamma(\\eta_1+1)-(\\eta_1+1)\\ln(-\\eta_2)\\right) \\\\\n&= -(\\eta_1+1)\\frac{1}{-\\eta_2}(-1) = \\frac{\\eta_1+1}{-\\eta_2} \\\\\n&= \\frac{\\alpha}{\\beta},\n\\end{align}", "9419f4dba1f0de05286607f63ff6c09e": "P_s(E) \\approx K_s(A)Q\\sqrt(3\\gamma_c(A)SNR_{norm}),", "9419f8b6b83ed8ad078677b9fea38749": "\\tan A=\\frac{\\textrm{tanh(opposite)}}{\\textrm{sinh(adjacent)}}=\\frac{\\tanh a}{\\,\\sinh b\\,}.", "941a45347480df7ffd8f88fa465273c5": "\\frac{d(-y)}{dx} = -\\frac{dy}{dx}.", "941a5218fcceeca126c56e2e1d71b825": "\\Chi^2 = 2\\sum O_{ij} \\ln \\frac{O_{ij}}{E_{ij}},", "941b1c579f0df0ae0db6916369fe1350": " \\sigma_y^2 \\le 2y_\\text{max} (A - H), ", "941b38adcf343213ed619288f57fdb56": "\\int_{-\\infty}^\\infty dp\\,P(x,p)= |\\psi(x)|^2", "941b3c6825b083440aa4d38954977bd0": "\\nu_c \\propto t^{1/2}", "941ba633dadcc33924cd6701650d1da3": "v = \\frac{\\Gamma}{2\\pi r}", "941c2901e45cb5100efe609e36a68929": "(2,X) \\subseteq \\Bbb{Z}[X]", "941c2cefd98b0e3bc2b85099f8499b4b": "\n L_0 = -\\rho\\, \n \\left\\{ \n \\zeta\\, \\left[ \\frac{\\partial\\Phi}{\\partial t} \\right]_{z=0}\\, \n +\\, \\int_{-h}^0 \\frac12 \\left[ \n \\left( \\frac{\\partial\\Phi}{\\partial x} \\right)^2\n + \\left( \\frac{\\partial\\Phi}{\\partial y} \\right)^2\n + \\left( \\frac{\\partial\\Phi}{\\partial z} \\right)^2\n \\right]\\; \\text{d}z\\; \n +\\, \\frac{1}{2}\\, g\\, \\zeta^2\\,\n \\right\\}.\n", "941c7e9de6f637140ccc0d312d4d5b54": " P''(\\rho) + (c-1) P'(\\rho) + d P(\\rho) = 0 ", "941cddbfff3bac5c19776b3a1c0d1dcc": "\n\\gamma_e^2B_1^2T_{1e}T_{2e} \\geq {1}\n", "941cf781cc626b97f9d293e06eb7a930": "I \\propto \\frac{d}{\\sqrt{L}}", "941d17e6772811dac4b1081cf055c82f": "x_3 = F(x_1,x_2,x_3)", "941d8bf836782390d5f31fd393dd24f1": " \\mathcal{F} \\left\\{ \\exp (-x^2/2) H_n(x) \\right\\} = (-i)^n \\exp (-k^2/2) H_n(k). \\,\\!", "941d90c914a97ec034e6d7df2a5c45c3": "I_A << I_B = I_C ", "941e08448f6533a0c8bbaf3ee7dbc895": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\left( \\frac{\\tan(\\theta+\\delta)-\\tan\\theta}{\\delta} \\right) .\n", "941e3c1a6e52e476ac963b17b537db15": "g(x, y) = x^2 + y^2 - 1. \\, ", "941e3eccb1bffc4e15f436042dbb3814": "{\\hat{q}_{{\\rm w}}}({r_{\\rm w}}),", "941e41126aa2b1440adc357cf2ed87b7": "\nf=\\frac{1}{T}. \\,\n", "941e71bea6672e7c5ea8b671b14c8e51": "p(x) = x^2 + x + 1", "941e7bd6671556a2723e38ed80b46511": "\\frac{\\ln\\, \\mathcal{L} (\\alpha, \\beta|X)}{N} = (\\alpha - 1)(\\psi(\\hat{\\alpha}) - \\psi(\\hat{\\alpha} + \\hat{\\beta}))+(\\beta- 1)(\\psi(\\hat{\\beta}) - \\psi(\\hat{\\alpha} + \\hat{\\beta}))- \\ln \\Beta(\\alpha,\\beta)", "941eab0a7b2fdfe346ad1dbc68e32fd6": "\n \\begin{matrix}\n 4\\uparrow 3= 4^3 = & \\underbrace{4\\times 4\\times 4} & = & 64\\\\\n & 3\\mbox{ multiplied copies of }4\n \\end{matrix} \n ", "941ec42923a058a42a287a3da592c260": "d_0\\not=0,", "941ec49b9d4dea346cf0b1ab16a7cf29": "(0,+\\infty)", "941ec8e287c10f4057b19b5ffb50d6f0": "\\mathrm{Pic}(X)\\ ", "941f22896695153c18a28ab6665cd2a6": "\\left(1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ -2\\sqrt{1/3},\\ 0\\right)", "941f4fdd886e327320367b1ffb3b3a79": "\\Gamma(x)=\\int_0^\\infty t^{x-1} e^{-t}\\,dt", "941f79e3c1e4f698628eae5fc745e233": "\\sigma_1 - \\sigma'_n l_0/2", "941f894a5c3b45de667c0939f63a004c": "\\surd, \\sqrt{2}, \\sqrt[n]{}, \\sqrt[3]{x^3+y^3 \\over 2} \\!", "941f950751d12eb9c27eb83737fcf72d": " z=0 \\,\\!", "941fca58a0f8b93bcfd476495624a49c": "N(p) < p", "94202bdfef9be196eb49cf9814249c23": "X=\\left(\\begin{matrix}x_1&x_2-ix_3\\\\x_2+ix_3&-x_1\\end{matrix}\\right)", "9421009ca716f8cfefabda46102767cd": " f_1 \\in O(g_1) \\text{ and }\n f_2\\in O(g_2)\\, \\Rightarrow f_1 + f_2\\in O(|g_1| + |g_2|)\\,", "9421415113490868dc8d06ab9e16a34e": "D_qx^*(q)=-[D_xf(x^*(q);q)]^{-1}D_qf(x^*(q);q).", "9421a5d2935494270c98ee0771b26157": "x=y^2", "9421ad0dfe6e4002506d47ebfaa5d81b": "x = \\frac{1}{2}\\left[\\lambda \\cos \\varphi_1 + \\frac{2 \\cos \\varphi\\sin \\frac{\\lambda}{2}}{\\mathrm{sinc}\\,\\alpha}\\right]", "9421b65ad0993df3aa5d54c2673771d6": " \\mathbf{B} = \\nabla \\times \\mathbf{A} \n ", "9421b7bf96cc07b07ebd5f299f07ca73": "RR=\\frac{D_{E}N_{NE}}{D_{NE}N_{E}}=\\frac{D_{E}/D_{NE}}{N_{E}/N_{NE}}.", "9421ec9ed5416b7f53a191a52047a161": "\n P(\\theta|x_1,x_2,\\ldots,x_n) = \\frac{f(x_1,x_2,\\ldots,x_n|\\theta)P(\\theta)}{P(x_1,x_2,\\ldots,x_n)}\n ", "94225726be8bb0f1a83b34dc89e2e50b": "\\beta_{\\sqrt{s}} \\alpha_{\\sqrt{s}}(x)", "9422839597d86d8e7312da9321dd43b0": "n_{\\rm film} > 1", "9422c11b339b86843835a61b1d025499": "\n \\begin{align}\n f := & \\sqrt{F(\\sigma_{22}-\\sigma_{33})^2+G(\\sigma_{33}-\\sigma_{11})^2+H(\\sigma_{11}-\\sigma_{22})^2 \n + 2L\\sigma_{23}^2+2M\\sigma_{31}^2+2N\\sigma_{12}^2}\\\\\n & + I\\sigma_{11}+J\\sigma_{22}+K\\sigma_{33} - 1 \\le 0\n \\end{align}\n ", "942386a0672deefb9cb0e1c56032fa19": "A(X,Y) = \\tfrac12\\left(T(X,Y) - T'(X,Y)\\right)", "9423e3599ce8e1c4426c7765c670b540": " x^2 \\frac{\\log^2 T}{T} + \\log x \\ . ", "9423f7203173a0f01e7a4f529f9d7c9c": "\\Theta = \\frac{1}{2} \\arctan \\left( \\frac{2\\mu'_{11}}{\\mu'_{20} - \\mu'_{02}} \\right)", "942465c737d89f6e28e1601aa9291f9d": "\\frac{1}{|\\mathbf{R}|^3}", "942483e7fbb85507bece12afec67b1b5": "\\mathbf{v}(\\boldsymbol{x},t)=\\big(\\,v_1(\\boldsymbol{x},t),\\,v_2(\\boldsymbol{x},t),\\,v_3(\\boldsymbol{x},t)\\,\\big)\\,,\\qquad \\mathbf{f}(\\boldsymbol{x},t)=\\big(\\,f_1(\\boldsymbol{x},t),\\,f_2(\\boldsymbol{x},t),\\,f_3(\\boldsymbol{x},t)\\,\\big)", "94252cd87fd5156344078b38709ed6da": "\\rho_{sample}", "9425d4aadaedc39f67dc22a6ff06fe1f": " \\rho^* = 1 \\,", "9425f0bba8270070104169fe9406166a": "\\frac{d\\mathbf{M}}{d t}= -\\gamma \\mathbf{M} \\times \\mathbf{H_\\mathrm{eff}} + \\lambda \\mathbf{M} \\times \\left(\\mathbf{M} \\times \\mathbf{H_{\\mathrm{eff}}}\\right)", "9426074e3b3af1667bdb7ed48d3e0307": "\\nabla \\cdot \\mathbf A = \\frac{1}{\\sqrt{g}} \\sum_{i, j} \\frac{\\partial}{\\partial q^i}\\left(\\sqrt{g} a^{ij} \\mathbf e_j\\right) = \n\\sum_{i, j} \\mathbf e_j \\frac{\\partial a^{ij}}{\\partial q^i}.", "94260e9d30e4566b68958d781fae1bad": "\\langle\\overline\\psi\\psi\\rangle", "94262b65d65ee7f7f1963bc9cf340c8f": "(\\mathbf{a}\\ \\mathbf{b}\\ \\mathbf{c})\n=\\mathbf{a}\\cdot(\\mathbf{b}\\times\\mathbf{c}).", "94262d8e0a2c1ea824b382e5f336bfe0": "\\mathcal{M} \\vDash \\phi", "942659582e02c9f5c59e3f63bec5f1e3": "f^{-1} = \\sum_{n \\ge 0} a^{-n-1} (a-f)^n.", "9426a078c2f8bb1c704e465284af396d": "r = f \\tan(\\theta)", "9426af537c56a630f60c50581cb7b22f": "dS_h", "9426b2cec5b8e829af9613ea064c7d79": "I\\!I(X_i,X_j)", "9426fc73f30d7e7aaa197224d6a75961": "I=\\frac{\\mathcal{E}}{R}e^{-t/\\tau_L}=I_0e^{-Rt/L}\\,\\!", "94272e67ceba633643363a68efc4ac25": "n=2^{k-1}QR", "94276e89cd7449274158b1f1e29b7d50": "G(z) = \\left[(1-p) + pz\\right]^n. \\, ", "94277972812888b6a44e243e89e4b705": " P_n(0) = P_n(1) = B_n", "9427c6a7da67d936c95de7f95afe887f": "a_P\\sin i=", "9427f9b9fd91fffae693ccdbbc85f6b5": "\\langle \\sigma^n\\rangle=(1-k'^2)^\\beta,\\quad \\beta=n(N-n)/2N^2.", "942826e0253a6bdb548a7c468fb023b0": "(\\tfrac{g}{f})", "9428aa9e37e3194be6709646b05adb46": "\\gamma_1 = \\varphi_1 \\gamma_0 + \\varphi_2 \\gamma_{-1}", "9428c77a5fbe019b00f84d44d9037d0f": "F \\colon C\\to D", "9428f78fdae7ba1ff47578cbe786a86b": "\\theta^\\mu = 0,\\quad \\mu=p+1,\\dots,n", "942a30ee78c470930b8d599f1da94a21": "\\tilde{h_2} \\leftarrow h_2^r rem P", "942a6d38913a15b2a49d6fd9a0fbb0da": "a \\le b < c", "942a6f06e75a143b3bfe203f3ce655a7": "\n \\rm{FWHM}\\sim\\sqrt{t}.\n", "942c3d2222b29c1761e169de9ac4c039": " \\tan \\theta\n= \\frac{\\mathrm{opposite}}{\\mathrm{adjacent}}\n= \\frac { \\left( \\frac{\\mathrm{opposite}}{\\mathrm{hypotenuse}} \\right) } { \\left( \\frac{\\mathrm{adjacent}}{\\mathrm{hypotenuse}}\\right) }\n= \\frac {\\sin \\theta} {\\cos \\theta}. ", "942c41239df2cbb834174b707ca16270": "y_{n + 1} = B", "942c551997d176f29ef3519ca4174bcd": "-2\\omega^2 \\mathbf{r_B}(t) \\ . ", "942c862c70f6a4a4b84f8ee8bcf66594": " H(X) = \\mathbb{E}_{X} [I(x)] = -\\sum_{x \\in \\mathbb{X}} p(x) \\log p(x).", "942cdea6553df9b2fa8fe4e41e8f441d": " \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = D\\mathbf{f}(\\mathbf{v})[\\mathbf{u}] = \\left[\\frac{d }{d \\alpha}~\\mathbf{f}(\\mathbf{v} + \\alpha~\\mathbf{u} ) \\right]_{\\alpha = 0}", "942ce6d770920288647d2d0302c966ef": "\\Pi^0_n \\subsetneq \\Pi^0_{n+1}", "942d0f064a194451feb612a4398a2bb7": "\\{a,b\\} = \\{b,a\\}", "942d3ec0e8dd7cbd880cafd5365e7d03": "\\operatorname{cl}(r)=c", "942d451955de6ba9cf08007295c2fbb0": "(M, [g])", "942d6f3d58bb2d7b31343dfbf6d5f412": "\\operatorname{Tr}_{\\mathcal X}(Z)", "942d77c28de385b1c5988bfb89f21edf": " f \\in \\mathbb{Z}[x]", "942d97690c4258dc381b441fd1a7017e": "U_2 \\sim \\Gamma(\\delta_2, 1)", "942d9ffc7ed8d6d4265824027594ebbc": " 2635 + \\frac{ 10} { 2 } [ 10.5 - 4.5 - \\frac{ 1 }{ 2 } \\frac{ 1620 }{ 200 } ] = 2644.75 ", "942df64e6c8297bfb4da1bf284c40a08": "L \\le \\lambda_{MFP}", "942e48193c42f7ce01558826097cbdfa": "\\{\\hat{\\mathbf S}^2, \\hat{\\mathbf{S}}_z, \\hat{\\mathbf{L}}_z, \\hat{\\mathbf{J}}_z=\\hat{\\mathbf{S}}_z + \\hat{\\mathbf{L}}_z\\}", "942e948d6cca7b236ab19a7fc55b3697": "lengthSupp(\\psi )=T(\\alpha ,\\beta )=(\\alpha +\\beta )\\sqrt{\\frac{\\alpha +\\beta +1}{\\alpha \\beta }}.", "942ea734f13a29ca909ba6d2be9ca73c": "\n\\sum_{\\delta\\mid n}\\varphi(\\delta)=\nn.\n", "942eb7982d71f4519cf684ec94f87f6d": "w_{GDj}=\\sqrt{\\frac{2\\epsilon_{Si}V_{GD}}{qN}}", "942f1bd555161af152af0f5f3aaa0d68": "\\left( {\\frac {\\partial ^{2}}{\\partial {x}^{2}}} + {\\frac {\\partial ^{2}}{\\partial {y}^{2}}} \\right)(A\\left( x,y,z \\right) e^{ikz}) + \\left( {\\frac {\\partial ^{2}}{\\partial {z}^{2}}}A \\left( x,y,z \\right) \\right) {e^{ikz}}+2\\, \\left( {\\frac {\\partial }{\\partial z}}A \\left( x,y,z \\right) \\right) ik{e^{ikz}}=0.", "942f276207f43e3abe2b6b76479cdd88": " \\frac{d}{dt}\\frac{\\partial T}{\\partial \\dot{x}} - \\frac{\\partial T}{\\partial x} = F_{x} + \\lambda \\frac{\\partial f}{\\partial x},\\quad \\frac{d}{dt}\\frac{\\partial T}{\\partial \\dot{y}} - \\frac{\\partial T}{\\partial y} = F_{y} + \\lambda \\frac{\\partial f}{\\partial y}. ", "942f89dbb988877a655ec2e4b61b5309": "\\{q_i,p_j\\} = \\delta_{ij}", "942ff0b078125e091f6dda746f9ca3a8": " \\nabla\\times(\\nabla\\psi)= \\mathbf{0} ", "943001b25aa86e83451ef89a162be99c": "n/2 - 1", "94301d720522c32b8d4e22d9c43ee9ce": "I_j", "94301e97c284f424d1f0e4ae46ec35ef": "\\int_{-\\infty}^\\infty f(x) \\, dv(x) = \\mathrm{E}[f(X)].", "943074307258c5e26fc3cd3544690ff6": "{v_p^2} = {h^2 \\over{{r_p}}^2} = {h^2 \\over{{a^2(1-e)}}^2} = {\\mu a (1-e^2) \\over{{a^2(1-e)}}^2} = {\\mu (1-e^2) \\over{{a(1-e)}}^2}\\,\\!", "94307d2dcd6a32bd1a596e061abf11e8": "\\mathrm{\\nearrow}", "9430a9c30d93a5c3c1f1138dfd5db1cc": " \\mathbf{k}=k \\hat{e}_z ", "9430ca7e163a554a30f5b58786883a89": "(1+n)a=a+na", "9430f7f732cbaa1d36fb7b652393094d": " \\tbinom n k", "9431568a7c4a306dee658ff0912989d2": "\\mathbf{f}_r(t)\\in \\mathbb{R}^3", "9431eedd871c3f23b4bc1f58e251a3ed": "\nA = \\left( 1 - {\\theta_E^4 \\over \\theta^4} \\right)^{-1}.\n", "943205be7380d8c9a549920d9a963c96": " \\bold x^{(m+1)} = \\bold D^{-1}(\\bold U + \\bold L)\\bold x^{(m)} + \\bold D^{-1}\\bold k. \\quad (7) ", "94323070c010aed839a6a1d567c63a9f": "\\scriptstyle r\\searrow 1 ", "94326bc7ca8d626d2b545699c17aaa28": "j = \\left\\lfloor {k \\cdot r} \\right\\rfloor = \\left\\lfloor {m \\cdot s} \\right\\rfloor \\,.", "9432957b00f4b3020a547b2905c9be6c": "\\left|f(z)\\right|=\\left|f(x+iy)\\right|=1\\;.", "94331cf547998593083b93448a0d978e": "S = \\{z_1 = (x_1,\\ y_1)\\ ,..,\\ z_m = (x_m,\\ y_m)\\}", "94332119e43a2c49e529ab4eb20151a9": "y = \\sqrt{\\rho^2-(\\rho\\cos\\alpha-x)^2}+\\rho\\sin\\alpha", "94335cc7aa529867a32c58dfa57d1603": "p(X) = \\prod_{i=1}^{\\deg(p)} (X - a_i) \\in L[X]", "943374ca3df74240ef49b278a5f1ce9c": "x^a(p).", "9433fedf27c52365d57188c6ef112528": "(1+4)^2 = 5^2 = 25", "943403c53f5a2a7e15cca550f750906b": "10Mcalories(city.veg.intake)/200kcalories(calories/truckload)=50truckloads", "9434116fb6362a4f77ae62af74075231": "GW_{g, n}^{X, A}(\\beta, \\alpha_1, \\ldots, \\alpha_n) := GW_{g, n}^{X, A} \\cdot \\beta \\cdot \\alpha_1 \\cdot \\cdots \\cdot \\alpha_n \\in H_0(Y, \\mathbf{Q}),", "943416aef080f184ae7b3bf6e7576794": "D_{12} \\Delta n_1(x) < 0", "943458d6eb7f5cff1cd198efd74d7699": "q(x)=(x-x_1^2)(x-x_2^2)\\cdots(x-x_n^2). \\, ", "943513bf4ad35bf9b1a1cdc6bb8b1fe9": "\\Bbb Z[t,t^{-1}]", "9435686b1b34833ff47f0b6924ac6e7c": "C_{pq}^+", "9435be21c3d23be85e2a42a297a16788": " \n(1-\\epsilon)\\int_{A_k} \\varphi \\, d\\mu \\geq (1-\\epsilon)\\int_E \\varphi \\, d\\mu - \\int_{A-A_k} \\varphi \\, d\\mu.\n", "9435c0f36149f018799584678b4fb2cd": "\\mathrm{d}\\bold{F}=0", "943613098e71e8ac2bd41ccdc1d1b0cb": "\\cos c", "94362d50882161e3807d65d9e46d342e": " E^2 = \\left ( pc \\right )^2 + \\left ( mc^2 \\right )^2 \\,\\!", "94363af7b773246b949aac91cb3cb2e2": " \\nu=\\alpha c_{\\rm s}H", "94368bbf0560c05f92f02e07b94e537b": "\\omega\\not=1", "9436e21cdc0400e7d4e3f58fe257c37f": "C_0(\\rho,q) = \\sum_{i=1}^{N_{sd}} m_i", "9436f7595994ef81ee283d80672a5530": "\\left\\lbrace c_i \\right\\rbrace", "943767da9b6c23b1b3c3fc7a6065db6b": "W_D", "9437683cec7900e8b2954723150c6fb8": "82.4\\pm 2.3%", "943794d2bdeaca41375169d834aa5b49": "\\psi\\in L^1(R)", "9437b86e4087c1e5b4dd435be3ae1042": "y_i \\succ_x y_j \\Rightarrow f(x,y_i) > f(x,y_j)\\,\\!", "9437c705d3c7073920356418a5cd1b10": " w^k - \\gamma \\nabla F\\left(w^k\\right)", "9437cf70ed79e99789a90d3ddacacea3": "f/2=0.039 Re^{-0.26} \\, ", "9437d4f6aeb9c42631c7a8cfd561a8c6": "\\|u\\|_{BMO}P[B \\ge x]", "943f8a71562cac406395d8683574f9d9": "\\mathcal{L}_{WWVV}", "943fca633bdedb8cb580b00312b2ce75": " \\nabla", "943ff9b5c965512057b788fec411001b": "\\left [\\hat{a}_i, \\hat{a}_j \\right] = 0 ", "94402115d506e894439f264a33d8bbe4": "\\mathit{CT}_f(a)", "944025c9d9bd70afa48a7084fd03d50d": "\n\\begin{array}{c|c|c}\n\\text{Info-gap Format}& \\text{MP Minimin Format}&\\text{Classical Minimin Format}\\\\\n\\hline \\\\[-0.18in]\n\\displaystyle \\min\\{\\alpha: r_{w} \\le \\max_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} R(q,u)\\} & \\displaystyle \\min\\{\\alpha: \\alpha \\ge \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})}\\ \\psi(q,\\alpha,u)\\} & \\displaystyle \\min_{\\alpha\\ge 0} \\ \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} \\psi(q,\\alpha,u)\n\\end{array} \n", "94403ef435bf67bd678fbdfcf119669e": "x \\equiv a_0 \\pmod{m_0}", "944053da88ae7bd3e7b103825d7567a9": "\\frac{4}{3}\\pi r_{s}^{3} = \\frac{1}{\\rho}\\ .", "94408da96cc496474b9ce4acc3723f3f": "\\color{red}\\lor", "9440bb2a043d1593dfcea15b1c4661fe": "\\Lambda = (-1 \\leftarrow 0 \\rightarrow +1),", "9440f3a5477a7ac0f875c3f4e29553ad": "\\psi(\\psi_1(\\Omega_2)) = \\psi(\\zeta_{\\Omega+1})", "9441b00977c2defafed20aaf3d1596bf": " \\langle f(x)-f(y),x-y\\rangle\\geq \\delta |f(x)-f(y)|\\cdot|x-y|.", "9441e4856eddcbfa108df20b6b7b00a0": "X \\subseteq \\bigcup_{\\alpha \\in A}U_{\\alpha}", "9441eef205cf9d743759e1ff5c828c14": "\\frac{X - \\mu}{\\sigma}", "94420125b112bf598efaa7ce9a4fa215": " K_t^{\\rm Schr} = K_{it+\\epsilon} = e^{-(it+\\epsilon)H} \\, ,", "944213075963cacadc873b3a9b95c2f3": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta = (0 \\times \\cos\\theta) - (1 \\times \\sin\\theta) = -\\sin\\theta \\, . ", "944236534df6cd9d1e08457971459703": "U = S + E_s ", "94423d05f694d95cb2352ce436b0794b": " \\begin{array}{ll}\n\\Delta x' = x'_2-x'_1 \\ , & \\Delta x = x_2-x_1 \\ , \\\\\n\\Delta t' = t'_2-t'_1 \\ , & \\Delta t = t_2-t_1 \\ , \\\\\n\\end{array}", "944256845d4e9bf9d660a1031adb2c7e": "\n\\begin{align}\n u_t + u u_x + p_x &= 0, \\\\\n p - u_{xx} &= 2 \\kappa u + u^2 + \\frac{1}{2} \\left( u_x \\right)^2, \n\\end{align}\n", "9442632ee9923a775aecc3fbc764b044": "\\textstyle (x_k)", "9442d985624a4b646291c0e37336e5b8": "\\{Y_i\\} | (X = k) \\sim \\mathrm{Multinom}\\left(k, p_i\\right)", "94430902e632251daf804d964526d21c": "S_X-S_Y", "9443363c31acf1f00b71e3d0c0355fd8": "X_1^n(1) \\, ", "944371e64adeed6aa022e7781f301313": "\\Sigma^0_{k+1}", "9443937e13512907287bd89c1f04643b": "\n\\dot{Q}_{m} = \n\\frac{\\partial Q_{m}}{\\partial \\mathbf{q}} \\cdot \\dot{\\mathbf{q}} + \\frac{\\partial Q_{m}}{\\partial \\mathbf{p}} \\cdot \\dot{\\mathbf{p}} = \n\\frac{\\partial Q_{m}}{\\partial \\mathbf{q}} \\cdot \\frac{\\partial H}{\\partial \\mathbf{p}} - \\frac{\\partial Q_{m}}{\\partial \\mathbf{p}} \\cdot \\frac{\\partial H}{\\partial \\mathbf{q}} =\n\\lbrace Q_m , H \\rbrace\n", "9443a9722c9b8758c570bc8773c7d871": "\\mathbb{E}_E[\\mathbb{E}_m [\\Pr_{e \\in BSC_p}[D(E(m) + e)] \\neq m]] \\leq 2^{-\\delta n}", "9443d58125eb18b246d12840a3048f0b": "r_1 = \\frac{T_1}{r_1}", "94441874e761cf83a4d1c2f6198c0454": "U = \\sum_{i=1}^N p_i \\,E_i\\ .", "94442acefd287314885e0fa998fc0127": "\\widehat P\\to \\widehat M", "94447a8e352e4625307661389229e5ac": " A^n = \\oplus_1^n A", "9444b57a9c53409ada76c49f36d58ee0": " p = m v", "9444ea8bb08a2f8e606c0f0f410c51a3": " c_{12},c_{14},c_{24}", "9444f2810212715036faee0ed3704c93": "\\frac{1}{1-\\mu}", "9444f76f996a64fb20b068c371412629": "valprefs", "9445570e810b226c13d10a3b9a944e6d": "\n \\quad (10) \\qquad \\frac{\\alpha \\Delta t}{\\Delta x^2} \\leq \\frac{1}{2} \n", "944588ca136692fa705b1194adf62868": "I_n = \\int \\frac{dx}{\\sin^n{ax}}\\,\\!", "9445c6149288347269614168fb10c7b4": "\n\\begin{array}{lcl}\n \\mathbf{b}_x &= & \\frac{\\mathbf{e}_z}{\\mathbf{d}_z} \\mathbf{d}_x - \\mathbf{e}_x \\\\\n \\mathbf{b}_y &= & \\frac{\\mathbf{e}_z}{\\mathbf{d}_z} \\mathbf{d}_y - \\mathbf{e}_y\\\\\n\\end{array}.\n", "9445cfe1051718bbd5488f69b7fcb1d8": " \\ CVI = A\\frac{2{\\sigma}a}{3\\eta}\\frac{\\phi}{1-\\phi}\\frac{\\rho_p-\\rho_s}{\\rho_s}", "9445f86fdcf0b67e2ab6e7707b5eeb62": " E_D. ", "94462be9fa2d05934a1f15710ba20f44": "\\textstyle(x, y, z\\pm1)", "9446ddd507286294d9d0d7caede46df3": "\\emptyset^{(m)}", "9446ef943b0a8291ddfd76b61857e558": "C=\\left\\{\\begin{array}{ c l } C_0 & \\textrm{if}\\ |\\xi|<1 \\\\ C_0\\xi^2 & \\textrm{if}\\ |\\xi|>1.\\end{array}\\right.", "9447c862ab7fd5a490cd5ee5b48f329d": "\\iota : E\\to C(E)", "9447d6379060d97e85d6aa5f4ba18e37": "g_i\\,=\\,h_i", "9447dab70f6fc9f9f3ecef2fd42931ce": "O(V^\\omega)", "9447ea32bc9d4cef6d75dac7fdc2a968": "(1 - q)^k", "9448077e3f6975ffb27ed175030bfcd2": " x^a", "9448112d293a4216080fbb90293bc34e": "L^{4k},", "94486cc6c131e85fd1d11fa00ab997e6": "n 2^{n - 1}", "94490ceba468e47de0b6497d3e9b251a": " x \\preceq y \\preceq z ", "94491bb0355c19a8cc13a0918e2ace60": "E(G) = \\sum_{i=1}^n|\\lambda_i|.", "94492a1a4edf984c46efe57eacaeff27": " Q_{sensible} = m c \\Delta T \\, .", "9449514f37bda4edd6636d31591e799c": "\\langle \\Phi_1 , \\Phi_2 \\rangle = \\int\\limits_{\\mathrm{ all \\, space}} d^3\\mathbf{p} \\, \\Phi_1^*(\\mathbf{p},t)\\Phi_2(\\mathbf{p},t) \\,,", "944968cbce3e99901ce901dce4c4d3c8": "\\displaystyle8.42", "94497734f8bcbfd6a739aa7edbaaab04": " E \\subseteq Q \\times (\\Sigma\\cup\\{\\epsilon\\}) \\times (\\Gamma\\cup\\{\\epsilon\\}) \\times Q \\times K", "9449911f22b76d98763bf17b716c9c67": " \\Phi(x) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^x e^{-\\frac{1}{2} z^2} dz. ", "944a41ec78354872f5f6e0a5b1312f5d": " f(x)g(y) = (1/2\\pi)e^{-(x^2 + y^2)/2}\\,", "944ad29f8f181f5510b5e4bbdb14d589": "= (P^+ 2 \\partial_{[\\alpha} \\omega_{\\beta]})^{IJ} + (P^+ [\\omega_\\alpha , \\omega_\\beta])^{IJ}", "944b42a28af6f2c9f035b32d1a87ddd6": " x\\mapsto g\\otimes x", "944b5c67bfc92d61d7a53736c84efecb": " \\langle G, * \\rangle ", "944b691bdf93e7deb9a14e7a0ac149f7": "\\mathcal {L}_X : B \\to B", "944b781499dc80ac1f71260c7bbb6be4": "COP_{heating}", "944b86bf5971a7aa8c3388156b6ef674": "r=\\frac{a_2}{\\cos (\\theta+\\alpha)} - \\frac{a_1}{\\cos (\\theta-\\alpha)}", "944ba7981ec15b0a95885ffd422c0ed4": "\\det\\begin{pmatrix}A& B\\\\ C& D\\end{pmatrix} = \\det(AD - BC).", "944bd77771d4cb05cdf570fa89a7bc5e": "r>R_{LR}", "944bfc21db79c05d1b59a2a5491cff9c": "\\mathrm{rad}", "944c12a7cc381629b39d640170e65bce": "i\\hbar\\frac{\\partial \\Psi}{\\partial t} = \\hat{H}\\Psi\\,,", "944c56110e1a3f3039a62785106d6e37": "v=wa", "944c6b4af2ce2c05b6a23f85d93ac050": "|{\\Phi}\\rangle =\\frac{1}{\\sqrt{3}}|{00}\\rangle + \\frac{1}{\\sqrt{6}}|{01}\\rangle- \\frac{i}{\\sqrt{3}}|{10}\\rangle - \\frac{i}{\\sqrt{6}}|{11}\\rangle", "944c7ad6f25ec165c4676e0b3c8d8a8f": "\n\\left[ Q_R(\\mathbf{p}^{\\prime}),\nQ_R(\\mathbf{p}) \\right] = 0, \\;\n\\left[ Q_L(\\mathbf{p}^{\\prime}),\nQ_L(\\mathbf{p}) \\right] = 0, ", "944ce5cd2086dfd4120e76694410bb65": " \\mathcal{M} ", "944ceef5236a3ab9626588121851ff56": "\\Omega_{({\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)} = \\Omega \\cap \\{\\boldsymbol{x}\\in\\mathbb{R}^n|\\langle\\boldsymbol{x}-\\boldsymbol{x}_0,{\\boldsymbol{\\hat{a}}}\\rangle>0\\} \\qquad\n\\Omega_{(-{\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)} = \\Omega \\cap \\{\\boldsymbol{x}\\in\\mathbb{R}^n|\\langle\\boldsymbol{x}-\\boldsymbol{x}_0,-{\\boldsymbol{\\hat{a}}}\\rangle>0\\} ", "944d29d214f7fdcdec1b736332d6ca06": "\\omega = \\sqrt{\\cfrac{\\gamma p {A}^2}{\\mathrm{m}V}} ", "944d5258c0cfa612183b3e76eebd2f13": "\\,\\phi(v_i)=\\left(1+\\sum_j v_{i,j} x_j \\right)^p", "944e04570d2efe31cd26e3a990ae2647": "\\chi (s_1,\\,\\,s_2)=\\frac{1}{\\sqrt{2}}\\,\\,\\left (\\alpha (s_1)\\beta (s_2)-\\beta (s_1)\\alpha (s_2)\\right )", "944e3625f05bee65d2633115b1461b91": "\\pm\\frac{1}{\\sqrt{1 + \\cot^2 \\theta}}\\! ", "944e4576d2dda5c5e33fef3897332cc8": "e^{ik_0t}\\,", "944e5c302ee4f79cb2da1238aaf2b311": "\\mathcal{R}_4", "944e6f204ae8b4999ec296da3cd2c905": "\\alpha \\in (0,2]", "944ed947b436e7de4a6cdec018fcd4c0": "u_n = u_{n-1} + u_{n-2},\\text{ for }n > 1, \\,", "944f0a6c03e138a793c9b5bba7825edb": "\\textstyle \\mathrm{GL}_n(\\mathbb{Q})", "944f0aa244ec088a86aede7e4acdfe74": " \\mathfrak{tri}(A)\\oplus\\mathfrak{tri}(B)\\oplus (A_1\\otimes B_1)\\oplus (A_2\\otimes B_2)\\oplus (A_3\\otimes B_3). ", "944f32303bc709ea7a81485408c8d93b": "z_0=f(r')", "944f7e1ec76ea0b33d3976f97f73946f": "x^x = \\exp(x \\log x) = \\sum_{n=0}^\\infty \\frac{x^n(\\log x)^n}{n!}.", "944f94f8a3bebf82dd19de58d72d5c89": "V = 2\\pi \\int_a^b x\\vert f(x) - g(x)\\vert\\,dx", "944fe2b28f7d8f35908ca761abc2f975": " \\rho_a ", "944ff2ce44532366912b44796a0d885f": "x_1(x,t)", "944ff7fa02e2d365d535f7080eba8a1c": " H_{\\rm{int}}", "945008ff4b8bb472b1dba712430baf5d": "K(z,w) = A(w)\\Psi(zg(w)) = \\sum_{n=0}^\\infty p_n(z) w^n\n", "94501cdaeea3d91c8cfab87fb9918ef3": "\\mathbf{s} + \\mathbf{t} \\in \\mathcal{S}", "945030c165ce38ac6e440a1a004387b8": "p = {1 \\over 1 + e^{-2\\beta JH} }", "9450704a269c5da1618b0497835153a7": "A(\\tau)", "9450a74369df0462dc1a58db0744b520": "e^{\\pi \\sqrt{163}} = 262{,}537{,}412{,}640{,}768{,}743.99999999999925... ", "9450bb89a46ce06a0e6dff6c38b01857": "y_{1}=r_{A}-r_{B}+\\varepsilon _1", "945134693c0f986de139e364769380f2": " \\frac{d}{dt } \\int_S F \\, dS = \\int_S \\frac{\\delta F}{\\delta t} \\, dS - \\int_S CB^\\alpha_\\alpha \\, dS", "94515e724b8aa631332e32560c427511": " [n + l + v - y]^2 - ", "94516f1a6fad08163cd810f2f0acb5c5": "\\theta_1+(\\theta_2+\\theta_4)=90^\\circ ", "94518c818d3ec07de1e78a60072551d2": "{n\\choose p}", "94518e2acab4e4b1dd1ab8a1f5d5cebd": "\\text{Ohms reactance} =\\frac{%\\text{ reactance}*kv_{L-L}^2 * 10}{\\text{kva base}}", "9451978e08006d57cabd253527423b37": "\\bar{C}(z)", "9451f93bd9b3b324bd20aae1c7df2b51": "\\left[\\widehat{U}(t_1), \\widehat{U}(t_2) \\right]\\psi(\\mathbf{r},t) = 0 ", "9451fb32831cd36e8f0b5fc11ef9d54c": "\\mathbf{S}_\\mathrm{AB}=\\int \\Psi_\\mathrm{A}^* \\Psi_\\mathrm{B} \\, dV,", "9451ff048dfccbbeec3555ed1b5424f7": "\\begin{pmatrix}e^{i\\,\\xi_1}\\sin\\eta & e^{i\\,\\xi_2}\\cos\\eta \\\\ -e^{-i\\,\\xi_2}\\cos\\eta & e^{-i\\,\\xi_1}\\sin\\eta\\end{pmatrix}.", "9452266b6123daea2a692806785bd6cd": "Y\\left(\\omega\\right) = iB(\\omega)", "94522bb08bc986b64a3156d7566d9afc": "dy\\,\\!", "94527c9a6ffc6abab12c06d0b31f169b": "x\\ne \\pi", "9452abb3ff33a76272fe3cdd3c9a1aa9": "c_1\\equiv a_1+b_1- a_0^{p-1}b_0-\\frac{p-1}{2}a_0^{p-2}b_0^2-...- a_0 b_0^{p-1}\\mod p", "9452d5ccd003410210b83d94aad3efe2": "\\vec{\\nabla}\\times\\vec{A}=\\vec{B}", "9452f772ce51c541870eee1749cb57ab": "K_i = \\sum_j P_{ij}K_j.", "945345c27cdc10094f6cdad4f3e55e7b": "\\mbox{cas}(t)=\\sqrt{2} \\sin (t+\\pi /4)", "9453465538ffe81b2ca956d9b1a2695d": "\\chi(D) = \\chi(\\mathcal{O}) + ((D.D)-(D.K))/2.", "94538612f609a5581d06a8f38afaf756": "\\varepsilon\\in\\mathbb{R}", "9453be740c782fc0311a752e3a04a488": "V \\cong {\\rm T}M", "9454423badd19cb146c6b7b66b842923": "\\{(\\tfrac{3}{2},2),(2,4)\\}", "9454579a19a8c1c89ada3fb8041f7409": "X_t=x_i", "9454589a9c9e9f1f929de96ab5cefdc2": "e_i = 0", "94545ed3aaca4773ff243d299fe203b6": "\n\\phi = \\frac{p + c}{p_c + c},\n", "945467e333176f63b1c130ed75f0f1a2": "\\operatorname{Log}(z_1) - \\operatorname{Log}(z_2) = \\operatorname{Log}(z_1 / z_2) \\pmod {2 \\pi i}", "9454c9023854e7493c0c9c560cfbeb1a": " TL_y = \\frac{\\sum_i (TL_i \\cdot Y_{iy})}{\\sum_i Y_{iy}} ", "94552d78e73fc44537898033c9545994": "\\varepsilon(u) = \\frac{\\nabla u + \\nabla u^{\\top}}{2}", "94558f10d20b10c1568689a78021d1eb": "x>px'", "9455e3cef06515ec65c399bff59a1595": "|(f^n)^\\prime|> 1.", "9456354dd1fc3f8ce810084645f113a9": "\\mathcal{F}_{\\tau^+}", "9456362cf2e41a3d2b311c5cf39c1065": "\\operatorname df_p:\\mathbb{R}^n \\rightarrow \\mathbb{R}^m", "945684d976fcfe3b34c853e54a85b87d": "\\mathfrak{e}_7", "9456fa62d0c5be9b37ec297e5450e6b9": "\\hat{\\alpha}\\neq\\hat{\\beta}", "945723fff04ad7f436f36a02d51f0908": "\\mu = \\ln (\\alpha)", "94575e832bb98b023ba7b692c054e942": "|\\phi(x_i)| = 1", "945760864f04ba9e74c1611ea4568c7c": "\n\\rho {\\partial^2 \\xi \\over \\partial t^2} = {\\partial \\over \\partial x} \\left ( P_1(\\chi){\\partial \\xi \\over \\partial x} \\right ) + \n{\\partial \\over \\partial x} \\left ( P_2(\\chi){\\partial^2 \\xi \\over \\partial x \\partial t} \\right ) - P_3(\\chi) - F_0 {\\partial \\chi \\over \\partial x} \n", "94580cc435bc2350ff03456f89355a3e": "\\left(\\left\\{a, b\\right\\}, \\left\\{\\left(a, 1\\right), \\left(b, 1\\right)\\right\\}\\right)", "94580e5e0d9a1bf915e65ee899bb5908": " |A_1-B_1| + |A_2-B_2| + \\cdots + |A_N-B_N| ", "94581544767a4ddeb180f2cb73eabf88": "\\varphi\\Vdash p_j\\land\\Box p_j", "94586b32519f2625a5f228a4f6fdd940": "\\rho_L\\,", "9458735c14a88cb6a817e34177eaeb8e": "\n R_b = 37.5 - 1.6 R_a + 0.04 M_c\n ", "9458873bd6ed8752a66770ecc05640d5": "x_0 = 6 \\cdot 10^2 = 600.000. \\, ", "94589e4e3841523271a5a7faae4e0012": "\\begin{pmatrix}w_1 \\\\ \\vdots \\\\ w_n \\end{pmatrix}=\n\\begin{pmatrix}c(x_1,x_1) & \\cdots & c(x_1,x_n) \\\\\n\\vdots & \\ddots & \\vdots \\\\\nc(x_n,x_1) & \\cdots & c(x_n,x_n) \n\\end{pmatrix}^{-1}\n\\begin{pmatrix}c(x_1,x_0) \\\\ \\vdots \\\\ c(x_n,x_0) \\end{pmatrix}\n", "9458db7f41b7346755d3d828ed69793d": "\\{ 0,2 \\}", "9458eb07818fbf2335f0fd80ed3e59b0": "n+3", "945910183e4e20f9ab7be06eeec6c6a9": "\\chi^2_{k-\\ell}", "945936f2842b1f5c800f7b6d61647df2": "\\beta_k \\leftarrow {\\hat{r}_{k+1} r_{k+1} \\over \\hat{r}_k r_k}\\,", "94598b84d4803bae7acfc1e8a85d7422": "\\alpha\\in\\mathcal{O}_m", "94598c73e0a7edd3f4ad668d67c9a2db": "\\operatorname{mod}\\sigma_y^2(n\\tau_0) = \\frac{1}{2n^4(M-3n+2)}\\sum_{j=0}^{M-3n+1} \\left\\{ \\sum_{i=j}^{j+n-1} \\left( \\sum_{k=i}^{i+n-1}\\bar{y}_{k+n}-\\bar{y}_k\\right) \\right\\}^2", "9459b62e771fece5a6537c61c15cdbc2": "|a_{1/2}|^2 + |a_{-1/2}|^2 \\, = 1.", "9459e226ba4b6967fb0d291e7a073470": "2^{2^{\\Omega(n)}}", "945a835cf9024b8123ec84502c6951c5": " \\mathbb{P}(S_{X_t+1} > x) \\geq \\mathbb{P}(S_1>x) = 1-F_S(x)", "945aa3a3e1685c8bb1887e37d15afefa": "e\\in A", "945aa80d30a2d3f2fdf3c23c81788222": "\\mathbb{Z}^+=\\{0,1,2,\\dots\\}", "945ac6faed6f0116e68c4004a03f1d93": "s_{X}=\\text{sgn}(X), r_{X}=\\text{sgn}(|X|-|X|^{-1}), x=\\psi (\\max (|X|,|X|^{-1}))=\\psi (|X|^{r_{X}})", "945acd14f61739696ce5a83d3772e51b": " \\mathrm{d}^n x \\equiv \\mathrm{d} V_n \\equiv \\mathrm{d} x_1 \\mathrm{d} x_2 \\cdots \\mathrm{d} x_n \\,\\!", "945aff587c65333c8984a38818314816": " T\\big|_{}=T_e\\big|_A \\, ", "945b00f095520b5160023fcbac1e567c": "S''(0) =\\ 0", "945b79c4cd24104cfab9b90092a6c234": "\\overrightarrow{l}\\cdot\\overrightarrow{j} = {1\\over 2} \\left(\\overrightarrow{j}\\cdot \\overrightarrow{j} + \\overrightarrow{l}\\cdot \\overrightarrow{l} - \\overrightarrow{s}\\cdot \\overrightarrow{s}\\right)", "945b8fe1dd020784f8541957e3fe9222": "~\\cos(\\cos(x))~", "945b911dbf5ca2a029fb59b7cad7aa0c": "\\bar U D^2", "945b9536a9521ee45b26cb44928d496d": "I(p) =(p, \\int_0^\\infty I(p,\\tilde{\\nu}) d\\tilde{\\nu}) =(p, \\int_0^\\infty I(\\tilde{\\nu})[1 + \\cos (2\\pi\\tilde{\\nu}p)] d\\tilde{\\nu}). ", "945b9989e63bb59ae2ec7b98d3c31548": "x_j= \\left(1-\\sum_{i=1}^{j-1} x_i \\right )\\phi_j.", "945bf59803c395f966e9db83bccd8a4d": "\\theta(x)=\\sum_{p\\le x}\\ln p", "945c61c8a308f59b925db529917330ff": "=\\{e,x,...,x^{n-1}\\}", "945d29c4664457c2494786d7063fa9c7": " u'(x) = 3u(x) + 2. \\, ", "945d352a6885655ac3e656bf6462f02b": "\\begin{align}\n\\frac{d\\omega_i}{dt} &= \\frac{\\partial \\omega_i}{\\partial t} + v_j \\frac{\\partial \\omega_i}{\\partial x_j} \\\\\n&= \\omega_j \\frac{\\partial v_i}{\\partial x_j} \n- \\omega_i \\frac{\\partial v_j}{\\partial x_j} \n+ e_{ijk}\\frac{1}{\\rho^2}\\frac{\\partial \\rho}{\\partial x_j}\\frac{\\partial p}{\\partial x_k}\n+ e_{ijk}\\frac{\\partial}{\\partial x_j}\\left(\\frac{1}{\\rho}\\frac{\\partial \\tau_{km}}{\\partial x_m}\\right)\n+ e_{ijk}\\frac{\\partial B_k }{\\partial x_j}\n\\end{align}", "945d3948ebaaf734c399bdf5bafd1774": "x^2\\equiv q \\pmod{n}.", "945d45621022eecb3ca0d9e1d36c7ecb": "\n\\frac{d}{dt}\n\\left(\n mr^2\\sin^2\\theta\n \\,\n \\dot{\\phi}\n\\right)\n=0\n", "945d6b3ae2f4ed02f3fd61f187a519e2": "L_n^{(\\alpha+1)}(x) = - \\frac{d}{dx}L_{n+1}^{(\\alpha)}(x)", "945d7df2b8c6c052773698d292d26ee2": "\\mathrm{2\\ Na_2O_2\\ \\xrightarrow {\\Delta T}\\ \\ 2\\ Na_2O\\ +\\ O_2}", "945da2ed4f7980a2c28f142c0aea1f5c": "\\sum_{n=1}^{\\infty} \\log a_n", "945e0a3ae5c38ec06350290071f7bd81": "\\pi_1(X)\\ :=\\ DX/\\sim", "945e1d12110b7c66f98ca606f8079ace": "1.172 \\mbox{mol}^{-1/2} \\mbox{kg}^{1/2}", "945e35c6b88639b137cf6fce1d327078": "|H(\\omega)|", "945e5f34cac81f144e4aea18e4d33545": "\\mathrm{Pulse~dispersion} = \\frac{k \\delta n\\ n_1\\ l}{c} \\,\\!,", "945e94ee04856ca29ac2dad056e67311": "\\hat\\sigma_{\\beta_0}=\\hat\\sigma_{\\varepsilon} \\sqrt{\\frac{1}{n} + \\frac{\\bar{x}^2}{\\sum(x_i-\\bar x)^2}}", "945ea95cf62bf2990cdab9b098046fd1": "\\eta = 1 - \\sqrt{1 - \\frac{I_{sp}^2}{c^2}} = 1 - \\frac{1}{\\gamma_{sp}}.", "945eca84a544191507e5f3ec458ecc41": "\\pi^4", "945ed46418cd592144887199cb6b34c9": "I(q) = 4\\pi\\int_0^\\infty p(r)\\frac{\\sin(qr)}{qr}\\text{d}r.", "945ed516311a687f54b7a23a1906af43": " \\begin{align} Transmitter \\\\ Characteristics \\end{align} \\begin{cases} Duty \\ Cycle = PRF \\times Transmit \\ Pulse \\ Width \\\\ \\\\ Pulse \\ Spacing = \\left( \\frac {C}{PRF} \\right)\\end{cases}", "945ef025ed132749d5b8e221be55ef7b": "{}\\pi = A_{192} + D_{384} + D_{768}+D_{1536}+D_{3072} + \\cdots \\approx A_{192} + F \\cdot D_{192}. \\, ", "945f0777274429464cd6433aed84f373": "\n\\begin{matrix}\n\\mathrm{is}\\ \\mathrm{to}\\ \\mathrm{choose}\\ \\mathrm{one}\\ \\mathrm{of} & \\mathrm{these}. \\\\\n& \\overbrace{ \\left\\{ AX, AY, BX, BY, CX, CY \\right\\} }\n\\end{matrix}", "945f1a73680e888dfabcaee293628787": "\\binom{n}{k}_F = F_{n-k+1} \\binom{n-1}{k-1}_F + F_{k-1} \\binom{n-1}{k}_F ", "945f5c340b2bf756a8abfaad5f883a1e": "\\begin{align}\\mathrm{E}[\\ln(X)] =& [1 {+} 1 / \\eta]\\!\\!\\int_0^\\eta \\!\\!\\!\\! e^{-X}[\\ln(X)]dX\\\\ &- 1/\\eta\\!\\! \\int_0^\\eta \\!\\!\\!\\! X e^{-X}[\\ln(X)] dX \\end{align}", "945f5d277eb14774e1d17dd6ae2ad243": "{k(t,x)}", "945f5dddfec81376b60a0236c068eabf": "\\dot{w} = -\\kappa E v", "945f669221c9eb6f6eb42daf910b610a": "\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}:\\mathbf b", "945fa245f44dedf4a365bf3f65164cc9": "\\text{return} \\colon T \\rarr \\left( T \\rarr R \\right) \\rarr R = t \\mapsto f \\mapsto f \\, t", "945fbedff37e8120ce5f295f23e7c85a": "\\varphi(\\sum_{P \\in C}{c_P [P]} + \\sum_{P \\in C}{d_P [P]}) = \\varphi( \\sum_{P \\in C}{(c_P + d_P)[P]}) = \\sum_{P \\in C}{c_P + d_P} = \\sum_{P \\in C}{c_p} + \\sum_{P \\in C}{d_p} = \\varphi(\\sum_{P \\in C}{c_P [P]}) + \\varphi(\\sum_{P \\in C}{d_P [P]})", "945fc6679c6783e4dc370b91aac721fa": "\nK \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{k=1}^{N} \\left| \\frac{d^{2} \\mathbf{r}_{k}}{ds^{2}}\\right|^{2}\n", "945fcb07d1ca5e1171c6fc8eae755a3f": " \\frac{\\partial u}{\\partial y}(x,0) = \\frac{\\sin (nx)}{n},", "94604d757befc5c3a3a9398a85f736bd": "A(x)B(y)=\\sum_{i}c_i(x-y)^i C_i(y)", "94606b4cbf1e421ece664f973c022d93": "\\textstyle G/N \\approx \\prod_{i=0}^\\infty \\mathbb{Z}", "9460c1f0b2ddeb7f5a58e089129a2753": "\\textstyle\\text{rate(termination)} = k_t[\\text{M}^+]", "9460d962c13f0ea4c1e0d437f54bfbd3": " S = \\frac{K_\\mathrm{sat}^{(1)}}{K_\\mathrm{mineral}-K_\\mathrm{sat}^{(1)}} ", "94616ba604a11d03433195ca1045d1ed": "q_{ii} = -\\sum_{j\\neq i} q_{ij}.", "9461d45c0081e3b95eafaef52f82581f": " R = 24 \\pi \\, T", "94623825c57ba848a7cb3912536fe98c": "\\scriptstyle y \\in L", "94628c32b68ff5ab21722a090cb585b0": "I_n = -\\frac{\\sin{ax}}{(n-1)x^{n-1}}+\\frac{a}{n-1}J_{n-1} \\,\\!", "946366a7489fa24a449ac6bd4e455bef": "P\\,\\triangle\\,P'=N\\,\\triangle\\,N'=(P\\cap N')\\cup(N\\cap P'),", "9463b30cba858b11013e751b06256291": "E_{n_x,n_y}=\\frac{\\pi^2 \\hbar^2}{2m}(n_x^2/L_x^2+n_y^2/L_y^2)", "9463e01f28afd8454ea418cfb8e88b4b": "\n\\lambda_{k} = -\\frac{4}{h^2}\\sin(\\frac{\\pi (k - 0.5)}{2n + 1})^2, \\ k = 1,...,n.\n", "946423a7b992eb2c6baeb1a9cc71b8ac": "\\mathcal E^*", "9464fd011b81cad5dfcbd54e02cca643": "f_n = \\frac{v}{\\lambda_n} = \\frac{nv}{2L} = n f_1\\,\\!", "94650b9887bcde31958b1cffdca34a8e": "b_n=\\sum_{\\pi=\\left\\{\\,S_1,\\,\\dots,\\,S_k\\,\\right\\}} a_{\\left|S_1\\right|}\\cdots a_{\\left|S_k\\right|}, ", "946523dcb7dc0cd1312ab117a4e6ad15": "x^i[\\mathbf{f}A] = \\sum_{k=1}^n \\tilde{a}^i_kx^k[\\mathbf{f}].", "946591abe03c3f3d691c185b559fe8dd": "u, v, l", "9465b0e6bea7e846cbc019515a2ff4af": "H(X) = H'(X)", "9465cd0fc9e37436ac77243f5095a9ce": "{}_{\\forall j>0: t_j}", "9466032285e606dffd2c033eb212fd6d": "d\\, {*\\textbf{F}} = \\textbf{J}", "94662ba9a4d11c6aa1244e4b00c28de9": "d\\tau\\ ", "946656ae2199352fb9f9abd10b1bdec1": "V(r) = 4\\epsilon_{ff} \\left[ \\left(\\frac{\\sigma_{ff}}{r}\\right)^{12} - \\left(\\frac{\\sigma_{ff}}{r}\\right)^{6} \\right]", "946696a33ce181ab7141bc57c8d206c3": "\\sigma(x,x')", "9466a3ec1445737e91b842c5bdf98d49": "a^2-b^2 = (a-b)(a+b)", "9466c3aa92f983a8b13ac206f06503f6": "\nP_\\text{auto} = \\frac{e^{v(x_\\text{auto} )} }\n{1 + e^{v(x_\\text{auto} )} }\n", "9466ddea545e72efcb5b41118d40558e": " \\widehat{m}_{ts} = \\left( \\odot_{u \\in N(t) \\backslash s} \\mathbf{K}_t \\boldsymbol{\\beta}_{ut} \\right)^T (\\mathbf{K}_s + \\lambda \\mathbf{I})^{-1} \\boldsymbol{\\Upsilon}_s^T \\phi(x_s)", "9467025710441f1e083e3c1b87997be2": " \\operatorname{build-param-lists}[n\\ (g\\ m\\ p\\ n), D, V, T_1] \\and \\operatorname{build-param-lists}[g\\ q\\ p\\ n, D, V, K_1] ", "946709967dcc16f61091804b2671419e": "p(x) = \\frac{1}{\\sqrt{\\nu}\\,\\mathrm{\\Beta}\\!\\left(\\frac{\\nu}{2}, \\frac12\\right)} \\left(1 + \\frac{x^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}},", "946722408125716c02cb281dec9b412d": "G(x)=0", "94674d15d181c880934a2c6881f4aa33": "\\lim_{T\\to\\infty}\\sum_{t=1}^T \\frac{1}{T}(v_i^t) > 0", "9467757943d7fc923ebeaa84a9a8f964": "E(g(T))=\\sum_{t=0}^\\infty g(t)\\frac{(n\\lambda)^te^{-n\\lambda}}{t!}=0", "94678c70c0101f4e0f859b02878a74fc": "\\frac{3}{4}\\sqrt{\\frac{5}{2}}\\sin(\\theta)\\sin(2\\phi)(7\\sin^2(\\phi)-3)", "9467c110ca506e3fca3c5457c5592ff3": "\\approx (10 \\uparrow \\uparrow \\uparrow)^2 (10 \\uparrow \\uparrow)^3 154", "94682e53fba0bc60c6be315be319495a": "t_e\\in [0,\\infty)", "9468e4849e31a3f7cc32267e68db7c26": "(x + y) - y = x", "94691fe8a98d92f7ae6136a5bba79a87": "\\Gamma \\approx 2 ", "9469938d3db2e32c62d185de35e0330e": "\\frac{1}{C}\\lambda^n(\\phi) \\le ||\\phi^n(w)||_X\\le C \\lambda^n(\\phi), ", "94699469cb13f47d93ea4c76de5f21d0": "\\hat L", "9469957aeb9f8942af9ca265a531df03": "n^n", "9469ef2b5fd03dd612bc4bf3a2815ef0": "\\operatorname{vl}_vw[f] := \\frac{d}{dt}\\Big|_{t=0}f(v+tw), \\quad f\\in C^\\infty(E_x).", "9469f4e225f693b065ecd7eb2fc70525": "s_n=\\frac{P_{2n}}{2}.", "946a06ed5366a71bb42fab35ba1e1664": "\\scriptstyle i_\\mathrm r", "946a8393d7c2994265c04d4d5b068c3f": "m\\{x: \\, \\omega(f)(x)> \\lambda\\} =m\\{x: \\, \\omega(f-g)(x)> \\lambda\\} \\le m\\{x: \\, 4(f-g)^*(x)> \\lambda\\} \\le C\\lambda^{-1}\\|f-g\\|_1.", "946a900276fdbb4a6b189acfc5e8f81a": "Lp(v)", "946b03febe9e6ce742f9277d02f8dcf7": "\n (10)(50) - (R_a)(40) - (R_b)(25) + (1)(15)(32.5) - 50 + M_c = 0 \\,.\n ", "946bd74fe255d3a3c95ea9c8e716fad2": "g(y_1, ..., y_n) = z", "946bdcb8d483ef2cc9fc62731099aca7": "T(X,Y) = \\nabla_XY-\\nabla_YX - [X,Y]", "946bf4b5127ff33a9e552b61ad04fc6c": "\n\\varphi(q,\\alpha,u):=\\begin{cases}\n\\quad \\alpha &, \\ \\ r_{c} \\le R(q,u) \\\\\n-\\infty &, \\ \\ r_{c} > R(q,u)\n\\end{cases} \\ , \\ q\\in \\mathcal{Q}, \\alpha\\ge 0, u\\in \\mathcal{U}(\\alpha,\\tilde{u})\n", "946c05b5190d7144960f98f39834f07a": "\\exists n(n \\epsilon_G. ", "9472a597402663b2ad3f5b2ffb7ff7b2": "[a_1, b_1]\\times [a_2, b_2] \\times \\cdots \\times [a_n, b_n]", "9472a74d13d301bf875bec62650e90a1": "h_\\gamma [A] = \\mathcal{P} \\exp \\Big\\{ - \\int_{s_0}^{s_1} ds \\dot{\\gamma}^a A_a^i (\\gamma (s)) T_i \\Big\\}", "9472c618d95bb96b22db41cedd9ed908": " \\alpha = 2.3 ", "9472f0e25199d3dc1cb514c703399221": "\\mathrm{O}^\\times_\\mathrm{O} \\Leftrightarrow \\mathrm{O}^{''}_i", "9472f2b7abb50b0e263441d0ea3717d2": "(x',y') = (x + a, y + b).\\,", "94733dc86169eef069703873c31626ce": "\\begin{align}\n & P(a_{i},d_{i},\\gamma ,\\,\\tau ;\\ A,B,C)=P\\left( \\lambda _{A},t_{0} \\right)\\cdot P\\left( \\lambda _{B},t_{0} \\right)\\cdot P\\left( \\lambda _{C},t_{0} \\right) \\\\ \n & \\times P\\left( X_{A,B}=x,Y_{A,B}=y|\\lambda _{A},\\mu _{B},t_{0} \\right)\\cdot P\\left( X_{A,C}=x,Y_{A,C}=y|\\lambda _{A},\\mu _{C},t_{0} \\right) \\\\ \n & \\times P\\left( \\lambda _{A},t_{1}|\\lambda _{A},t_{0} \\right)\\cdot P\\left( \\mu _{C},t_{1}|\\mu _{C},t_{0} \\right) \\\\ \n\\end{align}", "947363ef4b9686113502ebe296ad70aa": "(g,\\nabla,\\nabla*)", "94738cabd1acc5365dc5aa6c90ce86a5": " a_n=0\\ \\quad \\forall n", "947394964d25496034b7adc73efba8ce": "\\lambda/2", "947395b55fe25d1a959f7046a152b731": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\frac{\\partial \\mathcal{L}}{\\partial {\\dot q_i}} - \\frac{\\partial \\mathcal{L}}{\\partial q_i} = 0 \\,.", "94739a4a19bf19425a07b8d07d7f50fc": " D' = \\frac{\\text{Crd}\\ 2 \\theta \\cdot AH}{\\text{Crd}\\ \\mu \\cdot 2R} \n= \\frac{\\text{Crd}\\ (108^\\circ + 2 \\delta) \\cdot 600 \\cdot 5 \\cdot 650}{21600 \\cdot 2 \\cdot 3438} t", "9473ccc57ada4ed63918bc89217a9131": "+\\ \\sum_{j=1}^3 x_j \\ \\frac{d^2 \\mathbf{u}_j}{dt^2}\\ ", "9474427533becb6819f5fbd8128b5955": "\\delta_\\mathcal{S}", "94746a145d4789c0025386bc58aabd92": " \\int f(x) e^{\\lambda S(x)} dx ", "94749de9d2ad531dcad30d2d2029ae75": "2^7+17^3=71^2\\;", "9474c9c49a86d684645a60c3cb699dea": "E,\\,F", "9474d21c591decdd8ad8de4e849dbd32": "r(m) = \\{r + km | k \\in \\mathbb{Z}\\}", "9474fed7e60c2a9d2d7ddaa4741be029": "\\left(b, q, u\\right)\\succsim \\left(d, s, v\\right)", "94751776537523bdc3631c77504bc1ca": "\\mathbf \\psi_{STO-3G}(s)=c_1\\phi_1 + c_2\\phi_2 + c_3\\phi_3", "94751ae806974796049211ccaa0b7175": "\n\\widehat{\\sigma}^2=\n\\frac{1}{n}\\sum\\left(\nX_i-\\overline{X}\\right)^2. ", "94752a2ecae87206c55311f40bb81ef8": "P\\cup\\{C\\}", "9475887daf7e7ddc18c99bad9995e497": "S_\\nu (H\\otimes H)", "94758f6812551f78f9dbcdc4b8349321": "x^1=(1,0,0)", "9475b1080bb7bf6d371c5b05d79bf781": "+\\frac{R_1}{R_2}{V_s}", "9476be5dec9d09ae99161b5a60f2ab8d": "\\dot{u}_a = -\\phi_{,a}", "9476c7865a1177a4b8f7b294b18c56fb": "\\oint \\vec{v}_s\\cdot\\vec{\\mathrm{d}s} =\\kappa", "9476e82014af01379329fb2d4d412252": "\\gamma_\\mathbf{u}=\\frac{1}{\\sqrt{1-\\frac{|\\mathbf{u}|^2}{c^2}}}", "94770c381533f81a6d9adb5dc0f197b7": "\\sqrt{f'_c /\\mbox{psi}}", "94771e90b9cdb1492f434fa49f9658d6": "\\operatorname{Var}(X) = \\operatorname{Var}(\\operatorname{E}(X|Y))+ \\operatorname{E}(\\operatorname{Var}(X|Y)).", "9477a3a1e53b03d01de459d583a2bd23": "[(A\\to(B\\to A))\\to[([((C\\to D)\\to E)\\to F]\\to[(D\\to F)\\to(C\\to F)])\\to G]]\\to G", "947830fc309c4f1500eb68dabc92bf23": "V(B_\\mu B^\\mu \\mp b^2)", "947871f60d266f514d2316988ef2c49c": "g_{ab}=-\\omega_0\\omega_0+\\omega_1\\omega_1+\\omega_2\\omega_2+\\omega_3\\omega_3\\,,", "9478805f860831764d911fb3722cefe5": "\\omega \\cup \\mathbb{Z}'", "947897bfbe4d44b67ebeeaf4f50a76d5": " \\phi_C^{-1}(x,y,z) = {((x,y,z) - (x_0,y_0,z_0)) \\over \\Delta} \\begin{pmatrix} v_y w_z - v_z w_y & u_z w_y - u_y w_z & u_y v_z - u_z v_y \\\\ v_z w_x - v_x w_z & u_x w_z - u_z w_x & u_z v_x - u_x v_z \\\\ v_x w_y - v_y w_x & u_y w_x - u_x w_y & u_x v_y - u_y v_x \\end{pmatrix} ", "9478a6253211404ffd7fae8aebf3e77e": "l = 2k", "9479555156ab1e53287e4313744ac2a7": "j=\\frac{g_2^3}{g_2^3-27g_3^2}", "9479e4c95a0d3575b560cef066f8dd03": "\\theta'=0", "947a97ee5a6bda7d31079894e0d4cc33": "\\left\\|q\\right\\|^2 = a^2 + b^2 + c^2 + d^2 = 1.", "947ae4f64d270589aca6f8b9d843b991": "F_{\\Theta|S=s}(\\theta)", "947af2b43838162723b6c59f13a5aeb4": "k\\;{}_1F_1(a;b;z)+l z^{1-b}\\;{}_1F_1(1+a-b;2-b;z)", "947b14731588da3b4939d9c24e1b1296": "\n u_1(z,t) = U_0\\, \\text{e}^{-\\kappa\\, z}\\, \\cos\\left( \\Omega\\, t\\, -\\, \\kappa\\, z\\right)\n \\quad \\text{ with }\\; \\kappa\\, =\\, \\sqrt{\\frac{\\Omega}{2\\nu}}.\n", "947b19797a12451bfb183c7f365c4a45": " f(Y, X, \\varepsilon) = 0 ", "947b4f6658622a5425178ca4cc44134e": "(C,\\otimes,I)", "947bc4afe4d6144fcee59fe7b91aa93c": "\\not\\Rightarrow", "947c0868ba6f09cd74f8423a37183a71": "\\left(1-\\frac{1}{m}\\right)^k.", "947c1b049c8f3fe2343b678631fd3b63": " dy = 2s \\,ds ", "947c40c80bbace1f09c7209c66180dc3": "\n\\frac{d}{dt} \\left( \\mathbf{p} \\times \\mathbf{L} \\right) = \nm k \\frac{d}{dt} \\left( \\frac{\\mathbf{r}}{r}\\right) = \n\\frac{d}{dt} \\left( mk\\mathbf{\\hat{r}} \\right)\n", "947c5e8c55de3eeb30a196f6d1671792": "\\left\\{ z \\in \\mathbb{C}\\ |\\ \\mathrm{Im}(z) \\geq 0 \\right\\} ", "947c79e0ddae572fa76d149f707708fa": " d \\in \\mathbb{N} ", "947c83e5f021547cc995e9578567daff": "f(x) = f(y)", "947c9c8ba5430b5f5a09b5b17815aa28": "\\sigma_{\\mathrm{tr}} = \\int (1 - \\cos \\theta) \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\mathrm{d} \\Omega", "947cb5320d51eb78cc15325f5cdb6f0a": "\\frac{K \\cdot t}{V} = -\\ln (1-URR). \\qquad(8)", "947cbc9097bcdbd28a578f5a7e67c3dd": "\\frac {\\sin {\\theta_1}}{\\sin {\\theta_2}} = n", "947d034bdc3afe61aca311b4fe0e273a": "\\frac{m}{100} = \\frac{E(M)}{N}", "947d307fabeb584d8de1a3a09c585db4": " \\delta t = +24^s.349 + 72^s.3165T +29^s.949T^2 + 1.821B", "947d37be261d3ac074842895c1075499": "(x-1)^2 = x^2-2x+1", "947d73121182aa9a887d0effbd2d2d76": "I(u)", "947d7d61cd9ae4cb2ccd29438d7f9974": "TN", "947d9993c929d2e7a6fe48e642a69794": "x_n=\\frac{A_n}{B_n}.\\,", "947e4cb44120cbb7c3bdf619769ee516": "L(\\hat\\rho) = \\prod_j \\lang y_j|\\hat\\rho|y_j\\rang^{f_j}", "947e6f7474b6b73e383f66b3948c7a4c": "\n\\begin{align}\n1 \\,+\\, \\frac{1}{2} \\,+\\, \\frac{1}{3} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{5} \\,+\\, \\cdots\\, + \\,\\frac{1}{n} &= 1 + \\left( \\sum_{m=1}^\\infty \\frac{1}{n^{m+1}} \\left( \\sum_{r=1}^{n-1}r^m \\right) \\right)\\\\\n&= 1 + \\frac{1}{n^2}\\sum_{r=1}^{n-1}r + \\frac{1}{n^3}\\left( \\sum_{r=1}^{n-1}r^2 \\right) + \\frac{1}{n^4}\\left( \\sum_{r=1}^{n-1}r^3 \\right) + \\frac{1}{n^5}\\left( \\sum_{r=1}^{n-1}r^4 \\right) + \\cdots\\\\\n&= 1 + \\frac{1 + 2 + \\cdots + n-1}{n^2} + \\frac{1^2 + 2^2 + \\cdots + (n-1)^2}{n^3} + \\frac{1^3 + 2^3 + \\cdots + (n-1)^3}{n^4} + \\cdots\n\\end{align}\n", "947e983b03372042353c783d094878e7": " \\mathfrak{so}(6,\\mathbb C) \\cong \\mathfrak{sl}(4,\\mathbb C).", "947eaeb3c550eac20dc45d0d4c643605": "A^\\alpha = \\left(\\phi/c, A_x, A_y, A_z \\right), ", "947ebc4ecadc5fb59b0641ec9ccc1201": " 8^{th} ", "947ecb6aef715f5fdc025b6f3af4bb5d": "\\gcd(a, \\operatorname{lcm}(a, b)) = a.\\;", "947ed446de88df441707a790ddb11d6d": "\nf\\left( x \\right)=H_D \\left( {F\\cap \\left( {0,x} \\right)} \\right)=x^D,", "947f44ac5897c8dd6d9f019719550f61": "\\vec\\omega = \\frac{d\\phi}{dt}\\vec u", "947f493f69147dd891a4b20be4e22526": "\\rho^{\\otimes n_1}, \\rho^{\\otimes n_2}, \\ldots", "947f879e53a63334b6d79f5aae4b2860": "\\scriptstyle C_i", "947f9074b9dc69b9cbf865e53df4b119": "\\hat{b}^{\\dagger}", "947fac6f3206c0f45f6d6bc7d31d079e": "\\sqrt{(a-b)(c+d)} \\, ", "947fc2bfe2e25ded98d4adb6247d1aab": "Z^{m}_n(\\rho,\\phi) = R^m_n(\\rho)\\,\\cos(m\\,\\phi) \\!", "947feedd2eaf7ed16f3854764c22f3c7": "\\| \\mu - \\nu \\|", "947fff220f1761f65beebef22f164e74": " \\frac{\\partial W}{\\partial \\mathbf{x}} = -\\frac{\\partial U}{\\partial \\mathbf{x}} = -\\big(\\frac{\\partial U}{\\partial x}, \\frac{\\partial U}{\\partial y}, \\frac{\\partial U}{\\partial z}\\big) = \\mathbf{F},", "948034f5f9cc4ac30722cb984df60d97": "\nf_2^2=a^2-c^2=(a^2+\\lambda)-(c^2+\\lambda), \\, \n", "94811576ef20134a4272038e6210f619": "\\mathbf J_a", "948132b7281885fa73e4b49ca3713a26": "{F(x) = G(x) \\over F(A) = G(A)} \\qquad {A = B \\over F(A) = F(B)} \\qquad {A = B \\quad A = C \\over B = C}.", "94817470c45c45198e3e9c4bf27d059b": "\\displaystyle P(w|z)", "94819ccec7726975e0d313c7cbbf49d5": "restrict(F, x, 1)", "9481a1a0cd364ee6ac1e57bf0f2c6183": " S \\to conc(\\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle, \\langle \\epsilon \\rangle) ~|~ conc(\\langle a \\rangle, S, \\langle a \\rangle) ~|~ conc(\\langle b \\rangle, S, \\langle b \\rangle)", "9481a2fe56610adf2e51cea462fa9eca": "(a+b)3", "9481cab0380f4aaac49ce3081a579e22": "{\\mathbb R}^m[z]/(z^{k+1})", "9481eea3997d8ef9b50dbc715f0b6299": " c_2(X) = e(X) ", "9481fa5131cfaf3b112f8e97020c8fb3": "\\chi_{k\\mid k-1}^{i} = f(\\chi_{k-1\\mid k-1}^{i}) \\quad i = 0,\\dots,2L ", "9482105f7490c33fbec9ad5fd6e05e73": "\\Lambda_{\\chi'}{}^{\\psi}", "94821f56b2bcb821f4e88a38df659d30": "E_{i,j}^C", "9482a64c7edede644c55672011089b6f": "\\Gamma\\left (4.5 \\right ) = 3.5! = \\Pi\\left (3.5\\right ) = {1\\over 2}\\cdot{3\\over 2}\\cdot{5\\over 2}\\cdot{7\\over 2} \\sqrt{\\pi} = {8! \\over 4^4 4!} \\sqrt{\\pi} = {7! \\over 2^7 3!} \\sqrt{\\pi} = {105 \\over 16} \\sqrt{\\pi} \\approx 11.63.", "9482af00de2192da6fe9ca4f9233af6a": "\\delta_{ij} = q_{ab} E_i^a E_j^b", "9482b43073880265ff487f4d71432b82": "\\pi : \\tilde X \\to X", "9482de5b66de9a344b2bb3ee8ed486a8": " u^L_{i - \\frac{1}{2}} = u_{i-1} + \\frac{\\phi \\left( r_{i-1} \\right)}{4} \\left[ \n\\left( 1 - \\kappa \\right) \\delta u_{i - \\frac{3}{2}} + \n\\left( 1 + \\kappa \\right) \\delta u_{i - \\frac{1}{2} } \n\\right],", "9482f954afba23ae875f5ee2e96be46f": "f(\\vec{x},\\vec{e}_i,t)\\,\\!", "948342322953c42b12fd52a38ce15e74": "{\\frac {|BD|} {|DC|}}={\\frac {|AB|}{|AC|}}. ", "94839390bb1615d0ba7d08edba2f889a": " R = {4 \\pi G \\over {3 } } \\rho (r) ", "9483d7d3645f0c6666bbeab101c984c7": "k\\ln{S_0}", "9483e4c86d55ae24ea17e6d5163966cf": "\\int_0^t", "9483f03850e31b6340affb6141546a0a": "\\iint_D f(x,y)\\ dx\\, dy = \\int_a^b dy \\int_{\\alpha (y)}^{ \\beta (y)} f(x,y)\\, dx.", "9483fc0e6f05ff3c6235ece66643f52e": "\\Omega_b \\approx 0.2,", "9484057413879b2cfc3e1932df8a93b2": "S \\subseteq T \\implies S^* \\subseteq T^*", "94840f620a5441b42c17bd03d7526109": "\\Upsilon_\\odot", "94841a6e28e0438ba251361ac682ea26": "Q = - \\frac{1}{2} m \\mathbf v_S^2 - \\frac{1}{2} \\nabla \\cdot \\mathbf v_S", "94841b1f0134e0fec47b2c080634ab57": "N{\\phi_i}K", "94843d4310ede3484e5e751bc829b83e": "\n\\frac{\\partial \\beta_i}{\\partial \\theta_j} = \\delta_{ij} + \\int_0^{r_\\infty} dr \n g(r) \\frac{\\partial^2 \\Phi(\\vec{x}(r))}{\\partial x^i\n \\partial x^j}\n", "9484432986feef752f2e55e834bcc60c": "\\displaystyle f(x)\\,", "948452e8b8ca1750cf63f07c5d390bd9": "Y_{10}^{-5}(\\theta,\\varphi)={3\\over 256}\\sqrt{1001\\over \\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot(323\\cos^{5}\\theta-170\\cos^{3}\\theta+15\\cos\\theta)", "94849171c9dce0fb8b9088c2b735ee33": " A^2 = \\frac{a}{R^2\\, T^{2.5}} = \\frac{0.4278\\, T_c^{2.5}}{P_c\\, T^{2.5}}", "9484bd0fb5236147b736a787cbdb0097": "\n \\rho_0 c^2_{33} = \\lambda + 2\\mu + a_{33}e_1, \\qquad\n \\rho_0 c^2_{3k} = \\mu + a_{3k}e_1, \\quad k=1,2\n ", "9484d5a8dcfe26b52b405d1221a3a0be": "(y - 3)(2y + 6)(-4y - 21)", "9484e93d6c6d0dfd263ef9241fd979ec": "E^N_{\\hat{M}}", "948537a11760599d9e02ab4f16fe8a78": "X \\times_Y Z \\rightarrow Z", "948579e67059d3a0e5c92030aec440c6": "E = 57,000 \\mbox{psi}", "948582f96538f3fd4bfb93c3f00fccaa": " G_\\mu = \\frac{\\alpha N [1-(2\\frac{\\lambda(t)-\\lambda_\\mu}{\\delta\\lambda_g})^2] - \\alpha N_0}{1 + \\epsilon \\sum_{\\mu=1}^{\\mu=M}P_\\mu}", "94858325b1640b2cda6182c1f9a18afe": " T_i\\equiv 1 \\bmod N_i (is _1)", "9485f87f7f1713c17b2a684061c81528": "I(P_{t_0}, P_{t_m}, Q_{t_0}, Q_{t_m})", "9486656381ae6875e2d02d6dfb09f25c": "R(C)=\\lim_{i\\to\\infty}{k_i \\over n_i}", "9486d87ed703460373597198fc6acab9": "\\!\\mathcal T", "94870e2f357d66736f87bff258b06a44": "M(h)", "94873fb353e56abee76c5b0cd300e260": " H(|f|^2) + H(|g|^2) = - \\int_{-\\infty}^\\infty |f(x)|^2 \\log |f(x)|^2\\, dx - \\int_{-\\infty}^\\infty |g(y)|^2 \\log |g(y)|^2 \\,dy \\ge 0. ", "94876f4c79cd881a602a8951c8a33a70": "\\mathbb{K}=\\mathbb{R}\\,\\!", "9487bb6cb29b98ebad60aa1981b58ab4": "(c_\\max)^{2k} \\geq \\frac{1}{m-1} \\left[ \\frac{m}{\\binom{n+k-1}{k}}-1 \\right]", "94882f51f90b40243dc15ba4c8c10eb8": "\\frac{ds}{dT} = \\frac{W(s)}{b}", "94883a7c2d764472d101649f71f2c08c": "\ne^{x/y} = 1+\\cfrac{2x} {2y-x+\\cfrac{x^2} {6y+\\cfrac{x^2} {10y+\\cfrac{x^2} {14y+\\cfrac{x^2} {18y+\\cfrac{x^2} {22y+\\ddots}}}}}};\ne^2 = 7+\\cfrac{2} {5+\\cfrac{1} {7+\\cfrac{1} {9+\\cfrac{1} {11+\\ddots}}}}\n", "948872fb2bc7eaed63ea9a794b08b92f": "D\\gamma=-\\gamma D", "9488d3c0ef0522a17478c76bf634ca4a": " g_{\\alpha}(x)=\\Phi\\left[ \\Phi^{-1}(x)-\\Phi^{-1}(\\alpha)\\right]", "9488ed01565ef3fb618b3ccd71349138": "\\scriptstyle n", "948946fdcc28f18f8b566db1c15930aa": "W = aL^b\\!\\,", "948981c9a0cbb0e90344da3fe6d07d09": "\\scriptstyle t \\;<\\; n", "948a1e1baf8870926d499126e768bfd5": "f: M \\to X", "948a5e4f03daf03a84ce31137b6db32f": "f:\\mathbb{R} \\rightarrow \\mathbb{R}_{0}^{+}", "948a70a3f299ae9a8d1f6b2d7b2998bb": "\\frac{x_1 + \\cdots + x_n + x_{n+1}}{n+1} - ({x_1 \\cdots x_n x_{n+1}})^{\\frac{1}{n+1}}\\ge0", "948a730f26ea10ca177846b3540aec3c": "H=H_0+H_1.", "948abbac152e18715c94b81604444ba5": "[\\hat{x},\\hat{p}] | \\psi \\rangle = (\\hat{x}\\hat{p}-\\hat{p}\\hat{x}) | \\psi \\rangle = (\\hat{x} - x_0 \\hat{I}) \\cdot \\hat{p} \\, | \\psi \\rangle = i \\hbar | \\psi \\rangle,", "948abbd0283c19cb097cd6226b8253f9": "w_b", "948aed07222e244d06ca772a87aad3da": "l / 2", "948afff54b2d0cc1d1540bd00a644a02": "P(\\text{well}\\cap\\text{negative})=P(\\text{well})\\times P(\\text{negative}|\\text{well})=99%\\times99%=98.01%.", "948b13480f92545912bcd673040347e3": "2a'", "948b6f2a48e7597dedef0d4889403380": "\\Lambda_{\\mathrm{m}} = \\frac{\\kappa}{c}", "948b8e63be92414c45e5fb45a9cb8a6a": "\\hat{u}", "948c57d762e5ab02e2114ddd46e80075": "dx=\\frac{r\\,dr}{\\sqrt{r^2-y^2}}.", "948c8f8dc1015504c574ce8f8962cc34": "I_{xy} = \\frac{1}{24} \\sum_{i = 1}^{n} ( x_i - x_{i+1} ) ( 3 x_{i+1} y_{i+1}^2 + x_i y_{i+1}^2 + 2 x_{i+1} y_i y_{i+1} + 2 x_i y_i y_{i+1} + x_{i+1} y_i^2 + 3 x_i^2 y_i^2 )\\,", "948c9886607dd8e199611cfda2f1be72": "f_d = f_r-f_t = 2v \\frac {f_t}{(c-v)}", "948ced17f9939de1be0fe44e51e1041c": "y^2 + a_1 x y +a_3 y = x^3 + (a_2 + d a_1^2) x^2 +a_4 x + a_6 + d a_3^2. \\, ", "948d01e958bf21019d0a92358ac96178": "lk(L_i,L_j)=0; i,j=1,2,3; i0, \\Pr\\big(d(X_n,X)\\geq\\varepsilon\\big) \\to 0.\n ", "94926d0da8e9f67dced622fc2813beac": "\\log\\langle u,u'\\rangle = \\langle \\log(u) , u'/u \\rangle \\quad (u>0) ", "9492c830461baa3927924832a6c9d865": "\\left({\\left\\lfloor{\\frac{5}{12}}\\right\\rfloor+5 \\bmod 12+\\left\\lfloor{\\frac{5 \\bmod 12}{4}}\\right\\rfloor}\\right) \\bmod 7+\\rm{Tuesday}=\\rm{Monday}", "9492ca95d535734f44947f00359bed39": "(2, 1), (2, 4)", "9492daad417613cb1237e897aae5ffcf": " H(x) \\in \\mathbb{R}^{n \\times n}", "94935d0548004bdba93f86bb54572f6f": "\\mathrm{tr} (\\mathbf{P} \\mathbf{P}^H)^{-1}", "94940edcdee9010f9fd473a1f9143ee0": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi(x_1,x_2\\cdots x_N,t) = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\frac{\\partial^2}{\\partial x_n^2}\\Psi(x_1,x_2\\cdots x_N,t) + V(x_1,x_2\\cdots x_N,t)\\Psi(x_1,x_2\\cdots x_N,t) \\, .", "94942d6b1f72f56866130189187fbb7f": "|.|", "949457db7906036b731aa62225b19e76": "\\text{mode} =\n \\begin{cases}\n \\lfloor (n+1)\\,p\\rfloor & \\text{if }(n+1)p\\text{ is 0 or a noninteger}, \\\\\n (n+1)\\,p\\ \\text{ and }\\ (n+1)\\,p - 1 &\\text{if }(n+1)p\\in\\{1,\\dots,n\\}, \\\\\n n & \\text{if }(n+1)p = n + 1.\n \\end{cases}", "949494452c85b7756fec7dc42e48a02d": "9-j", "949553636b8f25053c2e8b158d9d0d6c": "S_B = 4\\pi a_B^2 - \\ ", "949557c8493d824c65cf6fe118815a6e": "d/\\lambda=0.3", "9495637e64ea441b29dcd339e057ca56": "\n\\begin{align}\n &= \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W\\left(\\nabla\\cdot\\mathbf{F}\\right) \\, dV\\\\\n &= 2\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W\\left(1+y+z\\right) \\, dV\\\\\n &= 2\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W \\,dV + 2\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W y \\,dV + 2\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_W z \\,dV.\n\\end{align}\n", "94958b133c8f6e0ee6b32f1d10846f1f": "\\frac{d\\vec{p}}{dt}=\\Sigma\\vec{f_o}+\\Sigma\\vec{f_g}=m(\\vec{a_o}+\\vec{a_g}) ", "949599c7e0674bfa67aa27b93f495c20": "D_\\mathrm N", "9495d294d33a43e91aa7c50be644372f": "\\frac{\\partial y}{\\partial \\mathbf{X}^{\\rm T}},", "9495d7c4c4f80de5883c52ed06a1e3b1": "\\tilde E_8", "9495eff075b4e6b4779cf9698d26df33": "\\mathbf{Mon}_\\mathbf{C}", "9496012cdc448a3599e7c12278657fc8": " \\mu_{1} ", "94962804d3c89c1083644b82d59bac6c": "-\\frac{km}{L^{2}}", "94962f8974c04c3d10bcaf0aafc121c5": "{\\partial u \\over \\partial x } = e^x \\sin y, {\\partial^2 u \\over \\partial x^2} = e^x \\sin y", "949630e2880dc1cb9f70bb6129e35060": "[ES] = \\frac{[E]_0 [S]}{K_d + [S]}", "9496fd8fae81fc60f0961f95fc579637": "\\neg loaded(2)", "949702c2ae3216079a1fa7c51da2093f": "U_{n - 1}(1) = n T_n(1) = n", "9497b5ab5171c80255f55809e090d298": "\\mathbb{N} \\!\\,", "9497c70b090db5311716c49f33a2d71b": "\n\\binom{n}{k}_q\n=\n\\frac{[n]_q!}{[n-k]_q! [k]_q!}.\n", "9497e5bb49e60b51d2edce3454fc0268": " CD = \\frac{ 1 }{ n } \\frac{ \\sum| m - x | }{ m } ", "94980a7af08d265526d2b7bd5d9ab1bc": "(\\gamma\\ < 0 ", "94981c82df60cba92a6d94b1628c3a9a": "h,m \\in \\mathcal{N}", "94982a8c2fbffe4f7ce3d7c0464b0c81": "\n F_m(x) = F_{m-1}(x) + \\sum_{j=1}^J \\gamma_{jm} I(x \\in R_{jm}), \\quad\n \\gamma_{jm} = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{x_i \\in R_{jm}} L(y_i, F_{m-1}(x_i) + \\gamma h_m(x_i)).\n ", "94986df20b8de459209de1a8fff1481b": "\\mathbf{W}_0(t) \\triangleq \\mathbf{W}(t) + \\int_0^t \\theta(s)ds ", "9498a8ec03d152ce10f67cb9d36f8ad8": "K_G(X)^{\\wedge} \\cong K(X_G). ", "9498c2fb9e8e4708c0be8b6fb3adf7a1": "{d \\over dx}(\\rho u)=0 ", "9498c99cc65843edccdb1cd314c38d7e": "\\Pr(\\theta, x|t) = \\Pr(\\theta|t) \\Pr(x|t),\\,", "94991b1052ff49556103a146eb76e3c3": "X(j\\omega) = \\int^\\infty_{-\\infty} x(t)e^{-j\\omega t}\\, dt", "94998a8ee8ae067f531d2df97e837b12": "\\exists^{\\rm P} \\mathcal{C} := \\left\\{\\exists^p L \\ | \\ p \\mbox{ is a polynomial and } L \\in \\mathcal{C} \\right\\}", "94998ea9d917c66c94c2a782910e3518": "A(h)", "9499b51c75de502c5f585e4509731a51": "u(x,t) = \\frac{1}{2}\\left[g(x-ct) + g(x+ct)\\right] + \\frac{1}{2c} \\int_{x-ct}^{x+ct} h(\\xi) \\, d\\xi.", "9499b63c29b5a9e8cd869afae24db5b6": "\\mathbb{P}^n_A = \\operatorname{Proj}\\, A[x_0,\\ldots, x_n].", "9499e2c6f336fdbe5cc2c0f8e0411dc1": "{\\bar{\\mathbb Q}}(x)", "949a22a32698a57ab0b12f68ae60f7e4": "\\sum_{\\lambda_1,\\,\\lambda_2\\in\\mathrm{Spec}(H)}\\lang\\psi_{\\lambda_1}\\mid \\psi_{\\lambda_2}\\rang=\\delta_{\\lambda_1\\lambda_2}", "949a24b0e63a6ef2b787ce7351a62446": "\\omega\\cdot4", "949a35b454c14ca628cff5f5d11e9342": " \\mathfrak{a}^{W \\theta(0)} = (\\varepsilon).", "949aa71fae4f0b3688c6be93318ddf78": "\\underline{\\times 62 \\mathit{9}}\\, ", "949aa975f36c467dc5caf80c650da25c": "[E_2]", "949abb0bd806f884fefaf828cd90657a": "x = {a \\over 2} \\cdot \\frac {\\sin [(m + n) p + \\theta_0]}{\\sin [(m - n) p + \\theta_0]}, \ny = a \\frac {\\sin [m p + \\theta_0] \\sin n p}{\\sin [(m - n) p + \\theta_0]}\\!", "949ac992c8b4c9785d2047d7fbb3e6a3": "\\sqrt{\\frac{g}{k}\\tanh\\left( k h \\right)}", "949ad072d20782e364232e43e3586089": " F = m(t) \\frac{dv}{dt} - u \\frac{dm}{dt},", "949ad960ca64cd3f0da22e6738a0dc71": "\n2 \\uparrow 16\n", "949b6de932089fdf434386255434de32": "(90^\\circ - z)", "949b807e2aa388b07a3b0f97a9c306a5": "\\nabla \\times \\mathbf{E}(x)", "949b9a57eee583392101b538fab4c71c": "x \\in_R [0, q-1]", "949bb7c7e9b3a738fa77493c1541d306": "(a, b)^* (a, b)\n = (a^*, -b) (a, b)\n = (a^* a + b^* b, b a^* - b a^*)\n = (|a|^2 + |b|^2, 0 ).\\,", "949c3399a7d9210c24e08ff52532d9af": " H_{R} ", "949ca039da96d4bfb4e50b1a72ae0871": "M(t)=2\\,\\frac{I_1(Rt)}{Rt}", "949cad6c2e5e598d9f873ffe3de1e5fb": "V[n] = \\int V(\\vec r) n(\\vec r){\\rm d}^3r. ", "949cbcc42ad8d29de50182a553efb410": "\\boldsymbol\\eta^T \\mathbf{T}(x)", "949d042bdb80f663cd5afeba75f3fa2f": "E_{lake},", "949d26a3a64bb4dc2861c78ec3478c49": " \\begin{align} \n\\Phi_E & = \\int\\!\\!\\!\\!\\int_a E dA\\cos 90^\\circ + \\int\\!\\!\\!\\!\\int_b E d A \\cos 90^\\circ + \\int\\!\\!\\!\\!\\int_c E d A\\cos 0^\\circ \\\\\n& = E \\int\\!\\!\\!\\!\\int_c dA\\\\\n\\end{align} \n", "949d7af4d0478c9431a69b5e2b702634": "\\bigl[ \\begin{smallmatrix} 2 & 1\\\\ 1 & 2 \\end{smallmatrix} \\bigr]", "949d7e960753a31fd67b2ff856137315": " \\mathcal{M}", "949daf437bf7d57c8151f84291be5864": "E(\\mathbf{r},t) = \\Re\\left\\{ E_0(\\mathbf{r},t)\\, e^{i\\,( \\mathbf{k}_0\\, \\cdot\\, \\mathbf{r} - \\omega_0\\, t )} \\right\\},", "949db4dac98218a50df8c71dbf34bf22": "\\begin{align} 2\\cdot R_*\n & = \\frac{(52.5\\cdot 10.47\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 118\\cdot R_{\\bigodot}\n\\end{align}", "949de712d4c19d9c80b8287d9a8cacbf": "\\displaystyle x = \\frac {ad}{{a^2+b^2+c^2}}", "949e4e96960fd5a9f30368707a76d85a": " H = K \\cdot \\mathrm{Sp}(1).\\, ", "949e76aeec5e631771866c3d953bbf3f": "1-l", "949e8ecc7df7d3eff395c412ec958828": "\\scriptstyle C^b(\\mathcal{A})", "949edce7798ffc4f1861765873c73081": " \\psi(t) = e^{i\\omega t \\boldsymbol{\\sigma}\\cdot \\mathbf{\\hat{n}}} \\psi(0),", "949f112ce6ad9287f27188d2dd51f4dd": "\\scriptstyle H_\\infty(X)", "949f606e86782aeef4f672f8253974b0": "{\\partial c_i\\over\\partial t}+{\\partial\\over\\partial x_j}(u_j,c_i) = D_i{\\partial^2 c_i\\over\\partial x_j\\partial x_j}+R_i(c_1,...,c_N,T)+ S_i(x,t)", "949f7f7cb2cc56b6aa7b5f4b67c012b5": "\\left\\langle\\operatorname{p.v.}\\frac{1}{x}, \\phi\\right\\rangle = \\lim_{\\varepsilon\\to 0^+}\\int_{|x|>\\varepsilon} \\frac{\\phi(x)}{x}\\,dx.", "949fd15ee52c2d616ccf3e4ee7b65789": "\\ln|y| = \\ln|f(x)|\\,\\!", "94a00a33ab64a46c01a634a32a7d68cd": "\\sqrt {(x+c)^2+y^2} = 2a - \\sqrt {(x-c)^2+y^2}", "94a036690bdc9221e2e38947d4066483": "i\\colon X \\to Y", "94a063b8fd6a31f03d1fd96bafe5f1dc": "\\int_0^\\infty e^{-x}a(x)\\,dx = \\int_0^\\infty e^{-2x}(1-x)\\,dx = \\frac12-\\frac14.", "94a09e364ee8135ee4ef1e5cf44ce0e2": "\\alpha_{11}=0", "94a142b2e3d754e10e3dbf00cf032312": "Z_0 = \\frac{1}{\\pi}\\sqrt{\\frac{\\mu}{\\epsilon}} \\ln\\left(\\frac{l}{R} + \\sqrt{\\left(\\frac{l}{R}\\right)^2-1}~\\right),", "94a14aaabae74944296994afeca4397b": "h_{n+1} = h_n + h_n r_n", "94a157ef192ec85358c84f41f49ae9ef": "f_* ([M]) = [X]", "94a1f4f63585b835bfbac6dabfb088e6": "\\tan(\\theta - \\phi) = -v/c", "94a215e37e859d12df8d141d03e60378": "\\mathbb{E}_S", "94a21c25f70c1bee8266106a869ee11a": "(1 \\pm i)^2 = \\pm 2i", "94a238c5c0f5daf07334d435a8680790": "(-1)^{n+1}", "94a2e22b0ccb58424d9fb97d358bbb82": "M = \\begin{pmatrix}I_p + A B & -A \\\\ 0 & I_n \\end{pmatrix} \\begin{pmatrix}I_p & 0 \\\\ B & I_n \\end{pmatrix}", "94a325d2fa9bff95e45424544ce8aab5": "XY. \\,", "94a3385b7073a1f7692a86698d6430fa": "\\partial^2+m^2", "94a355f13bf161cd66f696c78ed1584d": "x \\mapsto \\begin{bmatrix}\n 1 & x \\\\\n 0 & 1\n\\end{bmatrix}", "94a3dc1457b223c67ee13a653743bd1b": "\\int x^5 r \\; dx = \\frac{r^7}{7} - \\frac{2 a^2 r^5}{5} + \\frac{a^4 r^3}{3}", "94a3dda2b98df9465544778e456ae46d": "\n\\left\\{\nZ_{1},\\ldots,Z_{s}, Z_{s+1}|Z_{1},\\ldots,Z_{s+c}|Z_{c},\nX_{s+1}|X_{1},\\ldots,X_{s+c}|X_{c}\n\\right\\} .\n", "94a3e162ffb084c9126d4ea38d3a0f0d": "D = nF^{1/N} + P", "94a3f05486327a05b9d50d659092cb49": "P_{s}", "94a3f21a0a11158374cc2e55058fd06b": "w_1,w_2\\in A_1", "94a41f27092f05be86ebc1d05df36b67": "4pq^3", "94a46f3d36a8250e2cae0154361183ad": "|E| \\leq \\gamma|B| \\implies \\omega(E) \\leq \\delta\\omega(B)", "94a4b47754436287f6db571a273a803c": "\\beta_S=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial p}\\right)_S", "94a550c2223159f67f02b407e251a839": "511 \\text{ keV}", "94a565eb57ed151b9a8115a016b3fdac": "V_a={\\sqrt{R_aG_a}\\,\\lambda\\cos\\psi\\over\\sqrt{\\pi Z_\\circ}}E_b", "94a58eaa3dff521b4733d6ebabe8c991": "\\mathrm{C}^{\\beta}", "94a59dc32db1a1b1554fe3b474bc3c60": "f\\colon D^n \\to D^n", "94a5bd4aa2780d3615260c8ad57ddde8": "A = \\frac{11}{4}a^2 \\cot \\frac{\\pi}{11} \\simeq 9.36564\\,a^2.", "94a6015ddbfc24c75d7bbc6bef5440af": "Q = A^2 + G\\,", "94a61ff2604685bf5fa1614b85cea5e9": "\n\\langle\\chi_{k'}(\\mathbf{r};\\mathbf{R}) |\nT_\\mathrm{n}|\\chi_k(\\mathbf{r};\\mathbf{R})\\rangle_{(\\mathbf{r})}= \\mathcal{T}_\\mathrm{k}(\\mathbf{R})\\delta_{k'k} \n", "94a62b255b39325782fb12e045365cea": "\n\\mathbf{F}_{\\mathrm{Coriolis}} = \n-2m ( \\boldsymbol\\omega \\times \\mathbf{v}) =\n-2 ( \\boldsymbol\\omega \\times \\mathbf{p})\n", "94a6643cb7249ff4aa09b4d79d9bb67c": "c_n = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} f(x) \\mathrm{e}^{-inx} \\, dx.", "94a6dd89df342637a95b1f1d71f88d80": "96 \\leq i \\leq 126", "94a6fee984946838839425b9a1650bb9": "\\sum_{i=a}^bt_i,", "94a6ffe039194281209b088464c967de": "j \\mapsto \\begin{pmatrix}\n 0 & 1 \\\\\n -1 & 0\n\\end{pmatrix}", "94a72aa9a7204d1c5448a5a121ff29e8": "P_{\\mathcal{M}}, P_{\\mathcal{S}}", "94a75cf7232ba7286a48316101c9ed8e": " S_{Ti} = \\frac{E_{\\textbf{X}_{\\sim i}} \\left(\\operatorname{Var}_{X_i} (Y \\mid \\mathbf{X}_{\\sim i}) \\right)}{\\operatorname{Var}(Y)} = 1 - \\frac{\\operatorname{Var}_{\\textbf{X}_{\\sim i}} \\left(E_{X_i} (Y \\mid \\mathbf{X}_{\\sim i}) \\right)}{\\operatorname{Var}(Y)} ", "94a784325a9b6f39384aa6551d82c1f7": "\\mu=\\sigma_i^2", "94a7c223c763ff4d46632a7379feb697": "\\scriptstyle \\omega_c \\;=\\; 1", "94a7c677b3042582c1525354d53b8225": "\\beta_i^C \\underset {\\lambda_i < 1}{=} (1 - \\lambda_i )^{-1}. \\, ", "94a825f167d7d56baef9db0b670b9c77": "H_{Tide}", "94a8ef824cc817ee7627ba7448a0b14c": "k=\\frac{k_0 A}{a}\\sqrt{\\left\\{1+\\left(\\frac{1-n}{1+n}\\tan\\varphi\\right)^2\\right\\}\\frac{\\sigma^2+\\tau^2}{t^2+\\cos^2(\\lambda-\\lambda_0)}},", "94a92a6265fffd07ef29469c169a843e": "y_{n+1}=-bx_n+ay_n-y_n^3.\\,", "94a990462684a2fbb5dfc7fab1c8975d": "SF", "94a993b1b91b5e377f561b85d5620ab3": "\\rho:{\\mathbb Z}\\rightarrow Homeo\\left(F\\right)", "94aa13d6936718919ab0303f62c2d551": "\n \\kappa = 2 \\cdot D_1 ~;~~ \\mu = 2~(C_{01} + C_{10})\n ", "94aaef6e48a2f09fa4de94323a358f33": "2 T \\times C_5", "94ab000617d2a5e322381b950e063e75": "\\mathbf{d}=\\{d_i\\}_{i=1}^N", "94ab4b38b4bc849c49f3d94e935847c5": "\\det(\\Sigma)^{-n/2} \\exp\\left(-{1 \\over 2} \\operatorname{tr} \\left( S^{1/2} \\Sigma^{-1} S^{1/2} \\right) \\right).", "94ab885738e490109b5e80b5de42c2e4": "P_f=P_m\\theta(Y)^{\\frac{1}{1-a}}", "94abbcd2f4ccd5e1189ce667e794a01e": "\\operatorname{Ob}(\\mathcal{C}),\\quad A\\mapsto |A|=", "94ac8940ca471631db2f8f5b1c63c6e3": "2^{255} < q < 2^{256}\\,", "94acbcdf9bd1edfb5539164279895da7": "Y=X\\frac{Z_R}{Z_R+Z_C}=X\\frac{R}{R+\\frac{1}{j \\omega C}}=X\\frac{1}{1+\\frac{1}{j \\omega RC}},", "94acd0fb00d288838ea6fc9e2bac43e5": "\nE_\\mu (z)=\\sum\\limits_{m=0}^\\infty \\frac{z^m}{\\Gamma (\\mu m+1)}.\n", "94ad03e4cb75ce5c9abffbbb06709fcf": "-\\frac{1}{\\varphi}=1-\\varphi = \\frac{1 - \\sqrt{5}}{2} = -0.61803\\,39887\\dots.", "94ad1f9c2109ff05ccd1561164e329bc": "\\phi=-{\\delta\\over\\delta J}E[J]", "94adc1674de8fed023e2956611af6e51": "v: A \\to C", "94ae358552155c798cd17c5a88ffe90d": "N=\\Pi_{i=1}^l(m_i!\\,i^{m_i})", "94aead20a6649d08a620fafa6d856ff5": "\\mathbb F_5[x]", "94aeb13c2c31de7ebf8e8c2717517458": "M=\\,", "94aeb2dd092bcbc48c095ad77ef3bfa1": "c_1=1", "94aef70f7e3cf1a9bd12c62712883d9d": "\\,\\!Q_{m-\\frac{1}{2}}(\\chi)", "94aefe697379dcdde1e561545cdad5d4": "\nV_{BSL}= \\begin{cases}\n \\sqrt{\\mu^2 g^2 t_{prt}^2+ 2 \\mu g d_{ACDA(s)} } - \\mu g t_{prt}, & \\mbox{if } V_{ACDA(s)} \\le V_{ACDA(d)} \\mbox{ or } V_{cl}\\\\ \n \\frac{d_{ACDA(d)}}{t_g}, & \\mbox{if } V_{ACDA(d)}< V_{ACDA(s)} \\mbox{ or } V_{cl} \\\\\n V_{cl}, & \\mbox{if } V_{cl}< V_{ACDA(s)} \\mbox{ or } V_{ACDA(d)}\n\\end{cases}\n", "94af0aa8de5db7bba1a9cdc5e62918a6": "\\omega^{\\omega^\\alpha} \\,", "94af0da5cdc9bd49b5de6a7537e9bb35": "\n P_B(\\lambda) =\n \\frac\n {\\sinh^2\\left[\\eta (V) \\delta n_0 \\sqrt{1-\\Gamma^2} \\frac{N \\Lambda}{\\lambda}\\right]}\n {\\cosh^2\\left[\\eta (V) \\delta n_0 \\sqrt{1-\\Gamma^2} \\frac{N \\Lambda}{\\lambda}\\right] - \\Gamma^2}\n", "94af36b08a89271078d4a538585fca35": "a=0.055", "94af3ddeb8ec2333745f77d74c056901": " \\displaystyle{-\\Re (A\\xi,\\xi)\\ge 0}", "94af7da2001348c67d8f94f401e96e66": "u:\\mathbb{R}^n\\rightarrow \\mathbb{R}^+\\text{ in }W^{1,p}(\\mathbb{R}^n),", "94af8f7c7a275611df4add04a61bec89": "n^2 = \\sum_{k=1}^n(2k-1).", "94afea7330c482f778c17533749fc75d": "X=\\bigcup_{i\\in I}X_i", "94b044a366c735a59aa5c4a532b06e11": "\\Gamma(z+1) = (z+a)^{z+1/2} e^{-(z+a)} \\left[ c_0 + \\sum_{k=1}^{a-1} \\frac{c_k}{z+k} + \\varepsilon_a(z) \\right]", "94b07489ffa71a483fb5f6650731f2cf": "V_{\\mathrm{m}}", "94b0ad0bb9d74528ad77c3554d0bc352": " f(\\cdot,\\cdot): M \\times N \\to \\mathbf{R} ", "94b11008d379dda859645580112fc37c": "\n N_D=N_{P_2}+(x_D-x_M)k_j \\qquad (9)\n", "94b116fa0475a51d760fc7397e3c25d6": "1 = \\sum_{a = 1}^\\omega \\lambda^{-a}\\ell(a)b(a)", "94b14b15997755d75c80cbbdf1ab4011": "\\mathbf{D}^2_{xyz}", "94b15ae603276c7df6e5736a7fe2a5c6": "\\alpha_{01} = \\frac{p+1}{4},\\;\\alpha_{00} =\\alpha_{10} =\\alpha_{11} = \\frac{p-3}{4}. ", "94b17f3b91da8581bce3b19c3b18681d": "A_1, \\dots, A_m \\vdash B_1, \\dots, B_n", "94b184080acf2d51590d92a1944eea61": "\\int\\limits_{-\\infty }^{\\infty }{{{P}_{V}}f(u,\\xi )}.d\\xi =2\\pi {{\\left| f(u) \\right|}^{2}}", "94b1a4d657ee4fa28d358edd93446fe0": "L_{1}\\cup L_{2}", "94b1bfb608f624da516b28ce63d5b509": "~w_{t}", "94b1d15e78c8d6c64697d4915c447c95": "i \\inf_{x \\in C} \\phi(x)", "94b62b914ff3a75e5e02231e37bd8377": "C\\subseteq Q^n", "94b64ca659b8f666b7ea7e3a7198012b": "R_\\mathrm{ab} = \\frac{R_aR_b + R_bR_c + R_cR_a}{R_c}", "94b66b4f743be2d6244ef9189b448b31": "\\overline\\psi = \\begin{pmatrix}{\\overline\\psi}^*_1\\\\ {\\overline\\psi}^*_2\\\\ {\\overline\\psi}^*_3\\end{pmatrix}.", "94b6caff6bedd2c5b44d05383b137e3d": " d\\sigma^2 = dr \\wedge d\\theta", "94b71ff89a1f2b3b2954a0a1b817ca0a": "\\mathrm{LMMSE} = \\sigma_X^2 - \\sigma_{\\hat{X}}^2 = \\Big(\\frac{\\sigma_Z^2}{\\sigma_X^2 + \\sigma_Z^2/N}\\Big) \\frac{\\sigma_X^2}{N}.", "94b79fb8bb4c8bab26b7a925e2efa58f": " \\arccot z = \\frac {\\pi} {2} - \\arctan z \\ = \\frac {\\pi} {2} - \\left( z - \\frac {z^3} {3} +\\frac {z^5} {5} -\\frac {z^7} {7} +\\cdots\\ \\right) \n= \\frac {\\pi} {2} - \\sum_{n=0}^\\infty \\frac {(-1)^n z^{2n+1}} {2n+1}; \\qquad | z | \\le 1 \\qquad z \\neq i,-i ", "94b7d1f9d1b1fb6d0e375723fa93f9fc": " E = Y_1^3 ", "94b7d5be1f372d4671c07d19d407e6fc": "l_{y}", "94b7d7e0ce34f5fead775782895e875d": "c > 0 ", "94b842e883a56a7f076579be94cfadf2": " \\omega = \\sqrt{\\frac{U_\\text{eff}''}{m}} ", "94b88ae3087f87e2c42ddbd19cb060c2": "X \\sim NM(k_0,\\{p_1,\\cdots,p_m\\})", "94b8adca6b82a020986fec1cc073c1de": "V = \\frac{16}{9} \\sqrt{3}a^3 \\approx 3.07920144a^3", "94b8c2a863c7fd12deee477ba35f3442": "s = \\sqrt{ \\frac {5-\\sqrt 5}{2}} \\ ,", "94b91523b2c2696f0417d6d69dd18310": " \\frac{1}{(1-t)^n}.", "94b9295da39545bfb7cc09688351ff0c": "(y - k)^2 = 4p(x - h) \\,", "94b943f4898415522d9b33e446674a5c": "\\frac{\\mathrm{d} \\det(A)}{\\mathrm{d} \\alpha} = \\operatorname{tr}\\left(\\operatorname{adj}(A) \\frac{\\mathrm{d} A}{\\mathrm{d} \\alpha}\\right).", "94b94a7705ca265a375982c4b8e4196e": " U_a =\\alpha P_a + \\beta D_a + \\varepsilon_a\\, ", "94b9854517a7f56575a7e2e8bd3e95c8": "\\color{RubineRed}\\text{RubineRed}", "94b9c90544108c81ad18458c282e6f61": " -e^{-q \\tau} \\phi(d_1) \\frac{d_2}{\\sigma} \\, = \\frac{\\nu}{S}\\left[1 - \\frac{d_1}{\\sigma\\sqrt{\\tau}} \\right]\\, ", "94b9cfcf46705d33925fc9a7d3f61bd8": "s(x + 2\\pi k) = s(x), \\quad \\mathrm{for } -\\infty < x < \\infty \\text{ and } k \\in \\mathbb{Z} .", "94ba1e930444b944fb579a4561f66077": "v \\mapsto v / |v|^2", "94ba5f8ccd140adde0f697de28511a69": "a_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n = b_m \\,", "94ba7045e58b4d056621786e788d111d": " {r}_{k}, {r}_{l} ", "94ba7798a6d19955de51dea5b20caa82": " 0 1. ", "94cafc304ffe6ca03186618c324c9b56": "(I,+)", "94cb24044dbc77ad66d11d0dc2480565": "\\scriptstyle 1/p", "94cb578cf61029490cccb5914709f487": " A_w = f_{pm}(p - m)\\,", "94cb7abd2a146f7957ca5f0051181f08": "(\\mathbb{C}, F_n)", "94cb910072f6324ddbafcfee59d4d4b9": " F(00101) = f(0,0) + f(0,1) + f(1,0) + f(0, 1) + f(1, 0) = 0 + 1 + 2 + 1 + 2 = 6. \\, ", "94cbc4985daab9154e314d9f888bdbb1": "\\scriptstyle \\mathbf{q}", "94cbdc9851863dc720371efef5f4e09d": "\\frac14\\int_0^1\\frac{x^8(1-x)^8}{1+x^2}\\,dx=\\pi -\\frac{47\\,171}{15\\,015}", "94cc24e011ac060966e7756b6a5d4b49": "\\frac{dy}{dx} = y + e^x.", "94cc6db90f7d4e215b94de8c3bd0d028": "(...A(x)...B(x)...)", "94cc6fc7175bc4d89ca53dfddbe6ba31": "\\{x_i,s_i;i=1\\dots N\\}", "94cc7d08297a7242058f384edabf35f8": "\\! w=-1/3", "94ccbca929c31e220d1bb74f93a42adb": "k \\cdot \\nabla\\omega \\times \\frac{\\partial V}{\\partial p} ", "94ccc34b0566a88e8c6ac6f6bdf2c778": "\\alpha, \\beta\\in\\mathcal{O}_m,", "94cd1719bff070707b99bb3f91dc3611": "\\,\\!D_1 = \\overrightarrow{v_r}^2 - 2 \\overrightarrow{v_r} \\overrightarrow{v_i^1} + \\overrightarrow{v_i^1}^2", "94cd49a8b418cab1d89460bb20754c10": "L^{p_\\theta}", "94cdefe0fa2139b5d69344e2cf7cbdf7": " \\boldsymbol{\\omega} = ( 0, 0, \\omega ), ", "94cdf4f45c63f7ee3ebfd5875253b256": "ki=j=-ik", "94ce95976cb7d2fd4c764c47e76b8fa9": "J\\to 0", "94cea845ba9a5f5e8f9766a141f8c6fe": " \\left(x - a \\right)^2 + \\left( y - b \\right)^2=r^2", "94cee203fbf96ee2a96075679a631158": "\\{\\omega = H\\}", "94cefa2028b6d05997a1b27d8daab60d": "\\gcd(p,q)= \\gcd(q,p).", "94cf3cd7802a7b6a2eb629070b5efa55": " \\beta_{max} ", "94cf5cf8bb1b83fc932bee6054fe81df": "1 + \\frac{1}{2} + \\frac{1}{3} + \\cdots + \\frac{1}{n} = \\sum_{k=1}^n \\frac{1}{k}", "94cfcd2f5eb44cdf9eb4b521683d65ea": "e_1,...,e_k", "94cfd51932f851b5288e67a9e75de482": "\\mathbf{v}_{n+1/2}", "94d00a3e432a2620577dc4afafe2f810": " X = \\{ (p,q) \\in P \\oplus P^\\prime : \\phi(p) = \\phi^\\prime(q) \\}. ", "94d00f4568aba4ec637a84a5892a105e": "\n\\begin{align}\n m(t) \\quad \\stackrel{\\mathcal{F}}{\\Longleftrightarrow}&\\quad M(\\omega) \\\\\n \\sin(\\omega_c t) \\quad \\stackrel{\\mathcal{F}}{\\Longleftrightarrow}&\\quad i \\pi \\cdot [\\delta(\\omega +\\omega_c)-\\delta(\\omega-\\omega_c)] \\\\\nA\\cdot \\sin(\\omega_c t) \\quad \\stackrel{\\mathcal{F}}{\\Longleftrightarrow}&\\quad i \\pi A \\cdot [\\delta(\\omega +\\omega_c)-\\delta(\\omega-\\omega_c)] \\\\\nm(t)\\cdot A\\sin(\\omega_c t) \\quad \\stackrel{\\mathcal{F}}{\\Longleftrightarrow}& \\frac{A}{2\\pi}\\cdot \\{M(\\omega)\\} * \\{i \\pi \\cdot [\\delta(\\omega +\\omega_c)-\\delta(\\omega-\\omega_c)]\\} \\\\\n=& \\frac{iA}{2}\\cdot [M(\\omega +\\omega_c) - M(\\omega -\\omega_c)]\n\\end{align}\n", "94d0498b1e8cbef643a1685942e28da2": "\\frac {FP} {k_{B}T} = \\frac {1}{4} \\left ( 1 - \\frac {x} {L_0} + \\frac {F}{K_0} \\right )^{-2} - \\frac {1}{4} + \\frac {x}{L_0} - \\frac {F}{K_0}", "94d0fe36cac1423807b450677ac294ca": "\\operatorname{vec}(X)", "94d11f85bfd16dd05f100d65ff530e4c": "G_{N}=\\frac{2}{3}G_{M}.", "94d21206db83ed60c5163c0c0f027cf2": " \\Delta Y(1-c) = \\Delta I", "94d22d0f0248c1b96968d183aff2b4da": "x\\cdot(y + z) = x\\cdot y + x\\cdot z ", "94d29ae41b473719f9c48bb0f6186576": "L_i = \\ln\\left(\\frac{P_i}{1 - P_i}\\right) = z_i = \\alpha_1 + \\alpha_2 X_i.", "94d29b0989297fbe4efb774d93e25c10": " A_{i,j}", "94d2e0ff6cdcfa8219fd6c420b664d56": "C(S_0, t) = e^{-r(T - t)}[FN(d_1) - KN(d_2)]\\,", "94d3ff19149f3363e85280b9415e4e6f": "X=\\begin{bmatrix}\nx_0+x_1 & x_2+ix_3\\\\\nx_2-ix_3 & x_0-x_1\n\\end{bmatrix}.", "94d4083e45310961c31f06937789c38d": "C[u] \\in L\\Leftrightarrow C[v] \\in L", "94d43c4bea17ca8258bba4feaaf5baca": "\n\\int_0^{2\\pi}\\int_0^1 r [V_{nl}(r\\cos\\theta,r\\sin\\theta)]^* \\times\n V_{mk}(r\\cos\\theta,r\\sin\\theta)drd\\theta =\n \\frac{\\pi}{n+1}\\delta_{mn}\\delta_{kl},\n", "94d4622a52262ea9f969cc4a57120f7f": "\\displaystyle{g^\\dagger=\\begin{pmatrix} \\overline{d} & -\\overline{b} \\\\ -\\overline{c} & \\overline{a}\\end{pmatrix}}", "94d49f0ad2a3115dc36bad2ee5c1df69": "- A_{21}g_2 + B_{21}g_2F(\\nu) = 0\\,", "94d4fe2534d44f3931a6dd6efeb5d951": " \n\\left ( \\mu_z \\right )_{l,m;l',m'} = \\mu \\int_0^{2\\pi} \\mathrm{d}\\phi \\int_0^\\pi Y_{l'}^{m'} \\left ( \\theta , \\phi \\right )^* \\cos \\theta\\,Y_l^m\\, \\left ( \\theta , \\phi \\right )\\; \\mathrm{d}\\cos\\theta .\n", "94d52929fb18686bd70bce2af381a4c7": "f(x) + g(Tx)", "94d536ead135c6b5ae8b872b50740cd3": "+ \\left(2.625 - x+ xy^{3}\\right)^{2}.\\quad", "94d5777185081cf24707f44ea6966b1f": " F(\\varphi,k) = F(\\varphi \\,|\\, k^2) = F(\\sin \\varphi ; k) = \\int_0^\\varphi \\frac {d\\theta}{\\sqrt{1 - k^2 \\sin^2 \\theta}}.", "94d5879582b6817f0c6728a839d9e66a": "\\, t'=t \\qquad x'={x-vt\\over 1-v/c}\n\\,", "94d59b96344b7c436c5ae50ade49d72b": "\n\\begin{align}\nK & = \\frac{12}{N(N+1)}\\sum_{i=1}^g n_i \\left(\\bar{r}_{i\\cdot} - \\frac{N+1}{2}\\right)^2 \\\\ & = \\frac{12}{N(N+1)}\\sum_{i=1}^g n_i \\bar{r}_{i\\cdot }^2 -\\ 3(N+1).\n\\end{align}\n", "94d5bd8e8b8b5b046fba3e1a603001f8": " \\lambda^2 - \\operatorname{tr}(\\mathbf{M}) \\lambda + \\operatorname{det}( \\mathbf{M}) = 0 ", "94d5f60c8467a1c4e2b104cd936a980f": " V = V_0 e^{j \\omega t} = V_0 \\left (\\cos \\omega t + j \\sin\\omega t \\right ),", "94d650e48c4183a2277ad254a8b98a25": " S_i^+ ", "94d65c3b7b44d5bd1554812f484502dd": "P(z) = \\exp \\left ( I(z) + \\frac{1}{2} I(z^{2}) + \\frac{1}{3} I(z^{3}) + \\cdots \\right ). ", "94d666a6d2c20ecf510e419691acfba4": "Z(k,z)=e^{i|k|z}\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,e^{-i|k|z}\\,", "94d66eb0ce81fe4999f216a399a09d4b": "\ns(x) =\n\\begin{cases}\n \\varepsilon, & \\text{if } f(x) \\geq \\varepsilon \\\\\n 0, & \\text{if } f(x) < \\varepsilon\n\\end{cases}\n", "94d68458a40e19f52ac0dff5fd294cfb": "I_{DS} = I_{DSS}\\left[1 - \\frac{V_{GS}}{V_P}\\right]^2", "94d6ab94f630bccd25d5fb625a231e02": "Z(G)", "94d707ca8c49843793ab5b3a81080b0f": "P(E|H) P(H)", "94d76faa0224f5d81fad098008221209": "s_{c+1} = e_1 \\alpha^{(c + 1)\\,i_1} + e_2 \\alpha^{(c + 1)\\,i_2} + \\ldots", "94d7ccdcb5de74d450ec43f2dcc7b20b": " D(S,c) = \\frac{1}{|S|}\\sum_{f_{i}\\in S}I(f_{i};c) ", "94d908405884ba9c500a2026a14789d9": "x_{\\perp}", "94d92e008517aa2514f75d71c0fae552": "\\hat G = \\mathbb{Z}", "94d931f7b1106e80aab93b9f3600869f": "Q_c^{(i)}", "94d93462917ddedba2188df25fcb77bf": "48 = 6 + 6 + 9 + 9 + 9 + 9", "94d93586b9118822ce42012d0c032682": "\\hat{\\alpha} = 1 + n \\left[ \\sum_{i=1}^n \\ln \\frac{x_i}{x_\\min} \\right]^{-1}", "94d95b3b6a02747d5c10ae7394c85915": "\\mathrm{color1} = \\frac{21}{32} * \\mathrm{color0} +\\frac{11}{32} * \\mathrm{color3} \\approx \\frac{2}{3} * \\mathrm{color0} +\\frac{1}{3} * \\mathrm{color3} ", "94d9f37b2c783ff4e3fda9b0501b552f": "M_0,M_1,M_2,M_4", "94da0702d812e41c0864d5c28f359a00": "\\approx\n1-\\frac{\\zeta(\\alpha)}{2^{\\alpha\\!+\\!1}\\tau^\\alpha}", "94da0c930ef84f5cffd5632a8a26f120": "c=4P_cT_c^2V_c^3", "94da0c96900ea092fc25c818b3f4040f": "\\psi' = e^{i\\chi}\\psi", "94da943f68fbd9d0c37480a0df9cfd26": " A_{bcc} = \\frac{2J_{ex}S^2}{a}", "94dabcaa7d682137f01f5337553eeb45": "2^{\\aleph_0} = \\aleph_1. \\, ", "94db2e1b314b8eec56ccc2942aaedba1": " D_{\\mathrm{eff}} = D \\left( 1 + \\frac{1}{192}\\mathit{Pe}_d^{2} \\right)\\, , ", "94dc35d199878097ecb509e6c6de6367": "\\frac{c(n-np_fn+cp_f+pf)}{n^2-pfn(n-c)}", "94dc643c7ede6ce79980b87737fa1bbf": "\n\n\\dot C_{RW} (t)\\,\\,\\, = \\,\\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 \\,\\,\\left\\{ {{1 \\over \\lambda }\\,\\,\\, - \\,\\,\\,{v \\over {\\lambda ^2 L}}\\left[ {1\\,\\,\\, - \\,\\,\\,\\exp \\left( { - \\lambda {L \\over v}} \\right)} \\right]\\,} \\right\\}", "94dc928b795a881960225ec5a83270c8": "O(N^{2d+1} )", "94dccf2c46f8d1f4d208ff9d28c2d263": "r = n,\\,", "94dcddeea4e9348f73a29084d846a37d": "x_{\\perp }^{\\mu }", "94dcef1edd5e82c910c3f42be64328d5": "x^\\frac{\\alpha}{\\alpha+1}+y^\\frac{\\alpha}{\\alpha+1}\\leq \\left(\\frac{x}{s}+\\frac{y}{t}\\right)^\\frac{\\alpha}{\\alpha+1}\\Big(s^\\alpha+t^\\alpha\\Big)^\\frac{1}{\\alpha+1}=\\left(\\frac{x}{s}+\\frac{y}{t}\\right)^\\frac{\\alpha}{\\alpha+1}", "94dcf5411a8d23e6e8c48d4ead20f4f8": "T_0\\cup T_1", "94dcfc98acda6464e8a79dbb8a724a7d": "\\vec \\phi(a_i) =[\\phi_1(a_i),...,\\phi_k(a_i),\\phi_q(a_i)]", "94dd18a6ec014d4361ac63af50cb8e9d": " -\\sigma_3 \\sigma_1 = i \\sigma_2 = \\begin{bmatrix} 0 & -1 \\\\ 1 & 0\\end{bmatrix}.", "94dd2b3766080664b85d0dcf9556d377": "(\\Delta \\phi)(v)", "94dd3ab88f4395e7ef93132bb8a1f4a3": "\\sum_k c_k\\,y_k(t).\\,", "94dd7b94ae51a61a7d65c34aa3e2f8b3": "v_{i,j} = \\sqrt{\\frac{2}{n+0.5}} \\sin(\\frac{\\pi i (2j - 1)}{2n + 1})", "94ddec262ee378b0321991e5c1346be0": "U(l, m, P_1) \\equiv i \\lambda K(l, m, P_1, \\nu) \\sqrt{I(l, m)}", "94de2b838132e54447809049f62e6fd3": " \\theta_\\mathrm{left} ", "94de473e74ac580f7d1008556865c3c1": "\\varepsilon(w,z) = -\\overline{\\varepsilon(z, w)}.", "94de5866094d2537376748c6bd697154": "\\operatorname{Gode}(\\operatorname{coker}(d_{i-1}))", "94de6ef2317ea860ca1e3fa39dcaf7bf": "\\psi_X\\colon F(X)\\to N\\,", "94dea3d82882e406b00446331a98ddbc": "\\displaystyle{\\Delta \\Phi_\\lambda= (\\lambda^2 + 1) \\Phi_\\lambda.}", "94dee6d139abf7f1ef27bc9e128a71cb": "\\beta_0 < \\kappa \\,,", "94df41f480c6e97365857cad1e675265": " A \\cdot B = \\frac{1}{2}( AB + BA ) = 0.", "94df4b189cc2a11a0e89a3c346cf5ba2": "\\scriptstyle{TX}\\,", "94df6de6d726731eaa40937249588bba": "\\lim_{h\\rightarrow 0}\\frac{f(x+h)-f(x)}{h},", "94dfdb3ead42ba1d9fac93cc3ab45c9c": "{AE}_{4}", "94dfe05d913b41aa10cc24d823408e08": "\\textstyle r^2 = a \\mod n", "94dff40f2dd42d4a1b42701dc7c14412": "\\sin(x)=\\cos\\left(\\frac{\\pi}{2}-x\\right)", "94e016cd3b9874186c38dd6ed0319c46": "a+c=b+d", "94e04a25b72a2044a121c88c6638f935": " n_{o} = \\tfrac{1}{2}(e_- - e_+) ", "94e065fd78b7523c2dc81b63d9d22d03": "\nh_i^\\prime=1/h_i\n", "94e07a03bcebdd7fe52203dd39fb4aa4": " \\Delta H_{fus}", "94e0c5280e91e41179f0ff7535909a67": "h=\\left({M_i \\over M_R}\\right)^2 \\left({P_i \\over \\rho g}\\right)", "94e0f9b9b4405485ef9d02becf1c8c00": "\\int_C {e^{itz} \\over z^2+1}\\,dz.", "94e0ffa4513870046a7c400fec59c1d8": "r = \\lim_{n \\to \\infty} \\frac{1}{n} H(M_1, M_2, \\dots M_n),", "94e141dd38a9ffa123ca27beb64307d5": "c_1 x_1 + c_2 x_2\\,", "94e1439200787d1849f68897d49f68c7": "\\frac{\\partial \\ln{T_{eff}}}{\\partial \\ln{\\mu}} \\approx -26", "94e17a38690c9ed49121c0a4c081966b": "\\sqrt{\\det(V V^\\mathrm{T})}", "94e1a25ad2e1cefe2f99a6742f3bff87": "\\frac{dU}{d\\log\\xi}=-U(U+n(n+1)^{-1}V-3)", "94e1da5fe364e862d9b73b08116068f7": "\\int_{0}^{\\infty} xe^{-ax}\\sin bx \\, \\mathrm{d}x = \\frac{2ab}{(a^2+b^2)^2} \\quad (a>0)", "94e1e527187a0fa457e2cea865835824": " \\bar V_t = r_{t} + \\gamma \\bar V_{t+1} ", "94e20261590f15bf6e7d57a382f1fd20": "f(x,y)=\\begin{cases}a\\quad x<0\\\\b\\quad x>0\\\\c\\quad(x,y)=(0,1)\\\\d\\quad(x,y)=(0,-1)\\end{cases}", "94e2296a2637e2c9f03d3dff1dec7dd3": " \\epsilon^{\\mu \\nu \\rho \\sigma} \\gamma^5 \\gamma_\\nu \\partial_\\rho \\psi_\\sigma + m\\psi^\\mu = 0", "94e23035870f62b73ceb65f64b5e9ba7": "r_1 = \\begin{bmatrix}\\cos{2\\pi \\over n} & -\\sin{2\\pi \\over n} \\\\[8pt] \\sin{2\\pi \\over n} & \\cos{2\\pi \\over n}\\end{bmatrix} \\qquad s_0 = \\begin{bmatrix}1 & 0 \\\\ 0 & -1\\end{bmatrix}", "94e24af7692913070c94cfaa2e78be46": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0 \\right)", "94e2b2dbdd3057daacfa3cff3aa2dce4": "G/uv", "94e2ca0781f0126c8499b9f9fb1ce29d": " \\int\\limits_{\\mathrm{ all \\, space}} \\, d ^3\\mathbf{r} \\Psi^\\dagger (\\mathbf{r},t)\\Psi(\\mathbf{r},t) = 1", "94e2daab6f63bba66886685ca57bf0bd": "\\operatorname{mr}(G)", "94e30ee135c025b92cf00a462c1f0dc3": "(n/s) \\lg l = 4 n \\lg \\lg n / \\lg n", "94e34db535582663ebd3474578d8cf7f": " \\sigma \\!\\,", "94e3c3c5dfb106766ab3d60f54d92a59": "|U| = \\mathrm{card}(U) = \\inf \\{ \\alpha \\in ON \\ |\\ \\alpha =_c U \\},", "94e47fdde730f2e2a2051c0fa40f9823": "\\operatorname{sinc}(x)\\,", "94e521b66719f330f3f50423c7fbc908": "G^k(x,\\xi) = \\frac{g^{ki}}{2}\\Big(\\frac{\\partial^2 L}{\\partial\\xi^i\\partial x^j}\\xi^j - \\frac{\\partial L}{\\partial x^i}\\Big). ", "94e55731ef6fbc52546c5191ae4e6635": "T_f=\\frac{T_w + T_\\infty}{2}", "94e5633bdfc5054b8ccd30a4cb430b81": "\\scriptstyle{\\frac{R_1 R_3}{R_2}}", "94e56e538706d47a5a8bfadf5bd1a568": "{\\langle\\!\\mathrm{op}\\!\\rangle}", "94e5bb8e967dfd38c08d9df509063572": " V= a \\times cos(\\phi)", "94e5d16267ba092515c3e47fdc934fd4": " j \\in \\{1,..,p\\}", "94e602214cf26e1e68f401df613cb0f4": " \\ P = {{\\ 1 \\over\\Delta R \\sqrt{ 2 \\pi\\, } \\ } \\exp[{ \\ ( \\ln{R} \\ - \\ln{\\bar{R}} \\ )^2 \\over\\ 2 \\Delta R^2 \\ \\ } \\ }] ", "94e61f5e38eddaec08288dc5604b65f8": "e^{(1-t)z} L_n^{(\\alpha)}(z t)=\\sum_{k=0} \\frac{(1-t)^k z^k}{k!}L_n^{(\\alpha+k)}(z).", "94e6234d2eb00c673ee9f49db1c882c6": "L u =\\sum_{k=1}^{\\infty}-2\\frac{(-1)^k}{k}\\sin kx.", "94e63c07be024d4ce1c0e101cfa3538e": "x\\in E", "94e642ebcbd839fe614278c6765f33a8": "\\mathfrak e_7\\cong (\\mathfrak{so}_{12}\\oplus \\mathfrak{sp}_1)\\oplus \\Delta_+^{64}", "94e655c3f3571d49cd38eedba3f65eb3": "A_j(x) := \\sum_{k=0}^{\\nu_j - 1}\\frac{a_{j, k}}{k!}(x - \\lambda_j)^k", "94e65fa42269c11b12fa1e53684fcbf9": " L = \\vec r_{uu}\\cdot \\vec n, \\quad\nM = \\vec r_{uv}\\cdot \\vec n, \\quad\nN = \\vec r_{vv}\\cdot \\vec n. \\quad\n", "94e67f73df0db2b0291f36094fbc8a0b": "\\|h\\| \\le \\delta", "94e68cca0a0f7829d97e2a63db80fdd6": "D_{Bohm} = \\frac{1}{16}\\,\\frac{k_BT}{eB}", "94e695c7ee534daf185a26405cb3a2f2": "W(u_1,\\dots,u_d) \\leq C(u_1,\\dots,u_d) \\leq M(u_1,\\dots,u_d).", "94e6c1648753feddece041b3f6bb0e93": "i = r + \\pi^e", "94e7579a8279a69538e6f6f4426ab914": "\\tbinom{n}{2}p", "94e7ab951e9c35378ae14cff309ed3bc": "2\\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y)", "94e7fa3088ea00b6fd118553c4fcfee8": "X \\sim \\textrm{Pois}(np). \\,", "94e7fb8906bf0ec7a4862f1f062bcd5f": "(2-y)y", "94e81dea23e2ae489af0f53bb38d8757": "\\mathrm{\\phi} = 0.65\\,\\!", "94e854fdc48ff08976c82a01ad1fe457": "\\gamma^2 = \\frac{1}{1 - v^2}.", "94e89388b784944fdcd669135b8e10d0": "\\scriptstyle i\\omega ", "94e8963d80086f668a185d0f2a35dab2": "\\,f^n", "94e8cb5429fc8719e154ca7b7d6fc631": "(x^2>a^2)", "94e8f8129e5c841543fb8fb0bdf24356": "\n\\forall \\gamma\\ \\in \\Gamma\\ \\sum_{i \\in \\mathrm{N}} \\nu\\ _i (\\gamma\\ ) = const. ", "94e90d950117005a81d5af1468911517": "\\textstyle\\sum_{i=1}^k e^{\\eta_i}=C", "94e91b6214b002ce281cf9033f9b53e2": "\n{\\mathcal L}_{\\rm KS} = \\frac{1}{16 \\pi G} (R - 2 \\Lambda )\n- \\frac{1}{4} B_{\\mu\\nu} B^{\\mu\\nu}\n- V(B_\\mu B^\\mu \\pm b^2) + B_\\mu J^\\mu\n+ {\\mathcal L}_{\\rm M}.\n", "94e9780c666e2205821daf776147f04b": "((a+b)+c)'=(a+b)'+c'=(a'+b')+c'=a'+(b'+c')=a'+(b+c)'=(a+(b+c))',", "94e996cbb74997d7b2808d37ef68f920": "A(c)", "94e9c2380262b7c6a1f3a38a674eae14": "U = 4", "94ed94429acd341e844560fc38093da4": "\\textstyle{\\bigcap_{n=1}^\\infty G_n}", "94edaca77fbe1fd3539d8503ba805bb0": "\n M = -EI\\cfrac{\\mathrm{d}^2w}{\\mathrm{d}x^2}\n ", "94ede8cebab11c8517bceab98aaa9af0": "L_{ML}(\\theta)=\\sum_{i=1}^{N} \\|\\hat{R}-(VSV^H)\\|^2 ......(9) ", "94ee1527f7f96ea8d423013392db23aa": " R_t = \\max\\left (Z_1, Z_2, \\dots, Z_t \\right )- \\min\\left (Z_1, Z_2, \\dots, Z_t \\right ) \\text{ for } t=1,2, \\dots, n \\, ", "94ee45c39c3f2cb4222f86133a41f2d1": "\\liminf B = \\sup\\{ \\inf B_0 : B_0 \\in B \\}", "94ee56114b17c620de6a37b1b30f4a9a": " E =\n a_1 a_2\\int {d^3k \\over (2 \\pi )^3 } \\; \\; D\\left ( k \\right )\\mid_{k_0=0} \\;\n\\vec v_1 \\cdot \\left[ 1 - \\hat k \\hat k \\right ] \\cdot \\vec v_2 \\; \\exp\\left ( i \\vec k \\cdot \\left ( x_1 - x_2 \\right ) \\right )\n", "94ee736fce72b61a24ae42283c0feb76": " x = X(\\omega)", "94ee8ae1c1a99dce47a2f0b59c141f40": "(a - b) \\,", "94ef0c0d91a0ce8ffc10349e1cfb4732": " Z_v = \\frac{1}{1 - e^\\frac{-h \\omega}{2 \\pi k_B T}} ", "94ef0f35481baea7aa25fa6893556853": " | \\psi \\rangle = \\sum_i c_i | \\phi_i \\rangle .", "94ef1e0db8919a83ffd925edf1b07e60": " g_{\\mu \\nu},_0=0 \\,", "94ef234a38ce485a9339fef9fa05c144": "\\lim_{j\\to\\infty} \\sum_k a_{j,k} = \\sum_k \\lim_{j\\to\\infty} a_{j,k}.", "94ef771793439fe006cbf3716286e293": "T^{\\mu\\nu}_g = \\frac{1}{4 \\pi G} \\left [\\partial^\\mu \\phi \\, \\partial^\\nu \\phi \\, - \\frac{1}{2} \\eta^{\\mu\\nu} \\partial_\\lambda \\phi \\, \\partial^\\lambda \\phi \\right] ", "94ef7e199de783ba9625f8bd4348acc6": "A^\\text{WSM-score} _2 = 22.00,\\text{ and }A^\\text{WSM-score}_3 = 22.00. ", "94efcb693408778b6bc2f8f39f7ebaee": "N_n - N_p", "94efd40212b233c2e73433bae47029b8": "\\{a_N\\}", "94efe70b2e10f4e823ac642d62519385": "E^n = M^pL^qt^r = E^{p-q-r}\\,\\!", "94f064a41bee2d272593e638734de5f3": "Z(T,V,N) = e^{- \\beta A} \\,\\;", "94f0d601265e9395f9873926be5aa93e": "\\begin{align}\n y &= E \\left \\{ \\frac{1}{(-\\Delta + 1)_{\\Delta - 1}} \\ \\sum_{r = 1 - \\Delta - \\alpha - \\beta}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r (1)_{r - \\Delta}} (1 - x)^r \\right \\} + \\\\ \n&\\quad + F \\left \\{(1 - x)^{\\Delta} \\ \\sum_{r = 0}^\\infty \\frac{(\\Delta)(\\Delta + \\alpha)_r (\\Delta + \\beta)_r} {(\\Delta + 1)_r (1)_r} \\left (\\ln(1 - x) + \\frac{1}{\\Delta} +\\sum_{k=0}^{r-1} \\left(\\frac{1}{\\Delta + \\alpha + k} + \\frac{1}{\\Delta + \\beta + k} - \\frac{1}{\\Delta + 1 + k} - \\frac{1}{1 + k} \\right) \\right ) (1 - x)^r \\right \\}\n\\end{align}", "94f1295d5f77f0f55c687db9234db844": "z=0\\,\\!", "94f1ef8331ba59379eef9764a02f5d3e": "u''+{p(z) \\over z}u'+{q(z) \\over z^2}u = 0", "94f218c530a7b68e961406fc7fc879fe": "\\scriptstyle \\mathcal{R}_T", "94f24cb8522fef58e4d24f80dd2d90c7": "\\triangle_{n}-\\frac{n-2}{x_{n}}\\frac{\\partial}{\\partial x_{n}}", "94f27ece78522fed3c1250e527152d11": "[SU(3)\\times SU(2)\\times U(1)_Y]/\\mathbb{Z}_6", "94f2c73a7bdf610e8a4bac57d55b5a5e": "\\mathrm{[Fe^{II}(CN)_{6}]^{4-}}+\\mathrm{[Ir^{IV}Cl_{6}]^{2-}}\\rightleftharpoons\\mathrm{[Fe^{III}(CN)_{6}]^{3-}}+\\mathrm{[Ir^{III}Cl_{6}]^{3-}}", "94f2e7121c18c49032f03bc0ba18c9ed": "\\scriptstyle \\dots 0 0 0 1 1 1 \\dots ", "94f337893e5accd22867dfdb004ecf85": "(I_2)", "94f40cd3cba28dfb5e7a8285da547a22": "S = \\{x \\in V\\;:\\;|x| = 1\\}.", "94f429a49bca407cff37fb6cf1c8b613": "\\left[0,\\frac{1}{\\pi}-\\frac{1}{n}\\right] \\ \\text{and} \\ \\left[\\frac{1}{\\pi}+\\frac{1}{n},1\\right]", "94f4555ba262e5bf000946d298c0e43d": "-\\left(k_{ij}\\right)^{-1}\\mu\\phi u_j+\\rho g_i-\\partial_i P=0", "94f47b1a8caeef3d8b23c1c5f7f55dfa": " I(\\cdot) ", "94f4804663f0584e045b7468eb47477a": "\\{U_0,\\ldots,U_k\\}\\,\\!", "94f48a1af8c5c5261c5727ba4dd0b947": "v=\\alpha's + \\beta'n", "94f4983c4f7bd13cab25e50c83c7187f": "u(t,x,y)", "94f584c412f1186bbc4e223faeb900eb": "F_{i_1} \\cap \\dots \\cap F_{i_k}", "94f5e68071c112e19c7425ce424a5e80": "\\operatorname{Var}[Y\\mid X_1]=\\operatorname{E}(\\operatorname{Var}[Y\\mid X_1,X_2]\\mid X_1)+\\operatorname{Var}(\\operatorname{E}[Y\\mid X_1,X_2]\\mid X_1).\\,", "94f61fbdca104ad0ab44342baf52fa97": "c_{kj}", "94f641e14a3039eda46d305547d97d65": " \\mathbf{x}_{t+1}=\\mathbf{x}_t + s E_t, ", "94f64b56206e4b3802e8b6e5b86aa2a9": "{\\mathrm{d} f(z)}/{\\mathrm{d} z}\\neq 0", "94f6d7e04a4d452035300f18b984988c": "300", "94f6e14759ea27571013947ada7ba32a": " L(n,k):", "94f6f49866dad09189d2986238f1ebfa": "3\\omega_{p}", "94f6f7de3e8cf9bf8695a8ede87ec1d3": "U,f(U),f(f(U)),\\dots,f^n(U), \\dots", "94f725363c6fc65a64960bca83bc0826": "V_n={\\pi^\\frac{n}{2}R^n\\over\\Gamma(\\frac{n}{2} + 1)}.", "94f8106286c075ef5788a5c557ca1ce1": "E(2\\omega,z=0)=0", "94f819f7d72f3991359cdf2409adb932": "\\scriptstyle m_A", "94f8462caa69a2c4f048fe30caf0598f": "\\cos^2\\theta", "94f865aa6d5df1290237bf0322994eb1": "G/G_x", "94f8825bb40a03609fd92fa444bed029": " x=\\left( R+r \\right)\\cos \\theta -r\\cos\\left( \\theta+\\alpha \\right) =\\left( R+r \\right)\\cos \\theta -r\\cos\\left( \\frac{R+r}{r}\\theta \\right)", "94f8d490476e8f45c1b2f5fd884876ce": "a = \\frac{2}{3} \\sqrt{-m_a^2 + 2m_b^2 + 2m_c^2} = \\sqrt{2(b^2+c^2)-4m_a^2} = \\sqrt{\\frac{b^2}{2} - c^2 + 2m_b^2} = \\sqrt{\\frac{c^2}{2} - b^2 + 2m_c^2},", "94f91cb4575f47aa1de03d90ddb1402c": "g\\circ f:X\\to Z", "94f9917ff416ef498435f51ba10fb838": "\n\\sum_r {x_{ij}^r = T_{ij} } \n", "94f9a0ba61c954327bcc598293a9fad6": "\\mathrm{laea}_y", "94fae9202ffea920dfa35551c2cc9a8d": "W=\\mu mgd", "94fb1a522511f8f70bb3ff133adb8562": "\\dot{\\mathbf{v}}", "94fbaca872d19f3664a0d4082e632ded": "F_{t,T} > S_t e^{r (T-t)}", "94fbcf531eb6c213bc3aefe0eee4eb46": "\\frac 63 = 2", "94fbd5b0c51998ee40af555d201cc5ed": "\\overline{p}_{\\lambda}=\\frac{\\overline v_{\\lambda}}{\\overline g_{00}} ; \n\\qquad \\overline{F}_{\\lambda}=\\frac{1}{2}\\frac{\\partial\n\\overline{g}_{ij}}{\\partial\nx^{\\lambda}}\\overline{v}^{i}\\overline{v}^{j}.", "94fbecae730b38274ba4e62e38c8c84f": "u_{n}", "94fbf1a20613bd6d65be780df723a53a": "C_8 = 5 \\alpha_1 \\alpha_2 \\hbar \\omega", "94fbfd310d12c6b316bf832275c7eae3": "a_0^j = y_j", "94fca684d551214c2829f370cd118a8f": " S_{ab} = R_{ab} - \\frac{1}{4} \\, R \\, g_{ab}", "94fcd65c3f8b7a94f6ea7101fbe5a6d0": "D_4 \\overline{MR}", "94fddfafdecfe76116a92378b47c5b6c": "z \\; {}_0F_1(;a+1;z) = (a\\vartheta) \\; {}_0F_1(;a;z)", "94fe37cff4bfbd3741a8040e6d27fbd3": " \\quad K (x, y) = \\overline{K (y, x)}\\; , \\qquad K (x, x) > 0\\; ,", "94fe5a3c7870a5ba3a2d18d062f4a573": "m_{T}=m\\gamma", "94fe668c80113341416b96929a8f1024": "[1,4]", "94fe9afd9799b18ff8fe3e21bbfbb3b2": "{2}\\alpha + {2}\\beta =180^\\circ ", "94febd5008b59c1c317e57b3e292a8bf": "(6) \\ Q = \\frac{2\\pi\\lambda \\ell f^2}{n \\nu^2},", "94fef999f10de1790713a7382e592ed9": " \\scriptstyle (n = 5) \\displaystyle \\sum_{k=0}^{n-1} e^{\\frac{2 k^2 \\pi i}{n}} = 1 + e^\\frac{2 \\pi i} {5} + e^\\frac{8 \\pi i} {5} + e^\\frac{18 \\pi i} {5} + e^\\frac{32 \\pi i} {5}", "94ff1df27332d04c5cab8f882a95441c": "t _3[R - \\beta] = t _2 [R - \\beta]", "94ff30dac661dc6fb945525346794ee5": "W = N \\cdot x\\,", "94ff33754a1fcca7c2ee3b784d485689": "\\scriptstyle 2\\pi NQ", "94ff677b5f5851f9a4baf197b6ebefc6": "p(z) = a_0 + a_1z + a_2 z^2 + \\cdots + a_n z^n, \\quad a_n \\ne 0\\, ", "94ff6edad3c1f705dda60f6cfdc62dab": "Z=\\frac{Y}{y}(1-x-y)", "94ff8d015f5af6c530c0b91c37b6a548": "\\frac{e}{N} ", "94ffafbdb16f9986f6a1e6660cbeab10": " \\sqrt{S} \\approx \\begin{cases}\n2 \\cdot 10^n & \\text{if } a < 10, \\\\\n6 \\cdot 10^n & \\text{if } a \\geq 10.\n\\end{cases}", "94ffb57f1820c2869a658035199c11af": "F: \\mathbb{N} \\rightarrow X", "950017c8e2d898be92dba56d92b2e7d5": "\\ \\ R_S \\ll r_E ", "95006654f04639ecaa893942b47c742d": "\\left\\|g\\right\\|_d:=(g,g)^{1/2}_{d} ", "9500ed95d6805cefd94920415cd45c68": "P(X > x) = P(Y < k).", "950101ee4143580dadb235a7e5e216b6": "m = \\left\\lfloor \\frac{m}{r} \\right\\rfloor r + \\left[ \\frac{m}{r} \\right]_1 r = \\left( \\left\\lfloor \\frac{m}{r}\\right\\rfloor + 1 \\right)r - \\left( 1 - \\left[ \\frac{m}{r} \\right]_1 \\right)r", "950106ae4bcc1653a854db4e84a85528": "C = \\emptyset", "95019ee833539727760d0ec686722d62": "\\frac{P \\land Q}{\\therefore P}", "9501a279afc3c5096229fac41eb87d58": "[S]=[S]_0(1-(V_\\max [S]_0 / (K_M + [S]_0)/[S]_0))^{t}\\,", "9501d01ceb91886710782ba6ad649ea3": "H(X_2)", "9502081598911f1ffec27e2d470a7325": "\n\\begin{align}\n \\mathbf{u}_1 &= \\mathbf{a}_1, \n & \\mathbf{e}_1 &= {\\mathbf{u}_1 \\over \\|\\mathbf{u}_1\\|} \\\\\n \\mathbf{u}_2 &= \\mathbf{a}_2-\\mathrm{proj}_{\\mathbf{e}_1}\\,\\mathbf{a}_2, \n & \\mathbf{e}_2 &= {\\mathbf{u}_2 \\over \\|\\mathbf{u}_2\\|} \\\\\n \\mathbf{u}_3 &= \\mathbf{a}_3-\\mathrm{proj}_{\\mathbf{e}_1}\\,\\mathbf{a}_3-\\mathrm{proj}_{\\mathbf{e}_2}\\,\\mathbf{a}_3, \n & \\mathbf{e}_3 &= {\\mathbf{u}_3 \\over \\|\\mathbf{u}_3\\|} \\\\\n & \\vdots &&\\vdots \\\\\n \\mathbf{u}_k &= \\mathbf{a}_k-\\sum_{j=1}^{k-1}\\mathrm{proj}_{\\mathbf{e}_j}\\,\\mathbf{a}_k,\n &\\mathbf{e}_k &= {\\mathbf{u}_k\\over\\|\\mathbf{u}_k\\|}\n\\end{align}\n", "9502c1d2629c2e7e35a153c36895793b": "\\mathbb{F}_q = \\left\\{ 0,1, \\gamma, \\gamma^2, \\ldots ,\\gamma^{n-1}\\right\\} ", "9502d6851ff2b6dc5f94e265fbf6b7bc": "\\tau_t^s\\dot\\gamma(s) = \\dot\\gamma(t)", "950328d13cd7cc862477924968d24e77": " c: E^1 \\to \\{ 1 , \\ldots , k \\}", "950391d05a63931c6808f59ef4f2ab0b": "\\arctan x = \\frac{i}{2} \\ln \\left(\\frac{i + x}{i - x}\\right) \\,", "9504133973a5a23980b74d3e35768505": "\\mathcal{O} = \\{ab^{RC} : a, b \\in \\mathcal{B}, a", "9504176c5ca165628e5037f0ce40edd4": "\\mathfrak{g}_{-1}", "950438b8763943b0f335d9a7acee3b07": " S(\\rho^{12})- S(\\rho^1)\\geq 0 ", "950465266b9bad6f1c5b41c95ab58967": "-\\frac{\\hbar^2}{2m}ik (-A_r + A_l + B_r - B_l) + \\lambda(A_r + A_l) = 0.", "9504759252679b9b09240c1cd8f7385a": "-x^2 \\le x^2 \\sin(\\tfrac{1}{x}) \\le x^2 \\, ", "9504ec5507af082d375f84272dcf597d": "g(S^2)", "950506f1dda59aaa190addbb40c40f04": "\\lambda_n = - n \\left( \\frac{n-1}{2} Q'' + L' \\right).", "95050c6578efe1fc1aa7462643b4da42": "\\varepsilon_1 = \\frac{1}{E}(\\sigma_1-\\nu(\\sigma_2+\\sigma_3))", "950523ec72398cecdf4f77ba1192b7b0": "O(nd^2)", "950598baeaa595a374026e1bb03bc4d8": "\\sum a_PP + \\sum b_PP = \\sum {(a_P + b_P) P}", "9505c0801486ce75a184a042ab14366d": "\\omega_c=\\frac{1}{\\sqrt{LC}}.", "9505e2f03efb12421406110ba8457bfa": " 1 \\to H^0(\\mu_2;K) \\to H^0(\\mbox{Pin}_V;K) \\to H^0(\\mbox{O}_V;K) \\to H^1(\\mu_2;K).\\,", "9505ebd32ac64cd002df813a510dc762": " \\overline{z^2}: ", "950601d428c0d08fd7b83efcd75d5f62": "P_{em} = {speed \\times T}", "9506160bfa879e2a64c18086a6e80ace": "\\frac{\\partial v}{\\partial t} = - g \\frac{\\partial \\eta}{\\partial y}", "9506288a88f9d44592c7c6bd141526c5": "\\ell(y) = \\max(0, 1-t \\cdot y)", "950636e5fc3e851992fe86d9ccc4769a": "\\scriptstyle L^1_{loc}(\\Omega)", "9506ba7ab1c5616cf369c520d571633c": " x(t) = v t \\cos \\theta ", "9506c2eb7c1404854ec47e19b442a353": "k = \\frac{\\sigma_y}{\\sqrt{3}}", "95070514ee4e6bcd22c7013af2eb276c": "\\frac{1}{2 \\mathrm{NA_i}} = N_\\mathrm{w} = (1-m)\\, N, ", "95072d1be6caad2eabbffebe8d4fa563": "n_c(\\mathbf{k}) ", "95072d96d963acc1fc0b5b53e1a45520": "k^{m+1} \\psi^{(m)}(kz) = \\sum_{n=0}^{k-1}\n\\psi^{(m)}\\left(z+\\frac{n}{k}\\right)\\qquad m \\ge 1", "95073b039a5896bc9c509774e5af27e3": "V=\\begin{cases}\n4.500L & L < 0.018\\\\\n1.099 L^{0.45} - 0.099 & L \\ge 0.018\n\\end{cases}\n", "9507658cbe832f2c4ff9a26b839fbf02": " = \\operatorname{tr} \\left( (2 \\eta^{\\mu \\sigma} - \\gamma^\\sigma \\gamma^\\mu ) \\gamma^\\nu \\gamma^\\rho \\right) \\,", "9507755d2c05041bbdfbb620c09c4c3a": "\\text{primpart} (p) =\\frac{p}{\\text{cont} (p)}.", "9507890325ab1157770f4d994934b7e6": " B = \\bigcup_{k\\in\\mathbb{N}} E_{n_k,k}", "9507d30694b5f0f22430b97aea2e3be0": "K_{m,n}", "9507d68ab76cf5dc5bc2e9bc673cc267": "\\hat{X(t|t)}", "950892a7d4cde692fe3faea7713e327a": "\\displaystyle{L(a,b)^*=L(b,a)},\\,\\,\\, L(a,b)c=L(c,b)a", "95089c8711e9230b2590cc5e87848c06": "f(t) = F \\sin(\\omega t + \\phi)", "9508b990c9e552c67fb5d574a270816b": "\\omega(x, n)= \\limsup_{H\\to\\infty} \\omega(x,n,H).", "9508c61289b93db07e0623acc481b6c3": "(\\gamma(t),\\gamma'(t))", "9508cacc6b8e2cc750b6844fae9b1d16": " \\frac{\\pi}{4} \\prod_{p = 1\\,\\text{mod}\\,4} \\Big(1 - \\frac{1}{p^2}\\Big)^{1/2} = 0.764223... ", "9508f797f871052cfe8a901284497ab2": "1/100", "95096d945449f29bb390b9121f9852df": "\\int x r^3\\;dx=\\frac{r^5}{5}", "950970e3dbd6cc9c2cd86cd13ef2e668": " \\limsup_{n \\to \\infty} \\frac{\\# \\big( A \\cap \\{0,1,\\cdots,n-1\\} \\big)}{n} = 1. ", "950991c85f8cfc37f85e14319e00dba5": "a(\\xi,\\zeta) = \\operatorname{sech} (\\xi) e^{i \\zeta /2}", "9509a1cb71036a4da53e57a178fc7aa7": " \\mathbf{T}, \\mathbf{T}' ", "9509d7d87e47f299d4479a133656cfa8": "\\frac{\\partial V}{\\partial x_i} = \\rho_{ji} J_i \\, \\rightleftharpoons \\, J_j = \\sigma_{ji} \\frac{\\partial V}{\\partial x_i} \\,", "950a1e5d437d3e9e20d0d60448a56457": "\\color{SeaGreen}\\text{SeaGreen}", "950a28f62a34ec75118744df0cadb34d": "mk = n", "950a500043e959b70dfad8d01ff3fc81": "\\omega \\in \\left[ 0, 2\\pi \\right)", "950a5716fbfc226a8f9f010908b3b639": "a_{ij} = - {\\overline a}_{ji} ", "950a6a9dd294baa1c368e64f6ac7b90e": "\\hat{Y}(X)", "950a9259ba705e59a693049582d32a00": "X \\to Y \\to Z \\to", "950b3cd4ba72dffbc6ece6224728dfce": " G_2^{\\mathbb C}", "950b7fec96bbb269f1c9578c6d313fc6": "A_{n}=A_{n-1}\\left( 1-\\frac{r_{avg,n}\\Delta t_{p}}{A_{n-1}} \\right)", "950b92704aca313e91171eff99b3d27f": "\\sum_{p: \\, s_i \\rightarrow t_i}{f_p} = r_i \\; \\; \\forall (s_i,t_i) \\in \\Gamma.", "950bb96e3c2ea387b28d1f97edb3daa1": "k_2=K v^a H^{-b}", "950bbbdd426f4319819a9bf04a7918f7": "\\Delta J = L-1, L, L+1; \\Delta \\pi = (-1)^L, ", "950c0097fdbea7f8fb455bd142a7a6c8": " \\vec{\\nabla}\\phi = -\\vec{E} ", "950c633c6fda802593ac504b6b51f311": "\\frac{\\delta^n Z}{\\delta J(x_1) \\cdots \\delta J(x_n)}[0]=i^n Z[0] \\langle\\phi(x_1)\\cdots \\phi(x_n)\\rangle", "950c8e7cca68c424cc14bcb39e9b3332": "T: \\mathcal{B}(\\mathcal{H})\\rightarrow \\mathcal{B}(\\mathcal{K})", "950cab184d031c009665b30f72b200ac": "N(d_2)", "950cdadc9b6d4efdd3fe3a3cbad33c99": " M^{\\times}_M + e^{'} = M^{'}_M ", "950d28c88c8bb6fff872f4f78eb85add": " \\pi_1(X-D) \\to \\pi_1(X-C)\\,", "950d38efee53d239a1606846f0f0f6b9": "f(f^*(x)y)=f(y)\\; ,\\; f(xf_*(y))=f(x) ", "950d9b5f483580f0addbed9790a05fb7": "b\\setminus a", "950e2f3d6405aa114a2b3e350bf9940b": "a \\uparrow (a \\uparrow (b - 1))", "950e6bf9724ab0bb8c93d9d63bf2bf0a": " k_{GT} ", "950e8d269d6749753d951189a399c99b": "\\dot{y}", "950ef9a62796fdc59e1fc28e88cff9b0": "p \\supset q \\cdot q \\supset p", "950f6694945e1e5fa36d6dda630c82fb": "{\\rho}=\\sqrt{x^2 + y^2 + z^2}", "950fc81c3e732e94164250b3645d2a58": "H_1,(H_2)", "950fca35acbad28123513e8eec73e178": " j \\geq v ", "950fceb22a59374d72e68566d612f0fd": " 2\\left(\\frac{\\alpha V}{\\omega}\\right)^2 = \\frac{1}{\\sqrt{1+\\omega^2 \\tau^2 }}-\\frac{1}{1+\\omega^2 \\tau^2}", "950fdfc160d86db4c041cd7ba8b32d32": "{ \\frac{p_1}{q_1}, \\frac{p_2}{q_2}, \\dots, \\frac{p_n}{q_n} }", "95100ae874c61097da91d88acea635c8": "(\\cdot)", "951025a61f8ad9afaecd4d010ff79cc0": "\\log \\log n", "9510ac0577b6dd8740677bf0a1019761": "u_4 = u_3a_3-1=0.05\\cdot20-1=0 \\, ", "9510d9afdf23a9df94f0d2f50d92c102": " 1 \\lor 2 \\lor 3 \\lor 4 \\iff I/2 ", "951163dfa033b12c8e74ee9afbeb3dd0": "\n\\mathrm{CF}(M) = \\{[0;a_1,a_2,a_3,\\dots]: 1 \\leq a_i \\leq M \\}.\\,\n", "95116921d7255d9ad8901166b21d4de8": " \\mathbb{E} ( |Z| | X ) = \\frac2\\pi \\sqrt{1-X^2}. ", "951196ca354c5c72b8356494f97c3b5d": " A ", "95119efc6a8f84791b3b9cbd910d48c8": "w \\approx v", "9511c9134220b2c4b8382c6f84e096ad": "r= {d \\over 2}", "95124e7b2ced5b0af8345fb1a2932a9a": "j \\sqrt \\frac{19}{25}", "9512894c60424b0eb84e61bcc18092c7": "\\color{Emerald}\\text{Emerald}", "95129cbf55a8f30a6e3befd6d12a9c94": " \\delta W -\\lambda \\delta K = 0", "951355455d255e3df636f733bed0875b": "H^* \\simeq G^*", "9513e174e52693ed96390a84b22af321": "v^{(jam)}(t) = \\frac{q_{2}(t) - q_{1}(t)}{\\rho_{2}(t) - \\rho_{1}(t)}\\qquad\\qquad(3)", "95140d6a7298cec4fe464d3196d513a7": "\\scriptstyle P\\left( -{1 \\over z^\\star} \\right) ", "95143f5756bb7858fb7a4653ef2128a1": "\\vec L", "95146a52e1cc6e4f2d866c45f589dd73": "M \\vDash \\phi", "9514773c88ba25f9c24fa62e6a5f6d42": "\\Phi(\\tau)\\,", "95148e6eeeb0b8075df8db0ab6a5ded1": "\\int_3^6 \\int_2^4 \\ 2 \\ dx\\, dy =2\\int_3^6 \\int_2^4 \\ 1 \\ dx\\, dy= 2\\cdot\\mbox{area}(D) = (2 \\cdot 3) \\cdot 2 = 12", "9514e8839b577371e251fb05c8cb50c9": "\n\\rho=e\\int(f_i-f_e)d^3p,\\quad \\vec{j}=e\\int(f_i-f_e)\\vec{v}d^3p,\\quad \\vec{v}_\\alpha = \\frac{\\vec{p}/m_\\alpha}{(1+p^2/(m_\\alpha c)^2)^{1/2}}\n", "9514faab2af9f63b344180682e23c498": "\\zeta(4)=\\pi^4/90,", "95152585d04a6f4b3bb930994334d543": " z\\geq 0 ", "95154c39d60885db882d4666d3a2c2bd": "\\hat{u}_k(t):=\\frac{1}{2\\pi}\\langle u(x,t), e^{i k x} \\rangle", "951562af7545f28222393cb42d3b0fac": "\\Lambda < c ", "951647f3cc569dfc3834085129913337": "\\tau\\;", "951657ed56176ed640b574fd6b08c3c2": "\\scriptstyle a_1,a_2", "9516bbfe6210121f0685ab16ab81a226": "I(B,C,D) = C \\oplus (B \\vee \\neg{D})", "9516c48e490286a92c71d5dc966fd356": "\\{\\gamma_t\\}", "9517a6998a52acc2230fe57566494d28": "\\mathit{N} + 1 = \\mathit{pn}", "9517b20b36d87e6c14aaa3eba32882a1": " \\frac{\\mu l m}{4 r^3}", "9517eb8def9cf9abf84370b405930122": " H_K", "951816ca911a420525f381049b298691": "Z_2 \\cong \\{\\pm 1\\}", "95184fb5ab4463a9a220b5903d350f49": " p(\\cdot; \\theta)", "95188858401a991da713f6e088dd9cf4": "F_{t,T} < S_t", "9518c67fce1f390aca83305a86a61482": " N = n \\, dV ", "9518d43e5e5c8e6d5e42766e3099483c": "R\\langle X\\rangle:=\\bigoplus_{w\\in X^\\ast}R w", "9518de62e07ddfc8b9558e7b496847a7": "C(\\Sigma, S) = - \\frac1{(n - 2) \\sigma_{n}} \\int_{S'} \\frac{\\partial u}{\\partial \\nu}\\,\\mathrm{d}\\sigma',", "9518fadfc7a06addc37420c01810ee18": "j=c_4^3/\\Delta.", "951907edef620294385b879bcb9e2b9d": " q^2/(4\\pi\\epsilon_0 a)", "95196f9cff4b5f68b82cbf33a536620b": "\nr_{\\mathrm{inner}} = \\frac{a^{2}}{r_{s}} \\left( 1 - \\sqrt{1 - \\frac{3r_{s}^{2}}{a^{2}}} \\right) = \\frac{3a^{2}}{r_{\\mathrm{outer}}}\n", "9519b39050c5dc4a1dd49259a45026a2": "(a;q)_\\infty", "9519bc5581eceeb117242d1992b1fae9": "\\mathbf{y}_t", "9519e7e090b067075d5d738f0e696bc2": "L=aD_x^2+bD_y^2", "951a610a4e55b8d3a8b5a123215fc2e5": "E_\\sigma = \\frac{3}{2} \\lambda_s \\sigma \\sin^2(\\theta)", "951a80d6c4f29de22d18f1db74887c01": "\\partial^* E \\subseteq \\operatorname{support} D\\chi_E \\subseteq \\partial E", "951a87fa79db94190bd42afb0df3a2c7": "\\alpha n_{i}dt", "951a8c29f1713be6340ffac9a887f9d2": "|z| \\to \\infty", "951a8e7b3c14536979b1ad2724804845": " \\langle {\\Phi^{*}}\\vert e^{-T}He^{T} \\vert{\\Phi_0}\\rangle = E \\langle {\\Phi^{*}}\\vert {\\Phi_0}\\rangle = 0 ", "951ac8a3153f2dac9a60ef89aa057bc8": "f(A) = P \\begin{bmatrix}\nf(d_1) & \\dots & 0 \\\\\n\\vdots & \\ddots & \\vdots \\\\\n0 & \\dots & f(d_n)\n\\end{bmatrix} P^{-1}, ", "951ae4cd0b8c50c86e1bc2b677a9be44": "f(x,y) = -20\\exp\\left(-0.2\\sqrt{0.5\\left(x^{2}+y^{2}\\right)}\\right)", "951af0d6ffac901f64aa03834dd8a7d3": "P_n^{(\\alpha,\\beta)} (z)\n= \\frac{(-1)^n}{2^n n!} (1-z)^{-\\alpha} (1+z)^{-\\beta}\n\\frac{d^n}{dz^n} \\left\\{ (1-z)^\\alpha (1+z)^\\beta (1 - z^2)^n \\right\\}~. ", "951af877de9dbe9e28d0ced5d0e4aa6b": "R_{xx}(j) = \\sum_n x_n\\,\\overline{x}_{n-j}", "951b41a3c3f97e472b3e24943bd3e4d1": "\\theta_0(X) = X", "951b4c88b3d21b59e2be4efa6cdfc0b8": "c: V\\mapsto \\mathbb{R}^{+}", "951b7cf07964e80e3c01aba77ebfe658": "\n\\beta_\\text{max} = 0.072 \\left(\\frac{1+\\kappa^2}{2}\\right)\\epsilon.\n", "951bb299d8d81f0bbe05c4adf83033be": "\\frac{2 f_H}{n} \\le f_s \\le \\frac{2 f_L}{n - 1}", "951c1650b6694dbd85f9086c5c1e1b52": " \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} = 0 ", "951c41b884b37ae35f609dcb527a34ee": "dA = dX \\; dY.", "951c8613763e380264a329d3c8fb69b9": "F(x) - G(x) = C", "951c8cffe7e393a0058409c6689fb994": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} - \\boldsymbol{l}\\cdot\\boldsymbol{\\sigma} - \\boldsymbol{\\sigma}\\cdot\\boldsymbol{l}^T + \n \\text{tr}(\\boldsymbol{l})~\\boldsymbol{\\sigma}\n", "951cda72777692b9a0dc907a97162f61": "ANC=+2[Ca^{2+}] + [Na^+] - [Cl^-] ", "951cfd6b30a532831798450782c10fcb": " A \\geq 0 \\mbox{ and } B > 0 ", "951d08594a22146b99c6c36d681cdb8c": "A\\vee B = \\{ S\\cup T \\mid S\\in A \\wedge T\\in B \\}.", "951d584b83db19f70163433ef31bc4e0": "W_{t,d,n} \\sim \\textrm{Mult}(\\pi(\\beta_{t,Z_{t,d,n}}))", "951d5c3c19bff5f72931aa6fd2913785": "x = -\\frac{t^2-2t+5}{t^2-2t-3}, y = \\frac{t^2-2t+5}{2t-2}", "951d60d772357c534102cac16d250018": "\n s_i=\n \\begin{cases}\n 4 & \\text{if }i=0;\n \\\\\n s_{i-1}^2-2 & \\text{otherwise.}\n \\end{cases}\n ", "951d7ee6d03ca2e8ac360c6948a25ab3": "(x,y) \\ ", "951e0331c6d1193ff545058e885dc906": "D_* := U \\otimes_{\\mathbf{Z}[\\pi]} C_*({\\tilde X})", "951e65fbae4775a89966a77c7caca81d": " |y\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} ", "951ea7be187e56d989cc705383914a1e": "\\frac{\\partial\\mathbf{F}} {\\partial\\mathbf{X}}=\n\\begin{bmatrix}\n\\frac{\\partial\\mathbf{F}}{\\partial X_{1,1}} & \\cdots & \\frac{\\partial \\mathbf{F}}{\\partial X_{n,1}}\\\\\n\\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial\\mathbf{F}}{\\partial X_{1,m}} & \\cdots & \\frac{\\partial \\mathbf{F}}{\\partial X_{n,m}}\\\\\n\\end{bmatrix},\n", "951ebee7dd85abcb39094b64fc4284e9": "\\mu(n) = 0", "951ebf3d84016150c6241fb72a055f79": "1-x", "951f13da68a8e870355d914397ad4de6": " \\int_a^b \\omega(x)\\,f(x)\\,dx ", "951f4a7f74e1088a49b0875d97d22b79": " \\overbrace{\\smile \\smile \\smile\\smile}^{\\mathrm{Foot 5}} | \\overbrace{\\underbrace{-\\smile}_{\\mathrm{Brahma}}}^{\\mathrm{Foot VI}} | \\overbrace{\\smile\\smile-\\smile}^{\\mathrm{Foot 7}} | \\overbrace{-\\smile\\smile\\smile}^{\\mathrm{Foot 8}}", "951f5dbef7a2207020f071427596ecb6": "(\\nabla_{\\mathbf v}\\alpha)({\\mathbf u})=\\nabla_{\\mathbf v}(\\alpha({\\mathbf u}))-\\alpha(\\nabla_{\\mathbf v}{\\mathbf u}).", "951f827e2f7fa0a660a80c807d164d50": "X:=\\bigcap_{n=1}^{\\infty} X_n", "951f855975e2e841faa318b9c7681995": "S = \\frac {1000V_c}{\\pi D} \\, ", "951f9980940d1c7bd1c395fe141b7472": "\\frac{m_0}{m_1}", "951f9ba266ed386964ae9a7499248395": "bs^{-1}(x'')", "951fbd7facd3bad7608ca37011b3e280": "\\mathbb{U}", "951fc4cbb38f1dd27f2a5f67d78a27f6": " \\lambda_1 = 3, \\quad \\lambda_2 = 2, \\quad \\lambda_3= 1. ", "951ffe79901171d1af572ff9bb976175": "\n S = \\cfrac{M_{\\mathrm{max}}}{\\sigma_{\\mathrm{max}}} = \\cfrac{I}{c}\n ", "95202d0a82c5945113c124fbc56daf7f": "\\scriptstyle \\mathbb{R}^n ", "95207b1c8e97887651f379f06aac67a5": "[x_1, x_2] \\cdot [y_1, y_2] = [x_1 \\cdot y_1, x_2 \\cdot y_2],\\text{ if }x_1, y_1 \\geq 0.", "9520e205b3bea9c70b6af1e910522d71": "\\mathbf{v}(t) = \\frac{d\\mathbf{r}(t)}{dt} .", "95217d43c717bee7001fd74de9d63207": "\\sqrt{p}", "95220da8fc3c383dd7caacdcbfdf7710": "s_{iw} = 0", "9522885e1c3a51b7041f07084405d1e3": "\\displaystyle \\hat{f}(\\xi_x, \\xi_y)=", "9522c9a8344de2da29f62ae71e2a951d": "\\hbar/i", "9522d91069f03df9bf5dd2ef80c097c7": "{\\mathcal D}={\\mathcal F}[\\partial_x,\\partial_y]", "95239bd5b68aed104bb3da3eac524ac5": "2^q", "95243fbaf7e517e6aa4334b5358a92bb": "|\\Phi^-\\rangle_{AC}", "9524d53a6947bba9e46d1a9c0b58f759": "\\tau = \\mu \\left.\\frac{\\partial u}{\\partial y}\\right|_{y = 0}", "9524de1c54aca0180c2614da0695e9df": "\\hat{V}_{\\textit{ps}} = V^{\\textrm{loc}}_{\\textit{ps}} + \\sum_{i,j} D_{ij} | p_i \\rangle \\langle p_j |", "95256499ad3fc77357c34853c766d8c8": "y_{4i}=\\left [\\mathbf{P}^{2i}(\\mathbf{C}) \\right ] \\in \\Omega_{4i}^{\\text{SO}}", "9525fd6539d6714c01ddba544339fcdd": "Z=\\frac{p \\underline{V}}{R T}", "952621bde53905a97faf078d2716aaae": "\\sigma_{xy}\n=-\\frac{\\partial^2\\Phi_{xy}}{\\partial z \\partial z}\n -\\frac{\\partial^2\\Phi_{zz}}{\\partial x \\partial y}\n +\\frac{\\partial^2\\Phi_{yz}}{\\partial x \\partial z}\n +\\frac{\\partial^2\\Phi_{zx}}{\\partial y \\partial z}", "95263014fed3ae2bae87c85b202c487a": "u \\in M_{\\lambda}", "95263dac99ed4f47cf37ea4e31a4d679": "1280 Cx + 2048", "95266b7cefcf002fbe7b70b1c75d7f08": "\\frac{\\sigma_b dS'}{|\\bold{r}-\\bold{r}'|}+ \\frac{1}{4\\pi\\epsilon_0} \\iiint\\frac{\\rho_b}{|\\bold{r}-\\bold{r}'|} d^3\\bold{r'}", "9526995aceeb4065dfc341684b25ca2d": " \\mathbf{f} = - {4\\pi G \\over {3 c^2} }\\rho(r) \\mathbf{r} ", "95273e94057881239105da76acc7a1d5": "\\tan \\varphi =\\frac{F_{\\mathrm{Cfgl}}}{F_{\\mathrm{g}}} = \\frac { \\mathit{\\Omega}^2 r }{g} \\ ,", "9527ea2f66bbc1e7a32b60aa52547569": " \\mu = \\frac{m_e M}{m_e+M} \\,\\!", "95280c3f1a612ded2cf0616ee50a11ff": " \\Sigma_i(L_{p_i}(Dw))_{x_i} = 0", "95281658aceeeeb879664daae1533539": "{\\mathcal I}_\\theta", "95284c7e1146529a602ef241972069d1": "\n\\begin{align}\n P & = \\frac{5}{(1 + \\frac{4.5% + 50\\mathrm{bp}}{2})^{(2 \\times 1)}} \n + \\frac{5}{(1 + \\frac{4.7% + 50\\mathrm{bp}}{2})^{(2 \\times 2)}} \n + \\frac{105}{(1 + \\frac{5.0% + 50\\mathrm{bp}}{2})^{(2 \\times 3)}} \\\\\n & = 98.49861 \\\\\n\\end{align}\n", "95285d4270b697f3ba20579903e7e364": "Y = \\mathcal{F}", "9528c2feb396e5571106cf23ba5698e9": "7 \\times 6", "9529042961adc5fff5e49b9c963bb0c1": "h_{\\bar{a}}(\\bar{x}) = \\left(\\Big( \\sum_{i=0}^{\\lceil k/2 \\rceil} (x_{2i} + a_{2i}) \\cdot (x_{2i+1} + a_{2i+1}) \\Big) \\bmod ~ 2^{2w} \\right) \\,\\, \\mathrm{div}\\,\\, 2^{2w-M}", "95291b89fd062026373ece392d9908d2": "\\tilde\\gamma\\colon [0,1] \\to P", "952962dd06a0cdb8e55611d41053ec32": "\\nabla^2 f(x)", "952965720f287d12fe8991c5f8da06e2": "\\text{If } s(t) \\rightarrow |\\chi(\\tau,f)| \\text{ then }s(t) \\exp[j\\pi kt^2] {\\rightarrow} |\\chi(\\tau,f+kt)| \\, ", "952988da97fbd8f2ea65990c03eac425": " a", "9529b6644c37afe079edc9955be2d207": " Q_4 ", "9529b7b145b9af0aeb36ec5d07bdab9f": "\\pi _{1}^{2}-p^{2} =-\\left( \\varepsilon _{1}-\\mathcal{A}_{1}\\right)^{2}=-\\varepsilon _{1}^{2}+2\\varepsilon _{w}A-A^{2}, ", "9529d07e81ba45ca0777b0dc370f090e": "f \\neq f'", "9529eeefde889751f6e819a78267cb2b": "x \\mapsto g(x)", "9529fd6133c6778744bb5564a7cfadb7": "h(\\cdot , \\cdot)", "952a0bd8a7a600e07144589fd8445941": "\\pi_i(X) \\equiv 0~, \\quad 1\\leq i\\leq n ,", "952a2b1b8c4e345b46213473525e77a9": "\\mu Z.\\phi \\vee \\langle a \\rangle Z", "952a3cdeae9cb6ec3ec7afa461f72f2d": " A_i = G \\left( \\frac {R_{C2}} {R_{C2}+ R_{L}} \\right) \\ . ", "952a4da118da4cd38bbcc45169f08fe4": "b_i^{-1}", "952a79ce5da23ce1b7e0b7e05e26dedc": "\\Delta\\vec{F} = \\begin{bmatrix}F_1(\\vec{p})-F_1(\\vec{q})\\\\\\vdots\\\\F_m(\\vec{p})-F_m(\\vec{q})\\end{bmatrix}\\!", "952b270f9de758ea30bb34ce8b651884": "\n P^{\\text{old}}(i|s_j) = \n \\frac\n {\\exp\n \\left(\n -\\frac{1}{2\\sigma^{\\text{old}2}} \\lVert s_j - T(m_i, \\theta^{\\text{old}})\\rVert^2 \n \\right) }\n {\\sum_{k=1}^{M} \\exp\n \\left(\n -\\frac{1}{2\\sigma^{\\text{old}2}} \\lVert s_j - T(m_k, \\theta^{\\text{old}})\\rVert^2 \n \\right) + (2\\pi \\sigma^2)^\\frac{D}{2} \\frac{w}{1-w} \\frac{M}{N}}\n", "952b853ac9b18ad8554254b659b3430e": "\nZ_\\text{upper}(x)=Z(x)+\\frac{1}{2}T(x)\n", "952bb19c5446500fe56a5dc761a4da1f": "\\vec E", "952c4dcb6aeb7ffb721d4e1ceb95c378": " F_k = \\rho \\int_A \\sum_i (u_k v_i n_i - v_i u_i n_k) \\, \\mathrm{d} S. ", "952cab149c51b419b1684e3eda40ee90": "\\frac{\\mathrm{d}}{\\mathrm{d}\\alpha} \\varphi(\\alpha)=\\int_0^1\\frac{\\partial}{\\partial\\alpha}\\left(\\frac{\\alpha}{x^2+\\alpha^2}\\right)\\;\\mathrm{d}x=\\int_0^1\\frac{x^2-\\alpha^2}{(x^2+\\alpha^2)^2} \\mathrm{d}x=-\\frac{x}{x^2+\\alpha^2}\\bigg|_0^1=-\\frac{1}{1+\\alpha^2},", "952cd3807d9c9d8ce6644d211085afaf": "R-R_{0}=-d \\cdot \\cos\\left(l\\right)", "952cf2fa4599dab7e3b7a28add1f07df": "\nds^2 = \\left( 1 - \\frac{a}{r^4} \\right) ^{-1} dr^2 + \\frac{r^2}{4} \\left( 1 - \\frac{a}{r^4} \\right) {\\sigma_3}^2 + \\frac{r^2}{4} (\\sigma_1^2 + \\sigma_2^2).\n", "952dcac5ed759336e40e81f458c8c916": "\n\\begin{align}\nf(x) & = 2 - 2x \\text{ for } 0 \\le x < 1 \\\\[6pt]\nF(x) & = 2x - x^2 \\text{ for } 0 \\le x < 1 \\\\[6pt]\nE(X) & = \\frac{1}{3} \\\\[6pt]\n\\operatorname{Var}(X) & = \\frac{1}{18}\n\\end{align}\n", "952df011d537424a6d6d66cc5347d243": "\\mathrm{d}V = \\rho\\,\\mathrm{d}\\rho\\,\\mathrm{d}\\varphi\\,\\mathrm{d}z.", "952dfbc0f9dd5fa530cd21933bf12e43": "\\mu=10", "952e1c706c0e6a5ce65a00d3e7c52d94": "\\begin{align}\n \\|P\\|_2\n &= \\sqrt{\\max\\left\\{\\lambda_{\\mbox{max}}\\left[Z P^*(z)\\cdot Z P(z)\\right] : z\\in\\mathbb{C}\\ \\land\\ |z| = 1\\right\\}} \\\\\n &= \\max\\left\\{\\|Z P(z)\\|_2 : z\\in\\mathbb{C}\\ \\land\\ |z| = 1\\right\\} \\\\\n \\|P^{-1}\\|_2^{-1}\n &= \\sqrt{\\min\\left\\{\\lambda_{\\mbox{min}}\\left[Z P^*(z)\\cdot Z P(z)\\right] : z\\in\\mathbb{C}\\ \\land\\ |z| = 1\\right\\}}\n\\end{align}", "952e4a737e3428a1aa4d799edbf5b62c": " \\frac{4}{\\Omega^2}\\frac{d^2u}{dt^2} + [a_u-2q_u\\cos (\\Omega t) ]u=0 \\qquad\\qquad (3) \\!", "952e5f2c3fbf518e11927f587b087547": "\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} + N~\\cfrac{\\partial^2 w}{\\partial x^2} + m~\\frac{\\partial^2 w}{\\partial t^2} - \\left(J+\\cfrac{mEI}{\\kappa AG}\\right)~\\cfrac{\\partial^4 w}{\\partial x^2 \\partial t^2} + \\cfrac{mJ}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial t^4} = q + \\cfrac{J}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial t^2} - \\cfrac{EI}{\\kappa A G}~\\frac{\\partial^2 q}{\\partial x^2}\n", "952ea45d8ece3e044c5806d8503ebd79": "\\,f(u,v) = - f(v,u)", "952eb6c3b46057be412ea88ea026b7fd": "F_{2p+1}", "952eea0e57b525cf8d5512309d37adab": "\\ t_i ", "952f124b39355659bd0d11cdbaf82878": "\\displaystyle{\\nu(f(z))={f_z\\over \\overline{f_z}} \\, {\\mu^* - \\mu\\over 1-\\overline{\\mu} \\mu^*},}", "952f73b0b32698f894b3a2764a021254": "\\vec{r}_i", "952f7475b7e2396481f02f7143971b86": " \\frac{1\\, \\mathrm{s}}{1023 \\times 10^3} = 977.5 \\, \\mathrm{ns} \\approx 1000 \\, \\mathrm{ns} \\ ", "953011c0b82dd8f9df6c2d03ffa50bb0": " \\eta = \\sqrt{ {5\\over 2} + {\\sqrt{5}\\over 2} }\n ", "95307a01acb6215b0e61138ea4df3b88": "t_{ijl}", "95307abc76d67a2f13d4c6b9aa63071c": "\\varinjlim \\mathrm{Hom}_A(X', Y/Y')", "95309d8605513220e5ae24ca9a023bf6": "C_{e_2} = (I - K_2A)C_{e_1}.", "9530b39882d486f8935a69346ae28819": "c_n = \\sum_{k=0}^n \\frac{(-1)^k}{\\sqrt{k+1}} \\cdot \\frac{(-1)^{n-k}}{\\sqrt{n-k+1}} = (-1)^n \\sum_{k=0}^n \\frac{1}{\\sqrt{(k+1)(n-k+1)}}", "9530b422594ad6c18d34294f989b1d71": " C(u)= \\sum_{i=0}^{i=n} R_{i,p}(u)P_{i}", "9530dfc47558b3264a9b60af67d810df": "{\\mathrm{d}R\\over \\mathrm{d}\\theta}={2v^2\\over g} \\cos(2\\theta)=0", "9530fc111e87d1cd5e1d8f6b7bb2f762": "(\\lambda,\\alpha) \\in \\mathbb{C}\\times\\mathbb{C}^*", "95312ad1919c84b9a05318a81b8a138b": "\n\\frac{dK}{dt}= I - \\mu K, \\quad \\frac{dL}{dt} = \\left(\\bar \\lambda \\frac{I}{K}- \\mu\\right) L, \\quad \n\\frac{dP}{dt} = \\left(\\bar \\varepsilon \\frac{I}{K}- \\mu\\right) P.\n", "9531d6952a03460a51e4042eb8ce7dee": " \\boldsymbol{ \\nabla \\times B} = \\mu_0 \\boldsymbol {J_D} \\ , ", "953279f86ae4c1ab1a348e1f9254e5c3": "\\boldsymbol{U}", "9532833c1ceaa1a7c4db162ab9f237f7": "2^{6 \\times 8} + 2^{2 \\times 8} = 281474976776192", "953290db27871e7fa31bff1663de8e30": " R(\\theta,\\delta) = {\\mathbb E}_\\theta L\\big(\\theta,\\delta(X) \\big)= \\int_\\mathcal{X} L\\big( \\theta,\\delta(X) \\big) \\, dP_\\theta(X)", "9532dbc6ba11c5678a9c8d7d91649c42": "\\delta e = - e e_\\alpha^I \\delta e_I^\\alpha", "95330677dbcd9e96be0f2e36e141263f": "\\theta = \\arcsin |\\frac {2 \\phi} {\\pi}|", "953330e9cc8e5a6175a5a25e8c13567a": "\\rightarrow^2", "9533585f3f12aa634134f9a0435e4271": "0 \\le s_1 \\le s_2 \\le \\cdots \\le s_n", "9533ebe6a5cae5775a965655a7a069db": "H_{a,m}=H_{a-1,m}+\\frac{1}{a^m}", "95343e564a1f883cc4b2ecf58cfbc3a7": " \\mathbb{M} ", "953442d37b8b0dbf595840f0ddbdf85f": "\\sum_{k=1}^M \\lambda_k \\nabla g_k (p) = \\nabla f(p) \\quad \\Rightarrow \\quad \\nabla f(p) - \\sum_{k=1}^M {\\lambda_k \\nabla g_k (p)} = 0.", "953448fd60559bff6d24f7bbd52bea53": " F = FV[(\\lambda N.S)\\ L] ", "95344b96652e0509f0e3050211d66b70": "\\mathop{\\mathrm{im}}(L) \\cong V / \\ker(L)\\text{.}", "9534748fbfce9f7e54c0b3b4311783c7": "\\mathbf{X}\\!", "95351ff98ec58a9f1c83c5a4a967467d": "\\int_{[a, b]} K_X(s,t) f_i(s) \\,ds =\\beta_i f_i(t)", "953524a7eff0240eb713ffbe1043568f": " \\sin y = x \\, ", "953525b53c82d1886904573396180396": "\\sum_{a \\in B} w(a).", "953526b30af44f06c8399281ef2a273c": "\n \\begin{array}{ccc}\n t & \\approx & \\frac{0.693147}{r} \\\\ & & \\\\\n & = & \\frac{0.693147}{R%} \\\\ & & \\\\\n & = & \\frac{0.693147}{R \\frac{1}{100}} \\\\ & & \\\\\n & = & \\frac{0.693147 \\cdot 100}{R} \\\\ & & \\\\\n & = & \\frac{69.3147}{R} \\\\ & & \\\\\n & \\approx & \\frac{70}{R}\n \\end{array}\n", "9535908495a3ddac174e7398dee49341": "|\\Im(\\Gamma(z\\!+\\!1))| < \\pi ", "9535a3fcc8b71f61c3ddeac200dfc3ec": " \\sgn(x) = \\frac{x}{|x|}", "9535d053498d3a139924a84c64035aac": "R\\left[ n,k \\right]=E\\left\\{ Y\\left[ n \\right]Y\\left[ k \\right] \\right\\}=\\frac{1}{N}\\sum\\limits_{p=0}^{N-1}{f\\left[ \\left( n-p \\right)\\bmod N \\right]}f\\left[ \\left( k-p \\right)\\bmod N \\right]", "9535d8f1e81f36c25929f42a403c2b97": "v _{0}", "9535e3243309f855c5ccc9856ee6179e": " \\log_i(x) = {{2 \\ln x } \\over i\\pi}.", "953629f08cb48444be7d9b0255466eb2": "m = \\frac{1}{\\sigma\\sqrt{\\tau}}\\ln\\left(\\frac{F}{K}\\right) ", "95362be5bdef5ca56b07f97622d94804": "\\scriptstyle\\overline{\\mathrm{W}}(\\mu,\\cdot)", "9536325d2f29852e6924a01765a7d8c2": "\\{0,1\\}^{<\\omega}\\,", "9536899eb963d8b2b930b0cff4e2fc1a": " \\frac{Impeller\\ Tip\\ Diameter}{Eye\\ Tip\\ Diameter} ", "9536ece8b75cca0aeecc2ed66a6e8ec2": "\\ z^{\\tau}", "953771ffd5e19ab6e5c031f65d94a620": "\n\\begin{align}\n\\Delta \\bar{e}\\ = &\\frac {1}{V_0}\\ \\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ f_r\\ + \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ f_t\\right)\\frac{r^2}{\\sqrt{\\mu p}}du\\ = \\\\\n&\\frac {1}{\\mu}\\ \\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ f_r\\ + \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ f_t\\right) r^2 du \\end{align}\n", "9538b3e3986bfd12bdc18429df5bb2a7": "p_\\text{T}", "95391e70ee225ce7aa4dbcf60c831554": " d(x) ", "953922ff7d5d84c727b032965dc35c24": "\\lim_{n\\to\\infty}a_n = \\infty.", "9539a6cfbb357cf180ab99505e916d2d": "E^i_a = q_{ab} E^b_i", "9539d580e69484655c165842a7cb1d9f": "200-x", "9539d9d99374727d00037d269018017a": "\\phi=\\theta", "9539f818ffdd0bc87538a8bdf55f8bc0": "\\sum_i \\operatorname{Tr} \\; (\\rho \\otimes \\omega)(F_i \\otimes \\Psi_i ^*(O)) = \\operatorname{Tr} \\; \\rho \\cdot O.", "953a29636d290f6714a86cd45c669893": "Q_2(z,v) = \\frac{1}{2} Q(z,+1,v) - \\frac{1}{2} Q(z,-1,v) =\n\\frac{1}{2}\\left(\\frac{1}{1-z}\\right)^v\n-\\frac{1}{2}\\left(\\frac{1}{1+z}\\right)^{-v}.", "953a390b2f3d31e35589efdc7940ecc9": "\\Phi(1)=1", "953a464cac9fa0ba9ff00f2afc88adbb": "\\psi_{3} = 3x^{4} + 6Ax^{2} + 12Bx - A^{2}", "953a657b7e6f6cf0afa313b36fc8dc3b": "\\frac {\\sigma_B} {\\sigma_D}", "953adafe69af6b398bb690123b945e43": " apk^G ", "953b25b9839bf6fd8e8cb6beb7432e93": " R_{in} = r_E = \\begin{matrix} \\frac {V_T} {I_E} \\end{matrix} ", "953b286702081ad1b5d57efb9afcf735": "\\mathbf{\\hat{e}}", "953b328d4440db89a54905d75d8e8aa7": "{\\pi} = 20 \\arctan\\frac{1}{7} + 8 \\arctan\\frac{3}{79} ", "953b3d67e045b8fe47141e1b4d75a102": "\\dot{\\mathbf{x}} = F(\\mathbf{x})", "953bd2c576f82ae2fb1d36529d9380f1": "\\sum_{n=1}^{\\infty} \\frac{1}{F_n} = \\frac{1}{1} + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{8} + \\frac{1}{13} + \\cdots", "953bd4a2b1640a49428ea2ae8a1b114a": "P_X(x)", "953bf25f99b695b93a0701106dee23d9": " D(f) = ", "953c0eeee493ace5cd86639f79d8cf2c": "\\mathfrak{P}^{125}", "953c56d9896558fcc1b844004f45cad4": " i \\ ", "953ca02fe748bc497835cf8105fd8c09": "\\int_G f\\,d\\mu", "953cf27bedf5f667c71aa8a79e4f405c": "\\frac{f(t)}{m(t)} = V_{exh}(t) \\frac{\\dot{m}(t)}{m(t)}\\,", "953d8073fccccd04815ec3d623deb4de": "\\mbox{EOT = GAST}-\\alpha-\\mbox{UT}+\\mbox{Offset}", "953da5cd9f3ea67b45dabcc1d849da0c": " p_j = \\frac{\\partial \\mathcal{L} }{\\partial \\dot{q}_j}\\,.", "953dee393aaf4c9b778f54b197ffafed": "f(x) \\propto x^{\\alpha}e^{\\beta x}.", "953df6967f924ea2eede7883c08bd742": "\\hat{\\alpha}=(\\text{Inverse digamma })[\\ln \\hat{G}_X - \\ln \\hat{G}_{(1-X)} + \\psi(\\hat{\\beta})] ", "953e9569ff1c5aa05f5585a286856442": "T_{0}>0", "953f2329e8090d4c12bb4cba5eef48dc": " \\lambda f.\\operatorname{let} x\\ x = f\\ (x\\ x) \\operatorname{in} x\\ x ", "953f408a8e0567523bf96fb3a9b9502c": "\\limsup_{x\\to a} f(x) = \\inf \\{ \\sup \\{ f(x) : x \\in E \\cap U - \\{a\\} \\} : U\\ \\mathrm{open}, a \\in U, E \\cap U - \\{a\\} \\neq \\emptyset \\}", "953f4886214f1d8b64205c76295cd93f": "U \\subseteq \\mathbb P (V)", "953f4c2df9cd49c8a850200209087a9d": " \\forall a \\forall b \\forall c\\; a \\wedge (b \\wedge c) = (a \\wedge b) \\wedge c ", "953f6beb5a6ea3ed0a3cad330d813b49": "e\\epsilon(t)\\langle \\Psi_{1} |x|\\Psi_{0}\\rangle = i\\hbar c_{1}'(t)", "953ffaa1105fb5cb79a52064b00d02e1": " EPQ = Q = \\sqrt {\\frac {2KD}{F(1-x)}} ", "95401743e5f34aec1453f2707f9b86fd": "p_{0} \\in \\mathcal{M}_{0}", "9540187bfb450403481bf95ebdedfae0": "\\{x_1,\\ldots,x_m\\}\\in\\mathfrak X^m", "95402cf856628c26215c2cc70c83ffd0": " = \\frac {\\beta I_E} { \\beta +1 } \\frac {1}{\\beta} \\frac {\\beta_0} { V_A} \\frac {1} {(\\beta +1) }\n\n=I_{out} \\frac {1} {1+V_{CB} / V_A} \\frac {1} { V_A} \\frac {1} {(\\beta +1) }\n\n = \\frac {1} { ( \\beta +1 ) r_0} \\ ,", "9540352f444326933f6c7cf29d7d1256": "u(x) = \\mathbf{E}^{x}[ f(X_t) ].", "95404068414f74ed116023b39e295db3": "G=\\begin{bmatrix}A & B\\\\B^\\mathrm{T} & D\\end{bmatrix},", "95406191d502b8290746b19dde099a54": "4^5 - 424 = 600", "9540af9545002d8d2c46bf7dab607d3b": "k'\\Omega_0", "9540bad706b5f9f5d2263bf8649cd360": "(\\phi \\or \\neg\\phi)", "9540c088c6d770516a9e74029d107e3b": " \\{ 1 , \\{ \\mathbf{e}_1,\\mathbf{e}_2,\\mathbf{e}_3 \\} , \\{ \\mathbf{e}_{23},\\mathbf{e}_{31},\\mathbf{e}_{12} \\} , \\mathbf{e}_{123} \\}, ", "9540d79af2058d5b383c646b612a2c01": "\\displaystyle \\frac{1}{a + 2 \\pi i \\xi}", "9540dd08ecb04c19c52b0186ecdb178e": "a=-\\zeta_n\\sqrt{1-\\zeta_n^2}\\sqrt{1-x_m^2}\\sqrt{1-x_m^2/\\xi^2}", "95418b661ada6108eb5df1ac902d8f8b": "\\exists n\\;P(n)", "954199230a1489d77ad9e4822934c7dc": "\\Delta E = -4\\beta \\sin \\frac{\\pi}{2(n+1)}", "9541cd4248d8da925aa43eea1eb4e1ad": "-\\frac{(b-a)^5}{6480}\\,f^{(4)}(\\xi)", "9541cd8df048d103e9068c6bd33c3fa4": "\\Delta \\Pi = -\\Delta V + \\frac{\\partial V}{\\partial S}\\,\\Delta S", "9541d8108080b943aa0843e5ddb6ee1b": "\n\\frac{1}{\\xi}\\frac{d\\xi}{dt} = \\ln \\left(\\frac {L_0}{L} \\frac{P}{P_0}\\right) \\frac{d \\alpha}{dt}, \\quad \\alpha = \\frac{1 - \\bar{\\lambda}}{\\bar{\\varepsilon} - \\bar{\\lambda}} \n", "9541d82a7b1078fdc412a3f7abf45874": "c = c(\\varepsilon) > 0,", "95421d67db914ed49ff573a0a3960281": " v_y ", "954221b8b562e0f9ee66c3adf8c4fe84": "\\scriptstyle v(\\boldsymbol{x})", "9542231237d4c63299d53fe5801aa025": "A \\vee B \\equiv ((A \\rightarrow B) \\rightarrow B) \\wedge ((B \\rightarrow A) \\rightarrow A)", "954224715b708fde40cb00dfbf637c3b": " \\mathbf{X}_{n \\times p} = \\left[\\mathbf{x}_1,...,\\mathbf{x}_n\\right]^T ", "9542306af112360ff96365ba0ae85863": "\\left(\\partial_t + \\gamma \\right)^2", "95423ba65b98cd685a807d3215aed625": "\\displaystyle{Tf(x)=\\widehat{\\widehat f}(-x)}", "95426410b2cff25909aa940a7a281041": "I=\\left(\\frac{2rB_H}{\\mu_0 n}\\right)\\tan\\theta\\,", "954280824a63a00c9a3f3ab59bf9c38b": "\\sigma_y\n= \\frac{\\partial^2\\Phi_{xx}}{\\partial z \\partial z}\n +\\frac{\\partial^2\\Phi_{zz}}{\\partial x \\partial x}\n-2\\frac{\\partial^2\\Phi_{zx}}{\\partial z \\partial x}", "954285aa234e4724f49e61f4f95f0743": "\n\\begin{align}\n\\sqrt{114} & = \\frac{\\sqrt{114}+0}{1} = 10+\\frac{\\sqrt{114}-10}{1} = 10+\\frac{(\\sqrt{114}-10)(\\sqrt{114}+10)}{\\sqrt{114}+10} \\\\\n& = 10+\\frac{114-100}{\\sqrt{114}+10} = 10+\\frac{1}{\\frac{\\sqrt{114}+10}{14}}.\n\\end{align}\n", "954322810f486ac24f0d70218627d2a0": "\\vec{d}", "954367a9540b27997e400d2cb0fa42b6": "f(x,y) = e^x\\log{(1+y)}", "9544304b8da0d2cdc381df3955a934a2": "\\begin{align}\n{\\mathbf{A=LL^T}} & =\n\\begin{pmatrix} L_{11} & 0 & 0 \\\\\n L_{21} & L_{22} & 0 \\\\\n L_{31} & L_{32} & L_{33}\\\\\n\\end{pmatrix}\n\\begin{pmatrix} L_{11} & L_{21} & L_{31} \\\\\n 0 & L_{22} & L_{32} \\\\\n 0 & 0 & L_{33}\n\\end{pmatrix} \\\\\n& =\n\\begin{pmatrix} L_{11}^2 & &(\\text{symmetric}) \\\\\n L_{21}L_{11} & L_{21}^2 + L_{22}^2& \\\\\n L_{31}L_{11} & L_{31}L_{21}+L_{32}L_{22} & L_{31}^2 + L_{32}^2+L_{33}^2\n\\end{pmatrix}\n\\end{align}", "95449bf35634c3cba21e4248525277e9": "N^m \\doteq \\frac{OA + BA + (OA - 3BA) \\times{m}}{OA +BA - 2BA\\times{m}}", "9544b3f7b8efeea7115246105b6397df": " \\Delta H^\\circ_{form} = H(T)compound - \\sum \\left \\{ H(T)elements \\right \\} ", "9544e74684609192af838f629cc7aed8": "0\\to\\mathbf{R}\\to H(V)\\to V\\to 0.", "95451594e8b5b865dfe3a45a840155ca": "\\Omega'", "954560774669981f59eb7517403fdb5c": "\\scriptstyle v_\\mathrm r", "95458f5e7f6eede646909131e539d881": " B_{k}", "9545d6e6b798ee5c94b32e752cca956e": "U=\\{ z: |z-z_{0}| < r\\}", "954605b173a7521f553046b12b577840": "Rating=\\frac{Load\\ Waterline\\ Length+\\sqrt{Sail\\ Area}}2", "9546ac8bc70b7de9d8d945866604958d": "\n u(x,y,z,t) = 0 ~,~~ v(x,y,z,t) = \\hat{v}(x,z,t) ~,~~ w(x,y,z,t) = 0 \\,.\n ", "9546fbb87c6a489e0e22d53899986bda": "\\,C\\,", "95470b2ca9cf7afa23e6a915a02de180": "[x_1, x_2] \\cdot [y_1, y_2] = [\\min(x_1 y_1,x_1 y_2,x_2 y_1,x_2 y_2), \\max(x_1 y_1,x_1 y_2,x_2 y_1,x_2 y_2)]", "9547190b145cbb7dd198eabaeb3e3bd3": "\\widehat{M\\ N}=01\\widehat{M}\\widehat{N}", "95473e3a07d03a534a6b7001f76c8c58": "|V|^{m+1} = |V|^m + \\Delta |V|\\,", "9547f3a905183a0116952ce7fdb7d56c": "V\\times\\varepsilon_{ij}", "954806fc61e5d0f9045cf934a126e6e3": "\\left.\\right. A^2_0", "9548304356870380c590c9742af127c9": "f ' (c) = \\frac{f(b) - f(a)}{b - a}.", "95488ce31729b1228bab397ef0915a5c": "\\Delta E_\\text{kin} = \\frac{1}{2} \\Delta m\\, v_{2}^{2}-\\frac{1}{2} \\Delta m\\, v_{1}^{2}.", "95489c6139e663db028adae0c69a3eb1": "\\alpha^6 + \\alpha^3 = (\\alpha^2 + \\alpha) + (\\alpha^2 + 1) = \\alpha + 1 = \\alpha^5", "9548a51cde29028f5a3a65e689010c93": "\\ \\omega_{xy}^2=4D\\tau_D+t_{0}", "9548db3b420722d0e09313c39cdf59a2": "\\lambda_{mn}(-ic)", "9548ebb86c743d4e3bbe30a18d0fa110": "\n \\frac{1}{1 + \\frac{1}{n} z^2}\n \\left[\n \\hat p + \\frac{1}{2n} z^2 \\pm\n z \\sqrt{\n \\frac{1}{n}\\hat p \\left(1 - \\hat p\\right) +\n \\frac{1}{4n^2}z^2\n } \n \\right]\n", "9548f9d0e56e5505726e2eb32bf0de2d": "\\gamma=\\mu-\\alpha_J\\alpha", "9549f5c9258a7c0b825b663df48fa133": "V - \\{\\varnothing\\}.", "9549f8abd257abf7677382d3b5e4012a": "x \\equiv \\pm\\; a^{(n+1)/4} \\pmod{n},", "954a2afc8d1ec24e076799cbc3f6b71c": "1/r^n", "954a2c0883abc04c35e2038e4aa52bce": "B^n y^n", "954a393811129c879328a7c7d9d3bd44": "\\sum_{j=1}^{n-1}R_{j}^{2}=1.", "954a696dd680b6715b2677c0a773f521": "y_i = a + bx_1 + bx_2 + ... + bx_n + e_i\\ ", "954ac10c67c41c70308220dbea598cbb": " S= \\pi/2 \\int_0^\\infty N(d_p)d_p^2 \\,\\mathrm{d}d_p", "954af2052879342237ca7d22eee00f43": "W^6=SU(3)/T^2", "954b0f72e9fcb7e5703751210484ae42": "\\scriptstyle n = \\sqrt 2", "954b85abf47aeda9514de42a98e5d6c2": "o(t)", "954c07ff48d2aebf45ddd80705f5c009": "B_m(x,y)\\,", "954c1200895135d987a7597f3da0ec6c": "G = \\frac{N+1}{N-1}-\\frac{2}{N(N-1)u}(\\Sigma_{i=1}^n \\; P_iX_i)", "954c6d65d890243f0b3b9a9818fb29f3": "r_1,", "954c7a67c7d61db90a3f8e5b73d10a16": "n = \\frac{1}{3\\pi^2} \\left(\\frac{2m}{\\hbar^2} E_f\\right)^{\\frac{3}{2}} ", "954c81dbbc666e1bbfbe2f5f435a6128": "\\overline{\\mathbf{AB}} = (\\overline{\\mathbf{A}}) (\\overline{\\mathbf{B}})", "954ca902550a7c91b9e2b5b05a07c24e": "location(x,s)=location(x,s')", "954cb5aa2e2300a0d2e0478b8b2af9a6": "v \\mapsto B(v, w)", "954cb95d2a4a29cc6a3c698e98aa0c2c": "\\sigma_{zz} = M \\epsilon_{zz}", "954cc0c26b8357242d4aea4061df323c": "g'(x) = \\frac {2 {[f'(x)]}^2 - f(x) f''(x)} {2 f'(x) \\sqrt{|f'(x)|}}, ", "954ce38cde7a27f13993f1231ce4212f": " K(t) = \\log(M(t)) = a_1(e^t-1)+a_2(e^{2t}-1)", "954cf84af219221b34bb79e07aa76743": "\\lim_{x \\to +\\infty}{f(x)} = L", "954d1a94d0af2e9bd4080482bec6fa39": "r_L = \\left| \\frac{S_{12}S_{21}}{\\left|S_{22}\\right|^2-\\left|\\Delta\\right|^2} \\right|\\,", "954d1f95db451cce43ad5710e808957b": " E^2 - k^2 = m^2 \\,", "954d7ad704e526508ae64c05937ae9da": "\\left(\\pm\\sqrt{\\frac{5}{2}},\\ 3\\sqrt{\\frac{3}{2}},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "954d8597010de16eabed3a13982e6b91": " G,H,K ", "954d98553fd613a6f72cc4f9e27baaad": "0.3333...", "954de7ca9fddc3ea09e9fe534dfc7bce": "R_S=\\frac{\\sqrt N}{4}\\left(\\frac{\\alpha-1}{\\alpha}\\right)\\left(\\frac{k_B}{1+k_B}\\right)\n", "954e00d143c931ce750c6d77d09c3cd9": "\n \\dot{\\mathbf{F}} = \\frac{\\partial}{\\partial \\mathbf{X}}\\left[\\mathbf{V}(\\mathbf{X}, t)\\right] = \\frac{\\partial}{\\partial \\mathbf{x}}\\left[\\mathbf{v}(\\mathbf{x},t)\\right]\\cdot\\frac{\\partial \\mathbf{x}(\\mathbf{X}, t)}{\\partial \\mathbf{X}}\n = \\boldsymbol{l}\\cdot\\mathbf{F}\n ", "954e27c0bb4189c4fe71773a5c369552": "F_{O_2loop}=\\frac{(P_{amb}*K_{bellows}*K_E+1)F_{O_2feed}-1}{P_{amb}*K_{bellows}*K_E}", "954e29ffc1bdf5b846f1a8d59304dacd": "\\begin{pmatrix}\n\\lambda & 1 & 0 & \\cdots & 0 \\\\\n0 & \\lambda & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots& \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\lambda & 1 \\\\\n0 & 0 & 0 & 0 & \\lambda \\end{pmatrix}", "954ed2705410f01f14a210be343c835c": "2H^+ + 2e^- \\rightarrow H_2", "954ef165f81e2cb95c24accfb1654682": "\\mathcal{D}\\,\\!", "954ef9eafac6c221d2d29821a851efdc": "(\\eta,\\zeta)=\\int_M \\eta\\wedge \\star \\zeta = \\int_M \\langle \\eta, \\zeta \\rangle \\; \\mathrm{d} \\text{Vol} ", "954f4453b6ce81366ae50153bb159899": "1+\\max(|a|,|b|,|c|,|d|)", "954f44f7a5a0688c67aad3d5e83ed30a": "K = \\mathbb{C}", "954f55ee28d52ba4d58defaa46a133f5": "a=0;", "954fb16a2140b25bb2c9216119253d95": "S^{0},\\cdots,S^{T}", "954fcef4541a562a0df2a20bf3ec840d": " \\mathcal{L} = \\frac{1}{2}(\\partial_x \\phi(x))^2 - \\frac{1}{2}m^2 \\phi(x)^2 + \\phi(x) \\int{\\frac{\\phi(y)}{(x-y)^2} \\, d^ny}", "954fe54dff41e093262dc674e237eaec": " \\scriptstyle \\phi_0 ", "95502d30d8ba8cc46d73c76e3f8db8ce": "*\\mbox{-}", "95503964982fd6d65bdf8e8dc9a3570f": "\n\\begin{align}\nf(X_t)= & f(X_0)+\\sum_{i=1}^d\\int_0^t f_{i}(X_{s-})\\,dX^i_s + \\frac{1}{2}\\sum_{i,j=1}^d \\int_0^t f_{i,j}(X_{s-})\\,d[X^i,X^j]_s\\\\\n&{} + \\sum_{s\\le t}\\left(\\Delta f(X_s)-\\sum_{i=1}^df_{i}(X_{s-})\\,\\Delta X^i_s-\\frac{1}{2}\\sum_{i,j=1}^d f_{i,j}(X_{s-})\\,\\Delta X^i_s \\, \\Delta X^j_s\\right).\n\\end{align}\n", "95503a026cdb3a94fa9969960a8a2e01": "\\sin (x + iy) = \\sin x \\cosh y + i \\cos x \\sinh y,\\,", "955042bf0b9ecc50570bd721367ed800": "\\mathcal X_k = \\mathcal A_k\\otimes \\mathcal B_k", "9550441eb13ea84771933e89d0fb3e6c": " P(X_1^n(i')) \\geq P(X_1^n(i)) ", "95504782a4d5185e368ae94b39e74ab1": "\\operatorname{Gal}(l/k)", "95506771a179728e1e42a29bfc7064d5": " R = \\left[ \\frac {10^6 \\, \\mathrm{W} \\times 5000^2 \\times \\left(1.56 \\, \\mathrm{m} \\right)^2 \\times 1 \\, \\mathrm{m}^2} {12.57^3 \\times 10^{-16} \\, \\mathrm{W}} \\right] ^{1 \\over 4} \\,\\!", "9550abbd46aff113125345bd0a1149ed": "p(\\theta) \\log p(\\theta)", "9550ae5ab48a0d6e4005b856df560207": "\\Psi_M\\,", "9550dd94f2dce55d0cd4d79077dbd4ae": "Factor \\rightarrow (Expr)\\,|\\,Int", "9550e245290ba51db74c116a0749a297": "\\theta _i = \\theta _r", "955120cc511eae476e32dbd7020593ca": "\nf_\\mathrm{c} = {1 \\over 2 \\pi \\tau } = {1 \\over 2 \\pi R C}\n", "95512f787794c175e0bebedf5f572281": "\n \\int_\\Omega \\frac{\\partial }{\\partial t}(\\rho~\\eta)~\\text{dV} \\ge\n -\\int_\\Omega \\boldsymbol{\\nabla} \\cdot (\\rho~\\eta~\\mathbf{v})~\\text{dV} - \n \\int_\\Omega \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right)~\\text{dV} + \n \\int_\\Omega \\cfrac{\\rho~s}{T}~\\text{dV}.\n ", "95515a4d5d46092ed5a236a2e90dc915": " = \\frac {\\Gamma(\\tfrac13)\\;\\Gamma(\\tfrac{5}{6})} {\\Gamma(\\tfrac{1}{6})} = \\frac {(-\\tfrac23)!\\;(-1+\\tfrac56)!} {(-1+\\tfrac16)!}", "9551a37e6bbadea3889ac1c3ebca75a8": "\\phi_i(x)", "95524fe2a7ced2efd8dbf2dfcb027aef": " L", "9552c68dddddea8a09f74cb2d86f8f3e": " E_1 ", "9553586566f19978f92341d1fd27be21": "\n\\left.\n\\begin{matrix}\nx*y*z=x*(y*z)\\qquad\\qquad\\quad\\,\n\\\\\nw*x*y*z=w*(x*(y*z))\\quad\n\\\\\n\\mbox{etc.}\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\ \\,\n\\end{matrix}\n\\right\\}\n\\mbox{for all }w,x,y,z\\in S\n", "9553ac68f0ec3b2f979cd7d19d0904d7": "w\\,\\in\\,G", "9553af7f0574f14ec8fa6a7be962dddf": "f(T,M,L,g) = 0.\\,", "9553ee13c1eed5c41084f4fe2693ea58": " x^* = -Q^{-1}A^{T}\\lambda. ", "9554a231fded4b5420b2045719cde83d": "\\mbox{House Price-Rent ratio} = \\frac{\\mbox{House price}}{\\mbox{Monthly Rent x 12}}", "9554b72fcafbc252017cf040e7d4fbe8": "\\frac{V_1}{T_1} = \\frac{V_2}{T_2}.", "9554cded8eee21e8d3e62d9c03beab8e": "\\displaystyle \\cos{A}\\cos{B}\\cos{C}=0.", "9554d709889e8ba6f1e8b8e3d9f653e5": "d(x_m, x_n) \\leq q^n d(x_1, x_0) \\left ( \\frac{1}{1-q} \\right ) < \\left (\\frac{\\varepsilon(1-q)}{d(x_1, x_0)} \\right ) d(x_1, x_0) \\left ( \\frac{1}{1-q} \\right ) = \\varepsilon.", "9554e20b2fbd7ec42c053a3b0cc25ba4": "T = M_T * y' + v_T", "9555a1b0058627c4c3cdd63b408559bd": "a = \\frac{1}{\\sqrt{2}}(q + i p) = \\frac{1}{\\sqrt{2}}\\left( q + \\frac{d}{dq}\\right) ", "9555ae693e8b846ffe4873bc8c2b8722": "\\gamma=\\frac{1}{\\sqrt{1-\\beta^2}}", "9555cf4058663fc8eebb1f09d4dbadea": "\\Phi_{\\langle\\cdot,\\cdot\\rangle} : V\\to V^*", "95562b60934ac6af6d6bca7d1130e1f7": "\\mathcal{O}_{X,P}", "95562ebb97665f6dc1da3bf355fdf0fe": "\\, A_{(a_0,\\;a_1,\\ldots,\\;a_n)}\\,", "95568dcc2429d4e3bfaccfb6969a30eb": " H = \\int_X^\\oplus H_x d \\mu(x) ", "9556b628e57eb86feb4bc55961d49f15": "I=\\ln 2 \\approx 0.69314718", "9556fbc91a88ccc7b565581978f12676": "\\scriptstyle t \\;=\\; 0", "95572ea805c945b48e306de422fc6ef3": " V(t) = Vo(1-e^{-t/ \\tau}) ", "955770086534a41dbdace4219c8d4dcd": "\n{\\color{white}.\\,\\qquad)}\\sin\\phi = \\cos\\alpha_0\\sin\\sigma,", "9557b266233fe43618c964b9426cce56": "\\displaystyle (1-3xt+t^3)^{-1/2}=\\sum^\\infty\n_{n=0}P_n(x)t^n", "9557f17cbb659663181fd79297cc4264": "H^*=\\frac{(G^*_d-G^*_m)(19G^*_d+16G^*_m)+(4\\sigma/R)(5G^*_d+2G^*_m)}{(2G^*_d+3G^*_m)(19G^*_d+16G^*_m)+(40\\sigma/R)(G^*_d+G^*_m)}", "95582b32c13504f6604ad48151bf893a": " F = F_0 \\subset F_1 \\subset F_2 \\subset \\ldots \\subset F_\\infty ", "95582bc64ebc4393511a0fa3182448e8": "v_i(p + r,t) = v_i(p,t) + \\sum_j J_{i j}(p,t) r_j = v_i(p,t) + \\sum_j \\partial_j v_i(p,t) r_j;", "955836f1438169ea1ceb867c08f89b29": "16.9897\\,\\mathrm{dBW} = 10 \\log \\left(\\frac{50\\,\\mathrm{W}}{1\\,\\mathrm{W}} \\right)", "955843b178fed351cb5f78a6ed06102d": "T(G) = \\sum_{i=1}^{29} N_i(G)", "95584b169af99a7754cca4604b93153a": "\\mathbf{J}\\times\\mathbf{B} = \\frac{\\left(\\mathbf{B}\\cdot\\nabla\\right)\\mathbf{B}}{\\mu_0} - \\nabla\\left(\\frac{B^2}{2\\mu_0}\\right),", "9558c4d6bc0eb230368469dac46f2a97": "|\\mathbf{E}| \\sin\\theta ", "95590d208a0e1f7cb179b5c7ba6f5547": "C=\\{i\\}", "95591f4b1e07abc93be792d01a77633a": "\t\\begin{array}{|rrrr} \n 1 & \\text{-}12 & 0 & \\text{-}42 \n\\end{array}", "9559281efa2e5481eb590ef3cc1304e6": "\\begin{matrix} \\frac{1 \\;EVU \\;\\times \\;cosine \\;of \\;latitude \\;A} {1 \\;EVU \\;\\times \\;cosine \\;of \\;latitude \\;B} \\end{matrix}", "95596fe963c6eb7461476b47aeb37d5b": "\\tilde{e}_i u = \\sum_{n=1}^\\infty f_i^{(n-1)} u_n = \\sum_{n=0}^\\infty e_i^{(n+1)} v_n,", "955a140cfd878900f9431db90844875a": "I = I_{L} - I_{0} \\left\\{\\exp\\left[\\frac{q(V + I R_{S})}{nkT}\\right] - 1\\right\\} - \\frac{V + I R_{S}}{R_{SH}}.", "955a5db68cbde33a9197f944e2240a10": "\\eta_G = (-1)^{L + I}\\,", "955af497abf8a1815dc71c86c21bdab8": " x_n=(ax_{n-1}+c_{n-1})\\,\\bmod b\\,,\\ \\ c_n=\\left\\lfloor\\frac{ax_{n-1}+c_{n-1}}{b}\\right\\rfloor", "955b65bfbabb06a16b7a7ffa187bcdd9": "q \\le r \\le p ", "955b8dd0aeb14ccf3a92dfe178af33ea": "S^2\\times \\{pt\\} ", "955bb4132bd1b600253a0ecf64e02425": "\\beta = 2{\\rm cos}\\left(\\frac{1}{3}{\\rm arccos}\\left( {\\rm det}(B)/2 \\right) + \\frac{2k\\pi}{3}\\right), \\quad k = 0, 1, 2.", "955bfe0c6404ce187f17f43abe575abf": "\\prod_{(x,y):\\,P(x)=Q(y)=0} (x-y)", "955c3b4c72750bb95b3a02ed88596b0c": "X[k] = \\sum_{n=0}^{N-1} x[n] \\cdot e^{-i \\frac{2 \\pi}{N} k n} \\quad \\quad k = 0, \\dots, N-1", "955c4b63326529962529aad47c37af12": "\\mathbf{p} = (p_x, p_y, p_z) = p_x\\mathbf{i} + p_y\\mathbf{j} + p_z\\mathbf{k}", "955c9a763a4ee4c0d85dace7fc4e6376": "\\Psi(x,y)", "955cbecda23de7c1d1294af5e571e1a5": "S=[0,1]", "955cf37153833f479a6f6ac988680715": "\\begin{matrix}\n x_1 & = & x_0 - \\dfrac{f(x_0)}{f'(x_0)} & = & 10 - \\dfrac{10^2 - 612}{2 \\cdot 10} & = & 35.6 \\quad\\quad\\quad{} \\\\\n x_2 & = & x_1 - \\dfrac{f(x_1)}{f'(x_1)} & = & 35.6 - \\dfrac{35.6^2 - 612}{2 \\cdot 35.6} & = & \\underline{2}6.395505617978\\dots \\\\\n x_3 & = & \\vdots & = & \\vdots & = & \\underline{24.7}90635492455\\dots \\\\\n x_4 & = & \\vdots & = & \\vdots & = & \\underline{24.7386}88294075\\dots \\\\\n x_5 & = & \\vdots & = & \\vdots & = & \\underline{24.7386337537}67\\dots\n\\end{matrix}\n", "955d1ef1261fbbffabd8e93497c0175a": "T^\\nabla(X,Y) = \\nabla_X Y - \\nabla_Y X - [X,Y].", "955d77b0a4f32eab9715024ccef00fa4": "\\Sigma\\cup\\{\\#\\}", "955e313bff27aebac943eb4f4eefbd87": "\n A = \\cfrac{6 c \\cos\\phi}{\\sqrt{3}(3-\\sin\\phi)} ~;~~ B = \\cfrac{2\\sin\\phi}{\\sqrt{3}(3-\\sin\\phi)}\n ", "955ed62f239026bb434688c000d970e0": "\ny = a \\ \\sinh \\mu \\ \\sin \\nu \\ \\sin \\phi\n", "955f0809f4d5551e9b2ab3df8f3478cf": "x^2-2x-8=0", "955f8ff953694fa0e4408b62b8ccc2b3": " = \\langle \\frac{d\\psi}{dt} | Q | \\psi \\rangle + \\langle \\psi | \\frac{dQ}{dt} | \\psi \\rangle + \\langle \\psi | Q | \\frac{d\\psi}{dt} \\rangle\\,", "955fa755a491641f55bb85836ec18265": "B \\otimes_A R", "955fae56959decfd1cf9956aa32b598d": "p_{ij}=p", "956004a69c9b6df473645644269065a2": "v \\otimes w \\, ", "956020d92869d30a375013da57eb1e1a": "p (\\tau)", "95603942fbeaa2c7578c8851cd1d768e": "f(u,v)\\cdot a(u,v)", "9560e741e4621634a8ee584d623d0900": "(1/5)b_{1}-(1/4)a_{1}", "95616d5071a0397e61b37d026cc8d69c": "\nC = \\frac {|S_A-S_B|}{S_\\mathrm{ref}}\n", "9561915c8cc473d19924713925ca6cfc": "\\mathbf{v}_r = \\mathbf{v}_{p2} - \\mathbf{v}_{p1}", "9561ace10342aec2c1f411629b98ba6d": "\\alpha = \\theta + \\psi - \\frac{\\pi}{2}.", "9561b71ca607397bc70527773385ceae": " n \\!", "9561e77ebf49ddd67ca868ceddf4bf9c": "\\partial_{xy}", "956217c44f0329d9b7d8543c7c6133fe": "I: \\langle s,t \\mid s^2, t^3, (st)^5 \\rangle\\ ", "95621c9a54d6fd1bf216f9f094b54b04": "P(g\\gg1)= \\int dk k^{-\\gamma} \\delta (g-k^\\eta)", "95623c98a8fec95cfa90acebb7450e14": "T_{k,n}", "95625f90f92bad07fc8bc176c8bbf5ab": "VSWR = \\frac{1 + \\Gamma}{1 - \\Gamma}", "9562b958a0fa57336c5ab93a58e5ea52": "{2GM \\over z^3} \\times 10^9", "9562bfce6d1497452f773c3c6779308c": "\\therefore 2x^T M^T - 2\\lambda x^T = 0 ", "9562c8d756fdfa3df479d1fae594d02a": "i\\in\\{1,\\ldots,n\\}", "956320a6b7ab78eaf94d08b27e67c54c": "\\tilde{d}", "9563507be5fb55a5fc47511325a17cac": "\\frac{AF}{FB} \\times \\frac{BD}{DC} \\times \\frac{CE}{EA} = -1,", "95638b66018bffc548675cae3e09f4f8": " 3.2 \\cdot 10^{4}", "9563924c034d240bb9e1b786aa219037": "0\\le k^2\\sin^2\\phi\\le1", "95641b98dcff82f355054f73efbaeaac": "\\scriptstyle b_n", "95645966156eaead0ef00d6f4f682522": "\nj=1,2,....,n\n", "956490de97a0f12a3d8b5a3c42e12bb4": "\\phi^*(\\omega_j^i) = (A^{-1})_p^i\\, \\omega_q^p\\, A_j^q.", "9564d4eafdda9cc965d0954700c1dfe9": " \\vdash (q,1,AZ) \\vdash (q,\\epsilon,Z)", "956590b950dcd61dd81e280e6d0fca0b": "\\frac{5 \\cdot 17}{11}", "9565d446c0a7f8007828eadbc0236150": "y^2 = x^3 - px - q", "9565d48eca242cc6fbd4f96e84856fe2": "\\sigma_i\\sigma_j = \\sigma_j\\sigma_i \\mbox{ if } j \\neq i\\pm 1", "9565d51b3d38dba4b2afee1d4be78a6f": ">1", "9565ee914c05003491ce551d180d3ccb": "E_{||}", "956688f410d72ab7e5131b705c87ef10": "\n\\mathcal{L} = \n-\\textstyle{1\\over4} F_{\\mu\\nu}F^{\\mu\\nu}\n+\\textstyle{1\\over2} \\epsilon^{\\kappa\\lambda\\mu\\nu} A_\\lambda {(\\hat k_{AF})}_\\kappa F_{\\mu\\nu}\n-\\textstyle{1\\over4} F_{\\kappa\\lambda} {(\\hat k_F)}^{\\kappa\\lambda\\mu\\nu} F_{\\mu\\nu} \\ ,\n", "9566bdbf0b3209f2401dff7309b178c5": "\\mathsf{NP} \\subseteq \\mathsf{P/poly}", "9566c8a56090bc365572fd1508323945": "L<1", "9566d50dba85d223e349fdb7ecfc2f7e": "\\arccos (1/x) \\,= \\arcsec x \\,", "9566e28fde6dab715e943392baa18ba5": "D^{2}+ V^{2}\\le 1 \\, ", "9566e46b57dcaf4dd6bf01234c72196a": "10^4", "9566e9846533e0205234f54b4f55bedc": " a_{s,s-1} ", "9566efb796d95a5952755bfb10e6be73": "\\omega -\\overline{\\omega} = i\\sqrt{3}.", "956731daaa60bd2cf42b7b1f4fb6cc3d": "\\alpha = \\eta_1+1,", "9567febfe9244ca122011ee0c9dd47fd": "f= -k_BT \\log W.", "9568476cb9235be6508c188af1c7ce39": "2NH_3 (g) \\rightarrow \\; 3H_2 (g) + N_2 (g)", "95684b0666648c9b52fe1b13765eeaaa": " \\begin{bmatrix} 1 & \\mathbf a & c \\\\ 0 & I_n & \\mathbf b \\\\ 0 & 0 & 1 \\end{bmatrix} ", "95684dfc0807f46e4767bcb79d9de5da": "\\textrm{Ratio}_2 = \\sqrt{\\textrm{Uploaded \\,\\, Total} + 2} ", "956852c138a4729f99e185a7f700db98": "K(x,y) = (x^T y + c)^d", "95685601956df8acc82a317b1ad2fc29": "\\!\\ Re = 10^6", "95687afb5d9a2a9fa39038f991640b0c": "dp", "9569019689bce8196d6332079190a8f0": "SU(n)\\;", "95690b5c27e0b71d99c5457869db7753": "\\displaystyle{\\tau(a,T,b)=(-a^*,-T^*,-b^*).}", "95693a44fce2e709975e5b609184f9fc": " f(A \\mid m) = \\prod_p \\phi \\left(A_p \\mid A_\\text{base} + r\\sum_c C_{cp}, \\nu^{-1} \\right) ,", "9569bd20e9acb79386f0b3c250d330bd": "P^k", "956a18ae2b9cbf21942bc0f27aac1ce6": "\n \\kappa G h \\left(\\nabla^2 w^0 - \\frac{\\mathcal{M}}{D}\\right) = -q \\,.\n", "956a1fefa8261dad07cb85b3d9e91965": "= \\|x\\|^2 + \\langle x, y \\rangle + \\langle y, x \\rangle + \\|y\\|^2", "956a48c59646058c84bd98829805cc00": "e=1/\\cos\\theta\\,", "956a50a529a2c330e1322df6202aa756": "2\\cdot q^2 \\cdot q^3 = 2q^5", "956a5373889f09b677a39c1928175b00": "^\\mathbb{T}", "956b1a5d0b216ccd917a8bc64ca3f0cb": " r_0 ", "956b6b8e315e171c3770bd70a98afadb": " \\left(\\frac{N Q^{0.5}}{(g H)^{0.75}}\\right)", "956b7fce659d6ce78c2a15b80d858c2b": " \\forall x \\, P(x) \\equiv P(a) \\land P(b) \\land P(c) ", "956b9977a48652fdf1ed902d3d371001": "\\mathbf{B}=\\nabla \\times \\mathbf{A}", "956baa6902762056bd3aca827bbbe2f0": "R= R_1+R_2\\,", "956bc844b71d25e9f8e90c45c5b5c6fa": "x(x+1)(x^2-x-4)", "956bd528d9250e69356cc8397893320e": "\\Delta i = i-\\langle i\\rangle", "956c131ab71468f744b82b89508b58ba": "p^0 = \\sqrt{m^2+|p|^2}", "956c55dbcf5ecaeda48c5ed6141e92b7": "~P_{\\rm in}", "956c58461dcc870693eb6c1d26b8d718": "1/\\binom{8}{4} = 1/70 \\approx 0.014,", "956c66ee12958c1bb3797b4f1dba8c95": "\\int_{-\\infty}^\\infty |e^{-x^2}|\\, dx < \\int_{-\\infty}^{-1} -x e^{-x^2}\\, dx + \\int_{-1}^1 e^{-x^2}\\, dx+ \\int_{1}^{\\infty} x e^{-x^2}\\, dx<\\infty.", "956c98d56b52241ae50f22e5a4ef671f": "n\\neq 2,6", "956d49936af1a08da19ccd308071cbaa": "{\\part A_D \\over \\part z} = - {i\\beta_2 \\over 2} {\\part^2 A \\over \\part t^2} = \\hat D A, ", "956d55d4a61252f33fb5be41a083dd3c": "\\tau_{32} > \\tau_{21}", "956d8ad220759eb64e863aca89c3c7f8": "\\text{DFFITS} = {\\widehat{y_i} - \\widehat{y_{i(i)}} \\over s_{(i)} \\sqrt{h_{ii}}}", "956d92d620a178c148bbfed2597e9135": "a=1.6, \\, s\\approx\\sigma/\\sqrt{2.25} = 2\\sigma/3", "956daff695d7e55a2a4b606676c29121": "=j\\cos\\left(\\frac{1}{n}\\arccos\\left(\\frac{\\pm j}{\\varepsilon}\\right)+\\frac{m\\pi}{n}\\right).", "956ddb3363743fe5e8296dc2beed9922": "W_2", "956dfbd371ddbb85edfb0350fe4f567b": "\\frac{P}{K}=\\frac{1}{1+Ae^{-kt}}", "956e2a13ca5bc766655a1a1f98431d60": "S_\\mathit{wi}", "956e64727fdfaeda6d1fd8c9d4b21599": "d = (abc)^\\frac{1}{3}\\,\\!", "956e69f62bbefd36e2d505d810f4876b": "[M] \\in H_n(M;\\mathbf Z_2)", "956ee2172e0a617347a45cf7b89a9da1": "\\frac{\\part\\, \\ln G_X}{\\partial \\beta}, \\frac{\\part\\, \\ln G_{(1-X)}}{\\partial \\alpha} < 0.", "956ef31e39c42e96e2667cc2c999ec7c": "\\delta\\alpha=0", "956f3eb26cc912017216dd5f47f61038": "\\hat{H}'_0= \\beta E", "956f3fdba2b3db9ed2768e01dad93b2c": "S_{\\rm oblate}", "956f4c98791932efee87ac53058b31ef": " \\operatorname{E}[g(X_1,\\dots,X_n)] = \\int_{-\\infty}^\\infty\\cdots \\int_{-\\infty}^\\infty g(x_1,\\cdots,x_n)~f(x_1,\\cdots,x_n)~\\mathrm{d}x_1\\cdots \\mathrm{d}x_n . ", "956f4ca49edad0bc20481f41205e7af0": "d_p = \\sqrt{w_p^2 + h_p^2}", "956f67bc7102d3079f273b309502e408": "n =4", "956f780968ceab2dd692970d7ea90cef": "\n\\begin{align}\nx = \\frac{\\pi R(\\lambda^\\circ-\\lambda^\\circ_0)}{180}, \\qquad\\quad\ny = R\\ln \\left[\\tan \\left(45 + \\frac{\\phi^\\circ}{2} \\right) \\right]. \n\\end{align}\n", "956f80076f37a695d1bf51c1944795a0": "A-BK", "956fbecf738581e2244470f0ea4e27ba": "(B_0, B_1, \\cdots \\in H(A_n)) \\Rightarrow (\\cap_{n<\\omega}B_n \\in H(A_n))", "956fc806ddc7b66b98abe05e1bd01cf4": "G = (\\{S\\}, \\{a,b,c\\}, S, \\{r_1, r_2, r_3\\})", "956fdc79d4f842546f0b3b457882bbe4": "\\rho_C(x) = x \\,", "957028d91f8a558b323f3361fd8e8cb3": "N !", "95702e1219ba5d62c44399b78dd55dd7": "\\varphi_{F}(e,x,y)", "95706289486f987ae705b8be2dca5825": "2^{-23}", "957096a34aa194ee42b801738f22a421": " I-Q ", "9570a7a77dcda424b82eb3f70805ce66": "\n\\begin{align}\na_t& = 2s'_t - s''_t\\\\\nb_t& = \\frac \\alpha {1-\\alpha} (s'_t - s''_t).\n\\end{align}\n", "9570bb2541503c8162e1dff61eda0702": "E({\\mathit{He}}_n(X))=\\mu^n.\\,\\!", "9570c12788356e5fa787dded326d4958": "k = \\frac{{EA}}{{L}}", "9570d0c16f7607a9685a6a297a2628d9": "\\displaystyle{(\\Delta_2 -R_2)Sf=4S\\Delta_2f},", "9570ecb00df3c8804acd54c413291414": "J(\\mathbf{x}) = (\\mathbf{x}-\\mathbf{x}_{b})^{\\mathrm{T}}\\mathbf{B}^{-1}(\\mathbf{x}-\\mathbf{x}_{b}) + \\sum_{i=0}^{n}(\\mathbf{y}_{i}-\\mathit{H}_{i}[\\mathbf{x}_{i}])^{\\mathrm{T}}\\mathbf{R}_{i}^{-1}(\\mathbf{y}_{i}-\\mathit{H}_{i}[\\mathbf{x}_{i}])", "957144bc3ba7c0d15f8c2f1af960f800": " \\frac{1}{\\gamma } =\\sqrt{1-\\frac{2G M}{r c^2}} ", "95718ea0fd0609d46f5fea91cbbdd9bc": "(\\operatorname{Id},K)", "95722b74ceb2f8309c0fc81a63ba5f12": "\\Psi _{BETA}(\\omega ) =-j\\omega \\cdot M(\\alpha ,\\alpha +\\beta ,-j\\omega (\\alpha +\\beta )\\sqrt{\\frac{\\alpha +\\beta +1}{\\alpha \\beta}})\\cdot exp\\{(j\\omega \\sqrt{\\frac{\\alpha (\\alpha +\\beta +1)}{\\beta }})\\}", "9572442052f64f4176e7b12e24be8322": "\\mathcal{I}_{1/n}", "9572443f49b6e1a774fdbe1c02efdca8": "\nP(j, \\vec{x}) \\approx \\frac{1}{n}\\sum_{i=1, c_i=j}^n w_i\n", "957256ecd276b52097856028cf32f2ae": " c = 1/ \\sqrt{\\mu_0 \\epsilon_0}", "95728b2163750ccb57a1d01b9c80ce60": "r \\in R_n", "9572e38fad0a92a2d9d8bf0b2a229a1f": "c_{k-2,1}<0", "9573354eba351654bf6a0eaf8167cef7": "n \\left(x,t \\right)=n_0 \\left[ 1 - 2 \\left(\\frac{x}{2\\sqrt{Dt\\pi}}\\right) \\right] ", "95737afaaf75a1470693e6f5c3a9ed01": "\\operatorname{erf}(z)", "9573828d633a8d37b6183ed50e17fbb0": " O(\\frac{\\log^2 n}{k}) ", "9573fac63d4e18f0f8e86a991945a831": "\\ v = \\frac{R_p}{R_d}=\\frac{k_p[M][M\\cdot]}{2fk_d[I]}=\\frac{k_p[M]}{2(fk_dk_t[I])^{1/2}}", "957400a6370307d8c7c9f6e696f307d9": "\\mu>0", "95744afe37cc1bc1000ca8bea2e2c9de": "\n\\begin{matrix}\n1 2 1 2 1 0 \\rightarrow &\n1 1 1 1 1 1 \\rightarrow &\n0 0 0 0 0 0 \\\\\n\\end{matrix}\n", "9574b0288b4c478a0e3eeeb902c7f247": "\\sum_{n=1}^\\infty (-1)^{n} a_n\\!", "9574b2ae37f9c1baaa6c9c854b7cc90f": " \\frac{F_p}{F_s} = \\frac{1.9 \\cdot 10^{27} \\cdot (778.3)^2}{1.989 \\cdot 10^{30} \\cdot(1.883)^2} \\approx 163 ", "9574c55cf3ee56d946c7f99d7a9f596c": "\\psi\\, ", "9574cf4f8a94636393ad75324b57e20d": "\\Delta y = \\Delta G * 1", "9574fbcf92f0e04abcfb09020cc76c78": "f\\colon (a,b) \\rightarrow \\mathbb{R}", "957501ba3a0be0be421fb18c631ec0e0": "u(t) \\le \\alpha(t) + \\int_{[a,t)} \\alpha(s) \\sum_{k=0}^{n-1} \\mu^{\\otimes k}(A_k(s,t))\\,\\mu(\\mathrm{d}s) + R_n(t)", "95750bfa08c6f24408c94d943d2c52d5": "\\mu_2=\\kappa_2\\,", "95754f5aed494dd7614b63b92ce68b04": "(2,10,3)", "95758e8b85ac13abe276676ba3ad79fb": " \\partial_t^+ \\omega_{n,i} + \\partial_x^+ \\kappa_{n,i} = 0 \\quad\\text{where}\\quad \\omega_{n,i} = \\mathrm{d}u_{n,i+1/2} \\wedge K \\, \\mathrm{d}u_{n,i+1/2} \\quad\\text{and}\\quad \\kappa_{n,i} = \\mathrm{d}u_{n+1/2,i} \\wedge L \\, \\mathrm{d}u_{n+1/2,i}. ", "9575e8682c2da857accb2f07f9bb73a7": "ln(r) = ln(A) - \\Delta H/(R \\cdot T)", "9575fabc5c2289e19bac786223146d80": " g=C_1 - 2 b_{ij} \\, ", "957669f8c4f51ad461dff69b13c51a29": "\\sqrt{(x-y)^2} = |x-y|.", "95773bd1d4443494987302e2a896f740": "\\approx 1.7724538509055160273\\,,", "9577730b6b5b9c6b5bf401b2a1729d72": " g(\\theta|\\theta_m) ", "95778f490efea11c573a9dc683f540a0": " g(v) = \\int \\delta(y f(z) - z + x) g(z) (1-y f'(z)) \\, dz", "9577a1f651988b80312efb1d8fc3b628": " A+L-> A^{++} ", "9577d9b2cb01daa8771533ca7c4f33c4": "i_f = \\frac {R_D//r_O} {R_D//r_O +R_F +r_{\\pi}// R_S} \\ i_t \\ . ", "9577fabf56f3a52ca75fbc9fb9417ea2": "\\varepsilon \\left[ M \\right]=\\sum\\limits_{m=M}^{N-1}{\\left\\langle K{{g}_{m}},{{g}_{m}} \\right\\rangle }", "9578191851d32e25c1dd293d501d08a7": "{{\\left\\{ \\frac{1}{\\sqrt{N}}{{e}^{\\frac{i2\\pi mn}{N}}} \\right\\}}_{0\\le m 0", "959645e45fe75cfe06b1264204018d95": "\\scriptstyle x \\,<\\, y", "9596b2f45d30691929bff916798acdfa": "0.37V+0.27U+0.36EG=0.28SW+0.72BK", "9596e0f0914b08ca99ea5410b19b35ce": "F=Q_\\text{accept}\\subset Q", "9596f955e89807b1d9a3adad08115615": "\\gamma = \\alpha \\,G + kG \\,", "9597176468f929d5929e0d86af059883": "\\gamma(z,x)", "959754c6254575631db762e9f45b1f6f": "R_{abcd}^{}=K(g_{ac}g_{db}- g_{ad}g_{cb}) \\, ", "959795a81367a64f38e5645712bebe30": "(\\mathbf{C}\\otimes \\mathrm{End}\\, X^*\\boxtimes X \\boxtimes X^*\\boxtimes \\cdots)^{\\prime\\prime} \\subseteq (\\mathrm{End}\\, X\\boxtimes X^* \\boxtimes X \\boxtimes X^* \\boxtimes\\cdots )^{\\prime\\prime},", "9597c45be1c0f506ca717c0171377d7b": "\\omega_{n}=\\frac{\\psi_{n+2}\\psi_{n-1}^{2}-\\psi_{n-2}\\psi_{n+1}^{2}}{4y}.", "9597e276e9f8c55daf264472a1d43de7": " B,", "9597ebd3b7657c7f36a92b4b4162840c": " x^4 - y^4 = xy. \\,", "9598468b0d4e3b01da234bc2767f5f51": "N-\\tbinom{c_k}k", "95985cfd19b9c3b81a23822848dc2d3b": "\\theta_{max}\\,", "9598f974536d7d13d113366f72d1cf99": "V_{max} - V_{min}", "95990b29f66fcb710f2ec38f13fdf0de": " X \\sim {\\rm U}(0,1)\\, ", "95992192c188ee5254916d3bda230fd9": "\\kappa - 1 - \\frac{e^2 a \\kappa}{\\sqrt{p^2+(1-e^2) z^2 \\kappa^2 }} = 0,", "9599239b1fcf16410cf0b3fed4be1941": "V^{2} \\,", "9599513c3ad7ae4dd42e875a1113445f": "\\sigma_\\mathrm{1}\\cos(\\beta_\\mathrm{o})=\\sigma_\\mathrm{s}-\\sigma_\\mathrm{ls}-\\frac{\\tau}{r_\\mathrm{o}}", "959953cb37f5132f6e7b21ac57b707c5": "p_1 = q_1 = 1", "9599628d92f51271d3acd744b96ee99e": "E_b - {3\\over 5}\\epsilon_F \\sim 17\\;\\mathrm{MeV} ", "959984793c0c6fb8cb4215a3a3c80756": "K= \\frac{[S]^\\sigma[T]^\\tau}{[A]^\\alpha[B]^\\beta}", "9599a5ebf3d69adc823de6d474b9d8b2": "{}^q\\!D=\\left ( {\\sum_{i=1}^S p_i^q} \\right )^{1/(1-q)}", "9599bb3057b9450332f99e76c4363246": "\\det(i\\omega I_n-A) \\neq 0,\\ \\forall \\omega \\in R^{-1}\\text{ and }\\exists \\mu_0 \\in [\\mu_1, \\mu_2]\\,:\\, A+\\mu_0 BC", "959acabf6afbfb264a0b4347c030d9de": "\\operatorname{gr} U", "959af1edf224a4d58ead93a039a153ea": "Y \\,", "959b6324c9e61bb7e367eef6e26b0863": "\\Theta(\\min \\{n,m\\})", "959b69661b757ebcf58fda16294b48de": "1-q~", "959b6edaab6085415df2421bf9aaa76d": "0\\rightarrow H^0(\\mathbf{G}_m)\\rightarrow H^0(j_*\\mathbf{G}_{m,K})\\rightarrow\\bigoplus_{x\\in |X|}H^0(i_{x*}\\mathbf{Z})\\rightarrow H^1(\\mathbf{G}_m)\\rightarrow H^1(j_*\\mathbf{G}_{m,K})\\rightarrow\\cdots", "959b83e2ff0929673c2f1b390bac08e1": "I_K^\\Delta", "959bc929b535add4f3b34d068a475437": "p\\ f", "959bd2289d24a0fe7e41bd16ee9ec6e8": "y_u=au+b", "959c1d0f71b6bab482cb35072fae186e": " (x^\\lambda,\\sigma^m) ", "959c6b51fbbff0e7061c366def67126d": "\\mathbf{v} = c \\cdot \\mathbf{v}_\\textrm{op} = c \\left( \\mathbf{v}_\\textrm{D} +\n\\mathbf{v}_\\tau \\right)\n", "959c7546b9b53b7a142fc3ba8d0aa7d5": "(q=F(\\tau-\\theta))", "959ce66d6748b6dbbe1c60df4177bb6e": "\\mbox{tr}^2\\mathfrak{H} > 4.\\,", "959cf8e8a61d247084f2d23dcb6ab2bb": " C_{mp} = \\partial^2 Q/\\partial m \\partial T \\,\\!", "959d03e1fd08233de4150199964a6148": "RRR=R/MDD", "959d8a80302f67ce658f00073c6db841": "Tx + Uy = r", "959dd90546955629a88a204876a60843": "\\{e_i(t)\\}", "959de203fd3e2582a5caea79c49cd433": "z^2 + 2az + b^2", "959e635c5808cc3811d8fd99dcb34898": "U(1) \\times U(1)", "959e702e1e0ee2448366c20e89c9e49e": " \\tan\\frac{\\hat{\\gamma}}{2} \\mathsf{C} = \\frac{\\tan\\frac{\\hat{\\beta}}{2}\\mathsf{B} + \n\\tan\\frac{\\hat{\\alpha}}{2} \\mathsf{A} + \n\\tan\\frac{\\hat{\\beta}}{2}\\tan\\frac{\\hat{\\alpha}}{2} \\mathsf{B}\\times \\mathsf{A}}{1 - \n\\tan\\frac{\\hat{\\beta}}{2}\\tan\\frac{\\hat{\\alpha}}{2} \\mathsf{B}\\cdot \\mathsf{A}}.\n", "959ea10f2facd244529126e9cd6f01db": "L = \\{a^n b^n c^n | n \\ge 0\\}", "959eb989945a598bb0d89d1706e76eb1": "\\mathbf{T}(\\mathbf{X}) = \\sum_{i=1}^N \\mathbf{T}(x_i)", "959f424167e4c37ec21012943f69c8b1": "A^\\dagger = -A,\\;", "959f5f85a40513e5cf50a822620f3319": "^{99}Mo", "959f7793275525acf08ca39380038f2c": " {= 150}", "959f9db733601a359166cf64eb48ecf9": "\\|\\mathcal F\\|_{q,p} = \\sup_{f\\in L^p(\\mathbb R)} \\frac{\\|\\mathcal Ff\\|_q}{\\|f\\|_p},\\text{ where }1 < p \\le 2,", "959fb0abe6f15b0a58ea45a6a9d6a472": " O(|V|^2) ", "95a0002f51bfd1a88ad9d72a68fad795": "{\\mathbb L}_{x^2}(L)\\equiv{\\Big\\langle\\Big\\langle}\n \\partial_{xx}-\\frac{2}{x-y}\\partial_x+\\frac{2}{(x-y)^2},L{\\Big\\rangle\\Big\\rangle}", "95a04a0ea26f96af6c0832dbfac7475f": "L_\\beta \\beta + L_r\\frac{d\\mu}{dt} + L_p p = 0", "95a05757ded6196848cd19c9efb09549": "\\sum_{n=0}^\\infty a_n \\varphi_{n}(x)", "95a06b23e22389ef825be575ba4def65": "n/p", "95a0931e68eb007dc6d97c15f7df259d": "T(q,x) = \\left\\{\\begin{array}{lll}\n\t\t\t\t\t\t\t\t\t\t\tT_{1}(q,x) & \\mbox{if} & q\\in Q_{1} \\\\\n\t\t\t\t\t\t\t\t\t\t\tT_{2}(q,x) & \\mbox{if} & q\\in Q_{2} \\\\\n\t\t\t\t\t\t\t\t\t\t\t\\{q_{1}, q_{2}\\} & \\mbox{if} & q = q_{0}\\ and\\ x =\\epsilon\\\\\n\t\t\t\t\t\t\t\t\t\t\t\\emptyset & \\mbox{if} & q = q_{0}\\ and\\ x\\neq\\epsilon\n\t\t\t\t\t\t\t\t\t\t\t\\end{array}\\right.\n", "95a0c1d50dc9c43656dac43839d8addb": " F = \\frac{{\\displaystyle \\sum_{i=1}^n \\left(\\widehat {y_i}-\\bar {y}\\right)^2}/{k}} {{\\displaystyle {\\sum_{j=1}^{k}} {\\sum_{i=1}^{n_j}} \\left(y_{ij}-\\widehat {y_i}\\right)^2}/{(n-k-1)}}", "95a0f381e46d4da5e9e4996ad6663dd3": "V = {-e^2\\over 16\\pi\\varepsilon_0 Z^3}\\left(\\frac{x^2+y^2}{2}+z^2\\right)+ {3e^2\\over 32\\pi\\varepsilon_0 Z^4}\\left(\\frac{x^2+y^2}{2}{z}+z^3\\right)+O\\left(\\frac{1}{Z^5}\\right).", "95a11fed025a1a20abbcee0838161c4f": "r_c=\n\\frac{\n\\sum{XY}}\n\\sqrt{\\sum{X^2}\\sum{Y^2}}", "95a165e6f7fd167cee332bf24c848a3a": "T = N \\omega^2 = \\frac{U}{N M (N+M)}-\\frac{4 M N - 1}{6(M+N)} ", "95a18a9ac9108cc7105cc13ac5d8b63b": "= {1 \\over 2} \\epsilon^{MIJN} \\epsilon_{OPKN} F_M^{\\;\\;\\; K} G^{OP}", "95a18fbf8ca24d922509fdaed1a7d749": "r \\leftarrow UOWHash^{\\prime\\prime\\prime}(k^{\\prime},L_B(N),x^{\\prime},\\tilde{k}) \\in Z", "95a195a9e2e050dede34dc3f300b0f27": "\\frac{p-q}{p+q}.", "95a1a598c4118152d1ffc53db00ec77c": "V = x^6 + ax^4 + bx^3 + cx^2 + dx \\, ", "95a1c8e5c1a20e2ea3906a0d53c3b051": "\\sqrt[n]{1}", "95a1db9f076113c01e495014b25866ac": "J_\\alpha", "95a25a671956b36b5f705152c5cac980": "\\,e^{()}\\,", "95a2878652070e7db8394387b1e65a74": "\\mu = \\frac{q}{k\\, T}D", "95a2a2b29a2eb7c9608d9fc02ecb2226": "g_0=\\frac{g_{obs}}{1-\\beta_2 g_{obs} \\ln \\Lambda/m}", "95a2b08d0e8001bc303e1becf8c90b68": "dS/S = g \\cdot \\rho/\\sigma \\cdot dr", "95a3473e526ad751ffb88b56d19d8fd2": " \\sigma(X + Y) = \\sqrt{\\operatorname{var}(X) + \\operatorname{var}(Y) + 2 \\,\\operatorname{cov}(X,Y)}. \\, ", "95a376da1cb5ea7bd14968bb62ed555e": "E^r_{lm} = \\int \\mathbf{E}\\cdot \\mathbf{Y}^*_{lm}\\,\\mathrm{d}\\Omega", "95a38a9e97b2e63c77242c01b9aff4db": "(\\forall x Rx) \\to \\lnot \\exists x \\lnot Rx", "95a3ff500fb97710ab423c6bea01b441": "P_{i + \\frac{1}{2}} = \\frac{1}{2} \\left[ \nQ \\left( u_{i} , \\frac{u_{i+1} - u_i}{\\Delta x_i} \\right) + \nQ \\left( u_{i+1} , \\frac{u_{i+1} - u_i}{\\Delta x_i} \\right)\n \\right], ", "95a46944cd9f57442ffa84026a147854": "0 \\leq \\phi < 2 \\pi", "95a519da2ea0dad8da6ea38c99e21ea6": "X[p\\Delta_{F}]", "95a554dd74b72eb3daef8b462a4921ea": "c+v", "95a58fdf5f43cc746e5f26e87cf0e6cb": " f_{\\mathrm{FD}} ", "95a5fe6512a09878b9a1d81f2ba14c9a": "PV = nRT \\,", "95a61f2028ff30ec5a0fbc81a3d26585": "\\Delta H_{reaction}^\\ominus = \\sum \\Delta H_{\\mathrm f \\,(products)}^{\\ominus} - \\sum \\Delta H_{\\mathrm f \\,(reactants)}^{\\ominus}", "95a64fe5bf1502ffdb74ec5dfbfbd8f1": "(\\max_i R_{ij}), \\,", "95a6faf848e746e587b04e6e3eb465f5": "{\\mathcal F^\\bull}.", "95a74afb6d0bf9d8380f30b294d66e10": "K_N(X,Y)=K_M(\\tilde X, \\tilde Y)+\\tfrac34|[\\tilde X,\\tilde Y]^V|^2", "95a76349f52e35e9ddcdaf521ee1c622": "f \\cdot u = -{1 \\over \\rho}{\\partial P \\over \\partial y}", "95a7a8d71babf7ae2d154cea87c01775": " W = C_{XY}C^{-1}_{Y} .", "95a7b34c989f461b39ec7cb7d15b99d9": "R_A(x)", "95a7be092645b62bdc9f143388d8bc98": "f_{\\#}\\left(\\sum_tn_t\\sigma_t\\right) = \\sum_tn_tf_{\\#}\\left(\\sigma_t\\right)", "95a8301b0c14d0847a53082b0658dd80": " \\ddot{s}_{\\overline{n|}i}=s_{\\overline{n}|i}(1 + i)=s_{\\overline{n+1|}i}-1", "95a84a3f67d638d99766f5ad51d37576": "\\left[\\!\\! \\begin{array}{r} 3 \\\\ -5 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right],\\;\n\\left[\\!\\! \\begin{array}{r} -2 \\\\ 1 \\\\ 0 \\\\ -7 \\\\ 1 \\\\ 0 \\end{array} \\right],\\;\n\\left[\\!\\! \\begin{array}{r} 8 \\\\ -4 \\\\ 0 \\\\ 9 \\\\ 0 \\\\ 1 \\end{array} \\right] ", "95a8b861480e7bf2434c2ae24b15b823": "x + z\\leq y + z", "95a91caf6bad42b19655cbd32d91f03e": "J_{i,j}=0", "95a97edd59d9341311dd7f1ffdcd9b33": "{\\mathit l}", "95a99a3e907d33d0d9d484949b5cd151": "\\sin \\left ( \\frac{\\pi}{p} \\right ) \\sin \\left(\\frac{\\pi}{r}\\right) - \\cos\\left(\\frac{\\pi}{q}\\right)", "95aa3cecb057d21d8e43a142aa6de4ca": "-1 < a < 0:", "95aa50c39366101f4d1c42132da251fa": "z^{\\text{utopian}}", "95aa50f71a01c82354a7a2b385f1c4d8": "VQ", "95aafb3a310b77b2465b24ab77c5d3ec": "\\tfrac{h}{p}=\\tfrac{q}{h} ", "95ab28d23e63b0c098fe99937d93f34c": "\\Delta m_{body}", "95ab822bce35e52342eb2b111a84de09": " {e^{s^2/4} \\left(1 - \\operatorname{erf} \\left(s/2\\right)\\right) \\over s}", "95ab9dcf20de38384a12cbc51189f74b": "P_{k\\mid k-1}", "95aba465ee65ad381a77b53908ec8ef8": "M(n,t) \\leq \\frac{2^{n+1}}{\\sum_{j = 0}^t{\\left( \\binom{\\lfloor n/2\\rfloor}{j}+\\binom{\\lceil n/2\\rceil}{j}\\right)}}.", "95abc0dd677f4a476a60a231401b9d9b": " \\langle f, k(x,\\cdot) \\rangle_\\mathcal{H} = f(x) \\ \\forall f \\in \\mathcal{H}, \\forall x \\in \\Omega ", "95abed1ef8b0bd1fece80534934350ad": "\\theta=(\\sigma_\\epsilon^2,\\eta)=(\\sigma_\\epsilon^2,H,\\theta_3,\\ldots,\\theta_k),H=\\frac{\\gamma+1}{2}", "95ac8d12e14aa429edc44aed95a0b11b": "x_1(t)\\,", "95ac9266a711992a9b8c1f2962a9a4ff": "\\displaystyle\\iota_X \\alpha = \\alpha(X) = \\langle \\alpha,X \\rangle", "95acf6724dbde3d0bb41b5c29bbaf6ad": " w' \\in \\mathbb{M} and s \\in {\\{0,1\\}^*} ", "95ad3ca9a298c16bec5140a27677bffb": "5y^4\\frac{dy}{dx} - \\frac{dy}{dx} = \\frac{dx}{dx}", "95ad8e708a2733aafb1998cf6db7f403": "A^{(n+1)}", "95ad9c18df08a44cdcc7de1d5e04c318": "E_{\\mu }(z)", "95adbe6c7e92d6acc4d385e5d75d1776": "\\displaystyle c_n=\\sum_{[i,j]=n}(i,j)a_ib_j", "95add126763244ab08d9e69a655b6135": "g^{bc}", "95ae10911ccd94b57da5535ac94fec03": "\\ n", "95ae1a8a3c755889f9e4c47840d62d76": "\\nabla^*\\nabla", "95ae2bef170039a68b6e8cc4dbe964a9": "ATX=TX", "95ae38ddd94a74d4e42bae47894974a6": "A_n = \\sqrt{p_{n+1}}-\\sqrt{p_n}", "95ae40b6d960d85b6c18a74f5b58cb6d": " \\tilde{\\bold{v}} \\bold{u} = \\bold{v} \\times \\bold{u} . \\,\\!", "95ae64222c5f7d8dec967aec26f14e0b": " ai + k + 1 - l - i ", "95ae6f8f086c81dbe90b085907e3adcc": "f(x) \\geq f(c) - \\epsilon.", "95ae8c4c05dcb16032af512241102e78": "f'(3)= \\lim_{h\\to 0}\\frac{f(3+h)-f(3)}{h} = \\lim_{h\\to 0}\\frac{(3+h)^2 - 3^2}{h} = \\lim_{h\\to 0}\\frac{9 + 6h + h^2 - 9}{h} = \\lim_{h\\to 0}\\frac{6h + h^2}{h} = \\lim_{h\\to 0}{(6 + h)}. ", "95ae98c054a6aa389163782003faef86": "\\mathrm{EV}_{100} = \\mathrm{EV}_{S} - \\log_2 \\frac {S} {100} \\,.", "95aeabb4bac454d12f35928df57706fd": "R_y= -\\frac{1}{2+a+\\phi}", "95af0525fb73e0e72c8743c0148be747": "\\sum_{j=0}^n P_j(x)\\ge 0\\qquad (x\\ge -1).", "95af17c607d8a8ea46bf05040569c2d4": "M_1^* + M_2 \\xrightarrow{k_{12}} M_1M_2^* \\,", "95af413766b621a040421b0bc51074f5": "\\tfrac{\\pi^{12}}{12!}", "95af571d3e8dfe1e2c094f36ea687616": " \\theta(x) \\in \\mathbb{R}^{ 2^N } ", "95af6277870f893c510cd9b54f451a67": "\\frac{d}{dt} X^i(t) = \\nabla_{C(t)} X^i(t) = \\Gamma^i_{jk} X^j(t) C^k(t)", "95afacef2a1eb399c87ff7fca374c2a6": "\\psi \\to (\\forall x \\ \\phi )", "95b0012b3e069f53f4a7bd789102eb04": " \\Gamma_a^T ", "95b02a47fc7a9d81a75bdc66751fac3a": "|P_c^n(0)|\\leq 2", "95b0a0f81120272640b0379587a34444": "f(x+h)=\\frac{a_0+h}{b_0+b_1h+\\cdots+b_{d-1}h^{d-1}}+O(h^{d+1}).", "95b0b339684f2d7e845de90eb9881c92": "\\frac{1}{f} = \\frac{1}{f_1} + \\frac{1}{f_2}-\\frac{d}{f_1 f_2}.", "95b0ee54f9d1b7a8113fad43e7f908e6": "\n\\big( {a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\big)^m \\, \\big( {a^\\dagger}^{(\\mu')}(\\mathbf{k}') \\big)^n \\, \\big|\\,0\\,\\big\\rangle \\propto \\big|(\\mathbf{k},\\mu)^m; \\, (\\mathbf{k}', \\mu')^n \\, \\big\\rangle,\n", "95b0f00e5607f65da810a18302a31a51": "k=1, \\dots, n-1", "95b1041d84d90bf7c4857a7ae3b23de4": "X /\\!/ G", "95b164cbae819f32ee4a6ecac5a1753e": "\\Delta = h\\, \\sqrt{\\frac{4\\,h}{3\\,H}}", "95b1716f4aa2a34fcec6e7266aa21f6f": "n \\geq \\frac{p_f}{p_f - \\frac{1}{2}}\\left( c + 1\\right)", "95b243d07cb2eeae42593b11eeb364ce": "{Q(x) \\over E(x)} = P(x) = 5-x", "95b2597da32afcc2931c6d9d97b049e3": "\\mathbf{p^{n+1}=p^n} +f \\mathbf{\\delta p}", "95b273fc87c84814ce286c9fd6c24cc1": "K > 0", "95b30b86e97a8a367a315913078dbbc1": "P_{\\text{ph}}", "95b333ba5f17414eb626bf1adbe56559": "\\sum_{n\\ge 1} \\frac{\\mu(n)}{n^s} = \\frac{1}{\\zeta(s)}", "95b3397f932bb7eca6336869a3021093": "H_0M = V^* = (L\\oplus\\bar{L})^\\perp\\sub T^*M\\otimes{\\mathbb C}.", "95b3f24763abe325943613d27e386ee9": "(6,1,1)", "95b427d36d1c5a74bbcbe16becc45d5e": "D_x(t,f)=G_x(t,f)\\times W_x(t,f)", "95b47320145777e55e9b98faf423fa99": "A_i=\\begin{pmatrix}\ns_{i-1} & s_i\\\\\nt_{i-1} & t_i\n\\end{pmatrix} \\,.\n", "95b4d2111d86cf72ae40f754bfcc23bf": "\nE[L(t)] - E[L(0)] \\leq Bt - \\epsilon \\sum_{\\tau=0}^{t-1}\\sum_{i=1}^NE[Q_i(\\tau)]\n", "95b4ea1adc324e8383645d9ebbc95b81": "(x,y,z) ", "95b5248674f7c100916e37400e30841c": "g_1 \\ne 0.", "95b579ef30e30b29d988a6895e071b6b": " C(x+t) ", "95b5ab09bdab03f7471c815df4f6f758": "P(r) = \\frac{3}{a}\\left(\\frac{r}{a}\\right)^2 e^{-(r/a)^3}\\,.", "95b5c70994dee8f51f56d579ecbd6591": " \\left\\{ x \\mid ( x \\in x ) \\to Y \\right\\}", "95b5f4e3563bf845db52acd7ee20ae1b": "\\langle \\psi | A^\\dagger", "95b5f8798f08d66fca992efa3b595956": "\\forall \\epsilon > 0 \\,\\, \\alpha_n(\\epsilon) \\to 0 {\\rm \\;as\\; } n\\to \\infty,", "95b6802df4a95ccd35f01e466a9a7617": " \\frac{\\mathbb{P}( \\theta_{up}(X) < \\theta | \\theta) }{ \\mathbb{P}( \\theta_{up}(X) < \\theta | 0 ) } \\leq \\alpha' \\text{ for all } \\theta.", "95b6909657fc9aff11af70f5afd0e155": "f_{\\Delta}\\,", "95b7d14006e5d176a27b0c92242f872b": "\n\\Omega^{2}(t) = \\omega^{2}(t) - \n\\frac{1}{2} \\left( \\frac{d\\beta}{dt} \\right) - \\frac{1}{4} \\beta^{2}.\n", "95b8299c782507ff9e00a7f8c9c6fca9": "1/(ik_2)", "95b87b10971e00044fbee063da45f19a": "Te_k = \\sum_{j=1}^n a_{jk} e_j", "95b8ffef81360441078c756a44167692": " |\\Psi\\rangle = \\sum_{s_{z1}} \\sum_{s_{z2}}\\cdots\\sum_{s_{zN}}\\int_{V_1}\\int_{V_2}\\cdots\\int_{V_N} \\mathrm{d}\\mathbf{r}_1\\mathrm{d}\\mathbf{r}_2\\cdots\\mathrm{d}\\mathbf{r}_N \\Psi |\\mathbf{r}, \\mathbf{s_z}\\rangle ", "95b99abc512cf0cbc654090cbb0bdcf1": "V_{-\\Delta y}\\text{ and } V_{+\\Delta y} ", "95ba0a6acd833dcd2bc93eea31b1cbad": "(p-k')^2\\approx \\,", "95ba0edabf5e89e0ab63eb43a485a62b": "u_1 = v_1 + \\dot{u}_x", "95bab664ebe8e166d6ee472315de024c": "Y_{n + 1} = Y_{n} + a(Y_{n}) \\delta + b(Y_{n}) \\Delta W_{n} + \\frac{1}{2} \\left( b(\\hat{\\Upsilon}_{n}) - b(Y_{n}) \\right) \\left( (\\Delta W_{n})^{2} - \\delta \\right) \\delta^{-1/2},", "95bb1487dd3938ec83beae280f67e556": "\\bold{P}_i", "95bb734d697788e20f112580df9dac99": "\\bar{\\beta}=\\beta\\left ( 1-\\frac{K^2}{4\\gamma^2} \\right )", "95bb79a567a35607fc7f79061eef24ec": "Y = 0.299 \\times R + 0.587 \\times G + 0.114 \\times B + 0", "95bb96bce16e8443188cfc00b47ad1d5": " \\Omega \\times \\Omega ", "95bbd7d715d2d83cb04c72c5bec06386": "(x^{q^{2}}, y^{q^{2}}) = \\pm q(x, y)", "95bbdbb0b32b16864304637caa58a1ed": "\\ A+(B.C) = (A+B).(A+C).", "95bbf8acd48c0cedc743ac06fd2a8262": "\\sigma_i^2", "95bc177dd45ee0c99a4fdf5b94625468": " t > s ", "95bc27d7273c3a8ea233c94730870fc0": "\\mathbf{v_n}", "95bc413473f4983cf655690de9f7a6f6": " \\scriptstyle{\\phi(\\xi)=\\frac{1}{\\sqrt{2 \\pi}}\\exp{(-\\frac{1}{2}\\xi^2})} \\ ", "95bc5344e591c0df6cc290961b24122d": " d\\tau^2= \\left(1-\\frac{2M}{r} \\right)\\, dt_r^2- 2\\sqrt{\\frac{2M}{r}} dt_r dr - dr^2-r^2 \\, d\\theta^2-r^2\\sin^2\\theta \\, d\\phi^2\\,", "95bcde622d57467bd7e9931cbe8f8240": "\n\\begin{array}{cl}\n\\displaystyle\\frac{x:\\sigma \\in \\Gamma \\quad \\sigma \\sqsubseteq \\tau}{\\Gamma \\vdash x:\\tau}&[\\mathtt{Var}]\\\\ \\\\\n\\displaystyle\\frac{\\Gamma \\vdash e_0:\\tau \\rightarrow \\tau' \\quad\\quad \\Gamma \\vdash e_1 : \\tau }{\\Gamma \\vdash e_0\\ e_1 : \\tau'}&[\\mathtt{App}]\\\\ \\\\\n\\displaystyle\\frac{\\Gamma,\\;x:\\tau\\vdash e:\\tau'}{\\Gamma \\vdash \\lambda\\ x\\ .\\ e : \\tau \\rightarrow \\tau'}&[\\mathtt{Abs}]\\\\ \\\\\n\\displaystyle\\frac{\\Gamma \\vdash e_0:\\tau \\quad\\quad \\Gamma,\\,x:\\bar{\\Gamma}(\\tau) \\vdash e_1:\\tau'}{\\Gamma \\vdash \\mathtt{let}\\ x = e_0\\ \\mathtt{in}\\ e_1 : \\tau'}&[\\mathtt{Let}]\n\\end{array}\n", "95bcf83fe1260cb1a5de43fbc3ad0697": " [\\operatorname{E}(R_a) ] = R_f + [\\operatorname{E}(R_m) - R_f] * [ \\rho_{am} \\sigma_a \\sigma_m] / [ \\sigma_m \\sigma_m ] ", "95bd193330e7a10c32da7d263cffbee0": "\\frac {a} {b} \\times {bd} = \\frac {c} {d} \\times {bd}", "95bd1eea58f0486fc64db01d4317a4c4": "S_m - S_n \\le a_{m}", "95bd33d2be4963252ad5fea0a671f460": "T\\subseteq S", "95bd43c48afc1e0b5e97719181ada2d5": "\\vert\\{ n : \\phi(n) \\le x \\}\\vert = \\frac{\\zeta(2)\\zeta(3)}{\\zeta(6)} \\cdot x + R(x) \\ ", "95bd521b8e40e3c651d8b3374a7dbd53": "\\epsilon_{ab}", "95bd5841bce8266e10dc525eb2ddf6b0": "W = (w_{i,j})", "95bdc923237ae7157374b2d3e4c15876": "p = {d - c}", "95bdec5699059b0c12cbe325a31157eb": "\\mbox{NDVI}=\\frac{(\\mbox{NIR}-\\mbox{VIS})}{(\\mbox{NIR}+\\mbox{VIS})}", "95bdf69392f5c4806510423c51c754a8": "\n\\partial_t \\tilde{u}=D\\partial_x^2\n\\tilde{u}-U(x)\\tilde{u},\\quad U(x) =\n-R^{\\prime}(u)|_{u=u_0(x)}.", "95beb8271bda82b111e9f39f86cb6cdf": "\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx=\\frac{a_{n}}{a_{n-1}p_{n-1}(x_{i})}\\int_{a}^{b}\\omega(x)p_{n-1}(x)^{2}dx ", "95bee27768d124b99ca32653cda7e83c": "\\operatorname{DS}(t,\\hat t) = \\frac{100}{n-1}\\sum_{i=2}^{n}d_i,", "95bf331d8c41d3a64e50080a6904800b": "\n \\operatorname{E}[X] = \\int_{-\\infty}^\\infty x\\,f(x)\\,dx.\n ", "95bf3be733e5846b92739648cbf47046": "\\mathbf{v} = (r, \\angle \\theta, h)", "95bf516da9f607eed373ccda56899a9c": "\\left [\\begin{smallmatrix}\n1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{smallmatrix}\\right ]\n", "95bf5265c3c706e211bb505093e5cfeb": "V_{nk} = \\frac{1}{2\\pi} \\int_0^{2\\pi} V(x) \\ e^{2\\pi i (k-n) x} dx .", "95bfc9e6b9b61117961c66e808e8bdc7": "1 + 9\\left(\\frac{1}{4}\\right)^4 + 17\\left(\\frac{1\\times5}{4\\times8}\\right)^4 + 25\\left(\\frac{1\\times5\\times9}{4\\times8\\times12}\\right)^4 + \\cdots = \\frac{2^\\frac{3}{2}}{\\pi^\\frac{1}{2}\\Gamma^2\\left(\\frac{3}{4}\\right)}.", "95bfcd034051f7d8c1d7f371f2e243fe": " g(r) ", "95bff72d645a971db52ac6ad81a76456": "\\scriptstyle\\big\\lfloor \\frac{2}{2-(r-1)^2}\\big\\rfloor", "95c0459c1548c3300e314455ca204b53": "x^* \\in X^* \\backslash \\{0\\}", "95c077153509b7135e0a6f481848c5d5": "(x,y)=(11,5)", "95c095e50b144e04c27baa0dfb9dda89": "g \\frac{\\partial h}{\\partial x} + g (S - S_f) = 0 ", "95c165a6b4606c048308c3950764b345": " \\frac{\\Delta W}{2 U^2}", "95c188f6931b02d2a013e0b841ad3b2a": "O \\to O' \\equiv U O U^{-1} = \\pm I (O) \\pm I = O", "95c18d1585caf304a5f2dae630b03ff5": "N < n ", "95c1f22403bb5b706fd391a185027148": "m | b - a | \\leq \\int_{a}^{b} f(t) \\, \\mathrm{d} t \\leq M | b - a |.", "95c2007d978a793a64ac543fdf4e6725": "\\log_{10} (f_2/f_1)", "95c219d535f995d5ec6806163a01a1f5": "\\, v = v^i e_i = \\begin{bmatrix}e_1&e_2&\\cdots&e_n\\end{bmatrix}\\begin{bmatrix}v^1\\\\v^2\\\\\\vdots\\\\v^n\\end{bmatrix} ", "95c28f066d68d8ad7571072b70ae6710": "\\left( \\frac{\\pi}{H} \\right)^2 + \\left( \\frac{2.405}{R} \\right)^2", "95c2999fdc3abfdce0f4c0812dd2a32b": "a_{r(m)}, b_{r(m)}, c_{r(m)} \\in \\mathbb{Z}", "95c2c97d8d291694cd2cb476db9a3e3d": "\\int f \\, dg", "95c2e02f8649fef5e36096202dd6bb45": " \n\\nabla C(n) = -2E\\left\\{\\mathbf{x}(n) \\, e^{*}(n)\\right\\}\n", "95c30ddb7da0de570ba6caa8014cb953": "\\begin{matrix} {2 \\choose 2}{2 \\choose 2}{44 \\choose 1} \\end{matrix}", "95c36812cfa336d0e782b765a5a162f1": " \\frac{1}{\\sin \\theta}\\! ", "95c36ecab0ee49a8036c53b9b5ea539e": "\\prod_{i=1}^n (t - x_i) = \\sum_{k=0}^n (-1)^{k} a_k t^{n-k}", "95c3a09768d2ad5057a7919c6d40b0b1": " ESS = ||{\\hat y} - \\bar y||_2^2 ", "95c3c49da01d6133219e658914aacc32": " \\operatorname{I}: X \\cong \\operatorname{Prim}( \\operatorname{C}(X)).", "95c42f2a9d1ace63fe4cec401c0744f7": "c_2 = {1 \\over 2} c_0 + {1 \\over 2} c_1", "95c4bdcfbd400e64db3425ee2f108ed4": "g \\left( t \\right)", "95c4e943facd23bdbd33153bab0b19fc": " f(x+y)=f(x)+f(y). \\ ", "95c576e70bfd43666991e23e08ad3c6c": "(5 \\; 13 \\; 6 \\; 9)", "95c5ba33fdfe1a1cf5060d971a39bb25": " P / B", "95c5bca3224ceb69cdfe85089cc57a1f": "C\\in V\\iff f^{-1}[C]\\in U", "95c627600744f032d318d9c1146af79d": "[K_m,K_n] = -i \\epsilon_{mnk} J_k ~,", "95c6811425c96e570942713d75a1b5c3": "H=-\\sum_{n=-\\infty}^\\infty c_n \\phi_n", "95c698cd26b2f8644947d19303157a26": "r=a \\cos (k\\theta)\\,", "95c7151748fa1098b3b721c939b8fe8a": "\\mathfrak{m}_{\\mathbb C}", "95c768dcac92ab96ca8f273ebe0326ea": " m \\ddot x + c \\dot x + k x = 0,\n\\qquad x(0) = x_0,\n\\qquad \\dot x(0) = 0, \\qquad \\qquad (10) ", "95c776bbbcf273fd031574ef9721dca3": "O(pn(m+n\\log n))", "95c7c080f422e0f1954ea6a9659fcb4e": "\n\\begin{align}\n\\frac{\\partial f}{\\partial\\nabla\\rho} = \\frac{\\partial f}{\\partial\\rho_x} \\mathbf{\\hat{i}} + \\frac{\\partial f}{\\partial\\rho_y} \\mathbf{\\hat{j}} + \\frac{\\partial f}{\\partial\\rho_z} \\mathbf{\\hat{k}}\\, , \\qquad \n& \\text{where} \\ \\rho_x = \\frac{\\partial \\rho}{\\partial x}\\, , \\ \\rho_y = \\frac{\\partial \\rho}{\\partial y}\\, , \\ \\rho_z = \\frac{\\partial \\rho}{\\partial z}\\, \\\\\n& \\text{and} \\ \\ \\mathbf{\\hat{i}}, \\ \\mathbf{\\hat{j}}, \\ \\mathbf{\\hat{k}} \\ \\ \\text {are unit vectors along the x, y, z axes.}\n\\end{align}\n", "95c800b0c671c166db6ffa5504062d52": "\\langle\\psi_{jk},\\psi_{lm}\\rangle = \\delta_{jl}\\delta_{km}", "95c822406dffca3dc00a5d6057adb519": "Tf(x) := \\int_{\\mathbb{R}^n} m(\\xi) \\hat f(\\xi) e^{2\\pi i x \\cdot \\xi} d\\xi,", "95c87001f2ac78584f5666d4ccbc0859": "\\sum_{n=-\\infty}^\\infty\n\\frac {\\Gamma(a+n) \\Gamma(b+n)}{\\Gamma(c+n)\\Gamma(d+n)} = \n\\frac {\\pi^2}{\\sin (\\pi a) \\sin (\\pi b)}\n\\frac {\\Gamma (c+d-a-b-1)}{\\Gamma(c-a) \\Gamma(d-a) \\Gamma(c-b) \\Gamma(d-b)}.", "95c870f263c9945ba513f8106d8ff517": "\n\\mathbf{a} = \\begin{bmatrix}a_x\\\\a_y\\\\a_z\\end{bmatrix},\n\\mathbf{b} = \\begin{bmatrix}b_x\\\\b_y\\\\b_z\\end{bmatrix},\n\\mathbf{c} = \\begin{bmatrix}c_x\\\\c_y\\\\c_z\\end{bmatrix}\n", "95c8889983c78cfacb9543c3cb3b88e3": "\\phi_1\\colon \\alpha\\mapsto\\varepsilon_\\alpha", "95c8a050ed8a4244060f7a40a5e2fbfd": "\\mu \\left( \\Omega \\cap \\mathbb{B}_{r} (p) \\right) \\leq C \\sigma \\left( \\partial \\Omega \\cap \\mathbb{B}_{r} (p) \\right),", "95c8cad8387409dd07ed7bf62ae919a6": "ax+by+cz+d=0", "95c8d55251891eba0af2bb6c0cc912d8": "\\log(e) = 1 + 2 \\pi i k", "95c9602b09577da28c8bf144be101365": "\\phi \\colon D^{\\mathrm{op}}\\times C\\to\\mathbf{Set}", "95c98973a6b3b18a352f99dba024e94f": "N_2=\\frac{4n}{2m+2n-mn}", "95c98bcab5c6eed0b031ae098c6a4c42": "\\Pi^0_{k+1}", "95c9bfdac01b57e4d6e493e492124fe0": "\\sqrt{6} \\rho^2 \\cos 2 \\theta", "95c9ca3efc4360c6762b641838771cbc": " S = -\\sum_j \\eta_j \\ln \\eta_j.", "95ca02040b2e00e8517f5f7995cef768": "\\mathbf{V}^{-1}", "95ca3bbaf8437e5a828c2d4cad0fd0c9": "\\text{WAL} = \\sum_{i=1}^n \\frac {P_i}{P} t_i,", "95ca4d2efed60c62f42f12dca87f9b45": "\nd\\ (\\log\\rho)/d\\ (\\log r) \\propto -r^{\\alpha} .\n", "95ca70ebde097b600e331e657d6bdc36": "E_i^a E^i_b = \\delta_b^a", "95ca8caf531be68d1407d909b3091319": "W^s(f,p)", "95cab7aaab8fcf0e95c39d924d1dc4ef": " \\angle DVC = \\angle EVC - \\angle DVE ", "95cafe8da83c664a03133d23f5b44f3c": "\\Gamma_1 = \\frac{Z_1 - R_0}{Z_1 + R_0}", "95cb4b9d97f97c7a9539f185d7751d0f": "q^n(x^n) \\le (n+1)^{|X|} 2^{-nD_{\\mathrm{KL}}(p^*||q)}", "95cb739130a965deaa064c8e6c066fa3": "\\nabla\\times(\\mathbf{A}\\times\\mathbf{B}) = (\\mathbf{B} \\cdot \\nabla)\\mathbf{A} + \\mathbf{A}(\\nabla\\cdot \\mathbf{B}) - \\mathbf{B}(\\nabla\\cdot \\mathbf{A} ) - (\\mathbf{A}\\cdot \\nabla) \\mathbf{B} ", "95cb7beca932ae6483319e24784ebfea": "\\rho = \\,", "95cb801c616ef198acfb94448372c892": "\\begin{align}\n\\mbox{Maximize } & c^Tx \\\\\n\\mbox{Subject to } & Ax = b, \\\\\n & x\\geq 0,\\, x_i \\mbox{ all integers}. \\\\\n\\end{align} \n", "95cbda1a7df052653c62e3f2cc4a76eb": "{\\bar{R}}_3", "95cbdd59078c84b285f94840880d3221": "\\mathbf{e}^i\\cdot\\mathbf{e}_k=e^{ij}\\mathbf{e}_j\\cdot\\mathbf{e}_k=e^{ij}e_{jk} = \\delta^i_k.", "95cbddca4a4b04ae8cfd2109ce8202f5": "\\sqrt[12]{2}\\sqrt[7]{5} = 1.33333319\\ldots \\approx \\frac43", "95cc06eae97f22b21a836a8214bb8803": " N=\\frac{\\mu V}{2 D_m}", "95cc235648759eb9808b8facb2864bb5": "Z_2^6", "95cc3a94d45a0315ea3ea3ce28aed3db": "\\hat {f}[\\delta (x)] =1", "95cc5362ff480c7ccb3614744e3922f9": "\\frac{1}{\\pi} = \\frac{12}{640320^{3/2}} \\sum_{k=0}^\\infty \\frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}", "95cc659955067f61898836ae532a1c50": "L-\\Delta L = \\left ( R-r \\right )\\Theta ", "95cc714927b41adf64f75aedc8d717ed": "{{u(X)}^{*}} = \\sum\\limits_{i=1}^N \\alpha_i\\phi \\left( r_i \\right) + \\sum\\limits_{k=1}^M \\alpha_{k+N}\\gamma_k\\left(X\\right),\\qquad (5)", "95ccadac79bbb98a0454877dd5538766": "W_1 < K N_1 \\frac{1}{1 - \\rho}", "95cd011c5c4502ce625942b28cc39873": "\\displaystyle{I}", "95cd38d9ada2c6cd09ba2d73d8c066ef": "H = H \\supset V(H) \\supset V^2 (H) \\supset \\cdots = H_0 \\supset H_1 \\supset H_2 \\supset \\cdots, ", "95cd5e9b915b97fb887a4f7deab53f3a": "L_3 + L_1 \\rightarrow L_3.", "95cd901b23f7c3ad838e992109577d0d": "\n\\begin{align}\nK &= A_\\text{rect} - 2 \\times A_\\text{tri} \\\\\n&= \\left( (B+A) \\times H \\right) - \\left( A \\times H \\right) \\\\\n&= B \\times H \\\\\n\\end{align}", "95cda2cf0219cccee7ad1fb3f3b7318e": " (x(t),y(t),z(t)) ", "95ce726c8cffd20545b6b282af27fd8b": "{\\Pr}_{h \\in H}[h(a)=b, h(a')=b'] = \\frac {\\epsilon}{\\left\\vert B \\right\\vert} ", "95ceb7704216ce4faa053dc4a4634d5a": "x=1+\\sqrt[5]{2}-\\sqrt[5]{4}+\\sqrt[5]{8}-\\sqrt[5]{16}\\,.", "95cf2e64ff3fd1f08d7b93356585b945": "\\widehat{f}(n)=\\frac{1}{2\\pi}\\int_0^{2\\pi}e^{-inx}f(x)\\,dx,\\quad n=0,\\pm1,\\pm2,\\dots.", "95d002614c42d05783ffea9788f38ce2": "\\text{GDP}_t", "95d036546d6ab206d49a677772d3f644": "v_{th}=\\sqrt{\\frac{2 k_BT}{\\pi m}}", "95d03d00ce32c344617352d55c6a4614": "dz-E_t h(x,t) dt", "95d05faaa65c0b29f772572b51938d6f": " \\Box A^{\\alpha} - R^{\\alpha}_{\\ \\beta} A^\\beta = -\\mu_0 J^\\alpha", "95d0c563475c22acdc7590ab10fa211b": " \\left | \\alpha -\\frac{p}{q} \\right | < \\frac{1}{q^2} ", "95d0ee311465e79b6704370ac77742ad": "\\boldsymbol{\\Phi}=\\operatorname{diag}(f_1,\\ldots, f_{3N-6}) ", "95d1273d55c9c9d2a70069d8d44e47e4": "f \\mathrm{inv}(f) = \\mathrm{id}_y", "95d12c079a5a8545c82351a3da8e94b6": "P_e = \\sum_c \\frac{n_c^2}{n^2}", "95d163f6f935fe0d5246810ee64bee16": "\\frac{a-b}{a}\\,\\!", "95d16e1b24a0ad10ccf386d860fe615c": "e=u(1+t^2)(1+v^2)", "95d209cc7d30a418e94a688eaa95ba26": "\\int \\mu(\\mathrm{d}\\bar\\omega)Z_\\Lambda^\\Phi(\\bar\\omega)^{-1} \\int\\lambda^\\Lambda(\\mathrm{d}\\omega) \\exp(-\\beta H_\\Lambda^\\Phi(\\omega | \\bar\\omega)) 1_A(\\omega_\\Lambda\\bar\\omega_{\\Lambda^c}) = \\mu(A)", "95d24f81aa66cecf4b2e23015f646c70": "d\\omega = 0", "95d27fe4b554ec52635aadc2091f8113": "\\bar f(x)", "95d296f136652fd9a54c1bf0de989327": "E_G^0 \\varphi = \\varphi", "95d2a6d1bbada9d2bc794fd8e884f8e5": "\\, p_0 \\,", "95d2cd5a3b7a09f73ef0e82e170d7d9f": "10 \\uparrow^n 10", "95d2d086dc993419324b2fc98e7ee3b3": "D_K", "95d2e4140e5ac18e18ef2823090218f1": "\\gamma'p", "95d303be0697e9f05f749b751cca6d79": "\\scriptstyle \\frac{\\partial M}{\\partial x} = \\,0", "95d353534c009873bf23ec859de37d5c": "\\frac{L}{L_{\\odot}} = \\left(\\frac{M}{M_{\\odot}}\\right)^\\alpha", "95d3b369efe1bff50628efd68fe992b1": "\\|T\\| \\leq \\|T\\|_2 \\leq \\|T\\|_1 ,", "95d3c485f8e6780837f3b2b58b4e3eb8": "DP_{p,c} = \\frac{\\sum_{p,c}(DP_{T}^{V})}{count_{p,c}(DP_{T}^{V}<>NULL)}", "95d3e75f99f7b0c72b5b793e609eda41": "\\int \\theta \\, d\\theta = 1 ", "95d43a54fe14ba8c8569784c80437c21": "T(n) = a \\; T\\!\\left(\\frac{n}{b}\\right) + f(n) \\;\\;\\;\\; \\mbox{where} \\;\\; a \\geq 1 \\mbox{, } b > 1", "95d5e856ecee515fbae8d8a1e5f40b3e": "p_{CJ}\\,", "95d6114eb428c9a246207dfb45c28fdc": "\\theta(a) \\cap C_G(b) \\subseteq \\theta(b).", "95d63837585ac663a1be3843f25b8aed": "n_e^2 = 1", "95d64d7da4f63562324bcb932ffc5a8c": "\\Delta(X) =\n\\min\\{|G| : G\\neq\\emptyset, G\\mbox{ is open}\\}", "95d66c0b3d0f46602ca58ea37d7c8de9": "R_{\\alpha \\beta \\gamma}^{\\;\\;\\;\\;\\;\\; \\delta} V_\\delta = (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_\\gamma.", "95d699e31a95d29904baf1cd1ca3899a": "{\\iint F(p,q,z;x,y) dx dy} = \\text{Minimum} \\qquad \n\\left[ \\frac{\\partial z}{\\partial x}=p \\quad;\\quad \\frac{\\partial z}{\\partial y}=q \\right]", "95d6ab6a4bb68f190a33906dd1f4de1c": "\\dot{\\omega}\\,", "95d6d465551723fae46732361c46cbba": "\\le \\sum_{k=-\\infty}^{\\infty}{\\left|h[n-k]\\right| \\left|x[k]\\right|}", "95d6d8d34be5fbe4115f953aa6f2a3d7": "T_{n}(x)=e^{-x}\\sum_{k=0}^\\infty \\frac {x^k k^n} {k!}.", "95d6e99bbaf07852e2cd3eed0c42278e": "E(\\widehat\\beta_j)=\\beta_j\\,", "95d73adb7c2a596aaee3e2cc21092892": " h'(P) \\gets \\max(h(P), h(N) - c(N,P)) ", "95d7639b0fc0f16f5fad86a80a91b371": " F_\\theta = \\mathbf{F}_A \\cdot \\frac{\\partial\\mathbf{v}_A}{\\partial\\dot{\\theta}} - \\mathbf{F}_B \\cdot \\frac{\\partial\\mathbf{v}_B}{\\partial\\dot{\\theta}}= a(\\mathbf{F}_A \\cdot \\mathbf{e}_A^\\perp) - b(\\mathbf{F}_B \\cdot \\mathbf{e}_B^\\perp).", "95d79d5d6dd4ddc5e3bea634dd8b9233": "\n\\|x - y\\|_{A}^2 = (x-y)^{T}A(x-y)\n", "95d7bf960924ed0a99fa7b46b437b7aa": "\\sum_{i=1}^{n} x_i", "95d7f9511ddf8cb14f94989465fbc848": "{\\mathcal O}(n^{40})", "95d817eef2ed6a173e9b1fbf64137289": "\n\\varphi(e^{-\\pi x}) = \\vartheta(0; {\\mathrm{i}}x) = \\theta_3(0;e^{-\\pi x}) = \\sum_{n=-\\infty}^\\infty e^{-x \\pi n^2}\n", "95d819d15c33b32c73eaad7493fd4b60": "1 - \\epsilon ", "95d88458801bafa560c1bc65b92eae9f": "\nA(D) = \\iint_D\\left |\\vec{r}_u\\times\\vec{r}_v\\right |du dv.\n", "95d9145abd8e605f870532a92e4bd12f": "y \\geq 0", "95d93f069ef4343b287a633468555949": "0\\,\\mathrm{V} - V_S = -V_S", "95d964bf53931ca66998ee299cdf3461": "\\frac{r-1}{r}", "95d9d22779a81bfb65539bd56565edbb": " v_p = a \\overline{M} + b \\rho ", "95d9d7d9c9884eff031d975b13214021": " \\mathbb Q (\\zeta_n +\\zeta_n^{-1}). ", "95d9e7515214c96b704bb30ad07ee826": "\\{(+,-,-,-); l^an_a=1, m^a\\bar{m}_a=-1\\}", "95d9f456e48bf2c33fedc2132155522e": "I_b(u) = H_p(x)-E_z \\left[ H_b(x'|z,u) \\right].", "95da324ac2b1c1ce90000690ea392b87": "y = e^{\\sin {x^2}}.", "95da3cc99d89ddcccd8b52a516d913af": "\\frac{1}{0.3+\\frac{1-0.3}{4}} = 2.105", "95da8532a7ae9b81311d1441b2cf724f": " \\phi = || d - G m ||_2^2 \\, ", "95daf57dead122e31fe57e60b8a77a52": " r^{5} + r^{4} - 4r^{3} - 16r^{2} -20r - 12 = 0 \\, ", "95dafd34d0a9e4ddcde829d590b17ea6": "\\sqrt{\\frac{1}{3}}\\!\\,", "95db2c0e66fa356122fbd2c4a3e177fd": " x = a (1 + 2\\cos t + \\cos 2 t), \\,", "95db837f25537cd5a26de6f3571d044a": " \\pi - \\arccos{\\left( \\frac{1}{\\sqrt{3}} \\right)} ", "95dba8fc5c9b2fa2162dd96fdd6c1a61": "(T_w)_{w\\in W}", "95dc0ea622e0dc1369f861c17fbf7430": " \\sum_{k=1}^\\infty \\; \\frac {1} {k^p}, \\qquad ", "95dc3b453eb6595a6f30328d59e79301": "\\frac{d}{dx} f(x) = g(x)h(f(x))", "95dc53a13bdf9d367ed43b1d760db7b2": "z=\\dfrac{x-a}{b-a}", "95dc671efde300453964efc6ecfc5897": "\\{\\mathbf{X}_k:k=1,2,\\ldots,n\\}", "95dca28228af37b78df769d286b96896": " \\alpha (t - t') = \\frac{1}{2\\pi} \\int_0^{\\infty} J(\\omega) e^{-\\omega|t-t'|} d \\omega ", "95dd1129a45d91551d8a9f46b5d835c8": " X = \\int\\limits_0^\\infty e^{A \\tau} Q e^{A^H \\tau} d\\tau ", "95dd3b6cf0dcc19b3717955fe8752f49": "\n \\lambda = r\\,\\frac{p}{1-p} \\quad \\Rightarrow \\quad p = \\frac{\\lambda}{r+\\lambda}.\n ", "95de30db0a0bff55686c9e3087d4ba2e": "V=\\int w\\,dx", "95de3d0e64a9844a0f8a3a8da39cd355": "\\scriptstyle \\delta n_0 \\eta", "95de4e5f21e00955efaf4f832254237a": "(P \\or Q) \\leftrightarrow (\\neg P \\rightarrow Q)", "95de6c7c62456ede7ecfb3cceb4d476d": "\\Delta\\mu_1=\\Delta\\mu_1^{ideal}+\\Delta\\mu_1^{excess}", "95deb18c82c0a7dca761c9e957b0b6ab": "I\\cap N\\neq \\left\\{ 1\\right\\} .", "95deb492c01db495cd62d4543ae03225": "c_Ls_a + c_{L-1}s_{a+1} + c_{L-2}s_{a+2} + \\cdots = 0", "95dec44ca8d36b57b57b00076c9cd22c": "a*b=\\overline{ab}.", "95df490e67c2abbc73dd9075cefc3442": "\nt \\rightarrow \\lambda^{3}t , \\qquad \\mathbf{r} \\rightarrow \\lambda^{2}\\mathbf{r} , \\qquad\\mathbf{p} \\rightarrow \\frac{1}{\\lambda}\\mathbf{p}~.\n", "95df90448587c27ffdfe39e378efcb62": "A^2 +B^2 + C^2 > 0", "95dfbfe1a8e6bbe0685e3b12b90f6f3c": "\\mathbb{R}^{d+1}", "95e01c2110cd0ad49aa63254d94d8e29": "\\mathbf{v}_k = \\sum_{i=0\\atop ai+k < n}^{n/a} \\mathbf{e}_i", "95e040a63b9e5e28b05100d0f0876174": "\\mathbf{\\hat{u}}", "95e04d37dfd694ebfeb7be1473c182cf": "\\Delta\\downarrow{X}", "95e090c85b0a01c76498b1ea569812b2": "\\begin{pmatrix}\\!-3&\\,\\!2\\\\ \\,\\!3&\\!-4\\end{pmatrix}", "95e0a7d8e180ac8794b333dcff35cde5": "\\scriptstyle{1\\mapsto12}", "95e0b0defd5103305b44eb8b04aca7b2": "\\{\\mathcal{H}_n \\}", "95e0bdd28694e9701c200fbe35319be4": "\\cdots\\rightarrow H^{n}(X;G)\\rightarrow H^{n}(A;G)\\oplus H^{n}(B;G)\\rightarrow H^{n}(A\\cap B;G)\\rightarrow H^{n+1}(X;G)\\rightarrow\\cdots", "95e109d082600d931b3ea816065308cf": "\\alpha = \\lim_{k\\to\\infty} \\frac{\\log|w' - z^*|}{\\log|w - z^*|},", "95e11ce15d223d9777063404e0861d3e": "P\\quad=\\quad\\begin{matrix}1&1&2&2\\\\2&3\\\\3\\end{matrix}, \\qquad Q\\quad=\\quad\\begin{matrix}1&1&1&3\\\\2&2\\\\3\\end{matrix}.", "95e1a758f4dff1dc88fb0f56c27385a8": "f:\\mathbb{Z}^n\\mapsto\\mathbb{R}^n", "95e1ad7a07bf33b5916a9407259c4989": "b_1, \\ldots, b_n \\in \\{0,1\\}.", "95e2127077e21942b3f1ee9b9e16a8a3": "P_1 = \\frac{\\mathrm{outs}}{\\mathrm{unseen}\\,\\,\\mathrm{cards}}", "95e23ba5e46b30f59b8ebce46b5d2d16": "\\begin{bmatrix}\n\\cos\\,\\theta & -\\sin\\,\\theta \\\\\n\\sin\\,\\theta & \\,\\cos\\,\\theta \\end{bmatrix}=\n\\exp\\left( \\theta \n\\begin{bmatrix}\n0 & -1 \\\\\n1 &\\,0 \\end{bmatrix}\n\\right),\n", "95e2d860d977b7608666804ebc1a693d": "a_1b_2", "95e2df9879987998e391abfca52e6879": "y(w) = \\sum_{m\\in Z}\\frac{(t^mw)^2}{(1-t^mw)^3} + \\sum_{m\\ge 1} \\frac{t^mw}{(1-t^mw)^2}", "95e3221d98fc1e290fa0a28141e116e9": "(q, \\delta\\geq 0, \\epsilon\\geq 0)", "95e3770fca65b20b3d7a66057a948346": "S \\subseteq [n]", "95e3c28a14a13925fb4f44eea8b9b88c": "V_s(q,\\omega=0) \\equiv \\frac{V_q}{\\epsilon(q,\\omega=0)} = \\frac {\\frac{4 \\pi e^2}{\\epsilon q^2 L^3} }{ \\frac{q^2 + \\kappa^2}{q^2} } = \\frac{4 \\pi e^2}{\\epsilon L^3} \\frac{1}{q^2 + \\kappa^2}", "95e3cc800f52e685dc4a55247e6731bf": "<\\Delta R_i \\cdot \\Delta R_j > = \\frac{3 k_B T}{\\gamma}(\\Gamma^{-1})_{ij}", "95e4016abf557814e389b742d97a6e21": "\n\\frac{d^{2}u}{d\\varphi^{2}} = \\frac{r_{s}}{2} \\left[ \\left( u - u_{2} \\right) \\left( u - u_{3} \\right) + \\left( u - u_{1} \\right) \\left( u - u_{3} \\right) + \\left( u - u_{1} \\right) \\left( u - u_{2} \\right) \\right]\n", "95e406cd732f31ff07d9a5838f9c5261": " (V \\otimes W)_i = \\bigoplus_{\\{j,k|j+k=i\\}} V_j \\otimes W_k ", "95e45cb52ae446481e8eef295595bbb9": "q(A)=A^2-5A+10I_2=12I_2\\not=0.", "95e4a0e5a318bf25a30bd6d6afee92fe": "r_{B}", "95e51a7d9c4368137531f32f15f3f324": " H(\\mathbf{p},\\mathbf{q};V)=K(\\mathbf{p})+\\varphi(\\mathbf{q};V) ", "95e582e79a87f3524297a063a88d8683": "\\ f_{X,Y}(x,y) = f_X(x) \\cdot f_Y(y) ", "95e5a155c19327c877e1395141bd2675": "\\mathcal{G}(a)-\\Lambda a=\\kappa \\mathcal{T}(a)", "95e5a1a3e5df4757676d18a865629aec": "r_a r_b+r_br_c+r_cr_a = s^2,", "95e5b842e1ec06fcc57b22ebbc025b8f": "\\varepsilon_{ij}", "95e62150c93da8c84044af17075dbf4f": "L_{\\omega_1}(x) \\subset H_{\\aleph_1}", "95e6355af0f32c34908898da74087f51": "\\mbox{For } \\gamma = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\mbox{PSL}(2, 7) \\mbox{ and } x \\in \\mathbf{P}^1(7),\\ \\gamma \\cdot x = \\frac{ax+b}{cx+d}", "95e695e47ae6223e895376fff372df30": "\\boldsymbol{\\mathcal{A}}(\\omega)\\boldsymbol{u}=\\Delta_2\\boldsymbol{u}+\\omega\\nabla\\left(\\nabla\\cdot\\boldsymbol{u}\\right)", "95e6ba52a00eca601a763bce4fc38a0a": "{r \\choose 2}", "95e6de75621336a010421f5129475a47": "\\tilde p(x)", "95e6f98933ba3e216584ec17a3449c0e": "X^3 = X \\times X \\times X", "95e72b6be5c48d87c9bbc7f0c02c4489": "x_j + y_j", "95e7786dfa69e2d36432b9bbed0e62ff": "\\alpha / A", "95e79040a6ae01f8d1f54546ba300dc6": "e:S^d\\hookrightarrow S^{d+1}", "95e7c3d6b28247287b94aaa4c8dd8d4d": "x=\\infty", "95e7cabe3ff886fc5f21c89c6e53a610": " \n\n\\frac {v_{\\ell}} {v_a} = \\frac {v_{\\ell}}{v_i} \\frac {v_i} {v_a} \n", "95e7fbee6f9c308ed9764cfdf3408037": "\\int_0^\\infty t^b e^{-at} \\,dt = \\frac{\\Gamma(b+1)}{a^{b+1}}.", "95e82474cf46ebf4420c498a457de477": "\\bar \\xi", "95e8448ac35c5eec8f55777b00870173": "\ne_1 e_2 \\overset{def}{=} e_1 \\circ \\operatorname{lift}(e_2)\n", "95e87868e26e4b823e22c09dfb7a820e": "\\{c\\} \\cup \\lambda\\setminus\\mu", "95e8b7678e15339e82df8661aa49357d": "q = \\tfrac14 \\left(L'^2+ 27M'^2\\right),", "95e939ef694fb90e775069f6007d4600": "f\\in L^2(Y)", "95e9477180b800df39f1fe45cd5c290b": "S \\subset [a,b]", "95e9ee05f4084d3eb44b06a199c9a33c": "V(K) = -\\frac12\\mu^2\\phi^\\mu\\phi_\\mu - \\frac14g(\\phi^\\mu \\phi_\\mu)^2\\;", "95ea1cb80e1f235db205c275990fe1f5": "d_A = \\frac{s k - z}{r}", "95ea8963d89c542e7cc61edcd749f1ee": "(v_\\alpha)_{\\alpha<\\lambda}", "95ea9c3e3991b8aa9b06edb70e7bd393": "ax^2+bx+c\\,", "95ea9d05c90d8775e002494628191917": "(k,\\theta)", "95eac7c219630898ced8183d8c02379b": " \\lim_{x \\rightarrow c} M(x)", "95eb004e286d37603829d53197ea2208": "c \\,", "95eb337a5c1a9e4224180af984a33f0a": "a(u,u) \\ge c \\|u\\|^2", "95eb7093c8469bb72b4cd2d933ad57e8": "\\begin{align}\n \\frac{1}{r\\sin\\theta} \\left(\n \\frac{\\partial}{\\partial \\theta} \\left(A_\\phi\\sin\\theta \\right)\n - \\frac{\\partial A_\\theta}{\\partial \\phi}\n \\right) &\\hat{\\boldsymbol r} \\\\\n+ \\frac{1}{r} \\left(\n \\frac{1}{\\sin\\theta} \\frac{\\partial A_r}{\\partial \\phi}\n - \\frac{\\partial}{\\partial r} \\left( r A_\\phi \\right)\n \\right) &\\hat{\\boldsymbol \\theta} \\\\\n+ \\frac{1}{r} \\left(\n \\frac{\\partial}{\\partial r} \\left( r A_\\theta \\right)\n - \\frac{\\partial A_r}{\\partial \\theta}\n \\right) &\\hat{\\boldsymbol \\phi}\n\\end{align}", "95ebab760f202c1d95c6e81d50098093": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} = \\sum_{i=1}^k \\mu_i \\nu_i = \\Delta_rG_{T,p}", "95ebf0f28915c0e535893d3fd3386464": "ax^3+bx^2+cx+d=0\\,", "95ec30806e6856c1e235a39bff1c5b62": "\\varphi(t)=2\\,\\frac{J_1(Rt)}{Rt}", "95ec639d27abe95b298530c742d5a3ab": "c=s+1", "95ec9058e14870597f88fbc82b6446df": "\n\\mathcal{Z} = \\left( 0, 1\\right) \\cdot \\left\\{ \\prod_{j=1}^{N} \\mathbf{W}_{j} \\right\\} \\cdot \\left( 1 , 1\\right)\n", "95ec960a942b8f61f52ef2fe96633493": "\\cos^3\\theta = \\frac{3 \\cos\\theta + \\cos 3\\theta}{4}\\!", "95ec96eebb2e288416f79659af08e9ab": " \\Sigma=\\frac{1}{\\sum_{i=1}^{N}w_i}\\sum_{i=1}^N w_i \\left(x_i - \\mu^*\\right)^T\\left(x_i - \\mu^*\\right), ", "95ec9d5e0aeb60246bee82ff11d5b40e": "Z=Z_0", "95ecd6cf8cbe2947ec5cc86e5bbfc5e6": "y_i^{\\prime\\prime} \\ne ?.", "95ecd8fa8cf9237ec97be073f1a287aa": " \\frac{n!}{\\prod_{i=1}^{k} x_i!} ", "95ecff36c56fb83a0d4e3dac7c6d2c15": "HS_S(t)=P(t)\\,\\left(1+\\delta\\,t+\\cdots +\\binom{n+\\delta-1}{\\delta-1}\\,t^n+\\cdots\\right)", "95ed31aa02eebcc785abc3c56dfd6d43": "\\mathbb R^d : \\mathbb{S} = \\{u \\in \\mathbb R^d : |u| = 1\\}", "95ed542896def09dd0b6649a04c4f075": "(-\\infty, b) = \\{x \\mid x < b\\}", "95edfa23f4c32e157773d60cec8d5edc": "T^2 = c \\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2 = c n S^2", "95ee0817bf3392f6918ae14fcd7dd980": "\\mathbb Z\\times\\mathbb Z", "95ee1a59ff4776c606f638867b47ab1f": " d \\left( \\sin{\\theta_i} + \\sin{\\theta_m} \\right) = m \\lambda", "95ee1d597206958cafab803864efbccd": "g\\phi_0 \\bar\\psi\\psi", "95ee47aca78c502550e8e978f7fd9fd3": "w_{k+1}", "95ee5d21383caa596da0caf48291b4a2": "{\\tilde{B}}_{5}", "95ee62a310766fc11a3c4a1254c9e0da": "\\mathbf{M}=\\sigma^2\\left(\\mathbf{X^TX}\\right)^{-1}", "95ee6b09215da961f5b1f48e4e94e4c2": "(\\forall x,y)\\ (\\exists z)\\ {\\rm multiply}(x,y,z).", "95eea9d6b9edc59548d9cc1e6b9a35ef": " \\left(\\tan(x)\\right)' = \\left(\\frac{\\sin(x)}{\\cos(x)}\\right)' = \\frac{\\cos^2(x) + \\sin^2(x)}{\\cos^2(x)} = \\frac{1}{\\cos^2(x)} = \\sec^2(x)", "95ef3571561af572cf3300cd33184353": "F = \\mu_f R \\,", "95ef76a448ff49242ba73040938d3a03": " b=\\sum_{i=0}^n (-1)^i d_i, ", "95effa2594b3219d8ecd5dd150cc5c16": " 2^{\\aleph_0} ", "95f01f4a318a35937c449af94e5775fc": " \\operatorname{E}(R_i) = R_f + \\beta_i (\\operatorname{E}(R_m) - R_f) ", "95f025c5719145f969d269e604218800": "\\phi \\approx \\frac 1{G_N}", "95f061b0dc1c0d04543edadcd1762a4e": "|\\bigstar | \\bigstar | \\bigstar", "95f097b72428c21d7b2a25964dd955a4": "\\operatorname{Div}", "95f0a8545c205c6db393f916db80ae82": "\\sum\\limits_{i=1}^m \\sum\\limits_{j=1}^n t_{i,j} x_{i,j} ", "95f0aec70554ff7c48c6201e4851e95d": " D[n] = [F_2, S_2, A_2]::[F_1, S_1, A_1]::\\_]", "95f0c4b13c2192365d6f59613db9dd5b": "\\text{Sp}(1)", "95f0ce0b3ea490674ddd354ed6b53314": "0, \\pm \\tfrac{1}{2}, \\pm\\tfrac{\\sqrt{2}}{2}, \\pm\\tfrac{\\sqrt{3}}{2}, \\pm\\tfrac{\\sqrt{4}}{2} = \\pm 1", "95f10965a95ac36b722f9065e9c2b373": "B_c = B ", "95f116cff2b7a5b692c96b0adbf0181a": "(i_0, \\ldots, i_4) = (0, 1, 4, 1, 0)", "95f130552e540193b062509b3879509e": "s = 0.7", "95f13bf773bd905d0cd8fafe83a84b9a": "\\langle a \\rangle = \\{ a^{k} : k \\in \\mathbb{Z} \\} ", "95f1a5701e0f18d09ab7f1c46a08c25b": "{\\rm Area}(\\mathcal D)={\\rm Area}(w).", "95f1c4478dbaa2f47031b2d92ae9fd8d": "m \\cdot n", "95f1e279d3091760b8673f1a3a5f2995": "\\mathbf{R}^{m}", "95f1f7a2413be4757a6e85dbce1cd20f": "x^2-a^2=0,", "95f205d7a9bec14e8adb74c1c19940bb": "f(t) = a*sin(t), a = 8", "95f2655cbba85f83a0703e89dbbc1163": "\\begin{align}\np(\\mathbf{X}|\\mu,\\sigma^2) &= \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{n/2} \\exp\\left[-\\frac{1}{2\\sigma^2} \\left(\\sum_{i=1}^n(x_i -\\mu)^2\\right)\\right]\n\\end{align}", "95f271a016c900881441ada6c90ed7c1": "T=\\{x\\}", "95f28f329962de4a1df226369d630a76": "s_N[n] = \\frac{1}{N} \\sum_{N} S_k\\cdot e^{i 2\\pi \\frac{n}{N}k},\\,", "95f2aa3b8e1375c6cd7813694f6b8c9f": " \\mathbf{w} \\leftarrow \\mathbf{w}^+ / \\|\\mathbf{w}^+\\| ", "95f313d89fa54b8ce8c339b02f2bf9a6": "u = \\tilde{u}", "95f330b89a19f467b2ab1e3c226a64d9": " e = m_\\text{e} = \\hbar = 1 \\ ", "95f3408902b303517c5fee2702499bd7": "P = I_r\\oplus 0_{d-r}", "95f38d4f1fc602f11d2c06dacdfe08e9": "\\vdash_{M}", "95f3cc4328cf01b14fb2078f5dceb1b3": "n^\\text{right}", "95f3e18fbbabdbe7ab0d8b0e1d18f58b": "\n\\begin{align}\n\nP[\\text{Suicide}]= {\\color{Blue}P[\\text{Suicide}|\\text{Protestant}]} P(\\text{Protestant})+ {\\color{Blue}P[\\text{Suicide}|\\text{Not Protestant}]}(1-P(\\text{Protestant}))\\\\\n\\end{align}\n", "95f4688541c359ae3b0df01914222fe1": " {^{y} x} ", "95f4862bd205baf8703d3df36b29d7b4": "A = (a_{ij})_{i,j = 1, \\dots, n}", "95f4a27d8beb4ee89cc134d8c8ce0a2a": " (v-b^*)\\frac{\\partial \\Pr(b^*\\ \\textrm{wins})}{\\partial b}-\\Pr(b^*\\ \\textrm{wins})=0 ", "95f4f20d5f19182d8fd397a99b4c101f": "a_1 = 0", "95f503f0d959b9e6c3f61060cfc4af66": " C_f = C_f ( \\alpha , M , Re , P) ", "95f5277b54235a03b905ac398f9c7d74": "dr<0", "95f5486f99d3210663eed357de0e8ba9": "\\lim_{j \\to \\infty} f(\\hat{x}_{n_j},\\theta_{n_j}) = f(\\hat{x},\\theta) > f(x,\\theta) = \\lim_{j \\to \\infty} f(x_{n_j},\\theta_{n_j})", "95f55f3ef9b4d7e1ff14a3954688a544": "PFER", "95f5657177bf2fce4d84a7384f606013": "\\psi_\\sigma(\\mathbf{r}, t) \\rightarrow D(\\Lambda) \\psi_\\sigma(\\Lambda^{-1}(\\mathbf{r}, t)) ", "95f56a75ad5878921e647eeea2caf02b": "\n\\begin{align}\n\\textbf{P}_{k\\mid k} & = \\textbf{P}_{k\\mid k-1} - \\textbf{K}_k \\textbf{H}_k \\textbf{P}_{k\\mid k-1} - \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} \\textbf{K}_k^\\text{T} + \\textbf{K}_k (\\textbf{H}_k \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} + \\textbf{R}_k) \\textbf{K}_k^\\text{T} \\\\[6pt]\n& = \\textbf{P}_{k\\mid k-1} - \\textbf{K}_k \\textbf{H}_k \\textbf{P}_{k\\mid k-1} - \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^\\text{T} \\textbf{K}_k^\\text{T} + \\textbf{K}_k \\textbf{S}_k\\textbf{K}_k^\\text{T}\n\\end{align}\n", "95f5aa40658c24606861c355d5f35695": "f(z)= \\sum a_n z^n ", "95f5c23875f3efe3f21c95368c871e73": "\n\\chi = \\frac{C}{T - T_{c}}\n", "95f5ce6411a2e8c78c0f55b39512bbd1": "R(x_1,x_2)", "95f5d05f95e32297c9957c9291a8bedf": "B = \\frac{n}{\\left| \\alpha \\right|} - \\left| \\alpha \\right|", "95f6685e2a7072f7196fe432b7dcb5ca": "\\Delta = \\frac {2X_{max}} {M} ", "95f67e0c659cc7f4e82a7df944e7d4a5": "q(x)=\\frac{p(x)-y_1}{y_2-y_1},", "95f69f01327f95011eaf3a4833554746": " \\beta(M) ", "95f6bc072524b8759c457a87e225ca33": "P_2 = P_0(1+r)^2- c(1+r)- c", "95f78aae62a578afbe6eb603f4d0bf42": "\\sigma^2_c", "95f7937b6e852dc7bf38a0342b33e03f": "p(a) \\leftarrow", "95f79fbd8d6771477369341149a5aeaa": "X_b^0", "95f7cfb72059944c02d82e5c85dffaf8": "X: I \\to Spaces,", "95f7eacd89be511e1c0a9bda9b3ed7ca": " \\Delta f \n= {1 \\over r^2} {\\partial \\over \\partial r}\n \\left(r^2 {\\partial f \\over \\partial r} \\right) \n+ {1 \\over r^2 \\sin \\theta} {\\partial \\over \\partial \\theta}\n \\left(\\sin \\theta {\\partial f \\over \\partial \\theta} \\right) \n+ {1 \\over r^2 \\sin^2 \\theta} {\\partial^2 f \\over \\partial \\varphi^2}.\n", "95f7fb41b19760bf914d911f75d140ee": "\\frac{\\tan\\theta}{\\theta}", "95f80745b75034b8d82551fe27cbc665": " \\langle p | \\hat{x} | \\psi \\rangle = i \\hbar {d \\over dp} \\psi ( p ) ", "95f8221130be94188cac96abe0b5640e": "\\operatorname{x}(u) = -a(1-e)( \\operatorname{F}(\\operatorname{sn}(u,k),k) + \\operatorname{F}(1,k)) - a(1+e)( \\operatorname{E}( \\operatorname{sn}(u,k),k) + \\operatorname{E}(1,k)) \\, ", "95f86474199546e7538c70d85b007571": "\\frac{\\nu}{2}", "95f879d8aca2b91d87575b8fac3ff012": "\\eta'(1) = \\ln(2)\\gamma-\\ln(2)^2/2", "95f8e136ac0532600c7a45373e54c637": "(p+q) \\times (p+q)", "95f9263a1a1bec4e5d3ad1691b12a382": "1 + z = \\frac{1}{\\sqrt{1-v^2/c^2}}", "95f9606b50884c253e3ab7f27dc39bf8": "{12 \\choose 3} \\cdot \\frac{1}{79} \\approx 2.78", "95f9789d6b4f921a08c429e5a15dac58": "SO(1,3)", "95f97d63a0d427c4c53c39003f81726c": "\nR_1 = \\max_{f(X_1,X_2)} \\min \\{ I(X_1;Y_2|X_2), I(X_1,X_2;Y_3)\\} \n", "95f97dfa7f6c9aa2351f338553bc1e10": " l = \\arccos { r_x \\over { \\mathbf{\\left |r \\right |}}}", "95f9cb6581fc9e068b29d1aeaf7ae284": "y\\to 0", "95f9d572332cc31210d08f600f6dabfa": " \\ E", "95f9d637d39aa42e39b94288a8b11c3f": "F = 8E + 650", "95f9def470f6fa74325355c821364e55": "X_k = U_k + \\omega_N^k Z_k + \\omega_N^{3k} Z'_k.", "95f9ef0b6ac891bd9d9a0cae5672e820": "\\sum_0^N a_k \\binom{d-1+n - k}{d-1} = \\left(\\sum a_k \\right) {n^{d-1} \\over {d-1}!} + O(n^{d-2}).", "95f9f0c26730f28b2d207173537631d7": "\\beta(r,s) \\rightarrow 0", "95f9f5c5e82f47e4ab03ad222e2d1ab0": "2\\cdot3\\cdot4", "95fa50eda62e0e7acd8f10ef2f39e28e": "a+bi ", "95faa39d04a4ebb2371eea1a09721b9f": "\\Re(s) > 1 \\,.", "95faf8c32656054e4be5f85dcb9764fb": "f_y(x) := \\sum_{i=1}^{\\infty} x_i y_i \\qquad x \\in X.", "95fbd9f389c25760ccc4e982363f2439": "\\;\\sum_{k=0}^{t-1} r^k = 1 + r + r^2 + ... + r^{t-1} = \\frac{r^t-1}{r-1}", "95fc06bf1dc6f38803af18038945e138": "\\cot A = \\cos A \\cdot \\csc A \\ ", "95fc17332f2376f5c446389fbb088102": "\\ \\{q,r,\\sim r\\}", "95fc40b67270abdfacf78b23061455f9": "\\sim\\;\\subseteq S\\times S", "95fcccfa68553134c3ca99bb05565726": "-a^n u[-n-1]", "95fcd1080e18b6957e707d8ac91db403": "2^{409}", "95fd0ba43240173134de05723b50d288": " \\dot{\\phi} = L/r^2 ", "95fd29ff690cef5e56292d0c9197c3b5": " O(m \\log n)", "95fd4db44880ccb58a81270dc625f115": " \\operatorname{let-combine}[\\operatorname{let} p : \\operatorname{de-lambda}[p\\ f] = \\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x) \\operatorname{in} p] ", "95fd7b57cf80f63b7ea1d09ae74e0045": " H_0 \\approx {DF}_T \\frac{1}{N} \\sum_{\\omega\\in \\text{sample set}} H(\\omega)", "95fe0226766e3b7b2e65345aa7b99ded": "u_1 \\equiv v_1 \\wedge\\ldots \\wedge u_n\\equiv v_n \\Rightarrow f(u_1,\\ldots,u_n) \\equiv f(v_1,\\ldots,v_n)", "95fe109861a660e23906e2b9b5ff2fa5": "\\beta << H ", "95fe92a7a13abfbeffca3f8427286b68": "\\therefore\\phi(t)=\\int{\\frac{\\alpha}{c+vt}\\,dt}=\\frac{\\alpha}{v}\\log(c+vt)+\\kappa", "95fe9faf04c7c74922653c269d6841a0": " 2\\pi /d", "95fea1446b3734d4ebb90454a033d122": " \\uparrow_{S_{|\\mu|} \\times S_{|\\nu|}}^{S_{|\\lambda|}} \\left ( V_\\mu \\otimes V_\\nu \\right ) = \\bigoplus_\\lambda c_{\\mu \\nu}^{\\lambda} V_\\lambda, ", "95ff05a5275a9815f95da7ccdb85e5aa": "V \\otimes V^*", "95ff4f7451619fe183a6d7e989055027": "\n F(z) = \\cfrac{16\\gamma}{3 z_0}\\left[\\left(\\cfrac{z}{z_0}\\right)^{-9} - \\left(\\cfrac{z}{z_0}\\right)^{-3}\\right]\n ", "95ff6ef0871b7eb752ca59a52bd0af54": "R(v_i \\otimes v_j) = \\sum_{k,\\ell} R_{ij}^{k\\ell} v_k \\otimes v_\\ell ", "95ff9cb23f80bc3fa224b5a3f21007d1": "2 (1 - \\cos(A, B))", "95ffbbefc5c40f84a4f816a78059da21": "\\mathrm{4H_2 + CaCO_3 \\rarr CH_4 + CaO + 2H_2O}", "95ffd11dff2b5ef34a54685c1fe3f668": "V>U", "960018697a1d176505e35bd96a1dc5a1": "\\operatorname{VaR}_{\\alpha}(L)", "9600212b76a1582631a8ee04a7aaeba6": "supp(B \\Rightarrow 1) = P(B \\and 1) = P(B)P(1|B) = P(1)P(B|1)", "96010d77ee70e39251f6d5e6881f0196": "g_1=0;", "96014362658dce2aeede4940407851ec": " H \\oplus H .", "96015fa5483dbbe2896ecdb6f6cda322": "\\frac{1}{2}\\chi'^2 = \\mathfrak{M} (2\\chi)^{1/2}", "960177e0dfb70401ddb2ac847c69e8f0": "\\Psi_4 := C_{\\alpha\\beta\\gamma\\delta} n^\\alpha \\bar{m}^\\beta n^\\gamma \\bar{m}^\\delta\\ . ", "960180836892b890c0731a9c5df2a3e7": "k=2^{a}5^{b}n", "9601ac8bebef7cd5b6522a61cdc0db64": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x) = f(x)", "9601bdf23ce299b9356817ce0a3283f0": "X\\times [0,1] / (X\\times \\left\\{0\\right\\})\n\\cup(\\left\\{x_0\\right\\}\\times [0,1])", "9601c65c06552621b7e8b1ae158e0c82": "U_{\\epsilon I}", "96022131b842426e0deb9141108b8f91": " \\begin{align} \\hat{H} & = \\sum_{n=1}^{N}\\frac{\\hat{\\mathbf{p}}_n\\cdot\\hat{\\mathbf{p}}_n}{2m_n} + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t) \\\\\n& = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2 + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N,t) \n\\end{align}", "9602329192f99cf9c2f2526c4f03ab33": "D = Z (2S+1) \\frac{\\Phi}{\\Phi_0}.", "96023db05298f8f133abef23025a0da7": "[a*]p \\equiv p \\land [a][a*]p\\,\\!", "9602516576094cd7de5a731d26d4658f": "\\scriptstyle -e^2/(4 \\pi\\ \\epsilon_{_0}\\ r)", "960284cdb0fbebce26bd07771d128747": " e = 1+\\cfrac{2}{1+\\cfrac{1}{6+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\cfrac{1}{22+\\cfrac{1}{26+\\ddots\\,}}}}}}}.", "9602babdd1802a4880f768c8240faba9": "v(t) = r'(t) = (x'(t), y'(t), z'(t)) = (-a \\sin(t), a \\cos(t), b)\\,", "9602de181ef5562e7f47ad6359c304c9": "P^{(k)}(\\lambda_j) = a_{j, k}\\quad\\forall 1 \\leq j \\leq r, 0 \\leq k < \\nu_j", "9602f6b3b71845d883398b0d43e2ec5f": "su(2)", "960311aec44c786b15c9fe6ffcb2905f": "1/p+1/q=1", "960321b4cb768dab5d95c0346f254361": "(x:y:z)", "96032805afbcd9dde87e1f0d5a3531d9": "C_{01} = 0", "9603595dbe1b35f0583c0bd0bbe1a573": "\n\\begin{align}\nSV = EV - PV\n\\end{align}\n", "96035c8cab9b083357e6d2ce4d9cc4c2": "R(n,k)=\\frac{2k-1}{2n+1}.", "96038847c6b85e6748242ca87a751061": "\\nabla _{\\alpha }", "96045450b9ecf256e1b17146cc7ce234": "\\scriptstyle U_1\\in \\tau_1", "96046004440745e2a758382b68dd7404": "\n\\Phi(x) = A_{t}(x) = U^{-1}(x),\n", "9604bf5ccefe31d7e241a9b71e16d2ce": " K = 1 - \\frac{e}{c}\\,", "9605b2a6180fd636d5f9995f15718ce4": "dy = \\frac{\\partial Y}{\\partial u} du + \\frac{\\partial Y}{\\partial v} dv.", "9605d9eb99bf1571588bb7346ade48f8": "A=\\operatorname{Aut}(G)", "960616fde4baad78d8fc9f82a83ea745": " x \\cdot y = L_1(x)y + L_2(y)x - L_1(x) L_2(y) e. ", "9606d1b74e056cf7a1207d9af4df4e89": " e_x + 1/2", "9606f1d16f8586d3788318c63c70ff4c": "r=R", "96070f2a71e3bf1f3f1613bc39523e84": "\\widehat{f}(x)", "960721cd148128c7206fc13e6ee281e9": "f^*(x)\\approx f^*(y)", "96073d0fdc0b9909e1b2b252578d6281": "\\eta^{-1}(x),\\eta^{-1}(y) \\subset S^3", "960769ae5c4b8393f1252f0b69417428": "*S^{IJ} = S^{IJ}", "960772536e40d519c651864e66bbf8ec": "u = \\frac{\\partial \\psi}{\\partial y}; \\quad v = -\\frac{\\partial \\psi}{\\partial x}", "96077bb602ea5457fc79dfcff45ce81a": "\\frac{Y}{2\\log Y}.", "9607aa57b9138a7dc51f9db0532adf59": "{ E = \\tfrac12 k_\\text{B} T} \\ ", "9607bc5b861d98a2269297757c1a5156": "I^pH_i(X) \\, ", "9607d4aab4edc5c8ed2ecd380d942b24": "K(\\!(T^{1/n})\\!)", "9607dfa5790d7683c1a97cabbed632a4": "e = n J_q(\\delta)-1 ", "9607f9d15f91d9e5b03aafc94fab464f": "\\operatorname{lb} y", "96082b1a8d4350b65856ce9b86735c82": "x(t,k)=s(t)+n(t,k)", "960846b106285c247087e8da078c05f8": "k=a_i-a_j", "960859e5ad8641a39a981297902c31ee": "\\scriptstyle\\frac{\\Delta y}{\\Delta x}\\approx f'(x)", "96089b34e89ce92f4dba90b1845527bf": " \\mathbf{F}_{21} = I_1 \\oint_{C_1} d \\mathbf{l_1}\\ \\mathbf{ \\times} \\mathbf{B}_{2}", "9609035fc3fad8c2c8c127f69ff540ab": "\\{f(x) ~|~ x \\in X\\}", "960917f2305aa78a730aeacb84ba0d59": "\\frac{1 + 2T^2}{(1 - T)(1 - 2T)}", "9609587be793b05b9f7f43d868594567": " \\mathcal{L}(\\phi,\\nabla\\phi) = -\\rho(t,\\mathbf{x})\\,\\phi(t,\\mathbf{x}) - \\frac{1}{8\\pi G}|\\nabla\\phi|^2,", "9609adf5b0ba923b664c27c5cf0e77b7": "\\rho = v_0 - v_2 = 0 ", "9609f6d47235156f61fc268b630fd31d": "\\scriptstyle \\nabla_S", "960a109d6c5ebd342610e804c7529cc2": " t_s = t_0 e^{-aT} ", "960a7b22b43e37979637c7b637393602": " X(t)= \\sqrt{2E}\\cos(t) , \\qquad P(t) = \\sqrt{2E}\\sin(t) ", "960aaf1269c54b08d12fb1b1588b1619": "\\partial^\\alpha(fg) = \\sum_{\\nu \\le \\alpha} \\binom{\\alpha}{\\nu} \\, \\partial^{\\nu}f\\,\\partial^{\\alpha-\\nu}g.", "960ac15317cb5e03db4fa5bf6cc608e8": "0 \\leq d_{i} < 2^n", "960ad21420c80c5de42056a4ba52f8ef": "(7)~~ ~~ V'(p)=\\frac{V(p)}{p\\cdot (1+V(p))}", "960ae17e7e02fc4b04e6fdefd334df95": "q = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3 + \\cfrac{1}{\\ddots+\\tfrac1{a_k}}}}}= [a_0; a_1, a_2, \\ldots, a_k] ", "960af396d9351fb8726554f3245737df": "(N',P')", "960b0e58f970f0ec8a641b1f5a83a8f9": "\\pi \\approx 4 / \\sqrt{\\varphi} = 3.1446\\dots", "960b9b13825ba334467f4c0de16fd22d": "d_\\alpha \\overline{\\Lambda}=0", "960bcb646f982b98c0948ba8f8a2606a": "\\Delta_{S^{n-1}}f = -\\ell(\\ell+n-2)f.", "960be76e2806fced85d4b6374d31b75a": "\\lambda = \\lambda_{in} + \\lambda_o", "960c09c819d1dfacf139be1882c08c74": " \\lambda \\,", "960c72eea6d10d94265e8a00f8e0823d": "PV(D_k)", "960c7e1f05f98d01f67c8d22cb39465a": "\\leq^*", "960cbd1628d50576c7ec59d228757927": " \\mathrm{J} = \\mathrm{det} \\left( \\frac{\\partial \\tilde{\\sigma}^\\alpha}{\\partial \\sigma^\\beta} \\right) ", "960ce70e6c576be6f5b9b3059b9ac06d": "U \\subset Y", "960d272b08bf10fc3d37a7f373f85d6c": "(X,X^n) \\,", "960d576b8ffe8867dc85c31fb8157586": "\\nu_\\max = T \\times 58.8\\ \\mathrm{GHz}\\ \\mathrm{K}^{-1}", "960dc1cd11869917cedfd6f0286b38ad": "Q = \\iint \\Phi_\\nu \\mathrm{d} \\nu \\mathrm{d} t ", "960ddf3051ecaa2c2b7019ee932ed849": " f(m) \\le \\operatorname{Median}( f( x )) ", "960e52b1f0a95991f18a9307377d9ae4": "p=\\sqrt{2m_0\\Delta E+\\frac{\\Delta E^2}{c^2}}=\\sqrt{2m_0\\Delta E} \\sqrt{1+\\frac{\\Delta E}{2m_0c^2}}", "960e54e74604fdf81b135306cd2e0c68": "L = 4", "960e84cbabca021960510272a0b2521e": "\\langle\\vec x\\rangle", "960e94cb73cf903a790ca48d3cde1e9c": "\\Lambda \\subset \\Gamma", "960ef78556c7c5c7989e0df478bdf46e": "{{H}_{WCM}}\\left( s \\right)\\equiv \\frac{{{i}_{out}}\\left( s \\right)}{{{i}_{in}}\\left( s \\right)}", "960f2d39b9f80e4cc214e07f64183dcd": "\\frac{1}{P_{syn}}=\\frac{1}{P_1}-\\frac{1}{P_2}", "960f314dd81f158f39a33de560df89ba": "|\\psi|^2=|e^{ikz}+\\frac{f(\\theta)}{z}e^{ikz}e^{ik(x^2+y^2)/2z}|^2", "960f41e98cef748f2d3606729efbd7af": "\\left(\\frac{\\partial}{\\partial x^1},\\cdots,\\frac{\\partial}{\\partial x^n}\\right)", "960f4f3caaf90238911d44a59d9dca03": " \\int_0^T \\sum_{d=1}^D |\\theta_d(t)|^2 dt < \\infty", "960f566075b93ae9dc0c781c82c36d8b": "E = - \\frac{1}{2} \\sum_{\\langle i,j \\rangle} 4 J B_i B_j + \\sum_i \\mu B_i", "960fac3f3d3db3e2625e9b8e90b207e5": "U_{pre}", "960fd22f5abca7613e85cb664abcf4da": "\\nabla : \\Gamma(\\operatorname{Sym}^k(TM))\\to \\Gamma(\\operatorname{Sym}^{k+1}(TM))", "961008945f99a1b6350b1e4cfde650ce": "M_0=M(r_B)=\\int_0^{r_B} 4\\pi \\rho(r) r^2 \\; dr \\;", "96101480abb6a1080a8ed936c55c844f": "\\alpha =\\frac{m(m-m^{2}-\\sigma ^{2})}{\\sigma ^{2}},", "96103ec0bdb7749eec07ef41877e9869": " -z \\frac{dX(z)}{dz}", "9610628c057e523da6227ad5d9c0c925": "v_p= \\sqrt{ \\frac {K+\\frac{4}{3}\\mu} {\\rho}}= \\sqrt{ \\frac{\\lambda+2\\mu}{\\rho}} ", "9610a97ca1c9495eb939801030a22fc3": "\\mathcal{L}_\\mathrm{int}(x) = g\\overline{\\psi}(x)\\phi(x) \\psi(x).", "9611239a9440efb035543ee72404c356": "\n \\boldsymbol{Q}\\cdot\\boldsymbol{Q}^T = \\boldsymbol{Q}^T \\cdot \\boldsymbol{Q} = \\boldsymbol{\\mathit{1}}\n ", "96113b9978442692c9842e6ecbdccbda": "|h_i(x(0))|", "96115a2e99034632cae92833b037dd6c": "ST_x(\\top) \\equiv \\top", "961160adb27221c2e62b108f100af272": "XDH(a,b)=c", "96117a6293ef3f1dbbdeabbbe1e453dc": "\\bigtriangleup_{SO,dB}", "96119ccb0080850c344c6d902ee6a63c": "\\mathbf{f} = \\rho \\mathbf{g}", "9611a4ca98b9994da9a871f5d03081d0": "(z)_v \\leftarrow \\mathbb{D}((z)_v)", "9611c74d38735853aaee7aa6a1c30133": "k = 1.381 \\times 10^{-23}(Ws/K)", "9611f05822748f8e9d1140cded9127fd": " Q = \\frac{2}{3}\\left[\\left(n_\\text{u} - n_\\bar{\\text{u}}\\right) + \\left(n_\\text{c} - n_\\bar{\\text{c}}\\right) + \\left(n_\\text{t} - n_\\bar{\\text{t}}\\right)\\right] - \\frac{1}{3}\\left[\\left(n_\\text{d} - n_\\bar{\\text{d}}\\right) + \\left(n_\\text{s} - n_\\bar{\\text{s}}\\right) + \\left(n_\\text{b} - n_\\bar{\\text{b}}\\right)\\right]", "96120a384c6f4ee94fd623f4f812f0bb": "(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t \\in [0,T]},\\mathbb{P})", "96125219b2c7fea6a31e7241dbb75148": "s_{i_1 i_2 \\ldots i_N}", "9612fed06e5991f1281c6f6f0803437c": " \\mathop{\\mathrm{ker}} f := \\{a \\in A : f(a) = e_{B}\\}\\mbox{.} \\! ", "96131fe9a60640712dbcdab4d8b6565a": "\nR_A(v) = \\frac{\\sum_{i=k}^n \\lambda_i \\alpha_i^2}{\\sum_{i=k}^n \\alpha_i^2} \\leq \\lambda_k\n", "96132943e915524c685f7847868a06e8": "\\left(\\mathbb{Q}^*,\\cdot\\right)", "9613414a519d68c28404bde50f523e2a": "\n\\sum_y R_{x\\rightarrow y} = 0\n\\,", "96134783d9c5fea1b06142f9e108fb6b": "E=\\frac{m c^2}{\\sqrt{1-\\frac{v^2}{c^2}}}=\\frac{m c^2}{\\sqrt{1-Z^2 \\alpha^2}}", "961467f471cf36d5185432965ab63c41": "x_n=(7x_{n-1}+3)\\,\\bmod\\,10,", "96148a4f394a1119859aefac13129d32": "x_2 < x_3", "9614fa55b42a71c0b8a8a0480b391fd6": "f_{k-1}(x) \\neq 0\\,", "96155b1bc331454ce48057c12f9596a1": "\\ [x,y,z]", "96157422a7ab885cfd681a123c6d9ab8": " [T \\psi] (x) = f(x) \\psi(x) \\quad ", "9615966dbb0334773c89f32b1e38e98a": "\\bar Z = \\Sigma_i Z_if_i ", "961645a44bb9a23fb3944e28ceb9857a": "y_{n+1} = y_n + h \\left( \\tfrac32 f(t_n, y_n) - \\tfrac12 f(t_{n-1}, y_{n-1}) \\right) .", "9617353f48623168a7b2d1c781765c29": "\\begin{cases} \\dfrac{\\partial u}{\\partial t} (t, x) = F u(t, x), & t > 0, x \\in \\mathbf{R}^{d}; \\\\ u(t, \\cdot) \\to f, & \\mbox{ as } t \\to 0; \\end{cases}", "9617654cf09706d2a06aee078b6cb775": "\\scriptstyle F \\,\\otimes\\, \\delta", "96179a420a7e33f3b272906039960cc6": "\\rm{plus}(a,b,c)=(\\rm{DTC}_{v,x,y,z} \\rm{succ}(v,y) \\land\n\\rm{succ}(z,x)) (a,b,c,0)", "96180193037e3bd1c854864e4e3745c0": "E_2 = y_2 + \\frac{q^2}{2gy_2^2} = 0.59ft + \\frac{\\left(10\\frac{ft^2}{s}\\right)^2}{2\\left(32.2\\frac{ft}{s^2}\\right)(0.59ft)^2} = 5.06ft", "96183912ce317fd03f8e4fbdb06dd790": "S(\\rho_{A}) \\, + \\, S(\\rho_{C}) \\, \\leq \\, S(\\rho_{AB}) \\,+\\, S(\\rho_{BC})", "9618865551991a5bb9f5f7b94e380ef7": "\\Phi_{651}(x)", "9618a567a63a2e140ef99c00b1bd6c65": "7 \\frac{3}{4} - 7 \\frac{1}{2}i = 1(16) + 1(-8i) + 2(-4) + 1(2i) + 3\\left(-\\frac{1}{2}i\\right) + 1\\left(-\\frac{1}{4}\\right) = 11210.31_{2i}", "9618c404659ef5d1d710c4064ff59c77": "E^2 = (mc^2)^2 + E^2 {v^2\\over c^2},", "96190f158e0bb469cb8f4757f66dde40": " \\Delta Y= \\frac{\\Delta I}{(1-c)}", "9619599df5ae8300b9f93b77a731b194": "Difference~ratio = \\frac{ | Limit_{log-normal} - Limit_{normal} | }{Limit_{log-normal}}, ", "96197cad29b77d1589f6525c346f81ad": "J_{\\varepsilon} (S) = \\int_{S} e^{- F(x) / \\varepsilon} \\, \\mathrm{d} \\mu_{\\varepsilon} (x)", "961983623ffc63f374dc777b80983609": " \\tilde{Q}", "961986928955c3f318a88f119ee31fa9": "-685.81=10\\log_{10}\\left(e^{{-\\left(4\\pi\\right)}^2}\\right)", "96199e662a139cdac362bc77bc24373d": "w=\\sum_{i\\in K}\\alpha_iv^i", "9619dd84d740b0524630cfaaa4463caf": "K_1 \\neq 0", "961a0cc44c0aede9833feac159560833": "S \\rightarrow S/R", "961a1fe72a0eefe6f488550f508ad08a": "\\begin{align}\n \\sigma^2+\\mu^2-\\mu &= \\sum_n n(n-1)\\cdot\\Pr(N=n\\mid M=m,K=k)\\\\&\n = \\sum_{n=m}^\\infty n(n-1)\\frac{m-1}n \\cdot \\frac{m-2}{n-1} \\cdot \\frac{k-1}{k-2} \\cdot \\frac{\\binom{m-3}{k-3}}{\\binom{n-2}{k-2}}\\\\&\n = \\frac{m-1}1 \\cdot \\frac{m-2}1 \\cdot \\frac{k-1}{k-2} \\cdot \\frac{\\binom{m-3}{k-3}}1 \\sum_{n=m}^\\infty \\frac 1{\\binom{n-2}{k-2}}\\\\\n& = \\frac{m-1}1 \\cdot \\frac{m-2}1 \\cdot \\frac{k-1}{k-2} \\cdot \\frac{\\binom{m-3}{k-3}}1 \\cdot \\frac{k-2}{k-3} \\cdot \\frac 1{\\binom{m-3}{k-3}}\\\\\n& = \\frac{m-1}1 \\cdot \\frac{m-2}1 \\cdot \\frac{k-1}{k-3}\\\\&\n\\end{align}", "961a772ce886dea9a6ffe7424f571aa2": "n_B(a+b)-n_B(a-b)=\\mathrm{coth}\\frac{\\beta b}{2}+n_B^{\\prime\\prime}(b)a^2+\\cdots", "961a872bdaf3953cfae8f123de1c2e09": " s_{k+1}", "961aad981b02d9fc883763ec3cc9b9b1": "\\chi=\\frac{m_v}{m_l + m_v}", "961ac976c680e148348fe4076ceed3cb": "\\sum_{x \\in C} w_m(x) = n(\\lfloor\\frac{rB}{r+1}\\rfloor - \\lfloor\\frac{B}{r+1}\\rfloor)", "961ae210abba41f46f68105694649ccb": "\\scriptstyle {n\\choose k}.", "961b08dadb483afe824032530bce64b8": "x \\sim B(n,\\theta)\\,\\!", "961b1b270c119f69f43ac7598e1d858e": "\\operatorname{multiply}\\ m\\ n\\ f\\ x = m\\ (n\\ f) \\ x ", "961b3de79ef7d7e44cce3c1eb339ddd5": " T_B(t) ", "961b3e8bac507a90c8462488f8055a1f": "\\ M\\ ", "961b67f7d865c6316ac1a8435b33ca99": "L(x_0)", "961bb36d31d7a93ec820b0ee96060523": "y_1 + \\frac{q_1^2}{2gy_1^2} = 3.47", "961c08bb1d34a2fb552e160a9e80e43e": "P_{22}=P_{33}=0", "961c1cd1d5d28e08ab7689c517f108cd": "\\mathrm{VO_2\\; max} = {15{ \\mbox{HR}_{max} \\over \\mbox{HR}_{rest} }}", "961c7d54560eb68d928f3f142d55196c": "\\begin{align}\\mathrm{E}\\{[\\ln(X)]^2\\} =& [1 {+} 1 / \\eta]\\!\\!\\int_0^\\eta \\!\\!\\!\\! e^{-X}[\\ln(X)]^2 dX\\\\ &- 1/\\eta \\!\\!\\int_0^\\eta \\!\\!\\!\\! X e^{-X}[\\ln(X)]^2 dX \\end{align}", "961c83a09264b9257de7fba44c6f60a3": "U_n(1) = n + 1\\,", "961cb1cb1da5585c38888a34c174286b": "f_t(x)=(1/3)x^3-tx,\\,", "961ce24d6f2db69b18eda02d818953db": "\\frac{\\pi(x)}{1 - \\pi(x)} = e^{\\beta_0 + \\beta_1 x}.", "961cedb4ea405ed40fc667bedee85846": "I^-(\\varepsilon,t,f,dg)={1\\over\\varepsilon}\\int_0^tf(s)(g(s+\\varepsilon)-g(s))\\,ds", "961d670f2e5e1921c03a04e7e8e32baf": "\\displaystyle{\\mu_g(z)={z^2\\over \\overline{z}^2} \\mu_f(z^{-1}).}", "961da44dbdfc653f6eb6b832afb18049": "\\text{Weight} = \\text{weight density} \\times \\!\\, \\text{volume}", "961e6765c7c8ddccbb40cabe19ab09a8": "D = \\log{\\text{DOR}} = \\log{\\left[\\frac{TPR}{(1-TPR)}\\times\\frac{(1-FPR)}{FPR}\\right]} = \\operatorname{logit}(TPR) - \\operatorname{logit}(FPR)", "961eb17f4eca319cae4bc2bce6452531": "y_j \\in Y_j", "961ecc2845b37f0a38a945943f8a010e": "a_i = \\gamma_{x,i} x_i\\ = \\gamma_{b,i} \\frac {b_i} {b^{\\ominus}}\\, =\\gamma_{w,i} w_i\\ = \\gamma_{c,i} \\frac{c_i}{c^{\\ominus}}\\, = \\gamma_{\\rho,i} \\frac{\\rho_i}{\\rho^{\\ominus}}\\, ", "961edf361c5dffefbcc827b8cff51c25": "\\text{cont} (q) =\\frac{\\text{cont} (p)}{c},", "961ee396092cc73f0ee29ab25401594c": "0\\leq t\\leq 1", "961eef951a6f9d85bdfe609624418da3": "RPF = \\frac{U_x V}{P_a - P_v}", "961f61faf24beeb3270156be593056be": "\n\\begin{align}\n D_0 &= 1+\\frac{9}{4}n^2+\\frac{225}{64}n^4+\\cdots,\n\\qquad\\qquad&\nD_4 &= \\frac{15}{16}n^2+\\frac{105}{64}n^4+\\cdots,\\\\\n D_2 &= \\frac{3}{2}n+\\frac{45}{16}n^3+\\frac{525}{128}n^5+\\cdots,\n&\n D_6 &= \\frac{35}{48}n^3+\\frac{315}{256}n^5+\\cdots.\n\\end{align}\n", "961f909bee60b0bc674c5fe1b2dc43d6": "\\cos c = \\cos a \\, \\cos b + \\sin a\\, \\sin b \\,\\cos \\gamma.", "961fccc0be43db03f741d86c51c29c62": "\\Pi^{-\\top}", "961ffc99f1db5c22504160559c8a5d99": "C/C_0", "96203bbf88f8cb9966c4d32afb01b6f4": " \\beta_\\mathrm{R}", "9620b8ce8aed830a9d1de8032921ba34": "1 = (-1\\times -1)^\\frac{1}{2} \\not = (-1)^\\frac{1}{2}(-1)^\\frac{1}{2} = -1", "962162fe35f8419caea91a594b9dc394": "-\\hbar\\omega (D-2)/24", "96217cb485bcb135ee851fb418c4866a": "m_{i,j} = m_{j,i} \\in \\{2,3,\\ldots, \\infty\\}", "9621e00ab9d435a18fc89e31ff52d165": "S^4, {\\mathbb C}P^2,...", "96224c04dab731c4c7d8478b5f14ab3d": "Q(u), Q(u')", "96225581ba8caff9b3fa620cda64709e": "Z(\\beta) = \\mbox{tr}\\left[\\,\\exp \\left(-\\sum_k\\beta_k H_k\\right)\\right]", "96225fb9cc44c9de895c0dc565c98379": "\\{T_r, p_r\\}", "962267f1bf40d6d9a91434eb3406bffd": "\\scriptstyle{1/8}", "962273e8ad26927ed70f0dc5a8e7d3ae": "\\nabla \\times \\mathbf{B} = \\mu \\mathbf{J} + \\mu \\epsilon \\frac{\\partial \\mathbf{E}}{\\partial t}.", "9624629e1efab799afc4100ebba1a048": "1 \\leq i \\leq n", "9624794989eb56f81a11cf0f863f35c5": "\\mathbf{Y}_{11}= -\\sqrt{\\frac{3}{8\\pi}}\\mathrm{e}^{\\mathrm{i}\\varphi}\\sin\\theta\\,\\hat{\\mathbf{r}}", "962486089ebb655a57f03c3dc1ceac6c": "\\chi_s(G) \\leq 3", "9625343bcc1a32951eae60ae0430d453": "\\varphi(0)", "96254aa19e921c97893007767c127012": "R = (1 \\otimes S)(R^{-1})", "96254f70867ce45396a91c444ddd8612": "i+j =\\,{\\rm const}", "962573fb95edab38a701bb72676f4802": "\\hat{x}_{\\mathrm{MMSE}}=g^*(y),", "9625b70d58c00d0ba7e28045756cec57": "\\cos \\theta\\,\\!", "9625bb0bfd7850ecbcc5c57f9e66ed49": "\\frac{\\partial v_i}{\\partial x_i} = 0 ", "9625d4c1dede4b07f3848dc822d43557": "x^5(x^2-x-1)+(x^2-1)", "9625e26f4ad1e308b3f6d9723c6e1ba0": "{1,2,3,4}", "9625e339e4faafc6a1ee912f3bf2c8bc": " \\iint_R |N_u \\times N_v| \\ du\\, dv = \\iint_R K|X_u \\times X_v| \\ du\\, dv = \\iint_R K \\ dA", "96264ff5703af53fa468b5d2dab791e8": " P_i ", "9626983ea31c4c9216334e9b94a58c24": "\\text{Factor of safety} = \\frac{\\text{Ultimate tensile strength}}{\\text{Maximum stress}}", "9626e2f5b476e5d5c28cf38a53cc8f60": "G\\times X", "96271d8699c3bcf10772073a87d8d269": "2h \\times 2b", "9627369f010ea365dfb7d1461a3a7000": "A_{c_{\\beta}}", "96279821e5a0b4718bd1ac8bd2bdc92e": "\\mathfrak{P}^{29}", "96279e31f68b2c0a008264203eb8fa32": "L_A", "9627b8eefc46a8be7e1cec03864cb2bb": "E_3 = \\varnothing. ", "9627cb5a3b7c68533a5c74dac06cf16c": "\\vec{\\top}", "9627e382c11be8d2346e34051aaee1fd": "\n \\mathbf{r}_A = x_A\\mathbf{e}_x \\quad \\text{and} \\quad \\mathbf{r}_B = (L-x_A)\\mathbf{e}_x \\,.\n ", "96283b755021bcebb70d8022270a8f34": "i^{\\prime \\prime} (V) = \\frac{e^2 nS_z}{4m}\\frac {1}{V}f\\left ( \\sqrt{2eV/m}\\right ) ", "9628df349b1d7672b0da17739f5091e6": "\\deg r_{k+1}<\\deg r_k\\,", "96291a5191f26178d8ebd9b727699b71": "f(w)\\Vdash'p", "96292672d8695cf846fb9322a2782b10": "\\mathbf A_d = \\mathbf M_{11}", "9629ac612c97562348694e8a72452efa": " s^2 = (vt)^2 - (ct)^2 .\\,", "9629af93475f5e2792a780b37c8ca26e": "\\mathrm{^{249}_{\\ 97}Bk\\ \\xrightarrow {(n,\\gamma)} \\ ^{250}_{\\ 97}Bk\\ \\xrightarrow [3.212 \\ h]{\\beta^-} \\ ^{250}_{\\ 98}Cf}", "9629b3f8608c6a230af65247aaa4cdf9": "\\mathrm{prem}(p, T)\\neq0", "9629bf51f7be08d80f1f750b46e1ec40": "x\\in\\mathfrak g", "9629f1959c19ab3fdbce600802ec4cb5": " \\mathbf{F}=(0,-mg),", "962a80ecf722912f8023887c2118124b": "r=\\frac{1}{(1+kw)^4}", "962a921057c349dcad3bccebb69f7eec": "2 I \\times C_{29}", "962aa820562bce32695df4ba9f59ab58": "y/(x/2)", "962b17bf5538e2d7b8c61c78b4ae7571": "\\overline{K(x,y)}", "962b671710eae700491bd69348f137d8": " R^\\mu_{\\ \\nu\\sigma\\tau} = \n dx^\\mu((\\nabla_\\sigma\\nabla_\\tau - \\nabla_\\tau\\nabla_\\sigma)\\partial_\\nu).", "962b733264d4720bcb439d94254e8453": "\\cdots\\to H^0(\\mathcal O_U) \\to H^0(\\mathcal O_U^*)\\to H^1(2\\pi i\\,\\mathbb Z|_U) \\to \\cdots", "962b7a38de56c0a250b5ac1446f39b00": "A Q = Q\\Lambda ", "962b95fe9a890e5a01a424a30d6adaf5": "Q = C_d\\; A_2\\;\\sqrt{\\frac{1}{1-\\beta^4}}\\;\\sqrt{2\\;(P_1-P_2)/\\rho}", "962bb6b773f27ba37058314bb2adce3d": "\\sin x \\approx x\\textrm{\\ radians} = x \\cdot \\frac {180} {\\pi} \\textrm{\\ degrees} = x \\cdot 180 \\cdot \\frac {3600} {\\pi} \\textrm{\\ arcseconds} ,", "962c52012e21e885231ea9ea0ceaa41c": "\\|fg\\|_1 \\le \\|f\\|_p \\|g\\|_q", "962c5ab5c5068e9ab92835b1b08a8118": "\\tilde{y}_j", "962c7382c12d56f9d37faca6b3c72fd7": "\\bold{j}_{\\rm n} = n \\mathbf{u} ", "962c7c92083ea4ef467b1f8d2b39ac26": "-2x \\equiv y -2z \\pmod{7},", "962cb6c1b6286d2257e54083fa7ebfcc": "[S \\rightarrow XY], \\quad [X \\rightarrow aXb, Y \\rightarrow cY], \\quad [X \\rightarrow ab, Y \\rightarrow c]", "962d16c6d142e5c5431cb9ee7ac00110": "F(\\sigma) = \\int_{C_{0}} F(p) \\, \\mathrm{d} \\gamma (p) + \\int_{0}^{T} \\alpha^{F} (\\sigma)_{t} \\, \\mathrm{d} \\sigma_{t}", "962d61634121556e1faad546f5aba18f": "\\Delta h = \\frac{\\frac{1}{2}mv_1^2}{mg}", "962dba98b2fb84a9406abff13a87e745": " 1-f ", "962e06b2e5828b56cb8546eb7628f894": "\\varepsilon_{ij}=\\frac{\\alpha \\varepsilon_y}{E} \\left (\\frac{EJ}{r\\,\\alpha \\sigma_y^2 I} \\right )^{{n}\\over{n+1}}\\tilde{\\varepsilon}_{ij}(n,\\theta)", "962e511d041c6ac36e43fc8daa698fb2": "F_L = qvB\\,\\!", "962f72bfc96eb321f1894566bc2e86dd": "|\\psi_\\beta\\rangle", "962f96d88692a554c52920f3e16c487e": "\\left[\\frac{\\alpha}{\\pi}\\right] = 1.", "962fa0c7ef4cd2f8054b2262097cc87c": "f_* f^*", "962fb136d1ed9534f3e39b212bcc8639": "T_u\\cdot \\left(J\\left(R\\right)\\right)^i\\subseteq T_{u-i}", "962fde3b86737c80f702dfee15323cb2": "f(\\lambda x, \\lambda y, \\lambda z) = \\lambda^k f(x,y,z).\\,", "962fe77b22ee48c381b54c4b3a327c80": "\\lim_{t \\to 0}\\int_{\\mathbf{R}^d} K(t,x,y)\\phi(y)\\,dy = \\phi(x).", "963037c76ccacf84918b81278f4ab2e7": "\\varprojlim_{i \\in I} F(C_i) \\cong F (\\varprojlim_{i \\in I} C_i)", "9630499eaf52033da88a9c258f7348c1": "y = f(x_1, x_2, ...)", "96305848987d613a3896f16b78e8bebf": "H = \\frac{q L t}{N^2},", "9630824c297cc3c54badce748648dd9e": "\\Delta t = \\frac{2L}{v_x}", "963123d6ba15e531cac42d449320e07c": "S_t", "9631347cf1190b1ba764a381e4678c9c": "A_x=\\pi_Y[A\\cap \\lbrace x \\rbrace \\times Y]", "963196d638ff3615b426bf6ac76d5677": "\\ \\mu_\\alpha \\cos \\delta", "96319a3adb29b129e207abdb8cebd75c": "O(dn^2\\log(r)\\log(q))", "96319abe9acea9176583858c188c2aff": "(T,J)\\to (T',J')", "9631a7fddf1bcb5422cd0ad3994aff53": "\\sin ^2 \\theta + \\cos ^2 \\theta = 1", "9631c01adfec6329ba74945ddedfdb5e": " \\scriptstyle \\beta\\; \\sim f(\\beta | \\theta) ", "9631c67d2fddce5b1ca57380287c9be8": "r_{i,j}", "96321eb78fa78d2e9ed1ad14a0aeea8d": "m_i = M_n", "9632c913b01b736f2629d610437d83ca": "i_{Q} \\colon Q\\hookrightarrow E", "9632f0d5876df253adca00e2ca394c98": "\\mu\\left (S(t)\\right)=t.", "9632f6b6062954ff26e63d12196ee523": "[h_i](t)", "96330907975f080d3d40fdbe88f39bbb": "\\mathbb{Q} \\left( \\sqrt{d} \\right)", "963338f679f1ef399dd6032167964168": "\\int \\coth x \\, dx = \\ln| \\sinh x | + C , \\text{ for } x \\neq 0 ", "963386d4e68c608ab487301c8d329328": "H_p(x)", "96338e6c50ba2bb82b5ce5f98e56f202": "\\ d[\\mathbf{x}(1), \\mathbf{x}(4)]=|u(1)-u(4) |=|u(2)-u(5) |=0 0\\ \\exists \\ \\delta 2 > 0 : \\forall x\\ (0 < |x - a | < \\delta 2\\ \\Rightarrow \\ - \\varepsilon < h(x) - L < \\varepsilon).", "965481b5ca925db4998a281a854f46fc": "(2 + i)", "965497ae4830d136d9dcc772eb9e96c1": " \\begin{align}\n&\\mathrm{Sp}(2, \\mathbf C)\\cdot \\mathrm{SO}(n, \\mathbf C)&&\\subset\\mathrm{Aut}(\\mathbf C^2\\otimes\\mathbf C^n)\\\\\n&(Z_{\\mathbf C}\\,\\cdot)\\, \\mathrm{Sp}(2n, \\mathbf C)&&\\subset \\mathrm{Aut}(\\mathbf C^{2n})\\\\\n&Z_{\\mathbf C} \\cdot\\mathrm{SL}(2, \\mathbf C) &&\\subset \\mathrm{Aut}(S^3\\mathbf C^2)\\\\\n&\\mathrm{Sp}(6, \\mathbf C)&&\\subset \\mathrm{Aut}(\\Lambda^3_0\\mathbf C^6)\\cong \\mathrm{Aut}(\\mathbf C^{14})\\\\\n&\\mathrm{SL}(6, \\mathbf C)&&\\subset \\mathrm{Aut}(\\Lambda^3\\mathbf C^6)\\\\\n&\\mathrm{Spin}(12,\\mathbf C )&&\\subset \\mathrm{Aut}(\\Delta_{12}^+)\\cong \\mathrm{Aut}(\\mathbf C^{32})\\\\\n&E_7(\\mathbf C) &&\\subset \\mathrm{Aut}(\\mathbf C^{56})\\\\\n\\end{align}", "9654a2dbcc5ef010c2a16742eb85901e": " \\mbox{vec}(ABC)=(I_n\\otimes AB)\\mbox{vec}(C) =(C^{T}B^{T}\\otimes I_k)\\mbox{vec}(A)", "96557f3c1634b211236e53fdbcb4b73a": "\\pi\\in G^\\wedge", "9655b744f88af7a288c07c5c26a0b4f4": "n_{i, t} = \\frac{s_0\\cdots s_{i - 1}}{\\lambda^i}n_{0, t}. ", "9655d7a42b76279c711c143baf6979ab": "\\textstyle \\ell^1 \\times \\ell^\\infty", "9655e790f276b32dd53b2057d504a397": "\\frac {\\pi}{2N((\\pi+1)^l -1)\\prod_{j=1}^k {\\rm max}(1,h_j)}\\Bigg|\\sum_{n=0}^{N-1} e(\\mathbf{h}\\cdot \\mathbf{t}_n)\\Bigg|", "96560527132e123bef8c4ef10d75ea5b": "\\mathbf{U}=\\mathbf{F}/\\sqrt{N}", "9656477d87c4366b09836badd4b4b9d3": "N + 1", "9656648288173ec3f10e22b95728e6f5": "\\mathfrak{g} = \\mathfrak{a}_0 \\supset \\mathfrak{a}_1 \\supset ... \\mathfrak{a}_r = 0", "96566c2b8c519945095146e4b79d69b9": "z_i = y_i", "9656cb49420b61db050c953834170013": "\\int_{0}^{\\pi/2}\\sin^n x\\,dx=\\int_{0}^{\\pi/2}\\cos^n x\\,dx=\\frac{(n-1)!!}{n!!}\\cdot\\frac{\\pi}{2},", "9656d82ef19c18382e2ef37f915963f2": "\\frac{\\mbox{Earnings per Share}}{\\mbox{Dividend per Share}}", "965704090eff914b5b7cfec8b502e1be": "\n (\\forall R)\n ", "96573391827bb63632a1bb3664a0b87f": "D^{(0)} f^{(0)}_{n,i} (x)=\\lambda^{(0)}_n f^{(0)}_{n,i} (x) ", "96575e18267a51322868e5fb4ec790b1": "\\displaystyle{\\Delta_3 M_0 f=4M_0\\Delta_2 f.}", "9657f1fabdfbe142913624457c998b6d": "\n\\begin{pmatrix}\nq\\\\ p\n\\end{pmatrix}\n\\mapsto\n\\begin{pmatrix}\n q \\\\\n p - \\tau d_i \\frac{\\partial V}{\\partial q}(q)\\\\\n\\end{pmatrix}.\n", "96580e33c983fcd4b239eac3c141ed78": " \\mathbf{v}(E) ", "9658c66e0daf07b0e48ad737da9c157f": "\\displaystyle f(z)=z^{-1}+a_0+a_1z+\\cdots", "9658c6d884b2b46407fb9c08b2a485e4": "\\leq w(f^{*}) + \\frac{w(f)}{4},", "9658d0b2c95b560bdcec92786321544d": "FB_i", "9658dab4b20e85b46d4d1c6b2670068f": " \\nu\\ : \\Gamma\\ \\to \\mathbb{R} ^\\mathrm{N} ", "9658e067ddcee780a0a746f648a0a482": "\\mu^{(i)}_t", "96591e283f389131249f63616a64990f": "\\; (A - 1 I) p_1 = 0 ", "965986b16dc602300a6290e027f4bf6f": "\\min_{q_1,q_2,\\ldots,q_M}\\{P_\\mathrm{net}( q_1,q_2,\\ldots,q_M)\\}> ", "965987be0e47a7aa2ac71439a5c7017f": "\\varepsilon_1:\\ x+2y+z=1, \\quad \\varepsilon_2:\\ 2x-3y+2z=2 \\ .", "9659c74b45f88ba5988390569505d522": "\\begin{cases}\n \\tan^{-1} ( \\Psi_{xy}(f) / \\Lambda_{xy}(f) ) & \\text{if } \\Psi_{xy}(f) \\ne 0 \\wedge \\Lambda_{xy}(f) \\ne 0 \\\\\n 0 & \\text{if } \\Psi_{xy}(f) = 0 \\text{ and } \\Lambda_{xy}(f) > 0 \\\\\n \\pm \\pi & \\text{if } \\Psi_{xy}(f) = 0 \\text{ and } \\Lambda_{xy}(f) < 0 \\\\\n \\pi/2 & \\text{if } \\Psi_{xy}(f) > 0 \\text{ and } \\Lambda_{xy}(f) = 0 \\\\\n -\\pi/2 & \\text{if } \\Psi_{xy}(f) < 0 \\text{ and } \\Lambda_{xy}(f) = 0 \\\\\n\\end{cases}", "965a5016d84728c8c9061ece452fb0dd": "\\alpha_1, \\ldots, \\alpha_m", "965a55c047cbf9130a6217d61ca9b947": "\\tfrac{P}{8a^2}", "965a998e9f2ac7562255f3e147bc6d71": "\\left( {c_{2P} - c_{1P} } \\right){{dT} \\over T} = \\left[ {\\left( {{{\\partial v_2 } \\over {\\partial T}}} \\right)_P - \\left( {{{\\partial v_1 } \\over {\\partial T}}} \\right)_P } \\right]dP", "965a9aec35c1f469f041a34267577db1": "\\kappa=Cs", "965ac1f5ae7d9cf7eeec20ee915abdc3": "\\left\\langle 3{{\\cos }^{2}}\\theta -1 \\right\\rangle =\\left( 3{{\\cos }^{2}}{{\\theta }_{r}}-1 \\right)\\left( 3{{\\cos }^{2}}\\beta -1 \\right)", "965acca3c15ba0f77f8689a94bc1ad8b": "\\Sigma(8)", "965ae4535f6af6e31eca4800f0b868f0": "w\\in W^{\\mathfrak{p}}\\Leftrightarrow w(\\tilde\\lambda)", "965bf1da51cebba7e4f88ff3b155889e": " V_{ij} = \\operatorname{Var}_{X_{ij}} \\left( E_{\\textbf{X}_{\\sim ij}} \\left( Y \\mid X_{ij} \\right)\\right) ", "965bf955fb66f83d3f72026a2368bed9": "c'=c/n", "965c0790c72474af47b8d99119704f63": "\\alpha = \\omega", "965c4bcbdd21b7194152ed303242fcae": "\\exists W \\forall x [PxW].", "965c79aad019944ab4a5d162e071b4ae": "\\mathbf{g}(i)", "965ca10e517eee4b144ecc16e26d560f": "(y+x)^n=\\sum_{k=0}^n{n\\choose k}y^{n-k} x^k", "965cc4be43613d5d5ff254ae37a08dd9": "\\begin{align}\n \\boldsymbol{\\omega} &= (x,y,z) \\\\\n \\boldsymbol{\\tilde{\\omega}} &=\\boldsymbol{\\omega\\cdot A} = x A_{\\bold{x}} + y A_{\\bold{y}} + z A_{\\bold{z}} \\\\\n &= \\begin{bmatrix}0&-z&y\\\\z&0&-x\\\\-y&x&0\\end{bmatrix} .\n\\end{align}", "965cc6dbde2428002c566c2849837da9": "[SU(4)\\times SU(2)_L\\times SU(2)_R]/\\mathbb{Z}_2", "965d20e71c9594e7544838165809299b": "m(f,z_0)=\\lambda \\,", "965d36622745e48e14b57b744ceb7bfd": "\\left(\\tfrac{\\pi}{2}\\right)^n", "965d3c6d75be7a89531ad6499d79cc39": "v_M < c", "965df418cad8883df001352efc030b79": "\\chi(n)=\\left(\\frac{n}{p}\\right)", "965e27b6b42b298d69c07ca3f269448b": " Y_{i,j}", "965e7102c45e79025f4e9983ba30f419": "\\boldsymbol{B}", "965e7b281a2e46dd9dbca108ca6ccceb": "\\Delta\\mathbf{b} = \\mathbf{R}^T\\Delta\\mathbf{x} = \\mathbf{R}^T(\\mathbf{I - A})^{-1}\\Delta\\mathbf{y}", "965ea74be4bda4e92268521f8d364c2a": "\nK(a,b; m) = (-1)^{\\Omega(m)} \\sum_{v,w\\mod m,\\, v^2-\\tau w^2\\equiv ab\\mod m} e^{4\\pi i v/m}.\n", "965eb9528d74bffb93c1a8b640176c74": "c_n=\\Gamma\\left(\\tfrac{n+1}{2}\\right)\\pi^{-\\frac{n+1}{2}}.", "965edbe0b7963a390cab1878baece39e": "\\Phi (X(mech,x_1)) = H[p(X_0(mech,x_1)) \\parallel \\Pi p( ^k M_0(mech,\\mu_1))]", "965f8e09296bb709e0780df3417b25f5": "\n\\theta_M\\approx 2\\theta_E\n", "965f9d597b45b91d8f2bda52dd37e8e2": "\\tfrac ac\\ =\\ \\tfrac bd,", "966009d29295783e08f6b42b80b4104e": " \\beta = \\arccos \\frac{a^2 + c^2 - b^2} {2 a c}.", "9660776fea9a4b1eecac9f0323a15afd": " Y_i(S_{x}) ", "966085d51e5a69c16f8152a458a29c20": "\\Delta t=-\\frac{2GM}{c^3}\\log(1-\\mathbf{R}\\cdot\\mathbf{x})", "966087cec07b44eca067887eddd4b561": "w \\in \\left\\{ 1,...,q-1 \\right\\}", "96608f194313cd41b37ecddbb1885846": "{\\color{Blue}~6.7}", "9660bb71c85b3a9068a4440d50a407a7": "\\mathbb{E}[Z|\\mathcal{G}] = \\sup_{n\\in\\mathbb{N}}\\,\\mathbb{E}[\\min\\{Z,n\\}|\\mathcal{G}]\\quad\\text{a.s.}", "9660d2363cdb184201650163886f71ee": "l_c=\\frac{\\pi}{\\Delta k}", "9660f07da1525f80e37724095faaa061": "\\beta_{0j} = \\gamma_{00} + U_{0j}", "966176a75e93a58b872515b17d50244f": "d_w", "9661c40de9cb415e06dade065c3aa6bd": "\\| \\Phi(1) \\| = \\| V \\|^2.", "9661d9b76ba8d7c72ef6fed5a4dd5566": "\\mathrm{Pr}(x|\\mu) = f(x - \\mu)", "9662001d24e491450b72af7b6ccac47d": "M^{k+1}=M^k \\cdot M", "96621a1490f46066d950c03b3036b405": "\n\\lim_{k\\to 0} \\sigma_e = 4\\pi a^2\\;.\n", "966235f362523a35c073de2631b5457a": "\n\\begin{align}\nU_0(x) & = 1 \\\\\nU_1(x) & = 2x \\\\\nU_{n+1}(x) & = 2xU_n(x) - U_{n-1}(x).\n\\end{align}\n", "966243d85bfa1332be277d0a81d4afb0": "\\gamma^{k' k}", "96625560693a96b2b36dd8115fbf9af0": "S_n^{}", "966266d22d3c79d6f0d7d227262bf61a": "c_\\kappa c_\\lambda x=c_\\lambda c_\\kappa x", "966283214d6a88bc27f4d62168b63bc3": " Q_{s}=q_n q_{n-1}\\cdots q_s.\\,", "96628e2efbfff000f68f6e91f13b952c": "s-b=m(mn-k^{2}) \\, ", "9662b1f61473f9144d876d0950f5397b": "\\begin{align}\na_k&=\\int_{-1}^1\\varphi(y)\\cos(2k+1)\\frac{\\pi y}{2}\\,dy \\\\\n&= \\int_{-1}^1\\left(a\\cos\\frac{\\pi y}{2}\\cos(2k+1)\\frac{\\pi y}{2}+a'\\cos 3\\frac{\\pi y}{2}\\cos(2k+1)\\frac{\\pi y}{2}+\\cdots\\right)\\,dy\n\\end{align}", "96633bb730e5b646bb4cde4a0398ff84": "x_8", "9663498302b3e529058ec7c221053409": "U = {N^{\\prime}\\varepsilon\\over2} + q\\varepsilon,", "966350a193a59917dde41bb2712381b7": " \\det A_{33} < 0 ", "966363d759c0b2c72e6103dec652aaa1": "F_{709}", "96636e5da441a3d5160dad5218701ed6": "\\textstyle \\frac{A}{\\beta+\\gamma} = 1", "96638caafec46ffff1c34bddb5f4780e": "C(e) = \\bigcap_{n=1}^{\\infty} E^n(e).", "9663a339760a5543b0aaff9cdf95ff97": "\\scriptstyle \\int_t r(t)\\, \\dot \\theta(t)\\,dt,", "9663cd9de01ce0ae8b50eec03d1e6c60": " M = E -\\varepsilon \\sin E ", "9663e9dec3d4affef35c2b7ea6e5aefd": " 2 \\psi + 180^\\circ - \\theta = 180^\\circ. ", "9663ec2c113d8a9d14f8170b02a55f20": "z=\\cos(\\theta)+i\\,\\sin(\\theta)=e^{i\\theta}", "9663fde5392a6bc0326701a1d8a4f766": "\\ M \\times N", "9664663b2fa70da0ac448a73e694e122": "\\qquad \\varphi(f) \\ge 0\\text{ for any } f \\in C_0([a,b])", "966467b7077dd174914b525e4e62bd61": "\\mathbf{y}(x) = U(x)\\mathbf{y_0} + U(x)\\int_{x_0}^x U^{-1}(t)\\mathbf{b}(t)\\,dt", "9664850857ad2d2793946362d4a8a4fa": "pF = 2E = qV.\\,", "9664ae135761e2efd4a44f3c31043809": "\\frac{\\Delta}{\\Omega_{\\perp}}", "9664bf544e3f52cc1b21eec5e5072226": "0\\to 2\\pi i\\mathbb Z \\to \\mathcal O_V\\to\\mathcal O_V^*\\to 0", "966506e6d3bf705db181fddd113d9506": " \\log_b\\!\\left(\\begin{matrix}\\frac{x}{y}\\end{matrix}\\right) = \\log_b(x) - \\log_b(y) ", "966552f8e4d725acc38508d9b2dc72c9": "\\begin{align}\n\nH_1 &= -\\vec{d}\\cdot\\vec{E} \\\\\n&=-\\left(\\vec{d}_\\text{eg}|\\text{e}\\rangle\\langle\\text{g}|+\\vec{d}_\\text{eg}^*|\\text{g}\\rangle\\langle\\text{e}|\\right)\n \\cdot\\left(\\vec{E}_0e^{-i\\omega_Lt}+\\vec{E}_0^*e^{i\\omega_Lt}\\right) \\\\\n&=-\\left(\\vec{d}_\\text{eg}\\cdot\\vec{E}_0e^{-i\\omega_Lt}\n +\\vec{d}_\\text{eg}\\cdot\\vec{E}_0^*e^{i\\omega_Lt}\\right)|\\text{e}\\rangle\\langle\\text{g}|\n -\\left(\\vec{d}_\\text{eg}^*\\cdot\\vec{E}_0e^{-i\\omega_Lt}\n +\\vec{d}_\\text{eg}^*\\cdot\\vec{E}_0^*e^{i\\omega_Lt}\\right)|\\text{g}\\rangle\\langle\\text{e}| \\\\\n&=-\\hbar\\left(\\Omega e^{-i\\omega_Lt}+\\tilde{\\Omega}e^{i\\omega_Lt}\\right)|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\left(\\tilde{\\Omega}^*e^{-i\\omega_Lt}+\\Omega^*e^{i\\omega_Lt}\\right)|\\text{g}\\rangle\\langle\\text{e}|,\n\\end{align}", "966555aaca4bc7abc48dfac344149d85": "-261\\pm 20", "96656a9bc210b920c6b7b0d61e8e3e91": "s=q,t=q", "96657d3048038a322aefc8df061b050a": "_2F_1", "9665f7494cc2089544f64b1df22f5821": " u_i= u_i(p)", "966654529e9ff89c31e4b12edef5fda0": "\np_C=\\frac{a'}{27b'^2}, \\qquad \\displaystyle{v_C=3b'},\\quad\\hbox{and}\\quad kT_C=\\frac{8a'}{27b'}\n", "96666657e901f88f61035da78ad15458": "f_H(D)=\\frac{1}{1+kD}\\,", "966667be26b0b2a55dace7dd45749872": "\\dot{Q}/T", "9666a61631f878edd3461b8256989981": "O\\,", "9666b7a77934091e964c0c86ed24a7c0": " {h_1 \\over h_0} =\\frac{-1 +{\\sqrt{1+{\\frac{8v_0^2}{gh_0}}}}}{2} ", "9666fbb75fb9dc980be2c52f88f22b4b": "(4x^2+20x)+(3yx+15y)\\,", "96671781bd0f8c9ebd5c31fd9762c871": "\\forall x .\\ [D(x) \\wedge \\neg D(f(x))]\\, ", "96673359fbc142de5a069486f2343e04": "\\operatorname{Maps}_*\\left(X,Y\\right)", "9667358a02b27964b8fcfdca83f5ed60": "\\mathbb P \\left\\{ \\sup_{t \\in T} f(t) \\geq u \\right\\}.", "96674f77649d3b4850c07fcee88f7257": "A^{\\mu}=(\\phi / c, \\mathbf{A})", "9667d59c56cecb46ea099cb4f71aafcf": "\\mbox {id} \\le f \\circ f^l\\qquad\\mbox{(left unit)}", "96694323d94a9f7bf89df7fb9b723772": "\\begin{align}\n\\mathrm{d}\\mathcal{H} &= \\sum_i \\left[ \\left({\\partial \\mathcal{H} \\over \\partial q_i}\\right) \\mathrm{d}q_i + \\left({\\partial \\mathcal{H} \\over \\partial p_i}\\right) \\mathrm{d}p_i \\right] + \\left({\\partial \\mathcal{H} \\over \\partial t}\\right) \\mathrm{d}t\\qquad\\qquad\\quad\\quad \\\\ \\\\\n &= \\sum_i \\left[ \\dot{q}_i\\, \\mathrm{d}p_i + p_i\\, \\mathrm{d}\\dot{q}_i - \\left({\\partial \\mathcal{L} \\over \\partial q_i}\\right) \\mathrm{d}q_i - \\left({\\partial \\mathcal{L} \\over \\partial \\dot{q}_i}\\right) \\mathrm{d}\\dot{q}_i \\right] - \\left({\\partial \\mathcal{L} \\over \\partial t}\\right) \\mathrm{d}t.\n\\end{align}", "96695822a9d557b13d6bc2b7329b925c": "X\\in \\mathcal{C}", "9669feeaa36866dab8988782681d450d": "\\Delta \\langle 2\\rangle", "966a6c4c66cb2e79caf0d5a0f6304328": "\n\\beta \\,\\,\\, \\approx \\,\\,\\,{{3\\,k} \\over {\\mu _T^2 }}\\,\\left( {{{\\sigma _T } \\over {\\mu _T }}} \\right)^2 \\,\\,\\, \\approx \\,\\,\\,30\\,\\,\\left( {{{s_T } \\over {n_T \\,\\bar T}}} \\right)^2", "966a97e85f2e57b4dc0a6d9fec52c757": "\n T_{11}= 2C_1 \\left(\\alpha^2 - \\cfrac{1}{\\alpha}\\right) \n ", "966ad2387c70e65d5398931db73cdaef": "M/m", "966ae0f615acbea391fc82c71e01e4d7": "10\\frac{1}{2} ", "966b2e39ab8d6ede80ff012d01b37e36": "\n \\sup_{\\theta\\in\\Theta} \\big|\\,\\hat\\ell(\\theta|x) - \\ell(\\theta)\\,\\big|\\ \\xrightarrow{p}\\ 0.\n ", "966b37c688f7fad2706ed65907311072": "f = n_1 n_2 \\langle \\sigma v \\rangle.", "966bfb73ad955910a83516b1e7421530": "\\{f^{-1}(U_1) | U_1\\in\\tau_1\\}", "966c38c9a492522559817f4a8328d448": " (2j+1)^2 ", "966cebc419b0e2800311db946db95656": " Z(\\omega) = jL \\bigg( { \\omega^2 - \\omega_0^2 \\over \\omega } \\bigg)", "966d19c4d9d09128034876594f1c61c7": "\\lambda(q) = \\sum_{n\\ge 0} {(-1)^nq^{n}(q;q^2)_n\\over (-q;q)_{n}}", "966d38f9d689acafd39edaa02900416b": "\n F=\\frac{\\pi}{4}E^*Ld\n", "966d3f7fd291a71e001d039b85460c83": "0 + j\\infty", "966d42424bdccdf2c8a1aa1e404606dc": "\\psi_x = -\\varphi_y, \\quad \\psi_y = \\varphi_x.", "966dfeaf9b0b1a3afbd7e09a23977693": "( X^\\mathrm T X)^{-1}X^{\\mathrm T}", "966e24457cf6b31bed294c9b92af8a93": "\\mathcal{L} = \\overline{q}_L\\,i\\displaystyle{\\not}D \\,q_L + \\overline{q}_R\\,i\\displaystyle{\\not}D\\, q_R + \\mathcal{L}_\\mathrm{gluons} ~.", "966e3bd1f21091c718101900c1ac41dd": " \\psi_{2}(\\mathbf{q}) ", "966e42f1a0c13e9e912156b360797f05": "\\,g(X,Y) > 0", "966e62a4566e586d33f73ba92563b273": "\\gamma_e", "966eadc90272318de458c86ac1e9e56f": "C : y^2 = f(x)", "966eae0ded637802596ce570a3e29d8b": "p_t = p_0 + \\int_0^t \\sigma_s dB_s ,", "966eff2d2d654594a970c7763014bdf4": "a + b\\sqrt{ - 1} + c\\sqrt{ - 1}\\,^{\\sqrt{ - 1}}", "966f186b9564ef0395bccbca862f58f8": " {D \\rho \\over Dt} = {\\partial \\rho \\over \\partial t} + {\\nabla \\rho \\cdot \\mathbf{v}}. ", "966f921832fec44874fc663040612ff2": "\\sin \\left(x+y\\right)=\\sin x \\cos y + \\cos x \\sin y, \\,", "966fccf69f49358566c7705ec3820fb0": "\\rho = \\frac{1}{h^n C} e^{\\frac{\\Omega + \\mu_1 N_1 + \\ldots + \\mu_s N_s - E}{k T}},", "9670292ceb39c098ab16050cc36e42d3": "\\text{s.t.}\\sum_{i=1}^n a_{ij} x_i \\ge q_j, \\quad \\quad \\forall j=1,\\dots,m", "967050998eba765adec95031409d4c3b": "\\beta(s)=\\left(\\frac{\\pi}{2}\\right)^{s-1} \\Gamma(1-s) \n\\cos \\frac{\\pi s}{2}\\,\\beta(1-s)", "9670512ac7f21f1efed67a5d5075a3a6": "\\hat{q}_{{\\rm c}}", "9670b4b545ef5c5fb03b29a256ac872a": "u[4] := 2*atan(\\sqrt(a1^2+b1^2)*cot((1/2)*\\sqrt(a1^2+b1^2)*\\eta)/a1+b1/a1)", "96712cf8f3b4a13a96d847a603c82be1": "\\beta'<\\beta", "9672064eb3f735db22e42dadf75e5ead": "\\begin{align}\n &(V_i-V_o)DT -V_o(1-D)T = 0\\\\\n \\Rightarrow\\; &V_o - DV_i = 0\\\\\n \\Rightarrow\\; &D = \\frac{V_o}{V_i}\n\\end{align}", "9672153ece473063284006054daabcd5": "1.79\\pm 0.01", "9673a823fb41990c1fabc3573ba02f42": "R \\simeq \\operatorname{M}_n(C)", "9673d16a7cb5b2a852e005dadab0c950": "\nQ^m_\\ell \\equiv \\sum_{i=1}^N q_i R^{m}_\\ell(\\mathbf{r}_i),\\qquad -\\ell \\le m \\le \\ell.\n", "9673ef1ee0282a98a5b363d6ad195512": "G(N, P, T) = - k_B T \\ln \\Delta(N, P, T) \\;\\, ", "9674250bf63214c7750aeefee24ab2ac": "*(T^{IJ} + i U^{IJ}) = {1 \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} (T^{KL} + i U^{KL}) = i (T^{IJ} + i U^{IJ}) .", "9674307aaface96bd1c0d74c711e3da7": "S\\subseteq \\mathrm{Range}(\\mathcal{A})\\,\\!", "96743d3aa3aa0216aca32c67dd162e58": "0 \\subseteq (x^{n-1}) \\subseteq (x^{n-2}) \\subseteq \\ldots \\subseteq (x^1) \\subseteq (x^0)=R", "96745d77f7326e4859216e1ef74563d8": "\t\n\\begin{array}{lccccccc}\n\\text{Minimize} & f^T x & + & g^T y & + & h_k^Tz_k & & \\\\ \t\n\\text{subject to} & Tx & + & Uy & & & = & r \\\\ \t\n & & & V_k y & + & W_kz_k & = & s_k \\\\ \n & x & , & y & , & z_k & \\geq & 0\n\\end{array}\n", "9674734bb7106f12be1616273d868aa6": "x_0\\ ", "96748a493cd1ce13d2438ca51f4d2dd4": "X(u,v) = \\Re \\left ( c(w) - i \\int_{w_0}^w n(w) \\wedge c'(w) \\, dw \\right)", "9674ab36b62e6b308973ef92c2922b6e": "t_f", "9674b1b62bab169db7bb590e93b46d76": "\\operatorname{AMISE} (\\bold{H}) = n^{-1} |\\bold{H}|^{-1/2} R(K) + \\tfrac{1}{4} m_2(K)^2 \n(\\operatorname{vec}^T \\bold{H}) \\bold{\\Psi}_4 (\\operatorname{vec}^T \\bold{H})", "96750222fa33acf428b8df77d4fb9223": "z=\\frac{\\hat{p} - p_0}{\\sqrt{p_0 (1-p_0)}}\\sqrt n", "967514b3e05ac51c451138335f806d22": "r = a\\cos(\\theta-\\theta_0)", "9675628cd9a12f635ceeed9a87aa2a87": "VSWR = \\frac{R_\\text{L}}{Z_\\text{0}}", "9675b7a93ee422689f5538459c26ed13": "(p \\to q) \\land (r \\to s)\\ \\vdash\\ (p \\to s) \\lor (r \\to q)", "9675bcb566970eee4bf16ec93b2b36f3": "C_3=\\sqrt{\\frac{4 + c_3 + c_4}{12}}\\approx 1.14261", "9675d6eb11d256d7cd48000dabda972a": "\nU_{hy} = c_0 \\exp \\left (\\frac{c_1 - r}{c_2} \\right )\ncos(c_3(c_1 - r)\\pi ) + c_4 \\left (\\frac{c_1}{r} \\right )^6\n", "96765dcc59be3af1c5d493fc661ee3de": "\\scriptstyle\\hat{b}=x_{(n)}", "96766558f85ca5f4a222c10de8e08316": "\\mathbf{v}^{(0)}", "9676acf14e784286a063c3f460c8dee3": "{\\mathrm {Mp}}(n,{\\mathbb R}).\\,", "967708f1036c120b5b1bbe4378963207": "\\mathfrak P_2(K)", "96770a9c00677188e36e4df5a97cdefe": "a[n]b\\,\\!", "967717bfff073e99932801b5bbb40fd7": " \\oplus_{i=1}^m A \\otimes_B \\cdots \\otimes_B A", "9677245afe9ed217f7c84a8e454090bb": "\\exists R.C", "96774d1b82efe182412100590c68c5dc": "\\left(a, q, v\\right)\\succsim \\left(b, q, v\\right)", "967859eaa839f4dc5744660b84d2db0b": "2=(1+i)(1-i)", "967878d1da852d4b07a961e3168b0fff": "\\Delta", "9678b4824a47f751fe8cf4eec6b76791": " n =3 ", "9678cb4cbbdcf520dbf552dde3c30448": "\n\\mu = \\operatorname{Re} \\operatorname{arccosh} \\frac{\\rho + z i}{a}\n", "9678f532ec9ccc806707bb4ae739a56b": " E = { {\\hbar^2 k^2} \\over {2m_0} }( \\gamma _1 + {{5} \\over {2}} \\gamma _2 - 2 \\gamma _2 m_j^2)", "96792f1de2f183cf335b57d9c884f3ee": "E_\\text{k} = \\frac{p^2}{2m}.", "96796084acb556956043ce1222bb3747": "\\Delta=b^2-4ac.\\,", "9679760672faa6a2571177d0dab856ed": "\\nu_1, \\nu_2", "9679da780fdc6a47d149fb278e963ded": "\n\\begin{bmatrix}\n 16 & 11 & 10 & 16 & 24 & 40 & 51 & 61 \\\\\n 12 & 12 & 14 & 19 & 26 & 58 & 60 & 55 \\\\\n 14 & 13 & 16 & 24 & 40 & 57 & 69 & 56 \\\\\n 14 & 17 & 22 & 29 & 51 & 87 & 80 & 62 \\\\\n 18 & 22 & 37 & 56 & 68 & 109 & 103 & 77 \\\\\n 24 & 35 & 55 & 64 & 81 & 104 & 113 & 92 \\\\\n 49 & 64 & 78 & 87 & 103 & 121 & 120 & 101 \\\\\n 72 & 92 & 95 & 98 & 112 & 100 & 103 & 99\n\\end{bmatrix}\n", "9679e3f60682486934c7fa7009bc84e5": "D_{med} = E|X-median| ", "9679e9a734ff651cc804393f2122ac51": " a \\tfrac{b}{c}", "967a3276393be54ce765033f733aa210": "E_n(j)", "967a4097b361534ef5af9b986c01d65d": "\\mathcal{G}_{n,r}", "967ac39f974dac480c0c26541cff1fc1": "\\epsilon_{C}", "967af184bd82758f1ebf17af95726022": "y \\ne x", "967b0bb25731d4930212bb57e0d85c75": "C_{HbO2}", "967b2e5f8b831b7184816b93870deca8": "a\\ne b = c \\ne d, \\alpha \\ne 90 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ, \\zeta = 120 ^\\circ", "967b87f4296f667ae6de71a7f97e8f90": "< M/4", "967bc518741d82cd0c0de5d2b188644e": "n\\alpha", "967be9a5432205246240498859ef0864": "\\sin(\\pi \\nu)\\mathbf{J}_\\nu(z) = \\cos(\\pi\\nu)\\mathbf{E}_\\nu(z)-\\mathbf{E}_{-\\nu}(z)", "967c07469d5b1863abe91f319ebd81ae": "\\textstyle n \\le 2^{r-b+1}-2^{-b+1}", "967c187991ba19b7e14ed6997467e8aa": "dS\\,\\!", "967c49ec48ebc001e980a260774d5e0f": "\\varphi(n)\n=\\sum_{d\\mid n} d \\cdot \\mu\\left(\\frac{n}{d} \\right) \n=n\\sum_{d\\mid n} \\frac{\\mu (d)}{d} \n,\n", "967c4c9a8d53ef40a816f3ca3d47975e": "O(b^d)", "967c665e98f9c22181ee5df38e724851": "d(c,y)\\,", "967c6fd70d505ee4a9c0bd819f97b907": " \\widehat{\\boldsymbol{\\beta}}_{p} = \\widehat{\\boldsymbol{\\beta}}_{ols} ", "967c8bcfdffeacc73f76808eabc129ee": "g(n)=\\sum_{d\\,\\mid \\,n}f(d)\\quad\\text{for every integer }n\\ge 1", "967c92b31775a53bfede55d1074b1599": "x_{17}", "967cbe9ade666ab8c4c14b3d155e26c5": "\\textstyle \\lim_{n \\to \\infty}x_n", "967cc85f2c2a6e6fd32b0623c3037845": "d=\\sqrt{2f\\lambda}", "967d06b7924f820fdff4ce7b1e56753f": "F(x,y,p) = y + f^\\star(p) - px = 0~.", "967d24f295008a25637b26f243db1bf8": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 10.07089 \\log_e(T+273.15) - \\frac {6351.140} {T+273.15} + 81.14393 + 9.127608 \\times 10^{-6} (T+273.15)^2", "967d30d903b1db5b90d61a9586e55195": "M_{k} \\equiv q a^{k}", "967d3d588bb24619eba15f78a33b13dc": " r = b(1 + \\cos \\theta) = 2b\\cos^2 {\\theta \\over 2}", "967dbce70565ba563a16f88a84274abd": "\\mathbf{r}(s) = \\left[ x(s),\\ y(s) \\right] \\ . ", "967dca65521dbf1f2d315b9fea34cf5d": "f(x,y) \\ne 0", "967e2df9d95fe65f0daa8859d7c55623": "\\! J > 10", "967e6c81e91912fedb16995d112425b8": "A(a,b) < Q(a,b)", "967f0056c030af1d2ffb6a6faddea2e2": "\\rho_\\mathrm{in} = S_{11}\\,", "967f0a2d3069d4b61aa21533c1b2e5fe": "N(t)< 0", "967f24c63fe02631194740ddc0cbe10e": "\\int_{-\\infty}^\\infty f(t) \\delta(t-T)\\,dt = f(T).", "967f2efd752dd30872f001f150250795": "[00,\\beta]_{|\\beta} = \\left(g^{\\alpha\\beta}\\left[00,\\alpha\\right]\\right)_{|\\beta} = -\\frac{\\kappa \\, \\rho_0}{2}\\qquad\\qquad (*)", "967f50f9d50ff8a0999cfe16f7dee07e": "\\cap_{n<\\omega} W_n \\neq \\emptyset", "967fbd17acaf9c1557be8eef29fbf5bd": "(x_1,x_2)", "967fe4fdd948949e626cfa73e7a857b2": "\\epsilon_{1}(p) = E_{1}(p)-E_0", "967fe74f6bdda003893f4a99ca346652": "q_1,q_2,\\dots,q_n", "967fee8c0a56ce76a1364a19f6cff298": "C_n \\to \\{\\pm 1\\}", "967ff4b030ea4dc2618e23f4f4534d47": " \\Delta \\beta_j\\!", "967ffa3ca82c4b8aad1075067fb3fec5": "\\pm 1", "96803732a00275131eb3aecbda7f88ec": " \\mathbf{m} = \\mathbf{r} q \\,\\!", "96803e2f6de4ac5c25b21c8620d935d7": "\\frac{\\partial \\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y) }{\\partial c} = (\\beta- 1) \\sum_{i=1}^N \\frac{1}{c - Y_i} \\,- N (\\alpha+\\beta - 1) \\frac{1}{c - a} = 0", "968073386649cec40d41bf40defb176f": "P_b = Q\\left(\\sqrt{\\frac{2E_b}{N_0}}\\right).", "96812daf06eae6562e4f3f54dd3030db": "q_\\pm=1+\\tfrac12(wh)^2\\pm wh\\sqrt{1+\\tfrac14(wh)^2}", "968135f4b43778fdc8397bb4814b61c8": "\\frac{dTR}{dP} = f'(P) \\cdot P + Q", "968151aa978bd9e84a4b8c5765d07e1a": "\\partial \\mathbb T", "96816bb3d3d721aead0ae40973003c96": "\np_i(\\theta)= \\Phi \\left( \\frac{\\theta-b_i}{\\sigma_i} \\right)\n", "968186b6ab9aca642a02ef1c58e9e936": "v_{||} \\equiv \\vec{v} \\cdot \\hat{b}", "9681c4c09cea4ffd2b592ff03959bd2d": "Y = [y_1 , \\ldots ,y_N ]^T", "9681d9eb0e3256ee3746e617305b2814": "\\varepsilon(a,b,c,d) (cz+d)^k", "96821d4d479488000c42db656036bdb9": "\ny = 1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\cfrac{z}{1 + \\ddots}}}\\,\n", "96821e1eb7c016027276300f05f193e2": "(z_1, z_3; z_2, z_4) = 1-\\lambda\\,", "968264ddf6072f653d88f876d84c5d61": "\\pi(s)", "9682892a95f5d7f03207b801c7cf733a": "g(\\mathbf{x}^\\prime) = \\hat{\\mathbf{b}}\\hat{\\mathbf{a}} \\; g(\\mathbf{x}) \\; \\hat{\\mathbf{a}}\\hat{\\mathbf{b}}", "9682aa1a47e387241c60029a3bd44af1": " P( | X | \\ge k ) \\le 1 - \\frac{ k^2 } { 3 \\operatorname{ E }( X^2 ) } \\quad \\text{if} \\quad k^2 \\le \\frac{ 4 } { 3 } \\operatorname{ E }( X^2 ). ", "9682bc757933b70e5be76c9c5a369f2a": "\\beta>1,", "968346db823d203d5051c131add7226d": "f(x_{j})=0", "9683f6182ccef9526eddae8e77a03818": "\n \\frac{d}{dz}\\left[\\mu(z)\\,\\frac{dV}{dz}\\right] = [k^2\\,\\mu(z) - \\omega^2\\,\\rho(z)]\\,V(k,z,\\omega) \\,.\n ", "968400fed1e702832d2468ddb32f6730": "b=K\\cdot L\\cdot c_b=K\\cdot 100 k_e \\frac{Y/Y_n-Z/Z_n}{\\sqrt{Y/Y_n}}", "968431c743c23b95ffca658ce238b95a": "G(n, a, b)\\,\\!", "9684547889cd06f9125dc35bf9ee3f21": "\\chi _B", "96847aaa39796cbc787c5edb78c8ee05": "\\tau\\to\\widetilde\\tau", "9684d59cb68c6f82e03d9b34f9a6f3ae": "\\boldsymbol{ v} = \\begin{pmatrix} v_e \\\\ v_n \\\\ v_u \\end{pmatrix}\\ ,", "9685347ba6ca9a25beb6cb132b5da1af": "\\int x^m\\arccos(a\\,x)\\,dx=\n \\frac{x^{m+1}\\arccos(a\\,x)}{m+1}\\,+\\,\n \\frac{a}{m+1}\\int \\frac{x^{m+1}}{\\sqrt{1-a^2\\,x^2}}\\,dx\\quad(m\\ne-1)", "96853cbbaa6ac53bf3aa8929dea10661": "K_0 = \n\\prod_{r=1}^\\infty {\\left( 1+{1\\over r(r+2)}\\right)}^{\\log_2 r} \\approx 2.6854520010\\dots", "9685499f36fcfff4c680c1baf946b0d5": "I_{3}", "9685cd885497f379220ff3bd3bb218a2": "\\begin{align}\n &\\frac{\\partial}{\\partial x}\\left(2 \\mu \\frac{\\partial u}{\\partial x} + \\lambda \\nabla \\cdot \\mathbf{v}\\right) + \n \\frac{\\partial}{\\partial y}\\left(\\mu\\left(\\frac{\\partial u}{\\partial y} + \\frac{\\partial v}{\\partial x}\\right)\\right) + \n \\frac{\\partial}{\\partial z}\\left(\\mu\\left(\\frac{\\partial u}{\\partial z} + \\frac{\\partial w}{\\partial x}\\right)\\right) \\\\ \\\\\n & = \n 2 \\mu \\frac{\\partial^2 u}{\\partial x^2} + \n \\mu \\frac{\\partial^2 u}{\\partial y^2} + \\mu \\frac{\\partial^2 v}{\\partial y \\, \\partial x} + \n \\mu \\frac{\\partial^2 u}{\\partial z^2} + \\mu \\frac{\\partial^2 w}{\\partial z \\, \\partial x} \\\\ \\\\\n & = \n \\mu \\frac{\\partial^2 u}{\\partial x^2} + \n \\mu \\frac{\\partial^2 u}{\\partial y^2} + \n \\mu \\frac{\\partial^2 u}{\\partial z^2} + \n \\mu \\frac{\\partial^2 u}{\\partial x^2} + \\mu \\frac{\\partial^2 v}{\\partial y \\, \\partial x} + \\mu \\frac{\\partial^2 w}{\\partial z \\, \\partial x} \\\\ \\\\\n & = \\mu \\nabla^2 u + \\mu \\frac{\\partial}{\\partial x} \\cancelto{0}{\\left(\\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} + \\frac{\\partial w}{\\partial z}\\right)} = \\mu \\nabla^2 u\n\\end{align}", "9685d3aaf2aa496066db1f9d41ba24b3": "\\Phi \\times a = \\Phi", "9685df84db7fcb773c9e5e36eedd0434": "i \\ge -1", "968609e8fa7f2316e167943472efd016": " u_i \\ge y_i - a_0 - a_1x_{i1} - a_2x_{i2} - \\cdots - a_kx_{ik} \\,\\ \\,\\ \\,\\ \\,\\ \\,\\ \\text{for} \\,\\ i=1,...,n", "96864c8df702ab7c7ea08622627d388b": " f(x) ", "96864d5ee5d7575713e2d0dd7e1f5f75": "\\mbox{f(t)} =\\frac{1}{2}(\\sin(\\omega t+\\alpha) +\\sin(\\omega t-\\alpha))= \\sin(\\omega t)\\cdot\\cos(\\alpha) ", "96865bd46e9b758bb4323b5b468579ab": "\\sigma_{ax}(E)=1.3\\times10^{13}b\\frac{C}{E_p}", "9686ae57eaa20f31555dc90073abf1b9": "Z_{mn} =A_1P_n^m(i\\zeta)+A_2Q_n^m(i\\zeta)", "9686d3430fcb4b4ee4e9e390ad379b37": "{_uM_u}=\\frac{10^2}{(32.2)0.168}+ \\frac{0.168^2}{2}=18.500ft^2", "968773702e74f67f05f95ba4fcae2146": "\\phi_{W}\\,= \\phi_{w}", "96879e0de0fa6c160552c49679b48ba4": "\nII = -\\mathrm{d}\\mathbf{N}\\cdot \\mathrm{d}\\mathbf{P} = \\omega_1^3\\odot\\omega^1 + \\omega_2^3\\odot\\omega^2\n=\\begin{pmatrix}\\omega^1 \\omega^2\\end{pmatrix}\n\\begin{pmatrix}\nii_{11}&ii_{12}\\\\\nii_{21}&ii_{22}\n\\end{pmatrix}\n\\begin{pmatrix}\\omega^1\\\\\\omega^2\\end{pmatrix}.\n", "9687a51fb57dc48e8a5bda7493fec775": "\\displaystyle{T_c(\\partial_z D(\\varphi)) = \\partial_z D(\\varphi).}", "96880b56c4c8aa76a7511493b3fb16cb": "\\frac {a + ib}{c}", "96880bd32172be05d03a0c40ac43b137": "S_n(s) = \\sin^2\\left(n\\arcsin\\left(\\sqrt{s}\\right)\\right).", "96880dd0fd4363652ee10d28d8a32128": "\n\\nabla^2 \\varphi = \\cfrac{1}{h_1h_2h_3}\\frac{\\partial }{\\partial q^i}\\left(\\cfrac{h_1h_2h_3}{h_i^2}\\frac{\\partial \\varphi}{\\partial q^i}\\right)\n", "96881abbc0f35677efac7b58140d3adc": " Df = -((x^2-1) f^\\prime)^\\prime =-(x^2-1)f^{\\prime\\prime} -2x f^\\prime ", "96885603dcbfd96ba9544ddf16ea6f74": "\n\\Delta S = S_2 - S_1 = - k \\ln \\frac{c_2}{c_1},\n", "968891bf92bbcdc96d6573d651e58382": "\\alpha = (\\frac{A-1}{A+1})^2", "9688b42d9a8156e3c0bb7c1f797ccfd0": "\n\\Pr\\left\\{ \\left( A_{1}\\cap\\cdots\\cap A_{N}\\right) ^{c}\\right\\}\n=\\Pr\\left\\{ A_{1}^{c}\\cup\\cdots\\cup A_{N}^{c}\\right\\} \\leq\\sum_{i=1}^{N}\n\\Pr\\left\\{ A_{i}^{c}\\right\\} ,\n", "968904ee648f96b801ee32e371de95f5": "\\rho(\\boldsymbol\\beta|\\sigma^{2})", "96890808e2ecb7f0bc2de120d298cb82": "C(a) = \\int_1^a f(t)\\,dt+ \\frac{1}{2}f(1) - \\sum_{k=1}^{\\infty}\\frac{B_{2k}}{(2k)!}f^{(2k-1)}(1)", "96890d90cc97d33335e3f0647fba5853": "\\theta_0 = \\alpha_1 - q \\alpha", "968910f853d6e9b13936611ebf53a5f9": "s\\left\\{\\begin{array}{l}4\\\\3\\\\2\\end{array}\\right\\}", "9689166cda091c2d0bb53c0f5f9f5e04": "(8.d)\\quad \\gamma_{,\\,z}=2\\,\\rho\\,\\psi_{,\\,\\rho}\\psi_{,\\,z} \\,.", "96893d5cea9955d1f30c9ff9fd8d99a0": "\\sigma=E\\epsilon", "9689948e6f0dc7c9db39a55ab88a3c27": "\\textstyle\\{-(3,4),-(4,3)\\}", "9689ecfe0899fdf58e2301d548473d26": " \\lim_k \\int f_k \\, d \\mu = \\int f \\, d \\mu. ", "968a393c15807e24a5ead6c15a487e81": "p_n=\\frac{\\begin{vmatrix}1 & 0 & \\cdots && e_1 \\\\ e_1 & 1 & 0 & \\cdots & 2e_2 \\\\ e_2 & e_1 & 1& & 3e_3 \\\\ \\vdots&&\\ddots&\\ddots&\\vdots\n\\\\ e_{n-1} & \\cdots & e_2 & e_1 & ne_n \\end{vmatrix}}{\\begin{vmatrix}1 & 0 & \\cdots & \\\\ e_1 & 1 & 0 & \\cdots \\\\ e_2 & e_1 & 1& \\\\ \\vdots&&\\ddots&\\ddots\n\\\\ e_{n-1} & \\cdots & e_2 & e_1 & (-1)^{n-1} \\end{vmatrix}}\n=\\frac{\\begin{vmatrix}1 & 0 & \\cdots && e_1 \\\\ e_1 & 1 & 0 & \\cdots & 2e_2 \\\\ e_2 & e_1 & 1& & 3e_3 \\\\ \\vdots&&\\ddots&\\ddots&\\vdots\n\\\\ e_{n-1} & \\cdots & e_2 & e_1 & ne_n \\end{vmatrix}}{(-1)^{n-1}}\n =\\begin{vmatrix}e_1 & 1 & 0 & \\cdots\\\\ 2e_2 & e_1 & 1 & 0 & \\cdots\\\\ 3e_3 & e_2 & e_1 & 1 \\\\ \\vdots &&& \\ddots & \\ddots \n\\\\ ne_n & e_{n-1} & \\cdots & & e_1 \\end{vmatrix}.\n", "968a423d7fba066ba26de560f528b76b": "(x^{q}, y^{q})", "968a8e1d64bc42000cdbc306772b4173": "H_1(X) \\cong (H_1(A)\\oplus H_1(B))/\\text{Ker} (k_* - l_*)", "968ac2c09a0c32a3fc34da2ad83a3683": "-\\sin \\varphi\\mathbf{\\hat{x}} + \\cos \\varphi\\mathbf{\\hat{y}}", "968b213baf87bc496f067951f7d68e16": "\\, z = z_0 + r \\cos \\varphi \\,", "968b379b8ee1e9b962682a13853113c8": "\\Delta(C_{in}(C_{out}(m^1)_i), C_{in}(C_{out}(m^2)_i)) \\ge d.", "968b59c669de99eba6e3553970fb6752": "\\textit{ideas}", "968beece5c73b179430206bfef76bb9c": "\\mathbf{R_{01}} = \\frac {\\mathbf{W}} {\\mathbf{I_1}^2} ", "968c057a42f4cddaae2d8ee460ee8b23": "U_1, U_2, \\ldots, U_{n-1}", "968c51601583b608cd8c296da53fbbce": "\\kappa_2 =\\mu_2-\\mu_1^2,", "968d31b43fcb3dafc77cb75ba8b3d218": "(R^2/p^2)\\mathbf{p}", "968d559fcf8e2c853e401b959741c7b9": "\\scriptstyle a^2+b^2=c^2", "968dab8a0671a7033e8129b532ebc055": "l-1", "968e0ca8da8e7e8a8f31548cad7edd59": "f_{12}", "968e264713972a626b13735cb3b10bf1": " N \\sim c B^a (\\log B)^{t-1} ", "968ebeb0ff4dd6cae4d84cc60766fd43": "F= M a", "968ecb9b1ce3d6324fc755762c466269": "\\ y[n] = \\sum_{k=-\\infty}^{\\infty}{h[k] x[n-k]}", "968ee8a69bd79a87e8f8e3dbe0f30444": "a = \\frac{P+\\sqrt{D}}2\\quad\\text{and}\\quad b = \\frac{P-\\sqrt{D}}2. \\,", "968ef6aaf26db759af252c7b3a4af0cc": "ad_x", "968f4a9d30a08bf2f3854713d54ced9a": "5F_5^2=125\\equiv 4 \\pmod {11} \\;\\;\\text{ and }\\;\\;5F_6^2=320\\equiv 1 \\pmod {11}", "968f58d3c75e3696280532b6d3656a42": " \\sum_{i=1}^m x_{ij} = 1 \\qquad j=1, \\ldots, n", "968fe71b4174b2bc2e9c91749316fe81": "V\\colon M \\to TM", "9690616aaed115001ec30161dd600ae0": "\\Delta (mv) = \\rho q(v_2 - v_1)", "9690893e5e647853bf8c9d5bba180410": "\\alpha_{\\rm L}=k_3=k_{\\rm F}", "9690f50e038039cbf23b6512c841be66": "\\left( \\begin{smallmatrix} 1 & 3 \\\\ -1 & -2 \\\\ \\end{smallmatrix} \\right)", "96914054e894f933f098096f489c3c2e": "{2n\\choose n+1}.", "9691842d59d1d96e6fa19c1365c16ebb": "\\sum_{i=3}^\\infty P(E_i)=\\sum_{i=3}^\\infty P(\\emptyset)=\\sum_{i=3}^\\infty a = \\begin{cases} 0 & \\text{if } a=0, \\\\ \\infty & \\text{if } a>0. \\end{cases}", "96919d38756e8b57cc8624d95ae9b321": "v^* = \\frac{v}{V}", "9691b07e45502db979579fabd33d51b7": "\\chi \\propto \\tau^{-\\gamma}", "9691b1706e4e76436ff0d77b68a89f16": "O[n^2]", "9691d1660513ac6c056082c0042f5a7a": "f:X\\otimes U\\to Y\\otimes U'", "9691d444109f61208b17d852c0ab5871": "F^Q", "9691f3db999a2068fca7cd96df3f8692": "\\omega_T", "9691f5ed791d47f1588599cfdd3752e7": "V_t= \\sqrt{\\frac{2mg}{\\rho A C_d }}", "9692576f7e43da91597919d70255b6eb": "|S\\cup T| = |S| + |T| - |S\\cap T|.", "969292fd8b0d9ed55e8aafaf6a8a08c4": "\\nabla=\\partial_\\rho\\, \\hat{e}_\\rho +\\partial_z\\, \\hat{e}_z ", "969294d77f473ac9385e3b6a4ab0c943": " f = \\frac {{\\ell}}{{D}}", "9692a23cf75b604023e03c4b01a87cf8": "\\frac{d}{d s} u(x(s)) = \\nabla u(x(s)) \\cdot \\dot x(s) = \\xi \\cdot \\frac{\\partial H}{\\partial \\xi}", "9692b07b4810232ecfef23ce7480b242": "\nE(\\omega,\\vec{r}) = \\sum_{n=-\\infty}^{+\\infty} K_n^{E_y} e^{-j\\frac{2\\pi n}{a}z} e^{-j\\vec{k}\\vec{r}}\n", "96932e55c34d326d8c27a5a1a3ade953": "\n L_A = \\frac{E_w}{\\pi} \\frac{Y_b}{Y_w} = \\frac{L_W Y_b}{Y_w}\n", "969341eac9d73f4d58e301f922f1ef42": "\\| \\varphi_1 \\|_\\infty = \\sup_{t \\in I_a} | \\varphi(t)|. ", "969364518442e2b6cca0a14121173bae": " \\varphi (L) X_t = \\theta (L) \\varepsilon_t\\,", "9693b40407a9621623ae4f47d7842b96": "\\text{Muscle force} = \\text{PCSA}_2 \\cdot \\text{Specific tension}", "96948fa40c1fc7ad885149882221153d": "\\vec{v}_R + \\vec{v}_{\\nabla B} = \\frac{2\\epsilon_\\|+\\epsilon_\\perp}{qB}\\frac{\\vec{R}_c\\times\\vec{B}}{R_c^2 B}", "9694b64daebda61f05698755124be40c": "FGT_0=\\frac {H} {N}", "96955206e922958ad257b58e48a56b53": "\n\\left(\\frac{p}{q}\\right)_3 \\left(\\frac{q}{p}\\right)_3 = \\left(\\frac{(\\frac{L'M+LM'}{2M})}{p}\\right)_3^2.\n", "9695a269b6aa2adf3bbc1378323b5ef4": " a \\cdot \\tfrac{b}{c}", "9695dc150602cfcae9e9093eb2bd0e45": "\\begin{align}\nd\\Phi_{G} & = dU - TdS - SdT - \\mu dN - Nd\\mu \\\\\n& = - P dV - S dT - N d\\mu \n\\end{align}", "9696829c971e779f6eabbbd766bc3f69": "f_0,\\ldots, f_k\\in A", "969688b0cc8adeef267e4ef916fc491c": "\\supseteq, \\nsupseteq, \\supsetneq, \\varsupsetneq, \\sqsupseteq \\!", "9696be29631490e4ea6be0a3dacec040": "\\lambda = \\mu \\nu", "9696d72f662dc92d99a029ba23e0dfe9": "S = \\frac{E[R_a-R_b]}{\\sigma} = \\frac{E[R_a-R_b]}{\\sqrt{\\mathrm{var}[R_a-R_b]}},", "96970b3544ad2239e2d3038845d820c9": "\\scriptstyle (u_i,v_i)", "969740cdefb79f7015cc6ef4c827fff2": "\n \\nabla ^2 p^{n+1} = \\frac {\\rho} {\\Delta t} \\, \\nabla\\cdot \\mathbf{u}^*\n", "96974da7a973e7f053f9366edd01b137": "dT_D=\\bar{\\sigma}d\\bar{\\epsilon}^e", "9697aeee29a345b0779d3810cda3635d": "S_2(k_{\\lambda}) = k_{-\\lambda},\\ S_2(e_i) = - k_i e_i,\\ S_2(f_i) = - f_i k_i^{-1}", "9697e658e07a68aa8639cf7906ad7e13": "\\langle f,g\\rangle=\\int_{-\\infty}^\\infty f(x)\\overline{g(x)}\\, w(x) \\, \\mathrm{d}x.", "969893d382591d2a2699b2522c2dcfac": " M \\succ 0", "969898aec4e52289b07ced47d015ea8c": "(u_t)_{t \\in [0,T]}\\,\\!", "96998530500bd73fe57f83e12d136317": "\\mu_0 \\mu_r (1 + \\chi_0) (H_{ext} + H_{exc})", "9699a0b0a76888777d8b2c2b16b0230a": "\\mathbf{x} = (\\alpha, \\beta, \\gamma)", "9699a16d0a063757bb5104583f32fde2": " \\left( \\begin{matrix} 1\\\\ \\frac{c }{\\rho } \\end{matrix}\\right),\\left( \\begin{matrix} 1\\\\ -\\frac{c }{\\rho } \\end{matrix}\\right) ", "969a047b26737cf7ec382a18659c2218": " D_2(x,\\alpha) = x^2 - 2\\alpha \\,", "969a92d083e1d1be595e5bdb4b0a3e32": "\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\mathbf{j} = \\sigma\\,", "969aabded4b34f0dac6ead580e263f9b": "B_n^{(-p)} \\equiv 2^n \\pmod p,", "969aede0bd00e3ce46e300be551fedf2": "\n\\operatorname{Li}_s(e^\\mu) = -{\\Gamma(1 \\!-\\! s) \\over 2\\pi i} \\oint_H {(-t)^{s-1} \\over e^{t-\\mu}-1} \\,\\mathrm{d}t \\,,\n", "969b16c64825e8fec17b19dba8e1e196": " \\left \\{ f : x \\to \\sum_{k=-\\infty}^{+\\infty} c_k e^{i k x} \\right \\} \\ \\overset{P_N}{\\longrightarrow} \\ \\left \\{ P_N f : x \\to \\sum_{k=-N}^{N} c_k e^{i k x} \\right \\}", "969b98c8f0a5090ac790ac8e95bbaea1": "Pr[ f( A( f(x) ) ) = f(x) ] < \\frac{1}{p(n)}", "969b9e51d80af0979db9fbc7e5c88f15": "\\limsup_{\\delta \\downarrow 0} \\mu_{\\delta} (F) \\leq \\mu(F),", "969ba73148d1435b98bc300932d794c5": "\\Delta(p;\\mu^2)=\\frac{1}{p^2-\\mu^2+i\\epsilon}", "969bf8faf3a62e103c5a19f7029f2573": "\\frac{\\operatorname{d}E_k}{\\operatorname{d}t} = F \\cdot \\frac{\\operatorname{d}s}{\\operatorname{d}t}", "969c3d6e7eeeb50ed8f84bfddb484bf3": "eRPF = \\frac{U_x}{P_a} V", "969c407a9cb22091558f87821ca55d52": "\\forall p \\forall a \\, (p \\neq a)", "969c6e1fbcc0b5272419a94f85798ea3": "\\lim_{n \\to \\infty}{ P\\left( \\left| f\\left( \\frac{K}{n} \\right) - f\\left( x \\right) \\right| > \\varepsilon \\right) } = 0", "969ca748651e23cdf048b270d1e42663": "X_1, X_2, \\dots , X_n", "969ccd0711d1deaa1933300255665392": " \n w(x,y) = \\frac{a^2}{2\\pi^2 D}\\sum_{m=1}^\\infty \\frac{E_m}{m^2\\sinh\\alpha_m}\\,\n \\sin\\frac{m\\pi x}{a}\\, \\left(\\alpha_m \\coth\\alpha_m \\sinh\\frac{m\\pi y}{a}\n - \\frac{m\\pi y}{a}\\cosh\\frac{m\\pi y}{a}\\right) \\,.\n", "969ce6aa1e9d574ff024f2e19c0b16af": "\\scriptstyle{\\mathbf{H}}", "969cea4373af25b8ec64dff9ec2821bb": "gW = gW^{T} - f(U)", "969cf02095be996a04c19cc7fa8c5f00": "P_{compress} = \\frac{u}{3} = \\frac{4\\sigma}{3c} T^4 ", "969d5059825fc30ee8c921d1856efe16": "Z_{1}", "969d6bbbdfa6117195feddbfe8db952d": "\\textstyle 2^{l-2-r}", "969d7b3e5ae7e96077112c9903ec1bf0": "P_{LOSS}", "969d823689e1f6e4c0cf0ef0bcbeade8": "p'(x),", "969dc2b9313800ee68db050fd9a21e86": "e^{-iH_nT} = [e^{-iH_n\\delta}]^{T/{\\delta}} = [e^{\\frac{{\\delta}}{2}F}e^{{\\delta}G}e^{\\frac{{\\delta}}{2}F}]^{n}", "969e1c42faf3f0b6ad29f36e6e38d03a": "\\delta= \\sqrt{{2\\rho }\\over{\\omega\\mu}}\n\\; \\; \\sqrt{ \\sqrt{1 + \\left({\\rho\\omega\\epsilon}\\right)^2 }\n+ \\rho\\omega\\epsilon} ", "969ea9db062954fac0c0d11c5a375234": " LC_{float}=\\max(S_T - S_{min},0) = S_T - S_{min}, ~~\\text{and} ~~ LP_{float}=\\max(S_{max} - S_T,0) = S_{max} - S_T,", "969ebe52b9b48d16ab865e3f2fe0d098": "p(y|\\theta,\\xi)\\,", "969eeee73d9b9883e3f417515338cfee": "\\bold{j}_\\perp", "969efeeaf005d67657d67457827ce8be": "S_{B}=\\frac{{n}_{B}(t=0)-\\dot{n}_{B}(t)}{\\dot{n}_{A}(t=0)-n_{A}(t)}\\left |\\frac{\\mu_k}{\\nu_p}\\right|=\\frac{0-72}{100-10}\\cdot\\frac{1}{1}=0.8=80%", "969f0950b70f90d5a5cfce9f53a4afd1": " {E[\\vec{X}]^m}_m = R_{ab} \\, X^a \\, X^b ", "969f0ce896a06504c4b629f76d5b54a2": "\\underset{c}{\\diamond}", "969f50a4cfb4ba5be4bda5c9fe3fd9bc": "\\{O^i_n\\}", "969f75760d5eb01b725c10069292e361": "{Z_\\mathrm{iT}}^2=-(\\omega L)^2 + \\frac{L}{C}", "969f9b088578fbecae73f5f0be562dcc": " - \\Delta \\Phi = 4\\pi\\rho.\\,", "969fb8bece6c030816510a15264d06a1": "S_s=-\\textstyle\\frac12\\int[\\sigma^2 h^{\\alpha\\beta}\\phi_{,\\alpha}\\phi_{,\\beta} +\\textstyle\\frac12G l_0^{-2}\\sigma^4F(kG\\sigma^2)]\\sqrt{-g}\\,d^4x\\;", "969fd083d959f47ce748b7f234ee7e90": " 0 < \\mathbb{E}[S_i] < \\infty. ", "969fd9e021fc855a2f5d4fe46223332f": "i_C", "96a01e792339bcbdf5f15a7fe370d49c": "\\gamma_{F}^{-}", "96a090ced3d2e516c6749b29675971b9": "\\mathcal{L} = ln(\\Lambda) = K -\\sum^{\\infty}_{i=1}\\frac{y_i^2}{2} \\frac{\\lambda_i}{\\frac{N_0}{2}(\\frac{N_0}{2} + \\lambda_i)}", "96a0fa4e946af60cc176267f15d402da": "\\exp_2^{i-2}(n^{O(1)})", "96a167a861e2a51a547abe779bded4ae": " k=0,1,2,... \\!", "96a1ab560795cb4a1575d1ed2b429695": "\\operatorname{Var}[Y] = \\operatorname{E}[Y^2] - \\operatorname [E[Y]]^2", "96a1bd04239e3733fe76f3356e79d09f": "\\zeta(6)=\\sum^{\\infty}_{k=1} \\frac{1}{k^6}=\\frac{\\pi^6}{945}\\,\\!", "96a1c48c5a57bbc8223d54197ffbef83": "\\ (r,\\ \\theta_\\text{inc},\\ \\phi_\\text{az,right})", "96a1f05b2b0a6a382bd63ac52ab193ca": " \\frac{x^\\alpha \\cdot _2F_1(\\alpha, \\alpha+\\beta, \\alpha+1, -x)}{\\alpha \\cdot B(\\alpha,\\beta)}\\!", "96a21264748cb3457985ddf347bd7af3": " {\\vec{x}}_{(nr)}(t)={\\frac{{\\sum_{i=1}^{N}\\,m_{i}\\,{\\vec{x}}_{i}(t)}}{{\n\\sum_{i=1}^{N}\\,m_{i}}}}", "96a22fcdac211b7823de47f233fd9856": "\\Delta m^2>0", "96a261513b95fd6966f482e2f738166c": "\\textstyle= \\oint_{\\partial \\Sigma (t)}\\left( \\mathbf{E}( \\mathbf{r},\\ t) +\\mathbf{ v \\times B}(\\mathbf{r},\\ t)\\right) \\cdot d\\boldsymbol{\\ell}\\ ", "96a28067c145d3df67249225c465f3a7": "\\begin{smallmatrix}d_B = {\\left ( 1.87 AU \\right )} {\\left ( {\\frac {149,597,871 km}{696,000 km}} \\right )} = 280,000,000 km = 402 R_{\\odot} \\end{smallmatrix}", "96a2a7369020419629645c0b6ed8a692": "A^*\\,\\!", "96a2fe7f6d03e869e2a82f3206158c99": "\\text{Estimated Z-factor} = 1 - {3 (\\hat{\\sigma}_p + \\hat{\\sigma}_n) \\over | \\hat{\\mu}_p - \\hat{\\mu}_n |}.", "96a30d972a75033c0c9a10e079c6ac08": "\\scriptstyle \\mathbb{R} ", "96a33887ddee7b65b8e4ba79de014f9d": "m_1, m_2 \\in \\mathbb{N}", "96a3be3cf272e017046d1b2674a52bd3": "01", "96a432b4a8fe70b1c8ef0df02046b6c6": "S^\\mathbb{Z}=\\{ x=(\\ldots,x_{-1},x_0,x_1,\\ldots) : \nx_k \\in S \\; \\forall k \\in \\mathbb{Z} \\}", "96a46609431a337acdfc065a9e9e22d7": "\\psi \\circ f \\circ \\varphi^{-1}.", "96a477cc93ad181b53dcb8f124a3a137": "a_n=(-1)^n", "96a47b16dfbe805c7b90febda2fc02f0": "(\\mathbf{Z}/l^n \\mathbf{Z})^2", "96a47dd43c9f73dc600e444a14d88bf4": "P\\Psi(\\vec{r}) = \\Psi(-\\vec{r})", "96a51fa332e00efc63012c6323651273": "\\sigma(i)", "96a573d0a3c377ce21e4d20fad9d3aba": "\\alpha\\, = \\sum_{k=1}^N b_kv", "96a5b5d63ae606ca27eaffe36478e00d": "T^{*}", "96a5b753479e53fa0d71d171f5316a0d": "-\\hbar^2", "96a5b83a862dac2f927df6bb87dcbdc6": "\\xi\\in (-\\infty,+\\infty) \\,", "96a61f51252ca1c99d00eef5ee7aec90": "S_i=-n_tR[x\\ln x+(1-x)\\ln(1-x)].", "96a62a83f23fc98819b8a8958190107a": "P\\in {\\mathcal P}\\setminus (\\mathcal Q\\cup {\\mathcal R})", "96a6331d41a645c750e34f5e06cb457d": " P( X > \\omega ) = \\Phi( \\frac{ \\omega - n \\mu }{ \\sigma \\sqrt{ n } } )", "96a69b48fcb6c951addb36d08eb85b5c": "\\left[1\\times\\log\\left|{\\sqrt{5} + 1 \\over 2}\\right|, \\quad 1\\times \\log\\left|{-\\sqrt{5} + 1 \\over 2}\\right|\\ \\right].", "96a6d0d4af0e38f95e05fa3411ffc5bd": "Z = \\frac{X - i}{X+i}", "96a6f48371989be557bca6d765fcdef7": "\\int_{X} \\exp \\left( \\left( \\frac{| u(x) |}{C_{1} \\| \\mathrm{D}^{k} u \\|_{L^{p} (X)}} \\right)^{p / (p - 1)} \\right) \\, \\mathrm{d} x \\leq C_{2} | X |.", "96a74300b6fba6c5ee0137c54b2c0ce3": "\n \\left|+z\\right\\rang \\leftrightarrow \\begin{bmatrix}1\\\\0\\end{bmatrix}, \\quad\n \\left|-z\\right\\rang \\leftrightarrow \\begin{bmatrix}0\\\\1\\end{bmatrix}\n", "96a775ae19dbeb361b973fc4841e6d68": "2^{\\aleph_0} \\le \\mathfrak c", "96a79dd20d018e75d2868715f8cda3d5": "W = \\frac{1}{n}\\sum_{i=1}^n Y_i", "96a7bdbf8edd375219e7f635881076e3": "\\tilde{U} \\tilde{V} = 1", "96a8403762afcffbf6772ee21e55571c": "\n\\Delta P_L = \\frac{\\sum_{i} p_{i_{1}}\\, q_{i_{0}}}{\\sum_{i} p_{i_{0}}\\, q_{i_{0}}}\n", "96a851ed9e24aafd3502f18d6329661f": "Fr^2=\\left(\\frac{y_c}{y}\\right)^3 = \\frac{q^2}{gy^3}", "96a89eec101e3a441951eb1b188d3f6a": "x^3y + y^3z + z^3x = 0.\\ ", "96a916f6fa2e053c005f69ab31ec8578": "\\Re(s) \\ge b\\,", "96a935bcd0ca53a66b6c1cb887f8ee91": " X_\\epsilon := \\bigcup_{x \\in X} \\{z \\in M\\,;\\ d(z,x) \\leq \\epsilon\\}", "96a938db59928597a13f3693d3bf093d": "Q_{max}^r(R)", "96a941a7cb620f8a94ca747da663a7c2": "v_e=v_0 [Cl^-]_{0^{ }}/[Ag^+]_0", "96a99432f27638a8f57dd139dd96a2f5": "O(ln^2)", "96a9c75abd000127ef1098bcd27eb72c": "U[0,1]", "96aa74a2f5657684b27b0c9ba0180763": "{}^tu : Y^*_{\\mathcal{H}} \\to X^*_{\\mathcal{G}}", "96aa7eb34911c1ce1787225abe8c8602": "\\langle r^2\\rangle", "96aa93177b322f61c8bbbe4ef131820d": "E_4(\\mathbf{x}) \\propto E_3^*(\\mathbf{x});", "96aa9f45750dc24455f015fe6c76f73c": "\\displaystyle{(\\Delta u,v)= (u_x,v_x) + (u_y,v_y) - (\\partial_n u,v)_{\\partial \\Omega}.}", "96aac3f9170a3b40a7ffce50d822e7c4": "\\,x+t", "96aac846365d01fd36989ad9cde187d5": "\\hat{\\rho}(t)", "96aacc014d041f9ec8a5e37a1bc80b12": " q^ \\ast \\equiv 0 \\bmod \\ z ", "96ab1cc37f7e6c3cfa204146d81b42f1": "t^2\\,\\!", "96ab2d250b6068f3e5af326860cd4f84": "\\textbf{NC}^1 \\subseteq \\textbf{NC}^2 \\subseteq \\cdots", "96abed579adfacf25f5f38e425b11ab7": "\\operatorname{im} f", "96ac1488a0f40005f8a973e94ed7d788": "2(1-\\epsilon)\\gamma\\cdot n", "96ac3fb6365b81ec58b17d28ae16b2ef": " \\begin{align}\n \\int_{11}^{14} \\int_{7}^{10} \\ (x^2 + 4y) \\ dy\\, dx & = \\int_{11}^{14} \\left (x^2 y + 2y^2 \\right)\\Big |_{y=7}^{y=10} \\ dx \\\\\n & = \\int_{11}^{14} \\ (3x^2 + 102) \\ dx \\\\\n & = \\left (x^3 + 102x \\right)\\Big |_{x=11}^{x=14} \\\\\n &= 1719\n\\end{align}", "96ac547d12c7205909f6ddcbb520b97e": "L^* = \\frac{L}{n^2}", "96ac5b49c8f04b809bc46165b5033f6e": "f_u", "96ac65bb23f74a04666c7ca2cbdbe623": " Dp_{ ij }: d_{ ij } = [ | r_i - r_j | - ( m - 1 ) ]^p ", "96aceb4a44b3541b0d9bd9b7f2ddaff7": "w = -1", "96ad336211112f1327cdb73b90efa791": "a=\\frac{1}{2}s\\tan\\!\\left(\\frac{\\pi(n-2)}{2n}\\right).", "96ad6f4e5ebff385e12c1947be068dd4": "\\begin{align}\n \\sum_n n\\cdot(n\\mid m) &= \\sum_{n=m}^{\\Omega - 1} \\frac{1}{H_{\\Omega - 1} - H_{m - 1}} \\\\\n &= \\frac{\\Omega - m}{H_{\\Omega - 1} - H_{m - 1}} \\\\\n &\\approx \\frac{\\Omega - m}{\\log\\left(\\frac{\\Omega - 1}{m - 1}\\right)}\n\\end{align}", "96ad746828764bf53758473775e9f8e0": "V \\frac{dC}{dt} = -K \\cdot C + \\dot{m} \\qquad (1)", "96ad7e4617c84a830c616e6e9a3ee471": "\\oint_{\\Gamma} \\mathbf{F}\\, d\\Gamma =\\sum_{i=1}^{4} \\oint_{\\Gamma_i} \\mathbf{F} d\\Gamma ", "96adfe51906d481f116644342365ed3b": "\\alpha=\\beta=0", "96ae02a5d06e510adff868913f7370cc": "\\hbar = 1.0547 \\times 10^{-34}(Ws^2)", "96ae187a0acefb1a9c4781f8fbe1c790": "x_i = d'_i - c'_i x_{i + 1} \\qquad ; \\ i = n - 1, n - 2, \\ldots, 1.", "96ae7086be14df3c0902ed28e39311cb": "\\mathfrak{so}_{2\\cdot 4 + 1} = \\mathfrak{so}_9,", "96ae7554a42e6af2cedf50724f760738": "\\Pr(-A < T < A)=0.9,", "96aea8f9bf1c3900e3b22dc0caed0978": "\\mathrm{pv} \\sqrt{z} = \\sqrt{r} \\, e^{i \\phi / 2}", "96aefba0c0ab71a16985f3d24c09aa77": "\\frac{8! \\times 3^4}{24}=136080", "96af0cb0a92204363cb8a9344f237c48": " \\left\\vert u\\right\\vert _{W_\\infty^{m}(\\Omega)}=\\max_{\\left\\vert \\alpha\\right\\vert =m}\\left\\Vert D^{\\alpha}u\\right\\Vert _{L^\\infty(\\Omega)}", "96af3a4d797faf8bedaa951c6388d451": "\\operatorname{pd}_R k = 1 + \\operatorname{pd}_R (R/(f_1, \\dots, f_{n-1})) = \\cdots = n.", "96af5752e33c9efab5d3f49a4e2a5860": "\nR^\\ell{}_{ijk}=\n\\frac{\\partial}{\\partial x^j} \\Gamma^\\ell{}_{ik}-\\frac{\\partial}{\\partial x^k}\\Gamma^\\ell{}_{ij}\n+\\Gamma^\\ell{}_{js}\\Gamma_{ik}^s-\\Gamma^\\ell{}_{ks}\\Gamma^s{}_{ij}\n", "96b0160cc2d187064b2b146b58dc8497": "!p\\!", "96b0471de2850f2e247904d102968e30": "\\pi(\\theta) = 1", "96b0529e83d9fed678883b25a6b817b1": "1/\\varphi", "96b0a61d471f904dc2185b5b1ecc1650": "U_s(t) = \\mathbf{T} exp\\{-i\\int_0^t dt' H_s(t')\\}.", "96b0c1bee0975cb34bdf39a04e8481df": "\\begin{align}\n& BF=AE=BC\\cos\\hat{B}=a\\cos\\hat{B} \\\\\n\\Rightarrow \\ & DC=EF=AB-2BF=c-2a\\cos\\hat{B}.\n\\end{align}", "96b0d3466a10bc52ac293322ff478094": "\\frac{a^2 z}{((a-1) z+1)^2}", "96b0da967121b4e9920b0bdb40d4a3bf": "\\left\\lfloor \\log_2 n\\right\\rfloor + 1", "96b0dd8c14ec4d834b2ac21363f37474": "\\Pr(X = x)", "96b12e8ed3d1b2c82512fd6cbb963749": "P_f(f)df=\\left(\\frac{c}{f_0}\\right)\\sqrt{\\frac{m}{2\\pi kT}}\\,\\exp\\left(-\\frac{m\\left[c\\left(\\frac{f}{f_0}-1\\right)\\right]^2}{2kT}\\right)df", "96b184073e482f1833a201b2ab831b71": "\\mathbf{ii} = \\begin{pmatrix}\n 1 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{pmatrix}, \\cdots \\mathbf{ji} = \\begin{pmatrix}\n 0 & 0 & 0 \\\\\n 1 & 0 & 0 \\\\\n 0 & 0 & 0\n\\end{pmatrix}, \\cdots \\mathbf{jk} = \\begin{pmatrix}\n 0 & 0 & 0 \\\\\n 0 & 0 & 1 \\\\\n 0 & 0 & 0\n\\end{pmatrix} \\cdots \n", "96b1993a17e3c418ec9fc7267d68ef7a": "\n d = \\cfrac{a^2}{R} = \\left(\\cfrac{9F^2}{16R{E^*}^2}\\right)^{1/3}\n ", "96b1f019828e030adcdd9f98040a2780": "R_n\\,", "96b2206369ea087e57554fbc2b4c8e39": " I_C = I_P - mr^2,", "96b25260431924d31460a02bda8a317f": " - E \\psi = \\frac{\\hbar^2}{2m}{d^2 \\psi \\over d x^2}\\,", "96b25784f95a171b6af52b22bba48e30": "M_{1} =m_{1}+S_{1},", "96b25c4b5a85e8a20808c40941c0e6c3": "B_c", "96b276e286cf1660d25ce1dae57855ff": "\\int \\sin^2 x \\cos 4x \\, dx \\,=\\, -\\frac{1}{24}\\sin 6x + \\frac{1}{8}\\sin 4x - \\frac{1}{8}\\sin 2x + C.", "96b2a1f44893a3e68a38ff17ce45da8a": "B \\to K^* l^+ l^-", "96b2bfa9e87976447a128b1135dbc494": " \\begin{align} \n \\frac{v_1 (1- z)}{J_1+ (1 - z)} &= \\frac{v_2 z}{J_2+ z} \\\\\n J_2 v_1+ z v_1 - J_2 v_1 z - z^2 v_1 &= z v_2 J_1+ v_2 z - z^2 v_2\\\\\n z^2 (v_2 - v_1) - z \\underbrace{(v_2 - v_1 + J_1 v_2 + J_2 v_1)}_{B} + v_1 J_2 &= 0\\\\\n z = \\frac{B - \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} &= \\frac{B - \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} \\cdot \\frac{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\\\\n z &= \\frac{ 4 (v_2 - v_1) v_1 J_2}{2 (v_2 - v_1)} \\cdot \\frac{1}{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\\\\n z &= \\frac{ 2 v_1 J_2}{B + \\sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}. \\qquad \\qquad (3)\n\\end{align}", "96b2f99f9cd8ca26cb07f3a382d1d84c": "[t_{a}, t_{b}] = i f^{abc} t_{c}", "96b3360ffa986ed2aefa6926f860636c": "\n\\left. + \\left[Q_R^\\dagger(\\mathbf{p}) p_1 \\epsilon_1^{1*}(\\mathbf{p})\n+ Q_L^\\dagger(\\mathbf{p}) p_1 \\epsilon_1^{2*}(\\mathbf{p})\n\\right]e^{-i p x} \\right\\}.\n", "96b37b85bed53063499e47fdd877b779": "dt_{d}", "96b413aae7431dd5948a6d0e481a3f87": "X(y,t)", "96b41da48830148253100524782067ea": "H_*(E) \\to H_*(B)", "96b4225306390e325ccbe7aaac64042e": "\\textbf{R}^d ", "96b49f50383afa5d80244b672a461221": " I_1 = Y_{11} V_1 + Y_{12} V_2 \\qquad I_2 = Y_{21} V_1 + Y_{22} V_2 \\qquad \\text{with} \\qquad Y_{12} = Y_{21} \\, ", "96b49f5f1b79d018a8db8c6a2b81327d": "c_q(n) = \\sum_{d\\,\\mid\\,q} \\mu\\left(\\frac{q}d\\right)\\eta_d(n).", "96b4f58e3b8ec5553fc595f9c93f4e12": "j - PC(i,k|j)", "96b527a6447760b5e84579c46bc61fa4": " (m,\\tilde{m},t) ", "96b55f6979e0db446400c46b45d66f32": "\\mathbf{B}^{-1} = \\mathbf{B}^{*} \\mathbf{(B B ^ {*})}^{-1}", "96b5e3c7844afae91403d7c1ee7bc807": "Z_t", "96b607f8d5ecdb4a3a5c3efa12c8a55a": "R_\\text{ku} = \\frac{1}{n R_\\text{q}^4} \\sum_{i=1}^{n} y_i^4 ", "96b60e279874b8bf615b3390d90cd809": "\\begin{align}\nPDOP &= \\sqrt{\\sigma_x^2 + \\sigma_y^2 + \\sigma_z^2}\\\\\nTDOP &= \\sqrt{\\sigma_{t}^2}\\\\\nGDOP &= \\sqrt{PDOP^2 + TDOP^2}\\\\\n\\end{align}", "96b63043f7cb24fc9ee54fdefe496780": "\\textstyle ID", "96b681e9d0e2080ad9d3a6ef823f6299": "S_{v \\times v}=\\begin{bmatrix}s_c&s_{c+1}&\\dots&s_{c+v-1}\\\\\ns_{c+1}&s_{c+2}&\\dots&s_{c+v}\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\ns_{c+v-1}&s_{c+v}&\\dots&s_{c+2v-2}\\end{bmatrix}.\n", "96b6891934cdb3d3749e87483f6f62f3": "\\phi(\\omega) = \\arg(Y) - \\arg(X) = \\arg( H(j \\omega))", "96b6b717f2da2db0779a18a3b6ec4abc": "v_a=\\frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b}", "96b6c6b73ff5798334ec715a5c1776a5": "\\{f_1, f_2,\\ldots, f_n\\}", "96b6d1d74fe906be4dd75f5267f7aefb": "{}^0\\!D", "96b6dccef1542936f40458ece1d767fc": "v_f=v_i+a\\Delta t", "96b6e0b9c9fd96535db2cc122ed66760": "\\mu _{i,{\\rm liq}}^{\\star} = \\mu _{i,{\\rm vap}}^\\ominus + RT\\ln \\frac{{f_i^{\\star}}}\n{{p^\\ominus }}", "96b716d144a48efeab88fc2c7e46bc30": "\\mathcal{O}(d^2 \\log n) ", "96b7ca11459117a722fcdcaa2ac4a075": " z_{n+1} = A + B z_n e^{i K/(|z_n|^2 +1)+C} ", "96b7ff11e94e4ed158bd973ae36c1ae2": " \\phi_m = 90 ^\\circ - \\arctan ( f_\\text{0 dB} /f_2) \\, ", "96b8162a5f727c190e4c2b0f8ef3ff49": "(d,u,e,\\nu)_{L,R}^i", "96b83d2e9bcd73ddb84736db43b4c0f4": "P=\\frac{e^2 c}{4\\pi\\varepsilon_0 r^2}\\cdot Sum_n", "96b96d29d5d07d8edd8e32f0d12334fb": "\\operatorname{Cl}_{5}(2\\theta) = 16\\operatorname{Cl}_{5}(\\theta) + 16 \\operatorname{Cl}_{5}(\\pi-\\theta) ", "96b96f3016d5a4e7582643c2329bab10": " y= \\begin{cases}\n0~~ \\text{if}~~y^* \\le \\mu_1, \\\\ \n1~~ \\text{if}~~\\mu_10,\\exists N", "96d0b42e042e3bfcf51792dd416eab69": "\\mathbf{B} \\ = \\ \\mu_0(\\mathbf{H} + \\mathbf{M}). \\ ", "96d0d24dee8120bf0a1ea349fcc661db": "\\frac{x}{e^x-1}- \\log(1-e^{-x})", "96d100bc701a23f14a69a345f121d561": "\\vec\\alpha = \\begin{bmatrix} \\mathbf{0} & \\vec{\\sigma} \\\\ \\vec{\\sigma} & \\mathbf{0} \\end{bmatrix} \\quad \\quad \\beta = \\begin{bmatrix} \\mathbf{I} & \\mathbf{0} \\\\ \\mathbf{0} & -\\mathbf{I} \\end{bmatrix} \\,", "96d12ea1712799ecf70ef8828d5ae0a2": "Pmf = 1", "96d15c02782b06fe2ede77110a0f6dc6": "\\displaystyle{F(z)={1\\over 2\\pi i} \\int_{-\\infty}^\\infty {f(s)\\over s-z}\\, ds.}", "96d20dda5e5ee76a175ef79897ebbeee": "h_c(t)", "96d23660bce0ad07a9f82095547d9a01": "F = 6.456", "96d26f2b35a1da0cc6ce420e4bcc6b6e": "\\sum_{i\\neq j} x_i (1 - k_i) (1 - k_j)^{-1} = -x_j", "96d3bd0847a3895f35b2a6f367fb04c1": "\\sum_{p: s_i \\rightarrow t_i}f_p = \\sum_{p: s_i \\rightarrow t_i} f_p^{*} = r_i \\; \\; \\forall i.", "96d3c0a51fc4c5b36f5d296a7678d634": "Z_1 = \\sqrt{-2 \\ln(U_1)} \\cos(2 \\pi U_2) ", "96d3d844072c9bd4dbe1cad2173fbaed": "\\varphi(\\boldsymbol{x},t)", "96d3e3fcfbfbeb8ed6303113fb6573f8": "A=S^2-{n} ", "96d409fafa72ed7ab971a8dfbb09836f": "(0,y)", "96d4396a9bec8e757e4508cf4da64e09": "f(\\vec{v}+\\vec{w}) = f(\\vec{v})+f(\\vec{w})", "96d4cdff8ed57e93e3b3d843cffe3af7": "BP", "96d60559943c61762f866b45ce0a95a2": "I_*", "96d65af9c9e6237b38504ca14736d5cc": " y = \\Pi x \\, ", "96d6bd7f0ea0e3e9a688af6ee855e367": "\\{ \\mathcal{F}_{t} \\}_{t \\geq 0}", "96d6de6c9862e55e0793d7e7c1ef419f": "lb > lb_{computed}", "96d6fb2e81fa29eedfb40239c7145677": "c^2 = b^2 + 2bd + d^2 +h^2.\\,", "96d7325be472e3f124e92ba78944efd4": "\\tan\\alpha_{m} =0.5(\\tan\\alpha_{2}-\\tan\\alpha_{1})", "96d757196cdbbd7674384f137d53e067": "(-31764)^4 + 27385^4 + 48150^4 + 7590^4 = (-31764 + 27385 + 48150 + 7590)^4 \\, ", "96d775ac9272b065dd023e2dffb9492f": " B_H (t_j)=\\frac{n}{T}\\sum_{i=0}^{j-1} \\int_{t_i}^{t_{i+1}} K_H(t_j,\\, s)\\, ds \\ \\delta B_i.", "96d77b4c8f5638d8f7828d47a5aaa73d": "\\lim_{x\\to c} f(x) = f(c).", "96d80280e75f3028d5545aaa37936cb7": "\\theta \\in (1/2, \\theta_c)", "96d86b16075173ac45735c061a676ce2": " b=FF^T ", "96d926f309524dd990680f912f689711": "\\mu(\\mathcal{R})\\,\\!", "96d953f38845576bedcd7a1a52bf9009": "p_R=\\frac{p}{p_C},\\qquad\nv_R=\\frac{v}{v_C},\\quad\\hbox{and}\\quad\nT_R=\\frac{T}{T_C}", "96d99dfdc0c87afcf3c6d98407e30902": "\\mu y = \\int \\mu q(x)dx", "96da02b87aba20725ec87d8935f2184f": "u(\\mathbf{x}) = \\sum_{i = 0}^{N}{ \\frac{ w_i(\\mathbf{x}) u_i } { \\sum_{j = 0}^{N}{ w_j(\\mathbf{x}) } } },", "96da2435e657a33201c7d110f3b7ee10": "Kf\\left( t \\right)=\\int_{-\\infty }^{\\infty }{C\\left( t-s \\right).f\\left( s \\right)ds=C*f\\left( t \\right)}", "96da3cb6450e999e3df7b2f9654d4de1": "i \\hbar {\\partial\\!\\!\\!\\big /} \\psi - m c \\psi = 0", "96da4c4e995312e779c3aa4455cdce86": "I_C =\\frac{I_{CBO} + \\alpha I_B}{1-\\alpha - \\left(\\frac{V_{CE}-V_{BE}(I_B)}{BV_{CBO}}\\right)^{\\!n} }\\cong \\frac{I_{CBO} + \\alpha I_B}{1-\\alpha - \\left(\\frac{V_{CE}}{BV_{CBO}}\\right)^{\\!n} }", "96da5e09738e29b3fc6b19fb15f5b589": " G(s) = \\frac{Y(s)}{U(s)} = \\frac{\\sum_{k=0}^n b_k s^{\\beta_k}}{\\sum_{k=0}^m a_k s^{\\alpha_k}}", "96daa263394eaf4b135edb3c3d9af7fb": "P_{I, M''}", "96daf912e656d969373dc1ab35b04067": "(\\lambda(g) f)(x)=f(g^{-1}x),\\,\\,(\\rho(g)f)(x)=f(xg)", "96db3d66b37104be2bb6a75f0104316f": "\\hat e_3=(0.451272,-0.079571,0.888832)", "96dba35c2ef9f4c1bc01e31f860f0000": " 0 < v_i < v_\\max \\,", "96dbc3cb25d74cc6b9a4dd2fe99b6fde": "Lk", "96dc2f4593d9ed5c7cfb5271b843fdae": "G_P = \\frac{P_\\mathrm{load}}{P_\\mathrm{input}}", "96dc7df3d75950820289908dfe4d2af9": "\\omega_a=\\sqrt{\\frac{k_1M}{m_Am_B}}", "96dc824a9bda0e171a21766242a36193": "i=1,\\dots,q-1", "96dcee7e29fea164b6c146da1391372c": "[(a,b)] < [(c,d)]\\,", "96dcfbdfd3f9cd3571c3d793b7f6a626": "\\displaystyle{G_{\\mathbf{C}} =\\bigcup_{\\sigma\\in W} B\\sigma B,}", "96dd0559efe4c37e6ac617a403f6601a": "\\hat{\\theta}_{\\mathrm{MAP}}(x)\n= \\underset{\\theta}{\\operatorname{arg\\,max}} \\ \\frac{f(x | \\theta) \\, g(\\theta)}\n {\\displaystyle\\int_{\\vartheta} f(x | \\vartheta) \\, g(\\vartheta) \\, d\\vartheta}\n= \\underset{\\theta}{\\operatorname{arg\\,max}} \\ f(x | \\theta) \\, g(\\theta).\n\\!", "96dd2131b40c26f0f749d2702641538e": "\n\\hat{n}\\ =\\ \\cos i\\ \\hat{z}\\ + \\sin i\\ \\hat{h}\n", "96dd2397c0ddb4ea07d5fe3e15ed5775": "V\\left(x\\right)\\,", "96dd411a1478995a5c1508209fab3e34": "2 \\theta + \\sin \\left( 2 \\theta \\right) = \\pi \\sin \\left( \\varphi \\right) \\qquad (1)", "96dd5d35102eacecb9f1b8864d6bb7f2": "\\Pi_{H}(m)=max|\\{h\\cap D:|D|=m,h\\in H\\}|\\,\\!", "96dd5e096829c86094a7c35ef62e29f0": " X \\sim \\textrm{Kumaraswamy}(1,b)\\,", "96ddacc41f5929014c8bda0ae742d3b9": "\\omega_1 ", "96ddeff00b709a6ec0dedf366395dd7d": "\\Lambda_{ij} = (\\delta_{ij} - \\hat{n}_i \\hat{n}_j) \\cos\\theta - \\varepsilon_{ijk} \\hat{n}_k \\sin\\theta + \\hat{n}_i \\hat{n}_j ", "96de336bbd5d7b6831514b60791dfdb3": "\\left( H_0 + V \\right) | \\psi \\rangle = E | \\psi \\rangle. \\,", "96de47c0b232a35b0f01f3aa8e78e07e": "\\limsup_{n\\to\\infty}\\frac{\\log d(n)}{\\log n/\\log\\log n}=\\log2.", "96de61a62d1c35a3936dd65d28f60166": "\\sum_{n=1}^\\infty \\rho_n < \\infty", "96debb818b6189e09b4e646a9393dc82": "M_a", "96dec4821af0206c09bc73e78da8827b": "C_2=0.6601\\ldots", "96df28d37cc8136271db808a5c1c601f": "\\begin{align}\n & \\overline{x} = \\frac{1}{n}\\sum x_i, \\quad \\overline{y} = \\frac{1}{n}\\sum y_i, \\\\\n & s_{xx} = \\tfrac{1}{n-1}\\sum (x_i-\\overline{x})^2, \\\\\n & s_{xy} = \\tfrac{1}{n-1}\\sum (x_i-\\overline{x})(y_i-\\overline{y}), \\\\\n & s_{yy} = \\tfrac{1}{n-1}\\sum (y_i-\\overline{y})^2.\n \\end{align}", "96df382c49c269e94be4d934043ea2da": "\\mathbf E_J \\cdot \\mathbf e_i = \\delta_{Ji}=\\delta_{iJ}\\,\\!", "96df76e1c52facbcf1aec4b8ebd0f8d3": "L _ {0, 1} = S _ 0 e ^ {r (n/365)} - D e ^ {r(n-y)/365} - X", "96df7f5941664386d75e997bc915f3c6": "\\lim_{h\\to 0} \\frac{(x+h)^n-x^n}{h}=nx^{n-1}.", "96df96dd95bbf61a8224853c7a06c48c": "c>0", "96df9c915e0e35697fdb43683560e9e3": "\\mathfrak{P}^{48}", "96dfb3580731c02badb21ef2b4fd290b": " \\textstyle \\prod_{i=1}^r (2s-2 + s(p_i)^{-1/2})= O{(m^\\epsilon )} ", "96e027f168cc2a618f093cdaa865e41c": "\\gamma^{th}", "96e08895ea060723219fe0e6a2fe33b6": "a,b \\in \\{0, \\ldots, q-1\\},\\,", "96e09ac5fa3c3c7a59b8e05ebfdd1870": " \\boldsymbol \\beta^{(s+1)} = \\boldsymbol \\beta^{(s)} - \\left(\\mathbf{J_r}^\\top \\mathbf{J_r} \\right)^{-1} \\mathbf{ J_r} ^\\top \\mathbf{r}(\\boldsymbol \\beta^{(s)}) ", "96e0b63011bccb4b780ab9407fd57448": "G_x", "96e14f1c2fa3aa269a4fd722eca94b00": "a {\\,\\begin{array}{|c|}\\hline{\\!n\\!}\\\\\\hline\\end{array}\\,} b\\,\\!", "96e15d868494618705c22a2a2d218b12": "\\mathcal{O}(-D)", "96e180d6850c3eb67e4fff5cca292ce5": "G = (N, T, M, S)", "96e1a4ccbc9c246c7a4a89cac9c96c00": "G_{dBi} = 10 \\cdot \\log_{10}\\left(G\\right)", "96e1f1976e5d99cbc18d340859a32893": "p=p(t,x,y,z)\\;", "96e20324e8bc05195362c5a228d6dd97": "\\,\\operatorname{cr}(z_1,z_2,z_3,z_4)={{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}", "96e20cea718f6cd78ddf7842933eed48": " n_u ", "96e22239e5bd80513201210d56d21b23": "\nP_{ex}\\propto \\int d\\lambda_{ex}I(\\lambda_{ex})\\epsilon(\\lambda_{ex})<\\mid F_{in}\\cdot e_{ex}\\mid^{2}>\n", "96e28ac08dd68613d32cb057ee285176": "0 < s < t \\leq 1", "96e29803ff66cba03c3d9e3e7f0590d7": "\\displaystyle{F\\star G(z)={1\\over 2\\pi} \\int F(z_1)G(z_2-z_1) e^{i(x_1y_2-y_1x_2)}\\, dx_1dy_1.}", "96e2b892dd82ccfcc5ad7444d51cb287": "S(\\exists)", "96e2cf071a6da3207ff88d925006fcb8": "\\textbf{x}_{o}", "96e2e6988d845d3dc72837aa89a9314c": "(\\lnot b\\lor\\lnot c)", "96e3c168557196b5e1f8b178e2f5403f": "i\\sim j ", "96e3cc81a54f7955fb28fbec2ebe84bd": "x_1, x_2 \\in \\mathcal{S} (X)", "96e3fbad59fedc8bf57a788503f1baad": "\\varnothing= \\sqrt{\\frac{4.444\\times 10^{-6}\\cdot \\mathrm{denier}}{\\pi\\rho}}", "96e42c0c0a03c834622d65df7808844d": " var( \\alpha ) = \\frac{ \\alpha }{ \\ln( X )( 1 - X ) } ", "96e44f9490b78cce2265192a941bcf38": "A = \\frac{1}{2} b h ", "96e467d4c1f4be64171f4c134c5ac06a": "\\mathbf{v}_k", "96e474bec3c6d8d8e03124b799c32544": "F_Q = \\frac{\\hbar^2}{4m} \\left[ \\frac{\\nabla^3 \\rho}{\\rho} - \\frac{ \\nabla (\\nabla \\rho \\cdot \\nabla \\rho) }{ 2\\rho^2 } - \\left( \\frac{\\nabla^2 \\rho}{\\rho} - \\frac{ \\nabla \\rho \\cdot \\nabla \\rho }{ \\rho^2 } \\right) \\frac{\\nabla\\rho}{\\rho} \\right]", "96e4a4890f5a9605d3cd7ab2e89171d4": "\\cfrac{\\cfrac{stC \\qquad \\overline{s} D}{tCD} \\, \\operatorname{var}(s) \\qquad s \\overline{t} E}{sCDE} \\, \\operatorname{var}(t) \\Rightarrow \n\\cfrac{stC \\qquad s \\overline{t} E}{sCE}\\, \\operatorname{var}(t)", "96e4b8cdfea4d4ba6b779cb05ae374c0": "v=c/n", "96e4cf8778e4d99ce337e0e5df8a0469": "G_{\\theta} = 4\\pi\\left(\\frac{U_\\theta}{P_{\\mathrm{in}}}\\right)", "96e509891d9a91e97f03546554f727af": "A_{i,j} = p(c_i | x_j )=Kp(c_i) \\exp \\Big( -\\beta\\,D^{KL} \\Big[ p(y|x_j) \\,|| \\, p(y| c_i)\\Big ] \\Big)", "96e5b05bc12f74d63d70e1cb2fda4373": " 0\\subset W_0\\subset W_1 \\subset W_2=H^1(X),\\, ", "96e5ca1eee4b523de70e2255e0f12f7e": " c = \\left\\lfloor\\frac{y}{100}\\right\\rfloor \\quad \\text{and} \\quad g = y - 100 c,", "96e5d570fdadd08df013e688b9ef1375": "y^j\\,", "96e62bac535159e46c05e85222766470": "\\{\\xi_j^{(n)} : n,j \\in \\mathbb{N}\\}", "96e65a08d8d0b04214175cfdf64bc341": "\\langle X, \\mathcal{F}\\rangle ", "96e6b498755ce7c020be19be0b8e7c4e": "\\underline{u} \\in U.", "96e7006910da1da84afe39b27b9fbf00": "\n \\lambda^2 = \\pi^2\\left(\\frac{m^2}{a^2} + \\frac{n^2}{b^2}\\right) \\,.\n", "96e761b1976a1d933d838788a223c9fa": " H=\\frac1 2 (u^2+v^2+Ax^2+By^2 )+x^2 y-\\frac 1 3 \\epsilon y^3", "96e774e0c37817eb4f0e4cbb3a24fb3f": "d=h(T-t) = \\frac{2hv}{g}-ht", "96e7aa52f7145e6e3e959d5c9f8a4091": "<\\hat{a}_{j}^\\dagger\\hat{a}_{j}>", "96e806fc14009a9d5d0b375b8ab24d76": "\\scriptstyle\\sqrt{-1}", "96e8090935f9544d2e1f1cdf503f8c3b": "\\mathfrak{P}^{80}", "96e87c9decf480d5a332ba63ceec4dc2": "t_3=1.075 \\colon", "96e90fac758e0800b7b8af07127a38b0": " M(p) = \\int^\\infty_0 (1+\\alpha x)^{-\\gamma}x^{p-1}dx ", "96e9231af0cbb03efd38ec6119f2a4da": "\\Delta : \\mathcal{A} \\rightarrow \\mathcal{A \\otimes A}", "96e92cf8a58280db23f575afe80bcf33": "\\left\\{{3'\\atop4}\\right\\}", "96e9aaa99b3db1130b0aa93eab10248d": "{\\rm depth}({\\mathbb B})", "96e9d8325bfba58f203a732a590ddc82": "c_{ijk\\ell}", "96ea661e1bb1f22c5ab4a83a9efbbf8f": "P_b=\\rho g h \\,", "96ea75ba997c206d792b00fef6aafa2c": " B_{in}=B_{out}+B_{lost}+B_{destroyed} \\qquad \\mbox{(1)} ", "96ea886049b2a065c777ccb9235cf6e8": "|n_1 n_2 \\cdots n_N; S\\rang = \\sqrt{\\frac{\\prod_j n_j!}{N!}} \\sum_p |n_{p(1)}\\rang |n_{p(2)}\\rang \\cdots |n_{p(N)}\\rang ", "96ea9cbeab5d5092780b64b83c6b51bd": "P(c_j|h_i)", "96eaa4234048163da3dc9024ac43297f": "\\frac{B_{a+1}(x)}{a+1},\\,a\\notin \\mathbb{Z}^-\\,", "96eaf60748cd0e155fd1bdc608f1dc47": " \\mathbf{A} ", "96eb11105747e7a681c2411bdb8a181f": "H = \\Sigma_k H_k", "96eb185ebf6cc4ed1c45a23c3a466417": "a=e=i", "96eb5ca6f4a7343772e75b23382f22d3": "f(t-1) = \\int_{-2}^{0} s(t + \\tau)\\, d\\tau = \\int_{0}^{+2} s(t - \\tau) \\, d\\tau\\,", "96eb9bf5314b593783ee57983efbed9d": "\\cos x", "96ebd2d2a09cd33da2d2b0c03ebdd139": "C_\\mathrm{total} = C_1 + C_2 + \\cdots + C_n", "96ebed5acde4612e055213eaad68d8bd": " (z_1, z_2,...z_n)", "96ec51f2ff042f6996a9f9782d60eb4e": "p(x) = \\frac{1}{\\pi} \\int_0^\\infty\\! e^{-t \\log t - x t} \\sin(\\pi t)\\, dt.", "96ec594fba01637a34f13cbcb60adf0e": " y_{n+2} ", "96ec5d63e8dd140510d2cd14d4a7f604": "K_\\mathit{row}(S_{w}) = (1-S_\\mathit{wn}(S_w))^{N_\\mathit{o}}", "96eca2731a140fec836989d09f238904": "A=4\\,A_0={\\sqrt{3}}a^2\\,", "96eccb532d019d25dd8321b2dfa9fa34": " \\frac{\\partial u}{\\partial t}", "96ecd3d7749819dc03efa2bddd623206": "T = \\frac{\\omega^2 m}{2} \\left ( \\frac{\\mathrm{d} x}{\\mathrm{d} t} \\right )^2 = \\frac{m \\left ( \\omega A \\right )^2}{2}\\sin^2\\left ( \\omega t + \\phi \\right )\\,\\!", "96ecfdbb422f65673390d9c493dc1e0c": "y' = 1 + y^2\\,", "96ed28c4a3a2ba150c64854f7bef5d43": "S_0 / R = \\frac{5}{2}", "96ed7810c47441796412043d001ba54e": "L_{ab}(\\mathbf{x})", "96ed98e77da11d913b10aa3db1054896": " H^n(X;A) ", "96edff46ac34a8a9cdfdc91f5067489d": "\\mathbf M_y", "96ee187ae13075b8b74d8f944af37b44": "\\Pi(n,k)=R_F\\left(0,1-k^2,1\\right)+\\tfrac{1}{3}n R_J \\left(0,1-k^2,1,1-n\\right) ", "96ee3a1fc2fc3bc89e210ba2bac750a8": "\\Psi (x,t)", "96ee84e562f496e9c1072d0fe4e7460e": "k = \\delta n = n^{\\gamma -\\frac{1}{2}}n = n^{\\gamma +\\frac{1}{2}}", "96ee88593cb4537b530cd95e0aef3da1": " \\{\\varphi \\circ f\\} = \\{\\psi \\}", "96ee90a90cec78c8532cdc7b0bd67f50": "L(a) = a^2", "96ee952a921b3fecf810d3bec7debef9": "\\begin{align}\nY' &=& 16 &+& ( 65.481 \\cdot R' &+& 128.553 \\cdot G' &+& 24.966 \\cdot B')\\\\\nC_B &=& 128 &+& (-37.797 \\cdot R' &-& 74.203 \\cdot G' &+& 112.0 \\cdot B')\\\\\nC_R &=& 128 &+& (112.0 \\cdot R' &-& 93.786 \\cdot G' &-& 18.214 \\cdot B')\n\\end{align}", "96eeb429782d472fdb32b064af7a3abc": "a:=\\sqrt{\\frac{a_1^2}{4}-a_0}", "96ef651ccf22e81acdf0313928c1c699": "f(x;\\mu,b,n)=\\frac{ b^n {(x-\\mu)}^{-1-n} }{ \\left( e^{\\frac{b}{x-\\mu}} -1 \\right) \\Gamma(n) \\zeta(n) } ", "96efbc1e9c22b02166eacc34e02097df": "|\\{d' \\in D \\, | \\, t \\in d'\\}|", "96efe22514c11beb02872c7efbefa785": "i = 1, \\dots, n-2", "96f09625de168c2ec7ea5d1ade9d5b18": "C_d\\;", "96f16710a09bd36fbdc1108b184d5393": "\n \\Pr(N_t=k) = f(k;\\lambda t) = \\frac{e^{-\\lambda t} (\\lambda t)^k}{k!} . ", "96f18379d5361b93ba656a7cc991d2a2": "a_{\\mathrm{dual}}(Z)=2Z^d\\,\\left(\\frac{1+Z}2\\right)^A\\,q_{\\mathrm{dual}}(1-(Z+Z^{-1})/2)", "96f1b2d777b36883abf7672e40444898": "h/2", "96f1f4d9fa9404cb7d0664fcccbdd359": "(V\\oplus W)(x)=V(x)\\oplus W(x)", "96f2a64a091119f4ce34aeea87921dfa": "\\boldsymbol{\\delta}'=\\mathbf{Q}\\alpha_jsc_1-\\mu\\boldsymbol{\\epsilon}'c_0+\\alpha'_j\\big[-\\boldsymbol{\\alpha}c_0+2\\alpha_j\\boldsymbol{\\beta}s^3\\bar{c}_3+\\frac{1}{2}\\boldsymbol{\\delta}\\alpha_js^4c^2_2\\big]", "96f2b32890ef7c602b27b66b61ca5f28": "\\epsilon_a", "96f2b748f4f885b4ea5676e078c28d89": "\\hat H = -t\\sum_{\\sigma}\\left(\\hat a^\\dagger_{i\\sigma} \\hat a_{j\\sigma} + \\hat a^\\dagger_{j\\sigma} \\hat a_{i\\sigma}\\right)\n+\nJ\\sum_{}(\\vec S_{i}\\cdot \\vec S_{j}-n_in_j/4)\n", "96f2c5fbbf4181efdccafff20feca7ca": " |f_{R}-f_{Q}|\\leq C||f||_{BMO}", "96f310f79238798754d1afcfdf3cff46": "r(k)=\\text{E}(y_i,y_{i+k})/\\text{E}(y_i^2)", "96f31600ce73d879616256b9563c7ffc": "36=15+21", "96f31aef72b0b706d0468240212a5a2a": "F:Z\\to N=F(Z)", "96f37b5f119d315ddcc261403e15e5a8": "\nR = \\frac{V}{I} \\,\n", "96f3a9e23bd4d351efef8db0a9f0b432": "x_{2}^{1}", "96f3ab35c625f5723c95c3af7f13cb4a": "(+,+,+,+)", "96f3dc38388e61914215656078107b38": "\\epsilon_{abcd} = -\\epsilon^{abcd} = \\begin{cases} +1 & \\mbox{if } \\{abcd\\} \\mbox{ is an even permutation of } \\{0123\\}, \\\\ -1 & \\mbox{if } \\{abcd\\} \\mbox{ is an odd permutation of } \\{0123\\}, \\\\ 0 & \\mbox{otherwise.} \\end{cases}", "96f49075ad8d26e39dcb1579752af8be": " E[X^k]=\\prod_{i=1}^{k} \\frac{\\alpha+i-1}{\\beta-i}. ", "96f4beeab499132647e05f07eb12e036": "t g(\\frac{\\boldsymbol{y}}{t}) = 1", "96f4efc80d5ec97e8098f8801e8b10cc": "\\begin{bmatrix} 1 & F_1 & P_1 \\\\ 1 & F_2 & P_2 \\end{bmatrix} \\begin{bmatrix} y_L \\\\ y_F \\\\ y_P \\end{bmatrix} \\ge \\begin{bmatrix} S_1 \\\\ S_2 \\end{bmatrix}, \\, \\begin{bmatrix} y_L \\\\ y_F \\\\ y_P \\end{bmatrix} \\ge 0. ", "96f58fe65c1d67fc4652b02b0aa990b7": "\\Delta\\Omega/P=\\rho", "96f59208bddefadc209d4eb404a72554": "c_0\\,\\!", "96f59a064c2085d622abc1023fe66852": " l(t) = \\min_{t\\le s\\le 1} (1,h(s)-M),\\,\\,\\,\\,\\,\\, r(t) = 1 - \\min_{0\\le s\\le t} (1,h(s)-M).", "96f61a1113342b9d967e65e372e9cdd0": "w = \\frac{S e}{G_s}", "96f62c79c91a7380b8e6a6487e126bd4": " J=\\left[ \\begin{array}{ccc|c} I_{xx} & I_{xy} & I_{xz} & x_g m \\\\ I_{xy} & I_{yy} & \nI_{yz} & y_g m \\\\ I_{xz} & I_{yz} & I_{zz} & z_g m \\\\ \n\\hline\nx_g m & y_g m & z_g m & m \\end{array}\\right] ", "96f68cbc8e6cc630e31574bcb3b64cbe": "(x-1)(x^{10} -x^6 -x^5 -x^4 +1) = 0 ", "96f6c0684199ce09e26ee01a5571a7ec": "H_i(\\mathbf{RP}^n) =\n\\begin{cases}\n\\mathbf{Z} & i = 0 \\mbox{ or } i = n \\mbox{ odd,}\\\\\n\\mathbf{Z}/2\\mathbf{Z} & 00,", "96f7362eaaa825744141afe4d5c2d340": "\\textstyle 0", "96f74cf9128b4d46c01fc9f585de4eff": "\\sqrt{\\Delta x_1^2+\\cdots +\\Delta x_n^2}", "96f76846fa84360254f34da6b25d8967": "C_w", "96f7bdf24c818037f25fa9c27c56d46d": " r \\rightarrow 0\\,\\!", "96f7ff16c3ad7f824a85e96c326ba367": "(0, 0.5]", "96f8b8eef18f46f93735d668533b82a9": "\\ i=n ", "96f8cddfb4803f5877587fb7bff9f492": "E=P*T", "96f91b42bfa83802f7a6eb62c1c149ab": "FF = \\frac{P_{m}}{V_{OC} \\times I_{SC}} = \\frac{\\eta \\times A_c \\times E}{V_{OC} \\times I_{SC}}.", "96f930882c2f5248999c4cc80238fc6a": " t_2 = \\frac {2 \\pi R }{c + R \\omega}. ", "96f936d5fc27b596c87c1346973edad8": "\\mathbf{A}=\\mathbf{U}\\mathbf{\\Lambda}\\mathbf{U}^{*} ", "96f94da12216b20868da5a3f8d8901e3": "5\\times 2\\% = 10\\%", "96f9b6a0d841dfac190eb7f61d201526": " \\cot \\theta\\!", "96f9c1e6c8fbba3d1342e3166fbd2041": "c=v+e=m \\cdot B' + e", "96f9e66c5ae596172fbafd6f78b85a11": "\\beta,\\gamma<\\alpha", "96fa25802fbba9b4493d0c7e2f78641d": "k_0^2 = 4\\pi e^2 \\frac{\\partial n}{\\partial \\mu}", "96fa2adf88151ca315030d2faa2c83b4": "\\frac{dP(r)}{dr} = - \\frac{ \\left( \\rho(r) c^2 + P(r) \\right) \\left(c^4 r_s + 8 \\pi G r^3 P(r) \\right)}{2 c^4 r \\left(r - r_s \\right)} \\;", "96fa63aa2394df93cea0867b98ea4dad": " J:TM\\rightarrow TM ", "96fab4ec5018a61c5f316c87c67d71c1": "\\begin{pmatrix} e^{i\\theta/2} & 0 \\\\ 0 & e^{-i\\theta/2}\\end{pmatrix}", "96fae6786876f8942a18cd7bcafe8afb": "\n\\left\\langle {J(0)} \\right\\rangle _{F_e } = 0 ", "96fafac0c054b9eb47d3f630ed02c289": "c_i", "96fb1dae2c2739defa563ac1255e109d": "\\Delta g_h \\approx - 3.084 \\times 10^{-6}\\, h", "96fb57add4683e61418317d3e4b6da40": "p \\in particles", "96fc8dc4640d8fa6443f81dda79f2635": "4- 2\\sqrt{3}", "96fd0705693568ec7ac5c49b52d87341": "C=N/V\\, ", "96fd23a82fda7e60b938569d0b00855d": "\\sigma_{1t}", "96fd95ba9f1fc575e0a7443dc3e87891": "\\frac{\\alpha}{k_i} + \\frac{1-\\alpha}{N}", "96fdaa35db43205f20a70327c88bbddc": "\\Phi_{ab} = J_{ab} - \\frac{1}{3} \\, {J^m}_m \\, \\eta_{ab}", "96fdb527fa756afc3395ad74d80e1a55": "\\arcsin x= -i \\ln(\\sqrt{1-x^2} + i x)", "96fe503adc538c896fe04b5847a7d002": "V^{\\otimes n} ", "96fe525bda4bc556de4b7afc89260848": "m=\\frac{1}{RR}", "96fe5fa62c5ffe14004c4555c4306182": "0 \\leq r \\leq r_i: B(r) = \\frac{\\mu_i I r}{2 \\pi r_i^2}", "96fea1cb47320ce32503af0e24902043": "n_1\\sin\\theta_1=n_2\\sin\\theta_2", "96fece16c314dbd76e3969938026bd67": "D = \\frac{p}{2} / \\sin\\frac{\\alpha}{2}", "96ff72e65dc12b6cb9968dd6a2ea45fe": "{s_i \\over v_i} = \\sigma", "96ffac712269af50b91a77052cf6b098": "1.22 \\lambda N", "96ffb050672cc0f0561fdd02d9ac7a88": "(x-r)^2 = x^2-2rx+r^2.", "96ffb066378f040f524265511c10b500": "\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\,", "96ffb997100678259792dc424e12c248": "\\sqrt{\\lambda}", "96ffc74f081569e8d3dc59cbdc231f80": "\\int |D_n(t)|\\,dt \\to \\infty ", "96ffd3e535f81ea1dc6fbb085962df16": "p_0 = \\pm \\sqrt{\\vec{p}^2 + m^2}", "96ffd6a5e9fdfafd5b0e7dc175519835": "\\delta: Q \\times (\\Sigma \\cup \\{\\varepsilon\\} \\cup Q) \\to Q", "96ffffcbba45028804cb11f935d33f42": "13 ^ x\\,", "97007bce7707e4453bbecb8565c8f78c": "\\ k=\\frac{2 \\pi}{\\lambda}", "9700c48858e8f4e57e8a417e44191803": "\\nabla E = 0", "970132b30af7015149b32ed3352fe0a2": "\\begin{align}\n \\mu\\left(1 + k^{-1}\\right) - 1 &= E\\left[m\\left(1 + k^{-1}\\right) - 1\\right] \\\\\n \\Rightarrow \\hat{N} &= m\\left(1 + k^{-1}\\right) - 1\n\\end{align}", "9702062a1766918e9796ae32136d6669": "\\scriptstyle \\dot{s}^2", "970211e26cfe89d4953da6b0d664b058": "\\pm x^{7/22}", "9702485303c506149e46bccc5cd81b82": "\\tan \\alpha", "97025c22cce18aade6f228e545e0b200": "\\scriptstyle h :\\; \\mathcal{S} \\,\\times\\, \\mathcal{X} \\;\\rightarrow\\; \\{0,\\, 1\\}^m", "97028cc5da96974c09c3d4d7c7f05997": " \\frac{\\part \\psi}{\\part t} + H\\left(x,\\frac{\\part \\psi}{\\part x},t\\right) =0.\\,", "9702a6ff87753613b3cf061aa5df8b48": "\\rho_{min} \\equiv \\frac {- \\bar{a}} {1- \\bar{a}}", "970306f1afd44877c03bd096fc18f229": "\\mathbf{w}_n(x_n)", "970326594e8142d8566bdacd9246cc24": "(\\omega+dd'\\varphi_1)^m = (\\omega+dd'\\varphi_2)^m", "97032e324dbdf551770856131d705541": "P(s)=\\sum_{n>0} \\mu(n)\\frac{\\log\\zeta(ns)}{n}", "970391e65277a7af1cf7cecf80f8ab83": "a^{n-1}\\ \\equiv\\ 1 \\pmod n \\, ", "9703d63d1affb54e2757433126920210": "e^{in\\mu-|n|\\gamma}", "9703e98526ea90dac7be6aa0a888bd24": "E[|\\xi\\eta|]\\leq \\sqrt[p]{E[|\\xi|^p]} \\sqrt[p]{E[\\eta|^p]}", "97040713688e1ca5094f3a13fa7c6c82": "BE \\leftarrow BE \\cup \\lbrace e|e \\in C_i ", "97043ecaf6a1f4074962d512ad04281d": "\n\\Delta \\varphi = \\frac{L}{\\sqrt{2m}} \\int_{r_{\\mathrm{min}}}^{r_{\\mathrm{max}}} \\frac{dr}{r^{2} \\sqrt{E - U(r) - \\frac{L^{2}}{2 m r^{2}}}}\n", "97044da2fa00f2915a7e4fa3be6373e0": "\\# MCS = \\frac{\\#\\text{ attempts}}{\\#\\text{ monomers}}", "97044e051ba061f4a5c7c3d20873e824": "(\\alpha,\\beta,id)", "9704733c310aa00171190a5bba6e23ae": "\\int_{-\\infty}^\\infty \\mathrm{sinc}(t) \\, e^{-i 2 \\pi f t}\\,dt = \\mathrm{rect}(f),\\,\\!", "9704c3d4ef5ca28a4d32698158e9ef0b": "\n\\Gamma_{xy}(f)= \\mathcal{F}\\{\\gamma_{xy}\\}(f) = \\sum_{\\tau=-\\infty}^\\infty \\,\\gamma_{xy}(\\tau) \\,e^{-2\\,\\pi\\,i\\,\\tau\\,f} .\n", "9704fabe422b1e77c0011821aaea55e5": "\n \\begin{align}\n M & = D^{\\mathrm{beam}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2} - \\left(D^{\\mathrm{beam}}+2D^{\\mathrm{face}}\\right)~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\\\\\n Q & = S^{\\mathrm{core}}~\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x} - 2D^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^3 w}{\\mathrm{d} x^3}\n \\end{align}\n ", "970516ef46b1138bf9bd36fcf398e40d": "f(\\mathbf{x};\\mathbf{u})", "970528622a072036890cfb7e0fa83d92": "\n\\begin{align}\n\\det(\\mathbf{R} - \\mathbf{I}) =& \\det\\big((\\mathbf{R} - \\mathbf{I})^{\\mathrm{T}}\\big)\n=\\det\\big((\\mathbf{R}^{\\mathrm{T}} - \\mathbf{I})\\big)\n= \\det\\big((\\mathbf{R}^{-1} - \\mathbf{I})\\big) = \\det\\big(-\\mathbf{R}^{-1} (\\mathbf{R} - \\mathbf{I}) \\big) \\\\\n=& - \\det(\\mathbf{R}^{-1} ) \\; \\det(\\mathbf{R} - \\mathbf{I})\n= - \\det(\\mathbf{R} - \\mathbf{I})\\quad \\Longrightarrow\\quad \\det(\\mathbf{R} - \\mathbf{I}) = 0.\n\\end{align}\n", "97054dc34127a7a46d4d7ee124e6a33b": "e^{\\epsilon q(d,r)}\\times\\mu(r)\\,\\!", "970683c8a048d94b79314d468587d40b": "f:(X,\\tau)\\to Y", "9706c12677c8535d647a17e85d472a6b": "\nD \\int_x \\psi(x) | x\\rangle = \\int_x \\psi'(x) |x\\rangle\n\\,", "9706c957fe8c40b123a41f07f5518493": "\n\\frac{\\hat{p}^2}{2m_0} = -\\frac{\\hbar^2}{2m_0} \\nabla^2 = \n- \\frac{\\hbar^2}{2m_0\\,r^2}\\left[ \\frac{\\partial}{\\partial r}\\Big(r^2 \\frac{\\partial}{\\partial r}\\Big) - \\hat{l}^2 \\right].\n", "9706cd88ac4d12b1474e127e4d64170a": "\\textbf{R}^2", "97070fc12e346ad218efaae0424bcbe0": "I_{x_1}r=I_{x_2}r=1", "970710761f83a9a75e74313322b75f95": "\\operatorname{width}(R_0) / 2 \\sqrt{n}", "97077685710f8c8e728c01acf9bf038b": "I\\!\\left(x\\right) = x", "9707a4bc14d075404db955a5f47a65b0": "x_i: \\,\\,\\, p( c_k |x_i)", "9707c89e6556121f67386aa730c1a221": "\\bigcup V(a_i) = V (\\Pi a_i)", "9707e6eb8bdd83915904a299730ddd28": "\\ln(x) < \\ln(y) \\quad{\\rm for}\\quad 0 < x < y \\,", "9707f6a9fe49e5547c258bb24dcd72b5": "(A\\to(A\\to B))\\to(A\\to B)", "9708044799269d46367607f73151a0cb": " p(t) = \\alpha (t-t_0)^2 + \\beta (t-t_0) + \\gamma \\, ", "970815dd5f1d1c6e90c1d1bb219f1ed6": "m_0-m_1", "97084f242e22bcffdbd1803681d04120": "\\textbf{K}(t)=\\textbf{R}^{-1}\\textbf{B}^{\\text{T}}\\textbf{S}(t),", "970853612f67db2cc21453a3a28bfb65": "E_r=\\frac{1}{\\beta}\\frac{\\sqrt{\\pi}}{2}\\frac{S}{\\sqrt{A_p(h_c)}},", "97088837a0a9b2ec0d37d88ef3ad6deb": "T_a:A\\to A", "9708aac8083a4df94b996904c4d168aa": "\nf(x)\n=\n\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)}\nx^{-\\alpha-1}\n\\exp\n \\left(\n \\frac{-\\beta}{x}\n \\right).\n", "9708c6e710ddd46f057d583b12166a69": "\\begin{cases} u_{t}=ku_{xx}+f & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ u(x,0)=g(x) & IC \\\\ u(0,t)=h(t) & BC\\end{cases} ", "97091ffaba67802a8290994259926091": "2\\leq p \\leq dim_{\\Bbb C}M-2", "9709819046d3aaa2d2c2fba8c3e060da": " P(n>0) = 1 - P(n=0) = 1 - \\frac{m^0 \\cdot e^{-m}}{0!} = 1 - e^{-m} ", "9709824c08053b0c42b49e0ad1350320": "U_g(t) = U(t) - \\frac{1}{C_1}\\int\\limits_{t}^{0}I(t_1)dt_1 +U_0", "9709875a9ca4ec599eabc5ba0e6c4c7a": "k(T) = N_A^{2} \\sigma_{AB} \\sqrt \\frac{8 k_B T}{\\pi m_A} \\exp \\left( \\frac{-E_{a}}{RT} \\right)", "9709922eb72df9ad8094cfbf5b5ca590": "\\left\\langle R(t)R(t') \\right\\rangle = \\delta(t-t').", "97099aee08e809c5f4547ef2d8cb0bf9": "\\begin{pmatrix}\nA'(x)u_1(x)+B'(x)u_2(x)\\\\\nA'(x)u_1'(x)+B'(x)u_2'(x)\\end{pmatrix} =\n\\begin{pmatrix}\n0\\\\f\\end{pmatrix}.", "9709e7fdab06e68682cbb0593d25d3e1": "x_0, x_1, \\ldots, x_{N-1}", "970a245311ee671aede5e646d68f0cc9": "\n\n T^{\\alpha \\beta} = \\left( \\begin{matrix}\n \\rho & 0 & 0 & 0 \\\\\n 0 & p & 0 & 0 \\\\\n 0 & 0 & p & 0 \\\\\n 0 & 0 & 0 & p \n \\end{matrix} \\right).\n", "970a34a3e7e4b6dbdfb8ebb579a6a8d6": " r_\\pi = \\begin{matrix} \\frac {V_T} {I_B} \\end{matrix} \\to \\infty \\ ", "970ac8f4d3bfb13fdcc7ecff465c9891": "kO_*(X)", "970b07fc4b0e635e043167ae4b24dd60": " \\frac{\\partial u_i}{\\partial x_i} = \\frac{\\partial \\bar{u_i}}{\\partial x_i} + \\frac{\\partial u_i^\\prime}{\\partial x_i} = 0", "970b090d77bdc042e3b7361a563478cb": "\\frac{1}{h_1}+\\frac{1}{h_4}=\\frac{1}{h_2}+\\frac{1}{h_3}", "970b3c1b7597e0ae4d27b84f8bed5a10": " ~r'^2=x_0^2+y_0^2+z_0^2", "970b88e545be89d98f892481da82a6c3": "y'y^{-2} - \\frac{2}{x}y^{-1} = -x^2", "970bb8b3bae4c27761445c6aeef08559": "\\pi_1 = \\pi\\ ", "970bbd87da25bc68dc82efba5294b48b": "P/L", "970bcafdaa56e71e3e223a0972368b25": "L = M^\\text{T} M\\,,", "970c1505b7b4391d3b03315e7d129f5c": "\nT = \\frac{1}{2} \\left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \\cdots + u_{s}(q_{s}) \\right\\}\n\\left\\{ v_{1}(q_{1}) \\dot{q}_{1}^{2} + v_{2}(q_{2}) \\dot{q}_{2}^{2} + \\cdots + v_{s}(q_{s}) \\dot{q}_{s}^{2} \\right\\}\n", "970cb19ec345cb5c4b5c318348219977": "\\begin{align} \n\\lim_{(x,t)\\to (x^0,0)} u(x,t) &= g(x^0)\\\\\n\\lim_{(x,t)\\to (x^0,0)} u_t(x,t) &= h(x^0)\n\\end{align}", "970ced02ad95d304d05d3e96c23c9316": "\\omega_{ci} = eB/m_ic = 9.58 \\times 10^3 Z \\mu^{-1} B \\mbox{rad/s} \\,", "970d3da6fc449665176de7215e9d1867": "\\text{Gain}_i(\\sigma^*, a) = 0", "970d4b5608a71ba6c91bf4567ba21a91": " \\frac{d}{dt}\\hat{\\boldsymbol{u}}=\\boldsymbol{\\Omega \\times \\hat{u}} \\ .", "970da1e477739c17362129f0111d7a2b": "\\mathbf{Z} = \\{ Z[m,n] \\} ", "970dfae02f15488c101d7b063a780248": "T_0(\\varepsilon) > 0", "970e0c8f5d8ca4234d7ca8d3fca264e9": "\\Delta_0 = 0,", "970e8638a90283ea4c88eaabef5e941c": "d(\\Psi_g)_e=\\mathrm{Ad}_g\\colon \\mathfrak g \\to \\mathfrak g.", "970ee6ce51a16468c42f5c45fab4b081": "\\Gamma;\\lnot \\psi \\vdash \\lnot \\phi", "970ef525a8a64f241d236868c4f6c233": "\\mathrm{p.v.}\\; R_C(x, -y) = \\sqrt{\\frac{x}{x + y}}\\,R_C(x + y, y),", "970efba9c9aba129a1e81465c5b9d56b": " T_N(x) ", "970eff7fb04285b2173ac88a95969bb9": "\\mathfrak{P}^{76}", "970f0ebd9a2627e72f73690f54abc5d9": "\\hat{t}=-\\sin u\\ \\hat{g}\\ +\\ \\cos u\\ \\hat{h}", "970f59ca3ad1d46a37336a933b66acbe": "g\\ n = (n, n - 1)", "970f5be9c473a29425d66a8f1a5c0cd1": " \\Phi = v_p^2-\\frac{4}{3}v_s^2 = \\frac{K}{\\rho}. ", "970f6272916041e4955dcb430bbc637d": "L_f:=\n\\left\\{p\\left(\\tilde{b},f\\right): \\tilde{b}\\in\\widetilde{B}\\right\\}\\mbox{ for }f\\in F,", "970f6b1e623248e1c0e38c01d168f3e2": "\\Re(y)>0", "970fbc6c3c2ed45aa21ceb3b1161e93b": "x_0 = 1", "970fc38d3e1faed4fad256cf650ccde0": "K_m", "970fd991b2ff803c79e4f5243d67178b": "\\bar C = \\sqrt[3]{\\frac{\\Delta_1 - \\sqrt{\\Delta_1^2 - 4 \\Delta_0^3}}{2}}", "970fe2c5d278227583ba46de5187a283": "P = \\frac{n-\\Delta}{2} \\quad (17)\\,", "971047af525e217cfbcac3d469e6f464": "S_P", "971072f14755df77c6bafc8afb56d360": "\\,f(x)=\\alpha(\\neq 0)", "97109c5939087d3109da0649edeb41ef": "\\varphi_{xx} + \\varphi_{yy} = \\sin\\varphi,\\,", "9710a28b2bc9d8dae602e91a520071bb": "dy= F(y) \\, dx\\,\\!", "9710ae4e891d2d230a98c2c8c3e87c75": "t_i, i \\in \\{0,1,\\ldots,d\\}", "9710bee688097fc6a90571c246656106": "\\mathbf{P}^n,", "97112f45b2dd89407af1b6e7d88e8d19": "g=p+u(w)", "9711607638dddbbdc9e26f4bf8b63765": "O(|S_i|)", "9712ad4f83ff306214333da8c0a6bb83": "\n N^{\\mathrm{face}}_{xx} := \\int_{-f/2}^{f/2} \\sigma^{\\mathrm{face}}_{xx}~\\mathrm{d}z_f\n ", "9712c540135478fae037204834354116": "\\gamma \\left( 0\\right) =t_{0}", "97134dcd17b8161f3455ea3a9765c34d": "h(D) > C\\frac{\\sqrt{|D|}}{\\log |D|}.", "97137bcacbb5ba9c0ee8923a33649b0d": " x \\in (0,1) ", "9713aa6e7e32d6ae8edd240b60fc8e0c": "6F_{[ab,c]} \\, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0. ", "9713d4dc6a641bfc17e9f57dc50ebef2": "\\mathbf{Q}_{\\mathbf{X}} = \\frac{1}{n-1} \\mathbf{M}_{\\mathbf{X}}^T \\mathbf{M}_{\\mathbf{X}}, \\qquad \\mathbf{Q}_{\\mathbf{XY}} = \\frac{1}{n-1} \\mathbf{M}_{\\mathbf{X}}^T \\mathbf{M}_{\\mathbf{Y}}", "9713fc94f60a4fd2809f60ab71afc20b": "\nR(t) = r(t) e^{-D(t)} = r_{0} e^{\\alpha t - D(t)}\n", "971463c5a54dcf2020d289f03a8d134b": "x \\setminus L\\,= y \\setminus L", "971471713264404a0975de59b5ce7efc": "M_\\mathfrak{p}", "971530862fd98b833ba77c29309024ad": " \\arccos ( -\\frac{3 + 8\\sqrt{2}}{17} ) ", "9715718f5ffe3c7d6b021e7a41741569": "\\textstyle{\\partial\\over{\\partial\\epsilon}}\\big|_{\\epsilon=0}\\,", "9715c238376bf8284fbdb0fc241839c2": "f(g(a) + k) - f(g(a)) = f'(g(a)) k + \\eta(k) k.\\,", "9715f9bdbb3812ea627dabefead3706e": " M_2 (X,\\vec Y,Z)", "9715fbcd469c2544624ec821e85077cb": " E = -\\gamma m\\hbar B_0 \\ .", "9716123d92d334c5fe524c411bf0344f": "\\scriptstyle{\\boldsymbol{x}}", "971616cbbcf99973f5377fc5e18bfa6b": "V \\propto I", "9716219003bfdd9b80f2348368059244": "\\begin{vmatrix}\\cfrac{K-P}{P}\\end{vmatrix}=e^{-C}e^{-kt}", "971656f793614b65768523df4f9bb466": " \\displaystyle{U(s)V(t)=e^{-ist}V(t)U(s).}", "9716b28e752e2965e52dee172f7a71db": "\\left\\langle x-\\sum_i c_{i}p_{i},p_{j}\\right\\rangle =0.", "9716dfe0b8f2deae1c019632e9fd9aa5": "g\\in V_l", "97171ffe45ea51cbb343c2b55bf92617": "T = \\frac{a^{2}}{2(\\cot \\beta + \\cot \\gamma)} = \\frac{a^{2} (\\sin \\beta)(\\sin \\gamma)}{2\\sin(\\beta + \\gamma)},", "97174522d9e6d7882850b823e57f3bd3": "({\\mathcal P},{\\mathcal Z},\\in)", "97174aa315e6932afedcef78695dbc90": "(\\textbf{c}_N,\\textbf{c}_P)", "97175888070517c710e0dadc3efc9bea": "f(x,y) = - \\left(y+47\\right) \\sin \\left(\\sqrt{\\left|y + \\frac{x}{2}+47\\right|}\\right) - x \\sin \\left(\\sqrt{\\left|x - \\left(y + 47 \\right)\\right|}\\right).\\quad", "97175eb07516393b6d5580450aa1bfed": "\\{f(n+m)-f(m)-f(n): n,m\\in\\mathbb{Z}\\}", "97177703dadcb114da874f2e32322b79": "\\mathbf{r=\\left(I-H \\right) y^\\text{obs}}", "9717a331abb379ab122698e7bf1db4d3": "T = 4600\\left( {\\frac{1}{{0.92(B - V) +1.7}} + \\frac{1}{{ 0.92(B - V)+0.62}}} \\right) ", "9717b474f48571e177a4595bd6fe8137": "i_* i^{-1} \\mathcal G", "97182cbe9891c3675171a71d22cca00b": "A_m(2,3) = 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, \\ldots", "971850b2af746c527aac62de921cf489": "b_1=h(a_1),\\,b_2=h(a_2),\\,\\dots,\\,b_n=h(a_n).", "971895f652ce10266447ac53bd9c2267": " e^{s_4}=\\sqrt{\\frac{c+u_2}{c-u_2}} ", "9718b6c40252331ea2c81644de5f3d99": "\\textstyle\\mathbf{IPC}+\\bigwedge_{i=0}^n\\bigl(\\bigl(p_i\\to\\bigvee_{j\\ne i}p_j\\bigr)\\to\\bigvee_{j\\ne i}p_j\\bigr)\\to\\bigvee_{i=0}^np_i", "9718dfd41841ade7a4db93cc794c2e76": "{R^1}_{212} = \\frac{g'}{r \\, g^3} = {R^1}_{313} ", "9718ee498d7a54d311dd023dc757a1ef": "\\psi(x)=\\sum_{k_+} u_k (x)a_k e^{-iE(k)t}+\\sum_{k_-} u_k (x)b^\\dagger _k e^{-iE(k)t},\\,", "97190c3defffec4b4fc243e2dd7e77c9": "y_i \\in S\\left(Y\\right), i = 1,\\ldots, n", "971948873bd681dcad6c6cb8497b4c17": " \\theta \\boldsymbol{\\hat{\\theta}} \\,\\!", "97194cedd0002a97965a0dd0c2bd2ebd": "F_{\\alpha \\beta}^{\\;\\;\\;\\; IJ } = \\partial_\\alpha (P^+ \\omega_\\beta)^{IJ} - \\partial_\\beta (P^+ \\omega_\\alpha)^{IJ} + [P^+ \\omega_\\alpha , P^+ \\omega_\\beta]^{IJ}", "97196aabf7f4760156c44f1f267c0f71": "\\tfrac35+\\tfrac23", "97197271274a4966051ffa8406a43059": "A \\times A \\rightarrow A", "97199879df5356f0aa4ca80f5a5a3d6d": "\\nu=\\frac{2eV}{h}=\\frac{V}{\\Phi_0}", "971998fc7091a5ef14f00a70edc66a80": "R^*g(x) = \\int_{x\\in\\xi} g(\\xi)\\,d\\mu(\\xi).", "9719ac737e8fd648efe89c7d7a97df57": " (y_i,x_i) \\in C_{yx} ", "9719f2090f72493a9ac10142949800d5": "(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y.\\ ", "9719f25d0ae46224b025a069628a09e0": "\\mathcal{H}f = \\lambda f", "971a9a58bb3bb9951903a13bede127ee": " Q = a_S + b_S P \\,", "971aabd3c1ea06d4eac4ec936a190bb4": " \\dot{x} = x^2 + \\varepsilon ", "971af6ef8c426e99deac10c3f752f83b": "\nc_0^p+c_1 p\\equiv a_0^p+a_1 p+b_0^p+b_1 p \\mod p^2\n", "971b208e9c83a353afd07d802f29a66a": "x = (K+W)/2", "971b815cfcde707034ffe8f105eb04f1": "\\chi_\\epsilon (x,x) = 1", "971c3f50e06c1287a36fbe6c9fa31fa5": "\\text{CNR}_{\\text{dB}} = 10\\log_{10}(E_b/N_0) + 10\\log_{10}\\left(\\frac{f_b}{B}\\right)", "971c5805ca15c55f0312c44caf6e961c": "g(n)=f^{n}(1)", "971c6e0dbce97fb73a8772bacf6d15ba": "H(P,Q) = \\sqrt{1 - BC(P,Q)}.", "971ca819d22296113ebd77bc94e36ca6": "\\mathbf{112}_8 = \\mathbf{1} \\times 8^2 + \\mathbf{1} \\times 8^1 + \\mathbf{2} \\times 8^0 ", "971d3bcd528b5261b67a21981901e36c": "{4 \\choose 1}", "971d747151c113fb10a363daf7206336": " y_n = \\begin{cases}\n1, & if \\, U_n > 0, \\\\\n0, & if \\, U_n \\le 0\n\\end{cases}", "971df1dd381be21bb4ef0906054f5f33": "u(\\boldsymbol{x},t) ", "971dfff37234eb42e9e3260c62a3975c": "G \\,", "971e3f681dc279d7613052a5d5a9df61": "\\langle A, \\oplus, \\lnot, 0\\rangle,", "971e3fe1e1a25e4b488bad796ec61b79": "q=0", "971e8d3a6865a4fd680cada47fe07c4d": "(x,y),", "971e9cb8661612769ab59862abc61c76": "i = 2r+1", "971ec7465bd5bc2e61809fbf91050447": "\\{0,1,\\infty\\}", "971f2023c1f5f54b8bd389bb06fa6d86": "\\mathbf{y}", "971f3e9adbfd30148793ea021f6f0efc": "x,y \\in \\mathbb{R}\\,", "971f726e819204815cfb4f7365c7ddfa": "M_{d} =", "97201f3867fdf139e9ba41ca86933d88": "t_{up}", "9720295b482cf5b9ce1df54144801e33": "e^{\\mu}\\cdot e^{\\nu}=e^{\\mu+\\nu}", "97202a70c75282f1aec8500962a8d8e8": "\nj_{elec} = \nj_{ion}^{sat}\\sqrt{m_i/2\\pi m_e}\\,\ne^{-e\\Phi_{sh}/k_BT_e}\n", "9720e1689f4f060ba8c5b4ba48eb44d4": " \\displaystyle\nH \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int { P ({\\ln P}) \\, d^3 v} \n= \\left\\langle \\ln P \\right\\rangle\n", "9720fd4a183e369be6215b1d7c7cbed3": "V = \\frac{5}{3} \\sqrt{2}a^3 \\approx 2.3570226a^3.", "972105a92d28aab09958495e09330d62": "\\mu(T^{t}(A)\\bigtriangleup A)=0", "97217aa94c95fa9e31b5119b1f34d28e": "\n\\operatorname{erf}(z) = \\frac{2}{\\sqrt{\\pi}}\\int_0^z e^{-t^2} dt,\n", "9721a1c0818e848784e21bc19a0d8661": "\\frac{(C+F)r}{2}\\, .", "9721f9048b8be7f4b535221faf6080e6": "\\scriptstyle(x_1, x_2, \\ldots, x_n)", "972216bc21751fa00a880b95c4d64bdd": "i^*, i_*", "97226adfbbd9bf2a90ad3fbe447354ab": "\\bigcup_{\\alpha\\in I} x_\\alpha", "97226b4d85b0dc214d6d0c868b6b2997": "R_{jm}", "9722720ae21727d3c41211f6d21cdc17": "\\tilde O((\\log n)^6)", "972275f044044b651459eb60cbe6b386": "\\Delta S_{mix}\\,", "97232b2417d9c5d5f9f6871fd80d3cd1": "\\{z: r \\le |z| \\le R\\}", "97232f873db4ccfa40e66962dea2b68e": " \\overline{\\mathbf{A}} = \\frac {\\Delta \\mathbf{V}}{\\Delta t} \\ ,", "9723315add55cfbc0d1098c0a2d2d02f": "A(u,\\varphi)", "9723d90e797f2f27b713c0fa13cc384b": "a_{11}+(1/3)b_{10}", "97243ef80e172a3b7a41fa318737540d": "\\mathbf{G}=\\mathbf{R}-\\frac{1}{2}\\mathbf{g}R,", "972460f289d7cfefafbcc08a8389cad1": " \\gamma =0 \\,", "97248eeda49ed9f0d82638c0508d8f98": "SHS^\\dagger=H", "9724b8793392abf854159f0f7c459450": " \\tilde{\\mathbf{y}}_{k} ", "9724c2c89cdd1c43ced954108c494295": "\\text{if }n\\text{ is even}", "9724ea509399af4d897f4377361542f4": "P_N", "9724f309a74a5bd2bfabf54c8862d30b": "\\scriptstyle |\\zeta|\\leq \\max\\{1,\\|a\\|_1\\}", "97255bbf59c17bcdd6ba1a1b33d23007": " \\textstyle\\ \\mbox{rate(prop)} = \\frac{k_{init}k_{prop}[\\mbox{I}][\\mbox{M}]^2}{k_{term}[\\mbox{H-X}]}", "9725b01ded3c3664f71609229111f256": "Arity: F\\rightarrow\\mathbb{N}", "97265e534f182cebfe59f550b10c0593": "h_2 = c \\sin B\\text{; } h_2 = b \\sin C \\,", "9726625e4dfe79954ba234ec6d95d4ce": "x(t+\\tau)", "9726d4c8439bfe4ec96b275ddc152704": "\\xi_{inf}(\\alpha)\\geq \\xi_{sup}(\\alpha)", "9726ed8ca9db8867b382d4c39fa9a2b6": "\\mathbf{c} = \\mathbf{b} + t\\mathbf{a},", "97272d9fa6271df9f567ac9b0a552dd6": "g: Z \\rightarrow Y", "97278314f0cdf95209c1cacdd2630a7c": " \\ -p_a \\ln(p_a)-p_b\\ln(p_b)- {} \\ ", "972795f76fa59de97444bd4a99a8463c": "Ur=1/3", "9727c1726c869379dcd7088f23b80a8a": "X\\vee Y = \\{x\\vee y\\mid x\\in X, y\\in Y\\}", "9727dd1fa680ce84f00bff2a2edebd0f": " \\gamma (1) - \\gamma (0) = 4\\pi", "97282322a6f9d764ac280767b91ecc64": "\\mathbf {F}_{i}^{(T)} - m_i \\mathbf {a}_i = \\mathbf 0.", "97282d6495e69a22d183f7f631a08f5b": "a_k \\log^k(n) + \\cdots + a_1 \\log(n) + a_0. \\, ", "97284c49f3bf198a97880a38c667b571": "R (z)", "972896d2d2dc929f20a96519ac936601": "\\Lambda^k(T^*M)", "9728c21704f6f17fc153ac8cd9bd68d3": "Q = \\left[\\begin{array}{rrrr}\n2 & -1 & -1 & 0 \\\\\n-1 & 3 & -1 & -1 \\\\\n-1 & -1 & 3 & -1 \\\\\n0 & -1 & -1 & 2\n\\end{array}\\right].", "9728e93516d8d5d052c50d498aba63d9": "3 a_1 a_2.\\,", "9728f8940412a48f3d84b5cef8354769": "{\\rm Riesz}'(x) = \\frac{{\\rm Riesz(x)}}{x} - x\\left(\\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^4} \\exp\\left(-\\frac{x}{n^2}\\right)\\right)", "9729427c2129e119a0b584a04e3fec8f": "U(|\\Phi|)/|\\Phi|^2\\Big|_\\min \\frac{ 2 }{ \\sqrt { 3 } }", "97345d97262a2c1602571c1832d28139": "f = \\frac{1}{\\ln(2) \\cdot C \\cdot (R_1 + 2R_2)}", "97348381c1ff348739682a0cf5066bde": "Q(i)=0", "9734dafbfcf081a3370935e62c3a4670": "L(x) = (x^4 + x^2 + x) \\otimes (x^2 + x) \\otimes (x^2 + x).", "97351c6e0cbd04b0884d82ada763c298": "\\kappa < 2^{\\kappa}.\\!", "97356729f03e67c304c0c76aefa57b1a": "\\frac{\\partial}{\\partial x}(\\rho u \\phi)\\,= \\frac{\\partial}{\\partial x}\\left(\\Gamma\\frac{\\partial \\phi}{\\partial x}\\right),\\quad 0 0 ", "974360d76515ab8f4e873f08c3493354": "z(t) = x * y", "974365c64b66e0b585e387cbd1328860": "K,b", "9743b6309ed413bada4306a303db69fc": "\\scriptstyle D=R),", "9743ea8a5167914ab7017698a103a5ef": "\n\\begin{align}\n\\boldsymbol{\\Tau}_{\\boldsymbol{n}}(z)& = \\frac{(b_n+z)A_{n-1} + a_nA_{n-2}}{(b_n+z)B_{n-1} + a_nB_{n-2}}& \\boldsymbol{\\Tau}_{\\boldsymbol{n}}(z)& = \\frac{zA_{n-1} + A_n}{zB_{n-1} + B_n};\\\\\n\\boldsymbol{\\Tau}_{\\boldsymbol{n+1}}(z)& = \\frac{(b_{n+1}+z)A_n + a_{n+1}A_{n-1}}{(b_{n+1}+z)B_n + a_{n+1}B_{n-1}}& \\boldsymbol{\\Tau}_{\\boldsymbol{n+1}}(z)& = \\frac{zA_n + A_{n+1}} {zB_n + B_{n+1}}.\\,\n\\end{align}\n", "9743f687b547d7fa54b0175de6f3b052": "\\mathfrak T_{\\Phi} ", "9744077524ce325c899e089537430a44": "\\pi_1(H)/\\pi_1(G)", "97443bcf56609d12f383cc79e2a13f41": "\n\\begin{bmatrix}\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n\\end{bmatrix}\n", "97449dedba9fdb1da625f030469b4486": "H_1(G;\\mathbf{Z})=H_2(G;\\mathbf{Z})=0", "9744b5a3519969045ac2f4aad5f74f47": "C_{Cr} = \\frac { U_{Cr} \\ \\times \\ \\mbox{24-hour volume} }{P_{Cr} \\ \\times \\ 24 \\times 60 mins}", "9744b738c0d3f0a307f05e71bb0eb7d5": "L(\\varphi,\\partial_\\mu \\varphi) = \\int_\\mathcal{V} \\mathcal{L}(\\varphi,\\partial_\\mu \\varphi) dV \\,,", "97453e60361a3c3311714c9fa6b6809e": "\\nabla_\\alpha e^I_\\beta = 0.", "9745576ba4efc21ea21fc0e0f5fb24c5": "e^{\\pm i\\pi \\sum_{k=1}^{j-1}f^{\\dagger}_k f_k}", "974579b5b6cb05cb8dad80e10d8b03ac": "DP_{T/D}^{V/S}", "974583ff9ce355eec337b32d413428dd": " \\omega = 2 \\pi f ", "97458e6da7324a7163a9e07d53a1bb31": "\\mathbf{x}(t)", "9745bb9e7f25d61bf67c4838cb0fb9d2": "[1,2,3] + [10,20,30] == [11,22,33]", "9745fee5a4ce0b8402e3e44634691a63": "\\psi\\, =\\, 3\\, \\frac{\\eta}{h}", "974627d1eae403b8ea2d9ceebea6d8a4": "x_2 (t)=\\begin{cases}\n\\cos( \\pi t); & t <10 \\\\\n\\cos(2 \\pi t); & 10 \\le t < 20 \\\\\n\\cos(3 \\pi t); & t > 20\n\\end{cases}", "974655c674a579236be16786a04df166": "\\Omega_2 = \\{ \\land, \\lor, \\rightarrow, \\leftrightarrow \\}.", "974664a4a367e5ecd2e9d1534142ba11": "i=1,2,\\dots,n", "9746e0675ce25a183e7245f4a11e383e": " F_{ij} = \\exp [ \\beta_0 + \\beta_1 \\ln (M_i) + \\beta_2 \\ln (M_{j}) - \\beta_3 \\ln (D_{ij})] \\eta_{ij}^{\\ }", "9746e3bb8db70a7a5ec4c80eff005b3e": "I_\\alpha(x) = i^{-\\alpha} J_\\alpha(ix) =\\sum_{m=0}^\\infty \\frac{1}{m! \\Gamma(m+\\alpha+1)}\\left(\\frac{x}{2}\\right)^{2m+\\alpha}", "9746f27d9883ca687bf238f6885bbd50": "\\{a\\in F: |a|< 1\\}", "9747634c546a06a1128a82c71e16ce83": " \\sum_{i,j } p_{ij} \\log \\frac{p_{ij}}{p_i p_j }", "97478c2d5ede1b8d6490b670e7809d5d": " \\mathcal{P}_z c = \\mathcal{E}_c. ", "97478c4acea7b3ccf32580e7b5c4f710": "c_0=2^{m_\\alpha/2 + m_{2\\alpha}}\\Gamma({1\\over 2} (m_\\alpha+m_{2\\alpha} +1)).", "9747acea2d3170aea957e86fbd665482": "\\mathbf{v}=\\mathbf{GU}", "9747c18f0fa01d6de9995f398e1d1d73": "\\frac{RV^{(n)}-IV}{\\sqrt{2t\\int_0^t \\sigma_s^4 ds}},", "9747ec21237ee01fb3b5859f1de4b0d4": "\\tilde{n}=1-\\delta-i\\beta", "974813c13a10a5f6a4f6547406f11f2a": "{s_1}", "97481cc7fccfe38dc4ecda9b123b35c1": " \\frac{1}{PM} = \\alpha \\frac{2R}{x(180-x)} + \\beta", "97482eef78e470a6b76d81a8c68b7c23": "\\operatorname{cov}", "974842c3d9d0988bea6ad94061550150": " \\| \\tilde{H}_n y_n - \\beta e_1 \\| = \\| \\tilde{R}_n y_n - \\beta \\Omega_n e_1 \\| = \\left\\| \\begin{bmatrix} R_n \\\\ 0 \\end{bmatrix} y - \\begin{bmatrix} g_n \\\\ \\gamma_n \\end{bmatrix} \\right\\|. ", "9748544c4ef3e4bbec05348c4c46db38": "\\vec{x^*} \\mathbf{A} = \\mathbf{0}", "974864490d07370cb528c6f64636fc11": "\\Xi'(x)=\\alpha^{-7}+\\alpha^{5}x^2.", "97486f3750f386eef933ccd1d0a4469e": "w=w_1 w_2^{-\\delta}", "9748bbc5df04fe1048b156cb5a36b6e3": "n=6,", "9748d550d401cee28dee4f6c455c5772": "\\nabla \\cdot (\\vec u \\times \\vec v) = \\vec v \\cdot (\\nabla \\times \\vec u) - \\vec u \\cdot (\\nabla \\times \\vec v)", "9749803649d6667945e4060ddc746b2d": "\\aleph = \\aleph_1", "9749bcd7177c21c17bca09f32b79de60": "x^p_{ij}", "9749d2c5b016092fcb28e37372ca9c7b": "\\sum a_n e(x_n)", "974a03967f700b1ca5233a4755dae365": "(A_\\bullet, d_\\bullet)", "974a1a67a7aed4e3efa04018bb683036": " D= z{\\partial\\over \\partial z},", "974a9d59e0bc43a842691185496f2f4f": "h^2_2=\\frac{2L^2k_2C_o}{rD_c}", "974b06685e9ce4ed530b6514e55d44b3": "\\gcd(f(x),a(x)) = \\prod_{i \\in A} p_i(x),", "974b3f715da6868bada16fb51ea70b10": "\\overline\\Gamma", "974b5a3a670643897fd647af41d6327a": "r_p=\\frac{p}{1+e}", "974bc5bf2c8c869c6dfbd92256e30e60": "\\sum_{i=1}^n a_i\\log\\frac{a_i}{b_i}\\geq a\\log\\frac{a}{b},", "974cb3fef3b00829c21be3499898a005": "\\pi = (\\pi_1 , \\cdots, \\pi_n)", "974cb705b0c744f1855953c1086337df": "t_{2}", "974cbdfbdf088e6384370e791e132649": "u_r", "974cc379c9ada0f690f2c8023fd5c940": "\\hat{\\delta}=1.", "974d73275ffa72622d23d3d6ec7b4849": " \\mathbf{J}_{\\text{f}}+\\mathbf{J}_{\\text{D}} +\\mathbf{J}_{\\text{M}} = \\mathbf{J}_{\\text{f}}+\\mathbf{J}_{\\text{P}} +\\mathbf{J}_{\\text{M}} + \\varepsilon_0 \\frac {\\partial \\mathbf{E}}{\\partial t} = \\mathbf{J}+ \\varepsilon_0 \\frac {\\partial \\mathbf{E}}{\\partial t} \\ ,", "974d9c4c1db92947ebe22064e114c89d": "\\mathbf{\\ddot r}_{Sun} = G m_{Earth}{r_{{Sun},{Earth}}^{-2}}\\hat\\mathbf{r}_{{Sun},{Earth}} + G{m_{Moon}}{r_{{Sun},{Moon}}^{-2}}\\hat{\\mathbf{r}}_{{Sun},{Moon}}", "974d9dfeb3843c7d993fd06daf56b3f7": "x'=x+a.\\,", "974dbb0922238991cc856d316c13dd6c": "\\mathbf{J} = \\hat{O} \\mathbf{E}", "974dcb835d19403506db4f256e402d64": "C\\subset{\\mathcal P}(X)", "974e079d8e05c5bfe691720179a1352f": "g: [0,T] \\times \\mathbb{R}^m \\times \\mathbb{R}^{m \\times d} \\to \\mathbb{R}^m", "974e0d3014805b9f9098136c758d56a5": "C_{k+1}", "974e7cdb571922db1a56ce308bf67df1": "(A_x\\ ,\\ -0.6\\ A_x)", "974ec01bd48059fec54ec31124a2ae70": " y_{n+1} ", "974ecbc20a578952e780f116e5ce7696": "\\forall_{\\gamma < \\omega}{V_{\\gamma}:} \\neg \\and_{\\gamma < \\omega}{V_{\\gamma +} \\in V_{\\gamma}}.\\,", "974ee57a49f6de33820bcfe00c9ba497": "q \\in (1,2)", "974f189a0aaee6f2cf1a817f7c6c6bbe": "C_{X,Y}=\\frac {\\left \\langle (X_i (n)- \\mu_i) (X_j (n)- \\mu_j)\\right \\rangle}{\\sigma_i \\sigma_j} \n", "974f466234f99bf8d3eb54cadfefd55b": " \\forall \\epsilon >0 \\, \\underbrace{\\exists \\delta > 0 \\, \\, \\forall x \\in \\mathbb{R}} \\, \\forall h \\in \\mathbb{R} \\, \\left( \\, |h| < \\delta \\, \\to \\, |f(x) - f(x+h)| < \\epsilon \\, \\right) ", "974fac6ca8f02066db84397c41246fbd": "y_1, ..., y_d", "974fb6190fa2283ea17be0dc166aafdd": " p_i = n_i/\\sum{n_i}", "974feba567c3b9e14531067c8cbe5f2e": "k \\ge 2", "975059255488ee684973294a28c3d691": "\\textstyle \\delta > 0", "9750906a2916a760fba4e3057e4f7696": "\\frac{P_B(t)-P_\\infty(t)}{\\rho_L} = R\\frac{d^2R}{dt^2} + \\frac{3}{2}\\left(\\frac{dR}{dt}\\right)^2 + \\frac{4\\nu_L}{R}\\frac{dR}{dt} + \\frac{2S}{\\rho_LR}", "97509569b29af6ebed04664f5c432fe4": "S_{A}(\\lambda)=100 \\left(\\frac{560}{\\lambda}\\right)^5 \\frac{\\exp \\frac{1.435 \\times 10^7}{2848 \\times 560}-1}{\\exp\\frac{1.435 \\times 10^7}{2848 \\lambda}-1}", "9750b254689e8d7abf759cbc5bc95eed": "\nE_{k}(r_{k}^{A}) + \\sum_{l=1}^{N} \\min_{X} E_{kl}(r_{k}^{A}, r_{l}^{X}) > E_{k}(r_{k}^{B}) + \\sum_{l=1}^{N} \\max_{X} E_{kl}(r_{k}^{B}, r_{l}^{X})\n", "9750cb347a7014a808044e12bed5c74a": " \\tan \\theta =\\!", "9750ddec06ddc1fb1fafaa950bcc5d0e": "a, 2a, 3a, \\ldots, (p-1)a, \\,\\!", "9750ee4c9f9a7b8d3687229668f93b8d": "\nc(\\tau) = \n \\sum_k \\frac 12 (a_k^2 + b_k^2)\\cos (2\\pi \\nu_k \\tau).\n", "9751014c87d342acfa503a8967fc852f": "\\mathcal{P}_{=n}(S)", "975114279bcdacbdabe62e312e195009": "\\scriptstyle \\{|0\\rangle, |1\\rangle\\}", "9751df0668e358800d2d6b7a7393cf46": "\\operatorname{Ob}(M*):=M", "9751e6c428ce3f0deed9eaf3c20f618e": "g:\\mathcal{P}(\\kappa)\\to\\kappa^{+}\\,", "9751f941429c2cfc8382899829e9848f": "\\mathrm{SF}", "9752057b3c52987ebe212cd43b2fc731": "\\delta W=-P(dV-v\\,dM)", "975233e1233d03ae0afc2fc52a3c4c5d": "\n\\mathbf{\\epsilon^1}(\\mathbf{p}) \\times\n\\mathbf{\\epsilon^2}(\\mathbf{p})= -i\\mathbf{p} / p_0, ", "975259f63a873a0e5f9c2553618716c7": "K_{d}", "97526bab17dfbee946e63782adb4df84": "h_{inv}(n) \\, \\forall \\, n \\, \\in \\mathbb{Z}", "9752cd3f01281f746ddd8c7c8046a841": "d(fg) = (df)g + f(dg) \\neq f(dg)", "975395b38bd46414c9bcd9f3f788fa3f": "W^k", "97539a03815c1d98c548f5df7c5569ac": " (\\partial A)_T=-(\\partial T)_A=P\\left(\\frac{\\partial V}{\\partial P}\\right)_T", "9753a705480f0923ef5845500ffd4369": "\nw_{x,j}=f_{x,j}*\\frac{Idf_{x}}{max_{i}Idf_{i}}\n", "9753e6ca2588826f43f5dcf695dfb299": "X = \\frac \\sigma {\\sigma_\\mathrm{zen}} \\,.", "9754c74561cd8d8f8ab51da161b4adfa": "traces\\left(P\\right)", "9754f65c1b8666b325911210444e316a": "\n\\begin{cases}\nu_i n_i & > 0 \\\\\n\\sigma_{ik} n_i n_k & = 0 \\\\\n\\sigma_{ik} n_i \\tau_k & = 0\n\\end{cases}\n", "97553872e1bb8e03f4f0201f29a05ac6": " S_3 = {16 \\over 15} \\approx 111.731 \\ \\hbox{cents} ", "97553faf09c4e2fe56060f4d27721e6e": "E_{ij} = \\sum_{a=1}^m x_{ia}\\frac{\\partial}{\\partial x_{ja}},", "9755b24d83a024e8ed4ab8decdec5a84": "\nB= \\frac{1}{2} \n\\begin{bmatrix}\n1 & 1 & 0 \\\\\n0 & 1 & 1 \\\\\n1 & 0 & 1 \\end{bmatrix}\n", "9755dc6d8447fda195841c4942aa9372": " \\int_0^1 f^2(\\mathbf{X}) d\\mathbf{X} - f_0^2 = \\sum_{s=1}^d \\sum_{i_1<\\dots 0", "9757811325408a472234d304473e781a": "\\|f\\|_p = \\left(\\int_D |f(x+iy)|^p\\,dx\\,dy\\right)^{1/p} < \\infty.", "9757e4819017006f4406b5267dd7b03d": " \\and T_6 = [F_6, S_6, A_6]::K_5 ", "9758011dbc599d0cba664a3a46d3f299": "\\log_2(26^{n^2})", "97583fb8b5579d6dd5903c6be5129245": "\\left(-k_y^2-k_z^2+\\frac{\\omega^2n_x^2}{c^2}\\right)E_x + k_xk_yE_y + k_xk_zE_z =0", "97585dfc5a6f01e3d0e1475f89e7c4e2": "P\\left(x\\right)=D\\left(x\\right)s\\left(x\\right),\\text{ }D\\left(x\\right)=\\frac{{e}^{x}}{{\\left({e}^{x}+1\\right)}^{2}}", "975931ef749ae19739db475281b8e41d": "{\\scriptstyle \\partial \\Omega}", "975937f0c8ece1939cbd50b0b70426f9": "\\log{g(z^{-1})-g(\\zeta^{-1})\\over z^{-1}-\\zeta^{-1}} = \\log{f(z)-f(\\zeta)\\over z -\\zeta} -\\log {f(z)\\over z} -\\log {f(\\zeta)\\over \\zeta},", "975993a6a0cc779817e9f271de1d3bb3": " \\sum_{W \\in X} c_W v_W=0, \\quad v_{W} \\in W \\setminus \\{ 0 \\} ", "975999b56b0e96248e1201d48882e82e": "\\|\\cdot\\|_K", "975aadfef0c3d9d57ca5ef9e236b597f": "\\displaystyle 6.28=2\\pi", "975b052155f112def4ce68e461748435": "\\mathcal{P}(A)", "975b1a2ddb36a8f64eeb08fdd2d710a8": "\\overline{Z}_{n-k}", "975b24f60d4c34edebe7ae2150e6a16b": "\\vec{u} = ", "975b3135c2e245b046d42f9ddea243a9": "\\hat m_ - ", "975b354c5fd11a3cd0a8ebc4b968bd9d": "\nG_\\mathrm{dB} = 20 \\log_{10} \\bigg(\\frac{31.62~\\mathrm{V}}{1~\\mathrm{V}}\\bigg) \\equiv 30~\\mathrm{dB} \\,\n", "975b4b44ac5d4fbcd3df6df890d6c41b": "L = \\mu_0 N^2 (R - \\sqrt{R^2-a^2})", "975b9ff13afbc0018b41cca17dc4e784": "Rev_t = WACC_t \\times RAB_{t-1} + Dep_t+Opex_t", "975ba5c252049d3bc50ebbb5b399a3c4": "M \\sim N\\,", "975bd58fee0fb0c56d8921af0200e3d7": "\\rho^{T_B}", "975be0f353fcfd4d04c222c452a53db7": " A_{1} = \\frac{P}{\\sqrt{2}} ", "975c72cb0f480ff61c55df7d9376d145": "{L(\\hat{\\theta})}", "975ca8804565c1a569450d61090b2743": "1/2", "975d176fce3f6911d9da859107521785": "A_1 \\subseteq A_p \\subseteq A_\\infty\\text{ for }1\\leq p\\leq\\infty.", "975d187384c1415e72c88670269d85df": "v_1, \\dots, v_l", "975d449ce5bad222875aac15ee6865ff": "{6\\choose n}", "975d7d57944bd13cc9c70299981e9239": "\\rho_\\nu(\\nu,T)=F(\\nu)\\frac{1}{e^{h\\nu/kT}-1}", "975dd31c85469178f58788fa8f7e5e7a": "q_2=0", "975dd785fa469e797bdab9338d361b1e": " F_i, i=1,2,\\dots s ", "975e4c7eba255144be77dead6fe38a42": "e^{-\\delta\\tau}", "975e82ee46300a50d901d66c00fe64b1": "d_{1}", "975eac93bbc1433ca7401ccf929b91dc": "K = 8", "975f105a74c72613b480df2c8d7658ea": "\\rm{PC} = \\frac{\\rm{AC} \\sin x}{\\sin \\alpha}.", "975f14b8001cb8fc948b4ab65421ea04": "P_{\\rm abs} = P_{\\rm emt\\,bb}", "975f38c369a0d96de80d391c3d911f2e": " P(X \\leq x) \\leq \\frac{e^{-\\lambda} (e \\lambda)^x}{x^x}, \\text{ for } x < \\lambda .", "975ffbc4d9be82bbb2ddbd6fcd825b30": "c_{i}+(b_{i}-a_{i})", "976037a7a7aba1521bf2cf91c347bf0a": "C_1 \\approx 8.86", "976096bb515a33f7a60fa0053c9e0eff": "\\sum_{i=0}^h 3^i", "9760ab9b6f6d50f300b7b6c01205da92": " -\\frac{{\\hbar}^2}{2m} {\\nabla}^2 \\Psi= (E-V) \\Psi", "9760dab512995eddec331f6f14d615f0": "A^*,B^*\\,", "9761189504a091de95accd660d6bf684": "(a_1^2+a_2^2+a_3^2+...+a_n^2)(b_1^2+b_2^2+b_3^2+...+b_n^2) = c_1^2+c_2^2+c_3^2+...+c_n^2\\,", "9761217fbb1883fb5065870e096098d2": ") + z_k\\epsilon_k", "9761a153f397f3b81dc62b6949eafacd": "\\displaystyle\\epsilon", "9761a9ce88838d3cb8e8c1404b4bc695": "\\begin{align}\n\\left(a^2 + nb^2\\right)\\left(c^2 + nd^2\\right) & {}= \\left(ac-nbd\\right)^2 + n\\left(ad+bc\\right)^2 & & & (3) \\\\\n & {}= \\left(ac+nbd\\right)^2 + n\\left(ad-bc\\right)^2, & & & (4)\n\\end{align}", "9761c3703bdc27c98cf50beec9df0b7e": "\\nabla^{2}\\mathbf{a}=\\boldsymbol{\\nabla}\\left(\\boldsymbol{\\nabla}\\cdot\\mathbf{a}\\right)-\\boldsymbol{\\nabla}\\times\\left(\\boldsymbol{\\nabla}\\times\\mathbf{a}\\right).", "9761d1a6b83d48c546a72866e6ea0b16": "\\mathbf{y}_i < \\mathbf{y^*}_i", "9761f394f07d9ef3bf51536837fc7a1f": " A = \\sup_{|z| \\le R} \\operatorname{Re} f(z). ", "9762a0d19bdc9830acfdae72857c4b20": "\\sigma_h = \\displaystyle\\sum_{i=1}^n \\rho_i g h_i", "9762ca5378d0463b3fcc87c0faff8d39": "\\rho^+", "9763c3842eed4ae713aeec226896e380": "f^*\\mathcal G := f^{-1}\\mathcal{G} \\otimes_{f^{-1}\\mathcal{O}_Y} \\mathcal{O}_X", "9763c7ceecab7fbdc9e13cddcee9c71e": "\\, \\delta Q ", "9763dfe4f3fa6bf93bf85cbbdaba5e93": "\\rho_n ", "9763f0f983cba53771b30c999ab193fd": "\\mathbb{T}\\subseteq\\mathbb{R}", "9763f31463a222dddbf171d8c21c5a6d": " E=\\mu ", "97647eda3a1707fdde0f4c2b77bf71bf": "\\forall x \\in L", "9764a6fb095f9fd9899e294977a3bec3": "\\begin{bmatrix}\na & 0 & b & c \\\\\n0 & a & 0 & d \\\\\n0 & 0 & a & 0 \\\\\n0 & 0 & 0 & e \\end{bmatrix}", "9764c045fd419b7942738c272b8c1046": "\\overrightarrow{PZ} = \\frac {n \\times M_P^{gi}}{F^{gi} \\cdot n}", "9764d6570b65f6a0ae2ef5c1a43cc707": " n \\ge 1, k \\ge 0", "976599b7c830ad0e23e86303a671eb9b": "v_p = \\frac{1}{\\sqrt{\\epsilon \\mu}} ", "97666c8427303d79f958cb8366d5d8d3": "({\\Delta}{\\lambda}, {\\Delta}\\varpi)", "9766b7df89ccc0cd8e187a467d70eaf4": "\\Delta x = \\Delta y = \\Delta l", "97670f2013bd25e87c14ddb4eb4a6bec": "\n \\overline{\\boldsymbol{u}}_S\\, =\\, \\overline{\\boldsymbol{u}}_L\\, -\\, \\overline{\\boldsymbol{u}}_E.\n", "97677e229f99d4bfa244935ef2e697f5": "\\left({6 \\over 50}\\right)", "9767983a96605700b179c66952b52e63": "\\gamma = Currency Drain Ratio", "9767b6e34d898385237660c966219abc": "\\bar \\partial_{j, J} f = \\nu.", "97682e73bebd339a985f71f7a7b4d6cc": " a_i > b_i ", "976870880fd2b8d2c2bd0de8608e86a9": "Ly=Lclm(D+a(C))y=0;", "9768cbe16185aefad4084cac08edfaaf": "u(x,0)=s(x) ", "9768d494fc0a028ac45c4e6169fefd3d": "\\ \\displaystyle \\forall \\ ", "9769f13d47092deeac7184a0244d0b80": "~1/p", "976a8911093a2c3cb092647679e245d3": "NH_K (M)=(k_1 +_w m_1 )\\times(k_2 +_w m_2 ) \\mod 2^{2w}", "976ab4f1fd08c1e5ef0b182c5becf3b5": " \\frac{F}{F_\\text{P}} = \\frac{\\left(m_1/m_\\text{P}\\right) \\left(m_2/m_\\text{P}\\right)}{\\left(r/ l_\\text{P}\\right)^2}.", "976ad78943c92e5dd9ddbe34f2126aa3": " q^1,\\,q^2,\\,q^3 \\equiv \\alpha,\\,\\beta,\\,\\gamma", "976b27b76f870f7562fb9de34b4f266b": "d^2y = d(dy) = d(f'(x)dx) = f''(x)\\,(dx)^2,", "976b78e64129efbd908fa722e290dfb4": "x_1 = \\frac{1}{2} \\left(x_0 + \\frac{S}{x_0}\\right) = \\frac{1}{2} \\left(600.000 + \\frac{125348}{600.000}\\right) = 404.457.", "976bd585f13eda0ba0320b741e2fddd6": "U(t)=1-\\frac{i\\lambda}{\\hbar}\\sum_n\\langle n|V|n\\rangle t-\\frac{i\\lambda^2}{\\hbar}\\sum_{m\\neq n}\\frac{\\langle n|V|m\\rangle\\langle m|V|n\\rangle}{E_n-E_m}t-\\frac{1}{2}\\frac{\\lambda^2}{\\hbar^2}\\sum_{m,n}\\langle n|V|m\\rangle\\langle m|V|n\\rangle t^2+\\ldots ", "976c2260ff5025d698c9502b111f50f2": "\\mathit{FNR} = \\mathit{FN} / (\\mathit{FN} + \\mathit{TP}) ", "976c90d4c4ea00289fdf6d2221459f76": "\\mathfrak P_n(K), n\\ge 2", "976ca0efc08a3763ef0d4dcd67a95aa2": "n^2 - n + 1", "976caa5b3c1d5f84b089d5384876d523": "\\Re(s) > \\sigma_0", "976ce9ab1333e08d28447ac7127dfce3": "\n \\Pr\\!\\left( \\lim_{n\\to\\infty}\\overline{X}_n = \\mu \\right) = 1.\n ", "976cf6c05a704e9ce8dbba81d3f5d4d1": "C_S = k_O I / L\\;", "976d682061aedcf635a6fab5d46be11a": "\\scriptstyle \\mathbf{P}", "976de09f1b50738d2114f5c128d79cca": "1_{n \\neq 0}", "976dfb98de1bacc8f5cc3e4888212078": "\\exists x (P_1(x) \\land \\phi(x))", "976e86c258106aa369a873c1dc705318": "\\left\\lceil\\sqrt{84923} \\right\\rceil = 292", "976f139e4b143f6d5e50ad8ffecc0ca9": "M:=\\exp\\left(|\\alpha|^2+\\mathrm{Re}\\,\\alpha\\right), \\,", "976f1e356e896d7beb6c5edf9393992b": "\\gamma^{r}", "976f697a85f95b02b73b801f0f604a31": "W \\,\\ ", "976f7225c5398fffb2716a02f0edd252": " E\\Psi = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2\\Psi + V\\Psi ", "976f76d45a86e03304749bead6c33e6e": " (\\exists{x}) P \\qquad (\\exists x \\ . \\ P) \\qquad \\exists x \\ \\cdot \\ P \\qquad (\\exists x : P) \\qquad \\exists{x}(P) \\qquad \\exists_{x}\\, P \\qquad \\exists{x}{,}\\, P \\qquad \\exists{x}{\\in}X \\, P \\qquad \\exists\\, x{:}X \\, P", "976f78ba68c35653363ceaf4e77ec8e3": " \\qquad \\qquad \\mathbf{j}_e = \\mathbf{A}_{ee}\\cdot\\nabla\\frac{E_\\mathrm{F}}{e_c} + \\mathbf{A}_{et}\\cdot\\nabla\\frac{1}{T} ,\\ \\ \\mathrm{and}", "976ff7822475cf6100162cc0656f08e0": "\\scriptstyle M_{V_{\\odot}}=4.83", "977033eb81d5723c29968686e698ceaa": " \\dot y= V_d\\ \\frac{T_y-y}{\\sqrt{(T_x-x)^2+(T_y-y)^2}} ", "9770b7ae870f46b7611ad3fd5714023d": "\\scriptstyle{ (T + S)^\\prime = T^\\prime + S^\\prime }", "9770da1ed4dac12f974886caf5de7e9a": " I \\omega^2 \\left( r_{\\mathrm{axle}} / R_{\\mathrm{groove}} \\right) ", "977109e189e80bf103cac76e135de832": "\\pi (Y_{\\ell\\,m}) =(-1)^{\\ell} \\Rightarrow", "9771936cf3fa3111bed2b115cdf42f20": " r + t = m + n ", "97719e84a1d0aa4fd38b3721d0eb22e8": "\\mathbb{R} \\times \\mathbb{R}^n", "97726a585c67028f83e1dc4b3289e59d": "u^7+v^7+b=0 ", "977296fc37b49b00f50882b005afa469": "\\frac{1}{2}\\ln\\{(2\\pi e)^{N} \\det(\\Sigma)\\}", "9772b5c2a13afa27558057dbf9e13322": "2 a \\left( - \\frac{b}{2 a} e^{-\\frac{b}{2 a} x} \\right) + b e^{-\\frac{b}{2 a} x} = \\left( -b + b \\right) e^{-\\frac{b}{2 a} x} = 0.", "9772dcf5cd4855260a8862b555c6e6ad": " \\vec{r}_1, \\vec{r}_2 ", "97736ba66313e9c9eac004baad06d553": " \\|AB\\|_1=\\operatorname{Tr}(|AB|)\\le \\|A\\|\\|B\\|_1,\\qquad \\|BA\\|_1=\\operatorname{Tr}(|BA|)\\le \\|A\\|\\|B\\|_1", "97736c94cc15a46fc9da00152057d868": "\\frac{dU_1}{dt} = \\int_0^L \\frac{du_1}{dt}\\,dx = J_1(T_{1L}-T_{10})=\\gamma L(\\overline{T}_2-\\overline{T}_1)", "97740de606d75ab4ea861759548a961d": "E_6^{\\mathbb C}\\,", "97742a6a54d78e451774ce8af9d0ab71": "\\mu=16", "9774f042e139f370aa82dd886335395a": "t_1, \\ldots, t_k", "9775f7572c2055aec251005c5844e004": "a^n+b^n=c^n", "97766c435655865c32ff823faced0abc": "\\mathrm{d}U = \\delta Q - \\delta W + u'\\,dM,\\,", "9776bb3f35d25fb73b565513447ad22e": " I_z = \\int_{0}^{L} dz \\int_{x,y} dx dy \\, \\rho(x, y, z)\\,r^2 ", "9776d202a699b1775f618509d68eb9ad": "Ax^2 + Bxy + Cy^2 +Dxz + Eyz + Fz^2 = 0.", "9776dc6a6e9ab51db380239e8858554a": "\\mathbb{Z}_{26}^n", "9776ff4be626811b4e2dad37d08e929a": "\\lambda_c", "97776977662870e1344d255b8e1171f0": "\\Phi_{01}", "97776ae3cde737ad3f1b0e4d39710c8b": "\\begin{bmatrix}v_1 & \\cdots & v_k\\end{bmatrix}.", "977773d1ae3791c298b3cbed6d34e91d": " v = -e^x \\cos y. \\!\\;", "977782615e54462c805f44648c6a14cf": "z_{0}", "97778b7fc68ccef750bfbba950d23f21": "A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\\cdots A_n B_n A_n^{-1} B_n^{-1}", "9777ffe4b504a019caa608cdf639367a": "{x_0}\\times I", "977904693121ad40dfb95c6c1ce1202d": "p_O", "977981154373c02e2852e5622c2309ff": " [x_1, \\; x_2, \\; x_3] ", "9779d9cc1bdb6cb02a13921b02664fd6": "L_{sd}", "977a30f98ebc1ed300f80c62fc48eb96": " \\rho^{ABC}=\\bigoplus_j q_j\\rho^{AB^L_j}\\otimes\\rho^{B^R_jC},", "977a5d201e5a1f9bf3f5b09f05b908e3": "P=\\frac{K}{1+Ae^{-kt}}", "977a7402ed03f5153d48d477a3788d7d": "\\mathbf{a}+\\mathbf{b}\n=(a_1+b_1)\\mathbf{e}_1\n+(a_2+b_2)\\mathbf{e}_2\n+(a_3+b_3)\\mathbf{e}_3.", "977a86721a5184dc8536c060f1b5b896": "\\mathrm{gf}_i", "977aae53584695ff934544dfff8c0f05": "\\lambda(X) f(g)={d\\over dt}f(e^{-Xt}g)|_{t=0},\\,\\, \\rho(X) f(g)={d\\over dt}f(ge^{Xt})|_{t=0}.", "977b624b1bd406e0edc19ae6fe7cfa06": " C_a and C_i ", "977bb1c869d9e389fafc030006233d93": "\\Pr[\\text{miss any min-cut}] \\le \\binom{n}{2} \\cdot \\frac{1}{n^3} = O\\left(\\frac{1}{n}\\right).", "977bbe93ef864eca82e0810de710c61e": "\n \\begin{align}\n M'(\\omega, T) &= M' \\left(\\frac{\\omega}{a_{\\rm T}} T_0\\right) \\\\\n M''(\\omega, T) &= M'' \\left(\\frac{\\omega}{a_{\\rm T}} T_0\\right) . \n \\end{align}\n ", "977c1187a2b1b28ed054fd59c214dbb6": "+4586.45''\\sin (2D-l)", "977c8b15d5bf3bbbddeecd5db68fc894": "\\mathbf{L} = \\mathbf{r} \\times \\mathbf{p}.", "977c8c9941c7a81339a6c07966a8e3a6": " \\frac{b-a}{2} (f_1 + f_2) ", "977c972281e25188865ab27e860b58e8": "\\boldsymbol{E}", "977c984ae71eccd29a3e1c4a80b00cae": " \\mathbf{F}^2 = \\mathbf{E}^2 - c^2\\mathbf{B}^2 + 2 i c\\mathbf{E} \\cdot \\mathbf{B}", "977d040a251292062be0e20c32c87eee": "dX_t=\\sqrt{2D} \\, dB_t+\\frac{1}{\\gamma}F(x)dt", "977d4921e12179724aa5f696ceca9947": " \\nabla_jw^{(n)} ", "977d699a968f4339372cdc4c9e43e774": "m\\neq 1", "977d968d25cd5f1d88f7ce2931a78b03": "z = a + bi", "977db3cf24be8462308ee7c3c05dc99a": "R_{MC}", "977e0212295f57274ec56d41b21b86ff": " U(x):= \\sum_{i\\in I} u_i(x).", "977e1ff68c9b61ea0e99ef561f7098dc": " \\left\\langle Q | \\psi_n \\right\\rangle = \\sqrt{\\frac{1}{2^n\\,n!}} \\cdot \\left(\\frac{L\\omega}{\\pi \\hbar}\\right)^{1/4} \\cdot \\exp\n\\left(- \\frac{L\\omega Q^2}{2 \\hbar} \\right) \\cdot H_n\\left(\\sqrt{\\frac{L\\omega}{\\hbar}} Q \\right) ", "977e801ced95574cd57f45b011c8fe28": "p(x) \\propto L(x) x^{-\\alpha}", "977edb677f7f892334a80fd3a2ea389f": " V_i \\, ", "977ee0905b4c703140cc8eeb6bff3220": "\\begin{align}\n(r+c)(r+c-\\alpha -\\beta )+\\alpha \\beta &=(r+c-\\alpha )(r+c)-\\beta (r+c)+\\alpha \\beta \\\\ \n &=(r+c-\\alpha )(r+c)-\\beta (r+c-\\alpha ).\n\\end{align}", "977ee4ea9a22198ab2dd2d9be78024b6": "k\\to \\infty", "977eefcdbbda4dbb35add229617df423": " m_0 \\ ", "977ef1550c309fc70ee0da67d68260f8": "S(x,y,z) = (x^2 + y^2)(\\cos\\theta)^2 - z^2 (\\sin \\theta)^2.", "977ef34b801791f40acaa63975b7c6fa": "D \\succeq 0", "977f0d4b35b9c342bc0f775ac1b3bbe7": "v_\\lambda \\in V", "977f277075bf21ed2d2da358075c8670": "\n\\frac{\\exp{{\\sum_{k=1}^1 (\\beta_n} - {\\tau_{k}})}}{1+ \\sum_{x=1}^2 \\exp{{\\sum_{k=1}^x (\\beta_n} - {\\tau_{k}})}}\n", "977f8fbc8176c68311156c4c950385f9": "\\{r\\in L\\mid [r,S]\\subseteq S\\}", "977fb58b1601f64f0ad47640b3b3f72f": "u_i \\in (0, 1]", "977fe4809108682db95b8ef282a1bc0e": "\\frac{1 \\pm \\sgn(\\cos(kr^2))}{2}\\,", "978042eef62d4548cd126c7ab9a7fa35": "(2n - 2)", "97808b76697d664ec0001dd65a5c9c80": "\\gamma,\\delta\\in\\Gamma", "9780ec26320a5c91a7bb80d4c3f6c1f9": "k2^{n+1}+1", "9781079b4f90a16dd37f689678c724ae": "T>0", "97814529dc8678dec33cf384cf5fbadd": "\\bar{X}_n = 2n", "978153321c3ebccd201b02facbe654b5": "c_\\kappa x", "97816a422ddbc58d36a15a0cf98ef3fe": "2^m - 1", "9781c230f2ad190982d3fbda43fd7f03": "y(0) = 1", "9781e1bcb4a2b77872c84e7c55a9fb40": "k = \\frac{2\\pi}{\\lambda} = \\beta", "9781e47e629f7f2581059bc2c992f14b": "\\frac{(z+1)^n - 1}{((z+1)-1)},", "97825a1283f489d294bb5f5e7fbb58a9": "ax+b\\,", "978286554928581a09469192e755999f": "\\Beta(b,c-b)\\,_2F_1(a,b;c;z) = \\int_0^1 x^{b-1} (1-x)^{c-b-1}(1-zx)^{-a} \\, dx \\qquad \\real(c) > \\real(b) > 0, ", "9783596a45b8ca75a7e80b4f61646f5d": "\\textstyle {4\\choose 3,1,0} \\ {4\\choose 2,1,1} \\ {4\\choose 1,1,2} \\ {4\\choose 0,1,3}", "97836ce5359356ccb7f7973400331950": "X_1 Z_2 Y_3 = \\begin{bmatrix}\n c_2 c_3 & - s_2 & c_2 s_3 \\\\\n s_1 s_3 + c_1 c_3 s_2 & c_1 c_2 & c_1 s_2 s_3 - c_3 s_1 \\\\\n c_3 s_1 s_2 - c_1 s_3 & c_2 s_1 & c_1 c_3 + s_1 s_2 s_3 \n\\end{bmatrix}", "9783a15df58f9e5853e3595551c8814b": "\\begin{array} {l}\nf'(x_0)=\\frac{f\\left(x_0 + h\\right) - f(x_0)}{h} -\\frac{f^{(2)}(x_0)}{2!}h - \\frac{f^{(3)}(x_0)}{3!}h^2 - \\frac{f^{(4)}(x_0)}{4!}h^3 + \\cdots\n\\end{array}", "9783a8cfac293d41c7283f931ce9db06": " \\chi^2 ", "9783bc31459ae746f9486a66b8b5783d": "E^*", "97842995850784196089061ed8d552d6": " F^*(u \\; dy^1 \\wedge \\cdots \\wedge dy^n) = (u \\circ F) \\det \\left(\\frac{\\partial F^j}{\\partial x^i}\\right) dx^1 \\wedge \\cdots \\wedge dx^n ", "9784489478af63b1429aeb52b47deeef": " i^{th} \\ ", "97845d4ef2a4e660d248be6f18c11a16": " \nG(k)=\\langle e^{ikx}\\rangle\\equiv \\int_I e^{ikx}P(x,t|x_0)dx,\n", "978496e62ff2b3901e25c06729ddb20e": "a_1+\\cdots +a_{n-1} \\leq b_1+\\cdots+b_{n-1}", "9784cb4ae923be6c1c270ddadebbc602": "\\{X_i\\}_{i=1}^n", "9784d27587b72e6a174547f749509e1d": "\\left | f^{\\prime\\prime} (p) \\right | = 0", "9784e04f9e0e04baba6c37b5fb889cbb": " T_2", "9784f125d000b29951044247061d4335": " \\boldsymbol{\\hat\\theta }\n=\\cos (\\theta) \\cos (\\varphi) \\boldsymbol{\\hat{\\imath}} +\n\\cos (\\theta) \\sin (\\varphi) \\boldsymbol{\\hat{\\jmath}}\n-\\sin (\\theta) \\boldsymbol{\\hat{k}}\n", "9785b410b56fc262c44bd6f80c7c104c": "\\oplus_m f_*(\\mathcal O_X(mK))", "9785d9fff35eed23e05549b0de2b47c7": "\\mathbf B^{-1}\\,\\!", "9785f88dbe17191b86113930a8ee61fc": "\\text{Resolution (lp/mm)} = 2^{\\text{Group} + (\\text{element}-1)/6}", "97864b0c7cf6ef226035be9c2c5b1a4f": "F_{k\\times n}", "978651d45cace6fffc59b0de3e94c699": " \\hat{H}_D = 2g_I\\mu_\\text{B}\\mu_\\text{N}\\dfrac{\\mu_0}{4\\pi}\\dfrac{\\mathbf{I}\\cdot\\mathbf{N}}{r^3}", "97867cf4ca6e5b8639b24ad50b63976a": "P(h_j=1|v) = \\sigma(b_j + \\sum_{i=1}^m w_{i,j} v_i)", "978692a8399db421c7cd4faef17542b0": " d_{\\sigma y} = 2 \\sqrt{2} \\left( \\langle x^2 \\rangle + \\langle y^2 \\rangle - \\gamma \\left( \\left( \\langle x^2 \\rangle - \\langle y^2 \\rangle \\right)^2 + 4 \\langle xy \\rangle^2 \\right)^{1/2} \\right)^{1/2}. ", "978695b9152382a40907e3e448cb6186": "(P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2 \\or P_3^0) \\and (P_1^1 \\or P_2^2)", "97871c8eeab02628e2ea14968d749aa7": "h\\geq D_F", "978733d1b2e5a357f80b5da224005ea3": "f(a+b,c) = f(a,c) + f(b,c),\\quad f(a,b+c) = f(a,b) + f(a,c),", "97873c9c4595a6778c42cc3088a0515c": "\\begin{pmatrix}\n1 & 0\\\\\n0 & 1\n\\end{pmatrix}", "9787416d5668eaac5d02325e2b07b4b2": "f_i\\ ", "978746193f88641b61ec1be47600176b": "81.7\\pm 0.2", "97876e3dea655e092681631672ae1c5b": "H^i(X, \\mathcal{F}(n))", "978777a751420fc5f9484aa587bb0f91": "\n\\frac{P_A } {1 - P_A } = e^{v_A }\n", "97882d5076ea2f428ff5ab24f488ccc8": "\nh(x)=x^{\\{m\\}'}\\left(H+L(\\alpha)\\right)x^{\\{m\\}}\n", "97882e9e71b69c61badb7836a3da4af9": "\n{\\partial\\rho{\\bold u}\\over\\partial t}+\n\\nabla\\cdot(\\rho \\bold u)\\bold u+\\nabla p=0\n", "97883e8e420876a7f28b93f5c60b71e4": "10\\rightarrow (1,2)_{1\\over 2}\\oplus (1,2)_{-{1\\over 2}}\\oplus (3,1)_{-{1\\over 3}}\\oplus (\\bar{3},1)_{1\\over 3}", "9788bb8366af5c3555fac66d34d07e1a": "{1\\over 2|\\vec{k}_a|}", "9788dccf19b398cd01175bb0b5370f23": "nth", "97895a118078290a649ec60e7381ce0e": "\\frac{7\\pi}4\\!", "97897e674bce30faa98fa90de6390010": "=\\sum P (D, \\sigma |T,M)", "9789e9f64e9b7325d4de4259781d75cc": "\\{0, 5, 7, 13, 16, 17, 31\\}", "978a20c8d1e7538db879db3f4556292c": " + \\int_{\\partial \\Omega} |K(z_n,w)|\\cdot |\\varphi_n(w) -\\varphi(w)|\\, |dw|.", "978a3f3a6f1cd4f9a888fce4afa5ceda": "C=\\sum a_iC_i", "978a634e826ea674bdb07a5605bd5adb": "\\sum_{r=0}^{\\infty }{a_{r}(r+c)(r+c-1)s^{r+c+1}}-\\sum_{r=0}^{\\infty }{a_{r}(r+c)(r+c-1)x^{r+c}} +(2-\\gamma )\\sum_{r=0}^{\\infty }{a_{r}(r+c)s^{r+c+1}}+(\\alpha +\\beta -1)\\sum_{r=0}^{\\infty }{a_{r}(r+c)s^{r+c}}-\\alpha \\beta \\sum_{r=0}^{\\infty }{a_{r}s^{r+c}}=0.", "978a75697ff761a24de554edf6e628ea": "\\mathbf I_K\\,\\!", "978ada6e4ca428a8f42ca5cbc2637395": "\\alpha=\\frac{1}{1 + \\beta^2}", "978b1233be00a5400227f282c3f3e116": "f(x)=x+b", "978b2e917bc60bf8115579cd3b5173cf": "\\mathcal{S} ", "978b697963cfc5abfced8a4b611977c9": "An", "978b7883b0a0ba70b5c4bc19f43e2051": "\\operatorname{Out}(A_n)=S_n/A_n=C_2", "978bd9cceb9a35644ba67d19aab36a1a": "M(n,b,z)", "978c16472ffacca952849d8fbeae0572": "\\sigma_k=\\gamma_k\\gamma_0", "978c2d60d7b2d6f0f2acf458bcdb1d5b": "t_0 = \\frac{1}{\\sqrt{3}}", "978c39a0646c14ec42699e5ed609226c": "\n\\langle\\chi_{k'}|\\big(P_{A\\alpha}\\chi_k\\big)\\rangle_{(\\mathbf{r})}\n=\n\\frac{\\langle\\chi_{k'}\n|\\big[P_{A\\alpha}, H_\\mathrm{e}\\big] |\n\\chi_k\\rangle_{(\\mathbf{r})}} {E_{k}(\\mathbf{R})- E_{k'}(\\mathbf{R})}.\n", "978c640255fe1b451b60436c52baeace": "A - I = \\begin{bmatrix} 2 & 2 & 6 \\\\ 2 & 1 & 5 \\\\ -2 & -1 & -5 \\end{bmatrix}, \\qquad A + I = \\begin{bmatrix} 4 & 2 & 6 \\\\ 2 & 3 & 5 \\\\ -2 & -1 & -3 \\end{bmatrix}", "978ca157d1e51d60b46b2932e5bc210c": "m = \\sqrt{\\frac{hc}{2G}} = \\sqrt{\\frac{\\pi c \\hbar}{G}}", "978ca288152f056d28f99040eb86869d": "\\mathcal L(K):= (\\mathcal P,\\mathcal Z, \\in)", "978d430f7500bf5d68c3a78d54c4db2f": "\\epsilon = \\cfrac{\\partial{F}}{\\partial{X}} - 1", "978de55b5779e684b0441acb8a886bc8": "F_i= F_L^2/ (0.5 \\times\\rho \\times V^2 \\times e \\times b^2 )", "978e05a63f7235615c84f6bf74e04385": "\\mathrm{Hom} (\\varinjlim X_i, Y) = \\varprojlim \\mathrm{Hom} (X_i, Y).", "978e2084c62c53e6a0053355318a1cda": "\n\\ell _R((R/P)\\otimes(R/Q)) < \\infty.\n", "978e2d9c5ccfd83b1f4a5c2eada4bb8b": "\\frac {de} {dt}=-e^2", "978e3c42fc2033ba57c0c932f50cb1a6": " x_n = \\frac{1}{k_{n+1} + \\frac{1}{k_{n+2} + \\cdots}} .", "978e76e8e42121d1a6da2041f0400374": "g\\otimes e^{-in\\sigma}", "978ea309aa4e39a076d802a96bb62910": "\\alpha_3:=P", "978ea3cbc34f6be8c911710d1e5b7d26": "\\displaystyle{(Au,u)=(Tu,Tu)=\\|Tu\\|^2\\ge 0,}", "978ead696c16f10bf65cb25a6a9577a0": " t(x) = \\frac{1}{2}\\left[ E - \\sin( E ) \\right]. ", "978eba8c595dc7b52d06260a27dea579": "M_2 ", "978ecd1d4f9e0c416935eacb866d443d": "\\left \\{ \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} \\right \\}", "978ed184263956989074ecf8f964d299": "E\\left(r_j\\right) = r_f + b_{j1}F_1 + b_{j2}F_2 + ... + b_{jn}F_n + \\epsilon_j", "978ee20afcb9cb4c8d2f54f6ad38e8aa": "\n\\vec y = \\vec f(\\vec x) \\,\n", "978f35665f8799ce2460497b64427f29": "\\left(x - r_1\\right) \\left( x - r_2 \\right) = x^2 - \\left( r_1 + r_2 \\right) x + r_1 \\ r_2 = 0. ", "978f3febc823c1146b63996ed4ace4be": "(a \\cdot a)(b \\cdot b) - (a \\cdot b)^2 = (a \\wedge b) \\cdot (a \\wedge b).", "978f74f21757c04d61a67b67ced5fd61": "\\lambda x\\!:\\!\\sigma.\\lambda y\\!:\\!\\tau.x\\!:\\!\\sigma \\to \\tau \\to \\sigma", "978fbdc18b8ba708a554794fddf13d56": "\\tilde{H} = \\sqrt{det (q)} H", "978fec0a739a01ea26dcb1283fc0586d": "d_X \\colon X \\times X \\rightarrow \\mathbf R", "97901b0fc03418a2df024089824a6418": "\nK(x-y,\\Tau) = e^{-\\alpha \\Tau} \\int_{x(0)=x}^{x(\\Tau)=y} e^{-L}\n\\,", "97906a33387a9f4460c2a47ed50375f9": "= \\operatorname{E}\\!\\left[\\operatorname{cov}[X,Y\\mid Z]] + \\operatorname{E}[\\operatorname{E}[X\\mid Z]\\operatorname{E}[Y\\mid Z]\\right] - \\operatorname{E}[\\operatorname{E}[X\\mid Z]]\\operatorname{E}[\\operatorname{E}[Y\\mid Z]]", "9790aea8b2abf077c26f5a6cfc3bad7f": "\\begin{align} \\\\\nR_X(\\theta) =\n\\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos \\theta & -\\sin \\theta \\\\\n0 & \\sin \\theta & \\cos \\theta\n\\end{bmatrix} \n\\end{align}\n", "979141404ec5c47a99de1e8f02d16078": "\n N(\\phi) = \\frac{a}{\\sqrt{1-e^2\\sin^2 \\phi }},\n", "97914dd616d141e36d07ce78cd04c043": "\\{x_1,x_2,\\dots,x_n\\}", "97919d131faad76c3fd0ef3549dcf2e0": "c_n = \\sum_{k=0}^n a_k b_{n-k}.", "9791d4880802417b60e346eedf6145dc": "\\mathbf{S_z} = \\pm \\frac{1}{2}\\hbar", "97931ba8a3541d055c0c022a6a4a6cf9": "\\dot{\\mathbf{x}}(t) = A(t) \\mathbf{x}(t) + B(t) \\mathbf{u}(t) + \\mathbf{v}(t)", "97933019e409c35aa16f864a8a589d79": "\\mu(\\mathbf{x}_w^{(k)}, \\sigma_I^{k}, \\sigma_D)", "979334216b2aa48a1bee1336269f9ddf": "\\begin{align}\n \\nabla \\cdot \\mathbf{E} \\;&=\\; 0\\\\\n \\nabla \\cdot \\mathbf{B} \\;&=\\; 0\\\\\n\\end{align}", "9793420ba5ccfbc7fc95eae49e964258": "\\scriptstyle \\mathrm{det}\\left ( \\lambda I - T \\right ) = {\\prod}_i \\left ( \\lambda - T_{ii} \\right )", "979363cf768ee85125a84666f818befb": "\\tilde{v}", "97936ed8a0dcaca03d318453fb55d2b1": "a_i^2 \\mod N = \\prod_{j=1}^m b_j^{e_{ij}}", "97936f62cc8f145b18c5e5b89023445b": "\\mathrm{SINR}(x_i) {{=}} \\frac{|x_i|^{\\alpha}F_i}{\\sum_{j\\neq i} |x_j|^{\\alpha}F_j +N} ", "9793a168acf0018e5eed7d6f886d91f0": "\\omega_N \\,", "9793c58e72689eb9ba1593a516da95c3": "\\Phi\\rightarrow[0, 1]", "9793d5e6a698defabeaecf03bf99a974": "\nz = \\frac{1}{2} \\left(\\tau^{2} - \\sigma^{2} \\right)\n", "9793f15d7269f6d44a84d51282199c39": "\\alpha^{p}=X", "979405ba4f9c9d7eecdc88fc92da2bae": "S\\to J\\left(S\\right)", "97941ae162e01790b823e2551f4ad0c1": "a_i, b_j, c_k,...", "97942e48e2d0ea11be690460fdc91576": "c_L(s, x)", "979465dbe74b2a1610f23c033b5ac308": "F_{SE}", "97947778d9d902dc24568a86843530cd": "\\beta W", "97951bb192ea7b741a5f6f32879c209a": "P_{t,k}(x)", "97954d64f3072ccdff1ffb5ff8087c45": "T_m = -\\frac{\\Delta G^\\circ}{R\\ln\\frac{[AB]_{initial}}{2}}", "97955880ee942b07056a85675644335c": "\\mathcal{M}=", "979589cb03f599a60817b6db493e1779": "{\\beta} (3)", "97958f7b92529e554949c6adf3c69d65": "k(\\mathfrak{p})=R_\\mathfrak{p}/\\mathfrak{p}R_\\mathfrak{p}", "979594a234a797da62c460fa91d8d705": "\n\\begin{align}\n&\\sum (\\text{observed value} - \\text{fitted value})^2 & & \\text{(error)} \\\\\n&\\qquad = \\sum (\\text{observed value} - \\text{local average})^2 & & \\text{(pure error)} \\\\\n& {} \\qquad\\qquad {} + \\sum \\text{weight}\\times (\\text{local average} - \\text{fitted value})^2. & & \\text{(lack of fit)}\n\\end{align}\n", "9795fa3223a8e8fcb7d1a60a46ee0b89": "\\ C_1^2 (3)=\\frac{17}{50}", "97960734f6d1caf54110d2cd7e1dd6db": "\\frac{P^0_n(\\sin\\theta) }{r^{n+1}} \\quad n=0,1,2,\\dots", "979643882bc1090d1da4df1662a844b9": "\\int_{a}^{b} \\alpha f(t) + \\beta g(t) \\, \\mathrm{d} t = \\alpha \\int_{a}^{b} f(t) \\, \\mathrm{d} t + \\beta \\int_{a}^{b} g(t) \\, \\mathrm{d} t.", "979693143d72e50b44bb48b251ace18b": "f(V_i)\\subseteq W_i", "9797507116a4be06ed15ed9f5bfd6e2b": "f: Z \\to X", "979754db05397d414b1d66d3bd12b7be": "i \\in [n]", "97976f791a5bf77fe1e7943ee6e8598b": "S_{32} = \\frac{b_3}{a_2} = \\frac{V_3^-}{V_2^+}\\,", "97979219dac3618595f1fefc30f9b59a": "c_1 \\mathbf{v}_1+\\cdots+c_n \\mathbf{v}_n.", "979796fb558b15a9e966b64287d6aae0": " \\omega_{z} > \\omega_{xy}", "9797c1e237a9e3288b22227b6075c20b": "\\begin{align}dy_{\\text{1}}\\ =\\ (-ky_{\\text{1}}-wy_{\\text{3}}+vy_{\\text{1}}+I_{\\text{1}})dt\\ +\\ cdW_{\\text{1}}\\\\\ndy_{\\text{2}}\\ =\\ (-ky_{\\text{2}}-wy_{\\text{3}}+vy_{\\text{2}}+I_{\\text{2}})dt\\ +\\ cdW_{\\text{2}}\\\\ dy_{\\text{3}}\\ =\\ (-k_{\\text{inh}}y_{\\text{3}}+w'(y_{\\text{1}}+y_{\\text{2}}))dt\\end{align}", "9797eb4302cf303461e49b7c414a575c": "\\rho_i = \\alpha_{j_i} / \\alpha_i.", "979855991124227d24704fc6f856bcd4": " Y \\subseteq R^{n_y}.", "97989dbe102ef2f584d203c87954f7c3": "q = \\sqrt[3]{2}(\\cos \\theta + \\epsilon \\sin \\theta) \\qquad \\theta = 20^\\circ, 140^\\circ, 260^\\circ", "9798b277d88fbcca65e3c637cc759ba7": "W_{1-i}", "9798cd28a0308dea4e08b1cce1de60fa": "\\kappa(s) = \\|\\mathbf{T}'(s)\\| = \\|\\gamma''(s)\\|.", "97992ba2b1392c91dd95aac2b54510fe": "L[u] = \\sum_{i,\\ j =1}^n a_{i,j} \\frac {\\partial ^2 u }{\\partial x_i \\partial x_j} + \\sum_{i=1}^n b_i \\frac {\\partial u}{\\partial x_i} +c u ", "97997fb12b5fa44b96c8de2fb7db7211": "\\sigma_X^2+\\sigma_Y^2", "9799a3be72a4186d9eec44f02be76a6e": "T_\\frac{1}{2} = \\tau \\cdot \\ln 2 ", "9799b123a9becee7b161373ce30481d1": "\\frac{\\partial T}{\\partial t} + u\\frac{\\partial T}{\\partial x} - K_T h = 0.", "9799bfc0a6cf76045eff2bed25ce9ff9": "c=2\\pi r_s\\,\\!", "9799f1825fa2c64c128550081469dead": " (1 - \\varepsilon) \\mu(B_k) = \\int (1 - \\varepsilon) 1_{B_k} \\, d \\mu \\leq \\int f_k \\, d \\mu ", "979a12ff45cc159d3fb52a20d4324cdc": "\\approx m_0 c^2 \\left( (1 - (-\\begin{matrix} \\frac{1}{2} \\end{matrix} )v^2/c^2) - 1 \\right) \\ ", "979a32c9f74e05a8f78e6cbbc0993b1b": "\\mathbf\\Sigma_0", "979a383497d35f4ad6d88d8da9931ff0": "|jm\\rangle", "979a3c10eabf7ca3f6f5d7e208590918": "{\\mathfrak M}:= ({\\mathcal P},{\\mathcal Z};\\parallel_+,\\parallel_-,\\in)", "979a759189e7683cefee65b6b7fb1845": "\\gamma = \\frac{\\sum_i \\gamma_i c_{V,i} }{ \\sum_i c_{V,i} }, ", "979ae89de616268e6ef0bfc39358b8f2": "E_{m,Na} = \\frac{RT}{F} \\ln{ \\left( \\frac{ P_{Na^{+}}[Na^{+}]_\\mathrm{out}}{ P_{Na^{+}}[Na^{+}]_\\mathrm{in}} \\right) }=\\frac{RT}{F} \\ln{ \\left( \\frac{ [Na^{+}]_\\mathrm{out}}{ [Na^{+}]_\\mathrm{in}} \\right) }", "979b37724fa019cdab3846345e878036": "\\varepsilon_{i_1 \\dots i_n} \\varepsilon^{j_1 \\dots j_n} = n! \\delta_{[ i_1}{}^{j_1} \\dots \\delta_{i_n ]}{}^{j_n} = \\delta^{j_1 \\dots j_n}_{i_1 \\dots i_n} ", "979b59d08d6bbd59f46773ff2fa828de": " \\delta W_{g,p} = \\rho_s g t\\left(2b\\delta h + b^2 \\text{cos}\\left(\\alpha\\right)\\delta\\alpha\\right) ", "979b60fb7e15ae8f15f7cea541939fd2": "\\Delta E = {\\beta\\over 2}(j(j+1) - l(l+1) -s(s+1))", "979bdf6e2751c330e20b80aa2006666b": "\\Phi(x_1, x_2, \\ldots, x_m)=\\mbox{min}_{1\\leq i \\leq m}\\Phi_i(x_i)", "979c31f0e8f3a2e5d1ae46538509a328": "\\scriptstyle w_0\\left(n-\\frac{N}{2}\\right),\\ 0\\le n \\le N-1,", "979c6118a95454e71931222aa64968d8": "\\Phi [\\gamma] = \\hat{O}' \\Psi [\\gamma] \\qquad Eq \\; 2", "979c82904d305769e0f1b5442e272905": "{\\displaystyle dx_2/dt=0}", "979cf961e21223c19ee2e5ca8e799fca": " \\langle r_{\\alpha} | \\alpha < \\beta \\rangle ", "979d5bde8b31adc74a891c620afbf443": "\\scriptstyle \\equiv", "979d645d86231d69951b7187d30fffd4": "G = f^{64}(4)\\, ", "979d66b375e8918dc524340fcab7dbd5": "F = \\bold{E} + Ic\\bold{B} = E^k\\gamma_k\\gamma_0 -c(B^1\\gamma_2\\gamma_3 + B^2\\gamma_3\\gamma_1 + B^3\\gamma_1\\gamma_2),", "979d711839a59e96dd9df804333b76d1": "\nA^T A = I\n\\,", "979d8f047488169ba9597bea97011d07": "\\frac{1}{s_n}\\sum_{i = 1}^n \\left( X_i - \\mu_i \\right)", "979e4178bdbfb7f4e7f1bbef5a44805f": "{}+6\\kappa(\\kappa_2(X\\mid Y),\\kappa_1(X\\mid Y),\\kappa_1(X\\mid Y))+\\kappa_4(\\kappa_1(X\\mid Y))\\,", "979e5e760de455e64d451fbb27a0a088": "0.276486 \\pm 0.000008", "979ee2f2594441ab442727fcecd805b1": "E_0=\\{2\\}, E_{i+1}=E_i\\times E_i", "979ef09a688d48c8f5e1c0ec04c45e9c": " \\lambda = \\frac{hc}{E_\\text{i} - E_\\text{f}}. ", "979f09a544cb83831c27a0b2d7fb6391": "<1.5\\times10^{-6}", "979f339fd19030903b7ac05fdc462829": "c=\\frac{3RT_c}{8P_c}-V_c", "979fa249623ed03f4b4e9f0f4d1668a0": "x=[5%,60%,20%,10%,5%]", "97a00054d5341d1d7baf2df0bfbd9014": "\\|\\alpha\\|_2=\\left(\\sum_i\\alpha_i^2\\right)^{1/2}", "97a0240881784a23b9d5e3594e555b69": "f_j = x_j, j=1,\\dots,n", "97a0623f960595b23cb3ab36d1e3b32f": "\\int \\frac{ R}{x}\\,dx= R+ \\frac{b}{2} \\int \\frac{dx}{ R}+c \\int \\frac{dx}{x R}", "97a0926193a5198cc6b80bed3cdefe2f": "\\{x\\in V \\mid f(x,y) = 0 \\mbox{ for all } y \\in V\\}", "97a0ad879449aefde8178a5aba997234": "{\\rm Fructosamine} = ({\\rm HbA1c} - 1.61) \\times {\\rm 58.82}", "97a11669cba3ba9ccdfaa329a537ddd2": "R = \\begin{bmatrix} \\cos \\theta +u_x^2 \\left(1-\\cos \\theta\\right) & u_x u_y \\left(1-\\cos \\theta\\right) - u_z \\sin \\theta & u_x u_z \\left(1-\\cos \\theta\\right) + u_y \\sin \\theta \\\\ u_y u_x \\left(1-\\cos \\theta\\right) + u_z \\sin \\theta & \\cos \\theta + u_y^2\\left(1-\\cos \\theta\\right) & u_y u_z \\left(1-\\cos \\theta\\right) - u_x \\sin \\theta \\\\ u_z u_x \\left(1-\\cos \\theta\\right) - u_y \\sin \\theta & u_z u_y \\left(1-\\cos \\theta\\right) + u_x \\sin \\theta & \\cos \\theta + u_z^2\\left(1-\\cos \\theta\\right) \n\\end{bmatrix}.\n", "97a11f58cce00e0c4c82804bd2e311f0": "O^+ + H_2O \\rightarrow H_2O^+ + O", "97a13b29d1d629c3c236c6ba8fe3f517": "E^r_{lm}", "97a1684d3f0c68df51d3bc243e283015": "\\frac{1}{2} r^2 (\\theta - \\sin{\\theta}).", "97a1ad841a6267b4ce1e26562877a006": "y\\in F_m", "97a1fad7863e4c3f12065b753f20fd4a": "\\Delta S_v", "97a206a384e7fdb3739ef5b7db178153": "\\displaystyle (b)", "97a2136ef5da436bb3ddf579f640b366": "(1-t)^{-d} = \\sum_0^\\infty \\binom{d-1+j}{d-1} t^j", "97a22e7459b8f9fd84e60da787562af3": "\\frac{x^2}{1+c}+\\frac{y^2}{c}=1.", "97a2304bd62e943474311ac42219a44d": "\\tau_M(\\rho:\\mu)", "97a24e5d0034dfdd08129a2bbc6cb112": "T(\\mathbf{x}) = \\sum_{|\\alpha| \\ge 0}^{}\\frac{(\\mathbf{x}-\\mathbf{a})^{\\alpha}}{\\alpha !}\\,({\\mathrm{\\partial}^{\\alpha}}\\,f)(\\mathbf{a})\\,,", "97a28e3771468be3677ca4a80e2b12b6": "\\begin{smallmatrix}\\frac {h} {{h}_{\\odot}}={\\left( \\frac{{{T}_{\\odot}}_{\\rm eff}} {{T}_{\\rm eff}} \\right)^2} *\\frac \\sqrt{L} {a}\\end{smallmatrix}", "97a2f731890b77b619eb4183a7a7bcbb": "2.28\\sigma", "97a39e6c460c663651aa4d28744e6d4f": "F \\approx \\frac{k_B T}{L_p} \\left ( \\frac{1}{4 \\left( 1- \\frac{r}{L_c} \\right )^2} - \\frac{1}{4} + \\frac{r}{L_c} \\right ) \\,", "97a3b585df74e380222bdf1c85430fe0": "\n2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\sin i\\ \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right) \\ \\hat{g}\n", "97a3f24b9cfafa40e5b638981b6d1db9": "e^{\\mu z + \\delta (\\gamma - \\sqrt{\\alpha^2 -(\\beta +z)^2})}", "97a46948cbff959e32e9a62e618795c2": "L' = CR_0^2 \\!\\,", "97a488446a7d34389b1897c5d1144e72": "m\\rightarrow0", "97a48a5378586e1f7a695080503fb3b7": "f_x = 0", "97a4970daf5f3336f5e6675e74afdae1": " T=T_1 + T_2 + T_3 + \\cdots ", "97a4c8715e09a8d59f4f369632f3bc8b": "\n\\begin{matrix}\nI_p & = & A^2 + B^2, \\\\\nQ & = & (A^2-B^2)\\cos(2\\theta), \\\\\nU & = & (A^2-B^2)\\sin(2\\theta), \\\\\nV & = & 2ABh. \\\\\n\\end{matrix}\n", "97a4ecf5ad7c01b4dd79b095ddf0d0f0": "p_2 = 1", "97a4f0326c7b6cb9ba49251d6208b0cf": "J = \\int_{0}^\\infty \\left( x^T Q x + u^T R u \\right) dt", "97a518e874b53b61503cfb9c29a8de32": "\\dfrac{ \\partial u }{ \\partial y } = -\\dfrac{ \\partial v }{ \\partial x }", "97a521a45e396d50abb3435c8e957b1d": "\\mathrm{TT}", "97a569fa895f4c1150cfe040a1945fac": "\\varphi \\in X^*", "97a57aeac0ea9a420bb0073dfae0cc60": "aX\\,", "97a5a5df83e70235caa59811a60cdf4d": "\\frac{1-\\alpha}{N}", "97a5f77fae7814b953141db4690c1005": "\\frac{3\\pi}2\\!", "97a643f6105d0247b2ff45257e8e7767": "\\partial M=\\Sigma^*_0\\cup\\Sigma_1", "97a652c07e75d5f01faa63c6069a8ecb": "\\mathrm{nil}_\\alpha = \\mathrm{roll}\\ (\\mathrm{inl}\\ \\langle\\rangle)", "97a68a586a3cec30b66fa42029f71a9b": " \\langle s \\mid t \\rangle = \\int_X \\langle s(x) \\mid t(x) \\rangle d \\mu(x) ", "97a6f4c0571828a9f87cf7a7ef8e3c17": "T_G(x,y)=\\sum\\nolimits_{A\\subseteq E} (x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|},", "97a70e846d329113516012c4bf5f83f6": "\\frac{N_0}{2} ", "97a753267492f1b4b8769b4911277ae1": " \\rho = \\frac{m}{V},", "97a75e2cc304615f7763238a4db3db37": "\\nu:\\mathbf{P}^1\\to\\mathbf{P}^3", "97a75fdcad3c853c5e7151f7595b8837": "\\textstyle\\tilde{\\sigma^2}", "97a77754529e2af8dea093356e9b36f5": "T^2=I", "97a78673570ee368628928757eb7aa3d": "x_1 \\,", "97a7bb44057d47ace160491796fd269f": " \\textstyle \\nu=(D_2-D_1)\\frac{\\partial N_2}{\\partial x}", "97a7fb954381014dcef51222f447efef": "\\psi(\\psi(0)) = \\omega^{\\omega^2}", "97a7fe51e82d771beb904c1cd5a27f2c": "b>a=c", "97a81fb8fb85ac2ce37b09b7d415b064": " \\oint_S \\mathbf{H} \\cdot {\\rm d}\\mathbf{l}= I + I_d,\\,\\!", "97a842333acfac8d0139b68cc444f95a": "(gate6\\vee \\overline{gate8}\\vee gate7)\\wedge (gate8) = 1", "97a8a6e6133d49e04e6560f46776d11a": "{-{{\\overrightarrow{V_g} \\cdot \\nabla}{\\partial \\over \\partial p}({{f_o^2 \\over \\sigma}{\\partial \\Phi \\over \\partial p}})}-{{f_o^2 \\over \\sigma}{\\partial \\overrightarrow{V_g} \\over \\partial p}{\\cdot \\nabla}{\\partial \\Phi \\over \\partial p}}}", "97a8d7e2d1058b3a64eac199e2b663cc": "\\frac{d}{dt} \\left .{\\!\\!\\frac{}{}}\\right|_{t=t_1} (F^*_{t,t_0} Y_t)_p = \\left( F^*_{t_1,t_0} \\left( [X_{t_1},Y_{t_1}] + \\frac{d}{dt} \\left .{\\!\\!\\frac{}{}}\\right|_{t=t_1} Y_t \\right) \\right)_p", "97a8d823ec3ef0178b7bf22fbafadb78": "\\color{OliveGreen}\\mathbf{P}\\cdot\\nabla^{2}\\mathbf{Q}-\\mathbf{Q}\\cdot\\nabla^{2}\\mathbf{P}=\\nabla\\cdot\\left[\\mathbf{P}\\left(\\nabla\\cdot\\mathbf{Q}\\right)-\\mathbf{Q}\\left(\\nabla\\cdot\\mathbf{P}\\right)+\\mathbf{P}\\times\\nabla\\times\\mathbf{Q}-\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}\\right].", "97a8e38e00af840f933636c93bc46bf9": "f[x_0,\\dots,x_n] = f(0)\\cdot p_0[x_0,\\dots,x_n] + f'(0)\\cdot p_1[x_0,\\dots,x_n] + \\frac{f''(0)}{2!}\\cdot p_2[x_0,\\dots,x_n] + \\frac{f'''(0)}{3!}\\cdot p_3[x_0,\\dots,x_n] + \\dots ", "97a8e897ce6fedf2eff071e6386c9c77": "\\lambda_n = p_1 + \\cdots + p_n.\\,", "97a9005a4511ee782021fb2d8d44464d": "{\\int}_{\\ominus\\alpha} \\mathbf{F} d(\\ominus\\alpha)=\n-{\\int}_{\\alpha}\\mathbf{F} d\\alpha\n", "97a96a7f942802ae6cba2ea1a205dfce": " g_2 = \\frac{m_4}{m_{2}^2} -3 = \\frac{\\tfrac{1}{n} \\sum_{i=1}^n (x_i - \\overline{x})^4}{\\left(\\tfrac{1}{n} \\sum_{i=1}^n (x_i - \\overline{x})^2\\right)^2} - 3 ", "97a98921d4024bf2fbc29582247f7b48": "f : R \\times S \\rightarrow S", "97a9a0786e12f31be38b5261e2cce672": "(f_t, U_t)", "97a9e770762404564a33647fe96cdc5e": "\\mathbf{u} \\cdot \\nabla \\mathbf{u} = \\nabla \\left( \\frac{\\|\\mathbf{u}\\|^2}{2} \\right) + \\left( \\nabla \\times \\mathbf{u} \\right) \\times \\mathbf{u}", "97aa5242104a419ab5ae2b9867b3ae1e": "\\mathbf{r}_s", "97aa5ba36f7a92fd54c7ba703e114646": " x + y + z = 0\\text{ and }\\|x\\| = \\|y\\|,\\, ", "97aa9a4a36283be590311f02e2484e2b": "\\Delta\\otimes\\Delta^* \\cong \\bigoplus_{p=0}^n \\Gamma_p \\cong \\bigoplus_{p=0}^{k-1} \\left(\\Gamma_p\\oplus\\sigma\\Gamma_p\\right)\\, \\oplus \\Gamma_k", "97aaaa3c3dc50d96b217e7d68f1179b0": "i\\gamma^\\mu D_\\mu \\psi - m\\psi = i\\gamma^\\mu \\nabla_\\mu \\psi + \\frac{3\\kappa}{8}(\\overline{\\psi}\\gamma_\\mu\\gamma^5\\psi)\\gamma^\\mu\\gamma^5\\psi - m\\psi = 0,", "97aac98623f1915b6712ac944657419c": "\\exists x:~\\neg Q(x)", "97aaf171f18f552f7938d52c3338b6e1": " \\lim_{k \\to \\infty} \\frac{k}{2^{2k - 1}} = 0, ", "97ab1b9a8c4b3bb9efc48677decfb2c0": "\n X_n\\ \\xrightarrow{p}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{d}\\ X\n ", "97ab262ec7ec329637979f295faf29d0": "\\alpha_{n-1}(a) = \\partial_n'(b)", "97ab2f736f0feeedbf70c1fbaafbef2e": "[1]p \\equiv p\\,\\!", "97ab3e47fa57a01d90d6649d215d5fca": "C=\\{e_{1},\\ldots,e_{n}\\}", "97ab71e279d0c80277070ae22cc7244c": "\\delta R= R_{\\mu\\nu} \\delta g^{\\mu\\nu}+g_{\\mu\\nu}\\Box \\delta g^{\\mu\\nu}-\\nabla_\\mu \\nabla_\\nu \\delta g^{\\mu\\nu}", "97abdee5eb46f931241df44c91618224": "Z(s) = R_1 + \\frac {R_2R_3}{R_2 + R_3} \\ .", "97ac38803ae8998f68766b9f320a9d33": "L \\in \\mbox{PCP}(O(\\log n),O(1))", "97ac4867279e6ccc2f9999a0976166d6": "\\frac{| E |}{| \\pi_{j} (E) |}", "97ac6331242d8e06b913d518fee5cc58": "M,s \\models p", "97ac820eebd0e026bee42fdb68f50b54": " \\begin{align} \\mathbf{A} & = {1 \\over 2} \\Big | \\sum_{i=1}^{n-1} x_iy_{i+1} + x_ny_1 - \\sum_{i=1}^{n-1} x_{i+1}y_i - x_1y_n \\Big | \\\\\n & = {1 \\over 2}|x_1y_2 + x_2y_3 + \\cdots + x_{n-1}y_n + x_ny_1 - x_2y_1 - x_3y_2 - \\cdots - x_ny_{n-1} - x_1y_n| \\\\ \\end{align} ", "97ac89aa1e44897bbe1f2b0ada994489": "\\mathfrak{so}_{2n+1},", "97ace2d262dc17ce912b30505fe5102b": "\\pi_{j}(f) = f(j)", "97ad10c312b8586d5e8c8aa79d637343": "L_{0, 1} = [S _ 0 N(d _ 1) - X e ^ {-rn/365}N(d _ 2)] e ^{rn/365} ", "97ad1aeed25dfbfcfeec1596557c8ae7": "{d(x,y)}/{2} + \\varepsilon", "97ad2d88b9a2ba5526cac082a59abd08": "2 (B - 1) (B^{P-1}) (U - L + 1) + 1", "97ad70785c9a03e1c865b36439d5675c": "\\mathcal{R}\\,\\!", "97ad92e717faeb10d7b6f3aa21472504": "{x^2 \\over a^2} + {y^2 \\over a^2-c^2} = 1", "97add83efa4481e4d6fec2fa35af4af3": "\\left|\\frac{a_n}{a_{n+1}}\\right| = 1+ \\frac{h}{n} + \\frac{C_n}{n^r}", "97adf73be2f4e385b3d4869609a124a6": "\\scriptstyle \\infty", "97ae29bb5fe5c097a2e7e520dce04715": "a_k=0", "97ae5980f70213c319ef2abf3c18a737": "U(t+\\bigtriangleup t,w)=U(t,w)\\exp\\bigg(|\\frac{w}{w_h}|^{-\\gamma} \\frac{|w|\\bigtriangleup t}{2Q(w)}\\bigg) \\exp \\bigg( i|\\frac{w}{w_h}|^{-\\gamma} w\\bigtriangleup t\\bigg) \\quad (1.8.a)", "97ae7fda4e60e2fd62b1fbec58ca9b96": " \\int_V \\rho(\\mathbf{r}) (\\mathbf{r}-\\mathbf{R})dV =0, ", "97ae993949ba8610083181b44ebf8489": "S_w = \\frac{V_w}{V_v} = \\frac{V_w}{V_T\\phi} = \\frac{\\theta}{\\phi}", "97aeaf34295f3a664fd8405559c96f4f": " \\mu_0 v_0 + \\mu_1 v_1 + \\cdots +\\mu_n v_n + \\mu_{n+1} v_{n+1}. ", "97aece7918ed293ca291e5567033b86c": "det(J(\\theta_{D})_{({u}_{1},{u}_{2})})>0", "97aed6f54cfb188b70234f8faad7ffa2": "E[(x_1 - x_2)^2] = E[x_1^2] - E[2x_1x_2] + E[x_2^2] = (\\sigma^2 + \\mu^2) - 2\\mu^2 + (\\sigma^2 + \\mu^2) = 2\\sigma^2", "97aeedf9f5baa45164ccda40112451f3": "\\int x^2 \\phi(x) \\, dx = \\Phi(x) - x\\phi(x) + C", "97af1a307a8f568febdd8cf31310aaa1": "\\ O(V E)", "97af311bb42194b55aa2309cbc403081": "\n\\mathrm{d} \\mathcal{H} = \\sum_i \\left( - \\frac{\\partial \\mathcal{L}}{\\partial q_i} \\mathrm{d} q_i + {\\dot q_i} \\mathrm{d} p_i \\right) - \\frac{\\partial \\mathcal{L}}{\\partial t}\\mathrm{d}t\n.", "97af777f7c019b8f2893d12388d2c154": "\\begin{align}\n\\int 2\\sin(x)\\cos(x)\\,dx &=& \\sin^2(x) + C &=& -\\cos^2(x) + 1 + C &=& -\\frac12\\cos(2x) + C\\\\\n\\int 2\\sin(x)\\cos(x)\\,dx &=& -\\cos^2(x) + C &=& \\sin^2(x) - 1 + C &=& -\\frac12\\cos(2x) + C\\\\\n\\int 2\\sin(x)\\cos(x)\\,dx &=& -\\frac12\\cos(2x) + C &=& \\sin^2(x) + C &=& -\\cos^2(x) + C\n\\end{align}", "97afb618aecc7d38fa734ce476f21bd6": "T_c \\, = \\, T_b \\left[0.584 + 0.965 \\sum {T_{c,i}} - \\left(\\sum {T_{c,i}}\\right)^2 \\right]^{-1}", "97afdfccb6d19f32e5fede34e8de513e": "\\begin{smallmatrix}\\frac{M_{Uranus}}{M_{Earth}} \\ =\\ \\frac{8.68 \\times 10^{25}}{5.97 \\times 10^{24}}\\ =\\ 14.54\\end{smallmatrix}", "97b00ad614dc50f9d910919b16d04ce4": "f(xy) = f(x)f(y)", "97b0100b4d89765e2154fa43d667f44c": "v = \\sqrt{2gh}\\,", "97b018f8caf039d0ef5b22352412c78b": "-\\sqrt{E_b} \\phi(t)", "97b0310c914b9f6303d630c8889ae502": "O(P_{r}+P_{s})", "97b071a02a4c9add0da260a0002a069a": "\\Pr \\left[ X \\leq x \\right] = \\frac{2}{\\pi} \\arcsin\\left(\\sqrt{x}\\right).\n", "97b087a052dd68a76c757a0ab1159dfe": "[T_{j},T_{j+1}]", "97b0a23186040bca713c58285ea5aea1": "d=2e", "97b113d3159e24548fdce83096cb02ff": "T(A\\lor B)=T(A)\\lor T(B),", "97b122293e1c7f2ee5c0f03dd6f632ce": "f\\in V_k,\\; m\\in\\mathbb Z", "97b1225da9ffcfd8ebed6044f2d0ac4f": "\nv_{1}(q_{1}) \\dot{q}_{1}^{2} + v_{2}(q_{2}) \\dot{q}_{2}^{2} + \\cdots + v_{s}(q_{s}) \\dot{q}_{s}^{2} =\n\\dot{\\varphi}_{1}^{2} + \\dot{\\varphi}_{2}^{2} + \\cdots + \\dot{\\varphi}_{s}^{2} = F,\n", "97b1878b4fff48782edcd42102ab6830": "SvO_2 \\,", "97b1bb6507d0a2c5e92d12832c85c4a8": "c(y)=\\frac{2\\,S_w}{(1+\\lambda)b}\\left[1-\\frac{2(1-\\lambda)}{b}y\\right],", "97b1ec143431a5e5b8863268aa16742a": "S\\cup T \\in Q", "97b2440fc248cf6e8a4f9f67825af5c7": "W^{a\\mu\\nu}", "97b2829d16ed12781a2a3d1cdc40d814": "f: R \\rightarrow R, x \n\\mapsto x^2", "97b2cb2f8af7f9a131f808260aeebb55": "\nw(n_i,g_i)=\\frac{g_i!}{n_i!(g_i-n_i)!} \\ .\n", "97b2db5bf41b6924dbab073f40b3c376": " Z_t^2 = (X_t^2-Y_t^2) + 2 X_t Y_t i = U_{A(t)} ", "97b3150b23c7c187b20ede5262396b0a": " \\tfrac{\\ln n}{n}", "97b339befbd12ef82ca654f354b81159": " \\pi(\\mathbf{x}) =\\pi (\\mathbf{x}+\\mathbf{e}_i)\\mu_i(x_i+1)/ \\lambda_i\n= \\pi( \\mathbf{x}+ \\mathbf{e}_i- \\mathbf{e}_j) \\mu_i (x_i+1) \\lambda_j /[\\lambda_i \\mu_j (x_j)]", "97b33d99f485e8ac754b8f4e2c8dd672": "H(P,Q)", "97b3946fa377838fac45de222518cacd": "rate=\\frac{-d[COOH]}{dt}=k[COOH]^2[OH]", "97b3bd3500e0a478dda562f2629dc7fd": "k=\\sec\\phi=\\cosh\\left(\\frac{y}{R}\\right)=\\cosh\\left(\\frac{2\\pi y}{W}\\right).", "97b3ec6c7aad16cf1dfbff6ac2bf3a87": "\\hat{\\alpha} = 1/N \\sum_{i = 1}^N y_i, \\hat{f_j} \\equiv 0", "97b4afa4d5f5e09855423b441e8b78c6": "V^\\prime(\\lambda)", "97b503671db9b1d321b5fbe532159e2d": "S = \\int d \\tau \\Big[ {dx \\over d \\tau} p + {dt \\over d \\tau} p_t - \\lambda (p_t + C' (x,p)) \\Big]", "97b543dd9d1990359d7c14da383514a6": "\n {D_d \\over c} { (\\vec{\\theta} - \\vec{\\beta})^2 \\over 2} \n+ {D_{ds} \\over c} { \\left[ (\\vec{\\theta} - \\vec{\\beta} ) {D_d \\over D_{ds}} \\right]^2 \\over 2}\n= {D_d D_s \\over D_{ds} } { ( \\vec{\\theta} - \\vec{\\beta} )^2 \\over 2} .\n", "97b5841740fb141019495258ab7f0581": "\n\\begin{bmatrix} x_c \\\\ y_c \\\\ z_c \\\\ w_c \\end{bmatrix} = \n \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 1 & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\\\ z \\\\ w \\end{bmatrix}\n", "97b601cb89d503308c6f5ee63277b2dd": " N_{t} = \\frac{K}{1 + \\frac{K-N_0}{N_0} e^{-rt}}", "97b68e4075e5734ea802aa8ef628662d": "x \\in R", "97b690aeb2a0e8171577444a320e983c": "(2)\\qquad \\sigma\\bar{\\sigma}+\\frac{1}{2}R_{ab}l^a l^b\\,\\hat{=}\\,0\\,,", "97b6964ca8ca9981b74f72ed64736623": "Q = I - 2 {{\\bold v}{\\bold v}^\\mathrm{T} \\over {\\bold v}^\\mathrm{T}{\\bold v}} .", "97b6b7f1ee8bb81c79627f1df4081523": "T = \\left( \\frac { R_1 / (1 + sC_1 R_1) } {R_1 / (1 + sC_1 R_1) + R_2 + 1/(sC_2)} - \\frac {R_b} {R_b + R_f } \\right) A_0 \\,", "97b704db9a3634ce8c58b44737700742": "d\\omega(X,Y)=X(\\omega(Y))-Y(\\omega(X))-\\omega([X,Y]).", "97b753db67a6b0840db1d530d1b1f9af": "d(v) \\sim 1 + \\delta v^{2}/c^2 \\,", "97b770be2db8dc817d41b047527d4988": "\\dot{x}_i=\\lambda F_{p_i},\\quad\\dot{p}_i=-\\lambda(F_{x_i}+F_up_i),\\quad \\dot{u}=\\lambda\\sum_i p_iF_{p_i}", "97b7baf1366f00792c90957f9a7a5db2": "\\; W", "97b7f011cc0d4b59ba272a3d4baaa0d7": "p_0^\\prime(t)=\\mu_1 p_1(t)-\\lambda_0 p_0(t) \\, ", "97b800d546ff5a5e2b5f67c4b3d69559": "k_F", "97b878bff2b23bd30db3f2292e421950": "y^{l+1} e^{-y^2/2}", "97b917b076fe1e0e2fb7ccc0352f4cb4": "\\displaystyle{\\partial_{n\\pm} u =(\\lambda \\mp{1\\over 2})\\varphi,}", "97b92d407627e8ce26b601684d67846d": "p_0=h/\\lambda", "97b9479b494876e121f9bb26c1551f71": " x_{0,j}=j\\Delta", "97b95774e002abd9f683123b9b310660": "U^*\\Delta U = -{d^2\\over dt^2} + 1.", "97b9634a80ae2e51cda321c611b821be": "10\\uparrow\\uparrow\\uparrow\\uparrow 2=(10\\uparrow\\uparrow)^{8}(10\\uparrow)^{10}1", "97b99f401793e43eaa1a7569780624a0": "1 \\leqslant i \\leqslant n", "97b9ffcf32003894c585e7d58149091c": "q = q_s + \\vec{q}_v.", "97ba9216c73cee1b1a9bace10ad29351": "P(p, s)", "97bacf198f4dfd3a4b03324d572f5e95": "\\pi_G(k)", "97bb161352c948de12d2db1f1ac66b07": "\\because h{(-x)}=\\frac {f{(-x)}-f{(x)}}{2}=-\\frac {f{(x)}-f{(-x)}}{2}=-h{(x)}", "97bbf01e798970a6c85b6030dacc0965": "\\vec{F} = q(\\vec{E} + \\vec{v} \\times \\vec{B})", "97bbf991a7419214d26496493cfd9b3c": "\\textstyle |\\psi_j \\rang ", "97bbfa2f7c9745628306a24502d251e4": "\\ x=0", "97bc00cc52a9d377f4e5a081679124ce": "[x,x,y] = 0", "97bc658a06a39830a2340be078effc6c": "\\scriptstyle (0,\\, 1)", "97bc6a64728d0e9e1f31d16b44747059": " \\mathrm{Res}(f, \\infty) = {-1\\over 2\\pi i}\\int_{C(0, r)} f(z) \\, dz", "97bc7ac29c580a4315027ef913ef5e5d": "\\left(\\sqrt{\\frac{2}{5}},\\ \\frac{2}{\\sqrt{6}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm1\\right)", "97bcc8713ebd34ead985c7d23efcb66d": "\\zeta(s)=\\sum_{n=1}^\\infin \\frac{1}{n^s},", "97bce59cf047e05925c5bba1fd31c2ad": "x \\rightarrow 0", "97bd089786067328322b8d831007261d": "\\sgn(x) \\lfloor |x| \\rfloor", "97bd5e34cb5345015b9d1d67708832dd": "\nA = \n\\left[\\begin{matrix}\n1 & 2 & 3\\\\\n1 & 2 & 1\\\\\n3 & 2 & 1\\\\\n\\end{matrix}\\right]\n", "97bd937333b2489581eed632d7896abd": "G^{\\hat{a}\\hat{b}} = 8 \\pi \\epsilon \\, \\left[ \\begin{matrix} 1 & 0 & 0 & \\pm 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\pm1 & 0 & 0 & 1 \\end{matrix} \\right]", "97bdae68bcf65459a80debe432959e01": "\\pi^{ab} = \\sqrt{q} (K^{ab} - q^{ab} K_c^c)", "97bdb7017d48ce7aa9f440311e792443": "\\begin{align}\n \\rho(x, y, z) &= \\frac{20B(x^2 + y^2)}{(r^2 + x^2 + y^2 + z^2)^3} \\\\\n p(x, y, z) &= \\frac{-A^2B}{(r^2 + x^2 + y^2 + z^2)^4} + \\frac{-4A^2B(x^2 + y^2)}{(r^2 + x^2 + y^2 + z^2)^5}.\n\\end{align}", "97bdbe6b1d7ca3644743f5ead84a8e90": "\\left[\\hat{b}, \\hat{b}^\\dagger \\right]_- = 1", "97bdf6569ac011ff62a154ec6a7c520c": "r(t)-tr^\\prime(t)=(\\sinh^{-1}(t)-t/\\sqrt{1+t^2},1/\\sqrt{1+t^2})", "97be0183c03ae6fcb7b50557236729f3": "\\frac{\\boldsymbol \\sigma (s)}{ds} = \\frac{\\mathbf{u}(\\boldsymbol {\\sigma}(s))}{|\\mathbf{u}(\\boldsymbol{\\sigma}(s))|}", "97be0318b2113c13a9feec7ff1d54f2c": "f(\\vec{v})=\\mathbf{z}_{\\rm{l}} \\vec{v} \\mathbf{z}_{\\rm{r}}=\n\\begin{pmatrix}\na_{\\rm{l}}&-b_{\\rm{l}}&-c_{\\rm{l}}&-d_{\\rm{l}}\\\\\nb_{\\rm{l}}&a_{\\rm{l}}&-d_{\\rm{l}}&c_{\\rm{l}}\\\\\nc_{\\rm{l}}&d_{\\rm{l}}&a_{\\rm{l}}&-b_{\\rm{l}}\\\\\nd_{\\rm{l}}&-c_{\\rm{l}}&b_{\\rm{l}}&a_{\\rm{l}}\n\\end{pmatrix}\\begin{pmatrix}\na_{\\rm{r}}&-b_{\\rm{r}}&-c_{\\rm{r}}&-d_{\\rm{r}}\\\\\nb_{\\rm{r}}&a_{\\rm{r}}&d_{\\rm{r}}&-c_{\\rm{r}}\\\\\nc_{\\rm{r}}&-d_{\\rm{r}}&a_{\\rm{r}}&b_{\\rm{r}}\\\\\nd_{\\rm{r}}&c_{\\rm{r}}&-b_{\\rm{r}}&a_{\\rm{r}}\n\\end{pmatrix}\\begin{pmatrix}\nw\\\\x\\\\y\\\\z\n\\end{pmatrix}.\n", "97be14b2a23f7f1979b035107af00427": "\\delta(\\varepsilon / 2) \\le \\delta_1(\\varepsilon) \\le \\delta(\\varepsilon), \\quad \\varepsilon \\in [0, 2].", "97be1c5cb1b4195b40cc294ca5318bbe": "C_{D,0} \\times S", "97be2be1b4e6db0560c6b0b6620c9905": "\\varepsilon_M\\,\\!", "97be2c981e281d5b5bb74fbd6c24a7c7": "\\operatorname{Var}(T) =\n\\operatorname{E}(T(T-1)) + \\operatorname{E}(T) - \\operatorname{E}(T)^2,", "97be900d7d7fd0adb6c4268325f7ea14": "X^{(0)}=\\frac{1}{\\sqrt{p}}X^{(m)}", "97beb535661763f3da2afad8b91388ba": "\\frac{d}{dt} \\Lambda(t) = -\\frac{S'(t)}{S(t)} = \\lambda(t).", "97bf0de4724b66d445304fc55fe5de8c": " {\\mathbf r}(s) ", "97bf120d0aabaf93cd0a17ba05b4c21c": "\\|A\\|_{\\ell_1 \\to \\ell_1} \\leq M", "97bf3418cf10bb90cfb0b0cf78696e00": " \\kappa(A) = 1 .\\,", "97bfb309bc2bc184ca1273b0cc42e529": "d'(f(x),f(y))\\leq k\\,d(x,y)", "97c01553b7537b5260d44a63841a08fa": "\\scriptstyle \\frac{1}{\\Gamma(\\alpha)} \\gamma(\\alpha,\\, \\beta x)", "97c017dcdce3b4e38c972a84356e3827": " p = A_4 = A_7 = p ", "97c01bd4ec5456b30fa9cf741fc5999c": "S_\\ast(B)", "97c02911bee9a9c7320be3feb38bfd96": "\\wp(z;\\tau) = \\wp(z;1,\\tau) = \\frac{1}{z^2} + \\sum_{n^2+m^2 \\ne 0}\\left\\{\n{1 \\over (z+m+n\\tau)^2} - {1 \\over (m+n\\tau)^2}\\right\\}.", "97c02fd4f6ec8e23177745a16c8dd60e": "\\ F_R ", "97c0a69c487ec1dcd88cd659d89bb697": "f(x)=\\Omega_R(g(x))\\ (x\\rightarrow\\infty)\\;\\Leftrightarrow\\;\\limsup_{x \\to \\infty} \\frac{f(x)}{g(x)}> 0", "97c0a88e9761f2992e581b5aec2900e0": "483597.9 \\times 10^9 \\,", "97c0cda8db7ecfda1190681073916f9a": "(x+y)^n=\\sum_{i=0}^n{n \\choose i}x^{n-i}y^i,", "97c1007fe726219d7524e70bd98a385a": "33\\frac{1}{3}\\%.", "97c11cd3d2198f0dc6b152f0beb9abfe": "\\rho (\\ell (a), \\ell (b)) = d(a, b).", "97c12c8eb0d7af70cf2dbbe4b927bcd8": "f(S)+f(T)\\geq f(S\\cup T)+f(S\\cap T)", "97c151194a7d6db4aa2138530ab1abd3": "s\\in\\R\\,", "97c16cf76a2e635bf5a2f5d7b18dc552": "\\mathbf{STUVWXYZ} \\!", "97c1b5d02a7abda87943726349a297b4": "i\\geq n", "97c24599066db40096cbf2f2c306cbe9": "\\vec{f}_0 = \\partial_T - \\sqrt{2m/r} \\, \\partial_r ", "97c2780226cb799c6346bd6940c22420": "f_0=\\frac{V_s}{4H}", "97c2d2dd9732f8917d11f8bbf3aa9df5": "d= \\mathop{\\mathrm{rank}} \\mathbf{X} =\\max\\{i,\\ \\mbox{such that}\\ \\lambda_i >0\\}", "97c2db4db3e59ee0ed70ecafe23209b6": "\\chi_1(\\mathbf{r};\\mathbf{R})\\,", "97c30f0b34ec8c80664588a5dd49667e": "\\varepsilon ^{ijk}", "97c36e820d8af06e4ef1e906ad31be27": "B = \\frac{\\mu_0 I_{enc}}{2L}", "97c3b76f6e6b32f1eb7172a6ebae771b": "AFR = {1 \\over {MTBF / 8760}} \\cdot 100 ", "97c3e78679782a881e5d273da9d3ede7": "\n(x, y) = (r\\cos\\theta, r\\sin\\theta)\\qquad(r, \\theta) = \\left(\\sqrt{x^2+y^2}, \\quad \\arctan\\frac{y}{x}\\right).\\,\n", "97c4108ac09aac8a32f02399725b36f0": " \\lambda_1 \\tilde x_1^2 + \\lambda_2 \\tilde x_2^2 + \\cdots + \\lambda_n \\tilde x_n^2, ", "97c41ed252a73fcc833f6fb80eb2acf2": "W=L_w+L_t", "97c444bc746c101def2ce5343e68e962": "\n -5 < \\Re(t) < 5\n", "97c4823734da9da3117047b3206924b6": "b = 2^\\beta v", "97c520b52d0ad9d2c77c3545f43de1e0": "p, q \\in \\Omega", "97c55129dc5f7c6d8b3f323f8503920f": "\\frac{1}{f} = \\left ( \\frac{{n}_\\mathrm{lens}}{n_\\mathrm{med}-1} \\right )\\left ( \\frac{1}{r_1} - \\frac{1}{r_2} \\right )\\,\\!", "97c5777039f08550df242b97d31b2abc": "\\displaystyle{\\mathfrak{t}=\\mathfrak{z} \\oplus \\mathfrak{t}_1\\oplus \\cdots \\oplus \\mathfrak{t}_m,}", "97c5904dbbccc261f47cd00c8e323288": " R_s = \\frac{1}{2} \\left[\\left(\\frac{\\cos i - \\sqrt{m^2 - \\sin^2 i}}{\\cos i + \\sqrt{m^2 - \\sin^2 i}}\\right)^2 + \\left(\\frac{m^2 \\cos i - \\sqrt{m^2 - \\sin^2 i}}{m^2 \\cos i + \\sqrt{m^2 - \\sin^2 i}}\\right)^2\\right]", "97c5bcde06c50880075a529ef87775f4": "\n\\bar{\\kappa}_{ff} = 3.68 \\times 10^{22} g_{ff}(1 - Z)(1 + X) \\frac{\\rho}{\\rm g/cm^3} \\left(\\frac{T}{\\rm K}\\right)^{-7/2} {\\rm \\, cm^2 \\, g^{-1}}.\n", "97c5ca18588652a9dfff6114ab8dc563": "\\vec r=R \\hat u_R (t)\\ ,", "97c5cf106c2eb35613f007096ce354a6": "x=1-q, y=0", "97c5f0c5e1af7a513dcf989c4815d726": "\n \\boldsymbol{N} = \\boldsymbol{P}^T = J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma} ~.\n ", "97c5ff37b0dd80b209b2fedb8999f417": "(D_\\mu F_{\\nu \\kappa})^a+(D_\\kappa F_{\\mu \\nu})^a+(D_\\nu F_{\\kappa \\mu})^a=0", "97c6348976badcbefa045b597343ada2": "\\begin{align}\n \\delta_\\Phi\\mathcal{L}\\, &=\\, \n \\mathcal{L}(\\Phi+\\delta\\Phi,\\eta)\\, -\\, \\mathcal{L}(\\Phi,\\eta) \\\\\n &=\\, -\\, \\int_{t_0}^{t_1} \\iint \\left\\{ \\int_{-h(\\boldsymbol{x})}^{\\eta(\\boldsymbol{x},t)} \n \\rho\\, \\left( \\frac{\\partial(\\delta\\Phi)}{\\partial t} \n +\\, \\boldsymbol{\\nabla}\\Phi \\cdot \\boldsymbol{\\nabla} (\\delta\\Phi) \n +\\, \\frac{\\partial\\Phi}{\\partial z}\\, \\frac{\\partial(\\delta \\Phi)}{\\partial z}\\,\n \\right)\\; \\text{d}z\\, \\right\\}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t.\n\\end{align}", "97c64a1e09b74de35c5a1d94572a677a": "(\\lambda_1 - \\mu)^{-1},...,(\\lambda_n - \\mu)^{-1}, ", "97c6b8357ba568610aa11832339a0abe": "\\Phi(\\mathbf{u})=f(u)\\exp(-(\\sqrt{k}/4)u^{2})\\,", "97c6c6107120201456f035755c101b33": " k_1 = \\frac{k_\\mathrm{B}T}{h}\\left(1-\\exp\\left(\\frac{-h\\nu}{k_BT}\\right)\\right)\\exp\\left(\\frac{-E^\\ominus}{RT}\\right) ", "97c77e72029bcc92a597dab1099282ec": " D_{\\infty} ", "97c79e14f9aea47c5cdc175ab379ef5e": " \\int x^n e^{ax} dx = \\frac{1}{a} \\left ( x^ne^{ax} - n \\int x^{n-1} e^{ax} dx \\right ). \\!", "97c7ae3665d6968f6f41241de8e84019": "\\mathcal{K}_0(x; n) = 1", "97c7d4152fcd52c0221498ea1c874990": " { \\omega \\over s^2 + \\omega^2 } ", "97c8045b79fd0457915b41c1dba6d2f1": "\\alpha^{\\frac{\\mathrm{N} \\mathfrak{p} -1}{n}}\\equiv \\zeta_n^s\\pmod{\\mathfrak{p} }\n", "97c85b60c02491513eb3bb104505d3a5": "\\mathbf{F}_\\text{loop}=\\nabla \\left(\\mathbf{m}\\cdot\\mathbf{B}\\right) ", "97c88070ba675b9e7da9db91ecd97ff7": " \\left[ \\begin{matrix} \\cosh(\\beta) & 0 & 0 & \\sinh(\\beta) \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n \\sinh(\\beta) & 0 & 0 & \\cosh(\\beta) \\end{matrix} \\right] ", "97c8f3e320a561dd91bec3b2f92816f5": "\\mathrm{Tor}_1^R (N, M) = 0", "97c930e43de466516f8ff49f1cb7a0ff": "MCHC = \\frac{Hb}{Hct}", "97c99b146eb3a5bc110141b450b51841": "\\underbrace{\n\\sum_{n=-\\infty}^{\\infty} \\overbrace{x(nT)}^{x[n]}\\ e^{-i 2\\pi f nT}\n}_{\\text{DTFT}} = \\frac{1}{T}\\sum_{k=-\\infty}^{\\infty} X(f-k/T).", "97c9a8cc09c8f888b3f1754db8076848": " { \\mathit l^*} = { \\mathit l} + { \\mathit l^{\\prime} }. ", "97c9b64aab028733581e5859e3890253": "0 \\in L_{n} (\\pi_1 (X))", "97c9d71780c285aeb033fe56aec37456": "(t, r, \\ldots)", "97c9fdf7b8b666d72dce5c55ba633f5c": "\\lim_{n\\to\\infty}k_n=\\infty", "97ca138d004ec8e161929928fa38faa8": " e_{n+1} = y_{n+1} - y^*_{n+1} = h\\sum_{i=1}^s (b_i - b^*_i) k_i, ", "97ca1c4158012fd1a6074db2416f17ea": "f^*,", "97ca1d463a64e31e62bdbc8c7eff411c": "t_1, t_2 \\in B", "97ca34151b42c559d8af79a4c0a32925": "\\text{light}", "97ca6bff251f22b8d304679e9a82ec30": " P(Presentation~is~caused~by~condition~in~individual) = \\frac {P(Presentation~WHOIFPI~by~condition)}{P(Presentation~WHOIFPI)}", "97ca729bbf3e6f7873fda156b311bf28": "\\tau_2\\Big.", "97cae0c24b4d5abf824ab8a933768c7d": " \\{0,1\\}^\\infty \\,", "97cba4485b0ff2b972515637d75346d3": "V_\\varepsilon = \\Bigl\\{ f : \\mu \\bigl(\\{x : |f(x)| > \\varepsilon \\} \\bigr) < \\varepsilon \\Bigr\\}, \\ \\ \\varepsilon > 0", "97cbb2db97c564dfb16ae672451c7388": "R_0(f) \\leq R_2(f)^2 \\log N", "97cbcad4dac99924226954775b4032db": "A=X_0\\supsetneq X_1\\supsetneq \\dots \\supsetneq X_n=0", "97cbf02e0ff69315754bed8760dd73f1": "\\gamma_1 = \\frac{2\\sqrt{\\pi}(\\pi - 3)}{(4 - \\pi)^\\frac{3}{2}} \\approx 0.631", "97cc43f3c0ec6590af19b86cc938fc4e": "\\scriptstyle \\leq2\\times10^{-42}", "97cc9223bb9df2ff6efd146c968d16f0": " (1+z)^3 T^2 + (1+z)^2 T + z = 0 \\ ", "97cc9b317008c00d93182be45c011fec": " F_{1} = \\frac{4\\pi^2 m_{1}r_{1}}{T^2} ", "97ccd8bb7b43b6bfbdb5cb857bc62260": "Rim thickness = Module \\sqrt{\\frac{N}{2N_A}},", "97cce3be638bce58e2fac6c2f22ab322": "\\boldsymbol{F} = \\boldsymbol{\\nabla}\\mathbf{u} + \\boldsymbol{I}", "97cce74dc2edc9439aaadaffd95b85a5": "j 0", "97ded4fb2fd3e14817b8273ca4f4a375": "\\scriptstyle v_n = v_1 \\times \\cdots \\times v_{n-1}", "97dede559bcb3f9e3df134ad95281eb8": "x \\cong_{\\mathcal{B}} y", "97df5c85f37d8dbd4114e0ea8a27c200": " \\frac{1}{\\sqrt{2+2u_z}}\\begin{pmatrix} 1+u_z \\\\ u_x+iu_y \\end{pmatrix}.", "97df7eeb6d94f98b0387a5d43087a35e": " C_l ", "97dfa26a64db2d64731207ea29391a00": "\\frac{\\sigma^\\delta \\alpha}{\\alpha-\\delta}", "97dfbe9d87a86dc416088d3339a31155": "\\scriptstyle {\\frac{1}{2}}\\mathrm{LSB}", "97dfc29656066ca4fe1d7df83aec41e9": "G = \\{(g,[p_g]): g\\in SO^+(3,1),[p_g]\\in \\pi_g\\}", "97dff977cd36d76fe0107f11a0592402": "x\\mapsto -x", "97e013f7e01fc5c96c2b7b0b0a3f4318": "S(x){\\textstyle{\\{0,\\ldots,\\,d-2\\}\\atop =}}E(x)=\\sum_{i=0}^{d-2}\\sum_{j=0}^{n-1}e_j\\alpha^{ij}\\alpha^{cj}x^i.", "97e04e2e3a67092555addf5c5db8d1f9": "\\Delta p_{\\text{B}}", "97e06aa80e79b84cdb59e51561b85480": "\\mathcal{O}_{[(12)]}\\;", "97e06dec0a0e7dc0db3ebcd67887c83b": "(M, \\vec{0}) = (E_1, \\vec{p}_1) + (E_2, \\vec{p}_2).\\,", "97e0a63f9b2c6786d3aa7bfe329e0acb": "(X_\\infty, d_\\infty)=\\lim_\\omega(X_n,d_n,p_n)", "97e0e618bbf9ca1590b978695d601c2e": "\\scriptstyle t^2 - x^2 = 1 ", "97e0f5946261f9e667625ca038f16cfe": "\\operatorname{atan2}(y, x)=2\\arctan \\frac{y}{\\sqrt{x^2 + y^2} + x} ", "97e14305dcb25ed6a22c924d0a5f526a": "\\hat\\mathcal{F}", "97e15b3c9994b77baf963dd97ae6fec0": "T_{1B}", "97e2b9a93221b536161c7f2ea68c24a7": " Df(a) : \\mathbf{R}^n \\to \\mathbf{R}^m \\quad\\mbox{with}\\quad Df(a)(v) = J_f(a) \\, v, ", "97e2d61ce651d652ea31731688dc855d": " h_t(x,y) ", "97e331b92a4dda6af502cad6d7a8373e": "E=\\frac{1}{16}\\rho g H_{m0}^2,", "97e33938a27111f1b25a1a9cbf7ba0f5": "f\\colon (0,1) \\to \\mathbb{R} ", "97e37fd2c6848fff732f274413aa8626": "(x)^{(n+1)} = x(x+1)\\cdots (x+n-1)(x+n)=n(x)^{(n)}+x(x)^{(n)}.", "97e38adf2e60bfe4fec75d089434da37": "\\vec{F}=-\\frac{Gm_{1}m_{2}}{r^2} \\hat{r}", "97e38d20baa43b86b15f0a31efdc1794": "\\omega_A", "97e3973c7527b668d46e9da93daa616b": "\n\\begin{align}\n\\left[\\delta K\\right]\\mathbf{x}_{0i} & + [K_0]\\delta \\mathbf{x}_i + [\\delta K]\\delta \\mathbf{x}_i \\\\[6pt]\n& = \\lambda_{0i}[M_0]\\delta\\mathbf{x}_i + \n \\lambda_{0i}[\\delta M]\\mathbf{x}_{0i} +\n \\delta\\lambda_i[M_0]\\mathbf{x}_{0i} \\\\[6pt]\n& {} + \\lambda_{0i}[\\delta M]\\delta\\mathbf{x}_i + \n \\delta\\lambda_i[\\delta M]\\mathbf{x}_{0i} +\n \\delta\\lambda_i[M_0]\\delta\\mathbf{x}_i + \n \\delta\\lambda_i[\\delta M]\\delta\\mathbf{x}_i.\n\\end{align}\n", "97e39828a7ba03c85f24c91673acac51": "\\int_a^b u(t)\\varphi'(t)dt=-\\int_a^b v(t)\\varphi(t)dt", "97e3a31c511d8f185166bf856a8ec00c": "\\begin{align}\n Q &= v_1A_1 = v_2A_2\\\\\n p_1 - p_2 &= \\frac{\\rho}{2}(v_2^2 - v_1^2)\n\\end{align}", "97e3bb234ecedb7358ddd4d52070f612": " H_o = pK_{BH^+} -log \\frac{[BH^+]}{[B]} ", "97e3c4edbdbdf96926ad8b2c5a462f40": " \\eta_{xx} + \\eta_{yy} = Q(\\xi, \\eta)", "97e47a34c88e8d4de3dace419c28c7e1": "R_3 = \\frac{R_aR_b}{R_T}", "97e4b21cca067061a60ebe466d4c71d1": "1 < d_H(J_c) < 1 + {|c|^2 \\over4\\log 2} + o(|c|^2).", "97e4c40b3024c69edb23af269f20074c": "[HG]_{eq} = \\frac{K_a[G]_o[H]_o}{1 + K_a[G]_o}", "97e4c7f0cd5906a39d1451b2241646e2": "\\left(\\tfrac{5}{17}\\right)", "97e553625db66b4010ede6b1551c44aa": "BGL(R)^+", "97e5a8b7bd7857e6669a3f1f3ee72531": "F(x,y) = F_0 e^{ikx} e^{-ky}", "97e5b259ff0d55638064983fadd0196a": "F_Y(y) = \\operatorname{P}(g(X) \\le y).", "97e5dd14c04e26e10eee99ae0fe537fb": "\\lim_{\\varepsilon\\rightarrow 0+} \\left[\\int_a^{b-\\varepsilon} f(x)\\,\\mathrm{d}x+\\int_{b+\\varepsilon}^c f(x)\\,\\mathrm{d}x\\right]", "97e5e3d4b32e120065f04aeb6178ecc0": " L = P \\sum_{j=1}^n \\frac{1}{(1+i)^j} ", "97e67efc24b4bcab690f7cdbafffec96": "y=\\phi(x)", "97e6ed30105eba21f25e2fa3cfcb5ddf": "a = 1+r\\frac{delivery term days}{currency basis}", "97e6f33494ffdac39874deb220f4800c": " P(y \\textrm{~spikes}) = \\frac{\\left(\\Delta \\lambda\\right)^y}{y!} e^{- \\Delta \\lambda}", "97e701a51acf272f7342c296443dab66": "\\Delta x_{it}", "97e71ba8ecb948a65a845277bd24da7f": "n:=|z|-1", "97e72194a917c8325cfa619e1e9e4ffd": "\\,\\Sigma^2 = \\Gamma", "97e72e862ae3dde9128ad5812ddd8ba2": "I(v) = \\Omega (|v_0|) \\ ", "97e742645577c4c066d1f73d4d4947b3": " \\phi_A (x) := x_0 + L x ", "97e75908f62064de1277f97b6ac4ae5a": "\n \\begin{cases}\n 0 & \\text{for }x<1 \\\\\n \\sum_{j=1}^i p_j & \\text{for }x \\in [i,i+1) \\\\\n 1 & \\text{for }x \\geq k\n \\end{cases}\n ", "97e76b4d31b6e4e220e894ebaf3ff36c": "\\int 2x\\, dx = x^2 + C.", "97e795d360b4df064a334a8d170e1f58": " j = 1, \\ \\dots \\ , \\ n\\ .", "97e7d9def2be8e1a68749b3e8e1c35fd": " \\wp ", "97e7e4b590f6951c31b864a322dc0746": "p(x) = f(x_0) + \\frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0). \\,\\!", "97e838589416d24e91d2af40ed66a254": "V+\\,", "97e8569797b8a67f4fd71a53f46d2938": "\\scriptstyle \\frac{I}{a\\sqrt{p}}", "97e860f10f89d3b0905b451f6c11a511": "{\\rm tr}\\left(\\mathbf{U}^{-1}\\frac{\\partial \\mathbf{U}}{\\partial x}\\right)", "97e8b2b745aecc1ee03f84d05491f951": "x\\mapsto xc", "97e8b67ec336639595f2ad7445401efe": "p^{\\frac{2}{27}n^3+O(n^{8/3})}", "97e90f541d6bb27f2edf9de172a2848e": " a_{01} = \\mathcal{L}(p_8)+p_3p_8+p_2p_9,", "97e9636851fb5424dd15a2b3a2d82269": "\\rho (\\hat{D},D)\\le\\epsilon", "97e96ded2f7fc0a978d235968bec9223": "f(z)=z^8-9/10", "97e972bd98ceacee6e5202fa86c3eeb1": "\\begin{matrix}\\frac{4}{3}\\end{matrix} n^3 + O(n^2)", "97e999c58d59e5fe1aee62ef73ba3155": "T_a\\ :\\ y^2 = x^3 + 3a(x+1)^2 ", "97e9b3ae39c85b640dc8163d270b1347": "k_{i\\perp}", "97e9cf1047b2e5eed6dc985f656f26c5": "\\mathfrak{P}^{104}", "97e9ee716d4c6ea0103c78d811330e15": "\\frac{t}{N} = \\frac{2p}{(1-p)^2}.", "97ea0f83964bc715c0904690166581ea": "\\Psi_B", "97ea2ec5c1507c511786a961fa08f7e2": "F_i=\\mu_o\\int\\limits_V {\\vec M \\cdot \\frac{\\partial \\vec H} {\\partial x_i}\\, dV}\\,\\!", "97ea5aa3c543b7e1cd052ee923d83a79": " \\mu_B", "97ea6c5c8f9021bf3dd964f9a4aa3858": "\\left(\\frac{1}{0!}+\\frac{C_1}{1!}+\\frac{C^2_1}{2!}+\\dots\\right)\\left(1+C_2+\\frac{1}{2}C^2_2+\\dots\\right)\\dots ", "97eaa2be1b39b2b72ac2b6640014b23b": "\\delta(q_s, q_c) \\to q_t", "97eaa8babe623f1fb54595b6aecbe1a9": "\\Sigma _i w_i = p \\cdot \\omega + \\Sigma _j p \\cdot y^*_j", "97eabdf2a5b15a617fd2f51d08a2aebf": "\\langle\\hat{\\phi}_j\\rangle=0", "97eb2cc15212b0a743f9954797c1c9ee": "(y\\leq x)", "97ebc86a00fcd51aa57bda155c3b910a": "\\displaystyle \\frac{\\left(2\\pi\\right)^\\alpha}{\\Gamma\\left(\\alpha\\right)}u\\left(\\pm \\xi \\right)\\left(\\pm \\xi \\right)^{\\alpha-1} ", "97ec200258268436761a9756bfe2c6cf": "t_1=3, u_1 = 1", "97ec472d900517399818bf46a3391f79": "(A.2.d)\\quad \\gamma_{,\\,z}=2\\,\\rho\\,\\psi_{,\\,\\rho}\\psi_{,\\,z} \\,.", "97ecf3b9a805295c5b483ef4056c2202": "\\ell\\le\\int_{\\gamma_\\theta}\\rho\\,ds =\\int_{r_1}^{r_2}\\rho(e^{i\\theta}r)\\,dr.", "97ed9a0b3644464ce082d43df8451834": "\\frac{1}{2}(n-1)", "97edb00e959b59649b80b1f8c6227fca": "\\int_a^b f(x)\\, d\\rho(x)", "97edd8ce9d0ec4c9754052708be11121": "\\chi_{nlm}({\\mathbf{k}})= \\int d^3r e^{i{\\mathbf{k}}\\cdot {\\mathbf{r}}} \\chi_{nlm}({\\mathbf{ r}})", "97edeacff720fb8833f21cb17a531bd4": " |n\\rang = |n^{(0)}\\rang + \\lambda |n^{(1)}\\rang + \\lambda^2 |n^{(2)}\\rang + \\cdots ", "97ee3e55416a16de63005c2f869eea9d": "\\log \\Pr(X = x) = \\sum_i \\log \\Pr(X_i = x_i|X_j = x_j\\ \\mathrm{for\\ all}\\ \\lbrace X_i,X_j \\rbrace \\in E).", "97ef813c1c962ffd4a24d3ededcd8c65": " X_1,\\dots,X_n. \\,", "97efb88931c06e1764ad100bfa005f8d": "\\bar{x},\\bar{y},\\bar{u},\\bar{v}", "97efcc6d94bd54ebb9f68598a353edcd": "y_0=y_2=0", "97efd6317559272c55248dc24515f903": "\\tilde\\delta_{ijk}", "97f037dc9b85263af18b8a942cb0efcf": "f(0) = h(0,\\langle\\rangle),", "97f04eb2deb4795cd344676ff6660b52": "[a,\\infty)=\\{x\\,|\\,x\\geq a\\}", "97f0769cf3ad35973d2047b901e7eaad": "\\vec{\\xi}_5, \\, \\vec{\\xi}_6", "97f07724ba36f8f56f806bf6a8f1b3f8": "u(x,t)=\\frac{1}{2\\sqrt{\\pi t^\\alpha}} e^{-\\frac{x^{2 \\beta}}{4t^\\alpha}}", "97f07f6c3dd9fd5d4f262a3df9abf2c9": "\\begin{matrix} {3 \\choose 1}^2{40 \\choose 2} \\end{matrix}", "97f08245c7b7e19890b24fe4ce1fa2f4": "f : \\widehat{\\mathbb{R}} \\to \\widehat{\\mathbb{R}},\\quad A \\subseteq \\widehat{\\mathbb{R}}.", "97f0b7cd9785b479b6ad70a773bcaf7a": "\nI(R) = I_{e} e^{-7.669 \\left[ (\\frac{R}{R_{e}})^{1/4} - 1 \\right]}\n", "97f0ba4eeb48f5350e5af82741d27db8": " X^{(i)}Y^{(i)}=(XY)^{(i)}", "97f0d0420291a58adb4f4f727000eebd": "\\zeta (s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}<\\infty. \\!", "97f113a7fdaea5cca65f2715617289b7": "\\int_0^{2 \\pi} e^{x \\cos \\theta + y \\sin \\theta} d \\theta = 2 \\pi I_{0} \\left(\\sqrt{x^2 + y^2}\\right) ", "97f140be5e98e3adfb55f01788b39c05": "\\{ P \\in C(\\overline{K}) : \\hat{h}(P) < \\epsilon\\}", "97f19d71914732825264319edaa8fa26": "\\operatorname{sign}(p_{i-1}(\\xi))= -\\operatorname{sign}(p_{i+1}(\\xi));", "97f1cf7360e61336f8b83de059c4a0ca": " D(-11,2.2) ", "97f1e780d41d9b9807f97cd2e177f4a4": "\\alpha (s) \\ ", "97f22a70e2816a3a538153da856994ef": "\n\\widehat \\beta_{OLS} = (X' X)^{-1} X' y\n", "97f35c4db27a676222a835cff83c08ef": "\\hbox{SSIM}(x,y) = \\frac{(2\\mu_x\\mu_y + c_1)(2\\sigma_{xy} + c_2)}{(\\mu_x^2 + \\mu_y^2 + c_1)(\\sigma_x^2 + \\sigma_y^2 + c_2)}", "97f376936e6227842e5ef851444248b8": "H_{\\ast}(X;R)", "97f3dbe6973087d39ecd6d56bed77425": "h^2+ef+fe", "97f42f317d71def05d779d096e262d70": "\\pm 1/(1+ \\rho ^2) \\ ", "97f472bf7021f084fd60919c80d6708c": "n\\times(n-1)/2", "97f49b9fdd97652ab46b8dacc49e253b": "Q(M)= 1 + \\frac{M-1}{M} + \\frac{(M-1)(M-2)}{M^2} + \\cdots + \\frac{(M-1)(M-2) \\cdots 1}{M^{M-1}}", "97f4e3fe9f3cdb6ab8407fe6e371a447": "\\neg A \\or B", "97f59b6b6357ef59496544ec013b6bfd": "P^{-1}\\cdot\\det P =\n \\begin{pmatrix}\n g_{\\mbox{o}} \\rightarrow 1 & - h_{\\mbox{o}} \\rightarrow 1 \\\\\n -g_{\\mbox{e}} & h_{\\mbox{e}}\n \\end{pmatrix}\n", "97f5a045349e92c3433f5f8f26461169": "\\frac r R ,", "97f5b4a0811053e6fe11698533e5857d": "\\begin{cases} - \\Delta u(x) = f(x), & x \\in \\Omega; \\\\ u(x) = 0, & x \\in \\partial \\Omega; \\end{cases}", "97f5d47dcda7a5516de8e6b890883e03": "m_\\mathrm{HCl} = \\left(\\frac{52.0 \\mbox{ g }\\mathrm{H_2S}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{H_2S}}{34.1 \\mbox{ g }\\mathrm{H_2S}}\\right)\\left(\\frac{6 \\mbox{ mol }\\mathrm{HCl}}{3 \\mbox{ mol }\\mathrm{H_2S}}\\right)\\left(\\frac{36.5 \\mbox{ g }\\mathrm{HCl}}{1 \\mbox{ mol }\\mathrm{HCl}}\\right) = 111 \\mbox{ g}", "97f6045216e774de69ba02258ae32ffe": " y_n = y_{n-1} + h A(t_{n-1}, y_{n-1}, h, f). ", "97f615ef5205e29478163e09b20e8491": "\\vartheta^1", "97f619b877392a2ddc37b861612eb989": "C_G(k)= (-1)^{|E|+|V|+k(G)} T_G(0,1-k).", "97f74f9adab01373e3f087faafa324ac": "n = 1200 \\cdot \\log_2 \\left( \\frac{b}{a} \\right) \\approx 3986 \\cdot \\log_{10} \\left( \\frac{b}{a} \\right)", "97f769586b0c5456c0c03726692de7e6": "\\text{cl}(A)=\\text{cl}(B)\\,", "97f783d435acf3fa19355aa9dc8c32e1": " b = 1 ", "97f7b9f05151ad88b83f5b2ec0e654ca": "y = y_{1}wy_{2}", "97f7e2f05e7a388076e662abfad42873": "\\omega = -v_\\theta/r", "97f89e9e5bdcb15de451c76f2329b126": "v_j\\,\\!", "97f8b08fab373547274564a210d72143": "\\,_nq_x", "97f8c77a223e265c74e63306abc548e0": " \\frac{1}{s} \\left(1 - \\frac{1}{s}\\right) \\left(2 s^3 y' + s^4 y''\\right) + \\left(\\gamma - (1 + \\alpha + \\beta)\\frac{1}{s} \\right) (-s^2 y') - \\alpha \\beta y = 0", "97f9003aece40d04dccfcf8e3089e9da": "\\hat n_{21}=\\hat n_{2} - \\hat n_{1} = \\alpha^*_{LO} \\hat a + \\alpha_{LO} \\hat a^\\dagger", "97f9272f63a89552fb60a4b99a9d64c3": "(((ab)c)d)", "97f92c6a1936cccb42e48ee8332ff7d8": " k ={1,\\ldots,n}", "97f9ef4565c143101d9a1c5375b823b3": "f : M \\to N", "97f9fc45ef157f29cfd15dc94f5d75b7": "K(\\tau_1,\\tau_2)", "97fa5ae9c77d1b9b71f38e2e3d937808": " X_1, X_2 , \\ldots, X_n ", "97fa7403110ca44f1b2ae4f347631dc5": "\\tilde{P}_{r}=\\frac{\\exp\\left[-\\beta\\tilde{H}\\left(r\\right)\\right]}{\\tilde{Z}}\\,", "97fa742369345513784c347ce0780695": "\\hat{m}_{ij}^{(0)} = 0", "97fac62b08f0eebe2a894a15560fcd10": "p \\geq 1", "97faec4fefeb8bc6e2b6fcec00f0ce28": "I^2R", "97faf29c2f73135d6af000f28fcd13ba": "\\left[ D \\right]_{1/2}", "97fb5227dccf230d30e03a04b089bbfc": "TK_R(y^{2})=(0\\ 0\\ 1)", "97fba227e52c284d66290fde66d8aca9": "f(t) = \\sin(t), \\quad t \\in [-\\pi,\\pi]", "97fbadc776839871436825390835fa47": " T^{*}_{11}", "97fbe9736833da98ae71b5ae1650c666": "\\begin{align}\ne^{\\pi \\sqrt{19}} &\\approx (5x)^3-6.000010\\dots\\\\\ne^{\\pi \\sqrt{43}} &\\approx (5x)^3-6.000000010\\dots\\\\\ne^{\\pi \\sqrt{67}} &\\approx (5x)^3-6.000000000061\\dots\\\\\ne^{\\pi \\sqrt{163}} &\\approx (5x)^3-6.000000000000000034\\dots\n\\end{align}\n", "97fc0c21b009279ee57f5b77cf5ad932": " s \\,", "97fc30b856a8b71d35c2c2a33faba2b1": "z = A^+b", "97fc6a75585814e2ea18c9860694586a": " \\operatorname{lambda-named}[V] = \\operatorname{false} ", "97fc74b95e5f8dfd7bda935ae83ddf6a": " np\\, ", "97fc7f0dbce220cb51274e5c9542d5af": "1 \\leq i,j \\leq n, i \\neq j, 1 \\leq k, k_{1}, k_{2} \\leq r, k_{1} \\neq k_{2}, 1 \\leq l, l_{1},l_{2} \\leq s, l_{1} \\neq l_{2}", "97fca1089f2199a4657c1c02c09d92e5": "{\\mathbf{\\Sigma}}", "97fd1579c180f83f149ba9442010f930": "\\frac{1}{\\lambda_{\\mathrm{vac}}} = R_{\\mathrm{H}} \\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)", "97fd1aa3f2722faeee945a058bfc61d3": "g(x^*) = a + b \\ln\\big(e^{cx^*} + d\\big)", "97fd5b177af1e541068c98410da9df4d": "P^{-1}AP=J=\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 2 & 0 & 0 \\\\\n0 & 0 & 4 & 1 \\\\\n0 & 0 & 0 & 4 \\end{bmatrix}.", "97fd9187d859573240b123f1c5d76bea": "\naa = 0.953 - 0.002314 \\left( \\left( \\tau_2^T \\right)^2 + \\alpha_2^T \\beta_2^T \\right)\n", "97fdf90850f660f05349f4ad145b62dc": "x<0", "97fe3cced35bfec7d2119f30ee15275d": "10_{153}", "97fe8f1d4b74491482dfe5348e1f3436": "\n-p=\\left(\\frac{\\partial U}{\\partial V}\\right)_{S,\\{N_i\\}}\n =\\left(\\frac{\\partial F}{\\partial V}\\right)_{T,\\{N_i\\}}\n", "97feec89786f3de81a5a38dfe7a759ae": "B_r(p) \\triangleq \\{ x \\in M \\mid d(x,p) < r \\}.", "97fef7ed53e13ac4756567f47772f6e6": "\\frac{F(y)-F(x)}{y-x} = \\frac{p^{2n} - 0}{p^n-(p^n - p^{2n})} = 1,", "97ff1c272395aaf185806ced845167e0": "t=T\\,\\!", "97ff560ed573761680081c1221b7d198": "\\mathbb{Q}S_n", "97ff7008031e96fdaf216527fa4540f7": "\\frac{\\sin A}{\\sin \\alpha} = \\frac{\\sin B}{\\sin \\beta} = \\frac{\\sin C}{\\sin \\gamma}.", "97ff888a5f1ab50a0e3b73f5f85b7d1f": "\\left( \\frac{d^2}{dx^2}f(x) > 0\\right)", "980018029d6d8c41e0155172d093c8dc": "\n\\begin{bmatrix}\n a3 & b3 \\\\\n c3 & d3\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n a2 & b2 \\\\\n c2 & d2\n\\end{bmatrix}\n\\begin{bmatrix}\n a1 & b1 \\\\\n c1 & d1\n\\end{bmatrix}.\n", "98003a0a844dc972b8ae8b0e2755c734": "\\varepsilon_M=\\frac{\\varepsilon_{kk}}{3}=\\frac{\\varepsilon_{11}+\\varepsilon_{22}+\\varepsilon_{33}}{3}=\\tfrac{1}{3}I^e_1\\,\\!", "98004be34e39e6d766bfe14baef8c78e": "H\\vert{\\Psi}\\rangle = E \\vert{\\Psi}\\rangle ", "9800b72aa3eb7a0bd4b9c664ad34c10c": "{}_a\\tilde{D}^\\alpha_x f(x)=I^{\\lceil \\alpha\\rceil-\\alpha}\\left(\\frac{d^{\\lceil \\alpha\\rceil}f}{dx^{\\lceil \\alpha\\rceil}}\\right).", "98012654079f51167adde094423161f1": "1393+985\\sqrt{2}=2786.00035\\ldots", "98015a0c0600a1470d9c2c1e9270e736": "P(x,y,z)=0", "9801f14a4c98a83938e9029fc77f4ac3": "|\\langle x,y\\rangle|^2", "98020e52bb70ad961e3dcb1498703aec": "\\frac{\\partial \\lambda_i}{\\partial K_{(k\\ell)}} = \\frac{\\partial}{\\partial K_{(k\\ell)}}\\left(\\lambda_{0i} + \\mathbf{x}^\\top_{0i} ([\\delta K] - \\lambda_{0i}[\\delta M]) \\mathbf{x}_{0i}\\right) = x_{0i(k)} x_{0i(\\ell)} (2 - \\delta_k^\\ell) ", "9802b2cd97cdffd5f2586685dd473cdd": "(\\neg C)^{\\mathcal{I}} = \\Delta^{\\mathcal{I}} \\setminus C^{\\mathcal{I}} ", "9802ceaef0a897a3ce07b0719a27c887": "p'_c=\\tfrac{1}{\\langle k\\rangle}", "9802d15ff6be720cc1d282586b2919ef": "\\begin{Bmatrix} 3 \\\\ 3 \\end{Bmatrix}", "9802d28a8a8d05cd86ea283c4a021c5a": "D_{5h}", "9802f5e60748feded26cc3a9b5f71b1d": "\\bigg. J = - D \\frac{\\partial C}{\\partial x} \\bigg. ", "9803028e50db5e168e19c1836409a0af": "p + 2^y", "9803a4caa6786d489f4dd15648839d72": "\\|Z\\|_F \\le \\|X\\|_F", "980451309cef94c5a106924150864ae5": " y = y_1 + \\frac{1}{z} ", "9804b5497eb31684a912fb0d7cf178bf": "y_1(t)\\,v''+(2y_1'(t)+p(t)y_1(t))\\,v'+(y_1''(t)+p(t)y_1'(t)+q(t)y_1(t))\\,v=r(t).", "980517bf3dbbe7db4b9579210c4e5b76": "(x+1)\\partial_x(x\\partial_x((x+1)v(x)))+\\lambda v(x)=0", "980532fe270d6f0f06a4d5887fc71130": "\\mathbf{i}(x \\times y) \\leftrightarrow \\mathbf{i}x \\times \\mathbf{i}y,", "9805488c9f39b8b0e9dc4e9a836f8161": "~\\Phi_9(x) = x^6 + x^3 + 1", "9805674ce34aaa7e9d4905ada3a8ffc2": "D_M: H^p(M,A,Z)\\to H_{n-p}(M,B,Z).", "98059121306ef774a0d1f399e04500e7": "\\mathcal{H}/\\Gamma", "9805c419442e23f39124519294069da5": "x\\in \\Omega", "98060af4d464031d3e5bf9beb636a65e": " \\dot x= V_d\\ \\frac{T_x-x}{\\sqrt{(T_x-x)^2+(T_y-y)^2}}", "98060cca7f23feafc5e3406f63f8a6ca": "r = ", "9806209301b9eaedafb7c9c7b6caa35f": "\\scriptstyle Y_0=y ", "98065e1769a83a95083591902110766c": "x \\mapsto F(x, y, p)", "98068133036c10806dee69c62907c613": "\\phi\\,(t)", "98069b1656f9c80d2f03f698fb1de773": "\n XAY =\n \\begin{bmatrix}\n I_r & 0 \\\\\n 0 & 0 \\\\\n \\end{bmatrix},\n", "98069b9b1d2d279ce9278f88f21808dc": "\nH = \\frac {V_{out}}{V_{in}} = \\frac{Z_2}{Z_1+Z_2}\n", "9806e1309b492861a026984321868d2a": "\\varphi(q)", "980701f71332ec14fbfc7eaa7dd2fd8f": "\\alpha_0 \\,\\! ", "9807274cd4060b13d90613ae99cb2501": "\\text{extend}: (((S \\rarr A) \\times S) \\rarr B) \\rarr ((S \\rarr A) \\times S) \\rarr (S \\rarr B) \\times S \\,", "98079a917eabdb7aa1ea56d284c2d858": "[\\mathfrak{m}_+,\\mathfrak{m}_-] \\subset \\mathfrak{k}_{\\mathfrak{C}}", "9807dd4ef615dab4df56f5467b15d6f5": "x \\succeq_q y, \\, \\forall q \\in P", "98085af8a350d7172b6f9d45fb8c4029": " \\Sigma \\,", "9808ac353bba3096c5b395a9ccf9f705": "(3)~~~~~\n dy=\n \\left(\\frac{\\partial y}{\\partial u}\\right)_v du\n +\\left(\\frac{\\partial y}{\\partial v}\\right)_u dv\n", "9808d853042dc24fcdc4bdac4fdaf546": "RC = \\frac{(H+BB-CS+HBP-GIDP) \\times (TB+(.26 \\times (BB - IBB + HBP)) + (.52 \\times (SH + SF + SB)))}{AB+BB+HBP+SH+SF}", "9808decc9388049191d0f89bdf1530cf": "\\frac{d}{d t} \\int_{\\Omega} L \\ dV = -\\int_{\\Omega} \\nabla \\cdot ( L\\mathbf{v}) \\ dV - \\int_{\\Omega} Q \\ dV.", "98093483a39120b5edd3f2cb730c3535": "x_m(t) = x_r(\\theta_r(t)) x_c(\\theta_c(t))", "98096d3f5a14741211c3146498ba37a4": " \\operatorname{def}[F_2] \\and \\operatorname{ask}[S_2] \\and FV[A_2] \\subset V ", "980984afb406febf4552a1f3741a8544": "(\\bigvee_{i\\in I}{y_i})*{x}=\\bigvee_{i\\in I}(y_i*x)", "9809e75e32aca49e9e2c347fc1563d18": "\\ y(t) = h(t)*x(t) + n(t)", "980a19adde9c9068a349ae3e789cfa9a": "P(B|A) = \\frac{\\sum_{F\\subset A \\cap B}\\left| \\int \\mathcal{D}\\phi O_{in}[\\phi]e^{i\\mathcal{S}[\\phi]} F[\\phi]\\right|^2}{\\sum_{F\\subset A} \\left|\\int\\mathcal{D}\\phi O_{in}[\\phi] e^{i\\mathcal{S}[\\phi]} F[\\phi]\\right|^2}", "980a3731e13bac6d70fdfae23c52e841": "\\cos C = -\\cos A", "980adcff9c2bff0b95771864fce900af": "y=Sx", "980ade405a6dad9891133054492eb655": "Q^{(n+1)/2}", "980afb7ce139a62b7a5c71c47c122106": "1+\\sqrt2 \\, ", "980b3ca2d93f207f889b2d50aa7536f1": " \n \\textbf{(5)} \\quad \\hat{\\textbf{v}}_{k} \\leftarrow \\hat{\\textbf{v}}_{k} + ( \\beta / [ \\Delta \\textrm{T} ] )\\ \\hat{\\textbf{r}}_{k} ", "980b80b391f3c54d2c15ae4ab4e05cfa": "{\\rm tr}(A^n)-{\\rm tr} (A){\\rm tr}(A^{n-1})+\\cdots+(-1)^n \\det(A)=0.\\,", "980b88228c027a29be096ede6e5b375d": " Efficiency\\;Factor =\n\\frac{m_{o}\\;M_{o}}{C\\;(1-m_{o})}", "980b9545e9f5b864c5b8c881f1ab4e9d": "\\log_b \\left (b^x \\right) = x \\log_b(b) = x.", "980c056d439fc897667bbef8d596301b": " \n(s_i', t_{si}', t_{ei}')= \n\\begin{cases}\n(s_i', ta_i(s_i'), 0) & \\text{if } (x, x_i) \\in C_{xx},\\delta_{ext}(s_i, t_{si}, t_{ei}, x_i)=(s_i',1)\\\\\n(s_i', t_{si}, t_{ei} ) & \\text{if } (x, x_i) \\in C_{xx},\\delta_{ext}(s_i, t_{si}, t_{ei}, x_i)=(s_i',0)\\\\\n(s_i, t_{ei}) & \\text{otherwise}\n\\end{cases}\n", "980c4ecdf45c9a0659ea9a425b556262": "s(t)\\,", "980d372e0d28683d9930159973a98a82": "f(T_2,T_3) = \\frac{f(T_1,T_3)}{f(T_1,T_2)} = \\frac{273.16 \\cdot f(T_1,T_3)}{273.16 \\cdot f(T_1,T_2)}.", "980d8fb219164bc95d96af55b4038caa": "Spin(7)", "980df3d03e66550bc2083a001dcb9c59": "39^2", "980e1a439467ed5312538f41a1180fbe": "2 \\arctan \\frac{1}{5}", "980e383967a0f98c3ddb96e614b0484a": "(X_\\infty, d_\\infty)", "980ec2529720d8e68cd48db1abfc4921": "z \\gg z_\\mathrm{R}", "980ec3506d3ad60a952d85ad21178a06": "\\forall x\\in X, P(x) \\equiv \\{x\\in X\\} \\rightarrow P(x) \\equiv \\{x\\notin X\\} \\or P(x),", "980ec9de2f6a63e72c11986b4e26f79f": " r_N^k(n)>0 \\leftrightarrow n \\in N\\mathfrak{G}^k, ", "980eca2a6e2e86b5650258658e2f815a": "r_{AB}=|\\vec{x}_A-\\vec{x}_B|", "980f64e24e99f26e3ea95ead00b9fa93": "IV_0", "980fa2ecf87fd9306c188d659f257c01": " \\iff \\neg B \\to \\neg A ", "981024b70ebb40b7a06d569181234eb6": "\\sqrt{\\frac{Z_{I2}}{Z_{I1}}}", "98103293e65f5aa4f30ed69b4f5deabc": "\\mathbf{F} = - \\mathbf{\\nabla} E_\\mathrm{p} \\, .", "9810b041d54820f7680a83084ddbb17d": "2 n - 4", "9810b8df6e2b5ede53bebefdb6cc5b5f": "c_n = 1", "9810e79eb04d443f548707814fdba00a": " v_f^2 = v_i^2 + 2 a \\Delta d \\,", "9810f3409b0c8302ed8d54dcc26281a5": "e^{-10}", "98113093da156dc52a12acefe8f81694": " x_i ", "98116436516637bb74201dc2d9540379": "S (a,b) \\leq S (a,a)", "98118b8c244488b864a7b2816aabb523": "T_{mnk}", "9811ae0642f1814c3259ad036a0a27a2": "m_{\\mathrm{u}} = 1\\,\\mathrm{u} \\,", "9811c6d2fea5f279c4cf4ceff6571d47": " \\nabla \\cdot \\nabla u = -f,", "9811ec9bf42727885e6e5b6c9e47dab1": "\n\\xi = \\frac{v}{\\omega}\n = \\frac{p}{Z_0 \\cdot \\omega}\n = \\frac{a}{\\omega^2}\n = \\frac{1}{\\omega}\\sqrt{\\frac{I}{Z_0}}\n = \\frac{1}{\\omega}\\sqrt{\\frac{E}{\\rho}}\n = \\frac{1}{\\omega}\\sqrt{\\frac{P_{ac}}{Z_0 \\cdot A}}\n", "9812914d93c9308d7d12bdc565011d7a": "\\cot(\\beta l)", "9812c17f0b2adbe8e0d6005dd82962b6": "L_0(s)", "9813139229e6419292604b6fa5a43a25": " [x_{i-1/2}, x_{i+1/2}], ", "9813bc22397b9abe52e281c25fc131b1": "1+\\sigma t\\,e^{\\sigma^2t^2/2}\\sqrt{\\frac{\\pi}{2}}\n\\left(\\textrm{erf}\\left(\\frac{\\sigma t}{\\sqrt{2}}\\right)\\!+\\!1\\right)", "9813c432d7d27f1909281c61da68c201": "\\sum_{s=1}^S\\phi_s=1", "9813cccd14bacdd53944408b9c5eba48": "N\\times|\\omega_k|", "9813d56f93aa5960dfb8b2d720251f31": " (\\|\\Lambda+\\rho\\|^2 - \\|\\lambda+\\rho\\|^2)\\dim V_\\lambda\n= 2 \\sum_{\\alpha \\in \\Delta^{+}}\\sum_{j\\ge 1} (\\lambda+j\\alpha, \\alpha)\\dim V_{\\lambda+j\\alpha}", "9813f348a9e0fd2325e7d42d6d5ea9e4": "c_1,c_2,c_3,c_4", "9813ff20b6f7ffedf966479b9f76b5ad": "\\mathit{NPV} = \\mathit{TN} / (\\mathit{TN} + \\mathit{FN})", "98141e0f8d924bac0b94880407bdf34f": " \\Gamma_0 ", "98142442b16c3dd9271262b690a06ab3": "\n f(k;\\mu_1,\\mu_2)\n =\\sum_{n=-\\infty}^\\infty\n \\!f(k\\!+\\!n;\\mu_1)f(n;\\mu_2)\n ", "9814461729a808d576228e367d9f9ba3": "f^e_{\\mathbf{k}} f^h_{\\mathbf{k}}", "98144a3812a2af1ad77674a4ca83c588": "X_t=X_0+\\int_0^t\\sigma_s\\,dB_s + \\int_0^t\\mu_s\\,ds.", "98149dae33c934aa0c33a32c03d66bfe": "R_S=\\frac{v_{Bullet}^2}{g}\\, \\sin(2\\left(\\theta-\\alpha)\\right)\\left(\\frac{\\cos(\\alpha)\\cos(\\theta-\\alpha)+\\cos(\\theta)-\\cos(\\alpha)\\cos(\\theta-\\alpha)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "98150c90fb1e6b85ac9131291d5c28c9": "\\textrm{ad}_x", "9815567a831a79aec2cefe0289f9ff2a": "Q(f(x)) = \\frac{f(x+1)}{f(x)}", "98156395f6ad3ba84a404e6ea4d61ac0": "\\frac{a+b+c}{3}", "981577dd52beb81f1aa7a581370ad0b2": "\\textstyle [n,k]", "98158c014a1584094c3ddc00cbd16fc9": "\\frac{d\\mathbf{a}}{dt} = \\sum_{i=1}^{3}\\frac{da_i}{dt}\\mathbf{e}_i.", "98159088b20bda273ceafb6f887c2777": " [Z_i] = \\operatorname{Trans}_{Z_{i}}(d_i) \\operatorname{Rot}_{Z_{i}}(\\theta_i),", "98159ba61a45953c41f1efe77e7bfba0": "\\mu_i(x_1,x_2,\\cdots , x_m)", "9815adc74a99991d82e8398dd5fa73c7": "\\langle\\gamma,\\delta\\rangle", "9815e97201db088ea759560973f2a8ad": "k!", "981627b6dc277e27a6909415bc0fc8d3": "L_3 + L_1 \\rightarrow L_3", "98167fbb16988bf40812b683b123a4d4": " \\left( \\begin{array}{c} \\tau_u \\partial_t u\\\\\\tau_v\n\\partial_t v\n\\end{array} \\right) =\n\\left(\\begin{array}{cc} d_u^2 &0\\\\0&d_v^2\n\\end{array}\\right)\n\\left( \\begin{array}{c} \\Delta u\\\\ \\Delta v\n\\end{array} \\right) + \\left(\\begin{array}{c} \\lambda u -u^3 - \\kappa_3 v +\\kappa_1\\\\u-v\n\\end{array}\\right)\n.", "981694fc0b013122620ac592f00c4e00": "\n \\begin{align}\n M_{xx} & = -D\\left(\\frac{\\partial^2 w}{\\partial x^2} + \\nu \\frac{\\partial^2 w}{\\partial y^2}\\right) \\\\\n & = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty\\frac{16 q_0}{(2m-1)(2n-1)\\pi^4}\\,\n \\left[\\frac{(2m-1)^2}{a^2}+\\nu\\frac{(2n-1)^2}{b^2}\\right] \\,\\times\\\\\n & \\qquad \\qquad \\left[\\frac{(2m-1)^2}{a^2}+\\frac{(2n-1)^2}{b^2}\\right]^{-2} \n \\sin\\frac{(2m-1) \\pi x}{a}\\sin\\frac{(2n-1) \\pi y}{b} \\\\\n M_{yy} & = -D\\left(\\frac{\\partial^2 w}{\\partial y^2} + \\nu \\frac{\\partial^2 w}{\\partial x^2}\\right) \\\\\n & = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty\\frac{16 q_0}{(2m-1)(2n-1)\\pi^4}\\,\n \\left[\\frac{(2n-1)^2}{b^2}+\\nu\\frac{(2m-1)^2}{a^2}\\right] \\,\\times\\\\\n & \\qquad \\qquad \\left[\\frac{(2m-1)^2}{a^2}+\\frac{(2n-1)^2}{b^2}\\right]^{-2} \n \\sin\\frac{(2m-1) \\pi x}{a}\\sin\\frac{(2n-1) \\pi y}{b} \\,.\n \\end{align}\n ", "98170ae58985b1b74b3671dd8e26fb5c": "\\ln : \\mathbb{R}^+ \\to \\mathbb{R}.", "9817280ce5fab1fde47ea219a346936e": " E_{local} = E(R_i) ", "98174c2e8b502f7e020b7ab805cd15ff": "\\!Y", "98175b53de5e9dde7ceb7d09178ff6ce": "\\nabla_\\mu T^{\\mu\\nu}=0.", "9817864ca456270546313d5b4e21e7c2": "\\mathbf{J}_\\mathrm{D}=\\frac{\\partial \\mathbf{D}}{\\partial t} ", "9817934c3fb9eef0809310aa6667f6df": "\\omega(i)", "9817bb172eb37cc854c85fdbe309ec9b": "-\\boldsymbol{\\hat{r}_{21}}", "9817c3c1c7ad39716eaa82d230324e0c": "d_p, d_{pm}, t", "98182c08175a04bd0c61620cacd5fc25": "\\frac{\\sum_{k=1}^N k^2}{a}", "981936169c137b7edb89b8d864103e85": " \\prod_{\\mathbf{b}Q=0} \\Theta_{\\mathbf{a}}(P+R) = \\Theta_{\\mathbf{a}}(\\beta P) \\ . ", "981949099f671d730dcfb492c54137b3": "\n\\frac{\\partial }{\\partial s}\\left\\langle \\frac{\\partial f}{\\partial t}, \\frac{\\partial f}{\\partial t}\\right\\rangle\n=\\frac{\\partial }{\\partial s}\\left\\langle v+sw_N,v+sw_N\\right\\rangle\n=2\\left\\langle v,w_N\\right\\rangle=0.\n", "9819668d0dbef926d1e7136faf97ebe9": "\\textstyle X^+=\\{v_i:\\lambda_i>0\\}", "981979d573247691296352e81fc6a43e": "s_{(i)}", "9819cbc75fab8402118d73b0f1cd7fc5": "\\omega_{f} (\\delta) := \\sup \\left\\{ | f(s) - f(t) | \\left| s, t \\in [0, T], | s - t | \\leq \\delta \\right. \\right\\}.", "9819d1db003a85c9e610f63d0ea3578e": "X(t) \\triangleq G(t) + \\Gamma(t) ", "9819d22ff6c1101f07bef7e486184e39": " f(x) = \\int_{C^n} \\exp[-(\\overline{z} \\cdot \\overline{z} - 2 \\sqrt{2} \\overline{z} \\cdot x + x \\cdot x)/2](Bf)(z) \\, dz, ", "981a170908853cba52d2e62baac6886a": "(a_n)_{n\\in\\N} + (b_n)_{n\\in\\N} = \\left( a_n + b_n \\right)_{n\\in\\N}", "981a2c8048a3b711f6646da2e9931dd0": "(x_n)_{n\\in \\mathbb N}", "981a464ba7744ecd9195fa2e02369453": "(u(t),v(t))", "981a50931391a37d167afda06ee7d406": "B(x;r)", "981a75936e1c000f87edb0673b3ac620": "\n\\sum_{j=0}^n (-1)^j \\partial_{ \\mu_{1}\\ldots \\mu_{j} }^j \\left( \\frac{\\partial \\mathcal{L} }{\\partial f_{i,\\mu_1\\dots\\mu_j}}\\right)=0\n ", "981aad7763d36293962817a0b5c06657": " A = - \\log_{10}\\left( \\frac{I_l}{I_0} \\right) = \\frac{\\alpha'\\ell}{2.303} = \\alpha \\ell = \\varepsilon \\ell c. \\,", "981ac6be799b2c8e1d81792782b3c66b": "C^{(V)}_T(V,T)=p(V,T)\\,+\\,\\left.\\frac{\\partial U}{\\partial V}\\right|_{(V,T)}\\ ", "981b658df5fa54fd45a12fa92d0ab3eb": "X \\sim \\chi^2(\\alpha)\\,", "981b767f98fbdd3c7da70f7ebc99a182": " {d \\over dt} {\\partial \\over \\partial \\mathbf v_1} L\\left( \\mathbf r_1 , \\mathbf v_1 \\right) = \\nabla_1 L\\left( \\mathbf r_1 , \\mathbf v_1 \\right) ", "981c125ed62e3958455b9d14100b1970": " R_{t+1}=R=\\beta ^{-1} ", "981c35a4178bd9f9cb12511f54ddecc3": " \\mathbf{\\hat{r}} = \\mathbf{r} \\,\\!", "981c6a289f72da009eeac4c824fa2517": "\\frac{1}{\\sqrt{r^2+z^2}}\\,", "981c6e5c16b8ee76fc925a2690eb599f": " j \\colon U \\to X", "981cc9abb3e0dd276673af332995fcd2": "\n \\mathbf{D}^j(\\alpha,\\beta,\\gamma)\\otimes \\mathbf{D}^{j'}(\\alpha,\\beta,\\gamma)\n", "981d4103663a7a6c731c371007e40caa": "\\frac{\\dot{W}_{turbine}}{\\dot{m}}=h_3-h_4", "981d7d531abb1156b75d3ed733b1b4c6": "\\mathrm{kei}(x) \\sim -\\sqrt{\\frac{\\pi}{2x}} e^{-\\frac{x}{\\sqrt{2}}} [f_2(x) \\sin \\beta + g_2(x) \\cos \\beta],", "981dab5dda7796c3e30384ce070cae6a": "\\left( (1-w_i \\overline{w_j}) K_\\lambda (z_j,z_i)\\right)_{i,j=1}^N\\,", "981dabd9c8b1747a61a28dfcabf38ec0": "\\gamma^{\\langle1\\rangle}=\\gamma_{SS}", "981e1c11be1720ef9ffe5795265bd43a": "A_0=a_{0,i}", "981e2db226b8e194b68bfbdc7c4c603f": "X_{6}", "981e36c2395d06c0e7c9c5816a253393": "D_j\\cap C_n\\ne\\emptyset", "981e55476a75d07eef336ebfd6f7edf7": "c = \\sqrt{gh}\\, \\left[ 1 + \\frac{H}{m\\, h}\\, \\left( 1 - \\frac12\\, m - \\frac32\\, \\frac{E(m)}{K(m)} \\right) \\right]", "981e6f0c0b64b8d99349c85263a0775c": "f = a \\ln(\\pm bA)\\,", "981e71b0970f336e42d4936fc08a46cb": "\n\\frac{1}{\\sqrt{\\pi}}\\int w(z)\\,dz =\\frac{1}{\\sqrt{\\pi}}\n\\int e^{-z^2}\\left[1-\\mathrm{erf}(-iz)\\right]\\,dz\n", "981f03e19b887fbe8577506f68020161": "\\{ \\dots\\}", "981f0d89fcad17843a79a678faa8292f": "0 = \\operatorname{Tor}^R_{i+1}(M, k) \\to \\operatorname{Tor}^R_{i+1}(M_1, k) \\to \\operatorname{Tor}^R_i(M, k) \\overset{f}\\to \\operatorname{Tor}^R_i(M, k), \\quad i \\ge \\operatorname{pd}_R M.", "981f2c42a3a375c2f0ee056bad8d2ecb": " v = r \\frac{d\\theta}{dt} = r\\omega", "981f8c6b938fbb1ed0574c856efe2684": "\\frac{\\delta Q}{T}", "981fa28ef48f70fec83704ad84525195": "A = 10^{-Loss/20} \\qquad R_x = R_y = Z_S \\frac {1 + A} {1 - A} \\qquad R_z = \\frac {2R_x}{\\left ( \\frac {R_x}{Z_S} \\right ) ^2 -1} ]\\qquad \\, ", "981fecfa564563e05a6029a478adbfa2": "HF=\\frac{1^{st} HVL}{2^{nd} HVL}", "982008e96a774f89b272b02803b1aa9a": "(= \\hat{\\beta} \\hat{\\alpha})", "9820506b5eee95063ce493e12d8fff9b": "\\mathbf{v}_\\textrm{D} \\equiv \\left[ \\mathbf{D} \\right]\n\\mathbf{v}_\\textrm{op} \\textrm{~and~} \\mathbf{v}_\\tau \\equiv\n\\mathbf{v}_\\textrm{op} - \\mathbf{v}_\\textrm{D} = \\langle \\mathbf{D} \\rangle\n\\mathbf{v}_\\textrm{op}\n", "98207e205cf676d1bf06506b3efde628": "\\forall \\delta > 0,\\,\\mu\\{x\\in\\mathbb R|\\phi_1(x)\\ge\\delta\\} = \\mu\\{x\\in\\mathbb R|\\phi_2(x)\\ge\\delta\\},", "98209ce8a6309a945be5a206535af824": "O(k^{2-\\epsilon})", "9820ffabfa3fbb206eb89e8a1ea537a2": "\\mathrm{div\\,\\, curl\\,\\,}\\vec v\\,=0", "98212278e952a52cbe29d59275f744ad": "C(n) = 2n \\ln n = 1.39n \\log_2 n.", "9821351f157c739dac29e3b9070e1398": "A \\cap \\varnothing = \\varnothing\\,\\!", "982140121e59e2df5fafbb734e301125": "\\phi'(t)=\\frac{\\alpha}{c+vt}", "982150cbf22f8241bd8a3daa13079687": " Q = \\begin{bmatrix} 1 & \\textrm{i} \\\\ \\textrm{i} & 1 \\end{bmatrix}, ", "9821882f12f5000ec532db97e4133e00": " {x_1*x_2*\\dots*x_n} = \\sgn(\\sigma) ({x_{\\sigma(1)}*x_{\\sigma(2)}*\\dots* x_{\\sigma(n)}}) \\qquad \\forall\\boldsymbol{x} = (x_1,x_2,\\dots,x_n) \\in A^n", "9821f1ee4766f332c316ed86a20d52ab": "e^{-\\frac{\\lambda}{2}}\\frac{\\Gamma\\left(\\alpha + 1\\right)}{\\Gamma\\left(\\alpha\\right)} \\frac{\\Gamma\\left(\\alpha+\\beta\\right)}{\\Gamma\\left(\\alpha + \\beta + 1\\right)} {}_2F_2\\left(\\alpha+\\beta,\\alpha+1;\\alpha,\\alpha+\\beta+1;\\frac{\\lambda}{2}\\right)", "9821f61c2ec86873f00ab842dd54d007": "\\mathcal{F}_x", "9822132ced390a97eb70e8479b9c30aa": "V_T = - \\frac{3 G M }{2 d^3}\\Delta d^2 \\,", "98221dd1f2a9734318101514dcd4272d": "m\\{x:\\, x\\notin \\cup J_m^*,\\,\\,\\, |Tb(x)| \\ge \\lambda\\}\\le \\lambda^{-1}\\int_{(\\cup J_m^*)^c} |Tb(x)|\\, dx \\le \\lambda^{-1} \\sum_n\\int_{( J_n^*)^c} |Tb_n(x)|\\, dx.", "982230de43654f67e04c53aec4698709": "\ndF(u;\\psi)=\\lim_{\\tau\\rightarrow 0}\\frac{F(u+\\tau \\psi)-F(u)}{\\tau}=\\left.\\frac{d}{d\\tau}F(u+\\tau \\psi)\\right|_{\\tau=0} \n", "982244dc5e5453cfbc28a90bf44ef230": "\\approx 0.52 \\text{ deg}", "98225164ef3f5f5ba4d8edc5792be291": "1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\cdots =\\sum_{n=1}^\\infty\\frac{1}{n}.", "98229852d5c3e7ebef39170ea765e858": "\\Omega:{\\rm Th}\\mathcal{S}\n\\rightarrow{\\rm Con}{\\rm Fm}", "9822a33dbfa274d0df7323291b9414c3": " \\begin{align} n & = 1,2,3 \\cdots \\\\\n\\ell & = 0,1,2 \\cdots n-1 \\\\\nm & = -\\ell\\cdots\\ell\n\\end{align}", "9822f60f3b51ba661a194c95d79fd6f0": "|k^{(0)}\\rang", "982322e1a129ba8eaca8eda30dfacc89": "t>0\\,\\!", "98236cfef9f9709a01cdfc80e91e5604": " AQ_n = Q_{n+1} \\tilde{H}_n. \\, ", "98242a3a3e4ae53e5a32436641503305": " (id \\otimes \\Delta)R = \\Phi_{231}^{-1}R_{13}\\Phi_{213}R_{12}\\Phi_{123}^{-1}", "982475d0fd265f3458c06f54646547a5": "\\Phi_{00}=D\\rho -\\bar{\\delta}\\kappa-(\\rho^2+\\sigma\\bar{\\sigma})-(\\varepsilon+\\bar{\\varepsilon})\\rho+\\bar{\\kappa}\\tau+\\kappa(3\\alpha+\\bar{\\beta}-\\pi)\\,,", "9824a51aedf62fcd88063a1b8cd8ee26": "\\mathbf{E} \\cdot \\mathrm{d}\\mathbf{A} ", "9824b26a51714309aa4afd370035ce53": "C_{1}", "9825007aaa47f1fb731d2f50f02df4f7": "P=\\sum_i|\\phi_i|^2", "9825693aa6e3cea027476e62f0849c7c": "y_t = w_t", "9825a2ee67dc3923cf7d552448df1e2f": "T(\\Omega)\\cap U=\\varnothing", "9825b92cad74d6980036fd8c628e2a22": "(f,b)", "9825e517a55e684170d27bbc6a26e4c4": "\\sigma^2_{\\varepsilon_j}", "9825e5b5e873dcd6ec22c77364966473": "0^2+0^2=1", "98260098f26e8cae60be62a7fe2dad67": "e^{2i\\phi(\\omega) - i\\phi(2\\omega)} = i", "9826034ce62bbd8055c602cc6b4c4794": "\\exp(-\\tfrac{3 \\pi i}{4}) = \\exp(\\tfrac{5 \\pi i}{4}),", "98268c5cfdca3c09d41e1233b2092506": " \\epsilon \\;", "9826dedcfc4cd3df0207a8631bac0853": "$50,000{ \\left( \\frac{1}{1+I} \\right) }^n", "982700fc3310398481a8da1ddaed69d3": " \\lambda = \\frac{Q[u]}{R[u]}.\\,", "982752841dc93a15d6256ef4355ee419": " (x -\\xi)^2 + (y - \\eta)^2 = c^2 t^2, \\,", "98275a6bcc0b5f6f588f8c8fb1407e3e": "\n\\operatorname{Arg}\\left(\\frac{-1- i}{i}\\right) \\equiv \\operatorname{Arg}(-1-i) - \\operatorname{Arg}(i) = -\\frac{3\\pi}{4} - \\frac{\\pi}{2} = -\\frac{5\\pi}{4} \\equiv \\frac{3\\pi}{4} \\pmod {(-\\pi,\\pi]}\n", "98279b8b62c53e20b43182157a2dedd1": "\\psi:S \\longrightarrow \\mathbb{R}^{3}", "9827be707eeb96bf8a79d91151069503": "\nL(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{n_1, n_2, n_3}\n\\delta(\\mathbf{r} - n_1 \\mathbf{a}_{1} - n_{2} \\mathbf{a}_2 - n_3 \\mathbf{a}_3)\n", "982813d46431db3be672549d811631c0": "g \\geq (d-1)(d-2)/2", "9828515b6105e990930a4f4fccd7b05a": "[G:C_G(x)]\\frac{\\chi(x)}{\\chi(1)} ", "982852c6d25d5e6cb89c2a925d0d0ba5": "0 \\le r_1, r_2 < 2^m", "9828866b431bc4c883546b4a6f886940": "\\mathcal{E}_k", "98289d0124eeebbabd90161cc10a4bb2": "\\mathbf{x}(t)=\\mathbf{x}^*+e^{\\mathbf{A}t}[\\mathbf{x}(0)-\\mathbf{x}^*]", "9828dadff691c38d1c2f836a1ce6ed79": "\\mathbf{F} = q\\mathbf{E} + q\\mathbf{v} \\times \\mathbf{B}", "9829a554cea216592e4a7833e51478ed": " a(P) = \\prod\\nolimits_{1\\leq i 1 \\\\\n \\end{cases}\n ", "982e5d115f847ddffc151d60f58db5ee": "\\mathop{\\uarr}x = \\{y \\in X : x \\leq y\\}", "982e6045330acf29ad45766b97e1c870": "\\begin{bmatrix} -1 & 3 & -1 \\\\ -3 & 5 & -1 \\\\ -3 & 3 & 1 \\end{bmatrix},", "982f1877cdcf00febf7dbf67cc7a1229": "S_4", "982f390460b6aaa31fa30f5f1017ac9d": "\\phi_m=\\frac{\\phi_1+\\phi_2}{2}.\\,\\!", "982f4a9229f5b71fd101fcded6a6f69f": "V_{T+1}(k)=0", "98302a47bc58dcf28e838e142a210d47": "M_{CD}^{f}", "983061b7167533ab16f5cdfcd3c7e242": "b-l>g-h", "9830769a60664d1cb03a7f049b7c0e5d": "\\forall i = 0 \\ldots n , \\ H^i (X; R) = R", "9830838d76791816366842a9bd8722b7": "\nu\\odot v\\equiv zx^{\\prime}-xz^{\\prime}.\n", "9830b0ed5a957ff5ebfc7624a5ed45c5": "y=2", "983104064553cd8b24305447e992d8b5": "U_1(P,Q)=1, \\,", "983124eb099aa7af5dfcce1a969badad": "\n\\sum_n A_n\\delta q_n + \\sum_n B_n\\delta p_n = 0,\n", "98320343905593358359176e642e6136": "E_{KL}=\\frac{1}{2}\\left(\\frac{\\partial U_K}{\\partial X_L}+\\frac{\\partial U_L}{\\partial X_K}+\\frac{\\partial U_M}{\\partial X_K}\\frac{\\partial U_M}{\\partial X_L}\\right)\\,\\!", "983273fab3e2fd29be00842045e3bd36": " T(\\omega^i,\\ldots,\\omega^j, {\\mathbf e}_k \\ldots {\\mathbf e}_l) =\n {T^{i\\ldots j}}_{k\\ldots l}\n", "9832d98c2c78e5defd58d07cdef395c7": "e(p, u^*) = \\min_{x \\in \\geq{u^*}} p \\cdot x", "9832f9bb5ae5052a3084fe747674eeda": "\n \\mathrm{L}(\\omega) = \\frac{\\partial}{\\partial t} \\langle \\hat{B}^\\dagger_{\\omega} \\hat{B}_{\\omega} \\rangle = 2\\,\\mathrm{Re}\\left[\\sum_{\\mathbf{k}} \\mathcal{F}_{\\omega}^\\star\\, \\Pi_{\\mathbf{k},\\omega} \\right]\\,.\n", "98331d9cd796643da9407b38d8c4097f": "r = \\frac{dh}{dt}\\ , ", "98331fc41542b9e9b0b3d90ed2ee9084": "\\mu_j = m_j", "983339969d37a43e4b5d1d47e140bf9c": "\\langle x^{m} \\rangle = \\int_{x_\\min}^\\infty x^{m} p(x) \\,\\mathrm{d}x = \\frac{\\alpha-1}{\\alpha-1-m}x_\\min^m", "983340c663380eacde4a09812e2e3c2c": "f_s", "983361f002312a61d0c6b3aab9a577b4": "PVNB = (Z_1(1+I_1)/(1+r_1)) + (Z_2(1+I_1)(1+I_2)/((1+r_1)(1+r_2))) + ... + ((Z_n(1+I_1)(1+I_2)...(1+I_n))/((1+r_1)(1+r_2)...(1+r_n)))", "9833a85992fe23bab1fe2a851f883468": " Z. ", "9833d32db6cf43761c540ad13a5c43d6": "2^{126}", "98341f2fdcbc7c2051af7ed94f17e464": "[a,b] = \\{x \\in \\mathbb{R} : a \\le x \\le b \\}", "98343e817ef388f39844767985a8d710": "\n C_{AB}=r(v_{AB})\\Delta{t}=q_0\\Delta{t}-k_0\\Delta{x}=q_0(t_B-t_A)-k_0(x_B-x_A); for\\ v_{AB}\\in[-w,v_f] \\qquad (2)\n ", "98345e775b5a227b276155cd6f6945fc": "\\rm Si + O_2 \\rightarrow SiO_2 \\,", "9834d386803902f509a5716808629ee5": " g_{\\rm safe} \\leq g \\leq G ", "9834d7d8e4e4a548eed1783922096570": "t_l \\leftarrow \\max\\{t_{li}: i \\in D\\}", "9835506e5f09ba77163c5133965bc704": "\\limsup_{n\\rightarrow\\infty} \\frac{p_{n+1}-p_n}{(\\log p_n)^2} = 1,", "98355625931e96880f39b44098a0d743": "S_n / n", "9835803f8efe71276dd0d45631f912f5": "q(y)=\\sum_{k=1}^\\infty\\rho_{2k}(y)s_{2k}(y),", "983593b7cd6ce67a5ff0f7318d1312c2": "x^{ 5 }+x^{ 3 }+1", "9835af243882a950a4fa4e5a22eb3c8d": "C(t)", "98360d121f0092609b551744973f203b": "A \\rightarrow B: \\{N'_B\\}_{K_{AB}}", "9836341e4ad69e05db7fec7d7d74092e": "i \\in \\{ 1,2,\\ldots,I \\}", "9836c8962ab1d1c3c55b68981814d799": "x'=1.", "9836f423ad11be80030a554f79eb4688": "\\mathcal{P}\\boldsymbol{\\sigma}", "98373dc45f85305911a0d0543bcb28e2": "(y^i):M\\rightarrow{\\mathbb R}^n", "98375d02c8a1218f8f1224ff3881f5c3": "\\forall u\\in D_i\\,(w\\;R\\;u\\Rightarrow u\\Vdash A).", "98375ed7dc5f42693dee3e20f9353eb3": "f(\\langle x_1, ..., x_m \\rangle, \\langle y_1, ..., y_n \\rangle, ...) = \\gamma", "983798a2b93040f66c2b01c092a7868a": "(X \\times I) \\sqcup Y", "98379e04f70007bf6481dce7633beef1": "\\omega = 2\\pi f = 2\\pi / T \\,\\!", "9837a0395065fa2262612ae7a69083e9": "\\vec{X} = \\vec{e}_0 = \\frac{1}{f(r)} \\, \\partial_t", "9837a104d9c31fdfc4da138004ee54c7": "\\Sigma _{YY} ^{-1/2} \\Sigma _{YX} \\Sigma _{XX} ^{-1} \\Sigma _{XY} \\Sigma _{YY} ^{-1/2}", "98383ee3e1fee03554ae8a6875196cad": "\\Delta= \\begin{pmatrix}I&B_2+z_2&B_1+z_1\\\\J^\\dagger&-B_1^\\dagger-\\bar{z_1}&B_2^\\dagger+\\bar{z_2}\\end{pmatrix}.", "983842ae568f273fdd831740cb64c03c": "G_R \\to 1", "983847efcbb9bb5355fa2eec5ee0ccc0": "(a\\nu)^2 > 2gl ", "983887cbe4a6db5d96769735e44be2c7": "\\det(J_F(x,y))=2xy \\cos(y) - 5x^2.", "9838aa6efa59ba622d820f4355304fbb": "{\\lambda }_{\\mathrm{E}}={\\left(\\frac{k}{e}\\right)}^{2}{\\sigma }_{0}T\\left({I}_{2}-\\frac{{I}_{1}^{2}}{{I}_{0}}\\right)", "98392b443d03b51b3465110a97987793": "\\Phi_p(z)=\\frac{z^p-1}{z-1}=\\sum_{k=0}^{p-1} z^k. ", "9839731f4a5180fb23dffe0534ade313": "x(t) = \\sum_{k=-\\infty}^{\\infty} c_k e^{i 2 \\pi k f t}.", "983994a4225ed01288d4d6f1c7afa1b9": "D e_\\alpha = \\sum_\\beta e_\\beta\\otimes \\omega_\\alpha^\\beta(\\mathbf e)", "9839c2255d73a3249b2e9efd2d936173": " E_K =\\tfrac{1}{2} m|\\mathbf{v}(t)|^2 ", "9839c3008cb2805876f8f0288e6ab9a9": "T' = 1+4.939\\left( \\frac{d} {L} \\right)^2", "983a0531215fef235661166aae142d62": "{1\\over T} = {\\partial S\\over\\partial U} = {\\partial S\\over\\partial q}{dq\\over dU} = {1\\over\\varepsilon}{\\partial S\\over\\partial q} = {k\\over\\varepsilon} \\ln\\left(1+N^{\\prime}/q\\right)", "983a142e6a578596598a90489f4d6cf7": "\\partial_{t} u + u \\partial_{x} u = \\rho \\partial_{xx} u + f \\quad \\forall x\\in\\left[0,2\\pi\\right), \\forall t>0", "983a3c6a2a4df97e463942276a038bcc": "\\{x^0,x^1,x^2,x^3\\}", "983a679c7e98061bf1b5503646cb2136": "\\alpha_j(p,q)", "983abcb458239bc96ac6f019ded664eb": "u^2=(1.10,1.81)", "983b241ff68f24e417620a16d68328ff": "s+1", "983b7fe7239b04249147726d9e8e73a2": " \\lim_{n\\longrightarrow\\infty}\\alpha\\left(G_n,\\varepsilon\\right) =0", "983b81fd1b3621a8f3fded347b486d43": "v \\;", "983bf8d86fc5f249d6a24f50fd32b029": "\\scriptstyle n \\,", "983c33bec2e6ee8b8c616f4c92f44964": "\\Rightarrow \\psi = B e^{i \\beta x} + B' e^{-i \\beta x} \\quad \\left( \\beta^2 = {2m(E+V_0) \\over \\hbar^2} \\right). \\,\\! ", "983c4f49c12105adf67d32a7f4568906": "x_t = x(t \\beta \\hbar/P)", "983c5a7bb7dd97316984e641e9b04259": "\\tilde{V}_{N_j}", "983c8cf8acb92aa5623c627b33bdd751": "\\alpha_{t_n}", "983c979772ce2918622fb7f954b47b9a": " \\lambda(G)^\\prime= \\rho(G)^{\\prime\\prime},", "983ce29b9b47ef613562ac90146d4cff": "\\sigma_M", "983d0cde8da6daea789d084b423c8a5a": "\\frac{\\lambda_0} {\\sqrt{\\epsilon_r}}", "983d34dcde9a2407d2dd42201f2848b8": " k=\\sum_{i\\le l}k_i2^i ", "983d57f5e7e0dc31c4c3412a0844b3d0": "\\begin{matrix}{4 \\choose 2}{52 - 4r \\choose 2}\\end{matrix}", "983dab4c698bf0c54422aa898a115f19": "H_1, H_2, H_3", "983dbe56aba7502ca92077e54b714d64": " E_{ij} = \\frac{\\partial e_{ij}}{\\partial t} = \\frac12 \\left(\\frac{\\partial v_j}{\\partial x_i} + \\frac{\\partial v_i}{\\partial x_j}\\right).", "983dc81091e17df130a9d4d535260792": "M(x) \\cdot x^n - R(x) = Q(x) \\cdot G(x)", "983dff20177e5bad7e4c7c6f3111404b": "P(S|\\text{smaller coin})=P(S|\\text{slightly bigger coin})", "983e130290931215b3cb07c9d08b5388": "\\frac{S}{kN}=\\ln\\left( \\frac{VT^{\\hat{c}_V}}{N\\Phi}\\right)", "983e93f9ba42487599633c01085e9297": " \\lambda = (t)^{-1/p} ", "983eb4a6e4e0fb1305b5e75e214caba1": "t \\cdot r > D", "983ed0c3a1c73d65a6388497bf962419": "\\zeta_k", "983f1892b2cbdc180cb87cb4221cba63": "xy \\in P", "983f7651d6f8d5154b1c32d3b0fed41c": " m=[1\\ldots\\infty]", "983f960082f07492f48b05edc9ca4a33": "\\displaystyle -\\frac{\\pi}{\\left| \\nu \\right|} - 2 \\pi \\gamma \\delta \\left( \\nu \\right) ", "98406b34d0c582ad3089d901651af630": "f_P=\\delta_P= 0", "98406cbba4f207c4129dc1f70a91e810": "\\pi^0\\to e^+e^-\\gamma", "9840accc307dbbf8b5b60295521c333e": "= (\\|f\\|_p + \\|g\\|_p)\\frac{\\|f + g\\|_p^p}{\\|f + g\\|_p}.", "9840b337d2a877503c82435c0256057f": " \\pi^* = \\arg\\max_{\\pi}{E \\big[R |\\text{ treatments are chosen according to }\\pi \\big]} \\!", "9840bf5c3445b6dc1e6729fa228305dc": " \\varphi ( \\mathbf{x} ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{i=1}^N a_i \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) ", "98411ddd96cb02248492faef285ff75c": "n\\equiv(1-\\zeta_n)(1-\\zeta_n^2)\\dots(1-\\zeta_n^{n-1})\\pmod{\\mathfrak{p}}.", "98412dfa099dd1226798b1f41207e664": "-j\\infty\\,", "98413e3358688d05bfe07a638a5065ed": "x_3 \\in_R [0, q-1]", "9841e0dfd66b7f7197812e4ec8551bb6": "x*(y*z) \\longrightarrow (x*y)*z", "9841e0ec14e31061ff19b636e82ec451": "s \\not\\in \\mathfrak{m}", "98428093f95e7ed2141795e27af5328a": " \\mathbf{C}_{N} = \\mathbf{R} \\, \\left ( \\begin{array}{c|c} \\mathbf{I} & \\mathbf{R}^{-1} \\, \\mathbf{t} \\end{array} \\right ) = \\mathbf{R} \\, \\left ( \\begin{array}{c|c} \\mathbf{I} & -\\tilde{\\mathbf{n}} \\end{array} \\right ) ", "984291a1ae8b1d7e831a915c2cabcbc0": "x \\mapsto Mx + b", "9842a32ab1a90f710f06727efb757418": " \\ T_{h,i}- \\ T_{c,i} ", "9842c9fcac54bb2bb7cc06325f254bdb": "F = -\\Delta P A", "9842dba87dded3cb8be85ddadcf4cab0": "\\tan(\\alpha)", "9842f862de86aab3c3c5e7f764b71502": "MW=W_s-\\frac{m(m+1)}{2}", "984318bdf7a365fe9e05991d19be5375": "-I_\\text{p} = \\frac {V_\\text{out}} {R_\\text{f}}", "9843629a46e1299346155f7291356401": "S'(a) = u S(a)u^{-1}", "984378d14bdbdbe0c694101205e60b87": "A \\to aA, B \\to bB, C \\to cC", "9843a05833202af4bb15911b8ef1306a": "\\displaystyle{T_c f(w)=\\lim_{\\varepsilon\\rightarrow 0} {1\\over \\pi i} \\iint_{|z-w|\\ge \\varepsilon} {\\overline{f(z)} \\over (z-w)^2}\\, dx\\,dy.}", "9843c436cd97a6581345e8f78ca9d6d9": "w * \\lambda", "98444699e6ae61d068d336e595e4b628": " (a*b)(n) = \\sum_{k|n} a(k)b(n/k) \\ ", "9845198045ec71aa8304372c28b24630": " \\alpha ", "98453054332a86de5a2849623bd3d5bc": "N(L/K)", "984543e566f0650be5cc538091013b35": "z \\equiv x/k", "98454ba6da9864749d13bcdc3483670a": "N_{eq} = 10^4", "98457bcced6d830f9ebb2cb076c51d5b": "\\overline{e}_b(k,0) = \\overline{e}_f(k,0) = \\frac{x(k)}{\\sigma_x(k)}\\,\\!", "9845ab717b8590aca7867caa480a3169": " B_e = \\bar B(R_e) ", "9845b9f7adf30ef1ebf1d5d3224e4e76": "\\left [\n\\begin{smallmatrix}\n 1 & 2 \\\\\n 2 & 1 \\\\ \n\\end{smallmatrix}\\right ]", "9845ba004fe6680da0df3bba668941f2": " \\hat{f}_2^{(0)}= 0", "984647f190c9d194cb139e9f9d2fe16e": "v_n(x)=u_n(t)", "98464d6193e0bf3861514e163115ea1e": "\\Sigma = \\{ 1 \\}", "9846864eae2c31cb1a24f195afb6bca9": " \\psi (x,t) = \\int_{-\\infty} ^{\\infty}\\ dk_1 \\ A(k_1)\\ e^{i\\left(k_1x - \\omega t \\right)} \\ , ", "98470cc0c9a729f7d22812b042f39062": " H_n(x) = n! \\sum_{\\ell = 0}^{n/2} \\frac{(-1)^{n/2 - \\ell}}{(2\\ell)! (n/2 - \\ell)!} (2x)^{2\\ell} ", "9847165acd6f0723fe156bbf7d40b811": "\\operatorname{P(S/M)=\\sum_{x=1}^n{IMM_8(S_x)}}", "98477955fb52112ab857a49a420f162b": "\\langle k\\rangle", "9847d1f24e029b6d95ba91e80c66b058": " \\frac{d^2x}{dt^2} + 2\\zeta\\omega_0\\frac{dx}{dt} + \\omega_0^2 x = 0,", "98481bca8570cb698a8ffd751a32eb63": " r= 120 ", "98482e4064ac1cd20b3776163f7b11e6": "\\frac{d (\\eta + f) }{dt} = 0 = \\frac{\\partial \\eta}{\\partial t} + U \\frac{\\partial \\eta}{\\partial x} + \\beta v'", "9848561e2ca7fdee4495d1ca3aace8be": "\n\\mathcal{S}_{0}[\\mathbf{q}(t)] \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int \\mathbf{p} \\cdot d\\mathbf{q} \n", "984866475d3a6d9960bfc8bd74da0abd": "\n{1 \\over (n+1)!}\n", "98488678aa0fcf384a811ea7d7a5ed11": "p_1=1-\\frac{1+1/F}{1+\\epsilon}", "98488fb72cabe93bac78ae633d76bcb7": "f:X_0\\to X_1", "9848c66f1b9fb3400284ff011a94e2f7": " \\kappa(a,z) := \\overline{F_a(z)} ", "984973ea0e2b3a155ee1c04fee440fc3": "I_\\text{ball} = \\frac{2 m r^2}{5}", "98497705a7ab0f7e1b2b608e64ac60c6": "\\mathcal Z:=\\{\\{A_i,B_j,C_k\\} \\ | \\ i,j,k=1,2\\} \\ ,", "9849c5adf0aae260e7c8720cef648582": "{10^{-6} \\cdot K \\cdot m^2 \\over kg \\cdot s} \\equiv 1 PVU", "9849e2fddebee09560f28c8025a860df": "B^\\alpha_\\beta", "984a2b399fa42118c7c3049a70601e53": " f(t/|M|^{1/\\beta})\\approx 1+{\\rm const}\\times( t/|M|^{1/\\beta})^\\omega +\\dots\n", "984a5ebf4fb4c4fa6e26a661bc050dcd": "\\alpha_k = \\alpha_0 + N_k \\, ", "984a642ff59f41000f5c26efd97e4f8e": "\\mathrm{area}\\,(\\partial D)\\geq C\\,\\mathrm{vol}\\,(D)^{(d-1)/d}.\\,", "984b03816346c78c86c87324563b953a": "= {1 \\over 2} (F^J_{\\;\\;\\; K} G^{KI} + F^K_{\\;\\;\\; K} G^{IJ} + F^I_{\\;\\; K} G^{JK} - F^J_{\\;\\;\\; K} G^{IK} - F^K_{\\;\\;\\; K} G^{JI} - F^I_{\\;\\; K} G^{KJ}) ", "984b088cfde8ea1d7e9b76c1a2986962": " \\frac{q_c}{p} = \\left[\\frac{\\gamma+1}{2}\\mathrm{M}^2\\right]^\\left(\\frac{\\gamma}{\\gamma-1}\\right)\\cdot \\left[ \\frac{\\gamma+1}{\\left(1-\\gamma+2 \\gamma\\, \\mathrm{M}^2\\right)} \\right]^\\left(\\frac{1}{ \\gamma-1 }\\right) ", "984b2acb975f3de4d4a3aa1d8542adb3": "C^0, C^1, C^2, \\dots, C^\\infty", "984b68c02f780e5ad813b854b1f34e4f": "\\xi = \\sum_{\\alpha=1}^k e_\\alpha \\xi^\\alpha(\\mathbf e)", "984b7af1dc81ac4ce342e29f8f2ed5de": "\\frac{1}{\\sqrt{OSOI_{cas}}} = \\frac{1}{\\sqrt{G_{p,2}G_{p,3}G_{p,4}. . .G_{p,n}OSOI_{1}}} + \\frac{1}{\\sqrt{G_{p,3}G_{p,4}. . .G_{p,n}OSOI_{2}}} + . . . + \\frac{1}{\\sqrt{OSOI_{n}}}", "984b95bb7b5166619e9ac186311bf797": "u_{1}(j^{1}_{p}\\sigma) ", "984b9ed9a407a392e8909976f65ec088": "A=\\begin{bmatrix}1&1&0&2\\\\-1&-1&0&-2\\end{bmatrix}", "984ba84582548bda83811f63a0972538": "C:\\bold{Top}\\to\\bold {Top}", "984bb0f354ec5edc0d854f7777ae0996": "(a+b)^n", "984bbedd260af4612a6e3f9264f70dc9": " (u,v) ", "984bc2f31c666169cf83e5f0f129f2bc": "P_2 \\uparrow G", "984c30fe7984620598a3bff8c14e40f1": "S_{0.20}", "984c342f815fff268e2a34641bb76c7f": "T_M(\\rho;E)=\\tau_M(\\rho;\\mu)", "984c9a5a3debfc6a5649098407599ea4": "\\delta(\\mathbf{x}) = \\delta(x_1)\\delta(x_2)\\dots\\delta(x_n).", "984cb3b016f3495d5f791686f2dbc0c7": "\\mathcal{L}={1\\over 2}\\partial^\\mu \\phi_a \\partial_\\mu \\phi_a -{m^2\\over 2}\\phi_a \\phi_a +{1\\over 2}F^2-{\\sqrt{\\lambda /N}\\over 2}F \\phi_a \\phi_a", "984cda97b2183184d68a193f7083012e": "\\mathbf{J}_z\\Psi = \\hbar{m_j}\\Psi", "984cf8a0974316a99166e076f0ac9bd5": "c(g,h) = a_g^h a_h a_{gh}^{-1} ", "984d210851d597bc62b38684f3576178": "\\left(\\prod_{i=1}^c \\binom{m_i}{x_i} \\right) \\int_0^1 \\prod_{i=1}^c (1-t^{\\omega_i/D})^{x_i} \\operatorname{d}t\\,,", "984d252e85122b11042ab58b29f9a76e": "B^{G_K}", "984d2698ab80b9d66b53e031ae6f2d43": "\\alpha\\in x\\,", "984d33dcf507a0f49069214f089d5fe1": "k^{\\epsilon}(x)", "984d48caf52a59a60b441be0fe53e99e": "V=(\\frac{1}{3}\\sqrt{\\frac{61}{2}+18\\sqrt{3}+30\\sqrt{1+\\sqrt{3}}})a^3\\approx3.51605...a^3", "984d7dd4ae0d3ee4891e366654d23b8a": "(b, -)", "984d8da410eebc66c665b4609616882c": "O \\hookrightarrow G", "984d9dfeb4efabf08017fa4f41bc412e": "x \\rightarrow \\infty", "984dd709fa83c6ab7aec811f9a0c3937": "\\{|X_k - \\mu_k| > \\varepsilon s_n\\} := \\{\\omega \\in \\Omega : |X_k(\\omega) - \\mu_k| > \\varepsilon s_n\\}", "984e0b64e9c8afcfb945da819a0ff3d8": " \\begin{align} \\textrm{ad}_x : & \\mathfrak{g} \\to \\mathfrak{g} \\\\ & y \\mapsto [x,y] \\end{align}", "984e26bf9e73e7b5c2730a3ec0a8ecc3": "\\hat{\\mathrm{Td}}", "984e4a90fe0ee6f74f75dd2022ff6fdb": "\\bar{r}_s = \\frac{1}{K} \\sum\\limits_{k=1}^K r_k.", "984e5642a52fd4fd110e63c31070dc25": "E_0 = \\{\\, (0, \\sigma) \\in E \\, \\}", "984ef1b03447d069cb32b175ba627513": " g_r(f(A),f(B)) = f(C). \\ ", "984f80bc3e9b15fcbdb8d115bf34bd52": " \n\\int_{E}\\varphi \\,d\\mu\\leq \\liminf_{n\\rightarrow \\infty} \\int_{E}f_n\\,d\\mu\n", "984fb468e885aefcd2afa698e11f5aac": "0 = \\begin{pmatrix}0 & 0 \\\\ 0 & 0\\end{pmatrix}", "984ff070274a34c8c6dec71b81cab140": "\\sum_{i=1}^m x_{ij} \\leq 1,", "985083f3321a400d556171c10fbbb857": "\\Phi(t, x) = x\\,", "9850c192e535f424e31ab87c228dbd3a": "\\scriptstyle 2\\frac{1}{4}\\times 4\\times 8", "9850cbd6a4b021f8255a7924221ab13c": "L(\\theta;\\mathbf{x},\\mathbf{z}) = P(\\mathbf{x},\\mathbf{z} \\vert \\theta) = \\prod_{i=1}^n \\sum_{j=1}^2 \\mathbb{I}(z_i=j) \\ \\tau_j \\ f(\\mathbf{x}_i;\\boldsymbol{\\mu}_j,\\sigma_j) ", "9850dfaad8931b16ec5fd2e347525aa9": "\\sigma _x", "98511094ec8eff649c36585008dc17ac": "P_1 V_1 = P_2 V_2.", "985173a3bf8e40b19802cef544d1e301": "\\beta((X'_\\beta)',X'_\\beta)", "9851c7d58935de6763536adf3b11ea8a": "C_d \\mathrm{Re}^2 = \\frac{4mg}{\\pi\\rho\\nu^2}", "985225fe680902c0eb17ccaef5d91c2e": " \\operatorname{let-combine}[\\operatorname{let} p : \\operatorname{de-lambda}[p\\ f] = \\operatorname{let-combine}[\\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x)] \\operatorname{in} p] ", "985276b097c2211608678c82db17e455": "Z^\\mathrm{T}X", "98533d63fd17d48771216fbc7dd58d29": "\\displaystyle{E_a(z)=\\sum_{n\\ge 0} {(E_a,e_n)z^n\\over \\sqrt{n!}}=\\sum_{n\\ge 0} {z^n\\overline{a}^n\\over n!} = e^{z\\overline{a}}.}", "98539949ebfbd7a996f8135bf69e2bb0": "\\bold j = \\frac{1}{2m}\\left(\\Psi^* \\bold{\\hat{p}} \\Psi - \\Psi \\bold{\\hat{p}} \\Psi^*\\right), ", "9853de9b2ee7bfa3bc01593fb972c707": "k_B\\,T_c = 1.14E_D\\,{e^{-1/N(0)\\,V}},", "985492164bcef391b61761a4f122f4a4": "\\Delta y = f'(x)\\Delta x + \\frac{(\\Delta x)^2}{2}f''(\\xi)", "9854c019e69d995dc435367ad2bf4bb2": "w_{ij,kl}", "985546042cb542964d69fbd27dc7bd9b": "{S \\subset \\mathbb{F}_q^k}", "98556abe14f829f46ecb23354129c418": "[\\gamma^\\mu,\\gamma^\\nu]_{+} = 2g^{\\mu\\nu}", "985628f806055aaaa538a387ddee017e": "E(u,k)", "98568d540134639be4655198a36614a4": "Aa", "9856f83106c82c34de1010bf61297500": "m \\ge n", "9856f94bf7ef6aa8ac47bc2e412f9988": "\n\\begin{bmatrix}\nY_r \\\\ U_r \\\\ V_r\n\\end{bmatrix} \n= \\begin{bmatrix}\n\\frac{1}{4} & \\frac{1}{2} & \\frac{1}{4} \\\\\n1 & -1 & 0 \\\\\n0 & -1 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\nR \\\\ G \\\\ B\n\\end{bmatrix}; \\qquad \\qquad\n\\begin{bmatrix}\nR \\\\ G \\\\ B\n\\end{bmatrix} \n= \\begin{bmatrix}\n1 & \\frac{3}{4} & -\\frac{1}{4} \\\\\n1 & -\\frac{1}{4} & -\\frac{1}{4} \\\\\n1 & -\\frac{1}{4} & \\frac{3}{4}\n\\end{bmatrix}\n\\begin{bmatrix}\nY_r \\\\ U_r \\\\ V_r\n\\end{bmatrix}\n", "985706002227174f4a8458f783ecce59": " ( 1 - \\beta_k ) \\delta_0 ( x ) + \\beta_k f_k( x ), ", "985718fe2833f8692ffe44b9f73d88e4": " \\frac{dS}{dt} = \\mu N - \\mu S - \\beta \\frac{I}{N} S ", "9857330e13c3442d2cece063eecff459": "\\mathbf{M}_i", "985749254a8ce3d9b3d9a5a7a9e65686": "\\scriptstyle V\\left(1 \\,-\\, \\frac{1}{e}\\right)", "98576a6772f06d9a6886923a14b172db": "r_2=a \\frac {\\sin l_2(\\theta)}{\\sin (l_2(\\theta) - \\theta)} = a \\frac {\\sin ((l(\\theta)+\\theta)/2)}{\\sin ((l(\\theta)+\\theta)/2 - \\theta)} = a \\frac{\\sin((l(\\theta)+\\theta)/2)}{\\sin((l(\\theta)-\\theta)/2)}", "9857718a6c3bf1bdcdcdc464f4d8a09e": "\\mathrm{2\\ HSO_4^-\\ \\xrightarrow \\ \\ 2\\ H^+\\ +\\ S_2O_8^{2-}\\ +\\ 2\\ e^- \\ \\ \\ \\ E^0 = 2.123 V}", "9857ae8bf293c27dab916890f230bbdc": "R=\\displaystyle{\\text{Dom}(R)}", "9857d1458866b4db36534a2e0f7e111c": "U(\\theta) = \\underline{u}(\\theta_0) + \\int^\\theta_{\\theta_0} \\frac{\\partial V}{\\partial \\tilde\\theta} d\\tilde\\theta", "98582077ba6a186626412623e2c73348": "N = \\left(\\frac{Vf}{\\Lambda^3}\\right)\\textrm{Li}_{3/2}(z)", "98583556e5a5b0d1486e041cab8db03a": " \\theta_1 = \\angle DOE, ", "98584c0a9b81b80894426e0ce26d5e0f": "y \\prec z", "985856bec9904a669d0e4b8849653594": "\\Re^3", "98589afde4c564719ef9ade8649a84d7": " \\omega=Ae^{-\\lambda\\sqrt {-(x^2+y^2)}}", "9858b3c9e992cb795923e399b66dc156": "\\Box A_{a} = 0", "98590f2290e9b514027712a07b4106ea": "\\int_\\Omega \\dots\\, \\operatorname d\\omega_{\\text{i}}", "985921d5cad738c75937ec33632f268f": " k=k_0", "9859396e77ce317834e23fd6e454aa11": "n_r (t) = n(t) * h_r (f)", "985972acf56a43a4965fb331c4b2cd8a": "\\begin{align}\n \\frac{R_2}{R_1} &= \\frac{R_x}{R_3} \\\\\n \\Rightarrow R_x &= \\frac{R_2}{R_1} \\cdot R_3\n\\end{align}", "98597adf53743b4d7bab086c1f929198": "i = 1,\\ldots, n", "985985d48e32aab1ff4aac3a4ff94316": "C_{n}= (A_{n-1}^{xy }) (B_{n-1}^{xy }) (C_{n-1}^{xy+1}) ", "9859f6cedf225934352314e07a308f7d": "\\displaystyle{F_R(e^{i\\theta})=F(Re^{i\\theta})=\\sum a_n R^{-|n|}a_ne^{in\\theta}.}", "985a64d1f1454ad1c8b40b21525d9e40": " K\\backslash G/U = A_+.", "985a7297d105b9cf9ec79f8713fe71de": "\\mathfrak{B}=\\langle B, F\\rangle ", "985a82d87d20b156109ef0f55aff4b05": "\\Gamma(x+1)=x\\Gamma(x)\\,", "985a990ca37087e405641bda7b6430cf": " \\frac{1}{p!} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p} a_{\\lbrack \\mu_1 \\dots \\mu_p \\rbrack} = a_{\\lbrack \\nu_1 \\dots \\nu_p \\rbrack} ,", "985ad162733189c001ea898372ff5f32": "ax^2+bx+c=0,\\,\\!", "985ae50d5b00333ea37eb73056a4076d": "[Q_f,Q_g]=i\\hbar Q_{\\{f,g\\}}", "985af52dc328b0fb12b3c898883bae2a": "T=|t|^2=1-R=\\cfrac{1}{1+\\cfrac{m^2\\lambda^2}{\\hbar^4k^2}}= \\cfrac{1}{1+\\cfrac{m \\lambda^2}{2\\hbar^2 E}}\\,\\!", "985b3ebc6b1f2a8aa24c070892d7ba4a": "\\frac{dn(I)}{dt} = n_2 \\frac{dI}{dt} = n_2 \\cdot I_0 \\cdot \\frac{-2 t}{\\tau^2} \\cdot \\exp\\left(\\frac{-t^2}{\\tau^2} \\right).", "985ba55adb6149825bd3711a13aabd14": " \\hat N ", "985beab6163d4b8dd89d38f0a7c91a2d": "\\eta : k \\rightarrow H", "985c265ff157f5ac7ff0a381009524de": " |1 \\rangle", "985c6e978e993ca6fcf6e3e143dc18da": "f(x) \\ge x \\cdot b - \\beta", "985c8dfa90dc071229e021b6572577b6": "\\mathbf{X} \\geq 0, \\qquad \\forall i,j\\, x_{ij} \\geq 0.", "985cb7263f4f001e0c105600f10de85f": "\\mathit{l} \\lessapprox 10 ", "985cfd4d2661b045a996a653c8f4efa4": "\\hat{\\boldsymbol{\\imath}}(t) = (\\cos\\Omega t,\\ \\sin \\Omega t ) ", "985d06e3208ff19e6e31fa22a49e87ba": "y_i = a_i + F(b_i-c_i)", "985d560f3f99968ba8f338fb1e79f75a": "e^{j \\omega t}", "985e41a610a4a9ccbf40c26313310791": "p^e", "985edca3210ebb1e8c8458150d9a036c": "\\Sigma F = -kv_x = ma_x", "985ef0c612071d0b328dab543ae49dfb": "(I-T)^{-1}\\boldsymbol{1}\n=\\begin{bmatrix}2.75 \\\\ 4.5 \\\\ 3.5 \\\\ 2.75\\end{bmatrix}\\,,", "985f9e9b618128244ada7dbe74320a69": "\\frac{5}{12}\\left(3+\\sqrt{5}\\right)\\, s^3", "985faf6a4a0a69eacd8e03f92a6bbe68": "{\\overline{\\mathbf y}}", "985fb249ca1452fb3a304b2f30e6684d": "L_\\text{target}", "985fde0688ce97bcc8a3c5f0172f6f9b": "B = 1", "985fe6c1b6142ded3e92725e35c54db0": " D^2= \\sum_\\text{cyclic} a^2S_A\\left(\\frac {p_a}{K_p} - \\frac {q_a}{K_q}\\right)^2 \\, ", "9860c946ddcd8eabb543849996004706": "r:=r-sb;", "9861201ffb798826dcc72a15d08836c6": "x_n \\in \\mathbb{C}", "986191a3fd7ef20038c1ec49b1d8c5ab": "N\\rtimes H", "9861a4b635cee41697e4162b2dbd0dc2": "X_t = \\lbrace x_t^{[1]}, x_t^{[2]}, \\ldots , x_t^{[M]} \\rbrace", "98622470fda3410c76c56ffed02a85f2": "{dC_z\\over d\\alpha} >0", "986267d11200d5006c1d4b6f40f30f72": "L(s)=\\sum\\frac{a_n}{n^s}", "98627ac32dd68cb13e9749a705eab5c0": "\\left|\n\\begin{array}{ccc}\n x_1 & x_2 & x_3 \\\\\n x_3 & x_1 & x_2 \\\\\n x_2 & x_3 & x_1\n\\end{array}\n\\right|=\\left(x_1+x_2+x_3\\right)\\left(x_1+\\omega x_2+\\omega ^2x_3\\right)\\left(x_1+\\omega ^2x_2+\\omega x_3\\right),", "98627b96b34b4f3dfbf9762be4962bca": "10000^{-1/2}=10^{-2}=1/100", "9862fd60dbc85a38f33415f1b59d693f": "|\\Psi^+\\rangle ", "9863260b01a1aaffb0c60f5f9803818e": "\\lambda^{'}_{\\beta}|{\\Phi^{'[{{JC}}]}_{\\beta}}\\rangle=\\langle{\\Phi^{'[{DK}]}_{\\beta}}|{\\psi'}\\rangle=\\sum_{i,j,\\alpha,\\gamma}(\\Gamma^{'[{D}]j}_{\\beta\\gamma})^{*}\\Theta^{ij}_{\\alpha\\gamma}(\\lambda_{\\gamma})^2\\lambda_{\\alpha}|{{\\alpha}i}\\rangle", "986329e47617045174e319b4d024d465": "\\delta_{ij} = G_{ij}", "98634599cbff5fc152a537e14c03fae0": " \\{a^n b^n c^n d^n| n \\geq 1 \\} ", "986356584b4d18016c99e1a529e0af7a": "p_2\\,\\!", "98640f47806e7b7bc50a5098d1c35fda": "\\textstyle R_{x}=\\bold A \\bold R_{s} \\bold A^{*} + \\sigma^{2}I=\\sum_{k=1}^M \\lambda_{k}e_{k}r_{k}^{*}", "9864695c4d14b6dcef8aa10fd56f39f3": "R_{0202}=-1,", "98646e560286cca2db58b2931af1a54b": "U = \\left(1 - \\frac{r}{2GM}\\right)^{1/2}e^{r/4GM}\\sinh\\left(\\frac{t}{4GM}\\right)", "986485eaffaa570aa35409e1db7593a4": "(\\neg B \\to \\neg A) \\to (A \\to B)", "9864e2f174bc3a84b9989175dee72e44": "w_1,\\ldots,w_n", "986525eed918a316a70ff5c7ccc78581": "B^{(b-1)/2}\\equiv -1 \\pmod b\\;", "98657b8236c2a12790f9f59aba5ffda7": "\\vec{m} \\in M", "98658f88641759c4d5d6b8afc9741d85": "{\\rm Tr} \\left[ \\bold{F}\\wedge\\bold{F}\\wedge\\bold{A}-\\frac{1}{2}\\bold{F}\\wedge\\bold{A}\\wedge\\bold{A}\\wedge\\bold{A} +\\frac{1}{10}\\bold{A}\\wedge\\bold{A}\\wedge\\bold{A}\\wedge\\bold{A}\\wedge\\bold{A} \\right]", "986590e145334f8baf66d4dcd28c1ebe": "\\mathrm{C \\ \\xleftarrow {k_1} \\ A \\ \\overset {K} {\\rightleftharpoons} \\ B \\ \\xrightarrow {k_2} \\ D}", "9865d9e0ae0b1071ecc302697c8fa173": "\\sum _{n=1}^\\infty \\frac{p_n} {10^{n + \\sum \\limits_{k=1}^n \\lfloor \\log_{10}{p_k} \\rfloor }}", "9866380598810f2a512541392e253f77": "U(\\sigma)", "98664a764db791ab0fd078cf23463c22": "{{O}}({M\\cdot\\chi}^3)", "986696574ed62baf3bf82c5fbe353530": "c > \\log_b a", "9866ebb4fe65558e24b1f5e2e1a70d8b": "\\delta W = \\sum_{i} \\mathbf {F}_{i}^{(T)} \\cdot \\delta \\mathbf r_i - \\sum_{i} m_i \\mathbf{a}_i \\cdot \\delta \\mathbf r_i = 0", "986707cc3075a68e098d467718766716": "(\\sigma_1\\sigma_2\\cdots\\sigma_{p-1})^q.", "9867429f6693f89afc2186239d83fbde": "P(X\\mid A)", "9867614a8d8ecb6a349015e9bc7d6e82": "a \\in A\\;", "986761ad8cec98ef5765449d50dd7008": "[\\forall y (y = y) \\lor \\exists x ( x = x)] \\equiv \\exists x [ \\forall y ( y = y) \\lor x = x]", "9867734f0a505e40dec629403cab7bea": "\n\\begin{align}\ny-x^2&=0\\\\\nz-x^3&=0. \\,\n\\end{align} ", "98677ea1fb19d4e41cadc595417247d6": "a_i=a_{i+n}", "986799185fcafdb821029d0b63156d16": "{\\it N}", "98680560861e3766bada84c70172e206": "\\epsilon^0 = 0", "986809da3533b2ea27b85d7be65aed2f": "a^n=1", "98685ec6ca80b9c5614a2b201afad572": "\\int\\frac{r^3\\;dx}{x} = \\frac{r^3}{3}+a^2r-a^3\\ln\\left|\\frac{a+r}{x}\\right|", "9868942822e19cc34e70ba1ec993faac": "\\mathbf{u} = \\boldsymbol{\\nabla} \\Phi.", "98689cb1636200a7e48c4ddb947f9db4": "r= \\frac{R}{R+G+B},", "9868b1b4d25c5799a777d637213666d5": "\\|(\\lambda I-A)^{-1}z\\|\\leq\\frac{1}{\\lambda}\\|z\\|", "9868faf590eddc910f481242face5546": " R_{sp} = ", "9869538e2977e63e9a4020b93314de57": "e^{\\sqrt{\\ln{n} \\ln{\\ln{n}} }}", "986aa3b626b8090504cdd6a8a9ef6e5b": "\\gamma_x:=\\{\\Phi(t,x) : t \\in I(x)\\}", "986aa83d39f697cb8e658357f1e7858d": " (\\Delta \\otimes \\mathrm{id}_H) \\circ \\Delta \\circ \\eta = (\\mathrm{id}_H \\otimes \\mu \\otimes \\mathrm{id}_H) \\circ (\\Delta \\otimes \\Delta) \\circ (\\eta \\otimes \\eta) = (\\mathrm{id}_H \\otimes \\mu^{op} \\otimes \\mathrm{id}_H) \\circ (\\Delta \\otimes \\Delta) \\circ (\\eta \\otimes \\eta) ", "986abed9366ae6ea6379234ef443bf6e": "\\scriptstyle f_0 \\,-\\, \\frac{\\Delta f}{2}", "986acfd8a5df715f11971b0578e56b8f": " \\pi = \\lim_{n \\rightarrow \\infty} \\frac{2^{4n}}{n {2n\\choose n}^2} ", "986b1b3d73dcb268d64b7e64c9741b51": "s=s+\\Delta s", "986b3264186d4278533b6be3896fb5c2": "x_{1...\\lambda}", "986b48d5e763c1e00291cfcc64031374": "\\Delta[n]\\to C", "986b651dafc2a10b9569af2d34955524": " m^{-1}(x) ", "986bb652ba67e24f5ef27ecc8c60f346": "(X+Y)^\\pi = X^\\pi + Y^\\pi\\,", "986c22f151c46acac223b858e3fcf6fd": "\\in", "986c335dde2afe3654bfc4d3eccd4d02": "k>c+d-2.", "986c4442ade342dee445ee3078b8201f": "X^H", "986c6d7e62e40ea1c63e8cae47b6870a": " \\left( {j - \\sigma - {1 \\over 2}} \\right)^2 - {1 \\over 4} \\ge 0, \\quad \\forall j . \\quad \\quad (16)", "986c6df6807be96e1aa979500d3caf53": " p(z)T(z)^{-\\frac{\\gamma}{\\gamma-1}}=constant ", "986cadb6a657779837c1f2c2ada815d9": "\n -\\frac{\\partial^2\\zeta}{\\partial{t^2}}\n + \\nabla\\cdot\\left( c_p\\, c_g\\, \\nabla \\zeta \\right) \n + \\left( k^2\\, c_p\\, c_g\\, -\\, \\omega_0^2 \\right)\\, \\zeta\n = 0,\n", "986dad742911707c3ea22e7df880f618": "\\Delta p (2 \\pi rdr)= z\\Delta F_x", "986dc357a8ac6f34890e5ab3690209be": "= \\frac{1}{n\\!+\\!1}+\\frac{1}{1\\!+\\!2n}\\,_1F_2\n\\left(n\\!+\\!\\frac{1}{2};\\frac{1}{2},n\\!+\\!\\frac{3}{2};\\frac{-\\pi^2}{4}\\right)", "986dea43ff7a396f56c4bb4801dd6351": "1+\\sqrt 2", "986e4d9e6779dc92372aa5a375855bbf": "R U D B^2 U^2 B^\\prime U B U B^2 D^\\prime R^\\prime U^\\prime,\\,\\!", "986ec564a37b1bd92b85e5ee5e877e9b": "M = \\mu(\\mathbf{x}, \\Sigma_I, \\Sigma_D)", "986edf4ec525a682afed7c50fb616586": "\\mathfrak{o}_{2n},", "986f6aecb4406e2cb2e8c7d55d9ef1e3": "\nU_1 \\in [10V,11V]\n", "986f6c9a1aea4c11538eba46f603e8ff": "\\psi^\\alpha_i", "986f9299da8e9daeea943f38c3de2031": "\nf(t)q(t) = \\frac{f_{0}}{2} A \n\\left( \\sin \\omega_{p} t + \\sin 3\\omega_{p} t \\right)\n", "986fb20475cbbc1cce8b8908889d1b22": "\\textstyle s_{i}", "986fce6dc5b31ae3349b51264b0d242d": "a\\mathrm{N}p", "986fd5588a45298fcf8a29226853a09b": "\\phi = \\varepsilon\\delta", "986fe139f1b6ca29129f8cb8568a9cc5": "X^{\\dagger}", "987010df24d66a182524aecebbeeb4a1": "\\textstyle E\\left( x_{i}\\right) =0", "987024c6eb2a9a892fc4fb9337dcb857": "\\scriptstyle \\alpha_n", "987053eaba2f394160c37d26759b438b": "|{\\Phi^{[2..N]}_{\\alpha_1}}\\rangle", "987067e097bca33f019c63b1697e3c84": "\\omega.", "9870b2a64a5653448f3e9550920d61c1": "\\min(|S|,r)", "9870fdfe24352d7cbbd81b1f6dc55106": "2^m=3^n.\\,", "9871027d6077ee8dd51b8e00f2b66cf8": " \\log\\int \\exp(i \\text{tr} X^3/6)d\\mu", "98715274481718d357b873eeb2777577": " w^2 + w - 2t = 0 \\!", "98716b9321c75a1742681817ba4cd961": "(5,6)", "9871c9da26ba3c6e6d0fa7f3c532213d": " r-1", "98721063d9b22daab67f6494c3032243": " I = \\int e^{-ax^2/2}dx = \\sqrt{2\\pi\\over a} ", "98732c0a5cd98f4cd922122f52720a3f": "e^{2f}", "98738ffd6d89a9f2ac6aa6169e2cd93a": "\\partial_\\nu \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_\\nu A_\\mu )} \\right) = \\partial_\\nu \\left( \\partial^\\mu A^\\nu - \\partial^\\nu A^\\mu \\right), \\,", "9873d1b0487837434b6faa81899fa4a9": " \\cfrac{\\Gamma[\\alpha_1,\\ldots,\\alpha_n] \\vdash p[\\alpha_1,\\ldots,\\alpha_n]}\n{\\Gamma[\\tau_1,\\ldots,\\tau_n] \\vdash p[\\tau_1,\\ldots,\\tau_n]}\n", "98744d972fc93a4d59198f6da998ea45": "H=T^{0.5+\\varepsilon}.", "987479f2b5df6fca51ff0632fb80110a": " \\frac{\\left|F(\\omega)\\right|^2}{\\Delta f} = 4k_BT \\cdot 6 \\pi \\eta R ", "987481c3975cc6861f5864f9256ed959": "\\lambda_\\pm = \\frac{B\\pm \\sqrt{B^2+4M^2}}{2} \\text{.}", "9874b85f218dbda788aaa7e5d5636c9c": "x \\in \\partial \\Omega", "9874c72ce1a4c35cdd2b9eb260dc5861": "D\\tilde{u}=[u',u'',\\ldots ]^T", "9874df771ca47500239ee27fce4a928a": "\\theta^* = \\max_{\\theta} P(Y|\\theta)", "9875104a5b3df00a2ceb7795de530d5b": "\\begin{cases}\n y_t = \\textstyle \\sum_{j=1}^k \\beta_j g_j(x^*_t) + \\sum_{j=1}^\\ell \\beta_{k+j}w_{jt} + \\varepsilon_t, \\\\\n x_{1t} = x^*_t + \\eta_{1t}, \\\\\n x_{2t} = x^*_t + \\eta_{2t},\n \\end{cases}", "98756fa1ecd9026974e97dccbc42cbfd": "\\left\\langle \\partial^{\\alpha} \\delta_{a}, \\varphi \\right\\rangle = (-1)^{| \\alpha |} \\left\\langle \\delta_{a}, \\partial^{\\alpha} \\varphi \\right\\rangle = \\left. (-1)^{| \\alpha |} \\partial^{\\alpha} \\varphi (x) \\right|_{x = a} \\mbox{ for all } \\varphi \\in S(U).", "9875780a00f174be95de18746addc909": " d^*(A) = \\limsup_{N-M \\rightarrow \\infty} \\frac{| A \\bigcap \\{M, M+1, \\ldots , N\\}|}{N-M+1} ", "98757c7e3d0c892ee2e5b987be2cdf26": "{\\tau}_F ", "9875f007f03bae1d7c285721567c70ac": "X \\prec Z", "9875f9eef0b36c99e0b425d8a487a150": "o(i)\\,", "987617783c9c52704f0074dd5d316a4a": "S\\to S + \\int d^dx \\,\\alpha(x)\\partial_\\mu\\left(\\overline{\\psi}\\gamma^\\mu\\gamma^5\\psi\\right)", "9876188c2cb3ef108dc11ebb1ee50028": "H_3(x) = 8x^3-12x\\,", "98764333a8c4785af249dd18312b9379": "g_0 \\ll 1", "987643fc3f09bbdb24a5ac83ad8ae65b": "-x'", "9876d66778b4523cf82deaf60521a68b": "\\textit{e}_{ex}", "987708e23dbbf243cbca8202d39c347b": "\\widehat{\\sigma}^2={1 \\over n-m}\\sum_{j=1}^n \\widehat{\\varepsilon}_j^{\\,2}.", "987727a989beabfd731a6d045b23e06b": " [\\alpha]_\\lambda^T = 100\\alpha/l\\rho\\,\\!", "9877624ac9ded013ad7f035660ce7206": "k_*\\to\\pi_*\\circ F", "9877b04a500cbd475447e13f8853c517": "\\Sigma_2=\\{a,d,e\\}", "9877e29b821298b6b523b0daf4c5c395": "q_\\mathrm{cgs}=\\frac{q_\\mathrm{SI}}{\\sqrt{4\\pi \\epsilon_0}},\\quad \\mathbf E_\\mathrm{cgs} =\\sqrt{4\\pi\\epsilon_0}\\,\\mathbf E_\\mathrm{SI},\\quad \\mathbf B_\\mathrm{cgs} ={\\sqrt{4\\pi /\\mu_0}}\\,{\\mathbf B_\\mathrm{SI}}", "9878248efc9e4991cddf9efdd78cfbbe": "2 a_3.\\,", "987862e03f81e678c47f456c6ab2a07a": "\\approx 0", "987898c604af706ecfcabe04e6757aa6": "\\langle e,t\\rangle;\\qquad \\langle t,t\\rangle;\\qquad \\langle\\langle e,t\\rangle, t\\rangle; \\qquad\\langle e,\\langle e,t\\rangle\\rangle; \\qquad \\langle\\langle e,t\\rangle,\\langle \\langle e, t\\rangle, t\\rangle\\rangle;\\qquad \\ldots", "98789c0e04acf588f1e123366006e135": "\nV=\\frac{GM}{r}\\left(1+{\\sum_{n=2}^{n_\\text{max}}}\\left(\\frac{a}{r}\\right)^n{\\sum_{m=0}^n}\n\\overline{P}_{nm}(\\sin\\phi)\\left[\\overline{C}_{nm}\\cos m\\lambda+\\overline{S}_{nm}\\sin m\\lambda\\right]\\right),\n", "9878a52d6f5df5e1bb95f256471d956d": "\\text{liftM2} \\colon \\forall M \\colon \\text{monad}, \\; (A_1 \\to A_2 \\to R) \\to M \\, A_1 \\to M \\, A_2 \\to M \\, R = ", "9878ff6614a4d54bcea8d8c3d0940c0c": "\\hat{y}(k) = C \\hat{x}(k) - D K \\hat{x}(k)", "987934654add515401f6cfc2be01a497": " H H^{\\mathrm{T}} = n I_n \\ ", "987958784e969ac7e4709853a4cb18ab": "\\Phi \\left(\\eta,\\tau \\right)", "9879cc17b62156ebe60f9742ffc68646": "\\{p\\in M:\\varphi(p)\\le b\\}", "9879f3dff384a83550d35245f6ff854b": "d_{k+1}0\n\\end{align}\n", "98867d73f721ded278861d9f83b7c061": "\\sigma_8", "9886f797a0cc982c3f2b7bef230eeb8d": "\\frac{u_2 (m_2 - m_1) + 2m_1 u_1}{m_1 + m_2}", "98870c922978806190df9bde245bb0c0": "\\Delta G = -nFE \\,", "98871b4b4ae9c70d1f8960d17c77570d": "q(x,y)", "9887442ec41b6be7ad2d8c2370eaf96b": "C_1 \\supseteq \\cdots C_k \\supseteq C_{k+1} \\cdots, \\, ", "9887c74ffd2179adeaeba88800d1b5af": " s=\\frac{a+b+c+d}{2}.", "9888705c26479324135cb44e121aca97": "P(A\\mid B)", "988887c3ad8eb63a055e64513af49eff": "m(f) = \\operatorname{std}(f)\\prod_{p|2\\det(f)}{m_p(f)\\over \\operatorname{std}_p(f)}.", "988891b6a0f742985c3d9896e1cb24b9": "u(\\mathbf{x})", "988914d76bf937293ce876b0206dbea9": "\\operatorname{ord}_P(t)=1", "9889342650134a7faaed302cf384b34f": "|F=I+1/2,m_F=\\pm F \\rangle", "988941e61284130fbbe5d5e0f06e9a0b": "\\tan\\alpha = \\frac{C_{ym}}{C_{xm}}", "98894e743108b49b797de9714a0a2aed": "a_z", "9889874e46829d24bb12fd75a66a8355": "\\mathcal{P}\\left(X\\right)", "98899f26b88dd23f00d0350cf9abe94c": "(\\nabla_v w)^\\bot ", "9889c28dc5d410ffb5ed2cb68efc45cc": " g_w(\\zeta)={\\zeta\\over (1-\\overline{w}\\zeta)^2},", "988a19c218b641f7535693c222828588": "\\!\\mu_2", "988a3ed3b17a905938e21bb927272270": "\nc = (x^2 + y^2)^{1/2}\n", "988a42cb3f7cda3853b1ca0193d026a0": " TL = T \\times L\n = \\begin{bmatrix}\nPB & PG & PS \\\\\nHB & HG & HS \\\\\nDB & DG & DS \\\\\nJB & JG & JS\n \\end{bmatrix}\n", "988a4a88842999333799550d221a19f6": "T_m(T_n(1)) = T_m(1) = 1 = T_{mn}(1).\\,\\!", "988a71f795c8cf2e22303c6c8598248f": "B = A[\\![t_1, \\ldots, t_n]\\!]", "988a738cd8d47407edf4af045776ab89": "x^2-4x-1", "988a975eccd0989112fe4959994576b4": "\\begin{matrix} {1 \\choose 1}{11 \\choose 2}{4 \\choose 2}^2 \\end{matrix}", "988aa2bcffd35b246a85e947295f5b14": "\\lambda = \\mu/v", "988b169774fd1b7118ca8e27f38c1764": "T(t)\\ ", "988b8bac8f2f58eca9f12289f082080f": "-2\\boldsymbol\\Omega_0\\times\\boldsymbol v", "988b9436bc6d15e6f4ffefc01a2da8a6": "\n{\\mbox{LIVE}}_{out}[s] = \\bigcup_{p \\in succ[S]} {\\mbox{LIVE}}_{in}[p]\n", "988b975f9fde2860f98c9b18ec063428": "1 \\leq k \\leq m", "988bb684ca9a01d608b96e28104ce257": "\\frac{a}{b+c}+\\frac{b}{a+c}+\\frac{c}{a+b}\\geq\\frac{3}{2}.", "988be7868f30c98ec3251b26644b3699": " \\min_{x \\in X}\\;\\; f(x) ", "988c177254e98ba9ef4751fbbf9854c7": "R(N) = \\{ M \\mid \\exists w: M = M_0 + W^T \\cdot o(w) \\wedge w \\!", "988c826113888fcb8cd0ea35c085c5c9": "\\textstyle \\nu", "988d4681846aa6dcb76cac84dd74496e": " m_j\\frac{du_j }{ d \\eta }+l_jb_j = 0", "988d7e07ceac94bb4f9b786375501595": " B_m(x,y)\\,", "988d9d52b57617a9a2e4b9095f7eff9f": "\\lambda_1,\\lambda_2,\\dots,\\lambda_n", "988db5811b71f5eef6104227269fd9ca": "\n \\underline{\\underline{\\mathsf{C}}}^{-1} = \\begin{bmatrix}\n \\tfrac{1}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm yx}}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm zx}}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xy}}{E_{\\rm x}} & \\tfrac{1}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm zx}}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm xz}}{E_{\\rm x}} & - \\tfrac{\\nu_{\\rm xz}}{E_{\\rm x}} & \\tfrac{1}{E_{\\rm z}} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\tfrac{1}{G_{\\rm yz}} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm yz}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\tfrac{2(1+\\nu_{\\rm xy})}{E_{\\rm x}}\n \\end{bmatrix}\n ", "988e0a83d2ad0032359b1e904ce7ac00": "F_{13}", "988e66ec6124120fb23f9a25c306bdd6": "41^2", "988f1e03a9ad140dca0b3a43763992d6": "\\int\\operatorname{arcsch}(a\\,x)\\,dx=\n x\\,\\operatorname{arcsch}(a\\,x)+\n \\frac{1}{a}\\,\\operatorname{arcoth}\\sqrt{\\frac{1}{a^2\\,x^2}+1}+C", "988f3c2e3cac4d0c354f672761197434": "\\lambda(C_1 \\cup C_2) \\le \\lambda(C_1) + \\lambda(C_2)", "988f4f7a4411569d00f0520c4d348c4f": "f^*s=s\\circ f", "988f549c88f980edde7152c688549774": "|\\operatorname{intersect}(j_0/\\sqrt{n})| \\leq \\sqrt{n}", "988f5e049c9cc2af8c2e4407e0a18157": "\\frac{dv}{dr}=0", "988f969692e95cf8119258e1df333bb7": "\\scriptstyle \\psi^\\dagger \\nabla\\psi", "989049952fe42c1bd8322d8c6abaf2a5": "G = \\int_0^{\\pi/4} \\ln ( \\cot(t) ) \\,dt \\!", "989071d45b1dee1a8947144f5c57cd8a": "\\scriptstyle{{73+i43\\over 2}=36+i21}", "98908922aa44be47b4ff1781e2517cf2": "x = \\sum_{i}\\ x\\ v_{i}^{T} \\ v_{i}", "989099cdbcb8fb7acff403796e53d203": "(h(x_1),\\ldots,h(x_k)) \\in R^B_i", "9890b7093dc200b5212329bdb3ae1f2c": "x^2\\equiv y^2\\pmod{N}", "9890d3c4c289da6e871e76a972fd048e": "\\,\nP\\left( \\gamma+1;\\;\\gamma y_0,\\;\\gamma y_1+y_0, \\;\\gamma y_2+2y_1, \\;\\gamma y_3+3y_2,\\ldots,\\;\\gamma y_n+n y_{n-1}\\right)=0\n", "9891a8b725428a94924af3030a52866b": "\\ \\epsilon_1= n^2 - \\kappa^2", "9891b4e1a5fb6b75333bab87dabf392f": "(t)", "9891c1a01ba97a84485bacf71c8f2c5f": "\\int_0^t \\mathrm{ROI}(\\tau) \\, d\\tau = (-\\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{Q} +V_p) \\int_0^t C_p(\\tau) \\, d\\tau + \\mathbf{U}_n^T \\mathbf{K}^{-1} \\mathbf{A}", "9891de87f8b0b0803daf23cba9da7eef": "\n\\begin{align}\n \\partial_{t}u \n +\\partial_{x} & \\left\\{\n 35u^{4}+70\\left(u^{2}\\partial_{x}^{2}u+\n u\\left(\\partial_{x}u\\right)^{2}\\right)\n \\right. \\\\ & \\left. \\quad\n +7\\left[2u\\partial_{x}^{4}u+\n 3\\left(\\partial_{x}^{2}u\\right)^{2}+4\\partial_{x}\\partial_{x}^{3}u\\right]\n +\\partial_{x}^{6}u\n \\right\\}=0 \n\\end{align}\n", "9892b277376e5176bac1f6fc16859a00": "H_c(j\\Omega) < \\delta", "98930972c7487068d3419bb45cf7c20f": "\\frac{1}{r^4} P^3_3(\\sin\\theta) \\sin 3\\varphi = \\frac{1}{r^4} 15 \\cos^3 \\theta \\sin 3\\varphi", "98936fe3adb851724a8ae3e67be617f2": "H_\\alpha(A,X) = H_\\alpha(A) + H_\\alpha(X)\\;", "989377518ef47d97fbd72f6436512395": "\\left[\\begin{array}{ccc}\\textbf{diag}(Ax+b)&0&0\\\\0&t&c^Tx\\\\0&c^Tx&d^Tx\\end{array}\\right] \\succeq 0", "9893993b696aaff51888335f2d01757c": "\\theta_a - \\frac{\\Delta}{L_{ab}}= \\frac{L_{ab}}{3E_{ab} I_{ab}} M_{ab} - \\frac{L_{ab}}{6E_{ab} I_{ab}} M_{ba}", "9893c31e19c18425854f15fb3f98de25": "\\mu = \\mu_c^\\ominus + RT\\ln{\\left( \\frac{\\gamma_c c}{c^\\ominus}\\right)}\\,", "98940829e4cb44336dbe42c3e664b7f4": "T^{vap}\\,", "989426f5130ff8ed0573c8c0d84a1508": "\\varphi(x) = a x^2 + b", "989457a4e052ec340856ab796362e233": "\\displaystyle{h_w=h_za + a_z.}", "9894a32cf81747d518c09c6632389d06": "\\scriptstyle \\eta_t(x) ", "9894db45fc814a72cbb287e285340576": "\\tau_\\mathrm{n} = -\\frac{1}{2}(\\sigma_x - \\sigma_y )\\sin 2\\theta + \\tau_{xy}\\cos 2\\theta=0\\,\\!", "9894db52918dc87c80b82349c1154077": "V_{PL}", "9894e1782026d63bc5aa51469b293595": "E_{2}", "9895103a64f297520ccec4f2b1fb36b5": "(a+b)^2-a^2-b^2 = ab+ ba", "989535c8ec2826d2737947c9473ccbb4": "\\boldsymbol{ \\Omega} = \\omega \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix}\\ ,", "98953ae294d82fd3dd5f1f69964a7549": "\\frac{dL}{dt}v+L\\frac{dv}{dt}", "98955f2b63db193cd49ade2fc49940c7": " \n\\mathbf{J} \\rightarrow { 1 \\over {4 \\pi } } \\mathbf{J}\n", "98957a156823bd8b1d9113db6a48cdcf": "\nF(x) = \\frac{\\pi\\mu_0}{4} M^2 R^4 \\left[\\frac{1}{x^2} + \\frac{1}{(x+2h)^2} - \\frac{2}{(x + h)^2}\\right] \n", "9895b5379adcf1f8906ccfefcc355cd4": "x(i\\Delta \\alpha)", "9895dcba013d00b718854752b1ebfb86": "\\Delta\\mathbf{r}_i = \\mathbf{r}(t_i+\\Delta t)-\\mathbf{r}(t_i)\\approx\\mathbf{r}'(t_i)\\Delta t", "9895e161fd29fc6d841605625296e590": "\\mathcal{QS}", "98962bd276797fef43b66450a71d8270": "FK_i", "98964f886b4a3bcd98fac629f3cb6cc1": "b_1,\\ b_2,...,b_m", "98971628da69db9ce0224ba4f305d152": "\\text{var} = \\frac{(n-s)s}{(1+n)n^2}, \\text{ which for }s=\\frac{n}{2}\\text{ results in var} =\\frac{1}{4+4n}", "98972a641379ba4a152a930d9ffc51d2": "R^{-1}{R^{-1}}^T", "98977858d7386b6e43e13190eb4a5c9e": "1+1\\sqrt{2}=2.41421\\ldots", "98978ea44f659372c64095a122a8ff72": "\\alpha \\to \\gamma^{-1}\\cdot \\alpha\\cdot\\gamma", "9897cb7afdf3224093775e8db1273805": "B = 0", "989822040905e9c7b5c736fdc2d4d52a": " L_o ", "98988790ae5380b22502db8971a8d9c1": "2 \\cdot D, 3 \\cdot D,\\ldots ", "9898d57caa9ccea73e6a27bafd2fdc95": "\\mathbf{p}=m\\mathbf{v}", "9898dd4570ae7d562410f6e0f4f4f32a": " { P \\over E } \\ = \\ {1 \\over (i-g)}", "9898df723f0147786c1dac75113f3467": "\np^{-1}_*(X) = \\Big\\{ \\ X^k\\frac{\\partial}{\\partial x^k}\\Big|_v + Y^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_v\n\\ \\Big| \\ v\\in E_x \\ , \\ Y^1,\\ldots,Y^N\\in\\R \\ \\Big\\}.\n", "9898fb67e49331234de7bb1a2cadd955": "42^2", "98990e8b2efecbe6712d77ae37dc90ad": "f_0 = f_1", "989911c15a7b0294a0eb8daffc6ab849": " \\sum a_{nb} u'_{nb} ", "98993f59c3ca795418d7de483652c90f": "\\Delta G_v=\\frac{\\Delta H_v}{T_m}\\Delta T", "9899917b3410b7808e310501f2bdc2cc": "1,024 \\cdot D", "9899fbcaa294107e71e5ae1407d6ef0d": "\\textrm{var}(X_t) = \\varphi^2\\textrm{var}(X_{t-1}) + \\sigma_\\varepsilon^2,", "989a634873c53670208c84313591be3c": "\nH(I)= -\\sum_i P_I(i) \\log P_I(i) ,\n", "989a746fd6d2c2e57598f494b476fc6e": "\\mathcal{L}_\\mathrm{G}", "989abc54cf15456cedb6aed4c35c1de4": " \\equiv Kn = \\dfrac{\\lambda}{l}", "989af4a164f49a48b10eb904c1ae444a": "\\tfrac{1}{2}r(s-a)", "989b080f7bfec032317cdaa3327a717e": "\n C_{k+1} = \\underbrace{(1 - c_1 - c_\\mu + c_s)}_{\\!\\!\\!\\!\\!\\text{discount factor}\\!\\!\\!\\!\\!}\n \\, C_k + c_1 \\underbrace{p_c p_c^T}_{\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{rank one matrix}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!} \n + \\,c_\\mu \\underbrace{\\sum_{i=1}^\\mu w_i \\frac{x_{i:\\lambda} - m_k}{\\sigma_k} \n \\left( \\frac{x_{i:\\lambda} - m_k}{\\sigma_k} \\right)^T}_{\n \\text{rank} \\;\\min(\\mu,n)\\; \\text{matrix}}\n ", "989b254d5b38ec8c3d2f63f2c87dad8d": "\\varphi_x(x)", "989b3dfc65a718d0f44e29a76a64d090": " \\mathbf{J}_{u} = L_{uu}\\, \\nabla (1/T) + L_{u\\rho}\\, \\nabla (-\\mu/T)", "989b51f48c1e8217b4233458a7fc8fb4": " I_p = 0.5 + 0.5 ( \\frac { I_d - M_c } { k - M_c } ) ", "989b992c1d2ddaea9d4a22a05d2fb042": "R(u) - R(v)", "989b9eec77025ad856ca4c65bc7bf96e": "\\text{Var}\\left(Y\\right)\\ge\\text{Cov}\\left(Y,X\\right)\\text{Var}^{-1}\\left(X\\right)\\text{Cov}\\left(X,Y\\right) .", "989baea411beefbf6080cf192ef6ffed": "U_4(3)", "989bb5ac8af2ed7e6d1e4d85d097f132": "U_{thermal} = \\tfrac 1 2 N m \\overline{v^2} = \\tfrac{3}{2} N k T,", "989bc9932b20fd4622e1298838d957e2": " C_s = H P_g ", "989bd5bed0e9728b230c5d8f3266ce7c": "q_1^2+\\dots+q_k^2+p_1^2+\\dots+p_k^2 (+p_n)^2", "989c8ab5fb0c89c92ea32d8bea685849": "C_p-C_V=R", "989c91a076661069179f09ff2b26a545": "\\frac{1}{p^k},\\ \\frac{1}{q^\\ell},\\ \\frac{1}{r^m},\\ ", "989cbfa7900da880bd4aab8f76f68197": "\\hat{f}_2 \\,\\hat{f}_3 = -\\hat{f}_3 \\,\\hat{f}_2", "989d030e5bb193bc789678e2810ae27f": "\\frac{p(\\bar{S}_{2t})}{p(S_{t})} = \\frac{\\int_{\\bar{S}_{2t}}\\exp(\\epsilon q(d,r))\\mu(r) \\, dr}{\\int_{S_{t}} \\exp(\\epsilon q(d,r))\\mu(r) \\, dr} \\leq \\exp(-\\epsilon t) \\frac{\\mu(\\bar{S}_{2t})}{\\mu(S_t)}. ", "989da10fdae482c402029ed50f76986b": "\n \\int_{\\mathcal D}\n \\vert \\alpha \\rangle\\langle \\alpha \\vert\\; {\\mathcal N}(\\vert \\alpha\\vert^2)\\;\n d\\nu (\\alpha, \\overline{\\alpha} ) = I\\; ,", "989df3deacbf8e459d7917267ee1abc5": "\n\tE_3 = \\begin{pmatrix}-0.582075699497237650\\\\ 0.370502185067093058\\\\ 0.509578634501799626\\\\ 0.514048272222164294\\end{pmatrix}\n", "989e4ed84e7ae92cc0d82ac10e1de3d8": "\\max \\sum_0 ^{\\infty} \\beta^t u (c_t)", "989ef4940042465cb5fa65105868e89d": "\\alpha = -1 + \\sum_{k=0}^\\infty \\frac1{2^k+1} \\approx 0.2645.", "989ef915634940c36ff22d14c2b992bb": "\\cdots x_{n}.y", "989f7d75ddffa0471c9a9281daf7eb58": "AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2).", "989faea614757a7b9405bd3ba3997257": " \\scriptstyle \\zeta \\,", "989fb428cb3d87e78834054cb9268751": "\\frac{dT}{dt} = 0", "989ff4e9475f31abcf5d12d5471a2bc5": "\\{ ((x,y),[z:w]) \\mid xz + yw = 0 \\} \\subseteq \\mathbf{A}^2 \\times \\mathbf{P}^1.", "989ffb3bab26e9de4aab0950577c3ba7": "\\phi'_A = \\alpha \\phi_A + \\beta_A \\, ", "98a085a9df6888b7455f9f776db1fc56": " \\mathcal{C}", "98a0d078809739ab47d1b64b2c587641": "\\textstyle Z_{2}", "98a10ef2a5dacde934d5aa5592e1c315": " F(z)", "98a11e43c9363d652802074cc4bf0de7": " E = \\frac{m_0} {\\sqrt{1 - v^2/c^2}} c^2 ~, ", "98a16b5ea3fdb84ad5b59b22e63871e7": "i: Z \\to X", "98a17a314b7092ee5883ad4cb4bc09bb": "E_6, E_7, E_8, F_4, G_2.", "98a1da3c4f73e6250c3411179817b213": "f(x_2,a,g(z_2),y_2) \\{x_1 \\mapsto x_1, y_2 \\mapsto y_1, z_2 \\mapsto z_1\\} = f(x_1,a,g(z_1),y_1)", "98a1f046df07ac35104746d7c30f80e8": "g(x) = \\frac {f(x)} {\\sqrt{|f'(x)|}}.", "98a256f4475bc54464730c150fe7fdfa": "^3 3 = 3^{3^{3}} = 7,625,597,484,987", "98a2b7d46725073fce67a8dca0ae6398": "f^\\#(V(I)) = V(f^{-1}(I))", "98a2cad72e96b5e938d1e3a6d7b4d536": "\\bar{v}(T)=\\frac{0.317398726+4.22806245 \\times 10^{-5}T + 4.20481691 \\times 10^{-8} T^2}{1-2.89741816 \\times 10^{-5}T+1.61456053 \\times 10^{-7}T^2}", "98a32bee90be51f0e3aa4b904100079c": "(K,\\,\\lambda)", "98a34a42600b88233b28fb8707342ae3": "m_\\text{P} = \\sqrt{\\frac{\\hbar c}{G}}", "98a3823beb7602716c9f80ea28c03bc3": "\\int_{\\mathbf{R}^d} {f}(y)g(x-y)\\,dy.", "98a4b0016fe424d9b0a33ccb4dd494a0": "Z_n := \\frac{\\sum_{k = 1}^n \\left( X_k - \\mu_k \\right)}{s_n}", "98a4dcece72fe836f1148f6bad49b212": "\n\\langle \\psi_0| {\\delta S \\over \\delta x}(t) |\\psi_0 \\rangle = 0\n\\,", "98a51314a777f0ffa4c92ba5a92f0047": "\\sqrt[3]{1 + \\! \\sqrt[3]{1 + \\! \\sqrt[3]{1 + \\cdots}}} = \\textstyle \\sqrt[3]{\\frac{1}{2}+ \\! \\sqrt{\\frac{23}{108}}}+ \\! \\sqrt[3]{\\frac{1}{2}- \\! \\sqrt{\\frac{23}{108}}}", "98a55a42d729c90dcc61545be19e95ee": "\\nabla^2\\psi_j(r)=\\frac{d^2}{dr^2}\\psi_j(r)=\\kappa^2\\psi_j(r)", "98a55f60f958de89673c77f7698401e1": " y(n+1) = ay(n) + u(n)", "98a574c384efd0089c7165b975e3a371": " k = \\frac {\\sqrt{2 m_w E}} {\\hbar} \\quad \\quad \\kappa = \\frac {\\sqrt{2 m_b (V-E)}} {\\hbar} \\quad \\quad (4)", "98a585a94bc617aa2f84cb2329dc7c07": "\n\\vec n \\cdot \\left( \\mathbf\\Sigma_i \\nabla v_i \\right) = 0 \\,\\,\\,\\,\\,\\,\\, \\mathbf x \\in \\partial \\mathbb H\n", "98a5ae11711c5b4c75e3e06207db1656": "A,B,C \\in M", "98a5b0423087cef9b7c37524c64cbac7": "\\mathbb{R}^{m} \\ni u \\mapsto f(u) \\in \\mathbb{R} \\cup \\{ \\pm \\infty \\}\\ ", "98a5e0902d199243c0e2b78e03cb674c": "r = \\beta n \\log n", "98a5f98486a33374839a90381dc3a0ad": "L_{1,\\mathrm{loc}}(\\Omega)=\\bigl\\{f:\\Omega\\to\\mathbb{C}\\text{ measurable}\\,\\big|\\, f|_K \\in L_1(K)\\ \\forall\\, K \\subset \\Omega,\\, K \\text{ compact}\\bigr\\},", "98a63294cbbcf0fcf8a011e1c8330368": "\\kappa_o=\\frac{\\sum_{i=1}^{c}n_i(n_i -1)}{N(N-1)}", "98a682eedf5406e60fe920901ae8efd1": "\\rho(\\tilde{A}) = \\frac{\\rho(A)}{\\rho(A)+\\epsilon} < 1", "98a7d094e6aa4337dd8750c956a9d439": "a_{ij}u_{x_ix_j}", "98a7e2353918acb114577be0f42b901e": " H_a(s) \\ ", "98a81efebf937ae93fc075ca9fb58629": "\\scriptstyle X \\,=\\, \\bigcup_i X_i", "98a83a6f66eb0f7e8357bb4a127cd603": "{{\\mathbb I}\\left\\{{A}\\right\\}}", "98a872e9f2137648366c57a728cc771d": "\n\\int \\exp\\left[ \\int d^4x \\left ( -\\frac 1 2 \\varphi \\hat A \\varphi + J \\varphi \\right) \\right ] D\\varphi\n", "98a8b86385eaa89ff35f0179af0c3e37": "h = \\left(\\frac{4\\hat{\\sigma}^5}{3n}\\right)^{\\frac{1}{5}} \\approx 1.06 \\hat{\\sigma} n^{-1/5},", "98a9017bca2f1a51ba93ebc0cfd54bba": "\\mathcal{F}\\left[\\frac{df(t)}{dt}\\right] = i \\omega \\mathcal{F}[f(t)]", "98a902ed9c6382ee085d4e666b6440e6": "dI/dV", "98a927085d4972e8f9b03b4450b9fea0": " \\ln{(y)} = \\ln{(a)} + b x + u, \\,\\!", "98a99c161f16f767f882c34c3e239d44": "v^{\\circ}:= \\min_{d\\in D,\\,z\\in \\mathbb{R}} \\{z: z \\ge f(d,s),\\forall s\\in S(d)\\}", "98a9dee8a08bcda96cc30d87a806194f": "\\theta_E=\\frac{1}{\\displaystyle 1+\\sum_{i=1}^n K_iP_i}", "98aa2a405b2d52682e8595ef6b05e1f6": "\\mathbf{r}_\\mathrm{cg} = \\frac{1}{W} \\sum_i w_i \\mathbf{r}_i,", "98aa66a73f6babd331c978606da8f3fe": "\\mathrm{{}^1O_2 + intact\\ DNA \\ \\xrightarrow{} \\ {}^3O_2 + damaged\\ DNA}", "98aacd91319ec67ae27f6c95d568526d": "{c \\over {c+v}}", "98ab0c78c8748f3e9b5a0ae96a6c9141": " E=\\hbar \\omega \\,, \\quad \\mathbf{p}=\\hbar\\mathbf{k}\\,,", "98abd742a29ca39693248d12e4037eb2": "c = \\frac{\\lambda}{\\tau}.", "98ac1949d2699d3b121bca59ba8523ec": "\\frac{A}{A \\lor B}", "98ac6a2f86c527e3fe2b9450c3830b67": "P(D)y=f(x)", "98acc57763f2be7819e1aa1f0cec8541": "\n\\ E_{out} = { \\sigma * \\ T^4}\n", "98acd655ef2a5a075e6969ff49d9980b": "\nC\\frac{dV}{dt} + \\frac{V}{R}=0\n", "98acdc32a4152fb385c5f363dd6b1fca": "_{s.12 \\,}\\!", "98ad4a79c5570155ea83264c0fdf96ce": "x_1+x_2+ \\dots +x_n", "98adc7bcf5562baab2aa7df4e965fbec": "\\displaystyle{\\|W(F)\\| \\le e^{R^2/2}\\|\\widehat{F}\\|_\\infty.}", "98adf15497e69eeec9683f78dcbe63ee": "\n\\int_0^\\infty |f(r)|^2r\\operatorname{d}\\!r = \\int_0^\\infty |F_\\nu(k)|^2 k\\operatorname{d}\\!k.\n", "98ae4199d2fbe0e4780b47228e4484d8": "[a+(n-1)d]", "98ae5735dd842059bc170cd22b191bad": "\\operatorname{Var}\\left(\\sum_{i=1}^N X_i\\right)=\\sum_{i,j=1}^N\\operatorname{Cov}(X_i,X_j)=\\sum_{i=1}^N\\operatorname{Var}(X_i)+\\sum_{i\\ne j}\\operatorname{Cov}(X_i,X_j).", "98aee55ceea197a4be88b1a9a4b29e7b": " \\Delta A \\Delta B \\geq \\frac{\\hbar}{2} ", "98af198f8ede064a6cbcc2fda0e1b60f": "t_\\text{r} = t - R/c", "98af1f248b8501c40260acee5c92b83f": " \\rho_t+\\rho u_x+u\\rho_x=0 ", "98b08e8f42a46e1025b8738b86b85ab7": "\\sigma^{MAP}= arg\\underset{\\sigma}max P(D| \\sigma,T^ML, M) P(\\sigma|M)", "98b0b07ceab4307926dfcec2ed3f8e23": "V_{in} = A_{1}\\cos(w_{1}t) + A_{2}\\cos(w_{2}t)", "98b0b1a7c120bbe38754c4e1fcc8bea4": "e^{-i\\phi}", "98b0bcbab0feef0db5d86177112ca5f6": "-\\sqrt{\\frac{14}{45}}\\!\\,", "98b0ef6820196608aa565c0ff14e4417": "\nevidence = P(male) \\, p(height | male) \\, p(weight | male) \\, p(foot size | male) ", "98b0f576c4c49797c6fc2824c84628fa": "\\frac{\\partial^2 \\mathbf x_i(t)}{\\partial t^2} m_i = -\\frac{\\partial}{\\partial \\mathbf x_i} \\left[ V(\\mathbf x_i(t)) + \\sum_{k=1}^n \\lambda_k \\sigma_k(t) \\right], \\quad i=1 \\dots N.", "98b138f960a584c015cd9ce83d71ca55": "c_{ij}", "98b15fa6cc09d78d541ab98f6e77b640": "Pr[A \\cdot B \\cdot r=C \\cdot r]\\leq 1/2", "98b16779c729fa98659df8c713a78f6e": "\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\end{bmatrix}=\\begin{bmatrix} {\\color{red}e} \\\\ {\\color{red}f} \\end{bmatrix}.", "98b172254590d73ef7fc5e4413d71949": "\\Rightarrow \\Delta U < 0", "98b1ab9f7eb3eb1af06845af0a4376c7": "\\,x = \\ln y", "98b1b6aa6f5f3f3d95aa5c352b6c9112": "j(\\tau) = 32 {[\\vartheta(0; \\tau)^8 + \\vartheta_{01}(0; \\tau)^8 + \\vartheta_{10}(0; \\tau)^8]^3 \\over [\\vartheta(0;\\tau) \\vartheta_{01}(0; \\tau) \\vartheta_{10}(0; \\tau)]^8}", "98b1d867bc7bf7aa67888fae85a737df": "10^{-9}", "98b2076a5753815bf0b77faa4ab99775": " \\tau_p = \\frac{\\rho_p d_p^2}{18 \\mu}. ", "98b20ae70e5f302b8c519398ea149e4b": "\\frac{1}{T}=\\frac{1}{T_0} + \\frac{1}{B}\\ln \\left(\\frac{R}{R_0}\\right)", "98b2894c3fddfc534d1691f379cd429f": "DG=a_{\\text{print}}-a_{\\text{form}}", "98b29981a2a8a925ea31e60f06e997cb": "G_2.", "98b2a945d25f65d49f6307c48aa4396e": "i \\ne j", "98b2d9a33a7a71b0f715840a24c512f8": "y_c\\,", "98b39449324157c5f10441f3f3fa94de": " \\sqrt{\\lambda_{\\alpha }} V_{\\alpha i} \\,\\!", "98b39e2f6d65ef4450021ab04deb8a83": "\\mathbf{\\Delta X} = \\{1, -1\\}", "98b3f21c3f77bf3c4f017acf7bf6e3b2": "U_{L} = {K} \\cdot \\frac{\\nu \\cdot |z^+| \\cdot |z^-|}{r^+ + r^-} \\cdot \\biggl( 1 - \\frac{d}{r^+ + r^-} \\biggr) ", "98b41e40d6c7293ac2d5178673248dd7": "( 1 - T_s) ", "98b41efa8175f54eae61f4d3b01090b0": " f(x) = A \\prod (x - c_n)^{a_n} ", "98b4762c1cf7c8d734566db4e08bdbb8": " \\operatorname{let-combine}[\\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x)] ", "98b4786c298a6c8b3f9c4342172d284a": "84\\,", "98b47fe19317f5f8926192a8d064d0a0": "\\lim_{x \\to +\\infty}{|f(x)|} = +\\infty", "98b47ff486c74f8c8dc62c8e06f36742": "\\vec{f}_0 = \\vec{e}_0", "98b4a12172f2e3a49dba3b964d384a36": "F_{A\\alpha}(\\mathbf{R})", "98b4b0c0cdb22daa7e5accfedf6b3e06": " \\widehat\\phi(k) = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\phi(e^{i\\theta}) e^{-ik\\theta} \\, d\\theta ", "98b4c60650d8fb5ce1223676caaeaa6e": " \\overline{n} =\\frac{a_1 + a_n}{2}.", "98b4cd4732b039788ff7ad31e6f62eb5": "\\begin{align}\n a\\psi_1 + c\\psi_2 &= \\chi_1,\n\\\\\n b\\psi_1 + d\\psi_2 &= \\chi_2.\n\\end{align}", "98b4ff2fae8b12de53866504331be271": "\\cos x", "98b50672372d0c2e408c10b437d7d0c0": "\\mathrm{d}H = T\\left(\\frac{\\partial S}{\\partial T}\\right)_{P}\\mathrm{d}T + \\left[V+T\\left(\\frac{\\partial S}{\\partial P}\\right)_{T}\\right] \\mathrm{d}P.", "98b5179d67b8ce6740a37efa924dd063": "\\mathbf{M}^*\\mathbf{M}", "98b5208aff6987cac11e8be944d0acb9": "a \\perp v", "98b580ebdae6a66fd0401a9213ce0dd6": "Y'_j = \\frac{1}{12h} (1 \\times y_{j - 2} - 8 \\times y_{j - 1} + 0 \\times y_j + 8 \\times y_{j + 1} - 1 \\times y_{j + 2})", "98b5927b8c4350b75fd25a908859a145": " \\begin{matrix} c^{\\phi(n)/p_i} &\\equiv& x^{\\sigma \\phi(n)/p_i} g^{m\\phi(n)/p_i} \\mod n\\\\ &\\equiv& g^{(m_i + y_ip_i)\\phi(n)/p_i} \\mod n \\\\ &\\equiv& g^{m_i\\phi(n)/p_i} \\mod n \\end{matrix}", "98b59fbcc016959af7997786423fa941": "k_{i,i+1}", "98b5ef70a96cc530117a5ec2ec8e9efa": "\\frac{1}{\\sqrt{1-4a}}e^{\\frac{ax^2}{1-4a}}", "98b61ec717c4cfbe527aa2b9090d38fc": "\\iint_D \\left|\\vec{r}_u\\times\\vec{r}_v\\right|\\,du\\,dv", "98b65ec1fa17815f06abe024e6452ca4": "n_i=\\sum_{k=1}^i m(k)", "98b6b43e75262b3710c60603cd653848": "\\mathrm{proof}_{P_{i}}", "98b6f411f33d4e671ae69d1decc458d8": "r_{ij} ", "98b70b7119c93d3d672e5df16a5e1df3": "g^{k\\ell}\\Gamma^i{}_{k\\ell}=\\frac{-1}{\\sqrt{|g|}} \\;\\frac{\\partial\\left(\\sqrt{|g|}\\,g^{ik}\\right)} {\\partial x^k}", "98b718127be1055bc146a20f786c1de0": "C_{p} - C_{V}= V T\\frac{\\alpha^{2}}{\\beta_{T}}\\,", "98b731ff7060651949aa58ab26aa2c52": "\\chi_k\\,", "98b77c8f94317f3b9aa57d64cc9d6d8d": " x = u(1 - u^2/3 + v^2)/3,\\ ", "98b7a0b5e8a08eae31e8d82d07e6d61f": "\\mathbf{AB} = \\mathbf{BA} = \\mathbf{I}_n \\ ", "98b7a64cce65106ff07c3ef81632a6cc": "\\displaystyle{\\|f\\|_{(s)}^2 = \\sum_{n\\ge 0} |a_n|^2 (1+2n)^s,}", "98b7f4e3bd3e15efdacfd38885fe0f18": "\\pi_{n-1}(A)", "98b80e54906d6e807e23d9b7d7192c83": "Z^{M}_{0,j} = Z^{M}_{i,0} = 0", "98b83e3ed4a266a4b6feaa8f58f8e0d7": "f(ab)", "98b8403585e1af4ad657cc55288df5ea": "v^*", "98b89323a1b5d42c8266f7b2e11e118f": "\\pi=\\frac{\\sqrt{3}}{9Z} \\!", "98b8dc51e4ec9dba741494a85879a1d4": "q\\rightarrow 1 ", "98b8e40273cbf6c9ccc9f03f5cc57da8": "U(a,a+1,z)= z^{-a}\\,", "98b9325513e73c06ef8c788cf969c1c5": "C =\\{C_i\\}_{i\\ge1}", "98b9612e25b1fdf494a3b20e4ff510dd": "q{\\left({\\frac{\\alpha p -1}{\\beta p +1}}\\right)}^\\tfrac{1}{p} \\text{ if } \\alpha p\\ge 1\\!", "98b97425a420ab9426ad7c2a80fc28d5": "P\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)", "98b9b1cb0f14b9fdd98974767b1311b4": "\\Lambda_{\\mu'}{}^{\\nu} T^{\\mu'} = T^\\nu", "98b9b6ef47b467211ab4ef35168b67dc": "\\langle 2 \\rangle_\\mathrm{S} = \\langle 1 \\rangle \\langle 1 \\rangle", "98ba3b143f705a17c9d6f60a987d2a6b": "x_1,\\dots,x_k", "98ba594cbfe4bde258701f379fc5a006": "M=\\frac {Gm} {c^2}", "98ba791f49d24ae1c57acb9403cb11fa": " E = (x[n] - \\hat{x}[n])^2 ", "98bac6663c6226adcb65256f9c08b49a": "p(c,f,r) = \\mathrm{round} \\left( \\frac{c}{50} + \\frac{f}{12} - \\frac{\\min\\{r,4\\}}{5} \\right) ", "98bb4f4d13ebf3ec4f25c5a52dc3972f": "\\mathbb{Z}/m\\mathbb{Z}", "98bb94d9fdd8ab9e6c6cba984985373c": "L^2[0,1]\\,,", "98bbd8a58a401dd75f9e669bd50e784f": "1 = \\lambda f.\\lambda x.f\\ x ", "98bc12f32d0e9292adc9d9df5bd0b6f1": "z = 0.10 + j0.22\\,", "98bc2273f77a2002c61ad4f37e2ae246": "{1}/{m}", "98bc464b62a898de669fecd41981ae2c": "1+\\ln\\left(\\frac{\\sigma}{\\sqrt{2}}\\right)+\\frac{\\gamma}{2}", "98bc9ced8adb5119a71fdbcf3b4ec18e": "s_1, s_2, \\dots \\,", "98bcbd984cd508ed6866495c35712540": "A \\otimes_k k^{p^{-\\infty}}", "98bcce5e259ef31d55ddb2a11d604ccb": " \\infty\\ ", "98bcd6877a138ab2ec47c2e3760f275c": "d_w = 0.822 + (1 * 0.1 (1 - 0.822)) = 0.84", "98bd75281f1fb1b123e4bdf6d2bb8840": "\\{ (xx^{-1}x, x),\\; (xx^{-1}yy^{-1}, yy^{-1}xx^{-1})\\;|\\;x,y \\in (X\\cup X^{-1})^+ \\}. ", "98bdb3cb3f0ea055448ffe68c9c32444": "\\boldsymbol\\theta_{d=1 \\dots M}", "98bde94f2af7e89f6ade39a2af31611d": "n_{B}", "98be004eafc3bb9709f35580cbb4193e": "(P \\and (Q \\and R)) \\leftrightarrow ((P \\and Q) \\and R)", "98be07911e873dd22f1246e1caae79da": "x_3=5", "98be8ea56a74c5eb653eac5700dc1143": "\\bigwedge_{W \\in \\Sigma'} W = \\bigcap_{W \\in \\Sigma'} W", "98bef95e7284df45781e6421fbc79b9b": "\\overline{(A \\wedge C) \\vee (A \\vee B)} \\wedge (C \\vee A \\vee B))", "98bf195d35bec71a2bde6ddc4457ca4b": "y_2 = {y_1 \\over 2} \\left(-1 + \\sqrt {\\left(1+8{F_{r_1}^2}\\right)}\\right)", "98bf7dd67e227162ce354f74746d867b": "C_S", "98bf98b210af1cca650f23502d544642": "\\pi = 2\\left( 1 + \\cfrac{1}{3} + \\cfrac{1\\cdot2}{3\\cdot5}\n+ \\cfrac{1\\cdot2\\cdot3}{3\\cdot5\\cdot7} + \\cfrac{1\\cdot2\\cdot3\\cdot4}{3\\cdot5\\cdot7\\cdot9}\n+ \\cfrac{1\\cdot2\\cdot3\\cdot4\\cdot5}{3\\cdot5\\cdot7\\cdot9\\cdot11} + \\cdots\\right) \\!", "98bfb1b32f55635708f21745590d8f78": " Q=\\kappa \\epsilon_0 \\ E, ", "98c0036d6dfb5bcc7a255b99bbfd7244": "|w|_{a_i}", "98c02d49f59df3013d2a70dd947fa2bb": "\\color{BrickRed}\\text{BrickRed}", "98c033ce2257c03a66926e66d1bd79f0": "dy' = dy \\,", "98c086cbf485c4064e5ffbf61e0c73ca": "\\scriptstyle \\log n", "98c0acae4850eb7e5955fdd896785c7d": "N = \\frac{n_s}{n_c} \\times r_s", "98c0e71b057f5329be9b190bf3da2ee1": "a + nd,\\ ", "98c14ea1bc5778e6e5c387002633333c": "u_3 = \\tfrac{(x_1^2+x_2^2+ax_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{11} - 2x_3(x_1 x_9 +x_2 x_{10} +bx_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "98c1cfef60be0b536fa7bffd042a94f7": "e^{\\frac{2\\pi i}{7}}=\\frac{-1 + \\sqrt[3]{\\frac{7+21\\sqrt{-3}}{2}} + \\sqrt[3]{\\frac{7-21\\sqrt{-3}}{2}}}{6} + \\frac{i}{2}\\sqrt{\\frac{7 - \\omega^2\\sqrt[3]{\\frac{7+21\\sqrt{-3}}{2}} - \\omega\\sqrt[3]{\\frac{7-21\\sqrt{-3}}{2}}}{3}}", "98c219eeb594625680f1e5efd50c4b7c": "\\bigcup_{A\\in P} A = X", "98c2781902f4156f69ba986913940afe": "\\begin{matrix}\\mathrm{6\\; CO_2 + 6\\; H_2O \\quad \\longrightarrow \\;C_6H_{12}O_6 + 6\\; O_2} \\qquad \\Delta H^0 = + 2 870\\ \\frac{\\mathrm{kJ}}{\\mathrm{mol}}\\end{matrix}", "98c29aa27be9754368e060d619cff11c": "M \\mathrel{\\#} -M", "98c2a7a844b21d0bda863e36a6226797": "{h S}={0.05(D_{50})}", "98c2d11f9a152b9abf90ef13bca96dae": " A = \\int_{\\sigma(A)} \\lambda \\, d E_{\\lambda} ", "98c3752001775e24f1b802388ffde7dc": "H_\\alpha^{(2)} (x)= -\\frac{1}{\\pi i}\\int_{-\\infty}^{+\\infty-i\\pi} e^{x\\sinh t - \\alpha t} \\, dt, ", "98c3fd8c39879a95c1ebd086acbe32f6": "MMH^*_{32}", "98c425712205bbb106f9eeeac6cf36c6": "\\mu_{G}^{x} (F) = \\mathbf{P}^{x} \\left [ X_{\\tau_{G}} \\in F \\right ]", "98c4668467dc9072713e569c88c82aab": "n^2 = (n^2 - an + an - a^2) + a^2", "98c49846063e1e517cf77a95b0f21880": "f=g{\\,\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc\\,}h", "98c5001549874c9c17739a6517a4dbff": " A < -2\\nu", "98c50b832b3540c2562e3490572a970c": "\\frac{1}{N+1}\\sum_{i=0}^{N} i", "98c5511561057ab585f2dbf094f98447": "dS=\\frac{r}{\\sqrt{r^{2}-\\sum_{i\\ne k}x_{i}^{2}}}\\Pi_{i\\ne k}dx_{i},\\;\\forall k", "98c56a4b486a6d90df3654d0bb4ca42e": "(1-\\varepsilon)\\|u-v\\|^2 \\leq \\|f(u) - f(v)\\|^2 \\leq (1+\\varepsilon)\\|u-v\\|^2", "98c574183d6b5834d38a2402c3bfc3cc": " \\begin{bmatrix}-i&1&a\\end{bmatrix}.\\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix} = 0.", "98c57d7f34ebe19999e586dc1e838fbc": "A,B,C, D", "98c57e3032e0d12f5fa2d4f1d809e7b5": "(x,t)=(0,0)", "98c5e3ae84ae2e8178732b4871b069eb": "\\nabla \\times \\mathbf{B} = \\frac{4\\pi}{c}\\mathbf{J} + \\frac{1}{c}\\frac{\\partial \\mathbf{E}} {\\partial t}", "98c604a13c93aea41a6112400b443d77": " \\lambda", "98c6261a8281bb619809ec6fa9c20178": "\\hat{\\varphi}\\ ,\\ \\hat{\\theta}\\ ,\\ \\hat{r}", "98c65cb60c39868d5c543cd18d76b237": "\\mathbf{u} = u_1, \\ldots, u_d", "98c6868f431aa1ec65be41dd8f0ff4bf": "|C| \\le A_q(n,d) \\leq q^{n-d+1}.", "98c68f5e663e633ecc30e87665f1b5ad": "\\Phi^3 (3)=(49)^{-1} \\sum_{i=1}^{49} C_i^3(3)\\approx0.3336 ", "98c6e4b943af8b9705aebeadd1cdf6cf": "\\partial_t^2 + (\\coth t + \\tanh t)\\partial_t =\\partial_t^2 + 2 \\coth(2t) \\partial_t.", "98c6f2c2287f4c73cea3d40ae7ec3ff2": "1-2", "98c703b0dc25a822385f07fc3d35580a": "\\mathfrak X^m", "98c73e32035c6a72e0fdb06f4983c9f6": "\\sum_{k=m}^n f_k(g_{k+1}-g_k) = \\left[f_{n+1}g_{n+1} - f_m g_m\\right] - \\sum_{k=m}^n g_{k+1}(f_{k+1}- f_k).", "98c7a5df872c8530bd95b27d5a7e30eb": "w(match) = +2", "98c832b2e3d18520b56dc328293d5a77": " \\mathbf{F}_\\mathrm{Euler} = m \\mathbf{a}_\\mathrm{Euler} = - m \\frac{d\\boldsymbol\\omega}{dt} \\times \\mathbf{r}.", "98c8768fadab0d53867261f92a229b23": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) = 1", "98c8d38dabd0a85e14e070b0128ecc94": "0\\le \\lambda\\le 1", "98c8eddf12af1ef0bf6c36f3f23c9d26": "\\alpha > 2", "98c902026528bc8b58e51348b9f8cfe1": "F_{1}=F_2\\frac{Sin(\\beta )}{Sin(\\alpha )} \\,", "98c928c5faa941691c676e4c26fb7fea": "g_t = \\exp(tJ)=\\left(\\begin{matrix}e^{t/2}&0\\\\ \n 0&e^{-t/2}\\\\ \\end{matrix}\\right) \\quad\\quad\n\nh^*_t = \\exp(tX)=\\left(\\begin{matrix}1&t\\\\ \n 0&1\\\\ \\end{matrix}\\right) \\quad\\quad\n\nh_t = \\exp(tY)=\\left(\\begin{matrix}1&0\\\\ \n t&1\\\\ \\end{matrix}\\right)\n", "98c92da7155024f330ee2030c51760a6": "P(A|B) \\propto P(A) \\cdot P(B|A) \\ ", "98c944ea123946a128a6df9424447634": "\\displaystyle{u|_\\Omega = D(\\psi) - (\\lambda - {1\\over 2})S(\\varphi), \\,\\,\\, u|_{\\Omega^c} = -D(\\psi) +(\\lambda+{1\\over 2})S(\\varphi).}", "98c953318a5c0b9c7e4a26534cdcf029": "B_n(x) = \\sum_{\\nu=0}^{n} \\beta_{\\nu} b_{\\nu,n}(x)", "98c96c7123b4816908aa464357519e49": "e_J^\\gamma R_{\\alpha \\gamma}^{\\;\\;\\;\\; IJ} - {1 \\over 2} R_{\\gamma \\delta}^{\\;\\;\\;\\; MN} e_M^\\gamma e_N^\\delta e_\\alpha^I = 0", "98c9ca21124230980ba20f27fc673cca": "w_{n,i_n}(x_n) \\in [0,1] .", "98c9d400ce20bf10beca927398cd0589": "H(i\\omega)", "98c9f8463c205bdfc6255b543583fdc3": "4 \\arctan \\frac{1}{5}", "98ca42351e8a593caa634712837aa550": " Output(\\omega) ~ ", "98ca857558c4fb8681318ab71a2ac2ca": "\\displaystyle{L(a,b) = \\mathrm{ad}\\,[a,\\sigma(b)]}", "98ca9781f0737ba3a4da58c24ef590d0": "\\chi^{(2)}", "98cae89bd66797e75aefb43530661a5b": " q_1, q_2, \\ldots, q_n \\, ", "98cb4c5f2efe852e4717b88c9fc8c027": "\\int_X |f|\\, d|\\mu|<\\infty. ", "98cb75fe1f8b271c2dc5c770c58866c5": " \\frac{v_{k+1} -2v_k + v_{k-1}}{h^2} = \\lambda v_{k}, \\ k=1,...,n, \\ v_0 = v'_{n+0.5} = 0.", "98cbb51ccf6e04df6503d4df3e961d80": "(1,4), (2,3), (3,4), (4,1)", "98cbcd6f276ce07c5d6d1e1d1c21a33d": "f(\\textbf{e}_{1j_1},\\ldots,\\textbf{e}_{nj_n}) = A_{j_1\\cdots j_n}^1\\,\\textbf{b}_1 + \\cdots + A_{j_1\\cdots j_n}^d\\,\\textbf{b}_d.", "98cbf85605e0b2ff807944314290810e": "k = \\lfloor \\log_2 n\\rfloor", "98cc48c0541740bd3d1bd88a55a5248b": "a^\\ast \\in \\mathcal{A}", "98cc4915993f195e9e8b6729b572b5bf": "\n\\quad\n\\begin{cases}\nu_i n_i & = 0 \\\\\n\\sigma_{ik} n_i n_k & \\geq 0\\\\\n\\sigma_{ik} n_i \\tau_k & = 0\n\\end{cases}\n", "98cc4c9e5e102b80030a8df868869326": "\n\\{ \\beta, \\beta^p, \\beta^{p^2}, \\ldots, \\beta^{p^{m-1}} \\}\n", "98cc5febb9db6b1a8f6f892a88ed5451": "P(G, x)=0", "98cc69cbbcb470c9e53227bbd7ed4743": "\\frac{L_n^{(\\alpha)}(x)}{{n+ \\alpha \\choose n}}= 1- \\sum_{j=1}^n \\frac{x^j}{\\alpha + j} \\frac{L_{n-j}^{(j)}(x)}{(j-1)!}=\n\n1- \\sum_{j=1}^n (-1)^j \\frac{j}{\\alpha + j} {n \\choose j}L_n^{(-j)}(x)\n\n = 1-x \\sum_{i=1}^n \\frac{L_{n-i}^{(-\\alpha)}(x) L_{i-1}^{(\\alpha+1)}(-x)}{\\alpha +i}.", "98ccf08de7bb51a440c5573dac6ac6bf": "u=u_0", "98ccf195c5ca039cf07b2c7669383901": "\n\\begin{align}\n \\Psi_i &=& Ce^{i\\kappa z} \\left(\n e^{i\\pi z/a} + \\left[-\\frac{\\hbar^2 \\pi \\kappa}{m a |V|}\\pm \\sqrt{\\left(\\frac{\\hbar^2 \\pi \\kappa}{ma |V|}\\right)^2+1}\\right]e^{-i\\pi z/a}\\right)\n\\end{align}\n", "98cd0835c9eabdfd29d41bf1b7d4cb7a": "Pr[h(A) = h(B)] = J(A,B)\\,", "98cd568057dfd12afd87d1d3e03eb6be": "1 0 \\}.\\, ", "98d8a82af2f28795da7af68dd303c2b1": "\\rho\\,\\!C_p{\\operatorname{d}T\\over\\operatorname{d}t}{{=}}\\nabla\\left ( k\\nabla\\ T \\right )+ Q", "98d8d27a11f1aaf794d7408c2c68d391": "|S(\\omega)| \\approx k|W(\\omega)|. \\, ", "98d91b5ea14bdd360db5d2940bdfe315": "\\begin{align}\n \\mu_X &= \\frac{1}{\\sum_i { N_{X_i}}} \\left(\\sum_i { N_{X_i} \\mu_{X_i}}\\right)\\\\\n \\sigma_X &= \\sqrt{\\frac{1}{\\sum_i {N_{X_i} - 1}} \\left( \\sum_i { \\left[(N_{X_i} - 1) \\sigma_{X_i}^2 + N_{X_i} \\mu_{X_i}^2\\right] } - \\left[\\sum_i {N_{X_i}}\\right]\\mu_X^2 \\right) }\n\\end{align}", "98d9b802bdb09460331e78293a91bf18": "\\alpha = \\beta = 0.5 ", "98da1a89c33d2667af05ae51631862a6": "\\langle h|a,b\\rangle", "98da250dc6b7c3dce8f8506d199f1c70": "t\\left\\{\\begin{array}{l}r\\\\p\\\\q\\end{array}\\right\\}", "98da6c54483e3f616ba6aabf62f35d3a": "\n\\begin{align}\n \\epsilon(f) & = \\Big[ 1-G(f)H(f) \\Big]\\Big[ 1-G(f)H(f) \\Big]^* S(f) \\\\\n & {} + G(f)G^*(f)N(f)\n\\end{align}\n", "98daad559d06ebe4b5fa63c02bab810a": "\\Lambda f = \\int_U f\\,d\\mu_g = \\int_{\\varphi(U)} f\\circ\\varphi^{-1}(x) \\sqrt{|\\det g|}\\,dx", "98dbcb14fdfb31a67950c8282d677641": "P_{t+1} = P_t + \\delta \\cdot f(P_t,...)", "98dc321c0125cf5768bc50554b98ff3c": "D=(V,E)", "98dc36f7162ffac03d51f234ee2299a1": "\\{f_i \\circ \\pi_i\\}_i", "98dc7b84d060f9b382c4bfa68b3ff8b3": "{\\Theta}", "98dc92b39b0615ee4be59c7edb806074": "E_{zx,3z^2-r^2} = \\sqrt{3} \\left[ l n (n^2 - (l^2 + m^2) / 2) V_{dd\\sigma} +\nl n (l^2 + m^2 - n^2) V_{dd\\pi} - l n (l^2 + m^2) / 2 V_{dd\\delta} \\right]", "98dca4dc5ad2f883494c092664f156c9": "\\scriptstyle f(u, v)", "98dcd4d03c91c49340e6f3727cd9e5e2": "r=-2^{-k}\\log|c|", "98dcfdf61ad75a2204bc2d1b4bf434b8": "\\int e^{-\\frac{1}{2}\\sum_{i,j=1}^{n}A_{ij} x_i x_j+\\sum_{i=1}^{n}B_i x_i} d^nx=\\sqrt{ \\frac{(2\\pi)^n}{\\det{A}} }e^{\\frac{1}{2}\\vec{B}^{T}A^{-1}\\vec{B}}.", "98dd33a9ee83e0c475d5f22170247193": "U_{A_1} = \\frac {a e^{ikr}}{r} ", "98dddfa7483135a110a800d69386d2e1": "\\mathbb{F}_{2}[x]", "98ddf05aa238129611e0e50cacfb9478": "N_1 \\cap N_2=\\emptyset", "98de11f237c5655a2af29ea1250ad772": "\\lambda^A \\otimes_x \\lambda^B = f^{A,B,C}(x) \\lambda_C", "98de6e46123538147ecccd196d5cb3b0": "\\int p(x\\mid I)dx = 1.", "98de92aea014d45758436ef3e66c692b": "\n\\begin{align}\nI(\\theta) \n&\\propto \\left [\\frac{J_1(\\pi W \\sin \\theta/ \\lambda)}{\\pi W \\sin \\theta/\\lambda)} \\right]^2\\\\\n&\\propto \\left [\\frac{J_1(k W \\sin \\theta/2)}{(k W \\sin \\theta/2)} \\right]^2\n\\end{align}\n", "98dedfe9d7be4ae3ba80c62a37025fee": "\nR(90^\\circ) = \\begin{bmatrix}\n0 & -1 \\\\[3pt]\n1 & 0 \\\\\n\\end{bmatrix}", "98dee096120247d414beb54142d89883": "f_n(x_1,\\ldots, x_n) = \\sum_{i\\neq j} |x_i - x_j| / (n(n-1))", "98df0582824393bd53d1c432422065e3": "J^k_pf=f\\ (\\hbox{mod}\\ {\\mathfrak m}_p^{k+1})", "98df24ed706108a2fa02753109c1a3c8": "\\left(\\pm\\sqrt{\\frac{5}{2}},\\ 3\\sqrt{\\frac{3}{2}},\\ 0,\\ \\pm2\\right)", "98dfb6c82cf22643299ffe49c5514197": "\\overline{n^2}", "98dfd2c93a724ab4b9001568b4b8c415": "\n \\sum_{j_7 j_8} (2j_7+1)(2j_8+1)\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n \\begin{Bmatrix}\n j_1 & j_2 & j_3'\\\\\n j_4 & j_5 & j_6'\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\{j_1j_2j_3\\} \\{j_4j_5j_6\\} \\{j_3j_6j_9\\}}\n {(2j_3+1)(2j_6+1)}.\n", "98dfdb19afbefac674f18270739a52ef": "\\hat{x}=\\hat{a}", "98dff29732ef947f2b8db08302157d82": " [\\mathbf{\\hat Q,\\hat P}] = i\\hbar ", "98e045abbeaa2f5f054a92581b17c822": " M6 = K [ 1 - \\frac{ \\sum_{ i = 1 }^K | X_i - m | }{ 2 N } ] ", "98e0c2d05463f5afdc6343a9b66641bd": "\\ a < x < b ", "98e0dcd0514c85f57105a286e5ae1102": " [\\cdot,\\cdot] ", "98e1287e1feff8b55bcafe7ad8999d29": "~|\\alpha\\rangle~", "98e136b99f1533de6380517ddf680431": "\\displaystyle \\varepsilon\\rho_0", "98e142bf6dcc80953af7502808e99309": "y_2=-\\sin \\Omega \\cdot \\sin \\omega + \\cos \\Omega \\cdot \\cos i \\cdot \\cos \\omega", "98e143ffda593697700f7b971970a81f": "V_{k} (\\pi_{k} (K)) \\leq V_{k} (\\pi_{k} (L)) \\mbox{ for all } 1 \\leq k < n \\implies V_{n} (K) \\leq V_{n} (L).", "98e1516b44b01c2a19c03a73eaa8f0ee": "OPS = OBP + SLG \\,", "98e1787ddbb61d7eced1e20316d2d5f3": "\\tbinom n d", "98e186c35f2edc2d95884965003b848e": "\\scriptstyle |a\\rangle", "98e1aa0b10157c37fe279a958824e3c9": "\\displaystyle \\frac{1}{\\sqrt{2\\pi a^2}}\\cdot \\operatorname{tri} \\left( \\frac{\\omega}{2\\pi a} \\right) ", "98e1b2cb78c714c9f4863b2aa68e649a": "\\tan(x) + \\sec(x) = \\tan\\left({x \\over 2} + {\\pi \\over 4}\\right).", "98e22d93ce14c2613439fe85ac9c2f8c": "c = 2", "98e23b39ec3212219550fb96099c9f8b": "\\ C_z", "98e261a4db904bffc8ca5305155a4fda": "\\pi_{ij} \\leq \\pi_{ik}\\pi_{kj}", "98e2b2a6f61610669356524982f185fb": " f(x) = \\sum_{k=1}^K c_k x_1^{a_{1k}} \\cdots x_n^{a_{nk}} ", "98e306035074f2e8b8aa0dd0e56a5d6d": "\\{1, 2, \\dots, k^2\\}", "98e35d99c58fc1ccb09642fbfb51f936": " K = \\frac{[X]^u[Y]^v}{[A]^s[B]^t} = \\frac{k_1}{k_2}", "98e3721cb0001c485a96144c54f997d2": "K_g = C q_1 \\mu_g \\frac{P_1}{a} G (P_1^2 - P_2^2)^2", "98e3dcea0d476597f7eec34916627ea9": "\\sum_{k=0}^{n-1}\\left\\lfloor x+\\frac{k}{n}\\right\\rfloor=\\lfloor nx\\rfloor .", "98e407ba7bc2d47e6a90a39443b79f7d": "\\|y\\| = 1,", "98e41c8fb4d12aa00e3c7532851ee3cf": " \n(Eq. 3) \\text{ } E[\\Delta(t) + Vp(t) | Q(t)] \\leq B + Vp^* - \\epsilon\\sum_{i=1}^NQ_i(t)\n", "98e4209e0d745d1d679754367ce038e6": "{\\pi\\over 4}\\ {2\\pi\\over 3}\\ {\\pi\\over 2}", "98e45113aa47c95619bd6f0181488616": "\\mathbf{r} = R (\\cos\\theta, \\sin\\theta),", "98e4ca046cf383b77b4c2b57f9b4083e": "f(y; \\alpha, \\beta, a, c) = \\frac{f(x;\\alpha,\\beta)}{c-a} =\\frac{ \\left (\\frac{y-a}{c-a} \\right )^{\\alpha-1} \\left (\\frac{c-y}{c-a} \\right)^{\\beta-1} }{(c-a)B(\\alpha, \\beta)}=\\frac{ (y-a)^{\\alpha-1} (c-y)^{\\beta-1} }{(c-a)^{\\alpha+\\beta-1}B(\\alpha, \\beta)}.", "98e57a4ebea1abef0f9d8fe255b347af": " ELA = C_{Building}*\\sqrt{\\rho \\over 2}*{\\Delta}P_{Ref}^{n_{Building}-0.5}\\,\\!", "98e57e1a30633b7ce9ba22eaa99e1081": "u(v,w)", "98e643781cccb7f13537db150c02fb82": "\nw = f(z) = \\pm\\sqrt{z} = z^{1/2}.\\,\n", "98e6f6c06f45a209149383f810b49403": " T_{eq} = T_{ant} + T_{sys}", "98e70d5168018d2751dd64b69a44321b": "1-\\mu", "98e70f1f4db998448465918e36d3ef17": "\n y = \\beta'x + \\varepsilon, \\quad \\operatorname{E}[\\,\\varepsilon|x\\,]=0\n ", "98e7164f949961f5b8ee71859e3f80a0": "\\log y_j= -z_j^2 \\frac{0.51 \\sqrt I}{1+1.5 \\sqrt I}+\\sum _k b_{jk} c_k", "98e7501cdffd9ff699f9c86b2cf444c0": "A_k(n, r)", "98e75535f313c01f2edd10f8640d283e": "u(x,0)=g(x)\\,", "98e76edbc0335dff16fa25e4fa6577c0": "\\phi = \\tan^{-1} \\mu \\,", "98e7be09c5094d40245ba43127b654df": "{2\\pi\\over 5}\\ {\\pi\\over 3}\\ {\\pi\\over 2}", "98e8145109e9b9104d74a6e49eb888d4": "\n \\bar{d} = \\bar{a}^2 - \\cfrac{4}{3}~\\lambda \\bar{a}\\sqrt{m^2-1}\n ", "98e822e51eeee0e9e39b9c438aa353b8": "\\parallel {i}\\; {for}\\; {n} : P(i) ;", "98e870aef93015791ae7b9e7a0e5181b": "S(x)=S(y) ~\\to~ x=y,", "98e88314a6fc3758ade8e28cadf7d6c8": "\n\\times\n\\prod_{j=1}^{n}\n \\left\\{\n (-i)(2\\pi)^{-3/2} Z^{-1/2}\n \\left(q_j^2-m^2\\right)\n \\right\\}\n\\Gamma(p_1,\\ldots,p_n;-q_1,\\ldots,-q_m)\n", "98e88674f24919b0ea4a2d21a9be6206": "a^k a^m = a^{k+m}", "98e899a78b242316ae35bc2b97f6b9d7": "J_{k}^{(\\alpha )}(x_1)=x_1^k(1+\\alpha)\\cdots (1+(k-1)\\alpha)", "98e8c93c1156071371d23287542fce22": "\n\\rho_{uc}(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{\\mathrm{charges}\\ k} q_k \\delta(\\mathbf{r} - \\mathbf{r}_k)\n", "98e8cab4d77c22259ae0f637ac974e2f": "\\frac{x_0}{x_1}=\\frac{a_0}{a_1}\\times10^{b_0-b_1}", "98e8ffea06282e0c2979c3c2ae485b8c": "d_{n+1}=\\frac{S-m_{n+1}^2}{d_n}\\,\\!", "98e9603dfe3c9f6ca274d657d0f3248c": "\\textstyle 8 \\times \\frac{1}{3} r^2 \\times r = \\frac{8}{3} r^3 ", "98e97724a3e22da9046b2704a15b0d4e": "f(x) = g(x) - h(x). \\,\\!", "98e99015232d354ef67643cd01eb37d1": " \\int_U \\nabla \\cdot \\mathbf{F} \\, dV_n = \\oint_{\\partial U} \\mathbf{F} \\cdot \\mathbf{n} \\, dS_{n-1} ", "98e99f2306bc177afaf0c0794162133a": "\\pi = \\otimes' \\pi_v", "98e9a5fb94fefe233dd7ccdf1bba6cea": "d: D\\to J", "98e9be084fb06a12a410b02921bd6c7b": "(e-1) n!", "98ea11ebf2c9d7b6596e01a099c93074": "\\ s(t) = V \\cos(\\omega t)", "98ea63595123ab4e5c540eadcace7105": "t=\\Delta_{ij}/(SE\\text{ of }\\Delta_{ij})", "98ea8385f32fbbe6541171503fae531a": "u_n(X_1,\\ldots,X_n) = \\frac{\\partial}{\\partial z_1}\\cdots \\frac{\\partial}{\\partial z_n}\\log E(\\exp\\sum z_iX_i)\\big|_{h_i=0}", "98eaa00e36f823ad612188c7387a73a5": "\\begin{alignat}{7}\n2x &&\\; + && y &&\\; - &&\\; z &&\\; = \\;&& 8 & \\\\\n&& && \\frac{1}{2}y &&\\; + &&\\; \\frac{1}{2}z &&\\; = \\;&& 1 & \\\\\n&& && 2y &&\\; + &&\\; z &&\\; = \\;&& 5 &\n\\end{alignat}", "98eaa0ebdafd6e7e4cbd0bfaab43316d": "\\frac{\\sin\\theta_1}{\\sin\\theta_2} = \\frac{v_1}{v_2} = \\frac{n_2}{n_1}", "98eaf66e0a48eca1fa7f92847400989d": "x_i \\leq x_j", "98eb4777e424286e85f2b5777bf270e3": "\\mathbf x=(x_n)_{n\\in M}", "98eb6b1648735b6f0d9539025a41551d": "\\Phi_\\xi = \\int_S \\boldsymbol{\\xi} \\cdot \\mathrm{d}\\mathbf{A} \\,\\!", "98eb8c01baf3ad0cf3ce074316cc3b24": "e^{j2 \\pi f_0t}x(t) \\rightarrow S_x(t,f+1)", "98ec40ccec5abe118634b0d6fc6ff1fe": "\\nabla_u(ap)=u(a)p+a\\nabla_u(p), \\quad a\\in A, \\quad p\\in\nP.", "98ec5510088bb7fe3fd81478384bbb94": "\\partial\\Sigma", "98ecb80699c9e1ecf8c489ae5a9430e6": "N! \\approx N^{N+1}\\sqrt{\\frac{2\\pi}{N}} e^{-N}=\\sqrt{2\\pi N} N^N e^{-N}.\\,", "98ece31c238aaccbc5daab15867888dc": "P_\\pi", "98ecfd36a7407e79127036de99883d00": "C^1([0,1]) \\subseteq C([0,1])", "98ed67b53cb0ce4bbb23dbd3612d3820": "\\mathrm{FIS}(X)=\\mathrm{Inv} \\langle X | \\varnothing\\rangle=({X\\cup X^{-1}})^+/\\rho_X.", "98ed9299493cb6e6736b07507f982bba": "p_1,\\dots,p_n\\,", "98edaab1a40f45f5b0a2a47b64f8cab9": "\\textstyle p = 3", "98ee269e7eb1415e866f0274ab87326b": " \\nabla\\cdot\\left(\\psi\\mathbf{A}\\right)=\\psi\\nabla\\cdot\\mathbf{A}+\\mathbf{A}\\cdot\\nabla \\psi ", "98ee8959a29496365432f1523fb858fd": "M_r(f) = \\frac{1}{2\\pi}\\int_0^{2\\pi}|f(re^{i\\theta})|^2\\,d\\theta", "98eedeb0d18f2ff700c0f7464b34fe5d": "\\Delta_b", "98eee39c95c671968bb89fe7271d82d2": "\\partial_{vv}(L_v) \\leq 0.", "98efa94c6a0c960dcad8117c3f6ae7cb": "\\int_{\\mathbf{R}^d}\\int_{\\mathbf{R}^d}|f(x)||\\hat{f}(\\xi)|\\frac{e^{\\pi|\\langle x,\\xi\\rangle|}}{(1+|x|+|\\xi|)^N} \\, dx \\, d\\xi < +\\infty ~,", "98efb04ec6f08aa85c8a045046797bef": "P' = P \\oplus \\Delta", "98efd8cd91ff3e00cc4bee45ffdbcc72": "H_0=\\hbar\\omega_0|\\text{e}\\rangle\\langle\\text{e}|", "98efdc9761214dac39dcec28e12a0e5a": "p(a,x_1,\\ldots,x_n)", "98effea095a8a3cfd433a174d6b8b43e": "\\Delta \\bar l = \\Delta \\eta.", "98efffba539919d27497551169e7ae36": " \n(B.4)\\quad \\psi_{,\\,\\Phi\\Phi}+2 \\,\\big(\\psi_{,\\,\\Phi}\\big)^2-e^{-2\\psi}=0.\n", "98f05243ca7bcac17d4bcf9e48f7b360": "\\mathrm{Nu}_L=\\frac{hL}{k}", "98f0e74b60d93ea6376aaafd4edda5ab": "\\exists x (\\phi \\lor \\psi)", "98f0f0ac355cad587163f6953804083c": "\\tilde f(x):=\\inf_{u\\in U}\\{ f(u)+k\\, d(x,u)\\},", "98f1034dc087b7259439bc2bb7ac69ad": "\\sum_{i=0}^\\infty d_it^i=\\frac{P(t)}{(1-t)^d},", "98f12bb5898725705f95b50f50a76911": " i \\epsilon [n]", "98f13708210194c475687be6106a3b84": "20", "98f1657ecbadc6ed32586f8a077dd8f4": "1_{-R \\leq \\xi \\leq R}", "98f1a495c756e9cbdce03e02878776aa": "\\theta_{cr}=\\arctan((1-\\csc \\alpha)\\cot \\alpha) \\quad \\; ", "98f1c29862115d7941a03a7be906e126": "r_1, \\ldots, r_n", "98f1d302613782846708895f908ebf9a": "\\rho=\\frac{-r+2M}{2r^2}\\,,\\quad \\mu=-\\frac{1}{r}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\varepsilon=\\frac{M}{2r^2}\\,;", "98f23b3cd6cf4ae8b0bf71b5f593d497": "v_n = n / \\beta ", "98f248e61c5f760a8e895f5406e35a84": "\\|f\\|=\\sup_{x\\in U}{\\left|f(x)x^4\\right|}", "98f24f2101b5a29214e29c0ebe9726fb": "x_i = x_0 + ih", "98f295ee9bac2068742ead2be50bfc39": "\n\\mathbf{C}\n = \\Big\\{\\;\n \\big( p(a_1), p(a_2), \\dots, p(a_n) \\big)\n \\;\\Big|\\;\n p \\text{ is a polynomial over } F \\text{ of degree } = <\\Delta Y \\cdot \\Delta Y^T > = <\\Delta Z \\cdot \\Delta Z^T > =\\frac{1}{3} <\\Delta R \\cdot \\Delta R^T >", "991cf63ff3906d1312caea84506c7201": "W=\\frac{1}{2}\\cdot C_\\text{DC} \\cdot V_\\text{DC}^2 ", "991cfce1551e28c62efe5004347d465c": " x + \\Delta\\,x ", "991d0de5a195c602d9b3b043d337abdc": "l\\colon Z\\to Y", "991d1509482df586ca0101afa29d4ddf": "\\,k_1", "991d1f8bea19276fb2491d04c6822c66": "R_{\\alpha \\beta} - {1 \\over 2} g_{\\alpha \\beta} R = 0", "991d4ddb0093623cbda24af65c11083a": "\\psi_0=\\psi_1=\\frac{\\rho_1}{\\sigma_1^3\\sqrt{n}},", "991d53c26048920b29fd29eed07afc71": "\\bold c_\\mathrm w\\,", "991d5eab158c79b44aa7ed6d2eeb66e9": "B=(a,0)", "991d780a52a551ad3f801aad0e3d3bab": "oid(O_{j})", "991e182a821c5c214c3836bfa1a2e73d": "\\bigstar |||\\bigstar \\bigstar", "991e82ffc3b41e8ca369be5eed532bb5": "t < l", "991e95055651fa1f4e2eb9dda23f4b08": "\\delta t", "991eb2e4ebdc8325955efea9dd7ef268": "\\sigma_2\\Big.", "991eb64af5f32444ed49c1bd00b6c124": "\\, \\nu = \\nu_{\\mathrm{cont}} + \\nu_{\\mathrm{sing}} + \\nu_{\\mathrm{pp}}", "991ef1b4fba7684e6565bb9ce56a3661": "f(z)=\\displaystyle\\sum_{n=0}^{\\infty}\\frac{1}{n!}z^{-n}.", "991f5c873460a8219331a81e2b4e8d43": "\\rho(\\bold{r},t) = |\\Psi|^2 = \\Psi^*(\\bold{r},t)\\Psi(\\bold{r},t) \\,", "991f73f147e40b6797d90456267b5ffa": "\np_6(x) = x^6 + 4x^5 - 72x^4 -214x^3 + 1127x^2 + 1602x -5040.\n", "991f8344ff0bffefceaf36fd8241904e": "TX\\otimes T^*X", "991fadf8af2892b25dc52c8ada9f00d9": "\\tfrac{1}{17}", "991fc2854a56d508c981ea96916d8463": " A(D) = \\iint_D \\sqrt{EG-F^2}\\, du dv.", "991ff06ac90af29aea90d64b926e96ae": "\\partial_{\\underline{x_i}}=\\sum_j e_j\\cdot \\partial_{x_{ij}}", "991ff890bc50f354e98987b1d46891d0": "\\beth", "99200ff0e6b526cb26995a1f501ef679": " {d \\sgn(x) \\over dx} = 2 {d H(x) \\over dx} = 2\\delta(x) \\,.", "99203f4afbf0087de58b7c3836a1eea3": "\\{w:r^*<|w|10^{25}", "992e8471ae821b56ba538b50786915e1": "KP(x|\\ell(x)) \\leq \\ell(x)+188", "992ead9b1ae58667c8bffa66d5a10623": " \\text{pH(X)} = \\text{pH(S)}+\\frac{E_\\text{S} - E_\\text{X} }{z}", "992eb58f7bc2ad762a806ae090d0ac43": "{\\mathbb R} / {\\mathbb Z}", "992f06556bde14fec9d9b6ce2386cab5": "f = \\frac{1}{2 \\pi \\sqrt{LC}}", "992f1df4b3d4961bf5391efb0a56dc59": "x^{\\mu} \\rightarrow \\xi^{\\mu} = x^{\\mu} + \\delta x^{\\mu} \\!", "992f313c013298c4ff951a798d1f95ef": "\\Delta\\phi\\!", "992f4a8d981b4a329886f3db00110379": "H_f=\\int d^{d-1}x \\frac{1}{2}\\alpha^{-1}(\\pi_\\sigma,\\pi_\\sigma)+\\frac{1}{2}\\alpha(\\vec{D}\\sigma\\cdot\\vec{D}\\sigma)-\\frac{g^2}{2}\\eta(\\vec{\\pi}_A,\\vec{\\pi}_A)-\\frac{1}{2g^2}\\eta(\\bold{B}\\cdot \\bold{B})-\\eta(\\pi_\\phi,f)-<\\pi_\\sigma,\\phi[\\sigma]>-\\eta(\\phi,\\vec{D}\\cdot\\vec{\\pi}_A).", "992f5289e810f729e853dc2176d01f40": "(ax^2+bx+c)(dx^3+ex^2+fx+g),", "992f54bfd9007a484ddc2874af784d82": "cons(x,y) \\stackrel{?}{=} cons(1,cons(x,cons(2,y)))", "992f8b5d9d33c3c905d9065a2d375502": "2.\\overline{3}", "992f9431a1de96e5b4defa766a93fc8f": "\\frac{3^3 2^0 + 3^2 2^2 + 3^1 2^3 + 3^0 2^6}{2^7 - 3^4} = \\frac{151}{47},", "992fc63d5b310b63e12ea451500c15bf": "p^{(k)} \\gets {X^{(k)}}^T t^{(k)}", "992fd41f053d328db0ca0287eed0e2e9": "x^{(n)}", "9930023a655896aadc99e6f49660b286": "\\bar{P_e}", "993076e0ef4f42d6f221f200c4056e7d": "x_{i}^{1} = x_{i}^{0} + u_{1,i}^{0} + u_{2,i}^{0}", "9930a37fe665c34af11f39b96bcaac8e": "\\beta, s > 0\\,\\!", "9930de2c4d99a5711b4938249ba923ff": "\nC_s^2 = \\frac{ \n\\left\\langle \\mathcal{L}_{ij} \\mathcal{M}_{ij} \\right\\rangle\n}{ \n\\left\\langle \\mathcal{M}_{ij} \\mathcal{M}_{ij} \\right\\rangle\n}\n", "993186a770ae8b3ab8f2619c125ba298": "O_k", "99319b3c9f7e8eb2cd1858ef25002198": " \\mathbf{\\hat T}(\\lambda)|\\psi\\rangle ", "99324196a2d4728af135bca48697b890": "n = N(P_{n}, a)", "99328de1a09db94e5902f56d7cf39d84": "u < -|\\dot{x} + a(t,x,\\dot{x})|", "9932c4797e90aea7b31b5d2085ad0b0c": "\\tfrac{1}{p^k}", "9932e3acd25544301b09a3cf840c3a28": "\\textstyle a a = b \\, , \\quad b b = 0 \\, , \\quad a b = b a = 0 ", "9932e9612083b0c0e7e3e9a76b2282ca": "\\tau = t-z/c_0", "9933292e99a8f4bab47506aff765faf4": "y \\not\\in L", "993343a5b8f4308c1a497784729507c6": "\\mathcal{G}(n,0)", "99337acf5deee2e6bf4caf932a377bf7": "\\mathop{\\rm el}(F) \\to C", "9933a46dc9b07abc5e52fda06a4fb8b4": " S'_{ij} = \\cos(2\\theta) S_{ij} + \\tfrac{1}{2} \\sin(2\\theta) (S_{ii} - S_{jj}) ", "993412a1b54297c8eb278e92ddf02053": "Lu(x)=\\sum_{i,j=1}^n (a_{ij} (x) u_{x_i})_{x_j} + \\sum_{i=1}^n b_i(x) u_{x_i}(x) + c(x) u(x)", "9934447c08ed385b7454567308a2cad3": "\\mathrm{dim}_F(B)\\cdot\\mathrm{dim}_F(\\mathrm{C}_A(B))=\\mathrm{dim}_F(A).\\,", "9934a2b1108848ec2c2ecc1414387848": "\n \\mathbf{A}=\\begin{bmatrix} \n 4 & -7 & \\color{red}{5} & 0 \\\\ \n -2 & 0 & 11 & 8 \\\\\n 19 & 1 & -3 & 12\n \\end{bmatrix}", "9934a39ca3b22ddab56fc98a89495968": "\\sum_i T_{\\mathrm{amortized}}(o_i) = \\sum_i \\left(T_{\\mathrm{actual}}(o_i) + C\\cdot(\\Phi(S_{i+1}) - \\Phi(S_i))\\right) = \\left(\\sum_i T_{\\mathrm{actual}}(o_i)\\right) + C\\cdot(\\Phi(S_{\\mathrm{final}}) - \\Phi(S_{\\mathrm{initial}})) \\ge \\sum_i T_{\\mathrm{actual}}(o_i),", "9934c74d3df18564a79bd70e6c4ce380": "n=\\prod_{(q-1)|m \\text{ and } q \\text{ is prime}}q", "9934d09a1da8c34244cd8ffc56c2d66b": "\\psi(1) = \\omega^3", "9934e9a121fe58993aacc2ecd10f43de": "\\mathrm{Backoff Time} = \\mathrm{random}() \\times \\mathrm{aSlotTime}", "9934f67c148186aacee86498e2ebc8fd": "\\boldsymbol{\\mu}_\\ell = -e\\mathbf{L}/2m_e = g_\\ell \\frac{\\mu_B}{\\hbar} \\mathbf{L}\\,\\!", "9934fd894f942eaaf56ac83535504ecc": "\\vec{E}[\\vec{x},t] = - \\nabla\\phi [\\vec{x},t] - \\partial_t{\\vec{A}} [\\vec{x},t] ", "993530a1e51acd146e3c2f79dc018ec4": "a_n=\\tau(n)/n^{11/2}", "99353e0fd3f04def22583f1be82b9034": "-2\\tfrac{3}{4}", "9936085c831de6682ea89d53200a820b": "X_{m-1}", "993636e689c6fae001c76181df0d3582": "\\mathbf{{f}}=(0.3,0.7)", "9936da184217eeaf54422d47a5f7d4ee": " \\varphi \\left ( \\mathbf{x} \\right ) = \\frac { \\sum_{i=1}^N e_i \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) } { \\sum_{i=1}^N \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) } = \\sum_{i=1}^N e_i u \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) ", "9936da28d75f1a2e37f199d38239e27d": "2\\pi/\\lambda", "9936e4912c5fb7b4e97a89d82570b079": "-2\\pi\\le v <2\\pi", "9936f17a319321e0dbed74856f114aa7": "2+3i", "99370e84fa7db1c95ed85208819fa289": "\\pi_2f_2", "99370efde16939974018f3bcaea18fbf": "\\overline{\\mu_{k,i}}", "99374e075e297f34f9eee1f8d148acda": "ij = k\\,", "99380ab1cd8c20a6f2e835795b6e8c62": "\\overline{MR}=\\frac{\\sum_{i=2}^{m}{MR_i}}{m-1}", "993859933e5964f3588a40898f1b5920": "G(10,2)", "9938d538c3338f68d88aa3ac02fe23ca": "\\sum_{v \\in V(G)} v\\cdot\\text{deg }v", "9939371a611758915c73a9d40ba55e63": "\\sum_{n=0}^{\\infty}\\frac{(it)^n}{n!}e^{n\\mu+n^2\\sigma^2/2}", "993952c5a825cac3215ca59a10750cd2": "m > d", "9939606790abbe27961ac736732b8a23": " V(r) = -G\\frac{m_1}{r}. ", "993964028b2e3aa9148c3c6b1d11327c": "\\operatorname{Var}(z) \\approx \\frac1{n-3} .", "9939ab117ef6fdb073fa77e1a65cdd92": "p = \\frac{e}{70} + \\frac{f}{4}", "993a329be995ba6907ea194b03b551ab": "S(d),d\\in D,", "993ab1984340f7781e5911f836939056": "\\frac{1 + z + {\\scriptstyle\\frac{1}{2}}z^2 + {\\scriptstyle\\frac{1}{6}}z^3}{1}", "993ad079b6630ab922c333a267edfead": "v^k_1 = v^{k}_2 = \\cdots = v^k_k=0;", "993b52912a6b38f929d11dfc0cc04ba4": "S^1 \\to S^3 \\to S^2.\\,", "993ba2cd58866cb9ee5bb46d9b86b7e4": "\\ln y=\\ln\\lim_{n\\to\\infty}\\left(1+\\frac{x}{n}\\right)^n=\\lim_{n\\to\\infty}\\ln\\left(1+\\frac{x}{n}\\right)^n.", "993bff5055d5e496e4dfa456b8b4b32a": "e\\mapsto ne", "993c45c989b1b4e2318f8476443fc37c": "0+0<1.", "993c53dbcdea4cef6e2c6cc27ebbd1a2": " \\sigma \\approx G_{\\rm F}^2 E^2 ", "993c8d6801a871c1ead57eb43c67daf3": "\\mathcal{N} \\models \\theta \\qquad", "993cc81b44e488f67d963930bf896e9a": "P \\to (U = V)", "993d479f2ab51a8fed445921a7e6d2b7": "PV_\\text{fixed} = N \\times C \\times \\sum_{i=1}^n \\left( \\tilde \\delta_i \\times P^D(\\tilde t_i) \\right)", "993d6950fcc8861584255b6deec02bd7": "0 =\\frac{\\mathrm{d}f}{\\mathrm{d}t}", "993d9623f5cd280300cabb03e9b1c3c8": "\\dot{M_{T}}=\\frac{\\mathrm{turbine}\\ \\mathrm{work}\\ \\mathrm{required}}{W_{T}} ", "993d964ea5dfac770bca5c146006d247": " 5^{5^5} + 2 ", "993db7ce25fe321ca6ded7a7f8f7e2e8": " f(x) = a\\left(x + \\frac{b}{2a}\\right)^2 - \\frac{b^2-4ac}{4 a} ,", "993dbb37c31c6140701b744f68228e67": "X \\in \\emptyset", "993e3094efa4c1ce596c95ce6641f500": "H(s) = \\frac{ s^2 }{ s^2 + \\frac{ \\omega_0 }{Q}s + \\omega_0^2 }", "993e95839936b026eb49c73c63611dc6": "A(x_1, ..., x_n) \\to \\alpha", "993ee4819286265a4d759255f1502786": "\\forall p: \\forall q: \\mathcal{B}(p \\to q) \\to \\mathcal{B} (\\mathcal{B}p \\to \\mathcal{B}q)", "993f8cf57d231384c500678eb27c1e7b": "\\delta(R) \\ge d(R)", "993fb1c2bc23fa1058eeb4383e9449d8": " \\operatorname{sink}[(\\lambda p.(\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x))] ", "993fba9171dbcec3d708596be67675df": "|A_p|=s'\\leq s", "993fef1f62bae4c82e3c64ccd3854e0c": "W = (64k_B T\\rho_{\\infty } \\gamma ^2 /\\kappa )e^{-\\kappa D}", "99401c7cf29fd0a3930700781fb08f6c": "C_R=\\{z : z=R e^{i \\theta}, \\theta\\in [0,\\pi]\\}", "994063387fe12e8edb0e4c3eefe98676": "P_w", "994078cb2dd8f49960ee7b1f73d5c66d": "x^k+y^k=z^k", "99408b1a22f3d8efdb90f9c3d310b808": "\\| f \\| := \\sup_{t \\in [0, T]} | f(t) |", "9940ab609fd23fff89f23f9397a2d992": "K=\\langle t^2, tf_1,\\ldots, tf_m, (1-t)g_1, \\ldots, (1-t)g_k\\rangle.", "9940e04f9a3873b11256b80c5d4d67d9": "\\frac{\\pi^2}{16}", "9940e4ee99a54a79245ee3edbcd3d65a": " S \n\\approx \\frac{L}{\\left(\\frac{\\varphi}{\\theta}+1\\right) L \\sin \\theta - 1} \n= 1 \\left/ \\left( \\left( \\frac{\\varphi}{\\theta} + 1 \\right) \\sin \\theta - \\frac{1}{L} \\right) \\right.\n", "9940e8c1dc5ccf63588ce04fc573303d": "k_n/n", "99411b41b479e22deadc2027ef5df196": " {\\Delta}_{\\rho}(a,b) = ({\\Delta}_{\\rho}(a),b) - (-1)^{\\left|a\\right|}(a,{\\Delta}_{\\rho}(b)) ", "9941221dad0145a759ca32fd5c4978d6": "\nf(z) \n= \n\\frac{1}{\\pi i} \\int_{-\\infty}^\\infty \\frac{u(\\zeta,0)}{\\zeta - z} \\, d\\zeta\n=\n\\frac{1}{\\pi i} \\int_{-\\infty}^\\infty \\frac{Re(f)(\\zeta+0i)}{\\zeta - z} \\, d\\zeta\n", "99413b1c63416db0f6300e49409194df": "r_{k}\\leqslant R_{k}", "9941e14a7e54ac33cb39932055ce5fab": " D^{\\alpha\\beta}_{jj'} = \\frac{\\hbar^2}{2 m_0} \\left [ \\delta_{jj'}\\delta_{\\alpha\\beta} + \\sum^{B}_{\\gamma} \\frac{ p^{\\alpha}_{j\\gamma}p^{\\beta}_{\\gamma j'} + p^{\\beta}_{j\\gamma}p^{\\alpha}_{\\gamma j'} }{ m_0 (E_0-E_{\\gamma}) } \\right ]. ", "99420c7c459823c24f5de27c1df59b8f": "(X\\sqcup X^\\ddagger)^+", "9942219a40b9ff27389e874cd062d2d3": "\\left(\\operatorname{p.v.}\\frac{1}{x}\\right)[\\phi] = \\lim_{\\epsilon\\to 0^+} \\int_{|x|\\ge\\epsilon} \\frac{\\phi(x)}{x}\\, dx", "99424b5bd5d1934b034c5c98edc394d6": " S_{abcd} = \\frac{R}{n \\, (n-1)} \\, H_{abcd}", "9942935efe845d8a634a9c634160aaf6": "\\beta_i=\\mbox{The sum of all connected, irreducible graphs with one white and}\\ i\\ \\mbox{black vertices}", "99429ab055f66093f2fc0ecbae1f8e2b": " \\scriptstyle \\ell ", "9942a64279a410b728647ca1513cf734": "\\mathcal{E}_0", "9942d35ca69538ecbc65850f331d980f": "V_i = p_i^{-1}(U_i)", "9942ddc8c774d4df68edacdf891e8e6f": " 0 = \\Delta_rG^{\\ominus} + RT \\ln K_{eq} ", "9943544a6f59e44335d7c9fac42825ab": " C = 1.251\\sqrt{L} ", "99439996718b1fdac452a3974e6d94f3": "\\lambda~", "9943cfd3d97f341ed6be799b96023396": "A X A^{H} - X + Q = 0", "9943d2cc8658cfde570f6ed5b8fb13fc": "{\\bar{BP}}_4", "9943eac20c951b451cbd000dc4f1db21": " \\mathcal{E}_c = \\frac{1}{8\\pi} \\left [ \\mathbf{E}^2( \\mathbf{r} , t ) + \\mathbf{B}^2( \\mathbf{r} , t ) \\right ] ", "9943f3ad2c78565774ac938b381d5595": "\nKZFT_{m,k,\\nu_{0}} [X(t)] = \\sum\\limits_{s = - k(m - 1)/2}^{k(m - 1)/2} {X(t + s)\\times{a_s^{m,k}\\times{e^{-i(2m\\nu_{0})s}}}}\n", "99444c1c4527c4aae8b0ed94652dcac3": "\n\\mathrm{R} + \\mathrm{L} \\leftrightarrow \\mathrm{C}\n", "9944ef88b36e208c318e43fce7f0767f": "(x+1)^2 = x^2 + 2x + 1", "994548ad265ef9f39a11be7c091d4db0": "F_t : M \\to \\mathbb R", "9945d99461cf87e266579892db8e48ab": "1 = g_{\\mu\\nu}\\dot{x^\\mu}\\dot{x^\\nu} ", "9945e3bfb84c5d8b454dd4f016c450c4": "df:TN\\rightarrow TM", "9946185daff5b7229c9096ed1f5a1dba": "x' = x + ky", "994635269d0975fbca421d1222165a32": "Z = \\int \\mathcal{D} \\phi \\exp \\left(- \\beta H[\\phi] \\right)", "9946558b06586f8bfb20aacb2f11c266": "-2 = 2e^{i \\pi}", "99466e25aea72feed6a21c15ba50e90a": " |B| = 1 \\cdot \\begin{vmatrix} 5 & 6 \\\\ 8 & 9 \\end{vmatrix} - 2 \\cdot \\begin{vmatrix} 4 & 6 \\\\ 7 & 9 \\end{vmatrix} + 3 \\cdot \\begin{vmatrix} 4 & 5 \\\\ 7 & 8 \\end{vmatrix} ", "994672dc474c4a5b2f2179dd6cfe7849": "r, \\theta, \\phi", "9946d24458203757fe2fc8106892fecb": "h_t(x,x)", "9946fabd1272252b3571a5e85cfaff2b": "\n [\\boldsymbol{\\sigma}] = [\\mathsf{C}][\\boldsymbol{\\epsilon}] \\qquad \\text{or} \\qquad \\sigma_i = C_{ij} \\epsilon_j ~.\n ", "994707b7f5c2c2e2cad845eaa408cea8": "x = r\\sin\\theta\\cos\\phi", "9947503f8cea2aea159602ba649f3c82": "\\textstyle\\frac{1}{2.666 \\times 10^{21}}", "9947656bff89ceb077739b6852092fa6": "{\\eta_c}={\\eta_0}-({1\\over{cr}}).{{U\\,Ta\\,}\\over{Ic}}.({{Tr}\\over{Ta}}-{1})", "9947688cb300689b91a445a625f55949": "\nu(\\varphi) = -\\frac{\\alpha}{mh^{2}} \\left[ 1 - e \\cos \\left( \\varphi - \\varphi_{0}\\right) \\right] \n", "99478b72fd67419db02ebff836cd2a02": "L_S^{\\sigma\\prime}=a^2L_2-aM", "99478ed464809d2fa8a9f3c7d1a199d6": "(\\Gamma,L,M\\{\\mbox{·}|_B\\})", "9947fd985de8fc7ef99016216bf12225": "\\displaystyle l=\\frac{2(efg+fgh+ghe+hef)}{\\sqrt{(e+h)(f+g)(e+g)(f+h)}}", "99482c62f17de66df21237a08f36f2fc": "x^3+5004x^2+1169953\\frac{1}{3}x=41107188\\frac{1}{3}", "99483ffbe5010275201a1fc9fd13238e": "\\bar{\\partial}\\alpha=\\sum_{|I|,|J|}\\sum_\\ell \\frac{\\partial f_{IJ}}{\\partial \\bar{z}^\\ell}d\\bar{z}^\\ell\\wedge dz^I\\wedge d\\bar{z}^J.", "99484e13bedbaa3ac652b7aa480201a2": "P_i(x)=\\phi(P_{i-1},x)", "9948560fa610c39762ddadb16895b25f": "aRb :\\Leftrightarrow b \\in S \\cap a", "994865acb278376a5702538f82ee0d7d": "1 = (x^2 - x + 2) \\sum_{k=0}^{\\infty} a_k x^k", "99487d1e005ec04d274a0fa543a5135f": "\n\\begin{array}{ll}\nd\\in D & \\text{the decision being made, chosen from space } D \n\\\\\nx\\in X & \\text{the uncertain quantity, with true value in space } X\n\\\\\nU(d,x) & \\text{the utility function}\n\\\\\nf(x) & \\text{your prior subjective probability distribution (density function) on } x\n\\end{array}\n", "9948a8ac656a082bc20e59352026b128": "(3,\\bar{3},1)\\rightarrow(3,2)_{\\frac{1}{6}}\\oplus(3,1)_{-\\frac{1}{3}}", "9948dc584c92c415d0724d32c3f3b68b": "I_\\mathrm{cm} = \\int (x^2 + y^2) \\, dm.", "99498f3c3aec701591e08941e5be3d13": "V = \\int_R d^3 x \\sqrt{det (q)} = {1 \\over 6} \\int_R dx^3 \\sqrt{\\epsilon_{abc} \\epsilon^{ijk} \\tilde{E}^a_i \\tilde{E}^b_j \\tilde{E}^c_k}", "99499dc658ae8ee9c120566c0d95ea92": "Z_{P}", "9949b7cc087acb4217f48484312146d4": "\n\\begin{bmatrix}\n 17 & 89 & 71 \\\\\n 113 & 59 & 5 \\\\\n 47 & 29 & 101\n\\end{bmatrix}.\n", "9949e82ac963e72b68e7b367d2810c7c": "{\\tilde{D}}_{n+1}", "994a05a70c368f3367e9468e05eb4813": "G_{adv}(x,y) = -i \\langle 0|\\left[ \\Phi(x), \\Phi(y) \\right]|0\\rangle \\Theta(y^0 - x^0).", "994a08b7046b4efa26e61cbf043d0df9": "\\sigma_{x}=\\sqrt{\\langle \\hat{x}^{2} \\rangle-\\langle \\hat{x}\\rangle ^{2}}", "994a2aa798709ec6b5f159ecad54eb2c": " X_t ", "994a97758db24753044eeaa9357350d9": "c(m) = m^{17} \\; \\operatorname{mod}\\; 3233", "994a9b5fcb91a6ef39f94b60b657a793": "u(x, y, t),", "994aa4aef4b64a7f4f3ffc5feb943b6a": "\\ \\{x-2h, x-h, x, x+h, x+2h\\}.", "994add8da5a8eddf27bce7d3a80f8067": "\\Gamma_n \\simeq \\pi_0\\,\\text{Diff}^+(S^{n-1})", "994ae877486a4c490365e109d94b0044": " \\frac{\\partial T}{\\partial t} = \\kappa \\nabla^2 T +\\epsilon ", "994aedacbd438a6daaae8e380e589da3": " \\vec{v}_\\text{new}=\\vec{v} + 2\\vec{r} \\times (\\vec{r} \\times \\vec{v} + w \\vec{v})", "994b28acd69dcb031c8c87ae60dde784": "\nH_{SO}=\\frac{g\\mu_B}{2c^2}(\\bold{v}\\times\\bold{E})\\cdot \\bold{\\sigma}\n", "994b29cde30a57daa6bed7de7092fd18": "\\scriptstyle x(\\tau).", "994b621f3c71660bff50f3458fa9df66": "\\hat{U}_R=exp\\left(\\frac{i}{\\hbar}\\int dt' [\\mathbf{\\Omega}\\times \\hat{\\mathbf{r}}(t') ]\\cdot [\\mathbf{p_0}+m_{\\alpha}\\hbar\\mathbf{k}] \\right)", "994b654ce9711f18b5e9ed5b073f9a50": "\\sigma\\sqrt{\\ln(4)}\\,", "994b6993fe2bb95edcc1f10e615c56b5": "f_x(0,1) = p_x(0,1) = a_{10} + a_{11} + a_{12} + a_{13}", "994bdd22efed01ff2374872cbd0ec65a": "\\mathrm{Im}(\\gamma) = (a^2 + b^2)^{1/4} \\sin(\\mathrm{atan2}(b,a)/2) \\,", "994c34eae19242620c60c86f06458296": " \\begin{bmatrix}\n 0 & 1 \\\\\n 0 & 0 \n \\end{bmatrix}\n \\begin{bmatrix}\n 0 & 1 \\\\\n 0 & 0\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 0 & 0 \\\\\n 0 & 0\n \\end{bmatrix}\\,\n", "994c807718311270f8c7700bf1496cee": "\\pi\\colon F\\to M", "994ca27c9bb3410e9d31405fefd2c29b": "t=0, \\theta=\\pi/2", "994cb8bed13beeb3cfada7ad2af50b89": "\\frac{1}{4}, \\frac{1}{3},\\frac{1}{6}, \\frac{1}{5}, \\frac{1}{8}, \\frac{1}{7}, \\ldots", "994cf201a5a83b9101652e77934389e3": "\\operatorname{Var}(aX+bY) =a^2 \\operatorname{Var}(X) + b^2 \\operatorname{Var}(Y) + 2ab\\, \\operatorname{Cov}(X, Y).", "994cf43426f2b8e011938b0254dc4b03": "t=\\Phi", "994d52b3fad961218d14822c68116872": "{x\\over 2} < |M_x| \\qquad(3)", "994d5439e722b34c1cc996871871ac9a": "p_1, p_2,\\ldots,p_n", "994dc02b0470c7449dcf2b843532cbe5": "q+s", "994dd7f76bec19a3d40dd4fbad559305": "\\tilde{\\kappa}_{e-}", "994de5211ff9945aa1321601a2e0a94e": " S_j = S_k ", "994e29ea499240bfc2ebed3d4bdb1abb": "\\bigcap_v\\mathrm{ker}(H^1(G_K,A)\\rightarrow H^1(G_{K_v},A_v)).", "994e2b119a5a02a98ba3e3a16ce92ffa": "\\frac 12 =P(|X-\\mu|\\le \\operatorname{MAD})=P\\left(\\left|\\frac{X-\\mu}{\\sigma}\\right|\\le \\frac {\\operatorname{MAD}}\\sigma\\right)=P\\left(|Z|\\le \\frac {\\operatorname{MAD}}\\sigma\\right).", "994e311b6e7f1ef11842ad2b5fa4e499": "p\\colon Y\\rightarrow X", "994e36dbb8fea60e9da6fcdd25f7d991": "y(k)=Cx(k)+Du(k)\\,", "994e729ed87d4fad5772c0e94b3f93d0": "p_n(z).", "994e85645364bfc6f0a42dcf8466bfde": "H_{eff} = \\frac{H}{\\sqrt{2}}", "994ed8926d3c6cd93e6630861c44fe3e": " (\\mathbf{B} + 1)^m = B_m ", "994ee0e9b3aebf970a4389cbaf649838": "\\ \\exp x", "994ef21894bca1f460c53627ecbd1811": "{n \\choose \\lfloor{n/2}\\rfloor} \\approx 2^n \\, \\frac{1}{\\sqrt{n}}", "994fa1441ac318f24e3a06065d72d675": "C_i ", "994fc885aa7444526801497c99e6b5ff": "\\Phi_{mK}(M)", "994fd19ecedc8a3c7a58097fee9e2555": "{\\omega^1}_2 = -\\frac{d\\theta}{g}", "995026dd8f3f959feea8fb8bd2f44ca8": "\\rho_sdV", "9950385fbd0b58468fd1ef6c94173fed": "d \\Sigma", "99505328fd2579556594487d76f12a24": " \\boldsymbol{\\psi}(\\mathbf{x},\\mathit{t}) = \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} \\mathit{Q}(\\mathbf{x},\\mathit{t}|\\mathbf{x}',\\mathit{t}')\\boldsymbol{\\psi}(\\mathbf{x}',\\mathit{t}') d\\mathbf{x}'", "995143854599ec77a1911d36befa52b0": "\\int_0^\\infty e^{-x}\\frac{1-e^{-x}}{x} \\, dx= \\ln 2.", "9951b0f2a44e736d9b30b817f1137788": "H^0(X,L)", "9951dd7f1fb7585e118830c9262b64ca": "\\Delta w''=\\tau", "99523606e11c0a81a3574fe421872b83": " \\Gamma_{d-1} ", "9952f7fe82a4a48b0dce2712af61295a": "\\Theta(\\log^2 n)", "99539e8534d3c05195d7ec8ee5c69e82": "\\scriptstyle A \\oplus B \\oplus C", "9953d4d6392b0ce07343f424d8aa336f": "\\begin{smallmatrix}\\alpha_R\\ =\\ \\frac{138}{D}\\end{smallmatrix}", "9954034eb43d342b9a6205b3b6583cfe": " \\sum_{t=0}^\\infty \\beta^t u(c_t^{cert}) = \\sum_{t=0}^\\infty \\beta^t u(c_t^{vol}) ", "99542bab0977c876482148bb61833681": "\\scriptstyle s_i", "99542cad5e93a50dadab325b8baeca87": "\\mathfrak{gl}_\\mathfrak{g}", "99545005179e5c2610fd4d22e9f74e79": "Sq_p^I = Sq_p^{i_1} \\ldots Sq_p^{i_n},", "9954c0a54828e32a0c3762734f2ae12b": "{\\rm Riesz}(x) = const\\times x^{1/4} \\sin\\left(\\phi-\\frac{1}{2}\\gamma_1\\log(x)\\right)", "9954f661bdc2141b80cf91ba53c37ea0": " \\pi = \\pi_e - b(U-U_n) + v \\, ", "9954ff5329efdae128a82540f5615ed4": "\\exists D \\forall C \\, ( [ C \\in D ] \\iff [ P (C) \\and \\exists E \\, ( C \\in E ) ] ) \\,,", "99555e7590119eab5e922d04ef009ac1": "k(x) = (L_x, R_x)", "99556b2f83ada1746f70b748aa8e0728": "q^+", "995575881a5163c79ed5c7eeb05d03f8": "\\begin{align}\n \\left(\n A_r \\frac{\\partial B_r}{\\partial r}\n + \\frac{A_\\theta}{r} \\frac{\\partial B_r}{\\partial \\theta}\n + \\frac{A_\\phi}{r\\sin\\theta} \\frac{\\partial B_r}{\\partial \\phi}\n - \\frac{A_\\theta B_\\theta + A_\\phi B_\\phi}{r}\n \\right) &\\hat{\\boldsymbol r} \\\\\n+ \\left(\n A_r \\frac{\\partial B_\\theta}{\\partial r}\n + \\frac{A_\\theta}{r} \\frac{\\partial B_\\theta}{\\partial \\theta}\n + \\frac{A_\\phi}{r\\sin\\theta} \\frac{\\partial B_\\theta}{\\partial \\phi}\n + \\frac{A_\\theta B_r}{r} - \\frac{A_\\phi B_\\phi\\cot\\theta}{r}\n \\right) &\\hat{\\boldsymbol\\theta} \\\\\n+ \\left(\n A_r \\frac{\\partial B_\\phi}{\\partial r}\n + \\frac{A_\\theta}{r} \\frac{\\partial B_\\phi}{\\partial \\theta}\n + \\frac{A_\\phi}{r\\sin\\theta} \\frac{\\partial B_\\phi}{\\partial \\phi}\n + \\frac{A_\\phi B_r}{r}\n + \\frac{A_\\phi B_\\theta \\cot\\theta}{r}\n \\right) &\\hat{\\boldsymbol\\phi}\n\\end{align}", "995619c978e787e2439e04ba46324437": "g_{\\mu}", "995625673bc93a58f33fb52120089974": " \\delta(\\phi - \\eta) = \\int e^{ i h(x)(\\phi(x) -\\eta(x)d^dx} Dh ", "99569f63daa68b66d9f320444032834a": "V=\\frac{1}{2\\pi}\\frac{h}{2e}\\frac{\\mathrm{d}\\Delta\\varphi^*}{\\mathrm{d}t}.", "9957e533ab507960e586091602c48f96": "(a,b) \\cup [b,c] = (a,c]", "99594118e6f752ba6048131998396d52": "162 a^2 b^2 c^2 + 6 b^2 c^2 d^2 - 4 (b^6 + c^6) + 54 (a b^4 d - a c^4 d) + 81 (a^2 b^4 + a^2 c^4)\\ ", "99597c9508546a090569aa4cde1eab3a": "E = \\frac{3 e^2 Z(Z - 1)}{20 \\pi \\epsilon_{0} r_0 A^{\\frac{1}{3}}}", "9959874993e0f03d551101f9b5864836": "\\int_a^b f(x)\\,dg(x)", "995a155435ba7878d771bf63bdadfd44": "[\\sigma_1] \\mapsto R=\\begin{bmatrix}1 & 1 \\\\ 0 & 1 \\end{bmatrix}\n\\qquad [\\sigma_2] \\mapsto L^{-1}=\\begin{bmatrix}1 & 0 \\\\ -1 & 1 \\end{bmatrix}\n", "995a390de22867253298e5f057e1207f": "I(cat : N_0) = cat : E_0", "995a53dc292e1761bdcc28215206c9d3": "D=\\partial_{\\overline{z}}", "995ad108e37dbd4b033baa862048af1d": "\\gamma = \\int_0^{\\infty}\\frac{\\ln(1+x)}{\\ln^2 x + \\pi^2}\\cdot\\frac{dx}{x^2}\n = \\int_{-\\infty}^{\\infty}\\frac{\\ln(1+e^{-x})}{x^2 + \\pi^2}\\,e^x\\,dx.", "995afffd0020cf7200c310f027396372": "\\beta^5+\\beta^4-4\\beta^3-3\\beta^2+3\\beta+1=0", "995baf5f841b438bf4d9a2f8f7b37e49": "\\left(a_i, x\\right) = \\left(a_{i+1}, y\\right)", "995c046d32110746701be937164fbb8f": "\\mathcal{D}_\\alpha", "995c07f417bedef3f6ac1a6a6bb137ab": "\\tilde{f}(i)\\ne 1\\ ", "995c0f381001cf7f1b78f9c69f541c27": "x \\in p", "995c52b4c7f1a06459a3d32dda6ca415": "\\mathrm{Hom}_{\\mathbf {Sh}(X)}(f^{-1} \\mathcal G, \\mathcal F ) = \\mathrm{Hom}_{\\mathbf {Sh}(Y)}(\\mathcal G, f_*\\mathcal F)", "995c59decf017d7dcbfa44a38eb578dd": "\n\\left[Q^\\dagger(\\mathbf{k}),Q^\\dagger(\\mathbf{l})\\right] = 0, ", "995c7cb36bcafde4250b4bbab331b184": "\\beta > \\alpha", "995c7e5c27b9c24a90247052ac9c05cc": "i\\hbar", "995d28fb7d34c4531cbaa668bb5b0cf0": "\\mathrm{mul_c}(x)=c\\cdot x,~~~ ~ \\forall x \\in \\mathbb{C}", "995d2c008095bd2c8f18b921acddc7bc": "q^{k^2 - n k}{n \\choose k}_{q^2}", "995dad0c0bf4e976dd949cc5389f7401": "| \\xi - z | > \\varepsilon.", "995e05a0f4728989613704e2d3a2b2ed": "\\mu_g", "995e221b92b31368aedd40621957662b": "\\textstyle s=(c_1, c_2)", "995e53b9803672d75e97255645f6c6b0": "2\\le r 1", "9964b798c3e4cb47c9aa2fe4ca54c116": "\\tbinom{p+q-1}{p-1}-\\tbinom{p+q-1}{p}", "9964e3e45522d271449ae4706559dcf8": "pq=-\\frac{1}{4}.", "99656c0523e10f239df14dcb8b72e8c5": "\\ell i\\,", "99658c8a7dad86c3c30a754797f67cb4": "M_y\\ ", "99663c2bf152f2ad2437439fed9b9407": "RD(m)=RD(m_{ref})\\left (\\frac{m}{m_{ref}}\\right )^{1-D_s}", "996641e77d8408be9e24bb7d9cb199de": "\\begin{align}\n&\\operatorname{minimize}& & f(x) \\\\\n&\\operatorname{subject\\;to}\n& &g_i(x) \\leq 0, \\quad i = 1,\\dots,m\n\\end{align}", "9966706766342be847c6d24208639436": "Z_{base} = \\frac{V_{base}}{I_{base}} = \\frac{V_{base}^{2}}{I_{base}V_{base}} = \\frac{V_{base}^{2}}{S_{base}} = 1 pu", "996680cdf0591fa9332eb45e1a72b3f5": "\\begin{bmatrix}\n1 & 0 & 0 & 0 & 1\\\\\n1 & 1 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 1 & 0\\\\\n0 & 0 & 0 & 1 & 1\\\\\n\\end{bmatrix}", "99676cadf3004c8e7eb8daadbbf73340": "\\Psi_2 \\left( \\alpha; \\gamma, \\gamma'; x, y \\right)", "99677bfa78a9a3960cf4741fa5800992": "g_ig_jg_k^{-1}", "9967c04c977d26af695cca8a5239c3e1": "\\|\\mathbf{q} - \\mathbf{p}\\| = \\sqrt{\\|\\mathbf{p}\\|^2 + \\|\\mathbf{q}\\|^2 - 2\\mathbf{p}\\cdot\\mathbf{q}}.", "9968e14e005eedaf0223f409798cd3f7": " X_2", "9968fc4470c99482a3c991158a8e9448": "r_2", "996916155a55fa0e21ae65bf09f5f97f": " \\nabla_\\perp \\left( p +\\frac{B^2}{2 \\mu_0 } \\right) - \\frac{B^2}{\\mu_0 }\\vec{\\kappa}=0 ", "9969bf95e72ea055d66173ecdbf5828b": "V(x - v^b(1),1) = \\max_{\\alpha B_0 + \\beta S_0 = x - v^b(1)} \\mathbb{E}[1 - \\exp(-.1 \\times (\\alpha B_T + \\beta S_T + C_T))]", "996a27a157e826a62854faaab608af0b": "\\pi_{\\mathbf P}\\colon{\\mathbf P}\\to M\\,", "996a2d3b3cf3dc121a96c87b77a8d308": "\\mathbf{M}^T \\vec{u}_1 = \\sigma_1 \\vec{v}_1", "996a54dd1b254e0c467a1d4895f1ab93": "F_1\\cdot x_1+F_2\\cdot x_2\\leq F", "996a9ec3d869f1f0b368eab2c42fc54a": "\\mathrm{rem}(5, 3) = 2", "996aafe3a2bee9ed59d077dab4bf04a4": "O(\\log^4 q)", "996ac433e000f7a0c45eb9422e662f40": "\\Delta P =\\; C\\, a\\; h\\; \\bigg(\\frac {1}{T_o} - \\frac {1}{T_i}\\bigg)", "996ac45e73dfe4c510e030437a713c01": "b_{2}*b_{15} ", "996afa8824db13fea97ff321eed00a39": "P(\\{n_1,n_2\\}|N)= \\frac{[n_2 \\le N]}{\\binom N 2} .", "996b57e4097d834166dad4630b743ada": "\\forall v, w \\in V, \\forall a \\in A,\\; v + (w + a) = (v + w) + a", "996be524459babd460e22684ae275ede": "f(u)=u\\log u", "996c330cd460cb5dc312a438b7316587": "\\sin 54^\\circ=\\cos 36^\\circ=\\frac{1+\\sqrt{5}}{4}.", "996c5a1e9e6b1b66a7a888ad2eafff9f": " \\xi \\mapsto \\frac{ \\cos(\\theta/2) \\, \\xi - \\sin(\\theta/2) }{ \\sin(\\theta/2) \\, \\xi + \\cos(\\theta/2) }. ", "996c69a24ce3579b25aa9d5fcc36cd3e": "L := - \\delta \\circ \\nabla,", "996c8b5b83c19dfd1648772719ea028b": " \\Psi ( \\omega) := \\begin{cases}\n\\frac {1}{\\sqrt{2\\pi}} \\sin\\left(\\frac {\\pi}{2} \\nu \\left(\\frac{3|\\omega|}{2\\pi} -1\\right)\\right) e^{j\\omega/2} & \\text{if } 2 \\pi /3<|\\omega|< 4 \\pi /3, \\\\\n\\frac {1}{\\sqrt{2\\pi}} \\cos\\left(\\frac {\\pi}{2} \\nu \\left(\\frac{3| \\omega|}{4 \\pi}-1\\right)\\right) e^{j \\omega/2} & \\text{if } 4 \\pi /3<| \\omega|< 8 \\pi /3, \\\\\n0 & \\text{otherwise}, \\end{cases}", "996ca3af6fa16da84cc16722173264d2": "x\\in \\bar{X}\n", "996cc220e9872a9efd97f9018a0c75fd": " S(e,f) = \\{ h\\in M(e,f) : g\\prec h \\text{ for all } g\\in M(e,f) \\} ", "996cc968e0d303bededde71fb865f1e2": "\\Phi(\\mathbf{x}) = -\\frac{GM}{|\\mathbf{x}|}.", "996d0b8d8e97368d30f20de0da5b34ba": "N(A,t)", "996d81126554fc86c090f40232ae40df": "\\mathcal{B}_R", "996dd4b37e57da635117b92ea3f3a310": "F_\\alpha: (\\sigma^\\gamma F_\\gamma,v)\\to (F_\\alpha(\\sigma^\\gamma)F_\\gamma, \nF_\\alpha(v)), \\qquad I_a: (\\sigma^\\gamma F_\\gamma,v)\\to (I_a(\\sigma^\\gamma)F_\\gamma,I_av), ", "996e262e971d5827f98dfc80934e24b3": "v_2 = 1 - (R_2 + R_3 - Q_1)^2 / (4 R_2 R_3)", "996e4b573838f6c80b83fa4185fa802c": "\nx =\\frac{1}{\\pm\\sqrt{1+t \\,\\left(\\frac{1-X_2}{1+X_2}\\right)}},\\qquad\nX_2=\\frac{1}{\\pm\\sqrt{1+t_2\\,\\left(\\frac{1-X_4}{1+X_4}\\right)}},\\qquad\nX_4=\\frac{1}{\\pm\\sqrt{1+t_4\\,\\left(\\frac{1-X_8}{1+X_8}\\right)}}.\\qquad\n", "996e78b329729347cee2d035e5fae8c4": " N = \\left(-\\frac{g}{\\rho_0} \\frac{d\\rho_0}{dz}\\right)^{1/2}.", "996ea0fdf372828c4a59251eaa0c3f7e": "\n\\gamma_\\mu \\cdot \\gamma^\\nu = {\\delta_\\mu}^\\nu\n", "996ea54ae5ae88aaee42fecc77d8a836": "\\widehat{\\mathcal{H}}=\\frac{1}{2\\sqrt{\\gamma}}\\widehat{G}_{ijkl}\\widehat{\\pi}^{ij}\\widehat{\\pi}^{kl}-\\sqrt{\\gamma}\\,{}^{(3)}\\!\\widehat{R}.", "996ecd6b782ddcc6217ea4c0041eb31e": "\\mathrm{d} U = T \\mathrm{d} S - P \\mathrm{d} V + \\sum_i \\mu_i \\mathrm{d}N_i \\,\\!", "996ed0b00baff587718d8c72cac20f3a": "\\ \\Sigma", "996eff98eba360b1ecac06276b5645b7": "= (0.85 \\cdot 1 \\cdot 1.78 + 0.05 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.1036) = 1 \\cdot 1.78", "996f18b3c515b68071e021dd103deea6": "x_{opt}", "996f7ab681b4bb2bdfee85ab900a851e": " x^i", "996f91a1de4c5f0c76df4b463baf3dea": "\\operatorname{pmi}(x;y)", "996fda3d7d9ad6c068f16b8d4afcb2f2": "\\operatorname{E} (X \\mid I_1) = \\operatorname{E} ( \\operatorname{E} ( X \\mid I_2) \\mid I_1),", "996fe821acf619fce373a76186259442": " p(z)=\\lambda_1 + \\lambda_2 \\overline{z} + \\lambda_3 \\overline{z}^2 + \\cdots + \\lambda_N \\overline{z}^{N-1},", "996fee44fb124fa1539eae1fc77f4b53": "\\textstyle\\sum 1/n", "99707a917ac5468a1914d6f3669468aa": " \\cap ", "99709bb93c78ceaa25f37857205bf22f": "\\left | z \\right | = \\sqrt{z \\bar z}.\\;", "99709df67c9cb15688b32461d528a548": "\\mathrm{supp}\\,X.", "99709f3c20ef2755bc898f38372e1c08": "G(n)=\\prod_{k=1}^{n-2}k!=\\frac{\\left[\\Gamma(n)\\right]^{n-1}}{K(n)}", "9970a22c62cd6f23ed57d7f8e98549cd": "T^{\\bar k}p, \\dots, T^Kp", "9970e6db2aab5f8c2795e5b37a86d8cb": "\\prod B_\\lambda", "9970febe63ea5a939f4f16bb027a22d8": "\\rho_n \\to 0", "997159dfb23c8c675b2239748baa51ee": "T(\\lambda) = log^{-1}(-{SRM\\over 12.7}(0.018747e^{-{(\\lambda - 430)\\over 13.374}} + 0.98226e^{-{(\\lambda - 430)\\over 80.514}} +c_1 \\xi_1 + c_2 \\xi_2 + ...))", "9971947e4c9e8a2a847806335bf57be9": "\\epsilon_\\alpha.", "9971f415263e3f8e460fef97cb9578f4": "U(M) := \\{\\sigma \\in GL(M) \\ : \\ \\forall x,y \\in M, h(\\sigma x,\\sigma y)=h(x,y) \\text{ and } q(\\sigma x)=q(x) \\}.", "9972320ddddf976d791dc960c8887fbc": " (-1)^{n+m} n! \\; [z^n] [u^m] g(z, u)|_{u=1/u} |_{z=uz} = \n\\left[\\begin{matrix} n \\\\ m \\end{matrix}\\right]\n", "9972c63a7d49dfa97c04ba22860b4f98": "T(r,f) \\leq \\log^+ M(r,f) \\,", "9972d7cdbbe81693ac245061bae18e27": "M(\\hat{x}) \\triangleq\n\n\\operatorname{diag}( m_1(\\hat{x}), m_2(\\hat{x}), \\ldots, m_n(\\hat{x}) )\n=\n\\begin{bmatrix}\nm_1(\\hat{x}) & & & & & \\\\\n& m_2(\\hat{x}) & & & & \\\\\n& & \\ddots & & & \\\\\n& & & m_i(\\hat{x}) & &\\\\\n& & & & \\ddots &\\\\\n& & & & & m_n(\\hat{x})\n\\end{bmatrix}", "9972ebe22abc3ca4e3cae05e3c3828f1": "\\omega^*(x,n) = \\limsup_{H\\to\\infty} \\omega^*(x,n,H).", "99730c1a2951be6e2c06cc25940dd310": "p = O(\\exp(n^a))", "99731810c08e17f5f0c01e16550e98d0": "\\frac{\\hbar^2}{2m}\\frac{a}{A}=-\\frac{a}{4\\alpha}\\left[\\cot\\left(\\frac{k a}{2}-\\frac{\\alpha a}{2}\\right)-\\cot\\left(\\frac{k a}{2}+\\frac{\\alpha a}{2}\\right)\\right]", "99732ed060bfed5093ba013bd93fb3ac": "F^*(z)", "9973544c5aade33a092b94006590af12": " \\rho_1\\ :\\ f(x, y) \\rightarrow g(y) ", "99738a478da3c32873bd0296ec205ca9": "\\frac{1}{2}\\cot\\left(\\frac{x}{2}\\right) = \\frac{1}{x} + \\sum_{n=1}^\\infty \\left(\\frac{1}{x + 2n\\pi} + \\frac{1}{x - 2n\\pi} \\right)", "997409c61056cc3f40bdda1e41f89e2e": "W_i' = V", "99743fe92f6c1723ae7b706804b7fbfa": "\\sin\\theta_W = \\frac{g'}{\\sqrt{g^2+g'^2}}", "99747f134374d0a0b380c71b2bf75e02": "\n\\begin{align}\n(f - g)(z) &{}=(z - c)^m \\cdot \\left[\\frac{(f - g)^{(m)}(c)}{m!} + \\frac{(z - c) \\cdot (f - g)^{(m+1)}(c)}{(m+1)!} + \\cdots \\right] \\\\ \n &{}=(z - c)^m \\cdot h(z)\n\\end{align}\n", "9974a038bdb1bfbd6a286e94c9caed6d": "{\\rho_{air}} \\,\\!", "997538fe7f8a5f0d4846f07eebc9f1a7": " H(s) = H_0(s)\\frac{1 + \\frac{Z(s)}{Z_n(s)}}{1 + \\frac{Z(s)}{Z_d(s)}} ", "99758f2fe766ebb6a19bd5940e3f4339": "\\theta=\\int\\left(\\frac{M}{EI}\\right)dx", "99759af94baddcd3cb18b56f86385263": "v_\\mathrm{m}", "9975d9121d0c0ae1ff19e1a37f307366": "\\mathcal \\R^n ", "9976512bb2cf13017b5d961947f57d59": "Nc(n) = \\frac{(3n-2)(3n-1)}{2}.", "997694d57a4d3da1d185299ac0064020": "\n\\begin{align}\n\\begin{bmatrix}\nt'' \\\\ x''\n\\end{bmatrix} & =\n\\begin{bmatrix}\n\\gamma(v') & \\delta(v') \\\\\n-v'\\gamma(v') & \\gamma(v')\n\\end{bmatrix}\n\n\\begin{bmatrix}\n\\gamma(v) & \\delta(v) \\\\\n-v\\gamma(v) & \\gamma(v)\n\\end{bmatrix}\n\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix}\\\\\n\n& = \\begin{bmatrix}\n\\gamma(v')\\gamma(v)-v\\delta(v')\\gamma(v) & \\gamma(v')\\delta(v)+\\delta(v')\\gamma(v) \\\\\n-(v'+v)\\gamma(v')\\gamma(v) & -v'\\gamma(v')\\delta(v)+\\gamma(v')\\gamma(v)\n\\end{bmatrix}\n\n\\begin{bmatrix}\nt\\\\z\n\\end{bmatrix}.\n\\end{align}\n", "9976d416c00b6b3b22b621127d06fb0f": "F_D=C \\dot{U} \\, ", "9976e2e93659484e8fe108c4508368ea": " \\zeta(s) = \\frac{1}{\\left(1-\\frac{1}{2^s}\\right)\\left(1-\\frac{1}{3^s}\\right)\\left(1-\\frac{1}{5^s}\\right)\\left(1-\\frac{1}{7^s}\\right)\\left(1-\\frac{1}{11^s}\\right) \\ldots } ", "9976f8298ae3ed4b18e645423d53653d": "y = x^2 + 3x - 10", "99776f6e411fe29f942352d80484c04c": "R = A[X_1, \\ldots, X_n] / (f_1, \\ldots, f_m),", "9977a4110d179092268b1375954f8016": "PNPP+H_2O \\; \\overrightarrow{_{AP}} \\; PNP + P_i", "9978bb40507ce3dd9937c1ce85d3577a": " Hom(A,B)", "9979066504796421ef88f76f67b6eb3c": "g^{(n)}", "99791ec236159dafed38ff6dedbf7bbe": "(m+1,k+1)", "99793e7a28d8a28efb207cb8d460ab6c": "ax+by=1\\,", "997a10a0e57f2f37a3daddce539cab17": "\\frac{\\operatorname{tr}(\\mathbf{A}^T\\mathbf{R})}{\\operatorname{tr}(\\mathbf{\\hat{\\mathbf{M}} \\operatorname{diag}(\\mathbf{P}\\mathbf{1})\\hat{\\mathbf{M}}})}", "997a2e9d3e2bf531021618fdbdb5a081": "T^{r}_{r} = T^{\\theta}_{\\theta}", "997a4c9e7fb2d61a96f072ea0228de14": " I = \\frac{N} {\\sum_{i} \\sum_{j} w_{ij}} \\frac {\\sum_{i} \\sum_{j} w_{ij}(X_i-\\bar X) (X_j-\\bar X)} {\\sum_{i} (X_i-\\bar X)^2} ", "997ac70819cfddd1e99264c1cb5b8e27": "\\lim_{t\\to 0}\\frac{\\Delta^n_{t\\Delta x} f}{t^n}", "997acbbcec4f781a26aaa2fa5299f33c": "(X,d)", "997acbd7f8be35d2a48d23ebca593190": "\\frac{T^2}{\\langle\\sigma v\\rangle}", "997b2f985a8bc7e7049d8e6607aac44e": "s_j^2", "997b3b52e43f5f9fee48347a7b3bd6cb": " a_{N} ", "997b6e88c4c5b95e9f8198bc6438d7b9": "{\\bold v}^\\mathrm{T}{\\bold v} = (Q{\\bold v})^\\mathrm{T}(Q{\\bold v}) = {\\bold v}^\\mathrm{T} Q^\\mathrm{T} Q {\\bold v} .", "997b9bd5abba53d3b21c50cb6bcfafbe": " \\mathrm{area}(g) - \\tfrac{\\sqrt{3}}{2} \\mathrm{sys}(g)^2 \\geq \\mathrm{Var}(f),", "997bcbf6ad26adda8ddaba7c6ac61001": "Y_m = 2 P_{m-1} + 1", "997be7b5560abc21b1f80ff90ec2d318": "t_{e1} - t_{e2} ", "997c5cf28ae19283f7c8b239c3123028": "F_{buoyancy}", "997c98de252cba19ed4c2a7d043d6493": "\\lim_{n\\to\\infty} a_n = x.", "997ca154f245069a58d8ac2bf35e5d8f": " \\ln \\Gamma(\\eta_1+1)-(\\eta_1+1)\\ln(-\\eta_2)", "997cec03358b647481c057d2ec4659a7": "\\mathbf{x}\\cdot\\nabla", "997d436197da0fd0581bd2124b662245": "\\lim_{k \\to \\infty} (a_n)_{n\\in\\mathbb{N}}^{(k)} = (x_n)_{n\\in\\mathbb{N}}", "997d46517c331c224e433db0fa583258": "K\\neq 0", "997d99c896301356d39edbef48a7659d": "O(dn) = O(dmr)\\,", "997da8c69f337426d1613b88be62c1c8": "z(t) = (q(t),p(t))^{\\mathrm T}", "997dccfaed188b0cb3974443b1596a9c": "{\\mathbf{}}G_i,H_i", "997de22b9e4d07810d54b947c4feb7f0": " G\\left(X'_i \\beta \\right) = E\\left(Y_i |X'_i \\beta\\right) = E\\left[g\\left(X'_i\\beta_o \\right)|X'_i \\beta\\right] ", "997dea20daace3b1538d8e1f20d09a41": "\\left|\\vec a\\ \\vec b\\ \\vec c\\right|", "997e501b3a6f05486225291c2443f66f": "Z^e_2", "997e5bb9d7fe629b8f7ead112c68b793": " a = {\\mu \\over {2\\varepsilon}}\\,", "997ee8950b6564ed7912ae4a2a0b6cb2": "x'=\\gamma (x - v t), \\,", "997f07eef7733b824529c7013393de91": " x\\sqrt{\\frac{-2\\ln(s)}{s}}\\,,\\ \\ y\\sqrt{\\frac{-2\\ln(s)}{s}}.", "997f7787417cd35e7750c9a04b3bbb45": "\n\\mathrm{DR} = \\mathrm{SNR} = 20\\log_{10}{\\left(2^{16} \\sqrt{\\tfrac{3}{2}}\\right)} \\approx 6.0206 \\cdot 16 + 1.761 \\approx 98.09\\,\n", "997f885a82a8bb81d05f7b169ac3150d": "0.809\\pm0.024", "997fad9bb89d54d2357cc1a078163d1b": "A\\subseteq D+F", "997fd5395cca5567920ccdc4549c1268": "h^{-1}(L)", "998037e20f86a9c325339e850cbf3843": "\n\\langle \\phi + \\psi \\rangle = \\langle \\phi \\rangle + \\langle \\psi \\rangle, \\,\n", "9981050c7356db46c98dcac230d44f4d": "\\scriptstyle\\sigma^2_i", "99815edac1a14aac60d8697853c3e933": "\\bold{F} = d\\bold{A}+\\bold{A}\\wedge\\bold{A}.", "9981d329e03c1b108a5bcdb16016038b": "(z^2+c)^2 + c -z = (z^2 + c - z)(z^2 + z + c +1 ) \\,", "998200580551054a6a2f1461037d915d": "\\ell^2", "998213627361679eb4870f6e2fe0b80c": "F_{d} = 6\\pi r \\eta v_1 \\,", "9982194f9386b4f14a62e87c4aa8ff6f": "\\theta = \\tan \\theta - \\frac{\\tan^3 \\theta}{3} + \\frac{\\tan^5 \\theta}{5} - \\frac{\\tan^7 \\theta}{7} + \\cdots", "998276e3d4fe002920443d82805c6700": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{haversin}(x) = \\frac{\\sin{x}}{2}", "9982ea30a8a071138e3fdb0bc52b022f": " i\\omega C\\,\\!", "9982ebb763d0162cd9741a25993821d9": "GX\\xrightarrow{\\;\\eta_{GX}\\;}GFGX\\xrightarrow{\\;G(\\varepsilon_X)\\,}GX", "9982f0a7c43c99498dd2514d79ace426": "U={u_1,..., u_r}", "99833db02eca8dc8837c135b7cfb6e41": "_k\\mathbf{V}^i", "9983442cca4509438f4272bf0049bd18": "V_\\text{in}^+", "9983aee8787effad7929465b221ea1ba": "(0, \\infty)", "9983dae560e4cee428acbf71cbe14877": "B_{n}= (A_{n-1}^{xz }) (B_{n-1}^{xz+1}) (C_{n-1}^{xz }) ", "9983ead7072d5ccdb8e5c77ead64849f": "(a_1b_1 - a_2b_2 - a_3b_3 - a_4b_4 - a_5b_5 - a_6b_6 - a_7b_7 - a_8b_8)^2+\\,", "9983f5a4704e2eb2fd5be13371ff86fd": "(X_t)_{t\\geq0}", "9983fd6fa8878bf53be196d30dc4e0c7": "\n f(\\xi, \\rho, \\theta) = 0 ~.\n ", "99841bbda9fd7f73c71da13fed7de641": "\\gamma_k(X)", "99842a22fa81cc72af82e4f1be89110b": "t \\mapsto f(t)", "998435c5497cef4c98e3d9c284c67623": "\\mbox{efficiency} = \\frac{T_0}{T_{raise}} = \\frac{Fl}{2 \\pi T_{raise}} = \\frac{\\tan{\\lambda}}{\\tan{\\left(\\phi + \\lambda\\right)}}", "9984578bd18509f83565ad5197a1ed32": " G_1", "9984611dc8b5d0595a396b5c84cc23a7": "\\ell^2(A)", "99847ef6ce87ab90c4f59849d5b6cbea": "\n\\hat \\theta \\,\n", "9984c2a8fe131dd323141f3c63ce3ed7": "R \\hat{\\boldsymbol{\\beta}}= \\mathbf z.", "9984ca690ffde3d010932154271de914": "s_p^2=\\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2},", "99859f51b6217e0df3228684eac17f13": "f, g\\in \\text{Aim}(X)", "9985b6b2b37c12283d9cbaa5bb6638f6": " \\frac{dT_\\text{man}}{dt}=\\frac{3(-Q_\\text{surf}-Q_\\text{cmb})}{4\\pi\\rho_\\text{m} c_\\text{m}(R^3-R_\\text{c}^3)} + \\frac{Q_\\text{rad}}{V_\\text{m}\\rho_\\text{m} c_\\text{m}}", "9985cebd503f8ca2017451a966130864": "e^{ta}p_a+e^tbp_b+\\frac{1-p_a-p_b}{b-a-1}\\frac{e^{(a+1)t}-e^{bt}}{e^t-1}", "99860598cc663c646365313e513cb9e5": "H_{\\mathrm b}(p) \\,", "9986bd3d059e848632999eb2f6639325": "O(np)", "9986f6de818b60f28487541fd24d1de9": "\\approx\\frac{L_{L}-L_{T}}{\\lambda}\\left(\\frac{v_{A}^{2}-v_{B}^{2}}{c^{2}}\\right)", "9986ff838179ab127f1382b037d2a11d": "(ct)^2 - x^2 = (ct')^2 - x'^2 ", "998748cb0bd5cdd73d1a31a4db7d4cb7": "\\sum_n\\frac{1}{|z_n|^2}", "998846f736b0bff81c8bc1356f9b3a2d": "\\lceil n/r\\rceil", "9988b4ce426cc8f6aa9d70aede07e001": "x(t)=a\\cdot b^{t/\\tau}\\,", "9988c2642da80b5d97974a6e9d56591d": "{\\mathrm{excess}}", "9988c79ce13a766d996a30442322893d": "p(\\alpha)", "99890f30b46d8f1a299126f6d41e1f36": "A = B", "998912a24dbd088fa1bcce3e4a30e300": "b \\equiv a^{r/2} \\not\\equiv 1, -1 \\pmod{N}", "99899ac99c27f57ed067987ba13cd6e6": "\\begin{align}\nh_1&=1 \\\\\nh_2&=r \\\\\nh_3&=r\\sin\\theta\n\\end{align}", "9989abdcc7c1cf3f4fe122e733be1068": " \\operatorname{E}[Z^r] = \\zeta^r, \\quad \\operatorname{E}[\\bar Z^r] = \\bar\\zeta^r", "9989d5c0dd12c1453b48dab60b80ea37": "\\mathrm{d}U = \\delta Q - d W = T\\mathrm{d}S-p\\mathrm{d}V\\,", "9989e356fbac61cf1056b5d0957c88a6": "x^* \\in X \\subseteq X_R", "998a2592616b1488432ba34297a5fffe": "\\{X\\}_i, \\{Y\\}_i, f_i : X_i \\to Y_i", "998a291fe0f8a2b29c7053f8d222bbc5": "p=k_\\mathrm{B}TN \\left( \\frac{\\partial \\ln Q}{\\partial a} \\right)", "998a998c65a76a040533a26be7379119": " S_m(n) = \\sum_{k=1}^n k^m = 1^m + 2^m + \\cdots + n^m. \\, ", "998a9f1776e349461e96ef7a43fb3602": "d(x,y) = \\begin{cases}0 & x\\equiv y \\\\ 1 & x\\not\\equiv y\\end{cases}", "998ae8baaf38c56574f460fc7655b70a": "P = Fv", "998b218e080e079b27aba13294734d83": "(X_n , n \\geq 1)", "998b5cdfff7cff74587bc076ef94aade": "{1 \\over \\sqrt{\\mathit{f}}}= -2.0 \\log_{10} \\left(\\frac{\\frac{\\epsilon}{D}}{3.7} + {\\frac{2.51}{Re \\sqrt{\\mathit{f} } } } \\right) , \\text{(turbulent flow)}", "998b8c81d85b3dbf5c794944a5a8031a": "P_B(\\lambda)", "998b99e8508fae91dbbb2fd588f1801d": "\\mathbf{a_{C\\perp}} ", "998c4bf242e0f494966ebe7cf84b9bf3": "G_x = \\sum_{i = 0}^\\infty (-1)^i \\frac{1}{(2i + 1)x^{2i + 1}}", "998c8e26cfa1e1d7b1ab456e5d34985e": "r_{ad} = k_{ad} \\, p_A \\, [S]", "998c99e0ad8bd39cb711b6f7a485b3da": "\\delta(x-\\alpha)=\\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{ip(x-\\alpha)}\\ dp \\ . ", "998cc57f2973af3efbdc26d12e123457": "2^{-26.14}", "998d01a8d15e9010bd753258a8597dab": "N_p(x)=\\frac{1}{(2\\pi)^n}\\int_{[0, 2\\pi]^n}\\log|p(z)| \\,d\\theta_1\\,d\\theta_2 \\cdots d\\theta_n,", "998d1179fd34068051bd3863f2022a67": "\n\\sum_{n=1}^k \\, \\frac{1}{n} \\;>\\; \\int_1^{k+1} \\frac{1}{x}\\,dx \\;=\\; \\ln(k+1).\n", "998d4f4beb03f64ee3ebaaafb06eb3b9": "Q\\,\\!", "998d66daed9e9258f18134b65dca3699": "\\ b", "998df56350bb1bdfc31d9fb5e9ef45f9": "\\prod_{i < j} (\\alpha_j - \\alpha_i) = \\det V", "998e6f93367323bce3b7cea1f972ea3d": "w=s^{-1}\\bmod\\,q", "998f0aabed30daf82174e1d1f71b1b3e": "R_{21} = \\left. \\frac{V_2}{I_1} \\right|_{I_2=0} ", "998f19bd83d4a79682c5b60a3a905311": "\\xi (m) \\Delta m", "99901436e49df5a310510f98d3ec35c9": "\\mathrm{d}G\\,", "99904b5504f9f5c06dafd7542a40a2bf": "\\mathrm{d}\\vec v_s/\\mathrm{d}t=\\part \\vec v_s/\\part t + (1/2)\\vec \\nabla v_s^2-\\vec v_s\\times(\\vec \\nabla\\times \\vec v_s)", "999067561b346a6261fa97bf4b0731da": "U_i(I) = I_{L_i}", "9990e801c33f80c160007328aa6d0fcc": "{\\rm tr}", "999123f705014744cfeb7e8ae23a80b9": "1 - \\cos \\theta", "9991561c22f3d5eae21f05c9b144aaf9": "\\nu = 4", "99918f12a5c605661e21b183ee205ea5": " x= m^{-1}(x) ", "9991914823dd48b823ca7610a8f8baf1": " [i_0,i_1,i_2],", "99919a70000e2150f358889e2aa0584d": "\n\n\\begin{align} \n f_{1} & = \\frac {1} {2 \\pi (C_M+C_i)(R_A//R_i) } \\\\\n & = \\frac {1} {2 \\pi \\tau_1} \\ , \\\\\n\\end{align}\n\n", "9991a3f6555e5ec490e0fa9e5b785a56": " \\underline{y} = X \\underline{\\beta} + \\underline{\\varepsilon},\\quad (\\underline{y},\\underline{\\varepsilon} \\in \\mathbb{R}^n, \\beta \\in \\mathbb{R}^K \\text{ and } X\\in\\mathbb{R}^{n\\times K}) ", "9991b58758cac87c176639a3da276944": "\\displaystyle \\pi\\left( \\frac{1}{i \\pi \\nu} + \\delta(\\nu)\\right)", "9991bf9aef64739da0e84019c514a6ae": "L(N,M_0)", "99923d51c8ce92d1cb1668515569f94b": "\n\\frac{dA}{dt} = \\frac{\\partial A}{\\partial t} + \\{A, H\\}\n", "9992a0e83c2ecc74b34bec9dbbfe6752": "\\hat{\\rho}_A=\\sum_{j,k} c_{j,k}\\cdot\\hat{a}^j\\hat{a}^{\\dagger k}.", "99932c4efc7a58851273784f2d4e7e4e": "d+2", "99936da7f61d8c11cb485b9ad9e1940b": "-x^3-c_{8}x^2-b_{2}^2x+c_{8}b_{2}^2=0", "99938282f04071859941e18f16efcf42": "select", "99938abc5ec81cd24d64754f47e1dde3": "C^1(\\bar \\Omega).", "9993c0a08c6ed0c65a11a59a695eab17": "rpm = \\frac{\\upsilon}{C}", "9993e3ec6bdb545a51e438bb4d8a3f92": "H_{x}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[\\frac{m\\pi }{a}\\frac{1}{\\varepsilon _{r}}(A \\ e^{-jk_{x\\varepsilon }w}+B \\ e^{jk_{x\\varepsilon }w})+\\frac{jk_{xo}k_{z}}{\\omega \\mu }(C \\ e^{-jk_{x\\varepsilon }w}+D \\ e^{jk_{x\\varepsilon }w})]e^{-jk_{xo}(x-w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (42) ", "99942191354ac76ec3ab3f032b9db7d1": " \\psi^{(0)}_{-}", "9994357d59e50602b5716210c93d3ca7": "\\frac{{N-K \\choose n} \\scriptstyle{\\,_2F_1(-n, -K; N - K - n + 1; e^{it}) }}\n{{N \\choose n}} ", "99943e553844c71a4c276b7ca27b6048": " Q^{c} = M_A^{1/2}(AM_A^{-1/2})^+(b-AM_A^{-1}Q_b). ", "999450a60f23862a56160110656c92be": "w_\\max = 1", "9994881e43cedcac00ac682b64813314": "\\int \\frac{1}{h(y)} \\, dy = \\int g(x) \\, dx ", "99950792b234a7d66f0eb70bf1596b99": "\\begin{cases}\nR_{i,ser} = R_{i,a} + R_{source} \\\\\nR_{i,ser} \\cdot R_{i,b} / (R_{i,ser} + R_{i,b}) = R_{source} \\\\\nR_{i,b} / (R_{i,ser} + R_{i,b}) = Ratio_i\n\\end{cases}", "99955726084b4ba74040ca0b6bdd2e19": "\\operatorname{Var}(X) = \\operatorname{E}(X^2) - [\\operatorname{E}(X)]^2 .", "999582b35be55cd132091f84ab141c14": "10^{10^{10^{10}}}=10 \\uparrow \\uparrow 4=(10 \\uparrow)^4 1", "9995e7d22aed77f036a891c8c33cec1f": "0\\leq D_N({\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1})\\leq 1", "99960e9c160b16d210f7ad291ff8a910": "\\mathbb{C}^{2 \\times 2}", "99963293008ae7b67276c8f1c28763df": "\\partial f^{-1}\\left( Z \\right) = f^{-1}\\left( Z \\right) \\cap \\partial X", "999655e5198e14147f41f76655986072": "\\nu \\in [0, 2\\pi].", "9996b01fb89a25cc8e863a7d70582efb": "\\mathbf{V}_{r} = \\frac{\\mathbf{\\rho I}}{\\mathbf{2 \\pi r_x}}", "9996da8bc34153a6cd9e088d7914e023": "\\Gamma^k{}_{ij}", "9996df7f83b0f16f2ebecfb9f7c454df": "\\hat h_L(P) = \\lim_{N\\rightarrow\\infty}\\frac{h_L(NP)}{N^2}", "9997157806d6b9031627101b9a3b013f": "E(V/R)", "9997210146bb832ae195f65a02cd50dd": "\\mbox{Grade of Service}=\\frac{\\mbox{number of lost calls}}{\\mbox{number of offered calls}}\\qquad(1)", "99973d3a7ef9137cff49f3eb6cc86263": "\\frac{\\partial}{\\partial t} ( \\rho_0 + \\rho_0 s) + \\frac{\\partial }{\\partial x} (\\rho_0 u + \\rho_0 s u) = 0", "99974686391162b4872d2f57d90acbaa": " \\ = \\ \\frac{N-b}{\\max\\left(N-b-\\tfrac{r}{2}-\\tfrac12\\ , \\ \\tfrac12(N-b-1)\\right)}", "99974a11a9b14f1b77d248969bd32900": "W={2 \\pi r F}", "999759b44921e50b921eed1c4ee9eeff": "P_0(A) \\triangleq \\mathbb{E}[Z_0(T)\\mathbf{1}_A], \\quad \\forall A \\in \\mathcal{F}(T)", "99975ea66dd345107dc629348b95835d": "\\pi i", "99976ab8b2c32279ccf672c0335783ed": "\\left(\\sqrt{1/55},\\ \\sqrt{1/45},\\ -4/3,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "99979132b6724219a868c19463d30d32": "A + B", "99979c60d43d90641fc585e9ab1f5864": " 144 = 2 \\times 2 \\times 2 \\times 2 \\times 3 \\times 3 = (2 \\times 2 \\times 3) \\times (2 \\times 2 \\times 3) = (2 \\times 2 \\times 3)^2 = (12)^2.", "9997da23f8fe8a35457a7d7c093fc68a": " \\int\\!\\int\\!\\cdots\\!\\int\\; \\left|\\Psi^{(S/A)}_{n_1 n_2 \\cdots n_N} (x_1, x_2, \\cdots x_N)\\right|^2 d^3\\!x_1 d^3\\!x_2 \\cdots d^3\\!x_N = 1 ", "99986d3572f940a40be3143a0189f6a7": "\\neg(A\\lor B)", "99989456710db0268ec208045f961f10": "K \\to \\infty", "9998e248d8ddbaf53bcda94bc1534530": " p_2 := r_2.\\, ", "999901f1f1255d5317aef06ec3cefae4": "z \\mapsto z - R \\frac{z^{p}-1}{pz^{p-1}} + c,", "9999ef90d8421ae0242a032ea871d3eb": "X_{t-1}", "999aa1d102315e0647fe8764455c214c": "\\ x=s+v.\\,", "999acebc7eb3df8fc8e8de06679eba6b": " R =\\frac{\\omega_A}{\\omega_B}=\\frac{r_B}{r_A},", "999acfe55e0f21bb9691acdb3d53cf73": "\\scriptstyle M_{bol_{\\ast}}=7.416", "999af435830d539e3bac2e87457bfd8f": "\\lim_{n\\to\\infty} |S_n(\\beta)|^{1/n}.", "999b314f414bb99dbe05fd5c7aafec05": "\\alpha\\ominus\\beta:=\\alpha\\oplus(\\ominus\\beta)", "999b3ba138e02d724296190e54ba1de0": "C_J = \\frac{1}{L_J\\omega_J^2} = \\frac{\\Phi_0I_J}{V_0^2}\\cdot \\frac{\\cos \\phi}{2\\pi}. \\ ", "999b3da34f3f212f62fda07b600def28": "\\alpha_{2}(a) \\stackrel{\\mathrm{def}}{=} \\displaystyle\\sum\\limits_{d,\\,e \\in A} g(a,\\, d, \\,e )", "999b40bade62fbfc641b4f3fc623ad38": "\\scriptstyle \\{x:\\eta(x)=0\\}", "999b6430b0a10c7418543461495e7fe6": "\\hat 3", "999bd3fe53ec683ccb9caaf834a350b9": "(1-p^{m-1}){B_m \\over m} \\equiv (1-p^{n-1}){B_n \\over n} \\pmod{p^b}.", "999bf23c6b86a4b627546b3eecc7b441": "\\epsilon_{h,v}", "999bf3821f12816ce1eb52676803db5b": "\\phi_i\\,", "999c095b055c73baba7b00e5c4746c63": "\\forall y \\exists z ( F(y) \\vee \\neg F(z) )", "999c0e962cf1679db611142ca4b74c6f": " \\frac{d{M}}{dt}=\\gamma{M}\\times{B}-\\frac{M_x\\vec{i}+M_y\\vec{j}}{T_2}-\\frac{(M_z-M_0)\\vec{k}}{T_1}+\\nabla\\cdot D\\nabla{M} ", "999c5788138a55b4982f8daabc8f3f27": " \\boldsymbol{\\epsilon}^* ", "999c64bae753eda396100f7367a96567": "S^2_{n+1}", "999c7779f806cb8ab2313731d4fd41ab": "f_j=0", "999c7b581058ac6418918acc7e2bddb3": "C(h) ", "999c8ab234ae95e9c3fd85949fa17c09": "f[x_i, x_i, x_i, x_i]=\\frac{f^{(3)}(x_i)}{6}", "999c9af9e057c5302f53abd07778d129": " 0=\\frac{\\dot Q_L}{T_L}-\\frac{\\dot Q_L}{T_a}+\\dot S_i", "999ca9d6a276f04435ca8ad875e8b5f0": "i=1, 2, \\ldots, n-k", "999cfff761b595f2d86204d9a850d30b": "Y_{lm} (\\theta,\\phi)\\,", "999d16fde3564dd53d96345957583831": "f'(x_0), f'(x_1), \\ldots, f'(x_n)", "999d294cccfd7bc070c207f61f33c975": "f_\\alpha : X \\to Y_\\alpha", "999dbcc242ef89aaae3afb80561bd4b1": "\\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix} ", "999dc569d0e051acf816f550a8290ad6": "\\delta_1, \\dots, \\delta_N", "999df9fcd21b00b88392949ebbe9f071": "S\\downarrow T \\to \\mathcal A", "999e524fbfdad3435ef9e62d1aed16fa": "\\mathrm{Mg} = - {\\frac{d\\sigma}{dT}}\\frac{L \\Delta T}{\\eta \\alpha} ", "999e8d81d1daee5f6d05aaabb2594d53": "(A \\times 256 + B) / 10 - 40", "999f051c0204966c746fdf2893250ec9": "x\\in U_{\\gamma}\\subseteq V_{\\alpha}\\,", "999f0838b91f509b064f5f1ef87e0980": "\\int\\frac{s\\;dx}{x} = s - a\\arccos\\left|\\frac{a}{x}\\right|", "999f8b7253821b73d4ce705664ba8c76": "\\int_{0}^{\\infty}\\int_{-\\infty}^{\\infty} \\frac{1}{|a^3|}\\psi\\left(\\frac{t_1-b}{a}\\right)\\tilde\\psi\\left(\\frac{t-b}{a}\\right)\\, db\\ da=\\delta(t-t_1).", "999ff9207ca5a71d349ca31f6f581717": "\n D^{\\mathrm{beam}} = -M/\\tfrac{\\mathrm{d}^2 w}{\\mathrm{d}x^2}\n ", "99a0aa4813b46547b265f740e1539d26": "\nE_{\\mathrm{bend}} = \\frac {L^3 F}{4 w h^3 d}\n", "99a0acbc9304c72594bfd4d30afc8e15": "\\scriptstyle \\lesssim3\\times10^{-8}", "99a0b9456379b72d56362a22d4f7c225": " H^*_{\\text{eff}} | \\Psi_{E}^* \\rangle = E^* | \\Psi_{E}^*\\rangle", "99a0c1eb3b7db8fa116f8d0fe0a6312e": " |\\phi\\rang ", "99a0f1ed426c3fe7946fe8b342875d3e": "(\\ \\bar{r_0}\\ ,\\bar{v_0}\\ )", "99a0faa9e3ed438398e44c950d66ea00": "Z=S \\cdot \\sin\\phi", "99a18ca267e0289cd50c1db3dbd924ff": "V_\\text{E} = \\left( \\frac{R_\\text{f}}{R_1} + 1 \\right) V_{IOS}", "99a286c08a4afe7200fe5159a888c399": "f_\\theta(x) = f_\\theta(x,t) = f_{\\theta | t}(x) f_\\theta(t) ", "99a2a7bade9adc6c76546dd6afc8d657": "\\frac{\\pi}{4} = \\arctan\\frac{1}{2} + \\arctan\\frac{1}{3}", "99a2e35da11ef3bb2558c27b95b3af40": "D_\\text{in}", "99a313a1bc87ba3f91c17c7aa67b398b": "\np_w(\\vec\\theta)=\\sum_{k_1=-\\infty}^{\\infty}\\cdots \\sum_{k_F=-\\infty}^\\infty{p(\\vec\\theta+2\\pi k_1\\mathbf{e}_1+\\dots+2\\pi k_F\\mathbf{e}_F)}\n", "99a37a464bdc110455d3ddbe4724578d": "a=[t(u+v)+(1-uv)][u+v-t(1-uv)]", "99a4329474c64111f3342465ffa8f184": " \\epsilon_{0123} = +1 ", "99a4a77090760ac38c7cfc3f720698ee": "A\\to\\alpha", "99a4dc0079a63b195f345e0c6534d754": " \\vec{F} = \\mathbf{F}(\\mathbf{r}) = F( ||\\mathbf{r}|| ) \\hat{\\mathbf{r}} ", "99a4f23bd2d0579260f3c966dcdff8ea": "|x \\cdot y| = \\|x\\| \\|y\\| | \\cos \\theta | \\le \\|x\\| \\|y\\|.", "99a500fbc2936c9be6351411cc91aad7": "N_{l}^{k} E_{l}^{k} = \\left\\langle \\Psi_{m}^{(0)} \\left| \\left( V_{l}^{k}\\right)^{+} \\hat{\\mathcal{H}}^D\nV_{l}^{k} \\right| \\Psi_{m}^{(0)} \\right\\rangle = \\left\\langle \\Psi_{m}^{(0)} \\left| \\left(V_{l}^{k} \\right)^{+} V_{l}^{k}\n\\hat{\\mathcal{H}}^D \\right| \\Psi_{m}^{(0)} \\right\\rangle + \\left\\langle \\Psi_{m}^{(0)} \\left| \\left( V_{l}^{k}\\right)^{+}\n\\left[ \\hat{\\mathcal{H}}^D , V_{l}^{k} \\right] \\right| \\Psi_{m}^{(0)} \\right\\rangle \n", "99a503c9facb6d34c07a699dd2978617": "\\lambda_n > 0", "99a514baaa2a7e40159bde4e5a22e82e": " p = p_0 \\sin(k r - \\omega t)\\,\\!", "99a5244af1346e1e2ba92f5f1544e3f2": "OA = \\cos \\alpha \\cos \\beta\\,", "99a55ae0fae52a8225d02a56897eb0da": "{\\mathcal L}", "99a56a255bd8e2fd70a0f1254ce691ac": "(a, b)^*\\,", "99a61507c15c36de038e79d1ce6104da": "\nM = \\begin{pmatrix}\n1 & 4 & 7 & 1\\\\\n1 & 1 & 4 & 7\\\\\n7 & 1 & 1 & 4\\\\\n4 & 7 & 1 & 1\n\\end{pmatrix}\n", "99a61db07743c20d789aed2bc4438975": " r = \\sum_{m=1}^{n-1} k_m", "99a63d1d7c571e38eb838327a996a926": " \\mu(zI) = \\real\\,(z) ", "99a6496c76961b21137136338880a4a1": "\n\\underset{\\boldsymbol{x} \\in \\mathbf{S}}{\\text{maximize}} \\quad \\frac{f(\\boldsymbol{x})}{g(\\boldsymbol{x})},\n", "99a6859937beb68b53eca16af4554b94": "D_N({\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1})={\\rm sup}_J|F_N(J)- V(J)|", "99a6a18c09e188d76cbb03ee4ca2e522": "\\bold{r}_{vv}=\\Gamma^1{}_{22} \\bold{r}_u + \\Gamma^2{}_{22} \\bold{r}_v + N \\bold{n}", "99a6aeabe76b339537228cdc71738790": "X \\cdot f > 0,", "99a6e357edde524da8472788460431c9": "\\{\\psi^{\\langle1\\rangle}, \\psi^{\\langle2\\rangle}\\}", "99a6f2ce41c4cbe7283c2d79f72bf41c": "I(F_a;C)", "99a76085c3efe8cc1e073b1f4e1e32a5": "I_n = -\\frac{\\sqrt{ax+b}}{(n-1)bx^{n-1}}-\\frac{a(2n-3)}{2b(n-1)}I_{n-1}\\,\\!", "99a7695bb65fef318e3ba5c3d5b23663": " D \\left ( \\omega, k \\right ) = 0 ", "99a787efdeaedcb80eee83895807e074": "2^{k}2^{H(p + \\epsilon)n} \\ge 2^{n}", "99a7b23a85370fe21ecf6f8843e97e67": "x \\wedge ( y \\vee z ) = ( x \\wedge y ) \\vee ( x \\wedge z )", "99a8098c14473ba5c1deec7260de92db": "\n\\sum_{j=1}^\\infty 10^{-j!} = 0.110001000000000000000001000\\ldots\\,,\n", "99a82cba51951651f9376127b732dce7": "\\sum_{k=0}^d a_k \\binom{t}{k}", "99a836064cfb01422f0e40c173a72d61": "y'=a", "99a87a29a8eb229ee73dd2ca15c4280b": "\\hat{S} = 1 + \\sum_{\\mathbf{R},i,j} | p_{\\mathbf{R},i} \\rangle q_{\\mathbf{R},ij} \\langle p_{\\mathbf{R},j} |", "99a88df45cc8099605b57adb7839bd90": " \\hat \\hat R_{\\alpha,\\beta,\\gamma} (\\hat T_{l,m}) = \\sum_{k=-2}^2 \\hat T_{l,k} \\mathfrak{D}_{k,m}^{(l)}(\\alpha,\\beta,\\gamma)", "99a89112aac8d7372be0c1d7837c0966": "\\frac{m_s}{m_b} \\approx \\frac{1}{3} \\frac{m_\\mu}{m_\\tau}", "99a8c59aed64c5f7fe019303f32dfdbb": "-\\mu \\cdot \\frac {\\hat{r}} {r^2}", "99a8d0b3d2f85155f4cd295e6779acf5": " W(z) e^{W(z)} = z.\\,", "99a93918e709641133eb391d5c2ea296": " T_i", "99a99c00af9a975e8f4b74238ba827b8": "\\displaystyle{J=\\begin{pmatrix} 0 & I \\\\ -I & 0\\end{pmatrix}.}", "99a9c34f3a911b682465f3b3485c1e9a": "i=1,\\ldots,n", "99a9d6956210232c5959c6a866a0cf19": " \\alpha(t) = \\frac{ 7 \\ (C_2 + C_3) \\text{k}_{3(2)} [45 \\ {_2^0}S + 22 \\ {_2^1}S] } { 11 \\ [29 \\ C_1 \\text{k}_{3(1)} + 14 \\ (C_2 + C_3) \\text{k}_{3(2)} ] \\ {_2^1}S } ", "99aa1fa6913ad65638e2125c21744225": "\\rho \\left[ \\frac{\\partial \\overline{u_i}}{\\partial t} + \\frac{\\partial \\overline{u_i}\\, \\overline{u_j}}{\\partial x_j} + \\frac{\\partial \\overline{u_i'} \\overline{u_j'}}{\\partial x_j} \\right] = -\\frac{\\partial \\bar{p}}{\\partial x_i} + \\mu \\frac{\\partial^2 \\overline{u_i}}{\\partial x_j \\partial\n x_j}.", "99aa24ba3de032e55e12b8408ca4492f": "a=Q \\rho", "99aa2eb973b3181823b4d0f23e3262ab": "g^x", "99aa5b678b9f9ada335cffb303c2b68b": "\\mathbf{l}_b", "99aa7c2f7c8d216df383c6db4d4cb20d": "1/p", "99ab49f3df138f3f486d1ece90fc5c0c": "d \\; ", "99ab5292788e192c91e3c28a85440cd3": "v(x)=v_j(x)", "99aba72373b6dc95dfe27fcf6d593f31": "-1/k^3\\,", "99aba7daffa956576b477d2bdf8747b8": " \\text{Oxygen Content of blood} = \\left [\\text{Hb} \\right] \\left ( \\text{g/dl} \\right ) \\ \\times\\ 1.36 \\left ( \\text{ml}\\ O_2 /\\text{g of Hb} \\right ) \\times\\ O_2^{\\text{saturation fraction}} +\\ 0.0032\\ \\times\\ P_{O_2} (\\text{torr}) ", "99abae635fc4b53aed8c30656744c9bd": "(X\\sqcup X^\\dagger)^+", "99abd4a6ac912c676809ae08c43db189": "\\lambda(t)=\\lambda(0)", "99abe36b2c13733828716741df2fe800": "\\exp^{1}\\!=\\!\\exp ", "99abf0c081ca17a80d487ca453a62253": "\nJ = q \\mu E_\\mathrm{max} N_c e^{-\\Phi_B / V_t}(e^{V_a/V_t} - 1)\n", "99ac17f734fb064efd63111410cae91f": "\\langle k_-,\\,k_- \\rangle = - \\langle k_+,\\,k_+ \\rangle", "99ac180bb6d808e23b28d1cc09a60ba5": "s_i:N(C)_k\\to N(C)_{k+1}", "99ac2f443224f4d253a76f8a74e04cb3": "\\scriptstyle\\mathcal{F}(U)", "99ac4cb0f686005f4484037426a7a266": "\\theta_1(x,q) = \\sum_{n=-\\infty}^\\infty (-1)^nq^{(n+1/2)^2}e^{(2n+1)ix}", "99ac6a11c2f7eb55ba8e95d9256c37c8": " |\\phi_1\\rang, |\\phi_2\\rang, |\\phi_3\\rang ", "99acb9bcfbdf0966327e6062ac17abf4": "\\,\\sigma_i \\in \\Gamma^*", "99acce46711328ebd1665d5a8e5aa1e5": "w\\in L_{1}", "99acf990e70071883dc6673e3253e480": "\\ \\nu(t)=\\sum_{k=0}^{N-1}X_ke^{j2\\pi kt/T}, \\quad -T_\\mathrm{g}\\le t < T", "99ad642ef467036b3a89981f17be6791": "\\eta \\circ A \\neq G(A) \\circ \\eta", "99adc659689b5c4a349afe703a47809f": "p[\\sigma]", "99adda499059eca36e0f828310596dec": " s_\\nu = \\sum_{n=1}^N b_n z_n^\\nu \\ ", "99ae011e9ec9463fcffa2f4c2278c36e": "A_1 = B_1 = 0", "99ae05ad70ffc9b58ed45c15ae9f401a": "\\partial f(x)(h)= \\sum_s \\frac{{}_{(s)0}\\partial f(x)}{\\partial x} h\n \\frac{{}_{(s)1}\\partial f(x)}{\\partial x}.\\,\\!\n ", "99ae30646dd1dd76b542841109a850db": "V(a_0, b_0, c_0, d_0, \\dots)\\,", "99ae853ea21d9861618eb9511a85fb4d": " H(x_1, \\ldots, x_n) = \\frac{n}{\\frac{1}{x_1} + \\cdots + \\frac{1}{x_n}} ", "99aeef1a74cb1f6122018c0ee25256ba": "\\tfrac{1}{12}(b-a)^2", "99aefc48116e5cdd3506c4e10930969a": "\\mathbf{v'}_2 = \\mathbf{v}_2 + \\frac {j_r} {m_2} \\mathbf{\\hat{n}}", "99afb66755bddd40b29f2ee876a387a0": "\\displaystyle{h=g_z=T(g_{\\overline{z}})= T(f_{\\overline{z}})=T(\\mu f_z)=T(\\mu h) + T\\mu.}", "99afba4d4361f47900cd24a334c39d7f": "O(P_r P_s/M)", "99afd0f18fbbbc1fd0a636141806c90d": "P(x_3) - f(x_3) = + \\varepsilon\\,", "99afd78d09384627f69f88db7c76140f": "{D_H}", "99b03ce2fd0a4abed50be6bf209c1d98": "\\exp (i \\mathbf{q}\\mathbf{r})", "99b0ce850e6c7014fb6a4bb01787e355": "M_n(D_1) \\times M_n(D_2) \\times \\dots \\times M_n(D_r)", "99b1361e0346417cdf57e33eb091f8f6": "\\lim_{r\\to 1^-} h(re^{i\\theta})", "99b1492025d2ffb7a37d3a134e772ccf": "J(x) = \\sum \\frac{\\pi(x^{1/n})}{n}.", "99b16c2011b201243a407c3f96c882f6": "C_4 / C_2 \\cong C_2", "99b1af871bf5ae8a41f19b486631fc2a": "[H_{ij}^\\ast] = [H_i^+][H_j^-] +\n[H_j^-][H_i^{0+}] - [H_i^+][H_j^{0-} ] + (\\alpha_i - \\alpha_j\n)[H_j^-][Q_i^+]", "99b1dd2ed4af376366b5e6473ff839ea": "f^{-1}(\\mathrm{int}'(A)) \\subseteq \\mathrm{int}(f^{-1}(A))", "99b201acebf91b99c1b7e2d91a9a2ad5": "A = x(0)\\,", "99b236854bfa06c748a797611ada1891": "\\mathbf{u}=\\left( \\mathbf{z}|\\mathbf{x}\\right) ,\\mathbf{v}=\\left(\n\\mathbf{z}^{\\prime}|\\mathbf{x}^{\\prime}\\right) ", "99b2f0d7b2d5a0c5a3822bfdb80b6111": "r_0^2", "99b309881d1850f36337b9eb5f7c8f14": "a_0=0.35875;\\quad a_1=0.48829;\\quad a_2=0.14128;\\quad a_3=0.01168\\,", "99b32f504b30b2aa070ade6ad15c2926": "dx \\, dy \\, dz", "99b341f58a83c8ea77e03f911c0a1cb8": "\\scriptstyle z=-\\infty.", "99b40f1f404cc87358b16b5304582446": "\\left[\\begin{smallmatrix} a & b \\\\ b & d \\end{smallmatrix}\\right]", "99b4426b377f5b6fa0b2e67be894a756": "T_{\\textrm{eff}}(\\theta)\\sim g_{\\textrm{eff}}^{1/4}(\\theta)", "99b4942f662e023153881b37f17631f8": "\\psi(x,\\theta)=\\nabla_\\theta\\rho(x,\\theta)", "99b4d6c252a31dc69d56e1252c0aeffa": "\\mu_i = \\left(\\frac{\\partial H}{\\partial N_i} \\right)_{S,P, N_{j \\ne i}} or \\ \\mu_i = \\left(\\frac{\\partial U}{\\partial N_i} \\right)_{S,V, N_{j \\ne i}}", "99b5207f9fbea19915de905befbbf00f": "\\hat{y}=\\hat{b}", "99b536c6a66a2da464fe5953267ffb4c": " {3 \\over 2} = 1.5", "99b54c76c47fe4599f127e57d3ada435": "\\pi_i p_{ij} = \\pi_j p_{ji}\\,.", "99b5742ef65f961f6ac585eadce0fa25": "\\textstyle \\lambda(E)", "99b5763a660695ff63fff6c23411c7cd": "\\hat{a}^{\\dagger}_{\\mathbf{k},\\pi}", "99b579097351e3af9451c5c7d7fa610a": "T_{\\rm ex}", "99b58a039bc6bed21844094afc5de0c8": "~\\hat A = c \\hat a + s \\hat b^\\dagger,", "99b5e3419beae1fced8b61224dbcac10": "L^{-n} y_t = y_{t+n}", "99b63c269f9536266d01b5d3116882ba": "Jac(C)", "99b66f9471b877970145cfc253d305c1": "\\theta_o\\,", "99b69972664ba9de22bf0214874cefc9": "f(\\alpha)=\\gamma", "99b6b8fdee1b6dbcb4ebfa68c646043f": " \\frac{\\partial L}{\\partial \\mathbf{q}} - \\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} = 0", "99b6bda718633e6bad4f59e659f874c9": "\\displaystyle{T(x-a)g|_{x=a}=(x-a)Tg|_{x=a}=0}", "99b716122da7ce13d76e59c590ba11d6": "a_1, b_1, a_2, b_2", "99b749f11161d054b912dd1bf793bfb2": " a \\oplus (b \\oplus f(x)) \\equiv a \\oplus (f(x) \\oplus b) \\equiv (b \\oplus f(x)) \\oplus a \\equiv (f(x) \\oplus b) \\oplus a", "99b74f994fe57f8d36c168c5c8c94e70": "\\scriptstyle x_L[n],", "99b75d5f405a2f725056260b4897caf4": "S = R\\, \\zeta - \\tfrac12\\, g\\, \\zeta^2 + \\tfrac12\\, \\zeta\\, u_b^2 - \\tfrac16\\, \\zeta^3\\, u_b\\, u_b'' - \\tfrac16\\, \\zeta^3\\, \\left( u_b' \\right)^2 + \\cdots.", "99b77bdfefac436547044d148502524a": "[v\\;\\|\\;M]_h\n\\;\\|\\;[u\\;\\|\\;v'\\;\\|\\;N]_m\\!\\!\\rightarrow\\!\\![[w\\;\\|\\;M]_h \\;\\|\\;w'\\;\\|\\;N]_m", "99b7dcc917172e5b4cf04a8847341571": "(10 \\uparrow)^1 10^{1453}", "99b7fdc95a5dd0d7110c5a0a8138f307": "/ c(m)", "99b839f3bee235c4ceaa97f8002a54d6": "\\scriptstyle{\\theta}", "99b8485f0a9279aba25cb4473c8ce69d": "O(V^2 \\log V + V E)", "99b84dfd3e290d9fb9f54d093a5991de": "v = \\sqrt{v_x^2 + v_y^2 + v_z^2}", "99b89e6c1f58c0a085391e36a072bc76": "Z_3 = (Y_1+Z_1)^2-D-ZZ_1 = 2\\sqrt{3}", "99b8b34787e9ec61ed6bce4003f34910": "g_{00}", "99b8b3ab41240716fa08c159ee0a0857": "\\bigl(w(s, p1) + w(p1, p2) + ... + w(p_n, t)\\bigr)+ h(s) - h(t)", "99b9b7454f99aee7907af2918f48cbd7": "\n\\frac{dx(t)}{dt} = \\Pi_K(x(t),-F(x(t))).\n", "99b9d43259e09f6e1fab48f0d75703ea": "\\sum_i \\operatorname{Tr} \\; (\\rho \\otimes \\omega)(F_i \\otimes \\Psi_i^*(O)).", "99b9e03887bd31c1f5fcbbd7ae0d35dc": "B_2=2\\pi\\int_0^\\infty r^2\\left[ 1-e^{-V(r)/k_\\text{B}T} \\right] dr", "99b9e4c7359c802c8f4c7c95f1f3f3a5": "\\tilde{u}^2~=a_0+\\frac{a_1^2}{4} .", "99ba0683b1c8b6d0d0c83f75271532e2": "S={1\\over 64\\pi G}\\int d^4 x\\sqrt{-\\eta}\\eta^{\\mu\\nu}g^{\\alpha\\beta}g^{\\gamma\\delta} (g_{\\alpha\\gamma |\\mu}g_{\\alpha\\delta |\\nu} -\\textstyle\\frac{1}{2}g_{\\alpha\\beta |\\mu}g_{\\gamma\\delta |\\nu})+S_m", "99ba17ab5c1e212effedddb544f841d1": "\\pi(\\mathbf{r},t) = \\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\varphi}}\\,.", "99ba469673dd64a98c19c74d09aae0e2": "x_1 x_2 + y_1 y_2 = 0 \\quad", "99ba8764b2eab076096734651e5220f0": "\\frac{\\sum_{j=1}^{20}n(j)M^n(j,j)}{N} = \\sum_{j=1}^{20}f(j)M^n(j,j) = 1 - \\frac{m}{100}", "99bae734acfad05ed777e8c120b9f8b2": "(8m+61, 8+m, 3(m^2-61))", "99baed52271a9dad69a894b8a914079e": "\n\\left[ \\hat{g}_N(x)-z_{\\alpha/2} \\frac{\\hat{\\sigma}(x)}{\\sqrt{N}}, \\hat{g}_N(x)+z_{\\alpha/2} \\frac{\\hat{\\sigma}(x)}{\\sqrt{N}}\\right]\n", "99bb02a2b613469668398dcf6b751015": "0 \\le f(x) \\le g(x)", "99bb05e93feb63a4bd3ab6afb9cb28d7": " =\\sum_{i\\neq m}\\text{Tr}\\left\\{ \\mathbb{E}_{X^{n}}\\left\\{ \\Pi\n_{\\rho_{X^{n}\\left( i\\right) },\\delta}\\right\\} \\ \\Pi_{\\rho,\\delta}\n^{n}\\ \\rho^{\\otimes n}\\ \\Pi_{\\rho,\\delta}^{n}\\right\\} ", "99bb179593b898ca05f5a9b0d669b706": "R_t = R_p - R_v", "99bb1d563854dd04a707e9b4049114d8": "[T_{M/N}] = -[TN]", "99bb8c154cac25d79af5a67343224c1a": "Q_1 + Q_2 = Q_3.\\,", "99bbd9a9fe493143d8aecf18d872725e": "\n\\ddot{r}=-\\frac{G_NM}{r^2}\\left[1+\\alpha-\\alpha(1+\\mu r)e^{-\\mu r}\\right],\n", "99bbf9b733f449e9a602118c01ef19d8": "g_{\\kappa\\lambda}", "99bc05b985a6557303572f10feeb65e8": "D_S", "99bc4000b18a9360c0fd272ff0b05fc4": "\\tan\\frac{\\pi}{8}=\\tan 22.5^\\circ=\\sqrt{2}-1\\,", "99bc50bc918263bc0e8711b8378b0e7b": "\\beth_{n+1}(T)=2^{\\beth_n(T)}", "99bc7dd04af4ba920b05c96d9958fbab": "\\ \\varepsilon_E=\\frac{1}{2}\\left(\\frac{\\ell^2-L^2}{\\ell^2}\\right)=\\frac{1}{2}\\left(1-\\frac{1}{\\lambda^2}\\right)", "99bcb502fa84f4f933c906088b99482b": "k= \\cos(\\tan^{-1}(b/a))", "99bcde00f88d94b442cd7c17b2c0d400": "f,g\\in C(E)", "99bd2ab595b83c9b505a08420d902dda": "\\int_{M_0}^0 M^2 dM = - \\frac{K_{\\operatorname{ev}}}{c^2} \\int_0^{t_{\\operatorname{ev}}} dt \\;", "99bd4721f4f638bcff9210c0286a39d9": "\\Delta m_{ij}^{2} \\ \\equiv m_{i}^2 - m_{j}^2", "99be735333c5dd161f2ccbdfbc406a99": " -p \\delta_{ij} ", "99be815e896670c9f73f8af999107cd5": "\\mathbf{W},\\mathbf{B}", "99beb6f97800331e2c31ed8611827d34": "\\sum_{v \\in V} \\deg^+(v) = \\sum_{v \\in V} \\deg^-(v) = |A|\\, .", "99bec1e9425cf101c27f179413c4c364": "X = T + e", "99bed6791863a41573a3bc07a938efa1": "Z(X, t)", "99bf1b6d36aba6d03bef2bf58e9326a5": "\\gamma_{s} = \\frac{E_{s}}{N_{0}}", "99bf1da62e93a907a97af006b64bb9a1": "9_{46}", "99bf28ad774359a1eb6229de6604368d": "L_K=\\left(\\frac{m}{ne^2}\\right)\\left(\\frac{l}{A}\\right)", "99bf46a065063bacee578653d7b8b5db": "T^kV \\times T^k V^* \\to \\mathbb{F}", "99bf4795ddf5082470e666effd482a75": "i = 1, ..., N", "99bf635a3749dfb1a7072125a5fb9ad8": "\\lambda = \\frac{c}{f}\\,", "99bfbe6ac01128eb29bc5a3929497842": "s_j \\, s_k = r_{(j-k) \\text{ mod }n}.", "99bfc0c9ffe018ce2362f3c401551f79": "q_j^{*}", "99c046486e69a96715c0f3fd0d4ffc0f": "C\\frac{dr}{dt}-E\\frac{dp}{dt}=N_\\beta \\beta - N_r \\frac{d\\beta}{dt} + N_p p", "99c0944354c257ba459e762bd0d7e3c4": "\\vec{k} = \\vec{e}_0 + \\vec{e}_1", "99c09e39303d52803a0ff89ecbcb670b": "t \\rightarrow +\\infty", "99c0eb081868c5a290465bd0609121df": "\\delta _{mn}", "99c11b3da9f84a98d3b2bb20f6fe9eb4": "f'(z) \\ne 0", "99c221e029969cbbef503f8d17cec30a": "r_i (t)", "99c242b669172e7a7ce6bc7aa8ce8e5c": " \\omega = \\frac i2 (h - \\bar h ) ", "99c2493c26ae2dae8823cb9cd97c3781": " = \\operatorname{Tr}( M \\rho)", "99c2557cbf92da21310f60db33d049d4": "\\begin{align}\nq' \\colon [a, b] & \\to T_{q(t)}X \\\\\n t & \\mapsto v = q'(t)\n\\end{align}", "99c2b4f6a548a3ed1af8b41eac65df0b": "\\tau_{V \\otimes W} = \\tau_V \\otimes \\tau_W", "99c2e129a6c3dab2a9af86a398f59aec": "\\Omega - W^*\\,", "99c30be1cf39c02af31d95d82ecaca9e": " \\alpha_s(m_Z^2) = 0.1198 \\pm 0.0028(ex) \\pm 0.0040(th) ", "99c312cbe1045a0dcfdfdcb388cb1569": "o[hf_{xy}]=\\sqrt{d/N}", "99c324f8902e9510060392d65c191d8c": "\\beta = \\frac{\\lambda_{BN} - \\lambda_{NN}}{(\\lambda_{NP} - \\lambda_{NN}) - (\\lambda_{BP}-\\lambda_{BN})}.", "99c348a009b7381147e034f5dab12e9d": "Z_0(t) = \\exp\\left\\{ -\\int_0^t \\sum_{d=1}^D \\theta_d(t)dW_d(t) - \\frac{1}{2}\\int_0^t \\sum_{d=1}^D |\\theta_d(t)|^2 dt \\right\\} ", "99c36395841741bc40f2aa2146772820": " \\ G_F(x,y) ", "99c457ae4a4cd21f0bc531987ca58c61": "\\mathbf{r}_{N \\times 1} ", "99c528ebabb65f17be711d314bfa4fb4": "(x,y,z) = (|z_1|^2,|z_2|^2,|z_3|^2) . \\,\\!", "99c53ff9e52ace84e41b29e84b4ca38c": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } { \\exp \\left ( i\\mathbf k \\cdot \\mathbf r \\right) \\over k^2 +m^2 } = {e^{ - m r } \\over 4 \\pi r } ", "99c548ce78322a696ac865bc95c4fff4": "\\psi \\in \\Psi", "99c609e4e8b3f3a1c88a17d3dbb57837": "G_2 \\leftarrow ", "99c63cbe1190a3f9ae48579e6b8d2074": "\\delta g^{\\mu\\nu} = - g^{\\mu\\alpha} ( \\delta g_{\\alpha\\beta} ) g^{\\beta\\nu} \\,.", "99c6bbb2aa2451aeb643b34ccca80929": "=\\quad -{k_1 \\over n}(\\log k_1 - \\log n) -{k_2 \\over n}(\\log k_2 - \\log n)", "99c6c5a55e2ee0315e3df124db0d9663": "[x_1, x_2] - [y_1, y_2] = [x_1-y_2, x_2-y_1]", "99c6cd640fe64bcc302e530fbdff66f8": " \\frac{x^2}2\\cos x + \\int \\frac{x^2}2\\sin x\\, dx,", "99c7409666b11c399c7b8670fa7b1684": "\\begin{matrix}\n0 \\le x_1 \\\\\n\\vdots \\\\\n0 \\le x_n \\\\\n\\sum\\limits_{k=1}^n x_k \\le 1\n\\end{matrix}\n", "99c761abbe65441fae8948a2e79b1330": "\\textstyle \\lambda_{NP}", "99c768b41d4115a01fdb855f27387815": "N_\\mathfrak{g}(\\mathfrak{h}) = \\mathfrak{h}", "99c7c1527960673a2c68f40d07b61ea5": "E = PRMLUL^{-1}M^{-1}R^{-1}P^{-1}", "99c7d9586da9ffd9517cf404d7e3fcb1": "f'(x) = \\begin{cases} +2x, & \\text{if }x\\ge 0 \\\\ -2x, & \\text{if }x \\le 0.\\end{cases}", "99c80c68216d9467363d5ac89a4485ba": "\\sum \\dot{Q}_j/T_j,", "99c86d64afcf34ffe50d8ea9ee1cffed": "(\\vee) \\frac{A \\vee B}{A|B}", "99c8eb087be78463043b4c5bfb69558f": "\\lim_{x\\to 0^+} g'(x) \\;=\\; \\lim_{x\\to 0^+} \\frac{1}{\\sqrt[3]{x}} \\;=\\; {+\\infty}\\text{.}", "99c92a24bc69a9f7eeb742095dd5a5f1": "d = 8\\frac{V_{\\rm l}}{V_{\\rm g}}\\ell", "99c936b1f3862238cf11ceea717f5552": "\\int f(u)\\cdot g(x-u) du", "99c94fe2a07b198e41ee767ea3b06c16": "w = 99", "99c9b575310ff2957db09c853036151c": "\\text{SSMD*}= \\frac{X_i - \\tilde{X}_N}{1.4826\\tilde{s}_N \\sqrt{2(n_N-1)/K}}", "99ca08be18390eb189e43464482f3885": "14^3", "99ca4c641105b60134812466bea8ca0d": "\n\\begin{array}{ccc}\n{ L}(\\frac{l^2}{4} B_{\\mu\\nu} B^{\\mu\\nu}) &=& \\left|\\frac{l^2}{4} B_{\\mu\\nu} B^{\\mu\\nu}\\right|^\\frac{3}{2} \\,,\\\\\n{L}(\\frac{\\tilde{l}^2}{4} \\tilde{B}_{\\mu\\nu} \\tilde{B}^{\\mu\\nu}) &=& \\frac{\\tilde{l}^2}{4} \\tilde{B}_{\\mu\\nu} \\tilde{B}^{\\mu\\nu} \\,.\n\\end{array}\n", "99ca9538da40269030014d9c258a147c": "v = \\sqrt{\\mu(2/r-1/a)}\\,", "99cabc86695925e4598a126c54dcecb2": "e^\\pi \\,", "99cad4339efe2ec8c6f2da739c2255b3": " 1 \\rightarrow K^* \\rightarrow \\Gamma^0 \\rightarrow \\mbox{SO}_V(K) \\rightarrow 1.\\,", "99cadf50dc23929edb64bec41ae00f60": "d^\\star(v_i,v_g)", "99caedadb4a5440712b15cd77be922e7": "K \\wedge \\{\\neg f ~|~ f \\in F\\}", "99cb2b610c382798ba303904bb4fc60d": "\\scriptstyle \\|x \\,-\\, y\\|_2", "99cb348014058ef8eed7aeff61b2bc0e": "F_n = \\dot{m}\\;v_{e} = \\dot{m}\\;v_{e-act} + A_{e}(p_{e} - p_{amb})", "99cb3c30eb58222fceb75c11951e24ae": "N(t) = N_0\\,e^{-{\\lambda}t} = N_0\\,e^{-t/ \\tau}, \\,\\!", "99cb3d8bed01802e16eeba4746051e90": "p({\\tfrac{1}{2}}) = 8", "99cb4a5d02b92a666581c58b26627fd1": "U_n = \\frac{1}{n+1}\\,T_{n+1}'.\\,", "99cb57348c6af88d3ad17a0899b1002e": "v = \\sum_{i \\in I} a_i(v) e_i.", "99cb6d9e5fe9c36e687c7f48e3b65fb7": "f_P", "99cb9a92a6a37e7d926e227f1169f6a3": "\\operatorname{softmax}(k,x_1,\\ldots,x_n)", "99cbee4b3e43f6a57ff625cc70f3466b": "\\scriptstyle E(\\boldsymbol R) \\;=\\; \\boldsymbol \\omega(\\boldsymbol R) \\cdot \\boldsymbol L / 2", "99cbf4f9ba68e50ff651aa0e9923f993": "T_r\\,=\\,T_g+43", "99cc12e9da9cbede166f6f9dc370ff06": "\\frac{L_{\\rm A}}{L_{\\odot}} = {\\left ( {\\frac{9}{1}} \\right )}^2 {\\left ( {\\frac{26,700}{5,778}} \\right )}^4 = 36,933 L_{\\odot}", "99cc19ad212d3c16b528998ce2beb037": "dl'", "99cc363a36f9a067c43439e4744853fd": "\\pi : \\lbrace 1, \\ldots, m \\rbrace \\to \\lbrace 1, \\ldots, m \\rbrace", "99cc60d4e0fe92b724b70404fe63d1a3": "\\textstyle{\\frac {\\log(8)} {\\log(2)} = 3}", "99cc6d6e2b31db2f97fcd77da4c4c8ea": "\\log_a\\big( {[x_1, x_2]} \\big) = [\\log_a {x_1}, \\log_a {x_2}]", "99ccd06f89b69810c89db091595a21f4": "H_{inv}(z) = \\frac{1}{H(z)}", "99ccf96151f7717a249112e69cdeb1f5": "\\int_0^\\infty \\frac {e^{-x^2}-e^{-x}}{x}\\ dx=\\frac{\\gamma}{2}", "99cd4c065850f2aabcdf216025cc3573": "\\mathrm{O}(n)", "99cdf4b311386e4fe58fe750808e5304": "\\det(A) = \\det(P^{-1}) \\det(L) \\det(U) = (-1)^S \\left( \\prod_{i=1}^n l_{ii} \\right) \\left( \\prod_{i=1}^n u_{ii} \\right) . ", "99cdf5701ba5333a8a7668dd0fbc29bd": "a\\ ", "99ce56fae9431479881008392957ebda": "(b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15}) \\over 2", "99ce5b05b7aed44eb5b92e277ac9dfe9": " E_q^{x}=\\sum_{j=0}^{\\infty}q^{j(j-1)/2}\\frac{x^{j}}{[j]!}", "99cea2e5edbcd2647f6e9c39ac0259df": "\n\\left( R^T (X_R-T) \\right)^T T \\wedge X_L = X_R^T R T\\wedge X_L = X_R^T R S X_L = 0\n", "99ceeaadefbc2a6ac3bb079ff9e3792b": "0 \\le t < 1", "99cf2344a6367dec50b6ea9dba85ca12": "\\textstyle 1.\\ R_{x}= \\frac{1}{M}\\sum_{t=1}^M x(t) X^{*}(t)", "99cf3ed140d453f5d750d88b2fe4608e": "\\exists \\text{Phil} ( \\text{Phil}(a))", "99cf6848d4607df97656d466288e90fa": "\\frac{B_6}{{v_0}^5}", "99cf7f7e9fc866288f4bd487395f1b38": "S_gX_p\\, \\stackrel{def}{=}\\, g(X_p,-),", "99cf96d7a2251ce452151ad425cc664f": "\\bar{Y} = \\frac{Lc\\alpha(1+c \\alpha)^{n-1}+\\alpha(1+\\alpha)^{n-1}}{(1+\\alpha)^n+L(1+c\\alpha)^n}", "99cfde58d0adf917128896dd613542b1": "y''_n", "99cfe984aca2e2e638c970ad27668d4f": "\\varepsilon = +1,\\quad a = m^2 - n^2,\\quad b = 2mn;", "99cff7302de7254ea6a63c9e6b152ba7": " \\mathbf{AI} = \\mathbf{IA} = \\mathbf{A} ", "99d047c33d9abd802f5b2fc6049ea420": "\\mathfrak{sl}_2(\\mathbf{C})", "99d0575d8fb7bbbc543c782a67708dea": "f(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty\\ \\ d\\alpha f(\\alpha) \\ \\int_{-\\infty}^\\infty dp\\ \\cos (px-p\\alpha)\\ , ", "99d083cc94435dbfad22abe28ff92bf7": " \\Rightarrow \\theta _1 = \\theta _2 + \\tfrac{\\pi}{2} ", "99d08a35ada5951ca1f4a67f6e6908d0": "\\int_0^t {\\mathrm{d}t^\\prime} = \\int_0^v \\frac{\\mathrm{d}v^\\prime}{g-\\frac{kv^{\\prime 2}}{m}} = {1 \\over g}\\int_0^v \\frac{\\mathrm{d}v^\\prime}{1-\\alpha^2 v^{\\prime 2}}", "99d08c537596808afb8777742a0ba7df": " S_0 \\subset L^2 \\subset S_0' ", "99d0ce384f053387b1a5787389a3f51b": "\\scriptstyle \\times", "99d119b263cccbab8720131a013d2e4a": "x=\\sum_{i=1}^n b_i r^i", "99d138654fd4d8797b64eed5f9a3da2a": "\\mathcal{SHOIN}^\\mathcal{(D)}", "99d188c4130013f792a40c9c25a4e7f9": "\\bar{x} = \\frac{1}{2N} \\sum_{n=1}^{N} (x_{n,1} + x_{n,2}) ", "99d198e5c3724641482ba2ffff34c9b9": "f: E \\to QC", "99d1b701e9156dad424cd41d9aab17f1": " E_{k_f} = ", "99d1e9f0fbdeae9c947a929237ba5a91": "(-\\Delta)^{-\\alpha/2} \\delta(x) = c_{n,\\alpha} |x|^{\\alpha-n}", "99d1eed010d668848d9f5c08f3a9530d": " d : \\Omega^p(M)\\rightarrow \\Omega^{p+1}(M) ", "99d229ea9353c21fefa23b91377e27ea": "\n M_{\\alpha\\beta,\\alpha\\beta} = q \\implies M_{11,11} + 2 M_{12,12} + M_{22,22} = q\n ", "99d25c5113d6012e9285b0b593a6e14e": "V_{n+m} = V_n V_m - Q^m V_{n-m} \\,", "99d285d80f3a4a1267ba35b8fde3c0af": " v - \\hat{u} ( \\hat{u} \\cdot v) = \\frac{1}{{\\Vert u \\Vert}^2} ( {\\Vert u \\Vert}^2 v - u ( u \\cdot v))", "99d29ba16a956e98b507b8eb10fcbda9": "\n \\rho~\\dot{\\eta} \\ge - \\cfrac{1}{T}~\\boldsymbol{\\nabla} \\cdot \\mathbf{q} + \n \\cfrac{1}{T^2}~\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T\n + \\cfrac{\\rho~s}{T} \\qquad\\text{or}\\qquad\n \\rho~\\dot{\\eta} \\ge -\\cfrac{1}{T}\\left(\\boldsymbol{\\nabla} \\cdot \\mathbf{q} - \\rho~s\\right) + \n \\cfrac{1}{T^2}~\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T.\n ", "99d2a0ae3293a38f0900d70a5bc8bf91": "I_g", "99d2c219068a2de9f5af6b6425ac00b3": "Z^{-2}_2", "99d2f6a8b28ff13220bb120f178deba6": " I = - {I_\\text{0}}{\\sin({\\omega t}}) = {I_\\text{0}}{\\cos({\\omega t} + {90^\\circ})}", "99d36ec75067248d8ffc72518585465f": "(39/2, 5/2, -1)\\,", "99d3a85857f5038b66b34c32e4af447b": "\\left|{\\alpha \\choose k} \\right|\\leq\\frac {M}{k^{1+\\mathrm{Re}\\,\\alpha}},\\qquad\\forall k\\geq1", "99d3c4071f0a4026cf25dddc8c49f957": "\\;^{\\perp} \\subset", "99d410d46919b3892f15edc17a890a8f": "K(k) = \\frac{\\pi a}{2 \\, \\operatorname{AGM}(a,a \\sqrt{1 - k^2})}", "99d43041d4d1a6df6d0e8384a92f96c7": "[x, y] = xy - yx.\\,", "99d49f91888e640d5805e0c19bb7fb8a": "\\zeta's", "99d4bef2fbc4fdf8310af532105f4b99": "h = u + Pv \\,\\!", "99d4c9fba98daec66f933ce1dd3625c4": "K=\\dot{m}/C_B. \\qquad(6b)", "99d5184f4f1e132fb15d37e5d426a952": "({\\mathbf e}_r, {\\mathbf e}_{\\theta})", "99d548e6b0e0d109f40bc4279d13d547": "= -2\\sqrt{\\pi}\\,", "99d56372e2d007b227b2201a952cd164": "\\nu(A) = \\int_A g \\, d\\mu.", "99d58468ddf3d30eb966eefaf5bc536a": " g_0(x_i,y_j) = \\frac{\\displaystyle \\sum_k w_{ij} f_k(x,y)}{\\displaystyle \\sum_k w_{ij}}.", "99d585faf976ee1172ba58c2b4786c44": "R_0A", "99d588f5d4ab57bb5e8f9ad180079ac6": "\n\\begin{bmatrix}\n \\mathbf{f}_x \\\\\n \\mathbf{f}_y \\\\\n \\mathbf{f}_z \\\\\n \\mathbf{f}_w \\\\\n\\end{bmatrix}=\\begin{bmatrix}\n 1 & 0 & -\\frac{\\mathbf{e}_x}{\\mathbf{e}_z} & 0 \\\\\n 0 & 1 & -\\frac{\\mathbf{e}_y}{\\mathbf{e}_z} & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 1/\\mathbf{e}_z & 0 \\\\\n\\end{bmatrix}\\begin{bmatrix}\n \\mathbf{d}_x \\\\\n \\mathbf{d}_y \\\\\n \\mathbf{d}_z \\\\\n 1 \\\\\n\\end{bmatrix}\n", "99d6364c8ddedf850ef2a56390d02ebf": "\n\\mathbf{r}_1 = \\mathbf{r}_0 - \\alpha_0 \\mathbf{A} \\mathbf{p}_0 = \n\\begin{bmatrix} -8 \\\\ -3 \\end{bmatrix} - \\frac{73}{331} \\begin{bmatrix} 4 & 1 \\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} -8 \\\\ -3 \\end{bmatrix} = \\begin{bmatrix} -0.2810 \\\\ 0.7492 \\end{bmatrix}.\n", "99d6867c9fef327951455d4f53763eea": "v \\ne 0,", "99d69cce809038ce9a2c1776ac36a20d": "\\frac{d^2 \\Delta p_{\\text{B}} (x)}{dx^2} = \\frac{\\Delta p_{\\text{B}} (x)}{L_{\\text{B}}^2}", "99d805985969a2218aa274922d0849cd": "\\nabla_{\\dot{\\gamma}}\\dot{\\gamma} = k\\dot{\\gamma}", "99d841bdfeb53cb1dde4d20819b10174": "\\lfloor \\sqrt n \\rfloor", "99d8534b5d77844efc7b1ad9e0177cbc": " \\Delta ' (a) = F \\Delta (a) F^{-1}, a \\in \\mathcal{A}", "99d8aaeb06f5a0d76802750ad70fa6b7": "t \\leftarrow 1", "99d8b0ad83cdf5832403872cde065ea7": "{1 \\over 2}\\sum_{i = 0}^{n - 1} i!", "99d8fafa9c0b68167c1bb6519924ac85": "-265\\pm 30", "99d96892040d2e4054e9258919030079": "K^{m\\times n}", "99d96d530a72a3eedc9b4173e1250633": "L_\\sim", "99d994cfa4c593bd99fb8cef919fa28f": "\\tilde{f}(\\lambda)=\\int_{-\\infty}^\\infty \\int_0^\\infty f((a^2 + a^{-2} +b^2)/2)a^{-i\\lambda/2} \\, da\\,db,", "99d99c86d4c7c61816be44e11ad4e7b2": "\\mathcal{D}^{\\mu \\nu} = \\frac{1}{\\mu_{0}} F^{\\mu \\nu} - \\mathcal{M}^{\\mu \\nu} \\,", "99d9b9a91c5ab6508f78cc3543eb5497": "L\\in{\\Bbb H}", "99da07df4cb327d261627aa65f0328be": "2Y(\\omega)", "99da1ea640f7c7eb80a7c48a45fdfe6c": "p_j=jh,j=0,\\dots,N-1", "99da63519e9969f2de47d0f788fa3e03": "y \\leq \\frac{mr}{r+1}", "99da75e06f7695e78894ad44f1576a32": "\\scriptstyle a_{\\rm V}", "99da8b4c514c1cc86b568c73695f8215": "\\,\\!P_s", "99daa6413ca8b4bedc5fb5c540474499": "\nP(V) = \\frac{K_0}{K_0'} \\left[\\left(\\frac{V}{V_0}\\right)^{-K_0'} - 1\\right]\n", "99dad683002d359b09434764b9e3176a": "F<0", "99dae3d8fbf9d5ae1e419caf77668114": "S_{1,1} \\geq S_{2,2} \\geq \\ldots \\geq S_{r,r} > 0 \\quad S_{i,j} = 0 \\; \\text{where} \\; i \\neq j", "99db0460458a5c32592c2ccbadb55567": "\\displaystyle{H^\\varepsilon_h f(e^{i\\varphi})={1\\over \\pi}\\int_{|e^{ih(\\theta)} -e^{ih(\\varphi)}|\\ge \\varepsilon}\n{f(e^{i\\theta})\\over e^{i\\theta}-e^{i\\varphi}} \\, e^{i\\theta}\\, d\\theta,}", "99db0abab1539ae4570a50e88031943c": " {\\Gamma \\vdash M : (\\forall x:A . B)\\qquad\\qquad\\Gamma\n\\vdash N : A \\over \n{\\Gamma \\vdash M N : B(x := N)}} ", "99db4cf98b4b2bd7e90ca3274b0f6520": "\\alpha^n = \\sum_{d\\,|\\,n} d \\, M(\\alpha, d).", "99db6dad5fec64c075ceac057177830a": "\\dot X(t)", "99db8c6426862fbef6e4b16b804827f1": "1\\leq i\\leq m\\,\\!", "99dbb7ff3257d35b3161e16007eeb548": "e_{rs}\\,\\!", "99dbc4b2bed864f662e47cb976a362e6": "(abc \\dots xyz)=(ab)(bc) \\dots (xy)(yz)", "99dbeb50f406f301f5cfe05d9e7d1929": "k_z", "99dc2391f973c51557d66b0e1df3add7": "\\scriptstyle\\in\\mathbb{N}", "99dc249371cab89544e3c3f904795788": "W_B = \\hbar \\omega_B - \\ ", "99dc66ef3d5edc2b148ee0656cb81724": "L_G", "99dc771a2606dea95235fac082f49470": "\\beta_{1}, \\ldots, \\beta_{N}", "99dc79bbe149e40c6ef5834b2a52e3d1": "\\tfrac{a}{b} + \\tfrac {c}{d} = \\tfrac{ad+cb}{bd}", "99dc85b72defde2ceeda5a6e7fb23b28": " \\lambda=\\frac {1} {2} (.032)^2 4 = .002 ", "99dc8c42499b0365e287220bca4394e4": "\\{w_1, w_2, \\dots, w_n\\},", "99dc90ad02ee29c473530a8f10106c08": " 4 \\pi \\epsilon_0", "99dca0d89d6c5c53eb295aa8240d1b47": "\n \\frac{\\partial\\Phi}{\\partial z} = 0 \\qquad \\text{ at } z=-h.\n", "99dce7615f3e98299a04fb5d787441c1": "n = 2k", "99dcf2c6cf1a15baff77194a2b799a04": "f\\in\\mathcal{H}", "99dd4db91e599fd192b1488637259a1e": "X_m = \\frac{ \\displaystyle\\sum_j \\nu^j_m x_j}{\\displaystyle\\sum_j \\displaystyle\\sum_n \\nu_n^j x_j}", "99dd5001002578498b5a30cee60f2183": "Fr = \\frac{U^2}{gL}.", "99dd7eb532e368535c5d2334c3321358": " \\lim_{t\\rightarrow 0} \\rho_t(x) = \\delta(x) \\,", "99ddd66a42289d096ccacc790c6debf4": "4^3=64", "99de3462d3b95cba577ba1e87b0e7e99": "\\forall b \\in A", "99de434b119de80592e92ffd8c8c4422": " g_k(x) ", "99deb124a96bb9aa5c2c8a80fc75b121": "\\pm e^{C}\\neq 0", "99df1b90ceeb44dcd7322e1e5f32700b": "F _0 (x, y) = x+y,\\,", "99df62215781e890e315e0e3b5f23617": "h_{\\omega}(t)", "99df628726d083f839edd641e579fd80": "\\displaystyle{\\dot{a_2}=-2\\alpha} ", "99df8bc744a81009184b5ef00354bdf9": "\\phi(x=d) = \\phi_{DL}", "99dfadde51f6e53dc6fbdfce9e0473ce": " \\dot{x} + x = 0 ", "99dfcf2d5fff7bb9ab755d0f4326bb44": "\\scriptstyle\\mathbb{Q}\\left[\\,\\sqrt{5}\\,\\right]", "99dfe7832882b9a5882251d624a6f50f": "G_k \\subset G_{k + 1}", "99e0059f4cfeeeeada627e1335d7ab97": "Sym(\\mathbb{Z})", "99e06bbb87726699aa082c4a4bc8afc9": "f(v,u) \\leftarrow f(v,u) - c_f(p)", "99e0a8a981fce76cb3615ae9867f63ca": "w(n)=a_0 - a_1 \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right)+ a_2 \\cos \\left ( \\frac{4 \\pi n}{N-1} \\right)- a_3 \\cos \\left ( \\frac{6 \\pi n}{N-1} \\right)+a_4 \\cos \\left ( \\frac{8 \\pi n}{N-1} \\right)", "99e0e0509e4d04257f87a4d3ca535559": "\\scriptstyle\\pi_k", "99e11c1a3dea770ea165200b7da9dd15": "z(t) = z_0 - \\mu t", "99e11cd87c010d939ef03fc78dc06ae0": "f:\\,Y\\rightarrow Z", "99e1224a5621e4b59c6fd01a489734ea": " R = N \\int_{E_{min}}^{E_{max}} \\int_{T_{min}}^{T_{max}} \\phi(E_i)\\,\\sigma(E_i,T)\\,\\upsilon(T)\\,dT\\,dE_i. ", "99e137ae28afea5bf1dbbe0a5e7fccc3": " \\sigma(t)", "99e165ed63cdca3cf19f1ce26cfd8001": "|\\{Mf > t\\}| \\le {2C \\over t} \\int_{|f| > \\frac{t}{2}} |f|dx, ", "99e18465c90583b6389f1d63af32eac7": "2<\\sqrt{5}<\\tfrac{5}{2}", "99e1846f31dd5b19da626f11a537c7e8": " \nVP(x(t)) + \\sum_{i=1}^KQ_i(t)Y_i(x(t)) \n", "99e1fe940a565420bdee246e85878813": "{ 0.9/0.1 \\over 0.2/0.8}=\\frac{\\;0.9\\times 0.8\\;}{\\;0.1\\times 0.2\\;} ={0.72 \\over 0.02} = 36.", "99e203d9921b00e1aaeb6297ffa158d0": "\\Delta x = v_{x} \\Delta t ", "99e207295e5de6dd42ecf0d481806737": "O(n(n+m))", "99e23770c2b047b09201252d70777cb5": "i=1, 2, \\dots,", "99e2450bb16676dcae13180de4e97bb0": "\\tilde{Z}[0]", "99e24e204dd9b5d6702294c1dc21ca05": "\\left(\\frac{c+\\sqrt{c^2-4ab}}{2}, \\frac{-c-\\sqrt{c^2-4ab}}{2a}, \\frac{c+\\sqrt{c^2-4ab}}{2a}\\right)", "99e2c55ff1cfe80fb164d9b371cc2b9e": "\n\\langle O \\rangle_C \\equiv \\langle \\Psi_C |O|\\Psi_C\\rangle.\n", "99e2d5f793b15f0e522523a78c462bf1": " s = i \\omega \\,", "99e2f32436366be952f47f606e930076": "\\exist x[\\alpha (x) \\and \\gamma (x)] \\rightarrow (\\exist x \\alpha (x) \\and \\exist x \\gamma (x)).", "99e32b91a59fd821c6ec0f13a1b72827": "\\begin{align}\nH_a(s) &= \\frac{1/sC}{R+1/sC} \\\\\n&= \\frac{1}{1 + RC s}.\n\\end{align}", "99e34572797450c86407a4cc2a3ec824": " \\and T_4 = [F_4, S_4, A_4]::[F_3, S_3, A_3]::K_2 ", "99e365d9cddbdfa7c6008699d6b5971b": "R*", "99e38043d00847920981ac6f222a6bcc": "R_{load}", "99e3ab526fcb4da8b768405e15b3d8ff": "\\begin{align} \n R_\\Delta(N_1, N_2) &= R_c \\parallel (R_a + R_b) \\\\\n &= \\frac{1}{\\frac{1}{R_c} + \\frac{1}{R_a + R_b}} \\\\\n &= \\frac{R_c(R_a + R_b)}{R_a + R_b + R_c}\n\\end{align}", "99e444abf502c2e4c1b45fd021c3daab": "V(\\sigma) = -J_p s_0 s_1\\,", "99e456310186ec3b9a9a51aa507c9c22": " f(z) ={z\\over(1-e^{i\\theta}z)^2}. ", "99e458f982bd461dca99afa60e7b2e7f": " \\det\\left ( \\hat{A} - a \\hat{I} \\right ) = 0 ,", "99e48b693f3099d0282f93cc0b1ee395": "I_0 = \\frac 12(I_- + I_+)", "99e490d910a9f1d70c96f2c220ba635a": "s_{\\beta} = \\liminf\\{s_{\\alpha}| \\alpha < \\beta\\} = \\sup \\{ \\inf\\{s_{\\alpha}| \\delta \\leq \\alpha < \\beta\\} | \\delta < \\beta\\} \\,.", "99e49d314748f617cccb322a2152abd1": "A^{2-} + H^+ \\rightleftharpoons HA^- :\\beta_1=\\frac {[HA^-]} {[A^{2-}][H^+]}", "99e49f574687f1f20762de2b9dff2117": "\\pi \\otimes |\\det|^s", "99e4a2a2b348bd83bd72654d7a4dbaba": "\\mathbb R^n", "99e4a924bd8a5aa06ad2479b764606b0": " \\int_K | f|^p \\,\\mathrm{d}x <+\\infty,", "99e4e8824531e40a2fe22e7ee4aeb63c": "S(\\Psi_b)=(2m)^{-\\frac{1}{2}} \\sum\\nolimits_{b^\\prime\\in M_1/M} e^{- i m bb^\\prime} \\Psi_{b^\\prime},", "99e52548249041db739389dd4fe74b9d": "\\Delta_1^2 - 4 \\Delta_0^3 \\ge 0", "99e55a247efcfe881a0a895737ecfb2d": "\\int_7^{10} \\int_{11}^{14} (x^2 + 4y) \\ dx\\, dy ", "99e57d416e4c2d9425ae246c054e26de": " f_{ij}: A_i \\rightarrow A_j ", "99e59010121e1a8dcc3d1a2eb7dbec37": " \\mathbf{V}_P = \\dot{\\mathbf{P}} = [\\dot{A}(t)]\\mathbf{p}. ", "99e6c9ec3cb2d83389257a7004693337": "\\frac{1}{2}L\\cdot \\left(\\frac{I}{2}\\right)^2", "99e6d7b7e593a3f3b6ab8054efdc80cd": "D(n\\otimes n)=n.", "99e6fb5d3464d4257c9b1063b5171b78": "F^{ab}", "99e7018bfb1cc6a229a38a39258a601d": "\\left(d, r, x\\right)\\succsim \\left(e, s, u\\right)", "99e776c7439892178040a9d9684930a7": "\\lim_{n \\to \\infty} \\varphi(t_n, x) = y ", "99e7a6d26914f533df3c5d8be57bd0d3": "\\langle a \\rangle\\,\\!", "99e7c8f8683b4e1ef3f53863be34c43f": "\\delta(p,a,A)", "99e7d9502c0a1e9b69e426bf7f2a10fa": "\n\\sum_{n=-\\infty}^{\\infty} x[n]\\ z^{-n} = \\sum_{n=-\\infty}^{\\infty} x[n]\\ e^{-i\\omega n} = \\frac{1}{T}\\sum_{k=-\\infty}^{\\infty} \\underbrace{X\\left(\\tfrac{\\omega}{2\\pi T} - \\tfrac{k}{T}\\right)}_{X\\left(\\frac{\\omega - 2\\pi k}{2\\pi T}\\right)},\n", "99e813d5fd11f331052e81650137412e": "\\,\\,\\biguplus_{i\\in I}A_i\\,\\,", "99e8164513953b15a2af92d9069a4416": " \\delta = \\frac{2 \\pi d}{\\lambda} \\sin\\theta \\,\\!", "99e87d48f6d23498894a465a75b3526c": "\n\tT_{d,ave} = {\\rm ave}(T_d(\\theta,\\phi))\n", "99e894d2f3ca2f9857cf74f77042cb03": " \\beta A_{OL} \\left( f_{180} \\right) = - | \\beta A_{OL} \\left( f_{180} \\right)| = - | \\beta A_{OL}|_{180},\\ ", "99e8c5a54f16b4a6230b5b8d204e71c4": "\n\\begin{alignat}{2}\n\\|\\textbf{u}+\\textbf{v}\\|^2 &= (\\textbf{u}+\\textbf{v})\\cdot(\\textbf{u}+\\textbf{v}) \\\\[3pt]\n&= (\\textbf{u}\\cdot\\textbf{u}) + (\\textbf{u}\\cdot\\textbf{v}) + (\\textbf{v}\\cdot\\textbf{u}) + (\\textbf{v}\\cdot\\textbf{v}) \\\\[3pt]\n&= \\|\\textbf{u}\\|^2 + \\|\\textbf{v}\\|^2 + 2(\\textbf{u}\\cdot\\textbf{v}),\n\\end{alignat}\n", "99e8cbfc7bdd7de85ed1844cc87f4edc": "\\Gamma^{[0]i_0}_{\\alpha_{0}}", "99e91eeb32596d5d3c7a2f1cfbf92bad": "D\\xi = \\sum_{\\alpha=1}^k D(e_\\alpha\\xi^\\alpha(\\mathbf e)) = \\sum_{\\alpha=1}^k e_\\alpha\\otimes d\\xi^\\alpha(\\mathbf e) + \\sum_{\\alpha=1}^k\\sum_{\\beta=1}^k e_\\beta\\otimes\\omega^\\beta_\\alpha \\xi^\\alpha(\\mathbf e).", "99e9262e4e9d1e0a9bea7a8a567a2dcb": "d = S - N^2 \\,\\!", "99e92fb53ae1e7655bf30e45731c3d4a": "\\rho(\\mathbf{Y}|\\mathbf{X},\\mathbf{B},\\boldsymbol\\Sigma_{\\epsilon}) \\propto \\boldsymbol\\Sigma_{\\epsilon}^{-(n-k)/2} \\exp(-{\\rm tr}(-\\frac{1}{2}\\mathbf{S}^{\\rm T}\\boldsymbol\\Sigma_{\\epsilon}^{-1}\\mathbf{S})) \n(\\boldsymbol\\Sigma_{\\epsilon}^{2})^{-k/2} \\exp(-\\frac{1}{2} {\\rm tr}((\\mathbf{B}-\\hat{\\mathbf{B}})^{\\rm T} \\mathbf{X}^{\\rm T}\\boldsymbol\\Sigma_{\\epsilon}^{-1}\\mathbf{X}(\\mathbf{B}-\\hat{\\mathbf{B}})) )\n,", "99e9338c68dadbbee26635bbca0994b6": "f(x) = \\sum_{k=0}^{\\infty}f_k x^k", "99e934993baa4c0aae8ee07c97ba552e": "f:M\\rightarrow M", "99e98086115c015fe4d70101b44652b4": "\\textstyle\\varphi", "99e9ec8c96e885951ec5e4d362a103f4": "1 \\over 16", "99ea9cc0226634b99b4c858c2ed96bb6": "\n V_4 = 25 - R_a - R_b = R_c\n ", "99eaf9311ad8716e2be30f546532a367": "n^n ", "99eb02440936535c12ece0e6505f9699": "{\\tilde{A}}_{6}", "99ebf366ea6874f3fc764d14d1ce9e2a": "\\Gamma\\,\\!", "99ed0dd553ac2c3db559f44ac3ac0068": "m_{i\\to j}(r_j) = \\max_{r_i} \\Big(e^{\\frac{-E_i(r_i)-E_{ij}(r_i,r_j)}{T}}\\Big) \\prod_{k \\in N(i)\\backslash j} m_{k\\to i (r_i)}", "99ed5b780110345a0b9e5c3aa06739de": "(\\mathbf{H_1^\\parallel} - \\mathbf{H_2^\\parallel}) = \\mathbf{K}_\\text{f} \\times \\hat{\\mathbf{n}},", "99ed69abbeb8c0040d90d365e880ffe1": "S_2 = r(3^2) = 637,\\;S_3 = r(3^3) = 762,\\;S_4 = r(3^4) = 925", "99ed9bb23fa36c7a7af5f568291c10b7": "\\Lambda_{\\text{GUT}} \\approx 10^{16}\\,\\text{GeV}", "99edac224649e69b0950026ba270d732": "\nN C_v dT = - dW = - {N{k_B}T \\over V} dV\n", "99edbcd09c07820151ba7daa96621785": "d\\nu(\\tau) = y^{-2}dxdy", "99ee5a7c1177a2ca87e3c052035007c6": " 12 \\div x", "99ee6dee92c3fe22dff38971dbda7370": "f(x) = \\int_{-\\infty}^{\\infty} \\hat{f}(\\xi)\\ e^{2 \\pi i x \\xi}\\,d\\xi.", "99eeb240c576b2ea9b33a7252d4b9a76": "j\\in\\mathbb N", "99eeec1313f48eebb4f2a6d7b75c5ad3": " |\\phi\\rang \\lang \\psi| ", "99ef0f116e97a6c6307f32cd2ca46d42": "r \\geq \\sqrt{\\frac{1+c}{1-c}}", "99ef2a55b2ed4f7107ffa7feba3ab52b": "\\bigstar \\mathbf S", "99ef3341b87cd368e62083ab02061804": "\\begin{alignat}{2}\n M(\\theta,\\tau) & = \\phi(\\theta,\\tau)\\int s^*(u-\\dfrac{1}{2}\\tau)s(u+\\dfrac{1}{2}\\tau)e^{j\\theta u}\\,du \\\\\n & = \\phi(\\theta,\\tau)A(\\theta,\\tau) \\\\\n \\end{alignat}", "99ef53eb2e96091423088f4256548a41": "x^{'}_{i} \\leftarrow x^{'}_{i} + fw \\cdot u(-1, 1)", "99ef787ab2ebc712f0eb61b1f01798ee": "\\Rightarrow \\lim_{n\\rightarrow\\infty} \\frac{I(2n)}{I(2n+1)}=1", "99ef7895995d5e54a9f22ce78a09db06": " n_{jobs} ", "99ef7e228a7363093000e0f6e6034d4a": "(h,g)", "99eff02e839e393cc7e2249db3e5ba48": "\\int e^{-(x^2+y^2)}\\,d(x,y),", "99effdc8d12aa1ecd7902a181c553921": "\\psi_{1,m}", "99f08b3110c43f5a1bb16808a66531d8": "\\alpha(\\omega)", "99f0d6fb854ad3030124077111dcef90": "a(t_0)", "99f12191e76228b78b5fa75c038dd864": "y\\sim\\chi^2_1", "99f1227fd42f9246e0396ab3f6921e15": "\\mathcal{F} = N I", "99f1b2058c29dc995bf3574905c1f85c": "(f+g)[x_0,\\dots,x_n] = f[x_0,\\dots,x_n] + g[x_0,\\dots,x_n]", "99f1df3def9cc8f169254f2ed796b1b7": "\\| x \\|_{\\infty}", "99f2ae4a62a46644d702e19db45be2cb": "\\ L>>(x+a/2)", "99f30ac54cb237cbf8d0b073a4786122": "\nds^2 = 0 = c^2 dt^2 \\left(1 + {2 \\Phi \\over c^2} \\right) - \\left(1 + {2 \\Phi \\over c^2} \\right)^{-1} dl^2\n", "99f31d9be3017e8c16a3ff792b0b11a8": " y_2=l\\cdot x_2+m ", "99f3252f2d730946b1f52a98cfedddd6": "y \\in C", "99f37de3408898f3ae5cf7855471a3cb": "b_N(x) = 0", "99f39e2dbbe93846e73cf38ada60ca94": "\\mu = G*M = 1 * \\frac{DU^3}{TU^2}", "99f3de10e62461a8f9d20aa7daaee022": "\\hat\\theta_i= \\frac{x_i + \\alpha}{N + \\frac{\\alpha}{\\mu_i} } \\qquad (i=1,\\ldots,d),", "99f41f0216acfe04a19b16976a1f4950": "\\max_{x,R} R", "99f43f68d37fb6ce588acfd7d99335c2": "x:\\sigma", "99f454e4321293fcdc7043cfe20a4e47": "\\underbrace{1+\\cdots+1}_{n\\text{ terms}}<\\left|y\\right|\n\\text{ for every finite cardinal number } n.\\,", "99f4672684e7ff65857d73d2b49b3367": "z \\mapsto z^2 + c", "99f47fc20d09947336b95fda4a9be627": "\\log_{10} 2 \\approx 0.301029995663981195.", "99f48c9b2edf66f862a53fbc8b4527ef": "P_1^2,P_3^0", "99f4a15c390ecce1b2c57e01aa5acf49": "\\delta t =", "99f504ecf217b85dfba92c55f053af09": "(1 + \\sqrt{-5})", "99f50b8e5490502ea09fe7f84bf5c7dc": "\\omega_D = i^*(\\omega_X \\otimes \\mathcal{O}(D))", "99f51f8ba00137765d5c1708128862ab": "\\ {m^* = \\frac{\\hbar^2}{\\part^2 E / \\part k^2}}|_{k=0}", "99f520ab652e8c2a309261b30b319388": "F_{2n} = F_n L_n \\,", "99f5442cca064d889fdd488c97416a86": " V_1 = Z_{11} I_1 + Z_{12} I_2 \\qquad V_2 = Z_{21} I_1 + Z_{22} I_2 \\qquad \\text{with} \\qquad Z_{12} = Z_{21} \\, ", "99f5523642a519d340ef3362536f8571": "\\sum_{n=-\\infty}^{\\infty}c(n)e^{int}", "99f623196da82a4f07168cf04d92caf8": " -u[-n-1]", "99f6ace5a9b766da69ed48e1e2d143e8": "O(m\\sqrt{n}\\log^{\\frac{3}{2}}n)", "99f700312e3e9c0d8e24f624241fe3b3": "\\tau_0 = \\eta/E", "99f7659abe94b86acdfced133e03393d": "g^{\\mu \\nu} e_\\mu^I e_\\nu^J = \\eta^{IJ}", "99f80626ddfacfae62437b90d506b1c7": " n_in_i=n_1^2+n_2^2+n_3^2=1\\,\\!", "99f823441b3cdc5653d01876b816c9cc": "\\ P", "99f8d7e4b5fbddf6f50d0bd7b777a8e5": "\\lambda={2\\pi}\\sqrt{\\frac{rd}{2k}}", "99f91b2a7f528cc73c9b63859dff6da9": "f^\\left(-1\\right)", "99f9715406858646e27444d9cb209375": "y = (ce - ag)\\,", "99f981918d334dfc96a1bcf275d80c36": "f(x_{i:\\lambda})", "99fa756e4cac6ca268d4d6a611cdb329": "(2)~~ ~~ \\frac{\\mathrm{d} I}{\\mathrm{d}z}=-AI", "99fa9315357381cac79d6d95e147f799": "\\begin{align}\n\\mathcal{L}_{WWVV} = -\\frac{g^2}4 \\Big\\{&[2W_\\mu^+W^{-\\mu} + (A_\\mu\\sin\\theta_W - Z_\\mu\\cos\\theta_W)^2]^2\n\\\\\n&- [W_\\mu^+W_\\nu^- + W_\\nu^+W_\\mu^- + (A_\\mu\\sin\\theta_W - Z_\\mu\\cos\\theta_W) (A_\\nu\\sin\\theta_W - Z_\\nu\\cos\\theta_W)]^2\\Big\\}\n\\end{align}", "99fb54c066b724ea0b7bf0cf745f8232": "k > 0", "99fba8057b8a0d84d0a1eb3de0b1c365": " - \\infty < \\gamma < 1", "99fbc0e6ff2e19a9e500b20458fa6a25": "\\chi_{\\mathbb{P}}", "99fc68d15321dbfb9acf5a235ebd95f4": "(ababab^{-1})^3(abab^{-1}ab^{-1})^3=(ab(abab^{-1})^3)^4=", "99fc963f9a0b81445825a82337cf26da": "\\mathbf{\\nabla \\times E}' = -\\frac{\\partial \\mathbf{B}'}{\\partial t}.", "99fd1a7c664ed1a723180b89571a32b2": "P_{i - \\frac{1}{2}} = \\frac{1}{2} \\left[ \nQ \\left( u_{i-1} , \\frac{u_{i} - u_{i-1}}{\\Delta x_{i-1}} \\right) + \nQ \\left( u_{i} , \\frac{u_{i} - u_{i-1}}{\\Delta x_{i-1}} \\right).\n \\right] ", "99fd1c7d337b11ad76f9bf5890560aae": "\\gamma:= -\\frac{1}{2}\\big(n^a\\Delta l_a-\\bar{m}^a\\Delta m_a \\big)= -\\frac{1}{2}\\big(n^a n^b\\nabla_b l_a-\\bar{m}^a n^b\\nabla_b m_a \\big)\\,,", "99fd635776f2b48b043a47ca300f28c9": "\\{(k, x)\\ |\\ k \\in K \\wedge x \\in X \\wedge P(x) \\}", "99fdb36e1766a089b74afb652f4b34c4": "R_{\\mathrm{LR}} = 2[\\langle r_A^2 \\rangle ^{1/2} + \\langle r_B^2 \\rangle ^{1/2}]", "99fdbaaacef7810af6d195261d053bb4": "Pr(X y_k^{j+1}) = \\frac{r_{j+1}}{r}", "9a189af9d9b4530639a3f6276ec1532e": "F_Y(y; \\sigma) = \\frac{2}{\\sqrt{\\pi}} \\,\\int_0^{y/(\\sqrt{2}\\sigma)}\\exp \\left(-z^2\\right)dz = \\mbox{erf}\\left(\\frac{y}{\\sqrt{2}\\sigma}\\right),\n", "9a18a86bb735132e3550a064a1e1fc9e": "w=F(z|v)", "9a18ccdbe922dde8c46129b9ebf7bc73": "x_o = L - \\sqrt{ (\\rho - r_n)^2 - (\\rho - R)^2}", "9a19103ee469c1393865fe50febee6fb": "\\ {\\sum_{n=1}^\\infty \\rm Riesz(x/n^2) = x \\exp(-x)}.", "9a197f6ee7aabbe6e2aaba9c4406e2b3": " \\Psi(x_1,x_2,\\cdots x_N,t) = e^{-iEt/\\hbar}\\psi(x_1,x_2\\cdots x_N) ", "9a19800a3827b4056908d83f1fd4a1d1": "\\min_{w\\in\\mathbb{R}^d} \\frac{1}{n}\\sum_{i=1}^n (y_i- \\langle w,x_i\\rangle)^2+ \\lambda \\|w\\|_0, ", "9a19b4e6f3bec2bcf45aac553525d934": "I(\\lambda, T) = \\frac{2 h c^2} {\\lambda^5} e^{-\\frac{hc}{\\lambda kT}}", "9a19cb65679ce362e40a1e5403353076": "\\sigma_I=e\\sqrt{N}/t", "9a19dfac96a401700ea65ccb81daf310": "E_k \\varpropto \\sigma_tV ", "9a1a00d234bdedba574c9d9cbe5996c3": "p_{n+1} = p_n + K \\sin(\\theta_n)", "9a1a45d6a06815df82ed3b3c383adcf3": "m/s", "9a1aac0346dca5f7e82024a7bec41caf": "\\begin{align}\n u(t) &= \\operatorname{Re}\\{u_a(t)\\}\\\\\n &= m(t)\\cdot \\cos(\\omega t + \\phi) - \\widehat{m}(t)\\cdot \\sin(\\omega t + \\phi)\n\\end{align}", "9a1ae7ccd368f731d2b15dd7a9384121": "[D,P_\\mu]=P_\\mu,", "9a1af0403b85697cbea336969a9da8b3": "\\mathcal{C}^1", "9a1afbc914e11ec6f5b86a73e8f97083": "\\frac{dt}{ds} = 1", "9a1ba741141102c517ba642807eaa33f": "\nU(a,z)=\\frac{1}{2^\\xi\\sqrt{\\pi}}\n\\left[\n\\cos(\\xi\\pi)\\Gamma(1/2-\\xi)\\,y_1(a,z)\n-\\sqrt{2}\\sin(\\xi\\pi)\\Gamma(1-\\xi)\\,y_2(a,z)\n\\right]\n", "9a1c204541c46b682f89ac6a75da8e8d": "\\int \\frac{dQ}{T} = 0", "9a1c389a170d940ca59bf83c6f392d61": "=\\quad {k_1 \\over n}(\\log n - \\log k_1) + {k_2 \\over n}(\\log n - \\log k_2)", "9a1c3b698bc8125b4559140e0bd9561b": "d_i-d_{i-1}", "9a1cb160d862a77d6e8626b87bf63520": "\n\\langle R_d^2\\rangle\\sim \\sqrt{\\tau}.\n", "9a1cb3ece5ebeff67b747a701b1a96e4": "uLv-vLu = u \\frac {d}{dx} \\left( p(x) \\frac {dv}{dx} \\right)-v \\frac {d}{dx} \\left( p(x) \\frac {du}{dx} \\right). ", "9a1cb6d5217d6fd7f2e9bd79e6deecd3": "\\begin{align}\\tfrac{a}{b}<1-\\left(\\tfrac{1}{10}\\right)^b\\end{align}.", "9a1d31420b3d6b4b0c01d2d84c8a55a9": "wRw'", "9a1d9ffb0cba6b37be099843c6bba335": " \\begin{align}\nr_1(\\beta) &= \\beta + 1 \\\\\nr_2(\\beta) &= \\lambda \\beta^2 + \\beta - 1.\n\\end{align} ", "9a1e5046d03a1abbd1d02b6e2ff58c35": "S^*(e_0) = 0, \\ \\ S^*(e_n) = e_{n-1}, \\quad n \\ge 1. \\,", "9a1e58ce32a060ec84161f4c5a071b51": " K\\left(P(X), Q(Z)\\right) = \\langle \\mu_X , \\mu_Z \\rangle_\\mathcal{H} = \\mathbb{E}_{XZ} [k(x,z)] ", "9a1e7d0d189622f222dac35021aeaac3": "|F| = c", "9a1eae71d82a67412b9365a6b23ce49a": "d(\\gamma_0(t),\\gamma_\\tau(t)) \\,", "9a1ed50c899d0bb56a5d1bc76a9d1969": " Base~excess = 0.93 \\times [HCO_3^-] + 13.77 \\times pH - 124.58 ", "9a1fa35e66f9a654b05c6b86839596e8": "w=19.5", "9a1fbef90e78fff15cfc22c0fbd2747e": "|g(t,y_1,z_1) - g(t,y_2,z_2)| \\leq C (|y_1 - y_2| + |z_1 - z_2|)", "9a1fc062d7f6be1f7eefb4e4fa6dda65": "A_2=\\{n | (X_n,d_n)=(Y,d_Y)\\}\\,", "9a1fd4dd67e8eb0eb45d88eb15aae170": "f= \\frac{16}{Re} ", "9a20f2e10c504550cee38c91bc1c4372": " PV = k ", "9a211715fa020385056c01834bbfa5c8": "\\begin{align}\nf(a,b) &=\\sum_{n=-\\infty}^\\infty a^{\\frac{n(n+1)}{2}} b^{\\frac{n(n-1)}{2}} \\\\\n&= \\prod_{n=0}^\\infty (1+a^{n+1}b^n)(1+a^nb^{n+1})(1-a^{n+1}b^{n+1})\n\\end{align}", "9a214b4182d51e77ffeacb5f8d404e57": "\nW = W(\\boldsymbol{F}) - p~(J-1)\n", "9a2165620fbd2d66b39b4575e0437046": "tR = (T + mN) \\equiv T \\pmod {N}", "9a2177577373ebcdda87475f0e85e009": "\\pi_i=(1-\\rho)\\rho^i.\\,", "9a2192de3f75cba8bbbf457e1f0f463d": "H_3", "9a21ab1e327ebf8e9eb3bec2946c3494": "F_A(x) \\le F_B(x)", "9a21d49db3d2f77795a24163a66d0a91": "\\textstyle{\\frac {\\log(4)} {\\log(2)}}", "9a2210687c0d700a5a99f98fd1ad6c4b": "z\\mapsto\\pm z^{*}.", "9a2218fc7a417c0584f0ae5bc4016345": "{A}_{9}^{(2)}", "9a224394e3b1dd2e30e81eb8015b5b9f": "51984 = 228^2 = 37^3 + 11^3", "9a229788eb246431bfad7a1ce51550b3": "\\operatorname{Cl}(X) \\to \\operatorname{Pic}(X), D \\mapsto \\mathcal{O}(D)", "9a22f73fe937f0bd0dc5c2e309a396d7": " \\eta_{\\mu \\nu}=\\eta^{\\mu \\nu}=\\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\ \n0 & -1 & 0 & 0 \\\\ \n0 & 0 & -1 & 0 \\\\ \n0 & 0 & 0 & -1 \n\\end{pmatrix} ", "9a231c14a3416b1055b8ffb960151aee": "BS", "9a2325bb498b479bfda726dda1235afd": "p(X,A|\\theta)", "9a233f4be84c85090586b924ebac96a3": " Y_\\ell^m( \\theta , \\varphi ) = \\sqrt{{(\\ell-m)!\\over (\\ell+m)!}} \\, P_\\ell^m ( \\cos{\\theta} ) \\, e^{i m \\varphi }", "9a235f341d21a532ba4e1c194d1ef673": "F_4 = 4F = 4", "9a236c2b9325da31a3aefc7393c84665": "\\begin{align}\n\\mathit{HCOF}_{i,j,k} &= P_{i,j,k} - \\frac{\\mathit{SS}_{i,j,k}\\Delta r_j \\Delta c_i \\Delta_k}{t^m-t^{m-1}} \\\\\n\\mathit{RHS}_{i,j,k} &= -Q_{i,j,k} - \\mathit{SS}_{i,j,k}\\Delta r_j \\Delta c_i \\Delta v_k \\frac{h^{m-1}_{i,j,k}}{t^m-t^{m-1}}\n\\end{align}", "9a23f0ccba74970819177f1cbd76c8d9": "d_f=\\frac{-v_i^2}{2a}=\\frac{v_i^2}{2 \\mu g}", "9a246ca0e1a325ed676527a809e594cb": "E_h(\\mathcal{N}(\\mu,\\,\\sigma^2)) =\\mathcal{N}(\\mu + h\\sigma^2,\\,\\sigma^2).\\,", "9a24c9dbd54bbb9ff942784c684f8b4a": "a, b \\in {\\mathbb{C}}^n", "9a24f5f4ccc20ecf0f04a6f5232a59ab": " = \\alpha^{-p}B(\\gamma-p,p)\\, ", "9a24f6a8a27a96d819d72cd3fa2dad7e": "Q = C\\; A\\; \\sqrt {2\\;g\\;H\\;\\frac{T_i - T_o}{T_i}}", "9a2502ccc864c7e060000de29fbf3666": "x = 4", "9a253c76724a47a7475f70927bccd89b": " R_{abmn;l} + R_{ablm;n} + R_{abnl;m} = 0\\,\\!", "9a258cc50798e4f05a1941beeb9d40f1": " P=P(I),P^{\\prime}(I)>0.", "9a2592e4ca6e2921f48014cb56540529": "\\frac {P_1V_1} {T_1} = \\frac {P_2V_2} {T_2}.", "9a25bed936582d7a4fef62ba25c13fde": "q(\\lambda) = \\operatorname{perm} (\\lambda I_2 - A) = \\operatorname{perm} \\begin{pmatrix} \\lambda - 1 & -2 \\\\ -3 & \\lambda-4 \\end{pmatrix} = (\\lambda - 1)(\\lambda - 4) + (-2)(-3) = \\lambda^2 - 5\\lambda + 10.", "9a25de8c8e0ec7ca7a35c28783bc3725": "\nX \\ni x \\mapsto \\vert \\phi_n (x) \\vert^2\\,.\n", "9a25e39f624d0c5f4bd7cd1ae28c39bf": "\\rho_t = \\frac{M_t}{V_t}", "9a2602f62891f23433397e4c53a13333": "[x,y] = xy - yx", "9a2620753f6201fd359ad1143f8bd7dc": "p=0.1", "9a272511661d4db9cf291b9d0c146156": "\\begin{pmatrix}0 & a \\\\ a & 0 \\end{pmatrix}", "9a2738eea9750059342aaa30cc559521": "\\Gamma(z) = \\int_0^\\infty t^{z-1}\\,e^{-t}\\,dt", "9a2750cd45b8973d3fd645d6a7c52eb3": "\\ \\varepsilon_G=\\frac{1}{2}\\left(\\frac{\\ell^2-L^2}{L^2}\\right)=\\frac{1}{2}(\\lambda^2-1)", "9a276533a191f66e8e8e0e560dd789c5": "\\ LM = n \\cdot R^2 .", "9a28490ff4c785ba2d75c6bdb4b9ed43": "y[x := r] = y", "9a285d2fa91f182393a6022ddc1ecd70": "M\\!", "9a2949062e719c51f4418ebe451fb527": " t(n+m)= \\left (\\frac{1+\\sqrt{5}}{2} \\right )^{n+m} = O \\left (1.6180^{n+m} \\right ).", "9a295872213548684b6cddfa7286d60a": "~X~", "9a29ef552acbd93a1ddb1a8cffcb7d8d": "P_i \\not= O", "9a2a31d00393f80a43b4a039255cb9d6": " Y=\\max\\{\\,X_1,\\ldots,X_n\\,\\} \\, ", "9a2a4905029f6ba7015ccde0d3e98dc2": " \\text{PV} = \\text{FV}\\cdot \\exp\\left(-\\int_0^T r(t)\\,dt\\right)", "9a2a876e248936f2e5936c87a3f53742": "X_k^{(m)}=\\frac{1}{m^H}(X_{km-m+1}+\\cdot\\cdot\\cdot+X_{km})", "9a2abf758bef0f8b03c1c7ab6a651f74": "e^z = \\lim_{n \\rightarrow \\infty} \\left(1+\\frac{z}{n}\\right)^n ~.", "9a2ae4d90c2c4251ce7684843e98e61e": "\n\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}=\n\\begin{bmatrix}\n0.60974&0.20528&0.14919\\\\\n0.31111&0.62567&0.06322\\\\\n0.01947&0.06087&0.74457\n\\end{bmatrix}\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}\n", "9a2af49ee832a71077e895eb1602a317": "S(y)\\!", "9a2af62b5472f97fdbb6ca9488211f38": "0 \\le u_1,u_2,v<256^l", "9a2ba8e5e9367d4f9e70f4cfddad632a": "\\kappa = \\sqrt{\\frac{\\Sigma (z*e)^2 C_0^*}{\\epsilon_r \\epsilon_0 k_B T}}", "9a2bedafa5e6c8231e3f8c5cbfc1875a": "\\mathrm{auth}(p) = 1", "9a2c1e4ac4f22a5530d488128cb5e4c6": "p_w(\\theta')", "9a2c27b137826ed88019430afe3de947": "h(i,k)=(h_1(k) + i \\cdot h_2(k))\\mod |T|.", "9a2c4460987641101e5474935eaa579f": "f=O_NE_N ", "9a2cad5237e04536132befe1eac179c5": " D_H = \\frac {4 L W} {2 (L + W)} = \\frac{2LW}{L+W}", "9a2d2b97b3b5e8e99f0a0dcffe4ccdff": "I_1>0\\,", "9a2d2e52c80ad97a33ee04b76f8bd9e1": "Y=\\left\\{\\begin{matrix} X & \\text{if } \\left|X\\right| \\leq c \\\\\n-X & \\text{if } \\left|X\\right|>c \\end{matrix}\\right.", "9a2d3632e5fd32658a81b2b087212131": "b_{0 ^{ }}", "9a2d8c955517526c6a2ac3ac7acc1ec8": "\\displaystyle{\\tau(a,b) = \\mathrm{Tr}\\, L(ab).}", "9a2dc6301eb5b5287d02bce9b0b950a6": " S, T, P ", "9a2e0c3e9f64c77b436f11652c9c1463": "- 0.5772156649", "9a2e129f8a9e887bf17d5593a91933c3": "~\\gamma", "9a2e3983721474f18eaedbc0594dfa18": "k,", "9a2e45234c574ae9e17f10adc8c31713": "K(W) \\to K(V)", "9a2e4de00cd23890d158be16415957cb": "E_{a^{n}}", "9a2e75d7511b17918dd7df8995980af4": " F_0 \\sin(\\omega t)=m\\left(\\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} + 2\\zeta\\omega_0\\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\omega_0^2 x \\right),", "9a2eea9437f7239acd8b9d0e18f764f0": "x = l + r \\cos \\alpha \\,", "9a2fd7158d94861ba8992445e3e54abe": " a_i \\cdot x_i ", "9a2ff1f1b744ae53e5033961b942666c": "D(P):U\\times X \\to Y", "9a3013359c12fb86f871fd32c98d8f0a": "\\frac{d^2u}{d\\theta^2} + u = -\\frac{f(1 / u)}{h^2u^2}.", "9a301c56b62e588d7d0df2f029fc3cbf": "\\begin{pmatrix} 6 & 24 & 1 \\\\ 13 & 16 & 10 \\\\ 20 & 17 & 15 \\end{pmatrix} \\begin{pmatrix} 2 \\\\ 0 \\\\ 19 \\end{pmatrix} \\equiv \\begin{pmatrix} 31 \\\\ 216 \\\\ 325 \\end{pmatrix} \\equiv \\begin{pmatrix} 5 \\\\ 8 \\\\ 13 \\end{pmatrix} \\pmod{26}", "9a3044b03160124836a41c5affa6a7aa": "G =2e^2/h", "9a30512405c4138a689b66957d9e1601": "u_x", "9a30ab486ece5b1f562eb8097238188f": " \n\\frac{d}{dt}\\langle p\\rangle = \\int \\Phi^* V(x,t)\\nabla\\Phi~dx^3 - \\int \\Phi^* \\nabla (V(x,t)\\Phi)~dx^3.\n", "9a30c35263991f61d6e66172c7c5e653": "|\\lambda| > 1", "9a30d2425216e60f5a14119cb324c206": " u^*_{i + 1/2} ", "9a3128c1fef1eb9912701bf4c585745f": " \\frac{p(y|H2) \\cdot \\pi_2}{p(y|H1) \\cdot \\pi_1 + p(y|H2) \\cdot \\pi_2} \\ge \\frac{p(y|H1) \\cdot \\pi_1}{p(y|H1) \\cdot \\pi_1 + p(y|H2) \\cdot \\pi_2} ", "9a31e46122749a8297ab1c3a0e48cfe9": "\\rho=-\\frac{1}{r}\\,,\\quad \\mu=\\frac{-r+2M}{2r^2}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\gamma=\\frac{M}{2r^2}\\,,", "9a3203a24ee05253a64b27943a7bd2d0": "C(x_{\\infty}) = + \\infty", "9a324aefc66eafb09c2bbbbbcf76e91a": " \\frac{k(t)}{k} = s. \\frac{f(k)}{k} - n ", "9a32a79d0db0fabbadf6d0549a769aa5": " v=6m+3", "9a32b2482ace676cc9180179359daa9a": "n = 2m + 1", "9a32b99010587ea84ba6e91b5cc00bff": "R=\\Big(\\frac{3}{4}\\frac{1}{\\rho \\pi}M\\Big)^{\\frac{1}{3}}", "9a333c4fdd8152eb2d1415fd04398210": " \\{ \\, \\lambda_i \\, \\}", "9a3340e64a533a51aff98f1d1e9e45dd": "\\sum_{k = n}^{N(n)} \\alpha(n)_{k} = 1", "9a3391a5461315e2402bc397bf41466e": "g_{\\vec z}(X)=X^n+g_{n-1}(\\vec z)X^{n-1}+\\cdots+g_0(\\vec z).", "9a340682085d01fd464f5811b49ec562": "\\int_{-\\infty}^\\infty e^{-(ax^2+bx+c)}\\,dx = \\sqrt{\\frac{\\pi}{a}}\\exp\\left[\\frac{b^2-4ac}{4a}\\right]", "9a341d425d9bae88653d5419e8609afd": "Wr + Tw", "9a3524110efcd765b8e74cd500f1491b": "c = d", "9a3587c60872b5e08d0f7225814ba168": "EER=(89*wt)-80", "9a35a0140d1d7c240b5e0f912c12d26c": "a^m\\equiv a^n\\pmod{p}", "9a35fa36a170dbd79b75c612ddee421d": "\\lambda(\\zeta)", "9a361a026fc5b9f46ecf64cb16a9116a": "- \\sigma_{xz}\\sigma_{xy} - \\sigma_{yz}\\sigma_{xy} + \\sigma_{yz}\\sigma_{xz}", "9a361a6e82282de83b801159ce3a032a": "\\bar F(y) - bcode(y) \\le 2^{-L(y)}", "9a36359908800be938b0d1ad310772ff": " \\alpha \\psi + \\beta |\\psi|^2 \\psi = 0. \\,", "9a36486da8ed58485b9c98f1f09fb226": "\\ M_{pitch_{max}} > D_{pitch} \\times drag ", "9a364c9aa1b65f27026a93e78f995543": "P(x) = \\lim_{n_t\\rightarrow \\infty}\\frac{n_x}{n_t}.", "9a3660d76a0668e4681b19dba1271333": "\\scriptstyle \\epsilon_\\mathrm{r}>78(38)", "9a36653f466d846e6842012d95a3b638": "\\left(\\sum_\\alpha c_\\alpha X^\\alpha\\right)+\\left(\\sum_\\alpha d_\\alpha X^\\alpha\\right)=\\sum_\\alpha(c_\\alpha+d_\\alpha)X^\\alpha", "9a36f3a8c39809cf34b5c7c3b0f04e51": " L_z\\, | n, \\ell, m \\rang = \\hbar m \\,| n, \\ell, m \\rang. ", "9a37a0d5b23fdababfc6b8ea2d8ec0a7": "\n \\boldsymbol{E}^e := \\tfrac{1}{2}\\ln\\boldsymbol{C}^e \\,.\n ", "9a37aa9886ab1c2f60e356229edf583b": "s = 2\\left( r\\pi n \\right)", "9a37f6bb8caa52d5045b0875d15ac2d2": " r = { \\mathit{NOPAT} \\over K } ", "9a3832499964dac73e89d452e2cbe004": "\\color{BlueGreen}\\text{BlueGreen}", "9a38783c3c476606c5fb7c691da054c6": "\nh_{\\nu} = r \\sqrt{\\frac{\\mu^{2} - \\nu^{2}}{\\left( b^{2} - \\nu^{2} \\right) \\left( c^{2} - \\nu^{2} \\right)}}\n", "9a387b0f7a33bf761fd104c558ec9c28": "P \\in \\mathbb{R}^{m \\times n}", "9a390439f71844dfa14741cf0b470c01": " D = O^TAO = \n \\begin{bmatrix}\n \\lambda_{-} &0\\\\ 0 & \\lambda_{+}\n\\end{bmatrix} =\n\n \\begin{bmatrix}\n \\left({ {3\\over 2} - {\\sqrt{ 5} \\over 2} }\\right)& 0\\\\ 0 & \\left({ {3\\over 2} + {\\sqrt{ 5} \\over 2} }\\right)\n\\end{bmatrix} \n ", "9a3909af9f1338543ed3391951546642": "\\delta(A^T\\cdot) + b^T\\cdot", "9a3a3bfcf0ec9f6aa8780480f4a783d1": "X(\\omega)=Y(\\omega)\\qquad\\hbox{for all }\\omega.", "9a3a50dea82134bab54d5e60316a0bf1": "N_G(x) = Ax^{\\delta} + O(x^{\\nu}) \\mbox { as } x \\rightarrow \\infin.", "9a3a791a791ae218a47d87621277b524": "\\mathrm{^{226}_{\\ 88}Ra\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{227}_{\\ 88}Ra\\ \\xrightarrow[42.2 \\ min]{\\beta^-} \\ ^{227}_{\\ 89}Ac}", "9a3b0d54444c0b5d00b0c0187e103fd1": "(\\epsilon^{0123} = \\eta^{0\\mu}\\eta^{1\\nu}\\eta^{2\\rho}\\eta^{3\\sigma}\\epsilon_{\\mu \\nu \\rho \\sigma}= \\eta^{00}\\eta^{11}\\eta^{22}\\eta^{33}\\epsilon_{0123}=-1)\\,", "9a3b34aadc2c1ab5930d766d04dcdd60": "E=\\gamma mc^2 \\,,", "9a3b807dbd2201ad2faf3c3aa75a6027": "\n N_{\\alpha\\beta,\\alpha} = 0 \n ", "9a3b9a4f5ff2337e201697046662b0d6": "\\forall u \\forall v \\ldots\\exists a \\exists b\\dots.", "9a3bbfb270e764244360547068479920": "pn ", "9a3cc49f6d61df165ba007bd321cdf3b": " k_e \\approx \\frac{1}{4\\pi\\varepsilon_0}\\approx 8.987\\ 551\\ 787 \\times 10^9 \\;\\; \\mathrm{N\\ m^2\\ C}^{-2}. ", "9a3cfd05e3eb4c8d8bedb07d2877a482": " {\\mathcal S} =\\{f| \\sup_t |(1+t^2)^N(I+\\Delta)^M f(t)\\varphi_0(t)|<\\infty\\}.", "9a3e1f72cada05bfa3bed3e79ac6464b": "\\mathbf{v} = \\left[ \\begin{matrix} r \\\\ \\angle \\theta \\end{matrix} \\right]", "9a3e2954b39ab50bdc682932720a5ae5": "S_0(t) = \\exp\\left(\\int_0^t r(s)ds + A(t)\\right), \\quad \\forall 0\\leq t \\leq T. ", "9a3e595d4a7307d13f50382176ad1d5a": "\\frac{H^2}{P^2}=\\frac{2P^2+1}{P^2} ", "9a3e939863b9cf98cafd5afc4b47d7b6": "E\\!\\left[X_n^2 \\right ] = \\sum_{s=0}^n 2^{2n-s-1} p^{2n-s} = \\frac 12\\,(2p)^n \\sum_{s=0}^n (2p)^s = \\frac12\\,(2p)^n \\, \\frac{(2p)^{n+1}-1}{2p-1} \\le \\frac p{2p-1} \\,E[X_n]^2,", "9a3eb6a39edbf1978c33cb0a55d72d1d": "X_\\mathrm{L}", "9a3f0858e79cb51a47b657c5eb490aa5": "p+q^{\\sqrt{-1}}", "9a3f6c004a037a6ee046f3ff2ed55eac": " T(e^{f+g})T(e^{-f}) T(e^{-g}) ", "9a3f782e184df143ee0886793b123124": "\\operatorname{cov}[X,Y] = \\operatorname{E}[XY] - \\operatorname{E}[X]\\operatorname{E}[Y]", "9a3f8d9f2710675b9f9a9e6adf7754d1": "\\mathfrak{F}\\subseteq\\mathfrak{B}", "9a3ffd7a8a689538ebd2e806e71a2f5b": "s_{\\lambda+1}=s_\\lambda- \\frac{P(s_\\lambda)}{\\bar H^{(\\lambda+1)}(s_\\lambda)}, \\quad \\lambda=L,L+1,\\dots,", "9a4000fcc399777329f927287c40e78a": "\\langle X \\rangle_\\infty", "9a4038d7a9212c36e88e95b31d7eafe3": "FWER=P\\left( {V \\ge 1} \\right) = E\\left( {\\frac{V}{R}} \\right) = FDR \\le q", "9a404fc35fa5085271a39f490e6330f0": " \\mathbf{C}", "9a40563035cb98f5a232c68a195e9f05": "M_Z=\\frac{M_W}{\\cos\\theta_W}", "9a405989e441b60ae5bc6319e94acf85": " f^{(k)}(x) = \\begin{cases} \\frac{p_k(x)}{x^{3k}}e^{-\\frac{1}{x^2}} & x>0 \\\\ 0 & x\\leq 0\\end{cases}", "9a410a82f59b4a61f193c6c476839464": " \\chi(V^*)= \\sum_i (-1)^i \\operatorname{dim} V = \\sum_i (-1)^i \\operatorname{dim} H^i(V^*). ", "9a41b5f6024c8255d67130084d3298fc": "\\,\\!D = (\\overrightarrow{v_r} - \\overrightarrow{v_i})^2 = \\overrightarrow{v_r}^2 - 2 \\overrightarrow{v_r} \\overrightarrow{v_i} + \\overrightarrow{v_i}^2", "9a420a2ea8fdaec2eb23710c1555b1c7": "\\bar{q}(x, y)=-\\bar{q}(x, y)", "9a4239322d7151d0291bae4d6567942b": "H^1(X, {\\mathcal{O}_X}^*)", "9a426797182fd0741e1334f0b8661815": "\\Omega_k = 1 - \\Omega", "9a429c64a91f51d0b5c5fd58f308e5c0": "a\\in(\\mathbb{Z}/n\\mathbb{Z})^*", "9a42badeda0de57607e1e8a58b0cb381": "\\max_{x\\in\\mathbb R}\\; 2x", "9a42eaef246edf25064e754c93aaee97": "\\frac{\\partial {\\rm tr}(\\mathbf{AXBX^{\\rm T}C})}{\\partial \\mathbf{X}}:", "9a434e5764e44b9db2b0bb11e8e1565d": "\\mathfrak{so}(5)\\cong \\mathfrak{sp}(2)", "9a43565950f5659216b5c06027d6b239": "\\beta(\\alpha)=\\frac{2\\alpha^2}{3\\pi}~,", "9a4364ab57f2db2bbabdb15e0157047a": "\\begin{align}\nZ&=\\int_{-\\pi}^{\\pi}d\\theta_1\\cdots d\\theta_L\\;\ne^{\\beta J \\cos(\\theta_1-\\theta_2)}\\cdots\ne^{\\beta J \\cos(\\theta_{L-1}-\\theta_L)}\n=2\\pi \\prod_{j=2}^L\\int_{-\\pi}^{\\pi}d\\theta'_j\\;e^{\\beta J \\cos\\theta'_j}=\n\\\\\n&=2\\pi\\left[\\int_{-\\pi}^{\\pi}d\\theta'_j\\;e^{\\beta J \\cos\\theta'_j}\\right]^{L-1}\n\\end{align}", "9a43c44cb0e1b7b7934e013776b3d87d": "p_N(x)=1+x+x^2+ \\cdots +x^{N-1}", "9a44996680b6dbbf21b33308f6dea9e1": "q(x,y)= ax^2 + xy + by^2", "9a44aa7d9819ca89723114bb30ab2cd8": "Ab(S) \\Rightarrow Int2 \\wedge Obs2", "9a44d1aafc7b6144c63ed4891dd1d5ea": "\\exists x \\in A^c \\cup B^c", "9a451be942514acb3dee94b1a009bdb0": "\\nabla F(x, y, z).", "9a45995e35c39fc2f7beb90503598012": "Z_3 = 4X_1(XX_1+aX_1+1) = 24 \\, ", "9a45af9c23a5654070c1d2404251993c": "\\Sigma_{k-1}", "9a462982cfda5b23cc31efef85270ad3": "A \\otimes_R A = A[x] / f(x)", "9a46a897f4fcf87b2ed9561c1abd9bbd": "\n \\begin{align}\n n_\\alpha~N_{\\alpha\\beta} & \\quad \\mathrm{or} \\quad u^0_\\beta \\\\\n n_\\alpha~M_{\\alpha\\beta} & \\quad \\mathrm{or} \\quad \\varphi_\\alpha \\\\\n n_\\alpha~Q_\\alpha & \\quad \\mathrm{or} \\quad w^0\n \\end{align}\n", "9a46ca868e3441543af81206ccbe4b9d": "SU(2)_W \\subset SU(3)_W", "9a46d8e292569ad0b1dd9b12fd9bbdae": "\\Delta t = \\tfrac{4}{a} \\sinh( \\tfrac{a}{4} \\Delta\\tau) \\ ", "9a46eb9416d94d1c61c3a550d5f86bb0": "\\lbrace p_1,p_2,p_3,p_4 \\rbrace = \\lbrace 1,0.75,0.75,0.5 \\rbrace", "9a46ed0b1a4cc63960778c07da0dc194": "x \\in U.", "9a46f016bc1f846280b0c5b4f27a7518": "\\Gamma_{tot}", "9a46f6c660c918d69e0eaba2d4cbb36e": "\\delta_{max} = \\frac {F a (L^2 - a^2)^{3/2}} {9\\sqrt{3} L E I}", "9a47064cd2f4e4d726f8da93d0fc88a7": "H=\\{Q_1,Q_1\\}=\\{Q_2,Q_2\\}=\\frac{p^2}{2}+\\frac{W^2}{2}+\\frac{W'}{2}(bb^\\dagger-b^\\dagger b)", "9a473086530ca38c472dd132811509fc": " {G^a}_b \\, {G^b}_a = R^2 ", "9a4754dab265d4508d53256c243ec433": "\\mathbf{E} = \\hat{O}^{-1} \\mathbf{J}", "9a477d6b3489cbfa3db657810ca75fd2": "\\forall A_1 \\, \\ldots \\, \\forall A_n \\, \\exist C \\, \\forall D \\, [D \\in C \\iff (D = A_1 \\or \\cdots \\or D = A_n)]", "9a479ca9a3871c3e33c6eadaaf82bdeb": "~\\sqrt{\\frac{m \\omega}{2 \\hbar}}\\left(x+\\frac{\\hbar}{m\\omega}\\frac{\\partial }{\\partial x}\\right)\\psi^\\alpha(x,t)=\\alpha(t)\\psi^\\alpha(x,t)", "9a48036d34e8d8c8fbac487f8c1c3542": "P_n(x) =\n\\begin{cases}\n \\frac{2 }{L}\\sin^2\\left(\\frac{n\\pi x}{L}\\right); & 0 < x < L \\\\\n 0; & \\text{otherwise}.\n\\end{cases}\n", "9a481e2d3d4db39e0aa66f8c6c2416ba": "x \\equiv a_1 \\pmod {n_1}", "9a48308d091dbb363a58c7ab228ecb5a": "P(\\sigma)", "9a496933d383e9481f52fed3ced40590": "\\mathbf{H_1}\\mathbf{x_1} = \\mathbf{H_1}\\mathbf{x_2}", "9a4993253bead82a576fe279db379324": "x^w \\mod n", "9a4a3fb6e9285286837b437023a38008": "\\omega(k)= v(k)\\ k.\\,", "9a4a56d2765bf8758cb49c7ddb273ea4": "(\\mathbf{A}^\\mathrm{T})^{-1} = (\\mathbf{A}^{-1})^\\mathrm{T} \\,", "9a4a7e375cb06c810228c8491747fd4f": "\\frac{\\omega - \\omega_0}{\\omega_0}\\thickapprox -\\frac{\\iiint_{V}(\\Delta\\mu |H_0|^2+\\Delta\\epsilon |E_0|^2)dv}{\\iiint_{V}(\\mu |H_0|^2+\\epsilon |E_0|^2)dv}\\,", "9a4adcdd4f120c69f6003f408aa2e937": "y_q = 2", "9a4af7dbc3d871e8997a2f3c6817180d": "\\mathbb{E}[X_\\tau]\\ge\\mathbb{E}[X_0],", "9a4b7d3497c626b5c91f05e298df6d8e": "-121) = o(h)", "9a61c6e9fb987b700e9b1c9b18981510": "P_b(E)", "9a61dad35ce023d0880f45a3070491a4": "x0.", "9a62d2c84625bfdd137df7aba147383a": " H_c(t) ", "9a62e5f1a6d6c73caf6e689884d78d8e": "1/det(1-tM)", "9a62fe04a3f456ac9c62efd0fbe01c87": "f = a_n X^n + a_{n - 1} X^{n - 1} + \\cdots + a_1 X^1 + a_0X^0", "9a6326db75a3eabdc447d9cf7ab16ce9": "\n\\mathbf{F}_{\\mathrm{fict}} = - 2 m \\boldsymbol\\Omega \\times \\mathbf{v}_\\mathrm{B} - m \\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{x}_\\mathrm{B}) - m \\frac{d \\boldsymbol\\Omega}{dt} \\times \\mathbf{x}_\\mathrm{B}.\n", "9a638d91cdc6c1bbfc5d790335e0709c": "P_{4}^{0}(x)=\\begin{matrix}\\frac{1}{8}\\end{matrix}(35x^{4}-30x^{2}+3)", "9a639e2426d9f2385652f28458304bb2": "\nD = f(x_0+1,y_0+1/2) - f(x_0,y_0)\n", "9a64133fedf110daa63244a2894ece5f": "f = \\lim_{n \\to \\infty} \\sum_{k=0}^n c_k \\phi_k ", "9a643144ba4193e8eed854a8ff85961c": " |{\\rm det}(I+A) -{\\rm det}(I+B)| \\le \\|A-B\\|_1 \\exp (\\max(\\|A\\|_1,\\|B\\|_1) +1).", "9a645d00cb4a2a2e3128298627fac3ca": "\n\\begin{align}\n 1 &= \\sum_{n=1}^{\\infty}C_n\n = \\tfrac12 + \\tfrac1{12} + \\tfrac1{24} + \\tfrac{19}{720} + \\tfrac3{160} + \\dots,\\\\\n \\frac1{\\log2} - 1 &= \\sum_{n=1}^{\\infty}(-1)^{n+1}C_n\n = \\tfrac12 - \\tfrac1{12} + \\tfrac1{24} - \\tfrac{19}{720} + \\tfrac3{160} - \\dots,\\\\\n \\gamma &= \\sum_{n=1}^{\\infty}\\frac{C_n}{n}\n = \\tfrac12 + \\tfrac1{24} + \\tfrac1{72} + \\tfrac{19}{2880} + \\tfrac3{800} + \\dots.\n\\end{align}\n", "9a646da2f9c11f75f85d291456449b77": "\\mathcal L_v\\omega = 0", "9a6475102fb213504c5dec267ccce95e": " \\frac{\\partial z}{\\partial y} = x \\left( 3y^2 +2y(x+1) + x \\right) ", "9a648eb9989a1635b59a53b51593fb8a": "\\partial_t \\phi - 6\\phi\\, \\partial_x \\phi + \\partial_x^3 \\phi = 0, \\,", "9a649255d01cfe15798572a8f1457e87": "\\vec{R(t)} ", "9a64ded4573d9f50b38eea4c8728202a": "(1,0)^T", "9a6510b741007f615c09322ad004d045": "P_3(n) = \\frac{n(n+1)(n+2)}{6} = {n+2 \\choose 3}", "9a652079d99c844b530c53c6f181cc26": "{w=\\frac{\\frac{1}{2}\\dot{\\phi}^2-V(\\phi)}{\\frac{1}{2}\\dot{\\phi}^2+V(\\phi)},}", "9a65830e19ee0ca54bce2cee7b2ade9d": "\\Psi_m^{(1)} = \\sum_{kl} \\left| \\Psi_{l}^{k}{}^\\prime \\right\\rangle\n\\frac{ \\left\\langle \\Psi_{l}^{k}{}^\\prime \\left| \\hat{\\mathcal{H}} \\right| \\Psi_{m}^{(0)} \\right\\rangle}\n{E_m^{(0)} - E_{l}^{k}} = \\sum_{kl} \\left| \\Psi_{l}^{k}{}^\\prime \\right\\rangle \\frac{\\sqrt{N_l^k}}{E_{m}^{(0)} - E_{l}^{k}} \n", "9a65871c66027f14740e9b8ea170a16e": "\\left(\\frac{300\\times300}{2}\\right)^{15}", "9a65b0215c2e62f7d62c21cb0bcbc476": "g(\\gamma')", "9a65eca9f4ef813f5c5a0b7fa9b6365b": "\\mathrm{Ro}=\\frac{U}{Lf}", "9a6618708a78e3e5a8309df5befc55ab": "(1-2 / \\pi)(\\sigma_2-\\sigma_1)^2 + \\sigma_1 \\sigma_2", "9a6665644a6784f97381713b490de417": "\n\\hat{\\mu}= \\frac{\\sum_{i=1}^n w_i X_i}{\\sum_{i=1}^n w_i}, \\,\\,\\,\\,\\,\\,\\,\\, \\frac{1}{\\hat{\\lambda}}= \\frac{1}{n} \\sum_{i=1}^n w_i \\left( \\frac{1}{X_i}-\\frac{1}{\\hat{\\mu}} \\right).\n", "9a66a6382dddd41971e4f8551bcdc622": "D : C^1([0,1]) \\to C([0,1])", "9a66e63eb833cb33ab2da2241a2aadbf": "= \\left[ h(\\mathbf{x}) \\left( \\frac{1}{2\\pi} \\right)^{n/2} e^{-(1/2) \\mathbf{(x-\\theta)}^T \\mathbf{(x-\\theta)} } \\right]^\\infty_{x_i=-\\infty} \n- \\int \\frac{\\partial h}{\\partial x_i}(\\mathbf{x}) \\left( \\frac{1}{2\\pi} \\right)^{n/2} e^{-(1/2)\\mathbf{(x-\\theta)}^T \\mathbf{(x-\\theta)} } m(dx_i)", "9a66edb1c22655900fca78cc7c0fba0e": "P\\left(\\tfrac{p}{q}\\right) = a_n\\left(\\tfrac{p}{q}\\right)^n + a_{n-1}\\left(\\tfrac{p}{q}\\right)^{n-1} + \\cdots + a_1\\left(\\tfrac{p}{q}\\right) + a_0 = 0.", "9a672df13a8c75772461a4b3c962c34d": " \\hat{Z} ", "9a68086aa9aa98b94c922c31b3834937": "1-e^{-\\lambda\\delta}", "9a6839535f71d659fb0c3a5f8d607010": "a^{2} + b^{2} + c^{2} \\geq (a - b)^{2} + (b - c)^{2} + (c - a)^{2} + 4 \\sqrt{3} T \\quad \\mbox{(HF)}.", "9a685f148fbaf3d30529f81c75e176d2": "=\\quad -(p_1 \\log p_1 + p_2 \\log p_2)", "9a68ab7a70ef4f3396c11989ed701349": "\\nabla\\cdot\\mathbf{E} = \\frac{1}{\\varepsilon_0} (\\rho_\\text{f} + \\rho_\\text{b}) = \\frac{1}{\\varepsilon_0}(\\rho_\\text{f} -\\nabla\\cdot\\mathbf{P})", "9a68dbc80b792aa0630056df952f26b8": "\\mathbf{v} = \\boldsymbol{\\omega}\\times\\mathbf{x}", "9a68dca6e2e7e422cd1d0977d486542a": "/(x+0y) = /x + 0y\\ ", "9a68e30910b9158f730ccc63233fa83e": "M \\subseteq e(H')", "9a697e67f98c820172303a54deea73f4": "\n\\begin{align}\n& \\mathrm{Apriori}(T,\\epsilon)\\\\\n&\\qquad L_1 \\gets \\{ \\mathrm{large~1-item sets} \\} \\\\\n&\\qquad k \\gets 2\\\\\n&\\qquad\\qquad \\mathrm{\\textbf{while}}~ L_{k-1} \\neq \\ \\mathit{empty set} \\\\\n&\\qquad\\qquad\\qquad C_k \\gets \\{ a \\cup \\{b\\} \\mid a \\in L_{k-1} \\land b \\in \\bigcup L_{k-1} \\land b \\not \\in a \\}\\\\\n&\\qquad\\qquad\\qquad \\mathrm{\\textbf{for}~transactions}~t \\in T\\\\\n&\\qquad\\qquad\\qquad\\qquad C_t \\gets \\{ c \\mid c \\in C_k \\land c \\subseteq t \\} \\\\\n&\\qquad\\qquad\\qquad\\qquad \\mathrm{\\textbf{for}~candidates}~c \\in C_t\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad \\mathit{count}[c] \\gets \\mathit{count}[c]+1\\\\\n&\\qquad\\qquad\\qquad L_k \\gets \\{ c \\mid c \\in C_k \\land ~ \\mathit{count}[c] \\geq \\epsilon \\}\\\\\n&\\qquad\\qquad\\qquad k \\gets k+1\\\\\n&\\qquad\\qquad \\mathrm{\\textbf{return}}~\\bigcup_k L_k\n\\end{align}\n", "9a69bb94e3957813fd08ebfddc3e9f70": "\\lim_{x\\to\\infty}\\left( \\sup_n \\operatorname{P}\\big[\\, |X_n|>x \\,\\big]\\right) = 0;", "9a6a287121e2e50f91d1b09d25f1410c": " I = (10eV) \\cdot Z", "9a6aae70e9adccb7316e20f10b8293b3": "\\Delta Q_{ir}", "9a6ada764d8cdc0a47d307ec1707321c": "I_{o_{lim}} = \\frac{I_{L_{max}}}{2}\\left(D + \\delta\\right) = \\frac{I_{L_{max}}}{2}", "9a6aee25192f895d082b062b1b01c243": "W_\\text{in} = W_\\text{fric} + W_\\text{out} \\,", "9a6b0272b463f50cb4f535229dff2f22": "\\left( \\tfrac{1}{4} + \\tfrac{3}{8} \\right)\\sqrt{N} = \\tfrac{5}{8}\\sqrt{N}", "9a6b48a5dfe21ea2850b137936fff4af": "\\begin{align}\n\\mathbf{u} & =(u_1, u_2, \\dots, u_m) \\\\\n\\mathbf{v} & = (v_1, v_2, \\dots, v_n)\n\\end{align}", "9a6b50de3aa1199befd39634f4be74e9": "C_p/C_v", "9a6bb1ac1e2a0b4497b801a3a22ecd75": "G \\leq S_n", "9a6bb3cc74f28814de4c470b1730036c": " \\tilde{\\Psi}_\\alpha(p,t) := \\langle p |\\alpha\\rangle", "9a6c6ea768058f46de52f3012ecdf645": "\\widehat{D}^{\\dagger}", "9a6c91c3ad7ff3ea8c7400781aea7c7e": "y = R_1 + {(x - L_1)(R_2-R_1)\\over L_2}", "9a6d4585a5f2e7d1d5914f3c1f38f4ac": "\\Delta H_{ad} = \\Delta H^0_{ad}\\,(1-\\alpha_T\\,\\theta)", "9a6d6099ada1501343e02c08666534d1": "X^\\alpha = \\Pi_{i=1}^N X_i^{\\alpha_i}.", "9a6dc74e359699eeb3ef5d228daeaf15": " C_c(K\\backslash G /K) ", "9a6e11da4cf5f4050beeae16c5712094": "\\int_0^1 x^m (\\ln x)^n \\, dx=\\frac{(-1)^n n!}{(m+1)^{n+1}} \\quad m>-1, n=0,1,2,\\ldots", "9a6e93dab7fa35f061178a3c185674f0": "w \\mapsto (w, w)", "9a6ee852681203f95e41e0bf4e4456ea": "[T^a, T^b] = f^{abc} T^c", "9a6ef028045bcdd54f76e9f2e1fb5f03": "\\sin\\delta = \\sin\\phi_o \\sin a - \\cos\\phi_o \\cos a \\cos A", "9a6f28791444b4a2c76822c27299217f": "\\textstyle{\\omega = {2 \\pi f \\over F_s} = 2 \\pi f T,}", "9a6f630c3843dfb5b9f4c52e052201f1": " ||\\rho - \\sigma||_1 = \\max_{0\\leq Q \\leq 1} \\operatorname{Tr}(Q(\\rho - \\sigma)) ~.", "9a6f6e1b11044876b48dec675b0c04b6": "\n2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\sin i\\ \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right) \\ (1\\ ,\\ 0)\n", "9a6fa9d1446a6789acd6553e722d0a5a": "A_\\mu^a (x) = \\frac2g \\frac{\\eta^a_{\\mu\\nu} (x-z)_\\nu}{(x-z)^2+\\rho^2}", "9a705a6021693a3e8bfd44d721a8614e": "\\mu_G(x,y) = y^n m_G(x/y^2).", "9a70673d839dfd73c318267347328c42": "\\mathbf{0} = -\\nabla p + \\eta \\nabla^2\\mathbf{v}", "9a7085d94940badc2facd77686ba691a": "S \\rightarrow S/L", "9a7091fec4bf5abf43e26cb94b0b4cf2": "\\partial\\mathcal{L}/\\partial A_a = \\mu_0 j^a", "9a709cf6e34d6f9feb7529bbe6392a1a": " \\langle \\cdot \\rangle = \\frac{1}{Z} \\sum_\\sigma \\cdot(\\sigma) e^{-H(\\sigma)} ", "9a71047c661ea25c1d48fb8de0908d1e": "G_1 \\times G_2 \\times\\cdots \\times G_k\\,", "9a7114a5e0588cf249370d825098e0eb": "\\displaystyle{\\Delta f_n = \\lambda_n f_n,\\,\\, \\lambda_n>0,\\,\\, \\lambda_n\\rightarrow \\infty.}", "9a711b5652912519a3c7840683e68abd": "(10)\\quad T_{ab}=-\\frac{M(u)_{,\\,u}}{4\\pi r^2} l_a l_b\\;,\\qquad l_a dx^a=-du\\;,", "9a71c086bda3bab28aa41f2bc5e21398": " \nH(\\alpha) = \\frac{|Gi|}{|G|}\n", "9a71d19e3774e894cd41ef970f0a1a41": "{S_{p,q}}^\\mathrm{T}\\cdot\\begin{pmatrix}x\\\\y\\end{pmatrix} = \\begin{pmatrix}0\\\\0\\end{pmatrix}", "9a7234edfce8d28682873970bcbcce98": "p(\\gamma[1] \\gamma[2] \\ldots \\gamma[L]) = \\prod_{t=1}^{t=L} p(\\gamma[t+1] \\gamma[t])", "9a72405cfb0cb89dc7db5937d369b985": "\\alpha^A (x^{\\mu}) = \\phi^A (x^{\\mu}) + \\bar{\\delta} \\phi^A (x^{\\mu})\\,.", "9a7303378f5ec5d6ffe2155d3d0e06f8": "\\mathbf{g}_N", "9a7305f2ed3f3a099f53da8ea1536aa3": "\\delta()", "9a73247b37a98f59c251ddf0d025dc85": "\\approx 10^{-22}", "9a733ca8f957549f99781e7b215dc0fa": " E_{out} ", "9a7341150d0f3f0a6f189736ba980fe6": "\\bold{\\hat{r}} = \\sin \\theta \\cos \\phi \\bold{\\hat{x}} + \\sin \\theta ~ \\sin \\phi ~ \\bold{\\hat{y}} + \\cos \\theta \\bold{\\hat{z}}", "9a73531393ebda2339d59a1f518c5be1": "\\int p(x) \\, d\\mu_i(x) =0", "9a73584f2e5cb320bca25a83d4d7d0f3": "d(f,g) = \\sum_{n=1}^{\\infty} \\frac{1}{2^n} \\cdot \\frac{d_n (f,g)}{1+d_n (f,g)}", "9a7381955d4df4a6697a0776298413e4": " \\hat{X}=(W^H W)^{-1}W^{H}x \\, ", "9a7385013ff268c249d675b8411c9b05": "x = 5 + 2 \\epsilon_k", "9a73ba303b504b0c0c2ee3fa410d3e73": "\\operatorname{dCov}_n(X,Y)\\geq0", "9a73c28ae16e41d5ba0e61a71fcbf0b2": " \\sum_{x,y \\in C} d(x,y) \\leq \\frac{1}{2} n M^2.", "9a73cf39a4f2a8bd5159cad8661825a5": "H = -\\sqrt{g} \\left[ R + g^{-1}\\left(\\frac{1}{2} \\pi^{2} - \\pi^{ij}\\pi_{ij} \\right) \\right]", "9a73f1cc393c4b88ac64b31f5b48de64": "(\\mathbb{C},+,\\times,\\le)", "9a7416c068726a49c039da61670ff45f": "r=r(s)", "9a7473eb49bb0812c1df4ae2eed60d19": "\n\\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} = \\frac{a}{b}.\n", "9a7483390c4cfc7f506923f978485edf": "U_q(\\mathfrak{g})\\cong \\left(\\mathfrak{B}(V)\\otimes k[\\mathbb{Z}^n]\\otimes\\mathfrak{B}(V^*)\\right)^\\sigma", "9a7493cd1edd9db3cb147726e9cff2e8": " d= \\dfrac{P}{\\sigma} ", "9a754e3a18d2ad53b174b174193938f1": "\\overrightarrow{T}", "9a7560ed7c0fe9839eac375de636f596": " f(z) = \\frac{\\sin z}{z} ", "9a75794c91949b539bee2fce46e41c7c": "\n\\mathrm{det}(C) \n= \\prod_{j=0}^{n-1} (c_0 + c_{n-1} \\omega_j + c_{n-2} \\omega_j^2 + \\dots + c_1\\omega_j^{n-1}).", "9a762ee725a027932ce6b6fd5c5d3ef2": "\\displaystyle x^2=R^2-2Rr\\cdot \\mu", "9a762fe277b848734309b6b1940efbc4": "T_{em}\\approx\\frac{1}{\\omega_s}.\\frac{3V_{TE}^{2}}{(X_s+X_r^{'})^2}.\\frac{R_r^{'2}}{s}", "9a7639ebafc1e104973626263765bf12": "Q:\\ell^1 \\to X", "9a76925aca8ccf56528c84c7c61bc74d": " \\mathbf{K} = (\\omega/c, \\mathbf{k})\\,,", "9a76c89e90d9d35b29945775d61d1a7c": "\\left(\\sum_{n=-\\infty}^\\infty |\\widehat{f}(n)|^q\\right)^{1/q}\\leq\n\\left( \\frac{1}{2\\pi}\\int_0^{2\\pi}|f(t)|^p\\,dt\\right)^{1/p}.", "9a76eccc27c6007cc4a8e926be407317": "\n\\begin{array}{rccc}\n\\mathcal{D}: & (\\mathbb{R},+) \\times \\mathbb{R}^4 & \\rightarrow & \\mathbb{R}^4 \\\\\n& (\\hat{t},(t,x,a,b)) & \\rightarrow & \\left(t+\\hat{t}, \\frac{axe^{a\\hat{t}}}{a-(1-e^{a\\hat{t}})bx}, a, b\\right).\n\\end{array}\n", "9a772dc7726f08f58ae6f3bd1cc9efcb": "\\frac{1}{T}\\cdot\\frac{dT}{d\\varphi}=-\\mu", "9a773b0b85804bd6c2dc8ff637955e25": "V_i^p", "9a776e7a8af7e42dd35fde9134383412": "\\mu_L(\\bar{p}; \\Sigma_t, \\Sigma_s) = B^T \\mu_R(q; B \\Sigma_t B^T, B \\Sigma_s B^T) B", "9a77a5edac97e887123e561088184734": "Y\\times_X Y\\longrightarrow \\overline Y,\\qquad (y^i, y'^i)\\longmapsto y^i - y'^i, ", "9a77b57201036b727de6c40da3c0b50e": "\n(5.2)\\quad\n||f||_\\infty\\leq \\min\\{\\sqrt{\\pi/2}||h||_\\infty,2||h'||_\\infty\\},\\quad\n||f'||_\\infty\\leq \\min\\{2||h||_\\infty,4||h'||_\\infty\\},\\quad\n||f''||_\\infty\\leq 2 ||h'||_\\infty,\n", "9a77cd61fec6d9ee9281bd196b9bbe81": "u^R", "9a780fd514290758c8c5f4e2ac2bda7a": "k_{B}\\,", "9a78174de5a108b34d110fbd42f41b2c": "\\nabla I^T\\cdot\\vec{V} = -I_t", "9a783d33de27ff11d14a070cce2bfd25": "\\scriptstyle T_i < T_c < T_{i+1}", "9a78458406a8c132e751bc37c6a0eb51": "\\epsilon_{t}\\,\\!", "9a78b557c9e6e83caaf666f11fdcb127": "\\beta = \\sin^2(\\theta)", "9a7927f18c436b18171ee4fb0e67e998": "\\ln (2)\\,\\!", "9a79567b5b76ca31f1a4248103eba52d": " \\mathbf{D} ", "9a79972bbfae627dd78748416ecd57fc": " W = - \\alpha n R T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right) ", "9a79d967425ed945fc314e48c0a3d34a": "\\sum_{k=0}^\\infty {k+\\nu+1 \\choose k} \\left[\\zeta(k+\\nu+2)-1\\right] \n= \\zeta(\\nu+2)", "9a7afd2a34d04b6b6603e41f3834cbf7": "c_m =\\mathcal{O}(\\epsilon)", "9a7b0a5cca92baad53790bd15fe45bfd": "\n\\Phi_i= \\frac{x_i \\nu_i}{\\sum_j \\nu_j x_j}\n", "9a7b5c3135e6397f1f55dc828bb2bff5": "Prec(R_{ecall})", "9a7bce88f2675be0a7b94be106f93cfa": "\n \\boldsymbol{F} = \\begin{bmatrix} 1 & \\gamma & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ~;~~\n \\boldsymbol{B} = \\boldsymbol{F}\\cdot\\boldsymbol{F}^T = \\begin{bmatrix} 1+\\gamma^2 & \\gamma & 0 \\\\ \\gamma & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\n ", "9a7c7e9033f1126818c14717f88780bb": "\\exp(-|x/c|^p)", "9a7c8c1f00688cfca5303093692cb639": "\\sqrt{\\alpha_\\text{G}} \\,", "9a7ca06f811a74076b348a63cf2c25a5": "A(T,V,\\{N_j\\})=\\mathrm{max}_S(U(S,V,\\{N_j\\})-TS)\\,", "9a7cb7575198967a3e6e2d3e6d7ce27d": "\\mathbf F^+=\\langle V,\\cap,\\cup,\\to,\\emptyset\\rangle", "9a7d3543356abe268aa14d4094654ea2": "\\textstyle (AX, BY, ax, by) ", "9a7d3da6cad6684804afb32f80fdcd54": "(\\Lambda\\setminus\\Phi,i,M,C)", "9a7d73a8d99760e4368a683856e0994f": "\\det(A-xI)\\in K[x]", "9a7d9e951183316e9203f1f05531278f": " \\frac{d}{dx} x^n = nx^{n-1} , \\qquad n \\neq 0.", "9a7daeac8e48af8cb1c3cba210503292": "G_0(t,s)= \\begin{cases} 0 & t\\le s\\le b \\\\ X(t)X^{-1}(s) & a\\le s < t. \\end{cases}\\,", "9a7e60d313a181e27f7b1ff08483cf5e": "\\alpha_i = 0 ", "9a7e69ca23428a3bed7abcee5abc4719": " U_{ij} = \\delta_{i+1,j}, \\quad L_{ij} = \\delta_{i,j+1},", "9a7e6ad2c12dab9490943b4eea6261f1": "V_\\mathrm{IN}", "9a7e85321db3bab814444c6a0efc39a9": " u^2 = u_t", "9a7e903d82f65f0258b68f06d6afe9ff": "-D^{(0)}[\\partial_i||\\partial_j]", "9a7eac5387d43b23378401ecc9eb7446": " \\frac{1}{r_\\pi}", "9a7f298f4ec58ccaa336381123ff7e4b": "a = \\frac{0.42748\\,R^2\\,T_c^{\\,2.5}}{p_c}", "9a7f67a990a2a421cad03dadeb6681a1": " P = g^p ", "9a7f695c4f33e5eb226eefbae9f12898": "\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} + m~\\cfrac{\\partial^2 w}{\\partial t^2} - \\left(J + \\cfrac{E I m}{k A G}\\right)\\cfrac{\\partial^4 w}{\\partial x^2~\\partial t^2} + \\cfrac{m J}{k A G}~\\cfrac{\\partial^4 w}{\\partial t^4} = q(x,t) + \\cfrac{J}{k A G}~\\cfrac{\\partial^2 q}{\\partial t^2} - \\cfrac{EI}{k A G}~\\cfrac{\\partial^2 q}{\\partial x^2}\n ", "9a7fb6fc3aaee2478de5beab52c02bf3": "|\\mbox{ }|", "9a7fcedb27a744e988513390bd263fba": "g\\,\\!", "9a80120c1cc7266f675e5b4cd29a4290": "(1 + x)^r \\le e^{rx},\\!", "9a80f4c95fc7db6d0e0b7ae237795b71": "\\ \\mu ", "9a8109060e9767a4a8786df2ced8f422": "{ (close_{today} - low_{Ndays}) - (close_{today} - high_{Ndays}) = high_{Ndays} - low_{Ndays} }", "9a813bdfc7d704b6c8d45c3d0bebf17b": "\\int f \\, dx = 1.", "9a814d6ec97d5d6b013dfd910b44a739": "b_2 = \\tilde{f}_i b_1", "9a8155fa818bcce09e15280a81ee0ed8": "[0,+\\infty)", "9a817044c587f2b44f4833a26f16e4da": "\\ cdf_x(i) = \\sum_{j=0}^i p_x(j)", "9a81b2292fedd50323ae966382bd050c": "\\sec x = 1 + \\frac{1}{2!}x^2 + \\frac{5}{4!}x^4 + \\cdots = \\sum_{n=0}^\\infty A_{2n} {x^{2n} \\over ({2n})!}.", "9a81f1611c3148b356db92cddb946eab": "\\mathsf{L}(A;x)", "9a820f93253372efb418109f6da6ea44": " {D_{t}} = {(L/D)_{\\alpha}} ^{-1} \\times L ", "9a82215403b43417621f564ce7a76674": "(y_1, y_2, \\ldots, y_n)", "9a8267a82270dad7865ddabb98a33058": "{\\mathbb{R}}^n", "9a82835ccd17c1da0398346c7ce10dee": "N(M-1)", "9a829a5acd303106f1128fb64ff226fc": " v_F = \\frac{p_F}{m_e}", "9a82cc9dce9ab25749f1915b2047f746": "\n\\operatorname{Li}_s(e^\\mu) = \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} + \\sum_{k=0}^\\infty {\\zeta(s-k) \\over k!} \\,\\mu^k \\,.\n", "9a82e5031ce6c4441ada4ece0706a71b": " \\ddot y ", "9a82e91465a313c341db36192cb7b1e9": "M_p=\\sigma Z_p", "9a8390c0b86dc4f5b5a0ff72007c2b4f": "p(x)=e^{-x^2/2}, x\\in(-\\infty,\\infty) ", "9a83dc40dc3dc905aa08b1e43452920e": "\\varepsilon_{t-1}", "9a841903dbe338fe50072023448a7175": "\\begin{array}{c}\n\\bar{x}_j = x_i \\left(\\bar{\\mathbf{e}}_i\\cdot\\mathbf{e}_j \\right) = x_i\\cos\\theta_{ij}\\\\\n\\upharpoonleft\\downharpoonright\\\\\nx_j = \\bar{x}_i \\left( \\mathbf{e}_i\\cdot\\bar{\\mathbf{e}}_j \\right) = \\bar{x}_i\\cos\\theta_{ji}\n\\end{array}\n", "9a844e2c81b82889755dff8a81e55f15": "\\; k_1, \\ldots, k_l", "9a84abdf39bb067ae85af64c951b6bdf": " \\nabla^{(2)} \\cdot \\frac{\\partial f}{\\partial\\left(\\nabla^{(2)}\\rho\\right)} = \\sum_{\\alpha, \\beta = 1}^3 \\ \\frac {\\partial^{\\, 2} } {\\partial r_{\\alpha} \\, \\partial r_{\\beta} } \\ \\frac {\\partial f} {\\partial \\rho_{\\alpha \\beta} } \\qquad \\text{where} \\ \\ \\rho_{\\alpha \\beta} \\equiv \\frac {\\partial^{\\, 2}\\rho} {\\partial r_{\\alpha} \\, \\partial r_{\\beta} } \\ . ", "9a85893a590fc72d36f4d797398b17d8": "\n M(D) = atanh \\left( \\frac{D-4.9}{3} \\right)\n", "9a85902bab6b7ffb65fe92a9ba2bff8f": "\\exp(-(1/2)x-(9/4)\\tau) u(x,\\tau) ", "9a85b9c1ba485fa31a6b68c7f1eec218": "d(u,k)=\\frac{1}{2} [d(f,k)+d(g,k)-d(f,g)]", "9a85db1a3a8f0b083adfde8f1686edaa": "|\\Psi(x,t)|^2 \\sim \\mathcal{N}\\left( x_0 \\cos{(\\omega t)} , \\frac{\\hbar}{2 m \\Omega} \\left( \\cos^2{(\\omega t)} + \\frac{\\Omega^2}{\\omega^2} \\sin^2{(\\omega t)} \\right)\\right)", "9a85ef259212f2fc32fe05342a435c03": "g=\\rho(\\alpha))", "9a85f426d5506422d53fb48affe24fae": "E(T) = A + \\frac{a_1}{a_0} A", "9a865ee4cd82e99ea322fc93202c4ee5": "C=C_{in}+C_{out}", "9a86641362d8d75c8f0b97b1dd1e6e7d": " \\Phi(t)={1\\over t!}.", "9a866cfd83de10a003cea10d81696e45": "\n\\frac{1}{2} \\frac{1}{1-z} + \\frac{1}{2} (1+z)\n=\n1 + z + \\frac{1}{2} \\frac{z^2}{1-z}.", "9a869ee1428be6a8399b0b3a720b0b89": "A\\xrightarrow{f} B\\xrightarrow{g} C", "9a86a1d994565d1e239c704d8defb074": " \\sigma^{2}_{P} = \\sum^{n}_{i=1} x^{2}_{i} \\sigma^{2}_{i} + \\sum^{n}_{i=1} \\sum^{n}_{j=1, i \\neq j} x_i x_j \\sigma_{ij} ", "9a875d152c289cceafb114d9fe4e3c76": "\n \\limsup_{n\\to\\infty} \\frac{\\sqrt{n}}{\\ln^2 n} \\big\\| \\sqrt{n}(\\hat F_n-F) - G_{F,n}\\big\\|_\\infty < \\infty, \\quad \\text{a.s.}\n ", "9a87b939c6d654a114324bbda4bc4e67": " \\mathcal{M}_{g;k_1, \\dots, k_n} ", "9a88152b4c3d57956e422a034acb38fe": "z_0,\\dots,z_n", "9a885cd0d3f0d62b45125171081d9686": "S^1 0 ", "9a9631894bf68c7137af94bfa0722a76": "\\,\\!\\omega_r = \\omega_0\\sqrt{1-2\\zeta^2}", "9a96a5f59654172a7d84ffa428c26cfc": "\\sum_{k=1}^{\\infty}\\left\\vert\\left\\langle x,e_k\\right\\rangle \\right\\vert^2 \\le \\left\\Vert x\\right\\Vert^2 ", "9a96ac0bc340329e8b05d66ac04e2341": "\\operatorname{uxp}_a n, \\, a^{\\frac{n}{}} ", "9a96b7356cca569d83c088305a007e3f": "q(x)y-p(x)=0.", "9a9744c8e1006f1fa366a4219fb2f878": "p=p(r)Y_{10}\\,", "9a979263f58df90c5b858a62262cd178": " g(z) = f(z^{-1})^{-1}", "9a97b8aa4351ace810933c9412b39492": "\\mathbf{Q}_\\parallel = \\mathbf{G}_{hk} = h\\mathbf{a}^*_x + k\\mathbf{a}^*_y", "9a980905bbb5e88732ee1905ce6c1613": "p_{{\\mathrm{H}}_2}", "9a984a76657c068847e352d32752a168": "\\ln\\frac{1}{1-p}", "9a986c00612e1b03d304aa18424d1398": "(x-1)(x^4+x^3+x^2+x+1) \\rightarrow x^5-1.", "9a988e183389e85e050a42d40361b3e6": "\\log\\frac{g(z)-g(\\zeta)}{z-\\zeta} = -\\sum_{m,n>0}c_{nm}z^{-m}\\zeta^{-n}", "9a9898257646829b45316dd36ff9a019": "m: \\mathbb{R}^n \\to \\mathbb{C}", "9a990ac9198b834153ee3ed7ea6ac5ec": "Q^*", "9a990b65d0d20fecfa5a830ba972b5b2": "i 1", "9ac98637fe9c634be128b2dd869d94c4": "\\dot{Q} \\int_{r_1}^{r_2} \\frac{1}{r} \\mathrm{d}r = -2 k \\pi \\ell \\int_{T_1}^{T_2} \\mathrm{d}T", "9ac997010f68717e91934c4045b4227a": "\\mu_1,\\ldots,\\mu_c", "9ac9a5e9881810996e08e1226f561427": "\\vec{A}", "9aca147ae50164b8a2fbb38659b83f33": "\\xi = x - c t \\quad ; \\quad \\eta = x + c t", "9aca4a3640b6ef3713784bcf7a340953": " {D_i}", "9aca94cd286e2fa59805c31e7c6514a0": " \\frac{1}{|a|} F \\left ( {s \\over a} \\right )", "9acaa4def7c0bfea574c6c6199c80006": " \\Psi(r)", "9acac51f9d6288b462b79fe876cf554d": "\nx(t+\\Delta t)\n= x(t) + v(t) \\Delta t\n + \\frac{1}{6}\\Bigl( 4 a(t) - a(t - \\Delta t)\\Bigr)\\Delta t^2\n + O( \\Delta t^4)\n", "9acaf2bedfce075ddff5e54b335128a7": "m^{}_{}=1", "9acaf5609abce4c4d1d514aaaddec4d8": " \\int x\\phi(a+bx) \\, dx = -b^{-2}\\left (\\phi(a+bx) - a\\Phi(a+bx)\\right) + C ", "9acb4a8d9ee8e8aa8894765e69417a06": "A_{i j} = 0", "9acb8b235aaf9d57d9b87d5de13ba0bd": "\\frac{(\\mathbf{x}-\\mathbf{a})^{\\rm T}}{\\|\\mathbf{x}-\\mathbf{a}\\|}", "9acc32065e96b996a1db3d41ccb57062": "Happens(exit, 1)", "9acc609f3ccc8e519f03901f8cc4e897": "AA^H", "9acd49efa8fc290f8df3bdbd188d06d1": "\\overline{\\left [ \\left ( \\tau_s - \\bar{\\tau}_s \\right )^2 \\right ]}^{\\frac{1}{2}} = 1.4 \\sqrt{s},", "9acd5b885b4f79e936af6236470a00b1": "T_{2A}=W_{2A}(x)", "9acdc375c7242cc20961588642b4aa13": "G_1\\setminus G_0, G_2\\setminus G_1, G_3\\setminus G_2", "9acdce0ce62d1fba300e9838e7c3763e": "\\lim_{t\\to\\infty}\\|T(t)x\\|=0", "9acdedf41dadfb19974c7d484288b1d4": "T = \\int_\\mathbb{R} \\lambda\\, dE_\\lambda.", "9ace1bec15b783f2101c1897c86bcf9a": "\\mathbb{Q}(\\zeta_m),", "9ace75af2783424074c40e18f6b1c32b": " \\lim_{i \\to \\infty}\\sigma_i = 0 ", "9acecca72edc54b4f13773dd4eb7450c": "\\frac{1}{OSOI_{cas}} = \\frac{1}{G_{p,2}G_{p,3}G_{p,4}. . .G_{p,n}OSOI_{1}} + \\frac{1}{G_{p,3}G_{p,4}. . .G_{p,n}OSOI_{2}} + . . . + \\frac{1}{OSOI_{n}}", "9acf0830bc7bc7f7136ac22481b01377": " g = G \\left[ 1 + \\left(\\frac{5}{2} m - f\\right) \\sin^2 \\phi \\right] \\ , ", "9acf15c5742be702ed94d3b2886e5b63": "\\sum_n |\\langle x, e_n\\rangle|^2 = \\|x\\|^2.", "9acf9fdc431397aaebaed1b656f80ffc": "\\displaystyle J_{m+\\nu}(z) = J_\\nu(z)R_{m,\\nu}(z) - J_{\\nu-1}(z)R_{m-1,\\nu+1}(z) ", "9acfabcda71f6f55a9a4d582af702d23": "\\varphi^F_{d_1, d_2}(s) = \\frac{\\Gamma(\\frac{d_1+d_2}{2})}{\\Gamma(\\tfrac{d_2}{2})} U \\! \\left(\\frac{d_1}{2},1-\\frac{d_2}{2},-\\frac{d_2}{d_1} \\imath s \\right)", "9ad0278239c7eef23d5d5e7edfafcbad": "ds^2 = 0", "9ad0949ee4ed280e7e6f33aef514479d": "f(x)=q(x)(x-a) + r\\,.", "9ad099394c6c5cea2a84519998301c9b": "X_j", "9ad0a5a282950614fab7f88294cd815e": "a_{ij}=2 {(r_i,r_j)\\over (r_i,r_i)}", "9ad0dd5093a5ca8e3e6138d2b26f5cc0": "\\partial_{v}", "9ad1510c8e3b45fc87d2647ae5a598b6": "1/T_{b}", "9ad17d4ad03e772e68e98f802ab8befb": "\n \\nabla^2 \\varphi := \\boldsymbol{\\nabla} \\cdot (\\boldsymbol{\\nabla} \\varphi)\n", "9ad220ced005abe6da7fb2cf5edbd959": "dV = \\left|\\frac{\\partial (x,y,z)}{\\partial (u_1,u_2,u_3)}\\right|\\,du_1\\,du_2\\,du_3.", "9ad246898b7e6e95d11acf4ccbb175f3": "Y_{B}=\\frac{n_{B}(t)-n_{B}(t=0)}{n_{A}(t=0)+\\int_0^t\\dot{n}_{A,\\text{in}}(\\tau)d\\tau}\\left |\\frac{\\mu_k}{\\nu_p}\\right|=\\frac{72-0}{100+0}\\cdot\\frac{1}{1}=0.72=72%", "9ad24ec84bc451f8909b52b1d115a8ea": " 6\\sigma_{nX}=6\\sqrt{n}\\frac{\\Delta p}{6}=\\sqrt{n}\\Delta p ", "9ad271e8c972afd7642543fc109a0260": "g>G", "9ad31248e67991312093920529320aad": " X=\\begin{pmatrix} x & w \\\\ -\\overline{w} & -x \\end{pmatrix}.", "9ad354000cbbf06c0a1de124f47775f7": "\\operatorname{haversin}(\\theta)", "9ad3580f3b7bc02076c6beba5db35041": "''3x^3+4=28''", "9ad398d947c5ccbbf00c67628b1cd795": "v(n)=0", "9ad3c3a4062076b88503c83101784746": " P(X_i=x_i|\\partial_i) = \\frac{P(\\omega)}{\\sum_{\\omega'}P(\\omega')}, ", "9ad3ed6a17340417acb398f009042a89": "(1-\\alpha) E", "9ad3f8535c7328d895d1150513d52e95": " x_0 = 1.461632144968\\ldots", "9ad40b8e7a56d0dd0193b255b3008fb1": "a=0.51 m", "9ad4680e5d608e90637c9b2e2a868d2c": "t(y)= \\sqrt{ \\frac{ {y_0}^3 }{2\\mu} } \\left(\\sqrt{\\frac{y}{y_0}\\left(1-\\frac{y}{y_0}\\right)} + \\arccos{\\sqrt{\\frac{y}{y_0}}}\n \\right)", "9ad47b50990e27dfc2e451bae03fc9a2": "\\hat{\\mathbf {k}}", "9ad4dcbd8d279481f333ed3af0d198c6": "\\sin(M)", "9ad50b3d73e9117ed845666e2fdcef58": "10^{10^{10^{100}}}", "9ad50de7f8bafe1351c5fb8517a3ea98": "A = a_{ij} e_i \\otimes e^j", "9ad510947d7a4e6fc111e58e63a3df93": "\\frac{\\partial (\\mathbf{u} \\cdot \\mathbf{v})}{\\partial x} =", "9ad5246aa25ab3277f6fb8230cc7b669": "\\vec{v}(t_{n+1/2})", "9ad52fe3d43ab248de07e07ece73e20f": "{t}^2 = \\frac{30m}{sd^3l(1+l^2)}", "9ad55c751e3b2590c94e079e3b27d4f0": "A\\in \\mathcal{B}", "9ad57a59e38c1225abce13ee9aa2c522": "K_u", "9ad57d9bf30fd46529104cd3b6a77b5f": "\\delta_{2}^{0}(z) = \\delta_{2,1}^{0} \\times z^{1} = 3 \\times z^{1}", "9ad58690caa6567031184cc5b8edc3db": "H = n\\overline{u}y - nu\\overline{y}", "9ad5b8055038e4fdbf17c161187783ee": "\\triangleright", "9ad5f2969d87f1f1772ed4b2a8043620": "x_1=-\\tfrac{3}{4}", "9ad62b5cb080a349495f45501145dd6f": "\\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\frac{\\partial \\mathbf{f}_1}{\\partial \\mathbf{f}_2}\\cdot\\left(\\frac{\\partial \\mathbf{f}_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} \\right)", "9ad63a7d356e3affa42a692f2b2f3697": " n_{c}^{1/3} a_{H}^{*} = 0.26 \\pm 0.05", "9ad67e69aa7d1f14d5b276a299f3e0de": "A \\subseteq \\mathbb{N}", "9ad69835648676429fa53d468f064338": "n - l - 1", "9ad6d6802a508b863a5e79ab56c25f24": "\\xi\\in \\mathfrak g", "9ad75dc3ff12d1d67eee6a0fa83c4d85": "\\sum_{k=-n}^n r^k=r^{-n}\\cdot\\frac{1-r^{2n+1}}{1-r}.", "9ad78d94d9881a731b991a7c800c81ff": "\\mu(P) < \\varepsilon", "9ad7922466338fb2daac2e7bb233ba32": "\\scriptstyle G(n)=\\lfloor sn\\rfloor.", "9ad79ce8d3fafea958039b3f34cd4c59": "\\omega_{X}", "9ad865362280e944f81d791517fbf1b8": "\\operatorname{adj}(\\mathbf{A}) = \\begin{pmatrix} \\,\\,\\,{{d}} & \\!\\!{{-b}}\\\\ {{-c}} & {{a}} \\end{pmatrix}", "9ad8ae8188ab0e60c1ca4766792db163": "C_{\\mu\\nu\\alpha}=C_{\\mu\\alpha\\nu}=\\nabla^\\Gamma_\\mu g_{\\nu\\alpha}=\\partial_\\mu g_{\\nu\\alpha} +\\Gamma_{\\mu\\nu\\alpha} + \\Gamma_{\\mu\\alpha\\nu} ", "9ad8c4758fe263ea7de8e947bba58f40": "r=a+b \\cdot \\theta ", "9ad8ca243e620642d1504b8dda0fab16": "\\int\\limits_{-\\infty }^{\\infty }{{{P}_{\\theta }}f(u,\\xi )}.d\\xi =2\\pi {{\\left| f(u) \\right|}^{2}}", "9ad8f8602a7ac164a27bfdd8b2dc7b58": " S = ln\\frac{Re}{\\mathrm{1.816ln\\frac{1.1Re}{\\mathrm{ln(1+1.1Re)}}}}", "9ad8fa8917aa6d467adf82dc3bb63581": "\\frac{m_0}{m_1} = \\left[\\frac{1 + {\\frac{\\Delta v}{c}}}{1 - {\\frac{\\Delta v}{c}}}\\right]^{\\frac{c}{2v_e}}", "9ad93cf0b0b834ff8f997812cbf7f16f": " ms^{-3} ", "9ad9fe725ad6283a8b326094f9e5ef7d": "E_i=E_f+h\\nu\\,", "9ada2dfccbe9a52eca2e5f8f830f4540": "g(x; \\gamma_2) = f(x;\\; a=\\sqrt{2+6/\\gamma_2},\\; m=5/2+3/\\gamma_2). \\!", "9ada352cae128f26a38850b308fe869a": "m_2/(m_1+m_2)", "9ada9c0d5b7b549fd5a928a499a88892": "\\phi(x,t)=A\\exp[i(kx-\\omega t)]", "9adacc8e6497c212350609a6f0b9331d": "h=\\inf\\left\\{\\frac{|\\partial V|}{|V|}\\right\\},", "9adacea307b6bc057ae7a35031dfc4e5": "f_1, \\ldots, f_n ", "9adae05daa122146c53cfa3fe642bf39": " (A \\otimes B)(C \\otimes D) = AC \\otimes BD ", "9adb6a5638400de9516fc2feab7db7c4": "\\mathbf A\\cdot\\mathbf B = \\|\\mathbf A\\|\\,\\|\\mathbf B\\|", "9adb8bea5eba1b673687fc954b319846": " \\frac { \\text{density of object}} { \\text{density of fluid} } = \\frac { \\text{weight}} { \\text{weight of displaced fluid} } ", "9adb935e7b04e3101a8258702aba057a": "k=2\\pi /\\lambda ", "9adbc9e436a4f35bf85ce2922e1f8df1": "\\mu_k(\\lambda X) = \\lambda^k \\mu_k(X)", "9adbfd75d0b37946c7c9f2c2bb718505": "\\frac{2b^{2}}{a}", "9adbfee769ff8bcf320e461d6eac6b8e": " | s | = |s_{ \\pm } | = \\sqrt{ \\rho^2 +\\mu^2}, ", "9adc159bf521818aeb6aec46fa91d100": "\\tilde b_i", "9adc6376e6a5bc9db51cb7019fdd95b0": "k\\equiv 2, 3 \\pmod 4", "9adcb6151b1e6231af82b0bd1cff43a0": "\\delta_{n0}", "9adce64d0518b62039b61c161fd17647": "N(M)dM = \\frac{1}{\\sqrt{\\pi}}\\left(1+\\frac{n}{3}\\right)\\frac{\\bar{\\rho}}{M^2}\\left(\\frac{M}{M^*}\\right)^{\\left(3+n\\right)/6}\\exp\\left(-\\left(\\frac{M}{M^*}\\right)^{\\left(3+n\\right)/3}\\right)dM", "9add2e989cecbdec4d7ffa0d4093bbd5": "a=b=c=0", "9add56e7cbc1b6ea49d6f58c0cc676ef": "\\{a \\wedge b, \\neg a \\vee c, \\neg c \\vee \\neg b\\}", "9add7918342ca0cb15fc809283f32344": "(a - x)dx = b^2c", "9addccc4e32fcde7be63c2b28d08099c": "\\displaystyle{(Q(a)^{-1} a)a=a Q(a)^{-1} a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^2 =1,}", "9ade50714962b80d2bb4db84764ae262": "\\,\\operatorname{cr}(z_1,z_2,z_3,z_4)>1. ", "9ade661745ce1c0c4e3862e915ba54ee": "\\frac{CN}{(\\log N)^2}", "9ade8e3aca086483dde152886d05bdae": "\\begin{bmatrix}\nA & B \\\\\nC & D\n\\end{bmatrix}^{-1}\n=\n\\begin{bmatrix}\n (A - BD^{-1}C)^{-1} & -A^{-1}B(D - CA^{-1}B)^{-1} \\\\\n -D^{-1}C(A - BD^{-1}C)^{-1} & (D - CA^{-1}B)^{-1} \n\\end{bmatrix}\n=\n\\begin{bmatrix}\n S^{-1}_D & -A^{-1}BS^{-1}_A \\\\\n -D^{-1}CS^{-1}_D & S^{-1}_A\n\\end{bmatrix}\n,", "9ade9c4b02912684dedd8670fc394acd": "\ny''''(x) = - \\frac{d^2}{d x^2} \\left[ f(x) y(x) \\right]\n", "9adea4a410dd589fcc80e487f033cee2": "(x \\oplus a)\\{x \\mapsto b\\} = b \\oplus a \\equiv a \\oplus b", "9adee33ba0af824787487808de0aeebd": "H_k\\equiv\\frac{2K_uV}{\\mu}", "9adef3d8f2e6b42a96d76d946c158ebf": "\\overline{p + q}", "9adf2d9c458cbbf44d05f35ccdc463ef": "\\frac{d^2f}{dz^2} + \\left(\\tilde{a}z^2+\\tilde{b}z+\\tilde{c}\\right)f=0.", "9adf411ec3792a753fce64e915e1f218": "\\Pi_{a_1, ...,a_n}( R )", "9adf69337271d9ec50c4c6b0621bf776": "\\;^-T^{IJ}", "9adf87c024c5a512d504d541c1a5e8fb": "\n \\begin{align}\n D & = E^f\\int_w\\int_{-h-f}^{-h} z^2~\\mathrm{d}z\\,\\mathrm{d}y + E^c\\int_w\\int_{-h}^{h} z^2~\\mathrm{d}z\\,\\mathrm{d}y + \n E^f\\int_w\\int_{h}^{h+f} z^2~\\mathrm{d}z\\,\\mathrm{d}y \\\\\n & = \\frac{2}{3}E^ff^3 + \\frac{2}{3}E^ch^3 + 2E^ffh(f+h)~.\n \\end{align}\n", "9adfb724180f0998352cf46fa861c8e1": "\\left( X;\\ast\n,0\\right)", "9adfc336dce1352b652ddcd9aee71dc8": "eV_{ion} = E_g = \\frac{k M m}{r}", "9adfe44c6341485508286321f2ffe461": " r_2 = 1/g_{2,1} ", "9ae010dcce00b8c40d6cb29f7da07324": "\\mbox{Lift} = L = \\rho V \\tau", "9ae0b159612c668e2a35d933e59afc92": "\\sum_{k=1}^n k^m", "9ae0d35ed57fe63503d517533c90b5f5": " A = (a_{i,j})_{i=1\\ldots s;j=1\\ldots r} \\in \\R^{s\\times r}", "9ae0e97343500bf410b77a1bb504c1ed": "d^n C(u)/du^n", "9ae17d2c9bec4776bb16febe77f75597": "|dz|", "9ae2121e7d956c15a476e189d79b00d5": "W_\\alpha \\equiv \\overline{D}^2 D_\\alpha V", "9ae231f702519ddccb68ae8151befd8d": "p^\\omega s_1 s_2 \\cdots s_n p^\\omega", "9ae23b84b86293af9608a7625e32cce6": "\\nabla\\cdot\\left( c_p\\, c_g\\, \\nabla \\eta \\right)\\, +\\, k^2\\, c_p\\, c_g\\, \\eta\\, =\\, 0,", "9ae276e23d6918886f91dd0cfcc693fe": " \\mu", "9ae29d5e1d1f4abbaba4aec8882182de": " \\sin(x)=\\frac{i}{2} \\left(e^{-ix}-e^{ix} \\right) ", "9ae2a41a6f45ad0fcdfebf0e251c018a": "X = \\{v_1, v_2, v_3, v_4, v_5, v_6, v_7\\}", "9ae2f7ac679c829b31a55370cbfa9a95": " \\{ \\ln A_1, \\ln A_2, \\dots , \\ln A_n \\} ", "9ae32176247a480002dfa1b8eaa56033": "h_{t} = \\underset{h_{t} \\in \\mathcal{H}}{\\operatorname{argmax}} \\; \\left\\vert 0.5 - \\epsilon_{t}\\right\\vert", "9ae393820e4a13300437a074b2b681bc": " \\gamma\\ ", "9ae3a0433edeac5db418f4e8c86b5f4c": "\\sigma\\eta ", "9ae3a41df502267f067f86f037823c12": "P(X) \\,", "9ae3b30b216f583b59ad23c47e1910d2": "x_n \\approx -n + \\frac{1}{\\pi}\\arctan\\left(\\frac{\\pi}{\\ln n + \\frac{1}{8n}}\\right)\\qquad n \\ge 1", "9ae4473460caa3b1b0cdf1edfab12fee": "l \\in \\mathbf{Y}", "9ae44cccf6874876fab30a1f1dda387f": "{u}(r,\\phi,z)=\\frac{C^{LG}_{lp}}{w(z)}\\left(\\frac{r \\sqrt{2}}{w(z)}\\right)^{|l|}\\exp\\left(-\\frac{r^2}{w^2(z)}\\right)L_p^{|l|} \\left(\\frac{2r^2}{w^2(z)}\\right) \n\\exp\\left( i k \\frac{r^2}{2 R(z)}\\right)\\exp(i l \\phi)\\exp\\left[i(2p+|l|+1)\\zeta(z)\\right],\n", "9ae45b0461f86c59d8a2799a646ec8b4": " \\prod_{r=1}^7 \\Gamma(\\tfrac{r}{8}) = 4\\sqrt{\\pi^7} \\approx 219.8287780169572636207", "9ae4e9921cb0578a92b01359f80febc4": "\\sigma_y^2(n\\tau_0, N) = \\text{AVAR}(n\\tau_0, N) = \\frac{1}{2n^2\\tau_0^2(N-2n)} \\sum_{i=0}^{N-2n-1}(x_{i+2n}-2x_{i+n}+x_i)^2", "9ae4fbd38baa1ba02be31ec7ef11573d": "\n \\varepsilon_{xx} = \\frac{\\partial u_x}{\\partial x} = -z~\\frac{\\partial \\varphi}{\\partial x} ~;~~\n \\varepsilon_{xz} = \\frac{1}{2}\\left(\\frac{\\partial u_x}{\\partial z}+\\frac{\\partial u_z}{\\partial x}\\right)\n = \\frac{1}{2}\\left(-\\varphi + \\frac{\\partial w}{\\partial x}\\right)\n", "9ae5a6e5b96f4d4fffe454edf52c827f": "m_h \\leftarrow I_{W^{\\ast}}^{Z}(UOWHash^{\\prime\\prime}(\\tilde{k},M))", "9ae69178555d769ebb96fdf1c8c2f876": "\\sum_{n\\ge 1} f(n) x^n", "9ae753b5f919cc6a2ebd4f8fcaff9c1c": "\\theta (x) = 0", "9ae758e211de4e5228c2d3300143efa4": "z = (-RT/g) \\ln (P/P_0)", "9ae7ac77583c0caaef0c241c0597bf71": "\\phi_{a}\n\\left(\\mathbf{r}\\right)", "9ae7f8739cbb5f3fbfcc2d8f5c521b90": " m^2 ", "9ae81726cf0b896d6202dba28d412e3a": "T_x", "9ae8c8de3fc6062129b4ddd66cc954dc": "x_{11}'", "9ae8e6d6e64efedfde0c32f036c4b3da": "\\pi_z", "9ae8ff235005c8df4e73ab2ab74ede81": "\\frac{\\partial^2I(x)}{\\partial x^2} + \\omega^2 LC\\cdot I(x)=0.", "9ae90addfdf92704a9b1e4a77ea5012a": "T_{s,t} ", "9ae929de2ffde2acd6275e166769a1a6": "10000*100", "9ae92f2fa189ce9f7355676f41386862": "\\scriptstyle \\mathcal B", "9ae9bf90ad5d4135f55154eb5b25837a": "L_{f, P} \\le L_{f, P'} . \\,\\!", "9ae9c5644c1d4374b241634eb71df99e": "X^{ BS}", "9ae9ce407ef63736b5c2e6535ffa701b": " p_F = \\sqrt{2 m_e E_F} ", "9ae9d8f34cc53aed50b268f2392fced3": "B \\subset C", "9ae9de23afcd5e060b4fc85d990cd09a": "\\xi, \\rho, \\theta", "9ae9eb8a0c584e936063061cf4513d47": " Z_{12} ", "9aea2725b4be9316d3e0458129167a2f": "\\varprojlim \\pi_S^0(BG^{(k)}_+) \\to \\hat{A}(G).", "9aea851164eb66a05cfbe092bc924b13": "\\{M_i\\}", "9aea89a6543f927aedd4615ebf4bdfd9": "N = {1\\over8}{4\\over3}\\pi R^3\\,,", "9aea98e8e8edc0c424d5f7b6f88f8796": "\\frac{\\Delta F(P)}{\\Delta P}=\\frac{F(P+\\Delta P)-F(P)}{\\Delta P}=\\frac{\\nabla F(P+\\Delta P)}{\\Delta P}.\\,\\!", "9aeaaf090263d80940bb8541e2086550": "X \\longleftarrow U \\longrightarrow V \\longleftarrow Y", "9aeaebce84f43c2bcdf63064f86f3384": "\\scriptstyle\\vec{J}", "9aeafade2246d54f22068937884f8d37": "P*Q", "9aeb453587864c6457d5f5f8020381c9": " \\displaystyle (1+t)^z(1-t)^{-z}=\\sum_n g_n(z)t^n. ", "9aeb8105234e9e60d9f69d246d843c28": "\\scriptstyle \\tilde x'", "9aeb97bb275269ae1c5616d17ad1c906": "[X]^\\lambda=\\{Y\\subset X:|Y|=\\lambda\\}", "9aeb99cc5dfb8ea53538fde2e4a38480": "\\left(Im(\\Gamma(\\omega)) < 0\\right)", "9aec12c476b93db2648bf682b79cd05f": " t = a T + e x \\,", "9aec4668c6b2f6c7b8a66ddc744968a8": " 2d+1 > n ", "9aeca2e65505581a41677fd4e35f51de": " \\beta \\times \\{x\\} \\subseteq f[B]", "9aecb01c8c8cfab3e430fdb93f6a46ae": "x^y + y^x", "9aecbe75b4bfdf30495619b183d1b53b": "C_a O_2 = 1.36 * Hgb * \\frac{S_a O_2}{100} + 0.0031 * P_a O_2", "9aecddabf1b49f246085d15809487705": "-q_1\\ ,\\ -q_2\\ ,\\ -q_3\\ ,\\ -q_4\\ ", "9aece19e3f8d854b5603164919d06c79": "\\mathbf{x} = \\mathbf{y}", "9aed363aa94c6bfcd4f6782bd6a93e1d": "S_{p}=\\frac{\\dot{n}_{p,\\text{out}}-\\dot{n}_{p,\\text{in}}}{\\dot{n}_{k,\\text{in}}-n_{k,\\text{out}}}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "9aed7448c1dd15d11e7b6caa0e90c5f9": "\\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 \\geq 0", "9aed8220df6605cf66eb57c55b067d38": "\\forall x \\in S : \\phi(x)", "9aee7ca0c800ab284b57749cf7e90ef9": "2(x^2+3x-2)=2x^2+6x-4", "9aee9eab3ee9b29e6d371c446d980799": "y_t = y_{t-1} + c + u_t", "9aeea006c0c7dd257b808087657e5f4c": "\n\\begin{array}{c}\n\\left[ {{\\begin{array}{*{20}c}\n {\\Delta x_1 } \\\\\n {\\Delta x_2 } \\\\\n {\\Delta x_3 } \\\\\n \\vdots \\\\\n {\\Delta x_{n - 1} } \\\\\n {\\Delta T} \\\\\n\\end{array} }} \\right] = \\left[ {{\\begin{array}{*{20}c}\n {1 / x_1 } & 0 & 0 & 0 & 0 & { - \\frac{H_1^\\circ }{RT^{2}}} \\\\\n 0 & {1 / x_2 } & 0 & 0 & 0 & { - \\frac{H_2^\\circ }{RT^{2}}} \\\\\n 0 & 0 & {1 / x_3 } & 0 & 0 & { - \\frac{H_3^\\circ }{RT^{2}}} \\\\\n 0 & 0 & 0 & \\ddots & 0 & { - \\frac{H_4^\\circ }{RT^{2}}} \\\\\n 0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \\frac{H_{n - 1}^\\circ }{RT^{2}}}\n\\\\\n {\\frac{ - 1}{1 - \\sum\\limits_{1 = 1}^{n - 1} {x_i } }} & {\\frac{ - 1}{1 -\n\\sum\\limits_{1 = 1}^{n - 1} {x_i } }} & {\\frac{ - 1}{1 -\n\\sum\\limits_{1 = 1}^{n - 1} {x_i } }} & {\\frac{ - 1}{1 -\n\\sum\\limits_{1 = 1}^{n - 1} {x_i } }} & {\\frac{ - 1}{1 -\n\\sum\\limits_{1 = 1}^{n - 1} {x_i } }} & { -\n\\frac{H_n^\\circ }{RT^{2}}} \\\\\n\\end{array} }} \\right]^{ - 1}\n\n.\\left[ {{\\begin{array}{*{20}c}\n {\\ln x_1 + \\frac{H_1 ^\\circ }{RT} - \\frac{H_1^\\circ }{RT_1^\\circ }}\n\\\\\n {\\ln x_2 + \\frac{H_2 ^\\circ }{RT} - \\frac{H_2^\\circ }{RT_2^\\circ }}\n\\\\\n {\\ln x_3 + \\frac{H_3 ^\\circ }{RT} - \\frac{H_3^\\circ }{RT_3^\\circ }}\n\\\\\n \\vdots \\\\\n {\\ln x_{n - 1} + \\frac{H_{n - 1} ^\\circ }{RT} - \\frac{H_{n - 1}^\\circ\n}{RT_{n - 1i}^\\circ }} \\\\\n {\\ln \\left( {1 - \\sum\\limits_{i = 1}^{n - 1} {x_i } } \\right) + \\frac{H_n\n^\\circ }{RT} - \\frac{H_n^\\circ }{RT_n^\\circ }} \\\\\n\\end{array} }} \\right]\n \\end{array}\n", "9aeee312dc6b18525d65a11768fd54d6": "Q(x) = x^tx.\\,", "9aef074aee5230beca978843953e5a89": "(e^{-x}+1)^{-1},\\,\\,\\lambda\\,=\\,0", "9aef0f5b24097ccc29c7a77bbf6a03fb": "\\sigma = \\gamma_0 \\left( \\frac{\\Delta H^*}{N_A^{1/3}V_m^{2/3}}\\right),", "9aef28aff30b773183b6fec3e12b2294": "c \\rho\\left[\\frac{\\partial T(x,t)}{\\partial t} + \\epsilon u \\frac{\\partial T(x,t)}{\\partial x}\\right]=\\lambda \\frac{\\partial^2 T(x,t)}{\\partial x^2} + Q(x,t)", "9aef698aecf22681ee6fd09a29c9d108": "(b,a) \\in R", "9aef902b3cdc0dd7328b67f65eec3728": "\\Gamma_{ij,k}^{(1)}=0", "9aefd77cfbce82d00272d86d16b3ee20": " G_0 = \\frac { \\beta i_B} {i_S} \\ . ", "9aefe250fdccd53d9b61778213593119": "\\theta \\vdash \\phi", "9af01451fcabdf95749874ee3387ebac": "F_{0.5}", "9af06930bf3744ff6017f3220b4a56e0": "\\rho = \\sum_i \\varrho_i \\,", "9af0c5260ec8aa4b5e59452b5f3f1c95": "\\mathbf{n}_2", "9af0ce0b706274fea50930069bdaa62d": "\\mathbf{C}=\\begin{bmatrix}\nc(1,1) & \\cdots & c(1,n) &c(1,n+1)& \\cdots & c(1,m)\\\\\nc(2,1) & \\cdots & c(2,n) &c(2,n+2)& \\cdots & c(2,m)\\\\\n & \\cdots & & & \\cdots & \\\\\nc(n,1) & \\cdots & c(n,n) &c(n,n+1)& \\cdots &c(n,m)\\end{bmatrix}\n", "9af1234ea1763c9333e85a9ad5506de9": "(L_\\mathrm{W})", "9af1429db6d52379b9bcc49201df6f61": "a(y)", "9af1515820309180d4ee2157f6f85cfd": "d_p f : T_p M \\to T_{f(p)}N\\,", "9af15d6427d1f60fcc33f1505eb7e17d": "22A,22B\\;", "9af180c0f12b15b7b6d22815fe31177c": "TL_3(\\delta)", "9af19780925c70d331a74c8bf17c0fba": "\\prod_{i=1}^n f(x_i)", "9af1a4578c04a787c95756882538effc": " (u_i, u_{(i+1)\\bmod L(p)}) \\in E~; \\quad u_i \\neq u_{(i+2) \\bmod L(p)~}, ", "9af1d94743d95aea93374a631b32a3c0": " \\begin{cases}\n \\lambda_0 \\nabla f(x^*) = \\sum\\limits_{i\\in \\mathcal{I}'} \\lambda_i \\nabla g_i(x^*) + \\sum\\limits_{i\\in \\mathcal{E}} \\lambda_i \\nabla h_i (x^*) \\\\[10pt]\n \\lambda_i \\ge 0,\\ i\\in \\mathcal{I}' \\\\[10pt]\n \\exists i\\in \\left( \\{0,1,\\ldots ,n\\} \\backslash \\mathcal{I} \\right) \\left( \\lambda_i \\ne 0 \\right)\n\\end{cases} ", "9af22a9e9484c8f97c9d83e406fee17f": "\\lambda^\\theta", "9af24d5c7b30f9b77cd6f20b8279bed1": "ke^{-\\alpha}(\\alpha-1)", "9af2fb94e0c0d6753e5e896119cc6b8f": " ~\\tau_t=\\theta_t z_t ~", "9af3080a875b8efec205404a75bb8664": "Q_n^m(z)", "9af31041d7b068d8c0be5211b6471ba5": "\n\\left[ \\bar{X}_{i},\\bar{X}_{j}\\right] = 0\\ \\ \\ \\ \\ \\forall\ni,j,", "9af3107a066f6b0defb1cafc0499f6ed": "CW", "9af3147c5d67ec835fe4c7c7b4f26f8c": "x(x+b) = H(mU) \\mod n", "9af3530472c1b4092ce516cb2d8fb999": "\\int_a^b f(x) \\, dx\\approx \n\\frac{h}{3}\\bigg[f(x_0)+2\\sum_{j=1}^{n/2-1}f(x_{2j})+\n4\\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_n)\n\\bigg],", "9af36b9865db766d7a32ec7c742aaf72": "\\textstyle{\\int_M \\omega = \\int_N \\omega},", "9af40e54c3fa19798d698d9564a8062c": "\n\\frac{\\partial}{\\partial t}\\left[\\phi\\left(\\frac{R_S S_o}{B_o}+\\frac{S_g}{B_g}\\right)\\right]\n+\\nabla\\cdot\\left(\n\\frac{R_S}{B_o}\\vec u_o+\\frac{1}{B_g}\\vec u_g\\right)= 0\n", "9af4cecd719b863d39730b9f27e785a2": " C(s + \\epsilon\\,) = C(s) + \\epsilon\\,C'(s) + {1/2}\\,\\epsilon^2\\, C''(s) + ...", "9af51b0e1fba700fc7cba76e773a3efb": "\\bigvee L=a_1\\lor\\cdots\\lor a_n", "9af53f12c82c086e97b04e342c40351c": "\\begin{align}\n&2^6 \\cdot 3(-3^2+3 \\cdot 19 \\cdot 1^2) = 96^2\\\\\n&2^6 \\cdot 3(-39^2+3 \\cdot 43 \\cdot 7^2) = 960^2\\\\\n&2^6 \\cdot 3(-219^2+3 \\cdot 67 \\cdot 31^2) = 5280^2\\\\\n&2^6 \\cdot 3(-26679^2+3 \\cdot 163 \\cdot 2413^2) = 640320^2\n\\end{align}\n", "9af554154ebe16e2be1da18284e0c564": "i,j,k,", "9af57f6b7f53e9630e8340b964613a57": " T_0 \\cap T_1 \\cap T_2 \\cap \\ldots \\cap T_k \\cap \\ldots ", "9af59a2f1d30bcdd5e908a30a45df8fd": " S(\\rho^{12}) \\geq |S(\\rho^1) -S(\\rho^2)| ", "9af5ad7ab23a787796c6cf322e3740d6": "\\Bbb{Q}(y)", "9af5f145c6712331caa82a6ebddf833c": "2^{2^m}+1", "9af62b2d36a3757592c6f7ba550e22d4": "\\textstyle H(X_r|Y_r)", "9af658598a45d6b5f62040a38be94fc3": "[1].(z_1,z_2):=(e^{2\\pi i/p} \\cdot z_1, e^{2\\pi i q/p}\\cdot z_2)", "9af659b68abc19b0f53cf223f1a1250e": "L\\geq 0", "9af682a9da92c870c04adff8495ad5e6": "M_{g, n}^{J, \\nu}(X, A) \\to \\bar M_{g, n} \\times X^n,", "9af6846a59bde43b093edbbfa059e7c4": " (\\mathbf{v} \\times \\mathbf{B})_{t1} = (\\mathbf{v} \\times \\mathbf{B})_{t2}, ", "9af6a82c6974c4d09d4603a51b65011e": "\\zeta(z;\\Lambda)=\\frac{\\sigma'(z;\\Lambda)}{\\sigma(z;\\Lambda)}=\\frac{1}{z}+\\sum_{w\\in\\Lambda^{*}}\\left( \\frac{1}{z-w}+\\frac{1}{w}+\\frac{z}{w^2}\\right).", "9af6c402d08eebc5f0aafc5a034e162a": "\\mathbf F = m \\mathbf a_{\\rm cm}", "9af6e5791381a80812a050c79d54715b": "W(T)=V(T)\\setminus V(h)", "9af73d8ebb6c6921a619941b0c7f2273": "\\mathbf{L}=-i\\mathbf{x}\\times\\mathbf{\\nabla}", "9af7ec9418935ccd779b12f8764cd767": "r_\\text{max}", "9af7f617c3659e3cd4e0538a492357e7": "P(x_1,\\ldots,x_n)\\,\\!", "9af8842a45b3df52337f36fb1666481b": "\\{\\mathcal{L}^*g\\}(s) = \\int_{0^-}^{\\infty} \\mathrm{e}^{-sx}\\,dg(x).", "9af8894773aee5cee8ed12f2b6a4fa1a": " \\gamma_k = 1 - (1 - \\gamma_1)^{(b^{k-1})} ", "9af89d36693a35589991b3c83f7f942b": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{hacovercosin}(x) = \\frac{\\cos{x}}{2}", "9af90abd6a20fe471937393e28017f3f": "f(x_1, \\dots, x_i)", "9af990e43472d187deed2c269fdf63a5": " d_{p-1-i}= -d_i \\quad (\\bmod\\,p), \\quad i=0,1,\\ldots p/2-1", "9afa32775280426fbb8bbfd0be65be5d": "V_\\text{P} = \\frac{E_\\text{P}}{q_\\text{P}} = \\sqrt{\\frac{c^4}{G 4 \\pi \\epsilon_0}} \\approx", "9afa37eec229510183261b4523145006": "{\\rho}{\\omega}V\\,\\!", "9afa44650b6f178255cbbe2f4d89f87d": "\\sqrt{-7}", "9afa56642a83ec0084662b65c7471c28": " ma \\equiv p/t \\,\\!", "9afa9cfc23f447e8fcc20b9b626b1608": "\\alpha_5 = {{3\\alpha_0 + 4\\alpha_1} \\over 7}", "9afad86e995b96a2e5c30af36d5cdcfa": "\\mathfrak{P}^{110}", "9afb16f88c7f1ca4146a7e0d332e27f8": "P\\to P/H\\to X", "9afb279cb034f20494ad52bdac732ad1": "\\frac{7 \\cdot 13}{2 \\cdot 3 \\cdot 11}, \\frac{11}{13}", "9afb571b09921cab3c383223289b6fda": "H = \\begin{pmatrix}H_{AA} & H_{AB}\\\\H_{BA} & H_{BB}\\end{pmatrix} = M-\\frac i2\\Gamma, ", "9afb84b73296212f1c3150cd0f6416a2": "x=-7z-1\\;\\;\\;\\;\\text{and}\\;\\;\\;\\;y=3z+2\\text{.}", "9afbc0ce774ba2f498cfb345bf6561c1": "\\langle A,B \\rangle_\\mathrm{HS} = \\operatorname{tr} (A^*B)\n= \\sum_{i} \\langle Ae_i, Be_i \\rangle.", "9afbfc53ac57d320543f1e1ad26d3453": "\\Lambda_{\\mu \\nu} = g_{\\mu \\sigma} \\Lambda^\\sigma{}_{\\nu}.", "9afc2d3ce410b9fd06cb10a776aa4f8b": "\\mathrm{Ann}_R(S)=\\{a\\in A\\mid \\forall s\\in S, as=0 \\}\\,", "9afc59777a3b2b1d9a86a6dd0f4b5766": "P^{-1}AP =\n\\begin{bmatrix}\n0 & -1 & 0 \\\\\n2 & 0 & 1 \\\\\n-1 & 1 & 0 \\end{bmatrix}\n\\begin{bmatrix}\n1 & 2 & 0 \\\\\n0 & 3 & 0 \\\\\n2 & -4 & 2 \\end{bmatrix}\n\\begin{bmatrix}\n-1 & 0 & -1 \\\\\n-1 & 0 & 0 \\\\\n2 & 1 & 2 \\end{bmatrix} =\n\\begin{bmatrix}\n3 & 0 & 0 \\\\\n0 & 2 & 0 \\\\\n0 & 0 & 1\\end{bmatrix}.", "9afc85f02b7ca50787136e8bb3fd0cd3": "\n f\\left(k;\\mu,\\mu\\right) = e^{-2\\mu}I_{|k|}(2\\mu).\n ", "9afcf6dff4c30df1ac4b830b6a589ed0": "Y \\thicksim \\text{Inv-}\\chi^2(\\nu)", "9afcffdf0affb58787999be5a30989d4": "\\Gamma\\varphi(t)-y_0", "9afd56f40d4bad821211881bd495d143": "E_a\\,", "9afe22e3bc8f8fc9385e61afe87264e5": "2^i3^j", "9afe296bafe6430b8a025e696b3210bf": "-\\ell i\\,", "9afe2a3afdb626e873afc071f3401bd8": "\\mathbf{u}^\\infty", "9afe9318fda74aef47fa0a70e8fb07e4": "\\delta p = \\delta\\left(\\sqrt{2 m U}\\right) = \\sqrt{\\frac{m}{2 U}}\\delta U", "9aff112aed60246be79e1373ec0c2c41": "E_{kl}(r_{k}, r_{l})", "9affcab55e4aa002efc90e45fe664b99": "\\phi \\;", "9affdddb7064f043d58cbb079448afcc": "e_k = {\\partial \\over \\partial x^k}.", "9b005538b624e284abd44229ddf339a3": "\\mathrm{i}\\hbar\\frac{\\partial}{\\partial t}\\psi(x,t) = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2}\\psi(x,t) +V(x)\\psi(x,t),", "9b0061628fd9e715fdb5535592d8e59c": " e^{v\\hat{a}^\\dagger - v^*\\hat{a}} = e^{v\\hat{a}^\\dagger} e^{-v^*\\hat{a}} e^{-|v|^{2}/2} ,", "9b00a25872bbb9e6619de2f430f4e66b": " \\mu_{l} ", "9b00d357c3035f572ccecc90650d63cc": "[M-H]^-", "9b017020b9d48d433ec90c548e6b5c70": "-d", "9b01ad7c66dadf3af8945e5589f6d46f": "\\begin{align}\n \\sum_{j \\neq i} \\pi_i q'_{ij} &= \\sum_{j \\neq i} q_{ij} \\quad \\forall i\\in S\\\\\n \\pi_i q_{ij} &= \\pi_jq_{ji}' \\quad \\forall i,j \\in S\n\\end{align}", "9b01ddf91db6f027b1f555fffb456232": "\\pi=\\frac{5\\sqrt{5}}{2\\sqrt{3}Z} \\!", "9b01e11d7dd64184e18cf6b324a2c382": " E_\\ell = {\\hbar^2 \\over 2I} \\ell \\left (\\ell+1\\right )", "9b0201f3a6270ff23b9cdec5f6086255": "\\mathbf{F}\\left(\\mathbf{r}\\right)=-\\frac{1}{4\\pi}\\left[-\\boldsymbol{\\nabla}\\left(\n-\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n+\\int_{V}\\boldsymbol{\\nabla}'\\cdot\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n\\right)-\\boldsymbol{\\nabla}\\times\\left(\n\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\times\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\int_{V}\\boldsymbol{\\nabla}'\\times\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n\\right)\\right].", "9b022c4c2437fe2641fc688a2819eff0": " A \\circ B \\in {\\mathbb R}^{m \\times n} ", "9b02685796740d6701738359933eb1bc": "\\scriptstyle u=\\sqrt{y}", "9b02862675235f9d394e5180069df4cf": "d=\\tfrac{2km(k^2-m^2)}{k^2+m^2}, \\, ", "9b02a0e63270fa6c45611f597c8b3de3": "\\nabla^4 \\left({1\\over r}\\right)= {3(15-8n+n^2)\\over r^5}", "9b02ad213086836daa54f2723165847a": "{\\Theta}(n\\log^2 n)", "9b030341debfba8e6775ae79b195d2dc": "\\boldsymbol\\Sigma_{22}", "9b031e4f44b8e398682d1e92fefc19e7": "\\Sigma(\\vec{\\xi})=\\int \\rho(\\vec{\\xi},z) dz ", "9b034ff0f5e74cfdd636b18776f4c097": "\\int_0^\\pi \\cos mx \\cos nxdx=\\begin{cases}\n0 & \\text{if } m\\neq n \\\\ \n\\pi/2 & \\text{if } m=n \n\\end{cases}\n\\ \\ m,n \\text{ integers}", "9b037c4882b4c780c823d1d6153df845": "s_n = \\sqrt{\\frac{1}{n} \\sum_{i=1}^n (X_i - \\overline{X})^2}\\,,", "9b0386b785624023fb61a48c1ed73f37": "\\Z/N\\Z", "9b0491e9cfdb182e27910f7b0c9dae9c": " n = t | m - T |^{ -2 } p q ", "9b04a4d6d39081e48dbec2761915695f": " \\left( b_{k} \\right)", "9b04e56144ebc06fd5cd7bbfe441bd5a": "\\mathbf{\\Psi}(\\mathbf{x})", "9b052bb5a957fed1157116b4a989dbd2": "\\hat\\gamma(h)", "9b055025af03ff824e76f211bb9743b4": "u = 0.5 + \\frac{\\arctan2(d_z, d_x)}{2\\pi}", "9b056ac51c4d7efb8e5e4cbf5b57122d": "(\\hat{c}_v=3/2)", "9b058fd44556933751b476412453a154": "f(x) = 2x", "9b05c31dbb855d3152205dbf2310bfe9": "(\\hat{\\bold{x}}_1, \\hat{\\bold{x}}_2, \\hat{\\bold{x}}_3)", "9b05c8b9f1bc7a91d125a3fffd993070": "\\scriptstyle \\mathcal{O}_c(\\Omega)", "9b05de73d43f8c4ec1110c6bcc5312bc": "km", "9b05e30b531d50ce65459912b0b5418b": "T_{em}", "9b05f783636e1ff9ccac0e93fa28b7a8": "P(H_0|D) = \\frac{P(D|H_0)P(H_0)}{\\displaystyle\\sum_i P(D|H_i)P(H_i)} .", "9b05f9214dacbe97ac954e351f7df46b": "ds^{2}", "9b0605d42baad4bbd5724ccf208157b7": "r \\frac{\\partial u}{\\partial r} = r \\frac{\\partial u}{\\partial x}\\frac{\\partial x}{\\partial r} + r \\frac{\\partial u}{\\partial y}\\frac{\\partial y}{\\partial r},", "9b0609668022a5271c68b48e56518e15": "f[x_0,\\ldots,x_n] = \\frac{1}{n!} \\int_{x_0}^{x_n} f^{(n)}(t)B_{n-1}(t) \\, dt", "9b061391fe675fd221e8bbdf2e965724": "i\\overline{W} = W^{\\bot\\,\\bot}.", "9b06e5382baf8f73380fea9360ced858": "\\mathbf{C}_{1,1} = \\mathbf{M}_{1} + \\mathbf{M}_{4} - \\mathbf{M}_{5} + \\mathbf{M}_{7}", "9b0792da44a7d0fe9d3e8d2a8c7b1871": "= \\mathcal{L}_{V^{1}}(du^{\\alpha} - u_{i}^{\\alpha}dx^{i}) ", "9b07b873893752e53b5c5f87a80b5dee": "CandS_{k+1} := \\emptyset", "9b07f53c6d53c7452b4aca2b7207585c": "\\chi_{{\\scriptscriptstyle \\rm{Sym}^2} \\rho}(g) = \\frac{1}{2} \\left[ \\left(\\chi_\\rho (g) \\right)^2 + \\chi_\\rho (g^2) \\right]", "9b0894f15452a9a3c94c6955cf0fe2dd": " P \\in \\mathcal{P}, v_P ", "9b08a78513c5a8392d3a814fc2538b54": "\\mathrm{TAS}={a_0}\\sqrt{{5T\\over T_0}\\left[\\left(\\frac{q_c}{P}+1\\right)^\\frac{2}{7}-1\\right]}", "9b08c195dd673dd6b5181a078b537c48": "N = \\frac{e^{U(0, q]}}{c}", "9b08ca960795aed1448fb53751f3db7f": "{{2n}\\choose k} ", "9b08e951e8ccc63f28640f7f0a5150ef": "\\nabla F(x, y, z)", "9b090e65b5adc0fd2d81ddfe15d531c1": "A\\colon\\mathcal{D}(A)\\subset X\\to Y", "9b097250ca7637f8a789751dc5e627db": " A^\\alpha B_\\alpha \\equiv \\sum_\\alpha A^{\\alpha}B_\\alpha \\,.", "9b097295d8ab785433d9fbbdc0dcb54b": "C_m", "9b099b382eb59bdeac7bb9b613f21237": "|a| \\leq x", "9b099cbcac450ab347c9799376fa96ae": "x(t) = e^{-\\alpha t} \\left[\\cos{(\\omega t)}+\\left(\\frac{\\beta-\\alpha}{\\omega}\\right)\\sin{(\\omega t)}\\right]u(t).", "9b09e7bddda7288e88ea495873565bbb": "\n\\begin{align}\nf_\\text{e1,e2}\n& = f_\\text{P1,P2} \\left( 1 - \\frac{\\vec v \\ast \\vec e_\\text{e}}{c} \\right) \\\\\n& = f_\\text{1,2} \\left( 1 - \\frac{\\vec v \\ast \\vec e_\\text{1,2}}{c} \\right) \\left( 1 - \\frac{\\vec v \\ast \\vec e_\\text{e}}{c} \\right)\n\\end{align}\n", "9b0a0200327974f3c102de3270094f72": " \\nabla^2 \\phi = 0 \\; .", "9b0a0d82efa7801ed16f080ef80fee51": " \\{a \\in \\mathbb{R}: \\mu(\\{x: f(x) > a\\}) = 0\\} ", "9b0a1432c18dff4054f3cd37a1436d18": "G_{total} = \\frac{G_1 G_2}{G_1+G_2}.", "9b0a2ff5161387ccd80dc731add1602c": " a^2 - b^2 ", "9b0a33caac0facd169dbe42d70d171fc": "A_{p}", "9b0a49688b786596c4343e3a653d0d93": "\\textstyle[n, k]", "9b0a7ee9fbcfd50ec5dd05d804a82db9": "\\mathbf{r'}=\\boldsymbol{\\beta}c_0+\\boldsymbol{\\delta}sc_1", "9b0a88edf0e4249408d397f7cf3e8b0b": "\\mathbf{\\sigma}_{ij}= - p \\delta_{ij} + \\mu\\left(\\frac{\\partial v_i}{\\partial x_j}+\\frac{\\partial v_j}{\\partial x_i} \\right)", "9b0a952763bee2461c1a0234db3defff": "X_H", "9b0acd32492265c28575c7b6d0ff8186": "\n\\chi(\\mathcal{N}) = \\max_{\\rho^{XA}} I(X;B)_{\\mathcal{N}(\\rho)}\n", "9b0aed6e24719b6a6a6dd6e4f4cadace": "\\zeta(7) = 1 + \\frac{1}{2^7} + \\frac{1}{3^7} + \\cdots = 1.00834\\dots\\!", "9b0b3f052516501adaab37d75acb8d66": "\n\\frac{dr}{dt} = \\left( \\alpha_{\\mathrm{max}} \\cos 2\\theta \\right) r\n", "9b0b4a09454e2bb2c06ae4545a3756df": " \\textstyle\\int_\\omega\\mathrm{d}\\omega", "9b0b5934239b2b19f4c92969465f96a7": "\\langle f,g\\rangle_X", "9b0badb80e8532b81b139cefad0c2ee1": "f = \\begin{pmatrix}0&0&V_s\\end{pmatrix}^T", "9b0bbccd62abd8961e8b80b308fae0a7": "\\rho=5, \\ \\phi=20^{\\circ}, \\ z=3", "9b0bc871fe66d3f2a3d10dd04cc90607": "\\int{} , \\iint{} ,\\iiint{}, \\oint{} ", "9b0bd6ce2d8c1d672a53adf715fc7845": " \\mathfrak{g}", "9b0bddbe96f3b419114f307c7dd332c5": "r_O = \\frac{V_A +V_{CB}}{I_C}", "9b0be90ec7b7d0c45890abff664d73f3": "S^1 \\hookrightarrow S^{2n+1} \\twoheadrightarrow \\mathbf{CP}^n", "9b0c08a171fbdce8147b169abe425448": " \\frac{dA}{dt} = \\omega ", "9b0c1e95f53bb2e4a0128e6f01243a03": " r_n(t) \\ ", "9b0c74dd923f896df01149e13f4b9e4c": "p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\\cdots", "9b0ca12651d7d9684e7068ccdc3c3f94": "\\{ 0 \\}", "9b0ca524492ca234a2964bf65afde2c8": "\\mathcal{N}(\\mu(t),\\Sigma(t))", "9b0cb8f69c62f04f6dfa09f853915726": "P_\\mathrm{hp}={\\tau_\\mathrm{lb \\cdot ft}\\cdot\\omega_\\mathrm{rpm} \\over 5252}", "9b0cd0e45b0cb809b046a6de493ba83a": "z^3-2z+2", "9b0cec6f9fee73c2ddbd601b023d9a0d": "\nT = max(X) \\times max ( X \\setminus max(X) ) \n", "9b0d36ee2457e682f6557d8035855fc2": " a^4 - b^4 = (a^2 + b^2)(a + b)(a - b).\\,\\!", "9b0d973c886b3d4e4cc2cef6ab60f6f9": "p\\approx 2^k", "9b0ddbe2491f5fb05409436eae2c0f2e": "\\text{Lower fence} = Q_1 - 1.5(\\mathrm{IQR}) \\, ", "9b0e2d01144fdcc3af22131808166a3f": "\\vdash \\Box \\Psi \\rightarrow \\Box \\Box \\Psi", "9b0e453146f60e3cff8d1ab3e81edd3e": "v_f=U\\sin(\\beta)\\cos(\\psi)+U\\cos(\\beta)\\sin(\\psi)=U\\sin(\\beta+\\psi)", "9b0e894a0844275ca2de15e28de73b3f": "d = 9.2345 - 3^2 = 0.2345\\,\\!", "9b0e9033df5b3cbb9e0e56745f39a24f": " v_j(x) = \\sqrt{\\frac{2}{L}} \\sin(\\frac{(2j - 1) \\pi x}{2 L}) ", "9b0ebdc45113e3dce971e63f55308730": "p_{i,j}", "9b0ede7e9d55c0fe5ebd217d2773c13a": " y = kx ;\\,", "9b0ee7f62c5b54668538bee9b91ff686": "\\langle\\phi_{i}\\phi_{j}|\\hat{g}|\\phi_{k}\\phi_{l}\\rangle = \\int\\mathrm{d}\\mathbf{r}\\int\\mathrm{d}\\mathbf{r}'\\ \\phi_{i}^{*}(\\mathbf{r})\\phi_{j}^{*}(\\mathbf{r}')g(\\mathbf{r},\\mathbf{r}')\\phi_{k}(\\mathbf{r})\\phi_{l}(\\mathbf{r}').", "9b0ef327cc4b1235512b4cfd5a44ce30": "\\scriptstyle \\frac {1} {2} \\pi R^2 D ,", "9b0ef99aabf57803036aca717954db49": "\\displaystyle \\frac{-1}{\\sqrt{2 a}} \\sin \\left( \\frac{\\omega^2}{4 a} - \\frac{\\pi}{4} \\right) ", "9b0f46322a9fe67b98004b1e0e55af97": " T'(z)= \\exp(T(z)),", "9b0f7d9d6ebcb4f7de6df3fd0af216a1": "I_p(P,Q)", "9b0f9197094fa969a686a8b1b5482574": "\ny = A_0 + A_1x + \\ldots + A_{p_n}x^{p_n} - B_1xy - \\ldots - B_{p_d}x^{p_d}y \n", "9b1002ef3c3bda248630a0d066f81b75": "\n|0100\\dots0\\rangle\\equiv|f_2\\rangle=|u_1,t_1,s_1,r_2\\rangle\\equiv|100,100,100,010\\rangle\n", "9b10587335d66871abd3f336c218384c": "D(x,y)=\\det(H(x,y)) = f_{xx}(x,y)f_{yy}(x,y) - \\left( f_{xy}(x,y) \\right)^2 ", "9b106cf59f5feca258c2612442c6b235": "p',q'", "9b109d2b184301d18df96ab9bca2bbd8": "\\mu_{ex}= -k_{B}T \\ln \\int ds_{N+1} \\langle \\exp(-\\beta\\Delta U)\\rangle_{N}", "9b11b0438f1275a3e07756c6f4a8c8c6": "K = -\\frac{1}{E} \\left( \\frac{\\partial}{\\partial u}\\Gamma_{12}^2 - \\frac{\\partial}{\\partial v}\\Gamma_{11}^2 + \\Gamma_{12}^1\\Gamma_{11}^2 - \\Gamma_{11}^1\\Gamma_{12}^2 + \\Gamma_{12}^2\\Gamma_{12}^2 - \\Gamma_{11}^2\\Gamma_{22}^2\\right)", "9b11d52ee7d20b36dc85e540d4a63340": "u(z)", "9b11d7caef94a10ff1ec05f1a363124a": "\\begin{align} \\hat{H} & = -\\frac{\\hbar^2}{2m}\\left( \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2} \\right) + \\frac{m\\omega^2}{2} (x^2+y^2+z^2) \\\\\n& = \\left(-\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2} + \\frac{m\\omega^2}{2}x^2\\right) + \\left(-\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial y^2} + \\frac{m\\omega^2}{2}y^2 \\right ) + \\left(- \\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial z^2} +\\frac{m\\omega^2}{2}z^2 \\right) \\\\\n\\end{align}", "9b11dd21659d9a73a664baa5b339737b": "\\displaystyle{u=D(S\\varphi) - S(T^*\\varphi) + S(\\varphi)/2.}", "9b11fd7039128205e67004c3e500d0d7": "\\Omega(A(x), y)=-\\Omega(x, A(y))", "9b12108732b693b317c24fb6f7126d22": "4,000 - .02(50,000) = 3,000", "9b12190e010f57a78012e9cbe09b6a0f": "E[\\vec{X}]_{ab}", "9b12762a64b6b9031a060c7b82d929ac": " e^{\\pm ia} = \\cos(a) \\, \\pm \\, i \\sin(a) ", "9b12a6a7e5573baaefd7bfe0450ed3d0": "f(x,u)", "9b12baa224c93ba3d8d0d561610d2ffe": "m=\\frac{1}{1/5}=5", "9b12dfcdafadf013a4066304e9ce03d6": "p(e)=b", "9b12e60389adcaf59da7911472d09226": "\\exist ", "9b1311b7d59cfdc302ec37bbf1ece38c": " [X]_t ", "9b1352fa3f00709a06c8c7b873e17656": "\\sigma_{m}", "9b138a0098b8c62a52928db1383aa12a": " \\frac{ \\partial \\sigma }{\\partial x}+f=0 ", "9b13a54e5ad8365e43af37357bd386f0": "\\left |\\left | \\psi(T)\\right\\rangle - \\left |\\psi(0)\\right\\rangle\\right | < \\epsilon", "9b13e3c6ac35f21ca3705120cf887e78": "A = \\frac{\\sqrt{3}}{4}a^2", "9b13e8a967d571550e05779b4065655d": "f(x,t)=tx_{i}+p_{-i}\\cdot x_{-i}", "9b141347f6fdf2f9879a9fe775b08508": "\\{y_i,y_{i-1}\\} = -y_i\\, y_{i-1}", "9b148b773c67bcfeb08d6e1b56b5521f": "10^{-H}", "9b14ae65226531dc5d241ea4b9ac0741": "y^4-xy^3-8xy^2+36x^2y+16x^2-27x^3=0", "9b14f7f0a3e7b42dabdc5274fcd85fbf": "%\\text{ Error} = \\frac{|\\text{Experimental}-\\text{Theoretical}|}{|\\text{Theoretical}|}\\times100", "9b15276abe4c29ab95533a4efc8001a7": "{C_D}", "9b154508a0a58fed3dda45a693ae1a1e": " \\hat T_{2,0} = + \\sqrt{\\frac{2}{3}}( \\hat a_{z} \\hat b_{z} - \\frac{1}{4}( \\hat a_{+} \\hat b_{-} + \\hat a_{-} \\hat b_{+}) ) ", "9b15760e117f01a5ef4a424b91bede36": "(V(t)=V({\\vec{x}}_{i}(t)-{\\vec{x}}_{j}(t)) ", "9b15dea963dd6df7070d74212e7bac3a": "\\delta(\\mathbf{r})", "9b15f0d478ebb670ec619832042f7b1f": "\\mathrm{E_1}(ix)", "9b163634f63f0ae2c4f10a94c95749da": "\n\\frac{\\partial f}{\\partial N_i}=\\ln g_i-\\ln N_i -(\\alpha+\\beta\\varepsilon_i) = 0\n", "9b163cc87407bf1d82227981fb450581": "\\tau_{xy}^2 \\geq 0", "9b16d5e42d6f2c205a0048774a92bc30": "\\{a_1,...,a_n\\}", "9b16e2930f9090be5e66c6bd9f52cf57": "\\Gamma^i{}_{jk}=0", "9b172e28752c1635b23aa6fcc3dfddbe": "L_n^{(\\alpha)}(x)=\n{x^{-\\alpha} e^x \\over n!}{d^n \\over dx^n} \\left(e^{-x} x^{n+\\alpha}\\right) = x^{-\\alpha} ~\\frac{( \\frac{d}{dx} -1 ) ^n}{n!} ~ x^{n+\\alpha} .", "9b173ddac507ee5e58b484d559ba83e9": " \\lambda^q M(\\lambda^p t, \\lambda^q H) = \\lambda^d M(t, H) \\,", "9b1765c4cd5b3e83ff58765374e4ede9": "x^{\\frac{1}{n}}", "9b17bf0ce2377c1a2a8e31173bc88d49": "=\\frac{8e^4}{t^2} \\left(\\tfrac{1}{2} s \\tfrac{1}{2}s + \\tfrac{1}{2}u \\tfrac{1}{2} u \\right) \\,", "9b17d548ef28eee42da38aa6a7a5f115": "y=const", "9b17e9581121b21fe9615c0ab87a322a": "x_1 = 1.000000028975958", "9b180ff3869c01e4a0bf1464130202bc": "\\frac{\\partial \\boldsymbol{X}}{\\partial t} \\sim \\frac{\\ln(\\ell/a)}{4\\pi\\mu} \\boldsymbol{f}(s) \\cdot \\Bigl( \\mathbf{I} + \\boldsymbol{X}'\\boldsymbol{X}' \\Bigr)", "9b183a8c936ac85ec7f3df99237f65be": "ds^2=\\frac{dx^2+dy^2}{y^2}=\\frac{dz \\, d\\overline{z}}{y^2}", "9b18872e43445a6ea57a207ffac53a28": "S(v)", "9b18d04c9d41590da7d0021b5ce5d378": "Q(x_i) = E(x_i) = 0", "9b18e0b3ca9fc1274908314b900b8647": "h(t) = \\mathcal{L}^{-1}\\{H(s)\\} = \\frac{1}{\\beta-\\alpha}\\left(e^{-\\alpha t}-e^{-\\beta t}\\right),", "9b19091afebd5b1f5f4190a6054eaf17": "\\mathbf{D} \\equiv \\varepsilon_{0} \\mathbf{E} + \\mathbf{P},", "9b192c7749d624585f8b280f191e7982": " \nN = \\begin{bmatrix} \n0 & 2 & 1 & 6\\\\\n0 & 0 & 1 & 2\\\\\n0 & 0 & 0 & 3\\\\\n0 & 0 & 0 & 0 \n\\end{bmatrix}\n", "9b193895806cd7801bd9322f06e767da": "\\omega^k_{\\ j}(e_i)=\\langle \\nabla_{e_i}e_j,e_k\\rangle", "9b19672bb250ea0c5df4ca85b2ae1b38": "\\text{r}", "9b196df1bf3e5238e3107d71dcf299fd": "\\mathbf{S}^{n-1}", "9b196e6fed316a48e27ddb4701e92b85": "\\operatorname{U}(n)/(\\operatorname{U}(n_1) \\times \\cdots \\times \\operatorname{U}(n_k)).", "9b1979d3c44d2e2d3450fb078885b208": "G(x)=F(x,G\\vert_{\\{y: y\\,R\\,x\\}})", "9b199c7e579360f6780a8dea4f2c3350": " X - 23.3 | X | \\le \\mu \\le X + 23.3 | X | ", "9b19a6e84d86c486822ada9d213de76d": "\\phi(v;f_1,\\dots,f_n) := (\\phi(v);Af_1, \\dots, Af_n).", "9b19b2f542be98a74df990e0d7587f19": "h_1(n)", "9b19dbc1fd88dc5a45228277e12cdad7": "\\tfrac{25}{2}", "9b19fe4230ac04f0e68317d8653d591d": "\n \\mathcal{D}^{\\mu \\nu} =\n \\begin{pmatrix}\n 0 & - D_xc & - D_yc & - D_zc \\\\\n D_xc & 0 & - H_z & H_y \\\\\n D_yc & H_z & 0 & - H_x \\\\\n D_zc & - H_y & H_x & 0 \n \\end{pmatrix}.\n", "9b1a00f652a961b33639d4822051b513": "p+1 = 2^kn, k \\in \\mathbb{Z}, k \\ge 2, n", "9b1a4e150c386a3dea1ea9e4cc21a6f4": "\\nabla \\cdot \\mathbf{A}=0", "9b1a931660950c9334a0e9f81d3c57c7": " \\frac{Y}{K} ", "9b1a9edd408c5430408712134fb8bb07": "\\chi'(\\alpha):=\\chi(N_{F_s/F}(\\alpha))", "9b1ad7c44472364d655c4331987baed0": "1/|G|", "9b1ae70c8fd17a3b99c6fe3186f5230a": "\\pi_*\\mathcal{E}\\to \\mathcal{E}_*", "9b1c55208e2736bfebeedebaf985f8ea": "P_1 / P_3", "9b1c669a4de1fd7d0462bce92b33037c": "H_{ab}{}^c{}_{;c}=0", "9b1cdc4c1c45751838443c4c1e5f9cb2": "n\\geq 4, n\\neq 6", "9b1cef0e54b611ab14859abfd68f5dc3": "\\textstyle q^k |B( \\mathbf{c})| \\leq q^n", "9b1d2f5d891c253b762c60f32e9bb532": "a_{23}=\\frac {1}{{x_2-x_1}}", "9b1d719bd42ecf234a181add359f19e1": "2\\leq l> shamt + \\left(\\sum_{n=1}^{\\text{shamt}}2^{32-n}\\right)\\cdot \\left($t>>31\\right)", "9b1de4ce2d91bdc7b60d3648fdb94100": "A \\cup B = S", "9b1e6880ecb17a7a6f17fba58a1ce93a": "c_s \\approx \\sqrt{k_BT_e/m_i}", "9b1f51625be81df7b65184cafee1e69c": "\\rho\\propto a^{-4}", "9b1f95aa02d3ed68ead1085ed44a56c3": "\\frac{2.7 \\mbox{ mol }}{2.7 \\mbox{ mol }} = 1", "9b1fee1d679fbe9034c295c97ab2c1a5": "nA = A + \\cdots + A,", "9b2019bebf1e11e7af4736b97cf26128": "z = n\\,\\Delta z", "9b201df56bca8cf871c68453376127fc": " m ", "9b20814e4c256999a07bd658ad4bc4ed": " \\{\\mathcal{H}_2,\\mathcal{H}_1\\}=0 \\, ", "9b209ed26aaa8001b3fb5479dac114e8": "\n\\begin{array}{rlrrr}\n\\hat{g}_N(x)=&\\min\\limits_{x\\in \\mathbb{R}^n} & c^T x + \\frac{1}{N} \\sum_{j=1}^N Q(x,\\xi^j) & \\\\\n&\\text{subject to} & Ax &=& b \\\\\n&\t\t & x &\\geq& 0\n\\end{array}\n", "9b20a762d7997943a0d9f37a26df9985": "H \\subseteq 2^X", "9b20de474230fc1fc6376195451370c4": "\\Delta G_v = 0", "9b20e243103278817a12a7538c987223": " V_e = \\frac{n_e-1}{ n_{F'} - n_{C'}}", "9b2102167c3ff172f0ebddd7c8269e8e": "\\scriptstyle\\overline{n}\\,=\\,1+Q(M)\\approx 24.61658", "9b2102f29c32460898719035b4d830da": "x\\in B", "9b21144982fbbb4dd73e2af483e9b9cf": "\\int_{-\\infty}^\\infty f(x)\\delta\\{dx\\} = \\int_{-\\infty}^\\infty f(x) \\, dH(x).", "9b2155d2119218fd73aa051d0cd52f07": "{ {\\underbrace{a \\uparrow^3 (a \\uparrow^3 (\\cdots \\uparrow^3 a))...)}} \\atop{b} }", "9b21e901bc613be88693ed9278c5c5e6": "1_{-N \\leq n \\leq N}", "9b21fd6e8bbae55d99e162ecf70e8100": "\\tilde\\beta", "9b224e784b1cf1687cfa3264544c05a7": "\\mathbf{P} \\left[ S_{T} \\geq C \\right] \\leq \\frac{\\| X_{T} \\|_{p}^{p}}{C^{p}}.", "9b2251bd47e356b9111c59adb7096130": "x \\wedge \\bigvee S = \\bigvee \\{ x \\wedge s \\mid s \\in S \\}", "9b22f3ee7a01ea6d668118133211b694": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "9b23868c97334caa27713477cc0d1411": "\\sum_{j=0}^p\\binom{p}{j}\\frac{B_{p-j}}{j+1}=1", "9b23ad1ac3d76054ced1182f4d2cc8b2": "\\mathbf{P}( |X| \\ge a) \\le 2e^{\\frac{-a^2}{2n}}, \\qquad a > 0", "9b23b5d41f85669edaf5a6dfcfd3e4d5": "\\phi(s)=\\int_0^\\infty g(t) t^{s-1}\\, \\mathrm{d}t,", "9b23bcd6294febca1faecd2d1959a231": "\\nabla_{\\vec{e}_0}\\,\\vec{e}_0 \\neq 0", "9b24c1275b035668bd4d37f1f4d17a5b": "v_1\\,", "9b25df62a7b2d78e96aaa360c6adfe42": "(L_g)_*:T_hG\\to T_{gh}G.", "9b26288abc42f2c4b196353e92ebb09c": "ENH_{k_1,k_2} (m_1,m_2,m_3,m_4 )=(m_1 +_{32} k_1)(m_2 +_{32} k_2) +_{64} m_3 +_{64} 2^{32} m_4", "9b26a47013a30d4f576ffde4f8fbf1bd": "p_{X}(x) = \\int_y p_{X,Y}(x,y) \\, \\operatorname{d}\\!y = \\int_y p_{X|Y}(x|y) \\, p_Y(y) \\, \\operatorname{d}\\!y ,", "9b26b224e2d8b6560875836baf436289": "F:=1-\\frac{2M(u)}{r} ", "9b271eb95796250351096edfec73e538": " | \\gamma_\\mathrm{e} | = \\frac{|-e|}{2m_\\mathrm{e}}g_\\mathrm{e} = g_\\mathrm{e} \\mu_\\mathrm{B}/\\hbar,", "9b278f575975c0d02d37ca2544c31f6a": "\\lambda_{sp} = \\frac{2\\pi}{k_{sp}} = 2\\pi\\sqrt{\\frac{2\\kappa}{-(A+3B\\phi_{in}^2)}}\\;.", "9b283153b9890e64f1bb9fffe7b7b850": " \\beta_0 ", "9b2876259b3164ca0a0a7095b8647108": "\\Sigma_{j \\in S(a)} p_j = W(a)", "9b28901df162ff6e078763bc4443f258": "p\\neq 2,3", "9b28af977c814ae17f603f39fef1b3c2": "b_1,\\ldots,b_n", "9b28bd273228d3524a38b560225d0b15": "\\scriptstyle \\frac{1}{T}/\\frac{1}{NT} = N.", "9b28db638db3e2ab122308b4d4c1f74d": "\\frac{T'(t)}{\\alpha T(t)} = \\frac{X''(x)}{X(x)}.", "9b29140982a300c1733f2097e05d755e": "B^*=0", "9b29a7b3ce3b713f8aeaeb107a15c017": "P_{-\\frac12}(z)=\\frac{2}{\\pi}\\sqrt{\\frac{2}{1+z}}K \\left( \\sqrt{\\frac{z-1}{z+1}} \\right)", "9b29bdda49a95cebd375f6c2c8d88fe1": "y=\\lim_{n\\to\\infty}\\left(1+\\frac{x}{n}\\right)^n.", "9b29d7099d2d4b398d0a285e87e16a82": "TY=VY\\oplus HY", "9b29f979e77c25576e4681fd09b47cc3": "{\\frac{\\partial S}{\\partial r}}\\frac{1}{r}", "9b2a371f7e364618d744ba6a4e55fec6": "\\lnot a\\Rightarrow \\lnot c", "9b2a46a721c9a5dc49f1aa1fd374fdb5": "r_H = \\frac {e^2}{4 \\pi \\epsilon_0 m_H c^2},", "9b2a46ab9c11545cd280b792ade87064": " \\sigma =\\frac{\\int_E \\Phi (E) \\, \\sigma (E) \\, dE }{\\int_E \\Phi (E) \\, dE}=\\frac{\\int_E \\Phi (E) \\, \\sigma (E) \\, dE}{\\Phi} ", "9b2a9e1c035c0d0802566726be7f2902": " \\vec{S} ", "9b2afe92ec4368b03076287a2bb2fc2b": "\\vdash w : X", "9b2b0796cc4abc8c52ebe625ed4d1722": "x/y = x", "9b2b11fd5c7326af3e5a285bca3b1420": "\\exists V_0 :\\exists V_1 \\cdots (A_0)", "9b2b3d96e03b6cb6f2901bc45d5171ec": " |\\Phi^+\\rangle_{AB} \\otimes |\\psi\\rangle_C = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_B + |1\\rangle_A \\otimes |1\\rangle_B)\\otimes (\\alpha |0\\rangle_C + \\beta|1\\rangle_C). ", "9b2b601f69bb9310e55923ebf7a714e4": "\\zeta^*=\\sum_{J\\subseteq R}(-1)^J\\zeta^G_{P_J}", "9b2bbac0f0bd8d608d24d63bd5890249": "\n \\mathbf{F}^{-T}~\\mathbf{N}_i = \\cfrac{1}{\\lambda_i}~\\mathbf{n}_i ~;~~\n \\mathbf{F}^T~\\mathbf{n}_i = \\lambda_i~\\mathbf{N}_i ~;~~\n \\mathbf{F}^{-1}~\\mathbf{n}_i = \\cfrac{1}{\\lambda_i}~\\mathbf{N}_i ~.\n\\,\\!", "9b2bf537779546bbfe512d24dcfc73a4": "(\\delta\\mu)^2", "9b2c6991a9ca76970b4cbb7e63a3aae4": " \\gamma_i^{R} = 1 ", "9b2ca410e1ace3ecbb253cc99b50191c": "\\mathbf{X}={\\mathbf A}^{-1}", "9b2cbd7ec6f0ce001c493b512c142cb8": "R\\subseteq I \\times X", "9b2d46a4cc8082b1961310aaa2d98b81": "\\delta < \\min \\left \\{\\frac{\\varepsilon}{2r(m-1)}, (y_1 - y_0), (y_2 - y_1), \\cdots, (y_m - y_{m-1}) \\right \\}", "9b2d53b15c5f919930b93049c18bd797": "x\\cup y", "9b2d573d740a47e2eb3f13e6f43b131f": "\\textstyle h(z)=1", "9b2d79a12b3d9bec3703c1874d285ab4": "a^3 = \\frac{-\\sin(\\phi) a_1 + a_3}{\\cos(\\phi)^2}.\\,", "9b2d7c56afd188605822b9fba7b33440": "\\mathcal{Q}_{\\mathrm{Hur}}", "9b2d8851480e2833c03b668754e403f8": "a = \\frac{r_1 + s_1}{2} = \\frac{r_2 + s_2}{2} \\ = \\frac{r_m + E x}{2} \\quad (15)", "9b2da05afc9fff360480f7cff2774fd1": "y=8", "9b2da700a5998cfa419932eee55283fd": "\\ M^{max}_{V} = -9.96 - 2.31 \\log_{10} \\dot{x} \\,.", "9b2db9dc75c962dbc79f169029a1803a": "\\begin{matrix}\\ &light &\\ \\\\ 2Q + 2H_2 O &\\Longrightarrow & O_2 + 2QH_2\\end{matrix}", "9b2dc4b72d627f78506ae92c492890d3": "M(x) = \\begin{cases} \n \\frac{Px}{2}, & \\mbox{for } 0 \\le x \\le \\tfrac{L}{2} \\\\ \n \\frac{P(L-x)}{2}, & \\mbox{for } \\tfrac{L}{2} < x \\le L\n \\end{cases}", "9b2e2fae9165bd8532d30be15a2bcfbd": "mol Al = \\frac{grams Al}{g/mol Al}\\,", "9b2e3da604b15970ae8db9da0cecca40": "\\{\\mathbf{x}\\,|\\, \\exists \\mathbf{p} \\in [\\mathbf{p}], f(\\mathbf{x}, \\mathbf{p})= 0\\}", "9b2e49ba26be5a568b1888acdf38c8f2": "F_{net} = mv^2/r\\,", "9b2e670ebba49c7843deff91b2bb9dc3": "e \\leftarrow SEnc(k,s,1024,M)", "9b2e6819a6f8282c6993421a141212cf": "U(r,w) = U_0 exp(iwt) \\quad (1.6)", "9b2e8a34843a15b450c810fcad84274c": "\\sqrt{B}:=\\bigcap\\{ P\\subseteq R \\mid B \\subseteq P, P \\mbox{ a prime ideal} \\}\\subseteq\\{x\\in R\\mid x^n\\in B \\mbox{ for some }k\\in\\mathbb{N}^+ \\} \\,", "9b2eafcc67e5c37cc459cebe6d30354b": " (\\partial G)_S=-(\\partial S)_G=-\\frac{VC_P}{T}+S\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "9b2f172d6c78576aeffe44c0dc3c917a": "\\exists_f S =\\{ y\\in Y | \\mbox{ there exists } x\\in X \\mbox{ s.t. } f(x)=y \\}", "9b2f792bfceae439441c7b30bef3548c": "\\psi\\ \\stackrel{\\mathrm{def}}{=}\\ {\\phi + i \\chi\\over \\sqrt{2}}", "9b300c1db854098871ddeea1fb907aad": "2\\sum_{k\\in\\mathbb Z}\\left[{n+1\\choose 10k}_2-{n+1\\choose 10k+1}_2\\right]=f_n(f_n+1).", "9b30365d9afaa7b4ed52621f889360f6": "\\scriptstyle \\pi^2/6", "9b305b565b9bf54db015902b810de63f": "ae^{\\lambda x} + b > 0 \\,.", "9b30843bf9837abbcb67315f791232bc": "\\sqrt{\\varepsilon}", "9b30ec36d06f20882c406cc50e268952": " v = -{ \\left( a - \\lambda_{\\pm} \\right)u \\over c }", "9b30fdf9a09acbb5d41e727cbd07569c": "a<1", "9b3127eb2c0fe70bb4f3cad966630c2c": "(x^2+y^2)^2=x^2-y^2", "9b313f4efc428573ecbe19301fed54f1": "a_1,\\ldots,a_m \\in \\mathbb{Z}^n_q", "9b315089c4ec36eed2343db995b14b75": "C\\ell(E) = C\\ell^0(E) \\oplus C\\ell^1(E).", "9b3196557078b6affb58e0f13fbbb3e4": " M^{(1)}(B)=M^1(B)=E [{N}(B)], ", "9b31a2e64e1d8a48b5c8f1d4754a3c13": "\\frac{\\sin(x)}{x}.", "9b31afb39883982e842354b22eb5e09b": "\nE[X_i] = \\mu_i + \\frac{\\sigma_i}{a-1}, \\qquad i=1,\\dots,k, \n", "9b31d25f59dc465f29a02ed19c81df69": "\\mathbf{u} + \\mathbf{v}", "9b31f511e675c997beca0b02005dc03b": " L=I_\\text{ball} \\omega= \\frac{2 m r^2}{5} \\frac{2 \\pi}{T} ", "9b3253d15537639154d773509f3783e1": "s \\in \\mathbb{C}, \\operatorname{Re}(s)>0,", "9b3282b3b56672f8a5f8fdde8adb4cfd": "\\lambda_k\\geq 0", "9b32b1b8fd7bce5a52a21d676eb2b03f": "t_i\\in S", "9b32c463173b68b61d61013b0f25c449": "\\binom{1+a}{1+n}=\\frac{(1+a)\\binom{a}{n}}{1+n}", "9b33261205deef6d37cdd83d16f417e2": "O(c_{d,\\epsilon}log(n))", "9b336d37d8955cea671f19e9a5f78a53": " R = P + Q = (x_3:y_3:1)", "9b3413ea6963bb4f65dda46131376afa": "\\scriptstyle v(f({\\mathbf A}))", "9b341d6de4f6f4d59c8266e73d9828c7": "\\frac{dx_i}{ds} = a_i(x_1,\\dots,x_n,u)", "9b3435f1944da1ee1802c700032b6feb": " P^2 = P\\,\\!", "9b34f5d950c5128bc0592a9bc952118c": " \\frac{\\pi}{2} ", "9b3509e37cf07bcb81ccdcb1f80e09ea": "q_1 = R \\sin \\theta \\approx 1.22 {R} \\frac{\\lambda}{d} = 1.22 \\lambda N", "9b352f41599716332b0da88fc575f703": "B=|\\mathbf{B}|", "9b3530dec6276d68610cf9f11f63faf0": "\\beta_1 X,\\ldots,\\beta_k X", "9b358a45de2137b34300b4be3e1edfac": "u = x^\\ell e^{-x/2} L_{n-\\ell-1}^{(2\\ell+1)}(x)", "9b35c6891c7b4d4e41a315b6b72408c9": " \\frac{1}{\\sqrt{2\\pi\\,}} ", "9b35dd1fd8fb2e8ba4a972122aca50b4": " 2", "9b3685340bf566997eea3102e4f1f89f": "w_{n,i_n}(x_n),(i_n=1 \\ldots I_n)", "9b3691b4551ee73923afabe5dcec8092": "r_k = \\{x/\\beta^k\\}.\\,", "9b36a091c77d157e80de61274a281d28": "D(\\theta)", "9b36ebb6dfca3709a49ab064eb263353": " x = \\frac{A + B}{\\sqrt{a^2 - b^2}}, \\,", "9b370d076f0a130130ac2ebefd221783": "\\ell^{\\,2}(\\aleph_0)\\,", "9b370ebd478f6434ccfa4b64ba32d55e": "\\Delta(t) = (t - 1 + t^{-1})^2,", "9b37142904919a9e56dfe55fe69502b7": "\\frac{\\partial u}{\\partial t} = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x} + \\nu \\frac{\\partial^2 u}{\\partial z^2},", "9b377e5c59adce8798d80b0244c851d0": "g_{2}", "9b3783e7407ffb6592def1d9b43b5b0a": " z=re^{i\\theta}", "9b37dd021c49805ec8d1c075c5c42922": "d\\phi=\\frac{\\partial \\phi}{x_1}dx_1 + \\frac{\\partial \\phi}{\\partial x_2}dx_2 + \\ldots + \\frac{\\partial \\phi}{\\partial x_n}dx_n + \\frac{\\partial \\phi}{\\partial y}dy ", "9b37f39f9c9becbe7599fcd3a5c0a32a": "\\epsilon_{[t_l,t_1]},(z_1,t_1), \\epsilon_{[t_1,t_2]},(z_2,t_2),\\ldots, (z_n,t_n),", "9b38067e23298837802635d5172733d7": "MI", "9b3844e444b585fb05b11f1b8c891035": "0 \\le \\alpha \\le 1", "9b385a54224f5c9744ac429bd3596843": "\\begin{align}\n\\overline{I_a(t_0)}&=[t_0-a,t_0+a] \\\\\n\\overline{B_b(y_0)}&=[y_0-b,y_0+b].\n\\end{align}", "9b385fc61ed3cd93e0aa85149c65e337": "\\hat{\\rho}_{XY\\cdot\\mathbf{Z}}=\\frac{N\\sum_{i=1}^N r_{X,i}r_{Y,i}-\\sum_{i=1}^N r_{X,i}\\sum_{i=1}^N r_{Y,i}}\n{\\sqrt{N\\sum_{i=1}^N r_{X,i}^2-\\left(\\sum_{i=1}^N r_{X,i}\\right)^2}~\\sqrt{N\\sum_{i=1}^N r_{Y,i}^2-\\left(\\sum_{i=1}^N r_{Y,i}\\right)^2}}.", "9b38aa12608386b72307ce9ace7c07a3": "x\\in \\mathcal{X}", "9b38b5f76ea7afa1c7096ffed28022e4": " n_1 k_0\\sin\\theta_1 = n_2 k_0\\sin\\theta_2 \\, ", "9b39756c11b823181bda83c4419ae2e6": "b_{n+N} = e^{\\frac{\\pi i}{N} (n+N)^2 } = b_n e^{\\frac{\\pi i}{N} (2Nn+N^2) } = (-1)^N b_n .", "9b39772e1034df62c3126ed80162c349": "r\\!", "9b3985b9765078e7a437134987d5e6ca": "x^p y^q", "9b39906b2857d1935a6f95165453846c": "a(u,v) = \\int_\\Omega \\nabla u\\cdot\\nabla v,\\quad b(v)= \\int_\\Omega gv.", "9b39d61b7e34a5d26043fcdef9ab7b4a": "\\varphi_e(y)", "9b39ed4491f9dee4ed5c40ebb0b9b448": "\\mu \\le \\lambda", "9b3a591f99765dc76caed3d10c132cf0": " L_1 = -a_1 Z^{-1} \\, ", "9b3a9ace35e9662a2f4443efd401746e": "dV_\\beta^e = \\frac{4\\pi}{3} \\dot{G}^3(t-\\tau)^3V\\dot{N}d\\tau \\,\\!", "9b3b091cee8aa60321921fb0bec93133": "T = 2 \\pi \\sqrt { \\frac{L}{g} } \\qquad \\qquad \\qquad (1)\\,", "9b3bd99b8e077cd364b5008a2ea774ce": "\\theta_{AB} = {\\int_A}^B \\frac{M}{EI}\\;dx", "9b3bed06cc4ed705eddafdefb7244486": "\\displaystyle{a^{-1}= Q(a)^{-1}(a^{-1} - Q(a)^{-1}b)^{-1} + Q(a)^{-1}(a^{-1} - b^{-1})^{-1}= (a -b)^{-1} + (a-Q(a)b^{-1})^{-1}.}", "9b3bf64d08197d18db686e1111f3ff27": "+ \\ln\\Gamma_p\\left(\\eta_2+\\frac{p+1}{2}\\right)", "9b3c2a82e196ef44d1adf9e2d25f59b4": "h_3", "9b3c4847d5e7afa6edd39e36fdea6647": "A^2 = -n -\\frac{1}{n} \\sum_{i=1}^n (2i-1)(\\ln \\Phi(Y_i)+ \\ln(1-\\Phi(Y_{n+1-i}))).", "9b3d044e2d1fc9e4c78932cf4045fde3": " \\mathbf{u}_k = \\mathbf{v}_k - \\mathrm{proj}_{\\mathbf{u}_1}\\,(\\mathbf{v}_k) - \\mathrm{proj}_{\\mathbf{u}_2}\\,(\\mathbf{v}_k) - \\cdots - \\mathrm{proj}_{\\mathbf{u}_{k-1}}\\,(\\mathbf{v}_k), ", "9b3d848ee5d864927b0ff689f4656d2b": "-\\kappa\\frac{I_1(\\kappa)}{I_0(\\kappa)}+\\ln[2\\pi I_0(\\kappa)]", "9b3da3881a0c48308c12126a3e0bca5d": "H = L^2(\\mathbb{R}) ", "9b3ddfe767bf0477511c05a2ff9312ae": "\\sigma(z) = \\overline{z}\\,", "9b3de139c7b1c969c8069846f5910067": "\\tilde{H}_{\\mathrm{\\infty}}(W|E) \\geq m ", "9b3ea9e947270bb59ba432ed9f6ecd7a": "\nu_{n+1} = e^{L h} u_n + h^{-2} L^{-3} \\left\\{ \n\\left[ -4 - Lh + e^{Lh} \\left( 4 - 3 L h + (L h)^2 \\right) \\right] \\mathcal{N}( u_n, t_n ) +\n2 \\left[ 2 + L h + e^{Lh} \\left( -2 + L h \\right) \\right] \\left( \\mathcal{N}( a_n, t_n+h/2 ) + \\mathcal{N}( b_n, t_n + h / 2 ) \\right) +\n\\left[ -4 - 3L h - (Lh)^2 + e^{Lh} \\left(4 - Lh \\right) \\right] \\mathcal{N}( c_n, t_n + h )\n\\right\\}.\n", "9b3ecd4f5f0cc174717f19cec0743fcd": "\\mathbb{N}", "9b3ef4e728a9ef41ddafd2c511e45a32": "g_{\\mu\\nu}dx^\\mu \\, dx^\\nu =(-\\,N^2+\\beta_k\\beta^k)dt^2+2\\beta_k \\, dx^k \\, dt+\\gamma_{ij} \\, dx^i \\, dx^j", "9b3f38184baa4cbfa0dc40178fdb7aa1": "K_1 \\cdots K_m", "9b3f565fbac9ec994375e6f185746a31": "x^5 - 10Cx^3 + 45C^2x - C^2 = 0\\,", "9b3f9a6e18af7d8852c9478bfabeba75": "p_{\\text{wetting phase}}", "9b3fa646446010a0723d4fe451a8e415": "L_2\\,", "9b3fbc0e59fba6abd786b36f4ebf952c": "f(\\mathbf{x}_k+\\alpha_k\\mathbf{p}_k)\\leq f(\\mathbf{x}_k)+c_1\\alpha_k\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k)", "9b3fc1af4bdbdbb9ca7fbd04e8a09c0f": "\\mathcal{Z} \\, = \\, (1+x)^{N_{S}} ", "9b40299829099588e0e980dc38fb8777": "\\alpha,\\omega \\in\\mathbb{T}, \\lambda > 0", "9b403a5ce4a3040fa2f8013937a73ae4": "q=(ai + bj + ck)\\times(ei + fj + gk)\\,", "9b4043c2c44496dd5e11158835242d97": "\\ \\pi_0(X;x)", "9b409d1b6f79755ec93d441bd9e62f98": "\\mathrm{C'}", "9b40a144b4168b5467b4b1866375e6ea": "F(k,y)", "9b40a153394967559190e5c4a7d32cae": "\nL \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{2}{a b^{2} \\left( a^{2} + b^{2} \\right)}\n\\frac{1}{J_{\\beta}^{\\prime}}\n", "9b40c37f77e43928b8423bab902ac510": "\\min(c_f(A,B),c_f(B,C),c_f(C,D),c_f(D,F),c_f(F,G)) = ", "9b40c9a2d62622cc290ef849a8ce6a00": " b - \\sqrt{b^2-4ac} \\approx b \\left ( \\frac{2ac}{b^2} - \\frac{2 a^2 c^2 }{b^4} + \\cdots \\right ), ", "9b40cea730887643f47336210f388a46": "V_{ij}", "9b410282b6e8077036b8c3062f2f84b9": "\\langle r,s,t \\mid r^2 = s^3 = t^5 = rst \\rangle", "9b41269b5329219a0c193a035874be8d": "Y\\times Z", "9b415f97d359f44b917384bb72ae4bc6": "MN/M \\cap S \\cong 1", "9b41601ed443fc1a92b55c4510c6bbfe": "{\\ pV = nRT}.", "9b418fbbfec884c67f4e5cc04488b811": "d_H = 3 ", "9b41c8d1dacd0000f724e9617cc4abf4": "u(\\nu, T)\\partial \\nu = {8\\pi \\nu^2 \\over c^3}{h \\nu \\over e^{h v/kT}-1}\\partial \\nu.", "9b41ebfb364a3b99584637fd66f5e345": " \\mu_{Y \\mid x}^\\pi = \\mathcal{C}_{Y \\mid X}^\\pi \\phi(x) = \\mathcal{C}_{YX}^\\pi ( \\mathcal{C}_{XX}^\\pi )^{-1} \\phi(x) ", "9b41fd932beab7f15c38a7cb97d77207": "\\sum_{i=0}^n {n \\choose i} = 2^n", "9b423a0f46e4776ea64ccfa6379e2fe8": "|\\sum_{\\mathbf{k}} \\phi_\\lambda({\\mathbf{k}})|^2", "9b4243f36d11d23795e3fd7ca5b24214": "\\frac{\\operatorname{d}^2y} {\\operatorname{d}x^2} = -\\left(\\frac{2k}{rd}\\right) y", "9b42851159f256f01fada936909c022b": "g_m = \\begin{matrix} \\frac{\\partial I_D}{\\partial V_{GS}} = \\frac {2I_D} {V_{GS}-V_{th}} = \\frac {2I_D} {V_{ov}} \\end{matrix}, ", "9b42e45c6ff978aee07271528597e6a7": " \\textbf{Q} ", "9b42e9516487515e69b63f4840f730a2": "\\operatorname{Iso}(A,B),", "9b42f5600f6a46cc8d7079f8797759e2": "\\left|\\mathbf{B}\\right| = \\left|\\mathbf{a}\\right|\\left|\\mathbf{b}\\right|\\sin{\\theta},", "9b4305452b1dcf52924074f0054442fa": "S=\\emptyset", "9b433399029cc18177900deb2474e1db": "b_1=1", "9b43624f686e4cd187fe9be1a8f89473": "|\\mbox{Out}(G)|", "9b4386013deb3538444ec17d6c6f9af6": "A\\left( {\\vec x} \\right)", "9b43c06f1339d20be237eabdff1e1b8f": "F(v)=E-TS=kT\\log \\left(1-e^{-\\frac{hv}{kT}}\\right)", "9b43c9663fdf13aeaee2a308fc8cb1ed": "H=\\int dx \\left[{1\\over 2}\\partial_x\\psi^\\dagger\\partial_x\\psi+{\\kappa \\over 2}\\psi^\\dagger\\psi^\\dagger\\psi\\psi\\right].", "9b43da7e8a84c9ab9847202ea03fa22b": "q(y)", "9b440c7f2647ce9afa6231cc364be540": "2^{p-1}\\equiv 1\\pmod{p^2}.\\;\\;", "9b4469f1de63d5de06f8bde5a2e629ac": " \\prod_{p} \\Big(1 - \\frac{1}{(p+1)^2}\\Big) = 0.775883... ", "9b44aeb7a10c5cd7a8943e75fe15c8fb": "(a_1,a_2,\\dots,a_n)\\in R", "9b44c47488a738025d4f51cdf21b6065": "(X_1,X_2,", "9b44cef16cd1260de0078175596cc117": "\\Phi(\\mathbf{x},t) = \\Phi(x_1,t)\\Phi(x_2,t)\\dots\\Phi(x_n,t)=\\frac{1}{\\sqrt{(4\\pi k t)^n}}\\exp \\left (-\\frac{\\mathbf{x}\\cdot\\mathbf{x}}{4kt} \\right ).", "9b44fb13a1ce6b56a40ce38bb804c70c": " + \\underbrace{E_\\mathrm{sig}E_\\mathrm{LO} \\cos((\\omega_\\mathrm{sig}-\\omega_\\mathrm{LO})t+\\varphi)}_{beat\\;component}.\n", "9b45481a114e4774b2c6b21161777fc7": " \\psi(x_1, \\dots, x_N) = \\sum_P a(P)\\exp \\left( i \\sum_{j=1}^N k_{P\nj} x_j\\right) ", "9b4549afcc07ae32bec6e90473ef4537": "\\delta_2 \\in E_n(j)", "9b4550201d9330df8c077a31b2cea6f4": "11^2+2^2+2^2", "9b4570b0459dac2b6b7d7dad709029d4": "F_5(x)=x^4+3x^2+1 \\,", "9b45eb77a936fdba29c2bc4e9f60daff": " \\varepsilon >0", "9b460a75690c67596ec8d4c3c41c313c": "\\displaystyle u_t+uu_x=\\nu u_{xx}", "9b46142192e4e5973c4ba12f30d04e07": "m \\ge 2", "9b46696b822b6508bc8ba76268a5e751": "e^{-\\pi x^2}", "9b466d749c817cfc3a0a740a62c34427": "0 =\\frac {\\dot Q_H}{T_H} - \\frac {\\dot Q_a}{T_a}+ \\dot S_{i}.", "9b4674e98e056a3c7bb94989419037ee": " n\\geq1 \\!", "9b46a119340bf32f2ae4664a5e3b82b7": "\\sqrt{\\frac{2}{\\pi}} \\cos(ut)", "9b46e6fd5fe0e3c75390b157d418994c": "\\scriptstyle f\\cdot T.", "9b46ebe1e251d0651a1cb0fc6774cbec": "(x)_y", "9b46fcc0924cdfb3db1732cba011ffd0": " E_{ab} = \\sum_{i} ^{\\text{on }a} \\sum_{j} ^{\\text{on }b}\n \\frac {k_Cq_iq_j}{r_{ij}}\n + \\frac {A}{r_{\\text{O}\\text{O}}^{12}}\n - \\frac {B}{r_{\\text{O}\\text{O}}^6}\n", "9b477c8c0c754089e8f2355eefcaacc9": "\\frac{\\sqrt{7}}{2}\\sin(\\phi)(5\\sin^2(\\phi)-3)", "9b478c4ceadb4e9311299c61f75b242a": "w_r^+-w_r^-", "9b478c9964dd37582fd3c8a8312baa87": "ee=\\frac{[S]-[R]}{[S]+[R]}=\\frac{e^{-k_St}-e^{-k_Rt}}{e^{-k_St}+e^{-k_Rt}}", "9b47c99ca0fe134f1a722fb4a739d14e": " {\\rm tr}(t_at_b)= 2x_\\lambda g_{ab}", "9b47ee0ab122b3dc7a1a2614ab8e87f0": "DF(x_0) = \\begin{pmatrix} \n F & 0 \\\\\n 0 & 0 \n \\end{pmatrix} ", "9b482e83eb806a4ba9cbd28da256d1e0": "\\le 1", "9b48492b2f9503096c1bbd5ce9798422": "-9.09718(T_0/T-1)\\ -\\ 3.56654\\ \\log(T_0/T)", "9b4867908a2aaa574933c51755e90d85": "\\tan \\varphi = H\\cdot \\dot{y}/\\dot{x}.", "9b48c0f9bf61baf53bf47d120900822b": "Y \\sim \\textrm{Cauchy}(x_1,\\gamma_1)\\,", "9b48ec3946dab6d6a247e38799bbd2ba": "(\\overline{Y} + \\Delta \\overline{Y})/Y", "9b4903b7afd699bdee99f85d6ed239f1": "0.15 \\text{ mm}", "9b4914730d1baa5586028956a731973e": "S/k \\approx \\left(q+N^{\\prime}\\right)\\ln\\left(q+N^{\\prime}\\right)-N^{\\prime}\\ln N^{\\prime}-q\\ln q.", "9b491cd7f9ca054bcfd44b4872b74b76": "\\lambda \\wedge \\theta", "9b492aa843ad43ad944247fcec4a5ce4": "\\partial : H_{i}(X, A) \\to H_{i-1}(A)", "9b4935b4d73866a8dda91635a2d33ba6": " a_i (t+1) = a_i(t) + \\nu \\big [ x(t+1) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ] \\frac {u \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} {\\sum_{i=1}^N u^2 \\big ( \\left \\Vert \\mathbf{x}(t) - \\mathbf{c}_i \\right \\Vert \\big )} ", "9b49492d503dc5220b9c4d87e2a52a84": "f(s_i) = z_{i,m_i}", "9b495114ab6873500fd1b8ddf2289a35": " \\langle\\phi^* \\phi\\rangle = {1\\over Z} {\\partial \\over \\partial h} {\\partial \\over \\partial h^*}Z |_{h=h^*=0} = M^{-1} ", "9b49f01c5829c4424e041860381cfdfe": "b_0/b_{1^{ }}", "9b4a2bd532e963ca28769089f7f0f479": "\nG_k(u) = |G_k(u)|e^{i \\phi_k(u)} = \\mathcal{F} (g_k (x))\n", "9b4aac423a128eebd7483114445a32e0": "|2^\\wedge A|\\ge\\min\\{p,2|A|-3\\}", "9b4aeaedf5afb52ae9045f65b09da20f": "t=n\\delta t", "9b4aefdaa3f61081bf272bc8b8e871b9": "X_0 = ct, \\quad X_j =-x_j", "9b4b0a88ffa698b46bda559ff863e4df": "V^{\\otimes n} \\;\\overset{\\mathrm{def}}{=}\\; \\underbrace{V\\otimes\\cdots\\otimes V}_{n}.", "9b4b14e94ec2db2809af59be898d9737": "h \\,", "9b4b3d07c8920f554bf5d0fee4b22f81": "\\left(\\frac{\\partial T}{\\partial y}\\right)_x \\left(\\frac{\\partial S}{\\partial x}\\right)_y - \\left(\\frac{\\partial P}{\\partial y}\\right)_x \\left(\\frac{\\partial V}{\\partial x}\\right)_y = \\left(\\frac{\\partial T}{\\partial x}\\right)_y \\left(\\frac{\\partial S}{\\partial y}\\right)_x - \\left(\\frac{\\partial P}{\\partial x}\\right)_y \\left(\\frac{\\partial V}{\\partial y}\\right)_x", "9b4b69ce243da73bb64c58fb1c3aa78a": "\\widehat{h} = \\widehat{\\mathbf{S}}\\cdot \\frac{\\widehat{\\mathbf{p}}}{|\\mathbf{p}|} = \\widehat{\\mathbf{S}} \\cdot \\frac{c\\widehat{\\mathbf{p}}}{\\sqrt{E^2 - (m_0c^2)^2}}", "9b4b6b371b9dcd1c6bf227c00e6f511d": "e: A \\rightarrow A", "9b4b81ebf86c347a5c008fff4d79fc13": "k(\\boldsymbol{x},\\boldsymbol{y}) = (\\boldsymbol{x}^\\mathrm{T}\\boldsymbol{y} + 1)^2", "9b4bbcea45d16298912631740dc9271a": " x(t+1) = \\widehat{\\mathcal{M}} x(t)", "9b4bdc31a6cbc00ec6ac928e4040ce48": "a_{c,i} = \\gamma_{c,i}\\, \\frac{c_i}{c^{\\ominus}}", "9b4c02d7379924708140a9ad8f3238d9": "\\det(M) = \\frac{\\det(M_1^1)\\det(M_k^k) - \\det(M_1^k) \\det(M_k^1)}{\\det(M_{1,k}^{1,k})}. ", "9b4c1d62eb5974234f3437f8ac4ee7fe": "\\langle x, \\varphi \\rangle = \\int \\langle x , y \\rangle \\langle y, \\varphi \\rangle dy. ", "9b4c2735dd0b786f4b2262f51ee18e6e": "\\rho_q(\\bold{r}) = q |\\psi(\\mathbf r)|^2 ", "9b4c420d5c357dbe9cdbe293bfbd2811": "A \\otimes_k k_n \\approx B \\otimes_k k_m", "9b4c4743a39d54ace8cff4fee55f4a4e": "N_X", "9b4c7c12c99b64849325ac2d3a963e66": "\\Delta E = \\Delta E^\\circ - \\frac{R T}{n F} \\ln Q \\, \\,", "9b4cabb201ab66fef53a5e7a475bec7e": "u\\Vdash A", "9b4cb3259bc80fe521e9bf4c43567005": "\\operatorname{deg}\\big(\\sum_{\\alpha \\in \\Delta} \\lambda_\\alpha \\alpha\\big) = \\sum_{\\alpha \\in \\Delta}\\lambda_\\alpha", "9b4cd9d4e9cd08f813c2dda96c389a55": "a_1 v_1 + a_2 v_2 + a_3 v_3 + \\cdots + a_n v_n. \\,", "9b4d24aee6d072ad0a8ed85a0c3f26ff": " x^{\\underline n} = x(x-1)\\cdots(x-n+1).", "9b4d3049dbbf7f281f9ddf52ca60fda6": "\\Delta g_{ij}=f(T)=a_{ij}+b_{ij}\\cdot T +c_{ij}T^{2}", "9b4d5b24790c7b7dec1275776f64a812": "F(\\nabla)", "9b4da55a09717b5cbb70370f7740f361": "\\mathbb{R}_{+}", "9b4db478dbcdfb0c5012ea4f1dd936f1": " L(C)\\rho = C\\rho C^\\dagger -\\frac{1}{2}\\left( C^\\dagger C \\rho + \\rho C^\\dagger C\\right) ", "9b4e241d1af37d7ccc32fdde53fd9306": "\\begin{Bmatrix} 7 \\\\ 7 \\end{Bmatrix}", "9b4e5bf8ea6c58da5d37afdb077103cf": "\\mathrm{Var}[r_t] = \\frac{\\sigma^2}{2 a}(1 - e^{-2at}).", "9b4e95223102c050afba19c2f5771f50": " p_{ij} = m\\int(v_i- u_i)(v_j-u_j)fd^3v\n", "9b4e9bcd3e6e6b44c28e6ec9b0db45e5": "|x|\\ge\\xi", "9b4eb81e2b3ed9392463602f3b44d5cb": "\\forall B \\rightarrow U", "9b4ebd1bd08d58e8926f782b7e8b7c73": " \\langle s| s\\rang =N\\frac{1}{\\sqrt{N}}\\cdot \\frac{1}{\\sqrt{N}}=1", "9b4edf227623c00db856010bb7fcd334": "\\alpha_1,\\alpha_2,\\dots,\\alpha_n", "9b4eedddf8f274b5ef7c01ec5a07efc9": "F\\left(x, y, y', \\cdots, y^{(n)} \\right) = 0", "9b4f1e27fa27dcc2a967c4ea425acb52": "u_1 = \\tfrac{(ax_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_9 - 2x_1(bx_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "9b4f2c6c3d96b7b68bff3a13a966e777": "x(v)", "9b4f3862c69d57b30bc2093818742f80": "A^T W A v=A^T W b", "9b4f6995073b998bc2fc1869fcc68651": "F \\left( u_{i - \\frac{1}{2}} \\right) = f^{low}_{i - \\frac{1}{2}} - \\phi\\left( r_{i-1} \\right) \n\\left( f^{low}_{i - \\frac{1}{2}} - f^{high}_{i - \\frac{1}{2}} \\right)", "9b4f9e144f7048bcba96fa8be998695d": "v^{min}_{free}", "9b4fcf64d81e53833b9148c83e52dd50": "\\mathcal{R}^K _{\\theta} = \\{ z\\in \\mathbb{\\hat{C}}\\setminus Kc : \\arg(\\Phi_c(z)) = \\theta \\}", "9b500bb6537befd5159c85d84c60c721": "Z_{2m}", "9b507c2b1b8b8f5090f949416397db3a": "y_0 \\in \\mathbb{R}", "9b50944a9421ae3f1770afe7d60d61e4": "\\color{Black}\\tfrac{3}{m}", "9b50d4a9c27100d61c1298d720214fd0": "\n\\det\\begin{bmatrix}\n\\wp(z) & \\wp'(z) & 1\\\\\n\\wp(y) & \\wp'(y) & 1\\\\\n\\wp(z+y) & -\\wp'(z+y) & 1\n\\end{bmatrix}=0", "9b50d949c844a4e9b1d8897b920a8c9c": "\\mathbf{E} = \\sum \\mathbf{E}_i = \\frac{1}{4 \\pi \\epsilon_0} \\sum_i \\frac{q_i}{\\left | \\mathbf{r}_i - \\mathbf{r} \\right |^2}\\mathbf{\\hat{r}}_i \\,\\!", "9b5119282dca6053fa709184c7a8f65e": "B^* = [b_1, \\ldots, b_{nm}] ,", "9b5171375058695372b67706337cdd2c": "a = -2r, c = -r, d = r\\,\\!", "9b517a11e29d4bb319525ecdf2db2a19": "\\left( \\frac{3}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{4}{\\sqrt{3}},\\ 0 \\right)", "9b526776311bd041550ad174bd49d83e": " P (x) ", "9b52811b5e0f8ca08664dd707e329f22": "\\Psi_A(r_1,r_2)= \\frac{1}{\\sqrt{2}}[\\Phi_a(r_1) \\Phi_b(r_2) - \\Phi_b(r_1) \\Phi_a(r_2)]", "9b52c48615c3c449df3abe04e7d036d9": "M_{t} = f(X_{t}) - \\int_{0}^{t} A f(X_{s}) \\, \\mathrm{d} s,", "9b52fa5be1507289b74330166b6afb54": " \\bold c^Tx", "9b53120fb2128d01ff21b42cf4160292": "R_T", "9b53379375a597fa2a3a848d8e1f46a9": " T_\\mathrm{sat} = T_\\mathrm{wb} ", "9b53c244b586d3ac166df8d982f10a08": "Q_A(X)=q_{\\mathrm{prim}}(X)\\,q_{\\mathrm{dual}}(X)", "9b53e9dfabc65150420b93c2f063271d": "3K(1-2\\nu)\\,", "9b5439fc817093806eced7302543b843": "x \\lor y = y", "9b548240a212b5c14c17f7ef2b378dd7": "\\mathcal{O}(M \\log M)", "9b549179b8c38057008bb7937f6b7332": " a = |\\mathbf{r}_A - \\mathbf{r}_P|, \\quad b = |\\mathbf{r}_B - \\mathbf{r}_P|, ", "9b54e5f92a61a0bab68f7dc1bf539884": "\\varphi(m, n, 0) = m+n,\\,\\!", "9b54ef85eae41d76b829774b5f4c3612": "\\exists x\\,\\!", "9b557408acda8c5b51dd13469c8280f8": "\\alpha\\in Z_+ \\,,J\\subseteq N", "9b55bd06d5b60c33385a72e136740989": " \\Gamma_{q^n}\\left(\\frac {x}n\\right)\\Gamma_{q^n}\\left(\\frac {x+1}n\\right)\\cdots\\Gamma_{q^n}\\left(\\frac {x+n-1}n\\right) =[n]_q^{\\frac 12-x}\\left([2]_q \\Gamma^2_{q^2}\\left(\\frac12\\right)\\right)^{\\frac{n-1}{2}}\\Gamma_q(x).", "9b55c7810cf89f8c441ba2a3c20e88dd": "F_{t,T} = S_t \\cdot e^{(r-c)(T-t)} ", "9b55d52c7a86a704e97189769bf481e8": "\\scriptstyle \\hat{\\mathbf{e}} \\;=\\; [e_x\\ e_y\\ e_z]^\\mathrm{T}", "9b5607a5b8af7df9b783009490c3e8ff": "\\tfrac{d\\epsilon}{dV}", "9b5614a9e5c45654b4090cd9843c428a": "\\phi(x)=\\sum_{i=1}^p \\phi^{(i)}(x)", "9b5651404092cfeda1fccdd91ff8822a": "V(x,u)", "9b56566fdd2500b75d977fde8249cff2": "-\\frac{\\pi}{2}<\\beta<+\\frac{\\pi}{2}\\,\\!", "9b565f4393376050e13645dc987903d2": "\\operatorname{atan2}(y, x) =\n\\begin{cases}\n\\arctan(\\frac{y}{x}) & \\mbox{if } x > 0\\\\\n\\arctan(\\frac{y}{x}) + \\pi & \\mbox{if } x < 0 \\mbox{ and } y \\ge 0\\\\\n\\arctan(\\frac{y}{x}) - \\pi & \\mbox{if } x < 0 \\mbox{ and } y < 0\\\\\n\\frac{\\pi}{2} & \\mbox{if } x = 0 \\mbox{ and } y > 0\\\\\n-\\frac{\\pi}{2} & \\mbox{if } x = 0 \\mbox{ and } y < 0\\\\\n\\text{undefined} & \\mbox{if } x = 0 \\mbox{ and } y = 0\n\\end{cases}", "9b5686d0524af2a759ecdf01fa378258": "\\langle A_\\alpha: \\alpha \\in S \\rangle ", "9b56d0ec4d1552a782ce11dd63e5598d": "k_{eq} = k_1 + k_2 . \\,", "9b56fc3bf1cce260d3a15721edd72d92": "\\begin{pmatrix}\n W & -A^T \\\\\n \\Lambda A & C\n\\end{pmatrix}\\begin{pmatrix}\n p_x \\\\\n p_\\lambda\n\\end{pmatrix}=\\begin{pmatrix}\n -g + A^T \\lambda \\\\\n \\mu 1 - C \\lambda\n\\end{pmatrix}", "9b57106f0da41b08e27c07809ad17d48": " \\bar { \\mathcal{N} }(\\epsilon) = g(\\epsilon) \\ F(\\epsilon) ", "9b572f4fd8a58db56e4b9f03d14ba6b2": " \\frac{\\partial\\rho(x,t)}{\\partial t}=D\\nabla^2\\rho(x,t) ", "9b5746d991038630910a4cd23da5193a": "K = 1/N", "9b574d08cf5932ce7723ea6d23d31a39": "\\gamma_{ab} = \\xi \\left [ a_{ab} (x,y,z)+O(1/\\sqrt{\\xi}) \\right ], ", "9b575af44ef8c4d1008cfd6cfce50bdc": "\n\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}2\\left[ \\text{Tr}\\left\\{\n\\left( I-\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\right) \\Pi_{\\rho\n,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\}\n+\\sum_{i=1}^{m-1}\\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left( i\\right) },\\delta\n}\\Pi_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}\n^{n}\\right\\} \\right] ^{1/2}\\right\\} .\n", "9b577125d14f4a4343742956f98cc549": "\\mu \\in \\Re \\,\\!", "9b581b9d6a865c5373b6c035e48b257d": "\\delta\\left( q,a,x \\right) = \\emptyset", "9b58788a9ea9512dc1507139dc5a95a1": "y \\in \\mathbb{R}^p", "9b589cde95f31115a4be4e2d7dbd0808": "\\textit{mother}: \\textit{dog} \\longrightarrow \\textit{dog}", "9b58b2dc5f203c5d1b5c9b2e6d455780": "b = {1/2}", "9b5957b5c1628f2f9bccf022c34df5d4": "A_{m_1} \\cap \\cdots \\cap A_{m_p}", "9b5969eee327f220c0f1a628d1921736": "\\sum_{i=1}^3 w_{i} = 1", "9b59ba6e244edcee933957ad5fea8169": "id \\circ t = t", "9b59ed5760e229eec79aaef28df3a6dd": "L_*^{}", "9b5a65819f4d4502ef47070a72e367e0": "\\displaystyle \n \\left(\\frac{2\\pi}{T}\\right)^2=\\frac{2\\pi g}{\\lambda}\\tanh\\left(\\frac{2\\pi h}{\\lambda}\\right)", "9b5a8301c92c0732ff4c285aba01cfbd": " a,b,c \\in \\mathbb Z,n \\in \\mathbb N", "9b5aa19248b4d30e6e870b77fe81c756": "\\overline{\\mathcal{M}}_{g,n}", "9b5aaed50a7dde2d20b2f7d84299c876": "\\textrm{E}[\\xi]", "9b5ab32bc922bf6eb3160fdbc7b87a2e": "z^{T} = (q_{1}^{T},q_{2}^{T})", "9b5ab78abdbd33e29d9d8f285af431c6": "4.67 \\times 10^{1240}", "9b5b28b130fbf1d5a64ee7ffda2cfb28": "\\Lambda_k", "9b5bb035efca2ee0eb7872cfcb81f649": "\\phi(t)=\\frac{\\alpha}{v}\\log{\\left(\\frac{c+vt}{c}\\right)}\\,\\!", "9b5bb80cb201466476d785cb8e9f3a77": " L^{+-} [A] + L^{++-}[A] = L^{+-}[B]. ", "9b5c06017771e104f7c25e7a86fbee16": "\n\\begin{align}\n \\mathbb{E}_{\\mathcal{S}}\\left\\{p_{e}\\right\\}\n &= \\mathbb{E}_{\\mathcal{S}}\n \\left\\{\n \\sum_{a^{n}} \\Pr \\left\\{ E_{a^{n}}\\right\\}\n \\mathcal{I}\\left(E_{a^{n}}\\text{ is uncorrectable under }\\mathcal{S}\\right)\n \\right\\} \\\\\n &\\leq \\mathbb{E}_{\\mathcal{S}}\n \\left\\{\n \\sum_{a^{n} \\in T_{\\delta}^{\\mathbf{p}^{n}}}\n \\Pr\\left\\{E_{a^{n}}\\right\\}\n \\mathcal{I}\\left(E_{a^{n}}\\text{ is uncorrectable under }\\mathcal{S}\\right)\n \\right\\} + \\epsilon \\\\\n &= \\sum_{a^{n} \\in T_{\\delta}^{\\mathbf{p}^{n}}}\n \\Pr\\left\\{E_{a^{n}}\\right\\} \\mathbb{E}_{\\mathcal{S}}\n \\left\\{\n \\mathcal{I}\\left(E_{a^{n}}\\text{ is uncorrectable under }\\mathcal{S}\\right)\n \\right\\} + \\epsilon \\\\\n &= \\sum_{a^{n} \\in T_{\\delta}^{\\mathbf{p}^{n}}}\n \\Pr\\left\\{E_{a^{n}}\\right\\}\n \\Pr_{\\mathcal{S}} \\left\\{E_{a^{n}}\\text{ is uncorrectable under }\\mathcal{S}\\right\\} + \\epsilon.\n\\end{align}\n", "9b5c71b1698721694d92c154d4b15d2c": "z=[(X-b)/(k+\\lambda)]^{1/2}", "9b5c967beb8d03f985bd0aabdb07b33c": "1/(\\sqrt{2}+2)", "9b5cb72a1d81f94d427fbc44feef5aaf": " \\omega_0 \\,", "9b5ced0844dc358fcc9f7b3dfc221857": " \\int_{\\Gamma} \\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta = \\int_{\\Omega} \\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta .", "9b5d77488248578a9308069b4d042b66": "\\mathbf{A} = \\begin{pmatrix}\na_{11} & a_{12} & \\cdots & a_{1n} \\\\\na_{21} & a_{22} & \\cdots & a_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{m1} & a_{m2} & \\cdots & a_{mn} \\\\\n\\end{pmatrix}", "9b5d8c0eee8b2fbf25ceceaa2b9e524e": "A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 ", "9b5dd75032e9309c51c4203dc569b273": "S'=\\wedge^\\bullet W^*.", "9b5e4f24e72c2b011752b1549b94503e": "q = p_a q_b + p_b q_a\\,\\!", "9b5f1e36f7ca7ddf1618194870a56c24": "K_1\\Rightarrow K_0", "9b5f281a6b3611b7430608a3b55e2c13": "O\\left((nm)3 + m^3n^2\\right)", "9b603e7f3caf398e7a87261235119b3b": "\\operatorname{tr}(\\gamma_5 a\\!\\!\\!/b\\!\\!\\!/c\\!\\!\\!/d\\!\\!\\!/) = 4 i \\epsilon_{\\mu \\nu \\lambda \\sigma} a^\\mu b^\\nu c^\\lambda d^\\sigma", "9b60545d4f5ef4f7d5587600938839c6": "d_1 = e_1 u_1 + e_2 u_2", "9b60b7288e9e14b340af7161c38ca511": "\\mathrm{e}^x = \\sum_{n=0}^{\\infty} \\frac{x^n}{n!}", "9b61173557197c273256311a275966d8": "\\mathcal{F} \\left\\{ \\sum_{n=0}^\\infty \\exp (-x^2/2) H_n(x) \\frac {t^n}{n!} \\right\\} = \\sum_{n=0}^\\infty \\mathcal{F} \\left \\{ \\exp(-x^2/2) H_n(x) \\right\\} \\frac{t^n}{n!}. \\,", "9b6148bc5173f2878673c3686b4eb0ff": "u_{1}=H\\left(m\\right)\\cdot w\\,\\bmod\\,q", "9b616ba5bd4f7c25756d7d18870d0702": " f_1 \\propto \\frac{1}{\\sqrt{\\mu}}.", "9b616f22edfd53d4bcc621bb1b8c07a9": " L \\left(1- \\frac{1}{2}\\left(1-\\left(\\frac{L}{H}\\right)^\\alpha\\right)\\right)^{-\\frac{1}{\\alpha}}", "9b618f7e014ead9238faf773381e054c": "0 = F_{[\\alpha\\beta,\\gamma]} = \\dfrac{1}{3!} \\left(\nF_{\\alpha\\beta,\\gamma}\n+ F_{\\gamma\\alpha,\\beta}\n+ F_{\\beta\\gamma,\\alpha}\n- F_{\\beta\\alpha,\\gamma}\n- F_{\\alpha\\gamma,\\beta}\n- F_{\\gamma\\beta,\\alpha}\n\\right) \\,", "9b61da1ae5a5d939dd29b3a1aa184cd8": "{R-r\\over R+r} f(x_0)\\le f(x)\\le {R+r\\over R-r}f(x_0).", "9b6255b53ab6a07c64502c3df5838bdc": "y=y(x)", "9b625bba45b7707a8e2dbe945bc328f4": "y = mx+b_1\\,", "9b62c38d4cfb65b9629694c9437472dd": "b_{j_k}\\in{\\overline{b}}", "9b63208949089f68434f12d1a79ae2d6": "\\mathrm{pv}\\ z^\\alpha = e^{\\alpha \\mathrm{Log}\\ z}.", "9b635505c02fe106f2e426ff42c50baf": " \\frac {y} {Nk - 1} \\pm \\frac {y^2} {(Nk - 1)^2 (Nk - 2)} ", "9b63a176f1c4c6d7f13209c169a2a53d": "r_1 = r_2 >> 1 \\,", "9b63a1d9fa2a896d2e38a44abeed19b3": "m_{1/2}", "9b6466b58419f0f4001ca38f717bba88": "BW\\cong f_H\\,", "9b64abf11f6b83a0927bc0004f0d98b1": "\\phi=1/\\gamma", "9b64e1c075595256b6a38a28ce029907": "A=\\{a_1, a_2,\\dots,a_m\\}", "9b64eb044c54a1280bde8c98cabf5496": " \\frac {R_f} {R_b} = \\frac {2 A_0 + 3} {A_0 - 3} \\,", "9b64f40d875a6cb40ef703bc2d483423": " \\displaystyle \\bigcup_{n = 1}^{\\infty} A_{n} = \\bigcup_{n = 1}^{\\infty} A_{\\sigma(n)} ", "9b6524b80abbecdec2c4d0efcb37193d": " (-1)^k\\chi(M^{2k})\\geq 0. ", "9b658e341b79a8edff71e8406ed1b4c9": "|\\psi\\rangle_A |\\psi\\rangle_B |A\\rangle_C = [\\alpha^2 |0 \\rangle_A |0\\rangle_B + \\beta^2 \n|1\\rangle_A |1\\rangle_B + \\alpha \\beta (|0\\rangle_A |1\\rangle_B + |1 \\rangle_A |0\\rangle_B ) ] \n|A \\rangle_C", "9b65c28bcb59715e5f365989d4f7ca7a": "\\begin{matrix}\\mathrm{Cabtaxi}(10)&=&933528127886302221000&=&77480130^3 - 77428260^3 \\\\&&&=&41337660^3 - 41154750^3 \\\\&&&=&18421650^3 - 17454840^3 \\\\&&&=&10852660^3 - 7011550^3 \\\\&&&=&10060050^3 - 4389840^3 \\\\&&&=&9877140^3 - 3109470^3 \\\\&&&=&9781317^3 - 1318317^3 \\\\&&&=&9773330^3 - 84560^3 \\\\&&&=&8444345^3 + 6920095^3 \\\\&&&=&8387730^3 + 7002840^3\\end{matrix}", "9b65d9683f7420abd11f7c30221c5a7a": " |q\\rangle ", "9b6631d983ef868edf056d17cba0f8b5": " 0 < \\theta < \\pi ", "9b663e0420c76b83b87620b042c76087": " w = \\frac{1}{x} - \\frac{ v^2 }{ 2 \\mu } = \\frac{- \\epsilon}{\\mu} ", "9b665d88ffb5997950c290059874e109": "\n\\nu = \\operatorname{Im} \\operatorname{arccosh} \\frac{\\rho + z i}{a}\n", "9b669f1da241917fb13f7fdb8724b9e9": "\\mbox{Average trade debtors} = \\frac {\\mbox{Opening trade debtors} + \\mbox{Closing trade debtors}} {\\mbox{2}}\n", "9b66a4016393202d8b6a589f12a49589": "\n\\nabla^2 \\phi = \\frac{1}{h_1 h_2 h_3}\n\\left[\n\\frac{\\partial}{\\partial q^1} \\left( \\frac{h_2 h_3}{h_1} \\frac{\\partial \\phi}{\\partial q^1} \\right) +\n\\frac{\\partial}{\\partial q^2} \\left( \\frac{h_3 h_1}{h_2} \\frac{\\partial \\phi}{\\partial q^2} \\right) +\n\\frac{\\partial}{\\partial q^3} \\left( \\frac{h_1 h_2}{h_3} \\frac{\\partial \\phi}{\\partial q^3} \\right)\n\\right]\n", "9b677c436aecb4811b93f452c57e6361": "\\mu_0, \\lambda", "9b6783accf9a46fa7391bd93d84a6ef0": "\\vec{u} = \\vec{e}_0", "9b679436f6d5f208686a4207698de950": " D-CA^{-1}B \\, ", "9b67ab2700523500d4c4697e3d3f4204": "\\scriptstyle d_i/d_{i+1}<0", "9b67b9768379a422739874f16389265b": "K= \\sqrt{\\frac{(a+b)^2(a-b+2c)(b-a+2c)}{16}}.", "9b681978f11eccf601e03f20c6fd6490": "q=-\\exp(-\\pi\\sqrt{3})", "9b6822a06a919b95b4d2818f9973d330": "\\Delta f = \\nabla^2 f =\\nabla \\cdot \\nabla f =\\operatorname{div}\\operatorname{grad} f, ", "9b68e69828803f97a9eeeaed30fda081": "f^h_\\mathbf{k}", "9b68e775065e3fd7bc9b0feb871b423d": "P_{max}", "9b68f0ec0b88364a50fdc7f86d8b42aa": "\\omega_{jk} = \\omega_{kj}^{-1} = e^{2\\pi i \\nu_{kj}/N_{kj}}", "9b68fdf0cdcea41a68aa52275c3dde9a": " \\kappa_t(N)-N ", "9b690946f702163f84680f74cb73c081": "\n \\quad (2) \\qquad \\frac{u_i^{n+1} - u_i^n}{\\Delta t} + a \\frac{u_{i+1}^n - u_i^n}{\\Delta x} = 0 \\quad \\text{for} \\quad a < 0\n", "9b6937835d4328a01b108845f508ae65": "N^2\\,", "9b69852c8080f6831d36fcd3639f796f": "\\tau A^2", "9b69be9fe8f9292ed053bfae3b49e9fb": "\\mathfrak{p} = \\mathfrak{p} R_\\mathfrak{p}", "9b6a0377eea0778458adfad0e6f263b7": "R^i f_* \\mathcal{F}|_U = 0, \\quad i > \\operatorname{dim} f^{-1}(s).", "9b6a03d3c0aee3346784086147265899": "(U, \\mathcal{O}_U)", "9b6a06299fe462b936a8eacfe6b209f0": "\\arctan(z)=z-\\frac{z^3}{3}+\\frac{z^5}{5}-\\frac{z^7}{7}+\\cdots .", "9b6a22e23bcf198eb286dd6e563b35b8": " (x+1)^2-(x-1)^2= 4x", "9b6a95449c665da9c0ac97f1023afbe1": "M_{\\mathfrak{p}}(V):=\\mathcal{U}(\\mathfrak{g})\\otimes_{\\mathcal{U}(\\mathfrak{p})} V", "9b6aa3b3ec9855685b4f39b0d8821fe7": "k_s > \\delta_\\nu\\,", "9b6aaa033083b3704a21f09ff34da6ce": " S^* = S / Nk ", "9b6afc8cc26ecc3ba0ba88b576537e3c": "B_{SO}", "9b6b11bbb0bb54f44807d9bfd70a3def": "\\int_{[0,1]\\times[0,1]} f(x,y)\\,dx\\,dy", "9b6b2e1b7920b7225c674d8f5588eb75": "-A^{\\pm 3}", "9b6b3833bb42ef6a67e97e325ef79cc2": "r \\approx 1 ", "9b6b5386add442aea7e555596c0a1878": "0 \\subsetneq (y_1) \\subsetneq (y_1, y_2) \\subsetneq \\cdots \\subsetneq (y_1, \\dots, y_d)", "9b6b8d32573550c2044590e36bab56c2": "\nK = - V \\left( \\frac{\\partial P}{\\partial V} \\right)_T. \\qquad (2)\n", "9b6baa4ad18b310295f25bde60ae5f4d": "\\hat{T}_f (\\varphi)= T_f(\\hat{\\varphi})", "9b6bc9b19c651a670411c422575efdbb": "\n\\sum_{n=1}^{\\infty}(\\zeta(4n)-1) = \\frac78-\\frac{\\pi}{4}\\left(\\frac{e^{2\\pi}+1}{e^{2\\pi}-1}\\right).\n", "9b6bd5ea15ace568b848b74d737627e9": "\n (\\varphi_{1\\tau} + a'(\\varphi_0) \\varphi_1 \\varphi_{1\\theta})_{\\theta}\n = \\frac{1}{2} a'(\\varphi_0) \\varphi_{1\\theta}^2.\n", "9b6bdcc3cf4c15673f07d0767af8e99a": "m_{tot}", "9b6bff22665baa0b594c9f266f448806": "\\langle\\psi|0\\rangle_B=0", "9b6c30692cc75ce63a8b6186438ef76e": "\\textstyle\\binom{n-t}{k}/\\binom{n}{k}", "9b6c86d5c7e98cad39724f8818797673": "\\log W", "9b6c9539024dd61492cc5fda6c459289": "44.537^\\circ", "9b6cbc4be827addec0eddb6b361e3ce3": "P \\not\\equiv_{b} Q", "9b6d1718809d1df994a897d8dc463ef3": "\\mathbf A = A_x \\mathbf{\\hat x} + A_y \\mathbf{\\hat y} + A_z \\mathbf{\\hat z} \n = A_r \\mathbf{\\hat r} + A_\\theta \\boldsymbol{\\hat \\theta} + A_z \\mathbf{\\hat z}", "9b6d34b1dd9e670ef174ba83469dabbe": "\\textstyle (1/c)p(x)", "9b6d7bb950c95d65cc564002c9fae4dc": " \\frac{\\hbar \\omega}{2} \\left(-\\frac{d^2}{d q^2} + q^2 \\right) \\psi(q) = E \\psi(q)", "9b6dd9434092a2bc85f6c85387c2f8a3": " f_t(z)", "9b6e45fd9c31f39634c5b1354ce4ed0a": "a f^*\\left(\\frac{x^*}{a}\\right)", "9b6e513aea472140c0a8017fc86fd37a": "\\omega_0 = \\sqrt{k_F / m}", "9b6e53237ad22a4c0325c6cfc8b1818d": "\\lambda \\|B\\|,", "9b6e7f668061e29982e5edfb94e08199": "\n\\sigma_{ij} = 2 \\mu(T) S_{ij} - \\frac{2}{3} \\mu(T) \\delta_{ij} S_{kk}\n", "9b6e844cc5c462dcc608dd959e415e2c": "Decrease.space(units.space)=unit.space(unit.space)*(num.users.individual.cons(scalar)-num.max.rent.units.stocked(scalar))", "9b6ec1dbe936549f791b7b5e15daca67": "\\frac{2}{R(t)}", "9b6f14592997b8831d87e0d21470eddb": "r \\approx i - \\pi.", "9b6f5a4e79cbdea0362e056225196bdc": "\\vec{v},\\vec{w}", "9b6f5cc4c8e6f4669cc2196d20a70677": "I_0=\\left|\\frac{A e^{\\mathbf{i} k g}}{g}\\right|^2", "9b6f8132a993268c81fb79f372d0ab51": "P(H_1)=P(H_2)", "9b704ad65c1bc619457705fa60c5ebdc": "\\lambda_1\\geq\\lambda_2\\geq\\cdots\\geq\\lambda_j", "9b7074c57d612c65ff3d3f8b6c261d80": "(e'_p\\otimes e_q)e_q=(e_q.e_q) e'_p = e'_p\\qquad(1)", "9b70c5a7cde6fbda9dc65589d02a0f78": " H(i,j)=\\sum_{r=1}^{\\infty} r \\sum_{k \\neq j}((M_{-j}^{r-1}))_{ik} m_{kj}", "9b70c963bf74cdcecedf15ba1d4dfe53": "\\mathbf{DTIME}\\left(o\\left(\\frac{f(n)}{\\log f(n)}\\right)\\right) \\subsetneq \\mathbf{DTIME}(f(n))", "9b7119f3e705d6988db45dc6f5680609": "H(H(u))(t) = -u(t)", "9b71730e582006619b172082f1be042e": "y=f(x,i,j)", "9b71e240d0fd001c0120221373abb177": "f(x_1 ... x_n)", "9b71ee7335d1b2afb7bbc1df72fe35ea": "\\|x\\|_{X_0 \\cap X_1} := \\max ( \\|x\\|_{X_0}, \\|x\\|_{X_1} ),", "9b7220cd9092bce95d20e70a040ee23e": "q\\phi(\\mathbf{r},t)", "9b72229dc472a55f47eabc9043cb6c3a": "\n\\begin{align}\nV &= \\frac{R}{2}\\sqrt{I^2 - I_0^2},\\\\\n &= \\frac{R}{2}\\left(I^2 - \\left( 2I_c\\cos\\left(\\pi\\frac{\\Phi}{\\Phi_0}\\right)\\right)^2 \\right)^\\frac{1}{2},\n\\end{align}\n", "9b7274ca27f4c65816b8bdf03b3682bb": "x^{x^{\\cdot^{\\cdot^{\\cdot}}}}", "9b7359b1b5d8bc13da9fc91f13ba0686": " E_{G}=\\sum_{i=1}^{N}\\,{\\frac{{{\\vec{p}}_{i}^{2}(t)}}{{2m_{i}}}} +V(t),\\qquad {\\vec{P}}_{G}=\\sum_{i=1}^{N}\\,{\\vec{p}}_{i}(t), ", "9b738c6e0f6e4852cb805aab9298eed3": "E_k = \\alpha + 2\\beta \\cos \\frac{2k\\pi}{n}", "9b73bc55116ee801297746fa0164817d": "(p,a,b,G,n,h)", "9b73d074aa6ce717db03ad20abcb690e": "\\nu\\in \\mathrm{Tan}(\\mu,a)", "9b73f8d02b8b75f191ba1d4b55ada891": "6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509)", "9b743794ccb386ad7e0ee7805b553751": "b \\to c\\;", "9b756e95d4eaf34a66a667771362fc9b": "P(X_1=\\omega_1, X_2=\\omega_2,\\cdots, X_n=\\omega_n)= p^k (1-p)^{n-k}", "9b7603821987b9237adfc457ff48875d": "(\\lambda,P,Q)", "9b766c85022723173d5081c45710ecea": "\\tbinom {2n-1}{n-1}", "9b766ccee2d117ea5189015d35935a0d": "\\rho(x) = A e^{-U/ (k_B T)}", "9b76d3d6afb228e3979fe61acdd64890": "(y,y,\\ldots,y)", "9b7728b8794fd05b09e2245ef85da4cc": "x_0*w_0", "9b7733a6bea68ebd7353f893438c2191": "H(\\kappa)", "9b77e9d0619f175ce7e4d354a92dc2c4": " O(n \\log(n))", "9b781856c86d97f8a1a4f255703fc875": "\\operatorname{E}[\\,z_t(y_t - x_t'\\beta)\\,]=0", "9b785d60b8263ef9acc97b118679601a": "t/m^3", "9b787730d4f4ce7f4b5d88adf6c1d48e": "\\ \\sum_iw_i", "9b789802773f166428f687d68204d09f": "\\phi^{-1}(U) \\cap \\phi^{-1}(V) = \\phi^{-1}(U \\cap V)\\;", "9b78cbb858ab8b0285f7f2e98f02de29": "\\forall i \\in \\left( \\left[ 0, l - 1 \\right] - M \\right), {n_N} \\ge {n_i}", "9b78d9f79e7381959df5048cf8f015d1": "9 \\div x", "9b7913a00f7791968f3b5273e1c7578a": "\n\\frac{a r_2}{1 - b r_2} = g\\left( \\frac{\\theta_2}{k} \\right)\n", "9b79179593e1ff7d10d2cc3b7e11441f": "y_{I_{1}1}", "9b79802691b4749218be9f3b85d0b6c6": "\\mathbf{p} = (p_1,p_2,\\dots,p_K),", "9b7a49c5a3996812f6e052b033549506": "\\mathfrak{su}_8", "9b7a776c03c411164a1bc1d40b84d29d": "\\left [ 0,1 \\right ] ", "9b7a944c03e2e61078c68763c06b59d9": " y=C \\left \\{ x^{-\\alpha} {}_2F_1 \\left (\\alpha, \\alpha +1-\\gamma ; 1; x^{-1} \\right ) \\right \\} +D \\left \\{ x^{-\\alpha} \\sum_{r=0}^{\\infty} \\frac{(\\alpha )_{r}(\\alpha +1-\\gamma )_{r}}{(1)_{r} (1)_{r}} \\left( \\ln \\left (x^{-1} \\right )+\\sum_{k=0}^{r-1} \\left( \\frac{1}{\\alpha +k}+\\frac{1}{\\alpha +1-\\gamma +k}-\\frac{2}{1+k} \\right) \\right) x^{-r} \\right \\}", "9b7ad119f0f6d29277867721a681110a": " a(t) = \\frac{\\mathrm{d}^2 x}{\\mathrm{d}t^2} = - A \\omega^2 \\cos( \\omega t+\\varphi).", "9b7b3c7242be157c0bc12d3918714147": "log Y_i", "9b7b761ab0a8d8b3808975f0d13d7a03": "\\lambda^2 + 1 = 0 \\!", "9b7b7a918f085d28144ff6b9e1577793": " S_i = \\{ j \\in N: \\pi(j) \\leq i \\} ", "9b7b90b47d568297815be57c644d1add": " {|\\langle\\psi(0)|\\psi(t)\\rangle|}^2 = {||c_+|^2 + |c_-|^2 e^{\\frac{-i(E_{-}-E_{+})t}{\\hbar}}|}^2 = 1 - 4|c_+c_-|^2 {\\sin}^2(\\frac{\\omega t}{2})", "9b7bc618a8e16819ad33cab9264e1c5e": "G_{i+1}\\setminus G_i", "9b7bd7fcfe91b27d0dee0a0519a1bce6": " \\mathbf{Frob} (\\mathfrak P) ", "9b7c2c1b6480e2d310ea4ee911c21817": "\\Delta=b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.\\,", "9b7c639654a0cad9e71ae1a47a64ac21": "\n\\begin{align}\n\\frac{\\pi}{4} =& 183\\arctan\\frac{1}{239} + 32\\arctan\\frac{1}{1023} - 68\\arctan\\frac{1}{5832}\\\\\n& + 12\\arctan\\frac{1}{110443} - 12\\arctan\\frac{1}{4841182} - 100\\arctan\\frac{1}{6826318}\\\\\n\\end{align}\n", "9b7c6f63bf32bdd0611002604e345f08": "(\\log^6(n))", "9b7c73f8e3032870da8502c93255da8d": "\n4.71 \\left (\\frac{\\mbox{characters}}{\\mbox{words}} \\right) + 0.5 \\left (\\frac{\\mbox{words}}{\\mbox{sentences}} \\right) - 21.43\n", "9b7c80c803544e35961924866688a306": "\\phi_x (y) = \\left\\langle y , x \\right\\rangle \\quad \\forall y \\in H ", "9b7cb024a7f96d99d4ae6093b8b86fc3": "(x^i):M\\rightarrow\\R^n", "9b7cea20a741fddfed1de3f92c8cc0dd": "\\epsilon^{\\sigma \\mu \\nu \\rho}=0", "9b7d01936f8820d2a5679d15aa9dcc73": "\\frac{d}{dt}M(t,t_1) = -A(t)^{T}M(t,t_1)-M(t,t_1)A(t)-C(t)^{T}C(t), \\; M(t_1,t_1) = 0", "9b7d06bc4494aa252738356337f4d700": "\\mathbb{C}[t,t^{-1}]", "9b7d173b068dc4d5517bfae92d676437": "PL", "9b7d5d77d9d5c1ae7efd455b242b4634": "v_\\star", "9b7d7bc9132bc55e98f6152a02de17c8": "I = \\lim_{\\Delta s_i \\rightarrow 0} \\sum_{i=1}^n f(\\mathbf{r}(t_i))\\Delta s_i.", "9b7d8d6b6215e28c852329b45d221013": "L = \\ell^2(\\mathbb{R})", "9b7d914b2f458c7b3709d9826eb1f7aa": "\\scriptstyle A=\\begin{pmatrix}1 & 2\\\\ 0 &1 \\end{pmatrix} ", "9b7ddb09c69e97526499d6e69fb73546": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{r}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^2 u\\ \\cos u\\ du\\ = \n\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^2 u \\cos^2 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^3 u\\ \\cos u \\ du\\ = \\\\\n&\\hat{g}\\ \\left(\\int\\limits_{0}^{2\\pi}\\ \\sin^2 u \\cos^2 u \\ du\\ +\\ \n3\\ {e_g}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^4 u\\ \\sin^2 u \\ du\\ \\ +\\ \n3\\ {e_h}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^4 u\\ \\cos^2 u \\ du\\ \\right) \\\\\n&+\\hat{h}\\ 6\\ e_g\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^4 u \\ du = \\\\\n&\\hat{g}\\ \\left(2\\pi \\left(\\frac{1}{8}\\ +\\ \\frac{3}{16}\\ {e_g}^2\\ +\\ \\frac{3}{16}\\ {e_h}^2\\right)\\right)\n+\\hat{h}\\ \\left(2\\pi \\left(\\frac{3}{8}\\ e_g\\ e_h\\right)\\right)\n\\end{align}\n", "9b7e2d0a905527daf47039828af7d70b": "\n \\frac{ u }{{ u}^2} ( u v - u \\cdot v)\n= \\frac{1}{ u} ( u \\wedge v )\n= \\hat{u} ( \\hat{u} \\wedge v )\n= ( v \\wedge \\hat{u} ) \\hat{u} \n", "9b7e44ff9632948e0c5daa02bb9cf4de": "\\begin{bmatrix}\n c_3 c_1 - c_2 s_3 s_1 & s_2 s_3 & c_3 s_1 + s_3 c_2 c_1 \\\\\n s_1 s_2 & c_2 & - c_1 s_2 \\\\\n -c_2 c_3 s_1 - c_1 s_3 & c_3 s_2 & c_3 c_2 c_1 - s_3 s_1\n\\end{bmatrix}", "9b7ea90eba97f54cfd12949c1500d0fa": "\\sqrt{\\left\\langle \\left(\\hat{x} - x_0\\right)^2 \\right\\rangle} \\sqrt{\\left\\langle \\hat{p}^2 \\right\\rangle} \\geq \\frac{\\hbar}{2},", "9b7f0a322b95c24c00c0e346cc77c1a1": "\\sigma_a := \\sqrt{\\mathbb E\\|X - \\mu\\|_\\alpha^2}", "9b7f1ad2bc908b62d555cd356db0e797": "3 \\times (8 \\times 8)", "9b7f1fa32aa51cc4b1d57f80e9b2fcbc": "\\phi'", "9b7f5ce01def73823b933008e80bb818": "[\\mathbf{b}]", "9b7fbcfc9e5cc58ba7f30c712ccf6e3a": "\nx = \\frac{r\\mu\\nu}{bc}\n", "9b804ff4badf668f1fc9f494d4dd2bc7": "x\\langle y\\rangle", "9b8078a3a6cd293df6d935209f92151c": "{m2 \\over A1}", "9b8107598dac2901659c423e2fe94100": "\\scriptstyle \\frac{5+3\\sqrt{5}}{5}", "9b8122d9844de27fe4650507eabb92cf": "\\displaystyle TM", "9b81b0cf38649a1cac170601785908cd": "\n \\boldsymbol{S} = S^{ij}~\\mathbf{b}_i\\otimes\\mathbf{b}_j = S^{i}_{~j}~\\mathbf{b}_i\\otimes\\mathbf{b}^j = S_{i}^{~j}~\\mathbf{b}^i\\otimes\\mathbf{b}_j = S_{ij}~\\mathbf{b}^i\\otimes\\mathbf{b}^j\n ", "9b81ed74f7ad00aad4583624f0837865": "\\begin{bmatrix}\n1 & 0 & 0 & 0 & 0\\\\\n0 & 4 & 0& 0 & 0\\\\\n0 & 0 & -3& 0 & 0\\end{bmatrix}", "9b81f007419ceba649a8da5bcee11b3d": "\\mathrm{Da} = \\frac{ \\text{reaction rate} }{ \\text{convective mass transport rate} }", "9b82228ff4c92754db2d68fae9a00f5a": "Reliability = 0.5 \\times \\left( 1 + e^{ \\left( - \\lambda \\times Time \\ Between \\ Maintenance \\ Actions \\right)} \\right)", "9b826bb5d6651594f2494f608b28b32f": "\\lambda\\ ", "9b827de52f5b090557c79a88cd7d3fdc": "I_{in0} = \\frac{V_1}{Z}", "9b8290d12b1df4afe469e856d8c41146": "\\gcd(57-13,77) = \\gcd(44,77) = 11 = q", "9b829402b10bfe6eafbf6b68fbcdf450": "\\displaystyle\\Delta_{\\alpha<\\delta} X_\\alpha", "9b829f4acc0aae696d207f7439dbc6cb": "S_j ", "9b83001ec9a246ab2c3fb0a37a482cfb": "\n x_r = \\sqrt{ H_{rr}(y) } \\left( y_r + \\sum_{j=r+1}^n y_j \\tilde{H}_{rj} (y)\\right), \\qquad x_j = y_j, \\quad \\forall j \\neq r. \n", "9b833c9e946a4557648eeaed318caf0a": " A \\to X \\to X/A \\to \\Sigma A \\to \\Sigma X \\to \\Sigma \\left (X/A \\right ) \\to \\Sigma^2 A \\to \\cdots. \\, ", "9b8356e41f1eae8392aa101681126475": "\\,\\log\\left(\\Gamma(x)\\right)\\,", "9b837681da12a00c09766f08f0183b17": "_{P}(f) = 0", "9b83830ed2c48f47b1ba428afd90b9e5": "\\Pr(A|B_2)={95 \\over 100}", "9b83a1590d244712c6737117b1779d9d": "{}\\mathcal{F}", "9b83e91d6b08d617b8af16642c8427db": "\\displaystyle{}_{r}G_r(a_1,...a_{r};b_1,...,b_r;q,p;z) = \\sum_{n=-\\infty}^\\infty\\frac{(a_1,...,a_{r};q;p)_n}{(b_1,...,b_r;q,p)_n}z^n", "9b83e94e1df13775876760159216eceb": "\\propto g G^a_\\mu \\bar{\\psi}_i \\gamma^\\mu T^a_{ij} \\psi_j\\,,", "9b83ebdd47ffc684fa86a6292abf5f61": " \\frac{ X_1+\\cdots+X_k }{ \\sqrt{r_1^2+\\cdots+r_k^2} } ", "9b84120d06280cc4307742bb3b4f05ca": "\\{ a^n b^n c^n d^n : n \\geq 0 \\}", "9b845b6619005ee10642e2d31a4027ef": "R^m", "9b845b6619a2c8e97b86bc4b19d854de": "\\sigma_1 = 2\\mu\\varepsilon_1 + \\lambda(\\varepsilon_1 + \\varepsilon_2 +\\varepsilon_3)", "9b849f233a461cb3376974712d68c222": "\\scriptstyle \\dot q", "9b84b50b47ae9de41c6126233fca4f9f": "s(y)", "9b84ddd24ab51ff4262e061c685bdf9d": "D_{\\mathrm{KL}}(P(\\theta)\\|P(\\theta_0)) = \\Delta\\theta^j\\Delta\\theta^k g_{jk}(\\theta_0) + \\cdots", "9b85426e33dd9e6fd4fec6d6a3e34577": "\\; E_{jj}", "9b85eef10fac8c5f73862a2c3f19ec38": " k = \\omega \\sqrt{LC} = {\\omega \\over v} \\ ", "9b85f3a989d1dbbe6225c29ba05396c5": "\\prod _x \\tan x \\cot (x+1) = C \\cot x \\,", "9b864f53b95968bbe7eccc4ef83855b1": "\\scriptstyle{E(C_1^s + C_2^s)=1}", "9b865429335745fb847c6d3e7d3e3a2d": "n^{\\frac 23}", "9b865ebb1038aad9ca30898301ac647e": "U=h(u)", "9b86879ab4483975dfa792778b749d1a": "m\\colon Q\\to Q", "9b86a92a06893b2c17f9f2b0f0ca5ba9": " \\mathbf{r}(0), \\quad \\mathbf{\\dot{r}}(0). ", "9b86cea3a70b28dedacc4cd07a747a9f": "(V_{\\xi})_{\\xi\\in\\kappa^{+}}\\,", "9b8723d67f2fe8202b5f8e39f38c093e": "dB_t \\,", "9b87823bf3a40d33c110f7e7458c29cc": "f_{pn} = {1 \\over (2.0665 - 1.0665p/100) }", "9b87e189ef52e38b91148b40ea5cbe72": "f(t^*,t)=d_t\\;(d_t=\\sum_{(v,t)\\in E}f(v,t))", "9b87f91409c7a5ac7581452483cb77d3": "{\\mathbb P}\\biggl( \\bigcup_{i=1}^n A_i \\biggr) \\le \\sum_{j=1}^k (-1)^{j-1} S_j,", "9b880cff536c798a8702bd3ea0f12dfa": "\\psi_3", "9b8812b076a07be2b3b1bc3662faafa2": " K \\approx n_N-2.48", "9b886211f606e2ff14be4c879c616336": "\\sin x/x", "9b88a501ed8c4c9b603c044cbab50d2c": "t=0,1,2,\\ldots,T,T+1", "9b88c9d14181a0c6e5e3c7b959ee5fa6": "\\kappa^2", "9b88dd94a3541d8df527b77ecfb5b0b9": "\\mathbf{a b} \\equiv \\mathbf{a}\\otimes\\mathbf{b} \\equiv \\mathbf{a b}^\\mathrm{T} = \n\\begin{pmatrix}\n a_1 \\\\\n a_2 \\\\\n a_3\n\\end{pmatrix}\\begin{pmatrix}\n b_1 & b_2 & b_3\n\\end{pmatrix} = \\begin{pmatrix}\n a_1b_1 & a_1b_2 & a_1b_3 \\\\\n a_2b_1 & a_2b_2 & a_2b_3 \\\\\n a_3b_1 & a_3b_2 & a_3b_3\n\\end{pmatrix}.", "9b88ff71cd65f9cb79d392b718e56334": "{\\bar{S}}_6", "9b890bc22a183f6d80be0d7163cd038b": "\\chi_2(\\mathbf{r};\\mathbf{R})\\,", "9b894e719c4c417a05dc131140b55d40": " \\lnot ( \\lnot x \\oplus y)\\oplus y = \\lnot ( \\lnot y \\oplus x) \\oplus x.", "9b897ef20afadcf9cf7c45da5bea1090": "\\mathbf{x} \\prec \\mathbf{y}", "9b8a0fcb8660cf0254878de4ff976c91": "Q_{\\ell m}'=-\\frac{ik}{c(\\ell+1)}\\int d^3\\mathbf{x'} r'^\\ell Y_{\\ell m}^*(\\theta', \\phi') \\mathbf{\\nabla}\\cdot(\\mathbf{x'}\\times\\mathbf{M}(\\mathbf{x'}))", "9b8a15b4f61378566cf93ed355a7946f": "\\gamma(s, z) - \\frac{1}{s} = -\\frac{1}{s} + z^s\\,\\sum_{k=0}^\\infty \\frac{(-z)^k}{k!(s+k)} = \\frac{z^s-1}{s} + z^s\\,\\sum_{k=1}^\\infty \\frac{(-z)^k}{k!(s+k)},\\quad \\Re(s) > -1, \\,s \\ne 0", "9b8a165c4b33b80f20181d5e607b3033": "V_{\\rm permitted} = \\sqrt{ 2 \\cdot decel \\cdot (XG-dist)} ", "9b8a2020f18c0b8db558b50a738da3e6": " V_2 = k_2 [E_{2T}], ", "9b8a38f95020f39bc2ba67408341ff4e": " \\varlimsup_{n \\rightarrow \\infty} f(n) = \\infty ", "9b8a4b8404e5baed439c367df3f7ecb0": "\\tilde I(\\omega)=I(\\lambda)\\frac{-d\\lambda}{d\\omega}", "9b8a5f9ca5d889d523d2dd20d9f33414": "-\\frac{\\Theta''(\\theta)}{\\Theta(\\theta)}=L,", "9b8a8428efa57dd0098d0be6b4e0aaae": "\n\\begin{align}\n\\Delta_1(p) &= \\begin{vmatrix} a_{1} \\end{vmatrix} &&=a_{1} > 0 \\\\[2mm]\n\\Delta_2(p) &= \\begin{vmatrix}\n a_{1} & a_{3} \\\\\n a_{0} & a_{2} \\\\\n \\end{vmatrix} &&= a_2 a_1 - a_0 a_3 > 0\\\\[2mm]\n\\Delta_3(p) &= \\begin{vmatrix}\n a_{1} & a_{3} & a_{5} \\\\\n a_{0} & a_{2} & a_{4} \\\\\n 0 & a_{1} & a_{3} \\\\\n\\end{vmatrix} &&= a_3 \\Delta_2 - a_1 (a_1 a_4 - a_0 a_5 ) > 0\n\\end{align}\n", "9b8af5b2b785a807bcdfdb26768b8d2b": " V_\\parallel(\\beta) = \\frac 1 q \\vec e_s \\cdot \\int \\vec F_L(s,t) \\,\\mathrm{d}s = \\frac 1 q \\vec e_s \\cdot \\int \\vec F_L(s, t = \\frac{s}{\\beta c}) \\,\\mathrm d s ", "9b8b121c4c281456805418a6d9e68cb4": "f(0)^2=f(x)^2+f(-x)^2\\,", "9b8b14636854557842036b7d549002ef": "K_d = \\frac{\\alpha^2}{1-\\alpha} \\cdot c_0 \\approx \\alpha^2 c_0 ", "9b8b16ac5ecba505adc4087cc7e860f7": "\\gamma(t) = \\frac{1}{\\sqrt{1 - |\\boldsymbol{\\beta}(t)|^2}}", "9b8b37a220d274b2c1d49beae060409f": " a_r = - R\\omega^2, \\quad a_t = R\\alpha.", "9b8b3b01ccf0dd7a32c4c0d48287d0ba": "\\operatorname{Li}_2(1-z)+\\operatorname{Li}_2\\left(1-\\frac{1}{z}\\right)=-\\frac{\\ln^2z}{2}", "9b8b48021cb2d07fb7a763bc2d2cf546": "\\{\\pi-\\theta,\\pi+\\phi\\}", "9b8bba77d883964ad7b635f6f898cb8c": "\\ {D \\mathbf{u} \\over D t} = \\mathbf{F} - {\\nabla p \\over \\rho} ", "9b8bc511b96e3671d3a3c987fbc55796": "F \\in \\mathcal{A \\otimes A} ", "9b8bdb78ff7427c835b75a687aa18267": "\\tau = \\frac{(\\text{number of concordant pairs}) - (\\text{number of discordant pairs})}{\\frac{1}{2} n (n-1) } .", "9b8be1490c7e41eca4f54f67b4d079c9": " \\text{Maximize} \\,\\, pQ(p) - wL - rK \\,\\, \\text{with respect to} \\, L, \\, K, \\, \\text{and} \\, p", "9b8cd1f7cbd1eae64188cbcffce71239": "\\mu_{T,e_1}", "9b8cd28d857e45855a2529c07df59442": "\\partial=\\pi_{p+1,q}\\circ d", "9b8d1bb242bd6fc4dd4ce94aae88ecef": "(gZ)", "9b8d52ab86d5910e12b9d6d3f4352a58": "t_v(x)", "9b8d64d91aa8995f4cfa906c4ff9e96d": "P(A)=\\Sigma_1^* \\times \\Sigma_2^* \\times \\cdots \\times \\Sigma_n^*", "9b8db09c9041b491538805c784125131": "2^n/4", "9b8e565b136400ed9827b35f96da5283": "\\exists a\\, (a=a)", "9b8e89bd8db10100c353d6bc678ebe4d": "L_1(x) \\otimes L_2(x) = L_1(L_2(x))", "9b8eb1b2ea8ad6fa2426e8bb3173d056": "\\frac{V_3}{V_4}=\\sqrt{\\frac{Z_{I3}}{Z_{I4}}}e^{\\gamma_3}", "9b8ec91aa818bfb5f9321fc891d13f8f": "\\ D_{GR}", "9b8f3edb22dc9ade86528dd7bbfe65f7": "E_N(\\rho)", "9b8f6d085cdaa195562885f83e3ab0ff": "M(A,B)", "9b8f7bf18d48b1b0e3cc4a1e150b6235": "\\displaystyle x(p,w)", "9b8fb0a44a28b7f0c25bc9e36851e566": "\\hat{H}\\Psi(x,t)=\\left(E_{0}c_{0}(t)\\Psi_{0}(x,t)+E_{1}c_{1}(t)\\Psi_{1}(x,t) + e\\epsilon(t)x\\Psi(x,t)\\right)", "9b8fde46ae91e318ca4fbafcf7585a19": "m_1 \\,", "9b8fefe61f64ca48091b3d8f0b73433d": " \\left(\\frac{\\partial S}{\\partial V}\\right)_{T} = \\left(\\frac{\\partial p}{\\partial T}\\right)_{V} ", "9b904e305d13ab5f98d075a350c78396": "u(c,l) = \\frac{c^{1-\\gamma}}{1-\\gamma}- \\psi \\frac{l^{1+\\theta}}{1+\\theta}", "9b913d0c8e96a04af1dc67c02a719326": "S_+|+\\rangle=0", "9b914ffeca7bce8ca30d9147be7e2d85": "\\{V_1, \\dots,V_n\\}", "9b916705eba26724133c60ed60bbaa43": "\\mathcal{H}_{i}\\Psi\n=0", "9b916efa233e4937179b1b715a556024": "d\\le \\log_{\\varphi} n", "9b91819824f9c54e967f9a31ce53d294": " y^3 + {5 \\over 2} \\alpha y^2 + (2 \\alpha^2 - \\gamma) y + \\left( {\\alpha^3 \\over 2} - {\\alpha \\gamma \\over 2} - {\\beta^2 \\over 8} \\right) = 0. \\qquad \\qquad (4) ", "9b91b4b5f060608cb6869531e352cf87": "\\underline{u} : A", "9b91b920f08bec7999d810e0497624b7": "\\binom{m}{k}", "9b91f4fbd05bf4b350911b5cc212701a": "\\frac{1}{2\\sqrt\\pi}", "9b91fc241783a41914777521b10d15d4": "\\|f(x)-f(y) \\|^2 \\leq \\, \\langle x-y, f(x) - f(y) \\rangle.", "9b9229680e5569612579d80071ce52ae": " (\\mathbf{v}\\cdot\\mathbf{a}) \\mathbf{v} = v^2 \\mathbf{a}_\\parallel ", "9b928cf020f9faeb9fef4e28b1bb758a": "X \\to [0, 1]^{C}", "9b92aa91716d08a2731fda3815af10f2": " \\rho_f: G_\\mathbb{Q} \\rightarrow \\mathrm{GL}_2(\\mathcal{O}),\\ ", "9b92ce8f2684cc30140e67589adf4d7d": "(\\mathbb{R},\\leq)", "9b92d8ed72e29ac88c3cc8e21940c6bd": "{\\partial F(x,y,p)\\over\\partial p} = \\dot{f}^\\star(p) - x = 0~.", "9b930104219a7087d987d670fdd95d4a": "\\begin{bmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & -1 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & 0 & -1 & 0 & 0 & 2\n\\end{bmatrix}.", "9b93069ea632036e5207e8e15c03a20a": "n=0,1,2...", "9b93082564fdf4d29000bf28098d7598": "\\boldsymbol{r} = (\\sigma_1, \\ldots, \\sigma_i, \\ldots, \\sigma_N)", "9b9310f516e2d0378479f4edc57fac37": "~S~", "9b931a15e2ffa76bc66142954b6d2f22": "M = E/v^2,\\quad L = S v/E, \\quad t = S/E \\,\\!", "9b93495919cee82b61a3ecd36035d791": "\\displaystyle{\\Psi^m\\cdot \\Psi^m \\subseteq \\Psi^{m+n},\\,\\,\\,\\, [\\Psi^m,\\Psi^n]\\subseteq \\Psi^{m+n-2}.}", "9b93990d04b45413ea592eab6727cdfa": "\\Delta w''=w''(x+)-w''(x-)", "9b93e66877447c48bdbaccc07c0896e3": "\\textstyle T = \\left( \\begin{array}{cc} 3 & 4 \\\\ 2 & 3 \\end{array} \\right)", "9b94624cf33a39ad2292c210f1d956c7": " \\eta = \\mathbf e_{i} \\;,", "9b9523da6b4eac06242bfecb38b6876b": "f_m(x)\\,", "9b953d33691ebcce3c65da7c8dc82fe6": "\n=\n\\begin{bmatrix}\n (\\mathbf A^T \\mathbf P_B^\\perp \\mathbf A)^{-1} \n & -(\\mathbf A^T \\mathbf A)^{-1}\\mathbf A^T \\mathbf B(\\mathbf B^T \\mathbf P_A^\\perp \\mathbf B)^{-1}\n\\\\\n -(\\mathbf B^T \\mathbf B)^{-1}\\mathbf B^T \\mathbf A(\\mathbf A^T \\mathbf P_B^\\perp \\mathbf A)^{-1}\n & (\\mathbf B^T \\mathbf P_A^{\\perp} \\mathbf B)^{-1}\n\\end{bmatrix}\n", "9b954ed6a633844f20666082ac1879c3": " \\pi=P(Q)Q-C(Q)", "9b958cf7fd1eb9dee6826a5b0d9339fa": "\\frac{2m}{\\hbar^2}\\left(V(x)-E\\right)", "9b958f0d3c2aacd4e1747006dbba7674": " \\mu_{\\mathbf{v}} = {\\mathbf{0}} ", "9b95b5499a3ff061c61597f53c6b950a": "\\int_{\\Gamma \\cup \\Omega'} \\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta = 0", "9b960102386f91fed0c6361aae3a1c29": " p\\trianglelefteq q \\text{ if and only if } p_1+\\cdots+p_k \\leq q_1+\\cdots+q_k \\text{ for all } k\\geq 1.", "9b96757a356d8b6ad54ed973291c4573": "(9)\\qquad \\mathcal{L}_{\\ell}\\theta_{(l)}=-\\frac{1}{2}\\theta_{(l)}^2+\\tilde{\\kappa}_{(l)}\\theta_{(l)}-\\sigma_{ab}\\sigma^{ab}+\\tilde{\\omega}_{ab}\\tilde{\\omega}^{ab}-R_{ab}l^a l^b\\,,", "9b967a17145d99a889fc8ca5bd21160c": " 0 = (x^2+y^2+z^2 + R^2 - r^2)^2 - 4R^2(x^2+y^2) . \\,\\!", "9b9705ba86c1a60a09f7b43d87770361": "\nS(\\rho \\| \\sigma) = - \\operatorname{Tr} \\rho \\log \\sigma - S(\\rho) = \\operatorname{Tr} \\rho \\log \\rho - \\operatorname{Tr} \\rho \\log \\sigma = \\operatorname{Tr}\\rho (\\log \\rho - \\log \\sigma).\n", "9b9744106d7d439ebb333b9b660d09d5": "\\beta\\, = 1", "9b9761b03d959e30262cc193d538117f": " f(k;N,K,n) = f(k;N,n,K) ", "9b97a29b8f65340215df8549fa0249fd": "O(L)", "9b98430cbbd6e47f56795c6e000fe6bf": "\\sigma_{\\mathrm Y}", "9b98c3a1050191d36574b4d28de5dd61": "\\sigma_{31}=\\sigma_{23}=0\\!", "9b98c56deb73075ea4b8ab99b947c1e6": " C = M_aT= \\frac{P_0 rT}{1-e^{-rT}} ", "9b98f7ec1a815f2731b36523374ff5a7": "P(\\vec x|\\vec y)", "9b98fcfb462d5df40ebf75046a261e74": "10\\uparrow^3 3 = (10\\uparrow^2)^3 1", "9b991dda95aa591a9c45d002489a588d": "d(z)", "9b9a37ecccc20e8ec0e15603319518c0": "R_{uv}", "9b9ad34287af17a85ff11fcc595e3887": "f^\\sharp: \\mathcal{O}_X \\to f_* \\mathcal{O}_Y", "9b9afe4d220be9b1ab41dfbb02d7d69f": "\n\\mathbf{m}_i^{\\phi} = \\frac{1}{l_i}\\sum_{j=1}^{l_i}\\phi(\\mathbf{x}_j^i).\n", "9b9b47d99790091a345062b542dc5cd6": "k \\mathrm{height}(v)", "9bbcf0ff2b9212b70cf7afae83d5ca9f": "U=\\exp \\left( \\frac{iq\\Lambda}{\\hbar c}\\right)", "9bbd0961adbb5440b80098c7d9d0eb9c": "\\operatorname{char}K\\ne2", "9bbdc0f68831b812f4251e60cc2c51bb": "(1+x/100)+2", "9bbded3f753d52036e5889a42eae0746": "\\Lambda_1\\cap\\Lambda_2=\\emptyset", "9bbdf306dd47632a923f89c424c713c4": "(\\Omega, \\pi)", "9bbe29b84ebd9b78420c201dadc2454b": "\\left\\{\\begin{matrix} n \\\\ 1 \\end{matrix}\\right\\} = \\left\\{\\begin{matrix} n \\\\ n \\end{matrix}\\right\\} = 1. ", "9bbe3aaf96a577943d7c9e2c044f7df1": "Tr(ab)=0", "9bbe78fd6aaae89a19d6d4c286ffd552": "\\operatorname{Spec} K[x]/(x-c) \\cong \\operatorname{Spec} K", "9bbf4f2c94879d1c1184010141be4dc4": " m_{fu} = \\frac{f-f_{st} }{1-f_{st}}\\left(m_{fu, 1} \\right) , f_{st} < f < 1 , m_{ox} = 0 ", "9bbf5a31cc4d95ab4c5b6819d45d68dc": "\\mathrm{proof}_{D}", "9bbf679dfc3aca8e0694207f339a0b2e": "kU_*(X)", "9bbf7ba4abd94edcb6c36608695eb1ae": "{m \\choose m}_q ={m \\choose 0}_q=1 ", "9bbf8f11f624dd7185070d200d458d40": "\\lnot \\textit{fem}(m)", "9bbfa9e41449d6ff507b1c6583d87891": "U_{\\mathrm {spring}} = \\frac{mP}{2\\beta^2 \\hbar^2} \\sum_{t=1}^P | \\mathbf{r}_t - \\mathbf{r}_{t+1} |^2", "9bc002d2e2c2e59b55a4746f9c9ef3f0": "M = k \\sqrt{L_1 L_2} \\! ", "9bc027e31bb1beac9fcc73babc2ce6d1": "\n\\left( x_{s} - x_{2} \\right)^{2} +\n\\left( y_{s} - y_{2} \\right)^{2} =\n\\left( r_{s} - s_{2} r_{2} \\right)^{{2}}\n", "9bc02d11c2b2e42f1dd190b2a9169931": "p_\\lambda=0", "9bc07f469f73edb5a66cfc5a7420e920": "\\mathbb{Z} \\cap [1, n]", "9bc0a2e70cd7e99bb0c463eb133e9e15": "q_{ij}=\\begin{pmatrix}\n\\alpha_{ij}&\\beta_{ij}\\\\\n-\\overline{\\beta}_{ij}&\\overline{\\alpha}_{ij}\n\\end{pmatrix},", "9bc103d38ed212cfb28f62c4e6662ac8": "x \\Rightarrow \\exists \\kappa. x", "9bc10436b9944671f42e6471cf348343": "E_r", "9bc11c2763cb978f878535c93b8fb50c": "\\mathbb{I}_R(B)=(B:B)", "9bc1c1c71939b07ddf8d91cca42501f9": "\nV_\\mathrm{out} = \\frac{R_2}{R_1+R_2} \\cdot V_\\mathrm{in} \n", "9bc1fd264c4f4a1dbb0f792e432f0567": "\\theta_a", "9bc22ed1979ef24fd85458ce60f107b9": "\\scriptstyle\\tan(\\delta-90^\\circ)=-\\tan(90^\\circ-\\delta)=-\\cot\\delta", "9bc23ccd4ab2203ad5a911002a805e8a": "\\displaystyle{{1\\over \\pi} \\,\\iint_{\\Omega\\cup\\Omega^c}{\\overline{\\partial_z D(\\varphi)(z)}\\over z-w}\\,dx\\,dy= D(\\varphi)(w).}", "9bc27c67c1fcbf2293622a234e4176d8": "\\prod_{i=a}^b f(i)", "9bc2a1061b53f7c0decf30b4523c5183": " = (ac-bd) + (bc + ad)i \\ ", "9bc31d8b7eeb954d8271c85eb7cfb6ba": " y_i = M y_o \\, ", "9bc3a0541a2ae5edec53a6db92d066b8": "\\frac{2}{2n+1}", "9bc3dc1ae0249ff516d3a58349c540d5": "2\\times2\\times\\cdots\\times2\\times2", "9bc438d96a7db2f646094863fb9cb079": " \\operatorname d M\n= \\frac{\\partial M}{\\partial t} \\operatorname d t + \\sum_{i=1}^n \\frac{\\partial M}{\\partial p_i}\\frac{\\partial p_i}{\\partial t}\\,\\operatorname d t.", "9bc481d43156ff3685413f06999c649e": "\\tau^{a b}{}_{;c}\\equiv \\partial_c \\tau^{a b}+\\Gamma^a{}_{c d}\\tau^{d b}+\\Gamma^b{}_{c d}\\tau^{a d}", "9bc49159f33a2e980b97676d39026610": "H(u) = h*u", "9bc4b8ce50d0254cb39922feae49ea40": "6 \\sqrt{N}", "9bc4ddefc858be87854925871834d575": " Lu = \\sum_{|\\alpha| \\le m} a_\\alpha(x)\\partial^\\alpha u\\, ", "9bc4dfb634c957c19e87b688386047d7": "Q(p; x_0,\\gamma) = x_0 + \\gamma\\,\\tan\\left[\\pi\\left(p-\\tfrac{1}{2}\\right)\\right].", "9bc5135deacd43463c6d6192da353a6b": "\\left(\\frac{\\partial U}{\\partial y}\\right)_x = T\\left(\\frac{\\partial S}{\\partial y}\\right)_x - P\\left(\\frac{\\partial V}{\\partial y}\\right)_x", "9bc532559046c4b56762d32dddf9c99f": "\nP(X \\in A \\mid Y = y_0) = \\frac{\\int_{x\\in A} f_{X,Y}(x,y_0)\\,dx}{\\int_{x\\in\\Omega} f_{X,Y}(x,y_0)\\,dx} .\n", "9bc5c964d65bdd52d5faf9bcd8ca23b5": "a_{3}=(5/6)*d", "9bc5cb6be624ca3767c2ac8508ce8018": " \\alpha", "9bc6243a24b16890d445ded3bc3941bc": " Q=g^q ", "9bc68b692933487986a080516b7d4cbd": "P_i: [t_{i-1}, t_i] \\to \\mathbb{R}", "9bc698fad0ea612bb327dd7ec4877dcb": "x \\in y.", "9bc6b0a8afb6393add25aa4bcf06d71a": " \\mathbf{d} \\mathbf{c}^{\\mathrm T} = \\begin{bmatrix}\nc_1 d_1 & c_2 d_1 & c_3 d_1 \\\\\nc_1 d_2 & c_2 d_2 & c_3 d_2 \\\\\nc_1 d_3 & c_2 d_3 & c_3 d_3 \\end{bmatrix}\n", "9bc7326e39f9f7c119f3d06c642531ba": "p = p_0 + p_1 X + p_2 X^2 + \\cdots + p_{m - 1} X^{m - 1} + p_m X^m,", "9bc749309489bc9c1b239bff613c2f16": "Q_{rej}", "9bc751ec27d74c9c9e05699546325781": " c_s = -b/a ", "9bc7715115c5aef76b0445870408a7c3": "\\implies \\alpha = \\frac{1}{P(B)}", "9bc79d4808d8e35c999bb3cd35fcf07a": "D(G)", "9bc7b5003395e2c960939f4381b17d28": "\n\\begin{align}\n\\varphi_0^{-1}(X,Y,Z)&=\\left(\\frac{X}{Z+1}, \\frac{Y}{Z+1}\\right), \\\\[8pt]\n\\varphi_1^{-1}(X,Y,Z)&=\\left(\\frac{-X}{Z-1}, \\frac{-Y}{Z-1}\\right),\n\\end{align}\n", "9bc7ebeb5ed0be1e7be780cc15248f41": "\\operatorname{\\ker}(f)=\\{\\,x\\in V:f(x)=0\\,\\}", "9bc84c460f6e928b850b2ae10ffb1126": "\\eta=e^{-\\beta \\epsilon}", "9bc8581d0be000227c489b360d1c8c9e": "\\textbf{x}_{k} = \\textbf{F} \\textbf{x}_{k-1} + \\textbf{w}_{k}", "9bc8c7dd3e1baaad00354d8a76193c26": "rpm_{fan}", "9bc8cb83bd7749c5b575130549f599a0": "\nJ = \\frac{ q D N_c e^{-\\Phi_B/ V_t}\\left[e^{V_a/ V_t} - 1\\right]}{\\int_0^{x_d} e^{-\\Phi^*/ V_t} dx}\n", "9bc8cc6689f8c5851daa42e05fc00f09": " \\mathbf{\\bar v} ", "9bc8e2c8ba4b962a010169d92b7fc00d": "E_{a^{n}}\\equiv\nE_{a_{1}}\\otimes\\cdots\\otimes E_{a_{n}}", "9bc903ba35ba2bd0bc5309ccdd8f8298": "I_1\\,", "9bc944586e8f5f795ebb375505803d37": "T_\\mathrm{sat}", "9bc969e863ebe327c7759a9c21bf952e": "\\displaystyle{\\widehat{\\tau f}(n)=\\sum_m \\widehat{f}(m,n),}", "9bc9bc644e5feff9d37b78d1e3638c88": "v^k \\leftarrow b-Ax^k", "9bc9f247397e6161aae8f5759433c52d": "m = p_1 p_2 ... p_n \\in M", "9bc9f7ad10a0f15aae3387030e22235c": "\\left(\\frac{\\partial U}{\\partial S}\\right)_{V,\\{N_i\\}}=T", "9bc9f96d46c8ce225ff1d518970d31f6": "VF = { \\frac{1}{\\sqrt{LC}} } \\ ", "9bca3872203ea39dadcb34226b5bef75": " C(E) \\leftrightarrow \\forall x C(x) ", "9bca66e41c61995ceae2c2ce70e390ff": "\\{p_2, r_2\\}", "9bca6e9775798944a7ad7cde8f41820b": "\n-i \\hbar \\nabla\n", "9bca86a6cbd8609119157779c67fa396": " \\hat{A}_{m_j} ", "9bcb38447654e8811e7324767f1c9228": "D_i = \\frac{1}{2} \\rho V^2 S C_{Di} = \\frac{1}{2} \\rho_0 V_e^2 S C_{Di}", "9bcb6536de2c6809ab9fdd0c6f48ca34": "\\operatorname{div}\\,\\mathbf{F} = \\nabla\\cdot\\mathbf{F}\n=\\frac{\\partial U}{\\partial x}\n+\\frac{\\partial V}{\\partial y}\n+\\frac{\\partial W}{\\partial z\n}. ", "9bcb7c5f96512b0cfa2673e0e005d8c5": "d_1, \\dots, d_n", "9bcba9cc3fab93e4d2a0581f3148b079": "\\Gamma^{[C]}", "9bcc14ac5fa89d30508847b46dc6d816": "b \\triangleleft a = b^{a}", "9bcc4d852f462d322d94ca20c3c5cacf": "\\Delta\\mu=\\Delta\\mu_\\mathrm{o}-\\frac{4\\alpha\\Omega}{d}", "9bcc5d708dc9ec4808d451ffe2561163": " \\frac{f(x + h) - f(x)}{h} = \\frac{\\Delta_h[f](x)}{h}. ", "9bcca925311661e21030979000664c30": "\\textstyle \\beta_2", "9bccaa787dcb8763e4f94f21608eab3b": "\\chi_s(G)", "9bccad05a0bc180c488b9e7b8356d0ca": "C\\psi(q) = C\\psi(-q)", "9bccd294af4293b12ac5cb42ee9eb9f3": "\\mathfrak{X}(\\mathfrak{D}_0) = \\cup_{d \\epsilon \\mathfrak{D}_0} \\mathfrak{X}(\\{d\\})", "9bccece89346b7a38664d59cc9f289a6": " T' := [T,x] = Tx-xT = -\\operatorname{ad}(x)T,\\,", "9bcd40f14c5f1e30fdb5cd7ba3b2a041": "\\rho(y|x)", "9bcd66389a0fc51b28eb0f6e4908c134": "\\mathbf{A}^{n}=\\mathbf{Q}\\mathbf{\\Lambda}^{n}\\mathbf{Q}^{-1}", "9bcdd5899cbb0451e601ac1e74cd6738": "\\ ee = ((R-S)/(R+S)) \\times 100 ", "9bcdd6040c622c6efd44a37348566fd1": "\\left[J_z, V_q\\right] = q V_q ", "9bcdecfd7b6870ccbd8852e6f8fcd710": "k=k_n = {2n\\pi \\over Na}\n\\quad \\hbox{for}\\ n = 0, \\pm1, \\pm2, ... , \\pm {N \\over 2}.\\ ", "9bcdf6b2bc504aeea57fb948036b7834": "[f(X_i) - f(Y_i)]", "9bcdfce9c8554a64ca65638212250ff8": "a^{\\frac{x^2}{2}}\\,", "9bce04daa49c56f10cc3c88f4c528715": " \\frac{1}{1-z^{-1} }", "9bce25c2e723ac5ec1bc7db7fe1b0ce7": "\\frac{d f}{d t} = \\left(\\frac{\\partial f}{\\partial t} \\right)_\\mathrm{coll}", "9bce65c63044a5e52cd1fed145621a3c": "\\mathrm{E}_1(z) = \\int_z^\\infty \\frac{e^{-t}}{t}\\, dt,\\qquad|{\\rm Arg}(z)|<\\pi", "9bcf0f54cd0c133aaba941753968f0b7": "\\{ s_\\lambda : \\lambda \\in \\Lambda \\}", "9bcf2ac45cf0ccbadb7b01e57b48b212": "\\nu_{i}", "9bcf3ad7e5031d735ccb71b55cff5480": "(0, d)", "9bcf881f5c3bf2a97ed19f78361431d5": "h_i(v_{-i}) = \\max_{b \\in A} \\sum_{j \\neq i} v_j(b)", "9bcf905aa5669c73b77bfc581288328f": "2^3\\times 3^2=72", "9bcfa46d25b14e7acbb007c1dd07f612": "\\frac{4b}{3\\pi}", "9bcfa8baebc214c94fcb622f0be56e72": "F = \\frac{i}{2\\pi} \\mathcal{A}^2 \\operatorname{\\ln}\\frac{\\mathcal{A}^2}{\\Lambda^2} + \\sum_{k=1}^\\infty F_k \\frac{\\Lambda^{4k}}{\\mathcal{A}^{4k}} \\mathcal{A}^2 \\,", "9bcfb13b03e250998b2b0bcbd5016c4d": "\\rho_{nk}", "9bcfee74e7764f79fffbd3223242b2f5": " C\\{f(t)\\} = G(p)\n= \\int_0^\\infty pe^{-pt}f(t)\\,dt\\qquad(3) ", "9bd07e3ddf4297bccd455e3e4c8e52d0": " \\forall s,t \\in \\mathbb{R}: \\quad U_{t + s} = U_{t} U_{s}. ", "9bd0d6e8ab86f313d47627b9450d3f3a": "\n \\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}_{\\mathrm{vp}}}{\\mathrm{d}t} = -\\mathsf{E}^{-1}~\\cfrac{\\mathrm{d}\\boldsymbol{\\sigma}}{\\mathrm{d}t}\n ", "9bd0e42e760859e54b59405ee4ff6898": " \\mathrm{DCG_{6}} = rel_{1} + \\sum_{i=2}^{6} \\frac{rel_{i}}{\\log_{2}i} = 3 + (2 + 1.892 + 0 + 0.431 + 0.774) = 8.10", "9bd158f07aef95e67c1f7dba81848377": " (1 - \\omega^2 \\, r_0^2) \\, dt^2 = 2 \\omega \\, r_0^2 \\, dt \\, d\\phi + r_0^2 \\, d\\phi^2 ", "9bd18f3a352602b184ada541c6d985b9": "r = \\frac{g}{4k^2}\\,", "9bd1aa12289526ad6700b82d18e0364f": "(\\Sigma^{-1}E)_n = E_{n-1}", "9bd1ca94312b8fc1aba4bad75b5533df": "\\scriptstyle (\\Omega,\\; \\mathcal{F},\\; P)", "9bd1d39b5e7c2771cab37d9fab8a28e2": " \\begin{align}\n& \\text{maximize} && \\mathbf{c}^\\mathrm{T} \\mathbf{x}\\\\\n& \\text{subject to} && A \\mathbf{x} \\le \\mathbf{b}, \\\\\n& && \\mathbf{x} \\ge \\mathbf{0}, \\\\\n& \\text{and} && \\mathbf{x} \\in \\mathbb{Z},\n\\end{align} ", "9bd1ed82febf9228434b264342213fa1": "~~\\cos c = \\cos a \\cos b - \\hat{n} \\cdot\\hat{n}'~ \\sin a \\sin b", "9bd209955ed5cb2cc80d783c8bcbf1dc": "N_{NE},", "9bd2322b0d94ee85011e910f27960431": "\\overline\\pi:\\overline Y\\to X", "9bd24c064ac48a60666e66bf754a808d": "\nU(t) = e^{-iHt}\n\\,", "9bd28ad030be6fbffeb11f9a15dae649": " n \\to n + 1", "9bd2d590d348842a855c9a749d252214": "RMD(S) = \\frac{\\sum_{i=1}^n \\sum_{j=1}^n | y_i - y_j |}{(n-1)\\sum_{i=1}^n y_i}", "9bd2de0d80ac02b76ce0de0d73f93484": "\\varphi (t, \\vartheta_{s} (\\omega)) \\circ \\varphi (s, \\omega) = \\varphi (t + s, \\omega).", "9bd2ee655d6999aa0fc47fc7fa13534d": "y_t = c + A_1 y_{t-1} + A_2 y_{t-2} + e_t", "9bd30ae2bd104b53466e15c3b959cadb": "0 < \\alpha < 1", "9bd311d94f9f35adc547c1389a65cd4e": "\\alpha_{\\overline e}:A_e\\to A_{t(e)}", "9bd3d7edb011605d43d8bd13b2fd657d": "R_{xx}(j) = \\sum_n x_n\\,\\overline{x}_{n-j}.", "9bd4482d309b935a9db8040cf448991f": "\\frac{1}{1-K_\\text{max}} = \\beta_\\text{min} < \\beta < \\beta_\\text{max} = \\frac{1}{1-K_\\text{min}}", "9bd47112dd861dbb4981a83b21d0cfa3": "y.\\ y \\in \\{ 0, 1 \\} \\rightarrow x:=y", "9bd4797f274b1e375be37d8b101fceea": "A = k[t_1, \\ldots, t_n]", "9bd4b2acdc49fe6cc416f0a1a8d7388b": "y_i = (\\alpha_{i_1}, \\ldots , \\alpha_{i_k}, a_{i_1}, \\ldots , a_{i_d})", "9bd52a7a768df82522ac07828e28a21c": "2 \\leqslant k \\leqslant n", "9bd5ca8a6029057c1a5fbba8c7d3d542": "(v|u)\\in C", "9bd5da2b1abf670dba2754fff15bc2f2": "\\sum_{i=0}^{n} i!\\cdot{n \\choose i} = \\sum_{i=0}^{n} {}_{n}P_{i} = \\lfloor n!\\cdot e \\rfloor", "9bd61ea90245f18cb21daefcd2b1bacf": "n(n+1)(n+2)(n+3)(4n+1)", "9bd62cdc8e96491619b9505524604f76": "\\hat H = - {{{\\hbar^2} \\over {2 m_e}}\\nabla^2} - {1 \\over {4 \\pi \\epsilon_0}}{{e^2} \\over {r}}", "9bd6691332a3b8531140688700167a6d": " J \\colon \\pi_r(\\mathrm{SO}) \\to \\pi_r^S , \\,\\!", "9bd6f5815c3aa33192b6bcae3dbb438e": "\\mathrm{_{10}^{20}Ne} + \\mathrm{_2^4He} \\rightarrow \\mathrm{_{12}^{24}Mg} + \\gamma + Q", "9bd79a0d87bc4abdc06ab078bbe352d9": "E\\epsilon_x(y) = \\frac {-E\\epsilon_my}{c}", "9bd7e328536e999cff84a9ac7c5ccbcd": "G_f", "9bd85cf153cee04c03db69fa60de3858": " \\mathbf{F} = m \\mathbf{a}.", "9bd89b6749be9c8c6bcc9922c88c7c07": " \\alpha_2 = \\arctan \\left( \\frac{\\cos U_1 \\sin \\lambda}{-\\sin U_1 \\cos U_2 + \\cos U_1 \\sin U_2 \\cos \\lambda} \\right) ", "9bd89cd5c65ad6c3f4f81f7d3916a996": " X_t = X_0 + \\int_0^t\\sigma_s\\,dB_s + \\int_0^t\\mu_s\\,ds,", "9bd8acde5c9eff08183f4f9b7a58ecbd": "\\langle x - a_i\\rangle", "9bd8aff179dcea286a953241c479e626": "f(x;\\alpha,0,c,0)", "9bd90d724ab20995d2b3c41876b6fef0": "t_D = t_O = 0", "9bd91f2e49a18ee7f1ea14f4fa891ba2": "[g,h]^s = [g^s,h^s]", "9bd925f201d2b8a1b088e9771f7454c0": "\\sharp E(\\mathbb{F}_{q})", "9bd93ab6baad349da492c396c890948b": "|\\psi(t)\\rangle = U(t) |\\psi(0)\\rangle", "9bd94b2786e61aea54b9ce2a5d41e28b": "x = a \\sec(\\theta),\\quad dx = a \\sec(\\theta)\\tan(\\theta)\\,d\\theta, \\quad \\theta = \\arcsec\\left(\\tfrac{x}{a}\\right)", "9bda06c427027e2286034d5ea8627362": "\nk=1,...,n_c\n", "9bda2047269205b34f5bab2acfd7376e": "U_{es} = - 170 V", "9bdaaeb3f457d802799f4baa64c9b0d9": "0 \\le BC \\le 1", "9bdabfa58f97615d7746572d4e834273": "\\rho = \\frac {r - \\bar{a}} {1- \\bar{a}}", "9bdaf3c529bba44aa6c7fb7a5289c39a": "\\{A_k\\}_k", "9bdb34c7d1bd3c550a355a43edbe730d": "(a + b) = (b + a)", "9bdb84c0e1093a2c9f269eb0b85e2b36": "\\mathbf{S}:\\left( \\nabla \\otimes \\overline{\\boldsymbol{u}} \\right)", "9bdb9407deed9ff536b5cdc257d8d6f6": "c_\\theta", "9bdba68a0bc622e3824edf9521e3e50e": "\\scriptstyle Q", "9bdbb9cfc3d92577394f25a32189293b": "DCG_{6}", "9bdc4cd6f52cb17dd610848801ff0858": "SS(w)=s=syn(x_w)", "9bdc525a6157ca87badfb6fe587295d0": "\n \\hat{u}_j = \\frac{\\partial x_j}{\\partial q^m}~u^m ~;~~\n \\hat{v}_k = \\frac{\\partial x_k}{\\partial q^n}~v^n ~;~~\n \\mathbf{e}_i = \\frac{\\partial x_i}{\\partial q^s}~\\mathbf{b}^s\n", "9bdc7a3a98d80e743adb9b71f7de6f7a": " S_n=a_1+(a_1+d)+(a_1+2d)+\\cdots+(a_1+(n-2)d)+(a_1+(n-1)d)", "9bdcb4a0a4fa1fd6eb29ecb3a6bd95c3": " \\sum_{i,j} de_i de_j' u^2_x = -2dq dq' w_x w'_x ", "9bdd73097ff7050e2d4ad5ce4db7983f": "\\lambda>1", "9bdddc229d990af90f8322ed4ff46265": "\\ \\sigma ", "9bde107256332700a617eeed4aaa94ea": "\\mathbf e_i", "9bde7b8c13b006ad07a80193f12560e5": "x_1 = 1.000000014487979", "9bdec9333f8899622ab24c2fe5626eb4": "x=e^t.", "9bdece3a16a7e5b6f67f0957b2498f03": "z = x + i y", "9bdee7a392989b5e24e925df810d332b": " \\hat{x}_{i} ", "9bdf056ff86725ccd0580cddf67aa7b2": "\\beta = \\frac\n{\\left|P_1-P_3\\right|^2 \\left(P_2-P_1\\right) \\cdot \\left(P_2-P_3\\right)}\n{2 \\left|\\left(P_1-P_2\\right) \\times \\left(P_2-P_3\\right)\\right|^2}", "9bdf2b609a948c530cfd3cbf536fb6a3": "(\\sigma_{ij} - \\bar{\\sigma}_{ij}) \\mathbf{\\hat{n}} = - \\gamma \\mathbf{\\hat{n}} (\\nabla_{\\!S} \\cdot \\mathbf{\\hat{n}}) + \\nabla_{\\!S} \\gamma \\qquad ; \\quad \\nabla_{\\!S} \\gamma = \\nabla \\gamma - \\mathbf{\\hat{n}} (\\mathbf{\\hat{n}} \\cdot \\nabla \\gamma)", "9bdf34f4b2785298a6326a3d941ef162": "\\{w_{ni}\\}_{i=1}^n", "9bdf6b4f2f578fa8e9a14707a7ea4e5c": "\\alpha^{p^n}\\in F", "9bdf9b1433dcf0652f883307a61b015c": " f(x) = x^r,", "9be025323511d741f66219c27d3128ef": "\\mathbf{aaaaaa}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{281DAF40}\\,\\xrightarrow[\\;R\\;]{}\\,\\mathrm{sgfnyd}\\,\\xrightarrow[\\;H\\;]{}\\,\\mathrm{920ECF10}", "9be02ec7897cbf7f3bef3be16618da86": "h \\gg 1", "9be03925b5d8a1d5169b95450c93ae0d": "\\Bigl(\\coprod_i [0,1] \\times X_i\\Bigr) / ((0,x_i) \\sim (1,f(x_i)))", "9be07ad667045728b7c1889fa089126c": "v_\\perp", "9be087cb7a902b75f2c54c1508df2a2c": " \\chi^2 = \\sum_{i=1}^n {\\frac{(O_i - E_i)}{E_i}^2}", "9be0dd9e953397a9ad367d2a1f3f8f60": "x\\in U,", "9be1128c99baa23ec7352ae67508d021": "\\alpha_{fine}\\approx 1/137", "9be11abcf3c9e4dd0a2dc9ec81b6205c": "v_1 = 7", "9be12b6fc320cceb483d1966a712feb8": "C_\\mathrm{max}=\\frac{n_R^2}{\\mathrm{NA}_I^2}", "9be14844973ee8a8fbf7028993b0acdc": "\\mathcal{E}_{q}^{\\epsilon}(d)\\,\\!", "9be17ea6049cde5b23d1ed515a616cc3": "x^2\\frac{d^2y}{dx^2} + ax\\frac{dy}{dx} + by = 0. \\,", "9be1898b509fd499f9f6f616c0c52fb5": "\\ln b", "9be1b224bfce7a8159e1d5dcd44fc4cb": "(\\overline{m_1})", "9be1b8557e27b17995fa44bbbc932bde": "\\scriptstyle M\\ddot{o}\\times[\\frac{1}{2}-\\varepsilon,\\frac{1}{2}+\\varepsilon]", "9be1d0d31cd438a26abcdf5153d370cc": "g_{A,B}", "9be1e64582cc02f94d0c1a69fe1b23d8": "p(c_j)", "9be1ecc4acef99d1b9324cbff76de609": "\\log_2(1.0)", "9be1f3cdf2a52c79e941d650ad4b571f": "n_1\\sin\\theta_1 = n_2\\sin\\theta_2", "9be1f65939f14a7839e55a1218ee7e14": "(\\mu * \\nu)(E) = \\int\\!\\!\\!\\int 1_E(xy) \\,d\\mu(x) \\,d\\nu(y)", "9be219b7f2790dadf7644c3c72437cb7": "\\operatorname{Tr} (\\gamma^\\mu\\gamma^\\nu) = 4\\eta^{\\mu\\nu}", "9be26c68d5a748347364583f0db9ae10": "g(x) = f(x + x_0) - x_0", "9be29597d95fd5e9cbb281258f48cb14": "{\\textit{VAR}_\\text{tot} = SS_\\text{tot}/(n-1)}", "9be2b312aeb2085a6811d75d276a406c": "v_{k}", "9be360e0e7ffb2213033a008be77af51": "B \\wedge (p-q) = 0", "9be3b728f00b19b6f60a3f1bd06b5591": "E(s) = \\gamma(s) + R(s)\\mathbf{N}(s) = \\gamma(s) + \\frac{1}{k(s)}\\mathbf{N}(s).", "9be3ba48d9a3c8e8800071818ecb284e": "\\tfrac{5 \\times 17}{18 \\times 17}", "9be3f967374f56f1511bd7bf97b5a96c": "A^1 B_0{}^0 C_{00} + A^1 B_0{}^1 C_{10} + A^1 B_0{}^2 C_{20} + A^1 B_0{}^3 C_{30} + D^1{}_0{} E_0 = T^1{}_0{}_0 ", "9be42189d2e9e84cc0e93e174e492274": " g(\\tilde{\\nu}- \\tilde{\\nu}_{0}) ", "9be42207f2d4383d377211ec29eb8a1a": "V'_{in} = V_{in} - \\beta \\cdot V_{out}", "9be43fa03fa2cffb763c48a404c7df9f": "E_{F_B}", "9be440b8fd3704f66b4494f1aeb8b6ca": "\\alpha q > 2\\sqrt{n}", "9be444278afb0d9dc4c8f4bf0c6c1c68": " R_{\\mu \\nu}=0 ", "9be4551bbf805e56b30357e9b5e44dc7": "P_{I, M'}", "9be46e5d18b34352b39448bdd301b129": "\\sum_{i=0}^{n-1} {i \\choose k} = {n \\choose k+1}", "9be4a77afae92696cafff4adf4751a24": "\\lnot S", "9be51b6783876e8fc66d5f096de6c20a": " S_L \\; \\longrightarrow \\; \\alpha L -\\gamma +\\mathcal{O}(L^{-\\nu}) \\; , \\qquad \\nu>0 \\,\\!", "9be51ed11f90f5a9bfa0b05d051ca143": " \\langle H \\rangle = \\left[-2Z^2 + \\frac{27}{4}Z\\right] E_1 ", "9be56ca8588fe696c956e8d16eaebd92": "G\\;", "9be58d242cc8dfd9fe8861eb3506893d": "h(u_2,\\dots,u_n|u_1)", "9be5ddf0638c6f5f77a2e9923dfb24e0": "\nf \\colon [-\\epsilon, \\epsilon ]\\times [0,1] \\longrightarrow M,\\qquad (s,t)\\longmapsto \\exp_p(tv+tsw_N),\n", "9be63146b3724371e2396c1a8a4dcac1": " v_P = \\begin{cases}\n v_1, & \\text{for }k=0 \\\\\n v_N, & \\text{for }k=N \\\\\n v_k+d(v_{k+1}-v_k), & \\text{for }0 < k < N\n \\end{cases}\n", "9be631beb80a0e357fe0c78e2c9e2f4a": " i_{\\text{B}} = \\frac{I_{\\text{S}}}{\\beta_F}\\left(e^{\\frac{V_{\\text{BE}}}{V_{\\text{T}}}} - 1\\right) + \\frac{I_{\\text{S}}}{\\beta_R}\\left(e^{\\frac{V_{\\text{BC}}}{V_{\\text{T}}}} - 1\\right)", "9be6561584b5d226219c51127ce2024e": "\\scriptstyle x^*", "9be660651dc7d43715b18d2244719070": " e^{\\pm i 2 \\pi \\frac{r}{\\hbar} \\sqrt{2 m E} } = 1 = e^{i 2 \\pi n}", "9be692c45ab1b6eefad001dd871d5c82": "\n\\sum_{(k_1,\\dots,k_m)} G(k_1,\\dots,k_m) \\, t_1^{k_1}\\cdots t_m^{k_m} \\, = \\,\n\\frac{1}{\\det (I_m - TA)},\n", "9be6ef46630f67c094b2c50cff021dc5": "{210 \\choose 1} = {21 \\choose 2} = {10 \\choose 4}", "9be6f4339d2263742ed7dda47bf8bde0": "S^N \\equiv \\{ f\\colon N \\to S \\}", "9be72e179c5db0cfc4b4b17b951f84cf": " \\overrightarrow{S} ", "9be76f4ea7335b9fa169bd5f65a1987b": "P_{A,B}", "9be7840fde050ddc369c08d4ddf339ad": "Z_{GNM}=Z_X Z_Y Z_Z = \\frac{1}{\\sqrt{(2\\pi)^{3N} | \\frac{k_B T}{\\gamma} \\Gamma^{-1}|^3}}", "9be7c09e283c690f7bac7b6160cb42f9": "\n\\bar s = \\frac{1}{3} \\left( \\sin (355^\\circ) + \\sin (5^\\circ) + \\sin (15^\\circ) \\right) \n= \\frac{1}{3} \\left( -0.087 + 0.087 + 0.259 \\right) \n\\approx 0.086\n", "9be7c508561887418a1c0b5ffaa3b292": " x^2= n . ", "9be7dd5cc416135f195f93e3eb002754": "X(f) = \\mathrm{rect} \\left(\\frac{f}{f_s} \\right) \\cdot X_s(f)\\ ", "9be7de1b3baa9153ac51e399c7ab2715": "(R^q f_* \\mathcal{F})^\\wedge \\to R^p \\widehat{f}_* \\widehat{\\mathcal{F}}", "9be7f431da19011f34fc5276cc21ad5e": " {n \\choose k} + {n \\choose k+1} = {n+1 \\choose k+1},", "9be80c9d59ef9477a0ec16149f83e434": "\\bar P = \\frac{1}{N} \\sum_{u=1}^N \\sum_c \\frac {n_{cu}(n_{cu}-1)}{m(m-1)} = \\sum_c \\frac{o_{cc}}{mN}", "9be86427778f33a43ad7af4462a5e546": "\\scriptstyle U,V", "9be870615cb94d1b940846bf1f70b1d0": "Priority = \\frac{waiting\\ time + estimated\\ run\\ time}{estimated\\ run\\ time} = 1 + \\frac{waiting\\ time}{estimated\\ run\\ time}", "9be8a09235a01680744cac649fdda3d3": "\\ f^{\\prime\\prime}(x) = 0", "9be8f5e12cbb348a8b43f16b2ecf16f1": "(P \\and Q) \\to R", "9be91a8df2b7b9298f9e493878d1ffaa": "abc \\sqrt{1-\\cos^2\\alpha-\\cos^2\\beta-\\cos^2\\gamma+2\\cos\\alpha \\cos\\beta \\cos\\gamma}", "9be93bbe87e1a11ef417075c259d485b": " P(j,t,q) ", "9be96f3c27b9d90410961acdc9672539": "y'=A(x)y\\,", "9be997958039bc3e524f330a68e6a96d": "W = Fd = mad = ma \\left(\\frac{v_2^2 - v_1^2}{2a}\\right) = \\frac{mv_2^2}{2} - \\frac{mv_1^2}{2} = \\Delta {E_k}", "9be9a8198b851442f71d05b73892f34e": "\\frac{dx}{dt} = r x (1-x)", "9be9d82f6a0d2f7835e4553201fa6071": "\\{1,\\ldots, n\\}", "9bea062256715c2c287952b5ba8d564e": "a = d \\ne b = c, \\alpha = \\zeta = 120 ^\\circ, \\beta = \\epsilon \\ne 90 ^\\circ, \\gamma \\ne \\delta \\ne 90 ^\\circ, cos \\delta = cos \\beta - cos \\gamma", "9bea5687b6ea36a5c9142f10f2088d00": "n_1 = 1 = \\sum\\limits_{\\alpha_l=1}^{\\chi_c}(c_{\\alpha_{l-1}\\alpha_{l}})^2(\\lambda^{[l]}_{\\alpha_l})^2 + \\sum\\limits_{\\alpha_l=\\chi_c}^{\\chi}(c_{\\alpha_{l-1}\\alpha_{l}})^2(\\lambda^{[l]}_{\\alpha_l})^2 = S_1 +S_2", "9bea8051ee03173546f08487fd784c23": "0 < \\alpha ", "9bead34b72c4b0be067d04c4c819cebc": "f_0,\\ldots,f_m", "9beaee7b3cc155bb968b5683bd9f017e": "\\textstyle F_y", "9beaf95e986139267b181a20f928bc43": "\\left( \\frac{\\partial E}{\\partial v} \\right)_k", "9beb4f7f2ce571ba8f50960cf5bdbded": "R_1 ^ 2 = 1 - \\frac{\\sum (y - Y_r) ^ 2 }{ \\sum (y - Y_{a1})^2}", "9beba365dc3e598d6a700c8746be66a2": "ID = \\log_2 \\left(\\frac{2D}{W}\\right).", "9bebd2eb0e482c5296a88192e05b5b3b": "A = \\left(\\begin{array}{cc}0 & -1\\\\1&0\\end{array}\\right)", "9bebffe1438f30acaf58d38320264a9c": " q \\ ", "9bec423907d627838397d4ed20a0aff9": "\\dot S_i\\leq 0", "9bec59e2399dea16492fe2ac0f091db8": "b_0^\\mathrm{MSSM}", "9becc5119231063484d80d8964b75810": "\\boldsymbol {F} -m \\sum_{j=1}^{d} \\sum_{i=1}^{d}v_j{\\Gamma^k}_{ij}\\dot q_i \\boldsymbol{e_k} =m\\tilde{\\boldsymbol{a}}\\ , ", "9becf311fabe398d7e829f741463228c": "\\phi_{c_1,c_2,t}\\iff\\phi_{c_3,c_4,\\lceil t/2\\rceil}", "9bed0d7c4c8daa77d61565d86be9b937": "\\alpha_{A^-} = \\frac {K_a}{[H^+] + K_a}", "9bed386c08c03e9a798d6ee9e344faba": "\n\\mathbf x\n= \n\\begin{bmatrix}\n\\mathbf A^T \\\\ \\mathbf B^T \n\\end{bmatrix}\n^{+}\\,\n\\begin{bmatrix}\n\\mathbf e \\\\ \\mathbf f \n\\end{bmatrix}.\n", "9bed4bb67e5b1988ce91c4f33974094a": "\\mathbf{H^r}=\\bigoplus_{p+q=r}\\mathbf{H}^{p,q}", "9bed94fb96a894434374ed264f88cd7b": "\\Lambda.", "9bedb40d796ec0a04d1f10222118a4a2": "\\frac{x^n -1}{x-1} = \\sum_{i=0}^{n-1} x^i = 1 + x + \\cdots + x^{n-1}", "9bedd278d4b2a32477328f17bb74a9a8": "\nv'_{0.5} := \\frac{v_1 - v_0}{h}, \\ v'_{n+0.5} := \\frac{v_{n+1} - v_n}{h}\n\\,\\!", "9bedf3672126e4d7ff5707545a00a1b1": "F_1,\\dots,F_n", "9bee111503afaeed434e48b28214e8df": "\n\\sigma _z^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {2\\,x\\,y} \\right)^2 \\sigma _x^2 \\,\\,\\, + \\,\\,\\,\\left( {x^2 } \\right)^2 \\sigma _y^2 \\,\\,\\, + \\,\\,\\,2\\left( {2\\,x\\,y} \\right)\\left( {x^2 } \\right)\\sigma _{x,y}\n", "9bee7111d436ab278012301d9a0f6438": " \\lambda f.\\operatorname{let} x = f\\ x \\operatorname{in} x ", "9beef81e57261138beb39e93a8b48300": " t = \\frac{1}{g} \\operatorname{arctanh}\\left(\\frac{T}{X}\\right),\\; x= \\sqrt{X^2-T^2},\\; y = Y,\\; z = Z", "9beefa330e4c102e980c7b609e9388a1": "\\mathbf{x}|\\mathbf{d}", "9befcaa8a1ee114ecd501682adf276bd": "\\rho(v)", "9bf03b2925882c28a1597239fe6f57db": "a_j=c_j", "9bf06f76638df8f8925e2b9d65b47110": " k_z = k ~ \\cos \\theta ~ ", "9bf1574007c32ada78bea191f79a83c3": "\\, \\, S_0= \\, 2 , \\,\\, S_k= \\, 1+\\prod \\limits_{n=0}^{k-1} S_n\\text{ for}k>0 ", "9bf184ce697cf584954e135bc86facec": "(\\mathfrak{a}+\\mathfrak{b})/\\mathfrak{a}\\simeq\\mathfrak{b}/(\\mathfrak{a}\\cap\\mathfrak{b})", "9bf187462bd897177edf8f0a4af52c97": "\n \\left. \\frac{\\partial S(z)}{\\partial z_i} \\right|_{z=0} = g_i(0) = 0,\n", "9bf196512da7f2f76c40ab87f159c152": "\n W(z) =\n \\sum_{n=1}^{\\infty} \n \\lim_{w \\to 0} \\left(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}w^{\\,n-1}}\\ \\mathrm{e}^{-nw}\n \\right)\n { \\frac{z^n}{n!}}\\,=\\, \\sum_{n=1}^{\\infty}\n (-n)^{n-1}\\, \\frac{z^n}{n!}=z-z^2+\\frac{3}{2}z^3-\\frac{8}{3}z^4+O(z^5).\n", "9bf1e1a1c44095fb380ee0b3c2701d8d": " \\textstyle \\left \\langle {(\\Delta p)}^{2} \\right \\rangle = \\frac{1}{2}K^2n^2", "9bf1eb9a2cc5ada973de4d74eb2d69ad": "X(f)=\\mathrm{FFT}\\{x(t)\\}", "9bf213c14d84c1b66ae7f08df01b989d": " \n\\begin{bmatrix}\n \\mathbf{e}_1'(s)\\\\\n \\vdots \\\\\n \\mathbf{e}_n'(s) \\\\\n\\end{bmatrix} \n\n=\n\n\\begin{bmatrix}\n 0 & \\chi_1(s) & & 0 \\\\\n -\\chi_1(s) & \\ddots & \\ddots & \\\\\n & \\ddots & 0 & \\chi_{n-1}(s) \\\\\n 0 & & -\\chi_{n-1}(s) & 0 \\\\\n\\end{bmatrix} \n\n\\begin{bmatrix}\n \\mathbf{e}_1(s) \\\\\n \\vdots \\\\\n \\mathbf{e}_n(s) \\\\\n\\end{bmatrix} \n", "9bf223ac5b3902355189a8d2cd013101": "\\alpha = \\alpha_1 \\ldots \\alpha_n, \\forall_{i, 1 \\leq i \\leq n}: \\alpha_i \\in (V_n \\cup V_t)", "9bf23893a80ada975b1d251116f52b1b": "\\frac{d}{dt}\\langle A(t)\\rangle = \\left\\langle\\frac{\\partial A(t)}{\\partial t}\\right\\rangle + \\frac{1}{i \\hbar}\\langle[A(t),H)]\\rangle = 0", "9bf23b316d8d8807c09a4e58d037d999": "\\begin{bmatrix}\\cos(\\phi)&-\\sin(\\phi)\\\\ \\sin(\\phi)&\\cos(\\phi)\\end{bmatrix}.", "9bf27baf3546d099d958c9d33af9f82f": "\\tbinom{m+n}{m}", "9bf2e539b611c30a364fa7f3be6fdaf0": " N_q ", "9bf2eeaa0f24c68e42de7420b222344b": "f: \\; A \\to B", "9bf312eeef4faad5e874a6b850d2efd1": "X=\\{3,4,4\\}", "9bf31c7ff062936a96d3c8bd1f8f2ff3": "15", "9bf32967ad2b3322e4eb61c991dad68a": "J_{\\varphi_0}", "9bf37f1c5cf204cad292569c9e6e3fad": "\\gamma\\in [0,1)", "9bf3a6eff39a21201659a3c10b099740": " \\epsilon_r|\\mu|+ \\mu_r |\\epsilon| < 0,", "9bf3aca5c08318fafd3de930ea4f6b41": "x,x',open,open'", "9bf3d6ffffec5b4e8e510d07493f49e3": "C(\\epsilon)", "9bf3eb5425ade3243de42fe010ff8ee6": "\\tau_f^{\\sigma_j}\\,\\!", "9bf428efce6ab018a745c15bad236ce6": "(k^2 + h^2)(l^2 + m^2) = (kl - hm)^2 + (km + hl)^2 = ((a-c) - (a+c))^2 + ((d-b) + (d+b))^2 = (2c)^2 + (2d)^2 = 4n,", "9bf465a5bf1df62a39790d9112018091": "\\mathbf{F_g}", "9bf46e232508041a72f2621fd6d91c6a": "\\dot{\\varepsilon} = \\frac{1}{L}\\frac{dL}{dt}\\,\\!", "9bf48250bfc9e18985a1b69f1832c52e": "i_{exc}", "9bf5231767340fd1cf18bd162ea1787f": " c", "9bf54b7125bf51cb16c553f81863b8c3": " MV+mv=0 \\qquad (1)", "9bf54e5f0a16a0d4384b18e502561de3": " \\sum \\vec M = 0 ", "9bf555c171764e051c307bc8a5919cc4": "RxR", "9bf59e60b0f0d6f3066bff0acf372810": "\\scriptstyle X:\\Omega \\to \\mathbb{R}^n", "9bf6048ccb405c596894e7a0f0470403": "d W = V dq = V I dt ", "9bf606f9fa2fd18dac69b0f2a8cee9c0": "(x_1-\\bar{x},y_1-\\bar{y}),\\dots", "9bf655724b0827e1f1e39a5c8f9fe15e": "\\frac{1}{\\sqrt{2}} = \\sum_{k=0}^\\infty \\frac{(-1)^k \\left(\\frac{\\pi}{4}\\right)^{2k}}{(2k)!}.", "9bf6af7605e1119002a9f6574b24b038": "\\,q_x", "9bf7205e451dd222626b2d1581ab1d8e": " \\left(\\cos(x)\\right)' = -\\sin(x)", "9bf72af6866ad6cdcb93c9ef6470409d": "x = e^u", "9bf753ce94ab412f5e14b7bd203ed99b": "\n3 \\cos \\Omega = 1 - 4 \\cos^{2} \\left[\\left(\\phi + \\psi \\right)/2 \\right]\n", "9bf763ea13eacb48de1be5c62f605494": "V_\\mu ^\\prime = O_\\mu ^\\nu V_\\nu +P_\\mu \\, ", "9bf7aaaba24bfb36a6df3da3c1bf47b1": " \\psi \\rightarrow e^{i\\theta(\\vec{r} )} \\psi", "9bf7b170416a00dbcb89a102477808ab": "\\mathrm{CH_4 + \\begin{matrix} \\frac{1}{2} \\end{matrix}O_2 \\rarr 2 H_2 + CO}", "9bf8268cbda8f4de204edb242c120333": "E=E_{o}\\cdot e^{-j\\omega ( \\frac{x}{v} - t)} ", "9bf84867b5fd0afb00aacc26b76614f0": " \\boldsymbol{J_D} = \\epsilon_0\\frac { \\partial \\boldsymbol{E} } { \\partial t }", "9bf863354a74c95a61701c1943d61399": "\\forall x\\,P_1(x)\\lor\\cdots\\lor P_n(x)\\lor\\neg P'_1(x)\\lor\\cdots\\lor \\neg P'_m(x)", "9bf8df9827ebf3d392450efa90b6063a": "r_i=y_i-f(x_i,\\boldsymbol \\beta)\\,", "9bf92c1153d4443e9c892fab9f1dff1b": " A\\mathbf{v}_i = \\lambda_i B \\mathbf{v}_i \\quad", "9bf97bfb24522189c01ce1a49797232d": " \\omega, \\phi \\to \\pi^0 e^+e^- ", "9bf9e46dee78d1c4736e89e314b96204": "|U| > \\frac{1}{2}.", "9bfa1e9f9cf03e5b8651719486b8eb44": "\\sqrt{1-v^2/c^2}", "9bfa3f433a365ac0abca531f1907e87a": " z(\\infty) > 0 \\iff (1-Q) = (1-q)^L > 1-s ", "9bfa675a48e2226790c5f7973eef77eb": "\\operatorname{pred} \\equiv \\lambda n.\\lambda f.\\lambda x. n\\ (\\lambda g.\\lambda h. h\\ (g\\ f))\\ (\\lambda u. x)\\ (\\lambda u. u)", "9bfa88975daf72da556f86be7316a22e": "\\chi (X) = \\sum _0 ^n (-1)^j \\; \\mbox{rank} \\; H_j (X). ", "9bfad8996e04c8f8bf524022b78cdaaa": " \\frac{f_{\\theta_1}}{f_{\\theta_0}}(x) \\geq \\frac{F_{\\theta_1}}{F_{\\theta_0}}(x) ", "9bfb0c16dedf65ed3a93ee5d9a7c81f3": "(g \\circ g', h \\circ h')", "9bfb3c0bc11a402b4dbde487743c34ef": "\n\\begin{align}\n\\operatorname{tri}(t) = \\operatorname{rect}(t) * \\operatorname{rect}(t) \\quad\n&\\overset{\\underset{\\mathrm{def}}{}}{=} \\int_{-\\infty}^\\infty \\mathrm{rect}(\\tau) \\cdot \\mathrm{rect}(t-\\tau)\\ d\\tau\\\\\n&= \\int_{-\\infty}^\\infty \\mathrm{rect}(\\tau) \\cdot \\mathrm{rect}(\\tau-t)\\ d\\tau .\n\\end{align}\n", "9bfb4170db62f94222686bba876d7097": "I_\\text{z} = \\begin{cases} \\frac{1}{2} & \\text{up quark} \\\\ -\\frac{1}{2} & \\text{down quark} \\end{cases}", "9bfb66fde7aa881d188ab1f55d268d23": "\n\\begin{align}\nx_1 &= \\frac{\\zeta+\\bar{\\zeta}}{\\zeta\\bar{\\zeta}+1}\\\\\nx_2 &= \\frac{\\zeta-\\bar{\\zeta}}{i(\\zeta\\bar{\\zeta}+1)}\\\\\nx_3 &= \\frac{\\zeta\\bar{\\zeta}-1}{\\zeta\\bar{\\zeta}+1}.\n\\end{align}\n", "9bfb950d1c4988f1da640c63ee3102a6": "(p_1 + \\cdots + p_n)^c = \\sum_{k_1, \\ldots, k_n\\ \\in \\mathbb{N} : k_1 + \\cdots +k_n=c} {c \\choose k_1, \\ldots, k_n}\n p_1^{k_1} \\cdots p_n^{k_n} ", "9bfc1620bfb9a97f7199ca19f0d2296e": "\\lambda^{-\\Delta}", "9bfc373c0d1f88d89e801515078b0496": "\\mathbb{K}", "9bfc43a570969638760cb3bea5cebf6c": "\\|\\mathbf{x}\\| = |\\alpha|", "9bfc7e99f2272146f02939ed08a1efb9": "E = \\pi I", "9bfcbd64bc069f44b848fd2ab8076c47": " \\int ^T _0 R_N(t,s)\\Phi_i(s)ds = \\lambda_i \\Phi_i(t), var[N_i] = \\lambda_i", "9bfd2532924016199c3fe57a8fed35cd": " \nP_{ni}= \\int_\\beta L_{ni} (\\beta) \\, f(\\beta | \\theta) \\, d\\beta, \n", "9bfd282c92de420bfd85385de96f8554": "l_w", "9bfd307a08f934d95ffd8aa3332157ab": "[A]_i = ({V_{aq} \\over V_{org} K_D + V_{aq}})^i [A]_0", "9bfd86467602e501514b8bec3b83421f": "h_{ab} = g_{ab} + X_a \\, X_b", "9bfd898c61000753298d079638f0d5e5": "2^m / N", "9bfd9fe6bdd65de4a079cbe4fc9f28a0": "F[\\vec{g}]=\\sum_{j=1}^{N}f_j[\\vec{g}]e^{2\\pi i \\vec{g} \\cdot \\vec{r}_{j}}.", "9bfdbc158f0f55f698bbc1e1e52201b2": "R^n(X)", "9bfdeb83ee6380cbe6327c55962dd11c": "\\gamma (v) \\equiv \\frac{1}{\\sqrt{1-v^2/c^2}} \\ ", "9bfdf1f2b8b7ef727cb4f1bb8ed3470c": "G_i(f)=f-f\\ominus b", "9bfe1d9c71e341a41eef4ee2c1fa7555": " \\| S T\\| _{S_1} \\leq \\| S\\| _{S_p} \\| T\\| _{S_q} \\ \\mbox{if} \\ S \\in S_p , \\ T\\in S_q \\mbox{ and } 1/p+1/q=1. ", "9bfe320a234d0baf33bda282e91e5d9b": "F(\\vec{\\textbf x}, t)\\,", "9bfe5306e188034657e740f31cc253aa": "N=\\iiint_V n(x,y,z)\\;dV ", "9bfe8fdda88eb23dbde207df10b7c551": "y^2 = x(81x^5+396x^4+738x^3+660x^2+269x+48)", "9bff9b72f98cb2fc4e248c6b0ac089b3": "F_\\nu [e^{(ar)^2/2}] = \\frac {e^{-k^2/2a^2}}{a^2}", "9bffd5d69c3c874a247ec5de6d9625c4": "F_y = \\frac12 \\times \\rho \\times S \\times C_y \\times V^2", "9bffee21541e67818baf55a2dbb0ff70": " \\dot{x_{2}} = -x_{1} + \\varepsilon \\left( \\frac{{x_{2}}^{3}}{3} - {x_{2}}\\right). ", "9c00229c902508d582af0e18c0968dd0": "h(x)=1", "9c007d54bec0383e850893407494fe87": "|\\cdot|_\\ast=|\\cdot|_p^c", "9c00d3064ac161277a79412fdc6a67d7": "\\scriptstyle{\\Lambda_i}", "9c00e0acba2cf7292f185a9470ecf5e2": "R_{\\rho\\sigma\\mu\\nu} = {1\\over \\alpha^2}(g_{\\rho\\mu}g_{\\sigma\\nu} - g_{\\rho\\nu}g_{\\sigma\\mu})", "9c0113a72073cce65e90980055982c5e": "\\sum_j \\sum_i y_{ij}^2 - \\sum_j \\frac{(\\sum_i y_{ij})^2}{I_j}", "9c0118f092c19967e8b8a78995f83d48": "N=S^1", "9c01abeb9c92ac66866c7ab0f63029be": "z\\in\\mathbb{C}^{n}", "9c0220e3af35bc97cf96aff211484b2a": "\\mathrm{Mo} = \\frac{g \\mu_c^4 \\, \\Delta \\rho}{\\rho_c^2 \\sigma^3} ", "9c02a97b0464d3ee407ffe85c61aca65": "S(-k_\\mathrm{FE},-k_\\mathrm{PE}) = S^*(k_\\mathrm{FE},k_\\mathrm{PE}) \\,", "9c02ba8718466cf56e90f60cd3805650": "\\scriptstyle G = g(H)", "9c02ce359b7208a6cde5882583c22a49": "\\omega<\\omega_p", "9c02d8b0eb7d3369f2ac4eea61041ad8": "r=\\tfrac{1}{2}\\ :", "9c0306d7a4360de626566f4f08158a04": "\\scriptstyle \\tau,", "9c035b3c4c09c44ecc9c3fa9b40fc2f9": "\\mathbf{r}_4 = (a/4)(2\\hat{x} + 2\\hat{z})", "9c0397281945e319c66cb7f29ed431f2": "\\xi\\in F", "9c03af7beeb630e42d883901d61cc510": "\\psi_{2} : U_{2} \\to V_{2}", "9c03bcb48d00f55c3b36aa06d68ddb34": "\\tau = \\frac{d}{\\bar v} = \\frac{nd}{MFD(n)L}", "9c0439514602a0c2d57788445ff07b8d": "\\lambda_2\\!", "9c043c475810bda74bab1514cc5e65de": "N^{1/2}", "9c049ca080398c517cf3d4c881dbc14b": "\n \\mathbf{n}_i = \\mathbf{R}~\\mathbf{N}_i \\,\\!", "9c04a587c46e4dea1dcd99f33d0236b8": "(\\alpha-E)(\\alpha + \\beta - E)-\\beta^2=0\\,", "9c04d1c79461769fd237d92815a0d72c": "5.9 \\times 10^{-6}", "9c04de6b9841a4a903938b2f4ab52730": "e,m^*", "9c051ede88c79ad6119ff8787619d153": "f \\circ h = \\mathrm{id}_Y . ", "9c057f7e7e7e8a781beff7d4a3f30980": "\\Gamma \\,", "9c0589207a81b5e68703c47e65040221": "\\frac{\\langle E \\rangle}{A} = \n\\frac {-\\hbar c \\pi^{2}}{3 \\cdot 240 a^{3}}.", "9c058bca26745f38461b64e1599422f1": "\\omega=\\nabla \\times \\mathbf{v}", "9c058c49f28e23fb24735f0599fae4a9": "d_{\\lambda} d_{\\mu} = d_{\\lambda + \\mu}", "9c058e981971051059ba2e1b93c4d4fd": "\\lambda=10", "9c05d8354388774b673f769c5772632a": "(\\lambda x.x)[y := y] = \\lambda x.(x[y := y]) = \\lambda x.x", "9c068a704f5ee8b1116aaef9573c6bef": "m_{ij}", "9c068ba4b1b9f29b1cb4c64e9f13ba5a": "p \\cdot x_i > w_i", "9c06c07c98ba3e3f7e066051b825838a": "\\psi_L(x)= A_r e^{i k_0 x} + A_l e^{-i k_0x}\\quad x<0 ", "9c06c4fef7ec1bf3a7744158bd4df3ac": "\n\\xi( \\mathbf{p}) = {1 \\over \\sqrt{2}} \\left[ Q_L(\\mathbf{p})\n+ Q_R(\\mathbf{p}) \\right], ", "9c075b2be4478cbf8ea22c6b53ada6c4": " \\mathbf{y}' = A\\mathbf{y} ", "9c0783aaf209cfc70ba217efe04f2f48": "\\hat{V}^\\pi", "9c07881c2285eb321a4d7469316a0783": "\\alpha\\pm\\beta", "9c07b26f619d1ac41057a23dbd621a13": "\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial z}{\\partial x}\\right)_y =\n\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial z}{\\partial y}\\right)_x =\n\\frac{\\partial^2 z}{\\partial y \\partial x} = \\frac{\\partial^2 z}{\\partial x \\partial y}", "9c07c625dc24abea8521a16ddeaf0f6b": " f_1 \\Rightarrow f_2", "9c07ee79ef9f2cdbb3198e44c57fd14f": "D_k=2^{k-1}", "9c0813ee9bfec5a0c8ec33ab5840ae2f": "\\textstyle\\frac{4\\left(\\sqrt {2}-1\\right)}{3}", "9c08378ad4808bcd324fbda2d5a3a8d1": " \\Gamma \\vdash A; A \\vdash B \\Rightarrow \\Gamma \\vdash B", "9c090ccc43e9f38a8fc0bd76074f8e95": "\\frac{bh}{n + 1}", "9c093d09ad9fccd970e66e07daf2aea9": "\\beta_2(T_e)", "9c093f11cb3976551cbb8a329244e7f3": "U(\\bold{r}) = \\sum_{\\bold{G}} U_{\\bold{G}} e^{i\\bold{G}\\cdot\\bold{r}}", "9c0975dce2882be03522530d76325937": "f(\\alpha x, \\alpha y, \\alpha z) = (\\alpha x)^5(\\alpha y)^2(\\alpha z)^3=\\alpha^{10}x^5y^2z^3 = \\alpha^{10} f(x,y,z). \\,", "9c098b29a7c600d3b5f27a1b74d6c381": "\n \\boldsymbol{\\nabla}\\boldsymbol{S} = \\cfrac{\\partial \\boldsymbol{S}}{\\partial q^i}\\otimes\\mathbf{b}^i\n ", "9c09a3324697b5f4ebea82be77f0a011": " \\mathbf{B}^\\dagger = \\mathbf{B}", "9c09a40774233517536c628de798579d": " (f+g)(X) = f(X) + g(X) \\, ", "9c0a27cbe3afcf2958a09c42c0b4a38b": "\\text{Hom}_\\mathcal{C}(A\\otimes B, C)\\cong\\text{Hom}_\\mathcal{C}(A,B\\Rightarrow C)", "9c0ac2a7234ac6a026b983d80b71366e": " \\mathbf{y}' = \\begin{pmatrix} y'_{1} \\\\ y'_{2} \\\\ 1 \\end{pmatrix} ", "9c0b091f7bb545847d90d94215791cd7": "{\\scriptstyle\\sqrt{-1}^{1/3}}", "9c0b14ca397135607933c386a0830d0f": "e^{-y}= \\frac{1}{2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty} \\Gamma(s) y^{-s}\\;ds", "9c0b6d06270fa9b51f5982b886bb9a7a": "\\scriptstyle \\min(0.5/L,\\ 0.5/M)", "9c0b9bc79453a3e8ec6cd916673d240f": "\\Sigma(p)", "9c0ba86410ab8c2eaa94f1d901d74127": "H = E + PV ", "9c0baa7aa87f10fed8c9b3f128226c24": " X_v \\perp\\!\\!\\!\\perp X_{V \\setminus \\operatorname{de}(v)} \\,|\\, X_{\\operatorname{pa}(v)} \\quad\\text{for all }v \\in V", "9c0bbe98e76ab9dfda25c572331f5a4f": "H=(mR)^2+\\dot{x}^2+\\sum_n|\\dot{c}_n|^2+n^2|c_n|^2", "9c0bc8848413f6548be68612186e105c": " a_1 \\mathbf{e}_1 + a_2 \\mathbf{e}_2 + \\cdots + a_n \\mathbf{e}_n = 0 . \\,\\!", "9c0c2023c964b4db3818b1c83087fab5": "M(t)=M_0 \\left((p(t) g(t) \\mathrm{e}^{-t/T_{2b}}\\right)", "9c0c3029e2cd4ec592063fbe841a385d": "\\sum_{k\\geq0} (-1)^k m_k x^{n-2k},", "9c0c84f9ad82f830fde8618942c492eb": " \\gamma_1^{C, \\infty} = \\gamma_2^{C, \\infty}= 1 ", "9c0c96a73acca45f327234daf42d15a0": "U\\left(x,y\\right)>U\\left(x',y'\\right)", "9c0d18f02f00d61b055954019d1a9f53": "^nC_k", "9c0d3de21a513db3decb951bec8249c2": "N/\\sqrt{\\theta} = N/\\sqrt{T/288.15}", "9c0d51ab303d9bb4530e2a1b74605757": "\\left.\\right.\\omega_f(\\delta;t)=\\max_{|\\varepsilon| \\le \\delta} |f(t)-f(t+\\varepsilon)|", "9c0d5aed91f86da30eb2a372c122ec96": "T \\cup \\{\\beta_i\\}", "9c0d8caf50f4f9691b05389ebe505e47": "\\scriptstyle \\epsilon_r", "9c0dc117c005a8062fa4d9bc5c1f9406": "\\left(\\frac{15V}{8\\pi^2}\\right)^{1/5}", "9c0dde78aff1fd5fc70c7f25b27fe081": "Y_{8}^{8}(\\theta,\\varphi)={3\\over 256}\\sqrt{12155\\over 2\\pi}\\cdot e^{8i\\varphi}\\cdot\\sin^{8}\\theta", "9c0def4b955d8261549913a832c17c13": "x^n = 0", "9c0e13de3edb69eef51df172dcc9e7f0": "f\\left(t,T\\right)=-\\frac{1}{P\\left(t,T\\right)}\\frac{\\partial}{\\partial T}P\\left(t,T\\right)=-\\frac{\\partial \\textrm{log} P\\left(t,T\\right)}{\\partial T},", "9c0e7d7c282440179476915026dd93af": " \\begin{bmatrix} x_1, x_2, \\dots, x_m \\end{bmatrix} \\qquad ", "9c0ef895849787ee681634a2b7d7ba22": "\\mathbf F^+", "9c0f655b1860e05fc032c669afda2918": "f:n\\to K", "9c0fbc2b000c1d382fc576e604b50862": "L(z) = \\sum_{k=1}^\\infin \\frac{(-1)^{k+1}}{k}(z-1)^k", "9c0fc61f48ce7f03a9435238c2451ed1": "\\psi_{\\text{gr}}", "9c0ff62881069ecee2e0dafd8164b64e": "k_X\\to I^{\\bullet}_X = I^0_X \\to I^1_X \\to \\cdots ", "9c0ff821018195d64291d2eb5a2f5d98": "\\ TDOP = \\sqrt{d_{t}^2} = |d_{t}|\\ ", "9c1025b7d94546376406e8fdd883dd94": "a+1=b", "9c1074c42cb7e7253bb64466547571c6": "f(\\vec x,\\vec y ):=\\rho(\\vec x+\\vec y)-\\rho(\\vec x)-\\rho(\\vec y)", "9c109d4c5b5178dfcaf661ab12c874d2": "\\sup_{y^* \\in Y^*} -F^*(0,y^*)", "9c10e92c79d7a7af7db8beb49fd048c6": "|1 \\rangle", "9c113b7bf19347831961306b2aa5d729": "\\hat{\\mathbf{e}}_x, \\hat{\\mathbf{e}}_y, \\hat{\\mathbf{e}}_z", "9c1202821182fc9f3f5e59d66253d64c": "\\pi_i A \\times \\pi_j A \\to \\pi_{i + j} A", "9c122af39b1b241b829dfe4060921055": "P_1(T_{i+1})", "9c129fd00dfd62c14d1a220f2dc2c2c2": " \\vec r \\times \\vec F_R = \\sum_{i=1}^N ( \\vec r_i \\times \\vec F_i ) ", "9c12a5b1adc7fede61ce9ed1b4840da2": "\\textstyle (n+1)/2", "9c12cf3aa56843859e7af2363c1183a4": " u", "9c1319057e083bde1c45f0faf7df7633": "r_\\circ", "9c132b83123f31a6826dff2dee38882e": "AB = C.", "9c1368f751fc7dbd3af5dd738e45ddd1": "\\delta \\psi = -v \\delta x\\,", "9c137c52f073ab5f5393ee790ce18d96": " PA = LU, \\, ", "9c13a4687cc4d1ab6a6bc25346db9dac": "\\langle u,u'\\rangle +\\langle v,v'\\rangle = \\langle u+v, u'+v' \\rangle ", "9c13c1ad3eba0397fa521c91ad1c661c": "2y+h(x)=0", "9c14131b3e7f1adf590b041593e6b8fa": "U(a,b,z)", "9c145d1d4027869ad2535574724c8b6a": "t\\in [0,\\pi]", "9c147415fc864add46a9f0019dee726c": "\\ G_{GR}(\\tau)=1+\\frac{Diff_{GR}(\\tau)}{V_{eff}(+)(+)}", "9c14a689b2a22f84d67a8ca8b260cbeb": "I_{hollow} = \\frac{4m s^2}{7}\\,\\!", "9c14d916a431b4542a4cc89969968b97": "N_{FP}", "9c14e8c2eaf5e554dcf7605bb40b1c66": "\\left(\n\\frac{ \\left(3 + \\sqrt{5} \\right)^2 \\pi}{60\\sqrt{3}}\n\\right)^{\\frac{1}{3}} \\approx 0.939", "9c14fbe53fad3be5e87da97eabc0147c": " T(\\alpha) = {1 \\over 15} \\arccos \\left( {-\\sin(\\alpha)-\\sin(L)*\\sin(D) \\over \\cos(L)*\\cos(D)} \\right) ", "9c152c8ae0442f2522186ed2d1bf92ae": "d=p^r", "9c15d2a2bb958bf233fcace23b9e9dbf": "t_{TOF}\\dot= k_{TOF} \\cdot R_T", "9c1647f3fb3610d00ea07a24516ecf15": "\np \\xrightarrow\\alpha p'\n ", "9c167a10ebd19a37928aece1e6a9f456": "R^d", "9c1780f3180a609ea92e0ce57a1cca16": " f_X(\\mathbf{x}|\\boldsymbol \\theta) = h(\\mathbf{x})\\ g(\\boldsymbol \\theta)\\ \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(\\mathbf{x})\\Big)", "9c17f02432f8a4dd4df47e1a36394839": "\\textstyle \\frac{P(E|M)}{P(E)} > 1 \\Rightarrow \\textstyle P(E|M) > P(E)", "9c1863bc88ef8b19c7fa93b129b66244": "\\rho=\\frac{(r-M)^2}{2r^3}\\,,\\quad \\mu=-\\frac{1}{r}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\varepsilon=\\frac{M(r-M)}{2r^3}\\,;", "9c188190ec6087a5bd4472ffc1bf7fc3": " \\lambda_1 \\; ", "9c18aa578bf757a4754fc8d6feb9370d": "\\int\\limits_\\Omega\\left|\\nabla f\\right|", "9c18ad90275cb505bc73c31613c46af8": "\\omega = \\omega_0", "9c18b42e7b81009481f22290596312a1": "\nb_y = s_z a_z + c_z\n", "9c18b4bdf6942a142e938a623e4c5bed": "F(\\boldsymbol{v}) = \\int_A v_i f_i\\mathrm{d}x + \\int_{\\partial A\\setminus\\Sigma}\\!\\!\\!\\!\\! v_i g_i \\mathrm{d}\\sigma\\qquad \\boldsymbol{v} \\in \\mathcal{U}_\\Sigma", "9c18ccff9f850e1a4b802c6cee2105ac": "\\mathbf x = (x_1, x_2, \\ldots, x_n)", "9c18eea0d511687afc37491596cebffb": " a, b, c ", "9c190a48c290d098f239a7cc26322f70": "\n\\begin{align}\n\\frac{1}{2} m \\left ( \\frac{ds}{dt} \\right ) ^2 & = mg(y_0-y) \\\\\n\\frac{ds}{dt} & = \\pm \\sqrt{2g(y_0-y)} \\\\\ndt & = \\pm \\frac{ds}{\\sqrt{2g(y_0-y)}} \\\\\ndt & = - \\frac{1}{\\sqrt{2g(y_0-y)}} \\frac{ds}{dy} \\,dy\n\\end{align}\n", "9c193b89c9cfee2504c010bcf1d53695": "a,b \\in C", "9c1966942748f1f30ef38eada3824f33": "\\left(\\begin{matrix}1 & 0 \\\\ 0 & 1\\end{matrix}\\right)^{-1} =\n\\left(\\begin{matrix}1 & 0 \\\\ 0 & 1\\end{matrix}\\right).", "9c19746fa7a9fd385b083298195833ed": "\\nu(M_2) = \\nu(M_1) - \\theta \\,", "9c19a3f3bf3ea742af594dca7ac54c96": "\\frac{v-c}{c}=(0.7\\pm2.8\\ (\\mathrm{stat.})\\pm8.9\\ (\\mathrm{sys.}))\\times10^{-7}", "9c19b29138b7593b0535230d5bc5f022": "\\sigma_P=\\sqrt{\\frac{pq}{n}}", "9c19d9a38e8c00340d13e7eadaa5577f": "\nds^{2} = e_{1}^{2} + e_{2}^{2} + e_{3}^{2} - e_{4}^{2} .\n", "9c19fc0c9e3c0109ff5f7f3fcef0eb27": "\\ddot{\\psi}-\\psi''+\\sin\\psi+g\\sin(2\\psi)=0", "9c19ffb40151b4f72e6cbbc6418ec0ca": "\\psi/n", "9c1a4fd422cdd1f8e0ffcd468b781115": "\\mathbb{Z}^x", "9c1a5edb4480ba2a63c520d132324302": "\n\\gcd(a,b)\n=2^{\\min(a_2,b_2)}\\,3^{\\min(a_3,b_3)}\\,5^{\\min(a_5,b_5)}\\,7^{\\min(a_7,b_7)}\\cdots\n=\\prod p_i^{\\min(a_{p_i},b_{p_i})},\n", "9c1ab95591b12aa4c3fb113c8ebdeafb": "D_0(X)D_0(x)\\ge\\frac{1}{16\\pi^2}", "9c1af01af2af47edd9db94be8f20ac89": "H : A \\mapsto \\bigcup_{i=1}^N f_i[A],\\,", "9c1b06b2488226700430ffd3058e9d67": "\\left(\\tfrac{Q}{n}\\right)", "9c1b9af40be9b48795bdb105868085fe": " -y \\partial_x + x \\partial_y \\equiv i J_z ~, \\qquad -z \\partial_y + y \\partial_z\\equiv iJ_x~, \\qquad -x \\partial_z + z \\partial_x \\equiv J_y ~;", "9c1bc4788a0ad550ab9ae43d62e7e7c8": "\\partial_\\mu\\left[\\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)}\\right] = \\frac{\\partial \\mathcal{L}}{\\partial \\phi}", "9c1bd7a02b2dfa9e7e90ad2fcd3203b2": "\\textstyle{\\binom{4}{2,2}/2=3}", "9c1bdd16815ae37d2f16af641e80b0cb": " \\frac{2}{\\pi^2} n^2 ", "9c1bedf9d7245e35d9af692d6d098f8e": " S_n(t) = S_n(0)\\exp\\left(\\int_0^t \\sum_{d=1}^D \\sigma_{n,d}(s)dW_d(s) + \\int_0^t \\left[b_n(s) - \\frac{1}{2}\\sum_{d=1}^D \\sigma^2_{n,d}(s)\\right]ds + A(t)\\right), \\quad \\forall 0\\leq t \\leq T, \\quad n = 1 \\ldots N, ", "9c1c84e358bcee422adb9a475adc241c": "\\frac{1}{\\mu}", "9c1c8fb001af3b59971db94a285a5ca2": " L_z |\\psi\\rangle = m\\hbar |\\psi\\rangle", "9c1cca86e1b38572f4d0105e4a605634": "w[t]", "9c1cd0328fe28cd47bfec70ef9918760": "w_G'=0\\,", "9c1ce2cf0d9cf53916b6de3a3f6cd761": "\\mathcal P \\leftarrow \\{P_1, P_2, \\dots, P_k\\}", "9c1cf96fa36f76a2f9e048ae53258350": "qr(\\varphi) \\le n", "9c1d46a26e02bb1657e4ba525bf93c89": "u_\\infty(z,t) = U_0\\, \\cos\\left( \\Omega\\, t \\right), \\,", "9c1d4c053fd7d570f52a5636bf2c3398": "\\mathbf{x}^{k+1}=\\mathbf{x}^k-\\gamma^k P^{-1}(A\\mathbf{x}^k-\\mathbf{b}),\\ k \\ge 0.", "9c1d5c80443caff7568e7882479606ad": "f_{\\alpha}(\\vec{x},t;\\vec{v})", "9c1d60e4342c6e6dcbb1f0a8f28357a7": "\\, (1 - 2t)^{-k/2}", "9c1d73c82ef5e5520c25ce124cf2dc92": " S = A^T A", "9c1dd2632a740623397f7920ad65bddd": "W_1=\\hat{P} \\hat{D_B}^{-1/2}", "9c1e4458755648372b6e7948c615c3a0": "X \\sim \\textrm{Levy}(\\mu,c)\\, ", "9c1ed1cc2e2ac5fcd9a166f58f3fae30": "\\frac{\\partial \\bigg(a - (q_1+q_2)\\bigg) }{\\partial q_1} \\cdot q_1 + a - (q_1+q_2) - \\frac{\\partial C_1 (q_1)}{\\partial q_1}=0", "9c1ed8afb336abe0b3e9a819c4d22c67": "t\\left\\{\\begin{array}{l}p\\\\q\\\\q\\end{array}\\right\\}", "9c1ee54fe7e55b43560770e3bd817c69": "x \\mapsto a x + b y, \\quad y \\mapsto - b x + a y, \\quad a, b \\ne 0, \\quad a^2 + b^2 = 1,", "9c1f17cd4256f6f61d3a1d6f5a96457a": " m_p = \\mathrm{mass \\ of \\ plate} ", "9c1f28de2f0fefa15227651cc34263db": "ME-{P^2\\over 2}", "9c1f405c7cf43da2131139eca575a840": " a_{21} = p_2p_4+p_1p_5,", "9c1f6713b98063314dd466a5524c6e3c": "r_{1}=(g_{23}-g_{32})/(2sin\\Theta)", "9c1fa8ff231bbd4be98486e9121362da": "\\left(u_1\\left(\\sigma\\right),u_2\\left(\\sigma\\right),u_3\\left(\\sigma\\right)\\right) \\ ", "9c2084cac1482056a03b7fdd1b7b93be": "= \\left((0,1,1,1) \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}^{\\otimes 2}\\right)^{\\otimes |U|} \\left(\\left(\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}\\right)^{\\otimes 3} + \\left(\\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}\\right)^{\\otimes 3}\\right)^{\\otimes |V|}", "9c20b6e4f738b27e3c172f607fddcdc3": "\\displaystyle m \\big( ( A \\cap Z ) \\cup ( B \\setminus Z ) \\big) = p \\, \\operatorname{mes} (A) + (1-p) \\operatorname{mes} (B) ", "9c20c3c4f3a28dc91755658d6fe1f43f": "f:Z \\to P^n \\times P^n", "9c20daae8d160115418d91aa742f40f4": "\\mu_C(x) = \\inf \\{\\lambda > 0: x\\isin \\lambda C\\}.", "9c20e747591c532cc59e3654a86519d6": "\\forall A \\, \\forall B \\, (A = B \\Rightarrow \\forall X \\, (X \\in A \\iff X \\in B) )", "9c211f0a745edad4885c0debacfcc7e3": "\\Lambda:\\mathcal{X}\\times \\mathcal{X}\\to [0,\\infty)", "9c214b1cd5d2906def261d945b99395e": "r(B_p,\\ B_q).", "9c216262a9887204b05cdd3cac5b1948": "\\omega^j = \\sum_{i=1}^r f_i^jdg^i.", "9c2164c60ed0497569f077716215e0a3": "\\mathfrak c \\le 2^{\\aleph_0}", "9c21d0dc1b3a0ac5b4eb102b94b85031": " \\mathbf{v} = \\boldsymbol{\\omega} \\times \\mathbf{r} \\,\\!", "9c21e333d7f0fecd907e5c22cd03ff78": "\\pi = \\frac{(1+R)S - S^d}{S^u - S^d}.", "9c21eaf0d2a3775b5da4451d529b3935": " \\epsilon_p = \\Lambda^{-1}\\ ", "9c22025d5fd8dee543f774d55dc393b9": "\nr_0 = \\left [ 0.423 \\, k^2 \\, \\sec \\zeta \\int_{\\mathrm{Vertical}} C_n^2(z) \\, dz \\right ]^{-3/5} = (\\cos \\zeta)^{3/5} \\ r_0^{(vertical)}.\n", "9c221ad21d323f9221371f86dabfcc7e": "(M,\\omega)", "9c22836e0e475eded9475d911ab12ff0": "\\psi_0 = (1, 1)", "9c229c768b24beb965f83a06d6e9dd05": "\\log (x) = \\log (-x)", "9c22d3bfd4a8127a2394f9666f4ecb63": " \\ln(tu) = \\int_1^{tu} \\frac{1}{x} \\, dx \\ \\stackrel {(1)} = \\int_1^{t} \\frac{1}{x} \\, dx + \\int_t^{tu} \\frac{1}{x} \\, dx \\ \\stackrel {(2)} = \\ln(t) + \\int_1^u \\frac{1}{w} \\, dw = \\ln(t) + \\ln(u).", "9c2309929b5d4e6d89e1f48e3e3d6ebb": "a < b", "9c2328d767500dfe7185ddf7620f4135": "\nx_1^+ = x_1,\nx_2^+ = -\\gamma x_2\n", "9c2397e4e5a63b6a66e8ce44b59d7f49": "\\langle x, y\\rangle = \\tfrac{1}{4} (\\|x+y\\|^2 - \\|x-y\\|^2).", "9c23bca718096fef239fcae514a14c2c": "\\tau_{xy}=\\tau_{yx}", "9c245ac0f5215050afff5053e65fb294": "{x_1,\\dots,x_n}", "9c24834f74e5d40dc06d4d67d36cc37d": "K\\subset\\mathbb{R}^{m}.", "9c24983cfd2cd3d77c4f9928d30686a8": "c^2=\\frac{g}{\\alpha}\\frac{\\rho_L-\\rho_G}{\\rho_L+\\rho_G}+\\frac{\\sigma\\alpha}{\\rho_L+\\rho_G}.\\,", "9c249bfcef4e907cf6eb1dec8b2768f3": " k^m ", "9c24c6e0523f2b16e2a3045c6116d764": "\n v = -\\frac{1}{r}\\frac{\\partial\\psi}{\\partial z},\n \\qquad\n w = \\frac{1}{r}\\frac{\\partial\\psi}{\\partial r},\n", "9c24ef0a2ac42322a472a7a73a2e843e": "i,j\\in\\{1,2\\}", "9c2514e0ecd2858963a40f2cf1415c6b": "\\mathrm{\\tfrac{d\\bar{s} - s\\bar{d}}{\\sqrt{2}}}\\,", "9c25e61e9b7b79c0a6475e1e3e8e5575": "y(s)", "9c26291bc6543ee82e7595f3305ab95c": "\\phi = \\sum_n c_n \\psi_n. \\,", "9c2641d17ff21123f39628e577f49491": "\\underline{\\underline{\\mathsf{A}_\\varepsilon}}^T = \\underline{\\underline{\\mathsf{A}_\\sigma}}^{-1}", "9c2672a25b2a0a715d78e44368c4333c": "\\mathrm{LHS}=\\frac{dT(s)}{T(s)}\\cdot\\frac{r}{ds}=\\frac{dT(s)}{T(s)}\\cdot\\frac{1}{d\\varphi}=\\frac{1}{T(S)}\\cdot\\frac{dT(s)}{d\\varphi}.", "9c26d21ac3ad951ea3bed768cd0fe0fa": "\nPoss(a,s)\\wedge\\gamma_{F}^{-}(\\overrightarrow{x},a,s)\\rightarrow\\neg F(\\overrightarrow{x},do(a,s))\n", "9c26dd647ada5c362bf21458ed5e5139": "\\omega_i = \n\\begin{cases} \n 1/3 & i = 0 \\\\\n 1/18 & i = 1,2,...,5,6 \\\\\n 1/36 & i = 7,8,...,17,18 \\\\\n\\end{cases}", "9c26fd7bd9465e5ab1ab2be240df7edb": "(g,0)(h,0)=(gh,0)", "9c2705e9bb43a52aea61000f79de6e46": "\\kappa \\nleq \\aleph_0;", "9c270775ed54f1241699e3ccfd4cd732": "\\gamma_{\\alpha\\theta}\\cos{\\theta}+\\gamma_{\\theta\\beta}+\\gamma_{\\alpha\\beta}\\cos{\\beta}\\ = 0", "9c27b1f86cf967981d7a5c39e79bf096": "\n\\begin{bmatrix} S_1 \\\\ S_2 \\\\ S_3 \\\\ S_4 \\\\ S_5 \\\\ S_6 \\end{bmatrix}\n=\n\\begin{bmatrix} s_{11}^E & s_{12}^E & s_{13}^E & 0 & 0 & 0 \\\\\ns_{21}^E & s_{22}^E & s_{23}^E & 0 & 0 & 0 \\\\\ns_{31}^E & s_{32}^E & s_{33}^E & 0 & 0 & 0 \\\\\n0 & 0 & 0 & s_{44}^E & 0 & 0 \\\\\n0 & 0 & 0 & 0 & s_{55}^E & 0 \\\\\n0 & 0 & 0 & 0 & 0 & s_{66}^E=2\\left(s_{11}^E-s_{12}^E\\right) \\end{bmatrix}\n\\begin{bmatrix} T_1 \\\\ T_2 \\\\ T_3 \\\\ T_4 \\\\ T_5 \\\\ T_6 \\end{bmatrix}\n+\n\\begin{bmatrix} 0 & 0 & d_{31} \\\\\n0 & 0 & d_{32} \\\\\n0 & 0 & d_{33} \\\\\n0 & d_{24} & 0 \\\\\nd_{15} & 0 & 0 \\\\\n0 & 0 & 0 \\end{bmatrix}\n\\begin{bmatrix} E_1 \\\\ E_2 \\\\ E_3 \\end{bmatrix}\n", "9c280783147d89db0359b2f664a42733": "a\\frac{\\partial^2 u}{\\partial x^2} + b\\frac{\\partial^2 u}{\\partial x\\partial y} + c\\frac{\\partial^2 u}{\\partial y^2} + d\\frac{\\partial^2 u}{\\partial y\\partial z} + e\\frac{\\partial^2 u}{\\partial z^2} \\text{ + (lower-order terms)}= 0,", "9c281f9cd6684a72e23abb0145808fd5": "A\\in\\mathbb{C}^{n,n}", "9c283a5472557681d18fb43cac7c6cef": "S = {[A] \\over p} = {[B] \\over q} = \\sqrt[p+q]{K_{\\mathrm{sp}} \\over {(q/p)^q} p^{p+q}}\n= \\sqrt[p+q]{K_{\\mathrm{sp}} \\over {q^q} p^p}", "9c28cde4b43765397425c161afeb0383": "Ax^2 + 2Bxy + Cy^2 + \\cdots = 0", "9c28ce11426fe6a057d8ff28afa15cc0": "\n\\omega = 2 \\pi f = \\frac{2 \\pi}{T}. \\,\n", "9c28f0e5fcc8bb9f6874a788fcd67b2e": "P_t = 0.005 (V_t)^2\\,", "9c291ceecae605d6cd61acc1bdfb01ad": " \\psi(t_n)=\\sum_{l=-\\infty}^{\\infty}a_l(t_n)|l\\rangle ", "9c29271a1e59e25f67cfce49509afe7f": " D_t ", "9c2950fec9c4b72752d3e6acdfb3db49": "AB = \\sin \\theta\\,", "9c29ff912144b7546a8038b43670a9a4": "\n \\hat\\beta = \\frac{\\tfrac{1}{T}\\sum_{t=1}^T (x_t-\\bar x)(y_t-\\bar y)^2}\n {\\tfrac{1}{T}\\sum_{t=1}^T (x_t-\\bar x)^2(y_t-\\bar y)}\\ .\n ", "9c2a07301c81d0bbda185593b9982a38": "K(0)", "9c2a923cc203cebe9508abcb09ac2436": " \\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{M} f(- \\ln x) \\right\\}(s)", "9c2aae36a3c40c754d0012281e5bf14c": "\\nu(z) = n - \\log_P (\\log|z_n|/\\log(N)),\\,", "9c2aedd86db5021b6db300d94694f464": "S=\\wedge^\\bullet W", "9c2b172fdc5fa56f4ccfca7203fe9d2e": "\n\\mu(x+t) = \\frac{1}{\\omega - (x+t)}, \\qquad 0 \\leq t < \\omega-x,\n", "9c2be9cb3b60c78916beb0ebf1c0d48b": "\\frac{P}{A} = \\int_0^\\infty I(\\nu,T) d\\nu \\int d\\Omega \\,", "9c2c286e7844d9ee66fa31cc34e857f0": " g(s)= \\sum_{n=0}^\\infty a_n s^{-n} ", "9c2c5c7a415949ff19be043d198bb0cc": "\\beta ^{A}", "9c2cd4c78333427bbf0a80b75f56e69c": " \\frac{\\pi}{4} = 12 \\arctan\\frac{1}{49} + 32 \\arctan\\frac{1}{57} - 5 \\arctan\\frac{1}{239} + 12 \\arctan\\frac{1}{110443}\\!", "9c2d8275efa44c80c54f495d25e192a9": "\n\\bar{n}_i = \\frac{n_i}{g_i} = \\frac{1}{e^{(\\epsilon_i-\\mu)/kT}+1}.\n", "9c2d9126fa803b140b16120b8cd88b6c": "[d,r_od,\\delta] ", "9c2d9354b9764a795f73bf9308de983e": "\\overline{\\psi}\\psi\\,", "9c2daaf139bb88b63f04c91771d6621c": "\n \\cfrac{\\mathrm{d}\\varphi}{\\mathrm{d}x} = -\\cfrac{\\mathrm{d}^2w}{\\mathrm{d}x^2} -\\cfrac{q(x)}{kAG}\n ", "9c2db015fd0c00cd247ba8005a2914de": "m_{i+1}(\\hat{x}) \\geq |h_{i+2}(x(t))|", "9c2dced629256c5cac677ccb833b5e1e": "(k_{1(3)}\\equiv k_{1(2)}", "9c2e026a21ab9ba32c3c198819e91a87": "\\mathfrak{h}_i", "9c2e14278d00edc2e8e991557558d7ea": "\\Lambda_{(\\mathbf N)}^2=\\left (\\frac{dx}{dX}\\right )^2\\,\\!", "9c2e2d16cff282534e999defefa6ccb3": " \\langle T u, v \\rangle = \\langle u, T^* v\\rangle.", "9c2ea01d38a9cf8a6a054c818934ac95": "\\mathbf{a}\\cdot\\boldsymbol{\\nabla}\\psi=-\\psi\\left(\\boldsymbol{\\nabla}\\cdot\\mathbf{a}\\right)+\\boldsymbol{\\nabla}\\cdot\\left(\\psi\\mathbf{a}\\right)", "9c2f4eda95ed12ad1cc462b887c7a948": "o(\\tau)", "9c2f60517b023946a7e1ea84cf1e81ba": "L_n(R)", "9c2f8c16f66e7c5f0385b4d030e9ae8c": "\\left.\\right. H_2 ", "9c2fe52bb2b4c270b7e7ba7388c40bf5": "\\left( z_{i-k},\\ldots,z_{i+k} \\right)", "9c300ffd09627033fc14ba45ffc4ace6": "\\langle\\mathbf{p}_k,\\nabla f(\\mathbf{x}_k)\\rangle < 0", "9c302d457662db5063f0d3c337167235": "(\\neg p\\to q\\lor r)\\to(\\neg p\\to q)\\lor(\\neg p\\to r)", "9c305aee889466582eb28880b894380b": "\\{P_\\alpha[\\,\\cdot\\,]\\}", "9c308f697f7267334497991a558dedc5": "\\scriptstyle (X,\\tau_2)", "9c30b270ec9144d9c1d34ad708869682": "\\lambda = \\lambda_1 + \\cdots + \\lambda_r", "9c30d4512315c1ceb42688711cca5bc0": "y_3 = \\frac{(2x_1+x_2+A)(y_2-y_1)}{x_2-x_1}-\\frac{B(y_2-y_1)^3}{(x_2-x_1)^3}-y_1", "9c3175c443c898788bc7bfbb57d28d34": "-i\\frac{\\xi_j}{|\\xi|}", "9c318a131cff4f1a0f839ac38acb6625": "\\upsilon_{c}", "9c31a59b8436c82459d7ad49f7287d7d": "q^{ab}", "9c31ecc10bed35af3a83612738690fd8": "\\mathbf{x_0}", "9c3209d6b078304ed4447b70fa7bc8de": "PDP^{-1}", "9c3257da5a4f4256537216d841437a42": " n(M) \\ \\mathrm{d}M = 0.4 \\ \\ln 10 \\ \\phi^* [ 10^{ -0.4 ( M - M^* ) } ]^{ \\alpha + 1} \\exp [ -10^{ -0.4 ( M - M^* ) } ] \\ \\mathrm{d}M .\n", "9c3272b672cbe41988193e2a9ff1f80a": " {\\rm Jac}(a_{1},a_{2},a_{3}) := \n\\frac{1}{2} \\sum_{\\pi\\in S_{3}}(-1)^{\\left|a_{\\pi}\\right|}\n\\Phi^{2}\\left(\\Phi^{2}(a_{\\pi(1)},a_{\\pi(2)}),a_{\\pi(3)}\\right) . ", "9c327ca196edf5a4d68d68949eb9d26d": "v= {3 \\over \\big({{8.5 \\times 10^{28}} \\big) \\times \\big({7.85\\times 10^{-7}} \\big) \\times \\big({-1.6 \\times 10^{-19}} \\big)}}", "9c32b8526c26d98ae5ed85eb2411858c": "exp(ahr) = \\cosh(a) + hr\\ \\sinh(a),\\quad a \\in R. \\!", "9c32be98ec25392e9ef54a63888be016": "V_d = T\\,\\!", "9c32dda9fcf349890ea0437e880dab57": "(i\\gamma^2\\gamma^3,\\;\\;i\\gamma^3\\gamma^1,\\;\\;i\\gamma^1\\gamma^2) = -(\\gamma^1,\\;\\gamma^2,\\;\\gamma^3)i\\gamma^1\\gamma^2\\gamma^3", "9c3342040f174c3e43263ab09f1038b2": "\\sigma_w^2", "9c33481d0536ce60c2f57738eaa081e8": "\\frac{1}{N_A}\\frac{dF_2}{dN_2}B_2N_2", "9c341aa7f34344597ddf1ad4c4638edb": "\ndV = h_\\sigma h_\\tau h_\\varphi\\, d\\sigma\\,d\\tau\\,d\\varphi = \\sigma\\tau \\left( \\sigma^{2} + \\tau^{2} \\right)\\,d\\sigma\\,d\\tau\\,d\\varphi\n", "9c3429adf15a40c174a41b16c38f86b8": "a_i+a_{i+n}=9 \\, ", "9c345d1c0f393c21453226816acceee2": "X\\colon K\\to{\\mathbb{R}}^3\\,", "9c346d5a0a7076327c9eda234795dd06": "\\scriptstyle |V_\\parallel|", "9c347df453930547f8c3ae4db6f18e50": "F \\left( t \\right) =-kx \\left( t \\right) ", "9c348963a24bdd329bf149635b355771": " \\lim_{h \\rightarrow 0 } R(h, k) = \\frac{ k^n ( A^* + C h^n + o(h^{n+1}) ) - ( A^* + C k^n h^n + o(h^{n+1}) ) }{ k^n - 1} = A^* + o(h^{n+1}). ", "9c34f6a05b310ca20df692601ef40f5f": "z_{k-1}", "9c35451bd4867d0e160e09cbbb97abd2": " { s \\over s^2 + \\omega^2 } ", "9c35de1badb3e9bc08f760627c4ff7f0": "(e_1,e_2,\\ldots,e_n) \\in \\{-1,1\\}^n", "9c3624ebb57efae0d72122384ed0d96a": "A_m(p,r)=A_m(p,r-1)+A_{m-1}(p,p+r-1)", "9c3645264479f3f424f2e74d17195430": " \nH_L= \\frac{(l_1)^2+(l_2)^2}{2I}+\\frac{(l_3)^2}{2I_3}+ mgh n_3.\n", "9c368f2f71de1dbe0085e202a1f190db": "\n{} - \\frac{\\varphi_{i,j} - \\varphi_{i,j-1}}{h_{j-1}^y}\n\\left ( \\frac{h^x_i}{2} \\varepsilon^y_{i,j-1} + \\frac{h^x_{i-1}}{2} \\varepsilon^y_{i-1,j-1} \\right )\n", "9c36a7b516075672528840dc02fd6ff4": "m_p(x^\\mu )=m_0\\exp [\\Phi _N\\left( x^\\mu \\right) ], ", "9c375b5e1b0521b3ac7a52d734d9d70e": " \\Sigma=\\sum_{x\\in {N}}f(x) ", "9c376ec0af33559e6406516339ccddf3": " \\mathrm{det} ( \\tilde{h}_{ab} ) = \\Lambda^2(\\sigma) \\mathrm{det} (h_{ab}) ", "9c37a4a2eb693d092d6754cf8037640a": "O(V E \\log(V^2/E))", "9c37ac230e45d3a6aa6a8f3ad7a33db1": " v_i = K_i \\begin{bmatrix} 1 & -\\sigma_i 2^{-i} \\\\ \\sigma_i 2^{-i} & 1 \\end{bmatrix} \\begin{bmatrix} x_{i-1} \\\\ y_{i-1} \\end{bmatrix} ", "9c37b05943e5584827686728aa0b864d": "\\ell=1\\text{m}", "9c37b44815b8f85d17b757150b087fda": "n\\alpha=x_1 + \\cdots + x_{n-1} + \\underbrace{x_n+x_{n+1}-\\alpha}_{=\\,y},", "9c37b633c5d78d91b06e1b6460c6f4d5": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1^\\infty=\\left[\\tau_{21} +\\tau_{12} exp{(-\\alpha_{12}\\ \\tau_{12})} \\right]\n\\\\ \\ln\\ \\gamma_2^\\infty=\\left[\\tau_{12} +\\tau_{21}exp{(-\\alpha_{12}\\ \\tau_{21})}\\right]\n\\end{matrix}\\right.", "9c37f5df46575d0a8148788214293056": "\\quad D\\in", "9c37fc56dab5668cea2a373e80dee0dd": "F'(c)(b - a) = F(b) - F(a). \\,", "9c380718fb3e245cbc56e5af8fc9922a": "\\lambda=\\frac{\\mathbf{l}\\ \\mathrm{rot}\\ \\mathbf{l}}{v},\\ \\mu=\\frac{\\mathbf{m}\\ \\mathrm{rot}\\ \\mathbf{m}}{v},\\ \\nu=\\frac{\\mathbf{n}\\ \\mathrm{rot}\\ \\mathbf{n}}{v}.", "9c380b2c218300aefd3d6076402e7093": "a,b \\in M", "9c3858ed5e51c6a58d8ee9bfd3bd2e72": "(1 - f)", "9c3886feed84d239fd4aa42ce3a76ed8": "c^T x \\ ", "9c390b5eeb0384535bd319e9d59d4562": "H = H_0 + V_M,\\ ", "9c392cb91ea3959fe76a8170fe8eb318": "\\begin{align}\n\\mathbf{E} &\\rightarrow\\mathbf{B} \\\\\n\\mathbf{B} &\\rightarrow -\\frac{1}{c^2}\\mathbf{E}.\n\\end{align}", "9c3990fac3fe044e17f757dd9cb1680e": "\\sqrt{67}", "9c39a8188880108cb71bf05e37e2c54d": "Q(v) = B(v,v),\\,\\!", "9c3a42f194a1e4e348182aca98842bd0": " (\\exists z_1...z_{nm+1}) (B_1 \\wedge B_2 ... \\wedge B_n).\\ ", "9c3a5aacdd075dc947c3981404dfd063": "\\langle\\psi|\\psi\\rangle = 1", "9c3a9175f81613cdc07adb970dc2e7d3": "a_1\\geq 0,\\ldots,a_n\\geq 0", "9c3aa2f36da50d5adf2e96e6fcbb6966": "1.8675", "9c3aab71f28afdbe6393fa81895edf3a": "\\int\\limits_{-\\infty}^{\\infty} e^{-x^2}\\,dx = \\sqrt{\\pi}\\!", "9c3abcc6b3df431c0a5bdd4651fdb23d": "\\scriptstyle |x_1|^0 \\,+\\, |x_2|^0 \\,+\\, \\dotsb \\,+\\, |x_n|^0", "9c3ad6d51dba8ac1c1934c762fdadbc1": "\\hat{f}(x)\\pm w(x)", "9c3aee9ee5a639cda6989555d985f0ba": " {x}^2+2a{y}=0 \\,", "9c3b2f167f7ea095b13ca77043591491": "(\\alpha \\and Fx_1...x_n) \\rightarrow \\beta ", "9c3b4334004de46e28470bf3ca7921d7": "I_o=\\bar{I_D}=\\frac{I_{L_{\\text{max}}}}{2}\\delta", "9c3b9df39006337478b3a297e10c397b": "A+B+C = 0,\\,", "9c3b9fe06b93e453c31fc61bdb903e5a": "\\ O", "9c3bb1f0214df4c696972e901afb518a": " (1 - f) ", "9c3bc8390bd88bd86136dbe7b2e29fad": "G = [ I_k | P ]", "9c3bec920a8562b92639caf6b18eacea": "m\\colon L\\rightarrow G", "9c3c06fb26ff1990f4142ccf8cbfb2d6": "\\scriptstyle M_b", "9c3c0953b330239c21465f2fad86e59b": "\\sqrt{s}\\cdot\\sqrt{s}=s", "9c3c5ff89b3131c3b36373339864719c": "\\, (p_i)_i", "9c3c9eef970003517edb0848c30b65c1": "(1-p)^M + (1-p)^{M+1} \\leq 1 < (1-p)^{M-1} + (1-p)^M.", "9c3cf0f0b1aacab3333b71681c047a48": " \\overline{H_1 X} \\simeq \\mathrm{Hom}_{\\Bbb Z[t,t^{-1}]}(H_1 X, G) ", "9c3d9541220dcd7a8082112d822b6bc6": "\n B \\vdash B\n ", "9c3e2e0e6c48439f092245b9993b4ba1": " \\frac{z^{-1}}{( 1-z^{-1} )^2}", "9c3e55887d3c8daf7afb5e28b9d3b15b": " x^n = \\prod_{i=0}^{w-1} {x_i}^{n_i} = \\prod_{j=1}^{h-1}{\\bigg[\\prod_{n_i=j} x_i\\bigg]}^j ", "9c3e5e6ddf4b1633216a9727b7bc7c6e": "t_r \\approx 2.2 \\tau \\approx \\frac{0.35}{f_c}", "9c3e768b6e4d16d548480915239a43fd": " q(z) = z + q_0 = z + iz_\\mathrm{R} \\ .", "9c3e7c14306d4ebb176264c592b0f49d": "\ng'_k(u) =|g'_k(x)|e^{i \\theta'_k(x)} = \\mathcal{F}^{-1}(G'_k (u))\n", "9c3f2be45fbadc3d3ffb626277936a37": "\\phi_{e6}", "9c3f55c3bee9e2b7ba2105ee850f6ed0": " \\sum_{\\mu\\uparrow\\lambda}\\frac{e_\\mu}{e_\\lambda}=1.", "9c3f7c845f486df6c85321c3f48438e9": "10^{\\text{googol}}=10^{10^{100}}", "9c3fb2838e1a4d8b7b22bb1630135106": "m(\\sigma)=g\\sigma", "9c3fb7ebb478b256b8f1c19db25b2a3f": "\\frac{d^2J_\\nu}{dz^2}=3\\alpha_\\nu(\\alpha_\\nu+\\sigma_\\nu)(J_\\nu-B_\\nu)", "9c3fba9f884143daa4e6bb306c95103c": " = \\int \\left( \\frac{\\partial \\Phi^*}{\\partial t} \\right) A\\Phi~dx^3 + \\left\\langle \\frac{\\partial A}{\\partial t}\\right\\rangle + \\int \\Phi^* A \\left( \\frac{\\partial \\Phi}{\\partial t} \\right) ~dx^3, ", "9c402eb8db796d6a5585f1e8f0818a67": "\\overline{\\mathbf x}\\sim \\mathcal{N}_p(\\boldsymbol{\\mu},{\\mathbf \\Sigma}/n)", "9c409222e468dcd58c0ee702194707f1": "E = \\frac12 \\rho g a^2 = \\frac18 \\rho g H^2,", "9c417d2341de3033ecd13b858542fd3e": "\\left[v^b(k),v^a(k)\\right]", "9c419c57ca2e4c0aad45a025dcf62a6f": "\\, t_R = 10 - y \\,", "9c41b2c836a7661c6782e19d8dbf7be1": "\\mathbf{k} \\mathbf{k}^\\mathsf{T} = [\\mathbf{k}]_\\times^2 + I", "9c41d38e1af3783c694aa9affc9b4b10": " s > 0 ", "9c4237eba0eb4f5fc68dbe95efb11278": "\\alpha, \\beta > 0", "9c4248a6e2df372d42130bfb030cc958": " b_i = (\\lambda_i + r+1-i, n)", "9c42711c62772a58e0c593e12308a4cc": "\\rho\\colon G \\to GL(V)", "9c4271ac36eaeee95bcb8f4ea53e24ad": "5\\pi/4", "9c4291edf9bf2eadb3fb82cc3d810cd4": "b_1,\\ldots,b_m", "9c42b23ac9d3df6285faf559506fe986": "E\\in (0,\\infty)", "9c432a33b38803e1258589018e9607d6": "(d-b) = km", "9c434d7b20e4136655b20d2c07caf80f": "T^2 = \\frac{4\\pi^2}{G \\left ( m + M \\right ) }r^3\\,\\!", "9c43d099658d8991493acb0977e84c8e": "[x_0, x_N]", "9c443f9b76974a543be1889e2471895c": "\\mathrm{N}_{\\mathfrak{L}}(S)", "9c444b6ffcc62f986fbf743a3ab855f4": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = J^{-1}~J~\\text{tr}(\\boldsymbol{l})~\\boldsymbol{\\sigma} - \n \\boldsymbol{F}\\cdot\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{l}\\cdot\\boldsymbol{\\sigma} + \\dot{\\boldsymbol{\\sigma}} - \n \\boldsymbol{\\sigma}\\cdot\\boldsymbol{l}^T\\cdot\\boldsymbol{F}^{-T}\\cdot\\boldsymbol{F}^T \n", "9c445ff49d80aa988903588cab93c341": "\\mathbb{C}^{n \\times n}", "9c446280042139e7def54dcef13236b9": " f = \\sum_n \\left\\{ \\frac{A_n}{B_n} - \\frac{A_{n-1}}{B_{n-1}}\\right\\},", "9c446ab239b108b33ee756eeed772d50": " (\\lambda 14 (\\lambda 3 (\\lambda 1 2 3)))) (4 (\\lambda 4 (\\lambda 14 (3 1) (2 1)))))) (1 (2 (\\lambda 1 2)) (\\lambda \\lambda 5 (\\lambda 5 (\\lambda 2 (1 5))) 7 6)) (\\lambda 6 (\\lambda 1 3 2)))) (\\lambda 4 (\\lambda 1 3 2))) (4 4) 3))", "9c449532455af9b24478de42abc02b6d": "x^2 + (K_a +y) x - K_a C_0 = 0", "9c44beeb26d4238088ff7f39acfe1b6c": " a_1=40,014", "9c4523f23a32f3a7d122ce8b3657aa8c": " \\mathbf{s}_i ", "9c453a348bf017f7f51b1afb2b2d0a4d": "p_{i-1}", "9c457696cbd5fc2250a01cebe61d0193": "Z + o(Z)", "9c458c50a1e26f812c0062549e2ddc2e": "Q(g(x)) \\cdot \\frac{g(x) - g(a)}{x - a}.", "9c45ef7e83f060da71d4d66a8be78002": "U_i=F(X_i)", "9c46309b445bef1c5eabaabd26b6ddd0": " \\mathbb P [ R \\leq \\rho, \\Theta \\leq \\theta] = \\mathbb P[R \\leq \\rho] \\mathbb P[\\Theta \\leq \\theta] \\quad \\forall \\rho \\in [0,\\infty), \\, \\theta \\in [0,2\\pi].", "9c46884b0105b7c9e8d186d72540f6e0": "v_{\\infty} \\,\\!", "9c46982fa155cf3dab9f75cbc669f0a5": "\\vec{p}\\cdot\\vec{r}=0", "9c46cc69fa740f00bb7479cd38c4122b": "A_{S} = 4\\pi r^2.\\,", "9c474a60502e613819d2280824853de3": "\\cos x = (1-\\sin^2x)^{1/2}", "9c4790782771ef5f23cae642446a5a49": "\\operatorname{haversin}\\left(\\frac{d}{r}\\right) = \\operatorname{haversin}(\\phi_2 - \\phi_1) + \\cos(\\phi_1) \\cos(\\phi_2)\\operatorname{haversin}(\\lambda_2-\\lambda_1)", "9c47dd46f855c98d88f5956c38d0f887": "t_{\\mathrm{QCD}} = \\frac{\\hbar}{m_\\text{p} c^2}", "9c486ccd710edc3d001cb0c469f0c9e1": "E(e) = \\frac {(1 + 5%)} {(1 + 7%)} - 1 = -0.018692 = -1.87%", "9c48ab9e8aac0f826b7baed3e33ca72f": "\\mathbf{J} = \\mathbf{T}\n\\begin{bmatrix} g_1 & 0 \\\\ 0 & g_2 \\end{bmatrix} \\mathbf{T}^{-1},", "9c48d4267604281276f5e80f1c83302a": "g(x)=\\frac{f(x)-f(-x)}{|f(x)-f(-x)|}", "9c48d44d2b4f41a7fb93eb0ac4820697": " \\frac{dN_1}{dt} = \\frac{r_1N_1}{K_1}\\left( K_1 - N_1 + \\alpha_{12}N_2 \\right) ", "9c49377205d2cf2a5b56618e17a75bef": "^n y = x", "9c49559a12621a449e9d57ee7113014f": "\\eta_{th}", "9c4961d16fe55690586fac1420c44045": "x_{0} > 0", "9c496cd43d77db620978e694ccf541b6": " u=u(p) ", "9c49bda8e09eade471b3da685fb32755": "\\textstyle Q_{ID} = H_1\\left(ID\\right) \\in G_1^*", "9c49f6c170a39419627315430726bd41": "\\sum_{k\\in S} x_k", "9c4a0444b497ce47782006e86cdcafa9": " (R,\\cdot ,\\eta ) ", "9c4a079b1da64f67f28f98cdf40e7893": "dN_Q/dt \\le q_0", "9c4a24315345e92c402b6ba49cdef811": "\\mbox{dn}(u; k) = {\\vartheta_{01} \\vartheta(z;\\tau) \\over \\vartheta \\vartheta_{01}(z;\\tau)}", "9c4a2632c50ee808f12919c7768d23f9": " L\\frac{\\mathrm{d}^2q}{\\mathrm{d}t^2} + R\\frac{\\mathrm{d}q}{\\mathrm{d}t} + \\frac{q}{C} = \\mathcal{E} \\sin\\left(\\omega_0 t + \\phi \\right) \\,\\!", "9c4a990f2bd8c81e8272c96fe2aa1435": "\\mathfrak{k}", "9c4aa3d59466ae08450c1debfc3992ec": "\\frac{\\partial \\phi}{\\partial t} + \\nabla \\cdot ( \\phi \\bold{u}_p) + \\nabla_{\\bold{u}_p} \\cdot \\left(\\phi \\bold{A} \\right) = 0", "9c4aa6952032806c4adc0eb6a3f86bc2": "H(f)=\\mathfrak{F}(h(t))=\\int_{-\\infty}^{+\\infty}{h(t)e^{-j 2\\pi f t} d t}=\\sum_{n=0}^{N-1}{\\rho_n e^{j\\phi_n} e^{-j2 \\pi f \\tau_n}}", "9c4af4a68321c1077b388996cba87a02": "F(x,y)=(x^2+y^2,a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)", "9c4af57cb0e71b6fb5b4699b235acef6": "\\sum_{i,j = 1}^n (r_j-r_i)(s_j-s_i)= \\sum_{i=1}^n \\sum_{j=1}^n r_is_i + \\sum_{i=1}^n \\sum_{j=1}^n r_js_j - \\sum_{i=1}^n \\sum_{j=1}^n (r_is_j+r_js_i) ", "9c4afb25580073c67828cedcbc1ce4bc": "{1\\over 2} < \\left\\lfloor \\mathrm{mod}\\left(\\left\\lfloor {y \\over 17} \\right\\rfloor 2^{-17 \\lfloor x \\rfloor - \\mathrm{mod}(\\lfloor y\\rfloor, 17)},2\\right)\\right\\rfloor", "9c4b07d3c0bf962f102f362d27663351": "\n\\frac{CA}{CB} = \\frac{DA}{DB}.\n", "9c4b0bbd803fbff8099a133d9e31e2fe": "\\int x^n\\,dx = \\tfrac{1}{n+1}\\, x^{n+1} + C \\qquad n \\neq -1.", "9c4b92f97dbfe5bc6771638b8e8a4a03": "\\frac{\\partial}{\\partial b} \\left (\\int_a^b f(x)\\; dx \\right ) = f(b), \\qquad \\frac{\\partial}{\\partial a} \\left (\\int_a^b f(x)\\; dx \\right )= -f(a).", "9c4bc690c383524fe1c186bb2c43608c": "\\langle u,u'\\rangle *\\langle v,v'\\rangle = \\langle u v, u'v+uv' \\rangle ", "9c4be3c847ce5fd45105da0218705ef1": "min_{\\mu}=\\sum_{i=1}^{N} |x_i-\\mu|^{\\beta}", "9c4c57644856e74f0cd8a5024d0278de": "L_{\\mathrm{fiss}}=\\frac{L_D}{N}=\\sqrt{\\frac{\\tau^2_0}{|\\beta_2|\\gamma P_0}}", "9c4c8e52ca3235d2037f76466767b60b": "\\Rightarrow : C^{op} \\otimes C \\to C", "9c4ca312aa1c2cd730115b32fb88b024": "\\mu_0 \\nabla f(x^*) + \\sum_{i=1}^m \\mu_i \\nabla g_i(x^*) + \\sum_{j=1}^l \\lambda_j \\nabla h_j(x^*) = 0,", "9c4d673bb490191a889c526be37eeeb5": "L^{2} (B)", "9c4e24695ee10afe3d7118b604475516": "\\textstyle \\delta_\\nu>0", "9c4e3fd64d0c0a1df4be42cfa5883a5d": "\\frac{\\sqrt{Q(e)}}{K} \\leq \\| e \\| \\leq K \\sqrt{Q(e)}", "9c4e5d8b5814d581a586e0e875fa2c9a": "\\textit{dau}", "9c4e8e2571a75745077dc49bb154d2e4": "T_e", "9c4ed45bb899ccd0e9db1931e808326c": "f''(t)+4f(t)=\\sin(2t)", "9c4ed53e44190c680a79396c412edb8c": "H_{\\min}(A|B) = - \\inf_{\\sigma_B} \\inf_{\\lambda} \\{ \\lambda | \\rho_{AB} \\leq 2^{\\lambda}(I_A \\otimes \\sigma_B)\\}~.", "9c4ed7c7d09f2c0f7c6ea7f760e9ff9b": " r_\\text{cm} = \\frac{m_1 r_1 + m_2 r_2 + \\cdots}{m_1 + m_2 + \\cdots}.", "9c4eee2e4b25c4cbf9231e7d8a6fc2f3": "ca_i=\\overline{a_{n-i}}=a_{n-i}", "9c4f5325397e76e21ce691c44f46bd27": "\\,\\!f(x)", "9c4f6af9b897c27dc24941e75d5674fa": " \n\\left|{\\partial \\mathbf{x} \\over \\partial t}\\right| = \\sqrt{\\sum_i g_{ii}~\\left(\\cfrac{\\partial q^i}{\\partial t}\\right)^2} = \\sqrt{\\sum_i h_{i}^2~\\left(\\cfrac{\\partial q^i}{\\partial t}\\right)^2} \n", "9c50be31a51640c75e9073a79e46fd03": "\n\\begin{matrix}\nx &=& p\\\\\ny &=& p\n\\end{matrix}\n", "9c50daee5bc0444b32fd2c877c45072c": "\n\\begin{align}\n2i\\sin{\\textstyle \\frac{\\theta}{2}}\\left(S_p - S_q\\right) & = \n\\sum_{n=q+1}^p a_n \\left(z^{n+\\frac{1}{2}} - z^{n-\\frac{1}{2}}\\right)\\\\\n& = \\left[\\sum_{n=q+2}^p \\left(a_{n-1} - a_n\\right) z^{n-\\frac{1}{2}}\\right] -\na_{q+1}z^{q+\\frac{1}{2}} + a_pz^{p+\\frac{1}{2}}\\,\n\\end{align}\n", "9c510563a580814c414b63788732ae2b": "(x_2-x_1)", "9c5113b7a3833c51adcde98afbd73ea5": "W=Y_1+Y_2\\text{ and }P = \\frac{Y_1}{Y_1+Y_2} ", "9c512a992a6089e6295849b2a5ca4a1d": "E = 0:~~~~~~~~ R = \\left( \\frac{9 M (t - t_B)^2}{2} \\right)^{1/3}~;", "9c51818f568ad4de88f5c7436490e742": "E(x,y) = K(x\\mid y)", "9c519a78f102d5d6d6c4a2b8d1bdd8cf": "f'(x)\\approx\\frac{f(x+h)-f(x)}{h}", "9c51f1e8ed70eef0fe685e9850d8f4f4": "-\\infty = \\infty", "9c520032099a165e0775630fd262554e": "h(v,w) = \\frac{1}{2i}[v,\\bar{w}] \\mod L\\oplus\\bar{L},\\quad v,w\\in L.", "9c525520f697cf9d3db66b7ace598ad3": "\\Delta V^{n+1}_X = \\alpha_X \\beta (\\lambda - V_{tot})", "9c527a7c675d9c27b96c5fe5ff8e5b35": "\ns_i = \\sum_n^N p_{ni},\\quad i=1,\\dots,I\n", "9c528f15622b7cd5c6aec3f97fe58135": "\\int_{-\\infty}^0 \\phi(ax)\\Phi(bx)dx = (2\\pi a)^{-1}\\arctan(\\tfrac{b}{a}) ", "9c52a474000242b0a31b1a5ed45d4990": "V_F - V_R = Z_0I\\,", "9c52cbb1db791fa1c647cb3ea156398d": "t(d,n) \\geq \\Omega(\\frac{d^2}{\\log d}\\log n)", "9c52eafacc50b06e9cc4400bb80f0094": "\\sigma_{\\bar x} = 0.1/\\sqrt 10 = 0.0316", "9c5305b6b264f0b7e7b3bd1cee042e17": " H_n(\\theta)=\\beta ", "9c531444614a9e3d0146d338b7e5c80c": "\\mathbf{P} \\left[ \\max_{1 \\leq i \\leq n} \\left| M_{i} \\right| \\geq \\lambda \\right] \\leq \\frac{\\mathbf{E} \\left[ M_{n}^{2} \\right]}{\\lambda^{2}},", "9c53413a6236dc371ab2e61b75febb1d": "f(x)\\rightarrow x", "9c535d298dfb65f298aba593a91e279d": "\n \\cfrac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\int_{\\Omega(t)} \\mathbf{f}~\\text{dV}\\right) = \n \\int_{\\Omega(t)} \\frac{\\partial \\mathbf{f}}{\\partial t}~\\text{dV} + \\int_{\\partial \\Omega(t)} (\\mathbf{v}\\cdot\\mathbf{n})\\mathbf{f}~\\text{dA} ~.\n", "9c536bd1751299520c5476fd161b2fa4": "\\delta_t = \\frac{1}{2}", "9c537a3959944878acf4bafcefa3cee5": "\\omega^{\\varepsilon_0 + 1} = \\omega^{\\varepsilon_0} \\cdot \\omega^1 = \\varepsilon_0 \\cdot \\omega \\,,", "9c538bda00908c42553e01428a326000": " Spin(10,\\mathbb C) \\quad\\mathrm{on}\\quad \\mathbb C^{16}", "9c53ace36042313ebb83ad44aba060e5": "\\frac{42}{56}=\\frac{3 \\cdot 14 }{ 4 \\cdot 14}=\\frac{3 }{ 4}.", "9c53d2626c1546d9020a8be0d894e205": "G(x;\\sigma)", "9c53d2e8467008c8aa444b1a2af27dbd": "f=165.4(10^{2.1x}-1)", "9c53d516bf2dcf4670f4c343978410b3": "x^2-4x+2", "9c53ff4c8b0487935d35fc152a0e229c": "{h S}=R D \\tau_c*", "9c544d8043ef220297da60624d23738e": "\\;\\;\\frac{1}{y} \\; g\\left(\\frac{1}{\\sqrt{y}} \\right)\\;\\; ", "9c5479738555a14decb423ea3a9faa72": " \\tilde\\beta = Cy ", "9c54878b411fb7a36f62659d89585e52": "u_{tt}-c^2u_{xx}=f(x,t)\\,", "9c5488c23be22d94a52599fe10365791": "\\tau = \\langle t \\rangle = \\int_0^\\infty t \\cdot c \\cdot N_0 e^{-\\lambda t}\\, dt = \\int_0^\\infty \\lambda t e^{-\\lambda t}\\, dt = \\frac{1}{\\lambda}.", "9c54b43e3831c327451c918348d6d031": "1.9000", "9c553fd76ad0379a10c6c0b1198fdc0b": " I = P/A = \\rho v \\omega^2 s^2_m/2\\,\\!", "9c554353c1652495fd6e082f7015fa4f": "\n\\begin{align}\nr^2+r-1 & {} >0 \\\\\nr^2-r-1 & {} <0.\n\\end{align}\n\\,", "9c55577faaefe2a13253b453c4a9b8dc": "\\begin{align}\n \\sinh (-x) &= -\\sinh x \\\\\n \\cosh (-x) &= \\cosh x\n\\end{align}", "9c5587e59eb5b4a972a6a9ceaa7d6996": "\\lambda = \\frac{1}{e\\sqrt{|\\psi_1|^2 + |\\psi_2|^2}}", "9c55bae995d217c1212a7e49c9ea8275": "P^{-}(V)", "9c55e3060f70defa4c56e7bcd57aa12b": "y=\\beta_1+\\beta_2x+\\beta_3x^2\\,", "9c562f3ca14c1b7da4cced1dc6ba0edc": "f*\\eta_\\varepsilon \\to f\\quad\\rm{as\\ }\\varepsilon\\to 0.", "9c5632a230e1e49bd61b01c03bdeeff4": " y_{n+1}\\equiv a y_{n}^{\\varphi(m)-1} + b \\equiv m_{i}(a_{i}m_{i}^{\\varphi(m)}(y_{n}^{(i)})^{\\varphi(m)-1}+ b_{i})\\equiv m_{i}(a_{i}(y_{n}^{(i)})^{p_{i}-2} + b_{i})\\equiv m_{i}(y_{n+1}^{(i)}) \\pmod {p_{i}}", "9c56866d362547715d94cbaf5f0f9906": "\\ F \\gg 1 ", "9c57716e2829292c0634ddd67375acc1": "z^k", "9c57e608bc9facd6ededb28498faa8e1": "((A\\to B)\\to A)\\to A", "9c57fd8524df3e8da4733722a4c1cfea": "x' = x \\cos\\theta + y \\sin\\theta\\,", "9c5829a32ec1c8dae73b5279efb551f4": "y \\in \\mathcal{Y}", "9c583b721e936a61421c80c92751b002": "A_0,B_0,C_0,D_0", "9c586b74db41b6d3560ced51a09a2162": "\\cdots \\to F(X) \\to F(Y) \\to F(Z) \\to F(X[1]) \\to \\cdots.\\ ", "9c588de439ba31a957b27c9eb1d7ad8d": "n > 1,", "9c58c61b18b1fbfc523b0efbc144c2e3": "Seeded: BOD_5 = \\frac{(D_0 - D_5) - (B_0 - B_5)f}{P}", "9c58f1ac184e374dd99aa7fdb0e820f9": "\\begin{bmatrix}\na & b\\\\\nc & d\\\\\n\\end{bmatrix} ", "9c591a19a500bddea97e7e31567d9d00": "I(k+r)A_k+\\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \\over (k-j)!}A_j", "9c596854585a86ff6bb7726333f4036d": "N(\\mu,1).", "9c596a820b4edef56c517d9f6667da89": " H_2\\subset L^2 ", "9c5990872b1ff22630988a39d382ea5d": " d \\propto \\left| ln \\left|t\\right| \\right|", "9c5996958987779ad13a28ff26784330": "\\scriptstyle\\boldsymbol{u} \\in \\mathcal{U}_\\Sigma", "9c59aeb73d37f7396eed4b9ca9c39acc": "\\textstyle A \\in \\mathcal{F} ", "9c5a4b3234989a1d1a64a0965d10f664": " H = \\frac{\\Phi^2}{2L} + \\frac{Q^2}{2C} ", "9c5ae4958e29386746b68a2a7fffb10e": " f(t) = \\sum_{n=0}^{+\\infty} a_n g_{\\gamma_n}(t)", "9c5ae63f6e1bdff1221ae82521f497eb": "x = \\frac{\\sqrt{4ac+b^2}-b}{2a}.", "9c5af91bb85935ba95f4886dc4062a8a": "c_{n-1,x} \\neq 0", "9c5afe3a1fe8a6dd37594e5a945c1778": "\\epsilon\\,R_n(\\xi,1/\\omega_0)=1", "9c5b053e22b898cf92827b489dc082ef": "V_{012} = \\textbf{A}^{-1} V_{abc} ", "9c5b1abda1728c0f2eb09627c1d3a7ef": "Father(Mark, x) \\wedge Employed(x)", "9c5b32e56d488c68b786a2f354e7bb16": " y = c_i x^i \\,.", "9c5b453a04abbca9e85c1733046e9e43": "\\textstyle\\frac{r}{\\cos(\\phi)}", "9c5b52bd41f2768d5b42da85b190cb3b": "(A \\wedge \\bar{B}) \\vee (\\bar{A} \\wedge B) \\iff A \\neq B", "9c5b967888c92c2560e73be03ebae447": "\\mathbb{E}\\Phi(||\\mathbb{P}_n - P||_{\\mathcal{F}}) \\leq \\dfrac{1}{2}\\mathbb{E}_{\\varepsilon} \\mathbb{E} \\Phi \\left( 2 \\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n \\varepsilon_i f(X_i)\\right|\\right|_{\\mathcal{F}} \\right) + \\dfrac{1}{2}\\mathbb{E}_{\\varepsilon} \\mathbb{E} \\Phi \\left( 2 \\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n \\varepsilon_i f(Y_i)\\right|\\right|_{\\mathcal{F}} \\right)", "9c5be0c08824220180419d21d3a07e8a": "r=\\frac{pq}{2\\sqrt{pq+4R^2}}", "9c5bff5a97d31591f0fc5c0699bd6037": "A_{lu}", "9c5c52e11dc388788e0707f1b7f86057": "ab = ac", "9c5cabd287ead46b7e51e15f134b4934": "\\frac{6054}{86,318} = 0.0703", "9c5cb36b5c6f9f747623a3afb7deed15": "(f^*\\omega)_p(v_1, \\ldots, v_k) = \\omega_{f(p)}(f_*v_1, \\ldots, f_*v_k).", "9c5ceed905a5afa0d93af984fee1137e": "Y= F(K,G,N,Z)=ZK^{\\alpha}G^{\\beta}N^{1-\\alpha-\\beta}", "9c5d119a9b278823675caabf08307714": "2^p 3^q", "9c5d34f267e54ceec347e29c5a176147": "\\text{Let } R = \\{ x \\mid x \\not \\in x \\} \\text{, then } R \\in R \\iff R \\not \\in R", "9c5d835a359e51741f552c0da8c2bf58": " \\forall x \\forall y f(x, y) \\rightarrow \\forall z (f (x, z) \\rightarrow y=z ) ", "9c5d924f70a7bcf96c8382ba26327d7f": "X=x(0), x(1), \\dots, x(L-1).\\,", "9c5ddfd162d016ef140c9c57971ec229": " C_{12}=C_{23}=0 ", "9c5df4fd7b90f10dfa1b3431242cc13f": "\\sum_{m = -\\infty}^{\\infty} \\delta[n - M m] \\!", "9c5dfbb7e3d324fb509ca69d25df32ef": "\\epsilon_i\\,", "9c5e6fb978e2da0b61a70cfc3c29f597": "\\sigma_{\\bar{x}}^2 \\rightarrow \\sigma_{\\bar{x}}^2 \\chi^2_\\nu \\, ", "9c5e79edd7f9107a47271cb169d3c949": "P_0(m,n)=1", "9c5edc53379fa12561ba1f44db8e1208": " \\scriptstyle -\\sqrt{5}C_{20}=1.08262668\\times10^{-3}", "9c5ef2dc1b331c69aa5d626e9815832b": "\\begin{matrix} {r \\choose 3}{4 \\choose 1}^3{52 - 4r \\choose 1} \\end{matrix}", "9c5f27222a8fad7810fa003842f25de0": "\n\\mathcal{L}(\\lambda,\\nu|x_1,\\dots,x_n) = \\lambda^{S_1} \\exp(-\\nu S_2) Z^{-n}(\\lambda, \\nu)\n", "9c5fb4bc55212e2fe259078a592bc05b": "n_{eff}", "9c5fd9b81024a8feb6ab42eaffda97b8": "c_\\text{gravity-capillary}=\\sqrt{\\frac{g \\lambda}{2\\pi} + \\frac{2\\pi S}{\\rho\\lambda}}", "9c6007a85273c9cdb6aecf92dc7976ab": "\\mathbf{B} = \\mu_0 \\left(\\mathbf{M}+\\mathbf{H}\\right). ", "9c600a6c02757dc46585b1e5c23c0549": "\\frac{1}{C_\\mathrm{eq}} = \\frac{1}{C_1} + \\frac{1}{C_2} + \\cdots + \\frac{1}{C_n}", "9c6015a27b4d5e27cba6bbcccfa9050d": " \\left(\\sum_{i=0}^n a_i X^i\\right) \\cdot \\left(\\sum_{j=0}^m b_j X^j\\right) = \\sum_{k=0}^{n+m} c_k X^k ", "9c603a267dedbc53a2e718d898e7029d": "{a\\pi\\over 3}\\ {b\\pi\\over 3}\\ {c\\pi\\over 3}", "9c6061fa997b4c6325b1c110b9151b16": "\n\\begin{align}\n\\mathcal{S}[\\vec{x}] & = \\int \\mathcal{L}[\\vec{x}(t),\\dot{\\vec{x}}(t)] \\, \\mathrm{d}t \\\\\n& = \\int \\left [\\sum^N_{\\alpha=1} \\frac{m_\\alpha}{2}(\\dot{\\vec{x}}_\\alpha)^2 -\\sum_{\\alpha<\\beta} V_{\\alpha\\beta}(\\vec{x}_\\beta-\\vec{x}_\\alpha)\\right] \\, \\mathrm{d}t\n\\end{align}\n", "9c6080426a900dfb58f5efd7dd631d84": "\nt", "9c6085080b6aeb47942fe8add4c6ba74": "D_2 \\varphi", "9c609c25e3efdc0f211e27877ede2d1b": "\\dot m_{out}", "9c61935ade30210117f6685563c6ef5c": "[0,\\omega_1]", "9c61b99dec9e87f0fab2298bdb72014d": "p, q > 0", "9c61e5d3da81b846f8ff493deb851c06": " b(\\Theta) ", "9c622c218d165c892e371d77a7a5e334": "\\left| A \\right| = \\sqrt{\\frac{2 }{L}}.", "9c628665982eed01953da8d325095215": "\\scriptstyle S^3", "9c629f020369f2be3f52da4b6a885151": "\nQ_n^{(c)}(t) < \\sum_{b=1}^N\\mu_{nb}^{(c)}(t)\n", "9c62b00064902f6b9912f94373954409": "\\scriptstyle w_0(n)", "9c62bc19b53628eea097998ada1158f8": "\n\\lambda=\\frac{2 \\pi}{k}.", "9c62dbcca62508d93fbd11ca9e8f1172": " 2 H_3PO_4 + Fe_2 O_3 \\longrightarrow 2 FePO_4 + 3 H_2 O ", "9c63424e98d1049a2fd79a9b5c1bca71": "\\Delta p = P_{IP} - P_{EEP}", "9c6358675e310124f1c69e512211a5a5": "\nTi_0(z) = {z \\over 1+z^2}, \\quad Ti_1(z) = \\arctan z, \\quad Ti_2(z) = \\int_0^z {\\arctan t \\over t} \\,\\mathrm{d}t,\n", "9c63771aa0b73b4a4a7a6deac47c7780": " k-1 ", "9c637bfb2c2916c690548294b162e5a6": "\\{J_k\\}\\to\n\\{\\tilde J_k\\}", "9c63f6721d763427d6793abfc717365b": "A(u,\\varphi) = F(\\varphi)", "9c643b2a0dafb392a4241c61a5496883": "{8 \\choose 2} + {9 \\choose 1}{7 \\choose 2} = 217", "9c643bb46b661c4a2c10bb8fe1292b6c": "J(\\Gamma (T))^\\bot", "9c645fde921f5b86cf8c311b924099d2": "\\{X_n : n \\in [0,\\infty)\\}", "9c64980dfb522f7bc1df5c29d08f7923": "d\\Phi = \\sum_i \\frac{\\partial \\Phi}{\\partial X_i} dX_i", "9c64e60e3f6a58f72a8f7076cd6f04d3": "\\alpha z \\ll 1", "9c65012e798424ec863f998fc84b83ba": " k= \\frac{2\\pi}{\\frac{1}{r_{AM}}-\\frac{1}{r_{BM}}-\\frac{1}{r_{AN}}+\\frac{1}{r_{BN}} }", "9c6506ddc055aa9e1bc29e8c9f1ef99d": " \\sigma_{DC}(p) \\propto \\sigma_m (p - p_c)^t ", "9c652ee8e4a689badb7d0c6449376461": "{n_p}", "9c6542fbe9428d1cde26de1af0822fee": "\\mathbf{u} \\cdot \\mathbf{v} = \\cos(\\theta)\\ \\|\\mathbf{u}\\|\\ \\|\\mathbf{v}\\|.", "9c658c7eee0b01fefc91a17f7feca4d1": "\\Delta(\\tau) = L(\\tau+1)-L(\\tau)", "9c65af3c12bc42c7a8561974680b1a6e": "y''+p(t)y'+q(t)y=r(t)\\,", "9c65dcb147610e1ac3b271d16bfa5fdf": "2\\pi \\log\\left( \\frac{G(1-z)}{G(1+z)} \\right)= 2\\pi z\\log\\left(\\frac{\\sin\\pi z}{\\pi} \\right)+\\text{Cl}_2(2\\pi z)\\, . \\, \\Box ", "9c65e92f523c813b1e856ec5201aa9c2": " \\frac{\\partial^2}{\\partial \\phi^2} \\Phi(\\phi)+(\\nu^2-\\frac{\\lambda q m d^2}{2 \\pi \\epsilon_0 \\hbar} Cos[2 \\phi]) \\Phi[\\phi]=0", "9c66117c791350b8873b75504fff51d7": "\\operatorname{curl} \\, \\operatorname{grad} f \\equiv \\nabla \\times \\nabla f = \\mathbf 0", "9c665d0b29834e20353f2632f03e22ec": "u_i = \\alpha_j (v_i - p_i)", "9c66a23953cccbd7a7552d548e4e7d9c": "U_{A}(\\delta_{A})U_{B}(\\delta_{B})-U_{A}(\\delta_{B})U_{B}(\\delta_{B})\\leq\nU_{A}(\\delta_{A})U_{B}(\\delta_{B})-U_{A}(\\delta_{A})U_{B}(\\delta_{A})", "9c66d329e8cd00692c727b4eba45000c": " { Q_1 \\over \\ Q_2} = { \\left ( {D_1 \\over D_2} \\right )^3 } ", "9c6722dea686bb58601e08609fc3be6f": "Q_C=\\omega'_0 {R}{C}", "9c673f86fc4139eadadaa4f2ee9a4fbc": "\n\\frac{1}{\\sqrt{p}}\\sum_{n=0}^{p-1}\\exp\\left(\\frac{\\pi in^2q}{p}\\right)=\n\\frac{e^{\\pi i/4}}{\\sqrt{q}}\\sum_{n=0}^{q-1}\\exp\\left(-\\frac{\\pi in^2p}{q}\\right)\n", "9c6762a420b7dcd473a06d8ef2107b7b": " gn(\\Psi) ", "9c67b31d6561849e3571fcc1923a4958": "dy/dt=-3y+yz\\quad\\text{and}\\quad dz/dt=z+y^2.", "9c67f58c9076583e56a951e3f34d3611": " \\chi (M) = |V| - |E| +|F| ", "9c680bcfcca11570bb79b1465bf868d8": " LWC = (m_w * n) / N ", "9c684365f21abe56adcfceae6172ba84": "( a, b )_{\\text{reverse}} := \\{ \\{ b \\}, \\{a, b\\}\\};", "9c6867cd18fcaf43060f21f46ddd1f39": "\\langle R|x\\rangle\\langle y|R\\rangle,", "9c6873fce372b5eb2514cf0ac02ec711": "N = \\sum_{i=0}^{k-1} d(i) F(i+2),\\text{ and }d(k)=d(k-1)=1.\\!", "9c6877ff436f8857dbdf2e5d12023a5d": "x_1^\\prime = 0", "9c68c14cb873956bec4886d6789c522e": " H_{2k}(\\mathbb{CP}^n; \\mathbb{Z}) \\cong \\mathbb{Z} ", "9c6917bcc55d9ff2bf87a929f342bcef": "w(d)", "9c695a885eb656f1688ace024df67db3": "\\mathcal{DB}", "9c6971771f7ca4298c3d8235d3ef0315": "a_{1,2,\\ldots,n}x_1x_2\\ldots x_n \\!", "9c6982e6b4f47b0e55208e5f8a307781": "y=3x", "9c6994d6e9c5cec83fdfac605b96c481": "R_v = \\frac{V_{R min}}{I_{D min} + I_{L max}/(h_{FE} + 1)}\\ ", "9c69a801ef007d57dc1769f00befa86f": "V = I.R + \\frac{\\text{d}(L.I)}{\\text{d}t}", "9c69ad7397b3d50b4fa391cc7fdc4d40": " W =\\{w_j | j \\in J\\}", "9c69c8330a27175ea7ccf1a8d7af8518": " n_z = 1", "9c6a27c30e19dbd386aa7c19bd2265f4": "\\frac{\\partial \\ell(\\theta|X,Y)}{\\partial \\theta} = 0 ", "9c6a8455f94609f1ac259b7667ac1a1a": "c[n] = x[n] - x[n-RM]", "9c6b2a0c88139c9dab91f329eacf2e6a": "a_0,a_1,\\cdots,a_k", "9c6b599fd1a74bc18b6d3013799c1cfb": "\\hat{H}_{\\text{JC}} = \\hbar \\omega_c \\hat{a}^{\\dagger}\\hat{a}\n+\\hbar \\omega_a \\frac{\\hat{\\sigma}_z}{2}\n+\\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{\\sigma}_+\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_-\\right).", "9c6ba8769caa017620e140fd11aa6891": "\\Phi^{-1} \\ ", "9c6bcf5d4f1717a8796d90d5f2c41b0e": "\\frac{\\mbox{Market Price per Share}}{\\mbox{Balance Sheet Price per Share}}", "9c6be07bfe926f92a8ff9615bda0384d": "f : \\mathbb{X}^{I} \\to \\mathbb{X}^{k} : \\sigma \\mapsto \\left( \\sigma (t_{1}), \\dots, \\sigma (t_{k}) \\right)", "9c6c0fa4863b64d2c38621cf6b795e6d": "M(p\\,q) = M(p) \\cdot M(q).", "9c6d43e6bc39cd0877f995459b9576c3": "\\delta_{ij} = 1", "9c6d4440dc45a07ee133c2dc98b56213": " V_0 ", "9c6dac2529c0001feb0c1b9b2ce78739": "a_{\\pm} = \\left(19 \\pm 3 \\sqrt{33}\\right)^{1/3}", "9c6de4ec2029e0988ce4651304c844a0": "x_i\\le \\lfloor x_i' \\rfloor", "9c6e1096650b0b4d18267112046f1bf6": " V_{out} \\sim V_A \\equiv \\frac{B_{in}}{\\sqrt{\\mu_0 \\rho}} ", "9c6e3755031bf4062415865ad65cbe26": " \\frac{\\zeta^3(s)}{\\zeta(2s)}=\\sum_{n=1}^{\\infty}\\frac{d(n^2)}{n^s}", "9c6e587b4f242161bcb71cc52e000ccf": "\\cos(\\delta) = \\frac{1^2 + 1^2 - b^2}{2(1)(1)}\\ . ", "9c6e767cb7a4766b8531401ff0205f1c": "(x_1+\\cdots+x_m)^n = \\sum_{|\\alpha|=n}{n \\choose \\alpha}x^\\alpha", "9c6ef07c1d6c8a68fb2bcf2a443fb4f3": "\\alpha_0=(\\alpha,\\alpha)^{-1} \\alpha", "9c6ef6b2aede0753f9e6777a195cd0b0": "Z(t) = Z_0 + \\Delta Z_R + \\Delta Z_C ", "9c6f0ce45b954af4dc96cb8e059478ee": "{S \\over n} = {1 \\over n}\\sum_{i=1}^n (X_i-\\overline{X})(X_i-\\overline{X})^\\mathrm{T}", "9c6f5774727a7bf5e7e16024051becf3": "S = -mc \\int ds.", "9c6f8150333637b52150f547fcd01cae": "\n \\sigma_e = a + b~\\sigma_m\n ", "9c6f8cb4acf7c6de27e8f36cb39dfde9": "\n\\begin{pmatrix} x_c \\\\ y_c \\end{pmatrix} \n = \\begin{pmatrix} A & B/2 \\\\ B/2 & C \\end{pmatrix}^{-1}\n \\begin{pmatrix} -D/2 \\\\ -E/2 \\end{pmatrix}\n = \\begin{pmatrix} (BE-2CD)/(4AC-B^2) \\\\ (DB-2AE)/(4AC-B^2) \\end{pmatrix}\n", "9c6f9935afa27a93de6c13a7946b4f41": "\\overline{NE(X)} = \\overline{NE(X)}_{K_X+\\epsilon H\\geq0} + \\sum \\mathbf{R}_{\\geq0} [C_i],", "9c6fcec89fa6553be4454c74983b0c25": "\\bold{j}_\\perp \\times \\bold{B}_\\perp = 0", "9c7029c92c5be2b9065db283311f1172": " 1 = \\int\\limits_{R_N} d^3\\mathbf{r}_N \\cdots \\int\\limits_{R_2} d^3\\mathbf{r}_2 \\int\\limits_{R_1} d^3\\mathbf{r}_1 \\, | \\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N \\rangle \\langle \\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N | ", "9c705617706e48a01e2e96d01a04afb7": "\n\\begin{bmatrix} \\mathbf{A} & \\mathbf{B} \\\\ \\mathbf{C} & \\mathbf{D} \\end{bmatrix}^{-1} = \\begin{bmatrix} \\mathbf{A}^{-1}+\\mathbf{A}^{-1}\\mathbf{B}(\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\mathbf{CA}^{-1} & -\\mathbf{A}^{-1}\\mathbf{B}(\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1} \\\\ -(\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\mathbf{CA}^{-1} & (\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1} \\end{bmatrix}\n", "9c7096c695121dedf78d684fe69e8b2b": "D_L = \\frac{c}{H_0}\\left(z+\\frac{z^2}{2}\\right)", "9c70bf2913042e62c6d287351caaeb17": "P_{n+k+1}(z) = P_k(z)P_n(z^{2k+1}) + z^{2k}Q_k(z)P_n(-z^{2k+1}) ; \\, ", "9c712a63e1098f7121acb5301b390bf8": "(A,g)(B,h)=(A \\wedge gB, gh)", "9c714cad34e20346725d7bc01c76532b": "\\delta^n_h[f](x) = \n\\sum_{i = 0}^{n} (-1)^i \\binom{n}{i} f\\left(x + \\left(\\frac{n}{2} - i\\right) h\\right).\n", "9c7166eb1937d9bc224aa1c1192bcf78": "\\xi=\\left[\\begin{matrix}\\xi_1\\\\\\xi_2\\end{matrix}\\right],", "9c7173e64655da8c65e41b103182c8ba": "l_{21}\\cdot u_{11} + l_{22} \\cdot 0 = 6", "9c719fc6733f7e4b2339a1558f3d5907": "w\\phi(x,y)", "9c71c1987fe757c765b5886c6d6bd5ea": "\\partial_t{\\rho }_\\mathrm{rel}=\\mathcal{P}L{{\\rho}_\\mathrm{rel}}+\\int_{0}^{t}{dt'\\mathcal{K}({t}'){{\\rho }_\\mathrm{rel}}(t-{t}')}.", "9c7217982c4557609494ecb1ae9a11db": "\\scriptstyle - \\frac{11+4\\sqrt{5}}{41}", "9c721d3f9080706c13d84d7dd80110d4": " \\pi_U (f) = \\int_G f(g) U(g)\\, d \\mu(g)", "9c722629582d0b636f01b32f64a03db2": "\n{\\Vert \\mathbf u \\Vert}^2 = {\\mathbf u}^2\n", "9c723ca524e8ef90d335b5a23b3ec28f": "\\nabla \\times \\mathbf{H} = \\frac{4\\pi}{c}\\mathbf{J}_\\text{f} + \\frac{1}{c}\\frac{\\partial \\mathbf{D}} {\\partial t}", "9c726ca67ecd7cd8f5ca93991bda7164": " (c,b_c,a_{bc^b}) = (c, b_c, a_{cb_c}) ", "9c728fa1a156307708213209b955d4f0": "\\textstyle\\binom{n}{k}", "9c72b1c2908c5277678fcfe14de9bd47": "u_{l}", "9c736c2438bed96f656c4d051240b973": "k\\colon\\mathfrak{g}\\otimes\\mathfrak{g}\\to\\mathbb{R}: X\\otimes Y \\mapsto -\\mathrm{tr}(\\mathrm{ad}_X\\circ\\mathrm{ad}_Y)", "9c73d1f8308b7effa698d0f4d54b00c0": "V_c \\, ", "9c73e8b60d994832fcbcaf93f13492d3": "F_c = \\frac{(Q_{tc}F_s)}{Q_{ts}}", "9c73f59cb43d2729295ef92786119f14": "\n\\begin{align}\n1101.101_2\n&= 1 \\times 2^3 + 1 \\times 2^2 + 0 \\times 2^1 + 1 \\times 2^0 + 1 \\times 2^{-1} + 0 \\times 2^{-2} + 1 \\times 2^{-3} \\\\\n&= 1 \\times 8 + 1 \\times 4 + 0 \\times 2 + 1 \\times 1 + 1 \\times 0.5 + 0 \\times 0.25 + 1 \\times 0.125 \\\\\n&= 8 + 4 + 0 + 1 + 0.5 + 0 + 0.125 \\\\\n&= 13.625_{10}\n\\end{align}\n", "9c73ff02e855566bcf788617a75f7bb9": " l=\\frac{3\\cdot x^2-a_1\\cdot y}{2\\cdot y+a_1\\cdot x+a_3}=0 ", "9c74122460f595c8298645f3eae859b1": "w(t;\\delta) = -1 + \\frac{2(\\kappa E_0)^2}{(\\kappa E_0)^2 + \\delta^2} \\sin^2 \\left(\\frac{\\Omega t}{2}\\right)", "9c7459cc06ee802987b66f096b2802b3": "s.", "9c7460697b8eb0d2de9fd3db70d5a444": "f^*:\\mathcal{P}Y\\to \\mathcal{P}X", "9c74acbcdb74ea38f6f59a838ad67c1f": " u' = \\frac{\\partial \\psi}{\\partial y}", "9c756fde6ed91b902c7ce103c0d4690e": "\\Pr\\left(\\overline{X}_n - A \\frac{S_n}{\\sqrt{n}} < \\mu < \\overline{X}_n + A\\frac{S_n}{\\sqrt{n}}\\right) = 0.9.", "9c765ec69d1752728c6878810d4f4ba8": "v(y) = y^{l+1} e^{-y^2/2} f(y).", "9c765fb140969c171cb36474fc490b77": " y_3 = \\lambda(x_1-x_3) - y_1", "9c767ddbd4c0a69487422d17cdddef52": "s=2^{-1/n}", "9c76e30c8b30627bcf5f67710649d414": "d_{in}", "9c77633be705702565da134b6f5670e4": " \\overline{X}_n = (X_{n1},\\dots,X_{nk}) \\; ", "9c778aa8d0b53830d63677103d32d43e": "\\vec p=\\tfrac{1}{2}(1,0,1)^\\top \\ .", "9c7791b05f6458fbce59cce431c396fa": "A,B \\in \\mathbb{R}^n", "9c77ca9f4246d794a9469dea5c25f108": "{d^2x \\over dt^2}-\\mu(1-x^2){dx \\over dt} +x= 0", "9c77fe0b2c75cd11d053d20b8b50ae07": "p(t) = |s(t)|^2", "9c78720ff389704473371ee730ce3e6b": " \\frac{\\partial}{\\partial P}\\left(\\frac{\\partial H}{\\partial S}\\right)_P =\n\\frac{\\partial}{\\partial S}\\left(\\frac{\\partial H}{\\partial P}\\right)_S ", "9c78924a85c3cee0622600f635afe096": "\n {\\mathbf v} = \\frac{dx^i}{d\\lambda} \\frac{\\partial \\;\\;}{\\partial x^i} = \\frac{dx^i}{d\\lambda} {\\mathbf e}_i\n", "9c790aa8c612ef864109224d78e53d92": " c_1=0.75 ", "9c791758a2bb68dfcbd0b13e2b99f192": " {D^{\\mathrm{eff}}} = ", "9c792ececfe23a14910530a577c9158d": "x_1, ..., x_n", "9c7935811b1bde1f3add78bd65e0ec2b": "y=\\frac{m}{x}\\, ", "9c797d089c6106e5573103d5ad6a20db": "\\{\\xi | x\\in\\xi\\}", "9c798afaaa7c1b67bd43f91a2043e650": "T_{\\varepsilon }^{TM}=sin(\\frac{m\\pi }{a}y)\\cdot(Ae^{-jk_{x\\varepsilon}x }+Be^{jk_{x\\varepsilon }x}) \\ \\ \\ \\ \\ m = 0, 1, 2, ... \t \\ \\ \\ \\ \\ \\ (19) ", "9c79b95bd5c976488be3eb116502d690": "n+2", "9c79d13ae53ea88bafe36e7f87086812": " \\kappa = {0.407}", "9c79f0ef4b925b5daaba77c28057a284": " \\left| \\lang \\omega | U_s U_\\omega s \\rang \\right|^2 \\approx 9/N", "9c7a12d27bb54818843130f1b753b8ba": "\\chi^{-1}(+1)", "9c7a4a352ba6b05190211a0fdf442c7e": "\n g_{ij}(q^i,q^j) = \\sum_{k=1}^3 \\cfrac{\\partial x_k}{\\partial q^i}~\\cfrac{\\partial x_k}{\\partial q^j} = \\mathbf{b}_i\\cdot\\mathbf{b}_j\n", "9c7a5541411f54a537bf38881680db2d": "O(\\tau+\\sigma m) ", "9c7a68a3a1e2b550f43ba27545b04f69": "\\, \\varphi = \\tan\\alpha - \\alpha", "9c7ab20c6ebb11d47b418c2eb0b9963c": "\\displaystyle A_n = n A_{n-1} + 2 A_{n-2} - (n-4)A_{n-3} - A_{n-4},", "9c7acaeefdbdd4d429c8ec2bf9c101f5": "\\chi\\left(\\Delta_\\lambda\\otimes\\Psi\\right)=\\chi\\left(\\Delta_\\lambda\\right)+\\chi\\left(\\Psi\\right)", "9c7b1bb1f72a5c4647fa42ca9d98f1ef": "{-2 \\cdot \\ln{p(x|M)}} \\approx \\mathrm{BIC} = {-2 \\cdot \\ln{\\hat L} + k \\ln(n) }. \\ ", "9c7b6dcc53c261e1e643ea33326d4be7": "(\\kappa + 1)~r~\\theta", "9c7bddd89025e1af2f518527a7f376c4": "2\\sum_{i=0}^{k-1}(d-1)^i=1+(d-1)^{k-1}+d\\sum_{i=0}^{k-2}(d-1)^i.", "9c7bde3f5560d3fd1da5c0a3cbf90ff7": "\\alpha=", "9c7bea23f047786c2506b24ed0b39c1d": "\\, (\\hat{A}, \\hat{B}, \\hat{C}, \\hat{D})", "9c7c0d1d71e0ba2c69b183f97c5302aa": "Tr(g^k)", "9c7c283e8e9987110f10e9ee2238c868": "{C}_{7}^{(1)}", "9c7c2a1d9bd3058f48146f06c0a86256": " \\frac{\\partial u}{\\partial t}- 6\\, u\\, \n\\frac{\\partial u}{\\partial x}+\n\\frac{\\partial^3 u}{\\partial x^3} =0,\\,", "9c7c34ac61def7fed1b1dbf6f17f21c1": "(r/s,0)", "9c7c36d302b826b74337474d9ec8e0c2": "n_\\max \\approx 2\\epsilon^{-1} \\left| \\int_{x_0}^{x_{\\ast}} dz\\sqrt{-Q(z)} \\right| , ", "9c7c46f062e5670533baa7bfea0e47eb": " v = \\left |\\mathbf{v} \\right | = \\left |{\\frac {dx}{dt}} \\right | ", "9c7c9229c44d6bf4e2c0ae66ea470ecb": " \\rho_1\\ :\\ g(g(x)) \\rightarrow x", "9c7ccf364aca676894a5a9d5f3bc1c4d": "\\scriptstyle\\varepsilon", "9c7d0a439182da8b8ed6902cc912e3c2": "\\textit{par}(h,m) \\land \\textit{fem}(m)", "9c7d0f00f465aa322d7a797e8317e1b9": "i\\in \\mathbb{Z}/p\\mathbb{Z}", "9c7d364491e735669e21cb4a5535610c": "\\Omega^{}_{}=\\Omega^i_{\\ j}", "9c7db24062877033c33e2179923ffcf5": "z = e^{j \\omega}", "9c7df754b9e421a9d455049f79de9aef": "h_{1;k} = \\frac{(k^2-1) k^k}{24}.", "9c7df9d35e23644c781a3b950e935bdd": "P(\\sigma_k = s|\\sigma_j,\\, j\\ne k)", "9c7e0b08c3c8d95c3d41d3ce77ec0f4f": "\n\\begin{align}\n\\text{sum} & = {\\mathrm{obs}(\\text{AA}) + 2 \\times \\mathrm{obs}(\\text{Aa}) + \\mathrm{obs}(\\text{aa})}\n= {1469 + 2 \\times 138 +5} \\\\\n& = 1750\n\\end{align}\n", "9c7ebb4ac83853e81c197611c2ca7afe": "n<1", "9c7ed7fb9f02592bc8aa764147dc7ff8": "V(\\mathbf{r})", "9c7f0984daa05364d8bc25ff42f40344": "x^{ 17 }+x^{ 14 }+1", "9c7f2b00b92b572f957dad9d45ea616b": " \\ln \\Xi ", "9c7f312c06acb38988073dc63a9ff3c0": "[t\\backslash s]", "9c7f376b9873b840205f745ce54a74e9": " A \\in \\mathcal{A} ", "9c7f57fbea6153a8f7e47b1d49629acc": "\\phi \\land \\chi \\to \\chi ", "9c7f6138eb717994a2e73ff1e70c7b73": " \\begin{pmatrix} \n c_1 & d_1 & e_1 & 0 & \\cdots & \\cdots & 0 \\\\ \n b_1 & c_2 & d_2 & e_2 & \\ddots & & \\vdots \\\\\n a_1 & b_2 & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n 0 & a_2 & \\ddots & \\ddots & \\ddots & e_{n-3} & 0 \\\\\n \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & d_{n-2} & e_{n-2} \\\\\n \\vdots & & \\ddots & a_{n-3} & b_{n-2} & c_{n-1} & d_{n-1} \\\\\n 0 & \\cdots & \\cdots & 0 & a_{n-2} & b_{n-1} & c_n \n\\end{pmatrix}. ", "9c7f7e33431b808f66b39ba2afad38fd": "2^2 + 1 - 1", "9c7fca174ce15dd73f812d8641c6856a": "f_i(x+h) - f_i(x) = \\nabla f_i (x + t^* h) \\cdot h.\\,", "9c7fcf7e5959f49527a57cc1d60e3ffb": " \\sigma \\,", "9c8018af3e9ff30f78925397d98ce57b": " L_{n+\\ell}^{2\\ell+1}(\\rho)", "9c8042ac47ee09d04587d6a40dfc17ff": "f_0(q) = \\sum_{n\\ge 0} {q^{n^2}\\over (-q;q)_{n}}", "9c804e62d3703f5c0875bc6e9e3086fa": " 1-e^{\\alpha/2^r} \\leq 2^{-r}", "9c8064948fd466f63102f5918f4aa1cb": "b=x_m\\sqrt{1-\\zeta_n^2(1-1/\\xi^2)}", "9c8109f290ec35831401ebb7a82aa53a": "\\overline{\\overline{\\{0,1\\}}\\times \\overline{\\{0,1\\}}}", "9c817d690566294daf95cb5641b39b36": "[\\mathbf{AB}]_{I,J} = \\sum_{K} [\\mathbf{A}]_{I,K} [\\mathbf{B}]_{K,J}\\,", "9c818cef89c2efb3c4889fb1de3adbbb": "f_3", "9c81e50bae6d409b10317f0f69573cc1": "at=1 \\mod n.", "9c82055875657f5e6b18394a02e8266b": " \n \\mathbf{w}^+ \\leftarrow E\\left\\{\\mathbf{x} g(\\mathbf{w}^T \\mathbf{x})\\right\\} - \n E\\left\\{g'(\\mathbf{w}^T \\mathbf{x})\\right\\}\\mathbf{w} \n ", "9c822b41ce72f5f4c64d3ac4e52c33f3": "X_i=P_i/P", "9c8259b48fb352ca5136c1b407d558f8": "y \\in \\mathbb{F}_2^n", "9c82b421a4d38d8f4fe028661db3e755": "\\mathbf{\\Sigma}^0_\\delta", "9c82b98604abcb7f5a601e193676f89d": "g \\exp(t X)", "9c82d549df314f71ed60c3e3a2f89cc5": "r\\left\\{\\begin{array}{l}r\\\\p\\\\q\\end{array}\\right\\}", "9c82d9e4ac1916476363ec8ca8c9a19c": "\\ln p", "9c830eb79c2b495616365ca6ec3a3352": "\\rho_1,\\rho_2", "9c834f2b2029b6982ee266d8dfa664f0": "P'_{R_0}", "9c8362c3c2a2c5cda73a1ff3aa87032a": "L\\colon D(L)\\rightarrow X", "9c8389191f574e08d29dfcdb7694c970": "N(x, t)", "9c838acdf775b87f261b73f1be3ddb28": "\\sigma_0(p^n) = \\sum_{j=0}^{n} p^{0*j} = \\sum_{j=0}^{n} 1 = n+1,", "9c83d1d86df50e74014aa7dd32d6d385": "\n\\begin{array}{rcl}\nY' &=& (Y' + 128) \\gg 8\\\\\nU &=& (U + 128) \\gg 8\\\\\nV &=& (V + 128) \\gg 8\n\\end{array}\n", "9c83f2fabcaedfdb148b39c5ccb876a0": "\\forall x P(x)", "9c849a19f87eb185048c22e1dc6fa302": "\\operatorname{adj}\\begin{pmatrix}\n\\!-3 & \\, 2 & \\!-5 \\\\\n\\!-1 & \\, 0 & \\!-2 \\\\\n\\, 3 & \\!-4 & \\, 1\n\\end{pmatrix}=\n\\begin{pmatrix}\n\\!-8 & \\,18 & \\!-4 \\\\\n\\!-5 & \\!12 & \\,-1 \\\\\n\\, 4 & \\!-6 & \\, 2\n\\end{pmatrix}\n", "9c84b3773acb4248fc4be355b2a439b8": "\\mathcal{A}:=R", "9c856e7c85d3917fcc0420b56d904e3d": "\\textstyle{\\frac{\\log(6)}{\\log (3)}}", "9c857aaebe6e91588546167c4cf22d2e": " f_x(z, y) = \\frac{f(x + \\tfrac{1}{n}, y) -f (x, y)}{\\tfrac{1}{n}}.", "9c85835fb76e1b0558f18e9b75860564": "d_1, \\ldots, d_n", "9c85877ab2dc9e13fd719631119c6998": "\\prod_{1\\leq i0}(1+q^{n-\\frac{1}{2}}z)(1+q^{n-\\frac{1}{2}}z^{-1}).", "9c8c83da0e89f92196cc4f7374cbd896": " p \\times p ", "9c8ca943875d23ed1e35c9afa898ed8a": "\\Omega,", "9c8cffbe47f43af31512e1dde47548ff": "\\log_b(xy) = \\log_b(x) + \\log_b(y).", "9c8d08cd418c1479604a49708aaa20aa": "(x_{i1}-\\bar x_{i})", "9c8dc13010de02ae6652fb7194ed1284": "\\kappa>1/\\sqrt{2}", "9c8de841898dad453552dd8315ecb70f": "H_{inv}(s)", "9c8e248baaed7b559c7268bb558628b2": "X(t)=c_{off}", "9c8e8692be3cf62133ece85113c30bfb": "Loss_{EME} \\mathrm{(dB)} = 100.4 + 20 \\log(F) + 40 \\log(d) - 10 \\log(\\rho)", "9c8e96f33b00528a3e19673759dae887": " \\Psi (x,t) = \\frac{1}{\\sqrt{L}} e^{i(kx-\\omega t)}, \\quad 0 \\leq x \\leq L.", "9c8e9abc6841486e03da9a59f0db941f": "x^3-x^2-1", "9c8f024d61eb1897ab95791dbda1f7e8": "Pitch=1/Teeth Per Inch", "9c8f98fc0eec3e66b56863774b72aa24": "2{{C}_{\\mu 3}}", "9c8fb29979dd3db33cad24dd93904020": "\\begin{pmatrix}R & \\mp\\sqrt{1-R^2}\\\\\n\\pm \\sqrt{1-R^2} & R\\end{pmatrix}", "9c8fe464362c1b7ff08a6c62149c422a": "{\\bar{DV}}_3", "9c9038dbe2da60f62d2e469bb76e5274": "f(x)=\\int_0^1\\frac{g(t)-g(x)}{t-x}\\rho(t)\\,dt", "9c903a46b45eab8b3e93db5e68b01b48": "\\{3.5\\},\\{4\\}\\,\\!", "9c909dedf3b44565bdf2a24d9dea86d0": "f(g(h(j(x)))) \\, ", "9c90c32c0674a038b563193ce8243778": "\\sum_{n = -\\infty}^{\\infty}{\\left|h(n)\\right|} = \\| h \\|_{1} < \\infty", "9c90e4347e7653c2e26c29d3f070a7b0": "\\begin{matrix}\nr & = & ( y_p \\cdot p \\cdot m_q + y_q \\cdot q \\cdot m_p) \\, \\bmod \\, n \\\\\n-r & = & n - r \\\\\ns & = & ( y_p \\cdot p \\cdot m_q - y_q \\cdot q \\cdot m_p) \\, \\bmod \\, n \\\\\n-s & = & n - s \n\\end{matrix}", "9c90e60774cff39c84e5656dec565951": "\\frac{-a_1 - \\sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} ", "9c90f21fcdf3637b6b131162087d1240": " \\Phi _\\alpha \\left( {A_\\alpha ^1 , \\cdots ,A_\\alpha ^n } \\right)_{-} = \\mathop {\\min }\\limits_{\\begin{array}{l} W_{\\alpha - }^i \\le w_i \\le W_{\\alpha + }^i A_{\\alpha - }^i \\le a_i \\le A_{\\alpha + }^i \\end{array}} \\sum\\limits_{i = 1}^n {w_i a_{\\sigma (i)} / \\sum\\limits_{i = 1}^n {w_i } } ", "9c910210b8fe72e00e6d5b4298cec974": "\\begin{matrix} {9 \\choose 5}{4 \\choose 1}^5 \\end{matrix}", "9c9115fa32bb44c03cd41e220f351e4c": "\\{a_n\\}", "9c91261bee770e84780c92e44291f2da": "\\int_{0}^{1} (1 - x^2)^0 \\, dx\\text{ and }\\int_{0}^{1} (1 - x^2)^1 \\, dx", "9c913ecbee16649dffd114425c4ce70b": "\n\\begin{align}\n|\\mathbb{F}_q^n| & = |\\cup_{c \\in C} B(c,d-1)| \\\\\n\\\\\n& \\leq \\sum_{c \\in C} |B(c,d-1)| \\\\\n\\\\\n& = |C|\\sum_{j=0}^{d-1} \\binom{n}{j}(q-1)^j \\\\\n\\\\\n\\end{align}\n", "9c9150c867d510666be8316a917575b0": " X\\boxtimes X^* \\boxtimes \\cdots \\boxtimes X,\\,\\, X^*\\boxtimes X \\boxtimes \\cdots \\boxtimes X^*, \\,\\, X^* \\boxtimes X \\boxtimes \\cdots \\boxtimes X,\\,\\, X\\boxtimes X^* \\boxtimes \\cdots \\boxtimes X^*.", "9c915461d466e63d7c782bb966d0c8ab": "\\sigma(n) < e^\\gamma n \\log \\log n \\,", "9c9169d4344a213391502fe9fd9ffc70": "\n\\frac{\\partial}{\\partial t} \\nabla \\times \\vec{u} \n= 2 ( \\vec{\\Omega} \\cdot \\vec{\\nabla} ) \\vec{u}.\n", "9c91c2c94f794244e6c91fc1e888a208": "h_{\\epsilon}(p_{0})", "9c91c9508095ab1f92e6028d77323789": "S=I", "9c91dd8c045a26f292d623086cead468": "j \\notin I", "9c929819381b70c14bb9fd448fe7e096": " R^6_6(\\rho) = \\rho^6. \\,", "9c92a3821277e1cb2beccf98de2c2f27": "{\\widehat X}_1,\\ldots,{\\widehat X}_n", "9c92b3cdc1afc9e165c0a16c4990764e": "2d<3dR(p_j \\in I_0).\n\\end{array}\n\\end{array}\n", "9c95020cda654309c81b999cdd500cfb": "= {a \\over c} + {(bc - ad) \\over c^2}\\varepsilon", "9c95250e09ca1ffa78188dac7c84fdaf": "T_{ijk} \\equiv g_{kl} {T_{ij}}^l", "9c95786349e055c094fa46f732f1ca77": "\nx = \\frac{a + b\\phi}{c + d\\phi}.\\,\n", "9c9622c190ffecd19638094bd9d9c9d3": "\\operatorname{perm}^{(s_1,s_2,\\dots,s_n)}(A) := \\text{ coefficient of }x_1^{s_1} x_2^{s_2} \\cdots x_n^{s_n} \\text{ in }\\left ( \\sum_{j=1}^n a_{1j} x_j \\right )^{s_1} \\left ( \\sum_{j=1}^n a_{2j} x_j \\right )^{s_2} \\cdots \\left ( \\sum_{j=1}^n a_{nj} x_j \\right )^{s_n}.", "9c963fc0392dafa8eb0441f1ed81119b": "\\sin(1) = 0.1\\ 2\\ 0\\ 0\\ 5\\ 6\\ 0\\ 0\\ 9\\ 10\\ 0\\ 0\\ 13\\ 14..._!", "9c96af36125338894cf3d5e86116d111": "\\beta=0\\!", "9c9789434e3c809e2f3dc26751040a27": "\\lambda := K - 2/3 G", "9c97b9ba6a4eb20da5922e1b1a8d2565": "\\text{Percent weight of the solid phase} = X_s = \\frac{w_o - w_l}{w_s - w_l}", "9c97decea1c695286d14f3bef95f69e0": "\\sigma_N =A^\\alpha \\sigma_0", "9c97e0015691a626216e7a3026497a03": "\nW = {n \\choose 1} \\left|A_1 \\right| - {n \\choose 2} \\left|A_1 \\cap A_2 \\right| + {n \\choose 3} \\left|A_1 \\cap A_2 \\cap A_3 \\right| - \\cdots + (-1)^{p-1} {n \\choose p} \\left|A_1 \\cap \\cdots \\cap A_p \\right| \\cdots. ", "9c97e7329718151eface791298659f74": "\\Theta \\subset \\mathbb R^d ", "9c984ff5cb146685872d3f79c531d812": "\\omega_p \\,", "9c9887671fb3341fda458b7f67e487b4": "\\hat{y}(k) = C \\hat{x}(k) + D u(k)", "9c98aed7f1e098f28ab95131721bc827": " {\\stackrel{\\triangledown}{\\mathbf A}}", "9c98dcc1d090e8830eeff5cb5c191795": " \\mathbb P\\big( \\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha \\big) = \\int_\\Omega 1_{\\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha} \\, \\mathrm d \\mathbb P = \\int_\\Omega \\frac{\\|X - \\mu\\|_\\alpha^2}{\\|X - \\mu\\|_\\alpha^2} \\cdot 1_{\\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha} \\, \\mathrm d \\mathbb P \\le \\int_\\Omega \\frac{\\|X - \\mu\\|_\\alpha^2}{(k\\sigma_\\alpha)^2} \\cdot 1_{\\|X - \\mu\\|_\\alpha \\ge k \\sigma_\\alpha} \\, \\mathrm d \\mathbb P. ", "9c98e2f42709927cb9cf8d44ad67dba4": " L\\{f(t)\\} = F(s)\n= \\int_0^\\infty e^{-st}f(t)\\,dt\\qquad(2) ", "9c991b52cb8d32c426cce5a7a6c26939": "v^{(k)}(x)", "9c993486414f612701ce69ba4c0bd3fd": "f(h(x)) = (f(x))^c\\,\\!", "9c99517cddc306cb2cdc899bc5ccd982": "\\zeta(s,a)=G\\,_{s+1,\\,s+1}^{\\,1,\\,s+1}\\left(-1 \\; \\left| \\; \\begin{matrix}0,1-a,\\ldots,1-a\\\\0,-a,\\ldots,-a\\end{matrix}\\right)\\right.\\qquad\\qquad s\\in\\N^+.", "9c9a4af10a76ec0fd900f4291948c1c4": " M \\simeq F\\oplus T(M), ", "9c9a8c19f4e474f1f9da13a60b6237eb": "1, 2,\\dots, v-1", "9c9ae598a8b3efde5a380f8eccc49e64": "\\mathfrak{P}^{23}", "9c9b1027fa82bcf5003412a3b26eda32": "\\operatorname{Spec}\\,R", "9c9b27a7e88ed7a1809da297926e288d": "b_{n} = T_{n} \\cdot a_{n} = \\frac{3}{4} \\cdot {\\left(\\frac{4}{9}\\right)}^{n} \\cdot a_{0}", "9c9b32370ee9d599167ab719600a7c3d": "(-p_1,-p_2,\\dots,-p_n)", "9c9b42f3f54a5593ea46a2af88b35b5e": "\\text{Cov}(X_1,X_2)=0", "9c9bb019b0bc96f43de6ee8f1035f08c": "\\mathfrak e_6\\cong (\\mathfrak{so}_{10}\\oplus \\mathfrak u_1)\\oplus \\Delta^{32}", "9c9c1b706bc9b189d58d9a482d790833": "V-\\frac{\\hbar ^2}{2m} \\frac{\\nabla ^2 R}{R}", "9c9c2fdc5c4cc6b22748942d0e2692ef": " a_n = \\sum_{m=1}^d b_{m,n} u_m = b_{1,n} u_1 + \\cdots + b_{d,n} u_d.", "9c9c886b500d697d54a2e4365bf3657a": "\\underline{P}=\\frac{ax}{ax+by+cz}\\underline{A}+\\frac{by}{ax+by+cz}\\underline{B}+\\frac{cz}{ax+by+cz}\\underline{C},", "9c9c9c175913063eaffb20c099f3579a": "w^{j}_{i}=1\\,\\!", "9c9cdaa809a9b96e4965e2410f12bedb": "U \\subseteq X", "9c9cf5e8d7f14221ced37f5c0e4f5212": "{w_1,...,w_k}=argmax_w \\frac{w^TX^TCXw}{w^TX^TNXw}", "9c9d15a1a16e5bba21c886e19bc7de68": "(d,\\binom{d+1}{2})", "9c9d7e82f48dda166e4f661003cc899d": "\\mathbf{X_{01}}", "9c9da07a935603407d52df356c35f82b": "d_{yz} = N_2^c \\frac{yz}{r^2} = \\frac{i}{\\sqrt{2}} \\left(Y_2^1 + Y_2^{-1}\\right)", "9c9dbd7a80447e3622303ad5ce307a54": " P(Y_i=n)=p(Y_i=0)\\cdot \\frac{\\lambda_i^n}{M_i(n)}, \\quad (3)", "9c9e108428633dca8cf7fc7f3efc81e9": " \\lim_{n\\to\\infty} \\frac{1}{n}\\sum_{j=1}^n h(j) = 1. ", "9c9e5cb348247f362566b553c30ae7db": " \\hbar \\omega \\left( a^\\dagger a + \\frac{1}{2} \\right) \\psi(q) = E \\psi(q)", "9c9e99d4697dcc60ee9ea1ff21876984": "rank_n(f(\\mathbf{x}))", "9c9ea8693a33fcc4b3cd8ac164f8988f": "F(X) \\in D,", "9c9ecb40b95f44254e9ec38af981835e": "\\nabla\\times\\vec{B} = \\frac{1}{r} \\frac{\\partial}{\\partial r} \\left( r B_\\theta \\right) \\hat{z} = 0 ", "9c9ede0fcf1dfc84a8a1481b346a9649": "C_n=n^{1/\\alpha}", "9c9ee434fbc6ed4efc6124392ecc4153": "\\chi(\\mathcal{X})", "9c9f0bd3b0b405700e744d2b1df6b629": "J * 2 = 22; 22 - 13 = 9", "9c9f35cf1145245292998d17f19dbfc3": "s(x_i, x_k) = -||x_i - x_k||_2^2", "9c9f5f9391a5eb7456a1847fa044d8cd": "GY = TY\\;", "9c9f7877721f2902ce969fc7669b7c90": "\\ \\alpha = \\frac{1}{\\sqrt{s^\\mathrm{H} R_v^{-1} s}},", "9c9f85cc2ca2909b6af25510c8d749ad": "x^{iq}", "9c9fff7439f98e9393b5a79f1483d374": "(81/80)^{1/4}", "9ca01458c88528aad47052fbef849264": "S = m + 2.5 \\cdot \\log_{10} A.", "9ca0399b7a4d34211ceefd713901d285": "a^rba^{-r}=b^{k^r}\\;", "9ca1079a74af7b5e8868f445a9bfda52": " a^\\mu ", "9ca12fc8c348dd17b97b24a82a6e8450": " u\\left( {C,L} \\right) = \\ln {C_t} + v\\left( L \\right) ", "9ca142c3462f7f363f7f6a718947d82e": "IA\\cdot IC=IB\\cdot ID.", "9ca1815d7457520aae0cfabdba5716a8": "W=N_K(A)/C_K(A)", "9ca195eda9f48ef5338bc862566a0b36": "G_{Y_1}(z)=\\frac{\\ln(1-pz)}{\\ln(1-p)},\\qquad |z|<\\frac1p,", "9ca1aab4e89e4e7de35c45c02549dbe7": "y = c+\\int_a^t X^{-1}(s)g(s)ds\\,", "9ca1d234aff2cb88307892fcd0e48b8e": "\\underline{\\#(\\theta)}", "9ca1d56e96c7854466c11730f4d5e9fb": "\\tilde{V}(K) = \\frac{1}{a}\\int_{-a/2}^{a/2}dx\\,V(x)\\,e^{-i\\cdot K\\cdot x} = \\frac{1}{a}\\int_{-a/2}^{a/2}dx\\sum_{n=-\\infty}^{\\infty}A\\cdot \\delta(x-na)\\,e^{-i\\,K\\,x} = \\frac{A}{a}", "9ca20c4bf3aab796bfd29d2d344c9d8d": "D=\\left[ {1 \\over 2\\pi \\;\\left( \\sqrt2\\right)^{2n} \\; \\sqrt{n!} \\; \\left( \\sqrt{ N-1 }\\right)^{n+1} } \\right]^{N\\left( N-1\\right) \\over 2}", "9ca220d994e86b922958c03ca7ba573e": " \\scriptstyle \\begin{cases}\n \\scriptstyle \\Delta t_i (t_i,\\, E_i) \\;\\triangleq\\; t_i \\,+\\, \\delta t_{\\text{clock},i} (t_i,\\, E_i) \\,-\\, \\tilde{t}_i \\;=\\; 0, \\\\\n \\scriptstyle \\Delta M_i (t_i,\\, E_i) \\;\\triangleq\\; M_i (t_i) \\,-\\, (E_i \\,-\\, e_i \\sin E_i) \\;=\\; 0, \n\\end{cases} ", "9ca274e9731604dd5431de2ad526bb75": "[t_l, t_u]", "9ca29432cde2199797dee1f9be243be5": " F_{perf} ", "9ca2c51c75d201b64156d542b8fc3a0b": "\\mathcal P(E)^p", "9ca2f049cac64ab4c6268e18c4e28639": "\\phi=g \\circ f : \\pi \\to \\mathbb{Z}^b ", "9ca3007675baf514819c8ca0cb45acaa": "{\\Gamma\\vdash e_1\\Rightarrow \\sigma\\to\\tau\\quad\\Gamma\\vdash e_2\\Leftarrow\\sigma}\\over{\\Gamma\\vdash e_1~e_2 \\Rightarrow \\tau}", "9ca30663693892222ac12fb6e6fb13a6": "A_{\\infty} = \\bigcap_{n \\geq 1} A_n.", "9ca350a6374a9c288b7ecd230c6baf3d": "x \\mapsto [x]_R", "9ca359596a02033ed87a718e7290fb04": "\\lambda^{-4}", "9ca3596f61305b0530482174ec346a4b": "w_n", "9ca36e4469b2f0ae93cbd762fdaeeb29": "G_2 \\supset SU(3)", "9ca3ce7710f0a676becbe7057d7a2c65": "\\,(x_1,\\;f(x_1))\\,", "9ca403794d3e3e8e79f55b264c5a8dc0": "\\beta:Q\\to T^*Q", "9ca42e9d4da2fffe20c776cdebe16584": "J^{\\prime}", "9ca46e1eb3f8309ffe04240fee795541": "\\operatorname{csch}\\,x = {\\rm{i}}\\,\\csc\\,{\\rm{i}}x \\!", "9ca4995118a57b82d6b77f374fe41fb3": "\\rho_$ = \\rho_c", "9ca49b183c1647b32465a4b2da93539b": "\\beta_2= \\frac{\\mu_4}{\\mu_2^2} = \\frac{a_1+16a_2+3(a_1+4a_2)^2}{(a_1+4a_2)^2} = \\frac {a_1+16a_2}{(a_1+4a_2)^2}+3", "9ca4cd642c939b161de3c3d40c992c9e": "\\overline{792}", "9ca4f3e30421b1418f06545135a7e3ed": "0.9472 + 0.3208i", "9ca4f7a56376ea950dee600a7dd742e9": "\n\\beta_\\text{crit} = 5\\langle B_N\\rangle \\left( \\frac{1+\\kappa^2}{2}\\right) \\frac{\\epsilon}{q_\\star}.\n", "9ca513208ee9a93f37a70f05b510d431": "{}^qH = \\ln\\left ( {1 \\over \\sqrt[q-1]{{\\sum_{i=1}^R p_i p_i^{q-1}}}} \\right ) = \\ln({}^q\\!D)", "9ca51a0424ac0dc18ed7ac0facd74ad6": "(dy, -dx) = \\mathbf{\\hat n}\\,ds.", "9ca595619cda703a2bc2c21a0079f0be": "\\text{length} (\\gamma) = \\frac14\\iint n_\\gamma(\\varphi, p)\\; d\\varphi\\; dp.", "9ca6018652515139b6408f3164696802": "\\lambda_e \\approx \\frac{h}{\\sqrt{2m_0E\\left(1+\\frac{E}{2m_0c^2}\\right)}}", "9ca61f458c78bb5591d04aaaa14da0e7": "\\chi^2", "9ca638db4c196118315855394fec8d2e": "\\partial \\Delta_{n+1}", "9ca677e75885f2a3835cdfbefedbd451": "n_{i}dt", "9ca698e6161aeef0c8f703ba94759556": "A(l,\\alpha)", "9ca6ab8246dda8abd3c0267944719ffa": " \\ln(r) + \\varphi i", "9ca6f52b35547578a087baf91952231f": "k, \\kappa", "9ca718971dab107ba0bf1e8c5fe8cb24": "2^{nH(k/n)}", "9ca7637c2d9799a5ea18383a600f9e65": "\nposterior (female) = \\frac{P(female) \\, p(height | female) \\, p(weight | female) \\, p(foot size | female)}{evidence}\n", "9ca767b2a9cdeef63aa2ad871530f32d": "\\frac{n_1}{x_1} + \\frac{n_2}{x_2} = \\frac{\\left ( n_2 - n_1 \\right )}{r}\\,\\!", "9ca77dbd709ec2b99a5befd8e698484e": "\\frac{1}{2 \\pi} \\frac{d}{dt} \\phi (t), ", "9ca78284a4ce643608d80a885b1300d1": " E_1=2 q_4 q_1", "9ca7c98fb0676a80ba97264802e06596": "d\\sigma/d\\Omega=|f(\\theta)|^2", "9ca814237e36b7a5256ac2f4347b724d": "\\mathrm{W = \\frac{J}{s} = \\frac{N\\cdot m}{s} = \\frac{kg\\cdot m^2}{s^3}}", "9ca822321e86aadb590da4c945f5ffc5": "d\\geq 2", "9ca88c1959ecd6934bbbd3f994fe0234": "V_{\\text{ijkl}}=\\frac{e^2}{4\\pi \\epsilon _0\\epsilon _r}\\int d^3x\\int d^3x'\\phi _i{}^*\\left(\\overset{\\rightharpoonup }{x}\\right)\\phi _j{}^*\\left(\\overset{\\rightharpoonup }{x}'\\right)\\frac{1}{\\left|\\overset{\\rightharpoonup }{x}-\\overset{\\rightharpoonup }{x}'\\right|}\\phi _k\\left(\\overset{\\rightharpoonup }{x}\\right)\\phi _l\\left(\\overset{\\rightharpoonup }{x}'\\right)", "9ca8fdd8da8457637750e082cefcbc68": "L^n E_t [ X_{t+j} ] = E_{t-n} [ X_{t+j-n} ] \\, ,", "9ca92fcc66292aa85813685bb4010f4e": "u = \\frac{1}{2}\\left(\\mathbf{E}\\cdot\\mathbf{D} + \\mathbf{B}\\cdot\\mathbf{H}\\right)", "9ca97e9cf4430857c907e8a48dd21d05": "\\varphi(x) = \\int_{\\mathbf{R}^n}\\varphi(y)\\,dy\\,\\Delta_x^{\\frac{n+k}{2}} \\int_{S^{n-1}} g((x-y)\\cdot\\xi)\\,d\\omega_\\xi.", "9ca9863407f2a510b181d5479f7a8ce5": "*\\exp(1)*x^7+\\frac{379}{362880}*\\exp(1)*x^9+\\frac{149}{907200}", "9ca9bffe1dd847e918094fe4dfb083b7": "\\mu = 4 n^2 - 1 = 3, 15, 35, ...", "9ca9f0e04da767be8130f391f4b8a287": "C_{10}F_{21}", "9caa155feddba3aa40886a732c1f2373": "\\bar\\eta", "9caa649cb85151f17783a0d5806fa5e6": "{S' = \\ln W} \\; ; \\; \\; \\; \\Delta S' = \\int \\frac{\\mathrm{d}Q}{k_\\mathrm{B} T}.", "9caa6ec948a5293783130b5d51b530bc": " \\begin{align}\n& \\Re[Z(s)]>0 \\quad\\text{if}\\quad \\Re(s) > 0 \\\\\n& \\Im[Z(s)]=0 \\quad\\text{if}\\quad \\Im(s)=0\n\\end{align} ", "9caa948e0ad6ceda3a0a7cb69f7c55d9": "det q^{(2)} = q_{11} q_{22} - q_{12}^2", "9caaf739a8df79b435016b4e3d1b3400": "C_i\\subset X_i", "9cac1114f60abc5d9342076c0c47a11c": "\\mathbb{S}^n_{+}", "9cac1e87f2e697785a5e44a1592b87ee": " f = x+y+z ", "9cac5dfcc707a6b95dc9eaefd2c51bac": "\\mathbf{f}(\\mathbf{r})", "9cac97a540d8761e9c6178810709d80d": "\\int_{\\mbox{straight}}+\\int_{\\mbox{arc}}=\\pi e^{-t},", "9cadebff57901620d1812db23755e9cb": "[x,y]\\cdot v = x\\cdot(y\\cdot v) - y\\cdot(x\\cdot v)", "9cae01a9ee3f168c92703a8b759ebdfd": "S_0(m\\Omega)=m^*\\Omega", "9cae0a41eafa15c3fa4fc563f74d81e5": "\\left| r \\right| < \\left| n \\right|.", "9cae2744ed541a8458ed73f303e31d93": "\\forall a \\in T(s) \\forall b_1,...,b_n \\notin T(s) ", "9cae65b60577f6b0d5abbcd7b2d15bd9": "\\Delta t= {{\\Delta \\tau}\\over\\sqrt{1 - v^2/c^2}}", "9cae8bb2d7b5664795fbf5e4e7812a52": " \\sigma^2_X = \\sigma^2_T + \\sigma^2_E ", "9cae95250f16888018ce8de74df2de29": "\\mathit{rate\\,of\\,change} = {\\mathit{close}_\\mathit{today} - \\mathit{close}_{N\\,\\mathit{days\\,ago}} \\over \\mathit{close}_{N\\,\\mathit{days\\,ago}} }", "9caebe54dbf0686b5a79fc65a190c051": "\\frac{S(z=z_0)}{\\hbar}= k z_0", "9cafb199630dc31f7954a59a68e706eb": "P\\left(S^{T}|O^{0}\\wedge\\cdots\\wedge O^{T}\\wedge\\pi\\right)", "9cb06f3504876820e6f1cf0a5d1c6136": "x = -6 = 7 .", "9cb0a062ba2281dfec2b22cbd68418c4": "(\\vec{s}_a + \\vec{s}_b)^2 ", "9cb0af05cd8865439faba44153ecf15f": "\\mathrm{HA2} = \\mathrm{MD5}\\Big(\\mathrm{A2}\\Big) = \\mathrm{MD5}\\Big( \\mathrm{method} : \\mathrm{digestURI} : \\mathrm {MD5}(entityBody)\\Big)", "9cb0d0245371e4914d05e688a00179b1": "\n\\begin{bmatrix}y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ y_6 \\\\ y_7 \\end{bmatrix} \n= \n\\begin{bmatrix}1 & x_1 \\\\1 & x_2 \\\\1 & x_3 \\\\1 & x_4 \\\\1 & x_5 \\\\1 & x_6 \\\\ 1 & x_7 \\end{bmatrix}\n\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\end{bmatrix}\n+ \n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\\\ \\epsilon_7 \\end{bmatrix}\n", "9cb1128c5bccea2c2121ae69b895d4b9": " \\zeta(z) = \\exp\\left({\n \\sum_{m\\ge1} \\frac{z^m}{m} \\sum_{x\\in\\mathrm{Fix}(f^m)} \n \\mathrm{Tr}\n \\left({ \\prod_{k=0}^{m-1} \\phi(f^k(x)) \n }\\right)\n }\\right) ", "9cb12e5bed6e0f686d7523e7194f4abc": "X=D \\frac {10^m-1}{n10^k-1},", "9cb1dd7673ef73b2c0034fae5e2bf1a1": "\\sigma_{x'}", "9cb2135709a23a092f7b014114379310": "A = \\{ P \\or Q, \\neg Q \\and R, (P \\or Q) \\rightarrow R \\}", "9cb24db8990129e9062c6fd3376e7ea3": "\\det(A) = a_{1,1} a_{2,2} \\cdots a_{n,n} = \\prod_{i=1}^n a_{i,i}.", "9cb2a35c10c1318c21340f69711896fc": "\\lambda_n \\in \\R", "9cb2c3a31049173b281d88fefec09e31": "\\int \\frac{d \\theta}{\\sqrt{C_0 + 2 \\cos(\\theta)}} = t + C_1\\,", "9cb3341b0dd012f93ed5425bd9b82092": " V = -\\frac{W_{\\infty r }}{q} = -\\frac{1}{q}\\int_\\infty^r \\mathbf{F} \\cdot \\mathrm{d} \\mathbf{r} = -\\int_{r_1}^{r_2} \\mathbf{E} \\cdot \\mathrm{d} \\mathbf{r}\\,\\!", "9cb3727ef74eb0278b89ccc2402769b1": "d_{i,j} = \\Big| x_i - x_j \\Big |^2", "9cb388285615cbefc59b3466008355c2": "\\mathrm{d} u + P \\;\\mathrm{d} v = d h = T\\;\\mathrm{d}s \\Rightarrow \\mathrm{d}s = \\frac {\\mathrm{d} h}{T} \\Rightarrow \\Delta s = \\frac {\\Delta h}{T}=\\frac{L}{T}.", "9cb3895e064c17f39df6575e3ffffd90": "\\varprojlim_\\lambda \\mathcal{O}_{\\text{Spec} A/\\mathcal{J}_\\lambda}", "9cb3aaf5e3f784816c327e7b7f347bc5": "(\\forall x \\, \\alpha(x) \\or \\forall x \\, \\gamma(x)) \\rightarrow \\forall x \\, [\\alpha(x) \\or \\gamma(x)].", "9cb3abfa0ffb0ec164e3a70f8f5acefe": "e_J^\\alpha e_K^\\beta (\\partial_{[\\alpha} e_{\\beta] I} + \\omega_{[\\alpha I}^{\\;\\;\\;\\; L} e_{\\beta ] L}) = 0", "9cb4a8fd6190bc9f4d870f01d2b52089": "B_\\infty^{p,q} = \\bigcup_{r=0}^\\infty B_r^{p,q} = \\bigcup_{r=0}^\\infty (\\mbox{im } d^{p,q-r} : F^{p-r} C^{p+q-1} \\rightarrow C^{p+q}) \\cap F^p C^{p+q}", "9cb4ac070fdf3737c3674409344ad34b": "AA^T+BC=DD^T+CB \\, ", "9cb4d498e0027eba7e42f3014c13ce7a": "r \\, \\cos \\theta = x", "9cb58e35090a61d790479105484009a9": "z_1 = y", "9cb599f5b376f0633c262e71756474ce": "( i\\text{-th}, \\omega_k(i)) ", "9cb5a26772e131ebca6615730eb03fbe": "\n f(k;\\mu_1,\\mu_2)= e^{-(\\mu_1+\\mu_2)}\n \\left({\\mu_1\\over\\mu_2}\\right)^{k/2}I_{k}(2\\sqrt{\\mu_1\\mu_2})\n", "9cb5b73050aacbdbd6dbfa212f05339d": "\n \\int x^{m+n} \\left(a\\,B (m+1)-A\\,b (m+n (p+1)+1)-A\\,c (m+2 n(p+1)+1) x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^pdx\n", "9cb6401602bd9cf2ae31579e36cd9f23": "R = 2 {R_s \\over \\pi d}", "9cb6a478cc4b71ed499917558a216ab4": "t \\in \\mathbb{Z}", "9cb6c6e55dcf634fe9bc7f78a44a84b0": "\n\\bar{D} = \\frac{1}{n}\\sum_i (Y_{i2}-Y_{i1}) = \\frac{1}{n}\\sum_iY_{i2} - \\frac{1}{n}\\sum_iY_{i1} = \\bar{Y}_2 - \\bar{Y}_1,\n", "9cb6dc63ad4e61296e325f1f64e1c58b": "H (\\boldsymbol{r}) = \\sum_{\\boldsymbol{R_n}} H_{\\mathrm{at}}(\\boldsymbol{r - R_n}) +\\Delta U (\\boldsymbol{r}) \\ . ", "9cb6e492d5c1ed32afc4ea4bd6a30ced": "-\\mathbf{\\theta}", "9cb6ea89b642bca9eb2b447439e91ee4": "L(s,\\tau)=\\prod_p\\biggl(1-\\frac{\\tau(p)}{p^s}+\\frac{1}{p^{2s-11}}\\biggr)^{-1},", "9cb7275595557cf1bbef1ae86d3af4e0": "[x]E[y] \\leftrightarrow T([x])=T([y])", "9cb751192c8891a75bc321fcc045c2db": "\\frac{1}{s_0}+\\frac{1}{s_1}+\\frac{1}{s_2}+\\frac{1}{s_3}+\\cdots", "9cb797afaaa5946ddc9430e34e1e4f0b": "X*Y = A(X,Y). \\, ", "9cb7c1f1749b7904440b7214379980cc": "\n1-\\gamma=\\sum_{n=2}^{\\infty}\\frac{\\zeta(n)-1}{n}\n", "9cb7db3477a37f8ff5990ca6af492920": " \\vdash \\forall x \\, P(x) ", "9cb839f81348e3d95d4a8e2b0703ebec": "\n \\frac{S_n}{\\sqrt n} \\ \\stackrel{p}{\\nrightarrow}\\ \\forall, \\qquad\n \\frac{S_n}{\\sqrt n} \\ \\stackrel{a.s.}{\\nrightarrow}\\ \\forall, \\qquad \\text{as}\\ \\ n\\to\\infty.\n ", "9cb86c7791f6a00ee8f60858d367387d": "\\ E\\{ |y_v|^2 \\} = \\alpha^2 s^\\mathrm{H} R_v^{-1} s = 1,", "9cb892bf596e3893e28feac141b5aae6": "\\Phi=n\\Phi_0.", "9cb8ce33abb9181a406a92e3c0904689": " \\mu_q ", "9cb90c42879f8458a2f5e535ea533cb0": "f(\\mathbf{x})=\\sum_{i_1=1}^{I_1} \\sum_{i_2=1}^{I_2} \\ldots \\sum_{i_N=1}^{I_N} \\prod_{n=1}^N w_{n,i_n}(x_n) s_{i_1,i_2,\\ldots,i_N},", "9cb9c7e739fa296e16aaef938bf55720": "I_{2}(\\sigma_{xx}\\sigma_{zz} - \\sigma^2_{xz}) - I_{3}(\\sigma_{xx}+\\sigma_{zz})", "9cbaaab3e0114196c713e4a68f54231a": "G = (G_0, G_1)", "9cbac23e059ecd0fe3838d200f823f18": "[0,r_i]^n", "9cbb0c1c4294d0df82457967289ea2af": "\\phi(x) = \\frac{1}{\\sqrt{2\\pi}}\\, e^{- \\frac{\\scriptscriptstyle 1}{\\scriptscriptstyle 2} x^2}.", "9cbb0ebeaad9e4304d560dba491fb906": "C_c(\\mathbb{R}) \\subset C_0(\\mathbb{R}) ", "9cbb65dcdf192a94fa4d015fef53a831": "0 < p < 1\\!", "9cbb705dd4a9c74b69f9d8596888e45f": "\\,k_B", "9cbbe7556892ad30810c059bbde102b7": "x=\\sum_{b\\in B}{\\langle x,b\\rangle\\over\\lVert b\\rVert^2} b.", "9cbc0c83e4354e68bef7f9d361ea5c6e": " \\sum_{\\mathrm{spins}} |\\mathcal{M}|^2 \\,", "9cbc455e9ddad46042f0defde22c3ae8": "V = \\frac {abc} {6} \\sqrt{1 + 2\\cos{\\alpha}\\cos{\\beta}\\cos{\\gamma}-\\cos^2{\\alpha}-\\cos^2{\\beta}-\\cos^2{\\gamma}}, \\,", "9cbc4cc1cce61dabb819281c84399503": "\\pi\\!\\;\\!\\!\\!\\pi = 2 \\pi", "9cbc4ef47a358182d8f466548e41ebb5": " a_1 + 2 a_2", "9cbcb5a56648fef4d0b0014634b347e2": "\\arg\\min_{\\mathbf{w},\\mathbf{\\xi}, b } \\left\\{\\frac{1}{2} \\|\\mathbf{w}\\|^2 + C \\sum_{i=1}^n \\xi_i \\right\\}", "9cbcb7905606ef0bbcd2fd6a9f9a3697": " \\langle \\rangle ", "9cbcbf97d152523ad859564d7966ddc2": "\n\\omega(t) = \\text{random event on slot t (assumed i.i.d. over slots)} \n", "9cbce2a4548689beda5450a4289d1238": "d \\phi: \\mathfrak g \\to \\mathfrak h", "9cbd445f4bafb55c4de2262f06f06da7": "G \\oplus H", "9cbdcdbb6d0734839ca8474d3f72b948": "\n\\frac{\\partial}{\\partial t}\\left[\\phi\\left(\\frac{S_w}{B_w}\\right)\\right]\n+\\nabla\\cdot\\left(\n\\frac{1}{B_w}\\vec u_w\\right)= 0\n", "9cbdd2afccedac1cd936210c0f4b70e5": "\\mathcal D_B (\\rho)", "9cbde7bb1916995ac98d6776226eead1": "x^{16} + x^{15} + x^{11} + x^{9} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1", "9cbdf924c633de1a4f432503b8d644ee": "\nP \\in [124W,130W]\n", "9cbe25a2fd45d6f40edd997b4f6952ab": "\\tau_E \\approx R/\\sqrt{k_BT/m_i}", "9cbe3482b1e5114461fa6996e2780232": "W =\\int_{\\gamma} \\mathbf{F}(\\mathbf{r}) \\cdot d\\mathbf{r} = \\int_{\\gamma} \\bigg( kq\\sum_{i=1}^n \\frac{Q_i(\\mathbf{r}-\\mathbf{p}_i)}{|\\mathbf{r}-\\mathbf{p}_i|^3} \\bigg) \\cdot d\\mathbf{r} = kq \\sum_{i=1}^n \\bigg( Q_i \\int_{\\gamma} \\frac{(\\mathbf{r}-\\mathbf{p}_i)}{|\\mathbf{r}-\\mathbf{p}_i|^3} \\cdot d\\mathbf{r} \\bigg) ", "9cbe894d9cad6236598f10ed7ff76c63": " O(wl(log |A|) ", "9cbecc705bd5ed7e1bddcc5d90da975a": "\\mu(x)=0", "9cbefc3963e2ecf7a5e00dc4cc391349": "\n\\log L(\\theta_0 + h | x ) - \\log L(\\theta_0|x) \\geq \\log K.\n", "9cbf17984c1f9282d29a4c94d730d93b": "S = S'", "9cbf51766b1f84246eb4c17d5508ec25": "\\begin{cases}\n\\dot{q} = \\frac{\\partial H}{\\partial p}= \\{q, H\\} \\\\ \n\\dot{p} = -\\frac{\\partial H}{\\partial q} = \\{p, H\\}\n\\end{cases}", "9cbf80d35867710933d0b9c4b7e1d786": "x_{\\nu} = x_0 + \\nu h \\mbox{ , } h > 0 \\mbox{ , } \\nu=0,\\ldots,n-1", "9cbf862b9a1ba2fb3fc4282ee2489a45": "\\begin{matrix}(\\mathcal{E}B)(\\psi)= \\\\ (\\exists \\phi:\\phi_0=\\psi\\land B(\\phi))\\end{matrix}", "9cbf91b8d24ffb8d410434170fce4a7f": " (y_{n})_{n \\geqslant 0} ", "9cbfe3d652525390052297f5b5936db0": " (1 2 3),\\; (1 3 2),\\; (1 2 4),\\; (1 4 2),\\; (1 3 4),\\; (1 4 3),\\; (2 3 4),\\; (2 4 3)", "9cc03952a5d6f7a3e60fd05018ac051c": "A_{\\{1} A_2 A_{3\\}} + B_{\\{\\mu \\nu} B_{\\nu \\tau} B_{\\tau \\mu\\}} + B_{\\mu \\nu} \\sigma_{\\mu \\nu}^{\\alpha \\beta} \\psi^\\alpha_k \\psi^\\beta_k + A_\\mu \\Gamma_\\mu^{\\alpha \\beta} \\psi^k_\\alpha \\psi^k_\\beta + (sym.)", "9cc04eb6c7777c3da61c0a18edde3c8f": "U_{(i)}=F_X(X_{(i)})", "9cc0547a8cc1e000bb535d81b213dee2": "\\Gamma(1) = 1,\\,", "9cc05a7f71780399bd349eae136b5938": "\nE_\\theta (r) =\n{-jI_\\circ\\over 2\\pi\\varepsilon_\\circ c\\, r}\n{\\cos\\left(\\scriptstyle{\\pi\\over 2}\\cos\\theta\\right)\\over\\sin\\theta}\ne^{j\\left(\\omega t-kr\\right)}\n", "9cc080ca8e0112fa53cf622a882487ff": "\\boldsymbol{G}(\\boldsymbol{x},t):=(\\boldsymbol{F}(\\boldsymbol{x},t),1), \\qquad \\boldsymbol{y}(t) :=(\\boldsymbol{x}(t),t+t_0).", "9cc0cc83fdd03ba148d3f364d9e232d4": "\\mathbb{D}^{(p)}(x) := (\\sum_{i=1}^n x_i^p)^\\frac{1}{p}", "9cc1e0194947ac31cc8ebb15e79d6b30": "\\mu\\to 0", "9cc283ffc4b825c2e72330e85693aaa1": "R_{PV}", "9cc2f996f1ba9d58ee342f6d6ca92d71": "\\overline{OC}", "9cc35834239f0621083c69bd8de4987d": "\\operatorname{NWScore}(\\mathrm{AGTA},Y)", "9cc35bfb55ba2c4f0c885d763a53d265": "\\tau(\\nu), \\ \\forall \\mu, \\nu", "9cc38b67a6ae435bf2b913a13968bdae": "\n\\frac{d^3W}{d\\Omega d\\omega}=\\frac{e^2}{4\\pi\\varepsilon_0 4\\pi^2 c}\\left|\\int_{-\\infty}^{\\infty}\\frac{\\hat{n}\\times\\left[\\left(\\hat{n}-\\vec{\\beta}\\right)\\times\\dot{\\vec{\\beta}}\\right]}{\\left(1-\\hat{n}\\cdot\\vec{\\beta}\\right)^2}e^{i\\omega(t-\\hat{n}\\cdot\\vec{r}(t)/c)}dt\\right|^2\n", "9cc3ad266a14ad7e56377f265ff3415f": "\\boldsymbol{F}=\\boldsymbol{RU}=\\boldsymbol{VR}", "9cc3cbee7b4926d2f6d4e1c8d90a519c": "n_\\eta(a+b)-n_\\eta(a-b)=-\\frac{\\mathrm{sinh}\\beta b}{\\mathrm{cosh}\\beta a-\\eta\\,\\mathrm{cosh}\\beta b}", "9cc3d62f287f1f7d734441369b757065": "R_{x_1}\\dots R_{x_i}", "9cc3dcc2fda321a0bf195dfcab9da91a": "\\mathbf{s} = \\mathbf{u} \\mathbf{t} + \\begin{matrix}\\frac{1}{2}\\end{matrix} \\mathbf{a} \\mathbf{t}^2 ", "9cc4298ef39efa3e27c46c8f61eb8764": "\\varphi_\\beta", "9cc4b3043f235aa4918f6aa6d9dbff51": "5^o", "9cc4bd394e03b9d1f972310f54d00299": "\\hat{\\mathbb{C}} = \\mathbb{C} \\cup \\infty", "9cc5089e399a36417cf852af12d5a3df": "y_k[n] = \\scriptstyle \\text{DFT}^{-1} \\displaystyle (\\ \\scriptstyle \\text{DFT} \\displaystyle (x_k[n])\\cdot \\scriptstyle \\text{DFT} \\displaystyle (h[n])\\ ),", "9cc531e3a96df5cd0e5b93569ea9c892": "\\lVert \\land \\rVert", "9cc558b1661afbb25999b3b0322520d8": " \\operatorname{build-list}[x\\ (q\\ q\\ x), D, L_3] \\and D[p] = [q, \\_, \\_]::[x, \\_, \\_]::L_3", "9cc602917fab45067efdf940ca99a643": "\\boldsymbol{\\mu} = g(\\boldsymbol{\\eta}) \\, .", "9cc60486e74c9ac9e839819ea01f5a6d": "j \\le k \\cdot r < j + 1 \\text{ and } j \\le m \\cdot s < j + 1. \\, ", "9cc613a14e655bfc47a259e90269f367": "\\gamma = \\arccos(Y_3 / \\sqrt{1 - Z_3^2}).", "9cc615a4890df4e417a1fd2875d6bdfa": "\\xi - \\frac{\\alpha}{\\kappa} \\left( e^{\\kappa^2/2} - 1 \\right)", "9cc66692d405db5789afeef8ca78d6c2": "h(i) = i+1", "9cc6aa5afade31aaf6606ea961e68833": "\\varepsilon = \\varepsilon_{\\text{r}} \\varepsilon_0 = (1+\\chi)\\varepsilon_0 ", "9cc6e2a8c92ca920dcec514ee67d7f33": "A \\# H", "9cc751b632f3ec75e05ea2f320a0ecac": "g_{\\Omega} (\\hat{a},\\hat{a}^{\\dagger})", "9cc7fde81b37af956504c742ebcf9e9a": "\\mathcal F(|\\partial_j \\sigma|)", "9cc82f501cbf915c97b883c130fe0731": "\\int_a^b f(x) g'(x)\\,dx = \\left[ f(x) g(x) \\right]_{a}^{b} - \\int_a^b f'(x) g(x)\\,dx", "9cc873748fdc6486ae8e20920a2d803b": "\\scriptstyle k \\,=\\, 2", "9cc8aaba2d2e8f040b67713a92f89c47": "\\begin{matrix}0\\end{matrix}", "9cca388164a319ba4c563a9607f4bd3c": "F(\\{a_i\\},\\{A_j\\})", "9cca765bc609924e43b6995db018c4b3": "P_{12}", "9cca86062bb22cea0f4b92602eb05620": "x=\\mathit{0}.", "9cca97fdfedbba266567ca990d3c032a": "\n\\mathcal{G}(\\mathbf{k},\\omega_n) = G(\\mathbf{k},\\mathrm{i}\\omega_n)\n", "9ccade7d2f96b4cfc7d016e5a17aaa1a": "b_N", "9ccadf6594b70c1a2c3b22349e867c82": "\\dot{\\hat x}=f(\\hat x,u)\n +W(\\hat x)L\\Bigl(I(\\hat x,u),E(\\hat x,u,y)\\Bigr)E(\\hat x,u,y) ,", "9ccb2bd696eed31644cb9bcaf7276486": "T(Z,X,Y)+\\textrm{cyc~in~}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)", "9ccb75f72e96438911bc8dcfd53f5b59": "X_{(n-k(n)+1,n)}=\\max \\left(X_{n-k(n)+1},\\ldots ,X_{n}\\right)", "9ccb8f2e48667f123569eaf6b22ceda9": "\\mathcal{U}(\\mathfrak{b})", "9ccbb3f60208b7fa06f725e3ce36a657": "\nK_2 = 0.2 + 0.02 \\times \\log_{10}(V_c)\n", "9ccbcbf16121d0ece8afae6e01bf582e": "\\mathbf{B} =B_0 \\tanh(x/L)\\mathbf{e}_z", "9ccc0b919ac5cce0f1ebb87f0b5f3be7": " \\epsilon_{t}\\stackrel{\\mathit{iid}}{\\sim}WN(0;\\sigma^{2})\\, ", "9ccc0d010e53701f5fab5766a1172b68": "L^{B \\otimes_A C/A} \\cong (L^{B/A} \\otimes_A C) \\oplus (B \\otimes_A L^{C/A}).", "9ccc34c3e862386acdec2c6789f31b11": "\\scriptstyle \\nu(g+h) \\le \\nu(g) + \\nu(h)", "9ccce143a25947a8b629789c3e8591ef": "\\omega=\\omega_0+\\kappa A^2,", "9cccf2f20ad68153a8e60ddead8f08b8": " y' = \\tan(y)+1,\\quad y_0=1,\\ t\\in [1, 1.1]", "9cccfd4f584c42925b0838aae775249c": "\\sigma_u", "9ccd0e6b7a2b35e287b3d75352941c0c": "\\langle T_H \\rangle = \\frac{1}{\\Delta S} \\int_{Q_{in}} TdS ", "9ccd0f4382e0c1d2084fda3e2022e4e4": "1 - e^{-x^2/2\\sigma^2}", "9ccd29a186a12b9ed39811dabfadefc1": "\\overline{d}", "9ccd507efcbf400c71fca48f7333cdcd": "\n\\frac{V_{in} - V_C}{R} = C\\frac{dV_C}{dt}\n", "9ccdb83de01ddab39d0900cd28da5928": "\\pm p_{0} = \\pm \\sqrt{2m\\left| E \\right|}", "9ccdc34a102fc5e50ef86d47bb9470f0": "x*(\\bigvee_{i\\in I}{y_i})=\\bigvee_{i\\in I}(x*y_i)", "9ccea174e7a587ff98346c28684281e2": "\\eta_{th} \\le 1 - \\frac{T_C}{T_H}\\,", "9ccf623b747ee3dfeed8fe52a31b91f6": " H/H_{11} = H_{22} - H_{12}^\\ast H_{11}^{-1}H_{12}. \\, ", "9ccf9a0711faa7ad3b582da12f7ad4e2": "\n\\frac{\\Gamma(k-1)}{(4\\pi \\sqrt{mn})^{k-1}} \\sum_{f \\in \\mathcal{F}} \\bar{f}(m) f(n) = \\delta_{mn} + 2\\pi i^{-k} \\sum_{c > 0}\\frac{S(m,n;c)}{c} J_{k-1}\\left(\\frac{4\\pi \\sqrt{mn}}{c}\\right),\n", "9ccfd097fd03ee1f3587e46e3656716f": "\\frac{\\partial^2\\phi}{\\partial r^2} + \\frac{1}{r}\\frac{\\partial \\phi}{\\partial r} + \\frac{1}{r^2} \\frac{\\partial^2\\phi}{\\partial \\theta^2}=0.", "9ccfd50d98a372282d130a227cff891b": "D_\\gamma (L) = LF_\\gamma (L)", "9ccfd9e407ba28bfebd20d577c2319f5": "-\\sqrt{\\frac{3}{35}}\\!\\,", "9ccff925d508387ac06a1c34041837e2": "\\mathrm{tfidf}(\\mathsf{example}, d_2) = \\mathrm{tf}(\\mathsf{example}, d_2) \\times \\mathrm{idf}(\\mathsf{example}, D) = 3 \\log 2 \\approx 2.0794", "9cd0070ed6be6937f7a6bcd919b44ea4": "Q[x(t)]=\\dot{x}(t)", "9cd052668606f878ac1ff66c5ca10702": " Tr(x)", "9cd0767e41a01545d8dbb7f98a0dc3f7": "T(C)", "9cd090381766e683a3d3775ef3a36ed5": "\\varepsilon = \\sqrt 2", "9cd0959beb57399fbaacbfcc37d1f7b2": "\\scriptstyle t^*", "9cd0a5ba5c6ba440e6e4a69c1d1d25dc": "E \\in \\operatorname{FV}[G] \\and E \\not \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] ", "9cd0c76909e0b3e29ec2e56b5be10300": "\\,\\! f(x,y)=e^{x} \\sin y", "9cd10408e40ddf59f02b3c263901e57c": "m = n\\!", "9cd1169bc19dd326669197d61ff85374": "\n\\displaystyle\n\\text{Performance rating} = \\frac{1000 + 400 \\times (0)}{1} = 1000", "9cd159c138521fe80b96f70e1f05780e": " C:R->R^2 ", "9cd1ef3f7b199ce8c7e1ae4101190b43": "A S \\int \\mathrm{d}\\mathbf{r} f(\\mathbf{r}) \\mathrm{e}^{\\mathrm{i} \\mathbf{k}_{in} \\cdot \\mathbf{r}}\ne^{i \\mathbf{k}_{out} \\cdot \\left( \\mathbf{r}_{\\mathrm{screen}} - \\mathbf{r} \\right)} =\nA S e^{i \\mathbf{k}_{out} \\cdot \\mathbf{r}_{\\mathrm{screen}}}\n\\int \\mathrm{d}\\mathbf{r} f(\\mathbf{r}) \\mathrm{e}^{\\mathrm{i} \\left( \\mathbf{k}_{in} - \\mathbf{k}_{out} \\right) \\cdot \\mathbf{r}} ", "9cd2382d9b3f76143fe44a76bbea1a19": "Cl_3 = \\{good\\}", "9cd23af0508bf0c714e806fefe220beb": " n! = \\begin{cases}\n1 & \\text{if } n = 0, \\\\\n(n-1)!\\times n & \\text{if } n > 0\n\\end{cases}\n", "9cd240bc09612815660f7d9d6e8f3deb": " 1 = \\sum_{i \\in \\mathbb{N}} E_i ", "9cd25c0a723be06b6bb7cc0ea230c439": "f_Y(y|X=x) = \\frac{f_{X,Y}(x,y)}{f_X(x)} ", "9cd2b228abc6ccd63da0f3aad0b734d7": "FV = PV \\cdot (1+r)^t", "9cd2d1f44deaa185c10ba0e675737980": "{\\bold \\ D}", "9cd2f0c36b7e4099dc90c8a0d666fed3": " a_X(\\alpha,z):=z^{-1}\\ln(M_X(z)/\\alpha) ", "9cd2fcc81b702490ef7228dc9816bd5e": "\\rho_3(z)= -z \\,;", "9cd314009db00ac568967f94ebb820b8": "A \\equiv_1 B.", "9cd34323a017c6fa48f699e0ce0b1929": "x=y \\Leftrightarrow\\forall a\\, (a\\in x \\Leftrightarrow a\\in y)", "9cd356c38f47d4cf726bde2e7742ae7a": "t_{ij}^{(-1)}=\\delta_{ij}", "9cd3724a557a09cdf6c65c03b7fe3eb9": "A\\left(t,T\\right)=-r\\left(t\\right)+\\int_t^T\\mu\\left(t,s\\right)\\,ds~~~\\textrm{and}~~\\tau\\left(t,T\\right)=\\int_t^T\\xi\\left(t,s\\right)\\,ds.", "9cd3f2139c73d3d27b6a943d365824e5": " U^0 = \\left(\\begin{array}{c} u^{1,0} \\\\ u^{2,0} \\end{array} \\right)", "9cd4124791ac4a175116dc60eaba11f8": "G :\\mathcal{D}\\to\\mathcal{E}", "9cd4b92f87615103daee5d9b9282829e": "\\gimel(\\kappa)= 2^\\kappa", "9cd5543dd6af941ad4916d6d486cc00f": "Y = \\frac{1}{X}", "9cd568a1c186e7e20130a7bed3fe459a": "T \\sqrt{g h}", "9cd575cb18160f3ef9225daec1822e3f": "r=x\\hat{i}+y\\hat{j}", "9cd5be36f44906332a809ed221e5b13e": "\n{dJ\\over dJ } = 1 = \\int_0^T \\bigg( {\\partial p \\over \\partial J} {dx \\over dt}\n+ p {\\partial \\over \\partial J} {dx \\over dt} \\bigg) dt =\n H' \\int_0^T \\bigg({\\partial p \\over \\partial J}{\\partial x \\over \\partial \\theta} - {\\partial p \\over \\partial \\theta}{\\partial x \\over \\partial J}\\bigg) dt\n\\,", "9cd5f1aa4630fdf0eeed29c709965ecb": " x^2+y^2", "9cd60a2d1b93894071ea576dfa7fe72d": "\ne^{2} - 1= \\frac{2L^{2}}{mk^{2}}E ~.\n", "9cd61d6930b10e2ca1451994f4b775df": "\\tilde{x}_i", "9cd63662d8a4b381a91f6b06e67bb648": "(\\Omega,\\mathcal{A},\\mu)", "9cd6a5a3d8125da10e58e0fdf61b386f": "y^{j-v}", "9cd6fa5eb6c9dbe854a6c02673371e55": "p=3, 4,", "9cd702fbfc19f1d6a26617bf9e25c376": "p=29", "9cd772d417394e464ca19e9357509312": "\\neg(a = b)", "9cd78781d17495b990e2ab4ac92b6b74": "\\mathbf{u}, \\mathbf{v}, \\mathbf{w}", "9cd7b8be5645c50320505df0fbdbb63d": "t_{threshold}", "9cd8395d34d59de9eb36eecb504bbd39": "\\vec{\\phi}", "9cd83b83561965e16a263010c678a4eb": "t_s \\in [0,\\infty]", "9cd83f87c0cd1c23ee87d7258f54d35d": "\\scriptstyle\\Delta_2", "9cd8cd1d054fe53e2277ecc30348845f": "f_{0} \\ll 1", "9cd944abd8960948a4b138b23aa8e669": "a\\in M", "9cd9a37401a949a7c7aed0ef75b2cb0d": "4T = \\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}.", "9cda54896a77854a139fb7310046584b": "\\alpha_A = 2\\alpha_L", "9cda79fe8c3451d0056f544d8c386a75": "R_g^*\\Theta = g^{-1}\\cdot\\Theta", "9cda7d3e47c2c96728caac969b97c770": " C(a,q,x) = \\frac{F(a,q,x) + F(a,q,-x)}{2 F(a,q,0)}", "9cda9e51a7ea3c069e810c5b9d00d9dd": "F_\\text{seq}(a)", "9cdad64668e33d90f19cfe2d282a56b4": "(\\#V+1)", "9cdb3a8ed904a630c7fb7cc737609f09": " R_{TP}= R_{T} + \\frac{dR_{T}}{dt} \\cdot \\left( k_{TOF} \\cdot R_T+t_{Delay}\\right) \\,\\!", "9cdb4cd16d4b3a9f8de542ab3a8272f3": "x^4-6x^3+4x^2+6x-5=0", "9cdb5083dafcd6616c6afe1ed9c11e10": "E_{j0}", "9cdb70e06be8e99cda6d712a1f389ee1": "V\\subset \\text{Tp}(\\text{Prim})\\,\\!", "9cdb7275f0069e0992c56122ff24520b": " \\operatorname{get-lambda}[p, p\\ f = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x)] ", "9cdbac26b7b17771f04a229f8b17727f": "\\pi_{\\nu,k}(f)=\\int_G f(g)\\pi(g)\\, dg", "9cdbc7d1e6ec33a44279733d493fb2a1": " T = E - E_0 = \\frac{m_0 c^2} {\\sqrt{1 - v^2/c^2}} \\ - \\ m_0 c^2 \\ ", "9cdbcc2290203e0e1efdc0d29f2208e3": "O(\\log (n))", "9cdbce5352ef2774699620ebf4d90935": " \\mathbf{B} \\, \\mathbf{a} = \\mathbf{0} ", "9cdc017deaee28b9128b7acdb64d471d": "A > \\sqrt{N}", "9cdc4554b6fdc1a07f82b166c5136bef": " V \\equiv \\mathrm{VOL}~\\bold{tbsp}", "9cdc5a9dbb8badbf39cab18b2937aedf": "\\frac{\\partial}{\\partial t} f_\\alpha + \\vec{v} \\cdot \\frac{\\partial}{\\partial \\vec{x}} f_\\alpha + \\frac{q_\\alpha \\vec{E}}{m_\\alpha} \\cdot \\frac{\\partial}{\\partial \\vec{v}} f_\\alpha = 0,", "9cdc88e65c90a481aacc2615dc5784be": "\\lambda = \\Delta r/\\Delta N \\,\\!", "9cdc8ec40b9d67cdc39f86b4e761b66a": "P = \\frac{D_1}{r-g}", "9cdc9da58d5f2f18251ff182011994e0": " H_{\\text{max}, D} = \\frac{2\\left(\\sqrt{2}+\\beta\\right)}{\\sqrt{4+2\\sqrt{2}\\beta + \\beta^2}} ", "9cdcc6a784cb2550a0ccd5a22362d47e": "Q,q\\,\\!", "9cdd057f0fec309a7d3643dc39ef3dc0": "\\nabla \\cdot \\mathbf{F} = d", "9cdd090ab527f36d38a39b9ec6964072": "B_n(s)=\\prod_{k=1}^{\\frac{n}{2}} \\left[s^2-2s\\cos\\left(\\frac{2k+n-1}{2n}\\,\\pi\\right)+1\\right]\\qquad\\mathrm{n = even}", "9cdd8706398110fdfaa14091ea292b69": "T^{\\mu\\nu}_m = \\rho \\phi u^\\mu u^\\nu", "9cde428c9e238d8b408086b15318bce8": "\\hat{x}_k", "9cde57319c6626ba671950daf88e35ba": "|z|^{2} = \\Big(\\text{Re}(z)\\Big)^{2}+\\Big(\\text{Im}(z)\\Big)^{2} \\geq \\Big(\\text{Im}(z)\\Big)^{2}=\\Big(\\frac{z-z^{\\ast}}{2i}\\Big)^{2}. ", "9cde6366a766df927b6a98985ae82935": " \\mathrm{MA} = \\frac{W}{F} = \\frac{1}{\\sin\\theta}.", "9cdead6d2d30079acf7276865ed9712d": "X=1234567=e^{e^{e^{0.9711308}}}", "9cded9f5a6533b66d6ce861783e36d12": "h_k", "9cdef4d42c757f596a4caa598085a72a": "m_1= 1-m \\,", "9cdef5cd2dcd033b898358f390a7dbe5": "P^{\\prime}(A,B) = A \\oplus B", "9cdf49ef1a5ee45969840d110fbec699": "2^{\\frac{M_p-1}{2}} \\equiv 1 \\pmod{M_p}.\\,", "9cdf880434f1db93e4a1a419ee3ba83c": "N_k = \\left( \\frac{(3 + 2\\sqrt{2})^k - (3 - 2\\sqrt{2})^k}{4\\sqrt{2}} \\right)^2.\n", "9cdf94886cec4ad4b63ff9f480c46645": "\\alpha_{k}'\\alpha_{k}=1,", "9cdfa0be4bc0fa1e4daa301846341050": "o(i) > e(i)\\,", "9cdfca49a30ae217d09f467826061a31": "R_G = 0\\,", "9ce004104696e702972c19d426b7a5d3": " ~m ", "9ce0b4dfd65931e10e875e1eb9b42b16": "3^3 + 4^4 + 3^3 + 5^5 = 27 + 256 + 27 + 3125 = 3435", "9ce0baa1533818898ec12704797bb06c": "\\mathfrak{c}", "9ce0efb8bda4ce06d34f9b3245d74be6": "t \\to + \\infty", "9ce0f4e947f6ee511bbd1a25e0cc9b01": "T(n)=T(n-1)\\mu(T(n-1)) \\text{ for } n \\ge 1 \\, ,", "9ce10f5b659d1fd17510549f2ca61eb7": "\\Phi(t)\\rightarrow x_0\\quad \\mathrm{as}\\quad \nt\\rightarrow\\pm\\infty", "9ce12d9ba0d21f3a7cd9eb74633ccd52": "\n{\\rm Var}(T) = \\frac{R_1R_2}{N} \\left(\\sum_{i=1}^kt_i^2C_i(N-C_i) - 2\\sum_{i=1}^{k-1}\\sum_{j=i+1}^kt_it_jC_iC_j\\right),\n", "9ce138007f56165b69871130eddb1c9f": "{n\\choose \\lfloor n/2\\rfloor}\\le \\log_2 M(n)\\le {n\\choose \\lfloor n/2\\rfloor}\\left(1+O\\left(\\frac{\\log n}{n}\\right)\\right).", "9ce1a3f15598e23ff05d773f0c6df0fc": "= h^{2\\alpha}", "9ce1c4c267ace8e062475c78048049bd": " y_{mt} = \\left(1- \\frac{1} {R} \\right) A_t + y_t", "9ce25c04ffe548590f2f8d2d89d4c276": "E(\\{R_i\\}) = E_{el}(\\{R_i\\}) + V_{\\text{ion-ion}}(\\{R_i\\})", "9ce2725d6438b56dbd23ea764a265ed1": "M<10M_{e}", "9ce27d910104e845aee7a6e2b9c6246a": "\n\\begin{align}\n7_{10}& = d_{0}+d_{1}\\cdot b+d_{2}\\cdot b^{2}+d_{3}\\cdot b^{3}+d_{4}\\cdot b^{4}+d_{5}\\cdot b^{5} \\\\\n& = d_{0}+2id_{1}-4d_{2}-8id_{3}+16d_{4}+32id_{5} \\\\\n& = d_{0}-4d_{2}+16d_{4}+i(2d_{1}-8d_{3}+32d_{5}) \\\\\n\\end{align}\n", "9ce29173174c3a58d97eea37568955f9": "\\frac{d^2 f}{dx^2} = 0 \\, , ", "9ce29565993aeb0d50bcee39f4b2dfd7": "\\leftrightarrow \\!\\,", "9ce2ca65f2ce4e0156fecc311d94181e": "\\mathbf{a}_2 = \\mathbf{a} - (|\\mathbf{a}| \\cos \\theta) \\mathbf{\\hat b}. ", "9ce303799f182cb6faa6931570466ea5": "16 \\rightarrow 10_1 \\oplus \\bar{5}_{-3} \\oplus 1_5", "9ce306fbc4cc05c2fbf40b6dc9cf6b39": "Z_G(t_1,t_2,t_3) = \\frac{t_1^3 + 3 t_1 t_2 + 2 t_3}{6}.", "9ce335498a120167360a2caebf4f4d3d": "(i\\omega-\\xi)^{-n}", "9ce37f941ad9499eddb9f3e830e5eaba": "R = 1", "9ce383883d80c26006c653d14d1b8ef2": "\\lVert\\alpha q\\rVert = |\\alpha|\\lVert q\\rVert.", "9ce38e5d559a40685f987e371c6e3ed0": "{dy \\over dx}\\sqrt{1-x^2}=1", "9ce38e9ed3ceb4521ed1044a43730b74": "\\hat{H}(x) |\\psi\\rangle = 0", "9ce39006f950b129073e4058661fdc49": "f\\in {{L}^{2}}\\left( \\mathbb{R} \\right)", "9ce3af77c8a3621317d9537339d391e9": "\\exp{(- \\Delta^{1/2}w/ \\Gamma) }", "9ce4154ae6590c80e3cb01f3b7e4e6c6": "\\nabla_\\mu", "9ce45c46d666260fb0879614e5849da5": "\\mathbf{a} \\times \\mathbf{b} \\in NS\\left(\\begin{bmatrix}\\mathbf{a} \\\\ \\mathbf{b}\\end{bmatrix}\\right).", "9ce46512f34ba39acc6bedfa560251bb": " \\vec{F} = 2 m (\\vec{\\Omega} \\times \\vec{v}) ", "9ce479b58af871a56dc508f839a168be": " \\mu_l \\cdot \\left [ F_{l}^{max}-F_l(I_1,I_2,\\dots,I_{n-1}) \\right ] = 0 \\quad \\mu_l \\geq 0 \\quad k=1,\\dots,n ", "9ce4d41aae50bea463255c6875d8e5e9": "\\bar{16}", "9ce4d4611d93f3321dfd4b73a265103a": "V(\\mathbf{q})", "9ce4e3475cd0ff92908ef9b783b9fbb9": "\\mathbf{S} ", "9ce505117bb9fd8bea1d7c274bb7518c": "\\underline{x}^{-k} = \\int_{-\\infty}^\\infty \\frac{\\phi(x) - \\sum_{j=0}^{k-1}x^j\\phi^{(j)}(0)/j!}{x^k}\\,dx,", "9ce529275942510a9a38fa2d0be57548": "RE(x_i-1, y_i+1) < RE(x_i,y_i+1)", "9ce5fb934f082ab6e172d8d950e0dbd6": "\ns_{\\overline{n}|i} = \\frac{(1+i)^n-1}{i}\n", "9ce61b64cef5b41971f331e1dcb295f9": "n > 3", "9ce626253bb12cf3a5ee0dd9311f375c": "10+1.37218 \\frac{\\sqrt{2}}{\\sqrt{11}}=10.58510.", "9ce689f12ff8969407e9def019c0077d": "\\tilde E = E \\times \\mathbb{Z}_{n}", "9ce6a12e277aa7ea09a0d5bf574fecae": "\\delta_n \\leq \\Epsilon_\\text{mach}", "9ce7ab1ff3cc22d027f592f177f7c007": " m \\leftarrow \\tfrac{1}{2} (a+b) ", "9ce7ec4f49c9808cb19996be793f8b2e": "\\left(\\sqrt{1/28},\\ \\sqrt{1/21},\\ -\\sqrt{5/3},\\ 0,\\ 0,\\ 0,\\ 0\\right)", "9ce7f36bffd6be55cc81fd2a249d2c06": " Y\\to X", "9ce8181c49ee2e897360f92574d8815e": "S(n) < \\Sigma(3n+6) \\,\\! ;", "9ce82fa3c352d56a33c8053cc98542d9": "\\alpha' \\in Z^{*}(X)", "9ce83166c22b38b86ed00fa178029952": "X(t^{}_n)=f(X(t^{}_{n-1}),t^{}_{n-1},t^{}_n,\\theta,W) \\, ", "9ce86350662085bad54e46959a91dc3e": "F_{t,T} - S_t e^{r (T-t)}", "9ce86bfccc4e788106940953b2a20af2": "\\omega _0", "9ce874c1500ec2e30dbebf36d870e3dd": "k \\geq m_i-1", "9ce8df751740c69aef1322da16839050": "R_\\text{sk} = \\frac{1}{n R_\\text{q}^3} \\sum_{i=1}^{n} y_i^3 ", "9ce9078c3ddc85c803238e2996d0b7c1": "SW = \\sum_i \\alpha_i v_{\\pi(i)}", "9ce907cc45bb9f62871f8bff52cee51e": "\\gamma:[a,b]\\to\\R^2", "9ce92d0890d1d885dcafdae85c671b2e": "C_{4,1} = 0 + 1", "9ce96fb90cb3eb5e9cb38fcf5459bc18": "< \\alpha", "9ce9a24a82556abd096d6fd9c969f83d": "a_{x_1, x_2, \\dots , x_n} = \\left\\{\\begin{array}{ll}\n-2n, & \\text{if } s = n \\\\\n1, & \\text{if } s = n - 1 \\\\\n0, & \\text{otherwise}\n\\end{array}\\right.", "9ce9e1bf4ea85c3077c6f1c12091cc11": "C_0 = D/V_{blood}", "9cea181458534055fdd94c28169059d6": "S_+", "9cea1e2473aaf49955fa34faac95b3e7": "ax", "9cea7bbaf0279484acabb39fad170a0d": "\\begin{pmatrix} 3 & 3 \\\\ 2 & 5 \\end{pmatrix} \\begin{pmatrix} 7 \\\\ 4 \\end{pmatrix} = \\begin{pmatrix} 7 \\\\ 8 \\end{pmatrix},", "9ceadc967bbeb714daf654273fbbc21d": "\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}", "9ceae406da348f66ba8589b80493f3bd": "W_{\\mu} \\stackrel{\\mathrm{def}}{=} ~ \\tfrac{1}{2}\\varepsilon_{\\mu \\nu \\rho \\sigma} J^{\\nu \\rho} P^\\sigma ,", "9ceaea7586098ff14e0676454228be50": "R(x) = ax(x-1)(x-r)", "9ceaf77e650113d935372c5ba47eb575": " {R^{\\mu}}_{\\alpha \\nu \\beta } ", "9ceb2e87b5d629cec0784cfb7ac63909": "\\ 100(\\mbox{collateral price}) = 20(\\mbox{inverse price}) + 80(\\mbox{floater price})", "9ceb4824d465102de1ddeba808bdc3b3": "-\\log\\!\\left(1-(1-t)^\\theta\\right)", "9ceb88e06d5d133fd32cc0d5f368ac71": "\\mathbf n=\\frac{\\mathbf y- \\mathbf c}{\\left\\Vert\\mathbf y- \\mathbf c\\right\\Vert},", "9cec0e7cc7d25c2a0458f5182deecf5a": "\\theta_{(l)}=0", "9cec12139be9f7c7b81778e58ac641da": "H(\\mathcal{S}) = - \\sum_i p_i \\sum_j \\ p_i (j) \\log_2 p_i (j), \\,\\!", "9cec1391c5c9fd1cf218c3132e41f983": "E[\\vec{X}]_{\\hat{p}\\hat{q}} = \\frac{1}{2} \\, \\exp (-4 \\, f(u)) \\; \\left ( f'(u) ^2 - f''(u) \\right) \\, {\\rm diag} (0,1,1) ", "9cec51c7c18fd64dd66555fac6eb1cbf": "\\Delta\\Omega\\,", "9cec9a969dd5c62441a6c463fce97d06": " \\langle j|\\,\\rho \\, |i\\rangle \\,=\\,a_j\\, a_i^{*}.", "9cecd2efb7989fba95dee1933a20b803": "\\mathcal{W}^{-1}({\\mathbf A}+{\\mathbf\\Psi},n+\\nu)", "9ceceb51c89a3ee1367957b3ded81137": "B(t_0)", "9cecf48a34c989e9c4cd5248d4f06795": "\\operatorname{Tr}(\\sigma Q)", "9ceda57d1a603674b1f6e24114967715": "\\psi^{(0)}(z+1)= -\\gamma +\\sum_{k=1}^\\infty (-1)^{k+1}\\zeta (k+1)\\;z^k", "9cee5c2e5a344b2e540d13572e1cd858": "X \\to X", "9cef8b1a8c69f71e2ac568b12347c969": "f',", "9cefd96d40d0ee1256c8eff1def0312d": "-n", "9cefe72620668e8489be609e476fd856": "\\begin{smallmatrix}L_{\\bigodot}\\end{smallmatrix}", "9cf011685c72f4ad6dcf2789d81f8061": "\\phi_B^{}", "9cf0da63b5240a85a3abf50768bf921b": "f_i(\\vec{r},\\vec{p},t)", "9cf0dc9b1f5cf9762cbadb5a1156cf85": "C_{in}^1,C_{in}^2,..,C_{in}^N", "9cf1b1b368561dc777367a7288c65aac": "s_i=\\sum_n^N x_{ni}", "9cf1da93fbcad9e75108872d27c9e0f4": " a \\times \\tfrac{b}{c}", "9cf1e8f8ab00fb2580a18331dbf35a9f": "\\partial_\\mu = \\frac{\\partial}{\\partial x^\\mu}", "9cf21689bc12fbe02a43c9f39bfd3458": "\n\\sum_k \\frac{(-1)^k}{k!(j_1+j_2-j-k)!(j_1-m_1-k)!(j_2+m_2-k)!(j-j_2+m_1+k)!(j-j_1-m_2+k)!}.\n", "9cf2243b15e893877560837c46448142": "p(t) \\approx \\sum {p(\\tau ) \\cdot \\Delta \\tau \\cdot \\delta } (t - \\tau )", "9cf22cbf685c0740baffc51a108ae3ab": " \\omega(x,y) ", "9cf2423d3518c5e4895b61c6af24e1a6": "{v_0, v_1, \\ldots, v_d, \\ldots, (v_i \\wedge v_j), \\ldots (v_i \\wedge v_j \\ldots \\wedge v_r)}", "9cf2ea71ff29c9284af978ab59116c46": " L^2({\\mathfrak H}^3)", "9cf306108bc761fd6eea55401f20e909": "m=\\frac{(1+Currency Drain Ratio)}{(Currency Drain Ratio + Desired Reserve Ratio)}", "9cf30a1abc969756a285fcdc136bb911": "f_{i\\, ,\\, k}:C_i \\to \\R", "9cf31cb82e56607bccc8eba08463d6b7": "z_i := p(\\bar{x}_i) - f(\\bar{x}_i)", "9cf33c449f512204cc427016b9ad2205": "\\frac{\\partial c} {\\partial t} = \\left( \\left[ 1 + \\frac{ 2\\eta Y}{f''} \\right] \\frac{d^2 c}{dx^2} - \\frac{2KF}{f''} \\frac{d^4c}{dx^4} \\right) ", "9cf36021891ded4c90c435506123ed63": "\n\\frac{\\partial \\overline{\\rho}}{\\partial t} + \\frac{ \\partial \\overline{u_i \\rho} }{\\partial x_i} = 0\n", "9cf37206f0e34ba72ddced5edbb0dce8": "(\\mathbf A_+)^+", "9cf3c45e61b8c773c4cb4cde1c58923a": "2\\theta\\mu\\geq\\sigma^2", "9cf3e00429968a0bc2e478604dd6a4d7": " x^3-(l^2+a_1l)\\cdot x^2-(2\\cdot l\\cdot m+a_1\\cdot l+a_3\\cdot l)\\cdot x-m^2-a_3m=0 ", "9cf416ff0cd38b2d5e5a3a3625994a75": "k_L(f)=\\frac{\\dot{W}_{12L}(f)}{\\sqrt{[\\bar{W}_{11L}(f)+\\bar{W}_{11C}(f)][\\bar{W}_{22L}(f)+\\bar{W}_{22C}(f)]}},", "9cf41d980791209fe530ddf13e976f84": " O^*", "9cf450d3864153acb94221489b742c5a": "[\\gamma^{\\mu_1 \\mu_2 \\cdots \\mu_{2j}} P_{\\mu_1}P_{\\mu_2}\\cdots P_{\\mu_{2j}} + (mc)^{2j}]\\Psi = 0 ", "9cf4e49c5d31060e53680ca67acf610f": "\\mathbf{M} ", "9cf4ee56e2b4c41d330ac46f2582faf2": "R_\\text{zDIN} = \\frac{1}{s} \\sum_{i=1}^{s} R_{\\text{t}i}", "9cf5c696340dabef3e694ba4a81544dc": "c'_i =\n\\begin{cases}\n\\begin{array}{lcl}\n \\cfrac{c_i}{b_i} & ; & i = 1 \\\\\n \\cfrac{c_i}{b_i - c'_{i - 1} a_i} & ; & i = 2, 3, \\dots, n-1 \\\\\n\\end{array}\n\\end{cases}\n\\,", "9cf5e98622e5e0fe31ba0eb889b14f44": "\\vec{e}_0 = \\left( 1 + m/r \\right) \\, \\partial_t ", "9cf65d5b7f6cac031ef2467b4b96d867": "c_{1}=\\sqrt{\\gamma p_{1}/\\rho_{1}}", "9cf6672069af8ee0c3b4d8f06d6dcfed": "\\rho(x\\vec x)=x^2\\rho(\\vec x )", "9cf6868608de77b7d477533e703db2b0": " H(Y) = - \\int_{-\\infty}^\\infty P_Y (y) \\log_{2} (P_Y (y))\\,dy ", "9cf6dacb4ffa5908ac1862fc296cb176": "m>1", "9cf772302b3f25357226f6caa177e2aa": "| \\psi \\rangle = | \\psi_1 \\rangle \\otimes \\cdots \\otimes | \\psi_n \\rangle .", "9cf78c90eea0063d91fece735db4d40d": "T[\\Pi]", "9cf78f7d8551dbcfb837ea033c87eaf8": "Cl_1^{\\leq}", "9cf7cfeedbe0bb3b0173546d79f885b1": "\nC^{S_1}_{E_1} = (\\varepsilon^{3}_2 - \\varepsilon^{2}_2) / D\n", "9cf809284201a5b2c160e0db9ca3ffb5": "\\displaystyle{S=S(\\varphi_1)\\oplus S(\\varphi_1\\varphi_2)\\oplus S(\\varphi_1\\varphi_2\\varphi_3) \\oplus \\cdots }", "9cf84431271ef7052a6e83c08e92f56a": "B = 2W", "9cf87c0433d10b6568289d87dca4a672": "\\mathrm{E_1}(x)", "9cf881b32c07058c0778b5236f22db24": "\\begin{align}\nA_{[\\alpha_1\\cdots\\alpha_p]\\alpha_{p+1}\\cdots\\alpha_q} & = \\dfrac{1}{p!} \\sum_{\\sigma}\\sgn(\\sigma) A_{\\alpha_{\\sigma(1)}\\cdots\\alpha_{\\sigma(p)}\\alpha_{p+1}\\cdots\\alpha_{q}} \\\\\n& = \\dfrac{1}{(n-p)!} \\varepsilon_{\\alpha_1 \\dots \\alpha_p\\,\\beta_1 \\dots \\beta_{n-p}} \\dfrac{1}{p!} \\varepsilon^{\\gamma_1 \\dots \\gamma_p\\,\\beta_1 \\dots \\beta_{n-p}} A_{\\gamma_1 \\dots \\gamma_p\\alpha_{p+1}\\cdots\\alpha_q} \\\\\n\\end{align} ", "9cf882acd74a1261a619029625e71a0c": " P_n = \\frac{E_p^2}{2\\cdot R}\\,\\!", "9cf885b8cadd4ce82f0575e5338950b7": " \\gamma_{SC} ", "9cf8a6fc731d68a4cac495781cfd1e3d": "c_i(E)", "9cf91777692d66797b20de97a23a8290": "s_1 s_2=B", "9cf92c5857bda1bf38061b6ba374b626": "a (p-1)/2", "9d0baa73be2f0e4d9ee7447abd0cc999": "\\mu_n = (-1)^n\\frac{d^n}{ds^n}E[e^{-sX}].", "9d0bc98e2280cfab7d946458f74f8a00": "\\mathbb{F}_{q}^{n}", "9d0bcc197007d80b80680b93af8d5c4c": "X\\wedge Y = \\{x\\wedge y\\mid x\\in X, y\\in Y\\}.", "9d0bf3bf5a661f8da60f0c97a87655e5": " p_m = p - F. ", "9d0c066b16bd93d63fab98e85bc427b4": "\\mathbf{x}'_1 \\mathbf{\\beta} + b - y_1 \\leq k", "9d0c5b1e29429d36d3c8638c73a37151": "x_0 + \\Delta_\\text{B}", "9d0cf16cc25d00d155103f8d4b397ffb": "f_1, \\ldots, f_r = 0", "9d0d0fa69afa27850753bb552a424121": "\\pi_1(M^3)", "9d0d188a85ecc5501f2a3c456a28689d": " = \n- *\\mathrm{d}{*\\partial_i f \\, \\mathrm{d}x^i} = \n- *\\mathrm{d}(\\varepsilon_{i J} \\sqrt{|g|}\\partial^i f \\, \\mathrm{d}x^J)", "9d0d2aada68b847f75ab66232480a726": "f(r)", "9d0d335329ed344ba7e41c0293f34911": " a_1 < a_2 > a_3 < a_4 > \\cdots. \\, ", "9d0d788b20c5ca89f5c9adc3570f5928": "1/\\log \\log k", "9d0da7acbcd5f04e23346dc8feb18b55": "\\nabla H \\cdot \\mathbf f(\\mathbf r, t) = 0", "9d0dbcc767594fbfc6323029a45b5120": "\\begin{align}\\mathcal{Z} \\left \\{a^n x[n] \\right \\} &= \\sum_{n=-\\infty}^{\\infty} a^{n}x(n)z^{-n} \\\\\n&= \\sum_{n=-\\infty}^{\\infty} x(n)(a^{-1}z)^{-n} \\\\\n&= X(a^{-1}z)\n\\end{align} ", "9d0e3ab8c6d2942189535f1786cdc227": "(X(t),Y(t))", "9d0e8efcd56bf46d795360c95a04942e": "\\mathbf{v} = r \\frac {\\mathrm{d} \\mathbf{u}_\\mathrm{r}}{\\mathrm{d}t} = r \\frac {\\mathrm{d}}{\\mathrm{d}t} \\left( \\mathrm{cos}\\ \\theta \\ \\mathbf{i} + \\mathrm{sin}\\ \\theta \\ \\mathbf{j}\\right)", "9d0f067e4a968100589978f30622da76": "\\rho_\\text{air}", "9d0f15ae952f52d21d43275aa97d7d2c": "\\wedge^{m+1}_n = \\vartriangle^m_n", "9d0f1f68a746fdb0e38e22969c6cb99c": "Vol(B(y,(p+\\epsilon)n)/2^n \\approx 2^{H(p)n}", "9d0f687e00bf7d02cdcf660f248dd56b": "(w_0,w_1,\\cdots,w_n)", "9d0fa7b1a39a0a276966d9abe4e8526b": "1/h", "9d0fad6e3a63c1b95af4e5c96df0ae1c": "v\\ll c", "9d0fbf95b26bb80ae22db6736db6edb4": " |\\phi\\rangle = A |\\psi\\rangle", "9d0fcf38730ce8691dbbdad687cb4d7c": "|a|<\\delta .", "9d0fda2b9b06dcfe5142defd14d9b6a8": "H^1(X;\\mathbb{Z})=0.", "9d0fdb4c265295545197fe3efe238c2d": "D_{IS}(P(\\omega),\\hat{P}(\\omega))=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} \\left[ \\frac{P(\\omega)}{\\hat{P}(\\omega)}-\\log \\frac{P(\\omega)}{\\hat{P}(\\omega)} - 1 \\right] \\, d\\omega", "9d0fdf92a3a9d3d65d06d777e4fb6a43": "\n\\frac{{\\sigma _z }}{z}\\,\\,\\,\\,\\, \\approx \\,\\,\\,\\,\\sqrt {\\frac{{\\left( {a\\,b\\,e^{b\\mu } } \\right)^2 \\,\\,\\frac{{\\sigma ^2 }}{n}}}{{a^2 e^{2b\\mu } }}} \\,\\,\\,\\, = \\,\\,\\,\\,b\\,\\,\\frac{\\sigma }{n}", "9d0ff97b29489211a24dbe2a15d9ba7a": "\\langle O, \\mathbb{F}\\rangle", "9d104a3a80c5f48186162c4108540fa8": "\n\\ x(t + T) = x(t)\n", "9d107411d1a34af6616ebc09736167f9": "\\Gamma,\\Delta\\vdash A,", "9d10755b90ac37148711c6f774453660": "\\mathbf{M}_{\\text{effective}}(\\mathbf{x'})=1/2(\\mathbf{x'}\\times\\mathbf{J}(\\mathbf{x'}))", "9d10c3377ff4e17581cc8a307cb2e57c": " \\mathbf {P_{\\mu\\mu}} ", "9d10efd9fd402bcdbcfc9d0f8016a057": " \\frac{d}{dt} \\langle Q \\rangle = \\frac{-1}{i\\hbar} \\left( E \\langle \\psi | Q | \\psi \\rangle - E \\langle \\psi | Q | \\psi \\rangle \\right) \\,", "9d11604320b04ed6b47da880f6667a5e": "X_f= -mU\\frac{d(\\theta-\\alpha)}{dt}\\sin(\\theta-\\alpha)", "9d117696df1b5a60fd6b2c67a6ec4816": " k_{||} \\sim k_{\\perp}^{2/3} ", "9d11bca48b8d291e5ffe4b4a573d9f78": "g_k = k(3k-1)/2", "9d11f5c3e1130b6794d786feaf3eb82f": " \\int_{\\mathbb{R}} \\frac{d\\mu(\\lambda)}{1+\\lambda^2} < \\infty.", "9d120ea88bf889ebf4ba993b199680cb": "\\sum_{j=m}^{M-1}\\frac 1 {k\\binom j k}=\\frac 1{(k-1)\\binom{m-1}{k-1}}-\\frac 1{(k-1)\\binom{M-1}{k-1}}", "9d12237a1c2e90cf4b50d7b3c82f4ef5": "V_F(pol.)=V_F(unpol.)\\pm\\mu_N\\cdot B", "9d12cfc1b7a5baa5d79a55a36ccf2e33": "\\Phi_L = \\lambda_{\\rm Io} - 3\\cdot\\lambda_{\\rm Eu} + 2\\cdot\\lambda_{\\rm Ga} = 180^\\circ", "9d12d1a66eb1a2eb6211640e80aed089": "(\\mathbf{a}\\times\\mathbf{b})\\times\\mathbf{c}=(\\mathbf{a} \\cdot \\mathbf{c})\\mathbf{b}-(\\mathbf{b} \\cdot \\mathbf{c})\\mathbf{a},", "9d12d7b335d494209defc761605c4627": "\\alpha_2\\,\\!", "9d134e5828184c58fe6d8479d85fe064": "V^{\\prime}[\\sigma f] \\to V^{\\prime}[\\sigma]", "9d13539e16eb84abd9e0f18587015b6c": "\\lVert \\hat{y} \\rVert_\\infty \\leq 5in \\phi p^{1/m} \\rbrace ", "9d137895641222f903e58d2c3246b3aa": "(\\textbf{D}\\textbf{x} + \\textbf{e})^{\\rm T} \\textbf{C}^{\\rm T} \\textbf{A} + (\\textbf{A}\\textbf{x} + \\textbf{b})^{\\rm T} \\textbf{C} \\textbf{D}", "9d14416f80037c6685ca3ed90402ec70": "\\begin{bmatrix} I_k | P \\end{bmatrix}", "9d145877da916d6a9a7bc74e44eb78f1": "[S,T]:=ST - (-1)^{\\left|S\\right|\\left|T\\right|}TS ", "9d14a2a64da90789e0461d8052f47e7a": "R \\sim \\mathrm{Rayleigh}(\\sigma)", "9d14a66c156bd589f72a644e084ef5ec": " \\operatorname{E} \\operatorname{tr} e^{\\sum_k \\theta \\mathbf{X}_k } \n \\leq \\operatorname{tr} e^{\\sum_k \\log \\operatorname{E} e^{\\theta \\mathbf{X}_k} }. ", "9d14c9c1261ca6b0fdfbf2894dd499c4": "\\ \\kappa_t(\\cdot)", "9d1510ddc2486663b99e3bea4e262c77": "\\tfrac{3K-M}{2}", "9d1566ded3c39a8bd1f6f14540978aff": "MV_d", "9d15699799d18e6362847b2e62dd8c8d": "\\dot y(t)", "9d15a8f5ddd90017af285e2f880935fe": "A=A^T", "9d15c95a543abf8ba016667564c10adb": "L^{4k} \\cong L_{4k}", "9d1628a4a5fc760a5d81dc991361158c": "\nh^{\\mu}(x) = M^{\\mu \\nu}x_\\nu + P^\\mu + D x_\\mu + K^{\\mu} |x|^2 - 2 K^\\nu x_\\nu x_\\mu \n", "9d1661e5827a4fdd587dbb817ee62896": "f(n)=n+(n-1)+f(n-2).\\,", "9d16b7dec5ba1f1577a5e0a5a161b622": "du=u'(x)dx, \\quad dv=v'(x)dx. ", "9d16fd2c4897c066636897e6ed1d9c72": "x \\notin A", "9d17040d046a9c7baa08ece67c2fb7d1": "M_{21}", "9d177025af0fbf3aec39b6cb49f25ddf": "\\begin{align}\n\\Delta t^2 & = \\left[ \\int^{\\Delta\\tau}_0 e^{\\int^{\\bar{\\tau}}_0 a(\\tau')d\\tau'} \\, d \\bar\\tau\\right] \\,\\left[\\int^{\\Delta \\tau}_0 e^{-\\int^{\\bar\\tau}_0 a(\\tau')d \\tau'} \\, d \\bar\\tau \\right] \\\\\n& > \\left[ \\int^{\\Delta\\tau}_0 e^{\\int^{\\bar{\\tau}}_0 a(\\tau')d\\tau'} \\, e^{-\\int^{\\bar\\tau}_0 a(\\tau') \\, d \\tau'} \\, d \\bar\\tau \\right]^2 = \\left[ \\int^{\\Delta\\tau}_0 d \\bar\\tau \\right]^2 = \\Delta \\tau^2.\n\\end{align}", "9d17af93172ed4f436d1824166d72b20": "\\textstyle \\mathbf{v}_1, \\ldots, \\mathbf{v}_m", "9d17b9f8b796fc474284ba4dd44e025b": "\\{\\, S \\subseteq X : \\forall x,y\\in X\\ \\ x\\in S\\ \\land\\ y\\le x\\ \\rightarrow\\ y \\in S,\\}", "9d17fb61a2a9d954f6e810de5e1b22e1": "\\phi_{L}", "9d17fcc5c151e1cda9b19e55e24656cb": "\\frac{d}{dx} \\left(\\sum_{i=1}^k f_i(x) \\right) = \\sum_{i=1}^{k-1} \\frac{d}{dx}f_i(x) + \\frac{d}{dx}f_k(x)=\\sum_{i=1}^k \\frac{d}{dx}f_i(x) ", "9d189ded7bfcbe324b33bc8a2af30aff": "\\inf_{x: g(x) \\leq 0} f(x) = \\inf_{x \\in X} \\tilde{f}(x)", "9d18dc91e4328715a6499a1144c37d1e": "O(d!2^dn)", "9d193e725a212f4f02d6f177cbffef9c": "\\boldsymbol\\omega=[\\omega_x,\\omega_y,\\omega_z]", "9d194c347e5a810e65bf15c716d79f8d": "p_3=\\frac{m_2}{q_5}\\ ,", "9d1979d443ecb13257721a8bfa3db641": " \\frac{1}{4 \\pi \\epsilon_0} = \\frac{\\mu_0}{4 \\pi} = \\alpha \\ ", "9d19d10e560380ac43cad752d4620a2f": "0=A_n \\zeta(n) - B_n \\pi^{n} + C_n S_-(n) + D_n S_+(n)\\,", "9d19ecfdc7e4a7807fb2e6cf4a27373d": "D_{\\mathrm N}", "9d19fc82f910cf113fc0ed7d79b9e6f1": "\\,\\delta = \\lim_{m\\to\\infty}i^{(m)}", "9d1a78e18d602b216a68b643602d0046": "\\begin{align}\nh(x_1^n)= 1,\\,\\,\\,\ng_{\\theta}(x_1^n)= {1 \\over \\theta^n}\\, e^{ {-1 \\over \\theta} \\sum_{i=1}^nx_i }.\n\\end{align}", "9d1ab5875a4e79b1ac2ea12ed785e212": "A= -\\gamma v/c^2,\\,", "9d1adc62283473e59d2bbb5d3cff994a": "{dx}/{dt}=-ax(t-1)-2x^2-x^3", "9d1ae034926df43eac04d3a6d7ea7162": "S_t=S_0\\exp\\left(\\sigma B_t+(\\mu-\\sigma^2/2)t\\right).", "9d1aea929f3dbc690f4bd5528a25635d": "{U} = {\\omega*r}", "9d1af119c39d0a0505d38c858b7ec55f": "\\mathbb P", "9d1b1520b2fcb37b60a83f16648bf864": "x^{-6} \\cdot 2^4 = \\frac{16 \\sqrt{5}}{25} ", "9d1b7f0cb1019ebff9ac667db7f194f4": "\\ f_0(x)\\ ", "9d1bb1e1b5b1756a9c446b0e6501a0bc": "\\pi: EG\\longrightarrow BG.\\ ", "9d1bc401ebd415a3202e0fe1daa90d8a": "\\theta^{C_{T_0}} = \\frac {\\alpha_{T_0}\\,p_A^{C_{T_0}}}{\\frac{1}{K_{eq}^A} + p_A^{C_{T_0}}}", "9d1bd86e53a3f1ac03cfdde0d56bcb9f": " c' ", "9d1c2c345e26ae5a1f0306d6f54dd7bc": "8 \\times 6", "9d1c2fcb91c2374210518e4717a04ba3": "t^m\\,", "9d1c5eccd50e44dce59c3378ae674cc3": "\\mathbf{w} = \\mathbf{G}^{-1} \\mathbf{b}", "9d1c814e355c9d394648206d3d9f464e": "O(2k+1) = SO(2k+1) \\times \\{\\pm I\\},", "9d1caad88c9100e58b652f0219ec0d5d": "(p)^{n_p}", "9d1d954317fd6f73d6aa0b752211ea46": "\n\\operatorname{Li}_n(z) = z \\;_{n+1}F_{n} (1,1,\\dots,1; \\,2,2,\\dots,2; \\,z) \\qquad (n = 0,1,2,\\ldots) ~,\n", "9d1da2a04ab278eb5d5379e0840bbe9e": "e \\approx {163 \\over 3\\cdot4\\cdot5} \\approx 2.7166\\dots", "9d1df66cd1035a409cc16010aa004620": "B \\subset X\\times Y ", "9d1e26318fb6b995d628a82ea5f5b2d3": "\\begin{align}\n\\omega^{\\frac{M_p+1}{2}}\\bar{\\omega}^{\\frac{M_p+1}{4}} & = -\\bar{\\omega}^{\\frac{M_p+1}{4}} \\\\\n\\omega^{\\frac{M_p+1}{4}} + \\bar{\\omega}^{\\frac{M_p+1}{4}} & = 0 \\\\\n\\omega^{\\frac{2^p-1+1}{4}} + \\bar{\\omega}^{\\frac{2^p-1+1}{4}} & = 0 \\\\\n\\omega^{2^{p-2}} + \\bar{\\omega}^{2^{p-2}} & = 0 \\\\\ns_{p-2} & = 0.\n\\end{align}", "9d1e6845bcc83e07177d95565ed1ff5c": "\n d\\mathbf{f}_0 = \\boldsymbol{F}^{-1}\\cdot \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0\n = \\boldsymbol{F}^{-1}\\cdot \\mathbf{t}_0~d\\Gamma_0\n", "9d1ed64172643a3ba6c961c475f25417": "I_{\\text{c}}", "9d1ee4c686b0391d85ffc69125cfdaf5": " \\left [ W^{k,p}(\\mathbb{R}^n), W^{k+1,p}(\\mathbb{R}^n) \\right ]_\\theta = H^{s,p}(\\mathbb{R}^n),", "9d1f0ad2f7982c3a8fada08cad919bf6": "= i \\int_t \\left[ m \\left(\\frac{dy(it)}{dit}\\right)^2 - V(y(it)) \\right] dit", "9d1f11b3d461bde62a714de4668cd0d8": " Base~excess = 0.93 \\times \\left ( \\left [ HCO_3^- \\right ] - 24.4 + 14.8 \\times \\left ( pH - 7.4 \\right ) \\right )", "9d1f4edb86f0eaa429a42625fa7d2efa": "\\rho \\colon G \\to GL(V)", "9d1f6aa8f036e5bde9e4ddfb889e8d3d": "\\sigma(u,v)", "9d1f7fc2984ae9171f87f54ae38a6f73": "\\sigma_{\\mathrm f}", "9d1f8cbb63b2005d30fd66033e2fd15d": "aba^{-1}=b^k\\;", "9d1fe46ead4e2d99aa730ded81ab1722": "\\mathbf{D} = \\epsilon_0 \\mathbf{E} + \\lambda \\mathbf{P}", "9d1fe8ef87241fa72d6b571540f4a982": "P=\\frac{RT}{V_m-b}-\\frac{a}{TV_m^2}", "9d200fbc423231ad07ed3aab90da38ab": "\\frac{dx}{x}", "9d203b571494c97b95e5b7cc074f94aa": "\nT(a) = \\frac{1}{a} \\sum_{d | a} \\varphi(d) \\tau(d)\n", "9d203ba494f02cb51ab87a72c7a9afa1": "d(x, y) := \\| x - y \\|", "9d2072b32d2c52c9beb94ad64df73856": "K = -\\frac{1}{2\\sqrt{EG}}\\left(\\frac{\\partial}{\\partial u}\\frac{G_u}{\\sqrt{EG}} + \\frac{\\partial}{\\partial v}\\frac{E_v}{\\sqrt{EG}}\\right).", "9d207408bf4748066a4ee18b3b7c4fce": " \\beta_n", "9d209da2441a725ae3d8b998462ce481": "\\scriptstyle\\mathbf{H}", "9d20aa00d95a55ae037f9da3af685609": "\\ell_i", "9d20bb1d0633c20caa45b4f2d89e24b6": "-\\frac{N_{ AxBy(\\Delta)}}{1} = \\frac{N_{A(\\Delta)}}{x} = \\frac{N_{B(\\Delta)}}{y}\\,", "9d2150f657fec18d5c5bf4f34e988ce3": "\\overline{\\nu}", "9d21b56f4cd59236be3342f8faca454f": "\\Delta S_I = 0 \\,", "9d21e01ce4991f34a5811f75149e284b": "\\odot ", "9d21f6b24b6f604413391f7bde24d459": "\\scriptstyle f_A", "9d220fb83e9f59a7f398ee2c3cf35dce": "P_{\\text{ph}}^{2} = P_{i}^{2} + P_{f}^{2} - 2P_{i}\\cdot P_{f}", "9d22af204a333373d94e20b38fb1d1ab": "\n 2ID(E_F) > 1\n", "9d22ba863b5b44380fc71c59fd318086": "\\mathbb{C}S_n", "9d23126d770c275a80ff18df54036ea7": "\\beth_{\\alpha+1}(\\kappa)=2^{\\beth_{\\alpha}(\\kappa)},", "9d2352b8924fb20b583c67e3e2c18616": "r=|\\sqrt{(|6^2+4^2|)}|\\equiv|\\sqrt{|3|}|\\equiv", "9d2364c4b948c5e90ed98548421732d4": "C(a) \\nless C(b)", "9d236e4813cae15bc5c014879c5cef77": "x^n = \\sum_\\pi (x)_{|\\pi|}.\\,", "9d23a1a2e83f8c7409065f8227a9c514": "2^{m-r}", "9d2487b2b3a4ab6b117b41eccbc60e3c": "2%", "9d24d09391dc8becdd1cc83247524933": "H(s) = \\frac{9}{s^2 + 9},", "9d24da294b22798d2ca6fc35cc26b1cf": "(a,b) = \\{x \\in \\mathbb{R} : a < x < b \\}", "9d24ec6abb77a4ad988c7b204c477933": "{{N}\\over{N_0}} = {{e^{-E/k_BT}}}", "9d250f0c620229fc2f082923d2772f6f": "\n\\frac{\\partial(n \\langle{v_j}\\rangle)}{\\partial t} + n \\frac{\\partial \\Phi}{\\partial x_j} \n+ \\sum_i \\frac{\\partial(n \\langle{v_i v_j}\\rangle)}{\\partial x_i}= 0 \\qquad (j=1, 2, 3.)\n", "9d2566be783b090481ca6fd7df0fe02d": "b^{-(p-1)}", "9d2594622edb51ca4341114b52cf01e9": " N_{+} = \\operatorname{ran}(A + i)^{\\perp}, ", "9d25f5de4efa7a088343efd8461a758a": " Q_s ", "9d2619ef9f2843e796fcbc01a186761f": "P_2 = Q", "9d2640ec0a5c4a008572a6ce43612172": "\\approx 9 m^{3}", "9d26f31f2047f81354ce7ba424623474": " \\begin{align}\nV_1 &= E[X(1)^2|X(0) = i] - E[X(1)|X(0)=i]^2 \\\\\n&= (i-1)^2p(1-p) + i^2(p^2+(1-p)^2) + (i+1)^2p(1-p) - i^2 \\\\\n&= 2p(1-p).\n\\end{align}", "9d26f8352ad932daeab5f984951f72cf": "R_{41,23}", "9d27043dd55e80b1c6a5568749275d60": "Z_{i}", "9d27142dafbf7c3079077f24c2670507": "\\mathrm{auth}(p)", "9d2727653b18109ab995ef278211b42c": "dev(D)", "9d27396779e60a74acbecac598d0685a": "\\frac{dr}{d\\theta}\\cos n\\theta + r\\sin n\\theta =0", "9d2741f3de589733c9860032563a7c24": "\\bar{N} = 11101011", "9d277f2bfc74fc416d0c8749b8a96bd1": "b = h c k^{-1} / \\,", "9d27923d76e0221b6a6de205ca4d7c91": "\\cdots \\le \\lambda_k \\le \\cdots \\le \\lambda_1.", "9d27c02704277eade6edb088f184b900": "f(\\zeta) = \\int_{-A}^A F(x)e^{i x\\zeta}\\,dx", "9d27f958eda74aac3e8799efd0156eb9": "A,\\Omega^1", "9d2857f1bad6f8b2bfb05d419407846f": "X = \\frac{1} { \\sin (h + {244}/(165+47 h^{1.1}) ) } \\,,", "9d28b9de9c18ad7f2984a76c4a6bf5aa": "\\begin{bmatrix}x & y & 1\\end{bmatrix} . \\begin{bmatrix}A & B/2 & D/2\\\\B/2 & C & E/2\\\\D/2&E/2&F\\end{bmatrix} . \\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} = 0. ", "9d28e9a9d23e39ce1bbb86875b448e1f": "f(t) = \\frac{\\Gamma[(\\nu+1)/2]}{\\sqrt{\\nu\\pi\\,}\\,\\Gamma[\\nu/2]} (1+t^2/\\nu)^{-(\\nu+1)/2}", "9d295d4455b092ee39e7abde1b6502c9": "\\Sigma \\chi(F(n))\\,", "9d29611ecc821649d27e3f88b6265f6d": "\\textrm{VD}=\\frac{\\textrm{32}}{\\sqrt{\\left(\\frac{\\textrm{1920}}{\\textrm{1080}}\\right)^2+1} \\cdot \\textrm{1080} \\cdot \\tan{\\frac{1}{60}}}=49.94", "9d297542330da4b771842e7daf366a62": "\\log^32", "9d29b51e5cd4740012c3e14ca2892940": "\\alpha, \\beta \\in \\Omega(M, \\mathbb R)", "9d2a62e8b6a3885a060edb6e5224869f": " n = A_3 = A_6 = n ", "9d2a912164a8076c624c76e6f8350344": "d(\\mathbf{x}, \\, \\mathbf{y})", "9d2a9f58e4309e36dbca464c9107b9a0": "\\rho \\neq 0", "9d2aa0d7e4d3a26b113be09a61e1cba2": "C=(c_{i,j})_{1\\le i\\le n,1\\le j\\le m} \\, ", "9d2aa2c7b7888f346ed23f116de26734": "n,n_1,n_2 \\in N ", "9d2aaa1be0204560ce5893571b964438": "\\frac{11 \\cdot \\pi}{12}", "9d2abf063e0d578cd87bf4de6a45176f": "E_C'", "9d2af96d8388dbbaf61b4f3f9a4b242d": "L=(\\Sigma\\cup\\{\\epsilon\\}) \\times (\\Gamma\\cup\\{\\epsilon\\})", "9d2afb22a18055bfcbe9fb4dab3c464b": " A(t) \\triangleq \\int_0^t \\frac{1}{S^s_0(s)}dS_0(s), ", "9d2b3fb3fc41c1a0a3ff59a99f928b4d": "\\mathbf{v} \\cdot \\mathbf{g}", "9d2b559be0fa107c67aa155883a9d2eb": " \\begin{align}\n&\\lim_{\\beta\\to 0} H_{(1-X)} = \\text{undefined} \\\\\n&\\lim_{\\beta\\to 1} H_{(1-X)} = \\lim_{\\alpha\\to \\infty} H_{(1-X)} = 0 \\\\\n&\\lim_{\\alpha\\to 0} H_{(1-X)} = \\lim_{\\beta\\to \\infty} H_{(1-X)} = 1\n\\end{align}", "9d2b811c3d1c89d104c1d693ff44c2d4": " m \\ge \\sqrt{\\frac{(n+1)d}{2}} - \\sqrt{\\frac {(n+1)d}{2}} + \\frac{d}{2} - 1 = \\frac{d}{2} -1 ", "9d2bbeabaa9d0a4b880378ec77b45d30": "d_\\mathrm{i}", "9d2ca5d34e0cb5b2e1c1f07c0ff1f762": "m = \\frac{a + b}{2}.", "9d2cdff4336d7781c860dcf8feae99be": " K = C + S - F \\,", "9d2d18c5bb1f660ca47b988a4581b5be": "s (t) = \\sum_{n = -\\infty}^\\infty v[n] \\cdot h_t (t - n T_s)", "9d2d37de9a35d2ba864a9c98ae85b60d": "y = \\int \\frac{dy}{dx} dx.", "9d2d3fb587986da2ef78107431fea45e": "\\int x^5 r^3 \\; dx = \\frac{r^9}{9} - \\frac{2 a^2 r^7}{7} + \\frac{a^4 r^5}{5}", "9d2d81593de68715ea8438b2d7aca653": "q_0=2 \\pi /P_0", "9d2dab9ab45411248df80d3060899e99": "\\theta = \\frac{\\pi}{2}\\,\\!", "9d2e129f8ed99e5a9ab38ba8cbcbac99": "Pr(\\mathbf{p}) = W \\cdot m^{-N}", "9d2e198ed10ce90901dc2a63b7defc03": "\n\\omega_k = \\{i \\mid 1 \\le i \\le N , x^k_T(i) \\ne 0\\}. \n", "9d2e1da432b760bb05a96ed93316d00b": "{\\hat{x}}", "9d2e6a7519600ca6c0e9eae2edbd98e6": "h_{\\epsilon}^{*}", "9d2e6c3a54cd0ca8190a0cfe9d22d626": "Q_1 = (13)\\mathbf Z[i] + (i + 5)\\mathbf Z[i] = \\cdots = (2+3i)\\mathbf Z[i]", "9d2e94cb1a5cb1c5a6832e36a41952bf": "{2a_{13} \\times b_{13} \\over b_{13} + a_{13} -c_{13}}=d", "9d2ef800970582ec81b7091359ba503d": " B_1(t_1)\\cdot B_2(t_2)\\cdot\\dots \\cdot B_n(t_n)", "9d2fca390ef682d5315bcd601602a85c": "1\\cdot 1 + 2\\cdot 4 + 3\\cdot 1 = 12", "9d300d502a0d4f850e676164ee2faa06": "\\alpha = \\frac{1}{2}\\left[ \\frac{4DBM}{(F + BM - DM)(F - B - D)}\\right] ^2", "9d310a4645bae9590e7502dbecdd4bd9": "A(\\mathbf{V},\\eta_2) = \\left(\\eta_2+\\frac{p+1}{2}\\right)(p\\ln 2 + \\ln|\\mathbf{V}|) + \\ln\\Gamma_p\\left(\\eta_2+\\frac{p+1}{2}\\right).", "9d31ac331b992cc124718446d936333c": " \\mathcal{B} (\\mathcal{H}) ", "9d3242a3177c5fd44e2db4b0f327ebc2": "N_j\\log|u_i^j|", "9d3287da474299fd643e984d39574c81": " U=\\frac{1}{N-1}HA,\\quad V=HA, ", "9d328a234a3d2651cfbb1eb3fa79d4b8": "\\sum_{k=0}^{n-1}\\left\\lfloor x+\\frac{k}{n}\\right\\rfloor\n=\\sum_{k=0}^{k'-1} \\lfloor x\\rfloor+\\sum_{k=k'}^{n-1} (\\lfloor x\\rfloor+1)=n\\, \\lfloor x\\rfloor+n-k'\n=n\\, \\lfloor x\\rfloor+\\lfloor n\\,\\{x\\}\\rfloor=\\left\\lfloor n\\, \\lfloor x\\rfloor+n\\, \\{x\\} \\right\\rfloor=\\lfloor nx\\rfloor.\n", "9d32ecedd8a6e00579a65176abc6d0fc": "x_n=n+2", "9d33022ac4b5941d9c5915cd9320b0d4": "\n\\begin{align}\n\\mbox{y-intercept} = \\frac{ K_M^B}{v_\\max {[}B{]}}+\\frac{1}{v_\\max}\n\\end{align}\n", "9d33164912bf51a60af2de8a7cd1c20c": "n = 2^{k_2} \\cdot 3^{k_3} \\cdot 5^{k_5} \\cdot 7^{k_7} \\cdot 11^{k_{11}} \\cdots p^{k_p} \\cdots", "9d33383f0cd65cdc93053194c8b4ff1f": "\\mu_k\\ =\\ 0.3\\ k ", "9d335454cf4967778a3ccc97bcc29151": " \np(t) \\geq p_{min} \\text{ } \\forall t \\in \\{0, 1, 2, ...\\} \n", "9d339510656207765352e87c7968ada6": "\\lambda (U) > 0", "9d33b7b44c383d698306ac6f83bf38ae": " \\mbox{if }\\ell(yw) = \\ell(y) + \\ell(w) \\,", "9d33dcc1a7a6dd6ad517cb44890e361d": "\\mathcal{H}_2", "9d33e8233c305ba532f5e434f0a944db": " d\\nu_t = \\alpha_{S,t}\\,dt + \\beta_{S,t}\\,dB_t \\,", "9d344c23348ae1207b7eeec1e05e1dd8": "\\int_{V} |\\Psi(\\mathbf{r},t)|^2 \\, dV < +\\infty ", "9d347919b98fd079278bfc283ff23dff": " P = Se^{-r_{FOR}T}\\Phi(-d_1) \\,", "9d34c686622bafd46488b080bb1951b7": "\\frac {\\text{intake}-\\text{faecal excretion}}{ \\text{intake}},", "9d34cdbb94fcc091b5b16cb7faeac526": "x=5", "9d34e5e261699243b522b9fd3b29f584": "\\alpha = 2/n", "9d351ca0055550796179f97b6a3c46e4": "13 \\mid 143", "9d351e528c96f231d55a3dad8695463f": "T^+\\ ", "9d352942ab33bbfb3df3428413621dd0": "\n\\begin{align}\n& A= \\frac{1}{2}\\left(\\frac{\\Omega_{0} R_{0}}{R_{0}}-{\\Omega}|_{R_{0}}\\right)=0 \\\\\n& B=-\\frac{1}{2}\\left(\\frac{\\Omega_{0}R_{0}}{R_{0}}+{\\Omega}|_{R_{0}}\\right)=-\\Omega_{0} \\\\\n\\end{align}\n", "9d355212d675a820d934c2ab88faf169": "\\frac{d}{dt}W(t,t_1) = A(t)W(t,t_1)+W(t,t_1)A(t)^{T}-B(t)B(t)^{T}, \\; W(t_1,t_1) = 0", "9d3571acbb8542d6e2a504b24ca54d95": "\\phi(q)=\\prod_{k=1}^\\infty (1-q^k).", "9d35ad45206e248776a0fa9897140c7b": "y \\ll h", "9d35c37834e856a68ede8f1cf032d4dc": "\\ A {cos(\\phi)}^n ", "9d36269c2869ec7d8fe37dee810443a4": "n' \\to \\infty", "9d3640e802916a1013367a108ab8bae3": "H_{k}+\\frac {(\\Delta x_k-H_k y_k) \\Delta x_k^T H_k}{\\Delta x_k^T H_k \\, y_k}", "9d36414ec123bb8ed6d0f3e2d7ff7743": "(S, \\oplus)", "9d365d72122c3de9bfef221d4b06f158": "bp={2 \\pi N \\tau \\over 60}", "9d367ef908edf8edca20cbfb499e16e5": "e^{it} = 1 + it + \\frac{(it)^2}{2!} + \\cdots", "9d36beafae28a7a54af7c6b76c412fc4": "R = (\\lambda I - L)^{-1},\\, ", "9d36d7da07cb147aefc393dae12e1805": "\\forall n\\in\\omega\\,\\mu_{x\\upharpoonright n}(X_n)=1", "9d36e707a54b40ec63eac22dd7986041": "\\mu_s = \\mu_r \\cdot \\mu_l,", "9d36f89c96c84fab4ec67b762f48464b": "S \\sim \\sum_{i=1}^n \\sqrt { 1 + \\left(\\frac{\\Delta y_i} {\\Delta x_i} \\right)^2 }\\,\\Delta x_i ", "9d372238ba88f6a76a015dff9b3507cf": "\\hat{A}|a_i, b_j, c_k,...\\rangle = a_i|a_i, b_j, c_k,...\\rangle", "9d372614ec2ce6894befd3d2f0d0ffc4": "G=\\mathrm{SL}(2,\\mathbb{C})", "9d3729599b6fc4b92e59caba6d2a4966": "k_{1(i)}", "9d37c692d6498c5bd72936688fd529b2": "\\cos(z) + \\sin(z) \\!", "9d380205cd74315126db34ba337a75ce": " \\mu_{p} ", "9d3830238dd3b0d3c9916a742a36fd7b": "(1+x)", "9d384b29fe296463a9e82d03e4053cf3": "\\ell_2 = 0,...., s - 1", "9d388ee3223cd03d4a2904f03fe46d54": " \\psi_i = \\lang i|\\psi \\rang , \\psi_i^* = \\lang \\psi|i \\rang ", "9d3910fb3def3bccb587a8d3658e9bdf": "\\pi:M\\to Q", "9d3974c3a0619ac576256851dffd58da": "\\begin{matrix}(B\\,\\mathcal{R}\\,C)(\\phi)= \\\\ (\\forall i:C(\\phi_i)\\lor(\\exists ja)", "9d4d67cc7163a0fe4ded6abeadaa1a93": " t = \\frac{\\sqrt{2} \\cdot v}{g} ", "9d4d985e68ed6c76f311f9ae6cfe99e1": "F_n(\\theta) = \\begin{bmatrix}\n {cos[-(\\theta(t)-\\phi)\\frac{d}{D_n}]}\\;m_n\\;r_n\\;(\\frac{dN \\pi}{30D_n})^2\\;\\eta_x \\\\\n {sin[-(\\theta(t)-\\phi)\\frac{d}{D_n}]}\\;m_n\\;r_n\\;(\\frac{dN \\pi}{30D_n})^2\\;\\eta_y \\\\\n -m_n\\;g\\;\\eta_z \n \\end{bmatrix} ", "9d4da5a76f178eeb00157d04fa0e2ebd": " \\exists x \\left( \\left( F \\left( x \\right) \\land \\forall y \\left( F \\left( y \\right) \\rightarrow x=y \\right) \\right) \\land E \\left( x \\right) \\right) ", "9d4dc6e93e95b3c7243f8f83b3084d71": "|f(\\xi)|\\le M\\left(\\frac{|\\xi + \\lambda|}{y_0+\\lambda}\\right)^N", "9d4e1f5b3ef9c138151e6b5b922a49e5": " (\\sigma(g) \\xi)(k) = \\alpha^\\prime(g^{-1}k)^{-1} \\, \\xi(U(g^{-1}k)), ", "9d4e215f4116db6348eed2b5f01e2156": "dz = \\left(\\frac{\\partial z}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial z}{\\partial y}\\right)_x\\!dy", "9d4e4d02148178dda3a33f83dcb412f5": "\\sum{PV(D_k)}", "9d4ec08a737c4fb7a2e31c54ec0d6860": "0\\leq w \\leq 1", "9d4eee551bafc1ba33c9c21b4fa7142b": "\\lambda=\\frac{h}{p} \\,", "9d4f063dba0a669f9dafb4df8be30218": "\nsin\\theta = \\frac{opp}{hyp} \n", "9d4f3ce3f096de2d8db9cc864bde7a59": " p = {n R T \\over V} = {\\text{constant} \\over V} ", "9d4f4038a43fa9852d70180a5b4bc50e": "H_2 = H_1 - h_f", "9d4fa77aa1c3b843bfe223cbc17e50c5": "2 + \\epsilon", "9d508b2cdec24a07dcb392e542e17bb3": "\\mathcal{\\hat{H}}_{HB} = \\sum_{it\\} \\right ) \\,dt.", "9d68cf58d3cd0d30e3ecdf8a9e052631": "T_b=\\frac{h\\nu}{k} \\ln^{-1}\\left( 1 + \\frac{2h\\nu^3}{I_{\\nu}c^2} \\right)", "9d68e2b79e5b9a94c172dfdc7325b687": "f(D) \\subseteq Q", "9d692839648d2dac571a794bab76ff3f": "\\delta=\\frac{\\mu_t-\\mu_c}{\\sigma},", "9d6934a9fd2b4b37b1c28256d15362bc": "h = \\lambda^2 g \\, ", "9d6935ef3228357ef732da651d3d6373": "\\mathrm{root} = a - b\\,\\!", "9d698e6b517f72f5896ad6b78a6c930e": "A= \\begin{bmatrix}\n2 & 0 & 0 & 0 \\\\\n1 & 2 & 0 & 0 \\\\\n0 & 1 & 3 & 0 \\\\\n0 & 0 & 1 & 3 \n\\end{bmatrix},", "9d6a02ad54dd83f69c0ee7380f0aea51": " V_{d \\cdot 2^r} \\equiv 0 \\pmod {n} ", "9d6a3572060044d7f645d4a6c022c942": "\\scriptstyle{(r_{2},\\theta_{2})}", "9d6a362db362833071abb046ff434f48": "t(x,y)", "9d6a44a2333d08b1213c62169b7aa573": "R_G(x,y)=\\sum\\nolimits_{A\\subseteq E} x^{r(E)-r(A)} y^{|A|-r(A)}", "9d6a685b897ca14f74898758422627c8": "\\left[y^{(i)}\\right] = f \\left( \\left[x_{1}^{(i)}\\right], \\cdots \\left[x_{n}^{(i)}\\right]\\right)", "9d6ae26fb9b8392b687f22e025c4b839": "\\min c^T x \\text{ subject to } x \\in P", "9d6b7be81cf70661016f8fc5c58e0518": " \\beta \\,", "9d6b973d86465895d9b8bb5cc65899c8": "\\Phi_{3\\times 5\\times 7\\times 11\\times 13}(x)", "9d6c162882aa6c2c5b4c44d27fe8b4cb": "1\\le u \\le 5 ", "9d6c4a587d53863516ad8438c8c9e167": " \\frac{2r}{5} \\sqrt{25-10\\sqrt{5}} = \\frac{R}{2} \\left(\\sqrt{5}-1\\right) \\!\\, ", "9d6d0bd3d60a46708e0726d14db850c4": "\nE_\\mathrm{max} = \\left[\\frac{2q}{E_s} (\\Phi_i - V_a) N_d\\right]^{1/2}\n", "9d6d5d27c7333d579fafe2ef9bcf46d3": "\\textrm{% \\, Load \\, Regulation} = 100% \\, \\frac{V_{min-load} - V_{max-load}}{V_{nom-load}}", "9d6d61f5def01889b11afdd087484fa3": "\\mathrm{(SNR)_{O,FM}} = \\frac{A_c^2 k_f^2 P} {2 N_0 W^3}\n", "9d6d8d87e843ee41efe18c13a4ecd09c": " S_j \\Lambda_{\\nu} + S_{j+1}\\Lambda_{\\nu-1} + \\cdots + S_{j+\\nu-1} \\Lambda_1 = - S_{j + \\nu} \\ ", "9d6d9f7a931f349de193a1a2e252ad9f": "y = \\pm ax^{3 \\over 2}.", "9d6dab5662a5bfb11580217dd06f7b79": "Q_1(X)E_2(X)", "9d6df26147f4cab057f0109e32ad6dd2": "H^2(X, S^1)", "9d6dfd334c8746c767ebc31d278c890a": "u(x,\\dot{x})\n=\n\\begin{cases}\n |\\dot{x}| + k + 1 &\\text{if } \\underbrace{x + \\dot{x}} < 0,\\\\\n -\\left(|\\dot{x}| + k + 1\\right) &\\text{if } \\overbrace{x + \\dot{x}}^{\\sigma} > 0\n\\end{cases}", "9d6e5ec34f80b695676f1084c9cb8cce": "=\\int_{X^{2m}}\\frac{1}{|\\Gamma_{m}|}\\sum_{\\sigma\\in\\Gamma_{m}}1_{R}(\\sigma(x))dP^{2m}(x)\\,\\!", "9d6eb8492be0db33d48be341594c7a43": "\\exists R_0\\ldots\\exists R_m \\phi", "9d6ecdde2f3e21632294f85e11affe4c": "\\delta(s):=\\inf\\big\\{\\|f-u\\|_{\\infty,X}\\,:\\, u\\in \\mathrm{Lip}_s\\big\\}\\leq+\\infty.", "9d6ed3ce0f4394ea6843865851c57b0c": "\\delta_i^j", "9d6ef330701fc93d5357abe82f37679a": " S_j = \\sum_{k=1}^{\\nu} Y_k X_k^{j} ", "9d6efd053e1b887614352aaa5e5a05b1": "K = A \\otimes H", "9d6f6b124ad8ab997c93d04b432189ed": "\\mathcal G^+(2,0) \\cong \\mathcal G(0,1)", "9d6f73d49add92111b6cd3a912bb13ee": " \\{ a_1-\\lfloor a_1\\rfloor, a_2-\\lfloor a_2\\rfloor, a_3-\\lfloor a_3\\rfloor, \\dots \\} ", "9d6f832c137aad80191e7b5b5757da56": "SG_A", "9d6f83ee82e01fbffa070389d2233189": "\\lambda \\Pi", "9d700ceb605434cb29ea1d038c814fe9": "L \\subseteq \\{0,1\\}^*", "9d701b1a56341060938ae89780feb172": " \\left\\{ \\cdots, \\frac{1}{16}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{2}, 1 \\right\\}", "9d705ac74e6a1d434f55e792b6526538": "-\\sigma^2/2", "9d7078ecaf39a7ae8cb72ba32b7693a6": "\\int_X|f|d\\mu\\le \\liminf_{n\\to\\infty} \\int_X|f_n|d\\mu", "9d708162730e156ef61176e13678eb9d": "E = \\frac{m_{Z}^{2}}{2 m_{\\nu}}= 4.2\\times 10^{21} \\left(\\frac{\\text{eV}}{m_{\\nu}}\\right)\\text{eV}", "9d70a7d9077e29e8eadf872ab9693f86": "\\hat{v_i}'\\equiv i[\\hat{H}'_0,x_i]", "9d70d92b7cc89f905655c659754d469d": "\\overline{X}_{\\mu \\geq \\epsilon}", "9d7187ecc44ab1ac0a89d3a04d2551af": "\\frac{d}{dz} \\ln \\xi \\left(\\frac{-z}{1-z}\\right) = \n \\sum_{n=0}^\\infty \\lambda_{n+1} z^n", "9d725115de1b38fb7e520f88e302ec4c": "\nG_{\\alpha\\beta}(\\omega) = \\frac{\\delta_{\\alpha\\beta}}{-(\\omega+\\mathrm{i}\\eta) + \\xi_\\beta}.\n", "9d7253dd4092ab5ae576a4461ccd1e8e": "\n\\mathbf{k} = \\left[\\begin{array}{ccc}\nk_1 \\\\\nk_2 \\\\\nk_3\n\\end{array}\\right]\n", "9d727cad5e5b1148e81ceca024312e6d": " PIC_{ij} (l) = (p_{ij} (l) - P_i P_j (l))^2", "9d7294d5f6cc47e59e1f4f436f2d40ae": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n{1 \\over \\vec k^2 + m^2}\n", "9d72ad11b9f8848941c6c379fdabb0b4": "A_{o}^{C} = \\left( \\frac{ Total \\ Time }{Total \\ Time + Corrective \\ Down \\ Time } \\right) ", "9d72b75137fda963728288d9ab301237": "Hypo(\\lambda_{1},\\dots,\\lambda_{k})=PH(\\boldsymbol{\\alpha},\\Theta).", "9d72f27f1b9a18a0661dc2ae8b142a7e": " \\lim_{n\\to \\infty} \\frac{|x_{n+1}-\\ell|}{|x_n-\\ell|} = \\mu.", "9d73254ad46373d7a86e31f0b8f8686a": "\\langle f(\\theta)\\rangle=\\sum_{k=-\\infty}^\\infty \\int_{2\\pi k}^{2\\pi(k+1)} p(\\phi)f(\\phi+2\\pi a)d\\phi.", "9d732776d226044cbd8234c4c53090e4": "\\frac{1}{\\alpha}[X^m]\\left( \\frac{f}{X} \\right)^\\alpha=-\\frac{1}{\\beta}[X^m]\\left( \\frac{g}{X} \\right)^\\beta", "9d7381fd6580b9535b56cef712592cca": "\\begin{align}\n_2F_1(a,b;1+a-b;z)&= (1-z)^{-a} \\;_2F_1 \\left(\\frac a 2, \\frac{1+a}2-b; 1+a-b; -\\frac{4z}{(1-z)^2}\\right)\\\\\n&=(1+z)^{-a} \\, _2F_1\\left(\\frac a 2, \\frac{a+1}2; 1+a-b; \\frac{4z}{(1+z)^2}\\right)\n\\end{align}", "9d73e5556c0ad95464d0ff1d97717f55": " E_f ", "9d740a4ea1d9425ca1ba0b170a793b95": "a_{ji} = 0", "9d746ab959f98b40dda0846d9a44c2e8": "y=b", "9d749a161447dcdd3e93f6e2be83fc08": " b^x = \\lim_{r \\to x} b^r\\quad(r\\in\\mathbb Q,\\,x\\in\\mathbb R)", "9d74c11245a4c000ed27201bc3bde366": "f_{0} : M \\to M';", "9d74e34ef8961c98f5c4d367a15087ec": "\\scriptstyle{m=(8/3)E/c^2}", "9d7513a40d8b8aa87b07ec428fcc8233": "S_t = S_0(1 - \\delta)^{n(t)}e^{ut + \\sigma W_t}", "9d753a57847b915b2e9708209ae118c1": "\\vartheta(x)=\\sum_{p\\le x} \\log p,", "9d75ae5aa3ebe062dd7fcb39b977f589": "I\\propto \\frac{1}{r^{n-1}}", "9d7622fc954794691258bb3d6f43f778": "\\begin{align}\nT' &= \\frac{F'}{A'}\\\\\n &= \\frac{x^3}{x^2} \\times M\\frac{a}{A}\\\\\n &= x \\times M \\frac{a}{A}\\\\\n &= x \\times T\\\\\n\\end{align}", "9d765bca0fd391544fb137831b806e96": "s,t\\in\\mathbb{Z}", "9d768aea4936f4d519436949753c5feb": "\\nabla^2\\vec{G} = \\vec{\\nabla}(\\vec{\\nabla} \\bullet{} \\vec{G})", "9d76ab1edae105c99cd0a2605a83f4e6": "G: \\mathbb{N} \\rightarrow X", "9d76acf02bb8426f776b94424312389b": "f(x,y)= x \\lor y", "9d76d7f563b92e80ee8b6bb1920ae648": "\\scriptstyle \\hat B \\;=\\; i \\hat v", "9d7709c89d2b1084a3f072ca99d0685f": "y (\\theta) = r (k + 1) \\sin \\theta - r \\sin \\left( (k + 1) \\theta \\right). \\,", "9d7768b4a59af743420377386ab3a49b": "\\wedge\\forall", "9d776c5680022ff9036dd952904708ea": "\n\\begin{pmatrix}du \\\\ dv \\end{pmatrix} = \\begin{pmatrix}p & r\\\\ q & s \\end{pmatrix} \\begin{pmatrix}dx \\\\ dy \\end{pmatrix} = \n\\begin{pmatrix}p\\, dx + r\\, dy \\\\ q\\, dx + s\\, dy\\end{pmatrix}.\n", "9d7777f67d768d21fb4354181ad67e50": "{\\nu}", "9d77acbd56d687224a0dfda47502ed3e": "f_{xx}", "9d78427216bf0bb168adf0bf6a91270c": "^{\\circ}", "9d784810259bad7ddd2bb3b1ca34fe4e": "\\arccos x ", "9d78d05ca3f55c42edebf94b33d3ca79": "\\displaystyle \\mathrm{circ}(\\sqrt{x^2+y^2})", "9d7975865711f65389858113639568a5": "\\mathcal{O}_{X, x}", "9d79b95a1ad3be833725ffdd7d415ab7": "r\\approx\\frac{2}{T}\\ln{\\frac{M_aT}{P_0}}", "9d79c63bb955cddd1726ef1acfbcd779": "h = (b - a) / N", "9d7a0d1d624e971faac56b656471b05e": "{\\rm Hess}(G)=\n\\begin{bmatrix} \n-\\sin\\theta & 0 \\\\ 0 & 0 \\end{bmatrix}\n", "9d7a2a01371ab11d2b54ae913c59390b": "\\mathbf{r}_i = 1 ", "9d7a349218cfc5a74d116ac48790a400": "\\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{j=1}^n f(a_j)=\\int_0^1 f(x)\\, dx.", "9d7a6f68106dcb1a58e8fbeaa3c2f370": "\\Delta\\,G=\\Delta\\,G' + \\Delta\\,G_{sol}", "9d7ab56d00054f81fd85fc6118413d55": "x \\;\\in\\; (0,\\, \\infty)", "9d7ab6c9ff75f1c4f4c93d38d97427ec": "\\scriptstyle 1/(kT_{\\rm F}) ", "9d7b267eb9370e6affc6a50586ac80d8": "r\\sin \\frac{x}{r} = x - x\\cdot\\frac{x^2}{(2^2+2)r^2} + x\\cdot \\frac{x^2}{(2^2+2)r^2}\\cdot\\frac{x^2}{(4^2+4)r^2} - \\cdot ", "9d7b3f8d4c546bc06553411a2c2ff8d3": "\\oint_W \\,\\{d_{\\,total\\,}U\\}", "9d7b577800090bb567d437c6685af331": "\\frac{\\partial}{\\partial x}\\left(\\frac{\\rho h^3}{12\\mu}\\frac{\\partial p}{\\partial x}\\right)+\\frac{\\partial}{\\partial y}\\left(\\frac{\\rho h^3}{12\\mu}\\frac{\\partial p}{\\partial y}\\right)=\\frac{\\partial}{\\partial x}\\left(\\frac{\\rho h \\left( u_a + u_b \\right)}{2}\\right)+\\frac{\\partial}{\\partial y}\\left(\\frac{\\rho h \\left( v_a + v_b \\right)}{2}\\right)+\\rho\\left(w_a-w_b\\right)-\\rho u_a\\frac{\\partial h}{\\partial x} - \\rho v_a \\frac{\\partial h}{\\partial y}+h\\frac{\\partial \\rho}{\\partial t}", "9d7b7066055de6226cd0cfbc34ba3b46": "{\\mathfrak p < \\mathfrak t}", "9d7b74f8c223d048217102db8d99d020": "\\lfloor \\cdot \\rfloor", "9d7bcdf886f530d86b92b846365f4f7a": "1\\over{\\sqrt{2}}", "9d7bf075372908f55e2d945c39e0a613": "p\n", "9d7c110d4896cd079eca2907bcfd14df": " F_i > \\frac{C}{V_g} ", "9d7c1ce385116f9c8b24460975ebc9af": "\\frac{\\Phi(y)}{y} \\in [a,b],\\quad a} J_{ij} \\sigma_i \\sigma_j. ", "9d834bd6cde964c6b263843648f150e1": "\n\\mathbf{u}\\odot\\mathbf{v\\equiv}\\sum_{i=1}^{n}z_{i}x_{i}^{\\prime}-x_{i}\nz_{i}^{\\prime},\n", "9d8390ca24dce1c9c1d99f24ef463804": "6 \\times 9", "9d8398f7f956c3a3f893863ea2af0a9d": "K=\\frac{q B \\lambda_u}{2 \\pi \\beta m c}", "9d83ccb2750dd360f78c70bc42ef4def": " v_i\\in V .", "9d8401263e6c82c1ae050d4fa21e5b71": "p_K(x + y) \\le p_K(x) + p_K(y) + \\epsilon, \\quad \\mbox{for all} \\quad \\epsilon > 0.", "9d840e0710bb6e5e58d3769eb5cc0bc1": "\\vec{\\imath}", "9d844566d45ced5cbc9e1400e9a882fa": "v = v_0+at \\,", "9d8453590c2ef4ed39e7eac755657578": " S = ", "9d84e7e00d9e83800ccfd7107fbd04b3": " x\\, ", "9d84eb77783e89f879adc42d2d81429e": "W = \\int_C Fds", "9d84eb89c3fd467e999aa4c8cb1eebee": "\\lambda = h\\, \\sqrt{ \\frac{16}{3}\\, \\frac{m h}{H} }\\, K(m),", "9d8528f4378352c375d5c848bbf4cf7d": "r_n(t) = 2^{-n/2} \\sum_{k=0}^{2^n - 1} \\psi_{n, k}(t), \\quad t \\in [0, 1], \\ n \\ge 0.", "9d8538adaa769726cabb44ddbb650770": "x \\in \\Theta", "9d8596f23a842feec52e5b81f278c55c": "z \\equiv \\exp(\\mu/kT)", "9d85ff6865d75480f022bca44351ec80": " m = M - 5 ( 1+ \\log_{10}p).\\!\\,", "9d860186889e90d17ccae6b616235d99": "S\\subset X\\times \\{T,F\\}", "9d864faff4ba22cd5bfd54bd4d1bc347": "\\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}\n=\\frac{1}{k!}\\sum_{j=0}^{k}(-1)^{k-j}{k \\choose j} j^n.", "9d86694e336f28bfe88c91d5e49cedb7": "f_Z(z) = \\int^\\infty_{-\\infty} f \\left( x\\right) f \\left( z/x \\right) \\frac{1}{|x|} \\, dx. ", "9d876e5263c3da8150c9ab98fd25d3f7": "\\sum^n_{i=1}|x_i|\\le C \\|x\\|", "9d87728aee85be5bc115152447972bd0": "\n\\begin{align}\nc^2+\\frac{d}{2 \\varphi}\\frac{\\varphi d}{2} & = a^2, \\\\[8pt]\nc^2+\\frac{d^2}{4} & = a^2, \\\\[8pt]\nc^2+r^2 & = a^2,\n\\end{align}\n", "9d87844d3390b1e4cc92f24353145298": "n_i(n - n_i)", "9d87c02549507f175c6d805bffd283e0": "\n\\begin{align}\n\\left[\\frac{2}{p}\\right]_3 =1 &\\mbox{ if and only if } &L \\equiv M &\\equiv 0 \\pmod{2} \\\\\n\\left[\\frac{3}{p}\\right]_3 =1 &\\mbox{ if and only if } &M &\\equiv 0 \\pmod{3}\\\\\n\\left[\\frac{5}{p}\\right]_3 =1 &\\mbox{ if and only if } &LM &\\equiv 0 \\pmod{5}\\\\\n\\left[\\frac{7}{p}\\right]_3 =1 &\\mbox{ if and only if } &LM &\\equiv 0 \\pmod{7}\\\\\n\\end{align}\n", "9d884a9ce18fcf4da4d7f66c6e53ab78": "p < d^{3d^2}", "9d886397b649b964d7a7fea898fbf6ec": " 1\\le i,k\\le n ", "9d88a85020a8d0f7c4dd8fa6798a915a": "1+\\frac{(q^\\alpha-1)(q^\\beta-1)}{(q-1)(q^\\gamma-1)}x + \\frac{(q^\\alpha-1)(q^{\\alpha+1}-1)(q^\\beta-1)(q^{\\beta+1}-1)}{(q-1)(q^2-1)(q^\\gamma-1)(q^{\\gamma+1}-1)}x^2+\\cdots", "9d8913f708892c31f871052692e1b7a5": " a_s ", "9d891e5f2c0bbc5e97e0c2c22095dcb3": "\\Delta \\subset P^n \\times P^n", "9d896cfeb46a6d9191e7f69edc3bbc11": "p^\\mu = (E/c, p_x, p_y, p_z) = (\\hbar\\omega/c, \\hbar k_x, \\hbar k_y, \\hbar k_z) = \\hbar k^\\mu", "9d8980d95018cffda6b0d77684ba1523": "k=3", "9d8991b9c1f95a5f568673262068fbfc": "\\scriptstyle 20n", "9d8996cba87a4990f612226951205642": "\\Rightarrow \\beta\\cos\\theta (1-\\beta\\cos\\theta)^2 = (1 - \\beta\\cos\\theta) (\\beta\\sin\\theta)^2", "9d89d2c00db2c3d6d242dc29e7bd6ef0": " R(h,k) := \\frac{ k^n A(h) - A(kh)}{k^n-1} ", "9d89d57ff4e1163c9c4fd8bc39a5e545": "\\tbinom{p+q}{q}", "9d8a2a2363ffc840ea9e0caa0ff2674f": "A \\in K_{m,n} \\, ", "9d8a4c5723ffd5c1ae7e8a1a79e4891e": " y^3 \\rightarrow 1 ", "9d8aaa62e1c0eafe1e17e9d3613c1578": "(g_\\theta,\\boldsymbol z)", "9d8ab5ba81286ff4e2381927e56bbcb5": "E_t\\,", "9d8b1380661e744d80466e87079ca55e": "D(Y)=\\exp[-2F(\\theta_0)]", "9d8b4ed233aa3dd9fda392c456564c6e": " \\star : \\bigwedge^{k} V \\to \\bigwedge^{n-k} V", "9d8b6e22dda60016e8c64dacf500bc8e": " 2n+1 ", "9d8b7ca51c865bbcc8f49646dac448a9": "r = -p", "9d8b83afcb77f1fcd86d11e9e66f3c98": " B(s,t) = c_0(0) (1-s)(1-t) + c_0(1) s(1-t) + c_1(0) (1-s)t + c_1(1) s t. \\, ", "9d8bd056966f948ff78b58689d9ba2b2": "1\\le n\\le x", "9d8be2dfa20a7c3eb5b13a6ddaae0490": " \\begin{cases}\n\\text{Mesh 1: } -V_s + R_1I_1 + R_3(I_1 - I_2) = 0\\\\\n\\text{Mesh 2: } R_2I_2 + 3I_x + R_3(I_2 - I_1) = 0\\\\\n\\text{Dependent variable: } I_x = I_1 - I_2 \n\\end{cases} \\, ", "9d8bf4e532eb1432cde431ac38b2cd1a": "f = AB ", "9d8c008a2e8a1a023de060960ac38574": "\\frac{d [Z]}{dt}= 2k_{III} [A] [X] - k_V [B] [Z] ", "9d8c192baf146bd64afad2b15e7301cd": "g_\\ell^m", "9d8c36a0e68767e2fd5f2a76f7f0b9ae": "HA^- + H^+ \\rightleftharpoons H_2A; K_2=\\frac{[H_2A]}{[H^+][HA^-]}", "9d8c976372a35ef2c69e0db8afc7d890": "\\left(h\\pm c, k\\right)", "9d8cd8d60ed237d4c0ca21e60e92ef07": "\\vec{F}(\\vec{q})\\!", "9d8d1b80733389a3b51010842c8d592a": " |\\text{Tr}(A)^2 -4| + |\\text{Tr}(ABA^{-1}B^{-1})-2|\\ge 1. \\, ", "9d8d2d5ab12b515182a505f54db7f538": "Age", "9d8d336077b071fb27b13621968ec3ab": " \\prod_{n=1}^\\infty \\underset{p_n: \\text{ prime}} {\\left( 1-\\frac{3 p_n-2}{{p_n}^{3}}\\right)} = \\frac {6}{\\pi ^2}\\prod_{n=1}^\\infty \\underset{p_n: \\text{ prime}} {\\left( 1-\\frac{1}{{p_n(p_n+1)}}\\right)} ", "9d8d47f8544949680f3d110d72ad0895": "\\frac{\\partial uv}{\\partial \\mathbf{X}} =", "9d8df6dd336d5c49a151a507e1eb94a2": " \\sum \\|u_n\\|_p < \\infty ", "9d8e2e975f92410d907d4f234b384516": "3 \\uparrow^8 3", "9d8f01177ddbf6a32add79089275172c": "L=L_{0}/\\gamma.", "9d8f6d53ad070771a6910933810023c4": "\\Box ", "9d8f8b37c5bc990bf182f89cb9f45c69": "\\scriptstyle u", "9d8fbb00e8e3c4d7b5781ca6bfc262ed": "300% \\times 10% = 30%,", "9d8fed1928832aae5e9a4408af8df6ee": "\\alpha + \\beta = \\operatorname{mex}(\\{\\,\\alpha' + \\beta : \\alpha' < \\alpha\\,\\} \\cup \\{\\, \\alpha + \\beta' : \\beta' < \\beta \\,\\}),", "9d90550233ecc57d674d0a6fa4d26cc4": "D = V \\setminus T", "9d9097e22583bb90629456628898a15b": "\\bar{\\mu}", "9d91257355d40a55ca57305c3f8056a9": "(B_\\bullet, d_{B,\\bullet})", "9d914b4185acaea5642c67244fecb243": "\\frac{T(x,t)-T_\\infty}{T_i-T_\\infty}=\\sum_{n=0}^{\\infty}{\\left[\\frac{4\\sin{\\lambda_n}}{2\\lambda_n+\\sin{2\\lambda_n}}e^{-\\lambda_n^2\\frac{\\alpha t}{L^2}}\\cos{ \\frac{\\lambda_n x}{L}}\\right]}", "9d9152b008f6721d581657ceb1fd46f9": " |\\Re z| > {1\\over 2} |w|.", "9d915d739b3885ae2ebff1e7ad4a7d15": "e(n)", "9d917cd1c70ffa7b39545f2973012e0e": "x \\mapsto \\frac{1}{x}, \\, y \\mapsto \\frac{1}{y}.", "9d919595722ec5a417fc5f425227eacd": "\\alpha < \\beta < \\gamma \\,, \\quad \\delta < \\epsilon < \\cdots < \\lambda\\,,\\quad \\mu < \\nu \\cdots < \\zeta\\,.", "9d91a5b7cbc44db3ad0c4fc0842b0065": "\n\\boldsymbol{\\hat r}\n=\\sin (\\theta) \\cos (\\varphi) \\boldsymbol{\\hat{\\imath}} +\n\\sin (\\theta) \\sin (\\varphi) \\boldsymbol{\\hat{\\jmath}} +\n\\cos (\\theta) \\boldsymbol{\\hat{k}}\n", "9d91caa20188fc3075bb6288d778c2ed": "\\begin{align}\n \\sum_n \\mathcal{L}(n)[n < \\Omega]\n &= \\sum_{n=m}^{\\Omega - 1} \\frac{1}{n} \\\\\n &= H_{\\Omega-1} - H_{m - 1}\n\\end{align}", "9d9248f9efe40a514fb07751c668c933": "A_{SG} + A_{SL}", "9d92cb17749aecb2bdc2f4cdb917e832": "\\begin{align}\n x_{i+1} &= x_i + v_i\\, \\Delta t + \\tfrac{1}{2}\\,a_i\\, \\Delta t^{\\,2} , \\\\[0.4em]\n v_{i+1} &= v_i + \\tfrac{1}{2}\\,(a_i + a_{i+1})\\,\\Delta t .\n\\end{align}", "9d92d95072f3011c59b816dc369601fd": "\\sum_{i=1}^{n}b_{ip}*2^{n-i}", "9d92e9dda9995e00f1e102f85b6297aa": "\\mathcal{A}_q", "9d930b44aefe6af7d024d0d80f0655d3": "\\displaystyle{g=\\begin{pmatrix} A & B \\\\ \\overline{B} & \\overline{A}\\end{pmatrix}}", "9d9320a652f125a2914cf12e71ddbfc6": "p_A(t+\\Delta t) = p_A(t) - p_A(t)\\mu_A\\Delta t + \\sum_{x\\neq A}p_x(t)\\mu_{xA}\\Delta t ", "9d93962a15a78c4a71a833d8bab40adf": " g(t) = 2(t-0.5), \\, ", "9d93b6be68b022e3b1250a605badec42": " v = \\sqrt{\\frac{2GM}{r}}\\,\\!", "9d945642282d157aa11c33fef79055bb": "y_b = b_0 x^c \\sum_{r = 0}^\\infty \\frac{c(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^r ", "9d945850b29b9f5d3a098e9701c5500a": "x^2-1=0", "9d94723e3b69e67fb89340e1689650d1": "{\\bold \\mathrm v}", "9d94eee552d059b7b7e228dbb069349c": " E(\\varphi,k) = E(\\varphi \\,|\\,k^2) =\nE(\\sin\\varphi;k) = \\int_0^\\varphi \\sqrt{1-k^2 \\sin^2\\theta}\\, d\\theta.", "9d951c439b1688ac7746bcaca89eaba8": "p_n = \\left\\lfloor r^{n^2}\\alpha \\right\\rfloor - r^{2n-1}\\left\\lfloor r^{(n-1)^2}\\alpha\\right\\rfloor.", "9d953471cc663e5f654266d934c6dbd5": "\\nu_\\tau", "9d9569f9b8d4f26439cb1594416b0a8a": "\\frac{1}{R}=\\frac{1}{R_1}+\\frac{1}{R_2}", "9d95a57957510403380c2583bb3a7f67": "T: z\\mapsto z+1", "9d95beaccee58ad1333d3d74b0ac2668": " \\lang Ax , y \\rang = \\lang x , A^* y \\rang ", "9d95dacd085871959416ed6792122898": " w_{ij} ", "9d96361b7cddb1aa7b532850f5a16f89": "2. \\; \\; \\mathrm{O} + \\mathrm{O}_2 \\; \\xrightarrow{} \\; \\mathrm{O}_3", "9d9680edbc6abcd372945e810e2f432e": "\\begin{matrix} 4 & 2 & 1 \\\\ 3 & 1\\end{matrix}", "9d96d5fa1eb6ece52df5db0620fae6bb": " e^{i \\theta}", "9d974723bcc63d550dbd785279b926e4": "C : y^2 + h(x) y = f(x) \\in K[x,y]", "9d9792ae93b4d0d5b312cd4d80490208": "\\alpha \\tilde\\phi = {d \\tilde\\phi \\over d z} = 0 \\quad \\text{ at }z = z_1\\text{ and }z = z_2.", "9d98208ee6121703154d392b1c23b877": "\\operatorname{Spin}(3) \\to \\operatorname{SO}(3)\\,", "9d983068b4aeefe396149dac967c1798": "\\operatorname{pos}\\left(\\cup_{i \\in I} U_i\\right) = \\sup_{i \\in I}\\operatorname{pos}(U_i).", "9d98a53e248131831e83994e5ac5eb07": "\\mathcal{L} = K + \\frac{1}{N_0} \\sum^{\\infty}_{i=1} Y_i \\hat{X_i}", "9d98ef26e1e857a624c9442dc5c2bb31": "G^0(X)=\\{g(x)\\in G(X)\\quad : \\quad g(x_0)=1\\in \\widetilde P_{x_0}\\} ", "9d9902c1ae46a4d00a944c4d0b248659": "S = | \\psi \\rangle \\langle \\psi | , ", "9d990de3de26d1c883387107c731ad49": "\n da~\\mathbf{n} = J~dA~\\mathbf{F}^{-T}\\cdot \\mathbf{N}\n\\,\\!", "9d995e068568c1e1900ad7638025f4ea": "G(\\omega=0,\\,p) \\to 0", "9d9961590104e3f5336f889ef3e3e805": "a_k\\,", "9d996f5d69c5482d6991f13c5e21724a": " \\hat {\\mathbf i}, \\ \\hat {\\mathbf j}, \\ \\hat {\\mathbf k} ", "9d99a2bcc71408906885c38ad5584c18": " p_2 (x_1, x_2, \\dots,x_n) = x_1^2 + x_2^2 + \\cdots + x_n^2 \\, ,", "9d99ba4a3eea7080017b25d097275ec8": " p \\mapsto \\{ (\\alpha, q) \\in S : p \\overset{\\alpha}{\\rightarrow} q \\}.", "9d99c162fb97c8306dabc11afa625aea": "F_{net}", "9d99c186d58f28fb58b9fd1a87a827eb": "X \\in \\mathcal{L}^2", "9d9a44d4ff299774c9067dadcbe50193": "a_{i}+b_{i}", "9d9a72b15f4eea7214502ddd798a3571": "\\ G_G(\\tau)=1+\\frac{(Diff_k(\\tau)+Diff_k(\\tau))}{V_{eff, GR}(+)^2}", "9d9aa632c260be4d988773f6f29c00de": " q(f,g) = \\frac{|\\mathrm{ker} f:\\mathrm{im} g|}{|\\mathrm{ker} g:\\mathrm{im} f|} ", "9d9aae7e2de91eedc7963322a4e79480": "{\\nabla^2 A \\over A } = {1 \\over c^2 T } { d^2 T \\over d t^2 }.", "9d9ac2b484afb402729a19a3021c10a4": "\\operatorname{Sym}\\, T = \\frac{1}{k!}\\sum_{\\sigma\\in\\mathfrak{S}_k} \\tau_\\sigma T,", "9d9b06e4d8fea966225dec59e28d4b4e": "\\|f\\|_p = \\left( \\int |f(x)|^p \\;dx \\right)^{1/p}", "9d9b640ff78219e751863575fe2ee4ff": "X^2 = x^2 + 2 + 1/x^2", "9d9b6543da9877477280a9797943df4b": "\\lambda_I", "9d9ba29ee3da1012291a103960038ae8": "D_{f,\\varepsilon} = \\{(x,y)\\in X\\times X : |f(x) - f(y)| < \\varepsilon\\}", "9d9c64c69792fa209b289079f08e8a85": "{\\widehat{AU}}_5", "9d9c763a378f08958ef61dcbd41988fc": " f_\\alpha \\circ f_\\beta = f_{\\alpha+\\beta}. \\, ", "9d9cdfee61b6b43d6cacbf5edf68636d": "H_{inv}(z) = \\frac{D(z)}{A(z)}", "9d9d032973549e903815302edb3d0a1e": "\\Delta Y/Y = 3 - 2 \\Delta u.\\,", "9d9d0fc85078fe09a6f2c0b30c048807": "y=3.5+1.4x", "9d9d2e8422d9aadf2b530090b5ab04a1": "a\\le u(t)\\le b", "9d9d87edfea1ca55d8299193275354d4": "x_i= c_i \\cdot \\frac{\\sum x_j M_j}{\\rho - c_i M_i}", "9d9d9b20636cbb90cfb2244366754fe8": "\\min \\sum_{i=1}^n w_i |x_i - a_i|^p", "9d9dd89ad6ac856172f5f537da623d06": "V_{nm}\\equiv\\langle n^{(0)}|V|m^{(0)}\\rangle", "9d9dff9320e27082b15b4ed7a086ba83": "1 ", "9d9e036bf0ff12c38672bf7b4deb87a6": "a = \\frac{t \\times p}{p - t \\times p} = \\frac{t}{1 - t}", "9d9e231eeda99f2b5dbfbfe4c94c66a3": "\\left| \\frac{d \\sigma}{d \\Omega} \\right|", "9d9e27a1578928359c08ad726ee3ed5e": "E = E_0[e^{i\\omega t} + e^{-i\\omega t}]", "9d9e769ff2bc0433df215db1288d0735": "1-\\sin^2[\\arctan(x)]=\\frac{1}{\\tan^2[\\arctan(x)]+1}", "9d9eb7a37225e38ce34af42fbf2b42cd": "\n\\pi _1 (a) \\pi _1 (b) = \\pi (a) | _{K_1} \\pi (b) | _{K_1} = \\pi (a) \\pi (b) | _{K_1} = \\pi (ab) |_{K_1} \n= \\pi_1 (ab)\n", "9d9ec820891a8558c371c7cdcc82814b": "\\min_{I_k}\\; (-W)=\\min_{I_k}\\;\\left \\{ \\sum_{k=1}^n C_k(I_k)\\right \\}", "9d9f015bbd516c1a0ff069307866985c": "\\mathfrak c \\!\\,", "9d9f62abd985ecfc21b8b1c3af6204dc": "B_{j_{k+1}}", "9d9fd77acee35d8c0452bea060d860fe": "\\operatorname{Th}(\\R)\\models\\varphi.", "9d9fddd93c71fa181aee904c670d739b": "\n\\lim_{x^0\\rightarrow+\\infty}\n\\int \\mathrm{d}^3x \\langle\\alpha|f(x)\\overleftrightarrow\\part_0\\varphi(x)|\\beta\\rangle=\n\\sqrt Z \\int \\mathrm{d}^3x\n\\langle\\alpha|f(x)\\overleftrightarrow\\part_0\\varphi_{\\mathrm{out}}(x)|\\beta\\rangle\n", "9da03780840b8a6168fbafabc965c3ab": "\\scriptstyle 2n", "9da0fe08552067f78459a584a7a58c27": "(x + y)^2 = x^2 + 2xy + y^2.\\!", "9da0fffab8bd161a743c09babf3ec574": "\n \\operatorname{Pr}\\big(\\big|g(X_n)-g(X)\\big|>\\varepsilon\\big) \\leq \n \\operatorname{Pr}\\big(|X_n-X|\\geq\\delta\\big) + \\operatorname{Pr}(X\\in B_\\delta) + \\operatorname{Pr}(X\\in D_g).\n ", "9da10d89f6db3aa051b2369c31df41cd": "\n \\mathsf{I} = g^{ij}\\mathbf{b}_i\\otimes\\mathbf{b}_j = g_{ij}\\mathbf{b}^i\\otimes\\mathbf{b}^j = \\mathbf{b}_i\\otimes\\mathbf{b}^i = \\mathbf{b}^i\\otimes\\mathbf{b}_i\n ", "9da11072c762ece61daa6f471740f697": "\nP(J_{ij}) = \\dfrac{1}{\\sqrt{2\\pi J^2}}\\exp\\left\\{-\\dfrac{N}{2J^2}\\left(J_{ij} - \\dfrac{J_0}{N}\\right)^2\\right\\}.\n", "9da13c7e4f296c29eab4bcf29a43b00f": "x/x2 + y/y2", "9da19c9266ed4e30ab978b59d79f8645": "G = \\mathrm{erf}(\\sigma/2)", "9da201839ae7b080a6e6cf0eb812a243": "ds^2=\\left(1-{r_g\\over r}\\right)dt^2-{dr^2\\over 1-{r_g\\over r}} - r^2\\left(d\\theta^2+\\sin^2\\theta d\\phi^2\\right) \\;,", "9da2a28153574eeecf70024e5987e1ca": "1-\\zeta(2)+\\zeta(3)-\\zeta(4)+\\cdots=|\\frac{1}{2}|", "9da312541a3d1f8365169cac7c3d83e5": " N(m,q,n) = \\sum_{r=-\\infty}^\\infty N( m + rq, n)", "9da331640574fbf08631d5fa94e88f0e": "Q_k(z) = \\exp\\left(\\sum_{d|k} \\frac{z^d}{d}\\right).", "9da355ef779b010d1b050966cfdb5cdf": "T^{\\mu \\nu}\\,", "9da359dccf81c29b16e0c8e442a4dce3": "F \\times_G H", "9da36b2f548a8587074bf45d4e8b9eb0": "P^{m}\\{|Q_{P}(h)-\\widehat{Q_{s}(h)}|>\\frac{\\epsilon}{2}\\}\\leq\\frac{m\\cdot Q_{P}(h)(1-Q_{P}(h))}{(\\epsilon m/2)^{2}}\\leq\\frac{1}{\\epsilon^{2}m}\\leq\\frac{1}{2}\\,\\!", "9da39a7b31c96ed0605493ddc52582f6": "b^*\\,", "9da3bfaa8518cfb4b62f82dc7c85600d": "0.067=1.73/26", "9da42d519fc963c69ef64f05573caf4e": " \\mathbf{} v ", "9da4653c3212c414aeeb1342685b10f4": "Fo= \\frac{\\alpha t}{L^2} ", "9da46c0eaffdd6fa7a9158b2a7f972e9": "a_n = (-B)^{\\frac{n}{2}} \\left( E \\cos(\\theta n) + F \\sin(\\theta n)\\right),", "9da483a0a9cb7e19510bd49eb11c1191": "|\\mathbb{R}|", "9da4ff684c6645681533b4c6d56cf6e2": "E=(\\hat{y} - y)^2", "9da50136f5a91bcdbcf43978789133e1": "\\chi(M)=\\frac{1}{32\\pi^2}\\int_M\\left(|Rm|^2-4|Rc|^2+R^2\\right)d\\mu ", "9da559b4e9f5d93885a01ad8a10af2e4": "g_1, \\ldots, g_k", "9da571b6dccdc7ba5e08193342ada8f9": "v[\\alpha^{x_i}] = i", "9da581a1c7b97ba4d19d61545d73b147": "\\scriptstyle a_1 \\times a_2 \\times \\dots \\times a_d", "9da5e89300419efafcc132fd64e4da42": "Ra_L<10^9", "9da61e146b6aff4b9c0cc1993f7f6152": " B^1_1...B^1_k, ..., B^k_1...B^k_k ", "9da65c226182098ddc0def6636a087e8": "(\\mathbf{1},\\mathbf\n\n{3},0)", "9da73bb9f2a377bc5f79d4025078d345": "1 - 2\\rho", "9da73c45a3533dbbdcaf645069ef48c4": " W = w \\cdot p^{k} ", "9da76b704e007eca18b88578df73b6f3": "\n F \\;=\\;\n \\begin{bmatrix} F_1\\\\ F_2 \\\\ F_3 \\end{bmatrix} \\;=\\;\n \\begin{bmatrix} \n \\kappa_{1\\,1}& \\kappa_{1\\,2}& \\kappa_{1\\,3}\\\\ \n \\kappa_{2\\,1}& \\kappa_{2\\,2}& \\kappa_{2\\,3}\\\\ \n \\kappa_{3\\,1}& \\kappa_{3\\,2}& \\kappa_{3\\,3}\n \\end{bmatrix}\n \\begin{bmatrix} X_1\\\\ X_2 \\\\ X_3 \\end{bmatrix}\n \\;=\\; \\kappa X\n", "9da7702c0d48e939de6e978e1fa51440": "\n (\\rightarrow R)\n ", "9da7746eec9fc2c4d672c9be96b45e8a": "s_2^3", "9da775e7d1959ceeabe97ec199b22456": "a(x) = a_0 + x.\\,\\!", "9da796793ff03ac8d282be72158cf7fd": "\\int \\vec E\\cdot\\mathrm{d}\\vec l = -V", "9da7ca82d76218498216220a5d04004a": "|L_1(x)\\cdots L_n(x)|<|x|^{-\\varepsilon}", "9da814959dd76caca9802ea3d2447250": "P_{hs}= \\frac{R\\, T}{V_m}\\, \\frac{1 - \\eta^3}{(1 - \\eta)^4}", "9da85acd060915213387cffc957a3582": "h_{\\,+}", "9da88350c6231b67863affcaa4d5df88": "r\\to1", "9da8b705807c5fd5997b07f8936be38d": "f(n) \\simeq U( \\mu x\\, T(e,n,x))", "9da8d6292737fbd127cb3a55dd7fa02a": " Q \\ge 0,", "9da8dbc42147fc0998dac494fef4ec90": "\\sigma(X^*, X)", "9da946c0b8f81a9a936d4f9486227123": "\\dot x(t) = x", "9da9f0df462eaece7120a502c84650cb": "\\overline{C_n}=\\frac{1}{N}\\sum_{i=1}^N\\cos(n\\theta_i)", "9daa4d64f7d726295c51b936e097ca43": "s\\times s", "9daa9fd40e0985847ca94be9169b7164": "\\frac {p(x)} {q(x)} > 0\\,", "9daaa29b9b37a6d2522f91768ea72c50": "\\phi_{\\alpha\\beta}\\!", "9daac651165a3942493e2ece16366ba5": "{p e^{iq} \\over r e^{is}} = {p \\over r}e^{i(q - s)}.", "9dab4ad99d36e743657f35c4b68dd3d1": "d \\in \\{\\, -1, -2, -3, -7, -11, -19, -43, -67, -163\\,\\}.", "9dab6f0b7c0ca7cd870c895bf323201e": "\\forall \\left(A_i\\right)_{i\\in I} \\in \\prod\\nolimits_{i\\in I}\\tau_i \\ : \\ \\mathrm{P}\\left(\\bigcap\\nolimits_{i\\in I}A_i\\right) = \\prod\\nolimits_{i\\in I}\\mathrm{P}\\left(A_i\\right)", "9dabbac3263bd6c198f42141f09990ed": "N+P\\,", "9dabdc3f5425e11f9316baa82d39044e": "B1: = A1 - A2", "9dac468ab488452910b5161543715f78": "\\frac{d}{d\\mu}\\left(\\overline{g}_{\\lambda k\n}\\overline{v}^k\\right)=\\frac{1}{2}\\frac{\\partial\n\\overline{g}_{ij}}{\\partial\nx^{\\lambda}}\\overline{v}^{i}\\overline{v}^{j}", "9dacb7fc50c76fa796685054a0da9574": "\\begin{align}\n0\n&\\approx f(\\mathbf{a} + \\mathbf{v} + \\mathbf{w}) - f(\\mathbf{a} + \\mathbf{v}) - f(\\mathbf{a} + \\mathbf{w}) + f(\\mathbf{a}) \\\\\n&= (f(\\mathbf{a} + \\mathbf{v} + \\mathbf{w}) - f(\\mathbf{a})) - (f(\\mathbf{a} + \\mathbf{v}) - f(\\mathbf{a})) - (f(\\mathbf{a} + \\mathbf{w}) - f(\\mathbf{a})) \\\\\n&\\approx f'(\\mathbf{a})(\\mathbf{v} + \\mathbf{w}) - f'(\\mathbf{a})\\mathbf{v} - f'(\\mathbf{a})\\mathbf{w}.\n\\end{align}", "9dad4713ccc52207c9dd91a939b5e038": "(\\forall{Y_{1},Y_{2}}{\\in}{p}:Y_{1}\\ne Y_{2}\\rarr ({x}{\\notin}{Y_{1}}\\or{x}{\\notin}{Y_{2}}))", "9dae16e6fa944539e4edeba87128fb8b": " d = \\frac{v \\cos \\theta}{g} \\left( v \\sin \\theta + \\sqrt{v^2 \\sin^2 \\theta + 2gy_0} \\right) ", "9dae52fe3dba1ab22b2bcdad887584cf": " \\mathbf{p} = M\\mathbf{V},\\quad \\mathbf{L} = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}).", "9dae5b21f27d82b101c39ed8ff11fa11": "\\mathcal{I} \\models (a,b) : R", "9dae71529d4f48a7e03dfe64986d501a": "\\ a=|cz|,", "9dae773cde73a79c4102c69b976d82ad": "(2j_1+1)(2j_2+1)(2j_3+1)", "9dae7857bfd4b9bbed6a62c7f8386e26": "a_W=\\frac{a_S+2c}{3}\\,", "9daebbe9a0e453104eb1e4c65c1c0db8": "\\sigma(a)=b, \\sigma(c)=d", "9daed89a190ac5ef69a92cd4bdc3942e": "\nL = {nh \\over 2\\pi} = n \\hbar ~ .\n", "9daeefa9d49c1bf59b59e14da755f369": "d = \\sqrt{1.5h}", "9daeefdd6e15d41a1fd72b455a969c33": "f(z)=f(z+\\omega_2)", "9daf0c479bf650bfea65afa437efe3f8": "\\Gamma(qz;p,q)=\\theta (z;p) \\Gamma (z; p,q)\\,", "9daf5b565183dba547be12144c7451cb": "A = \\left( 2.457 \\ln \\left( \\left( \\left( \\frac {7} {Re} \\right) ^ {0.9} + 0.27 \\frac {\\epsilon} {D} \\right)^ {-1}\\right) \\right) ^ {16} ", "9daf7082dd12b28d7cc557273fee5102": "a \\uparrow \\uparrow \\uparrow \\uparrow b", "9daf9179c694414971793011e8d50a7f": "f(z) = 1/z^*", "9dafa1403439929eab5b642f317801eb": "z0 = z_{cr} \\,", "9dafa27cdf09e00da50f07bc3b16b916": "y \\in \\{ true , false \\}", "9dafd3b65a263b7ecf4b073aceff13bc": "-\\otimes_RN:\\mathrm{Mod}\\mbox{--}R\\rightarrow \\mathrm{Ab}", "9db02c9b211c2a64ee8f24c54d70163a": "G(x) = x - \\frac{(p-1)(p+1)}{3!}x^3 + \\frac{(p-3)(p-1)(p+1)(p+3)}{5!}x^5 - \\cdots. ", "9db072b31a62c4910eb32bf69e75cd18": "(v \\circ u) \\circ t = v \\circ (u \\circ t)", "9db086d5c5305c3412347cf06f64ae70": " \\frac{dx}{dt} ", "9db0cc94aa620ee18aecd8c69cf633be": "\\Box \\mathbf{A} = \\mu_0 \\mathbf{j}", "9db133d79448c6e6da50c9397b6faedb": "f(0) \\geq 0 ", "9db153c5fd057b802c79d8c001e97818": " f: \\{-1,1\\}^n \\to \\{-1,1\\} \\!", "9db188f63d2c2aad11f828aa26f8e5af": "D(Q) = \\{ \\psi \\in L^2({\\mathbf R}) \\,|\\, Q \\psi \\in L^2({\\mathbf R}) \\}.", "9db197477886eb399f1da1fe3488cf1a": "$20,000 - $1332 = $18,668", "9db1eb08d3f428b42985284110a87dca": " v(N) ", "9db267150e1ba3a1df1c66954ca6a519": "\\epsilon_{[t_n,t_u]}", "9db2afc1dfdc085073606fd9d6ef937d": "\\mathcal O (-1)", "9db2cbb9f49de33753e5632fba717bf7": "\\sup \\{ | f(x) | : f \\in \\mathbf{F} \\} < \\infty.", "9db387f0b1668cb15ab03e2689e706d5": "H_\\alpha ", "9db3967d7a4928f3be77c71a4dd35f7e": " e^x \\, ", "9db45733447f2b50b38d84ef11dcbe49": "A^+ = C^+B^+ = C^*(CC^*)^{-1}(B^*B)^{-1}B^*\\,\\!", "9db46f68c4eb4a8c2d857faa4d222df6": "\\frac{\\partial\\mathbf{x}_i}{\\partial K_{(k\\ell)}} = \\sum_{j=1\\atop j\\neq i}^N \\frac{x_{0j(k)} x_{0i(\\ell)}(2-\\delta_k^\\ell)}{\\lambda_{0i}-\\lambda_{0j}}\\mathbf{x}_{0j}", "9db4b616f5441ad8a3eef8ace147e5fa": "\n F(x) = \\begin{cases}\n 0 & \\text{if }x < \\mu \\\\\n 1 & \\text{if }x \\geq \\mu\n \\end{cases}\n ", "9db4e5bf076d252775aa31d2c5b0092d": "(w_1, w_2, \\ldots, w_k)", "9db57792978c2d25d9b66db378ecb005": "\\scriptstyle w(\\tau) \\isin R^4", "9db585ff4f60a9bfdc45e9ee1d046e7e": "\\sigma = \\hat f (\\epsilon)", "9db5ec3cdba31c46f096ab08a80b7790": "F(\\mathbf{k})", "9db60be666ee00d554b48dfb5aa2f80a": "64^2", "9db69b4143b72779b2d36d6cb9edeffe": "{a\\pi\\over 5}\\ {b\\pi\\over 3}\\ {c\\pi\\over 2}", "9db69d5e593037ce789f9befbb30b353": "m=2", "9db6eb93aa0d00ec05e28ed99b718419": "E = \\frac{4{ n \\choose 4 } + 2\\left( { n \\choose 2 } - n \\right)}{2} + n + n = 2{ n \\choose 4 } + { n \\choose 2 } + n.", "9db702eb16942885186122876b0a6b50": "\\{|e_1\\rangle, \\dots, |e_d\\rangle\\}", "9db77f6feb75f5f10dbca014ae766984": "\\langle x - x_n, x - x_n \\rangle = \\langle x, x \\rangle + \\langle x_n, x_n \\rangle - \\langle x_n, x \\rangle - \\langle x, x_n \\rangle \\rightarrow 0.", "9db8b67225f0880039c3fd02809c5529": " \\forall X \\exist Y \\forall uvw[(u,v,w) \\in Y \\leftrightarrow (u,w,v) \\in X].", "9db8fd531e38614b71b8f66ff550fe7a": "\\{C\\land I\\}\\;\\mathrm{body}\\;\\{I\\}", "9db907cc7d9bb92bb407f93658388619": "x^2-1 = (x-1)(x+1)", "9db934c2705d6e4c72cc9db9bdaf2c2f": "\\overline{e}(k,i+1) = \\frac{1}{\\sqrt{(1 - \\overline{e}_b^2(k,i))(1 - \\overline{\\delta}_D^2(k,i))}}[\\overline{e}(k,i) - \\overline{\\delta}_D(k,i)\\overline{e}_b(k,i)]", "9db9a9e60edaf446a3a3155739aff145": " \\omega = \\frac{1}{2 \\sin(\\theta)} \\begin{bmatrix} R(3,2)-R(2,3) \\\\ R(1,3)-R(3,1) \\\\ R(2,1)-R(1,2) \\end{bmatrix} ", "9db9c77e591feca9c8c30d42184d929e": "\\phi_n(p,t)=\\sqrt{\\frac{\\pi L}{\\hbar}}\\,\\,\\frac{n\\left(1-(-1)^ne^{-ikL}\\right) e^{-i \\omega_n t}}{\\pi ^2 n^2-k^2 L^2},", "9db9d2fb408102964b7b5408f6ae9c8c": " m_\\mathrm{s} = - \\tfrac{1}{2} ", "9db9f7e7c59a76e9912539a4c7fbed84": "C = x/(x+y)", "9db9ff8e327a0a21861e9c4a3b13175a": "J_A (\\rho) = S (\\rho_B) - S (\\rho_B|\\rho_A)", "9dba00147681383ec5a5e6d730e70792": "w = d + m + y +c \\ \\bmod\\ 7", "9dba0bdb2ab7d46120dd86d1a1efe1b5": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-5}{\\sqrt{3}},\\ \\pm3\\right)", "9dba1ca375de549dc36474adff079abf": "\\exists z'\\forall x[x\\in z'\\leftrightarrow \\Phi(x)]", "9dba97acb54921074ddaddb27e63ecfb": "\\frac {F_\\text{load}}{F_\\text{in}} = \\frac {2 \\pi r}{l} \\,", "9dbaaea94fb58ea87057dee6baa293bb": " \\equiv (a^{N-1})^{u}\\equiv a^{u(N-1)} \\equiv a^{uq((N-1)/q)}\\equiv (a^{uq})^{(N-1)/q}\\pmod{p}", "9dbad59edf26d5410f4a198aa211c63a": "I_{base} = \\frac{S_{base}}{V_{base} \\times \\sqrt{3}} = 2.09 \\, \\mathrm{kA}", "9dbb31dc21961c4386af6432ded5d606": "E_\\mathrm{L}", "9dbb8a1e5a1a4a96ae30a7cc709d72f6": " \\sum_{j,k\\ge0}m_{j+k}c_j\\bar c_k\\ge0\n", "9dbb903352805948a9c604734b7f4d3e": "\\theta = \\frac{n\\varphi}{m}", "9dbbc9d3f6d02d7212f7fe1cb714bbf7": "\\mbox{power} = \\mbox{torque } \\times\\ 2 \\pi\\ \\times \\mbox{ rotational speed} \\cdot \\frac{\\mbox{ft}\\cdot\\mbox{lbf}}{\\mbox{min}} \\times \\frac{\\mbox{horsepower}}{33,000 \\cdot \\frac{\\mbox{ft }\\cdot\\mbox{ lbf}}{\\mbox{min}} } \\approx \\frac {\\mbox{torque} \\times \\mbox{RPM}}{5,252} ", "9dbc27be7e63297fe1dfb79a1eb03f37": " \\frac{\\mu}{\\hbar}\\psi^\\dagger \\left(iI\\delta_{ij} + \\sigma_k \\varepsilon_{ijk}\\right)B_i\\psi + \\frac{\\mu}{\\hbar}\\psi^\\dagger \\left(-iI\\delta_{ij} + \\sigma_k \\varepsilon_{ijk}\\right)B_i\\psi = \\frac{2\\mu}{\\hbar} \\left( \\psi^\\dagger\\sigma_k\\psi\\right) B_i \\varepsilon_{ijk} ", "9dbc2b3813427a5b9c389b345273d821": "\\begin{align}\n E &{}= C - T \\\\\n &{}> G_n \\\\\n P_n &{}= C - G_n \\\\\n &{}> C - E \\\\\n P_n &{}> T\n\\end{align}", "9dbcc2d302bb63991f8dccf07ca9d515": "r_c = \\left\\{ \\frac{\\partial V}{\\partial J} \\right\\}_{V=0}", "9dbccc07f601b5ef173eae66cffc7a0e": "K|z-w|", "9dbcdad6c6c94715e9de1027e4b0cde0": "i \\leftarrow i+1", "9dbcf8f4523b910764da7f544a192a69": "q(x)", "9dbd81f7f206c6c4877e5366d21b7b09": "(\\lambda x.(\\lambda y.x))", "9dbdf0321bb82e14435fc8e05e0169ce": "B_k^{-1}", "9dbeede574900782d823fb0a43be28a8": "P(3)=P_0 2^3", "9dbfd0c7d3fb741050677052e19e742b": "{k_f}", "9dc00d61bff14b51621e4e3a1fb7e93e": "\\partial \\Sigma", "9dc068e1512da06ef3ce36fbf6cb42f0": "\\Omega\\left(1-\\frac{1}{m}\\left(\\frac{\\Gamma(m+\\frac{1}{2})}{\\Gamma(m)}\\right)^2\\right)", "9dc0873e6189840596881a2965577615": "\\! \\Delta=(\\delta+\\mathrm{d})^2 = \\delta \\mathrm{d} + \\mathrm{d}\\delta", "9dc0986d1d58ea2925d1cf0aeab29613": "\\sigma_A", "9dc0fa765787708032c7855fa8774b08": "{n \\choose k}.", "9dc1061e214ea333274e63f22fd38b76": "R_{(a)(b)} = \\frac{1}{2} \\left[ C^{cd}_{\\ \\ b} \\left( C_{cda} + C_{dca} \\right) + C^c_{\\ cd} \\left( C^{\\ \\ d}_{ab} + C^{\\ \\ d}_{ba} \\right) - \\frac{1}{2} C^{\\ cd}_b C_{acd} \\right]", "9dc10a97d9d17632024cdb2ad692975f": "\\sum_{r=1}^{m+n}{x_{r,m+j}}-\\sum_{s=1}^{m+n}{x_{m+j,s}}=b_{m+j}", "9dc1d38275e3fb84651f1d6e341bf72c": "\\mathrm{d}U = T\\,\\mathrm{d}S - P\\,\\mathrm{d}V\\,", "9dc24c63e445c40216ad3421433245f6": " \\textrm{U}(x) ", "9dc272642895342107ded50c948db451": "\\ x + y = a, x - y = b, xy = c, x^2 + y^2 = d, x^2 - y^2 = e, x^3 + y^3 = f, x^3 - y^3 = g", "9dc2af0b7bfd3e57d685b23899b188f7": " E\\left(u|X\\right) = 0 ", "9dc2b146105d80fac769c5cc8cae1cf4": "K (x, y) = \\langle x | y \\rangle", "9dc2c49a9fc20c6d3e76465732340602": "\\delta W = \\mathbf {F}_{1} \\cdot \\sum_{j=1}^m \\frac {\\partial \\mathbf {r}_1} {\\partial q_j} \\delta q_j +\\ldots+ \\mathbf {F}_{n} \\cdot \\sum_{j=1}^m \\frac {\\partial \\mathbf {r}_n} {\\partial q_j} \\delta q_j.", "9dc2c556efc9579fc9a4c473af106458": "z_5=\\chi_{\\psi_{5,5}}(z_5,\\rho_{\\psi_{2,5}}(z_2))=\\chi_1(z_5,\\rho_{1}(z_1))=x_1q_1", "9dc2e7316379de73a2575d6875fc9a86": " \\mu_0 > \\frac{ x + m }{ 2 } \\pm k | x - m |", "9dc36e73c15912521a4fc724e874de06": "TX=\\mathfrak F\\oplus T^0X", "9dc37e87100ef8bb2f6be91d28470764": "\\mathbf{a}_i = \\begin{pmatrix}A_{i1} & A_{i2} & \\cdots & A_{im} \\end{pmatrix}\\,,\\quad \\mathbf{b}_i = \\begin{pmatrix}B_{1i} \\\\ B_{2i} \\\\ \\vdots \\\\ B_{mi}\\end{pmatrix} ", "9dc386afc4dd78cee9489dcf38a5434c": " Q^* = -\\left(\\frac{d}{dt} \\frac{\\partial T}{\\partial \\dot{q}} -\\frac{\\partial T}{\\partial q}\\right).", "9dc3892a8c264ea9ad4364ca4c51946e": "L_q", "9dc3a5ebd4d587907367b0be34080ca2": " \\int f(x) d_q g(x) = \\sum_{k=0}^{\\infty} f(q^k x)(g(q^{k}x)-g(q^{k+1}x)), ", "9dc3e67dff8c97f55b3aa7c08edad097": " \\frac{\\omega_A}{\\omega_B} = R.", "9dc411357431633f00c8abd47f009e01": "\\varphi_x", "9dc453764a64947d817a07579b2447d0": "\\mathrm{bei}(x) \\sim \\frac{e^{\\frac{x}{\\sqrt{2}}}}{\\sqrt{2 \\pi x}} [f_1(x) \\sin \\alpha - g_1(x) \\cos \\alpha] - \\frac{\\mathrm{ker}(x)}{\\pi}", "9dc4a9014a65e20e737db333d14d1582": "\\langle i,\nc\\rangle", "9dc4d8d265e06afcfb4df3cf18927533": " V = \\frac23 A B C \\frac{4}{r t} \\beta \\left( \\frac{1}{r},\\frac{1}{r} \\right) \\beta \\left(\\frac{2}{t},\\frac{1}{t} \\right). ", "9dc4edce0abe70e00d5c15a0af2aaa72": "\\kappa=\\kappa_0e^{2\\Phi_0}=(8\\pi G_D)^{\\frac{1}{2}}=\\frac{\\sqrt{8\\pi}}{M_p},", "9dc4f1b61c774335318b1c44220ccfd6": "\\mathcal{O}(m \\log n)", "9dc58dfa9f0387ecdf054f93472f938d": "\\{x|\\phi(x,a)\\}", "9dc5de1fe72d5005106e27e51b39b28e": "0 = ( X_1 * M_1 + X_2 * M_2 + \\ldots + X_n * M_n ) - X_0 * ( M_1 + M_2 + \\ldots + M_n ) ", "9dc633cc83176c67b2f342c3bdbc6b50": "1+\\sum_{n=1}^\\infty \\frac{\\mu'_n t^n}{n!}=\\exp\\left(\\sum_{n=1}^\\infty \\frac{\\kappa_n t^n}{n!}\\right) = \\exp(g(t)).", "9dc68912157a903e5189c9bd7387974b": "\n \\begin{bmatrix} \\sigma_{xx} \\\\ \\sigma_{zz} \\\\ \\sigma_{zx} \\end{bmatrix} = \n \\begin{bmatrix} C_{11} & C_{13} & 0 \\\\ C_{13} & C_{33} & 0 \\\\ 0 & 0 & C_{55} \\end{bmatrix}\n \\begin{bmatrix} \\varepsilon_{xx} \\\\ \\varepsilon_{zz} \\\\ \\varepsilon_{zx} \\end{bmatrix}\n ", "9dc71cc5d79ef35a366dd38c800ef7d7": "\np_{X,A}(x,a) = p_X(x) p_A(a)\n", "9dc72acfa15166cbb453f160a87f5ceb": " g - 1 , g\\in G ", "9dc74ec1a4b0b489b870d1108b8fcf8d": "V_d/V_t = \\frac{PaCO_2 - PeCO_2}{PaCO_2}", "9dc831dad0881dd0104c37dfda3ba5e7": "\\mathbf{v}(t)", "9dc8a6fe5bae45190faa2ecc332e4995": "=0\\,", "9dc8d353a150e9500455bc730d0bcb27": "X_j(\\omega)", "9dc8d83998bb1381ba0acf8b5683084d": "\\Big( (\\mathcal{M}, s) \\models \\neg\\Phi \\Big) \\Leftrightarrow \\Big( (\\mathcal{M}, s) \\not\\models \\Phi \\Big)", "9dc90f0fc2f41cd9028648f43b0cf914": "\\mathrm{I\\!I} = b_{\\alpha \\beta} \\, \\text{d}u^{\\alpha} \\, \\text{d}u^{\\beta}. \\,", "9dc93c5d6e17ba30e1a41255eb991e79": " c_n = 2\\frac{s_n}{S_n} . \\,\\!", "9dc9457c9c52b99de5b2208f3bbf075a": "f(n,m) = n^2 + m^3 + O(n+m) \\text{ as } n,m\\to\\infty\\,", "9dc948cc53d6d9a7c2248984fd453f66": "D_i := \\prod_{1 \\le j \\le i} [j]^{q^{i - j}}", "9dc94d557d7c5e6b1de49cbcea184126": "\\theta^{in}_{1}", "9dc95e28dc1bf5dac67c1ff6bb106f1e": "\\Sigma^p_{ij}", "9dc962004795ddca634779d692a65b6b": " \\delta^3(\\vec{p'}-\\vec{p}) ", "9dc971e5e3449501684939d3b9d41b77": "Z_{st} = \\frac{\\lambda}{1+\\lambda} = \\frac{1}{1+AFR}", "9dc988d3fe2e79b3701ad066dbc18028": "S_n=\\sum_{i=1}^n X_i", "9dc9b3305ff029de31b58f7331c3ca0d": "{{\\int_{0}^{t} \\mathrm{ROI}(\\tau)d\\tau} \\over \\mathrm{ROI}(t)}", "9dca1a06fd7fc158e2c93b421deb15c4": "\\hat{\\sigma}_- = |g \\rangle \\langle e |", "9dca3a426f96ba6f07a4d3f4aa0f2b14": "s(n) \\in \\Theta (4^n n^{-{5 \\over 2}}).", "9dca3cdac9d02c75848cefa860bf1c60": "B[f \\circ g] = B[g]B[f] ~,", "9dca501333b1e0f3218b734606299ee1": "\n\tQ_{n+1}R_{n+1} = J_n Q_n\n", "9dca8f7cdfa3189f2a66de49d6c4e2b1": "W(y_1,y_2)(x)=W(y_1,y_2)(x_0) \\exp\\biggl(-\\int_{x_0}^x p(\\xi) \\,\\textrm{d}\\xi\\biggr),\\qquad x\\in I,", "9dcab8154bf52826b3231cad50f11f0e": "\\sum_{s\\in\\mathcal C} c(s) = \\sum_{s\\in\\mathcal S} c(s) x_s.", "9dcad6d389d0961b9c7bdc9ae0532576": "\\frac{12_{10}}{17_{10}}", "9dcae3ed723f027dee39d3497661ef9d": "p\\in\\mathbb{N}^*", "9dcb2cfca17026647f7c1480b43fa5a2": "\\scriptstyle \\mathcal{L}", "9dcb5a39670325449f257d0ce77f46ea": "\\chi_{k'}(\\mathbf{r};\\mathbf{R})", "9dcc110dd3fecf5880692d9eaa57f8f5": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n \\frac{B\\,x^{m+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^q}{b (m+n (p+q+1)+1)}\\,+\\,\n \\frac{1}{b (m+n (p+q+1)+1)}\\,\\cdot\n", "9dcc1a162c66aba01154b51f1e4e82da": "p(A)=0\\cdot\\operatorname{ev}_A(B)=0", "9dcc94fa6d7575c0fb1e0c3c0c4e109b": "\\textrm{U}(x)=\\begin{cases} 0, & x \\leq 0, \\\\ 1, & x > 0. \\end{cases} ", "9dcc99b69caf555b4003eb653a617564": " H_n^{(r)} = \\sum_{k=1}^n H_k^{(r-1)}\\quad(r>0). ", "9dccc0be4999e8991f81465c6c36db4e": "\\pm\\sqrt{1 + \\tan^2 \\theta}\\! ", "9dccdf8190f562fdc21d239aed4cf241": "\\operatorname{Ti}_2(z)=\\int_0^z \\frac{\\tan^{-1}x}{x}\\,dx = \\sum_{k=0}^{\\infty}(-1)^k\\frac{z^{2k+1}}{(2k+1)^2}", "9dccee67a275d5db9b37e583941eb977": "\\sum_{n=N+1}^{\\infty}\\left | c_{n}\\right |^{2}\\left [1-\\cos\\left(E_{n}T\\right)\\right ] \\leq \\sum_{n=N+1}^{\\infty}\\left | c_{n}\\right |^{2}", "9dcd0540ba2074f91915a0e78a3cc412": "\\begin{align}\n y_{v} & =my_{u} \\\\ \n & =-\\frac{v-f}{f}\\left( a \\, \\frac{vf}{v-f}+b \\right) \\\\\n & =-\\left( av+\\frac{v}{f}b-b \\right) \\,,\n\\end{align}", "9dcd295a7fc8bc5e7110f7ceedb46edd": "x = \\frac{\\alpha}{1-\\alpha}", "9dcd3562525fa066a2f0ec8178fa0905": "Pr[T(G,x)=1]=0.1, \\quad Pr[T(G,x)=2]=0.9(0.1)=0.09, \\quad Pr[T(G,x)=3]=0.9^2(0.1) = 0.081,", "9dcd7abd46a7910a8e7478f642bd3906": "\\Xi_e = \\Phi_e = \\int \\frac {U_e}{T^2} d T", "9dce7e4f44edb2ed30114918806ff066": "\\begin{cases}\n\\ln \\gamma_1^{C, \\infty} = 1 - \\dfrac{r_1}{r_2} + \\ln \\dfrac{r_1}{r_2} - \\dfrac{z}{2} q_1 \\left( 1 - \\dfrac{r_1 q_2}{r_2 q_1} +\\ln \\dfrac{r_1 q_2}{r_2 q_1}\\right) \\\\\n\\ln \\gamma_2^{C,\\infty} = 1 - \\dfrac{r_2}{r_1} + \\ln \\dfrac{r_2}{r_1} - \\dfrac{z}{2} q_2 \\left( 1 - \\dfrac{r_2 q_1}{r_1 q_2} +\\ln \\dfrac{r_2 q_1}{r_1 q_2}\\right)\n\\end{cases}\n", "9dce822ea65b342ee46440c83bbe4225": " \\lambda_{low}=\\widehat{\\lambda} \\left (1-\\frac{1.96}{\\sqrt{n}} \\right ) ", "9dce912eb1b5ce04031d18cee2dc6940": "\nq(x,y,t)=e^{-i \\omega t} \\hat q (x,t) e^{-(y/b)^2} \\Re \\left\\{ e^{i (k x - \\omega t)} \\right\\} + \\mbox{Random Noise}\n", "9dcf6a7901afe37e63d8900dc8f0ec35": "\\sum_{i \\in A_1} a_i - \\sum_{i \\in A_2} a_i \\in W, \\quad A_1, A_2 \\supset A_0.", "9dcf86a033d7eab3e9e9b8b682a9e5f3": "\\mathbf{A^{\\rm T}C^{\\rm T}XB^{\\rm T}} + \\mathbf{CAXB}", "9dcfa0909d0c540b216f470fe5211e7f": "I(R)", "9dcfe68ff36902ea739d49e0598a7276": "\\widehat{i}=1^i0", "9dcfea86c7d0bdc501174bdacfe245d7": "\\phi(T)=1\\,\\!", "9dd03f194be7eea02ea5433908bb24ad": " N \\in \\mathbb{N} ", "9dd0d65aff640f08ac57b50388fc1447": "\\zeta = \\frac{c}{2 \\sqrt{mk}}", "9dd0fcfc961197918a137d9cf4745651": " 2~r^{-1}~\\cos\\theta \\,", "9dd119e68d272c66a722bedf7b7c7fe0": "\\mathrm{ad}_x\\colon \\mathfrak g \\to \\mathfrak g", "9dd158ae23fbe31365e0ecf34abb1737": "z_i, i = 1, \\dots, r (z_1 = z^*)", "9dd15c1b227c5780970d5ae798a3d282": "r_{cf_{i}}", "9dd1e8124931d930c4db10c8f53e6689": " a_{\\gamma} = \\sum^{A}_{n} \\frac{ U^{A}_{\\gamma n} - H_{\\gamma n}\\delta_{\\gamma n}}{E-H_{\\gamma\\gamma}} a_{n} = 0, \\gamma \\in B ", "9dd247037c30b5319659b363b3527f44": "\\{e_{i_1}, \\, ... \\, e_{i_k} \\}", "9dd2b5da9dd3d90e2cbd178d8595a291": "F( L ) = \\left[ \\frac{1}{L}\\sum_{j = 1}^L \\left( Y_j - j a - b \\right)^2 \\right]^{\\frac{1}{2}}.", "9dd2ca71934c4023452250ab442a4cc8": "\\!\\gamma_j", "9dd2f125efa2876bb5abf064451704d2": "F:\\mathbb{C}^2\\to\\mathbb{C}^2", "9dd3206c81b91dacb16bc7df1623205d": "\\begin{align}\n r &= (g^k \\bmod\\,p) \\bmod\\,q\\\\\n &= (g^{u1}y^{u2} \\bmod\\,p) \\bmod\\,q\\\\\n &= v\n\\end{align}", "9dd345dcd01c96cc04bb72353ce7239c": "k \\in \\mathbb{N}~~", "9dd39a76bd16b29f11e7351b06226e98": "x \\mapsto \\frac{x^5-2x^4+3x^3-2x+1}{x^2(x-1)^2}", "9dd420206d748b30fbb4774ebaf868bf": "p \\rightarrow p + \\frac{\\Lambda c^4}{8 \\pi G}", "9dd42260972c21f12f0e1f8a71b9f9f0": "p:V\\setminus\\{0\\}\\to P(V)", "9dd42f12b84bab5e308886bfc6c3e2cf": " S = \\{ (x_1,x_2) | x_2 = 0 \\} ", "9dd482e91d253cb26878ed1a9a793764": "z_k+w_k", "9dd4a131d848cee4c966345ba8498f59": "p_a,p_b \\in [0,1] \\text{ and } a,b \\in \\mathbb{Z} \\text{ with } a\\leq b", "9dd4d234256c2f8bdaa996778ef0bc2d": " T e_k = \\lambda_k e_k \\quad 1 \\leq k \\leq \\ell. \\quad ", "9dd4e461268c8034f5c8564e155c67a6": "x", "9dd52abe9ca6f76ea53abfef58d7b80a": " \\operatorname E(g(X)) = \\int_{-\\infty}^\\infty y f_{g(X)}(y)\\,dy, ", "9dd52d34190077cb988631eca650a406": "\\tau_{th}", "9dd5320e4bf8df97a9df98b6cf3d0e1a": "x^2-c^2t^2=c^4/\\alpha^2", "9dd5414c60f8c3223341512d3b2fbb29": "Z_{21} = {-Y_{21} \\over \\Delta_Y} \\,", "9dd5c7217fadf07c05759b21952d5a33": "\\Omega^{SO}(\\text{point})\\to \\mathbb{Z}[t]", "9dd5d4b58bda1851f04812b03ee60b11": "\n N_P \\le N_L + \\sum_L(C_{AB}),\\ v_{AB} \\in [-w,v_f] \\qquad (3)\n", "9dd5d86c9c677ee3c70f861fef502312": "\\vec \\nabla \\cdot \\vec F = 0,", "9dd5f938f8a88a4ecad127360c75d0ae": " \\partial(...)/\\partial \\theta = 0 ", "9dd6749273e0ed119424b48cab0fd4fe": "f_{il}", "9dd689eed291aadac17f0b260cd67ba6": "h_{A}(x)=x \\cdot a", "9dd6bcfc544898ae3542c505875105ed": "O((\\ln n)^{c\\,\\ln\\,\\ln\\,\\ln n}).", "9dd6c05179403906880d2c028164cea8": "f\\circ g", "9dd6edc688101ce6d77af93f09d83333": "\\boldsymbol\\mu = [\\mu_1, \\dots, \\mu_p]^T", "9dd6fca28e0d12da80872a8bfe1414b5": "z(n,n;s,t)=\\Theta(n^{2-1/t}), ", "9dd728971490ad6b84da6b4fa3c77111": "\\tfrac12(z+z^2)", "9dd7526ed17783a072361b75adf42e9a": "(R,P)", "9dd7602cec7166481e2b7fdf26e485da": "a^{-\\frac{1}{x+x^2}}\\,", "9dd7f1927c28581c904c5df1059b4e6f": " \\begin{matrix}\\frac{2\\pi}{a}\\end{matrix} ", "9dd805abdb73a5a6eaca2246d515e5c7": "\\tan x=\\frac{\\sin x}{\\cos x}=x\\frac{f_{3/2}(x/2)}{f_{1/2}(x/2)},", "9dd80803e8c86b5b9996fc2841f80437": "(s_\\lambda)_{\\lambda=0,1,2,\\dots}", "9dd81a722309cdd620b8bdc197007c6e": "p a_r = \\frac{n^2 a^2b^2}{r^2}\\varepsilon \\cos \\theta - p\\frac{n^2 a^2b^2}{r^3}\n= \\frac{n^2a^2b^2}{r^2}\\left(\\varepsilon \\cos \\theta - \\frac{p}{r}\\right). ", "9dd8286822dfeca1bace4e6aa22d82d1": "Y_{c1}(u) = Y_{c2}(u)", "9dd89a081b284bf2ad196987b96690ac": "{\\widehat{RR}}_3", "9dd8afdacfd8fa9c28a1a38e3df62456": " 2 \\cdot 11^2 + 11 ", "9dd8c23497beff20739cc80d60a4e90a": "\\left( \\frac{1}{x} \\frac{d}{dx} \\right)^m \\left[ x^\\alpha Z_{\\alpha} (x) \\right] = x^{\\alpha - m} Z_{\\alpha - m} (x)", "9dd8eefe27898e89007888683346eee5": "C_G(O_{p',p}(G)/O_{p'}(G)) \\subseteq O_{p',p}(G)", "9dd8f26deb1a2de9c8040e3f3fb54aeb": "\\{(\\Delta_n)_i\\}", "9dd9651db6c8f9efb6f682c307156345": " x \\log x = 0 \\!", "9dd97a17ce42cc5660b4f10dff734f96": "s_k^T", "9dd9e4e1cf294388dc7db86ae8fc0692": "\\Delta := (C_{11} - C_{12}) [(C_{11} + C_{12}) C_{33} -2 C_{13}C_{13}]", "9dd9fd1e31638ff1ea951d48470a6808": " X \\to Y^I", "9dda099b00d8a9d044655e7e1dc46e7b": " \\qquad h = \\frac{2\\gamma \\cos\\theta}{r\\rho g} ", "9dda5f30fb0ccf3ad95f977a8c0dd920": "\\scriptstyle 2kn \\, + \\, 1", "9dda815f1d8c5cbdc40c346d471fbff9": "f(t) = a + b t +\\int_0^\\infty (1-e^{-t x})\\mu(dx)", "9ddab2f5d01d211fb3bd722204218b02": "\\begin{align}\n&\\underset{x}{\\operatorname{minimize}}& & f(x) \\\\\n&\\operatorname{subject\\;to}\n& &g_i(x) \\leq 0, \\quad i = 1,\\dots,m \\\\\n&&&h_i(x) = 0, \\quad i = 1, \\dots,p.\n\\end{align}", "9ddac02e2712f02e544a4aa1c54d610d": "\\scriptstyle V_0", "9ddb6e64af91969aca4f698ff676a04e": "\\alpha\\to\\infty", "9ddb75b49ecc0e95f870c5078ee8fbea": " x \\le y", "9ddb9b4e2439717fe070c784844e6b47": "G(k)\\le k(3\\log k +11)", "9ddbb71f9eff1c3f065d022225d58375": "2=(1+i)(1-i)=i(1-i)^2", "9ddbe6a804113d948ffc403432f99634": "e^{\\sigma}", "9ddbfa7cf4b872447f061a262ae9bc9e": "\n \\langle q_{j+1} | \\exp\\left( {- {i \\over \\hbar } \\hat H \\delta t} \\right) |q_j\\rangle =\n\\exp\\left( {- {i \\over \\hbar } V \\left( q_j \\right) \\delta t} \\right)\n \\int { dp \\over 2\\pi } \\langle q_{j+1} | \\exp\\left( {- {i \\over \\hbar } { { p}^2 \\over 2m} \\delta t} \\right) |p\\rangle \\langle p |q_j\\rangle \n", "9ddc89c560571436dd8bdec94297919a": "\\mathbf{a}_j", "9ddcb5e579fa6f58ebb0adf5283ed966": "\\,u_{ij},v_{ij}", "9ddcd56ff8faef12d1d11d6c2e5cf33a": "\\int_{-\\infty}^{c(z)} f(x') \\, dx' = \\Phi(c(z)).", "9ddce1fb364e652de745675751ea68ff": "{y}={r} \\, \\sin\\theta", "9ddce5d27149dfeed1ba258080b1c721": "{}^{ U_{nj}, \\quad j \\not= i \\,) \\\\\n & = Prob(\\, \\beta z_{ni} + \\varepsilon_{ni} > \\beta z_{nj} + \\varepsilon_{nj}, \\; j \\neq i \\,) \\\\\n & = Prob(\\, \\varepsilon_{nj}- \\varepsilon_{ni} < \\beta z_{ni}- \\beta z_{nj}, \\; j \\neq i \\,)\n\\end{align}\n", "9dde7941b2ce728309ca5cb2abf11e0f": "{DE}_{8}", "9dde8d03f0aa5b7bf97cd104c0f05568": "_{(q'+p)''\\,}\\!", "9ddf106fc21af8ea19c887c174d3a766": "y=h(t)", "9ddf77ae30d15a9375a82c64939b33fd": " M^{p,q}_m(\\mathbb{R}^d) = \\left\\{ f\\in \\mathcal{S}'(\\mathbb{R}^d)\\ :\\ \\left(\\int_{\\mathbb{R}^d}\\left(\\int_{\\mathbb{R}^d} |V_gf(x,\\omega)|^p m(x,\\omega)^p dx\\right)^{q/p} d\\omega\\right)^{1/q} < \\infty\\right\\}.", "9ddf922fa94551c23d79658eff2dae46": "H_1(X;\\mathbb Q) \\simeq \\overline{H^2(X;\\mathbb Q)}", "9ddfc2f5eed02726fa94e8a702496487": "|A_i| = q_i", "9ddfd38d1980c7d8602b7771bfa0ed57": "\\tau=\\sqrt{\\frac{\\rho_c\\mu_c}{\\pi}}\\int\\limits_0^t\\frac{\\frac{\\partial u_p}{\\partial t'}}{\\sqrt{t-t'}} \\, dt', ", "9ddfebb2d6779d201bd4a8c13f835875": "\\tfrac{n}{4}\\cot{\\tfrac{\\pi}{n}}", "9de0522c1b75dc3a879ac4aa6e277093": "b^2 - b - a", "9de061a889f3e2433e4863b95513b1f5": "K_\\mathrm{fluid}^{(2)}", "9de0c7691800a9277877f7cd6d8d1c11": "\\Sigma^E_k=\\mathrm{NE}^{\\Sigma^P_{k-1}}", "9de0c9c7053e1e2a7cef2a140374a0bc": " Q_k = Q_T{C_k\\over\\sum_{k=1}^N C_k} ", "9de0d5cb61ec9a444d5c8154609ab8a4": "p(q) = {q^{\\alpha-1}(1-q)^{\\beta-1} \\over \\Beta(\\alpha,\\beta)}", "9de15c1f80b8fc5db848911054dacf4c": " (\\mathbf{AB})^\\dagger = \\mathbf{B}^\\dagger\\mathbf{A}^\\dagger ", "9de1b3399413002778ff3f58be67cf53": "\\int \\mathrm{d}^Dx \\sqrt{-g}\\, f(R, R^{\\mu\\nu}R_{\\mu\\nu}, R^{\\mu\\nu\\rho\\sigma}R_{\\mu\\nu\\rho\\sigma})", "9de1f916ea385ee57c9bc33d49981fda": " n ^ 2 - (n - 1) ^ 2 = 2 n - 1 ", "9de20f98825ce769bcd1e1a167ebad3c": "H^1_{\\text{et}}(G\\times_KK^s,\\mathbf{Z}_p)", "9de211daedaa6543c2feb1cfc88b5e4f": "p_{ij} = p^{kl} . \\,\\! ", "9de21581c3bec715cf0e06982c9028ee": "\\boldsymbol{\\xi}_x(\\boldsymbol{x},z,t)\\,", "9de244817729df5b613fa23428d9e9c1": "j,k\\in \\mathbb{Z}", "9de25a5692d7dd6ccc7c058208f22445": "d = 5.597661 + 29.5305888610 \\times N + (102.026 \\times 10^{-12})\\times N^2", "9de2a3f2cfe23ca7b613f11e3ed0def6": "\\left|\\frac{a_{n+1}}{a_n}\\right|\\ge1", "9de2a707126d1a1afd30a59496577c2e": "\nS_z \\equiv -i\\hbar\\Big( \\mathbf{e}_{x}\\otimes \\mathbf{e}_{y} - \\mathbf{e}_{y}\\otimes \\mathbf{e}_{x}\\Big)\n\\quad\\hbox{and cyclically}\\quad x\\rightarrow y \\rightarrow z \\rightarrow x.\n", "9de2fd385dc4785dec26f16b96659e8e": " \\nu^{*n} = \\underbrace{\\nu * \\cdots *\\nu}_{n \\text{ factors}}", "9de2ff71d3f6bbd2624b3bba47097292": "\\,\\hat O", "9de3158e21d0e1b46a04a19d46e063d3": "A \\rightarrow \\beta_1A^\\prime\\, |\\, \\ldots\\, |\\, \\beta_mA^\\prime", "9de33d15b19b691fa149668f420061b0": "p \\leftarrow p(x + lb), M \\leftarrow M(x + lb)", "9de357150c34543baf6563be4c3445e6": "A \\oplus B \\oplus C \\oplus D \\oplus E", "9de41f65cc9eb87ffaa12bcc84ef9729": "N = g_s BA/\\phi_0", "9de42cfa3cf2dcb482fdbe277a2103a7": "\\mathbb{E}[\\max\\{X,0\\}\\,|\\,\\mathcal G]=\\infty,", "9de444e29afd2f9a9c25b807338165f4": "Bn=\\frac{\\tau_0}{k}\\left(\\frac{H}{V}\\right)^n.", "9de44ba014bfdbd36d7207eeab60fdc3": "\n \\boldsymbol{S}^T\\cdot\\mathbf{n}_0 = \\boldsymbol{F}^{-1}\\cdot\\mathbf{t}_0\n", "9de45057ab40a7d8bd34be01a92f896e": "\\{ 1, \\dots, k \\}", "9de4c265ddd95f10e330c730a24a9cb5": "H_2^+ + H_2O \\rightarrow H_2O^+ + H_2", "9de4d98f02d51cdd32b24ca78a124eee": "w(E) \\in H^{\\le \\mathrm{rank} E}(X)", "9de4ecda584f07ac9fa6fcf70eea3a33": "M_{xz}", "9de561b818f019730f1c85f17ad6469f": "\\begin{align}\nM_O &=\\int_S (\\mathbf{r}\\times\\mathbf{T})dS + \\int_V (\\mathbf{r}\\times\\mathbf{F})dV=0 \\\\\n0 &= \\int_S\\varepsilon_{ijk}x_jT_k^{(n)}dS + \\int_V\\varepsilon_{ijk}x_jF_k dV \\\\\n\\end{align}\\,\\!", "9de610fcaad499e7a77150c9e54acde6": " \\boldsymbol{\\omega} = \\nabla \\times \\mathbf{u}", "9de6207f0763b2b1b58049282dc580ca": "_{s.6 \\,}\\!", "9de657d95b6da3acf559ec7e41dd711d": "S_n\\,\\!", "9de66d68e599af6c25f16d02ccd41830": "-0.25", "9de6f25fba16220ce66415b3f89bba3b": "\\mu \\left( F^{-1} (A) \\right) = \\mu (A).", "9de7050c201647b4ba4b61ea10dd60fe": "\\Sigma\\ :=\\ \\Sigma\\ +\\ m_i", "9de7bcb8e1474984df4563fdd5cf1d7c": "\\begin{pmatrix} N_{t+1} \\\\ D_{t+1} \\end{pmatrix} = \\begin{pmatrix} -B & -E \\\\ A & C \\end{pmatrix} \\begin{pmatrix} N_t \\\\ D_t \\end{pmatrix} = J \\begin{pmatrix}N_t \\\\ D_t \\end{pmatrix}.", "9de7c11495f4d8c481c973de5a969365": "A' = A\\,\\bmod\\,Q", "9de875cc073c9a3c6c396d441e65cb87": "\\bigcup_{k\\in\\mathbb{N}} \\mbox{NSPACE}(n^k)", "9de8d4a01091c1c26f6a99e3b601bb6f": "\\frac{dy}{dx}-y=x", "9de97462b5d6fb7c6cc38608be1b369e": " Z(D_n) = \\frac{1}{2} Z(C_n) +\n\\begin{cases}\n\\frac{1}{2} a_1 a_2^{(n-1)/2}, & n \\mbox{ odd, } \\\\ \n\\frac{1}{4}\n\\left( a_1^2 a_2^{(n-2)/2} + a_2^{n/2} \\right), & n \\mbox{ even.}\n\\end{cases}\n", "9de994b4767d587272ca4cbd3debe54d": " \\frac{1-F_{\\theta_1}(x)}{1-F_{\\theta_0}(x)} \\geq \\frac{f_{\\theta_1}}{f_{\\theta_0}}(x) ", "9de9b73cdcbdd514984e2f8e0a7884d8": "z > 1", "9de9e0b54ad59404997b8a1cd039598e": "E=E_i-E_f=R_\\mathrm{E} \\left( \\frac{1}{n_{f}^2} - \\frac{1}{n_{i}^2} \\right) \\,", "9de9e10c31c1d9b42a77ce0164dac038": "O(d^2)", "9dea20f8e233be69c306d8ac28a42970": "\n \\begin{align}\n D_x & = D_{11} = \\frac{2h^3 E_1}{3(1 - \\nu_{12}\\nu_{21})} \\\\\n D_y & = D_{22} = \\frac{2h^3 E_2}{3(1 - \\nu_{12}\\nu_{21})} \\\\\n D_{xy} & = D_{33} + \\tfrac{1}{2}(\\nu_{21} D_{11} + \\nu_{12} D_{22}) = D_{33} + \\nu_{21} D_{11} = \\frac{4h^3 G_{12}}{3} + \\frac{2h^3 \\nu_{21} E_1}{3(1 - \\nu_{12}\\nu_{21})} \\,.\n \\end{align}\n ", "9dea3beed68885482775b505e7948b8c": "[[a\\;\\|\\;M_1]_h\\;\\|\\;M]_m \\rightarrow [b\\;\\|\\;M_1]_h\\;\\|\\;[M]_m", "9dea979c73d06e6af0fbe90ad0ed716c": "m = \\frac{1000}{\\log(2)} \\log(1 + \\frac{f}{1000}) \\ ", "9deaf91904f438cf11bd81d248f3f9c0": "\n\\varphi_{\\mathrm{in}}(x)=\\int \\mathrm{d}^3k \\left\\{f_k(x) a_{\\mathrm{in}}(\\mathbf{k})+f^*_k(x) a^\\dagger_{\\mathrm{in}}(\\mathbf{k})\\right\\}\n", "9debbaef72f82c8cfd830db6001dc902": " f(X-\\mu)=f([X+b]-[\\mu+b])=f(X^{(1)}-\\mu^{(1)}) ", "9debbd60fdc4e975fdef161909fed285": "\\bar x_i= \\leftarrow x_i - \\gamma \\Bigg\\{ w_{internal} \\bigg[ \\alpha \\frac{\\partial ^2 x}{\\partial s^2} (\\bar v_i)+\\beta \\frac{\\partial ^4 x}{\\partial s^4} (\\bar v_i) \\bigg]", "9debe6f353c0f2735dff2880e293e304": "(O-\\lambda I ) |\\psi \\rangle = |h \\rangle; ", "9dec1c2ea52fa3a7cbb2b82d6cedee82": "0.\\dot{0}1234567\\dot{9}", "9dec1ea664d6c883aa3428e929f700b2": "H_p(x)=-\\int p(x) \\log p(x) \\, dx.", "9dec3976efaff3629bed37afb33a7410": "(D_{ij}-T_{ij})^2", "9dec466c31be450f78a168387cafa419": "\\Phi_t\\left(D\\right)", "9dec5b778fb77f11635a6576bd1d2ee1": "q_i \\circ q_j^{-1} : U_i \\cap U_j \\to Homeo(F)", "9dec5c7d46c10e5e6e899636e7e77a1c": "\nX^{\\rm VV} = X ^{BS} + \\underbrace{\\frac{\\textrm{X}_{vanna}}{\\textrm{RR}_{vanna}}}_{w_{RR}} {RR}_{cost} +\n\\underbrace{\\frac{\\textrm{X}_{volga}}{\\textrm{BF}_{volga}}}_{w_{\nBF}} {BF}_{cost} \n", "9dec7cafd27a1baa8a94c13c27520206": "\\varphi : U \\rightarrow {\\mathbb R}", "9dec883554d3c6fe5ae4f31bcafc5d3f": "a_{Q}(t)", "9ded306e2ccf62140dbc436d2aa303fd": "\\sum_{k=1}^\\infty (\\zeta(2k+1) -1) = \\frac{1}{4}", "9ded54347c771d3a1745140165f4edb6": "\\vec{P} = A \\times (B \\vec{r}) - C \\vec{r} = (A \\times B)\\vec{r} - C\\vec{r} = (A \\times B - C)\\vec{r} = \\vec{0}", "9ded7825070b255e7bc092cdc2c8e98a": "a_{n}", "9dee188e86364d1e7af7f49c0cd6ea55": "V = V_2", "9dee1ced76ffefc50128435275315efe": "\\mathbf e =\\frac{1}{2}\\left(\\nabla_{\\mathbf x}\\mathbf u + (\\nabla_{\\mathbf x}\\mathbf u)^T - \\nabla_{\\mathbf x}\\mathbf u(\\nabla_{\\mathbf x}\\mathbf u)^T\\right)\\approx \\frac{1}{2}\\left(\\nabla_{\\mathbf x}\\mathbf u + (\\nabla_{\\mathbf x}\\mathbf u)^T\\right)\\,\\!", "9dee7988d6ee16639e1698f2be7f3f24": "\\omega: S \\times \\Sigma \\rightarrow \\Gamma", "9deeeda1a1d047d325583d9cb1f245fe": "[v][w]", "9def11440847c0b2a4e0b93477b97980": "\n\\varphi(t;\\alpha, 0, \\lambda,\\mu) = \n[1+\\lambda^{\\alpha}|t|^{\\alpha} - i \\mu t]^{-1}\n", "9def4eb7e3b09ef5c43e757b5511dfae": "\\displaystyle\\omega", "9def716ab076eb6866a5865db1586478": "\\sin(x \\pm y) = \\sin(x) \\cos(y) \\pm \\cos(x) \\sin(y)\\,", "9def726f08f8d5d174d0ca7b12ce20ff": " \\Delta f = d^* d f\\,", "9defa1d3c0c4df450f123a84cd96ea10": "Ax-b=0", "9defa32a9dc485e5a04de7d3c1a263be": " B = \\; - \\mathrm{dln} (i) / \\mathrm{d} (1/V) - \\kappa (1/V). ..........(51) ", "9defb55af70fb3ee6441ec7f4cc3526f": " y = b_0 + b_1t\\,\\!", "9defbaf04d2de6423ba7ae8e5d342bc2": " \\sigma (t)=E{D_t^\\alpha} \\varepsilon(t), \\quad 0<\\alpha<1. ", "9defbee339289f469bca237660ada733": "prog : I_{static} \\times I_{dynamic} \\to O", "9defec5ffd7906bd7f7c304aeb2fdcf4": " F = \\int \\left(\\rho \\mathbf{V} \\cdot d \\mathbf{A} \\right) \\mathbf{V} \\cdot \\mathbf{f} +\\int pd \\mathbf{A} \\cdot \\mathbf{f}.", "9deffa33e3f0b2a2baba9379e54e74a9": " F_\\theta = \\mathbf{F}_A \\cdot \\frac{\\partial\\mathbf{v}_A}{\\partial\\dot{\\theta}} - \\mathbf{F}_B \\cdot \\frac{\\partial\\mathbf{v}_B}{\\partial\\dot{\\theta}}= a(\\mathbf{F}_A \\cdot \\mathbf{e}_A^\\perp) - b(\\mathbf{F}_B \\cdot \\mathbf{e}_B^\\perp) = a F_A - b F_B ,", "9df0479d09f738e65a5eedc8fc9bd631": "\\Sigma_{k}^{\\rm P}", "9df0d1a135068a568f9dacfcefd5b120": " G_{j-1}", "9df0d995ad291f5fa39f831379b14e69": "\n \\log_{10}[\\log_{10}[\\nu + \\lambda + f(\\nu)]] = A - B\\,\\log_{10}(T)\n ", "9df19843f71c9e8e6124c235370ae264": "{\\color{white}^\\big|}r^{(k)}=\\,r^{\\overline k}=\\,r\\cdot(r+1)\\cdot(r+2)\\cdot(r+3)\\cdots(r+k-1)", "9df1a3e3b9e78d51bd216b03b5cdf713": "n\\le 2N", "9df1fdd9ee720a2d43248e2b5fa3ff02": "D_y f(x) = \\max_{z \\in Z_0(x)} \\phi'(x,z;y),", "9df291958980ff4bfba580c070df830f": "b_i.", "9df2b5e9b923768142e176461f2cf046": "{\\Delta h} = h_2-h_1 ", "9df2c93e3dcfc48391142111d4971af5": "\\tau > 3", "9df305f84f9cffa5decac285c84d62b0": "B(x,\\mu)", "9df31c3df9bc1001dbc2ac66e306cb20": "|vxy| \\le p", "9df321c3bf16004e6b14ddd0ed240a73": "\\nu Z.\\phi", "9df326edcd3302b2fc2016feae941838": "f:\\left\\{1,2,...,m\\right\\}\\to\\left\\{0,1,...,n\\right\\}", "9df373ab079592a36b68e29b3556d0ca": " \\scriptstyle x_{nj}, \\; j \\neq i ", "9df3ab31495d3d0f538df37b846cc79a": "a(n) = \\begin{cases}\n \\frac{2^{n+1}-2}{3}, & \\text{when }n\\text{ is even,}\\\\\n \\frac{2^{n+1}-1}{3}, & \\text{when }n\\text{ is odd.}\\end{cases}", "9df3b5e2ba8109f9aa33ca500392bd1d": "\\displaystyle{[H_m,F_n]=-2F_{m+n},}", "9df3bcfb80092914947f29b81259f3da": "E_{i} = \\sqrt{p_{i}^2 + m_{i}^2 }\\simeq p_{i} + \\frac{m_{i}^2}{2 p_{i}} \\approx E + \\frac{m_{i}^2}{2 E},", "9df41a20d7d08b710b65c15dba89c9dd": "\\frac{dP_{rad}}{dr} = -\\frac {\\kappa\\rho}{c}\\frac{L}{4\\pi r^2}", "9df41db03429d7c36575e3c84174068b": "\\scriptstyle{y = \\sin(t)}", "9df43c436cc71185de6ff3b8924231ec": "WTS(O_j) = \\mathrm{old}WTS(O_j)", "9df43f4536cee029c172847dc969139a": "\\beta' = (1+1 / \\beta)^{-1}", "9df45c53771700251df8feaef9be8487": "F({\\bold u}^{(1)},\\dots,{\\bold u}^{(d)})=\\frac{1}{d!}\\frac{\\partial}{\\partial\\lambda_1}\\dots\\frac{\\partial}{\\partial\\lambda_d}f(\\lambda_1{\\bold u}^{(1)}+\\dots+\\lambda_d{\\bold u}^{(d)})|_{\\lambda=0}.", "9df4d88731aed491a3fec77c6981990b": "(x^*,y^*,p)", "9df55e42c8787fbb5276e9763d24a471": " \\,Y ", "9df5a7cab97a9eeeca3dd2e97750deae": "\\|Du_\\lambda\\|_{L^p(R^n)}^p=\\int_{R^n}|\\lambda Du(\\lambda x)|^pdx=\\frac{\\lambda^p}{\\lambda^n}\\int_{R^n}|Du(y)|^pdy=\\lambda^{p-n}\\|Du\\|_{L^p(R^n)}^p", "9df5bb120c348ba24394b054cfa30c38": "\\Delta = \\omega_{light} - \\omega_{transition}", "9df5f1b7b76a8a34dea7c5415b026f60": "f_c^n(z_0)", "9df6416e07053649867c7decfc3f69d9": "x[n] = \\sum_{k} x_k[n-kL],\\,", "9df65489f0fc32bf29c77bee05bf6564": " A, ", "9df669bcceddb09ba37beec9d27537b9": "\\mathbf{\\gamma}", "9df6741fbd1c1cd3aaa78997faaab399": "N \\to \\infty", "9df6d301a30d1cbd9a93dba683b03b42": "\\Phi = \\frac{T^{3/2}\\Lambda^3}{g}", "9df6fc49ddb874011c0991d4de431fa2": "\\int_0^\\frac{\\pi}{2}\\sin^n{x}\\,dx = \\int_0^\\frac{\\pi}{2} \\cos^n{x}\\,dx = \\frac{1 \\cdot 3 \\cdot 5 \\cdot \\cdots \\cdot (n-1)}{2 \\cdot 4 \\cdot 6 \\cdot \\cdots \\cdot n}\\frac{\\pi}{2}", "9df712f167fa9b65744f8a6735d0c4b3": "V_\\mathbf{E} = - \\int_C \\mathbf{E} \\cdot \\mathrm{d} \\boldsymbol{\\ell} \\,", "9df71b24dbcb6a22b60cd56b6ac4e0da": "E_1 > E_2 > E_3", "9df788a0312a7eeb34c6596d147ed57e": "H(\\mathbf{x}) = \\mathcal{F}((1+i) \\mathbf{x}^*) / \\sqrt{2N}", "9df83164331ffba720dddaeac388a2ea": "f_x = \\frac{df}{dx} ", "9df8a26b94c23f0d0f699f9e4245471d": "X\\subseteq \\operatorname{cl}(X)", "9df8cab02b96cb53029ccb672a517633": "\\mu (A) := \\lambda (A \\cap (0, 1))", "9df8cdb01e279e08fd4cba9826ddb70a": "\\mathbf{v}_i\\mathbf{v}_i^{\\mathrm{T}}", "9df8f6f4879a4803b1b8ae17d6ff8051": "\n d\\boldsymbol{\\varepsilon} = d\\boldsymbol{\\varepsilon}_e + d\\boldsymbol{\\varepsilon}_p \\,.\n ", "9df95fbdf96537841f947f3bae44fda4": "\\left ( 1 - \\frac{\\dot{R}}{c} \\right ) R \\ddot{R} + \\frac{3}{2} \\dot{R^2} \\left ( 1 - \\frac{\\dot{R}}{3c} \\right ) = \\left ( 1 + \\frac{\\dot{R}}{c} \\right ) \\frac{1}{\\rho_l} \\left [ p_B(R,t) - p_A\\left(t + \\frac{R}{c}\\right) - P_\\infty \\right ] + \\frac{R dp_B(R,t)}{\\rho_l c dt}", "9df976ebb41ebce8fcbabb714a2dc22b": "\\kappa^{2}", "9df981a9e869e03b667479bbcb5e7bef": " \\mathbf{L} ", "9df9825cc9d63ab71f0f40fb16e94065": "\\omega(X)", "9df9efad4d89903b7f9a07dc427669da": " \\Delta(\\mathbf{c}_{k_1}, \\mathbf{c}_{k_2}) \\geq q - k + 1", "9dfa35f605b6a5271b4930badc7ef646": "f(I)\\subset I", "9dfa92bf06dc2fe66ffcb800a69fd849": "t_{d} = D^{-0.2}*(\\frac{S_{e}}{C_{te}})^{0.4}", "9dfae06603cc695a6802a142c211be1e": "Z_+ = Z_1 +\\ldots+ Z_n", "9dfae3aca05a31122a3cb3e3d41c312f": "\\displaystyle H_{ee}=\\frac{1}{2}\\sum_{n,m,\\sigma}\\langle n_1 m_1, n_2 m_2|\\frac{e^2}{|r_1-r_2|}|n_3 m_3, n_4 m_4\\rangle c^\\dagger_{n_1 m_1 \\sigma_1}c^\\dagger_{n_2 m_2 \\sigma_2}c_{n_4 m_4 \\sigma_2} c_{n_3 m_3 \\sigma_1}", "9dfaf3d5eca0ee46fc0cc629e4913935": "F(w)=\\frac{1}{w} \\int_0^\\infty f \\left(\\frac{t}{w}\\right) \\, d\\alpha(t).", "9dfb18641769567e268e23a9d6a66abb": "\n\\Phi(\\varphi)\\ =a\\ \\cos m\\varphi\\ +\\ b\\ \\sin m\\varphi\n", "9dfb6a57fb1b51b286a62711c9de505b": "v_{\\text{out}}", "9dfba0c9343def9a9c640b923bcb74be": "L^2|E_n, l, m\\rangle=l(l+1)\\hbar^2|E_n, l, m\\rangle", "9dfbf2ef345a194d3cd5ac987a2ac089": "\\mu\\,_0,", "9dfc66bb1d0ca79ca18d3c1ebba53685": "\\sqrt{(F)}=\\cap_{i=1}^{e}\\sqrt{\\mathrm{sat}(T_i)}", "9dfcaeb987734a6956b38c24b5ee162d": " D_1(x,\\alpha) = x \\,", "9dfcb71dad3ced71f805b7bb4cd9dfc0": "{2}", "9dfce81fb65917b51eabb10e3b2f9930": "\\widehat{D}\\,\\!", "9dfcea9294bf2748a80b7b47232befe9": "M=A^{T}B", "9dfd055ef1683b053f1b5bf9ed6dbbb4": "\\hbar ", "9dfde73e3591074239af54c5fade3370": "\\vec v_{M|T}", "9dfe0c8c386dcb5cf5615914ba268968": " S_x = {\\hbar \\over 2} \\sigma_x,\\quad S_y = {\\hbar \\over 2} \\sigma_y,\\quad S_z = {\\hbar \\over 2} \\sigma_z.", "9dfe12a08293ee7b20b1004964e0407b": "B_3 (N, M, r, \\mu) = \\frac{\\left\\langle\\sigma_y^2(N, M, T, \\tau)\\right\\rangle}{\\left\\langle\\sigma_y^2(N, T, \\tau)\\right\\rangle}", "9dfe1ab39808c5344bf3825e577a57a5": "\\begin{bmatrix}\n \\cos a \\cos A\\\\\n \\cos a \\sin A\\\\\n \\sin a\n\\end{bmatrix} = \\begin{bmatrix}\n \\sin\\phi_o & 0 & -\\cos\\phi_o \\\\\n 0 & 1 & 0\\\\\n \\cos\\phi_o & 0 & \\sin\\phi_o\n\\end{bmatrix}\\begin{bmatrix}\n \\cos\\delta\\cos h\\\\\n \\cos\\delta\\sin h\\\\\n \\sin\\delta\n\\end{bmatrix}", "9dfe3dce76672f1bc5d66caa0fb78285": "\\int_0^\\infty d\\mu^2\\rho(\\mu^2)=1", "9dfe42829c6746942548c8afa353b3f1": "\\left(\\sqrt{2}/2,\\sqrt{2}/2\\right)", "9dfe4a6ed6e39578c74894611558852c": "f(a+h) \\approx f(a) + f'(a)h.", "9dff192b8b507d23fb8e072c25b74c5b": "\\Delta C/\\Delta Y", "9dff415d3a6c0044a0a2e0539241b25b": "-m\\frac{\\mathbf{e_r}}{M_{\\mathrm{Pl}}^2 r^2} = -m\\frac{\\mathbf{e_r}}{M_{\\mathrm{Pl}_{3+1+\\delta}}^{2+\\delta}r^2 n^{\\delta}}", "9dff4764826b2110f75f6675bd356dec": "W^{A}", "9dff962aa8ba345ee3f1b4edc46a9485": " \\operatorname{E}[X] = \\int_0^\\infty (-x)(-f_X(x))\\;\\mathrm{d}x = \\left[ -x(1 - F(x)) \\right]_0^\\infty + \\int_0^\\infty (1 - F(x))\\;\\mathrm{d}x ", "9dffef902d81b22b9c3ec6e25bc4b70e": "c(u,v)=1", "9dffefd61f41d3b74e7b6f2ec26e69d4": "B \\in \\mathbb{F}^{p \\times n}", "9e002f9737cf681e9f3442b166770f8e": "\\lfloor\\sigma(G)\\rfloor", "9e00329622e8876db01bfb10ff394f94": "e_\\alpha = \\frac{m}{(m, r_\\alpha)}", "9e007dc3f4aaa456bd7173014346daf7": "\n \\begin{align}\n & bD \\frac{\\mathrm{d}^4w_x}{\\mathrm{d}x^4} = 0 \\\\\n & \\frac{b^3D}{12}\\,\\frac{\\mathrm{d}^4\\theta_x}{\\mathrm{d}x^4} - 2bD(1-\\nu)\\cfrac{d^2 \\theta_x}{d x^2} = 0\n \\end{align}\n", "9e00a0206ce5238db074fab9ee5b07a2": "\nE[X^{n}]=(-1)^{n}n!\\boldsymbol{\\alpha}{S}^{-n}\\mathbf{1}.\n", "9e0115698508185dae6fd7ad8a8930f5": "\\lim_{x \\to 0} \\frac{\\sin (2x)}{\\sin (3x)} =\n\\lim_{x \\to 0} \\frac{2 \\cos (2x)}{3 \\cos (3x)} =\n\\frac{2 \\sdot 1}{3 \\sdot 1} =\n\\frac{2}{3}.", "9e014b7410f46450ce95c4bc297a6292": "A=\\left[ \\begin{array}{cccccc}\n1 & 0 & -3 & 0 & 2 & -8 \\\\\n0 & 1 & 5 & 0 & -1 & 4 \\\\\n0 & 0 & 0 & 1 & 7 & -9 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\end{array} \\,\\right]. ", "9e0177dd3085265b1d6ad7c7a8d27cc7": "\\Delta E\\, =\\, \\Delta \\left( p\\, +\\, {\\scriptstyle\\frac12}\\, \\rho\\, v^2 \\right).", "9e019a112facb0bd52ee36255c5f2548": "B(E(m),(p+\\epsilon)n)", "9e0212ecfd4234ce348db9f461238198": "[\\mathbf L \\cdot \\mathbf L, \\mathbf L \\cdot \\mathbf n] = [L^2, \\mathbf L \\cdot \\mathbf n] = 0", "9e02b6e5141c753a09c2760961913d66": "S_{up} = S \\cdot u", "9e02c77103a940e88da553d2591019d1": "\\Delta{v} = {v}_1 - {v}_0 = \\int^{t_1}_{t_0} {a} \\, dt", "9e0334e9b559a3af48b8988458cf6b0a": "f_{IJ}^K", "9e03c07357e7ce0cc6d309859c3e4371": "\\exp({s t})", "9e03c82a982baa65464a8f091b583062": "\\frac{-\\$10,000}{\\$50,000} = -0.20 = -20\\%", "9e04613e86e330548491e6d31bc015f4": "\n C'_1 = \\frac{C_1 \\,C_2}{C_2 + (T'_0-T_0)} \\qquad {\\rm and} \\qquad C'_2 = C_2 + (T'_0-T_0) \\,.\n", "9e046235998d301fd9894e8fc13d0b68": "\\frac{1}{v(E)}\\frac{\\partial\\psi(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)}{\\partial t}+\\mathbf{\\hat{\\Omega}}\\cdot\\nabla\\psi(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)+\\Sigma_t(\\mathbf{r},E,t)\\,\\psi(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)=\\quad", "9e048761fd9f337b6b406f336d30b1ba": "\\begin{cases}\n g\\left( p \\right)=0 & \\text{means point satisfies constraint} \\\\\n \\nabla f\\left( p \\right)-\\lambda \\, \\nabla g\\left( p \\right) = 0 & \\text{means point is a stationary point}.\n\\end{cases}\n", "9e0542319491d49bc3607ff3991dcdf5": "F[", "9e05766cf54f8084581d932a8f0c3b0c": "\\bar{P}_{e} = 0.143^2 + 0.200^2 + 0.279^2 + 0.150^2 + 0.229^2 = 0.213", "9e057d2cbf4a57d06a8703a684e95b1c": "\np(\\mathbf{x}; A)\n=\n\\prod_{n=0}^{N-1} p(x[n]; A)\n=\n\\frac{1}{\\left(\\sigma \\sqrt{2\\pi}\\right)^N}\n\\exp\\left(- \\frac{1}{2 \\sigma^2} \\sum_{n=0}^{N-1}(x[n] - A)^2 \\right)\n", "9e058cee178eafde2489df18bcfc4cf9": " \\int_{\\overline{\\mathcal{M}}_{g,n}} \\frac{c(E^*)}{(1-k_1\\psi_1) \\cdots (1-k_n \\psi_n)} ", "9e0632b8bda33f07ebfaecd68f71f989": "D \\gg r_0", "9e0637587c0d4c0e36ee9c4249df45a4": "\n\\theta\\circ(\\theta_1\\circ(\\theta_{1,1},\\ldots,\\theta_{1,k_1}),\\ldots,\\theta_n\\circ(\\theta_{n,1},\\ldots,\\theta_{n,k_n}))\n=\n(\\theta\\circ(\\theta_1,\\ldots,\\theta_n))\\circ(\\theta_{1,1},\\ldots,\\theta_{1,k_1},\\ldots,\\theta_{n,1},\\ldots,\\theta_{n,k_n})\n", "9e06745915567c7eb4a29446335ad609": "K_i(R) \\simeq K_{i+1}(\\Sigma R)", "9e06d6706c3c9571cdbdc639384230f5": "\\ \\tau_{D,GR}", "9e07830575866420a44e771ae2ab1825": "\\sigma^2_W/\\mu_W", "9e07b2d79733fc30d3f4b915a8128229": "0! = 1, \\ ", "9e07c7e88f8f7ea148e6a3a50820e47a": "{}^\\dagger \\!\\,", "9e08177229a05f4e9d2f3b682543226f": "f:\\kappa^{+}\\to\\kappa\\,", "9e083b945b882d12ff638565cd3a1f7b": "X\\left( t \\right)", "9e085e7ac9e41da0f99c23ec72006758": "\\frac1x + \\frac1y = \\frac{x+y}{xy}", "9e08a485c0847a3a13f4738e36517efe": "x \\ne p", "9e08e3a1dd1eb59f21d83903c8577cb4": " \\textstyle g_{j+1}[n] = P(f_j^o[n];f_j^e[n]) ", "9e08ede4e4cafb63fff9fe40af2a15fd": " \\alpha = 1.187452351126501\\ldots\\, ", "9e09751071efeec32019cb5bcef4b5df": "\\epsilon_{xyz}", "9e0a09d641fc0628df598915134f5ad9": "\\pi({\\mathbb B})", "9e0a5538d736d2735d554679c7e28613": "\n a\\uparrow^n b=\n \\left\\{\n \\begin{matrix}\n a\\times b, & \\mbox{if }n=0; \\\\\n a^b, & \\mbox{if }n=1; \\\\\n 1, & \\mbox{if }b=0; \\\\\n a\\uparrow^{n-1}(a\\uparrow^n(b-1)), & \\mbox{otherwise}\n \\end{matrix}\n \\right.\n ", "9e0ac0bdb334f998d9c1e31a417bcfa3": "2 \\mu", "9e0b7472a0e11cb24b8edc58f0634db5": "D^+", "9e0c17aa90d38046134b0b3bc73ffe7c": " \\mathcal{A}(1^+\\cdots i^- \\cdots j^- \\cdots n^+) \n= i(-g)^{n-2} \\frac{\\langle i \\; j\\rangle^4}{\\langle 1\\;2 \\rangle \\langle 2\\;3 \\rangle \\cdots \\langle (n-1)\\;n\\rangle \\langle n \\; 1 \\rangle}", "9e0c3150044f204d064638059492d31e": "x_0=(1+M_0\\textstyle \\sum_{j=1}^{n}{b_j})^{-1}=1-\\sum_{j=1}^{n}{x_j}.", "9e0c6e0437361aaacd4d5c3374d054bb": "\\kappa_P(s) = \\pm\\sqrt{\\kappa(s)^2+\\tau(s)^2}.", "9e0c85ab9cfb30d543ff6df25c46f452": "\n\\left(\\frac{\\alpha}{\\mathfrak{p} }\\right)_n \n\\left(\\frac{\\beta}{\\mathfrak{p} }\\right)_n \n=\n\\left(\\frac{\\alpha\\beta}{\\mathfrak{p} }\\right)_n\n", "9e0ccbd768219081003a3f52c3de3433": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=30,\\cdots ,35", "9e0cd75d0b706076a5d74590facc2dea": "P_{in} = V^2 \\cdot Re \\left\\lbrace \\frac{1}{Z_{in}} \\right\\rbrace", "9e0ce74a10a17780953be739b5c9164b": "\n\\begin{align}\n\\delta S[g]&= \\int {1 \\over 2\\kappa} \\sqrt{-g}\\delta g^{\\mu\\nu} \\left(F(R)R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu} f(R)+[g_{\\mu\\nu}\\Box -\\nabla_\\mu \\nabla_\\nu]F(R) \\right)\\, \\mathrm{d}^4x.\n\\end{align}\n", "9e0cf5c036465d9fc02b167edd3b74b3": "W_0^2(r)^2:=1-\\sum_{j=0}^{\\infty}W(2^{-j}r)^2", "9e0d01d579431933523581ace2700f46": "p_i=p", "9e0d1902f69776e9bdaa8957600f14ca": "0 \\le r \\le s-1\n", "9e0d2ce46533eb0ff4615bacafd9d7f8": " L_\\text{cen} \\, ", "9e0d5b263bc3155ef2e40708a2566dc7": "\nN_i = \\frac{g_i}{e^{\\varepsilon_i/kT}/z} \n", "9e0db402374af8567d8e371953262be7": "\\rho(\\mathbf{r},\\mathbf{r}^\\prime) = \\sum_{i\\alpha j\\beta} \\phi_{i\\alpha}(\\mathbf{r}) K_{i\\alpha j\\beta} \\phi_{j\\beta}(\\mathbf{r}^\\prime)", "9e0e0bc4769116e4232837cbdfd34274": "h(y)/h(x) = \\frac{p(D|y)p(y)}{p(D|x)p(x)}\\ ", "9e0e1ab7c82f0693238905ffafb9d783": "\\frac{\\partial R}{\\partial x}=p\\left( 1+\\frac{1}{\\epsilon _{x,p}}\\right)", "9e0e567674694e13bf92e38f2f4a471a": "S_{y_B}", "9e0e8c3736e45b40d36e53bc2da83f36": " \\frac{dS}{dt} + \\frac{dI}{dt} + \\frac{dR}{dt} = 0 ", "9e0e8ea9f98cb142068e91dd5a2a17bb": "g_{i+1}", "9e0ee4bb3f0a8f63adb0cacea33e27ac": "z = (z_1, \\dots, z_m) \\in \\mathcal{H}^m", "9e102ef9f940d64c6dc07b6b475e5a6a": "p(\\ell)", "9e1039cf2fba47e7ec659822d1e2de06": "f: V\\to V", "9e103a3f359740aa0109fc94aab4e515": "\n\\begin{align}\nr&=\\frac{h}{1-\\frac{\\cos{\\theta}}{\\mu}}\\\\\nh&=r_0\\left(1-\\frac{\\cos{\\theta_0}}{\\mu}\\right)\n\\end{align}\n", "9e103ae876b224180dfdee657c65444f": "\n\\lbrack \\mathcal{S}_{1},\\mathcal{S}_{2}]\\psi =0,\n", "9e104ad4201606d6260102f969f0225b": "s = \\int_0^t \\sqrt { x'^2 + y'^2 + z'^2 }\\, dt", "9e1068b28f0815ac9f858e13c9a7c78e": "\n\\begin{align}\n\\mbox{slope} =\\frac{K_m}{v_\\max}\\times \\left(1+\\frac{[I]}{K_I}\\right)\n\\end{align}\n", "9e1089d021f8ecf7da2f55eac30509dd": "X = \\{a + b\\sqrt{3} | a, b \\in \\mathbb{Z}_q\\}", "9e10df4b36a9122615a7517d58ba847c": "\\gcd(p_1, p_2, \\dots , p_n) = \\gcd( p_1, \\gcd(p_2, \\dots , p_n)).", "9e112af3ceb0c95f480be457e3b05173": "\\phi(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}", "9e11407fb19cd6c8f8a3b2e3ee90a198": " C_{\\alpha \\beta} =\\begin{bmatrix}\n K+4 \\mu\\ /3 & K-2 \\mu\\ /3 & K-2 \\mu\\ /3 & 0 & 0 & 0 \\\\\n K-2 \\mu\\ /3 & K+4 \\mu\\ /3 & K-2 \\mu\\ /3 & 0 & 0 & 0 \\\\\n K-2 \\mu\\ /3 & K-2 \\mu\\ /3 & K+4 \\mu\\ /3 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\mu\\ & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\mu\\ & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\mu\\ \n\\end{bmatrix}.\n\\,\\!", "9e116ba94f65d5d094fcdef7e76cb0f4": "S^3 = \\left\\{q\\in\\mathbb{H} : ||q|| = 1\\right\\}.", "9e117330276be2906cb9cc0d7c5319e6": "\\alpha\\simeq 3/2", "9e11a09379e2725c8f59482d60b4bebc": "\nV_c = -2k_e QL\n \\ln \\left( r_e \\right) .\n", "9e11ac4926455519de4548014f66ee24": "v v^T", "9e11af9193ea84fa45055ceb4bfd7c5e": "({\\and}R), ({\\or}L)", "9e127e8ee6c275665818ea99e94e014c": "\n\\left(C-1\\right)\\sigma^*_i = \\text{Gain}_i(\\sigma^*,\\cdot)\n\\Rightarrow\n\\sigma^*_i = \\left(\\frac{1}{C-1}\\right)\\text{Gain}_i(\\sigma^*,\\cdot).\n", "9e12e69e05864f31c27ee7b22a2adaa9": "R=-4\\cdot8 \\equiv 7 \\pmod {13} ", "9e130e17c76e939c19ac31b41260fa9b": "S_q({p_i}) = {1 \\over q - 1} \\left( 1 - \\sum_i p_i^q \\right),", "9e1318d08dd6bd8627b52d4bf9e8b152": " \\lambda = \\dfrac{\\mu}{\\rho} \\sqrt{\\dfrac{\\pi}{2 R \\theta}} = \\dfrac{M}{Re} \\sqrt{\\dfrac{k \\pi}{2}} \\equiv ", "9e132670419049f68df50ae787ba7d48": "\\oint_{C} (L\\, dx + M\\, dy) = \\oint_{C} (M, -L) \\cdot (dy, -dx) = \\oint_{C} (M, -L) \\cdot \\mathbf{\\hat n}\\,ds.", "9e133c22138ded040c1c5a769acb9eb4": "N(a + b\\,\\omega) = a^2 - a b + b^2. \\,\\!", "9e13a17d04bd6f59e8ce7f20ea0d3a41": "\np_{k} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial L}{\\partial\\dot{q}_{k}}\n", "9e13cd9e16506b5a07c2428ba57a57f0": "v_{xc}^{\\mathrm{LDA}}(\\mathbf{r}) = \\frac{\\delta E^{\\mathrm{LDA}}}{\\delta\\rho(\\mathbf{r})} = \\epsilon_{xc}(\\rho(\\mathbf{r})) + \\rho(\\mathbf{r})\\frac{\\partial \\epsilon_{xc}(\\rho(\\mathbf{r}))}{\\partial\\rho(\\mathbf{r})}\\ .", "9e13f50927144b61714322f6ac28e604": "-\\nabla \\times \\mathbf{E} = \\frac{1}{c}\\frac{\\partial \\mathbf{B}} {\\partial t} + \\frac{4 \\pi}{c}\\mathbf{j}_{\\mathrm m}", "9e13fc6798e7dcb2b3510e2b1de6dc28": "\\mathbf{E} \\left[ e^{\\lambda X} \\right] \\leq \\exp \\left( \\frac{\\lambda^2 (b - a)^2}{8} \\right).", "9e142e0355b11936820c218adfc3eca5": "|f(y) - f(x)| \\le |\\nabla f ((1- c)x + cy)| \\, |y - x|.", "9e14b2c97586a4c21e21babc4d866e42": "d(x,y)< D_k", "9e14d3a967f4a59f437d5b817e56b695": "\\ M_B \\approx M_V \\approx -19.3 \\pm 0.3 \\,.", "9e1549efe9018210b9e325722b01a9eb": "\nE_{\\text{MP0}}\\equiv E_{\\text{HF}} = \\langle\\Phi_0|H|\\Phi_0\\rangle.\n", "9e15b92c53da6f9f5ba016f63b8aea96": "\\frac{\\partial f}{\\partial a}=b, \\frac{\\partial f}{\\partial b}=a", "9e15e7263e5b0c6edee6b1f7e8463215": " {\\overline{\\mathcal{M}}}_{g,n} ", "9e15f010f213c4a0ba0a28de28ebcf7e": "\\mathrm{MSE}(h) = E_\\mu \\|h(x)-\\mu\\|^2.", "9e163206f8327f2c48b3942951ec35f0": "P(spike|\\mathbf{s}^K) = P(spike)f(\\mathbf{s}^K)", "9e167fd486b1f7ca0b00505e6f1fc290": " \\mu={G}(m_1 + m_2)\\,\\!", "9e16dd3ae225620d90d1d1e0be0019e7": "\\{X,Y\\}", "9e16ebb8bd3b616a5c307af9b2e7094f": "\\begin{smallmatrix}\\sqrt{79.4^2\\ +\\ 62^2}\\ =\\ 100\\end{smallmatrix}", "9e16ec9876921dfda440a7c4c5ed32a4": "d(a \\wedge b) = (da)\\wedge b + (-1)^{\\operatorname{deg} a} a \\wedge(db)", "9e172968e7041a7fbeba29fb49dcb7a5": " S_X^2 = \\frac{1}{n-1}\\sum_{i=1}^n \\left(X_i - \\overline{X}\\right)^2\\text{ and }S_Y^2 = \\frac{1}{m-1}\\sum_{i=1}^m \\left(Y_i - \\overline{Y}\\right)^2 ", "9e175b2cf8d3a45111dd40c29ecbdf75": "u(c)=-e^{-a c}", "9e17f41eb53ebda6d2783418163f01c8": "K'=A_K(t) K", "9e17f9b31ecf4d4b4009a0545d1d1cde": "\\mathbf{F}_{\\mathrm{ext}} + \\mathbf{v}_{\\mathrm{rel}}\\frac{\\mathrm{d} m}{\\mathrm{d}t} = m {\\mathrm{d} \\mathbf v \\over \\mathrm{d}t}", "9e1805a3789e98ba54fe957f091340bc": "a,b \\in R \\cup\\{-\\infty,+\\infty\\}", "9e1828a8be987ab1455ca69689b92b5a": "\\theta m", "9e184900c2b43d28c864163ac6fddcc9": "\\vec \\sigma", "9e18923c32d62ded8d0518e3a0a0adf3": "r(x)=\\frac{4x^3 -3x}{16x^5 -20x^3 +5x}=\\frac{p(x)}{q(x)}.", "9e1921f6963bd7fed50638034d1c33f8": " [n, k + 1]_q ", "9e1927898758ae95f7029c3de3a4cfb1": "\\zeta(-5)=-\\frac{1}{252}", "9e193121d384693833308d22413b7295": "\\lnot (", "9e19530f69ee137e6fa1aac619d1815d": "e_n\\,", "9e1964a7a7ba9107ee4b9b846d6dde07": "(s)_n", "9e196bc5ad7b12df5367938ea77019dc": "2^{\\omega(n)}=\\sum_{\\delta\\mid n}\\mu\\left(\\frac{n}{\\delta}\\right)d(\\delta^2).\n", "9e19759ee26e1600b92e668d3f345a40": "(6)", "9e19a93eabee5e695af5c0446a2036f1": "\\boldsymbol\\varepsilon \\sim \\mathcal{N}(0,\\boldsymbol\\Sigma)", "9e19b9863eb3ea594832dce41d4eb99c": "X \\subseteq \\Sigma", "9e1a20dec8195998d876d921386f1aeb": "R_{12,34}", "9e1a2f714bb1eb5fd25317b6588d65cb": "x^\\alpha\\;", "9e1aaab87bdaf4dcaabce96530d7d951": "\\phi: X_i \\to U_i", "9e1ae9f84e0d54033f3b73eda28672aa": "\\scriptstyle\\underline{\\mathrm{W}}(\\mu,\\cdot)", "9e1aeb8ad06a6f3cdd060a9927f0d97e": "b_d=0", "9e1af3bb98bb03a2fcccf356eef4ce10": "\\mathbf{\\hat{a}} = \\frac{\\mathbf{a}}{\\left\\|\\mathbf{a}\\right\\|} = \\frac{a_1}{\\left\\|\\mathbf{a}\\right\\|}\\mathbf{e}_1 + \\frac{a_2}{\\left\\|\\mathbf{a}\\right\\|}\\mathbf{e}_2 + \\frac{a_3}{\\left\\|\\mathbf{a}\\right\\|}\\mathbf{e}_3", "9e1b230f181039c1d06f9f39dcbf78a7": "\nH = \\frac{1}{2\\mu R^2}\\left[p^2_{\\theta} + \\frac{p^2_{\\varphi}}{\\sin^2\\theta}\\right].\n", "9e1b40cef5a09f4b29de41c17b83d4e1": "\\mathcal{D}^{\\mu \\nu} \\, = \\, \\frac{1}{\\mu_{0}} \\, g^{\\mu \\alpha} \\, F_{\\alpha \\beta} \\, g^{\\beta \\nu} \\, \\sqrt{-g} \\, - \\, \\mathcal{M}^{\\mu \\nu} \\,.", "9e1bcd264f63c47227618076df0d2f68": " (\\mathbf{AB})^\\star = \\mathbf{A}^\\star\\mathbf{B}^\\star ", "9e1c04922d6cdc6e6c64c61e109377bd": "(V\\otimes_{\\mathbb R}W)^{\\mathbb C} \\cong V^{\\mathbb C}\\otimes_{\\mathbb C}W^{\\mathbb C}.", "9e1c0a9ea9a3ad537fa9ddb3c0817286": "\\boldsymbol{AV}_i=\\boldsymbol{V}_{i+1}\\boldsymbol{\\tilde{H}}_i", "9e1c1f94de4d7568340c3d448c28f04a": "(P \\or Q) \\equiv \\neg (\\neg P \\and \\neg Q)", "9e1c2ebed086fff062ac5a94344eb96d": " \\frac{1}{\\sqrt{\\varepsilon_0}}\\mathbf{D} ", "9e1c4ae960863b017029269a78973413": " \\rho''", "9e1d0f8268f0498b68507d1da2326d0b": "\\tau_{12}", "9e1d2d0c0a0aa37e51347deb9a04f339": "(a+b)(a-b)", "9e1d47d40de850041cce40eba2417381": "\\sigma_z", "9e1d97ea1943db5eda45965a8b0ce7b9": "\\mathbf{Y}_{lm}\\cdot\\mathbf{\\Psi}_{lm}=0\\qquad\\mathbf{Y}_{lm}\\cdot\\mathbf{\\Phi}_{lm}=0\\qquad\\mathbf{\\Psi}_{lm}\\cdot\\mathbf{\\Phi}_{lm}=0", "9e1e162955ef8d7c4ec85c5453ca9995": "(\\phi_1,\\lambda_1)\\,\\!", "9e1e1872dd3c39276c566202df166c17": "m \\leftarrow \\frac{a+b}{2} = \\frac{1+2}{2} = \\frac{3}{2}", "9e1e1f79696e8b83f482b7ea4610646c": "x'=x-vt", "9e1e42b7ee93560029437854db9f4f88": "{L_n}^m(x) = (-1)^m \\frac{d^m}{dx^m} L_{n+m}(x), \\text{ for } x \\ge 0", "9e1ed4e3e902828272bb812a92bdb693": "\\max (|x|, |y|) < \\exp\\left(\\left[10^6H\\right]^{{10}^6}\\right)", "9e1f2244af9ad1699a388cec0abc5b61": "C = g^r ~\\bmod~ p", "9e1f2e82b4b77a7b04f4f4d75d13394b": "\\vec g_{eff}", "9e1f36cef4b6276df252b1ca4a0accde": "\\mathfrak{p}_2 \\ne \\sigma(\\mathfrak{p}_1)", "9e1f67153b964b3558a9cf3c86687b64": "\\lim_{n \\rightarrow \\infty} \\frac{u(n)}{n} = \\ln(2) = 0.693147 \\dots\\, .", "9e1f7e033afda34f8c2398cdb08654d1": "\\Gamma(0,z) = \\lim_{s\\rightarrow 0}\\left(\\Gamma(s) - \\tfrac{1}{s} - (\\gamma(s, z) - \\tfrac{1}{s})\\right) = -\\gamma-\\ln(z) - \\sum_{k=1}^\\infty \\frac{(-z)^k}{k\\,(k!)}", "9e1f926d335526553c0d1d10b5442ecd": "h\\ll \\lambda_i", "9e1fe7535e6792dd819f2f1eb6ce46f4": "\\Delta Q=T \\Delta S", "9e20564a69ed63c65879713350d167b4": "\\{|{a_i}\\rangle\\otimes |{b_j}\\rangle\\}", "9e2058d8069647f44e48a2cdde0ff9e7": "R_\\text{x}", "9e205d7e5098a106a8b821d1b8270cd2": "r=g(\\theta)", "9e2095d4d9e0dfb899402b847f3c4094": "P = Q + A^T \\left( P - P B \\left( R + B^T P B \\right)^{-1} B^T P \\right) A", "9e20cba1e02475d953af25555a9264c3": " \\sum_{i=1}^n \\text{MB}_i = \\text{MC} ", "9e20e7200ba42afdfdd2640abf3bd0ff": "\\psi(\\bold r | \\bold r') = \\frac{e^{ik | \\bold r - \\bold r' | }}{4 \\pi | \\bold r - \\bold r' |}", "9e211cc9dc6000de182fa72e6fbfb5df": "f\\colon (X,x_0) \\to (Y,y_0),", "9e218c1484f97e048aa5055dca8972f2": "\\theta_m \\rightarrow \\theta^*", "9e21a6788cf7ccb232b17bb3fdb5b1f3": "\\tan \\alpha\n= \\frac{\\sin L}{(\\cos \\phi_1)(\\tan \\phi_2)- (\\sin\\phi_1)(\\cos L)}", "9e2232377b4f4bb921dff82eba85f161": "f_1(X)f_2(X) \\cdots f_k(X)", "9e2283a5d32dfaa5713086e216500519": "\\gamma_2=\\mu_4/\\sigma^4-3", "9e22cd6b1c039eba0a3271b6ad5a8d49": "|N'|\\leq \\chi(x,X)\\,", "9e2303340258f160aabca47943353f8c": "\\scriptstyle g\\colon \\Omega \\to {\\Bbb R}", "9e23be4a822f124aa16b0806ec999aaa": "HA + ECT_0 \\vdash (\\phi \\leftrightarrow (\\exist n \\; n \\Vdash \\phi))", "9e23dd2fed169561dd8abc47e747ecc1": "T(\\exp(ahr)) = 1 .", "9e2421ae6ad863392091b76d31217d6d": "\\lim_{n \\rightarrow \\infty } \\frac{ \\left | { p}_{n+1 } -p \\right | }{ { \\left | { p}_{n }-p \\right | }^{ \\alpha} } =\\lambda ", "9e24376771853f8096671692f621bee3": "\\displaystyle{[L_m,G_r^{\\pm}]=({m\\over 2}-r) G^\\pm_{r+m}}", "9e24cc620e3c89cc5120e341a66bab85": "\\Delta W_n = W_{\\tau_{n + 1}} - W_{\\tau_n}", "9e24f04c0e173ad6aefd1a6f37ea4860": "2^{2n}", "9e24fced012288612db871df98dcabed": "~G_0~", "9e2518d595c4fc4e68039e68ba3c434a": " 52.8\\,\\frac{\\mathrm{ft}}{\\mathrm{s}} =\n 52.8\\,\\frac{\\mathrm{ft}}{\\mathrm{s}}\n \\frac{1\\,\\mathrm{mi}}{5280\\,\\mathrm{ft}}\n \\frac{3600\\,\\mathrm{s}}{1\\,\\mathrm{h}} =\n \\frac {52.8 \\times 3600}{5280}\\,\\mathrm{mi/h}\n = 36\\,\\mathrm{mi/h}", "9e252ac8240dadd79d17f4ad817fef9b": "\\textstyle {4\\choose 2,2,0} \\ {4\\choose 1,2,1} \\ {4\\choose 0,2,2}", "9e258ee73330851950c78c51ece42cbc": "\\mathcal{A}\\cup\\mathcal{B}.", "9e258f5051c18029a564fb183ebb512e": "\\{(i_W,t) | t\\in T_I\\}", "9e26d7fb0e21eb02c5dcaef7051ec217": " C = \n \\begin{bmatrix}\n 0.500 & 0.000 \\\\\n -0.357 & 0.143 \\\\\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 11 \\\\\n 13 \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 5.500 \\\\\n -2.071 \\\\\n \\end{bmatrix}. ", "9e26ef6d734ea7f0889ad505509cae75": "\\mathbb{D}:\\mathbb{F}^d \\rightarrow C_o", "9e27c1121052c581be26aa38be28ddb0": "\\sim2", "9e27f2935dfdf21d6fb979f60f07701e": "\\vec r = (r_1,\\ldots,r_N)\\in\\mathcal S^1\\times\\cdots\\times\\mathcal S^N.", "9e280e6f1d0c848b9279e24776b2c14c": "f\\,g:(x_1,\\ldots,x_n)\\mapsto f(x_1,\\ldots,x_n)\\,g(x_1,\\ldots,x_n)", "9e2832a2e012e33879ba3c9bc36c6cb5": "2j+1", "9e28842523ad9b605d101bb6e5b48d32": "(T-\\lambda)(\\varphi)(x) = (x^2-\\lambda) \\varphi(x). \\quad ", "9e28bdd0194d93f687130451059f1208": "{f:} \\mathbb{R}_+ \\to C^\\infty(\\mathbb{R}^n)", "9e28e4f2b44cd140e8dfc1e8afae0f78": "(q_1,...,q_{d(t)}) \\in \\delta(q,V(t),d(t))", "9e2930cfb3be1da5ca28acd10038eba8": " \\theta = \\tan^{-1}\\left(\\frac{\\sum_{i=1}^k(w_iy_i-z_ix_i)}{\\sum_{i=1}^k(w_ix_i + z_iy_i)}\\right). ", "9e29c1e262b3faeca1f496a374de3447": "T(q) = g^{-1} q g \\!", "9e29c44f84c14b72644fca280dd030fa": "\\Gamma(n+1) = n!\\,", "9e29d473f2750d76c2a3cc1e4615d694": "R(u,v)w+R(v,w)u+R(w,u)v=0 ^{}_{}", "9e2a1fb3f629ab22d2f60a557ad6ca44": "M = N g \\mu_B J \\cdot B_J(x)", "9e2a2416fae263b88c3c811fbdadf502": "\n\\begin{align}\n&\\Rightarrow B^{-1}=\\pm B^{adj} \\text{ is integral.} \\\\\n&\\Rightarrow x_0=B^{-1}b \\text{ is integral.} \\\\\n&\\Rightarrow \\text{Every basic feasible solution is integral.}\n\\end{align}\n", "9e2a4ef9758956cc994927960521a0eb": "-1, -2, -3, -7, -11, -19, -43, -67, -163.\\ ", "9e2a561e66cb678d6ba841a8150b3760": " CZI = \\frac{ \\sum min( x_i, x_j ) }{ \\sum ( x_i + x_j ) }", "9e2a591006bc386e5513393ef952f31a": "\n \\operatorname{E}[\\hat\\sigma] = \\sigma\\cdot\\Big( 1 + \\frac{1}{16n^2} + \\frac{3}{16n^3} + O(n^{-4}) \\Big).\n ", "9e2a62b4e7f4dd774831d41a014afaac": "H_0\\colon r^2=0.", "9e2a755c3ddd9ada225acaf48fce9b1a": "X\\in D^{\\leq 0}", "9e2a91440753c4994d620ca0d72062a4": "d_1(x,y)<\\delta \\Rightarrow d_2(f(x),f(y))< \\varepsilon \\quad\\mbox{for all}\\quad x,y\\in M_1.", "9e2ab9efee5c10b2beb3c8f506affffa": "\\mathcal{F}_{d}=\\frac{1}{2}K_2\\left(\\frac{d\\theta}{dz}\\right)^2", "9e2aca3bc8a03c0982612ff3e21fce58": " \\sigma_{B}^{2} = \\langle(\\hat{B}-\\langle \\hat{B} \\rangle)\\Psi|(\\hat{B}-\\langle \\hat{B} \\rangle)\\Psi\\rangle = \\langle g|g\\rangle ", "9e2adcf83b8c2636552f114159ef9b16": "(1 + \\cos^2\\theta)", "9e2aed6d8ef540f7c8e7c9aebeb3f69b": "\\tan(z) = \\sum_{k=0}^{\\infty} (\\operatorname{PP}(\\tan(z); z = \\lambda_k) + \\operatorname{Res}_{z=\\lambda_k} \\frac{\\tan(z)}{z}).", "9e2b7ad90e269bca82e1dd6604e2da82": "1.8823", "9e2b8eac6b23c7d1192a9e205792bf27": "\\vec{c}_1(u)", "9e2bc61a8c33856018cc3f9ea7e2f101": "\\lbrace\\Psi_j\\rbrace", "9e2bd7395f790297a2dae65dc009c83f": "\\langle Y_i Y_j \\rangle = N_{ij}", "9e2bdde682a5e9930cbfea46649664f2": "\\textit{NOUNPHRASE} \\; \\textit{VERBPHRASE}", "9e2be54a7bd908a1f435b3c1a2bef624": "H^2(M)=0", "9e2be90629090ad664732d1d2ae54f32": "\\textstyle \\mathbf{\\bar{x}}", "9e2bfa29f76b190cc6db441b9a6dad70": "[k_X,k_X[n]]\\cong H^n(\\mathrm{Hom}^{\\bullet}(k_X,k_X))=H^n(X;k_X).", "9e2c7450ee26f627f218bfcd6cbdf46b": "\n\\begin{vmatrix}\nx&y&1\\\\\nx_1&y_1&1\\\\\nx_2&y_2&1\n\\end{vmatrix}\n=0\\,.", "9e2cf8179c3873a1cfa79ac5392c91d9": "\\frac{7 \\times 24}{5 \\times 8} = 4.2", "9e2d153a883f6ed885f2117005f3ad94": "mU_i", "9e2d23cdbf14fbdacee4779117bd213c": "\\begin{align}k_0 & = (v - x_0)^d\\mod N \\\\k_1 & = (v - x_1)^d \\mod N\\end{align}", "9e2d96216215dc800b1b914298444731": " \\Psi(t)= \\sum_n c_n(t)\\psi_n(t) e^{i\\theta_n(t)}", "9e2dc2cf73e9c9b0846d50326432ba85": " | v | = \\sqrt{v^2 - 2gx \\tan \\theta + \\left(\\frac{gx}{v\\cos \\theta}\\right)^2} ", "9e2dfe131314f133ad9aec0ebc1bb772": "\n(\\omega^2 m - 2 k)^2 - k^2 = 0 \\,\\!\n", "9e2e035a25b43bcc82389f10b72acb48": "\n\\psi^{(1)}(z)\n", "9e2e065cb78662f366e8cefbf35c8016": "\\mathcal{H}(X)", "9e2e4aa385c71dbd8ab4f773307fcd04": " \n=\\left(\\frac{1001}{449}\\right) \n=\\left(\\frac{103}{449}\\right) \n=\\left(\\frac{449}{103}\\right) \n=\\left(\\frac{37}{103}\\right) \n=\\left(\\frac{103}{37}\\right) \n", "9e2e9344cb016257eef8d838a046d790": "\\scriptstyle \\vec{B}", "9e2ece29ade82c561f67b0d9b92f1152": "\\mathrm{^{253}_{\\ 99}Es\\ \\xrightarrow [20 \\ d]{\\alpha} \\ ^{249}_{\\ 97}Bk\\ \\xrightarrow [314 \\ d]{\\beta^-} \\ ^{249}_{\\ 98}Cf}", "9e2ed7c50228eeba88b7359a0fc79e4c": "\n\\int_0^{\\infty} G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, \\eta x \\right)\nG_{\\sigma, \\tau}^{\\,\\mu, \\nu} \\!\\left( \\left. \\begin{matrix} \\mathbf{c_{\\sigma}} \\\\ \\mathbf{d_\\tau} \\end{matrix} \\; \\right| \\, \\omega x \\right) dx =\n", "9e2ee4445f129720b5cbd9fa457be340": " dA_{\\bold{x}} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & -d\\theta \\\\ 0 & d\\theta & 1 \\end{bmatrix}~ . ", "9e2ee840fbd050cdce2f4219ad7f3dff": "x \\leftarrow (x-\\mu)/\\sigma", "9e2f69fc3d880140106a5faa956cb669": " \\log_{10} F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1}]} = -m_R/2.5 - 5.45", "9e2f7bf890ca9f7de2a3107a0bf7b7ef": "Y_{p}=\\frac{\\dot{n}_{p,\\text{out}}-\\dot{n}_{p,\\text{in}}}{\\dot{n}_{k,\\text{in}}\\underbrace{-n_{k,\\text{out}}}_{\\text{only for Definition 1}}}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "9e2fae0774a406aaa06ccc4d608e6c1e": "\\nabla\\times(\\vec{E}+\\frac{\\partial\\vec{A}}{\\partial t})=0", "9e2fc12d9331e54a6824531150a4bacd": "T_a f(x)= \\exp(-aD) f(x).", "9e2fd8668ae4989f92c7524c707acce4": "\n\\frac{dr}{d\\tau} = \\frac{\\partial H}{\\partial p_{r}} = - \\left(1 - \\frac{r_{s}}{r} \\right) p_{r}\n", "9e2fe60aa0646574113704ba315e891f": "S = \\int d^4x \\sqrt{-\\tilde{g}}\\frac{1}{2\\kappa}\\left[ \\tilde{R} - \\frac{3}{2}\\left( \\frac{\\tilde{\\nabla}\\Phi}{\\Phi} \\right)^2 - \\frac{V(\\Phi)}{\\Phi^2} \\right]", "9e2ff5ca421cee777e6d9e80cb310138": "L_\\phi=\\phi R-{\\omega(\\phi)\\over\\phi} g^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi+2\\phi\\lambda(\\phi)\\;", "9e302ea00361ffabb2006c04c2dd15c6": "I'_x \\subset \\Omega_x", "9e303db43ce943e6495d7ffcce97b80c": "u_n = (4 \\pi^2 k^2 + q_n^2)^{1/2}, q_n^2 = \\frac {iw} {D_n}, \\gamma_n=\\Lambda _n u_n", "9e304c4b6a70bc36bff51d81bc180348": "\\ L_n", "9e306867cd3ac817a204e63c1dd738a9": "\\, D \\Omega = 0 ", "9e3091b6d85eb4157394340e1303d6a7": "\\mathbb{P} \\{expr_{1}, \\dots, expr_{n}\\}", "9e31569becd422d04fd403abfbc3af44": " \\lim_{k \\rightarrow \\infty} A^k/r^k = v w^T", "9e31c53bf9af48f224afd67ab6c48301": "\\mathrm1/2{O}_2 + {2H}^+ + {2e}^- \\rightarrow \\mathrm{H}_2\\mathrm{O}", "9e31de679d4175b9d27cc19d69929d4e": "Q_{MAX}", "9e31e6ae4b9fd52edd5d78cca3585f11": "\\{1,2,3\\} = \\{3,2,1\\}", "9e31f15d662929a91a15d5c3167e3564": "0.9999\\sqrt x<\\psi(x)-\\vartheta(x)<1.00007\\sqrt x+1.78\\sqrt[3]x", "9e31fe4a50f928bde5ea2225200f1e86": "26^2 + 27^2", "9e322ad0880aa3ce91546164ae9f23eb": "\\phi > \\lambda", "9e32ce33d6d31e415dba86f9aee469a7": " f, g \\in C(G) ", "9e330883e2f40fdb5529f2670675f4cf": "\\omega(X,\\mathbb{B}^n)(E)=\\int_E \\frac{1-|X|^2}{|X-Q|^n}\\frac{dH^{n-1}(Q)}{\\sigma_{n-1}}", "9e3315afe0e5f5bfe8b38f5233b8e005": "\\sigma_{k+1} = J_p^+(\\hat{x_k})\\delta p_k", "9e33411703269d116235f302d22d14e1": "\\rho < \\frac{1}{4}", "9e334638af85b083ee4ec7565e06df1c": "ml^2\\ddot\\theta(t)= - mlg\\sin\\theta(t) - kl\\dot\\theta(t)", "9e334d6357fe8e9e1904a6d2b5ef5141": "\\cos C=\\frac{a^2+b^2-c^2}{2ab}.", "9e335d0678001cc4f92f298377456aa0": "\\det \\boldsymbol{\\varphi}_w'(0) = 1", "9e336fa5ea0300f3007ef8889cfda2ec": "T = m*T_m + T_h + E", "9e338a15b275148581b8637b59402ad4": "\n\\begin{matrix}\\frac{\\partial f}{\\partial x}&+&\\frac{\\partial f}{\\partial y}&&&=0\\\\\n&&\\frac{\\partial f}{\\partial y}&+&\\frac{\\partial f}{\\partial z}&=0\n\\end{matrix}\n", "9e338bab233e926ee363d2dcabf2d930": "(\\exists R.C)^{\\mathcal{I}} = \\{x \\in \\Delta^{\\mathcal{I}} | \\texttt{there} \\; \\texttt{exists} \\; y, (x,y) \\in R^{\\mathcal{I}} \\; \\texttt{and} \\; y \\in C^{\\mathcal{I}}\\} ", "9e339a3eb6fbde2212e66888ca75bd32": "V = \\frac{2}{3} \\pi a r^2, ", "9e339b4ff3b48c3d477c6b879fff935f": "\\mathcal{L}(\\theta |x)", "9e33aae109ffa4ee8595ec5b6053860e": " \\dot \\gamma ", "9e33ac49822a52b98091da5f5829dd53": "\\bar F(B) = p(A) + \\frac 12 p(B) = \\frac 13 + \\frac 12 \\cdot \\frac 14 = 0.4583333...", "9e33b51d5bdd34cb87cc3c3f6ed844b7": "\\ c_i(q_i)=cq_i ", "9e3410bd84950b2f6f18299639cf30a2": "\\operatorname{E}_{\\theta_0} \\phi (T) = \\alpha", "9e34396c0e10432a742ecccf18d528fe": "\\frac{dS}{dt} = \\frac{\\dot Q}{T}+\\dot S_{i}", "9e34b59a6ddc482e03ed3e44800dc620": " \\varphi(A) = {\\rm f} (\\{ \\langle A e_n , e_n \\rangle \\}_{n=0}^\\infty ) ", "9e34cfcfef5a3681b62e6c4d0a41ce6e": "(-1)^s\\, i^{m-m'}", "9e34d81c6754f7b2f30898114c6d1860": "\\mathbf{P}[X_i = 1] = p, \\mathbf{P}[X_i = 0] = (1-p)", "9e34f029a3fb42ff223546a86708da86": "x = x_1 + hx_2,\\ y = y_1 + h y_2,\\ z = z_1 + hz_2, \\quad h^2 = -1 = i^2 = j^2 = k^2 .", "9e3501af742182ca0a79eeb222c2b594": "|vy|\\ge 1", "9e354f04adbe0daac1349cf338422467": "\nT = 300 \\mbox{MeV/k} =3.3 \\times 10^{12} \\mbox{K}\n", "9e358b8957397321e8d65e4098065179": "\\sum E = E_\\mathrm{k} + E_\\mathrm{p} \\, ,", "9e35b981b429b01bb39161d69bde2f3e": "\\operatorname{E}\\Big[e^{i\\theta X_t} \\Big] = e^{-t (|\\theta | + i \\beta \\theta \\ln|\\theta| / (2 \\pi))}. ", "9e35d13ebd312d8be2501b7014df5af4": "\\psi(x,y) = \\begin{bmatrix}\\chi(x,y) \\\\ \\eta(x,y)\\end{bmatrix}", "9e35d7ae87f10616fb20d13a273cce63": "M_p=2^p-1", "9e360bb855a32414a64724cc045dc0d1": "QA=Dc", "9e360f3c60f7e46fdda423c6b0f1c936": "\\text{GDP}_t = Be^{at}U_t", "9e36389150b70b2b3e758303395a7d05": "x=x(t), y=y(t)", "9e3669d19b675bd57058fd4664205d2a": "v", "9e369718f45e35503991ea707ed95780": "=\\frac{b^2}{4a} -\\frac{2\\cdot b^2}{2\\cdot 2a} + c\\cdot\\frac{4a}{4a}", "9e36c87198b458fba3d7bf5593a02f25": "D(\\textbf{x}) = D + \\frac{\\partial D^T}{\\partial \\textbf{x}}\\textbf{x} + \\frac{1}{2}\\textbf{x}^T \\frac{\\partial^2 D}{\\partial \\textbf{x}^2} \\textbf{x}", "9e36f554c3495cc2cab48877241e5f44": " -(2+\\tfrac{3}{4}) = -2-\\tfrac{3}{4}", "9e374d49d9479eedf14fb5271249f8f7": "\\frac{1}{9} + \\frac{1}{18} = \\frac{1}{6}", "9e37813b36993c160a937dd3db71edd5": "R_{5,2} = 165 r^5-360 r^4+252 r^3-56 r^2", "9e378e40f0543776dff70fabfcb3c902": "\\exp(\\sqrt n)", "9e37920df45c596195dd93d01cea9068": "\\Gamma(0,x) = -{\\rm Ei}(-x)", "9e3898efb819df394e5352a4b31fe3a5": "e_2,e_3 \\in (-1,0]", "9e38a1c41143ac93c2bc3d471563c4dd": "Z := Z + L_i*w_i;", "9e394672ec6c7f5105142e52e9eb63cd": "\\mathrm{A}_4 \\twoheadrightarrow \\mathrm{C}_3", "9e3959a35f22a1f73f6bc032a287be33": " P( A_1 / A_2 ) = (25/10) ^{0.20} \\times (20/30) ^{0.15} \\times (15/20) ^{0.40} \\times (30/30) ^{0.25} = 1.007 > 1. ", "9e39817f85ff28dbc22c1c97e955108f": " r= k [A]^2\\, ", "9e39f10ba5347de8cfe34a889aad75d2": "\\mathcal{N} \\models \\varphi", "9e3a163a7e489e64366471410977ea97": "\\kappa(X,X,Z)=E(X^2Z)-2E(XZ)E(X)-E(X^2)E(Z)+2E(X)^2E(Z),\\,", "9e3a8228b5f2a4fef60dddeeaa06c07b": "f(x_4)", "9e3addaf1f5d04086bd37b0492961b3f": "(P \\and (Q \\and R)) \\leftrightarrow ((P \\and Q) \\and (P \\and R))", "9e3b41ec05bc997d6888b1d79eb7dd7e": "q_1 \\otimes q_2 = q \\cong \\langle a_1b_1, a_1b_2, ... a_1b_m, a_2b_1, ... , a_2b_m , ... , a_nb_1, ... a_nb_m \\rangle.", "9e3b9dd7f52384ec57ebc467083257da": "\\theta = \\frac{T - T_e}{T_b - T_e}", "9e3be431fad56f23be324a519d0c8fe2": "g_{xx} \\approx \\cos(q u)^2", "9e3c2a548ec5c27b9381133974475667": "\nx = a \\cosh \\xi \\cos \\eta,\n", "9e3c48b13ec8a366315ce8c871fcfc22": "T_1 \\wedge T_2", "9e3c54d36b9a8af1b718b709df135fa2": "\\overline{X}_{\\mu \\geq \\epsilon} := \\nu^{-1}(\\epsilon) / U(1).", "9e3cceed6035ca2675883f1dd0cc5af5": "2^{7/12}\\approx 3/2;", "9e3d237eed72458a48c5a54bb73848fc": "79^2", "9e3d27dc92a2a593062a81d0498cfc47": " R_p:= \\biggl[ 1 + \\Bigl(\\sum_{0\\leq k0, ", "9e405714f641a1da5238758bdab474b2": "\\theta\\colon TM \\to V_oE,", "9e40e8b216c6beb91606539688f34021": "\n|\\langle\\alpha' |\\psi^\\dagger(\\mathbf{k})|\\alpha \\rangle|^2,\n", "9e412119f7e72cafefaf901b47f350c5": "S_{\\rho}(z)=\\int_I\\frac{\\rho(t)\\,dt}{z-t}.", "9e4143232e4dcb2dcde4d78eb62af523": "Ratio.of.indv.waste.to.coll.rent.waste=(numusers.individual.consump/num.max.rent.units.stocked)", "9e41594ce0933b7e160180d40a65ae2c": "u_w", "9e4159e3d3360e690c6bcf8175f9900b": "(f*g)[n]=\\sum_{m=-M}^M f[n-m]g[m].", "9e416d7d6ed01b788fb426dccaed0792": "i=k", "9e417e936daaaa9bd17df3252083a69b": "H(q) =\\sum_{n=0}^\\infty \\frac {q^{n^2+n}} {(q;q)_n} = \n\\frac {1}{(q^2;q^5)_\\infty (q^3; q^5)_\\infty}\n=1+q^2 +q^3 +q^4+q^5 +2q^6+\\cdots.\n", "9e4185df8ede92857917b99bb964303b": "\\scriptstyle \\bar\\psi(k)", "9e419aeed372ccace910cf28f920329a": "\\lim_{d \\to \\infty} \\frac{dist_\\max - dist_\\min}{dist_\\min} \\to 0", "9e421749233e8731cc61f895ddee22b9": "S = \\langle P_2(\\cos \\theta) \\rangle = \\left \\langle \\frac{3 \\cos^2 \\theta-1}{2} \\right \\rangle ", "9e4287d6e6072069e6028dbc8d44b917": "\\{\\mu_n\\}\\subset\\Pi", "9e429573678863a93e54745d1eb95cbd": "0 = \\Gamma^{\\alpha}_{\\beta \\gamma} g^{\\beta \\gamma} = \\tfrac12 g^{\\alpha \\delta} ( g_{\\gamma \\delta , \\beta} + g_{\\beta \\delta , \\gamma} - g_{\\beta \\gamma , \\delta} ) g^{\\beta \\gamma} \\,.", "9e42a1f816502b0e9bfba2bbd596b7a1": "\\begin{align}\\Pi(x) &= \\pi(x) +\\tfrac{1}{2}\\pi(x^{\\frac{1}{2}}) +\\tfrac{1}{3}\\pi(x^{\\frac{1}{3}}) +\\tfrac{1}{4}\\pi(x^{\\frac{1}{4}}) \\\\ &\\ \\ \\ \\ +\\tfrac{1}{5}\\pi(x^{\\frac{1}{5}}) +\\tfrac{1}{6}\\pi(x^{\\frac{1}{6}}) +\\cdots\\end{align}", "9e430f95f4a14bac0d0758f2072dbc28": "1/2\\pi R C", "9e43a66833340a9931f608defaba12bc": "\\ A \\approx\\pi R^2\\theta^2/4", "9e453624da7e6a8c0a6216d79b677b05": "\\mathrm{N}_p S", "9e45444a8119f7978552233c80a89dd5": "[H^+]_{0^{ }}", "9e4564ff457dea18698e7c4eea5e2c44": "C \\frac {\\pi D^2}{4} = C_s \\frac{\\pi D_s^2}{4}", "9e45cad46c88ec29165dbaa5aed52751": "\n f(\\boldsymbol{\\sigma}) = 0 \\,.\n ", "9e466395e1b2b9068568f0c1413fd8ce": " P[N=k]\\, ", "9e46846d1ff88f2af8c428d8867897cd": "\\langle Q (u\\wedge v),w\\wedge z\\rangle=\\langle R(u,v)z,w \\rangle.", "9e4684d810470b2d2a3def12943b9318": "\\{f,g\\} = \\omega(X_f,X_g) = -\\omega(X_g,X_f) = -\\{g,f\\} ", "9e46a97d5d525281ea47662e6f08d7a0": "\\sum_{i=0}^{n-1} \\frac{i}{2^i} = 2-\\frac{n+1}{2^{n-1}}", "9e46bd541e1fe909aed8adb08b09a44e": "\\Psi[U\\mathbf{A}U^{-1}-(dU)U^{-1},U\\phi]=\\Psi[\\mathbf{A},\\phi]", "9e46c7e42da419cd1a9caa8bb5a46639": "P(R_n | \\bar{W})", "9e4774d11e383d7805b7a8c1dffe13fd": "\\frac{{}_{(1)1}\\partial axb}{\\partial x}=b\\,\\!", "9e47b8e2cee49149a47e0943e245b300": " [K]^{-1} ", "9e47bd118a40081058d9372232dd8f2b": "{\\rho}=\\sqrt{r^2+h^2}", "9e48412271eff0650ed355cbedce7457": "\\Delta x = 1", "9e4876601ed31f29df7cf71a15538506": "n\\to\\infty", "9e48bc2175c39b08685df1e5dc09d9d1": "\\scriptstyle \\eta(y)+\\zeta(y)=1 ", "9e4904654ce8a2ed8e41956984d42686": "h_M", "9e498faa743d5e2c85f7684e6090567e": " \\alpha^i\\ne 1", "9e49b594ea86f73612287ac32e7bfc84": "\\chi(G) = \\text{min} \\{ \\chi(G+uv), \\chi(G/uv)\\}", "9e49c3a158fd3f2b31e33f4d0fae9094": "\\equiv M_{k'k}=\\frac{1}{V}\\int d\\bar{r} H'e^{i\\bar{r}(\\bar{k}-\\bar{k}')}=\\frac{1}{V}\\sum_{\\bar{q}}\\int d\\bar{r} H_{\\bar{q}}e^{i\\bar{r}(\\bar{k}-\\bar{k}'+\\bar{q})} = \\frac{1}{V}\\sum_{\\bar{q}} H_{\\bar{q}}\\delta _{\\bar{k}-\\bar{k}'},_{\\bar{q}}=\\frac{1}{V}H_{\\bar{q}} \\;\\; (7)", "9e4a6cf805ea330f5f741723658ad8c2": "h(t)=f(a+t(x-a)).\\,", "9e4ab72cb4834fe305611fe310d1db1f": "\\;q_c = P\\left[\\left(1+0.2 M^2 \\right)^\\tfrac{7}{2}-1\\right]", "9e4ba0ef130df9fa70bc7236d66a7774": "S/\\hbar", "9e4bd9734b2f7b6192042f72921a8ba3": "G = \\frac{V_\\text{out}} {V_\\text{in}} = 1 + \\frac{R_f} {R_g} ", "9e4c3b99c87f67a0a52d224c9829767a": "\\mbox{R}(z, 0) = 1", "9e4da59f8be63430b49fdb681ff3669e": " S(z+1;x)=\\cos(2 \\cdot 2^z \\arccos (x) )=2\\cos( 2^z \\arccos (x))^2 -1 =f(S(z;x))\\ ", "9e4e1cc3d369df18903152a53604a732": "\n\\begin{align}\n\\bar{H} &= \\frac{2\\mu}{2\\mu + \\frac{1}{2N_e}} \\\\\n&= \\frac{4N_e\\mu}{1+4N_e\\mu} \\\\\n&= \\frac{\\theta}{1+\\theta}\n\\end{align}\n", "9e4e2b156ced3a6a9cf3a5cf44cdf861": "\\begin{bmatrix} \n\\ \\ 1 & \\ \\ 2 & \\ \\ 1 \\\\\n\\ \\ 0 & \\ \\ 0 & \\ \\ 0 \\\\\n-1 & -2 & -1 \n\\end{bmatrix} = \\begin{bmatrix} \n\\ \\ 1 \\\\\n\\ \\ 0 \\\\\n-1 \n\\end{bmatrix} \\begin{bmatrix} \n1 & 2 & 1\n\\end{bmatrix} = \\begin{bmatrix} \n1 \\\\\n1 \n\\end{bmatrix} * \\begin{bmatrix} \n\\ \\ 1 \\\\\n-1 \n\\end{bmatrix} \\begin{bmatrix} \n1 & 1\n\\end{bmatrix} * \\begin{bmatrix} \n1 & 1\n\\end{bmatrix}\n", "9e4e3263de2ae07dec8a30ba827e6841": "\n \\frac{\\partial}{\\partial\\theta}\n \\left[\n \\int T(x) f(x;\\theta) \\,dx\n \\right]\n =\n \\int T(x)\n \\left[\n \\frac{\\partial}{\\partial\\theta} f(x;\\theta)\n \\right]\n \\,dx\n", "9e4e748ce4058a68f23693a6bc433a4b": "f(x; a, m) = \\frac{\\Gamma(m)}{a\\,\\sqrt{\\pi}\\,\\Gamma(m-1/2)} \\left[1+\\left(\\frac{x}{a}\\right)^2 \\right]^{-m}, \\!", "9e4e851340ac5f53acdd1e702bd246d5": "y=a \\cos(bX) + b \\sin(aX)", "9e4ecf1c713d43ad1a819af58c2a79ee": "\\begin{align}\n \\left | Y \\right | &= \\sqrt{G^2 + B^2} = \\frac{1}{\\sqrt{R^2 + X^2}} \\\\\n \\angle Y &= \\arctan \\left( \\frac{B}{G} \\right) = \\arctan \\left( -\\frac{X}{R} \\right)\n\\end{align}", "9e4ed5e36efdead4abddba5315ce02b2": "V(x) = \\text{True}", "9e4ef7f6fcf3f3f4206c5827c6f204d2": "\\int_0^\\infty \\frac{\\sin \\omega}{\\omega}\\,d\\omega = \\frac{\\pi}{2}", "9e4f0f9eceecadb5b235831a27d9f499": "=\\frac{1}{N}f\\Theta \\bar{f}\\left[ n-k \\right]with,\\bar{f}\\left[ n \\right]=f\\left[ -n \\right]", "9e4f19f8c49a7e125059884cc55177a9": "\\rm \\ 3 K_2NiF_6 \\xrightarrow{\\Delta} 2 K_3NiF_6 + NiF_2 + F_2", "9e4f4607f90fd4a4bc9c271fa9af627d": "\\frac{1}{4\\pi}\\frac{p_n(\\chi(t, \\zeta))}{(1 - |t|^2)^2} ,", "9e4f64b6829c1e2bdd97adbb8ca59d0f": "Bf: BC \\to BD", "9e4f9a21c23aa27262e6a62787c1b061": "[T(u), P(u)_{>}", "9e4fd11c24f523a957251c22e6a0e36d": " \\Rightarrow \\left|1,n\\right\\rang \\left|2,n\\right\\rang ", "9e501968dbbecf8c67076e203825b934": "\\begin{align}\n\\overline{BD} \\cdot\\overline{AC} & = 4R^2[\\sin(\\theta_3+\\theta_2)\\sin(\\theta_3+\\theta_4)]\\\\\n& = 2R^2[\\cos(\\theta_2-\\theta_4)-\\cos(2\\theta_3+\\theta_4+\\theta_2)]\\\\\n& = 2R^2[\\cos(\\theta_2-\\theta_4)+\\cos(\\theta_1-\\theta_3)].\n\\end{align}", "9e502587a07b4585f59236a9cb545c3e": "\\mathcal{F}_{0}", "9e5088baf88284441bbf395ff17bf85b": "f_\\textrm{sim} = \\frac{f_\\textrm{p}f_\\textrm{1}N_\\textrm{1}} {(f_\\textrm{p}f_\\textrm{1}N_\\textrm{1})+1}", "9e509596ff9c3db8908947d200ff1441": " b_n = T_{n,n} ", "9e50b9caffc6ac40df2f64f40bce003c": "\\scriptstyle y_{2k}", "9e50da412fe42f075b2cbcaab8455b0c": " \\kappa ", "9e513cfa5995cd60cb9793b28d8a2127": " \\sum_{m_1 \\le m_2 \\le \\dots \\le m_N} P_{S/A}(n_1, \\cdots n_N \\rightarrow m_1, \\cdots m_N) = 1 ", "9e51ca20c4fb24a2fa59aaa5d9a78a8c": "(a, b) = (c, d) \\iff a = c \\and b = d. ", "9e520a15311341981d02cdb994a0e76f": "- {\\partial \\phi \\over \\partial x^\\mu} = \\dot{u}_\\mu + {u_\\mu \\over c^2 \\dot{\\phi}} \\,", "9e5265e872e7c9b8507233dbc77b952b": "\\varepsilon \\rightarrow -\\infty", "9e5270eebe2ad6b07a24bc31103e439a": "H = T^{a+\\varepsilon},", "9e529c17fdf4e5587955384ef16c639b": " \\operatorname{d}^3 r =\\mathrm{d}x \\ \\mathrm{d}y \\ \\mathrm{d}z ", "9e52cba5d92db4442ebf19bbea4ddd8e": "g_{\\alpha\\beta}(\\mathbf{x}) = \\eta_{ij} \\, e_{\\alpha}^i(\\mathbf{x}) \\, e_{\\beta}^j(\\mathbf{x})", "9e52ce62a473ab1590d9ac1c13a8bf23": "(i\\gamma^{\\mu} \\partial_{\\mu} - m) \\psi(x) = 0.\\,", "9e52db1a1d16227187647b274bef7c4d": "\\hbox{AU}", "9e52facfd6e7928d5556908cd0e72f38": "f_V", "9e530be07e6ddf990726ec3706de5033": " f(i)M^{k+1}(i,j) = f(j) \\sum^{N}_{n=0} M(j,n)M^k(n,i)", "9e531ae0597237014a470e530b78b229": " \\left| \\nu_{i} \\right\\rangle = \\sum_{\\alpha} U_{\\alpha i}^{*} \\left| \\nu_{\\alpha} \\right\\rangle", "9e531e657e83a71bb089b18fc428fc06": "\n \\frac{U_p}{U_s} = 1 - \\frac{\\rho_0}{\\rho} = 1 - \\frac{V}{V_0} =: \\chi \\,.\n ", "9e53481d77e47d26cf4a93661a28219e": " \\frac{\\hbar^2}{M} \\nabla_{r_1} \\cdot \\nabla_{r_2} ", "9e5350a9d949da61f80120fee334aba2": "N(a_1, a_2) = (a_1-1)(a_2-1)/2", "9e5359e8d371b6d021b2075a621e42b1": "Q(\\alpha)= \\frac{2}{\\pi} \\int W(\\beta) e^{-2|\\alpha-\\beta|^2} \\, d^2\\beta.", "9e536599437721774a9b77f5fd237c56": "f(A) = \\sum_i \\sum_{j \\in C_i} p_{ij} = \\sum_i p_i", "9e53b35a21112402ad18d003bc6dba96": " \\mathbf{c}\\cdot \\mathbf{a} \\mathbf{b} = \\left(\\mathbf{c}\\cdot\\mathbf{a}\\right)\\mathbf{b}", "9e53f395dd5a70561ad9765086684122": "\nm\\ddot{y} = - \\frac{\\partial V}{\\partial y} - \\frac{q B}{c}\\dot{x}.\n", "9e541f6012ac73c4e843a8998678027c": "E=\\{((v_1,v_2,\\dots,v_n),(v_2,\\dots,v_n,s_i)) : i=1,\\dots,m \\}.", "9e54b124a8029b7a9cdcd1083d771205": "\\gamma _m \\ge 0,\\quad \\forall m . \\quad \\quad ( 3)", "9e54c1a5f8ba5c6829421f4bbcd689a6": "T_M(d)=\\frac{4T_{MB}}{H_fd}\\left(\\sigma\\,_{sv}-\\sigma\\,_{lv}\\left(\\frac{\\rho\\,_s}{\\rho\\,_l}\\right)^{2/3}\\right)", "9e54f42e67aabd17841e62b40f4a539e": " \\angle ZHA = 90^\\circ + \\frac{1}{2} (\\varphi_H - \\varphi_A) ", "9e5521e4da9c4ec203be19011b6bca93": "H_{\\mathrm{dR}}^{k}(M) \\simeq H_{\\mathrm{dR}}^{k}(S^1).", "9e5535d8afc7af90fb952a4c2fb1432f": "N_\\downarrow/N", "9e553b0bc716db1f03f650c95f62c27c": "\n\\begin{pmatrix}-(1)&\\alpha^{-1}+\\alpha^{6}x\\\\\n\\alpha^{3}+\\alpha^{1}x&-(\\alpha^{-7}+\\alpha^{7}x+\\alpha^{7}x^2)\\end{pmatrix}\n\\begin{pmatrix}\\alpha^{-7}+\\alpha^{7}x+\\alpha^{7}x^2&\\alpha^{-1}+\\alpha^{6}x\\\\\n\\alpha^{3}+\\alpha^{1}x&1\\end{pmatrix}\n=\\begin{pmatrix}1&0\\\\ 0&1\\end{pmatrix},\n", "9e555ffe635a2ad8d8c7f5e27beb09d1": "f_{\\varphi}(z) = \\varphi\\circ f(a+zb)", "9e5579ba345f6fb8d47c4af4d7d092a1": "x^\\star= \\nabla f(\\nabla f^{\\star}(x^\\star)),", "9e55a8645017b27964594815219cf4a9": "A_{reduced} \\equiv D^{-\\frac12} A D^{-\\frac12}", "9e55ac62c44e1d6c531d816ba2cb67fa": "r=\\frac{K}{s}=\\frac{ab}{a+b}", "9e55e23244ef33bb5913444e73406bc6": "P_3=(X_3:Z_3)=(9:24)", "9e568edcebb75bf7e972371023a4076d": "(\\pm 8)^2 \\equiv 64 \\equiv 18\\pmod {23}", "9e56d987a26f1354f9bef3323c9295a9": "\\alpha = \\left(1 + \\left(0.37464 + 1.54226\\,\\omega - 0.26992\\,\\omega^2\\right) \\left(1-T_r^{\\,0.5}\\right)\\right)^2", "9e56fc71047f25d1d28151f6749626d6": " \\mathbf{r} = (x, \\ y ) = r (\\cos \\varphi ,\\ \\sin \\varphi) = r \\hat{\\mathbf{r}}\\ , ", "9e573a393f1a731b25b10044cd7c4193": "(a^2b)^{1/3}", "9e573bb3e2312829d395abb10d94ea88": "\\lim_{\\varepsilon\\to 0}\\phi_{\\varepsilon} * f (x) = f(x).", "9e575df91d8e4122163cadd59dda5691": "-\\frac{1}{p}", "9e578d704286f2273b16fd3fbfc5331b": "\\left( -\\sqrt{2 \\over 5},\\ \\sqrt{2 \\over 3},\\ \\frac{-4}{\\sqrt{3}},\\ 0 \\right)", "9e57df00aac7fadf957963927234a1d4": "Z(\\mathbb{C} [S_n])", "9e58b8053ffc60b237d9ffb7be26db44": " \\sum_{n=1}^\\infty \\left(a_1 a_2 \\cdots a_n\\right)^{1/n} \\le e \\sum_{n=1}^\\infty a_n.", "9e58bbf1d9fdff58ce58ce16d5fd80f6": "\\pm g^{-1}", "9e58db7eab2900572064826c0c940b02": " a + L*(s_1 ... +s_{D-k}) + u *(s_{D-k+1} ... +s_p) ", "9e590164c7dd2434d2bb8ac1dd84c75b": "\\frac{\\delta}{2\\pi}\\lambda = a \\left ( \\sin\\theta + \\sin\\alpha \\right ) \\,\\!", "9e59340cdc1f9d4195f3e981e81ebdf0": "X_D", "9e5a0a339034fd58c1c43f13ec2eed4e": "\\frac{n+m+1}{6}", "9e5a26ba9e9b014dac60f93bebd7e4c6": " C_{1\\epsilon} ", "9e5a305becd770c5ffe8a3ba1deb3412": "\n\\begin{align}\nM \\geq |Z|\\\\\n\\end{align}\n", "9e5a312392fcddb1f7f14530efa74b7a": "A(T,V)\\,", "9e5ae4eebf5b09183f0e6b0fca090d5d": " G_{\\sigma}=C{\\sigma}^{-\\left(1/2\\right)}exp\\left(-|x|^2/4{\\sigma}\\right)", "9e5af93a631f05a6e10244bdebb026da": "\\sigma = {F \\over A} \\pm \\frac {My}{I} ", "9e5b86a8050640a464f9a6db199cfbf6": "\\frac{\\ln(1+\\sqrt{p})}{\\beta}", "9e5ba15dd76b904d522cd18c9f107e66": "P_s = \\frac{1}{Z} \\mathrm{e}^{- \\beta E_s}. ", "9e5bdc49f134e6891aa76a0f8f981c44": "\n\\phi = \\arctan \\frac{y}{x}\n", "9e5beb4c4e081d3c049cb7ee9e092fba": "{7n^2 - 7n + 2}\\over2", "9e5c24a132c06520a167791692987f1d": "{\\lambda}'", "9e5c2c0cbc1c846e9bc33b833fa518d4": "|\\psi\\rangle_A", "9e5c65347318a05a2537c5555e0d01cf": " e^{-idtk^2}", "9e5c70fd5e4f1dcd3dbb93783d24b87b": " \\hat u(t,\\xi) = (H(\\cdot,\\xi) \\ast \\hat F(\\cdot,\\xi))(t) ", "9e5c761e0d8d1b46c1e5590c16a12002": "(p_1 - a_1)\\frac{dr_1(t)}{dt} + (p_2 - a_2)\\frac{dr_2(t)}{dt} + \\cdots + (p_n - a_n)\\frac{dr_n(t)}{dt} = 0", "9e5c782ecba0452e70fa5f605cde989e": " C = \\gamma_0 + \\gamma_1 \\frac{u^2} {c^2} + \\gamma_2 \\frac{u^2_r}{c^2}+... ", "9e5ca5a33db4181cb069a8fa2b649f0e": "h(x',y')", "9e5cad2a5896ca7babc9057ca39eee31": "\\begin{align} 2\\cdot R_*\n & = \\frac{(116\\cdot 1.96\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 48.9\\cdot R_{\\bigodot}\n\\end{align}", "9e5cbcfb155e6f424500c60e00b2850b": " x_0 > z_1 ", "9e5cda3a22bcb44381755170d1469bc8": "a_n = \\frac{a_{n-1} a_{n-3} + a_{n-2}^2}{a_{n-4}}", "9e5d052fecbbe69d388bc4f7e560c4fe": "t_0 + 3", "9e5d23a309e352d907e181d5ea970ff0": "\\textstyle \\frac{ 4195835}{3145727} = 1.333{\\color{Red}739068902037589} ", "9e5da5607137b3d5fcdd3ba2bcd383d6": "\\textstyle\\vec{r}_k", "9e5da9d9f355fb631d189dc7889a9c0c": " \\lim_{t \\to \\infty}\\phi(t)_j = \\frac{1}{N} \\sum_{i = 1}^N c(0)_i ", "9e5dac401102804f2e7eb2b758ca9472": "d \\leq n - k + 1", "9e5dddd3cbbfd554b9fbc379fef523c6": "A(t)\\, ", "9e5df60ed465774b212ca5fcef545f02": "\\operatorname{End}(\\mathfrak{g})", "9e5e12ec73bc3b71991503b92077031b": "Xf(p) := \\left.\\frac{d}{dt}f(\\gamma(t))\\right|_{t=0}.", "9e5e2385b7e386e318676636c53d3142": "A = {\\cot\\alpha}.", "9e5e29d843bfe5f837f25aa3448f7595": "\\| Mf \\|_{L^{p}} \\leq C_{p, n} \\| f \\|_{L^{p}}", "9e5e52e92f60b6e47a6abf5eaf0d8e74": "1 - \\frac{1}{e} + o(1) \\approx 0.632", "9e5e64bc6f8585f39ce3266e6d8f53dc": "a_1, \\dots, a_{i-1}\\in F", "9e5ec90933da47fb206f1cff1289fb48": "1 \\ll \\frac{A}{B} \\ll 1", "9e5ed2a3a1fcd6ea4afcebc8507aea8c": "d_{x,y} = \\min (d_x,d_y) ", "9e5ed810ed90633a9be94fa0cb25c415": " \\mu m ", "9e5f00d7b5ea57d9a3c5a54dbde505c2": "\\tilde{x} \\sim F(\\tilde{x}|\\theta)", "9e5f73c915e54039eca60f57eae22525": "^2\\text{H}", "9e5f757d8a4e0d868c100eaa5db59871": "T=2", "9e5fb72f12371dd00668ac0423d3aec1": "P_{i}", "9e5fbe1872e4249f6522f1689e71e564": "f_p = \\frac{\\Delta p}{L} \\frac{D_p}{\\rho V_s^2} \\left(\\frac{\\epsilon^3}{1-\\epsilon}\\right)", "9e601f21a7ee79f0f9e7bea74202f82a": "\\ k(x, x') = \\langle \\phi(x), \\phi(x')\\rangle_\\mathcal{H} ", "9e605ce3bb6ad134bb55c54d861ceb6a": "1.4", "9e60658159e814d9ced6e19bfac47eee": "\\boldsymbol \\xi = \\boldsymbol v\\delta t\\ ,", "9e60ca7ca29f6c2e479435b7ea0422e7": "\\frac{a(1-r^m)}{1-r}", "9e60f795dfdc793327ff104a77aa2349": "n_{\\alpha}", "9e6106ce7d00c0b3bb5176da184d597d": " W^{k,p}(\\mathbb{R}^n) = H^{k,p}(\\mathbb{R}^n) := \\left \\{f \\in L^p(\\mathbb{R}^n) : \\mathcal{F}^{-1}(1+ |\\xi|^2)^{\\frac{k}{2}}\\mathcal{F}f \\in L^p(\\mathbb{R}^n) \\right \\} ", "9e6161f59e6fefd554a0064687a1767c": "= \\frac{\\sin \\theta}{\\theta} \\times \\sin \\theta \\times \\frac{1}{1 + \\cos \\theta}.\\,", "9e617e184358aa195869729c55e021f0": "s_L(n)", "9e61db34ea25dac53a84f4f1842d06e0": "U \\subset \\Omega \\cap V", "9e61e7fa5671c616e86ffc39ac1925e4": "\\mathrm{Ad}(h)", "9e6218ef67bfcbbd65dbf8c7bfcf14fa": "\\mathbb{E}[Y_k|\\mathcal G]\\le\\inf_{n\\ge k}\\mathbb{E}[X_n|\\mathcal G]", "9e62b791fc2be574f439b482aa7f12b2": "\nc_{j,r_j-k}=\\frac{1}{k}\\sum^k_{i=1}\\frac{(r_j-k+i-1)!}{(r_j-k-1)!}\\,R(i,j,p)\\,c_{j,r_j-(k-i)}\\,,\n~~~~~~ (k=1,\\ldots,r_j-1;\\,j=1,\\ldots,p)\n", "9e62e3a9938b39673e3b8d71f0496ac5": "P_{RX}", "9e62ebad7e62930c0cd41145e68b6475": " \\mathrm{Ar} = \\mathrm{Ri}\\,\\mathrm{Re}^2", "9e62fa4f03616de3fae8f8938b216162": "e = \\sum_{i=1}^n a_i \\otimes_B b_i", "9e631635c17caf2799657f2225c2cc6e": " \\sum{n_h} = n ", "9e631b8751a36a42b364fae752dbbbaf": "(2-\\sqrt{2})n + O(1)", "9e631fe034e5cca4672c18edcdc5363d": " M_s = \\log_{10}\\left(\\frac{A_{max}}{T}\\right) + 1.54\\cdot \\log_{10}(\\Delta) + 3.53 ", "9e63e548e0a3c2f4884b4938c1fab91d": "2n=p+q", "9e6423d7feae15a4e47e4f748adc4aca": "a^{-a}", "9e643d1fe01b7e0a5a8de742ef1e7ffa": "\\bar{\\lambda} = 1", "9e65352b7abee704f547840db0c7fe1e": "\\sum_{n,k} {n\\choose k} x^k y^n = \\frac{1}{1-y-xy}.", "9e657346fdc0b3b279ab41e27da81425": "\\hat{\\textbf{d}}_j = \\Sigma_k^{-1} U_k^T \\textbf{d}_j", "9e65999153aab7b0b657a4e67fcb07a2": " p(x) dx = \\psi^*(x)\\psi(x) dx", "9e65d0ae02e15d4a9f8744a69e8231a0": "f(x) = x + x^\\frac{4}{3}.\\!", "9e65f8baaf436a986c0f3a2e8a44acd1": "\\begin{bmatrix} \\dfrac{\\Delta \\mathbf{[g]}}{g_{22}} & \\dfrac{g_{12}}{g_{22}} \\\\ \\dfrac{-g_{21}}{g_{22}} & \\dfrac{1}{g_{22}} \\end{bmatrix}", "9e65fb3a8640c087ecd1f857f55b013e": "\\begin{align}\nd\\mathbf{x}^2&=d\\mathbf x \\cdot d\\mathbf x \\\\\n&= \\mathbf F \\cdot d\\mathbf X \\cdot \\mathbf F \\cdot d\\mathbf X \\\\\n&= d\\mathbf X \\cdot \\mathbf F^T\\mathbf F \\cdot d\\mathbf X \\\\\n&= d\\mathbf X\\cdot\\mathbf C\\cdot d\\mathbf X\n\\end{align}\n\\qquad \\text{or} \\qquad\n\\begin{align}\n(dx)^2&=dx_j\\,dx_j \\\\\n&= \\frac{\\partial x_j}{\\partial X_K}\\frac{\\partial x_j}{\\partial X_L}\\,dX_K\\,dX_L \\\\\n&= C_{KL}\\,dX_K\\,dX_L \\\\\n\\end{align}\\,\\!", "9e66ac95ebb4ee1725280e4ec5898dc8": "L(A, B)", "9e672c4963917c569e148278c0aabfdb": "K > 0,", "9e674dd7e34449909f5c89c7ed53d658": "r=a+bsc_1+\\gamma s^2c_2", "9e67523efd6fe547687d6b2dd898a1d3": "\\phi^{-}(a_i)\\le \\phi^{-}(a_j)", "9e67ff2af3d9af1be3b0903b9fcfb25a": "CG = 1/C_{rt}", "9e681e0d3632a26dffe2222d594c506f": "\\langle g \\rangle", "9e68d2451901dc68400a1aefaba1e79c": "x_0 \\rightarrow x_1 \\rightarrow x_2 \\rightarrow \\cdots", "9e68d297d428c821d63e3562f0560621": "\\displaystyle{\\psi(a)={1\\over 2\\pi}\\int\\widehat{a}(x,y) W(x,y)\\, dx dy.}", "9e691813ee461ce4eb24041b9ff8ecfb": " M=m_{0}+t ", "9e691d77ce5bddffb3c587cc53b32bb8": "j\\ \\stackrel{\\text{def}}{=}\\ \\sqrt{-1}", "9e694fc6891fa3807bfc40ab9346effb": "\\sigma(T)", "9e697d66734eb0758b73708417a95845": "\\Sigma(t)", "9e69972d976659536f5f8a63f17e92da": "\\Delta(B) = B\\left[\\frac{C}{2}(\\Delta(r))^2 - D\\Delta(r)\\right].", "9e69b4521918bfface05aa130d5fc7cf": "Np_i |\\phi_i\\rangle\\langle\\phi_i|", "9e69eb4599f2248ec2db65a2ad5831bb": "R(u)", "9e69ef2e2c03c040c411a10a426f3623": "K={LN-M^2\\over EG-F^2}, \\quad H={EN-2FM+GL\\over 2(EG-F^2)}.", "9e69fd4887e67db279d22225d775d2bd": "\\, \\mathcal{P}_B (A) = (A \\;\\big\\lrcorner\\; B^{-1}) \\;\\big\\lrcorner\\; B ", "9e6a06251e744cdfaf6a1a26b5a226a7": "\\displaystyle{C=\\gamma^{-1}}", "9e6a1862600e0b6c1d2a3707f2f8c67c": "\\mathrm{E}( \\mathrm{2} ) = \\sum_{d=1}^\\infty d \\frac{6}{\\pi^2 d^2} = \\frac{6}{\\pi^2} \\sum_{d=1}^\\infty \\frac{1}{d}.", "9e6a275fcb960bdfc467eccacaa92c76": "F_{i+1} = F_i - E_F(v_i,V)", "9e6a443c856e0b2b59170d28d6f94356": "O\\left(1\\right)", "9e6ab8250becb604866a46627d5a9bba": "x_{\\mathrm{ZOH}}(t)\\,= \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\mathrm{rect} \\left(\\frac{t-T/2 -nT}{T} \\right) \\ ", "9e6b1ea6b1626e59ebac6b25a79e6115": "\\epsilon \\left( \\sum_i n_i \\sigma_i \\right) = \\sum_i n_i ", "9e6b5374cf23f6c73b25df23fe1d388b": "n! < (n/2)^n", "9e6ba271fc4424f478a4fde895b6eb16": "p(x+\\varepsilon)=p(x)+\\varepsilon\\,p'(x)", "9e6c1e7e44aa5a829f8b691f84218d4b": "{\\Gamma}(\\tfrac12)", "9e6c53b8c6c37e88cf68060f257b54d9": " A \\rightleftharpoons B ", "9e6c98397e5dafa1a789902384dd2901": "|\\overline{A}_+\\cap\\overline{B}_-|=1", "9e6cc4e48dec65dab3e645242ca53a22": "F_{air} = -kv", "9e6ccdf117fdb552953123732c9d8e12": "\\mathbf v = (\\mathbf v \\cdot \\hat{\\mathbf u})\\hat{\\mathbf u} + (\\mathbf v \\wedge \\hat{\\mathbf u}) \\hat{\\mathbf u}\n", "9e6cdd8940313479b4c2b93e94afd4f1": "q\\lesssim k", "9e6cdf7e017a1642fbfc5fa741fad1aa": "0 \\to R \\overset{f}\\to R \\to R_1 \\to 0", "9e6ce2d882db84d42923c5eddae7e325": "B \\rightarrow A: \\{A,\\mathbf{N_B'}\\}_{K_{BS}}", "9e6d75e648bee7e554cc162ac0196fb2": "BD^{-1}", "9e6d913e209b29aef3b46adef1fe111a": "\\deg(g)/2", "9e6db25ec6cb44fd27679ba350bc4c39": "\\frac{\\lVert\\boldsymbol{x}_m-\\boldsymbol{x}^\\ast\\rVert_\\infty}{\\lVert\\boldsymbol{x}^\\ast\\rVert_\\infty}\\leq\\bigl(\\sigma\\kappa_\\infty(\\boldsymbol{A})\\epsilon_1\\bigr)^m+\\mu_1\\epsilon_1+\\mu_2n\\kappa_\\infty(\\boldsymbol{A})\\epsilon_2", "9e6dba01b87c3bb300745b5204cb7dfa": "DTIME(g(T(n))) = DTIME(T(n))", "9e6dcaf5b6530ad8a14029bc3baacf5b": "q = e^{2 \\pi i \\tau}\\ ", "9e6ede2e65ef31dc2cd627dae699cd9c": "\n\\cos{\\theta} = \\frac{\\mathbf{d_2} \\cdot \\mathbf{q}}{\\left\\| \\mathbf{d_2} \\right\\| \\left \\| \\mathbf{q} \\right\\|}\n", "9e6edec275706a301a7762ebb77e10ac": "\\lambda_\\mathrm{op,abs}", "9e6f37fc24893992eea30d7dda1b31f5": "\\mathbb{HP}^{2}/\\mathrm{U}(1)", "9e6f4769a02fd8b23c6aaa5e5dffe2fd": " T_{ij} - T_{ji} ", "9e6f95e906585f9df67bd41ce5af4ece": "f(x)b. \\end{cases}", "9e73fab53de6ec85b9570cf3f10f6b94": "\\mathrm{d}w/\\mathrm{d}x", "9e74812191a25ee4e1c8906df031e242": "50 \\approx \\sqrt {30 \\times 77} \\mathrm \\ \\Omega", "9e74ad84a459ed6b2aae7c6052a83cc7": "(m,f,a,b,G,n,h)", "9e74b2112db515e1214c6ffb4871d93f": "\\left(\\frac{x_\\mathrm{m}}{x}\\right)^\\alpha", "9e74d322452e9f6c03fe81ff9aae834c": " \\frac{\\partial v}{\\partial t} + fu = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y}", "9e74db3923fada16eb487a0a5760d265": "\\dot{L}=2\\dot{R} + \\dot{S}=0,", "9e74e0f68ea01412d921498503b26f67": "F = 2((X_1+D)^2-A-E) = 0", "9e74e3fceb59df825b1a0860397aaffc": " A^*\\,\\!", "9e7500069bcd941aacc381ea9d7e6768": "D_i^*=D_i\\frac{\\partial\\ln c_i}{\\partial\\ln a_i}.", "9e754508dc404adaf4e9090a223e72a9": "\\mathbf{v} = \\langle v_1, v_2, \\dots, v_{n - 1}, v_n \\rangle", "9e7597541971019ab16d9e67aaf9263d": " \\sum_{k=0}^{n}\\binom{n+1}{k}(n+k+1)B_{n+k}=0 ", "9e75c10d03250ad2416ee6f3b1adb7b6": " p'_\\infty(x,y)=^hp'_z(x,y,1).", "9e765389534b43a21dc902233b1009c5": "E=\\frac{1}{2N}\\sum_i^N e_i^2", "9e76ea127bcca44295ab886bfc140085": "\n \\log a_{\\rm T} = -\\frac{C_1 (T-T_0)} {C_2 + (T-T_0)}\n ", "9e76fa34e09bf4d6a8cecd3022c97745": "n\\ne 4", "9e77317dd4ba01e5628bd2f7ca052a6e": "\nP_{em}\\propto \\int d\\lambda_{em}\\Phi_{det}(\\lambda_{em})\\textit{f}(\\lambda_{em})<\\mid F_{in}\\cdot e_{ex}\\mid^{2}>\n", "9e7755604e3437edd5ceebc8c0a448ae": " B_n = [z^n] C(z) = \\frac{1}{n} [w^{n-1}] (w+1)^{2n}\n= \\frac{1}{n} {2n \\choose n-1} = \\frac{1}{n+1} {2n \\choose n}.", "9e775ffd78580df333a414f9111401c4": "n \\leq x", "9e77b6f210308fd99a32d9371c101a41": "\\displaystyle{Q(a)=2L(a)^2 -L(a^2).}", "9e77ea3a9ebfb535958aa141c748e9d1": "\\{ H_i\\}", "9e781f122024a6cb452e9b47d6d392f1": "\\Delta u=n^a\\partial_a u=1", "9e78b3847c7dc9220affa62301c03dd0": "\\Phi(E)=e^{\\beta(E-\\mu)}", "9e78d80ca7fe2308b713a804a53a08de": "\\ M_{heel} = D_{heel} \\times (lift \\times cos(\\beta) +drag \\times sin(\\beta) ) = D_{heel} \\times lift \\times (cos(\\beta) +{(L/D)_{\\alpha}} ^{-1} \\times sin(\\beta))", "9e78db25657646df95f0b3debe35ca69": "P(1) = 36.79%", "9e791750216c61a3e74d79bbd22e60c5": "\\begin{align}\n c_n &= c_0^n \\\\\n \\tau_m &= m \\cdot T \\cdot c_0^n\n\\end{align}", "9e7976ba54bbfbc8a63f193de9ac3aa1": "\\Xi(x)=a(x)\\Gamma(x)", "9e79aa038849a0e383ff935e13c0604c": "\\Omega^0_M(\\mathbf V)", "9e79e0f76f3c8f36e5a3decd304ca5cb": "L(P,t) = \\#(tP \\cap L)\\,", "9e79e607904e48d8852db89195be0de8": "y_{\\mathcal{P}}", "9e7a1148dc73e4c81d58fc4abb711227": " (1-\\varepsilon) g(n) \\le f(n) \\le (1+\\varepsilon) g(n) \\, ", "9e7aa8a726bd192820448b073b7d8d14": "\\mathbf{P}(\\omega)=\\varepsilon_0 \\chi_e(\\omega) \\mathbf{E}(\\omega).", "9e7b30ac5ba4d20534790c18cc89743e": "A[4]=S[2,7]=anana$", "9e7b57856461b0c2ecf6e144dbdc68a7": " U_i =\\alpha P_i + \\beta D_i + \\varepsilon_i\\, ", "9e7b7774d47de3d4f12c1fb4c311bb73": "C_P \\phi _P= \\left(\\dot m_s - \\frac {(\\dot m_w)^2} {\\dot m_s} \\right)\\phi_W +{\\left(\\dot m_w+ \\frac {(\\dot m_w) ^2} {\\dot m_s} \\right) \\phi _{SW}} + 0.\\phi_W \\text{ for }45<\\theta< 90 ", "9e7bad698f9b57e2cc4b108cb842bb53": "k= 2 \\pi / \\lambda ", "9e7bb50b274c37517c4a99ba8c323462": "\\psi(\\boldsymbol{r}) = \\sum_{m,\\boldsymbol{R_n}} b_m ( \\boldsymbol{R_n}) \\ \\varphi_m (\\boldsymbol{r-R_n})", "9e7c15d379a406a30d3b8fd9436848cd": "A=\\begin{pmatrix} \\cosh(\\phi) & \\sinh(\\phi)\\\\ \\sinh(\\phi)& \\cosh(\\phi) \\end{pmatrix}.", "9e7c6002038b5480c2abda1548512bca": "V_x = V_{tot} \\times \\frac{P_x}{P_{tot}} = V_{tot} \\times \\frac{n_x}{n_{tot}}", "9e7cdbe32c264f90ad24555c252793b1": "\ndz = {{\\partial z} \\over {\\partial x_1 }}dx_1 \\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_2 }}dx_2 \\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_3 }}dx_3 \\,\\,\\, + \\,\\,\\, \\cdots \\,\\,\\,\\,\\, = \\,\\,\\,\\sum\\limits_{i\\,\\, = \\,\\,1}^p {\\,{{\\partial z} \\over {\\partial x_i }}dx_i }{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(5)}}", "9e7d026a5a9c5b5ba51f253d298dd43a": "q = \\dfrac{q_0}{i}", "9e7d78e5be245147507c07ade9a37f37": "\\frac{{}^\\mathrm Nd\\mathbf a}{dt} = \\frac{{}^\\mathrm Ed\\mathbf a }{dt} + {}^\\mathrm N \\mathbf \\omega^\\mathrm E \\times \\mathbf a", "9e7d9ca196777f3fa3beb0b4f48eefaa": "\\lambda \\leq \\kappa", "9e7db320a72aa00318617f60d996ff88": " \\displaystyle{\\sum_{n\\ge 0} {r^n \\|D^{n \\over 2}v\\|\\over n!} < \\infty}", "9e7dc4cb31d98073c126b25b29295a1b": "A_m(3,3) = 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715,\\ldots = A_{m+1}(3,1)", "9e7de2299a588b01d1822b3cf1e6d3df": "\nP_{DBF} = \\int -d \\hat{z}. \\vec{E}\n", "9e7de2cdcf7d767f66f1a510b57c844b": "SSA = \\frac{A}{V\\rho} = \\frac{3}{r\\rho},", "9e7dedc3fbbf3108697bed4b6bb812fc": "A_1B_2", "9e7e28da7cc869241bc5313f21daa4a2": "x^{\\mu}(\\tau),\\; \\mu \\in \\{0,1,2,3\\}", "9e7e43e7e8009e2a83e2c4accd371e54": " d\\mu_n = \\prod_{i=1}^n (1+a\\cos (3^i 2\\pi x))\\,dx,\\;\\;\\; |a|<1.", "9e7e910b7f277ef29518ab53d26ec8af": "t_{k},", "9e7ed1a6b2b1ed2111b074a9bebbf13e": "Qf", "9e7effcef0af879b2c515e8cbde23bf5": "\\mathbf{x} = \\sum_{j=1}^k (\\lambda_j-\\alpha\\mu_j) \\mathbf{x}_j", "9e7f306b9ce7f5a756413f869be43255": "\\mathfrak{H}^2=\\{x+r i|r>0\\}", "9e7f745edd7288f70247e431753cba26": "p(\\alpha a)", "9e7fd172263685b2cc0051496b40dca7": "\\operatorname{\\beta'}()", "9e806a2a5db3b6769c978790e2ef82cb": "K_i \\varphi \\implies K_i K_i \\varphi", "9e807b61db0ee4440b5a984ab0b7980f": "\\wp\\left(z\\right)", "9e80f98e9fe8d443e5180f514c3c1286": "z^p\\equiv z\\pmod{p^3}.", "9e812a78639c9e392d111becd10d8925": "i \\not= j", "9e820581d2669c371b568e0977d73bc5": " x^1 \\ldots x^n", "9e826e96493d9b42d530819dd6f9b000": "\\Lambda > c ", "9e8296e29ed59ccc2a90d6f3b7363205": "\\int_\\Omega \\alpha \\equiv \\sum_i \\int_{U_i} \\psi_i \\alpha,", "9e829d002b49703f1dcc3fb6722aa132": " e^{- \\rho t} \\sin ( \\mu t) \\quad\\text{and} \\quad ", "9e82cd0a085c081ae95ad27331f07299": "r=a\\cdot e^{b\\theta}", "9e82fbf8c735888ff12fd7c8ac9212a3": "c^2 = a^2 + b^2 - 2ab\\cos(C) + O(c^4) + O(a^4) + O(b^4) + O(a^2 b^2) + O(a^3 b) + O(ab ^ 3) + O(a^3 b^3).", "9e82ff3880de03c30621f2cabf8d4025": " y \\in [q]^n ", "9e83283e81880ed3ea0e9e2e4e86c457": "E[Z] \\ge \\rho", "9e83af591a85e8a71e6d316c5f886733": "\\beta \\in \\{\\alpha:A\\}", "9e83dc67dffebcc7ccd39a14a2f65b6b": "\\begin{align}\n\\left[n_1^2\\sigma_1^2+n_2^2\\sigma_2^2+n_3^2\\sigma_3^2\\right]-2\\left(\\sigma_1n_1^2+\\sigma_2n_2^2+\\sigma_3n_3^2\\right)\\sigma_\\mathrm{n}+\\lambda\\left(n_1^2+n_2^2+n_3^2\\right)&=0 \\\\\n\\left[\\tau_\\mathrm{n}^2+\\left(\\sigma_1n_1^2+\\sigma_2n_2^2+\\sigma_3n_3^2\\right)^2\\right]-2\\sigma_\\mathrm{n}^2+\\lambda&=0 \\\\\n\\left[\\tau_\\mathrm{n}^2+\\sigma_\\mathrm{n}^2\\right]-2\\sigma_\\mathrm{n}^2+\\lambda &=0 \\\\\n\\lambda &= \\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\n\\end{align}\\,\\!", "9e843fb67bc6789da9c00cfa3a364d61": " E = 1 - {\\tau'_{\\rm D}}/{\\tau_{\\rm D}} \\!", "9e845bbb0b9b70dadbf79ed0df7ca715": " A_\\alpha B^\\alpha \\equiv \\sum_\\alpha A_{\\alpha}B^\\alpha ", "9e84c51f46863cdde6b2192a5f61988c": "\\mathbf{P} = \\gamma m_0 \\left(c,\\mathbf{v}\\right) = (m c,\\mathbf{p})\\,.", "9e84d4619310c7e0f161ee22be0f776a": "e^{2 \\pi b}", "9e84ed58888d86d03628e035bc9baca3": "D^{KL} (a||b)= \\sum_i p(a_i) \\log \\Big ( \\frac{p(a_i)}{p(b_i)} \\Big ) ", "9e85dfad8c9b44bed91cb247b639151d": "\\tan\\psi - \\tan\\theta \\tan\\phi \\tan\\psi + \\tan\\theta + \\tan\\phi = 0\\,", "9e86a2a3111a47307e8bf315cda8ea80": "\\langle \\text{Up}, \\text{Left}\\rangle", "9e874b6a4321f1ff18c4395854249f30": "\nB(t) = A(t) - D(t), \\forall t \\ge 0.\n", "9e8769019ed74f9386c957ae7c3f5e4e": "\\ln \\tau_{ij} = A_{ij} + B_{ij}/T + C_{ij} \\ln(T)+ D_{ij} T^2 + E_{ij}/T^2 ", "9e877dac827f07810800a82a317bbd54": "\\,f(u,v) \\le c(u,v)", "9e87827e00a25900272263e7cf17223f": "|x_n|_1 = \\sum_{i=1}^{2^n} 2^{-n} = 2^n \\cdot 2^{-n} = 1.", "9e879396c6dd943f06523c9699124545": "\\exists x \\phi(x)", "9e87ee1cab45b3f1503857b7cac24770": "H \\oplus H", "9e88022d1eec45b2ad38dec2ec872c57": "z=c \\left(a^2 d_a d_c + b^2 d_b^{\\,'} d_c^{\\,'} - c^2 d_c d_c^{\\,'} \\right),", "9e8825e8acb58db5d5341c6bedd8ca0c": "f_{xx}f_{yy} < f_{xy}^2", "9e891185abdde5534d0702d1520d8ae2": "C \\propto \\dfrac{1}{d} ", "9e8979bad6cc86577e4f2dff0e33f365": " ds^2 =r^{-2}(dx^2 + dr^2)", "9e8988b3d3935642f93d86c12fda784d": "\\sqrt[4]{x}_s", "9e89ba0af62e989a44137f6e752a0981": "\\scriptstyle{A = (x + y + z)/3}", "9e89bb5463d2b75e9a3aac037c4020e0": "b(w) = \\frac{w}{g(w)} \\frac {d}{dw} g(w)\n= 1 + \\sum_{n=1}^\\infty b_n w^n", "9e8a68dacabb5faa2761727de7ae52dd": " \\{ \\psi \\in X : T\\psi = \\lambda\\psi \\} ", "9e8a787c9c8fe4ab75e4e228ac5f4a8f": "\\frac {R_\\mathrm{E}} {y_\\mathrm{atm}}\n = \\frac {X^2 - 1} {2 \\left ( 1 - X \\cos z \\right )} \\,;", "9e8a8e3b6f0d5eae3542b43937d554ac": "F(x; \\mu, s) = \\frac{1}{1+e^{-\\frac{x-\\mu}{s}}} = \\frac12 + \\frac12 \\;\\operatorname{tanh}\\!\\left(\\frac{x-\\mu}{2s}\\right).", "9e8b1102014fbd51b3a464efd62f1e8e": "\\tau = \\sqrt{\\epsilon}", "9e8b2c611c008c9c5039eb4e02e21db7": " X_{\\{f,g\\}}= [X_f,X_g], ", "9e8b74603b5b4e8f19aa1e4aca83c45e": "(-\\infty,x)\\times(-\\infty,y)", "9e8bff824903de255509901a4f6e9e4f": "|J_1\\dots J_n|\\le C^p", "9e8c1e095a00b4d7cf9705fc1d9107bc": "\\mathbf{m} = I\\mathbf{A}", "9e8c5087c8eb79beb53458beb2e4375a": "x \\wedge y = 0.", "9e8c5f427f24ec417568a68a7f368697": "c_{rs}\\,\\!", "9e8d05754d262ce6229aaf88f20e11f9": "P = \\left ( \\frac {9.247 m} {1 ~ \\mbox {kg}} + \\frac {3.098 h} {1 ~ \\mbox {cm}} - \\frac {4.330 a} {1 ~ \\mbox {year}} + 447.593 \\right ) \\frac {\\mbox {kcal}} {\\mbox {day}}", "9e8e5882c3b9a51ca595677eb4368760": " \\mathbf{a_{av}} ", "9e8e5e7072dc31aa4243430a8e64c2cf": "\\left( 0,\\ \\pm1/2,\\ \\pm1/2,\\ \\sqrt{1/8},\\ -\\sqrt{3/8}\\right)", "9e8ee437910132df3e52edea297bc87f": "l_{m1}", "9e8f2028357da7082ea7d4f50d7a1cf9": " R_{\\|} ", "9e8f4dddaadd2a3c5869bbfc8ff6c658": "=-\\frac{a}{4\\alpha}\\left[\\left(\\sum_{n=-\\infty}^{\\infty}\\frac{1}{\\pi n + \\frac{k a}{2}-\\frac{\\alpha a}{2}}\\right)-\\left(\\sum_{n=-\\infty}^{\\infty}\\frac{1}{\\pi n +\\frac{k a}{2}+\\frac{\\alpha a}{2}}\\right)\\right]", "9e8f54b90e4590240fdc2d020e3db316": "K_1 \\subset U_1", "9e8f8a272bd0355b91c0f052494c2822": "\\exp_{*}\\colon \\mathfrak g \\to \\mathfrak g", "9e8fc8a26364ecb25933cc5f8abea463": "j = \\frac{\\Delta F}{m \\Delta t}", "9e8fcf021386faf940c6d2a9f95f511c": "\\frac{\\alpha(\\alpha+\\beta-1)}{(\\beta-2)(\\beta-1)^2} \\text{ if } \\beta>2", "9e8fede9a4e7f52a46f47ea510c073e0": "\\begin{align}\n z_{11} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_1}{I_1} \\right|_{I_2 = 0} \\qquad z_{12} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_1}{I_2} \\right|_{I_1 = 0} \\\\\n z_{21} \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{I_1} \\right|_{I_2 = 0} \\qquad z_{22} \\,\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{I_2} \\right|_{I_1 = 0}\n\\end{align}", "9e901f43caa0f5ba3027dc6670661925": "\n= \n{2} \\int_0^{\\infty} {k^2 dk \\over \\left ( 2 \\pi \\right )^2 } {1 \\over k^2 + m^2} \n \n\\left\\{ {1\\over kr } \\sin\\left( kr \\right) + 2 {1\\over \\left(kr\\right)^2 } \\cos\\left( kr \\right) \n- 2 {1\\over \\left(kr\\right)^3 } \\sin\\left( kr \\right) \\right \\} \n", "9e90311ae702955e6b3bf6ae58e844fb": " \\mathbf{F} = q \\left [ \\mathbf{E} + \\left ( \\mathbf{v} \\times \\mathbf{B} \\right )\\right ] ,\\,\\!", "9e904d3ec58f3b8091fd3688fa27c633": "v=\\dot{x}", "9e90cb84984f7bbb4e0c39b3b6b958a1": "(M,d)", "9e90d968b5133658f8078bffb4d95303": "D^5", "9e9131281bd2ddb484d5dc87ccd46021": "Pr[\\sigma \\gets \\mathrm{Setup}(1^k): \\mathcal{A}^{{\\mathrm{Prove}}(\\sigma,.,.)}(\\sigma)=1 ] \\equiv Pr[(\\sigma,\\tau) \\gets \\mathrm{Sim}_1: \\mathcal{A}^{{\\mathrm{Sim}}(\\sigma,\\tau,.,.)}(\\sigma)=1 ]", "9e917ea93868d73b4dec18ad75f50a2c": "f_c \\approx {1 \\over \\pi ({D + d \\over 2}) \\sqrt{\\mu \\epsilon} }= {c \\over \\pi ({D + d \\over 2}) \\sqrt{\\mu_r \\epsilon_r} }", "9e917f7bb35797815b565aafc87f4f2a": "\\langle\\mathbf{u}_1,\\mathbf{u}_2\\rangle = \\left\\langle \\begin{pmatrix}3\\\\1\\end{pmatrix}, \\begin{pmatrix}-2/5\\\\6/5\\end{pmatrix} \\right\\rangle = -\\frac65 + \\frac65 = 0,", "9e920f92e7f7d78b754a37c857da0401": "w_y(x)", "9e9231184031c67ccf5f12bf60cf6673": "O(n\\log k + k \\log n)", "9e92659aa7469e84406e8e042b0347cc": "(x_1, \\ldots, x_k).\\exists x_{k+1}, \\ldots x_m. A_1 \\wedge \\ldots \\wedge A_r", "9e92c4a0306fb02792a8b61f282591c1": "\\begin{bmatrix} d^\\prime \\\\ s^\\prime \\\\ b^\\prime \\end{bmatrix} = \\begin{bmatrix} V_{ud} & V_{us} & V_{ub} \\\\ V_{cd} & V_{cs} & V_{cb} \\\\ V_{td} & V_{ts} & V_{tb} \\end{bmatrix} \\begin{bmatrix} d \\\\ s \\\\ b \\end{bmatrix}.", "9e92d4d766338eae9280bd9d7d08b544": "{}_1F_1", "9e938ddd4a39720e7c15caaa0f304006": "\\varphi I_n \\cdot E= A^\\mathrm{tr}\\cdot E,", "9e94a8a6f145d849ecf4572c2fbdff08": " Var(Y) = V^*(\\mu^*) = \\nu^*_0 + \\nu^*_1 \\mu + \\nu^*_2 \\mu^2 ,", "9e94ebc811aacf8e25390afe018d814a": "2^2\\cdot 3", "9e950ea297830db4e627e9b4fb8d36a7": "\\rho(A)>1", "9e961435e8f8b66d5e14470c3010a512": "a \\parallel v", "9e968e46da5c22aebfdd330c23677c61": "H_{\\omega^\\omega + \\omega + 1}(1) - 1", "9e96a5747e7335de86f2b348992394ca": "\n \\delta W = \\delta U\\,\n ", "9e96cc5c7dc1abbcaa73932ba15f439f": "\n\\begin{align}\n\\Delta \\hat{z}\\ &=\\ 2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\cos i\\ \\left[e_g \\ (1-\\frac{5}{4} \\sin^2 i)\\ \\hat{g}\n+\\ e_h \\ (1-\\frac{15}{4} \\sin^2 i)\\ \\hat{h}\\right]\\quad \\times \\ \\hat{z} \\\\\n&=\\ 2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\cos i\\ \\left[\n\\ e_h \\ (1-\\frac{15}{4} \\sin^2 i)\\ \\hat{g}\\ - e_g \\ (1-\\frac{5}{4} \\sin^2 i)\\ \\hat{h}\\right]\n\\end{align}\n", "9e96f5510f56d86a6be4000fbbf59aa9": "S(tx) = t^m S(x)\\,", "9e96fb173263b2d006654735b63b9d7c": "E_u=100{ {mass\\ flow\\ rate\\ of\\ solids\\ in\\ the\\ undersize\\ stream} \\over {mass\\ flow\\ rate\\ of\\ solids\\ finer\\ than\\ screen\\ size\\ in\\ feed\\ stream}}", "9e97080ebc1cc8b3b914e671d19d38ee": " \\psi (b)", "9e97357651a96fb44c11d8b3bfa8f68c": "(1, 6),", "9e977ec9e726ec83f35ea6ad58ff7c38": "A^k=C^{-1}J^k C\\,", "9e979154574f8ad6f02f6d2655af5d06": "(\\mathrm{Ad}_g)^{-1} = \\mathrm{Ad}_{g^{-1}}", "9e97938e2842e903b88bb5b0b20e6e5e": " \\exist F^n =_{def} \\{x_2...x_n : \\exist x_1 F^n x_1...x_n\\}.", "9e97c92895ae6be27ba1020bff59c6d4": " L_2 = \\{ a^n b^n c^m : m,n \\ge 1 \\}", "9e9802c5bc6e92242dab6bffd3baf39c": "\\vec{\\imath},", "9e9854c22573eab9b5de19122633e0da": "\\langle s,t \\mid s^2, t^3, (st)^4 \\rangle\\,\\!", "9e98bb93f9468da8eb3144a74aa15aac": "\n\\beta _{\\,\\,1} \\,\\,\\, \\approx \\,\\,\\,\\,30\\,\\,\\left( {{{s_T } \\over {n_T \\,\\bar T}}} \\right)^2 \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\beta _{\\,\\,2} \\,\\,\\, \\approx \\,\\,30\\,\\,\\left( {{{s_T } \\over {\\bar T}}} \\right)^2", "9e98cf9caa83930410ea7ce8ab47e62a": "\\!\\mathcal A \\models_{\\{\\emptyset\\}}^- \\phi", "9e98d347df5c3cf45e81225eb7453f78": "\\dot{\\mathbf{x}}", "9e99319a45924a6a17918d62f7454af2": "-\\gamma^{-1}<\\delta<\\gamma^{-1}", "9e994a00383140976290ada4a03ed8af": "\\omega_D=\\omega.\\sqrt{1-\\xi^2}", "9e995290a71091036546bb8396c9e621": "\\{x : \\phi(x)\\}.", "9e9992d6bf50b7580f971487c466a8cb": "G=(V,E)", "9e99afddb32d1be7d9609c9a19b1aa7b": "d^2G = d\\Sigma \\cos{\\theta_\\Sigma} \\frac{dS \\cos{\\theta_S}}{d^2}= \\pi d\\Sigma\\left(\\frac{\\cos{\\theta_\\Sigma}\\cos{\\theta_S}}{\\pi d^2} dS \\right)=\\pi d\\Sigma F_{d\\Sigma \\rarr dS}", "9e99b599be494eb0322038e3556bc68c": "\\mathbf{N}=\\mathbf{n}\\, N = \\mathbf{n}\\, (\\mathbf{T}\\cdot \\mathbf{n}) = \\mathbf{n}\\, (\\mathbf{n}\\cdot \\mathbf{\\tau} \\cdot \\mathbf{n}).", "9e99f1745d892e1d1cdd782817046983": "G^v", "9e99fd209d79c711e5014b943c1ae3ef": " \\Delta P = {(v_2^2 - v_1^2) \\over 2}+\\Delta z g+{\\Delta p_{\\mathrm{static}}\\over\\rho}", "9e9a312d2503ae4fdf417bfe2e65fda6": "\\int_0^\\pi \\frac{\\sin t}{t}\\ dt = (1.851937052\\dots) = \\frac{\\pi}{2} + \\pi \\cdot (0.089490\\dots)", "9e9a40ba62a4be15a5cfa3c616f43084": " \\mu > 0", "9e9a5d460ca0f1dbe5c159fb70409972": "\\sigma_0^2", "9e9a65ef72258206fc5fcf3f9d31ab47": "\\mathcal{C} = -\\mathcal{C}", "9e9a77f058a7e2db4a160f1115751874": " f(x) = \\mathrm{sgn}(x-d) \\, \\Bigg(1-\\exp\\bigg(-\\bigg(\\frac{x-d}{s}\\bigg)^2\\bigg)\\Bigg), ", "9e9aa2355dbab42561fba569f3ed194b": "\\tfrac{1}{2}(q-1)", "9e9ac907dda9a6296a50731fc51b214e": "\\operatorname{atan2}(y, x) = \\begin{cases}\n\\arctan(\\frac y x) & \\qquad x > 0 \\\\\n\\pi + \\arctan(\\frac y x) & \\qquad y \\ge 0 , x < 0 \\\\\n-\\pi + \\arctan(\\frac y x) & \\qquad y < 0 , x < 0 \\\\\n\\frac{\\pi}{2} & \\qquad y > 0 , x = 0 \\\\\n-\\frac{\\pi}{2} & \\qquad y < 0 , x = 0 \\\\\n\\text{undefined} & \\qquad y = 0, x = 0\n\\end{cases}", "9e9b1ead031341020d52ac5756f626ed": "(t,x), Y_t(x)", "9e9ba3b9cf230a5c8e9e29c731ff19a7": "\\gamma_i = 1/r", "9e9bb1036927560d73aa80cf0d68a9fc": "\\mathcal{F} \\geq n", "9e9bc8e3f4bf72d060691021f755e56b": "\\overline {\\sum_{r=0}^n a_r\\zeta^r} = \\overline{0}", "9e9bd83534fd22e0944af9c397ab644e": "\n\\begin{matrix}\n|L| &=& \\sqrt{Q^2+U^2}, \\\\\n\\theta &=& \\frac{1}{2}\\tan^{-1}(U/Q). \\\\\n\\end{matrix}\n", "9e9bd8eb4e31eec1654f3ac0cda516d8": "\\kappa(X+Y,Z_1,Z_2,\\dots)=\\kappa(X,Z_1,Z_2,\\dots)+\\kappa(Y,Z_1,Z_2,\\dots).\\,", "9e9c05dcaa052088d8852dc6ecb1f737": "R = \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2} \\quad \\text{and} \\quad \\sigma_\\mathrm{avg} = \\tfrac{1}{2} ( \\sigma_x + \\sigma_y )\\,\\!", "9e9c15fc7edf6752a13780ccc626e1ca": "\\frac{m_{rel}}{m_{0}}=\\gamma\\!", "9e9c33e7ccb77adbb6517a3eeb6d0ff6": "e^2 = e", "9e9c7a8f2754487dcce4af92b7773074": "\\widehat{x}(n) = \\sum_{i=1}^p a_i x(n-i)\\,", "9e9c8ff0924858538094d0d477dcec46": "s\\ge 2", "9e9c9e5dfe6abf0124f8a3fc2fa72147": "G(\\mathbf X_1)' \\mathbf W G(\\mathbf X_2)", "9e9cc51619ea3ef07187eda44a44a1fc": "\\Delta_g", "9e9dec0eb71d9d283159cd4d00ba903c": "x_1, x_2, .. x_{2n}", "9e9e47106167e4bdca51ba073db458f1": " {R^{\\mu}}_{\\alpha \\nu \\beta } = { 1 \\over r^2 } \\eta_{\\alpha \\beta} {\\delta^{\\mu}}_{\\nu} ", "9e9ebf5c3ce58c8eab07563c0534b1f9": "M = {m_1, m_2,..., m_n}", "9e9ec3e688df00aadaee7e864a9ff9b8": "\\ w'", "9e9ec719351dd43e684eeda56ce2310f": "G=2\\pi/a", "9e9f2759b7b3c98dc27ee886186c75dd": "C_H\\;", "9e9f2bcb7c5e9309bbc84a216028db3b": "\\{x_{ij}\\}_{n\\times k}", "9e9f55138c6402369fda1a21819f6687": "w\\,\\Delta{z} = \\operatorname{cov}(w_i, z_i) + \\operatorname{E}(w_i\\,\\Delta z_i)", "9e9f5e0568d90d85d785007ff972d43f": " F = \\begin{bmatrix}\n 0 & 0 & 0 \\\\\n 1 & 1 & 0 \\\\\n 1 & 0 & 1\n\\end{bmatrix}.", "9e9f664107d49dc0f796fd5cf48b543b": "T^{(FRL)} = \\left[1 - f - e(1-b)^{-1}c \\right]^{-1}d + \\left[1 - f - e(1-b)^{-1}c \\right]^{-1}e(1-b)^{-1}a,", "9e9f7594a9738c275ad4d93a830cebe0": "s+b \\geq 3", "9e9fb6f1b7016da2937cd6120ae465c1": "| \\gamma' | (t) = \\| \\dot{\\gamma} (t) \\|,", "9ea028a4b61fbcb6b5039ca342f8a3be": "\\displaystyle{(f_1,f_2)=\\sum_{g,h} (\\Phi(h^{-1}g)f_1(g),f_2(h)).}", "9ea0304df8d159432044a219da51fd22": "f(E) = \\frac{1}{1 + e^{\\frac{E-\\mu}{k_{\\rm B} T}}}", "9ea049cf847bf8b78dadef2c0c7768fd": "|X| \\leq k", "9ea062a1eb66933f2bde16be26d7d0e1": "f(z)=\\sum_{i=0}^n u_i z^i,\\quad\\quad g(z)=\\sum_{i=0}^n v_i z^i.", "9ea06c2a6b8075a9c56fa5bdff663be2": "a=ga_1", "9ea0d587dfe9aec25624409c02f05c10": "(x_1,x_2,x_3)", "9ea0f6924c3564f91fa9efc4e7634695": "\\langle X, \\mathcal{F}, \\mu \\rangle", "9ea10f05e9035f71db3f0d5f014074b1": "B, f'_t, D_0", "9ea1167db3ca55351e9d543304ebf14a": " V = \n\\begin{bmatrix} \\cos \\theta & \\sin \\theta \\\\ \n\\ \\sin \\theta & - \\cos \\theta \n\\end{bmatrix}. ", "9ea12be004d107eec8391badfc80206f": "\\scriptstyle \\tilde{\\nu}", "9ea1b3153953a99139fd2fc0dbec77d2": "\\textstyle d(x) = 0", "9ea1b4b12fbbd75b6cb688f62e0230fe": "\n\\mathrm{var}\\left( \\hat{A} \\right)\n\\geq\n\\frac{1}{\\mathcal{I}}\n", "9ea1d8256dceeac9ed47a843430f4d27": "\\sqrt{2} \\approx 1 + \\frac{1}{3} + \\frac{1}{3 \\cdot 4} - \\frac{1}{3 \\cdot4 \\cdot 34} = \\frac{577}{408} = 1.4142...", "9ea251a8ca1895d2c1229b7c62f863d3": "\\Pi_{\\mathbf{f},0}|_{\\mathrm{Sp}(N)}= \\bigoplus_{\\mathbf{h}, \\,\\,\\mathbf{g},\\,\\, g_{2i-1}=g_{2i}} M(\\mathbf{g}, \\mathbf{h};\\mathbf{f}) \\sigma_{\\mathbf{h}}", "9ea2614326a4fe5c66001c064d3251e1": " \\lfloor nt \\rfloor ", "9ea281dd6237c349e8f4b249fdaaced3": "\\left\\vert \\Phi^{+}\\right\\rangle\n^{\\otimes n}", "9ea2ea3a09bd4c4960c10422b8d6d252": " 1\\longrightarrow N \\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\beta}\\ \\, G \\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\alpha}\\ \\, H \\longrightarrow 1", "9ea2ec72804c5ce84c737774b38b1739": "U{}^n_n", "9ea301e4a7e442cc08fcfa60db05320b": "\nf' = f { 1- v/c \\over 1- s/c }\n\\, .", "9ea31b5f640327ba19ddedcb73186dc4": "C_{ij} = \\sqrt{C_{ii}C_{jj}}", "9ea31d7c01bd9caec47fc6a721f819e4": " \\oint_{\\partial \\Sigma(t)}\\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{F}/q(\\mathbf{r},\\ t) = - \\frac{\\mathrm{d}}{\\mathrm{d}t} \\iint_{\\Sigma(t)} \\mathrm{d} \\mathbf {A} \\cdot \\mathbf{B}(\\mathbf{r},\\ t). ", "9ea3370156330d89daf8ad565f008495": "\\sin\\alpha\\sin\\beta={1\\over2} \\left( \\cos(\\alpha-\\beta)-\\cos(\\alpha+\\beta)\\right)", "9ea346810a1fdd5d81de085307e817d1": "\\implies U^'_i(\\frac{w_i}{\\sum_jw_j}*B)(\\frac{1}{\\sum_jw_j}*B-\\frac{w_i}{(\\sum_jw_j)^2}*B)-1=0", "9ea35880a80b5cabf0419f5b31714fb5": "\\int \\delta_\\epsilon \\left(\\mathcal{F} e^{iS}\\right) \\mathcal{D}\\phi = 0", "9ea38880f54db2137434669f99462c11": "\\mathbb E", "9ea3e18c1596db0608d5207bbd0334cc": "X_k", "9ea4200bf5269a30b09806146430abc6": "X_1|X_2", "9ea43a8d433c115900aaf69e822d412f": "e^{\\pi\\sqrt{163}} \\approx (2^6\\cdot 10005)^3+744", "9ea486a1c46e2d0288b88c95acdf3653": "\\Gamma(s, z)", "9ea4960edf8900b20b3530ca7507884c": "v\\sim w_n\\,", "9ea4a4930e538a6f288293b8ce9824ae": "{l_1}", "9ea529f5e43a4185cc059fbf4f858239": "f_{dp}=2f_1-f_2", "9ea59370def17a6317c4f1f76fcb44d5": " \\lambda_g ", "9ea5a0043e5094903e2cb832ece5ff3c": "|S|\\times n", "9ea60b7bd9ce87a7abde5f1ed39db081": "\\|S \\otimes T\\| \\leq \\|S\\| \\|T\\|", "9ea62e5d98eaa38b021271ba3753dd93": "\\Gamma(3) = 2,\\,", "9ea64207821b53534392097fd3a49f84": "|E_n| \\leq \\left[ 2\\varepsilon + O(n\\varepsilon^2) \\right] \\sum_{i=1}^n |x_i| ", "9ea6589870e95f314faa3bff99e09ae4": "f^{i}X_{i}\\,", "9ea6665ca58659b737d6ebeb3d8970b6": "L_{x' x'}(t,x(t),x'(t)) \\ge 0, \\, \\forall t \\in[a,b]", "9ea6fa26b2416c769dd9ebcf060c67c3": "\\begin{align}\\operatorname{arcoth}\\, x = \\operatorname{artanh} \\frac1x & = x^{-1} + \\frac {x^{-3}} {3} + \\frac {x^{-5}} {5} + \\frac {x^{-7}} {7} +\\cdots \\\\\n & = \\sum_{n=0}^\\infty \\frac {x^{-(2n+1)}} {(2n+1)} , \\qquad \\left| x \\right| > 1 \\end{align} ", "9ea7015bda2ef15217d1540663790d62": "\\scriptstyle \\epsilon", "9ea705e170a679cf6eea4866cb9ede15": "d = \\sqrt{h(2R + h)} \\,.", "9ea769907bf8778a3daf3627d821656f": "J^k_0 (\\varphi\\circ f)=J^k_0 (\\varphi\\circ g)", "9ea799abde7728a733dbda0f2206c558": "\\operatorname{BHN}=\\frac{2P}{\\pi D \\left(D-\\sqrt{D^2-d^2}\\right)}", "9ea7b5e6eea19dc2b54e967d75bd3244": "\\sigma_{t}^{2}", "9ea7c197f9a4a413e89de9ff3ddb6577": "\n (\\boldsymbol{\\nabla}\\cdot\\boldsymbol{T})\\cdot\\mathbf{c} = \\boldsymbol{\\nabla}\\cdot(\\mathbf{c}\\cdot\\boldsymbol{T}) ~;\\qquad\\boldsymbol{\\nabla}\\cdot\\mathbf{v} = \\text{tr}(\\boldsymbol{\\nabla}\\mathbf{v})\n ", "9ea7dc54b5fe1a6ab22a4a5fd52edbe7": "x = 3k + 1", "9ea7eb10b8bf795f8cd724bc7dcf453d": "O(m + \\log n)", "9ea7fdaaef1a3e4763957c73e99d4266": "\\pi \\approx 3.14", "9ea84568aa6de56e3180464628a514b6": "N^{3}\\ge \\mathrm{Re}^{9/4} = \\mathrm{Re}^{2.25}", "9ea88701598dbc4b7205bfa7983f6bcb": "s \\in \\mathbb{R}", "9ea888f58114108ed02fea555f01fe8c": "\\xi_i", "9ea8eecd32614712ebcb25d07892349c": "\\sum_{d \\mid n} \\frac{\\mu^2(d)}{\\varphi(d)} = \\frac{n}{\\varphi(n)}", "9ea91a972b6f4c82f69cdf4da0717aea": " \\mathbf{S}(\\mathbf{p}(t)) ", "9ea944d94bc4fc086c870ddf0e893424": "x^2>3", "9ea99a35954b8b91afc720657330ae43": "\\operatorname {somb} (\\rho) = 2 J_1(\\rho) / \\rho.", "9ea9c0e014484bb53ccbb8d319928e04": "g\\left( \\hat{x},t\\right) >0", "9ea9cb9ce42bf75cea56016fd7a5b66c": "y(x) = 0", "9eaa2726adbfc7e5edf6f35ae837bed8": " S \\subseteq N ", "9eaa337f1c54f63f4e7e0c1def9bfa1b": " \\lambda x.\\lambda y.p\\ x\\ y ", "9eaa475104da650bbea04b5df69e619f": "f_m(x) \\neq 0 \\,", "9eaa4d5d7dfae7280d077c0bd1a26b19": "\\forall a, b \\in X,\\ a R b \\Rightarrow \\; b R a.", "9eaae50dd58cb5eb4408719bdf1db8a4": "\n\\mathcal F_t^* = \\sigma\\left\\{ X_s^{-1}(B) : s\\in[0,t], B \\in \\mathcal E^*\\right\\},\n", "9eabbdc567c21bc00513c2edd97516e0": "\\displaystyle (c)", "9eabd4b901fec3bc64aadb06173a71fb": "\\cos\\theta = \\frac{g(\\mathbf{a},\\mathbf{b})}{\\|\\mathbf{a}\\|\\,\\|\\mathbf{b}\\|}.", "9eabfa8387cd9943aa421ca3007d08a2": " (2) ", "9eac648a02c786c8717ef781d4584db7": "\n\\omega_\\vec{p} =\n\\begin{bmatrix}\n\\phi \\\\ \\frac{\\vec{\\sigma}\\vec{p}}{E_{\\vec{p}} + m} \\phi\n\\end{bmatrix} \\;,\n", "9eac747796073d134dee5c604d6a10d3": "\\mu_1 \\neq \\mu_2", "9eac932668c1f7cd6c9d73cc50daa857": "= \\bar{x} - (\\text{SE}\\times 1.96) .", "9eacb74f953e94d3e37aeadfd3a222e6": "{1 \\over i \\alpha \\, Re} \\left({d^2 \\over d z^2} - \\alpha^2\\right)^2 \\varphi = (U - c)\\left({d^2 \\over d z^2} - \\alpha^2\\right) \\varphi - U'' \\varphi", "9eace4ea60b7b147ae4f31fdccbe5af3": "H(s) = \\frac{1}{p_L(s)}, \\qquad p_L(s) \\neq 0.", "9eace7db83a6f75b531903786f449bf9": "R^\\textrm{op}", "9ead22c4ee19ed873b6939d50cb0af2d": " \\bold{F} = \\frac{1}{2}F_{pq}\\bold{\\theta}^p\\wedge\\bold{\\theta}^q.", "9ead5466b934304e70e36ac8727ad023": " F = \\frac{\\phi_2-\\phi_1}{x_2-x_1} = \\frac{\\Delta \\phi}{\\Delta x}\\,\\!", "9ead80be4dca2866b87101c9157060da": "E_{n\\,j}", "9eada7e7b1091d9eff9a88f6ed825ba5": " K_s = \\gamma p = \\gamma \\rho R T = \\,\\rho a^2", "9eae58460933addf4e33a5853fa4673b": "\\eta(s) = \\sum_{n=1}^{\\infty}{(-1)^{n-1} \\over n^s} = \\frac{1}{1^s} - \\frac{1}{2^s} + \\frac{1}{3^s} - \\frac{1}{4^s} + \\cdots", "9eae6fefb8864711d6d92e4a0788cfda": "1+k_i4^{k_i}", "9eae795aaea86164c2bb1d29974ebb5c": "\\scriptstyle{G_a}", "9eae8058de2402d27136e423d693d967": "E = Z_1\\cdot Y_2 ", "9eaeac6712e368a4769b46b3a5b47cc6": "V_{TE}=\\frac{X_m}{\\sqrt{R_s^2+(X_s+X_m)^2}}V_s", "9eaeca06029333dea308cb403f4d0df7": "\\Omega (|v_0|)", "9eaf59ead43cb33a5980c257e8fd481a": "Nc/4", "9eaf5dd22ff36013ae07c423a9210fed": "4\\psi(a)a \\equiv 1\\ \\text{mod}\\ c ", "9eaf7fdaec4d94867dfde9f49db41008": "(1-c)(1-ee)=1-c(1-ee'')", "9eafab1e3ce5cec6b38725a91751c6b5": "\n \\displaystyle \n (\\omega)\n \\longleftrightarrow\n S(0,2)\n =\n \\varnothing\n", "9eb03b3c1af70ed97a733917fb62e43f": "\\quad =\\, \\frac{\\pi}{\\,2n\\,}\\sec\\frac{\\,\\pi\\,}{2n}\\!\\cdot\\ln \\pi + \n \\frac{\\pi}{\\,n\\,}\\cdot\\!\\!\\!\\!\\!\\!\\sum_{l=1}^{\\;\\;\\frac{1}{2}(n-1)} \n\\!\\!\\!\\! (-1)^{l-1} \\cos\\frac{\\,(2l-1)\\pi\\,}{2n}\\cdot \n\\ln\\left\\{\\!\\frac{\\Gamma\\!\\left(1-\\displaystyle\\frac{2l-1}{2n}\\right) }\n{\\Gamma\\!\\left(\\displaystyle\\frac{2l-1}{2n}\\right)}\\right\\} ,\\qquad n=3,5,7,\\ldots \n", "9eb06217fa8d212e077fe7a4f8b341cb": "\\displaystyle N_{m}\\Phi _{m}=\\sum\\limits_{n=1}^{K}L_{m,n}i_{n}.", "9eb09f73a17ba85715ec5cbac7b46689": "\\operatorname{SL}(2,5) \\cong 2\\cdot A_5 \\cong 2I,", "9eb0c264062980c6695b8de581253cfa": "b^2 = c^2 + a(2b\\cos\\gamma - a)\\,.", "9eb0d72f3ab37f7f7fba027a953dcbe1": "\\overline{Q}_N", "9eb101c644a7ee95b7b72b10b2556cfd": "\\frac{}{}|g\\rangle", "9eb107e6f431bfe141afa6260ced65cd": "(\\Omega_1,\\Sigma_1,\\mu_1)", "9eb13b5fbd8e1d28a98013b2917e9078": "y^*(p)", "9eb2241cf7251500c2da6f3203e714d8": "L = \\lim_{n \\to \\infty} x_n\\Longleftrightarrow \\forall \\epsilon>0\\;, \\exists N \\in \\mathbb{N}\\;, \\forall n \\in \\mathbb{N} : n >N \\rightarrow |x_n-L|<\\epsilon.\\; ", "9eb3356ac219d5dce0e27f677f3cbc9d": " L_k \\mathbf{z}_i ", "9eb3f337e025a6ce43da21c25cefced5": "(z\\bar{z}-a^2)^2 -4a^2(z-a)(\\bar{z}-a)=0", "9eb4b400a5517fdafd9a9cb9ebb4902c": "\\ (\\mathrm{ApEn})", "9eb4d1cd750f7aee0137d10e06f0f88c": "Ddf - df\\otimes df +\\left(|df|^2 + \\frac{\\Delta f}{n-2}\\right)g = \\operatorname{Ric}.", "9eb4e3d88d4a2b21b011b7ab2932d680": " \\dot {\\mathbf r} = (\\dot x, \\ \\dot y ) = \\dot r (\\cos \\varphi ,\\ \\sin \\varphi) + r \\dot \\varphi (-\\sin \\varphi ,\\ \\cos \\varphi) = \\dot r \\hat {\\mathbf r} + r \\dot \\varphi \\hat {\\boldsymbol{\\varphi}} \\ , ", "9eb4f598c35454e605159af475810834": "f(\\boldsymbol{x})=\\|\\boldsymbol{x}\\| = \\sqrt{x_1^2 + \\cdots + x_n^2}", "9eb5f3333b50948e82cab2c627f17f53": "\\nu = 1/3", "9eb62af67156096c0b9e8bb0f9e4174f": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} = RT \\ln \\left(\\frac {Q_r}{K_{eq}}\\right)~", "9eb63ffacb0cebac7339b0db91f7ad1b": "\\mathfrak{P}^{38}", "9eb6494f5038f19ffd2e98a977417528": "\\widehat\\mu", "9eb6bd4b6670d14861d211c71f28bc82": "f(z)=\\frac{1}{2\\pi i}\\int\\nolimits_{\\Gamma} \\frac{f(\\zeta)}{\\zeta-z}\\,d\\zeta", "9eb6c66879fdd00f2243619b3a163067": "\\vert \\psi\\rangle", "9eb6e32cbcada434fdd7ce7307956aa2": " \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}^{-1} = \\begin{bmatrix} E & F \\\\ G & H \\end{bmatrix}. ", "9eb702f5297f09e367caf2b198ce1355": "E_*(X,A) = E_*(X/A,*) = \\tilde E_*(X/A)", "9eb7356702bd3fddc17c638f111c26a3": "o_1, o_2, \\ldots, o_T", "9eb73dfc13ca77d1f8ae96b3987c5641": "O(k^3 n^2 \\log(B))", "9eb77c093b9466f2e13bf20cce47ccb7": "q_4:=\\frac {q_2-q_1+2m_2}{2q_2}", "9eb82f73e7de11b3f345bf29489e9054": "\n\\frac{p(h|\\Theta)}{p(h|\\Theta_{bg})} = \\frac{p_{\\mbox{Poiss}}(n|M)}{p_{\\mbox{Poiss}}(N|M)} \\frac{1}{^nC_r(N,f)} p(b|\\Theta)\n", "9eb83eb392653d7269a3cf45bd39cfd4": "\\Box \\mathbf{A} + \\nabla \\left(\\frac{1}{c^2}\\frac{\\partial \\phi}{\\partial t} + \\nabla\\cdot\\mathbf{A}\\right) =-\\left(\\frac{mc}{\\hbar}\\right)^2\\mathbf{A}\\!", "9eb92a82f5512948f07a820c24f568f9": "G-a", "9eb9a93f1f20ff52c9c027932f63c9de": "\\overline{N}(f)", "9eb9efcbfd3ded9f5d048f195eacf6d7": "h \\left( (1-\\lambda)x + \\lambda y \\right) \\geq f(x)^{1 - \\lambda} g(y)^{\\lambda}", "9eba22b1ca44141c77875b6f49ebfc11": "\\sin^2 \\theta \\, d\\theta.\\ ", "9eba439a50d0543ed918dcd1865c4c7b": "0\\leq \\epsilon\\leq 1/2", "9eba4b303ca32866270829f570fb78b3": "(+\\text{d}y,-\\text{d}x)", "9eba5ca8f45515a5b143d6adf069ee04": "T(\\omega_k)", "9ebae8f0f5f4947e03ced9c9c4e44d82": "\\dot{\\rho}_{f} = -3 H \\left( \\rho_{f} + w_{f} \\rho_{f} \\right) \\,", "9ebb07726e001c8f025b25986a308601": "x_i,w_i", "9ebb125b6985fc177aaae3b022f8671f": " -\\infty < t < \\infty, \\; 2 m < r < \\infty, \\; 0 < \\theta < \\pi, \\; -\\pi < \\phi < \\pi", "9ebb886ba2626a02dd8290694199a87e": "A \\succeq B", "9ebbb24e4f4103ba591e746f18b7bab2": "X \\rightarrow Y \\in T \\land Z \\subset X~\\Rightarrow~Z \\rightarrow Y \\notin S^+", "9ebbdeba8cd733369e042bd59d6ce5c7": "\n\\sum_{n = 0}^\\infty \\frac{(-1)^{n}}{2n+1} \\;\\;=\\;\\; 1 \\,-\\, \\frac{1}{3} \\,+\\, \\frac{1}{5} \\,-\\, \\frac{1}{7} \\,+\\, \\cdots \\;\\;=\\;\\; \\frac{\\pi}{4}.\n", "9ebc366b8714d1791326c494a365df35": "\\textstyle \\nabla_{\\perp}^2 \\stackrel{\\mathrm {def}}{=} \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2 }", "9ebc3a197aecdf076de462371de18993": "h = 6.626\\ 069\\ 57(29)\\times 10^{-34}\\ \\mathrm{J \\cdot s} = 4.135\\ 667\\ 516(91)\\times 10^{-15}\\ \\mathrm{eV \\cdot s}.", "9ebc78f5ad4c102b1e38a3052f1dc9d9": "3n^2 + 1, \\ n>1", "9ebc87b46234b4ce3daca61bd085e6d0": "M_{UT} = \\begin{bmatrix}1.823 & 0.043 \\\\0.043 & 0.012\\end{bmatrix}", "9ebcc38353b4be4842f4b8ae47f7730a": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} =\n \\int_{-h}^h x_3~\\begin{bmatrix} C_{11} & C_{12} & 0 \\\\ C_{12} & C_{22} & 0 \\\\\n 0 & 0 & C_{66} \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix}\n dx_3 = -\\left\\{\n \\int_{-h}^h x_3^2~\\begin{bmatrix} C_{11} & C_{12} & 0 \\\\ C_{12} & C_{22} & 0 \\\\\n 0 & 0 & C_{66} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} \\varphi_{1,1} \\\\ \\varphi_{2,2} \\\\ \\frac{1}{2}(\\varphi_{1,2}+\\varphi_{2,1}) \\end{bmatrix}\n", "9ebcd32d8538cedf5d75b222a28a6b99": "\n \\mathcal{E}^{ijk} = \\cfrac{1}{J}~\\varepsilon^{ijk} = \\cfrac{1}{\\sqrt{g}}~\\varepsilon^{ijk}\n", "9ebcfb3e7be9e12dd103e3f603d6c2ba": "w''\\notin W^{\\mathfrak{p}}", "9ebd434e1741d72c3def95ce7c26d57a": "\\operatorname{E}(|X^n|) = \\int_{-\\infty}^\\infty |x^n|\\,dF(x) = \\infty,\\,", "9ebd4b2599e5978133a0400fe7994b2f": "\\overline{P}(Cl_2^{\\leq})=Cl_2^{\\leq}", "9ebd6291739f2fe604f88c507ad86ea4": " u = \\sum_i f^i S(f_i)", "9ebd83865179e8571d526628f8841274": "\\frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2)", "9ebd8b8c9ccfb1a03b9defdfc86201e9": "x \\mapsto x^p. \\, ", "9ebdb74a67b5797ee770a1e024f09847": "\\mathbf{b}(x)", "9ebe20e92a653c5eb4f8cf32d11d267a": "d_t", "9ebe228ccbfe8ce3e18fa4511e4496d6": "(\\phi \\leftrightarrow \\chi) \\to (\\phi \\to \\chi)", "9ebe77a9bca7e04794bb0e0bd1463076": "\\mu_r'-1=(\\frac{\\lambda_g^2+4a^2}{8a^2})(\\frac{f_c-f_s}{f_s})(\\frac{V_c}{V_s})\\,", "9ebe8a94011e3d3a72663c21c32a86d2": "\n\\dot{\\mathbf{X}}(t) = \\mathbf{A} \\cdot \\mathbf{X}(t) + \\mathbf{B}\\cdot \\mathbf{u}(t)\n", "9ebea4467b2631dcd5a8e07bd7c1272f": " \\left(\\frac{\\rho N D^2}{\\mu}\\right)", "9ebf18866291b4816c5e1490997a0770": "\\phi_B(v)=[v]_B", "9ebf2b7fb95f1fb58112841818f8059e": "\\boldsymbol{p}_k", "9ebf2cbecabe93829c33957e5c7c79f8": "\\left\\{ {W_\\alpha ^i } \\right\\}_{i = 1}^n ", "9ebf4ee5b60c3ee2c0ae921bcb3ecd51": "L_0 = 1", "9ebf5a47467a70a34da8d47131c58b70": "Mor((X, \\alpha), (Y, \\beta)) := \\{ f : X \\vdash Y \\mbox{ }|\\mbox{ } f \\circ \\alpha = f = \\beta \\circ f \\}", "9ebf5d475c03b41fb462b7cfd530cef5": "\\exists_{x_1} \\dots \\exists_{x_n} (\\varphi_1 \\land \\varphi_2 \\land \\varphi_3 \\land \\varphi_4)", "9ebf8c26c0d3b9c54e1aa109590d34f2": "\\ e ", "9ebff33d1abc26faab83a8a4a0c79bc2": " \\mathcal A _\\infty", "9ec03ab6f18908e83456eb6190366904": "\\lim_{n \\to \\infty}(f(n))^{n^{-2}}=\\left(\\frac{4}{3}\\right)^\\frac{3}{2}=\\frac{8 \\sqrt{3}}{9}=1.5396007\\dots", "9ec04d87d905f6e0d33d6b68ad5ce4c2": "f(y_1, z_2 + y_1^r, z_3 + y_1^{r^2}, ..., z_m + y_1^{r^{m-1}}) = 0", "9ec0a630e6ff8ee9f8d3b26b32b79a02": "L : C \\to C[W^{-1}] \\to C", "9ec0d33fdaab4e73476d49b8aeacaf69": "BA=I", "9ec1024ef5b42194573124f01589fd3c": " W = -\\frac{1}{2n^2}\\quad \\hbox{with}\\quad n \\equiv k+l+1 . ", "9ec110fe7ec6db3f092e6ea0b2b3c875": " f^{-1}(y) = \\frac{(y-f_{n-1})(y-f_n)}{(f_{n-2}-f_{n-1})(f_{n-2}-f_n)} x_{n-2} + \\frac{(y-f_{n-2})(y-f_n)}{(f_{n-1}-f_{n-2})(f_{n-1}-f_n)} x_{n-1} ", "9ec11975a786cb0ade135f75d512b681": "w_t^{[m]} = ", "9ec1415d09c5730e7fb7e7cd9a94eae0": " A_i", "9ec17b5cc16d500362c7e64a26827e97": "\\left|\\nabla f(\\mathbf{x}_k)\\right|", "9ec189ad21dc84f1fddf3a8185cbcb62": "R_{T(\\alpha)}", "9ec1b4c564611ddac674c52dad206f2d": "\\textrm{RSS}=y^T\\left(I-H\\right)^T\\left(I-H\\right)y.", "9ec1d94a9a876ed1073a396135f362c7": "\n\\{f, \\phi_j\\}_{PB} \\approx 0,\n", "9ec1e16e346711af1d23ae4237c2b525": "V_{TN}", "9ec2252f75c466bef76f8fd9bfed2275": "J=D/(2-D)", "9ec241bfce26606b832825db74e27c56": " L_{p} = \\{ A \\in K(H) : \\left( \\sum_{n=0}^\\infty \\mu(n,A)^p \\right)^{\\frac{1}{p}} < \\infty \\}, ", "9ec259d86332c46f4af78f368b7f123e": "\\scriptstyle\\sigma^2=1/\\sum w_i", "9ec293e981d72ef6d8d4d5430026d184": "U: K_1 \\rightarrow K_2.", "9ec2a14562f126848c73cdedf4af4469": "x - y > c", "9ec2a31087295151f21f26fa3b9dc999": "\\mathrm{df}_i", "9ec2b1980de86ed8b875aca285ce40ca": "\\mathbf{B} = \\frac{\\omega^2}{4\\pi\\varepsilon_0 c^3} (\\hat{\\mathbf{r}} \\times \\mathbf{p}) \\frac{e^{i\\omega r/c}}{r}", "9ec2b7e9c44e4feb2ed127bcc28db61b": "\\delta_{s_i}(X) = \n\\begin{cases} \n1 & \\mbox { if } s_i \\in X\\\\ \n0 & \\mbox { if } s_i \\not\\in X\\\\ \n\\end{cases} \n", "9ec2c427e024ec4e9c0bfcee0494cd8e": "u(c(t))", "9ec30868fafa373d355cebf04cca6714": " \\begin{align} \n\\frac{\\partial \\rho}{\\partial t} & = \\frac{1}{i\\hbar } \\left [ -\\frac{\\hbar^2\\Psi^{*}}{2m}\\nabla^2 \\Psi + U\\Psi^{*}\\Psi \\right ] - \\frac{1}{i\\hbar } \\left [ - \\frac{\\hbar^2\\Psi}{2m}\\nabla^2 \\Psi^{*} + U\\Psi\\Psi^{*} \\right ] \\\\\n & = \\frac{1}{i\\hbar } \\left [ -\\frac{\\hbar^2\\Psi^{*}}{2m}\\nabla^2 \\Psi + U\\Psi^{*}\\Psi \\right ] + \\frac{1}{i\\hbar } \\left [ +\\frac{\\hbar^2\\Psi}{2m}\\nabla^2 \\Psi^{*} - U\\Psi^{*}\\Psi \\right ] \\\\\n & = - \\frac{1}{i\\hbar } \\frac{\\hbar^2\\Psi^{*}}{2m}\\nabla^2 \\Psi + \\frac{1}{i\\hbar } \\frac{\\hbar^2\\Psi}{2m}\\nabla^2 \\Psi^{*} \\\\\n & = \\frac{\\hbar}{2im} \\left [ \\Psi\\nabla^2 \\Psi^{*} - \\Psi^{*}\\nabla^2 \\Psi \\right ] \\\\\n\\end{align} ", "9ec3acba5f954faf6d423a4d9645909c": " \\frac{p^{\\gamma -1}}{T^{\\gamma}} = \\mbox{constant} ", "9ec3e65a53670d1e23fae033c83803bb": "L-1", "9ec3eff42c04d6d161746ea7547947c0": "A^{2}", "9ec455bd551ece07b97b5c828619c6f1": "f(z+1) = f(z)", "9ec46aaf900a0b35cbf1a9de73408663": "\n\\mathcal{N} \\int \\left[ e^{-\\beta H(p, q)} x_{k} \\right]_{x_{k}=a}^{x_{k}=b} d\\Gamma_{k}+ \n\\mathcal{N} \\int e^{-\\beta H(p, q)} x_{k} \\beta \\frac{\\partial H}{\\partial x_{k}} d\\Gamma = 1,\n", "9ec4c0afd450ceac7adb81c3bcfc9732": "L1", "9ec516d754471b4126028f80ad06aae3": "\\R \\mathbb{P}^3", "9ec53f5582ae8b65a89816b77080e3b1": "(\\mathbf{S}, \\mathbf{M}, \\mathbf{P})", "9ec5471e5812aba75e803dd82dd82e3d": "\\alpha_{11} > 0", "9ec57fe5dffd875c7722f30a88570b3c": "\\,\\!\\theta_{n}", "9ec5978078bee07c0666ea67d8e8934a": " \\frac{p}{(36+n)}", "9ec5a89935a70a2285b2bda9a8ff882c": "f(x, \\boldsymbol \\beta)=\\beta_1 + \\beta_2 x +\\beta_3 x^2", "9ec5be08e0b44a3e87b7cb05be0fa97c": "\\phi_V = \\int_S \\mathbf{u} \\cdot \\mathrm{d}\\mathbf{A}\\,\\!", "9ec5dafdd18b4a03aa23dc6eb2c2e307": "\\mathcal{M}_0 := \\left\\{z \\mapsto \\frac{uz - \\bar v}{vz + \\bar u} : |u|^2 + |v|^2 = 1\\right\\},", "9ec5f4e5b95df6c3db738a878938f24b": "\\pm g^2/\\Delta", "9ec5fcd8a53e82849093ecd2a8eca677": "\\mathcal{L}=\\frac{1}{2}(\\partial_\\mu\\phi)^2-\\frac{m_0^2}{6g^2}(2e^{g\\phi}\n+e^{-2g\\phi})\n", "9ec5ff4b3d6aa17289b251ffbac9b8bd": "M \\hookrightarrow D^4", "9ec6045bdc3e9e4c40ff0b9d28be99f8": "N = \\{b_1,\\ldots,b_n\\}", "9ec61336173129b39919128678b9b1df": "\\bar B\\subset \\bar A", "9ec615dde6abccb28965accfc9aa0fab": "\\theta= \\pi", "9ec645a9426afb40c36f31bb9b3fe2ee": " SU(4,H)_L\\times H_R = Sp(8)\\times SU(2) \\supset SU(4)\\times SU(2) \\supset SU(3)\\times SU(2)\\times U(1) ", "9ec652b10e94e27a9514581a5b21055f": "x^{n-1}+x^{n-2}+\\dots +x + 1 =(x-\\zeta_n)(x-\\zeta_n^2)\\dots(x-\\zeta_n^{n-1}).", "9ec6a0eeebcd01dea9ad911f7cbb4492": "\\Psi = z\\, u_b(\\xi) - \\frac{z^3}{3!}\\, u_b''(\\xi) + \\frac{z^5}{5!}\\, u_b^\\text{iv}(\\xi) + \\cdots,", "9ec6dac4cf160bde201143aee5759632": "\\sqrt{\\sqrt{10} \\cdot \\sqrt{100}} = \\sqrt[4]{1000} \\approx 6 \\,", "9ec7591768257656f7afdaaf32063325": " L(n,3) = (n-2)(n-1)n!/12", "9ec7994711d1cc780fafb518dd5fd4ad": "{\\mathbf v}= v^ie_i", "9ec7c5fe52709b02d582a134fc29eea2": "\\sum_ {} \\vec{F} = \\vec{T} + m\\vec{g} = 0", "9ec7d7e5fbc9079264c672c7db0c3adb": " \\mathsf{q}=(\\mathbf{q}-\\mathbf{p}, \\mathbf{p}\\times\\mathbf{q}),", "9ec80a65bfb3ad6538b299bfef8f6aac": " {d\\over dx} \\left( \\int_{f_1(x)}^{f_2(x)} g(t) \\,dt \\right )= g[f_2(x)] {f_2'(x)} - g[f_1(x)] {f_1'(x)} ", "9ec827580c017e965031afb6c6ff922b": " V_\\mathrm L = \\frac{V}{R} \\omega_0 L ", "9ec8e3d0df3d7b60a175ad282a21b70d": "0<\\lambda_1\\le\\lambda_2\\le\\cdots,\\quad \\lambda_n\\to\\infty,", "9ec8ea206a8d552546f320ea430b55b1": "V={\\pi D^3 \\over 12}", "9ec9142791a60667cba732de94dab591": "c_{-n}=c_{np}=c_n^p", "9ec92ae12f38b277604819edd1192ee7": "(\\log^{7.5}(n))", "9ec98a656b3500239d77f631b67811cc": "y_{ij}", "9ec9d597f82182c82555a9b056dcf373": "\\left(\\sqrt{\\frac{2}{5}},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "9ec9ff1c95e7fb69f985627b881c5dba": "\\sum_{i\\in F}N_i=M\\,", "9eca1222b764b162b9562d9abe6a9398": "\\frac{1 + {\\scriptstyle\\frac{2}{5}}z + {\\scriptstyle\\frac{1}{20}}z^2}\n{1 - {\\scriptstyle\\frac{3}{5}}z + {\\scriptstyle\\frac{3}{20}}z^2 - {\\scriptstyle\\frac{1}{60}}z^3}", "9ecb24f6b50d0b9e6717def36ce293d2": "1 + \\tan^2 \\theta = \\sec^2 \\theta\\,", "9ecb3ef5765ec25a6b2f45f865e34649": "\nI(\\mathcal{V})\\equiv -\\sum_{\\mathcal{T}\\subseteq \\mathcal{V}}(-1)^{\\left\\vert\\mathcal{V}\\right\\vert -\\left\\vert \\mathcal{T}\\right\\vert}H(\\mathcal{T}) \n", "9ecb4610c0e543249dbecab6d16ab388": "m_1 = \\overline{p_1 d_1} \\, ", "9ecb8371455cb67a9f94b54cc907b43d": "c x_1^{[i_1]} x_2^{[i_2]} \\cdots x_n^{[i_n]}", "9ecb9072c611d121cca0c8e06139ba2e": "\nU_0=62636860.850 \\ \\textrm m^2 \\, \\textrm s^{-2}\n", "9ecbb3369816bbfc307957f37c0778c5": "x \\in Z_n", "9ecbcc713a5474e732359c27d1160daa": "A_s/A_{ce f}", "9ecc2ca8bca731c6ced6133119f76b8f": "f(x)=g(x)+b(x)", "9ecc59620b5e0b62f78cb7d013ee8f1a": "\\mathrm{Re}(z)=\\frac{z+\\bar z}{2}\\;", "9ecc7cc46db7d84f1eb1c7e70dd8b9a0": "\\bigcap \\mathbf{M} = \\{x \\in U : \\forall A \\in \\mathbf{M}, x \\in A\\}.", "9eccb1c51bf59c29e089184bdd7d8f26": "\\epsilon^{ab\\ldots n}", "9eccd20badbd07e8ea2a3d0a55e4ffd9": " \\frac{\\omega_c'}{i\\omega} \\to\nQ \\left( \\frac {i\\omega}{\\omega_0}+\\dfrac {\\omega_0}{i\\omega} \\right)", "9eccd33dda84f2eec4d98e1fc23d45eb": "\\Delta v\\equiv v_{\\beta}-v_{\\alpha}", "9ecd0874416a6211bfa1c3b257124097": "|\\psi_1\\rang", "9ecd892e636593075ad534e6c84ca498": "\\hat{f}:\\Z_2^n \\to \\Z", "9ece3429d8de8c1aaec40deef0d65912": "n_c = \\frac{n_r}{n_s} ", "9ece784f7dc39e2051ec8075e112f1d4": "D^4", "9ece9d2b4b63bf46868ad102a3836d13": " \\chi(4,9) = q_2 q_4 + q_3 q_4 - q_4", "9ecee320ed9342dc457c416f42fceb10": "c=\\det \\begin{pmatrix} z_1 & w_1 & 1 \\\\ z_2 & w_2 & 1 \\\\ z_3 & w_3 & 1 \\end{pmatrix}\\, ", "9ecfe2ac84875164b7ae7611f85f2d88": "2^{42}", "9ecffc64846b4362b06d4b1813edc3f7": "C_3 = \\{\\pm S, \\pm V, \\pm SV\\},", "9ed0374d64e022945bb715bf367cf46e": " R_{n+1}\\,\\leftarrow\\,R_n - a_n g_{\\gamma_n}", "9ed03a3802438cfe8be94d2d39c2663c": "HM = \\frac{1}{\\frac{1}{n}\\sum_{i=1}^n \\frac{1}{a_i}} = \\frac{n}{\\frac{1}{a_1} + \\frac{1}{a_2} + \\cdots + \\frac{1}{a_n}}", "9ed052ce18f4750614ac5c7c1e2b4895": "\n\\mathfrak{C}_{\\operatorname{even}},\n\\mathfrak{C}_{\\operatorname{odd}},\n\\mathfrak{P}_{\\operatorname{even}},\n\\mbox{ and }\n\\mathfrak{P}_{\\operatorname{odd}}\n", "9ed055db8ad533ed4733f3d4a09de181": " y_1, y_2 \\in Y", "9ed0628facb0aad3aaa84a285056cab8": "D_{\\mathrm F}", "9ed0850c9e050063091a5a7291a1927e": " \\gamma = \\frac{1}{0.6}", "9ed1343c38614284c7b0fecaba1b936e": "(r,\\ \\theta)", "9ed156b8072a5592ea54e36170b4a0b1": "\\scriptstyle DEC_1", "9ed15eb08643449a4b1a076b08ebb7b4": "N_{eq} < N", "9ed1a111ff802bbb82a22063695b7b3b": "\\nu_b", "9ed1a1b5564e130974a73e76a51168c3": " C(a,q,x) \\approx \\cos(\\sqrt{a} x), \\; \\; S(a,q,x) \\approx \\frac{\\sin (\\sqrt{a} x)}{\\sqrt{a}}.", "9ed20c72eb3781444a6bde99473fd00b": "\\cos (\\arcsec x) = \\frac{1}{x}", "9ed2187d74de40690a20826246e5f505": "\\theta :[0,T] \\times \\mathbb{R}^D \\rightarrow \\mathbb{R}", "9ed21fdffd82c86d50e522428790db20": "\\scriptstyle k\\ge 4", "9ed23f3fbfe2b4a8d3941a37a532db7a": "\n\\Delta \\varphi = \\frac{4K}{\\sqrt{r_{s} \\left( u_{3} - u_{1} \\right) }}\n", "9ed29e2a71cd38ebf1d4bdb22c4845a0": "\\frac{1}{T_{v}} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{\\epsilon_{v}^{1/2}} \\frac{d\\epsilon_{v}^{1/2}}{dt}", "9ed2dbf2e75ca2befcd5d0c241777af9": "\\rho_c=\\frac{3}{8\\pi}H^2", "9ed3563b04289f4f8e4e054f59755a78": " \\hat{A}\\hat{C} = (A + \\epsilon B)(C + \\epsilon D) = AC + \\epsilon (AD+BC). \\!", "9ed3564fddd13c42f734e5ce4b26c45d": "\\langle \\phi | \\psi \\rangle = 1", "9ed3c736b418ac5b604a2805c491ebb5": "D = 2(Y_1-B)", "9ed4343f2b3bf4b5edb500f877223c18": "W_h", "9ed487991f02c0baf7f3b190e6e50fda": "p \\cdot x'_i < w_i", "9ed4a235e10600d99f37ea613003b599": " \\widehat{\\boldsymbol{\\beta}}_{k} ", "9ed4e313ceef0f025c531a84f8af0a1c": "PDI=1+p \\,", "9ed4fe133f1d454f694c72f71a55cb37": "{PL}", "9ed52591134853523b04b9bcd83455a6": "\\widetilde{\\mathcal{O}_P}", "9ed52e8ff4edff77afab6c3b1adc3d75": "s = s_0\\cos(k r - \\omega t)\\,\\!", "9ed60d6d7c37b85679d53524f38da87c": "\\omega_m = \\gamma \\mu_0 M \\ ", "9ed643d106c3c8de6d9e47b03f037559": "\\Phi(z, s, \\alpha) = \\sum_{n=0}^\\infty\n\\frac { z^n} {(n+\\alpha)^s}.", "9ed65f47e3c2bad0e469ff1d3e91977b": "o>j", "9ed6951997b57817257c717a97a7fbf4": "\nP(X|z,\\alpha,\\beta) = \\prod_{i=1}^T N(x_i|\\alpha+\\beta z,z)\n", "9ed6c7e98000e5e796c68004faa4e82d": "f(x; x_0,\\gamma) = \\frac{1}{2\\pi}\\int_{-\\infty}^\\infty \\phi_X(t;x_0,\\gamma)e^{-ixt}\\,dt \\!", "9ed6e5bd9cbd6015ef7921786c28d2ee": "Z=\\sqrt{R_{\\mathrm{ESR}}^2 + (X_\\mathrm{C} + (-X_\\mathrm{L}))^2}", "9ed70f2c47a07ee89c70432c99bf435e": "\\omega_{Y|X}", "9ed742ccd2ecbc122583b7077a136e87": "-\\frac{x}{\\|x\\|^{2}}\\in\\mathbf{R}^{n}.", "9ed74d599c9e9fb073d7f6ee9737de05": "\\gamma_{1,2} = \\frac{(a - d) \\pm \\sqrt{(a-d)^2 + 4bc}}{2c} = \\frac{(a - d) \\pm \\sqrt{(a+d)^2 - 4(ad-bc)}}{2c}.", "9ed76747f54d38d8ce46e592f790ace7": "k= 1,...,n", "9ed78a3bd81d8aa7008fc55070784486": "g \\in \\mathcal{C}", "9ed81f572f15a2f77e1739e8ddae11fb": "L = P (A|T=t)", "9ed833e7c1666e69e4e9cde7843f4a3c": "\\Pi(\\phi,n,k) = \\int_0^\\phi \\frac{1}{(1 - n \\sin^2(t))\\sqrt{1 - k^2 \\sin^2(t)}}\\,dt.", "9ed862d36cd76dea612c6e369974354c": "c = E(k_E; m)", "9ed86ba5253a5984bbd3452d7cfa5976": "\\operatorname{Kurt}\\left(\\sum_{i=1}^n X_i \\right) = {1 \\over n^2} \\sum_{i=1}^n \\operatorname{Kurt}(X_i),", "9ed87f4b26034993b2359675c8f5634d": " = (2 \\pi)^{-M N / 2} |R|^{-M / 2} \\exp \\left\\{-\\frac{1}{2} \\sum_{n=0}^{M-1}(\\theta^2 m^T R^{-1} m) \\right\\}", "9ed881b836d9e7d507d7216f39b0943d": "\\frac2{\\pi R^2}\\,\\sqrt{R^2-x^2}\\!", "9ed8dc4100221927c9ab0c050581f5aa": "\\scriptstyle U", "9ed90cd7cbd42f2793bbe8b2f2a9108d": "r=b\\cos\\alpha,\\ \\theta=\\tfrac{\\rho+\\lambda}{2}\\ \\mbox{where}\\ \\alpha=\\tfrac{\\rho-\\lambda}{2}.", "9ed92cb8e2267f57b322657745b12e41": "c\\in[x-2h,x+2h]", "9ed92cc619e005abcd5f29520da638be": "\nds^2 = \n\\left( 1 - \\frac{r_\\mathrm{S}}{r} + \\frac{r_Q^2}{r^2} \\right) c^2\\, dt^2 - \\frac{1}{1 - r_\\mathrm{S}/r + r_Q^2/r^2}\\, dr^2 - r^2\\, d\\theta^2 - r^2 \\sin^2 \\theta \\, d\\phi^2,", "9ed991f7161820b91a786b88d10fdf34": "\\vec{v'} = q \\vec{v} q^{-1} = \\left( \\cos \\frac{\\alpha}{2} + \\vec{u} \\sin \\frac{\\alpha}{2} \\right) \\, \\vec{v} \\, \\left( \\cos \\frac{\\alpha}{2} - \\vec{u} \\sin \\frac{\\alpha}{2} \\right)", "9eda16010c14ebf6cf78211988a30c8f": "T - A", "9eda1dde4047b2de436b487c99df13ee": "\n\\omega^2 = \\omega_p^2 + \\gamma\\left( \\omega \\right) {T_e\\over m} \\vec k^2\n.", "9eda4e867af297fa4c8053c41277ffa8": "a_{\\pi}>0", "9eda53882505edc0f1289ddb8ea6fc41": "\\sum_{n=2}^\\infty n^m \\left[\\zeta(n)-1\\right] =\n1\\, + \n\\sum_{k=1}^m k!\\; S(m+1,k+1) \\zeta(k+1)", "9eda6bb307d1ef0dcbdb33d5e625fd41": "x_1 \\ll x_0", "9eda8fd6cfe418e26d305b0c1d745ea4": "dV_{fb}\\leftarrow V_{fb}(t)-V_{fb}(t-dt)", "9edaa594ea7f4870ee36085f0a871e30": "\\rho_C", "9edadd1a69203842d55f8e1aafcf86ea": " I_0(a,b) = 0 \\, ", "9edaf8160f6e0e7f1e1576aed13eb67a": "\\displaystyle{\\Psi_{b}=\\sum\\nolimits_{x\\in M} \\delta_{x+b}}", "9edb3cf931b9f5acbc08073dfd0268cc": "\n \\sigma_1 = \\cfrac{1}{\\sqrt{3}}~\\xi + \\sqrt{\\cfrac{2}{3}}~\\rho~\\cos\\theta ~;~~\n \\sigma_3 = \\cfrac{1}{\\sqrt{3}}~\\xi + \\sqrt{\\cfrac{2}{3}}~\\rho~\\cos\\left(\\theta+\\cfrac{2\\pi}{3}\\right) ~.\n ", "9edb512bf5bc9e16c8ba4443f90ef8b2": "\\pi_1(X,w) = \\langle u_1,...,u_k, v_1,...,v_m | \\alpha_1,...,\\alpha_l, \\beta_1,...,\\beta_n, I(w_1)J(w_1)^{-1},...,I(w_p)J(w_p)^{-1}\\rangle", "9edb6ca520a6c938880c4e5387941074": "\\;^3R", "9edbbf9a3f760dc467ae3b5f0efc83ba": "n B^{n-1} y^{n-1}", "9edbc287daf69d90f30dbd0c507804c2": " \\vec \\sigma (\\phi, \\theta, t) = (v_x t + t s \\cos \\theta \\cos \\phi, v_y t + t s \\cos \\theta \\sin \\phi, v_z t + t s \\sin \\theta, t) ", "9edbc649365b4f70168eed9d3814de58": " \\int_0^\\infty \\frac{dx}{x^{2}+a^{2}}=\\frac{\\pi }{2a} ", "9edbee83b179b84e4de7e32a28b7c9bc": "\\frac{a_{2}}{a_{1}}={(\\frac{T_{2}}{T_{1}})}^{0.5}", "9edc1730eb03853698de7040682fe06a": "\\ \\beta = 180 ", "9edc2b60e20fb9f2ff56dd99ae04d5e5": "p^{\\prime}=(p-1)/2", "9edc3afc119541b33935d0d3b6ca9d48": " ||\\widehat{\\mu}_X - \\mu_X[P_X] ||_\\mathcal{H} \\le \\epsilon ", "9edca6ccdff997054bfa5e2c8f2d2aa4": "\n \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + (2\\mu + \\lambda)~\\nabla(\\nabla\\cdot\\mathbf{v}) -\n \\mu~\\nabla\\times\\nabla\\times\\mathbf{v} + \\rho\\mathbf{b}\n ", "9edcb64fe62f25d98982651b42af77db": "v(E) = \\sum_{H_i < E} 1.", "9edd38951a22fc0bcb9ef22705e4f2ce": "U=U(s)", "9edd62dbe98401a8692cc5dd1c8f7d8b": "\\mathcal{H_S}", "9edd89d181d1a34f4e979d364a6c4291": "\\frac{\\theta}{\\theta_b}=\\frac{\\cosh {m(L-x)}}{\\cosh {mL}}", "9edde1039bf8bf60b32330dff9a8fc94": "\\int_{-\\infty}^\\infty \\delta(x) \\, dx = 1.", "9eddecf4ecc3ab4462cefd21acf77e2e": "\\scriptstyle x^*\\in I", "9ede22b9b4f3f7ae5db5759e3dfe6bad": "\\tilde p \\,=\\, \\scriptstyle \\frac{X + 2}{n + 4}", "9ede2531f0d6c67f27084cf6506a96f9": " \\operatorname{E} (X | Y=y ) = \\sum_{x \\in \\mathcal{X}} x \\ \\operatorname{P}(X=x|Y=y) = \\sum_{x \\in \\mathcal{X}} x \\ \\frac{\\operatorname{P}(X=x,Y=y)}{\\operatorname{P}(Y=y)}, ", "9ede3cc3b84ad6a9803e4b3b5e68a702": "V_{y} = \\int_a^b \\pi \\, \\, x^2 \\, \\frac{dy}{dt} \\, dt .", "9ede533bbd1f120cf827104e651ef785": "(-++)\\,", "9ede70a041db0a4099228d6435f199f1": " Q_i^{(2)}=b_{i+2}b_{i+3} ", "9ede88376907b2bbc795ad2317dcbb05": "0 \\to \\mathcal{I}/\\mathcal{I}^2 \\to i^*\\Omega_X \\to \\Omega_Y \\to 0,", "9edea13c56e05bb270750add7b914aea": "D(af) = a(Df)\\,", "9edeb326946ad57f49da1da6ec1a7015": "p(\\langle X_1, X_2, ...., X_n \\rangle)", "9edebcc6d40361f86b4ed437789cda57": " = \\frac{-1}{i\\hbar} \\langle H \\psi | Q | \\psi \\rangle + \\langle \\psi | \\frac{dQ}{dt} | \\psi \\rangle + \\frac{1}{i\\hbar}\\langle \\psi | Q | H \\psi \\rangle \\,", "9eded0d8b2be7c5cf9508c4d16dafc7b": "a\\!\\!\\!/a\\!\\!\\!/=a^\\mu a_\\mu=a^2", "9edf449eda31bcc966bf841359ff2567": "\\frac{m_\\mathrm{NaCl}}{m_\\mathrm{Na}} = \\frac{m_\\mathrm{Na}+m_\\mathrm{Cl}}{m_\\mathrm{Na}} \\approx \\frac{23\\mathrm{u} + 35.5\\mathrm{u}}{23\\mathrm{u}} = \\frac{58.5}{23} \\approx 2.5", "9edf69517c18386df387967854fc2b48": "D[\\widehat{B}(\\varphi,\\hat{\\mathbf{a}})] = \\exp\\left(-\\frac{i}{\\hbar} \\varphi \\hat{\\mathbf{a}} \\cdot D(\\mathbf{K})\\right)", "9edfbb55d0d0a42014ed59f42e2be915": "H(Q|P) = E\\left[\\frac{dQ}{dP}\\log\\frac{dQ}{dP}\\right]", "9edfd16d8712c8f9dece0590e5991dbe": " ds^2= {4(dr^2 + r^2\\, d\\theta^2)\\over (1-r^2)^2}.", "9edfd365970345f9851adaa0bc2d6dac": "\\langle S(x)S(y)\\rangle \\propto \\langle H(x)H(y)\\rangle = G(x-y) = \\int {dk \\over (2\\pi)^d} { e^{ik(x-y)}\\over k^2 + t }", "9edfde556ac6b07edb28a526675a9678": "\\epsilon_i = D X_i - 1 \\,,", "9ee01b631ec3120d3d79449b701dd725": "D = U - 128", "9ee027505f5de414a6c8954004cc18a5": "|a \\wedge b|^2 = |a|^2|b|^2 - |a \\cdot b|^2. \\,", "9ee0594d4845c5c74edda33c5707d1c1": "P_\\mathrm{op}\\psi = mc\\psi. \\,", "9ee066a0b2387b6370b0aa68a42f93d1": "b_y", "9ee06d91df8c23fd59455a44da9d02a9": "j\\sqrt{\\frac{1}{2}}", "9ee07ea30b79d6e96ae3cf1e4ec49bb6": "\\displaystyle ax^4+bx^3+cx^2+dx+e", "9ee08f7ccae705e574109d981ace7432": "\\frac{r}{c} = nK_a - rK_a ", "9ee0af0acc6e7a32d578d90b76710b8d": "a_1 = |\\mathbf{a}|\\cos\\theta = \\mathbf{a}\\cdot\\mathbf{\\hat b}, ", "9ee0da4db08c2bcc60fcb534990eb0e0": "R_{\\Phi}=\\int_{0}^{\\pi/2}2\\sin \\theta \\cos \\theta R_F(\\cos \\theta)d\\theta", "9ee0fb7ca87fbad046611213de71e338": "\\mathbb{F}_{q^k}", "9ee1c86284a042a4c1d5de48b991fa8f": "z_1, \\ldots, z_4", "9ee1e2fd052652fdd4c39ceadb900b70": " [U_g \\psi] (x) =\\psi(g^{-1} x).\\quad ", "9ee1f153fbaaf507545c290c852603c2": "0\\le x,y \\le 1", "9ee1f8e9785a64d4d4ab3b49997962a4": "C_n\\left(X\\right)", "9ee28cbfef25e4a171a5955ff0a34868": "\\lfloor - \\rfloor", "9ee36d14d2057d2180f1f3f2dc8b4481": "\\scriptstyle{ TS = T \\circ S}", "9ee3fd1eb60823a7576dc2f76dfc3706": "r = r^{*}", "9ee4090fca419dbb4de3647121c40fc3": " T=1/\\beta", "9ee4426807508b0895f3ce47d520898b": "J \\sim Poisson(\\frac{\\lambda}{2})", "9ee449fc1589f23c726e9a28e2f51b71": "f_i \\left(x \\right)", "9ee4579ce75dfba2fb9ed016dd5c0ddb": "\\scriptstyle \\Omega_{\\text{E}}", "9ee45f62fd675798b245e35752f46729": " \\mathbf{\\alpha,\\, \\beta} ", "9ee4a43f17beb961a97d5a6d6633f4d4": "CDC \\rightarrow B: certificate(A, CA)", "9ee560767751c3b8f614e3dc16fd932e": "\\tau = rF_{\\perp},", "9ee660f8c5ac88dc78a6200d677a6b43": "H[1]", "9ee69ed407e11be0017334dbfa9c32eb": " k \\geq 3 ", "9ee6aaf533f520b38b5ae4b5ebd1c36c": "\\delta p = c\\rho _0[v_{Ar}\\cos(\\omega t-kx)-v_{Al}\\cos(\\omega t+kx)]. ", "9ee6c171882cefbfaebc4d0f11ed83be": "\\Lambda = {|\\mathbf{X}|/|\\mathbf{X}+\\mathbf{Y}|} ", "9ee7603844620eb4d9561962f001b6cb": "\\omega^{A}_{x\\cup y}=\\omega^{A}_{x}+\\omega^{A}_{y}\\,\\!", "9ee76c4938a5e3b729415ca021ccb917": "\\langle x,x\\rangle \\geq 0", "9ee76e960915992f83d185c078375931": " Z = \\sum_{x \\in \\mathcal{X}} \\exp \\left(\\sum_{k} w_k^{\\top} f_k(x_{ \\{ k \\} })\\right).", "9ee79a443bc3b9941bdcd0e7dd342a2f": "COLOUR ::= red | blue | green", "9ee8310697ed9843a09194c5fbef50ed": " V = \\mathbb{R}^2", "9ee870b886054764d3e493c2e81341e9": "\\ln T_\\text{hold} - \\ln T_\\text{load} = \\ln\\frac{T_\\text{hold}}{T_\\text{load}} = -\\mu\\cdot\\phi", "9ee880f6c1f5ed93a742c6b348d88863": "f^{-1}(B)", "9ee88e5e99d63fed74d34c1624a9c889": "r= \\sec^2\\theta\\,.", "9ee8960ecb9b7db1d8c878c3b499067b": "\\begin{bmatrix} {\\nu_e} \\\\ {\\nu_\\mu} \\\\ {\\nu_\\tau} \\end{bmatrix} \n= \\begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\\\ U_{\\mu 1} & U_{\\mu 2} & U_{\\mu 3} \\\\ U_{\\tau 1} & U_{\\tau 2} & U_{\\tau 3} \\end{bmatrix} \\begin{bmatrix} \\nu_1 \\\\ \\nu_2 \\\\ \\nu_3 \\end{bmatrix}. \\ ", "9ee8bae9deb2b3a88b8ef927e8be3df4": " R_0 > 1 , I(0)> 0 \\Rightarrow \\lim_{t \\rightarrow +\\infty} \\left(S(t),E(t),I(t),R(t)\\right) = EE. ", "9ee98c5575ca46149c9da7cddb548a86": "b \\uparrow\\uparrow X \\uparrow Y \\times Z + T", "9ee9d84932cf8f65da46953373e0256c": " \\ell_P =\\sqrt{\\frac{\\hbar G}{c^3}} \\cong 1.616 24 (12) \\times 10^{-35}", "9ee9e3b82293e798b30386e8bc96613f": "R_j P_\\varepsilon(x)= c_n \\frac{x_j}{\\left(|x|^2 + \\varepsilon^2 \\right)^{\\frac{n+1}{2}}}.", "9eea1fe4467889bebb1376b8561dfd0e": "= {1 \\over 4} (\\epsilon_{MN}^{\\;\\;\\;\\;\\;\\;\\; IJ} \\epsilon_{OP}^{\\;\\;\\;\\;\\;\\; KN} + \\epsilon_{NM}^{\\;\\;\\;\\;\\;\\;\\; IJ} \\epsilon_{OP}^{\\;\\;\\;\\;\\;\\; NK}) F^M_{\\;\\;\\; K} G^{OP} ", "9eea6a0ad37653cdb0160e12319c52d4": "J = \\rho I", "9eea78a491f0c1d9bffa0963abc11b98": "p_{ij}^k=p_{ji}^k", "9eea843039021fcacf736062534bd912": " \\bigoplus_{i \\in I} M_i. ", "9eeaa203a473fb30c36a0a35e32ef58b": "d\\epsilon_{i,j}=d\\epsilon_{i,j}^e+d\\epsilon_{i,j}^p", "9eeb1e55c855953fcfe1c8f6bd98801b": "f(n)=2^{-n}\\,", "9eeb2387daba808079227ef0b53db491": " \\Phi ", "9eeb25d48f7415209e4563196f2d2f92": "\\approx \\ln(1.23456) + 2 \\times 2.3025851. \\,\\!", "9eeb35ca4391514c3e5317750ce68e16": "\\sim \\!\\,", "9eeb80e81813e5de94d3921023504db8": "x \\rightarrow b \\equiv \\{\\{x\\}, \\{x, b\\}\\}", "9eeba8417e0ccf785f034649c7905a1f": " ds^2 = -dT^2 + dX^2 + dY^2 + dZ^2,\\; \\forall T, X, Y, Z", "9eebbffb7b8a1f068f3ed9ea2c63cae6": "x'\\approx x", "9eebc5f99d9ce121b559fbb4daa2b49b": "R_{out} = r_O \\left( 1+ g_m R_E(A_v+1) \\right) +R_E \\ .", "9eebee32f774d8629c08ad70d108d10b": "G_v", "9eec0d810c00dabc70888862e2686a0c": " \\Vert f \\Vert_{\\infty,\\,B_r}", "9eec174d2435bb5941629b83700205a2": " x_{n} = H^{T}y_{n}", "9eec3353096e5ef6a1ac6d44562087ec": "\\arg(X) \\ ", "9eec336455a41db7f8c01034cc5362fe": " \\pi_1(X,x) = \\varprojlim_{i \\in I} {\\operatorname{Aut}}_X(X_i),", "9eec44d431920b5aede9da19aae66ee4": "\\nabla \\times \\mathbf{H}_0 = -i\\omega\\mathbf{E}_0(\\epsilon + i\\frac{\\sigma}{\\omega})", "9eec5f6580a89c1456eed0c0cf2076b5": "M \\times N", "9eec84d513f2f9b63a58bdf2494e224a": "\nR_{n,l}(r) =N_{nl} \\, r^{l} \\, e^{-\\frac{1}{2}\\gamma r^2}\\; L^{(l+\\frac{1}{2})}_{\\frac{1}{2}(n-l)}(\\gamma r^2).\n", "9eec9dad010387ae8034a0c65329f777": "\n\\eta_{01} = \\langle \\mathbf{e}_0 \\bar{\\mathbf{e}}_1 \\rangle =\n \\langle 1 (-\\mathbf{e}_1) \\rangle_S = 0.\n", "9eecb13117499b990248f4e168e7a935": "1_{A_t}", "9eecb7db59d16c80417c72d1e1f4fbf1": ";", "9eecdd02a0f3b666cac2a25afa1ceb94": "\n\\alpha \\varepsilon^2\\partial_{t} \\phi =\n\\varepsilon^2\\nabla^2\\phi- f'(\\phi) - \\frac{e_0}{h_0}\nh'(\\phi)u+\\tilde \\eta({\\mathbf r},t)\n", "9eecef9bc43e5b2203f12e26e1b4457d": "R_X \\geq 5", "9eecf5bb02a51eed9771805ebc390ca9": "\n \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + m^2 }\n \\mathcal J_0 \\left ( kr \\right)\n= K_0 \\left( m r \\right) \n.", "9eecfcc1c26e17ebed0938096a4742df": "\\rm Si + 2H_2O \\rightarrow SiO_2 + 2H_{2\\ (g)}", "9eed2e28b1b322ddc56e722663f18281": "{P_{roll}= \\mu_{roll} \\cdot m_{vehicle} \\cdot v }", "9eed4e181b0183c9511e74aede04890b": " H(x,p,t) = \\frac{p^2}{2m} + V(x,t) ", "9eedacd29ad4d2fb88ddf8a05033a44f": "g(x) = k \\cdot f(x).", "9eedc94ad39a856b083af7b34657dc57": "v_{gu} = \\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} {7000} / m_{gu} ", "9eeea36f08b12c3845d0b665b5629cde": "c_2 = 0.52119", "9eeee7f4f29cd0eaa0a96511adcbdc3f": "\\Delta G^\\ominus", "9eef178191d576bf10e860f452a56848": "z \\in L \\implies \\Pr\\nolimits_x[\\exists y. \\phi(x,y,z)] \\geq \\tfrac{2}{3}", "9eefecc9dbb0b9ace2f4c6a2f73fae0f": "\\Phi^2 (3)=(50)^{-1} \\sum_{i=1}^{50}C_i^2(3)\\approx0.3336 ", "9ef015900ddd2040d3b9fce22ba549d8": "i^{4n+1} = i\\,", "9ef03d2ad0515cdbc1ad6be1cb362717": "P = (a_D-a_S ) / ( b_S-b_D)", "9ef0567e22a2bbb12fe0460f289d45f4": "W = \\int_C \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{x} = \\int_{\\mathbf{x}(t_1)}^{\\mathbf{x}(t_2)} \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{x} = U(\\mathbf{x}(t_1))-U(\\mathbf{x}(t_2)).\n", "9ef06cc026c85d535809191290894c54": "E(x)=E_0 \\exp{(-\\frac{x}{R_0})}", "9ef0703335d49fffb0fba016b08fbabf": "f'(a)", "9ef0994f2517007bac4cc18ef9c13117": "\\scriptstyle \\{S_i|j\\}", "9ef09bc044470181fb66719095450f60": "\\mathcal{A} f(x) = \\sum_i b_i (x) \\frac{\\partial f}{\\partial x_{i}} (x) + \\tfrac1{2} \\sum_{i, j} \\left( \\sigma (x) \\sigma (x)^{\\top} \\right)_{i, j} \\frac{\\partial^{2} f}{\\partial x_{i} \\, \\partial x_{j}} (x).", "9ef09dda0e8e7bb7b214fd7966a7ca95": "p = u_1X_1 + \\cdots + u_kX_k.", "9ef1311aa10743ecb69bf76abe169f2b": "\\gamma_{\\mu, \\sigma^{2}}^{n}", "9ef14e7a54e56704ac8640f21a2f5d8a": "(Q_Cx)(\\phi x)\\equiv (Q_Lx)(x=x,\\phi x)", "9ef1584f7b048333184d994e1bf870dd": "2^{n(\\bar{H_n}(X)+\\epsilon)}", "9ef1aeda867d957fb8d0b81589093a4b": "I^M_L", "9ef1af9a5324f8fd0e08ea16dbc17233": "s_{ij}=max(n_{ij}/n_{i},n_{ij}/n_{j})", "9ef1b07bd9d4ca0140c80fc11a459438": "X_1,X_2,\\dots\\,", "9ef1b573563a12eed17af33de8193ba5": "\\langle X \\rangle = \\sum_{N_1 = 0}^{\\infty} \\ldots \\sum_{N_s = 0}^{\\infty} \\int \\ldots \\int \\rho X \\, dp_1 \\ldots dq_n.", "9ef1c89a71f12a147f5b0fce14ee122b": " S(M) = O( \\sqrt{M}) + (\\sqrt{M}+1) \\cdot S(\\sqrt{M}) ", "9ef1ce1f9b35c658e988c4345646b1d5": "\\alpha \\cdot 1 = (x+y\\omega)(1 + 0\\omega) = \\left(x\\cdot 1 + 0 \\cdot 0 \\left(n^2-a\\right)\\right) + (x\\cdot 0 + 1 \\cdot x)\\omega = x+y\\omega = \\alpha", "9ef1d19a1b4be4cfad3824cbe1034048": "\\bar{B}_1^{p,q}", "9ef1faa6f86dedb67d870584e5c998d6": "\\{p_i, v_i \\} \\,", "9ef234d82aad225d5545eb3d4e65eb20": "\n\\boldsymbol\\mu\n=\n\\begin{bmatrix}\n \\boldsymbol\\mu_1 \\\\\n \\boldsymbol\\mu_2\n\\end{bmatrix}\n\\text{ with sizes }\\begin{bmatrix} q \\times 1 \\\\ (N-q) \\times 1 \\end{bmatrix}", "9ef23a4ad07cac4f0daed4cf58e341b0": "Z =\\operatorname{Ric}- \\frac{S}{n}g", "9ef2874651b6b23f058df3641ac47333": "(t-2)t - 1(-1) = t^2-2t+1 \\,\\!,", "9ef2a7dbb1600ca22023a385dbc77189": "\\text{for single-phase and three-phase systems:}", "9ef2e05efbf6803dccb6bcbddf00635e": "r=e^{\\cos \\theta} - 2 \\cos (4 \\theta ) + \\sin^5\\left(\\frac{2 \\theta - \\pi}{24}\\right)", "9ef34bfd2c659f2539c6c9c39fe85f76": " \\lambda_{in} = \\Delta E^* = \\frac{3 f_D f_A}{f_D + f_A}(q_{0,D} - q_{0,A})^2 ", "9ef37162e1f3b4d6ef7bce0cdc786163": "\\frac{y_n}{x_n}X-Y", "9ef394e713e227faae2af8c607f59aff": "D_m", "9ef3e816a834fed959febc7f9a56c012": "C_F=\\frac{3h^2}{10m_e}\\left(\\frac{3}{8\\pi}\\right)^{\\frac{2}{3}}.", "9ef45fcab12fc0fdaef73f5680ef1e59": "\\frac{dN(\\epsilon)}{d\\epsilon}\\propto \\epsilon ^{-p}", "9ef4d9afc2bf62f4f19ff0829971b6c5": "f^{-1}(L)=\\{s\\vert f(s)\\in L\\}", "9ef4e1e7ded59566f791c3a8b10f060c": "f (z_{0}) = \\frac{1}{ 2\\pi i } \\oint_\\gamma { f(z) \\over z-z_0 }\\,dz", "9ef532c69f3fb732d4b00ead67b1d646": "(E_{bi} -J_i)", "9ef537d7952ed5a98bfc9b3a3e716517": "f(t_i)(x_i-x_{i+1})", "9ef5722c330bcae0458b87c35c7a01cc": " u = u(\\mathbf{r},t)", "9ef5a4beedb5a8e632990a9dadccda30": "\\cos\\theta", "9ef5af4b172329d8d7a3cd9a4239dedd": "H(\\phi) = {1\\over 2} |\\dot\\phi|^2 + |\\nabla \\phi|^2 + V(|\\phi|).", "9ef5be03632f1776ea9ba03b71ea6dd7": "\n G = G_I = \\begin{cases} \\cfrac{K_I^2}{E} & \\text{plane stress} \\\\\n \\cfrac{(1-\\nu^2) K_I^2}{E} & \\text{plane strain} \\end{cases}\n ", "9ef5c3fe33e274ccad7b56db5abe73fc": "\\mathit{p_i}", "9ef5db61af0c0d69875ff753a249bd2f": "V_0", "9ef68a6912ce13f53a8537abd947835e": "r = Ae^\\frac{-E_a}{RT}[A]^m[B]^n,", "9ef6975c7b82ff79682dbcec2d043ea0": "\n\\operatorname{MAD} = \\operatorname{median}_{i}\\left(\\ \\left| X_{i} - \\operatorname{median}_{j} (X_{j}) \\right|\\ \\right), \\,\n", "9ef6b6162ecc65e0de1c33b31f8d0c58": "h_{21} = \\left. \\frac{ I_{2} }{ I_{1} } \\right|_{V_{2}=0} ", "9ef6c8180e1e0d3a09e6d250bbc1978a": " a_0 \\, = a_1 = a_2 = 0 ", "9ef782335e3936a51531c04e18114e28": "Initiates(break, broken, t) \\leftarrow HoldsAt(hashammer, t)", "9ef798e6b6d5564a0e32faecdb9c0402": "\\mathit{w} > 0", "9ef7ec34e0019333fa49dcda3178299d": "\\mathbf{D} = \\epsilon\\mathbf{E}", "9ef8170ea5bbbcf5975516942389553f": " C_t(x) ", "9ef841a6adf3a4edbd5bbda2f6d9d555": "f^{*}(t) = \\inf\\{\\alpha \\in \\mathbb{R}^{+}: d_f(\\alpha) \\leq t\\}", "9ef85fec425e7d41fb1a76a0522dc5ad": "r := |\\mathbf{r}_1 - \\mathbf{r}_2|", "9ef893eeb3207c07df0d2b853e185817": "\\lim_{x \\to a^{+}} f(x)=\\pm\\infty.", "9ef8d53241ae382b9bf0cea31a224881": " \\mathbf{T} = T^i{}_j \\mathbf{e_i e^j} ", "9ef92963af61ae2208447665f9753b13": "(e_1, \\ldots, e_n)", "9ef92a32bac676626cb8d0125e6cdc05": "\\phi_1(t) = \\sqrt{\\frac{2}{T_s}} \\cos (2 \\pi f_c t) ", "9ef92ced400e589c5743e103707063f3": "m_2=\\left.2\\mu+\\Delta^2\\right.\\,", "9ef9321ba58959f56d228e7919954a2c": "\\sum_{I=1}^{L}I = L(L+1)/2", "9ef978558b6788c214b78781e0147a2b": "M= {1\\over N} \\sum_{i=1}^{N} S_i.", "9ef991cff00fb133f719939e8b7e9b8b": " \\theta_A = \\alpha_F\\,p^{C_F}", "9efa0446484404363db6156c9668b564": "x_i=(x_{i1},x_{i2},\\ldots,x_{in})", "9efa2805a426e4c677b3103dc9f127da": "N = \\int\\vert\\Psi(\\mathbf{r})\\vert^2 \\, d^3r. ", "9efa8c37821f8501369d002bfe407e8e": "A_d(X,{\\Bbb Q})\\,", "9efadf62a5e3c30e0839f065bb356d4d": "\\displaystyle{v_i^2=\\varepsilon,\\,\\,\\, v_iv_j=\\varepsilon v_jv_i \\,\\, (i\\ne j),}", "9efb361b16a2eb8054346606b79fb54a": "{\\mathbb Z}_2", "9efb71dcb09bbe3368af48f961e939cf": "AV=VD", "9efb7324b21df52d73efaff3ed0640d7": "\nI(R) \\propto e^{-kR^{1/4}} \n", "9efb85d5edadb5ce07cdab962c4a909e": "\\left(i \\hbar \\frac{\\partial}{\\partial t} - q \\phi \\right) \\psi = \\left[ \\frac{1}{2m}{(\\boldsymbol{\\sigma}\\cdot(\\mathbf{p} - q \\mathbf{A}))}^2 \\right] \\psi \\quad \\Leftrightarrow \\quad \\widehat{H} = \\frac{1}{2m}{(\\boldsymbol{\\sigma}\\cdot(\\mathbf{p} - q \\mathbf{A}))}^2 + q \\phi ", "9efb8831b6751b268546044572a67c80": "\\approx 161.052", "9efbc6625de9b507ff461cc3765831e3": " \\{p\\}=X_1\\cap X_2", "9efc2328376c4df8dbff47ccc4ca9428": "q_1(x) \\le q_2(x).\\,", "9efc31d4baa556774b0c58bc8772c746": " \\operatorname{Hom}(E_x, F_x) ", "9efc3fcb9e85fe156f8d1985a03fcf02": "ea=ae=a", "9efc6d91506b6eaef8e9fade31dfe25c": "\n\\left.\\begin{array}{r}\n\\prod_{i=n-k+2}^n m_i\\\\\n\\alpha\n\\end{array}\\right\\}<\\frac{\\prod_{i=1}^k m_i}{m_0}\n", "9efcd1969a677a4b534ea36c5e477468": "\\lambda = \\frac{1}{2}(p\\pm \\sqrt{\\Delta})\\,", "9efd0f9c9f03af5b9ab2e500b79a4cec": " E_k=\\frac {1}{2} m_s v_r^2", "9efd186ce47c2fd68b0e62a9e9f8a2eb": " \\tfrac12 + \\tfrac16 \\sqrt3 ", "9efdcd7ae53e722f769229c3d246e037": "\\phi(\\zeta_p) = \\zeta_p^q;", "9efdda62a4c7e231085b7eb6547d08bf": "\nH = H_0 + \\tfrac{1}\n{2}J\\sum\\limits_{\\left\\langle {i,j} \\right\\rangle } {\\sum\\limits_\\alpha {(\\sigma _{i\\alpha } } - \\sigma _{j\\alpha } )^2 } + \\ldots. \n", "9efe0ebe1b3a315de20de2f978581914": "\\dot \\epsilon \\ll \\frac 1 \\lambda", "9efe6a694f1442cc3317bf003f34dda0": "\\tau(t) = FL\\,", "9efe77da609b472777930d1515fd7b7b": "\\operatorname{Li}_1(\\tfrac12) = \\ln 2", "9efe90b70f90a29d376d258dde6e4abd": "b_i = k_i", "9efeb068c68552ce4401d5594fba4ac6": "\n\\eta_q =\n\\frac{q}{q-1}\\frac{P_q P_{q-2}}\n{P_{q-1}^2},\n", "9efeb5f9fc051fb545976ca9f9da0a9f": " \\frac{2}{\\pi} \\cos^{-1}\\frac{t}{l} + \\frac{2}{\\pi} \\frac{l}{t} \\left\\{1 - \\sqrt{1 - \\left( \\frac{t}{l} \\right)^2 } \\right\\}. ", "9efec6f9ee39478bc77da80ce4d51f82": "T_{11} = \\frac{-\\det \\begin{pmatrix}S\\end{pmatrix}}{S_{21}}\\,", "9eff65c61cb9c2bf3a75e1072a4eff4d": "\\mathcal{\\tilde{H}}_{S}\\subset\\mathcal{H}_{S}", "9eff95d803d79bc426887e615edbd8c3": "\\mathit{w} \\sim v", "9effc2b1bae92459469cebc188c5bebb": "\nh = \\sqrt{a^2+2\\sigma^2}\n", "9f00449dc3a4a1fcffee4b86790816f3": "\\dot \\gamma^* = \\dot \\gamma e^{-i0}", "9f009cce6eb55340114ea0b626b5cf0d": "\\frac{\\rho}{\\rho+1}\\;{}_2F_1(1,1; \\rho+2; e^{i\\,t})\\,e^{i\\,t} \\,", "9f00a24b793909e0ddfbd3835a0956b8": " \\mathbf{z}^{\\pm} = \\mathbf{u} \\pm \\mathbf{b} ", "9f00a49b6cae2fdafe200fd4edbad34e": "X_{(i)}{=}(x_i^1,\\dots,x_i^p)^T\\sim N_p(0,V).", "9f00f0482fc5f0cc526cf717350f8ea8": "z_k=0", "9f00f1c8c7ee0f2eaaf604abca771150": " \\boldsymbol{w}\\in\\boldsymbol{W}\\subseteq\\R^n ", "9f01033234925725ccbdd342d100b176": "\\mathbf{Z_0} = \\sqrt {\\mathbf{R_0}^2 +\\mathbf{X_0}^2} ", "9f011af21d45175ccbda724bf05a3eb0": "X(e^{i (k\\omega)}) \\!", "9f011c2fce473efd382d6d65888c90c5": "\\begin{bmatrix}\n 3 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n 1 & 2 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 1 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 2 & 1 \\\\\n 0 & 0 & 0 & 0 & 0 & 1 & 3 \\\\\n\\end{bmatrix}.", "9f016149902101fbdfb2868068997de7": "W_{0}^{k, p} (X)", "9f01711ba92537f0188fa4d3ed513cad": "y_t = f(t) + e_t", "9f01858223d1e8d705e0c65e43c391bc": "N_i = N \\cap M_i", "9f01b7b73aa0b9f19effa8c8204e4c46": "1, 3, 7, 15, 31, 63, \\ldots", "9f01bf3279a4a5ec2ebacfcc7a9f38cf": " \\R \\times V \\rightarrow V ", "9f01cfacdc3747333bcefb2a08fe7631": "f_1(g)=\\int_K f(x^{-1}kg)\\, dk,", "9f01da37407aa84edb38d958f6280ead": "\\gamma_1,\\gamma_2", "9f01dfc900621a0ea145326f39ef93b5": "(l_1+l_2)^\\theta=l_1^\\theta+l_2^\\theta.", "9f02522e1f74b7cb7be0d7923418f632": "In=n", "9f02783790444bd614fe30fdcfad4940": "f:FY\\to X", "9f028a4b1f1849ed77341a70750a509b": " \\hat h_{V,\\phi,L}(P) = \\lim_{n\\to\\infty} \\frac{h_{V,L}(\\phi^{(n)}(P))}{d^n}, ", "9f02a42eea42a37f4389d4246b4f1c25": "\\operatorname{tr} \\left( \\gamma^\\sigma \\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\right) = \\operatorname{tr} (\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma). \\,", "9f02e11a5fe5ebf19faec5f980b1f84a": " \\vec v := 3xy\\mathbf{i}+y^2 z\\mathbf{j}+5\\mathbf{k} ", "9f02e7cf68616d85edbf8450ae145277": "\n\\begin{align}\nm(\\varphi) &=\\int_0^\\varphi M(\\varphi) \\, d\\varphi\n= a(1 - e^2)\\int_0^\\varphi \\left (1 - e^2 \\sin^2 \\varphi \\right )^{-3/2} \\, d\\varphi.\n\\end{align}\n", "9f030db26e31e17196afe53eece51639": " \\max_w \\mathcal{L}(w) = \\max_w (w{T}Aw-\\lambda w^{T}Bw)", "9f031591a81d9b1f7e8685a9af0e6668": "\\eta,\\;\\eta\\omega,\\;\\eta\\omega^2,\\;\\ldots,\\;\\eta\\omega^{n-1},", "9f0334b74adcc7bd861cd5aac15ebdcf": "\n\\varphi \\left( {t,x} \\right) = \\left( {{{x - ct} \\over {\\Delta x}} - {1 \\over 2}} \\right)^2 - {1 \\over 4} . \\quad \\quad (13)\n", "9f03362680fec1357205a134cb919ecf": "_{2}\\!", "9f043a40a7b2a60c7821d40ee215d9dd": "Pretrieved", "9f0477e202f9d1d3fbd51098b3c259fb": "I = [A \\ B][A \\ B]^{-1} \\begin{bmatrix}A^\\mathrm{T}\\\\B^\\mathrm{T}\\end{bmatrix}^{-1}\\begin{bmatrix}A^\\mathrm{T}\\\\B^\\mathrm{T}\\end{bmatrix} = [A \\ B](\\begin{bmatrix}A^\\mathrm{T}\\\\B^\\mathrm{T}\\end{bmatrix}[A \\ B])^{-1} \\begin{bmatrix}A^\\mathrm{T}\\\\B^\\mathrm{T}\\end{bmatrix} \n= [A \\ B] \\begin{bmatrix}A^\\mathrm{T}A & O \\\\ O & B^\\mathrm{T}B\\end{bmatrix}^{-1} \\begin{bmatrix}A^\\mathrm{T}\\\\B^\\mathrm{T}\\end{bmatrix}", "9f0485bfe713f8478f7ec364838786b0": " \\xi_k=1/d \\,\\!", "9f049413f1694a2e55abde410956444b": "A\\subseteq E^\\circ ", "9f04975f1ce9c24d41e9e19ed8a55b64": " \\frac{B}{w} = \\frac{h}{w_1} \\, ", "9f04ba9cdb8dc9d789ee8a1868a67bd1": "\\psi : C(E)\\to M", "9f04fffa317b05d1b68c53589fe949ce": "\n\\begin{align}\n&P(|X_t-\\phi(t)|\\leq\\varepsilon \\text{ for every }t\\in[0,T])\n\\\\[6pt]\n&=P(|X^\\phi_t|\\leq\\varepsilon \\text{ for every }t\\in[0,T])\n\\\\[6pt]\n&=\\int_{\\{|X^\\phi_t|\\leq\\varepsilon\\text{ for every }t\\in[0,T]\\}}\n\\exp\\left(\n-\\int^T_0\\dot{\\phi}(t) \\, dX^\\phi_t\n-\\int^T_0\\tfrac{1}{2}|\\dot{\\phi}(t)|^2 \\, dt\n\\right) \\, dP^\\phi.\n\\end{align}\n", "9f05187e896de711e786453e840d09fb": "A = 3.9083 \\times 10^{-3} \\; {}^{\\circ}\\mathrm{C}^{-1}", "9f051f0ac98691aa97f773691878a8ed": "\\frac{\\mathbf{p}^2}{2 m} = E.", "9f05be2c01ca5e0835030f94b3abfda7": "\\mathcal{B} = 0", "9f05fdbe22a3d2243ce715b744adc237": "\\ldots d_4d_2d_0.d_{-2}\\ldots", "9f0673af4913504b3c5a5937026ef125": "X|n", "9f069a698e4e66f9be97eb421da62789": "90^o", "9f07737989d998117e9c2f2f1a19fab4": "1-e^{-\\sigma^2/2}", "9f07950c124e47e737a14493955e3ba5": "\\pi(x) = (\\text{number of primes }\\leq x),", "9f07f8888469c28f0ec3526392c6f669": "Mf=Df+\\frac{n-2}{x_{n}}Q(f)", "9f084ce200f1759fb3c45d3e9ecfc480": "Z=\\{1,2,5\\}", "9f084dbe619ef07c9064ea65a3b65b48": "c_\\mathrm{max} = w+D_1,", "9f086828ce3509c4ad13eeb0b2a3bc28": "(\\forall x P(x)) \\lor (\\exists x Q(x))", "9f086bd5ea16a90c4f600fc8bffcab3d": "y = \\rho \\sin \\varphi", "9f08a51fedd628260f92241be1610fd1": "\\mu,\\lambda \\geq 0", "9f08d156344c39f20178a04be6fae4d4": " {\\left(\\frac{1}{e}\\right)}^\\frac{1}{e}", "9f08fe5ab8dd0d71b38c617302f3e30f": "2\\pi/f", "9f094b9ebf7cbee57a9860eab4afebed": "f \\le g", "9f09b0e444a8d8439ed71ea2ba4fbf01": " m_{1}, m_{2} ", "9f09fdf459367c11f2b4ce95b708bc48": "y=x(E/q^{2})^{-1/\\beta }", "9f0a024a1c0c2ab1eab48db1eda1be38": "\n N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 \\quad \\text{and} \\quad\n M_{\\alpha\\beta} := \\int_{-h}^h x_3~\\sigma_{\\alpha\\beta}~dx_3 \\,.\n", "9f0a6605d42fa1944156913921ec89aa": "\\mu(T,p_i) =g^\\mathrm{u}(T,p^u)+RT\\ln {\\frac{p_i^*}{p^u}} + RT\\ln x_i =\\mu _i^*+ RT\\ln x_i", "9f0a6d8372dd45f16938ae9532c77e54": "D=CG", "9f0aaa960da88849547cf3011f69f73c": " \\widehat{\\mathcal{C}}_{Y \\mid X}^\\pi = \\widehat{\\mathcal{C}}_{YX}^\\pi \\left( (\\widehat{\\mathcal{C}}_{XX}^\\pi )^2 + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\widehat{\\mathcal{C}}_{XX}^\\pi ", "9f0ab291e833ccd81bcd71d02198af01": "g \\, =\\eta+h", "9f0bb79ef8207f7b12ffde0a50704092": "L_{\\kappa , \\kappa}", "9f0bbb4aeb1358b349a08a6d775cf465": "\\displaystyle{Q(a)R(b,a)c=2Q(Q(a)b,a)c.}", "9f0c05a15e6b1c275ccfb6615cdfe488": "\\frac{1}{\\sqrt2} \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}", "9f0c5434b5ebbaaccde89816cddb83cd": "\\ell(x)", "9f0c8e23981259a8330a2ed8dab34b98": "{{P}_{Y}}\\left[ m \\right]={{{\\hat{R}}}_{Y}}\\left[ m \\right]=\\frac{1}{N}{{\\left| \\hat{f}\\left[ m \\right] \\right|}^{2}}", "9f0ca58befceb67bec70631b2273e9c6": "T^p_n (x) = x+n", "9f0cadd178c02635b5b0b0bf38844651": "r_{\\pm}= \\mu \\pm (\\mu^{2}-a^{2})^{1/2}", "9f0d015f5094240388e53a90c1ff998a": "f(t') < 0", "9f0d0e412073d03a80b51df96a4783f4": "\\mathbf Z", "9f0d2e91d884071b20c4e686f3c8503b": "{(\\mathcal{I}(\\theta))}_{i, j} = - \\operatorname{E} \\left [\\frac{\\partial^2}{\\partial\\theta_i \\, \\partial\\theta_j} \\ln (\\mathcal{L}) \\right ]\\,.", "9f0dcdaea165179ac8e09b61ac352173": "S_0\\colon \\Z^k \\to S", "9f0decbf6d2d06dff2371ec416b4fae5": "360\\left(1 - \\frac{1}{\\varphi}\\right) = 360(2 - \\varphi) = \\frac{360}{\\varphi^2} = 180(3 - \\sqrt{5})\\text{ degrees}", "9f0e6d90083729b1fb28efd93e15dffc": "\\Omega_2(t) =\\frac{1}{2}\\int_0^t dt_1 \\int_0^{t_1} dt_2\\ \\left[ A(t_1),A(t_2)\\right]", "9f0e80c3aa51dfeafaaca93e3934559f": "\\mathbf{g}(n)\\left\\{1+\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)\\right\\}", "9f0e86b73eb8f7eccee0f72111777f36": "g(t)=\\mathrm{e}^{-\\gamma^2 G^2 D_0 (\\Delta \\tau)^2 t}", "9f0e89a7e38dd6dce7d5cd84cdb93a84": "\\frac{1}{{{D}_{Ae}}}=\\frac{1-\\alpha {{y}_{a}}}{{{D}_{AB}}}+\\frac{1}{{{D}_{KA}}}", "9f0ec0bfe894cd349d61345eb3de06e3": "z \\approx 0.27", "9f0ef2ea8bf696d59250fcebce3d7439": "R_s = \\overline{\\rho} / x_j = (\\overline{\\sigma} x_j)^{-1} = \\frac{1}{ \\int_0^{x_j} \\sigma(x)dx } ", "9f0efa578c7ae45a2a79ea7fdc43c04e": " M_2 = \\sqrt{\\frac{M_1^2\\left(\\gamma - 1\\right)+2}{2\\gamma M_1^2 - \\left(\\gamma - 1\\right)}}", "9f0f114921dd8bbf20a50bf7fb08023b": "(A P_L)", "9f0f3c99241d200066112dccc9feabed": "Z = \\int \\mathcal{D}\\bold{g}\\, \\mathcal{D}\\phi\\, \\exp\\left(\\int d^4x \\sqrt{|\\bold{g}|}(R+\\mathcal{L}_\\mathrm{matter})\\right)", "9f0f4a5da877fd1d2fecf3ef42174388": "\\operatorname{pos}(U) = 1", "9f0f696d3fef3194ef5ac90c96806686": "\\scriptstyle{0.42 AU}", "9f0fcc2e4a3a3762f84107b93133bc1c": "y \\vee x = x", "9f0ff5044a177395157473ebf2b77996": "C \\subset [r^n - 1]", "9f0fffff8a68269fe407e045eea8c59d": "X_1,\\ X_2,\\ldots", "9f102deab28f906ffcc536c41f0b9a8a": " y_{n+s} = y_{n+s-1} + \\int_{t_{n+s-1}}^{t_{n+s}} p(t)\\,dt. ", "9f104298525a4047c67e3431eb00a48c": "\\tau (\\omega) := \\inf \\{ t > 0 | B_{t} (\\omega) \\geq a \\}.", "9f1066668cd350903f815b0db7879a7e": "\\lim_{x\\to\\pm\\infty}\\left[f(x)-x\\right]", "9f106a6a7932de0dc5120a94870217eb": " \\|r_n\\| \\le \\inf_{p \\in P_n} \\|p(A)\\| \\le \\kappa_2(V) \\inf_{p \\in P_n} \\max_{\\lambda \\in \\sigma(A)} |p(\\lambda)| \\|r_0\\|, \\, ", "9f1071d7901d1dfe30e13a1e9b168a3d": "\\sigma_{yy} + \\sigma_{yz} + \\sigma_{xy}", "9f10768991ef82007d227197982e8232": "\\mbox{Energy charge} = \\frac{[\\mbox{ATP}] + \\frac{1}{2} [\\mbox{ADP}]} {[\\mbox{ATP}] + [\\mbox{ADP}] + [\\mbox{AMP}]}", "9f10a8f98a220ed37ba01f2b4ffb32a0": "Q[L]=m \\sum_i\\dot{x}_i\\ddot{x}_i-\\sum_i\\frac{\\partial V(x)}{\\partial x_i}\\dot{x}_i = \\frac{d}{dt}\\left[\\frac{m}{2}\\sum_i\\dot{x}_i^2-V(x)\\right]", "9f10b5ef1a3d81ae745487de4ec590fc": "k = \\lceil 2 n^{1/3}\\rceil,", "9f10cefce3211e87f788f3a06453a0de": "f'(y)=\\frac{-2}{y^3}", "9f10df03dc333b627818cf2543bc9795": "\\nabla_\\sigma \\nabla_\\mu V_\\nu - \\nabla_\\mu \\nabla_\\sigma V_\\nu = (\\partial_\\mu\\Gamma^\\rho{}_{\\sigma\\nu}\n - \\partial_\\sigma\\Gamma^\\rho{}_{\\mu\\nu}\n + \\Gamma^\\alpha{}_{\\sigma\\nu}\\Gamma^\\rho{}_{\\alpha\\mu}\n - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma})V_\\rho", "9f11491aa0e7d051fa87c492f4c2dcef": "GPK_R", "9f11f337da36a021d1c86ace298fbc8e": "-\\log P(g)", "9f125b321821ab960906d0559937d175": "M_x = \\frac{M}{M_\\odot}", "9f12ad848d7ccfc99450a59f54dcaa90": "~ (k_x,k_y)", "9f12cbc7491c2342d1dc215b1171f071": "U-normalization", "9f12cd76e1e6f19e8ecfb396962bc1f6": "\\lambda(y_i) 0 \\,\\}\\right| ", "9f1cbd26482d15ff546373ec207d1b58": "(\\Omega, \\mathcal F, P)", "9f1cd4ad9e6daa3a7b32ffa9e6613ec3": "\n\\epsilon = \\frac{1-q}{1+q}e^{2i\\phi} = \\frac{a-b}{a+b}e^{2i\\phi}\n", "9f1d0115fb067b97827bf0cf2a864785": "{{f_o{\\partial \\overrightarrow{V_g} \\over \\partial p}}={\\hat{k} \\times \\nabla ({\\partial \\Phi \\over \\partial p})}}", "9f1d5447919424cc354df5cd77908100": "|0\\rangle \\rightarrow |0\\rangle , |1\\rangle \\rightarrow e^{i\\phi}|1\\rangle.", "9f1d6b313b8174e8219ec179c519cf01": "Z=(z_1,\\dots,z_n)", "9f1d7ba296990806ece2fd841b0cc197": "\\frac{\\pi}{3(\\sqrt{6}-\\sqrt{2})}", "9f1d944bdde3d960aff42c7b94da4476": "\\frac{1}{2}+(1-2\\delta)", "9f1db0f92bb49b303ecb9f7d6f3ea722": "\\omega\\, =\\, \\Omega(k)\\, +\\, k\\, U\\,", "9f1e05dbbbcf22e65bebe0cfb10b416d": "(t_0,t_1,t_2)", "9f1e48a05904ac87b6246c476b550937": "H + G \\rightleftharpoons\\ HG", "9f1e67eb406c0a11f41e6cff0e7e3c3f": " (v_1 \\wedge v_2 \\wedge \\cdots \\wedge v_k, w_1 \\wedge w_2 \\wedge \\cdots \\wedge w_k) = {\\rm det} \\, (v_i,w_j). ", "9f1eab2eff0c38e831f4081b801973a0": "\\frac{\\partial p}{\\partial x} = \\mu\\frac{\\partial^2 u}{\\partial z^2}", "9f1ee1e37060bfaea5223531411849c9": " 0 \\leq \\mathrm{BR} \\leq n ", "9f1efe2ab1bda8628e65b3035ca950f6": "CH_4 + 2O_2 \\rightarrow CO_2 + 2H_2O + electricity + heat", "9f1f30e87f93a02bd18ed470bc80f0da": "f_\\mathrm{img} = \\begin{cases} f + 2f_\\mathrm{IF} , & \\mbox{if } f_\\mathrm{LO} > f \\mbox{ (high side injection)}\\\\ f- 2f_\\mathrm{IF}, & \\mbox{if } f_\\mathrm{LO} < f \\mbox{ (low side injection)} \\end{cases} ", "9f1f33801ce58237bb3574f7bcceefdf": "|\\mathbf{k}|=\\omega_{\\mathbf{k}}/c", "9f1fc04f7a82ea9f199f9765a0c97829": "f(x)= \\{ \\mathcal{M}^{-1} \\varphi \\} = \\frac{1}{2 \\pi i} \\int_{c-i \\infty}^{c+i \\infty} x^{-s} \\varphi(s)\\, ds", "9f1feb0c009c19da1df6d0eb006af468": " x_{n+1}=f(x_n), n=0,1,2,\\ldots, ", "9f2013791ba9a8bd4bcdb807e2a26fbb": "\\bar{6}", "9f2025af416634b0f1916f9c9a138304": "\\kappa=100", "9f2059bc25bbff7a7f0a0184a1100423": "V(r)=-\\frac{g^2}{4\\pi} \\frac{1}{r} e^{-\\mu r}", "9f20686116f234e00d1b9333ccdbf521": "\\frac{P}{K}= \\frac{g_n}{(1-t_p)s'_c}", "9f2072ce22af571dae3580586fd9d8ec": "\\mathbf{T} = \\mathbf{T}^{(1)} + \\mathbf{T}^{(2)} + \\mathbf{T}^{(3)} ", "9f2087dd740fe6e263bc4b76fe411358": "\\scriptstyle\\pi(x)\\,>\\,\\mathrm{li}(x)", "9f20badf6fb424c5abfba4a9f8c3043a": "i_j\\equiv 0,1\\bmod 2(p-1)", "9f20bfa91cb3060acf11686015746746": "u(x, t)", "9f20f67d3504ec3c7fd9c928ce97d9c7": "v(x,\\tau)", "9f2108078b37d775e704c6920bdc8f31": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = -\\left\\{\n \\int_{-h}^h x_3^2~\\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} w^0_{,11} \\\\ w^0_{,22} \\\\ w^0_{,12} \\end{bmatrix} \\,.\n", "9f2132df90aa65452638031ccb43dd28": "S_T(v;W)", "9f213590e0c1410373d97db8c1344b6b": "\nK(\\alpha) = \\int_0^{\\pi/2}(1 - \\alpha \\sin^2\\theta)^{-1/2} \\, d\\theta .\n", "9f21406000ea2f5681267a406080d312": "H_0(X) = \\frac 1 {1-0} \\log \\sum_{i=1}^{|X|} p_i^0 = \\log |X|.", "9f21558d7622269cbf7b4fb06156d5fa": " C e^{-m\\vert a \\vert }", "9f216628bb15267ea14814e0b49b3f3a": "e^{-q_1 T}S_1(0) N(d_1) - e^{-q_2 T}S_2(0) N(d_2)", "9f21abbf226eb21aeb068f7a6ea597cb": "\\displaystyle \\frac{1}{{(2 \\pi)}^{n/2}} \\int_{\\mathbf{R}^n} f(\\mathbf x) e^{-i \\boldsymbol \\omega \\cdot \\mathbf x}\\, d^n \\mathbf x ", "9f21df17e91ed9863a3febf6496a6829": "a_{14}*a_{3} ", "9f224554e4c51cb76898772248ecea1e": "\n a(m,m')=\\min\\left(1,\n \\frac{p_{m'm}p_{m'}f_{m'}(m')}{p_{mm'}q_{mm'}(m,u)p_{m}f_m(m)}\\left|\\det\\left(\\frac{\\partial g_{mm'}(m,u)}{\\partial (m,u)}\\right)\\right|\\right),\n", "9f2267a25b7d7c8bfa92e476f2905cbb": "h \\begin{Bmatrix} p , q \\end{Bmatrix}", "9f228b541e17e65ca201249902552b23": "Y_i=(Y_{i1}, Y_{i2}, \\ldots, Y_{i n_i})", "9f22a1c4d1983c85aeda09acd14217db": "\\begin{align}\n L_c &= \\Big(\\frac{Y_w}{L_w} D + 1-D\\Big)L\\\\\n M_c &=\\Big(\\frac{Y_w}{M_w} D + 1-D\\Big)M\\\\\n S_c &= \\Big(\\frac{Y_w}{S_w} D + 1-D\\Big)S\\\\\n\\end{align}", "9f22d66bc0c1d31874bf5b6b9f52300d": "-\\frac{H^2\\chi_\\perp}{2}", "9f231c702d154c89fa9be794ab9ee371": "\\!\\exists R_1 \\ldots \\exists R_n \\psi(R_1 \\ldots R_n)", "9f234a2c3a644970a090ff88c7798537": "{\\mathcal O}_{\\exp}(G)", "9f237c79833b594ca1d954f018e90067": "\\phi_{-n} = \\overline{\\phi_n}. \\, ", "9f2389478742ec98f594ef69e07dc56b": "\\langle m \\rangle", "9f23b17192dbb330024098a07a9e4c00": " \\lambda_1 + \\lambda_2 + \\cdots + \\lambda_n = 1, \\lambda_i \\ge 0. ", "9f2434d5b2db6bb502e5e3aa787f2c37": " {}^\\mathrm{N}\\mathbf{a}^\\mathrm{P} = \\frac{^\\mathrm{N}\\mathrm{d}}{\\mathrm{d}t} ({}^\\mathrm{N}\\mathbf{v}^\\mathrm{P}).", "9f24530036e2ab53089f5b5793473fbf": "A_\\lambda = - \\log \\mathcal{T}.\\ ", "9f24c9a88f2fc249ffebb40b26c8a9b0": "\\nabla \\mathbf{v}.", "9f24fcff84f514f3434867eefd392357": "\\left(h {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} k\\right)_{ijkl} = h_{ik}k_{jl} + h_{jl}k_{ik} - h_{il}k_{jk} - h_{jk}k_{il}", "9f25367637549fe9a36080dc40fc9efb": "D[k,l] = 0 \\qquad \\text{for } k \\ne l.", "9f256655d9bb2a763246e73d67c8e2f2": " a^3 + b^3 = (a + b)(a^2 - ab + b^2),\\,\\!", "9f259571f256688b8aad1aa7335cc8ca": "SU(n+1);", "9f26603dc139a4b8153f17bfd2a37cfd": "T \\cup \\overline T", "9f269ac3fb46128642d8e435829431d5": "2^{n^{O(1)}}", "9f26aa9aeff0b5f551ee83df265ccaec": "3\\uparrow\\uparrow3^{3^3} = 3\\uparrow\\uparrow7625597484987 = \\underbrace{3^{3^{3^{3^{.^{.^{.^{3}}}}}}}}_{7625597484987{\\rm\\ threes}}", "9f26b5b5327dcf407d63d3b8aec46300": "m - r", "9f26c6c7792c0f1d98707c3b7e883808": "\\frac{d^2}{dx^2} D(p+x\\|p) = \\frac{1}{(p+x)(1-p-x)}\\geq 4=\\frac{d^2}{dx^2}(2x^2).", "9f26d8bd81ecdc0a1275c9bdd071f41b": "\\tilde{u}_k(K)=\\frac{\\frac{2m}{\\hbar^2}\\frac{A}{a}f(k)}{\\frac{2mE_k}{\\hbar^2}-(k+K)^2}=\\frac{\\frac{2m}{\\hbar^2}\\frac{A}{a}}{\\frac{2mE_k}{\\hbar^2}-(k+K)^2}\\,f(k)", "9f27311e661c168086a0ef1ee866daf0": "p=\\Delta p+p_0", "9f27410725ab8cc8854a2769c7a516b8": "green", "9f27418b437a9630878744474c6d4e19": "\\Upsilon \\, \\upsilon \\,", "9f274baa1f49105a45e140840984f5fb": " \\beta = \\frac{1}{2} ", "9f277fa55e3123653c992cb18acb4e61": "\n\\dot{m} = \\rho Av\n", "9f285288a42e3b1d9be7eeb7a941394e": "R+RL", "9f287419b9e0a01f6ffb16dcce51cbd1": " (\\mathrm{III}_T * f)(t) = \\sum_{k=-\\infty}^{\\infty} f(t-kT) ", "9f29538dd471d73325aa97fccd9ef5bb": "1 + \\cfrac{1}{1 + \\cfrac{1}{2 + \\cfrac{1}{1 + \\cfrac{1}{2 + \\cfrac{1}{1 + \\ddots}}}}}", "9f298578c5f52ae75d397900a004ec73": "\\begin{pmatrix} Z\\\\ I\\end{pmatrix}", "9f29880fba951e18edfc7fd94ef9d09b": "u_N^n = u_c(t^n).", "9f29a229665f538d20675be9df8876f2": "\\mu(f) = \\mu(g) = 4", "9f29abde1bb7db037da9d05ea02015db": "n_0", "9f29eabb9866fa96d51eb00a6207b058": "\\omega = 2 \\pi f\\,\\,\\!", "9f2a2dbac7d7cac3d38a635e63f4bf0d": "\\mathbb K(\\mathfrak g^*)", "9f2aa0d18995564357e6e4f6b141337d": "g = (1 + P)^N", "9f2ac401e5a6371c01a3a7c27a4d4727": " H(x) = [x > 0].", "9f2b131e7246db1fde2fd61338ea6805": "\n\\sum_{n=N_k}^\\infty\\frac1{n\\ln(n)\\ln_2(n)\\cdots \\ln_{k-1}(n)\\ln_k(n)}\n", "9f2b1d0cadb48fa57a0bdf2f05d33daa": " ~p^{a x + b} = c x + d ", "9f2b336a2896d88f4056f9bd3eb00cf6": "\\nabla^2 \\sqrt \\rho = \\nabla \\nabla \\rho^{1/2} = \\nabla (\\frac{1}{2} \\rho^{-1/2} \\nabla \\rho) = \\frac{1}{2} \\nabla (\\rho^{-1/2} \\nabla \\rho) = \\frac{1}{2} \\left[ (\\nabla \\rho^{-1/2}) \\nabla \\rho + \\rho^{-1/2} \\nabla^2 \\rho \\right]", "9f2b351356748da8f6ca1c8892524b2c": "A'_\\alpha (\\mathbf{r},t) = A_\\alpha (\\mathbf{r},t) - \\partial_\\alpha \\chi (\\mathbf{r},t) \\,", "9f2b4c10564970946c04ec7e722dcf76": "X\\equiv \\frac{m_\\mathrm{H}}{M}", "9f2b6b23113708ea6280c77030168277": "\\displaystyle Wg(3,d) = \\frac{2}{d(d^2-1)(d^2-4)}", "9f2b7349e9904f8161c704ab1545e139": " \\chi_0(a) = \\begin{cases} 1 & \\mbox{if } \\gcd(a,n) = 1, \\\\ 0 & \\mbox{if } \\gcd(a,n) \\ne 1.\n\\end{cases} ", "9f2b784ad7cc1c2bf34bc1432eaee67e": "F r_1^2=\\frac{y_2^2}{2 y_1^2}+\\frac{y_2}{2 y_1}", "9f2c01d2ecacc4ed70ebb6dede28be00": " \\mathrm{number\\ of\\ samples} = (2^n)^2 = 2^{2n}.", "9f2c05a674937913e66a345265d78003": "\\displaystyle I", "9f2c2ab520c49122e51d6643226264b1": "x \\vee z = y \\vee z", "9f2cdfacc763830f0a46609e39725c3f": "\\|\\mathbf{p}\\|=\\sqrt{p_1^2+p_2^2+p_3^2}=n", "9f2ce62d1267840cb03247ec4625800a": " \\partial_t \\omega + \\partial_x \\kappa = 0. ", "9f2d3453026f7fade86dd2bce1548a34": "M(n)=\\sum_{k=1}^n \\frac{\\lambda(k)}{k}", "9f2d63e2fbe38c33fc99677058b3388c": "\\sigma\\colon K\\to K", "9f2d66fca1c8382e672982c3eeac1b68": "\\begin{array}{rcl}\nx(t)&=&R\\left[(1-k)\\cos t+lk\\cos \\frac{1-k}{k}t\\right],\\\\[4pt]\ny(t)&=&R\\left[(1-k)\\sin t-lk\\sin \\frac{1-k}{k}t\\right].\\\\\n\\end{array}", "9f2dd13406f9a6309008925a53589410": "{n \\over p_2} - 1 =p_1 - 1 n] = \\sum_{k=n+1}^{2n} p_k.", "9f5f1b2b3c1d47a362d45ef2012d08d9": "\\forall t . \\textit{executeopen}(t) \\wedge \\textit{true} \\rightarrow \\textit{open}(t+1)", "9f5f285b30bda7e69f6620f489fa1f9a": "\n\\operatorname{E}\n\\left[\n (Z-\\mu)(Z-\\mu)^\\dagger\n\\right] ,\n", "9f5fd302e44fb47505b602e508005c63": "(\\boldsymbol{\\mathsf{L}}^{-1})_{ij} = \\mathbf{e}_i\\cdot\\bar{\\mathbf{e}}_j=\\cos\\theta_{ji}", "9f60108b9061e16dcd9d096e5206c07a": "\n \\tan\\alpha \\approx \\frac{R\\cos\\phi\\,\\delta\\lambda}{R\\,\\delta\\phi}, \\qquad\\qquad\n \\tan\\beta=\\frac{\\delta x}{\\delta y},\n", "9f603763fd1e87a18f6290f466b6a6c4": " SVR = 80 \\times \\frac{\\left (MAP - MRAP\\right)}{CO} ", "9f605a84c0ed3259f0d268363047238b": "\\int_0^1 x^n(\\log\\, x)^n dx", "9f606ba514ee56054a38067c188b1528": "E_\\mathrm{sig} \\cos(\\omega_\\mathrm{sig}t+\\varphi)\\,", "9f60a1ff13878be3f9ce1aae76675217": "H[\\xi]\\leq\\frac{\\pi\\sigma}{\\sqrt{3}}", "9f60bfe47143401e88e6066e8cd896f1": "\n\\frac{(256 - 3125a^4)}{1155}\\frac{d^4\\phi}{da^4} - \\frac{6250a^3}{231}\\frac{d^3\\phi}{da^3} - \\frac{4875a^2}{77}\\frac{d^2\\phi}{da^2} - \\frac{2125a}{77}\\frac{d\\phi}{da} + \\phi = 0\n", "9f60e647c3897acae0ac84ac22a8190f": "\\Gamma(U_i, \\mathcal{O}_{\\mathbf{P}^n_A}) \\to \\Gamma(X_i, \\mathcal{O}_{X_i})", "9f611fc076b07adadd1431b9f77f632e": "\n \\sigma_1 = \\cfrac{Mc_1}{I} = \\cfrac{M}{S_1} ~;~~ \\sigma_2 = -\\cfrac{Mc_2}{I} = -\\cfrac{M}{S_2}\n ", "9f613342760b7b57f5fb0d9cf58587e8": "\\operatorname{Cl}_2(\\varphi)=-\\int_0^{\\varphi} \\log\\Bigg|2\\sin\\frac{x}{2} \\Bigg|\\, dx:", "9f618650d4a92eec3739f83d84df4a3d": "j_2,", "9f621f36301ae55a59919f74d459b343": "\n \\mu_n = \\operatorname{E}[X^n] = \\frac{1}{\\lambda^n} \\sum_{k=0}^n (-1)^k {n \\choose k}\\, \\Beta(\\lambda k+1,\\, \\lambda(n-k)+1 ).\n ", "9f624c6081a59d6abb4bda4879a9a60e": "\\chi(E) := \\sum_{v\\in E} \\chi(v).", "9f625e0f75256e0b30b8cac64a701b8a": "577-408\\sqrt{2}=0.00086\\ldots", "9f62d9046d15045e8a605677552877ff": "L \\; = \\; 46.3 \\; + \\; 33.9\\log f \\; - \\; 13.82 \\log h_B \\; - \\; a(h_R) \\; + \\; [44.9 \\;- \\;6.55 \\log h_B] \\log d \\; + \\; C ", "9f62db2c4952cabceb1c04a1c64368bc": "J \\neq \\{\\}", "9f6311e5a6e393f090dcb7023b19bf91": " 110_2 \\rightarrow 10_2 \\rightarrow 1 ", "9f6326c80c952982ed1df2afbecd4fb9": "K^{+}", "9f637792833c788e320beb38142ad640": "\n G(x;\\sigma)\\equiv\\frac{e^{-x^2/(2\\sigma^2)}}{\\sigma \\sqrt{2\\pi}}\n", "9f637f36fd6176dab995c9be7c7a0cb8": "y(t)= A \\sin(kx + \\omega t).", "9f63ac73b5256811c35d9e9de06192b0": "\\overline{a}_n", "9f63b5ca49c9772de9fba0719b16977c": "\\frac{1}{1} + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{6} + \\frac{1}{24} + \\frac{1}{120} + \\cdots = e.", "9f63d2b96c8926c69075273978ad4a15": "\\mathrm{2Mg_3Si_2O_5(OH)_4}", "9f64044d6bd8d40113d4308b11e9dd0c": "\\frac{\\partial \\Lambda}{\\partial \\lambda}=x^2-1.", "9f640db2f2923fcf29f301847538bc87": " w = \\left\\lfloor \\frac{\\sqrt{8z + 1} - 1}{2} \\right\\rfloor ", "9f6458250f18323c1391a754a0fa413d": "x_1>\\cdots >x_n", "9f64772b499a9276440eaaa92885745e": "V_\\mathrm{out} = (V_+ - V_-) \\cdot G_\\mathrm{openloop} \\pm \\frac {V_\\mathrm{cm}}{ 10 ^ {\\frac {CMRR}{20}}} ", "9f651862c822574599408fddad23b101": "\\mathbb D\\setminus\\{0\\}", "9f6524955ca2c7486091794f44b88930": "\\lim_{\\delta\\rightarrow0^+}\\omega(\\delta,f)\\log(\\delta)=0", "9f653b56726cf3328ffd57f48a3463f8": "\np_{me} = {T n_c \\over V_d} {2 \\pi}\n", "9f657ccdd079ca73c1f4c16d2ac7fa54": "\n\tZ_t = W_0 + 2\\sqrt{W_+W_-}\n", "9f65a45240f6f72b9e2ee8b2fee5ef6b": "F_{-n} = (-1)^{n+1} F_n.", "9f65baa2fc43b590d82df02aaeaf253d": "\\neg p?\\,\\!", "9f65c0232349ca530b97b8f77896798e": " \\nabla^2 = {1 \\over r^2}{\\partial \\over \\partial r}\\left(r^2 {\\partial \\over \\partial r}\\right) ", "9f6612f2a75b8b932ef8e3e037c09b46": "T_{M\\mu \\nu }", "9f667a75489ff7e4c153dbb67b5bd84b": " (P_1P_5)", "9f667f49c11a199c9cc94a7a20f33717": "X(i\\omega)=H(i\\omega)\\cdot F(i\\omega) \\ \\ or \\ \\ H(i\\omega)= {X(i\\omega) \\over F(i\\omega)}.", "9f66b9472d6c50f772bbbf6d7e4d785c": " u''(x)-u(x)=0 ", "9f66ded90b35bd65ee46d5de7f918818": "v_A \\ll c", "9f676d36a8cefe1adec8252ea1a756a9": "a = 0.1 ", "9f67b7ff50cee30c3741904026993129": "z' = z_1 z_2 \\ldots z_{n'} ", "9f68238710060dd6ca59efefdbf91a5e": "\\psi (x)=\\sum_{k}u_k (x)a_k e^{-iE(k)t},\\,", "9f68322c72c295f51ec6f6829b56e888": "\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1", "9f68592d82effee53ac7404daef402db": " r(L) = \\{ xady : xabq, pcdy \\in L \\} \\ . ", "9f68706592270eb8d6a75652bbd18cdb": "\nC^{S_2}_{E_3} = (\\varepsilon^{1}_1 - \\varepsilon^{2}_1) / D\n", "9f687e3c2ff5f99fd48c2360800982d3": "-l \\le m \\le \\,l ", "9f6898c7475bf946831d442d73e5fe45": "\n(\\mathbf{f}_{k' k})_l\\approx\\frac{1}{d}\\left[\\gamma^{k' k}(\\mathbf{R}|\\mathbf{R}+d\\mathbf{e}_l)-\\gamma^{k' k}(\\mathbf{R}|\\mathbf{R})\\right]\n", "9f68a54bf5a90b1129c8f80862b25002": "C = \\prod_p \\frac{1-N(p)/p}{(1-1/p)^m}\\ ", "9f68b3e8065665f2aca501905e4cf48a": "D \\subset \\Bbb Z^n", "9f68b936886d2f4dfef5a2283d92a35e": "m=L_b(\\left\\lceil (L(M)+8)/64 \\right\\rceil)", "9f68d7a0f14ab115b28b30886ef62224": " d = S_{k} + C_1 \\ S_{k-1} + \\cdots - (d/b) (S_{j} + B_1 \\ S_{j-1} + \\cdots ).", "9f68e4f01b2ecdd0102bc8ec4438eb4b": "P_1 - P_2 = \\frac{1}{2}\\cdot\\rho\\cdot V_2^2 - \\frac{1}{2}\\cdot\\rho\\cdot V_1^2 ", "9f691b2fc80edfa367e52d7158618b27": "b\\sin\\alpha=a\\sin\\theta\\pm\\sqrt{c^2-a^2\\cos^2\\theta},\\,", "9f696678f765786e29818e97867d108a": "r_{SOI} = a_p\\left(\\frac{m_p}{m_s}\\right)^{2/5}", "9f69695ee5df05e4ca9d4b8b7c60fc8d": "(A,\\mathfrak{m}_A)", "9f698bbc529b19256e76b1d4aa99da35": " V \\to \\operatorname{Alb}(V) ", "9f6a0f550088edba8b64c44c022ca1f7": " \\frac{1}{4} + 4 \\left( \\frac{1}{8} \\right) = \\frac{3}{4} \\leq 1.", "9f6a4123a9f6f42dc528d157115b040e": "K_c = \\sqrt{\\frac{E G_c}{1 - \\nu^2}}\\,", "9f6a58c5d44e846907de1a0f99d2aae1": "\\displaystyle\\mathbf r", "9f6a9cda09697d60a93e006411d61123": "\\mathrm{st}(S,\\mathcal U)=\\bigcup\\big\\{U_i:i\\in I,\\ S\\cap U_i\\ne\\emptyset\\big\\}. ", "9f6b37d26d8c2b2274c19007eae4c6db": "(h, \\, h^{x+y})", "9f6b4e3e238b3381ca9344bb1405947f": "\\tan\\frac{c}{2} \\cos\\frac{\\alpha-\\beta}{2} = \\tan\\frac{a+b}{2} \\cos\\frac{\\alpha+\\beta}{2}", "9f6b59c4e4490a06686a104d602e38e1": "\\sigma_D = s\\sigma_I", "9f6b5fd3b79b914c825f6b02284fde13": "u_1=\\dot{\\lambda}", "9f6b99f7538f519b2f3d955a2413e84e": "{\\omega^2}_3 = -\\cos\\theta \\, d\\phi", "9f6bce0ad8957ec9a5a3ee4ded487f03": "\\tilde{f}_i \\tilde{e}_i", "9f6bfeff1fa67fdb07abd0a70b2bee67": "s_\\mathrm{in} = \\frac{1+\\left|S_{11}\\right|}{1-\\left|S_{11}\\right|}\\,", "9f6c2458bfd7f178e969a2079952d701": "\\mathcal{O}_L / \\mathcal{O}_L \\mathfrak{p}", "9f6c328a8672fcad1f0fc2fa3a930b09": "\\, =10[(x-5)+(y-5)] + (10-x)(10-y)", "9f6c4b5bf6e08460fbb349ec0b93f98e": "\\varepsilon_{kk}", "9f6c73928c0d447c1b4db71d2443bcdf": "S \\to \\alpha", "9f6d3906ad4edc7ade8baeeffaf65760": "\\text{parent} = \\frac{i - 1}{2} \\textbf{ or } \\frac{i - 2}{2}", "9f6e015e187e3d4fd7b87b4e685d07ba": "\\textstyle b > l_1 -1", "9f6e0422b4db32c3a267f325becbb044": "Q_1=(b_1- b_2)^2+(a_1-a_2)^2", "9f6e20cbb356d1c4767502d2fbd1c252": "\\sigma_1(W)\\geq\\gamma\\sqrt{pq}", "9f6e33ac2eddf928ab8f6a6011430cb7": "\\frac {F^\\prime(s)}{F(s)} = - \\sum_{n=1}^\\infty \\frac{f(n)\\Lambda(n)}{n^s}", "9f6e4989b6a34a97beae4bd75b5e72eb": "t_0=-2\\frac{|q|}{q}\\sqrt{-\\frac{p}{3}}\\cosh\\left(\\frac{1}{3}\\operatorname{arcosh}\\left(\\frac{-3|q|}{2p}\\sqrt{\\frac{-3}{p}}\\right)\\right) \\quad \\text{if } \\quad 4p^3+27q^2>0 \\text{ and } p<0\\,,", "9f6e7b18694b07fd848eabf4c9ac90be": "x_k \\succ x_1", "9f6e8f3c137aabf6e4cbb13e6252fccc": "\\scriptstyle{dx}", "9f6edaf8f9416cc1616e5a37c7f2e33b": "\\Phi(x)", "9f6f02e1ce1397c5f68f840236163e6d": "-1 \\in \\{\\pm 1\\},", "9f6f31fe800f8e0707d248fdc2909663": "h:\\mathbf{x} \\mapsto h(\\mathbf{x}) \\in \\mathbb{R} ", "9f6f62d35b4233ed799c592614415f2a": "\\tau_G(v) = C({k_i},2) = \\frac{1}{2}k_i(k_i-1).", "9f6f73abb25105e6ae5b383cda8567e0": "\\Gamma_i=\\Gamma(1+i/k)", "9f702253085d6b7b9cecd185f1b829cf": " \\left(\\varepsilon_\\text{r}, \\mu_\\text{r}\\right) \\equiv \\left(\\frac{\\varepsilon}{\\varepsilon_0}, \\frac{\\mu}{\\mu_0}\\right)", "9f702b72996b99f0b6c10d41d921e63f": "X \\hookrightarrow Y", "9f70758c87c7ea7801bfcd9e28c8b3f5": "m, Jn(m,x)", "9f70765227c060263e94fa66b463f6de": "\n\\frac{1}{\\phi}=-\\sum_{k=1}^\\infty\\frac{\\mu(k)}{k}\\log\\left(1-\\frac{1}{\\phi^k}\\right).\n", "9f70964b6056b31865a471786422f0b9": " \\begin{align}\n B_m(n) &= n^m-\\sum_{k=0}^{m-1}\\binom mk\\frac{B_k(n)}{m-k+1} \\\\\n B_0(n) &= 1.\n\\end{align}", "9f71050b635768789814d86eac80d899": "d=2 + \\epsilon", "9f713fc2a0f4aff67e7edecd6f56aba9": "dn-\\binom{d+1}{2}", "9f716652147c517c54f3d8bc66c46a1c": "x^{30} + x^{29} + x^{21} + x^{20} + x^{15} + x^{13} + x^{12} + x^{11} + x^{8} + x^{7} + x^{6} + x^{2} + x + 1 ", "9f7194f6b514eb305eef75ef9bf6bb76": "y_j=\\beta_0+\\beta_1 x_{1j}+\\beta_2 x_{2j}+\\cdots+\\eta+\\varepsilon_j \\, ", "9f719fb894ff1f587883d99ae5d7946e": "\\nu = 3/4", "9f71b7c134614fbc1ad874ffa78fb394": "f(x_1,x_2):\\{0,1\\}^n\\to\\{0,1\\},\\ x_1,x_2\\in\\{0,1\\}^{n'},\\ 2n'=n", "9f71cc687231b36ed03c141ee5efd9b2": "\\bar \\Omega", "9f71fc68a6c95e730e802353c9c6be97": "\\bar{X}_1", "9f7242a4480661f9d2d6527b3af2a433": "\nc = ( 3 + p/|p| ) / 2 \\,\n", "9f72c42e957addc9d229644ade08ed7d": "i_1,i_2,i_3>1", "9f7320a806192d5398112febe95f4e76": "\\theta\\in\\Theta", "9f73dbcd00c04987e0f407fd1d6dae8c": "\\{u_k\\}_{k=1}^\\infty\\subset H", "9f7402639e2ea96efe8c523823832573": "g_{00}+2g_{0i}u^i+g_{ij}u^i u^j=0", "9f74128659c16082bcf3d0db83ed8364": "\\rho={\\log P_1\\over \\log P_2}", "9f745e7a1a268f08ea23ed4dce8ce6ef": "R(M,x) = x^T M x ", "9f748c043c98cf3767f6391337099106": " \\frac{\\omega_A}{\\omega_I} = \\frac{N_I}{N_A}, \\quad \\frac{\\omega_I}{\\omega_B} = \\frac{N_B}{N_I}.", "9f75065653506cd25ce6e241778dee28": "0\\leq x \\leq 1, 0\\leq y \\leq 1\\,", "9f751a305ed0eb88c4a879e1e42990cf": "I_y\\,", "9f75685d51304706b4791e615a7927bf": "\\chi(\\tau,f_D)", "9f75ca8be96dcec5acd8ca80adeac857": "nr \\,\\!", "9f760229be40414df77885fe81615a64": "(c) \\qquad |K(x,y) - K(x,y')| \\leq \\frac{C|y-y'|^\\delta}{\\bigl(|x-y|+|x-y'|\\bigr)^{n+\\delta}}\\text{ whenever }|y-y'| \\leq \\frac{1}{2}\\max\\bigl(|x-y'|,|x-y|\\bigr)", "9f76023e477543f1aa07bab518b7db52": " \\vec x ", "9f7656ed1910c0ac1bd581833c092435": " W(\\varphi_\\lambda,\\theta_\\lambda) = \\varphi_\\lambda\\theta_\\lambda^\\prime- \\theta_\\lambda \\varphi_\\lambda^\\prime\\equiv 1,", "9f76955d18fa0bf93d77cbf2a7136a1c": "150\\leq A \\leq 165", "9f76bc88c6ec9ece478ebe6e18783619": "(x, v) \\mapsto x.", "9f772ee9a2d8c7843c8e237d3519d97f": "\nW = \\int_{t_1}^{t_2} v(t) i(t)\\, dt .\n", "9f77a21dc8f1620c1841bf899007ca59": "\\psi_{n00}(\\bold{r}) = R_{n0}(r) Y_0^0", "9f781b478488da3d579f9f881b2fd0e9": "r = 10.30 \\%", "9f786f1cef47b8a1e6a8f546fe4b7971": "BV = \\frac{PV}{1-HC}", "9f78a5c58424b2e44697c26b81f9d62d": "Y_x=\\varphi_*(X_{\\varphi^{-1}(x)}).", "9f78abaf1a963fd38470c07898b666cc": "\\dot{p}_i =\\frac{\\partial L}{\\partial q_i}=0.", "9f78b8f3fd78919dd89c74ec9d6fd005": "\\mathbf{F}\\times (\\nabla\\times\\mathbf{F})=0.", "9f7912c36cd8070734044e712c5c1215": " a_{ij}\\ge 0", "9f798e571764ac241f4b2c9f3b3006f0": "q\\equiv w^{2} \\pmod l", "9f799ab46f08299c185a31a7afc18111": "7^2+24^2 = 25^2", "9f79dcc11d8a0d17a27206bccdd6a7a0": "\nV \\to V + \\Lambda + \\overline{\\Lambda}\n", "9f7a0d3844ea65d9198d8b537732c057": "a^{p-1} \\equiv 1 \\pmod p", "9f7aba87f7c09e5b1fe8f901270cd2fc": "SU(4)\\times SU(2)_L\\times SU(2)_R", "9f7aec3ab630839ff02810eccaebb022": "E=v\\sqrt{2(n+\\alpha)eB\\hbar}, \\alpha=0", "9f7b03f0b6e1c96f80b5621c08bd0461": "\\scriptstyle \\mathbb{D}", "9f7b0f6b8d372044b909ed601bc33e13": "\\varepsilon_0 = \\omega^{\\omega^{\\omega^{\\cdot^{\\cdot^\\cdot}}}} = \\sup \\{ \\omega, \\omega^{\\omega}, \\omega^{\\omega^{\\omega}}, \\omega^{\\omega^{\\omega^\\omega}}, \\dots \\}", "9f7b23f951db559604eee4ff332330e6": "\\frac{-j}{-i}=k", "9f7b3bff98ba7b5a2d539d4905fcb379": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 1}{4 \\choose 2}{8 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "9f7b6494671409ffcefb4f5a46f9889d": "R_{\\rm d} = \\sqrt{r_{\\rm d}^2 + \\frac{3}{4}\\left(\\frac{m_{\\rm e}}{m_{\\rm d}}\\right)^2 \\left(\\frac{\\lambda_{\\rm C}}{2\\pi}\\right)^2}", "9f7b6b40f8129733360b513153dae5d0": "C \\otimes H", "9f7bd31dec019eca0a7b6f59d5792fa1": "\\left(1,\\ 1+\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+3\\sqrt{2}\\right)", "9f7be03e564050b6cf236abe5015d4c0": "\\int x^2\\arccsc(a\\,x)\\,dx=\n \\frac{x^3\\arccsc(a\\,x)}{3}\\,+\\,\n \\frac{1}{6\\,a^3}\\,\\operatorname{artanh}\\,\\sqrt{1-\\frac{1}{a^2\\,x^2}}\\,+\\,\n \\frac{x^2}{6\\,a}\\sqrt{1-\\frac{1}{a^2\\,x^2}}\\,+\\,C", "9f7bffcf58423289a08cfbcdc822fc10": "\\scriptstyle N_1=2 ", "9f7c2527cee06287848259aeba110cf9": "|M-m(\\phi_{n-1})|<0.01", "9f7cbfed48a231a52693793b9d344824": "V_\\mathrm{BE} = -V_\\mathrm{out}\\,\\!", "9f7ceffc3a9d214715bca59682699f46": "M_0=1", "9f7d705289d341dc4c3e6c7275cc948b": "\\begin{align}\n \\frac {\\mathrm{d}f}{\\mathrm{d}\\varphi} &= \\int_0^{2\\pi} \\frac{\\partial}{\\partial\\varphi}\\left(e^{\\varphi\\cos\\theta}\\;\\cos(\\varphi\\sin\\theta)\\right)\\;\\mathrm{d}\\theta \\\\\n &= \\int_0^{2\\pi} e^{\\varphi\\cos\\theta} \\left(\\cos\\theta\\cos(\\varphi\\sin\\theta)-\\sin\\theta\\sin(\\varphi\\sin\\theta)\\right)\\;\\mathrm{d}\\theta \\\\\n &= \\int_0^{2\\pi} \\frac {1}{\\varphi}\\;\\frac {\\partial}{\\partial\\theta}\\left(e^{\\varphi\\cos\\theta} \\sin(\\varphi\\sin\\theta)\\right)\\;\\mathrm{d}\\theta \\\\\n &= \\frac {1}{\\varphi} \\int_0^{2\\pi}\\;\\mathrm{d}\\left(e^{\\varphi\\cos\\theta} \\sin(\\varphi\\sin\\theta)\\right) \\\\\n &= \\frac {1}{\\varphi} \\left(e^{\\varphi\\cos\\theta}\\;\\sin(\\varphi\\sin\\theta)\\right)\\;\\bigg|_0^{2\\pi} = 0.\n\\end{align}", "9f7d8e91aa75ca505434af96df40d2ca": "\\scriptstyle{2\\pi i \\mathbb{Z}}", "9f7db3865bd1b009bd76b2e950928937": " A = X_1\\cdot Z_2 ", "9f7dd6f8283a35f233121aeada15d628": "dx = \\frac{2 \\, dt}{t^2 + 1}.", "9f7ded11a04ccbe03967416709b1e6c6": "\n\\dot{\\phi_j} \\approx \\{\\phi_j, H\\}_{PB} + \\sum_k u_k\\{\\phi_j,\\phi_k\\}_{PB} \\approx 0.\n", "9f7e021b6544c5145a2d40d74161e5f7": "\\Delta \\vdash A \\to B \\,.", "9f7e147855af0604bf661ac3b6dcb6da": "\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}=\n\\begin{bmatrix}\n 1 & 0 \\\\\n \\frac{1}{\\lambda R} & 1\n\\end{bmatrix}\n", "9f7e5ea5b7abbcf446a15276fea27e5e": "\\mathrm{d}\\mu_{\\alpha} = \\mathrm{d}\\mu_{\\beta}.", "9f7f5190495bf16acbbe32b9c9c2adfa": "C_Y^{-1}C_{YX}=\\left[\\begin{array}{ccc}\n4.85 & -1.71 & -.142\\\\\n-1.71 & .428 & .2857\\\\\n-.142 & .2857 & -.1429\\end{array}\\right]\\left[\\begin{array}{c}\n4\\\\\n9\\\\\n10\\end{array}\\right]=\\left[\\begin{array}{c}\n2.57\\\\\n-.142\\\\\n.5714\\end{array}\\right]=W^T.", "9f7f642dff202386048cf68844fe4b1c": "1_{n\\geq 0} - 1_{n<0}", "9f7fb76cdaff2b3963c7c2953643af69": "G(x,y)\\in\\overline{K}", "9f7fc8ca4fe8eaaaa9cb3b14f75d280c": "L_r,\\,L_t,\\,L_L/10", "9f7fe0cb4cde39545c41baffb6134d33": "\\ A_q(n,d)", "9f8000568dd2bb41f2fa65766f6464f3": "P_0 = \\frac{GM}{R^2}\\frac{2\\tau}{3k}", "9f804e16f3fd9dfe0553a048051e7318": "\\begin{align}\n\\sum_{n=1}^{\\infty} \\varepsilon_n &<\\infty, \\\\\n\\sum_{n=1}^{\\infty} \\delta_n&<\\infty, \\\\\n\\prod_{n=1}^{\\infty} (1+\\delta_n) & 0 \\\\ \n(x_0+\\frac{1}{y_0},+\\infty) & y_0 < 0 \\end{cases}", "9f815ed8f5570b1dec7960fad16d733b": "\\ w' = -\\xi' \\frac{\\part \\overline{w}}{\\part z}.", "9f816fa115990d50736e74cd1f7815ba": "n_2\\!", "9f81cbdde72cc4493b0aa34288d1a64f": " (\\bullet\\bullet)(\\bullet\\bullet)", "9f81d80cc52538f05a33b5d04548af5c": "\n\\begin{align}\n{{\\partial w }\\over{\\partial t }} &\\sim \\frac{W}{T} \\\\[1.2ex]\nu {\\frac{\\partial w}{\\partial x}} &\\sim U\\frac{W}{L} &\\qquad\nv {\\frac{\\partial w}{\\partial y}} &\\sim U\\frac{W}{L} &\\qquad\nw {\\frac{\\partial w}{\\partial z}} &\\sim W\\frac{W}{H} \\\\[1.2ex]\n{\\frac{u^2}{R}} &\\sim \\frac{U^2}{R} &\\qquad\n{\\frac{v^2}{R}} &\\sim \\frac{U^2}{R} \\\\[1.2ex]\n\\frac{1}{\\varrho}\\frac{\\partial p}{\\partial z} &\\sim \\frac{1}{\\varrho}\\frac{\\Delta P}{H} &\\qquad\n\\Omega u \\cos \\varphi &\\sim \\Omega U \\\\[1.2ex]\n\\nu \\frac{\\partial^2 w}{\\partial x^2} &\\sim \\nu \\frac{W}{L^2} &\\qquad\n\\nu \\frac{\\partial^2 w}{\\partial y^2} &\\sim \\nu \\frac{W}{L^2} &\\qquad\n\\nu \\frac{\\partial^2 w}{\\partial z^2} &\\sim \\nu \\frac{W}{H^2}\n\\end{align}\n", "9f81dbea995ad4bdb8191f9a403466bf": "\\mathcal M_Y", "9f82a03c9e9cd9c0acaeada633177160": "\\sqrt {S} = \\sqrt {\\vert S \\vert} \\, \\, i \\,.", "9f82ae8d9547391971e6846411e4d17f": "\\mu_{15}=2", "9f82c696e9f5135f205663cd67385c8c": "\n\\frac{\\partial f_N}{\\partial t} + \\sum_{i=1}^N \\dot{\\mathbf{q}}_i \\frac{\\partial f_N}{\\partial \\mathbf{q}_i} + \\sum_{i=1}^N \\left( - \\frac{\\partial \\Phi_i^{ext}}{\\partial \\mathbf{q}_i} - \\sum_{j=1}^N \\frac{\\partial \\Phi_{ij}}{\\partial \\mathbf{q}_i} \\right) \\frac{\\partial f_N}{\\partial \\mathbf{p}_i} = 0.\n", "9f82d90743dd1187f0e2790bd0ec2e43": "\n\\vartheta_3(z) = \\sum_{n=-\\infty}^\\infty q^{n^2} \\exp (2 n i z)", "9f82f07e35d8c244b7f8c7287ce7bd99": "A\\to B,B\\to C\\vdash A\\to C", "9f831388e3ddc498bac8a5bccc73c622": "\n\\begin{bmatrix}\n n & \\sum x_i & \\sum x_i^2 \\\\ \n \\sum x_i & \\sum x_i^2 & \\sum x_i^3 \\\\\n \\sum x_i^2 & \\sum x_i^3 & \\sum x_i^4 \n\\end{bmatrix}\n\n\\begin{bmatrix}\n a_2 \\\\ \n a_1 \\\\\n a_0 \n\\end{bmatrix}\n=\n\\begin{bmatrix}\n \\sum f_i \\\\ \n \\sum f_i x_i \\\\\n \\sum f_i x_i^2\n\\end{bmatrix}\n", "9f8347812f4af7f32fa13b1b6995c3bb": "Y \\xrightarrow{v} Z \\xrightarrow{w} X[1] \\xrightarrow{-u[1]} Y[1]", "9f835164b3f7b7554a4f54cb64729e0e": "\\begin{Bmatrix} 3 \\\\ 8 \\end{Bmatrix}", "9f83e4273561b00450a6bfd29b9eed57": " \\langle x | j \\rangle = \\Psi(r_{x,j}) e^{-i \\phi _j}", "9f83eb358994ea937c495e67d267343e": "\\frac{\\partial u}{\\partial t} = \\eta v - \\frac{\\partial \\Phi}{\\partial x} - c_p \\theta \\frac{\\partial \\pi}{\\partial x} - z\\frac{\\partial u}{\\partial \\sigma} - \\frac{\\partial (\\frac{u^2 + v^2}{2})}{\\partial x} ", "9f83f55e196967a0e7d3927d0d7b848e": "\\frac{1}{2} \\,", "9f83f7be9c33d4bb1da557777b431087": "u_b = \\frac{Q}{\\zeta} + \\tfrac16\\, \\zeta^2\\, u_b'' + \\cdots.", "9f843a773f8e65b0b8625711645374d0": "T_G(x,y)=\\sum\\nolimits_{i,j} t_{ij} x^iy^j,", "9f84a3914ed3f0dcc0978c58204ed293": " v_s ", "9f84a66d88d24c3b1bc91df5b5346a13": "O(n^2)", "9f8515b09106e6ae691054b72e1c81c6": " \\omega \\rightarrow 0", "9f8521aec8727eb99417b3c4cda676cb": "c(m)=1", "9f85743ad8200de18f85869c839b3892": "k=\\frac{1}{3}Cv_g\\Lambda=\\frac{1}{3}Cv_g^2\\tau", "9f8599871fb19faddcb8d10ae35e52c8": "EHS = HS \\times (1 - NPOT) + (1-HS) \\times PPOT", "9f85a84e5dc4dd2448cb22e460b1f3e6": "E(x_t)=x_0 e^{-\\theta t}+\\mu(1-e^{-\\theta t}) \\!\\ ", "9f85e122df10e556e4c820aedf8e07bf": " F=\\mu A \\frac{u}{y}", "9f85e1f5a58d63444114436d5ffd01be": "\\frac{e^{iut}}{\\sqrt{2 \\pi}}", "9f85e82d57609e0fec22589bb36fe33a": "p< 0 & \\implies \\quad S_{11} > 0 \\\\\n \\Delta_2 > 0 & \\implies \\quad S_{11}S_{22} - S_{12}^2 > 0 \\\\\n \\Delta_3 > 0 & \\implies \\quad (S_{11}S_{22}-S_{12}^2)S_{33}-S_{11}S_{23}^2+2S_{12}S_{23}S_{13}-S_{22}S_{13}^2 >0 \\\\\n \\Delta_4 > 0 & \\implies \\quad S_{44}\\Delta_3 > 0 \\implies S_{44} > 0\\\\\n \\Delta_5 > 0 & \\implies \\quad S_{44}S_{55}\\Delta_3 > 0 \\implies S_{55} > 0 \\\\\n \\Delta_6 > 0 & \\implies \\quad S_{44}S_{55}S_{66}\\Delta_3 > 0 \\implies S_{66} > 0\n \\end{align}\n ", "9f91b0c0a868592d92dbae8b875fb575": "\n\\alpha \\ge \\psi(q,\\alpha,u) \\ \\ \\ \\longleftrightarrow \\ \\ \\ r_{w} \\le R(q,u)\n", "9f91eeeafe9b396b3b3b0d635600e32f": "(D+\\omega L) \\mathbf{x} = \\omega \\mathbf{b} - [\\omega U + (\\omega-1) D ] \\mathbf{x} ", "9f9222ed36d98ceb2590af98be9f8e1a": "B(T,L)=b(\\epsilon L^{1/\\nu})", "9f922765e66ed7e0af0b1171de00680a": " {\\partial \\rho \\over \\partial t} + \\nabla \\cdot (\\rho \\mathbf{u}) = 0", "9f925d08d924943a07e338fbe2251df5": "\\tau'=\\frac{1}{r}\\mathbf{r}\\cdot\\mathbf{Q}s^2c_2+\\alpha_j'\\big[as^3c_3+\\frac{1}{2}bs^4c^2_2-2\\gamma s^5(c_5-4\\bar{c}_5)\\big]", "9f927c6b546997a7b6d7c9e5fbd7da22": "\nZ(S_n) = \\frac{1}{n} \\sum_{l=1}^n a_l \\; Z(S_{n-l}).", "9f92e2d0081f9656e190f070dd515cc8": "\\Omega_{1,1/2}\\propto\\binom{z/r}{(x+iy)/r}", "9f9324a13a508e7a86f9a0a0e318865f": "I_{\\text{C}} = \\alpha_F I_{\\text{E}}", "9f933a968277bfb3a6ad92ba9ebe02b1": "x,\\xi", "9f936ce1b797f059e3d3346a1692831d": "\\operatorname{st}(x) < \\operatorname{st}(y)", "9f937d76887de8a1bd71e6919bb5bdbb": "\\frac{W_m^r(S)}{V_g^f(S)} = \\cfrac{K_BK_v/D_m}{\\left(t_g^fs + 1\\right)\\left(t_ms + \\frac{K_m}{D_m}\\right)}", "9f9380b8042c226e453c5bf6a59f3a70": "\\dot{x} = -a \\cdot \\sin E \\cdot \\dot{E}", "9f93822d3a3c26ade927ac227fb386f9": "a, b\\in A", "9f9410bf676f6031d550738b46ce0eed": "f_Y \\circ c_1 = c_2 \\circ f_X", "9f944dff790292f9953aead4a8a9df36": "\\,\\epsilon \\in \\Sigma", "9f944eba78d51aba7cbc4e24b3287d81": "Q_{q} = \\sqrt { \\frac {\\hbar} {2M\\omega_{q}}}(a^{\\dagger}_{-q}+a_{q})", "9f947a02ce36b79b61a46cea8c41b26f": "\\scriptstyle \\pi r^{2}_{AB} c_A", "9f94a42d0591e20f1dcba6fe7625f971": "\\textstyle X \\sim N(\\mu, \\sigma^2)", "9f94d6e110af82416954564c2b8068b2": " b_1 = f(0,0) \\,", "9f9515f788f796607c9bbd892f27dfa9": "g: [0,1] \\to [0,1]", "9f953b722a75c1ced1eda33d852de41b": "(2\\ell+1)", "9f956eb7c275547ed84f9e9299c49928": "\\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B}}{\\partial t} \\qquad \\qquad \\ (2)", "9f9573347a7c66d0308bf0d907d7fc17": "\\displaystyle{f(x_0)={1\\over R^{n-1}\\omega_{n-1}} \\int_{|y-x_0|=R} f(y)\\, dy.}", "9f9574e89bf6ac7b5a182d261a96d283": "A_Q", "9f95c629046424b27bed82714a3b5204": "\\frac{2-p}{\\sqrt{1-p}}\\!", "9f9610005c9b453287df68746a6dc066": "|\\phi \\rangle", "9f9612b24e45989138e719938916dd36": "\\hat qq^{-1}", "9f968f9de8058fa75b412c51dff2ee81": "J(x):X'\\to{\\mathbb F}", "9f969509d4278c606018d3d77ec835c0": " {\\partial \\mathbf{x} \\over \\partial \\lambda_1}\\times {\\partial \\mathbf{x} \\over \\partial \\lambda_2} =\\left({\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial \\lambda_1}\\right) \\times \\left({\\partial \\mathbf{x} \\over \\partial q^j}{\\partial q^j \\over \\partial \\lambda_2}\\right) = \\mathcal{E}_{kmp}\\left( h_{ki}{\\partial q^i \\over \\partial \\lambda_1}\\right)\\left(h_{mj}{\\partial q^j \\over \\partial \\lambda_2}\\right) \\mathbf{b}_p ", "9f96db8bf244b3625ea30446c3ef1687": "\\mu\\in\\mathfrak{h}^*", "9f974cbeb176d473cdc5a11af346f37e": "S \\Rightarrow^{ac}_{f} AA \\Rightarrow^{ac}_{g} AA \\Rightarrow^{ac}_{h} SA \\Rightarrow^{ac}_{h} SS \\Rightarrow^{ac}_{k} SS \\Rightarrow^{ac}_{l} aS \\Rightarrow^{ac}_{l} aa", "9f978bb3ca5f9b25b71966171b80ac87": " w = \\tanh L ", "9f98286e6bf1df6da34007f9198391ab": "S \\,", "9f9880aa11f404714c8c03426292ac83": "N_{E}\\approx H_{E}", "9f988260cbfd3cf1b0fee90dc944e9a0": "c_\\infty^2/c^2=1+4(\\phi/c_\\infty^2)+(15+2\\alpha)(\\phi/c_\\infty^2)^2\\,", "9f98f66399c4d046ac3e4b006041a7c0": "(\\Omega,\\Psi,S,A,R,q,p)", "9f992702c1920fa2357fcda71e27e364": "\\frac{C_{P}}{C_{V}} = \\frac{\\left(\\frac{\\partial S}{\\partial T}\\right)_{P}}{\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}}\\,", "9f992e9efa063005aac0eaca27951518": "w_{1}\\leq N", "9f99744472869be3ff21be10eb7d334e": "\\{\\mathrm{length}(c_k)\\}_{k=1}^{M}", "9f999961c6ea70dd29b0cdce96196125": "xy \\equiv zz \\rightarrow x=y.", "9f99a4f28868e3034f681957c6ca4625": "(x, \\lambda)", "9f99d2bd58a8d202cd8405be2d8b5917": "\n\\begin{align}\n\\hat{H}_{I} &= \\hbar \\omega_c \\left(\\hat{a}^{\\dagger}\\hat{a} +\\frac{\\hat{\\sigma}_z}{2}\\right)\\\\\n\\hat{H}_{II} &= \\hbar \\delta \\frac{\\hat{\\sigma}_z}{2}\n+\\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{\\sigma}_+\n+\\hat{a}^{\\dagger}\\hat{\\sigma}_-\\right)\n\\end{align}\n", "9f99ddddd45a7f1d01efc7cfb7f109ea": "P=\\left(\\frac{D}{\\omega}+L\\right)\\frac{\\omega}{2-\\omega}D^{-1}\\left(\\frac{D}{\\omega}+U\\right),", "9f9a9b9c194614a3602ba911ddbf4394": " sinon ", "9f9b2b55130dbcd43c755f8a0e6aa6f0": "n^{(n - 1)^{(n - 2) \\cdots }}.\\,", "9f9b4c2b614e16b2612c8a8e4db3e48a": "x = \\mathrm{azeq}_x\\left(\\frac\\lambda 2, \\phi\\right)\\,", "9f9b843994f75e584c13baa74ba8907f": "{T^\\mu}_\\lambda", "9f9bb6769a26ac637a0ac15fd2386dc2": "(x^2-1)", "9f9c16bee1d419b730698369aa359626": "n_1n_2", "9f9c4768ab0213b68481c61fdec56663": "A^+ = \\frac{1}{2}\\left(\\sqrt{A^2}+A\\right).", "9f9c4abd2801a8b97899cbfa2e710657": "\\triangle\\delta = \\delta' - \\delta = -\\beta \\quad \\left(= -\\beta \\cdot \\sin(90^\\circ)\\right)", "9f9c6a11c8c786434a95332e6aad8b26": "A=(\\frac{3}{2}(2+\\sqrt{3}))a^2\\approx5.59808...a^2", "9f9c6cfcc8c65c969c204e40a91a0870": " X \\prec Y \\Leftrightarrow Y - X ", "9f9c6f22ff17b0ea0f20ad72731085e2": "\\mathcal{F}_t = \\sigma(\\{X_s^{-1}(A) : s\\leq t,A \\in S\\})", "9f9cbb7c20cb834b35b9f41c4af183e0": "\\Delta = \\Delta_1 \\times \\dotsb \\times \\Delta_N", "9f9ce7e55fdc869c20389fb4def68230": "{\\sigma }_{0}={e}^{2}/\\left(\\hslash {a}_{0}\\right)", "9f9d09875def174a12d15e1ea42dc717": "\\hat{g}_N(x)", "9f9d09bcac7daba622e4d38488efd415": "e_n=\\frac1{n!}\n\\begin{vmatrix}p_1 & 1 & 0 & \\cdots\\\\ p_2 & p_1 & 2 & 0 & \\cdots \\\\ \\vdots&& \\ddots & \\ddots \\\\ p_{n-1} & p_{n-2} & \\cdots & p_1 & n-1 \\\\ p_n & p_{n-1} & \\cdots & p_2 & p_1\n\\end{vmatrix},\n\\qquad p_n=(-1)^{n-1}\n\\begin{vmatrix}h_1 & 1 & 0 & \\cdots\\\\ 2h_2 & h_1 & 1 & 0 & \\cdots\\\\ 3h_3 & h_2 & h_1 & 1 \\\\ \\vdots &&& \\ddots & \\ddots \n\\\\ nh_n & h_{n-1} & \\cdots & & h_1\n\\end{vmatrix},\n\\qquad h_n=\\frac1{n!}\n\\begin{vmatrix}p_1 & -1 & 0 & \\cdots\\\\ p_2 & p_1 & -2 & 0 & \\cdots \\\\ \\vdots&& \\ddots & \\ddots \\\\ p_{n-1} & p_{n-2} & \\cdots & p_1 & 1-n \\\\ p_n & p_{n-1} & \\cdots & p_2 & p_1 \\end{vmatrix}.\n", "9f9d2373a293775b91faaef404f1e8fc": " U_{(\\pi_{s}, \\pi_{o})} = a*\\pi_{s} + b*\\pi_{o} - c*|\\pi_{s} - \\pi_{o}| ", "9f9d55e151fef03e771effd32b16f5af": "L^2 = l(l+1) \\hbar^2 \\,", "9f9da338d18d49ce4ced099f4aca7473": "a\\frac{dx}{dt} + bx = Af(t).", "9f9deaf91fe8a4fec608663245895567": "bcode(x) \\le bcode(y) < bcode(x) + 2^{-L(x)}", "9f9e417e44827b2bdc9bb74270f57532": "\\sqrt[n]{a^m} = \\left(a^m\\right)^{\\frac{1}{n}} = a^{\\frac{m}{n}}.", "9f9e4f97126600ea79294681832d3aef": "\\beta\\eta", "9f9e67ec5e1ec4f0b0c2420730d53af1": "V_R", "9f9f4b6dc5dda4fca94a25cd46e6d124": "Z(\\beta) = \\sum_{x_i} \\exp \\left(-\\sum_k\\beta_k H_k(x_i) \\right)", "9f9f521b833e94d2760351bcd2003439": " \\mathrm{div}(\\dots) ", "9f9f542a56bcc84bf5ef7f77f66c21c3": "S(A: B |\\Lambda)=0", "9f9f950e5e0a86104de3691fe7a3dc2c": "\\Delta(t) = t\\otimes I + I \\otimes t +\\sum_{s\\subset t} s\\otimes [t\\backslash s],", "9f9fb3c3de919eb8450c28d889334f9b": " M_i^* \\Delta_3= (\\Delta_2 + {3\\over 4})M_i^*.", "9f9fba2f045b78dcb09b9d17d55cdd0e": "w\\Vdash(\\forall x\\,A)[e]", "9f9fc260f2ec1fd15607656a84abe790": "0-0^2=0,\\ 1-1^2=0,\\,", "9f9fd7e419e9a6d2144072f9a83eef6e": "C_3 < S_3", "9f9fdc8d094bb576c50d609bad00932c": "\\mathrm{A}_n", "9f9fe94ce207c9da8d951041d0c7c45f": "FV = PV \\cdot (1+i)^t", "9f9febcb314054343f039e0e812dfd4e": "\\{z_1,\\ldots,z_m\\}", "9fa023ef474bce58bb52d353bdd99603": "\\tau = \\frac{\\omega_2}{\\omega_1}", "9fa03af89a50f88ea45150c3180d9f46": "= \\int_{-\\infty}^{\\infty}{\\left|h(t) e^{-s t} \\right| dt}", "9fa03b8e2777dc326dac395337a0ee89": "\ne^{\\theta(x)}=\\prod_{p\\le x}p.\\;\n", "9fa051e90a1bd01739cef336deecd86a": "plus(x,y,z)", "9fa067ce48a00d4cf33b6da99bdb47f8": "dV = r^{n-1}\\sin^{n-2}(\\phi_1)\\sin^{n-3}(\\phi_2) \\cdots \\sin(\\phi_{n-2})\\,\ndr\\,d\\phi_1\\,d\\phi_2 \\cdots d\\phi_{n-1},", "9fa08914e28217959603ab9a0f5d232c": "Z_{11} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \\over \\Delta_S} Z_0 \\,", "9fa09c7fad318d88f941b2e5c7cc77ee": "\\Delta \\vec{p}\\,\\!", "9fa0d32478639daed848dbd7f86082e5": "\\omega_r(\\phi) = \\left|\\left| {\\partial (x,y,z) \\over \\partial \\lambda } \\right|\\right| = r \\cos \\phi\\ .", "9fa16ccbae8f39de93c1e25066123914": "1 + v^T A^{-1}u \\neq 0", "9fa1731806eb914b2422fe3f0abe8606": " P_2 \\ ", "9fa180662823c1f2b9e8051703ec81dd": "D(P \\parallel Q)", "9fa1a741c2e96aea0dea4031f2ab52a4": "I(t) = \\frac{1}{R_p}\\left[ U_g(t) - \\frac{\\exp (-t/R_pC_p)}{R_pC_p}\\left( \\int\\limits_{0}^{t}U_g(t_1)\\exp (t_1/R_pC_p)dt_1 + C_{int} \\right) \\right]", "9fa1eadc1d2a60c10605c146092c24c0": "L(Y|\\boldsymbol{\\mu})=\\prod_{i=1}^n \\left ( 1_{y_i=1}(\\mu_i) + 1_{y_i=0} (1-\\mu_i) \\right ), \\,\\!", "9fa1ecde27405efa0955920141f5cbe9": "I(t)-\\frac{V_\\mathrm{m} (t)}{R_\\mathrm{m}} = C_\\mathrm{m} \\frac{d V_\\mathrm{m} (t)}{d t}", "9fa26426afd07aec6a84f51e42a7e395": "\\mathbf{B}=\\begin{bmatrix} 1+\\gamma^2 & \\gamma & 0 \\\\\n\\gamma & 1 & 0 \\\\ \n0 & 0 & 1 \\end{bmatrix}\\,\\!", "9fa26e21cbe061bd122b147ac31f3495": "F_{*} (\\mu) = \\mu.\\ ", "9fa27252b89127558f6d4a8711bf9e75": "U_{11}", "9fa27f96cbaee92a57e3e5f96ad9d78b": "\\mathbf{x}\\times\\mathbf{J}(\\mathbf{x})/2", "9fa2be5f579ac946b8f38d8dc1b7cd9d": "\\mathfrak{k}_0", "9fa2e88ba30875c7b53fce68ddff697a": " r_j = b_j^2 + \\tau_j^2 ", "9fa38a5fa9808538b61d6cfd848b5c76": "a_1 = b_3-b_1", "9fa3fb40267661722034087ba61ef635": " \\overset{\\text{Energy flux rate}}{\\lambda_v E=\\frac{\\Delta (R_n-G) + \\rho_a c_p \\left( \\delta e \\right) g_a }\n{\\Delta + \\gamma \\left ( 1 + g_a / g_s \\right)}}\n~ \\iff ~\n \\overset{\\text{Volume flux rate}}{ET_o=\\frac{\\Delta (R_n-G) + \\rho_a c_p \\left( \\delta e \\right) g_a }\n{ \\left( \\Delta + \\gamma \\left ( 1 + g_a / g_s \\right) \\right) L_v }}\n", "9fa405037af9ba95f6da88875a4480a8": "\\varphi_\\pm = (1\\pm \\sqrt{5})/2", "9fa43d67db94f99e53e4f023cf692872": "\n\\left[Q(\\mathbf{k}),Q(\\mathbf{l})\\right] = 0, ", "9fa444b3e845641f7528f7429155d95b": "R_i M_j \\subseteq M_{i+j}", "9fa49ba7b9ec684eb296e643841394ae": "F_1\\;=\\;\\big(\\frac{h_B} {30.48}\\big)^2", "9fa5279a16c8d3f04723492852e291bc": "C - D = \\frac{\\nu_c - \\nu_d}{\\nu_a - \\nu_b} (A - B) + k, ", "9fa59267a0152d93929e06a1dbbb60f7": "\\omega^{< \\omega}\\,", "9fa599feb9e3b284ab7bd2b8a1ee88ca": "\n T_{11} = \\cfrac{\\sigma_{11}}{\\lambda} = \n \\left(\\lambda - \\cfrac{1}{\\lambda^5}\\right)\\left(\\cfrac{\\mu J_m}{J_m - I_1 + 3}\\right) = T_{22}~.\n ", "9fa5b260c05be896b65b285e78d458e7": "\\sum_{1\\le k\\le n \\atop \\gcd(k,n)=1}\\!\\!k = \\frac{1}{2}n(\\varphi(n)+[n=1])", "9fa5d179887aeaf7d3fb442dbd9e620a": " {\\overrightarrow{V}} ", "9fa5ee3180883ccdaed2cb0a3f0dbd1f": "V \\otimes \\cdots \\otimes V \\otimes V^* \\otimes \\cdots \\otimes V^*", "9fa5f1ab80d3006cdd234bca87ce4e88": "G(\\theta)", "9fa6087615ec6d50e8e099db3ee31840": "f^{*} (E_{V}) = (f^{*} E)_{U}", "9fa63fe8b766562bacf40a35414b2378": "f(x) = \\frac{x}{\\sigma^2} \\exp\\left(-\\frac{x^2}{2\\sigma^2}\\right)", "9fa6727fe4602151ac67663ebc2fe4a6": "W = \\frac{2 \\pi}{\\hbar} \\left| M_{i,f} \\right|^2 \\times \\text{(Phase Space)} = \\frac{\\ln 2}{t_{1/2}}", "9fa706ce4637d814af536cb3aed27a4f": " \\frac{z}{y}=kd ", "9fa7251b1c441da01ca10cd3b137ea50": "\\scriptstyle{\\vec{d}_{i,j}}", "9fa72eb407de5c85e01c9206446f868b": "\n\\begin{array}{l|l}\nS = \\emptyset, T = \\emptyset & \nS \\neq \\emptyset, T \\neq \\emptyset, S \\subset T \\\\\n\\hline\nS = \\emptyset, T \\neq \\emptyset & \nS \\neq \\emptyset, T \\neq \\emptyset, T \\subset S \\\\\n\\hline\nS \\neq \\emptyset, T = \\emptyset &\nS \\neq \\emptyset, T \\neq \\emptyset, T = S \\\\\n\\hline\nS \\neq \\emptyset, T \\neq \\emptyset, S \\cap T = \\emptyset &\nS \\neq \\emptyset, T \\neq \\emptyset, S \\cap T \\neq \\emptyset, \\lnot (S \\subseteq T), \\lnot (T \\subseteq S), S \\neq T\n\\end{array}\n", "9fa7648b1701b19fe9770050b02583e2": "\n\\cosh \\mu = \\frac{d_{1} + d_{2}}{2a}\n", "9fa785f887913392f0e6d2db27c7b2aa": "D_T = \\{\\,w \\in (T \\cup \\overline T)^* \\mid w \\text{ is a correctly nested sequence of parentheses} \\,\\} ", "9fa7b7b59c7893ec9571970606593945": "I=- \\iiint\\limits_V\\left(\\nabla\\cdot\\mathbf{J}\\right)dV.", "9fa7cd3e769de341c6b45b083ac87a3a": "X \\left[ k \\right] = \\sum_{n=0}^{N-1} W[n] x[n] e^ { \\frac{-j2 \\pi kn}{N}} ", "9fa7d9870e051a2e6e4209a1b515f57c": "\\,_np_x\\!", "9fa833dd6224c153d0792b69829492cc": "\nE = \\frac{1}{2}M \\dot{r}^2+\\frac{1}{2} m \\left(\\dot{r}^2+r^2\\dot{\\theta}^2\\right) + Mgr - mgr \\cos{\\theta} = Mgr_0 - mgr_0 \\cos{\\theta_0}\n", "9fa8536c843cfda4dba239b1024d659c": " \\mathcal{L}_p ", "9fa8684e7ff3082bca11fe27b02debe1": " \\psi(0) = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}", "9fa8c66a100ee23f179279bd55ffac5c": "A + \\operatorname{core}B \\subset \\operatorname{core}(A + B)", "9fa8cb350d653b70f3a7d6f97e04dc56": "\\Epsilon_G", "9fa92525bf6157f6917ba2f4ca4cc098": " T_a = \\inf \\{ t \\geq 0 \\colon L^0_t > a \\}.", "9fa93a4cd0396f282cc961e5253ac379": "I_{2}(\\sigma_{yy}\\sigma_{zz} - \\sigma^2_{yz}) - I_{3}(\\sigma_{yy}+\\sigma_{zz})", "9fa959f0ef9f1da29ea17396c4ff80a5": "7^5", "9fa98254ea1c89cdbcede4f39ab87856": "L_x = \\partial_x L", "9fa993d4515e47959379a1723489a7d9": "2\\sqrt{d-1}", "9fa9a055ba9aaaaff30facccedfa9b1f": "\\varphi(\\vec x)", "9fa9bac11ef1fc8a77384d6a5410fc17": "a \\wedge (b \\vee c) = (a \\wedge b) \\vee (a \\wedge c)", "9fa9cc0a5cc17263ab13ae7a58b5166a": "\\frac{\\text{d} [{^0_2}S]}{\\text{d}t} = \\text{k}_{2(1)} C_1 -\\text{k}_{1(1)} {^0_2}S E ", "9fa9db03aec2d1018a53e44dab339f99": "N(a) = O\\left(\\frac{(\\log a)(\\log \\log \\log a)}{(\\log \\log a)^3}\\right),\\,", "9fa9ed21b5a1394df1f84d848d99a2cf": " \\Delta S_{\\mathrm{system}} = \\Delta S_{\\mathrm{compensated}}+\\Delta S_{\\mathrm{uncompensated}}\\,\\,\\,\\,\\text{with}\\,\\,\\,\\,\\Delta S_{\\mathrm{compensated}}=-\\Delta S_{\\mathrm{surroundings}}.", "9faa1419d00b692018bfd0d61ee349ca": "\n \\boldsymbol{V}\\cdot(\\mathbf{n}_i\\otimes\\mathbf{n}_i)\\cdot\\boldsymbol{V} = \n \\lambda_i^2~\\mathbf{n}_i\\otimes\\mathbf{n}_i ~;~~ i=1,2,3.\n ", "9faa8644419e0e7ed9f1da61b1fc1a3c": " P = {5^{1/4}}. \\ ", "9faabb235070ca7beb72190bf6b67ac3": "100 = 3^{3+1} + 2\\cdot 3^2 + 1.", "9fab1fb39c5fdc4d79f85765a062f4da": " \\mu = {m_1m_2 \\over m_1+m_2} ", "9fab4aa4aaab28f962b78832b1ec8b68": "\n \\alpha\nD_{\\alpha }\\left( \\frac{n\\hbar }{a_{n}}\\right) ^{\\alpha }=\\frac{Ze^{2}}{a_{n}\n}, \n", "9fab5c44b3661cdd7bc276202d1cbbe7": "x_{\\sigma(j)}y_j+x_jy_k\\le x_jy_j+x_{\\sigma(j)}y_k\\,, \\quad(3)", "9fabda3a05fbf193af15d1aa7e485fe9": "\\operatorname{Li}_{-2}(z)=\\sum_{k=1}^\\infty k^2 z^k=\\frac{z(1+z)}{(1-z)^3}\\,\\!", "9fac211e3e70f4604925dfb1ac4bea5a": "\\boldsymbol{Q}", "9fac622893b1064c9e2c2edcca210f92": "0, \\infty\\,", "9faca0cca35a9fb7640fd3d7900d614d": "\\begin{align} p_n &= r^n \\sum_{m=0}^{m=n} P_n^m(\\cos\\theta)(a_{mn}\\cos m\\phi +\\tilde{a}_{mn} \\sin m\\phi) \\\\\n\\Phi_n &= r^n \\sum_{m=0}^{m=n} P_n^m(\\cos\\theta)(b_{mn}\\cos m\\phi +\\tilde{b}_{mn} \\sin m\\phi) \\\\\n\\chi_n &= r^n \\sum_{m=0}^{m=n} P_n^m(\\cos\\theta)(c_{mn}\\cos m\\phi +\\tilde{c}_{mn} \\sin m\\phi) \\end{align}", "9facc3014341c17cba96901c71444705": "w=\\Pr \\{b1", "9fb5ed0c774dcb787d1124e7b07ecc8c": "\\|x-y\\|<\\epsilon.", "9fb5fb1bf31c305078b20fadd5c3cefb": "e^i, 1 \\leq i \\leq m", "9fb6360cd8cb0c9320903c845f293283": "\\scriptstyle f_{cyan} = f_\\mathrm{image}(1) = f_{gold} - 1\\cdot f_s.\\,", "9fb643c6e419a8a5602a95533179d986": " U(d,d_1) - P = CS \\,", "9fb65796688215ac0ee4452ca0986e05": " \\sum_{i=1}^n\\sum_{j=1}^n K(x_i, x_j) c_i c_j \\geq 0", "9fb684c65227ab478bdb134f6b288b95": "\\omega_L+\\omega_0 ", "9fb6c727b082391bfc23e75dc50bca26": "s_a(t)\\cdot e^{-j 2\\pi \\frac{B}{2} t},", "9fb6e7a49af7bd22fe53177e2dde09e8": "\\scriptstyle \\hat{\\mathbf{u}}", "9fb7941e83c3a4798760b763d00c2efe": "U_{ij} = U_i \\times_U U_j", "9fb7deb65d31001efb9ba593b389385e": "X\\oplus 0 \\cong 0\\oplus X \\cong X", "9fb7fbd729fef0dab3b99580f9e4ce6c": "g=m", "9fb84e0bb573ba381e07b56163e8b3a8": "\\sqrt{I} = \\operatorname{I}_{\\mathbb{P}^n}(\\operatorname{V}_{\\mathbb{P}^n}(I)),", "9fb86666733f46332dfbea6afa798619": "(x_1, x_2)", "9fb8c569f9e0625ff71fa887143435d6": "A_m(3,1) = 1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, \\ldots", "9fb8d40ce23806d2ec96a360c0bc1d81": "\\mu (E) = \\int_{[0, 1]} \\mu_{x} (E \\cap L_{x}) \\, \\mathrm{d} x", "9fb8db3f25898209e0b6f5b2098482d0": "w = 500 \\left ( \\frac{\\ell n}{n-1} + 12n + 36 \\right ) ", "9fb909ac69e01e351e29a38527803bc3": "x > c_2", "9fb91372290d4307156daa04ab95a0c2": "\\begin{align} \n a_r &= \\frac{(r + c - 1)(r + c - 2) + (1 + \\alpha + \\beta)(r + c - 1) + \\alpha\\beta} {(r + c)(r + c - 1) + \\gamma(r + c)} a_{r - 1} \\\\ \n &= \\frac{(r + c -1)(r + c + \\alpha + \\beta - 1) + \\alpha\\beta}{(r + c)(r + c + \\gamma - 1)} a_{r - 1}\n\\end{align}", "9fb9252fe65ecda262b557feaa96aead": "Q_e = \\sqrt{2\\alpha m_0c^2\\epsilon_0 \\lambda_0} = e. \\ ", "9fb9bffa3891191bb461eb1ab0fb62a2": "\\left(\\frac{\\partial S}{\\partial x}\\right)_{E} = \\frac{X}{T}\\,", "9fb9cb6771fd3512b984ca3d5f09fc1e": "\\Psi = A^G A^{G-1}\\dotsb A^2 A^1 |x\\rangle", "9fb9fb841e67d29ab34bc4645d3de0a7": "\\mathbf{1}_{X}", "9fba177959d982509d378203e15302d2": "\n\\boldsymbol{\\alpha}=(1,0,0,0,0),\n", "9fba4fea90d386d2e2a5f02a8e52e231": "\\Phi_{mn}", "9fbaca69661ab9ba1cbca5a9d9f7fcf2": "F_m = \\frac{R_{norm}}{R_{modern}}", "9fbb297fcaf00bfd8487c81fa672b0d0": "E_i = c F^{i0},", "9fbb2ea52220dce0beff26ebd4f7921e": " {Q_2} ", "9fbba522a4c31a233d1744cf81939855": "\\tfrac11", "9fbbb1c4e99d8399f474f7a5afdeb369": "\\{b_k\\}_{k=1}^{M-1}", "9fbc23622d51d50b7f5bb1e321374a90": "q2*=(5000+2c1-3c2)/4", "9fbc334765b0babc6313bd1ef114f148": "M_y= (-x^4-\\frac{8x^3}{3}+16x^2)|_0^2", "9fbc4c1d78908d1c0a6fba48c1a4862b": "w_\\min = q ", "9fbc503aed72193af706f61290e1ce01": "\n1-F(\\rho,\\sigma) \\le T(\\rho,\\sigma) \\le\\sqrt{1-F(\\rho,\\sigma)^2} \\, .\n", "9fbcb927c7b40403325ac77d06f804c5": "[sf,g] = s[f,g]", "9fbcccf456ef61f9ea007c417297911d": "a.", "9fbd2fd1456abac33d241ea1e4f6bb7b": "N\\propto D^{1-q}+\\text{a constant}.", "9fbd5999a048cb1cfe58de1ae706eaa1": "\nKZ_{m,k} [X(t)] = \\sum\\limits_{s = - k(m - 1)/2}^{k(m - 1)/2} {X(t + s)\\times{a_s^{m,k}}}\n", "9fbd74cfa0bac6befabf26f102b30553": "\\textstyle \\pi(\\theta \\in [0,1]) = 1", "9fbd7a15c0f269f369199dc55df9278c": " P = k_v\\, T \\,\\!", "9fbd89f25e0b3341568dcbbcb29a0336": "\\begin{matrix}\n y_{1}=r_{A}-r_{B}+\\varepsilon _{1} \\\\\n y_{2}=r_{C}-r_{D}+\\varepsilon _{2} \\\\\n ... \\\\\n y_{5}=r_{B}-r_{C}+\\varepsilon _{5} \\\\\n\\end{matrix}", "9fbdaeb63436fde681e4aff35611f124": " \\zeta (L,S)\\, ", "9fbdba87bafe6c3d75c9c519fcbd64ec": "\\frac{\\partial^2 fg}{\\partial x_1 \\partial y_2} - \\frac{\\partial^2 fg}{\\partial x_2 \\partial y_1}", "9fbdf99479e188ae1e84d6b949bae593": " P_B(t) ", "9fbe1ea0fb3fdad784b40cef6ec36a81": " E_d = -\\infty ", "9fbe53403fe20afa8056f78eb3df30be": "S = 10^{\\left( {S^\\circ - 1} \\right)/10}", "9fbe9aaa640af9980d6392588b75e2e2": "\\frac m2", "9fbec6efcaeb2f7271b0fc2ac120d0d8": "\\sqrt{2/N}", "9fbf133496275ebd51d166c8f0f04a1c": "-\\frac{\\lambda^2}{\\hbar^2}\\int_{t_0}^t dt_1\\int_{t_0}^{t_1} dt_2e^{\\frac{i}{\\hbar}H_0(t_1-t_0)}V(t_1)e^{-\\frac{i}{\\hbar}H_0(t_1-t_0)} e^{\\frac{i}{\\hbar}H_0(t_2-t_0)}V(t_2)e^{-\\frac{i}{\\hbar}H_0(t_2-t_0)}+\\ldots", "9fbf2cd97501e3b66d2924569afd2e79": "\\frac{\\sqrt{15}}{2}\\sin(2\\theta)\\cos^2(\\phi)", "9fbf41f741613afa9fd53c9b1fe89e92": "W^*_x(t,f)=W_x(t,f)", "9fc0091afa30ff904a6c8051ce5f7633": "\\delta (e_\\alpha^I e_I^\\alpha) = 0", "9fc0220e9bad6c7011ef56ea08f667de": "\n \\delta\\boldsymbol{\\epsilon} = \\tfrac{1}{2}[\\boldsymbol{\\nabla}\\delta\\mathbf{u}+(\\boldsymbol{\\nabla}\\delta\\mathbf{u})^T] ~;~~\n\\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\sigma}+\\mathbf{b}=\\mathbf{0} ~.\n ", "9fc0849eaccffc68764972362866bb3a": " \\langle x(t) \\rangle = \\int dx' \\int dx \\, x' P(x',t|x,0) W(x,0) . ", "9fc088357f065784055da3d27d444b8d": "c_0=1/4\\pi^{3/2}, \\,\\, c_k=1/(2\\pi)^{3/2}\\,\\,(k\\ne 0)", "9fc098ac1303a9b4a35232a89ecfa040": "\\mathrm{r.ann}(\\mathrm{\\ell.ann}(T))=T", "9fc0ae10769d23cc7f08dab7d083b02d": " \nM(\\vec X) = \\left( {\\begin{array}{*{20}c}\n {\\mu \\Sigma ^{ - 1} } \\\\\n { - \\Sigma ^{ - 1} } \\\\\n\\end{array}} \\right)\n", "9fc0e9de1ea89bc1192f0ee6bde6bdf8": "\\{y_n\\}", "9fc0efebdbcee60b44d954032bd83cf3": "\nU =\n\\begin{pmatrix}\n1&1&1&1\\\\\n1&i&-1&-i\\\\\n1&-1&1&-1\\\\\n1&-i&-1&i\n\\end{pmatrix}\n", "9fc1379460842b4bbf30365b419e7186": "\\left(\\sup_\\alpha f_\\alpha\\right)^*(x)\\le \\inf_\\alpha f_\\alpha^*(x).", "9fc1902d7230df7e5508b67a501b1d00": "u_i \\in \\Sigma", "9fc19bfa30d08b4990c05a05d57f2736": "N/N\\,\\!", "9fc1ab27228d5f00f349a8487d0c9144": "\\Phi(t) = \\int_0^{t}\\varphi(\\tau)\\mathrm{d}\\tau", "9fc1ace9f3c63dc6b701e1cb3879729b": "\\phi_0 = \\frac{h}{e} - \\ ", "9fc1eec9f87e07d25572fc7143a0bc10": "(\\tfrac{a}{m}) = -1", "9fc2043753658575093b2d88ae263c1e": "lb_{computed}", "9fc2223214aab442ffc7c3120ed92b8b": "\\nabla_b Z_a", "9fc272e24221fb55dc96d8dffae6daed": "\\infty - \\infty = NaN", "9fc28e2813e26d2e555ff6c3913a84a7": "\\phi(G) = \\min_{S \\subseteq V}\\varphi(S).", "9fc2b19f921ad6cc50e99f2c025d02e5": "\\quad \\sqrt{(x'_1-y'_1)^2 + (x'_2-y'_2)^2 + (x'_3-y'_3)^2} = c(s'-t')", "9fc2c384b3f3325c8052840a695e2d8d": "0 \\leq j < m", "9fc2c394f125a1747ef6110d13271328": " \\emptyset = K_0 \\subseteq K_1 \\subseteq \\ldots \\subseteq K_n = K ", "9fc2db97ef4174941ec69a64b8097db5": "1 + c + c^2 + \\cdots\n =\\frac{1}{1-c}\n =\\frac{1}{1-MPC} = \\frac{1}{MPS}", "9fc3015a2317e73e4b80cc12e3a3aaa6": "X \\leq 0", "9fc33b9158f0121599a2a68c641682fc": "(\\mathbf{A},\\mathbf{B})", "9fc3582077eb9dba989d54194429f133": "\n w(x,y) = \\sum_{m=1}^\\infty \\left[ \n \\left(A_m + B_m\\frac{m\\pi y}{a}\\right) \\cosh\\frac{m\\pi y}{a} + \n \\left(C_m + D_m\\frac{m\\pi y}{a}\\right) \\sinh\\frac{m\\pi y}{a} \n \\right] \\sin \\frac{m\\pi x}{a} \\,.\n", "9fc37eff96bee6f5c18d897d28308a05": "\nn\\lambda = 2L\n\\,", "9fc3827c05e18324d3f672c43205089e": "\\dot{Q}", "9fc3894c2eef4b62048faca044471e66": "\\phi_i(b) = \\max\\{ n \\ge 0 : \\tilde{f}_i^n b \\ne 0 \\}", "9fc3c181eb5aa0c0808be79fd27eedb2": "\\frac {1} {z}", "9fc4054922e885b407975751a36cd04e": "\\omega(x) = (x^2-1)/8.", "9fc41c93f1c1371ae2415f228a9952b0": "\\mathbf C = \\begin{bmatrix}\n-4 & 2 & 1\\\\\n1 & 6 & 2\\\\\n1 & -2 & 5\\end{bmatrix}\n", "9fc43a55adab0d135f7bf616c50f072d": "(T, {+})", "9fc482735f22310d72e204febd059740": "g = G(\\xi_1, \\xi_2, \\cdots )", "9fc5183763d21ad0decc6d093f0eba3b": " F(x;\\,m,\\Omega) = P\\left(m, \\frac{m}{\\Omega}x^2\\right)", "9fc52698b343a58f5f346441a774bb74": "(((\\infty),[c,d])\\notin I", "9fc537f966ed6fbea47ece58ff9cdd7b": "f\\in \\{\\max,\\min\\}", "9fc596a8d7211fa459469ad45120f13e": " \\langle xy | xy \\rangle = \\langle x|x \\rangle \\langle y|y \\rangle \\ ", "9fc5df2ed2dc9320803bea0ba0f34bdf": "\\mathbf{d}", "9fc5f410f806baffa3e914d572897e31": "\\vec F_h", "9fc5fe0d9295cf57b882efecb4a552ba": " \\Phi_{bh}", "9fc601e2f77d649b1eb05b340158b396": " \\chi(\\lambda) = \\left( \\lambda + 8 \\pi \\epsilon \\right)^2 \\, \\left( \\lambda - 8 \\pi \\epsilon \\right)^2 ", "9fc64667b371890d7a9ba7dffc2e53c5": "St=f_{st}D/U", "9fc69a4ef360883356709f0367890fd3": " I_L = \\sum_{i=1}^N m_i |\\Delta\\mathbf{r}_i^\\perp|^2= -\\sum_{i=1}^N m_i \\mathbf{S}\\cdot[\\Delta r_i]^2\\mathbf{S},", "9fc6a554afb17316bd86d5dcb79e9cf5": " \\frac{d^3}{\\left(dx\\right)^3} \\bigl(f(x)\\bigr)=\\frac{d^3}{dx^3} \\bigl(f(x)\\bigr)", "9fc6c916e140d47613ff3d83db1318d3": "~m_{max}=2", "9fc7795edc1e6d26d91680801aa856dd": "2\\lceil\\log_2k\\rceil", "9fc7887747314bb420e23f3b68a56683": "\\ \\mathrm{SNR}(f) = S(f)/N(f)", "9fc7934616c9f3d27f499f5e96d772b9": "\\int fd\\lambda = \\lim \\sum_{i=1}^n f(\\alpha_i)\\lambda (f^{-1}(A_i))", "9fc796b98367c1fc7f1ad879dd7c5d73": "D_0(x)", "9fc7a0ba1d0affa37ebcef72d5863021": "\\Rightarrow v_j ' v_i = 0", "9fc80b1e2522b792fe6187810df73cf2": "N \\setminus \\{b_i\\}", "9fc86698c5d76290b59ceb2ace1d6bc6": "\\log(\\exp X\\exp Y) = X + \\left ( \\int^1_0 \\psi \\left ( e^{\\operatorname{ad} _X} ~ e^{t \\,\\text{ad} _ Y}\\right ) \\, dt \\right) \\, Y, ", "9fc875e6a6ffa37e70b71e14397aafec": "1/r^4", "9fc8a5f36d7db899655c051e56a6d84c": " x^2 = -1 \\implies x = \\frac{-1}{x} \\implies f\\ x = \\frac{-1}{x} \\and Y\\ f = x", "9fc8ba9257587fc80cf373a3b50862e2": "h = \\frac{\\sqrt 5 - 1}{4} \\ .", "9fc91b6683c96fcbdcca27d2194a45ab": "\\int\\frac{1}{x^2(ax+b)} \\, dx = -\\frac{1}{bx} + \\frac{a}{b^2}\\ln\\left|\\frac{ax+b}{x}\\right| + C", "9fc91fce7204cab8ea40976ce8fe2186": "x=\\left( y,z\\right) ", "9fca0b024bdf9aba1a2adccd16b41948": "S \\subseteq A", "9fca93c04b008e97fa759b4ca8ab7ccf": "\\epsilon(x_0)", "9fca95b981901b9cf9258184355fdeb2": "L = \\{(a,\\mathit{in}),(b,\\mathit{out}),(c,\\mathit{out}),(d,\\mathit{in})\\}", "9fcab9dd13d443087332edf5a6334125": "M(p) = |a| \\prod_{i=1}^{n}\\max\\{1,|\\alpha_i|\\}=|a|\\prod_{|\\alpha_i| \\ge 1} |\\alpha_i|.", "9fcb30498bf75c78b2359c9174329056": "N_s\\times 1", "9fcb67b5320526584afed59d2b5ff381": "\\sum_{i=1}^n {r_i} = r_1 + r_2 + r_3 + \\cdots + r_n", "9fcb750af93c678dbf31f6c2d98d2243": "\\sum_{Treatments} I_j (m_j-m)^2", "9fcb8ac3e5c1e3ab474002d2e764e45f": "u(X)", "9fcbb3267eebd08b9e7f8b1ed9a13f26": "W\\left(e\\right) = 1\\,", "9fcc0937d221140b9a470a7111b53dfb": "L_1 = 45.23", "9fcc1c476d75839e88038269e5982d18": "\\theta_p = \\int\\!\\!\\!\\int \\!\\!\\! \\int\\phi\\Omega_p \\; d \\Omega_p d \\rho_p d \\bold{u}_p", "9fcc1f26682542cfa09db5249fb754d4": "E_c(u) = \\exp\\left[-\\frac{1}{2}\\left(\\pi\\lambda\\delta\\right)^2u^4\\right],", "9fcc20b835edd3460dde38ec9f47131d": "\nK = \\frac{3}{2}M \\sigma^2\n", "9fcc3a01e4a23a54c8bfb2633f9f6137": "\n V_3 = 25 - R_a - R_b = R_c\n ", "9fcc56e838fbf05eb056a64d20d3870a": "\\frac{\\mathrm{d}U}{\\mathrm{d}t} = \\dot Q + \\dot n H_m - p\\frac{\\mathrm{d}V}{\\mathrm{d}t}.", "9fccaf80445c68d665f73dd9a1bce0d6": "\\mathcal{L}(A)", "9fccbd84cdb876d46306ab49a0af03d3": "\\begin{matrix} 64 \\times {4 \\choose 1}{10 \\choose 1}{3 \\choose 2} = 7,680 \\end{matrix}", "9fcd5a49b3f190e03ed3471c5db111b9": "|n|,\\,", "9fcd5b510f53fd6a753f9149eef60f2e": "(MW/P_c)^{1/2} \\, = \\, 0.348 + 0.0159 * MW + \\sum_{j=1}^{35} n_j \\Delta_j", "9fcd9d5d39cca718980a307f659f2e54": "n \\to \\infty", "9fce2cebe420a6063e13a8f44cdc09f4": "a = a_0[ 1 + \\eta [c-c_0 ] + \\cdots ] ", "9fce765df49443e8a0c334e2182cafb2": "a^2+5b^2=3", "9fceacfe3881bedba47a277900c7beda": " [X_i]=\\operatorname{Trans}_{X_i}(r_{i,i+1})\\operatorname{Rot}_{X_i}(\\alpha_{i,i+1}).", "9fced3a4df73340e9971193bda27a4a9": "h(xy)=h(x)h(y)", "9fcf526519e587ddc6bd53960e8434f1": "g_m+g_v=\\pi+g_y \\,", "9fcf7876e46db4e31296da05e6412350": " \\sin^2\\left( \\left( r+ \\frac{1}{2} \\right)\\theta\\right)", "9fcfb03e799d3152dd6582c89423cfc2": " P(x)=y ", "9fcfc031d6d398d0e0f96566b6e05d14": "\\mathbf{\\hat z}", "9fcfe7a650e6a4e51a60ace36a081760": " f(\\vec{x}) = \\max_{(\\vec{a},b) \\in \\Sigma} \\vec{a} \\cdot \\vec{x} + b.", "9fd0259232b926ac775097c58cc54ef5": "\\gamma_\\xi=-\\frac{1}{4}\\xi+\\frac{1}{8}\\xi\\left(2\\chi_\\xi^2+\\operatorname{ch}2\\chi-1\\right).", "9fd044bc046b610ef103204695a16ee4": "\\sin(kt)e_i(t)", "9fd08af69e435990052273b7d002411e": " I_{16} ~,~ \\Gamma_{a} ~,~ \\Gamma_{a_1 \\dots a_4} ~,~ \\Gamma_{a_1 \\dots a_5} ", "9fd0fab9cbe80469963b2a145827a0a5": " d\\psi=\\chi\\wedge\\omega, \\,\\, d\\chi=\\omega\\wedge\\psi", "9fd11aaff79e6afd3341c8909bf6f3fa": "\\|\\mathord{\\cdot}\\|", "9fd1783acfcfb7a2c0ffc6df9b1e83b8": " \\langle \\varphi, e\\rangle = \\int_E \\langle \\varphi, f(x) \\rangle \\, d\\mu(x)\\text{ for all functionals }\\varphi\\in V^*.", "9fd17943bbef00424d62f45359735751": "g_l = 1", "9fd17c823ae666a251c7ab29dd25be71": " \\left(\\frac{\\Delta S}{\\Delta t} \\right)_{i=1}=I-O=q^{ss}-q^{trans}_{i=1}= 10.0 \\text { ft}^3/\\text {s}-7.1 \\text{ ft}^3/\\text{s} = 2.9\\text{ ft}^3/\\text{s} ", "9fd17ca1786d81a804bccaee8ec5b1b5": "\\delta \\subseteq Q \\times (\\Sigma\\cup\\{\\epsilon\\}) \\times (\\Gamma\\cup\\{\\epsilon\\}) \\times Q", "9fd1acb652a00484c72a037652459ab3": " 2 \\, {\\rm Im}(\\lambda) \\int_c^x |\\varphi +\\mu \\theta|^2 =-{\\rm Im}(\\mu).", "9fd23a551795b3f392ca206357ea2044": "\n\\begin{align}\nS(A,P) & = \\sum_{d\\mid P}\\mu(d)\\left\\lfloor\\frac{X}{d}\\right\\rfloor \\\\\n& = [X] - \\sum_{p_1 < z} \\left\\lfloor\\frac{X}{p_1}\\right\\rfloor + \\sum_{p_1 < p_2 < z}\n\\left\\lfloor\\frac{X}{p_1p_2}\\right\\rfloor - \\sum_{p_1 < p_2 < p_3 < z}\n\\left\\lfloor\\frac{X}{p_1p_2p_3}\\right\\rfloor + \\cdots\n\\end{align}\n", "9fd29a918f019f76eb45148554c817a6": "B_\\nu=B_{eq}-\\alpha(\\nu+{1\\over 2})", "9fd2f89517ccdb494a5f4a85ab633e99": "\\mathbf{p} = (\\sum_{i=1}^n m_i) \\mathbf{V},\\quad \\mathbf{L} = \\sum_{i=1}^n m_i(\\mathbf{r}_i-\\mathbf{R})\\times \\mathbf{v}_i = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times(\\omega\\times(\\mathbf{r}_i - \\mathbf{R})).", "9fd3081d451e649d96519d0f1c7efed4": "\\forall i \\qquad a_i |0\\rangle = 0", "9fd36fc457cb5d4353f7b5604782a189": "\\Gamma_q(n)=[n-1]_q!", "9fd3c5e47aa05ecf2dcb67e0338debe7": "\\oint_C {1 \\over 4!\\;z} \\,dz={1 \\over 4!}\\oint_C{1 \\over z}\\,dz={1 \\over 4!}(2\\pi i) = {\\pi i \\over 12}.", "9fd44d206664114f8a9445fe2e27e22f": "L^{4k+1}.", "9fd56f76e921360ea83dc93230380949": "\\frac{\\mathrm d\\|x\\|}{\\mathrm d t^+} \\leq \\mu(A)\\cdot \\|x\\|,", "9fd5a5ac5584fabc8d259d9443ab2114": " \\phi_{xt}=\\phi_{tx} \\quad \\Rightarrow \\quad U_t=-JU_{xx}+2JU^2U\n\\quad \\Leftrightarrow \\quad \n\\begin{cases}\niq_t=q_{xx}+2qrq \\\\\nir_t=-r_{xx}-2qrr. \n\\end{cases}\n\\,", "9fd5cec3dcf4da95e7ae6ee8f2c81a09": "[t, t + dt)", "9fd5d1e3b1eb137daa2f3e9ddc957f58": "K_i=\\frac{Q_i-Q^{min}}{Q^{max}-Q^{min}}", "9fd6b0a8eff2492cde49cb0d915fcee0": "S = -(n_s - n_{\\overline{s}})", "9fd6bc8c333f7b8f10a10080f6cc2276": " \\displaystyle \\sum \\limits_{n=0}^\\infty \\textstyle \\left(\\frac{(2n-3)!!}{(2n)!!}\\right)^2 = {1 \\! + \\! \\left(\\frac {1}{2} \\right)^2 \\! + \\! \\left(\\frac {1}{2 \\cdot 4} \\right)^2 \\! + \\! \\left(\\frac {1 \\cdot 3}{2 \\cdot 4 \\cdot 6} \\right)^2 + \\cdots}", "9fd6cf6560c21ebec23f67cc5fda83a2": "D\\textbf{y}=\\textbf{c}", "9fd6ff8cc03a4facb970ac63c06daecb": "\nL = \\tfrac12\\rho v_0^2 A C_L = mg\n", "9fd71cab4af781481626c64defa970fd": "\\mathrm{MA}_{compound} = \\frac {F_{out 1}} {F_{in 1}} \\frac {F_{out 2}} {F_{in 2}} \\ldots\\frac {F_{out n}} {F_{in n}}. \\,", "9fd781c960e95e0c9b405f45efbb304c": "P= \\frac{2}{3}\\frac{E_{tot}}{V}=\\frac{2}{3}\\frac{\\hbar^2 k_{\\rm{F}}^5}{10 \\pi^2 m_{\\rm{e}}}=\\frac{(3 \\pi^2)^{2/3} \\hbar^2}{5 m_{\\rm{e}}}\\rho_N^{5/3} ,", "9fd7a5326b3a6975e158a123f941af53": " \\mathbf{F} = \\sum_{i=1}^n \\mathbf{F}_i,", "9fd7b4f472f0687a7980eccec268b8f6": "\\omega^{CK}_1", "9fd7ecb7c2f7f03fa2ff6b0da7523bff": " S_d(r) = \\sum^{d-1}_{i=0}(-1)^{i}2^{d-i}{d \\choose i} { d+r-i-1 \\choose d-i-1 }. ", "9fd805984c87a84611536073c4cb66e2": " \\rho c (T_P - {T_P}^0) \\Delta V = \\int_t^{t+\\Delta t} [( K_e A \\frac {T_E - T_P} {\\delta x_{PE}}) - ( K_w A \\frac {T_P - T_W} { \\delta x_{WP}})] \\,dt + \\int_t^{t+\\Delta t} \\bar S\\Delta V \\,dt ", "9fd823c800d135775730c3187bfa0003": "\\Vert \\chi_\\alpha - \\chi_\\beta \\Vert_{BV}=2+|\\alpha-\\beta|", "9fd8340c9b15f2131ba14c93ac90c4b0": " \\int\\limits_0^T f(t)\\;\\mathrm{d}t ", "9fd845ad6f2bb75e69c7a439c49719cc": " \\phi \\to \\eta^\\prime \\gamma ", "9fd89f4d902d682536672d92b4479bc6": "\\frac{1}{18} + \\frac{1}{27} + \\frac{1}{54} = \\frac{1}{9}", "9fd8c45ca04307f77750a106248b2852": "1,499 \\cdot 15,540 = 23,294,460\\,", "9fd925630df3d087143ef717b4a4d411": "R = \\frac {\\textrm''{Isentropic \\,\\, enthalpy \\,\\, change\\,\\, in \\,\\, rotor}''}{\\textrm''{Isentropic \\,\\, enthalpy \\,\\, change \\,\\,in \\,\\,stage}''} ", "9fd94d72db1b82908080108b2df2aa9d": "\\alpha^*F", "9fd9f47c2ce4ba538c78c5342d99a7f9": "\\mathbf{w}^{\\text{T}}\\mathbf{S}_W^{\\phi}\\mathbf{w}", "9fda2927cf66ec56d53c50e9f180eb95": "\\begin{matrix} {2 \\choose 2}{2 \\choose 1}{44 \\choose 2} \\end{matrix}", "9fda337a44e463a9d30f97a644900de4": "A(q) \\dot q=0", "9fda66b2541a7ae87de1087a64e584b5": "T^\\mathrm{c}", "9fda74bb825b48428b9c32f85ad4cb27": "{\\left[ \\operatorname{ad}_{e^i}\\right]_k}^j = {c^{ij}}_k. ", "9fda8fe941047c2421a524dea9a91f9d": " i j = j i = k, \\quad i^2 = -1, \\quad j^2 = +1 .", "9fdad7bcd81b6805145859a5d9bbb759": " dt ", "9fdae432c9032474e1a0f22af2eaff4f": "[Z]_j", "9fdb77c5bec98c90e968fa0256fbe3ee": " m_{u,d,e}^i = \\lambda_{u,d,e}^i v/\\sqrt{2}", "9fdbab512227f12d658e413f0551bc9f": "\\widehat{T}(\\Delta \\mathbf{r})", "9fdbcfca7f72054286111724b830039c": "y-y_0=-c\\ \\frac{y_P-y_0}{z_P-z_0}", "9fdbf74bbb15cd2245737f3f1bdeb0c8": "mP_p", "9fdc8f73592218648582908230a77663": "\\eta_1 \\ge \\eta_2 \\ge \\eta_3.\\,", "9fdca6a007df796b924aa43bd2937fdc": "\\alpha_m", "9fdcfef4d2c99df35ce2a1d399943f44": "\\sum_{n=k}^\\infty (-1)^{n-k} \\left[{n\\atop k}\\right] \\frac{z^n}{n!} = \\frac{\\left(\\log (1+z)\\right)^k}{k!}.", "9fdd20cf80edfe6d0d3e3230b48664be": "\\langle 0|\\Phi(x)\\Phi^\\dagger(y)|0\\rangle=\\sum_n e^{-ip_n\\cdot(x-y)}|\\langle 0|\\Phi(0)|n\\rangle|^2.", "9fdd2887084c6af3df6ce8de9b5e2e39": "\\min_{x \\in \\R^n} f(x)", "9fdd440c34bf90a24a70ae01e2456c11": " v_{\\phi} = {\\omega_0 \\over k_0 } ", "9fdd48b4b598e9116e3a3ca7bafd8c1a": "p(\\mathbf{X},\\mathbf{Z},\\mathbf{\\pi},\\mathbf{\\mu},\\mathbf{\\Lambda}) = p(\\mathbf{X}\\mid \\mathbf{Z},\\mathbf{\\mu},\\mathbf{\\Lambda}) p(\\mathbf{Z}\\mid \\mathbf{\\pi}) p(\\mathbf{\\pi}) p(\\mathbf{\\mu}\\mid \\mathbf{\\Lambda}) p(\\mathbf{\\Lambda})", "9fdd5eddf807cc6038fac1237d7f3d53": "L'=\\gamma L", "9fdd86925ad70706047cd0ed57d502be": "\\zeta \\mapsto R(\\zeta,\\eta) \\xi.", "9fdde678e9bcbc9c9171f2bc08c6cb84": "\\scriptstyle V(\\cdot,\\Omega):BV(\\Omega)\\rightarrow \\mathbb{R}^+", "9fddfb39fd11f0b733207cae4b2dd484": "\\{\\boldsymbol{\\gamma}_j\\mid j=1\\ldots p\\}", "9fde47a78dc7352089f3ee2212caab34": "Y \\sim \\text{Lap}(\\lambda)\\,\\!\\,\\!", "9fde600bd398175c64a15a3c5fd0e839": " \\{\\lambda_a, \\lambda_b\\} = \\frac{4}{3}\\delta_{ab} + 2\\sum_{c=1}^8{d_{abc} \\lambda_c} ", "9fde73bb256a49e4b670d71565f3de39": "R=\\frac{3}{2}(B-\\sigma)L", "9fde73c5d76bc249de1370dfb2c18b69": "\\mathbf{B} = \\mathbf{\\nabla} \\times \\mathbf{A}, \\, ", "9fde854c0f73289961b31b01a66cbf74": "h = [\\![(c, \\oplus),(g, p)]\\!]", "9fde99bc550017b1cd342c05916f1d3b": "\\Delta^{gi}", "9fde9c812267406ba265cbf242e662d8": "0^{(\\delta)} = \\{ \\langle n,i\\rangle \\mid i \\in 0^{(\\lambda_n)}\\}", "9fdebe550c3dc381ef7dd7d342b62cb7": "\\alpha(x_{j+1}-x_{j-1})=2\\alpha hu_x(x,t)+\\left(\\frac{\\alpha h^3}{3}\\right)u_{xxx}(x,t)+O(h^5).", "9fdec3c387c0a8907693b2e8304f9533": "\\tfrac{0.5 \\times \\text{tf}_{t,d}}{\\text{max(tf}_{t,d})}", "9fdec498999f11c0826058d99877e3e2": "\\pi(x;q,a) = \\frac{x}{\\varphi(q)\\log(x)} \\left({1 + O\\left(\\frac{1}{\\log x}\\right)}\\right)", "9fdedb0f092f59850548613f5368e24d": "\\Pi_n^0", "9fdf0563f28c394fa3c9868d8bcf5eec": " \\langle Q \\rangle_\\psi = \\langle \\psi | Q \\psi \\rangle =\\int_{-\\infty}^{\\infty} \\psi^\\ast(x) \\, x \\, \\psi(x) \\, \\mathrm{d}x\n= \\int_{-\\infty}^{\\infty} x \\, p(x) \\, \\mathrm{d}x ", "9fdf17a75718a902768a898789967143": "d:=\\frac{M-\\mu_\\text{baseline}}{SD}", "9fdf21bcc0a058a2d9a6d43b66972324": "y_{1,t} = c_{1} + a_{1,1}^1y_{1,t-1} + a_{1,2}^1y_{2,t-1} +\\cdots + a_{1,k}^1y_{k,t-1}+\\cdots+a_{1,1}^py_{1,t-p}+a_{1,2}^py_{2,t-p}+ \\cdots +a_{1,k}^py_{k,t-p} + e_{1,t}\\,", "9fdfd5b59bc9fcaf9bb4799221b1bd96": "L_4", "9fdfffe6573ec4d722c263a5b1a89c7d": "\\tilde{U} \\tilde{V} = 0", "9fe02d841b037f1596dbee054ffe9dbb": "t_{k+1}=\\pm\\frac{a}{GCD(a,b)}", "9fe05340016c062d9a49a442cb4f4bd3": " Q_H ", "9fe08fa95edc831729c4551b71f21961": "(\\mathbb{Z}/n\\mathbb{Z})^* \\simeq (\\mathbb{Z}/p\\mathbb{Z})^* \\times (\\mathbb{Z}/q\\mathbb{Z})^*", "9fe0fc052ee7229d0a3ea770e538e72d": "\\hat{T}_{2}", "9fe101a294c5e0b231b754f5ce8f4c44": "L=\\int_{-\\infty}^\\infty g(x)\\ln(g(x))\\,dx-\\lambda_0\\left(1-\\int_{-\\infty}^\\infty g(x)\\,dx\\right)-\\lambda\\left(\\sigma^2-\\int_{-\\infty}^\\infty g(x)(x-\\mu)^2\\,dx\\right)", "9fe168ced7ae92a443518f778e54c096": "x \\to 0", "9fe1850ac5d9b224fcf7002cdbc767d4": "I/P_n", "9fe1c261ca4bb4e9296fc406366eb3c0": "\\Delta E_Q = (1 - a_0^2)(E_{\\rm CISD} - E_{\\rm HF}), \\ ", "9fe1e257787072df63f5e853efc4768b": " U(R)^\\dagger \\widehat{T}_{pqr\\cdots} U(R) = R_{pi}R_{qj}R_{rk}\\cdots \\widehat{T}_{ijk\\cdots}", "9fe1e9487c4a538f98d385e390785b8d": "S\\,\\!", "9fe20369d3d08ac128fc5d122c034905": "S_n := \\sum_{k=0}^n T^k", "9fe2a31039b7bc4f33aa35f8bba6b47b": "T^2 = P_H \\; V^2 | _H", "9fe2ac3b3e9364d490bb901ca79cf0e4": "\\Delta G^\\circ (\\mathrm{total}) = \\Delta H_{\\mathrm{total}}^\\circ - T\\Delta S_{\\mathrm{total}}^\\circ", "9fe2beba68acf3094f8cd82adc4db9b4": " t = \\frac{\\sgn(\\beta)}{|\\beta|+\\sqrt{\\beta^2+1}} . ", "9fe2e0f23576f6f7cfe1f96207caea50": "Y_{i-o}, ..., Y_{i-1}", "9fe30a8fcbd74bfbd6edc45c81a5bc97": "\\hat{\\textbf{x}}_{k\\mid k} = \\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k\\tilde{\\textbf{y}}_k", "9fe31142d028df6e896a0b18d1303d81": "\\mathbf{a}(t)\\;", "9fe319fee0924887c8c7f2272ba8a52a": "x_1, x_2, x_3 ", "9fe336c7b689892746df3c47a28c25ab": "G/\\mathrm{Tor}(G) = \\oplus_{i \\in I} \\mathbb Q = \\mathbb Q^{(I)}.", "9fe36a7b2f43d824e1b46febdea668d3": "\\frac{P \\to Q, Q \\to P}{\\therefore P \\leftrightarrow Q}", "9fe38d899784c440a021b82347e1fd8f": "\\textstyle (1 + x + \\ldots + x^{p-k-1})", "9fe399e781599094969439015d60a0da": "f(x)=\\sum_{n=1}^\\infty {a_n \\over n!} x^n", "9fe39bea13ff6c0c18f8c88a8f9b7c5e": "p(t) = Im[\\omega L\\mathbf{q}e^{-j\\omega t}] \\ ", "9fe406f7e3611c131d41e9569887d720": "t^2/2", "9fe4f6a929e86b5c5a7d19d4a18fc304": "\\theta_i", "9fe5378d14b976f83b74a6cf1c03ce0b": "\\ell^\\infty(\\{1,2, \\ldots, n\\}), \\quad n \\geq 1 ", "9fe590d201f3127fe4e56fe368d00c23": "\\epsilon (a,b,c,d)=e^{\\frac{b{\\rm{i}} \\pi}{12}}\\quad(c=0,d=1);", "9fe62cd493fbae342802141d399dcac2": "_{6}\\!", "9fe63b85074afc71b1ed9e2fbad80396": "\n f( c ) = f( c_o )\n + \\left( c - c_o \\right) \\left( \\frac{\\partial f}{\\partial c} \\right)_{c\\,=\\,c_o}\n + \\frac12\\, \\left( c - c_o \\right)^2 \\left( \\frac{\\partial^2 f}{\\partial c^2} \\right)_{c\\,=\\,c_o}.\n", "9fe64a6a388947aa9f2c618e84d030ec": "f_x", "9fe654248b8ac2ceea401b97859ebb8c": "e=\\frac{S-x^2}{2x+e} := \\frac{S-x^2}{2x} ", "9fe73a688e2d9888fe3d210e3e547c87": "f : R^{L} \\rightarrow R^{|\\textbf{x}|} ", "9fe77084dab9382276ffe4d0dc1e29a2": "{\\aleph_0}", "9fe781c474de35812804cab23693c161": "\\inf_h \\|\\varphi-\\psi\\circ h\\|_\\infty\\ ", "9fe7856a0d69ce21a8d351b469d23cc0": "\\mathcal{H} = \\{\\langle w, \\cdot\\rangle : w \\in \\mathbb{R}^d\\}", "9fe7c96193451148d0d7b5e2f379efc5": " u(r,t)\\propto exp\\pm(iqr-i\\omega t) ", "9fe7dc2ef15e110e32a18a4c7423e40a": "f(t) \\rightarrow f_e ", "9fe7deacdb66893cf2c6fd529b902749": "\\rho: F \\rightarrow K ", "9fe7f8e35f33f8ccc0774387a7872376": " \\langle V^2 \\rangle = 4Rk_BT\\,\\Delta\\nu. ", "9fe7fc7de9661570186e9ccd6ba13fc1": "\n\\begin{align}\n\\omega_\\phi &= {1 \\over r}\\left(-\\frac{1}{\\sin\\theta}\\left({\\partial \\over \\partial r} \\left(\\frac{\\partial\\Psi}{\\partial r}\\right)\\right) - \n\\frac{1}{r^2 }{\\partial \\over \\partial \\theta}\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial\\Psi}{\\partial \\theta}\\right)\\right) \\\\\n&= {1 \\over r}\\left(-\\frac{1}{\\sin\\theta}\\left(\\frac{\\partial^2\\Psi}{\\partial r^2}\\right) - \n\\frac{\\sin\\theta}{r^2 \\sin\\theta}{\\partial \\over \\partial \\theta}\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial\\Psi}{\\partial \\theta}\\right)\\right) \\\\\n&= -\\frac{1}{r\\sin\\theta} \\left(\\frac{\\partial^2\\Psi}{\\partial r^2} + \\frac{\\sin\\theta}{r^2}{\\partial \\over \\partial \\theta}\\left(\\frac{1}{\\sin\\theta}\\frac{\\partial\\Psi}{\\partial \\theta}\\right)\\right).\n\\end{align}\n", "9fe870dacc2401550d3e2f5ca2a877ab": "M(n) > (1 - 2^{-1/2} )n ", "9fe873d489686dee3cded04cd2181230": "R_{\\theta CH}", "9fe88e4433473a9762529daf1db11733": "(23,12)", "9fe8b67b43909011224c822b29ce1bc1": "Z_C(t) = \\frac{P(t)}{(1-t)(1-qt)}", "9fe8c3b9bf441ce4d294818777608bdc": "\\scriptstyle\\epsilon_0", "9fe8e1958199122acd41d47e2b839d84": "\\gamma_n \\le (2/\\pi)\\Gamma(2+n/2)^{2/n} \\ .", "9fe91c46c03f196fdb4d98d90a6fa52f": "I = ", "9fe947c769d4f8f44c4e5699e9080e0b": "\\{a^{\\,}_i, a^\\dagger_j\\} \\equiv a^{\\,}_i a^\\dagger_j +a^\\dagger_j a^{\\,}_i = \\delta_{i j},", "9fe9a1191cd994bf1ee33239574740e9": "\\gamma[t] \\in T(\\sigma[t])", "9fe9c6e0e3af10e532051e706f6a8279": "\\!\\rho", "9fe9cd726a2eaeff0321a619623aa239": " h\\circ f = g\\circ h ", "9fe9e3be56a0efce9758f1484c445429": "(u,u+du)", "9fea8b2849c68d5dcf7775569501d567": "m_H(\\Sigma)", "9fead515dab3c392d613278383e09137": "\\mathrm{lift}: \\mathrm{M} \\, A \\rarr E \\rarr \\mathrm{M} \\, A = a \\mapsto e \\mapsto a", "9feb90df72816067208e5aaa20e71639": "x_0 = S+1", "9fec422f1d96e7dce62ca1b28e71bef1": "\\mathbf{v} = (\\rho, \\angle \\theta, \\angle \\phi)", "9fec4def2978ff8d6f71eea0941f7bab": "\\omega= de_1 \\cdot e_2", "9fec5814bf72c8fd6db128fdbbc3cb9f": "\\sigma_8~", "9fecdee33dc73dc2b5b4c7384f50ce64": "2\\pi+1", "9fed583f3c36c455fd72e54c9806facb": "g^{(n)}(x)", "9fedabb385784178a4fcdd0c480f28ab": "\\; \\Sigma_3 \\Sigma _1 \\Sigma _3^{d-1} \\Sigma_1 ^{d-1} = \\omega ~.", "9fee13f8df6b5ba46b6dc5f24687f4b1": "T^{k}V=\\{0\\}", "9fee69a82f5aba3355cc9b0c77d21a43": "Net\\ Cash\\ Flows\\ from\\ Financing\\ Activities = ", "9fee73b94102c53025e57e1f6d402890": " \\bar r_1 \\times \\bar r_2\\ ", "9fee788cc9755cd9aabca99ea7c7195f": "E_e = m_ec^2.\\!", "9feecc7dc4d5df2f69622624f525cc52": "\\displaystyle \\theta", "9fef1d5de007622bf2d3b7b18b5a91cc": "\\det(I-tg)", "9fef2a59842f2eed7f3481eb08bb7349": "\\phi_e", "9fef453384fb0c4cc0f2098159f4eda0": " (1)\\qquad \\int_Y K(x,y)q(y)\\,dy\\le\\alpha p(x) ", "9fef94989f0aefed4c953823bd945e89": "\\alpha \\beta \\gamma \\delta \\epsilon \\zeta \\eta \\theta \\!", "9feff2c9420a74e3bf43171399b6c4b0": "\\scriptstyle X_L \\;\\ll\\; (-X_C)\\,", "9feffe7fda95a19f93c1cca33eeb9d6b": "H_2S", "9ff00c3ace569e8f38d31705577317be": "\n=\n\\frac{1}{\\sqrt{2\\pi}} \\sum_m e^{im\\theta_k}\\int_0^\\infty r\\operatorname{d}\\!r\\, f_m(r)\\int_0^{2\\pi}\\operatorname{d}\\!\\varphi\\,\ne^{im\\varphi}e^{i kr\\cos\\varphi}\n", "9ff021be18cbba379d70c031e51aa6b1": "d(\\gamma_0(t),\\gamma_\\tau(t))=\\sin^{-1}\\bigg(\\sin t\\sin\\tau\\sqrt{1+\\cos^2 t\\tan^2(\\tau/2)}\\bigg).", "9ff05bf4e4e1c8465f088395bbcae1b9": "C_V\\ ", "9ff1479961b027d811b9ce2fc4760c79": "(X_n)_{n = 0}^N", "9ff184ce309364c8757585ff5345a5d8": "\\begin{align}\n\\alpha > 2: \\quad \\operatorname{E}\\left [- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial a^2} \\right ] &= {\\mathcal{I}}_{a, a}=\\frac{\\beta(\\alpha+\\beta-1)}{(\\alpha-2)(c-a)^2} \\\\\n\\beta > 2: \\quad \\operatorname{E}\\left[-\\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial c^2} \\right ] &= \\mathcal{I}_{c, c} = \\frac{\\alpha(\\alpha+\\beta-1)}{(\\beta-2)(c-a)^2} \\\\\n\\operatorname{E}\\left[- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial a \\partial c} \\right ] &= {\\mathcal{I}}_{a, c} = \\frac{(\\alpha+\\beta-1)}{(c-a)^2} \\\\\n\\alpha > 1: \\quad \\operatorname{E}\\left[- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha \\partial a} \\right ] &=\\mathcal{I}_{\\alpha, a} = \\frac{\\beta}{(\\alpha-1)(c-a)} \\\\\n\\operatorname{E}\\left[- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha \\partial c} \\right ] &= {\\mathcal{I}}_{\\alpha, c} = \\frac{1}{(c-a)} \\\\\n\\operatorname{E}\\left[- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\beta \\partial a} \\right ] &= {\\mathcal{I}}_{\\beta, a} = -\\frac{1}{(c-a)} \\\\\n\\beta > 1: \\quad \\operatorname{E}\\left[- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\beta \\partial c} \\right ] &= \\mathcal{I}_{\\beta, c} = -\\frac{\\alpha}{(\\beta-1)(c-a)}\n\\end{align}", "9ff187104c6699296334c5a1dc15dc25": " \\{ {\\mathbb C} \\}_{x \\in X}.", "9ff1beb28091140aa5e9e529611c1454": "x_0,\\ldots,x_n \\,\\!", "9ff1dc4538eae05be0e45f4a1b19ee2a": "\\mathbf{v^1} = \\frac{1}{\\sqrt{N}} [1, 1, ..., 1] ", "9ff221bbab80b21c4ee62f0ccd554d7d": "\\xi=\\frac{x}{2 \\sqrt{t}}", "9ff239f59c19076a4174775107f18168": "\\mathbf{r}(t) = f(t)\\mathbf{i} + g(t)\\mathbf{j} + h(t)\\mathbf{k}", "9ff26efa39cd65446a7b0581060a4f54": "\\varepsilon\\rightarrow 0", "9ff29507725f4909ffffb597f6a1df76": "\\prod_{i=1}^\\ell(q+d_i-1)= \\sum_{w\\in W}q^{\\dim(V^w)}.", "9ff2ed5a79dad5b9b8815343cfdf670f": "VAS(x^3-x^2-2x+1,\\frac{x+2}{x+1}) \\cup VAS(x^3+6x^2+5x+1,x+2)", "9ff310bd76aea313b5014c5ae5356e75": "\\mathbf{\\Pi}^0_2", "9ff3349f1fe8f20b94e1cefe82a2f1d5": " y'_1 y_1 e_{11} + y'_1 y_2 e_{12} + y'_1 e_{13} + y'_2 y_1 e_{21} + y'_2 y_2 e_{22} + y'_2 e_{23} + y_1 e_{31} + y_2 e_{32} + e_{33} = 0 \\, ", "9ff3881d9db146a229632ce7a8a1cf93": " v_r=\\frac {1} {2} v_L + \\sqrt{v_d^2 + v_c^2} ", "9ff3c06e9ef4ba3a2ad22b1207000f12": "\\gamma_0=\\ln(2) / \\sqrt{3}\\pi", "9ff3c0c6e340dee79873f06eda3c34b0": "c = 6 v,\\ ", "9ff3ceb217789f6196dca89310d4febc": "h= e^{-2} (\\kappa^{-1} - {\\kappa_0}^{-1}) \\sqrt{p^2+ z^2 \\kappa^2 }, ", "9ff3dcaae48e2371bc942cab9591f08d": " \\frac {\\Delta H^0_1}{\\Delta S^0_1} < \\frac {\\Delta H^0_1 + \\Delta H^0_2}{\\Delta S^0_1 + \\Delta S^0_2} ", "9ff3e9c6b3277694e41beb06304e6907": "\\rho_m(r)=m_e N_m e^{-r^2/r_m^2}", "9ff42f81f22e73ff8606efd75d29f22f": "\\frac{n_n(T)}{n_p(T)} = \\exp\\left(\\frac{-\\Delta m}{T}\\right)", "9ff43a88e14b912539aceff2e0b4e0ea": "\n \\boldsymbol{\\sigma}^T\\cdot\\mathbf{n}~d\\Gamma = d\\mathbf{f} = \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0\n", "9ff43af41a928da6bd4a9cc9dda0d981": "n=10000", "9ff442f8095c56fcc1184ac2d7af73b1": "g\\left( h \\right) =g_{0}\\cdot \\left( \\frac{r_{0}}{h + r_0} \\right)^{2}", "9ff446d1ff611ec1d27bcc896b9046e1": "X = \\partial_v", "9ff4749bf71a97025e4c6a38e4b5e307": "H_{\\aleph_0}", "9ff50d4959d952f6c34bea3eb5393ff0": "(n^3-n)/6", "9ff54190c0b215f285fb00a9a6504d1f": " \\scriptstyle OP \\times OP^{\\prime} =R^2", "9ff561a626c36a9d8bcca2b93439ecf0": "g^{(2)}(0)=0.0", "9ff56904d4c09957790615021159f552": "\\displaystyle v_{ij}", "9ff5713d0d48e50708eea1e797113ff2": "E = F \\times d \\;", "9ff5b8a568d6183ba4eb6bc31448a3fa": "\\begin{matrix} {3 \\choose 1}{11 \\choose 1}{4 \\choose 2}{40 \\choose 1} \\end{matrix}", "9ff61eb55eb1b6c9488708e77681762d": "\\Delta t = \\pi \\hbar / 2 \\Delta E", "9ff7c46b045ae89286d49d8913e9ef81": "R = -V/I \\,", "9ff7ee41364b085129dfcd56be15e7be": "s(x) \\odot s(y) \\leq s(x \\cdot y)", "9ff7fad8afc66898ce96622deea44d97": " \\Psi_P(a,b)=\\Phi_P(a+b, ab+ba). ", "9ff81183c06e52bdc69328d80e3e4bdf": "\\psi\\!:[0,1]\\times\\Theta \\rightarrow [0,\\infty)", "9ff816e05f87b1b65b19b1413e394f2b": "\nE\\left\\{y(n)H^*_{\\overline{p}}x (n) \\right\\} = E\\left\\{\\sum_{p=0}^\\infty{G_p x(n)H^*_{\\overline{p}}x (n)}\\right\\}\n", "9ff83d4e4d35438e606f8fc9be828bfd": "\\Delta a^\\mu = {\\Lambda^\\mu}_\\nu \\otimes a^\\nu + a^\\mu \\otimes 1 \\,", "9ff84d6672a8b006738d56424a624faa": "\\langle f_i, f_j \\rangle=\\int_a^b f_i(x) f_j(x) w(x)\\,dx=\\|f_i\\|^2\\delta_{i,j}=\\|f_j\\|^2\\delta_{i,j}", "9ff870ebbe9a47381391708e90868653": "\\exp \\left( -z + \\sum_{k\\ge 1} \\frac{z^k}{k} \\right)\n= \\frac{e^{-z}}{1-z}.", "9ff89c2b6df9ab67661af54a6fc33cb1": "S_i^2", "9ff8ee54b02bf5425a0546573dcda792": "\nDG(f;s) = \\sum_{n=1}^\\infty \\frac{f(n)}{n^s}\n", "9ff8f9c420a203c3a3f309e565c365e8": "g(x) = \\begin{cases}0 & \\text{if }0 \\leq x < 1-\\alpha\\\\ 1 & \\text{if }1-\\alpha \\leq x \\leq 1\\end{cases}.", "9ff944b2522d802fd9370acaa84afa45": "c_\\text{deep}=\\sqrt{\\frac{g\\lambda}{2\\pi}}.", "9ff98720168635d2e5ca9dade9a3e4ef": "\\ N = pq ", "9ff9c8bcb1d085fbf34f69d5377dc935": "M_p \\equiv 3 \\pmod 4", "9ff9d385df301ad2de59da674077c4c6": "\\sqrt{5} (6 \\rho^4 - 6 \\rho^2 +1)", "9ff9ea2039bef899d8e9b00ec882189d": "((\\neg p \\to \\neg q) \\to (q \\to p))", "9ffa178dfb053819dafa2cee9bbcffd3": "\\lim_{T\\rightarrow T_H^-} Tr[e^{-\\beta H}]=\\infty", "9ffa6e6bc39dae02b878c88bdb0ddbea": "|\\eta|<=1", "9ffb091d7bc3d392cbc46c16174d68f2": "\\left( {\\partial c_A \\over \\partial y} \\right) _{y=0}=0.332 {c_{A\\infty} - c_{AS} \\over x} Re^{1/2}", "9ffb1198cd207d8006dba76fd086a28f": " b_1 + b_2 x + b_3 y + b_4 x y \\, ", "9ffb269dd6535b8148fe62bca25a71d7": "\\delta^{\\mu}_{\\nu} \\, B_{\\mu} = B_{\\nu} \\,", "9ffb8841a76d75cd514d1798a4f6ee83": "\\mathbf{B} \\cdot (\\mathbf{v} \\, dt \\times d\\boldsymbol{\\ell}) = -dt \\, d\\boldsymbol{\\ell} \\cdot (\\mathbf{v}\\times\\mathbf{B})", "9ffb957485e0e8d4a51a7d6d779227a2": "\nx_k = e^{-\\frac{2k\\pi {\\rm{i}}}{N -1}} - \\frac{t}{N-1}\\sum^\\infty_{n=0}\\frac{(te^{\\frac{2k\\pi {\\rm{i}}}{N-1}})^n}{\\Gamma(n + 2)}\\cdot \\frac{\\Gamma\\left(\\frac{Nn}{N-1} + 1\\right)}{\\Gamma\\left(\\frac{n}{N-1} + 1\\right)} ", "9ffbfb129748e331eb2bb619b3c72a26": "\nq \\,\\xrightarrow{a(x)}\\,q'\n", "9ffc07e1d009c9810642e9480fd85db2": "\\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\begin{bmatrix}\n\\varepsilon_{11} & \\varepsilon_{12} & 0 \\\\\n\\varepsilon_{21} & \\varepsilon_{22} & 0 \\\\\n 0 & 0 & 0\\end{bmatrix}\\,\\!", "9ffc30a3c7ba8aacd53c4197d9ae44c3": "\\delta/\\delta t", "9ffc481ed51cc661ca137dfbd7ac0b99": "\\mathrm{Net\\ profit\\ Margin} = {\\mathrm{Net\\ Profit}\\over\\mathrm{Revenue}}", "9ffce1d30997321c8bea39b96a126f68": "\\boldsymbol{T}(X)", "9ffd95f07f56d08833ad766f4efe0d1e": "\\det(t+E+(n-i)\\delta_{ij}) = t^{[n]}+ \\mathrm{Tr}(E)t^{[n-1]}, ~~~~ ", "9ffda1bb8daf3687878c81e5861ce0da": "\\scriptstyle (X,\\tau)", "9ffdc91d1bf8aa8a20c71594970b1dcc": "k'=k[B]_0", "9ffddc885a5e29c3ce350d68c83f5256": "\\mu^+", "9ffe327532cd2ba4c25a96e63334cad0": "\\Omega_{JKI} := e_J^\\alpha e_K^\\beta \\partial_{[\\alpha} e_{\\beta ] I}", "9ffe3658d0783a3a54241ffa60ea162e": "\\prod_v |x|_v = 1.\\ ", "9ffecad87b093a83f21cc678190f25b8": "\\hat{P}_\\mu = i\\hbar \\partial_\\mu", "9ffed6a8c30a42772aae2631b1d02cbd": "\\varepsilon\\,", "9fff00131383060529370961ba3dd3b6": "\\chi_+^x = {1 \\over \\sqrt{2}} \\begin{bmatrix}\n 1\\\\\n 1\\\\\n \\end{bmatrix} \n", "9fff15a1957622a1023429a9a7d6e49a": "\\ge \\frac{1}{r}", "9fff73e74b4371ce7e79e3efa46acfc0": "\\mathbf{\\dot{x}}(t) = \\mathbf{f}(t, x(t), u(t))", "9fff9596bb5cae5fc079104e02d5eaed": "\\mathbf{R}_i(t)=PR(p_i; t)", "9fff99ff88ee7c7238923463891284c8": "\\scriptstyle (x,\\, y)", "9fffa1b6b94cead2de4e0d500fa47402": "d\\Omega^2\\equiv d\\theta^2+\\sin^2\\theta\\,d\\phi^2.", "9fffa443ce80807c6db86fc9c1501052": "j: d_j=1", "9fffb54ea8707b8af642e198d66c35de": "\\int d^{d-1}S E_r = \\mathrm{constant}", "a0000905c35e2e6e3fbe66d8eb92f5c1": "x_1 < \\cdots < x_n", "a0000b85b25272afcbe6c0281fdbca2a": "t_x > 0", "a0001a3072f22780a488c5ecbfc07846": " \\hat{X} ", "a000583a647b8a6f68f32d3fd6b3bf03": "\\frac {\\Delta \\epsilon_p} {2} = \\epsilon_f '(2N)^c", "a0009510aefd36746cb39b045ad934f9": "x-(-1)", "a000b2250c13054c3e0477bafb4a5ce6": "\\int{r^2}dm", "a00106e29e28f96132baae215de3ac8f": "\\sim p(a,a),\\ \\sim p(a,c),\\ \\dots", "a0010d37b90ed86f331696353c136bc3": " \\textbf{b} = \\textbf{a} \\pmod p = \\textbf{f} \\cdot \\textbf{m} \\pmod p ", "a0016ceee10634ca50c56298bef583bd": "\\; \\Phi _E (f)", "a00190f61b2f928e171e9f1221232b50": "D_H = \\frac{4 A}{P},", "a001baea9050843e1c8c1c2d729fe204": "(a_1, a_2, a_3, \\ldots, a_n) = (a_1, (a_2, a_3, \\ldots, a_n))", "a001bcfd4a0fcdcb0ea68ad50c768b2c": "A_z=2D\\frac{1+R_{eff}}{1-R_{eff}}", "a001d0dba2cbb792b69767b9c0faed66": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{3 \\choose 2}{3 \\choose 1}^2{36 \\choose 1} \\end{matrix}", "a001fefbdf357048a319dd7452a8e0ae": "\n \\mathbf{b}^i = \\boldsymbol{\\nabla}\\psi^i\n ", "a0020daa3fec60004254eda5aa74adff": "\\phi(n)", "a0026009eed775c51233de11c2398730": "(t - x)^{k-1}_+", "a0026eb481f32b9578066046d0f0c9bd": "S_{base}= \\sqrt{3}V_{base} I_{base}", "a00292c2f1e9d3eada2edfd65d17e947": "x=\\sqrt n\\, \\tan\\, t", "a003215520bbb2525fc493b4e41aec8d": "\\biggl\\|x - \\sum_{n=0}^N x_n\\biggr\\|\\to 0", "a00328339485fa89f037af2d45e8aa76": "\\Gamma[\\Phi] = \\Gamma_{k=0}\\big[\\Phi, \\bar{\\Phi}=\\Phi\\big] ", "a0032b2b18d854531897637942d9f67f": "\\mathbb{S}_k", "a0037397a83fd14093974a23c41c80a6": "=\\sum P(D|\\sigma,T,M) P(\\sigma|M)", "a00385a29871cad7812aeda865aa43d9": "0\\leq\\lambda\\leq 1", "a0038636c735c338e58d109f7ddec581": "k[x_1, \\dots, x_n].", "a003b4ae321d02734a377e81d06c39fc": "\\bigcup \\{ X, \\bigcup X, \\bigcup \\bigcup X, \\bigcup \\bigcup \\bigcup X, \\bigcup \\bigcup \\bigcup \\bigcup X, ... \\}.", "a0045eeae00d945f87ecae01385783d1": "\\scriptstyle \\lim\\limits_{k\\to\\infty}\\mathbf{P}^k", "a0046e9594e8151980c203c20af1edf1": "\\int_0^1 p^s(1-p)^{n-s}\\,dp={s!(n-s)! \\over (n+1)!}", "a0048e16f1b92f49e507e5c3185cba26": "\\displaystyle{g=\\begin{pmatrix} \\alpha & \\beta \\\\ \\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix},}", "a004d7a319537eb7c8c8d1f27d564b99": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}}\\det(\\lambda~\\boldsymbol{\\mathit{1}} + \\boldsymbol{A}) = \n \\det(\\lambda~\\boldsymbol{\\mathit{1}} + \\boldsymbol{A})~[(\\lambda~\\boldsymbol{\\mathit{1}}+\\boldsymbol{A})^{-1}]^T ~.\n", "a005d5ea61be246f86e91606f69f967f": "\\mathbf{x} = A\\mathbf{t}\\;\\;\\;\\;\\text{where}\\;\\;\\;\\;A = \\left[ \\begin{alignat}{2} 2 && 3 & \\\\ 5 && \\;\\;-4 & \\\\ -1 && 2 & \\end{alignat} \\,\\right]\\text{.}", "a005ebc21afe95c9f60661740f7867a5": "\\sum_{i \\in I} (p(x)-f_i)^2\\theta(\\|x-x_i\\|)", "a0067462892182160d81ba862d3c6ced": "\\lambda_n = 2L/n", "a006c09f613e19f0d0cb9291415292bd": "\nV_2 = \\frac{(R_1 + R_2) R_3 V_\\text{B} - R_2 R_3 V_\\text{A}}{(R_1 + R_2) R_3 + R_1 R_2}\n", "a006f2f1dde7d3081f60811d1643079f": " A\\to\\Box A", "a0070227dd75e537e1f0365d0bf74182": "Re_b", "a0073f2a6f51d9cdf8ef7ed9710771de": "E_{7}", "a007623733f9369cab91e96d6158226c": " \\text{angle in degrees} = \\text{angle in radians} \\cdot \\frac {180^\\circ} {\\pi}", "a00770d94d44742679107be667ba81d4": "R \\xrightarrow[-f]{} RA", "a0077a7da97c15e51f01e7d6a71c5652": "m_{A}", "a007b067ba0ff9e7e743f000bcab2204": " \\operatorname{Im}(A^+) = \\operatorname{Im}(A^*)\\,\\!", "a007f707fee44018634a9aa9c6caf7e9": "A = I", "a007fb8ddfc4eea95a11abc08a4a0137": "\\scriptstyle \\tau_s = 1.3\\times 10^{-6}", "a0081e9b1f64aab7e2d24b51c1ca3af5": "P_0(0,\\rho)=\\ln\\rho\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,1\\,", "a008a224a17830237ee281919c412e3d": "E_p", "a00963babd1f6b85ec3e9973bf1e9bf8": "10^{9}", "a00976e6eeae767a5b4ec12feae568d6": "f(t-) = \\lim_{s \\uparrow t} f(s)", "a0097982b73bb1e542ff15610d715616": "P = \\{s_1,s_2,s_3,\\dots s_k\\}", "a00990d6b629021d15a2a2157db0062a": "\\frac{(-1)^\\ell}{1-4\\ell^2}", "a009aeba405ef9159f74e15621e56dac": "T=\\frac {\\hbar c^3}{8\\pi Gk_BM} \\ ,", "a009c813e70bfa08145fd26488228e61": "NTU \\ = \\frac{U A}{C_{min}}", "a00a060510df325b21ea4bb71e7e618a": "[S]^k", "a00a0fc013e0bb44e54af87b5c85c040": "p \\cdot (\\Sigma _i x_i^*) \\leq r", "a00a390f8027c63a5a45f39011843ec7": "\\mathbb K", "a00a53c1919fdd98fe2aa8fa2b3591eb": "L_1 \\subset S^n", "a00a6a2d1d3b06e34bb5cd2c6c6f652c": "P = S^*S", "a00af8fd7cf438cc409e652bdaa910ca": "K''\\,", "a00afe4cdffb28cf2740fd494cf1d010": "\\textstyle\\ I(C, F_{f})=-\\underset{C}{\\sum }\\underset{F_{f}}{\\sum }BEL(F_{f},C)\\log \\frac{BEL(C,F_{f})}{BEL(F_{f})BEL(C)}", "a00b2da08cd38b037e468c3440ff7760": "63973 = 7 \\cdot 13 \\cdot 19 \\cdot 37\\,", "a00b3160d8b9dcdc13cb578d4ed8900e": "M_{13}", "a00b36d3db086c72f188e35516c1f764": "\\boldsymbol\\theta_{i=1 \\dots N}", "a00b585be0a07fca8321fe210eef71dd": "e(\\hat{\\theta}) \\le 1.\\ ", "a00b629a6429aaa56a0373d8de9efd68": "\\frac{\\sqrt{2}}{2}", "a00b674809331cbea6897492429562a2": "a\\equiv\\pm b\\pmod p", "a00b9c5693b6102fd3004abdef0dbfff": "q_i : p^{-1}(U_i) \\to N \\times F", "a00bc468c3b02ea42715ba6a3401d1ee": "f\\circ g\\colon Y\\to Y", "a00c1add37663d8e8f7efb56c2c00049": "\\alpha_{k=1 \\dots K}", "a00c67749f5f9f24f03241437b55372e": "\\frac{\\alpha}{c+iv}\\geqslant\\frac{\\alpha}{ic+iv}=\\left(\\frac{\\alpha}{c+v}\\right)\\left(\\frac{1}{i}\\right)\\,\\!", "a00c7c7879e20b6d31c170d331a1020b": "m_s\\,", "a00c92d5df20efc8e08b9f08aec5597f": "nw(f(X))=w(f(X))\\leq w(X)\\leq\\aleph_{0}\\,", "a00c9fcdae510ef53629f00b989d9760": "\\left\\langle\\pm \\mathrm i \\sqrt{2},Z_2\\right\\rangle", "a00d067a2c18a3d4af0dc212f6a9dbeb": "z_T=\\frac{\\lambda}{1 - \\sqrt{ 1 - \\frac{\\lambda^2}{a^2} }},", "a00d173417c5176e3ca96ffaaeadb8b4": " \\lambda_y(f\\cdot\\varphi\\circ\\pi)=\\varphi(y) \\lambda_y(f) \\qquad \\forall y\\in Y, \\varphi\\in C_b(Y), f\\in L^\\infty(X,\\Sigma,\\mu)", "a00d5a319263c17f2e4e8e9abc8a6efc": "{}_0F_1(;a;z)", "a00d74c74c3820477c5a2387f0da0c31": "W_{it} = \\alpha + \\beta{x_{it}} + \\epsilon_{it}", "a00dc18e4a3e223f2d96084c881f4ee4": "\\mathrm{d}U = \\delta\\,q + \\delta\\,w\\,,", "a00dd61be8a3232ea18c8dc5b9868a1b": "\\alpha_4 = {{2\\alpha_0 + 3\\alpha_1} \\over 5}", "a00ea314aaa84dede070746ddeba1f20": "\\sum_{n=0}^{\\infty} \\frac{x^n}{n!}", "a00ec890a574defe4f9c4293d87e1613": "\\mathit{v}_{\\epsilon}=\\sum_{i-1}^{B} {\\left( m_{i,\\epsilon} - \\mu_{\\epsilon} \\right ) ^2 p_{i,\\epsilon}}", "a00ee1a31aeb166fc7d355a0780eae4c": "\\psi(\\mathbf{x},t) = Ae^{i(\\mathbf{k}\\cdot\\mathbf{x}- \\omega t)}", "a00ef83cb02730dad2e1186b94ccc3ba": "\\|x(t)-x_e\\| < \\epsilon", "a00f741a16b19c64d17442c6912ba761": "M\\otimes _R M^*\\cong R", "a00f89c294e35efa6756491f7bea0811": "F:X \\times [0, 1] \\to X \\, ", "a00f97ba4e530e0dcfba7c8d336b7dbf": "\\mathfrak{g} = \\mathfrak{k}+\\mathfrak{p}", "a00fcc3afb1b0961a7520702bf824ebb": "J=-D\\frac{d\\rho}{dh},", "a00ff1f15532a944e70cebf7405af2cf": "\\mathbf{Z}/n\\mathbf{Z},", "a01047d40abf1685013679545166993c": "\\delta_{xi},", "a0104c74a8d23edc6966135dc06dd43a": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i (\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i) \\cdot(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i)) + \\sum_{i=1}^n m_i \\mathbf{V}_C\\cdot(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i)) + \\frac{1}{2}\\sum_{i=1}^n m_i \\mathbf{V}_C\\cdot\\mathbf{V}_C.", "a0115d3840731e8297f6adfdcdee7bbd": " E ( [ X - E( X ) ]^{ 2k } ) ", "a011a319ee8fb43e8da1c20dbf63f3e8": "p_i/q_i", "a011bae3357676819acf8d4a9a9401f8": "\\int_0^\\infty x^{2n} e^{-\\frac{x^2}{a^2}}\\,dx = \\sqrt{\\pi} \\frac{(2n-1)!!}{2^{n+1}} a^{2n+1} =\\sqrt{\\pi}\\frac{\\left(2n\\right)!}{n!}\\left(\\frac{a}{2}\\right)^{2n+1}", "a011c925d2e52138996e210947d557f2": " V(f) ", "a011dc481375a77563e46bed9d71c06a": "X \\rightarrow S.", "a0123d143d3ff819c89e741f54156ffc": "\\mathcal{L}_{X} := \\left( \\Phi_{X} \\right)_{*} ( \\mathbf{P} ) = \\mathbf P \\circ \\Phi_X^{-1}", "a0127a010c864270e9ed62cf451a2c99": "rQ^2", "a012dbfd2387899944bad1f1139ff108": "n \\ge m ", "a012f210dfb7ccb17c0d9a07d3475bbc": "m_{rel}=\\frac{m_{e}}{\\sqrt{1-(v_e/c)^2}}", "a0133fab888d56b13e69cf0c211a9644": "u = x^{\\ell}e^{-x/2}L_{n+\\ell}^{[2\\ell+1]}(x).", "a01380776648e6fa7fd41455224d8f67": "D\\subseteq X", "a013c0288ad275c78c685339304d5f56": "\n4X_nZ_n = (X_n+Z_n)^2 - (X_n-Z_n)^2\n", "a013ca0140a8c195c606d7af8f34ad24": " |A \\rangle = A_1|e_1 \\rangle + A_2|e_2 \\rangle + A_3|e_3 \\rangle =\n\\begin{pmatrix} A_1 \\\\ A_2 \\\\ A_3 \\end{pmatrix},", "a01409644f14a63fcce4f9a41a9e1f06": " f \\sim_x g\\quad\\text{or}\\quad S \\sim_x T.", "a014168c0a6c96dcfcafa9e969e23ed1": "\\nabla_a", "a014e7ff483a1515515c9a3bd2629820": "C_\\beta(s) =\\begin{bmatrix}\\cosh\\sqrt{|k|}s&\\sqrt{|k|}\\sinh\\sqrt{|k|}s\\\\ \\frac{1}{\\sqrt{|k|}}\\sinh\\sqrt{|k|}s&\\cosh\\sqrt{|k|}s\\end{bmatrix}.\n", "a0150fbf761c67ad8dfb59324083e895": "CE = \\%C + 0.33 \\left( %Si + %P \\right)", "a01557ed9f77da1950f096b62e69bb65": "E_{0}=mc^{2},\\quad E=\\gamma mc^{2},\\quad E_{k}=(\\gamma-1)mc^{2},\\quad p=\\gamma mv", "a0157e0a970d89c64cba46f674e65e59": " \\displaystyle{|a_3|\\le 3}", "a015871b29bf82bc40d10516b5f419ab": "\nx_{1} - x_{2} = \\frac{r_{1}^{2} - r_{2}^{2}}{D} \n", "a015d8c9c8bdea9a9653a860579424b1": "m_{ij} = \\frac{k_{1j}+k_{2j}}{k_{1H}+k_{2H}+k_{1T}+k_{2T}},", "a015de8015f1142abaad105145ec80d4": "\nC_b = \\frac {V}{L_{WL} \\cdot B \\cdot T}\n", "a015fce67f651f2eb4bd205bc4a56c22": "\\nu_{kj}=1", "a016b61c4d76152173508930b2180e96": "p \\in \\mathbb{R}^n", "a016c6e2a3a0e6ac47614c3f50107ba1": " 1- O(R \\log(1/R))", "a016cadcc4a0339d0f8b2c47310e44a2": "(b - 4)b^{b - 1} + 2b^{b - 2} + b^{b - 3} + b^3", "a016d339926b79a531a7ca4dbe48976a": "\n \\qquad \\qquad a^+ = \\text{max}(a,0)\\,, \\qquad a^- = \\text{min}(a,0)\n", "a01755eed7517a520a59588bb084a670": "N_D", "a017897cad6df1c916b06ba55a0a0e74": "\\mathbf{J} = \\mathbf{r}\\times \\mathbf{p}", "a017b5edca8e6f3e0c90495796ffbe7e": "h_k (x) = e^{ikx}, \\quad k \\in \\mathbb{Z},\\;", "a017bb542e44994c4aaaf1046c4fce58": "(Y,d)", "a017ed0faa3f553200f62a571854797b": "yRx", "a017f7aae1031e67b3300574c7bd7c4f": " \\lambda = \\frac{1}{Mean \\ Time \\ Between \\ Failure}", "a01805b121555bc693fd736ffb24cd0f": "X_{n+1} \\equiv \\left( a X_n + c \\right)~~\\pmod{m}", "a01816a921f3e311a9d271a1c267815f": "\\mathbf{F}\\left(\\mathbf{r}\\right)=-\\frac{1}{4\\pi}\\left[-\\boldsymbol{\\nabla}\\left(\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\cdot\\boldsymbol{\\nabla}'\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\\right)-\\boldsymbol{\\nabla}\\times\\left(\\int_{V}\\mathbf{F}\\left(\\mathbf{r}'\\right)\\times\\boldsymbol{\\nabla}'\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\\right)\\right].", "a0181f809957fa07bab41af4834b137e": "\n0 = \\frac{d^{2}\\theta}{dq^{2}} + \\frac{2}{r} \\frac{d\\theta}{dq} \\frac{dr}{dq} - \\sin \\theta \\cos \\theta \\left( \\frac{d\\phi}{dq} \\right)^{2}\n", "a0182e1d24ac31df58b18393315f1bcc": "R_\\mathrm{S} \\,\\!", "a0186e73ceeb8d8fbdd38b597b226edd": "\\left\\lfloor 2^\\omega\\right\\rfloor, \\left\\lfloor 2^{2^\\omega} \\right\\rfloor, \\left\\lfloor 2^{2^{2^\\omega}} \\right\\rfloor, \\dots", "a0189044399e5f26709b5cd61582d8a2": "x (\\theta) = (a + b) \\cos \\theta - b \\cos \\left( \\frac{a + b}{b} \\theta \\right)", "a0193bf54e582754fd069d4b2bcd3cc4": "\\rho \\in L(H)", "a019e748bed862c2534d6bbf6b8da996": "c_{10}", "a01aa8a96dee2e55664aa635eced0c9c": "n = d_k^1 + d_{k-1}^2 + \\dots + d_2^{k-1} + d_1^k\\, ,\\text{ e.g. } 135 = 1^1 + 3^2 + 5^3 \\, .", "a01b007b79ac4ba3cf2c4aa405a96c4b": " S = \\int_{t=t_0}^{t_1} L(x, \\dot x, t) dt \\,", "a01b17b1d878c99772a783f8d283a217": "x = {y_2/y_1}", "a01b2da7767c3576860a2138b12a7d18": " \\frac{d\\psi}{d\\phi} ", "a01b83d7a33164700942b4b74732fd96": "r \\,\\!", "a01bc6421f1038f4f75f7f25826a7a91": " |\\psi \\rangle = | \\phi \\rangle + \\frac{1}{E - H_0} V | \\psi \\rangle. \\,", "a01bf5689c02b877a466a8e350386935": "a+b=c\\,\\!", "a01c1f4093d90ea52f501cf9f11674e0": "\\psi _{j}", "a01c7053dea5a3419fea88e08bed91a1": "\\Psi(x,x^{1/a})", "a01cd0549d5ba75d3d4addb84ba2a3c0": "\\mathbf{Y} = \\mathbf{X}\\boldsymbol{\\beta} + \\boldsymbol{\\varepsilon}", "a01cf03107a4275a3af10e010e642c9e": "a \\to b\\;", "a01cf63bf80b9ddada321a55a54c6b6b": "\\langle \\Sigma\\rangle = \\rm{diag}(2, 2, 2, -3, -3) f", "a01d18ee68c29633b2c75043501ae76f": "\\Delta \\phi = 4 \\pi \\rho", "a01d2aaededa9669cabc91092e138b32": "g^{(0)}_{\\mu\\nu}=\\mbox{diag}(-c_0,c_1,c_1,c_1)\\,", "a01d7e7f5d0ad4ad4e1374722770b4f3": "V = \\frac{Mp}{Mb}C D", "a01da2da8e1e37519635b2a1b679239c": "\\vec{u} = (u,v,w),", "a01dc7ee0f1f5878284d2cf13c48fe3f": "\\theta_{\\mathrm{f}} - \\theta_{\\mathrm{i}} = \\tfrac{1}{2} (\\omega_{\\mathrm{f}} + \\omega_{\\mathrm{i}})t", "a01e13cf72364a95c1969d1d9ce975cf": "F(a,b,c,x)=\\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(b)}\\sum_{n = 0}^\\infty\\frac{\\Gamma(a+n)\\Gamma(b+n)}{\\Gamma(c+n)}\\frac{x^n}{n!}", "a01e1b8dea8332ded58735aecb1641f5": "R_n", "a01e7ac1676c47aceb4e685b8f4541c9": "X(u,v) = \\begin{pmatrix} \\cos u \\sin v \\\\ \\sin u \\sin v \\\\ \\cos v \\end{pmatrix},\\ (u,v) \\in [0,2\\pi) \\times [0,\\pi).", "a01e98aeb73367327ab90cb2c4417aa9": "k:= \\nabla_x \\varphi", "a01e9a420ee9bf874c71770a1acfbcc2": "|\\mathcal{S}| = N", "a01e9aeb05fb183bf9cd36aa0d560a70": "\nh = \\frac{k}{\\left [ n P(\\vec x) \\right ]^{1/D}}\n", "a01ee5f666dedac3a0bc43c8d1c5e795": "\\text{d} R / \\text{d} Q", "a01f3301b24a249199c13fd425f58fa6": " R=\\tfrac{1}{2}\\sqrt{p_1^2+p_2^2+q_1^2+q_2^2} ", "a01f485d9d1adfba0792337c4240dfda": "\\Phi_n(z) = \\prod_{k=1}^{\\varphi(n)}(z-z_k) ", "a01f59be1130fba43a4996223eb213b5": "\\, F = E - V + 2, ", "a01ff26fa495dadce5a1ed8af3c97a12": " G = \\mathrm{det} \\left( G_{ab} \\right) = \\frac14 h \\left( h^{cd} G_{cd} \\right)^2 ", "a01ff7bc3b385dbdacf6843fd8615014": "\n\\frac{L + 2d + \\sqrt{S} - F}{2.37} \\leq 12 \\mbox{ metres}\n", "a020037b88beb370cca13324cb664bbf": "b \\div a", "a02008282802b60722dd457eaeade57e": "\\operatorname{Ad}:g\\mapsto \\operatorname{Ad}_g", "a0200d13a8ffe2a24d8ad0255a659053": "p_4=a_{30}, ", "a0202486e7a654476363a99f5d7e608a": "\\frac{1}{2}D_k", "a0202b5e7246a3aab9dab4ee85f5ee41": "ab = \\frac{(a + b)^2 - (a - b)^2}{4}", "a02068c0b253c2f4ba8900d89ad7bbde": "\\left(\\frac{b-a}{2}, \\frac{a+b}{2}, \\sqrt{\\frac{a^2+b^2}{2}}\\,\\right) = \\left(\\frac{b-a}{2},A(a,b), Q(a,b)\\right),", "a02071b6b1dc544534fcd42edc0bd6c0": "F_{\\mathcal{D}}:\\mathcal{B}\\to \\mathcal{D}", "a02088cf37d145eefde1d6c51ed588bc": "\\overline{16}_H", "a020b36239c9e746640f9007f91f786d": "l_{21} = 1.5", "a020ef8c107f3095869d1276c91fd7af": " \\alpha :H\\rightarrow G", "a0211b4d60cd58275077866bdb205f8a": " \\gamma :[0,1]\\rightarrow Z", "a0219a5ce79fdd6d43c3d226fb01c042": "\\text{mode}=-\\frac{\\ln(z^\\star)}{b}\\, \\qquad 0 < z^\\star < 1", "a021faa63a339c09190cc4f7871bc412": "\\lim_{n\\to\\infty}{ \\left|\\{\\,s_{k+1},\\dots,s_{k+n} \\,\\} \\cap [c,d] \\right| \\over n}={d-c \\over b-a} \\,", "a0220bd06870bff17ea5dc1a7dbe003b": "\\begin{cases} \\text{if } x\n\\begin{cases}\n \\text{if } y , 3\n\\\\\n \\text{if } \\neg y , 2\n\\end{cases}\n\\\\ \n \\text{if } \\neg x\n\\begin{cases}\n \\text{if } y \\text{ , } 1\n\\\\\n \\text{if } \\neg y \\text{ , } 0\n\\end{cases}\n\\end{cases}", "a0222080bf4628c41f6c68cf8bc9fe57": " |\\phi\\rangle", "a022309dc5d8fbc44e9569279dc09b11": "\\textstyle \\sum b_n", "a022af545afbdede932d59892e3bc9b9": " \\mathcal{T} ", "a022c35d83328e7f82c058d215185852": "X/\\sim", "a022e7582e6514cd2b0fa53b3ebec9fd": "\\mathbf{r^{\\prime}}", "a022f566cdb5bfba330c2789c3cc2d62": "A_{r} := \\{ x \\in \\mathbf{S}^{n} \\mid \\rho_{n}(x, A) \\leq r \\}.", "a02314a152f1a72c17f199ce0f33bc42": "(\\mathbf{B}-\\hat{\\mathbf{B}})", "a0235a1e0ab478f45b6aa0c03e01edd1": "\\tilde{W}_t", "a023afb42ea2b012e897d279d61fce53": "g:W\\to\\mathbb C", "a0240114fbf2cdae46dbed8b7440e82b": "\n\\delta\\psi^\\alpha(x) = h^{\\mu}(x)\\partial_{\\mu}\\psi^\\alpha(x) + \\partial_\\mu h_\\nu(x) \\sigma_{\\mu\\nu}^{\\alpha \\beta} \\psi^{\\beta}(x)\n", "a0242c951a25c1a6a3eda705e7b7128c": " \\big(\\langle A, B \\rangle = \\langle B^\\mathsf{T}, A^\\mathsf{T} \\rangle \\big)", "a024354e447d33f99bd5a3e41495bdd7": " (r(\\cos \\varphi + i\\sin \\varphi ))^{n} = r^n\\,(\\cos n\\varphi + i \\sin n \\varphi).", "a0243fbb5aacaecb00a513406298d08f": "\\frac{\\pi - x}{2} = \\sin x + \\frac{1}{2}\\sin 2x + \\frac{1}{3}\\sin 3x+\\cdots", "a02457a4bcd0633688d5ba9b7eab4e29": " E[{N}(B)]=E\\left( \\sum_{x\\in {N}}1_B(x)\\right) \\qquad \\text{or} \\qquad \\int_{\\textbf{N}}\\sum_{x\\in {N}}1_B(x) P(d{N}). ", "a0245a8636ab2ae4f2273df120b1a5f4": " \\langle q| \\mathbf{ \\hat T^ \\dagger}(\\lambda) = \\langle q + \\lambda| ", "a0247c868a8844cf237b6ba8a1dbb105": "\\frac{{\\mathbb E}[(X-\\mu_X)(Y-\\mu_Y)]}{\\sigma_X \\sigma_Y}", "a024fcb7547015e98f9812ee16bc0f9c": "f_{458} = f_{678} = \\frac{\\sqrt{3}}{2}, \\,", "a02501d53c7a8d5881917a9a5ec28cd6": " m = A_5 = A_8 = q ", "a0252d991df636281e5f15152a88a117": "t \\!", "a0256797490ce0a9865820e30a744b7b": "~\\vec e~", "a025b49b5588a147381f35de15dcf55e": "x_2 = \\gamma \\frac{v}{f^\\prime}", "a02610def093e3a799fbb24f99e3b144": "(-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b|-1)}[c,[a,b]] = 0.\\,", "a02632d8d9e6525b0b2650893a667348": "P^{(a,b)}_k(\\cos\\beta)", "a0267b58e9f9a785024d0ae9ef954bf4": "\\nu=c/\\lambda", "a026e87d2180129961756d220e3c5e8e": "n_e = N_C \\exp\\left(-\\frac{E_{\\rm C} - E_{\\rm F}}{kT}\\right)", "a026fc2c34578703185f3bfba895d681": "t^2-x^2-y^2-z^2", "a0270deac40fcae27641eff83d534b00": "{\\mathcal M}z^3=z^1.", "a0275467fdd506275bef127328d14809": "=\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)\\left\\{1+\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)\\right\\}^{-1}", "a02784ff2f8bdde51e83bc37f21a79ef": "O \\left( \\frac{n}{\\log^2(n)} \\right)\\, .", "a027bf07194405e95b65a25aeb95a2b5": "e^x - 1\\approx x", "a02809f324bf3fdd72bc8ea2a137511c": "\\sigma_\\theta = \\frac{p(r + 0.6t)}{tE}", "a0285a60c56cf7d304f68bb1fe5792d7": "\\psi = (x^i,y^{\\sigma})\\, ,", "a02864fd6173a92560b4f918b8c77136": "\\phi(z) = \\lim_{n \\to \\infty} (\\log|z_n|/P^{n}),\\,", "a0286e73c14cfb118f964718c834f261": "\\{(\\tfrac{3}{2},2)\\}", "a02901c4402d3bf17cc3b0934f6f0ed1": "\\psi_{\\nu}\\left(\\bold{r}\\right)", "a0293d1161010c03b482609c6c2d6eb4": " E^\\vee ", "a02942424af6cc357e6a59b037ea8110": "[J_n,J_{n+1}]", "a029452d1aacddbbc45d17dea9358f98": " \\dot{I}_i = 0 ", "a029bf126ece2aef2d5a3fc1551243d9": " K = K_0 \\cdot SU(2). ", "a029f89c8c53368df71d2a8b30db0dbb": "\\Lambda(\\varphi,\\hat{\\mathbf{a}}, \\theta,\\hat{\\mathbf{n}}) = \\exp\\left(-\\frac{i}{2}\\omega_{\\alpha\\beta}M^{\\alpha\\beta}\\right) = \\exp \\left[-\\frac{i}{2}\\left(\\varphi \\hat{\\mathbf{a}} \\cdot \\mathbf{K} + \\theta \\hat{\\mathbf{n}} \\cdot \\mathbf{J}\\right)\\right]", "a02a2d53af6aeff5c97c7b58136443e5": "L.W.L", "a02a662a4c0e4db30e026e718730c557": "\\vec{x}_{CM}", "a02a7317b16c0d2e8a048c276dcba625": "I_p = C \\cdot dv/dt ", "a02afc14bf6e8ada4eb27ce998fe89ce": "E_{\\theta s} = H_{\\theta c} \\qquad E_{\\phi s} = H_{\\phi c} \\qquad H_{\\theta s} = \\,-\\frac{E_{\\theta c}}{\\eta_o^2} \\qquad H_{\\phi s} = \\, -\\frac{E_\\phi c}{\\eta_o^2}", "a02b2fed88bbf6e6dca25d1021e2a7dd": "l^{-1}", "a02b301fd8f160ef264a526cdfa9b92e": " [x :\n\\varphi(x)]", "a02b440621ad19e9410977637b28cb80": "y = Q(x)", "a02b4e1ad2e5679918d35b79da0af5f9": "T(e) = x L_e(x)", "a02b8f9a49f0b6f28fcb9cae08ab8a70": " \\mathbf{x} \\cdot (\\mathbf{y} \\times \\mathbf{z}) = \\mathbf{y} \\cdot (\\mathbf{z} \\times \\mathbf{x}) = \\mathbf{z} \\cdot (\\mathbf{x} \\times \\mathbf{y})", "a02b942d61ff6e2c664749c74d76256e": " 2q_0 = \\left(1 + \\frac3{e'^2}\\right) \\arctan e' - \\frac3{e'}", "a02beef5012ef51f802998a803f4c555": "\\sqrt{N - |\\mathbf{Z}| - 3}\\cdot |z(\\hat{\\rho}_{XY\\cdot\\mathbf{Z}})| > \\Phi^{-1}(1-\\alpha/2),", "a02c308cbbe813d5672b9665686f7273": "C(x_1, x_2, \\dots , x_n) ={ \\left({ x_1^2+x_2^2+\\cdots+x_n^2 \\over n}\\right) \\over \\left({x_1+x_2+\\cdots+x_n \\over n}\\right)}, ", "a02c42ae2dbeeaa282b8bf8322db1718": " \\subseteq w_{1,\\beta_1}(X_{\\alpha_2})\\subseteq X_1 ", "a02c59c95fddaaaf17ce1fb6886ba33b": "(x_1, y_1)", "a02c830d0e99daa3ccf6b9a4a226a60f": "p = \\tan \\theta", "a02c896ebaeba830f89547892680f24b": "\\boldsymbol{r}_i", "a02c9079d054375ac9c2477d8b81d068": "\n \\operatorname{Var}[\\,\\varepsilon|X\\,] = \\sigma^2 I_n,\n ", "a02ca5988d7eea25d81580e52b119628": "e^{i\\theta} = \\cos \\theta +i\\sin \\theta", "a02ca71189965174ffcfc4fe3bcc1db9": "\n \\delta V_{\\mathrm{ext}} = \\int_0^T \\left[\\int_{\\Omega^0} q(x,t)~\\delta w^0~\\mathrm{d}A\\right]\\mathrm{d}t\n", "a02cd22138942ef1d8f6a6b954b30b0f": "-1/3", "a02d17a45b6cb47008c2a3897e941db5": "\\begin{align}\n\\int_{C_R} f(z)\\, dz\n&=\\int_0^\\pi g(Re^{i\\theta})\\,e^{iaR(\\cos\\theta+i \\sin\\theta)}\\,i Re^{i\\theta}\\,d\\theta\\\\\n&=R\\int_0^\\pi g(Re^{i\\theta})\\,e^{aR(i\\cos\\theta-\\sin\\theta)}\\,ie^{i\\theta}\\,d\\theta\\,.\n\\end{align}", "a02da61a77e946a2cb8f40f999fbca27": "E = K_uV\\sin^2\\left(\\phi-\\theta\\right) - \\mu_0M_sVH\\cos\\phi, \\,", "a02dc4369b228b1e1209bdee0af76489": "gate4", "a02e879ac956db9d1e1589ba60a5cdea": "S_i=S_{t2}-S_{t1}.", "a02e9886fee867269bf9c4742d61b09e": "AE=4", "a02f48f64e8373be67af0aa4ce428499": " \\delta A = A_{xx}-A_{yy}. \\,", "a02f4cc81e7e599e1a31d8260ac37ec7": "\n \\tilde{r} \\leftarrow \\left(\\omega\\sqrt{\\cfrac{\\rho_0}{\\kappa}}\\right) r = k~r\n ", "a02f4f881a0bd796f0bed42ba133043d": "O(n + u\\log u)", "a02f634a73d5397c237debe7976654bc": "A=\\begin{pmatrix}\n -1& 3&-1& 0&-2& 0& 0&-2 \\\\\n -1&-1& 1& 1&-2&-1& 0&-1 \\\\\n -2&-6& 4& 3&-8&-4&-2& 1 \\\\\n -1& 8&-3&-1& 5& 2& 3&-3 \\\\\n 0& 0& 0& 0& 0& 0& 0& 1 \\\\\n 0& 0& 0& 0&-1& 0& 0& 0 \\\\\n 1& 0& 0& 0& 2& 0& 0& 0 \\\\\n 0& 0& 0& 0& 4& 0& 1& 0 \\end{pmatrix}.", "a02f8fbafaae02086ded3b7e20b0d82d": "\\Psi^*", "a02fac8d51fc74c84dfeacd5d5e6660b": "\\nabla^2 \\mathbf A' - \\mu_0 \\varepsilon_0 \\frac{\\partial^2 \\mathbf A'}{\\partial t^2} = \\Box^2 \\mathbf A' = - \\mu_0 \\mathbf J", "a0305ed9bc92eb237a6ee303cc4d75b5": "r_1 = a-c", "a0306ff20beb87a46fc97a24569530a8": " UW = U \\times W\n = \\begin{bmatrix}\nUW_{11} & UW_{12} & \\cdots & UW_{1n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nUW_{m1} & UW_{m2} & \\cdots & UW_{mn} \n\\end{bmatrix}\n", "a030816d928358d1dda23c9d8fbc3a4d": "\\underline{E}[f]=\\min_{P(X)\\in K(X)} \\sum_x f(x) P(x).", "a030a34a1e5f3332ed4c54a8cc5f0c55": "\\frac {V_f (t)} {V_0} = \\frac {Q} {\\sqrt{2}} \\frac {\\Delta R (t)} {R}", "a030c70a95929f69b24bfd588bb44391": "\\frac{dU\\left(x_0,y_0\\right)}{dx}= U_1(x_0,y_0).1+ U_2(x_0,y_0)\\frac{dy}{dx}", "a030f4563cc5ba1b9a21bf22947d8112": "x - r", "a03118aea28ee34eca5b5ac31e288ea0": "1-R^{s/[s+1]}", "a0315f85942b803b18ed8a1bfa1c2b9e": "f_i=p(\\alpha^i)", "a0317802370888e5a52049f8a025b43b": "f(x), \\tilde f(x)", "a031975fdd2d09c7f79437f0ee48d8a5": " \\phi ^2 - t\\phi + q = 0,", "a03198b04ebc91afc14e951327ab34ff": " T_{\\mathbf{v}} = \n\\begin{bmatrix}\n1 & 0 & 0 & v_x \\\\\n0 & 1 & 0 & v_y \\\\\n0 & 0 & 1 & v_z \\\\\n0 & 0 & 0 & 1 \n\\end{bmatrix}\n", "a0319e82cb80f98ee37cfd6524e234dd": "\\Delta(t) = t - 1 + t^{-1}, \\, ", "a0320eec58a6a8a6fca12be9f04c1e84": " x = y = z = 2", "a0321e74821bd70d46d2ea143952b127": "\\displaystyle \\delta\\left(\\xi - \\frac{a}{2\\pi}\\right)", "a032586d244507d217a3d838ba7d0c06": "d {s_i} = {{c_{i P} } \\over T}dT - \\left( {{{\\partial v_i } \\over {\\partial T}}} \\right)_P dP", "a0325a0473175cbdc264fbe5e0d6573b": " m_1, m_2 ", "a032942acc90a56152bf18ee794f4af8": "\\lim_{n\\to 0} {Z^n-1\\over n}=\\ln Z.", "a0329fde7e93ae23c956fc62bb6cce09": "\\sum_{k=0}^n {\\alpha \\choose k}{\\beta \\choose n-k} = {\\alpha+\\beta \\choose n}", "a0331ae479c912cb86461f4b5f718d2d": "\\forall v \\ \\mathrm{Rot}_G(v,i)=(v[i],\\pi (i))", "a03338c2799a6b246a157b94450f44c9": " {dC \\over dt} = \\frac{G - Q' C}{V}", "a0334bab4d64b11e05e70e0a76a10e43": "\\mathit{RGB^\\prime=\\begin{cases}\n1.099\\cdot RGB^{0.45}- 0.99 , & RGB \\ge 0.018\\\\\n-1.099\\cdot (-RGB)^{0.45} + 0.99 , & RGB \\le -0.018\\\\\n4.5\\cdot RGB & -0.018 < RGB < 0.018\\\\\n\\end{cases}}\n", "a033983ff0f6703ed0ac7d9a1b322dc0": "f_i=-{\\partial}(\\overline{\\rho v_i v_j} )/{\\partial x_j}", "a033a812cecf061ab2dd01fda866f360": " V_0 \\ ", "a033b7cb3a4b0b2447257632db59335b": "h_{00}", "a033e331fa483d41f1933d056529be25": "\nm \\to 1\n", "a0344f81b74c341d4339c47f241ca238": "\\begin{align}123_{10} = 123 / 7 = 17\\text{ with a remainder of }(4)\\\\\n17 / 7 = 2\\text{ with a remainder of }(3)\\\\\n2 / 7 = 0\\text{ with a remainder of }(2)\\\\\n&= 234_7\\end{align}", "a034cc224e0dfada1763a9daa46eebe9": "\\mathcal{L}_f", "a034d545c9b9f59a5a20b589fd8ad807": "(A \\times B)^c = (A^c \\times B^c) \\cup (A^c \\times B) \\cup (A \\times B^c).", "a0351b852d9752285de2ea535a75457d": "H_n: \\mathbb{N} \\times \\mathbb{N} \\rightarrow \\mathbb{N}\\,\\!", "a0358e8bfb15bd51581c1d00744c629f": "C_{1} = 4 \\sqrt{2}", "a0359c8e15004e97172fd9a406288c8a": "F(x,y) = x + y.\\ ", "a035d985513e997173aee95e73264064": "V^1\\, ", "a035e8d72c63c499a6476f389d55d005": "I(x,y,t) = I(x+\\Delta x, y + \\Delta y, t + \\Delta t)", "a035f732bfcbf996eacb33aa39fa0db8": "N_k = \\frac{1}{k}\\sum_{d|k}\\mu(d)\\cdot m^{k/d},", "a0361eb672782befda113e34d7052076": "r = r_0 \\, \\sec (\\phi + \\omega \\, t)", "a036417076ef92de363dd2d122fabb90": "{{v}_{i}}={{E}_{{{x}_{-i}}}}\\{\\phi ({{x}_{i}},{{x}_{-i}})\\}", "a03664571a7205f5cdf1bec9a3191471": "-\\boldsymbol\\Omega\\times\\boldsymbol v", "a0366dc92ea949e5789e756b66986663": "\\scriptstyle P'_i \\;=\\; E_K(P_i \\,\\oplus\\, 2^i L)", "a03676779732e6847c6ca0bba9673fd4": " B(z, u) = \\exp \\left(u \\left(\\exp z - 1\\right)\\right).", "a036ee82872f363029bc12da9e6bcd18": "\\epsilon \\rightarrow 0^+", "a037e891e5441e8f39a91f8e85badd9a": "x \\cup y", "a03882149f1a21b823d82a71c4574219": "L_{\\rho_j}(\\Gamma_j)\\ge 1", "a038ba59cd27a27dd466e0feb044b44d": "\\log \\zeta(s) = s \\int_0^\\infty \\frac{\\pi(x)}{x(x^s-1)}\\,dx,", "a038bd5866270616a2a750c101032cd9": " T_{01} = T_{10} = \\dot{X} X' = 0 ", "a0393056c7643ee5557004ef83c70dfd": "3 \\times 10^5", "a03962eacd3047d335a7c979e287411d": "\n\\mathbf{L} = \n\\mathbf{I}(0)\\;\n\\boldsymbol{\\omega}\\quad\\hbox{or}\\quad L_i = \\frac{\\partial T}{\\partial\\omega_i},\\;\\; i=x,\\,y,\\,z.\n", "a039e19446a1a81cbb47d52739a4dd19": "(c, d, h)", "a03a7db04a6aefec3a9d8ab3f1c6af3f": "\\mathbf{x}_j', t_j'", "a03ab8e0b34613cbfc35469a16900ed8": "p_{m,k-1}, \\ p_{m-1,k-1}", "a03acd94efbab8429fea9c1cc78c6c13": "M V = X = P Q \\,.", "a03af056de92d431b8faf303bc01cf31": " \\frac{\\pi}{4} = 4 \\, \\arctan \\frac{1}{5} - \\arctan \\frac{1}{239}", "a03b12eeeb8dab85769d04ab01ad813d": "\n\\sum_{a< \\mu\\le b} \\varphi(\\mu)e^{2\\pi i f(\\mu)} = \\sum_{f'(a)\\le\\mu\\le\nf'(b)}C(\\mu)Z(\\mu) + R ,\n", "a03ba9738da6f418c55e0f5bb2196406": "\\scriptstyle x_L", "a03bc1215ce1101d3a012e20c8c2776f": "\\omega_{n-1}", "a03ce1088d2a4077c942fd7d866cf737": " J_g(0,0) = \\begin{pmatrix} a_{1,0} & a_{0,1} \\\\ b_{1,0} & b_{0,1} \\end{pmatrix}. ", "a03d0dc8c715ec3fde3fb8a7c69b2235": "EBP = \\frac{F_s}{Q_{es}}", "a03d4641dd88cc41219c283cb641a26b": "P(z) = \\binom{z+k}{k}", "a03d6e9f1747b8d6c9d3c8258109267c": "(\\textbf{q}_i^0)", "a03e54a56a77dad212016f65b641577e": " \\rho(g), \\,\\, g \\in G ", "a03e679c11058c88de72fb53765d089d": "(c, h, s)", "a03e74a9bf2aa8d44cf086978c45f949": "\\lambda_i = \\lambda_{0i}+\\delta\\lambda_{0i} \\, ", "a03efc75c8620560c35f6890810628d2": "d(f,g)=\\sup_{x \\in [0, 1]} |f(x)-g(x)|.", "a03efed352e58f78d8c38903947872d0": "B_2(t) = \\sum_{i=0}^n \\beta_i^{(n-i)} b_{i,n}\\left(\\frac{t-t_0}{1-t_0}\\right) \\mbox{ , } \\qquad t \\in [t_0,1]", "a03f188f7adf8a8d14f2cad4e2a2c37c": " \\sum \\{ \\lambda_d(C) : C \\in G_n, \\, n > N \\} < \\varepsilon. ", "a03f39218405b5b0cf8c3972d14856a5": "S^1 \\times S^3", "a03f43c3a8c0b8aad5837e0429a3b38b": " L \\, \\text{d}x^2 + 2M \\, \\text{d}x \\, \\text{d}y + N \\, \\text{d}y^2. \\,", "a040018b1aa9cc1320eec5557c808f8e": "x=-1/(2\\lambda)", "a04021e34b6275ad248d32f9841f3d8d": " {\\mathbf A}_{22\\cdot 1} ", "a0403e5b6d809cdf6225a12a1e6eddd5": " \\frac{19(b-a)^5}{90000}f^{(4)}(\\xi) ", "a040bc028fe0bd982a6dc3cf0fb432a6": " d = \\frac{v^2 \\sin 2 \\theta}{g} ", "a040eeda0ef0b55d06e876f2887a9bea": "B(\\rho,\\hat{p})", "a041925bcfc4a4b1d3ca833860e9873c": "\n\\frac{d^2x^{\\lambda}}{d q^2} + \\Gamma^{\\lambda}_{\\mu\\nu} \\frac{dx^{\\mu}}{d q} \\frac{dx^{\\nu}}{dq} = 0\n", "a041fdbdd6e46c600bb414c2eef84c73": "r_{k}", "a042166e4e5cd25b62d3fa8e813baaf8": "{\\Bbb R}^3\\ltimes{\\Bbb R}= ({\\Bbb C}\\times{\\Bbb R}) \\ltimes{\\Bbb R} ", "a042192ff366ef959b975c7d89394c72": "(d,\\zeta)", "a0421c2fdf745cc77b5b6c23ed32c139": "(P\\setminus\\widehat{1})\\cup(Q\\setminus\\widehat{0})", "a0421e43eff66d0d51768b0a1c2fc7fe": " F(t,(x,y)) = 3t^2x - y - 2t^3.", "a0427f6cbbef20e956d505f63ffcf96e": "Fe^{2+} + Ce^{4+} \\rightleftharpoons Fe^{3+} + Ce^{3+}; K=\\frac{[Fe^{3+}][Ce^{3+}]}{[Fe^{2+}][Ce^{4+}]}", "a042ee7e67dd5423e7d852350f48ef96": "F(A) = \\bigcup_{T \\in \\mathcal{T}} F(T)", "a042f32774a9b252510d27ea8460477f": "A=D+L+L^T", "a0433d7f2472bfb610d43bc14fbaad05": "\n \\begin{align}\n F &= \\int_h^0 f^2\\; \\text{d}z = \\frac{1}{g}\\, c_p\\, c_g \\quad \\text{and}\n \\\\\n G &= \\int_h^0 \\left( \\frac{\\partial{f}}{\\partial{z}} \\right)^2\\; \\text{d}z = \\frac{1}{g} \\left( \\omega_0^2\\, -\\, k^2\\, c_p\\, c_g \\right).\n \\end{align}\n", "a0434cc19258444d9091d823b05a7fc7": "V_\\beta^e = \\frac{\\pi}{3}V\\dot{N}\\dot{G}^3t^4\\,\\!", "a0435775cceed7c47e9faf8854db1d6f": "y_{k+1}", "a0435c2ae64afb1c40c14ae779bf9d8b": "\\scriptstyle <10^{-13}", "a0439d8bd046323ca027471f9e33f960": "C\\left(F_1(x_1),\\dots,F_d(x_d) \\right)", "a043c9477253d63b8ce9171e07e278ec": "\\gamma_0(r)", "a044021e0e9a162b08e344e032ca7069": "\\psi(\\Omega^{\\omega+1}\\,\\psi(\\Omega) + \\psi(\\Omega^\\omega)^{\\psi(\\Omega^2)}42)^{\\psi(1729)\\,\\omega}", "a04466d8915b3509bd947b9b7d5b3ed9": "\\vec{x^*}", "a04472b931dfdd9904383913933e1d69": "\\,Cov(Z)=I", "a0448f3afd6f8a27b06009793d99987e": "\\mathfrak{N} = \\langle\\N, 0, S\\rangle\\,", "a044c252b58dd8bbd2cbbf9145973b58": "h : \\begin{array}{rcl}\nSmProj(k) & \\longrightarrow & Corr(k) \\\\\nX & \\longmapsto & [X] := (X, \\Delta)_X \\\\\nf & \\longmapsto & [f] := \\Gamma_f \\subset X \\times Y\n\\end{array}", "a044cb1085149829cd2a5bffe90d5494": "\\mathrm{Lan}_FX", "a044cc8bbbb5b561e2ba32c7749de0e6": "x_0(t)", "a0450acc063f22b887ae3f39fcf7f3ca": "\\scriptstyle{X^-=X\\setminus \\lbrace x_0\\rbrace}", "a04528227cb63269daeead1d7e69343a": "F^\\mathrm{op}", "a04545418194b28e8dbb4f7c580908a3": "\\frac{d}{dx}(y e^{\\int_{s_0}^{x} P(s) ds}) = Q(x) e^{\\int_{s_0}^{x} P(s) ds} ", "a04562ab9e30778a03fdde99f29a46ff": "\n\\phi_n(z) = \\sqrt{\\frac{2}{L}} \\times\n \\begin{cases}\n\\cos \\left(\\frac{n\\pi z}{L}\\right) & n \\, \\text{odd} \\\\\n\\sin \\left(\\frac{n\\pi z}{L}\\right) & n \\, \\text{even}\n \\end{cases}.\n", "a045d78c10f76d1947a08eae1593b6aa": " \\bar k = \\frac{\\sum_{k=1}^n t_i(B)}{nTL} ", "a046032f4fb236a1a4b4d103abc54d4f": "w(x)=[F(x)\\; (1-F(x))]^{-1}", "a046125404d0ae802669c5a74278f8ac": "Y_{9}^{-2}(\\theta,\\varphi)={3\\over 128}\\sqrt{1045\\over \\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(221\\cos^{7}\\theta-273\\cos^{5}\\theta+91\\cos^{3}\\theta-7\\cos\\theta)", "a046efe63609100da8c252a90d805ddc": "f'(x) = 0", "a047728ab54a89b30c5f8cdc6b10ff2f": "S \\subset F", "a0477ff089a976bc49bfd2d5166693c9": "\\, 1-p+pe^t", "a047a866d6b55afabd61dc4d58e6d94d": "\\sigma_y^2(\\tau) = \\frac{3 f_H}{4\\pi^2\\tau^2}h_2", "a047fe6c584e6f89daef5cca069e48d0": "\\Gamma = \\frac{B \\exp(-\\gamma l)}{A \\exp(\\gamma l)} =\\frac{B \\exp(-j \\beta l)}{A \\exp(j \\beta l)}\\,", "a0482166c72e56b80acd478761ee09d1": "h:V\\times W\\to Z", "a0482dc6797c90cc5b50748f4ab44468": "x = p^{a}y", "a0483b47b1ba2aeea2ab3ee4e4cf525b": "\\operatorname{Ins}(y)", "a048559662d4b59eb8308cf4f404c163": "t = 2", "a048cb64967471ad283151176ab6b7e5": " g \\ = \\ i - {E \\over P}", "a048f7b2a8d4ef128102ad2e129bb51a": "\\mu^2\\;", "a04921742a992c7502c9cb9bf864873a": "I_0/I~=~1~+~K_{SV}~[O_2]", "a0495f22ec112085da46d4a3c218df38": "\\mathbf{R}\\mathbf{R}^T=\\mathbf{R}^T\\mathbf{R}=\\mathbf{1}\\,\\!", "a04a840bdd8826b1351ac8d0c167d6a8": "G_a(z) = (1-z)\\sum_{k=0}^{\\infty} s_k z^k.\\!", "a04a8ced9331257fac295403f8a7d3d5": "\nf(t) = (h*s)(t) = \\int_{-\\infty}^{\\infty} h(\\tau) s(t - \\tau)\\, d\\tau. \\,\n", "a04a9a169b101a756c72286a462e4434": "1/\\sqrt{\\mu_0 \\varepsilon_0}", "a04ac45173e958c94f38c7e89961dc44": "\\textstyle 1 \\le l \\le (n+1)/2", "a04b254f3940e7aa21b5978a07ff14cc": "\\theta H^2(\\mathbb{D},\\mathbb{C}),", "a04b4a8f6d0adb1592f5fd506a8606c5": "\\beta_i \\in \\{ 1, 2, \\ldots, n \\} ,", "a04b553e10c56083cfaa0c6da420a1f6": "n(x)", "a04b5ddaa2fb9f0de0f865b8725e7eeb": "\\frac{1}{2^{\\frac{k}{2}}\\Gamma\\left(\\frac{k}{2}\\right)}\\; x^{\\frac{k}{2}-1} e^{-\\frac{x}{2}}\\,", "a04b61fa76a4f2e49b558c33f7e819a4": "\\kappa, \\rho, \\sigma, \\tau\\,; \\lambda, \\mu, \\nu, \\pi\\,; \\epsilon, \\gamma, \\beta, \\alpha. ", "a04b6ee0b0351f79c5ed11cb014958b1": "X_j = (X_{1j}, X_{2j}), j = 1, 2, \\dots n", "a04b815558c1c9ca0b736d4b90a3efd2": "\\bigcup_{n \\in \\omega} J_{\\alpha, n}", "a04b947e73df7c7e23a306090c0bd4cc": "[E]_0 = [E] + [ES] + [EI]", "a04ba6fa966c5db7a68e2df675db2cd6": "=O(VCDIM(H)\\log(1/\\alpha))/\\alpha^{2}\\,\\!", "a04bec7835580fc4944be98f97f95a96": " (I+J)(I \\cap J) = IJ. ", "a04bf04255dd8cb9f4ddb61570f7aa15": "V_- = V_\\mathrm {in} \\frac{R_1}{Z_\\mathrm {in}+2R_1}", "a04c4a9954d373dd74759156243a6c4c": "P(\\overline{\\zeta}) = a_0 + a_1\\overline{\\zeta} + a_2\\left(\\overline{\\zeta}\\right)^2 + \\cdots + a_n\\left(\\overline{\\zeta}\\right)^n = 0.", "a04c4f7e5359213605838de86248f74e": "\\mathrm{Proof}_T(x,\\#\\rho) \\to e \\leq x.", "a04c90308a9d22bd7a4d91fdc865f64c": "vol(K)>2^nvol(R^n/\\Gamma)", "a04d25e3cb9128264302d5d691a9537c": "\\Bbb L=[{\\Bbb A}^1]", "a04d42cecbf77c7bc1a19ef4463de088": "\\left((10^8)^{(10^8)}\\right)^{(10^8)}=10^{8\\cdot 10^{16}}.", "a04d4e82ed4db1d22b4fd15f61167686": "\\scriptstyle\\frac{L_{V_{\\ast}}}{L_{V_{\\odot}}} = 10^{0.4(M_{V_{\\odot}} - M_{V_{\\ast}})}", "a04d684f1791c107fef04b5557f0f0ec": "sin (t)", "a04d8647d67bbdd4ed57cf2965a0d93c": "\\bold{X}(\\bold{u}_0)", "a04dad7319d6051241d6307a63debeec": "M \\mapsto A^{-1} M A.", "a04e346081c31f381bfaa45185ba2067": "\\Gamma_{\\alpha \\beta}^\\gamma", "a04e5d948893a1caa12dc46cbe72ef5e": " \\left \\{ 10 - \\left[ ( 3 \\cdot \\sum_{i=1}^{5} d_{2i-1} + \\sum_{i=1}^{4} d_{2i}) \\pmod {10} \\right] \\right \\} \\pmod {10} ", "a04e940974748cb233794b971b9b2063": "h\\;=\\;\\dfrac{\\delta y}{a\\delta\\phi\\,}=\\,\\sec\\phi.", "a04ea4c87322c62fb5478c06aef1a902": " B_q(n, k) = q^{-k(n-k)} \\binom{n}{k}_{\\!\\!q^2}", "a04ea8448e380f7767fb9a4f0b527bdc": "g(x, t) = \\frac{1}{\\sqrt{2 \\pi t}} \\exp({-x^2/2 t})", "a04ed2c0ea838160e36106356fe9b669": "\\rho_e", "a04f01b7bde88f14b9e325aa9d2e407b": "\\det( h_{ij}(0) ) \\neq 0", "a04f0ffd27b3c95ad9055dae20a352d9": "(n + g)", "a05074b13bc4426da6537c5d49acd64a": "(\\mathcal{C},\\partial_{\\bullet}'')", "a05135dc77431220953719d3cf204521": "q_{n+1} = p_1 q_n + p_n q_1\\,\\!", "a0517fb5ea3656ebec07a4dec9d9b46d": "\\beta _p(T,p)\\ = \\frac{\\left.\\cfrac{\\partial V}{\\partial T}\\right|_{(T,p)}}{V(T,p)} ", "a051a8266463cc8d3a39282f4c942daa": "\\min(3-0,2-0,1-0) = ", "a0521335019c432fb2f0c6c235934c68": " \\mathbb E \\log K_t = \\log K_0 + \\sum_{i=1}^t H_i ", "a05217a53b6a5b587fb8430c07fda712": "\\hat{\\alpha} = \\delta/\\sqrt{1-\\delta^2}", "a05266694759f9f652f3808bc89dad40": " g_i(x) < 0 ", "a052d1f43b3e49cccc8b8933f23756ae": "d \\approx \\sqrt{ 2\\cdot 6378 \\cdot h} \\approx 112.9 \\cdot \\sqrt{ \\cdot h} ", "a052fd552e0d802b56dc0ebfb475faa5": "\\log_c (g) = \\log_c (b) \\cdot \\log_b (g).", "a053229611edc3535e75d229b3cf45d8": "-5.74012\\times 10^{-3}.", "a053817b47d02f65420079340fd7f1bd": "\\widehat\\sigma", "a053a9b5baffc8dd0f3bff6d10d100ef": "\\frac{n!}{m_1!\\,m_2!\\,m_3!\\,\\cdots 1!^{m_1}\\,2!^{m_2}\\,3!^{m_3}\\,\\cdots}.", "a053b756b60ed7435462479b0aa14be4": "\\operatorname{add}(r) := \\max(|r|-\\max(|p|, |n|), 0) \\, ", "a053d56efa270505ad0e5a1fb9d51de2": "\n\\eta_{Carnot}=1-\\frac{T_{cold}}{T_{hot}}.\n", "a05415eec6ccf931b5d3abe3b5bdbafb": " \\mathcal{E}^0, \\mathcal{E}^1 - \\mathcal{E}^0, \\mathcal{E}^2 - \\mathcal{E}^1, \\dots, \\mathcal{E}^n - \\mathcal{E}^{n-1}, \\dots ", "a0546283682fa11b6f30a7ecf10030a4": " \\underline u = u(\\overline E,\\underline P) = \\frac{\\underline PL}{\\overline EA} ", "a0550c0b4ed321bd3ff601c6456ebf15": "\\scriptstyle -a", "a0553404307cf297fb1ef9fea236d8c0": "F_{n}=\\left(2^{2^{n-1}}\\right)^{2}+1^{2}.\\!", "a05537262da4f4cc564535a246f8ca18": "{x}\\mapsto{x}\\cdot N({x})", "a0555ab8a69e9b0834673cfe6ec7f41e": "O_{p1}", "a055abf03cec34631bdce2c9008824d5": "\\mathbf{X} = (x_1, \\dots, x_n)", "a055fe620e3606138a82aa01820b107d": "\\,\\overline{A}_x = 1-\\delta \\overline{a}_x.", "a0560828f515540f483168dd59a88ee3": "\\displaystyle{TF\\circ f= F\\circ f.}", "a056246066af623f71c4ebb9ca73f48d": "g^*\\circ_T f^* = (\\mu_Z \\circ Tg \\circ f)^*.", "a056420adb9f903416962ca39acb13a2": " \\sigma(E) = \\operatorname{Tr}(E S). ", "a0569c6f3181363fdb5f71ff1b321cf9": "\n{L}=\\frac{\\partial}{\\partial \\mu} \\left[ \\frac{(1-\\mu^2)}{(\\eta^2 - \\mu^2)} \\, \n \\frac{\\partial}{\\partial \\mu} \\right] - \\frac{1}{\\eta^2 - \\mu^2} \\,\n \\left[ -\\frac{s}{\\eta} \\, \\frac{(\\eta^2 + \\mu^2)}{(\\eta^2 - \\mu^2)} + \n \\frac{s^2}{1-\\mu^2} \\right]\n", "a056d2ab8167337e462d09e33cd90beb": "\\mathbf{cd}", "a056e472824f3bcc351e29818902d58f": "h = \\sum_{j=d+1}^n [\\min_{r_j}(E(r_j) + \\sum_{i=1}^d E(r_i,r_j) + \\sum_{k=j+1}^n \\min_{r_k} E(r_j,r_k))]", "a057058c4d6fc8fcdb09a687d7f02fac": "\\frac{1}{2} \\begin{pmatrix}\n1 & \\pm 1 \\\\ \\pm 1 & 1\n\\end{pmatrix}", "a0570a994d96a4dfe0f2e11c47e59818": "\\operatorname{Map}", "a057228401ec1378c348e8c726946fed": "{1 \\over R} = \\limsup_{n\\rightarrow\\infty} |a_n|^{1/n}.", "a05794fa44ad5f7eb6a0425dc6f11635": "\\eta_0=\\frac{g_0V_0}{h_{PR}}\\cdot I_{sp}=\\frac{\\mbox{Thrust Power}}{\\mbox{Chemical energy rate}}", "a057cf706148e5885c1a04bf7dbe5ff4": "\\sup\\{d(H)|H\\leq G\\}", "a057fa52b3a4c349c217961293be0e51": "\\gamma^{\\mu \\nu} = \\delta^\\mu_A \\delta^\\nu_B \\delta^{AB}", "a05806e6258ac4fcbf2586ee4a9c2ae7": "P(2\\omega)=2\\epsilon_0d_{\\text{eff}}(2\\omega;\\omega,\\omega)E^2(\\omega),\\,", "a0589da9ad3b60563c893598be68987c": "GF(q),", "a058b1f2e3b0e04269ea2d89b6b169b2": "\\langle s_n : n \\in \\mathbb{N}\\rangle", "a058d24d92caa8ec0b9688e4ca3c0270": "\\boldsymbol\\varepsilon=\\frac{1}{2}\\left(\\boldsymbol{F}^T+\\boldsymbol{F}\\right)-\\boldsymbol{I}\\,\\!", "a058e95976992d119801c7cc083bd1c0": "\\Delta_{F}", "a058ec99e4b4c5551866a13d9f357f60": "\\bar{g}_{\\kappa\\lambda} = \\eta_{\\kappa\\lambda} =", "a0591d3fc02a6889f10af3d959907c66": "(a_1,a_2), (b_1,b_2)", "a0599d34e0d33e26cf46a391d2f48f03": "\n\\begin{align}\n{}_a\\mathbb{D}^q_tf(t) & = \\frac{d^qf(t)}{d(t-a)^q} \\\\\n& =\\lim_{N \\to \\infty}\\left[\\frac{t-a}{N}\\right]^{-q}\\sum_{j=0}^{N-1}(-1)^j{q \\choose j}f\\left(t-j\\left[\\frac{t-a}{N}\\right]\\right)\n\\end{align}\n", "a059a3cb9904b7de6bd182731623aa26": "\\nabla X = \\omega\\otimes X ", "a05ae85d1577c7717a200bc43119c268": "\\color{red}\\rightarrow", "a05af582a64eacfcf91b8828034db10d": "\\Omega(f) = \\frac{1}{\\tau_f} (f_i^{eq}-f_i)", "a05b09e9639adcf520ae954d29133a37": " E_\\text{k} = E_i + \\frac{M V^2}{2}. ", "a05b198e0f56de8e49c7bc3be259bab1": "\\mathbf{x} \\in \\mathbb{R}^{m}", "a05b92482bca885d696eec8b75115956": "Y_2^* ", "a05b96f2b3ea695e8219919a57a69a10": "c_i=v_{PR,i}-v_{exp,i}", "a05bd6b72d54fab2cffc4ac4ba079e59": "{\\widehat{\\varepsilon}_i\\over \\widehat{\\sigma} \\sqrt{1-h_{ii}\\ }}", "a05bed892e79151781127fbc31a1ed6a": "G(t;\\mu_1,\\mu_2)", "a05c15f01aa711644ce3a7bf96c67b5a": "K(x,y) =\n\\left\\{\n \\begin{matrix}\n 1, & \\mbox{if} \\; x \\geq y\\\\ \n 0, & \\mbox{otherwise}. \n \\end{matrix}\n\\right.\n", "a05c28be08011f9038574b3dd7f46369": "Q=\\nu / \\Delta \\nu", "a05c7dfd2dbeaba7f07ec321b9c62b02": "ax+by+cz", "a05ce18dd44cc7a4efd94a7a68cc14f0": "I_n(x)", "a05d62497f86a4aa4076ed6e49bdb3e3": "\\nabla \\times \\mathbf{A}", "a05d6278e71893ae256bbfd736db07ad": "[H^+] = \\frac{C_i v_i - c_{OH} v}{v_i + v}", "a05d6a853122068fea97cebb11150e73": "I (k) = \\delta (|k|)", "a05d6ba623dbe07eaf58fcaab8bd77e1": "|\\mathbf{r} \\rangle", "a05d87e053123ffe47a94e949b1c1c62": "\\dot x(t)=A x(t) + B u(t)", "a05db7688a381d98041694e4a2b97fea": "(\\mathbf{e_1e_2})^2 =\\mathbf{e_1e_2e_1e_2}= -\\mathbf{e_1e_2e_2e_1} = -1 \\ , ", "a05de6bc3a971a55c5984ca3649519af": "\\tan (\\alpha + \\beta) = \\frac{\\sin (\\alpha + \\beta)}{\\cos (\\alpha + \\beta)}\\,", "a05ec6d0e6c5039d5876f78a28cb7dff": "(-b_1,a_1)", "a05fc46f95b004d221d068c762052b00": "\\,I^-(x) = \\{ y \\in M | y \\ll x\\}", "a060f17fcc401977952e4acc65f76262": "Al_{2}O_{3}", "a060f7ea03d115ac8c00953e492b5679": "m_1=3\\ ,", "a06146ca68978f5eae4b97d3f3b07150": "\\begin{align}\n x &= r(t - \\sin t) \\\\\n y &= r(1 - \\cos t)\n\\end{align}", "a0614f05dec9218016737546c240aa62": "K^2 = \\frac{1}{4} (pq + rs)^2 \\sin^2 A", "a061c96e34bee7d21f1f3fe180169179": "\\frac{\\partial}{\\partial y^i} = \\sum_{j=1}^n\\frac{\\partial x^j}{\\partial y^i}\\frac{\\partial}{\\partial x^j}.", "a061dcde343522f24f46d2d1f02fb943": "\\mathcal{N}(\\sum_{i}p_{i}\\rho_{i}) \\le \\sum_{i}p_{i}\\mathcal{N}(\\rho_{i})", "a061e72d336f767c1e96690910b0f2cd": "a_{s-1}=-1", "a0622970af32ec5b430ba3a915f174be": "\\operatorname{dist}(x,Z)^\\alpha \\le C|f(x)|. \\, ", "a0624c91c13f46765ab8a46e1b57814d": " \\theta_m ", "a0625f93c463aa0369f791a24590caf4": "f,f_1,f_2", "a062ba762f1714be9cced4e2aff0b2be": "F=\\tau/\\Phi", "a062bcd3b2e15824bb948b89a767138d": "n^{\\overline{k}}", "a0634764285a2f868c41c63299fb0efa": "x \\wedge S \\wedge x", "a06368854033bdad94e5cffd92896ef3": "\n\\begin{align}\nV(z)=\\left\\{\n\\begin{array}{cc}\n V(z+la),& \\textrm{for}\\quad z<0 \\\\\n V_0,&\\textrm{for} \\quad z>0\n\\end{array}\\right.,\n\\end{align}\n", "a063744a1700a7615498f7c2a2a293fd": "S^2\\cong \\mathbb{CP}^1", "a063827a5d75a944d0dd7ac50c8b49bc": "x\\in \\operatorname{Int} E_i", "a063cb2e858227c45ffafc0cb0202730": "E_{dp}", "a063eda67fd0bc4eead19f3967c6a3ef": "\\mathbf{i},\\mathbf{j},\\mathbf{k}", "a063f5ec8b04d54cbbead79823fcd5c5": "\\int_{S^2} {}_sY_{\\ell m}\\ {}_s\\bar{Y}_{\\ell'm'}\\ dS = \\delta_{\\ell\\ell'} \\delta_{mm'}, ", "a064d5ebef0f9206870e5c891b61deb6": "n_1 \\sin \\left( \\theta_\\mathrm B \\right) =n_2 \\sin \\left( 90^\\circ - \\theta_\\mathrm B \\right)=n_2 \\cos \\left( \\theta_\\mathrm B \\right).", "a064f2b3729022a00ad26e7a3a7cc43d": " \\mathbf{y}'_{1}, \\mathbf{y}'_{2} ", "a06568471d64146444357c59958ffc02": "k_\\mathrm{e} = \\frac{1}{4 \\pi \\epsilon_0}", "a0659b20d1ce957f80cb83723ae17b2c": " \\{ I_1, \\dots I_n, \\varphi_1, \\dots, \\varphi_n \\} ", "a065d1d89fc8544e5a42f0fe399dc139": "\\begin{align}\n\\operatorname{Var}(x) &= \\frac{\\partial^2 A\\left(\\eta_1,\\eta_2 \\right)}{\\partial \\eta_2^2} = \\frac{\\partial}{\\partial \\eta_2} \\frac{\\eta_1+1}{-\\eta_2} \\\\\n&= \\frac{\\eta_1+1}{\\eta_2^2} \\\\\n&= \\frac{\\alpha}{\\beta^2}.\n\\end{align}", "a065e15e25f2dbe55b6246efa6197629": "f \\in D(T^*)", "a065ef045d7d882e3eb139b103a4cb37": " u(x,t)= A e^{i(kx-\\omega t)} + B e^{-i(kx+\\omega t)},\\,", "a066464d35ddd606a1679c3a5b7ac198": "\\sum_{n=0}^{\\max\\{j,k\\}} S(n,j) s(k,n) = \\delta_{jk}", "a0665b4e0457cd501caca685e4eb4af7": "C V_{out}", "a0669de4ac43034752e47d0bb7706d9c": "\nVO_{A|B} = \\{ \\mathbf{v}\\,|\\, \\exists t > 0 : (\\mathbf{v} - \\mathbf{v}_B)t \\in D(\\mathbf{x}_B - \\mathbf{x}_A, r_A + r_B) \\}\n", "a066b0e6d40c68a7409b200b5d9f53b0": " \\tau = RC \\ . ", "a0673119deeb607315c9281a34d524ec": "g = \\tfrac{|f|}{I(|f|)}", "a0675e784f0838df026f31c3035a7d89": "P ::= (a,\\lambda).P\\,\\,\\, | \\,\\,\\,P + Q\\,\\,\\, | \\,\\,\\,P\\stackrel{\\triangleright \\!\\! \\triangleleft}{\\scriptstyle{L}}Q\\,\\,\\, | \\,\\,\\,P/L\\,\\,\\,|\\,\\,\\,A", "a067c6725b57b7e763698b1642ff8dc4": " q=(s, t_s, t_e) \\in Q ", "a067cfeb742d6fcdf89cbc24ee393b65": "i_j\\,\\!", "a0685cd12bf5e0f175cc130d8fca0747": "(T=g^{ab}T_{ab}=0)", "a0685f0bdc779a5396f3388747afb273": "g(r_0) \\, r", "a068e9d5cfdca492a9bbbcc8d296989f": "\\vec{v}", "a068f02da632572c8d6657796018736a": "1+ \\sqrt{5} \\over 2", "a0692b037229c343ad1cf23c961cbd29": "\\sigma_{11}= 6C_1 \\varepsilon = 3\\mu\\varepsilon", "a06939a8b2fe0e85727890a888a7e06c": "X_{it}", "a06942f33b08c63b1909175f948d1754": "\\mu = E[f]", "a06952be84dd450143c5b3bd50de0cd7": " u_{20}(\\mathbf{r}) = u_{SO}(\\mathbf{r}) = \\left | \\frac{1}{2},\\frac{1}{2} \\right \\rangle = \\frac{1}{\\sqrt 3} |(X+iY)\\downarrow\\rangle + \\frac{1}{\\sqrt 3} |Z\\uparrow\\rangle ", "a0697b66fb8662e7e66abb0bcbd3e612": "P\\left(\\frac{Z_n-a_n}{b_n} \\le x \\right) \\approx F(x) ,", "a069a6a1e520e335cd14b60ad6137e6e": "{\\underline P}X = \\{O_{1}, O_{2}\\} \\cup \\{O_{4}\\}", "a069edcde43990f97cc5fd58c45254c0": "\\mathrm{C^{\\prime}}", "a06a0cd245d76f68311bda2b534b22e7": "A = \\{ x \\in \\textbf{Q} : x < 0 \\or x \\times x < 2 \\}", "a06b4b873da6d7f9e38990d01aca51fc": "f(x|X \\leq y) = \\frac{g(x)}{F(y)}", "a06b799a42c5fe595de0f02b260a2b60": "\n\\begin{pmatrix}\n X_1 \\\\\n X_2\n\\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix}\n 0 \\\\\n 0\n\\end{pmatrix} , \\begin{pmatrix}\n 1 & \\rho \\\\\n \\rho & 1\n\\end{pmatrix} \\right)\n", "a06b82fdea2aeac88e1c2ec691cd375a": "C_{i_1i_2\\cdots i_n}(s_1,s_2,\\cdots,s_n) = \\langle X_{i_1}(s_1) X_{i_2}(s_2) \\cdots X_{i_n}(s_n)\\rangle.", "a06b941ca6029ffd3ef8c0653b7e280e": "x \\leq y", "a06ba026e4fcf65f7b13f36eeb62000d": "{\\gamma + \\zeta(2) = \\sum_{k=2}^\\infty\\left(\\frac1{\\lfloor \\sqrt{k} \\rfloor^2} - \\frac1{k}\\right) = \\sum_{k=2}^{\\infty} \\frac{k - \\lfloor\\sqrt{k}\\rfloor^2}{k\\lfloor\\sqrt{k}\\rfloor^2} = \\frac12 + \\frac23 + \\frac1{2^2} \\sum_{k=1}^{2 \\times 2} \\frac k {k+2^2} + \\frac1{3^2} \\sum_{k=1}^{3 \\times 2} \\frac k {k+3^2} + \\dots}.", "a06ba1c26b79fcb5016ca44e1ff2a16e": "\\mathcal{S}=\\langle{\\rm Fm},\n\\vdash_{\\mathcal{S}}\\rangle,", "a06ba5bfc45c94338e1686be4a57908a": "\\zeta(4)=\\pi^4/90", "a06bf637410c4915ee22ecaedc945485": "\\mathrm{nCH_4 + nFe_3O_4 + nH_2O \\rarr C_2H_6 + Fe_2O_3 + HCO_3 + H^+}", "a06c50150a1ec23c694c71ee53166492": "\\frac{d^2}{dz^2}f_z(z) + k_z^2 f_z(z)=0", "a06c6fac06d74d4f8b074561fbcf8c4a": "E_{s,3z^2-r^2} = [n^2 - (l^2 + m^2) / 2] V_{sd\\sigma}", "a06d0d70a29c68dedb655a8e6707e45e": " \\mathbf{F} = \\mathbf{e}_x\\frac{\\partial \\phi}{\\partial x} + \\mathbf{e}_y\\frac{\\partial \\phi}{\\partial y} + \\mathbf{e}_z\\frac{\\partial \\phi}{\\partial z}\\,\\!", "a06d567e9168e3a5b01ae2abd3bb00f3": "x \\mapsto gxg^{-1}", "a06d65f28e3ffb6baaba0897fc6c0fc1": "\\frac{d}{dx}a^x=\\lim_{h\\to 0}\\frac{a^{x+h}-a^x}{h}=\\lim_{h\\to 0}\\frac{a^{x}a^{h}-a^x}{h}=a^x\\left(\\lim_{h\\to 0}\\frac{a^h-1}{h}\\right).", "a06d8d9218faeff1a8f36bd28d4ebaad": "\\rho_{s} ", "a06df35694992f82ca296ce626afab5f": " V(f) = \\infty ", "a06dfde0434b893eedfb321b32eee785": "\\mathcal M f (s)= - \\frac 1 {s+a}, ", "a06e406750f04b22f983eb5bb2f0d604": " \\operatorname{lambda-free}[V] = \\operatorname{true} ", "a06eb4210b762d3d27264669cba61414": "R_G(p) = (1-p)^{|V|-k(G)} p^{|E|-|V|+k(G)} T_G \\left (1, \\tfrac{1}{p} \\right).", "a06ee96a8e2315aa404c0a0cfbebad1b": " GDP = f(M) \\,\\! ", "a06f3c57804496d591425613b27fddba": "\\nabla\\times\\mathbf{E} = \\sum_{l=0}^\\infty \\sum_{m=-l}^l\n\\left(-\\frac{l(l+1)}{r}E^{(2)}_{lm}\\mathbf{Y}_{lm}-\\left(\\frac{\\mathrm{d}E^{(2)}_{lm}}{\\mathrm{d}r}+\n\\frac{1}{r}E^{(2)}_{lm}\\right)\\mathbf{\\Psi}_{lm}+\n\\left(-\\frac{1}{r}E^r_{lm}+\\frac{\\mathrm{d}E^{(1)}_{lm}}{\\mathrm{d}r}+\\frac{1}{r}E^{(1)}_{lm}\\right)\\mathbf{\\Phi}_{lm}\\right)", "a06f60b391a0183b8649d80ba202198f": "\\boldsymbol{R_n}", "a06f626f71b63d86195ffadbb4c48345": "X=\\{0^2,1^2,\\dots,((p-1)/2)^2\\}", "a06fb6273ae70dcbff1fef825bee05fd": "c_i = y_i^2 x^{m_i}\\pmod{N}", "a06fc9dceea3766a85ca9995ad03e678": "S=\\{a,b\\}, (ab)^2=1", "a0704a1b9668f3c7f7235fe24095082f": "x_{n+1}=\\cos x_n\\,", "a0708ba6ba340ba16594a64997f2c4c1": "\\lambda, \\mu", "a070d71882edaed89a075e97a18ca828": "ZZ_3 = E", "a0714b939dae167c15b1ad666b4503a7": " y_i = \\begin{cases} \n y_i^* & \\textrm{if} \\; y_L a_4a_1", "a074c5e60ca6a29a05c7e48e003fb3b8": "\\{0,1,2,\\dots,n-1\\}", "a0752aa3933ef77806e930c48295609f": "\\rho(\\xi)_x", "a07558d1a7d2cbf48f3417374f4e5bea": "\\Phi_D = ", "a075b3b3f75209e0736f2e3c0a4ddb5b": "\\scriptstyle k \\;=\\; \\frac{2\\pi}{\\lambda}", "a075dac6f8778bbb80b83829a2911ae1": "\nR \\mbox{ metres} = \\frac{L + 2d + \\sqrt{S} - F}{2.37}\n", "a0761974e211deb90d445ca9511371ac": "O(EV)", "a0769e935916764075a584e3aa61dd03": "\\alpha\\omega = \\sum_{i_10", "a079f2b1e3d1332c82741f7ff99693c4": "x_3 = Bl^2-A-x_1-x_1 = \\frac{B(3x_1^2+2Ax_1+1)^2}{(2By_1)^2}-A-x_1-x_1=\\frac{(x_1^2-1)^2}{4By_1^2}=\\frac{(x_1^2-1)^2}{4x_1(x_1^2+Ax_1+1)}", "a07a0da2acc20533bb2fd214b0769d39": " \\mathbb{E} \\left( \\tfrac{3}{10} X \\right) = \\tfrac{3}{10} \\mathbb{E} (X) = \\tfrac{3}{10} \\cdot 5 = \\tfrac32,", "a07a4080554e13e3e4a2d13c8470a87c": "Q=\\begin{pmatrix}\n-\\lambda & \\lambda \\\\\n\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&&\\mu & -(\\mu+\\lambda) & \\lambda &\\\\\n&&&&\\ddots\n\\end{pmatrix}.", "a07a4e29e957d6dca15eed2283e77743": "I_0 = \\left(\\frac{\\pi}{8} - \\frac{8}{9\\pi}\\right)r^4 \\approx 0.1098r^4 ", "a07aaa956e33b2a95a8bc5a0f5c6a3a4": "Y_{1..j-1}", "a07ab620147ba0b0a731b98e75a6d1a5": " T\\,\\mathrm{d}S=\\delta Q+T\\,\\mathrm{d}S_{\\mathrm{uncompensated}} > \\delta Q .", "a07acae3154a37299d0b50a3f3920382": " \\sin \\theta = 1.22\\ \\frac{\\lambda}{d}", "a07ae25eda62557fb0ad40353584bd85": " \\vec n = \\vec v \\times \\vec w ", "a07af7c8692d679ab74bb2b44d2834e9": " s_i^2 = am_i^b ", "a07b17f2f9861233180ae3db1954122e": "D^\\ell_{mm^\\prime}(\\alpha,\\beta,\\gamma) \\approx e^{-im\\alpha-im^\\prime\\gamma}J_{m-m^\\prime}(\\ell\\beta)", "a07b70c890900907c24640350c9d302e": "G_N = G_S + \\sum_{j=1}^n A_{D_j} F_{dga_j}", "a07be19e8411103dd5b7a161fd9b630e": "\\Delta_n=\\left[\\begin{matrix}\nm_0 & m_1 & m_2 & \\cdots & m_{n} \\\\\nm_1 & m_2 & m_3 & \\cdots & m_{n+1} \\\\\nm_2& m_3 & m_4 & \\cdots & m_{n+2} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nm_{n} & m_{n+1} & m_{n+2} & \\cdots & m_{2n}\n\\end{matrix}\\right]", "a07c1c6a5452e53a9343ad5987e68f20": "\\approx 128.571", "a07c8c568818f2acec6739b4fd0a3521": " \\mathrm{Rot}(\\theta) = \\begin{bmatrix} \\cos \\theta & - \\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\end{bmatrix}, ", "a07ca05c3ebfd76e9a12827fef9906b1": "m\\frac{dV}{dt}=-v_e\\frac{dm}{dt}", "a07d0eb81a048f4d818458850085a196": "\n\\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x = - \\int_\\Omega \\langle\\boldsymbol{\\phi}, Du(x)\\rangle\n\\qquad \\forall\\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n)\n", "a07d14315a4091dc2abed452871a4d0d": "a_0=\\frac{1-\\alpha}{2};\\quad a_1=\\frac{1}{2};\\quad a_2=\\frac{\\alpha}{2}\\,", "a07d69deb3443ec0c8d52552cd779d28": " \\nabla_l G^{lm} = 0,\\,\\!", "a07d82e9649dc7b780ae526e388810b7": "0 \\to \\mathbb{Z} \\to \\mathcal{O} \\to \\mathcal{O}^* \\to 0", "a07e2264cbfacbfe487b424e330ae75f": "\\{1, 2, \\ldots, n\\}", "a07e2dec800d2de95549b5cd69b1bca8": "\\,\\Theta(N^2)", "a07e5743e38c4078f5e89c6b148b4b11": "\\iiint_\\mathrm{parallelepiped} 1 \\, dx\\, dy\\, dz", "a07ed89384341ef7a4c7557d616b9d93": " B^{-1}AB ", "a07efc582bd2456cb6f0c3486379ca28": " v = r \\omega", "a07f03a963afa95735f7b266796ce35a": "X_i'", "a07f41b4200122108f883edcf1867994": " \\arg(G(s))", "a07f54dd9a3024df60f8f98817420116": "q*", "a07f5e733b5c559353b16c35a6d6e270": "y_2=\\frac{y_1}{2}\\left(\\sqrt{1+8 F r_1^2}-1\\right)", "a07facf3e2d89cf89177b05c97dc091f": "T=v_1\\otimes v_2\\otimes\\cdots \\otimes v_r", "a07fc3a7dab9fa08160bf6d0c1f14ee2": "L \\left( x, y, k\\sigma \\right)", "a08007af2925c5b29d60422735d33e3a": "\\eta=1-\\left(\\frac{\\mathit{c}_{v}(\\mathit{T}_{4}-\\mathit{T}_{1})}{\\mathit{c}_{v}(\\mathit{T}_{3}-\\mathit{T}_{2})}\\right)", "a080463c4d2ccfa06f10052c754533ba": "\\mu=(3,2,1)", "a080a7fb918c9d3aea7b6bf5232987a9": "p((1+0.01x)(1+0.01y))", "a080b60b8fd7ae24deca8af406a4547f": "d\\mathbf{F}(\\mathbf{Y}) = \\operatorname{tr}\\left(\\frac{\\partial\\mathbf{F}} {\\partial\\mathbf{X}}\\mathbf{Y}\\right),", "a080ba3ea829445c6788070f71d008fa": "\\sigma=(6,3,8,1,4,9,7,2,5)", "a0815ee627c2628b814fe38705ecb688": "\\{E_{mb/c}T_{nac}g_c\\}_{m,n\\in Z}", "a081c8ec437288ee548293ff8b5240ea": "i_*", "a0820892f51f798f349e819df466b72e": "\n \\dot{\\varepsilon_{\\rm{p}}}^{*} := \\cfrac{\\dot{\\varepsilon_{\\rm{p}}}}{\\dot{\\varepsilon_{\\rm{p0}}}} \\qquad\\text{and}\\qquad\n T^* := \\cfrac{(T-T_0)}{(T_m-T_0)}\n", "a0821d07202e98b4373859c2644a9c11": "BA = A \\left( \\frac{\\pi}{180} \\right) \\left( R + K \\times T \\right)", "a0821d5f1618c7b60e32265b78d41bcc": "T = 2\\pi \\sqrt\\frac{L}{g} \\approx 2\\pi \\sqrt\\frac{6378100}{9.81} \\approx 5066 \\ \\text{seconds} \\approx 84.4 \\ \\text{minutes}", "a0822d2fc98f5228c6d60f160baf2a0a": "I^d", "a08282ba1448a0b44bcd785a020d116e": "\\begin{array}{ll}\n & P\\left(Spam|w_{0}\\wedge\\cdots\\wedge w_{N-1}\\right)\\\\\n= & \\frac{\\displaystyle P\\left(Spam\\right)\\prod_{n=0}^{N-1}\\left[P\\left(w_{n}|Spam\\right)\\right]}{\\displaystyle \\sum_{Spam}\\left[P\\left(Spam\\right)\\prod_{n=0}^{N-1}\\left[P\\left(w_{n}|Spam\\right)\\right]\\right]}\\end{array}", "a082fd062ccc2d8b4414ab120a3e679d": "\\lambda / \\mu", "a0835728598dc5a4b21adb7753062c50": "V_h.", "a0835b8b2fa7a7095c3002dffc7f506a": "S = \\exp(a h + b)", "a0836ce1eea68cff1b5d372328166d21": "L \\leftarrow \\operatorname{lcm}(L, M)", "a083c419436b0cf8fb4953e55adac379": " R \\ge 1 ", "a083c655cb3e56da49dddb1138bf6bb4": "\nv_{n} = \\sqrt{ k_B T / C }.\n", "a083ec63a10600e39e36d8e6af77f32f": "\\displaystyle{\\mathfrak{g}=\\mathfrak{g}_{-1}\\oplus\\mathfrak{g}_0\\oplus\\mathfrak{g}_1,}", "a0841575b2f99aa9cff0821d1830b593": "E_\\text{phase}", "a0847a84080039fb3c9791514bd03305": "\\frac{\\mathbf{\\Psi}}{\\nu + p + 1}", "a084ff244055d713a9341ddaf95a80d2": "\n\\sum_{k=1}^\\infty\\frac{\\varphi(k)}{k}(-\\log(1-x^k))=\\frac{x}{1-x}\n", "a085046a6332815a7efa1a759eb5cde6": " \\left(\\frac{221^2 + 67\\cdot27^2}{2}\\right)^2 - 67\\cdot(221\\cdot27)^2 = 1,", "a085301b67e37038c3bde953c6486b14": "f \\colon V \\to W\\,", "a0854a6069f70cd2b3e176ddfb822852": "\\left\\lfloor\\tfrac{n}{3}\\right\\rfloor + \\left\\lfloor\\tfrac{n+2}{6}\\right\\rfloor + \\left\\lfloor\\tfrac{n+4}{6}\\right\\rfloor = \\left\\lfloor\\tfrac{n}{2}\\right\\rfloor + \\left\\lfloor\\tfrac{n+3}{6}\\right\\rfloor,\n", "a08569f86ae82a371b24e3a9fa040fac": "(x^n,z^n)", "a085a720807d37ad45fed1a1c6589123": "B \\leq P A", "a085b6c0ca676eabf539e5ec6a578a23": "\\displaystyle M_{i}", "a086546b4308708120ae11eb33b5e5e4": " \\prod_{p b=a} g_b = g_a", "a0867a0b5cee4ea893fdb2354f562a97": "\nq(z) = a_4z^2+a_2z+a_0.\\,\\!\n", "a0868c7ac1cd2931c61b52dad7c638e4": "X^{\\operatorname{T}}", "a0868ecff88f88db93a6efce3d1169a7": "A_q(n,d) \\geq \\frac{q^n}{\\sum_{j=0}^{d-1} \\binom{n}{j}(q-1)^j}.", "a0873cb35f893b93214bc578752f9135": "=1\\cdot (1+q)\\cdots (1+q+\\cdots + q^{n-2}) \\cdot (1+q+\\cdots + q^{n-1}).", "a0878f61d0c56d57f49473c857d1462f": " P_{a1}=P_m(1-\\theta) ", "a087a33d473243b045df21873b8f0536": "L=4", "a087c27587560eab8c354bc4268ab2d3": "\\ln(f(x))=\\ln\\left(g(x)^{h(x)}\\right)=h(x) \\ln(g(x))\\,\\!", "a087cc88b1221cc1806b75ea3c888630": "\\Delta r=\\sqrt{n+1}-\\sqrt{n}.", "a087ffce4a6ca6263b2b625a8f0a2683": " \\sum_{j=1}^n \\gamma_j\\omega_j \\neq 0 ", "a08808adf0aec213066998d2cd9e0335": "\nu(x,t) = \\int h_t(x,y)u_0(y)dy.\n", "a0880e89595b38248b7d785ad8944f29": "\\rho^T", "a0888c9ddf1a56539e35c7982b0a1d8d": " \\mbox{Mat}_n = \\mbox{Sym}_n \\oplus \\mbox{Skew}_n , ", "a0889257868eff4a52d861edf423cbf2": "\\theta(a,b)", "a088d8426039b1e9d96e083f9254e66f": "\\tfrac{14}{B.23}", "a088e02e9ce71aed9537abfc426fdb1e": "f\\in\\mathcal{H}^\\infty(\\mathbf{C}\\setminus K)", "a089869098e0837af2572cfc894c7667": "\\frac{\\Diamond p\\land\\Diamond\\neg p}\\bot.", "a089e48b4de19e9fef40ee67f711b684": " \\int_{t_1}^{t_2} \\mathbf{F}\\cdot\\dot{\\mathbf{X}} dt = m\\int_{t_1}^{t_2}\\ddot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}dt. ", "a08a2c049bd2f908edfdf102372a32f9": "f(A_i A_i) = p_i^2\\,", "a08a5fb2d883cbef85cbc05fb33c274a": "\\lim f/g = 1", "a08a67c8ce1282cb00e538da0ba4cf85": "\\ \\ \\mathbf{dB} = 20 \\log_{10}(X)", "a08a733abd1a4f104fbf67d312d67ed0": "B_{\\mu\\nu}=\\partial_\\mu\\phi_\\nu-\\partial_\\nu\\phi_\\mu", "a08a82e5127e261dcb7801bf420e0d53": "1 - \\ln \\lambda \\, ", "a08b0549651cd06526d3d83011cce2c6": " \\psi_p ", "a08b2773e52961f37d7bc1eb74f39559": "\\sigma = \\frac{1} {\\tau}.", "a08b27b63b0e588fd3d52f36f7ca39a7": "\\cos a= \\cos b \\cos c + \\sin b \\sin c \\cos A, \\!", "a08b27ba7c45cb721800663c0d7078fd": "F^g_{X/S} : X^{(1/p)} \\to X \\times_S S \\cong X", "a08b8f8d291fd139d50682074d5b40e9": "\\mathrm{NA} = n \\sin \\theta\\;", "a08bbc2677cfb143358683bc554620b0": "\\langle Av,w\\rangle=\\langle v,Aw\\rangle", "a08bca470acc6576af69907ece917a75": "s_\\theta = +1, s_\\zeta = -1", "a08beeaf09c72e29c905e17e88d4b3ff": "\\sum_{\\lambda \\vdash n}(f^\\lambda)^2 = n!,", "a08c0ae987766fbc581624519a8af6c8": "\\rho_{earth}\\,", "a08c56e6f36954f53aeb887e0f136410": "\nT=\\frac{\\sum_{i=1}^n\\left(X_i-\\mu\\right)^2}{n+2}.\n", "a08c5b586afbeefa78b872548d278a78": " S = ( \\frac{1 - W}{W} ) ( \\frac{1 - \\epsilon}{\\epsilon} ) ( \\frac{\\rho_s}{\\rho_L} ) ", "a08c750e199a98f0676f1ee00861129d": "\\{n_t\\}", "a08cf3b727c3f959331a2fae20b5c382": "\\rightarrow \\subseteq S \\times S", "a08d35da319c04dc1514d86ee5e7930c": "T(X_1^n)= \\left(\\min_{1 \\leq i \\leq n}X_i,\\max_{1 \\leq i \\leq n}X_i\\right),\\,", "a08d736f09053f5d930829f102dd22f4": "\\cosh (\\gamma'd) = \\cosh (\\gamma d) + \\frac{Z}{2Z_0} \\sinh (\\gamma d)", "a08d94a0dbda0c7e51a5fc247bf442b3": "\\Lambda_{min} := \\{r \\in R \\ : \\ r-r^J\\varepsilon\\},", "a08e110fbfc2b310846607a273d9d0b8": "\\begin{align}\n& \\frac x {7} \\times 21 = \\frac {90} {3} \\times 21 \\\\\n& x \\times 3 = {90} \\times 7 = 630 \\\\\n& x = 210\\ \\mathrm {miles} \\\\\n\\end{align}", "a08e2c6b7e8770c3ef01b491decae529": "z' = z \\,", "a08e50c41081316b7c387edd5ca53238": "V_\\theta=\\frac{1}{r}\\frac{\\partial \\phi}{\\partial \\theta} = - U\\left(1+\\frac{R^2}{r^2}\\right)\\sin\\theta.", "a08e640359d1901e3f8bdea7ab082133": "\n\\begin{align}\n6x^2 + 13x + 6\n\\end{align}\n", "a08ecfa28d5fb4d1a809e40e9772ed47": "\\textbf{TVect}_K", "a08f5fcac50d54ea5dadad886d3f75f8": "Y \\sim B(n, pq).", "a08f6fd5ee7f72ac3ffbec2809ed9df9": "\\frac{5}{6}", "a08fd708e0a748479051f3b7675fc683": "a_n=G^{-1}\\left(1-\\frac{1}{n}\\right).", "a090d4996e00c264ce3d7f4abf606a08": " x_{\\mu} x^{\\mu} = \\eta_{\\mu \\nu} x^{\\mu} x^{\\nu} = (ct)^2 - \\mathbf{x} \\cdot \\mathbf{x} \\ \\stackrel{\\mathrm{def}}{=}\\ \\tau^2", "a090de8ffbdf8a48df20eab6b49197ca": "T_{ij}(K) = \\sum_{l=1}^K P_{ij}(l).\\log_{2} \\left ( \\frac{P_{ij}(l)}{P_{i}P_{j}} \\right )", "a090f9db9fab54325d57986236de2d2f": "c = 48436", "a09148e949d9179b902178505f5b578c": "P=(x_P:y_P:1)", "a091792b3bb0d1d4953c61f86d9605dc": "W^{\\pm,0}_L \\pi_T", "a091794487cc14833550671979043c4b": "(3)(5)", "a091815e8285b34a1ed7cea33e3c75d8": "u_1(\\mathbf{x},z_1)=-\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1(z_1-u_x(\\mathbf{x})) + \\frac{\\partial u_x}{\\partial \\mathbf{x}}(f_x(\\mathbf{x})+g_x(\\mathbf{x})z_1)", "a09184b8c01eaf83561004d6946a92a5": "F_{1}Sin(\\alpha )=F_2Sin(\\beta ) \\,", "a091ba93fd557ec5539ded474deafae9": "L(y, F(x))", "a091e4fd0ba2e9be62f69e18dc07dcba": "u_3", "a09201ede597907407718188a7472084": "\\ F = \\frac{\\partial x}{\\partial X} \\qquad \\textrm{and} \\qquad F^* = \\frac{\\partial x^*}{\\partial X}, ", "a09275a9ae8f59f1e17c0648bf4c945c": "\\frac{3\\alpha a + 1}{2d}-\\frac{1}{2}", "a092a42d311a52481e7c180963198945": "\\displaystyle{e^{-t(P^2+Q^2)} = (\\mathrm{cosech}\\, 2t)^{1\\over 2}\\cdot T_{Z(t)}}", "a0936a07a2378aa0e8bf152068de5a23": "C_{ijkl}", "a09385e6d0351edb75a1945214f14ab5": "\\left(\\frac{0.5036 \\mbox{ mol Ag}}{1}\\right)\\left(\\frac{107.87 \\mbox{ g Ag}}{1 \\mbox{ mol Ag}}\\right) = 54.32 \\ \\text{g Ag}", "a093fc5c5afb41b407981fbd34b2e74b": " x\\neq y ", "a0944f34f517d5cb667fa48383b842cb": "c_S \\in C", "a09481da876c166a47ffd581cea29527": "n_{air} = \\frac{A_{av} P}{RT}", "a09493b1f5a8faf6473042571d867d2c": "\\textstyle x^ia(x)", "a094bbcdc3e6f34d2ccf9ee7a8a48710": "\\mathbb{Q}[G] ", "a094c9a20b3ae5f24f583a18e0252150": "A(\\sigma_y, \\tau_{xy})", "a094ea64aab645ab77c2a6d58636f8af": "\\frac{1}{4}nl^2\\cdot \\cot(\\pi/n)\\,\\!", "a095231a03a9bea45084825b5ee47fd5": "z=e^{\\beta\\mu}\\,", "a0956c42a583e23688adb03888d57504": "A \\stackrel{\\rho}{\\to} A'", "a095bcbc1caa83e8fdad7fc1236f4645": "W_E = 4 \\int Y \\sigma^2~dx ", "a09606d6dd4ffeb5e6fc7d1e332efdd8": "tr : \\mathbb Q[\\mathbb Z]/\\Delta K \\to \\mathbb Q", "a0965ddf692f99c9d66f8e2b782bdda1": "{BE}_{6}", "a096aebcf3a3240f95022b6ae1502f46": "\\mathbf{x}\\cdot\\mathbf{y} = \\|\\mathbf{x}\\|\\,\\|\\mathbf{y}\\|\\,\\cos\\theta.", "a096d470254b147874a3c528abc22316": "{\\displaystyle\\varphi_t(x)}", "a0971f3c1fc2d4c967b3cd362cc29f48": "x y z", "a0972cfa6e3fcbc3dbc103fec3ca317b": "\\xi\\in\\mathfrak h", "a097add127cb8765a9689ee2e8ee6bd1": " P(\\alpha_i) = y_i \\quad\\text{for } i=1,\\ldots, m. \\, ", "a097cbe6619aaeb8c539bd481415de90": "f(\\boldsymbol{S})", "a097f9666ad4795724f43246d6b623ae": "\\mathcal M_\\omega \\models^+_{\\{\\emptyset\\}} \\tau('\\phi').", "a0981e942edfeb8e87fc5e6ed641f14b": "S_{xx} = \\left( 2 \\frac{c_g}{c_p} - \\frac12 \\right) E,", "a098378c3fecb393d7fdd908c580d878": " f_X(x_1 : t_1 ) = f_X(x_1 : t_1 + \\Delta), \\, ", "a0985011b30e33b1932b08fc9d725617": "\\sigma^2 C ", "a09867ec0eb8ed8bef153760a7b4d1a5": "\\lim_{x \\to c} \\frac{f(x)}{g(x)} = \\lim_{x \\to c} \\frac{f'(x)}{g'(x)} \\qquad \\text{ if } \\lim_{x \\to c} f(x) = \\lim_{x \\to c} g(x) = 0 \\text{ or } \\lim_{x \\to c} g(x) = \\pm\\infty", "a09918713f14944f2bce4aa39daccb50": " H \\left| \\psi_t \\right\\rangle = i \\hbar \\partial \\left| \\psi_t \\right\\rangle/\\partial t", "a099208118dfea3b7a277336b0f18e21": "\\begin{align}\nf(x,p) \\star g(x,p) &= f\\left(x+\\tfrac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{p} , p - \\tfrac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{x}\\right) \\cdot g(x,p) \\\\\n&= f(x,p) \\cdot g\\left(x -\\tfrac{i \\hbar}{2} \\stackrel{\\leftarrow }{\\partial }_{p} , p + \\tfrac{i \\hbar}{2} \\stackrel{\\leftarrow }{\\partial }_{x}\\right) \\\\\n&= f\\left(x +\\tfrac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{p} , p\\right) \\cdot g\\left(x -\\tfrac{i \\hbar}{2} \\stackrel{\\leftarrow }{\\partial }_{p} , p\\right) \\\\\n&= f\\left(x , p - \\tfrac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{x}\\right) \\cdot g\\left(x , p + \\tfrac{i \\hbar}{2} \\stackrel{\\leftarrow }{\\partial }_{x}\\right).\n\\end{align}", "a0992795a953604d57f1337cf3a77f56": "m_M = \\frac{g^{2}m_e}{e^{2}}", "a0992ed640df43c4e4a36ea83af7c8e2": "\nM_{9 \\times 12} = \\left[\n\t\t\\begin{array}{cccccccccccc}\n\t\t0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n\t\t1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\\n\t\t0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\\\\n\t\t0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n\t\t1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\\\n\t\t0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\\\\n\t\t0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\\\\n\t\t1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1\n\t\t\\end{array}\n\t\t\\right]\n", "a0992fa048c99a3b81a56143a93cb5d0": " \\frac{dv}{dr}", "a09941dfe45bc62e96e59bacd9089915": "f_{\\bar{\\Omega}}(\\alpha,\\alpha^*) = \\frac{1}{\\pi} \\int \\rho_{\\bar{\\Omega}}(\\alpha,\\alpha^*) |\\alpha\\rangle\\langle\\alpha| \\, d^2\\alpha.", "a0995481b086eb3e6a5797b48d61b296": " D_{F}(x-y) = \\langle 0| T(\\psi(x) \\bar{\\psi}(y))| 0 \\rangle ", "a0997c052633b68ff44befb0beb6df7f": "Q=\\begin{pmatrix}\nB_{00} & B_{01} \\\\\nB_{10} & A_1 & A_2 \\\\\n& A_0 & A_1 & A_2 \\\\\n&& A_0 & A_1 & A_2 \\\\\n&&& A_0 & A_1 & A_2 \\\\\n&&&& \\ddots & \\ddots & \\ddots\n\\end{pmatrix}", "a09989fdb8f7c88e6d6354c33268ee0f": " F_G = \\frac{Gmu}{r^2}", "a0998e8640ccf01b0d1447269441541d": "(6)~~ ~~ -\\mathrm{e}^t V'\\big(-\\mathrm{e}^t\\big)= - \\frac{V\\big(-\\mathrm{e}^t\\big)}{1+V\\big(-\\mathrm{e}^t\\big)} ", "a09a8c2759d8c672088db30174986f92": "A_1, A_2\\in \\mathbf{H}_n", "a09a94b6d45a2c03c8e97b3d18e6825c": "\\mathbf{F_C}", "a09aa1e03716630ba30ec1e69f02fc51": "R = \\frac{\\vec J_\\mathrm{refl} \\cdot -\\hat{n}}{\\vec J_\\mathrm{inc} \\cdot \\hat{n}} = \\frac{|J_\\mathrm{refl}|}{|J_\\mathrm{inc}|}", "a09ad6e746de48ed8e36d70ee186d175": "P_\\mathrm{prior}", "a09b01dd5aa3c86b3f177c29cd954338": "\\Delta G^\\circ", "a09b45e2bcb3654dfce89d9db85f8e13": "\\displaystyle \\sqrt{\\frac{\\pi}{\\alpha}}\\cdot e^{-\\frac{(\\pi \\xi)^2}{\\alpha}}", "a09b8afeb36acd8953ee8e7c3b767c17": "\\alpha _{g}=\\frac{m^{2}}{m_{P}^{2}}", "a09b95da7eafc3fc9e7cae009e47d611": "t_1 = \\varepsilon t\\,", "a09bb858da2536f1c2f8e788151fe191": "ind(P')", "a09c144be3b9ab3839aaa45494be4fec": "d_0=\\begin{pmatrix}\n1 & 0 & 0 \\\\ \n0 & \\alpha &\\delta \\\\ \n0 & \\beta & \\epsilon\n\\end{pmatrix}", "a09c3125017f9562fee517587fd15a19": " \\boldsymbol{.}: H\\otimes V\\to V ", "a09c46ec0792fc3b619b554100735ed5": "F_0(x) = \\underset{\\gamma}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L(y_i, \\gamma).", "a09cac5b993a11d16eede4e13023704b": "\\langle \\alpha_{\\iota} | \\iota < \\gamma \\rangle", "a09cafd559bd32ecf94e67797b5dadb5": "\\rho(E) \\equiv \\exp(S(E))", "a09d09fd6ab94b63cd5f716d8f20616c": "M_{i,j} \\in \\{0,1\\}", "a09d1c606bd32eb0fc9b11385c4cf798": "\\{z_i\\}_{i=0, 1, ..., N-1}", "a09d2859fcf0fbad859ef4ff9cd7e090": "X_{i}=\\frac{\\dot{n}_{i,in}-\\dot{n}_{i,out}}{\\dot{n}_{i,in}}=1-\\frac{n_{i,out}}{n_{i,in}}", "a09d9023c97f530fc95a696efd57bfb1": "\\frac{\\mathrm{d}^2 r}{\\mathrm{d}t^2}=0", "a09dc99fa3aa6bfab8ee96eb70d0bd0b": " f_W(x, z) = x^T W z ", "a09dc9e73e8b38c94e2ab1b6219d2ec9": "\\angle PAB + \\angle PBA + \\angle PCD + \\angle PDC = \\pi", "a09e02d988ee9f7db642e2a9a305b541": "\n w(x,t) = \\frac{2P}{\\rho\nAl}\\sum_{j=1}^{\\infty}\\frac{1}{\\omega_{(j)}^2-\\omega^2}\\left(\\sin(\\omega\nt)-\\frac{\\omega}{\\omega_{(j)}}\\sin(\\omega_{(j)}t)\\right)\\ ,\n ", "a09ee9c6f5b9bf53c60f03e42fa123fe": " \\delta(\\theta^{ML}) ", "a09f109b25a0b5a381789458004961d8": "\\operatorname{dist} (\\langle x_1, y_1 \\rangle, \\langle x_2, y_2 \\rangle) = \\operatorname{arcosh} \\left( 1 + \\frac{ {(x_2 - x_1)}^2 + {(y_2 - y_1)}^2 }{ 2 y_1 y_2 } \\right) \\,.", "a09f4ef45057354eeafedbf7bbf5d121": "\\int f \\,d\\nu + \\int g \\,d\\nu \\neq \\int (f + g)\\, d\\nu.", "a09f62bf3c65f7b27443f13eff18d46d": "F (A \\times B)", "a09f786077e170a91f252fc19a738c47": "C_1,C_2,C_3", "a0a02221cba526c979079a408e08044c": "\\beta \\leq 1", "a0a04ab87cce777febcdcdeceb2e8b45": "z=2m\\!", "a0a083e5faecb9b72ebca28fcda556d4": "\\nabla^2\\psi = 0", "a0a08931030340148237f4dfaeb53414": "\\pi_p(F_n(\\mathbf{C}^k))=\\pi_p(F_{n-1}(\\mathbf{C}^{k-1}))", "a0a1b7b0da617c5d9fbc6eb1620b0e27": "\\Delta^C_{n+1}", "a0a1bc2e878b93542e0f4adcdd4c761c": "k(\\mathbf{x},\\mathbf{y})=\\phi(\\mathbf{x})\\cdot\\phi(\\mathbf{y})", "a0a1c8cc0a8340afe3360eab3ae7b5a3": "\\frac{\\partial\\hat{t}}{\\partial V_z} = \\frac {1}{V_t}\\ \\hat{z}", "a0a1e70956e01a0a4833992ddec14cb8": "{(x_1, \\ldots , x_n)}", "a0a251e3b0793de6e08cb951011044a4": "\\sqrt{n}D_n\\xrightarrow{n\\to\\infty}\\sup_t |B(F(t))|", "a0a25ee13178fcf2ab5c5175e37885ef": "\\mathcal{L}_N", "a0a26fa2158788fe83767a429419447b": "2 \\over 5", "a0a2a9cab42820002dcff9086249f95c": "C(B) \\subset B", "a0a31165bc998d576c95fae13d6fc68c": "\n(q)_{i} = \\frac{\\Gamma(q+i)} {\\Gamma(q)} = q\\,(q+1) \\cdots (q+i-1).\n", "a0a31aebe1e756e0ca946828804a7365": "X(z^K)", "a0a31cad7d00434c1f2e0baed9094073": " \\lambda E = R_n - G - H \\,\\!", "a0a3218de7c73c4959192b31f1560669": "\\mathrm{cn}\\,(x)", "a0a32ac51fbb18a35e8169304d5c54f5": "A = \\frac{1}{2} ab \\sin C", "a0a3403d901e6a86604175e006c712f9": ".6 \\times .6+ .4 \\times .4", "a0a3883fdb894f2b94f55448fa4d0fcb": "M(x) \\cdot x^n = Q(x) \\cdot x \\cdot K'(x^n) + x \\cdot R(x)", "a0a446ee5ea3b6f7c410e3bdb689ce71": "\n\\text{minimize} \\quad \\text{over } \\widehat D \\text{ and } R \\quad \n\\operatorname{vec}^{\\top}(D - \\widehat D) W \\operatorname{vec}(D - \\widehat D)\n\\quad\\text{subject to}\\quad R \\widehat D = 0 \\quad\\text{and}\\quad RR^{\\top} = I_r,\n", "a0a466e3aac3630c6d6f9b9833deb813": " E[q(d,\\mathcal{E}_{q}^{\\epsilon}(d))]\\geq OPT-3t\\,\\!", "a0a4788f43a139be0df1ee1b8f65e534": "e^0 = 1\\,", "a0a47ed9496f1f5ec424d97533156ee0": "\\iota_v \\omega", "a0a49876046e7ee66c7cb7649da81054": "\\mathfrak{H}_1", "a0a49b7b7a8f9e7c95ca0e29e006c5a5": "|\\langle f|g\\rangle|^{2} \\geq \\bigg(\\frac{\\langle f|g\\rangle-\\langle g|f\\rangle}{2i}\\bigg)^{2}", "a0a4b016fbb9ebae642f41b3078ebd36": " \\underline{\\underline{\\epsilon}} ", "a0a4e705d95031dda7ddec49c13bccea": " Y^*_2 ", "a0a59d8151f95590d464ff214a1c3192": "R_{12} = \\left. \\frac{V_1}{I_2} \\right|_{I_1=0}", "a0a5a07e0b924ed9d048d4fe33935782": "\\frac {d M_{xy}'(t)} {d t} = \\frac {d \\left ( M_{xy}(t) e^{+i \\Omega t} \\right )} {d t} =\ne^{+i \\Omega t} \\frac {d M_{xy}(t) } {d t} + i \\Omega e^{+i \\Omega t} M_{xy} =\ne^{+i \\Omega t} \\frac {d M_{xy}(t) } {d t} + i \\Omega M_{xy}'\n", "a0a5b3e6e76dbb2f6dd5ecc17a89eb3f": "1 = \\det(\\mathbf I_n) = \\det(\\mathbf{A} \\mathbf{A}^{-1}) = \\det(\\mathbf{A}) \\det(\\mathbf{A}^{-1}),", "a0a5cb5992ff60f22cea2fe1cdeec14e": "m(x) = 0", "a0a5e752ad0d2922f804aa6d988aecd6": " \n\\begin{align}\nI(F_a;C) & = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i,c_j) \\log \\frac{p(v_i|c_j)}{p(v_i)} \\\\\n & = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\left [\\log p(v_i|c_j)- \\log p(v_i) \\right ] \\\\\n & = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\log p(v_i|c_j)- \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\log p(v_i) \\\\\n & = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\log p(v_i|c_j)- \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i,c_j) \\log p(v_i) \\\\\n & = \\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\log p(v_i|c_j)- \\sum_{v_i \\in F_a} \\log p(v_i) \\sum_{c_j \\in C} p(v_i,c_j) \\\\\n & = {\\color{Blue}\\sum_{v_i \\in F_a} \\sum_{c_j \\in C} p(v_i|c_j)p(c_j) \\log p(v_i|c_j)- \\sum_{v_i \\in F_a} p(v_i) \\log p(v_i)} \\\\\n\\end{align}\n", "a0a636da6ca8656e2b98592c7dd4a076": " \\Delta_{SL} ", "a0a63ef1d23de9585f7dc2357fd6dd8a": "\\partial_{\\mu} = \\partial/\\partial x^{\\mu}", "a0a65ebeb3b0776edc7298e3790e645c": "\\tfrac{50}{100}", "a0a67925f1ac5b3861e9cbe8429ef88d": " p < p_c ", "a0a7214f37c27c68a0f72576aded98ae": "(b,\\infty),(\\infty,c)", "a0a7218679740ef5f7646d9e08c8270a": "\\not\\supset", "a0a7533e30f5e2049fa169f803bcb1e2": "\\gamma_\\text{SG}\\ =\\gamma_\\text{SL}+\\gamma_\\text{LG}\\cos{\\theta} \\,", "a0a780e1ffbafea57a7d16a8a5551ca9": "i:X\\to K", "a0a7864e975b8e5d15fefa1101b80478": "A=(9+\\frac{5\\sqrt{3}}{2})a^2\\approx13.3301...a^2", "a0a7ae054c9b42a9822828b10eb760a2": " \\mathbf{s} ", "a0a7e30c9083b4eb093d4f9e7d8f0d69": " A_{m}(\\omega, \\gamma)=\\frac{4}{3\\pi}\\frac{1}{|m|^{!}}\\frac{\\gamma^{2}}{1+\\gamma^{2}}\\sum_{n>v}^{\\infty}w_{m}(\\sqrt{\\frac{2\\gamma}{\\sqrt{1+\\gamma^{2}}}(n-v)}e^{-(n-v)\\alpha(\\gamma)})", "a0a865024d41adb52e9f59507f09db05": " \\Bbb{C} ", "a0a8a8872584a3f014089033b25194a7": "x'=A(t)x+g(t)\\,", "a0a8b56719e13de01a910251f5e2bc90": "P \\le |S| \\le Vol_q(H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k,2k)", "a0a8da17a0a6213b23f0c884b4f43303": "\\left(G,C_o\\right) ", "a0a8e83310109662806f27fd8c59ab5b": " \\hat{P}_{Barlett}(\\theta)=V^HRV ......(5) ", "a0a941257e221301c14dd9b6e190c83e": "L_{}", "a0a9427f99fbb74da09c0a894e406738": "\\nabla \\cdot \\mathbf{D} = \\rho", "a0a95bd7216cb0a6b81b2d3afbfc0dfd": "\\hat{h}_E(Q) \\ge \\frac{C(E/K)}{D^2(\\log D)^2},", "a0a9633dd9ecfd73a6a26e85c59cf2c3": "\nL \\propto\\frac{\\sigma^4}{4\\pi G^2 B}\n", "a0a9709d1d4db3e32e350e73bbd7578f": "1+ 0.36 k^2\\le d_H(C) \\le 1 + 37 k^2.", "a0a9710d7f4daa7b8aedb6b67e24b5e1": "\\ (u_1 \\otimes \\dots \\otimes u_n)^* = Qu_1 \\otimes \\dots \\otimes Qu_n. ", "a0a9755e0e38c9471eb0107b682d8132": "\\alpha_{21}", "a0a9757cfd4d683f7035c1076c77eecc": " f : M \\times [0,1] \\to M \\times [0,1]", "a0a98a30f067b2efaf0d478afeeb9d70": "\\frac{d}{d\\theta}\\operatorname{Sl}_{2m+1}(\\theta) = \\frac{d}{d\\theta}\\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m+1}}=\\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m}}=\\operatorname{Sl}_{2m} (\\theta)", "a0aa0030ac9dfcfe51b66957013105fd": "0 \\le \\sum_{n = 1}^m (a_n + |a_n|) \\le \\sum_{n = 1}^m 2|a_n|\\le \\sum_{n = 1}^\\infty 2|a_n|", "a0aa4408057e1539753d77604c5ed39c": "V_{Y0}t - \\frac{1}{2}gt^2 = h - \\frac{1}{2}gt^2.\n", "a0aa6db306bdfe477174485b8e68d2f1": "\\Phi=\\tan(\\pi \\alpha/2)\\,", "a0aa7dc462a31054532d97a13f43fa5b": "\\mathbf{A}=\\begin{pmatrix}0 & 0\\\\ 0 & 1\\end{pmatrix}.", "a0aa7e2db2b0a3dff7df6b1995da90a0": "\\kappa(x, y) := \\textrm{Tr}( \\textrm{ad}\\,x\\, \\textrm{ad}\\, y )\\ \\forall x,y \\in \\mathfrak{g}", "a0aae2ed890825df0c6421354736f2ed": "\\xi-\\rho\\,", "a0ab05f0ae4f86c4776731593814bf7b": "\\epsilon_S = 0", "a0ab15c3961c5f7d9caaede119d0518d": "\\mu\\ll H^{m}", "a0ab83834fac660f52259ea2e41f3cf0": " 1 s \\sigma_g ", "a0ab9e9aa4458b6a71cbe12156c6e04f": "\\begin{align}\n\\frac{D\\vec\\omega}{Dt} \\equiv \\frac{\\partial \\vec \\omega}{\\partial t} + (\\vec V \\cdot \\vec \\nabla) \\vec \\omega = (\\vec \\omega \\cdot \\vec \\nabla) \\vec V - \\vec \\omega (\\vec \\nabla \\cdot \\vec V) + \\underbrace{\\frac{1}{\\rho^2}\\vec \\nabla \\rho \\times \\vec \\nabla p }_{\\text{baroclinic contribution}}\n\\end{align}", "a0abb16a23d63af4369d66ccf614d35c": "\\begin{align}\n F_{2n-1} &= F_n^2 + F_{n-1}^2\\\\\n F_{2n} &= (F_{n-1}+F_{n+1})F_n\\\\\n &= (2F_{n-1}+F_n)F_n ~ .\n\\end{align}", "a0abdf1bf18835d65d4f4849d0930005": "\\langle k^{(0)} | V |l^{(0)}\\rangle = V_{kl} \\qquad \\forall \\; |k^{(0)}\\rangle, |l^{(0)}\\rangle \\in D. ", "a0ac0d4ffb8900b7e1de04b8e3d41194": "\\langle a;b \\rangle p \\equiv \\langle a \\rangle \\langle b \\rangle p\\,\\!", "a0adb5060243e44242241b5d7e85a6fc": "\\begin{bmatrix}\n1 & 2\\\\\n3 & 4\\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\n0 & 1\\\\\n0 & 0\\\\\n\\end{bmatrix}=\n\\begin{bmatrix}\n0 & 1\\\\\n0 & 3\\\\\n\\end{bmatrix},\n", "a0ae7dd4bc16e2166cb51be7a498dbbc": " \\frac{d[D]}{dt} = k_2[B] = k_2K[A]", "a0aea04ac514ae95f0acc74e46602fa9": "f^* \\leftrightarrows f_*", "a0aea56b97ce450ffc02377b7bbde38d": "\\operatorname{Sp}(2) \\cong \\operatorname{Spin}(5).", "a0aee1cd87cc40211268d17cf729bcc2": "\\color{CadetBlue}\\text{CadetBlue}", "a0af12ccf24147bbe13646b9f3db6c1b": "a_j+\\varepsilon", "a0af2cba3e402a31cfef2ba1234d2f90": "\\frac{a}{\\sin \\alpha} = \\frac{b}{\\sin \\beta} = \\frac{c}{\\sin \\gamma}.", "a0af47ffd6fb589ce89f3bce5898c3a3": "\\delta ([x,y]) = [\\delta(x),y] + [x, \\delta(y)]", "a0afd019257e377ff294e9ad384b2edc": " {}^{(-1)}a = \\log_{a} \\left( {}^{0}a \\right) = \\log_{a} 1 = 0 ", "a0afd823f540bbd5988a1de6775c7a31": " \\alpha = \\arccos \\frac{b^2 + c^2 - a^2} {2 b c}.", "a0aff4ba81b73f6075a431f4b6920f6b": "\\exists\\,\\!", "a0aff870cedae87e793d6aaa5bcf3cd7": "|z_n| \\le 1", "a0b007052498ebaa2dfc29048afb4a11": "\\exists a\\, (x\\in a \\land y\\in a )", "a0b0354ec7a44f5f7afa382419530ffd": " \\mathbf{F} = m_0 \\frac{d(\\gamma(\\mathbf{v}) \\, \\mathbf{v})}{dt} = m_0 \\left( \\frac{d \\gamma(\\mathbf{v})}{dt} \\, \\mathbf{v} + \\gamma(\\mathbf{v}) \\frac{d\\mathbf{v}}{dt} \\right).", "a0b052e6dced042ff243c94e593706b3": "m = \\rho \\cdot dA \\cdot dz", "a0b05648c8ebf05ce35c23dd83546e09": "\\text{:} \\qquad H_0: p = \\tfrac 14", "a0b06177c5956ec95d748a5da66a8ec2": "C_d A,", "a0b08e4f510287b7cfa938cab37e00eb": "p c \\gg m c^2", "a0b0a71f6f8aca05006d2bd952ac6f93": "A,B,C,D,F,G", "a0b0d6f84156ce145c847501ee70e285": "\nx' = \\frac{x}{\\rho_s}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\rho_s^2 \\equiv \\frac{T_e}{m_i\\omega_{ci}^2}.\n", "a0b115bc4ac1ed0318412a90d6dc5001": "[-\\pi,\\pi)", "a0b12644e4a44c967086e31098b344ba": "R(r) \\,\\;", "a0b16bbe15249d1b83a4e17f1fee036b": " r_{i}^{t}=r_{i}^{t-1}+\\kappa\\cdot h_{i}^{t-1},\\quad i=1,\\ldots,N ", "a0b1939375e176998796684ff5bb3dd7": "F(x; k,\\lambda) = 1 - \\sum_{n=0}^{k-1}\\frac{1}{n!}e^{-\\lambda x}(\\lambda x)^n.", "a0b1b970b826a61d6b9cc0a76200dea7": "x^2 + y^2 - r^2 = 0", "a0b212e131a06b05650028bf7c4a607e": " [\\Pi] =\\begin{bmatrix} 0 & I \\\\ I & 0 \\end{bmatrix},", "a0b217a88550706c0ec6f7794652fbc3": "mE-{P^2 \\over 2}={-P^2 \\over 2}", "a0b22e479411c218c4ec0ae2ebc49705": " f_i(s)=\\tilde{f}_i(s_i,g(s_1,\\ldots,s_n))", "a0b259adc682b0cef3a3b0b7ec96d26e": "b=-k_2 (x_2 - x_1)+(y_2 - y_1).", "a0b25b24db51b4d709e6d639c647cc48": " B(f)B(g)+B(g)B(f)=2\\mathrm{Re}\\langle f,g\\rangle, \\, ", "a0b2aa480d0b31c8e5a44ad955dd3ffa": "\n \\mathbf{b}^i = \\sum_k g^{ik}~\\mathbf{b}_k ~;~~ g^{ii} = \\cfrac{1}{g_{ii}} = \\cfrac{1}{h_i^2}\n", "a0b2e383c2e48ca48d59ed57434c2968": "b(\\omega)", "a0b2ef97067d5463bee7d4e89afbe4af": "\\Pi = \\Gamma_{\\max}RT \\left(1+\\frac{C}{a}\\right)", "a0b319656dcaf081ae38925048546840": "\\mathbf{X} = \\mathbf{U}\\mathbf{\\Sigma}\\mathbf{W}^T", "a0b31f52e0e8d5ae891a7b046468545f": "x \\not\\in L \\Rightarrow \\mathrm{Pr}[A'\\,\\mathrm{accepts}\\,x] \\le 1/2 \\cdot (1-1/2^{f(|x|)+1}) < 1/2", "a0b324a7a05561f9e18e5b5916e6b567": "\\chi_a(G)", "a0b34829f41940808f51756c60ed6e35": "\\mathbf{T}\\alpha = \\sqrt{x^2+y^2+z^2}", "a0b3682845c1bef985fd9b9f4874a13d": "A_{ij} = \\langle \\phi_i | \\hat{A} | \\phi_j \\rangle,", "a0b3c2769a7c938b4f1c691c8b71e112": "\\begin{align}\n b_{n+1}(\\theta) &= b_{n+2}(\\theta) = 0,\\\\[.3em]\n b_k(\\theta) &= C_k + 2 b_{k+1}(\\theta)\\cos \\theta - b_{k+2}(\\theta)\\quad(n\\ge k \\ge 1).\n\\end{align}", "a0b3cb55dde9af3f89401a3e1fa9d1a6": " v/c=\\mbox{tanh}(s)={\\frac {e^s-e^{-s}}{e^s+e^{-s}}} ", "a0b3cb70d55cfa92380b6b082f28f53f": "\\lceil\\log_2(n+1)\\rceil", "a0b3e7e8142858519ba3394e0d37910a": " \n\\dfrac {\\dfrac{}{B \\leftarrow B} \\qquad \\dfrac{}{A \\leftarrow A} }\n {\\dfrac {B \\leftarrow (B/A), \\;\\; A} \n {(B/A)\\backslash B \\leftarrow A} }\n\\qquad\n\\begin{matrix}\n \\mbox{(Axioms)}\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad{ }\\\\\n {(\\leftarrow/)\\,\\,[Z=Y=B,X=A,\\Gamma=(A),\\Delta=\\Delta'=()]}\\\\\n {(\\backslash\\leftarrow)\\,\\,[Y=B,X=(B/A),\\Gamma=(A)]}\\qquad\\qquad\\qquad{ }\\\\\n\\end{matrix}\n", "a0b3ef3bd753f3fe3299112708b71213": "( +, \\ \\times, \\ \\uparrow, \\ \\uparrow\\uparrow, \\ \\dots)\\,\\!", "a0b456947c0c8fd4b14fce594354f0d8": "b \\pmod{v}", "a0b47e5d509eb6befeabfab66b9b225a": "1/x^c", "a0b52d556fe25e8c414b5362416f9808": "0 \\to K \\to \\Omega^*(E) \\overset{\\pi_*}\\to \\Omega^*(B) \\to 0", "a0b55330081fa96d480dee0e24903811": "r=a\\frac{\\cos 2\\theta}{\\cos \\theta} = a(2\\cos\\theta-\\sec\\theta)", "a0b5759944e6a8bb22ce3f2cf118426c": "\\mathrm{d} (U + pV) = T\\mathrm{d}S+V\\mathrm{d}p.", "a0b5c19189e49f2e887365801eb5cd3d": "S(A,P) \\geq \\pi(X) - \\pi(z) + 1, \\, ", "a0b5e8edaf3a9ed09cdb6d8f63066158": "\\vec E = {1 \\over \\rho}\\, \\vec j ", "a0b5f6038e6a4e54d527953f0a97abd5": "\\dot \\epsilon_{-\\infty} = -\\frac 1 {\\lambda}", "a0b60fcf8760097de1d8fae4516655f7": "\\mathbf{D}\\cdot\\mathrm{d}\\mathbf{S} = \\iiint_\\Omega \\rho_\\mathrm{f} \\,\\mathrm{d}V", "a0b647a4934d31f0bbe5e66089d54d25": "n=1,2,...\\infty", "a0b64d0acc6cf8d03d7190c23a697083": "f'(x) = \\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}.", "a0b6be9724d263f7e6109bdc56d73f05": "E\\in X", "a0b6c5d9a7a762d2575bf49b3425c343": "I = I_0\\,\\exp(-m(\\tau_a+\\tau_g+\\tau_{\\rm NO_2}+\\tau_w+\\tau_{\\rm O_3}+\\tau_r)),", "a0b6ee2fc09957911fe7bc391054edbf": "\\scriptstyle\\tau_l\\,=\\,4.7\\times 10^{-6}", "a0b7c9933d767be06644db47123ab630": "G = \\sqrt[n]{a_1 \\cdot a_2 \\cdots a_n} ", "a0b7cced762a8d12808d43ba3e77cca4": " \\Sigma_{t=1}^n \\kappa_t \\exp ( \\lambda_t x ) \\bold c_t ", "a0b7fb9c95948bdbfcb799ad35be7132": "\\sum_j \\frac{(\\sum_i y_{ij})^2}{I_j} - \\frac{(\\sum_j \\sum_i y_{ij})^2}{I}", "a0b84d4b19fbc6ca188b5c202d015122": "B/\\mathfrak{m}_A B", "a0b8702de701ac1063471a86e27bff7b": "v/c \\ll 1", "a0b88d264d38999b71b809a6bfce10f8": "\\begin{align}\n \\mathrm{MIPS}\n &= \\frac {\\mathrm{computational\\ complexity}}{T_\\mathrm{symbol}} \\times 1.3 \\times 10^{-6} \\\\\n &= \\frac{147\\;456 \\times 2}{896 \\times 10^{-6}} \\times 1.3 \\times 10^{-6} \\\\\n &= 428\n\\end{align}", "a0b88f51f67246b3697611a845422b2c": "H = {1 \\over 2} (\\kappa_1 + \\kappa_2).", "a0b8c75d26853b4b5a3445bc0fbfcf12": "dE^{\\prime}", "a0b8de5ecd42c5ef53645fa6a757ce91": "\\log (V_{out}/V_{in}) = \\log (\\prod_{i=1}^N Ratio_i) = \\sum_{i=1}^N \\log (Ratio_i) = a \\cdot (CodeValue + b )", "a0b93f329abf3d72429cedde4c6f8956": "v_1 = G x_2 \\quad v_2 = K G x_1", "a0b954a61698232a8a57514cd1fa51df": "f_0 \\,", "a0b97352d81940de4bc895d72b5d6731": "Q_{i+\\Delta}", "a0b9f78c4e616a893ad182070a9f805b": " R_\\mathrm{in} = \\frac{v_{S}}{i_{S}}", "a0ba4ef31aa144b92add8703d352db69": "\\operatorname{ip}\\langle y,x\\rangle = x-y", "a0ba59ee0f709ffd0a0d615cccbda765": " R \\oplus \\! R ", "a0ba64d80055f8060f3fb113edf6420c": "\n\\left[ {\\begin{array}{*{20}c}\n 2 & 9 \\\\\n \\bullet & \\bullet\n\\end{array}} \\right] \\to \\left[ {\\begin{array}{*{20}c}\n 2 & 9 \\\\\n 11 & 13\n\\end{array}} \\right]\n", "a0baf36ffd77a27a353b0068b1ef7862": "x>0;\\ 0 ", "a0baf3cbd1e4d01f9c161d6cc4f253f3": "(A \\and B) \\or C", "a0bb001db593da45d3f92caaeeccfa1a": "t = \\frac {Q_P} {I^k}", "a0bb15b91cd4d447fe4dffb21a1e9e91": "3a_{n+2} = 4a_{n+1} - 8a_n", "a0bb6331a74d87614ba38694b0cac00c": "g = 20\\log_{10}\\left|S_{21}\\right|\\,", "a0bb9c43a8223592352988b8a0c424e5": "P(q)", "a0bbb6a48b7117d582b6f38cd29880b0": "(D/N) = 1", "a0bbc458eb7be3ac5a9c4ec9fb6ebfa4": "T-\\lambda I: D \\to X", "a0bc471acb1933ecc3370cbd2bdd9e10": "\\,_2F_2()", "a0bc6f5e581a186c04adbbef89e1de6b": "\\neg P\\or Q", "a0bc93b49ca211e83b04ebb9fa127c39": " \\|M \\varphi\\|_{L^2(\\mathbf{T})}^2 \\le C^2 \\, \\int_0^{2\\pi} \\varphi(\\mathrm{e}^{\\mathrm{i} \\theta})^2 \\, \\mathrm{d}\\theta.", "a0bce4484a0925294722c22594c1b16e": "\\omega_{pe} = (4\\pi n_ee^2/m_e)^{1/2}", "a0bdde62311fb4c4224be53380db4a99": "L \\ = \\frac{A_s}{P}", "a0be46686195d34274c410fa35a32fdc": "X \\sim \\mathrm{Exponential}(\\lambda)", "a0be52dc81b5b40482f12b17a67b71e5": "\n\\begin{align}\ns_0& = x_0\\\\\ns_{t}& = \\alpha \\frac{x_{t}}{c_{t-L}} + (1-\\alpha)(s_{t-1} + b_{t-1})\\\\\nb_{t}& = \\beta (s_t - s_{t-1}) + (1-\\beta)b_{t-1}\\\\\nc_{t}& = \\gamma \\frac{x_{t}}{s_{t}}+(1-\\gamma)c_{t-L}\\\\\nF_{t+m}& = (s_t + mb_t)c_{t-L+(m\\mod L)},\n\\end{align}\n", "a0be5814a9e5c57e6bb8ee646da4dfbe": "G(\\mathcal K)/G(\\mathcal O)", "a0be8d39b33ebaf5541ae50969513462": "\\alpha:", "a0befa49f72a728b29d3dee61a60e0b5": "\\mathbf{X}(t)", "a0bf07c16f062e7dbf08dc0f1feefb4d": "\\Gamma^\\nu_{\\sigma\\mu}", "a0bf22094da5d1e9e28cd699c29ed85c": "BS \\to K(S)", "a0bf96e312a9264d0455b2af2ffe3ea0": "Y=F_X(X) \\,,", "a0bf993c398b5371258f03f71616af4d": "c\\in \\mathsf C", "a0bf9b0ebbcb00f90605e1b52c72cc46": "[ E_{ij}, E_{kl}] = \\delta_{jk}E_{il}- \\delta_{il}E_{kj}", "a0bfacb972886a5a417e590e3e5f29c0": "\\partial(uv) = u \\,\\partial v + v\\, \\partial u", "a0bfb59c5f34fb0667478d1d4687b9eb": "\\operatorname{sgn}(\\hat{x}_1-x_1)", "a0c0878e9f995d6859794dc46c818073": "O(1 / \\epsilon)", "a0c0ff3ecab2af4ce554fa5ae658f222": "g_{j} \\in L^{d - 1} (\\mathbb{R}^{d -1}).", "a0c12ca444425c53dda8d8f00b2b8d4c": "S_{Theil}", "a0c16c01ff56467aa3fa0e64769b11fb": "r := r + 1", "a0c18fc75b335653f1f20b09531591a2": "F_{t,T} = S_t e^{r(T-t)}", "a0c20dfd9aaa01e8dd9e239d62531ca7": "R_3(x)=\\frac{x^3-9x^2+9x-1}{(x+1)^3}\\,", "a0c20fa294d2d4ebc056f8a1bd9ec03d": "\\mathcal{D}^{\\mu \\nu} \\, = \\, \\frac{1}{\\mu_{0}} \\, g^{\\mu \\alpha} \\, F_{\\alpha \\beta} \\, g^{\\beta \\nu} \\, \\sqrt{-g} \\,", "a0c220c377721b8ad2c97fcfd5a5af8e": "\n\\begin{align}\n& \\int_{\\theta_j} P(\\theta_j;\\alpha) \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) \\, d\\theta_j \n = & \\int_{\\theta_j} \\frac{\\Gamma\\bigl(\\sum_{i=1}^K \\alpha_i \\bigr)}{\\prod_{i=1}^K \\Gamma(\\alpha_i)} \\prod_{i=1}^K \\theta_{j,i}^{\\alpha_i - 1} \\prod_{t=1}^N P(Z_{j,t}|\\theta_j) \\, d\\theta_j.\n\\end{align}\n", "a0c23ae33e972dd9ef131816692e19f8": " p(X, A|\\theta) ", "a0c293c9b421fb3a589736afeb36c52b": "x_o = x_t + \\sqrt{ r_n^2 - y_t^2} ", "a0c30acc1bfe9e58f3c7d975d3535bbf": "\\rho_{\\infty} (x) = \\left( \\frac{\\beta \\kappa}{2 \\pi} \\right)^{\\frac{n}{2}} \\exp \\left( - \\frac{\\beta \\kappa | x - m |^{2}}{2} \\right)", "a0c36a73645fce765dec921d0d4d8fb7": "\\max\\beta^{t}\\sum_{t=0}^{\\infty}Eu(c_{t})", "a0c39d8af532cf6016748240752c3cb4": "\\overline{A} = \\overline{U}", "a0c45aefa94332eb8150ed8280736f82": "\n\\begin{align}\n& {}\\quad (x_1^2+x_2^2+x_3^2+x_4^2+\\cdots+x_{16}^2)(y_1^2+y_2^2+y_3^2+y_4^2+\\cdots+y_{16}^2) \\\\[8pt]\n& = z_1^2+z_2^2+z_3^2+z_4^2+\\cdots+z_{16}^2\n\\end{align}\n", "a0c4c2ce7f9c78efeedd2bfb53ab9f3e": "\\neg ", "a0c4c5d72baedf23e457419c9ee0e105": "\n\\begin{align}\nH(z) & = \\frac{\\sum_{i=0}^P b_{i} z^{-i}}{1+\\sum_{j=1}^Q a_{j} z^{-j}}\n\\end{align}\n", "a0c552aa83b2a582215638bd83b7ee1b": "S=s_1,s_2,...s_n$", "a0c5bae1896b038a045429cad31199f1": " \\| \\mathbf{K} \\|^2 = \\frac{1}{\\hbar^2} \\| \\mathbf{P} \\|^2 = \\left(\\frac{mc}{\\hbar}\\right)^2 \\,,", "a0c5e24867f632f8734943972415a909": "\\left(\\sqrt{1/15},\\ \\sqrt{1/10},\\ -\\sqrt{3/2},\\ 0,\\ 0\\right)", "a0c6bc49fee37f5115ee7c7d36f468ef": "(Z_1,Z_2,\\ldots,Z_{n+1}) \\equiv \n(\\lambda Z_1,\\lambda Z_2, \\ldots,\\lambda Z_{n+1});\n\\quad \\lambda\\in \\mathbb{C},\\qquad \\lambda \\neq 0.", "a0c6ec4b2dfe1c3e878d5384f71fe0cd": "\n\\sigma_{j} = p + \\sum_{p=1}^N \\mu_{p} \\lambda_{j}^{\\alpha_p}\n", "a0c717136e5c0beb3620b764981b7f4a": "Fx_1...x_n \\rightarrow \\exist Fx_2...x_n.", "a0c72d8417f1e080f0ebeea9ef0824f3": "Q\\subseteq X^\\star", "a0c73e4391463cf68965738877fa6606": "\\mathcal{O}_X^n|_U \\to \\mathcal{M}|_U", "a0c7c2eb3dba249e9aa5aaa3ee1ff3db": "|Z_W| \\approx 60 \\frac {\\lambda}r", "a0c7da460e146a0edb186328a173f8b3": "dc/dl", "a0c7fab56f54eb24f4f72a765610da62": "R=g^{\\mu\\nu}R_{\\mu\\nu}\\,", "a0c831792c0dd1492e408d100b764738": " \\nabla \\cdot \\nabla v + \\lambda v = 0, \\,", "a0c8abcaf75eb60160099a08f16382ff": "D\\phi = \\partial \\phi - i A^k t_k \\phi", "a0c8fc224583f80834b21074926490e4": "m_{\\alpha}\\rightarrow \\infty", "a0c92cf23ccf7a37f32fa96136cce203": " y= u^2 \\,", "a0c947eb75a318ea6b661e0256a58e69": "\\scriptstyle TM \\cong M\\times {\\mathbb R^4}", "a0c9769125a5fa17efce6fb920cab875": "A_0:=\\alpha A", "a0c98097d4f70cf445a0b447a5ddeb28": "f_{t,a,b}(z)=e^{2\\pi it}z^3\\,\\frac{1-\\overline{a}z}{z-a}\\,\\frac{1-\\overline{b}z}{z-b}", "a0c986eb7b9f52b83c015ab196d78af0": "\\mathrm{high} = \\ln(2) \\cdot (R_1 + R_2) \\cdot C", "a0ca08c30211783b5365e58ba6d7d7a7": "f_{os} = 203 250 000 - \\frac{15626\\cdot 8}{12} \\approx 203239583", "a0ca3d27c1fe308ff0de282aed975375": "E_{1}u'(c_{2})=\\left(\\frac{1 + \\delta}{1 + r}\\right) u'(c_{1})", "a0ca5498db004d7a638c231956adf37e": " \\lambda_2", "a0ca719d12ef4f5102faeef0d0bf4e77": "k P := P + P + \\cdots + P ", "a0ca965bd71d1de765d4210e9a35bd73": "x_{min}", "a0cb21307cc82bc173814bd8bd1ba971": "F = \\frac{\\text{between-group variability}}{\\text{within-group variability}}.", "a0cb2d68d126f20e5fb34e86379120f6": "0 \\le y", "a0cb36d8069d5fdf0ef1a9cf162c969c": " P(N) = \\left\\{\\begin{matrix}\n\\frac{1}{N \\ln(\\Omega) }, \\;\\; N \\le \\Omega \\\\\n0, \\;\\; N > \\Omega \\end{matrix}\\right. ", "a0cb5cb03148a25e26c7b0972779d868": "n^{1.1}", "a0cb6eb61e00e9dbe2386d694d5dca64": " \\check{g} (x) = \\int_{\\widehat{G}} g(\\chi) \\chi(x)\\;d\\nu(\\chi),", "a0cb97b3899c4cb1a9dcd482722a22c9": "U:A\\to\\bold{Set}", "a0cc3b2a78541634c5da75e817319cc6": "\n\\gamma_2=\\frac{C''\\left(F_{\\text{mid}}\\right)}{C\\left(F_{\\text{mid}}\\right)}\n=-\\frac{\\beta\\left(1-\\beta\\right)}{F_{\\text{mid}}^2}\\;.\n", "a0cc48bf3a149f6b3b2227ceaa36c1d1": "\\sum_{k=0}^\\infty {2k \\choose k} z^k = \\frac{1}{\\sqrt{1-4z}}, |z|<\\frac{1}{4}", "a0cc84d1ed01fa44532f29e716253222": "u^2-vw^2=0", "a0cc98d47018671858e70119263c934b": "\\mu_i=\\left(\\frac{\\partial G}{\\partial N_i}\\right)_{T,P,N_{j\\neq i}}", "a0ccb5ed887b10cd4c3230ad7fe3e995": "O(n^2log(n))", "a0ccba96f26737a5c91ac68165a7eed9": "\\omega\\ .", "a0cd4869bdc4c5821744ea3800dfc8e1": "\\sum_{k=0}^{\\infty}\\frac{(-1)^{kq}}{(k+(j/p))^{2m}}=\\sum_{k=0}^{\\infty}\\frac{(-1)^{(2k)q}}{((2k)+(j/p))^{2m}}+ \\sum_{k=0}^{\\infty}\\frac{(-1)^{(2k+1)q}}{((2k+1)+(j/p))^{2m}}=", "a0cd8d73576ea8a973989e16220a4e1f": "\n D\\nabla^2\\nabla^2 w = - 2\\rho h\\ddot{w} \n", "a0cdaddad5a4a38447352b29895016fc": "U^{IJ} = - {1 \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} T^{KL}", "a0cde2271f8fd6e3562faaeab4c88f43": "T = \\sum_{i=1}^n(\\frac{1}{2}M\\mathbf{V}_i\\cdot\\mathbf{V}_i + \\frac{1}{2}\\vec{\\omega}_i\\cdot [I_R]\\vec{\\omega}_i),", "a0cde502bf1e6062dd048c35d4daacd8": "\\mathbb{Q} \\times \\Gamma", "a0ce68944b0fb9187db3a16eed550dd1": " \\begin{align}\n y_2 &= y_1 + \\tfrac32 hf(t_1, y_1) - \\tfrac12 hf(t_0, y_0) = 1.5 + \\tfrac32\\cdot\\tfrac12\\cdot1.5 - \\tfrac12\\cdot\\tfrac12\\cdot1 = 2.375, \\\\\n y_3 &= y_2 + \\tfrac32 hf(t_2, y_2) - \\tfrac12 hf(t_1, y_1) = 2.375 + \\tfrac32\\cdot\\tfrac12\\cdot2.375 - \\tfrac12\\cdot\\tfrac12\\cdot1.5 = 3.7812, \\\\\n y_4 &= y_3 + \\tfrac32 hf(t_3, y_3) - \\tfrac12 hf(t_2, y_2) = 3.7812 + \\tfrac32\\cdot\\tfrac12\\cdot3.7812 - \\tfrac12\\cdot\\tfrac12\\cdot2.375 = 6.0234.\n\\end{align} ", "a0ce921a4fc57968e564e7ff41b0a02e": "\\gamma=\\det(\\gamma_{ij})", "a0cea068051fc230627f01a9740bd075": "r_{k-1}", "a0ceacdcefe0ef0948bf5b2979743ad0": "\\eta^{IJ}", "a0cf80d8f2d050fdd723bf940e677bb7": "3 \\times 2\\tfrac{3}{4} = 3 \\times \\left (\\tfrac{8}{4} + \\tfrac{3}{4} \\right ) = 3 \\times \\tfrac{11}{4} = \\tfrac{33}{4} = 8\\tfrac{1}{4}", "a0cf84a79ee480d5d901a16372872f37": "\\mathfrak{p} \\cap \\mathfrak{g}", "a0cf888a16625d0d48d52b0387901475": " n \\geq 1. ", "a0cf901dcd86dba03e24e0d68963640e": "\\sum_i p_i = 1", "a0cf92e927a5710a9feda46015846ae5": "\\mathbf{P}(t+\\Delta t) = \\mathbf{P}(t) + Q\\mathbf{P}(t)\\Delta t ", "a0cf944e2b694b84e999c7e6c9726198": "\\mathfrak{so}(n, {\\mathbb R})", "a0cfbd1460a32ececdac9fbb632daaa0": "[c] \\in H_p(M)", "a0cfd16b5f1dd53c82fef8760d8545f3": "R \\times M \\to M", "a0cffec899eb034ccadec8468650c7ae": "\\gamma_{2},\\gamma_{3}\\,", "a0d058d663398eccd2b4724253c84d8e": "\\sigma_y^2", "a0d0d432dab07f6f48d729d730b2a497": "{\\beta }", "a0d0e6fd5aa282e3f4b0eaeb0cc69b93": "\\min_x \\frac{1}{2}\\|y-Ax\\|^2_2+\\lambda\\|x\\|_1.", "a0d10a6f0cf2cfadc28383ed7a289873": "\\frac{1}{{z+n \\choose n}}= \\sum_{i=1}^n (-1)^{i-1} {n \\choose i} \\frac{i}{z+i}.", "a0d15e04279db1f5b1fdb00eb1f68354": "Q[\\mathcal{L}(x)]\\approx\\partial_\\mu f^\\mu(x)", "a0d1627e211f1d56b0271aa7acc4b666": "|a(j)|+|b(j)| \\leq (Cj)^K; \\quad |z| \\ < \\ 1; \\quad K", "a0d16c149e752aa625b31280ade7f2f8": " \\frac{du}{dr} = -\\frac{1}{4\\pi r^2},", "a0d1b0632e3b5e85e9a65d8fd0cc4648": "a_0 + a_1x^1 + a_2 x^2 + \\cdots =\\frac{b_0 + b_1x^1 + b_2 x^2 + \\cdots + b_{d-1} x^{d-1}}{1- c_1x^1 - c_2 x^2 - \\cdots - c_dx^d},", "a0d1c208916283bd7533b9a9414121fe": " B^*.", "a0d1c867797f87fa2e33361d3229071b": "\\dot S_{ik} ", "a0d1f9692315675117b1fedb237f1dbb": "v^b(1) = \\frac{10}{1.1} \\log\\left(\\frac{3}{1+2 \\exp(-1)}\\right) \\approx 4.97", "a0d212c7b2b0ac4bf75838539c352ddd": "P \\equiv_{b} Q \\mbox{and } \\sigma \\equiv_{b} \\tau \\mbox{implies } \\sigma(P)\n\\equiv_{b}\\tau(Q)", "a0d265e94d714345d61bd23058a1ce61": "J_x^2,J_y^2,J_z^2", "a0d2709bd2437971f5934d87d0197f71": "\\csc^2\\left(\\frac{A}{2}\\right) : \\csc^2 \\left(\\frac{B}{2}\\right) : \\csc^2\\left(\\frac{C}{2}\\right)", "a0d288b2b8e7682f328d8b710f6c134e": "\nh = s^2 - r^2, \\,\n", "a0d2b084a4e02f0b75c45d0c87d76954": "v_{x_{K}}", "a0d30211d9901ad6fa181024ed94869d": "FIP=\\frac{13HR + 3BB - 2K}{IP}", "a0d31271a612ddde24e4baed7b3f632f": " u(1)=u_n ", "a0d3397e47b440d308649da7b4c275e4": "\\delta_\\epsilon S=\\int_{M^{d+1}} d\\Omega^{(d)}(\\epsilon).", "a0d379b80a94b667c0e92fe9d8c8bca9": "S^1 \\to G", "a0d4d4d57ebb60eaaa76962d3c4b4e71": "\n\\mathcal{A}_n(\\mathbf R)=i\\langle n(\\mathbf R)|\\nabla_{\\mathbf R}|n(\\mathbf R)\\rangle.\n", "a0d4fd1d0e1c73f88f6e6b3569da2364": " n = p q", "a0d561bed379ab6a0cc404df5500a4f7": " \\frac{a_{1}}{a_{2}} = \\frac{m_{1}}{m_{2}} ", "a0d5ea8a5066f282a3796ae79da3c4c4": "\\ln(c_n) \\sim 4\\pi \\sqrt{n} + O(\\ln(n))", "a0d6006573c4a05a7be42eac54e9fb93": "W=U\\sqrt{1+2\\lambda \\cos \\theta +\\lambda ^{2}}", "a0d62234f8ff0a037dc920b1ef79b361": "B_\\lambda(T) \\approx \\frac{2 h c^2}{\\lambda^5}\\,e^{-\\frac{hc}{\\lambda k_\\mathrm{B} T}}.", "a0d633b24e662bcd558dfd8e840fd117": " \\begin{cases}\n y_i \\left[ {w^T \\phi (x_i ) + b} \\right] \\ge 1 - \\xi _i , & i = 1, \\ldots ,N ,\\\\\n \\xi _i \\ge 0, & i = 1, \\ldots ,N .\n\\end{cases}", "a0d63f94077a6d7451f57fd232d692db": "\nP(r,t\\mid r_0 ) \\sim \\frac{1}{c_N} e^{-R_d^2/\\sqrt{2\\tau}},\n", "a0d664fdd9965ace52f10dd8d03aea2d": "u(t)", "a0d6b805ac9930896c2f8c3245daadbe": "S^1 \\times S^1", "a0d6c872c54bbe38b0225b33f32c8d09": "\\|\\nabla \\mathbf u\\| \\ll 1 \\,\\!", "a0d6dce53ae1c4bd3f512a3b0645ee66": "\\scriptstyle \\mathbf{\\hat{e}}_r, \\mathbf{\\hat{e}}_\\theta, \\mathbf{\\hat{e}}_\\phi \\,\\!", "a0d75375aea0d401f1f6cc1cab1fc50a": " |\\alpha(x,t)| = |\\alpha(x-\\omega'_0 t, 0)|, \\,", "a0d7f7e42a853f5ea8716f5a325fef59": "\\Bbb{C}", "a0d809425c78c0815d737b41873ab5d6": " \\mathbf{j} = \\mathrm{d} \\mathbf{a}/\\mathrm{d} t = \\mathrm{d}^3 \\mathbf{r}/\\mathrm{d} t^3 \\,\\!", "a0d819a46d3d2822c147b077585f0d04": "( I+wv^T )^{-1}=I-\\frac{wv^T}{1+v^Tw}", "a0d82758f8171fa0c6c71d3bd529b70c": "\\Gamma(2) = 1,\\,", "a0d86cf48a870f9727c0472d45c29c1e": "\\varepsilon = -{\\mu \\over{2a}} > -{\\mu \\over{R}}\\,\\!", "a0d8e27f351695f4217ea299910cd4ec": "\\operatorname{PI}(\\bold{H}) = n^{-1} |\\bold{H}|^{-1/2} R(K) + \\tfrac{1}{4} m_2(K)^2 \n(\\operatorname{vec}^T \\bold{H}) \\hat{\\bold{\\Psi}}_4 (\\bold{G}) (\\operatorname{vec} \\, \\bold{H})", "a0d8eb784edf0a22430d9a35306d40fe": "\\,K_1, K_2, K_3", "a0d921c0668adf32bbc7ad90603ba269": " \\dot{x_i} = \\sum_{j=1}^{n}{x_j f_j(x) Q_{ji}} - \\phi(x)x_i, ", "a0d932caf261ca9611ff6706d03c88bf": "LQ = \\frac{e_i/e}{E_i/E}", "a0d940426bc7836e7c774d9ee6a123c3": "u^0_{2,21}=u^0_{2,12}", "a0d972f269b8d75c41efa19e8af37a07": "s_\\nu := \\left\\{ \\begin{array}{rl} a & \\ \\text{if } z_\\nu \\text{ is a zero of order }a \\\\\n -a & \\ \\text{if } z_\\nu \\text{ is a pole of order }a. \\end{array} \\right. ", "a0d993a1df5ca87923503ec89934ab71": "\\forall (c,d)\\in P", "a0d9a1872978ec11e979507c7559e71c": "R=", "a0d9ef35fa67e64807ebda2d11a1fcba": "\\mu_d", "a0da1263e87b37697d78162bc8e7991f": "W_{t-1}", "a0da1a7adbf7a4a662b6d1d2ef79bd19": " \\sqrt{n\\log\\log n} ", "a0da23d29abbb0f4d60daa1739bf58d3": "u > t", "a0daa925b6402d372a2ffbd70c8e4bd1": "\\Delta(\\Delta G)", "a0dac30689be6a3d2dd0735a7cee99e9": "\\lambda^{(0)}_n", "a0db164fc91b3402ee11a2b194e46f59": "T_w(x) = \\begin{cases}\nx + w T_w(2x) & \\text{if }0\\leq x\\leq 1/2 \\\\\n(1-x) + w T_w(2x-1) & \\text{if }1/2 < x\\leq 1.\n\\end{cases}", "a0db9fe494e65b9b406fb85f67ad794a": "\\vec{e}_j", "a0dba23ae69fe416d83be30709dc9877": " e^+e^- \\to \\gamma \\gamma \\gamma \\gamma ", "a0dbcb91d511a45d2e3644d0d388fffb": "G(v,w;\\lambda)=\\left\\langle\\delta_v\\left| \\frac{1}{H-\\lambda}\\right| \\delta_w\\right\\rangle ", "a0dbedcd77f65402707474963f54581c": "\n\\psi _{\\mu }(\\tau )\\simeq \\nu \\tau ^{\\mu -1},\\qquad \\tau \\rightarrow 0.\n", "a0dbfd76f3e18de90b1e2549bdf7514a": "h(d)", "a0dc4279c28df4cb082890495347a87d": "\\mathrm{2 \\ NaAlSi_3O_8 + 2 \\ H_2CO_3 + 9 \\ H_2O \\longrightarrow 2 \\ Na^{+} + 2 \\ HCO_3^{-} + 4 \\ H_4SiO_4 + Al_2Si_2O_5(OH)_4}", "a0dc55049df59ea0399318d613d66fe7": "\n\\frac{1}{2}\\log\\left(1+\\frac{1}{n}\\sum_{i=1}^{n}\\frac{P_i}{N}\\right) \\leq\n\\frac{1}{2}\\log\\left(1+\\frac{P}{N}\\right)\n\\,\\!", "a0dca2871461205eedd732b057e3c3e8": "{{P}_{V}}g", "a0dcdeb610aeeced00758e96d033277b": "S/k = \\ln\\Omega = \\ln{\\left(q+N^{\\prime}-1\\right)!\\over q! (N^{\\prime}-1)!}.", "a0dcec8e3f39996b65d2b67d60c1a279": "C^{(p)}_T(p,T)=\\left.\\frac{\\partial U}{\\partial p}\\right|_{(p,T)}\\,+\\,p\\left.\\frac{\\partial V}{\\partial p}\\right|_{(p,T)}\\ ", "a0dd0dc84332cb1db96753146f2283f5": "\nP\\approx P_0 \\left(1 + X + \\frac{1}{3}X^2\\right)\n", "a0dd0e1518cb5b9abacd41a52f8709e5": " F^pH^k(X; \\mathbf{C}) = \\text{Im}(\\mathbb{H}^k(Y, F^p\\Omega^{\\bullet}_Y(\\log D))\\rightarrow H^k(X; \\mathbf{C})) ", "a0dd1905efe30418730bc9e13224f4d9": "x^2-ny^2=1\\,", "a0dd44ead7f06473d7f88e2bd3c4bd3b": "\n E_z= \\frac{1}{j\\omega\\epsilon} k_{xy}^2 \\sin k_x x \\sin k_y y \\cos k_z z\n ", "a0ddd185cbdaa82fc3cd9ca884cf351e": "\\scriptstyle \\sqrt{8} \\ = \\ \\sqrt{4}\\sqrt{2} \\ = \\ 2\\sqrt{2}", "a0ddd695aa9abb8ec4c50c4674b638ee": "t^2 - A t^2 = 0", "a0de2403e4323fdc35f2bbb2bfecd58d": " f^{*} = p/a - q/b . \\! ", "a0de2f05988722bd563b2fe7fbdeede7": "u(x,t)= A \\sin (kx- \\omega t + \\phi) \\ , ", "a0de6f1f1ebb584b88cf552eac8f89a3": "L=\\frac{L_{0}}{\\gamma(v)}=L_{0}\\sqrt{1-v^{2}/c^{2}}", "a0de927f487fef9a30b3e6fa6bb2e426": "|0 \\rangle_A |0 \\rangle_B |A\\rangle_C \\rightarrow |0\\rangle_A |0\\rangle_B |A_0\\rangle_C", "a0deb5284f206511a6d8f535015a2ed4": "student \\geq good", "a0df32821dfa34c4d7209365a0c1b0ea": " \\hat{b} \\, \\hat{b}^\\dagger - : \\hat{b} \\, \\hat{b}^\\dagger : \\,= 1.", "a0df38430425b0d1a6652f221c462546": "p \\approx \\mathrm{exp} \\left( -\\frac{\\sum\\limits_{i=1}^{N} N_i I_{r,A,i}}{\\left( \\overline{\\xi} \\Sigma_p \\right)_{mod}} \\right)", "a0df4728c6717640e06228f996c43a5c": "d(\\mu_1,\\mu_2)=\\sup \\left \\lbrace \\int u(x) \\, \\mu_1(dx) - \\int u(x) \\, \\mu_2(dx) \\right \\rbrace", "a0df7758e193583d2954054c0e4577c1": "\ni = (n-1)\\ (n-3)\\ (n-5)\\ . . . 1\n", "a0dfb73e8f38b5851925a69bb9bec549": "dS_{t}^i = \\mu_i S_{t}^i\\,dt + \\sigma_i S_{t}^i\\,dW_{t}^i", "a0dffe54e7a9c30b40f9bbe026fed090": "S=\\int dt L=\\int dt \\left[\\frac{m}{2}(\\dot{x}^2+\\dot{y}^2+\\dot{z}^2)-mgz+\\frac{\\lambda}{2}(x^2+y^2+z^2-R^2)\\right]", "a0e0362807953745eddb1e92aeb968e0": "\\frac{dN_i(t)}{dt}=-\\lambda_i N_i(t) + \\lambda_{i-1}N_{i-1}(t)", "a0e04c19989f6b78e0b01324e5f2277b": "b \\cdot \\nabla F(a) = \\lim_{\\epsilon \\rightarrow 0}{\\frac{F(a + \\epsilon b) - F(a)}{\\epsilon}}", "a0e050172d5f71203bab792f3f609a01": " \\langle f, g \\rangle = \\int_A \\overline{f(x)}g(x)\\, dx ", "a0e0838abef63af353dd8be71831e763": "\\lambda=(2,1)", "a0e08c8f42f9a474b98133e913e8357d": " u(t,x,y,z) = \\frac{1}{4\\pi c} \\iiint \\varphi(\\xi,\\eta,\\zeta) \\frac{\\delta(r-ct)}{r} d\\xi\\,d\\eta\\,d\\zeta; \\,", "a0e0ada134a343a9d21214efbed798a0": " \\begin{align}\nW &{}=\\frac{5}{6}B+Y \\\\\nB &{}=\\frac{9}{20}D+Y \\\\\nD &{}=\\frac{13}{42}W+Y \\\\\nw &{}=\\frac{7}{12}(B+b) \\\\\nb &{}=\\frac{9}{20}(D+d) \\\\\nd &{}=\\frac{11}{30}(Y+y) \\\\\ny &{}=\\frac{13}{42}(W+w)\n\\end{align}", "a0e0b1f54e4481b395bab6a8892fe634": "U \\otimes U \\otimes ... \\otimes U", "a0e0c2d23b687faf38f5f0fd49a67e31": "\\, f_u \\, (u=1, 2, \\cdots, 20) ", "a0e0cf7f6c75653fa2f3e2e889cb09c6": "\\scriptstyle{\\ell^1_n}", "a0e0f12036cf82e43b63c9ed2d874419": "\\lambda_1 < \\lambda_2 < \\lambda_3 < \\cdots < \\lambda_n < \\cdots \\to \\infty;", "a0e1cee07d9e236853f8deab11fd40ba": "\\frac{f'(x)}{f(x)} = \\frac{g'(x)}{g(x)}-\\frac{h'(x)}{h(x)}", "a0e1e9d6b45f1d2b5e956a246313b7d1": " x^{(7)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.8122 \\\\\n -0.6650\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8122 \\\\\n -0.6650\n \\end{bmatrix}. ", "a0e20360c34870a3381a3ac63e3a0eaa": "\n A_{\\bold{x}} = \\begin{bmatrix}0&0&0\\\\0&0&-1\\\\0&1&0\\end{bmatrix} , \\quad\n A_{\\bold{y}} = \\begin{bmatrix}0&0&1\\\\0&0&0\\\\-1&0&0\\end{bmatrix} , \\quad\n A_{\\bold{z}} = \\begin{bmatrix}0&-1&0\\\\1&0&0\\\\0&0&0\\end{bmatrix} .\n", "a0e205ef4f2987d288749be4311de25d": "d(u, v) = 2 \\arcsin\\left(\\frac{\\delta(u,v)}{2}\\right).", "a0e22fa83e42acbc405d7147c6879dac": "f(t) = \\sin(t), \\quad t \\in \\mathbb{R}", "a0e24c575c5595c26dcdf4f6f0486c8b": "\\sigma_n = \\frac{F}{A}\\,\\!", "a0e2a6e7a5191abdd4bc1ff145b94634": "\\begin{align}\n\n\\dot x&=u,\\\\\n\\dot u&=\\lambda x,\\\\[0.3em]\n0&=x^2+y^2-L^2,\\\\\n0&=ux+vy,\\\\\n0&=u^2-gy+v^2+L^2\\,\\lambda.\n\\end{align}", "a0e2b8c023271e34b171b23b7c68f2c6": "\\displaystyle \\sqrt{\\frac{2}{\\pi}} \\cdot \\frac{a}{a^2 + \\omega^2} ", "a0e2df2762310419318917b295e1a2f3": "K_p=\\left[\\sum_{k=1}^\\infty -k^p \n\\log_2\\left( 1-\\frac{1}{(k+1)^2} \\right)\n\\right]^{1/p}.", "a0e3a187ff128b8c513d9ee3a2a79b3a": " D_N(P) = \\sup_{B\\in J}\n \\left| \\frac{A(B;P)}{N} - \\lambda_s(B) \\right|", "a0e3a78531fc541e6c6026c299d299ee": "(\\tfrac{\\mathrm{mol}}{ m^2\\cdot s})", "a0e3bed9c03dd53a617eadb2ad1dcbd6": " M^2-(a+d)M+(ad-cb)1_{2\\times 2} =0 ", "a0e3c7e1c4f94020227886b7211ecf21": "\\operatorname{coim} f", "a0e3f7246ff1eed9ff9d7637e3b01b0d": " \\mu \\rightarrow 1", "a0e40cec227aba5041d5dfb3fd3b827b": "y(k)", "a0e4af44c596a5f734f0fc1c26d85636": "57000\\sqrt{f'_c}", "a0e4c3cfef5050583eeab0d1f9f7fc58": "n\\left(x,0\\right) = n_0, x \\le 0 ", "a0e4e78e5eb3b77a1c16e1478a1429ed": " v:T(TM\\setminus 0)\\to T(TM\\setminus 0) \\quad ; \\quad v := \\tfrac{1}{2}\\big( I + \\mathcal L_H J \\big).", "a0e4fd986707c15d148d5e9031327bca": "\n\\|f\\|_{d}=\\left(\\sum_{u\\subseteq D}\n\\int_{[0,1]^{|u|}}\n\\left|\\frac{\\partial^{|u|}}{\\partial x_u}f(x_u,1)\\right|^2 dx_u\\right)^{1/2}.\n", "a0e504eeeef06e703bb7e638d28e7ed7": "\\bigoplus_{n>0}S_n.", "a0e54d8296d8dee5795e8c2ab7c39481": "\\hat {G}", "a0e58bc7117288702827ab07bcbbb8a5": "\\pi =(\\pi_1, \\ldots \\pi_N) ", "a0e59f2e0c11c01243feb2728ed29c8f": " [0, \\infty) ", "a0e5b9e0875f701e5cf997322973aa5b": "\\langle p,q\\,\\vert\\; pq=1\\rangle", "a0e5ea5ca725b1863a998cb5d81dc73f": " f(x; a,b) = a b x^{a-1}{ (1-x^a)}^{b-1}.", "a0e63a4c7cd989ee99f6f37052769201": "[x_1 : x_2] \\cdot [y_1 : y_2] = [x_1 y_1 : x_2 y_2],", "a0e663cf0cb02e164be6e72218264664": "w_L'=0\\,", "a0e684008a31302bdcf4b2a9e7fed7e1": "U_{CE}", "a0e6978ffc1dc7ff965069c2f7e183f2": "\\mu _{i}", "a0e69ea5c56e240a32950d15a96bc54c": "\\epsilon = \\ell/m", "a0e714e3722ab1e73ecf205ef2bedf80": " x^3 - x^2 - x - 1 = 0 ", "a0e7484de77afc6b270e8d2258d0267c": " \\Xi(\\beta, z, \\Lambda):=1+\\sum_{n\\ge 1}\\frac{z^n}{n!} \n\\int_{\\Lambda^n}\\!dx_1\\cdots dx_n\\; \n\\exp[-\\beta V_n(x_1, x_2,\\ldots x_n)]\n", "a0e814744262616d04e301f6474c57db": "\\int \\arccot{x} \\, dx = x \\arccot{x} + \\frac{1}{2} \\ln { \\vert 1 + x^2 \\vert } + C , \\text{ for all real } x ", "a0e851f8add79446cb70fc07515ade6a": "E(X_1\\cdots X_n)=\\sum_\\pi\\prod_{B\\in\\pi}\\kappa(X_i : i\\in B)", "a0e89ba8533d0284a46a12ed9d091134": "CLV = \\frac{(close_{1} - low_{1}) - (high_{1} - close_{1})}{(high_{1} - low_{1})}\\!\\,", "a0e8fc1303769b9a63896d0f3e548cc2": "\n\\operatorname{cov}_W(X,Y). \n", "a0e9748d1c6e840bf71a79a545cc0e39": "\\ M^{reg} ", "a0e9997cb947ba28da08ec69e35d4ed8": "\\begin{align}\na &=\\sqrt{m\\omega \\over 2\\hbar} \\left(\\hat x + {i \\over m \\omega} \\hat p \\right) \\\\\na^{\\dagger} &=\\sqrt{m \\omega \\over 2\\hbar} \\left(\\hat x - {i \\over m \\omega} \\hat p \\right)\n\\end{align}", "a0e9ff87285e3d613723945a0069b1cd": "y_{1:t}", "a0ea6e64fb69228311bacdf21e271673": "Q \\in \\mathbb{F}_q[X,Y_1,\\cdots,Y_s]", "a0ea720aeb720f90d86452f1a4eb3312": "\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.", "a0eaa0ef4353eb57ffe0b04a613c66d4": " \\kappa^{F(n)}_r-\\gamma_r = \\frac{\\kappa_r}{\\sigma^rn^{r/2-1}} = \\frac{\\lambda_r}{n^{r/2-1}}; \\quad r\\geq 3\\,.", "a0ead68d04983d9515cf92ae2830843d": "\\beta\\alpha:f'f\\Rightarrow g'g:A\\to C", "a0eb0efa8fbda14617a5db7428ce9761": "f'(x) = e^x / (e^x+1) = 1 / (1 + e^{-x})", "a0eb3860f42820fc82e8d5a110f21f18": "\\int_0^\\infty A(x) e^{-xs}\\,dx", "a0eb7dcdbfd3ea444904c18a68188ff2": "\\scriptstyle M.", "a0ebb373113c3574907473a88181b3a9": "\\mu=\\frac{\\sum_{v\\in V} d(v)}{|V|}=\\frac{2|E|}{|V|}.", "a0ebf94ff3ee3ceaecc2997324e54efd": "\\mathbf{L}_\\alpha(x)", "a0ec14273b34d0dd5a64ddfdbcb6295c": " \\det \\mathcal O := e^{-\\zeta'_{\\mathcal O}(0)} \\;. ", "a0ec6b7d0f31ee5cac65e500682633ad": "G(x)=xx^*H^*(C_w+Hxx^*H^*)^{-1}", "a0eca626ef5a56af1df2b5a5c45017d3": "f:P\\to Q", "a0eca9ffc534237e68fbaa6af5c6a13f": "u_2 = \\begin{bmatrix}{\\ }1\\\\ +\\mathbf{i}\\end{bmatrix}", "a0ed31152ca3da305d11451016f2bcc8": "[x_m,\\infty)\\,", "a0ed84537d34c8226c89cbdaa013dfaf": "Q = \\left[ \\mathbf{e}_1, \\cdots, \\mathbf{e}_n\\right] \\qquad \\text{and} \\qquad\nR = \\begin{pmatrix} \n\\langle\\mathbf{e}_1,\\mathbf{a}_1\\rangle & \\langle\\mathbf{e}_1,\\mathbf{a}_2\\rangle & \\langle\\mathbf{e}_1,\\mathbf{a}_3\\rangle & \\ldots \\\\\n0 & \\langle\\mathbf{e}_2,\\mathbf{a}_2\\rangle & \\langle\\mathbf{e}_2,\\mathbf{a}_3\\rangle & \\ldots \\\\\n0 & 0 & \\langle\\mathbf{e}_3,\\mathbf{a}_3\\rangle & \\ldots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix}.", "a0edcceb515ce90f5fb91e090a7a5bac": " \\text{error} = \\int_a^b f(x)\\,dx - \\frac{b-a}{N} \\left[ {f(a) + f(b) \\over 2} + \\sum_{k=1}^{N-1} f \\left( a+k \\frac{b-a}{N} \\right) \\right]", "a0edd624d94ef996e3d0a72928cf6bd6": "f(x_0)\\ge f(x) \\, ", "a0ee4134e1cbb8ca7688ee4d250d1328": "\\phi(v_i) = \\tanh(v_i) ~~ \\textrm{and} ~~ \\phi(v_i) = (1+e^{-v_i})^{-1}", "a0eea1f68ade284d8a777ffbc0ba1bfb": "\\sum_{n=0}^{\\infty}\\frac{P(n)}{2^n} = \\frac{12}{5}.", "a0ef1a1dde876717842fe20ebc5da8f7": "\\left\\langle\\frac{F}{\\|F\\|},\\frac{T}{\\|T\\|}\\right\\rangle", "a0ef2c24e1dce00da0acb5a2816ab9ca": " E = 2 \\sin^{-1}(\\sqrt{ x }) ", "a0ef603f9046a9a940c031003c263710": "S^1\\times S^1.", "a0ef7341d871928ef5af790a5096306a": "p(p+1)/2", "a0ef9284df0db0b026d67a38e0d7b5bd": " b_n= m \\left( 1- n^{-\\tfrac{1}{\\alpha}}\\right) \\, ", "a0efae20328c082fa08afec0d18ecf9d": "\\nabla_\\beta A^\\alpha = \\frac{\\partial A^\\alpha}{\\partial x^\\beta} + \\Gamma^\\alpha{}_{\\gamma\\beta}A^\\gamma .", "a0f089342a147ced09a758d024115d5d": "\\hat{\\mathbf{e}}_1,\\hat{\\mathbf{e}}_2,\\hat{\\mathbf{e}}_3", "a0f0d4505020955e058b507c958837cb": "E\\pm = \\alpha + \\frac{1 \\pm \\sqrt{5} }{2} \\beta ", "a0f12b238d41a02d876dbb44b25dbf50": " y_{n+1} = y_n + \\sum_{i=1}^s b_i k_i, ", "a0f1507ce181e7367bb5496ba493d762": "\\mathrm{e}_i=\\partial_i", "a0f16ee2f158fc63db295230a54e7f71": " \\alpha_{k-1} ", "a0f1b6f02df7b7ff79aa20729cb1ba31": "\\vec F", "a0f1cf8b3d9d5bd302ca873d8c11ffbc": " U A U^* = B.", "a0f1eab467f4f9f2770ef8ad24305b32": "=2n\\sum_{i=1}^n r_is_i - 2 \\sum_{i=1}^n r_i \\sum_{j=1}^n s_j ", "a0f21088768323d42765f2369b1c1b5b": "Z_0 \\approx \\sqrt \\frac{L}{C}", "a0f221522b4a7d0f03d29631fb93d02d": "\\sqrt{N^2+d} = \\sum_{n=0}^\\infty \\frac{(-1)^{n}(2n)!d^n}{(1-2n)n!^2 4^nN^{2n-1}} = N + \\frac{d}{2N} - \\frac{d^2}{8N^3} + \\frac{d^3}{16N^5} - \\frac{5d^4}{128N^7} + \\cdots", "a0f2b0fbb9cc18a77a963adf7782a32e": "\\alpha(0) = 0", "a0f37c073298712302b883abc6ebc7fd": "\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} + g \\frac{\\partial h}{\\partial x} + g (S - S_f) = 0 ", "a0f39b66f7f1fb88df7d3e82ccb8eb7e": "4^7", "a0f427e772b942890660e89a00651d39": "|\\psi_k\\rangle", "a0f44edca03ef143933103b78bfbf709": "561 = 3 \\cdot 11 \\cdot 17\\,", "a0f4bc465acb3d43181a4660cf739b8d": "\\textstyle _5", "a0f4c2fe8f8a7da0a8031fab8a8e136b": "\\begin{matrix} \\frac{n - 1}{2} \\end{matrix}", "a0f5334af2a5be081ac89cc4246f3a9a": "\\begin{align}\n{\\bar{M}}^{\\alpha\\beta} & = {\\bar{X}}^\\alpha {\\bar{P}}^\\beta - {\\bar{X}}^\\beta {\\bar{P}}^\\alpha \\\\\n& = \\Lambda^\\alpha {}_\\gamma X^\\gamma \\Lambda^\\beta {}_\\delta P^\\delta - \\Lambda^\\beta {}_\\delta X^\\delta \\Lambda^\\alpha {}_\\gamma P^\\gamma \\\\\n& = \\Lambda^\\alpha {}_\\gamma \\Lambda^\\beta {}_\\delta \\left( X^\\gamma P^\\delta - X^\\delta P^\\gamma \\right) \\\\\n& = \\Lambda^\\alpha {}_\\gamma \\Lambda^\\beta {}_\\delta M^{\\gamma \\delta} \\\\\n\\end{align}", "a0f59c1a3a2397f9ccc372afedeb1d02": "\n\\Psi_x(x(T))=\\begin{bmatrix} \\frac{\\partial\n\\Psi(x)}{\\partial x_1}|_{x=x(T)} & \\cdots & \\frac{\\partial\n\\Psi(x)}{\\partial x_n} |_{x=x(T)}\n\\end{bmatrix}\n", "a0f5a912dad11cad4e1f78658401cd85": "|{-}x|_\\ast=|x|_\\ast=|x|^c_\\infty=|{-}x|^c_\\infty", "a0f5fafc0620390ae4e1c46accd43341": "\\delta_{ij} = 0", "a0f65be4e34dcbf196f871eb83256287": "\\frac{P(\\overline{Ba}|\\overline{H})}{P(Ra|\\overline{H})} \\ - \\ P(\\overline{Ba}|Ra\\overline{H}) \\ \\geq \\ P(Ba|Ra\\overline{H}) \\frac{P(\\overline{Ba}|H)}{P(Ra|H)}", "a0f67904d8e5741179401f103cc418af": "a_{n}, a_{n-1}, \\ldots , a_{1}, a_{0}", "a0f6e557ca12a0f0b9517d3f0eded56d": "\\psi_k(n) = n^k * \\mu^2(n)", "a0f706c7bb2a620fea851915f01c71e0": "\\lambda_k = R^n / n", "a0f7094f1fd57381b556557a09f92c5a": "\\Delta G^{\\mathrm{prot}}(\\mathrm{pH}) = \\mathrm{RT} \\ln10 ( \\mathrm{pH} - \\mathrm{p}K_{\\mathrm{a}}^{\\mathrm{HH}} ) ", "a0f7616d397474f7c701c1561adcdba1": "-1 < \\Re(s) < -\\frac12", "a0f7d8e44be9c48e59a6b0b0114d375c": "\\left| z\\cdot g_n(z) \\right|\\le C\\beta_n", "a0f803e7f3a62846324765f6e23b8a45": "V_Z-V_Y", "a0f82b960bc75a704c172627aa6844e2": " \\displaystyle{f(z)=R_1f R_0(z)}", "a0f86edfd2bd21ccd994807cca884ff4": "f_X(x;k,\\lambda)=\\frac{1}{2} e^{-(x+\\lambda)/2} \\left (\\frac{x}{\\lambda}\\right)^{k/4-1/2} I_{k/2-1}(\\sqrt{\\lambda x})", "a0f9168ff081d1e55fae3f19dab738ae": "0 < s < \\gamma n\\,", "a0f9b966c5a92141682e66c88515f77e": "G_1 = 1 \\Leftrightarrow L/K ", "a0f9e44f787d506f66a77ba095715138": "\\text{extract} \\circ \\text{duplicate} = \\text{id}", "a0f9ed899865e2a7c0669f46cb570792": "(L_\\mathrm{p})", "a0fa3b9b26f10f85407a7a47bbc82c0e": " S(T) \\! ", "a0fa481a96489fb7487b70ece5699c38": "\\lambda=\\frac{l}{l_0}", "a0fa5397e01f705b71b7717f67342eba": "c_f(u,v)=c(u,v)-f(u,v)", "a0fa6434e35ca97c22e3f59d2bbb6e5a": "(\\Psi)", "a0fa7344b67c0fc8e055de9087d957b9": "(k<0)", "a0fa8c5dee43f5359f4feef64c90a871": "\n\\frac{1}{\\sqrt{\\lambda}} = \\alpha - [ \\frac {\\alpha + 2\\log(\\frac{\\Beta}{Re})}{1 + \\frac{2.18}{\\Beta}}]\n", "a0fad49d2a565ba6abaa5587e7b89b57": " \\mu \\, ", "a0faf7b4c911b1fd4448c87db5067057": "3^4", "a0fafcff3e115a31b889268cbf760727": "\n\\begin{align}\nq & = 1 - p \\\\\n& = 1 - 0.954 \\\\\n& = 0.046\n\\end{align}\n", "a0fb15ba7d050969b24a454bce58ad0d": "Y = G + jB \\,\\!", "a0fb51cc25213155686d2149b444bb67": " x(t)=C e^{s t},\\,", "a0fb9141c753b209dfe7db577dc514ac": "\\mathcal{F}_T", "a0fbc9ddb8c077f1c4df78680a9a9ffb": "\\alpha_\\text{fw}=\\frac32.", "a0fbe08997b5d8e4a6a08715362822f6": "\n\\begin{array}{ccl}\n\\pi(\\tau)&=&P(\\sqrt{n}\\bar{D}/\\hat{\\sigma}_D > 1.64|\\tau) \\\\\n &=&P\\left(\\sqrt{n}(\\bar{D}-\\tau+\\tau)/\\hat{\\sigma}_D > 1.64\\right|\\tau)\\\\\n&=& P\\left(\\sqrt{n}(\\bar{D}-\\tau)/\\hat{\\sigma}_D > 1.64-\\sqrt{n}\\tau/\\hat{\\sigma}_D\\right|\\tau)\\\\\n\\end{array}\n", "a0fbe96b5f9a4c6a9a72b3ea4d7027f7": "k = 1.08 \\times 10^{10} \\cdot e^{-12,667/T}", "a0fc34cf66fc2f158b077875dde2fba8": " G_x(t,f)=\\int_{-\\infty}^\\infty e^{-\\pi(\\tau-t)^2}e^{-j2\\pi f\\tau}x(\\tau)\\,d\\tau ", "a0fc5ec8613036362d883dafb04723f6": "\\sum_{x\\in\\mathbb{F}^n}(1-f_1^{p-1}(x))\\cdot\\ldots\\cdot(1-f_r^{p-1}(x)) ", "a0fc681fabb5447c68c33fdfa6861409": "\\Delta E ", "a0fc7204ce8b771acd234759beda70c4": "J_a,~ a=1,2,3", "a0fc9acecafc8423c576ad9f49bf052a": " \\alpha_m \\leftarrow w_m \\cdot v_m \\, ", "a0fd0852d3dafbd9cb84b957f501e7e8": "\\frac{|z|^m}{|z-a|^n} = const.", "a0fd149cf7ccbc79a24e9a4830f2cbdf": "\\int\\cos^n ax\\;\\mathrm{d}x = \\frac{\\cos^{n-1} ax\\sin ax}{na} + \\frac{n-1}{n}\\int\\cos^{n-2} ax\\;\\mathrm{d}x \\qquad\\mbox{(for }n>0\\mbox{)}\\,\\!", "a0fd394bed1c21b2da802dec9299eae8": "\\mathbb{C} ", "a0fd520d74a8269ef19fbc9354c3b84e": "{n \\choose k+1} = \\left[(n-k) {n \\choose k}\\right] \\div (k+1) ", "a0fdb2112128c155553e223c5fbc7bed": "S(x)=\\sum_{i=0}^{d-2}\\sum_{j=1}^v e_j\\alpha^{(c+i)\\cdot i_j} x^i=\\sum_{j=1}^v e_j\\alpha^{c\\,i_j}\\sum_{i=0}^{d-2}(\\alpha^{i_j})^i x^i = \\sum_{j=1}^v e_j\\alpha^{c\\,i_j} {(x\\alpha^{i_j})^{d-1}-1\\over x\\alpha^{i_j}-1}.", "a0fdba021e2c7a526b2c30e6e5c0f8a9": "\\theta^i F_i=\\log p-C(v)+\\psi(\\theta)", "a0fdbbbaf81d2b168ad6afcb837715b0": "s \\mathbf{X}(s) = A \\mathbf{X}(s) + B \\mathbf{U}(s)", "a0fdd49feda973cf9ed2f668d3128019": "\\dot{v}=\\omega^2", "a0fe339bb7e91575506ef0669cc82567": "y = \\pm \\sqrt{3}", "a0fe3ce131e148928f48bd535c9c7560": "440 \\rm{ Hz}\\cdot (\\sqrt[12]{2})^{-24} = ", "a0fe59e14815632a61e049af42a839e3": "x*y = \\min(x,y)", "a0fe943e12cdfdae712a6077e8eaa5d0": "dG = \\frac{\\partial G}{\\partial x} dx + \\frac{\\partial G}{\\partial y} dy +\\frac{\\partial G}{\\partial u} du +\\frac{\\partial G}{\\partial v} dv = 0", "a0fe95fa429034ba5c2c4f61e15e8363": "\\langle \\cdot,\\cdot \\rangle_k", "a0fe9cadfc969daf9cb0eff189f57bab": "\\mathbf{y}_i^{\\rm T} = \\mathbf{x}_i^{\\rm T}\\mathbf{B} + \\boldsymbol\\epsilon_{i}^{\\rm T}.", "a0feec4701aebed6239bc68a18d911b5": "M_{A,B}", "a0fef0f04e53a5ff27e145e890cc2bff": "\\Delta \\Phi", "a0ff56495ff26d2779f4e9d428de85a8": "\n\\begin{bmatrix}\n 1 & 2 & 3 & 4 & 5 \\\\\n 2 & 3 & 5 & 1 & 4 \\\\\n 3 & 5 & 4 & 2 & 1 \\\\\n 4 & 1 & 2 & 5 & 3 \\\\\n 5 & 4 & 1 & 3 & 2 \n\\end{bmatrix}\n\\quad\n\\begin{bmatrix}\n 1 & 2 & 3 & 4 & 5 \\\\\n 2 & 4 & 1 & 5 & 3 \\\\\n 3 & 5 & 4 & 2 & 1 \\\\\n 4 & 1 & 5 & 3 & 2 \\\\\n 5 & 3 & 2 & 1 & 4\n\\end{bmatrix}\n", "a0ff95f10e22cc2536b3f39318297c1d": "\n\\frac{\\rm d}{{\\rm d}t}x(t)=f\\left(t,x(t),\\int_{-\\infty}^0x(t+\\tau)e^{\\lambda\\tau}\\,{\\rm d}\\tau\\right).\n", "a0ff98ed9b5288037b96af61e95a80a2": "S(\\alpha) = \\alpha \\cup \\{\\alpha\\}.", "a0ffb0c2df3ea1da365813c5736eb3cb": "P \\cup \\Delta \\models IC", "a0ffcb4df07c6a002f994a0b23135a9d": "x_{j} \\in X,\\, y_{j} \\in Y = \\{-1, +1\\}", "a0ffcd6d2e6e356d2d15476870a8861b": "e = [2; 1, \\textbf{2}, 1, 1, \\textbf{4}, 1, 1, \\textbf{6}, 1, 1, \\textbf{8}, 1, 1, \\ldots, \\textbf{2n}, 1, 1, \\ldots]. \\,", "a0ffeeec18cdbf4a7738fd291baae440": "b=1\\,\\!", "a1001bfb48aebaa6b616574546fb935e": "F = \\overline{F} \\circ Q", "a100652c548fcbfdf7296105fdfd245d": "v(i) = \\alpha v(i-1) + \\beta v(i-2)", "a100b40cfa24adc4385e82ccca578edc": "\n\\mathrm{A}^* + \\mathrm{Q} \\rightarrow \\mathrm{A} + \\mathrm{Q}^*\n", "a100dd38469b47e0b80403eb9929ac28": "\\mathrm{Im}(\\tilde{n}) = \\frac{c \\alpha_{abs}}{2\\omega}=\\frac{\\lambda_0 \\alpha_{abs}}{4\\pi}", "a100eb4e8f5e3e98152d66282cc39f31": " X \\,\\!", "a100edba96565efdeceb0ca088710093": "\n\\begin{align}\n&\\left\\langle x\\left|a \\right| 0 \\right\\rangle = 0~~~~~~~~~~\\Longrightarrow\\\\\n&\\left(x + \\frac{\\hbar}{m\\omega}\\frac{d}{dx}\\right)\\left\\langle x|0\\right\\rangle = 0~~~~~~\\Longrightarrow\\\\\n&\\left\\langle x|0\\right\\rangle = \\left(\\frac{m\\omega}{\\pi\\hbar}\\right)^{\\frac{1}{4}}\\exp\\left(-\\frac{m\\omega}{2\\hbar}x^{2}\\right)=\\psi_0 ~,\n\\end{align}\n", "a1014c7fd42f8392491125f5c986eb1f": "s^{k}(h( c_1 ))", "a1015024aac766e00623fdb23e00bd3b": " \\operatorname{var}(N) = \\frac{(M+1)(C+1)(M-R)(C-R)}{(R+1)(R+1)(R+2)}.", "a101774f4aa07a6048f1087e6ee17c98": " \\bar{H'} = \\frac{1}{r_{12}} - \\frac{Z}{r_1} -V(r_1) - \\frac{Z}{r_2} - V(r_2) ", "a10194d05cc3e6b4dca7ce7a5c1559d1": " \\frac{1}{\\|v\\| \\cdot \\| u \\|} ", "a101b2a5980cf0d3e57c8443f7b711d1": "\\{,\\}", "a101c38639ac660822f0b7eab0284893": " = (q^2;q^2)_\\infty\\,\\theta(-w^2q;q^2)", "a101cf5c69b80aa143a568b8c28afa1d": "{{z}_{O}}\\approx \\left( 2+{{g}_{m4}}{{r}_{O1}} \\right){{r}_{O4}}", "a101eeb22d27f352c5ac1cfbce182101": "\\begin{cases} u_{t}=ku_{xx} & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ u(x,0)=0 & IC \\\\ u(0,t)=h(t) & BC \\end{cases} ", "a10216b132906a679de34c859ad8d89d": "\\lambda=\\sqrt{xy}+\\sqrt{yz}+\\sqrt{zx}", "a1022f7b9654c6af8b45f65074faf8aa": "\\scriptstyle \\left(x,z\\right)", "a10257ca0fd59f595dbe6276e422b6d5": "\\nu_{t_{1} \\dots t_{k}} \\left( F_{1} \\times \\dots \\times F_{k} \\right) = \\mathbb{P} \\left( X_{t_{1}} \\in F_{1}, \\dots, X_{t_{k}} \\in F_{k} \\right)", "a102c0a64a779b9e16fd08090f91b806": "p(\\overline{z}) = 0", "a103bd8afda145bf324e43c38f6710f8": "R = \\frac{\\rho}{t} \\frac{L}{W} = R_s \\frac{L}{W}", "a10400fda4e780f507fe3749be834611": "X_1 \\to_{f_1} X_2 \\to_{f_2} X_3 \\to\\cdots", "a10408ea8026a8a402d8849cec226b10": "|f|^2", "a1040f656d593072f9dab6803f115d56": "x^i_n(p,m^i) = -\\frac{\\frac{\\partial v^i(p,m^i)}{\\partial p_n}}{\\frac{\\partial v^i(p,m^i)}{\\partial m^i}} = \\frac{\\partial f^i(p)}{\\partial p_n} + \\frac{\\partial g(p)}{\\partial p_n}\\cdot\\frac{m-f^i(p)}{g(p)}", "a104b0357d4533994aaf21e698155fad": "P(r,n)dr = 4 \\pi r^2\\left( \\frac{2 n b^2 \\pi}{3}\\right)^{-\\frac{3}{2}} \\exp \\left( \\frac{-3r^2}{2nb^2} \\right) dr \\,", "a104e4aa1542a634c894fcff3e3a574d": "D(f) \\leq Q_1(f)Q_2(f)^2", "a104ec7c7c15257d3934fa132274033d": "\\eta/s", "a1050cf335341e0b5059b6873b05deea": "\\mathbf{F}\\left(\\mathbf{r}\\right)=-\\frac{1}{4\\pi}\\left[-\\boldsymbol{\\nabla}\\left(\n-\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n+\\oint_{S}\\mathbf{\\hat{n}}'\\cdot\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'\n\\right)-\\boldsymbol{\\nabla}\\times\\left(\n\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\times\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\oint_{S}\\mathbf{\\hat{n}}'\\times\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'\n\\right)\\right]", "a105ec9947d6d3991ad28492b35ba254": "W=-\\int_{2V_0}^{V_0} P\\,\\mathrm{d}V =- \\int_{2V_0}^{V_0} \\frac{nRT}{V} \\mathrm{d}V=nRT\\ln 2=T\\Delta S_{gas}.", "a105f8cdf4b731d6d14f6689dbf19559": "Q = (u-v) \\cdot (u-v) - (u \\wedge v) \\cdot (u \\wedge v),\\,", "a106366dc9753e47dc045693b7dde187": "i_1 = \\frac{v_1 - v_2}{Z_1}", "a10677a9c8df76d48b104e95b6086e86": " dU = 0 ", "a1069489b3fa1a83cbf04393eeb683ce": "p_Z(z|a,b) = \\frac{ab}{\\pi^2(b^2z^2-a^2)} \\ln \\left(\\frac{b^2 z^2}{a^2}\\right).", "a1069b1ff911477815af12e9699559f6": "\\scriptstyle{f : [0,1] \\to \\mathbb{R}}", "a106d38b6807df4e0de200bb561293dd": " \\tilde{H}_{i}(\\operatorname{link}_\\Delta(\\sigma); k)=0\\quad \\textrm{for\\, all} \\quad\ni<\\operatorname{dim}\\, \\operatorname{link}_\\Delta(\\sigma). ", "a106d571357947ef0fd67dc0b8e4d185": "\\nu M \\equiv TM", "a106e11ac221c4bb171acc75a3d7b596": "\n\\pi_i^{\\mathbb{G}}(t) := \\mu_i(t) \\left( \nD_i \\log(\\mathbb{G}(\\mu(t))) + 1 \n- \\sum_{j=1}^n \\mu_j(t) D_j \\log(\\mathbb{G}(\\mu(t)))\n\\right)\n\\qquad \\text{ for } 1 \\leq i \\leq n\n", "a1075388ca948cb31beefff3fd0dd49d": "\\rho<\\alpha", "a1076bdf3f13d4ac09263e0a78e37f98": "\\begin{cases}\n \\left| (X_n, Y_n) - (X_n,c) \\right|\\ \\xrightarrow{p}\\ 0, \\\\\n (X_n,c)\\ \\xrightarrow{d}\\ (X,c).\n \\end{cases}", "a1079beba1d140aa3b5cd020482cd355": "A=D\\setminus\\bigcup_{n=0}^\\infty B_n.", "a1079c55432d4773d7cdd200a9da50da": "C<1", "a107b76f6c75f18adb5ee25bf063504f": "\\varphi(a)=\\varphi(b)=0", "a107caf11a94570e84376f7c998018f8": " TH_k ", "a1082e9fc97fecbf75ab053a8c9db9e5": "t=-{\\vec{P}.\\vec{C} \\over P^2}", "a108b0d4c0fd46e360810600a305c619": " O(\\mathrm{rad}(a b c) \\Theta(a b c)) \\ ", "a108dcb1ca028060975a038a90170d7e": "\\int\\mathrm{covercosin}(x) \\,\\mathrm{d}x = x - \\cos{x} + C", "a1090857ff3462caeb90eb8b65a49630": " [x + y, z] = [x, z] + [y, z], \\quad [z, x + y] = [z, x] + [z, y] ", "a10910b8e701ba6559a9fa156e160a63": " z.\\overline{z}+iw.\\overline{w}", "a10958be9d96fc599c98980a616b53e3": "\\langle f*S, \\varphi\\rangle = \\left \\langle S, \\widetilde{f}*\\varphi \\right \\rangle", "a10968668d29249b6849360e74b146c5": "\\mathrm{TMR} := \\frac{R_{\\mathrm{ap}}-R_{\\mathrm{p}}}{R_{\\mathrm{p}}}", "a1099061b65c56f1dac67cefe4c54635": "\\frac{d}{d\\theta}\\operatorname{Sl}_{2m+2}(\\theta) = \\frac{d}{d\\theta}\\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m+2}}= -\\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m+1}}=-\\operatorname{Sl}_{2m+1} (\\theta)", "a109b7e4b0823e4bf16802a3e8e253aa": "\\sum_{N=1}^\\infty P(\\{n_1,n_2\\}|N)\n= \\sum_{N} \\frac{[N\\ge n_2]}{\\binom N 2}\n=\\frac 2 {n_2-1} ", "a109bd974438c7c76b26c2e4e69be346": "Q^{max}", "a109c1da72a261aa48e36beada9c8d5a": "F_{1,i}(X)", "a10a51005fdd64905a819b5216251757": "M_X\\left(t\\right)=2\\exp\\left(\\xi t+\\frac{\\omega^2t^2}{2}\\right)\\Phi\\left(\\omega\\delta t\\right)", "a10a6828075bbc0d7ea7c6088210942b": "p_k=\\Pr[\\mbox{there is a cycle of length }k],", "a10b2095d58f73012ace9509380618b2": "[ABO]=[ABE]-[BEO] \\,", "a10b452679b79dd1be47f6ce4034cbee": " \\vec dF = -dE \\, dS \\, \\vec n", "a10b7d186c888c0246b3fde61183c6cc": "F = \\langle\\psi^{-}|M|\\psi^{-}\\rangle", "a10b89e06e5a88c12f172febb7c56816": "v<\\alpha", "a10bdd054e7e9be2f9b916052cdc6a17": "\\langle 1/n_1 \\rangle \\supseteq \\langle 1/n_2 \\rangle \\supseteq \\langle 1/n_3 \\rangle\\supseteq \\cdots", "a10bf107ebd50dd6e4685143971a72b4": "P_g=\\exp\\left(-\\frac{2\\pi W_m^2}{dW/dt}\\right)", "a10c29e6f3d036089219cbe1472de325": " (\\delta B_1, \\ldots, \\delta B_n)", "a10c4c0645d7962ce0ca20f475cf215f": "{\\bar{FR}}_4", "a10c5935929bd95f923729e126a43636": "\nV_{in}(s) = V\\frac{1}{s}\n", "a10c77f45310a0cdfe7c7206d9890c37": "\\widetilde{\\operatorname{Ric}}=\\operatorname{Ric}+(2-n)[ \\nabla df-df\\otimes df]+[\\Delta f -(n-2)\\|df\\|^2]g ,", "a10cabe53d0a895893a4c908604c09bf": "\\sin(\\delta_i+\\delta_j\\,)", "a10cc10b8de0a8db73d557f3e1bdb600": "a_{9}*b_{7}=r^2", "a10d03e9757297d8a4df3f9dd6f6aa52": "D(a, b)", "a10d0d60f898050f13c3a9dcd4e917c4": "\nrnb+sab = b.\n", "a10d132b177904271fb9ef67eca4e225": "\\phi(\\mathbf{x}, \\mathbf{c}) = \\phi(\\|\\mathbf{x}-\\mathbf{c}\\|)", "a10d1b6654e1a210e175608edc890a25": "\\operatorname{var}(u)=G", "a10d6a5b3b732943a6bca9f9d32a032d": " L_{ xy } = \\frac{ 1 }{ N } \\sum_{ i = 1 }^K \\frac{ X_i Y_i }{ X_{ tot } } ", "a10d74065cbcd2735d1c5033a0bbaaae": "\\tfrac{E}{c}", "a10e0bc7a9096adf1b11add5ab8b803c": "{\\Bbb C}P^1=S^2", "a10e7cb245cd143a0872604f1ee0d2b3": "log_{10}", "a10e8bc22f77b828fb672afd2e0f3351": "\\bigwedge^k V", "a10e921150e94152356e7a7017a8a9f7": "2LR_N(\\beta_{ML,1},\\beta_{ML,2})\\,", "a10ebbfb0f2758bc297c7031192aacc4": "\\displaystyle 2\\pi i^n\\delta^{(n)} (\\nu)\\,", "a10f35a615e145661a4da4ce1c590833": "E^2(\\mathbb{C}\\mathbf{P}^\\infty) \\to E^2(\\mathbb{C}\\mathbf{P}^1)", "a10f66a4bf9155b23e7052a2e351d3af": "G=U+PV-TS", "a10fee63c42b1f45610e0300e8289cd7": "y = {k \\over x}", "a110142b8c7299625f5a49ffcb13b6e9": "I(\\nu,T) =\\frac{2 h\\nu^{3}}{c^2}\\frac{1}{ e^{\\frac{h\\nu}{kT}}-1}.", "a11066295c24f314547dc3230d8a2c85": "\\mu_A", "a110c943f8dd2792e83b3d7bc447c6d9": "u'=\\frac{1}{EI}\\int M dx", "a110db244aa42bc8c6122178516f7138": " \\sigma_{ab} ", "a111295b461e0e96ce50b4fc700688e7": "\\frac{1}{\\sqrt2} \\begin{pmatrix} 1 \\\\ -i \\end{pmatrix}", "a1114f8074b4346ea743305756bc5ec8": "\\phi(nT) \\ = \\ \\phi((n-1)T) + \\arg(s_a(nT)\\cdot s_a^*((n-1)T)),\\,", "a1114fe7aa08c6d13de866939870e1e1": "\\begin{align}\n&{} D(X, Z) = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-z|f(x)h(z) \\, dx\\, dz\\ = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-z|f(x)h(z) \\, dx\\, dz \\int_{-\\infty}^\\infty g(y) dy\\ \\\\\n&{} = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |(x-y)+(y-z)|f(x)g(y)h(z) \\, dx\\, dy\\, dz\\ \\\\\n&{} \\le \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty (|x-y|+|y-z|)f(x)g(y)h(z) \\, dx\\, dy\\, dz\\ \\\\\n&{} = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-y|f(x)g(y)h(z) \\, dx\\, dy\\, dz\\ + \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |y-z|f(x)g(y)h(z) \\, dx\\, dy\\, dz\\ \\\\\n&{} = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |x-y|f(x)g(y) \\, dx\\, dy\\ + \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |y-z|g(y)h(z) \\, dy\\, dz\\ \\\\\n&{} = D(X, Y) + D(Y, Z)\n\\end{align}\n", "a111933b7a0e8d2ac96e852ea1d37fa3": "P(E) = \\delta(E - H[\\vec{\\sigma}])", "a111bc6e45fc0c08e2187d795cf6d020": " mR} = \\lbrace x \\in \\mathbb{R} \\mid x > R \\rbrace ", "a1211d6a12a2bab4c2dc647c28344f90": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) = \\left \\{ u : \\ \n|u - {\\tilde{u}} | \\le \\alpha \\right \\} , \\qquad \\alpha \\ge 0\n", "a12169828ab045cc443deee9b2522109": " \\Delta \\left(\\frac{1}{\\gamma }\\right) \\approx 5.307\\times 10^{-10} ", "a12204c80bf43080ca3e205d2c3eff0b": "S= \\int L \\, dx_3= \\int \\mathbf{p} \\cdot d\\mathbf{s}", "a1220b3814f53e3b7968d62bc1753cff": "\\{a_1,\\ldots,a_n\\}", "a12211305ace13fa691c1f7f956cda26": "\n\\left[ A\\right] =\\left\\{ \\beta A\\ |\\ \\beta\\in\\mathbb{C},\\ \\left\\vert\n\\beta\\right\\vert =1\\right\\} . \n", "a1225406223ee4d5cbab95763d84531b": "\\hat{H}_{0} \\equiv F + \\langle\\Phi_0 | (\\hat{H} - \\hat{F}) | \\Phi_0\\rangle", "a122d7786d8ab7f45e189bad773e3a68": "F(y)=0", "a123ca3bad01776f9a03643e249321bc": "l: (A \\to D) \\to A", "a123e70178082158db749cc06c50cdca": "\\frac{g_+}{g_0}", "a1240e4b0cf9f4fdfe433a5ec009c2d9": "(g \\circ f)(x) = 3x + 5", "a124cdb4ab4f12d65f39bf0bbafdf0bd": "\\mathcal{U}(\\mathfrak{g})", "a124d10870d81c36aa1841c3402ee4bd": "\nf(x) = \\frac{\\sqrt{\\frac{x-\\mu}{\\beta}}+\\sqrt{\\frac{\\beta}{x-\\mu}}}{2\\gamma\\left(x-\\mu\\right)}\\phi\\left(\\frac{\\sqrt{\\frac{x-\\mu}{\\beta}}-\\sqrt{\\frac{\\beta}{x-\\mu}}}{\\gamma}\\right)\\quad x > \\mu; \\gamma,\\beta>0\n", "a124de930ea74cbb3991c742ba7504d4": "x = 2", "a125051196b08525a098a781419e4c02": "g = \\frac{G M}{r_s^2} = \\frac{c^4}{4 G M} \\;", "a125500b7579cdd227e30b83f9d653f4": "\\frac{d}{dx} (\\mu(x)P(x)y')+\\mu(x)R(x)y=0", "a125d4cac20ea388561fad41a3d48e96": "\\phi_i\\;(1)", "a1262bd118ea48427732dc683ae24a88": "A e^{st}", "a1263c9f4f63c169b7c967a2c88d809b": "\\langle W,\\le,\\{M_w\\}_{w\\in W}\\rangle", "a12649ca994dffe62c3af12f37b72bb0": "\\;p(2) = p(1) r - A = P r^2 - A r - A", "a1268af03085ff85d8ace75fcfefdef1": "0 - G; 1 - T; 2 - A", "a126ea80ae4b7ce72d41e29738e7b91d": "N=cd", "a1270d6cd8f0e84a580e3ae8316c7711": "\\begin{align} \nP(0) = 0 \\quad & \\quad P(S(x)) = x \\\\ \nx \\dot - 0 = x \\quad & \\quad x \\dot - S(y) = P(x \\dot - y) \\\\\n|x - y| = & (x \\dot - y) + (y \\dot - x). \\\\\n\\end{align}", "a127695bb7ad3361b4301842a78dd2df": "\\frac{\\partial f_k}{\\partial x_i}", "a1279688409e969625072e6c4d4f16ae": "\\{2,3,4\\}", "a127a697b0e7151eda05aa725487dd33": "x_n .", "a127e8eef8f926585a9b4b025dd60b45": "a < b,\\,", "a1286bc18d5f35607146b023fe0c17cc": "m_p^2 \\equiv c^{\\frac{1}{2}(p+1)} \\equiv c\\cdot c^{\\frac{1}{2}(p-1)} \\equiv c \\cdot\\left({c\\over p}\\right) \\pmod{p},", "a128b9881bc940e2e69fdd901c8cbc95": " u(t) ", "a1290d2d687c65a6de9ccbffe9bf7c3c": "(\\hat{\\bold{x}}, \\hat{\\bold{y}}, \\hat{\\bold{z}})", "a12927c3d3e8e8ef29d643b8efea7316": " C = \\mathrm{Re}(N(i)) \\qquad\\text{and}\\qquad D = \\lim_{y\\rightarrow\\infty} \\frac{N(iy)}{iy} ", "a12931b678dc0bbc1b1ebeb554fcf0f9": " -u^b \\nabla^a F_{ab} = \\sqrt{g_{tt}} R_{00} u^a u^b = \\sqrt{g_{tt}} R_{ab} u^a u^b ", "a12953c18b0ec47013afd106b16a7b81": "e_k(t) = {1 \\over \\sqrt{2 \\pi}}e^{i k t}", "a129a32163d45f75b5f8db7140a53c66": "R_e^2=\\frac{N}{N-1}\\left(\\bar{R}^2-\\frac{1}{N}\\right)", "a129ac3cb09987f282ba3690daa2198d": "\np(\\infty) = s \\cdot p(0)\n", "a129f5bb69a07c40c1cd43c0d592f47b": "\nG(nz)= K(n) n^{n^{2}z^{2}/2-nz} (2\\pi)^{-\\frac{n^2-n}{2}z}\\prod_{i=0}^{n-1}\\prod_{j=0}^{n-1}G\\left(z+\\frac{i+j}{n}\\right)\n", "a12a3079e14ced46e69ba52b8a90b21a": "IP", "a12a6ec5fef71d4ecb25a3978bc83926": "\\{ 1 , ~i \\}", "a12a7401081ed9c856d3479915debb57": "\\Delta V_{0}/V_{0}=\\bigtriangledown u(r,t) ", "a12ac522615fa76760caf2e77b1b5ba2": "\\langle u, A u \\rangle = \\int_0^1 u(x) u'(x) \\, \\mathrm{d} x = - \\frac1{2} u(0)^2 \\leq 0,", "a12b96f42b2b794fe88bf79710721440": "\\lim_{h\\rightarrow 0}\\frac{\\sin h}{h}=1,", "a12bbc3b3c8766d99e8efccc383fbc35": "t,r,\\theta,\\phi", "a12bfa24c9bab2e9affa8a4e49044e0a": "X \\to", "a12c1c5f82be90ebbc66aaa1af088670": " \\delta W_{\\sigma} = 2\\sigma\\left( \\text{sin}\\phi\\delta h + b\\left(\\text{sin}\\left(\\phi-\\alpha\\right)\\delta\\alpha\\right)\\right) ", "a12c55bc1542d14e8bd80820ecc3fc2d": "CIRC(T)", "a12c9c96ea5b48617cef5f616477b967": "|a_k|\\leqslant\\lim_{r\\rightarrow+\\infty}Mr^{n-k}=0.", "a12d46063ea288c1fa980415639fbf48": "1~\\bold{tbsp} = 14.787~\\bold{cm}^3 ", "a12d6565f1b197ea239787df36d87cff": "k_\\pm", "a12dab1bc190f0e19a9eedec5ed60390": "\\mathit{6}\\, ", "a12dcb7824a2f758afb03dcccf477c36": "p(z)=a(z-z_1)(z-z_2) \\cdots (z-z_n).", "a12de5c28e2f974b1f714a73adcfe3c2": "V^\\infty_n(R) = (2R)^n.", "a12de7b05d9e28f1736c9194d0a6bf84": "t\\,\\!;", "a12e3ed89cf626f71bf2cd89134516da": " u (x) = \\frac{1}{(2 \\pi)^n} \\int e^{i x \\xi} \\frac{1}{P(\\xi)} \\hat f (\\xi) \\, d\\xi.", "a12ec5778a2ff556d960b628862e9c28": "p_1(u)=\\frac {-u}{1+u+u^2},\\ p_2(u)=\\frac {1+u}{1+u+u^2},\\ p_3(u)=\\frac {u(1+u)}{1+u+u^2}", "a12f1e3e77a0c944e7856e410a1e1850": "\\sigma_\\mathrm{n}", "a12f488d381362e0b9a9b2b398004b86": "Ax\\le b", "a12fb05f262b5c04455f036df01818e3": "\\mathbf{u} \\triangleq (u_1, u_2, \\dots, u_r) \\in \\mathbb{R}^r", "a12fe2daddf1c79ccec4591867704997": "L_{1} = \\left(1 - \\alpha - \\beta\\right)", "a1303e19c4ba053bdfb579673dbb8b86": "A=4r^2.", "a130504b1df5a91d37f9bec85e3cc394": " A = (I - Q)(I + Q)^{-1} \\,\\!", "a13051e753c019287f127858b070ff82": " P = \\sum_{i=(0,1,2,4,8)} \\mathbf{F}_i ", "a130527c522342327ad02d56b9f0271a": "(P \\and Q) \\to (Q \\and P)", "a130eb70320a40ba694d3d82fca8ff68": "\\varphi*\\psi", "a130ef38e4769612405f0dbb6be51ee4": "(x)_n = x(x+1)\\cdots(x+n-1) = \\frac{\\Gamma(x+n)}{\\Gamma(x)}", "a130f360afcb1d9cce1f7663ceb1180c": "p(\\theta|D)", "a1311d1256651043a4697f613660aeb5": "\\cot\\delta' = \\frac{\\cos\\delta}{\\gamma\\cdot(\\sin\\delta+\\beta)}", "a1315b5e1cbbf7c964f932d72acbcf58": "\n \\left.\n \\underbrace{\n\t (11{\\color{Red}\\underset{==}{33}}),\n\t (12{\\color{Red}\\underset{==}{33}}), \n\t (22{\\color{Red}\\underset{==}{33}}) \n }_{(\\gamma)},\n \\underbrace{\n\t (1{\\color{Red}\\underset{===}{333}}),\n\t (2{\\color{Red}\\underset{===}{333}}) \n }_{(\\delta)}\n \\underbrace{\n\t ({\\color{Red}\\underset{====}{3333}})\n }_{(\\omega)}\n \\right\\}.\n", "a1315c940d5f27a03d3641e1d76c78f5": "z*(\\mathbf{s}_j)", "a131a7b06f0c7ebed6998e1bc371f77d": "\\mathbf{\\epsilon}= \\epsilon_0\n\\begin{bmatrix} n_x^2 & 0 & 0 \\\\ 0& n_y^2 & 0 \\\\ 0& 0& n_z^2 \\end{bmatrix} \\,", "a131aa09a034a4d9d05f7b9161d53a89": "e_t", "a131b73467cb57adc1f3109028a51ee6": "X[x,y]=x-\\frac{x'\\int_a^t \\sqrt { x'^2 + y'^2 }\\, dt}{\\sqrt { x'^2 + y'^2 }}", "a132133de62db408fdb70547bc11e41b": " U = \\int_{V_n} \\mathrm{d} \\mathbf{m} \\cdot \\mathbf{g} \\,\\!", "a13219bc7b25b64b3138829ec1b21a5f": "DF_\\tau = \\frac{\\hat{\\gamma}}{SE(\\hat{\\gamma})}", "a132375055278e1c7b78b4a132c45ff3": "x^2 + y^2 = z^2", "a1323b1848ff28bc92db6823f745c92b": "\\lambda(V_k) = \\lambda(V)", "a132525997b3adfe8c46e2945cc6092c": "f:A\\rightarrow B", "a13255293d50e2a4ff487d2cddf50f31": "\\displaystyle{\\bigoplus_{k\\ge 0} S^k(H).}", "a132836ef797f5f63782ccacdb2ea4cd": "(\\mathbf{\\nabla}\\phi)^2 \\ge 0", "a13292d3652481eee39f674a8b7bd4c2": "H_2(C_n,\\mathbf{Z}) = \\begin{cases}\n 0 & n = 0, 1\\\\\n \\mathbf{Z}/2 & n = 2\\\\\n (\\mathbf{Z}/2)^2 & n = 3\\\\\n (\\mathbf{Z}/2)^3 & n \\geq 4 \\end{cases}.", "a13294cf60815d084bda04570092dd3b": "\\min\\{|z_i|\\} \\geq E", "a132e49f287e798da5d9bbb8e464166c": "x\\in X ", "a132f105cb99d0c00634ae833e5c046a": "\\lambda_{w}(t)= w^{|t|}t", "a132fb7ea8000ab5ca67af170e2e72c5": "\\pi+e", "a1332b132d378ed17e32b21a331e0d60": "\\hat{U}_{ee} = {1 \\over 2} \\sum_i \\sum_{j \\ne i} \\frac{e^2}{4 \\pi \\epsilon_0 \\left | \\mathbf{r}_i - \\mathbf{r}_j \\right | } =\n\\sum_i \\sum_{j > i} \\frac{e^2}{4 \\pi \\epsilon_0 \\left | \\mathbf{r}_i - \\mathbf{r}_j \\right | }\n", "a1337ece8437111a554beeaf1b0a9d1c": "\\boldsymbol {\\Omega}", "a133a1dfbbfef83dbed0a9bb71909eb7": "B = \\{ x \\in A | \\forall g \\in G, g(x) = x \\}.", "a1340326dae135e33d45bed8a2c41865": " \n\\Gamma_{jk}=p_i~p_l~a_{ijkl} ,\\quad\na_{ijkl}=c_{ijkl}/\\rho ,\\quad\np_i=\\frac{\\partial \\tau}{\\partial x_i} \n", "a1341fef6fcb0b295001fc38bece2121": "G_{M}", "a13422cb89e42bd0d9f379a9ddb6ac28": " e^{i \\sigma_z \\omega_r t/2}\\sigma_x e^{-i \\sigma_z \\omega_r t/2} = \\begin{pmatrix}\ne^{i\\omega_r t/2} & 0 \\\\\n0 & e^{-i\\omega_r t/2} \\end{pmatrix}\n\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0 \\end{pmatrix}\n\\begin{pmatrix}\ne^{-i\\omega_r t/2} & 0 \\\\\n0 & e^{i\\omega_r t/2} \\end{pmatrix}=\n\\begin{pmatrix}\n0 & e^{i\\omega_r t} \\\\\ne^{-i\\omega_r t} & 0 \\end{pmatrix}\n", "a134ef4abb891abf8945909baef109ca": " P(X > (1 + \\delta) \\mu) \\le e^{ \\frac{ -\\delta^2 \\mu }{ 2 + \\delta } }, ", "a134f18619fd5a3bd4ba759210dabc41": "|x|^{-n/2} J_{n/2}(2\\pi |x|)", "a134f23421760e2b93225eb95dc9e85c": "VAG (p, (a, m)) \\cup \\{[m, m]\\} \\cup VAG (p,(m, b))", "a135661731c1ac5eb42d2da0c7a04cec": "\\frac{dI}{ d \\Omega\\ I_{0}}=\\frac{4 \\sqrt{\\varepsilon_{0}}}{\\cos{\\theta_0}}\\frac{\\pi^4}{\\lambda^4}|t_{012}^p|^2 \\ |W|^2 |s(k_\\text{surf})|^2", "a135b7b89e5b4d6a305ebe6f779eb657": "4\\pi(n-1)", "a135e69da56f84aca82518498edeaa8d": "\\mathbf{A}^{-1} = (\\mathbf{L}^{*})^{-1} \\mathbf{L}^{-1} , ", "a135ea8dfe5762838e9ce7b53970b1bd": "y=f(\\mathbf{x}+\\Delta\\mathbf{x})\\approx f(\\mathbf{x}) + J(\\mathbf{x})\\Delta \\mathbf{x} +\\frac{1}{2} \\Delta\\mathbf{x}^\\mathrm{T} H(\\mathbf{x}) \\Delta\\mathbf{x}", "a1361c006935e7ea9a2c63b526314122": " L(n,n) = 1", "a1361f918c95feca9323fe4492957449": "(RU^2D^{-1}BD^{-1})", "a1364a35c67deabfbf65cfa1a0d8337b": "\\scriptstyle c(E) = c(\\mu) + c'(\\mu) (E-\\mu) + O[(E-\\mu)^2]", "a1368b09dcb6db4971a0737f4fdbcfc4": " V_0(q) - V_0(-q) = 4qB(q^2)", "a1368b9adbbd0034ed3eaa53077724f6": "w(x,y)~", "a13707266e7f070c520a02d23f4e4d4a": " \\qquad\\qquad = \\left( Z_{11} - {Z_{21}^2 \\over Z_{22}} \\right) \\, I_1 ", "a137154f7ec040f3de7366e60e6fafa9": "j = c", "a137736ad9eabc70eb61ec3e250a62e4": "\n-d(n)=\n\\frac{\\log 1}{1}c_1(n)+\n\\frac{\\log 2}{2}c_2(n)+\n\\frac{\\log 3}{3}c_3(n)+\n\\dots\n", "a137b5eab76046c79218410d51071b54": "\\mathrm{rect}(x) \\ ", "a137dccc030ec7ac40c4a97dce17c462": "IV_1'", "a137f5077af13cf17a1a8c2b432f58be": "(a+bx+cx^2)/(1-x^3) \\,", "a13819a38cc6cb7222b4b1c80f1706bd": "a_k\\not=0", "a13824eff9afefa8e34aa7aa8db038de": "I_{0.5}^{-1}(\\alpha,\\beta)", "a1384aa3ea4e1d4ca4707ed950caee26": "D_i", "a1386c3321da1596b3a52ab5c8ed39d2": " \\sqrt{4\\pi\\mu_0}\\mathbf{H} ", "a1387598f7a9b21263cf34977f3411ef": "M \\frac {d^{2}} {dt^{2}} u_{n} = -k_{0} ( u_{n-1} + u_{n+1} -2u_{n} )", "a1395bb8f14a21139cf49808a0fa3473": "C(x) = 0", "a13987fe97431abfaabd4ca386256b3b": "3\\uparrow\\uparrow\\uparrow\\uparrow 3", "a139d94ab201253aa2988ee4f431db34": "\ny^{2} = a^{2} \\left( \\sigma^{2} - 1 \\right) \\left(1 - \\tau^{2} \\right).\n", "a13a17237fff0e5abae0219af0899801": "\\tbinom26", "a13a1c74241d3b229965b7589c3e81b2": "x_0=34", "a13a64189babcbb1b7ddba68ffdf1ef4": "\\chi(G)=3", "a13ac2070ee8aa905e6208b5083c7016": "p_0 = \\frac{(\\left\\|\\mathbf{a}\\right\\|^2\\mathbf{b}-\\left\\|\\mathbf{b}\\right\\|^2\\mathbf{a})\n \\times (\\mathbf{a} \\times \\mathbf{b})}\n {2 \\left\\|\\mathbf{a}\\times\\mathbf{b}\\right\\|^2} + \\mathbf{C}.", "a13b10fdb6849f4fde970b5a3983aa0d": "\n\\mathbf{U} - \\mathbf{u}^\\infty = \\mathbf{u}' + b_0 \\mathbf{F} + \\frac{a^2}{6} \\nabla^2 \\mathbf{u}',\n", "a13b686f8a1b14aa5f74cb56591e7243": "0 = \\,g_{ik;\\ell} = g_{ik,\\ell} - g_{mk} \\Gamma^m{}_{i\\ell} - g_{im} \\Gamma^m{}_{k\\ell}. \\ ", "a13b9f7fc731a32170b3146426bba138": "x_{i,.} = \\sum_{j=0}^{3}{x_{i,j}}", "a13bd7d49343d8ebdde7a116c763b0dc": " a>0", "a13be08f985b892c77fd0393b2b8ed8a": "Y_{9}^{8}(\\theta,\\varphi)={3\\over 256}\\sqrt{230945\\over 2\\pi}\\cdot e^{8i\\varphi}\\cdot\\sin^{8}\\theta\\cdot\\cos\\theta", "a13c5016c0b9e0e02b28f1ee58e02c85": "\\theta = \\omega_i t + \\begin{matrix}\\frac{1}{2}\\end{matrix} \\alpha t^2 ", "a13c71a2f55b7df1b572c7c3a82f4d8a": "\\ [A]_t = [A]_0 - [B]_t ", "a13c97ff4195c83b46518830ad7b3d12": "G/N \\rightarrow G'/N'", "a13cb86c2440ebc601da60c34cba64b5": "{\\color{Blue}~5.2}", "a13ccbc25353f7ed790137672380af20": "x_{\\ast}", "a13d49b8344c03ec3e01af8552efb5fe": "T = 3.30 ", "a13d570ae53ce03c1e9e3ee8f764699c": "b = \\lambda", "a13d5758f4d7fecd51eecfb417899b15": "\n\\psi = \\left[\\frac{C_1 b^4}{r^2} + C_2 b^2 + \\frac{C_3}{b r^3} + \\frac{C_4 r^5}{ b^3})\\right] \\sin(\\theta)^2 \\cos(\\theta).\n", "a13d6c5e0ec602a88f30db20a6858504": "\\mathfrak{so}(4,1)\\cong \\mathfrak{sp}(1,1)", "a13d6e305c1f513572ef45350ce548f6": " \\lambda_1 , \\lambda_2, \\dots, \\lambda_{v} ", "a13d9611ee69c2388ece8f2821af7784": "\\begin{align}\n b^{m + n} &= b^m \\cdot b^n \\\\\n (b^m)^n &= b^{m\\cdot n} \\\\\n (b \\cdot c)^n &= b^n \\cdot c^n\n\\end{align}", "a13dd543d2a5f68852c91599ad0489c0": "RPF = \\frac{U_{PAH}}{P_{PAH}} V", "a13e1216bcda1bd9ef0f13d92ffc7ef7": "\\frac{\\pi}{4} = 5 \\arctan\\frac{1}{7} + 2 \\arctan\\frac{3}{79}.", "a13ee1e649b4ced24e70be52224e4f25": "[D\\rightarrow D^{'}] \\times D \\rightarrow D^{'}", "a13eeeed4d4e0dcef4787f738de4b2ba": "\\Gamma:\\mathcal{C}(I_{a}(t_0),B_b(y_0))\\longrightarrow \\mathcal{C}(I_{a}(t_0),B_b(y_0))", "a13f05dd49fb68b2f08096864fe996a8": "P(X)=(X-\\mu_1)\\dots(X-\\mu_N)", "a13f5233deb1ba391e88051a5cc58dc0": "\\mathrm{Da}_{\\mathrm{II}} = \\frac{ \\text{reaction rate} }{ \\text{diffusive mass transfer rate} }", "a13f5364254c59d742fdce256f5d8e31": " A(m, n) =\n \\begin{cases}\n n+1 & \\mbox{if } m = 0 \\\\\n A(m-1, 1) & \\mbox{if } m > 0 \\mbox{ and } n = 0 \\\\\n A(m-1, A(m, n-1)) & \\mbox{if } m > 0 \\mbox{ and } n > 0.\n \\end{cases}\n", "a13f5a36064d2adb1e12c2dcea6da361": "p = x^2 + 5y^2 \\Leftrightarrow p\\equiv 1\\mbox{ or }p\\equiv 9\\pmod{20},", "a13fe66b913216e218b19eaf08aaa571": "{(1+z)^2}", "a13fedd9eefa5618aeb950f05a29e782": "\\frac{4}{3} \\log(g(\\Sigma)),", "a13ff4dd9339fb297a81ef1d724d9e5e": "\\varepsilon = \\frac{t + u \\sqrt{d}}{2}", "a140483368e9a3e23f9fedd0e2ed0b82": "VC(x) < VC(y)", "a1405f93acff65e3be8d47d6f8832201": "\\begin{align}\n0 &= \\emptyset \\\\\n1 &= s(0) = s(\\emptyset) = \\emptyset \\cup \\{ \\emptyset \\} = \\{ \\emptyset \\} = \\{ 0 \\} \\\\\n2 &= s(1) = s(\\{ 0 \\}) = \\{ 0 \\} \\cup \\{ \\{ 0 \\} \\} = \\{ 0 , \\{ 0 \\} \\} = \\{ 0, 1 \\} \\\\\n3 &= ... = \\{ 0, 1, 2 \\}\n\\end{align}", "a14078fdbfd7f29ba759172632294d60": "\\overline{A \\cdot B}", "a14082f19f06fae49b01c9f53c7f1f3a": "\\sup_v |Av|_W/|v|_V", "a1409fff5786f4c3e9496bbbc0c4b928": "x_{n+1}=f(x_n), \\, n=0, 1, 2, \\dots", "a140d0056761cb430e140eac948accba": " \\sin x_1 \\neq 0 ", "a140f68b15dfdf1cf20c9903c679645e": "\n P = \\overline\\Gamma - \\overline{C}'\\Gamma^{-1}C\n ", "a1415e384bc95f0c0b8182881927b9f5": " \\mathbf{A}\\mathbf{A}^{-1} = \\mathbf{A}^{-1}\\mathbf{A} = \\mathbf{I} ", "a1417a23e3bf47ec44c1c29d7879db8d": "\\sqrt{\\mathrm{10}}=\\mathrm{1T.T1TT10T0000TT1100T0TTT011T0...}", "a1417ed6a4120ab2545b1414d5e1a5fa": "u_1(0,t) = u_0(t) = U_0\\, \\cos\\left( \\Omega\\, t \\right) \\quad \\text{ at }\\; z = 0.", "a141968fb9469a3f93d07767d94477af": "\\ \\epsilon(f) = \\mathbb{E} \\left| X(f) - \\hat{X}(f) \\right|^2", "a141a553c0ed0997e2ef6d7ddf64a7e6": "t / U ", "a141ad881d8fe457b94cbd2174a4630e": "(\\mathbb R^m, d)", "a141aedd8cc60ef3f082fd159a056705": "g \\; \\mapsto \\; U_g", "a141cbede29a4e8cb07777d2dcce16b0": " z = r ~ \\cos \\theta ~ ", "a14231edbbf92b223b4590197af90c1d": " (Y, P) ", "a142708a21c6d5dbfad29475773f33b5": "\\max (x_{i+1}-x_i), \\quad i \\in [0,n-1].", "a142b96726af8327932c571e34a469d1": "a=1.5", "a142cee5bb3ca05a1d3183fa7fb50bd5": "\\frac{24\\,\\bmod\\,5}{3\\,\\bmod\\,5} = \\frac{4}{3} \\neq 3", "a142ece71f5da44409c2e80f397a22ab": "\\left(\\frac{\\partial \\ln f_i}{\\partial P}\\right)_{T,x_i}=\\frac{\\bar{v_i}}{RT}", "a143241e807e6900965de8b795048ed8": "\\sqrt{2E_{1}E_{2}}", "a1436cb29965da1f0440319ecb8bbda2": "d: X \\times X \\longrightarrow \\mathbb{R}_{\\geq 0}", "a1441a4035dc28021108c8d1d2f16694": "\\mathcal M", "a1441ec876e1a5c8c340f25df65e12aa": "\\!\\models^+", "a14469d213b49dadf23440ffa39ad4d1": "e = \\frac{2}{R_\\mathrm{K} K_\\mathrm{J}}.", "a1449e8933117dd7b94e43b48af9c64f": "r_{\\pi} ", "a145099b0b10da6cfab0958aa43b19e1": "A\\subset B\\subset E", "a1451d4fde787747b865a303e2f2a57d": "x\\in u\\in U", "a145653d88173bb178f16d49701c7d00": "R_{ku} = \\frac{1}{n R_q^4} \\sum_{i=1}^{n} y_i^4 ", "a145836d1360aa251350710170493f9e": " S_{n} + C_1 \\ S_{n-1} + \\cdots + C_L \\ S_{n-L} = 0", "a145af525e10da70a0b48c694704de84": "\\frac{\\partial}{\\partial t}\\iiint Q\\, dV + \\iint F\\, d\\mathbf{A} = 0,", "a145bbd05d028af800bcefc5e0cbe983": "E_{ij} \\otimes F_{kl}", "a145d55311aa6f25d40422dbd22872e2": "k_{n+1} = 2\\cdot k_{n} - 1,\\!\\,", "a14612194c26b71c404312105ab2f255": "k\\in\\{1,\\dots,n\\}", "a1462370d83dc06eb45c4513d1e24d8a": "\\! (1 - 2it)^{-k/2}", "a1468094138bb3b0d3df6fc1f0347e00": "\\lceil\\rceil", "a146947853d4499dd0909ae6b3070a88": "\nL(F_e ) = \\beta V\\;\\int_0^\\infty {ds} \\left\\langle {J(0)J(s)} \\right\\rangle _{F_e }, \\,\n", "a14712291ac8fae521d27c52fbfee47d": " 2d-1 \\times 2d-1 ", "a14738a8843c8ae25aab5d54fe4fd130": "E = E^0 + f\\frac{2.303RT}{F} \\log[\\mbox{H}^+]", "a1474fdd2547f00f347548b3a2bf451f": "C_n < \\operatorname{SO}(3)", "a147611337099085e177227bcf01bb64": "I'[u_k]\\rightarrow 0", "a1478caca3ddc419be24970573d6e200": " \\nabla f(x_k+\\Delta x) \\approx \\nabla f(x_k)+B \\, \\Delta x", "a1485cd3418f2363a317619da5a197c8": "2^n 2^m = 2^{n + m}", "a14869704db14d75d74c465be3038793": "\\mathbb{Z}^{(-dn)}", "a148697a69c0878f5f13ebb7e75f0577": "S_N f(x) = \\sin(x) + \\frac{1}{3} \\sin(3x) + \\cdots + \\frac{1}{N-1} \\sin((N-1)x).", "a1487be19aa18fa628ba052fb70d6aa3": "A =\\pi R^2 + \\pi R S\\,", "a148bc84edc2d0ec60183070d4d57854": "U_{B, \\varepsilon}(y) = \\{x \\in V : p_\\alpha(x - y) < \\varepsilon \\ \\forall \\alpha \\in B\\}.", "a148bd68347a2bfc7f65f0fa89685228": "X=\\sum_{i=1}^r \\lambda_i Y_i + f Z_0 ,", "a149366ec572bda5c4da3b5c06058d4b": "\\sqrt{(k+\\tfrac{4}{3}u)/p}", "a149c4e446e4b92d47bfda9ff37ef3d3": "\\mathbf{E} = \\mathrm{Re}(\\mathbf{E}_0 e^{-i\\omega t})", "a14a9d09ea7d3e5f4a71a32cbc0f435c": "-\\boldsymbol{e}_k\\, a\\; \\text{e}^{\\displaystyle k\\, z}\\, \\sin\\, \\theta\\,", "a14aaf683edc9d2a20e238f01d4ce77b": "\\sum_{n=s}^t f(n) + \\sum_{n=s}^{t} g(n) = \\sum_{n=s}^t \\left[f(n) + g(n)\\right]", "a14ae981d21f0ad6147704b6c9e949d8": "m\\geq \\frac{4n}{\\log n}.", "a14b1a4080d9c6fc9fb4fa1b96831a73": "H(X_1,X_2,\\ldots,X_n)", "a14b99991d48e1eb52720d4858ddd43d": "B\\in \\mathcal S_n(\\mathcal B_{1\\cdots n}, \\mathcal D_{1\\cdots n})", "a14c15d27701ab9354ac9e55f433a843": "\\frac{1}{z} = \\frac{1}{r}\\left(\\cos(-\\varphi) + i \\sin(-\\varphi)\\right).", "a14c1ec7c9c8fce581cd9702e93beaa3": "x/y \\cdot z = z/y \\cdot x", "a14ca920f5352d28b13d0d44f21eb00b": "U_{ab}=-l_{[a}m_{b]}", "a14cf24b563b6c725251b043aa1a6584": "n\\, ", "a14da2e15d4060a40f432f6280faf639": " A(c) = 1/b = const ", "a14e4e57e655449a8d8f7f342ea5d906": " { p (r, \\theta ) } = { j \\omega \\rho_0 a^2 v_n { J_1 (k a \\sin \\theta) \\over k a \\sin \\theta } { e^{ j k r } \\over r } } ", "a14e6c23fafe1fb4c7f64e7fb275c139": "K=\\Q(\\sqrt{2})", "a14ea93dcc2b194b8562243dcf0bf069": "ds^2 = \\frac{\\alpha^2}{\\cos^2\\eta}(-d\\eta^2 + d\\Omega_{n-1}^2).", "a14ed1a29e3fc6343fc52ecc9e2fbf6d": "m \\frac{d v}{dt} = \\mathbf{F_{L}} + \\mathbf{F_G} + \\mathbf{F_P} + \\mathbf{F_D} + \\mathbf{F_T}", "a14f3f7fb3e08e01f6fd8e467f63d0ee": "\\delta \\subseteq \\left(Q \\backslash A \\times \\Sigma\\right) \\times \\left( Q \\times \\Sigma \\times d \\right)", "a14f45807bfaeaffa49c3fed4c855700": "\\left(\\frac{d}{dz}\\right)_q z^n = z^{n-1} \\frac{1-q^n}{1-q}\n=[n]_q z^{n-1}.", "a14f4980276513fccae790e3fda2f93d": "x_0 = 0.5", "a14f756c06aec9693eeca85c3c423270": "\nQ=\\left[\\begin{matrix}\nD_{0}&D_{1}&D_{2}&D_{3}&\\dots\\\\\n0&D_{0}&D_{1}&D_{2}&\\dots\\\\\n0&0&D_{0}&D_{1}&\\dots\\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\ddots\n\\end{matrix}\\right]\\; .", "a14fc365c7e8c0234ad9e315a1c93744": " {\\mathbf A}_{22\\cdot 1} \\sim \\mathcal{W}^{-1}({\\mathbf \\Psi}_{22\\cdot 1}, \\nu) ", "a15001de917ef09727d61c4164b4a183": " |N| ", "a15065075917af2f8642020bad1e6736": "{\\mathbf{}}I_n", "a150864d97c20fd2d8d56d1e059684f7": "\\zeta - \\eta = d\\beta\\,", "a1509be5a531abc02918293add8c3d1f": " f(r)=1-2a/r \\,", "a15114bc697eecba6a9633c2dcc482e2": " s_0'=(s_0,\\tau(s_0)) ", "a151bd0026f57524b00d2e0193026224": "\\textit{state}(s \\circ \\textit{open}, 1) \\leftrightarrow \\textit{state}(s,0)", "a151f0616a13ca9517ea3560a7ae3441": "\\mathbf{C^{j}} = \\begin{bmatrix}\nC_{11}^{j}\\\\ C_{12}^{j} \\\\ C_{13}^{j} \\\\ C_{14}^{j} \n\\\\ C_{21}^{j}\\\\ C_{22}^{j} \\\\ C_{23}^{j} \\\\ C_{24}^{j} \n\\\\ C_{31}^{j}\\\\ C_{32}^{j} \\\\ C_{33}^{j} \\\\ C_{34}^{j}\n\\end{bmatrix}.", "a152053150108f04a8deaca29da576f5": "\nx0,j=sign(j)\\Delta\\mid j\\mid ^{1/(1-a)};\\qquad 0\\le \\mathit{a}\\le1.\t\t\n", "a152597ad8f92495ee655bee763e145e": " \\left(\\frac{\\alpha}{\\beta }\\right)_m \\neq1 ", "a1527da5b2db10a7677576d7afcbee2c": "k_1'", "a152b4f838820c49235e134d7eb4b639": "m=\\phi(n)", "a152bd8f3a9f3995455f63ec4ff1b84a": "\\text{E}(X)\\leq\\text{EVaR}_{1-\\alpha}(X)\\leq\\text{esssup}(X)\\,", "a153314c2770680c076f98e5a708e7ab": "b_r", "a15363d4bc0c9d42d2230f5d7ccb92bc": "\n\\mathit{RD} = \\frac{\\rho_\\mathrm{gas}}{\\rho_{\\mathrm{air}}} \\approx \\frac{M_\\mathrm{gas}}{M_{\\mathrm{air}}}\n", "a153d395dfc980eceb79c9201dfc0081": "\\scriptstyle \\sum_{i=2}^{n+1} \\lambda_i/(1-\\lambda_1)\\, =\\,1", "a153e12a6830dd3e99d3e3f0af460096": " Q", "a1541404f0561bc69916034d6f617cd2": "R \\ \\stackrel{\\mathrm{def}}{=}\\ \\left|\\mathbf{r} - \\mathbf{r^{\\prime}} \\right|", "a15457aa5f7403582a4fc49d01fd180b": " |u^*|=|u| ", "a154767f717848b6bdfa0b7d6e14c2e5": "\\text{Aim}(X) := \\{f \\in \\operatorname{Met}(X) : f(p) + f(q) \\ge d(p,q) \\text{ for all } p,q\\in X\\}.", "a154c9fef6a51e08542321cf5e8542e8": " \\frac{1}{1}\\frac{y \\cdot r}{x}, \\quad \\frac{1}{3}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}, \\quad \\frac{1}{5}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2},\\quad \\cdots", "a154e66765d119633709484185a19354": "\\sqrt {m_1^2c^4 + p_1^2c^2} + \\sqrt {m_2^2c^4 + p_2^2c^2} = E", "a154e70c3a9fa577c79690c03925c8db": " \\, J_\\nu(x) \\, ", "a15509229bbcfe2ae5079b9691a2c219": "\\int e^x \\cos x \\, dx \\,=\\, \\operatorname{Re}\\int e^x e^{ix}\\, dx.", "a1553bf22ad6cbfdac7f7a1d7aaea848": "v_{(G; c)}(\\{1,2,3\\})=23.", "a1555463c361e7036a274a8b44e29192": "zw", "a15577a709e76a8461ff71665d4b9897": "\\hat{H}'_0= \\beta \\sqrt{m^2+|p|^2}", "a1557b0cad3146914d094902b959474b": "\\beta_{10\\ldots}= \\beta_{01\\ldots}\\ldots = 1\\,", "a1559911018dc1cff42c34fc2e767ea5": "s.t. \\quad a^1_jx_1+a^2_jx_2+...+a^n_jx_n\\leq \\sum_{i\\in S}b^i_j \\quad \\forall j=1,2,...,m", "a1560558b701ae55e943b7f49f1f9699": "\\Delta E_i", "a156301d3d451ad36fe9da1d6cb755df": "= \\frac{i \\Psi^\\prime}{z \\lambda} e^{-ikz} \\int_{-\\frac{a}{2}}^{\\frac{a}{2}}e^{-ik\\left[\\frac{\\left(x - x^\\prime \\right)^2}{2z}\\right]} \\,dx^\\prime \\int_{-\\infty}^{\\infty} e^{-ik\\left[\\frac{y^{\\prime 2}}{2z}\\right]} \\,dy^\\prime", "a15646d67e90545b57471e2cc7409dca": "(p,w,\\beta) \\in Q \\times \\Sigma^* \\times \\Gamma^*", "a1564d8ec3402978729a27eded420bd2": "wfe/({P}.\\sqrt{T})= [wfe/({\\delta}.\\sqrt{\\theta})] * (\\sqrt{288.15}/{101.325})", "a156537a059171ad484061479d21662f": "\\left \\{ \\cos \\left( \\frac{\\arccos x}{3} \\right) : x \\mbox{ is constructible} \\right \\}", "a1569ff0bb88868d3b819dc23423dad1": " z \\mapsto z^{k_1}, \\dots, z \\mapsto z^{k_n} ", "a1573e367005b1ff04ba480975a87948": "s_2 = fghhhkll", "a15805bb1ff38c82b58ab68568b73700": "L(\\lambda)", "a1584d0c220d06e60ae72e1cfbff8105": "a=\\mu(m_1-\\mu),\\; b=(n-\\mu)(\\mu+m_2-n)", "a15855504c4089c41d5eb7bfafca42cf": "w'=s_\\gamma w", "a1588678a478bcab9a52f8b9bc517410": "x(t_0)", "a15887850db7face073c1a3bfecf9763": "|A_i| = r^{\\ell_n-\\ell_i}.", "a158b67caf3690b1b2cb2af00920f6df": "x - 2", "a158ffc160154ad6a871a4060fcbba37": "E < \\operatorname{min}\\{ V( r \\to - \\infty ) , V( r \\to + \\infty ) \\}", "a1597f87c078588c57553e08b2678e79": "\\mathbf{v'}= \\mathbf{v} - \\mathbf{u} \\, .", "a159878f8fded82c252dcb8c2c9a4d3a": "\\sigma^2 = \\operatorname{var} (X_i)", "a1599134d2a053df0d92a608624c93bd": "Q_n = Q_i * MF * CTL * SF * SW", "a1599fb8a2f81751f1b85ae85405da19": "\n\\; _{p+1}F_{q} \\!\\left( \\left. \\begin{matrix} -h, \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) = h! \\; \\frac{\\prod_{j=n+1}^p \\Gamma(1 - a_j) \\prod_{j=m+1}^q \\Gamma(b_j)} {\\prod_{j=1}^n \\Gamma(a_j) \\prod_{j=1}^m \\Gamma(1 - b_j)} \\times\n", "a159cb3cdfadca21bf18a9ef1e22ebad": "I=[0,1],", "a159ffaa58f99aadcb2f0bb5c53082e4": " \\Gamma = P/S ", "a15a2ef67d9d954ac6b16d5af98a486c": "{d (\\rho \\phi ) \\over d t} + div(\\rho \\phi u) = div(R_\\phi. grad \\phi) ", "a15a31994bf7cc2f650428b98599ee74": "\\begin{align}\n \\psi &\\rightarrow e^{iq\\phi(x)} \\psi \\\\\n A &\\rightarrow A + \\nabla \\phi.\n\\end{align}", "a15a3cad57dcd2c1385fd62df1241ecb": "{\\bar{T}}_7", "a15ad4682f7d14c84099d1344a12d536": " {\\rm{CO}}_{\\rm{2}} + H_2 \\Leftrightarrow {\\rm{CO}} + {\\rm{H}}_{\\rm{2}} {\\rm{O}} ", "a15b0a09a1c0eb3a60eae808fa8862cb": "B_2(x)=x^2-x+1/6\\,", "a15b0c631dcdba269a4b2de9a2dcb618": "B \\times \\mathbb R", "a15b17b2261ac44359af93c938fd6e1f": "Y_{10}^{-3}(\\theta,\\varphi)={3\\over 256}\\sqrt{5005\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(323\\cos^{7}\\theta-357\\cos^{5}\\theta+105\\cos^{3}\\theta-7\\cos\\theta)", "a15b4beb95fd2adae888dfae2e19177d": "\\mathbb A^d_k", "a15bb04eed0d7df36609d07615239fcd": "\\psi_b", "a15bb16ffa1c91234eb4088a32247233": "(\\cosh x + \\sinh x)^n = \\cosh nx + \\sinh nx", "a15bdf344efb5b2d6b47f4c645fea21b": "K=-1/R^2", "a15c4507fee06845bcbde5f3da3d8d5c": "|z| = (z^* z)^{1/2}.\\,", "a15c65fdadb7d4a98ac8b638648e4a80": "\\bar{\\mathbf{A}}", "a15c794dc1b2adf66d07ce3ce302e8c0": "\\Delta G_v", "a15ca13a4416edfc7c7e162a189e5dfe": "\\operatorname{E}\\left(\\frac{k}{n}\\right) = \\operatorname{E}\\left[\\operatorname{E}\\left(\\left.\\frac{k}{n}\\right|\\theta\\right)\\right] = \\operatorname{E}(\\theta) = \\mu", "a15cccc53e6ba4189a8539c18541a18f": "\\lim_{t\\rightarrow 1}\\gamma(t)=e^{i\\theta}\\in S^{1}", "a15d0817749682d02a2101ddffa85d1c": "\\partial f_{\\#}\\left( \\alpha \\right) = f_{\\#}\\left(\\partial \\alpha \\right) = 0", "a15d23f7d6c42a7f793aae812a094b0b": "\n\\Lambda = \\sqrt{\\frac{h^2}{2\\pi m k T}}\n", "a15d5d0875dba6ef611a17f7cd98d1fe": "\\mathbf{S}_t=\\mathbf{S} \\times \\mathbf{S}_{xx}. ", "a15d69808c2359be3bc6836143abe9d9": "\\lnot F", "a15dc3fdb3addd8533cc5ace354b06e4": "r_g=2M", "a15de35f28ad1d2c47ec57ae1f4c1bec": "\\sum_{\\mathbf{k}} \\longleftarrow \\frac{N}{\\Omega} \\int_\\text{BZ} d^3\\mathbf{k}", "a15e1bfe69ffca1efba94d99a5553e35": "\n\\begin{align} \n& V_{\\text{obs, r}}=-R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}d\\frac{\\sin\\left(2l\\right)}{2} \\\\\n& V_{\\text{obs, t}}=-R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}d\\frac{\\left(\\cos\\left(2l\\right)+1\\right)}{2}-\\Omega d=-R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}d\\frac{\\cos\\left(2l\\right)}{2}+\\left(-\\frac{1}{2}R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}-\\Omega\\right)d \\\\\n\\end{align}\n", "a15e698ce58827e9cb0a40c562156bea": "s_2 = s_1", "a15e9a2d75dd3ad1e5b36a2a13641800": "\n M_{xx} = -EI~\\frac{\\partial \\varphi}{\\partial x} \\quad \\text{and} \\quad\n Q_{x} = \\kappa~AG~\\left(-\\varphi + \\frac{\\partial w}{\\partial x}\\right) \\,.\n", "a15eb0b2f703d36b3f51a48f4f234170": "\\{1, ..., S(c)\\}\\ ", "a15f1b31a842db5283bf17cfe527f5b0": "c:1{\\to}C", "a15f62293cea77edca4a810e87ed989e": " X + Y \\rightarrow 2Y", "a15fb0449c31a5c8d2b55ac60ae3cf57": "\\varphi-1", "a15fda5331218e41f76a749880c5c2f8": "s\\in S", "a15fe05da80d0e06fd7a5f2eb2888488": "e_n \\sim \\operatorname{Logistic}(0,1),", "a160696eedad76e68a11c97abccf5c5f": "\\mathbf{F}_i^*=-m_i\\mathbf{A}_i,\\quad i=1,\\ldots, n,", "a1608d7efc70d38a683ff073406f058f": "\\bar\\psi \\mapsto \\bar\\psi \\lambda^{-1}", "a1609ddcbf181d93245dca8567ffdc58": "\\Lambda = \\frac{M^2}{a^\\frac{3}{2}}\\,k", "a160d38ce7de6f88469d1828d3d83720": "\\kappa\\,\\hat{=}\\,0", "a160db50266cc1fd1d44af9f412d7dd2": "\\mathcal{L}\\{f^{(n)}\\}=s^n\\mathcal{L}\\{f\\}-\\sum_{i=1}^{n}s^{n-i}f^{(i-1)}(0)", "a161a2d593bf3259d526c2cab3b4ff5a": "S_1 = C^{-1}C \\setminus \\{\\varepsilon\\}", "a161b26a7b271fa0ffe64f41f82457b2": "X \\mathbf{\\operatorname{<}} Y", "a162083fad6239ae9fb613e37fd707ab": "C_h = s(1-\\cos \\theta)", "a162183c6abb1f6bcfadfec4abb33f2d": "\\mathbf{P}(\\mathcal G)", "a162355f47ccb4ce24bedf8e0252323f": "\\Gamma_i = \\left(\\Pi_{i+1}+\\cdots+\\Pi_p\\right),\\quad i=1,\\dots,p-1. \\, ", "a1627a9956cf861e19c02dbeed3126c2": "g'(x)", "a162a45c5fa42b06d8fb7acb1e08686a": "L \\left (x_1, \\cdots, x_N, t \\right ) =\\int_{u_1=-\\infty}^{\\infty} \\cdots \\int_{u_N=-\\infty}^{\\infty} f_C \\left (x_1-u_1, \\cdots, x_N-u_N, t \\right ) \\cdot g_N \\left(u_1, \\cdots, u_N, t \\right) \\, du_1 \\cdots du_N.", "a162a63dd85e7d8352ada6fe30db8192": "\\sigma_T = {\\sigma_{T_{cal}}[1 + \\alpha (T - T_{cal})] }", "a162dc9b0b943f6d9b5aa72c0669a8e2": "\\Delta^n x^m = \\sum_{k=0}^n (-1)^{n-k} {n \\choose k} (x+k)^m", "a162fec018acd037d033146222abbbaa": "x = \\Delta L. \\,", "a1632c3ac0635b1609d8ae53f3a80539": "s^2 = - c^2(\\Delta t)^2 + (\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2\\ .", "a1632e113a0def8bf47705f4044b8ed6": "n=|n_1|+|n_2|+\\cdots+|n_K|", "a1633bd57b24687fd39140c6fddb3633": "\\sin^2 x + \\cos^2 x = 1 \\ , ", "a1636564217b06e1226087fe9b9ead7c": "\\textstyle \\diamond", "a163805e4c1b1d1cdd882c01c49b0e70": "\nx=\\frac{-b \\pm \\sqrt {b^2-4ac\\ }}{2a},", "a163e418540c470b408e7107db0c2d00": "q(\\mathbf{x}^0)", "a1642569641aae7d15ff3a61d04360c4": "X^\\flat = g(X, -)", "a164b980499c55773d0455e6d525e7b7": "e^{\\ln(x)} = x \\qquad \\mbox{if }x > 0\\,\\!", "a164dc5efaad951f9f172617c02545f2": " Q = a_D + b_D P \\,", "a164ebd7187b486e7821fe6bb554f04d": "\\lambda(\\varnothing) = 0. ", "a164f4f389c63f975986eba99d9b4181": "\\sqrt{k/\\phi}", "a164f88eb1719c9d6da11ecb522dd966": "\\displaystyle F_t=\\lim_{z=0} \\gamma_-(z) \\lambda_{tz}(\\gamma_-(z)^{-1})", "a1650c301da79c31d34dde87612a6deb": " \\alpha=a_0+a_1\\mathbf{i}+a_2\\mathbf{j}+a_3\\mathbf{k} ", "a16528df4d5a25ace9ab959394b8918c": "n \\cdot (\\nabla\\times\\mathbf{u}) = 0", "a16559ed65b6e68649ba3cdb157eed55": "k_{sp} = \\sqrt{\\frac{-(A+3B\\phi_{in}^2)}{2\\kappa}}\\;,", "a1656cfaefa47bc524be6fb901aec009": "Q \\in S", "a1658d556ff004bf679816790371e485": "\\mathbf{B} \\equiv \\mathbf{B}_\\text{el} = \\mathbf{B}_\\text{el}^l + \\mathbf{B}_\\text{el}^s", "a165d8906bc7ae6f66a2491ba1ec7d91": " T = 2\\,\\Delta t = 9 \\times 10^{-3}\\text{ sec} ", "a165f63264ed74b141d1efadf17a2441": "\\partial/\\partial x^\\alpha ", "a1660c9413e36b52251baad91a5123e7": "x_1=0", "a1662538db656ac5e281ec0a761a054d": "0\\leftarrow\\lambda_w\\ll R_c", "a166348ec58327d3a28160c617029872": "T = 0", "a1663fd0852c8e4266679945451bad0d": "{\\rm Pr}_z \\Bigl( \\bigvee_i A(x,y_i \\oplus z) \\Bigr) \\le \\sum_i {\\rm Pr}_z (A(x,y_i \\oplus z)=1)\\le m \\frac{1}{3m}= \\frac{1}{3}.", "a1665a9874779464c045be0c85a68e09": "2y^2 = x^3 -x^2 + x", "a16675bfa745b59be79d49f97b9addfd": "M+dM", "a166cfec2ed75080a54df660d30179a1": "T+R=1.\\,\\!", "a166d4d3c800e72a8376f653f4eb6421": "\\langle df, dr\\rangle = \\frac{\\partial f}{\\partial r}", "a166e77659291a64872582e6d36d61c4": "g(\\tau)", "a167b15b1c5d8afbe58733670e8db73e": "S = \\frac{\\Pi}{T}, ", "a167b8366ce9e3f16ae6513461b2b51d": "0\\leq u(t_1, \\ldots, t_l) \\leq g(t_1 + \\cdots + t_l)", "a167bdbd41e716ac0b9e88b4d0298d27": "\\tau_{a b}\\,", "a167c7999f0eed84fde9cb036a1a38d6": "\\frac{1 + {\\scriptstyle\\frac{1}{2}}z + {\\scriptstyle\\frac{1}{10}}z^2 + {\\scriptstyle\\frac{1}{120}}z^3}\n{1 - {\\scriptstyle\\frac{1}{2}}z + {\\scriptstyle\\frac{1}{10}}z^2 - {\\scriptstyle\\frac{1}{120}}z^3}", "a167ecb72046d45d2d6959f8de9a2d72": "\\pi_1(Ff) \\to E(R)", "a168e383c71a7717196332759767dc59": "\\mathbf{R}=R_{ij}\\mathbf{e}_i\\otimes\\mathbf{e}_j\\,,", "a1690de184e98d20bf8809c92029f70a": "\\Omega=\\nu^2+2\\sigma^2", "a1699ea08f5f02305bcf97c824bb5ca8": "k_2 \\gg \\ k_1C_A, k_{-1},\\text{ so }r \\approx k_1 C_A C_S.", "a169d17c10d32d3fb2bb56fde0f9317f": "3^\\frac{5}{13}", "a16a15c34124bbfe4f78d4c0041ef1e7": " a^6 - b^6 = (a + b)(a - b)(a^2 - ab + b^2)(a^2 + ab + b^2).\\,\\!", "a16a2ef8a7afb5015089feb1606a89f3": "\n\\bar{h}^{i j} (t,\\vec{x}) \\approx\n-\\frac{4}{r}\\, \\int\\, x'^i x'^j \\nabla_k \\nabla_l \\tau^{k l} (t-r,\\vec{x}')\\, \\mathrm{d}^3x'\n", "a16a339dfcdfd37dc76d512d468dc371": "\\frac{2m-\\mu}{\\mu}", "a16a6971dff6192fccd7ddf413b22ebf": " \\Gamma_{a'} = \\gamma_{a'} \\otimes \\sigma_3 ~(a'=0, \\dots, d-1) ~~,~~ \\Gamma_{d} = I \\otimes (i \\sigma_1),~~ \\Gamma_{d+1}= I \\otimes (i \\sigma_2) ", "a16a92226f7af0f9d488623fddae400e": "F\\colon \\mathbb{R}^m \\to \\mathbb{R}^n", "a16ae2a0f08c2533356854d7cf4c02d2": "\\max[ (S-K) , 0 ]", "a16b158b690ee36b9efd4a1525c4345a": " \\vartheta(z,\\tau) = \\vartheta_3(z,\\tau) = 0 \\quad \\Longleftrightarrow \\quad z = m + n \\tau + \\frac{1}{2} + \\frac{\\tau}{2} ", "a16b37d0ae150eb81bb8721a75f32a9d": "T_d=\\frac{1}{G^o}.", "a16b3a96646e6edcf87c3e131b17698d": "r=2\\sqrt{\\frac{(\\sigma-uvx)(\\sigma-vxy)(\\sigma-xyu)(\\sigma-yuv)}{uvxy(uv+xy)(ux+vy)(uy+vx)}}", "a16b48e8f011c1f1d7b036e3f435adb3": "\\phi: R \\to S", "a16be74bc1d62ba8bd14d3970a31681b": "P(r = \\nu_1 k ; \\nu_1 n, p) \\ge P(r = \\nu_2 k ; \\nu_2 n, p)", "a16c6cd39f03268c342bb55d9a1bd7c1": " \\operatorname{Var}(X \\mid X8\\,", "a16ccfc971ec4d6709f8b4f971d60b18": "\\Delta y=f\\left(x_1 ,x_2 ,\\ldots ,x_{i,1},\\ldots,x_n \\right)-f\\left(x_1 ,x_2 ,\\ldots ,x_{i,0},\\ldots,x_n \\right)", "a16d10452955cc8929983439df1732d9": "0=A_{21}n_2+B_{21}n_2 B_\\nu(T)-B_{12}n_1 B_\\nu(T)\\,", "a16d109874cef99e5507e4917d67d18e": "\\mathrm{NPV}_{1,in}", "a16d18db728a71b72e86d3acd6ac838f": "T(w) = \\frac{\\int_{w/2}^{1} \\int_{0}^{\\sqrt{1-x^2}} f(x,y) f^*(x-w,y)dy dx}{\\int_{0}^{1}\\int_{0}^{\\sqrt{1-x^2}}f(x,y)^2 dy dx}", "a16d40584095dda4a9b9fb008aa3ca6c": "N \\le 10, 000", "a16d4109da6dbb49ecd6b5395161746e": "N_f\\,", "a16d878a43ddb7d9e5a15ecc74551789": " G_i(x,y) \\leq 0", "a16d8a7336469a149b3f9c56b022aa2f": "r = x - y^n", "a16d93131fcf8741894c4be40d807a21": "g_1 \\in \\left\\{ 2,...,P-1 \\right\\}", "a16da2e57b3a13ff943d86a436890e84": " P_{\\rm d} = \\frac{N}{d} = \\frac{25.4}{m} = \\frac{\\pi}{p} ", "a16db30e83b3324a21a857da14777492": " P(\\limsup_n \\frac{S_n}{\\sqrt{n}} > M) \\geq \\limsup_n P(\\frac{S_n}{\\sqrt{n}} > M) = P(\\mathcal{N}(0, 1) > M) > 0", "a16dbc33f9d081120289f4e792dbc069": "\\int_{T_0}^T \\frac {C(T^\\prime,X)}{T^\\prime}dT^\\prime = \\frac {C_0}{ \\alpha}(T^{ \\alpha}-T_0^{ \\alpha}).", "a16df1549c797a54a0da64abbd75bbfb": " \\frac{\\partial u}{\\partial t} = \\Delta u ", "a16e0e9121f46e4d1df70ba7cbe69652": "\\begin{align}\n(\\delta\\mu)^2=(\\mu_X-\\mu_Y)^2\n\\end{align}\n", "a16e47c7c307ea2ddc20898902b47a94": "(s+rb)a+(t-ra)b=c.\\ ", "a16e5947a14e8548323006dc5a8915b6": " ((12)(34)) (1 \\lor 3) \\to 2 \\lor 4 ", "a16e6a9591dfad421c04fcd8ed9c69a6": "{\\sigma_u} \\approx {\\H_V}*c \\approx {\\H_V}*{3.33}", "a16ea00e42aa166ac4edb42fe8f60ef7": "nF\\Delta E^\\circ = RT \\ln K \\,", "a16ea04acc40103203310be8efda3f54": "\\ln q^*(\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k) = \\ln p(\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k) + \\sum_{n=1}^N \\operatorname{E}[z_{nk}] \\ln \\mathcal{N}(\\mathbf{x}_n\\mid \\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k^{-1}) + \\text{constant}", "a16eda55499ee83efd61e48114ac8f5b": " P\\left ( \\mathbf{x} \\land y \\right ) = {1 \\over N} \\sum_{i=1}^N \\, \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) \\, \\sigma \\big ( \\left \\vert y - e_i \\right \\vert \\big )", "a16edce08d20dc6a81ba71fc8563f804": "\\mathrm{cf}(\\kappa) = \\inf \\left\\{ \\mathrm{card}(I)\\ |\\ \\kappa = \\sum_{i \\in I} \\lambda_i\\ \\mathrm{and}\\ (\\forall i)(\\lambda_i < \\kappa)\\right\\}", "a16f2c00d24a160e91b97893cb7b36a4": "C_{yy}(y_i)\\ne \\phi", "a16fa6228ac59e5f0f86455a2caa92ff": "\\Lambda= \\frac12 \\left|x_i-m_j\\right|^2 I(i-j=0) + \\gamma \\left|m_i-m_j\\right|^0 I(i-j=1)", "a16fe8be5e825693447b04d92e4bfce4": "\\pi : G \\longrightarrow \\mathcal{B}(\\mathcal{H}) ", "a17006754e7d270f6e0fea2ecb901b93": "\\overline{C}\\,", "a17022c3643548e48f666c66236fad49": "2i+1", "a1707c99d8476aad39f2f231590cea28": "E_0 = c_0 B_0", "a17149651ffb9d1e14d6a57aa077ec4d": "e_n, p_n", "a171ba258d82033aa87673431b737c0b": "h_{ab}=(\\mathcal{L}_X g)_{ab}=X_{a;b}+X_{b;a}", "a171f800ea98aa11bec5b9aef8a6b707": " \\frac{1}{\\cos(d/R)} ", "a1720284dd867a73f8b9e2cee35cc1df": " t \\in \\mathbb{T} ", "a1722f21b1cba4ed5abbd7540427d02c": "T(t,r)", "a172471dc0819462c82e5509847bb796": "D_{j}", "a1729b5b68cdd50e67c018fd8d66ef74": " \\frac{f'}{f} \\! ", "a172bff453526d53431080aba531938c": "i_{G/H}(\\sigma) = {1 \\over e_{L/K}} \\sum_{s \\mapsto \\sigma} i_G(s)", "a172c4b1b09e387ba2b6725cc78fa583": " \\rho\\, _{liquid} - \\rho\\, _{vapor} \\approx \\rho\\, _{liquid} ", "a172c92722ed41872336c3aaa6d35f7b": "n = -{v dP \\over P dv}", "a172d97c6ee5b99142e275591da7a32c": " m_k = \\sum_{i=1}^k \\lambda^i \\left\\{\\begin{matrix} k \\\\ i \\end{matrix}\\right\\},", "a172e635645a6e97d0c8a5f5f2c67c81": "G=\\,", "a172efada28d956ebd4596a8660aeac5": " {z \\choose m} {z\\choose n} = \\sum_{k=0}^m {m+n-k\\choose k,m-k,n-k} {z\\choose m+n-k}", "a172f9fcbb04485d3af59dde833cf93c": "\\xi\\in (-\\infty,\\infty) \\,", "a17303a29d6f054bda1945de2c05f82a": "\\xi\\in[0,1]", "a173956dd89a8629183bcca3dea3058a": "\\scriptstyle\\gamma'\\, =\\, \\pi/2 - \\gamma", "a173a6c37df692888a4b2019607e2330": "E_T= E_{\\rm x} = E_{\\rm y} = (C_{11}-C_{12})(C_{11}C_{33}+C_{12}C_{33}-2C_{13}C_{13})/(C_{11}C_{33}-C_{13}C_{13})", "a173d851b0c85dcda0c4081e7fd7cbac": "\n w(r) = -\\frac{qr^4}{64 D} + C_1\\ln r + \\cfrac{C_2 r^2}{2} + \\cfrac{C_3r^2}{4}(2\\ln r - 1) + C_4 \n", "a17406bfaeb7fe41faaa72ad3ee8a3c7": "d(x)+1", "a17418382aed99f32dde5c90427e9e8d": " \\Sigma _{22} - \\Sigma _{21} (\\Sigma _{11} )^{ - 1} \\Sigma _{12} ", "a1747bb7d476dab07eaeaa90c23672dc": "2.71< e <2.72", "a174e8543707b1afd56f9248c088c7d3": "U_i = [u_{i1},u_{i2},...,u_{im}]", "a175045213f8639b8f85bcff082860bf": "\\omega =\n\\left\\{\n \\begin{matrix}\n \\frac{q}{ \\sin( \\theta/2 ) }\n , & \\mathrm{if} \\; \\theta \\neq 0 \\\\ \n 0, & \\mathrm{otherwise}\n \\end{matrix}\n\\right.\n", "a1753f291e804c6d95beb1449e894d99": "\\ k \\ ", "a17551efa0f2598802708f7eb05f862c": "h=Nu_c \\frac{k}{d}", "a175f3084bfeb4387f9cbb212085dce8": " \\frac{d}{dr} B (r) = -DB.", "a1765d10b897f27e752e6fef030a8cc5": "\nCoDIAK_{t \\to t+1} = 1 + \\mathcal{B}\n", "a176a234ed2e83c028c09afa764de299": " n-1 ", "a176c491e05e4571ad4af9b0982f02a2": "p>10^{75}", "a1771445ada60f5e875172ff1de1c6cc": "L(x) = \\begin{cases} 1 & \\text{if}\\;\\; x = 0\\\\ \\quad & \\\\ \\displaystyle \\frac{a \\sin(\\pi x) \\sin(\\pi x / a)}{\\pi^2 x^2} & \\text{if}\\;\\; 0 < |x| < a \\\\ \\quad & \\\\ 0 & \\text{otherwise} \\end{cases}", "a17737f9980c6423b6c55e50343709af": " \\frac{1}{\\cos \\theta}\\! ", "a177d9723638a72fb6ef10283cfb31fd": " \\mathbf{[T]}=\\begin{bmatrix} 1 & 0 \\cdots 0 \\\\ T_{21} & T_{22} \\cdots T_{2n} \\\\ \\cdot & \\cdots \\\\ T_{n1} & T_{n2} \\cdots T_{nn}\\end{bmatrix}", "a177f7bc871650cfa5527063477cebd3": "k_{\\nu, s}", "a178198ddc9bb2276e2f68f8d746cb52": "\\int \\tan (x) \\,dx = -\\ln{\\left| \\cos (x) \\right|} + C", "a1786b2d1be91bd695d5512308298bcd": "z\\mapsto k\\,z^{k-1}", "a17871dda60cab40a545fc204eae9754": "GapSVP_{100\\sqrt{n}\\gamma(n)}", "a179005477169b4fca4eea6be48ae43f": "\\phi:F\\rightarrow F", "a179842721908ca8ff4af9aa26292708": "f \\otimes v \\mapsto g ", "a17987657ebe38c124185a1d3bc039c9": "p^\\prime_b = \\mathbf{H}_{ab}p_a \\, ", "a1798f3d0b6e2f048fae3b35ea90a336": "\\begin{align}\n\\text{sample mean} &=\\overline{y} = \\frac{1}{N}\\sum_{i=1}^N Y_i \\\\\n\\text{sample variance} &= \\overline{v}_Y = \\frac{1}{N-1}\\sum_{i=1}^N (Y_i - \\overline{y})^2 \\\\\n\\text{sample skewness} &= G_1 = \\frac{N}{(N-1)(N-2)} \\frac{\\sum_{i=1}^N (Y_i-\\overline{y})^3}{\\overline{v}_Y^{\\frac{3}{2}} } \\\\\n\\text{sample excess kurtosis} &= G_2 = \\frac{N(N+1)}{(N-1)(N-2)(N-3)} \\frac{\\sum_{i=1}^N (Y_i - \\overline{y})^4}{\\overline{v}_Y^2} - \\frac{3(N-1)^2}{(N-2)(N-3)} \n\\end{align}", "a17a7919a44a12b2d18d76956ae98c29": "M_{unit} = {q^2 \\over gy} + {y^2 \\over 2}", "a17a9f90499a16be77904a359e71db8b": "\\ F(K,L)", "a17aba1c8715cb8c7ac61e528db9fd24": " D^{\\epsilon}(\\rho||\\sigma) = - \\log \\frac{1}{\\epsilon} \\min \\{ \\langle Q, \\sigma \\rangle | 0 \\leq Q \\leq I \\text{ and } \\langle Q ,\\rho\\rangle \\geq \\epsilon\\} ~.", "a17b01f157db2f21311fc47c22b33f61": "\\frac{d\\tau}{dt} = \\sqrt{-g_{00}}", "a17b1a153a6cd7af50c6fcb4083bb0e7": "U \\setminus \\{a\\}", "a17b823e00512cd6b9a7c1c4d3b30eba": " \\{z^{-1},...,z^{-(M-1)}\\} ", "a17b920a152e2803eb5d533e7f68a360": "\\frac{1}{f} = \\left(n-1\\right)\\left( \\frac{1}{R_1} - \\frac{1}{R_2} \\right).", "a17bdd8f93181175714596ee5a863b9c": " \\scriptstyle \\beta_1", "a17bec83ab26ab9efa1c81f7d59d2919": "\\scriptstyle t=t'=0", "a17c27b0870e0e9b4a4aa1703a417988": "5/4", "a17c2c915af2e67667712681ca6b75a2": " \\left |\\{x\\in\\mathbf{R}^{n}:Mf(x)>\\lambda\\} \\right | \\leq \\frac{C_n}{\\lambda}\\|f\\|_{1}", "a17c75c16bba6fa3d7b36be929e2b266": "T \\boldsymbol{x} = \\boldsymbol{y}", "a17c8350b2ada63c11a69ba2e44bcea1": "\\displaystyle{\\widehat{f}(\\xi)= {1\\over \\sqrt{2\\pi}} \\int_{-\\infty}^\\infty f(x) e^{-ix\\xi} \\, dx.}", "a17c92695f2b90358dc4b34cefd96ff3": " A = S B \\bar{S}^{-1} \\, ", "a17cb0e68bcc366da5824f3e3885fd3d": "K(\\mathcal A) \\rightarrow D(\\mathcal A).", "a17cc77fdbd33e8ef9df566b9914d8ef": "g(\\mathbf{A}(a,\\theta, b, c)) = \\gamma(a^2 + b^2 + c^2)", "a17ce17a5144fb0ccaa2d20598cf0eb6": "X(\\omega) = \\pi [\\delta (\\omega +a)+\\delta (\\omega -a)],", "a17d1c01f95f9a9747f763fa57e0f8ab": "\\vec V_4", "a17e45de84a62743f7b943faf452093f": " V_t = Z_L I_t \\, ", "a17e65ed98d679ee2300cd70c74598c3": "\\Psi_3=\\bar{\\delta}\\gamma-\\Delta\\alpha+(\\rho+\\varepsilon)\\nu-(\\tau+\\beta)\\lambda+(\\bar{\\gamma}-\\bar{\\mu})\\alpha+(\\bar{\\beta}-\\bar{\\tau})\\gamma\\,.", "a17e77e4ad3f3795dfc0c74b83381a2a": "S=2\\pi rh+2\\pi r^2", "a17e9e6f808f662a6534bc79679b2f3a": " \\langle B\\rangle \\,=\\,\\sum_{i,j} a_i^{*}a_j\\, \\langle i| B |j\\rangle.", "a17ea4502864bfcc798366a43c590acc": "\\begin{array}{cl}\n\\bar{a}_p\\bar{b}_q\\bar{\\mathbf{e}}_p\\otimes\\bar{\\mathbf{e}}_q & = \\mathsf{L}_{kp} \\mathsf{L}_{\\ell q} a_k b_{\\ell} \\, (\\boldsymbol{\\mathsf{L}}^{-1})_{pi} (\\boldsymbol{\\mathsf{L}}^{-1})_{qj} \\mathbf{e}_i\\otimes\\mathbf{e}_j \\\\\n & = \\mathsf{L}_{kp} (\\boldsymbol{\\mathsf{L}}^{-1})_{pi} \\mathsf{L}_{\\ell q} (\\boldsymbol{\\mathsf{L}}^{-1})_{q j} \\, a_k b_{\\ell} \\mathbf{e}_i\\otimes\\mathbf{e}_j \\\\\n & = \\delta_k{}_i \\delta_{\\ell j} \\, a_k b_{\\ell} \\mathbf{e}_i\\otimes\\mathbf{e}_j \\\\\n & = a_ib_j\\mathbf{e}_i\\otimes\\mathbf{e}_j\n\\end{array}\n", "a17ed4b9948edf9f442be9e001d50b8c": "\\Lambda^{\\chi'}{}_{\\psi} \\,.", "a17eddd821cc2d1532e7e22e8b65f8f4": "\\psi=\\phi=\\frac{1}{2}\\sum{(\\theta^i)^2}", "a17ef7dd49ebe49d7a154248a6e5e12c": "A \\lor B := \\neg(\\neg A \\land \\neg B).", "a17f116ee93e3692d43191ae787cb6a5": "c = a\\downarrow = b\\downarrow", "a17f89ce4ae83a2973b113e51b8d25ec": "\nE = \\begin{bmatrix}\n 1 & 1 & 0 & 0 & 0 \\\\\n -1 & 0 & 1 & 1 & 0 \\\\\n 0 & -1 & -1 & 0 & 1 \\\\\n 0 & 0 & 0 & -1 & -1 \\\\\n\\end{bmatrix}.\n", "a17fed0d7372dc5045db4d75ddce23b8": "\nf = a_1 a_2^2 a_3^3 \\cdots a_n^n \\,\n", "a180300eaa7c3afc3d61589a5b2ef5a3": "\\displaystyle{f_m(t)=e^{imt}}", "a1803617833750fde04db9f739dcd95c": "\n\\alpha = \\frac{J}{Mc}\n", "a1807d9d4110b2c16dab672813a2baed": "\\frac{a-b}{a+b}", "a180d0b1e143e9ffffdbb62390164e14": "\\displaystyle \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{\\infty} f(x) e^{-i \\omega x}\\, dx ", "a180da24fa3013febcc9365126378a75": "\\ {F_{lat}} = {F_H} \\times cos(\\theta) ", "a180f0c218fd6b8d4021b9ffc7100de6": "value(x,a)", "a1815b6c454024b628bfcda540e4c0c7": "R_{\\mu \\nu}", "a1816436be55a5a21ca5a715b31a0e86": "(a,b) (c,d) = (ac, bd)", "a181a1934d49185a77e7f46801514681": "\nr_{pb} = \\frac{M_1 - M_0}{s_n} \\sqrt{ \\frac{n_1 n_0}{n^2}} = \\frac{M_1 - M_0}{s_{n-1}} \\sqrt{ \\frac{n_1 n_0}{n(n-1)}}.\n", "a181b9b781190073548aa3b66077f128": "\\psi\\circ(\\psi,1)=\\psi\\circ(1,\\psi).", "a181cd49c757ce52695424c2b47540c9": "(A\\mid(B\\mid C))\\mid[(E\\mid(E\\mid E))\\mid((D\\mid B)\\mid[(A\\mid D)\\mid(A\\mid D)])]", "a1823672e1f14525c9d35f53fbc45ef6": "Z_{in} = \\frac{V_1}{I_{in}} = \\frac{Z}{1-K}.", "a18297da3162aa9e49a608680202e863": "f(n) = \\Theta\\left(n^c\\right)", "a182d618bfeedd84ff09653c682a66f3": "|f_i|<1.\\,", "a18310702db426e01a37d78949aa884c": "_2^1\\text{S}^\\beta= {^{15}}\\text{N}^{14}\\text{NO}", "a1837073274c14f181e31e3eaa3214cd": "(X^i)_i", "a18372d75979387c06c11f5610d4aa83": "J,\\bar{g}", "a183fef8328268f9da87778548883175": "A_1BA_2", "a1841401e7c0174206028bfb8a7bdb1b": "j=1,...J", "a184195d7f802de7c2aaefdf12043421": "M_R:=\\max_{\\theta\\in [0,\\pi]} \\bigl|g \\bigl(R e^{i \\theta}\\bigr)\\bigr| \\to 0\\quad \\mbox{as } R \\to \\infty\\,,\\qquad(*)", "a18426c0989aaec3f526cf1ff8364203": " v \\wedge w \\mapsto \\tfrac14[v,w].", "a18427ac2960746a773db213ac00a2a3": "E =", "a1843c4b25404f1bbc2ba09f4693b317": "\nP(X \\in A \\mid Y \\in B) =\n\\frac{\\int_{y\\in B}\\int_{x\\in A} f_{X,Y}(x,y)\\,dx\\,dy}{\\int_{y\\in B}\\int_{x\\in\\Omega} f_{X,Y}(x,y)\\,dx\\,dy} .", "a184b3aa694307072500bd377b2f332b": " R_c = \\frac {R_x R_y} {R_x + R_y + R_z} \\qquad\n\nR_a = \\frac {R_z R_x} {R_x + R_y + R_z} \\qquad\n\nR_b = \\frac {R_z R_y} {R_x + R_y + R_z} \\qquad\n\n\\, ", "a184d98a79826b83c9c94016b2031774": "G(\\omega)= \\frac {1} {\\sqrt 2} \\sum_{k \\in Z} g_k e^{j \\omega k}", "a18506890a118fda7f0269d42d1e2959": "\\eta \\ll r \\ll L", "a1851e0ca89eb50c3293a390c9ef4aaf": "\\partial^2 u \\over \\partial t^2", "a1853727cbb4a65d97fdd819799b42bf": "E^\\mathrm{tot}(\\mathbf{x}_j,t)=\\sum_{n\\neq j}\n\\frac{E_n^\\mathrm{ret}(\\mathbf{x}_j,t)+E_n^\\mathrm{adv}(\\mathbf{x}_j,t)}{2}\n+\\sum_{n}\n\\frac{E_n^\\mathrm{ret}(\\mathbf{x}_j,t)-E_n^\\mathrm{adv}(\\mathbf{x}_j,t)}{2}", "a1856f91fe8ec67302e1b16dd6194bd9": "\\lim_{n \\to \\infty} \\inf_{G \\in \\mathcal{G}_n^d} \\lambda(G) \\geq 2\\sqrt{d-1}.", "a1859e5c0d2bb3c269a1eb1e13db9092": "(\\operatorname{arsinh}\\,x)' = { 1 \\over \\sqrt{x^2 + 1}}", "a185a6714700882c9cf160cb9ace7ffe": "r_{Airy} = 1.22 \\frac {\\lambda}{2\\mathrm{NA}} \\ , ", "a185c2964ea1b6d2a0cf494615c38a67": "B_0 = 100", "a18610ac26a264f59b7c12e2d4be67af": "\\alpha y_1(t) + \\beta y_2(t) = H \\left \\{ \\alpha x_1(t) + \\beta x_2(t) \\right \\} ", "a1862a9ec21a5288ee7d52d730a6a593": "\n\\sigma^2 = \\frac{n\\alpha\\beta(\\alpha+\\beta+n)}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}\n = n\\pi(1-\\pi) \\frac{\\alpha + \\beta + n}{\\alpha + \\beta + 1} = n\\pi(1-\\pi)[1+(n-1)\\rho]\n\\!", "a186aee5a98c1f9425bf430af1b0821c": "A_{12} > 0 ", "a186ec65e8284b6433a55899647cb4ac": "s(0)=t(0)=0", "a18737e37fb67adada40d0ae872b9c89": "\n \\begin{align}\n \\overline{\\bar a} &= \\bar a, \\\\\n \\overline{a + b} &= \\bar a + \\bar b, \\\\\n \\overline{a \\bar b} &= \\bar a \\bar b.\n \\end{align}\n", "a1876528dec88daa9de103b7030bda36": " x_i x_j = x_j x_i \\quad ", "a1877ccf85adee88970c7cdfb1b5fa6d": "\\mathfrak{P}^{69}", "a1879e846dd75b63cb825daf46038eb7": " X_i(t)", "a187a19b5e4502d0717f595949481cb2": "2^2 + 2 + 1 - 1", "a1882a13c137ddc8a241c99e6bce293b": "Y_{1 \\dots n}", "a1884a1698441e6f199e9d30ea0b48fd": " MV^2 \\approx m v_\\star^2 ", "a188bc680a2cc95b9f0266efd8be702a": "\\sum_{i\\in I} \\|x_i\\|^2 < \\infty.", "a18955dbd6c4e08aa7d762d0d8ecf190": "b_{\\nu, n}(x) \\ge 0", "a189770d55c3ff0c823a4c9eb300cde7": "\\frac{(r+1) S(N_{r+1})}{N}.", "a189abcfba85d4a5ed43ed4f9e07c055": " = \\left(\\frac{d^2s}{dt^2}\\right)\\mathbf{u}_t(s) - \\left(\\frac{ds}{dt}\\right) ^2 \\frac{1}{\\rho} \\mathbf{u}_n(s) \\ , ", "a189db771a798432d88bb1a3cf95d094": "P(\\omega_n)", "a189ed3f710cd51a5b2ff56291b81511": "A = (b \\cos\\gamma,\\ b \\sin\\gamma),\\ B = (a,\\ 0),\\ \\text{and}\\ C = (0,\\ 0)\\,.", "a189eedc878a6adfaaa56630228819ac": " C \\cap D \\,", "a18a7e6505a77c2108b20897c49fc9f2": "\\sqrt{\\sum_{i=1}^n(a_i-b_i)^2}. ", "a18a97f67c2558900250259415346bf5": "\\log w = \\log r + i \\theta", "a18a992052a37a8b382df8bc62fdb30b": "\\rho_{Out} = 0.07517*(1-{0.0035666*E \\over 528})^{5.2553}*({528 \\over T_{Out}+460})\\,\\!", "a18aa2137073fd22beeb7e03edc8ce9d": "\\mathbf b=0\\,\\!", "a18ab25a7bbf1713d7c0b9a37ed95bf1": "\\omega_s ", "a18b3bdf172c51bf662efee86c33ab3b": "\\epsilon_{eff}", "a18b6f588b4f525bedaf44bfdd45422a": "PR(A) = {1 - d \\over N} + d \\left( \\frac{PR(B)}{L(B)}+ \\frac{PR(C)}{L(C)}+ \\frac{PR(D)}{L(D)}+\\,\\cdots \\right).", "a18c0b49d087a1b37c26d87b1ac5b012": "\\Phi \\vdash \\phi", "a18caec2519a296b55cee025fab3f783": " u(t,x) = \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty e^{-\\alpha \\xi^2 t} e^{i \\xi x} d\\xi. \\,", "a18ceab7612c2202bbb823405bdc4156": "w_{i+1}", "a18cee7a87731916dc2852cd6042c5f3": "p_G\\left(z=\\eta\\right)-p_L\\left(z=\\eta\\right)=\\sigma\\kappa,\\,", "a18cf5f4f6678801e44ac803688a1b14": "I_\\nu(z)=\\sum_{k=0} \\frac{z^k}{k!} J_{\\nu+k}(z);", "a18d1f639c159183cee9941c33cdf91b": "\\tau_{i}", "a18d3cfd9ee5e53b691d1eb98076f65f": "G \\cong G_2 \\oplus \\bigoplus_{p \\equiv 1 \\bmod 4} G_p.", "a18d458c6b7dcbb026f4d74b1c28e3b2": "\\phi_0=M/B_0", "a18d6c39231dfdfb09dbfd37094544eb": "\\partial_n\\circ \\partial_{n+1}=0", "a18d759ce63add35f104d166291d1b22": "\\langle X, Y \\rangle = X Y - \\operatorname{E}Y\\cdot X - \\operatorname{E}X\\cdot Y+ 2(\\operatorname{E}X)(\\operatorname{E}Y) - \\operatorname{E}(X Y).\\,", "a18d979c6e4eca68a0c62777f99b58d4": "P\\mbox{acc}:\\mathcal{H}_Q \\to \\mathcal{H}_\\mbox{accept}", "a18dcbf7d8e4e8e86b1bbfec8fe55ef3": "x^5 - x + 1 = 0\\,", "a18df1b5738d6a8872fb66e9b3228b53": " \\displaystyle{m(g,z)=\\det (CZ+D)}", "a18e1e38de62b917d6932c0da501d9e9": "r_c", "a18e4f00fd0697ea1469e7569bd9bfea": "q \\in Q, a \\in \\Sigma \\cup \\left \\{ \\varepsilon \\right \\}, x \\in \\Gamma", "a18e8cc3c3de4df80eb78b730be146ff": " \\lambda_{u,d,e}^i ", "a18ef5af3e74c606ea68a4a63ff49a0b": "{\\mathrm{d}V}=A.x", "a18f230287b73d13111aef18fa54db51": "\\sum_{j=0}^\\infty \\frac{j\\lambda^j}{(j!)^\\nu Z(\\lambda, \\nu)}", "a18f73ffc6132d87fb4a66dd8a099055": "= \\operatorname{tr} (\\Gamma)", "a18f8b1a826b1629477bc1fd811104d8": "(v(x_1),\\ldots,v(x_n)) \\in R", "a18f8b7e574cc0f851c3d7872cc52aa3": "\\mu_{\\mathrm{eff}}=\\sqrt{\\frac{\\mu_a}{D}}", "a18fbed234dc2821852787466851b2b4": "\\beta_a\\,\\xi^a", "a18fe1a4ea920a3c8d322312e82c7f61": "\\chi_{\\text{1}}", "a18ff43a765fb6aded9a85334a4a9eae": "\\begin{align}\\Box((call \\lor \\Diamond open) \\to \n& ((\\lnot atfloor \\lor \\lnot open) ~\\mathcal{U} \\\\\n& (open \\lor ((atfloor \\land \\lnot open) ~\\mathcal{U}\\\\\n& (open \\lor ((\\lnot atfloor \\land \\lnot open) ~\\mathcal{U} \\\\\n& (open \\lor ((atfloor \\land \\lnot open) ~\\mathcal{U} \\\\\n& (open \\lor (\\lnot atfloor ~\\mathcal{U}~ open)))))))))))\\end{align}", "a1900135763534ae94ab4947a8915fb5": "\\eta = \\begin{pmatrix}-1&0&0&0\\\\0&1&0&0\\\\0&0&1&0\\\\0&0&0&1\\end{pmatrix}", "a190c3709d324bb5054b5d6b8b0cd76f": " \\mathrm{exposure\\ time} = \\frac{\\mathrm{shutter\\ angle}}{360^\\circ}\\times\\mathrm{frame\\ interval} ", "a19142a4513ae77f9091ac3178d1b3b3": "\\textit{state}(\\textit{open} \\circ s \\circ \\textit{open}, t)", "a191a9ef46e0dc291ff99eddc0909157": " \\det(H+1/4+s(s-1)) ", "a191bdd89bd3f6aecdb759b8ba9850bc": "\\hat{W}_{2}^{I}(z,x)\\geq 0", "a191c0e95754e7571266bf15ff3a0668": "\\nu = \\nu(q) = \\frac{1}{\\sqrt{1-q}} ,", "a191c62a74bbdaf5aa8fa0eb211ee940": "m \\over n", "a191f420f9022ce9a8cc49db77f0a4e4": "r\\in R", "a192056cc4e6cd663970eb8e1e1a7c12": "r^2=x^2+y^2+z^2", "a1929c3e686dd34520a5d88abb7fb202": " C_{\\infty} = \\frac {\\dot{m}}{K} \\qquad (10a)", "a193544dc6f8efb4f962cafb1f1618e4": "\\frac{1}{L_\\mathrm{total}} = \\frac{1}{L_1} + \\frac{1}{L_2} + \\cdots + \\frac{1}{L_n}", "a19368b252b1e8c31b83a0e74bf16c75": "\\lim_{n \\to \\infty}\\frac{\\pi((1+\\epsilon)n)-\\pi(n)}{n/\\log n}=\\epsilon,", "a193e83d8a2fa02ddceef33047bdbd66": "\\lambda^{*}", "a193ebc083b4745370f6f1343383d9cc": "m=n", "a193edff5f22508b5d0adba2125ab4ef": "\\bigcup_{i \\in T_1} M_i = \\cup_{i \\in T_2} S_{M_i}", "a193ef3aa688ef214e69d5d691659974": "a,b \\in F", "a193ef5293cd16938c63388c2ed28e3c": "W' = \\frac{L}{\\lambda} - \\mu^{-1} = \\frac{\\rho + \\lambda \\mu \\text{Var}(S)}{2(\\mu-\\lambda)}.", "a19415a0f5a11e6066f7dc4dfc338bd7": "(i,p)", "a194197e97909850aefde7bf569f24c1": "\\star \\mathrm{d}t \\wedge\\mathrm{d}y = \\mathrm{d}x\\wedge \\mathrm{d}z", "a1942a7f1f64d03b93e7e88d2f1575bb": "\nI(\\theta)=a_i^2 p_i(\\theta) q_i(\\theta).\\,\n", "a194b65f3bbbf85a029cedbdcf2ddabd": "\\frac {\\partial |V_\\mathrm T|}{\\partial x} = 0 ", "a195054e09ecf1a279621ba122f6e439": "{\\mathcal O}_{q_0}", "a195530c8dc4b889fe0638c30bb740f6": "s = dU_s/dU_p", "a195bd2d1eaeb0638ce5862913919bb0": "Q(x,y) = y^2 + 6x^2y - 4x^2y -24x^4", "a195d25f9bf041a903f5566e276322be": "|d(A,X)-d(B,X)|", "a19629bc79bc05702363e354f0f20867": "U_i=\\{ Z \\mid Z_i\\ne0\\}", "a19665eb1920df2ef8bb17c5f4f14fc8": "p(x|y)p(y)", "a196beee3be305ceceef648b4e343a83": "V_i(a,z)V_i(b,w) = V_i(b,w) V_i(a,z)=V_i(V(a,z-w)b,w).\\,", "a196ce3e52365664b7e9bf09fb2c3dd4": "x^\\prime = k\\ell\\left(x + \\varepsilon t\\right)\\!,\\;t^\\prime = k\\ell\\left(t + \\varepsilon x\\right)\\!,\\;y^\\prime = \\ell y,\\;z^\\prime = \\ell z,\\;k = 1/\\sqrt{1-\\varepsilon^2}.", "a196df1e72a658c1b109dfc5e5413ddd": "\\vartheta_{01} (z; \\tau) = -i \n\\int_{i - \\infty}^{i + \\infty} {e^{i \\pi \\tau u^2} \n\\cos (2 u z) \\over \\sin (\\pi u)} du.", "a196e08c291521e743fe3dc3b2711add": "I(z) = I_0 e^{-2z \\mathrm{Im}(\\tilde{k})}", "a196f8a6e74c8aa5894b001f5eac7d2d": "P \\subseteq [0, \\infty) \\times \\Omega", "a1970e91a403f003e14e7dce2aaa534f": "(4,1,2)", "a1972588690c1e9035dbf714cc8f641c": "A (\\mathbf{r}, t ) = A_o \\cos (\\mathbf{k} \\cdot \\mathbf{r} - \\omega t + \\varphi )", "a19767ddc856fa07a575c68f27b3e547": "\\Delta T(t)", "a1976ddabca1964cd3742fcc5d691f8e": "\\scriptstyle \\tan (\\delta'/2) = \\tan (\\delta /2) \\cdot \\sqrt {(1-0,5)/(1+0,5)} = \\tan (36,87^\\circ/2) \\cdot \\sqrt{1/3} = 0,19245\\;", "a19798fc38cd82c0512a9aa020226e12": "\\scriptstyle D_i(\\theta)", "a197ac4686a3f5e64a7dc4458a84661e": "c^2 = b^2 + a(a - 2b\\cos\\gamma).\\,", "a197cf15d0ba778199cb1e2ab584a6b9": "\\rho_m", "a197d6c6696bdd88d43a0dc0ecfa8b02": "P_1\\times_BP_2\\times_B\\times\\cdots\\times_BP_n", "a197eee0b04575799f15d02fc113eaa2": "\\displaystyle{Cf(w)={1\\over 2\\pi i}\\int_{\\partial\\Omega} {\\overline{f(z)}\\over z-w}\\, d\\overline{z}.}", "a1981bbbca2f44237987e36c895b3632": "\n [\\boldsymbol{\\nabla}f(\\mathbf{x})]\\cdot\\mathbf{c} = \\cfrac{\\rm{d}}{\\rm{d}\\alpha} f_\\varphi(q^1 + \\alpha~c^1, q^2 + \\alpha~c^2, q^3 + \\alpha~c^3)\\biggr|_{\\alpha=0} = \\cfrac{\\partial f_\\varphi}{\\partial q^i}~c^i = \\cfrac{\\partial f}{\\partial q^i}~c^i\n ", "a1982ac4d75c401f3fd67936b98caa5f": "\\kappa:\\Sigma^*\\to\\N", "a198614839918f2fe67e0013d2c1322a": "N\\gg K \\gg \\ln(N)\\gg 1", "a198cf094710c7ececc5e2aa1d03c6da": "\\begin{array}{rlll}\n\\text{mean} & = e^{\\mu + \\sigma^2 / 2} & = e^{0 + 1^2 / 2} & \\approx 1.649 \\\\\n\\text{mode} & = e^{\\mu - \\sigma^2} & = e^{0 - 1^2} & \\approx 0.368 \\\\\n\\text{median} & = e^\\mu & = e^0 & = 1\n\\end{array}", "a198db36c012fa148942e195216ab239": "a_{n-2}", "a198dd20cc68182791223929f7e37d25": "w(f) = w^{f}(f) \\leq w^{f}(f^{*})", "a199364fd914dfe59cd5eca14f65799f": "f(t) = 0", "a1999c8223d8ac50720000fbb3b367bd": "0\\le (a-b)^2=a^2+b^2-2ab,", "a199aa45ab237b944ee6fe8ec8e5936c": "\\overline{n}(f)", "a199ab91493a139bd71c75a2561af71a": "\\ldots,-2, -1, 0, 1, 2\\,\\ldots\\!", "a199bc70ca9d36787b070a18a6014bc6": "Z_{CPE}=\\frac{1}{Y_{CPE}}=\\frac{1}{Q_0\\omega^n}e^{-\\frac{\\pi}{2}ni}", "a199e70a308c05e2ea678f9b938cd2d2": "J_-|j\\,m\\rangle = \\hbar\\sqrt{(j+m)(j-m+1)}|j\\,m-1\\rangle = \\hbar\\sqrt{j(j+1)-m(m-1)}|j\\,m-1\\rangle.", "a19a2d1b34575c55c9ecbc203254525a": "\\bold{j} = \\rho \\bold{v}", "a19a5dca356260badffed8231a54bc1c": "VAG(x^3 -7x + 7,(1,\\frac{3}{2})) \\cup VAG(x^3 -7x + 7,(\\frac{3}{2},2)", "a19a9b1624a79239db4e6d54c37ab345": "A,B \\in z", "a19b29b8b99947424172ef77ee1b7b3f": "-1.15^\\circ\\sqrt{\\text{elevation in feet}}/60^\\circ", "a19b304a36c9ade2faa7d967b25f2712": "x = p^j\\frac{a}{b}", "a19b6a95a371beac1b2c26cd17bc85d5": "\\frac{dS}{dt} = b_0 + b_S S + b_I I + b_R R - \\lambda S - m_S S, ", "a19b700fe98fa376c03b5da4a3b82f1f": "P_f:=EP_f\\cap \\{x\\in \\mathbb{R}^S|x\\geq 0\\}", "a19b9f38b1ae81149d114d7b4049a548": "\\delta r", "a19bd7d8f70c13482a94e79744d77dd2": "y_3 = x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4", "a19c857c2066af2f237949a7eeda13c2": "e^{-x^2}", "a19ca751e84a75db6b73cacd4753f6a5": "D_n,", "a19cb3a39f7a04ec9589590535027078": "\\begin{align}\n \\mathbf{e}_1 & = \\cfrac{\\partial q^1}{\\partial x_1} \\mathbf{b}_1 + \\cfrac{\\partial q^2}{\\partial x_1} \\mathbf{b}_2 + \\cfrac{\\partial q^3}{\\partial x_1} \\mathbf{b}_3 \\\\\n \\mathbf{e}_2 & = \\cfrac{\\partial q^1}{\\partial x_2} \\mathbf{b}_1 + \\cfrac{\\partial q^2}{\\partial x_2} \\mathbf{b}_2 + \\cfrac{\\partial q^3}{\\partial x_2} \\mathbf{b}_3 \\\\\n \\mathbf{e}_3 & = \\cfrac{\\partial q^1}{\\partial x_3} \\mathbf{b}_1 + \\cfrac{\\partial q^2}{\\partial x_3} \\mathbf{b}_2 + \\cfrac{\\partial q^3}{\\partial x_3} \\mathbf{b}_3\n\\end{align}", "a19ccbdc5c2fe0464b711ad551bf6f28": "x=+a", "a19ce97d8fa2da03779886f6fc06b808": "D_{BA} = 0.2727", "a19d1da1d6237ec646b74e51e8f15ec7": "P_{4}^{-1}(x)=-\\begin{matrix}\\frac{1}{20}\\end{matrix}P_{4}^{1}(x)", "a19d2c76ca671f413dc5781e81e6389e": "T(n) \\leq T(n \\cdot 2/10) + T(n \\cdot 7/10) + c \\cdot n.", "a19d5c85d58aea89599bd8fcae3b8260": "\\varphi_t(x)=\\frac{2 (x-c_1)-tG(x)}{\\left((x-c_1)-t\\tfrac{1}{2}G(x) \\right)^2+t^2\\pi^2\\mu^2(x)}", "a19d7a03337b8a4c15821a03d26c268e": "\\bar{2}", "a19db29ed8d4d9f4abc7249960334d59": "(\\lambda_n^{(c)}) \\in \\Lambda", "a19db979b517ed6b04a10d53beb36c29": "A=id_{id_A}", "a19dbfa72753b248d0cc322bf6d09064": "c(V \\oplus W) = c(V) \\smile c(W);", "a19eafceab572b71bdb85740eefdb88c": "Z^i = Z^i \\left( t ,S \\right) \\, ", "a19ec8def9d5b02b110690299f74308b": " \\operatorname{ask}[S] \\equiv S \\in \\{ X : X = S \\} ", "a19eea87ef5c88a4d8c16b69a3d34b21": "2 \\theta", "a19f199f1ce3e2f6831be637c448ac4e": "\\lambda_R=700.0nm, \\lambda_G=546.1nm, \\lambda_B=435.8nm ", "a19f5cbaf4da2403f12d09d58fa7a018": " T_2 - T_1 = T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right) ", "a19f65f69d5ae486a7ecd8da66e69b83": "G\n", "a19fa4127a5f55ca099f0235c8b2afd1": "\\left (x,y,z \\right )", "a19fb8b04b643b9aa2adb77868b6d638": "X=(x_1,x_2)", "a1a01ea7f48d73ae6bc2126107e385f7": "f^{64}(27)\\, ", "a1a0892e8bd43146a2e719bdf41d1889": "C\\subset \\mathbb{F}_2^n", "a1a0ca68348c1b175fc25ac8f8764506": " \\dot{u}=g(u)", "a1a0d050eb1a395a00943f495d870000": "x_{j_1},\\ldots,x_{j_l}", "a1a0fa971560faba65256b0e269ce60c": "\n f := \\cfrac{1}{3z}~\\cfrac{I_1}{\\sigma_c} + \\sqrt{\\cfrac{2}{5}}~\\cfrac{1}{r(\\theta)}\\cfrac{\\sqrt{J_2}}{\\sigma_c} - 1 \\le 0\n ", "a1a14cd893296a12031294fa61aeda75": "GL(2)", "a1a1b6130c7d76991c1fcb5413136d1f": "I_D = \\frac{2I_{DSS}}{V_P^2} (V_{GS} - V_P - \\frac{V_{DS}}{2})V_{DS}", "a1a1bc23b922724175adda8be99e5f2e": "\\int_0^1 \\rho \\sqrt{2n+2}R_n^m(\\rho)\\,\\sqrt{2n'+2}R_{n'}^{m}(\\rho)d\\rho = \\delta_{n,n'}", "a1a1bdfa75540e436370e42ab52a0f13": " V_{Building}\\,\\!", "a1a1c890b210905d3b471bffa678ff46": "R^p f_* \\mathcal{F}", "a1a1f1fee04a0ff2ab27c04ab6d042e4": "\\nu_1=\\omega_a-\\omega_c, \\nu_2=\\omega_b-\\omega_c", "a1a2332fa57dea96d4c4fb11ea03dedb": "{\\Delta L } = \\alpha_L \\Delta T \\ L", "a1a248d90ecd774d61fd9ebd0dec7b1d": " {n+k\\choose k}= {n+k-1\\choose k-1}\\times \\frac{n+k}{1+k}.", "a1a2bd9d059b47f9ef670099826b0ddc": "\\tilde{u}(x,y,z,t)", "a1a2f22187440e5450e3236e02cbc911": "\\scriptstyle\\sharp ", "a1a2fa691f12c9927ffb22e26f0a109c": " \\{\\xi = 0 \\} ", "a1a30753b6c1fa4c6517a964e53ffa4f": "c'\\in \\mathsf C", "a1a335603cbd0e417b5f69eb4492252b": "k \\le \\log_{2}(N)", "a1a3bf54a0eaa31f20e7ad7748ac1082": "\\omega_{pi} = (4\\pi n_iZ^2e^2/m_i)^{1/2} = 1.32 \\times 10^3 Z \\mu^{-1/2} n_i^{1/2} \\mbox{rad/s}", "a1a3da0dc234bcd980a08836c59d24a3": "\\forall_{i,j \\in N_i,l, y_l\\neq y_i} ", "a1a44e71264ba1e07a1e4ae8225b0f8d": " \\sqrt{R_{xx}(\\infty)},", "a1a462ea95dc47553226b4dcbda6c90b": "t - t_0 = r_0\\ s\\ c_1(\\alpha s^2) + r_0 \\frac{dr_0}{dt}\\ s^2\\ c_2(\\alpha s^2) + \\mu \\ s^3\\ c_3(\\alpha s^2)", "a1a4827f56a7a95037eb530c5cfbbef0": "|Q_{P}(h)-\\widehat{Q}_{x}(h)|\\geq\\epsilon\\,\\!", "a1a504fa641491571ce5ed693245d4ef": " \\frac {R_{out}} {R_{out}+R_{L}} A_i \\ . ", "a1a50955d6a2b324269e00794c2c6f74": "\n\\begin{align}\n\\omega_r &= 0, \\\\\n\\omega_\\theta &= 0, \\\\\n\\omega_\\phi &= {1 \\over r}\\left({\\partial \\over \\partial r} \\left( r \\left(-\\frac{1}{r \\sin\\theta}\\frac{\\partial\\Psi}{\\partial r}\\right) \\right) - \n{\\partial \\over \\partial \\theta}\\left(\\frac{1}{r^2 \\sin\\theta}\\frac{\\partial\\Psi}{\\partial \\theta}\\right)\\right).\n\\end{align}\n", "a1a521e75a824b9a022e74cadccc02fe": "\\Delta S=Q_2/T_C", "a1a5b758eca29b45e0c114076fad15e1": "p = {{s \\over v} \\over {{c \\over v} + 1}}", "a1a5c70b1c16b0f477ef45e64aa3098d": "\n\\begin{bmatrix}\n\\begin{vmatrix}\n8 & -2 \\\\\n-4 & 6\n\\end{vmatrix}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n40\n\\end{bmatrix}.\n", "a1a60c9abe18c625685f4a3c974aa08d": "\\begin{alignat}{7}\n 2x_1 &&\\; + \\;&& 5x_2 &&\\; - \\;&& 3x_3 &&\\; = \\;&& 0 \\\\\n 4x_1 &&\\; + \\;&& 2x_2 &&\\; + \\;&& 7x_3 &&\\; = \\;&& 0 .\n\\end{alignat}", "a1a61a6f0a18fe257f98106ec6eba93b": "Z_{\\mu}(t) = X(t)", "a1a6589d510cbc952c46b53f5592d1ca": "S=\\mathbb R", "a1a66ef9ea27eb87828334a95ecaf1bc": "\\frac{\\text{d}[{^1_2}S^\\beta]}{\\text{d}t} \\simeq - \\frac{ \\text{k}_{3(2)} E_0 {^1_2}S^\\beta }{ ^1_2S^\\beta + K_2 \\left( 1+ \\dfrac{ {^0_2}S }{ K_1} + \\dfrac{ {^1_2}S^\\gamma }{ K_2} \\right) }", "a1a6816187967e016e2fa9e81c4bfbe0": "\\textstyle m=2", "a1a6f342c4e3edc4c9361f05e6cc1676": "U_-(F)(w)=\\frac{1}{\\pi} \\iint_{\\mathbf C} F(z) \\overline{z}e^{\\frac{1}{2} w \\overline{z}^2} e^{-|z|^2} \\, dx dy,", "a1a78d90824406319f19c3f299ea4747": "z_{11} = Z_{22} = R_2 + R_3 \\ ,", "a1a86b9288fdfeed3ad5644b715773f4": "\nY_i = X_i + Z_i\\sim N(X_i, n).\n\\,\\!", "a1a8d856e35e9b04b80e8820b85bac0b": "\n mat(\\begin{bmatrix}x_1x_2\\\\x_1y_2\\\\y_1x_2\\\\y_1y_2\\end{bmatrix})=\\begin{bmatrix}x_1x_2&x_1y_2\\\\y_1x_2&y_1y_2\\end{bmatrix}=\\begin{bmatrix}x_1\\\\y_1\\end{bmatrix}\\begin{bmatrix}x_2&y_2\\end{bmatrix}\n", "a1a9571eb3de29ac7fe36d98da3792ab": "i_{1} \\ne i_{2} \\ne \\ldots \\ne i_{l}", "a1a96b5502902bc69919addbedd0d5ec": "p(\\vec x|y=1)", "a1a98172ffad3f7753341b6fcb37cd30": "dst_A = 1", "a1a9b1d3d2898bd325f1adcd15712ffc": "\\;_{2}\\phi_1 \\left[\\begin{matrix} \nq \\; -1 \\\\ \n-q \\end{matrix}\\; ; q,z \\right] = 1+\n\\frac{2z}{1+q}\n+ \\frac{2z^2}{1+q^2}\n+ \\frac{2z^3}{1+q^3}\n+ \\ldots. ", "a1a9df77151553c9d2c92e8384e8402d": "f'_+,\\,", "a1a9f8fae993e28e0feb4528d33bd60e": "\n\\begin{bmatrix}\n 17 & 24 & 1 & 8 & 15 \\\\\n 23 & 5 & 7 & 14 & 16 \\\\\n 4 & 6 & 13 & 20 & 22 \\\\\n 10 & 12 & 19 & 21 & 3 \\\\\n 11 & 18 & 25 & 2 & 9\n\\end{bmatrix}.\n", "a1aa13bafc3af1cb15bbdc4dbc43c82a": "\\phi \\colon G_f \\xrightarrow{\\sim} F \\otimes_{A_f} \\hat{A}_f = F \\otimes_A \\hat{A}.", "a1aa1698a1f7d6857bd0fd73512ddfa1": "C_{Op}", "a1aa492b83d5507f1ae02d89fe4e42d3": "\\scriptstyle{R_3^0}", "a1aa6522210a8259d0e2b9965e6df0f0": " \\psi^-", "a1aa8d7262a4012378b54477724bc6d7": "\\begin{matrix}\n\\times & w & x & y & z \\\\\na & aw & ax & ay & az \\\\\nb & bw & bx & by & bz \\\\\nc & cw & cx & cy & cz\n\\end{matrix}", "a1aadea6ab4d703f86cb79ea2f089192": "\\sigma:E \\times_M E \\to E", "a1aae54f378e072011e62f31bbe6a882": " \\mathbf{E} = - \\nabla V ", "a1ab1ae22f223fd824130531e5c8c322": "1-p_i", "a1ab3630ab87a03eacb8ab8db1522786": "\\underline{1}\\in G", "a1ab5292b712146e280a4772c373994d": "\\frac{n}{k}", "a1aba374c5ce13b377d615bd63122cb4": "p_{cv}(t=0) = p_{cv,0}", "a1abaa6812da61b78b80d591898a0808": " x^2 - y^2 = 1 .", "a1ac158edb17f6ed8f62ef34aa39a9c3": "D \\in \\C", "a1ac8116b8f8fbdb174c37f62d5aa237": "\\omega(B)", "a1acc1699978bdbb0d7985ed7e768527": "(\\phi,x)^{\\downarrow y} = (\\phi^{\\Rightarrow y},y)\\text{ for }y \\leq x\\,", "a1ace06cec90603e15a5fc349a0801ac": " S_3 = \\frac {N^{-1} \\sum_{i} (x_i - \\bar x)^4} {(N^{-1} \\sum_{i} (x_i - \\bar x)^2)^2} ", "a1ace35560354fb7651d4b7c22b17153": "f = \\alpha_1 S_{xx,1} + \\alpha_2 S_{xx,2}.", "a1ad0d81cb01c125a078d8e3a287b040": "V_M = -\\vec{\\mu} \\cdot \\vec{B},", "a1ad0faf93869b09d45cb64623247103": "Y_{7}^{-3}(\\theta,\\varphi)={3\\over 64}\\sqrt{35\\over 2\\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(143\\cos^{4}\\theta-66\\cos^{2}\\theta+3)", "a1ad4d6655204a31b557afc28ae16467": "T = T_{i_1i_2\\dots i_k}e^{i_1}\\otimes e^{i_2}\\otimes\\cdots \\otimes e^{i_k},", "a1ad535130eb15158c68cba34d74a6c2": "\\neg x=(x\\Rightarrow 0).", "a1ad88914aa2cb4c5a7f736280c8a330": "\n\\frac{dE}{d\\sigma} = \\left.\\frac{dE}{ds}\\right/\\frac{d\\sigma}{ds} = \\pm\\mathbf{N}", "a1ad8be8cc0cc83c0a80bc358855efb9": "x_3 = \\frac{2x_1y_1}{1+dx_1^2y_1^2} = 0", "a1adfb25e2ffee55ad835b00af07958a": "{\\mathbf W}= \\frac{\\sum_{i=1}^{n_x}(\\mathbf{x}_i-\\overline{\\mathbf x})(\\mathbf{x}_i-\\overline{\\mathbf x})'\n+\\sum_{i=1}^{n_y}(\\mathbf{y}_i-\\overline{\\mathbf y})(\\mathbf{y}_i-\\overline{\\mathbf y})'}{n_x+n_y-2}", "a1ae248006fb7ccb3d1c65510aef5229": "\\varepsilon*1 = 1 ", "a1aea23381da37a96c673a5e4b04a948": "a \\to b \\to 4 \\to 2 = a \\to b \\to (a \\to b \\to (a \\to b \\to a^b)) = a \\uparrow^{a \\to b \\to 3 \\to 2} b", "a1aeaf9893c5886ef6f806ced4e00960": "0.45{K_u}", "a1aeded139ddf6d9eed727c8c08afd8f": "\\phi(a, b, 3) = a \\uparrow\\uparrow (b + 1)\\,\\!", "a1aefa7599a8c954d0a4a77ffa444fdf": "w(P,Q) \\in \\mu_n", "a1af2990e1a4886593ce7db2f3179f05": "R < 1", "a1af4d26b5fe5f8255cfd45ff61c0413": " F\\ f\\ n = (\\operatorname{IsZero}\\ n)\\ 1\\ (\\operatorname{multiply}\\ n\\ (f\\ (\\operatorname{pred}\\ n))) ", "a1af5d26fa4feabf332d4c9e7bf6b0d0": "d_\\mathfrak{g} \\circ d_\\mathfrak{g} = 0", "a1af884640be45f3a9954e7b96957359": "\\alpha\\in C", "a1af924aa8aa2e2ef94af3ecc94c8368": "\\frac{b}{3a}", "a1afa952dcd382b6576c5b212ddeed4d": "M = 3(N- 1 - j)+ \\sum_{i=1}^j\\ f_i, ", "a1b002eee1355003bbbad843cd150265": " \ne^{(k+1)} = e^{(k)} - \\omega A e^{(k)} = (I-\\omega A) e^{(k)}. \n", "a1b0094a5dad6fe3164bcc8eeb924d42": "-\\omega \\vec{e}_2", "a1b05a77f41794374b169ab016b57c78": " K_n = \\frac{d^{(n)}}{d\\theta^{(n)}} A(t) \\, .", "a1b05a9b8c7fa4f73cc6a4bcfbf2c73d": "\\frac{\\nu(^{16}O)}{\\nu(^{18}O)} = \\sqrt{\\frac{9}{8}} \\approx \\frac{832}{788}.", "a1b07afa61ccab8a39dfc2640df5654a": "\\{0\\} \\times (0,1)", "a1b0d23ba44cfb4e648368a43ee094a4": "ED(G)", "a1b0ee08fd8bbf731b8d8f838910e218": "N^T", "a1b10a6e6824702f94020b13439d7662": "G = {\\pi \\sigma \\over \\operatorname{arccosh}({D \\over d})}", "a1b129a43d335c74cd0d804c18b90ccd": "C^{-1} = \\omega A^{-1} + (1-\\omega) B^{-1} \\, ,", "a1b15c5100b2e29095c7b41d0d11fa32": "[L_{sr}] = \\begin{bmatrix} L_{Aa} & L_{Ab} & L_{Ac} \\\\ L_{Ba} & L_{Bb} & L_{Bc} \\\\ L_{Ca} &L_{Cb} &L_{Cc} \\end{bmatrix}", "a1b19be32027828ea8bb7bafac75c0d3": "\\Delta s", "a1b1aae8f2af2dfb76535d50f5253bee": " [\\Phi, Q] = i\\hbar ", "a1b1dfd10c1d6e540b77a1d02d4d140a": "\\alpha_t(x_t) = p(x_t,y_{1:t}) = \\sum_{x_{t-1}}p(x_t,x_{t-1},y_{1:t})", "a1b1fc943636b0b38d529a6da8df5a81": "(m_i,m_j) = 1", "a1b276eee4d1a268a6a5a85f85867bf3": "\\mathcal{G} = \\mathcal{Z} \\times \\mathfrak{G}\\{\\mathcal{G}\\}.", "a1b27990751820fc977fda79e1b83ee2": "\\psi(ua)=\\psi(vb)\\,", "a1b2a9e40055cc206d90b615675e7206": "f_\\lambda=e^{i\\lambda -\\rho}\\sum_{\\mu\\in \\Lambda} a_\\mu(\\lambda) e^{-\\mu},", "a1b347d2b7624600d16d639d0bf09acd": "r_{i+1}", "a1b363002c0f3e57c8dc5f1cf546e8fe": "\\overline{ B_r(p) } = B_r[p]", "a1b413b6b3f36e4324014f7b2ca375f5": "f(\\bar{u}) = f(x, y, z)", "a1b43f290d51f687cad023abc09f9be3": "a 0\\!", "a1b61c674d5851774d930b98f7385a20": "\\theta=u-\\omega\\,", "a1b682e100bb52a026851c4fcfb65ff2": "\\nu<0", "a1b68665d1949e531824d22accb5922b": "G = (C, F, L)", "a1b6aa0d1623b3976abc8ca92f886a47": "L(u)=\\frac{(\\tau-1)}{9}\\sum_{y_{i} t [ E( X - E( X ) )^{ 2k } ]^{ 1 / 2k } ) \\le \\min\\left[ 1, \\frac{ 1 }{ t^{ 2k } } \\right]. ", "a1cd8c44ca4b25590d264c392489a122": "k_m = \\pm\\sqrt{\\beta_2^2\\omega_m^4 + 2 \\gamma P \\beta_2 \\omega_m^2}", "a1cd9c86316cce7afb512bb6b14f9bc1": "\\bar {u} \\frac{\\partial \\bar{u}}{\\partial x} + \\bar {v} \\frac{\\partial \\bar{u}}{\\partial y} = -\\frac{1}{\\rho} \\frac{d\\bar{P}}{dx} + \\frac{\\partial}{\\partial y} \\left [(\\nu + \\epsilon_M) \\frac{\\partial \\bar{u}}{\\partial y}\\right]", "a1cdb3edd568f72fe31a0733f640619c": "\\begin{smallmatrix}V=\\frac{4}{3}\\pi r^3 \\end{smallmatrix}", "a1cddc528307692066a716233d0ef9e1": "\\begin{align}\n & H_1(t)=E_m(t-p)+\\alpha E_{sd}(t-p) \\\\\n \\end{align}", "a1cdfadb322cc91aca1efd65aa0165e9": "H_{NE}", "a1cea4d57c2d5c01d2f0103ad9957958": "A_{i,j} = 1", "a1cee79113487f81d10600e88132b1ab": "P(x, y)", "a1ceeef438a5540014c11988371b6b6e": "(Z_t)_{t\\geq 0}", "a1cf07f7219fca7a38eac393ac21de78": "M= {f_o \\over f_e}", "a1cfa663ca381d6827ad5146907b1961": " \\left \\langle i \\right \\rangle = \\left(\\frac{s}{q}\\right)\\frac{dq}{ds} ", "a1cfa7ad6b62f7c479e6e3fad39c7770": "\\,qP", "a1cfc59047a8ad4986a2902a1932f5d3": " i^i = \\left( e^{i \\pi / 2} \\right)^i = e^{-\\pi / 2} = 0.207879576 \\ldots.", "a1d030b8b8c0400c1e3d076809ddb0f6": " \\left\\{ h_p(t) \\right\\} ", "a1d04b53b2dd8f0bbe9fc7d431f61d47": "t_i \\in [u_{i-1}, u_i]", "a1d0515905b324626c88133c67ff7eff": "P\\left( X_{i,j}=x,Y_{i,j}=y \\right)=\\frac{\\lambda ^{x}\\exp (-\\lambda )}{x!}\\frac{\\mu ^{y}\\exp (-\\mu )}{y!}", "a1d072ba059bddcfebdd7f51b36a7ed2": "S = \\{\\, x \\in V\\;:\\; |x| = 1 \\,\\}", "a1d0bb739ebe8a67ba859d4fc577539b": " \\begin{align}\n&\\lim_{\\alpha\\to \\infty}( \\lim_{\\beta \\to \\infty} \\ln \\,\\operatorname{var_{GX}}) = \\lim_{\\beta \\to \\infty}( \\lim_{\\alpha\\to \\infty} \\ln \\,\\operatorname{var_{G(1-X)}}) = \\lim_{\\alpha\\to \\infty} (\\lim_{\\beta \\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}) = \\lim_{\\beta\\to \\infty}( \\lim_{\\alpha\\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}) =0\\\\\n&\\lim_{\\alpha\\to \\infty} (\\lim_{\\beta \\to 0} \\ln \\,\\operatorname{var_{GX}}) = \\lim_{\\beta\\to \\infty} (\\lim_{\\alpha\\to 0} \\ln \\,\\operatorname{var_{G(1-X)}}) = \\infty\\\\\n&\\lim_{\\alpha\\to 0} (\\lim_{\\beta \\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}) = \\lim_{\\beta\\to 0} (\\lim_{\\alpha\\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}) = - \\infty\n\\end{align}", "a1d0f1bd19196ec133fcff91a430aaec": "\\frac{1}{p} = \\frac{1-\\theta}{p_0}+\\frac{\\theta}{p_1},\\quad \\frac{1}{q} = \\frac{1-\\theta}{q_0} + \\frac{\\theta}{q_1}.", "a1d0f6356d0b0f48b6a8265daee93bc5": "X = A^T X A -(A^T X B)(R + B^T X B)^{-1}(B^T X A) + Q.\\,", "a1d111802ce164c62768d6ad8fe4ede1": " +E_{22}", "a1d12badee9a190ceacc73d68e0aa46d": " \\nabla \\mathbf{v} = \\begin{pmatrix} \\dot \\epsilon & 0 & 0 \\\\ 0 & -\\frac {\\dot \\epsilon} {2} & 0 \\\\ 0 & 0 & -\\frac{\\dot \\epsilon} 2 \\end{pmatrix} ", "a1d19043eccf77cba653d8edeeef8a07": "\\Delta _{{{\\dot G}_{0,\\Lambda }}}^{12}\\mathcal {V}_\\Lambda ^{(1)}\\mathcal {V}_\\Lambda ^{(2)}=\\frac{1}{2}\\left( {\\frac{{\\delta {{V}_\\Lambda }(\\psi )}}{{\\delta \\psi }},{{\\dot G}_{0,\\Lambda }}\\frac{{\\delta {{V}_\\Lambda }(\\psi )}}{{\\delta \\psi }}} \\right)", "a1d1a6bd3ca27f02b95c05e5304bae0a": "\\mu_{A} \\colon T(T(A)) \\to T(A)", "a1d2278dc83b772db232de86a0a49ba7": "\\frac{(-1)^g}{2}", "a1d2b7df52090e6c4302dd3d9c441749": "\\scriptstyle \\eta \\leq \\zeta ", "a1d2fa72a56f6ef8a102d85975a2dc8b": "\\alpha=\\frac{\\Delta w}{\\Delta t}=c \\frac{\\Delta \\eta}{\\Delta \\tau}=c^2 \\frac{\\Delta \\gamma}{\\Delta x}", "a1d366db76ca3ebe0d6141b680ec2202": " a=3, n=5, m=9 ", "a1d3698125aee73cc2d769a92a7f9798": "J_n(x)=2n(2n-1)J_{n-1}(x)-4n(n-1)x^2J_{n-2}(x).\\,", "a1d36dca7552cdb6653c66500445958b": "\\mathbf{p}_\\gamma = \\mathbf{p}_{\\gamma'} + \\mathbf{p}_{e'},", "a1d48426a19de814ac81d8c990d5cfc9": "\\scriptstyle\\mathcal{A}_q^n", "a1d4ecf38fa99034a063a03902504261": "(V_i,\\tau_i)", "a1d5139bb728088158ec96cd348be2b1": "C_{abcd} \\, k^d =0", "a1d5278eb608475b143eb87d33e21fcf": "x \\in W", "a1d52f9d97329667236322a232474993": " x + 1/x = 1", "a1d555cd5efea165a17ef0d20090467a": "[\\mathbf{x}_1], \\dots , [\\mathbf{x}_k]\\mbox{,}", "a1d56dd4bae8b4827d2ea4cff6f14e75": "S^7 = \\left\\{ x \\in \\mathbb{R}^8 : \\|x\\| = r\\right\\}.", "a1d5a8a802505a93e4cce95ee9f26ea9": "f(x,y,z) = \\left(\\frac{x}{\\sqrt{x^2+y^2}}, \\frac{y}{\\sqrt{x^2+y^2}}, z\\right)", "a1d5d2e46704138de55d710c0c882540": "A_0, A_1", "a1d614b2794e64ad089a8cc785832667": "|\\psi(t)\\rangle=c_1(t)e^{-i\\omega_1 t}|1\\rangle+c_2(t)e^{-i\\omega_2 t}|2\\rangle+c_3(t)e^{-i\\omega_3 t}|3\\rangle.", "a1d62c6a229a6e3ac948b8177639fa48": "\\Delta \\phi = \\pi \\;", "a1d649069264da0deb269039f5265e9b": "E=F[\\alpha]=F(\\alpha)", "a1d64c000f760b521028cb1707756770": "0 \\to I \\to L \\to L' \\to 0", "a1d6bcc016f3dcd1f043cdeb9e3ccb2c": " \\begin{align} \n C_{k+1} \n &= C_k + c_1(p_c p_c^T - C_k) \n - c_\\mu\\,\\mathrm{mat}(\\overbrace{[\\tilde{\\nabla} \\widehat{E}_\\theta(f)]_{n+1,\\dots,n+n^2}}^{\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{natural gradient for covariance matrix} \n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n })\\\\\n &= C_k + c_1(p_c p_c^T - C_k) \n + c_\\mu \\sum_{i=1}^\\lambda w_i \\left(\\frac{x_{i:\\lambda} - m_k}{\\sigma_k} \\left(\\frac{x_{i:\\lambda} - m_k}{\\sigma_k}\\right)^T - C_k\\right) \n \\end{align}", "a1d6fa800d794388f9d0e2466c58d834": " P(r,t) = \\chi \\vdots E(r,t_1)E(r,t_2)E(r,t_3) ", "a1d710b07699760222207ed7bf028771": "X \\sim \\Beta\\left(1+\\lambda\\tfrac{m-\\min}{\\max-\\min},1+\\lambda\\tfrac{\\max-m}{\\max-\\min}\\right)", "a1d731ed8b99c1abd6f6db5797442cab": "S^1 = \\{ e^{i\\phi} \\, | \\, 0 \\leq \\phi < 2\\pi \\}.", "a1d74d9bb81430b27302881769226bc4": "|0 \\rangle ", "a1d762fae82487985ef80d8d7e0d3008": " \\boldsymbol{\\beta}(x) = (\\mathbf{K} + \\lambda \\mathbf{I})^{-1} \\mathbf{K}_x", "a1d7649dd308baafdb20849c0b9c9369": "G(x) = F(Ax+h),", "a1d76c316bf5f07cb86cdb60f3444833": " x = \\frac{1}{k_1 + \\frac{1}{k_2 + \\cdots}}", "a1d770162dc171060bc528f0acf61a80": "\\mathbb{H}^n", "a1d7fd35a42c2a7fb722dd2f4483b3cd": "P_{LCU}", "a1d82f39111cea45e9473472cec02d7e": " \\begin{bmatrix} 1 & \\mathbf a & c \\\\ 0 & I_n & \\mathbf b \\\\ 0 & 0 & 1 \\end{bmatrix} \\cdot \\begin{bmatrix}1 & -\\mathbf a & -c +\\mathbf a \\cdot \\mathbf b\\\\ 0 & I_n & -\\mathbf b \\\\ 0 & 0 & 1 \\end{bmatrix} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & I_n & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}. ", "a1d8368f6df4fdd2b3f03694524a4da4": "\\delta(z) = \\varphi(z)/|\\varphi'(z)|", "a1d84829c8c39ad091050ffb31e6924e": "\\frac{x^3+x^2+1}{x^2-5x+6} = (x+6) + \\frac{24x-35}{x^2-5x+6},", "a1d88c817e26a704214f44509c640041": "D_2= [P] + [R] - 2[O]", "a1d8e4204cded6474a99248d8803ef7d": "\n\\zeta(s) =\n\\sum_{n=1}^\\infin \\frac{1}{n^s}.\n", "a1d94aab56803a512ce9f7ca777c6520": "\ni = \\frac{n!}{(n-2p)! \\ p!\\ 2^{p}} \n = \\frac{n!}{ (n-2(\\frac{n}{2}))! \\ { (\\frac{n}{2}) }! \\ {2^{ (\\frac{n}{2}) } } } \n = \\frac{n!}{ { (\\frac{n}{2}) }! \\ {2^{ (\\frac{n}{2}) } } }\n", "a1d983a6818f53d4acd9cb9baf9a31a9": " \\dim V_\\lambda =\\chi^\\lambda(e) = \\langle s_\\lambda, p_{1^{(n)}}\\rangle ", "a1d9d0bc372f65941c2ea60270266bbf": "E([Y_1 - c_0 - c_1X_1 - c_2X_2]^2)\\,", "a1d9e83e52843bdf221a2f812fd7b2e9": " \\left(r\\frac{\\partial}{\\partial r}\\right)^2 u + \\frac{\\partial^2 u}{\\partial \\theta^2} = 0", "a1da1dc0d6a42fd8fa495effbed95ce8": "\\mathbb{Z} f: \\mathbb{Z} X \\to \\mathbb{Z} Y", "a1da23e4901f1c5ec5752cc483994e47": "(\\mathbf{c}\\cdot\\mathbf{\\hat a})\\mathbf{\\hat a} + (\\mathbf{c}\\cdot\\mathbf{\\hat b})\\mathbf{\\hat b} = \\mathbf{c}, ", "a1da3395b84dd1bcdc98dd1cb3b04f63": "\\left(J\\left(R\\right)\\right) ^k=0", "a1da4b2d15eb9d804da5dbfb33b9d215": "\\{(-,+,+,+); l^an_a=-1, m^a\\bar{m}_a=1\\}", "a1da60329c3a12f65b9eb275a6ca2aca": "\\|X^\\top (Y-X\\beta)\\|_\\infty", "a1dafb7c8c44ae1c4f3e64fa75d8c978": "\\mathrm{pv}\\ \\log{z} = \\mathrm{Log}\\ z = \\ln{|z|} + i\\left(\\mathrm{Arg}\\ z\\right).", "a1db5ec3ed5c6937832c997da0dc33d1": "\\mathrm{Hom}_S(X,X')", "a1db61e14fb084fe370cd15843d5a49b": "\\mathbf{Q} = {1 \\over {N-1}}\\sum_{i=1}^N (\\mathbf{x}_i-\\mathbf{\\bar{x}}) (\\mathbf{x}_i-\\mathbf{\\bar{x}})^\\mathrm{T},", "a1db7a32d185347d88541414d39e20c9": "\n\\ H(z) = \\frac{z^K + \\alpha}{z^K} \\,\n", "a1db8d98dec36f106d63ba0275949313": "I(A;B|C) \\ge 0,", "a1dba054854cd63c96ee5236a11a7e83": "h_3(X_1,X_2,X_3)=X_1^3+X_1^2X_2+X_1^2X_3+X_1X_2^2+X_1X_2X_3+X_1X_3^2+X_2^3+X_2^2X_3+X_2X_3^2+X_3^3.", "a1dbed8ef0dfd6f13beee859b18814bd": "e_{ij}=\\frac{1}{2}\\left(\\frac{\\partial u_i}{\\partial x_j} +\\frac{\\partial u_j}{\\partial x_i}-\\frac{\\partial u_k}{\\partial x_i}\\frac{\\partial u_k}{\\partial x_j}\\right)\\,\\!", "a1dc7cbc2b5cfbb39436df763693e445": " \\begin{alignat}{2}\n \\bar{n}(\\epsilon_i) & = g_i \\ \\bar{n}_i \\\\\n & = \\frac{g_i}{e^{(\\epsilon_i-\\mu) / k T} + 1} \\\\\n\\end{alignat} ", "a1dce77ea3a1b38ee376706a2573000a": "y_1 = e^{(2+i)x}", "a1dcf1ac33a64fcfeadda00761d98612": "lm' = l'm", "a1dcf4b2c564a8ccb971624c12c071bf": "F_i + \\mu F_w \\cos \\theta - F_w \\sin \\theta = 0 \\, ", "a1dcf89a35de639503e81a09012f981a": "W_n (k).", "a1dd0a2bfa29e5b1f43b40573c007816": "\\mathbf{u} = \\mathbf{V} + \\mathbf{V}_d; ~~~\\mathbf{V}_d = -\\nu \\dfrac{\\nabla\\mathbf{\\Omega}}{|\\mathbf{\\Omega}|}; ~~~\\mathbf{\\Omega} = [\\nabla \\times \\mathbf{V}]", "a1dd837b383c74d1a8981ad778e5a055": "x \\vee y = y", "a1dd8943045dd62086e77decca350af7": "\\left[S\\rightarrow abc\\right]", "a1dda791e425d6da7eb8813ec2f91edc": "\n \\begin{bmatrix}\n -2 & 2 & -4 \\\\\n -1 & 1 & 3 \\\\\n 2 & 0 & -1\n \\end{bmatrix}\n ", "a1ddae9d7d637d073117d5ab480a747d": "\\ q= \\sqrt{\\frac{(\\sigma_1' - \\sigma_2')^2 + (\\sigma_2' - \\sigma_3')^2 + (\\sigma_1' - \\sigma_3')^2}{2}}", "a1ddb3d9d07dbd5631fddaadd83feb83": "y\\sim e^x\\,", "a1ddb7d71b9c3a405514ff5001d6d236": " \\text{Sl}_2(\\theta)= \\frac{\\pi^2}{6}-\\frac{\\pi\\theta}{2}+\\frac{\\theta^2}{4} ", "a1ddec133e184feb60d6ad97ab9d20bf": "[\\mathtt{Name}]", "a1ddf4cf9eb0416a0ac8f140ec437a25": "\\pm \\theta,", "a1de07ef67642f7a396de630f6677844": "f*(g*\\varphi) = (f*g)*\\varphi\\,", "a1de206e71f8d6eb1b098b9e03ce8443": "f : X \\to M", "a1de23d6c70eb1e85ae23226f4fbad1e": "\\frac{1+3\\gamma+\\ln(16\\pi c^2)}{2}", "a1de567ccb9fe4e7ec21e576714c88f0": "\\sigma_y=\\biggl( \\begin{matrix}\n 0&-i\\\\i&0\n \\end{matrix} \\biggr);", "a1de5fa89f175bbb9d87d5bc805f5a8d": "d \\Omega.", "a1de6414f5d93a3d78a749b76eb19605": " V \\not \\in \\operatorname{FV}[\\lambda F.E] \\to \\operatorname{de-lambda}[\\lambda F.E] \\equiv \\operatorname{let-combine}[\\operatorname{let} V : \\operatorname{de-lambda}[V\\ F = E] \\operatorname{in} V] ", "a1de841b39475e84d96a1d1037daaaa1": "n^2(n+1)(2n+1)=", "a1ded42f3d29e8ae96b3998837f2c86f": "I(t) = I_0 \\exp \\left(- \\frac{t^2}{\\tau^2} \\right)", "a1deed25fe60322218d7bfecbc49abc0": " p ", "a1df1b598634b1c5a0c82e703e1d5d46": "\\sigma_p^2 = \\int_{-\\infty}^\\infty p^2 \\cdot |\\phi(p)|^2 \\, dp.", "a1dfbd42d6e55b5d43db35c9298f0888": "k[f_1, \\cdots, f_d]", "a1dfc6635737b715edc1b04c7ba6c42a": "\\scriptstyle \\sqrt{c}", "a1e032db28045462c6701b6acf0706be": "b \\sin (A) = a \\sin (90^\\circ - A)", "a1e046e1430257509aba41b2b16b014b": "g=\\prod_{i\\in I}(X-\\alpha_i)", "a1e04e93294af094bcd7fc48adf77ca3": "\nN = A^2+4B^2+9C^2+16D^2+25E^2+....\n", "a1e0731348afcd877f8affcac77e16ea": "H_2(p_b)=- \\left[ p_b \\log_2 {p_b} + (1-p_b) \\log_2 ({1-p_b}) \\right]", "a1e0c9242758d95a72dd1b0268750043": " h << x ", "a1e0d0a6f64739da4aae42632043387b": "y^2 - x^2 = C,\\,", "a1e0dd30d3fd081ce0cb673b199f6bc9": "r_1, r_2, \\ldots r_n.", "a1e0e048cd675ce29f36811a2371271a": " t = w + y j \\ ", "a1e136ead99df76138d20e327ac674e2": "\\, L^{-1} X_{t} = X_{t+1}\\,", "a1e16a5f71e0298c06b70b1d9be4bd4b": "S\\,\\mathrm{d}T\\,", "a1e1984450742660856bf0a48aa28aa3": "\\scriptstyle n_0=\\frac{1}{h}", "a1e1e9dd7f0409288ee15e6aba423229": "c^*=\\arg \\max_c p(c|\\mathbf{w}) = \\arg \\max_c p(c)p(\\mathbf{w}|c)=\\arg \\max_c p(c)\\prod_{n=1}^Np(w_n|c)", "a1e1ec28846b00717c61e158359d0b0f": "W = F s \\,", "a1e21ef50829150821ab73db847f0b08": "\\displaystyle{B(a,b)=Q(a-b^{-1})Q(b).}", "a1e28ed55022971bd27e73e5563fb505": "M=\n\\begin{pmatrix}\n ap & aq & ar & as \\\\\n bp & bq & br & bs \\\\\n cp & cq & cr & cs \\\\\n dp & dq & dr & ds\n\\end{pmatrix}\n", "a1e29aa52d24a9ac5c9cdcf3c0fa2026": "V_1(\\mathbf{x},z_1) = V_x(\\mathbf{x}) + \\frac{1}{2} ( z_1 - u_x(\\mathbf{x}) )^2", "a1e2bd480d02025c0e0a921999c40feb": "\n \\boldsymbol{\\sigma} \n = 2\\left[\\left(\\cfrac{\\partial W}{\\partial \\bar{I}_1} + \n I_1~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right)~\\bar{\\boldsymbol{B}} - \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\bar{\\boldsymbol{B}}\\cdot\\bar{\\boldsymbol{B}}\\right] - p~\\boldsymbol{\\mathit{1}}~.\n ", "a1e2efffcdf8f2bcf13a215131abebe9": "=1-\\frac{2}{(r_a/r_p)+1}", "a1e3143817250dfcedaf81eb5873d10e": "x\\in j", "a1e332257f57736ed4511fa024e8110c": "2^{40}", "a1e36203ed648b27d027e2c17ee4b440": "-1 \\leq \\operatorname{psin}(\\bold{v}_1,\\dots,\\bold{v}_n) \\leq 1,\\,", "a1e38c0ce656bf3f7b8b591615b11620": "I_1 \\ldots I_m", "a1e3df927f6dabc92f02a60ac39913da": "m_{0}+t", "a1e3ef808a2938f538b0251f09afde17": "\\tbinom{c_k}k", "a1e411b53448dec8e10dc0b4dfe21be7": "X_1X_2\\cdots X_n. \\, ", "a1e430101f252701892668014de28e21": "\\frac{T_i ^f - T_i ^{f-1}}{\\Delta t} = a \\frac{T_{i-1} ^{f-1} - 2T_i ^{f-1} + T_{i+1} ^{f-1}}{h^2} - \\epsilon u \\frac{T_{i+1} ^{f-1} - T_{i-1} ^{f-1}}{2h} + \\frac{Q_i ^{f-1}}{c \\rho}", "a1e45624ff055e16b0578a981aa90d42": "\\hat{u}(f)", "a1e4837f2b451a73f759f77ac27c688c": "\\tilde\\nabla_{F_* X}F_* Y = F_*\\Bigl( \\nabla_X Y + X(\\varphi)Y + Y(\\varphi) X - g(X,Y)\\operatorname{grad}\\varphi \\Bigr)", "a1e494fc3800ee0c3311615c7f5afa14": "\\text{excess kurtosis} =\\frac{6}{3 + \\nu}\\bigg (\\frac{(1 - 2 \\mu)^2 (1 + \\nu)}{\\mu (1 - \\mu) (2 + \\nu)} - 1 \\bigg )", "a1e4ab6723f978eb244ac958abc9624e": "\\begin{align}\n I_1 &= E_1(u, v) \\operatorname{d}u^2 + 2F_1(u, v) \\operatorname{d}u \\operatorname{d}v + G_1(u, v) \\operatorname{d}v^2 \\\\\n I_2 &= E_2(u, v) \\operatorname{d}u^2 + 2F_2(u, v) \\operatorname{d}u \\operatorname{d}v + G_2(u, v) \\operatorname{d}v^2\n\\end{align}", "a1e4bc544e6ea87c684b998ca211baf7": "\\; s_i", "a1e4dd35a88a908e47d71e0b44155244": " N_{-} = \\operatorname{ker}(A^* + i) ", "a1e4e27293d70f71b5cbff30b1a89ee1": "| \\epsilon \\, | = 0", "a1e4efbbd2af49e91ea55f20f9507211": "\nV(\\mathbf{R}) = \\frac{1}{4\\pi \\varepsilon_0 R^3} (\\mathbf{P}\\cdot\\mathbf{R}) ,\n", "a1e50e44c0d4b3c9907b63ec62f1e608": "\n\\frac{d^{2}x}{dt^{2}} + b \\omega_{0} \\frac{dx}{dt} + \n\\omega_{0}^{2} \\left[1 + h_{0} \\sin 2\\omega_{0} t \\right] x = \nE_{0} \\sin \\omega_{0} t\n", "a1e5167dc1cfad3fe0eb4fb1a511fb0b": "a = b = c = d, \\alpha = \\gamma = \\zeta \\ne \\beta = \\delta = \\epsilon, cos \\beta = -0.5 - cos \\alpha", "a1e52f7139ba32a07f72d1c2488b6c75": "X_{-1}", "a1e58ab74fd38899833f627770d32713": "\\text{id}\\,", "a1e5c14daa85cd3461111a509c064ff4": "\\exists x\\,(Mx \\and Lx)", "a1e615d05f1209d1254eea45146f6499": "gcd(h^{\\prime},N)=1", "a1e6194a29ef98ddd22fafa712f5b7ad": "i\\frac{\\partial\\mathbf j(\\mathbf r,t)}{\\partial t}=\\langle\\Psi(t)|[\\hat{\\mathbf{j}}(\\mathbf r),\\hat{H}_v(t)]|\\Psi(t)\\rangle.", "a1e6b32de7fc447284af13ca22ba490b": "\\dot{\\theta} = (F\\sin\\alpha)/mv + L/mv + (g/v - v/r)\\sin\\theta,\\,", "a1e6d793a829c31e33a7f60580d3fb9a": "f(q_1, q_2, q_3,\\ldots, q_{n}, t) = 0", "a1e6ea3a99b049549e64b4fc48f4f1a2": "\\begin{smallmatrix}\\frac{M_{Neptune}}{M_{Earth}} \\ =\\ \\frac{1.02 \\times 10^{26}}{5.97 \\times 10^{24}} \\ =\\ 17.09\\end{smallmatrix}", "a1e7ed23d484a526ebb0ff53738e51fa": " b=|\\mathbf{E}|\\sqrt{\\frac{1-\\sqrt{1-\\sin^2(2\\theta)\\sin^2\\beta}}{2}}", "a1e7fad14067baf8d53c97a088d34f87": "\n\\hat{f}(\\mathbf{w})=\\int\\limits_{-\\infty}^{\\infty}\n\\int\\limits_{-\\infty }^{\\infty} f(\\mathbf{x})e^{-2\\pi i\\mathbf{x}\\cdot\\mathbf{w}}\\,dx\\, dy.\n", "a1e81385865dd53766e4a0abc667313d": "{{i}_{E3}}={{i}_{C2}}+2{{i}_{B}}={{i}_{IN}}-{{i}_{B}}+2{{i}_{B}}={{i}_{IN}}+{{i}_{B}}", "a1e85b4342a198c82c058788d02baebf": " \\mathbf{h}^m(\\cdot) ", "a1e861466ad480716363b970a8809546": " \\mathbf{v}_\\mathrm{g} = \\mathbf{\\hat{e}}_{\\parallel} \\left ( \\partial \\omega /\\partial k \\right ) \\,\\!", "a1e8d9f5d15f25687ce5252ac25797b9": "F_i = \\frac{q_i}{\\sum_j q_j x_j}", "a1e8da930f704ca4ace817080231348d": "P_m = P_2 + P_3 + \\cdots + P_n.", "a1e8ea656ed8cc0fcf5e3401939b2282": "\\left [i_{\\delta\\lambda}, s_B \\right ] s_B X = i_{\\delta\\lambda} (s_B s_B X) + s_B \\left (i_{\\delta\\lambda} (s_B X) \\right ) = s_B \\left (i_{\\delta\\lambda} (s_B X) \\right ),", "a1e9a49188fe1705a13d8fddfa1176d8": "\\mathbf{z}_{k+1} := \\mathbf{M}^{-1} \\mathbf{r}_{k+1}", "a1e9d030904015913e0d688668a38865": "\\,e\\cdot e", "a1ea1ec7341cafadc6ccb8e80b02c032": "y'-\\frac{2y}{x} = 0.", "a1ea21a074ecf89aab97352956f7bf1e": "\\psi_t+(u\\psi)_x+(v\\psi)_y+(w\\psi)_z=0.", "a1ea51d6a8b183b050f2caf4916a0a5a": " T_\\mathrm{max} = \\left({\\frac {IC}{\\sigma}} \\right)^{0.25} ", "a1eb243462ed39871da08562a846decf": " \\ge H_q^{ - 1} (\\frac{1}{2} - \\varepsilon )", "a1eb54f20705cff2d4a3b419bcc98fd8": "i_{j+1}=\\max\\{i:\\gamma(i)=\\gamma(i_j)\\}+1\\,", "a1eb56a282af2c862beac5d5f4e37a5f": "\\overline {AC}\\,\\!", "a1ebbb83f9565e8e1b1da1e1283baf73": "\\sigma' = \\sigma,", "a1ebdc9403ed0f9bd93b339b77d76624": "r(A)\\leq r(B)\\leq r(E)", "a1ebdeed66cccc072d095933cd51fc9a": "K^{n-i} (\\tilde M) \\cong K_i (\\tilde M)", "a1ec13c367ebbbd2fac53af11d418f26": "F \\in \\mathcal{A \\otimes A}", "a1ec1ea6973a12e50a5a60c452db0468": "\\frac{p_{2}}{\\rho g}\\, =\\, -\\, [z\\, +\\, \\frac{V_{2}^2-V_{3}^2}{2\\,g}]\\,", "a1ec734c2b3fc3d52ef2caccba98bb73": "\\ \\bar{X} ", "a1eca1d9256fbfb31a4086014d97015b": "\\frac{d^2 \\sigma}{d\\Omega d\\omega} = a^2\\left(\\frac{E_f}{E_i}\\right)^{1/2} S(\\vec{k},\\omega) ", "a1ecbf63301ee31d120045867a685dea": "2^{\\log_2 3}=2^{m/2n}", "a1ed06052bb1cfc9d78e1786710ad6e5": "s_i=p_0 q_i + p_1 q_{i-1} + \\cdots + p_i q_0.", "a1ed0c3bbc1f8738222f2f30ee117adb": " \\left(2\\frac{K}{N}-1\\right)W \\! ", "a1ed0eede5395db424da46fbed5cdad1": "\\sim N(0, A)", "a1ed242ead58e64b083c8d13f01ec7e1": "g_i = f_i / f_{i-1}", "a1ed28d6215f4cfb40300c13933c03fa": "E(k) \\propto k^{-p}", "a1ed59f680ba5e9ead4a7b601648c1a6": "\\frac{\\sqrt{a^2-b^2}}{b}", "a1edcc81035f9ec0c0505914037903b6": "n_{B}=0.75*1+1=1.75 mol", "a1edd8ba890eaca46f9f4d595304de87": " \\mathbf{u} \\wedge \\mathbf{v} = (u_1 v_2 - u_2 v_1) (\\mathbf{e}_1 \\wedge \\mathbf{e}_2) + (u_3 v_1 - u_1 v_3) (\\mathbf{e}_3 \\wedge \\mathbf{e}_1) + (u_2 v_3 - u_3 v_2) (\\mathbf{e}_2 \\wedge \\mathbf{e}_3) ", "a1ee0f16f6ab02496d8c45d0c081c349": " \\beta ", "a1ee11fd58cc1dda44f5329c1f6a4e48": "\\scriptstyle r \\gg d", "a1ee1a3b9ce9c391a610286abbc298a0": "K*", "a1ee228bcb4a68c6fb0446cba65348b9": " \\hat{G}(s, t) ", "a1ee2c677696eb2e7e5978d684154f15": "\\sigma (M) \\geq 0", "a1ee2e2673b86292bfaea6ba91aaf9ad": "\n\\begin{align}\n\\operatorname{Re}\\{(A e^{i\\theta} \\cdot B e^{i\\phi})\\cdot e^{i\\omega t} \\}\n&= \\operatorname{Re}\\{(AB e^{i(\\theta+\\phi)})\\cdot e^{i\\omega t} \\} \\\\\n&= AB \\cos(\\omega t +(\\theta+\\phi))\n\\end{align}\n", "a1ee6f5da40bcfe16b2994a189baca2d": "B_7", "a1ee8816f8eb5c117d5db7f4823dc41e": " c_{1} = \\tfrac{1}{2}i ", "a1eed8f9a6fd17e70ba4b483e0a0a9d7": "\\frac{y'}{x^2} - \\frac{2y}{x^3} = 0", "a1eef8f75e02424bc54509ed96fad21a": "\\nabla\\cdot\\nabla\\times\\nabla\\times\\mathbf{T}=\\mathbf{0}", "a1ef218f0f437b6122ba7dd5f1da8717": "k_\\mathrm{spec} = \\frac{\\exp{\\left(-\\tan^2(\\alpha)/m^2\\right)}}{\\pi m^2 \\cos^4(\\alpha)}, ~ \\alpha = \\arccos(N \\cdot H)", "a1ef467192e927310cd9aba174072ce7": "\\displaystyle{u(z)=\\int_{\\partial \\Omega} -K(z,w)u(w) + N(z-w)\\partial_n u(w) \\,|dw|.}", "a1ef98efedf57fe214ded2817e1868da": "\n\\int_{-\\infty}^{\\infty} \\frac{\\mathrm{d}\\omega}{2\\pi} \\rho(\\mathbf{k},\\omega) = 1,\n", "a1efa12a90db97032cf896e080777fcc": "W_s(x)", "a1f0085aa480e6303c10642e654d855c": "M_{cycles} = 901.12", "a1f0355aabb6b6ba9fa073f761c14b08": "e^{f(x)}", "a1f07a6dd02f55638bc592b0ced4e091": "\\displaystyle \\hat{f}(-\\nu) = \\overline{\\hat{f}(\\nu)}\\,", "a1f093f8334744ab5354f63db167fd8b": " p(\\theta_1, \\cdots, \\theta_m) = C \\prod_{1 \\leq i \\leq m}(1-\\cos^2\\theta_i) \\prod_{1 \\leq k < j \\leq m} (\\cos\\theta_k - \\cos\\theta_j)^2~.", "a1f0af177aee34eb2686bde967e57307": "\\delta\\Delta-\\Delta\\delta=-\\bar{\\nu}D+(\\tau-\\bar{\\alpha}-\\beta)\\Delta+(\\mu-\\gamma+\\bar{\\gamma})\\delta+\\bar{\\lambda}\\bar{\\delta}\\,,", "a1f0bcadaa2cda92a11c7e4378688ffd": " L \\approx 1 \\left/ \\left( ( 2.5 + 1 ) \\sin 0.277^\\circ - \\frac{1}{490} \\right) \\right. \n\\approx 67.203 \\approx 67\\tfrac{1}{3}\n", "a1f0cac4631d6315ee8543f1964d6d5e": " \\overline{BD}^{\\,2} = \\overline{BC}^{\\,2} + \\overline{CD}^{\\,2} \\ ,", "a1f0e6be226ffb5d189c760d4d2e61d8": " e^+e^- \\to \\rho, \\omega, \\phi \\to \\pi^0 \\gamma, \\eta \\gamma ", "a1f0f5c5edb4e266fa354e9d90ea7446": " X = \\left[ \\begin{array}{c} x_1 \\\\ \\vdots \\\\ x_n \\end{array} \\right] \\in \\mathbb{R}^{n\\times p},", "a1f1052f4c2442a8ba964fe46a87da82": "\\mathrm{j}_x", "a1f164643c3cb23e5ae22160d2c79938": "\\mathrm{kei}(x) e^{x/\\sqrt{2}}", "a1f19f1348e0f6319546e030c8243922": "V_k=V+q_k=\\{v+q_k : v \\in V\\}", "a1f1a3d353f2404a5bcf4e156721bb37": "\\langle x, y\\rangle = \\bar x_1 y_1 + \\cdots + \\bar x_n y_n", "a1f1b8559dafb2484f367e980407ae40": " E_k = m_k c^2 ", "a1f2112633e4f41b2edf6a4c93f4cb44": " (year + [year/4] + [year/400] - [year/100] - 1) \\mod 7 ", "a1f25a52a464f13ccf973a2eeef084e2": "=\\Pr(X_{n+1}=j|X_n=i)(1-e^{-\\lambda_i t}), \\text{ for all } n \\ge1,t\\ge0, i,j \\in \\mathrm{S} ", "a1f26b9a7856bc723baca9d3f7d8f6bd": "\\Phi_{105}(x);", "a1f2768eae671b3242447cb784d3fca1": "x_{r,s}", "a1f28e6d7cfbc5c0e6c66d543daceac4": "N_e = I(t)/2\\beta", "a1f3523de3b9de7779852a295407a76a": " \\mathbf{B}= \\mu_0\\left(\\mathbf{H}+\\mathbf{M}\\right)", "a1f36624dbf93c22239da21c9dcf79eb": "\\frac{1}{4}V_g", "a1f3a70743acc25ed0d6ba9d4e974db5": "\\ =", "a1f3eb5cf502f8be047e3e993ecc3786": "\\zeta(-7)=\\frac{1}{240}.", "a1f4c36935779132c4f20e3d3be79329": "d(\\mathbf{v}+\\mathbf{d},\\mathbf{w}+\\mathbf{d})^2 = (\\mathbf{v}+\\mathbf{d} - \\mathbf{w}-\\mathbf{d})\\cdot(\\mathbf{v}+\\mathbf{d} - \\mathbf{w} -\\mathbf{d})=(\\mathbf{v} - \\mathbf{w})\\cdot(\\mathbf{v}- \\mathbf{w}) = d(\\mathbf{v},\\mathbf{w})^2.", "a1f4ed1abb22e79d792a1497e5669b6e": "D = \\frac{1}{2} \\cdot c \\cdot t_0 \\cdot \\frac {S2} {S1 + S2}", "a1f539da83054a5d03d4b7cea9ba3160": "p(\\boldsymbol\\mu\\mid\\boldsymbol\\Sigma) \\sim\\mathcal{N}(\\boldsymbol\\mu_0,m^{-1}\\boldsymbol\\Sigma) ,", "a1f5ca296ae716dc1ee75ae000447fac": "\\frac{dx}{dy}=-\\frac{\\beta}{\\alpha}.", "a1f5e2bdb1af255077ba3e1cacce84a3": " U^N = (e^{i H \\Delta t})^N = \\int DX e^{iL}", "a1f60ea06aad618e18237b4bf5f68ed9": "\\Delta t=M+\\lambda_p-\\alpha", "a1f628cfb3ab7b2e4a3c5f32301799fe": "\\chi(\\pm 5)=-1", "a1f6796230bd83efae6d299d3332d361": " \\frac{dN(t)}{dt} = N(t)(r-\\alpha P(t)) ", "a1f6958dc450773d42bb3724bb4e46cc": "p=2a", "a1f6dc391f521c1c47eb7b7df30bf9fa": "[\\mathcal{L}_\\ell, \\mathcal{D}_a]=0", "a1f7243f801e50c7680766b87caeb9d6": "\\begin{matrix}4&4\\\\6\\end{matrix}", "a1f72769b36b83dbf77a62806ba563d1": "D = \\begin{vmatrix}\nx_1 & y_1 & z_1 \\\\\nx_2 & y_2 & z_2 \\\\\nx_3 & y_3 & z_3\n\\end{vmatrix}", "a1f76af9bf468aa26b4bf5a4c0c080ba": "\\mathbf{i}", "a1f7aa78055f10151dddf8cfe779afec": "{\\|x+y\\|^2-\\|x\\|^2-\\|y\\|^2\\over 2}\\text{ or }{\\|x\\|^2+\\|y\\|^2-\\|x-y\\|^2\\over 2}.\\,", "a1f7e1eb3e25b1fca81f5a2b0cb0426f": "(x_1,y_2)", "a1f7fa3bede849cacb530887644d059d": "t_S\\cong\\frac{Q_S}{I_R}", "a1f880ba02bb0265af1a22e7c19ac3fb": " DO_b ", "a1f8af39c9276dd0d007f8a1cd85932f": "e_1 e_2", "a1f8ee052e02f3d6c54f0ac101c15f7f": "(\\mathbf{q},\\mathbf{p})", "a1f902100e2d525bb92da382a9524479": "y_T = \\frac{1-\\Gamma}{1+\\Gamma}\\,", "a1f94c7a0cd17d95ab2f85381e943a96": "m_2 = 55 = 5\\cdot 11", "a1f9961b5264e3238e5598a15bc4116f": "\\textrm{advance}(t)", "a1fa36ec242464f882408ff283dc50a7": "\\omega_r(\\lambda) = \\left|\\left| {\\partial (x,y,z) \\over \\partial \\phi } \\right|\\right| = r \\ , \\quad\\mathrm{respectively}", "a1fa4570d59df3a87302c96ad367b7d1": "\\sum_{k=m}^n f_k\\Delta g_k = \\left[f_{n+1} g_{n+1} - f_m g_m\\right] - \\sum_{k=m}^n g_{k+1}\\Delta f_k,", "a1fb17953a3cc31a0b4993d229b3d5ce": "\\neg a", "a1fb79e5b6c9c74e6ad6f19208f5b812": "m_p(f) = \\prod_qM_p(f_q)\\times \\prod_{q a_{G1} = \\frac 1 n \\,", "a204787d8396468ca138a31193e314a5": " T + \\Delta T", "a2047a4c7416ffff5174f85e66b6ef8b": " \\left(A+UCV \\right)^{-1} = A^{-1} - A^{-1}U \\left(C^{-1}+VA^{-1}U \\right)^{-1} VA^{-1}, ", "a20480243f167fa1bba67bed0bcb91ab": "\n \\cfrac{1}{G(1+R_0)} = (\\sigma_1^y)^2 ~;~~ \\cfrac{F+R_0 G}{G(1+R_0)} = \\cfrac{R_0(1+R_{90})}{R_{90}(1+R_0)} ~.\n ", "a204d32207deebadc7416c06749b5390": " H(\\beta,\\lambda) = {c (2 a^2 + (c^2 - a^2) \\cos^2 \\beta) \\over 2 a (a^2 + (c^2-a^2) \\cos^2 \\beta)^{3/2}}.\\,\\!", "a20504c428c99c98b90979e9e5f1c12d": "q(t) = {1 \\over \\mathrm{R}} [\\Phi(0) - \\Phi(t)]", "a205227fd042b2e69886bb7566777acf": " \\varphi = \\varphi \\,", "a205651de6b933287292419bb5075b19": "F_W = 6\\cdot\\pi\\cdot\\eta\\cdot\\text{r}_{H}\\cdot\\nu", "a2058e951c96772b0b00c242a92a48da": "a_{n+m}\\geq a_n + a_m.", "a2059b8ef15bc1e34bc1eb32b00eea1f": " Z/pZ \\oplus Z/pZ ", "a205ca8f5f3e92c29e6150cc93ff66a8": "0\\leq x,y\\leq 1", "a20609e89e1bf112197882a3a4b56ff0": "\nf_{\\mathbf{w}}(\\mathbf{x}) = \\sum_{i=1}^p \\mathbf{w}^i \\Phi^i(\\mathbf{x}) = \\langle \\mathbf{w},\\Phi(\\mathbf{x}) \\rangle,\n", "a2064b855ad80536e6433f0fcef200c4": "\\cos_k(i+t)\\equiv \\cos_k(i)\\cos_k(t)-\\sin_k(i)\\sin_k(t),", "a20676841bdb17cf29bde98c1a799f2b": "\\int_{\\mathbf{R}^n} \\Phi = 1", "a206a0c6d8fdabb403281f173d46ab51": "p_i\\!", "a206bb7e8fcf57b856f3e1d2ef591001": "(D,V,s,R) \\models F(x_0, \\ldots, x_n)", "a206d3efad6a67de76150b5ef7d14d32": "\\begin{align}\np(\\mathbf{X}|\\mu,\\sigma^2) &= \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{1}{2\\sigma^2} \\sum_{i=1}^n (x_i-\\mu)^2\\right] \\\\\n&= \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{S}{2\\sigma^2}\\right]\n\\end{align}", "a206fa8a7e69ab5b2ff6807a68943c52": "f_y = \\frac{y_0 - y }{\\sqrt{{(x_0 - x)}^2 + {(y_0 - y)}^2} } \\quad (20)", "a20706cf83146ce58a3b709949bcd5bb": "(1+x)^2=1+2x+x^2 \\,", "a2071c94f3673b60ec80dfb71ad3aeb4": "F^a_a=0", "a2074dc314ff2d9f3da0cb7cccf6c076": "P_{\\mathbf{r}\\in R} (t) = \\int\\limits_R d^3\\mathbf{r} \\, \\left |\\Psi(\\mathbf{r},t) \\right |^2 ", "a2076d7beb219e9bbae0e83269a98030": "op_j", "a20773203e1066c1de28868da68cd75e": "z_{k-1}=0", "a2078dbfa39ae261ca6830cb9434563e": "Q_{1lm}", "a207cfec8f27b328398381614cb7fe52": "y = \\sin\\phi\\,", "a207d831b2ad325437fcdb29a2bf43f0": "q_1 = J(t',t') = \\lim_{\\delta \\to 0} J(t'+\\delta,t')", "a207dabf87b6aa48b161c50b90e794c4": "g(x)\\sim N(0,5)", "a2085510ab5adb512667155cc3e79ca4": "\\mathbf{\\alpha=\\alpha_1}\\,\\!", "a20889e875f10b6cd6b01dcaad9d4b5f": "P_{ATM}", "a2089aa1e4629ae638c6db2d489fe4db": " a_{\\lambda\\mu}^r = \\frac12(F_{\\lambda\\mu}^r + S_{\\lambda\\mu}^r)= \\frac{1}{2}(a_{\\lambda\\mu}^r + a_{\\mu\\lambda}^r\n - c_{pq}^r a_\\lambda^p a_\\mu^q) + \\frac{1}{2}\n(a_{\\lambda\\mu}^r - a_{\\mu\\lambda}^r + c_{pq}^r a_\\lambda^p a_\\mu^q), ", "a208de7dd39674520eb421a36e07387f": "\\left [ S_{3 \\times 3} | C_{3 \\times 1} \\right ] =\n\\begin{bmatrix}s_1&s_2&s_3&s_4\\\\\ns_2&s_3&s_4&s_5\\\\\ns_3&s_4&s_5&s_6\\end{bmatrix} =\n\\begin{bmatrix}1011&1001&1011&1101\\\\\n1001&1011&1101&0001\\\\\n1011&1101&0001&1001\\end{bmatrix} \\Rightarrow\n\\begin{bmatrix}0001&0000&1000&0111\\\\\n0000&0001&1011&0001\\\\\n0000&0000&0000&0000\\end{bmatrix}", "a2093b61be514b6bc687837db548f8ba": "=\\int_{\\mathbb{R}^n} f(x)e^{-2\\pi i x\\cdot \\nu}\\,dx \\int_{\\mathbb{R}^n} g(y) e^{-2 \\pi i y\\cdot\\nu}\\,dy.", "a20984197e2fff8ef4079258ffdb426c": " e\\ ", "a2099da13ea0d9a893be57871fbb639b": " U , V", "a209acf02ed20380ed07e18d1ccc320c": "\na = \\frac{n}{G \\plusmn \\sqrt{(n-1)(nH - G^2)}}\n", "a20a0bb8ce18afdb3246768a326dd13c": "\nI = \\int dq |q\\rangle \\langle q |\n", "a20a1c558983affe029edaf342cb750d": "\nx = a\\sigma\\tau \\!\n", "a20a5833955512dc9ec7f0744e8ad080": " (t^{k_0}-1)A = t^{k_0}A\\left(\\frac{h}{t}\\right) - A(h) + O(h^{k_1}) ", "a20a6987c90d4f52762cecb69c3442e4": "\n\\begin{align}\n\\hat {f} \\left[\\sum _{n=0}^{N} \\delta(x-nS) \\right]\n&=\\sum _{n=0}^{N} e^{-i f_x nS}\\\\\n&= \\frac {1-e^{ -i 2 \\pi NS \\sin \\theta/\\lambda}} {1-e^{-i 2 \\pi S \\sin \\theta / \\lambda}}\n\\end{align}\n", "a20a6d9f73a7d05f0236ccc2f4eab2dd": "\\operatorname{RMSD}= \\sqrt{\\frac{\\sum_{t=1}^n (x_{1,t} - x_{2,t})^2}{n}}.", "a20a8cc672b90b79354cd7f0dbcd35e4": "-1-\\eta", "a20a9a7fd89c31463ced40defbcfbd45": "\\|\\mu\\|=|\\mu|(X)", "a20ad65dab21d8c95fef6fde4c85e0c1": "a\\frac{2}{\\pi}", "a20b41e5f83c4406451c9e4356d32095": "\\det(I_n) = 1", "a20bb0b73819140a9a3ddd58ba2645e4": "{\\bold \\ V}", "a20c2b6483cae68af5ae7671567f53ff": "A=\\Bigl\\{\\sum^N_i \\alpha_i \\mathbf{v}_i \\Big| \\sum^N_i\\alpha_i=1\\Bigr\\}", "a20c5e7e3d0ae61492f136e01fbb85ac": "\\forall i=1\\ldots m", "a20c79b7dda5713b5cb727b91710e9ce": "\n \\lVert s(\\theta+h) - s(\\theta) - \\dot{s}(\\theta)'h \\rVert = o(|h|)\\ \\ \\text{as }h \\to 0,\n ", "a20ce2b380084bd8d4183d78e62267f0": "\\Sigma \\Tau \\Upsilon \\Phi \\Chi \\Psi \\Omega \\!", "a20d04651aee5041e9ab638513b9da64": "\nP(J_{i_1\\cdots i_r}) = \\sqrt{\\dfrac{N^{r-1}}{J^2 \\pi r!}} \\exp\\left\\{-\\dfrac{N^{r-1}}{J^2 r!}\\left(J_{i_1\\cdots i_r} - \\dfrac{J_0 r!}{2N^{r-1}}\\right)\\right\\}\n", "a20d25c823f8dd11863e37afc9156380": " \\nabla_\\alpha (\\sqrt{-g}\\nabla^{[\\beta} A^{\\alpha]} ) = \\mu_0 J^\\beta", "a20dbb907440e7acf6b668e268d5f6f7": "\\sum_{n=- \\infin}^\\infin\\frac{(-1)^n J_{\\alpha - \\gamma n}(z)J_{\\beta + \\gamma n}(z)}{n+\\mu}=\\frac{\\pi}{\\sin \\mu \\pi}J_{\\alpha + \\gamma \\mu}(z)J_{\\beta - \\gamma \\mu}(z).", "a20de1b3bb826551d1d3730c25c6b5ef": "\\xi_{[t]}", "a20df1fc62db8db3feb80ee773abb102": "\nu_{1} + u_{2} + u_{3} = \\frac{1}{r_{s}}\n", "a20e3fb1d24b4cb8adbe3dbb5db4cfbb": "H_1(z)=1+z^{-1}+z^{-2},\\,", "a20e42afa27b100264be11fb5d83ba79": "\\scriptstyle \\dim (T_p)^r{}_sM \\;=\\; n^{r+s}", "a20e527ecca3b719db0abbf90b57c82f": "F\\dashv G", "a20e65d8804ab6fa18b0b12f89b88f37": "-{dy \\over dx}\\sin y=1", "a20e7c5286ae7f5711816ed6a892c750": "C= N\\mu^2/k", "a20e7eeab75b67c63ae84946741f5550": "=({10^{2.0}})^{(3/2)}", "a20eac1c74cd535417f32b5d30b3275e": "r_k=(n-1)\\left|w_k\\right|", "a20ed9fdd8792dcfd378efa52ceeb8c2": "(G, *) \\cong (H, \\odot)", "a20f177b31b220ac530efe29a0c7c904": "O(V \\log V)", "a20f41e9221025a53e8f842d4bd04de3": " = e(p_0, u_1) - e(p_1, u_1) ", "a20f62abd034e87c7a72bdb90b7c29ef": "|\\mathcal{B}|=365", "a20fa46efc3829e2658967fc016db6a7": "A_{xy}(f)= (\\Lambda_{xy}(f)^2 + \\Psi_{xy}(f)^2)^\\frac{1}{2} ,", "a21022c7b05e4856f30b65117b05a346": "\\begin{align}\n0\\times 1 &= 0 \\\\\n0\\times 2 &= 0.\n\\end{align}", "a21025b4cf6aea38bcd4e85a103283b6": "|S_{ab}-\\delta_{ab}|^2.\\,", "a2106be8057f7a1a4f8679f56f89a151": "\\sqrt{|d_K|} \\geq \\frac{n^n}{n!}\\left(\\frac\\pi4\\right)^{n/2}", "a210bdb0651ff3a17d2e72e256671771": "\\hat{\\theta}_0", "a210f5f1c23853fa5ff665e32a003215": "F_{\\nu,max} = const", "a2110c7c92e1cfe302a2498a2384e5f3": "\\overline{W(T)}=V(\\mathrm{sat}(T))", "a2113bacd3c8993f3ba3a01f83e14bc5": "(x_1-a)(x-a)+(y_1-b)(y-b) = r^2.\\!\\ ", "a2116c1c9850267d37df5c8d3b49ed2f": "p_{\\rm HEL}", "a21196b4699eb5973bc99b38389a172a": "m_{tot}=E_{tot}/c^2", "a211d09ab0e80c0d448656b648e56d35": "N(v)=\\lambda(L)", "a211e088deda23a80cc9637290035354": "\\varphi \\circ f", "a211f37467a2c73ae39d7d5117af4e6f": "\\scriptstyle\\rho(u)", "a212313ff82837f7a573afd4e57160f4": " (x - 3)(x^2 - 4x + 29).\\,", "a2128ab5d052be2117c3986469aea36e": "(\\varphi_1\\otimes 1 + \\varphi_2\\otimes i) \\leftrightarrow \\varphi_1 + i\\varphi_2", "a2130bbc75966150d92d2055db6a4988": " \\psi_{\\alpha_1 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} (\\mathbf{r},t) ", "a21330cab77de0a9a12f082399926119": " \\sum\\limits_{i = 1}^M {p_i \\log p_i } \\le \\log \\sum\\limits_{i = 1}^M {p_i^2 } ", "a21367b8db09d280d24b7acd3bb0a3f2": "\\frac{V}{n}", "a2138838efdf3bafd1af49bb66081bab": " V_{CV}= V_{GS} (\\mathrm{for}\\ I_D\\ \\mathrm{at} \\ V_{DG}=0V) = f ^{-1} (I_D) \\ \\mathrm{with}\\ V_{DG}=0 ", "a213c36b50316b2ea6f5288f4294a5ec": "\\ H = \\frac{1}{2r}", "a213fb562173833a9a0901c9ec1a7aad": "\\ z_2", "a2141e424ccee8f2577b3134124a088b": "u(z) = \\frac{1}{2\\pi}\\int_\\mathbf{C} \\log|z-\\zeta|\\,d\\mu(\\zeta)", "a21488b43069023a1cf373cee83267a8": "\\lim_{s \\to t} F_{s} (x) = F_{t} (x)", "a214894958b05faec78685e1fba3114d": "\\phi_k = \\operatorname{atan2}(b_k, a_k)\\,", "a214a56b3081ec121ce1a7e2edbfb816": "H = \\log S^n = n \\log S", "a214a886ccc1001ce5ed92ce07a12512": "{f(x+h)-f(x)\\over h}.", "a214fa574b057248544455363cff0cce": "p_3(X_1,X_2)=\\textstyle\\frac32p_2(X_1,X_2)p_1(X_1,X_2)-\\frac12p_1(X_1,X_2)^3.", "a2151ed3c8c8857b5f6ce58f2779cd17": "(2m+1)^2+(2m^2+2m)^2 = (2m^2+2m+1)^2", "a21546903655580aab3bf33b02d75356": "T =T_0+T_1+T_2\\,\\!:", "a215d04dfd61f0154bad0e5b2c4db214": "m = \\frac{L\\left(C^{p-1} \\mod p^2\\right)}{L\\left(g^{p-1} \\mod p^2 \\right)} \\mod n ", "a215f2209109419d8c10677a5e721c6d": "H_n = \\begin{bmatrix} 2 \\times H_{n-1} & H_{n-1} \\\\ H_{n-1} & H_{n-1} \\end{bmatrix}", "a21617183c11d6f992be3d5bb5b973f2": "\\|f\\|_{A^2_\\alpha}=\\left\\{\\frac1\\pi\\iint_{|z|<1}|f(z)|^2(1-|z|^2)^{\\alpha-1} \\, dS\\right\\}^{1/2}<+\\infty\\text{ for some }\\alpha\\in(0,+\\infty),", "a2165b4cde66868b3d2d89b3cb57a07c": "[a]=\\{a\\}, ~~~~ [b]=[c]=\\{b,c\\}", "a2165ef6f3879356d9df3dba9372ebf0": "A(w)B(xw)=\\sum_{n=0}^\\infty P_n(x)w^n.", "a2167ff27a4760c5623a6bd200b2a218": "\\mathcal{O}_{Y'}", "a2168cfdaff5da890c6bd79fa658c863": "a\\ll 1", "a2169f39db49333d93d29702e079b340": "r = \\frac{L+1}{2}", "a216b1039c175f44d3cdfac0d69cb544": " \\sum_{i\\in I}a_i. ", "a216b41600930dba1588addcf7cd81cf": "= 1{\\rm\\ googolplex}", "a216c780802f61278566478e50c8ab6d": "\n\t\\Pi^+ \\approx \\Pi^{-} \\approx \\tau^{\\pm}_k E^{+}(k) E^{-}(k) k^4 \\approx E^{+}(k) E^{-}(k) k^3 / B_0\n", "a2170b3dff21073fa483fe5865c489e0": " \\begin{matrix}\nI&=&|E_x|^2+|E_y|^2, \\\\\nQ&=&|E_x|^2-|E_y|^2, \\\\\nU&=&2\\mbox{Re}(E_xE_y^*), \\\\\nV&=&-2\\mbox{Im}(E_xE_y^*), \\\\\n\\end{matrix}\n", "a21720642028f8a9fe4157c6800f3ba3": "\\ulcorner \\urcorner \\llcorner \\lrcorner \\,", "a2176a00ca481b589a248457a9df2feb": "Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0 \\,", "a21777e2814c1f885106e6129c6628fc": " \\mathbf{R} = \\frac{1}{m_1+m_2}(m_1 \\mathbf{r}_1 + m_2\\mathbf{r}_2).", "a2177ee30c5e8ddac0d30cb794df070c": "\\left| I, k_1\\ldots k_n \\right\\rangle", "a217ea1b182ddd3cec848170ec4bee2f": " a_1\\mapsto a_2\\mapsto a_3\\mapsto \\ldots \\mapsto a_k \\mapsto a_1.", "a217f31e196a64da93e075f49385c66b": "\\mathcal{P} = \\{true\\}", "a21807b5a70295e070062a20c9eb492a": "H_{(k)} \\ldots H_{(m)}", "a21809a50c2def560f70924061a136b2": " \\dfrac{M}{Re} \\gg 1", "a218444bacd576859d9dc6a182f93c86": "\\rightarrow \\ \\ldots\\,", "a21853fea86f26cebd0dfdd5c4db2d3e": "z'=\n2 \\cdot \\frac{{z} - \\mathit{near}}{\\mathit{far}-\\mathit{near}} - 1\n", "a218759e6651a8c8fe4358df4c53832c": "\\hat{p}_i\\leq \\hat{p}_{i+1};", "a218c3754263f2123b337c3c4ea0369d": "{m_e}", "a218d1208683a8113f8a8d3a7474db30": "\\varepsilon_\\mu \\bar{N}\\left(\\alpha(q^2) \\gamma^\\mu + \\beta(q^2) q^\\mu + \\kappa(q^2) \\sigma^{\\mu \\nu} q_\\nu \\right)N \\, ", "a218df4a4635e36b573abe37a886b453": "\\scriptstyle M^{-1}", "a218f59fb80dd353f0ead40ae6b77630": "l'= {\\rm ROL}(r' \\vee KL_{i,2},1) \\oplus l", "a21903769a850b7db4d2a2d41a5cd5ab": "p_{AB}(t)=p_{BA}(t).\\,", "a21903dc21729f3eb558234908bb4420": "\\Lambda(\\varphi,\\hat{\\mathbf{a}},\\theta,\\hat{\\mathbf{n}}) = \\exp\\left[-\\frac{i}{\\hbar} \\left(\\varphi\\hat{\\mathbf{a}} \\cdot \\mathbf{K} + \\theta\\hat{\\mathbf{n}} \\cdot \\mathbf{J}\\right)\\right] ", "a219af63052e99228622f7bbf11659ad": "x(i\\Delta \\alpha) = interpolation(\\{ x(j\\Delta t) , j\\Delta t \\} , t(i\\Delta \\alpha))", "a21a18be72ba69ff671117469aba6c1c": "cs(I,k)", "a21a26d70c530ea0f6ef03c03e32dc1e": "\\|f\\|_p = \\|g\\|_q = 1.", "a21ab770e9965f94c92da1f083dea05e": "n = ((\\tfrac{k}{2})^2 + (\\tfrac{h}{2})^2)(l^2 + m^2). \\,", "a21ace7aaa718efbca6de2988d95df89": "\\cos(\\theta_2+\\theta_4)=\\cos\\theta_2\\cos\\theta_4-\\sin\\theta_2\\sin\\theta_4 \\, ", "a21ae12d2c730fc2fd62fdd88e5ef629": "ds^2 = -c dt^2 + [dx-v_s(t)f(r_s)dt]^2 +dy^2 +dz^2 ", "a21b2945c7028e5dfb84ae076b6af94e": " \\bar{\\alpha} = 1-\\left( 1-\\alpha_\\mathrm{\\{per\\ comparison\\}} \\right)^n", "a21b2aa31c5e32ab976fcc5e62da92ad": "\\frac{X_{n+1}-\\overline{X}_n}{s_n\\sqrt{1+1/n}} \\sim T^{n-1}", "a21b54507e8a31f2b1eac70c5fc63ac6": "z \\in L \\implies \\exists \\pi Pr[V^{\\pi} (x) = 1] \\ge 1 - \\epsilon", "a21b736180aa2f8b450e13aeebaeb76d": "X_{ni}", "a21b8abb5cb070ce7b54d8a0938051af": "E(r) \\leq C(\\epsilon) r^{1/2 + \\epsilon}", "a21bc7dbffc173719735c9d265de10b7": "K_{X_n}", "a21bd880d572faa9e0e676de0ac02bac": "\\dot{p_{\\theta}} = -\\frac{\\partial {R}}{\\partial {\\theta}} = 0, \\quad \\dot {\\theta} = \\frac{\\partial {R}}{\\partial {p_{\\theta}}} = \\frac{p_{\\theta}}{mr^2}", "a21bd8b2d9c33a23b322fceb3cb7d22e": "\\langle\\psi(\\lambda)|\\psi(\\lambda)\\rangle = 1 \\Rightarrow \\frac{\\partial}{\\partial\\lambda}\\langle\\psi(\\lambda)|\\psi(\\lambda)\\rangle =0.", "a21cf968024d7a2fa5ab76c3a353a387": "\nS(E) = N\\left[\\log 2 - \\left(\\dfrac{E}{NJ}\\right)^{2}\\right]\n", "a21d0b4ff711a11ab829c47859e199df": "(\\phi, \\theta)", "a21d5583f0e46655b038af8b8c2a2d96": "\\tilde{\\boldsymbol{y}}_{k} = \\boldsymbol{z}_{k} - h(\\hat{\\boldsymbol{x}}_{k|k-1})", "a21da5b3f8fd9488b7888f465a034d46": " \\mathbf{H}_{3} = \\begin{pmatrix} 0 & -1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} .", "a21dc76811a531c98fdd7ae75f031f0e": "K_\\mathrm{sat}^{(2)}", "a21e2506b59562fbc53d922928f7c979": " \\Delta L ", "a21e3ed2da2cf44855c25c8eed6fddc3": " \\boldsymbol{\\varepsilon} ", "a21e63c8ee9ac3537a700c08e8e60b04": "g'(x) = \\lim_{h \\to 0} \\frac{k \\cdot f(x+h) - k \\cdot f(x)}{h}", "a21f233f5a0211f509aee2937512479e": "M_j\\,", "a21f58d7a09d1990bda61ec8b9340eaa": "r_j = a_j + b_{j1}F_1 + b_{j2}F_2 + \\cdots + b_{jn}F_n + \\epsilon_j", "a21f5f5d4b742b96961f5b29daf45f65": "a_1,\\ldots,a_n.", "a21f74b02fa33f9354d7e3e5e9795bf2": " \\text{gross margin} = \\frac{1}{1 + 1} = 0.5 = 50%", "a21f8b0e081e19cb116afc9339b7de68": "\nF(r) = Ar^{-3} + Br^{-2} + Cr + D\n", "a21fe3bdd6d464195b36962544680c7e": "P_{n+1}/I", "a22049855645eb974ac75654705c6f81": "V(\\mathbf{i},\\mathbf{j})", "a2206592050bc608de8af5ad0cffb074": "H^k(\\Omega)", "a220729ccb43200a2ba872a581a90280": "{\\bar{J}}_3", "a220ba910f974bdf74f71e7e2124aba6": "C_n\\left(C_m(x)\\right)=C_m(C_n(x))", "a220ca5e2a07357a5b0679a6e83e0430": "w=k(1-z)^{-a}", "a220cb3ac3a5568ae3c3d135f4d401c5": " C_r \\ = 0 ", "a220d4b929eb250096af8a16767a732b": "q = r_1 r_2 \\cdots r_m", "a220ddf46fcc54089ee83e73d0862801": "b_0=\\cos\\varphi\\!", "a220f2b7be7f82996efb4117524529f4": "\\bigcup G", "a2211d9ab97850c663b1d92879ccf75b": "\n \\mathbf{A}\n \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} =\n \\begin{pmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{pmatrix}\n \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} =\n \\begin{pmatrix} x' \\\\ y' \\\\ z' \\end{pmatrix} ", "a22126561517664d95d7d0a346772f29": " \\sigma [{\\mathbf u}] := \\langle \\sigma, {\\mathbf u}\\rangle\n", "a2216778757162625f16b6ab8eb6ecc6": "G= \\sum_i \\mu_i N_i\\,", "a221df3ba12b37a7d9d80ec9baa5c0e3": " \\alpha = (\\alpha_1, \\alpha_2, \\ldots, \\alpha_n) \\,\\!", "a2220771c47711bb304bd8be7b53e7bf": "\\! w\\ge-1 ", "a22216ce8c3cb665250799027d24751a": "y \\in \\mathbb{Z}", "a22225ec1e34f4cd2cb21f674a62a5fa": "\\displaystyle{\\mathfrak{h}=\\mathfrak{k}\\oplus \\mathfrak{p}_1,\\,\\,\\,\\ \\mathfrak{p}_1=\\mathfrak{h}\\cap \\mathfrak{p}.}", "a222d02f6290e8519ad1b4e8698d34aa": "f(\\alpha)<\\alpha", "a22303d66800966be31f98a035aa551d": "g\\left(x\\right) := e^{x} f\\left(x\\right)", "a2230944cacfcbf0a267a912aa0bc3fc": "f(\\vec r) =- \\frac{1}{i_n} \\oint_{\\partial V} G(\\vec r, \\vec r')\\; d\\vec S \\; f(\\vec r') = -\\frac{1}{i_n} \\oint_{\\partial V} \\frac{\\vec r - \\vec r'}{S_n |\\vec r - \\vec r'|^n} \\; d\\vec S \\; f(\\vec r')", "a223588873b7588a88073c7712211fd1": "\\Delta E = 0", "a223801d83fa060c77542a3cff2a7e9f": "\\alpha \\le \\beta \\quad \\Leftrightarrow \\quad \\alpha_i \\le \\beta_i \\quad \\forall\\,i\\in\\{1,\\ldots,n\\}", "a223aadc244b3038d95cdb79c5549a45": "\\begin{matrix} \\frac{3}{8} \\end{matrix}", "a2243141698cfb85b4540870db34c2a2": "\\{s\\in I:s\\le t\\}", "a2247dd6fa0594fffd62c7b498cf7f9d": "\n \\frac{\\partial J(\\mathbf{X},t)}{\\partial t} = \\frac{\\partial }{\\partial t}(\\det\\boldsymbol{F}) = (\\det\\boldsymbol{F})(\\boldsymbol{\\nabla} \\cdot \\mathbf{v}) \n = J(\\mathbf{X},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\boldsymbol{\\varphi}(\\mathbf{X},t),t) \n = J(\\mathbf{X},t)~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}(\\mathbf{x},t) ~.\n", "a224940de05fd999c65a47b09936a995": "\\,\n\\mathbf{S} = { \\mathbf{V} + {\\mathbf{U}}_{||} + \\sqrt{1- V^2 }\\, {\\mathbf{U}}_{\\perp} \\over 1 + \\mathbf{V} \\cdot \\mathbf{U} }\n", "a224a52a2ccb0cc477650989bd68d853": "W_{ij}=W_{i}\\cap W_{j}\\;", "a224f1459b269d15f2c7256b2d87c499": " H_p(t_{\\rm CMB}) = c\\eta_{\\rm CMB} = 284\\ {\\rm Mpc} = 8.9 \\times 10^{-3} H_p(t_0) ", "a22507e951dda2d35c84d33ab2427175": "H_r=-p^4/8m^3c^2", "a22537115e5d54b3b84f45447751e102": "\\psi_2 = A \\sin(kx) + B \\cos(kx)\\quad.", "a225487f82944eb21d7a2b5c47039cb5": "A \\to B", "a225630f382068d63f35b2636580a36c": " \\tau = \\mu \\frac{d u}{d x},", "a2257bb3b673115d4a62250179483f84": "S_B = - N k_B \\sum_i p_i \\ln p_i \\,", "a225b796738c0636b99a473afafcfbfe": "\\scriptstyle \\ z_i", "a225dda63b0f3db94dd157da58afeec7": "p(\\boldsymbol{x},t)", "a225ddfcb6a6540f7de56fc39cefc8a9": "f(x) \\in K[x]", "a225e4738b16f3ca575c13f7d8e0a171": " x={v-v_f\\over v_g-v_f}", "a225f0d9560cc462b527ed1e1f67f458": "V= \\tfrac{\\ln(\\hbox{Victims})}{\\ln(\\hbox{Monetary Losses})}", "a225fa51260c81fc1c295298fc11722a": "\\sum_{j=1}^K\\lambda_j\\beta_j", "a22617b2f2fce06bd0f9b836e5ceb1d5": "[\\lambda : \\mu]\\ ", "a2267cd4d5b7c1b4de466951fc1225a7": "\\mathbb{N} \\xrightarrow{a} G", "a226b07d53db1303e440bbe486798694": "-A = \\overline{A} + 1", "a226b5f59c9477412413940e56b0938a": "\\zeta(6)", "a226fc8301287c4aed9def992f952454": "{2^{2^{2^{2}}}}-3", "a2271b1fb29c12c594f0da522c000695": "\\{\\sigma,\\, \\tau\\}\\,\\!", "a2271b82195d1692331c2be94a18fdc3": "1 < q < 3", "a227263d17b38bb6935ae64262bab0e9": "F_{\\alpha\\beta} = \\begin{pmatrix}\n0 & E_x/c & E_y/c & E_z/c \\\\\n-E_x/c & 0 & -B_z & B_y \\\\\n-E_y/c & B_z & 0 & -B_x \\\\\n-E_z/c & -B_y & B_x & 0\n\\end{pmatrix}\n", "a2273e9b7980bdad46305b5e14bca6e4": "\\nabla\\cdot(\\rho(\\mathbf r,t_0)\\nabla u_k(\\mathbf r)) = 0,", "a2274f98c56e05f974ff5cfa09d6e1e4": "\\omega_{n} = \\frac{2 \\pi^{n / 2}}{n \\Gamma (n / 2)},", "a22759a82b1dda8ad44d4913571b0540": "y_N-y(t_N)", "a227bac1b0314ae4900b84ded35fe56a": " P( X \\ge \\epsilon ) \\ge \\frac{ C_0 }{ \\psi } - \\frac{ C_1 }{ \\sqrt{ \\psi } } \\epsilon + \\frac{ C_2 }{ \\psi \\sqrt{ \\psi } } \\epsilon ", "a228139a36aa3d8b84b2fb7073faf064": "\\delta(P,Q) = \\frac 1 2 \\int_\\Omega \\left| f_P - f_Q \\right|d\\mu\\;.", "a2285a8299a0ee0118dd3748d152a74a": "\\nabla _{\\vec A} \\vec B(X) = \\lim_{\\epsilon \\rightarrow 0} \\frac{1}{\\epsilon}\\left[\\Pi_{(\\epsilon,0,\\gamma)} \\vec B(\\gamma[\\epsilon]) - \\vec B(X)\\right] ", "a2292d2fbcd2d8c9ce46e1fd3506b63c": "X \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)", "a2294eef349991faded94d8d2f5a6c1d": "A \\land B", "a229b09f60632628dcc62d4a4928145c": "\\bar{q}, \\bar{abc}, \\overline{q}, \\overline{abc}, \\!", "a229e204158c805bb00be3bb88049d6c": "\\frac{d}{d t}N_2", "a229e86df744d698568356c62efa7c3d": "\n\\frac{4+17\\sqrt2-6\\sqrt3-7\\pi}{105} + \\frac{\\ln(1+\\sqrt2)}{5} + \\frac{2\\ln(2+\\sqrt3)}{5}.\n", "a22ae0f08d39f4d20a5d71b9c9ed3e96": "81.8\\pm 0.5", "a22bff10000d78d38085fcb93f51c515": " \\frac{dN}{dt} = rN\\left(1-\\frac{N}{K}\\right) - H ", "a22c6dde948b7aa91c7038a488099842": "f = \\tilde f i", "a22c77a7eec111f75b7898948fad93cc": "\\sqrt{ \\tfrac{2}{3}} \\sigma_y", "a22c84b3222975b6bd279b1354429a00": "v = k_{cat} [E_{T}] \\frac{[S]}{K_{M}+[S]}", "a22cd4cd57376fbe6d551c1d8ee23049": "A^+ = (A^*A)^{-1}A^* \\qquad \\Leftrightarrow \\qquad (A^*A)A^+ = A^* \\qquad \\Leftrightarrow \\qquad R^*RA^+ = A^* ", "a22d411cc4acc8082af4579460fea7bb": "\\frac{1}{i} = \\frac{1}{i} \\cdot \\frac{i}{i} = \\frac{i}{i^2} = \\frac{i}{-1} = -i.", "a22d5aadcbe5d3a58367914f7c908072": "\\max_{x_1,\\ldots, x_n} \\Delta_n(\\mathcal{C}, x_1, \\ldots, x_n) \\leq \\sum_{j = 0}^{V(\\mathcal{C}) - 1} {n \\choose j} \\leq \\left( \\frac{n e}{V(\\mathcal{C}) - 1}\\right)^{V(\\mathcal{C}) - 1}", "a22d6ee714afbc3e09acca6c8630e352": "\\forall a, b, c\\in X\\,(b\\,R\\, a \\land c \\,R\\, a \\to b \\,R\\, c).", "a22d99c28a54ba01144974874b0a4714": "\\mathbf{A}=\\left[\\mathbf{x_0},\\;\\mathbf{x_1},\\;\\mathbf{x_2}\\right]", "a22e0d8626e73660363ceebbeccd509d": "E(2)", "a22e1d50006931c203c95a8667d6d87f": "P_1\\cdot x_1 + P_2\\cdot x_2\\leq P", "a22e8ef66d80c838c5318f2b81ec386c": "a^2+d^2=e^2\\, .", "a22ed6286939ef2f8b04c8f25444ee75": " f^+= [f>0]f\\,", "a22f9cdc2c09cc7ef960afb9d8091701": "\\tfrac{E}{2(1+\\nu)}", "a22fb8bb335a3c13a1840b59c302fd15": "\\mathbf{H}_\\mathrm{eff}", "a230628a2115b30e42534f6ac70df974": "\\rho(\\pi,\\delta \\mid x)=\\operatorname{E}_{\\pi(\\theta \\mid x)} [ L(\\theta,\\delta(x)) ]. \\,\\!", "a2306af1dc4a8152db6f735ad33245be": "\\big| \\mu (x, t) \\big| + \\big| \\sigma (x, t) \\big| \\leq C \\big( 1 + | x | \\big);", "a2307f1270aadd2a61046bb987f96880": " \\sum_{j=1}^n{a_{ij} y_j} + f_i s_i \\le c_i", "a230854cce6ecee3e55c9217be5d23d0": "m\\vec{J}_p=\\rho _s \\vec{v}_s.", "a23092147e7ac0ab1d69f2de9ebeca1a": " L_k ", "a23095d730e93cbc80e420c542f5d37f": "w_2(M)\\in H^2(M,{\\mathbb Z_2})", "a230e1b872d6720d1cc617862626ffde": "f_i(x)", "a230f5cab9db2ecb30744dc04fad0908": "T_\\varepsilon f(w)=-\\frac{1}{\\pi}\\iint_{|z-w|\\ge \\varepsilon} \\frac{f(z)}{(w-z)^2} dxdy.", "a23173ea960752659e5270bea64367f1": " \\log_b(b^x) = x\\text{ because }\\log_b(\\operatorname{antilog}_b(x)) = x \\, ", "a231e7b170256288da1848bfa689d712": "k \\in \\{1, \\dots, t\\}", "a231f8e1eaa2903809623f8711637125": " X\\circledast Y\\cong Y\\circledast X,", "a232711b74ac5254d1e88ee3253a3e44": "x_1=x_2=x_3=-\\frac{b}{3a}.", "a232c10d83f37f32979df9dda0e87534": "M _{CD} ^f = - \\frac{PL}{8} = - \\frac{10 \\times 10}{8} = -12.5 \\mathrm{\\,kN \\,m}", "a23300d7c255451bb0b854b6ea43bd66": " V_S = 4\\pi - {8\\over 3}\\pi = {4\\over 3}\\pi.\\,", "a2331fb18f2b445da317f412864aeab6": "a_{\\text{print}}", "a23420cfc09f380405342e7b7881dcf1": "r_1 = -1, r_2 = 2, \\omega_1 = 2, \\omega_2 = 1", "a234292bbabdaeb08868b5647185b6f4": "y=r(1-\\cos(t))\\,", "a234465862d32cc16dede570461296ba": "100^{100^{100^{12}}}=10^{10^{2*10^{24}+0.3}}", "a23481d691e247ee9b5623173c204eee": "x_{j+1} = x_j^2 \\prod_{i=1}^k p_i^{m_{jk+i}}\\mod n", "a2351ac94ace4b0084541c496f6a116e": "\\mathbf{J}=-D \\nabla n \\ , \\;\\; J_i=-D \\frac{\\partial n}{\\partial x_i} \\ .", "a2351f3353d3a9643fef1584829f6a0b": "x_3=-2", "a2354755672e01465db6dafa4ab9c61d": "BV = V_i", "a235b4d68f5b0c7826e5d368da4e7a84": "\\sum_{n=1}^\\infty a_n^{-}", "a235d507ca862386df757cb4c59eec25": "M^2_{Pl}R(Q)", "a23696678670de8b6933186469b6baa6": "V(\\mathbf{r})=\\frac{-g^2}{(2\\pi)^3} \\int e^{i\\mathbf{k \\cdot r}}\n\\frac {4\\pi}{k^2+m^2} \\;d^3k", "a2370c456c0683d28087f5fafb7c7f6b": " \\nabla^2 f\n= {1 \\over \\rho} {\\partial \\over \\partial \\rho}\n \\left( \\rho {\\partial f \\over \\partial \\rho} \\right)\n+ {1 \\over \\rho^2} {\\partial^2 f \\over \\partial \\varphi^2}\n+ {\\partial^2 f \\over \\partial z^2 }.\n", "a23718a59154bfcfd99c49fe766042fb": " F : M \\times \\{0\\} \\to \\mathbb{R}", "a23826719711ffa34aa99c1eed9012a3": "P(x)y''+Q(x)y'+R(x)y=0", "a23868a37f61af02e01e18e4518d3190": "f(x) = \\frac{g(x)}{h(x)}", "a238b2fe5ce65e5444b65d1b0bc4cdd2": "\\mathrm{C^{\\alpha}}_{i}", "a238cf3d60ec9108d81437a59fae9612": "\\textstyle \\deg(p(x))", "a238fe7cea69de6c130dc99e5a925228": "Gcrd(I,J)", "a23949ed3f1e102b3937aa646aa706a5": "\\mathbf{\\tilde{U}}", "a239c82bdae8cd99719888f1329a4737": "i = 0, 1, 2, ...", "a239e93c740fc69b31fc69136df65a68": " \\boldsymbol{\\beta}_* \\perp \\{\\mathbf{v}_{k+1},...,\\mathbf{v}_p\\} ", "a23a33bed8f069d1471905e4aa83fbb9": " |\\langle\\mathbf{u},\\mathbf{v}\\rangle| = \\cos(\\theta)\\ \\|\\mathbf{u}\\|\\ \\|\\mathbf{v}\\|", "a23a463203e03af2282284138bc39776": "\\gamma_i= - \\frac{V}{\\omega_i} \\frac{\\partial \\omega_i}{\\partial V}. ", "a23a52c0e33a45307b5441c2808988fd": "e = \\textrm{H}(\\gamma \\parallel \\textrm{ID}_A) \\,", "a23aaa7b21ea9ff6bde8092d2d327931": "\\ 3.5 = \\frac{M_{pitch_{max}} } { M_{heel_{max}} }", "a23af8aa2e9f87f8188c5e6725b14de7": " \\lim_j \\int f_j \\, \\mathrm{d} \\mu \\geq \\int g_k \\, \\mathrm{d} \\mu.", "a23b5360ae1b71068635f66bf967d8c4": " C(t)=\\sum_{n \\ge 0} C_n t^n ={{1-\\sqrt{1-4t}}\\over {2t}} ", "a23bf6e9eb0857d0031673ef6548d9c1": "c_1, c_2, c_3, \\ldots, c_{32}", "a23c10c0bffc46c237cc892f51491ec7": "M=\\frac{1}{2}(P+Q)", "a23c6e737fc8360d78610573dcde250a": "x = \\sum_{i = 1}^{m} c_{i} (x \\cdot u_{i}) u_{i}.", "a23c7ab7c45271c6f3e080d5656150cb": "{\\Pi}_{\\infty}", "a23c8e1cc27a9974a33f4248dc102bb1": "\\left[\\theta_a\\right]_{qq^\\prime}=\\delta(q,a,q^\\prime)", "a23d0d77a436042917813f4a15ec3042": "\\sharp \\hat 4", "a23d495f2e878f87eb46e7938c099895": "on(box,t)=table", "a23d5ecdefae8b89845d9ef72a5c5e68": "\\neg e \\rightarrow \\neg f", "a23d9f61211137f2dc55dad8b0e6dca7": "\\mathbf{E}_{l,m}^{(E)}", "a23dcde5eeef7449d44f5ec6d6ba765c": "\n \\begin{align}\n \\cfrac{d\\epsilon^p_1}{d\\lambda} &= 2(G+H)\\sigma_1 - 2H\\sigma_2\\\\\n \\cfrac{d\\epsilon^p_2}{d\\lambda} &= 2(F+H)\\sigma_2 - 2H\\sigma_1\\\\\n \\cfrac{d\\epsilon^p_3}{d\\lambda} &= - 2G\\sigma_1 - 2F\\sigma_2 ~.\n \\end{align}\n ", "a23e1da74c76458cd51ffa6e61ccd2e7": "\\begin{align}\n\\bar{x} &= \\frac{1}{n}\\sum_{i=1}^n x_i \\\\\n\\mu_0' &= \\frac{n_0\\mu_0 + n\\bar{x}}{n_0 + n} \\\\\nn_0' &= n_0 + n \\\\\n\\nu_0' &= \\nu_0 + n \\\\\n\\nu_0'{\\sigma_0^2}' &= \\nu_0 \\sigma_0^2 + \\sum_{i=1}^n (x_i-\\bar{x})^2 + \\frac{n_0 n}{n_0 + n}(\\mu_0 - \\bar{x})^2\n\\end{align}", "a23e361baabb7c64df31385971c9ed7a": "\n\\begin{bmatrix} Y \\\\ I \\\\ Q \\end{bmatrix}\n=\n\\begin{bmatrix}\n 0.299 & 0.587 & 0.114 \\\\\n 0.595716 & -0.274453 & -0.321263 \\\\\n 0.211456 & -0.522591 & 0.311135\n\\end{bmatrix}\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n", "a23e7be68378e9887c32a661d2189ce9": "H=\\frac{1}{n}\\sum_{i=1}^{n} \\kappa_{i}.", "a23eb60cfa4c28f64a9cc9cc0eac62b6": " \\Phi(h) = \\phi_h ", "a23ec372f04d0184877f01fcc6640b2e": "8f(x+h) - 8f(x-h) - f(x+2h) + f(x-2h) = 12h f'(x) + O(h^4) \\,", "a23ec5153e137b93ec44b32892791404": "\\mathrm{Ste} = \\frac{c_p \\Delta T}{L}, ", "a23f2036baa199dc9c754abe02413493": "t_8 = 0", "a23f8dfffe7f1b0f055fc364cd0d26e9": "S \\rightarrow B: \\{T_S, K_{ab}, A\\}_{K_{bs}}", "a23faf10cbab821f1d46a1cd98073534": "\\begin{align}\n\\sigma_2+\\sigma_3&=2\\sigma_n \\\\\n\\sigma_2+\\sigma_3&=2\\left(\\sigma_2n_2^2 + \\sigma_3n_3^2\\right) \\\\\n\\sigma_2+\\sigma_3&=2\\left(\\sigma_2n_2^2 + \\sigma_3\\left(1-n_2^2\\right)\\right)&=0\n\\end{align}\\,\\!", "a23fd67479dee4b21bb36d626b8b6337": "p(O_{bg}|I, I_t) ", "a23ffc4fad0120aabd7d0a971fbad87b": "\\rho_1.", "a24021fa0059370d8fdeaf817880c76b": "\\displaystyle{\\mu_n=(1-\\varphi_n)\\cdot\\mu.}", "a240c7f68c8964e704800ff7003b32df": "1^2 + 2^2 + 3^2 + \\cdots + n^2 = {n(n + 1)(2n + 1)\\over 3!}", "a24121d26016843a5ea8803b09365444": "c_{\\mathrm{air}} = 331.3\\,\\mathrm{m \\cdot s^{-1}} \\sqrt{1+\\frac{\\vartheta}{273.15}}", "a24156e6311da7413250929d5de20be7": "\\begin{pmatrix} J_\\text{x} \\\\ J_\\text{y} \\\\ J_\\text{z} \\end{pmatrix} = \\begin{pmatrix} \n\\sigma_\\text{xx} & \\sigma_\\text{xy} & \\sigma_\\text{xz} \\\\ \n\\sigma_\\text{yx} & \\sigma_\\text{yy} & \\sigma_\\text{yz} \\\\\n\\sigma_\\text{zx} & \\sigma_\\text{zy} & \\sigma_\\text{zz}\n\\end{pmatrix} \\begin{pmatrix} E_\\text{x} \\\\ E_\\text{y} \\\\ E_\\text{z} \\end{pmatrix} ", "a241858f1dca930d74b30bdedab4a93c": " r = a\\ \\operatorname{sech} (n\\theta) ", "a241f36df0c96fa1e31da9ef224f9505": "x = \\zeta^{-\\frac{1}{N-1}}\\,", "a24251a8311e1e142582c1dd34750e74": "\\chi \\,", "a2428f9a21d81e944fa6595a22cdf308": "\\iint_A dx\\, dy= \\iint_B r \\,dr\\, d\\phi.", "a242ae966a8bb17c55b3ad8478164ebd": "\\Mu", "a242e7634cd39d35a6f2cfe9eaadfc49": "\\tan \\psi = \\frac {u'} {J} \\,;", "a242eb90f542bd41dfac799fa49f125e": "|S(a,c) + S(c,b) - S(a,b)|/15 < \\epsilon \\,", "a24357055b0bd120c61802b35a873aca": "\\pi(\\{1,\\dots,k\\}) = \\{1,\\dots,k\\}", "a2437a05e3eb8e55ef0a7e7d3fae582e": "\\scriptstyle\\boldsymbol \\nabla T", "a24394a24bd17becb5624e8199f84333": "\\nabla g_i (i\\in\\mathcal{I}')", "a2439c0678a30264a67d884e20569458": "\nX \\sim T^2_{p,m}\n", "a244795decc12a55a80f5d72e8e5effd": "\\varepsilon(t)", "a244830cdb3bf2510b5aefe26ab7fdd8": "k_{ET} \\varpropto J \\mathrm{exp}\\left [ \\frac{-2r}{L} \\right ] ", "a2451e9861f61edcc5f8f374a1be02a7": "\\dot{Y}(t)=\\frac{i}{\\hbar}[H_\\mathrm{sys},Y(t)]-\\frac{i}{2\\hbar}\\left[X,\\left\\{Y(t),\\xi(t)-\\int_{t_0}^tf(t-t_0)\\dot{X}(t^\\prime)\\mathrm{d}t^\\prime-f(t-t_0)X(t_0)\\right\\}\\right]\\,,", "a2455c7bb60b092e523d3f63531fed72": "a_P \\phi_P = a_W \\phi_W + a_E\\phi_E", "a245615d51b42f80144f4cb4b7373108": "\n\\begin{array}{lcl}\nK &=& \\text{number of mixture components} \\\\\nN &=& \\text{number of observations} \\\\\n\\theta_{i=1 \\dots K} &=& \\text{parameter of distribution of observation associated with component } i \\\\\n\\phi_{i=1 \\dots K} &=& \\text{mixture weight, i.e., prior probability of a particular component } i \\\\\n\\boldsymbol\\phi &=& K\\text{-dimensional vector composed of all the individual } \\phi_{1 \\dots K} \\text{; must sum to 1} \\\\\nz_{i=1 \\dots N} &=& \\text{component of observation } i \\\\\nx_{i=1 \\dots N} &=& \\text{observation } i \\\\\nF(x|\\theta) &=& \\text{probability distribution of an observation, parametrized on } \\theta \\\\\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& F(\\theta_{z_i})\n\\end{array}\n", "a245ca1a4fa36b18a95f0803e128af8c": "b^{(k)}x_k \\le n", "a245ebf36864f750c5cafadb121f2a61": "\\frac{\\bigvee_{i=0}^n\\bigl(\\bigwedge_{j=0}^k\\neg_{i,j}^0p_j\\land\\bigwedge_{j=0}^k\\neg_{i,j}^1\\Box p_j\\bigr)}{p_0},", "a2462a9f185ac62639e33e264e82081c": "\\forall n>p", "a246b6755bf4d3b2e38375cbca6b355d": "\\psi.", "a24776039e7beb5a3dd8a90c68cc1fc4": "ax+b \\equiv ay + b + i\\cdot m \\pmod{p}", "a247acdf87fe2432b3486ebdeaf7ae1f": "\\mathrm{gmul}_J \\lambda\\,", "a2480a9d441c941239801b37063781f3": "\\mathbf{\\nabla}\\cdot(\\phi\\epsilon \\mathbf{\\nabla}\\phi)= \\epsilon (\\mathbf{\\nabla}\\phi)^2", "a248814585aa123a6dcff57a84e92ec0": "B_{ab} = P_{cd}{{{W_a}^c}_b}^d+\\nabla^c\\nabla_aP_{bc}-\\nabla^c\\nabla_cP_{ab}", "a248a34d6a2834f1474d300f6c2d648a": "R_{up}", "a248b88ff1ecd9938d3b101cd9ed866a": "(F(Y),\\eta_Y)", "a249531d65573ea90bb050605de88aff": "\\displaystyle\\rho", "a2496c3b4f398504db22d447534ca6ce": "D_{\\mathrm F} = \\frac {H s}{H - ( s - f )}\\,.", "a24977db60c0a698f132e81e041632c2": "T(m) = \\Theta(m^{2})", "a249a479eda05c22053626786cdd3ba4": " h_{i}^{t}=F\\left( r_{i}^{t}\\right) +\\gamma_{i}^{t}h_{i}^{t-1}", "a24a7015a020199c697f1bc0a7f3f5f4": "3.7\\times 10^{-6}", "a24a8de201844181b0bb023ed78f68fe": "D_n\\circ D_n\\subset D_{n-1}", "a24a9289331245f60221fe9f098142c3": "12 \\times x", "a24ad4964beb61ec963c5026b9ef5300": "V_{ij} = {\\gamma\\over 2}{( s_{ij} - {s_{ij}}^o)}^2", "a24b67465bf476c1a6a3b1ea5ae06ed9": "0=SdT-VdP+\\sum_i N_i d\\mu_i\\,", "a24bb78263325b7eca8a2c9c24b60cac": "\\gamma \\! ", "a24bcab3055b92a2863732829de339a1": "T(z_1,z_2) = {Y(z_1,z_2) \\over X(z_1,z_2)} = {\\sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q} \\over \\sum_{p,q=0,0}^{m,n}a_{p,q}z_1^{-p}z_2^{-q}}", "a24c4162543c472ece06a1f9758ab803": "\n\\left( \\begin{array}{c}\\rho_R\\\\P_R\\\\v_R\\end{array}\\right)\n=\n\\left( \\begin{array}{c}0.125\\\\0.1\\\\0.0\\end{array}\\right)\n", "a24c9c13a8659679c42532470df1c80d": "f(\\star q(\\xi,\\tau ))", "a24ca86dea2d4456fe3f6d9a6e407335": "\\scriptstyle \\leq1.2\\times10^{-11}", "a24cb6d75f6590f8a7952e9ab760af47": "\t\\begin{array}{rrr|r} \n 1 & -9 & -27 & -123 \n\\end{array}", "a24cd8d4f60e77f9b9b34e9e78a88908": "\\tan 36^\\circ=\\sqrt{5-2\\sqrt5}\\,", "a24ce4046b3309e531d43b2de324110b": "x_{n+1} = \\frac{1}{2} \\left(x_n + \\frac{S}{x_n}\\right),", "a24d34835895df66a00ab66285d78406": "P[u](t,x) = [P(t,\\cdot)*u](x)", "a24d86fbf9edaffc6b4604454e1200d9": "4. \\quad \\overline{f\\star g} = \\overline{g}\\star \\overline{f}", "a24d8fb5b5ed8eae02877bb265866f4a": " \\frac {d}{dt}\\mathbf{F}( \\mathbf{C}(t) + \\mathbf{I}, t) = \\mathbf{F}_t(\\mathbf{C}(t) + \\mathbf{I}, t) + \\mathbf{v \\cdot \\nabla F}(\\mathbf{C}(t) + \\mathbf{I}, t) = \\mathbf{F}_t(\\mathbf{r}, t) + \\mathbf{v} \\cdot \\nabla \\mathbf{F}(\\mathbf{r}, t) ", "a24dd8d9ec362c1bfd46dd0cb28a4920": "A=\\sigma^2_x", "a24df545904278b9c364ade7ffce7dd2": "h_{ji}=\\begin{cases}\n\\boldsymbol{v}_j^\\mathrm{T}\\boldsymbol{Av}_i & \\text{if }j\\leq i\\text{,}\\\\\n\\lVert\\boldsymbol{w}_{i+1}\\rVert_2 & \\text{if }j=i+1\\text{,}\\\\\n0 & \\text{if }j>i+1\\text{.}\n\\end{cases}", "a24e0c63506010f91298abc4f839ec94": " P[\\text{Suicide}|\\text{Protestant}]", "a24e49b2c7601ee7faad8c5633b890d9": "\n c_p\\,c_g\\, \\left( \\Delta a\\, +\\, 2i\\, \\nabla a \\cdot \\nabla\\theta\\, -\\, a\\, \\nabla\\theta \\cdot \\nabla\\theta\\, +\\, i\\, a\\, \\Delta\\theta \\right)\\,\n +\\, \\nabla \\left( c_p\\, c_g \\right) \\cdot \\left( \\nabla a\\, +\\, i\\, a\\, \\nabla\\theta \\right)\\,\n +\\, k^2\\, c_p\\, c_g\\, a\\,\n =\\, 0. \n", "a24e4b3b8b9ac858f93f71d50098af91": "\\scriptstyle{E(C_2)=0.3}", "a24e76e8f0192d72cb72bdb3218deb1c": "\\sum_{n=0}^\\infty A_n {x^n \\over n!} = \\sec x + \\tan x = \\tan\\left({x \\over 2} + {\\pi \\over 4}\\right).", "a24e96aa9fc81c1caa77906554a4eaf6": "\n \\eta(x,t) \n =\\; H\\, \\Bigl( \\tfrac12 - \\tfrac{3}{512}\\, m^2 + \\cdots \\Bigr)\\, \\cos\\, \\theta\\;\n +\\; H\\, \\Bigl( \\tfrac{1}{16}\\, m + \\tfrac{1}{32}\\, m^2 + \\cdots \\Bigr)\\, \\cos\\, 2\\theta\\;\n +\\; H\\, \\Bigl( \\tfrac{3}{512}\\, m^2 + \\cdots \\Bigr)\\, \\cos\\, 3\\theta\\;\n +\\; \\cdots.\n", "a24e9fdc46fe461b68cb9896734d2ebc": "3\\ ", "a24ec636a64c66d595d99fd45e158a3e": "\n |\\boldsymbol{u}^{(1)}|<<|\\boldsymbol{u}^{(0)}||\n", "a24f196d0642d6d8718f691bdaafa6d8": "f(x_1, x_2, \\dots, x_n),\\,\\!", "a24f45f4dd4bf739694ad00c9e6c23d4": "4r\\le e+f+g+h \\le 4r\\cdot \\frac{R^2+x^2}{R^2-x^2}", "a24f791fccf9f843dc0a98ac0a10064c": "M_{rel}(\\lambda)=\\frac{M(\\lambda)}{M({\\lambda_o})}", "a24fa7816d23851edbe2baf362a61a86": "\\operatorname{tr}(P^{-1}AP) = \\operatorname{tr}(P^{-1}(AP)) = \\operatorname{tr}((AP) P^{-1}) = \\operatorname{tr}(A (PP^{-1}))= \\operatorname{tr}(A)", "a24fd5325e7bca1e741e6d82871a6ee5": "\\color{red}\\boxplus", "a24fe8c915fc63e8f481bebbe3f686c4": "S_N(f)", "a24fee86dcb6d7b3a40cbd3ddd6ea389": "\\tfrac{p^2}{12\\sqrt{3}},", "a25000a710068b52f2173d2c848a91a9": "\\mathcal{S} (\\mathbb{C} P^n) \\cong \\oplus_{i=1}^{\\lfloor (n-1)/2 \\rfloor} \\mathbb{Z} \\oplus \\oplus_{i=1}^{\\lfloor n/2 \\rfloor} \\mathbb{Z}_2", "a2500bc5f7a60fd63ed701cd82707416": "\\ C_L = C_o(f_L)^{k - 1}", "a2501476fb94f901bbd930cd294eab91": "g_{\\alpha \\beta} = e_\\alpha^I e_\\beta^J \\eta_{IJ}", "a25014956106db315ddb9e9244a06aea": "a: V \\setminus \\{ v^0 \\} \\rightarrow \\mathcal{A}", "a250221d0a712b1300de3e4d7039af8e": " r \\mapsto r \\otimes 1 ", "a250268912a25be37989a8172ed33670": "\\scriptstyle \\eta_x ", "a25061fdfc71a8d4a349f3a9090b096e": "\\delta(ab)=\\delta(a)b+a\\delta(b)", "a25088b43161e056cbec9974956e3973": "h[i] = 1 \\, \\qquad \\qquad \\text{for } i = 1, \\ldots, n ", "a250a2b7ee42caddc41ac7232db16b1b": "\\sin \\theta = \\frac {d} {J} \\,,", "a250daceca8217fd2956b90a08d8510c": "\\psi(0) = \\omega^2", "a25116a2b6f57215d321cd36ffbdf010": "10^{-12} W/\\sqrt{Hz}", "a2511dd251a56c28877ccc17bfa424ee": "U(x_1,x_2) = x_1^{a}x_2^{b}", "a2516f61b58545162e82247f1cd27f36": "GF(2^{8})", "a251ab4839c07054bc9c866edda633b1": "q=\\frac{-k}{\\mu} \\nabla P", "a251db26308e02e3bfe1debee2738571": "D f_s \\propto \\text{div}_{\\mathbf p} \\langle \\text{grad}_{\\mathbf q}\\Phi_{i,s+1}\\rangle_{f_{s+1}}.", "a251f2af49368a406eb269cade6ce52d": "G_1A", "a2525bdf0ce3b16207fa33eacbcec038": "k \\in \\mathbb{N}", "a2529636d9485ae4f1a1a42c28d80ec6": "\\hat{L}_a", "a252b12cd5c037a28a7dd6e1b4feea98": "j = \\sigma \\epsilon ", "a252b27300fffbb6d0f55f7cef6bf4bc": "Q=n^\\searrow.\\overline{n}^\\nwarrow(~)", "a252cc4ea37d5455fd4ecf8e6f0f7fa1": " \\frac{ X_n - \\mathrm{E} X_n }{ \\sqrt{\\operatorname{Var} X_n} } ", "a25371f1f49df220916192d0889aa853": "j \\geq n", "a253fe2efe800e53c82a7cf60cc02af0": "p_o", "a2544e50378baf8b191beb17cc3aed3c": " \\lim_{x \\to p}f(x) = L, \\ ", "a25496ebf095e4198da4088449c83ac6": "PH", "a254d32c247a29c42f81537c5e94a599": "2^{H(p)}=2^{-\\sum_x p(x)\\log_2 p(x)}", "a254ec7b03bdf2e0b8c475b36eb954d4": "\n \\liminf_{n\\to\\infty} \\sqrt{2n\\ln\\ln n} \\|\\hat{F}_n-F\\|_\\infty = \\frac{\\pi}{2}, \\quad \\text{a.s.}\n ", "a25525cf8a3f87a599484fe5484a6c26": "{\\eta}_{cathode}", "a2552b1f870af9225b7374b0bcfead64": "\\bar{\\mathbf{e}}_{j_1}\\otimes\\bar{\\mathbf{e}}_{j_2}\\cdots\\otimes\\bar{\\mathbf{e}}_{j_p}=(\\boldsymbol{\\mathsf{L}}^{-1})_{j_1 i_1}\\mathbf{e}_{i_1}\\otimes(\\boldsymbol{\\mathsf{L}}^{-1})_{j_2 i_2}\\mathbf{e}_{i_2}\\cdots\\otimes(\\boldsymbol{\\mathsf{L}}^{-1})_{j_p i_p}\\mathbf{e}_{i_p}", "a255512f9d61a6777bd5a304235bd26d": "x=1", "a2555a6fd3ecbd237f6409d1c8296027": "\\displaystyle{\\widehat{Tf}(z)= {\\overline{z}\\over z} \\widehat{f}(z).}", "a255affce57ca133bae379bb25ad6126": " \\operatorname{SO}(2, \\mathbf{R}) =\\left\\{\\begin{pmatrix} \\cos\\varphi & -\\sin \\varphi \\\\ \\sin \\varphi & \\cos \\varphi \\end{pmatrix}: \\varphi\\in\\mathbf{R}/2\\pi\\mathbf{Z}\\right\\}. ", "a255f1427bd9156c7f85127050797291": "f(b_1,b_2,\\dots,b_n)\\in B", "a255f77d4df2d75fd9d60a46ff33d047": "0.00304936208\\ldots", "a2564430bed5e6ff7bb46d7cac94961a": "\\mathbf{c} = [c_1,\\ldots,c_n]^\\top", "a256a0586b4dfc5cd91644200fd14072": "\\xi = \\frac{x - i y}{1 + z},", "a256df51e565e269ab58b151bcb58f5c": "f(x)=a_1 x+{a_2 \\over 2}x^2+{a_3 \\over 6}x^3+\\cdots+{a_n \\over n!}x^n+\\cdots\\,", "a256e8c613f6d5c5dc71265ddab1a9e6": "f(x) = P(x)", "a258042986088400d6aef2ea493456c5": " X^\\# ", "a258b5ba48cfd2c67a24f156514fac5e": "e^iF \\ne Fe^i", "a258e79c37d8aca3ac4406e4582afd87": "j=n-2", "a258fa5b190bed2dd3746745161870ed": "\\sum_{i=1}^{n} f(t_i) \\Delta_i ; ", "a25908afd852c3c8d4661b296c053160": "B\\text{d}x + C \\text{d}a \\,=0", "a2596afee8670c8ad332ca72d2d1b35d": " H \\,\\!", "a259795e96781ed72647cde9dbf1dae7": "k[t_1, \\cdots, t_n]", "a25979bce2a08c92b84c42f3702d527f": "m: 2^X \\rightarrow [0,1] \\,\\!", "a259834cfe7a89c03fb5f0a2b669af0b": "\\textstyle L_1 L_3 L_5 R_1 R_3 R_5 L_2 L_4 L_6 R_2 R_4 R_6", "a2598809cc90eced9e6263140fb4f470": "\\text{Recall} = \\text{Sensitivity} = \\frac{TP}{TP+FN} \\, ", "a25a7de6fbdb587659129f07573de319": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{sinc}^2 \\left( \\frac{\\xi}{a} \\right) \\,", "a25a8f1ec9c4901a51def1351d3b2d61": "l_{SO}", "a25a96d3d093064884609eefe1009031": "\\dot{Q} = \\frac{T_{hs} - T_{air,av}}{R_{hs}}", "a25ab064adcfaf5e3d0c768c529014d8": "\\scriptstyle \\geq2.7\\times10^{16}", "a25ab0b9395ec18d3e2d4d288df9d8ec": " \\phi = \\rm 3\\cdot\\lambda - \\rm 2\\cdot\\lambda_{\\rm N} - \\varpi ", "a25af39853cc67b300096fc1de11edde": "[s,t;x,y]_n=\\sum_{k=0}^n(-1)^k{s \\choose k}{t\\choose {n-k}}\nB_{n-k}(x)B_k(y).", "a25b111da776867ff388605e35a061c4": " \\bar v_i ", "a25b2c6db416896f1ae7ed785848e922": " S(y) = -ylny - (1 - y )ln(1-y)", "a25b33c99c49a24e04b3b7608a7137d8": "I_\\omega=\\{(x_n)\\in l^\\infty(R):lim_{n\\rightarrow\\omega}\\tau(x_n^*x_n)^{\\frac{1}{2}}=0\\}", "a25b5b2f945bd3fbd5eb54d0dc926375": "\\log_{_{g1}}h_{1} = \\log_{_{g2}}h_{2}", "a25bd0398e1ea1a5a517bb224c07b6bf": "\\langle x_{C} \\rangle_{C\\in D}", "a25c8d4883cc8d6fa0dea1792a0e8f44": "(1-x^2)\\,y'' + (\\beta-\\alpha-[\\alpha+\\beta+2]\\,x)\\,y' + {\\lambda}\\,y = 0\\qquad \\mathrm{with}\\qquad\\lambda = n(n+1+\\alpha+\\beta)\\,", "a25cdd1086bd0b7ce546d34dcb2f9b36": " N^{(i)}=\\binom{n_i}{i+1}+\\binom{n_{i-1}}{i}+\\ldots+\\binom{n_j}{j+1}. ", "a25cf2030a473f1a6ad43cc4673f4ccd": "m=l", "a25cfab4b9a9a1ec52f096b396826cc0": "a_j = f_{1j}c_1 + f_{2j}c_2 + \\cdots + f_{rj}c_r,", "a25d2453a6a67efc005911f1cfa9fb4c": " \\psi = \\psi^{+} + \\psi^{-} ", "a25d4051a86c58854c0fe7ee2bbf7e13": "\\|Df(x)\\|\\le K", "a25d473c0af83c3a8de535b8571e1c7c": "\\boldsymbol{S} \\Rightarrow_1 \\boldsymbol{aBSc} \\Rightarrow_2 aB\\boldsymbol{abc}c \\Rightarrow_3 a\\boldsymbol{aB}bcc \\Rightarrow_4 aa\\boldsymbol{bb}cc", "a25df123a0d8f30bde250a10cb8a104e": "f(x|\\theta)", "a25e91ae4f3cad28ee8b36a02c9df4d9": "\\scriptstyle\\nabla^2", "a25ebdf05b10eddf6af6d259a6b0283e": "r = r(t)\\quad\\text{and}\\quad \\theta = \\theta(t)\\qquad\\text{for }0 \\leq t \\leq 1.", "a25edb682811d89d9348374e32356e9f": "(B,+)", "a25f076221777aa548d6f0ab748f5f74": " x_i \\not \\le x_j ", "a25f20791ec32f0d0f56822dffdbed6d": "\\omega(\\phi)\\;", "a25f2e002f5c4c7a4f08447b9318d68d": "R<\\lambda", "a25f5ce67cbb209216126f762e0461b5": "x_j=\\frac{L_{(j)}\\cdot\\mathbf{b}}{\\det(A)}.", "a25f9ed67872971ab53ff505c42bbe34": "e_1 = 221, e_2=-935, e_3=715", "a25fa79a949e2b42ce5a9ea3f6aea374": "\\text{if}~n\\ \\text{is even}", "a25fe94d0634460cb273ba63b072575e": "\\frac{x}{m} = K c^{1/n}", "a2601aa9233af1c9068eaaad1a699144": "J = \\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\[2pt]\n0 & 2 & 0 & 0 \\\\[2pt]\n0 & 0 & 4 & 1 \\\\[2pt]\n0 & 0 & 0 & 4 \\end{bmatrix}.", "a2603fc0d756ac5872bb281c98a057d5": "w_i = \\frac{\\rho_i}{\\rho}", "a2606705149da2528ef8d00ebdee26f0": " v_i = \\sum_k D_{i k}p_k + \\sum_j C_{i j}-\\sum_j C_{j i}V_i", "a2607d32a193097911db2677c24692f0": "\\varepsilon = \\frac{\\Delta L}{L}", "a260a5e5053d50629014bee541c4e231": "\\sum_{i=1}^n (X_i - \\bar{X})^2\n = \\begin{Vmatrix} X_1-\\bar{X} \\\\ \\vdots \\\\ X_n-\\bar{X} \\end{Vmatrix}^2.", "a260d19d4bcdd9353b0c5a4492079fb0": "G\\subset\\mathbb{C}", "a260e1d4da98451b9837cc8470f8d772": "F_v(t) = \\frac{M_a}{r}(e^{rt}-1)", "a260f1315a85ff7985a6db362442a73c": "\n\\rho_0(\\mathbf{k},\\omega) = 2\\pi\\delta(\\xi_\\mathbf{k} - \\omega).\n", "a260fe606812b6b1d8b68f4dfbd71385": "~F=F(x)~", "a26154e82609b1e5b95da83f5e9e3865": "f(x_1,x_2,\\ldots,x_n) = g(x_2,\\ldots,x_n) \\oplus x_1 h(x_2,\\ldots,x_n)", "a2616d34ff5fcae6688afa44e7f44723": " (700,700) \\pm (100,100) \\, ", "a26196c0a624ebfc2070bccf6e696af6": " r = -e x - a\\,\\!", "a261a483d0a39368d9376e162d8099ad": "\\gamma_\\times", "a2622a54659a9a7fbfead4062764d897": "x^2 + ny^2", "a26241a631f8326ce94557a183602e7b": "n= m", "a26241e5d67574d73ca898eb787b19ce": "~\\sigma_{\\rm es} ~", "a262f44dc006abe40768a32a632af35f": " \\eta_{Qe} ", "a2630b62e5d5f75670ad600a8898b473": " \\begin{align} & \\frac{\\partial F^{\\alpha \\beta}}{\\partial x^\\alpha} = \\mu_0 J^\\beta\\\\\n& \\frac{\\partial G^{\\alpha \\beta}}{\\partial x^\\alpha} = 0 \n\\end{align}", "a26317c1d486a47542daa8b83cf430c5": " \\operatorname{de-let}[M\\ N] \\equiv \\operatorname{de-let}[M]\\ \\operatorname{de-let}[N] ", "a2636ecf0abf9ba6de6ebe595ca03d7a": " \\limsup_n \\frac{S_n}{\\sqrt{n}} > M ", "a263792f24431eedaec005a80696abb0": "N(0,1)", "a264978f6f084896db60223466255fdc": "\n\\begin{array}{c|cccc}\nc_1 & a_{11} & a_{12}& \\dots & a_{1s}\\\\\nc_2 & a_{21} & a_{22}& \\dots & a_{2s}\\\\\n\\vdots & \\vdots & \\vdots& \\ddots& \\vdots\\\\\nc_s & a_{s1} & a_{s2}& \\dots & a_{ss} \\\\\n\\hline\n & b_1 & b_2 & \\dots & b_s\\\\\n\\end{array}\n", "a265065666cda268a5728144fe126ed1": " C = P_1 \\text{diag} (I, P_2) \\cdots \\text{diag}(I, P_{r-1})Q_1R_1Q_2\\cdots R_{r-2}Q_{r-1}", "a2657b2dc7c96dbad5d13f536df5562e": "v_0\\dots v_kB(S'_1)\\dots B(S'_l)", "a265cdd7bc67c499b01f6dfd7bf9235d": "Q_\\text{rad}", "a265d75586154cbb24c558af1db3b8c0": "|\\widehat{\\tau f}(n)|^2 (1+n^2)^{k-1/2}\\le \\left(\\sum_m {(1+n^2)^{k-1/2}\\over (1+m^2+n^2)^{k}}\\right)\\cdot\\left(\\sum_m |\\widehat{f}(m,n)|^2 (1+m^2+n^2)^k\\right) \\le C_k \\sum_m|\\widehat{f}(m,n)|^2 (1+m^2+n^2)^k,", "a26654a7ed1030d9e7aef5df1cd5d08d": "e^{tX_i}", "a26659c97b5dc65a85ef7ad56d07adc4": "\\Pr(X < x \\and Y > y)", "a2665fa1906ad7fdb4a84a2c5902619b": "\\left\\{ \\hat{a}, \\hat{a}^\\dagger \\right\\} = 1", "a2668d50414f4d36a3ee8ecfc935a30f": "m=|\\mathcal S|", "a2670365fe48cf2809846c27e6c04603": "L_{1}' =(0.5, 0, 0.5)\\!", "a2670fccf1e59fda83ba2bc312c71c45": "\\sgn r, \\left\\vert s \\right\\vert \\!", "a26713ad42104ad6855cf9499d9f94b3": " f(E) = -\\frac {k} {m_w^*} \\tan(\\frac {k l_w} {2}) -\\frac {\\kappa } {m_b^*} = 0 \\quad \\quad (7)", "a26725407e4d8c9aa7d82a2d91baabce": "I=\\lambda I_s e^{\\eta V / U_T}", "a26778d7288da08bfeb8db92677fd6de": "\\ln 11", "a267bca28f70c87bcebf5f24123ffb08": " \\frac{\\partial N_x}{\\partial e} = \\frac{\\partial X}{\\partial e} - e\\frac{\\partial Q}{\\partial e} - Q ", "a267f5f692f7ad717461fa3ff9ae72a3": "\n \\frac{1}{r}\\cfrac{d }{d r}\\left[r \\cfrac{d }{d r}\\left\\{\\frac{1}{r}\\cfrac{d }{d r}\\left(r \\cfrac{d w}{d r}\\right)\\right\\}\\right] = - \\frac{q}{D}\\,.\n", "a26859b8a39fc6fbbc43eaa85f338f4f": "S={\\langle Q\\rangle}^{\\bot}", "a26905f0573782898d5987a5921b9eb5": "\\sigma = 0", "a2694fce967ae69e740d80dc132f77ab": "\\beta(X,X')", "a26959d62f79689c65fab671f38cf9f1": "\\mathbf{a}_1 = (\\mathbf{a} \\cdot \\mathbf{\\hat b}) \\mathbf{\\hat b},", "a2695fd9835ce306a958650959c676c9": "V_{RM}=K_E*V_{O_2}", "a26966167c3683ec69abea2c61be433a": " n~r^{-n-1}~\\cos(n\\theta) \\,", "a2698b98c05f804cc4b6bd038a413c0b": "s(N-2)", "a2698f2b12463c7ac5c814fb30676eec": "\\theta_\\mathrm B = \\arctan \\left( \\frac{n_2}{n_1} \\right). ", "a269b2143c7a7baa88ddc66af3ae8142": "\\scriptstyle\\mathbb{N}", "a269c0298a8f881a18ea67c7a767e6d2": "a=A", "a269d4350e4ad5f2d88e568cf85e35b9": "A\\,|\\!\\!\\!\\sim_L B", "a269f6fa34024a7e58c1530bd6804b55": "H = \\frac{L}{N} \\,", "a26ac018fbb3c45127d6f28d6bb15fcb": " A(\\beta , t ) = A(\\beta , 0 ) \\exp \\left[ R(\\beta)t \\right]", "a26acbc2d2f143f211503a2ccda1d11f": " i_m ", "a26ad3c88fec1be34219e68bf929eda1": "f^{\\Delta} = \\Delta f", "a26b47c70e5033f9dbca396f133cd2a4": " \\varphi\\left(\\sum_{i=1}^{n} g(x_i)\\lambda_i \\right) \\le \\sum_{i=1}^{n} \\varphi(g(x_i))\\lambda_i, ", "a26b53297518626e2814a4f36787af6a": "|a| \\ge 0 ", "a26bad3399b99f628509efb7c071e7f4": " V(\\hat q ) ", "a26bdeab8d153f64425bd0792eeff1df": " \\lambda_{\\pm} = g \\pm \\sqrt{g^2 - 1} \\, ", "a26c43f45a1813509de69969281d77a3": "F(x) = \\tfrac{1}{2} x^T Q x", "a26c51523a9eb1fd18ffba77efb71bb2": "\n G = \\cfrac{K_I^2}{E'}\n ", "a26c5bb4c469bbb793df14d3c427db4f": "= 4 + 6P(5,1) + 10P(4,2) + 5P(3,3) + P(2,4) ", "a26d83e8e2ed59bfc01449019c0b95eb": "\\mathbf{u}_y(\\mathbf{v}_x\\mathbf{w}_y-\\mathbf{v}_y\\mathbf{w}_x)-\\mathbf{u}_z(\\mathbf{v}_z\\mathbf{w}_x-\\mathbf{v}_x\\mathbf{w}_z)", "a26dd3f73ee90225de14a0c2ac6ce55f": "\\mathfrak{m}\\,", "a26de3236f020e05474ae484c42e845d": "\\vec x_i", "a26e51779ee77a4985bdad1ab036c6fb": "\\mathfrak k", "a26ea3495ca267e259fed0188e7f396e": "\\mu > \\mu_0", "a26edf03ae3d446eb2121b31fd568080": "w(x\\oplus y)", "a26edfdd9fd839028072c2eccd9f3142": " {\\rm ad}(X)\\cdot Y = XY -YX.", "a26f2c99a9402631298bab62ef752b21": "p_a", "a26f9c101150634236e5ee63ffdcdb49": "m= \\, ", "a26f9f6afaa543b1c37abc6260df12a3": " u_t + ( f(u) )_x = 0. ", "a2703fcc0a15c589b59ca3b46de57bb9": "\\mathbb{R}^n\\to\\mathbb{R}", "a2705160568d141516bee4494bbc9cb0": " (\\mathbf{B}^\\top \\otimes \\mathbf{A}) \\, \\operatorname{vec}(\\mathbf{X}) = \\operatorname{vec}(\\mathbf{AXB}) = \\operatorname{vec}(\\mathbf{C}). ", "a2707099f06f6acd6d0fa0531fcd07e6": "\\dotsb\\overset{\\partial_{n+1}}{\\longrightarrow\\,}C_n\n\\overset{\\partial_n}{\\longrightarrow\\,}C_{n-1}\n\\overset{\\partial_{n-1}}{\\longrightarrow\\,}\n\\dotsb\n\\overset{\\partial_2}{\\longrightarrow\\,}\nC_1\n\\overset{\\partial_1}{\\longrightarrow\\,}\nC_0\\overset{\\epsilon}{\\longrightarrow\\,} \n\\Z {\\longrightarrow\\,} \n0", "a2715eaa5cbb2b526cf79eb341515477": " \\lambda = \\frac {2 \\pi}{k} \\ , ", "a2718a6b9cb5f198259969088ef671a2": "C_{t-1} = C_{0} + cY_{t-2}", "a271ed9ad895cca68c5054a869dedde5": "[V,U_1]=U_2, \\,\\,\\,\\, [V,U_2]=-U_1, \\,\\,\\, \\, [U_1,U_2]=(K\\circ\\pi) V", "a2722d4bfcb736058db1a52d594770a2": "x(t)=\\frac{1}{2\\pi}\\int^\\infty_{-\\infty} X(s )e^{s t}\\, d s ", "a2724fd9c9c352bb560bb066f961e6ba": "f(\\zeta) = \\frac{1}{(2\\pi i)^n}\\int\\cdots\\iint_{\\partial D_1\\times\\dots\\times\\partial D_n} \\frac{f(z_1,\\dots,z_n)}{(z_1-\\zeta_1)\\dots(z_n-\\zeta_n)}dz_1\\dots dz_n", "a2727b8f4e3ff9dbd3c97eff2bd00b0b": " \\mathbf{u} \\cdot \\mathbf{w} = \n\\frac{1}{2}\\left( \\mathbf{u} \\mathbf{w} + \\mathbf{w} \\mathbf{u} \\right). \n", "a2728a8ad39af3d6aaba9813b5082196": "m\\frac{d\\gamma\\mathbf{v}}{dt} = q\\mathbf{E} + q\\mathbf{v} \\times \\mathbf{B}", "a27293af30850d16d0a0e37a7693330d": "L = -\\sum \\frac{\\partial (f_i p)}{\\partial x_i} + \\frac12 \\sum \\frac{\\partial^2 p}{\\partial x_i \\partial x_j}", "a272ea1c5e44426139be336be130a134": "y=x\\,\\frac{y_1-y_0}{x_1-x_0}+\\frac{x_1y_0-x_0y_1}{x_1-x_0}\\,.", "a27302fc57dcafccda4de3b7fdcfbb6f": "G_{\\mu \\nu} + g_{\\mu \\nu} \\Lambda = 8 \\pi T_{\\mu \\nu}\\,.", "a273480176c559f18970d563173e2f56": "\\frac{dy}{dt} = 0.85 y", "a27386d98b70da02b1e4359c48540dc0": "\\mathcal{N}^{PL} (X) \\cong [X,G/PL]", "a273e857d7fd14a77ddc6d410bbf66f7": "q_{l}", "a274035410c22c3a861f41743abe7098": " \\left\\{| \\nu_1 \\rang, | \\nu_2 \\rang, | \\nu_3 \\rang, ...\\right\\} ", "a2741361626ab5f21870b2e02977c7e0": "\\displaystyle{\\widehat{\\mu}(g(z))= {\\overline{g_z}\\over g_z}\\, \\widehat{\\mu}(z),}", "a274180fc18f974579f155b21e2204be": "A^{-1/2}", "a2744d8bc91a1551cd9a563bcd67aadc": "\\Omega =\\mathbb{R}", "a27489d5d571f5d824815e1c79760c0b": " y = \\Beta_{0} + \\delta_{0}d2 + \\Beta_{1}dT + \\delta_{1}d2 \\cdot dT ,", "a2750cbccf8d073c1247f9c358e0f6b1": "L_{[D]}", "a2751998aadf1d336cb09ea3a105d327": "\\begin{matrix} {3 \\choose 1}{11 \\choose 1}{4 \\choose 2}{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "a275b9910574160e735de1a46f407a75": "0<\\nu<\\infty", "a275c347e5860a885f403190848dddd6": "M_{r}=\\frac{(c)_{r}(c+1-\\gamma )_{r}}{\\left( (c+1-\\alpha )_{r} \\right)^{2}}", "a275d8d2b9b0ccf40c08985a40903c01": "y=ax^2+bx+c.", "a2760fc7c966a78cf748909647a3e732": "\\sigma_{t,T}=\\sqrt{\\frac{T\\sigma_{0,T}^2-t\\sigma_{0,t}^2}{T-t}}", "a27616cd9048dd79abb24ffcdd900ad0": "Q_A/Q_B", "a2764e3c0487ba652ff99f8229a4f0ab": "|r_{t}|", "a2766acfee167b0c30dbfcb3a765f7a7": "r_\\mathrm{atm}\n= R_\\mathrm{E} + y_\\mathrm{atm}", "a276ba9f769a49b67ccae13cbc373b21": "\\frac{I_S R}{n V_T} e^{V_s/(n V_T)}", "a276d69d89433385904492493320fe16": "x_1,\\ldots,x_k,x_i", "a276daeb82d50d36b160df42e87c0a02": "\\forall n, E_{n+1} \\subset E_{n}", "a276eae210276f65e1a9d511c1f9c6cf": "\\frac{dL}{dt} + LM - ML = 0. \\, ", "a2775d82b4084a06856e2420b1d817dd": "i\\partial_t^{} u + L_1u = \\phi u", "a2777f7b7c10b19540c90970fac1b785": "\\zeta(z) \\; \\Gamma(z) = \\int_0^\\infty \\frac{t^{z}}{e^t-1} \\; \\frac{dt}{t}.", "a2780cb6b8ff6ef086d4025893f4b654": " K", "a27835f2b7c4245cec82dbab64ba277d": "x(t)\\,", "a2785d4a62a14466ed931a4ea6f785c5": "\\gamma_s", "a2788fd1b99d2929b4db4deb50dab95f": " \\mathbf{a} = \\mathbf{\\hat{e}}_{\\bot} \\left ( \\partial^2 A/\\partial t^2 \\right ) \\,\\!", "a278a0e0e60a3f1c2b0e65d6249d31c1": " \\nabla f ", "a278ae011e97faea37a050b3437e7827": "\\dot \\boldsymbol{\\sigma_1}=\\boldsymbol{\\sigma_1} \\times \\bold B + \\boldsymbol{\\sigma_1} \\times \\boldsymbol{\\sigma_2}", "a278b58518e30080bc969a2427a18a1b": " V ", "a27911d09e99c20d32084611881eca68": "\\Big( (\\mathcal{M}, s) \\models A\\phi \\Big) \\Leftrightarrow \\Big(\\pi\\models\\phi", "a27922aae955d4c0acad0a1213bba3f3": " L \\simeq K[X]/(p). ", "a279270628eb221d406e481836d9af37": "C_2 = C_4 = 0", "a279ef8081bf0419a0aa0e506456fc83": "(V,E)", "a27a0464bf44c7af17da0463d972278f": " g=\\exp(F)\\exp(I), \\qquad F\\in\\mathfrak f, \\qquad I\\in\\mathfrak h. ", "a27a1659a8b334790b6a3e586bef8a15": "\\Gamma_g(N),", "a27a3313ffac1825d2dcb6a3cd31b090": "(x \\cdot y)^{-1} ", "a27a77e5687ab9fc13d168427cca29c9": "\\tfrac{B}{100} + 45", "a27a8f03fca948dfa9ee4aaf4da00e6b": "(\\mathbf{a \\times b} )\\mathbf{\\times} (\\mathbf{c}\\times \\mathbf{d})=\\varepsilon_{ijk} a^i c^j d^k b^l - \\varepsilon_{ijk} b^i c^j d^k a^l=\\varepsilon_{ijk} a^i b^j d^k c^l - \\varepsilon_{ijk} a^i b^j c^k d^l", "a27a9f4ba7794afb9f0b165b250aa64f": "(N,s)", "a27aa08004d27eeae6b2493ab10cfe3b": "R \\gg \\frac{D^2}{\\lambda}", "a27accbd7e03aefdd277303a71bb5e17": "X\\subseteq Y", "a27b9b13254a32dc3beae2c6bb414309": "\\frac{2(n+\\alpha)}{n+1}\\,", "a27bad802bf505f7a43994b43c061111": "\\cong", "a27be73d706937cdeed5c0eb70cef7cb": "\\ln y=\\lim_{n\\to\\infty}n\\ln \\left(1+\\frac{x}{n}\\right)=\\lim_{n\\to\\infty}\\frac{x\\ln\\left(1+(x/n)\\right)}{(x/n)}.", "a27ced62e51f46d55437ac28fce5f1a1": "x\\notin Y\\cup Z", "a27d10fb71ac75fea17876916b880027": "Ck_n = kT_{n-1}+1\\ ", "a27d18f8d84c40101947e28c8f2c98ac": "\\vert{\\Psi_0}\\rangle", "a27d830af8c8d35b372a10f4f33ba0e5": "[p,v]\\mapsto p(v)", "a27dafe68f4012cd1651eb31457fac68": "H_{2n}(x) = (-4)^{n}\\,n!\\,L_{n}^{(-1/2)}(x^2)", "a27dea0d1ba770c774c4cc2081d2bea3": "\\begin{align}\n n &= \\sum_{ij} n_{ij} = \\sum_i n_i \\\\\n z &= \\frac{1}{n}\\sum_{ij} z_{ij}n_{ij} = \\frac{1}{n}\\sum_i z_in_i \\\\\n w &= \\frac{1}{n}\\sum_{ij} w_{ij}n_{ij} = \\frac{1}{n}\\sum_i w_in_i \\\\\n n'&= \\sum_{ij} n_{ij}' = \\sum_i n_i' \\\\\n z'&= \\frac{1}{n'}\\sum_{ij} z_{ij}n_{ij}' = \\frac{1}{n'}\\sum_i z_i'n_i'\n\\end{align}", "a27deb2acca0f524c3c8108261f15dd6": "\\left(\\frac{\\partial S}{\\partial V}\\right)_T = \\left(\\frac{\\partial P}{\\partial T}\\right)_V", "a27e193ff626b3753a448007e699eb17": "\\mathbf{x}_1^\\mathrm{T} \\ell - \\lambda \\mathbf{x}_2^\\mathrm{T} \\ell = 0", "a27e4da4eb911d3553935e35f0bd0843": "\\langle x,x\\rangle \\cdot \\langle y,y\\rangle = |\\langle x,y\\rangle|^2 + |x \\times y|^2", "a27e75a0378d65495f666bcb10588e70": "n = g\\tau", "a27e7d7232af3898a36d119bb0901bef": "(t, x, u) \\in [0, \\inf) \\times D \\times R^p", "a27e84ad7dc3af470886e925eaa1c1a9": "R = I + W\\cdot dt + {1 \\over 2} (W \\cdot dt)^2 + ...", "a27ed4b3b11c2076d91423f7e46ed0c8": "{-i g^{\\mu\\nu} \\over p^2 + i\\epsilon }.", "a27f6447cb3826d61806806ee3437347": "\\Gamma , A \\vdash A , \\Delta", "a27f89cc5fc078c3428e94f227e24a71": " F(11100) = f(1,1) + f(1,1) + f(1,0) + f(0, 0) + f(0, 1) = 0 + 0 + 2 + 0 + 1 = 3. \\, ", "a27fbedf6bac8eadfc12c5ccd3724efb": "(\\exists x \\phi) \\lor \\psi", "a27fc12bc99d822b35ef0930d066c50a": "\\mathcal{J} (x_1, x_2, \\dots, x_n) = \\frac{( \\sum_{i=1}^n x_i )^2}{n \\cdot \\sum_{i=1}^n {x_i}^2}", "a27fe3349ec891daaae2b99146d42691": "\\frac{1}{1-z} \\exp\\left(uz - z + u^2 \\frac{z^2}{2} - \\frac{z^2}{2} +\nu^3 \\frac{z^3}{3} - \\frac{z^3}{3} + u^6 \\frac{z^6}{6} - \\frac{z^6}{6}\\right).", "a2802ea6c19b69d026d0ff2e922cb02a": "\\mu_k=\\mu_k(x)", "a280cdc7804d4c605decd0a933a4790c": "y=X\\boldsymbol \\beta +\\epsilon,", "a2814d08648a1c7ee86cbccd3f287c0a": "\\mathrm{d}(pV) = \\mathrm{d}H - \\mathrm{d}U = \\mathrm{d}G - \\mathrm{d}F ", "a28199b4072c21de9dcac0064635e517": "= \\sqrt{(x'_1-y'_1)^2 + (x'_2-y'_2)^2 + (x'_3-y'_3)^2}", "a2819aab660491f0d27171419c383fb2": "\\langle x_0, x_1, x_2, \\dots\\ \\mid\\ x_k^{-1} x_n x_k = x_{n+1}\\ \\mathrm{for}\\ kc(\\alpha)\\sqrt{\\frac{n + n'}{n n'}}.", "a2993c99447a79b91e44c7f5d85d668d": "\\{\\phi,T\\}", "a29989b5ba683422d5275a05d7ba69f4": "\\mathrm{III}_T(t)", "a29a417a1e2b54c2f62ca6b9b07efe90": " f_i(\\Delta) \\leq f_i(\\Delta(n,d)) \\quad \\textrm{for}\\quad i=0,1,\\ldots,d-1.", "a29a5b1d801c13b4511553f4dd5e413e": "\n\\begin{align}\n\\cfrac{\\mathrm{d}N_{xx}}{\\mathrm{d}x} + f(x) & = 0 \\\\\n\\cfrac{\\mathrm{d}^2M_{xx}}{\\mathrm{d}x^2} + q(x) + \n \\cfrac{\\mathrm{d}}{\\mathrm{d}x}\\left(N_{xx}\\cfrac{\\mathrm{d}w_0}{\\mathrm{d}x}\\right) & = 0\n\\end{align}\n", "a29a910ed291bf94290298ae1ef73487": "1+2\\cos(x) + 2\\cos(2x) + 2\\cos(3x) + \\cdots + 2\\cos(nx)\n= \\frac{\\sin\\left(\\left(n +\\frac{1}{2}\\right)x\\right)}{\\sin(x/2)}.", "a29a9fbd1b764ebd3f19de0d6d3cbfa4": "m_0, m_1", "a29abac1559162a68a0f46b2d03f690c": "C_P=\\left(\\frac{\\partial H}{\\partial T}\\right)_P", "a29ade35711294b8c1f10cfd90486cb0": " \\exp_2^{i+1}(x)=2^{\\exp_2^{i}(x)}", "a29ae62511e57bcae61e253e2811abe6": "\nT^2 \\sim \\frac{\\nu p}{\\nu-p+1}F_{p,\\nu-p+1}, \n", "a29b37f3009caab490dc37d6d4284646": "|\\Psi^-\\rangle_{AC} \\otimes (\\beta |0\\rangle_B - \\alpha|1\\rangle_B)", "a29b41520182dd901b6af910f0cb04fb": "{y_1/y_2}", "a29b869fe52c02e8849c12233cc8d545": "\\gamma^\\mu = x^\\mu \\circ \\gamma (t)", "a29bacabf7a1da2815402cc361172671": "s^* : map (B, W) \\to map (A, W)", "a29bdd4561f92e0aadee9198a928b31e": "V-V", "a29bf66701e37d0041fa489bd8bc34cc": "M_{\\sigma \\sigma'} = \\exp \\left( \\beta J_p \\sigma \\sigma' \\right)", "a29c247dabb539b437000285902387dd": "K^M_n(\\mathbb{F}_q) = 0", "a29c32638b1dfcae1a7df242c79c4a57": "\\pm \\phi(-x) R(\\pi)\\phi(x)", "a29c82e7bd57101933c7898d2bc1fc2b": "4r^4=k^2(j^2-1)", "a29ced9325709a39a0fe7cbce17ae30e": "10^{10^{10^{76.66}}}", "a29dbcf49f9ce91dc6e37ff82db76f6e": "A^\\nu", "a29dc760f73f05e929d57f2f57b55fb3": " {\\left[ \\prod (1 + R_i )^{t_i} \\right] }^{\\frac{1}{\\sum t_i}} - 1", "a29e0a3051ca824ae3add78fd16dee00": " g^{-1} : [x : y : z] \\mapsto [x : z : -y] ", "a29e18265d7a603f29df07b7a1cb4b9c": "f'(s) \\equiv f'(r) \\not\\equiv 0 \\pmod{p}", "a29e6edb24fb0ab2069836a5c62e78a5": "\\sec \\alpha = \\sqrt {1 + \\tan^2 \\alpha}", "a29f0d45d978dd964c2437afeeb93faf": "x^7 + x^3 + 1", "a29f13eab944fbf379e069fa563f881a": "\\sum_{k=1}^nk^p=\\sum_{j=0}^p\\binom{p}{j}\\frac{B_{p-j}}{j+1}n^{j+1}", "a29f2b900a9d3b13e4a1621b152ab43a": "S=\\{s_1, s_2,\\ldots s_N\\}", "a29f72009fcb1c53c628bed2e18d41a2": "m_{UT}=\\frac{1}{3}\\Sigma^3_{i=1}{m^+}_i", "a29f87b5f12ce04c4c6cbb4605ead1f9": " VR=\\frac{\\lambda_1\\lambda_2\\lambda_3}{\\mathbb E[\\lambda]^3} ", "a29fef07d7d5c3b292083ca0dc552021": " \\dot{\\rho}=-i[\\omega_c a^\\dagger a,\\rho]+L(\\sqrt{2\\kappa(\\bar{n}+1)}a)\\rho + L(\\sqrt{2\\kappa\\bar{n}}a^\\dagger)\\rho ", "a29ff19f30be5b80e975c4d94c845f69": "\n\\begin{bmatrix}\nt' \\\\ x'\n\\end{bmatrix} =\n\\begin{bmatrix}\n1 & 0 \\\\\n-v & 1\n\\end{bmatrix}\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix}\\;,\n", "a2a02bb60c40d925374f6721f02a8eea": "\\,m_x", "a2a03cb084189278cf8b4a19a2094eb8": "y=-\\infty", "a2a0a4e85fa85b2e8751982d18207526": " PQ ", "a2a0dd9bfcda910cc74babdce62df0f1": "\\begin{align}\n {} & \\frac{1}{5}\\left[(2051 - 2052)^2 + (2053 - 2052)^2 + (2055 - 2052)^2 + (2050 - 2052)^2 + (2051 - 2052)^2\\right] \\\\\n =\\; & \\frac{16}{5} = 3.2\n\\end{align}", "a2a0ef2bedf383e3d783c0fb72036876": "\\hat\\theta_i= \\frac{x_i + \\alpha}{N + \\alpha d} \\qquad (i=1,\\ldots,d),", "a2a1264ff3ba204dcfd5d11997ea1ed3": "I(A,B) = H(A\\cup B)+H(A\\cap B) - H(A) - H(B),", "a2a15a2bcfa6e8330b1ad61d72d58cc6": "\\, \\Delta P_{L}", "a2a21a8baba9dfc38799e908ebfb458f": " \\ell_p", "a2a291a1748931f89086bd0c5079b212": "f(x) = x^3,\\!", "a2a297cd1e42d334247200592dc54697": "\\partial f:\\partial X \\rightarrow Y", "a2a2b98a525a52cdc6d64293be379985": "g \\circ f = f \\circ g.", "a2a2ce383970b2e90e125e477f4164c3": " [T_t \\psi](x) = \\psi(x + t) \\quad ", "a2a2f9d897f4e1c14907d3cf5b66cd71": "L_y", "a2a36c2ad9135c77cd07e64afdba73c5": "\n \\boldsymbol{P} = \\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{E}} \\qquad \\text{or} \\qquad P_{iK} = F_{iL}~\\frac{\\partial W}{\\partial E_{LK}} ~.\n ", "a2a382395bd38ea45b61898ac478bccd": "\\mathbf{\\Phi}_{lm}", "a2a3b5717c989ab11971f65d7f42c80f": "r= \\frac{(M_2 - C_2) - M_1}{M_1}", "a2a3fe841216b4f97c2d24e92c10354c": " {\\vec X} = \\left \\{ x_1,...,x_N \\right \\} ", "a2a4273cd180821aa1a0f9c28507069c": "\\kappa = 1- \\frac{\\sum_{i=1}^{k} \\sum_{j=1}^{k}w_{ij}x_{ij}} {\\sum_{i=1}^{k} \\sum_{j=1}^{k}w_{ij}m_{ij}} ", "a2a42d82af9e643914d5a8969d768176": "FV \\ = C(1+i)^{n-m}", "a2a448f19aaeac2d9900b82480094150": "U_{\\mathrm{out}}(t)= \\frac{1}{T} \\int_{t-T}^t {\\sin\\left[2\\pi f_{\\mathrm{ref}}\\cdot s + \\varphi\\right] U_{\\mathrm{in}}(s)}\\;\\mathrm{d}s", "a2a46adab75088c6561b33cfb0971376": "x \\equiv \\pm y/2", "a2a4740419f30a41bf473555a91b2c70": "k_{pj}", "a2a48cfad14e0d19fcfbc81cc3a52e2d": "\\sigma = \\pi (r_a + r_b)^2", "a2a512ed5d987a9c7e3ca8f93ad66241": "J=\\sum_{i=1}^N w_{x_i} (r_i-x_i)^2 + \\sum_{i=1}^N w_{u_i} {\\Delta u_i}^2", "a2a5135a951b2cb1c61fa8e71af365d6": "\\frac{1+7x}{(1-x^2)(1-16x^2)}", "a2a5440d7e3d451eaad15cf1a43e4f78": "D_{\\mathrm N} = \\frac {H s}{H + ( s - f )}", "a2a55c3d73c884481aad52465078d193": "\\phi_{s_1}\\geq\\phi_{s_2}", "a2a5b3643aebb53689deaf17af1c6b81": " \\mathbf{H} ", "a2a5d8d03d3bc0555678c2e4eb12ffbb": "\\equiv_{RK}", "a2a5fa0e2c3220b71bb09c4cddd2d7f0": "J(y_1, \\ldots, y_n) \\approx \\sum^n_{k = 0}F\\left(t_k, y_k, \\frac{y_{k + 1} - y_k}{\\Delta t}\\right)\\Delta t.", "a2a62b7b223fdf698f4d7f6339f44017": "\n\\varepsilon'(\\omega) = \\left( 1 + 2 (\\omega\\tau)^\\alpha \\cos (\\pi\\alpha/2) + (\\omega\\tau)^{2\\alpha} \\right)^{-\\beta/2} \\cos (\\beta\\phi)\n", "a2a699cd211adc001818902fbaa2c308": "\\vert u \\rangle", "a2a6b70ed2c27a5811a952097723b0ff": "e_i = \\sum_{j=1}^m e_j p_{ji} \\text{ for }1 \\leq i \\leq m. \\, ", "a2a6cd31bd34d8a73010d69c3aab4bdf": " i < k ", "a2a7189c0444606bbbb9c94bad715b9f": "D(g\\cdot f)=g\\cdot (Df).", "a2a725f499bf01e9b3651c15a05c026f": "p(\\boldsymbol{r})", "a2a764f5985c285282b9eaec06180e9f": "\\scriptstyle y\\in U_2", "a2a7e846042770eb61379a67f7b99b2f": "\\le_{RK}", "a2a82b8cb5a8c1d1d68b53f91d1dcd5b": "a \\leq b \\Leftrightarrow \\ln(a) \\leq \\ln(b).", "a2a833ac8bec76afb057e244e8c5125e": "M,\\,N \\,\\to\\, \\infty", "a2a85385046807669e108ed91eaab732": " \\Psi(\\mathbf{r},t) = \\sum_{n=1}^\\infty A_n e^{i(\\mathbf{k}_n\\cdot\\mathbf{r}-\\omega_n t)} \\,\\!", "a2a869f88db5c70091c5e85757a1f631": "\\scriptstyle \\lfloor x\\rfloor", "a2a87eab7098e039809b6e150d921355": "P=P_\\max \\left(1-e^{-c\\cdot LAI}\\right)", "a2a880f03259b4f220349d7aa8076ae9": "\\theta = \\arctan{\\left(\\frac{X}{R}\\right)}", "a2a8e32f839cce19994aed2badb3f698": "\\bold{e}_z", "a2a9093a2dd51ab533bde09f0fb3ce60": " F(x ; k) = \\int_{0}^{x} \\frac{dt}{\\sqrt{(1 - t^2)(1 - k^2 t^2)}}.", "a2a93cca666cd17ef7542bef166a0e50": " ( a_1 + \\cdots + a_n ) \\mathbf{p} = a_1 \\, \\mathbf{x}_1 + \\cdots + a_n \\, \\mathbf{x}_n ", "a2a94cc5fdf63673aac1412465330667": "\\phi(4)=2", "a2a957593fda9edd18f522a0ac379ad9": "P(T) \\approx 10^{-17.73}~T^{-2/3}~~~(10^{6.3} < T < 10^{7} K) ", "a2a9e6be9d59a472a47593303f86a447": "\\sum_x f(x) = F(x) \\,", "a2aa64dafdfa43a3d759eb44f2732d53": "\\begin{matrix}\\frac {\\Delta L} {V_EL} \\end{matrix}", "a2aaf432caf2aa851bb438f78989dcfb": " S^{\\prime} \\to S ", "a2ab256dab28244b2238f991e4780e52": "\\operatorname{Cl}_2\\left(\\frac{\\pi}{2}\\right)=G", "a2ab50c24e143295bef79bfc039be141": " \\|T_n\\|_\\infty = T_n(1+\\sigma). ", "a2ab7d71a0f07f388ff823293c147d21": "\\sigma", "a2abb6139f4d10032a31dbb2cd0d27a3": "\\sigma_{j}^{+}, \\sigma_{j}^{-}, \\sigma_{j}^{z}", "a2abbf5ccc8447d54f928c47e5b7261d": "f v", "a2abc4e189f7f2019ca07fad837f8952": "\\textstyle k=n-r", "a2abfc0bf2d17cbafcf8e6f5a7bed0fb": "\n \\begin{align}\n \\lambda_i\\frac{\\partial W}{\\partial \\lambda_i} & =\n C_1\\left[-\\frac{2}{3}J^{-2/3}(\\lambda_1^2+\\lambda_2^2+\\lambda_3^2)\n +2J^{-2/3}\\lambda_i^2\\right] + 2D_1J(J-1) \\\\\n & = 2C_1J^{-2/3}\\left[-\\frac{1}{3}(\\lambda_1^2+\\lambda_2^2+\\lambda_3^2)\n +\\lambda_i^2\\right] + 2D_1J(J-1)\n \\end{align}\n", "a2ac93867cfe35ec1e679d9f717f453f": "a = g{m_1-m_2 \\over m_1 + m_2}", "a2ac94ae61863e96c16650d556002d5d": "s_i = \\omega^{2^i} + \\bar{\\omega}^{2^i}", "a2acdc24cdab411128de9eaad87152c7": " f(x \\and y) = f(x) \\and f(y),", "a2acef4a248d8cea8e90765ace174a0b": " GS = \\frac{ 1 }{ 2 } \\sum_{ i = 1 }^K | \\frac{ A_i }{ A } - \\frac{ t_i }{ T } | ", "a2ad14319266aa39d9b1c4ebfa5c45da": "R < r", "a2ad246980e1b9d24f3002d81fa9cdc5": "\\{ 0,4,7 \\}", "a2ad8907f8214d9cfde9b33844289200": "\\sigma_x^2(\\tau)", "a2adb2d42746385c2558f7e0bc14746d": "dU=-pdV+TdS+\\mu dN", "a2ae21048429eed474f134e1728021b8": "\\Pi (t,f) = \\delta_{(0,0)} (t,f) ", "a2ae7415f2dde9d36244b503ddeb3939": " v_{i2} ", "a2aeef99af012942999f330666a31b6a": "AB + CD \\longrightarrow AD + CB", "a2af45b8b9d017bed7078236d74e51de": " \\mathbf{r}_\\mathrm{com} = \\frac{1}{M}\\int \\mathrm{d}\\mathbf{m} = \\frac{1}{M}\\int \\mathbf{r} \\mathrm{d}m = \\frac{1}{M}\\int \\mathbf{r} \\rho \\mathrm{d}V \\,\\!", "a2af63bedfdf6ab2f08cc2f40aaf58a8": "x = \\frac{t^2-1}{t^2+1},\\ y = \\frac{2t(t^2-1)}{(t^2+1)^2}.", "a2afa25c0aedb3c2cb3688b847a49e52": " \\ddot{p} = -a p + b q - c p ", "a2b04a0c84123f4a0b6983f80a9f154f": "{m_P}", "a2b09f7a50db8dad8564065cadc86196": "y(x), ~ z(v)", "a2b0c6a7327f52630c68c0710c1d85e2": " |A \\cup B| = |A| + |B| - |A \\cap B|, ", "a2b0e6f402aa1d595f337eb446b5fa7f": "I_{PPP}=|f'_zf_zf_z\\chi_{zzz}^{(2)}+f'_zf_if_i\\chi_{zii}^{(2)}+f'_if_zf_i\\chi_{zii}^{(2)}+f'_if_if_z\\chi_{iiz}^{(2)}|^2", "a2b0e86bbcccd6b486e44b185e6f9e43": "\nF = \\frac{k(k-1)}{2}\n", "a2b0f5e383d44c526c08df1acb8a8d16": "\\rho(\\mathbf{x},t) = \\int_{-\\infty}^\\infty \\hat{\\rho}(\\mathbf{x},\\omega) e^{-i \\omega t}", "a2b0fe661f81678f753cba6653b9e7d1": " {^*\\! f} : {^*\\!A} \\rightarrow {^*\\mathbb{R}};\\,", "a2b10ddd9802fb6fb0b72672926cd445": "\\pm \\sqrt{x}", "a2b116b4325918026a9f434aba5a794b": "\\{ X \\leq x\\}", "a2b1d3a336d9bd8a845d78394d6c3b8f": "\\begin{matrix}\\text{If } y(t)=\\int_{-\\infty}^\\infty x(t-\\tau)h(\\tau)\\,d\\tau\\text{ then }\n\\\\ W_y(t,f)=\\int_{-\\infty}^\\infty W_x(\\rho,f)W_h(t-\\rho,f)\\,d\\rho \\end{matrix}", "a2b1f2c82e58393f45a4f1723beb8266": "\\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.", "a2b31a3c7003d59335822baf1335c1e0": "\nF(\\varepsilon)=\\frac{\\sqrt{\\varepsilon}}{2}\\int_0^1 \\left( \\frac{2}{u} - \\ln\\left(\\frac{1}{u}\\right)-2\\right)e^{-\\frac{\\varepsilon}{u}} \\, du\n", "a2b347ae03f94ce41b5e28e7e3e176ab": "\\pi_1(SO(m-1)) \\cong \\mathbb{Z}_2", "a2b361babaff66c8ffa323c24cb15602": "h\\begin{Bmatrix} p, q , r \\end{Bmatrix}", "a2b36efedf363d3113d71238ff0e2e6f": "S_{min}", "a2b3e16ee96bca457e66f7e3492b3b3e": "y(t) = \\cos \\left[ \\left( 1 + \\tfrac38\\, \\varepsilon \\right) t \\right] + \\mathcal{O}(\\varepsilon),", "a2b3eb775aa5e6edc6317559205713dd": " \\pi \\approx \\tfrac {355} {113} ", "a2b409d5fe1deaf8d6ef43e4fb241ded": "E\\, =\\, \\tfrac12\\, A\\, \\rho\\, g\\, h^2", "a2b466242c83e46e98ad3e5efe35a6b6": " P_1 \\ ", "a2b4b10a65a2f7cb822973bf43fbd65f": " y\\in A_j", "a2b4dd1bfa0c9242890222915204348a": "\n\\zeta(s)\\,\\sum_{n=1}^\\infty\\frac{\\mu(n)}{n^s}=1, \\;\\;\\mathfrak{R} \\,s >0.\n", "a2b50ca3aa3b62ac8eca66c24930fdc6": "a_{ij} = r_j-r_i ", "a2b534f6a0ecb02a86c778706ed77d14": "\\scriptstyle\\lesssim10^{-8}", "a2b53c5e74f0e8b701926240a6f8697e": "A_{k-1} \\rightarrow A_k \\rightarrow A_{k+1}", "a2b5500385fae8178588b9afd131af52": "f\\in L^1(\\mathbb{R}^k)", "a2b5575676aa452f167c31975c00a178": "\\mathfrak{P}\\{\\mathcal{B}\\} = \\prod_{\\beta \\in \\mathcal{B}}(\\mathcal{E} + \\{\\beta\\}),", "a2b5a545d5e511db98da65bfb675f9e9": " X = \\sum_{i = 1}^N X_i, A = \\sum_{i = 1}^N A_i ", "a2b5e5e3b21933fcdbac1439b54d44fc": "x^2+y^2-ax-by-c=0", "a2b5f9b9840daf4331498d0fbcf11ab5": "-\\phi", "a2b61b2ac5afb9f07cd25162aef84d0a": "\\displaystyle x=x_1", "a2b68b8d6a1e0cb48b637e21281363eb": "k^2-k", "a2b6e277376fe568f89002b1dbfdc577": "p = 2q + 1", "a2b6ef8816cded6b8dec2ce427a45713": " \\det(\\mathbf{AB}) = \\det(\\mathbf{A})\\det(\\mathbf{B}) ", "a2b6f2eabb4a74c6fa6dc3f6e2c1114b": "\\varphi_A", "a2b703b47bbbd77b8edb73662de0716a": "s_{jt}^{*}=\\frac{1}{2}(s_{jt}+s_{j,t-1}),", "a2b70e6f3bfdf24a94ce642162d98162": "\\nabla F(\\mathbf{x})=2A^T(A\\mathbf{x}-\\mathbf{b}).", "a2b7d17f6822b64016ed3b8ee00980ee": "\\lim_{x \\to a} f(x) = L.", "a2b7d225baaa3c74085cece626d3d5e2": "(\\rho(g)f)(x)=f(xg).", "a2b876534f37cbb2851a9a1d5c89dc09": "(V_k \\otimes I_{H_B})", "a2b8b70588f0a373da67823dee804791": "\\sigma_{k}", "a2b8d9ba5a30bfa0c1988bd86cefbc3f": "{dQ_b \\over dt} = F_b (C_{art} - {{Q_b} \\over {P_b V_b}})", "a2b9025ac8b11f2974925e481c146572": "d\\,\\mathbf{f}(\\mathbf{v}) = \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{x}} \\mathbf{v}.", "a2b917515f7e489707b238819f910020": "C = \\Psi^\\prime \\sqrt{\\frac{i}{z\\lambda}}", "a2b91ec15e3d488b157d88493e7477e3": "c = 1, k = 0", "a2b93d58abe8c0d38212b4ed161c0b54": "f(-x) = -f(x).\\,\\!", "a2b9684a0dd15c05cc572a502029d76d": "{{3n^2 + 3n + 2} \\over 2}.", "a2b97883a9c9447f8c295b25c6f6229f": "\n\\Delta G_{i}^{solv} = \\Delta G_{i}^{ref} - \\sum_{j} \\int_{Vj} f_i(r) dr\n", "a2b9a4d0faa98c4aac33c052bfb6125e": "\\mathrm{distance}_{\\textrm{(parsec)}} = \\frac {\\mathrm{v}_{\\mathrm{(km/s)}}} {\\mathrm{4.74u}_{\\mathrm{(arcsec/year)}}}", "a2b9d5fcd9cda1f800ed14d9c3d82771": "(x^{q^{2}}, y^{q^{2}}) = - \\bar{q}(x, y)", "a2b9e35d07e81b678cd0147dc7b7be24": "\\nu_t > 0", "a2ba7ddc69a736f391205206ac8fd183": "\\mathrm{rad\\,s^{-1}}\\,", "a2bb328a6d190e83998d348bf1ff2dca": "0 \\notin F", "a2bbbba54436330de58a460f99c53edd": " p^\\alpha || m ", "a2bbcd322058eb3810a7b33153b585a5": "J_\\pm = J_x \\pm i J_y \\,,", "a2bbe290bf0497082821bead95ddfbd7": " \\{O_n,f\\}=\\{E_n,f\\} =0 ", "a2bbef98b3b51382405018f5fa84a77a": "a\\gamma=\\gamma a", "a2bc005c7fa99320b9acea13353440bf": " \\overline{U}", "a2bc0b6707439a5310d823ab443c3698": "\\displaystyle{(f_m,f_m)_\\sigma=\\prod_{i=1}^{|m|} {i-1/2-\\sigma\\over i-1/2+\\sigma}= {\\Gamma(1/2 +\\sigma)\\Gamma(|m|+1/2-\\sigma)\\over \\Gamma(1/2-\\sigma)\\Gamma(m+1/2+\\sigma)}.}", "a2bd12c91b924cd23265f4a55b0c57da": "\\rho(\\mathbf{x})= \\frac{\\mathbf{x}^TA\\mathbf{x}}{\\mathbf{x}^T\\mathbf{x}},", "a2bd7e22a290a2cdbd87e0bbf230dcb9": "\\rho\\mathbf{v}\\otimes\\mathbf{v}-\\sigma+(p-c^2_0\\rho)\\mathbb{I}", "a2bdabb2001a0279b5a57b9658f2a954": "A_k := \\begin{bmatrix} a_{1,1} & a_{1,2} & \\dots & a_{1,k} \\\\\na_{2,1} & a_{2,2} & \\dots & a_{2,k} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{k,1} & a_{k,2} & \\dots & a_{k,k} \\end{bmatrix} ", "a2bdad88ff9f9873ebb131a174f07170": "p(X^{(1)}|\\sigma,I)={1 \\over a} {1 \\over \\sigma}f({X a \\over \\sigma a})= {1 \\over \\sigma^{(1)}}f({X^{(1)} \\over \\sigma^{(1)}})", "a2bdb62a5b31fac1b1c3db4d94410311": "\n\\begin{array}{rcll}\nu_{tt} - \\Delta u & = & 0 & \\mbox{ in } \\Omega \\times (0,T), \\\\\nu & = & 0 & \\mbox{ on } \\Gamma \\times (0,T),\n\\end{array}\n", "a2bde0f134e9fed9136c604e537f0714": "M_{KK}\\approx R^{-1} .", "a2bdf090b3dca38b255b1b63b23f7799": "d^n_{A[1]}=-d^{n + 1}_{A}", "a2bdf6f75a0987d6d13bf7d19e1c4462": "(9)\\quad g_{tt}=-(1+2\\psi)-\\mathcal {O}(\\psi^2)\\,,\\quad g_{\\phi\\phi}=1-2\\psi+\\mathcal {O}(\\psi^2)\\,,\n", "a2be6748708a38865b23039db0a67874": "c_f(u,v) = c(u,v) - f(u,v) ", "a2be78ae3d3cdd03c05a43d53fec7c50": "0 x_0", "a2e2fc07a24fee60afc975f87a1c8ef2": "\\begin{cases}\n y = \\beta x^* + \\varepsilon, \\\\\n x = x^* + \\eta,\n \\end{cases}", "a2e3517af1f277240bcbbf26f573c754": "\\phi \\to \\psi", "a2e366077e94f81362d89d2b515118f2": "\\left (\\partial_t-k \\partial_x^2 \\right )\\Phi=\\delta,", "a2e3f612d2a36576d75762ca888cf68a": "\\mbox{eGFR} = \\mbox{186}\\ \\times \\ \\mbox{Serum Creatinine}^{-1.154} \\ \\times \\ \\mbox{Age}^{-0.203} \\ \\times \\ {[1.210\\ if\\ Black]} \\ \\times \\ {[0.742\\ if\\ Female]}", "a2e3f68cc24f82d76561534eebd3a50c": "\\phi \\land \\chi \\to \\phi", "a2e4199bac1e0cae6efbd146fa9160e3": "\nI(X;Y) = h(Y) - h(Z)\n\\,\\!", "a2e47427e466709bf3611f6c68a4efad": "{\\mu}_s > {\\mu}_d", "a2e476699b1596fad8d7cb2cb50de55f": "\\nu\\,", "a2e4941fada692b642de8eeb2146c144": "\\sin(61\\tfrac78 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2-\\sqrt{2-\\sqrt{2}}}};", "a2e4f5db2723550f815d64666e8deced": "\\mathbb{Q}_p", "a2e504fb5d438221072467598223092b": "(\\mathbf{W}^i -\\mathbf{G})\\cdot(\\mathbf{W}^i -\\mathbf{G}) - (\\mathbf{W}^1 -\\mathbf{G})\\cdot(\\mathbf{W}^1 -\\mathbf{G}) =0,\\quad i=2,\\ldots, 5.", "a2e518dfcf166f4d6516e91cfe6f2d7d": " \\cdots \\to 0 \\to 0 \\to A^n \\to A^{n+1} \\to \\cdots \\to A^{m-1} \\to A^m \\to 0 \\to 0 \\to \\cdots ", "a2e5247c57d2b06a7291e84d1d5a1b2a": "\n|I| = { |V_\\mathrm{S}| \\over |Z_\\mathrm{S} + Z_\\mathrm{L}| }.\n", "a2e552290d9342e9d8a6a2f2bfbce9cb": " R = R_0 \\, \\sec (\\Phi-\\Phi_0 )", "a2e589596d2702afa2b2a3892f795b2d": "\\rho(A)<1.", "a2e5da693c0ed5e9195135f0eb9dbd4f": "E = \\hbar \\omega ", "a2e5f0b6253d248c6bd66a45ae94e7ab": "((bababab)^3cc^{(ab)^3b(ab)^6b})^2=1.", "a2e622e436766c2c4d283f9e9d732404": "i \\colon M \\to I", "a2e667af21289a6a188f41f6140dd8e0": "\\left(a \\rightarrow P\\right) \\left\\vert\\left[ \\left\\{ a \\right\\} \\right]\\right\\vert \\left(a \\rightarrow Q\\right)", "a2e6bf7be6235169f60be4c5e49899bd": "\\displaystyle \\frac{1}{|a|} \\hat{f}\\left( \\frac{\\xi}{a} \\right)\\,", "a2e6c19766fadbe1bce1842125fc1ebb": " 885.7 \\ \\mathrm{seconds} \\,", "a2e6c7266da623a0b8218ee761bfcbb0": "\\varphi_{s(p,x)}(y)", "a2e70cca865d75c7c6fc526e5aecd669": "\n \\varepsilon(p) \\equiv \\begin{cases}\n 1 & \\text{for bosonic operators,} \\\\\n \\text{sign of the permutation} & \\text{for fermionic operators.}\n \\end{cases}\n ", "a2e745a91b5e99627f105e977de3b6d5": "A^T\\textbf{x}", "a2e7578f903173c330c74b41a725b607": "m \\mid n", "a2e8256a37419328291ef14c9aa06c06": "\n\\begin{align}\nX_\\mathrm{a}(f) & \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\begin{cases}\n\\ \\ 2 X(f), & \\mbox{for } f > 0, \\\\\n\\ \\ X(f), & \\mbox{for } f = 0, \\\\\n\\ \\ 0, & \\mbox{for } f < 0,\n\\end{cases} \\\\\n&= X(f)\\cdot \\underbrace{2 \\mathrm{u}(f)}_{1 + \\sgn(f)} = X(f) + X(f)\\cdot\\sgn(f)\n\\end{align}\n", "a2e8a9a0f35ae6bbc66eb7f46b7a42f9": "\\operatorname{Ob}(\\mathcal{C})", "a2e8e70d2550283685195135302351d2": "\\scriptstyle f = f_\\mathrm{blue}", "a2e8fddcd732634e3f16dccc7e9fa324": "\\scriptstyle{E(C_1)=0.5}", "a2e9465971fac366892352cb2dd42f62": "c_i(e_j)", "a2e9701a1459997b4cf1c73b988d2484": "\\mathcal S", "a2e9c726167c0e8043a5d3c63a0cfdb6": "\\{q,N \\}", "a2ea2bd09049e6b862f5dc89936d42b5": "\\frac{(c-\\beta )(c)_{\\alpha -\\beta }(c+1-\\gamma )_{\\alpha -\\beta }}{(c+1-\\alpha )_{\\alpha -\\beta }(c+1-\\beta )_{\\alpha -\\beta }}s^{\\alpha -\\beta } ", "a2ea40aea2d9cbf13f1d32846156a676": "\\mathrm{chord}\\ \\theta = 2 \\sin \\frac{\\theta}{2}, \\, ", "a2eb376794df695b1b9e6de25f2d15a1": "E[W^{\\text{M/G/}k}] = \\frac{C^2+1}{2} \\mathbb E [ W^{\\text{M/M/}c}]", "a2eb3f343cd1884037e3e5981abf3455": "\\widehat Var (\\widehat\\beta_1)", "a2eb61d19ce1495ee83a4eb24b38fbf8": "\\rho_{\\operatorname{L}} = v / \\omega_{\\mbox{c}} . \\,\\!", "a2eb985b9f9c4824a58f900e104873c7": " \\mathbf {\\mu} ", "a2ec2f507f7c501e29185ccccd0455aa": "u_\\tau=\\sqrt{\\frac{\\tau_w}{\\rho}}", "a2ec4eae8277eee90ad6c8ceac9f3494": "\\| y_n - y_m \\|^2 = \\|y_n -x\\|^2 + \\|y_m -x\\|^2 - 2 \\langle y_n - x \\, , \\, y_m - x\\rangle", "a2ecd21281e7f8daf2b26047c22adad1": "H_n:\\bold{hTop}\\to\\bold{Ab}.", "a2ecd7b4a10055b8e647cc29e8e66a32": "x^\\prime, y^{\\prime\\prime}", "a2ecdb124649c70fdf6dae9b27cbbb2e": "L=n\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}\n=\\dot{x}_1\\frac{n \\dot{x}_1}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}+\\dot{x}_2\\frac{n \\dot{x}_2}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}+\\frac{n \\dot{x}_3}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}\n", "a2ecf65e98c266b08ebc67e347186dce": "(\\mathrm{Fe}^\\mathrm{III}_8)_A[\\mathrm{Fe}^{\\mathrm{III}}_{40/3}\\square_{8/3}]_B\\mathrm{O}_{32}", "a2ecff64841d048f50e12803a237b0ef": "S = g^{ij}R_{ij} = R^j_j", "a2ed218512aed1dfed615c64f8e5bfe2": "a_{10}", "a2ed243aa25aabdbdfb24955d13a137f": "F_e >0, F_w > 0", "a2ed75133e576063a9a62cbb3332dd81": " \\langle E \\rangle = \\Omega + \\langle N_1 \\rangle \\mu_1 \\ldots + \\langle N_s \\rangle \\mu_s + ST.", "a2ed7648b7bde6bdfa78cd0335e0240b": "\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = \n\\delta_{mn} \\Bigl(\\frac{1}{\\Sigma} \\frac{\\partial \\Sigma}{\\partial E}\\Bigr)^{-1} = \n\\delta_{mn} \\Bigl(\\frac{\\partial \\log \\Sigma} {\\partial E}\\Bigr)^{-1} = \\delta_{mn} k_{B} T.\n", "a2edc1fa21b6c09097297de938ebd72d": " \\mu_0 \\,", "a2ee03dbfd271cb56ac44d0f24f608fd": "\\frac{1+x_2}{x_1},\\frac{x_1 + x_3}{x_2},\\frac{1+x_2}{x_3}, \\frac{x_1+(1+x_2)x_3}{x_1x_2}, \\frac{(1+x_2)x_1+x_3}{x_2 x_3}, \\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3}", "a2ee24795f99ae6ede2e27dbab4f66ba": "\\sqrt{A}= \\begin{pmatrix}\\cosh ((\\ln 2)/2) & \\sinh ((\\ln 2)/2) \\\\ \\sinh ((\\ln 2)/2) & \\cosh ((\\ln 2)/2) \\end{pmatrix} =\n\\begin{pmatrix}1.06 & .35\\\\ .35 & 1.06 \\end{pmatrix} ~. ", "a2ee329712b15ca2031908e8e383c4d5": "\\delta w = PdV", "a2eea66bcb3df0d93d96467eb0a544c6": "y=\\left( Y(X_1),\\dots,Y(X_N) \\right)^T", "a2ef297a8f17f41d12b5608c45c8b003": "\\exists Z \\forall n ( n \\in Z \\leftrightarrow n \\not \\in Z)", "a2ef5f3a7a999deab78663401a183409": "Df_N=0", "a2ef681bc23611cc5c725b377681782a": "\\mathrm{eval}\\colon (Z^Y \\times Y) \\rightarrow Z", "a2efe124f8c0dc0193ad1497f0e8668b": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ 0.06\\end{smallmatrix}", "a2effcb19bfc81add92d6013ae7cc354": "\\partial \\varphi", "a2f03f8ecdf76bce9e2c08db914c78ca": "\\rho V g", "a2f04f05381d50c6b30a094fdae98b25": "f(y)", "a2f070a31330443ceb0dcf352fe50035": "n/2", "a2f0884dd3bb23699004e178b32c2629": " F = \\; k_{\\mathrm{e}} \\frac{Q^2}{r^2}, ", "a2f1383e73a601e5e4b8d3f6be272b6d": "aI_n", "a2f156d3d255644e8df4ffff8f0e7f60": " (S \\cap C) \\,", "a2f20b1e37950182b42d9a4effb75de4": "x_{k+1}^* \\leftarrow x_k^* + \\overline{\\alpha_k}\\cdot p_k^*\\,", "a2f251d296496418d5bffd98e4e343aa": "\\langle f|f\\rangle\\langle g|g\\rangle \\geq |\\langle f|g\\rangle|^2, \\, ", "a2f2626fab14a1fa7dc042d502d15f25": " \\boldsymbol{\\mu}_S=- g_S \\mu_B \\frac{\\mathbf{S}}{\\hbar}.", "a2f2c6ab59a313e433ee240e8c9e2556": "\\,y = e^x", "a2f3463d0f27215c28fa900ac10dd8a5": "f(f(x) + y - f(y)) = f(x).\\,\\!", "a2f35c56ecf8a76119582996ce046db2": "{a}\\,", "a2f3be952c1487055840e7e28241a7f1": "X_4 = \\,\\!", "a2f3cf4c90bed0aacafdb9af17c4daf8": "F_c / A", "a2f4951100c9b72e12e502b033bf5041": "j~", "a2f4a9ad364cd3d5ac78633256eccb88": "{x^2 \\over a^2}+{y^2 \\over b^2}+{z^2 \\over c^2}=1,", "a2f4f7e4d240ceeecb6cba4d3d8236af": " \\zeta(1-n,a)=-\\frac{B_n(a)}{n} \\!", "a2f524241f5749324a934146417775fb": "F: \\text{Mod}_R \\to C", "a2f5500235cae7b5d28c93cde5ecdfa4": "F_c(s) =F_a(s) F_b(s).\\;\n", "a2f556569fff231827b75cd653156bf5": "\\int_0^x \\! t^n \\, dt", "a2f59ddb876dba85b7726c3b6633c5b4": "\n\\frac{1}{\\Phi}\\frac{d^2\\Phi}{d\\varphi^2}\\ =\\ -m^2\n", "a2f5b11993679dafa520d1e60db8c27f": "c_2\\,", "a2f5e1f38209f32f94da248993914295": "T^{\\alpha \\beta}(\\mathbf{x},t) = \\frac{m \\, v^{\\alpha}(t) v^{\\beta}(t)}{\\sqrt{1 - (v/c)^2}}\\;\\, \\delta(\\mathbf{x} - \\mathbf{x}_\\text{p}(t)) = E \\frac{v^{\\alpha}(t) v^{\\beta}(t)}{c^2}\\;\\, \\delta(\\mathbf{x} - \\mathbf{x}_\\text{p}(t)) ", "a2f6164ef86581bc6276480ec0ace47f": " {n \\choose k_1, k_2, \\ldots, k_m}\n = \\frac{n!}{k_1!\\, k_2! \\cdots k_m!},", "a2f6220517943bc0ee9369f900d64398": "\\quad \\vec r: \\Bbb{R}^2 \\rightarrow \\Bbb{R}^3.", "a2f65e081daf629989e3b364ffa61b29": " g(M)\\equiv 0 \\bmod\\ \\prod N_i", "a2f69d00e1c6cd7225f04c7588965ed1": " \\frac {F_m}{\\psi g} ", "a2f6aa543bf3a70a3d6229d94b3246ff": "\\rho: \\mathcal{L} \\to \\mathbb{R}", "a2f6f1f0740b42e55c2af269c754f743": " \\varepsilon_5 < 2^{-47} < 10^{-14}. \\, ", "a2f7766b9a43ce900208a29702610529": "w=\\frac{1}{\\bar z}=\\overline{\\left(\\frac{1}{z}\\right)}", "a2f778ad446a45a85e233ced8063c5d3": "\n\\rho(r) \\propto r^{-7/4} .\n", "a2f778fee4e53d3867d3238f8ffdd726": "x^3 - 2x^2 - 4,", "a2f7a169d02a8b08ad4830e9e95879ce": "\\scriptstyle\\rightarrow", "a2f89504f03211d9ec27fa9ebd573d54": "1- \\tfrac{\\mathrm{B}(\\beta+n-k-1,\\alpha+k+1)_3F_2(\\boldsymbol{a},\\boldsymbol{b};k)} {\\mathrm{B}(\\alpha,\\beta)\\mathrm{B}(n-k,k+2) (n+1)}", "a2f8a24eade97099450e99ace4a9ad6d": " K = \\frac{m}{2} v^2 = \\frac{m}{2} \\dot{\\mathbf{X}}\\cdot\\dot{\\mathbf{X}}. ", "a2f8a3710722f98e104436faf71669dc": "(2~3)", "a2f8c4db0ab7ed7eb4900d500d1e3b0b": "F[\\alpha]\\supseteq F", "a2f93064732a053bcf5c22cc238ad6e9": " r = r_{1}+r_{2} ", "a2f96675c51f45a8645d9d6ebe19a768": "(i\\gamma^\\mu\\partial_{\\mu}-m)\\psi=0 \\;,", "a2f9a7a274efcdadcfaa426ac1f250a9": " F(s)= \\sum_{n=1}^{\\infty} f(n)n^{-s} ", "a2f9b3ccf43402ef03d0598972e5a94b": "C=\\frac{q}{V}", "a2f9def804e36b03bb8611da09083282": "M(M(x,y),M(z,w))=M(M(x,z),M(y,w))", "a2fa030f9b0f71cd66dd12b418f8b3d6": "\\sum_{k=0}^\\infty \\mid{b_k}^2\\mid < \\infty,", "a2fa71b44832b2c87d6d5dabcc167251": "\n{dy\\over dt} = y\n", "a2fa79ebeda65d6fd39faa8dd9623244": "D \\sqsubseteq C", "a2fa82f9dc120bfab79a5287040677d9": "s=c_1^x\\,", "a2fb2d9ec7733a36e258a956dd715da3": " m,n>N ", "a2fbfec1926ac93f9aa3694d453a6ee7": "\\displaystyle{\\|\\sum_{i=1}^n T_iv\\|^2 \\le AB \\|v\\|^2,}", "a2fc056464b458a120c9e0ac3f83b66c": " t \\rightarrow - \\infty ", "a2fc27803202a78772a61ae8a73b005b": "\\int x^2 r^3\\;dx= \\frac{xr^5}{6}-\\frac{a^2xr^3}{24}-\\frac{a^4xr}{16}-\\frac{a^6}{16}\\ln\\left(x+r\\right)", "a2fc58597f58caebf80c3921a98482d7": "\\operatorname{E}_{\\tau}[\\ln p(\\tau)]", "a2fcb09e30452f0b432c0d041f294f38": "\\textstyle l \\leq n-k", "a2fd3b2683f470884f2c12aba2f4b747": "x_3 = h = H/2", "a2fd97d57f343c0ad52beae0dc781fee": "{\\mathbf T}^*", "a2fdb7e4f5c926b4dacfb767277655bc": "\\tau,\\tau',\\tau''", "a2fdecb1d90cab71a82bf0074c2bb4bd": "\\omega_\\mathrm{sig}-\\omega_\\mathrm{LO}", "a2fdfc599dbf2aab923f275f87d32720": " F_4 = y, S_4 = \\_, A_4 = n ", "a2fe5ab4390697db9435b1e1f8926d80": "\n\\begin{bmatrix} d^\\prime \\\\ s^\\prime \\end{bmatrix} =\n\\begin{bmatrix} \\cos{\\theta_\\mathrm{c}} & \\sin{\\theta_\\mathrm{c}} \\\\ -\\sin{\\theta_\\mathrm{c}} & \\cos{\\theta_\\mathrm{c}}\\\\ \\end{bmatrix}\n\\begin{bmatrix} d \\\\ s \\end{bmatrix},\n", "a2fe9e39c1edf3aaa539ba3c1fc90eb6": "\ne^z = \\cfrac{1}{1 - \\cfrac{z}{1 + z - \\cfrac{z}{2 + z - \\cfrac{2z}{3 + z - \\cfrac{3z}{4 + z - \\ddots}}}}}\\,\n", "a2fed20264e6b33960434a76a061d953": "\n\\Delta g\\ =\\ \\frac{P}{2\\pi}\\ \\int\\limits_{0}^{2\\pi}\\left(\\frac{\\partial g }{\\partial v_1}\\ h_1\\ + \\ \\frac{\\partial g }{\\partial v_2}\\ h_2\\ + \\ \\frac{\\partial g }{\\partial v_3}\\ h_3 \\right)d\\theta\n", "a2fef264a77394c856f135213a081608": "\\mathcal{S}\\times \\mathcal{A} \\rightarrow \\triangle \\mathcal{S}", "a2ff591d7198df38d520c9ea8a011a8e": "\\mathfrak{g}\\otimes\\mathbb{C}=\\mathfrak{k}\\oplus\\mathfrak{p}^{+}\\oplus\\mathfrak{p}^{-},", "a2ff740427e9165a9f2fdb0174496256": " i^{th}", "a2ff99e9c4a5ef631dc42cb51491894c": "H_1: p=p_1", "a2ffb6c777cb7042550180de08664f54": "\\mathbf{e}_{12}^2 = -1, ", "a2fffeb8e2a4058c34deb53eb45ab374": "\\, B(x,y) = B(y,x) = \\tfrac{1}{2} (Q(x + y) - Q(x) - Q(y)) ", "a30029d82d5370adf83c32550ac2266d": "\\delta_{st}.", "a30034a922b23d6f7af7c56889ad8a4b": "K = \\frac{a+b}{4|b-a|}\\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.", "a300cac4b01b83c1cd1c46fa65c26f7e": " \\sum_{i=1}^{n} X_i \\sim \\textrm{IG}(n \\mu,n^2 \\lambda)\\,", "a300d05af551ec9bb8efb257199286cd": "S(N_j) = -e^{\\lambda P_0}N_j +i \\lambda \\varepsilon_{jkl} e^{\\lambda P_0} P_k R_l\\,", "a301045d7d7f827aaa88a408da4f3fe8": "\\ tan^{-1} ", "a3019ac7394b8e578153b464634a5bcc": "\\lambda=0,1,\\dots, M-1", "a301a68c7a2216000afb40650ab74eb7": "I_\\lambda=B_1 \\theta^{\\frac{3}{2}}e^{-\\frac{c}{\\theta \\lambda^2}}\\lambda^{-6}\\,,", "a301eac084bcabc0703577bae717b8f2": "f'''\\;", "a302530c7586fe0b6a8b205fb3ffffd5": "\\vec{h}_2 = \\frac{1}{r} \\, \\partial_\\theta ", "a302545c6d20afc00d571b44e675abe7": "\\rho=2u", "a3026e320c132de94f7c8ebb952bda60": "\\theta_n", "a302708cd7071a1a124b5f2f5204158f": "p(r,k)=Ae^{\\pm ikr} ", "a302871ece27b8050a1c5b0866c58966": "\\log_{10} F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1}]} = -m_R/2.5 + 23.9/2.5 - 6 - \\log_{10}(3.34 \\times 10^4 \\cdot \\lambda^2 / w) = -m_R/2.5 - 5\\times\\log_{10}(\\lambda) - 0.96", "a302b714c7f1aff1e5737ef0b81f267e": "z\\rightarrow y", "a30300d28a1c2f18f549d7bf4f1e7481": "\\begin{align}\nu^\\prime &= \\frac{4 X}{X + 15 Y + 3 Z} &= \\frac{4 x}{-2 x + 12 y + 3} \\\\\nv^\\prime &= \\frac{9 Y}{X + 15 Y + 3 Z} &= \\frac{9 y}{-2 x + 12 y + 3}\n\\end{align}", "a30307080530cc580646c65bd1e0a72b": "x=\\frac{-b\\pm\\sqrt{b^2-4ac\\ }}{2a}", "a30358adc31cd794576249021b29508a": "f^\\prime (0^-)\\leq f^\\prime (0^+).", "a30374364aae1366e5038acf2c70262e": "\n\\hat{L}(t) = \\Big( \\hat{K}(t)/\\pi\\Big)^{1/2}.\n", "a303ba0a2e727e9597615a560cacc2aa": "{\\bar{Q}}_5", "a303c8fc5e6a234f974ffb751df524e0": "ES_{\\alpha} = \\inf_{Q \\in \\mathcal{Q}_{\\alpha}} E^Q[X]", "a303d9a5e253e71e1a320d43b755b61c": "\\scriptstyle \\frac 1{2k+1}<10^{-10}", "a3040ccdacd3220368008a7a76760cf3": "\np = \\frac{a^3+b^3+c^3+3abc}{a^2+b^2+c^2}\n", "a3046bed26533db320bac4d5acfdcba5": "\\langle\\phi(0)\\phi(r)\\rangle\\propto\\frac{1}{r^{D-2+\\eta}}.", "a304952516f57ce665973f76475486e2": "\\omega_c = K_p K_v \\sqrt{2}", "a304ac2ac1308001584c0ee4abe713e6": "\nt_{\\textrm{lock}}\\quad \\approx\\quad 6\\ \\frac{a^6R\\mu}{m_sm_p^2}\\quad \\times 10^{10}\\ \\textrm{ years},\n", "a304c4e9b5dd6a617f8b57c6413ec639": "v_1 = U + v_1^', \\qquad v_2 = v_2', \\qquad v_3 = v_3'.", "a305bec24a31988801da1e1a498854c4": "\\ a_n = a_m + (n - m)d.", "a305c1191813fb24bc89eddc7945a1e7": "\\frac{\\mathrm{d}n_3}{\\mathrm{d}t} = I_{\\mathrm{in}} + \\frac{n_1}{\\tau_{13}} + \\frac{n_2}{\\tau_{23}} -\n\\frac{n_3}{\\tau_{31}} - \\frac{n_3}{\\tau_{32}}", "a305c5b0fcc465b93c227b6a6432f13b": "\\,_0F_1(;a-1;z)-\\,_0F_1(;a;z) = \\frac{z}{a(a-1)}\\,_0F_1(;a+1;z)", "a30649e3951267fd61a03a04c77fdeba": "m = 32, c = m^N \\mod N = 373", "a30679999b6457e6dfc94d721d4ce4a8": " V \\ = \\ V_1 \\ \\cup \\ \\cdots \\ \\cup \\ V_e. ", "a306d474c611e90e95f79cb590e7adb9": "\\langle f|g\\rangle+\\langle g|f\\rangle = \\langle \\hat{A}\\hat{B}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle+\\langle \\hat{B}\\hat{A}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle = \\langle \\{\\hat{A},\\hat{B}\\}\\rangle -2\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle ", "a3073c039dddb10ab25c76698e38deab": "v_\\text{e}", "a3073eae3f7a6c413e92e600edfb603e": "\n\\operatorname{Var}(X-Y)= \\operatorname{Var}(X) + \\operatorname{Var}(Y) - 2\\operatorname{Cov}(X,Y).\n", "a30755e48cefdf54f0ee55d0c6742878": "{\\left(\\frac{\\mathrm{e}^{it}-1}{it}\\right)}^n", "a307653b167da88da58b7073346bc72d": "r = r_{12}", "a3077ca245ff9b42895a52a8953c7594": "\\scriptstyle \\sqrt{18} \\ = \\ \\sqrt{9}\\sqrt{2} \\ = \\ 3\\sqrt{2}", "a307fe2b958535a70ba55c5d6a7d7a47": "\nu \\equiv \\frac{1}{r} = -\\frac{km}{L^{2}} \\left[ 1 + e \\cos \\left( \\theta - \\theta_{0}\\right) \\right]\n", "a308434cd5a32b37a257438417d8dd2d": "x_\\mathrm{a}(t) = \\cos(\\omega_0 t) + j\\cdot \\sin(\\omega_0 t) = e^{j \\omega_0 t}\\,", "a308893604ada26dedc96eeac2e53e41": "h = \\int_{t_0}^{t} v \\cos \\theta\\, dt", "a3088d223582c007b72f4435ad5381b1": "\\delta_{xy}=0", "a3089151e44409286f54f07078b83129": "f(x) = y,", "a308a7cc589eac6828c76b3ed6405a11": "\\Phi_{Y,X}:\\mathrm{hom}_C(FY,X) \\to \\mathrm{hom}_D(Y,GX)", "a308e577478aba38ec3ad1f4d2a713bf": "a_{\\mathrm{R}} = \\frac{v^2}{r} = \\omega^2r\\!", "a308ffe059a9df392637c08deffbcdac": " \\delta E_1 \\rightarrow 0 ", "a3096576e5908d6fbda9c727d701fb61": "\n u_x(x,y,z) = -z~\\varphi(x) ~;~~ u_y(x,y,z) = 0 ~;~~ u_z(x,y) = w(x)\n", "a30982a66305d3ad6d2e856b396ea99c": "J^\\mu (\\mathbf{r},t) = \\frac{\\partial \\mathcal{L}}{\\partial [\\partial_\\mu \\phi(\\mathbf{r},t)]}\\delta\\phi(\\mathbf{r},t) ", "a3099d95f33939cab04dc084fce1a3bc": "\\varphi(t;\\mu,c)=e^{i\\mu t-\\sqrt{-2ict}}.", "a309d523373ff09f4067390c55743f3c": "A(s) + B(s)=0", "a30a925d31fe3aee26c1cc391ec80023": "= -\\log\\lambda + 1\\,.", "a30ac77af7508a346fa30fb4bd2781f2": "\\displaystyle{f_t(\\lambda(t))=c(t).}", "a30b46fd36674258f218c1c318f10cd7": "Q(4,q) :\\ s=q,t=q", "a30b4bbc0220a703ad32ba4aa1aaf937": "\\mathbb{S}^n_R \\subset C_i", "a30b4e1e85e070047a7f208a3725d0a8": "R_1 = \\frac{V_{S} - V_D}{I_{D} + K \\cdot I_{B}}", "a30b6000c403b7125fe98d9c911aea7e": " |\\bold{r}_{12}| = |\\bold{r}_2 - \\bold{r}_1 | \\,\\!", "a30b6a7722f679fb987e077ad128b19b": "Y = g(X) = \\frac{1}{X}", "a30b76d71c033fed23122a9e6621aa24": "M_\\mathfrak{p} \\ne 0", "a30ba24d47bda66a48c3c2d0b2b79872": " f(\\mathbf{x}_k+\\gamma(\\mathbf{s}_k -\\mathbf{x}_k))", "a30bde98b0a7d5ba376bac7954042a6b": "\\rho \\, |dw|^2 = \\rho |w_{z}|^2 | \\, dz + {w_{\\overline {z}}\\over w_z} \\, d\\overline{z}|^2,", "a30c83df5444c98d2121df0ec5d6ef12": "A^{ab}_\\mu", "a30ca91be97a9cedc0d2426e54b0e782": "Father", "a30cc6ce33d8c71f9d2cfde41e76ba3d": " (x^2+y^2)(u^2+v^2) = (xu-yv)^2 + (xv+yu)^2 \\ , ", "a30cc6e07808e3b9716d2e2580fbe9c2": "\n\\left[ \\begin{array}{ccc|c}\n1 & -1 & 2 & 8 \\\\\n0 & 2 & -1 & -3 \\\\\n0 & 0 & -1 & -11\n\\end{array} \\right]\n", "a30ccc3e21af106bc95f0fdfbab70d9c": "ce(ab)<_s b", "a30d4bf14654c9b30833adb7abe2152b": "\n\\mathcal{H} (P,R,p_s,s) = \\sum_i \\frac{\\mathbf{p}_i^2}{2ms^2} + \\frac12 \\sum_{ij,i\\not= j} U \\left( \\mathbf{r_i} - \\mathbf{r_j}\\right) + \\frac{p_s^2}{2Q} + gkT\\ln\\left( s\\right),\n", "a30d59b3507546090e0a24131deebac4": "\\displaystyle{\\Delta u =f}", "a30d6c84b4ebe3a180ab59483d2aa80d": "T_{i-1,j}", "a30d82bbd10c31d5cc06884e2d59e519": "x_-^\\alpha = (-x)_+^\\alpha", "a30d8590d39f68a267e161cf673d0224": "\\mathit{bar}\\ 1", "a30d9456017e09319686512d8fc0b5ec": "\n\\sum_{k=0}^\\infty a_k z^k\\!", "a30deff363115705fac7be6baa15a716": "J^\\alpha{}_{,\\alpha} = 0", "a30e0a7ee40afd01c289389c8920dff1": " 2g_\\text{threshold}\\,l = 2\\alpha_{0}l - \\ln R_\\text{OC} ", "a30e22786fc0b1f3cf31216ec922b0b0": "g(x) > -\\infty", "a30e759845255f7f8051511196d13ee9": " Y_3 = (T_1Y_1)^2 - (T_1Z_1)^2 + (Z_1Y_1)^2 = 12", "a30ea8e2b2bc2cc5cb0dc37a5005692b": " y_{n+1} = y_n + h\\bigl( (1-\\tfrac1{2\\alpha}) f(t_n, y_n) + \\tfrac1{2\\alpha} f(t_n + \\alpha h, y_n + \\alpha h f(t_n, y_n)) \\bigr). ", "a30eea7db95dcc6dcdb77b11b48bc7cd": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 10.41840\\log_e(T+273.15) - \\frac {5778.024} {T+273.15} + 81.92460 + 1.178208 \\times 10^{-5} (T+273.15)^2", "a30efabe178e4f00e919dc70501b806d": "\\scriptstyle\\mathbf J = -\\sigma S\\boldsymbol\\nabla T", "a30f045f189a507b07031efb53a47f66": "\\mathbf{x}'", "a30f21c1c464df24c7812adcd8cba2fb": "D_n\\left(E\\right)", "a30f3dfaf9f9af094a98d2c64d8b1a0c": "\\gamma_\\text{go}", "a30f758a66154368472be92d73722426": "T_{}^{}", "a30fd6cfa590b0cb311d2d562c49bd37": "a_{14}+b_{15}+c_{13}-a_{14}", "a30ff61eb04119a1ac99e67c7829b401": "w(i)", "a310233dc2086d2823fff7272e7a8929": " 0 = n_0 < n_1 < n_2 < \\ldots < n_{q-1} < n_q = n,\\qquad n_i = \\dim \\ker L^i. ", "a310b0a07402c241cbc0f411af0a1d6e": "m_\\text{e}", "a310fb1241d74c21f181632cea5cd2bb": "\\displaystyle{\\frac{\\left[w+x+y+z+400\\left(2\\right)-400\\left(2\\right)\\right]}{4}}", "a3111d38c97da78deef1e8e12f782463": "-\\left ( \\frac{\\partial u}{\\partial x} \\frac{\\partial \\theta}{\\partial x} \\right ) - \\left ( \\frac{\\partial v}{\\partial x} \\frac{\\partial \\theta}{\\partial y} \\right )", "a311297bc8624d6f8b3beda5042d3e47": "\\rho\\ (x,y)\\,=2x+3y+2", "a311404c709506eefcc8d8cd4d16d89f": "\nC(\\alpha)=\\theta_E^2{(1-\\beta^2)}^2\\int_0^{\\alpha^2}dx\\frac{1}{{(x+\\theta_E^2(1-\\beta^2))}^2}\n", "a3115b36655901c33c8566f3bd18a876": " \\log S = k/e^{2} + \\log S^{0} \\,", "a311e60cf5cfcaec9307d87d26dcfe99": " S_0 = \\frac{2P_0}{\\pi R^2}.\\qquad(6)", "a31229af181500de1b9d4de08c1a0c9a": "n^{1/3}", "a312c2e0131d0ff5adad0d44be6fd0b3": "\\mathbf{\\Pi}_1^1", "a313395d82cfac747581f32466fc1e17": "\\psi = \\sum\\limits_{i}\\psi_ia^i,", "a313b08453bb9595556c0674ae19a6a4": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}~\\boldsymbol{F}\\cdot\\cfrac{\\partial W}{\\partial \\boldsymbol{E}}\\cdot\\boldsymbol{F}^T \\qquad \\text{or} \\qquad\n \\sigma_{ij} = \\cfrac{1}{J}~F_{iK}~\\cfrac{\\partial W}{\\partial E_{KL}}~F_{jL} ~.\n ", "a313db639a2ac1140a82278d34ebb8fc": " S_A = S \\cot A = bc \\cos A= \\frac {b^2+c^2-a^2} {2}\\,", "a3149689fa56972763d9cf11d309ecba": "f'_-(t).\\,", "a314b391d3b3df8442447d9961d600fd": " \\lambda_{i}=\\frac{(a + d) \\pm \\sqrt {(a - d)^2 + 4 b c}}{2}=\\frac{(a + d) \\pm \\sqrt {(a + d)^2 - 4(ad-b c)}}{2}=c\\gamma_i+d \\ .\n", "a314b71c1bc2dd5a6a1f78b524b6f3d0": "w'=C /u^2", "a314c517d4a98fe623f48dd84bd9f0de": "E_\\mathrm{p,m} = \\frac{1}{2}LI^2", "a314d00f70908d046b2591956200c6f7": "\\frac{1}{2^nn!}\\int_0^1(1-z^2)^n\\cos(xz)\\,dz=\\frac{A_n(x)}{x^{2n+1}}=U_n(x).", "a314e70f65bcff72751938059d40ed4d": "(x-1)^2+(1-x^2)=1", "a3152cbe24d2dc1f236e67f4b8a55e54": "\\mathcal{D},\\mathcal{E}\\subseteq\\mathcal{A}", "a3154cc3266a7760280762a7696e1eaa": "\\frac {dA(t)} {dt} = W \\cdot A (t)", "a315e405703cbce4298a08ec5bcc7b11": "H_2O \\rightleftharpoons H^+ + OH^-: K_W'=\\frac{[H^+][OH^-]}{[H_2O]}", "a315eb2717f78044e497e84c3b48229b": "D_k(x) = xP_k(\\log x)+\\Delta_k(x) \\,", "a316229e2290ddc11d423292f3a30398": " \\sum_{k=1}^{n} q_{k}=p ", "a31636563221c5b889bbb811a7a32c6b": "i>8", "a31640b38f358835b214a38b3031a6c5": "\\begin{align}\n b^0 &= {b^{1}}/{b} = 1 \\\\\n b^{-1} &= {b^{0}}/{b} = {1}/{b}\n \\end{align}", "a316c6d5574eb22c2c388bbfda1738e7": "\\cdot e^{-\\frac{\\omega^2}{2 a^2}} H_n\\left(\\frac \\omega a\\right)", "a316ddc2489763c32ce7127a6d1b1eba": "\na_n=\\frac{2-\\delta_{0n}}{N}\\sum_{k=0}^{N-1}\\cos\\left(\\frac{n\\pi\\left(k+\\frac{1}{2}\\right)}{N}\\right)\\log(1+x_k).\n", "a317026015749a134167b1a08429311f": "\nq_{13} = \\frac{q_1 q_2} {q_2 q_3}\n", "a31789b9dd0f04a857b8e4964a12314a": "\\sum,\\ \\prod,\\ \\bigcup,", "a317a195af3e4a487f381fe47a6e1ac5": "\\frac{e^{- \\beta E_m}}{\\sum_n e^{- \\beta E_n}}.", "a317d0594e5b6d5acb667596b1dcff2d": "G(J^\\prime, J^{\\prime \\prime}) = \\bar \\nu _{v^\\prime-v^{\\prime\\prime}}+B^\\prime J^\\prime (J^\\prime +1)-B^{\\prime\\prime} J^{\\prime\\prime}(J^{\\prime\\prime} +1)", "a3180f29760f4a863e4ce74966afa472": "C(\\theta)=B(p,w)=\\{x \\,|\\, px \\leq w\\}", "a318149479b617199ed7c1be77c7af5b": "\\scriptstyle \\log(e^x) \\,>\\, \\log(1 \\,+\\, x)", "a3183aa34749680f427e3b55f7487d6d": "\\sum_{k=\\alpha}^n {n \\choose k} (-1)^{k} f(k) = \n-\\frac{1}{2\\pi i}\n\\oint_\\gamma B(n+1, -z) f(z)\\, \\mathrm{d}z", "a31842a5ed31c1c416fb78f5e493d06a": "t_2 = Y^c h_{1}^{s_1}h_{2}^{s_2}", "a318729ca49ebe6a62d6a59517bf0e4f": " u=\\frac{x+t}2, \\quad v=\\frac{x-t}2, ", "a318784d7bf31d3a5564973eebcdffe5": "\\mathrm{rad}_t(p^e) = p^{\\mathrm{min}(e, t - 1)}", "a3194ae75b540310886e76e6b049a21d": "x = \\gamma^2 x - \\gamma^2 v t + \\gamma v t' \\,", "a31955fdf220c939daaef0cd23ebc1c1": "T = t(\\theta_0\\rightarrow0\\rightarrow-\\theta_0\\rightarrow0\\rightarrow\\theta_0),", "a319b28b6be09b0ed288be9bb9e9cb5f": " V_{\\text{out}} = -\\frac{R_{\\text{f}}}{R_1} ( V_1 + V_2 + \\cdots + V_n ) \\!\\ ", "a319b656a32d20a986e4075a3d767fb1": "\n\\left( \\frac{1}{\\lambda_{1}} J \\right)^{k} = \n\\begin{bmatrix}\n[1] & & & & \\\\\n& \\left( \\frac{1}{\\lambda_{1}} J_{2} \\right)^{k}& & & \\\\\n& & \\ddots & \\\\\n& & & \\left( \\frac{1}{\\lambda_{1}} J_{m} \\right)^{k} \\\\\n\\end{bmatrix}\n\\rightarrow\n\\begin{bmatrix}\n1 & & & & \\\\\n& 0 & & & \\\\\n& & \\ddots & \\\\\n& & & 0 \\\\\n\\end{bmatrix}\n", "a31a0a3eb36aa0abb4cb03273b8ebdec": "\\begin{matrix}{4 \\choose 1}^4\\end{matrix}", "a31a470a7fa7e5168987ec1f72ccdede": "\\hat{S_z}", "a31a4e2fe7da5f492797b0bbdb2fc52b": "\\Gamma^{\\sigma}{}_{\\mu \\nu}", "a31a860e7a59c7616c1515ec3ae652a6": "k+1", "a31aaee7a30183ceaf1a1e8380d4d1a2": "\\prod_{m=1}^\\infty\n\\left( 1 - x^{2m}\\right)\n\\left( 1 + x^{2m-1} y^2\\right)\n\\left( 1 + x^{2m-1} y^{-2}\\right)\n= \\sum_{n=-\\infty}^\\infty x^{n^2} y^{2n},\n", "a31af60d28cd1609423041b617df998e": "\\forall x \\in \\mathbb{S}\\ P(x)", "a31b2322cc7a76d0a9e4af89a783d334": "\\dot v=0", "a31b655bb164e04ce7cd9e8504d47b0e": "rate=\\frac{-d[COOH]}{dt}=k[COOH]^3", "a31b823cc301853eb3b7a650d48b4b0a": "p_x(i)", "a31ba938d80a25d0afa7a0e6a5b47acc": "H^n(X) = Z^n(X)/B^n(X), \\, ", "a31beb52a970e1b3dbc4164083d064de": "N=N_{\\hat{B}} +N_{\\hat{a}}", "a31cca132c5e95b4bdf7de231b1a9c90": "\\lim_{n \\rightarrow \\infty } \\!\\! \\left( \n\\sum_{p \\leq n} \\frac{1}{p} \\! - \\ln(\\ln(n))\\! \\right) \\!\\! =\n\\underset{\\!\\!\\!\\! \\gamma: \\, \\text{Euler constant} ,\\,\\, \np: \\, \\text{prime}}{\\! \\gamma \\! + \\!\\! \\sum_{p} \\!\\left( \\! \n\\ln \\! \\left( \\! 1 \\! - \\! \\frac{1}{p} \\! \\right)\n \\!\\! + \\! \\frac{1}{p} \\! \\right)}", "a31d51fab1bdf61a7795ce14d3fdf6f9": "\\frac{1}{A_{1}^{\\alpha_{1}}\\cdots A_{n}^{\\alpha_{n}}}=\\frac{\\Gamma(\\alpha_{1}+\\dots+\\alpha_{n})}{\\Gamma(\\alpha_{1})\\cdots\\Gamma(\\alpha_{n})}\\int_{0}^{1}du_{1}\\cdots\\int_{0}^{1}du_{n}\\frac{\\delta(\\sum_{k=1}^{n}u_{k}-1)u_{1}^{\\alpha_{1}-1}\\cdots u_{n}^{\\alpha_{n}-1}}{\\left[u_{1}A_{1}+\\cdots+u_{n}A_{n}\\right]^{\\sum_{k=1}^{n}\\alpha_{k}}}\n .", "a31d5c9cee95d5b0a458d1636280f5a1": "{\\rm{i}}K'(m) = {\\rm{i}}K(1-m).\\,", "a31d8d92a5c9e746b143fadfcb7d5dbe": " D_m^{(s)} ", "a31dae09ed6d513022029509ac4112d3": "\\sum_{i=1}^N c_i \\sigma^i \\,", "a31df4a4a6ebdbf450d01f62b17b4e81": " d\\Omega^2= d\\theta^2+\\sin(\\theta)^2 d\\phi^2", "a31e486e084ca7f49726385b8984a957": "\\begin{smallmatrix}d_S = {\\left ( 0.197 AU \\right )} {\\left ( {\\frac {149,597,871 km}{696,000 km}} \\right )} = 42.31 R_{\\odot} (rounded)\\end{smallmatrix}", "a31e55575ffbbe5bea8d14f94c36a67e": " {{LD} \\over {DS}} = {{GD} \\over {LD}} ", "a31e5c85f0b498938758bdbdff139714": "(x-3) (x-2)^8 (x+1)^7 (x^2+2 x-1)^6", "a31e682beea146e9825697aa9d2988ee": "f:A\\rightarrow\\mathbb{R}\\,", "a31e8d76922120d8b7d0de072a5b1449": "J_{\\hat{n}} \\equiv i \\hbar \\lim_{\\phi\\rightarrow 0} \\frac{R(\\hat{n},\\phi) - 1}{\\phi}", "a31f08324881f00e812675c195271c1a": " ~\\epsilon_t ", "a31f224dfde7e6686de54207d453e6a4": "\\overline{n_\\text{piv}} \\in n_\\text{clause}^\\text{right}", "a31fd86411ee7ff8872ae4e330d6741d": " A^\\mu = {R^\\mu}_{\\nu\\rho\\sigma} T^\\nu T^\\rho X^\\sigma.", "a31ff1426af070e0970dbc6054cbf273": "A^\\alpha", "a3206fb68a2f7cfa475d9b3551bb4c4f": "J_2 = 108\\,263\\times10^{-8}", "a32098fc752a7cf0bdbee799c86f1b9d": " A = \\sum_{n=1}^{N} A_n \\,\\!", "a320ce8b36e820165ca2830a16085445": " =2^{-n\\left[ 1-H\\left( \\mathbf{p}\\right) -k/n-3\\delta\\right] }.\n", "a320d359faf6d42e9c2f1a974651b246": "\\hat{h}_k", "a3210f81ebf253c8f7d165fe78b31ae4": "\\Delta V=A\\times ln\\left(\\frac{i}{i_0}\\right)", "a32135b42338327aae04f17fcdc63f03": "\\begin{smallmatrix} {\\delta} = \\frac{d_S}{D_S}\\end{smallmatrix}", "a32162248a5f496aaa7bc7adb05326e7": "\\Pr\\left(\\bigcup_i \\Omega_i\\right)=\\sum_i \\Pr(\\Omega_i)=\\sum_i\\Pr(X=u_i)=1.", "a3216f2e47394158ebb581842d1f00a9": "\n|K(a,b;m)|\\leq \\tau(m) \\sqrt{\\gcd(a,b,m)} \\sqrt{m}.\n", "a32187f7c3bfea5dc8318e8d4d0f6aef": " \\beta=b_0+b_1\\mathbf{i}+b_2\\mathbf{j}+b_3\\mathbf{k} ", "a321b2f55dc895ec05c8327223f8e8c8": "\\operatorname{cov}(W_{t_1}, W_{t_2}) = E \\left [W_{t_1}^2 \\right ] = t_1.", "a3220c85e5d8738e533e7d95726662a2": "I \\subseteq P \\times L", "a32221b88fc22b9b6eb2a9249e58132a": "\\mathcal{\\tilde{H}}_{S}", "a3224cd5533c4417a42b652118ebfd4b": "\nF^T\\mathbb{I} = \\mathbb{I}\n", "a3228616e0b1a3498d6877a7fd351a82": "f(ax+by)=\\bar{a}f(x)+\\bar{b}f(y)", "a322c621101b6f71b9ceebe84e2aae48": "i\\le j", "a322cf6b5c413cd4df6c36fbddffafec": " \\Delta E=E_{0}\\left(\\frac{w}{2R_{0}}+\\frac{\\alpha ^2}{4}\\right) ", "a323ac8239b889533e6d9c657631966c": "W_J", "a323b6b3b85e0b0b9836b68e2e3f6461": "\n\\begin{align}\n& {} \\qquad ((2+2)+(2+2))+(3+3) \\\\\n& {} =((2+2)+(2+2))+ 6 \\\\\n& {} =((2+2)+ 4)+6 \\\\\n& {} =(4+4)+6 \\\\\n& {} =8+6 \\\\\n& {} =14\n\\end{align}\n", "a323d51f7fa57a0a9d424ec887be440b": "A=(Q, \\Sigma, I, F, R, \\delta)", "a323daa3df5d33b962c1039490f5f8c7": "\\ell\\neq p", "a324060c627082a469f34defd8027b4f": "\\lambda_1 = i \\sqrt{\\alpha \\gamma},\\quad \\lambda_2 = -i \\sqrt{\\alpha \\gamma}.\\,\n", "a325259d6ded8a13246af8d56d10387f": "\\frac {dk/dt}{k} = g_k=g_q=s(1-u)(q/k)-\\delta.", "a3252d3535ecf87dcd8cfa4562c8665d": " F^{\\mu\\nu}= \\partial^\\mu A^\\nu - \\partial^\\nu A^\\mu ", "a3256ae843b783ce5a002170db5e1537": "A = \\begin{pmatrix} 0_{r,r} & B \\\\ B^T & 0_{s,s} \\end{pmatrix},", "a3258a4dd75faba43c295ac4f9d3b0d4": "{\\mathbf{k}}", "a3258bf337092e88426c44a486f3f254": " 2\\left(r - r_0\\right) - vt = v_0 t \\,\\!", "a325bafbedfbaf845b59aa3d40a880e7": "\\tilde{\\nu}_e", "a32628930982e46c4d8fb393359b7e4c": "\\alpha = 2 \\theta_{min} = \\frac{2\\lambda}{W}", "a3263e4b07fa07a5a9f17eef7f99c2fb": "h (0, x_1, \\ldots, x_k) = f (x_1, \\ldots, x_k) \\,", "a326636b625a9076b17cec968a8bae5a": " \\mathbb{T}^\\infty=[0,\\infty) \\cup \\{ \\infty \\}", "a3266d789d986685c71312e3bbeadde2": "s,t\\in I", "a3267561040fd83972e1faf6912c8cea": "e>\\frac {rc} {y-c}", "a326ac8e7a9f7b7dde1d16393ead28aa": "CK=(CK_0,\\ CK_1,\\ \\ldots,\\ CK_{31})", "a326cbe563995aa37211438f4dbdf12b": "\nI_A = T \\times P \\times R\n", "a326e3a90d5e1d05eda3015208b7a307": "a + b + c", "a3271c7a2b92f26f3d9b7b0beb1ed24e": "x^{\\mu} = \\delta^{\\mu}_{K} \\,", "a32743f29888430930fcd05258682110": "\\rho_{H}=f(T)", "a3277c2281bcb806732a887a2114dd52": "U = n S^2 / \\sigma^2 \\sim \\chi^2 _ {n-1} .", "a3277ce9a7ea7b7cc6cf9a3330b4ec06": "= 30.8", "a3279382ed2263e1c9a6d77451b161d6": "\n(1+x)^{s/t} = \\sum_{n=0}^\\infty \\frac{\\prod_{k=0}^{n-1} (s-kt)}{n!t^n}x^n\n", "a327cb7972e84fa0543f55fc5eb5af1e": "\\log_{10} p = A-\\frac{B}{C+T}.", "a327f7f3b91bcf02cdd131ce01410e70": "\n\\widehat \\beta_{FGLS1} = (X'\\widehat{\\Omega}^{-1}_{OLS} X)^{-1} X' \\widehat{\\Omega}^{-1}_{OLS} y\n", "a3285e9581326d49273bc9089218ba18": "p_X(x|a) = \\frac{a}{\\pi (a^2 + x^2)}", "a3286a51bd3498a411a5f83303090336": " \\Leftrightarrow V_\\mathrm{Mg}'' + V_\\mathrm{O}^{\\bullet \\bullet}", "a3287060654de7ac353af44997b99dd3": "{} \\vdash {}", "a3287a0316eb7aa0c7b75c82d2774e22": " \\gamma = (\\sqrt{\\nu^2 + 2r} - \\nu) / \\sigma", "a3287fa319bc6de5b5c69db370a6183b": "m\\times m", "a3288e96552167174ea5caafb8ede7d2": "C^{op}\\times D\\to\\mathbf{Set}", "a328fa1fcc192e710656d19ee9d8853e": "\\langle \\rangle,", "a32916fc540b90cacc52374c5e2dbe0e": "\\mathit{foo}", "a3292039d3a5fafce4cad5f0331cf4ea": "n_1 = \\sqrt{n_0 n_2}", "a329552e677cc60a841cf8d965067ab2": "\nx_n = b_0 + \\underset{i=1}{\\overset{n}{\\mathrm K}} \\frac{a_i}{b_i} = \\frac{A_n}{B_n} = \\boldsymbol{\\Tau}_{\\boldsymbol{n}}(0) = \\boldsymbol{\\Tau}_{\\boldsymbol{n+1}}(\\infty)\\,\n", "a32999104f088bdbd343b925c7a66e35": "H(u)[n] = u[n] * h[n]", "a32a72e73b7a84f60db99dd6de4e8621": "(\\mathbf{e}_r, \\mathbf{e}_\\theta)", "a32ab9264210ff864f666fafaecae002": "(m_L, m_{U}) = \\frac{1}{(1-g)} \\left[\\frac{a}{b} - \\frac{g\\nu_{12}}{\\nu_{22}} \\mp \\frac{t_{r,\\alpha}s}{b} \\sqrt{\\nu_{11} - 2\\frac{a}{b}\\nu_{12} + \\frac{a^2}{b^2} \\nu_{22} - g\\left(\\nu_{11} - \\frac{\\nu_{12}^2}{\\nu_{22}}\\right)} \\right]", "a32ad7474513b493c29cbc0dd03690f3": "k_n = \\frac{n\\pi}{a}", "a32af8cbfd109030c4e79634a3beaace": "H\\le 1", "a32b06c563a79cdadc3b370f8c45978e": " \\sum\\limits_\\alpha \\left\\langle {{\\sigma _\\alpha ^2 (0) }} \\right\\rangle = (n-1) \\frac{1}\n{{\\beta J}}\\int\\limits_{}^{1/a} {\\frac{{d^d k}}\n{{(2\\pi )^d }}\\frac{1}\n{{k^2 }}}.", "a32b682a804d57ccabd5448f2e97bb69": "{\\vec k}(t) \\equiv \\int_0^t {\\vec G}(\\tau)\\ \\mathrm{d}\\tau ", "a32c3d52fcfada3f410d4483e39a3d02": "\\scriptstyle{Z_{ij}\\,= \\,Z_{ji}}.", "a32c8ecbf10bc76424c3d221477f79aa": " (a_1,b_1) + (a_2,b_2) = (a_1 + a_2,b_1 + b_2), ", "a32ca572759747e59f9a5cdc93f223e4": " S = \\int {1 \\over 4}F^2 + {1 \\over 2}(D\\theta)^2 = \\int {1 \\over 4}F^2 + {1 \\over 2}(\\partial \\theta - He A)^2 = \\int {1 \\over 4}F^2 + {1 \\over 2}(\\partial \\theta - m A)^2", "a32ca9e22d42184b6af004feea41cb41": "\\gamma = 1", "a32cbeca84a47fdce8331ff64ca6511f": "1-s=\\frac{(a_1a_2+b_1b_2)^2}{(a_1^2+b_1^2)(a_2^2+b_2^2)}\\,", "a32d4cd0315c23a62635fe9bb54816d0": "(K_b\\varphi \\land K_a(\\varphi \\Rightarrow \\psi)) \\Rightarrow D_{a,b}\\psi", "a32da3089375ad3fd8f740d6dffcd4f1": "g(v) = r(e^v)", "a32e021ec29ca3597f63405d6ac281c3": "k\\sigma", "a32e3ead951974699e76827df6f92f93": "r^2 - 2 r r_0 \\cos(\\varphi - \\gamma) + r_0^2 = a^2.\\, ", "a32e6d8fa0f68956b57a2ad04c629704": "(1+\\delta)(1+\\epsilon) = 1 + \\delta + \\epsilon + \\delta\\epsilon \\approx 1 + \\delta + \\epsilon,", "a32ebc351929ca88befd03c364987eb2": "\\begin{align}\ny(k)& = h_0+\\sum\\limits_{m_1=1}^M h_1(m_1)u(k-m_1) + \\sum\\limits_{m_1=1}^M \\sum\\limits_{m_2=1}^M h_2(m_1,m_2)u(k-m_1)u(k-m_2) \\\\\n& {}\\quad{}+\\sum\\limits_{m_1=1}^M \\sum\\limits_{m_2=1}^M \\sum\\limits_{m_3=1}^M h_3(m_1,m_2,m_3)u(k-m_1)u(k-m_2)u(k-m_3) + \\cdots\n\\end{align}", "a32ebc4ea7cecbec66ee892eececd51b": "(7.c)\\quad \\frac{1}{\\rho}\\,\\gamma_{,\\,\\rho} =\\,\\psi^2_{,\\,\\rho}-\\psi^2_{,\\,z}-e^{-2\\psi}\\big(\\Phi^2_{,\\,\\rho}-\\Phi^2_{,\\,z}\\big) ", "a32f12a6d446ee4c9b14187b30c65d16": " \\varphi_X(-it) = M_X(t). ", "a32f89936d6ad9853a8e9aee1b8b9988": "P (D \\to (A \\wedge B \\wedge C))", "a32fb82707d24b0c133094c68a61b81d": " \\mu \\left(T,P,d\\right)= mu_{l}\\left(T,P\\right) + \\frac{\\Delta \\gamma}{n_{l}} \\frac{\\partial f\\left(d\\right)}{\\partial d} = \\mu_{1} ", "a3304d30d467b568e70c36e70abdc6a5": "\\mathrm{d}S_\\varphi=r\\,\\mathrm{d}r\\,\\mathrm{d}\\theta.", "a33083a1702f30b91f747be9f601899e": "{{\\rm SE} = \\sqrt{\\dfrac{1}{n_{11}} + \\dfrac{1}{n_{10}} + \\dfrac{1}{n_{01}} + \\dfrac{1}{n_{00}}}}", "a3310b02dcc57950c4ba3ac822f854bd": "\\bar{\\alpha} = \\arg\\left(\\frac{1}{n}\\cdot\\sum_{j=1}^n \\exp(i\\cdot\\alpha_j)\\right) ", "a3310f8590236f61cf8260c359e48b5b": "\\frac{d}{dt}A(t)=\\frac{i}{\\hbar}[H,A(t)]+\\frac{\\partial A(t)}{\\partial t},", "a33117ca1505fa60f2b4d20a1f04b0c2": "\\nabla^2 V=0", "a331328c1488c13ef46d97c8d0345c21": " F = x, E = \\lambda q.f\\ (q\\ q) ", "a331b9d3970b6fd00a8a4676603c39a7": "H(A)=\\frac{1}{2}(A+A^\\dagger)", "a331ef9b6d7f616456d408285a9a042f": " ( -e a\\ ,\\ 0)", "a3329be3fea84f0cf057a449b26e7884": " C_i^m (r)=(\\text{number of } x(j) \\text { such that } d[x(i),x(j)] < r)/(N-m+1) \\, ", "a332c69271a2fef1f21cf13d9afb1725": " G(\\tau)= \\sum e^{iaf(x)+ia\\tau n} ", "a332cf3cfa7c0ee62699907e87325674": " z_2(x,y)=2(y-x)G(x)+(y-x)^2G'(x)", "a332dcbdd004f6d8d24504be043a130b": "\\delta_s=1/\\sqrt{\\pi\\mu\\sigma f}", "a33313f9bf80f15da144406fd689e8f1": "Y(w_j) = {\\mu-j+1\\over 2}w_{j-2}", "a333295501de7ecee5d3b7866127500e": "H_i = E_{g(H_{i-1})}(m_i)\\oplus m_i.", "a3333a401829f2eff1de0c68af750ba6": "2M_{\\alpha\\beta} = \\delta_{\\alpha\\beta} M_{\\gamma\\gamma} + \\sum_i \\sigma^i_{\\alpha\\beta} \\sigma^i_{\\gamma\\delta} M_{\\delta\\gamma}", "a3334bbee187ca91dcaa40372cff6b93": "\\overline{D}_{\\dot{\\alpha}}f=0", "a33394a12879f955f428478cbafd71af": "v=\\sqrt{\\frac{2eU}{m_0}}", "a3339e591233e0920a60ff93225aebfc": "\\rm{Sym}^d V", "a333abbe7d8309724986d279bb886fb0": "f(x)=h(x,g(x))", "a333be460a9a33f65db2a3c57147288a": " \\mathbf{k} = [k(\\mathbf{x}_1,\\mathbf{x}'),\\ldots,k(\\mathbf{x}_n,\\mathbf{x}')]^\\top", "a333e1191d60707c2b7c396388dc06ea": "J = \\frac{\\text{tensile strength of external thread material}}{\\text{tensile strength of internal thread material}}", "a33420c50877aa50a9e77a44ee6e176b": "\\Omega(\\delta) = \\sqrt{\\delta^2 + (\\kappa E_0)^2}", "a334e486ba2de57aecb3418b781df711": "r,f:\\mathbb R\\to\\mathbb C", "a335258dc1bd28bd08a5680c8453e8ff": "\n ad - da = cb - bc,~~~ \\text{(cross commutation relation)}.\n", "a3353e7a02340e57e1cb8fb33fc609ff": "p(x) \\in \\mathbb{C}[x]", "a335430b553bee79a392f8e17ba9ab85": "V(t;T) = \\int_t^T f(u) b(t;u)\\,du.", "a3359686300b671a946e52996fa84752": "[f]'", "a335dbc09889946fe1c807e2228f76b0": "2^{1-s}\\operatorname{Li}_s(z^2) = \\operatorname{Li}_s(z)+\\operatorname{Li}_s(-z).", "a335f28bdfd9ca3f7f601e226e1aa33b": "\n \\sigma_{xx}^{\\mathrm{face}} = C_{11}^{\\mathrm{face}}~\\varepsilon_{xx}^{\\mathrm{face}} ~;~~\n \\sigma_{zx}^{\\mathrm{core}} = C_{55}^{\\mathrm{core}}~\\varepsilon_{zx}^{\\mathrm{core}} ~;~~\n \\sigma_{zz}^{\\mathrm{face}} = \\sigma_{xz}^{\\mathrm{face}} = 0 ~;~~ \\sigma_{zz}^{\\mathrm{core}} = \\sigma_{xx}^{\\mathrm{core}} = 0\n ", "a33612df71936c40753ed0d706e397dd": "\\scriptstyle c(x,\\eta)=0 ", "a33681855a1f3448d3eafaeb29837dca": "N = \\frac{c - b}{\\Delta x} ,\\, M = \\frac{d}{\\Delta t}", "a336ef3b98919e9c6da3ec98427059f5": "f:\\Omega\\rightarrow\\overline{\\mathbb{R}}", "a3374e269a97123e06bef07809dd824c": " \\phi = \\frac{1}{2} \\arcsin \\left( \\frac{gd}{v^2} \\right) ", "a3377246e8ef90e3be102cd82f8bfe9b": "\\sigma\\Gamma(1-\\gamma)\\Gamma(1 + \\gamma)", "a337d149075318eaff1563cf8443c6e7": " \\varphi(x) = f(x) + \\lambda \\int \\limits_a^x K(x,t)\\,\\varphi(t)\\,dt. ", "a33858d1c36ce3a28a87e538ad27730d": "A_i A_j = \\sum_{k=0}^d p_{ij}^k A_k . ", "a33878bde07f8154dc96ca42325c5bb9": "\\mathbf A^T", "a338904ea4ffd8fb7ee0c8d4ace21b18": "\\operatorname{Int} E_i \\cap \\operatorname{Int} E_j=\\emptyset", "a338ff2d43184145957d437336996148": " \n-V d\\Pi + N_+ d\\mu_+ + N_- d\\mu_- = 0\n", "a3393b8fc4acc7a4bfdad59bcdeb0ed3": "0/0 + x = 0/0\\ ", "a339a81c5dece305bd16ff4639eb7e76": "\\zeta_2\\;", "a339af3a5212ad184dac6385075fac11": "{v}=0.25.{g^{-0.5}}.{SL^{1.67}}.{h^{-1.17}}", "a33a417c5f056ba15854ca1ae4482987": "\\Rightarrow^{ac}_{h} SAAA \\Rightarrow^{ac}_{h} SSAA \\Rightarrow^{ac}_{h} SSSA \\Rightarrow^{ac}_{h} SSSS \\Rightarrow^{ac}_{k} SSSS", "a33a5bf5ac45964652c40d7100b50c82": "(\\mathbb{T}, \\curvearrowleft)", "a33ae6e3a9d2621ffc709af5aa79ca1f": "v^2=\\dot y^2-2 \\ell \\dot y \\dot \\theta \\sin \\theta + \\ell^2\\dot \\theta ^2.", "a33af4df5d119de0d63e008186093c36": "E:\\mathcal{B}\\otimes\\mathcal{B}\\to\\mathcal{B}", "a33affecff4f811e9f330a163077c115": " L(F)=\\frac{\\int_0^F x(F_1)\\,dF_1}{\\int_0^1 x(F_1)\\,dF_1} ", "a33b0d607772845803df709cc23046e3": "({I}^{2},{\\varphi}_{\\lambda},{S}_{\\lambda})", "a33b65d72dfef8e951bfe6ed86834ceb": "\\zeta(s) = 2^s\\pi^{s-1}\\ \\sin\\left(\\frac{\\pi s}{2}\\right)\\ \\Gamma(1-s)\\ \\zeta(1-s).", "a33b8ca44ded30abf82b7dca332e9a96": "(a,b]", "a33c16143501e7968c9f79aeb478e775": "d^D k", "a33c22990dd1b02a4940c443a44ec8e4": "\\Upsilon", "a33c547c9cb8077ff3df846cc83ed212": "x = r \\cos(\\lambda) / \\cosh(m (\\lambda-\\lambda_0)),\\,", "a33c85ae8a0a18582622bf45a11be7cf": "[7,4,3]_2", "a33cc620483d3885822d5d2eda4cbcc0": "J = \\lim\\limits_{A \\rightarrow 0}\\frac{I(A)}{A}", "a33cca0cec553384963b79cf2ad39c84": "\\langle x,x\\rangle=0", "a33d28ed569ed9857816ef92a73b082e": "T(X,Y)", "a33d314444d746f4c9f2abecaede8e7a": " \\phi={p}/{p_0} ", "a33e26d2f4110ad5cc748b7c34cf86ee": "\\mathbb{R}/(P\\cdot\\mathbb{Z})", "a33e6baa852502387fa46b4cbfbcd239": "a^d\\equiv 1\\mod n", "a33ecb9a9b6a3bb36f196c1e1cf4650e": "\\widehat{\\mathbf{a}}^{\\dagger}", "a33f08407294ccdc45ec453e6061f4cd": "\n\\mathcal{L}=\\frac{1}{16 \\pi G_{(n)}}(R-2\\Lambda) \n", "a33f08d02262b834dc67a49c655280cc": "SU(n,1)", "a33f0d6bbfcda1193dff111539cb1387": "p_{02}", "a33f2b7477db21c60b6e3e838abdd45b": "B_y = \\{x \\in X: (x,y) \\in B\\}", "a33f556418bb5cb8df5aae3bbfeb1355": "M=\\sup_i \\{\\|f(e_i)\\|\\},", "a33fc313a7f440541d4066ae47e8c8b9": "[z_{/\\cong_{\\mathcal{B}}}]_Y = \\{z\\in z_{/\\cong_{\\mathcal{B}}}\\mid z\\in Y\\},", "a34004d6bfb32388f3b6b2a72692039c": "Y_D \\colon D\\to\\hat D", "a34019b6124c427b70538400e60ea4eb": "\\mathcal{L}_n(f) = \\Delta^n f (\\beta n)", "a34058f70b0ec6a0f4179cc66060cd2e": "A(x,y)", "a3409eb0948383328c820e855bb0e64e": " {\\lVert x_{k-1}-x_k \\rVert}=\\langle x_{k-1}-x,Z_k\\rangle ", "a340e52ad9dbb8b13909e776415ed1da": "M((1-\\lambda) y_1 + \\lambda y_2) \\geq M(y_1)^{1-\\lambda} M(y_2)^\\lambda,", "a340efb0087c6460757eb48135f1dc8c": "\\|V\\|_2=\\|V^{-1}\\|_2=1", "a340fd31ae3f6a7d3c73d4deeb24cc15": "\\frac{P_t}{P_{amb}} \\leq\\left(\\frac{P_{t_1}}{P_{t_2}}\\right)_{M_1=M_m}", "a34134b26bdb26dfa79dee806cc764be": "c=\\sum_{(c)} \\varepsilon(c_{(1)})c_{(2)} = \\sum_{(c)} c_{(1)}\\varepsilon(c_{(2)}).\\;", "a34154c4c6b674c27f5e5ab39911d62f": " \\operatorname{fact}\\ n = (\\operatorname{IsZero}\\ n)\\ 1\\ (\\operatorname{multiply}\\ n\\ (\\operatorname{fact}\\ (\\operatorname{pred}\\ n))) ", "a341be79c5d446894ecf2bb64b858561": "\\begin{align}C\\left(T_{\\mu\\nu}-\\frac{1}{2} g_{\\mu\\nu}T \\right)&\\simeq C\\left(T_{\\mu\\nu}-\\frac{1}{2} g_{\\mu\\nu} T \\right) \\\\ & \\simeq C \\left[ \\begin{pmatrix} \\rho_0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 \\end{pmatrix} - \\frac{1}{2}\\begin{pmatrix} \\rho_0 & 0 & 0 & 0\\\\ 0 & -\\rho_0 & 0 & 0\\\\ 0 & 0 & -\\rho_0 & 0\\\\ 0 & 0 & 0 & -\\rho_0 \\end{pmatrix} \\right] \\\\ & \\simeq \\frac{C\\rho_0}{2} \\delta_{\\mu\\nu} \\end{align}", "a341f86bd89f96ad21661cce8918449a": "A_6,", "a3429297405527e1de47fd37ce626cb0": "V \\sin i=10-100km s^{-1}", "a342971f1f4c6ea2fd480532f667d28a": "\nP(x(t)) = \\sum_{n=1}^Nc_nx_n(t) \n", "a342f8b6835bce1b61811f7504be1cfe": "[y,z]\\subseteq B_{\\delta}([z,x]\\cup[x,y]),", "a342fd730e6310f279473567cb6022ed": "\\mbox{slope} = 100% \\cdot \\tan( \\mbox{angle}),\\, ", "a3434ec18111a7ce0ac1a6abdf7d294a": "(U,m).", "a3436724b5b0169498c86aae30119230": "w(z)=\n\\left(\\frac{z-a}{z-b}\\right)^\\alpha \n\\left(\\frac{z-c}{z-b}\\right)^\\gamma\n\\;_2F_1 \\left(\n\\alpha+\\beta +\\gamma, \n\\alpha+\\beta'+\\gamma; \n1+\\alpha-\\alpha';\n\\frac{(z-a)(c-b)}{(z-b)(c-a)} \\right) \n", "a343891bac6938a1a716b0cc465b7e86": " \\operatorname{isnil} \\equiv \\operatorname{first} ", "a343a299fa6ef0d8aea7ea6c689cb52f": "\\hat{\\Pi}_{n}", "a343ab8518dcceb29d31494077aedbbb": "\n\\begin{align}\n\\int \\frac{\\delta J}{\\delta\\rho(\\boldsymbol{r})} \\phi(\\boldsymbol{r})d\\boldsymbol{r} \n& {} = \\left [ \\frac {d \\ }{d\\epsilon} \\, J[\\rho + \\epsilon\\phi] \\right ]_{\\epsilon = 0} \\\\\n& {} = \\left [ \\frac {d \\ }{d\\epsilon} \\, \\left ( \\frac{1}{2}\\iint \\frac {[\\rho(\\boldsymbol{r}) + \\epsilon \\phi(\\boldsymbol{r})] \\, [\\rho(\\boldsymbol{r}') + \\epsilon \\phi(\\boldsymbol{r}')] }{\\vert \\boldsymbol{r}-\\boldsymbol{r}' \\vert}\\, d\\boldsymbol{r} d\\boldsymbol{r}' \\right ) \\right ]_{\\epsilon = 0} \\\\\n& {} = \\frac{1}{2}\\iint \\frac {\\rho(\\boldsymbol{r}') \\phi(\\boldsymbol{r}) }{\\vert \\boldsymbol{r}-\\boldsymbol{r}' \\vert}\\, d\\boldsymbol{r} d\\boldsymbol{r}' +\n \\frac{1}{2}\\iint \\frac {\\rho(\\boldsymbol{r}) \\phi(\\boldsymbol{r}') }{\\vert \\boldsymbol{r}-\\boldsymbol{r}' \\vert}\\, d\\boldsymbol{r} d\\boldsymbol{r}' \\\\\n\\end{align}\n", "a343cb88829c85c5c3049ed71c73cf29": " \\frac{1}{M_{\\mathrm{Pl}}^2 r^2} = \\frac{1}{M_{\\mathrm{Pl}_{3+1+\\delta}}^{2+\\delta}r^2 n^{\\delta}} \\Rightarrow ", "a343f56a9af730b9332839da561b561a": "\n\\begin{align}\n \\Delta ( \\Delta X_t ) & = \\Delta X_t - \\Delta X_{t-1} \\\\\n \\Delta^2 X_t & = (1-L)\\Delta X_t \\\\\n \\Delta^2 X_t & = (1-L)(1-L)X_t \\\\\n \\Delta^2 X_t & = (1-L)^2 X_t ~.\n\\end{align}\n", "a344af13cb869de455512e708deef0a6": "y=2qx-q^2.", "a34510f7cd5261e11a5b2651f305d51c": "\\tilde{\\mathcal{M}}", "a34525a338eb0974a34911e6af245b86": " s_0 = s_1 = 1 ", "a34534d7885eca9fdbe3cc9bd6c60956": " 1 = b \\cup b^\\perp ", "a34535209e46f319a24f7f8fcc002a0b": "C_x(t, f)=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}A_x(\\eta,\\tau)\\Phi(\\eta,\\tau)\\exp (j2\\pi(\\eta t-\\tau f))\\, d\\eta\\, d\\tau,", "a345620ba3a28c0637e1258745b37f5b": "\\nabla \\cdot ( \\nabla \\times \\mathbf{A} ) = 0 ", "a3456cfe44c9bf502fec7690d69a916e": " \\Delta w_i ", "a3457dd51410c2f5549a6b1a616faaf0": "\nm_{w}=m_{1}m_{2}/w, ", "a3458277c595d5292c55f7088d8112a5": "\\int^x_1\\frac{dt}{t}", "a345bdfc10943939be36925bab2b74b4": "z=\\langle f|g\\rangle", "a345df22de02d0ceec53dd77ba1634f3": "c_B(a,b)=\\frac{1}{4b}\\left(\\mathrm{coth}\\frac{\\beta(a-b)}{2} - \\mathrm{coth}\\frac{\\beta(a+b)}{2}\\right)", "a345fddc9e702f35acaff2900771262f": "\\Phi_{ij}=\\,2\\,\\phi_i\\,\\overline{\\phi_j}\\,,\\quad (i,j\\in\\{0,1,2\\})\\,,", "a34664f03737299494921949732f8c35": "[0,\\delta]", "a34673c2f12db99d73c20f384f58d4e3": "\\nu=p", "a3467d2dfac5361372164379b6440d42": "f(x|t)=Tr(x)", "a3468c04fd3f134497d3c511a40f286e": " \\scriptstyle{R_a = 73.13\\,\\Omega} ", "a346f7221f99eabfda4fc85a89c3f95f": "\\displaystyle \\frac{\\partial u}{\\partial t}=u(1-u)+\\frac{\\partial^2 u}{\\partial x^2}.\\, ", "a3471b1c08790ffcaf9d397a5c886d15": "\\phi_{i+1}", "a34745f6ce5940f9f22b7f5c47a53c90": "GL^{+}(n)", "a34775fe10ddc29123fc4a23ced81ae7": "\\left(1 - \\frac{t}{\\lambda}\\right)^{-1}\\, \\text{ for } t < \\lambda", "a347a8e835f93ab28808fccdf9579902": " f(\\theta, \\phi)= A\\cdot \\sin(\\theta)", "a347d6dbe61e0862ca4c1378fd08fc17": "\\gamma(b)=y.", "a34863577f06dbf16e5376a9e40d5fbc": "K(x,x_i ) = \\left( {1 + x_i^T x/c} \\right)^d ,", "a348f553bfe085c446e111a74b5c826a": " \\sigma_{J_i} \\sigma_{J_j} \\geq \\tfrac{\\hbar}{2} \\left|\\left\\langle J_k\\right\\rangle\\right| ~,", "a349163815060e9c2f9953f2b28f0e9b": "x\\mapsto \\log e^{x^2} = x^2", "a3492bd81acb68aa5afc9375cf172cf4": " V \\not \\in FV[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} L] ", "a34a3d1dcbbe5f8528b42b062579b247": "k \\cdot r < j \\text{ and } j + 1 < (k + 1) \\cdot r \\text{ and } m \\cdot s < j \\text{ and } j + 1 < (m + 1) \\cdot s. \\, ", "a34a4031cb971d117c23b808b756c7ee": " \\delta z^{\\pm} \\sim B_0 ", "a34a450df019f25370462be019c2eacf": " L(c) = \\int_a^b (E\\dot{x}^2 + 2F \\dot{x}\\dot{y} + G \\dot{y}^2)^{1/2}\\, dt ", "a34a49ad12646d6406ca8ccf2b46c11f": "\\mathfrak{q}_1 \\subseteq \\mathfrak{q}_2 \\subseteq \\cdots \\subseteq \\mathfrak{q}_n", "a34a60c7e8acdedbd76561b2dd6b8ee5": "\\omega_0 = 1/\\sqrt{LC} \\ ", "a34a614bfae7588135629d2604bc546d": "P(w,d) = \\sum_c P(c) P(d|c) P(w|c) = P(d) \\sum_c P(c|d) P(w|c)", "a34a75b38480f197d2822702ba510bb3": " \\beta \\, ", "a34ac2e2c655da7c975aa3a0b5cd4a79": "\\gamma_1,\\ldots,\\gamma_n", "a34adb0bb21ba212ca70c67b419146db": "\n\\left(\\mathbf{A}+\\mathbf{B}\\right)^{-1} = \\mathbf{A}^{-1} - \\mathbf{A}^{-1}\\left(\\mathbf{I}+\\mathbf{B}\\mathbf{A}^{-1}\\right)^{-1}\\mathbf{B}\\mathbf{A}^{-1}.\n", "a34b4e8c26ed869f5a9f8889a20aba62": "2 \\uparrow^6 2 \\uparrow^5 2 \\uparrow^4 2 \\uparrow^3 2 \\uparrow^2 65536.", "a34b803b01246b7142dd23c73650141f": "\\scriptstyle{\\mathbf{d}_{nm}}", "a34b90b015e719dd5c9b08b14ad0fcd8": " s=0 ", "a34bc4eb07f229f0eaa56e9b54b36da6": "( A~B )", "a34bf988abf1a9323ec0929c53413b44": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 1}{4 \\choose 2}{32 \\choose 1} \\end{matrix}", "a34c3b9d8da58a2486309dbf952e6ef3": "\\{D_3,D_4,D_5\\}", "a34c414040885adee7abeb5da1e29ed5": "l_1 (y) = \\frac{c_1 M}{M - y}", "a34c747a99a785f59228cac70f27cd57": "\\varphi_{h(e)} \\simeq \\varphi_{F(h(e))}", "a34cb38b0f24542a5306a008c670d149": "R= (5,0)", "a34d4316000145b5b19627563a515270": "\\hat{H} = \\frac{\\hat{p}_x^2}{2m} + \\frac{1}{2m} \\left(\\hat{p}_y - \\frac{qB\\hat{x}}{c}\\right)^2.", "a34d6df5a0560d6bc56cd84ad421135b": "0<\\Delta < \\tfrac{1}{3}", "a34d755946d701a0ded1801081df219e": "\\,\\mbox{D}(y, t) = \\exp\\left(- i \\frac{t}{2} \\sigma_y\\right)", "a34d7a1f4bad34bd5eb9511451d43378": "\\rho = \\frac{d q}{d V},", "a34e0501e64f219e8165919c9b337d6b": "(x(t)\\ ,\\ y(t))", "a34e46e64e128e0dadc9236b6fcf3268": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{hacoversin}(x) = \\frac{-\\cos{x}}{2}", "a34e6031ed01ce6a89850d987611b9fa": "R = \\left({a\\over 2}\\right)\\tan\\frac{\\pi}{q}\\tan\\frac{\\theta}{2}", "a34e6478c7c1d9b45ed591d82e303157": "n < u", "a34e6e649e65771d0f1f8d33022094b3": "\\begin{bmatrix}M\\end{bmatrix}^{-1}\\begin{bmatrix}K\\end{bmatrix}=\\begin{bmatrix}A\\end{bmatrix}", "a34e801ea676cddb74b22faa89e65370": " \\operatorname{Cov}[\\log(X_i),\\log(X_j)] = \\psi'(\\alpha_i) \\delta_{ij} - \\psi'(\\alpha_0)", "a34e92b41dec01e1382a6822d6068e9c": "U(p,T)\\ ", "a34ea418c33d587bed42ee482773469c": "\\frac{\\partial u}{\\partial t}= \\frac{k}{c_p\\rho} \\left(\\frac{\\partial^2u}{\\partial x^2}\\right)-\\mu u. ", "a34eaa1edec3edf493ccddef2fe28bcc": "\\begin{align}\n\\int_0^1 x^{-x}\\,dx &= \\sum_{n=1}^\\infty n^{-n} &&(= 1.29128599706266\\dots)\\\\\n\\int_0^1 x^x \\,dx &= -\\sum_{n=1}^\\infty (-n)^{-n} &&(= 0.78343051071213\\dots)\n\\end{align}", "a34efadbb586026193bd98eb0199ac14": "T=\\frac{Z+\\mu}{\\sqrt{V/\\nu}} ", "a34efd2d33fdd0008c9c0bad06b64c2a": "F_x\\!\\,", "a34f5ec7eb0b3a457b4b113426ef2766": "a^2 + b^2 + c^2 \\neq 0", "a34fa8872a18d9651885ad532a7f199d": "x^{4n}(1-x)^{4n}\n=(1+x^2)\\sum_{j=0}^{2n-1}(-2)^jx^{4n+j}(1-x)^{4n-2(j+1)}+(-2)^{2n}x^{6n}.", "a34fc963a1499abe37062c88abc1171d": " = a_0^2 + 2a_0 \\sum_{i=1}^{n-1}a_i + a_1^2 + 2a_1 \\sum_{i=2}^{n-1}a_i + \\ldots + a_{n-1}^2.", "a34fd12b7cfbf5bf79334aca7aa75045": " \\Gamma_{AB} \\star_{\\,\\Gamma_{ABC}} \\Gamma_{AC} \\rightarrow \\Gamma_A.", "a34fe48def3e3d3ff7255b70ff87a181": "eV", "a35011c0a233a58c93380497b095378d": "\\sinh(1) = 1.0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0\\ 1\\ 0..._!", "a3506063297b1a7def9dfd7395367f85": "\\bar V = \\frac{{RT}}\n{P} + \\Phi", "a3509bd918f1a80100df34ecf22bd7a5": "\\partial_y f|_{(x,0)}=x", "a350d54902c7d13266b225b57a5d27af": " \\dot{\\mathbf{r}}_i \\propto \\mathbf{p}_i,", "a350d5db41936bf97eb57d245f4fb40f": "(a, b) := \\left\\{ \\{a, 1\\}, \\{b, 2\\} \\right\\}", "a3513483c99d1f06c41196759f15aece": " \\frac {\\tau_{hom}}{G}=7.4\\times10^{-2}", "a35191c9812337a151536b88e0f1fdc3": "\\alpha = \\frac1{137.035999074\\dots}", "a351a4ea780552531f5157f7c524f211": "\n s = 100 \\sqrt {M / Q}\n", "a3524ab1530abe29ba875e5237bef417": "\\oint_{C} (L\\, dx + M\\, dy) = \\oint_{C} (L, M, 0) \\cdot (dx, dy, dz) = \\oint_{C} \\mathbf{F} \\cdot d\\mathbf{r}. ", "a3524d67218b1ee8df13018e9820392f": "\\boldsymbol{x}_0", "a35257154e09caca45deb59d87ea1ee0": "\\tfrac{A}{B}", "a35261ee9ef208c53c378b896e0acf76": "t=T-j", "a3529b58bbbf07f084dfee29fbdbafc5": "G_{24}", "a352ae14b54b5c5975a0a3496b5da3c8": "\\tau_c*", "a352d8208051a43295a7b68c8768b533": "\\frac{\\partial u}{\\partial t} = F\\left(u,\\, x,\\, t,\\, \\frac{\\partial u}{\\partial x},\\, \\frac{\\partial^2 u}{\\partial x^2}\\right)", "a352f01c064558c6081a56b5af33478d": "\\frac{f_o}{f_s}=\\sqrt{1-\\frac{v^2}{c^2}}", "a3535ebbdee57607efafcd82835d657c": "p_i,q_i", "a35394595a52a254a36cf4a57dc898cc": "r=\\frac{a_1}{\\cos (\\theta-\\alpha_1)}", "a353ac1ee5bc9a1a23574a9475d0e8f9": "\\mbox{DES-X}(M) = K_2 \\oplus \\mbox{DES}_K(M \\oplus K_1)", "a353b9630d3c8adecf6c226f06dadfeb": " \\Delta x\\, \\Delta p \\ge \\frac{\\hbar}{2}", "a3540debf7bd3a7e13a4bcc52ae8b322": " \\|f\\|^2 = \\int_K |f(k)|^2 \\, dk < \\infty.", "a354315fe65b80ceb7246a2a29bd7d44": "\\left ( \\pm \\sqrt{E_s/2}, \\pm \\sqrt{E_s/2} \\right ).", "a35442b384278e300266035120aab717": " \\sigma s^2 \\ ", "a3546ecd71786a8208291dd5ea225cd7": "A(x,y,z) dx + B(x,y,z) dy + C(x,y,z) dz", "a35490954b996865dcc9d992f7fd4bb8": "V(r)=1/2(m\\omega^2r^2)", "a354d38637c744b3a4a2bde53f8a6bdd": "\\iota_{i(oi)}[\\text{Q}_{oii}y_i] = y_i\\,", "a355052d4ce495e1eeeda28227b46b1d": "0 > x > y", "a3551fe853295d16f72f4737f81407b1": "\\scriptstyle g = (g_1, g_2, \\ldots, g_n)", "a35527a2b465ba6583b1d0edc80f28be": "\\begin{matrix} {10 \\choose 5}{4 \\choose 1}^5 \\end{matrix}", "a3553cec27a2c3698d9d203f8f8f8906": "\\sum w_i = 1", "a355511d3eda3ecd5f41aca6308c9a1a": "A \\cup R", "a355a10b4ffc9c0c31748be9c19206ae": "f(2,3,4)=12", "a3564db0313576c1ba9bacbb4ec9af18": "\\bigcup_{i=0}^{k} = V", "a3565f2b65174a99590fb78f7f38b4f5": "X_\\mathbf{k} = \\sum_{\\mathbf{n}=0}^{\\mathbf{N}-1} e^{-2\\pi i \\mathbf{k} \\cdot (\\mathbf{n} / \\mathbf{N})} x_\\mathbf{n} \\, ,", "a35663796ff411642e3bdaae9798dd5c": "I_C", "a35676c8174f4e6c37b4086982cd6f15": "-i\\frac{d}{dx}", "a356f2a553dcf1c3d52e6187faa5b939": "\\ \\omega = \\cos \\theta + i \\sin \\theta, \\quad i^2 = -1", "a357aa9a8b7029ff7c9d124561d8e6f9": "V=V_0 e^0", "a357ebf0f4cc6daa299470e0677a31ef": "x \\wedge (y \\vee z ) \\wedge w = (x \\wedge y \\wedge w) \\vee (x \\wedge z \\wedge w) ", "a357fec9ff16f0155e61d1ae4c17a49b": "R(x)=(x+1)(x-2).\\,\\!", "a35879eb36582c69e93555e037ff3506": "\\boldsymbol J = (c \\rho, \\boldsymbol j)", "a3589bc58939f78a8456c90b441f6d0c": " I \\subseteq \\mathbb{R} ", "a358f6c28c89ae5159f97c57365e0f44": "\\left ( \\mu_i= k_0\\frac{p_i}{p_0}\\right )", "a3590b577c06bcc8c29db7b142cdb119": " \\mathbb{F}=\\mathbb{R}", "a3591744aab1f6b85f51d838d39ad3ac": "A \\cap B \\in \\mathcal{B}", "a35929055263c9193cf3a857a0038391": "\\mu = \\tan \\phi \\,", "a35996bef32f49f39ddedd7609532e13": "\\scriptstyle \\mathcal{N}", "a359ac1747e520fc2c0da54779128bbc": "\\log _b \\sum\\limits_{i=0}^N a_i = \\log_b a_0 + \\log_b \\left( 1+\\sum\\limits_{i=1}^N \\frac{a_i}{a_0} \\right) = \\log _b a_0 + \\log_b \\left( 1+\\sum\\limits_{i=1}^N b^{\\left( \\log_b a_i - \\log _b a_0 \\right)} \\right)", "a359f19cf8b97439301369338160fa63": " k = 1 \\, ", "a35acfb82206d6cacb5cff2d98fd433a": "\\scriptstyle{R_{\\text{t}}} ", "a35b7be005c6d90bcb227b55dbb8f413": "h_{1},h_{2},...,h_{t}\\in H\\,\\!", "a35b85d0a8693dafd424bb4371257d78": "\\sum_j (A \\circ B)_{i,j} = (AB^T)_{i,i}.", "a35baee1f7bb223250ca98e01298945f": "X^i", "a35bcb94bd39ef15c4752d653892d0d5": "\\mathcal{F}_{d}=\\frac{1}{2}K_1(\\nabla\\cdot\\mathbf{\\hat{n}})^2+\\frac{1}{2}K_2(\\mathbf{\\hat{n}}\\cdot\\nabla\\times\\mathbf{\\hat{n}})^2+\\frac{1}{2}K_3(\\mathbf{\\hat{n}}\\times\\nabla\\times\\mathbf{\\hat{n}})^2", "a35bccbd8290faf130c2664dc09583f4": "p_1 = m \\frac{q_1 - q_0}{t_1 - t_0} - \\frac{t_1 - t_0}{2} \\frac{d}{dq_1} V\\left(q_1\\right)", "a35bdc870c4de3bf09edecf92e19a313": "H(x)=\\lim_{k \\rightarrow \\infty}\\frac{1}{2}(1+\\tanh kx)=\\lim_{k \\rightarrow \\infty}\\frac{1}{1+\\mathrm{e}^{-2kx}}.", "a35bdca57f41afcfd147ae80a3236c35": "\\scriptstyle 2\\sqrt{2}\\,\\approx\\, 2.8284", "a35be67bcd3d385916d3978bbe153ba6": " P_{m1}=P_m(1-\\theta)+P_m'(\\theta) ", "a35bea79c36501f7ecc8708c42f61866": "\\boldsymbol{Z_{-(m,n)}}", "a35bfab5ddde508afc780379eeb8a6c3": "\\nabla_\\sigma [\\partial_\\mu V_\\nu] - \\nabla_\\sigma [\\Gamma^\\rho{}_{\\mu\\nu} V_\\rho] ", "a35c05e2bba29fe2a8997b0cb90b4ff3": "\\textstyle k=p", "a35c438ce139c02e990bc81c7892adcb": "\\frac{\\partial}{\\partial x} = \\cos \\varphi \\frac{\\partial}{\\partial r} - \\frac{1}{r} \\sin \\varphi \\frac{\\partial}{\\partial \\varphi} \\,", "a35c53437b4357a493964538ebcfd883": "x \\geq {}^{*}\\![x]", "a35caf792fea0dfcd88cef7340d84601": "\nI =\n\\begin{bmatrix}\n \\frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 & 0 \\\\\n 0 & \\frac{1}{12} m (3({r_1}^2 + {r_2}^2)+h^2) & 0 \\\\ \n 0 & 0 & \\frac{1}{2} m ({r_1}^2 + {r_2}^2)\\end{bmatrix}\n", "a35ce5a9d22bf334a67a1a29ffa228f7": "P_R = P_{n_1,n_2,...} = \\frac{ e^{-\\beta (n_1 \\epsilon_1+n_2 \\epsilon_2+...)} }\n {\\displaystyle \\sum_{{n_1}',{n_2}',...} e^{-\\beta ({n_1}' \\epsilon_1+{n_2}' \\epsilon_2+...)} } ", "a35cefb6fe3dd1a7e583c4ed52aaa53b": " \\overline{(3-2i)} = 3 + 2i", "a35cf894d8624724f3c77129bec02b4f": " I^\\lambda ", "a35d0849cf2f61546584f03567a0bd18": "2^1 \\times 0.100_2 - 2^0 \\times 0.111_2", "a35d309363d01b99937f21be60c3f381": " \\frac{d^2u}{dt^2} = \\frac{\\Omega^2}{4} \\frac{d^2u}{d\\xi^2} \\qquad\\qquad (2) \\!", "a35d7123f4f8be47ad047273ab0bbec8": "u \\in X \\otimes Y", "a35d74227cfe569843c191dfc45248df": "R_x(t;\\tau) = E \\{ x(t - \\tau/2) x^*(t + \\tau/2) \\}.\\,", "a35dad191311552950e5b50529473fbb": "\\;[x,y]_\\tau:=\\mu((x\\otimes y)-\\tau(x\\otimes y))\\qquad \\mu(x\\otimes y):=xy", "a35defba3db4e47570d9b8bc7146e48a": "\\int_{-\\infty}^{t}\\alpha(t-t')[c_1 E(t')-c_2 I(t')+P(t')]dt'", "a35e116134bc4fe417ad4fbf1b740347": "\\frac{dx}{dt}=\\lambda_i", "a35e891ac57eef192122ff69e76a867e": "\\mathbf{r} ", "a35eac80d3b624e8c512ec9653d81f17": "\\zeta = { c \\over 2 \\sqrt{k m} }.", "a35ec6a8241d62e0f259edf320569b40": " \\tilde{p} \\pm z_{\\frac{\\alpha}{2}} \\sqrt{ \\frac{ \\tilde{p} ( 1 - \\tilde{p} )}{ n + z_{\\frac{\\alpha}{2}}^2 } } .", "a35ed62176997ee85c734d8312eb841c": "\\bar{H_n}(X)+\\epsilon", "a35f09f32a7a6d15bd745e8a3f76fa25": "\n\\langle (U^*)^k [ e_ {n+1} ], [ e_ {n+1} ] \\rangle \n= \\langle (V^*)^k [ e_ {n+1} ], [ e_{n+1} ] \\rangle \n= \\langle [e_{n+1-k}], [ e_{n+1} ] \\rangle \n= A_{n+1, n+1-k} \n= \\bar{\\alpha_k}.\n", "a35f507d4ce018660888571d561a9eb6": "l_k", "a35fdcf4cc94ed3ea4b8f9f98d073dbd": " E_j ", "a35fdda85a713feca661cb6fc9ae511c": "\n \\begin{align}\n \\sigma_{xx}^{\\mathrm{topface}} & = -z~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} -\\left(z - h - \\tfrac{f}{2}\\right)~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2} \n & = & -z~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} + \\left(\\tfrac{2h+f}{2}\\right)~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2}\\\\\n \\sigma_{xx}^{\\mathrm{botface}} & = -z~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} -\\left(z + h + \\tfrac{f}{2}\\right)~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2} \n & = & -z~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} - \\left(\\tfrac{2h+f}{2}\\right)~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2}\n \\end{align}\n ", "a35fef78e9034fa4aa6182be61b69f1d": "\nb^{2} = a^{2} \\left( 1 - e^{2} \\right) = a^{2} - \\frac{p^{2}}{4}\n", "a360a07bb644f8a4d11a6dec5d9b999b": "\\mathrm{\\frac{(SNR)_{O,AM}}{(SNR)_{C,AM}}}=\\frac{k_a^2P}{1+k_a^2P}\n", "a360e5f77103aaf87445c0a3731014f6": "(\\frac{1}{2}-\\epsilon)s", "a360f190a51bd5800609623a10355fd5": "R{{=}}\\frac{\\rho}{2a}", "a3611bb784f43fbbc08c5286596d75d7": " x_3 = \\lambda^2 - x_1 - x_2", "a3612c54b27fb8da16c6c05471036856": "\\scriptstyle x(s)", "a3617fa6a54f5b7366e888df056094c9": "\\begin{align}\n\\left|\\int_\\delta^T e^{-xt}\\phi(t)\\,dt\\right| &\\leq \\int_\\delta^T e^{-xt}|\\phi(t)|\\,dt \\\\\n&\\leq K \\int_\\delta^T e^{(b-x)t}\\,dt \\\\\n&\\leq K \\int_\\delta^\\infty e^{(b-x)t}\\,dt \\\\\n&= K \\, \\frac{e^{(b-x)\\delta}}{x-b}.\n\\end{align}", "a361bb5f5055839218c632b9c0565172": " D_n(1)=0 ", "a3624591afae87fdef9f9f969268154c": "\\overline{\\lambda}_{C} = \\frac {\\lambda_{C}}{2 \\pi} = \\frac {\\hbar}{m c}", "a36262df2c4925e57e8eaf7f470f1184": " H_n(X,A)", "a36269af98d07f6acd1dc70a7fa8e4b0": "\\ z_i", "a36279af130b0a2554fdcbbc0a6bf76f": "\\frac{\\partial {\\rm tr}(e^\\mathbf{X})}{\\partial \\mathbf{X}} =", "a3627d4d2f7328d4df2d47da78dc3572": "P\\left(\\sup_{f \\in \\mathcal{F}} |\\hat{R}_n(f) - R(f)| >\\varepsilon \\right) \\leq 8 S(\\mathcal{F},n) e^{-n\\varepsilon^2/32} ", "a362951d07b1dbb077005ee6b0903f51": "\\mathbf{H} = \\hat{\\mathbf{r}} \\times \\mathbf{E} / Z", "a3629c23dd68d856b7cafa7cdaa0b6da": " \\iint d\\mu(x) \\, d\\mu(y) \\prod_{j=1}^n (f_j(x) \\pm f_j(y)) \\geq 0. ", "a362f6f653e3538c71215410ef35e189": "2^{2^{2^{2^2}}} = 2 \\uparrow \\uparrow 5 = 2^{65,536} \\approx 2.0 \\times 10^{19,728} \\approx (10 \\uparrow)^2 4.3", "a36352dcd94a2a89bd5d842776f41b16": "x_1, x_2, \\dots", "a363559d293e8461bdb837a723cc9ed0": "\\hat{B} = \\mu^{1/2}", "a3635cf223081a74bfd241507cafd56b": "A_1 \\to A_2\\alpha_2 \\mid \\ldots", "a3635f9610c4b61e1094711a635a8abc": "|z| < \\frac{1}{1-p}", "a3637cb03b4885b97713c10770f9b15f": "M_V=-23", "a363864df170b03e13f5000ee0ba5bae": "a\\triangleright a=a", "a36399d38c9b5c7122cc55b959093c2f": "\n\\mbox{cas}(t) = \\cos(t) + \\sin(t) = \\sqrt{2} \\sin (t+\\pi /4) = \\sqrt{2} \\cos (t-\\pi /4)\\,\n", "a363ad53ef5e95e7d7e273769b3f0514": "C_{p_l}", "a36419b019665808fb6ed339b04ef0bf": "G_{\\mu\\nu}", "a36471d1461832166e7697bdef173e5b": "f_{\\text{critical}} = 9 \\times\\sqrt{N}", "a364b117f0568e0ebdb4983c4b9f0f6d": "a^{r/2} \\equiv -1 \\pmod{N}", "a3653bb3091ed2e107bcb68642f3a0aa": "\n \\boldsymbol{S} = \\frac{\\partial W}{\\partial \\boldsymbol{E}} \\qquad \\text{or} \\qquad\n S_{IJ} = \\frac{\\partial W}{\\partial E_{IJ}} ~.\n ", "a365e336923a403e09a3367cef9b6d40": " j: X\\hookrightarrow Y ", "a36646354fd26bd5a88d3d7186c183ab": "{(v,w)}", "a366dcdf47d1b3186b917c29efcbabe6": " \\sum_{k=0}^\\infty M(a_k - a_{k+1})", "a36704e5561b4faa9adbb240616bf300": "2 \\,", "a367208df13e5f29fb46f4a26096ee6e": "\nH|\\mathbf{k},\\mu\\rangle \\equiv H \\left({a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\,|0\\rangle\\right) = \n\\sum_{\\mathbf{k'},\\mu'} \\hbar\\omega' N^{(\\mu')}(\\mathbf{k}') {a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\,|\\,0\\,\\rangle =\n\\hbar\\omega \\left( {a^\\dagger}^{(\\mu)}(\\mathbf{k}) \\,|0\\rangle\\right) = \\hbar\\omega |\\mathbf{k},\\mu\\rangle.\n", "a36746fb5bc4241a8cea10d51335fd64": " j \\colon \\mathrm{Spec}(K) \\to \\mathrm{Spec}(R)", "a3674935ef1c06e278631e84857d043a": "b^2 = c^2 + h^2.\\,", "a36775db5ca2590c4d7257151cd524aa": "\\begin{array}{lll}\nu\n&=& \\cos\\frac{\\alpha}{2} + \\sin\\frac{\\alpha}{2}\\cdot \\frac{1}{\\| \\vec{v} \\| }\\vec{v}\\\\\n&=& \\cos \\frac{\\pi}{3} + \\sin \\frac{\\pi}{3}\\cdot \\frac{1}{\\sqrt{3}}\\vec{v}\\\\\n&=& \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\cdot \\frac{1}{\\sqrt{3}}\\vec{v}\\\\\n&=& \\frac{1}{2} + \\frac{\\sqrt{3}}{2}\\cdot \\frac{\\mathbf{i}+\\mathbf{j}+\\mathbf{k}}{\\sqrt{3}}\\\\\n&=& \\frac{1 + \\mathbf{i} + \\mathbf{j} + \\mathbf{k}}{2}\n\\end{array}", "a3678bc2381ed9fc6c39fe0f81f77a87": " (x^c,y^c) = (m_{10}/m_{00},m_{01}/m_{00}) ", "a367cea85d28e8c95d572574fca5b178": " x^{\\log_b(y)} = y^{\\log_b(x)} \\!\\, ", "a367d2053fa4c287f91b3e1541dd0e09": "\\frac{d[P]}{dt} = k(T)[A]^{n'}[B]^{m'}", "a367d8ed330e7a074dcd09b18fbad079": "\\text{var}\\,(Y) = a\\text{E}(Y)^b + \\text{E}(Y)", "a36851a94d99327aee130aa80a8ec286": "\\gamma_3=\\frac{Y_3-Y_1}{L_3-L_1}", "a368543b062c869adea351940ca094f1": "\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} n^v_{\\mathbf k} =\n- ( \\Omega_{\\mathbf k}^\\star \\, p_{\\mathbf k} - \\Omega_{\\mathbf k} \\, p_{\\mathbf k}^\\star ) \n+ \\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} n^v_{\\mathbf k}|_{\\text{corr}} \\, .\n", "a3686747d7705aad1baeec00e8898bc5": "I_1", "a368902db9ff296952fdc11ac4bb32ff": "\\lambda_m = \\frac{dm}{dl}", "a36898ad81b96de0f83e542cf10ced35": "\\Sigma_{y=0} = \\Sigma_{y=1} = \\Sigma", "a368d644eb1224a12b1ce81a28234aa9": "d\\mathbf{S}", "a368d7e1f051cb30c55b53d8e99abe64": "d\\gamma(1-2\\varepsilon)n\\,", "a368e2d821239aac89a109d17caa5abe": " h_1 ", "a368fb10967ddef09d8580311de570a1": "1_{A \\,+\\, B} (z) = \\sup_{x \\,\\in\\, \\mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),", "a368fb5faa85c79fcf4022f7410dc017": " \\Delta S_{tot} \\ge 0 \\,", "a369955ce5159bf223415dff9322e3d5": "\\gamma \\in \\Gamma-V", "a369b49a367395f3714605bea2a8310c": "F \\,", "a369d539578a35bb9f0c072cc0d307d0": "\\frac{CPI_2}{CPI_1}= \\frac{Price_2}{Price_1} ", "a369ff5a9382d02302b6e69c0fde73c6": "k_C.", "a36a359b8d0fbb8eecd53a12110f2798": "b = 2, 3, 5, 7, 11, \\ldots", "a36a3a296f51ceacf7702bc633cf05d9": "A=\\cup_{i\\in \\mathbb{N}} F_i", "a36a79ec79e17d9bc0d83d93c269bdf8": "c_n:=\\sum_{k\\in\\N, \\vert j\\vert=n} b_k a_{j_1} a_{j_2} \\cdots a_{j_k}.", "a36a7a47eb36a221b44e84cd34871042": "\\scriptstyle \\mathbf{A}_1", "a36aa799954406a09b9ccb2738cce2c6": "L=1", "a36ae2b277d45efbaff6839f8ec05ce5": "{}^{*}", "a36aea780cf3b06310bf15ee0ee50db6": "W = E/(Z^2 E_\\textrm{h})\\,", "a36aff89e271a14def453aacdcbee969": "EL(\\Gamma\\,')=h/w", "a36b3d2677404b81210d2b9dc12656d2": "\\frac{1}{\\tau^*}=\\frac{m}{m^*}\\frac{1}{\\tau}", "a36b8248138f0ba94392336103b00442": "eC_{Cr} = \\frac { \\mbox{(140 - Age)} \\ \\times \\ \\mbox{Mass (in kilograms)} \\ \\times \\ {Constant} } {\\mbox{Serum Creatinine (in } \\mu \\mbox{mol/L)}}", "a36b9034aec3ab2b12d7fc5af14ddfca": "q(v) = a \\in F^\\times", "a36bfada0462f6c0211ee82b5228f37a": "M_{xy}(0)", "a36c2242b942449a1009ba1f68b8c979": "~x_{\\min}~", "a36c53f361f4a44aff52237126687051": " \\Delta \\epsilon_{inelastic} = A_{pp}(N'_{pp})^{C_{pp}} ", "a36c5cf49ef32b893f5dc2960fe793b0": "\\frac{dE\\left[ \\Lambda(n+1) \\right]}{d\\mu} = 0 ", "a36c8e56ed254c39c026cece6c7b1a5a": "\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots = \\sum_{n=1}^\\infty \\left({\\frac 12}\\right)^n = \\frac {\\frac12}{1-\\frac 12} = 1. ", "a36cd320ab0b7d208bce8060593123e3": " P(Z \\le -u \\text{ or } Z \\ge v) \\le \\frac{ 4 + (u - v)^2 } { (u + v)^2 } .", "a36cede77c4835262e0e72d2485a43f0": "\n \\varepsilon_{ik,jl} - \\varepsilon_{jk,il} - \\varepsilon_{il,jk} + \\varepsilon_{jl,ik} = 0\n", "a36d1b4e162ebd85665c05961dd7dfc4": " D_\\mu ", "a36d4614f81aaf5f8be1e6dee8b996b1": "M:=|X_0|+\\sum_{s=0}^{\\tau-1}|X_{s+1}-X_s|=|X_0|+\\sum_{s=0}^\\infty|X_{s+1}-X_s|\\cdot\\mathbf{1}_{\\{\\tau>s\\}}", "a36dd9114cc9cb8a05ca3c8d1a768d4e": "x^{b_0}, ..., x^{b_{l_i}}", "a36ea64e3f9f2612e70aabf17872b5d5": "\\sigma_{ij} = c_{ijk\\ell}~ \\varepsilon_{k\\ell}", "a36ef42c9c00eccd2562c65451d18f71": "i^{\\prime \\prime}(V)", "a36f051ae16c879f277b0e0b1cbaea9f": "\\beth_2", "a36f130c6e75f2a3eb33f942a06b673b": "\n\\{\\vec{p}\\cdot\\vec{r},\\, H\\}_{PB} = \\frac{p^2}{m} - mgz+ \\lambda r^2 -2 u_2 r^2 = 0.\n", "a36f1901928993e16dd343c68c042d49": "Aa~Gradient=\\left(150-\\frac{5}{4}(P_{CO_2})\\right)-P_aO_2", "a36f1c0282d930cdd1147d993f1601a7": " BOP surplus = \\text{current account surplus} + \\text{narrowly defined capital account surplus}. \\,", "a36f2988fa92567f23e80cd46bfbacc7": " \\frac {MW} {N_0} = ", "a36f99d9d5f9a6fa49ace620d0ddfd0e": "P\\cdot Q", "a36fc4400691191d6275db2df847799a": "\\sum_{n=1}^\\infty b_n", "a36fd2a1cb43338193f02e91ae182360": "val_i", "a3700e647dc4e7fbb1cd79acb68c97b5": " \\scriptstyle i ", "a370388e249e6aa2f822fb549452f59a": "A + B \\rightleftharpoons AB; K_\\text{c} = \\frac{[AB]}{[A][B]} /\\text{M}^{-1}", "a3706626a9aeef99b243f46a4aa6f53f": "E_H = 1.229 V - 0.0591*pH \\ \\{ V \\} ", "a370e4d23cf2804f5358c1c841e53505": " C^\\infty(E,V)^K,", "a371889fd86260ddefe695d4cc34acef": " \\theta(\\xi)=a_0 \\frac{\\sin\\xi}{\\xi} + a_1 \\frac{\\cos\\xi}{\\xi} ", "a3718e6a4cee76857d940191e879e330": "\n\\begin{align}\n\\Pr(M_n \\leq z) & = \\Pr(X_1 \\leq z, \\dots, X_n \\leq z) \\\\\n& = \\Pr(X_1 \\leq z) \\cdots \\Pr(X_n \\leq z) = (F(z))^n.\n\\end{align}\n", "a371b723b110c3284a034ac9d59ea6fa": "\\; ||R_\\pi(\\varrho_{A_1\\ldots A_m})]||_{Tr}\\leq1", "a3720db398464d2b7e27fbd663ab57dd": " \\frac1z = \\frac1 {(z-1)+1} = 1 - (z-1) + (z-1)^2 - (z-1)^3 + \\cdots.", "a373178b6ea13840a4cd8f95fa4676c4": " [\\varphi \\star \\psi](x) = \\sum_{\\{u,v: u v = x\\}} \\varphi(u) \\psi(v)", "a3731d504cfa2a33f07b6ae86fa2ecda": "\\frac{H_{k,s}}{\\zeta(s)}", "a3732c4544d97f9dad52507567bc1de2": "w\\ne 1", "a373cf0e4336e41d466ab8583c064412": " t^\\prime=\\frac{t}{T} \\quad \\vec r^\\prime=\\frac{\\vec r}{L} \\quad \\vec v^\\prime=\\frac{\\vec v}{V}.", "a3742b173d6997381d545d1d5a4da95b": "x, y \\in (N \\cup T)^{*}", "a3743527c67cc81b6dd70f04a1c8d682": " \\mathcal L_X \\phi = X^a \\nabla_a \\phi = X^a \\frac{\\partial \\phi}{ \\partial x^a} ", "a37466935fe2af9d3f63d68cfa182e15": "K=\\tfrac{1}{4}\\sqrt{(2(a^2+c^2)-4x^2)(2(b^2+d^2)-4x^2)}\\sin{\\varphi}", "a374818aff6ce65550b3ab46fa5f70a9": "y = y_1 + s(y_2 - y_1).\\,", "a3749bc539973abbdd38e436fcea9bec": "\\varepsilon \\cdot \\mathrm{OPT_A}(x)", "a374b6eb370f884adbdb1310d07db65c": "\\tfrac{102GeV}{c^{2}}", "a3750aa6a68bc37b3e53ef5f51f7fa46": "x_\\alpha^{t+1} = x_\\alpha^t + v_\\alpha^{t+1}", "a3750d3be8dc6ac2de812197150ef24c": " p(\\textbf{x}_k|\\textbf{z}_{1:k}) = \\frac{p(\\textbf{z}_k|\\textbf{x}_k) p(\\textbf{x}_k|\\textbf{z}_{1:k-1})}{p(\\textbf{z}_k|\\textbf{z}_{1:k-1})} \n= \\alpha\\,p(\\textbf{z}_k|\\textbf{x}_k) p(\\textbf{x}_k|\\textbf{z}_{1:k-1})\n", "a37531d617d436e82142e3cb8b87e4ef": "X(t, \\omega; x_{0})", "a375324677450094742acf192afb802c": "g_{\\mu\\nu}\\!", "a3754b222790ea4d6c9f167a3e49b9ee": "\\begin{matrix} {4 \\choose 3}{3 \\choose 1}^3 \\end{matrix}", "a3755dc734a44af565853f1a2d487b3b": "\\,\\ \\cot x", "a37570068ee174d3e20661c98bc72756": "Z = \\frac{\\frac{P}{H[P/E]} - E12}{stdev(E12)} ", "a3757ce98d19bb24080577c92f879abf": "g(S)=f(\\Omega \\setminus S)", "a375a23d9f15a1069a9f6016547159a7": " \\simeq -79.0 ", "a376614fdf9d11d9de8247461cbc77ed": "a y_1^{\\alpha_1} \\prod_2^m (z_i + y_1^{r^{i-1}})^{\\alpha_i}, a \\in k,", "a37692355528a80dbc0a0b945b62062f": "S (\\theta, \\lambda, t) = \\frac{\\rho_i}{\\gamma} G_s \\otimes_i I + \\frac{\\rho_w}{\\gamma} G_s \\otimes_o S + S^E - \\frac{\\rho_i}{\\gamma}\\overline{G_s \\otimes_i I } - \\frac{\\rho_w}{\\gamma}\\overline{G_o \\otimes_o S }, ", "a376b64e67f65cf60d2e216bdd63523e": "\\{u,v\\}\\in E", "a376e2c8d3254f7f4db163ca8d26045f": "L_d^p = L_d^p(\\Omega,\\mathcal{F},\\mathbb{P})", "a3773f5409280ca4a4efe9d051b68c57": "x \\in \\{0,1,2,\\dots\\}", "a377433e4dc0f1bc05d07f7e7a8f2720": " A X A = A", "a3774eae6a410cf36e3920802bbfeb4d": "e_i e_i = \\pm 1", "a377b639c8afc878151b62bcf1a3b874": "q_i = a^{-s_i}/C", "a377e5101e23d97b490cc38494054af8": "\\left(F_0+K\\right)/2", "a378306f6a880704ea714bcd7ae4f198": "\\mathbf{X}_1", "a37840bdf610be50f747447ac95bc626": "\\begin{matrix} {11 \\choose 1}{4 \\choose 4}{44 \\choose 1} \\end{matrix}", "a3789351e638d5c1532664e4b25cbb59": "r^J \\Lambda r \\subseteq \\Lambda", "a378a531b048243f9af8cbd6197b472c": "as+bt=\\gcd(a,b)", "a378d0c5193ca7434ad516434a8072d6": "\\min_x \\|x\\|_1 \\;\\textrm{subject} \\ \\textrm{to}\\;\\;\\|y-Ax\\|^2_2 \\le \\delta", "a3791f7e2cb525c4d52d68456700dc03": "\\mathbf{\\nabla}\\times \\mathbf{B} = \\left(\\mu_0\\mathbf{J}+\\mu_0 \\epsilon_0 \\frac{\\partial }{\\partial t}\\mathbf{E}\\right) ", "a37939cc8a2d4ad532d32fd45f0eb1af": "Sp(2m,\\mathbb C)", "a3793d4dd62bba41311a38922cbebbf3": "\\scriptstyle{p(\\sigma^2|I)\\; \\propto \\;1/\\sigma^2}", "a379481bc82e39f0f2153dd5294d3888": "\\varphi(t;\\mu,c)=\\exp\\!\\Big[\\; it\\mu - |c\\,t|(1+\\tfrac{2i}{\\pi}\\log(|t|))\\Big].", "a379a701e6e31d030ac6af59129f408a": "0.4 \\leq x_{23} \\leq 0.5", "a379f2620276d78db6f465bdd7581477": "\\frac{dy}{dx} + \\frac{dx}{dx} + \\frac{d}{dx}(5) = 0;", "a379fc211e5841096d747ee8fe73f8c0": "\\frac{2}{3}n^3+n^2+\\frac{1}{3}n-2=O(n^3).", "a37a35fb95fafa45f879f6e83b2ee35b": "|\\mathrm{c}\\rangle = \\frac{1}{\\sqrt{2(1-e^{-2|\\alpha|^2})}}(|\\alpha\\rangle-|{-}\\alpha\\rangle)\n", "a37a4398ec4a15bfa7627435aabd0e43": "Q\\or\\neg P", "a37acc977dbad790770143eec247e7e1": "s_i \\subseteq s_j", "a37aef9ef5a219f39345f5f12e5dda8c": "q = \\frac{\\partial \\psi}{\\partial n}\\,", "a37b205730f8f2527360792d7818edfd": "\n\nF (t) = Ma (t) + Bv (t) + (K + iC) s (t) \\!\n\n", "a37b2df4717aaee2a72f2f52a7444cd8": "C_R =\\frac{v}{u}=\\sqrt{\\frac{v^2}{u^2}} =\\sqrt{\\frac{KE_\\text{after}}{KE_\\text{before}}}", "a37b7d3fa80bd6336ecacddae09f1fff": "N = N_0e^{-\\lambda t}\\,\\!", "a37c399b8bf03ca8d902c472fc833438": "\\frac{\\partial {\\rm tr}(\\mathbf{AX})}{\\partial \\mathbf{X}} = \\frac{\\partial {\\rm tr}(\\mathbf{XA})}{\\partial \\mathbf{X}} =", "a37c6038634691a28e74abeb46f698ec": "\\hat{\\beta}", "a37c9b292624bea07a7a922918e60d93": "\\frac{3}{4}\\sqrt{5}\\sin(2\\theta)(7\\sin^2(\\phi)-1)\\cos^2(\\phi)", "a37cb832b269c8b0ba120f7a8b69f60c": "\n\\mu_t =\n\\begin{cases}\n{\\mu_t}_\\text{inner} & \\mbox{if } y \\le y_\\text{crossover} \\\\ \n{\\mu_t}_\\text{outer} & \\mbox{if } y > y_\\text{crossover}\n\\end{cases}\n", "a37d22840cab701b7186a12b0ade8a7e": "Z[J] = \\int \\mathcal{D}\\phi e^{i(S[\\phi]+\\int d^dx J(x)\\phi(x))}", "a37dcdd2ad351280ecc6ab97cc779f00": "20 \\cdot b_n\\ dB ", "a37ddb2f86dcaba78c6fbea21d2760bc": "\\mathbf{a} (\\mathbf{b} + \\mathbf{c}) =\\mathbf{a} \\mathbf{b} + \\mathbf{a} \\mathbf{c}", "a37de4300be6f886c5489d23802f2d63": "v < c", "a37ea6324a91a1f4b6be2ec0bec0ecd4": "0 = A \\cdot 0 + B \\cdot 1", "a37ef2cea301ad46c774e15a854c5f14": "\\xi=1.", "a37ef9aacd8926163a4ae62426e1485d": "\\tfrac{(-1)^m}{m+1}.", "a37efbc02773ccd350d88ec4166ef232": "[0, t]", "a37f258b5dc16a2ab1fd8dcb68c143f4": "A_1, \\dotsc, A_n", "a37f2a3e831dc225d80a7e68149ec97e": " O((\\log n)^{5+\\varepsilon})\\, ", "a37f321af57afdd65665957ba0d41cf9": "R_T = R_0 \\left[ 1 + AT + BT^2 + CT^3 (T-100) \\right] \\; (-200\\;{}^{\\circ}\\mathrm{C} < T < 0\\;{}^{\\circ}\\mathrm{C}),", "a37f42457d452865a5eb01cdfebdfeab": "\\mathrm{ABCDEFGHI} \\!", "a37f4d0d7505f79ea2e28562287e7c23": "[23,12,7]_2", "a37f7455f2b926adbfcd61489567f95e": " \\begin{bmatrix}\n 1 & 1 \\\\\n 0 & 0 \n \\end{bmatrix}\n \\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 0\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 0\n \\end{bmatrix}\\,\n", "a37f79ae49cb8a1c05c1bba663f458b1": "\n\\begin{array}{rl}\n1. & P \\rightarrow Q \\\\\n2. & P \\\\\n\\hline\n\\therefore & Q\n\\end{array}\n", "a37fbfaa062563ef0c844762dfbc7d5c": " (M,m,\\tilde{m},t) ", "a38008e51395848f3db39a5b96d6cfda": "B_1 \\times B_2", "a380f00180c9e59288d229450fdaeab7": "\n f(k;\\rho,\\alpha) = \\frac{\\rho}{1-\\alpha^{\\rho}} \\;\n \\mathrm{B}_{1-\\alpha}(k, \\rho+1)\n ,\n \\,", "a3812af18422425cbcf5c0d6df03eba4": "\\hat{x}(k+1) = \\left(A - B K \\right) \\hat{x}(k) + L \\left(y(k) - \\hat{y}(k)\\right)", "a3816b1eba5b8f488398fe180371b3d8": "A=\\{x\\}", "a3819fbddfdaf17d3463dc53ee2e28ae": "\\frac{Du}{Dt} - f v = -\\frac{\\partial \\phi}{\\partial x}", "a3829e88e43b4cb02fa900016b72edf3": "f_c'(z_{cr}) = 0. \\,", "a382c0dd316069e3effe4651c2840f97": "[A(t),B(t),C(t),D(t)]", "a382dccdd751773ed85226633374b93e": "P(A_k)", "a38323bfa39e43d02fb77ff2bbb64f3b": "s \\in W^4", "a3833e86ecf66ace8139ab1019aaf178": "(B \\and C)", "a3833fa4db7c7aebfd777169bddd312a": "a,b,c,a,b,c,\\dots,", "a38396e1c38fde741037196803365613": "\\text{FO}(V_k,V_l,+)", "a383a3b55b89732c2298ed2e7350e528": "V\\cap Y", "a383bd8e8623847c9fb32e616ef7e243": "y^+ < 11.63\\,", "a384072a231c293c172dbdc7882c545c": "\\chi (s_1,s_2)", "a3841f99a64191feffd034aa11d1e881": "A := \\alpha_1 A^1 + \\cdots + \\alpha_d A^d", "a3846645fd387890f1b4ea4fc91d0df9": "L_1=\\ln x", "a38479a1d59d57875c96a3e0404e2061": "f_{y(3x^2-y^2)} = N_3^c \\frac{y \\left( 3 x^2 - y^2 \\right)}{2 r^3 \\sqrt{3}} = \\frac{1}{i \\sqrt{2}}\\left(Y_3^3 + Y_3^{-3}\\right)", "a3849283c165949660ff1d2cf52fb4e5": "\\mathbf{v}_1, \\ldots, \\mathbf{v}_m", "a384c4c4f19084e7c5d50bd424eed624": "\nD(X, Z) \\le D(X, Y)+D(Y, Z). \\,\n", "a384ddf11b28935c4968f32c7e9ab5ed": "r_B", "a3852b306dc76a5cb86b4d1df5ac06b7": "I_{\\mathrm{RMS}} = I_\\mathrm{p}\\sqrt {{1 \\over {T_2-T_1}} \\left [ {{t \\over 2}} \\right ]_{T_1}^{T_2} } = I_\\mathrm{p}\\sqrt {{1 \\over {T_2-T_1}} {{{T_2-T_1} \\over 2}} } = {I_\\mathrm{p} \\over {\\sqrt 2}}.", "a38565d07742f4ab471acd8de6c8dd78": "\\mathfrak{g}_0", "a38565d9f08316e97e9252aebfc90f06": "S(\\vec{q})=\\frac{4\\pi^2}{q_z^2}\\delta(q_x)\\delta(q_y).", "a385a079226b0c6646d7bf77cb2f7468": "\\varphi\\in \\mathcal{C}(I_a (t_0),B_b(y_0))", "a3862a2912543ccde1c8173bfac5d824": "f = f_c \\cdot \\mathrm{ARFCN} + f_b + f_o", "a38666e16111df3e97d2d2d802aae9db": "\\nabla_{\\bold{v}} (fg) = g\\nabla_{\\bold{v}} f + f\\nabla_{\\bold{v}} g", "a386e923c50f5c9b05548ffa46b90e29": " {\\mathbf v}[f] \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{df}{d\\lambda}= \\frac{d\\;\\;}{d\\lambda} f(c(\\lambda))\n", "a387112c42247d3e6a490da46be68543": "y(t) = \\int_{t_0}^{t} f(\\tau) d\\tau.\\,", "a3873ab56db0f87b9d5252666b309fe0": "y=-\\frac{1}{4}.", "a387d6aa33831bfa362acc216cb1b518": " \\mathbf{F}_{\\text{scat}}(\\mathbf{r}) = \\frac{k^4 \\alpha^2}{6 \\pi c n^3\\epsilon_0^2} I(\\mathbf{r}) \\hat{z}", "a387dacd617350b15e9b95f194a71a2a": "y_{it}-\\overline{y_{i}}=\\left(X_{it}-\\overline{X_{i}}\\right) \\beta+ \\left( \\alpha_{i} - \\overline{\\alpha_{i}} \\right ) + \\left( u_{it}-\\overline{u_{i}}\\right) = \\ddot{y_{it}}=\\ddot{X_{it}} \\beta+\\ddot{u_{it}}", "a388d496981384b1010bbb9cd6f50f20": "\\operatorname{ord} (a) = \\operatorname{ord}(\\langle a \\rangle).", "a388e0651233792272a0e16997cd9661": "\\frac{dV}{dt}=Q_0", "a3890a004ec017d90b4b5b69dc172da6": "\\alpha/h", "a3895d4c5c3d28588165ded5201fe082": "\\log \\left (\\frac {k_s}{k_{\\text{CH3}}}\\right ) = \\rho^*\\sigma^*", "a38ad1bcd38f51fbda189b69017cc234": "I = \\sqrt{Y^2+X^2}", "a38b080e2ff88b4141fc807a52b6758f": "s_\\lambda=s", "a38b394fb2e0aaeda3fc0ed9763f0bc5": "\\scriptstyle X \\;=\\; 0", "a38bbd5c5c7fb901eba194fdb0cf3611": "f(z)=1, g(z)=\\sqrt{z}", "a38c251957ef40947f97081294c764b5": "n_e(n - n_e) = 4", "a38c2ccdf413d7ad1566935ab9b5b9b7": "x = {3 \\over 4.0000}(2y)^{2 \\over 3} + {1 \\over 2}.", "a38c3736054402487dcc2c94a36d518d": "(\\mathbf{A} \\times \\mathbf{B}) \\times (\\mathbf{C} \\times \\mathbf{D}) = [\\mathbf{A},\\mathbf{B}, \\mathbf{D}]\\mathbf{C}-[\\mathbf{A},\\mathbf{B}, \\mathbf{C}]\\mathbf{D}=\n[\\mathbf{A},\\mathbf{C}, \\mathbf{D}]\\mathbf{B}-[\\mathbf{B}, \\mathbf{C},\\mathbf{D}]\\mathbf{A}", "a38c79d14f1cb319aa6542d1f8e48550": "RN \\subseteq N", "a38c7eb1b2da68fb9cbd0da006d7ac6b": "\\int \\frac{x}{\\sqrt{1-x^2}}\\,\\mathrm{d}x = -\\frac{1}{2}\\int \\frac{\\mathrm{d}k}{\\sqrt{k}} = -\\sqrt{k}", "a38d50028351f478228814324ab7a072": "c (S,T) = \\sum_{(u,v) \\in S \\times T} c_{uv}", "a38da14523ee5901c93efa438cb6b9ec": "\\frac{x_1}{x_2} = \\frac{k_2}{k_1}. \\,", "a38db09534474ab7b80b3819ede7bfef": " y_{t} = a_{1}y_{t-1} + a_{2}y_{t-2} + \\varepsilon_{t} ", "a38dbe819cd26b5066df29fcda127e9f": "d_{ij}", "a38df9bc7c869b90eae113ca2f60f7de": "U(r+\\bigtriangleup r,w) =U(r,w)\\exp (ik(w)\\bigtriangleup r) \\quad (1.2)", "a38e56ed35c49a1bd44d97b87c786ff2": "(X, \\cap)", "a38e8fb9a04e6db0e4e97756605084ed": "A \\geqslant B \\!", "a38e93c4e72f98c498f0f2d098767852": "\\log_{10} \\frac{\\textit{molar~mass}}{1000} = \\textit{distance~to~right~(decades)} ", "a38eb4df5d0e9b6598c04b0f80a0d48d": "\\Beta(x,y)=\\frac{\\Gamma(x) \\; \\Gamma(y)}{\\Gamma(x+y)}.", "a38ed687026ff73569f7cc5976159969": " \\mathrm{N}(\\alpha-\\beta)=(a_0-b_0)^2+(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2 \\leq \\left(\\frac{1}{4} \\right)^2+\\left(\\frac{1}{2}\\right)^2+\\left(\\frac{1}{2}\\right)^2+\\left(\\frac{1}{2}\\right)^2=\\frac{13}{16}<1. ", "a38ee50525885d3dc24c87e3d9408cd9": "\\delta(r)", "a38f4c421535a29cba89144598d4cf28": "f_i \\cdot i / (f_i \\cdot i + g_i (N-i) )", "a38f522e82cf4b723ef39e25299f9284": "\\Gamma_t", "a38f5886e8b243cf09b0b74dc3e2dd7a": " \\epsilon \\to 0 ", "a38fe4454522f01dd521be8102c9a1d6": "\\mathcal{E}(x) = g^x r^m \\;\\bmod\\; m^2", "a3902f61934b01014d7cb2ec2dacc720": "\\mathbf{1}_A(x) = \n\\begin{cases} \n1 &\\text{if }x \\in A, \\\\\n0 &\\text{if }x \\notin A.\n\\end{cases}\n", "a39037ba03412351ec1f3c8ca9e807fd": "y[n]", "a3904f143141153cc70766aff62f962c": "n\\!", "a3906d9d6b0efcdadf1efcc34d2e6a02": "\\mathbf Z_N", "a39083c01a39e8730170d6c9247e8c98": "\\partial \\Omega", "a39084686f8dc618a8153daf0300c0c5": "\n \\widehat Q_n(\\hat\\theta) \\geq \\max_{\\theta\\in\\Theta}\\,\\widehat Q_n(\\theta) - o_p(1),\n ", "a390d6c210165acb9aec64b4d8c463b0": "\\mathbf{U}^*(\\mathbf{x})\\equiv \\operatorname{argmin}_{\\mathbf{U}} J_0(\\mathbf{x},\\mathbf{U}).", "a390dda81822d11ec2295723de2c4d9e": "\\Re(\\zeta(it+1/2))", "a390e31b190ab108ec37be46c5e51396": "2 \\cos \\frac{2 \\pi n}{11}", "a390e8e7421b05b587c6fe8d2f7d66f7": "0\\rightarrow B\\rightarrow E\\rightarrow A\\rightarrow0", "a390eb7b2ddbc344e7fecbb1829ce07d": "k_{ad}", "a39107215fcc9bae6f70d26435e83511": "T = 273.16 \\cdot f(T_1,T) \\,", "a391212f980a9c71446e1cc4193647ae": "\n z := \\cfrac{\\sigma_b\\sigma_t}{\\sigma_c(\\sigma_b-\\sigma_t)} ~.\n ", "a391986844a4d642fc184d299d40ca44": "k^{2}", "a391c3e50421ee6f1fa2137206b1bcc6": "Q^{d}(P) = a + bP", "a392180735ad51113a12b7504fbd398c": "(a_n) \\stackrel{T_s}{\\longrightarrow} (s_n \\cdot a_n) .", "a3921b16079429a9708db4ec13703baa": "p_n[h]", "a39223ad6135f4591da141355d517806": "F(e_k)=d_k", "a3924eedcbe0d9d0c1a226e73b1fba81": "(1+\\alpha x)^{-\\gamma}=\\Sigma_{n=0}^{\\infty}(\\alpha x)^n \\frac{(-1)^n}{n!}\\frac{\\Gamma(\\gamma+n)}{\\Gamma(\\gamma)}", "a392a1c9b82077f4e4582542f49b8223": "\\Delta \\mathbf{y}", "a392b7ddf8ed9f821e2368b64401d97c": "t=\\sqrt{\\frac{r^2(n-k-1^*)}{1-r^2}}", "a39342bdf56c4b2eb69bc0117218762b": "\\bold{F}(\\bold{x}, t;\\nu) = \\oint_\\Omega\\ I(\\bold{x}, t;\\bold{\\hat{n}},\\nu) \\,\\bold{\\hat{n}} \\,d\\omega(\\bold{\\hat{n}})", "a39373f5fceac75c3f4111836d2a059f": " \\textstyle\\ \\mbox{rate(prop)} = k_p[\\mbox{I}][\\mbox{M}] ", "a39392ada25bb169f4de1dd7d15d0817": "(1 + \\epsilon)", "a393956797a17dee126e7cd35c705bc4": "g_L = -1", "a394a22640cb9d6c58566e961b218d93": "f_{r_k} \\to f", "a394a269c3878d7af58887abf4f2df32": "\nU(t,t)=I\n", "a394ee27a1dcb79cc7d38103338e6535": "d(t) = d_0 a(t)", "a39520eeb1d9fc3acca22a9e94126a10": "x_2(t_0)=x_1(t_0-t_0)=x_1(0)=x_0", "a3959f9959c1067fa6f5f96e25bddf61": " \\phi^{\\rm sat}", "a395d86ff42a92e7c427d8b16eca2494": "L=\\frac{P-MC}{P}", "a395f792ffed875a4c05ff28898d16fe": "\\rm{sinc(\\Delta_t\\omega)}", "a39608d3f4bcafe8919dd84f85728338": "\\langle 1,f\\rangle_X = \\langle 1,f\\rangle \\quad\\forall f", "a3960ee8eab1c63e4489fc190a9ae527": "\n \\boldsymbol{N}^T = J~(\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma})^T = J~\\boldsymbol{\\sigma}^T\\cdot\\boldsymbol{F}^{-T}\n", "a396e9f9a61ba579a1e2a19098c28049": "0^\\circ0", "a3a0512b39ffd10f7c7acdbfc3a42efe": "\\lim_{\\Delta x \\to 0} c = x_1.", "a3a0b666c4804d18af1600a8fc0ec10b": "N \\cup \\Sigma", "a3a0cf90f45b44d5e070c1039ecfdb30": "L_{j}", "a3a0ea0d923d7ca26577e10e8ca0f453": "\\varepsilon_{0} = \\sup \\{ \\varepsilon \\,:\\, \\delta(\\varepsilon) = 0 \\}.", "a3a0f6afec205644e0e029229006947e": "\n w_B = -\\cfrac{Pa^2b^2}{3LEI}\n ", "a3a169d66ca893c98ad4857be2e4b277": "\\lambda\\ x\\ .\\ e ", "a3a1b30f264409a97ae16d0b5489dc7f": " F\\left( J,K \\right) = \\tilde B J \\left( J+1 \\right) + \\left( \\tilde A - \\tilde B \\right) K^2 \\qquad\nJ = 0,1,2,... \\quad \\mbox{and}\\quad K = +J, ... 0 ... -J", "a3a1b7de356d1ac085a8412b7f510a69": "t \\in F", "a3a215c12175ad066fc712e1674652cd": "\\kappa=\\sigma_{\\rm T}/m_{\\rm p} ", "a3a244b9af0195bedbc352cd4337f5bf": "X_n \\ \\xrightarrow{d}\\ X \\quad\\Rightarrow\\quad g(X_n)\\ \\xrightarrow{d}\\ g(X);", "a3a269007dda8cb018ad34d863da7ff3": "h(u) = h(v)+1 ", "a3a2921df3080daa5b164d8bd664b84a": " T(\\neg A) = \\Box \\neg T(A) ", "a3a2a99c9d2ea83ceefb5f8450840ed5": "f(\\theta,\\phi) = C + C_i n^i + C_{ij}n^i n^j + C_{ijk}n^i n^j n^k + C_{ijkl}n^i n^j n^k n^l + \\cdots.", "a3a2d4b73a30d715d6a13222102b13c5": " D_i ", "a3a33a8003a5ba1b78f2c772780ea58c": " f_A(y)= f_A(y') \\ (mod \\ q) ", "a3a35af3f5e6c20f82084a2c64dab9e6": " y_{t-1}", "a3a379d47bbd69f4570cb2922b41b0ef": " \\Delta y=f'(x)\\Delta x +\\varepsilon \\Delta x = dy+\\varepsilon \\Delta x", "a3a39c65b8e691c550b8a02667b6ca85": "N = n/m", "a3a3b392bec935ff14199d07e6a86e26": "\nL(t) = \\frac{1}{2}\\sum_{i=1}^NQ_i(t)^2\n", "a3a3f952fba5fcdece66112dac7008ad": "s_{t}", "a3a40bfeaf41655424405cbae14ca0ae": "\n\\begin{align}\nx(t+1) &= x(t)\\left[1+b\\Big(y(t),z(t)\\Big) - d\\Big(y(t),z(t)\\Big)\\right],\\\\\ny(t+1) &=y(t)\\left(1+\\gamma-(\\gamma+\\eta)\\Big[1-z(t)\\Big]^{\\lambda}\\right),\\\\\nz(t+1) &= \\frac{g\\Big(x(t),y(t),z(t),p(t)\\Big)}{1+g\\Big(x(t),y(t),z(t),p(t)\\Big)},\\\\\np(t+1) &= p(t)(1-\\chi),\\\\\n\\ &\\ \\\\\n\\!\\!\\!\\text{where,} \\qquad &\\ \\\\\n\\ &\\ \\\\\nb(y,z)&= \\beta_0\\left[\\beta_1 - \\left(\\frac{e^{\\beta y}}{1+e^{\\beta y}}\\right)\\right],\\\\\nd(y,z)&= \\alpha_0\\left[\\alpha_1 - \\left(\\frac{e^{\\alpha y}}{1+e^{\\alpha y}}\\right)\\right]\n \\left[1+\\alpha_2(1-z)^{\\theta}\\right],\\\\\ng(x,y,z,p)&= \\frac{z}{1-z}\\,e^{\\,\\delta z^{\\rho}-\\omega f(x,y,p)}, \\ \\text{and}\\\\\nf(x,y,p)&=xyp.\n\\end{align}\n", "a3a42cc589d19f554b3d9579a36be357": " \\ f = \\ {\\left[{f_L}^m + {f_T}^m\\right]}^{1\\over m}", "a3a49a6049b2845548d6f05b3043b758": "\\mathit{V^*}", "a3a560e046b50c44d1472139e764241e": "\\frac{1}{\\nu+2}\\!", "a3a5975bf575267d48919aed12f80cb8": " \\sum_{n=1}^\\infty \\frac{N_1(n) - N_0(n)}{2n(2n+1)} = \\ln \\left ( \\frac{4}{\\pi} \\right ) ", "a3a5a1c4d728e49ec6d6fe82c30d08be": "(P_0,C_0) (P_1,C_1)", "a3a5bccc9435aa75f445c4a10057d85d": " \\forall n\\!\\in\\!\\mathbb{N}\\; P(n) ", "a3a5d60b1015bf4a1c688a6c751b3030": "|N - (q+1)| \\le 2 \\sqrt{q}.", "a3a5e9f800ed87fadbe3b5405fe4ddfe": "m \\geq \\alpha - 1", "a3a626d15ba1f499f27752a35e064618": "i_r=\\frac{V}{r_m}", "a3a62b7bef5013fce00bf0ca1c9bd7d0": "(\\alpha + \\beta)_i = \\alpha_i + \\beta_i", "a3a648bd8009716f5f84c52d64750cd5": "\n\\lim_{s\\to1} (s-1)\\zeta(s) = \\lim_{s\\to 1} \\frac{\\eta(s)}{\\frac{1-2^{1-s}}{s-1}}\n = \\frac{\\eta(1)}{\\log 2} = 1.\n", "a3a6a1dab1d0c3acdabed2523b9dafd4": "\\Rightarrow \\mathbf{h} \\approx \\left [\\sum_{\\mathbf{x}}\\left [G(\\mathbf{x})-F(\\mathbf{x})\\right ]\\left (\\dfrac{\\partial F}{\\partial\\mathbf{x}}\\right )\\right ] \\left [\\sum_{\\mathbf{x}}\\left (\\dfrac{\\partial F}{\\partial\\mathbf{x}}\\right )^{T}\\left (\\dfrac{\\partial F}{\\partial\\mathbf{x}}\\right )\\right ]^{-1},", "a3a6d69529e9b43bf590645c704b5e93": "[11,6,5]_3", "a3a7104b6f3b5120757c5fd263808c61": "\\real \\, (z)\\leq 0 \\, \\Rightarrow \\, |R(z)|\\leq 1", "a3a71f0a5d46494c8cfc397f2ef6e5a6": "2^2=4", "a3a7344263cad8f575751646aeed41a6": "B^{n|2m}", "a3a74dbb2c6a4d9df3336073e5bd4b9a": " Ran (- \\frac{i}{2} ( \\tilde{U} - 1)) = Ran ( \\tilde{U} - 1) ", "a3a758b1d64fb9a5515f47f56515600e": "\\mathfrak{sl}", "a3a7881432ae42066e39214c21827901": " \\operatorname{let} x: x\\ f = f\\ (x\\ f) \\operatorname{in} x ", "a3a7aa280f85b998a544fa78dfcfb3e5": "\\hat{\\mathbf{a}} = (a_1, a_2, a_3)", "a3a7b4450fca8ce75726a61ae2a77f47": "(p \\to \\neg q)", "a3a7bd0b15313f695f02e915c3ee8fef": "\\mathbb{W}^\\mathbb{T}", "a3a7d42cac1d5f7989044abf205b4579": "\\begin{align}\n t(\\vec{r}) & = \\int \\frac{p^2}{2m_e} \\ n(\\vec{r}) \\ F_\\vec{r} (p) \\ dp \\\\\n & = n(\\vec{r}) \\int_{0}^{p_f(\\vec{r})} \\frac{p^2}{2m_e} \\ \\ \\frac{4 \\pi p^2 } {\\frac{4}{3} \\pi p_f^3(\\vec{r})} \\ dp \\\\\n & = C_F \\ [n(\\vec{r})]^{5/3} \n\\end{align} ", "a3a87316e888dac81f360910b2933e72": "\\text{Calculate mutation rates.}", "a3a8ea5b064e82c6e63114780ffbebee": "=\\frac{\\sqrt 2}{\\pi}\\frac{Z_{DP}^{2} m^{*\\frac{3}{2}}kT}{\\rho \\hbar ^{4}c^{2}} \\sqrt{E-E_{CB}} \\; \\; (17) ", "a3a9ba8dba4d007b187a601629fa1d47": "R \\in\\sigma", "a3a9f33579b24ed7964e910ef138c25d": "SS_\\text{err}", "a3a9fb474840f8cdeee65d6bad12e9c0": " \\lambda_D = \\sqrt{\\frac{\\varepsilon_0 k_B/q_e^2}{n_e/T_e+\\sum_{ij} j^2n_{ij}/T_i}}", "a3a9ff0bb19b4e91c2ca869c37e10bfd": "y^2 = x^3 + ax + b\\pmod n", "a3aa2ab6d5e67fff3987616052d52f22": "\\dot{\\hat x}=F(\\hat{x},u,y)", "a3aa6c4464c4ba22fc9b81b0d33e9df4": "(bs^{-1}(x''), f'' - (d(x'') - d(bs^{-1}(x''))))", "a3aa8a2790cb35d128d338dca1f68806": "P(s)\\sim \\log\\zeta(s)\\sim\\log\\left(\\frac{1}{s-1}\\right)", "a3aaa5fe0cb032066154f8659e49a126": "\\scriptstyle \\vec F ", "a3aaa7edee323fdc7aa971e24e387f73": " S^4 \\cdot X^3 \\approx 1.4953 \\approx 696.6 \\ \\hbox{cents} ", "a3aae91801aed850279cac2ce59c1904": " r_i(\\sigma_{-i}) = \\arg\\max_{\\sigma_i} u_i (\\sigma_i,\\sigma_{-i}) ", "a3aafcfe4532dc1ab6447416801100e1": "\\operatorname{E}[\\mathbf{z}] = \\mu", "a3ab68b8be7a1ab22b8c2305f8f8345e": "x, 0 \\leq x \\leq 2^{n-1}", "a3aba657770eb47439760eea30fb005a": "RPM = {Cutting Speed\\times 12 \\over \\pi \\times Diameter} = {12 \\times 100 ft/min \\over 3 \\times 10 inches} = {40 revs/min}", "a3abfb854cdc29eb68fa2fdcb0651466": "=\\left(\\left( 1-\\frac{1}{\\sqrt{d+1}} \\right)^{\\sqrt{d+1}}\\right)^\\sqrt{d+1}+\\frac{1}{\\sqrt{d+1}}", "a3ac37ce413b6a35df333e031e20e0f9": "\\begin{matrix} {2 \\choose 1}^2{44 \\choose 1} \\end{matrix}", "a3ac5a160d79b9f91c765f98d6e1ed72": " a = a_u = { 0.8854a_0 \\over Z_1^{0.23} + Z_2^{0.23} }", "a3ac8a591a068c7521d8b33f75710240": " \\int_{-\\infty}^{\\infty} \\exp\\left[ -{1 \\over \\hbar} \\left( f\\left( q \\right) \\right ) \\right] d^nq \\approx\n\\exp\\left[ -{1 \\over \\hbar} \\left( f\\left( q_0 \\right) \\right ) \\right] \\sqrt{ (2 \\pi \\hbar )^n \\over \\det f^{\\prime \\prime} } ", "a3acb2a8b438f9f0d945570c34fc2654": " Y_1 ", "a3acb8bc177c73543e55bb9359a319bb": "AE = CE", "a3acbabf0baa98988748c3a733aed0d8": "\n\\int_0^\\infty f(r)g(r)r\\operatorname{d}\\!r = \\int_0^\\infty F_\\nu(k)G_\\nu(k) k\\operatorname{d}\\!k.\n", "a3ad068c726b388d505d50bc4496aa3e": "\\langle u, \\Delta u \\rangle = \\int_\\Omega u(x) \\Delta u(x) \\, \\mathrm{d} x = - \\int_\\Omega \\big| \\nabla u(x) \\big|^{2} \\, \\mathrm{d} x = - \\| \\nabla u \\|_{L^{2} (\\Omega; \\mathbf{R})} \\leq 0,", "a3ad10374b07f320b9315f45ffb90a42": "\\begin{align}\n 2U\\frac{ky}{rd} &= \\omega \\\\\n \\delta E &=\\frac{1}{8} m d^2 \\omega^2\n\\end{align}", "a3ad53ae8c02a511a3bc5609d5ce7404": "\n\\begin{align}\nd\\, {\\rm tr}(\\mathbf{AXBX^{\\rm T}C}) &= d\\, {\\rm tr}(\\mathbf{CAXBX^{\\rm T}}) = {\\rm tr}(d(\\mathbf{CAXBX^{\\rm T}})) \\\\\n&= {\\rm tr}(\\mathbf{CAX} d(\\mathbf{BX^{\\rm T}}) + d(\\mathbf{CAX})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}(\\mathbf{CAX} d(\\mathbf{BX^{\\rm T}})) + {\\rm tr}(d(\\mathbf{CAX})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}(\\mathbf{CAXB} d(\\mathbf{X^{\\rm T}})) + {\\rm tr}(\\mathbf{CA}(d\\mathbf{X})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}(\\mathbf{CAXB} (d\\mathbf{X})^{\\rm T}) + {\\rm tr}(\\mathbf{CA}(d\\mathbf{X})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}\\left((\\mathbf{CAXB} (d\\mathbf{X})^{\\rm T})^{\\rm T}\\right) + {\\rm tr}(\\mathbf{CA}(d\\mathbf{X})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}((d\\mathbf{X})\\mathbf{B^{\\rm T}X^{\\rm T}A^{\\rm T}C^{\\rm T}}) + {\\rm tr}(\\mathbf{CA}(d\\mathbf{X})\\mathbf{BX^{\\rm T}}) \\\\\n&= {\\rm tr}(\\mathbf{B^{\\rm T}X^{\\rm T}A^{\\rm T}C^{\\rm T}}(d\\mathbf{X})) + {\\rm tr}(\\mathbf{BX^{\\rm T}}\\mathbf{CA}(d\\mathbf{X})) \\\\\n&= {\\rm tr}\\left((\\mathbf{B^{\\rm T}X^{\\rm T}A^{\\rm T}C^{\\rm T}} + \\mathbf{BX^{\\rm T}}\\mathbf{CA})d\\mathbf{X}\\right) \n\\end{align}\n", "a3adc4537eb8004fa4853c1fcae6d387": "\\begin{align}\nx(t) &= A\\cdot \\cos( 2 \\pi f t + \\varphi ) \\\\\ny(t) &= A\\cdot \\sin( 2 \\pi f t + \\varphi ) = A\\cdot \\cos\\left( 2 \\pi f t + \\varphi - \\tfrac{\\pi}{2}\\right)\n\\end{align}", "a3adc89630b90ee802144a73fb41b695": "(\\hat{x}, \\hat{y})", "a3addd9893343be503dabb31a527bcc8": "(y-y')^2 = y^2 + y'^2 -2 y y' \\,", "a3ade16eea5b796484ec6692f7e8e481": "\\textstyle 6^2 = 36 \\equiv 1 \\equiv 1^2 \\pmod{n}", "a3ae540e675cd84a163d477ce85f5501": "B_n(x)=\\sum_{k=0}^n {n\\choose k}b^{n-k}x^k=(b+x)^n,", "a3ae80949e5e48ce6e7440811810a6d5": "0 < e < 1", "a3ae827d4a86e50fc322bf6c224642ef": " i:X\\to X^{\\star\\star},\\quad i(x)(f)=f(x),\\quad x\\in X,\\quad f\\in X^\\star ", "a3aebbe96544289337e2ecd39049ca56": " \\omega ", "a3aee9924f37b1fc41e25febb599b3c2": "\\Delta k=k_3-k_1-k_2", "a3af39badb9ca2ce07dfe96b41d8b27a": " 0.06/n ", "a3afdeadd2629e037232969da4ec7121": "(A, B, C, D, E, F)", "a3b004a6c8583a0a724d636d27a311f5": "f(\\mathbf{AA}) = p^2", "a3b03f9704912a0aab848ae36997e74b": "\\text{if} n \\text{is even}", "a3b090e09e4c7bf4444992091b9cacc8": " F = m(t) \\frac{dv}{dt} + v(t) \\frac{dm}{dt}. \\qquad \\mathrm{(wrong)}", "a3b094d2ea4e94e25f748f9fb949e06a": "L_1(G)", "a3b0a80e8f8a0981eb4a5e2add65d1b5": "f : F \\to M", "a3b0be12840772bf901b6be3eaf41464": "T(a)=\\{t_2,t_3,t_5,\\ldots\\}\\;", "a3b0dcc889efcfeb00f89e46838b8e60": "\\zeta(s)=s\\int_{-\\infty}^\\infty e^{-\\sigma\\omega}(\\lfloor e^\\omega\\rfloor - e^\\omega)e^{-it\\omega}\\,d\\omega.\n", "a3b0f2bb81f99c8fa0c911bf7ad335ea": "\\scriptstyle C_T", "a3b0f323500676a1ecb878427114313e": "\\frac{P V_{\\mathrm{m}}}{RT} = 1 + \\frac{B}{V_{\\mathrm{m}}} + \\frac{C}{V_{\\mathrm{m}}^2} + \\frac{D}{V_{\\mathrm{m}}^3} + \\dots", "a3b1088eef4d992a73ce82204cee65f5": "2000 C_{rr}", "a3b11802e7c2302f61897e15238ed391": "y^2-2Rx+(K+1)x^2=0", "a3b119d1f379727e2adafe12de2ad634": "(A \\land B) \\to C", "a3b1360f35b1e3e5d6aa453cccf81dda": " y_t", "a3b139105ce462635b34d769c4ced275": "\\mathbf{v}=v_1\\mathbf{i}+v_2\\mathbf{j}+v_3\\mathbf{k}", "a3b15ed8a55e52b881b02c73b9800a79": "R I \\subseteq I", "a3b1b0546ad65d0208f8809e233611b4": "U(\\xi)=\\int{p(y|\\xi)U(y,\\xi)dy}\\, ,", "a3b1b2fb1b995e4ba348fc82b2c8891f": "\\frac{Db}{G}", "a3b25bcc9bf1106146a178682229f004": "\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}", "a3b26124c9aad423ee9aeedd9cee9e9a": "q(x) \\ne 0\\,", "a3b291327878dc4ed0eccd2d2ba24c7d": " P_a = a a^\\mathrm{T} = \n\\begin{bmatrix} a_x \\\\ a_y \\\\ a_z \\end{bmatrix}\n\\begin{bmatrix} a_x & a_y & a_z \\end{bmatrix} = \n\\begin{bmatrix}\na_x^2 & a_x a_y & a_x a_z \\\\\na_x a_y & a_y^2 & a_y a_z \\\\\na_x a_z & a_y a_z & a_z^2 \\\\\n\\end{bmatrix}\n", "a3b2c884479e443ee5af4b0941e6d236": "Q = \\{0, 1\\} \\cup \\{q_1, \\ldots,q_m\\}", "a3b2d9cf4098d0681bc96db753316a49": "f(x)\\geq g(x)\\,\\!", "a3b2e802a74e256a0b8a5bddc01ea12c": "\\Delta mv = \\Delta m \\times \\frac{dy}{dt}", "a3b30ecf3a5e2581a2ee1c5d1da4a0f7": "\\frac{[S]}{K_M} = W \\left[ F(t) \\right]\\, ", "a3b3173e7876571298dd71e3d9497f10": "\\begin{align}\n \\sigma &= \\sqrt{\\frac{m-1}1 \\cdot \\frac{m-2}1 \\cdot \\frac{k-1}{k-3}+\\mu-\\mu^2} \\\\&\n = \\sqrt{\\frac{(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^2}} \\\\&\n\\end{align}", "a3b37ad33f5e572d6442d0805f50658e": "x = t_1, \\;\\;\\; y = 0, \\;\\;\\; z = t_2.", "a3b38d3c3aceab13c1dd84647b5d18b5": " \\theta_3=-36.53^\\circ", "a3b38eff78393b9260d79befbfb2633e": "\nP^T \\Delta_\\mu P=v\\Delta_v, \\quad(11)\n", "a3b3b2382bb48e3d3700d1e47a06a23b": "\n\\begin{align}\nA-A'\\;\\approx\\; B-B'\\;\\approx\\; C-C'\\;\\approx\\;\\frac13 E\\;=\\; \\frac13\\Delta,\\qquad a,\\;b,\\;c\\,\\ll\\, 1.\n\\end{align}\n", "a3b3dcc7635b072d3ca04c3039f757ba": " 2 x + 1 = 5 ", "a3b431f011437487fed485224d771edb": "\\scriptstyle \\Gamma(TM)\\times\\Gamma(TM) \\;\\rightarrow\\; \\Gamma(TM)", "a3b482bce965bc206dc166f0648eed8c": " \\mathbf{C} = \\sum_{i=1}^3 \\lambda_i^2 \\mathbf{N}_i \\otimes \\mathbf{N}_i \\qquad \\text{and} \\qquad \\mathbf{B} = \\sum_{i=1}^3 \\lambda_i^2 \\mathbf{n}_i \\otimes \\mathbf{n}_i\\,\\!", "a3b4fabf01cce018fdc4214509ee44c8": "N, e", "a3b4fb260e10569bd60236284ca560be": "4/18 \\approx 22%", "a3b5692dd9112331f8ac0e2c714eedb6": "f^{8} g h^{*} k", "a3b5f5ff8c3db8b769833275d4546f27": " \\frac{\\Delta r}{\\lambda} = \\frac{\\Delta t}{T} = \\frac{\\phi}{2\\pi} \\,\\!", "a3b70e7cbbca002b8d6083684ad9a6cf": " R_{ij} = |R_i - R_j|", "a3b7208e9967a13a693dfe129919ec3e": " a = {-\\infty}", "a3b73327844b5a48e1979d75965dacd9": "\\scriptstyle{RR}", "a3b734c039beb3de92d78006f8c77d62": "\\Delta V = \\sqrt{ V_{t,a}^{2} + V_{GEO}^{2} - 2 V_{t,a} V_{GEO} \\cos \\Delta i}", "a3b7444bfe020e2de320c635ea86b076": "\\frac{d^2E(Q)}{dQ^2}<0 .\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(4)", "a3b76694a37484bae752b3adfd32bad2": "y=\\left( R+r \\right)\\sin \\theta -r\\sin\\left( \\theta+\\alpha \\right) =\\left( R+r \\right)\\sin \\theta -r\\sin\\left( \\frac{R+r}{r}\\theta \\right)", "a3b7892dafa5afeab60ff497a67c1813": "\\left.\\frac{dy}{dx}\\right|_{x=c} = \\left.\\frac{dy}{du}\\right|_{u = g(c)} \\cdot \\left.\\frac{du}{dx}\\right|_{x=c}.\\, ", "a3b7e7f5c313e12caa0779b352487cf2": "y_i^{\\prime\\prime} \\leftarrow", "a3b86a09e2a7adb788217932af9cb90d": "\n\\frac{n \\,\\,\\mathsf{nat}}{\\mathbf{s(s(}n\\mathbf{))} \\,\\,\\mathsf{nat}}\n", "a3b87affd993d8f835e797443c05ab5c": "(-15, -26)", "a3b88dcebc7d2a07aed852a7303e02a8": " t \\cdot u(t)\\ ", "a3b8969b4d7e264c6d0d1b148e51beef": "[0,1]\\times[0,1]", "a3b8a5a3ed1332a27735cd742405352e": " \\{B_{j}:j\\in J\\}", "a3b8a82dd9da674cfecd360dc35347bb": " \\mathbb{E} ( \\mathbb{E} ( |Z| | X ) ) = \\int_{-\\infty}^{+\\infty} \\mathbb{E} ( |Z| | X=x ) f_X(x) \\, \\mathrm{d}x = \\mathbb{E} (|Z|), ", "a3b8e3f4898688680f41df3e276b5d85": " \\rho = \\frac{dE}{dJ}\\, ", "a3b948278d4337dcab0d0d2c5180fe34": "\\nabla\\circ\\partial=\\partial\\circ\\nabla", "a3b94f669a8db22ce8228c948878c169": "\\tau=RC", "a3ba545280e21e3bac2162342656c33e": "\\overline{3+4i} = 3-4i", "a3ba7c457b26743f736a15be6983d329": "~ \n\\frac\n{{\\rm d}n_2}\n{{\\rm d}t}\n=\nW_{\\rm u} n_1 -\nW_{\\rm d} n_2\n~", "a3ba8552b3f34fb369be3f10b8f34cb2": "I^-", "a3bab561aef5e0efa0f9a63bef69cf87": "(Fx_1...x_n \\and Ix_1y) \\rightarrow Fyx_2...x_n.", "a3bb053fd060e9198353929b0b31e4f0": "(X-\\alpha)^2", "a3bb064491e110d7b114a6a7e26512c0": "M_{21} = N_1 N_2 P_{21} \\!", "a3bb1f5b2fa1c8d254f6c7cd1287c2f1": "y=e^t.", "a3bb561c4fcf00c01a137b70bcb5c80c": "((p \\land q) \\Rightarrow r) \\iff ((p \\Rightarrow r) \\lor (q \\Rightarrow r))", "a3bb58c57ed1ea994b4230a7bdf9d2fe": "\\phi_j\\approx 0", "a3bb823f79249f6eb7207fc0fd2d1b8d": "E(z)=\\sqrt{\\Omega_m(1+z)^3+\\Omega_k(1+z)^2+\\Omega_\\Lambda}", "a3bb93d8fd45438fbab83f3d62ad091c": " \\frac{\\hbar}{i}\\frac{d}{dq}\\psi(q) ", "a3bbaea66df5346c8aca1b3b25eb8017": " \n R_{sp} = { C_p - C_v }", "a3bbf5bdb2eeda588bd4fb6068ad2d84": "\\mathbf{H}=\\{\\mathbf{h}_i \\vert^{N}_{i=0}\\subset \\mathbb{R}\\}", "a3bc436e965afdd78c88dcdbbbaeaf2b": "D=\\{(x_1,y_1),(x_2,y_2),\\ldots,(x_N,y_N)\\}", "a3bc7b3be7976500a1fef34770796729": "\\sum_i a_{ij}=\\sum_j a_{ij}=1", "a3bc8788054338b398a411943d5f6588": "c_{(uv)}", "a3bd28ee41c6badd83fb6c755c046ebd": "U(k)", "a3bd3f3bc862f688943ff630d65311bb": "\\sec A = \\csc A \\cdot \\tan A \\ ", "a3bd5d48e831b76f28e371d4eadba068": "n > n_0", "a3bd5f8ecff0d864456dfc103d103941": "\\cot\\frac{\\pi}{4}=\\cot 45^\\circ=1\\,", "a3bd712473e5bdfc9f4a03cfa1a7038f": "|A|=\\sum_{x\\in X} \\mathbf{1}_{A}(x).", "a3bdc4c0d2568b2a5af86214ccdb45c3": "KK^{-1}=K^{-1}K=I_2", "a3beb40f73406888d5e4014d36a347ca": "\\omega_{+}", "a3bec3f4d57c926b98eb640ce491e3c1": "e^{-x}\\,", "a3bee0cc490ea95caa6bf436679d1a3e": "P! ", "a3bee39db6270383afbacb5d778b7df3": "\\xi = \\frac{x}{X_0}", "a3bf04cf70f52267e2c86bab141ab3a7": "0_R", "a3bfac07abbcd8f8aebb9986dab9e952": "receipt\\;date - due\\;date", "a3c02fdf3dd1f861f44e1cc03b10df42": " \\sigma\\,", "a3c03ff2eb8762bf5fed118e5e3ba4a6": "\\phi(\\vec{x},t)=\\pm\\frac{m}{2\\sqrt{\\frac{g}{4!}}}\\tanh\\left(\\frac{m(x-x_0)}{\\sqrt{2}}\\right)", "a3c0dee472ec579ea47da1a80b60606e": "\\scriptstyle \\vec{F}_{1}", "a3c0eb14db513bf2c14856b4e23e7294": "dp / dt", "a3c0ee493e0b12d104fa91b10170488a": " 0\\leq u(x_0) = \\max_{x \\in \\overline{\\Omega}} u(x)", "a3c111540e49c50306523ffd4befa123": "\\left\\{\\mathcal{M} f\\right\\}(s) = \\left\\{\\mathcal{B} f(e^{-x})\\right\\}(s).", "a3c237718fcc1e4253be44e825ece0c2": "\\frac12n(n+1)", "a3c2456f2a72a0f76f4850854a514f8b": " n \\geq 3 ", "a3c263b4993c57aec8290771d187c3ff": "y^2,\\,\\!", "a3c2f48ca1baf5fd961702f7bc5401a6": " k = 1 \\ldots L", "a3c31adcb6c744b65d17590d58f927c7": "M_{i,j} = 1", "a3c31b4cc7752f6e0a5bca0382093344": " \\Delta=d^*d", "a3c327307b91756128ddc6cc6ee0d3bc": "\\scriptstyle \\left(\\alpha_i\\right)_{i\\in I}", "a3c3597e5189298bf78cf58cb0a79bdb": " \\tan \\theta = \\frac {\\mathrm{opposite}}{\\mathrm{adjacent}} = \\frac {a}{b}", "a3c395d5e460b073d696b9182c70fd53": "\\Gamma = V \\left(\\frac{dp}{de}\\right)_V", "a3c3da1a08fa09cc25191f27b09acaab": " \\Pr(T > T_i|\\theta) = 1 - F(T_i|\\theta) = S(T_i|\\theta) .", "a3c4299a3e455719219f8f53f06600e7": "10^2 - 92(1^2) = 8", "a3c43516f89e72ed8d28840529d0f4f8": "a_1,...,a_n", "a3c4387ca8f10503887fb5d0722ae55a": " S_n = 2 \\left(\\frac{2}{\\pi}\\right)^{n}\\sum_{k=-\\infty}^\\infty \\left(4k+1\\right)^{-n} \\quad (k=0,-1,1,-2,2,\\ldots) ", "a3c454837971b2b1c70709933bcb681b": " n_1\\sin\\theta_1 = n_2\\sin\\theta_2 \\, ", "a3c481b93100f51e21d265204ac52cbd": " \\Pi = \\prod_{i=1}^n \\|\\bold{v}_i\\|", "a3c4b8dde9862d2e43bcc86c68d5bac1": " w^{(2)}(r) ", "a3c4d3b142d1f8c039fc35ab2f73f85f": "R[x \\leftarrow 0] \\wedge R[x \\leftarrow 1]", "a3c5002e3eecb8c7ba8c601a3654b7f2": "\\frac{f\\left(x_1\\right) + f\\left(x_2\\right)}2", "a3c54a4379b55fa384439b33bcf58f5f": "\\tau_{ix} : \\pi^{-1} (x) \\to X", "a3c5792a6553b6d86da7350068b0371a": " E_1 = \\{(x,y) \\in \\R^2 : y = x^3 \\} \\ . ", "a3c5e0ade4112b6869b3b451a19f54f2": "\\hat{f}.", "a3c5edf9a6371b876781d64ade3f5318": " \\| f-X(f) \\| \\le \\| f-p^{\\ast} \\| + \\| p^{\\ast} - X(f) \\| \\,\\!", "a3c604721efc296b2972c035b905d1dc": " \\sigma^{2}( \\langle A \\rangle ) = \\frac{1}{M} \\sigma^{2}( A ), ", "a3c613c2d9f7d6416e60f686b006a273": " \\frac{d^2u}{dx^2} - x\\frac{du}{dx} + u = 0. ", "a3c616f85cd40487906672cee3bc2029": "1/d= \\frac{ \\sin(\\alpha + \\beta)}{\\ell\\ \\sin\\alpha \\sin\\beta }", "a3c63b464464a83078fb858659dce70f": "\\alpha < \\gamma", "a3c64d2768ecf6eb6310b4a84d26d1ef": " y_{ij} = \\mu_i + \\epsilon_{ij}", "a3c688e7da81a551df5623d32fc33dad": " {\\dot{m}}_B ", "a3c6a789df4017a40242383ae2ffa75f": "\\ln \\left ( \\frac{\\varepsilon^{(s+1)}}{\\varepsilon^{(s)}} \\right ) = 2 \\left ( k^{(s)} + x^{(s)} - 1 \\right ) \\delta^{(s)} \\Omega^{(s)} ", "a3c6bbbd5af14bb9c1033d88a508daf6": "\\frac{1}{\\sqrt{3}}(1,1,1)", "a3c6c93f7f759a2946112d2a268d6e26": "M_0(x_1, \\dots, x_n) = \\sqrt[n]{\\prod_{i=1}^n x_i}", "a3c6dc03c0e9aefc02fc1de2a98507de": "\\Pr(\\mathbf{x}\\mid\\boldsymbol{\\alpha})=\\int_{\\mathbf{p}}\\Pr(\\mathbf{x}\\mid \\mathbf{p})\\Pr(\\mathbf{p}\\mid\\boldsymbol{\\alpha})\\textrm{d}\\mathbf{p}", "a3c74f30a4add43b8d35da1a566187dd": "\\eta = 0\\,", "a3c7a1cda21e6ba252f1ff25f284a2d7": "Q_a", "a3c7b0562d966cd17cbeaea105f9468d": "\\chi_{\\rho \\otimes \\sigma} = \\chi_\\rho \\cdot \\chi_\\sigma", "a3c7cd6d147f99f38d330928bcd64a90": " s(0) = s_o ", "a3c8000d48ccd8c31e3ab08a7d25aed0": "\\partial_k=\\frac{\\partial}{\\partial x_k}", "a3c808647101f51f4f1f28d3bf520d7c": "a_0\\approx 1\\times 10^{-10} {\\rm m\\ s}^{-2}", "a3c83255fa0df26e704e1d62139cb740": "r_i=as_i+bt_i", "a3c8406cf9df8ff7fc2da4a2b76d53e9": "\\scriptstyle{\\pi(x)}", "a3c8b46e3d6bf2fffa254a26c039a8d7": " P = 1.58 \\times 10^9 / {100^{7/5}} = 1.58 \\times 10^9 / 630.9 = 2.50 \\times 10^6 \\operatorname{ Pa} ", "a3c90288fe268412045cbc13aa3d8b14": "\\mathbb{E}\\left [(H\\cdot M_t)^2\\right ]=\\mathbb{E}\\left [\\int_0^t H^2\\,d[M]\\right ].", "a3c90bf0bb930dba010a4b1baf55fc0d": " \\{ |f_{k_0} \\rangle, |f_{k_1} \\rangle \\} ", "a3c921783bc3bbdf63bb94cd293f20b4": " \\Omega = 2 \\left( \\arccos \\frac{\\sin\\gamma}{\\sin\\theta} - \\cos\\theta \\arccos\\frac{\\tan\\gamma}{\\tan\\theta} \\right) ", "a3c98432d5cf8209275f1c63ba850b7f": " \\nabla\\Psi = \\dfrac{i}{\\hbar}\\mathbf{p}Ae^{i(\\mathbf{p}\\cdot\\mathbf{r}-Et)/\\hbar} = \\dfrac{i}{\\hbar}\\mathbf{p}\\Psi ", "a3c9a0e35f77fd4d45becc7fe781c4b1": "\\Gamma(-n, z)", "a3c9a8f79a60f714b1ce06f5166f583d": "\\begin{align}\nD_{CH} = \\frac{2}{\\pi} \\sqrt{2(L-\\sum \\limits_{l}\\sum \\limits_u \\sqrt{X_{u}Y_{u})}}\n\\end{align}", "a3c9f74a1bd28259318f2f2728783fe0": "\\begin{array}{rcl}\n e(p) &=& e_{\\mathrm v}(p)\n\\\\ e(\\bot) &=& 0\n\\\\ e(\\top) &=& 1\n\\\\ e(A\\otimes B) &=& e(A) \\ast e(B)\n\\\\ e(A\\rightarrow B) &=& e(A) \\Rightarrow e(B)\n\\\\ e(A\\wedge B) &=& e(A) \\wedge e(B)\n\\\\ e(A\\vee B) &=& e(A) \\vee e(B)\n\\\\ e(A\\leftrightarrow B) &=& e(A) \\Leftrightarrow e(B)\n\\\\ e(\\neg A) &=& e(A) \\Rightarrow 0\n\\end{array}", "a3ca04f71eb07674b4b7fddd6ea37388": "|\\psi(x)-x|\\le\\frac{\\sqrt x\\,\\ln^2 x}{8\\pi}", "a3ca433870a89bef42741e02cd5442f4": "I\\subset k[x_0, x_1, \\ldots, x_n]", "a3ca5c10a50b0253333ba55fdbe7d604": "M(S)", "a3ca75f1587dacf6abb6a257c6f72b0a": "\nZ = \\frac{p}{vS} \\,\n", "a3cab56e2c110d03272b56edce03bf52": " S_{ab} = R_{ab} - \\frac{1}{n} \\, g_{ab} \\, R", "a3cb37f67447d008f83ed9318d29cfbb": " f'( q ) > 0 ", "a3cbb5ca8de1b41445eec252fc8c7879": "a=4mn \\, ", "a3cbe6d955b6e3d433b169094f7237b8": "T: D \\subset X \\to X", "a3cbe72badc6a5e482ba0d642aa689fa": "d(n)", "a3cc20a7ffddc1b645c432372dc6558e": "= \\frac{-1}{i\\hbar} \\langle \\psi | HQ - QH | \\psi \\rangle \\,", "a3ccea62eb4fe76cd504e7626475ff5c": "1. \\quad \\forall (a_n) \\in \\ell^2,\\ \\ c\\left( \\sum_n | a_n|^2 \\right) \\leq \\left\\Vert \\sum_n a_n \\varphi_n \\right\\Vert^2 \\leq C \\left( \\sum_n | a_n|^2 \\right)", "a3ccfe2b769c43a931932918f296fab0": " H_0 ", "a3cd042e92f72342ab3af8c13b9b44f0": " P / {\\rm W} = \\frac{ \\tau / {\\rm (N \\cdot m)} \\times 2 \\pi \\times \\omega / {\\rm rpm}} {60} ", "a3cd4baffe8cda15a4a29a4472518b41": "G(z_{t}, \\zeta, c)x= (1+exp(-\\zeta(z_{t}-c)))^{-1} \\zeta>0 ", "a3cd5386ae68b7868f50948d2db5e82e": "b = {\\ln{\\varphi} \\over \\theta_\\mathrm{right}}.", "a3cdd44db5dee129522167fed66e6a50": "\\scriptstyle x \\;\\in\\; U_n", "a3cde27b017f50f4d872859334ad2c70": " t \\propto B \\times \\lambda^3 ", "a3ce06fccc9010fe6f067266612bd50e": "\\epsilon(t)= \\frac { \\sigma(t) }{ E_\\text{inst,creep} }+ \\int_0^t K(t-t^\\prime) \\dot{\\sigma}(t^\\prime) d t^\\prime", "a3ce0965f4c5657e4317d09c1ae2da80": "i + j + 2ij \\le n", "a3ce25ca81d9d7dc5a25e0c2ef269909": "sin^{-1}", "a3ce441fe0a1eb505a86cc391a5d5e62": "\\bold{r} = \\bold{r}(q_i, t). \\, ", "a3ce6046592225da14494baa37116577": "\\frac{\\bar{P}}{N_o}=10^6", "a3ceac0c52b9944759ff1993e69bd639": "\\beth_n", "a3cf4f8fccf5137b3f8777d959e33887": "(b,a)", "a3cf50b0135fd406a85f4c04fa180326": "q = \\sum_{i=1}^N S^\\alpha_i S^\\beta_i \\neq 0", "a3cfb83a866333d027179a555a2bb9ff": " T(r,f')\\leq 2 T(r,f)+S(r,f),\\,", "a3cfe0be8992d6b2cb4fb89483b6757b": "Q^{(0)}", "a3d0018c7924764845ff57d1a725f559": "V_i(\\mathbf{x}_i) = V_{i-1}(\\mathbf{x}_{i-1}) + \\frac{1}{2}( z_i - u_{i-1}(\\mathbf{x}_{i-1}) )^2", "a3d023ceca8d4282f3f0a8e69e317c7c": " x_1 \\ge 1 ", "a3d04db8757e8ea3c14adb24b778f298": "(q/m)(\\vec E+\\vec v_s\\times \\vec B)=\\part \\vec v_s/\\part t + (1/2)\\vec \\nabla v_s^2-\\vec v_s\\times(\\vec \\nabla\\times \\vec v_s).", "a3d0d2ebdc09b90d4c84c88df361c803": " \\ Cxx.", "a3d15b3dd78538505faf0bc04875df98": "{\\theta}", "a3d15c5abc3371c0442888b95aab962a": "\\mathcal M_\\omega \\models^+_{\\{\\emptyset\\}} \\!\\phi", "a3d19505c0755c047883a7c367dfe3b8": "\\bar{\\partial} = \\frac{1}{2} \\left(\\frac{\\partial}{\\partial x} + i \\frac{\\partial}{\\partial y} \\right)", "a3d19f5e287a71ca221d4ad192c2b8ed": "T_h = 2T_{h-1} + 1", "a3d1a8c21258fd93d4b950964969e905": "B(\\widehat{\\theta}) = 0", "a3d1c090f74dd909847e2b4ff58c0ba7": "\\tbinom{n}{r}=\\tbinom{n-1}{r}+\\tbinom{n-1}{r-1}", "a3d260cc1c3de678f10018ae96423b22": "(d\\mathbf{X})\\mathbf{Y}+\\mathbf{X}(d\\mathbf{Y})", "a3d2a963cf55777b279257ca2f205d1f": "\\tau \\mapsto \\tau_V", "a3d2add141fdc99c8980214407345368": " D(2,1.2) ", "a3d2b541ff49fbd13abc6b8aa5a3fddb": "\\tilde{A}(\\omega, z+h) = \\exp\\left[{i \\beta_2 \\over 2} (\\omega-\\omega_0)^2 h \\right] \\tilde{A}_N(\\omega, z).", "a3d2f20f021de7a32aab9f7f4538e3c2": "\\left\\langle H', \\varphi \\right\\rangle = - \\left\\langle H, \\varphi' \\right\\rangle = - \\int_{-\\infty}^{\\infty} H(x) \\varphi'(x) dx = - \\int_{0}^{\\infty} \\varphi'(x) dx = \\varphi(0) - \\varphi(\\infty) = \\varphi(0) = \\left\\langle \\delta, \\varphi \\right\\rangle,", "a3d31c03801be21bceb008c2b76ffa0b": " dp = k \\frac{dS}{S}, \\,\\!", "a3d33971ffaf994c7db34f893782c0da": " \\psi\\left(\\frac{1}{8}\\right) = -\\frac{\\pi}{2} - 4\\ln{2} - \\frac{1}{\\sqrt{2}} \\left\\{\\pi + \\ln(2 + \\sqrt{2}) - \\ln(2 - \\sqrt{2})\\right\\} - \\gamma", "a3d34463921ea2dd2f448c7992d94281": "\\vec a_R= -\\omega ^2R \\hat u_R \\ , ", "a3d350ea73f3b908fd5fc05ee83864d2": "a^a + b^b \\ge a^b + b^a.\\,", "a3d371397b84fd1c7e455f8ddc7783e3": "\\hat{\\textbf{x}}_{k\\mid n}", "a3d3bad750d3c7cf03c4e2844b1d4192": "R>N", "a3d3cd6115286e3e97caaf93d95a93bf": " W(p,q) = S^p T^q ", "a3d4b22057e4d910ed66589edca38ae2": "\nC =\n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n", "a3d4bbde7701fa0672d116923fde54aa": " C^J_{E_1} ", "a3d54c93b4635195154c64d7f47a3390": "\n\\nabla_iu_i=0\n", "a3d598cf8d921c9485ed4b14e5b99c09": "x = h\\pm a \\; \\cos\\left(\\arctan\\left(\\frac{b}{a}\\right)\\right)", "a3d5e1d4506e027c99c49484f35cfa98": "\n\\begin{bmatrix}\n -26 & -3 & -6 & 2 & 2 & -1 & 0 & 0 \\\\\n 0 & -3 & 4 & 1 & 1 & 0 & 0 & 0 \\\\\n -3 & 1 & 5 & -1 & -1 & 0 & 0 & 0 \\\\\n -4 & 1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}\n", "a3d5fb84150ad8cc0a5e0841104bb49d": "w_2 = x_2", "a3d6450335f98671c651922ead8b4d7a": "K(L1,L2)", "a3d68091fbe52b38f8ab2a0a359ce5e2": "\\langle T, \\alpha\\rangle.", "a3d68693aa975353c1515e2dee7c5455": " x^n dx = \\frac{d ( x^{n+1})}{n+1} , \\,\\!", "a3d68f1ebfc1d0fcde13d4b822b7951b": "\\mathbb{P}(q(X) \\geq q^* | p_\\theta \\leq p_0^* , \\theta) = \\frac{\\mathbb{P}(q(X) \\geq q^* |\\theta)}{\\mathbb{P}(p_\\theta \\leq p_0^* |\\theta)}\n= \\frac{\\mathbb{P}(q(X) \\geq q^* |\\theta)}{p_0^*} = \\frac{\\mathbb{P}(q(X) \\geq q^* |\\theta)}{\\mathbb{P}( q(X) > q^* | 0 )}.", "a3d694d2032d10a397e83b384085f1b7": "f : (X, p, m) \\to (Y, q, n), \\quad f \\in Corr^{n-m}(X, Y) \\mbox{ such that } f \\circ p = f = q \\circ f", "a3d6d416a96786ca2e9eef7115205471": "C = \\infty", "a3d6de68fd14ba7b995428101cc3a0bf": "\nb=\\sum^t_{i=j}R_i\\Phi(x_i)\n", "a3d6f1ebf6aca899444b6e51296b4c19": "\\hat{C}^{(0)}", "a3d70dcdf982cefe433a93ec52502ce1": "\\left[P_a\\right]_{qq^\\prime}", "a3d7224b9d8a47e279caa6a42af8729a": "\\alpha_t(x_t) = \\sum_{x_{t-1}}p(y_t|x_t,x_{t-1},y_{1:t-1})p(x_t|x_{t-1},y_{1:t-1})p(x_{t-1},y_{1:t-1})", "a3d72831ea82f4ef851576a67b9616fa": "\\displaystyle{\\pi(g_1)f(x)=\\pm a^{-1/2} f(a^{-1}x),\\,\\, \\pi(g_2)f(x) =\\pm e^{-ibx^2} f(x),\\,\\, \\pi(g_3)f(x)=\\pm e^{i\\pi/8} \\widehat{f}(x)}", "a3d74bac5e41a27e1a7a5c1b6aa67f99": "\\begin{align}\np(s, t, u) = (\\alpha s+\\beta t+\\gamma u)^3 =&\n\\beta^3\\ t^3 + 3\\ \\alpha\\beta^2\\ st^2 + 3\\ \\beta^2\\gamma\\ t^2 u + \\\\\n&3\\ \\alpha^2\\beta\\ s^2 t + 6\\ \\alpha\\beta\\gamma\\ stu + 3\\ \\beta\\gamma^2\\ tu^2 + \\\\\n&\\alpha^3\\ s^3+ 3\\ \\alpha^2\\gamma\\ s^2 u + 3\\ \\alpha\\gamma^2\\ su^2 + \\gamma^3\\ u^3\n\\end{align}", "a3d758001cbcdee2834e219037ec0ba2": "P_3=(12-2\\sqrt{39},37\\sqrt{3}-18\\sqrt{13})", "a3d7597e8159173890b6d9c431199a22": "\n\\Delta H = \\underbrace{\\Delta G}_{\\textrm{elec.}} + \\underbrace{T\\Delta S}_{\\textrm{heat}}\n", "a3d76425a57eaba2f9f2bdb6674a8081": "2 \\lfloor N / 2^{k+1} \\rfloor + 1", "a3d79e7bd2fd7f560ef01d9dee97de21": "\\frac{128}{105}\\,\\sqrt{\\frac{2}{\\pi}}", "a3d7c35eade2c7174582cf95ddad471d": "\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\\n2 & 5 & 4 & 3 & 1\\end{pmatrix}=\\begin{pmatrix}1 & 2 & 5 \\end{pmatrix} \\begin{pmatrix}3 & 4 \\end{pmatrix} = \\begin{pmatrix}3 & 4 \\end{pmatrix} \\begin{pmatrix}1 & 2 & 5 \\end{pmatrix} = \\begin{pmatrix}3 & 4 \\end{pmatrix} \\begin{pmatrix}5 & 1 & 2 \\end{pmatrix}.", "a3d7eb58f2905086879695f1c187d53f": "Sq^i \\colon H^m(-; \\mathbf{Z}_2) \\to H^{m+i}(-; \\mathbf{Z}_2),", "a3d858e77ba520818d64d564bbecf2ae": "(m-2) \\sigma^2<\\|{\\mathbf y}\\|^2 ", "a3d867e066ebfc48cde1e14cb16d4848": "N_p(x) = \\log|a|+k_1x_1+k_2x_2+\\cdots+k_nx_n.\\,", "a3d873a421b7fb589e0cae374d0ce99e": "\nU = \\frac{1}{2V_{uc}} \\int \n\\frac{\\left( \\mathbf{p}_{uc}\\cdot \\mathbf{r} \\right) \n\\left( \\mathbf{p}_{uc} \\cdot \\mathbf{n} \\right)dS}{r^3}\n", "a3d8e2e7e16ee2750906808175b8d531": "\\pi(\\mathbf x)q'(\\mathbf x,\\mathbf{x'}) = \\pi(\\mathbf{x'})q(\\mathbf{x'},\\mathbf x)", "a3d9557eefa8e4c15d679c3b3487ba0b": "{4 \\choose 1}\\left[1 + {7 \\choose 6}{3 \\choose 1} + {7 \\choose 5}{3 \\choose 1}^2\\right] = 844", "a3da571d706285897fdee21f0ab48c12": "w \\mapsto \\frac{Rw}{w-2A}.", "a3da5d99e7dca65a5d03d9db86a32c99": "s = \\gamma^2 t", "a3dad9907add2afe37a66b4a75297954": "1.5370", "a3db13667ed9569f145ff306ab77badd": " C^k ", "a3db2653d2cbc4d000ab288c0bb92b74": "G,H", "a3db81be8b7f7f201b4ebcddc967b715": "\\scriptstyle\\boldsymbol{f}(\\boldsymbol{x})=\\left(f_1(\\boldsymbol{x}),f_2(\\boldsymbol{x}),f_3(\\boldsymbol{x})\\right)", "a3db866f56d852d36c500e0db7a7fa1e": "\nH_{p,q}^{\\,m,n} \\!\\left[ z \\left| \\begin{matrix}\n( a_1 , C ) & ( a_2 , C ) & \\ldots & ( a_p , C ) \\\\\n( b_1 , C ) & ( b_2 , C ) & \\ldots & ( b_q , C ) \\end{matrix} \\right. \\right]\n= \\frac{1}{C}\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z^{1/C} \\right). \n", "a3dc0766f4db34be1425ed64f0732c13": "P_{1},P_{2}...P_{n}", "a3dc9899eef0720647e88f974b139cbf": "(r-t)", "a3dcab0bfc1e75a89faae786ca2c9c9f": "\\Gamma(\\tfrac12) = \\sqrt \\pi", "a3dcab17541585f95f9cc1cc16f8e099": "\\overline{\\gamma}", "a3dcd6d2feda67cc010d700ad06c1907": "|\\sec(M_i)|\\le 1", "a3dd6d3c2e41a15579cbdb4bdb35dcb5": "~K(\\chi )= \\frac{1}{2}(1+\\cos \\chi)", "a3ddacf1491edb744ac2a3f37d87e920": "\\begin{align}\n\\hat{\\beta} ({r_{\\rm w}})\n &= \\min_{q \\in \\mathcal{Q}} {\\hat{\\beta}}(q, {r_{\\rm w}})\\\\\n{\\hat{q}_{{\\rm w}}}({r_{\\rm w}})\n &= \\arg \\min_{q \\in \\mathcal{Q}} {\\hat{\\beta}}(q, {r_{\\rm w}})\n\\end{align}\n", "a3ddada8cf38a39624d0db6a029114db": "\n\\delta = m + n - \\tfrac{1}{2} (p+q) ;\n", "a3ddcc344a09ff2ed74f8ce11ec90462": "\\sigma_{ji,j} + F_i = 0\\,\\!", "a3dde50d654ad1e9229bf639b20aae0b": "1,3<{\\frac{L}{m}}<1.6", "a3de00c1597600a387128a7add5b354f": "S_2", "a3de097afa855c9a4f2d06ceb97cf1b0": "M(n)=\\sum_{k=1}^{2^{2^n}} \\prod_{j=1}^{2^n-1} \\prod_{i=0}^{j-1} \\left(1-b_i^k b_j^k\\prod_{m=0}^{\\log_2 i} (1-b_m^i+b_m^i b_m^j)\\right),", "a3de34ef7ba1bb49e0ee07af425afd17": "\\frac{96}{95}", "a3de814cd383bb255318607c73f83282": "P = (x_0,\\ldots,x_n) , \\,\\!", "a3de9283f3c80a23e5d386e8c24df378": "\\displaystyle{B(\\alpha,\\beta)=(-1)^{(\\alpha,\\beta)}}", "a3de9361f354bd21ff0f218cd30633c5": "\\sum_{x \\ge 1}^{\\Re}f(x)=F(1)\\,", "a3de9fa4b4af191488e389f4dc9e4b60": " \\nabla^2 \\psi = \\nabla \\cdot (\\nabla \\psi) ", "a3dfaeca892b841a652228e72b3f41f7": "\\frac{1}{2}\\frac{23^2}{2} = \\frac{23^2}4 = 132.25", "a3dfb593666ccc8f75c0c11d921fac18": "e + 1", "a3e09f261966ce9d3ff3efa2e1346018": " m x\\mod p ", "a3e0b45f30c282b9ff8911d75336607e": "\\displaystyle{M(z)=\\sum_{j\\ge 1} {(-1)^j\\over W^\\prime(2^j) (z-2^j)}}", "a3e0b65217f5ceb8a6423cf7771cd97a": " F_a =\\frac{L}{N} \\frac{1}{\\textit{eff}}", "a3e0deecb4cd81495f2d5e07a613054c": " | \\vec x - \\vec{x_0}\\, |", "a3e13e9d3c4b2a8e5cc035774e787fc0": "\\mathbf{R}^{m \\times n} := \\mathbf{R}^m \\otimes \\mathbf{R}^n \\cong \\mathbf{R}^{mn}", "a3e15368db1869197b48474bbbe33382": "(x \\pm 2y,y), (x \\pm (2y+1),y), (x,y \\pm 2x), (x,y \\pm (2x+1)).", "a3e16d6891d5fcc118f31df6da6ef88a": " C_x = \\frac { a m^{ b - 1 } - 1 } { ( nm - 1 ) }", "a3e174e8aa4cb5911e3a1ef7dc43726d": "I(u) = \\int_0^1 (u_x^2-1)^2 + u^2dx", "a3e1768ef066c48bdbbcc591db9e5fe7": "\\Rightarrow z=\\frac{\\varphi d}{2}.", "a3e19b7b65c997e0d5c01ca0043309f6": "\\int_S {\\mathbf g}\\cdot \\,d{\\mathbf {S}} = \\int_S {\\mathbf g}\\cdot {\\mathbf{\\hat n}}\\,dS", "a3e19c40f2f752639937800f03c22740": "\\langle X \\rangle = \\operatorname{Tr}(\\hat X \\rho).", "a3e373fe464e4085234102dd034aedf3": "\\frac{\\gamma^{2\\lambda} | x - \\mu|^{\\lambda-1/2} K_{\\lambda-1/2} \\left(\\alpha|x - \\mu|\\right)}{\\sqrt{\\pi} \\Gamma (\\lambda)(2 \\alpha)^{\\lambda-1/2}} \\; e^{\\beta (x - \\mu)}", "a3e392da658e8878b7559acfb5f41a9c": "\\omega_{Z}", "a3e3995e08ad1ecba25f241b4f7286af": "M_A = \\tfrac{qL^2}{2}", "a3e3dfa6e0d08284451517e5c446443b": " \\begin{align}\n\\frac{\\mathrm{d}N_A}{\\mathrm{d}t} & = - \\left(\\frac{\\mathrm{d}N_B}{\\mathrm{d}t} + \\frac{\\mathrm{d}N_C}{\\mathrm{d}t} \\right) \\\\\n- \\lambda N_A & = - N_A \\left ( \\lambda_B + \\lambda_C \\right ) \\\\\n\\end{align}", "a3e45a00d77012e0f33f001935d8480e": "\\mathbf{v_3}' = \\mathbf{v_1}' \\times \\mathbf{v_2}' = (R\\mathbf{v_1}) \\times (R\\mathbf{v_2}) = (\\det R)(R(\\mathbf{v_1} \\times \\mathbf{v_2}))=(\\det R)(R\\mathbf{v_3}).", "a3e46a7383a76ff8746d387f9cebaa47": "S(B)_\\rho", "a3e4c5f9e33ee26dd5faa30f22719fea": "\\left(\\frac{\\partial S}{\\partial T}\\right)_{P}=-\\frac{\\left(\\frac{\\partial P}{\\partial T}\\right)_{S}}{\\left(\\frac{\\partial P}{\\partial S}\\right)_{T}}\\,", "a3e4e24e6806870406d2f3f2ac6f5659": "\nI_\\mathrm{total} = {15 \\mathrm{V} \\over 2\\,\\mathrm{k}\\Omega + (1\\,\\mathrm{k}\\Omega \\| (1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega))} = 5.625 \\mathrm{mA}.\n", "a3e4eda8fde9fa55ce877400c9798a89": " \\frac{-\\omega_c}{\\omega_a-\\omega_c}=R,\\quad \\mbox{or} \\quad \\frac{\\omega_a}{\\omega_c}=1-\\frac{1}{R},\\quad\\mbox{so}\\quad \\frac{\\omega_a}{\\omega_c}=1+\\frac{N_s}{N_a}.", "a3e505dc348ede19834d9dc863a14e97": "q \\equiv e^{i\\pi\\tau}.", "a3e5686ae3c8e5bda149d18a6c1b4380": "v_{cr} = 162.1 e^{-0.38 \\sqrt{m}}", "a3e587aa7f898a3c358a320df89ae48f": "H = X \\left(X^{\\rm T} W X \\right)^{-1}X^{\\rm T} W ", "a3e59dfb450f72101c9597c61497799b": "\\mathbf{K}_{k} = \\mathbf{P}_{k|k-1}\\mathbf{H}_{k}^{\\top}\\bigl(\\mathbf{H}_{k}\\mathbf{P}_{k|k-1}\\mathbf{H}_{k}^{\\top} + \\mathbf{R}_{k}\\bigr)^{-1} ", "a3e5a3ee8c1182d4f9beb54caa5a0467": "X \\sim \\textrm{Beta}(\\alpha,\\beta)\\,", "a3e64501790c5c8f8021336ea95d4910": " -2< Pe_{l} <2", "a3e671bdf038cbf0c196662811fe2d4f": "V_i(\\mathbf{x}_i) = V_{i-1}(\\mathbf{x}_{i-1}) + \\frac{1}{2} ( z_i - u_{i-1}(\\mathbf{x}_{i-1}) )^2", "a3e6aa626ad0b6ff69add85b01f6da43": "k/2", "a3e6ca977db81f0018e3efe4bcb31e9e": "0 = \\langle (e,0), (k, Tk) \\rangle = \\langle e,k \\rangle + \\langle 0,Tk \\rangle = \\langle e,k \\rangle", "a3e7071e52abc60c8cf71bfcf384e7d7": "\\mathbf{X}\\sim \\mathcal{W}^{-1}({\\mathbf\\Sigma}^{-1},\\nu)", "a3e78e46972452b9f31fa93c805246c6": "\\mathbf{K}(t)", "a3e7925725d5b8e21e737d96c7ec79cf": " L(\\lambda) z = 0 ", "a3e83eb12ae0bf25ec299780b3de09dc": "\\left.-\\frac{\\lambda^2}{\\hbar^2}\\int_{t_0}^t dt_1\\int_{t_0}^{t_1} dt_2e^{\\frac{i}{\\hbar}H_0(t_1-t_0)}V(t_1)e^{-\\frac{i}{\\hbar}H_0(t_1-t_0)} e^{\\frac{i}{\\hbar}H_0(t_2-t_0)}V(t_2)e^{-\\frac{i}{\\hbar}H_0(t_2-t_0)}+\\ldots\\right]|\\psi(t_0)\\rangle", "a3e88b98fec4e356d5b7ef51d05a26f0": "\\textbf R,T", "a3e909b4bbb5d4469a44dd2aca575bdf": "\\nu_e ", "a3e9275495b96539690d18c7744d3f26": "{\\rm ASPACE}(s(n))", "a3e92de7d0e1378b11ba6e8f1ec949ce": "g''(x,s) + k^2 g(x,s) = \\delta(x-s).", "a3e9555bc461bd5669ca92e7c09c08d5": " s=\\frac{1}{2} \\!", "a3e95f89dd110acde6086b8528922180": " \\left( {1 \\over 2} i a x^2 + iJx\\right ) = {1\\over 2} ia \\left ( x^2 + {2Jx \\over a} + \\left ( { J \\over a} \\right )^2 - \\left ( { J \\over a} \\right )^2 \\right ) = -{1\\over 2} {a \\over i} \\left ( x + {J\\over a} \\right )^2 - { iJ^2 \\over 2a}. ", "a3e9629367c07cb8a242f064bcb8f899": " d \\alpha_t = (\\zeta_t-\\alpha_t)\\,dt + \\sqrt{\\alpha_t}\\,\\sigma_t\\, dW_t", "a3e9e738dcde6343661b9c28bd7ecc1b": " |a_n|\\le {2\\over n-1} \\sum_{k=1}^{n-1} |a_k|,", "a3ea2f1b2ac7e372409ed5f0e8a0473a": " \\pi^+ \\, / \\, \\pi^-", "a3ea7037f35a3fc4fc756733a5abe159": " R_\\mathrm{out} =\\frac{v_{x}}{i_{x}}", "a3eaa063259d0a1445ecdc8a49aac90f": "\\begin{align}\n(\\bar{\\mathbf{a}} \\times \\bar{\\mathbf{b}})_i & = \\bar{\\varepsilon}_{ijk} \\bar{a}_j \\bar{b}_k \\\\\n& = \\det(\\boldsymbol{\\mathsf{L}}) \\;\\; \\varepsilon_{pqr} \\mathsf{L}_{pi}\\mathsf{L}_{qj} \\mathsf{L}_{rk} \\;\\; a_m \\mathsf{L}_{mj} \\;\\; b_n \\mathsf{L}_{nk} \\\\\n& = \\det(\\boldsymbol{\\mathsf{L}}) \\;\\; \\varepsilon_{pqr} \\;\\; \\mathsf{L}_{pi} \\;\\; \\mathsf{L}_{qj} (\\boldsymbol{\\mathsf{L}}^{-1})_{jm} \\;\\; \\mathsf{L}_{rk} (\\boldsymbol{\\mathsf{L}}^{-1})_{kn} \\;\\; a_m \\;\\; b_n \\\\\n& = \\det(\\boldsymbol{\\mathsf{L}}) \\;\\; \\varepsilon_{pqr} \\;\\; \\mathsf{L}_{pi} \\;\\; \\delta_{qm} \\;\\; \\delta_{rn} \\;\\; a_m \\;\\; b_n \\\\\n& = \\det(\\boldsymbol{\\mathsf{L}}) \\;\\; \\mathsf{L}_{pi} \\;\\; \\varepsilon_{pqr} a_q b_r \\\\\n& = \\det(\\boldsymbol{\\mathsf{L}}) \\;\\; (\\mathbf{a}\\times\\mathbf{b})_p \\mathsf{L}_{pi}\n\\end{align}", "a3eaa08ec6b40b3702e992062b3de1d3": "M \\times I,", "a3eaa21c78bdcb306172a267d9163b32": "u_*=\\sqrt{\\frac{\\tau_b}{\\rho_w}}=\\kappa z \\frac{\\partial u}{\\partial z}", "a3eabd7f8ed091b9cf0877234095a762": "A^4b4 + A^3b4 + A^2b4 + A^1b4 + A^0b4 + 63", "a3eaec227df15f6e14ce696ca3dc52d5": "\\frac{a}{a} = 1", "a3eb2e243e58fb8da543a22225dd1abb": " g(x,y) ~ = ~ h(x,y) * f(x,y) ", "a3eb942d847848571d433bcd37600987": " \\pi_R", "a3ebdb6082e9d60046071af8eb627097": "0B=\\{o\\}", "a3ebf9bc44ae50d4478e3f05d73514ad": "\\partial_i^{\\alpha_i}:=\\part^{\\alpha_i} / \\part x_i^{\\alpha_i}", "a3ec0f1fd0fcc43700c8d245adc297d3": "C_n = [n, \\infty)", "a3ec4ca6aca1cf0449cef4d36710d1df": "f(\\frac{x + y}{2}) \\leq \\frac{f(x)+f(y)}{2}", "a3eccb1ec4ce9663aba77dea50cf5ca1": "Z_\\mathrm{eff}=Z- \\sigma \\,", "a3ed0eb9bd0bc333a2e1883378a96337": "L_{4k+2}.", "a3ed1e892251edfa09f7dd4547699ecd": " (\\exists\\ m > 0) \\ (\\forall\\ i > m) (a_i = b_i) \\land (a_m <_m b_m)\n", "a3ed37f45f25f2dd24bd1956d16101c5": "G=\\langle a, t| (t^{-1}a^{-1} t) a (t^{-1} at)=a^2\\rangle ", "a3ed4520f1f6479c05c1fa9b646fd29f": "\\psi(x, t)", "a3ed47c5708d36978f5c632240dc45f5": "\\hat{\\sigma}_z\\approx 2 \\hat{b}^{\\dagger}\\hat{b}", "a3ed9c36261e210ef4a47ba3b1dd7786": " u^T A u + 2 v^T B^T u + v^T C v, \\,", "a3edc9cc18455be5b24c0c3533aee41c": " z \\mid c_i ", "a3eddf4f3329d5c0e21431853d27f6d9": " > 1 ", "a3ede7e570f858691130394f6d51095c": "\\delta_m ", "a3ee8166eec2a2b16f0a82e2fd16f184": "\\mathit{g(x)} = \\mathit{x^6} + 2\\mathit{x^5} + 2\\mathit{x^4} + 2\\mathit{x^2} + \\mathit{x}+ 1", "a3eea722d3d715dd117c7ef65f65b74c": "\\begin{matrix}\n \\xrightarrow{L^s} & \\underset{s>r\\geq1}{\\Rightarrow} & \\xrightarrow{L^r} & & \\\\\n & & \\Downarrow & & \\\\\n \\xrightarrow{a.s.} & \\Rightarrow & \\xrightarrow{\\ p\\ } & \\Rightarrow & \\xrightarrow{\\ d\\ }\n \\end{matrix}", "a3eecd24d70d83904d91c89a16b93b2a": "\\langle R^2\\rangle = Nb^2", "a3eed2ce21e03676722a15362b742765": "h(\\vec{x})", "a3eed450389dfc9781c164fedccd6ef0": "\\left [ \\frac{a}{b} \\right ] \\quad \\left \\lbrack \\frac{a}{b} \\right \\rbrack", "a3ef04c55d70b20b0dc091c9211e2fff": "\\left(30+\\left(2x-3y\\right)^{2}\\left(18-32x+12x^{2}+48y-36xy+27y^{2}\\right)\\right).\\quad", "a3ef41e95dd25061fe12c8b5e34ed2f2": "\\mathrm{GL}_n(\\mathbf{Z})", "a3ef552c552872515e7c584e9de1391c": "\\bar{\\nu} \\propto \\frac{1}{1+fK_d}.", "a3ef560c9799adf6fe63899fd9fa213a": "\\left(\\inf_\\alpha f_\\alpha\\right)^*(x)= \\sup_\\alpha f_\\alpha^*(x),", "a3ef7e2dd81826980cb1ae768fed2275": "\\mathbf{F}^4 = \\mathbf{I}.", "a3efb5cdd21746528638c7d6caff0b0f": "\\frac {1} {360 \\times 60 \\times 60} ", "a3efe475944e0fe60ac31dcbb26c6521": "\\tan\\gamma_4=\\cos\\beta/\\tan\\gamma_2=1/\\tan\\gamma_1=\\tan(\\gamma_1+\\pi/2)\\,", "a3eff641e484c99b59fbcdbaf27437c5": "\\gamma_{\\mathrm{la}}\\ =\\ \\gamma_{\\mathrm{ls}} - \\gamma_\\mathrm{sa}\\ >\\ 0\\qquad \\theta\\ =\\ 180^\\circ", "a3f04b0c8779e0e0899e958a81dc1a92": " (\\forall x\\in\\mathbb{R}):\\frac{x^2}{k(k+1)}f_{k+2}(x)=f_{k+1}(x)-f_k(x).", "a3f076ee78d80072b8dd011662d59314": " var(r) = s^2 ( \\log( r ) ) ", "a3f089aad3d3541016d2e834224df66c": "y\\sim Ax^p e^{\\lambda x^r}\\left(1+\\frac{u_1}{x}+\\frac{u_2}{x^2}\\cdots\\right)\\,", "a3f1159e6afc86a9d832501f129a7f6d": "R_o", "a3f1186196a47468deaa488bfab8b7fc": "\\lambda =\\exp[\\mu/(k_BT)] ", "a3f1b1a9b64b8163990c4eda822befcd": "\\begin{bmatrix} I & 0 \\\\ -VA^{-1} & I \\end{bmatrix} \n\\begin{bmatrix} A & U \\\\ V & C \\end{bmatrix} = \\begin{bmatrix} A & U \\\\ 0 & C-VA^{-1}U \\end{bmatrix}\n", "a3f1b57e5ada32345465f9867cce42fb": "a_{\\text{form}}", "a3f1d63e18f1e52fe76de3fa19c6b84d": "(\\forall R)", "a3f1eb0bbae43da3d4c467903e5a7e88": "\\left(\\frac{-3}{\\sqrt{10}},\\ -\\sqrt{\\frac{3}{2}},\\ 0,\\ 0 \\right)", "a3f207d4b8c15e85237cf87f74b54c78": "a_1+a_2 \\leq b_1+b_2", "a3f241168a11b5cf3d9ba0dd55f59533": "k > 0\\,", "a3f25cf972d6fc4a185cfd35cfb10b6e": "r^{-1}=\\lim_{n\\to\\infty}\\left|{a_{n+1}\\over a_n}\\right|", "a3f284e083611c9da5112c362c3bd051": "V_x = V_0 e^{-\\frac{x}{\\lambda}}", "a3f2bbfddaaa60af2f6d856c2077ebda": " Q(X) = \\int_\\Omega P(X \\mid Y) \\mathrm{d} \\pi(Y) ", "a3f2c9657247b3e25cb73a65c5e40f5b": "|I|", "a3f2fa97fff4c6f704e6c657aca1af1d": "R^{2} = 1 - \\left({ L(0) \\over L(\\hat{\\theta})}\\right)^{2/n}", "a3f3290dfc429e842304cba2ace30809": "\n \\frac\n {\\hat p + \\frac{1}{2n} z^2}\n { 1 + \\frac{1}{n} z^2}\n", "a3f359dd28d9b6f0497e77688a7877aa": "b \\simeq \\frac{a^2 - x}{2a}\\,\\!", "a3f36bb5f2eb89f2d411bb0542e620d8": " k_1[E][S]=(k_{-1}+k_2)[ES] \\,\\!", "a3f36c08d06dc4ec09e2da4bd6c1d825": "A(x, t) = A_0e^{2 \\pi i \\frac{x - v t}{\\lambda}}= A_0e^{i (k x - \\omega t)},", "a3f3b2691601893ec2088ab2e8d6e1f2": "4 \\uparrow^4 4", "a3f3d4aa5c40c2529e1e455cb9522717": "\\forall \\varepsilon > 0 \\ \\exists \\delta > 0 \\ \\forall x, y \\in M : \\left (d_M(x,y) < \\delta \\Rightarrow d_N (f(x) , f(y) ) < \\varepsilon\\right),", "a3f3e83c665ad8d5a9b7deda0a510c85": "U:H_1\\to H_2\\,", "a3f454d11f312834de41511a05ad3aa7": " \\frac { \\partial \\boldsymbol{e_j} } { \\partial q_k} = \\sum_{n=1}^{d} {\\Gamma^n}_{kj}\\boldsymbol{e_n} \\ , ", "a3f47a9af21e2a9b3c2ede4e168394de": " \\frac{a+b}{a} = \\frac{a}{b} \\equiv \\varphi\\,.", "a3f4892e5c0578dbedca224f5f5ccc6d": "=\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)-\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\mathbf{P}(n-1)\\mathbf{r}_{dx}(n-1)+d(n)\\mathbf{g}(n)", "a3f4923a8f0b8a8eb58c7609d8de5d19": "|\\psi_{nr}\\rangle, (r=1,2,...g)", "a3f4cc59cfd63b157eb67366a479a526": "U(t)=e^{-iHt/\\hbar}", "a3f5484c853ca81f793a296a58d8138a": " z_t ", "a3f5f0ef0fc686eef47a4696e95fd001": "h(x)=1/x-\\lfloor 1/x \\rfloor", "a3f60619134d91b269015821c24248c6": "12^m \\cdot (\\#\\phi)", "a3f62f252b0656edffa25af25332421f": "\\begin{align}\nK_\\mathit{row}(S_\\mathit{wi}) & = 1 & K_\\mathit{row}(1-S_\\mathit{orw}) & = 0 \\\\ \nK_\\mathit{rw}(S_\\mathit{wi}) &= 0 & K_\\mathit{rw}(1-S_\\mathit{orw}) &= K_\\mathit{rw}^o\n\\end{align}", "a3f666755b3871c2749abcdcdb2f2d61": "- \\leftarrow -", "a3f6e6b400653fe4418a650b60602039": "\\mathbf{A}(\\mathbf{r}) = \\frac {\\mu_0} {4\\pi} \\frac{\\mathbf{m}\\times\\hat{\\mathbf{r}}}{r^2}", "a3f6e8d38f9761af7cae35a5143af93d": "J_n=J_{n-1}+\\frac{\\Delta \\vec{F}_n-J_{n-1} \\Delta \\vec{x}_n}{\\|\\Delta \\vec{x}_n\\|^2} \\Delta \\vec{x}_n^T", "a3f712b18e9b57a9724596fefd3fa69f": "\\phi_1, \\phi_2", "a3f72965defce3521fe304e1bdef1a51": "\\mathbf{b}^{\\rm T}\\mathbf{A}", "a3f74cb93b23dc8f4f8a9b13bc894d1c": "\\displaystyle{|X(a)-X(b)|\\le C|a-b|.}", "a3f764ed645a59d7c742a48fb1f23d33": "a\\equiv b \\pmod{p}", "a3f7ca6ec8ba168307f8c00412b220a1": "E_{g}(t,x) = -0.302 + 1.93\\cdot x+(5.35\\cdot 10^{-4})\\cdot t\\cdot (1-2\\cdot x)-0.31\\cdot x^{2}+0.832\\cdot x^{3}", "a3f7d5b95f02a0edb44ac3e56705e8e3": " f_X(x| \\theta) = h(x)\\ \\exp\\Big(\\ \\eta(\\theta) T(x) - A(\\theta)\\ \\Big) \\,\\! ,", "a3f80b32a0e73d23cca77ea2885c93e2": "f_o = \\frac{1}{t_o} = \\gamma (1-\\beta) f_s = \\sqrt{\\frac{1-\\beta}{1+\\beta}}\\,f_s.", "a3f8816bfff7bc5462f6ef274c712060": "vh/c", "a3f8b2096a2b9164a4b94122339bbe54": "(G,q,g,h)\\,", "a3f8bce3fee2b94eaa6831e0b5f3417e": "l > t", "a3f8c7a3eb48b8634af36575a4fb4bc1": "se_r(\\omega,q)= \\sum_m A_{r,m} \\sin {m \\omega}\\text{ for }a=b_r(q)", "a3f94713c7e1e230375ce9adbb283fc6": "s^2 = \\frac{\\sum_{i=1}^N w_i x_i^2 \\cdot \\sum_{i=1}^N w_i - (\\sum_{i=1}^N w_i x_i)^2}{(\\sum_{i=1}^N w_i)^2 - \\sum_{i=1}^N w_i^2 }", "a3fa369b3512b43af66b2d0aa0eff9e2": "C_\\mathrm{cal} = \\frac{\\Delta{H}}{\\Delta{T}}", "a3fa5cbb825f379abc1600bf465cb55a": "Z_f=mU\\frac{d(\\theta-\\alpha)}{dt}\\cos(\\theta-\\alpha)", "a3fa6541d167632830ff2b0deb45298f": "\\text{HOMA-IR} = \\frac{\\text{Glucose} \\times \\text{Insulin}}{22.5}", "a3fa67f435b31a2c7d145447d105301c": "x \\neq 0.\\,", "a3fa735b968145dfa7e7ffe4aa506f85": "g(RD_i) = \\frac{1}{\\sqrt{1 + \\frac{3 q^2 (RD_i^2)}{\\pi^2} }}", "a3fa80989184daf63da0586b3bf2acf6": "dx(t) = \\mu(x(t),t)\\,dt + \\sigma(x(t),t)\\,dW(t)", "a3fa9a845e4b99e81d0e79d573390bbb": "N = 2^M - 1, \\,", "a3fb34ad830ac88aebea63ce9ea9b8f4": "\\omega = e^{-\\frac{2 \\pi i}{8}} = \\frac{1}{\\sqrt{2}} - \\frac{i}{\\sqrt{2}}", "a3fb752e52b934cfa61d32f2ceaf8ac9": "{13-r \\choose 2}{4 \\choose 1}^7 = {13-r \\choose 2} \\cdot 16,384", "a3fbbb6129e5af4ee1818e67923221c4": "\\sum_{p\\text{ prime }}\\frac1p = \\frac12 + \\frac13 + \\frac15 + \\frac17 + \\frac1{11} + \\frac1{13} + \\frac1{17} + \\cdots = \\infty", "a3fbbf27c30503e1476a79b926a7c958": "\\textstyle{f(x)=\\sum_{k=1}^\\infty \\frac {\\sin(2^k x)} {\\sqrt{2}^k}}", "a3fbd368df4ab79c033ec5fd2f50f151": "y'(t) = f(t,y(t)), \\qquad \\qquad y(t_0)=y_0, ", "a3fc123785e7c37dfaaa7544a064a656": "f^\\phi", "a3fc1d1166a75e45a2a67f15161e247b": "\\scriptstyle(-0.31\\pm0.73)\\times10^{-17}", "a3fc9f5161d0302b04ba5c2d91e03080": "h(r)", "a3fcf775fce47071fdc2e89ca2e1e9ab": "\\text{Base ohms }=\\frac{\\text{base volts}}{\\sqrt{3 } * \\text{base amperes}}", "a3fd14325a78373c0897514329c50bd3": "\\mathcal{T}_\\lambda = {I\\over I_{0}} \\qquad \\mathcal{A}_\\lambda = \\frac{I_0-I}{I_0}", "a3fd49efa84b3ee26e73dcca55873091": "\n {FE}_{T=2}= \\left[ (x_{i1}-\\bar x_{i}) (x_{i1}-\\bar x_{i})' +\n (x_{i2}-\\bar x_{i}) (x_{i2}-\\bar x_{i})' \\right]^{-1}\\left[\n (x_{i1}-\\bar x_{i}) (y_{i1}-\\bar y_{i}) + (x_{i2}-\\bar x_{i}) (y_{i2}-\\bar y_{i})\\right]\n", "a3fd4c39ff8d7ebd64e74e765efa4dd2": "||x_n|| \\rightarrow \\infty", "a3fd7097926ade7405b7390b2e5ec4af": "d\\,\\!", "a3fdf3af794aa8a6c1cfb7b03a903d25": "\\mathrm{Cl}^{'}_\\mathrm{i}", "a3fe24b0069233bd25da928970122ece": "B b = \\int_{\\Lambda<|k|<(1+b)\\Lambda} {d^4k\\over (2\\pi)^4} {1 \\over k^4}", "a3fe42b9831fd143df46f81991af72bf": "\\omega^{*} = \\Sigma ^M_{i=1}|S_{i}| - n", "a3fe7a74dba9405852a6ecc39d446c98": "s_i = \\arg{}\\min_{s\\in\\mathcal{A}}\\left(\\left|R_i - s\\right|^2 + \\left(-1 + \\sum_{k,l}^{}\\left|\\alpha_{kl}\\right|^2\\right)\\left|s\\right|^2\\right),", "a3feb95679e49c676a938fcbb2acf01d": " y(t)= \\sum_{m = - \\infty}^ \\infty \\sum_{n=- \\infty}^ \\infty C_{nm} \\cdot g_{nm} (t) ", "a3ff6a2d0c4367f296c10858829ae15a": "\\operatorname{E}(R(S)) = \\sum_{i=0}^n p^i (1-p)^{n-i} r_i ,", "a3ffb613a08a8df62fd2b5887f1d42ad": "f: |K| \\rightarrow |L|", "a4000e7ca4bdb1b7a0bf47b89e7b73ca": "f'(x+h) < 0", "a40034d46af14a4cf9f7d4ea1e713133": "\\begin{align}\n\\int_0^\\delta e^{-xt} \\phi(t)\\,dt &= \\sum_{n=0}^{N} \\frac{g^{(n)}(0) \\ \\Gamma(\\lambda+n+1)}{n! \\ x^{\\lambda+n+1}} + O\\left(e^{-\\delta x}\\right) + O\\left(x^{-\\lambda-N-2}\\right) \\\\\n&= \\sum_{n=0}^{N} \\frac{g^{(n)}(0) \\ \\Gamma(\\lambda+n+1)}{n! \\ x^{\\lambda+n+1}} + O\\left(x^{-\\lambda-N-2}\\right)\n\\end{align}", "a400817673606aa32d93378688ee685b": " | n_1 \\rang | n_2 \\rang ", "a400f7b6e8cefebb00c86229268ecc68": "\\rho\\widehat{a}_j \\rightarrow \\left(\\alpha_j - (1-\\kappa)\\frac{\\partial}{\\partial\\alpha_j^*}\\right)\\{W|P|Q\\}(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)", "a40159120929096f51d024e243b4e11f": "C_n = \\frac{1}{n+1}\\binom{2n}{n}.", "a40192c4b25716194d0b1597ff179bf9": "1/2^{2n}", "a401b72505dec97c7a2b7bcddfa443d1": "x(t) = e^{j \\pi \\frac{t^2}{a}}y(t) \\, ", "a402d4bc644a50b58706799d59aab3b9": " \\sin x \\approx \\frac{16x (\\pi - x)}{5 \\pi^2 - 4x (\\pi - x)}, \\qquad (0 \\leq x \\leq \\frac{\\pi}{2} )", "a402eb6a7d85e2c85e79cb8d46496175": "N_{d=1 \\dots M}", "a4031981085ac1d93b61197e596725d5": "\\lambda=\\frac{h}{p}=\\frac{h}{m_0v}=\\frac{h}{\\sqrt{2m_0eU}}.", "a4033ef25d14c21b1b5b5c56438062b5": "Nm", "a4036aab730420096109c243926cf685": "P(\\partial) = \\sum_{|\\alpha| \\le N}{}{a_{\\alpha}(x)\\partial^{\\alpha}}.", "a4037afedd5f292d8aa891658345d67e": "\\nu(S)=0\\,", "a403b939f44b82f30e8c641330843f15": "x^2+px+q=0", "a403daf5a9d6dec79eb055aa1e334874": " \\{ f_1,f_2 \\}(\\mathbf{x}) := \\langle \\left[(\\mathrm{d} f_{1})_{\\mathbf{x}}, (\\mathrm{d} f_{2})_{\\mathbf{x}} \\right],\\mathbf{x} \\rangle ", "a403e9c1b49d393e47c92dc9f80860c6": "\\mathbf{x}^{(1)}=0-\\gamma_0 \\begin{bmatrix}\n -7.5\\\\\n -2\\\\\n 209.44\n\\end{bmatrix}.", "a4041ede22f43dbd935b2aa4964b6f82": "((\\lambda x.y)x)[x := y] = ((\\lambda x.y)[x := y])(x[x := y]) = (\\lambda x.y)y", "a40452b0b16cc1f1eb07b839ffed08d6": "x=(x_e), e\\in E", "a40473cc6be8a2aa51410bea495bc14e": "\n\\psi_k(x)= \\frac{1}{\\sqrt{\\sqrt{2n}n!}}H_k(x) e^{-x^2/4}\n", "a404f332a2d53a33c7d11f5a485681f0": "\\Box P \\rightarrow \\Box \\Box P", "a404ff386cbca2a46de4b3149551451b": " \\mathcal{L} ", "a4050f98aed67ead0bb4ea9c9832baf4": "P_h\\approx{W}_hf\\approx\\eta{f}\\beta^{1.6}_{max}", "a4052a465047445a7f616c29055a7397": "\\nabla P=-\\frac{\\mu}{k}q-\\frac{\\rho}{k_1}q^2", "a4054ca51366c7fcc91b853f66bd828f": "(\\varepsilon_n)_{n=1}^\\infty", "a4057c5a40333ac6f02708b193eca597": "F_c = F", "a405ad235c85692e19d4b0bee30895b0": "m \\frac{\\mathrm{d}^2 x(t)}{\\mathrm{d}t^2} = F(x(t)),\\,", "a406063ca467b132dcbde33bd0a1fad8": "(M,\\omega,H)", "a406535f07f7a21d9353017ebc27b9cd": "i=1,...,m", "a4073ceace1bcb364645176811f72260": "H(\\alpha)", "a40761f884a1998baf8d3e1912f7890d": "R^m_n(\\rho) = \\! \\sum_{k=0}^{(n-m)/2} \\!\\!\\! \\frac{(-1)^k\\,(n-k)!}{k!\\,((n+m)/2-k)!\\,((n-m)/2-k)!} \\;\\rho^{n-2\\,k} \\quad\\mbox{if } n-m \\mbox{ is even}", "a40766490e3f3154994e94533e1446f6": "61:86 \\approx 1:1.4098", "a407adadd96e08930e7eec6726ad3985": "P(x) \\equiv A_j(x) \\pmod {(x - \\lambda_j)^{\\nu_j}}, \\quad\\forall 1 \\leq j \\leq r", "a407cced7246916e5800501ea4cdb9fd": " \\omega \\equiv e^{\\frac{2 \\pi i}{d}} ", "a407ffec61875c636639af3379b6111d": "D_B(V)=(B\\otimes_{\\mathbf{Q}_p}V)^{G_K}", "a4080fe073ae6bae5240332ae868a2ad": "\\tilde{t}_2 = e^{-i\\phi} \\cos(\\theta) \\tilde{t_R} - \\sin(\\theta) \\tilde{t_L}", "a408de07e314b79e0ebd3da3a43a175f": "\\int g \\left(\\frac{\\partial f}{\\partial t}\\right)_{coll}\\,d^3p=0", "a4092e3aedc8fb92ac57536995d825ba": "\\alpha^4+\\alpha+1=0", "a409623877d13f7bb3a9542b87c69e87": "\n\\underline{P}(Cl_t^{\\geq}) = \\{x \\in U \\colon D_P^+(x) \\subseteq Cl_t^{\\geq} \\}\n", "a409d3314d66c22d7d7f4e3a8d64c661": "\\frac{\\Gamma^3(1/3)}{4\\pi}", "a409e5f8d98c9dc9ddbf87deb86c7689": "f(g)= 1\\cdot s + \\sum_{g\\not= s}0 \\cdot g= \\mathbf{1}_{\\{s\\}}(g)=\\begin{cases}\n1 & g = s \\\\\n0 & g \\ne s\n\\end{cases}", "a40a000c605a7dc4c06ebed12d269d54": "\\forall\\ i \\ D_{**}(X_i, Y_i) > 0 \\,", "a40a2a559a6073e526fe78a33b2e21f9": "\\Psi(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp(-\\tfrac{1}{2}x^2)", "a40a6d3c6f1a72bf3fac4245b92b065d": "2 (2^{k-1}-1)/k ", "a40a7f9e65cf81c82c659d30d270d49f": "e^{t/2} \n \\prod_{i= 1}^{\\infty} \\cosh{\\left(\\frac{t}{3^{i}}\n \\right)}", "a40aa7845db2e47c1c91e9f2c5042ed6": "\\partial_{\\xi} = t^{\\gamma/2} \\partial_x, \\partial_{\\eta} = t^{\\gamma/2} \\partial_y", "a40ab5840a49ce3497472aa7c8672be8": "\\frac{2^{13347311}+1}3", "a40adcfbb6b543c5c979387ca5b1c980": "E\\left(\\epsilon(x_0)^2\\right)", "a40ae09f499c302192bacb9a19b0fad4": "u_1 = {g}_{1}^{k}, u_2 = {g}_{2}^{k}", "a40ae94b6307c0c1a194286d86d737ca": "[\\lambda]", "a40b533fb0b807e8504f88c04b3fd4f6": "\\Phi(x,y)", "a40b7710a85fbc035aeec12a2460aa99": "f(\\Omega) \\, ", "a40bcbf91000d85ebed7e35a273e1b38": "\\mathbf{Q}_{13} = \\begin{pmatrix} \\cos \\gamma_{13} & 0 & \\sin\\gamma_{13}\\\\0 & 1 & 0\\\\\n-\\sin\\gamma_{13} & 0 & \\cos\\gamma_{13} \\end{pmatrix}", "a40c1a75ac1000bf3b2a339e8ac0d32c": "D_{med} <= D_{mean} ", "a40c308340ce8e0f11df2167c38cf797": " C=(E,B,dim)", "a40c3ea291ac04d16ec2e28e73731aff": "\\omega_0=\\frac{R}{L}", "a40c64f5a1beb0d0a0d95111c4c5d1ae": "dm=-\\Delta m", "a40c739803bcbd72b90bccdd77bab42a": "D_i(a_j)", "a40ca8fb7af400b243e7afffed2adee9": "2x(x^2+y^2)=a(3x^2-y^2)", "a40cc5e34260e2395718da28d2f6386e": "V_{ix}\\left(t+ \\frac {\\Delta t} {2}\\right) = V_{ix} \\left(t- \\frac {\\Delta t} {2}\\right) + \\frac{\\Delta t}{M_i}R_{ix}(t)", "a40cefbb9241a6a1755006e7bdd068d1": "0,1,3,7,15,\\ldots", "a40d8a3a64002e014e2ade8933257a27": "H=(V,E)", "a40dae02cabc970c2503a9b22f83316d": "(A + B) \\cdot (A + C) = A + (B \\cdot C)", "a40dc55806a30033c066e36bd83acb29": "\\scriptstyle z^2< 1-y^2", "a40e22827114c14315c03010e9d4efb8": "\\begin{array}{cc} \\begin{array}{|rrrrrrrr} a & b & c & d & e & f & g & h \\\\ \\hline \\end{array} \\end{array}", "a40e4dce0296b97b20820edb36908986": "[[a\\;\\|\\;M]_m\\;\\|\\;N]_k \\rightarrow [[v\\;\\|\\;M]_m\\;\\|\\;N]_k", "a40e5c6d81abe6b7f25b816b012e240b": " x = y\\ f ", "a40e6074e3e1104eb74faea2eec5d736": "J = {M_{11} \\over M_{01} + M_{10} + M_{11}}.", "a40e90362648982ac44f754f22d4f677": "n/m", "a40eef6ac0a07de4b4763863cbd60688": "f:U\\to\\mathbb C", "a40f09fd737a64904e1aa65ffd5723cd": "\\mathrm{dist}(x,\\partial\\Omega) 0 \\mbox{, } 1 \\leq i \\leq n-1", "a414a82d0b5f0caf92ba0faeccfd5dc4": "r(\\theta)=\\frac{ab}{\\sqrt{(b \\cos \\theta)^2 + (a\\sin \\theta)^2}}", "a414c9b3a738940257dacf368ff18c07": "\\sum\\limits _{m}q_{m}\\nabla p_{m} = \\left(\\mathbf{Q}\\cdot\\nabla\\right)\\mathbf{P}+\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}.", "a41616ce57925a61c084d09b2c100e67": "\\Omega_z\\neq 0", "a416944f4b82c87bde574d963551d901": "v'= -v \\mod u' = -\\frac{72}{125}(17x-5)", "a416e22f03b3bbe5461b9a29cab161a0": "X_1,X_2,X_4", "a416facd759fd8e982f0a21518eaf02b": "y_{i}dx_i", "a4170b7389cf6ecdad1d83f18aad2861": "D(\\bar{\\theta})", "a4170e40d888d7ef604a06b22f2bbee5": "\\mathit{Fo}_m = \\frac{D t}{L^2}", "a4174dd1a80fece4e5cb89693984a301": "\\sqrt{3} \\times \\sqrt{7}", "a417655e890e7887367fc17c4fd83bbf": "G(n)", "a41799ad48d37b6acb66a597b4615733": "D x^k = k\\ x^{k-1},", "a4179eaf25a01f458b1603f924d1040c": "e^w = i\\,\\!", "a417aa5550488b12eef7db9e740f3a41": " |\\mathbf{A}\\otimes \\mathbf{B}| = |\\mathbf{A}|^p |\\mathbf{B}|^n.", "a417ae9de8404fb23f848162f1889463": "\\kappa\\ge\\aleph_0", "a417bef11ffb8f597c838924461d0d22": "\\ln(k) = 23.1 - 12,667 (1/T)", "a4180c2cf2c882c7fea6d5a28d80cb15": "-1.8417", "a418a37f7f0890c77f1f71426bae579b": "F = E - TS = (\\kappa - 2k_BT)\\ln(R/a).", "a418d2e4928ac4c2d18bfc87e639efe8": " (A^C)^C = A ", "a41972016206c0ba8505f123bebec60d": "\\sum_{j=1}^{j=N}{P_{ijkl}Q_{mnj}}", "a4197db1c7cc9459138918ded32e83ff": "(J_1)_z, (J_1)^2, (J_2)_z, (J_2)^2", "a419e6a52c0d41c8131102c373da71d6": "A'(x)u_1(x)+B'(x)u_2(x)=0\\,", "a41a26fbf7e70facf7518b7e205b6291": "\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty |\\chi(\\tau,f)|^2 \\, d\\tau \\,df=|\\chi(0,0)|^2 = E^2", "a41b269e8b3fc1287b0625183e3da7ab": " t_1 = {G^a}_a, \\; t_2 = {G^a}_b \\, {G^b}_a, \\; t_3 = {G^a}_b \\, {G^b}_c \\, {G^c}_a, \\; t_4 = {G^a}_b \\, {G^b}_c \\, {G^c}_d \\, {G^d}_a", "a41b2a9e0fc9423e8eb51c1abce87a07": "\n\\frac{dL}{d\\varepsilon}=\\frac{\\partial L}{\\partial y}\\eta + \\frac{\\partial L}{\\partial y'}\\eta'\n", "a41b84d2f1c5454a206f3638c8689ada": " \\left (s^3-s^2 \\right )y''+ \\left ((2-\\gamma )s^2+(\\alpha +\\beta -1)s \\right )y' - (\\alpha \\beta) y=0", "a41b99c06cbde41cc8179c4507c69c8e": "\\tau = \\cos \\nu", "a41baeff3b12e5761534abb87e98dccf": " N =", "a41bbceba6b5fb9bbe3ff43defdba005": "\\begin{align}\nF\\left(n_1,n_2,n_3,\\lambda\\right) &= \\tau^2 + \\lambda \\left(g\\left(n_1,n_2,n_3\\right) - 1 \\right) \\\\\n&= \\sigma_1^2n_1^2+\\sigma_2^2n_2^2+\\sigma_3^2n_3^2-\\left(\\sigma_1n_1^2+\\sigma_2n_2^2+\\sigma_3n_3^2\\right)^2+\\lambda\\left(n_1^2+n_2^2+n_3^2-1\\right)\\\\\n\\end{align}\n\\,\\!", "a41be0789c1332b1c970e8cbef40568a": "y(t) = \\frac{1}{1 + t}", "a41c0d551c5693c685da4e2666f71561": "\\scriptstyle \\beta l=\\pi/2", "a41c17c0214d7f1048447e6f9703ce49": "\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c}), ", "a41c251c6b19011d40b5e1c90f8250bd": "u \\to \\mathcal{E}(u,u)", "a41c3bfda0fbadbb7ae34225713dda1d": "I(l, m)", "a41d272055aaa453b86307fdab8f52d7": "\\sqrt[12]{2^7} = \\cfrac{3}{2} - \\cfrac{3\\cdot 7153}{12(2^{19}+3^{12}) + 7153 - \\cfrac{11\\cdot 13\\cdot 7153^2}{36(2^{19}+3^{12}) \n- \\cfrac{23\\cdot 25\\cdot 7153^2}{60(2^{19}+3^{12}) - \\cfrac{35\\cdot 37\\cdot 7153^2}{84(2^{19}+3^{12}) - \\ddots}}}}. ", "a41da0d7d546538f40f742c66f150073": "H^{\\sigma\\sigma_j}(\\vec{x},\\vec{x}') = \n\\begin{cases} \n h^{\\sigma\\sigma_j} & \\left | \\vec{x}-\\vec{x}' \\right | = c \\\\\nh^{\\sigma\\sigma_j}/2 & \\left | \\vec{x}-\\vec{x}' \\right | =\\sqrt{2c} \\\\\n 0 & \\text{otherwise} \\\\\n\\end{cases} ", "a41dd297a9b46c75c424bc12deef44ea": "g \\in \\Gamma(T^*M^{\\otimes 2})", "a41dd608d812f6dbf448390b507fbd4c": "A \\subseteq B \\subseteq K", "a41e488bc9c34839ad0ba2c2154ae521": "f'(x)=\\frac{dy}{dx}", "a41e6d9050a020b30db34defae0eab4e": "X_{(j)}", "a41eb2c6a46cb3d2ffe0486745dfbbb9": "\\frac{\\overbrace{x_2 + x_2} + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + \\overbrace{x_9 + x_9}}{10}. \\, ", "a41edd26ac98ad458731ed4d2bc21b43": " \\alpha=1 ", "a41f627e1871401d45c955a003f6a971": "\\left \\lfloor \\frac{(n+1)(K+1)}{N+2} \\right \\rfloor", "a41f6a08b01508282561c318e1577b42": "\\, a = \\sum_{\\mu} a_{\\mu} \\gamma^{\\mu} ", "a41fc59ba48213f0041ab2e154be1ce0": "x_1(t) = A + w(t)", "a42004b8cd46338e02b7fe222f3e1196": "\\frac{\\partial\\bar{e}}{\\partial V_z} = -\\frac {V_r}{V_0}\\ \\frac{\\partial\\hat{t}}{\\partial V_z}", "a4205af4ac48c0d37d655a6f61187f4b": "J(c)\\equiv \\int_c^\\infty \\frac{x^3}{\\exp x-1}dx\n=\\int_c^\\infty \\frac{x^3 \\exp(-x)}{1-\\exp(- x)}dx\n=\\int_c^\\infty \\sum_{n\\ge 1}x^3 \\exp(-nx)dx\n", "a420602a3b0f4b4ae6590f87e7a77731": " (\\mathbf{W} \\, \\mathbf{\\Sigma})^{T} = - \\mathbf{W} \\, \\mathbf{\\Sigma} ", "a42069d2232b9702a7e8135d2f44c0a6": "\\int (\\frac{x^2}{2} + yx) \\, dy = \\frac{yx^2}{2} + \\frac{xy^2}{2} ", "a420d632de37e56da659da5f77373eee": "\\| \\mathbf{x} \\| = \\sqrt{ \\mathbf{x} \\cdot \\mathbf{x}}", "a421070de686bd68a2bcfa4a2ba79be8": " 0 \\le j \\le n. ", "a42138bb1e45fad6119f911835824ec4": "\\beta=1/k_b T", "a421573a2aac0b0a61cde393c1edc674": "\\tilde A_n, \\tilde D_n, \\tilde E_k,", "a421d1ab083904007b06a999d5c69ccf": "\\cos 2p\\varphi\\,\\!", "a421f53c99c25249db8310d41607dee3": "(1,k)-", "a4227f64e3ee453747c730e8b09a5b04": " E_2 = E_{trans,c} = 6.00 \\text{ ft}", "a422a083677ebc8cbf994985296bd0ad": "t(n)^k", "a42311bb231bd76cbe36c67cba533d15": "\\textstyle b=0", "a4231349cce1998966cbe9905e540b25": "\\partial \\colon L_{n+1} (\\pi_1 (X)) \\to \\mathcal{S}(X)", "a4232745cb4a4dd4211348aa12d55935": "\\mathfrak{N}", "a4235d9881965ea4053fd8b27b1152f4": "M := (m^{(0)}_{ij}) \\in \\mathbb{R}^{I\\times J}", "a42376f2a91ab6f29779860962d4a86e": "V(r)=V_0", "a423834a50fbb7b364e8e5a154136de1": "e ::= x \\mid \\lambda x.e \\mid e \\, e", "a42384c2d59e8f363bc288cd611f6026": "AOP_m(x) = {x^{m+1} - 1\\over{x-1}}", "a423d4060514134b7f7dda2f34125bbc": "\n{\\beta \\over z}\\,\\,\\, \\approx \\,\\,\\,{{r\\left( {r - 1} \\right)} \\over {2\\,n}}\\,\\,\\left( {{\\sigma \\over \\mu }} \\right)^2", "a423f00b980b5e875d9a232a7797cdfc": "\\forall x, P \\Rightarrow wp(S,Q)", "a423f36174422743e3bb4b3e2fbb504b": "{\\mu }_{ph}\\sim T^{-3/2}", "a423fc7edea7371ca5f654e4a5bb0ca6": "|j|=\\sqrt{(\\Re{j})^2+(\\Im{j})^2}", "a42432b77bb362470709e00975ce1db8": "\\Pi_1 = q_1(P(q_1+q_2)-c)", "a42437f710efe7134c521dbcf942888e": "(\\textbf{a}\\textbf{b}^{\\rm T} + \\textbf{b}\\textbf{a}^{\\rm T})\\textbf{x}", "a4245961d3bbf25e0d0624242ce08f04": "\\textit{dau}(m,h) \\land \\textit{dau}(e,t)", "a424a06842071a10a39aac193f8b4d76": " a_{12} = p_2p_5+p_1p_6,", "a424bc3c4c287fe1da6d2856ff82a92d": "x(0)=x_0", "a424f0a4b8c26aa13c2d262d2063de5a": "\\left\\{ Q, Q \\right\\} = \\left\\{ \\overline{Q}, \\overline{Q} \\right\\} = 0", "a42504940b796a7fa4e0e8ea6ba49d95": "\\tau=0\\;", "a4254c9e22f3fdebab1cb0016e736bbc": "0\\leq \\delta \\leq \\frac{1}{4}", "a4254d57b973688a3ab8dffc2c76be9a": " a s f(s) f(f(s)) \\cdots f^{(n)}(s) \\cdots \\ ", "a425618205113a47c2a19815d131763d": "\n \\boldsymbol{\\sigma} = \\cfrac{1}{J}~ \\cfrac{\\partial W}{\\partial \\boldsymbol{F}}\\cdot\\boldsymbol{F}^T \\quad \\text{where} \\quad J := \\det\\boldsymbol{F} \\,.\n ", "a425ab3fe8c613dfcc40f375d6cd59fb": "cF(x)-cG(x)=\\int_{-\\infty}^x h(\\xi) \\, d\\xi + c_1.\\,", "a425b0bde387a58e44bd00787ae51df1": " p <_\\mathcal{O} q ", "a4262947fd9d86070fb475ffcb55499e": "\\lambda'-\\lambda = \\frac{h}{m_ec}(1-\\cos{\\theta}). \\, ", "a4278e49d21204912cbd486100c4b542": "\\frac{1}{L}\\sum_{\\ell=1}^L a_{\\ell} \\leq \\max a_{\\ell}", "a427b16961487c634f1e4f638e6d4f8a": "g(s(X_1, X_2)) = g(X_1+X_2)\\,\\!", "a427da7873c9c17ab4f283e5e740326e": "\n\\Gamma_k[\\Phi,\\bar{\\Phi}] = \\sum\\limits_{\\alpha=1}^N g_\\alpha(k) P_\\alpha[\\Phi,\\bar{\\Phi}] ,\n", "a427e2120a9b628cdc8d8e696c1efdcd": " d = |(I_t-\\mu_t)| ", "a427e3ba2c6d212ee61c924f7920f918": " m \\ge 1 ", "a427ee611df03b3553d48a12047eccac": "\n\\left(\\frac{\\partial^2 P}{\\partial V^2}\\right)_{C}=0\n", "a4287d70828931dcf88c4091be7de9f2": "H_n^{(2)}", "a428ad7ed343eff969e15e4d076d6cdf": "\\displaystyle{H=i(2E-I).}", "a42933051ca5c84902cc3ba7299a5266": " \\forall x \\left( \\phi \\right) \\to \\phi[x:=t]", "a42961dc4481905e852fbc76d67259bd": "\\approx-g_m R_{\\text{C}}", "a429de95824a184a509099e74fd7389a": "k=\\sqrt{2m E}/\\hbar", "a42a783f2781a285157bf24ce2b1b76b": "M\\{\\cup_{i=1}^m(\\xi \\in B_i)\\}=\\mbox{max}_{1\\leq i \\leq m}M\\{\\xi_i \\in B_i\\} ", "a42ad060b694fe226e9137ec7a609751": "x^2+y^2+\\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \\frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0.", "a42b0a7251d680043a8fabf515cdc52d": "\\mathbf{E} ", "a42b0ead02642047afd6c626b34fd67e": "(8,1,1)\\rightarrow(8,1)_0", "a42b2e0646f28a4c0456963da5144708": "\\scriptstyle E_{\\vec{p}} \\;=\\; +\\sqrt{m^2+\\vec{p}^2}", "a42b674f4f9a0727d7bc2574b88fd49c": "M_1=(-1.3515-1.7703x_D+5.9114y_D)/M", "a42b69776bd90a22b469bfcbdbb76613": " M(T,H) \\ \\stackrel{\\mathrm{def}}{=}\\ \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\left( \\sum_i \\sigma_i \\right) = - \\left( \\frac{\\partial f}{\\partial H} \\right)_T ", "a42ba11b51b429553c3444febbb56c46": "\\zeta = 0\\,", "a42bacc4890557dff6fb54738e05aafc": "\n\\frac{\\partial}{\\partial x} I(x,t) =\n-C \\frac{\\partial}{\\partial t} V(x,t)\n", "a42bbaf4f8583f94c1dbdceb4e62a583": " \\textbf{e} = \\textbf{r} \\cdot \\textbf{h} + \\textbf{m} \\pmod {32} = 14 + 11X+26X^2+24X^3+14X^4+16X^5+30X^6+7X^7+25X^8+6X^9+19X^{10} \\pmod {32} ", "a42bd4deaa95e85750eb1323b53f21a3": " (\\sqrt{n_x^2+n_y^2+n_z^2})^2 ", "a42c52d063cd40a948953bc3d2fb5f86": "ad-bc = \\pm 1\\;", "a42c5703684bb4d38fcfe3f1b17574c2": "\n(\\nabla \\phi)_k =\n\\frac{\\hat{ \\mathbf e}_k}{h_k} \\frac{\\partial \\phi}{\\partial q^k}\n", "a42c8f1a5c89754451107d008d320a20": " g_0\\gg 1 ", "a42c9e88b9f6d8d2c4597e4adb3d9b7d": " E_{i\\pm1} E_i G_{i\\pm1} =E_{i\\pm1} {G_i}^{-1}, ", "a42c9f07c576261de42a9f45aed73238": "f(x)=o(g(x))", "a42cf2cd92e6d1783e398c4b648caa74": "C_n^{1nn}(x) = Y_{(1)}", "a42cf36b3db2499709e851aa025f0fd7": "\\lambda\\in \\mathbb{R}", "a42d110172e0c71f4f084200fe878081": "y = A e^{-\\int f(t)\\,dt}", "a42d26077740c15958dd3784e8c35605": "\\Gamma={Q_{\\text{ADS}}M\\over nF}", "a42d389ed205795f3dd599d165fc3a35": "_2F_1 \\left(a,b;\\tfrac12\\left(1+a+b\\right);\\tfrac12\\right) = \\frac{\\Gamma(\\tfrac12)\\Gamma(\\tfrac12\\left(1+a+b\\right))}{\\Gamma(\\tfrac12\\left(1+a)\\right)\\Gamma(\\tfrac12\\left(1+b\\right))}. ", "a42d40eca02365cb760d6b5e96a7db7e": "r - v = \\sqrt{r^2 - (\\frac{L}{2})^2},", "a42d4e86ae5dab515263119c3eb1f11c": "p \\equiv q \\equiv 3", "a42d664e9e0df0c9571998a4d8457524": " f((n/2, n/2), (n/2, n/2), \\ldots (n/2, n/2)) ", "a42d83632d538d705ec00562f386de77": "\\Theta^i(\\mathbf e) = d\\theta^i(\\mathbf e) + \\sum_j \\omega_j^i(\\mathbf e)\\wedge \\theta^j(\\mathbf e).", "a42da59661abd4ad6434682b64f85176": "e(T)=\\frac{1}{2\\pi i}\\int_{\\Gamma} \\frac{e(z)}{z-T}\\,dz", "a42de1c1f1c2a356ab95a3c6255c78ea": "s_N(X_1,\\ldots,X_N) = \\sum_{\\sigma \\in S_N} \\sgn(\\sigma) X_{\\sigma(1)}\\dotsm X_{\\sigma(N)}=0~", "a42de351282e422bac6e04134e8d127f": "\\vert g, n \\rangle \\rightarrow \\vert e, n-1 \\rangle", "a42ea46e662a2f1c9541c52df1957897": " GL(4,\\mathbb R) \\to SO(4) ", "a42ec8a1931c93d466bf21fa0a00c929": "\\displaystyle{ea=ae=L(a)P(1)=P(L(a)1)=P(a),}", "a42f238f4a1b35acb0e815b273c58c94": " 1/\\sqrt{ k^2 - a^2} \\quad (a>0, k>a) \\,", "a42f565e66552964da135953c33d8a7c": "z_{P21} = 1.00 + j1.52\\,", "a42f67c2c41875ee6ebc21450f8edd12": "\n\\begin{align}\n\\operatorname{Var}\\left(h(B)\\right) & \\approx \\operatorname{Var}\\left(h(\\beta) + \\nabla h(\\beta)^T \\cdot (B-\\beta)\\right) \\\\\n\n & = \\operatorname{Var}\\left(h(\\beta) + \\nabla h(\\beta)^T \\cdot B - \\nabla h(\\beta)^T \\cdot \\beta\\right) \\\\\n\n & = \\operatorname{Var}\\left(\\nabla h(\\beta)^T \\cdot B\\right) \\\\\n\n & = \\nabla h(\\beta)^T \\cdot Cov(B) \\cdot \\nabla h(\\beta) \\\\\n\n & = \\nabla h(\\beta)^T \\cdot (\\Sigma/n) \\cdot \\nabla h(\\beta)\n\\end{align}\n", "a42f6e7a4fb0a6c967725af9f8b5146c": "B(t_0)=\\beta_0^{(n)}.", "a42fa013226e6ae81bb500bbe88f01b8": "\\frac{1}{2} \\le S <2", "a42fbefa2a90189e8482e88b32e428c0": "\n \\quad (3) \\qquad u_i^{n+1} = u_i^n - \\Delta t \\left[ a^+ u_x^- + a^- u_x^+ \\right]\n", "a42fd04695b065e5a461fd0011819c79": "\\left|q\\alpha -p\\right| < \\left|q^\\prime\\alpha - p^\\prime\\right|.", "a42ff0728ecee04f1d7a2ea914e9615a": "a[\\mathbf{f}] = v[\\mathbf{f}]^\\mathrm{T} G[\\mathbf{f}].", "a4302a916a0c2029a5d6f6f4e6b38450": "\\mathbf A:=A_1 \\, {\\rm d}x_1+A_2 \\, {\\rm d}x_2+A_3 \\, {\\rm d}x_3.", "a430682677afc53965cfb750d190c7b2": "U=U_1\\coprod \\cdots\\coprod U_k", "a430b5406c5bf104ef717e95d1909ac6": "C_{\\bold{k}} = 0", "a430e1cc8c82deb4cbd125a513793a1d": " A = \\int_a^{b} ( f(x) - g(x) ) \\, dx ", "a430f58b6a931060d1256c19d8027eba": "\\; Tr(W\\varrho_{A_1\\ldots A_m}) < 0 ", "a4314ae837f4f4120bafea0738744ed0": "GF(q^N)", "a4315d00f1fbcfa116dcf0be2e27c2f1": "\\left( \\frac{\\partial ( \\ln X_2 ) } {\\partial T} \\right) = \\frac {\\Delta H^\\circ_{fus}} {RT^2}", "a431961b7c4560b7de932be1928cbfe8": "\\theta > 0", "a432157655846a7f7649cfdb9018ccff": "\n\\begin{array}{rl}\n\\mbox{I.} & ((A\\; \\operatorname{f}\\; B) \\And (B \\to C)) \\to (A\\; \\operatorname{f}\\; C) \\\\\n\\mbox{II.} & ((A\\; \\operatorname{f}\\; B) \\And (A\\;\\operatorname{f}\\;C)) \\to (A\\; \\operatorname{f}\\; (B \\And C)) \\\\\n\\mbox{III.} & (A\\; \\operatorname{f}\\; B) \\leftrightarrow\\; !(A \\to B) \\\\\n\\mbox{IV.} & \\exists U\\; !U \\\\\n\\mbox{V.} & \\neg (U\\; \\operatorname{f}\\; \\cap)\n\\end{array}\n", "a43231d38de67d6e77d7cc7a52db822d": "(r\\bar{b}+b\\bar{r})/\\sqrt{2}", "a432483a345e51416d9990ed6210aba7": "\\Psi(\\mathbf{r}) \\ \\stackrel{\\text{def}}{=}\\ \\lang \\mathbf{r}|\\Psi\\rang", "a432b7bdbd6fea95206a4c15d2336aa3": "\\frac{(1 - \\cos x)(1 + \\cos x)}{x (1 + \\cos x)} = \\frac{(1 - \\cos^2 x)}{x (1 + \\cos x)}= \\frac{\\sin^2 x}{x (1 + \\cos x)} = \\frac{\\sin x}{x} \\frac{\\sin x}{1 + \\cos x}", "a432f6485b6ef769a7ae8b3f2d3e9165": "M_i \\ominus N_i \\sim M \\ominus N \\quad \\mbox{for all} \\quad i.", "a43318a7a28f42b988414cd8e05096a1": "u,\\,v", "a433236ac1589f47bf456b9675c9e8e5": "\nG(r) = {1\\over 2\\pi} \\log(r)\n", "a43328dcca6c94a8dd329db1e8a78901": "(b^n-1)", "a433370204ce12e91540831daba926a8": "f(x) = {1 \\over \\sqrt{2\\pi\\,}} x^2 e^{-x^2/2}", "a4333a1d88b809bb65ee5992e14d254d": "\nh(t) = 2\\, \\Theta(t) \\cdot g(t)\\,\n", "a4337bcaebe0a598471a77a304f220a0": "f(x) = \\frac{x}{x}", "a433ab992abf917402c8912cb7980ad6": "D = (u', v')", "a433df04716475f7c1c54feac8c88b25": "\\ x=1151, \\ y=120 ", "a43427676b2f9d6f8b5b4c9072a78c4f": "d = ", "a434494e9fb590a99e83cb8f507f2860": "\\alpha = \\frac{(^hX/^lX)_{AX}}{(^hX/^lX)_{BX}}", "a434680a1006a2fc69fb01102593d8b5": "\\{0, 6, 10, 15, 17, 18, 31\\}", "a434c984dd736d7a0cabbe92b847a2fb": "\\begin{align}\n \\oint_C \\frac{z^2}{z^2+2z+2}\\,dz\n&{}= \\oint_{C_1} \\frac{\\left(\\frac{z^2}{z-z_2}\\right)}{z-z_1}\\,dz\n + \\oint_{C_2} \\frac{\\left(\\frac{z^2}{z-z_1}\\right)}{z-z_2}\\,dz \\\\[.5em]\n&{}= 2\\pi i\\left(\\frac{z_1^2}{z_1-z_2}+\\frac{z_2^2}{z_2-z_1}\\right) \\\\[.5em]\n&{}= 2\\pi i(-2) \\\\[.3em]\n&{}=-4\\pi i.\n\\end{align}", "a434d9a0d942eb9ed514485fcb51fe00": "\\Sigma_2", "a434e6263b49ac3041c8cd71eb1f13ad": "\\mathbf{\\mathit{b}}_{\\alpha\\beta}", "a434f95202d06f021ac1363989f6ee2e": "\\left(\\dfrac{K}{N}\\right)\\geq 2r_o -1", "a4356945797dd8572c36ef3ebaf1a75a": "\\Delta_rG^{\\ominus} = - RT \\ln K_{eq}~", "a435a1cbb9cef9dea43048a4327ae2f2": " p = (w,x,y,z) \\in R^4", "a435abfc759ee09d0524aa9fac087b66": "{\\Gamma\\vdash e_1\\!:\\!\\sigma\\to\\tau\\quad\\Gamma\\vdash e_2\\!:\\!\\sigma}\\over{\\Gamma\\vdash e_1~e_2\\!:\\!\\tau}", "a435b0880c574aa49c2f3d4f6ba8c489": "(\\Delta A)^2 = \\langle\\hat{A}^2\\rangle - \\langle \\hat{A} \\rangle^2 ", "a435e2c2d174f1484e5ae4d933f798c2": "\\infty \\times 0 = NaN", "a43690c51cbe6e8965162770c40b1755": " \\Delta f(a) \\ \\stackrel{\\mathrm{def}}{=}\\ f(a+h) - f(a).", "a436e3c9fac04b1b7b9ef69ec319eaa7": " k=\\frac{\\sqrt{2m(E-U(z))}}{\\hbar}", "a436ece979d5469ba46ab2f748dcea6e": "P \\hat{B}+\\hat{B} P=0", "a43703ee6f5c673ac5cfdf4d79fcc5cc": "\\frac{\\;\\bot\\;}{},", "a4373091e517eb52837cfe2644290425": "\\overline{e}_b(k,i+1) = \\frac{\\overline{e}_b(k-1,i) - \\overline{\\delta}(k,i)\\overline{e}_f(k,i)}{\\sqrt{(1 - \\overline{\\delta}^2(k,i))(1 - \\overline{e}_f^2(k,i))}}", "a4375901e48d22452445f94bc2922ec9": "\nZ_{m+n} = X_{m-n}((X_m-Z_m)(X_n+Z_n)-(X_m+Z_m)(X_n-Z_n))^2\n", "a43759e09f845e3db839f079926084f0": "\\bigg\\{ \\Pr (h_2) > 0 \\bigg\\}", "a437ba35205d53613276d292ef2d5843": "e < 0.240", "a43827284e6eb5732148676411ee06a6": "AF = \\tfrac{PP - LP}{PP}", "a438673491daae8148eae77373b6a467": "n-1", "a438a873758773e0309bfb97468bc145": "\\vec{g}=\\frac{-G M}{r^2}\\hat{r}", "a438ef560dd81bc769a17e47ad559b06": "\\gamma(0) = \\begin{pmatrix} 0\\\\0 \\end{pmatrix}", "a439e095a208cc240794e14765fc90c7": "\\|L (v)\\|=2\\pi n\\to\\infty", "a439ebd920275fbee5abd1c10bbea988": "|{\\rm pcf}(A)|\\leq2^{|A|}", "a439f5fa0ab8919adbf6bb0c38350bb0": "= \\, \\partial_\\lambda \\partial_\\mu A_\\nu - \\partial_\\lambda \\partial_\\nu A_\\mu + \n\\partial_\\mu \\partial_\\nu A_\\lambda - \\partial_\\mu \\partial_\\lambda A_\\nu + \n\\partial_\\nu \\partial_\\lambda A_\\mu - \\partial_\\nu \\partial_\\mu A_\\lambda \\, = 0 \\,.", "a43a0aee08086038a09816b56054cdcd": "\\scriptstyle x_n \\;\\in\\; B(x_m, r_m)", "a43a2406782cb6a34ce88d428c9ad027": "C_{X\\tilde{Y}}", "a43a7e3959ff356338747cf05230796a": "\\phi^{+}(a)=\\frac{1}{n-1}\\displaystyle\\sum_{x \\in A}\\pi(a,x)", "a43b05d2c054dba07c34d291278d6dc4": "\\beta(t)", "a43b22ec1c4dc185019c3f9fa6b9bff3": "D[u] = \\frac{1}{2}\\int_M \\|du\\|^2\\,d\\operatorname{Vol}", "a43b2f2ccd40b1b0c6c3015469cee1a5": "0 = E \\{ (\\hat{x} - x ) y \\}", "a43b83deaa50ff75f21925ec16de722e": "\\frac{D_Fa^{\\tau}}{ds} = 2\\mu (F^{\\tau \\lambda} - u^{\\tau} u_{\\sigma} F^{\\sigma \\lambda})a_{\\lambda},", "a43bf26df4d39ea189be8e9621e0e958": " (\\forall x,y\\in V): \\langle A x ,\\, y \\rangle = \\langle x ,\\, A y \\rangle .", "a43c61d5696b1a15c86c4300429207a1": "\\frac{1}{4} < \\delta <1 ", "a43cb5fe0d25f584d4d6b1f184e36f67": "\\log |f(0)| = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log|f(re^{i\\theta})| \\, d\\theta,", "a43cf9b4bf59a44e90890b32bd9a6bce": " BzK \\equiv (z-K)_{AB} - (B-z)_{AB} \\geq -0.2", "a43d6cc45da67ba08cd05c0560a31de8": "{1 - H_q}(\\delta + \\varepsilon)", "a43d854136fd67df825b2acb62484d19": "\\begin{align}\n s \\cdot 0 &= s \\cdot (0 + 0) = s \\cdot 0 + s \\cdot 0 \\\\\n \\Rightarrow s \\cdot 0 &= s \\cdot 0 - s \\cdot 0 \\\\\n \\Rightarrow s \\cdot 0 &= 0\n\\end{align}", "a43d8c3cf6840106dc926d320046df09": " \\begin{align}\n \\alpha &= \\mu \\nu, \\text{ where }\\nu =(\\alpha + \\beta) >0\\\\\n \\beta &= (1 - \\mu) \\nu, \\text{ where }\\nu =(\\alpha + \\beta) >0.\n\\end{align}", "a43daa929dfbf4ea8f466db1caa31873": "\\pi _1 (a) = \\pi (a) | _{K_1}.", "a43e03e561928d0126c903b4480368e0": "\\operatorname{pd}_R M + \\operatorname{depth} M = \\operatorname{depth} R.", "a43e2d27c6a66db357979fe87a1c901d": "\\{ w \\in \\Sigma^* \\, | \\, f(w) \\in S \\}", "a43e499ebdf492e64a22f03b128f0526": "\\langle I,\\le \\rangle", "a43ea2de1449804b9a89fab5ff7c82ba": "L(t) = L(0)e^{nt}", "a43ecb8d44d914f65a8c14e63a28fe46": "Z_I=2Z_{8V}", "a43ed32de3773d8b00f6d1d11c2c99f7": " S(t) = P[T>t] = 1-P[T \\le t] = 1-F(t). \\, ", "a43ed71747b6e4ae31c2b188a13eda82": "AQ = \\sin \\alpha \\cos \\beta\\,", "a43ede37969c270770ae88efa1497801": "v=\\frac{L^2}{8r}", "a43ee64b3ea79fd091bf81a79eb57056": "n_{k,d}^{(-n)}", "a43f3e5ab3767e93523e875849a8bcf1": "K=\\frac{\\prod_k {a_k}^{m_k}}{\\prod_j {a_j}^{n_j}}\n=\\frac{\\prod_k \\left([A_k]\\gamma_k\\right)^{m_k}}{\\prod_j \\left([A_j]\\gamma_j\\right)^{n_j}}\n=\\frac{\\prod_k [A_k]^{m_k}}{\\prod_j [A_j]^{n_j}}\\times\n\\frac{\\prod_k {\\gamma_k}^{m_k}}{\\prod_j {\\gamma_j}^{n_j}}\n=\\frac{\\prod_k [A_k]^{m_k}}{\\prod_j [A_j]^{n_j}}\\times \\Gamma\n", "a43f5b6b35305ff52a043c24b838c751": "x^2=-2x", "a43f6fe50466889ccbe7e5f0cff5af23": "a^2\\ = b^2 + c^2 - 2bc\\cos(\\alpha)", "a43fb0889ec5cf97f02af75da1f8c613": " t = \\frac{d}{v \\cos\\theta} = \\frac{v \\sin \\theta + \\sqrt{(v \\sin \\theta)^2 + 2gy_0}}{g} ", "a43fb11de705863aa12bc85ca50e41ec": "= \\arctan \\frac{5*12 + 5*12}{12*12 - 5*5}", "a43ff185b4dfdcd246b4b6a4403bb4f7": "P_\\mathbf{k}", "a4401188055db500d51abba0147c9d9a": "K = mh\\,", "a44035960382e1f69f4ef719941f0c94": "2 T_1 \\ge T_2 \\ge T_2^*", "a44046ce62a0d15b2dbeff80fc20eeb1": "e ::= x \\mid \\lambda x\\!:\\!\\tau.e \\mid e \\, e \\mid c", "a440b3b24b8e4245f4cdcfccdd9d1d7b": "[S_i,S_j]=i\\hbar S_k\\varepsilon_{ijk}", "a440bf132e5fab74507f85bd4f1be165": "\\{p_2,p_1\\}", "a441139e880e9a52208c0bad8f6ab15a": "\\left(\\frac{1}{\\sqrt{10}},\\ -\\sqrt{\\frac{3}{2}},\\ -2\\sqrt{3},\\ 0\\right)", "a441bb1d7fd4a50db43eaf35a2637a35": "\\operatorname{tail} \\equiv \\operatorname{second} ", "a442159a5b4694807f6338dee59fdb4f": "0=\\delta L=\\int_{-\\infty}^\\infty \\delta g(x)\\left (\\ln(g(x))+1+\\lambda_0+\\lambda(x-\\mu)^2\\right )\\,dx", "a4422bd28b247c92a2c3fd757ccecb8a": "A = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\\\ 1 & 0 & 1 \\end{pmatrix}", "a442eb29c3d874f69a9710b52b3ee5f7": "m(X)", "a4430acca20a36c48b96c20bf6837139": "\\sum_{n=1}^\\infty {n^{-s}}e^{hn}", "a443cab2bf4e44468cb3c3123f425109": " \\langle \\cdot, \\cdot \\rangle_\\mathcal{H} ", "a44401077925d7a4f09057d3a6f5d07e": "C\\;2/c", "a444a8fdf5c0bf41c8b3124ee359362a": "A = F(A)", "a444b3ce9511fe84dce66901da56cf6d": "\\sum_{k=1}^{N}SNR_k", "a444df5377725a78acd1cbadfe394a10": "V \\cup \\{0\\}", "a44519b7ddc3f9d1c72feed799214acb": "[[x,y],z^x]\\cdot [[z,x],y^z]\\cdot [[y,z],x^y]=1.", "a4452be8741c8e711937420ca29651aa": "\\iota_{\\rho(\\xi)} \\omega", "a4452bef365b4248e9f6cc254589c423": "\\;\\vdash exp\\;:\\; \\tau", "a44543720db5927a39f7c428c2d8dffc": "\\ln\\left(\\sigma\\sqrt{2\\,\\pi\\,e}\\right) ", "a44625b70477e160186960c6de4e2afc": "\\scriptstyle v({\\mathbf A})", "a446920da2edb9e735c46f0f419c92e8": "\n \\overset{\\triangledown}{\\boldsymbol{\\sigma}} = \\mathcal{L}_\\varphi[\\boldsymbol{\\sigma}]\n = \\boldsymbol{F}\\cdot\\left[\\cfrac{d}{dt}\\left(\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T}\\right)\n \\right]\\cdot\\boldsymbol{F}^T\n", "a4472dfb9bb26424b9273eb553a999e5": "\\mathbf{d}_{x,y,z}", "a4474d2b18e8f02869585b683dd03d9b": "\nF(r) = - \\frac{GMm}{r^{2 + 4/243}}\n", "a4477f6378a9d24028beabf75324fe3a": "E_{tgu} = \\tfrac {m_{gu} \\cdot v_{gu}^2}{2g_c}\\,", "a447808c53570938fa44066fe3dbc635": " t = t_1", "a44792c0a94427e1ddc29a791042718f": "\\sigma'=1-\\sum_{j=1}^{3}2j\\beta_j\\cos\\left(2j\\xi\\right)\\cosh\\left(2j\\eta\\right), \\,\\,\\,\\tau'=\\sum_{j=1}^{3}2j\\beta_j\\sin\\left(2j\\xi\\right)\\sinh\\left(2j\\eta\\right),", "a447af381fd189a989da88d0896cdc90": "\\lim_{\\Delta x\\to 0} \\left ( f(x_0) \\frac{\\Delta g}{\\Delta x} \\right ) = f(x_0)g'(x_0)", "a447b4f828cff5f4178203263db5e29a": "q(x, y)", "a447cc3daf7e72a47bc7b8e1a9cbb035": "\\Gamma_w", "a44841eb7317ccfc28932e5e4d15fcbd": "\\varphi(\\vec x)=\\vec x", "a4487965fc476f3a2f2b316b8cc4c769": "\\rho=A\\xi+B", "a4488e9fcc6312340f7099981a12e11e": "\\Omega\\subseteq\\mathbb{C}", "a448b5126dcb1e23566b233311a55027": "\\Theta = \\Delta_c/\\kappa ", "a448f696270745a1c71042c51e22f440": "\\mathrm{root}^2 = x\\,\\!", "a4491371d008ae8c6659cb1adcf3b490": "\\boldsymbol{k}", "a4491e10b7c713b6fc610e8572561a82": "\\Box A(t)", "a4495acdf6cce086bbbb32c358524e3a": "f(x)=\\frac{1}{2\\pi} \\int_{-\\infty} ^ \\infty \\ e^{ipx}\\left(\\int_{-\\infty}^\\infty e^{-ip\\alpha }f(\\alpha)\\ d \\alpha \\right) \\ dp. ", "a44984a0f3acdd81ecd33dc0be547d84": " H_{jk} = H^*_{kj} ", "a449c37d1917aa4d92abd78f25948165": "~ \\sigma_{\\rm a}~", "a44a068f9ba168703a75569aa64a6df6": "\\displaystyle{c_0=\\varphi\\circ c_1\\circ \\varphi^{-1}}", "a44a88eea3f2273b6733c2424555874f": "b_{ij}", "a44a981cd802f265473ff91c33d4bb44": " DL_i ", "a44aa1a3e50d9c357fcb92758d08d3fa": "\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}(\\mathbf{x},t)~\\text{dV}\\right) = \n \\lim_{\\Delta t \\rightarrow 0} \\cfrac{1}{\\Delta t}\n \\left(\\int_{\\Omega_0} \\hat{\\mathbf{f}}(\\mathbf{X},t+\\Delta t)~J(\\mathbf{X},t+\\Delta t)~\\text{dV}_0 - \n \\int_{\\Omega_0} \\hat{\\mathbf{f}}(\\mathbf{X},t)~J(\\mathbf{X},t)~\\text{dV}_0\\right) ~.\n", "a44bc19f1c1ce09b2ed789073ac0b012": "T^{MN}(x)\\equiv \\frac{1}{\\sqrt{-g}}\\frac{\\delta}{\\delta g_{MN}(x)}\\Gamma[\\text{background}].", "a44bc381d1ea4dd3f1fef86cf3534c36": "g_k=k^{d-2} G_k", "a44be0f14dd5ccfb269f8378dd67fd33": "\\Im\\left[2|k_m|\\right]", "a44c0fe8b7a172b1b393c6829434fe1d": "\\Omega_j", "a44c56c8177e32d3613988f4dba7962e": "a,b,c", "a44c6674e3b5f11d18aa45c522d15f7e": " R=\\tfrac{1}{2}\\sqrt{a^2+c^2}=\\tfrac{1}{2}\\sqrt{b^2+d^2}. ", "a44c8c077f19eba1f2356fb7efe66515": "\\omega \\le \\omega_{max}", "a44cfec4a3701ee6e2335fef110bb6a6": "F = 0\\,", "a44d0350987491aff9f3bf3fc756ab7d": " N = \\frac{C}{\\tau} \\arccot \\frac{T_0-T}{\\tau}", "a44d0de5e12618d58a782913205f38f8": "{\\rho_{opt}} = \\frac{U}{V_1} = {\\cos \\alpha_1}", "a44d1c66be3b55914edf8d9213253f8b": "\\mathbf{\\hat T^ \\dagger} (\\lambda) =\\left(\\exp\\left(\\frac{-i\\lambda\\mathbf{\\hat P}}{\\hbar}\\right)\\right)^ \\dagger = \\mathbf{\\hat T^{-1}}(\\lambda) = \\mathbf{\\hat T} (-\\lambda) ", "a44d5b2b2bf97b58cdbe5780c2737461": "log_{10}[H^+]_{i^{ }}", "a44d5f17aa8f163691e8e81db8c5e0e8": "Z_\\text{in} = Z_0", "a44d66eb36e333cfd0e025ea40742495": "CEi = 6 Ph - 1 Ph - 0.5 Ph = 4.5 Ph", "a44d926c8c359a41856cc1dbe9167249": "N = (I - Q)^{-1} =\n \\left(\n \\begin{bmatrix}\n 1 & 0 & 0\\\\\n 0 & 1 & 0\\\\\n 0 & 0 & 1\n \\end{bmatrix}\n -\n \\begin{bmatrix}\n 1/2 & 1/2 & 0\\\\\n 0 & 1/2 & 1/2\\\\\n 1/2 & 0 & 0\n \\end{bmatrix}\n \\right)^{-1}\n =\n \\begin{bmatrix}\n 1/2 & -1/2 & 0\\\\\n 0 & 1/2 & -1/2\\\\\n -1/2 & 0 & 1\n \\end{bmatrix}^{-1}\n =\n \\begin{bmatrix}\n 4 & 4 & 2\\\\\n 2 & 4 & 2\\\\\n 2 & 2 & 2\n \\end{bmatrix}.\n", "a44da764742b0d94cadb4bcab603e151": "\n (a_1, a_2, \\dots, a_n) <^d (b_1,b_2, \\dots, b_n) \\iff\n (\\exists\\ m > 0) \\ (\\forall\\ i < m) (a_i = b_i) \\land (a_m <_m b_m)\n", "a44dd166376ed3046f8b7e8d3848252e": "\\Sigma_{XY}=E( X^TY) \\,,", "a44de743c010f3b59178ecc69033e85a": "\\tilde p(x)=\\sum_{k=0}^{i-1}\\tilde p_k x^k", "a44e5239774b6bc04dfcf1377551dcc2": "\\partial/\\partial x_j", "a44ef1a7a6efdf8a4b62da47ff7e4166": "N^k = 0\\,", "a44f02517c2efe3b1a80564cc7ef1ecf": "d\\mu (b,a ) = a^{-2}\\;db\\;da", "a44f5a12d8c8fb1701321cef8ade4794": "H > L", "a44f6f5b5ab73c4af594c217064105c1": "u_{l} = \\frac {1} {2 \\sqrt{N}} \\sum_{q} (Q_{q} e^{iqal} + Q^{\\dagger}_{q} e^{-iqal} )", "a45013346760642d67b7822f8aedcfc3": " X^3 + Y^3 + Z^3 + W^3 = 0", "a450208ed3d90b26256edce4e6ebaeb3": " A = A^{*} ", "a4502f91a0dcbdcb65799cdc6b7f162c": " \\psi(n) = n \\prod_{p|n}\\left(1+\\frac{1}{p}\\right),", "a45042a1d3bab156ca70dba5a14a4ab4": "R=|\\rho|^2", "a4504fe89ba566a18e33ef4e7bad599d": "t_\\frac{1}{2} = \\frac{0.5 A_{0}}{k_{0}} \\,", "a45055350ae5f7ee945066f396e5e099": "[x]_R", "a450b1afa80f8cc6e36568fc243f1b55": "\nP_{max,c}=C_{MP,C} \\times h_i", "a450bf0b6ba38e996c7ded3664de35ea": "(C)-(E)", "a450e961105619624e3fa97417f84be1": "\\varphi(s)", "a450fb0e3d3adf77df72805cf296bef5": "\\ C = \\sup_{p_X} I(X;Y)\\, ", "a4517bbc13d47da1907111f6a2ef0031": "X^o\\,", "a451a85d29a379decfbe42b8a78426e7": "X_m = X_{m-1} - qa_m - d_m, ", "a451c514abc5d07cb50fa1fd6c105578": "\\scriptstyle c(E) = A_{\\rm C} (E - E_{\\rm C})^{a_{\\rm C}} ", "a451dcc27b4e842f0ebb07a597484dfb": "F \\times_G B", "a451f61d0283e37cffa7365b066391be": "q q^* = 0 \\!", "a452021d61bbfd8292bb884f2d2429da": "\\vartheta_{\\textrm{AK}}=+1", "a4521ca93e3659fef801f0b7350ab988": "as_{k+1}+bt_{k+1}=0", "a45235313a0f59ddabb800330dfce807": "\\int_{0}^{\\infty}S(y)dy=1 ", "a4528741bfe67e1911a9e366e71bb6c3": "K_i(R) = \\pi_i(B^+\\text{Mod}_R), \\, i \\ge 0", "a452dd4b1036050905b55b61de420187": "\\mathrm{Carbon:} \\ a + b = 3", "a452f1aab19378c31f6f6427da2b4edb": "P(E_1\\cup E_2,\\Omega)\\leq P(E_1,\\Omega) + P(E_2,\\Omega_1)", "a4537e1af71296431957aa42f6597855": " Q(x)=(x^2+ux+v)\\left(\\sum_{i=0}^{n-4} f_i x^i\\right) + (gx+h). ", "a453a417ca32d0a63a0fd3827a9a9d67": "G(x_1,\\dots,x_n) = \\| x_1\\wedge\\cdots\\wedge x_n\\|^2.", "a45437e5451d542e6679cb845a4a4bc8": "\\Delta \\neq 0", "a4545f5cc11060632bb0f92b7533f53e": "\\Sigma Heat = \\mathit{Q}_{2-3}+\\mathit{Q}_{4-1} = \\left(\\mathit{u}_2 - \\mathit{u}_3\\right) + \\left(\\mathit{u}_4 - \\mathit{u}_1\\right) = -4 + 3 = -1", "a4546bcd382733f23282f745bd3e2e98": "k_j", "a454dbdd342648848eb39ee073468eb3": "r = 1111", "a454e0cd4cb60c841912f4fd67be7a98": "T^\\nabla(X,Y) = \\nabla_XY-\\nabla_YX - [X,Y] = 0", "a454e2adde1683c48fca861ec8891509": " \\sum_{\\ell \\in \\mathrm{leaves}(T)} 2^{-\\mathrm{depth}(\\ell)} \\leq \\sum_{\\ell \\in \\mathrm{leaves}(T')} 2^{-\\mathrm{depth}(\\ell)} \\;. ", "a4551ad0fcc0d6da78f5f32df877a42c": " z = x^3 - 3xy^2. \\, ", "a45528413d523664483f8cc8ca7aeffb": "\\gamma(1)", "a45561a55cdebb4131de16477ced85b8": "sI - A", "a4558be388618bf66e7d8023e8135067": "n_A n_B", "a455d8a373cad687bf41643bef274972": "V(g)", "a4562e5b3be6d7d045ce7b385fac8792": "f'(c) = \\frac{f(b) - f(a)}{b - a}.", "a4564b18f05b96d3424539b2e2af5b38": "\\{PS_k(X)\\}", "a456555fd71ff1fd37000652c127e57c": "x=t-\\mathrm{tanh}(t)\\,", "a4565c795f68f308d014ee60d76271b9": "a A + b B + n [e^{-}] + h [H^{+}] = c C + d D", "a456714325e1234d8716711bd3384747": "I \\leq \\frac{2 \\pi c R m}{\\hbar \\ln 2} \\approx 2.577\\times 10^{43} m R", "a456714ee1e739f4562bfa1669b77291": " \\left(\\frac{1}{2}\\right)^s+\\left(\\frac{1}{2}\\right)^s+\\left(\\frac{1}{2}\\right)^s = 3 \\left(\\frac{1}{2}\\right)^s =1. ", "a456db2e847df692d23c0a7496204d6c": "h''(t)", "a456e6f31c8756e539f6d7151ab438dd": " M^{(n)}(B_1\\times,\\dots,\\times B_n)=\\int_{B_1}\\dots\\int_{B_n}\\mu^{(n)}(x_1,\\dots,x_n) dx_1\\dots dx_n. ", "a4574fbd5242a3465e2100a6c796d44d": "\\lambda_1=\\frac{\\tau+\\sqrt{\\tau^2-4\\Delta}}{2}", "a4575d5e5ff75a0867989db261528e9c": " d^3 = at^2\\,.", "a45795c2f57ec9089a7b8124bc7fd405": " \\left( \\omega\\, -\\, \\boldsymbol{k}\\cdot\\boldsymbol{U} \\right)^2\\, =\\, \\bigl( \\Omega(k) \\bigr)^2 \\quad \\text{ with } \\quad k\\,=\\, |\\boldsymbol{k}| \\, ", "a457acd5fa1fca02e2d572079e899c06": "\\chi(S)", "a457c14d8e471b8ddcd768779655e427": " \\int_{P^t} f(x) \\, {\\rm d}x = \\frac{1}{p^t} \\int_{\\Bbb Z_p} f(p^t x) \\, {\\rm d}x", "a458159f1aa3d315faa426c6b462aeec": "f_\\mathrm{ls} - f_\\mathrm{sa}\\ =\\ -f_\\mathrm{la} \\cos \\theta", "a45823c40c254d7ddccc42464fabfcec": "j\\geq 3", "a4582fdfe1f147c3cb2b355209d1b0d5": "(21 - 1) \\cdot a + 2 \\cdot b.", "a4587731829b2cc5edc16cc66ff1258e": "f(x) = \\lambda e^{-\\lambda x} \\mbox{ for } x \\geq 0.", "a458a858d840ababc490a290f9302d01": "\\operatorname{tr}(xy) = 0", "a4596910628e305490065fe917ca4b1f": " \\lim_{t\\rightarrow\\infty} e^{-t}\\sum_{n=0}^\\infty \\frac{t^n}{n!}A_n(z). ", "a4596a89752866f7d02527d6c188ec03": "\\begin{align}\n a_2 &= \\frac{1}{2}(15 + 12) = 13.5\\\\\n g_2 &= \\sqrt{15 \\times 12} = 13.41640786500\\dots\\\\\n \\dots\n\\end{align}", "a4597e3ef86f56baadd3694e3f121b6f": "\\boldsymbol{S} \\Rightarrow_2 \\boldsymbol{abc}", "a4598c296220915b23529a8066061c6e": "\\rm \\ C_6H_5X + (CH_3BO)_3 \\xrightarrow[dioxane]{K_2CO_3, Pd(PPh_3)_4} C_6H_5CH_3 (X = Br, I)", "a459ad3c79ce493684df3eaea0b7c868": "V_{m} = \\frac{RT}{F} \\ln{ \\left( \\frac{ P_{K}[K]_{o} + P_{Na}[Na]_{o} + P_{Cl}[Cl]_{i}}{ P_{K}[K]_{i} + P_{Na}[Na]_{i} + P_{Cl}[Cl]_{o}} \\right) }", "a45a1f3fba2a06930f67b3e19c682704": "\\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \n \\; \\mathbf{\\hat{k}} \\mathbf{\\hat{k}} \\;\n{ \\exp \\left ( i\\mathbf{k} \\cdot \\mathbf{r} \\right ) \\over k^2 +m^2 } \n= \n{1\\over 2} {e^{ - m r } \\over 4 \\pi r } \\left[ \\mathbf{1}- \\mathbf{\\hat{r}} \\mathbf{\\hat{r}} \\right]\n+\n {1\\over 2} {e^{ - m r } \\over 4 \\pi r } \\left\\{ 1+ {2\\over mr} \n- {2 \\over \\left( mr \\right)^2 } \\left( e^{mr} -1 \\right) \\right \\}\n \\left[\\mathbf{1}+ \\mathbf{\\hat{r}} \\mathbf{\\hat{r}}\\right] ", "a45a4b2ef5b2b458c001387ca88e3b07": "= \\frac{ \\partial \\left ( \\frac{1}{2} \\left( t_j-y_j \\right ) ^2 \\right ) }{ \\partial y_j } \\frac{ \\partial y_j }{ \\partial w_{ji} } \\,", "a45a74742e522b3bd211f12968ad49fa": "\\sqrt{2} \\times \\sqrt{6}", "a45a8a342c975d562ff3aa58ed6f7a78": " f_*\\omega_Y^{\\otimes n}; ", "a45a8af24067207fba5daaac4a9def47": " H_{jj}^{ } = 0^{ } ", "a45a8e15fe60b8d16532b2ea54c5b26c": "\\sin(\\theta_3-\\theta_1) \\sin(\\theta_4-\\theta_2) = \\sin(\\theta_2-\\theta_1)\\sin(\\theta_4-\\theta_3) + \\sin(\\theta_4-\\theta_1)\\sin(\\theta_3-\\theta_2)\\,", "a45a9755bc7adc80e540ced18f815c9d": "\\vec{\\beta}", "a45abfb32512eb552899ea451aa070bc": "K = \\sqrt{(s-a)(s-b)(s-c)(s-d) - \\tfrac{1}{4}(ac+bd+pq)(ac+bd-pq)},", "a45ad299a6b3df5e0e1deebc311b70bc": "G_{nm}(\\mathbf{x}',\\mathbf{x}) = G_{mn}(\\mathbf{x},\\mathbf{x}')", "a45b1c1012fab7f25e4ead8f3473a2e8": "W_3", "a45b58139f65db93013867d74edc2f71": "\\ -87.55", "a45b652cc1e541cc11ba8b85acd6dd1a": "|\\zeta(s+it)-f(s)| < \\varepsilon\\quad\\mbox{for all}\\quad s\\in U.", "a45c03886d8340e0cd8be999aa4e78f9": "\ne(T_1,T_2)\n=\n\\frac\n {\\mathrm{E} \\left[ (T_2-\\theta)^2 \\right]}\n {\\mathrm{E} \\left[ (T_1-\\theta)^2 \\right]}\n", "a45c196d1ef7a7746983723dd535ef08": "\\scriptstyle\\mathbb{I}", "a45c79c75385285aceca1d4ab5d2d4ba": " \\Delta H_{SA} \\,", "a45c8937a6fe91aab7989610c03cc9b7": " A^+ A = I_n\\,\\!", "a45cfb05a343f550d7639f67e991960d": "e p (d+1) \\le 1,", "a45d0d68dca909c0ca27c8223a83e961": "\\scriptstyle x. \\; ", "a45d5763a97847403e42a48f8a63ca5d": "O(\\ell \\log n+k \\log^2 n)", "a45e0a079e383806f6e6d0bce6576f68": " s \\geq 1", "a45e1e84e8cf33d7924535d2d0b0fe06": "\\frac {dS}{dT} = \\left[S(T)\\right]^2 \\frac{A c_2}{C\\left(AT + B\\right)^2}\\exp\\left(\\frac{c_2}{AT + B}\\right)", "a45e8e79d9540bd32ecfebd177d7ebf9": "\\scriptstyle \\log_{10}P_{mmHg}=7.80307 - \\frac {1651.2} {225+T}", "a45eade2dbeb201bd183ecb1d3214337": " f_{a}\\;(1)", "a45ebd3079366669b63095bd7c69e469": "\\Bigg(\\frac{\\alpha}{\\beta}\\Bigg)_3 = \\Bigg(\\frac{\\beta}{\\alpha}\\Bigg)_3. ", "a45f573f58e93c45785eb4bcdc39cfc2": "J^1Q\\to TQ", "a45fb04ab171cf2566e4e64e12bdbd57": "u_{n\\mathbf k}(\\mathbf r)", "a45fc258ca478ae3c405a76bde0649cb": "u(x,t)=F(x+ct)+G(x-ct)\\,", "a460bab0c37a5e1f8703a82ca17f6ed7": "Q(\\lambda)=\\lambda^2 A_2 + \\lambda A_1 + A_0\\,", "a4610517fc4abfb22b4cc275d0245b3f": "\\textrm{c}(\\kappa,\\beta)\\,", "a4619636787fa4848e045bd0c7ff3f8c": "t(k) \\le 2^k\\cdot t(k/2)", "a461aa33ba7c2d12f060fdc49342aaf8": "G_i(F)", "a461c29244d58c1a6d38fbca60cb214f": "\\Lambda(x)=\\prod_{j=1}^t (x\\alpha^{i_j}-1)", "a461cad12ba770511324e7f7929f91ef": "\\lambda/n.", "a461e66f3d13a13ab57881df1247d123": "E^\\frac 1 2 ", "a462133a38488b2c7515da066f5aa215": "\\mathbf{v} = \\lim_{\\Delta t \\to 0} {\\Delta x \\over \\Delta t} ", "a4622864613b455552dde4c98eb86df5": " J = -D \\left[\\frac{\\phi (x + \\Delta x, t)}{\\Delta x} - \\frac{\\phi (x , t)}{\\Delta x}\\right]", "a4623f0c08a441fe74202c306ee72af3": "E_\\mu \\|h(x)-\\mu\\|^2 = E_\\mu \\{ \\mathrm{SURE}(h) \\} ", "a462eace462cac92cf094b264dfeed72": "y' = - x \\sin \\theta + y \\cos \\theta", "a4630d2a661b2ba8d764e9a189491673": "\\alpha = c_2 + 2r", "a4631b43c6ce0da87cb1a2374fc2d66f": "2\\frac{q_\\text{e}q_\\text{m}}{\\hbar c}", "a4633b7d6f839937e67a4ae57f517b88": "\\hat{H}\\Psi = i\\hbar\\frac{\\partial\\Psi}{\\partial t} \\,,", "a4639c726b920790b225b721e58e97d9": "q \\in \\mathbb{R}^n", "a463aa85e598fa857815aa7adb34ae3d": "X_G=\\{G(y) \\;\\vert\\; y\\in\\{0,1\\}^\\ell \\}", "a463abe65f7a78394f50ec2742a5c89d": "H_t = \\operatorname{E}_Q(H_T| F_t)", "a463d252086e60aec6dacc2d1f7b39d7": "(t_0-t)^{-\\alpha}", "a463d3d1acabcb800e97a4715ece958e": "X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} X[1]", "a463ec39e265b9ff3070c011cbd0c701": "\n\\begin{align}\nk & \\equiv H(m)s^{-1}+xrs^{-1}\\\\\n & \\equiv H(m)w + xrw \\pmod{q}\n\\end{align}\n", "a4641bfabdcc34d05c37fce57dbefdbd": "W(x_i, x_j) = x^\\prime_ix_j - x_ix^\\prime_j", "a4643c23e21caad7c3a7332ee70e6440": " \\| x + M\\| = \\inf\\limits_{m \\in M} \\|x+m\\|. ", "a4644073989b14afa324194829d7a184": "G: C \\to D", "a465005469a26bcf52545eb36fe1101d": " \\mathbf{a}_i = \\boldsymbol\\alpha\\times(\\mathbf{r}_i-\\mathbf{R}) + \\boldsymbol\\omega\\times\\boldsymbol\\omega\\times(\\mathbf{r}_i-\\mathbf{R}) + \\mathbf{A}_R.", "a466372541e515c191b8f1f549969e84": "10! = 3,628,800", "a46677e799d08db72d263c8f6c07a1d9": "\\hat \\beta \\pm 2.5se_{\\beta}", "a4668285186f23e7129478638a926cb7": "X_{ij},", "a466a8d634455906a97bc99a32ce62a3": " a\\circ b= \\{a,y,b\\}. \\,", "a466da58e2c1f3b0f9bc04a4b969c5e6": "R_{P_i}/P_i^{a_i} R_{P_i}", "a4674cd1624d3aa65082f6a8e251a3f0": "D_{\\max}(\\rho||\\sigma) = \\inf_{\\lambda}\\{\\lambda:\\rho \\leq 2^{\\lambda}\\sigma\\}", "a46753896c7947354bb387bd364d5c18": "en_e", "a46758f69c0813c46abe75fda82625d7": " \\begin{align}\n& && S \\mathbf{.v} = \\mathbf{0} \\\\\n\\end{align} ", "a467ccad2b11fb8272c97637c53bb398": "(1_A-1_{A_1})(1_A-1_{A_2})\\cdots(1_A-1_{A_n})\\,=\\,0,", "a467ee80e217ca2c1f21dfb19a4af21b": "\\theta(n)=\\frac{\\alpha}{c+v}+\\frac{\\alpha}{c+2v}+\\cdots+\\frac{\\alpha}{c+nv}\\,\\!", "a46831df6d0f85444c672f4bd138f2fb": "a_i^{\\rm eq}>0", "a468705426d8cdee9489e380404d1457": "\\forall N\\in\\mathcal{N}_{f(x)}\\exist M\\in\\mathcal{M}_x: f(M)\\subseteq N", "a4687b46c63da02dcb18b530b1d71ee5": "\\sigma_{3c}=6.3", "a468845158eeb6dd5b7a78b2cb047414": "A_{t+1} - A_{t} = 0", "a46921474a0d7c43a837a293eca81c73": "N = \\sum_{ij}{O_{ij}} \\; ", "a4695a81856165dc1445aa01479c0888": "\\Pi_{WW}(q^2) = \\Pi_{WW}(0) + q^2 \\Pi_{WW}^{\\prime}(0) + ...", "a469713ba52a463c73cb49228e544601": "\\frac{P_{\\rm min}}{\\dot m} = h_2-h_1 - T_a(s_2-s_1).", "a4699c14453eb298e004ad69816c832a": "N(k,d)\\geq \\left\\lceil N(k-1,d/2)\\right\\rceil +d", "a469a8245a154ea5bfbb3843c64dbe6c": "{\\mathfrak{T}}_{\\alpha\\beta}", "a469cbbeb4415f021f3e3858e98384ed": "u(r) = \\sum_{q} \\sqrt{ \\frac {\\hbar}{2M N \\omega_{q} } } e_{q} [ a_{q} e^{ i q \\cdot r} + a^{\\dagger}_{q} e^{-i q \\cdot r} ] ", "a46a4e1035aa22fca2a8a89c7fbb1a75": "W_q^{(H)}=\\frac{\\lambda(\\frac{1}{\\lambda^2}+\\sigma_B^2)}{2(1-\\rho)}.", "a46a950db05c5279bca40708b42444d0": "e^{-\\frac{A}{k T}} = \\operatorname{Tr} \\exp\\big(-\\tfrac{1}{kT} \\hat H\\big).", "a46aa2454419d285030188e52949f161": "C_{\\mathrm{sen}}", "a46ad80a5948ffa1287f8732a4c81cd1": "\\tfrac{2Ta}{a^2+2T}", "a46b8c80d2b61f78ba405b6b53822890": " N_{nl} = \\left [ \\frac{2^{n+l+2} \\,\\gamma^{l+{3 \\over 2}}\\,[{1 \\over 2}(n-l)]!\\;[{1 \\over 2}(n+l)]!}{\\;\\pi^{1 \\over 2} (n+l+1)! } \\right ]^{1 \\over 2}\n= \\sqrt{2} \\left( \\frac{\\gamma}{\\pi} \\right )^{1 \\over 4} \\,({2 \\gamma})^{\\ell \\over 2} \\, \\sqrt{\\frac{2 \\gamma (n-l)!!}{(n+l+1)!!}}\n .", "a46b9bd08378e676d12acbc8fa198c45": "a_n(x)y^n+a_{n-1}(x)y^{n-1}+\\cdots+a_0(x)=0 \\, ", "a46bd39af26a6ee5df8ae96e09250782": "C_p\\,\\!", "a46be189b8246f0f9657170a0b127665": "KD", "a46c4f39fee0be640e44711f1c596e63": "\\ln(c + \\alpha)_r = \\ln\\left((c + \\alpha)(c + \\alpha + 1) \\cdots (c + \\alpha + r - 1)\\right) = \\sum_{k = 0}^{r - 1} \\ln(c + \\alpha + k).", "a46c8b339622849f79364e38d5514947": "\\scriptstyle E[[X]]", "a46ccb832bc4ed140a922c2b50fd0ca2": "\\Lambda_K(s)=\\left|\\Delta_K\\right|^{s/2}\\Gamma_\\mathbf{R}(s)^{r_1}\\Gamma_\\mathbf{C}(s)^{r_2}\\zeta_K(s)", "a46cee42250b231f190764fc3e9dbe54": "\np(n) = |\\left\\{ (a_1, a_2,\\dots a_k): 0 < a_1 \\le a_2 \\le \\ldots \\le a_k\\; \\and \\;n=a_1+a_2+\\cdots +a_k \\right\\}|.\n", "a46da57a37409761e63cbb561029257e": "c = max | \\langle a_j | b_k \\rangle |", "a46dd58e9e372e5b90effdb9d4d26709": "G = \\oplus_i C_{p^{e_i}} \\ ", "a46e0cd572e528c7a42f0edf0d408eeb": " f(x) = 0.5 \\|Ax-y\\|_2^2 ", "a46e206d18434ab5c2345b74b5f474aa": "(x, k) \\in L", "a46e2a809387d88f6783f630746986d0": "\\lambda(mk+1) = mk", "a46e3464da54b53842fce23d8034f42f": "\ns=\\frac{1-(-1)^N e^{i \\Delta k \\Lambda N} }{1+e^{i \\Delta k \\Lambda}},\n", "a46e47baf223c30427b4b52eab19e67f": " M(x,x, \\ldots,x) = x ", "a46e90d89d5d81026f5ab52a61b7a2b6": "x\\;(x + 1)\\;(x^2 + x + 1) = x + 2x^2 + 2x^3 + x^4", "a46ebce79fc828f2477fe22bf5c2d6b8": " \\hat{X}, \\hat{Z}, \\hat{X} \\hat{Z}, \\hat{X} \\hat{Z}^2 ... \\hat{X} \\hat{Z}^{d-1} ", "a46ed42c4b50b62dd6e2dbcb6dfac94f": " \\langle Px, y-Py \\rangle = \\langle P^2x, y-Py \\rangle = \\langle Px, P(I-P)y \\rangle = \\langle Px, (P-P^2)y \\rangle = 0 \\,", "a46ee6d4792bffd961c1804034cad5fb": "\\frac {\\mu} {2 \\cdot a}", "a46eea8f46fedabb69902a600ee7b2a0": "\nu_l(x) = x^{l+1} e^{-x/2}f_l(x).\\,\n", "a46ef7909bcac7108ee6e1eb2a7398b0": "\\partial f_k/\\partial x_i", "a46f06d7c0aef7fb01d7d65b4146992a": "\ni{\\partial \\over \\partial t} \\psi = -{\\nabla^2\\over 2m} \\psi + \\left(\\int_y V(x,y)\\psi^\\dagger(y)\\psi(y)\\right) \\psi(x).\n\\,", "a46f6c25179b8387f834a364d1cb6dec": "S_L = 1", "a46fa7cb66fa31a39676665921a6c8fe": "\\!\\,\\gamma_1 :I \\rightarrow X", "a46fae98f58ece822c75a68906f22874": "n \\cdot \\delta p = \\frac{p}{2}", "a46fcc4052e34ee9f62f61f210b9eb5b": " p > \\sqrt{2n},", "a4709a89071fe3fbfbc11729d0cf6bcd": " g", "a470d9b7d2bb70a7d7821758b261a110": "\\frac{N}{2}", "a4711f707f705246f1c53e407789014e": "\\varphi(36)=\\varphi\\left(2^2 3^2\\right)=36\\left(1-\\frac{1}{2}\\right)\\left(1-\\frac{1}{3}\\right)=36\\cdot\\frac{1}{2}\\cdot\\frac{2}{3}=12.", "a4713536806b1e0ea5163a7bc489a9d8": " f(z) =\\sum a_n z^n.", "a47145bca25de17e8c08459e18f4b1e1": "a_u", "a471473b526c350c7da18debb98a7569": "y^2 = x^3 + 140x + 149 \\pmod{167}", "a47153368119fc7cc27de8ff7936b832": "\ny\\rightarrow \ny^9-36 y^7 (z^2+x^2)+126 y^5 (z^2+x^2)^2-84 y^3 (z^2+x^2)^3+9 y (z^2+x^2)^4 + y_0\n", "a471b371560c02cd8e84539de9038326": "\\textstyle 1 + 2\\sum_{i=1}^\\infty r^i = 3 + 2r", "a471c948b77d20304ea3452122333442": "r_{\\theta \\theta} = {d^2 r \\over d \\theta^2} = {2 a \\over \\theta^3}.", "a471e1eef7bdd14aff58bece0fae1aa8": "\n\\varphi = \\varphi_{0} + \\frac{L}{\\sqrt{2m}} \\int ^{u} \\frac{du}{\\sqrt{E_{\\mathrm{tot}} - U(1/u) - \\frac{L^{2}u^{2}}{2m}}}\n", "a471ec9a10832d4acde78bd221ecd626": "\\operatorname{pf}(BAB^\\text{T})= \\det(B)\\operatorname{pf}(A).", "a47207d7843c1aaa4860fc5f225152be": "\\log \\zeta(s) = s\\int_0^\\infty J(x)x^{-s-1}\\,dx. ", "a4723bc25686731fe3f635b59d630fe7": " = P_1 \\cdot P_2 \\times P_3 = 0 ", "a4728b083a7d76183de1e34565d652d4": "\\frac{\\partial x}{\\partial v}", "a473ad2c7372dcad0c5fae63286ebb19": "A_h(-\\theta,\\tau)", "a4741f5e04c2425d643b27b6e778001c": "\\pi:Y\\to X", "a47426b5182a769b556d83b4e5f83e6a": "\\Delta(v_1 \\otimes \\dots \\otimes v_k) := \\sum_{j=0}^{k} (v_1 \\otimes \\dots \\otimes v_j) \\otimes (v_{j+1} \\otimes \\dots \\otimes v_k)", "a4743974ebf96bbfa2d397984fc75633": "e^\\mathsf{T} = \\frac 3 2", "a4745b3db5b0c52d8fc318335e14dc32": "G\\ ", "a47513e4d8d445120beaf2103e9ce2f6": "\\mathbb H\\otimes\\mathbb O", "a4752a327c352bf33fa8e0c346245a4d": "\\begin{align}\nD_{m}=\\frac{(J_X+J_Y)}{2}-J_{XY}\n\\end{align}\n", "a4753bd2499f49f94ce82e5d1c9c6525": " {x_2 \\choose \\theta_2} = \\mathbf{S}{x_1 \\choose \\theta_1} ", "a47586d95019590a6ee219a030dbc1dc": " \\limsup_{n \\rightarrow \\infty} \\Pr\\left( \\sup_{\\theta \\in \\Theta} \\sup_{\\theta' \\in B(\\theta, \\delta)} |H_n(\\theta') - H_n(\\theta)| > \\epsilon \\right) < \\eta .", "a475c6d6aaa85f622374e99dd62fb0e5": "K(u) = \\frac12 \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "a475ed1ad8ed9b18e23767b61e926f6d": "\\Delta C_\\mathrm d\\,", "a4764f0292546e6a4b7d32a61d383fa2": " f_d, ", "a47681f0820af5599a3b69383b32dbb9": "{(A^{C})}^{C} = A\\,\\!", "a4769b0ce3e0f9e0f6c2bdbd41399977": "\\pi=\\frac{2\\sqrt{3}}{Z} \\!", "a4773b79f4927d79d75bef2991b6b320": "\\left | \\frac{d\\nu}{d\\lambda} \\right| = c/\\lambda^2", "a47742550cce89b633893d2a4d2d1ab7": "\n\\mathbf{r}_0 =\n\\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix} - \n\\begin{bmatrix} 4 & 1 \\\\ 1 & 3 \\end{bmatrix}\n\\begin{bmatrix} 2 \\\\ 1 \\end{bmatrix} = \n\\begin{bmatrix}-8 \\\\ -3 \\end{bmatrix}.\n\n", "a477438350eb0d6deeab7d9522f8a283": "E^2 = (pc)^2 + (mc^2)^2 \\,\\!", "a477b6f2b7c4f19f3a65a27ef96692ac": "PRI=\\frac{( \\rho 570- \\rho 531)}{( \\rho 570+ \\rho 531)}", "a47859a5f9e46e31c65540b13085060e": "\\frac{d\\tau}{dr} = k\\rho", "a4786f563fcfc3daadf96b40197f0e1a": " 30<\\varepsilon_r<50 ", "a478751cb53c6cce8a43e557bd09aa6e": "\\lambda^{(n-1)}", "a4788ab5dc66b5568b75044285c9c9de": "\\sigma >0", "a478b8f17ae12910b1ee562c04fb86a7": "D_{\\mu\\nu}^{ab}(p)\\stackrel{p\\rightarrow 0}{=}\\delta^{ab}\\left(\\eta_{\\mu\\nu}-\\frac{p_\\mu p_\\nu}{p^2}\\right)\\frac{Z}{p^2-M^2+i0},", "a478cfc27a0fa39c9c5346f5835c6eca": "V_f = V_i e^{rt}", "a478d2f99846127450e4cefabe6bb053": "O(n \\textrm{polylog} n)", "a4796b1dcde6c35926b2d278f8dd9683": "f_a'(y) = a + 2y. \\,", "a47998ece78b7428eaabbe97ddbacca5": "2M = I \\mathrm{tr}\\, M + \\sum_i \\sigma^i \\mathrm{tr}\\, \\sigma^i M", "a47aca0bc8353c3c8ca1361b3efaece6": " \\lambda m.\\lambda n.n\\ m ", "a47afca5081e9c05b8512fe8bc402837": "A =\\frac{p \\cdot q}{2}.", "a47b5450b3424e3f4786144e276c12a1": "k-\\mathrm{Alg}", "a47b5f599370139d844234713880dec3": "\nq _{n,\\nu}= \\frac{\\max\\{\\,x_1,\\ \\dots \\ x_n\\,\\} - \\min\\{\\,x_1,\\ \\dots\\ x_n\\}}{s} = \\max_{i,j=1, \\dots, n}\\left\\{\\frac{x_i - x_j}{s}\\right\\}", "a47b6946ae50bcba37a99d8fdba2439c": "M, w \\models \\Diamond \\Diamond \\varphi", "a47b6afed5bd1d596da871187dcfd666": "\\chi = \\left(\\!\\frac{d}{m}\\!\\right)", "a47c3cc70c102b9004fa9f84866e27eb": "\\frac{\\partial f}{\\partial t} = \\left(\\frac{\\partial f}{\\partial t}\\right)_\\mathrm{force} + \\left(\\frac{\\partial f}{\\partial t}\\right)_\\mathrm{diff}+ \\left(\\frac{\\partial f}{\\partial t}\\right)_\\mathrm{coll}", "a47c5ac1d6991ae2b8f4d6958a76bc17": "\n {k+r-1 \\choose k} = \\frac{(k+r-1)!}{k!\\,(r-1)!} = \\frac{(k+r-1)(k+r-2)\\cdots(r)}{k!}.\n ", "a47c738a2427a9060bad39241be08914": "\\displaystyle{M = \\begin{pmatrix} \\zeta_1 & 0\\\\ 0 & \\zeta_1^{-1}\\end{pmatrix}\n\\begin{pmatrix} \\cos \\varphi & \\sin \\varphi\\\\ -\\sin \\varphi & \\cos \\varphi\\end{pmatrix}\n\\begin{pmatrix} \\zeta_2 & 0\\\\ 0 & \\zeta_2^{-1}\\end{pmatrix}.}", "a47c749a1735a8071c7e8427a17e4042": "s=\\langle w_0,w_1,\\dots,w_n\\rangle", "a47ca7afd16b421465c4c755a7d79964": "dx^i", "a47cafa77bdd5bca8471710f46c10de1": "v_\\text{e-act}", "a47d144eda17756fb47a8ae5b02015a2": " X:\\Omega\\rightarrow \\R ", "a47d2bdcb8746f567f670bab10297d9c": "\\supset \\!\\,", "a47d312b6992421526846fff6f125e30": " t \\geq 0", "a47d5f48c1f9b4d671b0d05af06a1f5e": "\\lambda_8 = \\frac{1}{\\sqrt{3}} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2 \\end{pmatrix} .", "a47d843922bf2470feae398839a2c2e1": "p_g(X|B) = \\sum_x P_X(x)tr(E_x \\rho_B^x)", "a47db6dd57d43f6909189d68ed20b800": "p^{(i)}", "a47e2436a7e7a384fea0771bc817aa75": "\\overline{X} = \\frac{X_1 + \\cdots + X_n}{n} ", "a47e4cc77448cf5358d5b97b4e1989b0": " \\delta W = Q_1\\delta q_1 + \\ldots + Q_m\\delta q_m,", "a47e742f7db5d4f0d8ee416d765ebf48": " z_k(s) \\leftarrow \\alpha z_k(s) + (1 - \\alpha) x_j(s) ", "a47ea4d4ace4b396fdabc2e8ae6736e6": "(\\lambda_1 - \\mu),...,(\\lambda_n - \\mu). ", "a47ee103ae679e81af30673daa7bac34": "Y + \\delta Y", "a47f3a22211cd246f63f956a2d6df0e4": "\\frac{\\partial^2 h}{\\partial t^2} = c^2\\frac{\\partial^2 h}{\\partial x^2},", "a47f4ea02ca223db40e812d1340b9aaf": "a^{\\dagger}_{{\\mathbf{k}}_{l}}", "a47f7bad2abe12f1f33e8f79d4ee0081": " \\left[ \\begin{array}{c} \\dot{x}_1(t) \\\\ \\dot{x}_2(t) \\\\ \\dot{x}_3(t) \\\\ \\dot{x}_4(t) \\end{array} \\right] = \n\\left[ \\begin{array}{c} x_2(t) \\\\\n m^{-1} \\left[ f(t) - c x_2(t) - a k_i x_1(t) - (1-a) k_i x_3(t)\\right] \\\\ \n \\frac{h(x_3(t))}{\\eta(x_4(t))} x_2(t) \\left\\{A(x_4(t)) - \\nu(x_4(t))\\left[\\beta\\operatorname{sign}(x_3(t)x_2(t)) + \\gamma \\right]|x_3(t)|^n \\right\\} \\\\\n (1-a) \\omega^2 x_3(t) x_2(t) \\end{array} \\right] ", "a47fdb18d9855fcea6ff4a72efc330d6": "\n\\begin{matrix}\n\\qquad\\quad\\;\\, x^2 \\; + x \\quad + 3\\\\\n\\qquad\\quad x-3\\overline{) x^3 - 2x^2 + 0x - 4}\\\\\n\\;\\; \\underline{\\;\\;x^3 - \\;\\;3x^2}\\\\\n\\qquad\\qquad\\quad\\; +x^2 + 0x\\\\\n\\qquad\\qquad\\quad\\; \\underline{+x^2 - 3x}\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad +3x - 4\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad \\underline{+3x - 9}\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\;\\; +5\n\\end{matrix}\n", "a480710b5c00c561ef95b5ca7a19fe02": "m=26", "a480ea9d2faa287e82c0af9b2f7c296a": "\\alpha_q", "a48141549e23c737ed12480f8810d111": " r_s^{-1/2} ", "a481549c705ebea1358e80180bd2bac8": "a = \\frac{\\Delta y}{\\Delta x} = \\frac{\\Delta v}{\\Delta t}.", "a481acee793be8415e6b8ed12870af27": "\\mathbf{U}^{*}(x) := \\bigcup_{U_{\\alpha} \\ni x}U_{\\alpha}.", "a481b6334058de8565337bb139aaa21f": "\n[\\psi_A,\\psi_B] \\subset J_3(O)\n", "a481e1ea2fdd0874bfcdc7b7cf6742df": "(3 + i)", "a481ef197095988e965a25b16c63f5a7": "[(-{\\hbar^2 \\over 2M}\\nabla^2)+(-{\\hbar^2 \\over 2\\mu} \\nabla^2-{e^2 \\over 4\\pi\\epsilon r})]\\Phi^{n,k}(r,R) = [\\Epsilon-\\Epsilon_G] \\cdot \\Phi^{n,K}(r,R)", "a48226b9ceb435dd6a0f8520ef685b09": "\\psi(x,y,t)", "a4822a823f2d4daa32cc1e6595154303": "= {1 \\over 2\\pi} \\int_{-\\infty}^{\\infty} F(\\omega) e^{i\\omega t}\\;{\\rm d}\\omega \\ ", "a482919cf05a36e7c7b1612e441140ce": "E_{ex} = C - \\frac{1}{2}J_{ex} - 2J_{ab} \\vec{s}_a \\cdot \\vec{s}_b ", "a48293ce6722cb13f3dbfad522681143": "\\begin{cases} \\dot{\\mathbf{x}} = f_0(\\mathbf{x}) + g_0(\\mathbf{x}) z_1\\\\\n\\dot{z}_1 = f_1(\\mathbf{x},z_1) + g_1(\\mathbf{x},z_1) z_2\\\\\n\\dot{z}_2 = f_2(\\mathbf{x},z_1,z_2) + g_2(\\mathbf{x},z_1,z_2) z_3\\\\\n\\vdots\\\\\n\\dot{z}_i = f_i(\\mathbf{x},z_1, z_2, \\ldots, z_{i-1}, z_i) + g_i(\\mathbf{x},z_1, z_2, \\ldots, z_{i-1}, z_i) z_{i+1} \\quad \\text{ for } 1 \\leq i < k-1\\\\\n\\vdots\\\\\n\\dot{z}_{k-1} = f_{k-1}(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}) + g_{k-1}(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}) z_k\\\\\n\\dot{z}_k = f_k(\\mathbf{x},z_1, z_2, \\ldots, z_{k-1}, z_k) + g_k(\\mathbf{x},z_1, z_2, \\dots, z_{k-1}, z_k) u\\end{cases}", "a482a06f357c054f4deee99ee103350f": "\nG^{\\mathrm{T}}(\\mathbf{k},\\omega) = [1+\\zeta n(\\omega)]G^{\\mathrm{R}}(\\mathbf{k},\\omega) - \\zeta n(\\omega) G^{\\mathrm{A}}(\\mathbf{k},\\omega),\n", "a482c62c7f38151d0df85a5f046b6b7c": "P + Q", "a482ccf348a774c2a2b110d4125a6015": "u_n \\ne 0", "a482f7fbb3a4d9824a49d3a3a8389dbd": "\\lbrack\\mathbf z\\rbrack = \\lbrack\\mathbf z\\rbrack_1 + \\lbrack\\mathbf z\\rbrack_2", "a483898aa38f075e453e2b59c0202e3b": "g_{\\boldsymbol\\theta}", "a483c2dc6eec1843b497822b0db49879": "\\mathcal E^\\bullet=\\bigoplus_i \\Gamma(E_i)", "a484209bf85df420834ce932efac9e80": "\nV(t) = V_o e^{-t / \\tau}\n", "a484957204f42edcb883392bc0359761": " \n\\mathbf{v_i} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{d\\mathbf{r}}{dt} = \n\\left( \\frac{d\\mathbf{r}}{dt} \\right)_{\\mathrm{r}} + \n\\boldsymbol\\Omega \\times \\mathbf{r} = \n\\mathbf{v}_{\\mathrm{r}} + \\boldsymbol\\Omega \\times \\mathbf{r} \\ ,\n", "a485343c4a2206a58b3e32c666ed4a0d": "\\psi = \\omega_\\vec{p}\\;e^{-ipx} \\;", "a4853fbf2777c0631cb04aa88f751466": "d_1 = \\frac{\\ln(S_0/K) + (r_d - r_f + \\sigma^2/2)T}{\\sigma\\sqrt{T}}", "a48548f93dcda20b94d4b9559b99862e": "\\text{change open}_1", "a4858d3e400f3d65e5e73dee7ae3eadf": "1, \\frac32, \\frac75, \\frac{17}{12}, \\frac{41}{29}, \\frac{99}{70}, \\dots", "a48595efb1735b1e0d890e5e94f3e97d": "v_{(G; c)}(\\{2,3\\})=12", "a485a9bf4ea628d6489a36c523254f3d": "K_{2}", "a485bedf9bf31705cc4bae7a1f9e3071": " ~\\sigma_{t}^2= ~\\omega + ~\\alpha (~\\epsilon_{t-1} - ~\\theta ~\\sigma_{t-1})^2 + ~\\beta ~\\sigma_{t-1}^2", "a486115e2ad65bbcb3346bbee43b471a": "D\\{\\mathcal J\\}\\subseteq\\tilde D", "a48612abef644eb451a7e9f4145ff8a1": "(x-10)(x-2)^{35}(x+4)^{20}. \\, ", "a486884bd633a2612f7009dcdd840a2c": "X_{\\{F,G\\}}=[X_F,X_G]\\,", "a48697972d577ff48af8feab204fc51b": "\\langle Y_x, x\\rangle = 0, \\quad \\forall x\\in \\mathbf{S}^2.", "a486b898eae61c4d2ec0bc0c38f2ffa1": "N=5", "a486d9b322ad5fecc80258998e581160": "N = \\sum_{r=1}^\\infty r N_r.", "a486dcf004e79127d338751879da5864": "\\operatorname{Tr}(A \\cdot P \\otimes Q) \\geq 0", "a4875fde7831c625743d3e3ae2a2faf4": "\\rho(\\lambda X) = \\lambda \\rho(X)", "a487675590789d17738009ac3b078f83": "g_j \\colon A_j \\to B", "a487dc0eb3b907bce6ff5b599ca9fdfb": "C_{i} \\subseteq C_{i-1}", "a487fab76141cff18734bec44e9fa80d": "\ny^2 = x^3 + Ax + B \\, \n", "a488525fa08817df179e74fe4f202269": "\\{1,2,\\dots,n\\}", "a4885eb66d94f2db83302fb93422f4b0": "\n\\sum_{i=1}^n U_i^2=Q_1+\\cdots + Q_k\n", "a48875cbaec140f2ba063a3b98299568": "\\frac{V}{(I)}", "a488948a44b799db22cf0a9e11e8911a": "q=p^k.", "a488cf6039aa979e1f882fbb3d6929b2": "A + BC \\longrightarrow AC + B", "a4890f7bf875608eadf40fc08e3e33b9": "H_{rr}(0) \\neq 0", "a4894a79e702fe548d3083548ea9e8bf": "g(x,p)=\\int_{-\\infty}^\\infty ds ~e^{ips/\\hbar} \\langle x-\\frac s2|\\ \\hat G\\ |x+\\frac s2 \\rangle.", "a4895ac3d5f096638099a8ba757b35f4": " o_j ", "a4897a4824fa7708baef0ee528cb00aa": "0 \\le \\theta < 360^{\\circ}", "a4898afcf1041aa169f29d56261070b0": "\\{\\psi^*(x),\\psi(y)\\}=i\\delta(x-y). \\,", "a489c1baebf35883646f005370a07f29": " Z_\\mathrm{Eq} ", "a489c3ccb27b0dfb68c48227bb253355": "g_{ij} = X_i \\cdot X_j", "a48a3ae9fdc08f412aacf2c784a6794f": "[\\mathcal{H},\\mathcal{Q}]\\subseteq\\mathcal{Q}", "a48a6669e0da7fa0f25c46e0a0a20610": "\nj_\\kappa=\\prod_{(i,j)\\in \\kappa}\n(\\kappa_j'-i+\\alpha(\\kappa_i-j+1))(\\kappa_j'-i+1+\\alpha(\\kappa_i-j)). \n", "a48a77132ee368894658a9851b47ac7c": " T_m ", "a48aa2b9d69c31d5d1456f7065494fbf": " \\mathbf{\\bar x} \\sim \\mathbf{T} \\, \\mathbf{x} ", "a48ae756888364807b144614088a3dc4": " \\nu_0 ", "a48b3bf13cbfb98614f569f003ce24a2": " 2\\omega ", "a48b521c545ff480b14962a9b7bfe38a": "\\forall T_A^1,T_A^2,\\dots,T_A^n, [op_A(T_A^1,T_A^2,\\cdots,T_A^n)] = op_B([T_A^1],[T_A^2],\\cdots,[T_A^n])", "a48bdcd8b6b4a078b6f07902629d863a": "\\xi(2) = {\\pi \\over 6} ", "a48c848a78288ff4bcfe803b6db6ac7e": "\\mathbf{x}^\\mathrm{T}A\\mathbf{x}+\\mathbf{b}^\\mathrm{T}\\mathbf{x}=1", "a48cb12710293dbc503a1e086f6244e9": "F_j=f(\\alpha^j)=\\sum_{i=0}^lL_i(\\alpha^{jk_i})", "a48d97b4979a24ac3c2deae236db0a12": "z\\rightarrow -z^5 + z_0", "a48de4381030a67c51eb7b13edb5e980": "E(C)= \\sum_i \\sum_{j0 ", "a4a11f7005ea1ed79633bae02e342da4": "\\langle 0,1,+,\\times, <, =\\rangle", "a4a13b9becdffeb5cbd99a64962fa3ba": " \\bigoplus_{\\ell \\in I} (V \\otimes \\mathbb{C} f_\\ell) \\rightarrow V \\otimes W", "a4a18c6136e98e78f0f5c676caa56599": " \\frac{D_g T}{Dt} - \\frac{\\sigma p}{R} \\omega = \\frac{J}{c_p} ", "a4a196e9f6f22152c3be013629349f7d": "\\Longleftrightarrow B > R \\;", "a4a2c71716e12fb94998c684f60f406c": "\\beta =\\mathrm{\\frac{[ML] } {[M] [L] }}.", "a4a2fac764bc21cb0dcc43cba04a6557": "(c_1, c_2)", "a4a31ee9ab9bf9d7a5453af57dc2a033": "\\forall X \\subseteq U_p [CUM_p (X) \\Leftrightarrow \\exists x,y [ X(x) \\,\\wedge\\, X(y) \\,\\wedge\\, \\neg (x=y)] \\;\\wedge\\; \\forall x,y [X(x) \\,\\wedge\\, X(y) \\Rightarrow X(x \\,\\oplus\\, y)]]", "a4a349849bf639be98407a548ee50b55": "nT", "a4a351a6e799acf2713be8ca57ae53bf": "\n\\oint\\limits_{H(p,q)=E} p_i \\, dq_i = n_i h\n", "a4a3b49f6e5d4ec060860b74735a1566": "\\langle 0 | \\hat{a}^\\dagger = 0 \\qquad \\textrm{and} \\qquad \\hat{a} |0\\rangle = 0", "a4a3cdc76c011776daa51b12331d3b86": "\\scriptstyle \\lim_{k\\to\\infty}\\mathbf{P}^k", "a4a409fcc23ae715bfd2ad00fe68feea": "\\sum{{{x}_{n}}}", "a4a46d8c6c35bcf2023ea5c822278375": "a>b>c", "a4a4e3ab2c78d57289e6c31146cdd38e": "\\mathcal{H}=\\frac{1}{2}\\gamma^{-1/2}(\\gamma_{ik}\\gamma_{jl}+\\gamma_{il}\\gamma_{jk}-\\gamma_{ij}\\gamma_{kl})\\pi^{ij}\\pi^{kl}-\\gamma^{1/2}{}^{(3)}R", "a4a513446b0288b18919e385a890741d": "D = \\{ (x,y,z,w) | x^2+y^2+z^2\\leq r_1^2,\\ w^2\\leq r_2^2 \\}", "a4a528d0168d62a87f178d641a05d56e": "\\eta_H = n_H/(n_H+n_D)", "a4a54226ffb44caee33b8571d54a46cb": "17\\cdot18\\cdot19\\cdot20\\cdot21=(21)_{5}=21^{\\underline 5}=17^{\\overline 5}=17^{(5)}", "a4a581c345d2fb94d871b4ded425adbd": "\\|A\\|_2\\le\\|A\\|_F\\le\\sqrt{r}\\|A\\|_2", "a4a584f15042ca79ab37f1b13a8a5a28": "R_{2k}(V) = \\frac{(k!V)^{1/2k}}{\\sqrt{\\pi}},", "a4a587fd2a5e1a3ad8e61f7fb5d32758": "v_1, v_2, \\cdots, v_{m+1}", "a4a592098a0837428058aa90f8f38222": "\\textbf{Q}_k \\equiv \\textbf{Q}^{a}_k", "a4a5a21fca069671a087c11e40340aa5": "\\zeta_i", "a4a5b01bcc34667685b3e63ef09acff1": "\\nabla_XY", "a4a5f5f0f7e925d8a846a370a0befe80": "x_K[n] = \\begin{cases} x[r], & n = rK \\\\ 0, & n \\not= rK \\end{cases}", "a4a60ee7917d7efd96302d2c4807660c": "G_C = G_R = \\frac{1}{\\sqrt{2}}", "a4a648c0aab8e11f80d76807b399ec93": "Y | N=\\sum_{n=1}^N X_n", "a4a6c195063b763259dd5192c2e24659": "\\bar {10}_{10}", "a4a74464d4cabc217f416d59610d6dc9": "\\frac{\\partial^2 \\epsilon_z}{\\partial x \\partial y} = \\frac{\\partial}{\\partial z} \\left ( \\frac{\\partial \\epsilon_{yz}}{\\partial x} + \\frac{\\partial \\epsilon_{zx}}{\\partial y} - \\frac{\\partial \\epsilon_{xy}}{\\partial z}\\right)\\,\\!", "a4a75bc5034ee26843b6732aa24e85b2": "\n\\begin{align}\ns_P(nT) &= \\overbrace{\\tfrac{1}{N} \\sum_{N} S_k\\cdot e^{i 2\\pi \\frac{kn}{N}}}^{\\text{inverse DFT}}\\\\\n&= \\tfrac{1}{P} \\sum_{N} S_{1/T}\\left(\\frac{k}{P}\\right)\\cdot e^{i 2\\pi \\frac{kn}{N}}\\,\n\\end{align}\n", "a4a79772d976ea6ea8c92db22af18d2b": "\n \\ \\ a_1\n", "a4a7b5d4f77725ba1530f1347e6f757f": "\\sum_{i=0}^n \\binom{n}{\\lfloor n/2\\rfloor}", "a4a810bdde3df8bd7db278bb1a619c6d": "M=\\frac{PV}{RT}", "a4a8246dd3786af27de4822bf146722d": "L_\\mathrm{B}", "a4a82cc9f925d96fd93e628ad9eaeef8": "\n \\delta K = \n \\int_0^T \\int_{\\Omega^0} \\left[\n J_1\\left(\\dot{u}^0_\\alpha~\\delta\\dot{u}^0_\\alpha \n + \\dot{w}^0~\\delta\\dot{w}^0\\right) \n + J_3~\\dot{w}^0_{,\\alpha}~\\delta\\dot{w}^0_{,\\alpha}\\right]\n ~\\mathrm{d}A~\\mathrm{d}t \n", "a4a83d6545c66a51ab6c23df9bafd718": "y = C x \\, ", "a4a85f4885a5b963658100c3631c03b7": "\\infty\\,", "a4a89e74fb7a4c1356c43c113a664931": "(\\mu_{nb}^{(c)}(t))", "a4a8b2f2f2331dd416e2b02b4815e747": " \\delta(x) = \\lim_{\\varepsilon \\rightarrow 0^+}\\frac{1}{(2 \\pi)^n} \\int_{\\mathbb{R}^n} e^{i x\\cdot \\xi} e^{-\\varepsilon |\\xi|^2/2} \\mathrm{d} \\xi = \\lim_{\\varepsilon \\rightarrow 0^+} \\frac{1}{(\\sqrt{2 \\pi \\varepsilon})^n} e^{-|\\xi|^2/(2 \\varepsilon)}. ", "a4a8c11ae4aa433b232b874349a59702": "\\omega = e^{2\\pi i/3}", "a4a8c73d613d96e2f25f27c162d755bd": "\\mathbf{\\mu_s} = -\\frac{e}{2m}gS", "a4a8c98465e37d115cc54c9d451b09f1": "\\scriptstyle(0.20(0.21))\\times10^{-11}", "a4a9f57aef19fd352f6ebd105cbbc5d4": "\\mu_P = \\frac{1} {4 \\pi} \\iint\\limits_S\\left(\\frac{V_\\infty \\cdot \\mathbf{n}}{R} \\right) dS_U + \\frac{1} {4 \\pi} \\iint\\limits_S\\left(\\mu \\cdot \\mathbf{n} \\cdot \\nabla \\frac{1}{R} \\right) dS", "a4a9f6143b0c5b212552836dd61aeeb8": " \\sum_{n=1}^\\infty {4! \\over {n(n+1)(n+2)(n+3)}} = {4 \\over 3} ", "a4aabcd2eb0e677ab7c75741733db929": "\n \\mathbf{u}\\times\\mathbf{v} = [(\\mathbf{b}_m\\times\\mathbf{b}_n)\\cdot\\mathbf{b}_s]u^mv^n\\mathbf{b}^s\n = \\mathcal{E}_{smn}u^mv^n\\mathbf{b}^s\n ", "a4aabef93f14613b69fa61ae400adaac": "f(x_{p})", "a4ab4f62dec44b83e1bb5f21ff3b9a90": "G_r(\\theta_r,\\phi_r)", "a4abcca9cf92b6df4ee2f7b1d9f4f3ed": "I(X_1;\\cdots;X_n|X_{n+1}) = \\mathbb E_{X_{n+1}}\\big(I(X_1;\\cdots;X_n)|X_{n+1}\\big).", "a4ac1a204d888c7cf8ba62dd674db83d": "\n \\Delta H^\\prime = 2 \\sqrt{C_1^\\prime C_2^\\prime} \\sin (\\Delta h^\\prime/2), \\quad \\bar{H}^\\prime=\\begin{cases}\n (h_1^\\prime + h_2^\\prime + 360^\\circ)/2 & \\left| h_1^\\prime - h_2^\\prime \\right| > 180^\\circ \\\\\n (h_1^\\prime + h_2^\\prime)/2 & \\left| h_1^\\prime - h_2^\\prime \\right| \\leq 180^\\circ\n \\end{cases}\n", "a4ac7d2bf57bbe745b7c02349f4ccefb": "PV=nRT,", "a4ac8418aea115019b12346bffe6b1f2": " \\mathbb{T}^\\infty=[0,\\infty]", "a4aceb37c0dc6f03cb5f06d81f4f2e70": "f_\\varepsilon: X \\rightarrow Y", "a4ad04724c6ae472394f83d759c47f7a": "\\exists x \\, P(x) \\lor \\exists x \\, Q(x) \\Leftrightarrow \\exists x \\, (P(x) \\lor Q(x)) ", "a4adcaa6eefc557d772b2608a223369b": "\\mathbf{S} = \\frac{1}{\\mu} \\mathbf{E} \\times \\mathbf{B},", "a4add55da860b828a906f67948bfa70d": "\n\\frac{20}{20 + 30} = 0.4\\,\n", "a4ae0c93e8b73c57735ab44517aa36dc": "\\operatorname{div} (V) (\\sigma) := - \\int_{0}^{T} \\dot{V}_{t} (\\sigma) \\, \\mathrm{d} \\sigma_{t}.", "a4ae261b04ff790bb8bc5d589bb77a00": " \\int_{\\textbf{N}} f(x) {N}(dx) ", "a4ae7351a598535a23d3f2347c3b9de7": "E(r_A)={i\\over L^3}\\sum\\limits_{\\lambda=1}^2\\int d^3k [{{c k}\\over {2\\epsilon_0}}]^{1\\over 2}\n[e_\\lambda(k)a_\\lambda(k)\\exp(ikr_A)-H.C.]", "a4ae78b560deffc49107c5e7841da88a": "\\epsilon(v) = d", "a4aece1645f8649e169509e5fddec9a2": "x_n = z_n \\times z_{n-1}", "a4aed99ab32cf149ebfcdb18bd0b423a": "+22639''\\sin(l) +769''\\sin(2l) +36''\\sin(3l)", "a4af46dd06288a77d882555c2eb8cfa2": "\\frac{\\partial f}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = Df(\\mathbf{v})[\\mathbf{u}] = \\left[\\frac{d }{d \\alpha}~f(\\mathbf{v} + \\alpha~\\mathbf{u})\\right]_{\\alpha = 0}", "a4af779489f502f8df617c9321083107": "u_{\\bold{k}}(\\bold{r}) = u_{\\bold{k}}(\\bold{r}+\\bold{T})", "a4afb5316e53bb66421cbc9c5de27b91": "R_1 = -\\frac{2DF}{F - B}", "a4afe0405138f9533f209fe785bd24c8": "|\\tilde{f}(\\lambda)|\\le C_N (1+|\\lambda|)^{-N} e^{R|{\\rm Im}\\, \\lambda|}.", "a4aff446f2c107719231cb13f6808501": "\\begin{align}\nk_\\text{e} &= \\frac{1}{4\\pi\\varepsilon_0}\n\\end{align}", "a4b045c0bd3061abc451f99f4aa5e326": "T_0\\boldsymbol{x} = \\boldsymbol{x}", "a4b0a59c0406b92d6550ee987b7344f7": " V_r = -Z_c I_r \\, ", "a4b0d94119ccdf590c1ba8d8ec657bd5": "\\mu\\left([x_0, x_1,\\ldots,x_n]\\right)=\n\\mathrm{Pr}(X_0=x_0, X_1=x_1, \\ldots, X_n=x_n)", "a4b0f057136c40bbaa524f1ee267f816": "C_n(X)", "a4b1432432038fbb6d340407982580cb": " N ", "a4b1724a5293e20eb7179747bdd8987c": "\\sigma_2^{ }", "a4b1daf0b1933d6b40fcc6482c8c5255": "d \\Phi = d S - \\frac {1} {T} dU - U d \\frac {1} {T}", "a4b2190faa0625382f377bdfe4b7e50c": "P_{TNL} \\approx \\frac{1}{1+{L_{th}}^2 {B_g}^2}", "a4b2295e87af2774b2e5404e7a2d6bcd": "\\hbox{MaxScope}=\\left(\\tfrac{0.7*\\hbox{Population}}{\\hbox{Population}} + \\tfrac{0.5*\\hbox{GNP}}{\\hbox{GNP}}\\right)^V", "a4b2c70ad8d1f760ef963aa2472da217": "\\langle\\sigma(\\langle\\rangle),a_1,\\sigma(\\langle\\sigma(\\langle\\rangle),a_1\\rangle),a_3,\\ldots\\rangle", "a4b2c7ee02a961b2b392e790e5b0ca39": "R_H = \\frac{h}{e^2} = 25.812813 k\\Omega \\ ", "a4b3009d32447352852aa816b0535b3e": "p^e \\vert (v - 1) ", "a4b31bfd24899edd7c752f93bb6f5e92": "ab+ac", "a4b333aaf2af6276be752c5fd06e43a7": "\\alpha^p", "a4b48d7751e60e79f7e56c63446c3ecb": "f\\quad", "a4b4b6bc4a046a4e09afcd5109000b40": "\\epsilon_{\\mathrm tB}", "a4b4cb2d20c12dbb0cf581db05975994": "{ds}^2 = g_{\\mu \\nu} dx^\\mu dx^\\nu \\,", "a4b4e1fc8ecc92d05febfab2e7d0686c": " \\left|-\\frac{(b-a)^5}{2880} f^{(4)}(\\xi)\\right|.", "a4b4f005a1ebe1c645d8afec65d1a8de": "\\left(\\frac a0\\right)=\\begin{cases}1&\\text{if }a=\\pm1,\\\\0&\\text{otherwise.}\\end{cases}", "a4b4fbdf477fb2f21aa6018bb4c3687f": "V_0 = \\int E(s)\\,\\mathrm d s", "a4b5019814fa2aa257fe08dc37328f66": "G_f = G \\otimes_A A_f \\xrightarrow{\\beta^{-1} \\otimes 1} M \\otimes_A \\hat{A} \\otimes_A A_f = M_f \\otimes_A \\hat{A} \\xrightarrow{\\alpha \\otimes 1} F \\otimes_A \\hat{A}.", "a4b54c0aa6d0e5b9f292e564389f2dd7": "\\mbox{f(t)} =sin(\\omega t+\\theta+\\beta) ", "a4b55c472143a96ee7fe89e3b984df35": "\\,\\!\\ ^{4}2 = 2^{2^{2^2}} = 2^{\\left[2^{\\left(2^2\\right)}\\right]} = 2^{\\left(2^4\\right)} = 2^{16} = 65,\\!536", "a4b56a0f1804ad7e9726dd5810862456": "\\mathfrak {sp}_{2n}", "a4b5ca955e59f49f06d282fda4768308": " curry(T)(s*t) = curry(T)(s)\\circ curry(T)(t).", "a4b6069c7398807fe33ce4448045071a": "1/r_s", "a4b63df96906b957283a106855a05c6b": "\\ B=\\tfrac{3}{2}A", "a4b683c8d3c5d5702975979dde0367ab": "-\\Delta", "a4b6aec8257328cb51bdb64123c2d5f4": " D=|P_A-P_B|, \\, ", "a4b733a66c0167becfb3cc8832972db5": "g_{ij} = \\tau \\text{im} \\int \\tau i^*(\\nu \\cdot \\kappa \\tau).", "a4b75e4ff7ef3e12f6e81b53827ce6a7": "\\varkappa_{\\alpha}^{\\beta}=\\left ( \\frac{2 \\dot a}{a} \\right )l_{\\alpha}l^{\\beta}+\\left ( \\frac{2 \\dot b}{b} \\right )m_{\\alpha}m^{\\beta}+\\left ( \\frac{2 \\dot c}{c} \\right )n_{\\alpha}n^{\\beta}", "a4b7b599596971c3c3dee455c4a97c85": "\n1-\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\hat{\\Pi}_{\\rho_{X^{n}\\left(\nm-1\\right) },\\delta}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta\n}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho_{X^{n}\\left( m\\right) }\\ \\Pi_{\\rho,\\delta\n}^{n}\\ \\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta}\\cdots\\hat{\\Pi}\n_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\Pi_{\\rho_{X^{n}\\left( m\\right)\n},\\delta}\\right\\} \\right\\} ", "a4b7db2890b44cabfc356772444829ef": "\\,S_b |b+\\rangle = \\frac{\\hbar}{2} |b+\\rangle", "a4b82ee0457e48196472101c4c5665c4": "k=0, 1, \\ldots", "a4b836cd362ae4a363588bd9ceec17cc": " |{\\rm det}\\, (I+ zA)| \\le \\exp (|z|\\cdot \\|A\\|_1). ", "a4b883eba79b39dcf19b55dc96bd583a": "(15)\\quad T_{ab} =\\frac{M(v)_{,\\,v}}{4\\pi r^2}\\,n_a n_b \\;,\\qquad n_a dx^a=-dv\\;.", "a4b92f73d0637152f41614b2a9e956c1": "\\scriptstyle F_\\tau", "a4b96d628fc5fa0976042b63a28a370b": "N_1 +N_2t^1 + N_3t^2 +\\cdots \\,", "a4b9f423fa4ee9b735eb4d58509cc808": "\\widehat{C}=\\mathbf{Set}^{C^\\mathrm{op}}", "a4b9fb9dffe3bb8080434439f45e934a": "\\langle S * T,\\varphi\\rangle = \\langle S, \\psi\\rangle.", "a4ba02ad1540c694ab32113287279b78": "\\text{bind} \\colon (S \\rarr T \\times S) \\rarr (T \\rarr S \\rarr T' \\times S) \\rarr S \\rarr T' \\times S", "a4ba547763a41d6df4aa67a2e8188d03": "\nDv=1 \\,,\\quad \\Delta v=\\delta v=\\bar\\delta v=0\\,.\n", "a4ba96e43024949fa0df9c2e469f43ad": " \\Phi = \\int E \\left ( \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} \\right ) ", "a4bab34477226ba25674433fa61eee6f": "\n\\lambda < B < \\mu < A < \\nu\n", "a4bae17171f5301fccfa676afc944114": "\\eta=\\Re\\left\\{B\\,\\exp\\left(\\sqrt{\\mathcal{A}g\\alpha}\\,t\\right)\\exp\\left(i\\alpha x\\right)\\right\\}\\,", "a4bb0353e35ebeffc20f7541442b657a": "w^{T}q + w^{T}(D - \\lambda I)^{-1} w(w^{T}q) = 0", "a4bb0f63a117d006eb9db322818372c4": " \\ v_{ \\bar{x} } = \\frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}", "a4bb27f57c33c19edbbba473940dd0da": "\\mathfrak{d}_{L/K} = \\prod_{\\chi \\in \\mathrm{Irr}(G)}\\mathfrak{f}(\\chi)^{\\chi(1)},", "a4bb35305c5503e8c218ab69e7628c59": "\\Re(x) > 0", "a4bb7a06f73eb72654309ebf078856b3": "\\beta^*", "a4bb957a1c50fdae33c72611ad0e44f2": "\\frac{\\boldsymbol\\Sigma \\otimes \\boldsymbol\\Omega}{\\nu+p-n-2}", "a4bbefc03ef9074cb49840eb791fa639": "c = x_0 + i y_0\\ ", "a4bc0985744caf49a4893e42cc5c24b2": "\\Delta F", "a4bc14a65b0315f55327b36b91228385": "\\operatorname{det}(\\mathbf{A}+\\mathbf{UWV}^\\mathrm{T}) = \\det(\\mathbf{W}^{-1} + \\mathbf{V}^\\mathrm{T}\\mathbf{A}^{-1}\\mathbf{U})\\det(\\mathbf{W})\\det(\\mathbf{A}).", "a4bc8d957c23703a56478afc4dfcdd26": "\n \\text{RADF} = \\frac{I}{F} = \\pi~A_M~\\mu_0^k~\\mu^{k-1}\n ", "a4bc962b6472bd3fac0f915a0ff0c336": "(l)", "a4bcf73c4c5f0b6301c0a2967d57ac2a": " (S UDU^{-1} S^{-1}) h = -h^{\\prime \\prime} + V h,", "a4bd2cf6873622fceb38f2ef3a9be329": " \\theta_2 ", "a4bd6cf771fb4606567a0561aa528182": " \\mathbf{U}^*\\mathbf{U} = \\mathbf{U} \\mathbf{U}^* = \\mathbf{I}.", "a4be5f534113adaea92605cd64f1ffab": "\n (7) \\qquad \n \\cfrac{1}{\\kappa AG}~\\frac{\\partial^2 q}{\\partial t^2} -\\cfrac{m}{\\kappa AG}~\\cfrac{\\partial^4 w}{\\partial t^4} + \\cfrac{\\partial^4 w}{\\partial x^2\\partial t^2} \n= \\cfrac{EI}{J}~\\cfrac{\\partial^3 \\varphi}{\\partial x^3} + \\cfrac{\\kappa AG}{J}~\\left(\\frac{\\partial^2 w}{\\partial x^2} - \\frac{\\partial \\varphi}{\\partial x}\\right)\n", "a4befd172f53513674ba74f17e060dd1": " \\sqrt {-g}_{, \\rho} = \\sqrt {-g} \\Gamma^{\\sigma}_{\\sigma \\rho} \\,.", "a4bf27c690044aa69b358dadb8c12f6f": "Q^\\mathrm{T} Q = Q Q^\\mathrm{T} = I, \\,", "a4bf5384d3b8fa1812c84853e718d701": "F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)", "a4bfa62cc115e7eafd6740ed471acee8": "R |\\varphi \\rangle = (\\lambda I - L)^{-1} |\\varphi \\rangle = \\sum_{i=1}^n \\frac{1}{\\lambda- \\lambda_i} |e_i \\rangle \\langle f_i | \\varphi \\rangle. ", "a4c02a45a26f4866f9689cc972985135": " a_1 ", "a4c05fe67cb6feb3d8431cc4b2f0457d": " \\epsilon_{i}^{mix} = \\pm sgn(\\pm \\epsilon_{i}^{\\mu_{1}} \n\t\t\t \\pm \\epsilon_{i}^{\\mu_{2}}\n\t\t\t \\pm \\epsilon_{i}^{\\mu_{3}})\n", "a4c066459b52c48dae726353a85435c2": "\\dot{V}(x) < 0 \\quad \\forall x \\in \\mathcal{B}\\setminus\\{0\\}", "a4c0900210b5a0edbb2be1c49bc04d8c": "G=KAN", "a4c15f10e6e6a4a0ad0ffd4221856907": "\\displaystyle\\Delta_{\\alpha<\\delta} X_\\alpha,", "a4c168310e44e433a6665d85cabe7496": "f_{yy}(a,b)=-\\frac{e^x}{(1+y)^2}\\bigg|_{(x,y)=(0,0)}=-1\\,,", "a4c17358d777da081968637bf252ae5f": "\n\\begin{align}|0\\rangle_{1}\\otimes|0\\rangle_{2} & \\longrightarrow |0\\rangle_{1}\\otimes|0\\rangle_{2} \\\\\n |0\\rangle_{1}\\otimes|1\\rangle_{2} & \\longrightarrow e^{i\\phi}|0\\rangle_{1}\\otimes|1\\rangle_{2} \\\\\n |1\\rangle_{1}\\otimes|0\\rangle_{2} & \\longrightarrow e^{i\\phi}|1\\rangle_{1}\\otimes|0\\rangle_{2} \\\\\n |1\\rangle_{1}\\otimes|1\\rangle_{2} & \\longrightarrow e^{2i\\phi}|1\\rangle_{1}\\otimes|1\\rangle_{2}\n\\end{align}", "a4c1923892aab224e63c90ca544674d6": "\\partial_\\mu \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_\\mu \\psi )} \\right) = \\partial_\\mu \\left( i \\bar{\\psi} \\gamma^\\mu \\right), \\,", "a4c1990b05054d577473ea4f74ff91e8": "|\\mathbf{a}_\\text{c}| = \\omega^2 r", "a4c1a04b6dc6f23202e108fe4921aef8": " - \\Gamma_{ST} = \\Gamma_{TS} = {Z_s - Z_c \\over Z_s + Z_c} = \\Gamma_S \\, ", "a4c1dfb3b037453132bb93c91146acdf": "SU(4)\\times SU(2)_L \\times SU(2)_R", "a4c1e9aad704e68a0cd6a7d84809e3b5": "(x,y)\\in \\R^2", "a4c1ed1ffa5d1c4be59e7b9527c75a6f": "2^{31} -1", "a4c1fcd5e9a541cb152466e7ddd7b049": "B = a_2ZZ_1(X_1+ZZ_1)", "a4c226844d3519ce214c9630323931d1": "\\mathbf{C} \\to \\mathrm{End}(V)", "a4c25f3e2ce2443228066e2c13823619": "w_{n+1} = (2a-1)w_n + u_{n+1}", "a4c2915abae78cebc43a7fb5ed875ef1": "\\sqrt{I}, \\sqrt{J}", "a4c2d7a7b94ac11b25e1a3729b89979e": "L_2=1/2", "a4c2f80b29b0f0361c8ecf82176422ec": " { \\partial V \\over V_m } = { -1 \\over \\ \\gamma } {\\partial p \\over p_m } ", "a4c358538453e21d1f25905a22e4c8e8": "0111 = \\alpha^{-5}.", "a4c3a6d7f6e39e56e8d21ff73882b324": "Q(x)\\approx \\frac{1}{12}e^{-\\frac{x^2}{2}}+\\frac{1}{4}e^{-\\frac{2}{3} x^2} \\qquad x>0 ", "a4c3d5d95e38f22c1464cf2f170c53f1": " \n\\Delta E = E_{-} - E_{+} = \\frac{4}{e} \\, R \\, e^{-R} \\left[ \\, 1 + \\frac{1}{2R} + O(R^{-2}) \\, \\right] \n", "a4c43e4c4ceb70fc0f2fd5d05df1a5bb": "A_{Floor}\\,\\!", "a4c48e9563495bc4a9f64b3ad5c76c6f": "\\delta(x-y)=\\sum_n \\psi_n^*(x) \\psi_n(y).", "a4c514bceda5235533ed4ad4ef15367e": " d_\\lambda = \\frac{n!}{\\prod h_\\lambda (i,j)} , ", "a4c514d81e76d14f15b9f95d87321276": "P \\equiv_{b} Q", "a4c596b343b081c8798cd4d8a02c970e": "\\wp\\left(z\\right)=\\frac{1}{z^{2}}+\\sum_{\\omega\\in\\Lambda\\smallsetminus\\left\\{ 0\\right\\} }\\left(\\frac{1}{\\left(z-\\omega\\right)^{2}}-\\frac{1}{\\omega^{2}}\\right)", "a4c59db5f2a8be5c5b5853ac6bf1247a": "\\sigma=\\sigma_c", "a4c59dea4e1afcd6bf0aaa89509e78ce": "\\sum_{n,k} {n+k\\choose k} x^k y^n = \\frac{1}{1-x-y}.", "a4c5a90958203e8fc4949d27556379f8": "\\begin{pmatrix}hv & w+hx\\\\-w+hx & -hv\\end{pmatrix}", "a4c5b75c0c19ab554d0bd81eef4ab2d0": "\\omega_0 \\rightarrow 0", "a4c5c5af92bdd880227bff1a95bf27b3": "y_k = k \\cdot \\Delta", "a4c678140ee1106278da46e7609cef28": "\\gamma_*: D_* \\to D_{*+1}", "a4c6b9a11b7626db12250831dcda06a2": "\\mathbf{y}(t) = \\mathbf{x}(t)", "a4c7805d77bf267c7db371d7a436cf20": "T \\left|j,m\\right\\rangle \\equiv \\left|T (j,m)\\right\\rangle = {(-1)}^{j-m} \\left|j,-m\\right\\rangle ", "a4c7805e7092462b2648042f010747fc": "Place interwiki and category links on the documentation subpage, please.", "a4c7ae1c9acadaa5d6f301269032272f": "\\scriptstyle \\mu S(t)\\Rightarrow \\rho\\delta_1+(1-\\rho)\\delta_0\\quad\\text{as}\\quad t\\to\\infty ", "a4c7c5610ba7de467fc3c338aef6ffb1": "\\begin{matrix}{13 \\choose 4} - 432 = 283\\end{matrix}", "a4c7d53eee29f7843b468cf955ac79c6": "\n\\frac{d}{dq} \\left[ c \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\right] = 0\n\\,.", "a4c7dd7fc0981d267836aedac6711071": "{TL}_y", "a4c844fc394e757eb0139de8d13ad12b": " \\aleph_0 < |S| < 2^{\\aleph_0}. \\,", "a4c884a246218e6e6b33393ddc5a13bb": "\\mathrm{Hom}(\\mathrm{colim}F,N) \\cong \\mathrm{Cocone}(F,N).", "a4c8d64398f3859e7b6ed28eb1429475": "\n\\int_0^{2\\pi} L d\\theta = 2\\pi L = n h.\n", "a4c940d54bb3a0ee75ad92bd2a82744e": "f(n) = g(n)h(n)", "a4c994c2b271c9398798515606594cc1": "\\overline{\\overline v} \\mapsto v.", "a4c9f54a51e407edae616b87b2b92914": "(y_n/z_n)Z", "a4c9f7b1534b44d23b83e0951a596893": "x \\not\\in C^*(\\theta)", "a4ca0387f5f8dd8f1ee04a52bd692cf1": "\n\\left( i\\Delta + \\frac{\\partial}{\\partial t} \\right) \\psi(x,t) = 0\n", "a4ca29e9414df4e5fd3635c5acf93a2e": "\\Omega(\\sqrt n)", "a4ca8807781e7402b4350bc754b080cf": "\n \\varphi(\\theta_r - \\theta_c) = \\frac{A_c A_r}{2}\\sin(\\theta_r - \\theta_c)\n", "a4ca8f968066dd35856e2c3baa749aaa": "\\mathbb{P^{*}}", "a4cacccb6ebc07bfb83736b95479bb6c": "Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{t} - C_{t-1})", "a4cad38c243819959828c17f2ed97cb9": "A=\\frac{m}{P_y}", "a4cad509a1c8a66ad91d608f15e985e0": "H = (S, E)", "a4cae863b228b4cb99a1a895f9021c15": "\\bar x_{t+1}=\\bar x_t+\\gamma \\nabla f(\\bar x_t)", "a4caf509a17abb0d80f4de21010b9bba": "|x(t+1) - x(t)| < \\epsilon", "a4cafdcb2246f376014cf6bb72a3e2de": "J[\\rho] = \\frac{1}{2}\\iint \\frac{\\rho(\\mathbf{r}) \\rho(\\mathbf{r}')}{\\vert \\mathbf{r}-\\mathbf{r}' \\vert}\\, d\\mathbf{r} d\\mathbf{r}' \\, .", "a4cb1a69ef8cca385be8a1c352b80326": "V(\\bold{r})", "a4cb209edd965a8c783bd989fafdce6a": "x \\in Cl_t^{\\leq}", "a4cb34d6e13c55d6a1cbe33a78ac0a4b": "= \\frac{24!\\cdot 32!}{2}\\cdot \\frac{16!}{2}\\cdot 2^{23}\\cdot (3!)^{31} \\cdot 3 \\cdot {\\left( \\frac{4!}{2} \\right)}^{15} \\cdot 4", "a4cb45def8814b2c00ce77b8b70246da": " a_P = \\frac {a_W + a_E} {2} + {a_P}^0 - \\frac {S_P} {2} ", "a4cb4aa7fc401cbe8f07945212a07e54": "\\mathrm{Bi}_m=\\frac{h_m L_{C}}{D_{AB}}", "a4cb50a054aa2cda78fe09e653523c0d": "H_\\infty", "a4cb9a9a8500ac44ca1e7de283204fbb": "C_{4,4} = 25", "a4cba4b9b1f74a79e78c24febcf65a59": "c=\\tfrac{q^{2}(14u^{2}v^{2}-u^{4}-v^{4})}{4} \\, ", "a4cba9495a52048ab9ded26c8b311443": "\n\\mathbf{R}\n", "a4cbd1f5de48545de646c3cc494e16fe": "{\\boldsymbol\\Sigma},{\\boldsymbol\\mu},\\nu", "a4cc424ed9a471e38b18c11d0342258d": "Uxp_a(x)", "a4ccb022e0d649721a262219ce25625f": "\n\\mathfrak C = \\begin{pmatrix}A & B \\\\ C & D \\end{pmatrix} = \\mathfrak C ^\\dagger.\n", "a4ccf7cb830e026efcb7293566b021bb": "\\mathrm{d}S_\\varphi= \\mathrm{d}\\rho\\,\\mathrm{d}z.", "a4cd54dd5686c34ac02dc99742a2b53d": "r = (r_i(\\sigma_{-i}),r_{-i}(\\sigma_{i})) ", "a4cd9f83306cfd4bdc2e0d3d27b1fe19": "\\frac{(x+1)^p - 1}{x} = x^{p - 1} + \\binom{p}{p-1}x^{p - 2} + \\cdots + \\binom{p}2 x + \\binom{p}1,", "a4ce58db8db761b9e21e3a64690bcf57": "\\,E[X|Y=y]", "a4ce915885e6e00df32223d2f2aab187": "\\left|z\\right|\\leq R", "a4ceed32027821cdc471eb73dff05f8a": " f_1 =\nf_1(z) = \\sum_{j=0}^{\\infty}\\frac{a(j)}{b(j)}z^j , ", "a4cefdcc20eff7ebb84f3dad672d46b5": "V_n(R) = \\frac{2\\pi^{n/2}R^n}{n\\Gamma(\\frac{n}{2})} = \\frac{\\pi^{n/2}R^n}{\\Gamma(\\frac{n}{2} + 1)}.", "a4cf4993cc9bfee4902da3646877a279": "e_\\mu", "a4cf6e2dc9e9e03521dc5eb9bc36cd5a": " n_+(V) = \\operatorname{dim}\\ \\operatorname{dom}(V)^{\\perp}", "a4cf7ce5aec5685562a3222dbc5e6b7c": "P_\\lambda(z) =\\frac{1}{2\\pi i}\n \\int_{1,z} \\frac{(t^2-1)^\\lambda}{2^\\lambda(t-z)^{\\lambda+1}}dt", "a4d003e91883623f90411c27e7358db6": "\n{\\rm E}\\left[ s \\right]\\,\\,\\, = \\,\\,\\,\\sigma \\,\\,\\theta \\sqrt {\\,\\gamma _1 } \\,\\,\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\,\\hat \\sigma \\,\\, = \\,\\,{s \\over {\\theta \\,\\sqrt {\\,\\gamma _1 } }}\n", "a4d0c3f65edcf0843f0203d89a695bfa": "\\begin{array} {l}\nf'(x_0)=\n\\frac{f\\left(x_0 + h\\right) - f\\left(x_0 - h\\right)}{2h} - \\frac{f^{(3)}(x_0)}{3!}h^2 + \\cdots\n\\end{array}", "a4d129205b3e80ed46888233a894253c": " K(\\beta) ", "a4d18ae23af49adc261adfd6a5592df7": " \\vdash \\ \\ \\left( B \\rightarrow A \\right) \\ \\rightarrow \\ \\left( \\lnot A \\rightarrow \\lnot B \\right) ", "a4d1dc80e304ab3d90468c2f37cba861": "C_A", "a4d20de6397d0b0b1f47fdad013db18e": "\\eta(x,0)=\\Re\\left\\{B\\,\\exp\\left(i\\alpha x\\right)\\right\\},\\,", "a4d2955c1fc73de6b31aec6021edfe3d": "C(\\alpha)", "a4d2e7618712957c53436dabff3ff2be": "P \\not\\equiv_{b}Q", "a4d306a1e5477a3c9c6e5df3b788612e": "Prob_{pure}", "a4d344edbe7f6d7a7e0b880638536bb5": " K(u,v) = {4 \\over a^2 b^2 \\left(1 + {4 u^2 \\over a^4} + {4 v^2 \\over b^4}\\right)^2} ", "a4d37ab5dd95e8904cd6455807fc7cf8": "A \\in \\mathcal{F}_t", "a4d3b35140040c79466bee22ee1fd970": "q_\\infty = \\alpha \\phi_0 + \\frac{4 \\phi_0}{\\pi_r} ", "a4d3b477031b25ce83e56094beac9c8f": "y_j=g(h_j) \\,", "a4d433615e271d19dc68cfa20a3fd257": " (x_1,\\ldots,x_m) ", "a4d4bc1160aed5bcff02cf59b128f28a": "V_1(\\mathbf{x},z_1)", "a4d4d38807993acaf03c7cf23212c8b8": "{0^2 \\over 2}+g(0)+{P_\\mathrm{atm} \\over \\rho}=\\mathrm{constant} ", "a4d5068dc760ddf24e92572b9e5cf32a": "k_BT,", "a4d50fdd92fdbd654784714790fdb67b": " E(z\\mid x_A)=0", "a4d553bac83c056191255489bbe1eb33": "\\mathfrak{t}_{0}", "a4d5bad8db632c708fe9b47dcaa35eaf": "L(G)", "a4d5cbdcd567dcfeccaa41614b6c00d2": "(A = B) \\simeq (A \\simeq B)", "a4d605165eec400753af74588673f819": "{(\\eta_b)_{max}} = 2(\\rho\\cos\\alpha_1-\\rho^2)(1+kc) = \\frac{\\cos^2\\alpha_1 (1+kc)}{2}", "a4d6101c0ea391a3576b9f2f31d574b2": "N(\\varepsilon, U) = \\{f | \\operatorname{osc}_U f < \\varepsilon\\}.", "a4d613180a614c8376a3ddb514e8763a": "2^{2f(n)G(n)}", "a4d656da3d4c6b9218619c4827ed8f93": "fg-gf=i(f,g) \\, ", "a4d6613fce6a31523647b0971d32e700": " SG_\\text{apparent} = \\frac {W_{A_\\text{sample}}}{W_{A_{\\rm H_2O}}} ", "a4d68e336849cee32f5f72b45b8b13e7": "X\\circledast Y", "a4d6bd3a265f575dd131254d047b7a24": "{\\mathbf{\\dot{p}}}_{\\text{crystal}} = -e \\left( {\\mathbf{E}} -\\frac{1}{c} {\\mathbf{v}} \\times {\\mathbf{H}} \\right)", "a4d6d79d1e9e318b68f6a4a1c879626b": " X = \\sum_{k=0}^{\\infty} A^{k} Q (A^{H})^k ", "a4d6e93ef3b4ee0e402aea2cccb3c3b4": "\\frac{d}{d\\omega}Q(\\phi_\\omega)<0,", "a4d6f985b2cff9fbf44fd620f58a4fba": " k \\ge 1", "a4d7112663ade25798d9053043117b62": "{\\mathbf{}}K(t)", "a4d801892fd7a4458429c5cc55e32dbb": " {\\rm li} (x) \\sim \\frac{x}{\\ln x} \\sum_{k=0}^\\infty \\frac{k!}{(\\ln x)^k} ", "a4d8071c8c24a539a278258fd5cd4382": "\\theta_2 = - \\; \\theta_{\\rm S} + \\frac{\\theta_E^2}{\\theta_2}", "a4d8831c3b8e4f73d835508c3fc9c58a": "\\frac{3 e^{\\kappa^2} - e^{3 \\kappa^2} - 2}{(e^{\\kappa^2} - 1)^{3/2}} \\text{ sign}(\\kappa) ", "a4d8ce6594a856351ce74391221ff9da": "x^*\\, A = b^*\\,;", "a4d90232051c54179d12a79399310e03": "M_x[x] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&1&0& \\cdots \\\\\n0&0&1& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "a4d90eeef6f8569f1d2dda5ac4a9e669": "\n\\frac{\\partial}{\\partial x} V(x,t) =\n-L \\frac{\\partial}{\\partial t} I(x,t) - R I(x,t)\n", "a4d91a6bf6428fb1b6a31532ba24dcd8": "\n w(x,t) = \\text{Re}[\\hat{w}(x)~e^{-i\\omega t}] \\quad \\implies \\quad \\cfrac{\\partial^2 w}{\\partial t^2} = -\\omega^2~w(x,t)\n ", "a4d94e80843e54611af58338241dbce0": "{r_s / 2} < \\alpha ", "a4d94ff6f96a8588768dda04a7f3868f": "B(x)\\, dA = E(x) \\, dA +\\rho(x) \\, dA \\int_{S}B(x') \\frac{1}{\\pi r^2} \\cos\\theta_x\\cos\\theta_{x'} \\cdot \\mathrm{Vis}(x,x') \\,\\mathrm dA'", "a4d9889843e6df5f5ae8e12ab059bcf8": "ds^2=g_{ij}(q) \\, dq^i \\, dq^j", "a4d9be0dd5567f1daafb1c5972697e7e": "V = q\\phi ", "a4da25338bf4e6390a44a60efdeeb32b": "C_{k \\pm}|\\mathbf 0>=0", "a4da71b65d5c568e3e9916af61dbf4f9": " \\overline \\sigma = \\sigma (\\overline P) = \\frac{\\overline P}{A} ", "a4da8a039574f00847823ecdf9537a7c": "\\beth_\\omega", "a4da94e6876cca5fe20742304954eb12": "f_{\\Gamma, R}", "a4da9f55419cb8baadf8b268222906fa": "|{\\Phi^{[1..k]}_{\\alpha_k}}\\rangle=\\sum_{\\alpha_1,\\alpha_2..\\alpha_{k-1}}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}\\cdot\\cdot\\Gamma^{[k]i_k}_{\\alpha_{k-1}\\alpha_k}|{i_1i_2..i_k}\\rangle", "a4dad6e4188ef51959d56c07313fac0a": "\\left(\\mathbf{A} + \\mathbf{UBV}\\right)^{-1}", "a4dae485d3b9dff2b68583c536cce145": "S_c", "a4dae57af4357e36727b2ff94a94fbc4": "{\\mathcal C}_n", "a4dbc5557accd10bc2e983c6ff349f12": "\\gamma_x:=\\{\\Phi(t,x) : t \\in I(x)\\} \\subset M", "a4dbe1906ad966123f8c88dfc8c1406f": " \\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} .\\,\\!", "a4dc23b5777f783da947f88fd7f25142": "\\mathbf{j}_1", "a4dc6ab73cd2864a511ee5f9ca903661": "F_t\\colon X \\rightarrow Y", "a4dc6eb95566f3b86e302b022084069d": "\\lambda_m = \\frac{m}{\\sum_{k=1}^K \\frac{\\frac{m-1}{m}L_k(m) + 1}{\\mu_k} v_k}", "a4dc840a4a4c454756a0c017442fc789": "Var(X) = V(\\mu) = \\mu^2/r", "a4dc915725ffc0200fc55282c29d46fa": "as+bt=\\operatorname{Res}(a,b),", "a4dcad36b9207cdc4e3b8d817dc249e6": "\\cos \\theta = \\sin \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\sec \\theta}\\,", "a4dcae72583fc2d7bdb3d66bc0a5570f": " \\begin{align} \n\\operatorname{psin}(c_1 \\bold{v}_1,\\dots, c_n \\bold{v}_n) & = \\frac{\\det\\begin{bmatrix}c_1\\mathbf{v}_1 & c_2\\mathbf{v}_2 & \\cdots & c_n\\mathbf{v}_n \\end{bmatrix}}{\\prod_{i=1}^n \\|c_i \\bold{v}_i\\|} \\\\\n& = \\frac{\\left(\\prod_{i=1}^n c_i \\right)}{\\left(\\prod_{i=1}^n c_i \\right)}\\frac{\\det\\begin{bmatrix} \\mathbf{v}_1 & \\mathbf{v}_2 & \\cdots & \\mathbf{v}_n \\end{bmatrix}}{\\prod_{i=1}^n \\|\\bold{v}_i\\|} \\\\\n& = \\operatorname{psin}(\\bold{v}_1,\\dots, \\bold{v}_n) \\\\\n\\end{align}", "a4dccaaaf217fd29bf1e06b8f6748bb2": "\\cfrac{\\partial x_i}{\\partial q^j}", "a4dcd311abe918e776d1de3190ffdf7a": " E_n = -{Z^2 R_\\mathrm{E} \\over n^2} ", "a4dcffd1baced1d31bb6e29a8f775ede": "\\overline{Y}_i", "a4dddb6a4b7710ccf77f345528147312": "{u} = \\frac{u_\\tau}{\\kappa} \\ln\\, \\frac{y}{y_0}\\ ", "a4de1679dc9040e0d26e7c1a45c3db85": " BSC_p ", "a4de1ca79293882b44c917e02a8daefb": " d=1,\\dots,15,19,24,35,48 ", "a4de484fa76708fe9f7b2bde4cb3a0f7": "0\\le p<1", "a4df121c398dcae37c7afada75ab605a": "B e^{st}", "a4df4c5aaf8f092e03a00f7109e5de74": "\\frac{x}{y}", "a4dfcf52b7f7d958a4b884548323b7c4": "\\mathbf{F}=dA", "a4e03abb65e9b97a85119ab7eacbb1ca": "S[g]= \\int {1 \\over 2\\kappa} R \\sqrt{-g} \\, \\mathrm{d}^4x ", "a4e0451883f9b35bf5bef4792f2f6efc": "\n\\lambda = 8[(\\frac{8}{Re})^{12} + \\frac{1}{(\\Theta_1 + \\Theta_2)^{1.5}})]^{\\frac{1}{12}}\n", "a4e0de1ea96bb512ac8431f3964e90c5": "\\langle \\phi | \\psi \\rangle = \\langle \\phi | \\psi \\rangle ^2. \\, ", "a4e114cfdaafcc91285e496bc625dc8a": "\\sum_{a\\in\\Sigma}V(a)\\langle A\\otimes B,R_a\\rangle = \\langle A\\otimes B,R\\rangle", "a4e15f79bea9aef74a8705b37fdec175": "\\begin{matrix}\n {{D}_{0}}=\\sum\\limits_{i=1}^{K/2}{{{I}_{i}}\\delta t} & {{D}_{1}}=\\sum\\limits_{i=K/2}^{K}{{{I}_{i}}\\delta t} \\\\\n\\end{matrix}", "a4e18630dcb3bc0a804cf374ca6c8d5a": "m=k", "a4e26fd3b52836be8bd8f22906fcf210": "Z_{i,j} = Z^{M}_{i,j} + Z^{D}_{i,j} + Z^{I}_{i,j}", "a4e27601a887763d05d304ed16d1d4a7": "a,\\beta_\\pm", "a4e292a80789944ca8f3a47b11e733e2": "\n\\mathcal{D}_\\alpha V_I = \\partial_\\alpha V_I + \\omega_{\\alpha I}^{\\;\\;\\;\\; J} V_J . \n", "a4e2bf45b730a5303dd6d7ad57f5e382": "\\Phi_v = n\\frac{h}{2e}.", "a4e2c18ed7c30dc5a79f8d54c1e7f5bb": "x \\wedge y \\wedge (x \\vee y) = (y \\vee x) \\wedge y \\wedge x", "a4e2da52fb0e2611e61e9ed8c55b38e5": "\\nu(d)", "a4e2e3d1f8a20403e6f40e51d2ab3454": "\\ f(n) = n^{-a} (\\log n)^{-b} (\\log \\log n)^{-c}.", "a4e2e424c2788589363188510f069977": "\\scriptstyle Q(x)", "a4e30d886364b2541544453f24ebdcee": "n=0,1,2,\\dots", "a4e39d74fbdd4a4dfccd52002bcedb65": "\n C_s=\\frac{3\\lambda_3}{\\lambda_1 + \\lambda_2 + \\lambda_3}\n ", "a4e3ece41478a6bd8da190ad160525bc": "|\\mathcal{C} \\cap B(y, pn)| ", "a4e4088fd1079e87000195111381644c": " f \\mapsto \\int_a^b f(x) \\; dx", "a4e435d4d078e7df1fa07e13d4a32ebb": "m_2", "a4e46872eb692d6b3f83994c86cbbb47": "R_n=|E(z^n)|=\\sqrt{C_n^2+S_n^2}\\,", "a4e48293e209a2a59a3c4818106c159a": "u'_1, \\ldots, u'_n", "a4e4a64acd770ea43c1c5a8c56579501": "\n\\langle T \\rangle_\\tau = -\\frac{1}{2} \\sum_{k=1}^N \\langle \\mathbf{F}_k \\cdot \\mathbf{r}_k \\rangle_\\tau = \\frac{n}{2} \\langle V_\\text{TOT} \\rangle_\\tau.\n", "a4e4c7afe0c2e76381cfae772e39c841": "\\sqrt{2} \\cdot \\sqrt{3}", "a4e4e451fc0cb7fdfe61f57c3fd7a18d": "g\\left(x\\right)", "a4e51daed1d6636fe02186441ab533f3": " \\kappa(A) = \\frac{\\sigma_{\\max}(A)}{\\sigma_{\\min}(A)} ,", "a4e52d03343277acadf86e036378ca94": "\\Phi_{2D}(\\mathbf{x},\\mathbf{x}')=\n-\\frac{|\\mathbf{x}-\\mathbf{x}'|^2}{8\\pi}(\\ln|\\mathbf{x}-\\mathbf{x}'| - 1),\\quad\n\\Phi_{3D}(\\mathbf{x},\\mathbf{x}')=\n\\frac{|\\mathbf{x}-\\mathbf{x}'|}{8\\pi}", "a4e6179926ec29538bb3c3743a9db006": "[P,Q]=PQ-QP=-i\\hbar ~ \\operatorname{Id}_\\mathcal{H},", "a4e669be6847679767fe1c83bc77e9a5": "\\mapsto, \\longmapsto \\!", "a4e6dc17bcb444c89e91d6e5d923a3a4": "\n\\mathrm{d}s^2 = \\mathrm{d}w^2 + \\mathrm{d}r^2 + r^2 \\mathrm{d}\\phi^2.\\,\n", "a4e6e274100532ac4c7cc9f66881d4dc": "\\int_0^\\infty \\frac{x^{p-1}dx}{1+x}= \\frac{\\pi }{\\sin p\\pi } \\ \\ , 02 \\\\\n \\ \\infty & \\text{otherwise}\n \\end{cases}", "a505271afa308c030e499bb9d2e889f2": "f\\colon M \\to X", "a50595685707dd7cd3d06145d9ea6306": "\ny^{2} = \\frac{\\left( b^{2} + \\lambda \\right) \\left( b^{2} + \\mu \\right) \\left( b^{2} + \\nu \\right)}{\\left( b^{2} - a^{2} \\right) \\left( b^{2} - c^{2} \\right)}\n", "a5059abe695d7dddbb2f2b9d05d41797": "\\textstyle Q=\\prod_{j=1}^{r}(x-\\lambda_j)^{\\nu_j},", "a5059e19941a71ae9e36399a85f8f211": "\\nabla_\\theta J \\leftarrow \\frac{1}{\\lambda}\\sum_{k=1}^{\\lambda} u_k \\cdot \\nabla_\\theta\\log\\pi(x_k | \\theta) ", "a505e216e1e19ef44d6ee0fc69673e30": "\\hat{Y}(X_{0})=\\frac{\\sum\\limits_{i=1}^{N}{K_{h_{\\lambda }}(X_{0},X_{i})Y(X_{i})}}{\\sum\\limits_{i=1}^{N}{K_{h_{\\lambda }}(X_{0},X_{i})}}", "a50680e2dfe267c0b14932429ec28e22": "A_\\epsilon^{(n)}", "a506a228b09e5c818d866ee95fd6c926": " \\Omega = \\operatorname{E} \\left[X^2 \\right]. ", "a50758c94add9dfe1e982aa544b9cc4b": " p(\\mbox{diabetes}=1|\\mbox{glu})\n = \\frac{p(\\mbox{glu}|\\mbox{db.}=1)\\,p(\\mbox{db.}=1)}{p(\\mbox{glu}|\\mbox{db.}=1)\\,p(\\mbox{db.}=1) + p(\\mbox{glu}|\\mbox{db.}=0)\\,p(\\mbox{db.}=0)}\n", "a5075d2fdd9644126e0ab22eda64dd21": "r_0=x^{m+n+1},\\;r_1=T_{m+n}(x)", "a50774bd052ea892222b3e0e7595058f": " \\mu_i = \\frac{1}{r} \\sum_{j=1}^r d_i \\left( X^{(j)} \\right) ", "a507a78e5c547b74f51d05eab27909a9": "\\operatorname{tr} (\\gamma^\\mu) = 0. \\,", "a507be781c66dc3381d748de641b8184": "dry\\;basis\\;concentration = (wet\\;basis\\;concentration)/(1-w)", "a5086cb7205d130ff0334fe5a8372fea": "\\mathbf{R}^o", "a508807f96dced464fd71beee3906273": "\\phi_1", "a508cfe0703bc962382bd03f9b7f2ddd": "\\frac{\\mu}{\\mu^2+1} \\to \\frac{\\mu^2}{\\mu^2+1} \\to \\frac{\\mu}{\\mu^2+1} \\mbox{ appears at } \\mu=1", "a5091383f66c9b0700b134d7b1926f75": "\\Delta = (r^TV^{-1}r)(1^TV^{-1}1) - (r^TV^{-1}1)^2 > 0.", "a50959005f9cca92f4ab536edeb29e88": "\\nabla_\\ell \\omega_m = \\frac{\\partial \\omega_m}{\\partial x^\\ell} - \\Gamma^k{}_{\\ell m} \\omega_k.\\ ", "a5095bc1cba4622f6b8239af17560c12": "C_1,\\ldots,C_N", "a50969a1b08056f33377353d92db21e4": "\\sqrt{\\frac{\\text{effective bulk modulus}} {\\text{density}}}", "a50984542418de625e121a2f98eea2d8": " \\int\\sec x \\tan x \\ dx= \\sec x + C", "a509d777c11b8ed680d19c4853ecefd1": "\\hat{x} - x = W(y-\\bar{y}) - (x-\\bar{x}).", "a50a028f4ed8766e2380d125aa754f5a": "\\mu(T)\\,=\\,\\mu_0 \\exp \\left( \\frac {-C_1 (T-T_r)} {C_2+ T -T_r} \\right)", "a50a2dc3f40ce4037b67e94afd47ec73": "\\{r\\}", "a50a67ccc63a8bd4bb106c674e3140bd": "P=p_0+p_1 X+\\cdots +p_m X^m,\\quad Q=q_0+q_1 X+\\cdots +q_n X^n.", "a50ae26f031e03c6cefc47bf73161141": " Q_{a}^{(c)}(t) - Q_b^{(c)}(t) ", "a50af9c7da2e5876a31ae9c540f7e6f7": "\\exp\\left(O\\left(n^{\\frac{1}{k-1}}\\right)\\right)", "a50b1a5010ba76b125d11e6d3fc47aa6": "\n(H_\\mathrm{sat} - H_0) \\cdot \\lambda = (T_0 - T_\\mathrm{sat}) \\cdot c_\\mathrm{s}\n", "a50b56a55f4c85e06f5edba6c62d59f1": "\\left(x_n:n\\in \\mathbb{N}\\right)\\;", "a50b6b495290ccb1eec56868582d7a2c": "\\lambda(G) = \\min_{f\\perp G} {(Df,f)\\over (f,f)},", "a50b7c43786d463ed4632791d225ded3": "A^\\alpha = \\left( \\phi / c , \\mathbf{A} \\right)\\,\\!", "a50bbbc4ece13f6d53171fba6a729ee5": "E_{B} = a_{V} A - a_{S} A^{2/3} - a_{C} \\frac{Z^2}{A^{1/3}} - a_{A} \\frac{(A - 2Z)^{2}}{A} - \\delta(A,Z)", "a50bbc1c13e3c54c5fc6b73d38aafae8": "\\begin{align}\n\\sin 2\\theta &= 2 \\sin \\theta \\cos \\theta \\ \\\\ &= \\frac{2 \\tan \\theta} {1 + \\tan^2 \\theta}\n\\end{align}", "a50be7582cf68218913799df25e65c58": "S^3.", "a50c9392a2bdf95af322201ea7e28523": "\\scriptstyle 0 \\leq k \\leq n/2+1", "a50c99ff26fe257ac2d58843151e4009": "\\tau = (1 - \\theta) \\hat{\\tau}_{-} (t) + \\theta \\hat{\\tau}_{+} (t).", "a50ca865bc2bba71421f175ab9991d67": "\nV_R = IR = C\\frac{dV_C}{dt}R\n", "a50cfa2a03590f5b91fc50c4c46bb393": " [F_N]_{jk}:= \\exp[(2\\pi i(j - 1)(k - 1) / N] \n{\\quad \\rm for \\quad} j,k=1,2,\\dots,N ", "a50d63039f03e66210c9cc1fefd8745f": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n P^{-1}(A\\mathbf{x}_n-\\rho(\\mathbf{x_n})\\mathbf{x_n}),\\ n \\ge 0.", "a50d83bcc7ef04fb5f41365a8cf50378": " W_S = W ", "a50e1e1283a092e7bfd9c04c72df4cd7": "g_{\\mu\\nu}\\delta g^{\\mu\\nu}=-g^{\\mu\\nu}\\delta g_{\\mu\\nu}", "a50e3d7c8b1f4369fe0e31e6d2685a86": "G = \\frac{v_{out}}{i_{in}} = -R_f \\frac {g_m R_D}{1+g_m R_D} + R_D \\frac{1}{1+g_m R_D} \\ .", "a50ea90dc8adbe5cd935c25a969f3692": "\\lambda_{i}=\\lambda\\text{ and }\\mu_{i}=\\mu\\text{ for all }i. \\, ", "a50eb53c3b1d9a9ce9c21e3ca3e94061": "t\\widehat{\\otimes} s = \\frac{1}{(r+p)!}\\sum_{\\sigma\\in {\\mathfrak S}_{r+p}}\\operatorname{sgn}(\\sigma)t^{i_{\\sigma(1)}\\dots i_{\\sigma(r)}}s^{i_{\\sigma(r+1)}\\dots i_{\\sigma(r+p)}} {\\mathbf e}_{i_1}\\otimes {\\mathbf e}_{i_2}\\otimes\\dots\\otimes {\\mathbf e}_{i_{r+p}}.", "a50ebb359e8eff6e5e016e3888c91c38": "M = \\mathcal{}MU_*(X)", "a50ecd51605e55dfbf6424539e81e379": "x\\left(1 - \\frac{x}{K}\\right)", "a50ee6e69e4766af61e8d817ff3dacd2": "h_p(\\zeta,\\bar\\zeta) > 0", "a50efa10fe8a88102ed7600bb12f2888": "\\text{SL}_2^{\\pm}(\\Z)", "a50f11f1bdb9762de197a7bc912c703c": "\\mathfrak{g} = \\mathfrak{s}_0 \\oplus \\mathfrak{z}(\\mathfrak{g}).", "a50f173f6f9ee16be87f27b141f291ad": "G(x,f) =\n\\begin{cases}\n|J(x,f)^{-2/n}D^Tf(x)Df(x)&\\text{if }J(x,f)\\not=0\\\\\nI &\\text{if }J(x,f)=0.\n\\end{cases}", "a50f549a86c0c8af5b3a955b0a5d5e3e": "\\sqrt{n}(F_n(x)-F(x))", "a50f8021054cac76c81eb078ceac1ba1": "n^{-\\frac{1}{4}}", "a50f9141d5c87e6f913991f412f60cec": "\n B \\or C \\vdash C , B\n ", "a50fab5849aafdafd31f5d8ddc97db2e": "[ABE]=[ACE] \\,", "a50ffd2725cef166ec6e23ca7751b9ce": " \\begin{align}\n\\frac{{\\rm d}\\bold{R}}{{\\rm d}t} & = \\frac{{\\rm d}}{{\\rm d}t}\\left(\\frac{m_1\\bold{r}_1+m_2\\bold{r}_2}{m_1+m_2} \\right) \\\\\n& = \\frac{m_1\\bold{v}_1 + m_2\\bold{v}_2 }{m_1+m_2} \\\\\n& = \\bold{V} \\\\\n\\end{align} \\,\\!", "a51028979611a5ea755f801da6ca424e": "Nx^2 + 1 = y^2", "a510291175c1bde73fdf241069d176a3": "\\scriptstyle f(n) \\;=\\; 3 \\rightarrow 3 \\rightarrow n", "a51051ad32f37594976c9e91be26231a": "\\!\\Big[1+\\lambda^{\\alpha}|t|^{\\alpha} \\omega - i \\mu t]^{-1}", "a5106f5d592f9088cdd19058d259b892": " Q = \\begin{pmatrix} -\\mu_A & \\mu_{GA} & \\mu_{CA} & \\mu_{TA} \\\\\n \\mu_{AG} & -\\mu_G & \\mu_{CG} & \\mu_{TG} \\\\\n \\mu_{AC} & \\mu_{GC} & -\\mu_C & \\mu_{TC} \\\\\n \\mu_{AT} & \\mu_{GT} & \\mu_{CT} & -\\mu_T \\end{pmatrix}", "a51096a5062a032071cfff96244b3ca8": "F(s,t,u) = \\left(u \\tan s \\cos t, u \\tan s \\sin t, u \\right)", "a5109cf730679dee3ce2d571771300e6": "=\\frac{2v^2\\cos^2\\theta}{g} \\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right) \\sqrt{1+m^2}", "a510b3ee19fbacd1e7b3454beeb6601f": "\\scriptstyle X \\to (Y \\to Z)", "a510ebbeaa88d5492834f4581478a2fd": "y = 0.8707 \\times 0.52 \\times d(\\varphi) \\times \\pi", "a510f1b8b6f9aa92c2f2ece26922f02d": "S_{\\lambda/\\mu} = (h_{\\lambda_i - \\mu_j -i + j}), 1\\leq i,j \\leq l(\\lambda)", "a5113886623d7872fdf4c075ea4b58b6": "\\Omega_{\\rm mic} ", "a5114a1a9d5e6f747658ab57a41e9156": "\\begin{align}\n\\int_a^b f(t)dt &= \\int_a^b \\big[ u(t) + i v(t) \\big] \\, dt \\\\\n &= \\int_a^b u(t)dt + i \\int_a^b v(t) \\, dt. \n\\end{align}", "a5115214a96d33306a35d95f95b8fbcb": "\\frac{\\partial \\mathbf{a}}{\\partial x} =", "a5117be46c5e0ac9814f706f1e5c2b0a": "\nA X = L U X = B.\n", "a5117c4ab851b65bbea4d8ca003a893a": "cx^{1/2}-3x^{1/3}\\le k(x) \\le cx^{1/2}, c=\\zeta(3/2)/\\zeta(3)=2.173\\cdots", "a5128149f4f350c587c02ed6dd82e0ee": "\\tfrac{j}{i+j} = \\rho", "a512b14dc02cb6c51cfc2bdbeb5c59e2": "\\mathrm{R_{PB} =\\frac{V_{oxide}}{V_{metal}}= \\frac{ M_{oxide} \\cdot \\rho_{metal}} {n \\cdot M_{metal} \\cdot \\rho_{oxide}} }", "a512c671eb221675dcae3a7dd0d7384f": " A = \\bigoplus_{i \\in \\mathbb{N}} A E_i ", "a512c7263ff9668dae313818bc2c99a7": "\\mathbf{} \\eta ", "a513072154d3aedbcd5ec5e17c4350e5": " GM=3\\pi V/P^2", "a51359ed8e1b1a9cf4ccd27155e535bc": "Y \\in \\ker(I- \\Delta S)", "a5137bfbe70312833d293bbcfb841345": "f = {1\\over 2 \\pi} \\sqrt {g\\over L} ", "a513a1644869b4d3332ec5d2873e61ff": " \\frac{ V_{geostrophic} }{ V_{anticyclone} } = 1 - \\frac{ V_{anticyclone} }{ V_{inertial} } < 1 ", "a513a172ca4701861a24ea5354dd546e": "r(T_{m,n}) = \\min\\{m,n\\} + 1", "a513cf94983961047cbc8a3a0598959e": "\\mathbf{u} \\times \\frac{\\partial \\mathbf{v}}{\\partial x} + \\frac{\\partial \\mathbf{u}}{\\partial x} \\times \\mathbf{v} ", "a513f8f21ffd78208a92a6ecf9b40bf8": "\\beta_{n+1}^\\xi > \\beta_{n} \\,,", "a513fbcc86945a217db9b76a0583bf8f": "\\lbrace \\vec x: \\vec y \\rbrace", "a5141b34e1afa5cedc8fdc0ca8c1ebaa": "2n_R", "a5142f6b0c303d3dcce536076354f90a": "x = y - y^* \\,", "a5144dd548bd2006571039f098c70fb1": "\n \\left (1 - e^2 \\sin^2 \\varphi \\right )^{-3/2}\n =1+b_2 e^2s^2+b_4 e^4s^4+b_6 e^6s^6+b_8e^8s^8+\\cdots,\n", "a5145a7b32b540db8ab2b3733140cbbc": "F(t,T) = S(t)\\times (1+r)^{(T-t)}", "a514d146912a1e1dab71ddd92e43f877": "\\displaystyle{E:C^\\infty(\\mathbf{R}^+)\\rightarrow C^\\infty(\\mathbf{R}),}", "a514d2165b2bafc0969adf7fca8ce0cc": "2^1 \\times 0.001_2=", "a516a89643942c8a83775bcec9314b13": " y_{1,t} = a_1 y_{1,t-1} + a_2 y_{1,t-2} + \\dots + a_n y_{1,t-n}", "a516e08808706e9c5e64b5548f7a2e3f": "\\vec{u}=\\nabla\\phi", "a5175048cc25d637e743f819134acd82": "2^m-2", "a5176f6943cea018e0945eabda8b6ce5": "S = \\prod_{j=1}^m \\frac{1}{k_j!}\\,", "a517bf5641ba1809caa1b85eca3d9c8d": "\\tfrac{3K(1-2\\nu)}{2(1+\\nu)}", "a517e5719b155722c8986e9557d6d239": "V(N)", "a51807c774cc40ed47d4254c5eff1a59": "\nF(z) = \\sum_{k\\ge 1} \\varphi(k) \\sum_{m\\ge 1} \\frac{1}{km} f(z^k)^m =\n\\sum_{k\\ge 1} \\frac{\\varphi(k)}{k} \\log \\frac{1}{1-f(z^k)}", "a5183a3b90d04a71a9ba35ed78290ef6": " -\\frac{1} {4 \\pi} \\iint\\limits_S\\left(\\frac{\\mathbf{n}\\cdot \\nabla \\mathbf{U} }{R}\\right) dS_Q", "a518ab20063d96525b577d1e599ddb21": " {d \\bar h^k \\over ds} + \\Gamma^k_j \\bar h^j \\ \\stackrel{\\mathrm{def}}{=}\\ {D \\bar h^k \\over Ds}", "a518bd6cd40556bddfde0cea6efc2761": "a,b,c>1", "a518efeb67839aa079738db787d2962d": "Fm_s = \\frac {Fm_{ms}C_{ms} - Fm_{pb}C_{pb}} {C_s}", "a5190d6f080b9d7746abac7a012b0730": "m_{y}", "a51912b521cc149e7fc3df7a5a4daae2": "i\\in\\mathbb{Z}", "a5191c4c9f11a52608717910c5ffc130": "R_n = R_o + K G (W - W_e)", "a5191fafb1caf4a7d58e84c92c77fd15": "\\phi^{-}(a_i)", "a5194f9428818c75c8bcfa3f305b5c9a": " \\operatorname{build-param-list}[\\operatorname{let} V: E \\operatorname{in} L, D, V, \\_] \\equiv \\operatorname{build-param-list}[E, D, V, \\_] \\and \\operatorname{build-param-list}[L, D, V, \\_] ", "a51981474bac7c253f18a045d062401e": "\\displaystyle\\Delta(G)", "a51982398b14d1b49499f940a53b948c": "w\\left(r,r'\\right)=\\sum_{k=0}^{N-1}x_{k}^{\\left(rr'\\right)}\\left(s_{r}s_{r'}^*\\right)^k", "a5199a3c4bcd95fcfbbbdf3e9309cc59": "\\bold{A}", "a519c746138622c1dbd8edee810a5f56": "ax^2+bx+c=0", "a519cb2adbcd599cbd0e1cdb07d8532f": "\\textbf{R}_{x}(\\tau) = \\sigma^{2}e^{-\\beta |\\tau|}.\\,", "a519d5d7b6559037e75463cd39bf0f68": "2|\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|\\mathbf{AX}(\\mathbf{X^{\\rm T}AX})^{-1}", "a519dde58fb11df34dd8650bcc86edc7": "\\begin{cases}\\begin{align}4x + 2y & = 12 \\\\\n-2x - y & = -6 \\end{align}\\end{cases}\\,", "a519e6a47e289b4f28084f1b19891b6f": "e^{1/n} = [1; n-1, 1, 1, 3n-1, 1, 1, 5n-1, 1, 1, 7n-1, 1, 1, \\dots] \\,\\!.", "a519f0f8771974c5c780c8c276df27ec": "f(x_0^+)", "a519fb38b960b092966b42a46ed47107": "7/8+O(1/\\sqrt{B})", "a51a29d2812f0eacf8a8e8cd08ccf6c5": "\n \\sigma_{11} = \\sigma_{22} = \\cfrac{2C_1}{J^{5/3}}\\left(\\lambda^2 - \\cfrac{J^2}{\\lambda^4}\\right)\n ", "a51a6cd134103dcd9093d5835ddf20ba": " \\Phi(M) = Arf(H_1(M,\\partial M;\\mathbb{Z}_2);\\mu) \\in \\mathbb{Z}_2 ", "a51a901dfa26d9732f16228575ae120f": "\\frac{\\partial I}{\\partial x}\\frac{\\Delta x}{\\Delta t}+\\frac{\\partial I}{\\partial y}\\frac{\\Delta y}{\\Delta t}+\\frac{\\partial I}{\\partial t}\\frac{\\Delta t}{\\Delta t} = 0", "a51abcb481004c6f1a7eae6573b17e46": "\\displaystyle f'(x_n)", "a51afe8c783faef32cb2fa1cdbc920fb": "A^*=-A", "a51b12084eeda0668c6b1c4707f34bf0": "\\Delta p_{w} = H_{w} (1- \\epsilon_{w}) (\\rho_{s} - \\rho_{f}) g = [M_{s} g / A] [(\\rho_{s} - \\rho_{f}) / \\rho_{s}] ", "a51b22433a5e2dbedd64756e2595a115": "(n_1n_2 > 0 \\ \\and \\ m_1n_2 \\le n_1m_2) \\ \\or \\ (n_1n_2 < 0 \\ \\and \\ m_1n_2 \\ge n_1m_2).", "a51b2bbd8a8d288130058a34ce732440": " \\langle n' | n \\rangle = \\delta_{nn'}. ", "a51b39bcce57a391cd529f2f5436a795": "m = m'", "a51b3ae464b0bf220738caa710a77bb9": "Scenario \\quad I: \\qquad \\frac{L_{\\rm B}}{L_{\\odot}} = {\\left ( {\\frac{1,204}{1}} \\right )}^2 {\\left ( {\\frac{3,300}{5,778}} \\right )}^4 = 154,000 L_{\\odot} (rounded)", "a51b4fe00de2e6a25d4aff1b6b0bf9db": "\\langle f\\rangle = \\{f^0,f^1,\\dots,f^{n-1}\\}", "a51b870d37bc2405548e5adefb6becb6": "\\nabla \\cdot \\mathbf v = 0 \\quad \\Rightarrow \\quad \\nabla \\cdot (\\nabla \\times \\vec \\psi) = 0 \\quad \\Rightarrow \\quad 0 = 0", "a51b87cf0891e38dbcc72ac2b34c7c7c": "N=n/m", "a51bb8e14cd35f1dac62992d6866cd97": " \\nabla^2 1/r = - 4 \\pi \\delta(r) ", "a51bc23dfa7ebad26c3faac606494f12": " \\mu = \\left (m_1m_2 \\right )/\\left ( m_1 + m_2 \\right) \\,\\!", "a51be819389a57a4f249d4582f38c35b": "V^2=V_r^2+V_\\theta^2,", "a51c17eb57e082e286ae9ed41781d77d": "\\xi=o", "a51c23e92ca2fa493da58996bff9275c": "i = 1 .. N", "a51ce1ec1c5c37ecc691d94aa60c1014": " \\theta j ", "a51d0d6008a27a3a2bfba8092619dc03": " J^1Y", "a51d16b3deb9e8ef029df40521911cee": "f= ax + by + cz + d", "a51d39b702897d9cfe042464e529821d": "C_{F0} = \\frac{\\epsilon_0}{\\lambda_0}\\cdot S_{F0} = 5.1805\\cdot 10^{-7} \\ ", "a51d4db0dd4ab7c9181e29424f2b2964": "\\operatorname{minus} \\equiv \\lambda m.\\lambda n. (n \\operatorname{pred})\\ m", "a51da1b371da3560a6598d3d7e878010": "z_{c}=0", "a51dd7698488f9abf676c2d577249943": "(f + g)(x) = f(x) + g(x) \\,", "a51e2fbd733e9b3a4d7ab495758f1d20": "e^{+i\\omega t}", "a51e88229b7677a5648437f1698c7f3d": "(\\det\\Phi)'=\\sum_{i=1}^n a_{i,i}\\det\\Phi=\\mathrm{tr}\\,A\\,\\det\\Phi", "a51eba7b3acb0a792f28de5fe3a4206b": " {\\mathbf C}^g ", "a51ec8af972bf2bbbd8245e39f989db8": "\\chi(\\mathcal{X}) = m_0 - m_1 + \\ldots \\pm m_{\\dim \\mathcal{X}}", "a51ecf84b7caa514fb0b6c1d27ffbcaf": " h_{jk} ", "a51ef3d93f5bd80a7ce8299448061b9c": "\\scriptstyle A^a \\;=\\; \\ddot{x}^a", "a51f392861d445d6bce9897df5f42d80": "{\\sqrt{\\pi}}", "a51f4af0a622e325bb64cd051aa928d4": "s^{-1} arcmin^{-2}", "a51f6865f41ff8a6329bb6e9bc9e047b": "A=B^TB", "a51fa6030e8fce068e4de468016ff116": "\\scriptstyle V", "a51fc1949606195f05492d8602e26749": "\\neg P, Q", "a52065563d92a6743c2673349fcd90fa": "\\forall \\epsilon>0", "a520ff4fa28049a080f54f145ce49285": " T \\delta S/\\delta r \\,\\!", "a52170e5d261c752680db127a50d61f8": " T^2 \\sim q F_{p,\\nu}, ", "a521c00b38b1c20df24d8b29e25b07b4": "13n", "a521d24636c1ce850a41291bba660be1": " Y = \\frac{\\chi^2_n}{\\chi^2_n + \\chi^2_m(\\lambda)},\n", "a52223ba0a066c955ccf74aac6b081c5": "\\Delta^{*}", "a522359b42c74afd5f912c00aa4559ac": "s_{Ox}\\,", "a5225b7b0824b35e809792a4b6b069fa": "a=1,\\dots, l, b = 1, \\dots, m\\}.", "a5228d60441e02409bec61905e42b718": "Nu =\\frac{1}{2} u \\frac{\\mathrm{d}^{2} u}{\\mathrm{d}x^{2}}", "a522c204e298576a5cb90059ba672a43": "E_P", "a522e2b93dda422f4d9a2fa96d1a4bab": "E\\left(t,{\\mathbf r}\\right) = E_0\\,\\cos\\left(\\omega\\,t- {\\mathbf k} \\cdot {\\mathbf r} \\right),", "a522f60a8454d70ac693856473d90920": "51.7 \\times 10^6", "a5234eeb80395ec7ae386ab6b358b54d": "t\\in\n\\lbrack 0,1]", "a5237f67dc93c8dd07ce022c32eb8fd1": "c_n = \\frac{1}{P}\\int_{x_0}^{x_0+P} s(x)\\cdot e^{-i \\tfrac{2\\pi nx}{P}}\\ dx,", "a5238a2f6181623c04c44b35f6eece05": "x,y,z\\in X", "a523ab9d818d15877d1d315fb192769f": "e:M\\to G: p\\mapsto (p,p)", "a523b0f09f35d9ecba450aa8bec4284e": "(\\mathrm{Iso}(\\mathcal{A}),\\oplus)", "a523b87e540636d8ff848d0fbdceecdb": "\\textstyle (m+1)", "a523ba7c5c9f9cd78a6c83df3eb8c239": "G_\\Psi", "a523bd04ad6781c8bc1c8fa68872bd78": "\\mathbf{u}_1", "a523d0351b04f06e8d289a212e0d1eb4": "a, b_1,...,b_n", "a523d3f3218baf025540eb95226f2a7f": " q, n, m, d ", "a52424112748e5a3ffd367a088467713": "f(t, \\bar{u})", "a52424893a856eab31b1ca8f9753ab60": "\\displaystyle p", "a52425f2f429da06fae77f11ceb0f05f": "c(A) = \\sum_{i=1}^n c(a_i)", "a5244fdb7693354564595c636b8c8b28": "D_H = D.", "a524835f4aa01f5c363ed193d86c536f": "\\lambda e_j", "a524d182a6f576c41944e68658b2c57e": "\\mathbf{\\nabla}\\varphi_1=\\mathbf{\\nabla}\\varphi_2", "a524d441e54fbbd508663a79b61a027e": "\\textstyle d(x)(x^{2l-1}+1)", "a524f1ce80e4b5e51a669412585b4ec0": "(A-\\tilde\\lambda+vv^T)^{-1}", "a524f7d6a7f4f8704ac0a88aeca9665e": "r_\\nu=\\alpha_\\nu^1+\\alpha_\\nu^2\\mathrm i", "a5250d55f11ad87941070ab544d3a1cf": "2^{\\kappa}=\\kappa^{+}\\Leftrightarrow\\neg\\texttt{AX}_{\\kappa}\\,", "a52511e1a49561144f3a106026b87dd5": "2e^{it} - e^{2it}", "a52527c6131b005b7df224a3f4aa585a": "\\widehat{D}^nf(x)(y)", "a52543f038168e0487e4234550f4f2e0": "\\mathcal{N'}_k(x)", "a5258de8152805cb88bca53829b2b369": " \\overline{v_x^2} = \\overline{v^2}/3 ", "a525a38e796ca5c8f1d6ff0583c35b69": "\\|Tf\\|_q\\le C\\|f\\|_p.", "a525b3565e525d1ce63e160e5a36cea7": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{3 \\choose 1}{3 \\choose 1}{36 \\choose 1} \\end{matrix}", "a525bb76f1ec89247a8efb07b97fa61e": "s_{pm}=\\frac{a+jb}{c}", "a525f2cecbefd7c7bcfe18c9cb502b50": "L_{QA} = \\frac{L_{QL}}{n_B} \\ ", "a525f3bd39782a8ac9415524b6392118": "j=1,\\dots,n", "a5264ab7b5b640473e8e3ee92752d7af": "\\overset{\\frown}{AB}", "a5269df0a166ab199c5ae8949bad69ee": " Rate Ratio = \\frac{Incidence Rate 1}{Incidence Rate 2} ", "a526d37f89b3d50dba4a902b95ab3471": "a\\geq0", "a526d4bb54e2f3962b1d64ceebcd7dd7": "\\sqrt{2E_1}", "a526e2891eca62974b9f6e8b8ff80111": "p = \\left( 1-e^{-(m/n\\ln 2) n/m} \\right)^{(m/n\\ln 2)}", "a52716532b8504637dec3092ae7a110e": "{ E = \\hbar \\omega } \\ ", "a527355c7705e5a824a8b319df80773c": "\nH=I+\\begin{bmatrix}\n0 & j^T\\\\\n-j & Q\\end{bmatrix}\n", "a5273f5bcff2ab098fa6ec74db274edb": " T = \\frac{1}{4} \\sqrt{- \\begin{vmatrix} \n 0 & a^2 & b^2 & 1 \\\\\na^2 & 0 & c^2 & 1 \\\\\nb^2 & c^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 0\n\\end{vmatrix} } ", "a52750978c77c4ac20260e358ca2a302": "c = 1 + \\tfrac12\\, \\left( \\eta_1 + \\eta_2 + \\eta_3 \\right) = 1 + \\frac{H}{m}\\, \\left( 1 - \\frac12\\, m - \\frac32\\, \\frac{E(m)}{K(m)} \\right). ", "a5275f5d8198555041c037a4affda5e1": "\\{0,1\\}^\\infty", "a527816face5961a991ff0a8204f2203": "\\sum_{a \\in A} f(a);", "a527aebff0171d95ac3e8c7090fc6421": "m \\le \\frac1I\\int^b_aG(t)\\varphi(t) \\, dt\\le M.", "a527b45ade1c33c10ed8117c73c81dad": "17^{35}\\ \\equiv\\ 70\\ \\not\\equiv\\ 1 \\pmod {71}", "a5280894c7deebef88a736fe6c00c34b": " F(d_1,d_2,\\ldots d_k, \\mathbb F)", "a52814494d6ca8408e5d5d70a7d8d9c9": " \\left( c_p \\right)_{air} ", "a52894d721cc8af6aaf41ef6a0be5fb7": "\\omega = 2 \\pi - \\omega\\,", "a528aea71c645eaed6ca9ef649d5fef8": " W_{p} = \\mathbf{X}V_{p} = \\mathbf{X}V ", "a528afd6bfc211735945a7d5ced0c677": "\\xi=\\mathrm{arcsn}(x, k)", "a528b030e67fb70badd2d3782313568f": "\\omega_S = \\dot \\phi ' ", "a5290de398c66d41e18acf125eff5ded": " I_x(a,1) = x^a \\, ", "a5290f8d381504eed721f2f2ac68c0bb": "a_i\\in \\omega_{e_i}(A_{e_i})", "a5293a7a2f587ae8df957c2698f7829d": "\\mathbb{R}\\setminus\\mathbb{Q}", "a5293b9b3352f3b76e416f747f3a8064": "WCI=(10\\sqrt{V}-V+10.5) \\cdot (33-T_{\\rm a})", "a52979443fa68006625278ce3570ff67": "n_{xx} = (1 + \\chi_{xx})^{1/2} = (\\varepsilon_{xx})^{1/2} .", "a52a5865b1ecc2aa959fc4dc324674f7": "\n\\Phi (r, \\theta ) \\propto \\frac{1}{R} = \\frac{1}{\\sqrt{r^{2} + a^{2} - 2ar \\cos\\theta}}.\n", "a52ab9d12093ea463f74e80ea39d44eb": "B_i (0,2)", "a52b6072fe9e132107f4a0eff3afb395": "x = a + \\frac{1}{x'},\\quad x' = b + \\frac{1}{x''},\\quad x'' = c + \\frac{1}{x'''}, \\ldots", "a52b85d48c1f3e3d3f530ab42140253f": "(1-c)\\mathrm{OPT} + c\\mathrm{WORST}", "a52b9156a6eac910b757442581076fae": "\\textstyle f(t)", "a52b963e286fc7d2faccd76e1563e521": "(-p/2)", "a52b9bebfce978fed48e1126696858b4": " \\rho(m) = a m^{-5/2} exp(\\beta_o m) ", "a52bf26469a6c349f58f7c8043cf167e": "\n \\frac{d^3 y }{d x^3}\n = \\frac{d^3 y}{d u^3} \\left(\\frac{du}{dx}\\right)^3\n + 3 \\, \\frac{d^2 y}{d u^2} \\frac{du}{dx} \\frac{d^2 u}{d x^2}\n + \\frac{dy}{du} \\frac{d^3 u}{d x^3}\n", "a52c1b78f6868afdc07061a77da2fc5f": "\\bigoplus_i M_i", "a52c2fd205fedabb49e3e6b7c297172c": "P \\propto {E_o}^2 {|\\mu_{if}|}^2 ", "a52c57a669ccd06958aeaf8a5bb07419": "R_\\mathrm{E}", "a52c679f8e98ac808e54608d42761ef1": "\n\\mathrm{Var}(z)=1-R\\,\n", "a52c7021c2c6b38befc743e7fa17947a": "\\widetilde{X}", "a52ca2a09d08b0406f09a17deaa39bad": "\\scriptstyle \\delta g(\\tau)", "a52d1739ef8b25960af44e599bdd17f9": "\\begin{array}{rcl}\nc_n & = &\\displaystyle \\sum_{k=0}^n a_k b_{n-k}=\\sum_{k=0}^n (-1)^k (-1)^{n-k} \\\\[1em]\n & = &\\displaystyle \\sum_{k=0}^n (-1)^n = (-1)^n(n+1).\n\\end{array}", "a52d2bdc8713725ab262ccd5cfb3b8ec": "\n\\mathbf{P}= \\left( \n\\begin{array}{ccc} \n x_{11} & \\cdots & x_{1P} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n x_{F1} & \\cdots & x_{FP} \\\\\n y_{11} & \\cdots & y_{1P} \\\\ \n \\vdots & \\ddots & \\vdots \\\\\n y_{F1} & \\cdots & y_{FP} \\\\\n\\end{array}\n\\right)\n", "a52d3b93c29db379b40e972d88b9d150": "\\mathbf{V}^* = \\begin{bmatrix}\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n \\sqrt{0.2} & 0 & 0 & 0 & \\sqrt{0.8} \\\\\n \\sqrt{0.4} & 0 & 0 & \\sqrt{0.5} & -\\sqrt{0.1} \\\\\n -\\sqrt{0.4} & 0 & 0 & \\sqrt{0.5} & \\sqrt{0.1}\n \\end{bmatrix}\n", "a52d45cc0fd7a1cd06d656ebdd1393cc": " \\mathrm{skew}(X) = \\frac{( \\mu - \\nu ) }{ E( | X - \\nu | ) } ", "a52da08fd50ea2886ab09ffab9669092": "\\left\\{X_t\\right\\}", "a52dba033b6888080cf67426827451b4": "O(A_1:A_2|B) = \\frac{P(A_1|B)}{P(A_2|B)}", "a52dd1c217c5fe6a8ce7e541e9cf6cb8": "\n\\xi = r + x \\,\n", "a52e4af295d18079a1305a3e933b8eb6": "(1-f)p_{00}/(p_{01}+p_{00})", "a52ea335dac05838c47f411cf7e68bc2": "\\mathbb R^6/\\ker(A+\\sqrt{5}I)", "a52ec608c6db29203080cdbae614aa9c": "\\begin{align}\n& \\mathbf{(D-L)^{-1}} = \\frac{1}{120} \\begin{pmatrix}\n20 & 0 & 0 \\\\\n5 & 30 & 0 \\\\\n13 & 6 & 24\n\\end{pmatrix},\n\\end{align}", "a52f10600512678ad957d8b701d8adc4": "\\frac{A}{m} \\,", "a52f59fc5ba01001f8c5ae646ef7bed6": "\\prod_{i=0}^\\infty a_i\\,", "a5300b3171fd5a9483f5bfd6f80e919e": "S_v", "a53050a8329e4dbb40d5419647bf28dc": " \\Leftrightarrow V_\\mathrm{Ti}'''' + 2V_\\mathrm{O}^{\\bullet \\bullet}", "a53055c5061e1e6fdee7ddae2788e11f": " \\sin{z} = \\sin{1} + \\cos{1}(z-1)+{-\\sin{1}(z-1)^2 \\over 2!} + {-\\cos{1}(z-1)^3 \\over 3!}+\\cdots.", "a53067c2173149dc14c4b8ed11071129": "\\mathrm{T=Wb\\ m^{-2}=kg\\ A^{-1}s^{-2}}", "a530a573b38a63a3be55d6a9410920bb": "dg=\\frac{1}{2}\\,fn^2\\,dn=\\frac{f}{(\\hbar\\omega\\beta)^3}~\\frac{1}{2}~\\beta^3 E^2\\,dE", "a530c2be989d029954c7ef8ad13917f2": "F_{j}( )", "a530cfbe52fd2adee2296411476971ca": "f = f(x) = \\sum_{k=0}^\\infty a_k x^k = a_0 + a_1 x^1 + a_2 x^2 + a_3 x^3 + \\cdots", "a530d02d8c6c2390cb7bffe0a6b9f8ca": "g(x, y)", "a530f59a3ad33380b42cb12ebc082101": "k_{\\rm A}", "a530f9ea1b7cb96101a237775e4a2ce1": "\\mathrm{colim}_J\\,\\mathrm{lim}_I F(i, j) \\rightarrow \\mathrm{lim}_I\\,\\mathrm{colim}_J F(i, j).", "a53116ffd44f4bcf51aac72c34406a90": "R_{k}=kR\\left( 1-\\frac{p^{2}}{6n^{2}}\\right)", "a53129e25393ef8b1fc50a8b805b8c66": " Av = \\lambda B v.", "a5318ce8cfc2574948f4687314f373fb": "q_2, q_3, q_4", "a531f4754003e7dc7d2ef650f91b09fd": " a = \\frac {1} {2 n + 1} ", "a5322a102355c6aa0fca4cff89551a0f": "\\mathcal{E}^{n-1}", "a5324d2d9d73e4cbaa9ac03f0a196c75": "\\frac{\\omega_s-\\omega_c}{\\omega_a-\\omega_c}=-1,", "a5325065b777940378ed099c53aab3b7": "\\sum_{i=1}^n \\frac{w_i}{\\|x_i-y_j\\|} \\|x_i-y\\|^2", "a532b873c85c720ef2c0c574814faf21": "\\forall x Lxx", "a532d4b6ca2cffef4fb9df0d02bf3b0c": "{2n-1 \\choose n-1} \\equiv 1 \\pmod{n^3}", "a533343169e841426ff0851819aef24b": "\\mathrm{MSE} = E\\left[ (\\widehat{\\theta}(x) - \\theta)^2 \\right],", "a533847a06ad8fc0e67bbac83dbb750a": "A = -\\left(\\frac{\\partial G}{\\partial \\xi}\\right)_{P,T}.", "a533a339f09aa5d4f8465c2183807ae1": "\\dfrac{N V_{in}}{R_{i}} = -\\left ( \\dfrac{N_{p}V_{ref}}{R_{p}} - \\dfrac{N_{n}V_{ref}}{R_{n}} \\right )", "a533d7dcb5180377925b751f6bb3fcdd": "p \\vdash (p \\lor q)", "a5349b17d6be49e964993a0c67dcbfcb": "\\sigma^2_T", "a5353e7dbece88b2820391f19e2f9350": "(\\Omega_2,\\Sigma_2,\\mu_2)", "a5354dd7bc2d78b04d7897eca775f043": "L(\\sigma)", "a535c1ea7ab66a22114ed440a7b07ac3": "C^\\infty_0(\\Omega)", "a535cc711481e0551418ea9f30fdd2f5": "y_i \\in Y_i", "a536089d822b4a1b54be82ad9b048be4": "\\textstyle \\mathbf{b}", "a5364aa712ada91301299bcef41dc73b": "\\frac{dS}{dt} = \\sum_{k=1}^K \\dot{M}_k \\hat{S}_k + \\frac{\\dot{Q}}{T} + \\dot{S}_{gen}", "a53659aaf2c054de858463293aaa9192": "A_0 \\cong B_0 \\cong C_0 \\cong D_0", "a53666a2cdaf9c388bb9a9a55c53783e": "(\\mathbf{I})\\,\\mathrm{d}a=-\\frac{i}{\\hbar}[a,H_\\mathrm{sys}]\\mathrm{d}t+\\gamma\\left((N+1)\\mathcal{D}[c^\\dagger]a+N\\mathcal{D}[c]a\\right)\\mathrm{d}t-\\sqrt{\\gamma}\\left([a,c^\\dagger]\\mathrm{d}B(t)-\\mathrm{d}B^\\dagger(t)[a,c]\\right)\\,,", "a53668fe498637e7c7014b0d2b4ff009": " Y = \\widehat X \\beta + \\mathrm{noise}.\\,", "a5367fc390663c0e1a61b187f6a813cc": "b \\equiv a \\, \\bmod{\\mathfrak m}.", "a536aee71eb168e42130b3905a6d0a1c": "\n X_i \\cap X_j = \\varnothing, \\quad \\forall\\ i0 ", "a56378fb4a33abc41f04f995005d5ec6": "\\xi_t(x): \\mathbb{Z} \\to \\{0,1\\}", "a563a3be2469e07cbf6bb4dfd5220bc9": "\\mathbb{Z}/n\\mathbb{Z} \\rightarrow e^{2 \\pi i/n}", "a563d9e20327eaa9aa787320804d6f79": "(0,b)+(x,0),", "a56405004958201e78b10052513a4adf": "TS(T_i) = NOW()", "a564169cead87f9e46e01e493662ebd0": "P(W^+_1 \\in dy ) = y \\exp \\{ -y^2/2 \\} \\, dy, \\quad y > 0,", "a56418685e9a6e1f7c23daf14856f48d": " l^{-1} = B \\left(\\frac{h}{d}\\right)^2 \\frac{1}{d} \\left(\\frac{\\omega}{\\omega_D}\\right)^2N(\\omega) ", "a56437c7d88fb31058d114b82c7a6af9": "5 < 12", "a5646160e865586c89c64077a5fd5ab2": "A_{40}", "a5646a3abb821d8273287bc2035c9b73": "\\geq n", "a565153df227b0510b1c789d5cdc39ec": "PV", "a5652d40d2c7b2c7e1b795ac1b8bdfd9": "\\mathbf{x}_0 = \n\\begin{bmatrix}\n2 \\\\\n1 \\end{bmatrix}\n", "a56565e29ee41704733b59182c2bfe26": "f_t:U_t\\to\\mathbb C", "a5660b28f80ae0033fe118a2b443d678": "M \\underline{A}", "a56641db6c73749e9cb099251018cbce": "\\frac{y(s)}{f(s)} = \\frac{2.02(s+0.1)}{s+0.2}", "a566720ff086b22c8f0888946d4ed955": "\\ \\partial_\\mu (G \\Phi) \\neq G (\\partial_\\mu \\Phi) ", "a566800abbb7234a8d0efcbbeb1e6d3f": " \\begin{pmatrix} y_1 \\\\ y_2 \\\\ 1 \\end{pmatrix} = \\frac{1}{x_3} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_{3} \\end{pmatrix} ", "a566b76d50e3391d4b1642fad0912576": "q_{t-1}", "a566eec40af31a0dd506e2783c4d198e": "\\frac{a \\and b}{\\therefore c}", "a5671eaffb69a515fca5379f6bd935e0": "|\\mathcal Z|=n^3.", "a56722610b44de085ce9c97931e368a7": "C: y^2 + h(x) y = f(x)", "a5674c174dec1271caff1196f0f64d52": " \\text{d}_x f(x) = f(x+\\Delta x/2) - f(x-\\Delta x/2)", "a567886ae729a1e597fb041f9acea25c": "\\forall y((y>0 \\and y^2>2) \\implies y>x)", "a567b59fc2f1ce846460a8f33d315519": "\\scriptstyle A_\\infty(\\Omega)", "a567b8dc0ec7b54014257659959d4f9d": "\\frac{ab}{a+b}", "a567def81fd181dc92ccecc831e54289": "\\langle Tx \\mid y \\rangle_2 = \\langle x \\mid z \\rangle_1, \\quad (x \\in D(T))", "a56835acbcf2d4bccb9aba9328b4e949": "\\mathbf{J} (\\mathbf{k}, \\omega) = \\sigma(\\mathbf{k}, \\omega) \\; \\mathbf{E}(\\mathbf{k}, \\omega) \\,", "a568bf104397bd8311073893dff24222": "y_t", "a568c36cd6472ad5443bd5adac6aa488": "\\partial f(x)(h)=xh+hx", "a568cb90047f6aaa343cb977a518bf9e": "[H_0+\\lambda V(t)]|\\psi(t)\\rangle=i\\hbar\\frac{\\partial |\\psi(t)\\rangle}{\\partial t}", "a56902f245398b0a916df501cba5639e": "J_1(x)= {x} /{2 \\sqrt{2}} ", "a56985911f1dc5d4718733c91697c454": "\n\\begin{align}\n\\frac 1 p + \\frac 1 q + \\frac 1 r & > 1 \\text{ : Sphere} \\\\[8pt]\n\\frac 1 p + \\frac 1 q + \\frac 1 r & = 1 \\text{ : Euclidean plane} \\\\[8pt]\n\\frac 1 p + \\frac 1 q + \\frac 1 r & < 1 \\text{ : Hyperbolic plane.}\n\\end{align}\n", "a56a083df37f9982d1c51e1382bcec6a": "f(X)=X^3-aX+b", "a56a0c8a90efc6bb54a9dd5f5d1c5480": "u(x,0) = 0,", "a56a7c09d21ab24be984c86387bfcf5c": " v_b = 2.512^x = 2.512^{14.00} \\approx 400,000 ", "a56a9f9a663f271eee1fcb3da0c48783": "\\frac {ds} {d\\theta}", "a56aa620c9cc0e80c8460442bdf3d174": "A_{\\mu} = \\frac{Q}{R}k_{\\mu}", "a56abbaf16dba53b9b0115a2259e9ccf": " \\Delta^1_1 ", "a56ade5de7be947aeaa93f9be585523c": "m_\\mathrm{h} = \\frac {Nc} {f} \\,.", "a56ae83091660fc95345cffd5c44af8d": " \\vec S_m = \\{ S_m^a \\}, a = 1..A.", "a56b2d1771d72ddbeb2775b4250dd1b4": "e^{-ix} = \\cos x - i\\,\\sin x.", "a56b725030232e64c85c453fbed295c4": "y_0 = x_0 + x_1 \\, ", "a56ba40a6f58d0c740c11d572bd78f9d": "P \\vdash Q", "a56bcd157ed77cde49dea2470d3f0459": "\\frac{\\mathrm{D}\\mathbf{u}}{\\mathrm{D}t} = \\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{v}\\cdot\\nabla \\mathbf{u},", "a56bd294212dde88b576996054ad949e": "K = K_+ \\oplus K_-", "a56be101775132065049807b70c0f4bf": "X,Y\\in L", "a56bea8cb9f879aa720d18b2f846c438": "(b^2-4ac).", "a56c22f421d1f66a898bdd607a5c12d1": " (a, b \\approx a \\centerdot s) \\in R_q \\times R_q ", "a56c3c8e2c28dc6beb000c4457d4e55e": " \\overline T. ", "a56c558a5f1402333b39af3cdbe29350": "\n L^{-1}(x) = 3 x + \\tfrac{9}{5} x^3 + \\tfrac{297}{175} x^5 + \\tfrac{1539}{875} x^7 + \\dots\n ", "a56c56949f10a4e366ca7b082b76986b": "\\epsilon_d = \\frac{k_\\mathrm{cat}}{k_r} \\ll 1", "a56c9224f2d373d30fed864f56a728d4": "tv_1,\\dots,tv_n", "a56d5ec18ed499eaaf0e6ef4a5ad5a15": "G = 6.67428 \\times 10^{-11}\\,\\frac{\\mathrm{m}^3}{\\mathrm{kg}\\cdot\\mathrm{s}^2}.", "a56d671a45bd68160fa232238d0bb9fd": "B(T)", "a56d7f74c3ea809770ef21f469d421f5": "= {ac + \\varepsilon(bc - ad) \\over c^2}", "a56d8b347b871e900a02d35d60010078": "\\Sigma X \\cong S^1 \\wedge X", "a56e0c1a6a2bd85e72d62e5ef4ae1130": "F(5)", "a56e1ef2bafb746581030bcddd6acc18": " : \\hat{b}_3 \\hat{b}_2 \\, \\hat{b}_1^\\dagger : \\,= \\hat{b}_1^\\dagger \\,\\hat{b}_2 \\, \\hat{b}_3 ", "a56e20d41fb152b3837f6617b5f00a51": "A^2 = -n -\\frac{1}{n} \\sum_{i=1}^n\\left[(2i-1)\\ln\\Phi(Y_i)+(2(n-i)+1)\\ln(1-\\Phi(Y_i))\\right].", "a56e52fcfa304c343629678a3eb464e0": " W = -\\mathit{p} \\cdot \\mathrm{(V_2-V_1)}", "a56e71ebfa2fcaed33f59a4079477639": "(A.2.a)\\quad (\\psi_{,\\,\\rho})^2+(\\psi_{,\\,z})^2=-\\gamma_{,\\,\\rho\\rho}-\\gamma_{,\\,zz} ", "a56e85b0977c62f254848d15d6c9d002": " \\chi^2 = \\sum_{i=1}^{k} {(x_i - E_i)^2 \\over E_i}", "a56eb28f08d849efea44aed27cd26fdf": "\nf\n=\\frac{1}{2}R^{*}H_s\\frac{d}{ds}Rf.\n", "a56ef07ac21f3ba95b8b0daf8dedaf82": " \\mathbf{g} = \\frac{1}{c^2}\\mathbf{E}\\times\\mathbf{H} = \\frac{\\mathbf{S}}{c^2}\\,,", "a56f093e56c6e39e41afd06b8ae9b31c": "W + W_R^2 = (1,1,\\ldots)", "a56f0ea21185abe6756c691f7aad1bb7": "\\lim_{n \\to \\infty} \\left|\\frac{a_{n+1}}{a_n}\\right| = r.", "a56f63d6d5216864b4b3624804d0751d": "z=\\zeta", "a57028be8b91ca281a3bb84eae23ba74": "W = M^{-1}", "a57033e8529c4e3042982c63b119df0e": "\\chi^2_p", "a570a5ec751d73653df1524071bbb73c": "\n(\\mathbf{\\hat{f}_{0:1}})^T =\nc_1^{-1}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.5000 \\\\ 0.5000 \\end{pmatrix}=\nc_1^{-1}\\begin{pmatrix}0.4500 \\\\ 0.1000\\end{pmatrix}=\n\\begin{pmatrix}0.8182 \\\\ 0.1818 \\end{pmatrix}\n", "a570a7cf71d82ec9814d616a46058130": " \\mathbf{k} ", "a570aca40036d4388c1c1913476d5feb": "(2k)!!= \\prod_{i=1}^k (2i) = 2^k k!.", "a570cd0c5c5664e2d85582bf41446f91": " a(b-r_t)", "a5710d683bc18b8668724d6ba508036c": "O-O=0", "a57163c93bdc5b7036faaad93f4bce34": " \\sum_{n=0}^\\infty \\frac{e^n}{n!} = \\lim_{n \\to \\infty} \\left(\\frac {1+n}{n} \\right)^{n^{-n}(1+n)^{1+n}} ", "a57178492a0e0bff2dff5f61d7996e3e": "d=a+b\\times c", "a571ad94b0c5bc1b849215569a4cd4ba": "\\log_{10}(\\log_{10}(10^{10^{10}})) = 10", "a571b0087236feb4aee2df335ad3a741": "c(\\nu)", "a571bf8e0537514961e31582a4cb8be0": " \\dot{z}(t) = A\\dot{u}(t) - \\beta|\\dot{u}(t)||z(t)|^{n-1} z(t) - \\gamma\\dot{u}(t)|z(t)|^n ", "a571c27af036f52a614a7bead13a51f7": "P(X=x) = \\frac{1}{Z(\\beta)} \\exp \\left( - \\beta E(x) \\right).", "a571de7773f655b6a75ce6c9cd43fe6a": "= x+\\cfrac{2x \\cdot y} {3(2z-y)-y-\\cfrac{2\\cdot 4y^2} {9(2z-y)-\\cfrac{5\\cdot 7y^2} {15(2z-y)-\\cfrac{8\\cdot 10y^2} {21(2z-y)-\\ddots}}}}.", "a571e26542b522c14383a307dae5fe82": "p \\gg k", "a5722039b0d2991604d82f7856c15010": "([X],[Y]) \\in P^n \\times P^m", "a57246f4c980957e8f79c0ddcaff210c": "\n\\frac{1}{\\epsilon_r} = \\sum_{m=-\\infty}^{+\\infty} K_m^{\\epsilon_r} e^{-j \\vec{G}.\\vec{r}}\n", "a57276c9d921e6a2c03d9f3805ef3898": "p_{\\mathrm{sat}} = 6.1078 \\times 10^{\\frac{7.5 T}{T+237.3}} ", "a572d4679770d6141fff4208e9e5aa54": " \\left.\\frac{\\partial}{\\partial u} g(z, u)\\right|_{u=1} =\n\\left. \\frac{z+z^2+z^3+z^6}{1-z} \n\\exp\\left(uz - z + u^2 \\frac{z^2}{2} - \\frac{z^2}{2} +\nu^3 \\frac{z^3}{3} - \\frac{z^3}{3} + u^6 \\frac{z^6}{6} - \\frac{z^6}{6}\\right) \\right|_{u=1}", "a572eaee9b5251189eb8a7349f6fd8e7": "F_{H\\beta}", "a57301d0d74b7bc9caecbdf2dfab0e44": "\\left( f_{n(k)} \\right) \\subseteq (f_{n}) \\subset \\mathrm{BV}([0, T]; X)", "a573203d3f87f3610d4479b688881361": "\\mathbf{v} = \\nabla \\times \\mathbf{A}", "a573421296283b9b8a8943c63673c55b": "2r,", "a57347cb82feebf0077e0619315814ca": "\\rho^2 e_\\rho ", "a573908f3858e7c787bef83bc197a378": "\\sigma (J) = J", "a573915f4ea18f865d1e9c75a5917723": "\n\\left( \\left(1-\\frac{\\lambda}{N}\\right)\\delta_0 + \\frac{\\lambda}{N}\\delta_\\alpha\\right)^{\\boxplus N}", "a573948fd2a6e0f26d8f4df1d5cb8c86": " \\left | \\mathbf{r} \\right |\\left | \\mathbf{r}_0 \\right | \\ll \\lambda \\,\\!", "a57402444090328807a377cfe8063e2e": "\n P_{RX} = P_{TX} + G_{TX} - L_{TX} - L_{FS} - L_M + G_{RX} - L_{RX} \\,\n ", "a5742252d801110a9922341389c38eac": "{\\mathbb{C}}^2", "a5749d6f62113b572d72199af4a97082": "Y_t = \\int_0^t H\\,dX\\equiv\\int_0^t H_s\\,dX_s ,", "a5749ec33f2c95fe8c19d702d76d4968": "T_{1}", "a574bdf974faaacfe20ffb588eb6dffc": "r\\simeq (L/2\\xi)^3", "a57504a8c22e0939f14f9f50c5209323": " Z_\\beta = \\sum_\\sigma e^{-\\beta H(\\sigma)}", "a5750f59916196e8d4eb95cd6d21b017": "a_{t,j_t} \\nmid a_{k,j_t}", "a575af6c10f51a093f1ec921249c4ff2": "\\underline{\\lnot \\varphi \\vdash \\lnot \\psi}\\,\\!", "a575beefe32e1a6c9ec0dcb5a896d8db": "\\left|\\{1,1\\}\\right|=2 \\,", "a5767e67eb1f4ec9f38b09f37937cf50": "\\hat{R}_I", "a576842550cfaf326ff32f8c86d8bece": " \\int_{\\Bbb Z_p} f(x+m) \\, {\\rm d}x = \\int_{\\Bbb Z_p} f(x) \\, {\\rm d}x+ \\sum_{x=0}^{m-1} f'(x)", "a576a867f1e8ce775919b91aa6d9c2e5": " J_\\nu^{(1)}(x;q) = \\frac{(q^{\\nu+1};q)_\\infty}{(q;q)_\\infty} (x/2)^\\nu {}_2\\phi_1(0,0;q^{\\nu+1};q,-x^2/4) ", "a576b4cc025883764174408df77322ee": "\\mathcal{H} = \\sum_i {\\dot q_i} \\frac{\\partial \\mathcal{L}}{\\partial {\\dot q_i}}- \\mathcal{L} = \\sum_i {\\dot q_i} p_i - \\mathcal{L} \\,.", "a576cae719e92626d2e5427b77ab6e92": "\\displaystyle{\\mu(z)={z^2\\over \\overline{z}^2}\\overline{\\mu(\\overline{z}^{-1})}^{-1}.}", "a576d0bb9c46a162ec86864f4a37d300": "\\left(1-\\frac{1}{10^6}\\right)^{10^6}.", "a576d89e57674cfc55cf10e61926bd7d": "{\\lambda} = ", "a577405c211a4bc931d4c385bd2817cf": "\n\\frac{E(Y\\mid X)-EY}{\\sigma_y} = r\\frac{X-EX}{\\sigma_x},\n", "a5775281474de4910cd1b05b5ca01200": "\\sigma(\\omega)=\\frac{ne^2}{m^*}\\frac{\\tau^*}{1+\\omega^2\\tau^{*2}}", "a5775ba74d703fc0462bb3a70ba6364a": " u(\\vec{p}, s) = \\sqrt{E+m} \n\\begin{bmatrix} \n \\phi^{(s)}\\\\ \n \\frac{\\vec{\\sigma} \\cdot \\vec{p} }{E+m} \\phi^{(s)}\n\\end{bmatrix} \\,", "a5777b57d07587610418bde61072533b": "j(x)", "a577b96c8ac12d134697c54489b03b5f": "G_{\\text{Aff}}", "a577dd8909dfef8ca5a32dd838120414": "R_2 =(57-12) \\times 0.0362 ", "a577e35bc559814979a232f3c21f6fd1": " M \\times N ", "a5781fd4db92526a4f930087676733a3": " \\Delta S_{\\mathrm{overall}} = \\Delta S_{\\mathrm{compensated}}+\\Delta S_{\\mathrm{uncompensated}}+\\Delta S_{\\mathrm{surroundings}}=\\Delta S_{\\mathrm{uncompensated}} .", "a578905d96770d30a5040bbb9ff01cde": "x_0=0", "a578ee4fd70f3edf1489d24d47a14bd5": " X \\mapsto P X P^* ", "a578f93d86a9111f5900e4681f14ce16": "t_{i}", "a5790e9cd3ace6babfd46e8d14f56f98": "N_t = N_0 + \\int_0^t \\phi_s\\, d M_s.", "a57a35a159b9343149a23e2c5417135b": "C_r", "a57ad0460db9cb5e6f2ae4178c2f3740": "\\mathbf{E} \\cdot {\\rm d}\\mathbf{A}", "a57b38a708455c264b7609746c7ee7cd": "H^*(F)\\otimes H^*(B).", "a57b3f5812d19521f057e7c91f7c3e8d": " \\and S_5 \\implies A_5 = m ", "a57bc75771843cff3993d5ba81e3e1a1": "\n\\mathbf{F}(z,m_1, m_2) = -\\frac{3 \\mu_0 m_1 m_2}{2 \\pi z^4}\n", "a57bd7e98cd65377b59ec174a2d10ec3": "\\frac{\\partial f_{s}}{\\partial t} + \\vec{v} \\cdot \\vec{\\nabla} f_{s} + \\frac{Z_{s} e}{m_{s}} \\left( \\vec{E} + \\vec{v} \\times \\vec{B} \\right) \\cdot \\vec{\\nabla}_{v} f_{s} = \\sum_{s'} C\\left[f_{s},f_{s'}\\right]", "a57ccaf7e75c04580f34c76d73bd3671": "\\ \\mathbf x", "a57cdcf2ffa6c200fd054079044384fe": "(b,d,u)\\,\\!", "a57d32489c097dca61137462278b293c": " h(r_{12})=c(r_{12}) + \\rho \\int d \\mathbf{r}_{3} c(r_{13})h(r_{23}) \\, ", "a57d643d8d8a4aec0c9edeacd6e0498e": "\\sqrt{R^2 - r^2}", "a57d6f75e5288dcd0335c90f66fcc57c": "T_{m,n}", "a57d7d986e62a53674574ce6558749e7": " \\sup_A \\left| \\mu(A) - \\mathrm{mes}\\, A \\right| \n \\leq C \\left( \\frac{1}{n} + \\sum_{k=1}^n |\\hat{\\mu}(k)| \\right),\n", "a57d8175e80cdd9e9ad73d6140d9a1e3": "\\scriptstyle (g \\,\\circ\\, f)^{-1} \\;=\\; (f^{-1}) \\,\\circ\\, (g^{-1})", "a57dbaecb2612bf7a34769b35412848a": "\\xrightarrow{D}", "a57dcc54d055c66581dd177a7f8d1c80": "\\Delta(x)p_{1^{(n)}}", "a57f039bd1c714874434a7b5454401ae": "=\\max_{\\lambda\\in\\sigma(A)}\\frac{1}{|\\lambda -\\mu|}\\ =\\ \\frac{1}{\\min_{\\lambda\\in\\sigma(A)}|\\lambda-\\mu|}", "a57f19a2780a2b81a34f72dab727e592": " \\Delta= - \\partial_t^2 - \\coth t \\partial_t.", "a57fb8501013f0c29c542c2440c783db": " PostCaP~PRP_{PH} \\approx \\frac {PostCaP~ICO_{PH}}{ PostCaP~ICO_{all}} = 0.806 = 80.6%", "a57fe5d7d049842b427e279b02f5c420": "y_t = \\frac{x_t R}{L}", "a57fff15159032e420f95e0746938aef": "f=u d", "a58031368526035b51f51826ad7852a2": "\\displaystyle0~\\mbox{dB}", "a5803edd800b8531b64b762692dac537": " \\cos\\theta_2\\sin\\theta_3+\\sin\\theta_2\\cos\\theta_3=\\sin(\\theta_3+\\theta_2)\\times 1 \\, ", "a58085c78de05697e8d430e0b938bb8f": "\\operatorname{R}(n) = \\bigcup_{d\\,|\\,n}\\operatorname{P}(d),", "a580c52a058024c6220125ec1f9229ad": "u_i(S)=E_{ \\omega \\sim p_i}[u_i( \\omega ,s_1(\\tau_1( \\omega )),\\dotsc,s_N(\\tau_N( \\omega )))]", "a580d1156dd81055243352410511075a": "D_{IS}=-\\frac{\\mu_0\\gamma_I\\gamma_S h}{(2\\pi r_{IS})^3} BA\\!", "a580d642eb7517ad2e06daa6a2fe4cd2": "S'_{r^*}(r^*) = 0", "a580de9adf7a7263e77e31e516b798e2": "f\\left ( \\sqrt{2eV/m}\\right ) ", "a581689b491bcccad91cabf47cb1c862": "R = \\rho \\frac{\\ell}{A}. \\,\\!", "a58185e6b4d8efea054163ca1e68e7d6": "a_{1:T}", "a58189e5141ebf50ca6f992c3c4efa0f": " \\text{Gl}_2 ", "a581c5bd19a51e2df6c55db5563fcc56": "\\mathrm{Con}(\\mathcal{A})", "a581ebb9b536ffe79f900e5d78e45ce0": " \\left|{\\partial \\mathbf{x} \\over \\partial t}\\right| = \\left| \\sum_{i=1}^3 {\\partial \\mathbf{x} \\over \\partial q^i}{\\partial q^i \\over \\partial t}\\right|", "a581f38423456f6618200850be032659": "\\textstyle x\\mapsto ax^2", "a581f40559531e4429ee4480555fbee7": "\\mathcal{S}^{DIFF} (S^n) = \\Theta^n", "a581fd357881f47c513ebc9dc869387e": "f(z)=1/z^m, g(z)=z^m", "a5827c143f7d49ac84e4a10aac2b490c": "q(z)", "a582b853b1a8143cbe082dc30c5e6da1": "\\textstyle where\\ the\\ noise\\ eigenvector\\ matrix\\ E_{n}=[e_{d}+1, .... , e_{M}]", "a5835de40fc591ad28be0d589956f32b": "\\|f\\|' = \\sup \\{ |f(x)| \\,:\\, x \\in X, \\ \\|x\\| \\le 1 \\}.", "a583e9d870a55379d67c0925e0244d99": "i, j\\in A", "a5845fea6a9beac94375693268025c7f": " B^\\prime = -(n_b - n_{\\bar b})", "a5847393bb47d4f15323020f8c9a9219": "a,b=-\\frac{1}{2},\\frac{1}{2}", "a5850ab09afe550a2e2ffbfd152e0eca": "\\theta_0=\\frac{1}{6\\pi\\epsilon_0}\\frac{e^2}{m_0c^3}\\ ", "a58568eb0aeaf74d156e061796061281": "\\Delta\\nu=1/T", "a585e3fd1d3e7b93826fb33a1fb2d86a": "T\\subseteq \\mathbb{R}^{K}", "a585f49c6e231ddceb4736f60b8ec3d1": "T(A)\\,|\\!\\!\\!\\sim_{Grz}T(B).", "a58616fd9acb6777f5eabe7c7680b989": "p_{i}<\\frac{\\alpha}{m}", "a5861b0b78de969287d2289803c294a5": "\\begin{align}\n\\det(\\mathcal{I}(\\alpha, \\beta))&={\\mathcal{I}}_{\\alpha, \\alpha}{\\mathcal{I}}_{\\beta, \\beta}-{\\mathcal{I}}_{\\alpha, \\beta}{\\mathcal{I}}_{\\alpha, \\beta}\\\\\n&=(\\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta))(\\psi_1(\\beta) - \\psi_1(\\alpha + \\beta))-( -\\psi_1(\\alpha+\\beta))( -\\psi_1(\\alpha+\\beta))\\\\\n&= \\psi_1(\\alpha)\\psi_1(\\beta)-( \\psi_1(\\alpha)+\\psi_1(\\beta))\\psi_1(\\alpha + \\beta)\\\\\n\\lim_{\\alpha\\to 0} \\det(\\mathcal{I}(\\alpha, \\beta)) &=\\lim_{\\beta \\to 0} \\det(\\mathcal{I}(\\alpha, \\beta)) = \\infty\\\\\n\\lim_{\\alpha\\to \\infty} \\det(\\mathcal{I}(\\alpha, \\beta)) &=\\lim_{\\beta \\to \\infty} \\det(\\mathcal{I}(\\alpha, \\beta)) = 0\n\\end{align}", "a5862950c7a8b2dd9da8dffc0256903c": "\\nu(A)=\\int_A f_n\\,d\\mu", "a58636582972b3769715a081b07c285b": "\\neg \\neg \\psi\\,", "a5864f1a245c820e0b97673736f13628": "b_n = nb_{n-1}", "a586757141e7e91030618f240fd8afd3": " u^*_{i + \\frac{1}{2}} ", "a5872ceaf50c691cdd3638d670160695": "I_n = \\int \\sin^n{ax} dx\\,\\!", "a58786cdf02fa981c785480b1145ff29": "t=\\pi", "a5879dab3f9d9e365969109950afc272": "c_{11}", "a587dd9d8bd13bb133fa4e7e97efe27f": "\\vec{t_2}\\langle s''\\rangle=\\vec{t_2}\\langle s'\\rangle", "a588c3de6be9a5d06734bf11a3d27456": " X=\\{\\mathbf{x}_1,\\mathbf{x}_2,...,\\mathbf{x}_m\\}", "a588cbba9250f43512d6ae25be83b02f": "H^1", "a588d0414c40f9de665d6d1a67bc5a1f": "R[x \\leftarrow E]", "a588ec722602d56f1036d91f0e60fbfc": " V = \\int_0^H \\left[R \\frac{h}{H}\\right]^2 \\pi \\, dh ", "a588fd68cb0be14c29c19a1077ef0b99": "RM = GZ\\cdot\\Delta", "a5893fd35d613ee50fc17749fe02252d": "\n \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\tan\\theta\n = \\lim_{\\delta \\to 0} \\left[ \\frac{\\frac{\\tan\\theta + \\tan\\delta}{1 - \\tan\\theta\\tan\\delta} - \\tan\\theta}{\\delta} \\right]\n = \\lim_{\\delta \\to 0} \\left[ \\frac{\\tan\\theta + \\tan\\delta - \\tan\\theta + \\tan^2\\theta\\tan\\delta}{\\delta \\left( 1 - \\tan\\theta\\tan\\delta \\right)} \\right] .\n", "a5898b6db4d5d9bb1da5bdea2ca083d5": "\\operatorname{dCov}^2(X, Y) := \\operatorname{E}\\big[d_\\mu(X,X')d_\\nu(Y,Y')\\big].", "a58a4f79c97d9db5890579d154549f71": " p_4=a_{20},", "a58a9ee85bbd5228a82741046ec1c681": " 1 - \\varepsilon ", "a58aa2b695940d746bdef68457ac8e91": "\\tau_l=\\frac{B \\left ( l^- \\rarr e^- + \\bar{\\nu_e} +\\nu_l \\right )}{\\Gamma \\left ( l^- \\rarr e^- + \\bar{\\nu_e} +\\nu_l \\right )},", "a58aab0d9b354dea89304de58c6203aa": "\\lim_{x \\to+\\infty} \\log_a x= +\\infty \\quad \\mbox{if } a > 1", "a58aeda67132ff2b2f87bb022b089c93": "\\log \\frac{x}{m} = \\log K + \\frac{1}{n} \\log c ", "a58b1282c16ec73e01d87a0ed4a0dafb": "c = 2a\\,", "a58b24f379b4c028f417c8f77ba0150d": "\\frac{\\partial {\\rm tr}(\\mathbf{AXBX^{\\rm T}C})}{\\partial \\mathbf{X}} = ", "a58b3360897a36e7d70a84bae54de4d1": "\\mathit{MSE} = \\frac{1}{m\\,n}\\sum_{i=0}^{m-1}\\sum_{j=0}^{n-1} [I(i,j) - K(i,j)]^2", "a58c5972fa98d9d5b358bb595e8d0d48": "\\circ_1", "a58c854e20c794525063bc5aab1393fc": " P_n^{(\\alpha,\\beta)}(\\cos \\theta) = n^{-\\frac{1}{2}}k(\\theta)\\cos (N\\theta + \\gamma) + O(n^{-\\frac{3}{2}})~,", "a58c8cb4008f5cdc05d8967163d04493": "\\langle u_n, n=1,2,\\ldots \\rangle", "a58c97b277ddbfc4667a70c92436e482": "s(x)=\\frac{\\ln(1-(1-p)e^{-\\beta x})}{\\ln p},", "a58d12a3e9b83cf1006419393536b8fa": "\\lambda_1,\\dots,\\lambda_k", "a58d24e8e6e5153de87ab4b134a10528": " \\psi (x,t) = A e^{i \\left( kx - \\omega t \\right)} \\ , ", "a58d3006d7049d49c8d8f6f237a42df6": " \\Omega \\left( \\big[ \\rho (R) \\big] ; \\mu , T \\right) =A \\left( \\big[ \\rho (R) \\big] ; T \\right)- \\int d^3R \\big[ \\mu - V(R) \\big] \\rho (R),", "a58def526faea587be365851935e7be5": " s_0=\\frac{P_a}{P_a+P_d} ", "a58e343d1ef2e27daef7094ec19db267": "\\mathit{E}_\\mathit{G}", "a58ecd5552f76a5614bf6288b339da68": "_P^{(0)} = 2 ", "a58ee9bd4d3ecc72e4d95fccadb41cd2": "\\alpha'(0):=v\\ ", "a58f1225671e973ac7d0f0c8f3fcff4c": "k(b-1)^k\\, ,", "a58f416cb466a45c2e94fd8be8f31807": "t_{LL}^{\\mu \\nu} ", "a58f8a2e9ace3150a44dae629915dd33": "\\lang B,--,(-),() \\rang", "a58f90d1004c4934b2a03209e73f43f2": "\\frac{\\partial s^{\\ast }(p)}{\\partial p}.", "a58f9879eb35ce61c21c2fd3fb25b361": "\n\\Gamma_p(n/2)=\n\\pi^{p(p-1)/4}\\Pi_{j=1}^p\n\\Gamma\\left[ n/2+(1-j)/2\\right].\n", "a58f9e4b3445deedaef0efbcb6a89c5d": "rx \\ne 0", "a58fa0c558c329bcf4822751edf397f6": "\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{accumulation of}\\\\\n\\text{unconsolidated deposit}\n\\end{array} \\right]=\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{deposition}\n\\end{array} \\right] -\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{re-entrainment of}\\\\\n\\text{unconsolidated deposit}\n\\end{array} \\right] -\n\\left[\\begin{array}{c}\n\\text{rate of}\\\\\n\\text{consolidation of}\\\\\n\\text{unconsolidated deposit}\n\\end{array} \\right]\n", "a58fa1aed593625bc74d577577de26cf": "\\langle \\eta(\\vec x,t) \\eta(\\vec x',t') \\rangle = 2D\\delta^d(\\vec x-\\vec x')\\delta(t-t')", "a58fa9bd5f3ef1441fa519dabe6fd3cb": "\\boldsymbol{v}_i", "a58ff2886054dd75fd882a3e1e8313e6": "\\widehat{\\theta} = \\widehat{\\theta}(x)", "a59028635a7eac335567b299c624582d": "q = (266^{17}", "a590468af82457e21ed0a984964c7c74": "\\mathbf{r}-\\mathbf{r'}", "a590492384dce5061d181304c9a3ad45": "ax+by+cz+d = \\begin{cases} > 0 & \\text{front facing} \\\\ = 0 & \\text{parallel} \\\\ < 0 & \\text{back facing} \\end{cases} ", "a590a8e2881cf21de09ec170d408d203": " H(x) = -\\frac{d \\log[f(x)]}{dx} ", "a5912e90d9eb7a991c746e89a46d99c5": "x = \\frac{\\pi}{2} - t", "a5915f5b10b86e43830bf8cd084096a5": " 1 = \\frac12 + \\frac13 + \\frac1{3^2} + \\dots + \\frac1{3^{n-2}} + \\frac1{2 \\cdot 3^{n-2}}", "a59166b742d507cb026a1524b4121ee7": "|x(t)| \\le ce^{-\\delta t}|x(0)|,\\ \\forall t \\ge 0.", "a59166fc0d160e0411efe90b23b4489b": "\\mathcal{T}\\{[((\\partial^\\mu \\partial_\\mu+m^2)\\phi)[f],\\phi[g]]\\}=-i\\int d^dx f(x)g(x)", "a5917bafd76db33a70b77f13e717cdf5": "\\ t_\\text{P}", "a5918ec0866ad01c93619064ec449b24": "\\displaystyle s=\\frac{3\\sqrt{3}}{2}R", "a5919e5e4f63ee2ee94391f79b401058": "f(\\Theta) = \\frac{2}{\\pi}\\cos^2\\Theta, \\qquad -\\pi/2 \\le \\Theta \\le \\pi/2", "a591c62e816c767c9d73859bfbbadbf7": "u, v\\in\\mathbb{R},", "a592385d237f5f07c03b3e418ca8bcb7": "-2=c^2,", "a5929f166d918344face88b0f0a90a63": "b \\equiv a^{r/2} \\pmod{N}", "a592b33fafe88f36a146060a8450adf7": "\\left\\{B_i\\right\\}_{i=1}^{n}", "a5933f552abd91912a11c1395359cb4b": " (i_x, j_x) \\cdot (i_y, j_y) \\cdot (i_x, j_x) = (i_x, j_x)", "a59345753fd8cce9664d7b3518e251b9": "\nI_{k} = \\left( \\frac{1}{k} \\right) \n\\int d\\theta^{\\prime}\n\\int d\\rho^{\\prime}\n\\left[ \\frac{\\cos k\\theta^{\\prime}}{\\left(\\rho^{\\prime}\\right)^{k-1}} \\right]\n\\lambda(\\rho^{\\prime}, \\theta^{\\prime}) \n", "a59398b8026a12ae7236fbfdb1d7358f": "\\theta = \\frac{1}{2} \\arcsin\\left(\\frac{2}{3}\\right) \\approx 20.9^\\circ.", "a5940aeddfbcb4657f7d9ef2d09f61b1": " U = \\int_{0}^{L-L_o}{k\\ x\\ dx} = \\tfrac{1}{2} k (L-L_o)^2 ", "a59467f44f12e9166fe4fb400579b3b4": "\\mathbb{Z}_{2}", "a5950fceb08ea0af597ad30d0986f10a": "y_i=h(x)+s_{i+93}+b_{i+2}+b_{i+12}+b_{i+36}+b_{i+45}+b_{i+64}+b_{i+73}+b_{i+89}", "a5957b62c5442f8a2cdef351435f7f5d": "x^{y^z}=x^{(y^z)}.\\,", "a5958e3e066d872ee9e26b2ef8c11bcd": "\\{y_i\\}", "a595e42f8c191b8470c7270cd95f032b": "\n u_\\epsilon (x) \\xrightarrow[\\epsilon \\rightarrow 0]{ } u(x), x \\in D \\,\n", "a596030e368bd7574cdd1673e8153766": "ax^2 + bx + c.", "a59632375e48ab854b35a2d49b246f00": "e_m(P,Q) = 1", "a596441e88a33fa46ade406e00754dc3": "\\|f(x)\\|= \\left\\|\\sum^n_{i=1}x_if(e_i)\\right\\| \\le \\sum^n_{i=1} |x_i|\\|f(e_i)\\|.", "a596d93a6026ab16ef04168a01e72256": " 0\\le i < j \\le n. \\, ", "a596de6dd2a91db56da4a75abe2726d9": "s_r(x)", "a5975b642056ea89e302ddc395059031": "x_t = Ax_{t-1} + Bx_{t-2}", "a597795a36823ea16d62e041dccd4d63": "P(S^{t+1} \\mid h^t) = \\mathbb{E}_{S^t \\mid h^t} [P(S^{t+1} \\mid S^t)] ", "a597d95085846edd55fde00b5ed29520": "\\mathit{w_{max}} = (\\mathit{n} - \\lceil d_{min}/2 \\rceil )/2", "a597f9ab15e348cbc2d604e2e0f1553c": " G = 2 \\cdot N \\cdot MI(row,col) \\, ,", "a5987504bbd6a1c16c4ca7ccf4c28beb": "b=f(z_0)", "a598fafa13493e7ef721b5b888fa2950": "f(x_1,x_2,\\ldots,x_n):\\{0,1\\}^n \\to \\{0,1\\}", "a5991f59dffa468aa100e771f406ef53": "2^{|k_{f_1}|}+2^{|k_{b_{n+1}}|}+2^{|s_1|}", "a5992527220dd94b222a6ce87be56baa": "(\\sin(\\alpha))^{-1}", "a5993b8d9447af6eefa12dccd739f668": " \\langle \\cdot, \\cdot \\rangle : E \\times E \\rightarrow A ", "a5994d6cb0c9395ca2e0b6ed186ad8e7": "f \\colon Y \\to X", "a599626d5261b9bc8d0a3416f0f15492": "\\operatorname{span} \\left \\lbrace \\varphi(x_1), ..., \\varphi(x_n) \\right \\rbrace", "a5996d75e6b5a6121486d81929dc1688": "V= \\frac{12\\xi^2(3\\tau+1)-\\xi(36\\tau+7)-(53\\tau+6)}{6\\sqrt{3-\\xi^2}^3} \\approx 37.61664996273336", "a599d64ba7ff27892b57b4e2a30b6b86": " \\sigma_m ", "a599e0df3e531b73ef93c570b25924f5": "\\nabla\\cdot\\hat{\\mathbf{B}} = 0\\quad\\Rightarrow \\quad\\frac{\\mathrm{d}B^r}{\\mathrm{d}r}+\\frac{2}{r}B^r - \\frac{l(l+1)}{r}B^{(1)}=0", "a599f5fe0ec2014228b548291962279c": "\\hat P", "a59a013c584da0617752812c0c0cbd83": "O(n \\log^2 |G| + tn) ", "a59a0342ffbb660676a4c576381cd634": "\\scriptstyle \\left(-\\frac12f_\\mathrm{s},\\frac12f_\\mathrm{s}\\right),", "a59a1758847e33ee0f4161110292a3a2": " i^* = Select(IMM(s)).", "a59a591a300a53e448f85338d65971d5": "F_H = p_cA", "a59ac68d799d2cbdd3df7d23256fa4b3": "\\Delta t \\, \\Delta f = \\frac{1}{N},", "a59afb3868a54fe8919f95fcd8b9509e": "y_j(t) = f[\\mathbf{w}(t)\\cdot\\mathbf{x}_j] = f[w_0(t) + w_1(t)x_{j,1} + w_2(t)x_{j,2} + \\dotsb + w_n(t)x_{j,n}]", "a59afbc1d249a6f9191cf117ba570cab": " -2 \\le S_{CHSH} \\le 2", "a59b0e2f5092447587e69fb005b57983": "\\scriptstyle\\tau", "a59b7e9f0daa78b595f0989d8260472c": "|\\psi_I(t)\\rangle=\\left[1-\\frac{i\\lambda}{\\hbar}\\sum_m\\sum_n\\int_{t_0}^t dt_1\\langle m|V(t_1)| n\\rangle e^{-\\frac{i}{\\hbar}(E_n-E_m)(t_1-t_0)}|m\\rangle\\langle n|+\\ldots\\right]|\\psi(t_0)\\rangle.", "a59b8a15c4329382643f06ef08f01884": "X^n", "a59bb772ee58295738a8d2f79de8c0e1": "\\widehat H=\\widehat{OS}p(n|m;\\Lambda) ", "a59bff3a64a57500d2e0202e085868a6": "f:\\,\\{1,\\ldots,n\\}\\rightarrow\\{1,\\ldots,n\\}", "a59c3cfd0961f47366cb044d927af31b": "\\frac{x+y}{x-y} = \\frac{x+y}{x+(-y)}", "a59c50d51c3687bc0538fc11d2b4b746": " \\left[\\int_{S_2}\\left|\\int_{S_1}F(x,y)\\,d\\mu_1(x)\\right|^pd\\mu_2(y)\\right]^{1/p} \\le \\int_{S_1}\\left(\\int_{S_2}|F(x,y)|^p\\,d\\mu_2(y)\\right)^{1/p}d\\mu_1(x),", "a59cb2174cd7787335db9e19fd59b29f": "\\ lift >> drag ", "a59cf6967012f8fae76e152dbd556bbb": "U_\\text{C} = -\\dfrac{e^2}{4\\pi\\varepsilon_0r}", "a59d2906e4b3d9d89fe6c758b593eab1": "\\mathbf x + \\mathbf y = (x_1 + y_1, x_2 + y_2, \\ldots, x_n + y_n)", "a59d35d0332ceb12907765e69354e3fb": "c_0 > c_1", "a59d6560c339a88cd8a7884e78adf925": "\\textrm{Br}(K) \\cong H^2(\\textrm{Gal} (K^s/K), {K^s}^*).", "a59d8554ea456223f4e739f6b1d4daac": "(k+1-1)^2 A_k - A_{k-1} =k^2A_k-A_{k-1}=0", "a59e0d0fe089261db35d1277f48dfb5a": " \\oint_C f(z)\\,dz = \\oint_C {1 \\over (z^2+1)^2}\\,dz = 2 \\pi i \\,\\mathrm{Res}_{z=i} f = 2 \\pi i (-i/4)={\\pi\\over 2}\\quad\\square", "a59e139a490ae25ced1519a8a4af1661": "f_0(3) - 2", "a59e392981962df7a6a7aec1645f12b8": "\n2\\pi \\frac{27}{8}\\ \\sin^2 i\\ e_h\\ -\\ 2\\pi \\frac{3}{2}\\ e_h\\ -\\ 2\\pi \\frac{3}{2}\\ \\sin^2 i\\ e_h\\ +\\ 2\\pi \\frac{3}{8} \\sin^2 i \\ e_h\n\\ =\\ 2\\pi\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ e_h\n", "a59e8053570943373a5fae9aa43e0069": "\\partial/{\\partial t}", "a59e852504fdb1a2aaab9408118ae64b": "f\\in C_2(\\mathbb{R}),", "a59f12e672310643bf0576d660b4a21a": " \\sum_{i=1}^\\infty x_i\\,p_i = c\\,\\bigg( 1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dotsb \\bigg)", "a59f4c009885bf090892fc58f6f7ce50": "c_{(1)}\\otimes c_{(2)}=c_{(2)}\\otimes c_{(1)}\\;", "a59f7d99b40296819bda3335d5b66d13": "\\lambda=\\lim_{t\\to\\infty}{1\\over t} \\log{\\|C(x,t)u\\| \\over \\|u\\|}", "a59f8afbb03e782dc213bbb8b743348a": "J = CX", "a59f9c03407cf39ca2dc9bff9d1935ef": "\\Gamma (E)", "a59fb0b6c409ad28c113ac4e5d1d1875": "y_3 = \\frac{y_1^2-x_1^2}{1-dx_1^2y_1^2} = -1", "a59fb16a18b804f981fd7aa9994bb790": "\\frac{\\partial L}{\\partial t} + \\nabla \\cdot (L\\mathbf{v}) + Q = 0.", "a59fc12a134820917ee79e716a865375": "(a, f(a))", "a59ffd9037f626bfa08f5c591f993f26": "N(r)=\\pi r^2 +E(r)\\,", "a5a0020493eb67dd2bb2b1dfce5243ec": " e_1 = (1,0,0,0,0)^T,\ne_2 = (0,1,0,0,0)^T,\ne_3 = (0,0,1,0,0)^T,\ne_4 = (0,0,0,1,0)^T,\ne_5 = (0,0,0,0,1)^T", "a5a0406f40aa56d878c3c55c0f019d58": "g\\circ f", "a5a0b34db688c1ce9e3034271ae7b4f7": "\\hat{\\boldsymbol\\theta} =\\frac{\\partial^2 {\\mathbf{r}}}{\\partial r \\, \\partial \\theta}= (-\\sin\\theta\\ ,\\cos\\theta) \\ .", "a5a0c61fff4070c1439a1a69f7b97756": "t(\\delta_{x})=(\\frac{n}{n-1})\\delta_{x}", "a5a1394f9c54c88b481a27850833e620": "{{D_g \\zeta_g \\over Dt} = f_o ({{\\partial u_a \\over \\partial x}+{\\partial v_a \\over \\partial y}}) - \\beta v_g }", "a5a17ce45f65044e381ceceb8c735e78": "P' = Q_L\\cdot P\\cdot Q_R.\\, ", "a5a19c30f6832813fc9b92d3c19b75b0": "R_{\\mu\\nu}", "a5a1c189e6999f1e31e89a103d765b3d": "\\{ e^i \\}", "a5a1ccb3ce2a72660a417b7f1f779d2c": "O(1.9999^n)", "a5a1fcc95e3ace7be4958030a6cdd425": "p\\in[1,\\infty)", "a5a27310dd0392d605dba6e5f5870692": "D=x^{(0,1)}", "a5a2929bc80363d4ffe77ca4cf1911b8": "A=1", "a5a2b661f2677d083272bb14f868e144": "N(E)= \\begin{cases} {E^\\alpha \\exp \\left( { - \\frac{E}{{E_0 }}} \\right)}, & \\mbox{if }E \\le (\\alpha - \\beta) E_0\\mbox{ } \\\\ {\\left[{\\left( {\\alpha - \\beta } \\right)E_0 } \\right]^{\\left( {\\alpha - \\beta } \\right)} E^\\beta \\exp \\left( {\\beta - \\alpha } \\right)}, & \\mbox{if }E > (\\alpha - \\beta) E_0\\mbox{ } \\end{cases}", "a5a2be926e06b137fcf77797691233ad": "\n\\begin{align}\n\\mu_{X1} \\ge \\mu_{X2} \\\\\n\\mu_{X1}^2 \\le \\mu_{X2}\n\\end{align}\n\\qquad\n\\begin{align}\n\\mu_{Y1} \\ge \\mu_{Y2} \\\\\n\\mu_{Y1}^2 \\le \\mu_{Y2}\n\\end{align}\n", "a5a2beb888f693c1815347039aff2c7c": " \\ln \\Gamma\\left(\\frac{\\nu}{2}\\right)-\\frac{\\nu}{2}\\ln\\frac{\\nu\\sigma^2}{2}", "a5a40ae4133ad186fcdea32fb536a559": " \\max_{p(x_1)p(\\hat y_1 | y_1)p(x_2)} I\\left( x_1; \\hat{y_1}, y | x_2 \\right)", "a5a462cc4a0c37b92922e1f6851d0ca5": "\\frac{d^2 \\theta}{d t^2} + \\sin(\\theta) = 0\\,", "a5a477cdaa376009041a127789e0912d": " SU(5) \\supset SU(3)\\times SU(2)\\times U(1)", "a5a48e607388c898f46b463a16d0de24": "\nL_{2} = m r^{2} \\omega_{2} = m r^{2} k \\omega_{1} = k L_{1} \\,\\!\n", "a5a4ee818515fa2520f3280a9d37af61": "\\mathbf{a} = \\mathbf{c} \\times \\mathbf{d}", "a5a4f076051fc257651c34168c00fd4b": "~\\hat a^\\dagger = X-iP~", "a5a4f5a990951be36647a60053668dfd": "{\\Theta}=\\cfrac{1}{2K}p", "a5a55ff34cbf997718a182960602a720": "ad_{\\mathcal{O}_{[g]}}\\mathcal{O}_{[h]}=[\\mathcal{O}_{[g]},\\mathcal{O}_{[h]}]", "a5a5d1a410662c08c40ddd984b0f80ca": "\\leq_p", "a5a600103580ef6d045dc387d5bf74dc": "-a_{m}\\le S_m - S_n ", "a5a600974ce49eaa2433c661912d9d82": "\\ +bKK*ln(K)*ln(K)+bLK*ln(L)*ln(K)", "a5a6093339c7550fd5cd62fbeef1cb74": "(3)\\qquad \\dot{m} = \\rho_1\\;Q = C\\;Y\\;A_2\\;\\sqrt{2\\;\\rho_1\\;(P_1-P_2)}", "a5a62901c6c89d88513961cb39584cc4": "C_{V,m}=\\frac{C_V}{n}=\\frac{3}{2}R", "a5a655864e5a65ceef1587298623fe26": "\\gamma _p < 0 \\ ", "a5a68d784c756e081b1356cb6014d99c": " p_{2,2}(x) = y_2 \\, ", "a5a6b25ef87bcab7bf62fcb7a860294a": "\\Box\\Box A\\to\\Box A", "a5a6f8a01c13d065e47eae300df0f38f": "\\lambda=\\sqrt{\\sum_1^k \\left(\\frac{\\mu_i}{\\sigma_i}\\right)^2}", "a5a741172a637ec222154ebe5d6d1db1": "F_{ab}=F_{\\bar{a}\\bar{b}}=0.", "a5a7bc8de00f9c246d3d100ee76793bb": " \n\\begin{align}\nx_1^2 x_2 x_3 + x_1^2 x_2 x_4 + x_1^2 x_3 x_4 + x_2^2 x_3 x_4\n+ x_1 x_2^2 x_3 + x_1 x_2^2 x_4 + x_1 x_3^2 x_4 + x_2 x_3^2 x_4 \\\\\n{} + x_1 x_2 x_3^2 + x_1 x_2 x_4^2 + x_1 x_3 x_4^2 + x_2 x_3 x_4^2. \\,\n\\end{align}\n", "a5a7dabefcb396633ce914760afcff77": "\\xi (m) \\Delta m= 0.158 (1/m) \\exp[- (\\log(m)-\\log(0.08))^2/(2 \\times 0.69^2)]", "a5a7e87c2d9dd29eab4ae1c2b4655514": "x\\mapsto \\phi(x) = \\begin{bmatrix}a + b & \\mathbf a + \\mathbf b \\\\ -\\mathbf a + \\mathbf b & a - b\\end{bmatrix}.", "a5a7f33a3ea5c67b73ab39613c3bad8c": "(a - b)(c - d)=ac + bd - ad - bc", "a5a8461e6ea7c18903963649ba50ebd4": "{{P \\over Q} + 1} = {{X + Y + \\sigma (100 + \\alpha + \\beta)} \\over {X + \\sigma (100 - l_2 + \\beta)}}", "a5a8a281420e861644f9b7809345f8e2": "* \\to BQC", "a5a97701d886f6e71c896861e3f8945f": "\\textstyle \\gamma:=[-\\beta,\\beta]", "a5a985e4efed031ea0f08e7c6c130003": "d^* \\in \\mathbb{R}", "a5a99b14374cad7b452d553315e442db": "\n \\Pr(0.45 < r <0.55)\n = \\int_{0.45}^{0.55} f(r | H=7, T=3) \\,dr\n \\approx 13\\%\n \\!", "a5a9a1ab8a11dce9346b263ad9a6f4d4": "\\frac{\\partial x}{\\partial u} = - \\frac{\\left(\\frac{\\partial\\left(F, G\\right)}{\\partial\\left(u, y\\right)}\\right)}{\\left(\\frac{\\partial\\left(F, G\\right)}{\\partial\\left(x, y\\right)}\\right)}.", "a5a9b7eb31d29b76b9351919882b5ec3": " = \\frac{1}{2b}\n \\left\\{\\begin{matrix}\n \\exp \\left( -\\frac{\\mu-x}{b} \\right) & \\mbox{if }x < \\mu\n \\\\[8pt]\n \\exp \\left( -\\frac{x-\\mu}{b} \\right) & \\mbox{if }x \\geq \\mu\n \\end{matrix}\\right.\n ", "a5a9e6087b647fd6f309b4e9843b8f60": "\\sqrt{2} = \\sum_{k=0}^\\infty \\frac{(2k+1)!}{(k!)^2 2^{3k+1}} = \\frac{1}{2} +\\frac{3}{8} +\n\\frac{15}{64} + \\frac{35}{256} + \\frac{315}{4096} + \\frac{693}{16384} + \\cdots.", "a5a9fbd3e98c9a124670a2a799a9fe57": "H=\\sum\\limits_{A=1}^N(a_A^+\\cdot a_A)\\hbar \\omega\n-{e\\over{{\\sqrt{2}}\\beta}}(a_A+{a_A}^+)\\cdot E(r_A)+\\sum_\\lambda\\sum_k\na_{\\lambda k}^+a_{\\lambda k}\\hbar c k", "a5aa3c4308f9098f375c25a31bb71168": "V_{avg} = (V_1+V_2)/{2}", "a5aabbcf4a31560256033bedc3551625": "p=p_d+\\cdots+p_0", "a5aad24c448e80504ed4b4c8bec5c4db": "2xB_{n-1}^\\lambda(x;k) = B_{n}^\\lambda(x;k) + B_{n-2}^\\lambda(x;k)", "a5ab6c750d09c6c6884c92c3661f5822": "T = 2\\pi\\sqrt{\\frac{mL^2}{2\\kappa}}\\,", "a5ab76c8f6cc764842da82ae74ec72e7": "dS =\\frac{\\delta Q}{T}", "a5ab84aefc9e75bd825ed3e750f84994": "40 x 55 x 40 cm^{3}", "a5abc65a50df5124c262ad372678574b": " (abcd)' = a'b'c'd' = (-mam^{-1})(-mbm^{-1})(-mcm^{-1})(-mdm^{-1})\n= mabcdm^{-1} .\\,", "a5ac1a64d7eea95da2bb0a7f8a231fe7": "\\epsilon L_n=\\alpha", "a5ac1f719ec93ad43008022bd060ba20": "t_n=n\\,\\Delta t", "a5ac41ec27225e5c4519ac0c78db4b03": "n=\\sum_{i=0}^ka_ib^i", "a5ac4d5ce04db706f8d526b778f3cdf0": "r_s(t)=\\sqrt{(x-x_s(t))^2+y^2+z^2},", "a5ac4f6bddfe24190295cd971c8d6b40": "\n|\\langle f|g\\rangle|^{2}=\\Big(\\frac{1}{2}\\langle\\{\\hat{A},\\hat{B}\\}\\rangle - \\langle \\hat{A} \\rangle\\langle \\hat{B}\\rangle\\Big)^{2}+ \\Big(\\frac{1}{2i}\\langle[\\hat{A},\\hat{B}]\\rangle\\Big)^{2}\\, .\n", "a5ac5a45c31714394b52ee2b25fe5870": "\\operatorname{score}(\\mathbf{X}_i,k) = \\boldsymbol\\beta_k \\cdot \\mathbf{X}_i,", "a5ac671b44af6e9f505e50a624d5e408": "\\dot{\\theta}(t) = - k_g(t)", "a5acadeaafce666cd372263f2723fbdd": "k_1, k_2, \\dots, k_n", "a5acda913755e23332dc64b89deb1fcd": "\\alpha = (\\alpha_1,\\alpha_2,\\ldots)", "a5ad4c07e5203cb16fac75f7067357aa": "(Z_{i,j})", "a5ad7b3f8d6e48cbfec0b5e07875a60f": " \\frac{\\omega^4}{c^4} - \\frac{\\omega^2}{c^2}\\left(\\frac{k_x^2+k_y^2}{n_z^2}+\\frac{k_x^2+k_z^2}{n_y^2}+\\frac{k_y^2+k_z^2}{n_x^2}\\right) + ", "a5ad84c47f13993da095402c3a150857": "F(x) = \\frac12 + \\frac{1}{\\pi} \\arctan\\!\\left[\\operatorname{sinh}\\!\\left(\\frac{\\pi}{2}\\,x\\right)\\right]\n\\! ,", "a5ad94fe1ab79084e6afd79e0e811719": "\n \\frac{d}{dz}\\begin{bmatrix} V \\\\ T \\end{bmatrix} = \n \\begin{bmatrix} 0 & 1/\\mu(z) \\\\ k^2\\,\\mu(z) - \\omega^2\\,\\rho(z) & 0 \\end{bmatrix} \n \\begin{bmatrix} V \\\\ T \\end{bmatrix} \\,.\n ", "a5adbad9e9d24d0c2b28fe3c4f258c6b": "\\ P_{ij\\ldots}", "a5adcb101d47acb8c6543a495b2c968a": "\\alpha/(1-\\beta)", "a5addd73c744e109f74fe1cd261a1a6f": "(w,\\ k+\\alpha(e))", "a5ae24052a91ecfcae4eb29d5e80fd80": "\\beta\\leq\\beta'<\\alpha", "a5ae52c5d9781521f2849ddc0e48ac05": " f^*(-x) ", "a5ae79e9dcc2599bc6b7acf96867f942": "\\nu > 2", "a5ae7ebf475039749a1a51e1efe0a8ac": " \\psi_n ", "a5aebc51ac86862741133d8438597b5b": " F_y(t) = Y cos(\\omega t) ", "a5aefe583f2fa0df565c11b023d254ec": "\n\\mathbf{P}(\\tau,\\mu | \\mathbf{X}) = \\text{NormalGamma}\\left( \\frac{n_\\mu \\mu_0 + n \\mu}{n_\\mu +n}, n_\\mu +n ,\\frac{1}{2}(n_\\tau+n), \\frac{1}{2}\\left(\\frac{n_\\tau}{\\tau_0} + n s + \\frac{n_\\mu n (\\mu-\\mu_0)^2}{n_\\mu+n}\\right) \\right)\n", "a5af0cdbbec11d1d24c497f85d9b1661": "D \\subset R^n", "a5af21c4c6b8510d78b58cd4e033ad76": "\\sigma_3 = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}", "a5af2c31d540ebe997ee05ac0eaf815c": "|\\pi(x)-{\\rm li}(x)|\\le\\frac{\\sqrt x\\,\\ln x}{8\\pi}", "a5af4eb52773125414520234f7f6ab1c": " d: K^{i,\\cdot} \\to K^{i+1,\\cdot} ", "a5af6ffb557bf466b383e814dd1e411f": "h\\ ", "a5afb2ee3303594c54715dfe1fc569a6": "\\mathrm{so}(3,1) \\,", "a5afb4e00a331c2c3e4148bb4a48519a": "P_\\text{avg} = \\frac{1}{t_2 - t_1}\\int_{t_1}^{t_2} V(t)I(t)\\,\\operatorname{d}t", "a5afd6721a970ee3a3910a7aebfc001a": "g(\\vec{x},\\vec{e}_i,t) = \\frac{(\\vec{e}-\\vec{u})^2}{2}f(\\vec{x},\\vec{e}_i,t)", "a5afddfae10f20382e2a1c711d8e1312": " \\frac{\\partial u_i}{\\partial t} + \\frac{\\partial u_iu_j}{\\partial x_j}\n= - \\frac{1}{\\rho} \\frac{\\partial p}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 u_i}{\\partial x_j \\partial x_j}.\n", "a5afde2953561b30e820f77301f41705": "\\int_0^\\infty x^\\alpha e^{-x} L_n^{(\\alpha)}(x)L_m^{(\\alpha)}(x)dx=\\frac{\\Gamma(n+\\alpha+1)}{n!} \\delta_{n,m},", "a5b009b5e1177d2acf7ea2cc3adc00fc": " \\lambda_{a;bc}-\\lambda_{a;cb}=R^d{}_{abc}\\lambda_d", "a5b0391565412636ee17b1ed4f9edc40": "g\\in K[T_1,T_2,\\cdots ,T_r]", "a5b0b1013ec448c41ace640b5b917cde": "\n\\bar{Y}=\\frac{1}{n}\\frac{K_1[X] + 2K_1K_2[X]^2 + \\ldots + n\\left(K_1K_2 \\ldots K_n\\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \\ldots +\\left(K_1K_2 \\ldots K_n\\right)[X]^n}\n", "a5b0f3e1579d8b39c75d9e3cb7085bd0": "S = -k_{\\mathrm B}\\sum_{j}P_{j}\\ln\\left(P_{j}\\right)", "a5b145059e4008a58c3887b0c5b78a72": " [p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m]^2 - ", "a5b14db8a4500a9aabb123516c8dc01e": "GF(2^m)", "a5b1679413438f453cea44fac8ac53da": "F: C[0,1] \\to \\R", "a5b170d7f75231043f30f6687d60b21b": " \\mathcal{O}(\\log n)", "a5b17628160fce5ded7931c75ed4c0f4": "z \\mapsto -1/z", "a5b17f6f468e63a7682fe116ce6ed140": "f(x) = \\sum_{n=0}^{\\infty} f_n x^n", "a5b1a49a09548dcd5496c1b201985eaf": "Y'\\subset Y", "a5b1c12c1a532c02d6f2d4aa34f3a2e1": "\\beta_{i}~~", "a5b1fb5cbeda7d49fdcf11624b0f178b": "N(a + b\\sqrt{D}) = a^2 - Db^2", "a5b226347fb4e90b65c49afa4e1e8e4f": "\\lim_{n\\to\\infty} b_n \\ne 0", "a5b25b37d694e84a9d9da74b2081ce19": "(p?;a)*;\\neg p?\\,\\!", "a5b2705945e99483415dd976c4bef4aa": "F(r_1,r_2,r_3) = A ", "a5b277541058c07be93daf75cad6767d": "f(x)=\\sum_{k=1}^n f_k v_k(x)", "a5b27a29753284d25ebe6dd555b2928d": "\n\\left. -\\lim_{x^0\\rightarrow+\\infty}\n\\int \\mathrm{d}^3x f_p(x)\\overleftrightarrow\\part_0\n\\langle \\beta\\ \\mathrm{out}|\n\\varphi(x)\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n|\\alpha\\ \\mathrm{in}\\rangle\n\\right\\}\n", "a5b3003b48df112a6f3ef485a8807361": "y\\sim x^{a-c}e^x\\,", "a5b302e73670bfa8e65927cfad0f60f0": "x = \\operatorname{sn}(u,k)", "a5b329aaa71b3bcb591e4f4a22eca787": "\\lambda v'.\\oplus\\left(v',v\\right)", "a5b36e63d3dde4566787e3fbb5e66602": "\\cos \\theta + i\\sin \\theta = e ^{i\\theta } \\,", "a5b37a98c88f8acbc6826f665e6e04b5": " v'_{n+0.5} := \\frac{v_{n+1} - v_n}{h}. ", "a5b3d5e45648437923bb14c162b2fa17": "\\#X(\\mathbb F_{p^n}) = 1 -\\sum_{i=1}^{2g} \\alpha_i^n+p^n,", "a5b3f76a6ab23e9eb5d5d3e4a08d7d2b": "p_{fg}(z)\\,", "a5b499f166f94ca1ce93e3d8c2505d4d": "\\mathbf{AB} = \\begin{pmatrix} \na & b \n\\end{pmatrix} \\begin{pmatrix} \nx \\\\\ny \\\\\n\\end{pmatrix} = ax + by \\,,\n", "a5b4b67115615ee20eb4343cd5163375": "d(x,y)=x+y", "a5b4b95e5bf1e6aa491b5636b01e073a": "p \\ge 1", "a5b4ccfdc80af0f2626312a569eeaf2b": "\\lambda \\ge 2\\sqrt{d-1} - o(1)", "a5b58e66afae3c55b5b1aff7d2c10e5c": " E\\Psi = i\\hbar\\frac{\\partial \\Psi}{\\partial t} \\,\\!", "a5b6047c632fccb09289e93c4e5eaffc": "(\\Sigma,\\Mu,\\mathit{L},\\mu,\\nu)", "a5b61665d1e90d22cf6bf130ac3c933d": "Q_M (a,b) = \\exp \\left( -\\frac{a^2 + b^2}{2} \\right) \\sum_{k=1-M}^{\\infty} \\left( \\frac{a}{b}\\right)^{k} I_{k} \\left( a b \\right) ", "a5b65e21b9e7f9ac08684b9e1f018fea": "f(x) = \\log x", "a5b6abac91dada85d8a4e952b8d88cad": "(b+1)x=1+\\frac1{b+2}+\\frac1{(b+2)(b+3)}+\\cdots<1+\\frac1{b+1}+\\frac1{(b+1)(b+2)}+\\cdots=1+x,", "a5b78a435edcdbe6316a96dc5e56c5a1": " -1= \\iiint_V \\nabla \\cdot \\nabla u \\, dV = \\iint_S \\frac{du}{dr} \\, dS = \\left.4\\pi a^2 \\frac{du}{dr}\\right|_{r=a}.", "a5b7da2a0eb978b23a309c524e74f4f6": "\\frac{\\part f}{\\part y}(x,y) = x + 2y.", "a5b7e992e95a0db9f63512cc93951914": "\\scriptstyle |\\psi\\rang ", "a5b8488ffebeae9b17174dc969f514eb": "\\int_A\\left(\\int_B f(x,y)\\,\\text{d}y\\right)\\,\\text{d}x=\\int_B\\left(\\int_A f(x,y)\\,\\text{d}x\\right)\\,\\text{d}y=\\int_{A\\times B} f(x,y)\\,\\text{d}(x,y),", "a5b84cc5c0274be6beb9679b725d1a37": "[1]:[2..N]", "a5b84d9a65128755e88f5add129221c7": "\\lim_{n\\to\\infty} \\frac{1}{n}\\sum_{k=1}^n s_k = A.", "a5b88cb60d4e67baf5aee3a1b97971be": "X\\rightarrow\\{s,x=\\log_b(|X|)\\},", "a5b8961e0f943025a7930bdbf179f4e2": "R = k_{\\rm B} N_{\\rm A} = 8.314\\,472(15)\\ {\\rm J\\,mol^{-1}\\,K^{-1}}\\,", "a5b8d95862d93337875e4af6abc780c6": "\\frac{27}{25}", "a5b8e188e65cb458e5f00414611cb713": "\\bigg( (\\mathcal{M}, s) \\models \\phi_1 \\Leftrightarrow \\phi_2 \\bigg) \\Leftrightarrow \\bigg( \\Big( \\big((\\mathcal{M}, s) \\models \\phi_1 \\big) \\land \\big((\\mathcal{M}, s) \\models \\phi_2 \\big) \\Big) \\lor \\Big( \\neg \\big((\\mathcal{M}, s) \\models \\phi_1 \\big) \\land \\neg \\big((\\mathcal{M}, s) \\models \\phi_2 \\big) \\Big) \\bigg)", "a5b96ecacb7ab958f63c5288287674ef": "\n\\frac{dr}{d\\varphi} = - \\frac{r^{2}}{ac} \\left(1 - \\frac{r_{s}}{r} \\right) p_{r}\n", "a5b9adfcc6f0c3602f4c1a3038fed5e6": "X \\sim \\operatorname{EV}_1(a,b)", "a5b9bfd67e107541862d3368da6751df": "\\pi_i f_i(x)", "a5b9dccc4a85781b72fe6252d8354951": " b = \\frac{c.}{\\delta - n / {\\delta a} } ", "a5ba5060c12a3185e32e3cf7b9f202ac": " \\varphi = \\sin^{-1} \\left[ \\frac{2 \\theta + \\sin \\left( 2 \\theta \\right)}{\\pi} \\right], \\,", "a5bad75ac2e68682c4f4609b92e762b5": " E \\,\\!", "a5bae9f14699779b8c33f7e7b9f280f4": "x^{d-1}.", "a5bafab6c4926cd1c3e7a92799a95eda": " A = \\pi r^2.\\ ", "a5bb06052afb1351f83b00de6efd8b70": "\\left(\n\\partial x_{i}^{\\ast }\\left( t\\right) /\\partial t_{j}\\right) _{i,j=1}^{L}", "a5bb203c08ba0d85b198f99121f42fec": "\n \\begin{bmatrix}\n 1 & 3 & 2 \\\\\n 2 & 3 & 1\n \\end{bmatrix}\n\\oplus\n \\begin{bmatrix}\n 1 & 6 \\\\\n 0 & 1\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1 & 3 & 2 & 0 & 0 \\\\\n 2 & 3 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 6 \\\\\n 0 & 0 & 0 & 0 & 1\n \\end{bmatrix}.\n", "a5bb3e324d5467d5d9ce265d1c68ee9f": "B_{j_1}\\cup B_{j_2}\\cup\\cdots\\cup B_{j_k}", "a5bb653b936a4193207fa573ccdcd293": "\\mu(\\alpha,\\beta)", "a5bb7712bb1954646a081adf37494610": "t f(x_1) + (1-t) f(x_2),", "a5bb899b6380734dc65966d1a9e0aac8": "\\displaystyle \\frac{1}{\\sqrt{2\\pi a^2}}\\cdot \\operatorname{rect}\\left(\\frac{\\omega}{2 \\pi a}\\right)", "a5bc064aff74c0ccae9de7224baaa438": "s_k=\\prod_{i=1}^m x_i", "a5bc07278ec0898c35d6c651dc4b9164": "x^3 > x^2z^2 > xy^2z > z^2", "a5bc15fbcfed9b6851f1946667633acb": "T_e=\\frac{T^2}{1-R^2}\\left(\\frac{\\sinh\\gamma}{\\cosh\\gamma-\\cos\\delta}\\right)", "a5bc8145520b3fa708a74c32bf031486": "\\,\\! {^{n}a} ", "a5bd2cd7acd47f19cb27dd3b079f4875": "-175\\pm 50", "a5bd310cc1a047590e92d39cd133515a": "b_{i}-a_{i}", "a5bd38fda89175b8efc7f98c471aa72b": "~t_{\\rm r}~", "a5bd7808c49e8d6816d5835f09f84989": "K_q(n,n-2)\\geq q^2/(n-1)", "a5bd8254c95b1c67f6d632846ba15cb1": "2^m-1\\mid\\lambda", "a5be537d31c0c519e64045d9769e0194": "\n (A \\cap B)^o \\supseteq A^o + B^o,\n ", "a5be5d0a18441a6bef4155b9a981aa05": "T = e^{t\\Delta}.", "a5be9e3b446c1340a947b34fe23fbc1c": "\\{\\boldsymbol{v}_1,\\boldsymbol{v}_2,\\boldsymbol{v}_3,\\ldots\\}", "a5bedf10932fbb8fcd5f114363e1e32f": "\\displaystyle{U_{g}^{-1}f(x) = (cx + d)^{-1} f\\left({ax + b \\over cx + d}\\right),\\,\\,\\,g = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}}", "a5bef5adac21d229a77ba825dff73dbe": "\\phi_n = A_n \\cos \\left( \\frac{2\\pi n x}{L}\\right) ", "a5bf1b6528163470ca4878b3bc165856": "f_H(h)=\\begin{cases}\\lambda_1 e^{-h\\lambda_1}, & h\\le\\tau \\\\ e^{\\tau(\\lambda_0-\\lambda_1)}\\lambda_0 e^{-h\\lambda_0}, & h>\\tau\\end{cases}", "a5bf34b54b60c37a8d3ff6eb2106e16b": "\\tfrac12 a b \\sin(C)\\,\\!", "a5bf83f7a10eebace6baec26fbe1de10": "P = \\tau \\times 2 \\pi \\times \\omega", "a5bfb4b596993661fe6c82e1340e4d90": " 1.1 (100W)^{1/3},", "a5bfc9e07964f8dddeb95fc584cd965d": "37", "a5bfdc9a394e9eb69fadbbddc26c2d95": "\nz = re^{i\\theta}\\quad\\mbox{and take}\\quad w=z^{1/2} = \\sqrt{r}\\,e^{i\\theta/2}\\qquad(0\\leq\\theta\\leq 2\\pi).\\,\n", "a5bfe4dfc89a14fbed8f928f867c0a96": "Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0", "a5c0086593db6d992fb701652aa56c47": "\\lambda_1 = 1011.", "a5c03bc8fb270c7b8a9408b301f51781": " \\vert V \\vert ", "a5c040866377e8a4ca46c9e9074419fa": "\\frac{1}{r^3} P^1_2(\\sin\\theta) \\sin\\varphi = \\frac{1}{r^3} 3 \\sin\\theta \\cos\\theta \\sin\\varphi", "a5c04591bf9d4dee5e3e8db477798644": " ~s'^2=x^2+y^2+z^2", "a5c14c3f75722251a87dd8d0384fa56c": "dz \\approx c t", "a5c158cddc54f3f183ebe232bf27a274": " u=\\lambda \\phi^m \\left[\\frac{\\partial^2 u}{\\partial z^2}-(1-\\phi)\\right],", "a5c15bcfd5cc7cdf317dc351e946e8c3": "F_{out} < 1", "a5c1771b27f9f6bf028b48e22ed774a5": "m_i/(m_\\mathrm{tot}-m_i)", "a5c24f157effa0f1eb778e318911db2e": "~t>0~", "a5c25d8d035877c37a2c33b8211a16f6": "F_Y(y) = \\operatorname{P}(X^2 \\le y).", "a5c2d59d0925c9938800a53032e1db97": "N\\cap P'", "a5c368613df6e77f3973506f41173640": "\\operatorname{NP} \\subseteq \\operatorname{DTIME}(2^{poly(\\log n)})", "a5c36e0acd8a8d53867b5c9540197fc5": " \\lambda_1,\\lambda_2,\\lambda_3 ", "a5c3ac8273a7e55985ec73441ca24719": "\\frac{1}{2}Q_n = Q_{n-1}^2", "a5c3c8edf24a74366244891794fbfaf0": "\\mathbf{U}=\\gamma(c,\\mathbf{u}) \\,", "a5c3d528c625bddbe1fc55569f26bc06": " r=\\sqrt{x^2+y^2+z^2}\\approx z+\\frac{x^2+y^2}{2z}. ", "a5c3f420421f5dfe0ed1a70cddabdc37": "\\scriptstyle m \\, < \\, n", "a5c45f681f68a1f2b83c68dbc83eea46": "P_1(k)", "a5c4769628d466f06f350cae2422c588": "\\left(\\int \\left|f(x)+g(x)\\right|^p\\,dx \\right)^{1/p} \\leq\n\\left(\\int \\left|f(x)\\right|^p\\,dx \\right)^{1/p} +\n\\left(\\int \\left|g(x)\\right|^p\\,dx \\right)^{1/p}.", "a5c4b5135482a13e57e3856ba4ad8b9c": "\n\\begin{align}\n\\Pr(Y_i=1) &= \\frac{e^{(\\boldsymbol\\beta_1 +\\mathbf{C}) \\cdot \\mathbf{X}_i}}{e^{(\\boldsymbol\\beta_0 +\\mathbf{C})\\cdot \\mathbf{X}_i} + e^{(\\boldsymbol\\beta_1 +\\mathbf{C}) \\cdot \\mathbf{X}_i}} \\, \\\\\n&= \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i} e^{\\mathbf{C} \\cdot \\mathbf{X}_i}}{e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} e^{\\mathbf{C} \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i} e^{\\mathbf{C} \\cdot \\mathbf{X}_i}} \\, \\\\\n&= \\frac{e^{\\mathbf{C} \\cdot \\mathbf{X}_i}e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{e^{\\mathbf{C} \\cdot \\mathbf{X}_i}(e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i})} \\, \\\\\n&= \\frac{e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}}{e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} + e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i}} \\, \\\\\n\\end{align}\n", "a5c4d6e2737ab04b6d9e596f3fcb64ca": "\\Omega>\\omega", "a5c516c9a9d4eca92e1db17295cd7727": "q_n(w) = \\sum_{k=0}^{n-1} \\bigg \\langle \\! \\! \\bigg \\langle \n\\begin{matrix}\n n+1 \\\\\n k\n\\end{matrix} \n\\bigg \\rangle \\! \\! \\bigg \\rangle (-1)^k w^{k+1}", "a5c56dd5fab9c463ebd6a5be609243a5": "\\displaystyle iu_x+v+u|v|^2=0", "a5c5abd7514ebaca07f63d5e7b214d67": "B_n = \\sum_{k=0}^n b_k,", "a5c5d52ec825026668cf943151872892": " \\theta_a \\, ", "a5c5eec57e6c7550bd539e2e073268f6": "\n\\mbox{grade} = 1.0430 \\sqrt{\\mbox{number of polysyllables}\\times{30 \\over \\mbox{number of sentences}} } + 3.1291\n", "a5c6575b5aa368ad9c5b22c16ed4c411": "f = 1 - erf (\\eta / 2)", "a5c6cc82148f37695410e37867f3c1e7": "p>p_c", "a5c713196c71b30eb9c5024c823afa20": "{P}=P_b \\cdot \\left[\\frac{T_b}{T_b + L_b\\cdot(h-h_b)}\\right]^{\\textstyle \\frac{g_0 \\cdot M}{R^* \\cdot L_b}}", "a5c750b820524cde3fa15aed7027fad9": "D_{m,n} = \\left|\\begin{matrix}\nc_m & c_{m-1} & \\ldots & c_{m-n+2} & c_{m-n+1}\\\\\nc_{m+1} & c_m & \\ldots & c_{m-n+3} & c_{m-n+2}\\\\\n\\vdots & \\vdots & & \\vdots & \\vdots\\\\\nc_{m+n-2} & c_{m+n-3} & \\ldots & c_m & c_{m-1}\\\\\nc_{m+n-1} & c_{m+n-2} & \\ldots & c_{m+1} & c_m\\\\\n\\end{matrix}\\right|\n", "a5c7daf9ab11104ff093df4de31143ca": "H_{0}^2-2P_{0}^2=1 \\, ", "a5c7f973a5b8e741e47736649aa76175": "I=F\\cdot F^*", "a5c806701f92137877a6d689eba00fe6": "\\Delta : H \\rightarrow H \\otimes H", "a5c81e11dc180a2b52e14e8ec986e164": "W_i^*", "a5c89b6dc1210ccf1589928028c47563": "L^2=L_x^2+L_y^2+L_z^2= 2\n\\begin{pmatrix}\n1& 0& 0\\\\\n0& 1& 0\\\\\n0& 0& 1\n\\end{pmatrix}", "a5c8a11cd91c15635771b79622e7585d": "r = \\sqrt{\\pi/(4-\\pi)}", "a5c8ab8ea5aba984ea73c59b398b494f": "\n|\\psi \\rangle \\approx \\alpha |\\psi \\rangle\n", "a5c8bbb36fa677e7d9d98319262fb509": "D^+_{ij}", "a5c8f228f39e48972227b542929cb3da": "m_k = \\tau_k \\alpha_k \\Delta_k", "a5c93ec7034203e9260d05dbda7ad179": "\\,\\! {^{(-1)}1} ", "a5c94d01a8c2d6e73e34d9ed65a17ebf": "\\mathfrak{P}^{122}", "a5c96a4491566fb8a7794a955a2bfe45": "P_1, P_2, \\ldots, P_n", "a5c9753ac2b8a3fab3a28b3c694cf9a5": "\\mathbf{J \\left(J^TT \\right)^{-1}J}", "a5c9993930f7e32fe571a14508e5a2e3": "k[t_1, \\dots, t_n]", "a5c9ec458a890e9fcfeb79412de7cc23": "a^{p-1} \\equiv 1 \\pmod p.", "a5ca65f45d7b1795d471a6c6cfd1077f": "\\textstyle \\varsigma", "a5ca7acb336d18dc95a0d4f2460b87ec": "\\tan\\frac{13\\pi}{60}=\\tan 39^\\circ=\\tfrac14\\left[(2-\\sqrt3)(3-\\sqrt5)-2\\right]\\left[2-\\sqrt{2(5+\\sqrt5)}\\right]\\,", "a5ca8709aadb209edff46c5d967a519e": "\\forall A>0", "a5caee2744d779071215d9be5e8e870d": "\\mathrm{Rot}_{G \\circ H}", "a5cb32a1b19e5d25d628fc1948155048": " [EI] = \\frac{K_m k_3[I][ES]}{k_{-3}[S]} ", "a5cb3eb94ad6b248a6b5294892824568": "2n + 1", "a5cb5c38e6f2053caa17c97bab5b9988": "S.", "a5cb75b96e96cf524819137dc513c631": " \\lambda_{\\max}(A)", "a5cbc84c681a3d7d2597478cf92bacc0": " \\mathfrak{g}_0 = \\mathfrak{m}_0\\oplus\\mathfrak{a}_0\\oplus_{\\lambda\\in\\Sigma}\\mathfrak{g}_{\\lambda} ", "a5cbd2ddd196d0e983997e1426766ba9": " x = \\frac {D}{P} ", "a5cbe779bc36f8925f411e314a32a344": "01.\\end{cases}", "a5d95996082250bbdb90a713393af79d": "\\tilde{V}", "a5d96ed14df31e6ed600788b8068d371": "W_r/N = J\\sigma_r", "a5d988133492b2582e7b12d97eb4ddf3": " = 2 r \\arcsin\\left(\\sqrt{\\sin^2\\left(\\frac{\\phi_2 - \\phi_1}{2}\\right) + \\cos(\\phi_1) \\cos(\\phi_2)\\sin^2\\left(\\frac{\\lambda_2 - \\lambda_1}{2}\\right)}\\right)", "a5d9de1003e9966e16c200e8d79c765f": "SRH = \\int{ \\left ( \\vec V_h - \\vec C \\right )} \\cdot \\nabla \\times \\vec V_h \\,d{\\mathbf Z}\n\\qquad \\qquad \\begin{cases} \\vec C = Cloud\\ motion\\ to\\ the\\ ground \\end{cases} ", "a5da15c57c5ddd589317531a2cae898b": "s \\ge H", "a5da30dbd96f2444f2c80f2dc4b3354c": "(1-\\sqrt{2})^n", "a5da51cce79100ec4587889fc3d0923c": " \\qquad \\qquad \\mathbf{j}_e = \\alpha_{ee}\\mathbf{e}_e - \\frac{\\alpha_{et}}{T^2}\\nabla T \\qquad (\\mathbf{e}_e = \\alpha_{ee}^{-1}\\mathbf{j}_e+\\frac{\\alpha_{ee}^{-1}\\alpha_{et}}{T^2}\\nabla T), ", "a5da7a1f67ed14690d7840a8b371ff8f": "f(n) = p_i(n)", "a5dab9b612816218dd9dcc81673efcad": " r_1=k_1 [\\mathrm{A}]^a[\\mathrm{B}]^b ", "a5dae942082cb3c6239c00007cb4f88a": "G_N", "a5db23b5497d989385983f805530e4aa": " \\sum_{i=1}^{n}\\|\\mathbf{x}_i - V_{k}\\mathbf{x}_{i}^{k}\\|^2 = \\sum_{j = k+1}^{n}\\left(\\lambda_j\\right) \\; ", "a5db5af2b7885de8eef316270f1c1200": " \\mathcal{D}^{\\mu\\nu}", "a5db9dbbcd106caf702f047974d7c3b7": "\n1_{y_t=v_k}=\n\\begin{cases}\n1, & \\text{if } y_t=v_k\\\\\n0, & \\text{otherwise}\\\\ \n\\end{cases} \n", "a5dba2ab441d6bc9722f44743b6b4530": "f(2) = 2^2 - 5 \\cdot 2 + 6 = 0 \\quad \\textstyle{\\rm {and} }\\quad f(3) = 3^2 - 5 \\cdot 3 + 6 = 0.", "a5dbe1d15c8da839c449be2f70a23988": "\\sigma^2_j", "a5dc045b390b01b151c74854a7a585ff": " e_n \\ge 0, p_n \\ge 0, e_n\\cdot p_n = 0\\,\\!", "a5dc991a52533d0725903e4af7ba5c70": "\\mathrm{D} f (u_{0}) = \\left( \\mathrm{D} g (u_{0}) \\right)^{*} (\\lambda),", "a5dcb498ae724dc25cd38eff76731715": "h_0(x) = \\begin{cases} 0&\\mathrm{if\\ }|x| < 1\\\\ \\frac{1}{\\pi x} &\\mathrm{otherwise} \\end{cases}", "a5dd1344119c83b3c7bd56e81525b20e": "\n\tZ_{CO}^j = \\sum_{i=1}^n D_t^j e^{-\\hat{y_i}\\alpha_t^jg_t^j(\\boldsymbol{x_{j,i}})}\n", "a5dd6d63bbf39124889d38702f07c50f": "\nh(\\mathbf{x}) = \\frac{x_i}{|\\mathbf{x}|^2}.\n", "a5ddc4ab342cde8679aa66bebe23e998": "\\frac{x}{v(1-x)}=\\frac{1}{v_\\mathrm{mon}c}+\\frac{x(c-1)}{v_\\mathrm{mon}c}.", "a5de048c0af5e69add3698bf1143a579": "P=1", "a5de065e745631becf6ee5305d2f7404": "\\prod\\limits_{j=1}^n (1-{p_j}+{p_j}{e^{t}})", "a5de38c32a6185211c74d2bec9b77835": "\\tilde\\epsilon= (\\epsilon_{xx}, \\epsilon_{yy}, \\epsilon_{zz},\n \\gamma_{yz},\\gamma_{xz},\\gamma_{xy}) \\equiv (\\epsilon_1, \\epsilon_2, \\epsilon_3, \\epsilon_4, \\epsilon_5, \\epsilon_6),\n", "a5de3dd6fc427f2895a4ac4ed634dbaf": "x = \\sin u \\left(7+\\cos\\left({u \\over 3} - 2v\\right) + 2\\cos\\left({u \\over3} + v\\right)\\right) ", "a5dedd30af92118dd3c8aebdd0891506": "[[x, y^{-1}], z]^y\\cdot[[y, z^{-1}], x]^z\\cdot[[z, x^{-1}], y]^x = 1", "a5def100fbbcd5e60d87ca1cf6110baa": " \\langle \\phi_1(x_1) ... \\phi_n(x_n)\\rangle = {\\int e^{-S} \\phi_1(x_1) ...\\phi_n(x_n) D\\phi \\over \\int e^{-S} D\\phi}.", "a5df01fd19bf7c0fa037eefc086ec6d9": "a^{\\dagger}(\\phi_i)a(\\phi_i)\\,", "a5df22cc2d5cdc01dfa08afb45f1f6a3": "\\begin{align}\\langle f|g\\rangle-\\langle g|f\\rangle &= \\int_{-\\infty}^{\\infty} \\psi^*(x) \\, x \\cdot \\left(-i \\hbar \\frac{d}{dx}\\right) \\, \\psi(x) \\, dx \\\\\n&{} \\, \\, \\, \\, \\, - \\int_{-\\infty}^{\\infty} \\psi^*(x) \\, \\left(-i \\hbar \\frac{d}{dx}\\right) \\cdot x \\, \\psi(x) dx \\\\\n&= i \\hbar \\cdot \\int_{-\\infty}^{\\infty} \\psi^*(x) \\left[ \\left(-x \\cdot \\frac{d\\psi(x)}{dx}\\right) + \\frac{d(x \\psi(x))}{dx} \\right] \\, dx \\\\\n&= i \\hbar \\cdot \\int_{-\\infty}^{\\infty} \\psi^*(x) \\left[ \\left(-x \\cdot \\frac{d\\psi(x)}{dx}\\right) + \\psi(x) + \\left(x \\cdot \\frac{d\\psi(x)}{dx}\\right)\\right] \\, dx \\\\\n&= i \\hbar \\cdot \\int_{-\\infty}^{\\infty} \\psi^*(x) \\psi(x) \\, dx \\\\\n&= i \\hbar \\cdot \\int_{-\\infty}^{\\infty} |\\psi(x)|^2 \\, dx \\\\\n&= i \\hbar\\end{align}", "a5df591c696ddf526b64d66f902f25ee": "\\lambda p_k(t)=\\mu p_{k+1}(t)\\text{ for }k\\geq 0. \\, ", "a5df5b0fddac00b9b3ef7fb6d511dc0e": " A_w ", "a5df6eb8e14d1fd23a6946096d0e8ad4": "\\mathbb{M}", "a5dffa203dc7a25d5724caeca91a9b58": "(A \\wedge (\\lnot B \\vee C) \\wedge \\lnot C) \\vee D", "a5e009993483e201ce992cdb64b6697b": "A_2 = {-1 \\over (2)(1)}A_0={-1\\over 2}A_0,\\, A_3 = {1 \\over (3)(2)} A_1={1\\over 6}A_1", "a5e10407f99e3ecab08184957806df4b": "|x_i| < 2.4", "a5e122d39a01ce2c4e9495b6201bfcd8": "\n \\dot{\\boldsymbol{F}^{-1}} = - \\boldsymbol{F}^{-1}\\cdot\\boldsymbol{l} \\quad \\implies \\quad\n \\dot{\\boldsymbol{F}^{-T}} = - \\boldsymbol{l}^T\\cdot\\boldsymbol{F}^{-T}\n", "a5e1bff585b3d4cdd1a74ed42c46475e": "r^n + a_1 r^{n-1} + \\cdots + a_{n-1} r + a_n = 0.", "a5e1c05cdb83f2e2b01f0ac7567e278f": " {{\\alpha \\times \\left(1+(1-\\alpha)+(1-\\alpha)^2+\\cdots +(1-\\alpha)^N \\right)} \\over {\\alpha \\times \\left(1+(1-\\alpha)+(1-\\alpha)^2+\\cdots +(1-\\alpha)^\\infty \\right)}}= 1-{\\left(1-{2 \\over N+1}\\right)}^{N+1}", "a5e1ce3b16394887147d5acb3f7ddb86": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 10.08632\\log_e(T+273.15) - \\frac {6030.610} {T+273.15} + 80.87786 + 9.812512 \\times 10^{-6} (T+273.15)^2", "a5e1cf1ec9d7aef90204dd006ee17dd9": "u(x,t) = \\frac{1}{\\gamma_n} \\left [\\partial_t \\left (\\frac{1}{t} \\partial_t \\right )^{\\frac{n-2}{2}} \\left (t^n \\int^{\\text{average}}_{B_t(x)} \\frac{g}{(t^2 - |y - x|^2)^{\\frac{1}{2}}} dy \\right ) + \\left (\\frac{1}{t} \\partial_t \\right )^{\\frac{n-2}{2}} \\left (t^n \\int^{\\text{average}}_{B_t(x)} \\frac{h}{(t^2 - |y-x|^2)^{\\frac{1}{2}}} dy \\right ) \\right ] ", "a5e2bade06f9ec73dc286a21fe219ef1": " d \\mathbf{l}", "a5e2d85a61a6092162396ba2af0769af": " f_{M_t}(m) = \\int_{-\\infty}^{m} f_{M_t,W_t}(m,w)\\,dw = \\int_{-\\infty}^{m} \\frac{2(2m - w)}{t\\sqrt{2 \\pi t}} e^{-\\frac{(2m-w)^2}{2t}}\\,dw = \\sqrt{\\frac{2}{\\pi t}}e^{-\\frac{m^2}{2t}}", "a5e3691195d00e80a2e25e434fa027a0": " \\mathbb{Q}(\\sqrt{\\tau}) ", "a5e3ba66f20ae3e69c03474f00024993": "\\mathbf{F} = \\frac{G m_1 m_2}{\\left | \\mathbf{r} \\right |^2} \\mathbf{\\hat{r}} \\,\\!", "a5e3c8b1ef82bd8b8122fece17bf01d8": "T = 2 \\pi \\sqrt{\\frac{L}{g}} \\,", "a5e3f512aee8e9212b409d594e69dd19": "D_{2d}", "a5e40050e99072d8760f506ed711bb9e": "\n\\lambda(t|X) = \\lambda_0(t) + \\beta_1X_1 + \\cdots + \\beta_pX_p = \\lambda_0(t) + \\beta^\\prime X.\n", "a5e411742477099756d9f5e4cbc68de8": "\n S_{xx}(f) = \\int_{-\\infty}^\\infty r_{xx} (\\tau) e^{-2\\pi if\\tau} d\\tau.\n", "a5e4601e4218913c60c6df93a9ab0eed": "z^2-4z+5=0", "a5e494bbe5075ef1cb62a00a66e0cf5c": "X \\rightarrow Y \\in T~\\Rightarrow Y", "a5e4d95f1c74438fda52bb9005e649bc": "\\begin{align}\n\\mathbf{i}&=\\mathbf{j}\\times\\mathbf{k}\\\\\n\\mathbf{j}&=\\mathbf{k}\\times\\mathbf{i}\\\\\n\\mathbf{k}&=\\mathbf{i}\\times\\mathbf{j}\n\\end{align}", "a5e50760ad41a1c6eed5b5691f047bef": "A^\\mathbb N", "a5e53e4c7dd2a4e0303114e4f2a4f797": " {P} = \\frac{1}{f} = (n-1) \\left[ \\frac{1}{R_1} - \\frac{1}{R_2} + \\frac{(n-1)d}{n R_1 R_2} \\right],", "a5e559c928dcbcc9e67c19b2c1b1044b": "\\hat{x}_{k\\mid k-1}", "a5e5905ea8b662918d1cf718671b69f0": "Df\\,(x)", "a5e5a1e5fb079d22320e27749d48c100": "d(x,y) = \\lVert y - x \\rVert", "a5e5c371ac667cdd075430d130bc6d9f": "\\mathbb{E}\\Phi(||\\mathbb{P}_n - P||_{\\mathcal{F}}) \\leq \\mathbb{E}_{X} \\mathbb{E}_{Y} \\Phi \\left(\\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n [f(X_i) - f(Y_i)] \\right|\\right|_{\\mathcal{F}}\\right)", "a5e616ddda4ad9b7af3a4d7586bd0076": "\\chi_+^y = {1 \\over \\sqrt{2}} \\begin{bmatrix}\n 1\\\\\n i\\\\\n\\end{bmatrix} \n", "a5e63f843efcc64b1a244ec955fbc055": " S_{21} \\equiv \\frac{p_2 c_2}{p_1 c_1}\n", "a5e681f4076a82fa4a19aa2b9dc57971": "K_n (X) \\rightarrow \\oplus_{p \\geq 0} H_D^{2p-n} (X, \\mathbf Q(p)).", "a5e6867d138336b6ad2495a6280a52e5": "(a,b)\\in D \\Rightarrow b < a^a", "a5e7064354f32ebb0c245a7eb37df621": " \\sum_{n=a}^b e(f(n)) \\ ", "a5e70d6032687d2fc94c2293f7377fd2": "\\begin{bmatrix}r_0\\\\r_1\\\\r_2\\\\r_3\\end{bmatrix} = \\begin{bmatrix}\n14&11&13&9 \\\\\n9&14&11&13 \\\\\n13&9&14&11 \\\\\n11&13&9&14 \\end{bmatrix} \\begin{bmatrix}a_0\\\\a_1\\\\a_2\\\\a_3\\end{bmatrix}", "a5e76aeebc31477a2fc5c4a816e21fba": "p_0 + p_1(x + x'\\varepsilon) + \\cdots + p_n (x + x'\\varepsilon)^n", "a5e7ae2081bae3c589d735f7d12b3313": "\n \\begin{align}\n N_{11} & = A~\\cfrac{\\mathrm{d}u}{\\mathrm{d} x_1} \\quad \\implies \\quad\n \\cfrac{\\mathrm{d}^2 u}{\\mathrm{d} x_1^2} = 0\\\\\n M_{11} & = -D~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x_1^2} \\quad \\implies \\quad \\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x_1^4} = \\cfrac{q}{D} \\\\\n \\end{align}\n ", "a5e7de8c170ed7842a19aba927c68bac": " \\widehat{\\beta}_\\mathrm{GMM} = (X^\\mathrm{T} Z(Z^\\mathrm{T} Z)^{-1}Z^\\mathrm{T} X)^{-1}X^\\mathrm{T} Z(Z^\\mathrm{T} Z)^{-1}Z^\\mathrm{T} y", "a5e7f202ff3e971ba3817a0e237b748b": " n \\ge 1 ", "a5e83c859929ccd73bd7d56eadd30b2f": "\\sigma_3\\sigma_1=\\sigma_2\\sqrt{-1}", "a5e84decac494b4aa6f3fd7e9f0920db": " \\frac{1}{n} \\sum_{k=1}^n \\binom{n}{k}B_k B_{n-k}+B_{n-1}=-B_n \\quad \\text{(L. Euler)} ", "a5e8620bb8711b43dfea1d99415c23ab": " \\nabla q ", "a5e9112391086b8cd43577d75922d90e": "\\left|\\begin{matrix}\nA & B & C \\\\\na & b & c \\\\\nX & Y & Z \\end{matrix}\n\\right|", "a5e9317f335c0dbbfe577372ddcdfe21": " \\prod_{r=1}^k \\frac{1}{1-rx} = \\sum_{n=0}^\\infty \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\} x^{n-k}, \\quad \\text{ or } \\quad \n\\sum_{n=0}^\\infty \\left\\{\\begin{matrix} n\\\\ k \\end{matrix}\\right\\} x^{n+1}= \\frac{1}{(k+1)! {\\frac{1}{x} \\choose k+1}}.", "a5e944dc9ee54599bdc6ebb248781d36": "R(u,u')=[\\nabla_u,\\nabla_{u'}]-\\nabla_{[u,u']} \\, ", "a5e9ba0b9f465089b7ffd8e3c917551f": "S^2 \\subset \\mathbb R^3", "a5e9d8445e2cfe1eef2e803bdb74546b": "0<\\limsup_{r\\downarrow 0}\\frac{\\mu(B(a,r))}{r^s}<\\infty", "a5ea0a425a1a95eb298fa0281f7947b3": "v_R=\\frac{\\boldsymbol{v} \\cdot \\boldsymbol{r}}{\\left|\\boldsymbol{r}\\right|}", "a5ea122a738d71b23fe6c54eebd8c72e": "A'\\in\\mathcal{I}", "a5ea3f338fec3d64180c5cf8022ed449": "TE_{mnp}", "a5ea6f08bfdf609d6002b575c6d0941b": "f_i(x) \\le 0, i = 1,\\ldots,k,", "a5ea73ad8c02c25b3df0ca78c32b3b1f": "\n\\left(c_1|\\psi_1\\rangle + c_2|\\psi_2\\rangle\\right)^\\dagger = c_1^* \\langle\\psi_1| + c_2^* \\langle\\psi_2| ~.\n", "a5eaa7c4a9c7c53fc873479bb4286083": "\\tau_{oct}", "a5eacc17fb1a2fbdb164d26c7fdad9d7": "\\left[S 1 \\right]_0", "a5eaec57b2f37e694a9f7a922c142e8e": "\\iota\\varepsilon", "a5eaef1ad6ffc586599ae66ef8a94988": "\\delta'(-x) = -\\delta'(x)", "a5ebf1e4bc95bfa1eac588321b44da0e": "q_\\mathrm{QCD} = e/\\sqrt{4\\pi\\alpha}", "a5ec734437ed34ff576796baa4e1c8fd": "\\sqrt{\\frac{2\\pi}{\\sqrt 3}}\\left(\\frac{1+\\sqrt 3}{2\\sqrt2}+o(1)\\right)\\sqrt n\\approx 1.84\\sqrt n.", "a5ec82e21973f8609293e7d09feb83e1": "t {\\rm\\; unbounded} \\;\\;\\;{\\rm if}\\;[Av]_i = 0, [b-Ac]_i > 0 ", "a5ecdfffde9285acb284f68d47fe39fd": "\\text{Gain}=10 \\log \\left( {\\frac{P_{\\mathrm{out}}}{P_{\\mathrm{in}}}}\\right)\\ \\mathrm{dB}", "a5ecf1bba7757a481b7bbcc825a8c554": "c^{\\mathfrak T_{\\Phi}}:= \\overline c", "a5ecf62507064d3e044c35fcec3f085b": "Z' = i \\omega L' \\!\\,", "a5ee0a94b2925b7d969a7bf938a0d3c2": "d(x,y) + d(x,y) \\ge d(x,x)", "a5ee15ccf9380e6ae01965ef7cbd69ca": "\\textstyle \\left\\vert u+v\\right\\vert _{W_p^m(\\Omega)}", "a5ee798431d136719ba93f7de9dbb3d4": "\n\\times \\sum_{m_A=-\\ell_A}^{\\ell_A} \\sum_{m_B=-\\ell_B}^{\\ell_B}(-1)^{m_A+m_B} I_{\\ell_A+\\ell_B}^{-m_A-m_B}(\\mathbf{R}_{AB})\\;\nQ^{m_A}_{\\ell_A} Q^{m_B}_{\\ell_B}\\;\n\\langle \\ell_A, m_A; \\ell_B, m_B| \\ell_A+\\ell_B, m_A+m_B \\rangle.\n", "a5ee857c3997e2e3f7d77f8ed9b4e900": "x' \\pmod{m}", "a5ee92ac7903a5b228dc4c44ff1893da": "\\xi_{ij}(t)=P(X_t=i,X_{t+1}=j|Y,\\theta)=\\frac{\\alpha_i(t) a_{ij} \\beta_j(t+1) b_j(y_{t+1})}{\\sum_{k=1}^N \\sum_{l=1}^N \\alpha_k(t) a_{kl} \\beta_l(t+1) b_l(y_{t+1})}", "a5eeaf16b04fe75021725f3307fa7f85": "10 \\equiv 1\\pmod{9},", "a5eed39d69b095f61d691e1340d29ea1": "\\scriptstyle{| d \\rangle , \\ | s \\rangle}", "a5eeed6164a1f2266935b6cc6859ae99": "\\mathbb{P}", "a5ef2b2ae202a7f86abfd6e3602b2ea5": "\\begin{matrix} {1 \\choose 1}{3 \\choose 1}{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "a5ef2fbb3fa0c0c592adbc6627145ae5": "\\nabla \\times \\mathbf{B} = \\mu_0\\left(\\mathbf{J} + \\varepsilon_0 \\frac{\\partial \\mathbf{E}} {\\partial t} \\right) ", "a5ef8ef758a9b790952393eafc0ef370": "X=\\lbrace 1,2,\\dots,n\\rbrace", "a5efdc146312fe0a656079603d19de16": "\\langle E_n : n \\in \\mathbb{N}\\rangle", "a5f0366ba2d64981623212891f16049c": " \\Delta = 1 - ( L_1 + L_2 + L_3 ) + ( L_1 L_2 ) \\, ", "a5f05b310bbd6bdec39aae44f5542392": "A^{k}_{s_{1}, s_{2}, \\ldots, s_{N}}", "a5f0c549fcaaa58ee5906ea318c9258c": "\\tau_2 = C R_2", "a5f0e79ae4e238815089788d67767d1d": "dS_{tot} \\ge 0 ", "a5f0f1cf03be05f9f91a7414bdfab085": "\\frac{\\partial f}{\\partial y} = \\beta", "a5f14485c4e55e50067b4feac594550f": "\\scriptstyle A_0\\supseteq A_1\\supseteq A_2\\supseteq\\dots", "a5f15518ea873e6fcdb58b8c2ddf42da": "\\;^+T^{IJ}", "a5f16dd70223314f33323efdca897a13": "{E} = -{{\\partial A} \\over {\\partial t}}", "a5f1b302df7af0646192f7c44aafc3a4": "a_1 = 1.", "a5f1c8bb80a2fb64abc8a1a2ea932d6d": "s_{ab}=\\frac{C^*_{ab}}{L^*}=\\frac{\\sqrt{{a^*}^2+{b^*}^2}}{L^*}", "a5f1f66e769bc038ee4e71b034e867fa": "E(S_{T}) > F_{0}", "a5f24b7581d9bca5d9d8c8890c762ae1": "\\scriptstyle{\\bar{\\xi}=2.1, \\quad \\bar{\\xi}^2 = 6.8.}", "a5f24bccea5d2693e99bee71365853f9": "E_X = \\frac{RT}{zF} \\ln \\frac {X_o}{X_i}", "a5f34bf2bc2823f3165595f819e108f2": "C^k", "a5f3544c4d793779c7e5638562f04b78": "\\varprojlim{}^n\\cong R^n\\varprojlim.", "a5f3969f653eb5ca656a1971c532c315": "\\geq 1/10", "a5f39c625eb5b7dd41ceb97e76684ffb": "\\phi(\\mathbf{x};\\mathbf{c}) = \\phi(||\\mathbf{x} - \\mathbf{c}||) = \\phi(\\sqrt{(x_1 - c_1)^2 + \\ldots + (x_K - c_K)^2})", "a5f3c6a11b03839d46af9fb43c97c188": "K", "a5f3f13be36051c17aa762494415b490": " \\hat{B} |\\Psi \\rangle", "a5f45350519e8db3679eb0b41611bc28": "d(p,q) \\le R", "a5f4bebdf84597ee100a1d99a7b4892e": "v \\mapsto \\hat\\sigma (v):=\\overline{\\sigma(v)}\\,", "a5f4e9f9a2a58373d6662d74eef2c618": "P(n) = \\sqrt {I^2(n) + Q^2(n)}", "a5f520267ea8a6d3245d1b58d70a1c09": "\nM_c' = 2 \\frac{\\operatorname{det}(A)}{\\operatorname{trace}(A) + \\epsilon},\n", "a5f566d65b815af8ae80d8eecdbe4aa4": "\\{t1, t2\\}", "a5f5c084f2110dbcfa67a681af902496": " |B \\cap [1,n]| \\geq n^{1/h} ", "a5f5db7a6d65a43c0c6b2f3d5cdd7325": "TM\\,", "a5f63aaf9ea9a98b3a4af620d00a168b": "[x,+\\infty)\\times[y,+\\infty)", "a5f6a96000033def3d71f216e1d7486c": "A=-log(%T/100%)", "a5f6ac5ea1e0bd9ea281c0a9ea4c8584": "v_{2k} \\in H^{2k}(M;Z_2)", "a5f6e57516ab8728d3c1227f7a5e0962": "q\\ =\\ (1-t)\\ y_1 +\\ t\\ y_2\\ +\\ t\\ (1-t)\\ (a\\ (1-t) + b\\ t)", "a5f7159f158dc969a611ef29f1b207ee": "\\begin{alignat}{7}\nx &&\\; && \\;&& &&\\; \\;&& = \\;&& 2 & \\\\\n&& && y \\;&& &&\\; \\;&& = \\;&& 3 & \\\\\n&& && && &&\\; z \\;&&\\; = \\;&& -1 &\n\\end{alignat}", "a5f726567d330ac3c42095a3fd611e35": "\n\\frac{\\pi}{4} = \\cfrac{1} {1+\\cfrac{1^2} {2+\\cfrac{3^2} {2+\\cfrac{5^2} {2+\\ddots}}}} \n= 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + - \\dots\n", "a5f74711a364dcf6a3895da3544d81e4": " k-(\\operatorname{rank} G-\\operatorname{rank} H)\\leq 5. ", "a5f75f2551f8e36a7fa20c5bea6e4374": "\\mathrm{Cl}_2 \\; \\xrightarrow{UV} \\; {\\mathrm{Cl} \\cdot} + {\\mathrm{Cl} \\cdot}", "a5f7ee1872afdb5189d9a08720f2df15": "\nQ_{SW} = S (1 - 0.62 cc)(1 - a) \\cos \\theta(\\phi, t)\n", "a5f823170fea55581fb8bc7abd79582d": "h(u_0)=0", "a5f8667333d678e52dc5b97377a6a778": "S\\downarrow T \\to \\mathcal B", "a5f875315990b136fb88ad025924bd35": "\\partial R", "a5f8b958dc587593a618c7213e446a3c": "{(n-1) + (m-1) \\choose (m-1)} + {n + (m - 2) \\choose (m - 2)} = {n + (m - 1) \\choose (m - 1)},", "a5f8ca46d7b2e086d13ab54e732a94d0": " \\hat{D}^\\dagger(\\alpha)=\\hat{D}(-\\alpha)", "a5f91f732797be775d105723e14a4d36": "i_k^2 \\in \\{ -1, 0, +1 \\}", "a5f921b304c29220cf6c25dc9857fb1d": "\\frac{c}{n}\\sum_{k=0}(p_f\\frac{n-c}{n})^{k} = \n(\\frac{c}{n})(\\frac{1}{1-\\frac{p_f(n-c)}{n}})", "a5f92fb4a719ee35b858c37cfa629ad1": "\\begin{alignat}{7}\n2x &&\\; + \\;&& y &&\\; - \\;&& z &&\\; = \\;&& 8 & \\qquad (L_1) \\\\\n-3x &&\\; - \\;&& y &&\\; + \\;&& 2z &&\\; = \\;&& -11 & \\qquad (L_2) \\\\\n-2x &&\\; + \\;&& y &&\\; +\\;&& 2z &&\\; = \\;&& -3 & \\qquad (L_3)\n\\end{alignat}", "a5f99951042f788008aef9dc1dc29eed": "M = M_0 \\supset N_0 \\supset M_1 \\supset N_1 \\supset M_2 \\supset N_2 \\supset \\cdots .", "a5f99ae1a188299dbef797d1d943f8fd": "n:=n_1=n_2=\\cdots=n_K", "a5f9b85668630025fb930498dcd05be4": "\n\\begin{align}\n\\mu &= 0 \\\\\n\\frac{\\nu}{\\nu-2} s^2 &= \\frac{\\pi^2}{3} \\\\\n\\frac{6}{\\nu-4} &= \\frac{6}{5}\n\\end{align}\n", "a5f9e0d2a2196d42070ade8bf9d5f661": "\nx^{2} = \\frac{\\left( a^{2} + \\lambda \\right) \\left( a^{2} + \\mu \\right) \\left( a^{2} + \\nu \\right)}{\\left( a^{2} - b^{2} \\right) \\left( a^{2} - c^{2} \\right)}\n", "a5fa04a618a73c5b73f3aaddc0273d3c": "U_2\\left(x,y\\right)=(1-\\alpha) \\left(x/y\\right)^{\\alpha}", "a5fa3259cd77bde961d939f7bfc0b915": "a_{n}y^{(n)} + a_{n-1}y^{(n-1)} + \\cdots + a_{1}y^{'} + a_{0}y = 0", "a5fa633e476ad25eb7045e369cdadb8f": "(\\widehat{X})", "a5fa9dfc15c25624f4651f7635cb9f2a": "\\exp (-\\beta \\hat{H})", "a5faa41fc217dda8dfbe1d81c2c19f42": "\\lambda_d", "a5faf798c420a92248d8c5f6c2141262": "\\phi(F_2)=G_2", "a5fafd490219bdd5e1fd2b4841004896": "\\mathbf{\\nabla} \\cdot \\mathbf{E} = \\frac{\\rho_f}{\\varepsilon}", "a5fafda57c1ec5616c198e3b76e381ce": "\\frac{\\partial \\mu}{\\partial n} = \\frac{\\hbar^2 \\pi}{m} \\frac{1}{1-e^{-\\hbar^2 \\beta \\pi n / m}}", "a5fb6fc7521e21073de4983abeb804ea": "G[X]", "a5fb786c9c87a677abe88c1b3368e681": "\\log (\\det(A)) = \\mathrm{tr}(\\log A)~. ", "a5fbe50ccd10f86bdcfba115bdbe85f3": "\n\\mathrm i^{2m} \\operatorname{erfc} (-z)\n= - \\mathrm i^{2m} \\operatorname{erfc}\\, (z)\n+ \\sum_{q=0}^m \\frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!}\n", "a5fbfe425732e27c50e71df047a11aaa": "|\\mathcal P|=(n+1)^2 \\ ", "a5fc01fc36722b7d035541d297de1830": "C_*({\\tilde f}): C_*({\\tilde X}) \\to C_*({\\tilde Y})", "a5fc2c2c9648f55e8d071e976ee5a189": "\\underbrace{y_a, y_b, \\ldots}_\\text{n}", "a5fc53853a829e849128b42815b09961": "a_{s-1} = -1", "a5fc946b9b7c5162b09e6e9886481784": "Lg^* \\circ Rf_! \\to Rf'_! \\circ Lg'^*,", "a5fe95105588483575ed1550d89bb9b2": "g^{ij}g_{jk}=\\delta^i {}_k\\ ", "a5feddd336f95a73db3d846da3a050ea": "l_0", "a5ff1f6c098e73644a6ec0f4952ea767": "e^1 e^{4 \\pi i n} e^{-4 \\pi^2 n^2} = e", "a5ff62d663d45a966836168e58865ba8": "Q_{feed}=K_{dosage}*V_{RM}", "a5ff66d72171c792acf19896a97d0bb3": "\nA_2=-\\frac{i A_1 \\chi_0}{\\Delta k} \\sum^{N-1}_{n=0} (-1)^n (e^{i \\Delta k \\Lambda (n+1)}-e^{i \\Delta k \\Lambda n})\n", "a5ffc6b45136a83e60faa2c2ecaef8af": "b^\\dagger(t)", "a5fff50b88542aba568dca4382a5b1a6": "\\xi^a = \\left( 1, 0, 0, 0 \\right) ", "a600330069981d1c1a5cb90e8b501902": "\\alpha_{\\mathrm{sun}}", "a6005a92b977f56ac85d1fc0528fe355": "g(x_1, x_2, \\dots, x_n) = c,", "a600f6a85e8b41c82961dad93211b339": "f_i(r_1,\\dots,r_{i-1},1)", "a600f7fa5f827b75c370c23e88df264e": "M=\\frac{y_1^2}{2}+\\frac{q^2}{g y_1}=\\frac{y_2^2}{2}+\\frac{q^2}{g y_2}", "a601090ff37296190629eb58bf7a0060": "\\begin{array}{c}\n2x^2 + 7x + 6 \\,=\\, 0 \\\\[6pt]\nx^2 + \\tfrac{7}{2}x + 3 \\,=\\, 0 \\\\[6pt]\n\\left(x+\\tfrac{7}{4}\\right)^2 - \\tfrac{1}{16} \\,=\\, 0 \\\\[6pt]\n\\left(x+\\tfrac{7}{4}\\right)^2 \\,=\\, \\tfrac{1}{16} \\\\[6pt]\nx+\\tfrac{7}{4} = \\tfrac{1}{4} \\quad\\text{or}\\quad x+\\tfrac{7}{4} = -\\tfrac{1}{4} \\\\[6pt]\nx = -\\tfrac{3}{2} \\quad\\text{or}\\quad x = -2.\n\\end{array}\n", "a60171f0fcd3886c88772cbe44f1f32f": "(1,3)_0", "a60192c44fa83cf567046a6a0e08af9a": " H = \\sum_{i_1} \\sum_{i_2} ... \\sum_{i_n} G_{i_1,i_2...i_n} ", "a601a88f53a15487e82a1c569d54991a": " \\text{If } |\\Delta P| > \\mathit{ \\iota}: \\quad \\frac{\\Delta F(P)}{\\Delta P}=\\frac{DF(P)}{DP}=F[P,P+\\Delta P].\\,\\!", "a601c66f2625e8f1345e153f3894d2d6": "\\scriptstyle W_p(\\mathbb{D})", "a602a2b653ecdca9796ad1aa2ccbbd13": " \\frac{\\delta}{2\\pi} \\lambda = 2d \\sin\\theta \\,\\!", "a602bbe2cebba1fae55e505f47f1587d": "\\scriptstyle 2\\,x+b\\;\\sim\\;b.", "a6030098624e9f4ac3806cd2e925a257": "\\displaystyle{E=E_1(e)\\oplus E_{1/2}(e)\\oplus E_0(e),\\,\\,\\,\\, A=A_1(e)\\oplus A_{1/2}(e)\\oplus A_0(e).}", "a60304f666f7d719fe7e7e39a1daafc9": "\\scriptstyle U_i\\subset X", "a6033118ccbb35600b58ff0104398d9a": "h_n = 2h_{n-1}+1", "a60398c646115eb3079d698cdcfd15b8": "k_{3(3)}\\equiv k_{3(2)})", "a603bc6f48d87708d65ddb9f0463eae9": "m<\\infty", "a60430d4f9bf8ff04d5ad41bba74e967": "x[m] = \\delta[m],", "a6045ed54a2828d775ec62fe3e255577": "\\tfrac{13}{17}", "a6045fbcf3a2a439d4fb39c826796097": "f(\\lambda_{1},\\lambda_{2})<\\lambda_{1}+\\lambda_{2}+\\lambda_{2}^{2})", "a6054dfb0dd20da3a3d09f6ba5a6c854": "\\begin{align}\n & \\left( \\exists D \\right)\\Leftrightarrow \\left( \\exists i,o \\right) \\\\ \n & \\left( \\exists S \\right)\\Leftrightarrow \\left( \\exists p,w \\right) \\\\ \n & \\left( \\exists R \\right)\\Leftrightarrow \\left( \\exists c,e \\right) \\\\ \n & \\left( \\exists P \\right)\\Leftrightarrow \\left( \\exists \\rho ,v \\right) \\\\ \n\\end{align}", "a605872c62f96167d6debfcef4c2944f": "A_n \\subset \\mathbb{R}, n=1,2,\\ldots", "a60589cd615586355dcb7dc549913c72": "\\mathrm{MA}=M_o \\times M_e", "a605936f310ec481f0421f239c014014": " \\mathbf{v} = v_1 \\mathbf{e}_1 + v_2 \\mathbf{e}_2 + v_3 \\mathbf{e}_3 ", "a605d8737a94068afeef707d74ad7b44": "S \\to A", "a60682763e99a41c216c8a610d23e1a3": "(M, \\cdot)", "a606c1a60e57f1fe2f7967ca8b6640cd": "\\sgn (x) = \\begin{cases}\n-1 & \\text{if } x < 0, \\\\\n0 & \\text{if } x = 0, \\\\\n1 & \\text{if } x > 0. \\end{cases}", "a606e25d090c290284fbd46755427e55": " \\lim_{s\\to 1} (s-1)\\zeta_K(s)=\\frac{2^{r_1}\\cdot(2\\pi)^{r_2}\\cdot h_K\\cdot \\operatorname{Reg}_K}{w_K \\cdot \\sqrt{|D_K|}}", "a606ea81ea11e1df4d6c1d4ed1ee4c1e": "[A,B] = AB - BA. \\, ", "a606f03b4356d840d98a83bb80695bdc": "10 \\lesssim \\mathit{l} \\lesssim 100", "a6071f3abcface7477f9b0778b370268": "y_2 = y_c = \\frac{2}{3}E_c = \\frac{2}{3}(3.47) = 2.31", "a6084e035c3993e1c31728a6c61baea7": "f(y) = x \\odot y", "a608792044ca1d9f3f039f38f6b43aed": "\\frac{P_1}{P_0}", "a608931838c69dca89fbfad09b356515": "q_{n+1}", "a608a3dcc815cfc8fcef6639d3295ddc": "\\rho_{solute} = \\frac{\\rho\\,b\\,M}{1+b M},\\ b=\\frac{\\rho_{solute}}{M (\\rho-\\rho_{solute})},", "a6091ec67847f738841fd4e9db9f6a61": "x_k\\,", "a60949547bdb4e47d79813c5494860e1": "\\vec x(t_n)", "a60987558fdbf082af4cd5e629b7c2f1": "x^2+a^2y^2-1=0", "a6098a64789a63d6f453ba373459950a": "\\sigma^2 = 6\\,D\\,t", "a609abcf824dffdb9cc7b31b5ac426a6": "7^2 + 8^2 + \\cdots + 16^2", "a609b8a10bfe0650c9d8b6e12311ae4a": "\\displaystyle i^2 = j^2 = k^2 = ijk = -1", "a609bbb4c90c2fc58a4e16c5b5dbe243": "P(r|s) = \\frac{1}{\\Zeta(s)} exp \\left ( \\sum_{i} h_i(s)r_i + \\frac{1}{2} \\sum_{i\\ne j} J_{ij}(s)r_ir_j \\right )", "a60a41de0ea5bac1fa52345336733d09": " C-\\text{vertex} = \\csc^2\\left(\\frac{A}{2}\\right) : \\csc^2\\left(\\frac{B}{2}\\right) : 0", "a60aad2fa038bb4be0eaa5aeebde0e2a": "\\zeta' \\approx \\zeta", "a60ae22a66c27997a7303095a2193e38": "\\nu = {c \\over \\lambda}", "a60af7c958840c1abd69f998673e6b37": "B = \\begin{bmatrix} -1/2 & 0 & -1/2 & \\sqrt 2/2 \\\\ 1/2 & \\sqrt 2/2 & -1/2 & 0 \\\\ -1/2 & 0 & -1/2 & -\\sqrt 2/2 \\\\ -1/2 & \\sqrt 2/2 & 1/2 & 0 \\end{bmatrix}", "a60b1e9ea3156c1b03971e82f9803b62": "|\\mu|", "a60b4d41ccfd6f0657b577ac32d1565b": "\n\\begin{align}\np_t & {} = {E} [1(X \\ge t)] \\\\\n& {} = \\int 1(x \\ge t) \\frac{f(x)}{f_*(x)} f_*(x) \\,dx \\\\\n& {} = {E_*} [1(X \\ge t) W(X)]\n\\end{align}\n", "a60b5af88cac2927bc7cff33a28989d9": "{\\rm as}_n(\\mathbf{x})", "a60b68e7944d4f73e0233c236840c343": "G = G_1 \\times \\cdots \\times G_r \\times H_{r+1} \\times\\cdots\\times H_l\\,", "a60b728620b81286c3eac76c4498d668": "\\Phi_{,i i} = 4 \\pi G \\rho \\,", "a60b9cc28bb7a04a7ebb975e258b3a8f": " V_{\\mathrm{eff}}(r) = V(r) + {\\hbar^2l(l+1) \\over 2m_0r^2},", "a60bb9b3f52422fca47bb04eaa9a5daa": "\\mathbb{Z} \\setminus \\{ -1, + 1 \\} = \\bigcup_{p \\mathrm{\\, prime}} S(p, 0).", "a60be2137ff7974a12c94e1a254eadac": "\\vec{r}(s=S,t)", "a60bf2389171f53ef216a41c1ec2f794": "\\neg A \\equiv A \\rightarrow \\bot", "a60c7b940e95afee73b53550859273c7": "K(\\alpha) = K_+(\\alpha)K_-(\\alpha)", "a60c855c8fb376f144df3a569005357e": "4.97 \\approx 5", "a60c855dd9f46d2613cb91b2c15ead5c": "\\bar{\\boldsymbol{u}}_S", "a60cfb9f9b6be7cea3f4c7300ee43a8d": "\\begin{align}\n\\hat{J}_x & = \\hat{L}_x + \\hat{S}_x\\\\\n\\hat{J}_y & = \\hat{L}_y + \\hat{S}_y\\\\\n\\hat{J}_z & = \\hat{L}_z + \\hat{S}_z\n\\end{align}", "a60d1b9936c78acf267d2e72cb5ae327": "A(s) = \\lim_{z \\rightarrow 1^{-}} \\sum_{n=0}^\\infty a_n z^n.", "a60d1f5efdb3aef33eaac8d60dfc50f5": "U \\in (0,1]", "a60ed8745a7694873504c727cb491628": "S_{t+\\ell} > H ", "a60f1c38bebafb94a2906a31d30ca402": "d(\\boldsymbol{x}, \\boldsymbol{a})< \\delta.", "a60f29683e65ac1db60924529dac5940": "X_n - E(X_n) = o_p(a_n)", "a60fbd461938a5eb68c758970a89ba8e": "(S\\otimes T)(v\\otimes w)=S(v)\\otimes T(w).", "a6100aa391ce3746a2314af312cc2795": "f^*(x) = h^*(x) + b", "a6103caa841ac65a4eac1259be77be1d": "V^2 = -(\\mathbf u \\wedge \\mathbf v \\wedge \\mathbf w)^2\n= -\\left(\\sum_{i0", "a61f6c5a1a6688f7cbe09a90780c723a": "\\vartheta(z+a+b\\tau;\\tau) = \\exp(-\\pi i b^2 \\tau -2 \\pi i b z)\\vartheta(z;\\tau)", "a61f8f02b1e33ca353ae32b2824891fd": "\\sin x=\\Omega_\\pm(1)\\ (x\\rightarrow\\infty)", "a61fda06946799cd1ebd25eb05a9cce4": "\\textstyle H_1: \\left\\{0,1\\right\\}^* \\rightarrow G_1^*", "a620129bcff8aa66d4bca5a0338f62c5": " 3.1415926 < \\pi <3.1415927", "a62045fb921259443a8f4c7567399417": "\\mathbf{A} \\cdot \\mathbf{B} = A^{\\mu} \\eta_{\\mu \\nu} B^{\\nu} ", "a620e98539e8f5bbae461fa59f29c285": " x_i , y_i", "a6212953490f9aa74cdfcdddf232528c": "\\{\\lambda_i,\\phi_i\\}_{i\\in\\{1,\\ldots,N\\}}", "a62145d9979e92ca6bff4245d2c95de1": "t_n= \\sin i \\ \\cos u\\,", "a6215dfcefb9e15fe13fa97d2b93299f": "L \\in H(\\mathbb{X})", "a6219109ae488b83e46f308c45cd97ed": "\\frac{d^3}{dx^3}[f(x)]=\\frac{d}{dx}[f''(x)]", "a621ad4922c3bdc555ce94f80e83c469": "\\int \\bold{x} \\bold{x}^T K(\\bold{x})^2 \\, d\\bold{x} = m_2(K) \\bold{I}_d", "a621d28a86e249a5ea89e644d8459812": "\n pq - \\varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \\,\n", "a62204cd6f23feea4e3e71e295201570": " \\int_{-\\pi}^{\\pi} \\rho(\\theta, \\omega, t) \\, d \\theta = 1. ", "a622407721fc62f3dd5ef594e8cda16b": "\nT_{ss}^{\\rm max} = \\sqrt{\\frac{2}{1-e}}\\ T,\n", "a622d56cfa47042909ba326bfe11e8b4": "\\boldsymbol{\\mu} = (\\mu_1,\\ldots,\\mu_N)", "a6232ae466f390462dc1dfd0349ad672": " \\nu_\\mathrm{1}", "a6232b9a3df1838307340c5266f9e014": "\n\\mu=\\frac{D}{\\sqrt{M}},\n", "a6232c4bad95edf92e91ac14fdf1e4de": "\\varphi_{rotor}=[L_{rs} ] i_{stator}+[L_{rr}]i_{rotor}", "a6233bab46d46858d4b8cec12f3055a7": "g(f_i)_{i \\in m}", "a62360449ed6d9f3983e152e6656835f": "\\gamma_{RN}", "a62378887830b05bfe5a0b8c50cd46d0": "|0\\rangle_{1}\\otimes|1\\rangle_{2}, |1\\rangle_{1}\\otimes|0\\rangle_{2}", "a62425146460b7fde21ac3498fb6d962": "(\\neg B \\to \\neg A)\\and A", "a62447fad9925a48f319dd9cb0a1caaf": "0=-\\frac{\\dot Q}{T_a} + \\dot m s_1 - \\dot m s_2.", "a62457a8156ff81ecff34708e193ccfd": "T(V_i) < V_i", "a624a7cb7e78b35b855d67a8a3c62d18": "\\min_{\\tau}\\left(\\sum_{t = 1}^T {(y_t - \\tau _t )^2 } + \\lambda \\sum_{t = 2}^{T - 1} {[(\\tau _{t+1} - \\tau _t) - (\\tau _t - \\tau _{t - 1} )]^2 }\\right).\\,", "a624b4280023c8b7e751d758a71a1461": "A = \\{ x \\in \\mathbb{Z} \\mid x^2 < 2\\}", "a624e892e158dad29ae51ecbcae96b35": "size\\ in\\ bytes = \\frac{size\\ in\\ bits}{8}", "a62564c59a5461f0461757302641505f": "\\partial_j", "a625d8775ea9046bfb2dc9f1d8468155": "\\frac{D \\mathbf{v'}}{D t'} = -\\nabla' p' + \\frac{\\mu}{\\rho D V} \\nabla'^2 \\mathbf{v'} + \\mathbf{f'} ", "a6260d4cf20f5b0fc30c30379532836b": " E_{i=2,us} = y_{i=1,us} + \\frac{q_{ss,us}^2}{2gy_{i=1,us}^2}=3.57 + \\frac{10.0^2}{2(32.2)3.57^2}= 3.69 \\text{ ft}", "a6260ed53b0305288150a30a198e91f3": "\\kappa x{:}1{\\to}\\tau\\;.\\;e", "a626def8a9c1a6a7b40dae32d4f34ac9": "(b,\\varepsilon)", "a627324c99d4aaa69ac98d8722e79889": "S = \\bigcup_{i=1}^N f_i(S).", "a62758c9cf361ed22e43663cb95b7844": "\\begin{matrix}\n\\hat{t} (t,\\omega) & = t - \n\t\\frac{\\iint \\tau \\cdot W_{x}(t-\\tau,\\omega -\\nu) \\cdot \\Phi(\\tau,\\nu) d\\tau d\\nu}\n\t\t{\\iint W_{x}(t-\\tau,\\omega -\\nu) \\cdot \\Phi(\\tau,\\nu) d\\tau d\\nu } \\\\\n\\hat{\\omega} (t,\\omega) & = \\omega - \n\t\\frac{\\iint \\nu \\cdot W_{x}(t-\\tau,\\omega -\\nu) \\cdot \\Phi(\\tau,\\nu) d\\tau d\\nu}\n\t\t{\\iint W_{x}(t-\\tau,\\omega -\\nu) \\cdot \\Phi(\\tau,\\nu) d\\tau d\\nu}\n\\end{matrix}", "a6275bfc377c77d15d48a39a7d8fe31a": "T^{\\mu\\nu}", "a6277367a3c8bffab71080c2d795b602": "q\\in \\{0,1,\\ldots,N-1\\}", "a627a803724448f3490f37171f09bd16": "\nRGD_i = \\left( e_i^t - h_i^t \\right) \\times \\left( g - G \\right)\n", "a627ba5654004a88b839b16f4a1d59b6": "\n\\begin{bmatrix} \n\\dot{x}_1 \\\\\n\\dot{x}_2 \n\\end{bmatrix}\n= \n\\mathbf{h}(x_1, x_2) \n:= \n\\begin{bmatrix} \nx_2 - F(x_1) \\\\\n-g(x_1)\n\\end{bmatrix}\n", "a627daaf148e020d8dda963f6ce2237c": "\\mathbf{a} \\times (\\mathbf{b} \\times \\mathbf{c}) = \\mathbf{b}(\\mathbf{a}\\cdot\\mathbf{c}) - \\mathbf{c}(\\mathbf{a}\\cdot\\mathbf{b})", "a627e52ad1454273f74e59bc8d97d518": " E'' = \\frac {\\sigma_0} {\\varepsilon_0} \\sin \\delta ", "a627eab405efc56773f1fb6ebd49c237": "Z(\\mbox{SU}(n)) = \\mbox{SU}(n) \\cap Z(\\mbox{U}(n)) \\cong \\mathbf{Z}/n", "a627ed2d321a283d338de1f3e68af768": " \\vec{r}_2 = \\vec{0} ", "a627ef1e09a3b37d611714ba3c53b586": "A_r\\subseteq M", "a6286455612f1407bdb26d3f6ab91ba1": "\n\\mathbf{a} =\na_1\\mathbf{e}_1 + a_2\\mathbf{e}_2 + a_3\\mathbf{e}_3 =\nu\\mathbf{n}_1 + v\\mathbf{n}_2 + w\\mathbf{n}_3\n", "a6286d77afebbafbbabf4dff2e1e3f65": "\\delta Q=dU-\\sum p_idV_i", "a628714dd2d5964c5d0bb56a94f0fd9a": "|v|:=\\sqrt{-\\eta_{\\mu \\nu}v^\\mu v^\\nu}.", "a6288e658fb3652482a4260f48dee65c": "f(x) = {\\sqrt{\\beta} \\over C_q} e_q(-\\beta x^2) ", "a62898dcd98c2c0ac3e318bd929ff07a": "c_1=g^y\\,", "a628cd17c7612a75c2362995439305f9": "\n2 c^{2} \\frac{d\\tau}{dq} \\delta \\frac{d\\tau}{dq} = \n2 \\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} \\frac{dt}{dq} \\delta \\frac{dt}{dq}\n\\,.", "a628f499b80467b09bccec62320d346c": "(w,z) \\in \\C^2", "a6290c166c6ffdc6e52d2922feb3e776": "\\, \\begin{matrix} {52 \\choose 5} = 2,598,960 \\end{matrix}", "a62945a7386f2efe69b547821188b952": " S(a,q,x) = \\frac{F(a,q,x) - F(a,q,-x)}{2 F^\\prime(a,q,0)}.", "a629889cff13ebf04a8ed1b40e375fd3": "n 0: \\pi(x+\\sqrt{x}) - \\pi(x) \\ge \\frac{c_2\\sqrt{x}}{\\log^{c_1}x}", "a629f2cab322552cadff1e1cd4193136": "M_r=\\frac{2}{\\pi}\\int_{0}^{\\frac{\\pi}{2}}\\!M(\\varphi)\\,d\\varphi\\!", "a62a3c7d0f6d19a64eb6432d71c7d235": " p(x|m) = \\exp (V(x)) ", "a62a41a8c06dcc0ab921c5ee040c18d0": "z^n - 1 = \\prod_{d\\,\\mid\\,n} \\Phi_d(z).", "a62a9a4e4213aa7fe4a6e410d90e6605": "\\mathcal{G}=\\mathcal{H}\\oplus\\mathcal{Q} ", "a62aa92af16fa26a3fda86127cd9def7": "\\Delta\\langle\\mathbf{p}\\rangle= q \\mathbf{E} \\tau.", "a62ababc7190e978578c0a97fc2ada44": "(x_2,y_2,z_2)", "a62b3dbd5886bc0a5b4e9ffad061d3a4": " A_{2}(n,d) \\leq 2 \\left\\lfloor\\frac{d+1}{2d+1-n}\\right\\rfloor. ", "a62b40d14e97e9911298ab5fddb0ace3": "n^{3/2}", "a62b4a2af26cc497cefd7cf78fd4fe7f": "\\{ (-1)^\\pi \\}", "a62be0d88c16a9d4cb9f15ec03858599": "a \\in s", "a62cae29d8789acfb199490cfc4080a1": "2.9558", "a62cb610b7b2a10f37cec224a6d2c907": "x^2 - 1 = 8", "a62cf5e774adad20e13a74e616d949f5": "a\\ne b = c \\ne d, \\alpha \\ne 90 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "a62d81de09f834de7323944b7854e8df": "{}^4_2", "a62dbc0fca23c10092f7f837640a1f86": " f = \\frac1T \\approx 110 \\text{ sec}^{-1}", "a62dd1cb75e473ae8ab42fced22f9331": "p_y=m\\dot{y}", "a62e09616ab0ab44fdcb238175b898b4": "i_{fX} \\omega = f\\,i_X\\omega", "a62e31add811b820795ba7723d795b3f": "\\scriptstyle x ", "a62eb26afddbc302560d66b4c445ae2c": "\\ N(f)", "a62f60c8a6cc2fb3793b6d90e1f1d952": "\\operatorname{cov}[G(s), G(t)] = E[G(s) G(t)] = \\min\\{F(s), F(t)\\} - F(s)F(t). \\,", "a62f67774304bd188bc43a7a8724ab93": " \\Omega(\\log n / \\log\\log n) ", "a62fbf6e850bbb53f41e19151120f926": "8^n-1=\\left(4^n+2^n+1\\right)\\left(2^n-1\\right)", "a62fe067cb7872b5d2d2dc538a8bae4f": "\\Omega_0^{-}", "a6300af8825a9dffcff9ae6be6fa5f46": "x_{n0}", "a6303e2ca49f70fa9e65808a0d2d5626": "e^z", "a630425d09c6ea77df4e662442c16d53": "\\displaystyle{B(a,b)=Q(a)Q(a^{-1}-b)=Q(b^{-1} -a)Q(b).}", "a63055797cd5476d872d1d52e0493262": "k = A e^{-E_a/RT}", "a6309e8f89cdf56fa402e3b97bd7d25e": "\\left|\\frac{1}{\\left(z-\\omega\\right)^{2}}-\\frac{1}{\\omega^{2}}\\right|=\\left|\\frac{2\\omega z-z^{2}}{\\omega^{2}\\left(\\omega-z\\right)^{2}}\\right|=\\left|\\frac{z\\left(2-\\frac{z}{\\omega}\\right)}{\\omega^{3}\\left(1-\\frac{z}{\\omega}\\right)^{2}}\\right|\\leq\\frac{10R}{\\left|\\omega\\right|^{3}}", "a630aefe6e3e91d75b3535dc324a31ac": "X_g", "a630dc56ab681e9862e2e508a8d710c3": " 0,1,2,\\ldots,n", "a6316c0df567fcb0001caa9b4966cd67": " efgh(a+c+b+d)(a+c-b-d) = (agh+cef+beh+dfg)(agh+cef-beh-dfg)", "a63171943556aeb22b5720bcfb1f5f67": "\\mathit{h} \\,", "a63270c988612bcf46b8308530280c70": " S_B = S \\cot B = ac \\cos B= \\frac {a^2+c^2-b^2} {2}\\,", "a632712a79ad1f1ae31b4b1f24210826": "\\mathbf{H}^{(M)}(\\mathbf{x})=-\\frac{i}{kZ_0}\\mathbf{\\nabla}\\times\\mathbf{E}^{(M)}(\\mathbf{x})", "a632ad3ba1eb127405db53a848f611e3": "d(x,y) + d(y,x) \\ge d(x,x)", "a63323bc3a8d1166b9ee386f509d1d9a": "\\cos[(k-\\Delta k)x-(\\omega-\\Delta\\omega)t]\\; +\\; \\cos[(k+\\Delta k)x-(\\omega+\\Delta\\omega)t] = 2\\; \\cos(\\Delta kx-\\Delta\\omega t)\\; \\cos(kx-\\omega t),", "a6333f7bcd0131a659e11c89794e32d0": "a_i\\geq 0", "a633a461d1b74b3109275b9ba95c672d": "\\begin{align}\nds^2 &= \\frac{|\\mathbf{Z}|^2|d\\mathbf{Z}|^2 - (\\bar{\\mathbf{Z}}\\cdot d\\mathbf{Z})(\\mathbf{Z}\\cdot d\\bar{\\mathbf{Z}})}{|\\mathbf{Z}|^4}\\\\\n&=\\frac{Z_\\alpha\\bar{Z}^\\alpha dZ_\\beta d\\bar{Z}^\\beta - \\bar{Z}^\\alpha Z_\\beta dZ_\\alpha d\\bar{Z}^\\beta}{(Z_\\alpha\\bar{Z}^\\alpha)^2}\\\\\n&= \\frac {2Z_{[\\alpha}dZ_{\\beta]} \\overline{Z}^{[\\alpha}\\overline{dZ}^{\\beta]}}\n{\\left( Z_\\alpha \\overline{Z}^\\alpha \\right)^2}.\n\\end{align}", "a633f1f7791db45fa3a463b2b441d976": "(a-i)^2=2(a^2+i^2)-(a+i)^2=4e^2-4e^2=0", "a63405538d0c945533455e5c66437e08": "M={S\\over R}={3n^2(6n^2+1)\\over 2n}", "a6341003325e8af39c6f571e644292ee": "\\overline{B(x,r)}", "a6343031d6093659a1dac37934c2b7c2": "f(x;a, k)= \\frac{a}{k}\\left(\\frac{x}{k}\\right)^{-a -1} I_{[k,\\infty]}(x)", "a6343cd969b11a2ecb1771deccf79bff": "\\forall x \\in A: Q(x)", "a634483974037b444b41e892db06398b": "\\Re(s) = 0", "a634f3f00914e812ee247f953699c9ff": "\\frac{1}{2} + \\sgn(x-\\mu)\\frac{\\gamma\\left[1/\\beta, \\left( \\frac{|x-\\mu|}{\\alpha} \\right)^\\beta\\right]}{2\\Gamma(1/\\beta)} ", "a6355bca8658bdd89e90f83de4dc02eb": "f(x+1)=xf(x) \\text{ for } x>0, \\,", "a6356253aabb6e233931ca86572f2690": "\\nu\\ll mc^2/h ", "a635817f508cbc292f3236869016faba": "AB\\,", "a635a6ae61114bde0e4ad01f240808f3": " \n\\int_{-\\infty}^\\infty W_x(t/2,f) e^{i2\\pi ft}\\,df =x(t)x^*(0) \n\\ \\ \\ \n\\int_{-\\infty}^\\infty W_x(t,f/2) e^{i2\\pi ft}\\,dt =X(f)X^*(0)", "a6364bdc0a6c0f2ae9644ff4d47993c6": "\\langle N\\rangle = 3.04\\times 10^{22}", "a63655af4937f3a6e482aac4bc296782": "{(x_\\lambda - m')}/{\\sigma})", "a6365961c99e6f584392ed4a4fcb7c40": "\\mathfrak P(M | X=x) = \\lim_{U \\ni x}\n \\frac {\\mathfrak P(M \\cap \\{X \\in U\\})}\n {\\mathfrak P(\\{X \\in U\\})}\n \\qquad \\textrm{and} \\qquad \\mathfrak P(M|X) = \\int_M d\\mathfrak P\\big(\\omega|X=X(\\omega)\\big),", "a636670ad411c390af4015ce8464d4f3": "T^{IJ} = {1 \\over 2} (T^{IJ} - {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\;\\; IJ} T^{KL}) + {1 \\over 2} (T^{IJ} + {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\;\\; IJ} T^{KL}) =\\;^+T^{IJ} +\\;^-T^{IJ}", "a636ab4d3db8ccfcda2fd45ee951f57c": "T_{2} ", "a636cd6da09f0d35dc267fbde04889cd": " \\{x_1, x_2,\\dots \\}=\\{x\\}_i,", "a636ef343eebe403491d0f5e7cb80fa3": "\\tilde c_n = \\langle \\tilde f, \\phi_n \\rangle", "a63709a2ae378e9b9b8dd621891409b8": "X \\in V \\to \\operatorname{make-call}[H, V] \\equiv \\operatorname{make-call}[H, V \\cap \\neg \\{X\\}]\\ X ", "a6372628f1018b5d1bb077bfcd547f6d": " \\hat{p} = + \\hat{z} \\,", "a6374e4ce21c12047b7062809fc6c02a": "x\\to p", "a637525e27481254665d4390800a7fa8": "G(s) = \\frac{S_{x,s}(s)}{S_x(s)}e^{\\alpha s}.", "a637d5bca51438132eda1837367be1bb": "\\theta(t) \\sim \\frac{t}{2}\\log \\frac{t}{2\\pi} - \\frac{t}{2} - \\frac{\\pi}{8}+\\frac{1}{48t}+ \\frac{7}{5760t^3}+\\cdots", "a63818a2ca5454d27e5b93f5437248c1": "m(\\mathbf{x})=-m(-\\mathbf{x})", "a6389506000bb27d099a211659cf7ee5": "B_{t}^{\\tau} (\\omega) \\equiv a ", "a6389dfbc67e910fc0d98b8e9b4603bd": "n=2^i\\,3^j", "a63910426222dcc3516008fd60ccda0a": "D_p\\;", "a6393a7553a21df6c79c9ee564318660": "{\\rm Homeo}(X)", "a63a1ed5f4a86e2388ef1ef572d17b78": "H^k_{DR}(V)", "a63a287af2d26b41739e57f701d9f209": "a, c", "a63a2a2477a5214477f88230c4eeb946": "T(s,\\mathbf{x})=\\frac{\\displaystyle \\sum_{i=0}^n a_i(\\mathbf{x}) s^i}{\\displaystyle \\sum_{i=0}^m b_i(\\mathbf{x}) s^i}=K\\frac{\\displaystyle \\prod_{i=1}^n (s-z_i(\\mathbf{x}))}{\\displaystyle \\prod_{i=1}^m (s-p_i(\\mathbf{x}))}", "a63a3e601ae69f792c9346c84c85a240": "\\mathbf{C}[[x^{1/n}]]", "a63bde43d24b236e3a40b797c36bb94b": "=\\prod_{i}\n\\sum_{m_{i}} (z_{i}/2)^{2 m_{i}} \\frac{(2 m_{i})!}{m_{i}!^2}=\n", "a63bf96d2f9217eac3a2efaa4217d415": "\\xi (m) \\Delta m= \\xi_{0}\\left(\\frac{m}{M_\\mathrm{sun}}\\right)^{-2.35}\\left(\\frac{\\Delta m}{M_\\mathrm{sun}}\\right).", "a63c14ba2b602e4afec8d823fb8db0d0": " \\bold x^{(m+1)} = (\\bold D - \\omega \\bold L)^{-1}[(1 - \\omega) \\bold D + \\omega \\bold U] \\bold x^{(m)} + \\omega (\\bold D - \\omega \\bold L)^{-1}\\bold k. \\quad (9) ", "a63c2b89f3af74c950a45f8a9d60ec1c": "i_r = i_s\\,", "a63c75557105c6d618d7aa77cd600e23": "\\Delta S\n= \\hat{c}_VNk\\ln\\left(\\frac{T}{T_0}\\right)+Nk\\ln\\left(\\frac{V}{V_0}\\right)\n", "a63c8678aed14aa07cadea35bfba35ca": "\\Delta E = E_2 - E_1 = hc \\left ( \\nu_2 - \\nu_1 \\right ) = hc \\left ( \\frac{1}{\\lambda_2} - \\frac{1}{\\lambda_1} \\right ) \\,\\!", "a63c990052d3d2e7d07311adee1452a6": "G=feed(lactose)", "a63cb295817e50dd292baab3004ed71e": "p(x), p^{(1)}(x),\\ldots,p^{(n)}(x)", "a63d4a26a033a226cdd2c5cce33fe4cb": "Wage=\\frac{Capital}{Population}", "a63d638c95461cb5b944b79f618a645c": "\\neg B_i \\bot", "a63d6f1d97ac53e5f53577037fe311c4": "\\omega^{\\beta_1} c_1 + \\omega^{\\beta_2}c_2 + \\cdots + \\omega^{\\beta_k}c_k", "a63d6f7c29f5b7244db12f2665e11b33": "C = \\partial U/\\partial T", "a63da340c0f9fc103710836533dd5f16": " \\frac{70}{r}", "a63db0d955dd43de63e564956aac30f1": "\\displaystyle{\\begin{pmatrix}\\alpha & 0 \\\\ 0 & \\alpha^{-1}\\end{pmatrix}(a,T,b)=(\\alpha^2 a,T,\\alpha^{-2}b),}", "a63e2f5add9c2913716cccba1572e942": " z \\partial_x - x \\partial_z. \\,\\!", "a63e336b90c6edeb3d1cf9531b4641f2": "Z^{D}_{i,j} = Z^{D}_{i-1,j} \\cdot e^{\\frac{\\beta}{T}} + Z^{M}_{i-1,j} \\cdot e^{\\frac{g(1)}{T}} + Z^{I}_{i-1,j} \\cdot e^{\\frac{g(1)}{T}}", "a63e48de6dc459b4b4d7ad160530ae24": "K \\ge \\chi^2_{\\alpha: g-1}", "a63e60eb500e82bc383a077f6f5a2234": "\\eta_{\\alpha\\beta} = \\eta_{\\mu'\\nu'} \\Lambda^{\\mu'}{}_\\alpha \\Lambda^{\\nu'}{}_\\beta \\!", "a63e9707b6d7836cc4c0682d3feae522": "G^{\\operatorname{ab}}", "a63e9a50d75d421ca2b8572834ea2d18": "\\eta_Y(y) \\in \\operatorname{Hom}_S (X, Y \\otimes_R X)", "a63ea5439519d751febee20ff684f4ed": "x^{\\alpha} = x_1^a\\, x_2^b\\, x_3^c", "a63ecb099414b69d52702e6cfe10a0fc": "y = AK\\,", "a63ee99602d44878baf773c87469a47c": "a \\le x < b", "a63eff842422c5e28828c9dd701471eb": " \\pi_x \\ ", "a63f493401e5bb6b5db59ba4bfa3442a": "2^{k-l}", "a63f571d19fa6557865fb0a88ef87bb1": "\\sum_{n=1}^{\\infty} \\frac{3^n - 1}{4^n}\\, \\zeta(n+1) = \\pi\\!", "a63f89da49d64eb2a27d944643bb95d3": " \\frac{V_D}{V_T} = \\frac{PaCO_2 - P\\bar{E}CO_2}{PaCO_2}", "a63f8c9f7e74b5099baf5d8e9fffac83": "\\boldsymbol\\theta .", "a63fdee10e010e4a77d24b6734b1c09c": "g^2(q;\\tau) = \\frac{\\langle I(t)I(t+\\tau)\\rangle}{\\langle I(t)\\rangle^2}", "a6401d28f6a727919c11490c312e0246": "\\left[\\hat{f}_i, \\hat{f}_j^\\dagger \\right]_+ = \\delta_{ij} ", "a6403b4c928b784a3a5327d59700c831": "\\,a^2 + b^2 - 2ab\\cos(\\gamma) = c^2.", "a6407f509633762eeb79033210bb6d37": "\\rho_B", "a640a957aef3d9a4babdf8d00cfb2b66": " X(a,b) = \\frac{1}{\\sqrt{a}}\\int_{-\\infty}^{\\infty}\\overline{\\Psi\\left(\\frac{t - b}{a}\\right)} x(t)\\, dt ", "a640c14b022db91941b475d498031f94": "\n\\partial_t \\rho+\\partial_i(\\rho u_i)=0\\,\n", "a640e4c267713e50c20105c7d6be5ce3": "I_s\\approx{0.7}LRC", "a640f8797246adef2fd8dbc4d35912ab": "M\\hookrightarrow M\\times M", "a640f99c21ab96bb346297df4f3cbd90": "\\lambda = \\frac{kT}{mD}", "a6412d285dbf7c24e1598636318fad95": "\\mathbf{p}^{\\prime} \\ne \\mathbf{p}", "a64144bc27ab5c5727f2cde048ef1f81": "K \\le r(r+\\sqrt{4R^2+r^2})", "a641468e2e02bdda244270441a79d195": "= a(x, \\sigma(x), \\sigma'(x),\\sigma''(x))dx + b(x, \\sigma(x),\\sigma'(x),\\sigma''(x))d(\\sigma(x))+ \\,", "a6414ce24d1213d445005a4f85ff9042": "\nv_1 = 1.\n\\,\\!", "a641ae442b91f81a23da6050773b0281": "g^{x+r ~\\bmod~ (p-1)} ~\\bmod~ p", "a64229feb32051ae1e7521a2c3f9618e": "f(x;\\mu,\\sigma)=\n\\frac{1}{\\sigma\\sqrt{2\\pi}} \\, \\exp \\left( -\\frac{(-x-\\mu)^2}{2\\sigma^2} \\right)\n+ \\frac{1}{\\sigma\\sqrt{2\\pi}} \\, \\exp \\left( -\\frac{(x-\\mu)^2}{2\\sigma^2} \\right)\\qquad(x \\ge 0\\,)", "a64260ce4ed110422623818fcc35ad7a": "\n\\frac{\\partial}{\\partial x} I(x,t) =\n-C \\frac{\\partial}{\\partial t} V(x,t) - G V(x,t)\n", "a642ac044d7a54b3d809a3695608dd47": " \\sum_{w\\in S_n} \\sgn(w)\\frac{n!}{(l_1-w(1)+1)!(l_2-w(2)+1)!\\cdots (l_k-w(k)+1)!} ", "a642c2beabc2de28ea20ca3e1f89c26c": "x_r", "a642d2a2763206f65ab8e89d6dd21e09": "c_{\\mathrm{ideal}} ", "a64305218c305666c478f0e4ef65fcbd": "S_n=\\int_{t_0}^t{dt_1\\int_{t_0}^{t_1}{dt_2\\cdots\\int_{t_0}^{t_{n-1}}{dt_nK(t_1, t_2,\\dots,t_n)}}}.", "a6437dcfbb52cbdd059c54b615b66fb9": "\n \\epsilon^p_i = \\lambda~\\cfrac{\\partial f}{\\partial \\sigma_i} \\qquad \\implies \\qquad\n \\cfrac{d\\epsilon^p_i}{d\\lambda} = \\cfrac{\\partial f}{\\partial \\sigma_i} ~.\n ", "a643a0ef5974b64678111d03125054fc": "\\Sigma ", "a643a58cbd99e83af968b0ca0854cf94": "Z = Y (1- x - y)/y\\,", "a643aa6f069d5c154e2a58e23a56cab7": "h^0(X,K)=g", "a643d43dd44fa05bce9bb98f08cfe992": "\\scriptstyle\\mathbb{R}^n", "a6441f0242954464393cf80fccecdc9b": "\\omega_2", "a64483bce976587544390d060795e0ec": "Y=F(A,K,L)", "a6448e6723084417cd3c425a63e221ae": "x \\equiv \\pm (p+y)/2", "a64491314a0e79b22762380bd6175afd": "= S_n - \\frac1{b_n}\\sum_{k=1}^{N-1}(b_{k+1} - b_k)S_k - \\frac{b_n-b_N}{b_n}s - \\frac1{b_n}\\sum_{k=N}^{n-1}(b_{k+1} - b_k)(S_k - s).", "a644935855a9716fc1435c290d6ee0c0": "x \\leftarrow y|_x", "a6449d047e9465eaa83128739f3eaf41": "\\text{E}[-e^{-aW}]", "a644ea02f3c908141dfb0c4e817b0b72": "\\alpha = (\\alpha_1, \\alpha_2,\\ldots,\\alpha_n)", "a64571af8419f1c45035511a309ab253": "\\bar{v} = \\sqrt{\\frac{8 R T}{\\pi m} }", "a645850910d53246c0e6691e336f0f7d": "[a]\\oplus[b] = [a+b]", "a645df10a46de0e3de3b981b44683e7b": " p_Z(z)= \\frac{b(z) \\cdot c(z)}{a^3(z)} \\frac{1}{\\sqrt{2 \\pi} \\sigma_x \\sigma_y} \\left[2 \\Phi \\left( \\frac{b(z)}{a(z)}\\right) - 1 \\right] + \\frac{1}{a^2(z) \\cdot \\pi \\sigma_x \\sigma_y } e^{- \\frac{1}{2} \\left( \\frac{\\mu_x^2}{\\sigma_x^2} + \\frac{\\mu_y^2}{\\sigma_y^2} \\right)} ", "a6465c0244621c63e7e1e96eb55aad7a": "\\leftarrow ", "a646ac00363e48a887bf64b3baa9ebf1": " \\rho(X,A,C)", "a646e57ee08b548063f4d10076ae52cd": "\\textstyle E\\left( \\left\\vert x_{i}\\right\\vert ^{p}\\right) <+\\infty", "a64754b720993e4b7dd4c0d902bf2986": "I^+(\\varepsilon,t,f,dg)={1\\over\\varepsilon}\\int_0^t f(s)(g(s)-g(s-\\varepsilon)) \\, ds", "a647745fbf2cc674cf09dfce3a18017a": "a = \\sqrt[n]{x^m}", "a647e3290ee5c3810bbfa7f3c3be8263": "\\left\\langle -2, Z_2 \\right\\rangle", "a648031e2549f6eb4306535adf82d80a": "\\mathbf{x}_\\mathrm{com} = \\frac{\\sum_n m_n\\mathbf{x}_n}{\\sum_n m_n}", "a648077f46de43208cd3e45580e11caf": "||y||_V \\leq 1", "a64840e1ab7d20c68f33d443ef074f9f": "r \\geq 0", "a648883aee73ee28918078668c891957": "I^+(E)", "a6490f2b4f73da80dd799c0b8d7d0bd0": "r = 0111 \\oplus 0001 \\oplus 1001 = 1111", "a6496c80ca8c7ca3548879647fb7cfa8": " y=\\int_{-\\infty}^t \\exp(s-t)\\,dW(s) ", "a6497cff8ed2bd9e7571279f356fd3f7": "u\\cdot v=0, \\,\\,\\|u\\|=1,\\,\\,\\|v\\|=1.", "a64985db472b69433082b1ac7140d869": " V_\\lambda = \\{\\,v \\in V: A v = \\lambda v\\,\\}", "a6498b375f0ab78b16f2d56da34ddc9b": "j=-\\sigma \\nabla \\phi", "a64a0189a7035f3dbe75904352bddea1": "p(t) = \\det(t I_n - A) =\n\\begin{vmatrix}t-a_{1,1}&-a_{1,2}&\\cdots&-a_{1,n}\\\\\n-a_{2,1}&t-a_{2,2}&\\cdots&-a_{2,n}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n-a_{n,1}&-a_{n,2}& \\cdots& t-a_{n,n}\\\\ \\end{vmatrix} = t^n+c_{n-1}t^{n-1}+\\cdots+c_1t+c_0,", "a64a1bbfe6e65f90705a65f172626d16": "(X - 1)(X + 1) = 0", "a64a2cdceb3a515e80f25ae0d37425f0": "\n|\\langle \\mathbf{s}_i\\cdot \\mathbf{s}_j\\rangle|\n\\ge\\frac{C(\\beta)}{1+|i-j|^{\\eta(\\beta)}}\n", "a64a48ce7ac931ae8af0d4576a417ad6": "P.MPL", "a64a7cb5f63f78fc97a1371bd3c72404": "I_s = I_c \\sin(\\phi)\\,", "a64b06b383ebf41f1099686f37035da2": "n=\\int f\\,d^3p", "a64b3da4f4069eef67dcaa9a99e7fe07": "\\rho_c(\\mu^2)", "a64b4c388756c9a2c6ea37a0c95a5ae2": "\\alpha = \\frac{2\\omega k_0} {c}={{\\omega\\kappa_2} \\over{n_0c}}.", "a64bb0b388305ca5706aec352759f279": "\\mathcal{} X", "a64c0c7937f3d37503ed0ead3ec5e018": "H^{0}_k", "a64c10474c56d81201789bb634bf8b76": " -e^{-q \\tau} \\frac{S \\Phi(d_1) \\sigma}{2 \\sqrt{\\tau}} + rKe^{-r \\tau}\\Phi(-d_2) - qSe^{-q \\tau}\\Phi(-d_1)\\, ", "a64c1445f11cf1dab40a96055d38747e": " Q = \\frac{U Ar \\int^{\\Delta T(B)}_{\\Delta T(A)} \\frac{1}{K}\\,d(\\Delta T)}{\\int^{\\Delta T(B)}_{\\Delta T(A)} \\frac{1}{K \\Delta T}\\,d(\\Delta T)} ", "a64c6879098a0bf17518fb0d784b7832": "\\mathrm{j}_z", "a64c6884190164a4d47c240652b3bb51": "\\left \\{ a_n \\right \\}, \\left \\{ b_n \\right \\} > 0", "a64c731347e25f2570a3c796e413126e": " \\Big( \\frac{\\partial}{\\partial t} + \\frac12 \\frac{\\partial^2}{\\partial x^2} \\Big) f(t,x) = 0. ", "a64cad7a1ab3550c023b287fa928f7b0": "I = I_0 \\left ( \\frac{2 J_1(ka \\sin \\theta)}{ka \\sin \\theta} \\right )^2", "a64caf0ab60b0519dfc0fa263523febd": "\\ell(Y)", "a64cbd5ca7f73d9b7e27a0ff1de2a09d": "\\textstyle \\{(x,y) : x \\geq 0, \\vert y \\vert \\leq x / \\sqrt{2}\\}", "a64cf5823262686e1a28b2245be34ce0": "He", "a64d5a92cf7b26e1b095924794faf1ec": "\\nabla \\times \\mathbf{B} = \\mu_0 \\mathbf{J} + \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}}{\\partial t} \\ ", "a64e660e5236422c6ee6a5536e523a0c": "A_h=dx^\\lambda\\otimes[\\partial_\\lambda+((A^i_m\\circ h)\\partial_\\lambda h^m\n+(A\\circ h)^i_\\lambda)\\partial_i] ", "a64e8d5a94e92ef9fdf4bfdf4018e1bb": "B=e", "a64ec403cfd0831ba5ff946cebcf986a": "\n \\frac{\\partial^4 w^0}{\\partial x_1^4} + 2\\frac{\\partial^4 w^0}{\\partial x_1^2 \\partial x_2^2} + \\frac{\\partial^4 w^0}{\\partial x_2^4} = 0 \\,.\n ", "a64f571756a3e754dcea7616882c0559": "a_{n+2} = 2 a_{n+1} - a_n", "a64f590ad422a0738202480cc1842853": "\\frac{{10\\sqrt{a}}}{\\sqrt{a}^2} = \\frac{10\\sqrt{a}}{a}", "a64fee598e86052e00f6d876b0689274": " s \\, ", "a65002864990aaee55d7102f563a8982": "x_n<\\Lambda\\quad\\forall n\\geq n_0.", "a6500b9d12aa7272b0299048f4a0dc55": "\\text{FOV} = 4 \\cdot \\arcsin \\left(\\frac{\\text{frame size}}{4\\cdot \\text{focal length}}\\right)", "a650399f26f5dacd47c50707f410fbc6": "\\gamma,\\alpha,{\\it{i,j}}", "a650505836492a0313c662403ebdcc9d": "\\sqrt{T_HT_L}", "a65065e9eeb0b0855c5720f5a54ef912": "\nW_p = \\sum_{x_i \\in \\Omega}{f_r(\\|I(x_i)-I(x)\\|)g_s(\\|x_i-x\\|)}\n", "a650abd3fe79bb15c4e7dc96d09127d6": "\\begin{align}\n\\frac{dy}{du} &= f'(u) = e^u, \\\\\n\\frac{du}{dv} &= g'(v) = \\cos v, \\\\\n\\frac{dv}{dx} &= h'(x) = 2x.\n\\end{align}", "a65106c6a9a0354a201727c78cdb0857": "T(t,\\sigma)", "a6512a3fcc8f9a83b1cbe7b8d9b9b032": " \\begin{pmatrix} 1 & 0 \\\\ -\\frac{1}{f} & 1 \\end{pmatrix} ", "a65198468a7df4d2cd10e180b1f1487b": "\\sqrt2.", "a6519dba922437008a8af0a047802282": "\np_i({\\theta})=c_i + \\frac{1-c_i}{1+e^{-a_i({\\theta}-b_i)}}\n", "a651e45f3dba378516c58a9959018d2a": "00", "a66c1842e6875258a182236029eaf8bf": "q_i = 1-p_i", "a66c90a3e92d26c899f0ed5e261d83a3": "H_0: \\hat{\\rho}_{XY\\cdot\\mathbf{Z}} = 0", "a66d10d09dbe25236e870a7917ff0751": "\\frac{-d}{2c}(1-i\\sqrt{3}).", "a66dd920631b4dbdcdbfd1e603f00cd1": "\\mathbf{P}\\cdot\\nabla^{2}\\mathbf{Q}-\\mathbf{Q}\\cdot\\nabla^{2}\\mathbf{P}=\n\\nabla\\cdot\\left[\\mathbf{P}\\left(\\nabla\\cdot\\mathbf{Q}\\right)-\\mathbf{Q}\\left(\\nabla\\cdot\\mathbf{P}\\right)-\\nabla\\times\\left(\\mathbf{P}\\times\\mathbf{Q}\\right)+\\mathbf{P}\\times\\nabla\\times\\mathbf{Q}-\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}\\right].", "a66e22070f529af8c6beb04a22c2d431": "\n\\text{Prob}(x_j)\\text{ is selected for emigration} = \\frac{\\mu_j}{\\sum_{i=1}^N \\mu_i}\n", "a66eb896da032ac9e3f1b88fad3aca38": "\\, \\left(\\begin{bmatrix}A_{r\\overline{o}} & A_{12}\\\\ 0 & A_{ro}\\end{bmatrix},\\begin{bmatrix}B_{r\\overline{o}} \\\\ B_{ro}\\end{bmatrix},\\begin{bmatrix}0 & C_{ro}\\end{bmatrix}, D\\right)", "a66ee9ee7212ac090fab711b14aa686f": "\\displaystyle (A_\\alpha,X)=\\alpha(X)", "a66f0b11d2b011a888003268ace1af5b": "x^{\\alpha} = \\inf\\{x \\in \\mathbb{R}: \\Pr(X \\leq x) > \\alpha\\}", "a66f474f31658f3d5d320fe446b0cf3b": "v_a=\\frac{m_a u_a + m_b u_b + m_b C_R(u_b-u_a)}{m_a+m_b}", "a66f7de1c059bd6e88900e18f5f4e2b9": "G_{k+1}", "a66fdc05b05623db756209af7ebb50bd": "\\left\\{\\mathcal{F} f\\right\\}(-s) = \\left\\{\\mathcal{B} f\\right\\}(-is) \n= \\left\\{\\mathcal{M} f(-\\ln x)\\right\\}(-is).", "a66fe09bd064ecc42c208f21ed8d7b72": "\\forall_f\\colon \\mathcal{P}X\\to \\mathcal{P}Y", "a66ff898b9cc294d387d15970dc48986": "=-\\omega^2 R e^{i\\theta} + \\dot \\omega e^{i\\frac{\\pi}{2}}R e^{i\\theta} \\ .", "a67011ffca36d611c75cc64b61dc4160": "u''+ (1/2) fu=0.", "a6706b673fd1da9e9d8401938a5bbd06": "x \\isin \\Sigma", "a670aba6fc772bef3c4d939f7a79c0e8": " S \\mapsto \\sum_\\lambda \\operatorname{E}_A(\\lambda) S \\operatorname{E}_A(\\lambda)\\ .", "a6711d60d9e5b9ea4b6ff179845be48c": " y(t) = \\cos \\left( 2 \\pi ( f_{c} + \\Beta \\cos \\left( 2 \\pi f_{m} t \\right) ) t \\right)\\,", "a67127a74c6f744f9ea34cd51206472f": "y=ax+c", "a671a3cf4649f5d06d2e4bbc583d1d31": "1 + 8 + 30 + 80 + \\cdots + {n^2(n + 1)(n + 2)\\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\\over 5!}", "a671e6ad640aff328953720ca8459d5d": "\\, \\frac{d^3}{dx^3}G(x)\\geq 0\\, ", "a6720251cf16fd2a48ab81c369bedeac": "d < |\\mathbb{F}|", "a672100b9ee9cfbb91f208c7568f92a8": "\\ln \\frac{x_m}{\\alpha} + 1 + \\frac{1}{\\alpha}", "a6728239bf09267b27bff5fdb6ac058e": "10^{10^9}", "a67325f52aa4cf7eb45f39b262d5614d": "N_k(n)", "a6732e3d8196a1ea571ac339f7014a3a": "\\left| R \\left( x \\right) \\right| = R\\left( x \\right)", "a673647292150eaaf675a562669d408c": "M_{UT}=\\frac{1}{4}\\Sigma^4_{i=1}({m^+}_i-m_{UT})^2", "a67389b1e6109ae284d1d433dc85c049": "\\hat{a}_i^\\dagger", "a673ec3b2d75e947b7413bd28b96c356": "\n\\begin{align}\n\\\\\nc_i& = \\frac{1}{N} \\sum_{j=1}^{N} \\frac{x_{L(j-1)+i}}{A_j} \\quad \\forall i& = 1,2,\\ldots,L \\\\\n\\end{align}\n", "a673ed74499879f9647f91bccb2b8472": "\\operatorname{Pref} (\\operatorname{Pref} (L)) =\\operatorname{Pref} (L)", "a6741a3de9d4c7baad764697e3d84f5f": "\\begin{array}{ccl}\n\\vec \\nabla \\cdot \\vec E & = & 4 \\pi k_{\\rm C} \\rho \\\\\n\\vec \\nabla \\cdot \\vec B & = & 0 \\\\\n\\vec \\nabla \\times \\vec E & = & \\displaystyle{- \\alpha_{\\rm L} \\frac{\\partial \\vec B}{\\partial t}} \\\\\n\\vec \\nabla \\times \\vec B & = & \\displaystyle{4 \\pi \\alpha_{\\rm B} \\vec J + \\frac{\\alpha_{\\rm B}}{k_{\\rm C}}\\frac{\\partial \\vec E}{\\partial t}}\n\\end{array}", "a674b977ea55e441a0ab0f1721e0c623": " I'(x) = f(x), \\quad I(a) = 0. ", "a674d276f4f65a0e80a9887ab79f06bc": "{(y \\cdot r^e)}^d = y^d \\cdot r^{e\\cdot d} = y^d \\cdot r", "a67592c42bd8f5a98ba7752f0e3305b4": "\nX=\\sum_{i=1}^N Y_i \\phi_i=\\sum_{i=1}^N \\langle \\phi_i,X\\rangle \\phi_i\n", "a675de7358bcbe58cf38b4929b8aa2bc": "|P_1(V)|<|P(V)|\\ll|V|", "a675dfd78200814963f628a7d8e3bb27": "Z = R + jX \\,", "a6760a698454c62e6600989fd42cc19f": "(k_{AF})^\\kappa", "a676102195c43f5deeef4a685a802d0f": "U_\\alpha e_\\beta= \\varepsilon(\\alpha,\\beta)e_{\\alpha+\\beta},", "a6762a77eaa01fd777ed5c50ac33a11a": "x(t) = \\delta(t-t_1) \\,", "a6765c237eecb1c2e4b085aadf527304": "\\arccos z = \\frac{\\pi}{2} - \\arcsin z \\quad z \\neq -1, +1 \\,", "a6766312968d8639122fee8df4546728": "L(x,\\lambda) = \\tfrac{1}{2} x^{T}Qx + \\lambda^{T}(Ax-b). ", "a676bb7d0eeeeda51f857540158ad559": "\\Omega^G_n(X)", "a6770af25692b5ee1da207796d8db1dc": "\\theta(\\bold{\\hat{n}})", "a6770c126a04ef8a73b58adadf55025a": "\\varphi(t):=\\int_B \\operatorname{det} D g^t(x) dx", "a6770d397eeb8ea8b95ddd8c8d1052dc": "-0.978", "a677113d5fcee81678113dc4428f45e7": "r_i = U_i^T \\Sigma_{XX} U_i.\\,", "a6771975d8d8c2571eed38089c879db5": "V_1=\\sum_{i=1}^m{w^{i-1}} = \\frac {1-w^{m}}{1-w},", "a67732f7c83afe009d6ab4fbf8413552": "\\scriptstyle \\vec r", "a677373dba1aa2b19556444cffd5c87b": "P\\left\\{\\sup_{0\\leq t\\leq 1}|\\alpha_{U,n}(t)-B_n(t)|>\\frac{1}{\\sqrt{n}}(a\\log n+x)\\right\\}\\leq b e^{-cx}", "a67755d4ce31db344f7ce0cf1a45c86a": "\\mathbf{E_{1}} = \\mathbf{E_{2}} ", "a677609bf6a2245f0a49d7b67e6f8373": " f \\colon \\Omega \\rightarrow \\mathbb{R} ", "a6776590d85708d78398dac7b2b30891": "H_i(X; \\mathbf{Z})", "a67790e4c975f2895973f5b8fa1a9e89": "\\left(\\dfrac{dn}{2}\\right) \\left(\\gamma^2 + (\\dfrac{\\lambda}{d})\\gamma \\left(1-\\gamma\\right)\\right)", "a6779e2b9c2dd444f0d1aac4a3418fd0": "\\psi( \\mathbf r )= \\sum_{\\mathbf k } F( \\mathbf k ) e^{i\\mathbf{k\\cdot r}}u_{\\mathbf {k}}(\\mathbf r ) \\ , ", "a677fc4251a9d32ff1a63ac963f36aea": "\\int x\\cos x \\, dx.\\!", "a6780b2e87a9e07c5eb747f835908e48": "x_{22}=p_2q_2+D", "a67822518d9bd046c95571ae0308e9fe": "(t=1,2,...,n)", "a67823fe277766c4543c4acb4d407cb4": "{\\mathbf t}_0 ,\\dots ,{\\mathbf t}_{N-1}", "a67844e37571c4778ffc6838aa695a6a": "\\ln q^*(\\mathbf{Z})", "a678611e1e59eab6e28d72b856101995": "V = AS\\sqrt{t},", "a6788b54de9224a62279425e8184bab3": "\n \\psi(x,y,z,t) = \\psi_{0}e^{i \\left(\\omega t - k_{z} z - k_{x} x - k_{y} y\\right)}.\n", "a678c774a1df97681047273196edecbe": "\\Pr(v\\in S) \\geq \\frac{1}{k}", "a678f80e71fa80fff691c98179e5a5bc": "\\frac{1}{Nf(j)} = \\frac{1}{n(j)} = \\frac{m(j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)}", "a678fa90db5a86c46b391eb2604dd21d": "\\overline{f}([x])=f(x)", "a678faeb7ec503ddeb7ad5e8afd12cfc": "\n\\begin{array}{l} E(f)=A(f)X(f) \\\\ X(f)=A^{-1}(f)E(f)=H(f)E(f) \\end{array}\n", "a679137254568522865a69639171af2f": " x = \\Phi^{-1}( \\Phi(\\alpha) + U * (\\Phi(\\beta)-\\Phi(\\alpha)))\\sigma + \\mu ", "a679382423780e66b841385f1e3e93e7": " \\Delta P = \\frac{32 \\mu L ~{u_z}_\\mathrm{avg}}{D^2}. ", "a67a029df5a1fdbcb13acd5c31f816de": "p(r_1,\\dots,r_m)", "a67a08a3d2312f05f9a2bca169b3df99": "w=w^0", "a67a61d4b95569fd122eb83481db38e7": "(P_1, ..., P_n)", "a67a983fc61162252125a99921aa5a29": "f(\\Theta=D) = \\frac{1}{Z} \\exp\\left\\{-\\frac{1}{2}(D-\\mu)^T\\Sigma^{-1}(D-\\mu)\\right\\}", "a67aff9a46b875db144cedd8607ce665": "1-2^{1-s}", "a67b0cba12863b1eb3942186b7e70711": "\\chi_{\\varphi}(g) = \\mathrm{Tr}(\\varphi(g))\\,", "a67b1c16fea0e75d333f0725add3bdb6": "H[1,n]", "a67b6fb7c84a3a4f4c6bbe030915c015": "C(3, 3) = 1", "a67ba6f08f095019841438802100fada": "\\operatorname{tr}(\\gamma^\\nu)=\\frac{1}{\\eta^{\\mu\\mu}}\\operatorname{tr}(\\gamma^\\nu\\gamma^\\mu\\gamma^\\mu)", "a67bb9c7854cc9ad653c07e7d3e473d1": "{\\color{Red}\\bar{\\infty}m} = \\tfrac{\\infty}{m}m", "a67bc5248bf6285d974b11cd6c22ef43": "\\,\\operatorname{lcm}(a,b)^{-1} = \\gcd(a^{-1},b^{-1}) ", "a67bef2546f863d8f7a845128649b63e": "X''=(X')'", "a67c33eb22be4df9e53187bc98209029": "\nc \\int_{t_\\mathrm{then}+\\lambda_\\mathrm{then}/c}^{t_\\mathrm{now}+\\lambda_\\mathrm{now}/c} \\frac{dt}{a}\\; =\nc \\int_{t_\\mathrm{then}}^{t_\\mathrm{now}} \\frac{dt}{a}\\,\n", "a67c3e7d36b503341ec5beae6cb752f2": "v_f\\,", "a67c78796049ddbb01efcfa3a71483e6": "F_6", "a67c8402dc732199efe4dc055637c3e4": "\\scriptstyle{ x \\approx y }", "a67ccb0cf6c5444233e2fba250df3ffa": "\\varphi_e", "a67d552621cf03d1187e1d08755cc72b": "\\lambda'_i = \\lambda_{\\pi^{-1}(i)}", "a67d56dcf1700db959833d519553c9ca": " \\langle S^{2}\\rangle_{\\mathrm{ROHF}} = \\langle S^{2}\\rangle_{\\mathrm{exact}} =\\left(\\frac{N_{\\alpha}-N_{\\beta}}{2}\\right)\\left(\\frac{N_{\\alpha}-N_{\\beta}}{2}+1\\right).", "a67d9c490856e573b7f3b61a5eeb9ed9": "\\chi_e = n^2 - 1", "a67dc70b6ec59b2c1c52115bd2b76fe4": "a_1 = |\\mathbf{a}| \\cos \\theta = |\\mathbf{a}| \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{a}| \\, |\\mathbf{b}|} = \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{b}| }\\,", "a67deab68cb7569b238ca7b691d1f05f": "P' \\ne P", "a67e96e50e7709695ca9d1a0f29d8399": "x^4 + y^4 = a^4", "a67ee7ea40586248bab61274d2bc6fcf": "S^{-1}A S=\\Lambda_{A}", "a67f0b0ca0de2f107720a40f7f9de537": "O(A_1:A_2|B) = \\Lambda(A_1:A_2|B) \\cdot O(A_1:A_2) ,", "a67f10fafb9245df8bb55656e0ca13b2": "E_\\mathrm{cell}^\\ominus = E^\\ominus\\left( \\mathrm{Cat} \\right ) - E^\\ominus\\left( \\mathrm{An} \\right ) ", "a67f8061519dc5f6e36d54d4aa5fdc4c": "p^0", "a6801f99c1780c614a96af3f2af45428": "f(L) = L'", "a68073aef279593b52d59b7a4d2e01a7": "\\tau=\\omega_1/\\omega_2", "a680bf1d1a675fcd8d5989fd6afc1cd8": "\n\\operatorname{corr}(\\hat{\\beta},\\hat{\\theta})= \\frac{\\operatorname{cov}(\\hat{\\beta},\\hat{\\theta})}{\\sqrt{\\operatorname{var}[\\hat{\\beta}]\\operatorname{var}[\\hat{\\theta}}]}\n", "a681819113f23590665299609c6d8341": " S_B \\equiv S \\backslash S_A,", "a6819e96f4fd1aacf748defae51fb441": "\\frac{1}{\\beta}\\sum_{i\\omega_n} \\ln(\\beta(-i\\omega_n+\\xi))=\\frac{1}{\\beta}\\ln(1-e^{-\\beta\\xi})", "a681a2e13f602f4da4f1373744e2ebe3": "(A \\cap B)^C = A^C \\cup B^C\\,\\!", "a681cd16547d700744919f9324fcd314": "{g(x)}", "a681d71864e165f2aaef3fe802243e27": "\\pi^{-1}(b)", "a6821b96922ae5ec1fc54f446a913125": "f_\\mu:=\\mu\\min\\{x,\\,1-x\\},", "a68226db91213c17bae92519600d6ae8": " P(E|S_i) \\leq P(\\bigcup_{i \\neq i'} A_{i'}) \\leq \\left ( \\sum_{i \\neq i'} P(A_{i'}) \\right ) ^ \\rho \\leq \\left ( \\frac{1}{M} \\sum_{i \\neq i'} \\left ( \\frac{P(X_1^n(i'))}{P(X_1^n(i))} \\right ) ^s \\right ) ^ \\rho \\, .", "a68250072e0059eee3a2287e7faf7a08": " \\simeq 24.6 ", "a6827d520078266fc870bc92a5fda90f": "Dom(A_{\\alpha}) = \\{ f + \\beta (\\alpha \\phi_{-} - \\phi_+) | f \\in Dom(A) , \\; \\beta \\in \\mathbb{C} \\}.", "a682f74e424c2e4b2cef7e17f5a9a614": "\\mathbf{u} = \\mathbf{v}_{k-1/2} + q' \\mathbf{E}_k,", "a68355c5c25131f95bfd715a515d738a": "\\Delta(p)=\\sum_{n=1}^N\\frac{Z_n}{p^2-m^2_n+i\\epsilon}+\\int_{4m_N^2}^\\infty d\\mu^2\\rho_c(\\mu^2)\\frac{1}{p^2-\\mu^2+i\\epsilon}", "a683a562cd75c915aff06c6b9743b77f": " \\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{M} f(-\\ln x) \\right\\}(s).", "a683adc7c073c8e249d39b11977bbb94": "F s =\\tfrac{1}{2} mv^2", "a683c653e38b3c9698b414b23da13771": " Q = a_D + b_D P + d Z \\, ", "a683c7ea75d74d1d6f22ee3c360b4782": " w", "a683ec4083597feb62d1095e91d6bc64": " p_{Total}= p_{Static} + \\rho \\cdot \\frac{v_{Water}^2}{2} \\,\\!", "a684a5fc49da84cb377972e8e8ee2f97": "\\mathcal{W}", "a684aa2f74bf141cc3a03dcffc2403d2": "Z_\\mathrm{source}", "a6851ad0265e04e2801d455f0832c1fc": " \\sum_{q=1}^\\infty f(q) \\frac{\\varphi(q)}{q} = \\infty. ", "a6852b82c1d1e602abac79f87d96f163": "s_{BA}\\circ s_{AB}=1_{A\\otimes B}", "a6856de68ae257c2d1149d618670dfdf": "\\mathbb{C}_m", "a686285419927ec1ac6eb7e2d108e8a0": "P(H_0|D) = \\frac{P(D|H_0)P(H_0)}{P(D)}", "a68653e0a6c01d64fde838e375681fd9": "P = p^2", "a686cdc12337db1740cc23aadab19c44": "\n\\begin{cases}\n1, & \\text{if } p>c+k \\\\\n0, & \\text{if } p>c-k \\text{ and } p1", "a6912ee2498a8f3c28046712f96993fe": "|Z| = \\sqrt{Z Z^*} = \\sqrt{R^2 + X^2}", "a69148c1f1ccb36896ec3a24aecd21f3": "\\delta W = (3-n) C, \\, ", "a6916f334389be009343f1d7b7aa068c": "\n\\vec{V} = \\nabla \\phi + V_\\infty \\hat{x}\n", "a691c0606b9757a21fad1d3725b60e6b": " \\lbrace X_i,X_j \\rbrace \\notin E ", "a69201c72caaf560c2fd6cc09edcf411": "B = \\mu H + (\\chi + i \\kappa) \\sqrt{\\varepsilon \\mu} E", "a69203a7e705b158263c56395a2a0f17": "DPA_{n} \\subsetneq DPA_{n+1}", "a6922764189ad9779e81b777e9de9b42": "F = N\\,I = \\oint \\vec{H} \\cdot \\operatorname{d}\\vec{l}", "a692395159b5a2b42f71703278db510f": "0 \\le r \\le m", "a692c941c5d7dd3ed3558e8a3628e4cd": "h(0) \\ne 0", "a69308c9e1d1e61287ad35a32c549a27": "\\sum_{Observations} (y_{ij}-m)^2", "a693310caa6d0748ce11b59090aafbc0": "\\mathfrak{P}^{13}", "a6937f1ecf605be2824089bc8e371047": " L = \\frac{(1 - \\Delta_x)^k}{(1 + |\\xi|^2)^k} ", "a693c466c268e856fc27c2de493e987e": " \\langle \\mathbf{S}^2 \\rangle ", "a693c996212e83f7c6fde2de0bbbf9d4": "m_t", "a693d3d473504b30230c09fa2abc31a6": " \\frac{1}{x^n} ", "a6948c015a58a50c08cfd15a92453c80": "n=\\lim_{x\\rightarrow+\\infty}(f(x)-mx)=\\lim_{x\\rightarrow+\\infty}\\ln x", "a694c6efb0727cdbfcd4381a858ed89e": "\\{A_1, A_2\\} \\subseteq P' \\cup L' ", "a694ea4f7f0372c4caaf40954d9789b4": "r=u_1\\dots u_m", "a695ac4100c5a1e89a6daf0a4b29c346": "x^2 + px + q = 0", "a695e557d7d3d5aaf602a18db1be7a30": "(a_1 \\otimes b_1) \\cdot (a_2 \\otimes b_2) = (a_1 \\cdot a_2) \\otimes (b_1 \\cdot b_2).", "a69616a5f5dade5b8f3bc091582f3c17": "V_n \\le 3", "a69637ffcf892cdfaf504ed6c45d6e0a": "C_{HbO2} \\,", "a6968c0de4edbc79cb42165af4b24825": "\\scriptstyle e_1 \\times \\cdots \\times e_{n-1} = e_n", "a69753b0834972c3ebf4095a91c08d35": "\\sin z\\,", "a697a043b9e5f5c5cf52c5eee6c7a581": "x \\mapsto \\{x\\}", "a697b9c9bf469067caf7beddb71f3840": "\\,t_r=t_s-t_e", "a6987df0cd9d5eec127a86b676ddf2e1": "3k_B", "a6988f3072d681771f933cb9d9facc74": "2^{10_{dec}}", "a698a5c07fe89856afd77b3e13799419": "[m] = \\{0, \\dots, m-1\\}", "a698d7e0343aac350519573ad0c5d2b5": "R_p - R_b", "a698ea93f7b622beb281210bec60440e": "0.5 < N < 1.5", "a698edc0e89718fc0b7dd4e54a45cf5c": "K\\to L", "a69913f66f2cfd4bd3f8ea75954ac476": "nd", "a69976836725a79805cacd7dc7d93226": "z z^\\ast = (x + yj) (x - yj) = x^2 - y^2 \\!", "a69984b3e1aae8e5228086dd441fc592": "t_0,t_1,\\dots,t_n\\,", "a699bdcec8dae308c7860793d71bdebd": " Y() ", "a69a00c5a4851e72ca46819e8fe5458d": "y'_1(x)=y_1(x)-\\frac{y_2(x)}x=\\frac{x\\,y_1(x)-y_2(x)}x=\\frac{c_1}x,\\qquad x\\in I.", "a69a01ad35e621c29c19809a9bfd1077": "\\ f:V \\times V \\rightarrow \\mathbb{R}", "a69a4f337eda65efd9d9f2599e08a6f8": "k_d\\,\\!", "a69a8ceec7059d9e1265d009c4b5913e": "\\begin{bmatrix}\nc_2 c_1 &\t-s_2 &\tc_2 s_1 \\\\\nc_3 s_2 c_1+s_3 s_1 &\tc_3 c_2 &\tc_3 s_2 s_1-s_3 c_1 \\\\\ns_3 s_2 c_1-c_3 s_1 &\ts_3 c_2 &\ts_3 s_2 s_1+c_3 c_1\n\\end{bmatrix}", "a69adc044b5a2c467e6d19f8a1257913": "[y,x] = [x,y]^{-1}.\\,", "a69b0e6d0b486346791bd5ff967199d4": "\nP(Y = y | \\theta) = \\frac{\\exp(\\theta^{T} s(y))}{c(\\theta)}\n", "a69b0e907a3fbc96affc9837a01e523a": "\nx^2 \\equiv q \\pmod p \\text{ is solvable if and only if }x^2 \\equiv p \\pmod q\\text{ is not solvable.} \n", "a69b54ef25519d52f82ca88732d19f01": "y=x^2;", "a69bd6f444809f9cb5683ff7e3ff6876": "y=a \\sin({m\\theta}) \\sin({\\theta})", "a69c564e00e577605fef2e9e9de628b0": "h(m,m_b) = H^\\triangle (m) + m_b ", "a69c8d6877970254fd900fa334e50efc": "G^i = \\mathcal{D}_a \\tilde{E}_i^a = 0", "a69cd300390c909ff3a899ce93e76c3e": "(X + 124)^2 - 15347 \\equiv 0 \\pmod{2}", "a69cdedcbcc3ca3887a43179165de62a": "(x_k,y_k)", "a69d6d8b11b9841230c6b010bf324bc7": " \\tfrac{1}{2}m \\overline{v^2} = \\tfrac{3}{2} k_\\mathrm{B} T", "a69d9b890e608e6b21245241b7ab98fd": "V \\approx g\\bar\\Psi \\phi \\Psi", "a69da13474f76471ac82d74ee880bc80": " \\sigma _c > 1 ", "a69dfd00c5bad8a95094c78c18f21bd6": "p,q\\,", "a69eac4fbd2ca30efa653b90037e408c": "\\mathfrak{P}^{121}", "a69eb7b23c06ee0e7d6292edd035bbf5": "U^+", "a69ee338f970c01e10ae6b26d4922b8d": "\\tilde{F}:=1-\\frac{2M(v)}{r} ", "a69f3e45105e10ce5be86ab9f7683973": "m\\ddot{r}=-\\frac{p_{\\theta}^2}{mr^3}+\\frac{k}{r^2}", "a69f522ac9a8c5999821c5cc5ac275c6": "m s^{-1}", "a69f556b342aa8f747863561b323692d": "P(A > B | A=a) = 1/2. ", "a69f7c5ebd53ae3a77ff7f2b62f24b6d": "[D_{\\mu},F^a_{\\nu\\kappa}]=-igD_{\\mu}F^a_{\\nu\\kappa}", "a69f893efa2ada1f29d3a0b36ccefb97": "\n\\begin{align}\n\\frac{\\partial\\rho}{\\partial t} + \\rho_0 \\frac{\\partial u}{\\partial x} & = 0 \\\\[8pt]\n\\frac{\\partial u}{\\partial t} + \\frac{a^2}{\\rho_0} \\frac{\\partial \\rho}{\\partial x} & = 0\n\\end{align}\n", "a69fadf9a8afcb6bfef9bac8a93340a3": "\\langle b^\\dagger_\\mathrm{in}(t)b_\\mathrm{in}(t^\\prime)\\rangle_{\\rho_\\mathrm{in}}=N\\delta(t-t^\\prime)\\,.", "a69fb8bb350133d8f79ad23bdbbafa93": "(X+2)^n \\equiv X^n + 2 \\pmod{n, X^r - 1}", "a69fc6d1ac35053c6c174b3ecc4f75e2": " \\Delta t = 2 m ", "a6a00eede46f09fbbd7666a920e356ee": "\\sqrt{g\\over \\ell}", "a6a01c94b95434cd80b6a295317cf33b": "\\prod_{i=1}^n u_i \\sim \\Lambda(p,m,n).", "a6a07488a7c6a4567f83a03a57d302b9": "\\left\\langle1,\\vec c\\right\\rangle", "a6a0766ef0f5da08957c9bdc3fdbee7b": "p(t,B)", "a6a0a3eb843dff1f95ae7658eeec6056": "\nS_a \\left( {v_a } \\right) = t_a \\left( {1 + 0.15\\left( {\\frac{{v_a }}\n{{c_a }}} \\right)^4 } \\right)\n", "a6a0cb5ad3279bbcf63c9a92e03a3758": " \\omega = 0 \\ ", "a6a0fa5217e343a49b02f37ce7546485": "\\Psi_{\\alpha,\\beta}(u) = ", "a6a1738716474b4a4a39fa6aff71a210": "\\mathbf{r}=(x,y,z)", "a6a188b78b3927a650d8415dd54c1e60": "\\varepsilon_{\\nu_1 \\cdots \\nu_n} = \\delta^{\\,1 \\,\\cdots \\,n}_{\\nu_1 \\cdots \\nu_n} \\,.", "a6a1e1a262462e9ed426492ec80b9930": " \\gcd(a,b) = \\sum_{k|a \\; \\hbox{and} \\; k|b} \\varphi(k). ", "a6a20589d61783697f80fd9f60b36b2b": "C = \\epsilon \\frac{A}{d}", "a6a22d9d5bd791277dde2c42e037dfae": "\\begin{matrix}{r \\choose 3}\\end{matrix}", "a6a24d9b0ad67df51f1827731412b2d3": "|f|_2 = \\sqrt{|S|}", "a6a25d8b77ec5db75fa1b23f88ff46f6": "t\\!:\\!\\tau", "a6a27cb286341a9f98ece8120a261fcb": " f(k) = \\left\\{\\begin{matrix} 1/2 & \\mbox {if }k=-1, \\\\\n1/2 & \\mbox {if }k=+1, \\\\\n0 & \\mbox {otherwise.}\\end{matrix}\\right.", "a6a2ebf7f8c131739aef8a2a8e887bc3": "{\\mathrm{d} \\over \\mathrm{d}t}{\\partial{V}\\over \\partial{\\dot{q_i}}} = 0.", "a6a33303a167661137783eceb9455277": "H+\\vec{u}+\\vec{w}", "a6a36b26d6328bf152b7cfd546eda391": " \\mathbf N = \\mathbf J - \\mathbf L ", "a6a3cdbe53aaa34e4cbc0130536c2a2d": " N_A = N_{A0} e^{-\\lambda t} .", "a6a408ee2006e445fe95cec707b0c3c8": "{1\\over 2}\\sum_m", "a6a42d267e26fc573823983c7e9bf71f": "Contrast = \\frac{K}{Area}.\\ ", "a6a45d61e93d2ca47ba95ac59fa4cf0d": "\nb_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{a_i}{b_i}\\,\n", "a6a4999d5afe486fb2d017d26eb4a55a": "u_i^n", "a6a506f6a8fb698ecfabda7556f975e2": " Q=\\frac{1}{V^N}\\int \\prod_i d\\vec{r}_i\\exp\\left\\{\n-\\beta \\sum_{i=1, i", "a6acd81f11bdeb8b2e45c6936b1fd82c": "\\text{false}", "a6ad29ea5f32391e1704ebce36353d02": "\\lambda_B=\\left( \\frac{ \\eta }{Sc^{1/2}} \\right) = \\left( \\frac{\\nu{}D^2}{\\epsilon} \\right)^\\frac{1}{4} ", "a6ad4a46f04e47122c32ccdee9123cd8": "\\left|\\Delta\\right|<1\\,", "a6ad73d5ea36dd500f19058bfd0cd01c": "\\left\\lbrace { \\lambda, \\frac{1}{1-\\lambda}, \\frac{\\lambda-1}{\\lambda}, \\frac{1}{\\lambda}, \\frac{\\lambda}{\\lambda-1}, 1-\\lambda } \\right\\rbrace", "a6ade9c4dc289ef839aec12c469cec8b": "\\ell(k, \\theta) = (k - 1) \\sum_{i=1}^N \\ln{(x_i)} - \\sum_{i=1}^N \\frac{x_i}{\\theta} - Nk\\ln(\\theta) - N\\ln(\\Gamma(k))", "a6ae2a2eb574d71285c77fc0ed5da6f4": "\\omega_\\mu = \\nabla_\\mu \\omega.\\,", "a6ae795f35c19e7c455b508e48227f85": "1-2^{n}+3^{n}-\\cdots = \\frac{2^{n+1}-1}{n+1}B_{n+1}", "a6aeb6f0dae31ebb6af139bf9b8d7a0e": " u_{i + 1/2} ", "a6af1a90667c5a3ad3af67ea30a623aa": "G_{\\infty} = \\frac{v_{out}}{i_{in}} = -R_f\\ .", "a6af26b2f955b083b3acac12732ed1da": "1-\\sqrt{R} ", "a6af369dcaf884bb50a9a9acd259ee7a": "\nw_0(n) = 0.5\\; \\left(1 + \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right) \\right)\n", "a6af87a643c9e7af9ae8a84ba0b041c6": "\\left(\\frac{\\mbox{Net Income}}{\\mbox{Net Sales}}\\right)\\left(\\frac{\\mbox{Net Sales}}{\\mbox{Average Assets}}\\right)\\left(\\frac{\\mbox{Average Assets}}{\\mbox{Average Equity}}\\right)", "a6af9ba5e0adb7fcb708086e8d60389d": "\n\\frac{{x}^{2}}{a^{2}} - \\frac{{y}^{2}}{b^{2}} = 1\n", "a6afc3381d4f0f0ad1c599b7d37013e0": "\\hat t ", "a6afc9fe729be78fb2e055703b221a0d": "w \\subseteq \\mathcal{H}", "a6afea246580b168a7083604d1eab57e": "s_{j} = 1", "a6b010b660bfa0256af9c8a02493f267": "f_{partial}(2, 3)", "a6b0b8ec3de191868bd3455efcc07dc3": " P_j = B ", "a6b0cd57bea837d83adb5e6e3a4f64bf": "\\phi(n) < e^{-\\gamma}\\frac {n} {\\log \\log n} ", "a6b11fc726fe6217a53a76012cd55ca7": "m_{\\lambda}= \\text{dim}\\,\\mathfrak{g}_{\\lambda}", "a6b1666a95311969180d430966cf7855": "\\frac{d}{dt} f(y_t, t) = \\frac{\\partial}{\\partial y} f(y_t, t) \\dot y_t + \\frac{\\partial}{\\partial t} f(y_t, t) = f_y(y_t, t) \\dot y + f_t(y_t, t) := \\ddot y_t", "a6b16b84c4a4a280f09a9fd30bf8eba4": "\\phi(q) = (q;q)_\\infty=\\prod_{k=1}^\\infty (1-q^k)", "a6b176e8da5bb38febf4273f3c05b2b4": "p_{\\rm osc}", "a6b1770adcb20294fd63caa66c68523e": "|\\mathrm{c}\\rangle = \\frac{1}{\\sqrt{2(1+e^{-2|\\alpha|^2})}}(|\\alpha\\rangle+|{-}\\alpha\\rangle)\n", "a6b1d416d8259464464c80aca625aa05": "i\\hbar{\\partial \\Psi \\over \\partial t} = - {\\hbar^2\\over 2m} \\left ( {\\partial^2 \\Psi \\over \\partial x^2} + {\\partial^2 \\Psi \\over \\partial y^2} + {\\partial^2 \\Psi \\over \\partial z^2} \\right ) + V(x,y,z,t)\\Psi.\\,\\!", "a6b1dc22249713c9d2e86ccde81d072f": " \\mathbf{U} ", "a6b253d0be9768d8e1d8769edf270426": "N = 2", "a6b27f4c3074ce0e338e295ce3786512": "n=2, 3", "a6b29386e4a44ca99111a57c32798471": "\\Pr(X>0)=1-\\Pr(X=0) \\approx 1-e^{-0.6932} \\approx 1-0.499998=0.500002.", "a6b2b3b713f2b3fad3f2c5fd894cc127": "(x_3-x_1)", "a6b2da8e1089e63fe8c3bbc17bec17e9": "\\alpha = \\alpha_1 \\wedge \\dots \\wedge \\alpha_k", "a6b2ff415f62f7a28a376d2492c34632": "x:y:z=\\frac{a}{b+c-a}:\\frac{b}{a+c-b}:\\frac{c}{a+b-c}", "a6b30aa6b07c1f72f163304d8f016f43": "\nf_\\mathrm{rms} = \\lim_{T\\rightarrow \\infty} \\sqrt {{1 \\over {T}} {\\int_{0}^{T} {[f(t)]}^2\\, dt}}.\n", "a6b3963bb9799158160759ac8e35ca6f": " (k+2) (1 - ", "a6b3b616101a7c94b13ade5db8e9438b": "E_k = {3 \\over 5} (N_p {\\epsilon_F}_p + N_n {\\epsilon_F}_n)", "a6b452a5a0bf0a0666ca1fce98dc0d76": "a\\in M_u", "a6b4a889e0f2c935a7ddd17875d03ae8": "L=\\prod_{i=1}^n \\left(\\frac{1}{2 \\pi \\sigma_i^2}\\right)^{1/2} \\exp \\left( -\\sum_{i=1}^{n}\\frac{(y_i-f(x_i))^2}{2\\sigma_i^2}\\right)", "a6b4cc4774ccb968494ec0a55a28eef1": "Q = \\frac{ 2\\Delta m F}{M_{\\rm Hg}}", "a6b55e25838dcbf72155b5cab16349cf": "\\scriptstyle{|\\phi_1(t_0)\\rangle}", "a6b57031d3c93d43512e5c97e50d41a4": "\nL_{p_{\\alpha \\beta}} = \\frac {\\mu}{4 \\pi}\\frac{1}{a_{\\alpha}\na_{\\beta}} \\int_{v_{\\alpha}} \\int_{v_{\\beta}} \\frac {1} {|\n\\vec{r}_{\\alpha} - \\vec{r}_{\\beta}|} d v_{\\alpha} dv_{\\beta}\n", "a6b5794a6b8ca338dee9f46d76465447": " \\scriptstyle \\mathbf{k} \\,=\\, ( k_x, \\,k_y, \\,k_z) ", "a6b5de6268c9f8bcb4feb7a87d3a622a": "(C,\\Delta)", "a6b5e3a683caede3bd779a5a06c0aa31": "\\Pr_{e \\in BSC_p}[D(E(m) + e) \\neq m] = \\sum_{y \\in \\{0,1\\}^{n}} p(y|E(m))\\cdot 1_{D(y)\\neq m} \\leq \\sum_{y \\notin \\text{Ball}} p(y|E(m)) \\cdot 1_{D(y)\\neq m} + \\sum_{y \\in \\text{Ball}} p(y|E(m))\\cdot 1_{D(y)\\neq m} \\leq 2^{-{\\epsilon^2}n} + \\sum_{y \\in \\text{Ball}} p(y|E(m)) \\cdot 1_{D(y)\\neq m}.", "a6b604f883843218fa98abd4f6c10f06": "\n\\arcsec(z)\n", "a6b65afafcca979154f1c83ca5824037": " E_1", "a6b6ec30c456fa5564340ee40f0f87c2": " G_3\\cup G_3+H\\cup G_3+H\\cup H=P\n", "a6b727cc95c8fb76fe80869f58c3bb17": "\n \\rm{MSD}\\equiv\\langle \\left( x(t)-x_0\\right)^{2}\\rangle, \n", "a6b73c99f4f4156a91c92db1848f25d0": "\\hat{\\mathbf{x_i}}", "a6b7a0f8c104881be60654333b2e7266": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left( \\frac{\\partial L}{\\partial \\dot{x}_i} \\right) = m \\ddot{x}_i.", "a6b7ceb1fee100f7ba8f483253970031": "SL(3,\\mathbb{C})", "a6b88819b7bdf0946dc3fb89141fe46f": "\n \\mathbf{u}\\times\\mathbf{v} = \\varepsilon_{ijk}~{u}_j~{v}_k~\\mathbf{e}_i\n", "a6b89f9dd3b2e7d3f992c30fd91c7bfa": "p_{\\tfrac{1}{2}1} \\leftarrow 64x^3+192x^2-256x+64", "a6b925cd061264a39758e447c52a65b4": "T^k", "a6b93b7e4e016374f44e24b55b594f45": "1 + z = \\frac{a_{\\mathrm{now}}}{a_{\\mathrm{then}}}", "a6b98450c195fd56107482beb344494a": "2\\mathbf{x}^{\\rm T}\\mathbf{A}", "a6b9849b72e5becf583ba408aa8247ac": "ad = bc \\quad \\mathrm {or} \\quad a = \\frac {bc} {d}.", "a6b9a87b8208865ad7dffdc50daebdd1": " (M,\\omega) ", "a6b9d001855030e19d1cacec802662a2": "\\Sigma = \\{a, b, c, ..., y, z, A, B, ..., Z, ...\\}", "a6b9f46018ea2a7add8e907bfe76955a": "C^{k,\\alpha}", "a6ba337c5538bb8006457a3b79954b04": "L_{rec}", "a6bbb4f7e28b1d4b2c081f5b33440f42": "ZY = U(\\omega) + iV(\\omega)\\,\\!", "a6bbbed62511a9100baf14e6fb3c69bd": "1 \\times 1 = 1.\\,", "a6bbcfcd6fc9013820c7a12b4cd09997": "K_a (v_0 \\frac{[HA]_0}{[OH^-]_0}-v_i) \\approx 10^{-pH_i} v_i", "a6bbe4fe9753ebb94951963ba87753fd": "g(x) = \\begin{cases}\\frac{x}{1-\\alpha} & \\text{if }0 \\leq x < 1-\\alpha,\\\\ 1 & \\text{if }1-\\alpha \\leq x \\leq 1.\\end{cases}", "a6bc20d933094c65b0c04665b90d6c35": " G=1+\\frac{R_\\mathrm{b}}{R_\\mathrm{a}}", "a6bc5e8bf1458ec25eb3a99b4a1def22": "[\\sigma^{\\mu\\nu}, \\gamma^{\\rho}] = -i\\gamma^{\\mu}\\eta^{\\nu\\rho} + i\\gamma^{\\nu}\\eta^{\\mu\\rho}.", "a6bcbbbf1b9032905f162e351e359584": "x=v\\cos u, y=v\\sin u, z=h(u) \\,", "a6bcd1eddcf2923b077bd5e08d5731c6": "\\mathbb{R}^3", "a6bcfbf9b340b92e9a426e58c0cebb1d": "L(\\vec{r},\\hat{s},t) (\\frac{W}{m^2 sr})", "a6bd153706469fb921a137439930c182": "\\mathit{p}|\\mathit{n}", "a6bd4c796c58311413a84e5e1ff7976f": "m_{k}=0", "a6bd7ab10c17bc74d03e2105c6958b23": "\\mathcal{N}[u(x)] = 0", "a6bd8af3ee10a16c431dcdcae5b4931e": "p = \\frac{x^3 - y^3}{x - y},\\ x = y + 2,\\ y>0.", "a6bdfb6fbebf91ba11ac42fee4f65e14": "G_{ret}(x,y) = i \\langle 0| \\left[ \\Phi(x), \\Phi(y) \\right] |0\\rangle \\Theta(x^0 - y^0)", "a6be0ec68f8a599188a2f8567554a061": "g\\circ f=g", "a6be16e1b354b5e30bac433631b86175": "f(x) \\mapsto \\int_a^b K(x,y) f(y)\\, dy", "a6be917e110bc25c235d07b2c218e942": "{\\Bbb C}^n", "a6beb1225af916d3d9b6b524c6a9694f": "\n\\begin{align}\n \\sin^2\\!A &=1-\\left(\\frac{\\cos a - \\cos b\\, \\cos c}{\\sin b \\,\\sin c}\\right)^2\\\\\n &\n =\\frac{(1-\\cos^2\\!b)(1-\\cos^2\\!c)-(\\cos a - \\cos b\\, \\cos c)^2}\n {\\sin^2\\!b \\,\\sin^2\\!c}\\\\\n \\frac{\\sin A}{\\sin a}&=\\frac{[1-\\cos^2\\!a-\\cos^2\\!b-\\cos^2\\!c+2\\cos a\\cos b\\cos c]^{1/2}}{\\sin a\\sin b\\sin c}.\n\\end{align}", "a6beb48f8c10503f11ca7542a738e130": "g(z) = \\exp(\\exp(z))", "a6bee29e1867f902e5aace2c04014e14": "{6 \\choose 0}+{5 \\choose 2}+{4 \\choose 4}=1+10+1=12=P(10).\\,", "a6beff1b56dd99e17190399dc85239f7": "\\sigma_m = 0", "a6bf6df3240093ab2719eebadc4589b6": "\\frac{d\\mathbf{A}}{dt} = \\mathbf{KA} + \\mathbf{Q}C_p(t)", "a6bfb328278245b1efe9fd85fa969567": "\\Re(s)>1.", "a6bfbb820feabbc3bffd36c6ca7a71af": "\\mathbf{n}=(n_1, \\ldots, n_d)", "a6bfc2e5c2659f08576b990adea39b60": " \\widehat{D}^{\\dagger}(\\delta\\alpha)\\widehat{a}\\widehat{D}(\\delta\\alpha) = \\sum_{i,j}(\\delta\\alpha^{*}\\widehat{a} - \\delta\\alpha\\widehat{a}^{\\dagger})^{i}\\widehat{a}(\\delta\\alpha\\widehat{a}^{\\dagger}-\\delta\\alpha^{*}\\widehat{a})^{j}/i!j!", "a6bff16112d929e894fd63c8cc40fd2e": "r_{I2}=\\frac{R_2-Z_{I2}}{R_2+Z_{I2}}", "a6c02aa6634293203cc65202bd4dd957": "\n\\left(\\begin{matrix}\\lambda_1 \\\\ \\lambda_2\\end{matrix}\\right) = \\mathbf{T}^{-1} ( \\mathbf{r}-\\mathbf{r}_3 )\n\\,", "a6c05bc0d22f1bf13d339da7a676a080": "\\overbrace{I_{\\text{b}} R_{\\text{b}}}^{V_{\\text{R}_{\\text{b}}}} = V_{\\text{cc}} - I_{\\text{b}} (\\beta + 1) R_{\\text{c}} - V_{\\text{be}}.", "a6c0c181f15039daffa4ed76f4ebca99": " r = \\frac{s}{2}\\sqrt{d}.", "a6c0d5f12ce2f9c946a6febd7ba530b0": "n^{th} ", "a6c0e13e9d1aa8211bb428a71d0dfac0": "max(S)", "a6c11b0870665ffd55d8d2820a768ea1": "\\frac{32}{45}", "a6c126b906a2b675f34ea6639aec71a2": "T = T^{i_1\\dots i_n}_{j_1\\dots j_m}\\; \\mathbf{e}_{i_1}\\otimes\\cdots\\otimes \\mathbf{e}_{i_n}\\otimes \\mathbf{\\varepsilon}^{j_1}\\otimes\\cdots\\otimes \\mathbf{\\varepsilon}^{j_m}.", "a6c148b84b44bec564f14d37b19c821f": "\n\\langle H_{\\mathrm{kin}} \\rangle = \n\\langle \\tfrac{1}{2} m v^{2} \\rangle = \n\\int _{0}^{\\infty} \\tfrac{1}{2} m v^{2}\\ f(v)\\ dv = \\tfrac{3}{2} k_{\\rm B} T,\n", "a6c180dc615db69dc856c4b4e6463b37": "\nR_A(v) = \\frac{\\sum_{i=1}^k \\lambda_i \\alpha_i^2}{\\sum_{i=1}^k \\alpha_i^2} \\geq \\lambda_k\n", "a6c1f3e09b04f11727f53da789c66e93": " \\mathrm{Re} = {{\\rho {\\bold \\mathrm U} L} \\over {\\mu}} = {{{\\bold \\mathrm U} L} \\over {\\nu}}", "a6c2be7dd689740c46fe93927027965c": "DPA = \\bigcup_{n} DPA_{n}", "a6c2e6499075a2a18e6ebf4d20eafda8": "\\sum_{m \\in t_j} p(n,m) = \\sum_{m \\in t_j} p(n',m) ,", "a6c30c79e69c5231432c43a9f12154dc": " M^2(B)=(E [{N}(B)])^2+\\text{Var}[{N}(B)], ", "a6c3638cbe8b696ec92e8f2ee66a5bb8": "(A,AB)", "a6c36782c1748c7a1d14aaa0e5dd662a": "for\\, each\\, cand \\in CandS_{k+1}", "a6c37255ea0fb8c5b93d4652e1034fab": "F\\subset F'", "a6c3b05cd4a3931c4856a7a86f16374c": "\\lambda^6 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix}", "a6c426a816e4cb90be3dd41384b5c272": "(A + 1)^{-1} \\ge (B + 1)^{-1}", "a6c431d3c27a23b1b828c3526fd00159": "\\sqrt{2}+1:1", "a6c4430abce4224ef36a19bf02c48a86": "=(1\\text{eV}^{-1})\\hbar c ", "a6c49b168ca50ec21574adedc8a182bd": "\\hat{y}=\\hat{e}", "a6c49decf17e82cf642b9dbd4ef59e26": " x_1=x_2=\\frac{9ad-bc}{2\\Delta_0},", "a6c49e17037a04943093123820207085": "\\mathbf{A} i = \\left(A_1 \\mathbf{e_2e_3} + A_2 \\mathbf{e_3e_1} +A_3 \\mathbf {e_1e_2}\\right) \\mathbf {e_1 e_2 e_3} \\ ", "a6c4bab5df112dc554fb197b08a1d15a": "\\lambda = \\frac{h}{p}.", "a6c4bc6f4da1c86ed781c56e7bee415c": " \\frac{d}{d r} \\left( p +\\frac{B_z^2+B_\\theta^2}{2 \\mu_0 } \\right) +\\frac{B_\\theta^2}{\\mu_0 r} =0 ", "a6c4e8063b456cdea3484e52fd17b5ef": "\\sin (A \\pm B) = \\sin A \\ \\cos B \\pm \\cos A \\ \\sin B", "a6c53b0c30babadf1e1cd31baace41c8": "\\lim_{t\\to 0^+}\\frac{f(g(t))-f(x_0)}{t} = u(g'(0)).", "a6c550e4490e571957c1cbd6d7c037d9": "\n Q(\\theta) = C_\\alpha~e^{i\\alpha\\theta} + D_\\alpha~e^{-i\\alpha\\theta}\n ", "a6c567366059e72f6f7cba2ec25eb7c4": "\\sup_\\alpha f_\\alpha", "a6c5742793aa79a71eb35cc4152dd597": "\\mathrm{PS}_1", "a6c58eac50ea98f676c6384523e8e5ab": "\\frac{M(v)_{,\\,v}}{4\\pi r^2}", "a6c5ba5b626c78747fe6badfd6f2af49": "= \\pm\\frac{\\tan \\theta}{\\sqrt{1 + \\tan^2 \\theta}} ", "a6c5e5193f72b343fffd637cc5c35fca": "P_\\mathrm{Br} [\\textrm{W/m}^3] = {Z_i^2 n_i n_e \\over \\left[7.69 \\times 10^{18} \\textrm{m}^{-3}\\right]^2} T_e[\\textrm{eV}]^{1/2} ", "a6c5e8ad3e0f4c87c8a5315e77cf8dd2": "r_s = \\frac{n_1 \\cos \\theta_\\text{i} - n_2 \\cos \\theta_\\text{t}}{n_1 \\cos \\theta_\\text{i} + n_2 \\cos \\theta_\\text{t}}", "a6c65ec9796bcc9f10fd0df4d25fb501": "dE/dx", "a6c68dd3693b268bd8f93b5a9c584183": "\\boldsymbol{m}_k = \\frac{\\boldsymbol{p}_{k+1}-\\boldsymbol{p}_{k}}{2(t_{k+1}-t_{k})} + \\frac{\\boldsymbol{p}_{k}-\\boldsymbol{p}_{k-1}}{2(t_{k}-t_{k-1})}", "a6c72b4982cdc5b0c56c634ba2dc2cdb": "\\scriptstyle\\boldsymbol{\\sigma} = \\boldsymbol{\\sigma}(\\boldsymbol{u})", "a6c7565ae80d484f3e8f1a263039f906": " S=\\{s_1,s_2,\\dots,s_K\\} ", "a6c7719908e2853980a1050eb925fcdb": "2(\\sqrt{xy}-\\frac{x+y}{2})", "a6c7acfc672d8535b8d9d849465f458e": "F = ma", "a6c7b48b1715748d2591cc4a3afe01af": "\\scriptstyle\\mathbb{R} \\setminus X", "a6c81687daa68883da1c478531a5e8bd": "B_3 \\bar S", "a6c83ba808e786acebecda8f27852fad": "r_1,\\ldots,r_d", "a6c84a8ce514fc942ce5c507945ee58f": "\\frac{1}{4\\pi}\\int\\left (G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu}\\right )\\sqrt{-g}\\,dS^{\\nu} = {2G \\over c^4} \\int T_{\\mu\\nu}\\sqrt{-g}\\,dS^{\\nu}", "a6c857f8d7a8f9180531ce1146d98af8": "p \\!", "a6c8a6a7d832c5a3f048b0eb7a8f9ee6": "u(S,T) = T", "a6c8f0017e763ba3a24241d063b81c9b": "Z_{I2}\\,\\!", "a6c9345b24b12554578a084b250217b7": "T_{\\mathrm{rot}} = \\begin{cases}\n T_1(x, y) & \\text{if } x, y \\in (t, 1] \\\\\n N(R_{T_1}(x, N(y))) & \\text{if } x \\in (t, 1] \\text{ and } y \\in [0, t] \\\\\n N(R_{T_1}(y, N(x))) & \\text{if } x \\in [0, t] \\text{ and } y \\in (t, 1] \\\\\n 0 & \\text{if } x, y \\in [0, t]\n\\end{cases}", "a6c9ae5962baeea28dd3c3ec8fa8e9d1": "\\phi(x) = \\mathbf{e}_x", "a6c9d6e4f972cb4c26603ef9a861ded5": " \\tau_1 \\tau_2 = C_1 C_2 R_1 R_2 \\ . ", "a6ca1e72f015f3f9326124f81a98406b": "\\beta < \\alpha^\\omega.", "a6caf1ad72a069caeb21c8b955d94426": "\\scriptstyle \\mathbf{I}", "a6caf32fc29b18509b0b904ca4e1f135": "F:A\\times B \\to L(X,Y)", "a6cafe35c8f1c3503f1939639f270f0d": "\n w_{\\mathrm{max}} = -\\dfrac{P(L/2)b[L^2-(L/2)^2]^{3/2}}{9\\sqrt{3}EIL} = -\\frac{PL^3}{48EI} = w(L/2)~.\n ", "a6cb1e4158586a03b732b37e5edb766d": "X(x) = 0", "a6cb2db71a206c824baa97c289e3d2ad": "(s-a_p)", "a6cb43cdd34a84106198bf9f5e15ce98": "x\\text{ AND }y = \\sum_{n=0}^{b}2^n\\left(\\left\\lfloor\\frac{x}{2^n}\\right\\rfloor \\bmod 2\\right)\\left(\\left\\lfloor\\frac{y}{2^n}\\right\\rfloor \\bmod 2\\right)", "a6cb8e54e1506da3a6d23adacc1f0b06": "V_b", "a6cc2b2b7af18820e029d51a4565dd4d": "Z(N_i, V, T)", "a6cca435ff57ce33a81c5000eb0e20ce": "R(k-s)=(-1)^{k/2}R(s),", "a6ccb4b2e4c39b6369660d05d8207912": "U(r)\\propto r^k", "a6ccd95cc03653f91422dd0bed3c37a2": " d = \\frac {v^2} {g} ", "a6cd8a96ae92e1eae2ba9c7cfec04011": "x a^m b^n y c^m d^n z", "a6cd8cb3d5237339e6284a993841e079": "(\\mathbf{a},\\langle \\mathbf{a},\\mathbf{s} \\rangle /q + e)", "a6cd9753a9bbc426094977456c86dd27": "O_{6}", "a6ce080dd93d0caa556c98e02cbf6817": " \\int_0^T P \\;dX = n h ", "a6ce7e3fbc15e6a09f964fa11db6d51f": "TB_{max} = \\tfrac{1}{M}\\cdot\\tfrac{1}{2} = \\tfrac{0.5}{M}.", "a6cf8910bb53aa2586d80ea294df707d": " d = \n\\begin{bmatrix} \nd_1 \\\\\nd_2 \\\\\nd_3 \\\\\nd_4 \\\\\nd_5 \\end{bmatrix}, ", "a6d0f9c47736086181b3e464d0ab4aa0": "\\lambda^2\\left[\\Gamma\\left(1+\\frac{2}{k}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{k}\\right)\\right)^2\\right]\\,", "a6d1099516e476d9cd78f4fa4d86dae5": "\n\\begin{matrix}\nU = \\begin{cases}\n1 & \\text{with probability } p, \\\\\n0 & \\text{with probability } 1 - p,\n\\end{cases}\n& \\mbox{and} &\nV = \\begin{cases}\n1 & \\text{with probability } q, \\\\\n0 & \\text{with probability } 1 - q,\n\\end{cases}\n\\end{matrix}\n", "a6d110be335e38f7f01deacf5998fac2": "\\frac{{}^\\mathrm{N}d\\mathbf{a}}{dt} = \\sum_{i=1}^{3}\\frac{da_i}{dt}\\mathbf{e}_i + \\sum_{i=1}^{3}a_i\\frac{{}^\\mathrm{N}d\\mathbf{e}_i}{dt}", "a6d1a24b479142e3ae94ce9c777a61f3": " \\operatorname{cov}(x_s,x_t) = \\frac{\\sigma^2}{2\\theta}\\left( e^{-\\theta(t-s)} - e^{-\\theta(t+s)} \\right). ", "a6d1bf1f6ad56b6b9592a1078d560d7d": "n=15", "a6d1deba26403a26b2e9848f0b0f55c4": "\\operatorname{Var}(\\mathbf{1}_A (\\omega)) = \\operatorname{P}(A)(1 - \\operatorname{P}(A)) ", "a6d22d7b97d9ebb0733f02ddc4d44358": "C_1 = C_2 = C \\,", "a6d25a74170fb65ecac304a5e0cd490f": "v_i(k) = \\frac{\\delta_D(k,i)}{\\xi^d_{b_{min}}(k,i)}", "a6d262f780a87d2aea387e6fd082d0e7": " \\pi _ t", "a6d2758ea7ba1399e4f0ef6ee9083915": "f_0(x): \\mathbb{R}^n \\to \\mathbb{R}", "a6d29152fcb9d0964b3b16496a7a8dc0": "F_{\\alpha\\beta}=0", "a6d29c2b20fdcd9b31b3b70e421135e4": "S_q = \\langle |g(t + \\tau) - g(t)|^q \\rangle_t ", "a6d2a68f59b7cb7df35b0ea2bd59bab3": "\\phi^{i} ", "a6d2e4c451c0187f54792d6096de1883": "f_y(a,b)=\\frac{e^x}{1+y}\\bigg|_{(x,y)=(0,0)}=1\\,,", "a6d36b27f9462b8594a94a5520f6b926": "\\psi|_F = \\phi \\quad \\text{and} \\quad \\psi(x) \\geq 0\\quad \\text{for} \\quad x \\in K.", "a6d3811176644e3850240f23c6293098": "\\sum_{e\\sim v} f'(v) = 0 \\ ,", "a6d3c9b9ac4b0acc9054d227bc54b650": " [[[\\Delta,L_{a}],L_{b}],L_{c}]1 = 0 ", "a6d3ccccf587ec2aeacd3ee13aaabacf": " (f \\le g ) :\\iff (\\forall x, f(x) \\le g(x)).", "a6d3d75fb2422551929c0494d12698ca": "open\\;m.P", "a6d3dd83a3776bacefa7309af55f4cd9": "A\\left(V\\right)", "a6d40c3457acd34048d60ae0c4c8fd4b": "|\\psi(t)\\rangle=\\left[1-\\frac{i}{\\hbar}\\int_{t_0}^t dt_1H(t_1)-\\frac{1}{\\hbar^2}\\int_{t_0}^t dt_1\\int_{t_0}^{t_1} dt_2H(t_1)H(t_2)+\\ldots\\right]|\\psi(t_0)\\rangle.", "a6d4113ce990779df524a80519cc0a16": " \\mathbf{X}= \\left(ct, \\mathbf{r}\\right) ", "a6d44c21c9b3fecf8cba1a641842fb20": "P_{e|H_0}", "a6d44e979eac481d954ba3a06a355d9b": "X \\rightarrow Y \\rightarrow Z", "a6d47cbfa3d753e7a90ec62f3ab255ae": "\\ x^*=c(t)+Q(t)x \\quad \\text{where} \\quad c(t)=x_0^*-Q(t)x_0 \\quad \\text{and} \\quad \\alpha=t^*-t=t_0^*-t_0. ", "a6d485cba167cd76f1cf6f883d90a411": "\\min \\limits_{(S, \\overline{S})} \\operatorname{ncut}(S, \\overline{S}) = \\min \\limits_y \\frac{y^T (D - W) y}{y^T D y}", "a6d4899adfb51d11dc8b9e8fd18ebd61": "S = {V^{1.85} \\over k^{1.85}\\, C^{1.85}\\, R^{1.17}} ", "a6d4a02f59a7ae2fc29c1444c44e77ba": "\\Rightarrow \\sin\\theta_{max} = \\sqrt{1-\\cos^2\\theta_{max}} = \\sqrt{1-\\beta^2} = \\frac{1}{\\gamma}", "a6d521fecf11a9c49a862fcbb8f02d61": "\\lim_{b \\to 0^+} {a \\over b} = +\\infty", "a6d5a078fcb380300f5fdddd8e150f6d": "n=m+(m+1)+(m+2)+\\cdots+(2m-1)=\\frac {m(3m-1)}{2}=\\frac {k(3k-1)}{2}", "a6d5c4a06f5549b0cdab7876f7f1144d": "-b_k", "a6d5cbb06b57c080b1ed7c2e5db8815e": "g(t-u)", "a6d607145f59b3d2c2bd7e869cfaa178": "[\\phi[f],\\phi[g]]=0", "a6d611e1ef86c5133b5a616dd21fe09e": "p_{\\perp }=p-p\\cdot PP/P^{2}", "a6d62199e8476503e45106764d42513d": " \\ Fr > 1", "a6d6c90bfa49da100eaed3b02a476d2a": " -1", "a6d6ff6e0cb4c36679f59d07e3e9faad": "f^\\rightarrow(A) = \\{ f(a)\\;|\\; a \\in A\\}", "a6d71491b792e487b38dc8f467d4ec9b": " u \\wedge v = \\sum_{1\\leq i0", "a6e289deb232c0bb1d73ef1fd24884a8": " T(\\rho^{12} ) = N^{-1} \\sum_{j=1}^N (1_{\\mathcal{H}^1}\\otimes U_j) \\rho^{12}(1_{\\mathcal{H}^1}\\otimes U_j^*), ", "a6e29d1c4649d8c2df461c39ba775f6d": "m_n > \\frac{-2n!}{(2\\pi)^n}", "a6e2be9bee7bc5fe6f4a6b3f7106bc7b": "n[k]", "a6e3140b91ef61c93a674bf114ac512b": "A\\stackrel{f}{\\leftarrow} C\\stackrel{g}{\\rightarrow} B", "a6e339f51bcb2e8b1769e57688f54415": "\n\\begin{align}\n\\sum^\\infty_{n=0}\\frac{2n+3}{(n+1)(n+2)} & {} =\\sum^\\infty_{n=0}\\left(\\frac{1}{n+1}+\\frac{1}{n+2}\\right) \\\\\n& {} = \\left(\\frac{1}{1} + \\frac{1}{2}\\right) + \\left(\\frac{1}{2} + \\frac{1}{3}\\right) + \\left(\\frac{1}{3} + \\frac{1}{4}\\right) + \\cdots \\\\\n& {} \\cdots + \\left(\\frac{1}{n-1} + \\frac{1}{n}\\right) + \\left(\\frac{1}{n} + \\frac{1}{n+1}\\right) + \\left(\\frac{1}{n+1} + \\frac{1}{n+2}\\right) + \\cdots \\\\\n& {} =\\infty.\n\\end{align}\n", "a6e3491390fdaa1a67cbe45a944d0e6d": "{x_2}^3 \\,\\!", "a6e37768756feb01f14350be2bb4c5a0": "\\sum_{k=0}^n {k \\choose m} = { n+1 \\choose m+1 }", "a6e4dffc8ed2499ae920f4f0520637eb": "\\int_{-\\infty}^{\\infty} f(x)\\,dx = 2\\pi i \\sum_{k=1}^n \\operatorname{Res}(f, z_k)\\,.", "a6e4ee127c60dd6a5e0d6740c1128d2b": "\\left (a/q-cr_0/qx,a/q+cr_0/qx\\right )", "a6e51b89b49c8f3b717440f6b0e283e1": "{a^2}I_{n,m}= I_{m-2,n}-I_{m,n-1}\\,\\!", "a6e51d915654d8d11f1d7cdc38bc9009": "\\mathbf{U}^{-1}\\left(\\frac{\\partial \\mathbf{U}}{\\partial x}\\mathbf{U}^{-1}\\frac{\\partial \\mathbf{U}}{\\partial y} - \\frac{\\partial^2 \\mathbf{U}}{\\partial x \\partial y} + \\frac{\\partial \\mathbf{U}}{\\partial y}\\mathbf{U}^{-1}\\frac{\\partial \\mathbf{U}}{\\partial x}\\right)\\mathbf{U}^{-1}", "a6e554768c28db9104637dd49c0a9b03": " {dn \\over dt} = C_o({D \\over \\pi t})^{1/2} ", "a6e593a2f557d90b65673c0295cb8a46": "x \\in rowgroups", "a6e5cbcb768e009a2335bf9169d173be": "x=x_0 + \\delta", "a6e6206a58be4898ae0491440ba29ba5": "Q(R) = R", "a6e6570dd8e2b99eeafb59c585f820e8": "B_{\\mathrm {v}}", "a6e665bcad7ede9b2d46b0a6d08b4acd": "X(u,v) ", "a6e6b279a44733320a0994e4b0830570": "\\text{ER}_{g,\\beta}(X):=\\sup_{Q\\in\\Im}\\text{E}_Q(X)\\,", "a6e72951d6fa64e99c1799007c13308b": " W(\\lambda,x)_{energy} = \\frac{1}{2} ~ \\frac{\\lambda^2}{L(x)} ", "a6e7642713ceec82181f8fa708069e2f": "\\textstyle\\tau_{{(Q)}_{[\\epsilon]}}", "a6e79f5d7d546e6e85a04468d8ad7ad3": "\\text{if }m\\mid n\\text{ then }a_m\\mid a_n,", "a6e7c46c17fa907fc18d94782fbf94ad": "C \\in \\mathcal{C}.", "a6e7df0ebb20ce189a89a24eba8224cc": "b^k", "a6e7feb6817c709392529284284d81bd": "[zero,succ] : 1+N \\longrightarrow N", "a6e8628e0c35b8f8bff20c133e14245a": "\\bar{Z_i}=\\left( \\frac{\\partial Z}{\\partial n_i} \\right)_{T,P,n_{j\\neq i}}.", "a6e8fb3667adb68e39c6eaaa9e0e7c5c": "f(x;\\alpha,\\beta) = \\frac{1}{\\Beta(\\alpha,\\beta)} x^{\\alpha-1}(1-x)^{\\beta-1}", "a6e999c37646e0706a1be48410216bbd": "\\mathbb C^{2m}", "a6e9a748a3cb0da6c245e06324b7aa3d": "\\color{RawSienna}\\text{RawSienna}", "a6e9d644874d80e96c90f54051bfa00e": "\\begin{align}\nY' &= K_R \\cdot R' + (1 - K_R - K_B) \\cdot G' + K_B \\cdot B'\\\\\nP_B &=\\frac12 \\cdot \\frac{B' - Y'}{1 - K_B}\\\\\nP_R &=\\frac12 \\cdot \\frac{R' - Y'}{1 - K_R}\n\\end{align}", "a6e9e209f04aeaeb9cb7fdaa8cf979ab": "\\dot{y} = g(y) \\,", "a6ea8a9e5ed1f23c984267eb0fea1f62": "\\displaystyle{{1\\over \\pi} \\,\\iint_{\\Omega\\cup\\Omega^c}{\\overline{\\partial_z S(\\psi)(z)}\\over z-w}\\,dx\\,dy= -S(\\psi)(w).}", "a6ea9ec31fca9f829a817fb0e26ce4c8": "\\Gamma = \\{ \\gamma_{i} | i \\in I \\},", "a6ead187e4beef7391594fdb4286f2f4": "\\sum_{e\\in\\Gamma} c_e p^e", "a6eaee25cf193ad3c59e9e17e3ac94b2": "\\tilde{d} = \\sigma^2 / \\bar{x}", "a6eaf1e1aa9ebeb769de4a2f4dbb2640": "t_1 + \\tau, \\ldots, t_k + \\tau", "a6eb2eade76432bfda9ebab0d4895426": "[0,A)", "a6eb4a4693b03a9dac49813a81da248b": "\\lambda_- \\approx -\\frac{M^2}B.", "a6ebc09a1e83fc2458db5e45fca84b6d": "R( \\mathbf{p,z}) -C( \\mathbf{z}) =\\Pi^*", "a6ebd95daf417730ddb71ed6cc56aa11": "|R_{k+1}(x)|\\le\\sup_{x\\in U} |f^{(k+1)}(x)|.", "a6ebeb96e97ad5782bd29dd8f15abcc1": "\\langle x, \\ y \\rangle \\,", "a6ec0050cf25c3abbc8fde4e1b85163c": "B/{aB} \\simeq a^l B/a^{l+1} B \\subset (a^{l +1}B + A)/a^{l+1} B \\simeq A/{a^{l +1}B \\cap A}.", "a6ec2af110b54977aacec4619a37008b": "X_i^w", "a6ec4e8494541d4c8896ae4b9c531c0b": " x_1^2 + x_2^2 + \\cdots + x_n ^2 \\le 1.", "a6ec60fdea6f984f75d77820b8bdb050": "-\\int_0^{2\\theta} \\log\\Bigg| \\left(2 \\sin \\frac{x}{4} \\right)\\left(2 \\cos \\frac{x}{4} \\right) \\Bigg| \\,dx=", "a6ec74a7724d82e87953ea775965d1df": "D^T", "a6ed4197d08ee9dcabc87c75aa51190c": "A=\\begin{bmatrix}\n 1& 2 & 0 \\\\\n0 & 3 & 0 \\\\\n2 & -4 & 2 \\end{bmatrix}.", "a6ed87d15c816ff9a36e3d37642234ca": "m=-\\infty,0,1,2...", "a6edd8bcff427560ea70396406cc26ad": "{\\tilde{D}}_{n-1}", "a6ede89e3457c2c40246efdd613c6969": "T^{\\otimes |E|} (T^{-1})^{\\otimes |E|}", "a6ee538d6550a7794b12fd497f19493b": "x^{32} + x^{28} + x^{27} + x^{26} + x^{25} + x^{23} + x^{22} + x^{20} + x^{19} + x^{18} + x^{14} + x^{13} + x^{11} + x^{10} + x^9 + x^8 + x^6 + 1", "a6ee7a5b2bc24be5998d5530d80d7b4b": " P\\,\\!", "a6ee85c963a1ee763e8d4bddebe2585e": "\n\\mu_i(t) := \\frac{X_i(t)}{X_1(t) + \\cdots + X_n(t)}\n", "a6ee95f4f3ff2635f85b5571c34dce5a": " Qf(x,\\lambda)=0 \\, ", "a6eea1be7b34f5145065375884f72572": "|\\alpha \\rangle \\langle \\alpha|", "a6eec4de34d022ece2ab2563a76fd0ce": "P\\left(L_{k}|R_{k}\\wedge\\delta\\wedge\\pi\\right)", "a6ef1e4a794f33a3f118e376643c000d": "\\max\\left(d+5, \\frac{3d}{2}+1\\right)", "a6ef41a37d0ee2ac56ca0742084dedcb": "\\omega_\\alpha^{\\;\\; IJ}", "a6ef66f970736c055f4b4da51984dcb5": "\\mathcal{B_A} = (\\mathcal{A}, \\Delta, \\varepsilon, \\Phi)", "a6f031a90c63071c0fd5e44089ef20f8": "\\tfrac {1}{2} m v^2", "a6f03fa499a78f0aab7ac0f8380ade7e": "u =\\frac U{U+V+W}= \\frac{4X}{X + 15Y +3Z}", "a6f0d09d3b592155278aaf8678ce785e": " \\langle b, c | b^2 = c^2 = e, bcbc = cbcb \\rangle. \\, ", "a6f117005a8b05f3fde5d2c7cbd5ede9": "p \\equiv 3 \\pmod{4}", "a6f11baed0f3c99c4a26460807a9dce6": "t_r-t'_r", "a6f11d66bdb2bbfd5512f7be3cd68604": "R_a = \\frac{R_\\mathrm{ac}R_\\mathrm{ab}}{R_\\mathrm{ac} + R_\\mathrm{ab} + R_\\mathrm{bc}} ", "a6f13ba045b9224a96824f0524c60a49": " v_j", "a6f165d4e1cf2b0fddc1b441fa52ac26": " 2 \\times q. ", "a6f174afd0ff8da72abf2dce91ac5359": "\\vec e_i = \\begin{bmatrix}\\lambda_i^{n-1} & \\cdots & \\lambda_i^2 & \\lambda_i & 1\\end{bmatrix}^T,", "a6f1bf4fc88dd72b232f3ffe960dc615": "\\delta^a_c", "a6f1c3402a3f40fb673bdf98a0031b0a": "\\mathfrak{D}_{ij}", "a6f1f02b694f862f20c68d741494c374": "\nx_n = b_0 + \\underset{i=1}{\\overset{n}{\\mathrm K}} \\frac{a_i}{b_i}\\,\n", "a6f212f7fc16c1aa703f838297a30b3f": "x^2 y^{-1} z y z^3 x^{-2}. \\,", "a6f2bc3a6f66115e8751dbd03a9bb42d": "\nR(\\theta,d') = n - 2\\alpha(n-2)\\mathbb{E}_\\theta\\left[\\frac{1}{|\\mathbf{X}|^2}\\right] + \\alpha^2\\mathbb{E}_\\theta\\left[\\frac{1}{|\\mathbf{X}|^2} \\right].\n", "a6f2c8b487e0cd49e7e4b2fefda2580f": "P(N\\mid n) = \\frac{P(n\\mid N) P(N)}{P(n)}.", "a6f317b268ae825d94f832f970af607c": "\\tau", "a6f31bcdccadb8c772c593ecc0f28f5a": "\\exp^{\\!-1}\\!=\\!\\ln ", "a6f3873a513f0bc59ffeb99e70437443": "\\vec{X}=X_1, X_2, ..., X_n", "a6f3a4f745373cdcb2598c7b28ca9f31": "T^{\\mu\\nu} = \\left( \\rho + \\frac{p}{c^2} \\right) \\, U^\\mu U^\\nu + p \\, \\eta^{\\mu\\nu}\\,", "a6f4bbd1e44c4d88637240f151f15365": "EG\\phi \\equiv \\phi \\land EX EG \\phi", "a6f4bcf7c98bac7a23550b11521d3202": "\\displaystyle{Z^+=\\begin{pmatrix} \\overline{D} & \\overline{B} \\\\ \\overline{B} & \\overline{A}\\end{pmatrix}.}", "a6f4c50f684ebd286eda7a6fd039e936": " T_0(z)=44+k ", "a6f4e53b9a233e420f049805b06a6cf8": "\\rho(\\mathbf{B},\\sigma_{\\epsilon})", "a6f508417b2beb3472a3186cca03b479": "\\color{DarkOrchid}\\text{DarkOrchid}", "a6f54ea021cc96558b4be244537ce5a0": "U(t) \\; = \\exp \\left(\\frac{-i H t }{\\hbar}\\right)", "a6f57f5740aea3d12e53ffdef2abcd2f": "x=\\frac{-b \\pm \\sqrt {b^2-3ac}}{3a}", "a6f5da841b4315c71d1560394d4260af": "Z_{I1} = \\sqrt{\\frac{AB}{CD}}", "a6f5f6194a0c864f7020f0d3d8d65638": "E[B_H(t) B_H (s)]=\\tfrac{1}{2} (|t|^{2H}+|s|^{2H}-|t-s|^{2H}),", "a6f60a948f2dcf25d5624b8392a98456": "\\mathbf{v}_{AB} = \\frac{d}{dt} \\mathbf{X}_{AB} = \\mathbf{\\Omega \\times X}_{AB} \\ , ", "a6f67eaeaa1c27b7658ebfe95e964534": "\\rho(t)< t", "a6f735e413782d17e41639c3e4d84273": "\\phi(n_1)", "a6f77427be122b6404f1cfdc3dc2cfe0": "\\psi_n(x) \\equiv \\lang x | n \\rang ", "a6f797d7b615ebab1a8dd368f8c33502": "2c^2=4a^2+4a+2", "a6f7a950eb9a490fcd46146be05f0638": "p_n'(0)=a_n.\\,", "a6f7a9aad37d9bdc80d8c1dfd2f43dd9": "\\scriptstyle V_{in} \\;=\\; 0", "a6f7afa5a163c78f3f7dd90b6ea9b58e": "\\left(\\frac{7}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "a6f7b3d686a3f51157396588341c1477": "a(t) = (4\\cdot 4^t-1)/3", "a6f7bb596e7613c9232a6960d03257b9": "\\mu Z. \\phi", "a6f7ec0f4a33754e0ae58387df27101e": "\\zeta_f(z)=\\exp (\\sum_{n=1}^\\infty \\textrm{card} \n\\left(\\textrm{Fix} (f^n)\\right) \\frac {z^n}{n}),", "a6f81124cd41fd40888ccaa48ce799e5": "\\mathrm{SO}(4)\\,", "a6f82e2b941169cb64698afda62fea82": "M\\times {\\mathbb R^n}\\,", "a6f84cb9c75951bb551b18f9439734ac": "\\mathrm{[M(H_{2}O)_n] + L \\leftrightharpoons [M(H_{2}O)_{n-1}L] + H_{2}O}", "a6f88f45fc63ec28de8b3c7879f0eec3": " \\operatorname{int}(\\operatorname{int}(A)) = \\operatorname{int}(A) \\! ", "a6f8b3f0b847d023fb6e111bed74e8e8": "a_{ij} = \\langle Ae_j, e_i\\rangle ", "a6f926e86ebd177e146eb7fc5ddbd43a": "|\\mathbf{v}|", "a6f97027621531dd96abc94a40926e34": "m_c = \\sqrt {\\frac{2 a^2 + 2 b^2 - c^2}{4} }, ", "a6f979e947df2ba4c04d0d5a333c77ec": "\\pi_n(X)=\\pi_{n+1}(BX)", "a6f995fb4869fec82569f399c4ca959f": " c \\rho - \\bold{J} = c \\rho - J^k\\sigma_k", "a6f9c1924a67e930883575ad1a4b4c6c": " u(0,x,y,z) = 0, \\quad u_t(0,x,y,z) = \\phi(x,y,z). \\,", "a6f9e8365e2b4ec1bbcccc425733fdd2": "s * (hi + ho + 1) * 2 - s", "a6f9fe5680416ba9e52888204dc5f75a": "\\displaystyle x(t)", "a6fa765e0e72c8876b7da979bf0e40a0": "\\mathrm{d} G = T\\mathrm{d}S - p\\,\\mathrm{d}V + \\sum_i \\mu_i \\,\\mathrm{d} N_i + p \\,\\mathrm{d}V + V\\mathrm{d}p - T\\mathrm{d}S - S\\mathrm{d}T\\,", "a6fa912c29d2b5fdfb658fd7c07dcbbd": "x_{n+1} = \\frac{x_n}{2} \\cdot (3 - S \\cdot x_n^2).", "a6fa9d61ced92d78e931e200fcb2beae": "{n \\choose k}={n(n-1)(n-2)\\cdots(n-k+1) \\over k!}", "a6fab1636285a75f54d705ebabdddfce": "\n[A] \\ \\ \\ \\ \\tilde{S}(\\omega) \\mapsto \\lambda \\tilde{S}(\\omega) \\text{ implies } G \\mapsto \\lambda G\n", "a6facf04ebfa42d805298c4060556c4f": "S_n = \\sum_{k = 1}^n \\left[a + (k - 1) d\\right] r^{k - 1} = \\frac{a - [a+(n - 1)d] r^n}{1 - r}+\\frac{dr(1 - r^{n - 1})}{(1 - r)^2}.", "a6faefec49177bb0e817266bd984f354": "5 (36-52)^2 + 4 (33-52)^2 +6 (78-52)^2 = 6780", "a6fb27e934a60cf1c1518104066f0494": "(X)_- = (-X)_+", "a6fc2aff73621be4cefbd97f1eee80cf": " [L^{\\pm}_{u,v},L^{\\pm}_{w,x}] = L^{\\pm}_{w,\\{u,v,x\\}_\\pm}-L^{\\pm}_{\\{v,u,w\\}_{\\mp},x} ", "a6fc3bb2f205844d7bb1ba93954920e9": "f'(v)=-f'(L_e)", "a6fc6137649c35441586d6d3286bd3d8": "\\frac{M_0^\\mathrm{act}}{M_0^\\mathrm{pass}} = \\frac{M_1^\\mathrm{act}}{M_1^\\mathrm{pass}}", "a6fc913bbdcc2b4f7a2f0cb10788b6c7": "jx_L = j\\omega L_M", "a6fcb05b70684fc0937a984aa34b6485": "a = \\frac{\\|O_1-O_2\\|}{L} ", "a6fce5f697cf12f14f931ffc5456bfb6": "\\mathbf{\\Phi}(t, \\tau)\\equiv\\mathbf{U}(t)\\mathbf{U}^{-1}(\\tau)", "a6fd02cf370446fbeb65518266424a35": "(\\pi(a_1),\\ldots,\\pi(a_m))\\in A", "a6fd0dc5182854613f004293db76013a": " \\lambda_l = \\lim_{q\\rightarrow 0} \\operatorname{P}(X_2 \\le F_2^{\\leftarrow}(q) \\mid X_1 \\le F_1^{\\leftarrow}(q)). ", "a6fd36747f702c86713bd35306216122": " ~\\chi^v_j = 1 - \\psi^v_j ", "a6fda867fe8563d1f5edc7c2273c9e0a": "\\ r \\ll R_{1} or R_{2}", "a6fe0a2c9b0e0f705c3dbde14b7b5131": "\\mathbf{w} (i) = \\mathbf{w}(i-1) + \\mathbf{g}(i) e(i)", "a6fe51d7bbf2dea8b037f3d5a91b2143": " \\operatorname{false} ", "a6fe80953ab5aa36215983cadfcd5a63": "P,Q \\in E(K)[n]", "a6fe9547c6117d86b95c6cefc382e742": " \\int_0^1 \\sqrt x \\,dx = \\int_0^1 x^{1/2} \\,dx = F(1)- F(0) = \\frac{2}{3}.", "a6feaf586cd7963d0e983c6f4205b539": "d_b(z)", "a6fee44bdb59ab639e65cc24305c3d3d": "b_1=b_2", "a6ff86d6a7b09b1c76baa61b93a966dd": "H_k(V(M)) = 0", "a6ffe33d0d6e1295c3861a44017ae9f8": "\\hat{f}(n)", "a7003349db1e5bf5509152568645b46e": "P=\\mathbb{P}_{2}D", "a700338f29d5610eb5ebe8593fe28d9e": "f(\\cos \\theta) \\sin \\theta", "a7005a3b7b5b4a871e9c7e4d334b8729": "\\frac{\\left(1-D\\right)D}{2\\left|I_o\\right|} = 1", "a70064fe2cd11f22840e6e3e4a926e7e": "E_{ij} \\colon V \\to V", "a700bae2dd9c74dd17fab13cacdbd661": "d/c", "a7015ab3ccb17cdf18909c03c1d12e76": "CandS_{k+1} := CandS_{k+1} \\cup \\{s_1 \\cup s_2\\}", "a70162bbd4c97d5abaea0d8865945a30": "log(K_d)", "a7019dc36304865aab6ad8357fb6aba8": "\\int\\frac{x\\;\\mathrm{d}x}{1-\\cos ax} = -\\frac{x}{a}\\cot\\frac{ax}{2}+\\frac{2}{a^2}\\ln\\left|\\sin\\frac{ax}{2}\\right|+C", "a701bd4c3f37d1fc6f482ceb5bb1c064": "\\mathbf{P}_{11} = \\begin{bmatrix}\n1 & 1 \\\\\n1 & 1 \\end{bmatrix}, \\mathbf{P}_{12} = \\begin{bmatrix}\n2 & 2\\\\\n2 & 2\\end{bmatrix}, \\mathbf{P}_{21} = \\begin{bmatrix}\n3 & 3 \\\\\n3 & 3 \\end{bmatrix}, \\mathbf{P}_{22} = \\begin{bmatrix}\n4 & 4\\\\\n4 & 4\\end{bmatrix}.", "a701eb15aaebdd365911d0df1da9c8f7": "D_2", "a70249c5a57e0db3f9a8cf10d0667989": "\n\\tau = \\ln \\frac{d_{1}}{d_{2}}\n", "a7026ea3b0153678f88001532a85998e": "V=\\frac{1}{6}(45+17\\sqrt{5}+15\\sqrt{5+2\\sqrt{5}})a^3\\approx21.5297...a^3", "a7026f733e6c9adefb8a53e82b6bfaa7": "\\| \\sigma \\|_{C_{0}} := \\sup_{t \\in [0, T]} \\| \\sigma (t) \\|_{\\mathbb{R}^{n}},", "a70278ab7af5de4ea4c1fbb6752f3fde": " {A_{i}} = {i_\\mathrm{out} \\over i_\\mathrm{S}} \\Big|_{R_{L}=0} ", "a702982fd417a2f147a8f638d1b42697": "2\\cos \\left( \\frac{\\pi}{9} \\right)", "a702a775a24e054f89e541a1826e0c07": "E_K", "a702fa13d35e45a5fdba2ee92f21d82e": " f'={\\frac{1}{1- {\\frac{2M}{r}}}}\\sqrt{\\frac{2M}{r}} ", "a703634879e5e9b70829217859822b8f": "h(x)=[0;a_2,a_3,\\dots].\\,", "a70396983b53e871e9480437d66726e7": "\\displaystyle\\Delta=-\\partial_r^2 - \\coth(r)\\cdot \\partial_r,", "a703a2b1efa150f27b881298d17e89af": "\\widehat{\\lambda}_\\mathrm{MLE}", "a703a791ab5ed050c3487919ce093ceb": "p_k = \\frac{{{2n}\\choose k} \\cdot \\frac{k!}{k} \\cdot (2n-k)!}{(2n)!} = \\frac1k.", "a704071522278180f3c8d93f16cef191": " -e_1=<-1,0> ", "a704768550448188d5e07111775db679": "k\\in \\mathbb N", "a704a04f68effb68982227b7bd3251e5": "[\\Psi_j]_{\\Phi_j} \\leftarrow QR\\left(I_{\\langle\\Phi_j\\rangle}-[\\Phi_{j+1}]_{\\Phi_j}\\left([\\Phi_{j+1}]_{\\Phi_j}\\right)^*, \\epsilon\\right)", "a704bc5e0df4871d96e935930f69a0e2": " \\mathsf{S}=(\\mathbf{S}, \\mathbf{V}), ", "a70548273086df7f613beea6e1a67c6f": " |\\downarrow \\downarrow \\rangle ", "a705ceb2a64f657c0dd3010f9e8c928a": "\\lim_{c \\to \\infty} \\frac{1}{c^{2}} \\log \\mathbf{P} \\big[ \\| B \\|_{\\infty} > c \\big]", "a706f6821d6d49d4a3ef8ee630e8f138": "\\nu_1=2C", "a7070c18ab0fabcfd1dd45efdb3a20e8": "O(lgq(x))", "a707159b2d1aa11f3c9e7e27aa87b9f3": "\\max_{f \\in \\Delta {\\mathcal A}} |x(f)|=\\max_{f \\in F} |x(f)|", "a707a6a1d109725d420639c2c944c7e5": "S=\\{1,2,\\ldots,n\\}", "a707a92c60e630c3eb135924515f6c31": " \\left\\langle {\\exp [ - \\overline \\Sigma_t \\; t ]} \\right\\rangle = 1,\\quad \\forall t ", "a707bc65541c7d93f68ac6ec9c3ee33b": "\\mathfrak{e}_{6(2)}", "a707ec24eea3d14b25e9b19f58258044": "BS(m,n)", "a708251c9c711883e34322fa53ec8706": "(*) \\qquad \\forall U \\subseteq V, \\quad o(G-U)\\le|U|", "a70849b8773d8336a94038f6ca296cb1": "\\sigma_x : G \\to M, g \\cdot x", "a709265c1b4002c16d4646565dd110db": "\\int_0^\\infty \\frac{e^{-w/t}}{1-w} dw \n= \\sum_{n=0}^\\infty t^{n+1} \\int_0^\\infty e^{-u} u^n du", "a7095a7d94037d67a11b4752d4903a40": "\\alpha=3V\\frac{\\omega_p^2}{\\omega_p^2-3\\omega^2-i\\gamma\\omega}", "a7095eb93b8c78eb8f16ca40592f52bd": " \\int_X f d \\mu", "a70a4da8072ff7fbe2d82ab941daad9f": " G=\\alpha S + (1-\\alpha)e e^{T}/N", "a70aeaa08239d2bb4d66aa404a22da18": "F(2)+F(4)+F(6)", "a70b0a91833f59a700b0da46fce2bf07": "n_0 = \\frac{p_0}{k_{\\rm B}T_0}", "a70b363c2a3cb52b7e563ae41b2dde58": "\\scriptstyle\\varphi:M\\,\\longrightarrow\\, [0,\\infty]", "a70b9cb69dfef12291b77a42c5c7bf73": "10^{-20}", "a70bac5408e47e248e27089f7f76f3db": "\\vec{f}_3 = \\frac{1}{r \\sin(\\theta)} \\, \\partial_\\phi ", "a70c31e8313d3bc611f33d0db514894d": " u_p + u_o = (\\mu_p +\\mu_o) E", "a70c4034ed85dece836bb25be897b532": "\\forall x \\forall y[\\forall P(Px \\leftrightarrow Py) \\rightarrow x=y]", "a70c4b8b1aef8f2e8cbeeb2124faba1f": "\\begin{bmatrix}\n\\begin{matrix}R_1 & & \\\\ & \\ddots & \\\\ & & R_k\\end{matrix} & 0 \\\\\n0 & \\begin{matrix}\\pm 1 & & \\\\ & \\ddots & \\\\ & & \\pm 1\\end{matrix} \\\\\n\\end{bmatrix}", "a70c5c9874f18d75a029110489636c15": "\\sqrt{z} = (\\sqrt{y})^{-1}", "a70c65e55d9636e4a259da2fb000fb61": "a_n=\\lfloor\\alpha(n+1)+\\rho\\rfloor -\\lfloor\\alpha n+\\rho\\rfloor-\\lfloor\\alpha\\rfloor", "a70d0c9b2e529c999ec05569e1638668": "\\omega \\,", "a70d3d262a5633d262ec80153879c8ee": "A_{\\mathbf{s},\\chi}", "a70daea1c14ab7fe350135410248d556": "U^2= T-n( \\bar{F}-\\tfrac{1}{2} )^2,", "a70db7ab2a737ec4886ef17c4b039d66": "\n1.~~~~ds^{2}=g_{ij}dx^{i}dx^{j}\n", "a70e110de5c0c792a2e44807e8b0bcda": " \\hat{k} , \\hat{l}", "a70e77625819793fa4140f8f058200b1": " \\gamma", "a70f0472ef4f03d91304a739b569e5ef": "\\gamma V^{2/3} = k(T_c - T)\\,", "a70f10ba17bfa6e60a66d5e224021416": "\\mathbb{R}^1", "a70f2fe89b3d58e24bfe2e43b20f6b1c": "F_i\\,\\!", "a70f43fdc49dc53416f65d6f3b885eb7": "\\begin{matrix} {12 \\choose 5}{4 \\choose 1}^5 \\end{matrix}", "a70f688dde4f81d24bd36481ae3ca8c9": "\\begin{align}\nu &= 273+180\\sqrt{2}\\\\\nw &= 538359129\\sqrt{2}-761354780\\\\\n\\end{align}\n", "a71023bcb181b0ee4fb70e7ff7287718": "D^{*}", "a7105094394d32e841ff890d61e085c9": " D = \\ | a x_1 + b y_1 + c z_1+d | .", "a710538535fdd7dbf936496158beb7ce": "|U| > \\frac12", "a710d730817612216938500ad49d7084": "\\epsilon (t)", "a710f23cd4be69c1c93787a21f726a4d": "B=\\sigma", "a711183dd33efbd327e17d4a9584d9ad": "=(A \\cap A) \\cup (A \\cap A^C)\\,\\!", "a7111fd557b78c77a59799cbac4e3fef": "w\\log(n/w)", "a711264abd837ea77e10f17268482b2b": "\\frac{1}{2}\\left[\\ln(-g)\\right]_{|\\mu|\\nu|}-[\\mu\\nu,\\beta]_{|\\beta} = \\frac{\\kappa \\, \\rho_0}{2} \\, \\delta_{\\mu\\nu}", "a7115f35399e8c4d022c783f4297ced3": "s(1)=12", "a71305b43406abc247827444a92b38fa": "\\Pi = \\{ 0 = t_{0} < t_{1} < \\dots < t_{m} = T , m \\in \\mathbf{N} \\}", "a713718167f380c32eebb21ab46323ad": "\\forall i: \\quad \\eta_i({\\boldsymbol \\theta}) = \\theta_i.", "a7138ab38a595c2d3014f07cdc06647b": "R \\propto V_x^a\\,V_y^b\\,g^c.\\,", "a713978a4b62b71e0fac34a36d3ceacf": "\n u_1(x_1,x_2,x_3) = -x_3\\,\\frac{\\partial w}{\\partial x_1} ~,~~\n u_2(x_1,x_2,x_3) = -x_3\\,\\frac{\\partial w}{\\partial x_2} ~,~~\n u_3(x_1, x_2, x_3) = w(x_1,x_2) \n ", "a713c2e565f50b7e2c9f0dbf0b73da47": "P_{>} :=\\{p>0\\mid p\\in P\\}", "a7142423b2bca04b99ac48ba4286a132": "\\Omega _{ij} = \\phi (x_i )^T \\phi (x_j ) = K(x_i ,x_j )", "a71464a368940d6785d421e3f5ec1c12": "+I", "a714777fbf84e55d314f03c1729eec80": " \\prod_{p>2} \\Big(1 - \\frac{1}{(p-1)^2}\\Big) = 0.660161... ", "a714a37be463fb3a3b5121852e84b36e": "\\underline{F}", "a714de9878d707a3f6ab6e48bf15e9ac": " y={{c+di}\\over{c-di}}={{c^2-d^2}\\over{c^2+d^2}}+{2dc\\over{c^2+d^2}}i\n", "a7154fdccddffc30f9c15ec16aefd6ec": "2^{2-2}=1", "a7157c8db025d842f67e0dcfc4189bb8": "\\!\\mu_1(v_3)", "a7158b0d16b3ccf549a5cf2a774f0fab": "f(a,b) = \\sum_{n=-\\infty}^\\infty\na^{n(n+1)/2} \\; b^{n(n-1)/2} ", "a715a6a488a552e2f039b6e1d81b2032": "\\scriptstyle N\\rightarrow \\infty", "a715a9865241b36b9163e23d87866e26": " \\and (S_2 \\implies (\\operatorname{equate}[A_2, p\\ p\\ f] \\and V[F_2] = A_2)) \\and D[F_2] = K_2 ", "a715c58eed34d98029da920110c5b747": "a = b = c", "a715f4f9fe9384f2fef99a5bfec4cf8d": " \\sin^2 (2\\theta_{13}) = 0.092 \\pm 0.016 \\, \\mathrm{(stat)} \\pm 0.005\\, \\mathrm{(syst)}. ", "a716001ab476f9271c78dbbc9caae5f5": "{\\widehat{AF}}_4", "a716df22151c82daf1863519a06bbf42": "(k>0)", "a7173f5f029b214af8acbaa9966e4adc": "\\mathbf{H}^+_m \\times \\mathbf{H}^+_n", "a71783526252273494fcbb8ba61a55c1": "\\ a+b=a \\quad \\Rightarrow \\quad b=0, \\, ", "a717fc999f5afcb96a9a04da0a46f592": "p_{i+1}", "a717fca4a540385e0f1afb9e520e8ad4": " a,a^{-1}", "a7183124bcbc21a8209145b3f51e546e": "|{\\tilde{\\psi}_{t+\\delta}}\\rangle=e^{-i\\frac{{\\delta}}{2}F}e^{{-i\\delta}G}e^{\\frac{{-i\\delta}}{2}F}|{\\tilde{\\psi}_{t}}\\rangle.", "a71853660a2f4882b434953706aea133": " c(x,y,z)=\\frac{2\\sin \\angle xyz}{|x-z|}", "a71862c3832d2ca6d0d94e476406dcdc": "(x-3)", "a718e5aa1e81ce3c5d31c06d5b07ebb1": "\n R_{K\\_i',k} = - \\nu_{i',k} M_i A_k T^{\\beta_k} \\exp{\\left( -\\frac{E_k}{RT} \\right)} \\prod_{j'=1}^{N} \\left[ C_{j'} \\right]^{\\eta_{j',k}} = K_{i',k} M_{i'} \\prod_{j'=1}^{N} \\left[ C_{j'} \\right]^{\\eta_{j',k}}\n", "a718f593cb3d1a4c72949ce3925471e9": "F(x) =\\int_0^\\infty P_{it-1/2}(x)f(t) dt", "a719933619a752a7417afc15a4ecf5d0": "\\phi \\colon U_\\alpha\\to \\phi(U_\\alpha)\\subseteq\\mathbf{R}^n.", "a719e2c6e741b255a2b794f59cd3f88e": " f_1 \\in O(g) \\text{ and } f_2 \\in O(g) \\Rightarrow f_1+f_2 \\in O(g) ", "a71a06b467b4eb1cd1eb2ce38f9bf590": "k(\\lambda,\\,\\phi)", "a71a08aeb2899e59830c20996edbe4d9": "\\{(3,6,1), (3,4,2), (4,1,5)\\}", "a71a7bd9152232b3bdcecae1e5440be7": "\\scriptstyle k_1 = 0.01", "a71a9d11b912e50b3023a1e31fc12d97": "\\gamma \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\frac{1}{\\sqrt{1 - v^2/{c}^2}}", "a71ab8a6202ef77c65f955f9e1c3c38c": "E_d > \\Delta E", "a71add16a0f2ca131bb6dff4d8c9c699": " y \\cdot u = y \\cdot v = 0", "a71b2301af4081126f95d1b3eb328e30": " R_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu}R + \\Lambda g_{\\mu\\nu} = {8 \\pi G \\over c^4} T_{\\mu\\nu} \\!", "a71b4063f19aa972306f616189e785c0": "A=\\frac{1}{2}(b+d)a", "a71b472b99130c725c6a25e3d53eaacf": "\n\\begin{align}\n\\varphi\\left(\\sum_{i=1}^{n+1}\\lambda_i x_i\\right) & = \\varphi\\left(\\lambda_1 x_1+(1-\\lambda_1)\\sum_{i=2}^{n+1} \\frac{\\lambda_i}{1-\\lambda_1} x_i\\right) \\\\\n& \\leq \\lambda_1\\,\\varphi(x_1)+(1-\\lambda_1) \\varphi\\left(\\sum_{i=2}^{n+1}\\left( \\frac{\\lambda_i}{1-\\lambda_1} x_i\\right)\\right).\n\\end{align}\n", "a71b9ae19e509388f5ab7bc3559a9109": "P_t", "a71c21d68a733d43fd435df853a88f86": "V_{ijrs}^{(0)}", "a71c30beb160a141083b5afd4425d4a8": "A\\cup B=\\{1,...,N\\}", "a71c84ffd4c8d8547f2edff8fd5ac682": "\\rho:L\\otimes V\\rightarrow V", "a71c8b57f48cbd022d68a3b58137d32a": "\n\\langle p_1,\\ldots,p_n\\ \\mathrm{out}|q_1,\\ldots,q_m\\ \\mathrm{in}\\rangle=\n\\prod_{i=1}^{m}\n \\left\\{\n (-i)(2\\pi)^{-3/2} Z^{-1/2}\n \\left(p_i^2-m^2\\right)\n \\right\\}\\times\n", "a71c9a0cb0b8e83221630349cf929fad": "f(x)\\!", "a71d7f6a3e251dec9a82d3f00a7c8dc1": "A_n = \\frac{a_n}{\\prod_{k=0}^{n-1} f_k},", "a71e637932d13f222cfe3361f719e672": "(\\Z/2\\Z)^n", "a71e9bd6d63620abde9e536bb1034043": "\\mathbf{F}' = q\\mathbf{E}' = q\\mathbf{v} \\times \\mathbf{B}.", "a71f13fb7ec7e3aebc527f551052495a": "G^p", "a71f1a1e519ad3e787793b08bb86be31": "R \\approx S \\rarr P[S + (S \\times R)]", "a71f211f3c5f36a482ed9cc4bb788dcf": "\\text{CR}", "a71f235e16e5097b2a9ff096d05f021f": "2\\pi\\;", "a71f491218d8f9961ddba93dbe62af5f": "\n\\mathrm{var}(T)\\geq\\frac{1}{I}.", "a71f5ed9f3e0978cfd7bceab386fa339": "\\overline{Y}_{i\\bullet} = \\frac{1}{n}\\sum_{j=1}^n Y_{ij}", "a71f8c6f168ca87c6430f32ee6c581bd": "\\begin{bmatrix}\n c_2 & - c_1 s_2 & s_1 s_2 \\\\\n c_3 s_2 & c_3 c_2 c_1 - s_3 s_1 & - c_2 c_3 s_1 - c_1 s_3 \\\\\n s_2 s_3 & c_3 s_1 + c_1 c_2 s_3 & c_3 c_1 - c_2 s_3 s_1\n\\end{bmatrix}", "a71fbc9c854251cee780099bafb4ad8b": "p^2 + q^2 + r^2 + s^2 = 1", "a71fc80a000c469c1e1ba448382c2039": "\nD\\left(\\zeta\\right)=\\log\\left(\\frac{\\sqrt{1-2\\rho\\zeta+\\zeta^2}+\\zeta-\\rho}{1-\\rho}\\right).\n", "a71fce0901e4c38e607683deaae1ce0c": "\\sigma\\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix} = \\begin{pmatrix} \\overline{d} & \\overline{c} \\\\ \\overline{b} & \\overline{a}\\end{pmatrix}.", "a71fdddb8f010c376474d38431432309": " |\\psi\\rangle = \\begin{pmatrix} c_1 \\\\ c_2\\end{pmatrix} = c_1\\begin{pmatrix} 1 \\\\ 0\\end{pmatrix} + c_2\\begin{pmatrix} 0 \\\\ 1\\end{pmatrix} ;", "a71fea95719ea87af836d67f6e5aaafe": "\\hat\\theta_i(\\theta_i) \\in \\arg\\max_{\\theta'_i \\in \\Theta} \\sum_{\\theta_{-i}} \\ p(\\theta_{-i} | \\theta_i) \\ u_i\\left( y(\\theta'_i, \\theta_{-i}),\\theta_i \\right)", "a71ff4ac8a0e85777289859b35accf3d": "f(p) =0", "a71ffae62e338720654dda537eb30c69": " \\sum_{i=1}^n \\frac{x_i - x_0}{\\gamma^2 + [x_i - \\!x_0]^2} = 0", "a720097a7112c4f538306b1f89bc1bd2": "f(w,z) := w^p + z^q.", "a720439551e4cd1115c0b4e0d3c972e8": "S=\\sum_{p\\le P}\\exp(2\\pi i f(p)).", "a72063f818eb4b28cd8fee85c6d16ece": "Z_{iT mm'}=\\frac{ \\sqrt{1-\\omega^2} \\left(1-(1-m^2)\\omega^2 \\right)}{1-\\left(\\omega/\\omega_\\infin\\right)^2}", "a72077f2ddc9c5436672d7b509a52394": "\\mathrm{Herm}(a_1,a_2)\\,", "a7207ea529afca28d84813894392b3bc": "{m \\choose r}_q\n= \\begin{cases}\n\\frac{(1-q^m)(1-q^{m-1})\\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\\cdots(1-q^r)} & r \\le m \\\\\n0 & r>m \\end{cases}", "a7209e2dad21599f858f3b03890639c6": "0q", "a721f3af10cee3257abb4c080510c730": "|s\\rang = \\frac{1}{\\sqrt{N}} \\sum_{x=1}^{N} |x\\rang ", "a7226f9f6e81fce6ee5037eb55389693": "\\begin{align}C_{abcd}&= \\Psi_0U_{ab}U_{cd} \\\\\n&\\, \\, \\, +\\Psi_1(U_{ab}W_{cd}+W_{ab}U_{cd}) \\\\\n&\\, \\, \\, +\\Psi_2(V_{ab}U_{cd}+U_{ab}V_{cd}+W_{ab}W_{cd}) \\\\\n&\\, \\, \\, +\\Psi_3(V_{ab}W_{cd}+W_{ab}V_{cd}) \\\\\n&\\, \\, \\, +\\Psi_4V_{ab}V_{cd}\\end{align}", "a722a3acfc4acc88e1f249345cc052a5": "(\\overline{x3}\\vee gate6\\vee \\overline{gate5})\\wedge (gate7\\vee \\overline{gate2})\\wedge (gate7\\vee \\overline{gate4})\\wedge ", "a722edc7e6f03fa7e0a31a0d1540a279": "H = U + pV\\,\\!", "a72317bf0085f66d48774af76ac79884": "\\sqrt [13]{3} = 3^{1/13} = 1.08818...", "a72353169ebd821314004631bf6780f3": "\\Lambda(x) = \\prod_{i=1}^\\nu (1- x \\, X_i) = 1 + \\sum_{i=1}^\\nu \\lambda_i \\, x^i", "a723973c90dc54325bbfcc996da886db": "\\begin{align} \n\\lim_{s \\to a} \\frac{(s-a)P_1(s)}{P_2(s)} & =\\lim_{s \\to 0} \\frac{(s-0)((2-\\gamma )s^2+(\\alpha +\\beta -1)s)}{s^3-s^2} \\\\\n&= \\lim_{s \\to 0} \\frac{(2-\\gamma )s^{2}+(\\alpha +\\beta -1)s}{s^2-s} \\\\ \n&= \\lim_{s \\to 0} \\frac{(2-\\gamma )s+(\\alpha +\\beta -1)}{s-1}=1-\\alpha -\\beta. \\\\ \n\\lim_{s \\to a} \\frac{(s-a)^2 P_0(s)}{P_2(s)} &=\\lim_{s \\to 0} \\frac{(s-0)^2( -\\alpha \\beta)}{s^3-s^2}=\\lim_{s \\to 0} \\frac{( -\\alpha \\beta)}{s-1}=\\alpha \\beta. \n\\end{align}", "a723c6b8a134626e7b0752706b73a289": " Z(v;T)", "a724112ea91294fb59d4d714ddeffaec": "\\mathbf{F}_{13^2}", "a72461bca40defa99f38d439b1942448": "\n g(z) = a\n + \\sum_{n=1}^{\\infty}\n\\left(\n\\lim_{w \\to a}\\left(\n{\\frac{(z - f(a))^n}{n!}}\n\\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}w^{\\,n-1}}\n\\left( \\frac{w-a}{f(w) - f(a)} \\right)^n\\right)\n\\right).\n", "a724a66366a6f35091d54d64784231c1": " \\sigma^2_0 = <", "a724ec676d72793d2c5b9805f5f58cc5": "\\{\\phi_e: \\textrm{dom}\\, \\phi_e \\in C\\}", "a725186f7c6159119d725d28e3c915cb": "\\mathfrak{p}_1 \\supseteq \\mathfrak{p}_2 \\supseteq \\cdots \\supseteq \\mathfrak{p}_n", "a725255766506dd6c05a221d3c87128a": "\\Gamma_q(x)=\\frac{(1-q)^{1-x} (q;q)_\\infty}{(q^x;q)_\\infty}", "a7253c83b7214dffe7d672aea1c50ea8": "\\text{R-X}^+\\text{A}^-\\,+\\, \\text{M}^+ \\, \\text{B}^- \\rightleftarrows \\,\\text{R-X}^+\\text{B}^- \\,+\\, \\text{M}^+ \\,+\\, \\text{A}^-", "a72546b6416fb13652e3ff8c7ce48aa9": "p(x)=a_n x^n+\\cdots \\,\\! ", "a725599053c63a74f6a142765d7945f5": "y_1=\\varepsilon(x,\\bar{C})", "a7258fbc7aacf20ce9677831de23a05c": "\\mathrm{1 \\, sb = 10^4\\,\\frac{cd}{m^2} \\approx 0.3048^2 \\cdot 10^4 \\cdot \\pi \\,\\, fL = 2918.6...\\, fL}", "a725b47e7479c1f38ac4b00a0c18b96b": "\\scriptstyle x\\in [0,a]", "a725ca4ce1f7f1ff5acf6f2fdbe73f48": "\\left(\\begin{array}{c}\nx-a\\\\\ny-c\\end{array}\\right)^{T}\\left(\\begin{array}{cc}\nA & \\frac{B}{2}\\\\\n\\frac{B}{2} & C\\end{array}\\right)\\left(\\begin{array}{c}\nx-a\\\\\ny-c\\end{array}\\right)=G", "a726041ab1e3253e8c1681c93a19bb95": " \\mathrm{In}(H) = \\left( \\pi(H), \\nu(H), \\delta(H) \\right) \\, ", "a7260bb8c3dec2bdeb897c1846bdcbef": "\\pi_i(\\mathbb{HP}^\\infty) \\otimes \\mathbb{Q} = 0 ", "a7262cd09f721310eff849b0cb78578f": "\\,\\delta\\,", "a72687e32b0d839fe35833198a7c7059": "\\displaystyle{H^\\varepsilon f(\\varphi) = {i\\over \\pi} \\int_{\\varepsilon\\le |\\theta| \\le \\pi} {f(\\varphi-\\theta) \\over 1-e^{i\\theta}} \\, d\\theta={1\\over \\pi} \\int_{|\\zeta-e^{i\\varphi}|\\ge \\delta} {f(\\zeta)\\over \\zeta-e^{i\\varphi}}\\, d\\zeta,}", "a7269881217f735c0124ae6b706ca05a": "1\\leq p<+\\infty", "a726cd72581d3c1de5476c1d1e83ae4f": "T /\\, A", "a726d9e04d83e1d689456734ae08746d": "\n \\cfrac{d}{dt}\\left(\\boldsymbol{F}\\cdot\\boldsymbol{F}^{-1}\\right) = 0\n \\quad \\implies \\quad\n \\dot{\\boldsymbol{F}}\\cdot\\boldsymbol{F}^{-1} + \\boldsymbol{F}\\cdot\\dot{\\boldsymbol{F}^{-1}} = 0\n", "a72706c934d72822b07663ce3e18996d": "\\pi_2=p_x\\rho r^5/\\dot{m}^2", "a727b3abb09c0ecf8898343a5bbeffb1": "x_{s}", "a728326e86972dfec9616a861e7f69c0": "\\; I_k", "a7286304a1525b80ef7ac331c93a3c2b": "\\omega^{A}_{x\\land y}=\\omega^{A}_{x}\\cdot \\omega^{A}_{y}\\,\\!", "a728877741d8dce3a9253fba4128305c": " c = \\frac{4 h}{\\sqrt{a^2-b^2}}. \\,", "a728941d7442af37887322ce284ce62d": "p-1 = Q2^S", "a728ae670c41d636377a38a645159fff": " \\begin{bmatrix} V_2 \\\\ I'_2 \\end{bmatrix} = \\begin{bmatrix} A' & B' \\\\ C' & D' \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_1 \\end{bmatrix} ", "a728baef7e126bf8d23549bf03ebdc72": "D (\\mathbf{\\hat n},\\phi) = \\exp \\left( -i \\phi \\frac{\\mathbf{\\hat n} \\cdot \\mathbf J }{ \\hbar} \\right)", "a728ca98c906e0425361b8a6b6ed1ce0": "\\Pr(S - \\mathrm{E}[S] \\geq t) \\leq \\exp \\left( - \\frac{2t^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right),\\!", "a728cb4f4598fff48801a52f069d75ff": "(M_\\Phi f)(x)=\\sup_{t>0}|(f*\\Phi_t)(x)|", "a728e0dab3753759fef4526a8742134e": "S_0\\subseteq S", "a72938fba01fc6d0b7a8307ddbb099fe": " L \\, ", "a729665ff152d8b6cf5f0d5306dab155": " N = \\frac{E_\\max-E_\\min}{\\Delta},", "a72973fc867c182e5df8ff5bf1e1e599": " 0 \\rightarrow K^\\prime \\rightarrow X \\rightarrow P \\rightarrow 0. ", "a729c458304a4b6d56a4f4a40b85f3f3": "Xf(L) = \\int_L f = \\int_{\\mathbf{R}} f(x_0+t\\theta)dt", "a72a0daec991a8cc763b8445137b2a52": "\\textstyle 1 \\cdot 1 = 1 \\, , \\quad 1 \\cdot a = a \\, , \\quad a \\cdot 1 = a \\, . ", "a72a60e434f73779ea8a18f8f8dc486f": "P_{t+1} = N_t(1-e^{-aP_t}), \\,", "a72a69be1a36206c4cabfa5c83a80df3": "\n{\\hat{\\beta}}(q) = \\min \\{ \\alpha: \\ \\mbox{windfall is achieved for at least one } u \\in \\mathcal{U}(\\alpha,\\tilde u)\n\\}\n", "a72af565f8767ca9995bec07f8e2abda": "{(a_n)}_{n\\in\\N}", "a72af90fa920e65f5548bdd88868ada6": "-\\log(b-x)", "a72afd227dc511def48c73ed6d358168": " E_0(\\mathbf{R}) \\ll E_1(\\mathbf{R}) \\ll E_2(\\mathbf{R}) \\ll \\cdots \\text{ for all }\\mathbf{R}", "a72b18c11198b7bf9f0fea1af1863fd2": "\\frac{5}{12}", "a72b2594b5e47ae982702c4f65b40325": "\\langle f , g \\rangle := \\int_\\mathrm{F} f(\\tau) \\overline{g(\\tau)} \n\n(\\operatorname{Im}\\tau)^k d\\nu (\\tau)", "a72b8e3ac79f3f40f54a9e7ba5995b87": " \\tilde{P}(X_1, \\ldots, X_{n-1})", "a72c0a955916443dbf06785ccbf305fc": "s|t", "a72c544b07ab4483c7dd4bf44ae00279": "S=\\{\\,s_1,s_2,\\ldots,s_n\\,\\}\\,", "a72c6b752e57467e0de0d8346834bb23": "\n[x_2]:=[x_2] \\cap ( [x_3]-[x_1]) \n", "a72cf0d1879c2ec4c8d85d01836873ed": "N > 0", "a72d0b77599e366ac2d33be0bd96f7c7": "m(f)=-M(-f)", "a72d95719b9f459a24e749fe8c57a608": "\\sum_s c(s)\\lambda x_s^*=\\lambda c\\cdot x^*", "a72dc03bbcb96c7ec7e4946745b4f1f7": "=\\left ( \\frac{3}{331}\\right ) \\left ( \\frac{5}{331}\\right ) \\left ( \\frac{7}{331}\\right ) \\left ( \\frac{23}{331}\\right )", "a72e95a2c7f4eeff340ff5208d0f55b0": "o(f) + o(f) \\subseteq o(f)", "a72eea8fdebc0a1b14aa9c2f6b37525a": "\n{T}^{-1} = \\frac{d}{dE} S(E)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(10)", "a72f0bb7b3117479a2077461e3e03349": " \\dot{V}_D ", "a72f0ddba8fa4afce5df06f83dc4d2ff": "C_G(k)=k^{-1} \\chi_{G^*}(k).", "a7301f3810357ddb90a0a0a26892eb7f": "\\bigvee_{W \\in \\Sigma'} W = \\mbox{span} \\bigcup_{W \\in \\Sigma'} W \\;.", "a730232d1b13d10336215e3ccdc61ae4": " \\sigma_{kk} = \\sigma_x + \\sigma_y+\\sigma_z = \\sigma_{11} + \\sigma_{22}+\\sigma_{33}\\,", "a7303e2405d3d3baeb954176b77fb98d": "\n(\\theta*t)\\circ(\\theta_{t1},\\ldots,\\theta_{tn}) = (\\theta\\circ(\\theta_1,\\ldots,\\theta_n))*t; \n", "a730479a9242e6293d030e8c068e87ee": "G:=\\left\\{\\begin{pmatrix}\na & b \\\\\n0 & 1 \\end{pmatrix}\\bigg| a>0,b\\in\\mathbf{R}\\right\\}", "a7306137cc9d9fcdf8aa4fbcce0830de": "V_T, V_U", "a7309ea24c489ccd16ded69c69491f40": "T_{64} = 1 + 2 + 4 + \\cdots + 9,223,372,036,854,775,808 = 18,446,744,073,709,551,615", "a730c8caf63ec87d1e650b1104f6af67": "\\bold k", "a73187daacbd7fa02ca656dd399192d1": " \\mathrm{N} = \\frac {B^2 L_c \\sigma}{\\rho U} = \\frac{\\mathrm{Ha}^2}{\\mathrm{Re}} ", "a731bb2b6faa5f0ed049bde907a53778": "j \\in \\mathbb{R}^3", "a731dca3a53c8dbf065f295f30663a3f": "(P_K - 5*P_C)/P_K = 0.0051", "a732326406cc49d78d7754fb2fcfdf9d": " \\, \\frac{|SD|}{|CD|}=\\frac{|SB|}{|AB|}", "a73265613217d657e84c4f3225738e68": "\\frac{b^{p-1}-1}{p}", "a732f2ab88a4a0770fbda6e00561720f": "m_i = 1 ", "a7336a7ff0a21db56785b7c1f2d9ebf2": "\\begin{array}{rcl}\n\\ddot\\varphi_0 = \\ddot\\varphi - \\ddot\\xi &=& -(g+a~\\nu^2\\cos\\nu t)\\frac{\\sin\\varphi}{l} - \\frac{a}{l}\\left(\\ddot\\varphi_0 \\cos \\varphi_0 ~\\cos\\nu t -\\dot\\varphi_0^2\\sin\\varphi_0 ~\\cos\\nu t - 2\\nu\\dot\\varphi_0\\cos\\varphi_0~\\sin\\nu t - \\nu^2\\sin \\varphi_0 ~\\cos\\nu t \\right) \\\\ \n &=& -\\frac{g}{l}\\sin\\varphi_0 -(g+a~\\nu^2\\cos\\nu t) \\frac{1}{l}\\left(\\xi\\cos\\varphi_0 + O(\\xi^2)\\right) - \\frac{a}{l}\\left( \\ddot\\varphi_0 \\cos\\varphi_0 ~\\cos\\nu t -\\dot\\varphi_0^2\\sin\\varphi_0 ~\\cos\\nu t - 2\\nu\\dot\\varphi_0\\cos\\varphi_0~\\sin\\nu t \\right)\\;.\n\\end{array}", "a733a6108b1bd75577a09328f285519d": "\\pi\\over 5", "a733c5493329d9ab626a906c13b75181": "f = \\frac{1}{T}\n\n= \\frac{1}{\\ln(2) \\cdot 2RC}\n\n\\approx \\frac{0.72}{RC}", "a734067f85280e3723d262460fbfafbf": "\\Gamma^{a} {}_{bc}", "a73431b40d0cbd4e2e39ae7772c51448": "\\displaystyle{H(\\varphi)=-\\partial_n D(\\varphi)|_{\\partial\\Omega} =-\\partial_t(S(\\partial_t\\varphi)|_{\\partial\\Omega}).}", "a7343b494863a9c7224a691fd3917a25": " \\alpha \\wedge (\\text{d}\\alpha)^k \\neq 0 \\ \\text{where} \\ (\\text{d}\\alpha)^k = \\underbrace {\\text{d}\\alpha \\wedge \\ldots \\wedge \\text{d}\\alpha}_{k-\\text{times}}.", "a73449d82c9530707418fe8efceb0b08": "|U| \\ge c \\, |B|", "a73468187b488be9b5fbf4626a85dbb6": "\\theta_\\mathrm{A}", "a734b8db74897a9a42895a7454b1a156": "C_{D,0} = C_D - C_{D,i}", "a734c929068c8b5d93efd11f8d7e93d6": "I(X_1;X_2)=H(X_1)-H(X_1|X_2).", "a7352941798f1390c401f4bde5420d92": " y(t) = \\sum\\limits_{k=0}^{K} |H_k(\\omega_0,\\gamma)|\\gamma^k \\cos\\big(k(\\omega_0t + \\varphi_0) + \\angle H_k(\\omega_0,\\gamma) \\big) ", "a7352a9c6e3d8e6ee2b760a566774fa4": " 2f(x) = 2E_0 + 2E_1 x + 2\\sum_{n=1}^\\infty E_{n+1}\\frac{x^{n+1}}{(n+1)!} = 2E_0 + 2 E_1 x + \\sum_{n=1}^\\infty \\sum_{k=0}^n \\binom n k E_k E_{n-k} \\frac{x^{n+1}}{(n+1)!}. ", "a7353252689af94a92a178a46d87cc21": " \\mathbf{x} = A^{-1} \\mathbf{b} = \\begin{bmatrix} 0.8122\\\\ -0.6650 \\end{bmatrix}. ", "a7357b8b3b2cd72791f0b58ef21b9641": "E_{\\alpha, \\beta} (z) = \\sum_{k=0}^\\infty \\frac{z^k}{\\Gamma(\\alpha k + \\beta)}.", "a735c6d3fd2e597188542587642064bc": " u(0,x) = f(x), \\quad u_t(0,x)=g(x), \\,", "a7364d1051ec166c2f1ec9d238129bd7": "\\Big( (\\mathcal{M}, s) \\models \\Phi_1 \\land \\Phi_2 \\Big) \\Leftrightarrow \\Big( \\big((\\mathcal{M}, s) \\models \\Phi_1 \\big) \\land \\big((\\mathcal{M}, s) \\models \\Phi_2 \\big) \\Big)", "a7368095cb644e151663e187589f34f1": "e^- + H_2O \\longrightarrow H_2O^+ + 2e^-", "a736ba63c2c75a9155dd1b04075cb75a": "\\hat\\beta = \\frac{\\alpha}{\\bar{x}}.", "a736d8a7503c6634aa0a80c9ec2e830a": "L_n^{(\\alpha)}(x)^2- L_{n-1}^{(\\alpha)}(x) L_{n+1}^{(\\alpha)}(x)= \\sum_{k=0}^{n-1} \\frac{{\\alpha+n-1\\choose n-k}}{n{n\\choose k}} L_k^{(\\alpha-1)}(x)^2>0.", "a7384f4a9bf0de78bb0f64e516eb6f8b": "V_{out-up} = -\\dfrac{V_{in}}{RC}t_{u}", "a738df9c253fef97c65f00a78909e3e3": "ar^2+br+c", "a738f928c19f2e016bd2de4f969b4f58": "3x_1^2x_2^4x_3^5 < -7x_1^4x_2^5x_3^5", "a7392ccb9dad0a393f3ccd2f0d873ed8": "\\sum_{k=-\\infty}^{\\infty}{\\left|h[k]\\right|} = \\| h \\|_1 < \\infty", "a73930a8ec5d1c90a5882d70f1ec9527": " TV^{\\gamma-1} = \\mbox{constant} \\,", "a739410492d8da3d324aac2a8dbbd985": "P = b+2h", "a739aa82d137869911b257c060f2136e": "\\rho_{\\Phi} = (\\Phi(E_{ij}))_{ij} \\in L(H_1) \\otimes L(H_2) ", "a739cedd068a13d55337da6dfb245456": "\\Bbb{R}\\times T_x M", "a739f263b9860e6fed9f8e47afed9c30": "\\int_{-\\infty}^\\infty |\\hat{f}(\\xi)|^2\\,d\\xi = \\int_{-\\infty}^\\infty |f(x)|^2\\, dx.", "a73a1ef047cd21338223af0cf3511054": "\\operatorname{cov}[I(x,y)] = \\begin{bmatrix} \\mu'_{20} & \\mu'_{11} \\\\ \\mu'_{11} & \\mu'_{02} \\end{bmatrix}", "a73a46f4bc17e01e06fe787427b6d4fe": " \nu = \\alpha + \\beta x + \\gamma x^{2}/2 - L^{-1} N u\n", "a73a8b86fb1c3c0ae3d1b0ab648aecb0": "(\\nu n)A", "a73ad513d7d90e3a333c921591f61952": "C(K,\\varepsilon) = O((2/\\varepsilon)^n)", "a73ae1e455149c1efd3ae61d9e8e15ae": " \\int_{-\\infty}^{\\infty} [{\\mathrm{e}}^x / (1+ {\\mathrm{e}}^{wx})] \\mathrm{d}x = \\int_{0}^{\\infty} [u/(1+wu)] \\mathrm{d}u = (\\pi/w) / \\mathrm{sin}(\\pi/w). ..........(27) ", "a73ae837d92469d88d22713044003391": "\\nabla \\times \\mathbf{B} = \\mu_0 \\varepsilon_0 \\frac{\\partial \\mathbf{E}}{\\partial t} = \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}}{\\partial t}", "a73b36e9ca010c15f9ee39e1e341d5fb": "0\\rightarrow B\\rightarrow E^\\prime\\rightarrow A\\rightarrow 0", "a73b62d90a19ecc2e1e114dd65d595d4": "(2^{|k_{b_2}|}+2^{|k_{f_3}|}+...))", "a73ba2babc47fe1114bb8d25eee05374": "\\lfloor y/x\\rfloor+1", "a73bfddde8ef902839c82045965f43ee": " \\operatorname{build-list}[\\lambda x.\\lambda o.\\lambda y.o\\ x\\ y, D, V, D[g]] \\and D[g] = L_1 ", "a73c8d6e61184c298f405a738b42cf9d": " \\delta \\mathcal{S} = T \\int \\mathrm{d}^2 \\sigma \\eta^{ab} \\partial_a X^\\mu \\partial_b \\delta X_\\mu = ", "a73cbae0811c0c09a1667d2a0f38f838": "\\varphi = \\frac{c^2}{2} \\varepsilon \\gamma_{00}", "a73d1f6970fd0f49339a364f2776f7c7": "t=f(a(x,t),b(x,t))", "a73d6e610487ab9df4b9fce5563db09a": "(170, 75_2, 71, 56_2, 50, 30, 14_4, 6, 1)", "a73d8fec097d99eada5e59509fe69494": " r_{ij} \\equiv |\\mathbf{r}_i -\\mathbf{r}_j|\n = \\sqrt{(\\mathbf{r}_i -\\mathbf{r}_j)\\cdot(\\mathbf{r}_i -\\mathbf{r}_j)}\n = \\sqrt{(x_i-x_j)^2 + (y_i-y_j)^2 + (z_i-z_j)^2 } .\n", "a73dab3ba179e9b77f475c949f2225aa": "\\mathbf{x}_k=\\mathbf{x}", "a73dba769a62fa82fd56503a2f17742f": "2\\pi \\hbar=E^2/P", "a73dc4f2b1acf2b6c9ce87d3c5bfa0ff": "\\Gamma^b_{ac}", "a73dd5f9ce517001ae62dd70b2e38228": " 2^n + 2^n = 2^{n+1} ", "a73def4f7754b29908b6d810429a476f": "G = H C_G(X) ", "a73dfd2686279b962dbeaa0484f0f175": "\\rho_{xy} [n] = x[-n]^* * y[n] \\!", "a73e835692eb2e8a7bdc2a62b124d55e": "\\mathbf E_{1s} = <\\left (\\frac{\\zeta^3}{\\pi} \\right ) ^{0.50}e^{-\\zeta r}|-\\left (\\frac{\\zeta^3}{\\pi} \\right )^{0.50}e^{-\\zeta r}\\left[\\frac{-2r\\zeta+r^2\\zeta^2}{2r^2}\\right]>+<\\psi_{1s}| - \\frac{\\mathbf Z}{r}|\\psi_{1s}>", "a73ead24cf683a0159b13c332b59e28a": "\n \\left( \\frac{\\partial^2}{\\partial x^2} - \\nu^2 \\right)\n \\left( \n \\frac{\\partial \\eta}{\\partial t}\n + \\frac32\\, \\eta\\, \\frac{\\partial \\eta}{\\partial x}\n \\right)\n + \\frac{\\partial \\eta}{\\partial x}\n = 0.\n", "a73eaf5489faf04d09f834dfac7317b5": "\\mathbf{x}_1 \\ldots \\mathbf{x}_n", "a73eb0d9dcc01b464f1d18de81a45a2d": "\\begin{align}\nF(x; \\alpha, \\beta) & = { 1 \\over 1+(x/\\alpha)^{-\\beta} } \\\\\n & = {(x/\\alpha)^\\beta \\over 1+(x/\\alpha)^ \\beta } \\\\\n & = {x^\\beta \\over \\alpha^\\beta+x^\\beta}\n\\end{align}", "a73ed989195c3917bf15de7f0b675a18": "x \\leftarrow h^a rem N", "a73eea006c5d078408ce2faaabffabc1": "X,X^\\dagger", "a73f075663e376e36fbcb9b02c2843b5": "X \\to U(F(X))", "a73f7c2afa87ad6efc774fb03aa7c341": "\\hat f(\\xi) := \\int_{\\mathbb{R}^n} f(x) e^{-2\\pi i x \\cdot \\xi} dx", "a73f9e4a2762ac5c8dcc1fbdb1eef928": "(S+O)+O'", "a73ffd125570141013922e8d51e30921": "a_{W}\\,= D_{w} ", "a740184176c36d39e379ecc7bbab2a9e": " W_0=X_0\\,", "a74037ed26c39667b92e8df82ee0493f": "\\mathrm{curl}\\,\\mathbf{A}\\,", "a740519bbedc9bf2a9595b9fae2c1724": "\\scriptstyle |\\chi_n\\rang", "a740686c465f7887233873348af820f9": "y=(x-x_0)\\,\\frac{y_1-y_0}{x_1-x_0}+y_0", "a74095cf6c6c088ac4ebe9aa04f2a798": "d_P = d\\text{ (mod }p - 1\\text{)}", "a740d9a9f5676a8712bc5a556353fade": "Q=Y/2 + T_3/2", "a741a45bfd31a3c52ec4a4e81d8d6929": "\\mathbf{L}\\mathbf{c}=\\mathbf{X}", "a741fa6c594db666d4278fe770d3d0a1": " X \\dot = Y \\iff \\begin{cases} A \\to \\alpha X Y \\beta \\in P \\\\ A \\in V_n \\\\ \\alpha , \\beta \\in (V_n \\cup V_t)^* \\\\ X, Y \\in (V_n \\cup V_t) \\end{cases} ", "a742768a47b5b7c8c26980bfd4a65eb9": "\nv_o = \\frac{2\\pi a}{T}\\left[1-\\frac{e^2}{4}-\\frac{3e^4}{64} - \\dots \\right]\n", "a7429524d4f389aa7481a254c09b6662": "(2^1/1!!)\\pi^0 ", "a742dfffa43f09d52c7abf2923e3240b": "\\gamma'q", "a743114a417cffa16ca733612c5053d5": " H = - \\sum_{i=1}^r p_i \\ln (p_i)", "a7432dce1fcaf0d86f24104c54d50d28": "L^{3-}+2H^+\\leftrightharpoons LH_2^-:[LH_2^-]=\\beta_{12}[L^{3-}][H^+]^2", "a7435f73b820acdabfbcc2a2c3a5a72f": "\\tau(h_*) \\in {\\tilde K}_1(R)", "a74360d95246cf108d1b1ece34bea644": "d_0(p_1) = d_1(p_0)", "a74378535dc30cbf16e8894e09ea6b3d": " P( t ) = 1 - e^{ \\frac{ t } { T_E } } ", "a743a8d712254e52be0e68fa07bdcdcc": "f(\\gamma,u) \\ne \\empty", "a743aa5ddcf5837f6d4eba89b3c0860f": " \\langle A_\\mu(k) A_\\nu(k') \\rangle =\\delta(k+k'){g_{\\mu\\nu} - \\lambda{k_\\mu k_\\nu \\over k^2} \\over k^2}.", "a743b850f8eb938a4877552888702d67": "\\mathfrak{g}_-:=\\oplus_{j<0} \\mathfrak{g}_j", "a743c5322730629ab1dee077105fcd00": "\\widehat{E} \\rightarrow \\widehat{E} - q\\phi \\,, \\quad \\widehat{\\mathbf{p}}\\rightarrow \\widehat{\\mathbf{p}} - q \\mathbf{A} \\quad \\rightleftharpoons \\quad \\widehat{P}_\\mu \\rightarrow \\widehat{P}_\\mu -q A_\\mu", "a743d7cba5c8dbbff69831617febbf30": "\\|\\mu\\|_\\infty < 1", "a743ec8ab6924c92b0ca9487849b1fa2": " \\mathbb{E} (Y|X=x) = \\frac3{10} x ", "a74447f7ce0c6190bbe6416805682bdd": "x*y = x\\cdot y", "a7449b1dab145acc67af4c29a212c96c": "\\textstyle \\mu=a(\\theta_{k})^{*}R_{x}^{-1}a(\\theta_{k})", "a744a3b3f92aff891cb449657daf0f1a": "\\boldsymbol{R_p}= \\boldsymbol{R_n} - \\boldsymbol{R_\\ell}", "a744ad3abda65f25a0855602860346a6": " \\dot{\\theta}(t) = - k_g(t) ", "a744e6cd2fe0be549c39995f1892a48e": " (a_1, ..., a_{t-1}) ", "a7451514fbdd0d69fdeff303dfd0d453": "\\tilde{G}^{liq}", "a745b56a366bb3c5f4bc46cbe3abfc1a": "k-\\lambda", "a745d2be9745401487f52d912592c657": " \\delta W = \\sum_{i=1}^n \\left(\\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}} + \\mathbf{T}_i\\cdot\\frac{\\partial \\vec{\\omega}_i}{\\partial \\dot{q}}\\right)\\delta q = Q\\,\\delta q,", "a745e99e80dc72e381c9605f43bf83dd": " C : [0,1]^n \\rarr [0,1]", "a745fdf00dc04b320439bf26afae3806": "\\vert \\Psi \\rangle \\in {H_A \\otimes H_B}", "a7462ebc0d171ccfced664e3b84d1981": "\\alpha_{t} \\in \\mathbb{R}", "a7473435f0666c54248a550fffb2bcf1": "\\mathcal{N}(\\rho) \\equiv \\frac{||\\rho^{\\Gamma_A}||_1-1}{2}", "a74767ae7125474220d00628be86ca36": "w \\not \\models P", "a747e9824e7a8dd37c525d93156f12be": "\\tau_\\mathrm{n} = -\\frac{1}{2}(\\sigma_x - \\sigma_y )\\sin 2\\theta + \\tau_{xy}\\cos 2\\theta", "a7483020f43641409b7468b583322747": "q(T)(x)", "a748bff63db9e132bb2ea9db98dbf3c5": "S_2 = \\sum_{i=1}^n \\log x_i!", "a748cfea9eab9ebc58b423a1262cb7e0": "\n\\frac{1}{|\\mathbf{r}_{j}-\\mathbf{r}_i|} = \\frac{1}{|\\mathbf{R}_{AB} - (\\mathbf{r}_{Ai}- \\mathbf{r}_{Bj})| } =\n\\sum_{L=0}^\\infty \\sum_{M=-L}^L \\, (-1)^M I_L^{-M}(\\mathbf{R}_{AB})\\;\nR^M_{L}( \\mathbf{r}_{Ai}-\\mathbf{r}_{Bj}),\n", "a748d7fb2a506a8c12f6cee23801dd11": " \\prod_{p} \\Big(1 - \\frac{3}{p^3}+\\frac{2}{p^4}+\\frac{1}{p^5}-\\frac{1}{p^6}\\Big) = 0.678234... ", "a748f21d82cb758af078ebe3110ba87b": "~[T]_{\\beta'}=Q^{-1}[T]_\\beta Q", "a749a52b2274af7f7a77c663c93fa8ff": "\\mathrm{Lie}_q\\,\\mathcal{F}", "a749a5984fa6f88a9b9cd31086d81786": "g_{ij} = g_{ij}(\\mathbf{r})\\,.", "a749c1827c13ffe30901b6beaaac904f": "\\mathbb{R}^{\\mathbb{R}}", "a749d95a3b5a9f182d98683ec807656c": "\\sum_{i}c_{ji}[\\langle m_k|\\hat{V}|m_i\\rangle-\\delta_{ik}(E_j-E)]=0", "a74a1d96a42be10f263d7e288399a40f": "(\\Sigma, \\Gamma, S, s_0, \\delta, \\omega)", "a74a6dba5345945e6760aea2562685ce": "\\langle K\\rangle_N=\\lim_{q\\to e^{2\\pi i/N}}\\frac{J_{K,N}(q)}{J_{O,N}(q)}.", "a74a6f45f9c274503dc8563c5550380e": "O(n \\cdot \\log{n})", "a74aa0aa94d6c57d58f4e7bf39afd4c9": " \n\\Sigma _3 =\n\\begin{bmatrix}\n1 & 0 & 0 & \\cdots & 0\\\\\n0 & \\omega & 0 & \\cdots & 0\\\\\n0 & 0 &\\omega ^2 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n0 & 0 & 0 & \\cdots & \\omega ^{d-1}\n\\end{bmatrix}.\n", "a74b0562df0a52bd0e73d8fe3c224fb0": "\\forall A \\, \\forall B \\, ( \\forall X \\, (X \\in A \\iff X \\in B) \\Rightarrow \\forall Y \\, (A \\in Y \\iff B \\in Y) \\, )", "a74b13f822d0c55a1c9beb0961bbd3c5": "n(n-1)/2m", "a74b39de1b3b570232ae3bb040996198": "d_I(x,y)=d(x,y)\\, ", "a74b486dca1c49ee7e9818d709871843": "\\mbox{STRUC}[\\sigma]", "a74b4d711b6ae7edae73b1f53a64def9": "S_{j_k}", "a74c3008b99d5a80db2121fa488cdaaa": " (1.00, 0.00);", "a74c4395f6ca33134fc2fc1f93a1a9c0": "\\rightarrow_R", "a74c43a1fc76b3adae9b0ca9e94e9f59": "\\tanh \\frac{E}{2}", "a74c46875b0f719750702172ff2f8a3e": "\\left[ \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix} \\right].", "a74cdeb8e41e93955bac30bee24e18fa": "\\frac{dy}{dt}=?", "a74cf055806f78da758e18a1a34984c0": "\na_n = \\sum^{n-1}_{i=1} {1 \\over i}\n", "a74cfde39b81cb2bc66b38a84c043716": "p(y)=p(y|x)p(x)", "a74d12a2d4e569d9464c894672c9fe7f": "S \\approx {e^{-\\sigma^2}}", "a74d57c803194092ca5e0780138fab66": "|m|", "a74d6211c66511e4a31e7b866e5709e0": "J^1P\\to P", "a74d645fe0d45bdcc7c7eaad01dbec43": "\n\\begin{matrix}\nx = \\lambda_{1} x_{1} + \\lambda_{2} x_{2} + \\lambda_{3} x_{3} \\\\\ny = \\lambda_{1} y_{1} + \\lambda_{2} y_{2} + \\lambda_{3} y_{3} \\\\\n\\end{matrix}\n\\,", "a74d77a6993707ddbc34e4acb08a7b1a": "X = \\sum_{r=1}^{R} D^{(r)} = \\sum_{r=1}^{R} a^{(r)} \\otimes \\cdots \\otimes z^{(r)}", "a74db5101dbe8b415072cda92074295c": "\\Gamma (p,m)=\\{x\\in X \\mid p(x)\\leq m\\}.", "a74e93267aa0feb1b9fc4f727a098b4b": " 6~r~\\cos\\theta \\,", "a74eb2e6f62799acd329b4894b25e48c": "\\operatorname{rank}(A) = \\operatorname{rank}(\\overline{A}) = \\operatorname{rank}(A^T) = \\operatorname{rank}(A^*) = \\operatorname{rank}(A^*A).", "a74ebd084851ab27ec25c96e1a753f5a": "\\Delta V_m", "a74f14b731d1c33da58e4d81dc1d9126": "\\displaystyle \\|J_{n} - J_{n-1}\\|_{F}", "a74f9d0414b418cd3e2cb0ff3f39dbe2": "j \\,", "a74ff6213aa602a81707c681503b74d6": "\n EI~\\cfrac{\\partial^4 w}{\\partial x^4} + m~\\cfrac{\\partial^2 w}{\\partial t^2} = q(x,t)\n ", "a750a86b653a555061babc52eb3eb715": "\\pi = ne \\pm e \\,", "a750f336e3de62b4c19c6447beaa35b1": " \\left\\{ \\frac{1}{1+n} \\sum_{k=0}^n \\left( \\mu(k,A) - \\mu(k,B) \\right) \\right\\}_{n=0}^\\infty \\in j ", "a750ff08d06c6dd6d0274e2029ca0868": "\\begin{align}\n D[\\partial_i\\parallel\\cdot]\\ &:\\ p \\mapsto D((\\partial_i)_p \\parallel p), \\\\\n D[\\partial_i\\parallel\\partial_j]\\ &:\\ p \\mapsto D((\\partial_i)_p \\parallel (\\partial_j)_p),\\ \\ \\mathrm{etc.}\n \\end{align}", "a7512da6fdc73eccdc29fe17f2eeea1c": "\\mathbf {du}\\,\\!", "a75165b181281627a23cf8bf0ea7cbe4": "\\bot\\to A", "a751f53d4d11a9c4d25d5393fce492e8": "\\int_{0}^\\infty f(t) e^{-st}\\,dt \\; | \\; s \\; \\in \\mathbb{C}", "a75203b6044c744f0f355bddf50e1d23": "nn^{*}\\equiv 1(\\mod m)", "a75265f907e2f45f8a2535a12e3bb9f0": " \\hat{M}_I=\\varprojlim (M/I^n{M}). ", "a752812f463ccc2456392f40251c5ff7": "D_y^2", "a752f61a76ba93370db46250a33110da": "\\, (A, B, C, D)", "a752fb3faa6725f1592c0cdcf6240f95": "\\frac{F(x+1)}{F(x)} = f(x) \\, .", "a753581563271446774557c014fadfe7": "H_{so}=\\frac{1}{2} \\left(\\frac{Ze^2}{4\\pi \\epsilon_{0}}\\right)\\left(\\frac{g_s}{2m_{e}^{2}c^{2}}\\right)\\frac{\\vec L\\cdot\\vec S}{r^{3}}", "a753e61b9c31252a249fffb33429a1f9": "\\, e^{\\lambda(e^t-1)}", "a7541560d939d9f6a479d3d64ff941d7": "f_{n,k}(r)=A\\rho^\\gamma e^{-\\rho/2}\\left((\\gamma-k)\\rho L_{n-|k|-1}^{2\\gamma+1}(\\rho)+Z\\alpha\\frac{\\gamma\\mu c^2-kE}{\\hbar cC}L_{n-|k|}^{2\\gamma-1}(\\rho)\\right)", "a754160e2398c8d297bf893a9c733ae5": " E,F \\in \\mathcal{C} ", "a754168dc11dc8b57fff6b2d5b4d6084": "w\\notin\\{u,v\\}", "a7542463cfbee03625b2659e0cb67dbb": "2\\pi\\sum_{k=-\\infty}^{\\infty} \\delta(x+2\\pi k)", "a75501a65fb9e89a08695bdaa5f93e32": "H=\\omega_c a^\\dagger a + \\omega_a \\sigma^\\dagger\\sigma+ig(a^\\dagger\\sigma-a\\sigma^\\dagger)+iJ(a^\\dagger e^{-i\\omega_l t}-a e^{i\\omega_l t})", "a7554c6c6f372142b577b7df99de0140": "f_p", "a755f88ffb1969424ed2f180d8de165d": "\\ln(X) \\sim \\mathcal{N}(\\mu, \\sigma^2)", "a7562f1df40aec713d17318a2efbfe5b": "\\mathcal{P}(n-g_i) \\ni \\lambda : n-t_i = \\lambda_1 + \\lambda_2 + \\dotsb + \\lambda_\\ell", "a756377962750f5b978fe708a9060126": "\nd\\mu(t) = a \\mu(t) dt + \\Sigma(t) c^\\top \\eta^{-\\top}\\eta^{-1} \\left(dz - c\\mu(t) dt\\right).\n", "a7566849d8c2af3d692ad215aefea4e0": " \\log_{ 10 } \\left( 1 + \\frac{ 1 }{ 12 } \\right ) + \\log_{ 10 } \\left( 1 + \\frac{ 1 }{ 22 } \\right )+ \\cdots + \\log_{ 10 } \\left( 1 + \\frac{ 1 }{ 92 } \\right) \\approx 0.109 ", "a756bd42f1c1e05df45e56d6221c6edc": "{\\rm non}({\\mathcal L})", "a756bd9670bf0aff43f3644058b32918": "xf(x)=1", "a756c8efc907b85c5cb444e36f85d9f6": " I(f)(x)I(g)(x) = I(fI(g)(t))(x) + I(I(f)(t)g)(x) \\; , ", "a756e9f3b8a408f90c264a3e69aeb94e": "\\mathbf{\\operatorname{\\textbf{CAT}}(k)}", "a756f04ef4da0540972ca855fa569415": "\\mathbb{T}^2", "a75707024f355a0ae7b07970a62388a9": "N_f", "a7572facd738c624ebdb520e5c35ff43": "\\mathcal{O}_X^n|_U \\to \\mathcal{F}|_U", "a7573ee7abe774b2c37a52dab5991527": "\n\\begin{matrix}\nI(X;Y;Z) & = & I(X;Y|Z)-I(X;Y) \\\\ \n\\ & = & I(X;Z|Y)-I(X;Z) \\\\ \n\\ & = & I(Y;Z|X)-I(Y;Z) \n\\end{matrix}\n", "a75766ab082a4afa69fe31f1bfe369bf": "2|E|\\times 2|E|", "a7578d3acc251cf028c07fda83f26354": "_{s.10 \\,}\\!", "a757d13868e041fdfe1b580f232eccc5": "X = \\operatorname{spec} A, A = k[x_1, \\dots, x_n]/(f_1, \\dots, f_r)", "a757d497ff4e92b11638312b9887f2a0": "[Q,b^\\dagger\\}=\\frac{dx}{dt}-i\\Re\\{W\\}", "a75892f1dc54386c481fe7ddab82deaf": "\\overline{\\varphi_1 + i\\varphi_2} = \\varphi_1 - i\\varphi_2", "a758caf990e4617db1a72d47b07dd39e": "S(T)", "a758ed0fc34db57bba230918a257e17c": " \\psi (x) ~ = ~ A e^{-d \\left|x + \\frac{R}{2}\\right|} + B e^{-d \\left|x - \\frac{R}{2} \\right|}", "a75903b78ddd3fbc1b42289fe4af5b27": " p_i(r) \\rightarrow fitting \\rightarrow p_f(r) ", "a759110b97440fb03e4f102396aa61d7": "\\theta \\log{\\tan \\theta} - \\frac{1}{2}\\int_0^{2\\theta}\\log\\left(2\\sin \\frac{x}{2}\\right)\\,dx+ \\frac{1}{2}\\int_0^{2\\theta}\\log\\left(2\\cos \\frac{x}{2}\\right)\\,dx=", "a759247082db5e981f07492c7ca7b704": "Q_{\\bold{z}}(\\theta) = \\begin{bmatrix}\\cos \\theta & \\sin \\theta & 0 \\\\ -\\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 1\\end{bmatrix} , ", "a75a11c113a3e173e7ebbca4f749457b": "{\\hat{q}_{{\\rm c}}}({r_{\\rm c}})", "a75a5fc46d0243159e89e6b560af3c91": "x_1, \\cdots, x_n", "a75a69c649ef8a368538abea4457e81f": "\\psi(\\rho,z)", "a75a77bf3854d17e655fbd2773544f4b": "{i-k}", "a75a77eecb521857ecb0e649c5f3fdcb": "N_{in}", "a75a933bd588d2e913db95e2efdceca4": "n^2 U ", "a75ab6625df931cecf5c0b2a716223a2": "\n\\Pr(n_1,n_2,\\ldots,n_S| \\theta, m, J)=\n\\frac{J!}{\\prod_{i=1}^{S}n_{i}\n\\prod_{j=1}^{J}\\Phi_{j}!}\\frac{\\theta ^{S}}{(I)_{J}}\n\\sum_{A=S}^{J}K(\\overrightarrow{D},A)\\frac{I^{A}}{(\\theta) _{A}}\n", "a75ad86befe3c5d9d5e1305a5cbdb087": "4 \\times x", "a75b5715d6fc01cf9139b26652cbe0d9": "\\ 1 ", "a75b840dbdf5861b2d10c8eaff23c2e7": "\\hat{\\Phi}", "a75b9ed8e2a4e101c353101150a0d373": "TL_i", "a75c10ecb16fa67cc79abe9faa008c2e": "\\varphi_{AB} = \\frac{q}{\\hbar} \\int_P \\mathbf{A} \\cdot d\\mathbf{x}", "a75c44e3d42170ac089f282167772f4a": " \\int_{\\gamma} g(\\zeta) d \\zeta = f(b) - f(a).", "a75c5ec641780f57bea9b00e16104e1c": "\\sigma_\\theta = \\rho_\\theta - 1000 ", "a75d3ba7e9d796728ab00b7d082a0f7f": " F(X) = \\alpha \\left( 1 - \\left(\\frac{X}{K}\\right)^{\\nu}\\right) \\Rightarrow F(0)=\\alpha < +\\infty ", "a75d86b461f6231c3e313df31b8ad6af": "a_0 < b_1 \\leqslant b_0", "a75dcd77df3eb7e2523f69f115a63723": "\n\\left.\\begin{align}\n X \\perp\\!\\!\\!\\perp A \\mid B \\\\\n X \\perp\\!\\!\\!\\perp B \\mid A\n\\end{align}\\right\\}\\text{ and }\n\\quad \\Rightarrow \\quad\nX \\perp\\!\\!\\!\\perp A, B\n", "a75de08df138638cfc34bc7961587681": "\\left( p \\to q \\right) \\to \\left( p \\to q \\right) ", "a75e0018298d98cb6875c3b4d6bd1a0a": "X_f=X\\cos(\\psi)-Y\\sin(\\psi)", "a75e24b56faf54f4106e1819a659e1a5": "f_*: H_*(X, L) \\to H_*(Y, f^*L)", "a75e4495c5cd8971c69a59ea305ee2f9": "F_l(\\mathbf{a}_{0,k})=\n\\begin{cases}\na_0, &k=1\\\\\nf(F_l(\\mathbf{a}_{0,k-1}), a_k), &k>1\n\\end{cases}.", "a75e68e125cc96fb7c9490c9cbbc0574": "hkl", "a75ead5782550e8ee963baff1703c189": "1 \\,-\\, \\frac{1}{3} \\,+\\, \\frac{1}{5} \\,-\\, \\frac{1}{7} \\,+\\, \\frac{1}{9} \\,-\\, \\cdots \\;=\\; \\frac{\\pi}{4},\\!", "a75f9c31e8fd42ee247d2e1807fd38db": "\\omega_\\alpha\\pm \\omega_\\beta", "a75fd040fad0f50c1d3fddcf543148e0": "\\rho = m_i/V\\,\\!", "a7601611fdebb06ac2c729c9b90e59bb": " \n I[f] = \\int_{\\Omega} \\mathcal{L}(x_1, \\dots , x_n, f, f_{x_1}, \\dots , f_{x_n})\\, \\mathrm{d}\\mathbf{x}\\,\\! ~;~~\n f_{x_i} := \\cfrac{\\partial f}{\\partial x_i}\n", "a7601c9a847a7468eab96a615af91f7d": "\\tilde{U}_k^i=R_k^i/n", "a760244942718c451a0f32d9bb36ab8e": "100(1 - \\alpha)%\\text{CI}: \\rho \\in [\\operatorname{tanh}(\\operatorname{arctanh}(r) - z_{\\alpha/2}SE), \\operatorname{tanh}(\\operatorname{arctanh}(r) + z_{\\alpha/2}SE)]", "a760434abdbc8d5c0c12aff657c0050b": "x := \\sum_{i} a_i \\frac{N}{n_i} \\left[\\left(\\frac{N}{n_i}\\right)^{-1}\\right]_{n_i}", "a7604cae46cddcf9a49dfb76e2fe2b03": "k_*\\to h_*", "a76053a60e7daa8857bcc65e83970344": "n\\le2^{h+1}-1", "a760bcb876fb1021e213789ef08e857e": "v(t) = \\exp\\biggl(\\int_a^t \\beta(s)\\, \\mathrm{d} s\\biggr),\\qquad t\\in I.", "a7611462b713266961900e29da5a8676": "P_{FNL}", "a7612ecded46d6f020008368ab0f7080": "m 1 ", "a77b5720b67a335c353c23a282df67b0": "W_C = \\int_{C}\\bold{F}\\cdot \\bold{v}dt =\\int_{C} \\bold{F} \\cdot \\mathrm{d}\\bold{x},", "a77b7f1887c624f34c59f0d8ada5badd": "\\langle\\Phi_{i_{1}\\ldots i_{n}}^{a_{1}\\ldots a_{n}}\\vert(H-E_{0})e^{S}\\vert\\Phi\\rangle = 0", "a77badc0c64d15da6250c5962a99fef2": "\\scriptstyle\\hat f", "a77bb6469ff38f541f3542207ccb1389": "\\mathrm{P}\\left(\\bigcap_{i=1}^n A_i\\right)=\\prod_{i=1}^n \\mathrm{P}(A_i).", "a77be94f473a088af58acd65b9cc8b57": "T \\cup \\{\\gamma\\}", "a77bf95b260048399420127ece2ca994": " \n\\begin{align}\n&\\operatorname{tr} \\, A = \\operatorname{tr} \\, B, \\quad \n\\operatorname{tr} \\, A^2 = \\operatorname{tr} \\, B^2, \\quad\n\\operatorname{tr} \\, AA^* = \\operatorname{tr} \\, BB^*, \\quad\n\\operatorname{tr} \\, A^3 = \\operatorname{tr} \\, B^3, \\\\\n&\\operatorname{tr} \\, A^2 A^* = \\operatorname{tr} \\, B^2 B^*, \\quad\n\\operatorname{tr} \\, A^2 (A^*)^2 = \\operatorname{tr} \\, B^2 (B^*)^2, \\quad\\text{and}\\quad\n\\operatorname{tr} \\, A^2 (A^*)^2 A A^* = \\operatorname{tr} \\, B^2 (B^*)^2 B B^*.\n\\end{align}\n", "a77c391c46729114df7869940e0de9b8": "\\tfrac{7}{16}", "a77c746e086ef4006c4cc38110f5a0b5": "X^{\\ast }(t)\\neq \\varnothing ", "a77c84d7aab2a093f3dbcefc6580fbcb": "E[J] = i\\ln Z[J]", "a77c9533b8ebde53c9c9d1f91b7cab19": " 4N = a^2 + |D|b^2", "a77c9b9a1977ede7268684a77275ac97": "n \\mid q - 1", "a77ca8f7e90472dcc989ea16b64a2ef9": " g( y ) = y^{-2} \\frac{ 1 }{ b-a } ,", "a77cfde36bd0d322b4b8c0c0ef2491d5": "\\le \\leq, \\lneq, \\leqq, \\nleqq, \\lneqq, \\lvertneqq \\!", "a77cff658eab23d14ee25566f1187916": " \\langle S^{2}\\rangle_{\\mathrm{UHF}} = \\langle S^{2}\\rangle_{\\mathrm{exact}} + N_{\\beta} - \\sum_{i,j}^{\\mathrm{all}}|\\langle\\psi_{i}^{\\alpha}|\\psi_{j}^{\\beta}\\rangle|^{2}.", "a77d0cecfb9c509c3042ee0c6f8247e6": "\\ A", "a77d20166ebc595ad43df998e5688f84": "\n\\dot{\\mathbf{Q}} =~~\\frac{\\partial K}{\\partial \\mathbf{P}}\n", "a77d466dd0af5d2510ed26740a3373be": "16_6", "a77d768be0bad5b0af1fe8bb45bd8fd6": "E = 0.0165\\, ", "a77de61a59f814effcd0035282d5b6bd": "\\displaystyle A(s,\\lambda)\\pi_\\lambda(g) =\\pi_{s\\lambda}(g) A(s,\\lambda).", "a77e1531d3f0ae64f2416008c2654875": "\\widetilde{S}_{\\lambda\\gamma }(w_g) = \\begin{cases}\nw_g-\\lambda\\gamma \\frac{w_g}{\\|w_g\\|_2}, & \\|w_g\\|_2>\\lambda\\gamma \\\\\n0, & \\|w_g\\|_2\\leq \\lambda\\gamma\n\\end{cases}", "a77e2fd1941175c5841a5d737a12b150": "G_{LCU} = G_{12} + G_8 \\cdot P_{12} + G_4 \\cdot P_8 \\cdot P_{12} + G_0 \\cdot P_4 \\cdot P_8 \\cdot P_{12} + C_0 \\cdot P_0 \\cdot P_4 \\cdot P_8 \\cdot P_{12} = C_{16}", "a77ea7ee590019f545f0b83a0733e275": "\\mu = 1+\\chi_\\text{m}", "a77f4a9a592e0bd99371554fa78ce48d": "a=N_P/N_S=v_P/v_S=i_S/i_P=\\sqrt{L_P/L_S}", "a77f4acf21e20554b3a5ec8bd7ca0951": " (X_\\alpha)_{\\alpha} ", "a77f9cb540e44038946cfb069cfa995b": "a \\cdot DF = \\mathcal{P}_B (a \\cdot \\partial F) = \\mathcal{P}_B (a \\cdot \\mathcal{P}_B (\\nabla F))", "a77fd08568e6010f4cdd16b80ced1ed7": "\\zeta(3)=\\frac{7}{180}\\pi^3 -2 \n\\sum_{k=1}^\\infty \\frac{1}{k^3 (e^{2\\pi k} -1)}", "a7800228d4549c0a3cdf24981babfbd9": "d\\alpha(t)=\\alpha^{\\prime}(t)\\,dt", "a7801a6e0160f24d2ad62a18e3a5dd54": "\\begin{align}\n\\gamma^{0} &= -i\\biggl(\\begin{matrix}\n0 & I\\\\ \nI & 0\\\\\n\\end{matrix}\\biggr),\\\\\n\\gamma^{i} &= -i\n\\biggl(\\begin{matrix}\n0 & \\sigma_i\\\\ \n-\\sigma_i & 0\\\\\n\\end{matrix}\\biggr), \\quad i = 1,2,3\\\\\n\\end{align}.", "a78104737f87519eeb22f14de8f5e779": "y_0 = B \\sqrt{{ \\left(\\frac{x_0}{A}\\right) } ^2 - 1} \\quad (12)", "a7811aabf6614e4dd32cc214f167ba80": "C_V(V,T)\\ ", "a7816b7ec73dd67d57c716dce66f5943": "\\frac{n_e^2}{n-n_e} = \\frac{2}{\\Lambda^3}\\frac{g_1}{g_0}\\exp\\left[\\frac{-\\epsilon}{k_BT}\\right]", "a7819a56ca412334e101b4fe6f899cdd": "H_I = \\sum_{k,q} V_q (a_q + a_{-q}^\\dagger) c_{k+q}^\\dagger c_k,", "a781a29400c98676f714f11a8ff5f63d": "QC_x \\subseteq C_x", "a781dfbab3614a09e3474836c6a50615": "F = \\frac12 \\times 1.2 \\times 10 \\times 1.5 \\times 8.3^2 = 620 \\ newton", "a781eec3b3b50de5ca3ad08520c108f4": " C\\ell_{p+1,q+1}(\\mathbf{R}) = M_2(\\mathbf{R})\\otimes C\\ell_{p,q}(\\mathbf{R}) ", "a781fa90aa7c59de6742756b55c7affd": "\\vec\\beta", "a7826299a80ef1639d5a65a640d6cf9e": "H_n(P) = 0", "a78288ad5830d102203c954e5f985c04": "{}_e\\lceil X \\rceil_N", "a782933768ab94023d56242e3c0c0e91": "f_r=\\sqrt{\\frac{1-v/c}{1+v/c}}f_e.", "a782b457667bbddd3b6e7cc8847ad571": " \\begin{align} \nf(x; d_1,d_2) &= \\frac{\\sqrt{\\frac{(d_1\\,x)^{d_1}\\,\\,d_2^{d_2}} {(d_1\\,x+d_2)^{d_1+d_2}}}} {x\\,\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)} \\\\\n&=\\frac{1}{\\mathrm{B}\\!\\left(\\frac{d_1}{2},\\frac{d_2}{2}\\right)} \\left(\\frac{d_1}{d_2}\\right)^{\\frac{d_1}{2}} x^{\\frac{d_1}{2} - 1} \\left(1+\\frac{d_1}{d_2}\\,x\\right)^{-\\frac{d_1+d_2}{2}}\n\\end{align}", "a782c694beacc69fa202a35240995981": "\\mathit{\\Omega}_{\\mathrm{P}} R_{\\mathrm{groove}}", "a7831ab4c67f65d8cb12a22a9324ee01": "\\kappa = 2D_1", "a7832cd0f01f1ac545fe59aaf590c9ce": "F_{ij} = F_{ji}", "a7835d2fdd8c34aaf6fd25df664c28d4": "\\nu \\in [0, 2\\pi)", "a7835d31b20fd61a1a55b0a63273610c": "[0,a] ", "a7836dbf8b51ade0defb918d684cfde3": "\n(\\leftarrow\\star) \\quad {Z\\leftarrow \\Delta X Y \\Delta'\n \\over\nZ\\leftarrow \\Delta (X \\star Y) \\Delta'}\n", "a783bdf753ad27b06e9bf15852cedda4": "\\| \\cdot \\|_{a}", "a783d8f4b2317474d10c9738d00572e8": "\\neg \\Box F", "a783f4bbc7ad635a8e65c2b1e1c3f922": "C_{0}^{\\infty}(\\mathbf{R}^{n})", "a7841a608347afbd6f65734207bdde6e": " J\\subseteq I", "a784243d8211e519a1071acd55f1f3b0": "\\frac{3}{7}", "a784260be2d5c45e1ba69142690f47e9": "\\frac{v_1}{2} \\cdot v_1 = \\frac{v_1^2}{2}", "a784883cb3a22e6fb5b50febaf4e9c95": "((x,y),z)", "a7848fee9939bc0ff7c9d64e0be884c2": "n_a \\approx 1.003", "a784f34d67081685dacae645bf59f272": "H_{3} (x|q) =8x^3 - 4x(1-q^n)", "a7852e8928d7a3e16a7c58f618c0f0ce": "\\begin{align} x_{n+1}^2 = r^2 - y_n^2 - 2y_n - 1 \\end{align}", "a785aa5725838a8042b20c0e09e0416d": " 2r\\sqrt{5-2\\sqrt{5}} = \\frac{R}{2}\\sqrt{10-2\\sqrt{5}} \\!\\, ", "a785b4b7fa7269de9c05a3c7a718c3f3": "\\lambda_A\\,", "a785b6a0d0529a0e51f38efc794bbb50": "U = \\left( \\begin{array}{cccc} 1 & \\cdots & 0 & c_1 \\\\ \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & \\cdots & 1 & c_{n-1} \\\\ 0 & \\cdots & 0 & c_n \\end{array} \\right)", "a7861baf3f73858e9dc786c826783a42": "p_1 D_L[F_1(K,L)]=p_2 D_L[F_2(K,L)]\\,", "a7865ced1d58fe62a011813ddcf6fcdf": "\\sigma_{90}", "a78665ff4c2bc3992f282cfd76f6ddff": " P( X < k ) \\ge \\frac{ E( X )^2 - 2 k E( X ) + k^2 }{ E( X^2 ) - 2 k E( X ) + k^2 } \\text{ if } E( X ) \\le k \\text{ and } E( X^2 )< kE( X )", "a78678b9b11e3cb4c9a2ed204ccfc051": "Sc = \\nu / D_{AB}", "a786882146647bbca6826b9bb9934823": "\\mathrm{18 Mg_2SiO_4 + 6 Fe_2SiO_4 + 26 H_2O + CO_2}", "a7870aec0fcb7be22292db829c4b1d94": "\\gamma(X^*, X)", "a78755824f20df4f1e78c04c511b49c4": "\\textrm{Efficiency} = \\frac {w_{cy}}{q_H} = \\frac{q_H-q_C}{q_H} = 1 - \\frac{q_C}{q_H} \\qquad (1)", "a787dbc2f9902096a8fb13903dd63428": "V,", "a787eacc4bc60d5a544c13e5a21b819c": "\\rho = \\rho(P,T,S_1,S_2,...) ", "a78800751d31534c9db795e75459f775": "m_{rel}\\,", "a7881e55cf55a26998844f140c9e94bf": " E(e)", "a788b8ab36260a924bf681b141ae90a8": " u \\wedge v = u \\wedge (a u + x) = u \\wedge x", "a78983ff4cdaec3823d5f5584e63ef97": "\\frac{T-T_c}{T_c}", "a7898d805e21fa2f2f0992febbbcd025": " \\scriptstyle \\zeta < 1 \\,", "a78994e183b43c0079a58c530feb5c28": "A(z) = B(z) \\cdot C(z).", "a789af69cb2d9d1fbd55fca3b189027b": "\nx = 1 \\cdot x \\equiv x^q x^{q - 1} = x^{3n + 2} x^{3n+1} = x^{6n + 3} = (x^{2n+1})^3 \\pmod{ q}\n", "a789bbe91a224345d35dec842b2da34c": "\\begin{align}\n\\|Tf\\|_p^p &= p\\int_0^\\infty a^{p-1} m\\{x:\\, |Tf(x)|> a\\} \\, da \\\\\n&\\le p \\int_0^\\infty a^{p-1} \\left ( 4a^{-2}\\|T\\|^2 \\|f_a\\|_2^2 +C a^{-1}\\|f^a\\|_1 \\right ) da \\\\\n&=4\\|T\\|^2 \\iint_{|f(x)|(1-\\varepsilon)\\|f\\|_\\infty\\bigr\\}.", "a79a37678bab8fd71ea63bf5d921533c": "\\pi_x ((x,y))=y,", "a79ab03915a920e929fdaa561e1370c7": "(\\mathcal{A},\\mid,0)", "a79ab78383f04c72069d686249e8799e": " \\displaystyle{ds^2=E\\,dx^2 + 2F\\, dxdy + G\\, dy^2}", "a79acbaf50b3f9f15d9005dc0848f402": "\\forall E \\in \\Sigma : \\mu\\!\\left(E\\right) \\geq 0", "a79ad0615a430548eaaf08ba9efcf706": " \\mathbf{v}_0 = v_{0x}\\mathbf{i} + v_{0y}\\mathbf{j}", "a79aeb868336ac4e263758a3871c0914": "j^{ij}", "a79b0b150432a2669a3d07ecec791bba": "-1/n", "a79b0ceb358833b5e31a7ffdacd9592f": "\\psi=x_0+i\\gamma", "a79b1121f7ecc322feca8e8d1c5f7361": "\\mathbf{y=u_2G_2+c_{s2}}", "a79b2572a2b2d7e7d6e59a23c0c887d5": "\nT_i \\uparrow - \\tau_i (1-R_i) T_{i+1} \\uparrow - \\tau_i R_i T_i \\downarrow \n = (1 - \\tau_i) T_i \n", "a79b47f5839b40ab93fc9e226e5be6d4": "\\frac{E_1}{E_2} = \\frac{\\frac{1}{2} k_1 x^2}{\\frac{1}{2}k_2 x^2} \\,", "a79b81acd84530f19d00e65f0facbfac": "t(G)=p^{q-1}q^{p-1}", "a79b85418d89769f507f293fd3893c00": "U_{bias}^{LE}(\\mathbf{Q};t=0) = 0", "a79b8b36ee6a4f27f1ce693690d86a3a": "U(|\\Phi|)=m^2|\\Phi|^2-\\lambda|\\Phi|^4/2+\\gamma|\\Phi|^6", "a79b93476a03b9065e60463f8129b239": "\\frac{1}{2}\\left(\n -2v(\\ln(S),\\frac{1}{2}(T-t))\n -\\frac{\\partial v(\\ln(S),\\frac{1}{2}(T-t))}{\\partial\\tau}\n +\\frac{\\partial v(\\ln(S),\\frac{1}{2}(T-t))}{\\partial x}\n +\\frac{\\partial^2 v(\\ln(S),\\frac{1}{2}(T-t))}{\\partial x^2}\\right)=0.\n", "a79b97a1750aea00b96cc43c640a8ad5": "\n{{\\sigma _{\\hat g}^2 \\,} \\over {\\hat g^2 }}\\,\\,\\, \\approx \\,\\,\\,{1 \\over {\\hat g^2 }}\\,\\left( {{{\\partial \\hat g} \\over {\\partial L}}} \\right)^2 \\sigma _L^2 \\,\\,\\, + \\,\\,\\,\\,{1 \\over {\\hat g^2 }}\\,\\left( {{{\\partial \\hat g} \\over {\\partial T}}} \\right)^2 \\sigma _T^2 \\,\\,\\, + \\,\\,\\,\\,{1 \\over {\\hat g^2 }}\\,\\left( {{{\\partial \\hat g} \\over {\\partial \\theta }}} \\right)^2 \\sigma _\\theta ^2\n", "a79bfa1a02c8674e22f3c3ce917aab0b": "\\begin{align}\n x_i &= \\frac{z_i}{1+\\beta(K_i-1)}\\\\\n y_i &= K_i\\,x_i.\n \\end{align}", "a79bfe29584b084247ab4fb8996a60d3": "\\frac{(-2)^n\\,n!\\,\\Gamma(2\\alpha)\\,\\Gamma(n\\!+\\!1/2\\!+\\!\\alpha)}\n{\\Gamma(n\\!+\\!2\\alpha)\\Gamma(\\alpha\\!+\\!1/2)}", "a79c127b8325d1760ed13b146b1f7392": " \\Delta R \\approx \\Delta t [{\\beta I^2 \\over N}+{\\beta IR \\over N}] ", "a79c213b4207995cdf782eb329089b01": "\\textstyle H_2\\left(g_{ID}^r\\right)", "a79c53c860e3476c705198621857b9c2": "\\mathbf{J} = \\left( \\frac{n q^2 \\tau}{m} \\right) \\mathbf{E}.", "a79c6d818b89af3c2fced0afc40933b3": "\\scriptstyle\\R^+_0 \\;\\equiv\\; \\left[0,\\, +\\infty\\right)", "a79d653486b6e1a9b98091ca46350549": "A\\begin{pmatrix}r\\\\f\\end{pmatrix}=\\begin{pmatrix}r+f(-\\tau)\\\\f'\\end{pmatrix},~~~D(A)=\\left\\{\\begin{pmatrix}r\\\\f\\end{pmatrix}\\in X: f\\text{ absolutely continuous }, f'\\in L^2([-\\tau,0])\\text{ and }r=f(0)\\right\\}.", "a79da3a123e9dac56e72dc4d98f717ad": "V(u,\\Omega)", "a79dac1c99415605fa17cda8af69c233": "\\begin{matrix} {r \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "a79ddf81671ccd6cb8dbea9b44da67bb": "\\varphi(x)={1 \\over \\sqrt{2\\pi}}e^{-x^2/2}", "a79e7083d07f8af76bb98e542db8cff8": "\\log X!", "a79ea2e610c979e73107afd2e82e7ea4": "{\\bold x}\\cdot{\\bold x} = x_1^2+x_2^2+x_3^2=0.", "a79eb6c60f0549915955da45f3a564c2": "\\Pi\\in\\Gamma(\\wedge^2 TM)", "a79ec6da14f2a503b9eebe15c8b8544c": "O(\\log^7 q)", "a79f0df56c96aa290eace5eda574ebb3": "N(d_-)", "a79fac38dfdb5bac4a6c1ef3d73840d0": " F( x ; a,b) = \\frac{ \\log_e( x ) - \\log_e( a ) }{ \\log_e( b ) - \\log_e( a ) } \\quad \\text{ for } a \\le x \\le b.", "a7a016f67eb3da932f3d04c2570778da": "f(\\theta) = (1-\\cos\\theta) + I(1+\\cos\\theta)\\;", "a7a0bc968fd46960afc83e3623e4ebda": "c=\\frac{W(-\\ln(z))}{-\\ln(z)}", "a7a0c5e25d06ce2fd07ba288e2f2dc04": "\na\\ \\texttt{<=>}\\ b\\ \\ \\ = \n\\begin{cases}\n -1 & \\mbox{if }a < b, \\\\\n 0 & \\mbox{if }a = b, \\\\\n 1 & \\mbox{if }a > b, \\\\\n\\texttt{undef}\n & \\mbox{otherwise.}\n\\end{cases}\n", "a7a1072f6fd63598c9972e6db057ce00": "cdt", "a7a12c301e6e361712d9d315871eebc5": "\nx_n=\\theta+w_n,\\quad n=1,\\dots,N\n", "a7a136cede60ba896a292f7b93139f4f": "p_i \\equiv 1 \\pmod{2^i}", "a7a146e7cdd3ec5c510ecc2eababcb53": "H_p(t) = a(t) \\int_{0}^{t} \\frac{cdt'}{a(t')}", "a7a16f5dd4fa56b2c9e225defc34bc07": "K \\subset \\mathfrak{m} M", "a7a1e150a818f120e44b5ca2d5ff07a6": "E_\\pm(\\alpha,z) =\\exp -\\sum_{\\pm n>0} {\\alpha_n z^{-n} \\over n}.", "a7a286b78fc895212df88f131eff013a": "\nf_n = {1\\over {2 \\pi}} \\sqrt{k \\over m}. \\!\n", "a7a30f23a37857fcae21377765ce969d": "{1\\over (n+1,q+1)}q^{n(n+1)/2}\\prod_{i=1}^n(q^{i+1}-(-1)^{i+1})", "a7a341e59ef59e7552da050e92b7be73": "\nX_{i}^p=\\{x_{i1}^p,x_{i2}^p,...,x_{in}^p\\}\n", "a7a37fca5b714d75488c23d92bbf4867": "p_1, p_2, p_3, \\ldots", "a7a385ae6fa6bb24a01524d71b5b0c89": "K^*(BG)\\cong R(G)\\hat{_I}", "a7a3b7ce6e6baf246fffb6a001a9341f": "\\rho_g(X) := \\mathbb{E}^g[-X]", "a7a3c9439515f3067f10578b4b610cec": "h = A \\to bA", "a7a4231e4f8f23513bf2f43ab4fe06ac": "(2(1-Cij))^{0.5}", "a7a44770e3410ecd41cb22c210b87f8b": "\\frac{dU}{dt}=-U-u+2\\,,\\quad \\frac{du}{dt}=100(U^3-u).", "a7a44bd903681a7110148351449b249f": "a = {1/2}", "a7a46f4ff1ce9ba56d3ecb991e416f15": "a_n, b_n > 0, \\ \\ \\sum_{n=1}^\\infty \\min(a_n, b_n) < \\infty.", "a7a4fb1e0125a88bc2035425a480a88e": "\\mathbf{\\hat{\\imath}} = \\begin{bmatrix}1\\\\0\\\\0\\end{bmatrix}, \\,\\, \\mathbf{\\hat{\\jmath}} = \\begin{bmatrix}0\\\\1\\\\0\\end{bmatrix}, \\,\\, \\mathbf{\\hat{k}} = \\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix}", "a7a51c7d03d39d73ad99f7c730c2bc79": " \\rho_{\\mathit{sys}} = \\mathit{Tr}_{\\mathit{env}}( \\sum_{i,j} \\psi_i \\psi_j^* | i \\rang \\lang j | \\otimes | \\epsilon_i \\rang \\lang \\epsilon_j | ) = \\sum_{i,j} \\psi_i \\psi_j^* | i \\rang \\lang j | \\lang \\epsilon_i | \\epsilon_j \\rang = \\sum_{i,j} \\psi_i \\psi_j^* | i \\rang \\lang j | \\delta_{ij} = \\sum_{i} |\\psi_i |^2 | i \\rang \\lang i | ", "a7a5235bcd8929dc65faea84c8cc9c8f": "f(g(a) + k_h) - f(g(a)) = f'(g(a)) k_h + \\eta(k_h) k_h.\\,", "a7a54676076aead2496d8e2d78adbded": "\\operatorname{E}[|X - E[X]| ] (\\Beta(\\alpha, \\beta))=\\operatorname{E}[| X - E[X]|] (\\Beta(\\beta, \\alpha))", "a7a55fa1e4c8fdc8e6956fc158fcab5b": "[\\vec{X}]=([X_1], [X_2], ... ,[X_N])", "a7a56b1fafdb74687fa394627b28d7fa": "T_\\sigma \\quad ", "a7a57c805a704f675ae3d5daac812a9c": "(G, \\star)", "a7a5fa69e0881ac3a12345f0f7365996": "P(x_1,\\cdots x_n)={^h\\!P}(1,x_1,\\cdots x_n).", "a7a60786408d2c6685546abb3f95c366": "R_+ = \\bigoplus_{d \\ge 1} R_d", "a7a62f2de46c333f99d309f874491173": " K_a = \\cos\\beta \\frac{\\cos \\beta - \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}{\\cos \\beta + \\left(\\cos ^2 \\beta - \\cos ^2 \\phi \\right)^{1/2}}", "a7a67f2a492b32d001772f3046df0642": "vL[u]-uL^*[v]=\\nabla \\cdot \\boldsymbol M, \\ ", "a7a6ee377e45edc6b86a543783417448": "dp = 0", "a7a724e6cf6f29a1373d0decc29aaf7e": "\\Gamma(a^{-1})", "a7a7627fb0603c0ead9b808641497473": "\nI =\n\\begin{bmatrix}\n \\frac{1}{3} m l^2 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & \\frac{1}{3} m l^2 \n\\end{bmatrix}\n", "a7a80a841ea92cf36e0d8f1624e7432e": " K \\tau e^{-r \\tau}\\Phi(d_2)\\, ", "a7a8a0d477f76d12ece7b0fa045dc609": "\n\\left[ N\\left( u+v\\right) \\right] =\\left[ N\\left( u\\right) \\right]\n\\left[ N\\left( v\\right) \\right] .\n", "a7a907713840a1b3f5987ada4511c019": " \\scriptstyle\\overline{X}_n", "a7a953d0157c8d259df8edf68e03865f": " W_{c} ", "a7a98a0f166ab2e286ba4283761218bd": "\\mathfrak{P}^{25}", "a7a9a31af999ab9c70f3439406f8de11": "I_z", "a7a9cc4f4e299901c886b63492e5c064": "U(t)V(s) = e^{-i st} V(s) U(t) \\qquad \\forall s, t \\qquad ,", "a7aa213e1bed188e6c0056394808e57c": "\\partial_\\nu T^{\\mu \\nu} + \\eta^{\\mu \\rho} \\, f_\\rho = 0 \\,", "a7aa31ec608cb8f1ca2a2bd319c0e28a": "x^{n+1 \\over 2} \\notin \\mathbb{Z} \\big / n \\mathbb{Z}", "a7aa77e48951163559fe570c5541aa6c": " x(t) = a(t) \\cos(\\omega t + \\theta) \\ ", "a7aa8e3f5a31e07da0c67773fddc7db3": " R_{\\text{int}} = ({\\frac{ V_{\\text{NL}} } { V_{\\text{FL}} } - 1 } ) { R_{\\text{L}} } ", "a7aadc1146d050d9689e827d993f1729": " dQ ", "a7ab12a3f608d4cb4636f4f7f41b98a8": "{{n^2-n} \\over 2} + n - 1 = {1 \\over 2}n^2 + {1 \\over 2}n - 1.", "a7ab344433eaa75bc0a61a2b00c5aa4c": "\\scriptstyle\\,\\Gamma", "a7ab727d1a36a9f639af508b65ee0a5a": " r_{i} = \\frac{u_i - u_{i-1}}{u_{i+1} - u_i}.", "a7abb64a678b8ef743e30617545c12ee": "\\bigg/ \\bigg/ \\frac{RP, GA \\qquad \\mathrm{k}}{SSE, KA \\qquad \\mathrm{k}~vs. \\mathrm{x}} \\bigg/ \\bigg/", "a7abca26fd3160b1884f2fb400473616": "\\scriptstyle\\overline{BE}", "a7abd395c995152d345c23f6b6971eaf": "f(x,\\theta)=\\frac{\\partial F(x,\\theta)}{\\partial x}", "a7ac0702e7fc2b16a7ec1616b74ee414": "0.9 \\times tol", "a7ac2d319387b90d30e79ba22a2e6d09": "\\int_{\\partial V} \\nabla u \\cdot \\mathbf{n}\\, dS = 0,", "a7acec21602e5b77c81d185e4118ad3d": "e=\\rho c^2+\\rho e^C", "a7ad0b54a085c6b855179a5334b05443": "E(\\tilde{m}\\tilde{x}_i) = p_i, \\quad \\forall i.", "a7ad0f80182bff4f99e0f2220d05cc53": "Z_{m,m}", "a7ad3c63c7a775a4201fa74afb32481e": "\\textstyle{1\\over r}", "a7adce6580e04cda0020c0590e06122e": "\\begin{array}{rcl}\nl_0 & = & - {x_0}/{r'} \\\\\n m_0 & = & -{y_0}/{r'}\\\\\nl & = & {x}/{s'}\\\\\nm & = &{y}/{s'}\n\\end{array}", "a7add03e2e8159bf249d4257bd29619c": "\\frac18\\int_0^1 x^8(1-x)^8\\,dx=\\frac1{1\\,750\\,320},", "a7ae077f320c83c59c2d3c0077bf9cc7": "f = -g", "a7af261a8444fc2a40db35cfc9ff6a92": "T_{load}", "a7af3187bb367e4c8217ea38fda14133": "M_{ii}=\\sigma^2_{y,i}+\\beta^2 \\sigma^2_{x,i}", "a7af9051a91cba5aef88ba87388cf6a0": "\\Gamma^s_{ij}", "a7af94580aa71603afda2771bd9fd272": " \\int\\!\\!\\!\\!\\int_{A\\,\\subset\\mathbb R^2} \\left (\\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y}\\right)\\, d\\mathbf{A}=\\oint_{\\partial A} \\left ( L\\, dx + M\\, dy \\right ) ", "a7afa8a813c3c56be30e171a944dbd76": "V=\\dfrac{1}{2}kx^2", "a7afb2be332843deeef2817290c79a22": "y = \\sqrt{\\gamma}r \\quad \\hbox{with}\\quad \\gamma \\equiv \\frac{m_0\\omega}{\\hbar},", "a7afba864c89871cbe1a84e6bbdcc75e": " \\text{var}(Y) =\\text{var}(X)(c-a)^2 =\\frac{\\alpha\\beta (c-a)^2}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}.", "a7b02a2086309568263e940f86c0e588": "\n\\left( \\frac{\\partial ^{2}}{ \\partial\n u^{2}} + a^{2}\\frac {\\partial^{2}}{\\partial v^{2}} \\right)^{2}\nX(u,v) = 0.\n", "a7b042cc35010b11bb43b7b3f9e0daee": "A x= b.\\,", "a7b0579a7a7227c57f805afb03364f03": " v \\sigma_i \\sigma_j", "a7b0e8ee5e66f45ce02471206a59aec0": "y[n] = \\sum_{k=0}^{K-1} x[nM-k]\\cdot h[k],", "a7b0f59118cdde60a1f457950cd5a1ea": "= -\\log \\lambda \\int_0^\\infty f(x)\\,dx + \\lambda E[X]", "a7b1073fe28050b6e75b35cd4ebceca7": "V_\\text{E} = \\left( \\frac{R_\\text{f}}{R_1} + 1 \\right) \\left( V_{IOS} + I_{BI} \\left( R_\\text{f} \\parallel R_1 \\right) \\right)", "a7b18ec0e8af68a361789dbcf3ea49e4": " G^* P^*=P^*", "a7b21e966dcb0e2008024bf5f8855658": "8192 x + 4096", "a7b27602aee5a6e50915f9bda44d33ff": "w_i^{\\mathtt{KED}}", "a7b276502ed98ce7dbc142f7aa77c029": "f(0;x,y)=1", "a7b2ddc03b9480f923c02fdcab834435": "\\text{gcd}_{R[X]}(\\text{primpart}(p_1),\\text{primpart}(p_2))=\\text{primpart}(\\text{gcd}_{F[X]}(p_1,p_2)).", "a7b2e5775dfe22c2540eb1d13e92df22": "(r<1)", "a7b3263e0a4ad3a9dc62316a529641ae": "T_{D,max}", "a7b36a7f6fcb210d5c91297ca4e5aab1": "\\gamma_{xy}= \\alpha + \\beta = \\frac{\\partial u_y}{\\partial x} + \\frac{\\partial u_x}{\\partial y}\\,\\!", "a7b42a9a8ab443a95fccb989cf520d16": "i = 2 \\times (\\text{parent}) + 2", "a7b45835ccd46c5372b50e630a72a1bf": "\\sum_{j=1}^m (T_j)", "a7b473778010046a4cf3f76bf7f053b6": "(b_{7}-a_{7})+(b_{8}-a_{8})+(b_{14}-a_{14})+(b_{15}-a_{15})=2*(b_{12}-a_{12})", "a7b48f01adce40e314a7107c1b16de6b": "r=3.", "a7b4beaa82083fb6173213742593112f": "\\gamma \\lambda_n(L)", "a7b5011f3986c4b5e02bb460f1d5f3b1": "[0, \\infty],", "a7b539f6a832c6494c994fd22e89def1": "\n\\bar{h}^{i j} (t,\\vec{x}) \\approx\n-\\frac{4}{r}\\, \\frac{\\mathrm{d}^2}{\\mathrm{d}t^2}\\, \\left\\{ M_1 x_1^i(t-r) x_1^j(t-r) + M_2 x_2^i(t-r) x_2^j(t-r) \\right\\}\n", "a7b576b1c8d07831a4e2ef5ce7923008": "\\pi(x) - \\pi(\\tfrac{x}{2}) \\ge 1,\\,", "a7b581d4ca8d04db3dde64ee69bb49c2": "(r, \\theta, z) = \\left(R \\sqrt{1 - \\frac{R^2}{4}}, \\Theta, -1 + \\frac{R^2}{2}\\right).", "a7b5bbd5f784604a885e6dbc3c9ca1e3": "a \\land a = a", "a7b5f48686ba18f45ae106e9da9fc4e5": "\\pagecolor{Black}\\color{Yellow}\\text{Yellow}", "a7b62a507be485fddd9ca8463a7b75be": "\\scriptstyle Z", "a7b62bbb17743d3be977e301a3a0ffb8": " \\mathbf{e}_{123} ", "a7b62cf041c5d6120ae24bab6b04c95a": "\\omega(Lu, Lv) = \\omega(u, v).", "a7b671415ab67324231ea803ca619e00": "\\mathbf{F}_{kj} = -\\nabla_{\\mathbf{r}_j} V", "a7b67bba86d47a42eec8768d80a74197": "\\lang 2,1,0 \\rang", "a7b68e3d2813923a615d926b8b0173a8": "\\ \\mathcal{F}_{n}(\\mathbf{c} \\star \\mathbf{x}) = \\mathcal{F}_{n}(\\mathbf{c}) \\mathcal{F}_{n}(\\mathbf{x}) = \\mathcal{F}_{n}(\\mathbf{b})", "a7b6a0509b5950647e7f38f0b80e2378": "r=r_g", "a7b71e94dd1c8b3891bf38386483b8f5": "P \\rightarrow \\neg Q ", "a7b725ceb6292ec39ab06d3dd92b603a": "\\boldsymbol{\\beta}_s = \\mathbf{v}_s/c", "a7b748850f3b84d2ee80b7d251309799": "dF", "a7b75735508864552f50f47a8af02575": "\n\\ H(\\omega) = 1 + \\alpha e^{-j \\omega \\tau} \\,\n", "a7b791810a031150287a106f095ea229": "d(w) = (d(w^\\smallfrown 0) + d(w^\\smallfrown 1))/2", "a7b79d2f5107a55b9d6c85364a807686": " B_{i+1} = X_{i} < (Y_{i} + B_{i})", "a7b79f69ad6ddce0461461d1f800ca9d": " F = x, P = x, E = f\\ (x\\ x) ", "a7b7cf795fafc1cf510953e4a869eef2": " X_1,X_2,\\dots. \\,", "a7b810eb517c9af6b0e70590df210ed8": "\\forall A, \\exists B, \\forall C, \\forall D, C \\in A \\rightarrow D\\in B.", "a7b82a3016ff175cf7d85b06a389bd95": "x = h(y)", "a7b8326f42d3bfbae47068e72a54502f": "\n\\sum_{n=1}^{\\infty}\\frac{\\zeta(2n)-1}{2^{2n}} = \\frac16.\n", "a7b84bda8919f4202d03eb08e8c9751e": "\\Phi(E)=\\frac{e^{\\beta E}}{z}-1\\,", "a7b8b8707254713b6545b47ea61f546d": "\\psi_3(x_3)", "a7b8bb8241ec761e5aece365d118e389": "\\operatorname{E}_{\\theta_0}\\phi(X)=\\alpha", "a7b8c8ca95d11e52550e7c591c87460d": " z = \\frac{x - \\mu}{\\sigma},", "a7b8d6eb7c3add955b391154215f0c31": "\\arctan\\alpha \\pm \\arctan\\beta = \\arctan\\left(\\frac{\\alpha \\pm \\beta}{1 \\mp \\alpha\\beta}\\right)", "a7b9260123ce9ec2609469e5e5b190df": "c_{1r}", "a7b955a6d437a7bd5eb4c170824d3344": "\\begin{matrix} {10 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "a7b95d263e33ecfbbade76204170be51": "U\\, =\\, \\frac{H}{h} \\left(\\frac{\\lambda}{h}\\right)^2\\, =\\, \\frac{H\\, \\lambda^2}{h^3},", "a7b974ffb37b8cb3398e7bd2794f56cd": "\\Omega + 1 < \\Omega", "a7b98851fb0bd522e77ad140cc82e2b4": "F\\left(x,w\\right)\\equiv\\phi_n\\left(x\\right)w^n+\\cdots+\\phi_1\\left(x\\right)w+\\phi_0\\left(x\\right), \\, ", "a7b9e4616f7b9141255031b21807f601": " P \\in {\\mathbb\\{0,1\\}^*} ", "a7b9e5348e5ad97386ee8d20513fe911": "h = g^x\\,", "a7ba1ac1850b898784271189a3a7de36": "\\alpha^{\\mathrm{N} \\pi - 1} \\equiv 1 \\pmod{\\pi}", "a7ba371308ed57750780cea51eb903a5": "F(M) = F(M-e)+F(M/e)", "a7ba5605e65bb2ebc5a291acd47c7949": "j(-i) = k\\,", "a7bac87e42bcc224ba65becb701a1a67": "\\frac{(a+b)(1-p_a-p_b)+2ap_a+2bp_b}{4}", "a7bb07eb4ba21064e09dab5d2d94bde3": "f: X \\mapsto Y", "a7bb87e368dcc511541bb0125c3af0d9": "W: L^2(\\mathbb{R}) \\rightarrow H ", "a7bbb6b71fe130a4c49db2af6d7ddd42": "\n\\begin{Bmatrix}CS:\\\\DS:\\\\SS:\\\\ES:\\end{Bmatrix}\n\\begin{bmatrix}\\begin{Bmatrix}BX\\\\BP\\end{Bmatrix}\\end{bmatrix} +\n\\begin{bmatrix}\\begin{Bmatrix}SI\\\\DI\\end{Bmatrix}\\end{bmatrix} +\n\\rm [displacement]\n", "a7bc748c35f9dc11d37d5f5b9667bac5": " t = -\\tau \\; \\ln\\left(1-\\frac{V(t)}{V_0}\\right)", "a7bcc7cd57c48dc842ab7b5aed8b9ae0": "\\mathbf{a}\\cdot\\mathbf{b} = \\bar{a}_j \\bar{b}_j = a_i \\mathsf{L}_{ij} b_k(\\boldsymbol{\\mathsf{L}}^{-1})_{jk} = a_i \\delta_i{}_k b_k = a_i b_i ", "a7bd39c0a109ced1ec4083ccf6f9df53": "\\boldsymbol{\\omega} = (\\omega_1,\\ldots,\\omega_c) \\in \\mathbb{R}_+^c", "a7bd540a38092ff9fedbb8ca49121a6a": "\\phi~", "a7bdb88cae0bbe2a53360f6820689809": "x^3+Ax+B", "a7bdc7c21aa87bb8357358f4447d9470": " K_t(x,x') = K_t(x-x') \\, .", "a7be127554520c495226bdfd872904ec": "L \\,", "a7be33c9699b7c86ffe5b4f779a3b4a5": "g\\circ f=h\\circ f", "a7be384decd5985723b86cd6c874a52f": " \\Gamma^{p^n} ", "a7bed85216afdcdd8e0b4f8a696c6642": " \\overline{g}_{ij}=\\frac{g_{ij}}{g_{00}}", "a7bf6d664357fb49fe266ac7ae35cd13": "\\scriptstyle\\eta\\,\\perp\\,x^*,", "a7bf6ea35ef28796e028ff78037de92f": "A\\models \\phi_1\\land\\dots\\land\\phi_n \\iff \\forall\\phi_i (1\\leq i\\leq n), A\\models \\phi_i", "a7bf948a081ddd663403dcdba840504f": "\n A_M \n", "a7bff484029080271bb71902add8b9a9": "\\begin{align}\n\n \\tanh x &= x - \\frac {x^3} {3} + \\frac {2x^5} {15} - \\frac {17x^7} {315} + \\cdots = \\sum_{n=1}^\\infty \\frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \\left |x \\right | < \\frac {\\pi} {2} \\\\\n\n \\coth x &= x^{-1} + \\frac {x} {3} - \\frac {x^3} {45} + \\frac {2x^5} {945} + \\cdots = x^{-1} + \\sum_{n=1}^\\infty \\frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \\left |x \\right | < \\pi \\\\\n\n \\operatorname {sech}\\, x &= 1 - \\frac {x^2} {2} + \\frac {5x^4} {24} - \\frac {61x^6} {720} + \\cdots = \\sum_{n=0}^\\infty \\frac{E_{2 n} x^{2n}}{(2n)!} , \\left |x \\right | < \\frac {\\pi} {2} \\\\\n\n \\operatorname {csch}\\, x &= x^{-1} - \\frac {x} {6} +\\frac {7x^3} {360} -\\frac {31x^5} {15120} + \\cdots = x^{-1} + \\sum_{n=1}^\\infty \\frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \\left |x \\right | < \\pi\n\n\\end{align}", "a7c002de1afa283aeeb4e62bdf6a58de": " \\ \\alpha", "a7c00cd94e860f99c3386e2e3df862b0": "c(i,x):=p \\left(x_i|x_{-\\infty}^{i-1} \\right ).", "a7c024be4078c0881756368e3db1a95c": "k_\\mathrm{m}, k_\\mathrm{e}", "a7c072e0a572e65a7929b0742ff45cfd": " \\Pr(x \\le m - k \\sigma) \\le \\frac { 1 } { 2 k^2 } .", "a7c1280bdebcf3dffd6ccf2c8cd2e615": "P=(X_1:Y_1:Z_1)", "a7c141eb092a3d1b484672eb2b53e215": "H_n(a, b)\\,\\!", "a7c178c06289036c1976cd178907d05c": "E_{ij} = x_i \\frac{\\partial}{\\partial x_j}.", "a7c234cc3516f8eabfa5b09deb210fea": "\\ell = 1", "a7c24923139d4455d25391e6b5e12d48": "H_{\\infty}(X) \\geq k", "a7c2687a9617fe76a0373e9ff07eb37c": "DF_\\tau", "a7c2a2b22b24f30a14dac794e0fa1cf5": "\\nabla \\times \\nabla \\varphi = \\mathbf{0},", "a7c2bd45fab38f9d7d86f6488188b528": "\\sin^2\\left(\\frac{x}{2}\\right) = \\frac{1 - \\cos(x)}{2}.", "a7c2d143c3dffecf0baee33620e89f4e": "\\mathrm{Force}=- k Q \\!", "a7c2d931e89f8ed46962efd808745a7f": "R[x] \\otimes_R R[y] = R[x, y].", "a7c2ed4ed191fcd8c6c8562d79df4709": "n_e", "a7c2fd3fa5ed6022943bc40058fbd4fa": "\\mathbf{L}=(L_x,L_y,L_z)", "a7c3459899f6285c3ffba390c23e8e57": "\\sum_{S=1}^{N-n}\\prod_{j=1}^{n-1}(N-S-j)\\approx \\int_1^{N-n}(N-S)^{n-1} \\, dS = {(N-1)^n-n^n \\over n}\\approx {N^n \\over n}", "a7c395a68fea8cdd5d5574b65b348567": " = \\mathcal{S} + {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-h} h^{ab} \\left( \\omega_{\\mu \\delta} + \\omega_{\\delta \\mu } \\right) \\partial_a X^\\mu \\partial_b X^\\delta + O(\\omega^2) = \\mathcal{S} + O(\\omega^2) ", "a7c3ef3424c2ba6684dc0b7863d2c683": " 2 - 1 = 1 ", "a7c43bc402bc5b472a7cc55f1e6a6dbc": "\\scriptstyle N \\;=\\; n_1n_2 \\ldots n_k", "a7c46cb1f57504490e4e7a19d5d01904": " \\frac{4}{(k+1)!} < 10^{-5} \\quad \\Leftrightarrow \\quad 4\\cdot 10^5 < (k+1)! \\quad \\Leftrightarrow \\quad k \\geq 9. ", "a7c4a8697d58042f90a745eee9a9be2b": "j\\in [1,N]", "a7c4d2e816039a42748f808c7161984a": "g \\mapsto -g,", "a7c506eaeb5aef6660382bab7f732f1d": "S_k = S_{k-1}^2-2.\\ ", "a7c52de3726a1bf726739325ebaae345": "R_1+R_2=R_3+R_4", "a7c5cc508f2668cde1137cd8cc46fbb5": "\\bold{h}", "a7c5f396d35440fbd8f9e2a8b0af02bf": "\n\\mathrm{M} = \\frac{v}{v_\\text{sound}}\n", "a7c660e9412b122ff26c94c268404824": "n^{\\alpha}", "a7c672888ed4a23796381afed0af43a9": "\\sum_{m=0}^n P(m)P(m+2)=P(n+2)P(n+3)-1.", "a7c6a34dd0459657f038d592d0fc932c": "Q(x) = (c_2x^2+c_1x+c_0)(d_2x^2+d_1x+d_0).", "a7c6bd41f189c0b68d1f1e37848d01a8": "\\bar{x}_{B}", "a7c6c783c5d03fc91d0594b217f56580": "D\\,\\!", "a7c6cd92d6a78be1cc08d9f2ac652fed": " \\omega_d = \\frac{2\\pi}{T}, ", "a7c705c0ddaa8d46a5c959b4f7f8aa37": "{\\alpha \\choose \\alpha} = 2^{\\alpha}", "a7c713a614bc5d0c68aa329e77e6dcdc": "f'(x_n) \\approx 0", "a7c7161303497ef51c6b25ffe3752236": "\n\nA = \\left(\\begin{array}{cc} 0.5 & 0.2 \\\\ 0.4 & 0.1 \\end{array}\\right) \\text{ and } d = \\left(\\begin{array}{c} 7 \\\\ 4 \\end{array}\\right).\n\n", "a7c7375b7a32f838f4616dd408f477da": "\\Delta_T", "a7c77387d680258a73e3498eb119a6aa": "\\begin{align}\n V_L &= G_{L}V_{in} e^{j \\phi_L}\\\\\n V_R &= G_{R}V_{in}e^{j \\phi_R}\n\\end{align}", "a7c7e3e4492033f02c335c940163b63b": " \\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{x}} \\right) - \\frac{\\partial L}{\\partial x} = 0 \\qquad \\Rightarrow \\qquad m\\ddot{x} = 0 ", "a7c7ff6d51e1cd38264164952b751920": " \\phi _{q, \\mathbf{0}} (z) ", "a7c8b9c8d0384b9aedf4325722e98c26": " \\min_{\\mathbf{M}} \\sum_{i,j\\in N_i} d(\\vec x_i,\\vec x_j) + \\sum_{i,j,l} \\xi_{ijl}", "a7c8c00d08e4713b5164e3b31df72c17": "\\tau^{(t+1)}_j = \\frac{\\sum_{i=1}^n T_{j,i}^{(t)}}{\\sum_{i=1}^n (T_{1,i}^{(t)} + T_{2,i}^{(t)} ) } = \\frac{1}{n} \\sum_{i=1}^n T_{j,i}^{(t)}", "a7c9383aa6cabec7b105832db03d7e49": "\\mathbf{\\epsilon} = \\mathbf{Bq} \\qquad \\qquad \\qquad \\qquad \\mathrm{(4)}", "a7c940c56afa57e1bafe72843e29882a": "v^{*}:= \\max_{d\\in D}\\min_{s\\in S(d)}\\ \\{f(d,s): g(d,s) \\le 0, \\forall s\\in S(d)\\}.", "a7c9449c2ce64aece8a71ded3aa739a6": "\\alpha_1x_1+\\alpha_2x_2+\\cdots+\\alpha_nx_n", "a7c945f1fba0a4342b383176fd4a8890": "F(x) = \\int f(x) \\, dx.", "a7c96b412ee2b1ccf876490bbf28616b": "{A'}^\\mu = \\Lambda^\\mu {}_\\nu A^\\nu \\,,\\quad {A'}_\\mu = \\Lambda_\\mu {}^\\nu A_\\nu", "a7c99f91d2c0c6a5543d2fd977b828a8": "K = -1", "a7c9c40fc01c9532e2da2f2ec46d3b44": " \\mathbf{h}(n)", "a7c9cef5cd6a995efab41289f1317255": "X = \\mu_p", "a7ca05f48ff56afc3e599ee1054b8346": "f\\!\\left(x\\right) > f\\!\\left(y\\right)", "a7ca13ace5ca7dc6cfa5cc7dff586b78": " \\vec{e}_{\\beta} ", "a7ca18e0dd63e7d4ffc277aa84f458df": "x_1, x_2, y_1,", "a7ca34a79be90be19f770233b8e4bd48": "A(x-1)(x-2) + Bx(x-2) + Cx(x-1) = 1,\\,", "a7ca683fa07f7eb3f8d2f786eef6fb2e": "\\gamma N", "a7ca9ab1b6f53ddf3700573725c15e10": "[+,-,-,-]", "a7caba1d464623a5fc9ffadb8753059a": "\\frac{60499999499}{490050000000} = \n0.123456789\\overline{101112\\ldots9900010203040506070809}\n,", "a7cac7bcaeac4c66279909a69511af1a": "A_\\phi = 0", "a7cafb6f8789840d1443871574ce64cf": "s_i = \\lceil - \\log_a p_i \\rceil ", "a7cafdf40992d658b70a20c47c0b95aa": "\n Y = X\\Beta\\Gamma^{-1} + U\\Gamma^{-1} = X\\Pi + V.\\,\n ", "a7cb62ed2b26026832e87d3c1c29ac1c": "V_-=A+Be^{\\frac{-1}{RC}t}", "a7cba85f8a25cd82208e0ebe7e501f4c": "(\\pi-A-B-C) R^2{}{}.\\!", "a7cbf99178ee462d9ab86bd624b14926": "\\mathbf{a} \\times \\mathbf{b} \\times \\mathbf{c} = (\\mathbf{a} \\wedge \\mathbf{b} \\wedge \\mathbf{c}) ~\\lrcorner~ (\\mathbf{w} - \\mathbf{ve}_8)", "a7cc19a60200c377f5eabf46a7f830a0": "G(A, B) \\subseteq G(A', B') \\mbox{ whenever }A \\subseteq A' \\mbox{ and } B'\\subseteq B", "a7cc31a43c0ed398db64ce99c502274c": "\\pi/8", "a7cc8258791acdbb85a7850dc2b26edb": "\\varphi_{\\mathrm{M}}(p)(J) = a_0 + a_1\\cdot J + \\dots + a_n\\cdot J^n", "a7cce560f6581f81f75957d2b89cef4c": "P = D\\cdot\\sum_{i=1}^{\\infty}\\left(\\frac{1+g}{1+k}\\right)^{i} = D\\cdot\\frac{1+g}{k-g} ", "a7cce5a3552bc64b8da229a5043ef595": "A_0=QAQ^T", "a7cd13581686b71c7759851be01c8f9c": "\\doteq \\!\\,", "a7cd2b46b033ee2f11c22a57795e3bc6": "= \\frac{s_\\mathrm{a}(t)}{|s_\\mathrm{a}(t)|}\\,", "a7cd2f982731745d89e7f196bdd74306": " P(A_{i'}) = \\frac{1}{M} P(P(X_1^n(i')) \\geq P(X_1^n(i)))\\, . ", "a7cda236a2aefdc6f67775d2c3749f8b": "\\frac{{\\rm d}W}{{\\rm d}z}=\\frac{W(z)}{z(1 + W(z))}\\quad\\text{for }z\\not\\in\\{0,-1/e\\}.", "a7cda7057d6f20afb29c5f2d382e2a5b": "C\\approx W\\log_2 \\frac{\\bar{P}}{N_0 W} ", "a7ce4016125440bb76c70c422e6847b9": "pV^n", "a7cebfa23fc53af28cd66c9046b45745": "{4 \\choose 1}^7 = 16,384\\,", "a7cf086b2bdc2e561b6985c09c49d5ec": "\\boldsymbol{\\varphi}", "a7cf483cb7b0d02d0e4e3f4ad292713b": " X = \\operatorname{cl}(X)", "a7cf856280cc97f964189c2deb28e82d": "\\mathbf{f} = \\mathbf{f}(\\mathbf{x},t)", "a7cf8caadcbcc261361da399c5fd554f": " \\nabla^2 = \\frac{1}{r^2} \\frac{\\partial^2}{\\partial \\theta^2} ", "a7cf9022fffc82db5c76f01674ad13bf": "u(t) = -B(t)^{T}\\phi(t_0,t)^{T}\\eta_0", "a7d025e43bf1f799b5e121aca2a78a4f": "1\\leq\\kappa\\leq r", "a7d03627eeefd2a975be27ad66fbcead": "\\gamma_i(t)=P(X_t=i|Y,\\theta) = \\frac{\\alpha_i(t)\\beta_i(t)}{\\sum_{j=1}^N \\alpha_j(t)\\beta_j(t)}", "a7d07b2146eac3cb275b67e805ffa5c9": "\\forall x . R(x,f(x))", "a7d0d714fbd3fd6f6878c1444d4f437a": " \\nu > p-1 ", "a7d108a7cac03cc74f4ddf5ce9311eb4": " y_i = \\begin{cases} \n y_i^* & \\textrm{if} \\; y_i^* >y_L \\\\ \n y_L & \\textrm{if} \\; y_i^* \\leq y_L.\n\\end{cases}", "a7d1552d5323038f3e6da0818ac144eb": "\\,\\nu", "a7d1c06fcf3999d119ff35658256802b": "(1/0!)\\pi^0 ", "a7d2239fb42c8371306d29550b3fa842": "\\hat{X}\\vert_{\\mathrm F_{SO}(M)}\\,", "a7d26aab3a1ec9116b56aefee6c506e9": "\\Lambda = \\frac{h}{\\sqrt{2mE_K}}", "a7d288fb91b0dae22a8dbf14b2fd88bf": "\\eta_A:I\\to A^*\\otimes A", "a7d28c2f213d99265c6b0dc9e8a353bf": "\\frac{d}{dt}(v'(t) y_1^2(t) e^{\\int p(t) dt})=y_1(t)r(t)e^{\\int p(t) dt}", "a7d2a02e355aed7677c9c141a035dabe": "\\nabla\\times(\\alpha\\mathbf{B})= \\alpha(\\nabla\\times\\mathbf{B})=\\alpha^2 \\mathbf{B} ", "a7d2ce711184629ec0d433a7383f76d6": "A_1, A_2, \\ldots, A_n", "a7d2d31d681abb6163c7e3a5aa38351f": "\\begin{bmatrix}\nll(1-\\cos \\theta)+\\cos\\theta & ml(1-\\cos\\theta)-n\\sin\\theta & nl(1-\\cos\\theta)+m\\sin\\theta\\\\\nlm(1-\\cos\\theta)+n\\sin\\theta & mm(1-\\cos\\theta)+\\cos\\theta & nm(1-\\cos\\theta)-l\\sin\\theta \\\\\nln(1-\\cos\\theta)-m\\sin\\theta & mn(1-\\cos\\theta)+l\\sin\\theta & nn(1-\\cos\\theta)+\\cos\\theta\n\\end{bmatrix}.", "a7d2f2c3de681ad0bbbdd8ae6b8e46e2": "SO(2,2)", "a7d36f3473c548c5967b1c437a2273e7": "\n\\sum_{k=0}^\\infty \\kappa^k = \\frac{1}{1-\\kappa}\n", "a7d3f31b74eabb0d2a5f1bc2aa3b784c": " A_1, A_2", "a7d45141063d405ca088a089847f0be6": "s_{1}(t) =\\overset{\\cdot }{s}_{0}(t)+\\alpha _{1}(t)s_{0}^{\\gamma _{1}}(t)", "a7d45e3ff5c3af0453a90912357563df": "\\neg P \\vee P.", "a7d4b3dd719e66940b0739611b54b2ba": "\\forall \\alpha\\, \\varphi\\,\\!", "a7d4d11ef599763de93dbeceb0233ac6": "\\Phi(p) = 1/m \\cdot v", "a7d4d952f8a428fccdb142d7f51f5b5a": "\\sum\\limits_{A\\in F_k} \\prod\\limits_{i\\in A} p_i \\prod\\limits_{j\\in A^c} (1-p_j)", "a7d5172d3fb61787e6ad9da6aacc07e4": "\n-2\\pi \\frac{9}{8}\\ \\sin^2 i\\ e_g\\ +\\ 2\\pi \\frac{3}{2}\\ e_g\\ -\\ 2\\pi \\ \\frac{3}{2}\\ \\sin^2 i\\ e_g\\ +\\ 2\\pi \\frac{3}{8} \\sin^2 i \\ e_g\\ =\\ -2\\pi\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ e_g\n", "a7d54839263f4a4b12f411e10cb85af2": "\\mathbf{R}^{*T} \\mathbf{r} = \\mathbf{Q}^{*T} \\mathbf{q} ", "a7d56623c9b151780f26d1f637d2b403": "\\phi(t) = \\omega t -\\begin{matrix} \\frac{\\pi }{2}\\end{matrix}\\,", "a7d5820c80bd7aeb40ed64e52bb1b0d2": "A^0 B_1{}^0 C_{00} + A^0 B_1{}^1 C_{10} + A^0 B_1{}^2 C_{20} + A^0 B_1{}^3 C_{30} + D^0{}_1{} E_0 = T^0{}_1{}_0 ", "a7d585b45645205dd42c286dde12ac15": "\\Pr(X(t+h) = j | X(t) = i) = \\delta_{ij} + q_{ij}h + o(h)", "a7d59e95c8b0af58483fd357d818767b": " \\alpha \\in \\mathbb{N}^N", "a7d5ee5fd17960cca5ff7e0214e4f36b": "\\mathbf{W}=\n\\begin{bmatrix}\n1&0&0&0\\\\\n0&1-k&k&k\\\\\n0&k&1-k&k\\\\\n0&k&k&1-k\n\\end{bmatrix}\n", "a7d67da368fd318dffb55aaf0161a37e": "(x, v) \\mapsto (x, v, 0, v)", "a7d6931f72c806c3c7706c3a5b9e0cbc": "\\frac{T_1}{T_2} = e^{\\mu\\alpha}", "a7d6db7f6784b011ef4430ede863ec61": "2^{2^{4}} + 1 = 2^{16} + 1 = 65537.", "a7d6fc5a5f66a95d27d76459ee2b9d15": " \\!\\ S_m^5 = S_{(m^5 + 5m^3 + 5m)} ", "a7d7326d8afde5daaa47898768442af1": "\\mathbf{p_i}", "a7d77784bf15463ecbe0174af0bb5e4e": " T_H ", "a7d7c1e1e0328be6158148d8bb5c3a28": "\\frac{d^2N}{dz d\\Omega} = n_{prop}(z) \\frac{d^2V_{prop}}{dz d\\Omega}\n= \\frac{n_{prop}(z) a_0^2 r^2(z)}{H(z) (1+z)^3} ", "a7d8a6fa40bc9f0a42af05f5a08cb262": "e = \\frac { \\sqrt{{(x_0 - x)}^2 + {(y_0 - y)}^2}} {2 a} \\quad (17)", "a7d8cbbaab33d2d4d2d903aec81f42b4": "\\textrm{Supp}", "a7d8dac3772fe5d27d7a6555ef848b5a": "\n\\lim\\inf\\frac{\\varphi(n)}{n}\\log\\log n = e^{-\\gamma}.\n", "a7d90b8b7f9ccecec235654edabc520a": "\nR(M) = \\begin{cases}\n g - n(M) & \\mbox{if}\\ n(M) < g\\\\\n 0 & \\mbox{otherwise,}\n \\end{cases}\n", "a7d90d29e9471440455980774216ed62": "x_{k} \\in S_{i} \\cap S_{j}", "a7d9545cbee89df4cae5769adb825a5a": "\n\\begin{align}\n& \\pi(i):=0~\\mbox{for all}~i\\\\\n& \\mbox{For}~i:=1~\\mbox{to}~n\\\\\n& \\qquad \\mathcal{L}_i := \\mathcal{A}_i\\\\\n& \\qquad \\mbox{For all}~j~\\mbox{such that}~\\pi(j) = i\\\\\n& \\qquad \\qquad \\mathcal{L}_i := (\\mathcal{L}_i \\cup \\mathcal{L}_j)\\setminus\\{j\\}\\\\\n& \\qquad \\pi(i) := \\min(\\mathcal{L}_i\\setminus\\{i\\})\n\\end{align}\n", "a7d9e5b9fd1517fd8eb32eb7d2761d35": "\\gamma ( \\rho , \\infty) = 2 \\gamma (\\rho)", "a7d9eb946da1d3be5d916544cfd633c0": "0 = \\delta\\int{L[t] \\, \\mathrm{d}t} = \\int{\\delta L[t] \\, \\mathrm{d}t} ", "a7da30e6fd761cf376f5bc9f23737e8b": "\\mu_{3,2}", "a7da39c4910e930b66fe3cc20f8d26b3": "c_{i\\,j\\,k\\,\\ell}", "a7da440d00870eacc8ea0da8f7a08099": " u'(t) = \\Delta_Du(t) ", "a7da89ef1d7604826b741d353f80a642": "f(k)n^{O(1)}", "a7daa2cf658a836e80890fc58ac6c101": "[0,\\theta)", "a7dad04a356eeb516fcd221587649856": "M_a=\\lim_{N\\to\\infty}N\\cdot x(N)=\\lim_{N\\to\\infty}\\frac{P_T\\cdot r}{(1 + \\frac{r}{N})^{NT}-1}=\\frac{P_T\\cdot r}{e^{rT}-1}. ", "a7db115677bae7442ebceccc5fa10090": "{x} = \\sqrt {r^2 + a^2} \\sin\\theta\\cos\\phi", "a7db5fcb31d8a4646adb72ba8bdd7c52": "\\Delta x = \\frac{\\lambda}{\\sin \\varepsilon/2}.", "a7db6ed7033b49b88da6ec8719a02b49": "\\nabla:SM\\rightarrow T^*M\\otimes SM.", "a7dbda7588cfd107a64f71686b18a5bc": "x^2+x+1=0", "a7dc3eebdc39f2502148f958aa62ba75": "p_A = 1", "a7dc5d7286a1803d13ff44fa586b0267": "\\int \\frac{dx}{\\sqrt{x^2+1}} = \\mbox{arcsinh}(x) + C ", "a7dc6729afb939f6bda9f482e1ee521f": "K_\\text{limb}=\\frac{F_\\text{max}}{\\Delta L}", "a7dc8812ecf2a448e1acee84b3cf88f8": "e^{-1/e}\\eta_L", "a7ecb533f10f3fcea9c04cf06b3c1e4a": "{x^2 \\over a^2} - {y^2 \\over b^2} - z = 0 \\,", "a7ecb57ff7a5c7dd1b8d107621e43ea3": " f = 0\\,", "a7ecbfe55f776cd57b8cf612bded214c": "F(x) = \\int_a^x\\!f(t)\\, dt.", "a7ed3ced99be43411d1c2338b11ebed5": "A(1, 2)", "a7ed837dd7b6ff1d31860c0b7e5a56b3": "\n{\\upsilon_w \\over \\upsilon} = {f_w \\over f} = {(1+\\alpha V_a)}^{1 \\over 2}\n", "a7ed9a2fcbd80ba07537e3bbf07169ba": "\\textstyle\\binom{16\\sin^{\\scriptscriptstyle 3}t}{13\\cos{}t-5\\cos2t-2\\cos3t-\\cos4t}", "a7ee2d5bf34213a62d24a3c8edfb6fe6": " C^1_c", "a7ee33a15e60d4484e4756a9a65a5610": "M(j,j) = 1 - \\lambda m(j)", "a7ee8698b6664000dd6dc28ab72064c2": "(a,a+\\Delta)", "a7eea41ab1dd0fe34261c25165f2133f": "\\frac{E_{p}}{E_{s}}=\\pm j\\sqrt{1-(\\gamma /\\theta )^{2}}+\\gamma /\\theta ", "a7eeeda36e32834fc78c1406ac8594ee": "x_1(t),x_2(t),x_3(t)\\,", "a7ef2f96fb832c18f6662dced7ce10cf": "\n\\left(\\begin{matrix}p(0) \\\\ p(1) \\\\ p(-1) \\\\ p(-2) \\\\ p(\\infty)\\end{matrix}\\right) =\n\\left(\\begin{matrix}\n0^0 & 0^1 & 0^2 \\\\\n1^0 & 1^1 & 1^2 \\\\\n(-1)^0 & (-1)^1 & (-1)^2 \\\\\n(-2)^0 & (-2)^1 & (-2)^2 \\\\\n0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}m_0 \\\\ m_1 \\\\ m_2\\end{matrix}\\right) =\n\\left(\\begin{matrix}\n1 & 0 & 0 \\\\\n1 & 1 & 1 \\\\\n1 & -1 & 1 \\\\\n1 & -2 & 4 \\\\\n0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}m_0 \\\\ m_1 \\\\ m_2\\end{matrix}\\right).\n", "a7ef3d7656cf930a591e03cd4a8af365": "g(x) - L\\leq f(x) - L\\leq h(x) - L\\, ", "a7ef57f7502930593aed4a1b37a5e69b": "A_{i+1} = A_i\n\\,.\\,\n\\begin{pmatrix}\n0 & 1\\\\\n1 & -q_i\n\\end{pmatrix} \\,.\n", "a7ef97b883ab602ad4cfd4874682f106": " 3B_{j_{k}}", "a7efc91ea510bc31160d5a8cd081f75a": "\\Delta(n)=O\\left(\\frac{1}{n\\sqrt{\\log\\log n}}\\right).", "a7efea78c9fdcab91612277b5a529bd2": "\\chi^{-1/4}\\chi' = 2^{3/4} \\mathfrak{M}^{1/2}", "a7eff83f45d7585093386067fee497a3": " 3(a_2^2+b_2^2) = s_1^2+t_1^2, \\, ", "a7f0bb2c63727266db89a0a1e72c841a": "\\displaystyle{u(z)=\\int_{\\partial \\Omega} K(z,w)u(w) - N(z-w)\\partial_n u(w) \\,|dw|,}", "a7f1211daec7ac4ca823e854be3afe61": "\\rm{PB} = \\operatorname{sqrt}(\\rm{BC}^2+\\rm{PC}^2 - 2*\\rm{BC}*\\rm{PC}*\\cos(\\pi-\\beta-y)).", "a7f1738ca365d6775179f902b21cafaf": "\n\\frac{\\omega_L^2}{\\omega_T^2} = \\frac{\\varepsilon_{st}}{\\varepsilon_{\\infty}}\n", "a7f21e50e4316df5bc13e821ae3a59d0": "\\delta Q\\ =\\left [\\left. \\frac{\\partial U}{\\partial p}\\right |_{(p,T)}\\,+\\,p \\left.\\frac{\\partial V}{\\partial p}\\right |_{(p,T)}\\right ]\\delta p\\,+\\,\\left [ \\left.\\frac{\\partial U}{\\partial T}\\right|_{(p,T)}\\,+\\,p \\left.\\frac{\\partial V}{\\partial T}\\right |_{(p,T)}\\right ]\\delta T", "a7f279fcf05b1330b6de8da455963e71": "\\textstyle\\frac{2}{3}", "a7f29aa0c2df4dfc2032e0d8bf760927": "n = d_k^{d_k} + d_{k-1}^{d_{k-1}} + \\dots + d_2^{d_2} + d_1^{d_1}\\, ,\\text{ e.g. } 3435 = 3^3 + 4^4 + 3^3 + 5^5\\, .", "a7f2e01fdbf30321761e0684cc6aff69": "p(y) \\propto y^{m_1} (1-y)^{m_2}, \\!", "a7f301f41160371a579fdf0b582d2bec": "\\mathbf{F} =q_e \\left ( \\mathbf{v}\\times\\mathbf{B} \\right ) \\,\\!", "a7f38cbd31f76406f7c03b3ef94114d2": "B_\\lambda(T) = \\frac{2 h c^2}{\\lambda^5}~\\frac{1}{e^\\frac{hc}{\\lambda kT}-1},", "a7f46b9a2dc23eae301a139e7653c7c2": "M_{s_\\alpha\\cdot\\lambda}\\subset M_\\lambda", "a7f48a795d67caffa7098e470d49a489": " w_{ii}^{-1} = \\left(\\frac{\\partial\\eta_i}{\\partial\\mu_i}\\right)^2\\text{var}(y_i),", "a7f4db11135df30ae6cffd475d355cf4": "c \\cdot \\sum a_\\alpha X^\\alpha = \\sum c a_\\alpha X^\\alpha,", "a7f4fd653a21886dadcd35880fbc1e20": " Ax = \\lambda B x ", "a7f544dc06de7ab133fc1ad99d355a4d": "\\forall y\\in Y", "a7f5d0b72947bbd5f7839e31351bd0ab": "\\mathbf{v} = \\left[ \\begin{matrix} v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_{n - 1} \\\\ v_n \\end{matrix} \\right]= \\left( \\begin{matrix} v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_{n - 1} \\\\ v_n \\end{matrix} \\right)", "a7f5e91e2a311b6db274ecae7a00dc95": "\\gamma = \\zeta / m", "a7f6274c51c519d26bae214d513c74c2": "\n\\begin{align}\ni \\hbar \\epsilon g \\partial_g | \\Psi_\\epsilon \\rangle &= \n\\frac{1}{\\langle\\Psi_0| U |\\Psi_0 \\rangle} (H_\\epsilon-E_0) U|\\Psi_0\\rangle \n- \\frac{U|\\Psi_0\\rangle }{{\\langle\\Psi_0 |U| \\Psi_0 \\rangle}^2} \\langle \\Psi_0 | H_\\epsilon-E_0 | \\Psi_0\\rangle \\\\\n\n&= (H_\\epsilon-E_0)|\\Psi_\\epsilon\\rangle - |\\Psi_\\epsilon\\rangle \\langle \\Psi_0 |H_\\epsilon-E_0|\\Psi_\\epsilon\\rangle \\\\\n\n& = \\left[ H_\\epsilon - E \\right] |\\Psi_\\epsilon\\rangle.\n\\end{align}\n", "a7f64699785af28c1e499733c98453f4": "V(a) = \\max_{ 0 \\leq c \\leq a } \\{ u(c) + \\beta V((1+r) (a - c)) \\},", "a7f67dd60a16f41a2c45849aaad228c6": "\\sin(\\alpha)", "a7f6b54c6dfe96331e9de34ec1189289": "n' = B n^{d/(d+4)}", "a7f6d8ea1543b6726d661870fea421be": "d(x,(p,n))^2=(x-p)^\\top (\\hat n \\hat n^\\top) (x-p).", "a7f70b40dd82325dda1627bf47d954f6": "\nP =\n\\left(\n\\begin{array}{cc}\n Q & R\\\\\n \\mathbf{0} & I_r\n\\end{array}\n\\right),\n", "a7f7311affed57d9d8d23ebaafba8bcb": "G(s)", "a7f769374ff336271954846bcddd9dc8": "1/p + 1/q = 1", "a7f7b1c9fe6a89c7c815b9ee34098ec2": "\\rho = \\tfrac{1}{2}, \\quad A_{20}^* = B_{20} = iAe^{i\\xi_0}, \\quad A_{1n} = A_{2n} = B_{1n} = B_{2n} = 0 \\quad (n \\neq 0). ", "a7f7cecc8178c2596612a0b6950ed125": "\\sin{18^\\circ}=\\frac{1}{1+\\sqrt{5}}.", "a7f7d014f24dfc4d82c0c022e0ebe372": "y_j = 1", "a7f7ed3c034f733e5914c85c1435e7e3": "f \\sim g\\ ", "a7f84ae167f25dc6ea531f1deb26c8b1": " P = -\\alpha\\;R \\frac{d}{dR} \\left (\\frac{\\gamma\\;_s}{R} \\right ) \\,\\! ", "a7f8bdbc70cd9035e1726d9bdf65c1cf": "I'_0", "a7f907de07ee9c89ef910d9b723dcf25": "\\{X_n\\}", "a7f98d3b07e8c7842c17a9ac12440c83": "1200 \\log_2 {3 \\over 2} \\ \\hbox{cents} \\approx 701.955 \\ \\hbox{cents}. ", "a7f9aadceca55666b4d1b1c46b822ad9": "r 0 \\\\\n \\end{cases}\n\\end{array}\n", "a8216082bce8178ba71feb1bbe84585b": " \\dots \\to F_n\\to F_{n-1} \\to\\dots \\to F_0\\to \\mathbf Z.", "a8218f54bd70e1332e3afaadd46c720b": "X_{k+1} = 2X_k - X_k A_{k+1} X_k,", "a8221cc499d785ccdb74f0d806a0026e": "x_2 = n_2 p_2", "a8222334c621ee446e5353e77bf46daf": "\\neg p \\wedge q, r \\rightarrow p, \\neg r \\rightarrow s", "a8222bf00c52ee6b42cdb800d8285935": "\\dot{Q}=\\frac{T_{surf}-T_{surr}}{\\left ( \\frac{1}{h_rA_{surf}} \\right )}", "a82244dd723ebc51e34bcf9f63e37669": "-5\\,X^4+X^2-3,", "a82256d572ca4e656797a58d6262cf14": "M_{HNO_3}=0.063,\\ M_{HF}=0.020,\\ M_{H_2O}=0.018.", "a82260a0ebf562f85a6a5332ef303f27": " \\dot{\\bold{r}} = \\dot r \\hat {\\mathbf r} + r \\dot \\theta \\hat {\\boldsymbol{\\theta}} ", "a822ac556ee9dc89d3bc58cf6c5387dc": "-\\nabla p +\\rho \\mathbf{g} = 0\\,.", "a822e328b92fdc273feb5dc177dfdb88": "\\phi = \\frac{1}{\\lambda}", "a822f64c2425e907e9f84658c64b7117": "\\cot\\frac{13\\pi}{60}=\\cot 39^\\circ=\\tfrac14\\left[(2+\\sqrt3)(3-\\sqrt5)-2\\right]\\left[2+\\sqrt{2(5+\\sqrt5)}\\right]\\,", "a8233707e68edf6c22fa6abbf3ce9740": "\\operatorname{let} f : E \\operatorname{in} L ", "a823c5f3de6a7c1d17988d5526d02f06": "F^{\\mu\\nu}=\\partial^{\\mu}A^{\\nu}-\\partial^{\\nu}A^{\\mu}.", "a823cc78b3364a54359275956b50e58d": "\\tau_{xy}:= \\sqrt{ (x^0 - y^0)^2 - (\\vec{x} - \\vec{y})^2}", "a8240cb50b61375fac39732776d03291": "\n\\int_a^b \\left[R(f_m\\ddot{f}_n-f_n\\ddot{f}_m)+\n\\frac{R}{Q}L(f_m\\dot{f}_n-f_n\\dot{f}_m)\\right] \\, dx\n+(\\lambda_n-\\lambda_m)\\int_a^b \\frac{R}{Q}f_mf_n \\, dx = 0\n", "a82443a884f371f65db092b58166ebf2": "\\delta(k_1, k_2)", "a824465d389da87750ffa5e99f1a5678": "\\mathrm{Ext}^{s,t}_{A}(\\mathbf{F}_p, \\mathbf{F}_p).", "a8249373debda47b684d505f8f536b0d": "(P_1 \\star P_0)", "a824d3da908a199cfa27bd67cdcf5114": "z = \\frac{x - \\text{mean}}{\\text{SE}} = [F(r) - F(\\rho_0)]\\sqrt{n - 3}", "a82507fc2e5409283b6efa6c361f3867": "\\langle x,y,z \\mid z=xyx^{-1}y^{-1}, xz=zx, yz=zy \\rangle\\,\\!", "a8251ffa9ba5cc5d022244188ba36b77": "\\phi (B)X_t= \\varepsilon_t \\,", "a8255fc55225e78e8d34304aa88ecdb7": "G_b", "a8257c25161f2ba4bc2ba1e00e4211f7": " r_i, ", "a825854638993ba3f453343afc65015b": "(\\bold{x},t) \\mapsto (\\bold{x}+\\bold{a},t+b)", "a825b8d1b4901c6793c609959e929da5": "F_n-1=2^{2^n}", "a825f313b9c9c9e0b2c0b8f991cb7cf9": "\\displaystyle{(ST_g x,ST_gy)=(PT_gx,T_gy)=(Px,y) =(Sx,Sy).}", "a8263e4d2e92aee091d84707850873ae": "q(0)=0", "a8268f592f4065ef2db9361c0a5d2994": "1-(1-r)^{n}", "a826a20faba085def2d83d52d5759a5d": "R_n(t)\\le\\int_{[a,t)} \\alpha(s) \\mu^{\\otimes n}(A_n(s,t))\\,\\mu(\\mathrm{d}s) +\\tilde R_n(t)", "a826ae7099e36b8ad910b33cc0394d3c": " P_{(n-1)} = M_{n-1,n} P_{(n)} = \\left[ \\begin{array}{c} x_{n-1} \\\\y_{n-1} \\\\z_{n-1} \\\\ 1 \\end{array} \\right] = \\left[ \\begin{array}{ccc|c} X_x & Y_x & Z_x & T_x \\\\ X_y & Y_y & Z_y & T_y \\\\ X_z & Y_z & Z_z & T_z \\\\ \n\\hline\n0 & 0 & 0 & 1 \\end{array}\\right]\n \\left[ \\begin{array}{c} x_{n} \\\\y_{n} \\\\z_{n} \\\\ 1 \\end{array} \\right]", "a826c83d88e5b28489401060489ce169": "(p \\lor q)", "a8278960b2db85cd60857f52a76b6b43": "S: x \\mapsto \\frac{x}{x+1}", "a827dcae7ab1e1bb1d353478969f744a": "P(X\\leq x)=P(X0, \\\\\n0 & \\text{otherwise},\n\\end{cases}\n", "a828e45afa2781b0a1115ecbd2d2dac0": "\\rho = \\frac{1}{h^n C} P,", "a828f39edba9fef6c6b4e9f7b51eecb8": "Radius\\ of\\ turn\\ in\\ feet = \\frac{velocity^2}{11.29 \\times \\tan(bank)}", "a82914d66da783a9534a1ab79d7604b8": "\\boldsymbol{\\omega}\\times\\mathbf{\\hat{r}}", "a8297180d9b0135a2ab65fab88e80352": "\\int\\frac{\\mathrm{d}x}{\\cos^n ax} = \\frac{\\sin ax}{a(n-1) \\cos^{n-1} ax} + \\frac{n-2}{n-1}\\int\\frac{\\mathrm{d}x}{\\cos^{n-2} ax} \\qquad\\mbox{(for }n>1\\mbox{)}\\,\\!", "a829d3f9d2968301849d2bc9557091bf": "\\mathbb{H}", "a829e2709a6af2f3696a00cd5dd6fbc4": " \\langle A \\rangle_\\psi = \\| A \\psi \\|^2", "a82a047e6b503a7af501fa1cec129852": "\\log\\bar{\\omega}=\\frac{1}{M}\\sum_\\alpha \\log\\omega_\\alpha,", "a82a1bf9ccdb03b165f98c91dab4cd88": " U_g \\pi(A) U_{g^{-1}} = \\pi(g \\cdot A). ", "a82ad95ed559e79368214db243f47233": "2^{nd}\\,\\!", "a82adf633caa5de7d8f79b3f81346071": " \\frac{\\partial \\mathbf{u}}{\\partial t}=0.", "a82b0888da0ae2597eee41848ac9d561": " B_{2n} + \\sum_{(p-1)|2n} \\frac1p", "a82b0bc119c775d83c15ed3dad5b8a0e": "m^2_\\mathrm{h} \\ll m^2", "a82bd1e6458eb30effd683721da66fc7": "\\begin{align}\n\\frac{p}{p^*} &= \\frac{1}{M}\\frac{1}{\\sqrt{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)}} \\\\\n\\frac{\\rho}{\\rho^*} &= \\frac{1}{M}\\sqrt{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)} \\\\\n\\frac{T}{T^*} &= \\frac{1}{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)} \\\\\n\\frac{V}{V^*} &= M\\frac{1}{\\sqrt{\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)}} \\\\\n\\frac{p_0}{p_0^*} &= \\frac{1}{M}\\left[\\left(\\frac{2}{\\gamma + 1}\\right)\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right)\\right]^\\frac{\\gamma + 1}{2\\left(\\gamma - 1\\right)}\n\\end{align} ", "a82be86dbe93dcd2e6c7464a9d08cd0a": "\\omega \\in \\Lambda^{k+1}(M)", "a82bf4d46c2db0cc17d265d789759f66": "1 + \\ln \\frac{\\sigma}{\\sqrt{2}} + \\frac{\\gamma_E}{2}", "a82c86374800e7b5064d7f5445e9dc72": "\\rho = \\rho_m (a)+\\rho_{de}(a),", "a82ca4ff68213b87937cf4118e0fbb1b": "Y=\\lambda X", "a82ca8083f1b20aee6ded5d1376a9d27": "f: \\mathbf R^n \\rightarrow \\mathbf R^n,", "a82cf30b9b63e7d06968e8d82410123c": "\\delta = \\alpha \\beta \\varepsilon", "a82d184ac35614dd5888c0372b603819": "\\delta Q=0", "a82d1e988bde7e3cea83d14c3a451a77": " \\mathcal{A}_f = \\mathcal{O} / \\langle x, y \\rangle = \\langle 1 \\rangle . ", "a82d45b7ead51941de223e2a61b13129": "\\left(\\frac{0.2518 \\mbox{ mol Cu}}{1}\\right)\\left(\\frac{2 \\mbox{ mol Ag}}{1 \\mbox{ mol Cu}}\\right) = 0.5036\\ \\text{mol Ag}", "a82d5df10acf22b60b0da67d7f7feeae": "2x H_n(x|q) = H_{n+1} (x|q) + (1-q^n) H_{n-1} (x|q)", "a82d9041f6d7cd3f34346379bfc38e8f": "A_\\alpha^{\\;\\;\\; IJ} = {1 \\over 2} \\big( \\omega_\\alpha^{\\;\\; IJ} - {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\; IJ} \\omega^{KL} \\big).", "a82deafb6771a43bfac50738b33d975f": "\\kappa(\\alpha X_1 + \\beta X_2) = \\alpha \\kappa(X_1) + \\beta \\kappa(X_2)", "a82e442b8b29f42aadf64b747513911a": "v_{\\rm e}", "a82e4a6b509d5ecac8a164555ddc693c": " m+1 ", "a82e6ac5380602b22fa1556ec3538f56": "C\\cdot U", "a82ebd8733758c3f028b618f5f6debdd": " L_1\\;=\\;\\pi G ", "a82f19557b99f920a2df534787c95aaa": " \\lambda = \\sqrt{\\frac {r \\times \\rho_{m}} {2 \\times \\rho_{i}}} ", "a82f7eb56ee9e514de3fbc2bd0c65ddb": " \\mu = (2 \\pi)^{-n/2} \\times \\mbox{Lebesgue measure}", "a82f9ebdb5e66a57dcf6975146342146": "p(\\vec{r}) = \\frac{1}{\\Omega(E_\\vec{r})}", "a82fc8947175f816292b3f94fc87a397": "Z_w", "a82fd6ac6a01ff389646e165ceef2fc4": "\\lang \\epsilon_i|\\epsilon_j\\rang \\approx \\delta_{ij}", "a8301ed4544eda295170a4b40d8717ad": "\\Delta^{0,Y}_n", "a8309073a0eba3349bcc3f4f888539de": "\\mathcal{V}(x) =\\left\\{ V \\supset B~:~ B \\in \\mathcal{B}(x)\\right\\}", "a83093d6e0b28aad69bb8da2748d66fb": "\\frac{n \\pi x_0}{L}", "a830ecfee56933b2b4763857d3e4ab75": "c = -ax_1 - by_1", "a831646c1c64c1acfa518afc3bddb018": "e^{i \\pi} + 1 = 0\\,\\!", "a831874ed93c47bcec2a5dc6d565e45b": "E\\!\\left[X_n^2 \\right ] = E\\!\\left[\\sum_{v\\in T_n} \\sum_{u\\in T_n}1_{v\\in K}\\,1_{u\\in K}\\right] = \\sum_{v\\in T_n} \\sum_{u\\in T_n} P(v,u\\in K).", "a831a64b2c2bcb3024fdf6ac2756747c": "g(x) =\nP(y|x)", "a831aa4659bda1665429b4cf36034e66": "y = \\{y_1, y_2, \\dots, y_N\\}", "a83207cf4f681f1748aba5981cfdd2f3": "\\mathfrak{S}\\left(R(X,Y)Z\\right) = \\mathfrak{S}\\left(T(T(X,Y),Z)+(\\nabla_XT)(Y,Z)\\right)", "a8322deee1209880b270a33b3ccce711": "\\dot{\\mathbf{x}}(t) = \\left(A + B K \\left(I - D K\\right)^{-1} C \\right) \\mathbf{x}(t)", "a8325e54cc839bcaff61d32a121a0b4d": " \\hat{\\textbf{a}}_{k} \\leftarrow \\hat{\\textbf{a}}_{k} + ( \\gamma\\ 2 / [ \\Delta \\textrm{T} ]^\\textrm{2} )\\ \\textbf{r}_{k} ", "a83289779d25ef816e275b2a95794eee": "I \\frac{d\\theta}{dt}=h\\int_0^t F(t)\\,dt = h m_g V_g(t)=h m_b V_b(t)", "a832c2dab2d0aad33ee7af910780923f": "c=\\sqrt{\\frac{K}{\\rho}}.", "a83339d93e945318edc65707daaf2e05": "{\\mathbf{}}F_r(t)=H(t)\\left( A(t)-P(t)C'(t)W^{-1}(t)\nC(t)-B(t)R^{-1}(t)B'(t)S(t) \\right)G(t)+\\dot{H}(t)G'(t),", "a833592cef067499a9beb668567b05a2": "\\frac{d}{dx}\\, \\operatorname{arsinh}\\,x =\\frac{1}{\\sqrt{x^{2}+1}}", "a83377b5b107e1b967fde1db93b43b04": "^1_{n+1}", "a8338249fbb73529b4df9e8b18f74752": "\\hat{\\beta}_{FD} = (\\Delta X'\\Delta X)^{-1}\\Delta X' \\Delta y", "a833c697705a917f0c3ab9c3433ae11b": "y^2 + h(x) y = f(x)", "a833f1514e2876b870402c5fd739b788": "s^i \\in S", "a833f3593183d5963ab5349441234ee9": " \\mathbf{r} = r \\mathbf{e}_r . ", "a833fa90951f8709197dbe47052a6415": " F(x) ", "a834697114c57ca35f513d60710a81ef": "\\frac {L} {E} = \\frac {K} {C}", "a8346bf6109c8c1583b84c0441993d9e": "2 + 5 + 8 + 11 + 14", "a834719e7977b3848001c3f4b1ebdd11": "\\left[T_a, T_b \\right] = i \\sum_{c=1}^8{f_{abc} T_c} \\,", "a83482d152e434d32e36ccf0fd8c987f": "\n{\\rm E}[\\hat g]\\,\\,\\, \\approx \\,\\,\\,\\,k\\alpha \\left( {\\mu _\\theta } \\right)\\,\\,\\, + \\,\\,\\,{1 \\over 2}\\,\\,{k \\over {32}}\\left[ {9\\cos \\left( {\\mu _\\theta } \\right)\\,\\,\\, - \\,\\,\\,\\cos \\left( {2\\mu _\\theta } \\right)} \\right]\\sigma _\\theta ^2", "a83485b8d55a45ab22336ad8e45a8040": "p(x) \\equiv 0,", "a83510b3938c41c8c64bba8b1eea2a4d": " A[i]= j \\Leftrightarrow A^{-1}[j]= i ", "a8358933efd7f7ff2e4edfc40132e552": "\\neg x", "a8359fd9810482adc6a06c3d0b550a7c": "\\nu + p - n > 2", "a835c0c991448e48ea5ed841467a8691": " \\Delta \\mathbf{v} ", "a83602e8d6904af5072c41a7e7e19562": "\\mathbf{m} = q_m \\mathbf{a}\\,\\!", "a8361f08b85b42f4a0b565e01fc745a5": "(n - 1)!\\ \\equiv\\ 0 \\ ({\\rm mod}\\ n)", "a8366f59e37d9cf70ce2e7d6fcffc148": "K^\\ominus=\\frac{[R] ^\\rho [S]^\\sigma ... } {[A]^\\alpha [B]^\\beta ...} \n\\times \\frac{{\\gamma_R}^\\rho {\\gamma_S}^\\sigma ...}{{\\gamma_A}^\\alpha {\\gamma_B}^\\beta ...}\n\\times \\frac{\\left ({c^\\ominus_A}\\right ) ^\\alpha \\left({c^\\ominus_B}\\right)^\\beta ...}\n{\\left ({c^\\ominus_R}\\right ) ^\\rho \\left({c^\\ominus_S}\\right)^\\sigma ...} = Q^E \\Gamma C^0", "a8366f5fef9b837a7ca8d34e714451a6": "1^2+4^2 = 17", "a8367fc9fdaf07c0fde8ff948b32523d": "l\\geq 0", "a83684ee0b80c5232a5696d754b786da": "D = DO_{sat} - DO", "a836b469012bc772708b4c4f2727f2c5": "\n \\oint_C g(z)dz\n =\\oint_C \\left(1-\\frac{1}{z-z_1}-\\frac{1}{z-z_2}\\right)dz\n =0-2\\pi i-2\\pi i\n =-4\\pi i\n", "a836eb486fee898fbee1c88e2f777000": " \nQ(t) = (Q_1(t), Q_2(t), ..., Q_N(t)) \n", "a837335481bceade8861ee3da984a012": "X = x", "a837518ae4e4d101681c74bca319d3d0": "13 = 8 \\cdot 1 + 5", "a837575fb791bd7660f25b0a993e55d1": "\\scriptstyle2\\,\\frac13", "a83774164cad1528622bad9c7b974525": "M(l)", "a8377fc7e1f0a8eb4c19537de79a61e9": "\\displaystyle T=r_ar_b=rr_c", "a837984dc8ba89792a0edef353b8e1e8": "\\Pi \\, \\pi \\, \\varpi \\,", "a837e4474e6e81e8b67787f94d80508a": "p(\\lambda)= \\lambda^n \\exp \\left( -\\operatorname{tr} \\sum_{m=1}^\\infty {({A\\over\\lambda})^m \\over m} \\right),", "a837fcd26c3eab8e3c8e91b448b9e81e": "{\\color{Blue}~2.30}", "a837ff2b957aa47e875c04c4bf0345e7": "\\begin{matrix} \\frac{velocity \\;vector \\;at \\;a \\;given \\;latitude} {velocity \\;vector \\;at \\;equator} \\end{matrix}", "a838614d55e67eedd3104804bb212ff3": "\\lambda_{i_1}, \\dots \\lambda_{i_n}", "a83889577e67aca78253272a68541fb7": "\\lim_{x \\to c}f(x)=\\lim_{x \\to c}g(x)=0 \\text{ or } \\pm\\infty", "a838a324d15e4066aa02beaefa7e2b12": "\\lim_{x \\to 0} \\frac{\\ln(1+x)}{x} = 1 \\,", "a838e3742f77259f7402fd4e306d36cd": "\\dim Z_r = \\dim \\mathbf{Gr}(r,m) + nr = r(m-r) + nr", "a838eb1f7b4b5350de546130d99782e3": "\n\\left[ \\begin{matrix} t \\\\ x \\\\ y \\\\ z \\end{matrix} \\right]\n\\rightarrow\n\\left[ \\begin{matrix} t \\\\ x \\\\ y \\\\ z \\end{matrix} \\right]\n+ Re(\\alpha) \\;\n\\left[ \\begin{matrix} x \\\\ t-z \\\\ 0 \\\\ x \\end{matrix} \\right]\n+ Im(\\alpha) \\;\n\\left[ \\begin{matrix} y \\\\ 0 \\\\ z-t \\\\ y \\end{matrix} \\right]\n+ \\frac{\\vert\\alpha\\vert^2}{2} \\;\n\\left[ \\begin{matrix} t-z \\\\ 0 \\\\ 0 \\\\ t-z \\end{matrix} \\right].\n", "a838efeefa96624e2a8110dd059d0cd4": "f(x) = x^3 + x", "a8390a8b73089adac1e1d0cf059be458": " \\boldsymbol{\\phi} ", "a83918871d496d3bece012805e317328": "\\textstyle W ", "a8397827e3e5a9ffa078492e5b78b18c": "q^{2t}", "a8398b9ad4cf4f60cae7e82290bff0d3": "ES_{1.0}", "a839b1500c95a31fd962ed131d75a1f4": "(2,0,0)", "a83a44ac3cd22f8168aad1ef7b3b81f0": "L_{0}^{'}=x_{2}^{'}-x_{1}^{'}", "a83a818f7791eea85f6a33a033b1d44f": "D>2", "a83aad230c25e77beaed3c52cc8e8b21": "x_{j,m}=\\delta_{jm}\\,\\!", "a83ac6096ed5b1caeca97fd2ce5e2d1f": "\\alpha_r m_{i_r}= \\sum_j \\beta_{rj} m_j \\text{ for some } m_j>0 \\text{ and all } r , ", "a83ad7df7ccb6b1d0037fc79277eeaf7": " |\\text{GTE}| \\le \\frac{hM}{2L}(e^{L(t-t_0)}-1) \\qquad \\qquad ", "a83b1457645e18c5d534783b816569d2": "\nf(x) = h^{-1}(h(x)+1) ,\n", "a83b1857d75505e935c2fbdd760ba09c": "\\langle u, u' \\rangle", "a83b3b5c938670abfd4613e3f2255f8e": "V_\\max - V_\\max \\cfrac{[I]}{[I]+K_i} ", "a83b55bddc9aefbc14fa227bc471e5b0": "\\left\\{ \\Pi_{\\rho,\\delta}\n^{n},I-\\Pi_{\\rho,\\delta}^{n}\\right\\} ", "a83b5932a0486a31f60663e141420f18": "\\quad\\sum_{i=1}^n\\frac{x_i^n}{\\Pi_i(x_1,\\ldots,x_n)}=\\sum_{i=1}^n x_i", "a83b91ccb86d845f8314f2ac8f00db9b": "\\Gamma^{[l]i_l}_{\\alpha_{l-1}\\alpha_{l}}, l=1,2k-1", "a83ba07c54bafa38b351b31b7d8b1423": "\\mu_{eff}=g \\sqrt{\\vec{J}(\\vec{J}+1)}; g={3 \\over 2} +\\frac{\\vec{S}(\\vec{S}+1)-\\vec{L}(\\vec{L}+1)}{2 \\vec{J}(\\vec{J}+1)}", "a83bc3c09fc90779b0d3fe780ac110f2": "x\\in\\ker(T\\upharpoonright S-\\alpha)", "a83bc93e3827dadd84f5f0ba77911346": "R_m=\\frac{{}_{2}}{\\frac{1}{M}+\\frac{1}{N}}\\,\\!", "a83c2819175d7f331a08e42aec05af25": "\n\\sup_f \\inf_g \\iint K\\,df\\,dg=\\frac{1}{3}\n", "a83c89cf0aa58c24a2ed7e80cbceafae": " E(\\varphi \\setminus \\alpha) = E(\\varphi, \\sin \\alpha) = \\int_0^\\varphi \\sqrt{1-(\\sin \\theta \\sin \\alpha)^2} \\,d\\theta.", "a83ce44201f60861c543a79c5fe03da8": "R \\approx \\frac{8.3 \\ \\mathrm{J}}{\\mathrm{K} \\cdot \\mathrm{mol}}", "a83d19ce4bd49c63714afb8437080c8f": " \\bigg| \\mathbb{P} ( a \\le c_1 X_1+\\dots+c_n X_n \\le b ) - \\frac1{\\sqrt{2\\pi}} \\int_a^b \\mathrm{e}^{-t^2/2} \\, \\mathrm{d} t \\bigg| \\le C ( c_1^4+\\dots+c_n^4 ). ", "a83d1fcabb7e2a11f3ae268bbef4d161": "y \\in U", "a83d6b8ea6358c4f5f7a8d5505ecce4c": "\\{(i,a) \\mid a \\in A_i\\}", "a83d878757422c9e6b02dacb8105cbe4": "(-A)", "a83e012e6f306259e642240431c819bc": "-177\\pm 8", "a83e0a62e2216a8cbf5c5f2d0dd674ff": "{\\bold x}\\rightarrow X=\\left(\\begin{matrix}x_3&x_1-ix_2\\\\x_1+ix_2&-x_3\\end{matrix}\\right).", "a83e0e0eadb5417a3de2f4b2e87526a2": " \\frac {\\alpha^k d^{2k}} {k!} \\left( {\\frac{w}{d}} \\right)^{k-2}", "a83e2dee338accc523b3f423c2636694": "\\cos\\angle K_iOK_j = 1-2\\sin^2\\frac{\\angle K_iOK_j}{2}=1-2\\cdot \\left(\\frac{\\overline{K_iK_j}}{2R}\\right)^2 = 1 - \\frac{\\overline{K_iK_j}^2}{2R^2}", "a83e5a85faac12e58878220fc65f4db1": "\\textstyle v(x) = x^ib(x)(x^{j-1}+1).", "a83e5d3fa85aa845bbf1d04abf86b749": "\\textstyle \\alpha_1+\\alpha_2+\\cdots+\\alpha_n", "a83e7d89e6a38a653aa7a9831d2ff47c": "\\pm \\sqrt{-1}", "a83ef51fa84728342a20e1bac05bf8bb": "(**)", "a83f407dd53f474dfd3cff75b51dbfce": "dz = e^{i\\theta} dr + ire^{i\\theta} d\\theta\\!\\,", "a83f72dab9bbf15475a276db7e7321ce": " S - S_M = -{1 \\over 8\\pi G} \\int[\\beta (\\nabla\\phi)^2 + \\alpha (\\nabla \\hat{\\phi})^2 - a_0^2 \\mathcal{M}((\\nabla\\phi - \\nabla \\hat{\\phi})^2 / a_0^2)] d^4x ", "a83fb6d0a053bc878647075f4109b1a6": "dU/dt", "a83fba7329bfe144624cafc0e9433e6a": "=-\\frac{1}{\\eta^{\\mu\\mu}}\\operatorname{tr}(\\gamma^\\mu\\gamma^\\nu\\gamma^\\mu)", "a83fd5aef579b6f685866b7d4a6d9828": "\\mathsf{2Ce + 6H_2O \\ \\xrightarrow{90^oC}\\ 2Ce(OH)_3\\downarrow + 3H_2\\uparrow }", "a83fec1ed83ab1b5e880fa1ccdc0d945": "\\alpha-1/2, -\\beta-\\lambda\\mu^2/2, \\lambda\\mu, -\\lambda/2", "a8400ff94cd7e3feed57c95e0d6d25f8": "E_T=\\sum_{\\sigma\\in Q_T}\\epsilon(\\sigma)\\sigma(T)", "a8401a57774c923ba24e89adffa4f73b": "\n\\begin{align}\nb_N &= b_0 + \\frac{1}{2} \\operatorname{E}_\\mu \\left[\\sum_{n=1}^N (x_n-\\mu)^2 + \\lambda_0(\\mu - \\mu_0)^2\\right] \\\\\n &= b_0 + \\frac{1}{2} \\operatorname{E}_\\mu \\left[ (\\lambda_0+N)\\mu^2 -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\mu + (\\textstyle\\sum_{n=1}^N x_n^2) + \\lambda_0\\mu_0^2 \\right] \\\\\n &= b_0 + \\frac{1}{2} \\left[ (\\lambda_0+N)\\operatorname{E}_\\mu[\\mu^2] -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\operatorname{E}_\\mu[\\mu] + (\\textstyle\\sum_{n=1}^N x_n^2) + \\lambda_0\\mu_0^2 \\right] \\\\\n &= b_0 + \\frac{1}{2} \\left[ (\\lambda_0+N)(\\lambda_N^{-1} + \\mu_N^2) -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\mu_N + (\\textstyle\\sum_{n=1}^N x_n^2) + \\lambda_0\\mu_0^2 \\right] \\\\\n\\end{align}\n", "a840717d5b3e49e316fff62b0807c365": "h(\\mathbf{Y}) \\leq h(\\mathbf{X}) + \\int f(x) \\log \\left\\vert \\frac{\\partial m}{\\partial x} \\right\\vert dx", "a8407d37d6a664167304613dd9e66abf": "W^{3}", "a840e4877aa5e1678d8486a08ac866af": "\\kappa_{2k}=\\left.2\\mu\\right.", "a840f974d1131719a900083f7c411946": " U_E = \\frac{1}{2}QV = \\frac{1}{2} CV^2 = \\frac{Q^2}{2C}", "a840fb0a615b5f9986fb25018a57fc21": "D\\phi = \\begin{bmatrix}\n\\frac{\\partial\\phi^1}{\\partial x^1}&\\frac{\\partial\\phi^1}{\\partial x^2}&\\dots&\\frac{\\partial\\phi^1}{\\partial x^n}\\\\[1ex]\n\\frac{\\partial\\phi^2}{\\partial x^1}&\\frac{\\partial\\phi^2}{\\partial x^2}&\\dots&\\frac{\\partial\\phi^2}{\\partial x^n}\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\n\\frac{\\partial\\phi^m}{\\partial x^1}&\\frac{\\partial\\phi^m}{\\partial x^2}&\\dots&\\frac{\\partial\\phi^m}{\\partial x^n}\\\\\n\\end{bmatrix}.\n", "a8411366d99aeb371b0fe959d7287a54": " m = M + {\\mu}.\\!\\,", "a84118caaae8f5118ea05a4e64c0009e": "S_1\\approx \\infty", "a8412b5659847d7c2450e4e318bf4d11": "\n\\Sigma\n= \\begin{bmatrix}\n \\mathrm{E}[(X_1 - \\mu_1)(X_1 - \\mu_1)] & \\mathrm{E}[(X_1 - \\mu_1)(X_2 - \\mu_2)] & \\cdots & \\mathrm{E}[(X_1 - \\mu_1)(X_n - \\mu_n)] \\\\ \\\\\n \\mathrm{E}[(X_2 - \\mu_2)(X_1 - \\mu_1)] & \\mathrm{E}[(X_2 - \\mu_2)(X_2 - \\mu_2)] & \\cdots & \\mathrm{E}[(X_2 - \\mu_2)(X_n - \\mu_n)] \\\\ \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\\\\n \\mathrm{E}[(X_n - \\mu_n)(X_1 - \\mu_1)] & \\mathrm{E}[(X_n - \\mu_n)(X_2 - \\mu_2)] & \\cdots & \\mathrm{E}[(X_n - \\mu_n)(X_n - \\mu_n)]\n\\end{bmatrix}.\n", "a8415d76bc73c1adbec857e67384be3f": "\\{(0,1),(1,0),(1,1)\\}", "a84171326a2788d608049b1a7882670c": "c_n \\rightarrow 0", "a8419adc80d2bd4e3e07c7f6512d3863": "\\mu_2= 2\\sigma^2+\\nu^2-(\\pi\\sigma^2/2)\\,L^2_{1/2}(-\\nu^2/2\\sigma^2) .", "a841cb8c36b7c2e2e3c175323b674eca": "x^{\\alpha}e^{-x}\\,", "a84203149def80dc8b3fec2e73619842": " (1)\\quad e \\cdot v = v ", "a8422facccaa6833941a2b8e3842389f": "T_{SL}=", "a842380f1081324b47a87c288b70951b": "P(Y) = P(y_{32} \\mathcal{k} y_{31} \\mathcal{k} \\dots \\mathcal{k} y_1) = y_{\\varphi(32)} \\mathcal{k} y_{\\varphi(31)} \\mathcal{k} \\dots \\mathcal{k} y_{\\varphi(1)}", "a84248bd58f06c3f620adfb43706cd8b": "H \\cdot H = d\\ ", "a8425091f0cef0055d68cbba62b7bfd1": "\\frac{3^3 2^1 + 3^2 2^2 + 3^1 2^4 + 3^0 2^5}{2^7 - 3^4} = \\frac{170}{47}", "a842a070db74987a18299e4fa39c5573": "\\begin{cases} f : H \\to \\mathbf{R} \\\\ f(x) = \\langle T x, x \\rangle \\end{cases}", "a842a194293d4ed4eb1bca6dd6e0c8eb": "\\partial\\sigma", "a8430975224656439916831b6be3289b": "\\int_{0}^T | \\sum_{n=0}^N\\pi_n(t)| \\left[|r(t)|dt + dA(t) \\right] < \\infty ", "a8434652d887decc4d6d028851d101c3": "1\\le k \\le K", "a84366f44cffcb425b90a9fa03f125e4": "E_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\varepsilon _{o}\\varepsilon _{r}}(A \\ e^{jk_{x\\varepsilon }w}+B \\ e^{-jk_{x\\varepsilon }w})e^{-jk_{xo}(x+w)}sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ (47) ", "a8439dfe1ec4ba8ceb87b80dcf6e4980": "\\partial(u\\cdot v)=\\partial u\\cdot v+u\\cdot\\partial v\\,.", "a843ad3a4924f8dfe9bbc9c525efdfe3": " m \\geq 1", "a843ae4931302d9a9d70df7a81617d0e": "M^m", "a84400d7ffd2da87083ecd4a8f77aec4": "b_i(x)^{q^d-1}=1", "a84438246c7db614da32f20e16b5ef38": "\\psi^\\dagger ", "a8448ef37d978800a2479ec8d32a45b8": "c_s=\\sqrt{\\frac{\\gamma R T}{M}}=\\sqrt{\\frac{\\gamma k_BT}{m}}", "a844a4603202b43812ac6593ad9db722": "Z_{TE}", "a844c8fc675fbbffc756521aa1cf21c4": "\\alpha = \\frac{Q_{tc}^2}{Q_{ts}^2}-1", "a844f92f7f41e46c263c37524717db76": "\\frac{}{}L", "a845891899be689f18a376e3fd3c8432": "K(n,S)=K^n", "a845cede7f1df438179eafdc93f7c8d1": "k = e^{\\pm i\\theta} \\neq 1", "a845e447dc73e298f6e226e434a37e18": "\\epsilon_\\| = \\frac{1}{2}mv_\\|^2", "a84612e0dcd9170adc8876300fff1645": "c_p = \\sqrt{gh} \\qquad \\scriptstyle \\text{(shallow water),}\\,", "a8463da462f20ed93061afd468157a9e": "\\begin{align}\nK_1 &= \\{ z_1=x_1+iy_1 | a_2 < x_1 < a_3, b_1 < y_1 < b_2\\} \\\\\nK_1' &= \\{ z_1=x_1+iy_1 | a_1 < x_1 < a_3, b_1 < y_1 < b_2\\} \\\\\nK_1'' &= \\{ z_1=x_1+iy_1 | a_2 < x_1 < a_4, b_1 < y_1 < b_2\\}\n\\end{align}", "a8467cb3cc94959c2fb3857863127c02": "r = O(\\log N)", "a8469b3a99e7309ad5ee95f4d72a436d": "R_{K\\_i',k}", "a846a9371385b9d0e0d98273eb3c5ee2": "k\\colon K\\rightarrow G", "a8473e2cc2c84e90126d98fb64a50cda": "\\Delta \\omega\\propto A^2", "a8474790bd9b28e6ca9e9b398b37b8a3": "v(x,x)g(x\\mid x)-g(x\\mid x)W^{A}(x,x)+G(x\\mid x)W_{1}^{A}(x,x)=0,", "a8475bb6826360e1617a693c295a318d": "~\\Phi_{20}(x) = x^8 - x^6 + x^4 - x^2 + 1", "a8477f7291c33030dd5e2e23c83e63c1": " \\frac{\\partial}{\\partial t} S(q_i,t) = H\\left(q_i,\\frac{\\partial S}{\\partial q_i},t \\right) \\,\\!", "a8479e89666ae75863c649671c051d16": "\\langle u,v, \\langle u,w,x \\rangle\\rangle = \\langle u,x, \\langle w,u,v \\rangle\\rangle", "a847bcf226a2554b49c73b7a1b3a98a5": "y = k\\pm a \\; \\cos\\left(\\arctan\\left(\\frac{b}{a}\\right)\\right)", "a847d6f0e52340bd2a0c411ffb879f15": "\n\\frac{2t}{e^t+1}=\\sum_{n=1}^{\\infty} G_n\\frac{t^n}{n!}.\n", "a847dd379f9fcff16f9c0d429a6d835d": "\\exist^{\\le n}", "a84812d8b265d97037d01cbea83b6983": "\\zeta(h_1,\\cdots,h_{n-k})=\\zeta(i_1,\\cdots,i_k)", "a8489ee490fae6cedcfdd8f3d063c230": "S[\\Lambda_{boost}] = \\left(\n\\begin{array}{cc}\ne^{+\\chi\\cdot\\sigma / 2}&0\\\\\n0&e^{-\\chi\\cdot\\sigma / 2}\n\\end{array}\n\\right)", "a848ac00278113fb1016fd3789f3ff5a": "r_{12} = a\\sec\\alpha\\,|\\phi_1 - \\phi_2| = a\\,\\sec\\alpha\\;\\Delta\\phi.", "a8490b524bcbbf52e099692bcbdbbe8b": " x^2=-1", "a8490f01ec3689ae2fd1a3d7cb07c5f8": "\\scriptstyle\\overline{X})", "a8494c59b108b6fb0f6dd5a63eab2628": "{V_{D}} = \\frac{\\mathrm{total \\ amount \\ of \\ drug \\ in \\ the \\ body \\ per \\ body \\ weight \\ unit \\ (Kg) \\ (i.e. Dose)}}{\\mathrm{drug \\ blood \\ concentration}}", "a8499b8bff15b7cddb8d0a79dec9222c": "\\displaystyle{}+ \\iiint_{\\scriptstyle V} \\rho \\mathbf{f}_\\text{body} \\, dV + \\mathbf{F}_\\text{surf}", "a8499f85d52ffe319ce8e0c1272bb2b8": " \\left \\| u_m - u \\right \\|_{W^{k,p}(\\Omega)} \\to 0.", "a84a9e13ae9772a48a4d647b588347b4": "U=\\pi (V)\\,", "a84b0aae4c1ddb409cd63800d2244472": " A = B ", "a84b70536b1221c38da94c24766f975a": "\n\\sigma(X) = \\sqrt{E[(X-E(X))^2]} = \\sqrt{E[X^2] - (E[X])^2}.\n", "a84ba9fd83c060ae8ee498521ce64da3": "\\varepsilon_{Hb} \\,", "a84bec58d0566338c981aae3b2eca069": "H(s) = \\frac{1}{(s+\\alpha)(s+\\beta)}.", "a84cdc580949c5e3dbf113254bf804f0": "P_F, P_G", "a84cecd88f1706c2dfec7596cfb94561": "(H,2\\mathrm{Im}\\langle\\cdot,\\cdot\\rangle)", "a84d14db156a366b03ace3d2a686cf49": "x\\in L \\Leftrightarrow\\exists y\\in\\Sigma^{*}", "a84d6ecca2a358ef88cd3329320c244d": "\\lbrack\\mathbf b\\rbrack = \\lbrack\\mathbf b\\rbrack_2 \\cdot \\lbrack\\mathbf b\\rbrack_1", "a84dc9f17432e5d00f0dd588dafa0f4a": "n^i", "a84df34bfb1ef3c646a187bd2a21f72f": "u{1 \\over 2}(t+{1\\over t})+v{1 \\over 2i}(t-{1\\over t})=w", "a84dff37b7a2a8229f8f641bcdcff2c0": "i\\hbar \\frac{\\partial \\psi (\\mathbf{r},t)}{\\partial t}=D_\\alpha (-\\hbar\n^2\\Delta )^{\\alpha /2}\\psi (\\mathbf{r},t)+V(\\mathbf{r},t)\\psi (\\mathbf{r},t)", "a84e6df57867cf50f6a669120d7c33c0": " f(x,\\boldsymbol\\beta)= \\frac{\\beta_1 x}{\\beta_2 + x} ", "a84e6e1f23f110548a971a3afa3a47bb": "\n\\times \\sum_{m_A=-\\ell_A}^{\\ell_A} R^{m_A}_{\\ell_A}(\\mathbf{r}_{Ai})\nR^{M-m_A}_{L-\\ell_A}(\\mathbf{r}_{Bj})\\;\n\\langle \\ell_A, m_A; L-\\ell_A, M-m_A| L M \\rangle,\n", "a84ecd8b25e8e88de2772e2841764220": "\n\\frac{R}{r} = 1 + \\frac{2 \\sin\\theta}{1 - \\sin\\theta} = \\frac{1 + \\sin\\theta}{1 - \\sin\\theta} = \\left[ \\sec \\theta + \\tan \\theta \\right]^{2}\n", "a84eea4b5f27951b2dd3af745c38efea": "\\begin{align}\n\\hat{G}_X &= \\prod_{i=1}^{N} (X_i)^{\\frac{1}{N}} \\\\\n\\hat{G}_{(1-X)} &= \\prod_{i=1}^{N} (1-X_i)^{\\frac{1}{N}}\n\\end{align}", "a84f37fba3620b95ca196e593ba1cb2f": " \\operatorname{build-param-lists}[g\\ m\\ p, D, V, T_3] \\and \\operatorname{build-param-lists}[n, D, V, K_3] ", "a84f4b77ebfa8488531d4197881ffcc3": "V_n(k)", "a84f65bb2a4ded28a6f21d4080b99010": "\\sum_{k=0}^\\infty \\frac{1}{(z + k)^2} \\sim \\underbrace{\\int_0^\\infty\\frac{1}{(z + k)^2}\\,dk}_{= \\frac{1}{z}} + \\frac{1}{2z^2} + \\sum_{t = 1}^\\infty \\frac{B_{2t}}{z^{2t + 1}}", "a84f92dc64c31abde95cdedf221c16f2": "\\sigma_{SS}=E(\\sin^2\\theta)-E(\\sin\\theta)^2 =\\frac{1}{2}\\left(1 - C_2 - 2S_1^2\\right)", "a84faa313638c439fb88c697c48e0ab7": "\\operatorname{det} H L(x, y; t) = (L_{xx} L_{yy} - L_{xy}^2).", "a84fc1845721dbf301410b81dc989b27": "c=\\langle p'q'^{-1}|\\alpha,\\beta\\rangle", "a84fef64312d60edefcd3936bf0f8701": "\\epsilon^{0123} = \\eta^{0I} \\eta^{1J} \\eta^{2K} \\eta^{3L} \\epsilon_{IJKL}", "a850403b06d8260493350142c0f301e2": "e^{\\pi \\sqrt{163}}=640320^3+744+O\\left(e^{-\\pi \\sqrt{163}}\\right)", "a85069a56ea22613d5bd5d8e742c37ad": "\\triangle \\propto \\frac{iM}{A\\rho}", "a850a0e809073fd3f011731541062105": "R/\\sqrt{N}", "a850c8b9b2a6034dea96cf6c36fb1b0e": "\\lambda n.\\lambda f.\\lambda x.f\\ (n\\ f\\ x) ", "a850dd47c03141e55808cc32696948ba": "\\mathrm{slog}_b(z) \\approx \\begin{cases}\n\\mathrm{slog}_b(b^z) - 1 & \\text{if } z \\le 0 \\\\\n-1 + z & \\text{if } 0 < z \\le 1 \\\\\n\\mathrm{slog}_b(\\log_b(z)) + 1 & \\text{if } 1 < z \\\\\n\\end{cases}", "a850ddbcd70880670c3c5b8e4390d4f8": "F_x", "a850ea1f26bed21f2ccd8a43b63fcb92": "\\bigl|f_k(x)-1\\bigr|\\leqslant\\sum_{n=1}^\\infty\\frac C{k^n}=C\\frac{1/k}{1-1/k}=\\frac C{k-1}.", "a850ebd55705da5ca9dda0f28b1d637b": "\n I_1 = \\lambda_1^2+\\lambda_2^2+\\lambda_3^2 = 2~\\lambda^2 + \\cfrac{1}{\\lambda^4} ~.\n ", "a851012497624e4e119f82fdc9342f35": " M_A \\ddot{q}:=(M+A^+A)\\ddot{q} = Q + A^+b := Q_b, ", "a8510bfed6f52964096f020eafefeb00": "S = \\operatorname{End}_R(U)", "a85180b0ec06642618489c808f562b8c": "\\textstyle\\varphi\\,\\!", "a851865368561158618177d95c7a0b62": "\\begin{align}\n &\\frac{\\mathrm{d}u}{\\mathrm{d}\\theta }=\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\frac{1}{r}\\right)\\frac{\\mathrm{d}t}{\\mathrm{d}\\theta }=-\\frac{{\\dot{r}}}{r^{2}\\dot{\\theta }}=-\\frac{{\\dot{r}}}{h} \\\\ \n & \\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}=-\\frac{1}{h}\\frac{\\mathrm{d}\\dot{r}}{\\mathrm{d}t}\\frac{\\mathrm{d}t}{\\mathrm{d}\\theta }=-\\frac{{\\ddot{r}}}{h\\dot{\\theta }}=-\\frac{{\\ddot{r}}}{h^{2}u^{2}} \\\\\n\\end{align}\n", "a851b5a0cc89a65669b6937ca5851551": "\\nabla = \\mathbf{e}_\\text{x}\\frac{\\partial}{\\partial x} + \\mathbf{e}_\\text{y}\\frac{\\partial}{\\partial y} + \\mathbf{e}_\\text{z}\\frac{\\partial}{\\partial z} ", "a851bd0adab1b4fc8affb02bc0fe363f": "\\mu_i = m_i(1-e^{\\omega_i\\theta})", "a851f11180109338d95dca175f3186b3": " D = \\frac{\\mu_q \\, p}{q \\frac{dp}{d\\eta} } = \\mu_p p \\frac{\\partial V}{\\partial p} = \\mu_p \\frac{k_B T}{q}", "a8521edd6bb426b3f2a276410912883e": "\\omega^{2}(t)", "a852b0caaea6f7235087e36c24d0cc1b": "\\scriptstyle X \\;=\\; \\gamma(0)", "a852c64c932709e0f389ab475933db44": "\n\\begin{array}{lllllllllll}\n \\Lambda(\\alpha^{i+1}) &=& \\lambda_0 &+& \\lambda_1 (\\alpha^{i+1}) &+& \\lambda_2 (\\alpha^{i+1})^2 &+& \\cdots &+& \\lambda_t (\\alpha^{i+1})^t \\\\\n &=& \\lambda_0 &+& \\lambda_1 (\\alpha^i)\\,\\alpha &+& \\lambda_2 (\\alpha^i)^2\\,\\alpha^2 &+& \\cdots &+& \\lambda_t (\\alpha^i)^t\\,\\alpha^t \\\\\n &=& \\gamma_{0,i} &+& \\gamma_{1,i}\\,\\alpha &+& \\gamma_{2,i}\\,\\alpha^2 &+& \\cdots &+& \\gamma_{t,i}\\,\\alpha^t \\\\\n &\\triangleq& \\gamma_{0,i+1} &+& \\gamma_{1,i+1} &+& \\gamma_{2,i+1} &+& \\cdots &+& \\gamma_{t,i+1}\n\\end{array}\n", "a852dbb37ad98f369b0cec656a41a781": "\\mathbf{V}(t) = \\int_0^{t} \\mathbf{A} dt = \\mathbf{A}t + \\mathbf{V}_0.", "a8531f5e71e0f6978c28b9bbbecf5e9b": " p(x) = {{\\alpha \\lambda^\\alpha} \\over { (x+\\lambda)^{\\alpha+1}}}.", "a8533bf6b6a045218adfbdb05f2944af": "\\tfrac{2}{4}", "a85364d488c2df43430f18a183c14079": "\\ c f: x \\mapsto c f(x),\\quad c \\in {\\mathbb R}", "a853d1b9804da74fff3f88e5c60aaefb": "A \\le_{tt}^P B", "a853ef4a49048daed81b92b901de8d20": " x \\neq -x ", "a8542f85d295b287693e8b44401faff9": "\\!F", "a8547f5d957d173450d5af57628fb47f": "|\\psi(t)\\rang", "a854d6147115abddab416fabc3810f43": "\\psi(\\bold{r})", "a855730aa75e32d0feb50b4e3949e848": "\\partial_\\beta F^{\\alpha\\beta} = \\mu_0 J^\\alpha", "a856219d5fda9a7e731e938e0dd29136": "\\frac{dS}{dt} = - \\lambda S, ", "a8563f85b25b418a7c4a4a0bbf411c06": "\\underline{\\mathbf{d}}(\\ell) = \\mathbf{F}\\left[ \\mathbf{0}_{1xN}, d(\\ell N),\\dots,d(\\ell N-N-1) \\right]^T", "a856425d806306c37642570859461970": "X_s+X_r^{'}", "a8565e887e9e87f8d63648da08f4cccb": "a\\in M_w", "a856b093afbffbd2006e299058ab3836": "\\frac{d\\mu}{dx}=\\mu p(x)", "a856b2f2b33a4dddd0dc0cdacad63019": "\\ \\mu", "a856ecdec31e85c11eeb09357e7879d1": " -\\sum_{k,l}\\nu_k a^{kl}\\frac{\\partial u}{\\partial x_l} = c(u-g),", "a857006dfebacd75082c0ce4f031b789": "d(x,y)=\\sum^\\infty_n\\frac{1}{2^n}\\frac{p_n(x-y)}{1+p_n(x-y)}", "a8573d7bcd10cd08beee321329168acb": "\\vec v_0", "a85757a252baa333395443cbb7f70de8": "x^n+y^n = z^n", "a857fcea8a3221747f9d41ffb58edbad": "Z_0 = \\sqrt{\\frac{R + j \\omega L}{G + j \\omega C}}.", "a8580749c845278a8c9c63c9f0c040cb": "\\frac {1}{1-\\beta} \\log\\left(\\int_{\\mathbb R} |g(y)|^{2\\beta}\\,dy\\right)\n \\ge \\frac\\alpha{2(\\alpha-1)}\\log\\frac{(2\\alpha)^{1/\\alpha}}{(2\\beta)^{1/\\beta}}\n - \\frac{1}{1-\\alpha} \\log \\left(\\int_{\\mathbb R} |f(x)|^{2\\alpha}\\,dx\\right).\n", "a858439c2ff3cece1701687c94fd2990": "\\frac{1}{s-c}+\\frac{A}{(s-c)^2}+\\frac{2B}{(s-c)^3}", "a85867905dc8465d10012d813e9e3eab": "a \\equiv b \\pmod{n}", "a858ddefedd84d492d169075dd365b2b": "H^I_p(P_\\bull \\otimes N) = \\mbox{Tor}_p(M,N)", "a8599901497ae1e67a7bcc6c1e4eb189": "g_{[\\nu\\mu]}\\;", "a859e5b06c101e8f1f267d7b0219eeb6": " L_1 ", "a85a031f0981ad2a15fff31c6a57b787": "p(x=i)=p_i", "a85a215244146fd33357af498bffd41b": "y \\in V", "a85a24bc7a906f8cce0fbcb292468b9b": " \\operatorname{lift-choice}[V] = \\operatorname{none} ", "a85a66929bcc6ffaf9511270b15cff5d": "(D - W)y = \\lambda D y", "a85aad5fa1cbe09d313f89c16de81ed7": "\\sigma = \\frac{\\pi^2 k_B^4}{60 \\hbar^3 c^2} \\;", "a85ae8308986712d0a383bffb1bca937": "D\\geq N\\left(\\dfrac{(\\delta-(\\dfrac{\\lambda}{d}))}{(1-(\\dfrac{\\lambda}{d})})\\right)^2", "a85af22ca2d75ac9ac17c7d9e9db6cd7": "N = N_0e^{-\\lambda t}\\,", "a85bbdbda757a1389190e8cb6770f5cc": "\\displaystyle{G_{\\mathbf{C}} = G\\cdot \\exp i\\mathfrak{g} = G\\cdot P = P\\cdot G.}", "a85be02d68dae21511b9c4b636a17e7b": "\\alpha(w) = g(v, w) ", "a85be8ab0bcc45b3fde2c5f3fa4858b6": "Zn_{(s)} + Br_{2(aq)} \\leftrightarrow 2Br^{-}_{(aq)} + Zn^{2+}_{(aq)}", "a85bfc8c164f9c58334a38fba9836b25": "F(x;\\hat{\\theta})", "a85c248d733d28f37211bef0c379e0f5": "\n \\operatorname{E}[\\,\\mathbf{x}_i\\varepsilon_i\\,] = 0.\n ", "a85c7cfbf8595873dd64bdf9a642a071": " \\mathbb{S}_{ij} = \\left\\langle \\Phi_i^{SO} | \\Phi_j^{SO} \\right\\rangle ", "a85c9021811c3b8e791dabfb81d52758": "\\langle v,w\\rangle = v_1w_1+v_2w_2.", "a85d07fee2d2326e2bc101c7988dfd15": "10^{20}", "a85d5e6870275005088859765fbd856b": "S_0,S_1,S_2,\\dots\\ ", "a85dc78fe97f03dcd70d48937c8f4608": "\\neg \\exists x_1,\\ldots , x_k [p(n,x_1,\\ldots,x_k)=0]\\,", "a85e16be70d50e3ae866a36bf9e517d7": "a_n \\rightarrow \\sqrt{S}", "a85e1ff5e09fa064bb4e75a2cc1e6c5d": "\\begin{align}\\text{ where } &\\gamma \\text{ is the Euler constant: }\\,\\!\\\\ &\\gamma=-\\psi\\left(1\\right)=\\text{0.5777215... }\\end{align}", "a85e2cf51b92d74e6c7b7ecdac6b4ef1": "R \\in \\mathbb Z_m ^*", "a85e6d37fe95739b747610ea41e43dce": " (a + b)^n ", "a85ec04ac3cca39cad8b0ec3648b907d": "w\\Vdash(\\exists x\\,A)[e]", "a85f3d3870945624633eca363b408c4a": "\\phi_{12} : H \\otimes H \\to H \\otimes H \\otimes H", "a85f4b5101a774d76a2f624a209da3ae": "H^2-2P^2=1 \\,", "a85f6dccedf7171411fdc67b8ba7b9ea": "f'(c) = 0.", "a85f9bba545eeabbc2fb536a2a5ec928": "\\dot x = \\frac{ \\partial H }{ \\partial p } ", "a86013cbb5c6fa3af5f12e389ad20c11": " k \\times k ", "a8601d3dccc61a2ed7a96e2cec36bff1": "\\| \\varphi_1 \\|_\\infty \\le b.", "a860362b93ae4efcd586cb731b47db2e": " I_t = \\frac{\\sum_{i=1}^N Q_{i,t}\\,F_{i,t}\\,f_{i,t}\\,C_{i,t}\\,}{d_{t}\\,} ", "a8603983d81a20bd2f2ef0bb315b2830": "\\vec{\\xi}_j", "a860ce894c4efc8fea2aaee6d17867b7": "s > 2^{T-1}", "a861663451d5d1a0d84a2595a60ec891": "K(a,b;m)=\\sum_{0\\leq x\\leq m-1,\\ \\gcd(x,m)=1 } e^{2\\pi i (ax+bx^*)/m}.", "a8618fadef9373bf2265233713abef34": "s_1 = - (\\triangle a)_0 = -a_1+a_0", "a8619581003ce010c7bbeb10515462b4": " \\phi:U\\times L\\to p^{-1}(U)\\,", "a861c074b438433b12b349002c4ac113": "Q = A^+A\\,\\!", "a8621289bb4cb1ed57f208d64c1ec91b": "\\frac{\\pi r^2}{2}", "a86213f433210dfdb17e75c39dcca1c5": "{\\mathbf{}}n_r1", "a866f30971a1794a51efa371d09419fd": "\\vec \\rho", "a867137c8808eb2f85f0d4be83a399bd": "\\; f \\;", "a8678866bebd401d2e894d4220be8d9c": "{}^\\ast\\mathbb{R}", "a867a722720eba59818bccd06cec5372": " 2.0 = y' + \\frac{1}{2y'^2}", "a867b84182f478f9e9c55a03b3ca4fdd": "c_{0}+c_{1}e+c_{2}e^{2}+\\cdots+c_{n}e^{n}=0, \\qquad c_0, c_n \\neq 0.", "a867c62c3efbfc75650a112f3a28d623": "2.2 = y_1 + \\frac{2^2}{2(9.8)}", "a867cf18dc20a92327e68d0a8a71d3e8": "s_{0}=+1, r_{0}=+1, x=0.0 \\, ", "a867d9acf82a3eabe4a281777a3bc379": "\\operatorname{dCov}(X,Y)\\geq0", "a867e8597f35f73efb3917f3b5b0f1ba": "\\ P_f ", "a8681d0b75801db0180b8b888fb04fd1": "\\Delta_{n+1}", "a8688cedb8d327a40611bd9f92649334": " \\{ \\mathbf{e}_{k} \\}", "a868bb557d9bd2cf908dca98cf39600a": "Im(a)", "a869bdb232bcf51e330090185e00fa04": "\\sigma(t)", "a869cb2c4942b7b4cde9fe61a078df99": " K_{var} = \\frac{2}{T}\\ \\left ( rT- \\left (\\frac{S_{0}}{S^{*}}\\ e^{rT} -1 \\right ) - \\log\\left ( \\frac{S^{*}}{S_{0}}\\ \\right ) +e^{rT} \\int\\limits_{0}^{S^{*}} \\frac{1}{K^2}\\ P(K)dK + e^{rT} \\int\\limits_{S^{*}}^{\\infin} \\frac{1}{K^2}\\ C(K)dK \\right )", "a869f3b69c76ad2338e88a686ff2a024": " s\\cdot x = t\\cdot x\\text{ for all }x\\in X,", "a869f43d8e87787a8f265043534a0778": "K(G,i)", "a869ff85b5bb8a3c5a613d057b7f7619": "\\left(1+\\frac{r}{k}\\right)^k\\leq \\mathrm{e}^r,", "a86a5d1ae0cd978c88d7abf1617e5093": "\\rho\\, ", "a86a5e43b86f94de2cbca6f39cc50acb": " RH = \\frac{p_s}{p_0} \\times 100% = S \\times 100%", "a86a6001f6d7bd32f2beb24e4a53c309": "\\operatorname{E}[X^k] = \\frac{\\alpha + k - 1}{\\alpha + \\beta + k - 1}\\operatorname{E}[X^{k - 1}].", "a86adad47328d80dd2690160cb619813": "\\mathbf{F(\\omega^k)} = \\mathbf{P}\\mathbf{F(\\omega)}.", "a86b0a0a5b5849bbc22a78e36489bc2a": " \n p = \\frac{\\rho_0 C_0^2 (\\eta -1)\n \\left[\\eta - \\frac{\\Gamma_0}{2}(\\eta-1)\\right]}\n {\\left[\\eta - s(\\eta-1)\\right]^2} + \\Gamma_0 E;\\quad\n \\eta := \\cfrac{\\rho}{\\rho_0} \\,.\n ", "a86b2d3cdcf0c2ae90b3372424115c3c": " \\lim \\int \\limits_{E_n} \\phi = \\int \\phi", "a86b476d942a21c30901ddada8bc5f95": "\\mathcal{F}", "a86b6cce47c5bdd51978459566decf58": "v = \\frac{w}{|\\vec k|}", "a86b79c528582023ba331108fea5e867": "\n\\langle \\beta\\ \\mathrm{out}|\n\\mathrm T\\left[\\varphi(x)\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n|\\alpha\\ \\mathrm{in}\\rangle=\n\\eta(x)\n", "a86bc4a05746eac4dafe0484540cdc53": "K = h \\cdot S_a (Y + \\bar{Y}B) = h \\cdot S_a (yP + \\bar{Y}bP) = h \\cdot S_a (y + \\bar{Y}b)P = h \\cdot S_b S_a P ", "a86cced3650d391058fa96f504e1a32d": "4m_p=2\\pi A_0", "a86d0aebea3efbbf0868587a6ba96195": "\\sum_i W_i= 0", "a86d11b57c033a1b69300893b1325bb0": "(d \\mathbf{l} + \\mathbf{l_0} - \\mathbf{p_0})\\cdot\\mathbf{n} = 0", "a86d1add3eb8b66d4b16a282ef494bca": "P \\lor \\forall x \\, Q(x) \\Leftrightarrow \\forall x \\, (P \\lor Q(x)) ", "a86d80c1f14c6bcd436b1dba68c700c1": " \\nabla^2 f(x) \\succeq m I", "a86d8516fae7d34fadb10cf3857e4d82": "Q_{nxi}=1-P_{nxi}", "a86da76864a5cabaf222316b2ece8481": "Attr_i(U)^0 := U", "a86e37dbfb06cac3662eb4d2f0613bd2": "DP(i\\rightarrow k|j) ", "a86e7e9216914134ca9e8fbb8a8f6ec2": "I(\\rho^A:\\rho^B) \\ \\stackrel{\\mathrm{def}}{=}\\ S(\\rho^A) + S(\\rho^B) - S(\\rho^A,\\rho^B)", "a86ed95d115a4a64a49130459d64da79": "\\Pr(\\mathbb{Z}\\mid\\boldsymbol{\\alpha})=\\int_{\\mathbf{p}}\\Pr(\\mathbb{Z}\\mid \\mathbf{p})\\Pr(\\mathbf{p}\\mid\\boldsymbol{\\alpha})\\textrm{d}\\mathbf{p}", "a86f746e55276c6ac1dbec5e1b95167e": "\\Gamma_p(a) = \\pi^{p(p-1)/4}\\prod_{j=1}^p \\Gamma(a+\\frac{1-j}{2}),", "a86f772a30404e63c80a081265c46395": "~f_a", "a86f7fd00a31a9483529318812c01353": "A = 1 + \\beta_1 + \\beta_2 = 0.", "a87062a44c7ee792d432df6b6980575d": "2R^a{}_{bcd} = 2(\\Gamma^a_{bd,c}-\\Gamma^a_{bc,d}) \n= \\eta^{ae} (h_{eb,dc}+h_{ed,bc}-h_{bd,ec} - h_{eb,cd}-h_{ec,bd}+h_{bc,ed}) =", "a870d25f5cd9206ce7815ba7ef591054": "A_{n+1}(R) = (2\\pi R)V_n(R).", "a870ea3a7fd65f9f0c81009a3ac13b6f": " \\widehat q = \\tfrac 1 {\\sqrt 2}(\\widehat a^\\dagger + \\widehat a)", "a87118ed0b1aeb45bff550e1e877aed1": "\\mathrm{End}(_D V)\\,", "a871ba90c77126ae0d3c613d454bb4ac": "A[\\mathbb{N}_0]", "a871e2147047e7a8497f699cc46c45d8": "x^y \\mod z", "a8721642523b68279407f83a057660d4": "\\left(H^\\dagger H\\right)^2", "a8722886f6f19816db3b96bae669603b": " j< h", "a872a354403f24b7d2c2ac67fda70797": " a_1\\ge b_1 \\ge a_2 \\ge b_2 \\ge \\cdots \\ge a_N \\ge b_N=0", "a872c473c428743af1cd26ffc0737093": "n \\times m ", "a872ebe1c43ae29843ba3f893c366967": "a_{0} + a_{1} p^1 + a_{2} p^2 + ...", "a8734ce12412d1a0c7b5b4854bbb6e31": "f_{i*}\\alpha_i", "a87388d26b6a51225c8cafe698a1e100": "T(\\mathbf{false})\\ \\Leftrightarrow\\ \\mathbf{false}", "a873b4632701307c694fd71ddb35729b": "\\frac{d}{dx}f(x) = f(x)\\cdot(1-f(x)).\\,", "a873b5ea4a51a1cccd6af374316d2c3c": "\\sigma(N\\mathfrak{G}^k) > 0", "a873d4b60e78254abcd51c59024ae0d1": " G = 2\\sum_{i} {O_{i} \\cdot \\ln(O_{i}/E_{i}) }, ", "a873d733fe681f3f13632846892a6d27": " a_0 \\in \\Gamma (x_0), \\; x_1=T(x_0,a_0). ", "a8746a0f6fb4befcb4dadf53651ce53c": "R_n^m(\\rho)=\\sum_{k=0}^{(n-m)/2}(-1)^k \\binom{n-k}{k} \\binom{n-2k}{(n-m)/2-k} \\rho^{n-2k}", "a874e29b8331b6c1d060853d602b8318": "\\mu-\\beta \\ln\\left(\\ln 2\\right),", "a8755aa8c49a817878f9f2e71131c9ec": "\\scriptstyle{E_{4}}", "a8758bbab29cc6198808782813e1b0bc": "s, t \\in G", "a875df90aa176c3afcf41e3b076b2bb0": "(\\operatorname{div} X)\\omega = L_X\\omega = d(X\\;\\lrcorner\\;\\omega)", "a8760b0a733715fd8c0ef0d98970c176": "f = 4 \\chi_{[-5, 1)} + 3 \\chi_{(0, 6)}\\,", "a8762f04aef3648435adfcabeddec198": " \\left[0, 1\\right].", "a87633a1bf4e0a24b070cb09093891c1": "S=\\sum_F -\\Re\\{\\chi^{(\\rho)}(U(e_1)\\cdots U(e_n))\\}.", "a8772f15e205905f25f656887d497435": "\\dot{J}_{ab} = -\\frac{\\theta^2}{9} \\, h_{ab} - \\frac{2 \\theta}{3} \\, \\left( \\sigma_{ab} + \\omega_{ab} \\right) -\\left (\\sigma_{am} \\, {\\sigma^m}_b + \\omega_{am} \\, {\\omega^m}_b \\right) -\\left(\\sigma_{am} \\, {\\omega^m}_b + \\omega_{am} \\, {\\sigma^m}_b \\right) - {E[\\vec{X}]}_{ab}", "a8773714fe9d5a69fdb7824b8e819615": "wOBA=\\frac{(0.72*NIBB) + (0.75*HBP) + (0.90*\\mathit{1}B) + (0.92*RBOE) + (1.24*\\mathit{2}B) + (1.56*\\mathit{3}B) + (1.95*HR)}{PA}", "a8779276dd04b806ab98b851a2c95147": "t^\\ast\\le t^{**}", "a877cf58b9d84a4e23804229ca08f5bc": "\\forall t_0 \\in I", "a877f3c76508a0ccf6fbaaa64ea62d49": "h(x)=0", "a8782e73e4bc2741602dcb86f06e8bc7": "M_{\\pi \\oplus \\sigma}", "a8783153bb3f1d6d4994d1e228b2b921": "F_0 = (S_0+U-I)e^{(r-y)T}", "a878626745dfb1cbd52ad5af811c9a0e": "\\left|A_1 \\cap \\cdots \\cap A_p \\right|", "a878cfe147b480aa1de3fdc88995b128": "\n\\displaystyle\nx\n", "a878d3a01018218c2d6b6fe487a5d86b": "h_1,h_2,\\dots,h_n", "a878d5c8af1c64d7b76f555c0a558bc4": "r^n=a^n \\cos n\\theta", "a878eb552e0cb6625d9342ef6646b284": "S = 100", "a87957a0870bd12b7fcb194c0579df98": "r=\\infty", "a8796fb9c738e34c6b83c54a85b8e7ba": "R^*g(x) = \\frac{1}{2\\pi}\\int_{\\alpha=0}^{2\\pi}g(\\alpha,\\mathbf{n}(\\alpha)\\cdot\\mathbf{x})\\,d\\alpha.", "a8798d95d12905340d6ea4f33281e2ab": "(A\\equiv B)\\equiv((C\\equiv A)\\equiv(B\\equiv C))", "a879ec2764b825bed9c2e568d29b8640": "E \\exp(i u^T X)=\\exp\\{-(u^T\\Sigma u)+i u^T \\delta)\\}", "a87a2af0c94f4814a259ccbe490a2b6a": "r \\rightarrow r_s", "a87a52c123e341198c3e47760ce0a088": "M_z(0)=0", "a87b28633d4fb839de9ecdb7e921e70d": "\\eta+3 \\vartriangleleft O_K", "a87b4e43b800b1205d11eabc1be87d2c": "\\phi_i=\\sum_{j = 1}^n p_{ij}Q_j \\mbox{ (i = 1, 2, ..., n)}, ", "a87b54af41f2eb27b0f88b5f4ae552d5": "x\\in F(c)", "a87b810b0c588c23c34045784e9e4c4f": "\\mathbf{H}=-\\nabla\\psi.", "a87b96bc5225638dbe6aca796f122cf7": "H_{12} = H_{21} = \\beta \\,", "a87ba6b41d6cfc1eba5d596bfde4fe3f": "F:\\R^n\\supset X\\to\\R^n", "a87be6d11ff411ea31f842063c715f4c": "\\blacktriangleright \\!\\,", "a87c43c92e59301803e1cd7eae575b7c": "\\langle E_i \\rangle = \\frac{k_B T}{2}\\frac{\\int dx\\,\\,x^2\\,\\, e^{-\\frac{x^2}{2}}}{\\int dx\\,\\, e^{-\\frac{x^2}{2}}} = \\frac{k_B T}{2} ", "a87c7a43df43a142e28fa36dfe92b654": "\\frac m n = \\frac{p+r}{q+s},", "a87c99670bb1da44300ddf4006c36ca6": " \\arctan x = y \\, ", "a87cccb1577457282e0c86b5f4a60980": "T_x(\\phi) = \\phi(x).\\ ", "a87d05e5aa220e43f699dcf14742657e": "\\kappa^2 = -4B\\Omega", "a87d07d5533b5f3d15f8139c290e9736": "\\gcd{(m,n,k)}=1 \\, ", "a87d0d45234f095aaa3f5a320b3ef796": "(\\Omega^\\bullet_{\\mathrm c}(X),d)", "a87d75f1a1a0da7d912090085fd882ec": "c_j>0", "a87d75ff612ad6b4d042b16f9317e7c9": "\\alpha^n\\alpha^{m-n} = -\\left( b_{n-1} \\alpha^{n-1} + \\cdots + b_1 \\alpha + b_0 \\right) \\alpha^{m-n} = -\\left( b_{n-1} \\alpha^{m-1} + \\cdots + b_1 \\alpha^{m-n+1} + b_0 \\alpha^{m-n} \\right) ", "a87d9882e9bf594fb69a1507717b9502": "|\\mathcal{M}|^2 \\,", "a87d9eaf08cf19b439259ffb1d9d30ae": "H(\\varphi,\\partial_\\mu \\varphi) = \\int_\\mathcal{V} \\mathcal{H}(\\varphi,\\partial_\\mu \\varphi) dV \\,,", "a87da307f69912c01bb68b7a5756e6c6": "\\scriptstyle 1\\leq j\\leq k", "a87dc83e448ab9201209ac34273734e9": "l(\\Gamma)", "a87df36f22476aded8bcae1fd5add454": "\n\\times\n\\prod_{j=1}^{n}\n \\left\\{\n \\mathrm{d}^4y_j\\ \n i\\frac{e^{+ip_j\\cdot y_j}}{(2\\pi)^{3/2} Z^{1/2}}\n \\left(\\Box_{y_j}+m^2\\right)\n \\right\\}\n\\langle 0|\\mathrm{T}\\ \\varphi(x_1)\\ldots\\varphi(x_m)\\varphi(y_1)\\ldots\\varphi(y_n)|0\\rangle\n", "a87e28af02575493c1fabd447c4abb1f": " \\beta_{1} = -2.5, \\ \\beta_{2} = 20, \\ c = 0.75 ", "a87eccd98440b10713e8ec2eaba602f8": "\n\\left( \\frac{dr}{d\\varphi} \\right)^{2} = \\frac{r^{4}}{b^{2}} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{r^{4}}{a^{2}} + r^{2} \\right).\n", "a87f114dfcbe169a8e8255fa37b37c3f": "\\scriptstyle c_2 = (k_2L)^2", "a87f57a78bc8acad498068025d544212": "A_2B_2", "a87f81a14b37375310d32fa636d26ee9": "P_\\mathrm{kW}={\\tau_\\mathrm{N \\cdot m}\\cdot\\omega_\\mathrm{rpm} \\over 9549}", "a87fb4bc1f0e0191668183f604b03c26": " \\mathrm{d}S =\\frac{\\delta Q}{T},", "a87fe482ed5340f93af98f243ae0f616": "\n K_{\\rm Ic} = Y\\sigma_c\\sqrt{\\pi a} \n ", "a87ff679a2f3e71d9181a67b7542122c": "4", "a87ffe1fe60a8ab83e3f2a7320f02d87": " [ M^{\\mu \\nu} , Q_\\alpha ] = \\frac{1}{2} ( \\sigma^{\\mu \\nu})_\\alpha^\\beta Q_\\beta ", "a8800929e480839a817f52866586a793": "\n\\begin{bmatrix} X_1 \\\\ X_2 \\\\ X_3 \\\\ X_4 \\\\ X_5 \\\\ X_6 \\end{bmatrix}\n=\n\n\\begin{bmatrix} 0 & 0 & d_{31} \\\\\n0 & 0 & d_{31} \\\\\n0 & 0 & d_{33} \\\\\n0 & d_{15} & 0 \\\\\nd_{15} & 0 & 0 \\\\\n0 & 0 & 0 \\end{bmatrix}\n\\begin{bmatrix} E_1 \\\\ E_2 \\\\ E_3 \\end{bmatrix}\n", "a880953683e278531ca223950d55e166": "(4,1,2)\\oplus(\\bar{4},2,1)", "a881280aa5d52ec6a4ca53a307d9bdd2": "p^* \\geq d^*", "a8812ee11e9dc6d7e9740762363a35e0": "AP = \\frac {D} {10} + 1", "a88194e4ed836d50e15adbeebd840d69": "m= \\left(\\!\\!{n + 1 \\choose d}\\!\\!\\right) - 1 = {n+d \\choose d} - 1 = \\frac{1}{n!}(d+1)^{(n)} - 1.", "a881c2f97b2b949d98a0ba8232dfa094": "\\begin{align}I(X;Y) & = D_{\\mathrm{KL}}(P(X,Y) \\| P(X)P(Y) ) \\\\\n& = \\mathbb{E}_X \\{D_{\\mathrm{KL}}(P(Y|X) \\| P(Y) ) \\} \\\\\n& = \\mathbb{E}_Y \\{D_{\\mathrm{KL}}(P(X|Y) \\| P(X) ) \\}\\end{align} ", "a881cfecc9af68472cf4a47132a73521": "c\\frac{\\log n}{\\log \\log n}", "a881d830d3f63f710fdae086ad8de655": "\\tfrac{18+\\sqrt{30}}{36}", "a882612279e33528511774c1314b419e": " R' = \\langle R \\rangle + \\left( \\frac{d \\langle R \\rangle}{dQ} \\right) Q ", "a88292e6b6f5db13d72f518c254ad352": "C^{(2)}_{abcd}", "a882a32e2e85643ed8b7ab468652207f": "(C_\\beta|\\beta \\text{ a limit point of }\\kappa^+)", "a882fc6f4be17d8d002b1db7ef2a6fd9": "\\mathcal{L}=\\sqrt{-g}\\bigl(\\overline{\\psi} \\left(i\\gamma^\\mu D_\\mu-m \\right) \\psi\\bigr),", "a883448a0e2d2ee91bb89b3eeb49619e": " 2S + 1 \\ ", "a883833f664551c752907626e7cf1031": "\\frac{P \\to Q}{\\therefore P \\to (P \\and Q)}", "a883a0beddd0e80ef23ae186b3ccd30a": "f(a+\\eta b)", "a883b09a60f2d70d207bca2243c78c4a": "\\varepsilon(N_j) = 0\\,", "a88427605b79f4eb5e90741be6612b4b": " X_i = \\Omega \\phi_i + \\Omega^{1/2} \\xi_i. ", "a884603ac4ffe9ca43f2e6de04121dc1": " \\langle f, g\\rangle = \\int_{a}^{b} \\overline{f(x)} g(x)w(x)\\,\\mathrm{d}x.", "a88482aecc057016595d4cb0ba0c3620": "{\\mathbf Q}\\,", "a884fc68f2f00ead02130fad389e4157": "AD line = today's\\ advancing\\ stocks - today's\\ declining\\ stocks + yesterday's\\ AD\\ line\\ value", "a885212fbfabe2443465773caa140142": "\\alpha(p_{1}, \\, p_{4})", "a8854219d0466c2792105ef2cc0dae8b": "y_{k+1}=(1-f(y_k))/(1+f(y_k)) ~,~ a_{k+1} = a_k(1+y_{k+1})^4 - 2^{2k+3} y_{k+1}(1+y_{k+1}+y_{k+1}^2) ", "a88570aa93f5bb3a24b050e00ba83aa3": "\\sin (\\alpha +\\beta)=\\sin \\alpha \\cos \\beta + \\sin \\beta \\cos \\alpha", "a885c416ca411a938d41db54bc779c7f": "\\{B, \\tilde{C}\\}", "a885d78de25e38d73ac8ef86da26b1d2": "(x, y) \\mapsto xy", "a886a48246a23f44c4d0193388a1e96e": " \\sum_n e^{iu E_{n}}=Z(u) = e^{u/2}-e^{-u/2} \\frac{d\\psi _0}{du}-\\frac{e^{u/2}}{e^{3u}-e^u} = \\operatorname{Tr}(e^{iu\\hat H }),", "a886c11f4d028d28099f7dfab3ac0d56": "\\mathbf{(J^{T}J + \\lambda I)\\boldsymbol \\delta = J^{T} [y - f(\\boldsymbol \\beta)]}\\!", "a88700e0ab2ba65b7649c564dd4e62e6": "\\mathfrak{k}^*", "a88708ecc751ac66a3c3ee0c1753c992": "\\rho \\overline{u_i' u_j'}", "a88736b245958f4a990e7ab1a7662de0": "\\mu _\\infty", "a88821e27bf812c2543db115fac1feda": "Mass = Sum[25] \\; of \\; { EMA[9] \\; of \\; (high-low) \\over EMA[9]\\,of\\,EMA[9] \\; of \\; (high-low) }", "a8886f155fcf705be45536999ca6366a": "\n\\left\\langle \\frac{dG}{dt} \\right\\rangle_\\tau = \n2 \\left\\langle T \\right\\rangle_\\tau + \\sum_{k=1}^N \\left\\langle \\mathbf{F}_k \\cdot \\mathbf{r}_k \\right\\rangle_\\tau.\n", "a8888708db9213e4f76f0e3ec0d982dc": "\\bigcup_{i=1}^n A_i", "a88887a8cfa6513d5a05fba2bec17004": "\\operatorname{det}", "a888915663d7487690359816440d7bcd": " d_{iw} = \\sqrt{\\sum_{j=1}^{n}(t_{ij} - t_{wj})^2}, i = 1, 2, . . ., m ", "a888b9291a212dc8d7bf664345c27308": "{\\bar{K}}_3", "a888cc045550ba3ab393de604a23c1cb": "Y(\\omega, z) = \\sum_{n\\in\\mathbf{Z}} \\omega_{(n)} {z^{-n-1}} = \\sum_{n\\in\\mathbf{Z}} L_n z^{-n-2}", "a888d011adf22f3682afe4d7dc54d8fb": " \\lambda\\frac {d^2T \\left(x \\right)}{dx^2}-u\\rho_a c_a\\frac {dT \\left(x \\right)}{dx}=0", "a8894c20f6a1c182f07b1b8e70e7a1a8": "\\sigma_3\\,\\!", "a88987b0b381e194ff8b50845178991c": "\\displaystyle{ w_s(z)=-zp_s(z)}", "a88a058af5ba78f6abcc7aac143b7c73": "\\ \\phi_i = c_{1i} \\chi_1 + c_{2i} \\chi_2 + c_{3i} \\chi_3 + \\cdots +c_{ni} \\chi_n", "a88a2668f8112605312b6a98178e1bba": "\\alpha=\\tfrac{1-F}{F}p", "a88a517fc3d3ccc232f6d3579d46a820": "{\\hat{\\alpha}}(q),\\, {\\hat{\\alpha}}(q^\\prime), \\, \\ldots ", "a88a691c761bd09a310a97dfbc518e33": "\\frac{dD(L^{S})}{dL^{S}} = \\omega", "a88a822dba766708ae9e9c3228f1131b": "W^{1, p} (\\Omega) \\hookrightarrow L^{p^{*}} (\\Omega)", "a88ab37cb9836ce35a1c60942d0b8b27": "T^{\\mu a}=e_\\nu^a T^{\\mu \\nu}", "a88ad2b5cb59917cfacb69d0783b9fe5": "\\tau=1", "a88ae0f9c56a8662bf1f74fe2e9d76bf": "\\det(I+A) = \\exp(\\mathrm{tr}(\\log(I+A))). \\,", "a88b5c0ad82c7e22b89a681e38917df9": " RT \\ln \\frac{\\{S\\}^\\sigma \\{T\\}^\\tau} {\\{A\\}^\\alpha \\{B\\}^\\beta} = RT \\ln Q_r ", "a88b85a21028be170594ce46cef02dcd": "(\\Sigma \\cup N)^{*}", "a88ba773bf2ddca09c590c63deca2f8a": "\\tilde{\\boldsymbol{a}}", "a88bc786b5447517bd6fc7e8ea2b2e52": "\\sum_i X_i=\\sum_i Y_i=1", "a88c0ccb17f8930c139b1fa04adac293": "[t_-, t_+]", "a88c1ac992444c0463f81430ab3e267a": "x' \\equiv x \\pmod{w}", "a88c355d7f9d41fd5a1493933601bf38": " z = {-b\\over 2a} + i {\\sqrt{4ac-b^2}\\over 2a}.\\ ", "a88c3caca3d951862f0c5f75c300653d": " \\sum_{k=0}^{\\infty} p(5k+4)x^k = 5~ \\frac{ (x^5)^5_{\\infty} } {(x)^6_{\\infty}}", "a88c4c6d56917486ba586529b55b186f": " \\int \\frac{dx}{1+x^2} ", "a88c58df8a96a946e23933961f9cb34d": "\\vec y", "a88c70b4765a674cc174ad99a0788512": "\\Omega=\\rho g Q S", "a88c849719f260199e49946541fd97b5": "W'", "a88c972836c9af492c87b007d8d5a6fc": "{\\partial u}/{\\partial y}", "a88cf97d07bcd8779e0cd72f9551ff18": "VU = e^{2\\pi i \\theta}UV.", "a88db99d3887708cd27a76c18a834149": "g=30 ( = 2 \\cdot 3 \\cdot 5), 60, \\dots", "a88dba71bb3cf8ce4017afca4837ae5b": "Y=S \\cdot \\sin\\theta\\cos\\phi", "a88dbd6adae72aa79e4bc88de5a76f01": "a(\\cdot)", "a88e003246840307a89b06a66df0c1dc": "\n\\int_0^1 e^{x\\cdot \\ln a + (1-x)\\cdot \\ln b}\\;\\mathrm{d}x =\n \\int_0^1 \\left(\\frac{a}{b}\\right)^{x}\\cdot b\\;\\mathrm{d}x =\n \\int_0^1 a^{x}\\cdot b^{1-x}\\;\\mathrm{d}x =\n \\frac{a-b}{\\ln a - \\ln b}", "a88e054a72b82247255a55bf43ea96ac": "B(t)=(1-t)D + tA.\\,", "a88e652c1bd340f06919581eb99a6010": "A \\leq B", "a88edfdc55dc0f414349f6de68740ad4": "g_p(X_p,-) : Y_p \\mapsto g_p(X_p,Y_p)", "a88ee67a4cc1fa29afa21d9848992a7a": "|2,0,0\\rangle", "a88f9c9b1b6252a3d15017b40f381f8f": "= |(\\mathbf I_p-2 i \\theta \\mathbf I_p)|^{-\\frac{1}{2}},", "a88fc326ae40063153e22c773c193d74": "x_{n+1} = x_n + [I - L(F(x_n), x_n)]^{-1}(F(x_n) - x_n),\\ ", "a88fe40482b1dd9cbff3198cf5c9c223": "\\mathbf{X} = \\{X^1,X^2,X^3\\}", "a88fffd2097dbdd9411932dcfabd8913": "K \\supseteq \\mathbb{F}_q", "a8900252481a43289b3ecbdb2e84983f": "\nE[X^{n}]=(-1)^{n}n!\\boldsymbol{\\alpha}\\Theta^{-n}\\boldsymbol{1}\\; .\n", "a8903c174fa6d7c94fad14de3a597c7e": "\\hat\\beta\\ \\xrightarrow{p}\\ \\beta + M_{xx}^{-1}\\cdot 0 = \\beta", "a890566e9afaba4606defecb41907f13": "{|c_1|}^2+{|c_2|}^2 = 1", "a89073e48d46024b032f5a4954a9c5de": "\\Sigma(\\mathbf{x})", "a890b5b6e0bf28c4b503f71c3614ff82": "*22\\infty", "a890dd38a6edad042d340c4d7181f3c8": "x=\\frac{X}{Z}", "a890faf837a14f39ff30682802aff135": " g_t\\cdot A=\\begin{pmatrix} a \\cosh 2t +y \\sinh 2t +b & x+i(y\\cosh 2t + a\\sinh 2t) \\\\ x-i(y\\cosh 2t + a\\sinh 2t) & a\\cosh 2t +y \\sinh 2t -b\\end{pmatrix}.", "a8913d7ff3ed3d60c582025b1aec466a": "\\tfrac{1}{45}", "a89147ec956b0b7b8d2822d92f81fdc1": "p_A = 1/2", "a891a7170d8596348dddf7f7964be4e3": "\nT: RS(O,X,Y) \\mapsto (O'=S,X',Y') \\ \\stackrel{\\mathrm{def}}{=}\\ \n \\left\\{\\begin{align}\n \\vec t &= \\overrightarrow {OO'} = S\\\\\n \\alpha &= \\operatorname{arccos} \\frac{\\vec a_1 \\cdot {1 \\choose 0}}{|\\vec a_1|}\n \\end{align} \\right.\n", "a891f39b9a94dfaafccd4256a6f5b1d6": "x_{i+1}-x_i", "a893a2128a0c35577e509debeb1c6e88": "P^*=\\frac{1}{3}", "a894124cc6d5c5c71afe060d5dde0762": "0.9", "a8944bd780b3684bdac1b3465032bf11": " V_{IN} = V_{L1} + V_{C1} + V_{L2}", "a89486d0a90318168a7790dca39fc308": "K\\times\\mathfrak{p} \\rightarrow G", "a8950cbce222d7a2f02e2be581828680": "n_F=\\left(\\frac{3 N}{\\pi}\\right)^{1/3} ", "a895296afde6b2fd94fe167ab72b2756": "\\rho(X+Y) = \\rho(X) + \\rho(Y)", "a8956d046c9b53ae796585b384d6895a": "k = \\frac{\\pi \\Delta \\nu_\\circ}{2^{1/2}} \\sim 2 \\Delta \\nu_\\circ", "a89584721754b1f7fd61197004c7fede": "\n r \\le \\left(\\sqrt q+1\\right)^2 \\quad\\text{and}\\quad t \\mid q-1.\n", "a895b8061ee53e00529e39963871c79f": " \\mathbf{E}(z,t) = e^{-z / (2 \\delta_{pen})} \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)})", "a8966d612edf723372f8c03ed5ce878b": "S(a)=b.\\ ", "a896b22344ac1f9ef026e1320d4304f0": "B = \\left[ \\frac{\\rho_m L}{ \\left( T _m-T_o \\right )} \\right ]^2 \\left[ \\frac{\\pi }{4 k \\rho c} \\right] \\left[ 1 + \\left( \\frac{c_m \\Delta T_s}{L} \\right)^2 \\right] \\left(\\frac{1 min}{60 sec}\\right) \\left(\\frac{1 m^2}{10,000 cm^2}\\right)", "a896b717f2926e76404712621acafee2": " \\cfrac{\\Gamma, B \\vdash \\Delta}{\\Gamma, A \\and B \\vdash \\Delta} \\quad ({\\and}L_2)\n ", "a896c52197d3d70a24028687642986c2": "{(u,v) \\to 0}", "a896df66b4d1b0791d1b07b2255dfeeb": "\\mathrm{1\\,Fr/s = 1\\,statampere = 1\\,esu\\; current = 1\\,(cm/s)\\sqrt{dyne}=1\\,g^{1/2} \\cdot cm^{3/2} \\cdot s^{-2}}", "a896e782f60eeb3361650ffa0b7bd1ae": "conc(\\langle a \\rangle, conc(\\langle b \\rangle, \\langle bb \\rangle, \\langle b \\rangle), \\langle a \\rangle)", "a896f774334eca8cff6113814aa42c9e": "\n-\\infty \\leq \\operatorname{pmi}(x;y) \\leq \\min\\left[ -\\log p(x), -\\log p(y) \\right] .\n", "a8970ef403a272c8e65103bc1e792b24": "R_n=K[X_1,\\ldots, X_n]", "a89770cbe5c963c7ba420091e7084314": "\\hat K_j(i)", "a897749d2b81847e3a8b920dbf859412": "s, s", "a89799b6e2bb557df913d5a666b257fe": "\\frac{\\sum_{k=1}^n \\delta_{x_k}}{n}\\Rightarrow \\mu", "a897ad6510d6276bce133fe34376b906": "\\psi_{1,8}=12", "a897c5dc261d9005da43faceb67332fa": "f_k(x)=ay\\text{ and }f_{k+1}(x)=by", "a897c604e431abd620ba40791c7626e4": "{\\mathbb N} \\times {\\mathbb R}", "a8981c8ac56df6f21605f0294112c9d0": "Q = \\infty", "a8983e1d89439f3dc53d88eafad3c2d7": "K =\\mathrm{\\frac{[M(OH)] } {[M] [OH] }}", "a89849df789ecea5084e708e736e0094": "\\mathbf{r}_5 = (a/4)(2\\hat{y} + 2\\hat{z})", "a8987b4c689fa402cac6a6772ad5fbab": "{0\\ A\\choose 0\\ 0}", "a8987fe0b0a2c07ca52ac633f2eebd82": "\\mathbb{Z}/n\\mathbb{Z} = \\left\\{ \\overline{a}_n | a \\in \\mathbb{Z}\\right\\}. ", "a8988ce0f88f5292aa28b6e49f114d45": "f(n)", "a8988ec674fd6f73a891bf78f5eec2ce": " \\langle A | = \\begin{pmatrix} A_1^* & A_2^* & \\cdots & A_N^* \\end{pmatrix}", "a898aac1678031b7203de075d5bd9cb4": "j=r-1, r-2, ..., 0", "a898df239b65b002d7693a204555adc3": "T=(I-\\tilde P_\\star)(A-\\tilde\\lambda_\\star I)^{-1}(I-\\tilde P_\\star)", "a899cc44d00e9ac09861f9a4e1ccd382": " \\int{p(X,A | \\theta) p(\\theta | X_t, A_t, O_{fg})} d\\theta ", "a89a1ce400d6a0b141653b2b28440a37": "(\\mathbb{Z}/8\\mathbb{Z})^\\times \\cong \\mathrm{C}_2 \\times \\mathrm{C}_2,", "a89a26316951efdf3666f1f74d92b594": "t_G", "a89a7cab300440b1798d37fda2a37e37": "\\omega_1(t)=\\Sigma_0^t p(i)", "a89a85a5e28f54ecceca3340beacdf7b": " D = {\\mu_p \\, k_B T} ", "a89a995dcc3c88481ab6f97b9332a0c8": " V^*(s)= R(s) + \\max_a \\gamma \\sum_{s'} P(s'|s,a) V^*(s').\\ ", "a89adfc1b75b4a76c3fabff3e9429df8": "10-1.37218 \\frac{\\sqrt{2}}{\\sqrt{11}}=9.41490.", "a89b155519785cd504a1727947a1dbf4": "\\omega_{H} = \\sqrt{\\gamma\\frac{P_0}{\\rho} \\frac{A}{V_0 L}}", "a89b2e368f3939a0480ee106d3c2ee90": "M(E) = \\sum_{F \\subseteq E} (-1)^{|E \\backslash F|} g(F).", "a89b33b9448f3c641fcb3c894f64a2ed": "x=\\{x_1=5,x_2=1,x_3=5\\}", "a89b8a26f8cbe9ca16129c2ba9218aa0": "\\pi(\\mathbf{x})", "a89b9c0fb805c932ad0b7bd330274744": "\na_{\\overline n|i} = \\sum_{k=1}^n \\frac{1}{(1+i)^k} = \\left( \\frac{1}{1+i} - \\frac{1}{(1+i)^{n+1}}\\right) \\sum_{k=0}^\\infty \\frac{1}{(1+i)^k}\n", "a89bb872282ce46501627a67c473172c": "\n \\cfrac{\\partial W}{\\partial \\lambda_i} = \\cfrac{\\partial W}{\\partial I_1}\\cfrac{\\partial I_1}{\\partial \\lambda_i} = 2\\lambda_i\\cfrac{\\partial W}{\\partial I_1} \\quad\\text{and}\\quad\n \\cfrac{\\partial^2 W}{\\partial \\lambda_i \\partial \\lambda_j} = 2\\delta_{ij}\\cfrac{\\partial W}{\\partial I_1} + 4\\lambda_i\\lambda_j \\cfrac{\\partial^2 W}{\\partial I_1^2}\\,.\n ", "a89bf616ce831e8a426e5461e97c903b": "g_1 \\circ f = g_2 \\circ f \\Rightarrow g_1 = g_2.", "a89c0deb449f5a68576e17a006c1c7e4": "A\\in \\mathrm{RAT}(N)", "a89c32d3549916f7c4292d8e7b16217f": "\\{1,2 \\dots q\\}", "a89c4ab112c423681caa90c9e867c28e": "\\lambda e^{\\frac{i}{\\hbar}H_0(t-t_0)}V(t)e^{-\\frac{i}{\\hbar}H_0(t-t_0)}|\\psi_I(t)\\rangle=i\\hbar\\frac{\\partial |\\psi_I(t)\\rangle}{\\partial t}", "a89c5496d014905f4995fb24e0f6b537": "H^i(V,\\mathbb{Q}_\\ell)=H^i(V,\\mathbb{Z}_\\ell)\\otimes\\mathbb{Q}_\\ell", "a89cce39e9f8957e21f40c0d90b52833": " c^2\\, =\\; g h\\, \\frac{ 1\\, +\\, \\frac{1}{6}\\, k^2 h^2 }{ 1\\, +\\, \\frac{1}{2}\\, k^2 h^2 }, ", "a89d5e58ef72bdc3ca3e38ca63d198cb": " \\begin{align} \\langle\\psi_{\\varepsilon}|\\mathbf{\\hat P}|\\psi_\\varepsilon\\rangle & = \\int_{-\\infty}^{\\infty} \\, \\psi^{*}(x') \\, \\left(-i\\hbar\\frac{d}{dx'}\\right) \\, \\psi(x') \\, dx' \\\\\n& = \\langle\\psi|\\mathbf{\\hat P}|\\psi\\rangle \\end{align} ", "a89dd52319d478c6c51b7cb40cdf6079": "\\forall x, x \\ge y \\implies F(x) \\ge G(y)", "a89de6ce4ae796ebfaea5bb7030f3297": "\\sigma_1(W_{r,s})", "a89e18873975c03975d21dbfbb1aa7fb": "(a^{\\mathcal{I}},b^{\\mathcal{I}}) \\in R^{\\mathcal{I}}", "a89e20e6507fadcd899ae8de98f443fb": "m > 2", "a89e376c23ce4121d9ef30498ea75182": " \\left( \\begin{array}{c} K \\\\ k_1, \\ldots , k_n \\end{array} \\right)\n \\equiv \\frac{ K!}{k_1! \\cdots k_n!}", "a89ea86fbe4d7f7995e851d1f65ace7d": "\\frac{1}{\\sqrt{a}}=\\sqrt{\\frac{1}{a}}", "a89eb9f6320cd5ac72fe43d916b3afaf": "\\varphi_{ij}(x)=\\varphi_{j}\\left(\\varphi_{i}^{-1}\\left(x\\right)\\right).", "a89ee733739e6662599cd31e587c8a18": "f_{X,Y}(0,0)=f_{X,Y}(0,1)=f_{X,Y}(1,0)=f_{X,Y}(1,1)=1/4.", "a89f0fa7cd7619219575d0ee6ade1011": "\\int \\tanh ax\\,dx = \\frac{1}{a}\\ln\\cosh ax+C\\,", "a89f2ed640eb35a3cdf13a4bbf240e4a": "\\tau^{a b}{}_{;c}\\equiv (\\nabla_{{\\mathbf e}_c}\\tau)^{a b}", "a89f385703d5d06e2b9a5e946a599dde": "|\\mathcal{X}|=|\\mathcal{Z}|", "a89f50c96c0804c4c75ebc16816e2315": "\n\\begin{align}\n\\mathrm{II}_A & = \\frac{1}{2} \\left( (\\mathrm{tr}\\mathbf{A})^2 - \\mathrm{tr}(\\mathbf{A} \\mathbf{A}) \\right) \\\\\n& = A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}^2-A_{23}^2-A_{13}^2\n\\end{align}\n", "a89fc9373446ff51e7181f57976676ca": "l_\\omega", "a89ff39167034b9a74d250fb0abc938c": "\n\\operatorname{Cov}(x_{ij},x_{kl}) = \\frac{2\\psi_{ij}\\psi_{kl} + (\\nu-p-1) (\\psi_{ik}\\psi_{jl} + \\psi_{il}\\psi_{kj})}{(\\nu-p)(\\nu-p-1)^2(\\nu-p-3)}", "a8a003502e12d9f8da7c69bf1ee0b808": "y(0)=0", "a8a04597f96df270e66e8fe95912794f": "\\displaystyle{\\int_{\\partial\\Omega} u \\, \\partial_nv = \\iint_\\Omega u_x v_x + u_y v_y - u \\, \\Delta v,}", "a8a0fac17a12b624711bf04465414dc3": "\n \\mathbf{v}(x,y) = \\frac{1}{x^2 + y^2}\\begin{pmatrix} Ax + By \\\\ Ay - Bx \\end{pmatrix}, \\qquad\n p(x,y) = -\\frac{A^2 + B^2}{2(x^2 + y^2)}\n", "a8a13e0959ad779ebab8044a2919014f": "3+{\\frac {2} {n}}", "a8a13e0de24c354d5d2ed618a50f8b2e": "\\scriptstyle \\frac{x^5}{120}\\,-\\,\\frac{x^3}{6}\\,+\\,x\\,=\\,0.", "a8a14e687d3fd591e47cea4991f80905": "\\rho_0^{\\sigma}\\,\\!", "a8a176bc59eea44e859be36c11574a65": "\\tau_M \\oplus \\varepsilon^k", "a8a1bdf1b66a10249044202c76ee6dff": "\\operatorname{PSL}_2(\\mathbf{Z}[i])", "a8a283487d4cc2e1ad590c2efd3d6467": "A=\\omega-1", "a8a2abcb94aa5f5c6d6f0c4dad2c07cd": "g(T)v\\in W", "a8a2af72dfaa1504b1c9bee60a19af4a": " \\bold {-\\nabla} \\phi_< = \\frac{3}{\\kappa +2} \\bold{ E_{\\infty}} =\\left( 1-\\frac {\\kappa-1}{\\kappa+2} \\right)\\bold{ E_{\\infty}} \\ , ", "a8a2d64882da916e6490105b13ab084d": "K\\to R", "a8a2f37a51469b2699b3780e9ab2f82a": "S^{n-1} \\to S^n", "a8a332dff74668dfdfb2a848a672619e": "\\frac{\\partial u}{\\partial t} + \\sum_{i=1}^N \\mu_i(x,t)\\frac{\\partial u}{\\partial x_i} + \\tfrac{1}{2} \\sum_{i=1}^N\\sum_{j=1}^N\\gamma_{ij}(x,t) \\frac{\\partial^2 u}{\\partial x_i x_j} -r(x,t) u = f(x,t), ", "a8a3627da4fd411a9a380ba18ebc86e6": "f \\colon \\N \\to \\N", "a8a36704b5d1c811f102b79cb086b432": " H = \\frac{\\left(\\mathbf{P} - q \\mathbf{A}\\right)^2}{2m} - q\\phi \\,\\! ", "a8a39608d0c59ad93d6857aaa55bc49e": " e_i ", "a8a3c81939159738b36e82fb7190e1cf": "\\frac{1-F(t)}{1-G(t)}", "a8a4179dd695830c8c733fe52e1e1128": "U{}^2_n", "a8a48fe88a9834bbfaa425c5401a1143": "W'_{1-i}", "a8a490fb202d0ca7c272d06ca7931ea7": " \\mathbf{B} = \\frac{1}{c} \\oint\\frac{I d\\mathbf{l} \\times \\mathbf{\\hat r}}{r^2}", "a8a4c78a1e0fd28b4a97eb4fb599fff2": "=2f", "a8a4ddf8e7ad1dfb9870a1c6e7b3044f": "d(f(x), a)\\le M", "a8a4fe24100aa978b728a764cd94a402": " \\bold {\\nabla \\cdot (D-P)} =\\varepsilon_0 \\bold {\\nabla \\cdot E}=\\rho_f +\\rho_b = -\\nabla ^2 \\varphi \\ . ", "a8a500d9afa35f299bf6ec8177d2c0ab": "f_s\\left(x \\right) = y", "a8a5022a6f5200ea428329c467e3212f": " n = 10 {\\rm \\ years} \\times 12 {\\rm \\ months \\ per \\ year} = 120 {\\rm \\ months}", "a8a539d4ca636566ba567df10a9b1267": " \\mathbf \\zeta ", "a8a5fe46d21d33ac480d724d489c54c4": "d (\\mathrm{pc}) = 1 / p (\\mathrm{arcsec}).", "a8a60b953e2f8efdff5fb2cefa51a0e4": " y = y_0 + x \\tan \\theta - \\frac {gx^2}{2(v\\cos\\theta)^2} ", "a8a667fe3ce13d07b4dd8e50a84ce8bb": "M = \\begin{bmatrix} 1 & 1 \\\\ -1 & 1 \\end{bmatrix},", "a8a67315e212eaf2ebd1be4e201d1ad1": "\n \\int x^{m+n}\\left(a\\,B\\,c (m+1)-A (b\\,c+a\\,d) (m+n+1)-A\\,n (b\\,c\\,p+a\\,d\\,q)-A\\,b\\,d (m+n (p+q+2)+1) x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx\n", "a8a68e9e6e02a02c730e3b0e8d29517e": " \\frac{d[C]}{dt} = k_1[A]", "a8a6ebc7f5742ffc3f810761aba87cdb": "\\{\\cdot,\\cdot\\}:L_1\\otimes L_1\\rightarrow L_0", "a8a70ec6ab9ee3bc75286c48bc581e08": "0.9999\\ldots \\;=\\; \\frac{9}{9} \\;=\\; 1.", "a8a73b29d6d09266e4f5b4b9637f4acc": "\\begin{align}\n 0 & = \\dfrac{\\partial E}{\\partial h} \\\\\n & \\approx \\dfrac{\\partial}{\\partial h}\\sum_{x}\\left [F(x)+hF'(x)-G(x)\\right ]^{2} \\\\\n & = \\sum_{x}2F'(x)\\left [F(x)+hF'(x)-G(x)\\right ]\n\\end{align}", "a8a784c3a1d3440602f5e7c4c8241b47": "\\gamma_i = - \\frac{\\partial \\ln \\nu_i}{\\partial \\ln V}", "a8a7f845df5e831506d3df0ce1bb6b13": "\\mathbf{\\Pi}^1_n", "a8a808c7b16257a2621786501147a349": "\\textstyle s:\\,t=x^\\frac{1}{1+\\alpha}:\\,y^\\frac{1}{1+\\alpha}", "a8a83b149005a085c42e206bbdf9cd4a": "\\check H^n (X, F) := \\varinjlim_{\\mathcal U} \\check H^n(\\mathcal U, F).", "a8a84fabb9c7c29603d85ab6e7d22604": "\\lbrack RH^+\\rbrack", "a8a86f4f71bc82cbf71dc76b38c1f339": "\\beta = {\\frac{R_1}{R_2}} < 1, ", "a8a87263f1e759c7259c852acdc521e2": "(A\\to B)\\lor(B\\to A).", "a8a9127679c6e682c88b8a34269701a0": "\\frac {D(t)}{D(t_0)} = \\frac {R(t)}{R(t_0)} \\ , ", "a8a92e0f23de34c7f51c88101453a2e0": "V_{in} = A\\cos(wt)", "a8a9479cb3177e5a363693ddc334936c": "\\text{Base ohms}=\\frac{\\text{base volts}}{\\text{base amperes}}", "a8a949e5201b717dc44e4415bc1a3de1": "a = 80", "a8a96959910a49464cf68b8602a7e227": "[M+2H]^{2+}", "a8a97ad35f432e2ca7889e922901bf0c": "H = -\\int_\\Gamma f_{WN}(\\theta;\\mu,\\sigma)\\,\\ln(f_{WN}(\\theta;\\mu,\\sigma))\\,d\\theta", "a8a9923226a28d781795b447358cd2a9": "\\sqrt[3]{1} = \\begin{cases} \\ \\ 1 \\\\ -\\frac{1}{2}+\\frac{\\sqrt{3}}{2}i \\\\ -\\frac{1}{2}-\\frac{\\sqrt{3}}{2}i. \\end{cases} ", "a8a9e8d3b9edd56bd6c2d2ff8877ac8a": "R = \\frac{\\sigma^{(0)}(e^+e^-\\rightarrow \\mathrm{hadrons})}{\\sigma^{(0)}(e^+e^-\\rightarrow \\mu^+\\mu^-)},", "a8aa16e4d361d83147f8655ed26feec4": "\\vec f = (f^1, \\ldots, f^d)", "a8aa2accbcf8949a90570da9b4f9910a": " u_{t^i} = u^i \\, ", "a8aa73b3943f518f3ed8e12074339268": "E = \\frac{\\Delta l}{l}", "a8ab104df9d1aefdf2ecf38bc30ae71b": "{\\mbox{Div}}^0(E').", "a8ab346e046c6cba5699c9a6709f52c6": "F_i(K_i,L_{i-1})=FO(KO_i, KI_i, FL(KL_i, L_{i-1}))\\,", "a8abd6306ba0370a8f88a32484d28794": "\\mathit{x_{i}}\\,", "a8abf0e086ab74777228384fe58bdce5": " \\rho( \\bold T ) < 1 ", "a8ac1e6c8402af1cf434371d03badea3": " A(C;x,y) = \\sum_{i=0}^n A_i x^i y^{n-i} ", "a8ac397363bc7d86aba52d14abc7f324": " H = U+pV\\,\\!", "a8ac521465082fa67a200f1e1367a3ac": "\\begin{align}\nt' &= \\gamma \\left( t - \\frac{vy}{c^2} \\right) \\\\ \nx' &= x \\\\ \ny' &= \\gamma \\left( y - vt \\right)\\\\\nz' &= z\n\\end{align}", "a8ac6da74cb20963ba9efc3d0519ad01": "(k\\times n)", "a8ac6e4893c50e83345659ed9ce3893d": "x + (1 + 1)", "a8aca8e13f13bd6553dd4d55d677153e": "d\\varepsilon|S|\\,", "a8acb8e6fc6a0fea8d1d03ad43add1ac": "h_1+h_2", "a8acb9a79a52a9240fef14434a0a4505": "q = y + \\frac{(z - y) (y - x)}{y + p}.", "a8acd705c10a79f780c93ea9a98f0425": "{\\bar{V}}_3", "a8acd89570f4e9a6fb2199430152ce8f": "F_z = \\gamma m a_z \\,", "a8ad2e27edcea59a5e36e7d4a4a280ff": " \\zeta(s)", "a8ad41a1bb1f22ca55f5ae09391f4652": " x^{n-1} + x^{n-2} + \\ldots + x + 1 = \\frac{x^n -1}{x-1}, ", "a8ae113743a017546a4a974b00238f05": "\\,x \\ll y", "a8ae405b138f43bd31d9fa9ec8870b6a": "-\\frac{2 \\text{polylog}(3,1-p)}{\\beta^2\\ln p}", "a8ae4b8e405946c3be87d88800bf1060": "P_{i+1}(x)=-P_{i-1}(x)", "a8ae77cfde246e910e6a398eff11f5d6": "\\theta t", "a8aea7844b6fd89a272186e72786049d": "- (\\sqrt{-g}g^{\\mu \\alpha }),_{\\alpha }(\\sqrt{-g}g^{\\nu \\beta}),_{\\beta} +\\frac{1}{2}g^{\\mu \\nu}g_{\\alpha \\beta}(\\sqrt{-g}g^{\\alpha \\sigma }),_{\\rho }(\\sqrt{-g}g^{\\rho \\beta }),_{ \\sigma }-", "a8aed703340d75c89d41645853479030": "\\Phi(s) = \\gamma(s) F(s)\\,", "a8af3acb15d987779c5bbfe424452127": "\\mathcal G(4,1)", "a8af9f15f250e7b548ae7c57594f6127": "\n1 \\mapsto I, \\quad\ni \\mapsto - i \\sigma_1, \\quad\nj \\mapsto - i \\sigma_2, \\quad\nk \\mapsto - i \\sigma_3.\n", "a8afe578e199a1f9da8aa8f7effcca42": "u_1 \\not\\in \\mathcal{U}(1, \\tilde{u}).", "a8afea1d030bde57123f388a0bf8f34b": "f_B = \\frac{3}{100} E^{-4/5} yr^{-1}\\;", "a8b06bafd4b7abd5894c5ce1edd0ca2d": "GZ = GM\\cdot sin\\phi", "a8b0c6b668c07fed77a304bbcedd9d3d": "2^{|V|}", "a8b0e47ec9f574e99551e4c1ad0796bd": "\\alpha_n > 0", "a8b100d9bfc1381183525223dfd2c68c": "\\textstyle S(\\mathbf{q}) = 1/N \\langle \\sum_{ij} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{r}_i - \\mathbf{r}_j)} \\rangle", "a8b1019476817e0fdadbbedab560d5fc": "\\chi {{=}} \\chi_{nr} + \\chi_r", "a8b10f205a0efa9db8bc3640a8d56d38": "\\lambda=N-1", "a8b12782582a167991fa4d69040708c0": "\\kappa=|\\boldsymbol{\\kappa}|.", "a8b129a6f85b6f71aae8288b10fcc0cf": "\\partial_x u + \\partial_z w", "a8b153c92bdde70b5dec3d82aabf818f": "\\sup\\nolimits_{T \\in F} \\|T (x)\\|_Y < \\infty.", "a8b17ec99654e15ffeebd339d8a5dead": "S_C = S_L = S_q \\ ", "a8b1b85183954291d15a98c6005b3d6d": " \\theta_m", "a8b1ba8560175d4ee0d511bcab1d701e": "A_\\epsilon = \\left\\{ x \\, | \\, d(x, A) < \\epsilon \\right\\} ", "a8b1c861f8f9e6f73cf49c30a161c1e4": "\\Sigma_{A} = \\left\\{ (\\ldots, x_{-1},x_0,x_1,\\ldots):\nx_j \\in V, A_{x_{j}x_{j+1}}=1, j\\in\\mathbb{Z} \\right\\}.", "a8b1ec44c155e82ff00909a315e1caa6": "\\Psi = \\int_V \\left[{\\hat\\sigma}(\\xi) \\right]^m \\, \\mbox{d} V(\\xi)", "a8b20665bca567477350f0c7601c0bb9": "m_p = \\sqrt{c} \\, \\bmod \\, p", "a8b2304c9b0f160a40b5bbbe08310b14": "\n p(\\xi) = \\begin{cases}\n k_1~\\xi + k_2~\\xi^2 + k_3~\\xi^3 + \\Delta p & \\qquad \\text{Compression} \\\\\n k_1~\\xi & \\qquad \\text{Tension}\n \\end{cases}\n ", "a8b23ddaae2c6b4224fc2f7565e83908": "\\varrho(e)=m", "a8b2c01e54b1cdb3f6c763430cbcd768": "h_{0} = h_{1}^{2} \\frac{J^{\\prime\\prime}(u_{0})}{4\\beta^{2}}", "a8b32b1472fecde76ae71d2e8f939255": "(*) \\qquad e^0=1, \\qquad \\frac{d}{dx} e^x = e^x, \\qquad e^x>0, \\qquad x\\in\\mathbb{R}.", "a8b340317c2afa30d2750d9bd7ba80be": "\\boldsymbol{\\beta}'=\\mathbf{Q}c_0+\\mu\\boldsymbol{\\epsilon}'sc_1+\\alpha'_j\\big[\\boldsymbol{\\alpha}sc_1+\\boldsymbol{\\beta}s^2\\bar{c}_2-\\boldsymbol{\\delta}s^3(2\\bar{c}_3-c_1c_2)\\big]", "a8b38f74622c8330b1fd2744eaf3a012": "\\mathbf{\\Pi}^0_{\\alpha_i}", "a8b3a3005c223b21c36e61959c9e9b09": "\\theta = \\frac{l}{r} \\,", "a8b3b8d4ee8a24ead2708097523e1a3a": " \\left| A_{\\varepsilon}^n(X,Y) \\right| \\leqslant 2^{n (H(X,Y) + \\epsilon)} ", "a8b3bbeb0a2ccffa33218768520cbf8d": "X_i \\in [0,L_i],\\, i=1,2,3", "a8b417c116b28d46aeb4697cc632c9c7": "y(i,a)", "a8b4478932c6eeb7a024843766849bff": "\\nu\\!", "a8b4de66954a83554ee0aa39254d47aa": "\\operatorname{ht}(\\mathfrak{p})", "a8b4f529cd5fc32fdbab226c4dafc45b": "(id\\otimes\\tau_{A,A})\\circ(\\tau_{A,A}\\otimes id)", "a8b51e82ea80740efb48b6d481d6d79a": "f(x,y)=f(x)f(y)\\,", "a8b53bfea22757adfbb6190d396fd24f": "Z^n", "a8b58b8422efcec8504737476040eb07": "f''(x) = 6x.\\!", "a8b5bc64e5f8508b96c701f0fbb83a9a": "\\ T' ", "a8b5cb69544453b4629f85057bed447a": "L(x) = \\sum_{i=0}^n a_i x^{q^i}, \\text{ where each } a_i \\text{ is in } F_{q^m} (\\text { = } GF(q^m)) \\text{ for some fixed positive integer }m. ", "a8b5efa2eed9e27182cdad44e86906fe": "W_{ab}(t)", "a8b612279de7a4b0150ed5160c7ec401": "\\pi [R \\sqrt (R^2+h^2)+h2* \\ln (R + \\sqrt (R^2+h^2)/h)]", "a8b634d4f07b31d2abe8432264813ef6": " f(x)=\\sqrt{x} ", "a8b643ec9e158f0c1d842175c5dc07b7": "p_1, p_2, p_3, p_4", "a8b69776a5b9fcbd5b8d791029d05b38": " \\tau \\mapsto \\frac{-1}{n_G\\tau}", "a8b6a402b2a1756839cbc0547cec6557": "\\phi_\\lambda({\\mathbf{k}})", "a8b6b9f741799e167f92fdfd39a468aa": "c=(c_0,\\dots,c_{n-1})", "a8b721b7bb4a637c2d1a4df912a36a1a": " e^{{-i}\\frac{\\delta}{2}F_{2k}}, e^{{-i\\delta}{G_{2k+1}}}.", "a8b77b70c15097ca2e8280124058dd0d": "\\sigma_{\\rm long} = \\frac{pr}{2t}", "a8b7f885d9728b97428326ea09484df2": "D(X,Y)=\\min_{x\\in X, y\\in Y} d(x,y),", "a8b8a58046c81244b4f08c8efe0a55ca": "z=p+m\\omega_1+n\\omega_2", "a8b907830145ca169511fa9714c6aa56": "e_{0}", "a8b952d9b41d859f1f61554731db9527": "= \\dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}} ", "a8b9a489a0d2142844e56dfae6dca5e7": "\\le (3/2)^k", "a8b9bd5204599937f7a3724f3f2933b3": "p_k(x)", "a8ba6135ebc179fc4dae522bcb4112f4": "\\mathcal C = (\\mathcal C^1,\\dots,\\mathcal C^n)", "a8ba6a1a121be6030ad419f30996de75": "(j=1,\\dots,p)", "a8baaf04fa426735df913d01f7d2194a": "\n\\lambda\\left(x\\right)=\\frac{h}{\\sqrt{2m\\left(E-V\\left(x\\right)\\right)}}\n", "a8bae9c02a6f7abd8d3a8fcf2e7f3b44": "\\nu_{Te} = (eKE/m_e)^{1/2} = 7.26 \\times 10^8 K^{1/2} E^{1/2} \\mbox{s}^{-1} \\,", "a8bb7942b54a12440f1527288937071b": "e_i e_j = \\Bigg\\{ \\begin{matrix} +1 & i=j, \\, i \\in (1 \\ldots p) \\\\\n -1 & i=j, \\, i \\in (p+1 \\ldots n) \\\\\n - e_j e_i & i \\not = j. \\end{matrix}", "a8bbe3b048f42ed9cd4697e007e9367e": "\\scriptstyle >1.19\\times E_{\\mathrm{Pl}}", "a8bc0e9765641195fc3d410077da315e": "w(x) = \\mathrm{e}^{-x^2}\\,\\!", "a8bc142f1c637379fc8aa32421270a53": "\\mathrm{sin} \\, x", "a8bc4be247bfd482e2744c4720910861": "L(\\theta,\\delta) \\,\\!", "a8bc9d2f125c9a85ac9aa6cf650fb00d": "C=\\beta P \\, ", "a8bca1d8a5949e8b8ec730892e916f95": "t I-A = \\begin{pmatrix}\nt-2&-1\\\\\n1&t-0\n\\end{pmatrix}\n", "a8bcb5a29ec9555b4bfdc43627dabe70": "LF = \\sum_i \\alpha_i F(\\mathbf{v}_i)", "a8bcc2e6f0cd881fb7e8979883380752": "w(\\overline{0})=0", "a8bcd0daa951cd2a0bc8034548ee7897": "\\Delta\\,G_s = \\Delta\\,H_s - T\\Delta\\,S", "a8bd142ff50c917129b5088c2bc62fbe": "x'_i", "a8bd184eb0018ba8c59b4a6e2163db2c": "b=\\frac{\\mu_e}{\\mu_h}", "a8bd1ae454a3506c874be77a670c4189": "h(f^A(a_1,\\ldots,a_n))=f^B(h(a_1),\\ldots,h(a_n))", "a8bd2b716d65415361f35824ce164b5d": " (g(T)\\xi,\\eta) = \\mu_{\\xi,\\eta}(g) = \\int_0^1 g(\\lambda) \\, d\\rho_{\\xi,\\eta}(\\lambda).", "a8bd8961c1fed2a467f96a8eb394a790": "\\Sigma^2", "a8bde6cd54c1ed784807d2f5ca88f018": "H=\\frac{p^{2}}{2m}+\\frac{m\\omega^{2}x^{2}}{2} ", "a8be2778b2e4e46706582b9eec38899b": "A(t,z)", "a8bea8cfa38b484c0784ffed3b0ec6ef": "y_1, y_2, ..., y_m", "a8bee933eaf60043698529bef4be0f1c": " I_2 = \\sqrt{3}I_{23} \\angle (phase_{I_{23}}-30^\\circ) = \\sqrt{3}I_{23} \\angle (-120^\\circ-\\theta) ", "a8bf205359a2f0e1d5a9365bf4a35b37": " [ T_f ~, T_g ] = T_{i\\hbar \\{ \\{ f,g \\} \\} }. ", "a8bf62165838560e4982aa8f054cb3d0": "N(m,q,n) = N(q-m,q,n)", "a8bfa675f035514fe0793d95726dc5a8": " 2 m_1 m_2 (\\mbox{cosh}(s_1) \\mbox{sinh}(s_2)-\\mbox{cosh}(s_2) \\mbox{sinh}(s_1)) = 2 m_1 m_2 (\\mbox{cosh}(s_3) \\mbox{sinh}(s_4)-\\mbox{cosh}(s_4) \\mbox{sinh}(s_3)) ", "a8bffdd14f94834663a83109beb8f684": "1+\\sum_{n=1}^{N} p_n \\le \\prod_{n=1}^{N} \\left( 1 + p_n \\right) \\le \\exp \\left( \\sum_{n=1}^{N}p_n \\right)", "a8c0475b63d2a1ec0a77209595339e32": "m_g\\;", "a8c08edec86a0cf79eab04bd4114a4a2": "X \\cap (\\cap_{n<\\omega} J_n) \\neq \\emptyset", "a8c15de7c018682549f5666e346f277f": "\\angle \\frac{dG(s)}{ds}{|}_{s = j\\omega_c} = \\angle G(s){|}_{s = j\\omega},", "a8c1c377c9383b2da6fc4356219e9809": " D_{l} + \\frac{F_{l}}{2} ", "a8c1ccce71bfa1242974a510f4150c80": "2 Au(CN)_2^- (aq) + Zn (s) \\rightarrow Zn(CN)_4^- (aq) + 2 Au (s)", "a8c1db9e7bdea68e360f1e3845c8bf5f": "\\beta + 1", "a8c205cc579a4c659059c4991b15cc8c": "a > b^k", "a8c20f68c3cc1c2426070fff95047884": "\\begin{align}\nD_B &= + 3.059 U\\\\\nD_R &= - 2.169 V\\end{align}", "a8c242985dffe439880c31fd2ecd4a5b": "v_\\mathrm{B|A}=\\frac{v_\\mathrm{B}-v_\\mathrm{A}}{1-\\frac{v_\\mathrm{A}v_\\mathrm{B}}{c^2}}", "a8c28c76176dba0f0120951c3315b1ec": " \\sum_{ i \\in S } x_i \\leq v(S) ", "a8c2e0be63455d3092704dd33f8fc860": " O_t ", "a8c3352cf92d071ad256e419c5016246": "V = M\\,\\!", "a8c37701a8f47110d9ca0510160388ef": "3\\cos((p-q)\\phi)", "a8c3799cca4ee5e0f23b07833ef69cc9": "\\psi(x,t) = \\frac{1}{\\sqrt{2\\pi}} \\sum_n c_n(t) e^{2\\pi i n x} .", "a8c385f46a153b8729b05ae0648372be": "\\operatorname{P\\Gamma L}(V),", "a8c3b4fe88017f2f0be40dca32a73da6": "\\Gamma = dx^\\lambda\\otimes (\\partial_\\lambda + \\Gamma_\\lambda^i(x^\\nu, y^j)\\partial_i) \\qquad\\qquad (3)", "a8c3c6236fbdade9b2a8014f5463260c": " R= \\frac{D}{\\theta}\\,\\!", "a8c3ccf513d8ad6722f82cc9060e7fe7": "c_k", "a8c3ce50fb43f7220aac6d04b7b86340": "n>0, \\,", "a8c3e806ec590286e2028c6d55038398": " \\Pi ", "a8c41ddbfb57d613dcc37712144ddf49": "\\gamma_{\\rm CMB}+p\\rightarrow\\Delta^+\\rightarrow n + \\pi^+.", "a8c43ffa174a895f1d61c6518bc6a823": "H_{n}\\left({x}\\right)", "a8c457dda5d50aa0a0f1f8ed05b4187b": "A(x,G):=\\{a\\in F\\colon \\forall n\\in\\mathbb N,B(x,n\\cdot a)\\subseteq G\\}", "a8c49fd5c3365a232f260a46fb48233e": "(0.5\\times0.8)\\times100=40%", "a8c4a08f2c9ea5ab575eca73163ef40f": "xy=z", "a8c52e38df5331542009f0d9960bb8cc": "\\Delta E^{\\mathrm{tot}}=\\Delta E^{\\mathrm{kin}}+\\Delta E^{\\mathrm{pot}}+\\Delta U\\,\\,.", "a8c56763d881ce6e226345b447e418d1": "\\deg(r)<\\deg(b),", "a8c57a5b0ad4cf125536032deef42459": " \\frac {H^2}{\\mu}", "a8c5d2545eb826f73951fcfb92314bf8": "E_L(z) = E_0 e^{ik_Lz} + r E_0 e^{-ik_Lz}\\,", "a8c62087d22374fb0807f66d183c41ef": "\n M \\approx -\\cfrac{f[3(2h+f)^2+f^2]}{6}~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\n ", "a8c6215417c708b6acd5dfbfd969ac07": "\\frac{A}{a^2} = \\frac{B}{b^2} = \\frac{C}{c^2}\\, ,", "a8c6448c6445b5d8ae295e4973e89b73": "\\theta_m:T_mM \\to \\mathbb{R}", "a8c6c35859320ea4f7a5b3aab82d790e": "\\sigma :\\{\\;1, \\ldots ,n\\;\\} \\to \\{\\;1, \\ldots ,n\\;\\}", "a8c6fcd307b68987bd0ff5cff7964dfd": " \\lambda_\\mathrm g = \\frac {\\lambda} {\\sqrt{1- \\left ( \\frac {f_\\mathrm c}{f} \\right )^2}} ", "a8c74ef4567931ba418cb55f074d9039": "\\mathrm{height}(u) \\leq \\mathrm{height}(v)", "a8c7525325328c189040a9bef86e6092": "\\mathbf{F}_\\text{loop}=\\mathbf{F}_\\text{dipole} + \\mathbf{m}\\times \\left(\\nabla \\times \\mathbf{B} \\right)", "a8c7f25f6307fffbee8d22bcc4df596b": "S(\\omega||\\tau)", "a8c7f95309ecda347e4de7faa7f1caad": "f(a + b) = f(a) + f(b).", "a8c7fb15cb2f536c262af52b8b0ae9dc": "\n\\begin{align}\n\\bigg\\langle\\psi_{nl}\\bigg|\\frac{1}{r^{2}}\\bigg|\\psi_{nl}\\bigg\\rangle &= \\frac{2\\mu}{\\hbar^{2}}\\frac{1}{2l+1}\\bigg\\langle\\psi_{nl}\\bigg|\\frac{\\partial \\hat{H}_{l}}{\\partial l}\\bigg|\\psi_{nl}\\bigg\\rangle \\\\\n&=\\frac{2\\mu}{\\hbar^{2}}\\frac{1}{2l+1}\\frac{\\partial E_{n}}{\\partial l} \\\\\n&=\\frac{2\\mu}{\\hbar^{2}}\\frac{1}{2l+1}\\frac{\\partial E_{n}}{\\partial n}\\frac{\\partial n}{\\partial l} \\\\\n&=\\frac{2\\mu}{\\hbar^{2}}\\frac{1}{2l+1}\\frac{Z^{2}\\mu e^{4}}{\\hbar^{2}n^{3}} \\\\\n&=\\frac{Z^{2}\\mu^{2}e^{4}}{\\hbar^{4}n^{3}(l+1/2)}.\n\\end{align}\n", "a8c8143933cfab47c4cc2d103a5297c7": "L \\subseteq Y\\,\\!", "a8c880aaa65ff0d14e4ca33da2c9c398": "T \\ \\cos \\theta_1 =mg \\,\\!", "a8c89eeabed67f8048c4142cea48e043": "\\mathrm{STA}_{ridge} = \\tfrac{T}{n_{sp}} \\left(X^TX + \\lambda I\\right)^{-1}X^T \\mathbf{y},", "a8c8ca645191182667783e3282591687": " \\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{({\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x}) = \\!\\!\\!\\!\\!\\!\\!\\lim_{\\overset{\\boldsymbol{x}\\rightarrow \\boldsymbol{x}_0}{\\boldsymbol{x}\\in\\Omega_{(-{\\boldsymbol{\\hat{a}}},\\boldsymbol{x}_0)}}}\\!\\!\\!\\!\\!\\!\\!u(\\boldsymbol{x})\n", "a8c96bf050f5c13e2183931f752cef9f": " \\delta_{ext}:Q \\times X \\rightarrow S \\times \\{0,1\\} ", "a8c972b776217d5a65ebe028e7752b07": "\\partial/\\partial t = 0", "a8c9785e739f83fb9e81593187dfa9c8": "\\frac{k(k-\\lambda-1)}{\\mu}.", "a8c98ec729b6f86f73fb683923e29d70": "e(x)=x^ib(x)\\mod (x^n -1)", "a8ca06438b7b400a1c898aaf115a670c": "\n \\begin{align}\n \\delta U & = \\int_{\\Omega^0} \\int_{-h}^h \\boldsymbol{\\sigma}:\\delta\\boldsymbol{\\epsilon}~dx_3~d\\Omega\n = \\int_{\\Omega^0} \\int_{-h}^h \\sigma_{\\alpha\\beta}~\\delta\\varepsilon_{\\alpha\\beta}~dx_3~d\\Omega \\\\\n & = \\int_{\\Omega^0} \\int_{-h}^h \\left[\\frac{1}{2}~\\sigma_{\\alpha\\beta}~(\\delta u^0_{\\alpha,\\beta}+\\delta u^0_{\\beta,\\alpha}) - x_3~\\sigma_{\\alpha\\beta}~\\delta w^0_{,\\alpha\\beta}\\right]~dx_3~d\\Omega \\\\\n & = \\int_{\\Omega^0} \\left[\\frac{1}{2}~N_{\\alpha\\beta}~(\\delta u^0_{\\alpha,\\beta}+\\delta u^0_{\\beta,\\alpha}) - M_{\\alpha\\beta}~\\delta w^0_{,\\alpha\\beta}\\right]~d\\Omega \n \\end{align}\n", "a8ca4423773626c80770e77079f76f4c": "(1-t)g_1+t f_1=0, \\ldots, (1-t)g_n+t f_n=0", "a8ca72df2bde47b38551b612575fdfaf": "y=\\tfrac{1}{x}", "a8ca8111819292a673a6f28a1d719d4e": "x_i \\in [u]", "a8ca9b762687adbc16666794ddf46034": "n = 27225\\, ", "a8cafd1e5aeddc18a5a0991286b4e1e8": "\n\\begin{align} \n& V_{\\text{obs, r}}=\\Omega R\\cos\\left(\\alpha\\right)-\\Omega_{0}R_{0}\\sin\\left(l\\right) \\\\\n& V_{\\text{obs, t}}=\\Omega R\\sin\\left(\\alpha\\right)-\\Omega_{0}R_{0}\\cos\\left(l\\right) \\\\\n\\end{align}\n", "a8cb1496309f2aec867f017dfb3b9ff4": "S_{y_l}", "a8cb1b2059a6d4c986e0a7cb064d3421": "[D_\\mu, D_\\nu] = -igT^aF_{\\mu\\nu}^a.", "a8cb3b3015de91708e2d7ee5ddab1f56": "K=\\left|\\frac{m^2-n^2}{k^2-l^2}\\right|kl", "a8cb4eb76ea41fc70a33c83892c60aa8": "\\frac{d u_i}{d t} + \\frac{1}{\\Delta x_i} \\left[ \nF \\left( u_{i + 1/2} \\right) - F \\left( u_{i - 1/2} \\right) \\right] =0, ", "a8cbbde250e0e37363f635742b9cd9e2": "G = \\frac{11+N}{8}", "a8cc1d66f511247a3683ff6c36f5c293": "\nq_\\star = 1 + \\left(\\frac{3}{4}\\right)^{4/5} \\approx 1.8.\n", "a8cc73c0fb63323173330cac6e73a14c": "dU/dt=AU", "a8cd054826c9398bb7b2d5f8151a6319": "O(h^{p+1})", "a8cd1d71169634c5ea56baf141dc5dd6": "\\theta_1 = \\kappa_1 t + \\alpha_1", "a8cd209436a0d2e7826df47c72e0235a": "\\alpha(s,s) = 0", "a8cd6562825d18e26b095f98e93124d5": "\ny = X\\beta+\\varepsilon.\n", "a8cd6bdf74b6d6b05199ffd32a63ab24": "\\frac{A^{k+1}b_{0}}{\\|A^{k+1}b_{0}\\|}", "a8cd7433fd1d989cfc06f8a187e6acd5": "\\begin{align} & \\hat{A} \\psi = \\hat{A} \\psi ( \\mathbf{r} ) = \\hat{A} \\left\\langle \\mathbf{r} \\mid \\psi \\right\\rangle = \\left\\langle \\mathbf{r} \\mid \\hat {A} \\mid \\psi \\right\\rangle \\\\\n& a \\psi = a \\psi ( \\mathbf{r} ) = a \\left\\langle \\mathbf{r} \\mid \\psi \\right\\rangle = \\left\\langle \\mathbf{r} \\mid a \\mid \\psi \\right\\rangle \\\\\n\\end{align} ", "a8cd8810c9d936343892e62dbff9b9df": "\\gamma^I e_I^\\mu (x) = \\gamma^\\mu (x)", "a8cdd2a99529fba1a9f987eacb747672": " \\delta(u) = \\int_0^\\infty u_t\\, d W_t ,", "a8cde200c9c5c3aed16d425320d1aea6": " m_{0} ", "a8ce22593cf66464a54c2d894aa34fb1": "q_p(1/a) \\equiv -q_p(a) \\pmod{p}", "a8ce2553aac6abf734216d4c225358d9": " \\ Q = \\frac{ \\pi R^4 \\Delta P}{ 8 \\mu_{e} L } ", "a8ce324961bbe781972a111de014970a": "j = 1, 2, 3", "a8ce46e729f6f4d1017f8101092d18f3": "t = [t_0 : t_1]", "a8ce4dad6af9f9d77b08ebf94aa68f19": "(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +\\,", "a8ce74ee528ef7e920c36f8d8d5efd21": "{\\mathbf{}}S(t), 0 \\leq t \\leq T", "a8ceb2e187284b6903b18809f83eef6f": "\\omega_{ab}", "a8cecb890e5938da55e9f821264074b0": "\\overline{\\gamma}_s ~=~ \\frac{1}{n}\\sum_{i=1}^{n}\\gamma_s ~=~ \\gamma_s", "a8cf219dd39c1badb83a9aaac2fa7645": "\\big\\{ \\Big\\{ \\bigg\\{ \\Bigg\\{ \\dots \\Bigg\\rangle \\bigg\\rangle \\Big\\rangle \\big\\rangle", "a8cf61b8ba4ab1d14a9ab98fff566bd7": "\\mathbb{T}=\\mathbb{Z}", "a8cf80e47954398c290f982249c378bd": "E_{edge}=-\\left | \\nabla I(x,y)\\right \\vert ^2", "a8cfa0183e27f2d8928735a3633bb1a3": "{{\\left\\{ \\left| \\left\\langle f,{{g}_{m}} \\right\\rangle \\right| \\right\\}}_{m\\in \\mathbb{N}}}", "a8cfe51734ee8d3d2573266af2c1d7fe": "\\hat{\\lambda}_\\text{UMVUE}\n = \\sqrt\\frac{K}{n-1}\\frac{\\bar{V}}{s_V } ", "a8cfeec44f1261afaaa3eb8ba487e775": " \\phi = \\rm p\\cdot\\lambda - \\rm q\\cdot\\lambda_{\\rm N} - (\\rm p-\\rm q)\\cdot\\varpi ", "a8d03cda6ea34797ccd3d2638076b6a3": "P()", "a8d03d8a7147d7e72879743254e9b653": "F_n \\Bbb R^m", "a8d08006e9411a6dd970a39ff43660c2": "(m+1)(l+1)+d \\begin{pmatrix}l + 1\\\\2\\end{pmatrix} > n ", "a8d0d86ceac6cf5b892282845ddcdfb1": "\\sum_{k}(n)=\\{\\sigma \\in \\sum_{k}:\\sigma(i) \\sim \\forall i\\}", "a8d15a55326920df8d8a9839a5cf9a29": " \\cfrac{\\qquad }{ \\vdash t = t}", "a8d1a4db914c8e3bc1c5ba1bae496680": "\\int_M \\langle d\\alpha,\\beta\\rangle_{k+1} \\,dV = \\int_M\\langle\\alpha,\\delta\\beta\\rangle_k \\,dV", "a8d21f078db525a344c34b83b21d65b6": "\\hat{H} |\\psi_2\\rangle = E |\\psi_2\\rangle", "a8d24e5292489415e12747e6a946417f": "\n \\int_0^b q(x,y)\\sin\\frac{\\ell\\pi y}{b}\\,\\text{d}y = \n \\sum_{m=1}^{\\infty} \\sum_{n=1}^\\infty a_{mn}\\sin\\frac{m \\pi x}{a}\n \\int_0^b \\sin\\frac{n \\pi y}{b} \\sin\\frac{\\ell\\pi y}{b}\\,\\text{d}y = \n \\frac{b}{2}\\sum_{m=1}^{\\infty} a_{m\\ell}\\sin\\frac{m \\pi x}{a} \\,.\n", "a8d2559be3835b7c15c16697923b5456": "\n- \\frac{2 f'(a) f'''(a) - 3 f''(a) f''(a)} {12 [f'(a)]^2}.\n", "a8d290bdbfa3ff04d3e0f2feb69fca47": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 44.7\\cdot 3.30)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 32\\cdot R_{\\bigodot}\n\\end{align}", "a8d2d465fcf31f880f9fb7593dd7f2b5": "m\\leq s", "a8d32afea9a0b3952f64761c5d96969c": "H=\\left\\{ z\\in\\mathbb{C}|Im\\left(z\\right)>0\\right\\} ", "a8d33dd0cb098a850ff82509c385f78e": "I(\\theta) = 4\\int \\dot{s}(\\theta)\\dot{s}(\\theta)'d\\mu", "a8d360139ab46e5f66430400243e467b": "|v|>v_{max}", "a8d38ed4e1a1a0a9b0ee5de4b2e2602c": " p^n\\sim q^n ", "a8d3e5bf5207fa7f5ace3abfd8f34f1e": "\\scriptstyle{{1\\over 2}R_sI^2}", "a8d40bf6dec20fea5fe424ff23f15215": "\\mathcal{E}_\\mathrm{rms}=\\mathcal{E}_\\mathrm{m}/\\sqrt{2}\\,\\!", "a8d413814db86844d7214e6cbe74274b": " \\vec{E},\\vec{H} ", "a8d42b7d20cf36624b1e6cfefee856ab": " \\langle n\\pm 1|H|n\\rangle=-\\Delta ", "a8d43d26c7dea16efccaf6fe0089caa8": "s = -\\tfrac{1}{2}(1+i+j+k) \\qquad t = \\tfrac{1}{\\sqrt 2}(1+i).", "a8d44b7a7d71ff3647f4a71d74fc8b95": "\\mathrm{NCzS}\\!\\!\\!\\Vert", "a8d45e24562f2e9fc935ab6aecba6bfc": "{\\color{Red}\\tfrac{5}{m}}", "a8d493ef8fc549d678af84e7ea277577": "D = \\frac{\\varepsilon^2}{6 \\delta t}", "a8d4fa2838c815d211ce612cbc2ff4e4": "\\sigma_0(n)=\\prod_{i=1}^r (a_i+1).", "a8d50a42790b8cf303b70df396e5f1ee": "R_f", "a8d544fd32e01b659b38ce9ae5d68ec7": "\\omega(X_1,X_2,\\dots,X_n) > 0.", "a8d58e97f497ab82aa6cd1c182438fab": "(L_0,R_0)", "a8d5c0d23d0169c1a90ce601b736c4a8": "\\textstyle \\mathbf{c}_1", "a8d6030eb4c02f365bb3350955952a4e": "q^{2l}", "a8d65f5cacf71cec0878344ca60d4515": "\\mathcal{A}\\cap\\mathcal{B}", "a8d70b80383fda21b50822fa0ea26af4": " \\langle \\psi(t) | \\psi(t) \\rangle = 1", "a8d73641d4dcd4e987ce0cf7a89028f4": "\\operatorname{GEM}()", "a8d745f1882ef673900cbc324fd5bdbf": "\\iota_X\\colon \\Omega^p(M) \\to \\Omega^{p-1}(M)", "a8d77738367c4e7269e178deceb8544f": " {648 \\over 625} ", "a8d7a346012899e5d502ebc5fe93534b": "\\Bbb{Z}_p", "a8d7b9e9f1621de83e0b105bd5a42757": "\\int_{-\\infty}^\\infty |F(x + iy)|^p\\,dx < K ", "a8d7c30545a590fff761bdf106025542": "U-TS", "a8d7c3a4a8d86abd24f335c4650c4f7f": "1\\pi", "a8d82ba09ea67422ba520ba6f02d34c5": "\\sqrt{125348} \\approx 354.045 \\,.", "a8d83d8f899d325ca448cf31e17c9b79": " \\frac{J}{m^{2}sK} ", "a8d884804a69907b628dcd5434562739": "a_G = \\left( \\frac 1 n , \\ldots , \\frac 1 n \\right)", "a8d88e4cd21edf5fed2cdb2beb7f9af9": "x_0 + CS \\times \\frac{A}{N} = 175.5+3\\times -55 / 100 = 173.85", "a8d8cd3041261c3b405d7586ad283c7e": "\nK(x,y;T) \\propto e^{i m(x-y)^2\\over 2T}\n", "a8d8d2155dc37852d817dbecc9a41634": " P_J[f](x):=\\sum_{n\\in\\Z} s^{(J)}_n\\,\\phi(2^Jx-n)", "a8d8dba1c2dfd5d4427fdda40a289413": "R_{\\mu\\nu} - \\frac{1}{2} g_{\\mu\\nu} R = \\frac{8 \\pi G}{c^4} T_{\\mu\\nu},", "a8d8dbe1eebe949529bb070361e99ebb": "x=M", "a8d9501235f2cfca559d6399436aedb5": "\n i \\hbar \\frac{\\partial \\psi}{\\partial t} =\\left( -\\frac{\\hbar^2}{2} \\sum_{i=1}^{N} \\frac{\\nabla_i^2}{m_i} + V(\\bold{r}_1,\\bold{r}_2,\\cdots\\bold{r}_N) \\right) \\psi\n ", "a8d97237962b1118923790dbd2d00d6d": " \\theta_{up} = \\hat\\theta + \\sigma\\Phi^{-1}(1 - \\alpha'\\Phi(\\hat\\theta / \\sigma ) ) ,", "a8d9cbe8d15457ce37ef3a074515b2df": "\n = \\sum_{a \\in A_i} \\sigma^*_i(a) \\text{Gain}_i(\\sigma^*, a) \\quad \\text{ by the previous statements }\n", "a8d9e2eb301defff1dabe0bb0c1eb3e2": " polydeg(f) = \\max (deg(P), deg(Q)) \\,", "a8da0002d7c1e0162ba68a21162312d9": "\\zeta(2) = 1 + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots = \\frac{\\pi^2}{6} = 1.6449\\dots\\!", "a8da18c98d6c5ab86ff616d3fd34686b": "\\left( s x_i,s y_i \\right)", "a8da37013ac87a23211fffa1a55598ff": " {n\\choose k}= {n\\choose k-1}\\times \\frac{n+1-k}{k}.", "a8da38b4e56b3468e5771a33a51bceb9": "\\begin{matrix} \\Delta J = 0, \\pm 1, \\pm 2 \\\\ (J = 0 \\not \\leftrightarrow 0, 1;\\ \\begin{matrix}{1 \\over 2}\\end{matrix} \\not \\leftrightarrow \\begin{matrix}{1 \\over 2}\\end{matrix})\\end{matrix}", "a8dabe394586038f3dbf746610a14452": "\\kappa'", "a8dac7d04bf7f9eda001322e329182d4": "m' A(j)\\}", "a8dd624c1dbc30662a1c161136cefa91": "i = n + 1", "a8dd865e3cfc28c9b054ccb2e6f03d5b": "\\alpha =4", "a8ddc728688cfd420fde904556c4e48b": "V_t = \\frac{g d^2}{18 \\mu} \\left(\\rho_s - \\rho \\right)", "a8ddf7a7b7a262a920dc6d2ce55607ab": " \\vdash \\ \\ \\left[ \\ D \\rightarrow \\left( B \\rightarrow A \\right) \\ \\right] \\ \\rightarrow \\ \\left[ \\ B \\rightarrow \\left( D \\rightarrow A \\right) \\ \\right] ", "a8de1dea199591ff415899c0b9f1d44b": "\\partial f / \\partial z_1, \\ldots, \\partial f / \\partial z_n", "a8de1ea797af7fa3e5f7eaa91f4ffa09": "\\Gamma[\\phi]=S[\\phi]+\\frac{1}{2} \\mathop{\\mathrm{Tr}}{\\left[\\ln {S^{(2)}[\\phi]}\\right]+\\dots}", "a8de4171148883e9eb9595511024a1c3": "\\frac{d}{dy} e^{-2\\pi iy\\xi} = -2\\pi i\\xi e^{-2\\pi iy\\xi},", "a8de493c33f9c21726690fcc39a1a33e": " \\forall A\\subseteq\\mathbb{R}\\dots\\text{ or }\\exists A\\subseteq\\mathbb{R}\\dots\\ .", "a8de4d72820ce6e6cd091a14cbd3dd2a": "\\lim_{m\\to\\infty}\\frac{1}{\\sigma\\,\\sqrt{2\\,m-3}\\,\\mathrm{\\Beta}\\!\\left(m-\\frac12, \\frac12\\right)} \\left[1 + \\left(\\frac{x-\\lambda}{\\sigma\\,\\sqrt{2\\,m-3}}\\right)^{\\!2\\,} \\right]^{-m}", "a8de7501d623f0b2e6cd886bd0a02d9f": "=\\mathbb{E}_\\theta \\left[ V(x(\\theta),\\theta) - \\underline{u}(\\theta_0) - \\frac{1-P(\\theta)}{p(\\theta)} \\frac{\\partial V}{\\partial \\theta} - c\\left(x(\\theta)\\right) \\right]", "a8de7841c8a7ad79df7bb84005851ae7": " \\left \\{ 10 - \\left[ ( \\sum_{i=1}^{5} d_{2i-1} + 3 \\cdot \\sum_{i=1}^{5} d_{2i}) \\pmod {10} \\right] \\right \\} \\pmod {10} ", "a8deaca08b59552acb29a4a52fbfaede": "\\mathbb{C}^1", "a8dedad4fe37a2d1a0881da4bf5b4a18": "\nN=\\int n \\; \\mathrm d s.\n", "a8df3ef78aae57582ba85061fcbf36e8": "\\sum_jM_{j-1}^*/M_j^* = \\infty.", "a8df46cf613a974cb168b8bba526241b": "{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "a8dfe5876d0816fd0af7109919bcb3ca": " t_h = {v_0 \\sin(\\theta) \\over g} ", "a8e007443c4ebeb1e9883be9de653404": "\\begin{vmatrix}\n 1&2 \\\\\n 2&9 \\\\\n\\end{vmatrix} = 5", "a8e0882ec71d378d008490a3827c9a26": "S={1\\over 16\\pi G} \\int d^4 x\\sqrt{-\\eta}[2h_{|\\nu}^{\\mu\\nu}h_{\\mu\\lambda}^{|\\lambda} -2h_{|\\nu}^{\\mu\\nu}h_{\\lambda|\\mu}^{\\lambda}+h_{\\nu|\\mu}^\\nu h_\\lambda^{\\lambda|\\mu} -h^{\\mu\\nu|\\lambda}h_{\\mu\\nu|\\lambda}]+S_m\\;", "a8e093d0ace0a48306d584bc0df78e15": "[n] = \\{1, \\ldots , n\\}", "a8e0bdf02f97d32502418e3cdb67b690": "[X]=[Z]+[X\\setminus Z]", "a8e0d5b054a223e4dfec3cc09581b5a8": "\\{-\\mathcal{P}_i\\}", "a8e1047b5170b067992860641e5f33b3": "\\scriptstyle f (x_1,\\dots,x_k)", "a8e112ea57ffa82928703acb91337883": "t\\upharpoonright n", "a8e117828b5b3ca053b44d0cf6fae5cd": "p\\in \\mathbb{R}^{L}", "a8e136c13c7fa501c017dd37e7657e05": "U^{(n)}=S^{(n)}-T^{(n)}", "a8e13a964310965b7242ca34bc7a7e2d": "d\\omega = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i} dx^i \\wedge dx^a.", "a8e165228c048484ef6778af9bdebb6e": "\\{Ay_n\\}", "a8e19ed8c184a7f40f299800d84beb0c": "\\mbox{div}\\,(\\mbox{grad}\\,f ) = \\nabla \\cdot (\\nabla f)", "a8e1a8ea2e47a77fe3c9526f5a28a01e": "{d \\over dx} p_n(x) = np_{n-1}(x)", "a8e1d5a7f484c1463893b847e5d96cde": " m_A ", "a8e1e8f9e46ab9fb2c685e1107204f3a": "c_n = n^{-1/3}", "a8e1fe9890f69d5c33511d364385b614": "V \\times A", "a8e28f613f5016fd6c758196b19773a7": "b_Y(\\mu_{X1},\\mu_{X2})\\, ", "a8e2da31dc33313da25509461b15de73": "\\Gamma (z;p,q) = \\prod_{m=0}^\\infty \\prod_{n=0}^\\infty\n\\frac{1-p^{m+1}q^{n+1}/z}{1-p^m q^n z}. ", "a8e2ee431194c5a2a7198f2091fb929d": "\\sup_{k \\in \\mathbf{N}} \\mathbf{E} \\big[ \\big| M_{k} \\big|^{p} \\big] < + \\infty", "a8e30a018153e29a0636b2e00b8d6336": "y=\\frac{x_0-x}{m}+y_0", "a8e318fc3c787d887c4234396fecdaee": "\\mathbf{A} = \\begin{bmatrix}\n1 & 1 & 0 & 0\\\\\n1 & 0 & 1 & 0\\\\\n0 & 1 & 0 & 1\\\\\n0 & 0 & 1 & 1 \\end{bmatrix},", "a8e333682f36d94b19dd39792a4b3cd1": " \\overline{\\mathbf{V}} = \\frac {\\Delta \\mathbf{P}}{\\Delta t} \\ ,", "a8e3d83dec1bc26caffc32f5e9e6ff35": "\\vec{x} \\in \\mathcal{S}", "a8e3dc0b83fe6599ad821a23ddfd1f69": " N_B", "a8e47fd635fa4664245035eb7b5c69cd": "((P_1 - N_1) P_2 \\cdots P_n) = (P_1 P_2 \\cdots P_n) - (N_1 P_2 \\cdots P_n)", "a8e48570f6c64e16c6ca6ed48a1e4f21": " d_i(m\\otimes a_1 \\otimes \\cdots \\otimes a_n) = m\\otimes a_1 \\otimes \\cdots \\otimes a_i a_{i+1} \\otimes \\cdots \\otimes a_n ", "a8e485ab775ab8da50a904d29ef42b93": " -2 p \\cdot p' \\,", "a8e49b63f9f7d1629970eee52571fc0d": "{{S}_1=(\\varphi +2(\\varphi^2 +\\varphi^3 + \\varphi^4 \\ldots))(1 +\\varphi + \\varphi^2 + \\varphi^3 + \\ldots)}", "a8e4ca1471e4eafbe4769588e4ae0e50": "=(\\tfrac{95-5}{2})", "a8e4ea27d784e9db028f97c9f167de72": "A^{*}", "a8e52f5942acdaa6a73f46dc49238284": "X_{\\alpha}(0, e) = e", "a8e54100983db2cc4d74e5d08c6f7002": " G\\,\\!", "a8e570839ea7f35b19451276b361a5e1": "\\mathcal{P}", "a8e57754bf1e85b51cf2548186929017": "f=3n_a \\,", "a8e5b25d1bd27faee3ec8a1d8a2a82d1": "\\beta \\le -7", "a8e5baf686bab3055be4615f0c58dfe5": " \\nabla \\cdot \\mathbf{F} = \\star {\\mathbf d} {\\star {\\mathbf{F}^\\flat}} ", "a8e609debc7a90ee15ac309c2e2987c7": "4 \\pi \\left(\\frac{1}{\\pi}10\\,800\\right)^2 = \\frac{1}{\\pi}466\\,560\\,000,", "a8e61bb4e5d1f88b15829523ea7d65df": "\n\\mathbf{R} =\n\n\\begin{bmatrix}\n{(1-d)/ N} \\\\\n{(1-d) / N} \\\\\n\\vdots \\\\\n{(1-d) / N}\n\\end{bmatrix}\n\n+ d\n\n\\begin{bmatrix}\n\\ell(p_1,p_1) & \\ell(p_1,p_2) & \\cdots & \\ell(p_1,p_N) \\\\\n\\ell(p_2,p_1) & \\ddots & & \\vdots \\\\\n\\vdots & & \\ell(p_i,p_j) & \\\\\n\\ell(p_N,p_1) & \\cdots & & \\ell(p_N,p_N)\n\\end{bmatrix}\n\n\\mathbf{R}\n\n", "a8e622381630f610d80ee88ef19ee69d": "L = N\\,l", "a8e66abb26ae091400a109e948431a4d": "\\alpha_j = t_{jj}", "a8e66d4c0b71b03c40eacc3a83ee9352": "\\displaystyle 16(p^{4}+q^{4}+t^{4})=11a^{4}.", "a8e672a34214278fa3298c818c8e8a8d": "\\gamma_2=\\mathrm{atan2}(\\sin\\gamma_1,\\cos\\beta\\,\\cos\\gamma_1)", "a8e68e3baf239fe750d9048cc2075a59": "N \\le 30", "a8e69ebad4da049e37fc208c1e5e2191": "A_f(j_f)=\\dim(j_f)", "a8e6f8654eaadb91e98da97d9ad7b3e5": " X \\sim \\textrm{Kumaraswamy}(a,1)\\,", "a8e6fedc4faa9bc19231dfc0e73dca36": "\\frac{d^2w}{d\\rho^2}+\\left(1-\\frac{2\\eta}{\\rho}-\\frac{L(L+1)}{\\rho^2}\\right)w=0", "a8e73d69efcbcfa4721054acaa019d6d": "\\left( \\begin{matrix}\n E_{p} \\\\\n E_{s} \\\\\n\\end{matrix} \\right)", "a8e7514930b81a8b27eae15b9c802171": "(\\sigma_1 - \\sigma_2)^2 + (\\sigma_2 - \\sigma_3)^2 + (\\sigma_1 - \\sigma_3)^2 = 6\\sigma_{12}^2\\,\\!", "a8e831b2d7106ee8895a6d7afb8a9b6d": "n > 2", "a8e844a60655b295486dfc50af32e7fa": "D^n f : U \\to L^n(V\\times V\\times \\cdots \\times V, W),", "a8e894fb9bacc9a3df571341585f014c": " s_{xy}\\ ", "a8e89af4b7ff5fee3ee8b8332094c191": " f_X(x|\\theta) = h(x) \\left (\\eta(\\theta) \\cdot T(x) -A(\\theta)\\right )", "a8e8b1f9e39be0bdc78b4d905df01f36": " F = { n \\choose 4 } + { n \\choose 2 } + 2.", "a8e8e87248affda940dfa2fc65d05a08": "\n \\begin{align}\n u_r & = -\\cfrac{1}{4\\pi\\mu}\\left[F_1\\{(\\kappa-1)\\theta\\sin\\theta - \\cos\\theta + (\\kappa+1)\\ln r\\cos\\theta\\} + \\right. \\\\\n & \\qquad \\qquad \\left. F_2\\{(\\kappa-1)\\theta\\cos\\theta + \\sin\\theta - (\\kappa+1)\\ln r\\sin\\theta\\}\\right]\\\\\n u_\\theta & = -\\cfrac{1}{4\\pi\\mu}\\left[F_1\\{(\\kappa-1)\\theta\\cos\\theta - \\sin\\theta - (\\kappa+1)\\ln r\\sin\\theta\\} - \\right. \\\\\n & \\qquad \\qquad \\left. F_2\\{(\\kappa-1)\\theta\\sin\\theta + \\cos\\theta + (\\kappa+1)\\ln r\\cos\\theta\\}\\right]\n \\end{align}\n ", "a8e8ff0fb6fa5cb56645d48cf8786e76": "(X_i - X_j)", "a8e9805e624705531acc8d04f8a2ec3f": " \nr e^{i \\psi} = \\int_{-\\pi}^{\\pi} e^{i \\theta} \\int_{-\\infty}^{\\infty} \\rho(\\theta, \\omega, t) g(\\omega) \\, d \\omega \\, d \\theta.\n", "a8e98bac5c15e8d87f8584a806d77dfb": "f^*(\\omega\\wedge\\eta) = f^*\\omega\\wedge f^*\\eta,", "a8e995d0c440ef94144b36d755067849": "a_2 \\equiv b_2 \\pmod n,", "a8e9d5ebf08df36980680992f162bd9f": "X^2 + 1,\\ Y^2 + 1,\\ XY - YX", "a8ea4f2176c77900dd999ef6a27a70a4": "e^{-E(v,h)}", "a8eb55317eab3f507545ac91eb1a91f2": " y(x) = x \\cdot p + (p)^2. \\,\\!", "a8eb6a984583bd577ef472dbc974f762": " a_1b_1 + a_2b_2 = a_3b_3 ", "a8eb889a61509549b7a52458a90a0d75": "\\frac{\\frac{p+1}{2}}{p} = \\frac{1 + \\frac{1}{p}}{2}", "a8eb94e80c8f4694f53abeca3808d83e": "X = \\bigcup_{i\\in J} U_i.", "a8ebc070757eb3637b71f6313dab255d": "(a\\le x \\le b)", "a8ebc3a7ff869f005bb60ca86a6d8517": "\\operatorname{Li}_2\\left(\\frac{3+\\sqrt5}{2}\\right)=\\frac{{\\pi}^2}{15}-\\frac{1}{2}\\ln^2 \\frac{\\sqrt5-1}{2}", "a8ec178ad425b26a201ec9667b67e4e4": "{{|z_1-z_3|\\cdot |z_2-z_4|}\\over{|z_1-z_4|\\cdot |z_2-z_3|}}", "a8ec45f837cff9f6277f343a6bbef67e": " g \\ \\stackrel{\\mathrm{def}}{=}\\ { \\operatorname{tr}(\\mathbf{M}) \\over 2 } = 1 - { d \\over 2 f } ", "a8ecb127dbcd64e78443d1d3f11e6867": "\n\\varphi_1(x,y)=x+y,\\psi_1(x,y)=\\bar{y}-x,\n {\\mathcal E}_1(x,y)=\\exp{\\Big(\\frac{2y}{x+y}\\Big)}", "a8ed131cc1be0ff0da7cef598bce4c5e": "\\textstyle\\beta=2", "a8ede1329590d42967cd98a0799ca896": "y_D = -3.000 x_D^2 + 2.870 x_D - 0.275", "a8edee0ffc4b1d4a0e00c89173914acb": "L= \\left \\{\\lambda\\in\\mathbf{C} : \\ e^\\lambda\\in\\overline{\\mathbf{Q}}^\\times \\right \\}.", "a8ee167f93f649fba98886ef05d7279b": " x_i^{(k+1)} = {{\\left( {b_i - \\sigma } \\right)} \\over {a_{ii} }} ", "a8ee19b4ab41e8cf490398f4f391b91f": "\\alpha_{T}", "a8ee8fce0cf3c5228a2af10476de6b5d": "S (1 + c_n) = a_n^2", "a8eeae01b381cbe9abe32ee52d7b1983": "\\displaystyle{g_{\\overline{z}}=T^{-1}h,}", "a8eeca6dc9309021ba28db0ff0cf3fa0": " \\dot x=x^2,\\quad \\dot y=y.", "a8eedd2d161328945869eeaaa464dbe8": "fH^{n-i} M \\equiv \\mathrm{Hom}(H_{n-i} M; \\mathbb Z)", "a8ef35ff5fee5c808a23f0e9f6c22d87": "=P(X_1=0)P(X_2=0)+P(X_1=1)P(X_2=1)\\ ", "a8ef54a031aed3fa5005f7375daf832e": "x_m-\\varepsilon", "a8ef8efd82d388b74f2029594f258776": "\\Omega^k(\\mathbf{R}^3) = 0", "a8ef9f21afb690d5122864b8cf639042": "q,r \\in S", "a8f0295adac1f515c29af97617e1d92e": "a \\approx l", "a8f02c28d784e89431bbede4e72b779b": "J_{2n} = \\begin{bmatrix}0 & -I_n \\\\ I_n & 0\\end{bmatrix}.", "a8f03022d180a06d732723d7328e0c09": "\\begin{align}\n& (-i\\hbar\\gamma^\\mu \\mathcal{D}_\\mu + mc)_{\\alpha_1 \\alpha_1'}\\psi_{\\alpha'_1 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} = 0 \\\\\n& (-i\\hbar\\gamma^\\mu \\mathcal{D}_\\mu + mc)_{\\alpha_2 \\alpha_2'}\\psi_{\\alpha_1 \\alpha'_2 \\alpha_3 \\cdots \\alpha_{2j}} = 0 \\\\\n& \\qquad \\vdots \\\\\n& (-i\\hbar\\gamma^\\mu \\mathcal{D}_\\mu + mc)_{\\alpha_{2j} \\alpha'_{2j}}\\psi_{\\alpha_1 \\alpha_2 \\alpha_3 \\cdots \\alpha'_{2j}} = 0 \\,.\\\\\n\\end{align}", "a8f05421e9f17a4cd6f8826ba7f26236": "L A = \\frac{i}{\\hbar}[A,H]", "a8f08dfab07b991da8c48c3e7013f9a7": "P_0=Q_0", "a8f0d1aea89f33b435add8fd43534d79": "\\displaystyle{\\int_{\\partial\\Omega} \\partial_n u = 0,}", "a8f0dd358921e259cd77b256de94c1ae": "u(t) = \\begin{cases} b, & \\phi(x,\\lambda,t)<0 \\\\ ?, & \\phi(x,\\lambda,t)=0 \\\\ a, & \\phi(x,\\lambda,t)>0.\\end{cases}", "a8f1ff528095b4bae118421ea256d474": "\\sin\\sin\\sin\\frac1x^k+v-b", "a8f2029dd8b49e05fadf20a0edc045ef": "\\varphi_{\\beta+1}(\\gamma+1) [n+1] = \\varphi_{\\beta} (\\varphi_{\\beta+1}(\\gamma+1) [n]) \\,.", "a8f214dcc8d8b8bf7cc6952864499751": "T_m(T_n(x)) = \\{m\\{nx\\}\\}.\\,\\!", "a8f27c113e472ea3d749a2e0ee0e1451": " B^{\\lambda}", "a8f286910f6e036480e45d6cd66496aa": "\\frac{\\hbar^2}{2 m} \\frac{\\mathrm{d}^2 \\Psi(x)}{\\mathrm{d}x^2} = \\left[U(x)-E_{\\mathrm{n}}\\right]\\Psi(x) = M(x)\\Psi(x), \\qquad \\qquad (1)", "a8f2dea225b68d34c558457c0d3254e2": " n^{*}=\\sqrt{2E_i}/Z^{2} ", "a8f2e8b501391dfd8f46000b18e546c3": "\\mathbf{i}x,", "a8f314ada9f611e68d29be940e941377": "n_n(x) = \\left(\\frac{\\pi}{2x}\\right)^{\\frac{1}{2}}N_{n+\\frac{1}{2}}(x), \\text{ for } x \\ge 0", "a8f3716a45a3a6e52cef10980c42d58d": "_{q\\tilde{\\leftarrow}p=q'p\\,}\\!", "a8f384aef2bc328820b26ac548f5aaf7": "\nI_{0,0}(\\mathbf{R}_{AB}) = \\frac{1}{R_{AB}},\n", "a8f3d04a571182861994ed236739a4d4": "\\Phi_n^A \\Phi_m^B\\quad ", "a8f4075de646e79ad8493fcf11cc5c4e": "\\mathcal L \\left\\{Jf\\right\\}(s) = \\mathcal L \\left\\{\\int_0^t f(\\tau)\\,d\\tau\\right\\}(s)=\\frac1s(\\mathcal L\\left\\{f\\right\\})(s)", "a8f4bb755d141c72ddfd79c0e4b1041e": "\\delta_1=\\operatorname{E}(\\delta_0\\mid S_n=s_n).\\,\\!", "a8f4ce288d32694d6f86ae7cfa787132": "\\{x\\mid Ax=b \\,\\mbox{and}\\, x\\ge 0\\}", "a8f4d56773991aad19b0ced0c3bef239": "\\psi_\\text{q}(x)\\rightarrow e^{i\\alpha/3}\\psi_\\text{q}", "a8f4ea4f4a993e41d69c6f774acfd552": "pK_\\text{a}=-\\log K_{\\text{diss}} = \\log (1/K_{\\text{diss}})\\,", "a8f5045d642538ab1637812c40f9c589": "Z^0\\to b+\\bar b", "a8f514a6365dd75663afd5a88c19b65a": "dX = \\varepsilon\\,dW \\quad\\text{and}\\quad dY=-Y\\,dt.", "a8f529ad5a4b2a755771c430f010faf8": "\\times \\frac{K_{\\lambda - 1/2}\\left(\\alpha \\sqrt{\\delta^2 + (x - \\mu)^2}\\right)}{\\left(\\sqrt{\\delta^2 + (x - \\mu)^2} / \\alpha\\right)^{1/2 - \\lambda}} \\!", "a8f534d7c23c6d53be3e3e2d59334f97": "\nR=|m_1|\\,\n", "a8f5f50d75411365b643542c796a0c75": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{5}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "a8f665f0a38ed31a8a5cc1652c3adfce": "\\dot\\gamma_u = 0", "a8f67ad77ba1b1689784e03fbcd808b1": "[b^{3.1},b^{3.2}]", "a8f684b09bf00e2e71a6962928868278": "n!\\ge \\sqrt{2\\pi n}\\, n^n e^{-n}", "a8f73bcaa9e9c12fd35e12450a3bb73a": "x-a\\quad", "a8f76279126139058c578e1dd1befc0a": "\\scriptstyle x^4 + y^4 = z^2", "a8f7a4b9f2f6fc1213b49028bea32012": "\\forall i \\in \\{1,\\ldots,n\\}", "a8f7c780154a0d64304aba39961bc6a0": "\\omega_\\mathrm{sig} ", "a8f7fa698e8085ce1363724c4485fea2": "(\\xi)_+ = \\begin{cases} \\xi, & \\mbox{if } \\xi > 0 \\\\ 0, & \\mbox{otherwise}. \\end{cases}", "a8f83abb89a32eef6afbdf2c4c1048d5": "I([0,\\infty))", "a8f851e621f260bfb7adfe22a5ffba93": "\\ell_{P}", "a8f8b7049b89a2c6b2d4a628bab53aa5": "z=\\tfrac{1}{2}\\kappa(x^2+y^2)+\\tfrac{1}{3}(a x^3 + 3 b x^2 y + 3 c x y^2 + d y^3)+\\ldots", "a8f8bfebf526e8c34661c7d98090ea15": " p(x) = \\sum_{k=0}^{2K} y_k \\prod_{m=0,m\\ne k}^{2K} \\frac{\\sin\\frac12(x-x_m)}{\\sin\\frac12(x_k-x_m)}. \\, ", "a8f8d41992c6c9e6b8775fb7b6305e96": "(y + \\lfloor \\frac{y}{4} \\rfloor) \\, \\operatorname{mod} \\, 7.", "a8f90816b90179afeb75c8671a2a0b07": "= \\frac{I_{|n|}(\\kappa)}{I_0(\\kappa)}e^{i n \\mu}", "a8f909544cd11a8c8f5cd49e11215b52": "= \\frac{1}{T} \\int_{t_0}^{t_0 + T} \\mathrm{III}_T(t) e^{-i 2 \\pi n t/T}\\, dt \\quad ( -\\infty < t_0 < +\\infty ) \\ ", "a8f90f3b4e565b292b05aa2277d245ad": "\\sum_\\lambda P_{A,B,\\Lambda}(a, b,\\lambda)=P_{A,B}(a,b)\\,", "a8f939a6b89bd9c95919af3edff72bb1": "\\mathbf{L}=\\sum_i \\mathbf{r}_i\\times m_i \\mathbf{v}_i = \\left(\\mathbf{R}\\times M\\mathbf{V}\\right) + \\sum_i ( \\mathbf{R}_i\\times m_i \\mathbf{V}_i ).", "a8f970af0af4adfe87821c9764b7a940": "\\ u_*", "a8f9994fc3cfdbace4786fb5149af30f": "\n\\Gamma_{\\sigma_1,\\sigma_2}(x)\n=\nI*\\frac{1}{\\sigma_1\\sqrt{2\\pi}} \\, e^{-(x^2)/(2\\sigma^2_1)}-I*\\frac{1}{\\sigma_2\\sqrt{2\\pi}} \\, e^{-(x^2)/(2\\sigma_2^2)}.", "a8f9f8a9d6c4255a2a921e6ad43e4455": "\\Delta \\left(mc\\right) \\Delta\\overline{\\lambda}_{C} \\ge \\frac{\\hbar}{2}", "a8fa171c2e85bc96a853a1e26cf65ade": "-j \\sqrt{\\frac{1}{2}}", "a8fa281d70abc6f51658c477ac002fa4": "\n\\mathbf{\\lambda }_\\mathbf{0}^\\mathtt{KED} = \\left\\{ w_1^\\mathtt{KED} (\\mathbf{s}_0 ), \\ldots ,w_n^\\mathtt{KED} (\\mathbf{s}_0 ),\\varphi_0 (\\mathbf{s}_0 ), \\ldots ,\\varphi _p (\\mathbf{s}_0 ) \\right\\}^\\mathbf{T} = \\mathbf{C}^{\\mathtt{KED} -1} \\cdot \\mathbf{c}_\\mathbf{0}^\\mathtt{KED}\n", "a8fa3a1ad97bc25ba16fd8ab39e8142c": "f={{f}_{1}}+{{f}_{2}}", "a8fa431240c859f0722f46c543154569": "(E_\\delta,\\partial E_\\delta) = p^{-1}(P_\\delta,\\partial P_\\delta) \\cap E", "a8fa55c7de554c96a975304724875504": "a = \\frac{27(R\\,T_c)^2}{64p_c}", "a8fa588c143be6e1bcc1f085b6769cc2": "p_1 = 2, p_2 = 3, p_3 = 5,\\dots ", "a8fa71c778372b62574f604abd697dee": "H^p(X) \\otimes H^q(X) \\to H^{p+q}(X)", "a8fa87344c451ff7171db3aca6db19a8": " | a+bi | | c+di | = | (a+bi)(c+di) | \\,", "a8fa9089d87e6d56c8001c61309b3647": "\\nabla^{2}\\mathbf{B} = \\nabla(\\nabla \\cdot \\mathbf{B}) - \\nabla \\times \\left( \\nabla \\times \\mathbf{B} \\right),", "a8fa996d424374f56b15e90b6d1f42d6": " TM_{01} ", "a8faa90a0526647d1bbc5f0ebce22da3": "P(H|D) = \\frac{P(D | H)\\, P(H)}{P(D)}.", "a8fab316c8c0edad4222430e51c0bcc0": "X^\\mu - Y^\\mu = \\frac{1}{m_0^2} P_\\nu J^{\\mu\\nu} ", "a8fb8cf83ef7678ed59382043cea3f96": "\\bar{\\varepsilon}_i=-\\varepsilon_i=n-\\frac{3n}{2i}.", "a8fc34e7103de41b072030691ac2da2f": "\\rho_\\text{earth}", "a8fca295e8d3f396a15790a746ea5086": "y + z = z + y \\qquad\\mbox{for all }y,z\\in \\mathbb{R}", "a8fca3565439fb064fb5b83f8b582226": " \\delta R_{\\mu\\nu} \\equiv \\delta R^\\rho{}_{\\mu\\rho\\nu} = \\nabla_\\rho (\\delta \\Gamma^\\rho_{\\nu\\mu}) - \\nabla_\\nu (\\delta \\Gamma^\\rho_{\\rho\\mu}).", "a8fce659c88591adc3311986e6976d32": "\\int x\\,\\operatorname{artanh}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{artanh}(a\\,x)}{2}-\n \\frac{\\operatorname{artanh}(a\\,x)}{2\\,a^2}+\\frac{x}{2\\,a}+C", "a8fd2fa629fd5465aee701ca491bd264": "x^{*} M x \\leq 0", "a8fd31d8b371c4b5c67d038ad1300304": "\\int_2^5 x^2\\, dx. ", "a8fd61fb601e8405370fd60789163db9": "\n(P1) \\min_{F_l,l=1,\\dots,k \\text{ are q-flats}} \\min_S \\sum_{l=1}^k \\sum_{a_j \\in S_i} \\|a_j - P_{F_i}(a_j) \\|^2,\n", "a8fe221e3ca05ac7c2614487131e2830": "(p+q)^2-(p-q)^2=4pq.", "a8fe400411235b20dbea04f4afaa4116": " \\alpha^{(i)} = \\textbf{W}^{-1} \\tilde{a}^{(i)} \\ mod \\ q ", "a8fe9fb83c708bf828e66159517dcd74": "f^\\prime(x) = x^*(x):= \\arg\\sup_{x^\\star} {\\langle x, x^\\star\\rangle} -f^\\star(x^\\star)", "a8fec5a07954ebbe33e4153284e3984d": "\n \\mathbb{C}^{nm\\times nm}\n\\cong\\mathbb{C}^{nm}\\otimes(\\mathbb{C}^{nm})^*\n\\cong\\mathbb{C}^n\\otimes\\mathbb{C}^m\\otimes(\\mathbb{C}^n\\otimes\\mathbb{C}^m)^*\n\\cong\\mathbb{C}^n\\otimes(\\mathbb{C}^n)^*\\otimes\\mathbb{C}^m\\otimes(\\mathbb{C}^m)^*\n\\cong\\mathbb{C}^{n\\times n}\\otimes\\mathbb{C}^{m\\times m}.\n", "a8fee1a94a7432dbb23c7c9684cbd315": "n\\equiv k \\pmod{2k};", "a8ff72fa56f6faa8d166b6ef06d66909": "GL(n,q^2)", "a8ff837a5ef75e5ed19eb35275b2aba5": " L = |\\mathbf r \\times \\mathbf p| = rmv = n\\hbar", "a8ff9bdc58090c743f4cb10b0d3b27fe": "\n\\bar{\\kappa} \\propto \\rho T^{-7/2},\n", "a8ffa0c0181091b725d79e472c5bbaf4": "u = \\sum_{j = 1}^{n} F_{j} h_{j}", "a8ffc74805e6f33b43c92b48b286a3be": "F(\\mathbf{Q}-\\mathbf{Q}_n)", "a90044c33bef5469d717d8dc88e2b266": "(M_1,d_1),\\ldots,(M_n,d_n)", "a9006d4272e212cc211fad85d4ac35b5": "B[i]", "a901281c64b52159f6fd12183ad07f3c": "A = \\{(x,y) : x \\le 0\\}", "a901528dc1f39d77c12037c6992b891c": "F_1, F_2, \\dots", "a901b2c807ed1cc450dd2514c20307ae": "\\{ e_k \\}", "a901cfc38968ba2341a8482cfa978556": " e^{-H(\\sigma)} = \\prod_{B} \\sum_{k \\geq 0} \\frac{J_B^k \\sigma_B^k}{k!} = \\sum_{\\{k_C\\}_C} \\prod_B \\frac{J_B^{k_B} \\sigma_B^{k_B}}{k_B!}~,", "a901f9d30fdd576f94edb0815f280c30": "\\rho_t: L^{\\infty}\\left(\\mathcal{F}_T\\right) \\rightarrow L^{\\infty}_t = L^{\\infty}\\left(\\mathcal{F}_t\\right)", "a9020a7ddfe45f021961b8cf3a292ac5": "\\mathbb{N} \\setminus T", "a9021ddc046eebe59511f6aeef86ccc0": "\\alpha T=\\frac{T}{V}\\left(\\frac{\\part n R T / P}{\\part T}\\right)_p = \\frac{nRT}{PV} = 1", "a9023f6407fee30e13195425d6f6b905": "r=a_0.a_1 a_2 a_3\\dots.\\,", "a90255b6c2951c05ddd032a940af8043": "a \\oplus b = T(1,a,b)", "a90263eac3cca8eef4c99cdf29bed3e2": " | \\phi \\rangle = {1 \\over \\sqrt{2}} \\bigg( | 0 \\rangle + | 1 \\rangle \\bigg) ", "a90285b02369a28027e7287fcb81b6fe": "\\mathcal{R}_T = \\mathcal{R}_1 + \\mathcal{R}_2 + \\ldots", "a902875a5ccb5f2103f3b9df6fdf51b7": "\\, \\frac{\\partial^2 p}{\\partial z \\partial \\tau} = \\frac{c_0}{2}\\nabla^2_{\\perp}p + \\frac{\\delta}{2c^3_0}\\frac{\\partial^3 p}{\\partial \\tau^3} + \\frac{\\beta}{2\\rho_0 c^3_0}\\frac{\\partial^2 p^2}{\\partial \\tau^2}", "a902fa069ea1235bd72660a4600c35f9": "PoS = \\frac{\\min_{s \\in E} C(s)}{\\min_{s \\in S} C(s)}", "a903131dd1dc310c105ff49c0dbe2a34": "F \\leftrightarrows G", "a903675c940b1c837cd83426e0c2509f": "\\sum_{i=0}^{n-1} f(t_i) (x_{i+1}-x_i).", "a904a00f6c9e5367ad56840a630edfec": "[2^{r}-1,2^{r}-r-2,4]", "a904d3df1670b54d825b8092dbe07b63": "\\,\\Delta \\mathbf{w} ~ = ~ \\mathbf{w}_{n+1}-\\mathbf{w}_{n} ~ = ~ \\eta \\, y_{n} (\\mathbf{x}_{n} - y_{n}\\mathbf{w}_{n}),", "a904e2a690601531f40a4454f4ef9cde": " \\mathit{EVA} \\ = \\ ( r - c ) \\cdot K \\ = \\ \\mathit{NOPAT} - c \\cdot K ", "a9056acff70ea4b577dc30781f1f7e2e": "0\\rightarrow B\\rightarrow E^\\prime\\rightarrow A\\rightarrow0", "a905ba1f9c383b82794b1416b9138403": "G_1\\Rightarrow G_0", "a905d263118cbb7de07623a177eaa5c6": " \\mathcal{R}(\\alpha,\\beta,\\gamma) = e^{-i\\alpha S_x}e^{-i\\beta S_y}e^{-i\\gamma S_z}", "a905dd5b14f1750e52432783abeec5cf": "(Id \\otimes \\Psi_i)(M_i \\otimes I)(\\rho \\otimes \\omega)(M_i \\otimes I).", "a90655a75d80a409202d3cbb3797f286": "\n \\langle j_1, j_1; j_2, j_2 | j_1+j_2, j_1+j_2\\rangle = 1.\n", "a9070bf432f6b44f3685dcdb83f95836": "f_s =\\frac{1}{T} > 2f ", "a9077ec8ed20d3f4b3a42b1d4c736d80": "K^*(s)=\\log[\\text{E}(e^{sZ})]=\\lambda[\\kappa(\\theta+s)-\\kappa(\\theta)]", "a9078cdeab78eed84446fa5bc6de4703": "\\neg P(a)", "a907d345d6ee2e8ee75cd4bfb164787f": "p_i = \\sum_{k = 1}^n d_{ik}r_k =\n\nd_{i1}r_1 + \\cdots + d_{ij}r_j + \\cdots + d_{in}r_n = d_{ij}r_j + y", "a907d4e740aa558228cdf525e67d3829": "|N(S)| \\geq |U(S)|\\,", "a907de7fc97fc7eed42bbfc3edb1a581": "\\tfrac1n", "a90838e4a7e57bd49951644912003300": " R \\simeq 140 \\left ( \\frac{C}{\\lambda} \\right )", "a908580d41690aa6458459118e1be76d": "p_1\\,", "a9085e44fd2c4b5176869c6a68c4bf0d": "\\mathbb{F}_p^{k+d}", "a9087cd777063ba33d9f36cd39abc4f5": "L(x, y; t)", "a9089e05157237686abfd146a64e1632": "T_A^1", "a908d82a5193b93f80b3fd25c299db93": "(0,1)^\\kappa", "a908e4d4a4e160cf743c7c7acf10a401": " g = \\frac{G}{R+G+B}", "a908f139426c5440fe9f9ce455ee3a8d": "\\mathrm{SE}\\left(\\log{\\text{DOR}}\\right) = \\sqrt{\\frac{1}{TP}+\\frac{1}{FN}+\\frac{1}{FP}+\\frac{1}{TN}}", "a908f182c6c844ced748f718f53116db": " \\phi_2 = \\arctan \\left( \\frac{\\sin U_1 \\cos \\sigma + \\cos U_1 \\sin \\sigma \\cos \\alpha_1}{(1 - f) \\sqrt{\\sin^2 \\alpha + (\\sin U_1 \\sin \\sigma - \\cos U_1 \\cos \\sigma \\cos \\alpha_1 )^2 } } \\right) \\, ", "a908ff6ebdd2df23230fd518146523d4": "\\begin{align}\n\\text{social cost}(Z') & \\leq 2 E(Z') \\\\\n& \\leq 2 E(Z) \\\\\n& \\leq 2 \\text{social cost}(Z)\n\\end{align}", "a90959c4a1713fc21e0f9a52da77c3a1": "S_n = S_{n-1}S_{n-2}", "a909f229ac33a4f23c994c0719ea8b26": "\\frac{1}{x \\ln a}\\,", "a909fea5cb5a09b46c27dcc7e1cb7d2c": " p_{T} ", "a90a1ad021cb9bf44135a2ddd584835f": "\\displaystyle{\\|P_k(I+\\Delta)^k f\\|_{(-k)}=\\sup_{\\|g\\|_{(-k)}=1} |((I+\\Delta)^kf,g)_{(-k})|\n=\\sup_{\\|g\\|_{(-k)}=1} |(f,g)|=\\|f\\|_{(k)}.}", "a90a23110f8c9a6340e0f4029f6acca3": "G_0^{}", "a90a53460d05029ad2632aa0b007463d": "\\neg \\forall x \\neg \\phi(x)", "a90a57b888c6d45c04e9d8c98558eb56": "\\overline{\\varphi{(\\beta / \\alpha)}}\\!", "a90b3daee8dd070f6883a270c4fa35be": "R_c=\\frac{{}_{1}}{\\frac{\\cos^2\\alpha}{M}+\\frac{\\sin^2\\alpha}{N}}\\,\\!", "a90b43570cab298b14c781ca29a20669": "Y = \\frac{\\Delta F}{\\Delta t}", "a90b454c1eb3b48bb79674c479876136": " y^2 = x^3 + ax + b, \\, ", "a90b717f962c29fffde7177137c1d385": "\\mathfrak{A} \\subset \\mathfrak{P}", "a90b83513a297c770e2e2813dca6e1ff": " - \\mathcal{L}_X(A)\\,", "a90b9006b5320e0bfa279d6243be1a85": "g' = g{\\rho_1-\\rho_2\\over \\rho}.", "a90b9582fb3d7e290c96a2063dd0aa07": "f: \\R \\to \\R", "a90bbf29a9d8930393d891f0180b4a90": "f_v = [\\frac{v}{2800}]^{1/3}", "a90c043c40b1b28e0afada41ae5a00f2": "\nB(t) = \\frac{1}{2}\\sum_{i=1}^N[a_i(t)^2 + b_i(t)^2 - 2a_i(t)b_i(t)]\n", "a90ca31e220656ec9f51725b5553af15": "\\beta_{\\tau}", "a90d2a23ee3680a6a0de7027c33c6a06": " Chl = antilog(0.366-3.067\\mathsf{R}+1.93\\mathsf{R}^2 +0.64\\mathsf{R}^3 -1.53\\mathsf{R}^4) ", "a90d73338b85d85b864d2b3471cd0415": "4\\times 4", "a90dac3c56d7d72768d9c77ef4407108": "r^\\tfrac{n}{n+1}=a^\\tfrac{n}{n+1} \\cos \\tfrac{n}{n+1}\\theta", "a90dc61274bf9d448d87f40b90c7e984": "\\mathfrak{p}^2", "a90ef098f8196d44993056de8eae5e12": "I_L = \\mathbf{S}\\cdot(-\\sum_{i=1}^N m_i [\\Delta r_i]^2)\\mathbf{S}=\\mathbf{S}\\cdot[I_R]\\mathbf{S}=\\mathbf{S}^T[I_R]\\mathbf{S},", "a90efd2208ad4f20cd83c83fa3108cac": "p^\\mu", "a90f04287ac1ec0a274d7f204ce78793": "{{n^2-n} \\over 2} + (n - 1) - 1 = {1 \\over 2}n^2 + {1 \\over 2}n - 2.", "a90fdbb96304a6b78212ae3c431a15d8": "\\log_{10} \\left( p \\right)= ", "a91059939ff7dfb1226a154f4c80ad62": "\\Delta S = K y \\Delta x", "a91099bc3e7ad50038c989653a962a53": "s = t", "a910ae60ae3c1a60fd77de01b18ed124": "\\phi\\left(x + K\\right) = \\left(x + \\mathrm{Ker} F_i\\right)_{i \\in I}", "a9111af59a02969dddb10fb1607239ea": "\\cos h_0 = -\\tan \\phi \\tan \\delta.", "a91142b5e884c253e120a4e3d4402e19": "\\scriptstyle \\sqrt{12} \\ = \\ \\sqrt{4}\\sqrt{3} \\ = \\ 2\\sqrt{3}", "a91187eb2e0cebcb8b4c7c62301aa87e": "ab = g^{\\log_g(ab)} = g^{\\log_g (a)+\\log_g (b)}", "a911f2d242ffac341d5230bcb59096f7": "{\\mathbf x}_t= t {\\mathbf v} + {\\mathbf x}_0,", "a912056ab3883b245a1304010d12ab33": "\\Gamma_1\\cup\\Gamma_2", "a912497919b445191b1226ef530c6e9e": " \\ N-1 ", "a9126561da9983cf876cffd06dd7b48e": " z\\overline{z}=1", "a912a23f52da45155ae80d036d9431c5": "\\wedge^m_n", "a912b0e1359c4749e61fdb4869dcd938": "w(n)=a_0 - a_1 \\left |\\frac{n}{N-1}-\\frac{1}{2} \\right| - a_2 \\cos \\left (\\frac{2 \\pi n}{N-1}\\right )", "a912c944935d4899bfabca4773c17896": "\n\\mathrm{DTF}^2_{j\\rightarrow i}(f)=\\frac{ \\left| H_{ij}(f) \\right|^2}\n{\\sum_{m=1}^{k}\\left| H_{im}(f) \\right|^2}\n", "a912cae0197cf6ae4b5af0b2437f6024": "T_C\\,", "a912ecf7ea5b67db1a871879cb062dc7": "O(|w_k|^2)", "a9139a0df88e3d7cf4863d46e97b007e": "\\mathbf{q} [t] \\rightarrow \\mathbf{q}' [t'] = \\phi [\\mathbf{q} [t], \\epsilon] = \\phi [\\mathbf{q} [t' - \\epsilon T], \\epsilon]", "a913ad4c216751d5605c102958e041c8": "K_{X_i} \\cdot C", "a913ecf83d86a7fbcf24753b0364d2ac": "i\\quad", "a91411549cf762929c73aa6123b805a7": "\nb_n = \\frac{2 \\gamma_n^2}{ b^2(c^2+\\gamma_n^2-\\alpha^2)J_\\alpha^2(\\gamma_n)} \n\\int_{0}^b J_\\alpha(\\gamma_n x/b)\\,f(x) \\,x\\,dx\n", "a914176190bc116fc09a9b90eb78fdbd": "T_e ", "a9142a027d8f71fb23b7c85c899118a1": "B = 2 ", "a9142ffc326619a5cbd296b1cfb7f18e": " \\frac{\\partial f}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = \\frac{\\partial f_1}{\\partial f_2}~\\left(\\frac{\\partial f_2}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} \\right) ", "a9145fc0a39d29e0a3e6f286bd51d4a0": "| \\mathbf v|", "a914d2e1345b29a61da856d7e6d1fb1f": "\\sqrt{k/m}", "a914eea5c5b023dbcf29955c5e9f1587": "R_{Hn} = -\\frac{1}{nq}", "a914f3ddc7c9f9e08317644f2674073c": "{\\tilde{C}}_6", "a914f45e54ba40a0b145650c69aff4ff": " q_{jk}=\\frac{1}{N}\\sum_{i=1}^N \\left( x_{ij}-\\bar{x}_j \\right) \\left( x_{ik}-\\bar{x}_k \\right) ", "a9151a266b2df400ad7b25e9e1eb9f65": "\\cos\\theta_o=0 \\,", "a915effa0b38db86648762e71effd99d": "a_{12}", "a91617496395c73d0648d7bd2b045e54": "\\,A_x = 1-iv \\ddot{a}_x", "a91631e4b626c4def5dc1a5ace083f54": " P(S^{t+1} \\mid h^{t+1}) ", "a9166fc6cf417f3bbe5b31de939c9367": " F \\left( u_{i + \\frac{1}{2}} \\right) \\ ", "a916aa3801b72d6d2a470cf0355d3b84": " \\forall x \\in \\mathbb{K}^n\\,\\!", "a916f153bfb193276527412552e9f7ec": "k = \\sqrt{\\frac{i\\omega L}{i \\omega C}} = \\sqrt{\\frac{L}{C}}", "a9174c40f1eb531689ee091408c0fe35": "r_2 < \\infty", "a91753bc324ce3d1fb5b1214e1c45f75": "key=(a,b)", "a9175829ec7d590182a5672b1f6295f7": "e^{-k_{eff}r}/r", "a9176e9d9221e7fd5e108d6bc3d2cfa4": "R(t) = K \\int_0^t C_p(\\tau) \\, d\\tau + V_0 C_p(t)", "a917adee1895111b9eeee28b084ba78e": "\\displaystyle i \\partial_t^{}u + \\Delta u = -A u", "a91802785367e41af1abae582283c589": "f^*EG\\longrightarrow M", "a9180ffb5b43b9f58824be97e3081c16": "15795\\, X^2+30375\\, X-59535,", "a91867707d4ddd663f54a96dd0a4d7fe": " \\operatorname{build-param-lists}[g\\ m, D, V, T_4] \\and \\operatorname{build-param-lists}[p, D, V, K_4] ", "a9188058e1aae8dda2b036486798ea72": "e \\in [0..1]", "a91887edab3f767dc2624cbbf18a7a6b": "|\\chi_G(-1)|", "a9188a0b2b6f382df2230b28dbf69d94": "F^{-1}(p)=\\frac{-\\ln(1-p)}{\\lambda}.", "a918c4d45e7fdcb669fb501938afc91d": "\\begin{cases} \\pi_{r, 0}: J^{r}(\\pi) \\to E \\\\ j^{r}_{p}\\sigma \\mapsto \\sigma(p) \\end{cases}", "a918e9c38f2bc3158c9c8f3d897479e2": "\\int_0^{2\\theta}\\log\\left(2\\cos \\frac{x}{2}\\right)\\,dx= \\operatorname{Cl}_2(\\pi-2\\theta)- \\operatorname{Cl}_2(\\pi) = \\operatorname{Cl}_2(\\pi-2\\theta)", "a91907d37cfd52d0c312e8b6b945845c": "\\overline{\\Delta M} = \\overline{M}- M_0 ", "a919286d31e0a9742a809c46b97b6d05": "\n=\\int_1^{a} \\frac{1}{x} \\; dx \\; + \\int_1^{b} \\frac{1}{t} \\; dt\n= \\ln (a) + \\ln (b).\n", "a9194b81952dbf40391f49568d1c189a": "\\alpha = 1/2", "a9197fa7e1d6f84569d8b4ab1f39343f": "\\det(A) = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i = 1}^n a_{\\sigma(i), i}", "a9199729d32a78ed7f79973b462c08d4": "C(i,p,q)", "a919d88d9c2eff28f495c08b96039383": " x_1^2 + x_2^2 + \\cdots + x_{m+k+1}^2 = 1 ", "a91a08ed3e23ca73f194dc4093f70e2c": " \\{ f_n: S_n \\rightarrow T_n \\}_{n=0}^{\\infty} ", "a91a42f0287704c35f826f89db6f8add": " \n H_{ph} = \\sum_{q,v} \\omega_{q,s} a_{q,v}^{\\dagger} a_{q,v}\n", "a91ac26d175c1e176323e0279d6420f1": "0<\\phi<2\\pi", "a91aeab3dac94a7c1533c103e87b2f7a": "u_* = \\min (\\max(u, 0) , 1) \\in D", "a91af40b06da5a013cdc091bf3b50a76": "O^*(Z)", "a91ba114c3d6873be803189d1f85e24d": "\n\\begin{align}\n\\mathcal{L}_\\text{SG}(\\varphi) & = \\frac{1}{2}(\\varphi_t^2 - \\varphi_x^2) - \\frac{\\varphi^2}{2} + \\sum_{n=2}^\\infty \\frac{(-\\varphi^2)^n}{(2n)!} \\\\\n& = \\mathcal{L}_\\text{KG}(\\varphi) + \\sum_{n=2}^\\infty \\frac{(-\\varphi^2)^n}{(2n)!}.\n\\end{align}\n", "a91bdd713b2d1cc86e336dd0bee8e0be": "P=\\frac{dT/dx\\cdot d_z}{|B_z|\\cdot I_y}", "a91bee067c0e81f60fde1ed44a4fd8fd": "\n\\mathbf{a} =\n\\begin{bmatrix}b_x\\\\b_y\\\\b_z\\end{bmatrix} \\times\n\\begin{bmatrix}c_x\\\\c_y\\\\c_z\\end{bmatrix}.\n", "a91c736fb6fccf0accef87962b487362": "(t - 1)^2,\\,t^2 -1", "a91c8d4ea5fa1aede46677082ccfc1b6": " \\frac{1}{2}v_a^2 = GM \\left( \\frac{r_p - r_a}{r_ar_p} \\right) \\left( \\frac{r_p^2}{r_p^2-r_a^2} \\right) ", "a91cfceb7f06e10d0f37819b6df6b99c": "b=F^{-1}(c)", "a91d1a339b31885c908ad63f38cecc93": " A_i = 2 \\pi a_iL_i ", "a91d2982ef45ccb955a7da7149234cdf": "a=2.0", "a91d5de327735580a05abbe8241960c9": "x_{k-1} < x_* < x_k", "a91dab0d5313b5dec73ec9f5e85f51a9": "C\\beta", "a91dcc329c9105293362b0e63b00e29a": "\\mathrm{v}_\\perp=r\\,\\frac{d\\phi}{dt}", "a91ded8586171b2ef6a8be45a80972cd": "1 + w^{T}(D - \\lambda I)^{-1} w = 0", "a91df2d1ff9885cce9b85f451ba7dc3f": "(x_1, y_1, k_1)", "a91df3ab9a04ae0297dd9eb12afc4dab": " e^{-\\varphi} \\partial_x e^{\\varphi}= \\partial_x+\\varphi_x, \\quad e^{-\\varphi}\\partial_y e^{\\varphi}=\n\\partial_y+\\varphi_y,", "a91e58667552aa099607ae1889c78abc": "g\\ge 1", "a91e87cead3205662d02cf512e3ec3dc": "\\operatorname{Li}_2(x) \\,", "a91ec7d594454b37299b1d8f698a391f": "M_1\\prec_K M_3, M_2\\prec_K M_3", "a91ee3bf77a84dca888c0a0fb96b9737": " \\log_{10} \\left ( K_{eq} \\right ) = -\\Delta G^\\circ_{form} /(19.1448T) ", "a91ee693912454496a60aa45cb8f98d0": "X = (X_t)_{t = 0}^T", "a91f1c9b1ecd2d371d5471d84e51ef0f": "E_{rot} = \\frac{L^2}{2 I} \\,", "a91f54acf336b40fe792965b662996d4": "CW=\\text{Crosswind}", "a91f5a43d9e07c6a41c165b23f0bb0a3": "\\vartheta(z; \\tau) = \\sum_{n=-\\infty}^\\infty \\exp(\\pi i \\tau n^2) \\exp(\\pi i z 2n) = \\sum_{n=-\\infty}^\\infty w^{2n}q^{n^2}. ", "a92052cceb40c9b4b7403168451fd691": "k_{\\rm C}=1", "a920589303df57d313875de68c3af3ae": "\\alpha>0.5", "a9205986e240a225eb2a1bc76287130c": " \\alpha N_{vs} >> 1", "a920b5b81daa6c6ece1f0880932a7fb2": "2.5 \\theta = 1.385^\\circ ", "a920c6ca918eacc179a5ef3bd0817850": "\\mu=\\frac {1} {H}\\sum_{i=1}^H (\\frac {z-y_i} {z}) ", "a9214aee6ea57c28a5332eabbb920bce": "\n\\begin{align}\n\\ln \\Pr(Y_i=1) &= \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i - \\ln Z \\, \\\\\n\\ln \\Pr(Y_i=2) &= \\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i - \\ln Z \\, \\\\\n\\cdots & \\cdots \\\\\n\\ln \\Pr(Y_i=K) &= \\boldsymbol\\beta_K \\cdot \\mathbf{X}_i - \\ln Z \\, \\\\\n\\end{align}\n", "a9216a226b45380b28ef259a362c0606": "X(0) = 0 = X(L),", "a9217127399baac5cfca4d0d1467e1d6": "W'=\n\\begin{vmatrix}\ny_1 & y_2 & \\cdots & y_n \\\\\ny'_1 & y'_2 & \\cdots & y'_n \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\ny_1^{(n-2)} & y_2^{(n-2)} & \\cdots & y_n^{(n-2)} \\\\\n-p_{n-1}\\,y_1^{(n-1)} & -p_{n-1}\\,y_2^{(n-1)} & \\cdots & -p_{n-1}\\,y_n^{(n-1)}\n\\end{vmatrix}\n=-p_{n-1}W.\n", "a921992cb4b193b73322c264dbce484e": "P = I ^2 R ", "a921d09e2da11cc0a91ab067d5773fe9": "c_1(E), \\ldots c_{n}(E)", "a921e6034c35dce0208afd4d67009897": "e_{H_i}", "a921e8ce106a2951c282169db0c09d72": "\\mathrm{OTF}(\\nu)=\\mathrm{MTF}(\\nu)e^{i\\,\\mathrm{PhTF}(\\nu)}", "a92217000684d169b1f018017245c652": " \\frac{1 + \\sqrt {1+4*L}}{2}", "a9227b4b63e0b187c6152e99d49dd9d5": "{d \\over dt} \\left[ \\left( 1 + {1\\over 2} { v_1^2\\over c^2 } \\right)m_1\\mathbf v_1 \\right]=0", "a922dd58c77c5f256ccfda9412a42fe2": " \\beta(a) ", "a922e4b44a9bb5424c4ddffa2e95e7fc": "\\operatorname{E} \\left[\\left|N\\left(\\mu, \\sigma^2 \\right)\\right|^p \\right]= \\left(2 \\sigma^2\\right)^\\frac p 2 \\frac {\\Gamma\\left(\\frac{1+p}2\\right)}{\\sqrt \\pi}\\, _1F_1\\left(-\\frac p 2, \\frac 1 2, -\\frac{\\mu^2}{2 \\sigma^2}\\right),", "a9231d4bfb7f734255293861cb4328f7": "\\pi=p_1\\phi", "a92342f6400384ed1042b134432d9fa2": "\\, -i", "a923ccb7e74a5259e39c38265d75fa83": "X_1 = f(N_1)\\,", "a923d3ce9b9159333a4d3d4ec27a1802": "u_{i}^{n + 1}\\,", "a923e4d0304ddcf023c67b79abeafac8": "N'(x)", "a924230f6349efd996a556c946e2e505": "f(r_1)=3^2-2=7", "a92428c76e8f2769c47f9e8d3dfb0934": "\\dot\\gamma(0) = V.", "a92485ba53dc92cc272dba0305f02a78": "\\frac{1}{2\\pi}\\sqrt{\\frac{\\mu}{I}}\\,", "a9248d3a798e73949401836daadf25dc": "\\begin{align}\n &&N_{X \\cup Y} &= N_X + N_Y - N_{X \\cap Y}\\\\\n X \\cap Y = \\varnothing &\\Rightarrow &N_{X \\cap Y} &= 0\\\\\n &\\Rightarrow &N_{X \\cup Y} &= N_X + N_Y\n\\end{align}", "a924992c62cc9742a23b2b31383ec24a": " P(x;\\;y_0,\\;y_1,\\ldots,\\;y_n) ", "a924a24e4263aa5e3b39d44993410ae1": "\\tilde{U} = V - U", "a924a760f32917f88838ca45286c7fe2": "p_a(p_b(s))=p_b(p_a(s)).", "a924ba98b486e15878e0bb00b9d7fcd6": "\\int_{D} f(y) \\, G(x, \\mathrm{d} y) = \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau_{D}} f(X_{s}) \\, \\mathrm{d} s \\right].", "a92514ea0778e4fb9efcda3d318680c6": " \\Delta E = h\\Delta\\nu = h c \\Delta \\left( \\frac{1}{\\lambda} \\right ) = hc \\frac{\\Delta\\lambda}{\\lambda^2}", "a9252f186c674b62fcee746418de3c6b": " \\theta^{*} = \\theta^{MAP} ", "a9255917ad9aedd520f03fe592994b54": "[I \\land C ]\\;S\\;[I].", "a9258f195a5886c23d6f2c4f363a649d": " { c_B } = { \\left( { D \\omega^2 \\over \\rho h } \\right)^{1 \\over 4} } ", "a925a87de6ac0e2855fb412f8fed8ba7": "log_2(M_o) + log_2(M_e)", "a925cae72899beb404aa1186081ada53": "\\sum_{i=1}^3 u_i = 1", "a925da06fc230acda2f82e4a42dce125": "f(X)\\,\\,\\left( X\\in \\mathbb{R}^{p} \\right)", "a925fe7afcca9a599b51711d346ee1b2": " \\{\\varepsilon_{t},t=0,\\infty\\}", "a9262be4e122a8ec40ab6b4b6582594a": " \\lambda = \\left( \\dfrac{1}{ \\dfrac{1}{\\lambda_1} + \\dfrac{1}{\\lambda_2} + ... + \\dfrac{1}{\\lambda_n} } \\right) = \\left( \\dfrac{1}{ \\sum_{k=1}^N \\dfrac{1}{\\lambda_k} } \\right)", "a92658a6efac0c2a0887541d3fe8d083": "\\mathcal{O},", "a9268d302a97670214f6032e4e879394": "F(8)", "a926d9d697994ef648154f10c275710e": "R_i=\\sum_{j=1}^i r_j", "a926e2734cf3b6de27a41f0c8e58a713": " N \\ ", "a9271014af2d8454785f759ce772e41f": " L_2 ", "a92723ef85d003eb654d98e550495355": "\\mu _{X}(k) =\\left( \\frac{d_{2}}{d_{1}}\\right)^{k}\\frac{\\Gamma \\left(\\tfrac{d_1}{2}+k\\right) }{\\Gamma \\left(\\tfrac{d_1}{2}\\right) }\\frac{\\Gamma \\left(\\tfrac{d_2}{2}-k\\right) }{\\Gamma \\left( \\tfrac{d_2}{2}\\right) }", "a9276f63355a52cda1b904a54b6cc08c": "\\frac {[A_{ad}]}{p_A[S]} = \\frac{k_{ad}}{k_d} = K_{eq}^A", "a927ea42a590451fbd72398711c25990": "= \\begin{vmatrix}\\! \\boldsymbol{i}& \\! \\boldsymbol{j}& \\! \\boldsymbol{k} \\\\ 0 & 0 & \\omega \\\\v \\cos \\alpha\\quad &v \\sin \\alpha\\quad &\\quad \\\\ \\; + \\omega t \\ v \\sin \\alpha & \\; -\\omega t \\ v \\cos \\alpha & 0\n\\end{vmatrix}\\ \\ , ", "a928cd25a883a864f31f45e8ad87191b": " \\operatorname{J}: (\\xi, \\eta) \\mapsto (-\\eta, \\xi). ", "a928cd78c329d1784d340d9a4a546569": "E + iB \\rightarrow e^{i\\theta}(E + iB)", "a928edfb512de996b7b89f6de88e607b": " \\frac{d h}{d r} = \\frac{m \\omega^2 r}{m g} ", "a928ef5d76376855648c5c062ac03bc5": "\\mathbf{D} = \\varepsilon \\mathbf{E} = \\varepsilon_{\\text{r}} \\varepsilon_0 \\mathbf{E}", "a928fed5497665ffaace77a120945c2e": "\\operatorname{Hom}(V, W)", "a929590dcbb123441cb43c9de4002294": "xx^{-1}=x^{-1}x", "a929820a9c749c53bfce89ee6126a7da": "B\\left(z\\right)=\\frac{\\hbar k}{\\mu'}v-\\frac{\\hbar \\delta}{\\mu'}=\\frac{\\hbar kv_{i}}{\\mu'}\\sqrt{1-\\frac{2a}{v_{i}^{2}}z}-\\frac{\\hbar \\delta}{\\mu'}", "a929c803a186e54f9330821447bc7b1b": "\\mathcal N(m_k,\\sigma_k^2 C_k)", "a929d44a5221cf609f263984cd4c5d42": "V_0 T", "a929fbe99839de2676ef179aad301d14": "\\color{red}y", "a92a41f4d650343731d454958a8492be": "\\tau(X,Z)", "a92a5e88057522faaf29f3d081e9ea68": "MP=O", "a92a606843384cd749ea0038bbeb20a5": "\\epsilon'_1,\\epsilon'_2", "a92aa784f266167c0a945921d4aaeaea": "(\\mathcal{H},F_1,\\Gamma)", "a92ae61b92bad52558ef2df01a1f7816": "r = 2 a\\cos(\\varphi - \\gamma)", "a92b34eacf588764d6524423a6deea83": "\\operatorname{Tr } K = \\int K(x,x)\\,dx", "a92b449313e199c1b06b8f221c158069": "\\underset{i}{\\overset{k<3}{x_j}}(t_0)", "a92b8cf080cbb8b3eead613c22308071": "\\frac{\\beta_T}{\\beta_S} = \\gamma", "a92be15ae499b8219f21b83bff2df96c": "A x= \\lambda B x.", "a92c11c6fa06099255322ba39f5fb571": "\\mathrm{th}^n_k(x_1,\\dots,x_n)=\\begin{cases}1&\\text{if }\\bigl|\\{i\\mid x_i=1\\}\\bigr|\\ge k,\\\\\n0&\\text{otherwise.}\\end{cases}", "a92c1a7339914aac595e9c002074fde4": "G \\triangleright Sp(2a,q)\\text{ and }p^d=q^{2a};", "a92c4b0a676ef94c934a87cf91a59163": "\\mathit l^{\\prime} \\hbar", "a92c6c307adf1868b54e2e02e21d26b0": "\\vec{t_1} \\subseteq \\vec{t_2}", "a92cf6aa1aa5726c5cded701cdd83f49": "f(z) \\; dz", "a92cfa332f62bc6a056a088e16799e87": "\\sum_{\\mathrm{edges\\,}e} u_e w_e d_e", "a92cfc7375f716d709041bb6c4c0d250": " a \\leq x \\leq b ", "a92d4811d7e6f7e1f2fb997c498b5fd4": "\\langle p,x\\rangle-f(x)=\\langle p,x \\rangle-\\langle x,Ax\\rangle-c", "a92d5442651a4ed53be36f6dd9d107ba": "\\gamma_{xy}(\\tau) = \\operatorname{E}[(X_t - \\mu_x)(Y_{t+\\tau} - \\mu_y)],", "a92d6dda73c4747969caa973b8cafd2b": "T_s = T_a + T_o", "a92d7a901bd33121dc2efe932b89e980": "\\mathbf{j}\\times\\mathbf{B} = 0", "a92e4297ed6656db4bdaf6d55768c0a4": "\\|x\\|=\\|(x+y)-y\\| \\le \\max \\left\\{ \\|x+y\\|, \\|y\\|\\right\\}", "a92e58554792c3770a4b2dfde4f51932": "\\sigma_N /f'_t", "a92e86d2129849d08264a7db5445d070": " U_\\omega |s\\rang = (I-2| \\omega\\rangle \\langle \\omega|)|s\\rang=|s\\rang-2| \\omega\\rangle \\langle \\omega|s\\rang=|s\\rang-\\frac{2}{\\sqrt{N}}|\\omega\\rangle ", "a92e899a92757d7210024ea8f30a563e": "\\sqrt{1-\\beta^4}", "a92ecd55512f4d72022ae6d617a71852": "\\left \\{ z=(z_1, z_2, \\dots, z_n) \\in \\mathbf{C}^n \\ : \\ \\vert z_i \\vert < 1, \\mbox{ for all } i = 1,\\dots,n \\right \\}.", "a92f28044c7d86d62b8b5c862d13af0d": "n := \\sqrt{\\epsilon_r\\mu_r}", "a92f6e38173c5801589a161ac39bcdd6": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^1 = \\frac{1}{1} - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9} - \\cdots = \\arctan{1} = \\frac{\\pi}{4}\\!", "a92f96c8e9f0a7067d1d06b5660ee62e": "non(\\mathcal{N}) \\geq \\kappa", "a92f988075016b0c08c7e95aae8925c4": "\\mathbb{CP}^1 \\subset \\mathbb{CP}^n", "a92fa004558805c4f9cd8dca69276447": "\\lim u, \\liminf v, \\limsup w \\!", "a92fc281f954bcdbf144f9ebffba6d52": "X = R_0 \\cos \\left( \\frac{\\omega \\, \\Delta s}{\\sqrt{1-\\omega^2 \\, R_0^2}} \\right), \\; Y=R_0 \\sin \\left( \\frac{\\omega \\, \\Delta s}{\\sqrt{1-\\omega^2 \\, R_0^2}}\\right)", "a92fcc2567bb678b88c506cd0a475dbc": "\\prod _x \\operatorname{sexp}_a(x) = \\frac{C\\, (\\operatorname{sexp}_a (x))'}{\\operatorname{sexp}_a (x)(\\ln a)^x} \\,", "a92fe77193841c419cd44533f88e9024": "\\left [ \\hat{O}_{ij} , \\hat{O}_{k \\ell} \\right ] = \\delta_{jk} \\hat{O}_{i \\ell} - \\delta_{i \\ell} \\hat{O}_{kj}.", "a93035c7c6f1d0f913a55ccba8bdd22d": "\n\\sigma^2_\\mathrm{weighted} = \\frac{\\sum_{i=1}^N w_i \\left(x_i - \\mu^*\\right)^2 }{V_1}\n", "a930526b49bd8f5c6d8c6dbedaa78298": "\\theta_0 = (\\arg(a)+\\pi-\\arg(c_k)) /k", "a93058d6863ccd80352b7f5c8c6ad40b": "=\\kappa_1(\\kappa_4(X\\mid Y))+4\\kappa(\\kappa_3(X\\mid Y),\\kappa_1(X\\mid Y))+3\\kappa_2(\\kappa_2(X\\mid Y))\\,", "a930828240cc892e3a54a704b340f60f": "\\rho(X,Y)\\,\\stackrel{\\text{def}}{=}\\,\\operatorname{Ric}(JX,Y)", "a930b327b0f12473315dba0cc12214b4": " {N \\hbar \\omega \\over V} = \\mathcal{E}_c = \\frac{\\mid \\mathbf{E} \\mid^2}{8\\pi}. ", "a930bc7f8fb5f334cfcae85fa116cb70": "\\iota: S \\hookrightarrow X", "a930fa3c6e25c91c3229ab5efb8f83ea": "n^{0.1}", "a93139b50e71e0128a9be8a7a6fbe7b5": "\\mathfrak{P}^{99}", "a931c2d5d3d069f87004984afc40303c": "r_{433}", "a931efb0af085b66edc77cf82ae2b3af": "> 10^{29} \\ \\mathrm{years} \\,", "a93217a88a7bf073d91a7f35f0c4f50d": "6 \\cdot \\left[64 - {2 \\choose 1}\\right] + 24 \\cdot (64 - 1) + 6 \\cdot 64 = 2,268\\,", "a93237094d6a6eed9e7d35297d32e153": "\\frac{1}{\\sigma^2+\\tau^2}\n\\left[\\frac{\\ddot{S}}{S}+\\frac{\\ddot{T}}{T}\\right]+\\frac{\\ddot{Z}}{Z}=0\n", "a932585b0bf894bf10fdf12e791aec6f": "n=\\frac{c}{v_\\text{phase}}", "a93339105a7ba7ba6b4514adc5425098": "y = x + \\frac{b_1}{2\\,b_2} \\!", "a9334b733539861f9a80b51215f122c2": "\\sum_{j=1}^m a_{ij}N_j=b_i^0", "a9336b4e8d6fbd115a0f539e0315be21": "\\kappa_j'=\\mu_j'", "a933eb22bfb9acbf730ea9ba1b67ffda": "\\left(\\tfrac\\cdot n\\right)", "a933fc50b0ff0624968cf87e5884646e": "U_S (t, m,a^*(m))", "a934bcffc2e7418c27de32c63334d817": " IMM_8(S_x) ", "a934c53329a4339f6d6ef2a5453bf461": "\\binom{t}{i+1} = \\binom{t}{i} \\frac{t-i}{i+1}", "a935052f2906d5c0c37052a763e28a88": " \\textstyle n", "a9350a4134a7f86123c8a1ee36408d1b": "a_{227} = 7", "a9353bff922c5f9e0c482da1d034f8bf": "\\partial_t |f_t(z)|^2 \\le 0.", "a935c3018c970005a7c0412a9462416a": "\\phi=\\phi_{intf}", "a9360781e067df4ff07f7746f63ecb12": "z = 7.6", "a936372c13b2bc6662d40f584429c870": "\\hat{F}^i_{ab} \\dot{\\gamma}^a \\dot{\\gamma}^b", "a93640d72f4297e74e63993709b0a28f": "\nI_{l} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\int d\\mathbf{r}^{\\prime} \n\\frac{\\rho(\\mathbf{r}^{\\prime})}{\\left( r^{\\prime} \\right)^{l+1}}\nP_{l}(\\cos \\theta^{\\prime})\n", "a936decceee1bcc631e065e495bfe472": " \nP = \\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix} : \\begin{matrix}W \\\\ \\oplus \\\\ W' \\end{matrix} \\rightarrow \\begin{matrix}W \\\\ \\oplus \\\\ W' \\end{matrix}.\n", "a936e302358ad9716138bc44c8c07e41": " p + l(a - n - 1) + b(2an + 2a - n^2 - 2n - 2) - m ", "a93727f92f7958905223842e3d8c74f0": "P_2 = \\frac{1}{14(14 - 1)} \\left(0^2 + 2^2 + 6^2 + 4^2 + 2^2 - 14\\right) = 0.253", "a9372a7fee1523bac8e3308bc800c633": "t_s = \\frac{2 n_1 \\cos \\theta_\\text{i}}{n_1 \\cos \\theta_\\text{i} + n_2 \\cos \\theta_\\text{t}}", "a937a9844d10b280455deaa88d272630": "f_\\mathrm{V}\\approx 0.5346 f_\\mathrm{L}+\\sqrt{0.2166f_\\mathrm{L}^2+f_\\mathrm{G}^2}.", "a937cd7e035210b599a4ae1d3e3e722c": "h \\; = \\; h_L \\; - \\; h_O", "a937ddb790c39d986e27718a8982c6b7": "x=-1, x=1;", "a938a751fd22d9c195a3330a2280f7f7": "(Z+\\mu)\\sqrt{\\frac{\\nu}{V}}.", "a938bd89ae4720de8ee8ed535fd64db4": "\\forall xy\\, [ x\\neq y \\rightarrow \\exists z\\neq y\\, C(x;yz) ].", "a9391a8df01ca5b31c48da2afae894e0": "\nD \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{E^{2} + R_{tot} K_{d}}\n", "a939b835b7be3ea31a616c8d09926581": "(8) \\,", "a939ba0e497b03479a34d4ab3888074d": "\\nabla \\cdot F = \\langle \\nabla F \\rangle_{r-1} = e^i \\cdot \\partial_i F", "a939c46dcc0d7d70874d569c890cb02a": "= \\frac{2 h_t h_r }{d}", "a939cd06f5a4b77c4e242e750630e9dd": " ||p||_{\\tau} =\\left( { \\frac{1}{2\\pi} \\int_{0}^{2\\pi} |p(e^{i\\theta})|^\\tau \\, d\\theta } \\right)^{1/\\tau} \\,", "a939dbecb36b032f4fad09a3eac78dc5": " \\mathbf{x} = \\sum_{i=1}^n x_i\\mathbf{e}^i ", "a93a05b328ab8679b8a48f297646ba3e": "a > b\\,\\!", "a93a495198538449466aea78c1b22f29": "f(x)={f(x+1) \\over x}\\,\\!", "a93a65945666ebee16c56d5d4b26ac24": " I(\\mathbf r) = \\int U (\\mathbf r,t) U^* (\\mathbf r,t) dt \\propto A_1^2 (\\mathbf r)+ A_2^2 (\\mathbf r) + 2 A_1 (\\mathbf r) A_2 (\\mathbf r) \\cos {[\\varphi_1 (\\mathbf r)-\\varphi_2 (\\mathbf r)]}", "a93a6e353c5c0dce06ca0c455d82922f": " K\\times K ", "a93a8bca4138c130156d07a31c009d6f": "u_t-\\nabla^2u_t+\\operatorname{div}\\,\\varphi(u)=0.\\,", "a93abecf45a0e98a65aabf79d43cd945": "\n\\left(-\\frac{d^2}{d\\zeta^2}\n- \\frac{1 - 4L^2}{4\\zeta^2} + U(\\zeta) \\right)\n\\phi(\\zeta) = M^2 \\phi(\\zeta),\n", "a93ac9fa3490790d4d1c0415b6409fa1": "V(f(\\vec{x},y)) = \\theta(- y f(\\vec{x}))", "a93b19c6d3e949b459166aa4af10ea75": "L(s,\\chi) = \\sum_n\\frac{\\chi(n)}{n^s} = \\prod_{p \\text{ prime}} \\frac{1}{1-\\chi(p)p^{-s}}", "a93b1fa5ecd4861c1b8618f07496368c": "A_{2}(p,1) = p", "a93b40db3ac629742ba58d3e014920c1": "a (1-e^{2})", "a93b47e7a317c644e8725dcaa1d4faf7": "\\theta \\mapsto e^{i\\theta} = \\cos\\theta + i\\sin\\theta.", "a93b75145f03b87f6af93581e879e3c0": "Au(x):=\\sum_{i, j} \\partial_{x_i} a_{i j}(x) \\partial_{x_j} u (x)", "a93bd04ecda56c7bdb1e2ae900b5c02a": " K(h) = K_s \\mathrm{e}^{\\alpha h}", "a93bd6f99ab8198e09b7c5e6176bd7c2": "\n \\Pr(N>n\\mid M=m\\ge k,K=k\\ge 2) = \n \\begin{cases}\n 1 &\\text{if } n < m \\\\\n \\frac {\\binom{m - 1}{k - 1}}{\\binom n {k - 1}} &\\text{if } n \\ge m\n \\end{cases}\n", "a93bf908d4fb5a97a7a4ceef63644186": "y = \\frac{Y}{X+Y+Z}", "a93c088eedc8d86d569ae0945aff44f8": "(C\\times\\alpha){\\to}\\beta", "a93c20de762060e79f6446038da66d99": "\n\\Bigg[\\frac{\\mu}{\\nu}\\Bigg]_2 \\left[\\frac{\\nu}{\\mu}\\right]_2 = \n\\Bigg[\\frac{\\mu'}{\\nu'}\\Bigg]_2 \\left[\\frac{\\nu'}{\\mu'}\\right]_2. \n", "a93c2d983398927913beafec6d75a289": "v_{rms}^2 = \\frac{3RT}{\\mbox{molar mass}}", "a93cad1d75bc24dfb08a35f8efeb3530": "J_{\\alpha} = (\\mathrm{id} + \\alpha A)^{-1};", "a93d163f1c916287d4fcbcdd7a398c12": " I_B = I_C, I_A=0 ", "a93d257902434a51fcbe4e000c0f4e5d": "\\mathbf{r}(t) = r cos(t)\\hat{i}+r sin(t)\\hat{j}", "a93d4402019fbdc16f3db2b335215d89": " = \\sum_{k_1+k_2+\\cdots+k_{m-1}+k_m+k_{m+1}=n}{n\\choose k_1,k_2,\\ldots,k_{m-1},k_m,k_{m+1}} x_1^{k_1}x_2^{k_2}\\cdots x_{m-1}^{k_{m-1}}x_m^{k_m}x_{m+1}^{k_{m+1}}\n", "a93d50e84c264415b9e7724d6ba57bbc": " = \\mathbb{E}_\\theta\\left[ |\\mathbf{\\theta - X}|^2 \\right] + 2\\alpha\\mathbb{E}_\\theta\\left[\\frac{\\mathbf{(\\theta-X)^T X}}{|\\mathbf{X}|^2}\\right] + \\alpha^2\\mathbb{E}_\\theta\\left[\\frac{1}{|\\mathbf{X}|^2} \\right]", "a93d6365ec19872a5342028a9eb2b1ef": "j=1,\\dots,n.", "a93da2f831b346ebacfe4f00d08f90ae": "\\mu (E) = \\lim_{\\delta \\to 0} \\mu_{\\delta} (E),", "a93daff78a59a923782caaf325049f7b": "\\varphi=\\frac{\\mu_A^*-\\mu_A}{RTM_A\\sum_i b_i}\\,", "a93dc2f34bba73709cabb7da01d3fdd4": " \\mathrm{Im}(\\tilde{k}) = \\alpha_{abs}/2 ", "a93dd03fecdb53de7cfc2e5f96998851": "\\therefore \\exists x:~\\neg P(x)", "a93df87495a612c2963b59a8c483c311": "f(x)= x^2-2", "a93dfe26576a8cf22c640b1b24e997bd": "\\Gamma_0 [n+1] = \\varphi_{\\Gamma_0 [n]} (0) \\,.", "a93e20c8d843418a3a69cf23922e52cd": "a (+\\infty) = + \\infty", "a93e685fc16249a548618f292b6e490f": "q\n=e^{-\\frac{\\pi K'}{K}}\n=e^{\\frac{{\\rm{i}}\\pi\\omega_2}{\\omega_1}}\n=e^{{\\rm{i}} \\pi \\tau}\n\\, \n", "a93e706a56f268138f2f9d8d75847a0d": "\\vec{a}_{shell} = \\vec{a}_o - \\sqrt{\\frac{r}{r-r_s}} \\frac{G M}{r^2} \\hat{r} ", "a93e85a5c4b9ab6d4725db20a1708dff": "\\begin{align}\n y &= E \\left \\{ \\frac{1}{(\\beta + 1 - \\alpha)_{\\alpha - \\beta - 1}} \\sum_{r = \\alpha - \\beta}^\\infty \\frac{(\\beta)_r (\\beta + 1 - \\gamma)_r}{(1)_r (1)_{r + \\beta - \\alpha}} x^{-r} \\right \\} + \\\\\n & \\quad+ F \\left \\{ x^{-\\alpha} \\sum_{r = 0}^\\infty \\frac{(\\alpha - \\beta) (\\alpha)_r (\\alpha + 1 - \\gamma)_r}{(1)_r (\\alpha + 1 - \\beta)_r} \\left (\\ln \\left (x^{-1} \\right ) + \\frac{1}{\\alpha -\\beta } + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{\\alpha + k} + \\frac{1}{\\alpha + 1 + k - \\gamma} -\\frac{1}{1 + k}-\\frac{1}{\\alpha + 1 + k - \\beta} \\right) \\right ) x^{-r} \\right \\}\n\\end{align}", "a93e907301211c719b5cea188a444bb9": "\\mathrm{p.v.}\\frac{1}{s}", "a93eb98e36884009d5f5f66ed2440e08": "P\\equiv_{b}Q \\mbox{implies } P \\circ R \\equiv_{b} Q \\circ R", "a93ee17b1bbc42d3b58e627f0b048c01": " B_{4,3}(x_1,x_2) ", "a93ef4a051286cd7e4f6dd40aa8aeaa1": "u(x,t) = \\frac{1}{2c}\\int_0^t\\int_{x-c(t-s)}^{x+c(t-s)} f(\\xi,s)\\,d\\xi\\,ds.\\,", "a93f23728cc14990e9d06100c2af93d6": " (\\vec x- \\vec \\mu_0)^T \\Sigma_{y=0}^{-1} ( \\vec x- \\vec \\mu_0) + \\ln|\\Sigma_{y=0}| - (\\vec x- \\vec \\mu_1)^T \\Sigma_{y=1}^{-1} ( \\vec x- \\vec \\mu_1) - \\ln|\\Sigma_{y=1}| \\ < \\ T ", "a93f65f9cb3f72331e737f686590b9b8": "\\neg M \\neg \\underline{A}", "a93f685b76a808096e96c9e32819c500": "f'(a)\\ne0", "a93fa6ef54465deb9ee09b7ed7f9a078": "\\scriptstyle{\\tilde{\\kappa}_{o+}}", "a93fcd8928863bc6ad410a19caf784c5": "\n \\begin{align}\n \\mathbb{S}\\, &=\\, \\begin{pmatrix} S_{xx} & S_{xy} \\\\ S_{yx} & S_{yy} \\end{pmatrix}\\,\n =\\, \\mathbb{I}\\, \\left( \\frac{c_g}{c_p} - \\frac12 \\right)\\, E\\,\n +\\, \\frac{1}{k^2}\\, \\begin{pmatrix} k_x\\, k_x & k_x\\, k_y \\\\[2ex] k_y\\, k_x & k_y\\, k_y \\end{pmatrix}\\, \\frac{c_g}{c_p}\\, E,\n \\\\\n \\mathbb{I}\\, &=\\, \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n \\quad \\text{and} \n \\\\\n \\nabla \\boldsymbol{U}\\, &=\\, \n \\begin{pmatrix}\n \\displaystyle \\frac{\\partial U_x}{\\partial x} & \\displaystyle \\frac{\\partial U_y}{\\partial x}\n \\\\[2ex]\n \\displaystyle \\frac{\\partial U_x}{\\partial y} & \\displaystyle \\frac{\\partial U_y}{\\partial y}\n \\end{pmatrix},\n \\end{align}\n", "a93febcaf751d705ab8fefe98d1a6b79": "\\mathcal{O}_k,", "a94064a45acaf8d1416ae0c96f6ed924": "n = d\\sqrt{\\frac{ G( M \\! + \\!m ) }{4\\pi^2 a^3}}\\,\\!", "a9409c7a02b8158c08cbef7f366c42cf": " \\frac{dx}{dt} = -x + ay + x^2 y, ~~ \\frac{dy}{dt} = b - a y - x^2 y. ", "a940c52ada6f617440173ba74a04302e": "B_0\\ \\left(\\frac{(\\mu m)^2}{hr}\\right)", "a9413b8ed7d7229812de612ba1c980fe": "\\pi = \\frac{\\Gamma\\left({1/4}\\right)^{4/3} \\mathrm{agm}(1, \\sqrt{2})^{2/3}}{2}\\!", "a941a6835fe1d817872ccb3e2d47d68c": " T_s = -\\frac{\\ln (\\text{tolerance fraction})}{\\text{damping ratio} \\times \\text{natural freq}} ", "a941e051b9fc0f7de3423eddae5250d5": "\\begin{align} \\sin^2A + \\sin^2B &= 1\\\\ \\cos^2A + \\cos^2B &= 1\\\\ \\tan A &= \\cot B \\\\ \\sec A &= \\csc B \\end{align}.", "a9420c5d468c2d063b8a476aacbb507e": "\\tau(3)", "a9422806c6e94ec22184e7656dd6ffeb": "\n\\frac{d^{2}u}{d^{2}\\theta^{2}} = \\frac{2}{r^{3}} \\left( \\frac{dr}{d\\theta} \\right)^{2} - \\frac{1}{r^{2}} \\frac{d^{2}r}{d\\theta^{2}} ", "a9422de0436be9047afd4fa13adf25a2": "V = bh\\left(\\frac{a}{3}+\\frac{c}{6}\\right),", "a94249dd726c1c25548509e0a4d4f820": "x^n = x_1, x_2, \\ldots, x_n", "a9425301c7bd2ae2c3887094da74a4b6": "\\bar{M}_w=\\frac{\\sum_i N_iM_i^2}{\\sum_i N_iM_i}", "a9428afee935ba4539baad0267664fed": "\n\\begin{align}\n\\tan \\phi_1 & = \\frac{\\mathrm{Im}\\{H_{p,q}\\}}{\\mathrm{Re}\\{H_{p,q}\\}}, \\\\[8pt]\n\\tan \\phi_2 & = \\frac{2 H_{p,q}}{H_{p,p} - H_{q,q}}.\n\\end{align}\n", "a94295a3c81c3b088527340bf6e488ae": "f(1,1) = 0", "a942d4a2f77c4baa0de9a021d950adf8": "\n P_t = \\int_{-b/2}^{b/2}\\left(q_x(y)\\,w(x,y) - m_x(y)\\,\\frac{\\partial w}{\\partial x} + m_{xy}(y)\\,\\frac{\\partial w}{\\partial y}\\right)\\text{d}x\\text{d}y \\,.\n", "a9434d1d7cc7d6d100994bc35800320e": "\\frac{R_{1}^{2}}{\\varepsilon _{1}}+\\frac{R_{2}^{2}}{\\varepsilon _{2}}+\\frac{R_{3}^{2}}{\\varepsilon _{3}}=1", "a943686b66e0f05fb0cab6c0bbedc1f1": "e^x = \\lim_{n \\rightarrow \\infty} \\left(1 + \\frac{x}{n}\\right)^n", "a9437b1be8425c7212703daa469ad5ff": "\\, (1 - it\\theta)^{-k}", "a9437b600d2d19993c724296c89362e1": "\\frac{1}{2} I \\omega^2", "a9437da2f07d8e544bc83325a340fd71": "WBGT = 0.7T_w + 0.2T_g + 0.1T_d", "a943b5654b3bdc032d99ffbbe64b8ad1": "V \\,\\!", "a943d89d116e297bd0b4e3c32700abe3": " \ng_{\\mu \\nu}\n", "a94426f43242f363c0160e78016a0cfd": "G+uv", "a944451280530d368e2551a077c2d5ad": "\\dot{k}(t) = s\\frac{Y(t)}{A(t)L(t)} - \\delta\\frac{K(t)}{A(t)L(t)} - n\\frac{K(t)}{A(t)L(t)} - g\\frac{K(t)}{A(t)L(t)} = sy(t) - {\\delta}k(t) - nk(t) - gk(t)", "a9444af1387df91d497faade38615abd": "\\det(\\lambda I_n - A)", "a9444e409a663e2a887ed84aabd3bd26": "\\beta_1 = \\alpha_1 \\alpha_2 + \\alpha_3 \\alpha_4", "a9448eddfd87c650192dccd15b2557a4": " \\frac{p_a}{p_b} = \\frac{b}{a}, \\ \\ \\ \\ \\frac{p_b}{p_c} = \\frac{c}{b}, \\ \\ \\ \\ \\frac{p_a}{p_c} = \\frac{c}{a} \\,", "a9448f30b2009b183193471647352fd5": "f_i(x) = \\frac{1}{2\\pi i}\\oint_{\\partial\\Delta_i} y\\frac{p_y(x,y)}{p(x,y)}\\,dy", "a944ac64edb5a690594736b2f6e11614": " 4 \\left(\\frac{1}{4}\\right)! = \\left(-\\frac{3}{4}\\right)! ", "a944e957c9b6b8f2ea7ef8ac22c8cfba": "\nE_\\lambda\n=\\int\\left[(\\nabla\\theta_\\lambda)^2+f(\\theta_\\lambda)\\right]d^3 x\n=I_1/\\lambda +I_2/\\lambda^3.\n", "a944f51df50e0f6efc675347aa0b1ee5": "T''_n(1) = n \\frac{\\lim_{x \\to 1} \\frac{n T_n - x U_{n - 1}}{x - 1}}{\\lim_{x \\to 1} (x + 1)} =\n\\frac{n}{2} \\lim_{x \\to 1} \\frac{n T_n - x U_{n - 1}}{x - 1}\n.", "a9455010dba5f3cc71a7c6ec130aa36d": "\\Phi [f_1 \\star f_2] = \\Phi [f_1]\\Phi [f_2].\\,", "a945afad0d832a7aba36b129c6641f52": "\n \\beta := \\left(\\cfrac{m}{EI}~\\omega^2\\right)^{1/4}\n", "a9462e0a31c7422d9870b261ce9965bd": "\\vec{e}_1, \\vec{e}_2,\\dots,\\vec{e}_n", "a9464fc7b6c253de9970c49e5c373145": "P, Q \\in \\mathcal{A}", "a947205628f9f73817811749d02ae996": "L\\ ", "a947441cfef91321030382a350cdc987": "T_n(x) + U_{n-1}(x) \\sqrt{x^2-1} = (x + \\sqrt{x^2-1})^n. \\,\\!", "a947ac394454fc5ce3ad03ba56e035cd": "K_{n,m}", "a947b28ed134d749ff9a2b2344c83cd8": "x=2\\pi[t-1+\\cos(4\\pi nt)\\cos(2\\pi nt)], \\quad y=\\cos(2\\pi t)+ 2\\pi \\cos(4\\pi nt)\\sin(2\\pi nt)", "a947fd91c6121a88159b1d76415996aa": "\\delta=2^{k/2} \\epsilon", "a9483a6e8477308bf082d87ba4149eea": "\\mathbf{\\alpha_1 \\alpha_2 \\alpha_3 =1}\\,\\!", "a948587ecdb3ccf5c5f0dae0d4d9dd44": "\\operatorname{sup}_N", "a94886e5e44d969a8ecbffab0149c61f": "\nw = f(z) = \\frac{a + bz}{c + dz},\\,\n", "a948a4219373dd053bb5bbc36fb18f4c": "\\dim C\\ell(V,Q) = \\sum_{k=0}^n\\begin{pmatrix}n\\\\ k\\end{pmatrix} = 2^n.", "a948a70d6aaa2ee3882c32973ab42aa7": "\\left[n/n+1\\right]_{g}\\left(x\\right)", "a948bfc3b8f5a6fc3089b533d1139f46": "\\mathbf{Y}=[\\mathbf{y}_1,\\ldots,\\mathbf{y}_N] = \\mathbf{H}\\mathbf{P} + \\mathbf{N}", "a948de4eb0cfd1cd25f79a1b80369837": "{\\Bbb E}(e_H) = p^2 e", "a949021ad7d2afe59ce983e9e4090544": "(\\Omega^{-1} - 1)\\rho a^2 = \\frac{-3kc^2}{8 \\pi G}.", "a949343a4172bc24ee997908f83f58e5": " \\lim_{k\\to\\infty} \\frac{\\left(\\prod_p p\\right)^k}{k!} = 0, ", "a9493dbe71ec41ef4000d53127cb83cd": "\n T_{11}^{\\mathrm{eng}}= \\left(2C_1 + \\frac {2C_2} {\\alpha} \\right) \\left( \\alpha - \\alpha^{-2} \\right)\n", "a9493de0c6b91bd5970ad881539b9ea4": "(m+ni)^2 = (m^2-n^2)+2mni.", "a94955a25d17783b4352dc05fe4068ce": " {} \\sim 10^{28} \\,\\!", "a949beced57cc35acdfcfdd2e336f0a8": "(P \\land Q) \\to P", "a94a3457f68da25402d13bb5de9a7df2": "c = 0,", "a94a8547902ed08d8364aa027cd9073c": " H = DQ \\,\\!", "a94ae5942fbe7163611fcdc3433ba32f": "A_5 \\cong I,", "a94ae6af07b04e6432a63cb0effba470": "\\lambda=\\sum_{i=1}^k \\left(\\frac{\\mu_i}{\\sigma_i}\\right)^2.", "a94afdfd9729e731b318870b1947f250": "L(t), P(t)", "a94b0498ee71835da49af255dd64f100": "E_n=2^nE_n(1/2).", "a94b61e9671879a28b898d7dc3df5775": "SO_{n-1}", "a94b8e212b4e58b6370e4e4882153e66": " \\frac{dr_i}{dt}=l_i\\frac{du_i}{dt}", "a94bb346574d0c3f36fc660c56fff5ee": "\\mathcal{G}\\subseteq \\Sigma", "a94bc302cc123f9974bbbcfc064073e0": "k_s \\approx 3.5 D_{84},\\ ", "a94c37ae504f7255bb484b10f2101410": "p=r+k\\cdot q", "a94c50095159b1ac6974e8871e9f45c4": "\n\\left( y(1-y) \\frac {\\partial^2} {\\partial y^2} + x(1-y) \\frac {\\partial^2} \n{\\partial x \\partial y} + [c - (a+b_2+1) y] \\frac {\\partial} {\\partial y} - b_2 x \n\\frac {\\partial} {\\partial x} - a b_2 \\right) F_1(x,y) = 0 ~.\n", "a94c62c5b05b044421f637f959b08313": "C= \\frac{1}{2} (C_{i}^{j+1} + C_{i}^{j})", "a94d265106dd1c6e59d4de348b516fbf": "\\frac{\\theta_o}{\\theta_i} = \\frac{K_p K_v F(s)} {s + K_p K_v F(s)}", "a94d3bb1da898b649280437c1666c1fe": "W_{2D}", "a94d63e231dc19480ba5a7dd87d79b66": "L \\mapsto U^{[p]}(L)", "a94d9d5822e85912c84c89410c42544c": "i:=1:(l+k)", "a94da0a64c47ffbd5bf11b30a6e5d581": " B=(B_1,\\dots,B_m) ", "a94dca8cd7be0ba28b6546da1a0f0448": "\\tau_2.", "a94dcab7264d8fb60c3d630b013eead6": " J\\, p(z,n) = z\\, p(z,n), \\qquad p(z,1)=1 \\text{ and } p (z,0)=0,", "a94df4085d277c5142aa3621e3b96343": "n < 10^7", "a94e014feddec186215d36892226b22e": "\\phi_{k}(a_i)\\in[-1;1]", "a94e2aa0ecbcf9bbad374b95ad158e55": "x(t-t_0) \\rightarrow S_x(t-t_0,f)e^{-j2 \\pi ft_0}", "a94e46b7edba25d931803117f1a3f696": "\\lambda_{m+1}-\\lambda_m \\geq c(\\sqrt{\\lambda_{m+1}}+\\sqrt{\\lambda_m})", "a94e88ee0b9f38f2f0906486420ba4b4": "h \\equiv h(t) = A \\sqrt{R(t)} = h_0 \\sqrt{a(t)}", "a94f64ca0519d2e2bf61be68365ee9db": " L = {1 \\over 2}m_\\text{red} \\mathbf{\\dot{r}}^2 - V(r), ", "a94faa5ac80af709d8eb6bd197a9b4fb": "\\textbf{k}_{||}=(k_x,k_y)", "a94fb1eebedb7244d191b91517d31d88": " -\\delta\\ \\mathbf{r}^T \\sum_{e} (\\mathbf{Q}^{te} + \\mathbf{Q}^{fe}) \\qquad \\mathrm{(17b)}", "a94fdf64e62cdb12dc2031b2c32c669c": "\n\\lambda_{\\perp}", "a94ff5c0d358addeb83f1bd7f967925e": " A=\\iint_D \\left|\\frac{\\partial\\mathbf{r}}{\\partial u}\\times\\frac{\\partial\\mathbf{r}}{\\partial v}\\right|\\,du\\,dv. ", "a9501bd1c7906476fe7c49c8f7ffa6f1": " c = G = e = k_\\text{B} = 1 \\ ", "a950392aeab909b0a1541223f1ef7124": "\\partial A(Z)", "a950ebffbc43021d142da8d37bb95a04": "\nP_{\\mathrm{A}} = \\frac{D_{\\mathrm{A}}}{L}\n", "a95160541a3300651a1a64e28bb4c5dc": "y' = y(1-y)", "a951687cdaa63202dc70693c5cb1a9f5": "r_p=1", "a95176ee546494472d1a70ea622b3c3c": " \\delta d = 0.2 \\ln(10) 10^{0.2\\mu+1} \\delta\\mu = 0.461 d \\ \\delta\\mu", "a9517729bb79f545aabdfcc83d354efb": "\\tilde{\\lambda}_k = -\\textbf{H}_k^T \\textbf{S}_k^{-1} \\textbf{y}_k + \\hat{\\textbf{C}}_k^T \\hat{\\lambda}_k", "a951a52b42ef96d4e7ddb16c7048cb71": "\\scriptstyle x(b-x)=bx-x^2", "a951cd8a7e7f846de1b2e4c53422f59e": "\\gamma_{SG}\\ ", "a951d8647fa6896bdc1b5ae024fd9464": "L_{2k}(R)", "a951fbeff91123aa3f88803d9f5ba03a": "p(x_1^n)p(y_1^n|x_1^n)", "a9522c500cdc25b441654cf4d98c7740": "H'(X) = \\lim_{n \\to \\infty} H(X_n|X_{n-1}, X_{n-2}, \\dots X_1)", "a95230eb6ba7aceae5f25e1481a7ddd3": "I(\\lambda) = I_0(\\lambda) + I_1(\\lambda)", "a952337a385c29f61291ab5abd49f5ae": "P = cf", "a9529a0ca51e81899eed8551934cecf0": "[\\cdot, \\cdot] : \\Omega^k(M,\\mathrm{T}M) \\times \\Omega^\\ell(M,\\mathrm{T}M) \\to \\Omega^{k+\\ell}(M,\\mathrm{T}M) : (K, L) \\mapsto [K, L]", "a95361c8adb1161a033c90de80a8d1cd": "A^{-1} ", "a953b2fca0e2ad49a071fe7802fe0f29": "\n u_i^{n+1} = u_i^{n+1/2} - a \\frac{\\Delta t}{2\\Delta x} \\left( u_i^{\\overline{n+1}} - u_{i-1}^{\\overline{n+1}} \\right)\n", "a954117685fa51d5b3a5e0bac40b4369": "\\sigma_x(\\tau) = \\frac{\\tau}{\\sqrt{3}}\\operatorname{mod}\\sigma_y(n\\tau_0)", "a95427c33d99618d676ec681334d3928": "(c_i)_{i\\in I}", "a954755763ebe679d1eef9b4b0589874": "\\lim_{x\\rightarrow c}f(x)=\\lim_{x\\rightarrow c}g(x)=0", "a9547d62cf8cc6270fedfd4657c1fd41": "\\sigma_{ij} ", "a9548e629144f6241f6f10dca05ea06b": "L = T - V \\,", "a954cdeaf31d7b775627da1da83a02b9": "K' \\subset S^1 \\times D^2", "a95501e39534a38b855fa0d49f0d5430": "R\\approx1-2\\sqrt{\\frac{2\\epsilon_0\\omega}{\\sigma}} ", "a955041f9a6a5cb250a8198b17de1499": "S_{12} = \\frac{\\det \\begin{pmatrix}T\\end{pmatrix}}{T_{22}}\\,", "a955536e76a38890792cd3bed5276f4e": "Y=V", "a9556add680b255c405ccde5dd85c19a": " P_{d1}=P_d(1-\\theta)+P_d'(\\theta) ", "a9557f103bf329681f9ace3f0215c3ef": "\\lambda = h\\, \\sqrt{ \\frac{16}{3}\\, \\frac{m h}{H}\\, \\frac{c}{\\sqrt{g\\,h}} }\\, K(m),", "a955b5ae3192c8f94d59f47272e6853c": "z_{m,n}(\\rho,\\phi,\\tau)=A_{m,n}*Jn(m, k_{m,n} \\rho) * cos (m*\\phi + \\phi_{m,n}^0)", "a955bcb6ff39cabc8eb0dcc63569850d": "(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1)^2 +\\,", "a955df96459d69ce25f19e2e897d3be6": "b \\triangleleft T", "a95615f1e9cdf7c9c6b88f891a788666": "D^j_{mk}(\\alpha,\\beta,\\gamma)", "a9568af042c1faccbe786c86fde89d16": "D_{\\mathrm{KL}}(Q || P) = \\sum_\\mathbf{Z} Q(\\mathbf{Z}) \\log \\frac{Q(\\mathbf{Z})}{P(\\mathbf{Z},\\mathbf{X})} + \\log P(\\mathbf{X}),", "a956befe09f8e95b0cd1b9ae986b20e7": "\\wedge^m_1", "a956c2864052f56f5482b7b08b786d4d": "(G,e)\n=(G,p_0(p_1p_2p_3)^*(p_4p_5p_6))", "a957095e409b7a3ef3ad8c57827f9a53": "(A \\rarr \\mathrm{M} \\, R) \\rarr \\mathrm{M} \\, R", "a957122787d3b5229ce361f8cb9a8395": "\\Delta h_{uv}", "a957404c96e59f1746f97ab668c8e1f8": "n\\,", "a9574d270ee2cd106ff35e51ada59c2e": "\\overline{O_iO_j}^2=\\overline{OO_i}^2+\\overline{OO_j}^2-2\\overline{OO_i}\\cdot \\overline{OO_j}\\cdot \\cos\\angle O_iOO_j", "a957bf386b108a6dfed500e695345d38": "\n\\mu \\ddot{\\mathbf{r}} = \\mathbf{F}\n", "a957e28999e2a92380809aa5d29cd46d": " \\bar{X}=X\\cup \\{ \\infty \\} , \\bar{Y}=Y\\cup \\{ \\infty \\} ", "a957f07aa1633034f08d2f0a28c8b655": " \\mathbb{Z}_q ", "a9581e1443c5196420b4d87eed0bd085": "\\begin{alignat}{7}\n3x &&\\; + \\;&& 2y &&\\; - \\;&& z &&\\; = \\;&& 1 & \\\\\n2x &&\\; - \\;&& 2y &&\\; + \\;&& 4z &&\\; = \\;&& -2 & \\\\\n-x &&\\; + \\;&& \\tfrac{1}{2} y &&\\; - \\;&& z &&\\; = \\;&& 0 &\n\\end{alignat}", "a958993978d1aa147804436c14bc6b27": "PV(i)", "a9597c06fbcc4e2ec1851a4ed343438f": "\\mathrm{Pr}\\left(n\\vert\\psi_{n}\\right)=\\eta_{n}", "a959a4c329366063328c6efbf1e964aa": "\\setminus \\!\\,", "a959b0b12bc7801856a855adddc9151f": "m \\le \\sigma(n)", "a959c911c913f776ba8db8b09c585ce8": "(G^p,H^p)", "a95a155d6ee17e690c47bebbe052484d": "r_{k}.", "a95a1d3985cbaa4da00ae48d7e7a0107": " -\\mu\\boldsymbol{\\sigma}\\cdot\\mathbf{B} \\psi = i\\hbar \\frac{\\partial \\psi}{\\partial t}.", "a95a9314bc2f118c72f3c9414c81b2ea": "|u_n(x)-u_n(y)|^{1-\\frac{\\alpha}{\\beta}}\\le \\left(2\\|u_n\\|_\\infty\\right)^{1-\\frac{\\alpha}{\\beta}}=o(1).", "a95a9d82b7058a1f17c9b9f2f68a66b2": "O(N^{2K} \\, T)", "a95b3b2b159f1a24a965c4d0c3e36cc0": "m\\leq n^2", "a95b60ee82729067dfe2750e43fc5ca7": "Q_L = \\frac{X_L}{R_L}=\\frac{\\omega_0 L}{R_L}", "a95b80e83a8e53648f4f78e6d553b145": "B_0'", "a95bd402ad240f0e186b31eff8fe2b21": "\\alpha \\wedge \\beta = (-1)^{kl} \\beta \\wedge \\alpha. \\, ", "a95c2c791c51f8b32cd91d332f170e5f": "\\beta^*_{1-1}= \\frac{K}{K_\\text{w}} =\\frac{[\\text{M}(\\text{OH})] } {[\\text{M}] [\\text{H}]^{-1} }", "a95c5dcbc30187ab176d08f2620c2375": "t_{1/2} = \\frac{2^{n-1}-1}{(n-1)k[A_0]^{n-1}}", "a95cd76183add3a37b48349011475d10": "x^{n+1 \\over 2}\\,\\bmod\\,\\big(n,x^2-bx-c)", "a95cf307f48b957f0ba0a1e151b7b369": "S^{m - 1}", "a95d2a65841750f0bf4ea7453ee0a47f": "\\mu=1.555", "a95d70496737b8a2832a9164eca166f4": "w\\not=v", "a95d98fc00159faf9e2cf8280ee4d1a2": "\\xi^a \\xi_a = -1", "a95dab3c31d412aeaab0edfb63008eb1": "s \\in \\Sigma^*", "a95db3f9fe83f1e7151e9b610b168ce6": " \\{ a^{2^i} : i \\geq 1 \\}", "a95dbbdc6aa702a3a4a3a124286942e4": "\\sin \\theta=|\\bar{E}|", "a95df434336d0648d8b35bc765f5326f": " \\{\\sigma_i\\}_i ", "a95e19943f1b081ecbbcc3adf48d657f": "\\scriptstyle s_0,\\ldots,s_{m-1}", "a95e3a538b09847a391b6aefb9b7975b": "\\scriptstyle a\\in\\mathbb{R}", "a95e750f10d7b290766d38bfc29cd83a": "\\operatorname{var}(X) = \\frac{1}{4(2\\beta + 1)},", "a95e7610e05a77e7ea49630c96a2c765": "\ndf(t,\\mathbf{X}_t) = \\frac{\\partial f}{\\partial t} dt + (\\boldsymbol{\\nabla}_\\mathbf{X}^{\\mathsf T} f) d\\mathbf{X}_t + \\tfrac{1}{2} (d\\mathbf{X}_t^\\mathsf{T}) (\\nabla_\\mathbf{X}^2 f) d\\mathbf{X}_t,\n", "a95e7a2b2c7d2d92f7f0e26d9e49047d": "I = \\langle \\mathbf{S} \\rangle = E^2_\\mathrm{rms}/c\\mu_0\\,\\!", "a95ee62db59051b9404ee3152ac700c7": "F_\\rightarrow(x,y) = \\min\\{1, 1 - x + y \\}", "a95eef54cf862c5591802a5e5c60c9b7": "\\lambda(\\ln(2))^{1/k}\\,", "a95ef2f97af75365028ba004b5658632": "\\Delta{v_i}= {2\\sin(\\frac{\\Delta{i}}{2})\\sqrt{1-e^2}\\cos(w+f)na \\over {(1+e\\cos(f))}}", "a95eff962d71f79b834d166e89490aba": "\\kappa=0\\,", "a95f1f3d418df17e07131a15f824f40f": "= \\mu", "a95f3201d8668f964fe1aa0be09d910c": " [X,Y]=-i \\ell_B^2 ", "a95f3cc24cde716526d2b69d73dc8e2f": " \\hat{H}=\\hat{H}_0 - \\Delta (\\boldsymbol{\\sigma} \\cdot \\mathbf{m})/2 ", "a95f4e6b714d4697f0bcb1c2b0fd51d0": "\\bold{p}\\rightarrow -\\bold{p}(-t)", "a95f6cea1e2526ea17fe2d0cac23184a": " M^1(B)=\\lambda|B|, ", "a95fb8b1e305e055441d3341b8f2fee6": "(R_2,G_2,B_2)", "a9601a72910c172a9e130cb9f7165c5c": "(8)\\quad \\hat B_{ab}=\\hat\\theta_{ab}+\\hat\\omega_{ab}=\\frac{1}{2}\\hat\\theta \\hat h_{ab}+\\hat\\sigma_{ab}+\\hat\\omega_{ab}\\;,", "a96062fd16e7241359574dc9b56e0411": "\\begin{align}P(\\text{Rare}|\\text{Pattern}) &= \\frac{P(\\text{Pattern}|\\text{Rare})P(\\text{Rare})} {P(\\text{Pattern}|\\text{Rare})P(\\text{Rare}) \\, + \\, P(\\text{Pattern}|\\text{Common})P(\\text{Common})} \\\\[8pt]\n&= \\frac{0.98 \\times 0.001} {0.98 \\times 0.001 + 0.05 \\times 0.999} \\\\[8pt]\n&\\approx 1.9\\%. \\end{align}", "a960c4878fa3d98db4c96f4a170cc091": "\\left. \\frac{du_c}{dr}\\right|_{r= 0}=0", "a960d68f99b320b2be1c338612dd4d7f": " z : X \\rightarrow \\mathbb{R} ", "a96113fd0668d774134e9cbe488c657e": "J = 1", "a9615cf3c8da1cb5d547d1faa58d3edf": "\\mathbf{J} = \\sigma \\mathbf{E} \\, ", "a9615fcb1362745bd99a20b155f8c772": "\ng^{(2)}(0)={{\\langle (a^\\dagger)^2 a^2\\rangle}\\over{\\langle a^{\\dagger}a\\rangle^2}}.\n", "a961ac49bea0cadf395af2528932264f": "V(\\mathbf{r}) = \\sum_{\\mathbf{K}}{V_{\\mathbf{K}}e^{i \\mathbf{K}\\cdot\\mathbf{r}}}", "a962b2c933ccb2252b43933d677caf13": "\\sin^{-1}\\alpha", "a96302f0ec8c192a545fce8ffc0c1a17": "\\int_{A}f\\ \\mathrm{d}\\lambda,", "a963114a12e43dea32b3734411513ecd": " \\rho n ", "a96347462e161129554460f96ba1ffdc": "\\mathrm{sect}^{}_{}(\\Pi_p) = f(p)", "a963744a1c4ad0e12adcaa8899ce20b3": " \\dim\\!\\left\\{J^k(M,N)\\right\\} = m + n + \\dim \\!\\left\\{B_{n,m}^k\\right\\} = m + n\\left( \\frac{(m+k)!}{m!\\cdot k!}\\right). ", "a963810e9e3f4c1f7cfa9a03082e6f75": "t(BP) = \\frac{1}{\\lambda} {\\ln \\frac{N}{N_0}}", "a9638f9675f2368ba339be00883925aa": "P(g) \\approx g^{-\\delta}", "a9639ce5415d35665a25d6da04fa9225": "D_{\\mathrm{KL}}(P\\|Q) \\neq D_{\\mathrm{KL}}(Q\\|P)", "a9642baaf9a0780eecf2a8420472874c": "\n\\begin{align}\n\\mathcal{L}_\\mathrm{QCD}\n& = \\bar{\\psi}_i\\left(i \\gamma^\\mu (D_\\mu)_{ij} - m\\, \\delta_{ij}\\right) \\psi_j - \\frac{1}{4}G^a_{\\mu \\nu} G^{\\mu \\nu}_a \\\\\n& = \\bar{\\psi}_i (i \\gamma^\\mu \\partial_\\mu - m )\\psi_i - g G^a_\\mu \\bar{\\psi}_i \\gamma^\\mu T^a_{ij} \\psi_j - \\frac{1}{4}G^a_{\\mu \\nu} G^{\\mu \\nu}_a \\,,\\\\\n\\end{align}\n\\,\\!", "a9642f89925f6aee5d5683a28557d8dc": "\\scriptstyle C^1", "a964669df73ab4fe79dabd01b66b3f53": "L_2=\\partial_b^2 +\\partial_x^2, \\,\\,\\, R_2=b\\partial_b + x\\partial_x", "a96466ecb9ab45c7333fa1cc61959555": "\\left( \\frac{3}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "a9649364f532ddcfcb86f6d15a3ebf05": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) = \\left \\{ u(x): \\ \nu(x) \\in U(\\alpha,{\\tilde{u}}(x)), \\ \\mbox{for all}\\ x \\in X \\right \\} , \\ \\ \\ \\alpha \\ge 0.\n", "a964a9b934f6c7e4679b8158d77bfdfa": "MV=PY", "a964ace72be1b6d3d641c5e1213f4e51": "f(\\mathcal{E}) = \\frac{1}{1 + \\mathrm{exp}[(\\mathcal{E}-\\zeta)/k_{\\mathrm{B}} T]}. ", "a964f1c9551740ef73bef9b35116da59": "\\operatorname{Tr}(Q\\rho^{\\otimes n})\\geq \\epsilon ~.", "a964fc6e9490c1c0dda42c880c001eb2": "R_1\\,\\leftarrow\\,f(t)", "a9651c04c9bbbc6459e3458e1d1eec89": " \\text{E} \\; = \\; k_0 \\nu \\left [ {\\tanh}^{-1} (c \\sin \\omega ) + z \\left ( 1 + \\frac {\\omega^2}{10}(36 c^2 - 29)\\right ) \\right ] ", "a9651cbae41df9168686d3e4371e492c": "E^* = \\frac{E}{1 - \\nu^2}", "a96527faf58673939c36215b1448be75": "V \\rightarrow V+C", "a96533f747de9eaa9f4869c6894de9ff": "l=L(M)+16 \\left\\lceil L(N)/m \\right\\rceil", "a965604263fd097e5534cf81ca8e976a": "t=.5,1,2,4.", "a9659887b6561d77b86677b591936933": " O(\\sqrt{\\log n}/\\epsilon)", "a965baf258a910806c83a7e59b5f4c86": "|0\\rang", "a9666bbe75ffb4922164778408564b64": "q_f(A,i,j)", "a9669bf6cdff5408ba3813923483974f": "\\left(\\frac{p_{n+1}}{p_{n}}\\right)^n < p_{n}", "a966b79b8b19e8d040f0f78e3c8c49f6": "k=k_n", "a966c943408b693bf5ff84dfc4f723c9": "\n\\mathbf{T} = 8 \\pi \\mu a^3 \\left[ \\frac{1}{2} \\left( \\boldsymbol{\\nabla} \\times \\mathbf{u}' \\right) - (\\mathbf{\\Omega} - \\mathbf{\\Omega}^\\infty) \\right],\n", "a966f05e4aba98f2226d81c885b7d11c": "BoP=CA+KA \\, ", "a96702308291b8e0d3cd4604caa371a5": "\n\\left[ (-1)^{p - m - n} \\;z \\prod_{j = 1}^p \\left( z \\frac{d}{dz} - a_j + 1 \\right) - \\prod_{j = 1}^q \\left( z \\frac{d}{dz} - b_j \\right) \\right] G(z) = 0.\n", "a967800d7d8efe6d28b2a147d7e8e610": "\\displaystyle{J(x)=ix}", "a9679dd89cea223931b1d11dd696f719": "\\textstyle \\int K^2(u)du", "a967a57a5b4f87cbc94ca70b31ff7bb1": "\n1 = \\frac{1}{\\mathcal{Z}} \\sum_\\alpha \\langle\\alpha |\\mathrm{e}^{-\\beta(H-\\mu N)}[\\psi_\\mathbf{k},\\psi_\\mathbf{k}^\\dagger]_{-\\zeta}|\\alpha \\rangle,\n", "a967c907d45802273dbc2e529381af15": "p(x)=\\int_{-\\infty}^\\infty f(x,y)\\,dy.", "a967d1bb303191d7cb8ca7a9882f7594": "\\textstyle \\diamond=P", "a9682e9212bfe0f054eef184141d46a0": "f:X\\to\\mathbb R^k, \\ f(x)= (f_1(x),\\ldots,f_k(x))^T", "a96870a4b4052b59d3abc61f7b1a651f": "Y\\left( \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix},z\\right) = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix}", "a96921c819fa485473a63c46ad8dfce8": " \\forall A \\in \\mathcal{A} : \\Pr(A) \\leq x(A) \\prod\\nolimits_{B \\in \\Gamma(A)} (1-x(B)) ", "a96922c71f3e27febc0c7c822004b9cf": "\\alpha = {12 \\over \\left ( b-a \\right )^3}", "a969363ee4e1f089c6591536af7fec63": "k_q = {8RT}/{3\\eta}", "a9694250e1edb9985a8b95089a336967": "y_1,y_2,\\dots,y_N.", "a9697c94427231348184791670a997d5": "\\mu \\gtrsim 0.24", "a969cae429e0dfaba3680e17085b093e": "\\varphi(n) = n \\prod_{p|n} \\left(1-\\frac{1}{p}\\right)\n=n \\left(\\frac{p_1 - 1}{p_1}\\right)\\left(\\frac{p_2 - 1}{p_2}\\right) \\ldots \\left(\\frac{p_{\\omega(n)} - 1}{p_{\\omega(n)}}\\right)\n.", "a969cb33b067dacf456010596cd6f048": "\\mathbf{TU}\\alpha = 1", "a969fa342a4cc8717ff5f2bdd4649965": " \\gamma = \\frac{\\alpha+1}{\\alpha} ", "a96a0a4c2997e619def13656834db74b": "\n\\begin{align}\n\\Gamma(s,z) &= \\frac 1 {t^s} \\sum_{i=0}^{\\infty} \\frac{\\left(1-\\frac 1 t \\right)^i}{i!} \\Gamma(s+i,t z)\n\\\\\n &= \\Gamma(s,t z) -(t z)^s e^{-t z} \\sum_{i=1}^{\\infty} \\frac{\\left(\\frac 1 t-1 \\right)^i}{i} L_{i-1}^{(s-i)}(t z).\n\\end{align}\n", "a96a1a1bad8ec153d4cf78837f5e3d88": "xp'_x(a,b)+yp'_y(a,b)+p'_\\infty(a,b)=0.", "a96a34f5f185a831563718d60962c267": "\\frac{1}{(1-x)^2} = \\sum^{\\infin}_{n=1}n x^{n-1}\\quad\\text{ for }|x| < 1.", "a96a47dd5abad27c72e56132317353fa": "=A \\cap (A \\cup A^C)\\,\\!", "a96a51c29cd7129e5c102f21179b2c2e": "V= \\frac{n h}{12} (a_1^2+a_1a_2+a_2^2)\\cot \\frac{\\pi}{n}", "a96a54f33a1a0b9e5f60b7a9bc415821": "\\left(\\sigma_{ij}- \\lambda\\delta_{ij} \\right)n_j =0\\,\\!", "a96b112ac607421fb40a9bd4872e4568": "n \\approx 2", "a96b5915f30e4cedcf28d21454dd9646": "j \\in \\{ 0,0.5,1,1.5, \\ldots \\} ", "a96b6604f8a56ffc35b002eac714c87f": "P(x,y)=0", "a96b93bc9d5ed3d8c73377ae92a33b7f": "\\sigma_i \\in \\{ \\pm 1\\}", "a96ba2f55322687b1c8e24234fe7c5de": "\\alpha_3 \\le\\frac{43}{96}\\ ,", "a96bb420961259f0025a023ab7f102c9": "\ni{ \\partial \\psi \\over \\partial t } = - {\\partial^2 \\psi \\over \\partial x^2}\n", "a96bfce5b374d846d962ce42962d3d38": "m=\\gamma^0\\hat{H}_0+\\gamma^jp_j", "a96c033f31435a4f2db3bc469d46530a": "c(t)=2\\pi t + i \\cos(2\\pi t),\\quad 0 \\leq t \\leq 1. \\,", "a96c1d4f069fb32010f0983e8ab96be2": "P_{heat} = \\eta_{heat} \\cdot f_{recirc}\\cdot \\eta_{elec}\\cdot (1-f_{ch})\\cdot P_{fus}", "a96c2ae8f5e0083790b06510a2ce8ac8": " \\delta (x-y) ", "a96c5d97bd8196118e9ee44b0145184b": " \\Phi(x) ", "a96d462538e4c5edd08d1246264d10ef": "n_y=2", "a96d5b209c2b66e7015376d7ccd0ff43": "V(r) = \\frac{Z e}{r} e^{-q r}", "a96dbdb5f4973d37748743dfc5e56dfd": "H^1(X, \\mathcal O_X^*) =0 ", "a96dc18301d1415661fb3f0efd3b112e": "d((P_1,\\ldots,P_k),(Q_1\\ldots,Q_k))=\\max_{1\\le i\\le k}\\|P_i-Q_i\\|", "a96eae593ebfe695b9e3a6e31363f7cc": "f : M \\to M'", "a96eb8a7d643ab2769fe2bde7d291997": "~v~", "a96ec83de040e7935320cdfcc2561472": " \\alpha \\approx \\frac {1.22 \\lambda} {W}", "a96f60ac3e6b549a1409d2903e485c53": "\\gamma^{1,2,3} = \\begin{pmatrix} 0 & -i \\sigma^{1,2,3} \\\\ i \\sigma^{1,2,3} & 0 \\end{pmatrix}, \\quad\n\\gamma^4=\\begin{pmatrix} I_2 & 0 \\\\ 0 & -I_2 \\end{pmatrix}, \\quad \n\\gamma^5=\\begin{pmatrix} 0 & -I_2 \\\\ -I_2 & 0 \\end{pmatrix} ", "a96f6733d179564ebd8baa53f2254851": "N_2O + H_2 \\rarr N_2 + H_2O ", "a96f687ff3caaaa2750de21747d6aa54": "-\\log(b_i-x_i)", "a96f697dca0c5d3acb71b4bcfbf447a6": " \\tfrac{|X-\\mu|}{|Y-\\mu|} \\sim \\operatorname{F}(2,2) ", "a96ff2b6430f6b60c60bd225ccde5339": "f(a+h) = f(a) + f'(a)h + R_1(x),", "a9706c64134c65c5a3e1628ff0315bee": "e^{ix} = \\cos x + i\\sin x \\,\\!", "a97118fb9e8d7e006a466bfc0771f888": "X_i", "a971333e66be7c744c4e4b950e4001a1": "\\int_0^T e^{-xt}\\phi(t)\\,dt \\sim \\sum_{n=0}^{\\infty} \\frac{g^{(n)}(0) \\ \\Gamma(\\lambda+n+1)}{n! \\ x^{\\lambda+n+1}}", "a9717cc2d5ed043c236bff1093b876ee": "Z(FG)", "a971a739fcb79a4e42c1fa9c496f5fab": "a>b\\ge d>c", "a9725a64b7e3eaa528671c3a623bfab7": "\\forall i \\,\\forall j\\, [R_i \\bigcap R_j \\ne \\empty ]", "a9729ef43d2493f2ffe5548c0b802304": "\\mathfrak C", "a972acf5731c2b872fbd5db001ee1f1c": " - \\left\\lfloor \\frac{m}{2} \\right\\rfloor \\le r < m-\\left\\lfloor \\frac{m}{2} \\right\\rfloor ", "a972beaade702fb9e90b292ea72d3441": "\\lambda < [sup(X) \\cup \\alpha]^+", "a972cbab54e90f3977b3a7d60a3ea4ce": " \\Delta \\Phi + k^2 \\Phi=0", "a972f9c9805e943e2fc967abaa820b96": "\n\\varphi = \\varphi_{0} + \\frac{L}{\\sqrt{2m}} \\int ^{u} \\frac{du}{\\sqrt{E_{\\mathrm{tot}} - V(1/u) - \\frac{L^{2}u^{2}}{2m}}}\n", "a9735774147ac95d253664eb66d19f6f": "\\frac{4}{\\frac{1}{15}+\\frac{1}{13}+\\frac{1}{17}+\\frac{1}{100}}=18.83", "a9735b77b9794ff87ca1d9126cfb4272": "R=Q_2Q_1A=Q^T A=\\begin{pmatrix}\n14 & 21 & -14 \\\\\n0 & -175 & 70 \\\\\n0 & 0 & 35 \\end{pmatrix}.", "a97378fdd474db1a06422e35f49cd093": "\\frac{E_s}{N_0} = \\frac{C}{N}\\frac{B}{f_s}", "a973ece7811aceac67f82b890f12388a": "\\mathcal F(|\\sigma|).", "a973edcd86eea06d9e194798e8ab8d3b": " V(u,\\Omega):=\\sup\\left\\{\\int_\\Omega u(x)\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x\\colon\\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n),\\ \\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\\le 1\\right\\}", "a973f2a4bc4a2a8beddff0113394d0fd": "\\cos A = \\sin A \\cdot \\cot A \\ ", "a9740868551c20e750b76d7f3a249063": "t = r \\sqrt{\\frac{n-2}{1-r^2}}", "a97495b87e3be2a9f4a7688b8ea21789": "[X]^n", "a974c6e833e84f28d6aa9e4abe6d5dde": "I = b (1 + {c \\over d} \\,) ,", "a974e50d42edb07aea7728b5af868df7": " B_{ij}=O_{ij}^2 .", "a974e9e6f0ebb5027948837ea1846051": "\nD^{j}\\left( \\mathbf{A}\\right) =I^{\\otimes jn}\\otimes\\mathbf{A.}\n", "a9753b9ea11e3e52a368d3788fe70df0": "I(X;Y|Z) = H(X,Z) + H(Y,Z) - H(X,Y,Z) - H(Z)\n = H(X|Z) - H(X|Y,Z)", "a975fa26eee83625600c9d0454db0d82": "\\displaystyle{(F_f\\circ g)_z= [(F_f)_\\zeta \\circ g]\\cdot g_z, \\,\\, (F_f\\circ g)_{\\overline{z}}=[(F_{f})_{\\overline{\\zeta}} \\circ g] \\cdot\\overline{g_z}.}", "a97643c318ee7e434016aa2ced7b8646": "\n\\widehat{m}_{PC}(x) = h^{-1} \\sum_{i=1}^n (x_i - x_{i-1}) K\\left(\\frac{x-x_i}{h}\\right) y_i\n", "a9764e264f7f787759867ca95d2698c2": "\n\\begin{matrix}\n J_{p} & = & \\sum_{i=1,2,3,5}{(x_1(t_i) - x_1^m(t_i))^2} \\\\\n\\end{matrix}\n", "a976a11183334d993553246b0eda28ce": "\\mathbf{C}^n = \\mathbf{R}^n \\otimes_{\\mathbf{R}} \\mathbf{C}", "a976b98c9648918bbe4600eeabf96610": "h=x-x_0", "a976fe36978f04a4f158db6c2aac0d6b": "\\displaystyle{\\|v_{1}\\|^2=(L_{-1}^\\prime v_0,v_{1})=(v_0,L_1^\\prime v_{1})=\\overline{c}\\|v_0\\|^2.}", "a977238ab792b2154048d0e24b11e716": "f_{\\text{r}}(\\omega_{\\text{i}},\\, \\omega_{\\text{r}})", "a9778a8ed784ed63d1c31145e746d3ef": " \\begin{cases}\n\\mathrm{ev}: \\overline{\\mathcal{M}}_{g, n}(X, A) \\to Y \\\\\n\\mathrm{ev}(C, x_1, \\cdots, x_n, f) = \\left(\\mathrm{st}(C, x_1, \\cdots, x_n), f(x_1), \\cdots, f(x_n) \\right).\n\\end{cases}", "a977f5afa2ba56ff9c37f89d155b166c": "h_f = \\frac{8 f_D L Q^2}{g \\pi^2 D^5} ", "a977fe3eeedc3a559f66f62d593a1d73": "|A|r!(n-r)!\\le r(n-1)!", "a97816c3e9983288328918d1444414c7": "\\bar{X}_3", "a97822b18edb476a18cb29dcab3e0496": "A_{\\mu} = \\eta_{\\mu \\nu} A^{\\nu} \\,, ", "a9782e48baec11a92b1781a2ee39be50": "\\xi_1,\\dots,\\xi_T", "a9785688dfc5fd723e6d49df98159eae": "\\rho(\\alpha)", "a978e0c9f4c0486f1af58a86d7387381": " \\left({1 - {1 \\over { (1+i)^n } }}\\right) ", "a978f441d9e1d6c8962f8a8532c87c1d": "\\text{Posterior Probability}(p=x|s,f) = \\frac{x^{s}(1-x)^{n-s}}{\\Beta(s+1,n-s+1)}, \\text{ with mean = }\\frac{s+1}{n+2},\\text{ (and mode= }\\frac{s}{n}\\text{ if } 0 < s < n).", "a97959346d98e0e18b76957b77b869ac": " 1,\\ 2,\\ 3,\\ \\sqrt{2},\\ \\pi,\\ e\\ ", "a9797fa6adc352ab3d0c9e34bb537323": "\\nu = 1/q", "a97997dd086d096b4819a3fe596276bc": "\\mathcal{F}_0", "a979e79edf8672557891b79549904b5e": "\\mathfrak{m}_B/\\mathfrak{m}_A B", "a97a300b695dab0442db239db1545f85": " T : W \\rightarrow P ( \\Gamma )", "a97a45119fe6f6238bec10d52583efd5": "E()", "a97a7c3948b7dbc3d4180bbb19a1d206": "\n\\langle C, X \\rangle - \\langle b, y \\rangle\n= \\langle C, X \\rangle - \\sum_{i=1}^m y_i b_i\n= \\langle C, X \\rangle - \\sum_{i=1}^m y_i \\langle A_i, X \\rangle\n= \\langle C - \\sum_{i=1}^m y_i A_i, X \\rangle\n\\geq 0,\n", "a97ac50991e9ccc50b18adeaca41c020": "\\operatorname{W}(A) \\colon \\operatorname{ran}(A+i) \\rightarrow \\operatorname{ran}(A-i)", "a97ac70ec046695a97564e43e7e8fbc3": "G_{(1-X)} = e^{\\operatorname{E}[\\ln(1-X)] } = e^{\\psi(\\beta) - \\psi(\\alpha + \\beta)}", "a97adf5994bae4c3aee9a96966a46825": "{\\partial^2 \\over \\partial t^2} u(x+h,t)={KL^2 \\over M}{u(x+2h,t)-2u(x+h,t)+u(x,t) \\over h^2}", "a97b2ca39cf9976fb1f839fb5e73c517": "n\\quad(f'')\\,\\!", "a97b36666e58ee932081dc21e8adddd3": " \\mathbb{Z}^{n}_{q} ", "a97b882eb266928fd84b5d16a53c0e93": "{\\mathbf{v} \\div c}", "a97bb47096651741cc562540adfd961a": "18^\\mathrm{o}", "a97be29f564e2f9b60e87c8f9e42b184": "\\frac{Country}{World}", "a97c1ad8d5575cf3f8e489ea2fe4073d": "E(v,h) = -a^{\\mathrm{T}} v - b^{\\mathrm{T}} h -h^{\\mathrm{T}} W v", "a97c29e729928324fa4dd8fe63494b3a": "f(f(a, b),c) = f(a, f(b, c)).\\,\\!", "a97c8648d7366da3dfa26731e6d9ab45": "3.141592653 ", "a97cd9dd7ef44bdd5c4aee3debfe7c9e": " c_n = \\frac{\\rho}{\\sqrt{\\rho^2+\\sigma^2}} \\quad\\mbox{and}\\quad s_n = \\frac{\\sigma}{\\sqrt{\\rho^2+\\sigma^2}}. ", "a97d190a05a50f777d114f48f9fe968f": "|\\theta| \\le 1.", "a97d41038e9b10877abfbabcf1ede1a0": "C = V_w \\frac{\\Delta t}{\\Delta x} ", "a97d5b5914b847964d8aecb9d9169e38": "Tr(g^b)", "a97d6614be3fe39a498a0316b9dc8b74": "x = ( \\lambda - \\lambda_0 ) \\cos \\phi_0\\,", "a97d7719ebc99b7a2c338ec72368e023": "x,y\\in A, x\\cong_{\\mathcal{B},\\epsilon} y", "a97d7e7fa1314bf9a0b3cba5d1bbf5b4": "a = -g \\sin\\theta\\,", "a97dd272fd5cc549cd06327bf476e68b": "P_{out} \\le \\frac{(V_{br} - V_k)^2}{8Z_o} ", "a97dd7bcf7bc70e25bd422490bc4d821": "s_2 = 00", "a97e0e3a640d8668e7b68ed9acc2ded7": "s_\\gamma = O(M(n)\\log^2 n). \\, ", "a97e2ac9c73e48880f10bf3a1b4eb9c7": "P = K \\rho^{((n+1)/n)}", "a97e5bc5d2145a6321d409db900d25bc": "f(x)=ax^2+bx+c", "a97e5fec699e321562aa2186c67ab291": " \\gamma_{yy} ", "a97e7715b966213f9aa9896e0765d17a": "\\frac{\\partial L(t,\\mathbf{x},\\mathbf{v})}{\\partial x_i} = -\\frac{\\partial U(\\mathbf{x})}{\\partial x_i} = F_i (\\mathbf{x})\\quad \\text{and} \\quad\n\\frac{\\partial L(t,\\mathbf{x},\\mathbf{v})}{\\partial v_i} = m v_i = p_i,", "a97efdf97c7e1b82244e138f8419d84d": "\n\tf(\\boldsymbol{x}) = sign\\left(\\sum_{j=1}^2g_j^T(\\boldsymbol{x_j})\\right)\n", "a97f75ed768ddd3c978e10eb504cb532": "m_{x}", "a97f7efdad2adc4b8449e84d131243cb": " d_{ib} = \\sqrt{\\sum_{j=1}^{n}(t_{ij} - t_{bj})^2}, i = 1, 2, . . ., m ", "a9800c249bbb9543d8d7615a2dfc9389": "\\,\\text{ not} (a \\text{ or }b) = \\text{ not } a \\text{ and}\\text{ not } b", "a980298cf02b2d6422506d5dc948e918": "\\mathbb{Z}[\\eta]", "a9802c1ccf328744c539098f9f43ba10": "\\mu_\\mathrm{B} = {{e \\hbar} \\over {2 m_\\mathrm{e} c}}", "a980b2de11e3718f4c1fbcf85a85b043": "d = -4^4(4a^3+99a^2-34a+467)^3\\,", "a980c4fd54850f59c5934e9a6c49dd61": "s \\times s", "a980d198474d14d26eb39dfd43568dc6": "\\hat H = - {{{1} \\over {2}}\\nabla^2} - {{1} \\over {r}}", "a980df7d96cd700acf7f42e5ee4621c6": "x = \\tfrac{e-c}{b-d}", "a980e3ac5958a39bcaec91a261983649": "\\mathbf{B}(\\mathbf{r}) = \\frac{\\mu_0}{4\\pi} \\iiint_V d^3r' \\mathbf{J}(\\mathbf{r}')\\times \\frac{\\mathbf{r}-\\mathbf{r}'}{|\\mathbf{r}-\\mathbf{r}'|^3}", "a980e4c8eb9584f9daf441369edab263": " |\\mathbf{R}|^2 = \\mathbf{R}\\cdot\\mathbf{R} =\\operatorname{tr}[\\mathbf{R}\\mathbf{R}^T],", "a9810a91f8669b096568b6eb4a6a6360": "\\alpha = \\sum_{m=1}^\\infty p_m r^{-m^2}.", "a9811cec730bd57d0665daec2050b149": "c_{12} = \\frac { E_\\nu} {(1-2_\\nu)(1+\\nu)}", "a9812194f2ea58829b86a16b3b5865ac": "\\lambda I_n - A", "a981ee6af471bfa732e289d6283c2a0a": " \\int_0^\\infty \\frac{\\gamma x+\\log\\Gamma(1+x)}{x^{5/4}} \\,dx = \\sqrt{2} \\frac{4\\pi}{5} \\zeta\\left(\\frac 5 4\\right) ", "a98217db16e93128ab57bd1079d957de": "n! = \\Pi(n) = \\Gamma(n+1),", "a9829297bbd3a0620f7aa36656248d79": "F(s) = p_\\mathrm{int} + s\\cdot\\mathbf{w}.", "a982b93a2e1e092545ca69bcd09d8147": "v_t", "a9831b50a023054b7f1057a7d6a61e64": "\nH_{TC} = - J\\sum_v A_v - J\\sum_p B_p, \\,\\,\\, J>0.\n", "a98367813048ae3a15e3e543470a0b96": "{\\color{White}\\sdot} = A_v \\frac {R_i} {R_i+R_A}\\sdot \\frac {R_L} {R_L+R_o} \\ ,", "a983a7cb6d3e6aca2d81f03ce9eda4a2": "x(y^{-1}N)\\neq\\{0\\} \\,", "a984b67eb552f0eff0a215b728102e9c": " P_L = \\frac{1 - \\gamma^5}{2}", "a98506120343eabfa83338605f15742b": " \\text{ } (2) \\text{ } U_{n+1} \\equiv 0 \\pmod {n}. ", "a9855e38ab7ffd09cbf09d117950051c": "c/a", "a985d98806ec879a78c8e002ba15e74d": "L:H_1\\times H_2\\to K", "a9862e9caf889aaddf5455c4c222e9b9": " V(S,t) = v(x(S),\\tau(t)) ", "a98669c986812d92cdd52972f014c2f8": "dx = Adt + cdW\\ ,\\ x(0) = 0 ", "a9868687897287dd9572272db7461d67": "a_0,\\ldots,a_8", "a986a625f0cfe21adf9c2cabaaacb724": "\n\\nabla (\\varepsilon \\nabla \\varphi) = \\rho\n", "a9875a31e9e2b8d05ec178a1d6ad708e": "\\left(-\\dfrac{\\hbar^2}{2m}\\dfrac{\\partial^2}{\\partial x^2} + \nV(x)\\right)\\Psi(x,t)=i\\hbar\\dfrac{\\partial\\Psi(x,t)}{\\partial t}", "a987c5c7ea434fb0c6a98a973a4610e2": "\n 1 = \\omega_{pe}^2 \\left[\\frac{m_e/m_i}{\\omega^2} + \\frac{1}{(\\omega - kv_0)^2} \\right],\n", "a987e44d306193fc1b069fc46a387c5e": "C_c", "a98828a8ea7f5c8f6de8875657223ebb": " !x(y).P", "a988497d61b95d4ad0dc801a6a4e3106": "\\sigma _3", "a98849cbed4753d01ee4f53fe5ef1c76": "R_{abcd}, \\, C_{abcd}", "a98851c0149eb64c70342c005f4406ac": "\\bigcap_{i\\in I} S_i=0", "a988582fcfd570934ab49ead9fc01f7a": "(i+1,i+1)", "a988938f77e41e2fa144ee9de2e7828f": "z(r)=0", "a988975a0cb1e99e685c317187af749a": "\\bold{j}_\\mathrm{trans} + \\bold{j}_\\mathrm{ref}=\\bold{j}_\\mathrm{inc}.", "a9889af5324fb96281e878bcabbaad27": "\n\\Delta_{\\psi} \\hat{l}_z \\, \\Delta_{\\psi} {\\theta} \\ge \\frac{\\hbar}{2}, ", "a988a15a5fca9810ec17ca1f70339f1e": " X ( - \\sin \\theta, \\sin( C + \\theta) , \\sin( B + \\theta) ) ", "a988acfed12fbc185a8ed92acc10cf95": "\\rho = \\frac{1}{2}\\left(I +\\vec{a} \\cdot \\vec{\\sigma} \\right)", "a98904984e9e5fb6176ebf46ad800d3b": "\\begin{align}\n\\bar{J}^{\\mu} \\, & = \\, \\frac{\\partial}{\\partial \\bar{x}^{\\nu}} \\left( \\bar{\\mathcal{D}}^{\\mu \\nu} \\right) \\, = \\, \\frac{\\partial}{\\partial \\bar{x}^{\\nu}} \\left( \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\right) \\\\\n\n& = \\, \\frac{\\partial^2 \\bar{x}^{\\mu}}{\\partial \\bar{x}^{\\nu} \\partial x^{\\alpha}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial \\bar{x}^{\\nu} \\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\\\\n\n& + \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\beta}} \\, \\frac{\\partial \\mathcal{D}^{\\alpha \\beta}}{\\partial \\bar{x}^{\\nu}} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\frac{\\partial}{\\partial \\bar{x}^{\\nu}} \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\\\\n\n& = \\, \\frac{\\partial^2 \\bar{x}^{\\mu}}{\\partial x^{\\beta} \\partial x^{\\alpha}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial \\bar{x}^{\\nu} \\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\\\\n\n& + \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial \\mathcal{D}^{\\alpha \\beta}}{\\partial x^{\\beta}} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial \\bar{x}^{\\nu}}{\\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\frac{\\partial \\bar{x}^{\\rho}}{\\partial x^{\\sigma}} \\frac{\\partial^2 x^{\\sigma}}{\\partial \\bar{x}^{\\nu} \\partial \\bar{x}^{\\rho}}\\\\\n\n& = \\, 0 \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\frac{\\partial^2 \\bar{x}^{\\nu}}{\\partial \\bar{x}^{\\nu} \\partial x^{\\beta}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\\\\n& + \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, J^{\\alpha} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\frac{\\partial \\bar{x}^{\\rho}}{\\partial x^{\\sigma}} \\frac{\\partial^2 x^{\\sigma}}{\\partial x^{\\beta} \\partial \\bar{x}^{\\rho}} \\\\\n\n& = \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, J^{\\alpha} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\, + \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, \\mathcal{D}^{\\alpha \\beta} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\left( \\frac{\\partial^2 \\bar{x}^{\\nu}} {\\partial \\bar{x}^{\\nu} \\partial x^{\\beta}} \\, + \\, \\frac{\\partial \\bar{x}^{\\rho}}{\\partial x^{\\sigma}} \\frac{\\partial^2 x^{\\sigma}}{\\partial x^{\\beta} \\partial \\bar{x}^{\\rho}} \\right) \n\n\\end{align}", "a989274994405c2eff632a3e52bc67f8": "\\mathrm{CmCl_3\\ +\\ \\ H_2O\\ \\longrightarrow \\ CmOCl\\ +\\ 2\\ HCl}", "a989b9e0f7637a70a6040502d2c7301e": "\\mathbb{Z}[i].", "a98a8d5c0ee22a112b990e114463df6e": "\\mathcal{F} \\,", "a98a99c3b84e4a7a4a74a526e756d2d8": "\\rho^{'[DK]}", "a98ae59098726a4828dec3455775f297": "E=V_0", "a98af9edcc97188831e110c8eca5d33e": "\\lim_{n\\to\\infty}x_n=x, \\ \\ \\mathit{i.e.,} \\ \\ \\lim_{n\\to\\infty} \\|x_n - x\\|_X = 0.", "a98b0d15f04c2d0400b2d83a112fa7da": "\\phi=\\frac{1+\\sqrt5}{2}\\approx 1.618\\,", "a98b1483c2c0f89dd280752407655eb5": "\nh_{\\phi} = a \\sinh\\mu \\ \\sin\\nu\n", "a98b7707efe8ed1b61a5990485a9ed0b": "I(r,z) = { |E(r,z)|^2 \\over 2 \\eta } = I_0 \\left( \\frac{w_0}{w(z)} \\right)^2 \\exp \\left( \\frac{-2r^2}{w^2(z)} \\right)\\ , ", "a98b83b203e433731c8d389c4dc54460": "I_M(\\tau) = \\int_{-\\infty}^{+\\infty}|E(t)+E(t-\\tau)|^2dt", "a98c42ea5ba51fc00c91c537de84b71c": "2(AB)^2+2(BC)^2=2(AC)^2\\,", "a98c80b4d5c464b1fe2e0371b122317d": "S(f)S^*(f) = \\int_{-\\infty}^\\infty \\chi(\\tau,0) e^{-j2\\pi\\tau f} \\, d\\tau ", "a98c896ad289954ec8d338e726897f22": "\\phi=12.92321", "a98d1f2b24157885b3863e899d89f6a3": "{R^\\alpha}_\\alpha - \\frac{1}{2} \\, {\\mathrm{g}^\\alpha}_\\alpha \\, R = {\\kappa \\, T^\\alpha}_\\alpha~", "a98d2d7e9364c1ecfe153842b47aa173": "\\gamma_B", "a98d52b06b6c4636356f9944322f6a57": "h' \\in H", "a98d60fbc44f0ed9ff05710f93f17b03": "\\sum_{x \\text{ infinitely near }p}m_x(C)m_x(D)", "a98dbe64945f5e4e592c55e0436325e7": "n=2,", "a98de3713c931b5570b465248c9c65a0": "{{\\mathbf{k}}}", "a98e00bac73d36dbc4d255d734fff1c4": "n_{prop}(z)", "a98ed711c4a96b7ebf4de77556fbc77a": "u=V\\big(-\\mathrm{e}^t\\big)", "a98ed959d58d959335eb6130dd266551": "~p^\\alpha", "a98ede16a0ca4e4ce07df6ce5d778947": "\\begin{matrix} {1 \\choose 1}{3 \\choose 2}{44 \\choose 1} \\end{matrix}", "a98f4e6d3ac1fc16edc6be4c5d97d810": "y^2+x^2-z^2=0", "a98f7e217ddd032adeeeb5f0cbe2de20": "\\sigma_t = \\sigma(S_t,t)", "a98fa2235105c8d47ad6efc8e5d323d1": "\\sqrt{\\log{n}}", "a98facfef0bf7857c6d387890cd39df5": "BV \\leftarrow BV \\cup \\lbrace v\\rbrace", "a98ff12f9c38ee82f2c09ed0ba2123c5": " x^2 +y^2-1 = 0.", "a98ffe0b0cffdbf0e10329f5909bc7be": "URR \\,", "a99029ff9a61110f8474bcf4e0c8c9d4": "\\frac{\\partial \\Psi}{\\partial J}", "a9902d9212665da7a919deb16c7185c2": "\n{\\partial\\rho\\over\\partial t}+\n\\nabla\\cdot(\\rho\\bold u)=0\n", "a99052bc250d97db2ed3826dae8383a0": "(15)\\quad L+M=r\\,,\\quad l_+ + l_- =2M\\cos\\theta\\,,\\quad z=(r-M)\\cos\\theta\\,,", "a9905404668c669dfb481427d761fc85": " b_i = \\frac{{c_i}}{{\\rho - \\sum c_i \\cdot M_i}} \\,", "a9906329337dd5c7bc765c62eb49d0a1": "\\log F_p(s)=\\sum_{n=0}^\\infty \\frac{b_{p^n}}{p^{ns}}", "a9907ee221ef2acadcfb2ceb49963a98": " ( m r\\dot\\theta '^2\\hat{\\mathbf{r}} -m 2\\dot r \\dot\\theta '\\hat{\\boldsymbol\\theta})", "a99094f4dcf4d50700fa6e8efc0bca78": "\\sigma_e^* = \\frac{F^*}{A_0}", "a990ca480326690ed355ee5656133996": "C_{Y}", "a990eb75b99a25535e8929bd1887c7ab": "O(n^d)", "a9912b68d3851d8555922bd2ccc4b17a": "I'_x", "a99144207a7ba5e46539b70148b1b6a4": "O_1,O_2,O_3,\\ldots", "a9914b15d311c7853574f5d81517c750": "a^{d} \\equiv 1\\pmod{n}", "a9925021b622d279191f2cd407792fac": "0 = x - 12", "a992af9e72244fe39e9b5bb8972add17": "e^{\\hat{\\mu}=0}\\wedge \\cdots \\wedge e^{\\hat{\\mu}=3} \\wedge e^{\\hat{\\alpha}=1} \\wedge e^{\\hat{\\alpha}=2} \\wedge e^{\\hat{\\dot{\\alpha}}=1} \\wedge e^{\\hat{\\dot{\\alpha}}=2}", "a992afeaf0582aa572d131b4e8e11da9": "\nG_\\mathrm{dB} = 10 \\log_{10} \\bigg(\\frac{0.001~\\mathrm{W}}{10~\\mathrm{W}}\\bigg) \\equiv -40~\\mathrm{dB} \\,\n", "a992f61960c2d56b8983eb9bb60e6c67": "G_{min}", "a99311fd6494763829dfeab362fc89b7": "V_\\mathrm{Th} = I_\\mathrm{No} R_\\mathrm{No} \\!", "a99356664d60279e42c4f42a08392678": "\n \\frac{d^2 u}{d y^2} = 0,\n", "a9937ec739f3855f09f0ee073a3c4371": "\n -0.1 q_1 + 0.5 q_2 = 0\n", "a993ad638f572a1a2f133b6ba54db2c4": "\\omega_o", "a993b37b2b21d605083901f0933b4d8a": "J_{\\text{mag}}(\\mathbf{p}_z) = \\frac{1}{\\mu}\\iint_\\infty^\\infty (n_{\\uparrow} (\\mathbf{p}) - n_{\\downarrow}(\\mathbf{p})) d\\mathbf{p}_x d\\mathbf{p}_y", "a993cb188c57a56c682d279bdcbc4cfc": "{\\color{Blue}~2.6}", "a99419fe7f437a24c681831f9207a25d": "\\left|\\frac{a_n}{a_{n+1}}\\right| = 1 + \\frac{1}{n} + \\frac{\\rho_n}{n\\ln n}", "a994571ef1d6f9e9ad5ac867f6925cf3": "\\varphi f", "a9945843684a0e3b966ef4f253dee647": "M \\sqcup (-M)", "a99492d0dd9c49172f170e28dbca63ba": "\nf_X(x;n)=\\frac{1}{2\\left(n-1\\right)!}\\sum_{k=0}^{n}\\left(-1\\right)^k{n \\choose k}\\left(x-k\\right)^{n-1}\\sgn(x-k)\n", "a994b6af8a6f95cb108b7547284abdcd": "T_\\text{hot}", "a994cce228ead88536ddaa394e7ae021": "\\langle A, B \\rangle := \\mathrm{tr}(AB^\\mathsf{T})", "a994f01375d4c8549bd5de4a9a40f00d": "D(x_1x_2\\cdots x_n) = \\sum_i x_1\\cdots x_{i-1}D(x_i)x_{i+1}\\cdots x_n. \\, ", "a994f81b177304f2af88961aa3934dd6": "f_i(X_i) = E[Y - (\\alpha + \\sum_{j \\neq i}^p f_j(X_j)) | X_i]", "a995103f001e7687e4b91c460fc4f04e": " ~ \\omega ", "a99564fc502d1d88b0a9c157afc2df05": "B=\\left(\\begin{smallmatrix}\\frac{n}{m}&0\\\\0&1\\end{smallmatrix}\\right)", "a995817335fe2cdf86af5aea1a41b2df": "2^\\lambda", "a995884ab1e24b880523644555a92397": "\\ m= \\epsilon\\sigma p/\\rho Ac_p", "a9958ac925f7708d842ef3c98535d717": "x_\\min", "a99595708f375948a8363e6c0dd7edef": "\\nu_d\\colon \\mathbb{P}^n \\to \\mathbb{P}^m", "a9959fc52bc09aed55acc5e7ea8e1d3d": " \\lVert z \\rVert = \\langle z, z \\rangle .", "a995f2729c812a4642c8523cf64ab5f0": "X=2, X=3", "a9965dba8149e359d80e06ed9ea0afc4": "d(x,y) := |y - x|, \\quad x,y \\in \\mathbb{R}", "a9968003eb42a8bf915a641e74dfd7c4": "1 + r \\leq \\liminf_{n \\to \\infty} \\| x_{n} - x \\|", "a99681bf278007acfb10710edf7c3282": "|ST| = \\frac{|S||T|}{|S\\cap T|}", "a996ab20c95a38c70317cd25e0a37e07": "\\Sigma_t", "a9971fc3b7fe6574072386ab565f9bd1": "(l^2-3)/2", "a9972554fa7a2a575bed90e14b066206": "\\frac {s - D_{\\mathrm N}} {D_{\\mathrm F} - s}\n= \\frac {f^2 - Nc(s - f)} {f^2 + Nc(s - f)}\\,.", "a9973bd9b965d747a0e58afde7f61780": " S^* ", "a9978cd4709ed7ccbfbb34718ed38492": "X_C = \\frac{1}{2\\pi fC}", "a99796a0820f515d44be2eb54018b061": "g= \\frac{G}{R+G+B}.", "a997ad84524730a035a140990c230797": "{R^2}_3 = (C)-(E)-(F)", "a997af65686d70180bded9866835934b": "g=\\frac{G*M}{R^2}", "a997cdfb942d57a19a82b761eab36fcd": "T_4", "a997d098ee45e4c95a40dab20de9a258": "\\, ax_1 + by_1 + cz_1 + d = 0", "a998293949c79241c8e0f7c4fda8f7ce": " \\widehat{\\beta}_\\mathrm{GMM} = (X^\\mathrm{T} P_Z X)^{-1}X^\\mathrm{T} P_Z y,", "a9983ca5d8c48853a30991492fd0775e": " (0\\ ,\\ 0.001285)\\,", "a99867c396d32d3756fb5e08498e4f08": "10^{2^{n+2}}", "a9986c32cfef6e0521b4fbc3bbe8f980": "\\frac{}{} I = I_0 \\sin \\delta ", "a998865a73ca387426ad3f2d5dba8fd2": "\\frac{1}{F} = \\frac{1}{S_1} + \\frac{1}{S_2}", "a998a049dc9eab0e1aaca89baa16c6d9": "c = 14", "a998aa010ab460ea6656ea5f13ae6b8e": "X_k=\\sum_{n=0}^{N-1} U_{kn}x_n", "a998cf13febae3828d6a30f223e4e81d": "p(n,x_1,\\ldots,x_k)=0\\,", "a998e8a7051e4f7674106107849c362a": "b^2 - 4 a c", "a998f59ea6ac1f363d9864a4e3d2a3c9": "\\hat{f}(\\Delta_t n) = -\\sum_{m=1}^{M} \\hat{f}[\\Delta_t (n - m)] P_m.", "a999cd7194e269954472e6e134287e9f": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{J} + \\mu_0 \\epsilon_0 \\frac{\\partial \\mathbf{E}} {\\partial t}\\ ", "a99a11344ebc33eb0e8921ec62ddad89": "\\mbox{PEG Ratio} \\,=\\,\\frac{\\mbox{Price/Earnings}}{\\mbox{Annual EPS Growth}} ", "a99a13729df022c904f024d833c7de55": "P(X_{n+1}=1 \\mid X_i=x_i\\text{ for }i=1,\\dots,n)={s+1 \\over n+2}.", "a99a1a31db1542611cc3c96c2bece270": "\\mbox{Out}(G) \\to \\mbox{Aut}(G^{\\mbox{ab}})", "a99a365e51a4552059a9d83027824267": "Y_k=\\inf_{n\\ge k}X_n.", "a99a7562279ebe55ac033b85d242e76d": " (n-i)\\delta_{ij} ", "a99ab0c4987c8f45fbf58068029f6563": "\\frac{d\\psi}{dn}(s)=\\mathbf{n}\\cdot\\nabla\\psi \\ ", "a99b38009aa6adafc748a8bda7e5eaea": "m_t = a + br_t + cy_t + u_t,", "a99c6b93fcb4e0ac68a34792ad159719": " (r_{i-1}, r_i)", "a99c6f42942448f0e87bed53ade31627": "f(a)=a(a-1)(a-2)(a+1)(a+2)(a+3)", "a99ca0b780fa22387a9d09eb721e0c0f": "\n\\ln N! \\approx N\\ln N - N\n", "a99ccfac0056c0f36c8a11e4e08336de": "XY = I", "a99d60a680aa940338d179183eb1ca9a": "\\beta\\in[-1,1]", "a99d7dc08f3a58701d067d98c3798cdb": "b+\\varepsilon", "a99db32de5ae7252d87b156035d8ee5b": "\\Phi _1 =\\Phi _2 \\equiv \\Phi (x_\\perp )", "a99ddfe8427cd0ed1c72c28b55ede999": " \\nabla \\times \\vec{B} = \\lambda \\vec{B}", "a99df0661dac3c718311dd15beaa09ed": "L'_{old}=L/\\gamma", "a99e0859d2abd8c66fa9b17a3ee5739e": "{x \\leftarrow x}", "a99e9cec75e0ac2ec0f4c35d8e9ef6ad": "f_X(x|Y=y) = \\frac{P(Y=y|X=x)\\,f_X(x)}{P(Y=y)}.", "a99eb5482889609d1549e89082acd9e3": " = \\frac{2}{\\pi} \\arctan\\!\\left[\\exp\\left(\\frac{\\pi}{2}\\,x\\right)\\right] \\! .", "a99ecee4142ee1a8abaf45f6cf61c7f0": "g(x,u)\\le b + \\beta \\cdot dist(u,\\mathcal{N})", "a99f793b494e1b666acc119d9d2ad6a9": "P^+ = \\int_{\\frac{A g(0)}{L-1}}^{\\infty} \\frac{1}{\\sqrt{2 \\pi} \\sigma_N} e^{-\\frac{x^2}{2 \\sigma_N^2}} d x = \\frac{1}{2} \\operatorname{erfc} \\left( \\frac{A g(0)}{\\sqrt{2} (L-1) \\sigma_N} \\right) ", "a99fadf22f3fd634970fccefba5c7dde": " K(s,t) = \\sum_{j=1}^\\infty \\lambda_j \\, e_j(s) \\, e_j(t) ", "a99ffc4ccb3fce8e38ae84bb8ed7de9b": "median(A,i,j)", "a9a01a771795ecfaa1a3a798eca2375d": "\\scriptstyle \\hbar\\omega", "a9a07635dc18b8fc87e06f6bd7796c09": "\ne = \\sqrt{1 + \\frac{2 E L^{2}}{m_\\text{red} \\alpha ^{2}}}\n", "a9a0956b9ac87252fc7bea4f26c31fae": "\\nabla \\cdot \\mathbf{D} = \\rho_\\mathrm{f}", "a9a0a91f1bd971ed04ba89f310e357e6": "E(\\gamma)=\\frac{1}{2}\\int g_{\\gamma(t)}(\\dot\\gamma(t),\\dot\\gamma(t))\\,dt.", "a9a110605b10751526ba6eab775a5f57": "\\{ (x,y) | \\sqrt{(x-c_x)^2 + (y-c_y)^2} + \\sqrt{(x-a_x)^2 + (y-a_y)^2} = \\sqrt{(c_x-a_x)^2 + (c_y-a_y)^2}\\}", "a9a1912e942af616cdbd3d692c87ecb3": "X_n\\ \\xrightarrow{as}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{p}\\ X", "a9a1cc8ad7d692a7e228c92f6b1bf168": "i,j\\in I", "a9a1e1bc4bf0237c1260d1e60d20fb28": "O(f(n))", "a9a2199068916817a3b7b58d40db1105": "\\mathrm{A=C\\ s^{-1}}", "a9a2479e0415b2da42a7bf968b275072": "h = m", "a9a3ac865a643ea93b99044bdde60ec5": "\\lambda_2 = \\begin{pmatrix} 0 & -i & 0 \\\\ i & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}", "a9a3f08681b3481c30a7a221d1e93016": " \\mathbb{R}P^2 ", "a9a4004952c5d43087c875fc1af28e1e": "S:SO(3;1)^+ \\rightarrow \\mathrm{GL}(U);\\quad \\Lambda \\rightarrow e^{i\\pi(X)};\\quad e^{iX} = \\Lambda, \\quad X = \\omega_{\\mu\\nu}M^{\\mu\\nu}\\in \\mathfrak{so}(3;1)", "a9a42f0a94d435aa993e0ecd32deeedf": "(f+g)(c) = f(c) + g(c)", "a9a466dc3dc9b76e2a7596c9915efd25": "{\\alpha}>0.3", "a9a467ef41a534ae72d63c8c3a41c4ff": "f(x)=(a_1x_1+a_2x_2)^n=(b_1x_1+b_2x_2)^n=(c_1x_1+c_2x_2)^n=\\cdots.", "a9a47987e0317c2d630ad70a5c048843": " m(x,\\omega) = 1 ", "a9a479ead9ae7d6ac6e982a3f3aa1e41": "\\begin{align}\n\\langle f\\rangle \n& = \\sum_{x_i} f(x_1,x_2,\\dots) P(x_1,x_2,\\dots) \\\\\n& = \\frac{1}{Z(\\beta)} \\sum_{x_i} f(x_1,x_2,\\dots) \\exp \\left(-\\beta H(x_1,x_2,\\dots) \\right)\n\\end{align}\n", "a9a4bc872503916942a6171f5660e4da": "(\\omega_c + \\omega_a)", "a9a4c3b2f60edb8a465db04c263ca64d": "Ra ", "a9a4f8928e1e191f15a24c10c292fa72": " N_q = \\frac{ e ^{ 2 \\pi \\left( 0.75 - \\phi '/360 \\right) \\tan \\phi ' } }{2 \\cos ^2 \\left( 45 + \\phi '/2 \\right) } ", "a9a501c8f92e7c2b551c57ce38452952": "\\displaystyle{F(z)=e^{U(z) + i V(z)}}", "a9a5288d711e6588660c0e413e607220": "\\beta = \\frac {X_\\text{Ci}} {R_\\text{f} + X_\\text{Ci}} = {1 \\over 1 + R_\\text{f} C_\\text{i}s} ", "a9a56fff2e51e9c9c26bd77faec9e3b8": "(\\neg (I \\land R))", "a9a5fb99de5a67ef8788ac3982f11120": "\\xi(x(0)) = \\nabla u(x(0))", "a9a5fe2dca730972f5a8928783121c16": "R_{N} (t, s) = E[N(t)N(s)]", "a9a62f7aced9d382b91566d490928dd2": "\\sin \\left( \\frac{\\Theta}{2} \\right) = \\frac{1}{\\epsilon}", "a9a6320b1003840423706a3c9629c6dd": "\\sqrt {2\\psi \\, \\Delta\\theta Kt}", "a9a676329bfd81c4031f704b66a2d0dc": "L = \\frac{1}{5} l \\left[\\ln\\left(\\frac{4l}{d}\\right) - \\frac{3}{4}\\right]", "a9a6965d0c9ee90b432fa1a5e197d19e": "E(e) = \\frac {(i_$ - i_c)} {(1 + i_c)} \\approx i_$ - i_c", "a9a701df71649f506d6d63771f033356": "G=\\langle A | R\\, \\rangle", "a9a738ef9d4e46360dd9b87b39c691bf": "N=1", "a9a83175ecb3ae990ce54c55a8067865": "\\gamma_2\\in \\Gamma_2", "a9a8705774f07b98ae5d1807b21418ab": "\\sqrt{\\frac{15}{56}}\\!\\,", "a9a89e0bbf14f7b047ff160dea2ce2a2": "\\vdash\\text{Provable}(p)", "a9a8aa2b8a22234b8c83b57dd3d3280f": "1\\to G_0 \\to G \\to \\pi_0(G) \\to 1.\\,", "a9a8af7642b5475b04b30c8c62f52248": "x^3-(b_{2}-2a_{14})x^2+a_{14}^2*x+a_{14}^2*b_{2}=0", "a9a8b92038e99cefad787ebe9c455f61": "\n \\begin{align}\n \\varphi &= A_0~r^2 + B_0~r^2~\\ln(r) + C_0~\\ln(r) + D_0~\\theta \\\\\n & + \\left(A_1~r + B_1~r^{-1} + B_1^{'}~r~\\theta + C_1~r^3 + \n D_1~r~\\ln(r)\\right) \\cos\\theta \\\\\n & + \\left(E_1~r + F_1~r^{-1} + F_1^{'}~r~\\theta + G_1~r^3 + \n H_1~r~\\ln(r)\\right) \\sin\\theta \\\\\n & + \\sum_{n=2}^{\\infty} \\left(A_n~r^n + B_n~r^{-n} + C_n~r^{n+2} + D_n~r^{-n+2}\\right)\\cos(n\\theta) \\\\\n & + \\sum_{n=2}^{\\infty} \\left(E_n~r^n + F_n~r^{-n} + G_n~r^{n+2} + H_n~r^{-n+2}\\right)\\sin(n\\theta) \n \\end{align}\n ", "a9a90b2de3495735c3b0b21123d3b8a8": "P_2(c) = c^2 + c \\,", "a9a947d09b532413326cc415f966b02e": "\\alpha = -14", "a9a9a80f440ca70c341b423b32af2c91": " \\|f-X(f)\\| \\le (\\Lambda_n(T)+1) \\|f-p^*\\|. ", "a9a9e543bc93b76c2b0e14bdc6cb079d": "\\sqrt{ \\frac{1 + v/c}{1 - v/c} } f_0 = \\gamma \\left(1 + v/c\\right) f_0 \\qquad \\text{and} \\qquad \\sqrt{ \\frac{1 - v/c}{1 + v/c} } f_0 = \\gamma \\left(1 - v/c\\right) f_0, \\,", "a9a9eeafa6ce22478b818e143ece9687": "\\scriptstyle \\lambda \\;>\\; 0", "a9aa1af2fe63e5e2dd37798c7b6cde77": "\\Psi_w", "a9aa2b527ea47ec4ac7d44f9d0ab35e1": "\\mathbf{B}_{l,m}^{(E)} = \\sqrt{l(l+1)} \\left[B_l^{(1)} h_l^{(1)}(kr) + B_l^{(2)} h_l^{(2)}(kr)\\right] \\mathbf{\\Phi}_{l,m}", "a9aa2ba3d4f3d068fba3e2476902e5c7": "E \\exp(i u^T X)=\\exp\\{-\\gamma_0^\\alpha+i u^T \\delta)\\}", "a9aa3b3cd86320f9df508608af36f592": "\\frac{1/(k+q)^s}{H_{N,q,s}}", "a9aa6a2a59f20dd7f87819c8c4a3d822": "(a,1]", "a9aab73200bb992bf7d2e04ffa15b1a4": "\\int \\tanh x \\, dx = \\ln \\cosh x + C", "a9aab93067fae0cdf1b7d87d28416f43": " \\langle x^2 \\rangle = \\frac{1}{P} \\int{I(x,y) (x - \\langle x \\rangle )^2 dx dy}, ", "a9aabd3727c7f03435bceb6989a89771": " S^{*} = F_T = S_0e^{rT} ", "a9aac84a908b3b376e465fb60a100bef": "(ds)^2 = g_{ik}\\ dx^i\\ dx^k \\ , ", "a9aafe67e856d46b37f71b449614d272": " R\\left(x,u,u_x,\\ldots,\\frac{d^n u}{dx^n}\\right) ", "a9ab17a163196620b9056afabe626e51": "\\int {1 \\over x}\\,dx = \\ln \\left|x \\right| + C", "a9ab189890b8bc38616cb2dd7cf4fbfe": "\\sigma^+ - \\sigma", "a9ab49efc31c2c944fbb6e5814116dc0": "\n W_{ri+j} = W_j\\cdot A^{ij} \\cdot B^{j^2}\n \\quad\\text{for all}~i \\ge 0~\\text{and all}~j \\ge 1.\n", "a9ab571843835aabd50371caf1804233": "A \\models \\bigwedge \\Phi(\\bar{a})", "a9ab588d707a756969df2a7ef21e219f": "x_{0} \\geq 0", "a9ab5d804739d47cb31c0d8ad0fa41db": "\\sum\\limits_{i=1}^I\\beta_i=1;\\;\\;\\;\\;\\beta_i>0\\;\\;\\forall i.", "a9ab784ef832c68239dc6834fa18157d": "\\iota \\in Q", "a9ab975c175eb5fe88a919cb592b811b": "K = \\operatorname{round}\\left(\\frac{T}{s}\\right).", "a9abe0a79caf1c26a6999967aaec7a0c": "\\int U_n\\, dx = \\frac{T_{n + 1}}{n + 1}\\,", "a9ac09e4fea7641658892df09d187c8a": "\\scriptstyle V_2 = -\\frac{D}{1 - D}V_1", "a9ac0c63b7d2db0ba14fcda91d3078ad": "\\Gamma(x)\\psi(x)\\,", "a9ac14851b8cd7fc88b1563a27cb1088": "f_{\\text{r}}(\\omega_{\\text{i}},\\, \\omega_{\\text{r}}) \\ge 0 ", "a9ac28f41222c691d6539e4e888e2a47": " \\Pr\\left[\\,\\overline{A_1} \\wedge \\cdots \\wedge \\overline{A_n}\\,\\right] \\geq \\prod\\nolimits_{A \\in \\mathcal{A}} (1-x(A)). ", "a9ac33735337f8aa0e42d82c1302d860": "\n\\frac{\\rm d}{{\\rm d}t}x(t)=f(t,x,y),\\quad \\frac{\\rm d}{{\\rm d}t}y(t)=\\cos(\\beta)x+\\alpha z,\\quad \\frac{\\rm d}{{\\rm d}t}z(t)=\\sin(\\beta) x-\\alpha y,\n", "a9ac42190c31d8b41e0196235adf447f": " y(t_n) ", "a9ac8cc2e34f0aaff40daad347530671": "\n\\vartheta_{00}(z, q) = \\sum_{n=-\\infty}^\\infty q^{n^2} \\exp (2 \\pi i n z)\n", "a9ac95d258f1cdf282b31ef2e211859c": " x[n] \\ \\stackrel{\\mathrm{def}}{=}\\ x(nT) = x \\left( { n \\over f_s } \\right) ", "a9acb365afa7242e24a822b894b5331e": "I_p/I", "a9ace7784b8ec1dc4e0ee95f47f93b23": " S(A,P,z) \\le \\frac{X}{V(z)} + O\\left({\\sum_{\\begin{smallmatrix} d_1,d_2 < z \\\\ d_1,d_2 \\mid P(z)\\end{smallmatrix}} \\left\\vert R_{[d_1,d_2]} \\right\\vert} \\right) .", "a9ad1161fcfd3561aed2d5b45e231183": "k \\ge 3", "a9ad1845a933583a8f58b2ca6a96687c": "D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.\\ ", "a9ad24d5b29104834625b16c84be7b14": "A \\leq_{\\rm m}^{\\rm P} L", "a9ad31acb3fad0c3b2df28445938322b": "= \\sum_{m=-\\infty}^\\infty f[n-m]\\, g[m].", "a9ad6b4db1e03917346dbf338cca12a4": "\\begin{vmatrix}\n\\left(-k_y^2-k_z^2+\\frac{\\omega^2n_x^2}{c^2}\\right) & k_xk_y & k_xk_z \\\\\nk_xk_y & \\left(-k_x^2-k_z^2+\\frac{\\omega^2n_y^2}{c^2}\\right) & k_yk_z \\\\\nk_xk_z & k_yk_z & \\left(-k_x^2-k_y^2+\\frac{\\omega^2n_z^2}{c^2}\\right) \\end{vmatrix} =0\\,", "a9ad9d257425683369cad430c32b5e0d": "q=\\frac{2ab}{\\sqrt{a^2+b^2}}.", "a9ae201af005f1419716ac68ce498e9a": "\\mu\\notin\\sigma(A)", "a9ae6a283fd70439835775e9b7537203": "\n\\alpha_c = Prob\\{Demand~during~replenishment~lead~time\n\\le Inventory~on~hand~at~the~beginning~of~the~lead~time\\}\n", "a9aeaabf3a0c4bdb1f2344ce324e5fbc": "X = \\{O_{1},O_{2},O_{3},O_{4}\\}", "a9aebd8e8840aea82c658b3206ac5f7b": "\\sum_{i=1}^\\infty x_i", "a9aec39ac101d1fd5958f7eb86c480a6": "\\log\\Gamma(x) = \\zeta_{H}'(0,x) - \\zeta'(0),", "a9af2af5e847b31c3c184c7b59b9e4cd": "c(\\theta) \\mbox{ is the canonical index of } \\theta", "a9af41bff9aa46d61f67b876af9ace37": " \\frac {d \\vec V } {dt} = \\left[ \\frac{ d v_1 }{dt},\\frac {d v_2 }{dt},\\frac {d v_3 }{dt}, \\cdots \\right] \\ . ", "a9af69cc1628379fd88bf62e72903849": "D^{++} \\equiv u^{+i}\\frac{\\partial}{\\partial u^{-i}}", "a9af7b2850a6adece6066ce81077954d": "L(a)=|a|", "a9af81831005629d8a1fe4b44c8d9d08": "\n\\begin{align}\n\\langle R^{2} \\rangle & = \\langle \\vec R \\cdot \\vec R \\rangle \\\\ \n & = \\left\\langle \\int_{0}^{l} \\hat t(s) ds \\cdot \\int_{0}^{l} \\hat t(s') ds' \\right\\rangle \\\\ \n & = \\int_{0}^{l} ds \\int_{0}^{l} \\langle \\hat t(s) \\cdot \\hat t(s') \\rangle ds' \\\\ \n & = \\int_{0}^{l} ds \\int_{0}^{l} e^{-\\left | s - s' \\right | / P} ds' \\\\ \n \\langle R^{2} \\rangle & = 2 Pl \\left [ 1 - \\frac {P}{l} \\left ( 1 - e^{-l/P} \\right ) \\right ]\n\\end{align}\n", "a9af83e4371eaf9fafc671f3575b9b39": "\\pm 2 \\pm 4\\cdots \\pm n = 0.", "a9afe8dd0a909db53f8655efdec23c95": "(3 + 5) = (5 + 3)", "a9b0031a7a044b81882e4b655e444842": " c\\left(\\|\\nabla I\\|\\right) = e^{-\\left(\\|\\nabla I\\| / K\\right)^2} ", "a9b04b08f90f45656db1318b044e2cc9": "\\varphi'(s)", "a9b0511b4a2ac8ae6ecf4b0af66ce851": "\\sup_{P\\in \\mathcal{P}(S,A)} \\mathbb E \\|P_n-P\\|_\\mathcal{C}\\to 0;", "a9b0518d67da79cfadfed2007ed87c04": "\\frac{1}{1 - z}", "a9b06fa3d9a299be115d55bde3b70563": "c(h,k)^g c(hk,g) = c(h,kg) c(k,g) . ", "a9b079d656bd4766b4cf053fcf7880f8": "P_{Tx}", "a9b1112f53bd5f809eb6ee7a030218cd": "\\frac{\\partial u}{\\partial t}=0", "a9b1280b06b462eb656598ea46441b00": "\\left [ \\mathbf S \\right ] = \\begin{bmatrix} S_{11} & S_{12} \\\\ S_{12} & S_{11} \\end{bmatrix} ", "a9b128bd1fc33a289791c2f1b8a96c8e": "\\text{Distance} = c \\frac{\\Delta t}{2n},", "a9b1290a1a1e86fd38b3a23c238294cd": "\\alpha > 0, \\beta_A \\, ", "a9b131843d2a696efe57bd78848f90a0": "\\nabla^2 \\mathbf A' - \\mu_0 \\varepsilon_0 \\frac{\\partial^2 \\mathbf A'}{\\partial t^2} = - \\mu_0 \\mathbf J + \\mu_0 \\varepsilon_0 \\nabla \\left ( \\frac{\\partial \\varphi'}{\\partial t} \\right )", "a9b13bf9473f2ea03b11e4dfc3c45c3b": "\\mathbf{T}(s)\\,", "a9b16232560e80a80e9e86d1e3eb14ee": " \\mathrm{perceived\\ speed} = \\frac{\\mathrm{projection\\ frame\\ rate}}{\\mathrm{camera\\ frame\\ rate}}\\times\\mathrm{actual\\ speed}", "a9b171a2a6f214a07fe10d561ead3825": "x_\\lambda)", "a9b1b02380a1783095a93bc8c8464f78": "L_e(e_x)", "a9b266ebe20289ef9e2322f056cb1c39": "\\!\\mathcal A \\models_X^+ \\lnot \\phi", "a9b2d831dc141825a997b87579874126": "\\tan\\left(\\frac{1}{z}\\right)", "a9b2e4d40c7ced266d64aff7eeea85c0": " p_1^{\\mu_1} \\, p_2^{\\mu_2} \\cdots p_k^{\\mu_k} ", "a9b32bba12837a80c0b4d0b1e84bf97c": "\\displaystyle{\\partial_{\\overline{z}} (f\\circ g_n) = 0}", "a9b340a059552eef16229ebd3fbe67ac": " \nf(x,t)", "a9b36179a8652f707b3817f6d14c82dc": "\n\\frac 1{2\\pi \\sqrt{1-\\rho^2}}\\exp\\left(\\frac{(x^2+y^2)- 2\\rho xy}{1-\\rho^2}\\right)\n", "a9b39a082888b5fe0437271b724587bf": "R \\mapsto ", "a9b3ab1f2729f83e36b6d0a32a7ea6ac": " \\int {1 \\over x}\\,dx = \\ln|x| + \\begin{cases} A & \\text{if }x>0; \\\\ B & \\text{if }x < 0. \\end{cases} ", "a9b3cb8df0b09c72d34c21117658bc11": "\n\\Gamma = 2\\pi - \\sum_j \\theta_j\n", "a9b3edbfa127b0c88c49609e7955af29": " [x_1:y_1:1].[x_2:y_2:1] = g^{-1} ([x_1:y_1:1] \\times [x_2:y_2:1]) ", "a9b40f5637c410a3cd4a7d25910dc2e5": "dx \\approx dX\\,\\!", "a9b44cf75c44b984c55f08eebedb38f5": "d_Y(f_0(x),f_0(y))\\leq d_Y(f_0(x),f_n(x))+d_Y(f_n(x),f_n(y))+d_Y(f_n(y),f_0(y)).", "a9b49397c7223c4a741da1c0175ad880": "(1)\\qquad ds^2 = - e^{2 \\Psi(\\rho,\\phi,z)} dt^2 + e^{-2 \\Psi(\\rho,\\phi,z) } \\Big(d \\rho^2 + d z^2 + \\rho^2 d \\phi^2 \\Big)\\;,", "a9b5799c4e54c67a566e3bb296d7a48b": "\\displaystyle{F(z) =e^{G(z)+iH(z)}= z e^{f(z)}}", "a9b58a83b1f52879ee23332dbfcdf36b": "\\begin{array}{llll}\n1:&\\Gamma \\vdash id : \\forall\\alpha.\\alpha \\rightarrow \\alpha &[\\mathtt{Var}]& (id : \\forall\\alpha.\\alpha \\rightarrow \\alpha \\in \\Gamma) \\\\\n2:&\\Gamma \\vdash id : int \\rightarrow int & [\\mathtt{Inst}]&(1),\\ (\\forall\\alpha.\\alpha \\rightarrow \\alpha \\sqsubseteq int\\rightarrow int)\\\\\n3:&\\Gamma \\vdash n : int&[\\mathtt{Var}]&(n : int \\in \\Gamma)\\\\\n4:&\\Gamma \\vdash id(n) : int&[\\mathtt{App}]& (2),\\ (3)\\\\\n\\end{array}\n", "a9b5bd97d59d72dc3becdfe7b5e0607f": "0=X_0 \\leq X_{t_1} \\leq \\cdots . ", "a9b5c32df00866f09f36a186d97fc6bb": "\\mathbf{x}\\cdot\\mathbf{y} = \\sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\\cdots+x_ny_n,", "a9b637d7d5e950e2d23bd79dd4034f66": "\\tau = 0 ", "a9b645427327ca1a2299acb103f0a5de": "p(x\\mid y,I) = \\frac{p(y\\mid x,I) p(x\\mid I)}{p(y\\mid I)}", "a9b6b33fbc64160b86f6dc31f3fee8b8": " Y = (U,V) ", "a9b6c221d98e6b7780d67e9707e82835": "S\\sim_x T \\; \\Longleftrightarrow \\; \\mathbf{1}_S \\sim_x \\mathbf{1}_T.", "a9b719d3c3d3004778296542516f6a8b": "N_{B/A}(\\mathfrak{b})", "a9b7534943661c2a8e7a6feb4d063a46": "V_f", "a9b77e9311b7b8b7e0e24c86e80ceef3": "\\alpha_2 = c\\omega_1+d\\omega_2.\\,", "a9b7c706e61f2104bd3896fdaec60322": "\\cos(x- y) = \\cos x\\cos y + \\sin x\\sin y\\,", "a9b83a189072a155c697f9de9f3b63a0": "\\{v_2,v_4\\}", "a9b8cb4af4cedf9514fdac45f475170e": "I(\\theta)\\, ", "a9b8de663f8fdfe4e784f1b5b254cf28": "\\vec \\psi", "a9b93726e287e80bb23f0430f605d214": "\\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2}) \\,", "a9b9559061811e047236d9019bf95c72": "\\{ x ~|~ x \\in \\mathbf R \\land x > 0\\}", "a9b95e832a80a97f6d3a099d8dcbd1b9": "\\scriptstyle p=0.9^4", "a9b974a1a56d210971b5e6838ad6fae4": "f=q(u+v)-q(u-v), ", "a9b9f575a46003d96a67bccf778d4b6f": "\\textstyle \\frac{3\\sqrt{3}}{4} \\left( 1 - \\frac{1}{2^2} + \\frac{1}{4^2}-\\frac{1}{5^2}+\\frac{1}{7^2}-\\frac{1}{8^2}+\\frac{1}{10^2} \\pm \\cdots \\right)", "a9b9f617edd603c72b4e4e4fbbcf22c7": " \\Rightarrow u(0 x_1", "a9cb8196c3528061dedfd5393bbfaac6": "P_{< \\beta}", "a9cc280e235b3a1276923b1575571ad0": "X_i |(Z_i = 2) \\sim \\mathcal{N}_d(\\boldsymbol{\\mu}_2,\\sigma_2)", "a9cc5cfab17900484c4f3b84b3e304a5": "B_H (t) - B_H (s)\\; \\sim \\; B_H (t-s). ", "a9cd095644c1f00185b5ef8735e506f6": "\\frac {r^2 u} {2} ,", "a9cd647e97fb207c67a69477467807ce": "w_{n+1} = \\mathcal M w_n = w_n^2 - \\mu.", "a9cd7d6013a7791d25ab56057bb2994b": "g \\in [0,1]", "a9ce66c21a4503f0c23ff2aae5a15d2f": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 &\\qquad \\text{ ( by Lyapunov function } V_x, \\text{ subsystem stabilized by } u_x(\\textbf{x}) \\text{ )}\\\\\n\\dot{z}_1 = z_2\\\\\n\\dot{z}_2 = z_3\\\\\n\\vdots\\\\\n\\dot{z}_i = z_{i+1}\\\\\n\\vdots\\\\\n\\dot{z}_{k-2} = z_{k-1}\\\\\n\\dot{z}_{k-1} = z_k\\\\\n\\dot{z}_k = u\n\\end{cases}", "a9ce8e9625840d7bd81c4b3239c4dc2f": "(\\tfrac{1}{2},1)", "a9ceabb32d5c9e6a59b8dd311580e170": "\\tbinom{4}{2}", "a9ced8b404919cc76662b39f82fe6c67": "(xy)^n = x^n y^n [y,x]^{\\binom{n}{2}}.", "a9cee337a780f221122dbb9b7e8a2eef": "\\begin{cases}\\dot{\\mathbf{x}} = (f_x(\\mathbf{x}) + g_x(\\mathbf{x}) u_x(\\mathbf{x}))+g_x(\\mathbf{x}) e_1\\\\\\dot{e}_1 = v_1\\end{cases}", "a9cf185b41e9f94085c07dec39c895e6": "\\textstyle \\zeta_G(\\alpha)=\\frac{8}{3}\\zeta(\\alpha-3)+\\frac{16}{3}\\zeta(\\alpha-1)", "a9cf2dd6375dc7b2a95546552a0c39a8": "\n\\dot S + c_0 S^2 = c_1 \\left| \\dot T \\ln h + T \\dot h / h \\right|\n", "a9cf6ad0de317f597dff59c1d1357ac6": "Q\\gg 1", "a9cf6f7f45c0734545b186acd5d10a32": " S_i = \\frac{V_i}{\\operatorname{Var}(Y)} ", "a9cfcf4d4aaf8188b9ce0ad4e4a29bc4": "\\dot{v}_4", "a9d02776f2f1a5bdae881ceb155884c5": " r_{n,n+1} = \\frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}} ", "a9d07dcd3b05d4bd6a53d72840858f01": "\\Omega_0 = \\{ 0, 1 \\}.\\,\\!", "a9d0a33de7c2b1f1a525557521656523": "\\,Y", "a9d0a703089315ac15348e9ff3fce136": "\\Phi_n(x) = \\Phi_q(x^{n/q}).", "a9d0af277be6f011d6e509684e76c07e": "m * n ", "a9d0c468416888b946e33c00d0a1b28f": "i|1\\rangle", "a9d0cd01d7c3094ab2b386a646fcfd54": " W\\mathbf{r} = \\omega \\times \\mathbf{r}", "a9d0cf15ab37f00de4e7de4c11877bc5": "\\frac{dS}{dt} = \\frac{1}{2} k \\sum_{ \\alpha,\\beta} \\nu_{\\alpha\\beta}(\\ln p_{\\beta}-\\ln p_{\\alpha})(p_{\\beta}- p_{\\alpha}).", "a9d1ccfb8aa916b923ce7bac049241ee": "\nLM=\\left (\\frac{\\partial l}{\\partial\\theta} \\right )'\\left (-E\\left [\\frac{\\partial^2 l}{\\partial\\theta \\partial\\theta'} \\right ] \\right )^{-1}\\left(\\frac{\\partial l}{\\partial\\theta} \\right ).\n", "a9d1d5014ede7d08c5fc4aa96e5f52af": "M_2=\\left.2\\mu\\right.,\\,", "a9d1e98f6e78b7dacd19679b57209770": "f(n | \\lambda) = e^{-\\lambda}\\frac{\\lambda^n}{n!},", "a9d204827c054a47c1930b694c423654": "A' = x^2 A", "a9d28d0e5bbbb3861c91370432fa8311": "E_n=\\hbar\\omega_c\\left(n+\\frac{1}{2}\\right),\\quad n\\geq 0. \\, ", "a9d30157b59a5b074b309419357fd9a6": "a^2 + b^2 + c^2 + d^2 = 1", "a9d3216938d6e0603c841798acd30637": "\\nabla_{\\bold {r}_0} \\frac {1}{|\\bold r - \\bold{r}_0|} = \\frac {\\bold r - \\bold{r}_0}{|\\bold r - \\bold{r}_0|^3}", "a9d3a1a64e443952a7b0934ac3657be2": "\n \\left(\\frac{\\partial f_i}{\\partial t} \\right)_{\\mathrm{coll}} = \\sum_{j=1}^n \\iint g_{ij} I_{ij}(g_{ij}, \\Omega)[f'_i f'_j - f_if_j] \\,d\\Omega\\,d^3\\mathbf{p'}.\n", "a9d3e7567162b082c27fdae2f74cd38f": "\\frac{\\partial}{\\partial \\theta}", "a9d40bccca238b59fb4ca226ed3b6476": "\\mathcal{S}\\times \\mathcal{A} \\rightarrow \\mathbb{R}", "a9d44ac321b1543447f5e1d7d5f0b136": "\\sigma_L=\\sigma_L^D + \\sigma_L^+ + \\sigma_L^- ", "a9d476de097031812765676b236f0aba": "\n\\int e^x = f(u_n)\n", "a9d47f8c983998ddc786534396cb127f": " \\frac{\\partial}{\\partial z} \\left(f\\circ g\\right)= \\left(\\frac{\\partial f}{\\partial z}\\circ g \\right) \\frac{\\partial g}{\\partial z} + \\left(\\frac{\\partial f}{\\partial\\bar{z}}\\circ g \\right) \\frac{\\partial\\bar{g}}{\\partial z}\n", "a9d4b97f46466b3990f4c48109b1d1f0": "\\left|\\frac{f(x,\\alpha+\\Delta \\alpha)-f(x,\\alpha)}{\\Delta \\alpha} - \\frac{\\partial f}{\\partial\\alpha}\\right|<\\varepsilon\\,", "a9d4d01133b0ca92240c182a879fe193": "\\oint_{\\part V}\\mathbf{g}\\cdot d \\mathbf{A} = \\int_V\\nabla\\cdot\\mathbf{g}\\ dV", "a9d4df8514907a8268315c99f346df60": "\n e^{i\\mathbf{k}\\cdot\\mathbf{r}} = e^{ikr\\cos\\theta}=\\sum_{l=0}^\\infty i^l (2l+1) j_l(kr)P_l(\\cos\\theta),\n", "a9d54264ed4a5c8c06dac0ff47314a88": "V=\\begin{bmatrix}\n\\nu \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix}\n", "a9d5572e72c15e07bbdfb1c0e5abe436": "(N S_x)", "a9d5b413fc61413759479f14f54547db": "M_\\mathrm{e} = \\pi L_\\mathrm{e}", "a9d5d39d599e39199c7d780441354397": "m_3,", "a9d5de3891328a78c99c6ff830d3dc03": "\\mathbf{u} = u_\\theta\\ \\mathbf{e_\\theta}", "a9d5f9d50a045fd7cf1aa2a749c462ad": "\n \\cfrac{\\partial\\langle p \\rangle}{\\partial t} = 0 ~;~~ \\cfrac{\\partial\\langle \\rho \\rangle}{\\partial t} = 0 ~;~~\n \\cfrac{\\partial\\langle \\mathbf{v} \\rangle}{\\partial t} = \\mathbf{0} ~.\n ", "a9d602ab90724837661efd7012dd61d5": "a_n=a/n", "a9d63eb1968a130d187948aa310b4388": " \\theta =\\frac{\\alpha \\cdot P}{1+\\alpha \\cdot P}", "a9d65cfa28206bb92b13502f6506f90d": " \\mathfrak{I}\\,|\\, S\\, \\rangle= \\pm |\\, S\\, \\rangle", "a9d671d9239b20aeea3e432335edc757": "\\delta S_{H-P} = \\int d^4 x \\; e \\; \\Big( (\\delta e^\\alpha_I) e^\\beta_J \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} + e^\\alpha_I (\\delta e^\\beta_J) \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} - e_\\gamma^K (\\delta e_K^\\gamma) e^\\alpha_I e^\\beta_J \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} \\Big) ", "a9d67fff2f1358dcc43ecb6b30069bb5": "\\varphi + 1 = \\varphi^2", "a9d6abf657b819ae8165c37f12984d38": "\\ PV = \\frac{m}{M}RT ", "a9d6c4e2267b1ef9d058dc79a05a2598": " \\int_{0}^{\\infty} \\frac{dx}{x+a} ", "a9d6d7347d794ebf787130b298ad5f24": "p(x) \\ge 0 ", "a9d6ee84294f3e28942236af6d48845e": "p =\\frac{RT}{V }\\Rightarrow \\left(\\frac{\\partial p}{\\partial V}\\right)_{T}=\\frac{-RT}{V^2 } = \\frac{-p}{V }", "a9d75b96e88cd160cf0702b3af688445": "0.61 \\to 0.585 \\to 0.6225 \\to 0.56625 \\to 0.650625 \\ldots", "a9d76443dfe68746961b7f4fee3a9a8f": "\\textstyle (nd+1)(t-1)+1", "a9d7974adc99f8799257369ef7b43581": "\\frac{\\delta^n Z}{\\delta J(x_1) \\cdots \\delta J(x_n)}[J] = i^n \\, Z[J] \\, {\\left\\langle \\phi(x_1)\\cdots \\phi(x_n)\\right\\rangle}_J", "a9d7b5b1c20cba9f2dcd092e864bb936": "\\mathbf{\\rho}_{i},\\mathbf{\\rho}_{j}", "a9d7dcfbbda2c0a8aa38d473b5c6301d": "\\tan\\varphi =2\\cot\\theta\\frac{M^{2}\\sin^{2}\\theta -1}{2+M^{2}(\\gamma +\\cos 2\\theta )}.", "a9d820c2d463652aee434ee37a9e7705": "\\{f_1\\equiv w_{xx}-2z_{xy}-\\frac{1}{2x}w_x+\\frac{1}{2x^2}w=0, f_2\\equiv w_{xy}-\\frac{1}{2}z_{yy}-\\frac{1}{2x}w_y-6x^2z_x,", "a9d827016f8c57b66490b880b7f8668d": "\\mathfrak p", "a9d84c9d7fe2c54836b90c4110c85f09": "\\mathbf{c_{s1}, c_{s2}}", "a9d87c03a97766c581c24802d2aa0de1": "\\mathcal{F}_{n}\\left(\\psi_{tar}\\right)\\simeq 1", "a9d8f7c22f1c77eb85535faf96cd3625": "\\left(\\cos x+i\\sin x\\right)^2 = \\cos^2 x + 2i \\cos x \\sin x - \\sin^2 x,", "a9d963d96388ad104c2401b4c6390241": " \\operatorname{E}(X)= \\mu ", "a9d9a15e62aa41c7884428421550a837": "a,b,\\ldots", "a9d9b6aa634b431f7f53c8068ca48bc9": "\\frac{4\\times 2EI}{L}", "a9d9b9c8f011608c630f773402ff597a": " T \\rightarrow 0, S \\rightarrow C ", "a9d9e9ca4b31de59aa17fcc5184a63bc": " g(\\nabla_X Y, Z) = \\frac{1}{2} \\{ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(X,Y)) + g([X,Y],Z) - g([Y,Z], X) - g([X,Z], Y) \\}. ", "a9da1ba41586544b75602976ddc60383": "(p_i(t) x_i^\\prime)^\\prime + q_i(t) x_i = 0, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\, x_i(a)=1,\\,\\, x_i^\\prime(a)=R_i\\,", "a9da9723c416fa9d12999b55221eb0f4": " v'(r,r ')= \\frac{r '}{r } v \\quad \\Leftrightarrow \\quad \\rho = \\frac{ r ' }{ r } = \\frac{ v ' }{ v } = \\left( \\tfrac{r ' \\theta}{t} \\right) \\diagup \\left( \\tfrac{r \\theta}{t}\\right)", "a9dad8bb7d48ddbdc3760456a756c753": "f(n) = 2f((n-1)/2)+1=2((2l_1)+1) + 1=2l+1", "a9daf6b67e221597dc86f71563ef4cb5": " f(x) = \\int_A \\, w(a) \\, p(x;a) \\, da ", "a9db2ba80adfde428ae0b2b5542e5422": "k(t)=k(0) - \\frac{eE}{\\hbar} t", "a9db66fbbda0e63383c46fa72d931dbc": "\\Phi(x)\\,\\!", "a9db8dcb3c5a424fb36377aa1c912095": "g_i = 1 - \\sum_j \\frac{p_{ij} \\log p_{ij}}{\\log n}", "a9dbb030abda6c0e3c0065d44aed6f49": "\\displaystyle \\mathbb{E} Y_t = \\begin{cases}\n 1/\\sqrt{2\\pi} &\\text{for } 0 \\le t < 1,\\\\\n 0 &\\text{for } 1 \\le t < \\infty.\n \\end{cases} ", "a9dbcaca5e8ad88f62dbd09d66db57cb": " \\int w_i^\\alpha(\\vec r)w_j^\\beta(\\vec r)w_k^\\alpha(\\gamma r)w_l^\\delta(\\vec r) d^3\\vec r=0", "a9dbe053170b7841c12f8fadd622dc42": "C_{Building}\\,\\!", "a9dc17fe88af491d07406d059ad3cdba": "\\scriptstyle (u'(x,z,t),w'(x,z,t)).", "a9dc2f02ee57ec74d9afc21d773091b9": "0 - j\\infty", "a9dc3a0695cd1356f489b689a4d167b1": "w(k)=\\frac {2 \\pi} {\\hbar} \\frac {L^3} {(2 \\pi)^3} \\int d^3k' |\\lang\\ k'|H_1|k\\rang |^2 \\delta(E_{k'}-E_k)", "a9dc44bbfac31b5b3c23dde9acfc6cda": "f^{-1}(b) = \\{x \\in X : f(x) = b\\}.", "a9dcdc1f99d03dfdc15915a4de79ec3f": "A+A^*=2~\\mathrm{Re}~A", "a9dd16d999355e438379a7c86505c2a8": " V_p ", "a9dd2fe8e4e3c8737e36849501b529ec": " T = \\pi R_{\\rm eff}^{3} V(h) \n\\left( \\frac{1}{R'_{1}}-\\frac{1}{R''_{1}} \\right)\n\\left( \\frac{1}{R'_{2}}-\\frac{1}{R''_{2}} \\right)\n\\sin 2 \\varphi,", "a9dd3115c49e6468069f35b14238d48e": " K=\\sigma Y \\sqrt{\\pi a} ", "a9dd3f39303750a3aca450dd2855f02c": "\\mathrm{C}(f):= \\{\\varphi_i \\in \\mathbf{R}^{(1)} | (\\forall^\\infty x) \\, \\Phi_i (x) \\leq f(x) \\}.", "a9ddf4fbfad48ecbd78fb6c50bfaeb91": "\\Gamma(\\gamma)_s^t : E_{\\gamma(s)} \\rightarrow E_{\\gamma(t)}", "a9de2ec09bb784de56d010122dc8de10": "((p \\land q) \\to r) \\vdash (p \\to (q \\to r))", "a9debbdcf4afff2b7b6ccabce08f05ea": "\\mathcal{E}\\subset\\Pi^{n}.", "a9deca047c9982dfec569454fd7279a5": "r_a^2 + r_b^2 + r_c^2 = (4R+r)^2 -2s^2,", "a9def9c73a117f85daecdca764861b4f": " {m} = {\\iiint\\limits_V\\! \\rho \\,\\mathrm{d}V}. ", "a9df088dec64233a308139798e5a222b": "\\Psi_l^k = \\hat{\\mathcal{P}}_{S_l^k} \\hat{\\mathcal{H}} \\Psi_m^{(0)}", "a9df0ea7e485b8a72bb1df2b6658cf0f": "\\displaystyle{\\sigma\\begin{pmatrix} \\alpha & \\beta\\\\ -\\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix} = \n\\begin{pmatrix} \\alpha & -\\beta\\\\ \\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix}}", "a9df2285bf407febad7d00cac2690a13": "D_F(U\\xrightarrow{e}V)=Nw(e)-P_U+v-u", "a9df38ab9f9c9a2fe2eb2825992c9995": "x^2 - 2", "a9df7af46627a890ed0d058be4bdf90e": "\\overline T", "a9df826fdef408ab18b593873614f20f": " \\operatorname{Var}(S) = \\left( \\frac { Q } { Q - 1 } \\right)^2 \\frac { P_a ( P_a - 1 ) } { n - 1 } ", "a9dfab1207436b2cd6094a90ddf67657": "r \\times (n/w)", "a9dfdb5eda38f9dfaab834a9883d4a8a": "\\pi_1 M", "a9e0ad51e79683d552c6a846b83477ad": "W_2 = -\\frac{1}{16} \\, \\left( {C_{ab}}^{cd} + i \\, {{{}^\\star C}_{ab}}^{cd} \\right) \\, {C_{cd}}^{ef} \\, {C_{ef}}^{ab}", "a9e105fe1200ff9989d1519c57e81828": "\\mathrm{pH} = - \\log_{10}(a_{\\textrm{H}^+}) = \\log_{10}\\left(\\frac{1}{a_{\\textrm{H}^+}}\\right)", "a9e1524704325822c6eb67f63f850188": "f_{k-1} \\le \\sum_{i=0}^{d/2} {}^* \\left( \\binom{d-i}{k-i}+\\binom{i}{k-d+i} \\right) \\binom{n-d-1+i}{i},", "a9e156d594bcd2f67b35e639d4fa7f36": "\\ v_i = gt ", "a9e1867507118b6d7e417f20017eec26": "w_{k+1},\\ldots,w_n", "a9e1c24a86a3ab8bd00ed426903e1b21": "\nn^{-2/3} \\ln \\mathrm{PL}(n) \\sim 2.00945 -0.69444\\ n^{-2/3}\\ \\ln n -1.14631\\ n^{-2/3}\\ .\n", "a9e1ea5a0302462992a87ceca1b564c1": "r_j \\, r_k = r_{(j+k) \\text{ mod }n}", "a9e1ec0ad43c39ac9624e762c2a995bb": "[\\forall y \\; ([\\forall x \\in y \\; \\phi(x)] \\to \\phi(y))] \\to \\forall y \\; \\phi(y)", "a9e210221a8faadd9ec6702f7068aa2b": "\\gamma\\subset\\partial\\Omega", "a9e24d1ab11964cfa59da912da3a0111": "\\nu_\\text{max} = \\frac{\\pi}{2} \\bigg( \\sqrt{\\frac{\\gamma+1}{\\gamma-1}} -1 \\bigg)", "a9e2b554d76ee5471032f0df2d1a4058": "\\forall^\\mathrm{st}u_1\\dots\\forall^\\mathrm{st}u_n\\,(\\forall^\\mathrm{st}x\\,\\phi(x,u_1,\\dots,u_n)\\to\\forall x\\,\\phi(x,u_1,\\dots,u_n))", "a9e349760d0d45bd8923bb0e23f183f0": "z_{\\mathrm{S}} f_{\\mathrm{FD}} D \\mathrm{d} {\\it{\\epsilon}} \\mathrm{d} K_{\\mathrm{p}} ", "a9e398920a9a6d45ddfa88b211a359fe": "\\mathbf{q}(\\mathbf{x},t)", "a9e3bce820fd0dd04efdbcfd2de1dad4": " n \\times k ", "a9e3ead302010b7215f80670d3f76702": "\\tbinom nk=0", "a9e40d12286929d750df83b5c20fc216": "\\scriptstyle \\xi=(1+\\sqrt{5})/2", "a9e4426d468b0d95b2de9e5a05dd7747": "\\sin x+1=\\Omega(1)\\ (x\\rightarrow\\infty)", "a9e48539c1b130ccbc7823633ffabb1e": "(i+1)^{th}", "a9e4e24228130eeae49d618d6a2310ca": "c_{a,b}", "a9e52b431c187539c923c85f563a8256": "\\displaystyle{f(z) = CT^*h(z) + z.}", "a9e5a929d2ea6552dc091f5d4e822a68": "\\boldsymbol{x'}", "a9e609c5a34a52fad64ef3b719bcae58": "L+1", "a9e6401da8c99d9964e89462fb512da9": "\\Big( (\\mathcal{M}, s) \\models \\neg\\phi \\Big) \\Leftrightarrow \\Big( (\\mathcal{M}, s) \\not\\models \\phi \\Big)", "a9e648b0258e1e38ee9288f7ff33daeb": "p(\\tilde{x}|\\mathbf{X},\\alpha) = \\int_{\\theta} p(\\tilde{x}|\\theta) \\, p(\\theta|\\mathbf{X},\\alpha) \\operatorname{d}\\!\\theta", "a9e7a3f0b9bbc6aa0916f94bb7428297": "X \\xrightarrow{u\\,} Y \\xrightarrow{j} Z' \\xrightarrow {k} ", "a9e7d6c56c84a99a9775431c6d78836b": "(d\\Psi)_{e} : T_{e}G \\to T_{\\Psi(e)=e}\\mathrm{Aut}(G) ", "a9e7e4bd8f02be265bb63720d2db6bf2": "\n p = \\frac{\\rho C_0^2 \\chi}{(1-s\\chi)^2} \\left(1-\\tfrac{\\Gamma_0}{2}\\chi\\right) + \\Gamma_0 E \\,.\n ", "a9e85c13755428e22880e33b28c064bd": "\\alpha = e^2/(4\\pi)", "a9e8868bbf532324dc92307b8e442c7c": "\\rho=0\\,", "a9e8b330a8086fefc57aa7179dcfcb43": "S_n = \\sum_{k=0}^\\infty {k+n \\choose k} \\left[\\zeta(k+n+2)-1\\right]", "a9e8c80a5220d4483d6645f2af362266": "\\nabla S(z^0) = 0", "a9e8d4cb110dda2e7828254b62a9ba2b": "n > \\frac{3^2 (1 - \\bar p)}{\\bar p}", "a9e943bac35adcd6deed0caf576c49f9": "\\sum_{n=1}^\\infty \\frac{1}{2^{2n}(2n+1)}\\zeta(2n) = \\frac{1}{2}(1-\\ln 2).", "a9e975965c0a02183676cd2df252c859": "\n\\alpha_A=\\frac{1}{A}\\,\\frac{dA}{dT}\n", "a9e9ffef02aa020afd4dd841ced7abf2": "k^{\\mathrm{th}}", "a9ea0250851742b8262f8657bd5fcce9": "\\mathcal{H}_{0}=\\sum_{i=1}^{N}h_{i}\\left( \\xi_{i}\\right)", "a9ea6e01b5833ef19c732cfd6bcb3447": "a_i, b_i, c_i", "a9eab96e86adf80cfd5f217a8b264adf": "\\mathbf{W=(\\Sigma^y)^{-1}}", "a9eb09f5696f679e3fcb798ba6b58eca": "\\textrm{Prize\\ gained}=\\left [ \\frac {\\textrm {HK$200,000}} \\textrm{No.\\ of\\ opponents} \\right ] \\times \\textrm{No.\\ of\\ wrong\\ answers} ", "a9eb30f9f82072293970f963d52fbed6": "~V = 1/\\tau~", "a9eb63783849e358b372548f80f472d0": "\\exists x\\,Ux", "a9eb70d0f4b7a17edbbeb73879534d0c": "P(A \\cup B) = P(A) + P(B) - P(A \\cap B).", "a9eba14d7b184ae06570d13e2631a58a": " D(\\mathcal{A}) = H^2(\\Omega) \\cap H_0^1(\\Omega) \\times H_0^1(\\Omega) ", "a9ebaedf3aeac1d6f50369732a07982a": "\n\\mathbf{e} = \\frac{\\mathbf{A}}{m k} = \\frac{1}{m k}(\\mathbf{p} \\times \\mathbf{L}) - \\mathbf{\\hat{r}} ~ .\n", "a9ebdf373010e052b3df3047bd0fc0f9": "\\eta_{tot}=1", "a9ec092bcccf0a0e68892ce2376eb877": "\nn' \\leq \\frac{n \\log_k n}{k}\n", "a9ec21364bfc83b8be608d74557123c0": "y_k=\\nabla f(x_k+s_k)-\\nabla f(x_k),", "a9ec3e72320eef9cdbdc05b131595234": "f:R(X)\\rightarrow R(S_R)", "a9ec6fa43f97383bc57359c49a6e3a76": "\nx(t) = \\frac{2E_{0}}{\\omega_{0}^{2} \\left( 2b - h_{0} \\right)} \\cos \\omega_{0} t.\n", "a9eccc0b3bbe4e63aba443a8268887bf": "\\gamma=\\frac{3t_b^2E_1t_{w1}^3E_2t_{w2}^3}{16L^4(E_1t_{w1}^3+E_2t_{w2}^3)}", "a9ed1b526312b06e8d991bdd4dacac62": "3\\uparrow\\uparrow 3=3^{3^3}", "a9ed25793df688a58ea6016ac43f080c": "R_{\\text{E}}\\,", "a9ed2b7df4c816e1661ce879648aa8c0": "a^2(3x^2-y^2)=0\\,", "a9ed91ae365d2b46c1dcc349687b57ea": "\n k_{y} = \\frac{m \\pi}{b},\n", "a9ed941cfc31f115ad0a89268848fd3c": "(m, 1, m^2-61)", "a9edb37ed260475514012c506d851132": "Q=Q'", "a9edb3901da8a3c50e47d6ff0c1ea1c9": "p_{r+1}", "a9edcbcd80153cd2384c8fb44de34b05": " \\tau(x) = \\prod_{n=0}^{\\infty} ( 1 - x^{2^n} ). ", "a9edd6bd8f16422efd2fe98b4a4a7387": " p = \\left\\lfloor\\frac{n+b}{d}\\right\\rfloor c - a", "a9ee37699b6978c1fd573aee35bbd265": "\n\\frac{p(C_1|X^o)}{p(C_0|X^o)} \\propto \\frac{\\sum_h p(X^o,h|C_1)}{p(X^o,h_0|C_0)}\n", "a9ee8449f422ab595baa60a69e7f99e9": " \\omega_d = \\sqrt { {\\omega_0}^2 - \\alpha^2 } = \\omega_0 \\sqrt {1 - \\zeta^2} ", "a9ee9975592d83e90d0253815c99044f": "\n Y_{ij} = \\mu + \\beta_1 \\mathrm{Sex}_{ij} + \\beta_2 \\mathrm{Race}_{ij} + \\beta_3 \\mathrm{ParentsEduc}_{ij} + U_i + W_{ij},\\,\n ", "a9eedd94e4252416fa5ce0a21f62cfdb": " z = re^{i\\phi} \\,\\!", "a9ef46facb3d20c19511ad2e49a339f5": "T'_{pq}(e'_p \\otimes e'_q)", "a9ef4cd3f28fc9d39b5652b7751798c6": "(\\textbf{A}_N,\\textbf{b}_N,\\textbf{c}_N,\\textbf{A}_P,\\textbf{b}_P,\\textbf{c}_P)", "a9ef5684df77e00e751610304187b48a": "E_s(f)", "a9ef64dbaadaafb3ad696fa058aea658": "\\Theta(N^2)", "a9efc5aac127189170827aabec792c5e": "\n\\zeta(s) =\n\\sum_{n=1}^\\infty n^{-s} =\n\\frac{1}{1^s} + \\frac{1}{2^s} + \\frac{1}{3^s} + \\cdots \\;\\;\\;\\;\\;\\;\\; \\sigma = \\mathfrak{R}(s) > 1.\n\\!", "a9efe372dc79324440b82067cf9ac755": "\\vec y = \\mathbf{X} \\vec a + \\vec\\varepsilon. \\,", "a9eff25f072727446d391723702cecae": "\n\\mathbf{F}_{\\mathrm{fict}} = - 2 m \\boldsymbol\\Omega \\times \\mathbf{v}_{B} \\ ", "a9effea1c5e998524520fe30238c401a": "\\mathbf{J} (\\mathbf{r}, t) = \\int_{-\\infty}^t \\mathrm{d}t' \\int \\mathrm{d}^3\\mathbf{r}' \\; \\sigma(\\mathbf{r}-\\mathbf{r}', t-t') \\; \\mathbf{E}(\\mathbf{r}',\\ t') \\, ", "a9f01c553123a2a6e76cbe65c4152240": "\\Phi: X \\mapsto \\mathcal{P}(X)", "a9f088d9bb063af1398d8eb6c7364661": "\\gamma=\\frac{1}{\\sqrt{1 - \\frac{v^2}{c^2}}}, ", "a9f0896b52f3e3f53293f30bafed1870": "\\gamma \\leftarrow \\frac{2}{k+2}", "a9f0a4d822cb992b9aae6458cba6ebaf": " \\beta = \\mathrm{RM} \\lambda^2 ", "a9f0b2f6bf49255a41881065704ac779": "\\{\\ ;\\ \\} \\!\\,", "a9f0c7a93c02aa5b8bd36213c6a32d5b": "reject=1 \\,", "a9f0cbfeee70bd7d53d3848325460462": "{1 \\over \\varphi} = \\varphi - 1 = 0.618\\dots.", "a9f0cdc16339c43937fea2ca8973e3b6": "\\partial_\\mu A^\\mu=0", "a9f0cf93c67f2548b7b4dcfc9d51087b": "\\mathrm{I,\\;II,\\;III,\\;IV,\\;V,\\;VI,\\;VII,\\;VIII,\\;IX, and \\;X.}", "a9f111e4fc4b3d3ec018eadc8742034c": "\n\\frac{1}{2}\\mathrm{NADH} + \\mathrm{cyt~c_{ox}} + \\mathrm{ADP} + P_i \\iff \\frac{1}{2}\\mathrm{NAD^{+}} + \\mathrm{cyt~c_{red}} + \\mathrm{ATP}\n", "a9f17725b108df26c1a550fa528c4afd": "\\bar{x} = \\frac{x_1+x_2+\\cdots+x_k}{k},\\quad \\bar{y} = \\frac{y_1+y_2+\\cdots+y_k}{k}.", "a9f19296d046188c788dc8059291d1ef": "K_{rad}=\\frac{\\alpha^2 \\hbar^2}{2m}\\left(\\frac{3}{2} - \\frac{1}{1+(t/\\tau)^2}\\right)", "a9f1b2faf2703c5579682a26ffd56283": "\\mathfrak{P}^{41}", "a9f1b771a6c5c3076d35d11d7ab9e2f1": " \\left( \\sqrt{x} - \\sqrt\\frac{ab}{x}\\, \\right)^2 + 2\\sqrt{ab\\,{}}.\n", "a9f1e1311a605cddf851e11cfca07c74": "\\{A,B\\} = AB + BA.", "a9f26b787cc4f702d6188f9a600e46d8": "\\,\nA(t) = e^{iHt} A(0) e^{-iHt} .\n", "a9f28139e18b134da04c6134d1a495ae": "\\theta_1,\\theta_2, \\cdots \\theta_n ", "a9f29a4831f69b23015928648f9940d5": "H(\\nu) = \\int_{\\mathbb{R}^n} f(x)\\left(\\int_{\\mathbb{R}^n} g(z-x)e^{-2 \\pi i z\\cdot \\nu}\\,dz\\right)\\,dx.", "a9f2e283ab6604e5e0d7eae46a190c4c": "F (BS^{-1} f)", "a9f31d5392138ac96af51fc0c6f8acc4": "\\scriptstyle d^dk", "a9f3a5fcd540b95783d745d1fc7de417": " w_m \\leftarrow A v_m \\, ", "a9f3d125b407dad2de7bedc1bb2f76c7": "\\ln N = -\\lambda t + C \\,", "a9f3dbbd862c3a0f2b30752708f210e7": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} >0~", "a9f40d13207d447d9a7edf4c0f60e832": "A\\models \\phi_1\\lor\\dots\\lor\\phi_n \\iff \\exists\\phi_i (1\\leq i\\leq n), A\\models \\phi_i", "a9f44bee9216148dac99a8fee92ad483": "\\nu(A) = \\int_{g\\in G}\\mu(Ag^{-1}) \\, d\\mu^-.", "a9f4ab058b6c10acbf514613c5260f92": "y_0 = -0.64", "a9f5624895edc33e194906f1c28bd90d": "C_2(G) = N_c", "a9f5dd7f0ff3af6082a3921407ad6ada": "\\exists x (\\phi \\rightarrow \\psi)", "a9f623ab6977da835fc1900f5d257f17": "\\frac{h}{4 E}", "a9f64af509c0609c749d226ae7eb48cc": "U_{\\ell}", "a9f68b3914b9cea231a82c3872eb7558": "r:X\\to A", "a9f6b1172a258d7d9bd82245fec18e91": "J_{\\theta,q}(X_0, X_1) = K_{\\theta,q}(X_0, X_1),", "a9f6da1b1731964f7ae466c414702317": "\\det P=\\prod_{i=1}^n \\lambda_i \\le \\bigg({1 \\over n}\\sum_{i=1}^n \\lambda_i\\bigg)^n = \\left({1 \\over n} \\mathrm{tr} P\\right)^n = 1^n = 1,", "a9f74d168977ef91b359f80825be2d27": " \\langle \\mathbf{e}_0 \\bar{\\mathbf{e}}_k \\rangle_V = -\\mathbf{e}_k ,", "a9f787846f5b5d45341fa6743f6b63e1": "\\left(\n\\frac{2}{3}\n\\right)^{\\frac{1}{3}} \\approx 0.874", "a9f7bebb295959086768fb347d232ea3": "\\neg \\langle 0 \\rangle p\\,\\!", "a9f7c538d6802b111c42fbb3e7973913": "\\Bbb{R}^\\kappa", "a9f80eaa3f1bab45393b576b21f0be61": " \\%N_f ", "a9f82eb01490e611248753d99ed4767f": "f \\circ h", "a9f8376af74470adf22711a5e7ddf730": "= S_n - \\frac1{b_n}\\sum_{k=1}^{N-1}(b_{k+1} - b_k)S_k - \\frac1{b_n}\\sum_{k=N}^{n-1}(b_{k+1} - b_k)s - \\frac1{b_n}\\sum_{k=N}^{n-1}(b_{k+1} - b_k)(S_k - s)", "a9f8394a4ad79a7b018f3f431aa59719": "q(x):=f(x,x) \\in R/\\Lambda", "a9f8a4b2b87c9cadbcffcaa1d2572fc9": " K^{2n}(X) \\cong K^0(X) ", "a9f8a5a5c6062a753d974cde7a4560b9": "\\begin{array}{cc}\n \\begin{array}{c} \\\\ 3 \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & -12 & 0 & -42 \\\\\n & 3 & -27 & -81 \\\\\n \\hline \n 1 & -9 & -27 & -123 \n \\end{array}\n\\end{array}", "a9f8b85bf25a0aa45afd37a50484f8f5": "(\\mathbb{F}_{q^k})^K", "a9f927ecb9fda350898ff3d09e0f1e90": "z=\\{x,y\\}", "a9f9305c8c8b04c10138e73687effd4e": "F^\\omega_{<:}", "a9fa08baa644343a615778816e81b2ce": "\\scriptstyle n=n_0 - {4n_0 \\over m^2}", "a9fa2c577c44262e336999f06b2b7979": "1 = c_1d_1 + c_2(v_1+v_2+ h)", "a9fa5e5491b3f390bb2db5a97c1f4aed": "U_R", "a9fa5ed8d9ba09235198c17abd7a6a4d": " \\frac{\\mathrm{m^3}}{\\mathrm{kg}} ", "a9fa60e883d0320d617b0dcdb7c36f67": "\\ u=t", "a9fa65f87c3420ead92e2e8f7f55ac9e": "\n\\sum_{j = 0}^i \\left((j + 1) (j + 2) c_{i - j} c_{j + 2} + 2 (i - j + 1) (j + 1) c_{i - j + 1} c_{j + 1}\\right) + i c_i + (i + 1) c_{i + 1} = 0\n", "a9fb1311ad489332c57b63f2f4830cee": "R\\subset\\Sigma^{*}\\times\\Sigma^{*}", "a9fb18906b5be98dc680f416afa7436f": "|\\alpha\\rangle ", "a9fb27a847a8c764e6266f1e10df6e04": "\nQ(\\omega) = \\omega \\times \\frac{\\mbox{Maximum Energy Stored}}{\\mbox{Power Loss}}, \\,\n", "a9fb778e3303208d3bf16343be20c5e2": "p(m\\mid z_{1:t}, x_{1:t}) = \\prod_i p(m_i\\mid z_{1:t}, x_{1:t})", "a9fb8feb3f4f5e7fd5d47147c855a9cf": " \\Delta=g_2^3-27g_3^2. \\, ", "a9fb96fcbcc3f675bbaf2adffc9123c2": "\\operatorname{V}_{\\mathbb{P}^n}(I).", "a9fbca62ac135f146ab6e9d3d4fe10ec": "S(k)=-(A+i kB)^{-1}(A-ikB).\\,", "a9fbcc90fac81412c00ad1cafa4379c0": "\\mathrm{d}\\mu_i\\,", "a9fbe475d6527c332b24b8f676114bde": " \\exp(i\\gamma) ", "a9fbf50bc962e267b04352ad064829e0": "|\\zeta(x+iy)|=\\exp(\\sum_{n,p}\\frac{\\cos ny\\log p}{np^{nx}}).", "a9fbfb555812368c6cbecbd27e519c5d": "O(\\log k)", "a9fc1b4e8ee2227f4902cc7d2743b104": "e^{ikx} = \\cos kx + i\\,\\sin kx", "a9fc3d031abed0220364aa6e449cba9b": " -0.5", "a9fc41b5c2e19e46e141ef203f4e08fc": "\n\\sigma_{A}\\sigma_{B} \\geq \\sqrt{\\Big(\\frac{1}{2}\\langle\\{\\hat{A},\\hat{B}\\}\\rangle - \\langle \\hat{A} \\rangle\\langle \\hat{B}\\rangle\\Big)^{2}+ \\Big(\\frac{1}{2i}\\langle[\\hat{A},\\hat{B}]\\rangle\\Big)^{2}}\\, \n", "a9fc56d57d487fb8a6bdb9151bd5969a": "T_B^1~|~T_B^2", "a9fc71db855122595d40d71eea5889a2": "-\\frac{\\hbar^2}{2m_b^*} \\frac{\\mathrm{d}^2 \\psi(z)}{\\mathrm{d}z^2} + V \\psi(z) = E \\psi(z) \\quad \\text{ for } z > + \\frac {l_w}{2} \\quad \\quad (3)", "a9fca52913ef8fc41ac1920ce29ab38d": "P(s,x)", "a9fd0d41fd05d1c1422427a96ef798f7": "c_n\\lambda^{(0)}_n + \\epsilon \\sum_m c_m \n\\int f^{(0)}_n(x) D^{(1)} f^{(0)}_m(x)\\,dx = \\lambda c_n", "a9fd45345d22e0e5e50ce364eb6df551": "\n\\left[{\\hat{A}},{\\hat{B}}\\right] \\ \\stackrel{\\mathrm{def}}{=}\\ \\hat{A} \\hat{B} - \\hat{B} \\hat{A}\n", "a9fd578891972b6fb49d6e089c54c9a4": "ds^2(\\text{flat})=- du^2-2dudr+r^2 (d\\theta^2+\\sin^2\\theta\\, d\\phi^2 )=- dv^2+2dvdr+r^2 (d\\theta^2+\\sin^2\\theta\\, d\\phi^2 )=-dt^2+dr^2+r^2 (d\\theta^2+\\sin^2\\theta\\, d\\phi^2 )", "a9fd60d12730a7889c61957496c5b713": " | e^{-} \\rangle \\to \\frac{|v^{-}\\rangle + i |w^{-}\\rangle}{\\sqrt{2}} \\to i |c^{-} \\rangle. ", "a9fd7fc928445531296b4235aaf118e1": "f(x) = x + 1", "a9fd8cb4c2a6338bf9de6d22510cb078": "P_\\mathrm{vapor}", "a9fd957db19a8567d5b37750e17c7a8c": "\\displaystyle{f=f_1^{-1}\\circ f_2.}", "a9fdf8f2ce9b31ec222e3fbd82abfa36": "F=G\\left[\\mathcal{A},\\mu\\right]", "a9fdfb9cf8f76a15b4c6b25c43f646b6": "p = \\sqrt{\\frac{(ac+bd)(ad+bc)}{ab+cd}}", "a9fe247d23dfa59d46678cb7d8310bc5": "\\ \\bar{f} :(\\bar{X} , \\bar{X} \\setminus X )\\to (\\bar {Y} , \\bar{Y} \\setminus Y) ", "a9fe30a2fb1aabf64972f5625e2c475f": "\\displaystyle{\\varphi_{0,1}(z)=\\psi(z).}", "a9fe7892ef18f947d388b4cd7ab4462b": "R_s = R / N^2", "a9fe85d7579ed11632c7177b15573b89": " i\\hbar\\frac{d}{d t}\\left|\\psi(t)\\right\\rangle=H\\left|\\psi(t)\\right\\rangle", "a9fe9d2221b18fe9c49900353f70b742": " \\frac{1}{\\sqrt{2(1+c)}}\n \\Big((1+c) \\cos (\\theta ),\n a \\sin (\\theta )-b \\cos (\\theta ),\n a \\cos (\\theta )+b \\sin (\\theta ),\n (1+c) \\sin (\\theta )\\Big) . \\,\\!", "a9fea37243f124907703d460470e1f8e": "a^{f} \\in L^{2} (B)", "a9fefed3d8a26cd9253bd0abd3b57726": "\\mathrm {J\\hbar} \\,", "a9ff1633c50cfaf2ac63df282fe780db": "\\Im(z)", "a9ff1e5fb561f6200afdab0e81a54b74": "F(\\{a_i\\},\\{\\alpha A_j\\})=\\alpha F(\\{a_i\\},\\{A_j\\}).\\,", "a9ff370588705ee43fc1b024ec29609a": " \\eta = \\nabla \\times (u' \\mathbf{\\hat{\\boldsymbol{\\imath}}} + v' \\mathbf{\\hat{\\boldsymbol{\\jmath}}}) = -\\nabla^2 \\psi", "a9ff3f0037414f7331099cacd4d24cc4": " \\mathbf{e}^0 = \\begin{pmatrix} 1 & 0 & 0 & 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}^1 = \\begin{pmatrix} 0 & 1 & 0 & 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}^2 = \\begin{pmatrix} 0 & 0 & 1 & 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}^3 = \\begin{pmatrix} 0 & 0 & 0 & 1 \\end{pmatrix} ", "a9ff5c7aeab3a598d6ee332217228402": "vw \\mapsto \\frac{1}{2}(v\\otimes w + w \\otimes v)", "a9ff682955c780250c619436faa43b51": "B\\geq 0 ", "a9ff7721ebdea309ec5141eb9dc2961b": "\\frac{(b+c)^2}{bc}t_a^2+ \\frac{(c+a)^2}{ca}t_b^2+\\frac{(a+b)^2}{ab}t_c^2 = (a+b+c)^2.", "a9ffbc631e07289ce71dca66c80b1a6b": "x=c+vt\\,\\!", "a9ffebe8dd04296d710647c8b2f8a06d": " (2n)!/(2^nn! )", "aa0076a7e51df61a63f37bb018a4ef05": "\\bold(u) =", "aa00a9c4977931c040fec3c5e14c5761": "(1.8,2.8)", "aa00c4c0f44c5cb7cad68df40e8f8877": "0/0", "aa010f7dcbb3699c55841e76f4211237": "y = Y/T", "aa0118476a7814b563e47726cc74f8d8": "\\|\\mathbf A \\|^2 = (\\mathbf {A \\cdot A}) \\ ", "aa0128593204e00445b4f5f7df8651b8": "SS_\\text{tot}=\\sum_i (y_i-\\bar{y})^2,", "aa012a8d8289828426fd88bdbb549eb3": " (y_1...y_k) ", "aa019a6f646c50f7b89e93b24b02f011": "{_3\\text{F}_2}(a,b,c;d,e;z) = \\sum_{k=0}^\\infty\\{(a)_k(b)_k(c)_k/[(d)_k(e)_k]\\}z^k/k!", "aa01f9525b8421ce806fcaca39aa0043": "(id) \\frac{X \\cup \\{p, \\neg p\\}}{closed}", "aa01ff135f4df2096b8912e8619accc3": "x \\mapsto A (x + b)", "aa022ed48ea80faa9e665782b512a286": "\n\\left [ {\\partial^2 \\over \\partial r^2} - {l(l+1) \\over r^2} + { 2\\mu \\over \\hbar^2} \\left( E - V(r)\\right) \\right ] u(r) = 0\n", "aa02916c35e236e06528a5ba5898612e": "|\\zeta| = \\sqrt{\\chi^2+\\eta^2} = 1 \\quad \\text{which gives} \\quad \\chi^2+\\eta^2 = 1.", "aa02e39e16fbedefeb944c42ed8f2ff4": "\\sigma\\simeq0", "aa02eb1202a677f1513928c17bfa3035": "\\left(x+k\\frac{b}{\\gcd(a,b)},\\ y-k\\frac{a}{\\gcd(a,b)}\\right),", "aa03208a056d264f3230d7550597cf9e": " A = \\varepsilon \\ell c", "aa033077fb0fbb33b739a895a9be21f9": " m_a = m e^{ ( MSE / 2 ) } ", "aa033d050628dfa17d9a9aaafa739e4e": "D_{BC} = \\frac{\\frac{4\\times 2EI}{L}}{\\frac{3EI}{L}+\\frac{4\\times 2EI}{L}} = \\frac{\\frac{8}{10}}{\\frac{3}{10}+\\frac{8}{10}} = \\frac{8}{11} = 0.(72)", "aa0401947a1a511d1a03706b7d67076c": "{}= O_t\\{\\delta(u-\\tau);\\ u\\}.\\,", "aa0420b6b41265487401435f831778ba": "\\bar V \\otimes V \\to \\mathbf{C}.", "aa042f1f2ff302b89688e5f69e507a92": " t_s", "aa04339e588bec0906b7263516ff441a": " \\frac{d}{dx}\\arctan(x)= \\frac{1}{{1+x^2}}", "aa04446535c0dc31da6fc2c53c92c2a6": " \\C ", "aa049ed8fbac7457262a45957acc2778": "{\\alpha \\choose 0} = 1,", "aa04efb38101a36cf52cf4d92175708e": "\n\\begin{align}\n\\sum_\\text{sym} x^3 y^2 z^0 & {} = x^3 y^2 z^0 + x^3 z^2 y^0 + y^3 x^2 z^0 + y^3 z^2 x^0 + z^3 x^2 y^0 + z^3 y^2 x^0 \\\\\n& {} = x^3 y^2 + x^3 z^2 + y^3 x^2 + y^3 z^2 + z^3 x^2 + z^3 y^2\n\\end{align}\n", "aa0523f51978693c863413f35c901acb": "k_m", "aa056f2572405beaa7f16cbe46c1af2f": " r_0 = R_\\mathrm{earth} \\,\\!", "aa05dd77ba1bbd37ef6066b3d2c2dbac": "\\forall A: A \\cup \\varnothing = A\\, .", "aa05f5262f6847dea6de11ce35cb4f77": " \\left|1,V\\right\\rang ", "aa0639d45c8bc83e3ab3b90909e8df8e": "\\Sigma_1 \\otimes \\Sigma_2", "aa06473736a13fbc87e852467e29864a": "\n\\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} = \\mathbf{r} \\times m \\mathbf{v}\n", "aa0698b903fdc2e97ce0b2fe6e0c91e2": "f\\;", "aa06b7f50be5cfdd5e5b0aa347b2c932": "\\int_a^b f(x) g'(x) \\, dx,", "aa06bef20635d05d42081276c5802a45": "B_{1,2} = 0\\ ", "aa06c273646cb8483fd30cc8414c1323": "\\nu ,\\, \\mathbf{V}", "aa06de8a244f967c710cde0b8c1ce438": "\\lambda/2: \\quad \\nu_n=\\frac{c}{\\sqrt{\\varepsilon_{\\text{eff}}}}\\frac{n}{2 \\ell} \\quad (n=1,2,3,\\ldots) \\qquad \\lambda/4:\\quad \\nu_n=\\frac{c}{\\sqrt{\\varepsilon_{\\text{eff}}}}\\frac{2n+1}{4 \\ell} \\quad (n=0,1,2,\\ldots)", "aa06eb9ef6b399b985da80b9fa55db5b": "j_1,j_2,\\ldots,j_n", "aa0709ea089792390842d1ef3c960f63": " \\mathcal{L} = - \\frac{1}{4}\\mathcal{F}_{\\mu \\nu}(1+\\frac{m^2}{\\partial^2})\\mathcal{F}^{\\mu \\nu}", "aa077fba6bdac634212af6f5106a5b94": " {dP_\\mathrm{Br} \\over d\\omega} = {8\\sqrt 2 \\over 3\\sqrt\\pi} \\left[1-{\\omega_p^2 \\over \\omega^2}\\right]^{1/2} \\left[ Z_i^2 n_i n_e r_e^3 \\right] \n \\left[ { \\frac{(m_ec^2)^{3/2}}{(k_B T_e)^{1/2}} } \\right] E_1(y),\n", "aa0788aae2f4e3619cac95e02d9489b0": "A_{\\alpha ; \\beta ; \\gamma} g^{\\beta \\gamma} = A_{\\alpha ; \\beta , \\gamma} g^{\\beta \\gamma} - A_{\\sigma ; \\beta} \\Gamma^{\\sigma}_{\\alpha \\gamma} g^{\\beta \\gamma}", "aa078f801b31665e44992c647979c871": "\\left ( \\frac{d\\phi }{dx} \\right )_e=\\frac{\\phi_E - \\phi_P}{\\delta x_{PE}}", "aa079ef32d674c43461654ef8fcf28db": "i=i+1", "aa07bbf0b9929a147fced8dfb3703b63": "nhf/(2e)", "aa07d7b484c17fc14f751d5ff54fb410": "H \\approx H_\\mathrm{Heisenberg} - J \\sum_{} \\left[\\theta_{ij}J_{ij}^{(s)} - \\cfrac{1}{2}\\theta_{ij}^2 T_{ij}^{(s)}\\right]", "aa081b71b4e8aff6191a863bd08a4b10": " Y(t)=\\exp \\left( \\Omega (t,t_0)\\right) \\, Y_0 ~,", "aa085ac1d47d978bed18df2d56cb9a21": "1 = e_1 + \\cdots + e_n, \\quad e_i \\in \\mathfrak{a}_i.", "aa08988265a772f1b0c14f94498d86cd": "f_i(t)", "aa08ca0c44a796b6e8c43b47f343a962": "\\left(\\frac{a}{p}\\right)", "aa091763edc36a54792c0323ab147188": "P(A') = 1 -\\dfrac{365!}{365^N(365-N)!}", "aa0919ffd628af51a8619bc522c022dc": "S\\ ", "aa096833b5e934e2aa9828d26f9a62f7": "{}s(w,\\vec k)=\\iiiint\\limits\\,s(t,\\vec r)\\cdot e^{-j (wt-\\vec k\\vec r)} d\\vec x dt,", "aa098ad99e87d7eb477802cb3c6288b5": "\\mathbf{y}^{(n)} = \\mathbf{F}\\left(x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n-1)} \\right)", "aa09e9696d95b4192aa9011c64c66f6c": "y=\\frac{Y}{Z} ", "aa0a4972f7d45518aafa848fe565e21b": "\\mathrm{C_6H_6}", "aa0a5d2fc8f9b23bac3affe35b98255c": " u_t(0,x) = g(x),", "aa0a68456091646cf9300d01cfd0f67b": "V_k\\subset V_l,\\; k 1 ", "aa0cd3045cceff1e641897d889b10c77": "2\\pi/k", "aa0cd4de497fc5fcb8eab7e50fd3848b": "\\psi_{3}= \\phi_{3}-e_{C}", "aa0cd7a7ac946adc4b84d17785c37f50": " u^2/c^2 ", "aa0cfc54f2df7414c4c47dc15679b4cd": "ZZ = Z^2", "aa0d9aafd61231200d852e99c24ae8a2": "(\\mathit{KPR})\\qquad\\frac{\\neg p\\to q\\lor r}{(\\neg p\\to q)\\lor(\\neg p\\to r)}", "aa0d9d1375b3e7afd4f5f411f7325040": "\nW(t) = \\inf \\{w : A(t) - D(t+w) \\le 0\\} , \\forall t \\ge 0.\n", "aa0dae9cfd8d80066dc379fc1064575b": " 1 \\lor 3 \\iff | + \\rangle ", "aa0dbd3bcae31d128d07b663d2c084af": "R_s = \\frac{\\sqrt{N_2}}{4} \\left ( \\frac{\\alpha - 1}{\\alpha} \\right ) \\left ( \\frac{k'_2}{1 + k'_2} \\right )", "aa0e16d06b8fff6eb64b1e937ed30f71": "\\phi \\in C_c^\\infty(\\Omega)", "aa0ec6e11dc0886592f5d1750ec3cd2a": " \\frac{\\Delta E_M}{E_i} ", "aa0f0203e7ad06be9279b80280da889e": "=a \\langle T_v\\exp_p(v), T_v\\exp_p(v)\\rangle + \\langle T_v\\exp_p(v), T_v\\exp_p(w_N)\\rangle=\\langle v, w_T\\rangle + \\langle T_v\\exp_p(v), T_v\\exp_p(w_N)\\rangle.\n", "aa0f02c40483b898892268c6d242b280": "A \\subseteq S", "aa0f0a1e3daa18a1e18a593847664179": "\\forall x \\phi \\to \\phi^x_t", "aa0f3900012e525ee1105556f4db4855": "G = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\n & R(A) \\\\\nW(B) & \\\\\nCom. & \\\\\n & W(A) \\\\\n & Com. \\\\\n &\\end{bmatrix}", "aa0f6f8b7675611a0dc527f42c8b3dd5": "X_0:=X_1 \\cap X_2", "aa0f7cbcc4039720c5901c97fe7c87e7": "mv_{e}r=n\\hbar", "aa0fb321f27e9aa7d8f7f3693c5d3e68": "1 \\rightarrow K \\rightarrow G \\rightarrow H \\rightarrow 1,\\,", "aa1006054c8d067cce46bc12ebef7497": " {\\rm W} = \\frac{1}{N} \\sum_{j=1}^{N} \\ln \\left( \\frac{z}{y_j} \\right)", "aa1026f0c4eee752ef3368e2a85117f1": "\\tfrac{X}{1-X} \\sim {\\beta^{'}}(\\alpha,\\beta)", "aa102df34483deb1ff82da970da7756f": " R_s = \\,r_n + r_{n-1} + r_{n-2} + \\cdots + r_s. \\, ", "aa103bf5a519cd3e36e3a49302761987": "\\textstyle a_c=\\frac{1}{2W_S}v_c^2\\rho C_L -g,", "aa10621ce54553480d05e9011645003d": " M_1 (\\vec X,\\vec {\\rm E},Y)\n", "aa10892d25e0b429127c6c1bd33d8d9f": "\n\\langle x \\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_0^\\infty x dP(x) = \\int_0^\\infty \\frac{x}{\\ell} e^{-x/\\ell} dx = \\ell \n", "aa10d3dc958e0051fc5b7bb3943caa59": "\\Delta U=Q-W+m_{i}(h+\\frac{1}{2}v^2+gz)_{i}-m_{e}(h+\\frac{1}{2}v^2+gz)_{e}", "aa10dbc57bb0c76dab5659ef2ba24440": "q_{ir} = \\frac{\\partial C_i}{\\partial C_r}", "aa11058bfcd1f5af3b8a7da5e630b0f6": "\\scriptstyle n=n_0 - \\frac{C_0}{m+m'}", "aa112b6f23c5618ab25c973fd3b6d416": "\\mu_i = {\\mu_i}^o + RT \\ln a_i\\,,", "aa12b861336b4b072cb91065be4f7df5": "\\mathbf{\\tau}_{jk} = \\langle\\zeta_{j} | \\nabla\\zeta_{k}\\rangle", "aa12cb8f49c28e89ebec169c761f66df": "LEL_{mix}=\\frac{1}{\\sum \\frac{x_{i}}{LEL_{i}}}", "aa13353eb7efdb781acd632834b12a8e": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n -\\frac{(A\\,b-a\\,B) x^{m+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^{q+1}}{a\\,n (b\\,c-a\\,d) (p+1)}\\,+\\,\n \\frac{1}{a\\,n(b\\,c-a\\,d)(p+1)}\\,\\cdot\n", "aa135e67321926f181d788c1a35afdf2": "0=1 ", "aa201a8e65761086ff5c80de3400133a": "\\alpha_{n-1}^{}", "aa20340d81e52e59e1a6dd5a82c52f31": "\\mathbb{Q}(\\zeta_p)", "aa208ec15e4d6fc28f59e7686b91dd24": "h[e[x]]=(\\alpha +1) e[x]", "aa211173352b6dd3bbb94fb033e94177": " \\frac{dp}{dz} = -\\rho g_n = - \\frac {mpg}{RT}", "aa2120aa42db5a25cfb60da0962d20b8": "\\begin{align}\n\\mathbf{B}(\\omega) & = \\begin{vmatrix}\n\\mu_{1} & -i \\mu_{2} & 0\\\\\ni \\mu_{2} & \\mu_{1} & 0\\\\\n0 & 0 & \\mu_{z}\\\\\n\\end{vmatrix} \\mathbf{H}(\\omega)\\\\\n\\end{align}", "aa220063db50db01b83f9c783d4f542c": " M^{2}c^{2} = (P^0)^2 - \\vec{P}^{2}, ", "aa2213b79014b684b5a04daaf50846f4": "M_1+M_2^*", "aa224f6d72c914c508326b06a2adf094": "T_B(i)", "aa22ded7f6b9ebb5ba228f6a8f589bf7": "\\alpha_{11} < 0", "aa22f09ce720f488d9666e506cc6fdff": "\\rho = | \\phi \\rangle \\langle \\phi |", "aa2349a3a8048658f897efa9e9d47850": "{\\tilde{E}}_{7}", "aa2358066ef0f4d16a81f4f3104a33e2": "b_0.b_1b_2b_3b_4b_5 \\dots.", "aa235ed0736f1e1a9427e54f81d03830": "z \\in x", "aa23a6ed70eeceb0e0e016db8d62a2a1": "40\\sqrt{47115 / 25314} = 40\\sqrt{1.86} = 40*1.36 \\approx 54", "aa24521eb636438e606ff15631954dfb": "m_{ship} \\ c^2 = \\frac{dm_e \\ c^2}{\\sqrt{1 - \\frac{v_e^2}{c^2}}} + \\frac{(m_{ship} - dm_{fuel}) \\ c^2}{\\sqrt{1 - \\frac{(dv_{ship})^2}{c^2}}}", "aa252c1f9b039fe5ed5776c3761b2eed": "E_{\\rm corr} = 1-2/\\sqrt{3} \\approx -0.1547", "aa25341239b62a48e8ca337cbe94e4a7": "{\\Gamma,x\\!:\\!\\sigma\\vdash e\\Leftarrow \\tau}\\over{\\Gamma\\vdash \\lambda x.~e \\Leftarrow \\sigma \\to \\tau}", "aa254d5c1c8061982890a0ea7abc5265": "(T-\\lambda I)^{-1}", "aa25665b358e2a9c7c7d2163d38cd863": "\\mathbf A^{-1} = \\sum_{n = 0}^\\infty \\left(\\mathbf X^{-1} (\\mathbf X - \\mathbf A)\\right)^n \\mathbf X^{-1}~.", "aa2575de974ef88afe3bbf7f6546dd3d": "(\\begin{smallmatrix}M_\\odot\\end{smallmatrix})", "aa25859cb8605531ec149d9739933158": "i>j", "aa25c7a898c4195f8066d931fd07380b": "132 = (1)(23)", "aa25e33165178036a2957a7201efcc20": "82\\times 10^{-6}\\Omega \\cdot \\text{cm}", "aa25e7f9953cef09fbe77292cfa0035e": "\\theta_\\mathrm i", "aa25ee35dc7efd417a40c991a2548267": "H(A,\\,j\\omega)", "aa2615cba6dfa5d91ac4231e9544e183": " P^t ", "aa264a8ce114f1e86dda79b1f6205717": "G/N", "aa266c99de5db0ef60355fd70444c9d5": "\\displaystyle{h(\\theta+2\\pi)=h(\\theta) +2\\pi},", "aa26a92fec55364f9d2c6bd809404cfa": "{x =\\frac{\\sqrt{5} -1}{2}, \\frac{-\\sqrt{5} -1}{2}}", "aa26ef387de6926d9062813bfa6a74df": "N = (V, E)", "aa26f36250640d722f1be9ff013b49b4": "\n \\sigma_{11} - \\sigma_{33} = 2C_1(\\lambda^2-1) - 2C_2\\left(\\cfrac{1}{\\lambda^2}-1\\right) ~;~~\n \\sigma_{22} - \\sigma_{33} = 2C_1\\left(\\cfrac{1}{\\lambda^2} -1\\right) - 2C_2(\\lambda^2 -1)\n ", "aa26fe6fd9c947620da9408a9691106e": "\\zeta\\in S", "aa26ff5fc170853340e838b7d7db5304": "\\displaystyle w", "aa272b0511e4aada4b1f36270b5fc89e": "f\\colon B \\to A", "aa2740709949c80fd160377b928303c2": "\\tau_i\\colon \\Omega \\rightarrow T_i", "aa274fe8690ce5f79235ae1321afe069": "F_{\\mathrm{cf}} = \\mu r \\dot \\theta ^2 = \\frac {\\ell^2}{\\mu r^3}. \\, ", "aa27884bb8c504a893a6c4ca6a2f1b05": " \\pi = pq - [F_n + wx + g_0 w - g_1 w] \\qquad\\qquad (2) ", "aa282e035b78710b2d6448c3f2d1e2e7": " A = \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix} . ", "aa283c5231aab7eef9d3bc2703fb468f": "X \\rarr f_1 \\times f_2 \\times \\ldots \\times f_n", "aa2872c2fc6894b21e1d0db39ce57c26": "D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)", "aa2881221ebd0b5a7e4d6d634c10e7b5": "{\\theta}_{D}({u}_{1},{u}_{2})=\n\\left(\n \\begin{array}{c}\n {u}_{1}({b}_{1}-{a}_{1}) +{a}_{1}\\\\\n {u}_{2}({b}_{2}-{a}_{2}) +{a}_{2}\n \\end{array}\n \\right)\n", "aa288ff2c8fd2920d3d481821c2077d4": "\\pi(x) - \\pi(cx) \\ge n", "aa289fdf922f03271403fd919d6854a4": "x^3 + y^3 + z^3 + 3xyz \\geq xy(x+y) + xz(x+z) + yz(y+z)", "aa28e30a737f0a4afc9ec65526a08f67": "E_\\theta= i\\frac{Z \\,I_0 \\delta \\ell}{4\\pi} \\left(\\frac{k}{r} - \\frac{i}{r^2} - \\frac{1}{k r^3}\\right) e^{i(\\omega t-k\\,r)}\\,\\sin(\\theta)", "aa2937c34aced90feae20d02273ba8d5": "~\\sigma_{\\rm a}(\\omega)=\\sigma_{\\rm e}(\\omega)~", "aa29b3b8ab9689e671149d1deb4f9cd1": "J_i=-\\sum_{j=0}^{np-1}\\sum_{t=0}^{r-1}c(t+1)\\left(f_i^{(j)}(\\alpha_{n_t+1})+\\dots+f_i^{(j)}(\\alpha_{n_{t+1}})\\right).", "aa29ca68a4f14ed64e95d355be3d45e8": "x(t)=\\int{v(k(t))}{dt}= -\\frac{A}{eE}\\cos\\left({\\frac{aeE}{\\hbar}t}\\right)", "aa29cdbc3d67196415ac03b67678f868": "\\vert confidence_{i} - confidence_{j}\\vert \\leq 1 ", "aa29df8e70da1fd6c1825450f2b87ffa": "A:B:C\\,\\!", "aa2a0750ee1f613fbeb0aedc1e92c898": "\\Delta^{\\mathcal{I}} \\times \\Delta^{\\mathcal{I}}", "aa2a2856fbf9ceedaa9a7f993af1d6d5": "p(x_1,\\ldots,x_n) \\in \\mathbb{C}[x_1,\\ldots,x_n]", "aa2a4a47959cb6553db986d17b417482": "X_t=\\sum_{i=1}^t (x_i-\\langle x_i\\rangle)", "aa2a5b701b7594a118c477e9802436b5": "C^{1,\\alpha}", "aa2a858390a61228b82b5ce85fd7e6f1": "\ni = \\frac{n!}{(n-2p)!\\ p!\\ 2^{p}}\n", "aa2ad5d6185910a912a9cecaa3516a96": "\\sum_{k=0}^\\infty \\frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\\csc z, |z|<\\pi\\,\\!", "aa2ad7ebbc767874824e45641386a5f9": " Lu = a_{ij}(x)\\frac{\\partial^2 u}{\\partial x_i \\partial x_j} + b_i(x) \\frac{\\partial u}{\\partial x_i} + c(x)u, \\qquad x \\in \\Omega.", "aa2af73ab17d4f16b082a4cec70874c3": "\\textstyle Mt+1", "aa2b36dabd254260f584015a9e0c5c98": "KN", "aa2b48b47d6adf0dba51fb9999482dd6": "\\vec{r}(s,T)", "aa2b5c5db657321eafa7dc39b5abb231": "[a_0; a_1, a_2, \\ldots]", "aa2b601de56aaa2e243c8099377f88a6": "\\cup_{S \\in T} \\Delta \\cap DS", "aa2b85460d74528d605398e68b55c4e0": "f(x)=2 x^2-1", "aa2bbd354111aa31a768029e754b6d3b": "\\scriptstyle c_{ij} \\neq c^A_ic^B_j.", "aa2bd0ee2c98e5792861c22545a8db0b": "\\zeta(2n+1)", "aa2be13567d866aadd069e193415827f": "f\\circ\\mu = \\mu'\\circ(f\\otimes f)", "aa2c1f6bc5256e18385083e06ace8071": "B_i = -\\frac{z_i q \\kappa}{4 \\pi \\varepsilon_r \\varepsilon_0} \\frac {1}{1 + \\kappa a_i}", "aa2c3d66c6f93d21610cf8afba5abc90": "\\langle\\lambda_q \\rangle = \\frac{Q}{\\ell}\\,,\\quad \\langle\\sigma_q\\rangle = \\frac{Q}{S}\\,,\\quad\\langle\\rho_q\\rangle = \\frac{Q}{V}\\,.", "aa2c856335a6cf2de8e91751fe919bb7": "t_2\\ge\\sqrt n", "aa2ccecea6d3cac4e66000ad096d81a0": " m(E,a_E,a)=\\left[ERR(a_E, a)M_c(a)+(1-v)EAR(a_E,a)\\right]{\\frac{\\sum_LQ(L)F(L)L}{DDREF}}", "aa2d040b8fedac3edd035922882e6ece": "N=q^{\\alpha} p_1^{2e_1} \\cdots p_k^{2e_k}, ", "aa2d2f299b7b20073801d416bc8e1571": "A_m(0,4) = 1,4,6,4,1", "aa2d7a3fdac520a76e749d260dd8e056": "\\sqrt{ \\frac T{t_1}}", "aa2dc9148a9a5bc4b7b5006796f391bf": "\n \\operatorname{cost}(\\theta, \\sigma^2)=\\frac{1}{2\\sigma^2} \\sum_{j=1}^N \\sum_{i=1}^{M+1} P^{\\text{old}}(i|s_j) \\lVert s_j - T(m_i,\\theta) \\rVert^2 \n + \\frac{N_\\mathbf{P}D}{2}\\log{\\sigma^2}\n", "aa2e0f0aed6601dfdbffa26e191dbda4": "r\\in\\C", "aa2e62d0d88379289fc3f76bf7e306dc": "\\varnothing\\vdash A", "aa2e719f7ff942af7f704a5455ad547e": "\\mathbf{M} =\\int d\\mathbf{M} = -\\mathbf{e_y} \\frac {E} {\\rho} \\int {e^2} \\ dA = \\ -\\mathbf{e_y} \\frac {EI} {\\rho} ", "aa2f54df31ae8e191e2f4d0c619e065b": "\\pm 1.58 \\times IQR \\div \\sqrt{n}", "aa2f5f1c87629b1c53cb76f88d82234d": "\\mathrm{Perrin}(n)=P(n+1)+P(n-10).\\,", "aa301648108f87405c4eab984ca52d90": "\\mathsf{SAT} \\in \\mathsf{P/poly}", "aa3017acbc1bd20035f8e6ca08dfc58b": "|\\psi_F(t)\\rangle=\\left[1-\\frac{i}{\\hbar}\\int_{t_0}^t dt_1 e^{\\frac{i}{\\hbar}\\lambda V(t_1-t_0)}H_0e^{-\\frac{i}{\\hbar}\\lambda V(t_1-t_0)}\\right.", "aa3028e9a4173ef4b1c02289c680f186": "A(x) = \\exp\\left({ \\sum_{m \\ge 1} \\frac{P(x^m)}{m} }\\right) \\ . ", "aa304fd95ac851f59c3860f80040edfb": "\\kappa_4=\\mu_4-3\\mu_2^2\\,", "aa309c65d99ce303d914002e8d937f8f": "\n \\int_{\\Omega} \\boldsymbol{F}\\otimes\\boldsymbol{\\nabla}\\boldsymbol{G}\\,{\\rm d}\\Omega = \\int_{\\Gamma} \\mathbf{n}\\otimes(\\boldsymbol{F}\\otimes\\boldsymbol{G})\\,{\\rm d}\\Gamma - \\int_{\\Omega} \\boldsymbol{G}\\otimes\\boldsymbol{\\nabla}\\boldsymbol{F}\\,{\\rm d}\\Omega\n ", "aa30d4e538ff70e1ec7f2c3452d855f1": " \\mathbf{E} = \\mathbf{R} \\, [\\mathbf{t}]_{\\times}\n", "aa311f7de268be15187fbc48f6680248": "f \\colon G \\to {\\mathbb{R}}\\cup\\{-\\infty\\},", "aa31487536f90ece04c3841d2cc278bd": "c_0 = 0", "aa318b7b767333195b7f995ae3957e6c": "E_n=h\\nu_n={hc_s\\over\\lambda_n}={hc_sn\\over 2L}\\,,", "aa31ac719965cc8f8e8d301bb5f6c9f9": "P_\\infty(t)", "aa31db8e3adb23b29ffeca823d8c4986": "(t_1, 0, t_2) = t_1(1,0,0) + t_2(0,0,1)\\text{.}\\,", "aa31f2dd1f5c41a6651c00642045afb7": "\\bold{j}=\\frac{\\mathrm{d}\\bold{a}}{\\mathrm{d}t}=\\frac{\\mathrm{d}^2\\bold{v}}{\\mathrm{d}t^2}=\\frac{\\mathrm{d}^3\\bold{r}}{\\mathrm{d}t^3}", "aa31f363825e6471403839e9ff46e7be": "t=t_{0}", "aa3201baf0a4b0402852deea3848a6aa": "\\dot{\\mathbf{x}} = F(\\mathbf{x}) \\qquad \\text{where} \\qquad F(\\mathbf{x}) \\triangleq f_x(\\mathbf{x}) + g_x(\\mathbf{x}) u_x(\\mathbf{x})", "aa321a67792c69249b8186c04283028d": "\\varphi _j^{n + 1} = \\sum\\limits_m^{M} {\\gamma _m \\varphi _{j + m}^n }. \\quad \\quad ( 2)\n", "aa3226a320f7c5d7d0bf22cc468e9352": " \\qquad \\qquad \\mathbf{a}_{e} (\\mathbf{x},t) = \\sum_\\alpha(\\frac{\\hbar}{2\\epsilon_\\mathrm{o}\\omega_{ph,\\alpha}V})^{1/2}\\mathbf{s}_{ph,\\alpha} (c_\\alpha e^{i \\boldsymbol{\\kappa}_\\alpha \\cdot \\mathbf{x}} + c_\\alpha^\\dagger e^{-i\\boldsymbol{\\kappa}_\\alpha\\cdot\\mathbf{x}}), ", "aa32557ef32e9f7d1a9ed75fde6ce230": " \\mathbf{c}\\cdot\\boldsymbol{S} = c_m~S_{mj}~\\mathbf{e}_j\n ", "aa330d214058b1ca93a48e1d3edd0138": "g_t=\\begin{pmatrix} \\cosh t & i\\sinh t \\\\ -i\\sinh t& \\cosh t\\end{pmatrix}.", "aa334575e8be76613df573ca13a65933": "C_p = 1 - \\frac{|\\vec{V}|^2}{|\\vec{V_\\infty}|^2} ", "aa334b3a8c541b19db73b4cc8e08c2e5": "\\sum_{y \\in \\text{Ball}} p(y|E(m)) \\leq 1", "aa3361f902ce0d8923fc2e6c28f54da5": "1+0-x^2+x^3+0-x^5+x^6+0-x^8+\\cdots,", "aa33671bbe1b16929e49bdc111883e53": "\\left(\\pm\\sqrt{10},\\ \\pm\\sqrt{6},\\ 0,\\ \\pm2\\right)", "aa336948fee3304c75ba25349c9f9898": " \\Omega_{conf} \\, = \\, \\frac {N_S!}{N! (N_S -N)!} \n", "aa336f596005e6df3e4847f7df9eb57a": " i = C { dv \\over dt} ", "aa3385be6ab92062e4701e7d072fc9d4": "dT_\\text{core}/dt", "aa3397a9c5d673f0c27352a319153ef2": "h = \\sqrt{(1-e^2)\\mu a}", "aa33b9f5a1d1d5e0eb5689dcee8203c8": "c'(s)\\neq 0", "aa33d3ff51d0ceabca1142a7c86c692d": "\\omega^p = (x+yi)^p \\equiv x^p+y^pi^p \\equiv x + (-1)^{\\frac{p-1}{2}}yi \\pmod{p},", "aa346132ed55da3fdd2865d89d8c5c26": " w \\leq \\frac{\\sqrt{8z + 1} - 1}{2} < w + 1 ", "aa34f8a0bc83cdae96e0e60d99fd79c5": " Q(m,n) = a|mz+n|^2\\ ", "aa3545b3cb8cfc8f3cdf477f2d89287a": "CS(M)=\\int_{s(M)}\\tfrac{1}{2}Tp_1\\in\\mathbb{R}/\\mathbb{Z}", "aa3558aa6b36b092ecc4aec4e86e15ca": "\\Omega=(X,\\Sigma,\\mu)", "aa357b5415bfc503469f0d8ec6bd9cae": "\\tfrac12bh \\,\\!", "aa357e5c021d217137af4e9a2d2cfbdd": "\\nabla \\times \\mathbf{B} = \\mu_0 \\epsilon_0 \\frac{\\partial \\mathbf{E}} {\\partial t} + \\mu_0 \\mathbf{j}_{\\mathrm e}", "aa3593f1a98c80e7522d29661f42a741": "\\|f\\|_A:=\\sum_{n=-\\infty}^\\infty |\\widehat{f}(n)|<\\infty.", "aa35ea75b67b6215e472520333dcd4f0": "T_M(d)=\\frac{4T_{MB}}{H_fd}(\\frac{\\sigma\\,_{sv}}{1-\\frac{d_0}{d}}-\\sigma\\,_{lv}(1-\\frac{\\rho\\,_s}{\\rho\\,_l}))", "aa35eea86e45bc1178120840f957b93c": "v(\\mathbf{p},w)", "aa364983fc827e12ab0662b6e3f9db79": "x_i = \\left( x_0^{2^i \\bmod \\lambda(M)} \\right) \\bmod M", "aa365be9236783d071318c17199e59ef": " M = \\begin{bmatrix} x_0 & y_0 \\\\ x_1 & y_1 \\\\ x_2 & y_2 \\\\ x_3 & y_3 \\end{bmatrix}", "aa369c2371119d7069f684a48eae107b": " a=b=|\\mathbf{E}|/\\sqrt{2}", "aa36dd687ac3ae0bbe8e20008983cf9f": "P(\\alpha_i,\\beta_i)", "aa36dfaf6de22965a7fb802751e911a9": "\\Omega BQC \\simeq B(S^{-1} S)", "aa3762071f16cc372a14aa3ffb14a748": "\n \\mathbf{M} + \\mathbf{M}_{xz} = \\mathbf{0} \\quad \\text{or} \\quad -\\frac{F x_A}{L}(L-x) + M_{xz} = 0 \\quad \\text{or} \\quad M_{xz} =\\frac{F x_A}{L}(L-x) \\,.\n ", "aa377179fdbefd197f7f949419d1e349": "E \\approx \\mu \\pm kT", "aa3790abab39a09bb7f6e719af95bde5": "\\frac{2\\sqrt{3}}{3}", "aa37a8465bed5bff3a2519d0cf351aa6": "s K = 0", "aa3875bc40d41dbc923bcd7327fc9a13": "\\mathbf {r}_i\\,", "aa388ca71d2964942a78deae2197fa50": "(\\hat{r}\\ ,\\ \\hat{t})", "aa38af2fcf764bbed12c1be1b2d25aa2": "(3n+1)_{th}", "aa38b7f797d8125627ad6e6158d4f13e": " P= 4\\sup_f \\frac{\\left( \\int\\int\\limits_D f\\, dx\\, dy\\right)^2}{\\int\\int\\limits_D f_x^2+f_y^2\\, dx\\, dy}.", "aa38ba1f534d8775c0371574286c3a24": "\\frac{PV_i} {V} ", "aa38f107289d4d73d516190581397349": "w_i", "aa3906152f7f8aa1ba5fab38570148f3": "|\\psi(t)\\rang ", "aa3908c1bcae5e10f958eba80a4c80eb": "\\rho(x,y,z) = 3B\\left(a^2+x^2+y^2+z^2\\right)^{-1}", "aa3916e758a9d0f25d53103ac0d78859": "\\scriptstyle \\ddot{\\phi}", "aa396a47621099c5d613edc2471628c7": "| \\psi \\rangle \\in H_A \\otimes H_B", "aa39effb8bada3ca1962c7f832de9e4e": "G = G(L/K)", "aa39fec082cac1c1960daf06cd782cd3": "\\alpha=\\alpha_e:C\\to H, \\omega=\\omega_e:C\\to K", "aa3a62e5051abc7e5f63fd07b5fac9af": "R[x; \\sigma]", "aa3b3a37b792e031557cffbf4f0d1d2d": " \\log_b(x^d) = d \\log_b(x) \\!\\, ", "aa3b5102e889c2d85eea46411d782eda": " y'(t) = f(t, y(t)), \\quad y(t_k) = y_k. ", "aa3bf21292b2949c42202586f6567132": "\\qquad z=\\sin 3t", "aa3c3b3e9ab42ae27692f43244bae88c": " + \\mathbf{A^{\\rm T}X}(\\mathbf{X^{\\rm T}A^{\\rm T}X})^{-1})", "aa3c4ac1a90191fa9636cf8e1a173d0d": " \\Phi_{G} = - \\langle P\\rangle V", "aa3c6369613597e7fe47e39e44f44a94": "\\int_1^{\\infty}\\frac{dx}{x}", "aa3cf2be174d98f0a48781046dc4fe93": "\\sum_{m=-\\infty}^\\infty a_{cm+d}\\cdot x^{cm+d}", "aa3d46772f38d1311be43fb8aec7aebb": "\nA^\\circ=\\{ y\\in Y:\\quad \\sup_{x\\in A}|\\langle x,y\\rangle|\\le 1\\}\n", "aa3d5f20426d5f38577f68045de816bb": "S_C(N)<2^N", "aa3d7fc3d0a1ba96e9c8031ef44916eb": "c + n \\mathbb{Z} \\in \\mathbb{Z} / n \\mathbb{Z}", "aa3db1efb7b02f6e2b2a835462486fc3": "z_{n_{k}} (t) \\rightharpoonup z(t) \\in E;", "aa3df44aeb2867b5159c2807af1f4201": "p?\\,\\!", "aa3e6ee56561511f87d385f49e65c844": "\\mu^{-1}(0) / G", "aa3e81330fb7019ed5e1bc3afe474503": "\\textstyle{f(x) = \\sum_{n=0}^\\infty {s(2^{n}x)\\over 2^n}}", "aa3ed441d2b4365d10000b936f548ba4": "\\scriptstyle \\epsilon_0 = p_0 / p_{0m} \\,", "aa3edd782f1dbe15dd5d33be7c6bacb0": " \\left(\\varepsilon, \\mu\\right) ", "aa3f03821fccb4c8be780b4dbd5ee319": "R\\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ R_{1/2}\\ \\cup\\ S_c\\ \\cup \\ R_0 \\,", "aa3f1b248de38e6bdddbafdc8dc5632c": " \\frac {A}{H} = \\frac {1}{({\\frac {\\omega}{\\omega_0}})^2-1}", "aa3f362ea36289c18226538ff8f9da41": "W_{c'}", "aa3f3c971ddc1467af9608ed9b833326": "\\vec{D}^2_x=\\begin{bmatrix}1 & -2 & 1\\end{bmatrix}", "aa3f447d7c07446b37cb16fa79217cc3": " p(x)=p_nx^n+p_{n-1}x^{n-1}+\\cdots+p_1x+p_0 ", "aa3f6900db6e41e7aab66e5442b784e8": "c_c", "aa3f7ee88fdf16f3576fd090aba95ac1": "\\mathbf{V}=\\frac{1}{M}\\sum_i m_i \\mathbf{v}_i.\\,", "aa3fe1595545bb218d8bb06cbd067611": "J_\\mu^a", "aa403272b3a9a1d2414bf9585a5e9bc3": "\\frac{\\textrm{d}[\\textrm{CO}_3^{2-}]}{\\textrm{d}t}= k_2[\\textrm{HCO}_3^-] - k_{-2}[\\textrm{H}^+][\\textrm{CO}_3^{2-}], ", "aa403b0ea2aacbe08bf5f8f9101f2f43": "\\log{\\frac{|D|}{|\\{d' \\in D \\, | \\, t \\in d'\\}|}}", "aa40782e40638498debfa21ff8300f8f": "\\mathcal{L}_{D} = \\bar{\\psi}(i\\gamma^{\\mu} \\partial_{\\mu} - m)\\psi\\,", "aa40858b3b39e82656164d7719f9b312": "N_{e}", "aa40b99554dfcd23255fa2ab71df2031": "\\begin{pmatrix}T_T\\end{pmatrix}\\,", "aa40d38d1f7622ea369c49c80367e170": "2 k \\cdot p \\approx\\,", "aa40dde05d1b59ce66048f62603f3457": "k(b-1)^k 0} \\int_{\\{ \\omega \\in \\Omega | N_{t} (\\omega) > C \\}} \\big| N_{t} (\\omega) \\big| \\, \\mathrm{d} \\mathbf{P} (\\omega) = 0;", "aa459989ad7107acebb79c038fa4466b": "p_k(X_1,\\ldots,X_n) = X_1^k + X_2^k + \\cdots + X_n^k .", "aa45c6ff5fac11d03eaeea8e287ffd41": "ML+L \\rightleftharpoons ML_2: [ML_2]=K[ML][L]=K\\beta_{11}[M][L][L]=K\\beta_{11}[M][L]^2", "aa460246f08affeba70ef8be865562d7": "e^{\\pi \\sqrt{163}}", "aa4618745a555996c85c5f56d7dfc679": "\\tau_i^{}\\tau_{i+1}\\tau_i = \\tau_{i+1}\\tau_i\\tau_{i+1}", "aa462d8a639ad862f32e394dcfbc98fb": "ds = \\sqrt{dx^2 + dy^2}.\\, ", "aa463434a9381490eb1e60ac546ca17c": "P(T) = P(T|G)P(G) + P(T|B)P(B)", "aa463cb3a690817dd1c2f7e51ca6d19d": "\\psi_{2m+1} = \\psi_{m+2} \\psi_{m}^{ 3} - \\psi_{m-1} \\psi ^{ 3}_{ m+1} \\text{ for } m \\geq 2", "aa4653b2953fd6f86b591eb3d7f9d39a": "0 = J_\\mathrm{drift} + J_\\mathrm{diffusion} = -\\rho(x) \\mu \\frac{dU}{dx} - D \\frac{d\\rho}{dx} ", "aa465b64a3888ecd6075a7eaf66d1886": " 2\\frac{dZ_\\alpha}{dx} = Z_{\\alpha-1}(x) - Z_{\\alpha+1}(x)\\!", "aa468088f5733c5c3e87aa4c3a05de4f": "\\frac{\\tau_\\lambda}{\\tau_{\\lambda_0}}=\\left (\\frac{\\lambda}{\\lambda_0}\\right )^{-\\alpha}", "aa468c9ad16efc6ff19888756115890e": "X \\rightarrow T(X) ", "aa46c5e0d842218ffa59c4e42bfc4dba": "\\int\\limits_D F(x) \\, dx = 0 ~ \\forall D \\subset \\Omega ~ \\Rightarrow ~ F(x) = 0 ~ \\forall x \\in \\Omega", "aa46e9f140f257bae9bee1a43c6417aa": "[L'_{ij},P'_k]=i[\\delta_{ik}P'_j-\\delta_{jk}P'_i] \\,\\!", "aa473df3459ea318a13e662d18d60a1d": "\\Delta t = 2 T_c + 4 T_a\\,", "aa47ab50aa11e5e361046388609b25cd": "u_1=-\\tfrac{i}{2}\\int\\sin(kx)e^{(-2-i)x}\\,dx =\\frac{ie^{(-2-i)x}}{2(3+4i+k^2)}\\left((2+i)\\sin(kx)+k\\cos(kx)\\right)", "aa47b67b0a8611eff178cc65a35e9c57": " \\sum_{j=0}^d h_j^\\ast \\neq 0 ", "aa47c80b1f411bf8b5e77a7d7cccb33a": "R_M = \\frac{\\partial R_T}{\\partial Q_1} = M - Q_2 - 2Q_1", "aa47fa19b67316cd180a645dec5bf0b7": "p\\geq 1", "aa4804b319586930643fc12970bc3bf2": "0< p_0 < p_1 \\leq \\infty", "aa481968ec6704417e2e892dd534a5b4": "=\\langle \\hat{A}\\hat{B}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle\\, ", "aa482abb418c5a633911d21a41426648": "l=\\sqrt{r^2+h^2}", "aa485ff127c719dffb9b38e4d1bdbf92": "|s(t)|^2 {( \\frac{\\lambda}{4\\pi}} ) ^2 \\times ( \\frac{G_{los} e^{-j2\\pi l/\\lambda}}{l} + \\Gamma(\\theta) G_{gr} \\frac{e^{-j2\\pi (x+x')/\\lambda}}{x+x'})^2 ", "aa486aedb28768e8ce7689dff8c0e18c": "\n {\n \\rho~\\dot{\\eta} \\ge -\\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) + \n \\cfrac{\\rho~s}{T}.\n }\n ", "aa486c7a2224d694e46ac7411099a0e5": " - \\log( P \\ge \\varepsilon ) \\le \\sup_t( t \\varepsilon - K( x , t ) ). ", "aa489968c7df862b8c45a8fec4709623": "\nK_{21}x_{1}+K_{22}x_{2}=0\n\n", "aa491a5b6d921132cee2c2a58ebe4047": "(\\mathbf{x}_n)", "aa494992bb18db0f1c8ca7dc30815250": "x(t) \\to x(t+T)", "aa4962b681f6c5f429394587353a3c2e": "\\lim_{k \\to \\infty} u_{k} = u \\mbox{ in } L^{1} (\\Omega; \\mathbf{R}^{m})", "aa499098e24523dee54a48040af2a036": "g^c", "aa49a516ae3e53924800340d8a99bcd2": "A = \\begin{pmatrix} 0 & X \\\\ -X^t & 0 \\end{pmatrix}", "aa49e93ea0f7a5c8915cb6fc69345d29": "N^2 + KN", "aa4a251c3e5de17eba74cb286894644b": "w = \\bigvee W", "aa4a94fc8f9b7a930bbd4ee77ed2b97c": "\\tau_{D} := \\inf \\{ t \\geq 0 | X_{t} \\not \\in D \\}.", "aa4a9601556afa2fada2ff985db56948": "\\ I1 ", "aa4af9d974ab0d775c537bd1106af44b": "-i \\gamma^\\mu \\partial_\\mu \\psi + \\left( \\frac{m c}{\\hbar} \\right) \\psi = 0 \\,", "aa4aff7a3abb5e9f822242ce39ccabc3": " (\\forall x \\, P(x)) \\rightarrow P(y) ", "aa4bf0189cccf4a2ad07baccb882e660": "\\{|s\\rang, |\\omega\\rang\\}", "aa4bf34ec9c3593e6ff926b59e3a44b5": "(\\lambda + (k-1)/2)^{1/2}", "aa4bf8c3a4b6b97c4f4d687e53ff6d8b": " \nX = {{-q' \\text{ ln}_{q'}(U)} \\over \\lambda} \\sim \\mbox{qExp}(q,\\lambda)", "aa4c0da82bebf5e9d9c44fb4631d6d2d": "P(K)", "aa4c2d59780d1183641936f17dde9fec": "I_{C2} = \\frac{V_T}{\\left(1 + 1/\\beta_2 \\right) R_2} \\ln \\left(\\frac {I_{C1}}{I_{C2}}\\right)\\ . ", "aa4c3cacc7e074f671e5e931947d9c5f": "z \\in \\mathbb{C} \\setminus \\{0, 1\\}.", "aa4c44309da15fd3eabda9f7ee31645f": "\\Sigma K_j(p_1,p_2,\\cdots)z^j = \\Sigma K_j (q_1,q_2,\\cdots) z_j\\Sigma K_k (r_1,r_2,\\cdots)z_k", "aa4cb0aaabf178b003631bdfa626347f": "H = H(\\mathbf{q},\\mathbf{p},t) = H\\left(\\mathbf{q},\\frac{\\partial\\mathcal{S}}{\\partial \\mathbf{q}},t\\right)", "aa4cced160b7672954f408850b7c319c": "p(x,t)", "aa4cf35ba53b99738dc9c82ef74de1fb": "L \\approx 10\\log \\left(\\omega^2\\right)= 20\\log \\omega \\ \\mathrm{dB}", "aa4d1a6d2cf29f00e3a91e1b86842170": "\\Phi_i\\in\\Omega^i(A,\\mathrm{Hom}^{-i}(E,F))", "aa4dd2226b259245c25d6e1dbf04e1fe": "A\\,\\subseteq Q\\,", "aa4e0bb7781a0f72fb0e975ab813d330": " \\sigma_\\theta = \\dfrac{F}{tl} \\ ", "aa4e3cfb024c7ff30a8846913966dfb1": "\\sqrt{5}", "aa4e929fc8f415cad4845d0af60b06f8": " \\forall x \\, (P(x) \\rightarrow Q(x)) ", "aa4f1ec1939aace068df82e98525aadf": "\\sum\\nolimits_{1\\leq i0 ,", "aa5bf614c2a069fe76adcc9064e644fd": "\\omega_{p} = \\omega_{n}", "aa5c7bfd140acffc601dcea800ca72ae": " \\sin{\\frac{B}{2}}=\\sqrt{\\frac{efg + fgh + ghe + hef}{(f + e)(f + g)(f + h)}},", "aa5c876faeacc5d0fe6cf2fe2b647c8c": "\\forall\\ i \\ D_{**}(X_i, Y_i) = D_{**}(Y_i, X_i) ", "aa5c9e1a39c2dcaacc55b77487def46c": "\\frac{\\operatorname{d}V}{\\operatorname{d}t} \\leq -\\mu (\\sqrt{V})^{\\alpha}", "aa5ce6581c9303c68a05f79b7737142f": "\\delta\\mathbf{u}", "aa5cf4028e22d7bfed18160bf9e901a4": "\\beta_0+\\sum_{i=1}^n \\beta_i\\log\\alpha_i,", "aa5d0e312a8ec60726202ac7dc465100": "\\widehat{T}(\\mathbf{r},t)\\psi = -\\frac{\\hbar^2}{2m} \\nabla^2 (\\mathbf{r},t)", "aa5dc22552c6454790c0b85cbb3992e5": " \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\left(\\frac{\\partial^2 \\mathbf{S}}{\\partial x^{2}} + \\frac{\\partial^2 \\mathbf{S}}{\\partial y^{2}}\\right)+ \\frac{\\partial u}{\\partial x}\\frac{\\partial \\mathbf{S}}{\\partial y} + \\frac{\\partial u}{\\partial y}\\frac{\\partial \\mathbf{S}}{\\partial x},\\qquad (1a)", "aa5e3db0ff422995491d1c1c18ba5b59": "R_{\\mu \\nu} + \\left ( \\Lambda - \\frac{R}{2} \\right ) g_{\\mu \\nu} = \\frac{8 \\pi G}{c^4} T_{\\mu \\nu}\\,\\!", "aa5e4f040eee4a8901a563d46997eb70": "|f(x) - P_{n-1}(x)| \\leq \\frac{C_1(r) \\omega(1/n, f^{(r)})}{n^r}, \\quad 0 \\leq x \\leq 2\\pi ", "aa5eefd5dd859f7f7d2ce655872fcac6": "\\mathbf{y_1}=\\mathbf{x_1}+\\mathbf{e_1}", "aa5f01c1ed287aedd2ce3300760fbdeb": " {h_1 \\over h_0} = 1 ", "aa5f0cd8b331924a1df3af179be1c13c": " \\nabla = \\hat e^i \\,\\partial_i", "aa5f6480d86fbca48d9c3bdc5f10f22f": "\\mu = 25", "aa5f75a057e22cc43d8f2d9147934a03": " \\aleph_\\omega = \\bigcup_{n < \\omega} \\aleph_n ", "aa5f91577c5fc897ba47a100b59916c4": " X_i \\smallsetminus X_{i-1} ", "aa5fe6799bdaf15f814ade9c70382319": " \\qquad \\qquad \\mathrm{H}_e = - \\frac{\\hbar^2}{2m_e}\\nabla^2 + \\varphi_c(\\mathbf{x}),", "aa5ff660b32b2a32d6dcc383ad8e38fc": "\\displaystyle{\\mathbf{v}(t+h)-\\mathbf{v}(t)=h \\mathbf{t}(t) + {h^2\\over 2} \\kappa(t) \\mathbf{n}(t) -{h^3\\over 6} \\kappa(t)\\mathbf{t}(t) + {h^4 \\over 24}(\\dot{\\kappa}\\mathbf{n} -\\kappa^2 \\mathbf{t}) + \\cdots , }", "aa5fff696ec692f987c50000df6d9355": "M_{sl}", "aa60dca27921d7cc325a0c4c594b5923": " 1,1,1,0,0 ", "aa60ec218c6ace9d7a6be8acc77fdb35": "\\hat\\gamma(h)=\\frac{1}{2}\\cdot\\frac{1}{n(h)}\\sum_{i=1}^{n(h)}(z(x_i+h)-z(x_i))^2", "aa61368a34949a7dd7b3b5f8c7abc4e3": "\\frac{\\vec{F}}{V} = - \\vec{\\nabla} P", "aa61c99969c449d7fabf80f80fbb1d05": "u, v, ", "aa61dbf6ca5086d1dc43fcbe78b16a53": "O(n m)", "aa6212d02e625882d760ae4886a463e8": "d((a_n), (b_n)) = 2^{-k},\\,\\!", "aa6278092a71d174b93ae80db59e22c4": "S_Q = -\\int Q(x,p) \\log Q(x,p) \\, dx \\, dp ~.", "aa6285afac217807b6f44e688d981ac1": " m = \\frac{S_{yc}}{S_{yt}} ", "aa6287379bb4b73f3783999c589dec78": "f(x_n) , \\dots , f(x_{n+k})", "aa62a09b759d008405a42a22e91c65d3": "E(X[n]) = \\mu \\quad \\text{for all } n .", "aa62be9f7a6f005e2719744cbe056862": " MVA = V - K \\, ", "aa62c3dff9039a04c4035b7fb8039247": " \\delta_x ", "aa62d49ca4c0b8a0dca4ede8d307d78b": " \\pi = pq - [F + vx + g0w - g1w] \\qquad \\qquad (4) ", "aa62df456344767129f5ffe342d6be6b": "\\left|\\arg {zf^\\prime(z)\\over f(z)}\\right| \\le \\log {1+|z|\\over 1-|z|}.", "aa62fd4a2905bfbb30a900719b593f67": "\\scriptstyle n=0", "aa6328d7ba3019478e71ce30fa05827b": "\\langle \\bar{R}^2\\rangle=\\frac{1}{N}+\\frac{N-1}{N}\\,\\frac{I_1(\\kappa)^2}{I_0(\\kappa)^2}.", "aa635c563c3949dcfa60802b5bb36379": "=\\dot{x}_1 p_1+\\dot{x}_2 p_2+\\dot{x}_3 p_3", "aa6376be86a73dee24285f9d9b7d026f": " G_\\text{I,n} = e^{-(K_\\text{g}/T \\Delta T)}\\, ", "aa63afbf58c138dcbddb264db6634190": " P(n) = \\frac{m^n \\cdot e^{-m}}{n!} ", "aa640bb1b61d5377f846f0328f07544e": "O(2)\\,", "aa6413f7bc8ae76bca3da050c133f30a": "N(d_1,...,d_n)\\Big((x_1,\\ldots,x_n),(y_1,\\ldots,y_n)\\Big) = N\\Big(d_1(x_1,y_1),\\ldots,d_n(x_n,y_n)\\Big),", "aa642072be79747b865fc793d84ac949": "\\lambda = h\\, \\sqrt{\\frac{16}{3}\\frac{m\\,h}{H}}\\; K(m),", "aa642d72e8d5123e33906b5f2bdfe4ed": "A={5}t^2(1+\\sqrt{5}+\\sqrt{5+2\\sqrt{5}}) \\simeq 31.56875757 t^2.", "aa64c7eed0fdb76e4802b9934a10b63b": "\nT_{grav} = \\sum_{k}rmg\\cos(k\\Omega_0t)\n", "aa6598a633bca53bf1f0191e08d511d0": " Q[\\varphi] = \\iiint_D p(X) \\nabla \\varphi \\cdot \\nabla \\varphi + q(X) \\varphi^2 \\, dx \\, dy \\, dz + \\iint_B \\sigma(S) \\varphi^2 \\, dS, \\,", "aa65dd1b8bb3732c5f0cbb85f80b85e3": "t_i = x_{i+1}", "aa65e219f372f51753ea065d5e61f6f3": "\\text{var} = \\frac{(n-s+\\frac{1}{2})(s+\\frac{1}{2})}{(2+n)(1+n)^2} ,\\text{ which for }s=\\frac{n}{2}\\text{ results in var} =\\frac{1}{8 + 4n}", "aa66160d4074bedc3013efc3dcd9b012": " \\lambda f.\\operatorname{let} y\\ f = f\\ (y\\ f) \\operatorname{in} y\\ f ", "aa6634076c9cded6ce6ba99a038a5e80": "p(y,y_1|x_1,x_2)", "aa664cbda7c608ea4570fffc5ab4aadc": "M=\\oplus_{\\lambda\\in\\mathfrak{h}^*} M_\\lambda", "aa664f505fe125fe4f2c3ee3ae083324": "(* \\rightarrow *) \\rightarrow * \\rightarrow *", "aa665c7a760e1aa052293f2f07448f22": "\\! h: X \\to Y", "aa66687899a1268697d4859fc93aee79": " a^3", "aa66703136b08291fb7702e0a9ea302a": "\\mathcal{M}_{0}", "aa66723b570c5efe65e86feec63ec98f": " a^\\ast d - bc^\\ast = 1 \\qquad\\text{and}\\qquad \\mathrm{Im}(b^\\ast d) = \\mathrm{Im}(a^\\ast c) = 0 ", "aa66b397315cb475ef9ebe3d9b31de32": "\\sigma_1\\ge\\sigma_2\\ge\\sigma_3\\,\\!", "aa671e49422a11c78c15138d6c6f2e84": "\n\\begin{align}\ne^{tA}\\mathbf{z}'(t) & = \\mathbf{b}(t) \\\\[6pt]\n\\mathbf{z}'(t) & = (e^{tA})^{-1}\\mathbf{b}(t) \\\\[6pt]\n\\mathbf{z}(t) & = \\int_0^t e^{-uA}\\mathbf{b}(u)\\,du+\\mathbf{c} ~.\n\\end{align}\n", "aa674c61a9526ffef0559609278a549b": " K(p) u = Q(p), \\ \\ \\ p \\in {\\mathbf p} ", "aa676c4d430b6980e7f98061523da793": "\\gamma = \\frac{\\alpha K_T}{C_V \\rho}", "aa67d29a2314e36a15675ae0792a0c13": "\\psi(\\Omega^{\\Omega^2 (\\omega^2 6 + \\omega)})", "aa67f96669da6bf4f079e3b8576ec403": "\\bigoplus_{e\\in E} L^2([0,L_e])", "aa68416dc9a76ea0788895328b933bf7": "\\Chi^2 = \\sum_i{\\frac{(x_i-\\mu_i)^2}{\\mu_i}}", "aa684bc9a5f6e1a6dd0e3bdfb31f801d": "\\frac{1}{-\\ln p} \\times \\frac{\\beta(1-p) e^{-\\beta x}}{1-(1-p) e^{-\\beta x}}", "aa687da0086c1ea060a8838e24611319": "x_1", "aa6892b94db6311910a514610daaca68": " K(x,y) =\\sum_{k=1}^{\\infty}\\varphi_k (x) \\overline{\\varphi_k (y)}. ", "aa6a0c380f0a92aea8257507bcfe705a": "\\textstyle \\{(1, \\alpha), (1,0)\\}", "aa6a0f6246fed45240d2e56ec07211e2": "\\Re(s)=1/2,3/2,\\dots,n-1/2", "aa6a6c066e3307fbbc85898732951629": "n\\tbinom {2n-1}{n-1}/2", "aa6ab46f8ced6bf4dd35221f7a0cfe68": "\\mathbf{E} = \\frac{1}{4\\pi\\epsilon_0}\\frac{q}{r^2} \\mathbf{\\hat r}", "aa6ad0e33db7022d15216e8c7723691b": "b(f)b(g)+b(g)b(f)=0, \\,", "aa6afe4d0b767abf781b475588524b8e": "xy \\vee \\bar{x}z \\vee (x \\vee \\bar{x})yz", "aa6b1629167660cc8dc6f4d12977e60a": " x/2 \\text{ or } {x \\over 2} ", "aa6b2a9147ca38778ef046d6754e252d": "{u}_{2} (\\mathbf{q}) ", "aa6b4d3184bd5eb3d5c2326bf8bb4038": "\\omega_m", "aa6b4ea3e660b518a0c3d605d7e60d9b": " \\Delta_d", "aa6b569eff9f0f54984139a3a01ee641": "z^{-1} = \\frac{\\overline{z}}{{\\left| z \\right|}^2}", "aa6b833af08b314a341085d82269b944": "\\delta_\\odot = \\arcsin \\left [ \\sin \\left ( -23.44^\\circ \\right ) \\cdot \\cos \\left ( \\frac{360^\\circ}{365.24} \\left (N + 10 \\right ) + \\frac{360^\\circ}{\\pi} \\cdot 0.0167 \\sin \\left ( \\frac{360^\\circ}{365.24} \\left ( N - 2 \\right ) \\right ) \\right ) \\right ]", "aa6baf08fbf3d0cad7b2582fb470d55d": "\\tilde{h}\\triangleq\\sum_j u_j X^{\\deg(h)-\\deg(f^{(k)}_{j})}f^{(k)}_{j},", "aa6bb8970c7b861ec106141c59c7fbd2": "\\Psi = \\sqrt{\\left(\\frac{1}{k-1}\\right)\\frac{\\Sigma(\\bar{x}_j-\\bar{X})^2}{MS_{error}}}", "aa6c13c7f9f8216c59b5c5f94ba8396a": "T_yY\\,\\!", "aa6c3cd2b57caeccb10a2e8a9e238a5b": "L_{[\\omega]}: H_{DR} (M) \\to H_{DR} (M), [\\alpha] \\mapsto [\\omega \\wedge \\alpha]", "aa6c5b7830975fcf9516371f2ee87d8f": "f = F^{1/N}", "aa6cb5dc2882c9e4819fb79f9e1e1590": "\\displaystyle{S(t)N=NS_0(t),}", "aa6e06313027aa61903ccb77fb2a4ec2": "\\frac{1-p}{p}", "aa6e075bbe87ce64a307102e197877c9": "\\varphi(N)= N-p-q+1 ", "aa6e4d1ebe94f87801d7ed849427f5bc": "H_\\epsilon=H_0 + e^{-\\epsilon |t|}gV", "aa6f0165fed539cbc9c78a8ebd14ab3a": "\\frac{\\partial C_v}{\\partial \\sigma}", "aa6f1503b55c1f2797f6437b713a188e": "\\sum_{k=0}^n F_k t^k/k! = Z(G; 1+t, 1, 1, \\ldots, 1).", "aa6f54fb33437ff4da499e3525f89f9c": "\\mathbb{P}[X_i = H, \\ i=1,2,\\dots,n]=\\left(\\mathbb{P}[X_1 = H]\\right)^n = p^n", "aa705c16896840aaad7a99e560490695": "\\psi = \\int_A^P \\left( u\\, \\text{d}y - v\\, \\text{d}x \\right).", "aa7099e7cb4e0c758cd28509de277dd9": "X \\sim \\mathrm{F}(\\nu_1, \\nu_2)", "aa70ca60bfe022a956f19af44d7b6719": "\\vec{B}=\\frac{1}{R}\\nabla\\psi\\times \\hat{e}_{\\phi}+\\frac{F}{R}\\hat{e}_{\\phi}", "aa70feb503bb6f9056e62218932fe57e": "\nE = E^0 - \\frac{kT}{ne} \\ln \\frac{[\\mathrm{Red}]}{[\\mathrm{Ox}]}\n= E^0 - \\frac{RT}{nF} \\ln \\frac{[\\mathrm{Red}]}{[\\mathrm{Ox}]}\n= E^0 - \\frac{RT}{nF} \\ln Q.\n", "aa7130e1f1f61dc594d2b0b41b4f6a71": "\\bar K_0", "aa7177823a819140209f76f30d360fc7": "r = 1.224\\cdot w(z) \\, ", "aa719ab7a122842585f090862a6cd203": "\\hbar\\vec J", "aa71ce546b55e32cea424524dcfb84e7": "\\dot{v}^\\text{free}_\\alpha = a\\,\\left( 1 - \\left(\\frac{v_\\alpha}{v_0}\\right)^\\delta \\right)\n\\qquad\\dot{v}^\\text{int}_\\alpha = -a\\,\\left(\\frac{s^*(v_\\alpha,\\Delta v_\\alpha)}{s_\\alpha}\\right)^2\n= -a\\,\\left(\\frac{s_0 + v_\\alpha\\,T}{s_\\alpha} + \\frac{v_\\alpha\\,\\Delta v_\\alpha}{2\\,\\sqrt{a\\,b}\\,s_\\alpha}\\right)^2", "aa72294e1d16f896a5bcf8f14a931d61": "\\frac{\\Delta y}{\\Delta x}=\\frac{\\Delta y}{\\Delta u}\\, \\frac{\\Delta u}{\\Delta x}", "aa722c9e6a865069486e2562dc8fc896": "P_r = P_t*G_t*G_r*Loss", "aa72c96ecb511e9d1d061503922cb12f": "x^i \\to x^i + \\alpha^i x^+", "aa72fe66ad82b8c635407ea090a06c05": "\\sqrt S", "aa73360675a4acf97544cf6a3610a2d7": "f(z) = \\prod_{n=1}^\\infty E_{p_n}(z/a_n)", "aa73c1ea07e567c155233581303f3882": "\\displaystyle{B((z,w),(z^\\prime,w^\\prime))=\\sum z_iw_i^\\prime - w_iz_i^\\prime.}", "aa748f1cca0def54d6328ed4090829b5": " \\sum_{ i = 1 }^K p_i^2 ", "aa7491cd7fa16349e8932382ec47d908": " \\mathbf{f} : \\tilde{V} \\rightarrow V ", "aa74dfabca6300aef592e091f3771540": "\\exp(-\\epsilon t)/\\mu(S_{t})\\,\\!", "aa74f6b90cc211f3ce2dc95563cacca3": " \n\\Delta(t) = L(t+1) - L(t)\n", "aa75e00d2ce5be75c508c79aba81073e": "\\nabla \\cdot \\mathbf{F}_\\nu = 0 ", "aa75e7e65620d1d5ea14809016d1c09d": "F_{ax}", "aa76309610042081b2f65cd13f9ba784": "\\scriptstyle Q\\,\\sim\\,\\rm{W}(k,\\beta)", "aa767d13d754fc590e26bc69e519594d": "\n{} - \\frac{\\varphi_{i,j} - \\varphi_{i-1,j}}{h_{i-1}^x}\n\\left ( \\frac{h^y_j}{2} \\epsilon^x_{i-1,j} + \\frac{h^y_{j-1}}{2} \\varepsilon^x_{i-1,j-1} \\right )\n", "aa768c939060400a9e727401d53c5f0f": "\\sum_{k=0}^\\infty {k+\\nu+1 \\choose k+1} \\left[\\zeta(k+\\nu+2)-1\\right] \n= 1", "aa76c988cdd1eb5f8880715c4ab8713c": "\\begin{bmatrix} -2c & 0 \\\\ c & d \\end{bmatrix}, [c, d]\\in \\mathbb{R} ", "aa7711cee71ccddc4efa828a254e25eb": "(x-4) (x-1)^4 x^2 (x+1) (x+3)^2 (x^2+x-4)", "aa772ea50071161bbe66b3b61913be99": " \\Vert Mf\\Vert_{L^p (\\mathbf{R}^d)}\\leq C_{p,d}\\Vert f\\Vert_{L^p(\\mathbf{R}^d)}.", "aa777d8daedf4cd85fc94d2eec7e6265": "\\chi_\\mathrm{red}^2 < 1", "aa77a8d3a5c7dd1be42a54ab0b0243cf": "F(r) = {1 \\over 2}\\ln{1+r \\over 1-r} = \\operatorname{arctanh}(r).", "aa77be7b238af58c397d99b291b7243d": "\\| \\mathbf{K} \\|^2 = K^\\mu K_\\mu = \\left(\\frac{\\omega}{c}\\right)^2 - \\mathbf{k}\\cdot\\mathbf{k}\\,,", "aa7805a0b8063e9cfe3186a571f03079": "x<\\sqrt{m}", "aa784335c060e1de35fd31d84e386b71": "\nF L \\sin \\theta_2 = k_\\theta ( \\theta_2 - \\theta_1 )\n", "aa7871abbc4bd03a6b56b70173649c2c": "\\delta=\\frac{p(q-r)}{r(q-p)}", "aa789aa305b7f1c09584a6e94e982bcd": "[Q,f\\}=\\frac{\\partial}{\\partial\\theta}f-i\\bar{\\theta}\\frac{\\partial}{\\partial t}f,", "aa78b29cf57edc51e2d1f8bdd8feac2d": "c \\in S", "aa78d5b4c162ba5eeb7223eb8b0cdc94": " a_{20} = \\mathcal{L}(p_4)+p_3p_4+p_1p_7,", "aa793ad5c1827dc95f3e04265205a871": "\\textbf{G}(\\textbf{r}, \\textbf{r}^{\\prime})\\,", "aa793f8925c738ad8cfc4971b434a993": "\\sum_{n=0}^\\infty \\frac{f^{(n)}(0)}{n!}x^n=\\sum_{n=0}^\\infty \\frac{0}{n!}x^n = 0,\\qquad x\\in\\mathbb{R},", "aa79a456c119c4b35709911d92614df7": "\\mathbf{w_p}", "aa79f9d42a113b95b60088b9e7bc9140": "\\mathbf{B}(t)", "aa7a255924f70dfa2ecf1579e5e14e60": "\\textstyle l = 7", "aa7a46f242c8ff93edcbba688a4a87e4": "F_{\\infty}", "aa7a58dfe4e8dacd243ce4d44c4b36ea": "\\mu(A)=\\zeta\\left(1_A\\right)", "aa7a6ee2f0811a64dedbec2259e7da05": "f_e(0)=f_e(L_e)=0", "aa7a83a75d705001bf0eba4b61d75a36": "\\mathbf \\nabla \\cdot \\mathbf A = 0", "aa7a8661d8cbe6030115cb6f63949f7a": "r = \\limsup_{n \\to \\infty}\\sqrt[n]{|a_n|},", "aa7a9df5da1a8b48a1075285348c82cd": "Z_0 = 0", "aa7aa6b0270f5dfadb18f989c6dbb8a3": " \\boldsymbol{\\theta}^* ", "aa7acb9528d93bec55b1b7825e387606": "\\scriptstyle{F^-:X^-\\longrightarrow \\mathrm{R}^-}", "aa7ae60703752f38980cd9df652afc26": "\\mathbf{\\dot A} = \\dot A_\\rho \\boldsymbol{\\hat \\rho} + A_\\rho \\boldsymbol{\\dot{\\hat \\rho}}\n + \\dot A_\\theta \\boldsymbol{\\hat\\theta} + A_\\theta \\boldsymbol{\\dot{\\hat\\theta}}\n + \\dot A_\\phi \\boldsymbol{\\hat\\phi} + A_\\phi \\boldsymbol{\\dot{\\hat\\phi}}", "aa7aeff09bda645becd26ba8a7ecc4bc": "Q^T = G_3G_2G_1", "aa7af22fe7eb1afebe67af8f0afdb1cb": "\n\\langle z \\rangle=e^{i\\mu-\\sqrt{c}(1-i)}\n", "aa7b94983b115f29f597246a82c9452a": " V_\\text{model} = V_\\text{application} \\times 21.9 ", "aa7bcb17b3944fa86aad142851b1d448": " \\ell^1 t^{-1} = \\ell/t ", "aa7be65ee99b6e7b406d5d606952f226": " \\det(A) = \\det(A^\\mathrm{T})\\quad \\Rightarrow \\quad \\det(A^\\dagger) = \\det(A)^* ", "aa7bfa59f363881d97a53b2481e518d8": "\\frac{15 mol O_2}{2 mol C_6H_6}=7.5 mol O_2", "aa7c6b6970ce66c82a1827f632f74df2": "k=1,2,\\ldots,n", "aa7c8abaf376fa6659b7e58a81fb8d13": "\\int_{-\\infty}^\\infty G^{-1}(\\alpha) d H(\\alpha) = -\\int_{-\\infty}^a H(G(x))dx+ \\int_a^\\infty \\hat{H}(1-G(x)) dx,", "aa7cb716f463f802f729d4e142aefe86": "[-\\infty, \\infty]", "aa7ccb312ac523ebaf6bb3bd788617c5": "\\Pr[V\\text{ accepts }w\\text{ starting at }M_j] = \\max\\nolimits_P \\Pr \\left [V \\leftrightarrow P\\text{ accepts }w\\text{ starting at }M_j \\right ] ", "aa7cfac2e7286f42ed9ea5a9fcccc757": "S\\propto\\nu^\\alpha.", "aa7dc35cb591adf7dba34387d347cb74": "\n(1)\\cfrac{\n (2)\\cfrac{\n (1)\\cfrac{C_1 (1,3)\\qquad {C_8}^*}{C_3 {\\color{red}(3)}}\n \\qquad\n C_4 (1,-2)\n }\n {C_7 (1,3,5)}\n \\qquad\n (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{C_8 (-1)}\n}\n{\nC_9 (3,5)\n}\n", "aa7dce13c4c25404481fef7d3a343741": "\\lambda = 1/n_T", "aa7e32ecf09596f041e99a85d74a2d46": "u(x)\\leq u(y)", "aa7ea730018f3861218ce18f10189b45": "M = m_e^* + m_h^*", "aa7ec64b09d02391ac4d963bc43c49e0": " \\boldsymbol{S}_{k} = {\\boldsymbol{H}_{k}}\\boldsymbol{P}_{k|k-1}{\\boldsymbol{H}_{k}^\\top} + \\boldsymbol{M}_{k} \\boldsymbol{R}_{k} \\boldsymbol{M}_{k}^{T}", "aa7eee7179f492ed3ca76f28f1d2d0c8": "\\mathbf{\\ddot r} = GM \\mathbf{\\hat{r}}/r^2", "aa7f19c210b6462bdfea0d8554acbf54": "Y_2=\\frac{1}{T_2}", "aa7f86516048b7782604d2d3ca05d9a7": "V_w = \\frac{\\frac{dQ}{dh}}{T} = \\frac{1}{T} \\frac{d}{dh} \\left(\\frac{1.0}{n}AR^{2/3}S^{1/2} \\right)", "aa7f8660ece92994d98f0351590e2de2": "0\\leq k\\leq n", "aa7fa8728f0db14133e91671d0b64879": "a, b \\in A; p, q \\in P", "aa7fce49980e6f556aa0fedda1fb9e5b": "\n \\chi' = \\sum_{i=1}^n k_i s_i^2.\n", "aa7fd158cc0e88c7c050fbe002c6845f": "\\frac{1}{\\pi}\\frac{1}{u-t}", "aa8002eac024ed8e2ff1d75a2e5126fe": " I_{31} = \\frac{V_{31}}{|Z_\\Delta|} \\angle (150^\\circ-\\theta) ", "aa8014d4e25c3b8d0f89b6f9a6286f51": "t-", "aa8069c40dc85578f6824ad02d7b4d78": "\\sqrt{s/(n h)}", "aa8082fdf84b709ba57a6b67a54a8279": "z > y", "aa808ced41d1fa31b1101c87e2ee8ab5": "T(t) = A e^{-\\lambda \\alpha t},", "aa808cf9af95e3e345c3f161626bec73": "x = x n_2 [n_2^{-1}]_{n_1} + x n_1 [n_1^{-1}]_{n_2}", "aa809645b31213005d73b587cf36d2c2": "\\cdots\\to H_n(D^n)\\rightarrow H_n(D^n,S^{n-1})\\rightarrow H_{n-1}(S^{n-1})\\rightarrow H_{n-1}(D^n)\\to \\cdots.", "aa80a4b99200e4d6a14ca6e6aae575fc": "\\frac{d^n H_x}{dx^n} = (-1)^{n+1}n!\\left[\\zeta(n+1)-H_{x,n+1}\\right]", "aa80be9bc81215ff40988f465746cdaa": "S^{n+2}", "aa80cf5f3849537faabe71917dfd64e2": "\\pi(a)=(a_1,a_2,\\ldots,a_n)", "aa80d3d11d8db1adcbd8e1bbf68d230d": "N = \\begin{bmatrix} A & B\\\\ 0 & C\\end{bmatrix}", "aa812be1c84350ad1f7557a8a549e563": "f'(0) > \\mu \\,", "aa812dc7b9a8b441de078d4dffba283b": " \\mathbf v_1 =\n\\left( 1- {1\\over 2} {p_1^2 \\over m_1^2 c^2} \\right) {\\mathbf p_1 \\over m_1}\n- {q_1 q_2\\over 2m_1m_2 c^2} {1 \\over r } \n\\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right]\n\\cdot \\mathbf p_2", "aa815724e9492bbcf9eac973d28aea61": "h = E_m^{-1}(0)", "aa817b381f5fb4400cf9fba41ff49e71": "x = \\frac{1}{\\alpha^2}\\frac{V}{\\mathcal {E}_p/e}", "aa82167b310a566c4a1add848bc62e98": "\\overline{2} \\times \\overline{2}=\\overline{4}=\\overline{0}", "aa823990452d68a819d12a11653b7813": "x_0, x_1, x_2, x_3,\\dots", "aa823caeaf15da5c548e2d3af9f05b0c": "W_I", "aa824824e1d825e27210a232b8fc8b15": "\\gamma_n(x + y) = \\sum_{i=0}^n \\gamma_{n-i}(x) \\gamma_i(y)", "aa82c710931deb434f1a0315840da470": "k = 2\\pi \\,n/\\lambda_0", "aa830a960056e25d347cd8c9be83245f": "\\alpha_i \\in M_i", "aa83105f6ecd4442a51d8fca4c42c0a2": "\\forall x \\forall y [Rxy \\rightarrow \\exists z (Rxz \\land Rzy)]", "aa8343e2c42f74e3c0f9381eb46a073c": "X \\leftarrow Z \\to Y", "aa838ef0a6fb7ffda90d83725ca6d39c": "F_GE = \\left.\\coprod_U U\\times G\\right/\\sim", "aa83eca22a62c706c128aa00b7f0e2de": "\\mathrm{2 \\ RSH + Na_2PbO_2 \\longrightarrow (RS)_2Pb + 2 \\ NaOH}", "aa84650a27888056602ca66917263a6e": "\nC_n = {n \\choose 3} + {n \\choose 2} + {n \\choose 1} + {n \\choose 0} = \\frac{1}{6}(n^3 + 5n + 6). ", "aa847d6751de625658d275d7fa4bfd3b": "g(\\vec r)\\sim |\\vec r|^{-d+(2-\\eta)}\\,\\!", "aa847d8dc9a92e8e41442f53732b32dd": "\\scriptstyle \\delta t_{\\text{meas-err},i}", "aa8480cbe18c89f2a9664768cdbbd7c2": "0 \\to \\mathbf{Z}/p \\to \\mathbf{Z}/p^2 \\to \\mathbf{Z}/p \\to 0.", "aa84c0983223e67fb3e4117ace9b55ac": "\\int_0^x\\sqrt{1+y^\\alpha}\\,\\mathrm{d}y=\\frac{x}{2+\\alpha}\\left \\{\\alpha\\;{}_2F_1\\left(\\tfrac{1}{\\alpha},\\tfrac{1}{2};1+\\tfrac{1}{\\alpha};-x^\\alpha \\right) +2\\sqrt{x^\\alpha+1} \\right \\},\\qquad \\alpha\\neq0.", "aa84c0cfc276b7254f58bc6e1c2b01c3": " (\\mathbb R,+) ", "aa84caa35afb0b555ca19e777a9f2ccb": " T_2 = T_1\\left(\\frac{V_1}{V_2}\\right)^{(R/C_v)}", "aa84e580b971171410e0eadfe9b955f0": "e^{x+iy} = e^x(\\cos y + i \\sin y).", "aa854de9b1bf035a847d61abba31f9e9": "\\begin{pmatrix}b_1 \\\\ a_1 \\end{pmatrix} = \\begin{pmatrix} T_{11} & T_{12} \\\\ T_{21} & T_{22} \\end{pmatrix}\\begin{pmatrix} a_2 \\\\ b_2 \\end{pmatrix}\\, ", "aa857cad42a3ae31110a7e089bd9a177": "\n\\frac{1}{F_{ax}} = 1.0 + \n\\left(\\frac{4}{5}\\right) \\left( \\frac{\\xi^{2}}{1 + \\xi^{2}}\\right) + \n\\left(\\frac{4 \\cdot 6}{5 \\cdot 7}\\right) \\left( \\frac{\\xi^{2}}{1 + \\xi^{2}}\\right)^{2} + \n\\left(\\frac{4 \\cdot 6 \\cdot 8}{5 \\cdot 7 \\cdot 9}\\right) \\left( \\frac{\\xi^{2}}{1 + \\xi^{2}}\\right)^{3} + \\ldots \n", "aa8584ece76032ec2baf4d003a474b8a": "\\frac{\\partial \\mathbf{x}^{\\rm T}\\mathbf{A}\\mathbf{x}}{\\partial \\mathbf{x}} =", "aa85b6f10e304bce9373c7fcd6747665": "\n\\sigma_1 = \n\\begin{bmatrix}\n0&1\\\\\n1&0\n\\end{bmatrix}\n\\quad \\quad\n\\sigma_2 = \n\\begin{bmatrix}\n0&-i\\\\\ni&0\n\\end{bmatrix}\n\\quad \\quad\n\\sigma_3 = \n\\begin{bmatrix}\n1&0\\\\\n0&-1\n\\end{bmatrix}\n", "aa85edc9df7dd9d66b26c17b788da4a0": "I=\\int \\ln (x) \\cdot 1 \\,dx.\\!", "aa861db8a9301361bb35037cbf7d7461": " \\mathfrak{P}(\\mathfrak{C}(\\mathcal{Z})) = \\mathcal{P}.", "aa863802a40d43bd65ae051c264607e1": " T_0 = \\frac{P_0}{k_v(V_0)} \\quad and \\quad T_0 = \\frac{V_0}{k_p(P_0)} ", "aa867c32cee305ec51cf3d65fde06d8b": "\\text{Arbitrarily selecting from ohm's law the two base numbers: base voltage and base current}", "aa868607a3741da5cb4def847b07b700": "K-P=K \\cap (K*\\neg P)", "aa872a4353c7e37373a2d7f08c4d7399": "3 + 5 \\times 2", "aa872d9ca3d3dfdf435099121cb21113": "V(r) = -\\frac{1}{4 \\pi \\epsilon_0} \\frac{Ze^2}{r}", "aa873a1848d1480932ff71951ddf8ef3": " r = 2/9 ", "aa87ba97c5a437e4f37ddebac2e28e4a": " \n\\int_0^{\\infty} {k\\; dk \\over k^2 +m^2} J_0 \\left( kr \\right)\n=\nK_0 \\left( mr \\right)\n . ", "aa87c3fa1f55a0af8b4eaa1517f6037d": "x^\\mu=0", "aa8881c187103ab6a6891d272069a900": " f_X(x|\\boldsymbol \\theta) = h(x) \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(x) - A({\\boldsymbol \\theta}) \\Big) ", "aa888f36c4dd7a4ba80eb98b3e8cb5a8": "m\\times (d*3)", "aa88d1a9c65e482d60ad3e856fcda18f": "-y^2x+4y+2x-x^2+4z+xz=0", "aa88fc6964a43c1ddb74fe103874ee30": "J = 0 \\overrightarrow{\\to} 1", "aa8955324870cf27afe5a7d1252597ac": " E_r(R) = \\max_{\\rho \\in [0,1]} \\left[ \\rho R - E_0(\\rho) \\right]. \\, ", "aa8961b856db7827428ee94f419cb42b": "\\rho c_p\\,", "aa89674ee72272667bcf2bd6a238b891": "Ax+b\\geq 0, \\, \\frac{(c^T x)^2}{d^Tx}\\leq t", "aa898891b2613a206fa42c1b909d6e64": " \\mathbf{F}_i = -\\nabla V_i + \\mathbf{N}_i,", "aa89eabe56c93f13eb7ccbcf6dc3e4b4": "\\displaystyle{f_-(z)=a_0 + a_1 z + a_2 z^2 + \\cdots,\\,\\,\\,\\,\\, f_+(z) =z + b_1z^{-1} +b_2z^{-2}+\\cdots,} ", "aa8a35f12a01f5296241a35ed9ba052f": "\\frac{10}{\\sqrt[3]{b}}", "aa8a521383a26162e21b2649a3304135": "3(d-2)", "aa8ad40d6c464d111a70647498980190": "\n\\begin{align}\n\\operatorname{E}_{\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k} [(\\mathbf{x}_n - \\mathbf{\\mu}_k)^{\\rm T} \\mathbf{\\Lambda}_k (\\mathbf{x}_n - \\mathbf{\\mu}_k)] & = D\\beta_k^{-1} + \\nu_k (\\mathbf{x}_n - \\mathbf{m}_k)^{\\rm T} \\mathbf{W}_k (\\mathbf{x}_n - \\mathbf{m}_k) \\\\\n\\ln {\\tilde{\\Lambda}}_k &\\equiv \\operatorname{E}[\\ln |\\mathbf{\\Lambda}_k|] = \\sum_{i=1}^D \\psi \\left(\\frac{\\nu_k + 1 - i}{2}\\right) + D \\ln 2 + \\ln |\\mathbf{W}_k| \\\\\n\\ln {\\tilde{\\pi}}_k &\\equiv \\operatorname{E}\\left[\\ln |\\pi_k|\\right] = \\psi(\\alpha_k) - \\psi\\left(\\sum_{i=1}^K \\alpha_i\\right)\n\\end{align}\n", "aa8b1620d135b6f950f09d1379dc6678": "\\mathfrak{H} = \\mathfrak{H}_2^{-1} \\mathfrak{H}_1.", "aa8b1dacc5fe07c2c617206c21a2b524": "\\frac{\\partial \\boldsymbol{\\hat{\\theta}}} {\\partial \\varphi} =-\\cos \\theta \\sin \\varphi\\mathbf{\\hat{x}} + \\cos \\theta \\cos \\varphi\\mathbf{\\hat{y}} = \\cos \\theta\\boldsymbol{\\hat \\varphi}", "aa8b32f38c521cd61d8497adbce653ff": "T_{\\mathrm m}", "aa8b649acf6265dca940076fa5d4a2b5": "L'=T'_{0}v", "aa8be0a0210ad03ab09f946ba07fbb92": "\\scriptstyle \\mu\\!", "aa8c21fde95850a8e9451a149d7d224d": " W_4=X_1^{4}+2X_2^2+4X_4", "aa8c26aea02aa620e16d9744e22443cf": "X(L) = (-A^3)^{-w(L)}\\langle L \\rangle ", "aa8c5fb4a2be2e8918b5911dd665cf37": " y(t_0) = 0.\\,", "aa8c8dff15ae2b44b859189a420462f3": "\\sqrt{Q_i}", "aa8cafd1bb5a31f4ff46b6a09c4b0891": "| \\alpha \\rangle", "aa8cb53a3e988a0c5b57845ea9323386": "C_{D_{max}}", "aa8d3010bbbbaf9abecb2806068ad98a": " 4 \\pi \\, r_0^2", "aa8df07693d06e8125465c1f8bddc862": "\\text{Speed} = \\frac{\\partial\\Gamma}{\\partial S} = \\frac{\\partial^3 V}{\\partial S^3}", "aa8df295693d422ccd1b3f90fbdff383": "T^3 = S^1 \\times S^1 \\times S^1", "aa8e95fd502498585dd45f54bbc9b248": "\n\\begin{array}{c}\\textit{Opportuneness\\ Model}\n\\end{array}", "aa8eaf3ab1ef63a286a2ee087436baf0": "u \\equiv \\frac{1}{r} = \\frac{km}{L^{2}} \\left[ 1 + e \\cos \\left( \\theta - \\theta_{0}\\right) \\right]", "aa8eedd5692aad8424d5261b0b64f6d7": "\\displaystyle{|(w,i; z,-i)| \\le 16|z-w|,\\,\\,\\, |(f(w),i; f(z),-i)| \\ge |f(z)-f(w)|/8,}", "aa8efc129a6c71df3ccead2c7aeb1b22": "x = \\frac{\\sqrt{609}-7}{28} \\approx 0.631354477\\dots\\qquad y = \\frac{3\\left(\\sqrt{609}-7\\right)}{280} \\approx 0.189406343\\dots\\,", "aa8f14b9fb48b339418b82f867a97286": "\\lambda_1 \\mu_2 - \\lambda_2 \\mu_1 = \\sin y, \\quad s_1 + s_2 = 1. \\,", "aa8f297eb8be05ee6a3b5103654643f9": " F(a) = \\langle F_a\\mid F\\rangle. ", "aa8f361a048cdd2eae91464b11b8d3ec": "F_r(t)+F_{cr}(t)=0", "aa8f68cea10ddf9157cb69ec18b614f3": "X[f_1 \\dotso f_i f_{i+1} \\dotso f_j f_{j+1} \\dotso f_n] \\to \\alpha Y[f_1 \\dotso f_i] \\beta Z[f_{i+1} \\dotso f_j] \\gamma W[f_{j+1} \\dotso f_n] \\eta", "aa8fb14add13ae15157e86cc3f7fd5ca": "\\psi = A", "aa8fcc67a478f1d2bdb94b83d8e4473c": " \\theta=-\\theta ", "aa909784e43691239220ca4210a7aeb6": "\\Pi(\\mathcal Q)", "aa90aefdf6ef28a1a38936548f8dfa5a": "\\tfrac{1}{a}", "aa90bf2accb8efa3adc3e910cc6cd352": "b = \\frac{-d}{D} \\begin{vmatrix}\nx_1 & 1 & z_1 \\\\\nx_2 & 1 & z_2 \\\\\nx_3 & 1 & z_3\n\\end{vmatrix}", "aa90f2bba75a7cde3b864d86829e418b": "H + G \\rightleftharpoons HG", "aa911e3e5830a8102239fd42d619d13c": "f = 2 \\omega \\sin \\varphi \\,", "aa9179dfbbb4ec910e58e606db880d91": "\\int_0^\\infty x^2 j_\\alpha(ux) j_\\alpha(vx) \\,dx = \\frac{\\pi}{2u^2} \\delta(u - v)\\!", "aa91b5f2b902e94588bee5e2a34c3534": "A = \\{a^n b^n c^m \\mid m, n \\geq 0 \\}", "aa91ce944a30438289bb5207af209034": "D_{t} \\,", "aa922a30ce1173792a5bbaaeec5dd22c": "I=[a,b]", "aa928be49bff40398185a09139be2973": "\\psi(t) = {2 \\over {\\sqrt {3\\sigma}\\pi^{1 \\over 4}}} \\left( 1 - {t^2 \\over \\sigma^2} \\right) e^{-t^2 \\over 2\\sigma^2}", "aa92a6093476b07aef6016573a11481c": "\\exp\\!\\Big( i\\boldsymbol\\mu'\\mathbf{t} - \\tfrac{1}{2} \\mathbf{t}'\\boldsymbol\\Sigma \\mathbf{t}\\Big)", "aa9314fb903ee6d7f215e1b4655b25c5": "\\frac{5}{121}=\\frac{1}{25}+\\frac{1}{757}+\\frac{1}{763309}+\\frac{1}{873960180913}+\\frac{1}{1527612795642093418846225},", "aa9348bb96fb4a8524e87135ef0e807c": "\\forall \\varepsilon \\, \\forall x \\, \\exists \\delta \\, \\forall y \\, ( \\, |y-x|<\\delta \\, \\Rightarrow \\, |f(y)-f(x)|<\\varepsilon \\, ),", "aa9397870afa4f10f318b94d37141e76": "\\sum_{i=0}^m(-1)^ie_i(X_1,\\ldots,X_n)h_{m-i}(X_1,\\ldots,X_n)=0,", "aa93d153a548940a0f3dff78e7c701ad": "\\begin{bmatrix}1 \\\\ 0\\end{bmatrix}", "aa9406a1a220ae3223b14a7f0b689401": "\nV_\\mathrm{b} = \\frac {Bpd}{\\ln Apd - \\ln(\\ln(1 + \\frac {1}{\\gamma_\\mathrm{se} }))}\n", "aa941e60908807cc9bb8ceb6ff3b97b7": "\\!\\, b ", "aa945c24cbd277fc8178c8f252face20": " (\\exists x) A(x)\\ \\equiv \\ A(\\epsilon x\\ A) ", "aa946b1303c5bddaae1bf283c39b022b": "(x_i, y_i) = (M x_o, M y_o)", "aa94a7acf59f12a1d999414103abe421": "K_{TE}", "aa94b7bc8dc35a4b92ffce9b05055c08": "\\hat{b}_i", "aa94e9d99bff0e313b31a566fefaf100": "M = {1 \\over c(c-1)} \\sum AUC_{k,l}", "aa9510203504a4acec3fcad8114e7ec1": "\\left\\langle-R,A_R^2\\right\\rangle", "aa951af5ac2cdf4a0ff8abe65893f5a6": " e \\le {D \\over 2}", "aa9531390648e35e64863e18f4c364a8": "I_nt-A", "aa9582cbf1cae400764beade4bbeb245": "S_2 = a_1 \\circ a_2 \\circ \\ldots \\circ a_{n} (S_1)", "aa95c695b2e0d5001ffd8e898a7e4652": "\\mathbf{e}_{23}", "aa95e69f97dc7edab9c17a319a0f80a7": "<\\epsilon", "aa9625100e512fc19c4a1797e62fecbc": "{\\widehat{BH}}_3", "aa962d2373deff15c05c716ef1e3c082": "\\hat{m}", "aa965480bf403424a097da6f736fc046": "(C_{\\beta I}^{\\;\\;\\; J} e_J^\\beta) e_\\gamma^I = 0,", "aa968370efc9797533259ffe70ad1192": "w(x)", "aa96d7f0a335d2a150e482638231bba8": "\\mathcal{D}\\subseteq\\mathcal{E}\\left[\\mathcal{A},\\mu\\right]", "aa97076ba3eb3ce19b53ccab39e0affe": "B_{n} = H_{n}A_{n}H_{n}^{T}", "aa970f0ef16da4af2ef3c3e984561819": "\\pi\\colon X\\times EG\\to X", "aa97258c2520c178299ddd182d5ec83c": "s:U_{\\alpha\\beta}\\to U_\\alpha", "aa9784496a596b00de63e581d78577a2": "\\frac{1}{T} \\int_0^T Z(t)^{2k} dt = o(T^\\epsilon)", "aa97c081a6302869b2d36aa6fdf7f227": "\\scriptstyle T_{output} ", "aa97fbee0b66b96644a046e8559b2a76": "\\xi_i\\rightarrow \\xi", "aa9865e49980e0d45df9f4033ffd4067": "K=\\sqrt{3\\left( a_1 a_2+a_2 c_1+c_1 a_1- \\left( \\frac{a_1+a_2+c_1+1}{2} \\right)^2 \\right)}", "aa986807c1fbde7353f714250ac63730": "e_a(P)=0 \\Leftrightarrow P\\in V(a)", "aa98c1b8ebd92bf0ab1e2200391e91aa": " \\beta= \\frac{k\\Delta t}{2}", "aa98e030a12675d9da4cf4b80cc8cd80": "i \\hbar \\part_{X_i}", "aa9989c3368e0e1bee82eecf3341f21c": "\\int \\sec^n{ax} \\, \\mathrm{d}x = \\frac{\\sec^{n-2}{ax} \\tan {ax}}{a(n-1)} \\,+\\, \\frac{n-2}{n-1}\\int \\sec^{n-2}{ax} \\, \\mathrm{d}x \\qquad \\mbox{ (for }n \\ne 1\\mbox{)}\\,\\!", "aa99cf2e7b3439ae376a6582c9f5c7f1": "M_a = rP(t) - \\frac{dP(t)}{dt}", "aa9b30d041b3b82b601a145b9a940d5e": "k^a=n^a", "aa9b6ecc5ea4d43e78783cdc669f6db4": "p(n)\\lambda^n", "aa9bcb7ff8a7284049a0fdb2b8e38728": "\\theta=\\theta_0+h", "aa9bd40b8ae20a6465fa058fdbf3f6be": "(d - t) - c'", "aa9beac5caf7c900be8a5cdbe73078c2": " v(a) \\le 1 \\Rightarrow v(1+a) \\le 1\\ ", "aa9c234cf1d727c912a4c8b5528c35ae": "|b_{33}| < |b_{31}| + |b_{32}|", "aa9c283f26e8ffb66b5c826c480e705c": "KK(A, B \\otimes E) \\times KK(B \\otimes D, C) \\to KK(A \\otimes D, C \\otimes E).", "aa9c415502e6fb9a32c8a1895a2bc55e": "\\sqrt{1-x^2} \\,dx", "aa9c58a6a1c0f2e34feea12d066d8ad5": "A + L^p_d(K) \\subseteq A", "aa9ca933fb861344450a657c1388a8f1": "\\frac{3/36}{3/36 + 6/36}=\\frac{1}{3}", "aa9cb574a4d38a768902982d2557a549": "\\rm HCO_3^- + H^+ \\leftrightarrow H_2CO_3 \\leftrightarrow CO_2 + H_2O", "aa9cdf4d44b45c2dc1eb0e66fd44af72": "D(h)", "aa9cf2fde42f104c35a8ea4354eff7c1": "N^3", "aa9d3c3404bcef58965e4bb1fe9fb23c": " f", "aa9d401dcaba9104a5e5c958db383dd4": "\\begin{cases}\nf : \\mathbf{R}^2 \\setminus \\{(0,0)\\} \\to \\mathbf{R}^2 \\setminus \\{(0,0)\\} \\\\\n(x,y)\\mapsto(x^2-y^2,2xy)\n\\end{cases}", "aa9d6425bef877d9e683174eb1e1c821": "\\frac {p}{r}\\ =\\ 1 + e \\cdot \\cos \\theta\\ =\\ 1 + e_g \\cdot \\cos u + e_h \\cdot \\sin u", "aa9d757252a68d84638149792166fcdc": " a(z) U^{\\prime\\prime}(z) + b(z) U^\\prime(z) +(c(z)+\\lambda)U(z)=0.", "aa9de374da2beb2a4f8e00c854b62b1d": "\\sqrt{\\frac{3}{4}}\\!\\,", "aa9e0b7485ce284faad9d898630b8d19": "s \\ \\stackrel{\\mathrm{def}}{=}\\ m_{b} / f = v_{term}/g", "aa9e0dcc5ae0d70b76582482ed288275": "Q = C_\\mathrm{th} \\Delta T\\,", "aa9e5141cca76b1e9ae5ffb0f273362b": "=vu'+uv'", "aa9e9fc4a2ea21314a5ae2aae845fcd9": "\nD(A) = \\sum_{1\\le k_i\\le n} A(1,k_{1})A(2,k_{2})\\dots A(n,k_{n}) D(\\hat{e}_{k_{1}},\\dots,\\hat{e}_{k_{n}})\n", "aa9f5885cdc1a379632ff478b1755015": "\\frac{\\partial r_i}{\\partial \\beta_1}= -\\frac{x_i}{\\beta_2+x_i},\\ \\frac{\\partial r_i}{\\partial \\beta_2}= \\frac{\\beta_1x_i}{\\left(\\beta_2+x_i\\right)^2}.", "aa9f8e0aa922b9bd92027249e1457ebd": "ECI=AD \\times ECD\\!", "aa9fd1bddefc9932166466573ddd13dd": " \\mathbf{D} = \\varepsilon_0 \\mathbf{E} + \\mathbf{P} ", "aa9ff7a2bb42e6a99fab49a0756442f8": "\\frac{dr}{dt}=-r", "aaa0252719d2b450bb24d7fcb7886093": "x^2 - x - 1 = 0", "aaa0f946b575da3897c86ca09d73e1c3": "F=-kr", "aaa121e77b795ef70431b37843e9791f": "0 = \\sum_i f_i^{(k)} \\qquad s.t.\\ k=1,2\\,\\!", "aaa133f0005cb1e435b2b822ac087a22": " w_{m}(x)=e^{-x^{2}}\\int_0^x (x^2-y^2)^m e^{y^2}\\,dy ", "aaa15dafbb12573fce9c4d979149527b": "\\frac{1}{2m} {(\\nabla S)}^2 - \\frac{i\\hbar}{2m}{\\nabla}^2 S= E-V", "aaa1809f430687b7b5d7d85ade1add2d": "C^{0,\\beta}(\\overline{\\Omega})", "aaa1a05eb45484b9c520cc1b05c5288d": "ad-bc \\neq 0", "aaa25391b9393552332c1d4e9f369190": "n_{53}", "aaa2548ce0549cd47fcd03b8ed897f6d": "c \\hbar = G m_\\text{P}^2.", "aaa2a5463ca6e235caf06defb0231cfc": "Z^{-4}_4", "aaa2ac98b6969a2ee777a5a29ca4e36b": "\nh(x,t)=H\\exp\\left[i\\left(kx-\\omega t\\right)\\right]\n", "aaa2c224041117e1cb5b6ddc9cc21c2d": "\\mathcal{F}^4 [f] = f", "aaa30814605ba4236e94220c1ed644cb": "f(x)=|x|^p", "aaa31ba6ba655444dfcdaa91224775fe": "{\\sigma_{ab}}^c", "aaa3e14493c3483b20ff005de20a46b1": "v=c\\sqrt{1/2}", "aaa3e2a3336b3cb74332679d6ee2ff94": " I = \\frac{4 \\pi e}{\\hbar}\\int_{-\\infty}^{+\\infty} [f(E_f -eV + \\epsilon) - f(E_f + \\epsilon)] \\rho_s (E_f - eV + \\epsilon) \\rho_T (E_f + \\epsilon)|M|^2 d \\epsilon ", "aaa41c3b48e58efec587b76bea9bbe62": "T^*M", "aaa4291dc03d13bbc4fd84de0b3b9b63": " {A_{v}} = \\begin{matrix} {v_\\mathrm{o} \\over v_\\mathrm{i}} \\end{matrix} \\Big|_{R_{L}=\\infty} ", "aaa42b7701765ff65c393e681239d810": "n_{\\alpha,R} = n_{\\alpha,0} e^{- \\frac{e \\phi_{\\alpha}}{k_B T_{\\alpha}}}.", "aaa43bb525e4946c509881732f1b4365": "{\\rm cov}({\\mathcal K})", "aaa43c3cb6bd15e07abd181bd287c0b3": "\\psi_{\\nu} \\left( \\bold{r} \\right)", "aaa4762b5093cb10fb95c597c0b526b3": "((\\Omega, \\pi),P_i)", "aaa4a2501c9c54df4a622552e67a2ca0": "H=T^{0.25+\\varepsilon}", "aaa51a7eb8ff1cdd218ce1fd3323efc5": "\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}", "aaa54a2f160170858fbe1f4ac3f96900": "a_1, \\dots, a_n", "aaa58810388fdbaca2e5b1aff9b66c2c": "\\frac{\\partial \\rho \\phi }{\\partial t} ", "aaa58dc5cc263c16b9500d4c0e05f291": "y(k(t))", "aaa59208a5739c5879c694f4f5841b7f": "Rg'_* \\circ f'^! \\to f^! \\circ Rg_*.", "aaa599e9a8eb1de5e8956adfd228d11a": "f^{*}=p-q", "aaa5a6b9a8e1154e6eb5122db98d6428": "n=2j", "aaa5c845a04a1df8652241896dab82a9": "\\varepsilon_{13}\\,\\!", "aaa5cf1491345eae114286e4aecf0110": "\\mu(f)", "aaa60ac65eb445a52185850d9e1cec40": " \\left(d\\otimes \\mathrm{id}\\right)\\circ d = \\left(\\mathrm{id}\\otimes d\\right)\\circ d+\\left(\\mathrm{id} \\otimes \\tau\\right)\\circ\\left(d\\otimes \\mathrm{id}\\right)\\circ d ", "aaa615ff92d8600c8f18afeb67a9aadf": "\n\\mathbb{E}[X] = \\bar X\n", "aaa6854c6cf5c8fbb7044cf97ef8c79c": "f \\mapsto {\\hat f}(\\gamma) = \\int_G \\overline{\\gamma(t)} f(t) d \\mu (t)", "aaa691fe1e0a69ceb76dc5952c595948": "V(r) = D_e ( 1-e^{-a(r-r_e)} )^2", "aaa69b21c22192e1f22e57485acea41c": "m^*", "aaa6a4bb9e5df9689cc64305254f440a": "\\hat{y}_d=\\hat\\alpha+\\hat\\beta x_d ,", "aaa6f9dc286b5e7e1b286e8a3f4a8f5f": "\\scriptstyle{(1/3)^s + (1/2)^s + (2/3)^s = 1}", "aaa740791460cfe68b7dbdd3d3a1c130": "FWHM (\\Delta\\tilde{\\nu}_{D}) = \\tilde{\\nu}_{0} \\sqrt{\\frac{8kT\\ln 2}{mc^{2}}} = \\tilde{\\nu}_{0} (7.1623\\mbox{x}10^{-7}) \\sqrt{\\frac{T}{M}} ", "aaa75d66217bc8383b6bbfb7324c3638": "\\left(1-2^{-1/b}\\right)^{1/a}", "aaa7b2a3d55cde5bf88bffc5576b3d9e": "V=\\frac{m_{fullliquid}-m_{fullgas}}{\\rho_{liquid}-\\rho_{gas}}\\,", "aaa7b60f096ce9a99195fdc7b2079e23": "\\begin{align}\nH(X_1,X_2) &= \\int_0^1 - f(x;\\alpha,\\beta) \\ln (f(x;\\alpha',\\beta')) dx \\\\\n&= \\ln \\left(\\Beta(\\alpha',\\beta')\\right)-(\\alpha'-1)\\psi(\\alpha)-(\\beta'-1)\\psi(\\beta)+(\\alpha'+\\beta'-2)\\psi(\\alpha+\\beta).\n\\end{align}", "aaa8008b43be18bc100de028af9c2833": "\\frac{\\mbox{Net debt}}{\\mbox{Equity}}", "aaa866eb905260a644b4437c424e01a6": "p = 69 + 12\\times\\log_2 { \\left(\\frac {f}{440} \\right) }", "aaa87c1011c99e2f42fd87e1d5dc9f89": "\n\\mathbb{E}\\,\\mathbf{X}_k = \\mathbf{0} \\quad \\text{and} \\quad \\mathbf{X}_k^2 \\preceq \\mathbf{A}_k^2 \n", "aaa89d79844a124811535bcdaf3e1fb8": "LEX", "aaa915842b477eff461fc6ed6ac67346": " \\frac{\\pi}{180}M_r\\cos \\phi \\!", "aaaa0b632fc1085d727b13b0dbc97d85": "+z", "aaaa5b0f244ce714a1ad435d802a312d": "\\mathbf{\\tau} = \\frac{\\mathrm{d}\\mathbf{L}}{\\mathrm{d}t} = \\frac{\\mathrm{d}\\mathbf{r}}{\\mathrm{d}t} \\times \\mathbf{p} + \\mathbf{r} \\times \\frac{\\mathrm{d}\\mathbf{p}}{\\mathrm{d}t} = 0 + \\mathbf{r} \\times \\mathbf{F} = \\mathbf{r} \\times \\mathbf{F} ", "aaaa80f239c317ee57086255ef0b2bde": " \\vec{B}\\left(\\vec{r}\\right) ", "aaaaf95b188c25368ba7d04d485b93ed": "V_{\\!-} \\,\\, = \\frac{1}{R_{\\text{f}} + R_{\\text{in}}} \\left( R_\\text{f} V_{\\text{in}} + R_{\\text{in}} V_{\\text{out}} \\right) ", "aaab197dd39bc163b8e89d02c1b8b870": "\\textstyle M", "aaab331c79d319960cd90972757d155a": "A \\oplus B = B\\oplus A = \\bigcup_{a\\in A} B_a", "aaab68ab4f7dbb8c9989139f0d04d461": " W= N_K(T)/T", "aaab877f63a4903c06d47dd5a910584a": "h(v) = \\pm 1", "aaabaa172c68f2dbb6b0a7c6e2d1e77e": "\\eta(\\omega)", "aaabd1b3db947faa2f4b803999f8556e": "\\overline{n}^{\\nwarrow}.\\tau\\mid\\tau_0(n^{\\nwarrow}.\\sigma\\mid\\sigma_0(P)\\circ\nQ) \\rightarrow_{b} P \\circ \\sigma\\mid\\sigma_0\\mid\\tau\\mid\\tau_0(Q)", "aaac1c56b3240215eb22552e592ce392": "\\lambda =1 ", "aaac3503d31e7b6b4050353569133bd2": "A_\\infty", "aaac73966737524aa2cbe3d23131a3cf": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1} \\end{matrix}", "aaad8e985ad5facf45c30518692fb668": " G_m ", "aaada467eab4c9b12177a2f4296eae92": "\\vec\\omega=\\frac{\\vec{r}\\times\\vec{v}}{|{\\vec{r}}|^2}", "aaadcdb6bfdb52e9660545d31c5b9159": "\\frac{N^2_{nl}}{2\\gamma^{l+{3 \\over 2}}} \\cdot \\frac{\\Gamma[\\frac{1}{2}(n+l+1)+1]}{[\\frac{1}{2}(n-l)]!} = 1.", "aaae0f00138584b74095b4c7cb92c9a2": "f \\circ g ", "aaae1724f6a10aa44e3533aa8e7aed69": "(x - a) f'(v_x) = f(x) - f(a)\\,", "aaae9b660352e61257cc1c89d421a8c6": "_k\\!", "aaaed04d567bdba1e6ccc96dbc585b36": " j(\\epsilon) \\mathrm{d} \\epsilon = z_{\\mathrm{S}} f_{\\mathrm{FD}} \\left[ \\int D \\mathrm{d} K_{\\mathrm{p}} \\right] \\mathrm{d} \\epsilon =\nz_{\\mathrm{S}} f_{\\mathrm{FD}} D_{\\mathrm{F}} \\mathrm{exp}(\\epsilon / d_{\\mathrm{F}}) \\left[ \\int_{0}^{\\infty} \\mathrm{exp}(-K_{\\mathrm{p}} / d_{\\mathrm{F}}) \\; \\mathrm{d} K_{\\mathrm{p}} \\right] \\mathrm{d} \\epsilon...........(20) ", "aaaf1912591186cbf439fdc8f9f4297a": "\\{\\hat{A},\\hat{B}\\}=\\hat{A}\\hat{B}+\\hat{B}\\hat{A}.", "aaaf1a9562a0900d5de742e36d7d0667": "x^2 = \\langle x, x\\rangle ", "aaaf3a2fb3edf345788a6049cc91c18c": "\\sqrt{hPkA_c}\\theta_b", "aaaf57a8962ec98de0c64817ea2a4626": "q_\\text{P}", "aaaf62f39baf2ee0a993220bfc14d50c": " \\min_{u\\in\\R^n} c^T u ", "aaaf9e017d728128f112df68f2aeffd5": "a^n = \\underbrace{a\\times a \\times \\cdots \\times a}_n", "aab03efdd230066b7d1fc2f0b7e15260": "(h-r) ( \\tan \\gamma\\,_{n+1} - \\tan \\gamma\\,_n ) = r ( \\sec \\gamma\\,_n + \\sec \\gamma\\,_{n+1} ).", "aab070fc9397e253999803064f20a4ce": "\\Pi^n(f_1,f_2)= \\sum_{k=0}^n (-1)^k {n \\choose k}\n\\left(\n\\frac{\\partial^k }{\\partial p^k}\n\\frac{\\partial^{n-k}}{\\partial q^{n-k}} f_1\n\\right) \\times \\left(\n\\frac{\\partial^{n-k} }{\\partial p^{n-k}}\n\\frac{\\partial^k}{\\partial q^k} f_2\n\\right) ", "aab08d99413db52f8fa7c4511e195c0e": "(6)\\; h_j=\\frac{0.5\\sqrt{1+8(4.5)^2}-(3*0.5)}{2}", "aab0eb6ebf24a0cb69319d10048d768e": "\\operatorname{perm} (A) = (-1)^{m}\\Sigma_{U\\subseteq \\{1,\\dots,m\\}} \\det(A_U).\\det(A_{\\bar U}),", "aab1274dfe58c50515436b5ff6c2ee37": "\\sum_{n=1}^{\\infty}f(n) \\leq \\sum_{n=0}^{\\infty} 2^{n}f(2^{n}) \\leq 2 \\sum_{n=1}^{\\infty}f(n). ", "aab12d2e4274bd4ece32a4fb30283081": "B_{\\nu}(T)", "aab157666966765af84f732ea7c015db": "T_{64} = 2^{64}- 1. \\, ", "aab15aa2fab55413f9061332a16f15f1": "H(P)", "aab1adbea325ee2373b79421c163a01e": " [q + y(a - p - 1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2 - ", "aab1d022ec81a1a047a81e32ca111c19": "P=\\{1,2, \\ldots , m\\}", "aab1fa543133ca2cc9cf562a56afc3cf": " T:L^\\infty(X,\\Sigma,\\mu)\\to \\mathcal L^\\infty(X,\\Sigma,\\mu)", "aab22388c783b39807a928fa2422b223": "x\\in V_{\\alpha}\\setminus\\bigcup_{\\xi<\\alpha}V_{\\xi}\\,", "aab2374072aa95262774930b4f385b0a": "\\frac{(p\\to q)\\to p\\lor r}{((p\\to q)\\to p)\\lor((p\\to q)\\to r)}", "aab267d3954ab3dfb474dfa8bf7de6e1": "|q_i - q_{i+1}| < 2^{-i}\\,", "aab26ec5c8fef2a46ad8cf63ad5173bd": "\\bigwedge^i C^n \\mapsto \\bigwedge^{n-i} C^n.", "aab28130cf15b8b21d838681652d6861": "\\begin{align}\n \\left(\\Delta A_r - \\frac{2 A_r}{r^2}\n - \\frac{2}{r^2\\sin\\theta} \\frac{\\partial \\left(A_\\theta \\sin\\theta\\right)}{\\partial\\theta}\n - \\frac{2}{r^2\\sin\\theta}{\\frac{\\partial A_\\phi}{\\partial \\phi}}\\right) &\\hat{\\boldsymbol r} \\\\\n+ \\left(\\Delta A_\\theta - \\frac{A_\\theta}{r^2\\sin^2\\theta}\n + \\frac{2}{r^2} \\frac{\\partial A_r}{\\partial \\theta}\n - \\frac{2 \\cos\\theta}{r^2\\sin^2\\theta} \\frac{\\partial A_\\phi}{\\partial \\phi}\\right) &\\hat{\\boldsymbol\\theta} \\\\\n+ \\left(\\Delta A_\\phi - \\frac{A_\\phi}{r^2\\sin^2\\theta}\n + \\frac{2}{r^2\\sin\\theta} \\frac{\\partial A_r}{\\partial \\phi}\n + \\frac{2 \\cos\\theta}{r^2\\sin^2\\theta} \\frac{\\partial A_\\theta}{\\partial \\phi}\\right) &\\hat{\\boldsymbol\\phi}\n\\end{align}", "aab2960bf3240774cec6d408bc8500aa": "\\vec R", "aab2b2126bf2d1bd35148f82dd7a659a": "L'=L_{0}/\\gamma.\\qquad \\qquad \\text{(4)}", "aab2ecf42a700bf61f5bb2b8af43d62a": "B_\\delta(x)", "aab30b1e7a4f4def23887a706693b541": "-kA\\frac{\\partial }{\\partial y}{{\\left. \\left( T-{{T}_{s}} \\right) \\right|}_{y=0}}=hA\\left( {{T}_{s}}-{{T}_{\\infty }} \\right)", "aab3238922bcc25a6f606eb525ffdc56": "14", "aab33a788898424779fbb58ba0e915c8": "\\alpha=-1", "aab33dfed2bc68429aff2d0d466d469a": "\n\\mathbf{a} = {d\\mathbf{v}\\over dt}\n", "aab35221b229d5f82242ca268992c238": "\\ell = 4", "aab39778c9d2a6e2c4f81d92c139bd93": "\\mathrm{P}(X \\le x, Y \\le y\\;|\\;Z = z) = \\mathrm{P}(X \\le x\\;|\\;Z = z) \\cdot \\mathrm{P}(Y \\le y\\;|\\;Z = z)", "aab410912fca4ad42da5880f7173debc": "p(\\boldsymbol{x | \\theta})", "aab45a53b0812bfae693c60a1bb73475": "\\cot\\frac{2\\pi}{15}=\\cot 24^\\circ=\\tfrac{1}{2}\\left[\\sqrt2\\sqrt{5-\\sqrt5}+\\sqrt3(\\sqrt5-1)\\right]\\,", "aab479b4ad84267c195a2001a9ecb4bd": " w = 4 \\omega_a \\frac{E_a}{\\left|E\\right|} \\exp\\left[ -\\frac{2}{3}\\frac{E_a}{\\left|E\\right|} \\right]", "aab48d7a08d75fc9c11c81b1ba031fbd": "G_1 G_2\\, ", "aab4cfd101f085556fee0a3eda61ab93": "A=BQ+R,", "aab51540b88c9a60f20ddac5c5e19f66": "+y", "aab5da33475c7312452c0fdc93a0f99b": "1+\\sqrt{3}\\,", "aab5e222f013e8edeb1836fcaabdfbb9": "\n \\mathbf{x}(\\mathbf{X},t) = \\boldsymbol{Q}(t)\\cdot\\mathbf{X} + \\mathbf{c}(t)\n", "aab5ffb5cc39aa62b564a30b1578e521": "\\omega _c", "aab63a00c42b03f343b7aaf8a10ae6dd": "A > 0", "aab644d2d09a575ab48c89f22f593c07": "J^\\alpha = \\begin{pmatrix} c \\rho & J_x & J_y & J_z \\end{pmatrix}", "aab651404c91ccc9669f5fbc3ceed0b6": "E_d = - f'(P) \\cdot \\frac{P}{Q}", "aab661979918c9ed011108c97f7236a1": "K=\\frac{[HA]}{[H^+][A^-]}\\times \\frac{\\gamma_{HA}}{\\gamma_{H^+}\\gamma_{A^-}}", "aab6979359fed51fb0b37b4064ed4bf0": "x = a/2 +/- sqrt((a/2)^2 - b^2c/d)", "aab6e7dae80f1af910a6570797ecf5ed": " \\bar \\eta_{a \\mu \\nu} = \\epsilon_{a \\mu \\nu 4} - \\delta_{a \\mu} \\delta_{\\nu 4} + \\delta_{a \\nu} \\delta_{\\mu 4} ", "aab7257627ec079063ef6e759e57beb2": "a + bi + c \\varepsilon + d i_0", "aab7cf1c6f3e338fe7bbe98785b8fd64": "A_{ij} = \\begin{cases} x_{ij}\\;\\;\\mbox{if}\\;(i,j) \\in E \\mbox{ and } ij\\\\\n0\\;\\;\\;\\;\\mbox{otherwise} \\end{cases}", "aab83343c30eb9a02666fec9534da703": "\\nu \\in \\mathcal{P}_p(X)", "aab84c61a9342b9c96f1635274679e52": "E_N(\\rho) := \\log_2( 2 \\mathcal{N} +1)", "aab8b7cfa76b97b50af6cd6d7dd2a8ab": "\nJ_x = \\frac{\\hbar}{\\sqrt{2}}\n\\begin{pmatrix}\n0&1&0\\\\\n1&0&1\\\\\n0&1&0\n\\end{pmatrix}\n", "aab8cdf4c3a3dc82cac2f60da18ef623": " \\vec w \\cdot \\vec x ", "aab9032a0e8b6c45b69048f0a02e5d14": "x^2+y^2 = l^2 = const", "aab9042e2b3f0075588576cc83748959": "\\mathfrak{P}^{61}", "aab9266f4604da6ac6328ccc44fa17c3": "\\sqrt{\\frac{1}{70}}\\!\\,", "aab9ab8327b97e8541986ecd6424e176": "p_m^0=", "aab9ad3961ad57e8f83a19d9a8ba8329": "P,Q \\not= \\overline{R},\\overline{S}", "aab9b57facfbed3378653734e091b75f": "F(x) = \\sum_{i=1}^M \\gamma_i h_i(x) + \\mbox{const}.", "aab9d78592f2c3e0a3bf741d99d780a8": " { \\left( a - \\lambda_{\\pm} \\right)\\over c } = {c \\over \\left( b - \\lambda_{\\pm} \\right)}.", "aaba326b3f295a8068fd2ea8de22e95d": "x\\in \\Omega, \\xi \\in \\mathbb{R}^n", "aaba7e1b874260a6bbafd0183f4d8e37": "\\mathfrak{f}(L/K)", "aabae18467d8fbb99e15b87be5611fef": "\n\\frac{\\partial^2}{{\\partial t}^2} V =\n\\frac{1}{LC} \\frac{\\partial^2}{{\\partial x}^2} V\n", "aabb0e3993b2db9a1b205bfae3ceeb0c": "(n+1)(n+2)", "aabb0f8e40ea24a2d5fc1227663f1d4f": " 0 \\,\\!", "aabba8f7ef6cb846f261f30c3518a798": "\\alpha^\\prime", "aabbb1d387d15f99f8260d38700aa09f": "\\frac{L}{L_{\\odot}} = \\left(\\frac{M}{M_{\\odot}}\\right)^a", "aabbbd8f9f6c1616db9cfae3350669dd": "\\mu'=\\mu+\\frac{\\beta|c|\\ln(|c|)}{\\pi}", "aabccd7156f1de7cfab343b9541b7a9e": "\\scriptstyle s, t \\in V", "aabccf4b623c1c31cb48d5443b0f8663": "\\bold{\\hat{P}}", "aabce87e755bda494dc1668b79eb274c": " \\hat{p} ", "aabd023a84344e0753e8f7baaf602a7d": "\\frac{\\partial E}{\\partial x}= \\frac{\\rho}{\\epsilon \\epsilon_0}=\\frac{e_0 ((p-p_0)-(n-n_0))}{\\epsilon \\epsilon_0} = \\frac{e_0 (p_1-n_1)}{\\epsilon \\epsilon_0}", "aabd2253355e41e7ed20214e3e36095d": " P_{\\mu} = -i \\partial{}_{\\mu}", "aabd541d5fd4103afbb955b78063478c": "\\scriptstyle K^{+/-} \\rightarrow \\pi^{+/-} + \\pi^0", "aabd7c6de40b8b60407e8e61f87b7497": "(-1)^{2j-m'-m} = (-1)^{m'-m}", "aabd90dd303cb42f54ebfa8aa57b5d02": "p_4(x)=x^2-4/9\\,=(x-2/3)(x+2/3)", "aabdefa8a65e4e2d267035df997f27d9": "\\beta_{L} = \\beta_{U}[1+(1-T)\\phi] \n \\qquad(1)", "aabe388f99806f97ec451f701442b872": "\\scriptstyle y(n)=ay(n-9)+x(n)", "aabe465a02055a1f08f1a7b6d36b3995": " b_{j\\nu} = \\int_\\Omega \\ell_\\nu(y) \\psi_j(y) \\,dy. ", "aabe5e443b06a050a443e18053490f91": "h_n(x;q) = \\sum_{k=0}^n\\frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}x^k", "aabe6802ab55505fdc2b11e13fbb5f6d": "\\in \\mathbb F_{p_j} ", "aabe9eed8a4865ccc1870cae0f897f6f": "\\alpha(\\cdot)", "aabea90392e3e1eb2bf6f450ec2ae511": "m^2 c^4 = E^2 - p^2 c^2", "aabece01f56f79049f4386be15cfa5c7": "\\scriptstyle{b}", "aabf1eeeafba492fa8c7343c13f259a4": " 5", "aabf2f6b74348af2d0e144562f2b5b16": "p^0 =~0", "aac04616acafccf1d5c6c2bede6b8d63": "S_{33} = \\frac{b_3}{a_3} = \\frac{V_3^-}{V_3^+}\\,", "aac0cc7b259e530e31923219428c2435": "\\tau = \\tau(X, X')", "aac0ffe41d4fd8de3952d909e36226b2": "i_3 = a_3 + i_2", "aac10433328db312d8288d3abf6077c6": "\\lim_{N \\rightarrow \\infty} \\sum_{|n| \\leq N} \\hat{f}(n) e^{inx} = f(x)", "aac1494e5a244bdb8829fbe1d7f6a2f5": "\\operatorname{std}(f) = 2\\pi^{-n(n+1)/4}\\left(\\prod_{j=1}^n\\Gamma(j/2)\\right) \\zeta(2)\\zeta(4)\\cdots \\zeta(n-1)", "aac1774a4b812acce7578b2bd152000d": "\\mathcal{N}^{DIFF} (S^n) = \\Omega^{alm}_n", "aac17e718c7901d7eb9e6158ace7b97b": "H=-\\frac{1}{2}\\frac{d^2 }{dx^2} + V(x)", "aac1bc66b8eec0849d244fc9323ddfaf": "d_t=\\frac{\\sqrt{3}a_{C-C}\\sqrt{m^2+mn+n^2}}{\\pi}", "aac1dcbcb0293273be9fdcfab424607e": "2) KDC \\rightarrow : S_{KR_{KDC}}[ID_B||KU_B] ", "aac1e91052e9e428e864a970b13b273d": "A \\rightarrow u_1 u_2 \\dotso u_k, k \\ge 3, u_1 \\in V \\cup \\Sigma", "aac20fa9b580bfae8126d3abebc3eaf5": "L_4(\\xi)=\\left(\\sqrt{\\xi}+(\\xi^2-1)^{1/4}\\right)^4\\left(\\xi+\\sqrt{\\xi^2-1}\\right)^2", "aac264fafde1719b2079f15ca49f2e44": "|\\mathbf x|_\\infty := \\text{sup}_i |x_i|", "aac292036336830a97f3fc511a5f74af": "ab = \\frac{(a + b)^2 - a^2 - b^2}{2}", "aac31cee8b36bc2722bb6df5be08fd5a": "\\forall a\\in A", "aac34b074df42bdb186164e56f1c0a42": "y = h(x).", "aac380e168faebd9f2f1ddbe02c9f3af": "u = Px", "aac4031c0b250aa41be0a77257bfbbdf": "H(x) = H(y)", "aac42dfa17aca9218c97fb519d387524": "Q = m\\int_a^b \\! C \\, dT.", "aac42e38633d5eb400463d214c59ebbe": "|\\mathbb R|^ {|S|} = 2^{\\aleph_0}", "aac452dd74cc546c336bf2f3618b9e37": "\\!P(-z\\le Z\\le z) = 1-\\alpha = 0.95.", "aac45f9a0af3afc1e922e8e57ae89944": "-S_z \\otimes S_x", "aac47385ea9d07c13739f9b011c9403d": " G X= Y ", "aac4b7e2876ba6e229ead4145ec8054a": "\n \\hat{X}(z)=\\hat{X}_{Bayes}\\left( (\\Pi\\Pi^\\top)^{-1}\\Pi\\mathbf{P}_Z\\odot\\pi_z\n\\right)\\,.\n ", "aac4ecb7241063226c5ab59cef5e33c6": "P = \\frac{\\frac{X*R}{K}}{K} + \\frac{D}{K}", "aac4f39eb129fd685fc095a238b8cf87": " F_1 = \\frac{K_\\mathrm{fluid}^{(1)}}{\\phi (K_\\mathrm{mineral}-K_\\mathrm{fluid}^{(1)})}\\ \\ \\ \\ F_2 = \\frac{K_\\mathrm{fluid}^{(2)}}{\\phi (K_\\mathrm{mineral}-K_\\mathrm{fluid}^{(2)})} ", "aac520f18e4bb8edf750b0e030770f6d": "\\Pi_0", "aac556e0c902f28727b5e506950d9ff8": "\\operatorname{tr}(x[y, z]) = \\operatorname{tr}([x, y]z)", "aac56afa8d4ca3577673f07e3ec8c046": "\\left(n-k\\right) S", "aac57568c3b5d5a1b6e508ac8f32950e": "1 \\times \\sqrt{2}", "aac604d98b168eba381eb45a605f028b": " \\tan(\\theta/2) = b/a ", "aac66ee57b3648668078241721483cdd": "k+d \\leq n+1", "aac69a8ffe9930726205b9653731b1e5": "2=( 1 + 1 )", "aac6f041689e1131413fd0f05e2c6910": "\\mathcal{F}_{L^1}", "aac6feae6e32d1b30cb4fbcf6fc92ecb": "\\lambda_D = \\sqrt{\\frac{k_B T }{4 \\pi n e^2 }}", "aac71480a28a260c381111a5bd2cd595": "s_{n+1}=s_n^{d_n}s_{n-1}\\text{ for }n>0", "aac728e315b94e8bb0893297da1c9062": " H= {{\\hbar^2} \\over {2m_0}} [(\\gamma _1+{{5} \\over {2}} \\gamma _2) \\mathbf{k}^2 -2\\gamma_2 (\\mathbf{k} \\cdot \\mathbf{J})^2]", "aac75ae1518ac72cf37268cd573e4177": "MRT = \\left[ \\left(GT+273 \\right)^4 + 2,5 \\cdot 10^8 \\cdot v_a^{0,6}(GT - T_a) \\right]^{1/4} - 273", "aac7603e5247bdf83a49474fd9d8f57b": "\\sgn\\sigma'=\\sgn\\sigma", "aac761ffd3b93157a60a9e0c85219a8f": "d(w)\\geq d(u)", "aac79e37bb92ab4135db6d6b296b855d": "F :\\mathcal{C}\\to\\mathcal{D}", "aac7a1945ef776f2177dc0e8ccd1a578": "\\frac{a}{b+c/n} \\to \\frac{a}{b}", "aac7ba99b8bc6d483f519bc9a5adcab9": "t_{ij} = U_i \\cap U_j \\to G\\,", "aac7f91c30dc31ae3877a3eb0d15e44f": "u_1\\equiv\\,u_2,", "aac8116cc98a12d7c5c99a1e780cd6de": "q=\\pi(m)", "aac82cbd3bea6b11fd33eb2a23a91b41": " \\frac{\\partial c}{\\partial t} = D \\left( \\frac{\\partial^2 c}{\\partial x^2} \\right) ", "aac85d1e4fe1b73d544caf7ce4bdfe82": "(\\tilde{A}+\\tilde{\\delta A}-I)\\mathbf{v}=\\mathbf{0}", "aac87cd7aed21ddbafafe0be7d6c0afc": "G_{poles}", "aac8d3f431b46719ecbf1022dccd1966": "\\frac{1}{U_{pre}}+R_f", "aac8d7ec93a67285e62954a93ea82f4c": "(Bxuz \\and Byvz) \\rightarrow \\exists a\\, (Buay \\and Bvax).", "aac90091b9ae59eae063ce7b9249331c": "t' = t+c", "aac90441796426bc0a03735011519ad2": "{\\chi}^2{\\cdot}M(N-2) + 2{\\chi}M + (N-1)\\chi", "aac92b4611af0c29f2b7a2cb6a088c34": "tf'(r_1) \\equiv -(f(r_1)/p^{k-1})\\,\\bmod{p},", "aac9c74c513a06c5360c275c81ff4638": "P\\cap K", "aac9f4660a503245eff2c458f07a1abc": "8)\\,", "aac9f991a6602c5b84e997ae3ed89049": "\n w = w^K + \\frac{\\mathcal{M}^K}{\\kappa G h} \\,.\n", "aaca69253da32d355c0c819318c45d60": "A = CF", "aaca8baf9bd06f125c136f85799091be": "\\tan(\\delta'-\\alpha) = \\frac{\\sin(\\delta-\\alpha)}{\\gamma\\cdot\\left(\\cos(\\delta-\\alpha)+\\beta\\right)}", "aacaa0e8822ea959aef2775dc2d84947": " e_R^-, \\mu_R^-, \\tau_R^- ", "aacaa8f97e55bf08e46cd2fe7235bbd0": "P_{1,1}", "aacb00e6cbe854641ac937d07f8312a5": "\\ p_x(i) = p(x=i) = \\frac{n_i}{n},\\quad 0 \\le i < L ", "aacb24d36e239fe7b6c476d251a2c961": "(19)\\qquad \\Phi_{01}=\\overline{\\Phi_{10}}=\\,2\\,\\phi_0\\,\\overline{\\phi_1}\\,\\hat{=}\\,0\\,,\\quad\n\\Phi_{02}=\\overline{\\Phi_{20}}=\\,2\\,\\phi_0\\,\\overline{\\phi_2}\\,\\hat{=}\\,0\\,.", "aacb2c52449f401b3ad90702e56661c3": "f(x) = \\delta(x|C) = \\begin{cases}0 & \\text{if } x \\in C\\\\ +\\infty & \\text{else}\\end{cases}", "aacb6af37da11009535718ac7c8e5a62": " r = ( \\lambda x.x\\ x \\to y ) ", "aacbc7bf386f9215ee5f1c99ed05379d": "\\lim_{t\\to 1}\\gamma(t)", "aacbffec5e51970a8b642fca303cb606": "(8 \\pi / 3)r_{\\mathrm{e}}^2", "aacc05c22cccdcde62cd5bb863e784d5": "u\\notin\\Phi(V)", "aacc179f37758ecbdec8b59495a9bd67": " A x_1 = b_1 - B x_2. \\, ", "aacc351c2ec8494b9b3de88cc434a4bb": "\\Delta H\\,\\!", "aacca271d8dfb282c2806cca3ac08725": "L^n(R)", "aacd2c482b86dd39eec7240c74d26229": " pU=A*V*\\dot sU ", "aacdd60627f1b6cb1f67b5b2504993dd": "\\log(z) = \\ln(|z|) + i \\arg(z)", "aacde33bdc658313829c7718dc2bfd42": "\n S(b) = \\sum_{i=1}^n (y_i - x'_ib)^2 = (y-Xb)^T(y-Xb),\n ", "aacdf3af8bc445e5e841f2c4607dad3d": "= \\exp\\left(- \\frac{i}{h}\\ l_z dt\\right) = 1 - \\frac{i}{h} l_z dt + ...", "aace04e4e0429db25ccf5dd3ed64d6b3": "w''=\\pm\\tau", "aaceaae3d46d79b1d3c74063fafaf3e7": "\\sin^2(x)", "aacf12ad37ceb101970ca27078b9338a": "\\scriptstyle dt \\;=\\; 0", "aacf787d5ba874b7bcd55a355cb21128": "\\int_{-\\infty }^{z}v(x,y)g(y\\mid x)dy-G(z\\mid x)W^{A}(z,x)", "aacfcfc79891502b5feae91dab8e910e": "|1 m_1\\rangle", "aacff12dc2bbe2d74d1385a47258be92": "| x_{k+1}-x_{k}| < 1", "aad0cdad9d3f78fdfbf13829f26bd0f7": "\\exp(i \\omega_n \\sqrt{1-\\zeta^2}t)", "aad0f239648e567cd0aa6f902dffb2b1": "0.136 \\le \\Delta t/T \\le 0.238", "aad0f7a6142bb4bed17d77312471c5fa": "(a^2+b^2)(p^2+q^2) = (ap+bq)^2 + (aq-bp)^2\\,", "aad0fa23112189cc69d4cfc32b50df70": " \\delta_{int}(s)=s'=(\\ldots,(s_i', t_{ei}'), \\ldots) ", "aad15d4119f323974d4bf0d164258f4a": "(58\\cdot 21\\cdot 29)^2\\equiv 2^1\\cdot7^1\\pmod{91}", "aad18c0a88969b4c1bdc3711475796c2": "-\\infty", "aad19a9f323af49c08ab31c5923b6773": "\\{\\neg P(f(c)), \\forall x . P(x)\\}", "aad1ba1e9f4445122a984006bfd7bb76": "F_t = {s \\choose 1}{45 \\choose 1} - n_{42}", "aad1bf7c99534d1d04dfb5d746a7b268": "\\not\\leftrightarrow", "aad1d28637dca17d1f7a6048539f6a17": "\n(3) \\qquad -\\dot{\\lambda}'(t)=H_x(x^*(t),u^*(t),\\lambda(t),t)=\\lambda'(t)f_x(x^*(t),u^*(t))+L_x(x^*(t),u^*(t))\n", "aad1dd6c5fce4c5fb5f14d7b4c4d840d": "r_{k}^{A}", "aad200e0d0431634bf404c5092b72490": "\\log (r).", "aad201d8edc2ed596c565cf21c6caedf": " ((12)(34)) (1 \\lor 2) \\to 1 \\lor 2 ", "aad3003c9e07fef4d6ef8ca703802d90": "E(v_2~|~v_2 < v_1)P(v_2 < v_1) + 0", "aad35b9cb1dd26b0010aecf291278172": "\\frac{1}{c}\\frac{\\partial}{\\partial t}I_\\nu + \\hat{\\Omega} \\cdot \\nabla I_\\nu + (k_{\\nu, s}+k_{\\nu, a}) I_\\nu = j_\\nu + \\frac{1}{4\\pi c}k_{\\nu, s} \\int_\\Omega I_\\nu d\\Omega", "aad37a1533fa2991ea1299f90a05dce9": "w_i = x_i \\cdot \\frac {M_i}{\\sum_i x_i M_i}", "aad38dc2da3508e61e40c454e82cda3a": " u(0,x) = 0, \\quad \\frac{\\partial u}{\\partial t}(0,x) = f(x), \\quad \\mathrm{for} \\quad f \\in \\mathcal{S}'(\\mathbb{R}^n).", "aad4147e035d35240c8904f08fa03dee": "\\Phi({\\infty}) = 1= 100\\%", "aad446a8d8da5fce92d662dcd1952666": "\\mathbb Z", "aad47bfc9ca2c05b561f18d0a4acb838": "P\\quad=\\quad\\begin{matrix}1&1&2&2\\\\2&3\\\\3\\end{matrix}, \\qquad Q_0\\quad=\\quad\\begin{matrix}1&2&3&7\\\\4&5\\\\6\\end{matrix},", "aad5a5214bd620d49408f21970d0a053": "\\mathrm{0.1\\overline{6}}", "aad5aaec473f592eaa7fbfe1f67d72d6": "Slope= -\\frac{\\Delta H}{R}", "aad610d3a2a964cd4189ae23347e157f": " r^3+r^2+r-1>0 \\,", "aad6204c20fdc17fc7e528a8892ddb0c": "\n \\begin{align}\n R_0 = \\cfrac{H}{G} & \\implies\n (1+R_0)\\cfrac{1}{(\\sigma_3^y)^2} - (1+R_0)\\cfrac{1}{(\\sigma_2^y)^2} = (1-R_0)\\cfrac{1}{(\\sigma_1^y)^2} \\\\\n R_{90} = \\cfrac{H}{F} & \\implies\n (1+R_{90})\\cfrac{1}{(\\sigma_3^y)^2} - (1-R_{90})\\cfrac{1}{(\\sigma_2^y)^2} = (1+R_{90})\\cfrac{1}{(\\sigma_1^y)^2} \n \\end{align}\n ", "aad64d7361c3edffb37bac536f0aac1c": " \\prod_{m=1}^n \\left(1-\\frac{z^2}{((m-\\frac{1}{2})\\pi)^2}\\right)", "aad6dfdbcb4181423e7b2d24242f4ad8": " \\text{A} = \\lambda^+, \\text{B} = \\lambda^- ", "aad6f9b78807efeeb982d27545f9aa3c": "\n\\begin{align}\n E_{\\text{MP2}} &=\n \\frac{1}{4} \\sum_{i, j, a, b}\n \\frac{\\langle\\varphi_i \\varphi_j | \\hat{\\tilde{v}} | \\varphi_a \\varphi_b\\rangle\n \\langle\\varphi_a \\varphi_b | \\hat{\\tilde{v}} | \\varphi_i \\varphi_j\\rangle}\n {\\varepsilon_i + \\varepsilon_j - \\varepsilon_a - \\varepsilon_b} \\\\\n &= \\sum_{i < j, a < b}\n \\frac{2 \\langle\\varphi_i \\varphi_j | \\hat{v} | \\varphi_a \\varphi_b\\rangle\n \\left(\\langle\\varphi_a \\varphi_b | \\hat{v} | \\varphi_i \\varphi_j\\rangle\n - \\langle\\varphi_a \\varphi_b | \\hat{v} | \\varphi_j \\varphi_i\\rangle\\right)}\n {\\varepsilon_i + \\varepsilon_j - \\varepsilon_a - \\varepsilon_b},\n\\end{align}\n", "aad72cdda5ea01d87e8816cf7f1993ea": "\\frac{\\mbox{Market Price per Share}}{\\mbox{Present Value of Cash Flow per Share}}", "aad75dc707210bf71e31a154e374bf14": "1-n/p+n/q\\not = 0", "aad76c1d956a7d626cf90fa96559aff2": "|b|= \\frac {a}{2}|<111>|= \\frac{\\sqrt 3a}{2}", "aad76fc42a532fa93512e6d900e6fe51": "f^\\dagger=f^{-1}", "aad7787f39ec01944586af7b73e0e3c9": "M \\subset S", "aad79ec5124eaddc405183876c7504e3": "\\left|f_n(z)-z \\right|0\\,", "aaec95931238993b4b8d5da87b25b4e3": "\\Pr(R \\cap B \\mid \\Sigma) = \\Pr(R \\mid \\Sigma)\\Pr(B \\mid \\Sigma)\\ a.s.", "aaed0a31110784817b3fa0bcd53dcd28": " (p_i,p_j)", "aaed53f14c0e355dd036bdedf7d1160d": "T_{a+b} = T_a + T_b + ab\\ ", "aaed5ca56ef77279fe2fda4346ebbf54": "(W, p)", "aaed7858cecb1680c7d8c80316dc7767": "j = 3/2", "aaed8f47f70f2592616530f972190858": "b \\neq 1", "aaeda483ac9368a7e888d3042db223aa": "\\cos a \\,\\sin^2 c = \\sin a \\,\\cos c \\,\\sin c \\,\\cos B + \\sin b \\, \\sin c \\, \\cos A", "aaee392c2bfd5fc15de836b5bca8bd1b": "div\\left (\\frac{v_{t}}{\\sigma_{k}}\\nabla(R_{ij})\\right )", "aaee6988bf56a0159d52f2d8a5553965": "\\tilde{X}", "aaee71e611344e15b2c712992ccf5db0": "disc(\\mathcal{H}) \\leq 2t - 3", "aaeecd221c18247ff80da53e49f6ddc0": "\\overline{\\overline{z}} = z \\!\\ ", "aaeed370b29ad04d85e26e9b8e9d92f9": "-i \\hbar \\gamma^\\mu \\partial_\\mu \\psi + m c \\psi = 0 \\,.", "aaef02f9c7ad7ed117714d2e3aa4ce71": "F(\\xi) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^\\infty f(x) e^{-i \\xi x}\\, dx. \\,", "aaefb93a75005dccfc32bd5f6cc5e348": "\n\\begin{align}\n\\sin(kx) & = P(D)y \\\\[8pt]\n& = P(D)(c_1y_1+c_2y_2+c_3y_3+c_4y_4) \\\\[8pt]\n& =c_1P(D)y_1+c_2P(D)y_2+c_3P(D)y_3+c_4P(D)y_4 \\\\[8pt]\n& =0+0+c_3(-k^2-4ik+5)y_3+c_4(-k^2+4ik+5)y_4 \\\\[8pt]\n& =c_3(-k^2-4ik+5)(\\cos(kx)+i\\sin(kx))\n+c_4(-k^2+4ik+5)(\\cos(kx)-i\\sin(kx))\n\\end{align}\n", "aaf010d1c574e5abf65d644d8abe05ea": "-\\infty \\leq a \\leq b \\leq \\infty ", "aaf071ea579b54c9b0bc139fa59d82bd": "\\mathcal{M}( \\phi (p) + ... \\ \\rightarrow ...) ", "aaf0ab670dad98b10f4e74ec9e9b4fb2": " \\ C_l ", "aaf0c97168987453ed01181936621ee9": "\\Re(s)>L", "aaf0dd0f206ee902daef4c3ee3104659": "q + (D - \\lambda I)^{-1} w(w^{T}q) = 0", "aaf0e656c8593eedaa4f0f6d38723cf3": " \\mathrm{adj}(\\mathbf{A}) = q(\\mathbf{A}) = -(p_1 \\mathbf{I} + p_2 \\mathbf{A} + p_3 \\mathbf{A}^2 + \\cdots + p_{n} \\mathbf{A}^{n-1}), ", "aaf0f67edfc36a26132055f116956ef2": "(n+\\ell)!", "aaf152c605b7c7764f1f2b3d1d71dc4a": "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}=x_n-\\frac{x_n^2-S}{2x_n}=\\frac{1}{2}\\left(x_n+\\frac{S}{x_n}\\right)", "aaf168980bc9ef2e2906c74764bf0584": "\\neg (\\neg (\\neg A\\lor B)\\lor (C\\lor (D\\lor E)))\\lor (\\neg (\\neg C\\lor A)\\lor (E\\lor (D\\lor A)))", "aaf16c3aed48f36b41020f7df9707e98": "Ae^{\\mbox{ } j \\omega t}", "aaf1a476c121e1bff20719793cd2bf3c": "x^2 + n \\cdot y^2", "aaf1cc067d1c7597965ef0a2199e47d1": "\\sum_{i=1}^\\infty {1\\over p_i} < \\infty", "aaf322a4a94ef99cb993667bf097072c": "R(M,x) = \\frac{\\sum _{i=1} ^n \\alpha _i ^2 \\lambda _i}{\\sum _{i=1} ^n \\alpha _i ^2} = \\sum_{i=1}^n \\lambda_i \\frac{(x'v_i)^2}{ (x'x)( v_i' v_i)}\n", "aaf342028159357355a2159f72cd161d": "\n\\overset{4\\times 2 \\text{ matrix}}{\\begin{bmatrix}\n\\color{BrickRed} a_{11} & \\color{BrickRed} a_{12} \\\\\n\\cdot & \\cdot \\\\\n\\color{BurntOrange} a_{31} & \\color{BurntOrange} a_{32} \\\\\n\\cdot & \\cdot \\\\\n\\end{bmatrix}}\n\n\\overset{2\\times 3\\text{ matrix}}{\\begin{bmatrix}\n\\cdot & \\color{RedViolet}b_{12} & \\color{Violet}b_{13} \\\\\n\\cdot & \\color{RedViolet}b_{22} & \\color{Violet}b_{23} \\\\\n\\end{bmatrix}}\n\n= \\overset{4\\times 3\\text{ matrix}}{\\begin{bmatrix}\n\\cdot & x_{12} & \\cdot \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\cdot & \\cdot & x_{33} \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{bmatrix}} \n", "aaf34a6acb75d70a0cbc1e0d3d413653": "\\mathfrak m_A", "aaf357c0b599e07174349773982e31e5": "H_n \\sim \\ln{n}+\\gamma+\\frac{1}{2n}-\\sum_{k=1}^\\infty \\frac{B_{2k}}{2k n^{2k}}=\\ln{n}+\\gamma+\\frac{1}{2n}-\\frac{1}{12n^2}+\\frac{1}{120n^4}-\\cdots,", "aaf35f56b9679f44eb5360cd8147c3df": "\\psi_R = \\langle R|\\psi\\rangle = \\frac{1}{\\sqrt{2}}(\\cos\\theta\\exp(i\\alpha_x) - i\\sin\\theta\\exp(i\\alpha_y))", "aaf364ec4e5e1963f28c7efbb1159de3": "\\scriptstyle \\pi^{\\!*}F", "aaf373afab69d753ef0d02bfd224e4c2": "\\begin{pmatrix} 1 & 0 \\\\ \\frac{n_1-n_2}{R \\cdot n_2} & \\frac{n_1}{n_2} \\end{pmatrix} ", "aaf378212e94fe71f4bfcfb4a9db127d": "A=\\phi+\\mathbf{A}\\,,", "aaf3790d966a9585f48d9e7dd834eb48": "N = \\sum_{i=1}^c K_i", "aaf38bf8b34bcc0122bf1d5bfe4d947a": "\\exp(z):= 1+z+\\frac{z^2}{2\\cdot 1}+\\frac{z^3}{3\\cdot 2\\cdot 1}+\\cdots = \\sum_{n=0}^{\\infty} \\frac{z^n}{n!}. \\,", "aaf3aa84569c974a5fbbb799e70c533f": "\\overline{R}\\,", "aaf3b373e2fb7352d302d7d79c90afcf": "\\rho_2", "aaf3b8cdef4c8aa48634b20d6e877174": "\\tilde{c}", "aaf3b8d09e5aa3bd968ce7d1a5c8486b": "Q(s)|_{s = \\alpha_n} = 0", "aaf3dd48d90e83abe1b7b0fb55633ef5": "\\sigma=5.7\\times10^{-8}", "aaf4018e2a40d4a1677f1847bfdc265f": "t \\not\\equiv 1 \\pmod p ", "aaf484c42cc489167b4ea03344bc2340": "\\Psi _{beta}(t|\\alpha ,\\beta ) =\\frac{(-1)^{N}}{B(\\alpha ,\\beta ) \\cdot T^{\\alpha +\\beta -1}} \\sum_{n=0}^{N}sgn(2n-N)\\cdot \\frac{\\Gamma (\\alpha )}{\\Gamma (\\alpha -(N-n))}(t-a)^{\\alpha -1-(N-n)} \\cdot \\frac{\\Gamma (\\beta )}{\\Gamma (\\beta -n)}(b-t)^{\\beta -1-n}.", "aaf4b6206322a9b0ac2f619fe4509b95": "\\xi=x", "aaf4eb5cbef94a63f067623920e28aaf": "\\tfrac{\\omega}{2\\pi}", "aaf4ee770b5e64e97468720501729ea7": "\n\\sqrt{N} [\\hat{g}_N- g(x)] \\xrightarrow{\\mathcal{D}} Y_x\n", "aaf4fd3fea25ceac00abdf0f21bd96fe": "S:=\\operatorname{Autoreduce}(S)", "aaf525f1d5ee0982789965ee95ccb0bb": "\\scriptstyle \\pi(x)\\ln x/x", "aaf5d1a593ab06fc4e562365a683b695": "\\overline\\theta^\\pm", "aaf6079b72b0191b25cc8a89a3def849": "y=(1 + x)^r", "aaf660a4077a4d1a1dd2f7a35d02a5d8": "(1 - p)^{k}\\,p\\!", "aaf66321b04a26a0ee78ed2bc0c58c79": "\n\\rho_{x}=\\sum_{y}p_{Y|X}\\left( y|x\\right) \\left\\vert y_{x}\\right\\rangle\n\\left\\langle y_{x}\\right\\vert .\n", "aaf6870f52de05661e89955d6a7c4cb2": "(X,Y) \\mapsto \\nabla_{X} Y", "aaf6ecf893fb0dd7282e89f47c2df07b": " ( J f ) ( x ) = \\int_0^x f(t) \\; dt ", "aaf75b0ae8a2d3429dde75614d84b012": "x \\not\\in A^c", "aaf764cc3bed7753bca1be15b475e4e0": "y = a + b\\cdot x", "aaf77de040bfd14d8dd70fd0ee8505fc": "B:Y\\to X", "aaf78bcc96509604849ca35dd33b0dae": "0y) ", "ab00519509b1cf114e46925be2ef7bde": "1 \\mbox{ gaul} = 1 \\mbox{ atkinson} \\times \\frac{10^6 \\times \\left( \\frac {metres}{feet} \\right )^8}{\\frac{kilograms}{pounds}\\times g} \\equiv 1 \\times \\frac{10^6 \\times 0.3048^8}{0.4536\\times 9.80665} \\equiv 16.747 \\mbox{ atkinsons}", "ab00a38f69dc0d3c75b524e1a851b1e5": "u \\in Y", "ab00ad1aefd5228b400a8fd608235a7b": "\\mathbf{x}(t)=\\mathbf{\\Phi}(t, \\tau)\\mathbf{x}(\\tau)", "ab00dd2045a73863cf7cc69ff3daea9b": "\\omega_1 + \\omega_2", "ab00e53b7b25ed999eb21eec164e6761": "J^a", "ab012273ad34ba81126baa1baa0f1c21": " \\Pi, A \\vdash B", "ab0124cb74b8bf8cd61d7bf2932389e5": "= \\frac{1}{1 + RC \\left( \\frac{2}{T} \\frac{z-1}{z+1}\\right)} \\ ", "ab013425f02e3278accc6299d035a888": " \\mathcal{D}(\\mathit{k}) = \\int_{0}^{\\eta_0} \\dot{\\tau}e^{-[\\mathit{k}/{\\mathit{k}_\\mathit{D}(\\eta)}]^2}\\; d\\eta. ", "ab019310c4a2446ab0679fc7fa5598f4": "\\tau (\\gamma, T)", "ab01b5d758c49e26e7d3a23124bd313f": " f^\\downarrow( q_l, q_r ) = f( q_l ) ", "ab01eb31e11251272e26efa86188edf1": "\\Gamma_\\phi", "ab01ee48d301adb447b39ea379fb3712": " \\beta = \\tanh\\phi = { e^{\\phi} - e^{-\\phi} \\over e^{\\phi} + e^{-\\phi} } .", "ab02574263c12ea9ee1e66ea22c71834": "d_1+\\ldots+d_m=n", "ab0299f19c4de14f67662dc23e6fba0f": "\\log(r)\\sin\\left(\\frac{1}{2}\\theta\\right)=z\\cos\\left(\\frac{1}{2}\\theta\\right).", "ab02fe067a63d6804f7361d11a452e61": " g \\cdot E = U_g E U_g^*. \\quad", "ab0326e4fd6e33648f90466596f40028": "A= \\{a_10 ", "ab1333fe139f61f5df16786fe3f9f0d5": "\\rho = \\sum_{i =1} ^n p_i | i \\rangle \\langle i |", "ab1349736618e8a7c10f93e5cb2a7c33": "h=\\sqrt{\\frac{3}{4}-(\\!{\\color{Blue}R}-\\frac{1}{2}\\cot\\!\\left(\\frac{\\pi}{n}\\right)\\!)^{2}} - \\sqrt{{1}-(\\!{\\color{Blue}R}-\\frac{1}{2}\\csc\\!\\left(\\frac{\\pi}{n}\\right)\\!)^{2}}", "ab1382a6ff12af86944a3ebdf327197a": "V_n = -V_n,", "ab13f04200614d087810eaf5344f7c34": "\\gimel\\colon\\kappa\\mapsto\\kappa^{\\mathrm{cf}(\\kappa)}", "ab13fa46d332e027bd30bbb09c89625e": "1 - 5\\left(\\frac{1}{2}\\right)^3 + 9\\left(\\frac{1\\times3}{2\\times4}\\right)^3 - 13\\left(\\frac{1\\times3\\times5}{2\\times4\\times6}\\right)^3 + \\cdots = \\frac{2}{\\pi}", "ab1404ac5293559faf4bee8ba91cefd5": " \\vec{F}_{12} = \\frac {\\mu_0 I_1 I_2} {4 \\pi} \\int_{L_1} \\int_{L_2} \\frac {(dx_1,0,0)\\ \\mathbf{ \\times} \\ \\left[ (dx_2,0,0) \\ \\mathbf{ \\times } \\ (x_1-x_2,D,0) \\right ] } {|(x_1-x_2)^2+D^2|^{3/2}}", "ab14103eaa4306ab32ecd9368f279f39": "\\scriptstyle{ \\delta ' = S_h }", "ab14383c929f4d06b5f6ab0a78981c75": "k \\approx a\\Phi^b(T_{2lm})^c", "ab150d97e3d73cfe144115741d051fd5": "\\, h_2 - h_3 = {1 \\over {2}}( V_{r3}^2 - V_{r2}^2) + {1\\over {2}}( U_2^2 - U_3^2)", "ab151ec09e11883720a014a381729f92": "\\frac{1}{\\phi}(x_2-x_1)=x_4-x_1=x_2-x_3=\\phi(x_2-x_4)=\\phi(x_3-x_1)=\\phi^2(x_4-x_3)", "ab1534d8a667f675d1fa05a513cdc521": "2 f_m\\,", "ab155b02121570a920973d038ab4c8f9": "g_{\\rm centrifugal} = -\\left(\\frac{2\\pi}{T}\\right)^2 r \\sin\\theta ", "ab158301648a3e1be10a18b5fb0ac50a": "\\frac{f(y)-f(x)}{y-x} \\approx f'(x)", "ab169c40a368fb92f8c100acdfaede98": "\\beta_F = \\frac{\\alpha_{F}}{1 - \\alpha_{F}}\\iff \\alpha_{F} = \\frac{\\beta_F}{\\beta_F+1}", "ab16b9da166ec348132aa80d24214cec": "S=\\{a,b\\}, a^2 = b^s =(ab)^3 = 1, ", "ab16bff7bd420f1edc2b9f366eaa10b1": "\n \\omega_{c} = c \\sqrt{\\left(\\frac{n \\pi}{a}\\right)^2 + \\left(\\frac{m \\pi}{b}\\right)^2}\n", "ab16e67d95ec06cdae51ae6be9d4bec3": "(a + b i)^2 = (c + d i)^3 + 7", "ab17075bcf07c5a79f173fa7945011c8": "{\\rm APSPACE}=\\bigcup_{k>0}{\\rm ASPACE}(n^k)", "ab1751a84ab23180346c91a8231b6388": " |G : H| ", "ab17661923ab91abf1da743c9a8621b6": "p^*_n(x)", "ab1794cd63ef61404a90512c0e4e4312": "D_{k} = \\left(1 - \\alpha - \\beta - \\gamma\\right)^k", "ab17eaeb4489ef863243a97ef9ee46d0": "\\gamma = 1 + \\frac{1}{1- \\alpha}.", "ab18444ca5700e75c5cd51b731b9722b": "\\arccos\\left(\\frac{3q}{2p}\\sqrt{\\frac{-3}{p}}\\right)", "ab18b534e67b87a6b57ad8f142591390": "k_\\mathrm{spec}", "ab18dffdd47b44d0147faeed67c918f3": "{\\rm det}(-\\lambda I+A_0+A_1e^{-\\tau_1\\lambda}+\\dotsb+A_me^{-\\tau_m\\lambda})=0", "ab190c846f135e2c4a993c7d623423b7": " h_2 ", "ab19341302a7f6971ced514f5a00aa03": "A\\,\\triangle\\,B = (A \\smallsetminus B) \\cup (B \\smallsetminus A),\\,", "ab193ad6b21e9150af50f6c7d5af5703": "x \\mapsto x^p", "ab193f568b34f5281a7d8da0d5a35cb1": "\\,f(x) = x^2 - 612", "ab194629973d7bb8c863a889fed1dcdc": "\\left( 1-f^e_{\\mathbf{k}}-f^h_{\\mathbf{k}}\\right)", "ab196860800b0b3275658fb3c0f9e38e": "\\log(1-e^{-x})", "ab19f0805cc921cdf2f5a662bafff68d": " p > p_c ", "ab19f7f3c9a4d60dd88ca1c7b6234d56": " s = v t - \\frac{1}{2} a t^2", "ab1a230c0da5cfc8e260f29515b2610e": "\n\\begin{align}\nF(A)& = F\\left(\\sum_{k_1 = 1}^n a_{k_1}^1 E^{k_1}, \\dots, \\sum_{k_n = 1}^n a_{k_n}^n E^{k_n}\\right)\\\\\n& = \\sum_{k_1, \\dots, k_n = 1}^n \\left(\\prod_{i = 1}^n a_{k_i}^i\\right) F\\left(E^{k_1}, \\dots, E^{k_n}\\right).\n\\end{align}\n", "ab1a6863add6ab1ebb27ca917f51e91d": "\\{\\dots, -2\\omega, -\\omega, 0, \\omega, 2\\omega, \\dots\\}.", "ab1ad4944eb9d003a9e901267c92bb1d": "\\mathbf{\\bar{n}}", "ab1ad8db8e91bc533de5c009b1b5119e": "\\hat{\\bold{H}}_{\\operatorname{PI}}", "ab1ae7002c4bf8852ae1a53846fc328c": "a \\rightarrow \\infty", "ab1aef721296b13d96f4e29ebebdf55a": " \\frac{\\partial^2 (x p(x) - C(x))}{\\partial^2 x}={\\partial^2\\pi(x,t)\\over \\partial x^2},", "ab1b21bad1911212125eb9ae45bef70a": "\n\\ln y(r_{12})=\\rho \\int \\left[h(r_{13}) - \\ln g(r_{13}) - \\frac{u(r_{13})}{k_{B}T}\\right] [g(r_{23})-1] \\, d \\mathbf{r_{3}}. \\, ", "ab1b2df0b4963824b066fe0e795305c4": "\\scriptstyle y \\mapsto z", "ab1b4d8fe70bdbe08ff1136aa3896856": " c > 0 \\ ", "ab1ba7fe80f5ec66c93f50346aad1aca": "S_v(f) = \\frac{A^2}{2}\\frac{cf_0^2}{c^2 \\pi^2 f_0^4 + (f - f_0)^2},", "ab1bbff9a126b4b0fcd144f3a2ad5ada": "ABC \\doublebarwedge abc", "ab1bdf7297874665dc659ccdef8dd4e1": "\\langle f_1, \\ldots, f_n \\rangle", "ab1c5c5eab645800d64bc751c3ae291d": "F_2^{\\ast}", "ab1cab235f295eb3676fa305e601adb2": "g(t) = 1", "ab1cda0cea1702ebd68efd6672ddeb38": "\\begin{smallmatrix}F(\\alpha,m)\\,\\!\\end{smallmatrix}", "ab1ce57e2700080a5bcef72d448fa682": "(i{\\partial\\!\\!\\!\\big /} - m) \\psi = 0\\,", "ab1d50ffa1d01142dd736cf8721b644d": "\\mathbf{x} = \\mathbf{U} \\mathbf{D} \\mathbf{V}^T", "ab1d5c8316cde488ed0b8a0d89db4eda": "{\\Sigma }_{def}\\propto {\\left \\langle v\\right \\rangle}^{-4}", "ab1d61aceda524418ba11d1961b21324": " \\frac{\\gamma(s,x)}{x^s} \\rightarrow \\frac 1 s", "ab1d7c39636bce1bbb867bd02ac92d7d": "\\sum_{n \\in \\mathbb{N}}", "ab1d885e94f2ce9984417e327ff5ce64": "w \\in\\mathbb{R}_+^E: \\forall S \\subseteq E, \\sum_{e\\in S}w(e)\\ge r(S)", "ab1dfa7e3c72f6ef1771a0d5c5c0cb72": "W=S \\cdot \\frac{1}{\\sqrt{2}}", "ab1e3f03b0e97bc492e70af06a5c4c64": "\\displaystyle h^2>0.", "ab1e3f8773705c2ee9ee88dbf0215a4b": " d\\mathbf{M} \\ , ", "ab1e55041e32d8ceb27a7a4bf0bb95dc": "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1 \\rangle ", "ab1e82c0a5707763b162dd2c552ada87": "\\frac{1}{\\Gamma(z)} = \\frac{i}{2\\pi} \\oint_C (-t)^{-z} e^{-t} \\, dt,", "ab1efd83ff0428586016cb4678af35e4": "(\\dim C_k-1)", "ab1f48f45a47ffd37b900dafd05cf238": "S \\, = \\, S_{configurational} \\, + \\, S_{vibrational}", "ab1f67a19ae7d013fb9216433953f2e1": "p_{{\\mathrm{NH}}_3}", "ab1f73c8462b672de79218ab6d7e3ba6": " \\mathcal{A} \\setminus (\\{A \\} \\cup \\Gamma(A)) ", "ab1f877c8096b65a50418ffcbeaeb99a": "f(z)=\\frac{1}{(z-2)(z-5)}", "ab1fe42ed90e19841084936cf6c2a725": "D_{E}/H_{E}\\,,", "ab2008ef7ee706e65398ba8f2fbf88c8": "dS_w", "ab205df155b708a819acc2a411259eb1": "H_o = \\frac{\\sum\\limits_{i=1}^{n}{(1\\ \\textrm{if}\\ a_{i1} \\neq a_{i2})}}{n}", "ab206a3e9b0d571ec214519776b8ca6a": "\n\\begin{align}\n {52 \\choose 5} &= \\frac{n!}{k!(n-k)!} = \\frac{52!}{5!(52-5)!} = \\frac{52!}{5!47!} \\\\\n&= \\tfrac{80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000}{120\\times258,623,241,511,168,180,642,964,355,153,611,979,969,197,632,389,120,000,000,000} \\\\\n&= 2,598,960.\n\\end{align}", "ab20e29003f0b3ed3a9b3bcf714ee0bb": "T,V,P", "ab215296914a87293621922155633770": "m \\to \\infty", "ab217947a18792261f4ba565987cacbc": "\n\\frac{S}{kN} = \\ln\\left[\\frac{V}{N\\Lambda^3}\\right]+\\frac{5}{2}.\n", "ab21b071692257684d1351fa5da95111": "a_0 a_1 a_2", "ab21bbd0eb4c1e6473b8d30eb134d639": "A B B^{-1} A^{-1}", "ab21bdc86fd2296e770ee405587017c5": "u(x(\\theta),t(\\theta),\\theta) \\geq u(x(\\hat\\theta),t(\\hat\\theta),\\theta) \\ \\forall \\theta,\\hat\\theta \\in \\Theta", "ab21c2f08812ebbd3cc4d87dbba42fb9": "P_{\\rm k}=\\eta_0\\frac{R^2}{Q\\beta^3}~", "ab21d32ba600c3c8484b901a6400f180": "g_{\\phi}=9.780327 \\left(1+0.0053024\\sin^2 \\phi-0.0000058\\sin^2 2\\phi \\right) \\frac{\\mathrm{m}}{\\mathrm{s}^2} ", "ab21dcbe6b5dc490060c53d60c8d98ca": "f_2=11x^6", "ab22613f901894f41d51265f9ae49db4": "\\textit{mother}(\\textit{lassie})", "ab22a67d26e1eede05852d10920ab7ee": "\n\\frac{\\ddot{P}}{P}+\\frac{1}{\\rho}\\,\\frac{\\dot{P}}{P}+\\frac{1}{\\rho^2}\\,\\frac{\\ddot{\\Phi}}{\\Phi}+\\frac{\\ddot{Z}}{Z}=0\n", "ab22ff892a9eff1d8d156ce5285b9e38": "\n\\int_{t_\\mathrm{now}}^{t_\\mathrm{now}+\\lambda_\\mathrm{now}/c} \\frac{dt}{a}\\; =\n\\int_{t_\\mathrm{then}}^{t_\\mathrm{then}+\\lambda_\\mathrm{then}/c} \\frac{dt}{a}\\,.\n", "ab236c095e7f505a9f2b38b1815958e2": "D = -\\frac{2\\pi c}{\\lambda^2} \\frac{d^2 \\beta}{d\\omega^2} = \\frac{2\\pi c}{v_g^2 \\lambda^2} \\frac{dv_g}{d\\omega}", "ab236f5d2a6fdb7703bb16d720aaebf7": "B_1^{p,q} = (\\mbox{im } d_0^{p,q-1} : F^p C^{p+q-1} \\rightarrow C^{p+q}) \\cap F^p C^{p+q}", "ab239f7b5657a525d4f0b4d48526a081": "\n\\begin{align}\n&-\\int\\limits_{0}^{2\\pi}\\ 2\\ r_h \\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\ du\\ =\\ \n-6\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}\\ \\left(1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u\\right)^2\\ \\cos u\\ \\sin^2 u\\ du\\ =\\ \\\\\n&-12\\ \\sin^2 i\\ e_g \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u \\cos^2 u\\ du\\ =\\ -2\\pi \\ \\frac{3}{2}\\ \\sin^2 i\\ e_g\n\\end{align}\n", "ab23a2022c785b06202b6e3289d8fa75": " \\text{Cov}(X(s), X(t)) = G(s, t) ", "ab242c6d2d88eda434bbe8bc56067ed7": "\\ i_\\mathrm{M} = j \\omega C_\\mathrm{M} v_\\mathrm{GS} = j \\omega C_\\mathrm{M} v_\\mathrm{G}", "ab244622703198f43f01c6113283aaef": "=\\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} + \\frac{1}{2} + \\cdots", "ab244cd0d40d01f9a4e6a082ae1e6431": " \\tfrac{1}{2}(l^2+m^2-n^2),\\quad \\tfrac{1}{2}(l^2-m^2+n^2),\\quad \\tfrac{1}{2}(-l^2+m^2+n^2). ", "ab2476fa308b250d80d3f6c6e04c14f7": "\\sum d_i", "ab248e9c6332b8fad8aece426662ee2a": " \\hat H\\psi=\\lambda\\psi, \\qquad\n \\hat H=-D\\partial_x^2+U(x),\n", "ab249c783df2c8a022374c90db10ba6e": "Q = v A \\cos\\theta ", "ab24e5929432e6e69ece303a758a0da3": " \\operatorname{Tr}_W\\left(\\sum_{k,\\ell} T_{k \\ell} \\, \\otimes \\, | k \\rangle \\langle \\ell |\\right) = \\sum_j T_{j j} .", "ab24f4f7abe3facd7d0bb9ea791f80c2": "\\Gamma(\\tfrac14) \\approx 3.6256099082219083119", "ab252cb8a17d2525adc915c6934f6f7a": "d\\rho^2 = g(r)^2 dr^2 + r^2 d\\phi^2, \\;\\; r_1 < r < r_2, \\, -\\pi < \\phi < \\pi ", "ab252dc2ed9919550fe6d92d1ffbf431": " \\Delta^{(k)} \\, w_{ij} = \\Delta^{(k-1)} \\, w_{ij} \\left ( \\frac{\\nabla_{ij} \\, E^{(k)}}{\\nabla_{ij} \\, E^{(k-1)} - \\nabla_{ij} \\, E^{(k)}} \\right) ", "ab25b50e3193f9710b94bedde15e0352": "e^i=(-1)^{i-1}(e_1 \\wedge \\cdots \\wedge \\check{e}_i \\wedge \\cdots \\wedge e_n) \\epsilon^{-1},", "ab25f3b91f995055a9178b9e661a22f2": "f(x) = x^3 - 9x", "ab264f925979ea24d63162ecfb526985": "{\\mbox{chord} \\,\\sin A \\over \\sin C} = x\\text{ or }{\\mbox{chord} \\,\\sin B \\over \\sin C} = x.\\!", "ab26569fa0075c156065699ef2a8c102": "N_U", "ab267250e11b25c7070650fa97d042fc": "\\begin{align}T(x,y) &= 0 + 0(x-0) + 1(y-0) + \\frac{1}{2}\\Big[ 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \\Big] + \\cdots \\\\\n&= y + xy - \\frac{y^2}{2} + \\cdots. \\end{align}\n", "ab26f0923f59816df3462778be75932c": "BU_n\\cong Gr_n(\\mathbb{C}^{\\infty})", "ab26fd3a183f19f405916050606d1b3f": "\\lambda_q = \\frac{dq}{dl}", "ab2702c5e6b4b4b2f81a5221ff79f35b": "R_2(x) = f(x)-P_2(x) = h_2(x)(x-a)^2 \\ ", "ab27223ad2421541b86fbb987f1c99ea": "\n\\max_G \\text{ trace }\\left(G^TG\\right).\n", "ab27845f9b1e15c85ae1a31e4c214576": "0\\to \\oplus_{w\\in W,\\,\\, \\ell(w)=n} M_{w\\cdot \\lambda}\\to \\cdots \\to \\oplus_{w\\in W,\\,\\, \\ell(w)=2} M_{w\\cdot \\lambda}\\to \\oplus_{w\\in W,\\,\\, \\ell(w)=1} M_{w\\cdot \\lambda}\\to M_\\lambda\\to V_\\lambda\\to 0", "ab27ed80ead6d0e714508fae727947a6": "\\begin{align}\\omega(u,v) &= g(Ju,v)\\\\ g(u,v) &= \\omega(u,Jv)\\end{align}", "ab2817665abd6c097e18c16be7d7e37d": "\\gamma : [0, T] \\to M", "ab2833679bd684eb2d891807fcef09b0": "\\! d\\psi", "ab28349b939549e6794ef2e163546905": "{\\rm d}A", "ab2843da33d2602332cae69efea3f4a6": "PRF * Duty Cycle", "ab2857b41f7cb0135685663c13592034": "\\Delta M < 0", "ab285f91ca89c64fad107857aa10ca37": "F_{U,n}(t)=\\frac{1}{n}\\sum_{i=1}^n \\mathbf{1}_{U_i\\leq t},\\quad t\\in [0,1].", "ab288ca2ad232d411afc5d8edf74c5e4": "n'\\geq \\left\\lceil N(k-1,d/2)\\right\\rceil", "ab28916dfaadba6cc21b52d0d9e4a775": "E_{T}^2 = m^2 + (\\overrightarrow{p}_{T})^2", "ab28c13288f61ec82571478c7db54da9": "s_\\lambda=0", "ab2937fec0d9f8084254e46dc9cb411b": "\\begin{align}\n \\vec{f}_0 &= \\frac{1}{x}\\cosh(t) \\, \\partial_t - \\sinh(t) \\, \\partial_x\\\\\n \\vec{f}_1 &= -\\frac{1}{x}\\sinh(t) \\, \\partial_t + \\cosh(t) \\, \\partial_x\\\\\n \\vec{f}_2 &= \\partial_y, \\; \\vec{f}_3 = \\partial_z\n\\end{align}", "ab294cc518b59c7a4ead9f9f2b8415f1": "\\sum_{k=0}^\\infty \\left| \\Pr(S_n=k) - {\\lambda_n^k e^{-\\lambda_n} \\over k!} \\right| < 2 \\sum_{i=1}^n p_i^2. ", "ab2950ed436cd32763d0624931b60cbc": "V_0 \\equiv V_{a,0} = V_{b,0} = V_{c,0}", "ab29624b77212436ae7c0a6b1e56b4bc": "\\displaystyle \\hat{f}(\\nu)\\,", "ab296a0499a2bb05b2db57700a9f1c07": " \\sin \\theta = \\frac{v}{v_s}\\,\\!", "ab29f8f4cdb6e66745fbb7518006abd5": "\\int_0^r \\!\\sqrt{1+z'(x)^2}\\,dx \\, = a", "ab2a09c66a84a77d822700d496136b24": "V = \\frac{1}{4!}\\epsilon_{\\mu\\nu\\alpha\\beta}\\gamma^\\mu\\gamma^\\nu\\gamma^\\alpha\\gamma^\\beta.", "ab2a568b145d2cf2d85cc499bd9f1d92": "W_t = W e^{-\\delta g_t}", "ab2aafdd1c4f81bb0b074adb4407ef1f": "-z\\log z-\\frac{z}{2}\\log 2\\pi +\\frac{z}{2} +\\frac{z^2}{2}- \\frac{z^2 \\gamma}{2}- z\\log\\Gamma(z) +\\log G(1+z)", "ab2ad0d4f0c587f20e68946cc91bfcd5": "\\Rightarrow 1 \\le \\frac{I(2n)}{I(2n+1)} \\le \\frac{I(2n-1)}{I(2n+1)}=\\frac{2n+1}{2n}", "ab2ada0948870844f84b5bb3a6d182f8": "\\textstyle(x\\mp1, y\\mp1, z\\mp1)", "ab2ba33b29ec0f96235ab21885da8022": "P=(x^2+a)(x^2-a).", "ab2bccf69c04484f5e7e6cdc7873ff55": "dU=\\delta Q-\\delta W\\, ", "ab2c2741cf6557cd44a10c8c43055088": " \\ \\ p\\leq n-1, \\ b_{2p-1}\\leq b_{2p+1}", "ab2c6015fd0e1a0949ff9caf25dcbc81": "I_0 = \\frac{5\\sqrt{3}}{16}a^4", "ab2cb2c75fd1a5f3edd9856e0515d6d7": "Q_k,", "ab2cbfaebc76e1c02ca35953bc48c369": "\\left(a \\rightarrow P\\right) \\sqcap \\left(b \\rightarrow Q\\right)", "ab2ccb613617658548bab2652064bf93": "f(z) = \\sum_{n=0}^\\infty c_n (z-a)^n, ", "ab2d4459a72dc721cf3a9438bde151ee": "\\textstyle K_m", "ab2d53539e4749647af88947cbecc0dc": " m=y_1-l\\cdot x_1 ", "ab2d8d73cf95b6bc5417d746f071740a": "n_{\\rm air}0", "ab4c7d1e3e0701e910ee002f37733c31": "Q=\\frac{\\omega_0}{\\Delta\\omega}", "ab4cda9472dc3fcec85a378f80c99371": "f:R\\to\\mathbb{D}", "ab4d13bc96724b361ff20d09fc61aa47": "\\sum_{k=-a}^a(-1)^k{a+b\\choose a+k} {b+c\\choose b+k}{c+a\\choose c+k} = \\frac{(a+b+c)!}{a!\\,b!\\,c!}\\,,", "ab4d69b8f9d55278d59dca956d30c7a1": "(2) \\,", "ab4d6fd75289f6605ccafd33902ce83a": "\nz = a \\ \\cosh \\mu \\ \\cos \\nu\n", "ab4d7f64dc78c6ac2a23562e24463330": "(k^2-1) \\frac {\\mathrm{d}} {\\mathrm{d}k} \\left[ k \\;\\frac {\\mathrm{d}E(k)} {\\mathrm{d}k} \\right] = k E(k)", "ab4d96367c03802cf6f8535ca48ce163": "(Z,(\\mathcal{O}_X/J)|_Z)", "ab4d9c2e003ee2cfbeda7be6b4dd4e61": "\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p\\ ", "ab4dd58892d99302deba5398eef06dc3": "\\textbf{Q} = w\\textbf{q},\n\\textbf{Q}^0 = w\\textbf{q}^0 + w^0\\textbf{q}", "ab4e0b45271f44f1b0f0b02b511e8a8b": "\\frac{d}{dt} \\det A(t) = (\\det A(t)) \\, \\mathrm{tr} \\left(A(t)^{-1} \\, \\frac{d}{dt} A(t)\\right).", "ab4e20d0fa9738b10fcb8855aa237b05": "\\frac{{\\rm d}P}{{\\rm d}Q} ", "ab4e3a698a467d2881e6f0f0327c7a3c": "\\exists x. \\bigwedge_{i=1}^n L_i", "ab4e8a9881e38ce3a94351d0d6144b53": "w_1 = x_1", "ab4eb656d1e6cf9febc140783ed6a1fd": "\\frac{c}{b}=\\frac{m^2+n^2}{2mn}, \\ \\,\\frac{a}{b}=\\frac{m^2-n^2}{2mn}.", "ab4ed32a0ebcefb46bbd672ef972dab8": "n \\times (m-n)", "ab4f0587722825632b994686020b1b27": "A=l^2+l\\sqrt{l^2+(2h)^2}", "ab4f1963a6dd39d55a1614f4ad814a29": " \\Psi(x,B) \\sim \\frac{1}{\\pi(B)!} \\prod_{p\\le B}\\frac{\\log x}{\\log p}. ", "ab4f6cf0229eee0e6358b7646f3cd112": "\\begin{array}{lll} \n p_n(x,y) = p_0 \\sqrt{1-r^2/a^2} & \n r = \\sqrt{x^2+y^2}\\le a & \n a = \\sqrt{R\\delta_n} \\\\\n p_0 = \\frac{2}{\\pi} E^* \\left(\\delta_n/R\\right)^{1/2} & \n F_n = \\frac{4}{3} E^* R^{1/2} \\delta_n^{3/2} & \n E^* = E/2(1-\\nu^2)\n \\end{array}\n", "ab4f6de48133de4d602862603fd8c8d8": "v_2(t) = \\{m_1(\\hat{x}) \\operatorname{sgn}( e_1 )\\}_{\\text{eq}}", "ab4fa5513032097c175edb1b53c66255": "\\delta g_F", "ab4fc7097880a47a3c8db15b20f8ff3d": "[X,Y]", "ab4ff4c8a73b376dcda79c01e07ee906": "\\Delta y_1^n", "ab4ff6c4cb810395fcaf80ade2e5e59e": " x^2 - 2y^2=1 ", "ab4ffa55f688360e0c12aef543c18351": "T_0", "ab502c093b1ebbd64296fa89c924d0bc": "g(z, u, v)", "ab503feaea977322a6a43453e1a6f81b": "\\zeta(3) = 1 + \\frac{1}{2^3} + \\frac{1}{3^3} + \\frac{1}{4^3} + \\cdots", "ab51124924c0013f22d375cf05d052ff": " D^m ", "ab5165d2b2e021245d8a6a4e95aba6f0": "\\mu-s \\le x \\le \\mu+s", "ab5184eacd96010be07bfa310a74a0c8": " mg ", "ab51de2dc8e050a74f72e3031a8ca408": "\\mathcal{L}_{V^{1}}(\\theta^{\\alpha}) \\,", "ab52a11250db21a5994543ba2aa36d86": "\\frac{\\partial S_d}{\\partial q_1} = 0 = \\frac{\\partial}{\\partial q_1} L_d\\left(t_0, t_1, q_0, q_1 \\right) + \\frac{\\partial}{\\partial q_1} L_d\\left( t_1, t_2, q_1, q_2 \\right)", "ab52c60c031c7363a92c4570ae8e4cfd": "\\left({{1}\\over{4}}\\right)^r = O\\left({{1}\\over{N}}\\right)", "ab52f37a5da438803bcb0314de9ee166": "S(A,P,z) = \\left\\vert A \\setminus \\bigcup_{p \\in P(z)} A_p \\right\\vert . ", "ab53095a98c5a7159226509cf1bbeb37": "\\text{Loss} = 20 \\log_{10} \\left(\\sqrt{\\rho - 1} + \\sqrt{\\rho }\\right) \\quad \\text{where} \\quad \n\\rho = \\frac {Z_1}{Z_2} \\quad\nZ_m = \\sqrt{ Z_1 Z_2} \\text{ } \\, ", "ab5349822271b54905928d4aac8f7a9c": "\\mathbf F_B=\\int_V\\mathbf b\\,dm = \\int_V\\mathbf b\\rho\\,dV ", "ab5358cfc0375bdceffc8b06554f7760": "\\displaystyle w(3,2)", "ab5382f5fc717d0ce64a22b0e3f100b9": "\\scriptstyle{1/r}", "ab541a21f2ce05fab33f4c9e29d07d02": "\\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} | D_N(t) | \\, dt \\ge \\int_0^\\pi \\frac{\\left |\\sin\\left ( (N+\\tfrac{1}{2})t \\right )\\right|} t \\, dt \\to \\infty.", "ab547504d26ecee5c04c4968f900b452": "\\begin{align}P_{s:m}(T)&=a\\exp\\bigg(\\left(b-\\frac{T}{d}\\right)\\left(\\frac{T}{c+T}\\right)\\bigg);\\\\[8pt]\n\\gamma_m(T,R\\!H)&=\\ln\\Bigg(\\frac{R\\!H}{100}\\exp\n\\bigg(\\left(b-\\frac{T}{d}\\right)\\left(\\frac{T}{c+T}\\right)\\bigg)\n\\Bigg);\\\\\nT_{dp}&= \\frac{c\\gamma_m(T,R\\!H)}{b-\\gamma_m(T,R\\!H)};\\end{align}", "ab548adb10d956588ed3c4d1eab7a702": "X(f) = FT(x(t)) \\, ", "ab549f5a0bc7a3e9aab69f40c0aeed89": " \\langle d \\rangle", "ab54d21f6652d789a9a61d2b92c3ff1d": "\\bar{r}:\\bar{K}\\to X", "ab550fe78e78b5da102fd415b0643c87": "m_i \\geq 0, integer, i = 1, 2, ..., N_{sd}", "ab55a656419bb29f97f94f92a3a3b683": "C_{out} = (A \\cdot B) + (C_{in} \\cdot (A \\oplus B))", "ab55cd333391df7a1a22c714c72013bd": "{I} = {P}\\cdot U_a^\\frac{3}{2}", "ab55e62e01963100e2b1b41a930ef160": "f_2(\\cdot)", "ab5600e22f742cf94e47176d5df92d8b": "\\mathbf{F} = F_r\\hat{\\mathbf{r}} + F_{\\theta}\\hat{\\theta} + F_z\\hat{\\mathbf{z}}", "ab566d0441f54a64282651b76a20109c": "U=\\{ z \\in \\mathbf{H} : \\Im z > 1 \\} ", "ab56d7b3a14af31183bcade44f7279d6": "\\theta_c = \\theta_i = \\arcsin \\left( \\frac{n_2}{n_1} \\right), ", "ab572fcf6d36dd044ea129db16241f68": "\n\\begin{bmatrix}\nII & IB & IE \\\\\nBI & BB & BE \\\\\nEI & EB & EE\n\\end{bmatrix}\n", "ab575763e78a021f31cf930ab18d67a5": "\\scriptstyle<3.7\\times10^{-33}", "ab57a99c1d707005c43b353e5f3998b7": "G_{S_2}, G_{S_3}, ..., G_{S_L}", "ab57f1e535c329b31258f2e6a290d0c4": "f(a)=c", "ab5819063d2c7824ca2b71787630d4bc": "\\{0\\}\\hookrightarrow[0,1]", "ab58681d6befec03b44521b5d391d87e": "s_1, s_2, s_3, ..., s_k", "ab58b4ba5b0a6a1c2e6ec9a2e036d0b0": "rK+wL = C\\,", "ab58b4bbb4ba5834bab42e3bd33849e1": " V_{\\,alveolar\\,dead\\,space}= 24\\ ml.", "ab5911e5ba54198bdfd78fdfef4c27cd": " \\frac{\\sigma_t}{\\rho} ", "ab59ba051bf8ee4b6ead660c95e83602": "x^2 = n", "ab5a440a6b4dc09698bc2500b3ca68f8": "\\begin{align}\n B_{n} &= (-1)^{\\left\\lfloor \\frac{n}{2}\\right\\rfloor }\\left[ n\\ \\operatorname{even}\\right] \\frac{n! }{2^n - 4^n}\\, S_{n}\\ , \\quad (n= 2, 3, \\ldots) \\\\\n E_{n} &= (-1)^{\\left\\lfloor \\frac{n}{2}\\right\\rfloor }\\left[ n\\ \\operatorname{even}\\right] n! \\, S_{n+1} \\quad\\qquad (n = 0, 1, \\ldots)\n\\end{align}", "ab5a7ff367f4715ba865de25c50a83a7": "\\dot{q}^i", "ab5a82c3a010b3e74a1ac31bb26c367f": "\\lim_{t\\to\\infty} P(t) = K.\\,", "ab5ab85d3d13f42b7ca56a2e8cb47d72": "R_n = R_o + K(S - S_E)\\,", "ab5ae19fa7d7de85dfb97e735a851470": "g(x+y)-g(x) = \\int_x^{x+y} f(x')\\, dx' = \\int_0^y f(x+x')\\, dx' = f(x)g(y).", "ab5ae937f2bef4f20c1bc2eb9d48c92c": "Z_3 = (Z_1+C)^2-ZZ_1-F = 2", "ab5b8dfe5852cc3b99c592e1b6ef3b2f": "\\frac{\\partial \\langle H \\rangle}{\\partial a_{n'}^{*}}\n= \\sum_{n} a_n \\langle n'|H|n \\rangle\n= \\langle n'|H|\\psi\\rangle\n", "ab5bb554fc5fc49748bb04a3eebf51e0": "\\varphi:E^{\\times\nn} \\to M", "ab5bbdd7f05f07be4e22722e124bc300": " h_n = g_{n_1} g_{n_2} \\cdots g_{n_t} ", "ab5bde96ceee06a0ecbde9204363dd9a": "T^a \\square^b F^a ", "ab5be6803a222bbfff658d360f2a405a": "\\begin{align}\n A &= 63365028312971999585426220 \\\\\n &\\quad + 28337702140800842046825600\\sqrt{5} \\\\\n &\\quad + 384\\sqrt{5} (10891728551171178200467436212395209160385656017 \\\\\n &\\qquad + 4870929086578810225077338534541688721351255040\\sqrt{5})^{1/2} \\\\\n B &= 7849910453496627210289749000 \\\\\n &\\quad + 3510586678260932028965606400\\sqrt{5} \\\\\n &\\quad + 2515968\\sqrt{3110}(6260208323789001636993322654444020882161 \\\\\n &\\qquad + 2799650273060444296577206890718825190235\\sqrt{5})^{1/2} \\\\\n C &= -214772995063512240 \\\\\n &\\quad - 96049403338648032\\sqrt{5} \\\\\n &\\quad - 1296\\sqrt{5}(10985234579463550323713318473 \\\\\n &\\qquad + 4912746253692362754607395912\\sqrt{5})^{1/2}\n\\end{align}", "ab5c4755283d703a3e94531f234aff0a": "{\\int}_{\\alpha\\oplus \\beta} \\mathbf{F} d(\\alpha\\oplus \\beta)=\n{\\int}_{\\alpha}\\mathbf{F} d\\alpha\n+{\\int}_{\\beta} \\mathbf{F} d\\beta\n", "ab5c766abea2e38e83bc62fd6229a53e": " \\frac{n_{\\rm C_2H_6}}{n_{\\rm O_2}} = \\tfrac{1}{1} = 1", "ab5cd1d8c1c17e90e421d7c32b5a88fa": "\\sum_{p^k} f(p)\\;", "ab5cf738f0bd3ee41086e55f471549f6": "(1+X)^{p^i}\\equiv1+X^{p^i}\\text{ mod }p.", "ab5d86139d2d1d0afce467a056ee7e15": " \\nabla \\cdot \\nabla G = -\\delta(x-x',y-y',z-z') \\qquad \\hbox{in } V,", "ab5df459de6e0d5312e680a8af2a5434": "\\textstyle \\lambda \\times n", "ab5e47c3fedebde7e44ecabe4ce9b85b": "\\mathbf{J}_s=(1/T)\\mathbf{J}_u+(-\\mu/T)\\mathbf{J}_\\rho=\\sum_\\beta \\mathbf{J}_\\alpha f_\\alpha", "ab5e6cb3a913a1c9f32ee2da6379e745": "(p^0, w^0)", "ab5e7699ca9f6b07c89ef0dd373c1abc": "(x-3)(x-2)x", "ab5ebd08a0a9009c6081cb117a87e3d0": "\\frac{36}{25}", "ab5f1296b3aa2b550b467502e4f2cd4e": "\\displaystyle Wg(2,d) = \\frac{-1}{d(d^2-1)}", "ab5f9a4d12238b60465611175281ec97": "(\\Omega, \\mathcal F, \\mathbb P)", "ab5fa742d83a5283e9d3eb0c4f7453d8": "\\tau^\\alpha", "ab5fc12589921e7eb35b382d22b397d8": "x_{l+1},\\dots,x_{l+u}", "ab5fd0374c76a850bfc499a58d8ee291": "\\langle T_i: i \\in d \\rangle", "ab60371048d1654aa171a6f992390e76": "p^{i}_a = \\partial L / \\partial (\\partial_i y^{a})", "ab6056f4b479783567d46f8d97074499": " (x_m, y_m) \\,", "ab609731144ebc1c0cb0fbbb1d97cea7": " \\displaystyle{U(s)=e^{iPs},\\qquad V(t)=e^{iQt}.}", "ab60db09b15b1c1bf8d36ce4cf29b59f": "d'(\\mathbf{x}) = \\mathbf{x} - \\frac{\\alpha}{|\\mathbf{x}|^2}\\mathbf{x}", "ab60dbf1d922d7db3fccda2a7b826024": "\\tilde{\\mathbf{x}}_{1,2,3,4}", "ab60e290e828467e7656c68f8c43c85f": "\\lambda = \\delta t_{\\mathrm{MPC}} \\sqrt{kT/m}", "ab60f55c8551a2f707fb2ffefeccf093": "J(\\mathbf X,t)=\\det \\mathbf F(\\mathbf X,t)\\neq 0\\,\\!", "ab6134685063c54a10d80e0c6ecff34c": "(X^*,\\le)", "ab616be1fe8c9ccae20fc82ee7595124": "I_K(t)=\\bar{g}_K n(V_m)^4(V_m-E_K).", "ab61a819569bd16109fdb5c357cb55e2": "\\textbf{x}", "ab61b92dfd37350e52b7649cff91c032": "H(A)", "ab61f0a0dacb9454b02f17c896cc40cc": "\n \\boldsymbol{M} = \\frac{\\partial W}{\\partial \\boldsymbol{E}^e} = J\\,\\frac{dU}{dJ} + 2\\mu\\,\\text{dev}(\\boldsymbol{E}^e) \n ", "ab61f847355e839c48e2839915119a5e": "(+\\alpha, +\\alpha)", "ab62562a32c96150b10fc0c9656f884c": "\\sigma_x\n= \\frac{\\partial^2\\Phi_{yy}}{\\partial z \\partial z}\n+ \\frac{\\partial^2\\Phi_{zz}}{\\partial y \\partial y}\n-2\\frac{\\partial^2\\Phi_{yz}}{\\partial y \\partial z}", "ab629e7571667247e0fe868b865fc978": " \\lim_{z \\rightarrow a^+} \\int_z^b f(x) \\, dx", "ab631be95897db6972d915f280eebecc": "M_\\mathrm{w}", "ab631d020b8379fe555325cb2e04ee96": "F \\subset \\overline{NE(X)}", "ab63737ef319a7a7c3d578e23cc4c7f2": "m( \\sum_i r_i \\otimes_S t_i) = \\sum_i r_it_i", "ab637ec3236df826a7f50855e18b5099": "\\sin(x) = \\frac{e^{ix} - e^{-ix}}{2i} ", "ab638b3715cc6c3fd4c286148d33f455": "3/\\tilde{4}", "ab63dc02cb5117ad03dbf78b552b570b": "\n [\\boldsymbol{\\nabla}f(\\mathbf{x})]\\cdot\\mathbf{c} = \\cfrac{\\rm{d}}{\\rm{d}\\alpha} f(\\mathbf{x}+\\alpha\\mathbf{c})\\biggr|_{\\alpha=0}\n ", "ab63e77442297ccaa1aeb1a3a4a4ca0a": "1\\cdot(2i)^6 + 3\\cdot(2i)^4 + 3\\cdot(2i)^0 = -64 + 3\\cdot 16 + 3 = -13", "ab63fa42497cb3c740c320e44b98d0d2": " \\frac{d v^2}{dt} = \\frac{d (\\mathbf{v} \\cdot \\mathbf{v})}{dt} = \\frac{d \\mathbf{v}}{dt} \\cdot \\mathbf{v} + \\mathbf{v} \\cdot \\frac{d \\mathbf{v}}{dt} = 2 \\frac{d \\mathbf{v}}{dt} \\cdot \\mathbf{v} = 2 \\mathbf{a} \\cdot \\mathbf{v}", "ab63fb57875c1be7f29db56ab6422f00": " d h = h_x dx + h_y dy = \\exp(-p) h_x \\, \\sigma^1 + \\exp(-p) h_y \\, \\sigma^2.", "ab63fe1155ce2a84ea6a15301cf81b20": "(1:1:0:0)", "ab64117a59a993956ea17e1eaf7b2059": "\\vec x^N = \\vec x", "ab64383b0dcf51a9b833f077becdb774": "U = \\frac{1}{2}kx^2.", "ab645b22fe1700e0672de147de3f1d7d": "(\\tfrac{a}{b}) = -1.", "ab647000f18bf3585962b72dbf67f193": "\\tau_{yx}", "ab64879b18d340f2ad246676009c34df": "\\mu_{\\Lambda}(d\\lambda) = \\int_{\\phi \\in \\Omega_1} \\mu_{\\Phi,\\Lambda}(d\\phi, d\\lambda)\\ .", "ab6495c93e54648c3e90f8bc84767d79": "S(\\omega = \\omega_k)", "ab64eebf6e86299a2b4b816d51b9e745": "\\phi_{n}=x\\psi_{n}^{2} - \\psi_{n+1}\\psi_{n-1},", "ab653d7886729b97bd87bfd6b2f67d01": "f(\\alpha \\cdot x) = \\alpha^k \\cdot f(x)", "ab65555e6e783588285a3e8ceddb85c0": "V_0 = \\sqrt{\\frac{\\mu}{p}}", "ab658810de55e601a62ba7655d253997": "\\omega = f\\,dx", "ab660628518f3610d8267dce5fedfcef": "disc(\\mathcal{H},\\chi) := \\max_{E \\in \\mathcal{E}} |\\chi(E)|,", "ab66f7333517332b6789f2687cda43b7": "a_{V} A", "ab674198262a5210cf651e7e12819a12": "\nTE = HE \\oplus VE\n", "ab6742fa124a8f3a71879fc5807bc13d": "f(1,1) = p(1,1) = \\textstyle \\sum_{i=0}^3 \\sum_{j=0}^3 a_{ij}", "ab676fb08439b21e8d0c67825e947f80": "\n \\begin{align}\n \\varepsilon_{11} & = \\tfrac{1}{E}\\left[ \\sigma_{11} - \\nu(\\sigma_{22}+\\sigma_{33}) \\right] \\\\\n \\varepsilon_{22} & = \\tfrac{1}{E}\\left[\\sigma_{22} - \\nu(\\sigma_{11}+\\sigma_{33}) \\right] \\\\\n \\varepsilon_{33} & = \\tfrac{1}{E}\\left[\\sigma_{33} - \\nu(\\sigma_{11}+\\sigma_{22}) \\right] \\\\\n \\varepsilon_{12} & = \\tfrac{1}{2G}~\\sigma_{12} ~;~~\n \\varepsilon_{13} = \\tfrac{1}{2G}~\\sigma_{13} ~;~~\n \\varepsilon_{23} = \\tfrac{1}{2G}~\\sigma_{23}\n \\end{align}\n", "ab679fc46ce849d967a546d103ce2cf3": "12\\# = p_{\\pi(12)}\\# = p_5\\# = 2310.", "ab67a5414dc3b58140ebf0459c0d5b66": "\\{0,1\\}^S", "ab67b60053efd6b46ba98cce7ad5bba6": "i_i(U_g)=eZ_i n_i S_F\\int\\limits_{\\sqrt{2eZ_i U_g /M_i}}^{\\infty} f(v)vdv", "ab67dae76d8a3194ec4bbff4aa42994a": "\n\\begin{align}\n\\int \\frac{\\delta F}{\\delta\\rho(x)} \\ \\phi(x) \\ dx \n&= \\lim_{\\varepsilon\\to 0}\\frac{F[\\rho+\\varepsilon \\phi]-F[\\rho]}{\\varepsilon} \\\\\n&= \\left [ \\frac{d}{d\\epsilon}F[\\rho+\\epsilon \\phi]\\right ]_{\\epsilon=0},\n\\end{align}\n", "ab67e6bcc783962b8238c39ef387cee8": "\\pm 1/ \\sqrt 2=\\pm 0.70...", "ab67f1dadb2d4a80eb7743d484256c59": "\\nabla^2 B_x = {\\partial \\over \\partial x}(\\nabla \\cdot \\mathbf{B}) = 0.", "ab682b48989b6d42427d309a3cb0e4a5": "\\ \\Delta^t(\\alpha_{i,j,k}) = \\alpha_{i,j,k+1} - \\alpha_{i,j,k} ", "ab6844b7925885a5816c03dab3b62e93": "P(r|s) = \\prod_{} P(n_{ij} | s) ", "ab6892fddbb4b2ddf4425500c0654ff2": "\\epsilon = \\mu E", "ab68bcd269012e52d383c9add372a7c7": "\n \\varphi(x) = -\\frac{P}{2EI}\\,(L^2-x^2) \\,.\n ", "ab68ca95be9dc4e7877a797a16cd487f": "0=\\dot m h_1 - \\dot m h_2.", "ab68ea7eaf90f100923586be8e9ce496": " \n\\begin{align}\nf(a) & = f(x_r) + \\frac{t}{h}\\times\\text{sphuta-bhogya-khanda}\\\\\n& = f(x_r) + \\frac{t}{h} \\frac{D_r + D_{r+1}}{2} + \\frac{t^2}{h^2}\\frac{D_{r+1} - D_r}{2}.\n\\end{align}\n", "ab6931caf67427a9e7613ead9b132f68": "\\psi_n,\\chi_n", "ab6937eeebd8ad9299a9e4d3955fbc7a": "l^\\infty", "ab693812129e8b580102d8d5e7e7ac70": "\\sigma (\\Omega) = \\bar{\\Omega}", "ab6965d2555c6ca106f4a4f48e71d3e1": "OA\\equiv O(A/\\top)", "ab69a11c7f91c6cb90837da2abbf3813": " N_t^t = (1+n)^t ", "ab69ec63ca31bc40de600c16c0e98476": "\nf(t) = A e^{j \\omega t }, \n", "ab6a092c6f5d275a5920a3e626f2e981": "\\mathbf{L}_{\\nu}(z)", "ab6a871368d0065a5c8516cd52b6fb4c": "S_q^{BE}", "ab6aa75836d36a2d60e51d4cf551df75": "h \\in \\mathbb{N}", "ab6aef4fdd8e5832cf2c7bd998fa568e": "\\cos t = \\cos(2\\pi k+t) \\,\\!", "ab6af3ee594c32d8ccbcbdefa7b8d155": "O((\\ln p)^6).\\,", "ab6afabaf8a3c8ad97dc8f2cd61f4ede": "\\alpha_{A^-}+\\alpha_{HA}=1", "ab6b7156e2a68e10a52b50543d15779c": "\\begin{bmatrix}\nf_0 & f_1 & f_2 & f_3 & \\ldots &f_{\\omega - 1} \\\\\ns_0 & 0 & 0 & 0 & \\ldots & 0\\\\\n0 & s_1 & 0 & 0 & \\ldots & 0\\\\\n0 & 0 & s_2 & 0 & \\ldots & 0\\\\\n0 & 0 & 0 & \\ddots & \\ldots & 0\\\\\n0 & 0 & 0 & \\ldots & s_{\\omega - 2} & 0\n\\end{bmatrix}", "ab6bcdf0a22fad05cd2ac70cf92306cf": "\\boldsymbol{\\Pi}^0_1", "ab6c2d9ab9229916fff5e3daaf5caaa2": "\\| x \\|_{Y} \\leq C \\| x \\|_{X}", "ab6c55e9225c19db1a6cd1478c0a3e78": "M(j,j) = 1 - \\sum_{i=1, i\\neq j}^{20}M(i,j)", "ab6ca3e044b8003494f8b4570a6af60d": "\n C \\vdash C\n ", "ab6ccd54482d6a32c15bc84f61484602": " \\Delta Y - c\\Delta Y = \\Delta I", "ab6cf4a63da04521700483696274f9eb": "\\aleph \\!\\,", "ab6d1f3cea35f7300d4cb795e945ea36": "\\alpha[\\mathbf{f}] = \\left[\\alpha_1\\ \\ \\alpha_2\\ \\ \\dots\\ \\ \\alpha_n\\right].", "ab6d2dcf173971806a97e756a00c6d65": "(1-z^2) \\frac {d^2y} {dz^2} - 2z \\frac {dy} {dz} + n(n+1)y=0.", "ab6d56d44a9cf1ac4a368a0af3db6718": "\n\\begin{array}{lrclr}\n\\max\\limits_{x_{T-1}} & E[U(W_T)|\\xi_{[T-1]}] & \\\\\n\\text{subject to} & W_T &=& \\sum_{i=1}^{n}\\xi_{iT}x_{i,T-1} \\\\\n &\\sum_{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\\\\n\t\t & x_{T-1} &\\geq& 0\n\\end{array}\n", "ab6d5994586468f6ab6058c6857be180": "\\varphi+\\theta_0", "ab6d702e153772ec715f5703cc57500e": "P(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0", "ab6d85032631dbb12665bbe826b06972": "\\int f(x) dx", "ab6de0acdeb2f3b632be2e58b7f54c4c": "{(\\hat k_{AF})}_\\kappa", "ab6e07f79eab6ac382295f429afa0213": "i_s=2i_1\\sin(\\Delta\\varphi_b^*+\\pi\\frac{\\Phi}{\\Phi_0})\\cos(\\pi\\frac{\\Phi_a}{\\Phi_0}).", "ab6e0e3d82e24568a20ae08308db8f19": " a_0 \\!", "ab6eb391935e49fa38afae4961f56dd8": " f~ ", "ab6ee9392f3020c3db33e69452499f11": "\\hat{D}(\\alpha)=\\exp \\left ( \\alpha \\hat{a}^\\dagger - \\alpha^\\ast \\hat{a} \\right ) ", "ab6f0740828557d3e089b35633ea728d": " \\nabla = \\gamma^\\mu \\frac{\\partial}{\\partial x^\\mu} = \\gamma^\\mu \\partial_\\mu .", "ab6f0c69e759637f57d16fca3f0d3813": "2x + 4 - 4 = 12 - 4", "ab6f147768b077841ac4b3963140d895": "\\int_0^\\infty \\frac{dx}{\\sqrt{a^{2}-x^{2}}}=\\frac{\\pi }{2} ", "ab6f27aa3ef5a90a187869f216e14319": " \\mathrm{MA} = \\frac{W}{F} = \\frac{L}{H}.", "ab6f2e523f511e64f0d76affbced415d": "\\operatorname{Var}\\left(\\log_e \\frac{a}{b}\\right) = \\frac{\\operatorname{Var}(a)}{a^2} + \\frac{\\operatorname{Var}(b)}{b^2}", "ab6f6b90e399a494128ecce597325aca": "S_x(f) = \\frac{1}{4\\pi^2}h_{\\alpha}f^{\\alpha-2} = \\frac{1}{4\\pi^2}h_{\\alpha}f^{\\beta}", "ab6fe5f86f97a02f52c910f341473811": "\\hat{\\boldsymbol{\\imath}},\\ \\hat{\\boldsymbol{\\jmath}},\\ \\hat{\\boldsymbol{k}}", "ab7045d3df71bb5fb5d3b9afb293e122": "\\scriptstyle{\\mathbf{R}_\\alpha}", "ab705af051db363725143be965b03122": " LI = \\frac{w-PL}{LL-PL}", "ab705db8932bbbff3ea59229b5b362a8": "f = \\frac{q B}{2\\pi m}", "ab7070ee74213a3724f2f8775d8e6877": "\\frac{x^2}{A^2} - \\frac{y^2}{B^2} = 1 \\quad (1)", "ab70b21543e72fe555746ca645c2adb5": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "ab713aa2a1dadceaba643f52074c50e7": "x+2", "ab7142b8cc7f4c8857793ba8c5caf318": "\\left\\|\\int_Ef\\,d\\mu\\right\\|_B \\le \\int_E \\|f\\|_B\\,d\\mu", "ab716e0fd1af79ba32700743207ab1a9": "U(P)\\propto { \\int_S a_0 (\\mathbf{r'}) e^{i\\mathbf{(k_0-k)} \\cdot \\mathbf{r'}} dr' }", "ab71b77b6d1693780a7e831501d6a1b1": "\\ x_{32} = x_{31} = 0", "ab71cf41dc19a14d2a53c65a69f874c5": "K_m(S1, A), K_x", "ab71dc96d19ab410c69379b5e458c54c": "T_{22} = \\frac{1}{S_{21}}\\,", "ab71e7956ace237189bb4c11510c99e5": "\\lambda\\left(s,f\\right).\\left(s,g\\circ f\\right)", "ab71f1429f1ac991ea0e24fa9de4827b": "{\\mathbf a}_i, \\; {\\mathbf b}_i", "ab71f6f7cb633791a05e60f340a36728": "\\operatorname{E}[X_1+\\dots+X_N]=\\operatorname{E}[N] \\operatorname{E}[X_1]\\,.", "ab724b9a07403d722a0edb400b202f56": "\\boldsymbol{\\nabla}\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}=-\\boldsymbol{\\nabla}'\\frac{1}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|},\\,", "ab7273e04f5eabfae34b5c9b5117d9d3": " RDS= 2 \\frac{r_{\\alpha}-r_{\\beta}}{r_{\\alpha}+r_{\\beta}} ", "ab729bc605a35618a0267907680b37b7": "\n\\int_{-\\infty}^{+\\infty} f(x) \\, dx = \\int_{-1}^{+1} f\\left( \\frac{t}{1-t^2} \\right) \\frac{1+t^2}{(1-t^2)^2} \\, dt,\n", "ab72f8b4db43bae94adfafd7a949fd8f": "{\\mathcal L}f", "ab7360175835eb9ee92da8515ac854b5": "\\psi\\left(t\\right) \\approx \\operatorname{Re}(e^{i\\omega t})", "ab73b047147f4321ab3884ba6a3120d2": " M = \\sum_{i=1}^j\\ f_i ", "ab73c2ac64fc68bd4f8173b1e6ed95c8": "\\kappa = \\aleph_\\lambda", "ab73f0ea6881f87d759bf43dc6020dec": " \\mathbf{l} = \\int \\mathrm{d}\\mathbf{r} \\,\\!", "ab73f1751d6289f4d67f9dae5d4b6be2": "\\phi_k(\\mathbf{y}_{m+1},t_{m+1})=\\phi_k(\\mathbf{y}_m,t_m)", "ab7463cafe5ebead56743a3ca5738f0a": "V'_z=\\frac{ V_z }{ \\gamma \\left ( 1 - \\frac{V_x v}{c^2} \\right ) }", "ab754e74a5bab6d4ed99be5052289fa8": "e_i \\sim (1-\\varepsilon) N(0, \\sigma^2) + \\varepsilon N(0, c\\sigma^2).", "ab766e3e5e0013638ca23a9dbf5a10a3": " \\mathbf{A} + \\mathbf{B} = \\mathbf{C} ", "ab767f026774f64c85bddd904e378876": "n'_i = n_i\\sum_j w_{ij}\\,", "ab768899d83d81d855a5599bfd45e6c9": "\\scriptstyle (f_n)_{n=1}^\\infty", "ab769613f20593c8342d5477ddc9c00f": "\\nabla^2 \\Phi = 0", "ab76c4b2c01a9655fb4f058954138568": "{\\mathcal C}=\\{(-\\infty,t]:t\\in {\\mathbb R}\\}", "ab772599bfd7524676b169ae5cf2d3fb": "(q^m - 1)", "ab77553ace5e9a059f4e1cb50c5a3059": " \\left(\\prod_p \\frac{p \\gamma_{c,p}(n)}{(p-1)^c}\\right)\n\\int_{2 \\leq x_1 \\leq \\cdots \\leq x_c: x_1+\\cdots+x_c = n} \\frac{dx_1 \\cdots dx_{c-1}}{\\ln x_1 \\cdots \\ln x_c}", "ab77ac0130fb62a163c9de6f2a5d0c52": "\\epsilon/c", "ab781c3e71bd7904fde9e35b81a6aae4": "N^a (x)", "ab7851ab789acab7ad7ef1fe9a344ebf": "\\begin{array}{lcr}\nc_1 = (1cos(-45), 1sin(-45)) & = (1/\\sqrt{2}, - 1/\\sqrt{2}) & = a_1/\\sqrt{2} - a_2/\\sqrt{2} \\\\\nc_2 = (1cos(45), 1sin(45)) & = (1/\\sqrt{2}, 1/\\sqrt{2}) & = a_1/\\sqrt{2} + a_2/\\sqrt{2}\n\\end{array}", "ab787d93c3266ccc0a67898080e7e575": "\\begin{matrix}-\\frac{ \\beta R_{\\text{C}} }{ r_{\\pi} + ( \\beta +1 ) R_{\\text{E}} }\\end{matrix}\\,", "ab78c7b1f60165d9bf0073afc583c6b9": "\nU = \\frac{-Q_{1}}{2\\pi\\epsilon} \\int \\rho d\\rho \\ \\lambda(\\rho, \\theta) \\ln \\rho \n+ \\left( \\frac{1}{2\\pi\\epsilon} \\right) \\sum_{k=1}^{\\infty} k \\left( C_{1k} I_{2k} + S_{1k} J_{2k} \\right)\n", "ab78ccfbcd04b1ba22eb9427251cb20d": "n = 1", "ab78e1920700db13b4625e679bea6816": "\\rho_S (t) = \\mathrm{Tr}_E [U_t \\rho_{SE} (0) U_t^\\dagger] ", "ab78e99b3852fb6ad20e4eae50b5a24e": "\\mathop{\\rm supp}\\hat{f} \\subseteq \\mathbb{R}_+.", "ab79266292897899558033d900b3138c": " \\log_b(||X|-|Y||)=x+d_b(z),", "ab79415e96e2b2267d63433aa9ffd5f4": "\\vert\\vert Q\\vert\\vert_{\\mathcal{F}} = \\sup \\{\\vert Qf \\vert: f \\in \\mathcal{F} \\}", "ab79a16a1c8e2e7a3d798fa12c194163": "\\leqslant, \\nleqslant, \\eqslantless \\!", "ab79d7523586896bcd47fab4c084278b": "\\int_0^1x^{1/m}\\,dx.", "ab7a57acc56a0ce99692510110b92ac2": "\\varepsilon_k=\\pm 1", "ab7ae9d238316281ec6246deaa210973": "\\begin{align}\n g_{\\mathrm{Alice}} &= DI_{\\mathrm{Alice}}\\\\\n g_{\\mathrm{Bob}} &= DI_{\\mathrm{Bob}}\n\\end{align}", "ab7b640e1070908f381cc4f3e2d0a81d": "S^{a_1} R S^{a_2} R S^{a_3} \\cdots", "ab7b703414f2683f4b17171d434b54a6": " DO_0 ", "ab7b7829c30e2d2a992d7a16266ca389": "\nP_\\mathrm{avg} = V_\\mathrm{rms} \\cdot I_\\mathrm{rms} \\,\n", "ab7b7c6a126d213df7f7e00ba6eb5837": "\\psi\\to 0", "ab7b7fc3baba37ad67ed11a24942123b": "\\operatorname{ass}M", "ab7bbc29b0efd87f5cf124823a7ec1cd": "\\sigma = (s_0~s_1~\\dots~s_{k-1})", "ab7bbe704affd112737adb3f8a59398b": "\\frac{\\mathrm{d}}{\\mathrm{d} \\alpha} \\det(A)= \\operatorname{tr}\\left(\\operatorname{adj}(A) \\frac{\\mathrm{d} A}{\\mathrm{d} \\alpha}\\right).", "ab7bcc8ae0dde9f04cc79f04a18ba530": "U(c_1,c_2)", "ab7bff7f7cd8e9b479cdafb8d2422b14": "S_{abcd}", "ab7bff8ca1bc90b2a3228386166fbd69": "\\frac{\\pi_4}{\\pi_2 \\pi_3} = \\frac{a^2 q^2 I}{\\varepsilon_r \\varepsilon_0 k_b T} = (\\kappa a)^2", "ab7c005a87b94c2f37fe938231fef7ea": "\\Delta E=E_j-E_i-e^2/r_{ij} ", "ab7c0f7cd13adcd615b91e6e40a8f169": "\\min\\{f(x)\\ :\\ x \\in \\mathbb{Z}^n,\\ Ax=b,\\ l\\leq x\\leq u\\}\\ .", "ab7c260d134916f21624786e1978af5b": " U = \\begin{bmatrix} A & I - AA^* \\\\ 0 & - A^* \\end{bmatrix}.", "ab7c819e754f75e5a9bfc32715b38291": "A \\times (B \\cup C) = (A \\times B) \\cup (A \\times C),", "ab7d10fef9f9bf854a49e61a73db52bb": " C_\\varepsilon( v ) ", "ab7d3b49f735a4b5889038f0ef390da7": "a_1=\\frac{\\Gamma}{2\\pi i}.", "ab7d51316d46fba751ce0a4b95f39879": "\\Delta^{\\mathrm{op}} \\to \\mathbf{Set}", "ab7d992aa730738fa65cf67955d5a5f1": "{3\\pi\\over 5}\\ {\\pi\\over 3}\\ {\\pi\\over 3}", "ab7e26f3e3b41b7162d1431de4a11988": " s_1, \\ldots, s_k ", "ab7e73d485ee5cfb8f74abf8f0a7d333": "V(x)/I(x) = -Z_0", "ab7e942df78a2092e5efff7249cc26eb": "z\\rightarrow z^{4m+1} + z_0", "ab7eb5321fc3f115cf43d486fd78b9c1": "I_{Mm}", "ab7ee9e0dcd037ec124ea24d41408806": "\\|f\\|_p:=\\left(\\int_\\Omega |f(t)|^pd\\mu(t)\\right)^{1/p}", "ab7f04f6e9dcc85308f6072978202615": "\\frac{1}{10^7\\pi}", "ab7f1773941108311dc5548637ab138d": "\\omega \\approx 9.5 \\cdot 10^{-7}\\, \\mathrm{arcseconds} / \\mathrm{day}.", "ab7f34495f2e0a5cf33761505bcd322f": "\\displaystyle{\\Delta_2 \\varphi_\\lambda= (\\lambda^2 + {1\\over 4}) \\varphi_\\lambda.}", "ab7f34d9b7f6fd0fe8e936cd94fb74df": "\\mathbf{v} = {d\\mathbf{r}\\over{dt}}", "ab7f51b272ef0e15b32ce4822cac6038": "k_\\mathrm{cat} = \\frac{V_\\max}{[E]_T}", "ab7f90193dc048dd6158c3600529c137": " \\mathbf{A}_i = \\alpha(\\Delta r_i\\mathbf{t}_{i}) - \\omega^2(\\Delta r_i\\mathbf{e}_{i}) + \\mathbf{A}.", "ab7fa5f984af21b7ee70d2c8e4e73ed3": "R(T) = N(T')^\\perp=\\{y\\in Y | \\langle x^*,y\\rangle = 0\\quad {\\text{for all}}\\quad x^*\\in N(T')\\}", "ab7fd18acb93fcf3a15deefcb46c9e88": " |f_k \\rangle \\ ", "ab7fdc9644aea78cc702f0d69bfcc368": "PV_\\text{fixed} = PV_\\text{float} \\,", "ab7ff5059555859f87d1bd8199e202e9": "S(X, Y, \\mathfrak{E}, H)", "ab7ffcf04ee654f79afdb3aade4fe782": "O(T\\times\\left|{S}\\right|^2)", "ab805e4d15e134964330e6d7bc6ce916": " \\hat{S} ", "ab80783c5e44f3b460b8a6db8274cb35": "\\scriptstyle \\vec{F}_{2}", "ab80854a7ea28f935e4d6eeaf0b36972": "\\textstyle\\sum_{n=0}^\\infty a_n", "ab815ff90f828858621bb621f8c38451": "M_{+} \\equiv \\sup_{0 \\le t \\le 1} e(t) \\ \\stackrel{d}{=} \\ \\sup_{0 \\le t \\le 1} W_0 (t) - \\inf_{0 \\le t \\le 1} W_0 (t) ,\n", "ab818c4140d7c9c70af4eb3cfd9f572b": "\\mathbb{W}_n", "ab81b2ed3c8a81c7a79ee4e611042526": "SL(2, \\mathbf Z) = \\left \\{ \\left ( \\begin{array}{cc}a & b \\\\ c & d \\end{array} \\right )| a, b, c, d \\in \\mathbf Z, ad-bc = 1 \\right \\}", "ab820413024f0615da291ed063177c47": "k'=k+vp", "ab822673f114ec4ffa8a2fc5b1b5f0da": "V_c = \\sqrt { \\frac {2 T} \\delta }", "ab822bf6dcc288e50ad85481c676b1b6": "ax + by +c = 0 \\,", "ab822cdc72179f573d03276eb0468a02": "F_{cr}(t)\\,", "ab82300084790ebe295919b416c5afe3": "\\left\\{ \\theta \\bigg| y \\le \\frac{\\hat p - \\theta}{\\sqrt{\\frac{1}{n}\\hat p \\left(1 - \\hat p\\right)}} \\le z \\right\\}", "ab823176458865d42cc0327e3d495757": "K\\ll1", "ab8233422cf9af8ba59c07f2197178d4": "0 \\rightarrow M \\rightarrow E_0 \\rightarrow E_1 \\rightarrow E_2 \\rightarrow \\dots", "ab824407e0deafb6764473e2d41a7601": " 1 = \\int_{-\\infty}^{\\infty} \\psi^* (x, t) \\psi (x, t) dx ", "ab82765418cb3ea58834933e3676f71e": "p(\\tfrac{1}{2}) \\neq 0", "ab82a24715c65e80081a658b6b37d17b": "| 1 \\rangle ", "ab82c30de243bcd81ad328bfffc28747": "- \\mathbf{E} = \\nabla\\varphi + \\dfrac{\\partial \\mathbf{A}}{\\partial t}\\,, \\quad \\mathbf{B} = \\nabla \\times \\mathbf{A} ", "ab833eb5871cf8dc6660ea671d04026c": "\n\\Delta \\frac{1}{2}|\\nabla u| ^2 = |\\nabla^2 u|^2 + \\mbox{Ric}(\\nabla u, \\nabla u)\n", "ab8381b7dd8ffb6fbd0c49722b3b948b": "(\\mathbf{D}_2 - \\mathbf{D}_1) \\cdot \\mathbf{n}_{12} = \\rho_{s} ", "ab8443aac94025cd15733a129e0f408b": "A_3 = \\begin{bmatrix}\nb_{1,n} & b_{2,n} & b_{3,n} & \\cdots & b_{m-1,n} & b_{m,n} \\\\\nb_{1,n-1} & b_{2,n-1} & b_{3,n-1} & \\cdots & b_{m-1, n-1} & b_{m,n-1} \\\\\nb_{1,n-2} & b_{2,n-2} & b_{3,n-2} & \\cdots & b_{m-1, n-2} & b_{m,n-2} \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\nb_{1,2} & b_{2,2} & b_{3,2} & \\cdots & b_{m-1,2} & b_{m,2} \\\\\nb_{1,1} & b_{2,1} & b_{3,1} & \\cdots & b_{m-1,1} & b_{m,1} \\\\\n\\end{bmatrix}", "ab845cba77a7acbd3cda17c2e16608b7": "\\eta_0 = \\left(\\frac{4 \\cdot \\pi^2 \\cdot F_s^3 \\cdot V_{as}}{c^3 \\cdot Q_{es}}\\right)\\times100\\ %", "ab847cd5c8eb67f9ad02fc9d90032195": " m_2", "ab84a5c24b84eb4e03b1b32436603cf4": " \\Rightarrow \\beta b \\rightarrow 0 \\ ; \\ \\sin(\\beta b) \\rightarrow \\beta b \\ ; \\ \\cos(\\beta b) \\rightarrow 1. \\,\\! ", "ab84d6b3c7daa0fb9e191e9eb7c3257d": "\\mathit{prob}_{\\mathit{after}}(\\psi \\rightarrow \\phi) = \\sum_j|\\lang \\mathit{after}| \\phi, \\epsilon_j \\rang|^2 = \\sum_j|\\sum_i \\psi_i^* \\lang i, \\epsilon_i|\\phi, \\epsilon_j\\rang |^2 = \\sum_j|\\sum_i \\psi_i^* \\lang i|\\phi \\rang \\lang \\epsilon_i|\\epsilon_j \\rang |^2", "ab850824b1e8926e131e7fac881e5161": " \\bold{A} = \\bold{B} - \\bold{C} \\quad (5) ", "ab8551b779fcddc3e28b7513b306f9fc": "\\omega_i = - D_{$,i} := \\frac{\\partial V}{\\partial r_i}. ", "ab868764b27821d98c71af81919297c4": "S_m=-S_g", "ab86bf4a4f8d4f0a992bb161fd60fff2": "P_6 = (\\frac {1.2260} {1.2238} - 1) \\frac {360} {30} = 0.021572 = 2.16%", "ab878e961ffa37be32e17f29f080d2d1": "\\gamma\\ =\\,", "ab87e143abda81cda5a88af7beac65bc": "\\rho(g)\\cdot e_h = e_{gh}", "ab87e99607661766689c98b34ed51aeb": "g_{th} = \\frac{\\alpha_{wg}+\\alpha_{mirr}}{\\Gamma}", "ab87f41a6cef114782bc7e23fd3daa63": "R_2+R_3 = \\frac{R_a(R_b+R_c)}{R_T}", "ab88341501de92c668f381de0a592122": "\\pi^{-1}(y)", "ab88690d0d85639c89b68baba73c6985": "= T \\sum_{n=-\\infty}^{\\infty} x(nT) \\delta(t - nT) \\ ", "ab88878600bf1a7f8f58122043523787": "\\cos \\theta \\approx 1 - \\frac{\\theta^2}{2}", "ab88893724cba13c289a087c3cd8142a": "\\frac{\\partial \\phi^{\\alpha}}{\\partial u^{k}_{i}} - u^{\\alpha}_{l} \\frac{\\partial \\rho^{l}}{\\partial u^{k}_{i}} = 0\\,", "ab88a4d1d1de192f78ac3db80e6a5dda": " \\int_{t_1}^{t_2}F_x v dt = \\frac{m}{2}v^2(t_2) - \\frac{m}{2}v^2(t_1). ", "ab88f3bb03bb5dbe7f436c620b1a798b": "G_{dB}=10 \\log G_{W/W}=10 \\log 100=10 \\times 2=20\\ \\mathrm{dB}.", "ab88f41ad4766673176360281c58bc60": "\\,a_{65}", "ab892e45fa345df1dadc8ef927138f98": "c_i+c_j = x\\cdot g_i + x\\cdot g_j = x\\cdot(g_i+g_j)", "ab89316b510b799ad666709e5339486f": " |k \\rangle", "ab895d9ff7fdeb4dfa861787404eaf22": "p_{i_1,\\ldots,i_{n-1}}(f_1,\\ldots,f_{n-1})=\\int_{-\\infty}^{\\infty}p_{i_1,\\ldots,i_n}(f_1,\\ldots,f_n)\\,df_n", "ab89e6b2eef4bd435dc3f736358717ca": "\n\\mathcal{A} = \n\\left( \\frac{\\partial \\xi}{\\partial u} \\right) \\left(\\mathbf{p} \\times \\mathbf{L}\\right) +\n\\left[ \\xi - u \\left( \\frac{\\partial \\xi}{\\partial u} \\right)\\right] L^{2} \\mathbf{\\hat{r}}\n", "ab8a00bb5d04e9835047d0e129733242": "P_{rl}", "ab8a0810b676d7e5388858a07b29e08f": "O\\left[m + n\\log n + \\sum_{j=1}^m \\log(t(j) + 1)\\right]", "ab8a122bf5f978d1f3cabba694546915": "W_{out} = C_d \\rho |\\mathbf{u}|^3", "ab8a13dfae1c07ba44c554f2059c4b42": "\\int_a^x \\!\\!\\!\\int_a^s f(y)\\,dy\\,ds\n= \\int_a^x f(y)(x-y)\\,dy", "ab8a22f4e9ba286378e1316959caf322": "E_n(x,\\alpha)=\\sum_{p=0}^{\\lfloor n/2\\rfloor}\\binom{n-p}{p} (-\\alpha)^p x^{n-2p}. ", "ab8a24078b6d4c1c9875a34eb13851ba": "\\gamma_s(t)", "ab8a436989d6ced8ef1d409fceef7b69": "p(\\sigma^2|I^\\prime, \\mu) \\propto \\frac{1}{\\sigma^{n_0+2}} \\; \\exp \\left[ -\\frac{n_0 s_0^2}{2\\sigma^2} \\right]", "ab8a7af6ee144c3bf2b461942b0b305f": "\\mathcal{L}_{aux}=-\\frac{1}{2}(f(\\varphi),f(\\varphi))", "ab8ab25e297d3a61b9cb366a5752e78a": "\n\\nu \\quad \\approx \\quad \n {{\\left( \\; {s_1^2 \\over N_1} \\; + \\; {s_2^2 \\over N_2} \\; \\right)^2 } \\over\n { \\quad {s_1^4 \\over N_1^2 \\nu_1} \\; + \\; {s_2^4 \\over N_2^2 \\nu_2 } \\quad }}\n", "ab8ad58d279c4d1c42b9cc7c0c113b41": "\\sum_{n=0}^{\\infty} a_n. ", "ab8adf7e03106fe628ad4b5e1f39d481": "L^2 \\, \\ge \\, 4\\pi A - A^2/R^2.", "ab8b07156879b4ee8ea3ec25f94e891f": "m = \\frac{f(a+h) - f(a)}{(a+h) - a} = \\frac{f(a+h) - f(a)}{h}.", "ab8b316fa0e35b9fc07902e45c1d0270": "\\log \\|x\\|", "ab8baa90421af95046ff67f7ef8106de": "u = u(\\vec x, t)", "ab8bd1e97335e5697d0cdecc7ac3551b": " \\mathbf{E}(z,t) = \\mathrm{Re} (\\mathbf{E}_0 e^{i(\\tilde{k} z - \\omega t)})", "ab8c0e40b75cd0bc02a1fef0e603fe29": "A=\\frac{d^2}{2}.", "ab8cd6be4a960fb78a0ba5c369ca50b3": "\\begin{align}\n\\operatorname{lb}\\,x &= n + 2^{-m_1} \\left( 1 + 2^{-m_2} \\left( 1 + 2^{-m_3} \\left( 1 + \\cdots \\right)\\right)\\right) \\\\\n&= n + 2^{-m_1} + 2^{-m_1-m_2} + 2^{-m_1-m_2-m_3} + \\cdots\n\\end{align}", "ab8d75ee1f35453edaa47c3f1d1816de": " \\left(\\theta\\right)", "ab8db5070a6229f4b9038321af0a51af": "s = \\{ a_m:\\mathbb N\\to\\mathbb R, n\\mapsto n^m ;~ m\\in\\mathbb Z \\}", "ab8dbf36f4df1bf339205343f1c0d5dd": "\\mathrm{O^{\\eta}}", "ab8e056d7b221fb800cd39a0bd27ffed": "\\omega^2=\\omega_p^2+3k^2v_{th}^2", "ab8e12dc56ff66f8a391916693657f6c": "R_n^2(\\xi,x)\\le 1", "ab8e93fd8c9ba79ec1af2b98b916aec3": "\n\\int_0^\\infty |f(r)|\\,r^{1/2}\\operatorname{d}\\!r\n", "ab8ea3cef81be4f8e620cc2b3e82a044": " ( A_N \\otimes ( A_{N-1} \\otimes \\cdots \\otimes ( A_2 \\otimes A_1) \\cdots ) ", "ab8eb411b58a16e88fbd3869310e5f5a": "\\text{in single-phase systems:}", "ab8ed97d1b4a3aa629fdbba853c37010": "{-p_2}", "ab8f1fbcac98415bc868219e22cd75ae": "\\prod_{p|n} f(p)\\;", "ab8f43dc81d1c104279ef0468cad0399": "\\frac{S_1}{\\sin\\theta_1}=\\frac{S_2}{\\sin\\theta_2}=\\frac{S_3}{\\sin\\theta_3}=\\frac{S_4}{\\sin\\theta_4} = \\frac{\\overline{AC}}{\\sin(\\theta_3+\\theta_4)}=\\frac{\\overline{BD}}{\\sin(\\theta_3+\\theta_2)}=2R. ", "ab8f6159745f22fc2e77f32d3504f372": "L= 96.58+20\\ \\log_{10} (d) +20\\ \\log_{10} (f) ", "ab8f92bc8cdf1a9c2a928fe0d56622ea": "{{m+3}\\choose{m}}B_m=\\begin{cases} {{m+3}\\over3}-\\sum\\limits_{j=1}^{m/6}{m+3\\choose{m-6j}}B_{m-6j}, & \\mbox{if}\\ m\\equiv 0\\pmod{6};\\\\\n{{m+3}\\over3}-\\sum\\limits_{j=1}^{(m-2)/6}{m+3\\choose{m-6j}}B_{m-6j}, & \\mbox{if}\\ m\\equiv 2\\pmod{6};\\\\\n-{{m+3}\\over6}-\\sum\\limits_{j=1}^{(m-4)/6}{m+3\\choose{m-6j}}B_{m-6j}, & \\mbox{if}\\ m\\equiv 4\\pmod{6}.\\end{cases}", "ab8ff11de9604af03a59cef57d14d7a0": "R(p,p')", "ab9017d7913ca3ff82e35ed6cfb43b6e": "E(B)", "ab9027176cef2ce4ebf615e4b0ded37f": " [H_j^{0-}] = \\left[\n\\begin{array}{rrrr}\n0 & 0 & 0 & -q_{j,1}^0 \\\\\n0 & 0 & 0 & -q_{j,2}^0 \\\\\n0 & 0 & 0 & -q_{j,3}^0 \\\\\n0 & 0 & 0 & q_{j,4}^0 \\\\\n\\end{array} \\right],\n", "ab902bc29639a19a32690115ef556a3c": "D(b)", "ab902c3b6f3f71200f908c2e9dedfeb8": " G=\\left(V,E\\right) ", "ab9066ed7968bfef3f2a63c49b5a7a26": "R^{(p)} = A[X_1, \\ldots, X_n] / (f_1^{(p)}, \\ldots, f_m^{(p)}),", "ab90bd64300f8e07adf57e1012a62921": "dS/dz \\ne 0", "ab90e391ee784f91b370fa82bfe10f38": " v_{A|O'}= v_{A|O}-v_{O'|O}\\Rightarrow v_{A|O} = v_{A|O'} + v_{O'|O}", "ab910a78c7ed68457952b0a91c4a52d6": "f \\in k(x) \\backslash g(k(x))", "ab910fe1178c8bb746ac0a1a66317508": "\\sqrt{\\frac{\\pi}{2}}L_{1/2}^{(k/2-1)}\\left(\\frac{-\\lambda^2}{2}\\right)\\,", "ab912c4610710be547733f5634c06c46": "\\mathit{k_{tr}}", "ab9139d586bd83878980024d6a86a034": "\\begin{align}Q(\\delta\\mathbf{x},\\delta\\mathbf{u})\\equiv &\\ell(\\mathbf{x}+\\delta\\mathbf{x},\\mathbf{u}+\\delta\\mathbf{u})&&{}+V(\\mathbf{f}(\\mathbf{x}+\\delta\\mathbf{x},\\mathbf{u}+\\delta\\mathbf{u}),i+1)\n\\\\\n-&\\ell(\\mathbf{x},\\mathbf{u})&&{}-V(\\mathbf{f}(\\mathbf{x},\\mathbf{u}),i+1)\n\\end{align}\n", "ab9139fff06a40a45abf38c73f33c9ff": " \\frac{d p^\\mu}{d \\tau} = q F^{\\mu}_{\\nu} u^\\nu ", "ab914af63b1ae694fd6c6f69bb11e1a0": " p, q \\in \\mathbb{R} ", "ab9185551c2eb418c724a9fd82034fde": "n \\in T_1", "ab91acf94021c751eb3179d1c06ebb47": " \\sigma_e", "ab91ff48d8818acff59c6c4b72bd18a2": "\\mathrm{mg}/\\mathrm{m}^3 = \\mathrm{ppmv}\\cdot \\frac{M}{(0.08205\\cdot T)}", "ab923d45cc02687c30b990701a2891b4": "s_{i}^{2}", "ab92421c29feb3d564a273a9d7e68c03": "(\\mathbf{p}=q \\mathbf{d})", "ab92432d8f5840f4acbb52f68bc1afe5": "\\ \\Delta\\Theta", "ab925fc403138a719108cad1ac9e1f64": "u'' + \\frac{2}{x}\\,u' + \\left[\\frac{\\lambda}{x} - \\frac{1}{4} - \\frac{\\ell(\\ell+1)}{x^2}\\right]\\,u = 0\\text{ with }\\lambda = n.\\,", "ab9294572d02a5567e3a2cf6f7bf1d50": "\\frac{P_1V_1}{T_1} = \\frac{P_2V_2}{T_2} \\Rightarrow\n\\frac{V_1}{V_2}=\\frac{T_1}{T_2} \\frac{P_2}{P_1} \\Leftrightarrow\nCR=\\frac{T_1}{T_2} PR", "ab92a407a91d482c13255e47cf6dff32": "\n\\begin{align}\nF_{g,x} &= - m \\omega^2 x \\\\\nF_{g,y} &= - m \\omega^2 y.\n\\end{align}\n", "ab930b4f42659348af7ddc93a4035a3e": "\\tilde I = (R : (R : I)),", "ab941d29e489576fda06f604860987ab": "\\langle\\mathbf{v}\\rangle = 0 ", "ab944822ce250cb61b83496a7caf146a": "{a\\pi\\over 5}\\ {b\\pi\\over 3}\\ {c\\pi\\over 3}", "ab948158eaddb9a9b31f74bcb08ce3d2": "DPW = \\left(\\frac{\\displaystyle \\pi d^2}{4S}\\right) \\exp(-2.32^{*} \\sqrt{S}/d)", "ab94b80cb4c7232ff08deddf4184ed02": " \\max(\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4) \\le z \\le \\min(\\beta_1,\\beta_2,\\beta_3). ", "ab94bedffff89d2a36249b3168dff4c4": "x \\in X \\,", "ab94d91e920441fb99aaedc90edd7a61": "[\\phi_k,\\pi_k^\\dagger] = [\\phi_k^\\dagger,\\pi_k] = i\\hbar", "ab950ef99cfc60cc784e055dce60dc43": "\\operatorname{tr}\\left(\\mathbf{A}\\right) = \\sum\\limits_{i=1}^{N_{\\lambda}}{{n_i}\\lambda_i} \\!\\ ", "ab9529e0defc4fb20ec9dbaf8eced88a": "n(\\vec r) = N \\int{\\rm d}^3r_2 \\int{\\rm d}^3r_3 \\cdots \\int{\\rm d}^3r_N \\Psi^*(\\vec r,\\vec r_2,\\dots,\\vec r_N) \\Psi(\\vec r,\\vec r_2,\\dots,\\vec r_N).", "ab95379baac718a8e2cd6cdcc78f7f01": "\\mathbf{j}=-D(\\phi)\\,\\nabla\\phi(\\mathbf{r},t)", "ab953eeeb0acd21a3693f652f4dd801a": "\\tau_\\mathrm{PN} = \\frac{2G}{1-v}e^{-2{\\pi}W/b}", "ab955d681df9dccbaa79a333cf836f4d": "\\lim_{q \\rightarrow \\infty} I(q) \\propto S' q^{-(6-d)}", "ab95d6439d36e771a7d981f3ee35d047": "A_i,B_j,C_k,D_l", "ab95ff1554af3872d0ee2715ee4cba41": " \\scriptstyle 5^4+53\\sqrt{89}", "ab9606cb3146b1eea7857691d0bf3588": "s_{ij} = \\operatorname{sat}(i,j) - \\operatorname{unsat}(i,j)", "ab963d3b87d55072c0f0c36922142362": "\\displaystyle{{f(x)-f(rx)\\over 1-r}={-f(rx)\\over 1-r} \\ge -{1\\over (1+r)^{n-1}} f(0)> -{ f(0)\\over 2^{n-1}} >0.}", "ab96975871b5ba9c0d4cf0afeee1df8a": " M_{2,3} = \\det \\begin{bmatrix}\n\\,\\,1 & 4 & \\Box\\, \\\\\n\\,\\Box & \\Box & \\Box\\, \\\\\n-1 & 9 & \\Box\\, \\\\\n\\end{bmatrix}= \\det \\begin{bmatrix}\n\\,\\,\\,1 & 4\\, \\\\\n-1 & 9\\, \\\\\n\\end{bmatrix} = (9-(-4)) = 13", "ab96b264c013f34bff55c36c5c587361": "x_{jk}", "ab96ee824e5e5a85aa0bb5ccea73f385": "0\\to U\\to M\\to V\\to 0", "ab970c4a6944d29279be67e17a3d82c7": "p(y\\overline{\\|}x)", "ab971edfc23ce547d0a93158dbbdb36f": " \\frac {1}{c(w)} = a_1|w|^{-\\gamma} +cot(\\frac{\\pi \\gamma}{2}) \\quad (1.2)", "ab9722e0043a0c2f0829584154cef541": "\\ddot x=F(x)", "ab974e5a3d34ed3ee88f3f4992f827f7": "4 T^2 = (c h)^2 = c^2(b^2-d^2) = (c b)^2 - (c d)^2", "ab97780b2ad832fcdfb9715fcef3cab7": "\\tfrac{322-13\\sqrt{70}}{900}", "ab97ceb53613c83375217577483bfdcc": "w_y(y)=0", "ab983531eed3c5f59b9fd8fbe275c435": "p(x + 1)", "ab987445d7ba8ea5e2c74ad9fd78c2cc": "c = -1.54368901269109 \\,", "ab988b9cbbcaaa8de1e192c61b79655b": "\\mu_{T} = \\left( \\pi_{ST} \\right)_{*} (\\mu_{S}).", "ab98952c2d07762830dfe08050d2c5dd": "P = \\frac{0.2345}{2 \\times 3} = 0.0391", "ab98bb46a643ccc24bfe9cdfaccfddc5": "\\pi{}/2", "ab98bb8e9dd1e9ce9979c553a2a92923": "BQ", "ab98d59f47755eca9c984b00a9a6646d": "Q(x) = x^T L x = \\sum_{e_{ij}} w_{ij} \\left(x_i - x_j\\right)^2", "ab9966a7357f2276c1b3a4764c1c34fb": "w'_2 = 0", "ab99b5836fa47a28c28b22274b40b454": "\\scriptstyle (\\mathcal Y, \\Sigma) ", "ab99b998126f30af89121d21106de0c2": "\\phi(2,0,0)", "ab99e5a98f1e89ed6ba3d38532affe38": "Range_{max unambiguous} = \\left( \\frac{c}{2 \\,PRF} \\right) ", "ab99ec47eaa24047c198e67a2f117409": " \\hat{A}_0\\left[ f \\right]^2 ", "ab99f935ad363328f484856dc9c20a54": "g_J = 2/3", "ab9a0f4878fe83ecc4ccb881f1d81feb": "\\operatorname{cov}(\\mathrm{f}) = J \\operatorname{cov}(\\mathrm{x}) J^\\top", "ab9a24c99c0c70577c4972e9689ec22b": " \\mathcal{V}=\\frac{(C\\cdot E)^m}{\\Gamma(m+1)},", "ab9a66de0c5fc871529f0e8660dec7c7": " a**b ", "ab9a69c2c3263f648cfe1d55d0d5a6be": "f(n) \\in \\omega(g(n))", "ab9a6b109b060c98d36ddbc1a1dc8f09": "S_n(c)", "ab9a798ab3f7d571131ebba53c532d4e": "\\gamma_\\text{ow}", "ab9abfdf5111a2eda80555bef9e6331d": "\\det(\\mathbf{A}) = \\sum_{i=1}^3 \\sum_{j=1}^3 \\sum_{k=1}^3 \\varepsilon_{ijk} a_{1i} a_{2j} a_{3k}", "ab9afab7db626e6ab50493e84e2cdf6e": " \\sum_{\\lambda\\vdash n} \\left(f^{\\lambda}\\right)^2 = n! ", "ab9b360b828d2f25e4698e089b45b107": "\\varphi(x) = \\begin{cases} e^{-1/(1-|x|^2)}& \\text{ if } |x| < 1\\\\\n 0& \\text{ if } |x|\\geq 1\n \\end{cases}", "ab9b55bf6fdebe92b66acee87874acb6": "\\stackrel{\\text{H}\\ddot{\\text{o}}\\text{lder}}{\\le} \\left( \\left(\\int |f|^p \\, \\mathrm{d}\\mu\\right)^{1/p} + \\left (\\int |g|^p \\,\\mathrm{d}\\mu\\right)^{1/p} \\right) \\left(\\int |f + g|^{(p-1)\\left(\\frac{p}{p-1}\\right)} \\, \\mathrm{d}\\mu \\right)^{1-\\frac{1}{p}} ", "ab9ba840f2647d36efd57d3dba846828": " \\mathbf {\\nu} ", "ab9bce0eb6eff85312fe8fdc5d0d9273": "U(\\sigma)=\\lambda/4(\\sigma^2-\\sigma^2_0)^2", "ab9c1ca41ac44329f54cba7029b759ac": " h_n = \\frac{b-a}{2^n}. ", "ab9c6816cbe94a9c6caeaca079cd480e": "GapSVP_{\\zeta,\\gamma}", "ab9cef08371219e9c7aaa69aae7b4dba": "\\mu\\,(x)=\\frac{f_X(x)}{1-F_X(x)}=-\\frac{S'(x)}{S(x)}=-{\\frac{d}{dx}}ln[S(x)].", "ab9cf8b97d536d0394a102eae507dd54": "\\psi(\\Omega^3)", "ab9cfe0fe933686d0337c573428f1d37": "O(dn^4M^2)", "ab9d47644b4fff79d3a846edf9d74a1d": "A = \\left\\{Z \\in \\mathcal{L}^0: Z \\geq 0 \\; \\mathbb{P}-a.s.\\right\\}", "ab9d54c1d1b44c0d08e685b7900dcea2": "C_{yy}: \\bigcup_{i \\in D} Y_i \\rightarrow Y^\\phi", "ab9d58e5ee7c8db9f0b1fdd8dc0b971c": "M=(NB)/3", "ab9d679ae7ec21c35d850c7859e44977": " Y = \\{x_{11}, x_{14}\\},", "ab9d8a60b69f3571fc5705a8fb89d0ce": "N_\\mathrm {TOT} = 10^a, \\ ", "ab9deb12ef1451a83d28e59b58758baa": "\n\\operatorname{Li}_s(z) = \\tfrac{1}{2}z + \\sum_{k = -\\infty}^\\infty {\\Gamma(1 \\!-\\!s, \\,2 k \\pi i - \\ln z) \\over (2 k \\pi i - \\ln z)^{1-s}} \\,.\n", "ab9e116a1d6dc1480ef806b005f121b4": "\\tan(u \\pm v) = x \\pm t", "ab9e505f6c6ac1324c6c82b66b9e0521": "\\exp\\left(\\frac{1}{\\hbox{Vol}(U)}\\int_U \\log f\\right).", "ab9ef29225e40b965a96c65f1acc5efb": " ax^4+bx^3+cx^2+dx+e", "ab9f26139794aff297f2a6092aa4fb67": "\\langle g|f\\rangle = \\langle \\hat{B}\\hat{A}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle.", "ab9f3da784262b3073cb05510a732928": "C_{ox}", "aba009b428b25993f315b8f91c8e242c": "J_{ij}, S_{i}, S_{j}", "aba03c53208f8d57cb053219a07919a9": "\n\\sigma_{\\text{impl}}=\\alpha\\;\n\\frac{\\log\\left(F_0/K\\right)}{D\\left(\\zeta\\right)}\\;\n\\left\\{1+\\left[\\frac{2\\gamma_2-\\gamma_1^2+1/F_{\\text{mid}}^2}{24}\\;\\left(\\frac{\\sigma_0 C\\left(F_{\\text{mid}}\\right)}{\\alpha}\\right)^2+\\frac{\\rho\\gamma_1}{4}\\;\\frac{\\sigma_0 C\\left(F_{\\text{mid}}\\right)}{\\alpha}+\\frac{2-3\\rho^2}{24}\n\\right]\\varepsilon\\right\\},\n", "aba0866a802150b464c9a22c93b77832": "I_V", "aba0d5eb0dbf1dc1e03705f50a2cdf76": "f(x,y) = x^2 + y^2", "aba0de5847fc0803f39447278e80c2e1": "\\begin{array}{cc} \\begin{array}{rrrr} j &k & l & m \\\\ \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} a & b & c & d & e & f & g & h \\\\ \\hline & & & & & & & \\\\ \\end{array} \\end{array}", "aba1482a6171a263c251272878c0dc09": " \\pi \\frac{N^{1/2}}{4} \\left( 1+ \\frac{1}{\\sqrt{2}}+\\frac{1}{2}+\\cdots\\right) ", "aba1605a49f408f37b841afe2188194e": "a_{i+1}^j = a_i^j + m_i r_{i+j}", "aba18641e43f994bcd3a5bd3dbb877dc": "\\frac{x^x}{e^x}\\,", "aba1c0434ce46fd45a67bf1494d01e3f": "\\Delta^{+}", "aba1d96b86c187a81e3103ba2a144624": " \n\\kappa_1\\ge\\mu_1\\ge\\kappa_2\\ge\\mu_2\\ge\\cdots\\ge\\kappa_{n-1}\\ge\\mu_{n-1}\\ge\\kappa_n\n", "aba211eb2a7dc3a65f2fec8fd9111c86": "\\langle \\varphi, x \\rangle = \\varphi[x]", "aba240fa56fbc12c6e03310d820f4c86": " \\overline{BD}\\perp\\overline{AC},\\overline{EF}\\perp\\overline{BC} ", "aba2780f2f9e733ad03060b2df83f25e": " \\Delta V = 0 ", "aba2cbbe1147aa73ce94e95f29f7a122": "\n RR_\\mathrm{total}={T_s\\frac{\\gamma -1}{2}eM_a^2}\n", "aba304ce16cf90ffbe3bea7dba0905b3": "\\frac 1 r=1-\\Big(\\frac 1 p-\\frac 1 q\\Big)", "aba3239e9bbfde0b3d42c831a6f3e63a": " H(\\Sigma,q)", "aba3378d6f6628e8cf49f8daa9511e21": "\\zeta(s)=\\frac 1{s-1}\\sum_{n=0}^\\infty \\frac 1{n+1}\\sum_{k=0}^n {n\\choose k}\\frac{(-1)^k}{(k+1)^{s-1}}", "aba3765f8b52d4bd371e18f85a1edfa9": "\\ F_i ", "aba3830bc1c94e8abfe8b208a0fe1c06": "n^{(i)}", "aba3b34f8d5263ee708a80a79431c781": "d_{max}", "aba3fb25b2ee12839e260ccc3720df3f": "{\\Delta}x", "aba4348a16e0179ba04162e0b3dada00": "\nZ_{j,\\ell,m} = \\sum_{i:Y_i\\ge t_j}\\theta_iX_i - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_iX_i.\n", "aba46811510bb3a1c2f397ef707b6cde": "\\left(\\frac{d}{dt}e^{X(t)}\\right)e^{-X(t)} = \\frac{d}{dt}X(t) + \\frac{1}{2!}[X(t),\\frac{d}{dt}X(t)] + \\frac{1}{3!}[X(t),[X(t),\\frac{d}{dt}X(t)]]+\\cdots ", "aba46ebd7cb7e344aa5a37448d8fa500": "", "aba4acad3c928e95279a767a36d6cff6": "n\\ge r", "aba4b1828697a6ba5a62091f33e6f803": "\\kappa\\neq\\lambda", "aba4f0e0ede5ffbf2f955a84dd3ab81b": "M=\\langle W,R,{\\Vdash}\\rangle", "aba5287e9121f09b1c3ff746803c65ff": "S_{\\mu_1} \\times \\cdots \\times S_{\\mu_s}", "aba530a10179c2dfc6806d7c85e24398": "\\mathcal{L}_{EW} = \\mathcal{L}_g + \\mathcal{L}_f + \\mathcal{L}_h + \\mathcal{L}_y.", "aba5a254cec9578e9c8d4359f97b3bf5": "\\beta(i,j,v)", "aba5cc7b77230846cfa1401e3b6570f8": "\nD \\frac{\\partial c}{\\partial z} + s g c = 0\n", "aba5cf6a80af00f6084b0f9b83e46cbd": "\\Delta V_X", "aba6ad64a8934d8397db91c76909f68b": "\\omega^* \\in \\Omega", "aba6b1ea1173c79898311d28c56bf372": "(f(x_n))", "aba6bf9fc2edbcd29cf40fec31e1885e": " \\operatorname{E} \\left[ s^2_{\\text{biased}} \\right] = \\sigma^2 - \\frac{\\sigma^2}{n} = \\frac{n-1}{n} \\sigma^2 ", "aba6c6a0027d3f4b446f9c76b729c272": "\\mbox{Asset Turnover} = \\frac{\\mbox{Net Sales Revenue}}{\\mbox{Average Total Assets}}", "aba6f8f35d608af6d2ab698a7e82c969": "a^2-N = b^2", "aba756a18a06bbdf829e99160c276faa": " \\exp(\\psi(x)) \\approx \\begin{cases} \\frac{x^2}{2} &: x\\in[0,1] \\\\ x - \\frac{1}{2} &: x>1 \\end{cases}\n", "aba7591b62360187a286f93333417861": "=D_1F(x,y)\\cdot (s_2,s_1) + D_2F(x,y)\\cdot(F(x,y)\\cdot s_2,s_1)", "aba7768c4f98bdfc458ec33eb240c0dd": "(\\xi,\\eta,\\phi)", "aba80c4ef3df8b62e3cbf81eaa27b70b": "X + XY = X", "aba843294cfcaa3dc0eef9651f41b69c": "\\Delta_{\\pi} := (-1)^{\\left|q^{i}\\right|}\\frac{\\partial}{\\partial q^{i}}\\frac{\\partial}{\\partial p_{i}} ", "aba8959f88857d784defff0b9f5f2199": "\\mathfrak{sl}_{n+1}(\\mathbf{C}),", "aba8af8ae7ef56ecea7385e46347b177": " {\\rm diag}(a) = \\sum_{n=0}^\\infty a_n | e_n \\rangle \\langle e_n |, ", "aba8b398eb2367805b58c0af2c60f59d": "D_{(Q)} = {\\lim_{\\epsilon\\to0} { \\left [ \\frac{ln{I_{{(Q)}_{[\\epsilon]}}}}{ln{\\epsilon^{-1}}} \\right ]}} {(1-Q)^{-1}} ", "aba8d2bf2e0d6a41fc0678019835d778": " \\frac {d \\mathbf{x}_\\mathrm{A}}{dt} =\\frac{d \\mathbf{X}_\\mathrm{AB}}{dt} + \\sum_{j=1}^3 \\frac{dx_j}{dt} \\mathbf{u}_j + \\sum_{j=1}^3 x_j \\frac{d \\mathbf{u}_j}{dt} \\ . ", "aba8f7af8f8d4a9c731242c50d3d422e": "\\omega=2 \\pi f", "aba910e1b0f70449955e8a8e493dd333": "Y_m = 0", "aba93ca02cd9bab0741b81eafcff6537": "\\mathrm{ord}_{(A-sI)^{-1}}\\lambda=\\mathrm{idx}_A \\lambda", "aba949242c006457993c6ee4cec5c298": "S_1,S_2", "aba97b90155be814bece8a47a7199954": " \\alpha / 2 = \\frac{1}{\\delta_e}= \\frac{1}{2\\delta_p} = \\frac{\\omega}{c} \\; \\mathrm{Im}(\\tilde{n}(\\omega)) ", "aba99367b51a9b168cbc9df5c049621d": "\\psi = \\exp(-\\mathrm{i}kx)\\,", "aba99d23308af18847cedcfcd3f18f23": "\n \\Psi_L = \\bar{\\Psi}^\\dagger P_3 \n", "abaab08dc1f459ac4f9cb571aa97008d": "D(t)", "abab04e994c2caaba32c143678b2f116": " I(X;Y|Z) = \\sum_{z \\in Z} p( Z=z ) D_{\\mathrm{KL}}[ p(X,Y|z) \\| p(X|z)p(Y|z) ], ", "abab6ef5a63e7f64e3cd46e3ca5736a0": "r_0 e^{i\\varphi_0} \\cdot r_1 e^{i\\varphi_1}=r_0 r_1 e^{i(\\varphi_0 + \\varphi_1)} \\,", "abab80e16ef2674d1d5986c4998ae9db": "\nB_{0}= ({1 - \\epsilon_B - \\epsilon}) {B_\\text{max}}.\\,\n", "ababa8be97eeedcfe33979b85289ac1a": "\\hat{\\Theta}=(\\mathbf{Z}'\\mathbf{\\Omega}^{-1}\\mathbf{Z})^{-1}(\\mathbf{Z}'\\mathbf{\\Omega}^{-1}\\mathbf{Y}), \\, ", "ababb94434abdfd7de1527d05b3ba7c4": "\\lbrack RH^+\\rbrack = \\lbrack H_3O^+\\rbrack_0 \\left (1-e^{-k\\lbrack R\\rbrack t}\\right )\\approx \\lbrack H_3O^+\\rbrack _0 \\lbrack R\\rbrack kt", "ababc3ae722e621933af49c39551eda4": "|x|= 1\\,", "abac10866fa5b1bdcf30962a7bc7aea0": "(1+0.50)(1-0.20)(1+0.30)(1-0.40)-1=-0.0640=-6.40%", "abac65e8edd4d0d51289b39a54c3ec93": "\\operatorname{Cart}(R)", "abac718f98a995d7690f2a9e3444f981": "\n X_n\\ \\xrightarrow{p}\\ X,\\ \\ Y_n\\ \\xrightarrow{p}\\ Y\\ \\quad\\Rightarrow\\quad (X_n,Y_n)\\ \\xrightarrow{p}\\ (X,Y)\n ", "abac726c2004202295033754b192d324": "\\mbox{votes needed to win} = \\left({{\\rm \\mbox{valid votes cast}} \\over {\\rm \\mbox{seats to fill}}+1}\\right) + 1", "abad2361b5039975a6a90f4e2036f739": " r_\\mathrm{ corr } = \\bar{ r } +\\frac{ n }{ n - 1 } \\frac{ m_y - \\bar{ r } m_x }{ m_x } ", "abadc1a79a09e588ca1096529ec43235": "{CR}={{aperture\\,area\\,of\\,concentrator}\\over{receiver\\,surface\\,area}} = {Ac\\over Ar}", "abadee54964c5af84c8f5c8b8e9770d0": " f_s(x) = \\sqrt{\\frac{\\theta}{\\pi \\sigma^2}}\\, e^{-\\theta (x-\\mu)^2/\\sigma^2}.", "abadf5cc50fcc05067a797072a3ced3d": " \\nabla \\cdot (\\nabla U) = \\nabla^2 U = {\\partial^2 U \\over {\\partial x}^2} + {\\partial^2 U \\over {\\partial y}^2} + {\\partial^2 U \\over {\\partial z}^2} = 0.", "abae258308bc10fe111a83d70afc597b": " \\mathrm{Li}(X) + O_K(X \\exp(-c_K \\sqrt{\\log(X)}) , \\,", "abae80a4500b977ac43b2c4decc4c22b": "W\\left(-\\frac{1}{e}\\right) = -1", "abaf314b224c1be537e3fcc4af95af42": "{\\rm rank Hess}(G)=1", "abaf4d3d19d4c86cd5acbe1437b7117d": "C = aW+bY,", "abaf6207226ffc5a63c1b24d2498e59c": "\\hbar\\ = \\frac{h}{2\\pi},", "abafb7349346564544de626343287ed3": "k^i_x = \\frac{k_x}{M} ", "abafd36738c615689a2ce84cd65ae692": "v_1\\otimes\\dots\\otimes v_r,\\quad v_i\\in V.", "abafd4abd941f811f26c2bb79e088859": "f(x)=\\log(1+(\\cos x-1))\\!", "abafecd41af3a66e14e6a08772977c86": "P_t = \\sum_{n = t+1}^{\\infty} \\left(\\frac{1}{1 + r}\\right)^{n - t} \\mathbb{E}[d_{n} \\mid \\mathcal{F}_t]", "abb000caa3f4ebf825e5a29a11b59da2": "\\sum_{n=1}^{\\infin} n^0", "abb03b0e78d8644526b33724050ba4cd": "dx=\\frac{v dy}{\\sqrt{v_m^2-v^2}}", "abb0674ecb598ff6d4a8a47406d53b41": "\\forall s \\in S \\mbox{ . } s \\leq A", "abb0999c48590076959dfb6abe588fd6": "O(n \\sqrt{n\\log n})", "abb0f350d71681b86f0dd6b166901244": "V_1, \\cdots, V_k", "abb0f9e0c78fa3e81e6038522db0b437": "\\mathrm{e}^{-\\mathrm{i} (\\epsilon_c - \\epsilon_v) t/\\hbar}", "abb13f123067da7b960f82d4cd460aa4": "F^{-1}(p) = \\mu - b\\,\\sgn(p-0.5)\\,\\ln(1 - 2|p-0.5|).", "abb1460fa288feb30e17338a6d781496": "\\alpha_0+n\\alpha,\\, \\beta_0+\\sum_{i=1}^n x_i\\!", "abb1a0b0f9070802a6b572832fe5fedc": "[t-1,t+1] \\in \\mathrm{Supp}(\\rho(t)) ", "abb1a498015cff77ec68d975b4a12e3a": " d\\ln(r) = [\\theta_t + \\frac{\\sigma '_t}{\\sigma_t}\\ln(r)]dt + \\sigma_t\\, dW_t ", "abb207146785108cc5a4fcc1dfb06487": "M_n=2^n-1\\,", "abb24cc6839db298b4f5ef4a7b7359be": " H(Y) = \\frac{1}{2} \\log \\left( \\frac{ \\pi \\sigma^2 }{2} \\right) + \\frac{1}{2} ", "abb254e68d654134dc199c622510057f": "V(\\vec{r})", "abb2784a77af2bc076b249a1f86764a3": "\\frac{1}{\\zeta(s)} = s\\int_1^\\infty \\frac{M(x)}{x^{s+1}}\\,dx", "abb2a4ad4eebc540e566237e3e96ad95": "E_n = n K T ", "abb2ba5e4941122aa6964e2c6068e371": "\\operatorname{de-let}[\\operatorname{let} p : p\\ f = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) \\operatorname{in} p] ", "abb2cf3c0c42242ea4d9f93b5f8fb2c6": "S(-1) = \\log_b 1 = 0, ", "abb31d76dbe8d5631ff58ba86e9efff9": "c = { 1 \\over \\sqrt{ \\mu_o \\varepsilon_o } } = 2.99792458 \\times 10^8 ", "abb32e7b3d8f573b2b24e341afe34605": " 1\\leqslant j\\leqslant k-1", "abb364e34de8a5d81473e30dc7c0964d": "u(z) = \\frac{1}{2\\pi}\\int_0^{2\\pi} u(e^{i\\psi}) \\operatorname{Re} {e^{i\\psi} + z \\over e^{i\\psi} - z} \\, d\\psi\\text{ for }|z| < 1.", "abb37f6b2d47e063a3b100a49397a028": "\\left(r_1\\right)^2+\\left(r_2\\right)^2+x^2=\\left(r_1+r_2-x\\right)^2", "abb3e3898cec8df8367209c5f46424b3": "f(z)=0", "abb3f3b355436e37f813a517caebf8e4": "E[W_{t_1} \\cdot W_{t_2}] = E\\left[W_{t_1} \\cdot ((W_{t_2} - W_{t_1})+ W_{t_1}) \\right] = E\\left[W_{t_1} \\cdot (W_{t_2} - W_{t_1} )\\right] + E\\left [W_{t_1}^2 \\right].", "abb4958942a1662c86cd089402ac1449": "\\begin{pmatrix} 2 & 1\\\\1 & 2 \\end{pmatrix}\\begin{pmatrix} Ae^{t}\\\\Be^{-5t} \\end{pmatrix} = \\begin{pmatrix} x\\\\y \\end{pmatrix}. ", "abb52558d7a9d7717e90bd9da0f519a2": " p \\succ_P r", "abb5a743f7f634308cab91b6168bd0ea": "\\sqrt{\\frac{g}{k}} = \\frac{g}{\\omega} = \\frac{g}{2\\pi} T", "abb62f800aab07e0b6342d418a35c973": "u,v \\in K[x]", "abb6bd27def12d55d8ddc83c65ccd0de": " p \\leq 10 ", "abb6cd178e7f9ee79424d014da617e1a": "V = \\frac{1}{3} h(x_1^2 + x_1 x_2 +x_2^2).", "abb705a614780d72038c19f9b8dcd04b": " {d^2 X^\\mu \\over dT^2} = 0 .", "abb706cb6180a35ac4317eecc09ba811": "\\alpha=\\beta=\\epsilon=1/2", "abb73932b69c3b8d21a2fde5abcf2326": "f(v+v') = f(v) + f(v')\\quad\\text{and}\\quad f(\\alpha v) = \\overline\\alpha \\, f(v)", "abb775eda009a5e6d67666917bdd2e48": "\\; A_1, \\ldots, A_m", "abb77c506bfe573617ec9afc40f0ca82": "2-\\eta", "abb7c9b3aa09c0c710cf97daf4c33655": "\\left \\{(1+x)^{\\deg(p)}p\\left (\\tfrac{1}{1+x} \\right ),M(\\tfrac{1}{1+x}) \\right\\},", "abb801c32484b556e0755fde0c6eb45e": " P = 1 - x - y", "abb811391e2fadbc6275babe6f4fa708": " \\mathcal{W}({\\mathbf \\Psi}^{-1}, \\nu) ", "abb8445ae49af993daec9a14dc92e5ae": "\\beta \\ge 1", "abb86a90e78e5b28573c9b46734fbd71": "\\mathbf{x}=(0,0,\\dots 0)", "abb86c120e630c749eaaee7da37094b6": "\\int_{0}^{+\\infty}(\\omega-\\omega_0)^2|X_\\mathrm{a}(\\omega)|^2\\, d\\omega.", "abb86e782995775981ae4284dc8c161f": "\\!a+(n+1)b = (a+nb)+b", "abb8bfb58841b3e930f43d9790a5be8d": "\\frac{1}{|B|} \\int_B \\omega(x) \\, dx \\leq C\\omega(x), ", "abb8d1ebc0846abf92718e617a70b001": "\\begin{array} {l}\n0=\n\\frac{f\\left(x_0 + h\\right) - f(x_0)}{h}\n+\\frac{f\\left(x_0 - h\\right) - f(x_0)}{h}\n- 2\\frac{f^{(2)}(x_0)}{2!}h - 2\\frac{f^{(4)}(x_0)}{4!}h^3 + \\cdots\n\\end{array}", "abb8dc4424bd2ea6a7c3e4e4653c4ce2": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 2}{4 \\choose 2}^2 \\end{matrix}", "abb94fc8ee4e767b2891cc317c854916": "\\mathcal{H}_n", "abb9dea485c8def862e5173930d5e7ea": " \\vec n_1,\\vec n_2, \\vec n_3", "abb9f110d59e7aebee80d0df0ee73a2b": " Q_2 = \\left[ \\begin{matrix} \\cosh(\\beta) & 0 & 0 & \\sinh(\\beta) \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n \\sinh(\\beta) & 0 & 0 & \\cosh(\\beta) \\end{matrix} \\right] = \\exp \\left ( \\beta\\left[ \\begin{matrix} \n 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 \\\\\n 1 & 0 & 0 & 0 \n\\end{matrix} \\right] \\right )~. ", "abba020a0111529b700d4930e05448d4": "t=t_0+dt", "abba247a0b98f8e6cc674818b7fd1ee5": "f_i: (X,\\tau) \\to Y_i", "abba4a67fd7c784934b2e473e87abb04": "I_{r-1,r-2}", "abba7796e96b1271f9d4b89080d01854": "1+f(x)g(\\theta)", "abba7baf58d0dcc82ab7b49ad67233a3": " n = ( \\frac{ t }{ D } )^2 ( \\frac{ a + 1 } { m } + b - 1 )", "abbaf6ece5c8869846657918307c0501": " \\mathbf{S} = \\mathbf{W}\\mathbf{X}", "abbb4819d673c4ef29f8e480d4cbfac6": "\\boldsymbol{S}", "abbb5b0032d4ba129a61e50515c64813": "\\alpha=5\\,", "abbb8e57a3f674000054ea2fb9c4a20b": "{\\scriptstyle\\frac{-\\sqrt{3}}{2}}+{\\scriptstyle\\frac{1}{2}}i", "abbbc49d078b5dafb3e8e3d8e458ab29": "+ \\ R_2 \\left( \\frac { ( R_1 \\parallel r_E ) + r_{\\pi}} {( R_1 \\parallel r_E ) + r_{\\pi} + R_2 } \\right) \\ . ", "abbbf46d62abca6943a048491b7a2567": " \\dfrac{\\partial \\Psi}{\\partial t} = -\\dfrac{i E}{\\hbar} Ae^{i(\\mathbf{p}\\cdot\\mathbf{r}-Et)/\\hbar} = -\\dfrac{i E}{\\hbar} \\Psi ", "abbc0cea9d5f4d8d977dd6a2df1e0df6": "(20)\\qquad \\mathcal{L}_{\\ell}m=[\\ell,m]\\,\\hat{=}\\,0\\;\\Rightarrow\\; \\delta D-D\\delta=(\\bar{\\alpha}+\\beta-\\bar{\\pi})D+\\kappa_{}\\Delta-(\\bar{\\rho}+\\varepsilon-\\bar{\\varepsilon})\\delta-\\sigma\\bar{\\delta} \\,\\hat{=}\\,0 \\,,", "abbc4e04dbd4bdcfefbcfa43e59d3cc6": "\\mathfrak{P}^{45}", "abbc61a61089f8364b8a7f65ebc324e6": "\\eta,\\tau", "abbc9c3f2b9dad37124621e20eaed451": "g_\\varepsilon ", "abbce60dde4ab0cb84464bbacf0e4044": "\\Gamma_{ij,k}^{(\\alpha)}=-D^{(\\alpha)}[\\partial_i\\partial_j||\\partial_k]\\; (D^{(-\\alpha)}[p||q]=D^{(\\alpha)}[q||p])", "abbd116714a6d03f00ded2b11568ca12": "\\sigma=\\begin{pmatrix}\n1 & 2 & 3 & 4 \\\\\n2 & 3 & 4 & 1 \\end{pmatrix}.", "abbd2872dc935c3f7dcabfdd9ac47ef7": "\n(y - y_3)(y - y_1)(y - y_2)(y - y_5)(y - y_4)\n", "abbd4000cbf35b612e8ea69dcbbea617": "C_{\\Delta x, \\Delta y}(i,j)=\\sum_{p=1}^n\\sum_{q=1}^m\\begin{cases} 1, & \\mbox{if }I(p,q)=i\\mbox{ and }I(p+\\Delta x,q+\\Delta y)=j \\\\ 0, & \\mbox{otherwise}\\end{cases}", "abbd4aa536d7cb45061e96b900698f63": "c_0 = a_1[v_1] + a_2[v_2] + a_3[v_3]", "abbd7748f2f9aaa905e8614a575ece5d": "y_5", "abbd82143314837875c6607cc6d9027e": "\\{\\ldots \\; , \\; J_- J_- | j,m \\rangle \\; , \\; J_- | j,m \\rangle \\; , \\; | j,m \\rangle \\; , \\; J_+ | j,m \\rangle \\; , \\; J_+ J_+ | j,m \\rangle \\; , \\; \\ldots \\} ", "abbda177b1c77bd82b3ccd4c387cd221": "R\\sqrt{2\\pi e / n}", "abbdbc2a9058cd7ae6a04222e52f8b86": "\n(dE_\\lambda/d\\lambda)\\vert_{\\lambda=1}=-I_1-3I_2=0\\,", "abbdbcdda900911295f20278844d6897": "\\Phi(b) = p^*(b) \\smile U.", "abbddff45ab6b8caca6846cf7a8cb2b2": "\\frac{f(k;\\mu_1,\\mu_2)}{f(-k;\\mu_1,\\mu_2)}=\\left(\\frac{\\mu_1}{\\mu_2}\\right)^k", "abbdf190ea2a5ab14d484022c25f0d6b": "\\sqrt[53]{2}", "abbdf95ea546f5b7a6972fde1a2ea348": "Q\\,(N_f,1)_{1/N_c,(N_f-N_c)/N_f}", "abbe593421e6470d19b431d2a6d1ee5f": "\\{o_n, r_1, r_2 ... r_{t-1}\\}", "abbe7e9b36c799f614bed9620f411ac3": "\\psi(x,y) = \\psi(y,x)", "abbe81ba9c027b561f9dcf8809c6a391": "V(q) = -q^3 + 2q^2 - 2q + 3 - 2q^{-1} + 2q^{-2} - q^{-3}. \\, ", "abbe9560ebb3910933677a258425d2fa": "\\eta_\\varepsilon(x)=\\frac{1}{\\pi x}\\sin\\left(\\frac{x}{\\varepsilon}\\right)=\\frac{1}{2\\pi}\\int_{-\\frac{1}{\\varepsilon}}^{\\frac{1}{\\varepsilon}} \\cos(kx)\\;dk ", "abbedca92e3a86f7378988d8b73f95a2": "p_2(x,t) =\\,", "abbeec8a36712a4e52f465441a289d40": " \\gamma_x ~ (\\gamma_y ) ", "abbf68163a685c67896339095a173399": "I(\\nu,T)\\,", "abbfb587effcdb51a3db8df39c3eddef": "f=\\frac{1}{2\\pi RC}", "abbfbcd73c10c3fe1ae0a0c847fe4c07": " SI = \\frac{ 1 }{ 2 }\\sum_{ i = 1 }^K | \\frac{ A_i }{ A } - \\frac{ t_i - A_i }{ T - A } | ", "abbfcdcd5eb1b179ec9d3fe13beef716": "F/A", "abc0793b8609d660a561b876b952746c": "Var(\\bar{X}_n)\\to cn^{2H-2},~\\forall c>0", "abc085586d6ad7810cc7aba1d67f1e95": "2\\alpha h", "abc0cd9d716ce0de5e9523fa134c6233": "\\omega^{A:B}_{x}=\\omega^{A}_{B}\\otimes\\omega^{B}_{x}\\,\\!", "abc11d4c766a2f49aeb3c3cb43c65277": "\\rho _k = \\,\\,\\left( {\\,1\\,\\, - \\,\\,\\alpha \\,} \\right)^k", "abc15b4a9cec96c9ecb703d3454fb8ef": "\\textstyle n\\in\\mathbb{N}", "abc1622a3c6fb16f5f44d04598a06f5a": "k(L+x) \\approx n\\pi", "abc19befbf177faa35ce73da47825791": "\\beta ^{A}(z)", "abc1a99337a390ca61f1f6c35adab006": "Sp(n,\\mathbb C)\\times Sp(2m-n,\\mathbb C)", "abc1fdb443ac5b410042af773738ed30": "\\Delta x = L / (N+1)", "abc2075ed6a54a440c43d72497a79533": "P^{T}QP = \\begin{bmatrix}\nR_1 & & \\\\ & \\ddots & \\\\ & & R_k\n\\end{bmatrix}\\ (n\\text{ even}),\\ P^{T}QP = \\begin{bmatrix}\nR_1 & & & \\\\ & \\ddots & & \\\\ & & R_k & \\\\ & & & 1\n\\end{bmatrix}\\ (n\\text{ odd}).", "abc2b78322ff9572931442fca609c8f6": "\\scriptstyle\\Lambda", "abc2c9df5b4d5455e77cc074663f2d8b": " A = k[x,y]", "abc2d21f9454799fedbf5b38bc1c0906": " {\\Gamma \\vdash A:B} \\over {\\Gamma' \\vdash C:D} ", "abc340b412ca4d7a3ecb59e1d4eaf8f9": "s_x, s_y, s_z,...", "abc37b6749e00e03d17c631e30939a74": " \\equiv ", "abc410bda05f16183651903688a0cf02": "d \\Gamma_n = \\frac{S \\left|\\mathcal{M} \\right|^2}{2M} d \\Phi_n (P; p_1, p_2,\\dots, p_n) \\,", "abc420ec9c53fa7a2144cb5d7620ce56": "\\begin{align}\nH \\star W &= \\left(\\frac{1}{2}m \\omega^2 x^2 + \\frac{p^2}{2m}\\right) \\star W \\\\\n&= \\left(\\frac{1}{2}m \\omega^2 \\left( x+\\frac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{p} \\right)^2 + \\frac{1}{2m}\\left(p - \\frac{i \\hbar}{2} \\stackrel{\\rightarrow }{\\partial }_{x}\\right)^2\\right) ~ W\\\\\n&= \\left( \\frac{1}{2}m \\omega^2 \\left(x^2 - \\frac{\\hbar^2}{4} \\stackrel{\\rightarrow }{\\partial }_{p}^2 \\right) + \\frac{1}{2m}\\left( p^2 - \\frac{\\hbar^2}{4} \\stackrel{\\rightarrow }{\\partial }_{x}^2 \\right) \\right) ~ W\\\\ \n&\\, \\, \\, \\, \\, + \\frac{i \\hbar}{2} \\left(m \\omega^2 x \\stackrel{\\rightarrow }{\\partial }_{p} - \\frac{p}{m} \\stackrel{\\rightarrow }{\\partial }_{x}\\right) ~ W \\\\\n&= E \\cdot W. \n\\end{align}", "abc450606c7adbf02dc26c360e5009a1": "i_r = \\frac {1 + i_n} {1 + p_e} - 1\\,\\!", "abc4f0910b114955d2046e6ae6b84363": "{r_{critical}} = {k \\over h}", "abc53d05884f6cf8d542d6b024ee8e7e": "S(x)\\Lambda(x)=\\lambda_0\\sum_{j=1}^v e_j\\alpha^{c\\,i_j} ((x\\alpha^{i_j})^{d-1}-1)\\prod_{\\ell\\in\\{1,\\dots,v\\}\\setminus\\{j\\}} (\\alpha^{i_\\ell}x-1).", "abc590bc44915572f835550d26040eff": "m^2s^{-1}", "abc5abbed2dd23fe741b020f87408a1c": "m(i)=[\\varrho(e)]_i=e(b(i))", "abc5c30c310f1540fb7d7e11e7f8e97a": "\\frac{\\left(\\mathbf{B}\\cdot\\nabla\\right)\\mathbf{B}}{\\mu_0} \\, (\\text{S.I.}) \\qquad\n\\frac{(\\mathbf B\\cdot\\nabla)\\mathbf B}{4\\pi} \\,(\\text{c.g.s.})", "abc5cab56dfcc876fe7d5449010fddac": "\\left(-\\sqrt{2}/2,-\\sqrt{2}/2\\right)", "abc658947b5bd9acd817cb6f09b2fa88": "d_\\nabla(\\omega\\wedge\\eta) = d_\\nabla\\omega\\wedge\\eta + (-1)^p\\,\\omega\\wedge d\\eta", "abc6790a30c627a994d8333eb5cfb2f0": "i\\hbar\\dfrac{\\partial \\Psi}{\\partial t}= -\\dfrac{\\hbar^2}{2m}\\nabla^2\\Psi +V\\Psi", "abc6f6b85902e07af8550732d8fee12d": "\\bar X.", "abc73e5a559c20018aed8d67053a7791": "r(0)=r(2\\pi)", "abc7bdfca301a9532de964c84af0987a": " y' = z x = r^2 \\, \\cos \\theta \\, \\sin \\theta \\, \\cos \\phi, ", "abc81c0d587de44082332981de07d150": "[A\\to B]", "abc84d7b4edafd9b20aa035f894147ab": "b_0, b_1, ...b_n", "abc901725d0c4b7abb7d3441a5c6c7de": " \\frac{\\partial \\Pi (\\mathbf{\\xi}, t)}{\\partial t} = \\sum_{ik} A_{ik} \\frac{\\partial (\\xi_k \\Pi)}{\\partial \\xi_i} + \\frac{1}{2} \\sum_{ik} [\\mathbf{BB}^T]_{ik} \\frac{\\partial^2 \\Pi}{\\partial \\xi_i \\, \\partial \\xi_k}, ", "abc945802485774057693132921be29f": "\n\\begin{bmatrix}\nf_{x1} \\\\\nf_{y1} \\\\\nf_{x2} \\\\\nf_{y2} \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nk_{11} & k_{12} & k_{13} & k_{14} \\\\\nk_{21} & k_{22} & k_{23} & k_{24} \\\\\nk_{31} & k_{32} & k_{33} & k_{34} \\\\\nk_{41} & k_{42} & k_{43} & k_{44} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nu_{x1} \\\\\nu_{y1} \\\\\nu_{x2} \\\\\nu_{y2} \\\\\n\\end{bmatrix}\n", "abc94606fafb9297fdb6c78e3b38d365": " \\ +j \\omega C_C (R_A//R_i)\\,\\! ", "abc9cf42f7524af7d95c275bcb2ff600": "\\alpha \\ ", "abc9d36b6e20d6cb5e80a60fcc8e8aff": "\\gamma^k = -\\gamma_k", "abca039ff7a37305fe2f8f6dcb6522c4": "m\\frac{d}{dt}\\langle \\mathbf{r}\\rangle = \\langle \\mathbf{p} \\rangle", "abca6523e50e9467a40dccd7581f6882": "\\succ^p_W", "abcac7b29cb6825ed2579b9f52a232fc": "(0,-1,g), (g,0,1), (0,1,g),", "abcafb64d7d90bdbdd3271396d4b7340": "b' Y", "abcb1e70e200129423775b97e065ee9f": "P_{-\\ell} ^{m} = P_{\\ell-1} ^{m},\\ (\\ell=1,\\,2,\\, ...)", "abcb7a843aa5534c03d5cb001c2339ab": "\\text{Spindle speed (RPM)} = \\frac{SFM}{\\pi \\times \\frac{1}{12} \\times \\text{stock diameter (in)}} \\approx \\frac{SFM}{0.2618 \\times \\text{stock diameter (in)}}", "abcb939b1e5cf22bbbdcc21740abd3eb": " P = \\mathrm{d}E/\\mathrm{d}t \\,\\!", "abcbccb32bc5ce48b42e9593d0a18ebd": " \\alpha_i\\ge 0,~i>0", "abcbf89f35f9bde3e4d7ac0d376c4e34": "\\operatorname{Pr}_i(\\{t \\geq 0 : X_t = i \\} \\text{ is unbounded}) = 0.", "abcc05bba4be9ee7d433692723e88d21": "L(s\\otimes s)=L(n\\otimes s)=L(n\\otimes n)=s", "abcc0ecdb540831d09b5198bada23677": "\\frac{l-\\mu}{\\sigma}=-z,\\frac{u-\\mu}{\\sigma}=z,", "abcc4c1a4fcfa370f72effe982e46333": "S_{i+1} = C^{-1}S_i\\cup S_i^{-1}C", "abcc63a60a7674b4127d74303f2b26fc": "\\frac{\\delta S}{\\delta \\phi(x)}\\left[-i \\frac{\\delta}{\\delta J}\\right]Z[J]+J(x)Z[J]=0.", "abcc6a389147504575d201200462a691": "\\exp \\colon \\mathbb{R} \\to \\mathbb{R}^\\times_{>0}", "abcc856b9d8e0fef77e26d38f1468741": "v=r+\\gamma Pv. \\,", "abcc9007c39b3e8be6176557e7995583": "(A, 1, \\cdot, -^l, -^r, \\leq)", "abcc91f6eb9ce1c79b9dcd8632654e10": "C_1-C_2=-\\frac{n^2}{n_1n_2}D_{12}\\left\\{ \\nabla \\left(\\frac{n_1}{n}\\right)+ \\frac{n_1n_2 (m_2-m_1)}{n (m_1n_1+m_2n_2)}\\nabla P- \\frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2)+k_T \\frac{1}{T}\\nabla T\\right\\}", "abcc9c1262bdbd391a37321281c9193d": " F(x;\\mu,\\sigma,\\xi)=\\begin{cases} e^{-(-(x-\\mu)/\\sigma)^{\\alpha}} & x<\\mu \\\\ 1 & x\\geq \\mu \\end{cases}", "abcc9e8aecb1be3bf700beb3caada771": "=\\tfrac{1}{2}((3x_1-\\cos(x_2x_3)-\\tfrac{3}{2})^2 + (4x_1^2-625x_2^2+2x_2-1)^2 + (\\exp(-x_1x_2)+20x_3+\\tfrac{10\\pi-3}{3})^2) ", "abcd1a42f2d38e2f578eb8bef6ea4453": "G(\\infty)", "abcd1c221b5c5b924ac7779f5f8ee54d": "g(x)=\\frac{e^x}{\\cos x}.\\!", "abcd38a580e87bca72656b4e6f59b5af": "\\hat{\\rho}(t)=\\hat{U}^{\\dagger}(t)\\hat{\\rho}(0)\\hat{U}(t)", "abcd758905d8653b2634b1ecb714cf78": "k^2 = \\frac{1}{\\hbar^2}2mE_n", "abcd9df4da7f7f36f25606c92b7472f9": "Fx_1...x_n.", "abcdc1fdd631d8acadce27cde43f2084": "I^{i + j}/I^{i + j + 1}", "abcdd0fa362a0e55bff8c83656eb40ed": " G=\\pi(T) ", "abcdd9af668bc1dfd8b993b29c0493ea": "i=k-m, k-m+1, \\ldots, k-1", "abcde271ee06deb99af756595b6512a8": "u_0+\\langle u_1,\\ldots,u_m\\rangle=\\{u_0+t_1u_1+\\ldots+t_mu_m \\mid t_1,\\ldots,t_m\\in\\mathbb{N}\\}", "abcde5e5056c6f8585fb5d45b4e1bc23": "G(x) = \\exp(-|x|)", "abce5fea778228f0eef81eb7e20460e9": "B2: = A2 - A3", "abce6561547cc84ee413d9404a794c96": "\\beta l = {\\pi \\over 2}\\ ,", "abceb0245d877876dfce8e0e30d0ab2b": "(f_1 f_2 f_3 \\cdots f_n)' \\!", "abcedfdaf9d98379b9194921eda626af": "\\scriptstyle(-0.9(2.6))\\times10^{-12}", "abcf3a3ee7c61889fabbcf45e91f96e8": "({\\color{red}p_1}, {\\color{red}p_2}, {\\color{blue}d_1}, {\\color{red}p_3}, {\\color{blue}d_2}, {\\color{blue}d_3}, {\\color{blue}d_4}, {\\color{green}p_4})", "abcf3af17a8c38c7619d04653fce8c43": "\\text{cov}[X, Y] = E[X Y] - E[X] E[Y],", "abcf4d55557aebef87c9c0add3a78758": "T=0", "abcfca49f6a2a79075a4bbd35dee55c7": "\\log_{10}\\left(\\frac{Z/X}{Z_\\mathrm{sun}/X_\\mathrm{sun}}\\right) = A*[\\mathrm{Fe}/\\mathrm{H}].", "abcfef73fe9ee326f7dc728673a13673": "\\begin{align}\n\\gamma & = \\dfrac{1}{\\sqrt{1 - \\beta^2}} \\\\\n& = \\sum_{n=0}^{\\infty} \\beta^{2n}\\prod_{k=1}^n \\left(\\dfrac{2k - 1}{2k}\\right) \\\\\n& = 1 + \\tfrac12 \\beta^2 + \\tfrac38 \\beta^4 + \\tfrac{5}{16} \\beta^6 + \\tfrac{35}{128} \\beta^8 + \\cdots \\\\\n\\end{align}", "abd0424ae5c61a6e68e27096e87e2b81": "BP_1=f_1BP", "abd091e7dc157e232aa677e453a44420": "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ \\frac{\\sqrt{8}}{3} \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ y_2 \\\\ z_2 \\end{pmatrix}, \\begin{pmatrix} -\\frac{1}{3} \\\\ y_3 \\\\ z_3 \\end{pmatrix}", "abd0936690e20033b4f21af07fecfc97": "x\\in F", "abd0dd392054044cb99c7ddc379f63b8": "S_{i,t}=S_{i,t-1}-Z_{i,t}", "abd0e043e7ac4d90a16e57cd9b4085a3": " u = \\left ( \\gamma , \\gamma { \\mathbf{v} \\over c } \\right ) ", "abd0e6c88761b4b4901122a208b65bab": "(P^2-4Q) U_n = {V_{n+1} - Q V_{n-1}}=2V_{n+1}-P V_n \\,", "abd1554390bc09e0938caa910192acca": "K^{+}\\hat{g}", "abd17207cc5b5590640fada50f8fec9c": "R(k) = d \\cdot sin \\{ c \\cdot arctan [ b(1-e)k+e \\cdot arctan(bk) ] \\} \\,", "abd1c86a290f6ba2d0f4263826ebd82c": "\\tilde{g}_{\\mu\\nu}=\\Phi g_{\\mu\\nu},", "abd1e130a2bbebe96023dadec282af95": " I_8 ~,~ \\Gamma_{a_1 a_2 a_3} ", "abd214782895716b63cfbb50d0df2207": "\\mu_1 = \\arcsin \\left( \\frac{1}{M_1} \\right)", "abd229c1d6ef35d90dd64f73bd1f6020": " \\sum_{p \\in P_z} \\frac{w(p)}{p} < D \\log\\log z + E; ", "abd250999e807d1b3fa872e57dac5448": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}}\\det(\\boldsymbol{A}) = \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T ~.\n", "abd28f28d4cd679b347a68869749bf61": "MTF_{smear}(u) = \\frac {sin(\\pi \\alpha u)} {\\pi \\alpha u}", "abd2e10b57260ad7408214fe793d4c65": "R_{\\mu\\nu} = T_{\\mu\\nu}\\,", "abd2f0e5539f963edd3fef82b29b55eb": "\\scriptstyle \\tfrac{s^2}{l^2}", "abd319b87d4e09464ab16ec7e773026d": "f(\\{x_n\\})=\\{f(x_n)\\}\\,", "abd373c5b3160398b29b5a7115ee8104": "PV = \\frac{$1000}{(1+0.10)^{5}} = $620.92 \\, ", "abd3f083112df5b98ad23d77116f6005": "B_\\theta\\in D", "abd472e27f80ef4323f3dbddec3d092d": "0<\\beta<1", "abd51bf21852ef807dc9b9fd2e4e70a5": " T_e = 255 ~\\mathrm{K} = -18 ~\\mathrm{C} ", "abd520c263238d477470356387d4cea9": "{n\\choose k} = {n^{\\underline{k}}\\over k!}.", "abd5288c5a9a9d828dde59de97a99f2a": "E_\\mathrm{k,molecule} = {{1}\\over{2}}mv^2", "abd5413c52ba903404685586fceff85d": "P \\to (P \\or Q)", "abd5597d42a9b8017bb0bb66893d4fc1": "\nt(x) = x + \\sqrt{x^{-1} + 1}\\,\\sinh^{-1}\\!\\sqrt x,\n", "abd592f9d2869bed9c9c230bbe48c90a": "{\\widehat{HH}}_3", "abd5b37083d43eb16754ae7454c50864": "S=p^{\\mathbf{N}}=\\{p^{n}:n\\in\\mathbf{N}\\}", "abd5b5d7fd61cf02dfeaa7f738fe6a4a": "e = \\sqrt{1+\\frac{2\\epsilon h^2}{\\mu^2}},", "abd5c8f9401ac23d16b750a301885c05": "\\tbinom{n}{0}=1", "abd5e9490ece485e510f03244d61d842": "\\mathcal P:= \\overline{K}^2,\\qquad\n\\mathcal Z:= \\{\\{(a_1,b_1),(a_2,b_2),(a_3,b_3)\\} \\ | \\ \\{a_1,a_2,a_3\\}=\\{b_1,b_2,b_3\\}=\\overline{K}\\}", "abd67c7b80d58f7150fe83c907b34ab3": "\\delta_{\\sigma \\sigma'}", "abd6aa69ecb6b1ee5f789f205d2bf47b": "V=(\\frac{5}{3\\sqrt{2}})a^3\\approx1.17851...a^3", "abd6ce38e414844044469d17618c462c": " \\lambda_1, \\lambda_2, ..., \\lambda_n", "abd70fcb11dcb80b23041a208d88e0b4": "x = a-a+a-a+\\cdots = a-(a-a+a-\\cdots),", "abd715f5eaf1dc70adfe59d40088ad4f": "\\rho = 2 \\frac{1}{(2\\pi)^3} \\frac{4}{3} \\pi k_F^3 \\quad , \\quad E_F = \\frac{\\hbar^2 k_F^2}{2m}\\quad , \\quad\n\\rho \\propto E_F^{3/2}.", "abd71888526b3509e09029e014e6434a": "(a+b+c)\\left(\\frac{1}{b+c}+\\frac{1}{a+c}+\\frac{1}{a+b}\\right)\\geq\\frac{9}{2}", "abd76b962cd11953fb780e4cad2c9515": "T_2 \\,", "abd77290d03cdc9d8ce802654d2eb34e": "S_{\\text{0}},S_{\\text{1}}, ..., S_{\\text{n}}.", "abd780cb821c095b693b315ae685862c": "\\alpha^n = -(b_{n-1} \\alpha^{n-1} + \\cdots + b_1 \\alpha + b_0).", "abd78e3bcc58a7354ae94282d054c2bc": "\n f(u) \\propto \\left \\{ \\begin{array}{lr} u^{\\beta} & \\ u\\ll 1 \\\\\n 1 & \\ u\\gg1\\end{array} \\right.\n", "abd829668430424bf471a61f761f95f3": "\\tfrac{n}{\\pi}\\tan{\\tfrac{\\pi}{n}}", "abd82dc8b537b3d273d7882fbb2f19c9": "\\kappa(\\mathbf{M}^{-1}\\mathbf{A})", "abd83bf922a77a387a7486cbdef5727f": "\\omega_r=\\omega_a", "abd842f9a57d7902dd51fdd98335e468": "v \\approx \\frac {gh}{c}", "abd8afb12bd6782b8ec1466f6752e400": "\\mu_{v \\to u}", "abd8c3a08024c4be7be3beba4190c4ba": "A_e=A_{\\overline e}", "abd8cc33fea5e14047eef8eedcdef17a": " K_I", "abd8dbe52476cdad51607b8a676c0e05": "E_\\mathrm{res}", "abd91552b757e3175addf8f4be9a407f": "E(X^m) = \nE\\left( \\sum_{k=0}^m \\left\\{ \\begin{matrix} m \\\\ k \\end{matrix} \\right\\} (X)_k \\right) =\n\\sum_{k=0}^m \\left\\{ \\begin{matrix} m \\\\ k \\end{matrix} \\right\\} E((X)_k),", "abd9195116cf3b212a9c164efe95a0b7": "H_2L \\rightleftharpoons HL+H:pK_2=-\\log \\left(\\frac{[HL][H]} {[H_2L]} \\right)", "abd9481b4bea1cf3a6c85d7cc17d76bb": " \\psi_c := i \\psi^*.\\ ", "abd96164e4e6d216cf1c4aabbce0b679": "r_j = z \\left(1 + \\frac{\\left(x - x^\\prime - j d \\right)^2 + y^{\\prime2}}{z^2}\\right)^\\frac{1}{2}", "abd983d9da70af55f151447168df26b0": "c_1: X_1 \\rightarrow Y_1", "abd9ce174c74b1e234a6a1be1934a8db": "x\\geq -1, z\\geq -\\frac{a}{b}", "abda031353b20880f407a5ee90ade6b2": "\\textstyle{3}", "abda0c4299f273d27a927a457ea64178": "=\\prod_{k=1}^n [k]_q", "abda651f4001205017700d14c235694a": "c(i,j)\\ge c(i,k)", "abdaa97692d9b310ce2a75dc05976f0d": " \\alpha_1 = 1;\\alpha_m\\neq 1 ", "abdabdaa7f0183809bad3f01e3cee43f": "d \\mathbf{S}\\times \\mathbf{F}.", "abdaf22d3e48b0e1b3a6dc5870474072": "\\frac{dy}{dx}= \\text{st} \\left(\\frac{\\Delta y}{\\Delta x}\\right)", "abdb018c77a88e4194fb9a57693b9df5": "\\frac{\\partial u}{\\partial t}+\\frac{\\partial u}{\\partial x}=0 \\quad \\quad (1)", "abdb0c0abb2b095b8297d690be2d9918": "\\begin{bmatrix}\\mathrm{odd} & \\mathrm{even} \\\\ \\mathrm{even}& \\mathrm{odd} \\end{bmatrix}", "abdb8030de2d730da5bd63247fbc910b": "P \\cup \\Delta \\models G", "abdb93f470c3b9f7a009e04fd3bb40ce": "\nG_a(t) = min[1-(1+\\beta t) exp(-\\beta t), \\gamma] \\quad for\\ t \\geq 0\n", "abdb998864efd2d51c6064f8cf93cc08": "115.8\\pm 2.3%", "abdc9b3d5823de15453a8ffd4f20b014": "\\cdots\\rightarrow\\tilde{H}_0(A\\cap B)\\,\\xrightarrow{(i_*,j_*)}\\,\\tilde{H}_0(A)\\oplus\\tilde{H}_0(B)\\,\\xrightarrow{k_* - l_*}\\,\\tilde{H}_0(X)\\rightarrow\\,0.", "abdd207c6254eb7b3f3505d3feb78c6e": "\\mathcal{E}", "abdd34cbaa58b3fc8472cae9f0656872": "t_{i,1}", "abdd3bcdfac5393f228a03abdfda4d10": " e^{-S} = e^{-S_F} ( 1 + X + {1\\over 2!} X X + {1\\over 3!} X X X + ... ) ", "abdd4feb7d8367f876cd78d49037d4e3": "s((X_1, X_2)) = X_1 + X_2\\,\\!", "abdd8cfa509c517c54dd678b067b9981": "(n-2)\\times \\frac{180^\\circ}{n}", "abddb92a0bc3db8c2f9a90c34e011949": "B[u, v] = \\int_{\\Omega} \\nabla u(x) \\cdot \\nabla v(x) \\, \\mathrm{d} x,", "abde2d571b87171f70037154d435be6c": "\\scriptstyle TM_{mnl}", "abde4821cad9b35bd53028f5be6a87e2": "z'_k", "abde537f99ed8b022d9ce4989d7a984e": " N-N'=\\frac{D}{\\lambda}-\\frac{D}{\\lambda'}=1 ", "abdf0113fc11c5df8593e02fc4344d38": "V_{j(\\kappa)}", "abdf0a7975c71cb03accc073ca604ea6": "V V^*", "abdf70e36e46b927516bd2b1651893df": "y=\\sqrt{x} \\, ", "abdf7aec7cd864f209b272581d8aa34a": "2 \\sigma_1 / D", "abdfcdf2cfa38cf95f8c1522a569d84c": "\\ p\\mathbb{Z}_{p} ", "abe0371433e00012394cb32a2697181e": "G=S_3", "abe08ed521f10358b5624aa5f31f377e": "\\gamma=1/\\sqrt{1-v^2/c^2}", "abe098205f322b268f0504ae199853f8": "WTS(O_j) > TS(T_i)", "abe09ca8cecc2a1d5e2224379d1d1b48": "\\scriptstyle \\ z", "abe110ff55afaf17c67c4a0a462a0044": "\\,d^{(m)}", "abe124c8df7cec2ebc6512fefe37e147": "\\mathrm{If}\\; \\alpha \\ge 0 \\; \\mathrm{and} \\; Z \\in \\mathcal{L} ,\\; \\mathrm{then} \\; \\varrho(\\alpha Z) = \\alpha \\varrho(Z)", "abe12b73663c9dee2f1f5e7ae7435526": "h_k = \\lfloor 2.25 h_{k-1} \\rfloor", "abe13714548edeef266cb5575abf29cf": "p(A)=A^2-5A-2I_2=\\begin{pmatrix}0&0\\\\0&0\\\\\\end{pmatrix},", "abe13dae3ec64d3ca21add667f307322": "\\mathcal C_k,\\mathcal D_k", "abe1b2b8b6e475ab3b910c8026eba9fb": "\\tilde E_6", "abe1dca3d46ab06064fc68bc91862b7e": "S_3 = 4\\pi", "abe20e08a0a379a06c9442ef6fc151e5": "\n\\nabla^{2} \\Phi = \\frac{1}{a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right)} \\left( \\frac{\\partial^{2} \\Phi}{\\partial \\mu^{2}} + \\frac{\\partial^{2} \\Phi}{\\partial \\nu^{2}} \\right) + \\frac{\\partial^{2} \\Phi}{\\partial z^{2}} \n", "abe23e4a67cc054f0ac3ee775082b359": "H = z+y+\\frac{v^2}{2g}", "abe24417e259877b5318c6f6eb70fd0a": "(q_i,p_j)", "abe2547af192cbce532cb594df02e854": "\\boldsymbol{\\sigma}=\n\\left[{\\begin{matrix}\n\\sigma _x & \\tau _{xy} & 0 \\\\\n\\tau _{xy} & \\sigma _y & 0 \\\\\n0 & 0 & 0 \\\\\n\\end{matrix}}\\right]\n\\equiv\n\n\\left[{\\begin{matrix}\n\\sigma _x & \\tau _{xy} \\\\\n\\tau _{xy} & \\sigma _y \\\\\n\n\\end{matrix}}\\right]\n", "abe2cdca559084f15d95f189830e683e": "\\left(\\frac{\\partial S}{\\partial X}\\right)_U=0", "abe2f695ed7297c0f6f14665d2113343": "\\begin{align} 2\\cdot R_*\n & = \\frac{(38.6\\cdot 3.94\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 32.7\\cdot R_{\\bigodot}\n\\end{align}", "abe31cd769f0511939eed618570bff10": " \\xi\\nu_t", "abe32b21e7c20ec51ee71496bf292f12": "\n q_1=\n \\begin{cases}\n q_2 = \\cfrac{Q_b}{2m} & Q_b\\ \\mbox{even} \\\\\n q_2+1 = \\cfrac{Q_b+m}{2m} & Q_b\\ \\mbox{odd}\\\\\n \\end{cases}\n", "abe33a15829642ab2c82fdf967040044": "(\\vec x_i,y_i)", "abe38f85a2c5563847485243acb3ce0d": "\\mbox{mex}(\\left \\{0, 2, 4, 6, \\ldots\\right \\}) = 1", "abe3bd0e272216195f1125b9f87a785f": "\\bar{r} = \\frac{1}{n}\\sum_{i=1}^n {r_i} = \\frac{1}{n} (r_1+\\cdots+r_n)", "abe3c868efd8e0743ed9f6308a4de578": "\\sigma_{0.25,\\,0.5}=\\sqrt{\\frac{0.5\\cdot 0.166^2-0.25\\cdot 0.18^2}{0.25}}=0.1507\\approx 15.1%", "abe3dcf088f8b089ce0b7a0c653fc855": "\\gamma(Y^*, Y)", "abe412a8b577a6828da13fa785e4c7e4": "\n\\begin{align}\n\\frac{d^2\\sigma (E_+,\\omega,\\Theta_+)}{dE_+d\\Omega_+} =\n\\sum\\limits_{j=1}^{6} I_j \n\\end{align}\n", "abe4b1d72a44845a564c189e364b9023": "z=a \\cos({m\\theta})", "abe4b859157e4acd6502a7904191c0e7": "\\hat{\\mathbf{k}} = \\mathbf{k} / \\sqrt{\\mathbf{k}\\cdot\\mathbf{k}}\\,\\!", "abe4cb33a54376063c4e7eccb5361ccf": "\\hat{\\mathcal{H}}^D_v \\left|\\Psi_{\\mu}^{v+k}\\right\\rangle = E_{\\mu}^{k} \\left|\\Psi_{\\mu}^{v+k}\\right\\rangle", "abe4df2bf2e6a1474a868b4d832fe289": "K_{G}^{(a)}", "abe552c330bc06acdb85278297e2bf0c": "V = \\pm x^2 + a x", "abe5b248adb9e03cec7126d9368ec62a": "d:P(E)\\rightarrow P(E)\\,", "abe5be673680098ed77c84069d8cb639": "\\frac{M}{m}=\\frac{p}{mv}=\\frac{E}{mc^{2}}=\\gamma", "abe5cab46dee5829713b58aa519caf6d": "\\Pi^1_3", "abe5dc393eea113c42baed93e28c091c": "\\rho^{123}", "abe5f469f9f57ec4aaf9bb67201f1585": "\\alpha =6", "abe6341979a836058c71d21bc6518acc": "o_3", "abe68bb9af3cd46f6420a5421865805a": " \\frac{\\partial n}{\\partial t} + \\frac{\\partial n}{\\partial a} = - \\mu(t,a) n. ", "abe6cacfda25837ad0f0f9d0ba10e8a0": "\\frac{3}{3-2\\sqrt{5}} = \\frac{3}{3-2\\sqrt{5}} \\cdot \\frac{3+2\\sqrt{5}}{3+2\\sqrt{5}} = \\frac{3(3+2\\sqrt{5})}{{3}^2 - (2\\sqrt{5})^2} = \\frac{ 3 (3 + 2\\sqrt{5} ) }{ 9 - 20 } = - \\frac{ 9+6 \\sqrt{5} }{11}", "abe6f375324d7d50c571514e6e58baba": "e^A=\\begin{bmatrix} e^{a_1} & 0 & \\ldots & 0 \\\\\n0 & e^{a_2} & \\ldots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\ldots & e^{a_n} \\end{bmatrix} ", "abe7905c3c99854a8c943d78696a6919": "T(Z,X,Y)=-\\frac12\\langle Z,(\\nabla_{X}J)Y\\rangle ", "abe79a9b6e121a2c0a6af34cf1712256": "\n\\operatorname{Im}(p-\\mu)", "abe79c6a9edd37b08f42696866b3dd43": "(ab)(a c_1 c_2 \\dots c_r b d_1 d_2 \\dots d_s) = (a c_1 c_2 \\dots c_r)(b d_1 d_2 \\dots d_s)", "abe7a4a1eb62e21a90bbd13ff01e882d": "D_\\mu=I\\partial_\\mu-igT^aA^a_\\mu ", "abe7eeaf26a6d1b3327ad53d186ec492": "{\\left( b^k x \\right) }_{k=0}^\\infty", "abe8e491cf5a3c28e79760f900f02a16": "\\bar x = \\frac {\\sum_{i=1}^m \\sum_{j=1}^n x_{ij}}{mn}", "abe8f929271aad2134c0496cdc5b5a7e": "V_\\mathrm{out} = K \\ln\\frac{V_\\mathrm{in}}{V_\\mathrm{ref}}", "abe90947ce6cf99d194aaad45fd3f1e9": "\n \\int (d+e\\,x)^{m+1} ((A\\,c\\,d-A\\,b\\,e+a\\,B\\,e) (m+1)+b (B\\,d-A\\,e) (p+1)+c (B\\,d-A\\,e) (m+2 p+3) x)\\left(a+b\\,x+c\\,x^2\\right)^pdx\n", "abe927437497e6790db2681273ce1ba9": "\\mathbf{S} = \\mathbf{Y} - \\hat{\\mathbf{B}}\\mathbf{X}", "abe933070ccdd02112d7fa4674dd6b1b": "\n{1 \\over m}\n", "abe94b32d451ef6370721129c57e52a4": "f(z)=(z, z_2; z_3, z_4). \\, ", "abe973b2894d787500da55074734d00a": "\\mathcal{C}^1 \\,", "abe99914634528b14e4f470eca6623f0": "\n p = \\frac{1}{V_0}\\,(B + 2C\\chi + 3D\\chi^2 + \\dots) + \\frac{\\Gamma_0}{V_0} \\left[e - (A + B \\chi + C \\chi^2 + D \\chi^3 + \\dots ) \\right] \\,.\n ", "abe9995b949c36bc8768f5c1e1f83a85": "f(t)=\\sum_{n=1}^\\infty\\frac{\\kappa_n}{n!}t^n", "abe9a99b72e9fabe5699fd9d569abbfd": "M_a.t = \\int_0^t rP(t)\\,dt \\, - \\int_0^t \\frac{dP(t)}{dt}\\,dt \\, ", "abe9d9bdfb2a46c7be74d8a118a8a24f": " \\delta t_o = \\frac{x^*}{\\left(c - v\\right)}", "abea400a2e638fe5d4f7118837825213": "g_{eff}=c_s|\\frac{\\nabla B}{B}|\\sim 1/R_0", "abea5609c8c83c34dab3d644f2f8adf9": "0 \\to A \\to B \\to C \\to 0\\ ", "abea63d3555d9722746118b486c03d75": "{\\overline P}X = {\\underline P}X", "abeae5b434ae3de7472ba7676545cc84": "\\int \\mathbf{J}_1 \\cdot \\mathbf{H}_2 \\, dV = \\int \\mathbf{H}_1 \\cdot \\mathbf{J}_2 \\, dV.", "abeb39ae28f2cb9fbd32abd2df4ef059": " r = r_0 + \\left( \\frac{v+v_0}{2} \\right )t \\quad [3] \\,\\!", "abeba55836d0d73f5ac83796eca14913": "\\scriptstyle{\\dot\\gamma_m(t)=i\\langle\\psi_m(t)|\\dot{\\psi_m}(t)\\rangle}", "abec0ce35b551758bc6d5d1a8a869f88": "f = 0", "abec83ced6cb79a8d5c997d2e83c6eda": "\\{ true, false \\} ", "abec84b38327efdd99430a7d05e596e7": "\\dot{\\textbf{x}} = A\\textbf{x}", "abec963f64d8e950ff77132728c1565a": "{1}/{\\sqrt{2\\pi}}", "abecc6a8e2fcddf00cc7ff57028ddfa5": " p \\in [0, 1]^n ", "abecf7fbaf6d603484c9f30695e56580": "\\scriptstyle E\\{\\cdot\\}", "abed6aa54ce778c8fd3e3222bda1a626": "T_b = \\frac{1730.63}{8.07131 - \\log_{10}P} - 233.426", "abed94fb5108c1d83460069c86ad65af": "K(x,y) = \\left(\\sum_{i=1}^n x_i y_i + c\\right)^2 = \\sum_{i=1}^n x_i^2 y_i^2 + \\sum_{i=2}^n \\sum_{j=1}^{i-1} \\sqrt{2} x_i y_i \\sqrt{2} x_j y_j + \\sum_{i=1}^n \\sqrt{2c} x_i \\sqrt{2c} y_i + c^2\n", "abee1a55f7085b9ee557b07120c6f262": " \\int G_{bip}\\, dt = 0 ", "abee6a327d042c0d73f7d4e638e2eb28": "\\mu(X_t,t)", "abee6ad68e8904202adbaaa761a6b374": "\\nu + p - n > 1", "abee71b9be56336e33e72c6f0b781f45": "\\scriptstyle0\\leq\\mu(A)-\\mu(A_m)\\leq\\varepsilon", "abeed406ee304a42bef73adc9adb9e5a": "\\lambda_1, \\ldots, \\lambda_n", "abeed86bd1c11dd5ffa499c2b3ccfd9e": "\\delta=0", "abeef60aa86dbfe719e64d86678bfa31": "\\{y_{n}\\}_{n\\in \\mathbb{N}}", "abef24526d3a709bf207e0f4b72c86ee": "a\\in I", "abef3c90b263c805d7ea6411094122b5": " G^2 = \\left ( \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} x^2}\\,dx \\right ) \\cdot \\left ( \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} y^2}\\,dy \\right ) = 2\\pi \\int_{0}^{\\infty} r e^{-{1 \\over 2} r^2}\\,dr = 2\\pi \\int_{0}^{\\infty} e^{- w}\\,dw = 2 \\pi.", "abef4c9eec34191d39e93f81304bd58f": "T\\vdash_{\\mathcal{S}}\\psi", "abef579df8a0bf7fac660394a3d53e3c": "\\mu=1+s", "abef5bce185c02b31f62a6ce7b9b7e30": "r = \\frac{d}{\\sqrt{3}}.", "abeff5efe418e6b27c4b0695a57d1ef2": "\\mathcal{L}(n) = [n \\ge m]\\frac{\\binom{m - 1}{k - 1}}{\\binom{n}{k}}", "abefff3e95bdefe5c25cae75e0b8474e": "F(x,y,y',\\ \\dots,\\ y^{(n-1)})=y^{(n)} \\quad x \\in [0,+\\infty)", "abf003211136252010e067dc6e6fc2c9": "((x,z),Q)", "abf07d7630ed20b7d1ab41c62c7f3afe": "\\left \\{ \\bot,\\top \\right \\}", "abf08d1092898eaa4eff494b26a3ef40": "X \\times \\mathbb{C}", "abf0bc5ce82702a1634f426844e268a0": " - 1 < E_d < 0 ", "abf0ee790657867792ef25cc8ef4d953": "[X^{(n)}]", "abf115bc664acd2e1cbf1428891ea86e": "f:\\mathbb{R}^{N}\\to\\mathbb{R}^{N}", "abf15afcc087c1df3336b0a66d5f99f4": "\\cos\\,\\alpha\\,\\cos\\,\\beta = \\frac12[\\cos(\\alpha+\\beta) + \\cos(\\alpha-\\beta)]", "abf18ff0c87bebe670838515030af9b1": " \n\\frac{\\partial}{\\partial x_i}\n\\left[c_{ijkl}\\frac{\\partial U_k}{\\partial x_l}\\right]=\\rho\\frac{\\partial^2 U_j}{\\partial t^2}\n", "abf1dfa3f54cfc89b6ce1082a8d92e89": "\\forall m [m |\\lambda_j|", "abfd6bf37cc633326c639fe38ac04516": "i \\in \\{1, 2, \\dots, m\\}", "abfd7e8d6ae8ba17d8bb60c4aba93716": "T_{min}", "abfd93c410438e9250bd01b14e989dba": "\\chi(\\pm1) = 1", "abfdf45775f6e3e491d20b585a70a16e": "\\{d^k\\}_{k \\in K}", "abfe00c8e71944a1b967b5deb988f40f": "P_{AT}(\\nu,\\kappa,\\pi) = \\pi_T\\left(1.0 - e^{-\\beta\\nu}\\right) ", "abfe325c7cb29a4ce3153a07aeb6b2e2": "\\theta_i\\;", "abfe460b06463f7faad47946c33520b7": "\\sum_{n=-\\infty}^\\infty a_n\\overline{b_n} = \\frac{1}{2\\pi} \\int_{-\\pi}^\\pi A(x)\\overline{B(x)} \\, dx,", "abfe46d2c6d0216291661d5b27752288": " \\ell=v_F\\tau = \\sqrt{2\\pi n} \\frac{\\hbar\\mu}{e} \\approx 5.2\\ \\mu\\mathrm{m}\\times \\mu\\ [10^6\\ \\mathrm{cm^2/Vs}]\\sqrt{n\\ [10^{11}\\ \\mathrm{cm^{-2}}]}\n", "abfe569d58b54900f5382681efa203d8": "\\ \\mu_{\\alpha} = \\alpha_1 - \\alpha \\ ", "abfe7afb7ffc2f4208964080191867d2": "\\partial f / \\partial x_i", "abfe7b8d235939e96e3bebd7da66bbed": "\\nabla \\cdot \\mathbf{v} = 0.", "abfeeb0786e472b7094cfb5f3b890e16": "\n\\begin{array}{cl}\n \\displaystyle\\frac{x:\\sigma \\in \\Gamma}{\\Gamma \\vdash x:\\sigma}&[\\mathtt{Var}]\\\\ \\\\\n \\displaystyle\\frac{\\Gamma \\vdash e_0:\\tau \\rightarrow \\tau' \\quad\\quad \\Gamma \\vdash e_1 : \\tau }{\\Gamma \\vdash e_0\\ e_1 : \\tau'}&[\\mathtt{App}]\\\\ \\\\\n \\displaystyle\\frac{\\Gamma,\\;x:\\tau\\vdash e:\\tau'}{\\Gamma \\vdash \\lambda\\ x\\ .\\ e : \\tau \\rightarrow \\tau'}&[\\mathtt{Abs}]\\\\ \\\\\n \\displaystyle\\frac{\\Gamma \\vdash e_0:\\sigma \\quad\\quad \\Gamma,\\,x:\\sigma \\vdash e_1:\\tau}{\\Gamma \\vdash \\mathtt{let}\\ x = e_0\\ \\mathtt{in}\\ e_1 : \\tau} &[\\mathtt{Let}]\\\\ \\\\ \\\\\n \\displaystyle\\frac{\\Gamma \\vdash e:\\sigma' \\quad \\sigma' \\sqsubseteq \\sigma}{\\Gamma \\vdash e:\\sigma}&[\\mathtt{Inst}]\\\\ \\\\\n \\displaystyle\\frac{\\Gamma \\vdash e:\\sigma \\quad \\alpha \\notin \\text{free}(\\Gamma)}{\\Gamma \\vdash e:\\forall\\ \\alpha\\ .\\ \\sigma}&[\\mathtt{Gen}]\\\\ \\\\\n \\end{array}", "abff05ee665c75c7abd7076a75a28156": "\\scriptstyle PP'\\ =\\ PC", "abff44b75b117f0d1a48e0d77ab7bcf5": "\\chi_G(\\lambda) = \\chi_{G\\setminus e}(\\lambda) - \\chi_{G/e}(\\lambda).", "abff6d8509b0141da911a32bd627f6c0": "\\scriptstyle Re\\, ", "ac003d21493f03e728dcf6e5f3ea57af": "\\boldsymbol{p}_{i+1}^\\mathrm{T}\\boldsymbol{Ap}_i=(\\boldsymbol{r}_{i+1}+\\beta_i\\boldsymbol{p}_i)^\\mathrm{T}\\boldsymbol{Ap}_i=0", "ac00eb88ee4ea52200065f1b3fe54359": "a = [1,2]", "ac01076899b2af4a499dee36b4d7ff03": "\\pi:M\\to M", "ac019df1afb0b59be5668a3a8084db2a": "u/v", "ac01d9317a46c92e16240b3e6258a6d5": "K=-{\\beta^2\\over ((s-\\alpha)^2 +\\beta^2)^2} ,\\,\\, K_m=-{r[(s-\\alpha)^2 +\\beta^2)] +\\beta_t(s-\\alpha) + \\beta\\alpha_t\\over\n [(s-\\alpha)^2 +\\beta^2]^{3/2}}.", "ac020649653a3f68c3b6fc7371913f2a": "\\,d_i", "ac023d50fab033be1bcd1f357c8e14ca": "\\textrm{dim}(d\\mathcal{O}_s(x_0)) = n", "ac02c1a1679e73519f597dcbc0196492": "y(t)=e^{W_k(a)t}", "ac04013fca763d6a0f2fd38d5aa709bd": "\ndu = \\frac{d\\xi}{\\sqrt{E \\cosh^{2} \\xi + \\left( \\frac{\\mu_{1} + \\mu_{2}}{a} \\right) \\cosh \\xi - \\gamma}} = \n\\frac{d\\eta}{\\sqrt{-E \\cos^{2} \\eta + \\left( \\frac{\\mu_{1} - \\mu_{2}}{a} \\right) \\cos \\eta + \\gamma}},\n", "ac040f2a7b0e2ecbaa70e89e9faa33d2": " n = n_0 e^{-(aR)^2} \\, ", "ac042c57667e134e332a2a1ba8b2fb4f": "\\mathbb{R}^{p,q}", "ac04525f7dc4b522c878e1eb9a8ef9f7": " \\mathcal F", "ac04a356a5eff9ce3c69e4f3eb944c61": "H_3.", "ac05084dade062f19aca746986a1b1a3": "2\\sin\\left(\\frac{108^\\circ}{2}\\right) = \\phi ", "ac050f4530c15728e3b4a88d4e9e583b": "\n\n\\begin{matrix}\n23G(2n-1) &=& 4u^2 + 3v^2 + 9w^2 + 18uv - 12uw - 4vw \\\\\n23G(2n) &=& - 6u^2 + 7v^2 - 2w^2 - 4uv + 18uw + 6vw\\\\\n23G(2n+1) &=& 9u^2 + v^2 + 3w^2 + 6uv - 4uw + 14vw \\\\\n23G(3n-1)& = &\\left(-4u^3 + 2v^3 -w^3 + 9(uv^2+vw^2+wu^2) + 3v^2w+6uvw\\right)\\\\\n23G(3n)& = &\\left(3u^3 + 2v^3 + 3w^3 - 3(uv^2 + uw^2 + vw^2 + vu^2) + 6v^2w + 18uvw\\right) \\\\\n23G(3n+1)& = &\\left(v^3-w^3+6uv^2+9uw^2+6vw^2+9vu^2-3wu^2+6wv^2-6uvw\\right) \\end{matrix}\n", "ac05678f2b05da02c3a57539705428a5": "\\operatorname{Bun}_G(\\mathbf{F}_q)", "ac059825b8bcb4a2535e16ff5fbfde6a": "y = 5 - x", "ac05c4b4d9a8b63f05b11fe1f093701c": "A/\\Psi", "ac05d64c772b644e423b2700e44b890e": "\\delta \\leq t \\leq T", "ac05da8a124ba69150e0e0bd74d520b1": "\\boldsymbol{\\phi}(\\mathbf{R})", "ac06102b69e2ce2fb9c0ef91fce1521c": "\\phi(s,a)", "ac0618c0ad27e027c3c8cebad09b2b38": " \\mathbf{u} = \\mathbf{M} \\, \\mathbf{x}' ", "ac06fb7bd68933e08c0920b0700f0f9e": " (\\lambda x.\\operatorname{de-let}[f]\\ (\\operatorname{de-let}[x]\\ \\operatorname{de-let}[x]))\\ \\operatorname{get-lambda}[x, x\\ q = f\\ (q\\ q)] ", "ac072381b7a3df92c0d4ba506011d946": "x_\\text{V} = \\frac{3.912}{b_\\text{ext}}", "ac07354ed5393a08e93b1e1968bc440f": " 10^{18} ", "ac074192d96881869cccc1d900fe1ec4": "\n\\begin{align}\n y & = & \\frac {R}{2} \\ln \\left[ \\frac {1 + \\sin\\phi}{1 - \\sin\\phi} \\right] \n & = & {R} \\ln \\left[ \\frac {1 + \\sin\\phi}{\\cos\\phi} \\right] \n & = R\\ln \\left(\\sec\\phi + \\tan\\phi\\right) \\\\[2ex]\n & = & R\\tanh^{-1}\\! \\left(\\sin\\phi\\right)\n & = & R\\sinh^{-1}\\! \\left(\\tan\\phi\\right) \n & = R\\cosh^{-1}\\! \\left(\\sec\\phi\\right) \n = R\\;\\mbox{gd}^{-1}(\\phi) .\n\\end{align}\n", "ac075af03ae4bde96c13bfbc663c8f11": "d_1\\,", "ac075f6b525ea018e2f529a414f159a8": "G' (t,a,b) = \\frac{a}{(2\\pi)^{-1/2}}(t - b) e^{\\left(\\frac{-(t-b)^2}{2a^2}\\right)} \\,", "ac0773e8974ac764b4a1e62f7169f5de": " \\hat H = \\hat\\vec\\mathbf{a}\\cdot\\mathbf{A}\\cdot\\hat\\vec\\mathbf{b} = A_{xx} \\hat a_{x} \\hat b_{x} + A_{yy} \\hat a_{y} \\hat b_{y} + A_{zz} \\hat a_{z} \\hat b_{z} ", "ac077a867b9dd9e4988a96566956c346": "\\scriptstyle S=Z^1 ", "ac079aa6bf67d2b06ea0c5e52b0dde7d": "a^{2^rd} \\not\\equiv -1\\pmod{n}", "ac07a6e00a707d998a2772b3ca63ae4a": "\\mathbf{\\mathit{p}}(\\phi)", "ac07b45038261c0690e6b06bf8f06679": " [A,H] \\leftrightarrow i\\hbar\\{A,H\\} ", "ac07bc62d3ea6543d3d515236cb2cb90": " a = F_n F_{n+3} \\, ; \\, b = 2 F_{n+1} F_{n+2} \\, ; \\, c = F_{n+1}^2 + F_{n+2}^2 \\, ; \\, a^2 + b^2 = c^2 \\,.", "ac0806cc537ea585de31649622dc6b84": "|\\mathcal{X}|\\leq|\\mathcal{Z}|", "ac08275464479636be64acf5ac39aeee": "\\frac{\\partial f}{\\partial \\sigma}=0", "ac0829dd8b6646d615827f3ad165db0b": "\\textstyle \\mathbb{P} (A|B) = \\lim_{n\\to\\infty} \\mathbb{P} ( A \\cap B_n ) / \\mathbb{P} (B_n) ", "ac08db0396d59e967080df6cff1275ec": "{M_x}", "ac08f400966e1ee87030c08b31542af4": "\\textstyle p(z) = \\sum _{k=0}^na_k z^k", "ac09029773f12cc15242f66d15826857": "\\mathrm{^{239}_{\\ 94}Pu\\ +\\ ^{4}_{2}He\\ \\longrightarrow \\ ^{240}_{\\ 96}Cm\\ +\\ 3\\ ^{1}_{0}n}", "ac09efecc3881223de4bedb35567e592": "\\varphi_m( \\boldsymbol{r} )", "ac0a541bf35f96eee36dd1ec1495f15c": "2^{4^n}", "ac0a8379fc0d99e407a7f11ab023dca0": "\\pi(x;4,1)\\sim\\pi(x;4,3)\\sim \\frac{1}{2}\\frac{x}{\\log x},", "ac0aecad2e5ca0564be8c298bd4f1e06": "0 < \\varepsilon \\le 1 - H_q (\\delta )", "ac0b4d7c9bcfd4531f5ee64e839e1aa4": "f(z) = z^m e^{\\phi(z)} \\prod_{n=1}^{\\infty} \\left(1 - \\frac{z}{u_n} \\right) \\exp \\left\\lbrace \\frac{z}{u_n} + \\frac{1}{2}\\left(\\frac{z}{u_n}\\right)^2 + \\cdots + \\frac{1}{\\lambda_n} \\left(\\frac{z}{u_n}\\right)^{\\lambda_n} \\right\\rbrace ", "ac0b8444abb08219313c534332628368": "\\scriptstyle \\overline{BV}(\\Omega)", "ac0ba022a0c2210f4acdb94f35607f3a": "U \\cap \\Omega", "ac0c0c604ea7891f4630caae7e688907": "d(f(w), f(z)) = \\rho (w, z) \\,", "ac0c3198f1f3c371796e3545849092c5": "1 - 2^{rel_{i}}", "ac0c6fa2ed3dc35d70af0264eee38d31": "\nq_{ij} = \\bar{\\phi} \\overline{u}_j - \\overline{\\phi u_j}\n", "ac0ca363cf43af3fcd00f5011971437b": " \\%N = \\left ( 1 - \\frac{\\%N_f}{\\%N_F} \\right ) \\times \\left ( \\frac{A_F}{A_f} \\right ) \\times 100 ", "ac0cb6c6bb671d4b3d9db28753942641": "\\frac{q^2}{g}\\left(\\frac{1}{y_1 y_2}\\right)=\\frac{1}{2}(y_2+y_1)\\qquad\\text{where }q_1^2=y_1^2 v_1^2=y_2^2 v_2^2", "ac0cc0f227f6f2e60572d2d9a13d3041": " \\Delta v = \\pm 1 \\ (\\pm 2, \\pm 3, etc.", "ac0cf41a71cfb213644c47dbb57c30b7": "~\\Phi_{18}(x) = x^6 - x^3 + 1", "ac0d5138a6373cb66d6eb22d2290a2df": "\\scriptstyle H(\\text{pwd})", "ac0db0f9024ab06c567e78242e1aa93a": "\\left ( x+ \\alpha e_1 \\right ) \\succ \\left ( x \\right ), \\alpha>0", "ac0dc0bb6f63596fbce15dca6acc33d5": "\\displaystyle{J}", "ac0e191157299e66562d5b4e56a171a7": " f(\\tau)\\cdot g(\\tau-t),", "ac0ebf32a1f4763d8b6153ffdf246c96": "\\sigma_W^2", "ac0ecf552a619f32200256ce0ff0f57e": "E_{T} = | \\overrightarrow{p}_T |", "ac0ed8bfbd78de19f9082699fca4c0c8": " \\begin{align}\n \\boldsymbol{\\nabla}\\phi & = \\cfrac{\\partial\\phi}{\\partial x_i}~\\mathbf{e}_i \\\\\n \\boldsymbol{\\nabla}\\mathbf{v} & = \\cfrac{\\partial (v_j \\mathbf{e}_j)}{\\partial x_i}\\otimes\\mathbf{e}_i = \\cfrac{\\partial v_j}{\\partial x_i}~\\mathbf{e}_j\\otimes\\mathbf{e}_i\\\\\n \\boldsymbol{\\nabla}\\boldsymbol{S} & = \\cfrac{\\partial (S_{jk} \\mathbf{e}_j\\otimes\\mathbf{e}_k)}{\\partial x_i}\\otimes\\mathbf{e}_i = \\cfrac{\\partial S_{jk}}{\\partial x_i}~\\mathbf{e}_j\\otimes\\mathbf{e}_k\\otimes\\mathbf{e}_i\n \\end{align}\n ", "ac0f216aaa4285e6d2f141d9f9c183e5": "R_c, G_c, ", "ac0f5484e8fd8d882f83f988f46524fe": "\\varphi_1^2 \\varepsilon_1", "ac0fa30e8b27a3cb82145334ebcc6737": "\\Delta \\alpha", "ac0fc737b91cdbe23d1c6040f68e15d1": "\n\\begin{matrix}\n\\mathrm{I} & a_1 & \\quad & a_3 & \\quad & a_5 & \\quad & \\cdots\\\\\n\\mathrm{II} & \\quad & a_2 & \\quad & a_4 & \\quad & a_6 & \\cdots\n\\end{matrix}\n", "ac0fc870bab49a8b4c4ef904d2254cca": "PreCaP~ICO_{PH} = \\frac{PreCaP~ICP_{PH}}{1- (PreCaP~ICP_{PH})}", "ac0fd3667503b7b84a9b44b5621a737e": "P=\\binom{10}{1} \\times 0.02^1 \\times 0.98^9", "ac102fa3fc19e9ce685e51664a012bf1": "A = (\\varepsilon_{\\mathrm X} c_{\\mathrm X} + \\varepsilon_{\\mathrm Y} c_{\\mathrm{Y}} + \\cdots)\\ell", "ac10513d7d47808f2ba33c642bdad58d": "c_B(a,b)\\equiv c_+(a,b)", "ac1052c8c41fa0e8d67714e0723a068b": "w_0", "ac105b61fbb2a7a9684ed86215366425": "\\ln\\tau, \\tau, \\tau x, \\tau x^2", "ac108a13a04ed90773b580c6d0f3f490": "\\int\\limits_{-\\infty }^{\\infty }{Pf(u,\\xi )}.d\\xi =2\\pi {{\\left| f(u) \\right|}^{2}}", "ac10de059153b4aedb69547baf3473d2": " \\frac{\\frac{\\$10,000}{4 \\%} - \\$ 5000}{\\$30,000 \\cdot 75 \\% \\cdot 75 \\%} = 14.5 \\; years ", "ac1110375643fdc21f1daf70db0ad666": "X^2 = \\sum_{a \\in A}{\\sum_{b \\in B}{\\frac{(O_{a,b} - E_{a, b})^2}{E_{a, b}}}}", "ac114663f0399f1101d424796cfd2294": "\\Lambda^{\\mathrm{top}}M", "ac11c496359637d7daf6b36c5dce2334": "\\textstyle \\mathbb{R}^2", "ac11d43b7dc728392e44ac348b4a97c6": " \\{A,BC\\} = \\{A,B\\}C - B[A,C]", "ac11e4024ed856d2210d33248d76202c": "R \\frac{dQ}{dt} + \\frac{Q}{C} = V(t) \\Rightarrow \\frac{d \\chi}{d \\tau} + \\chi = F(\\tau)", "ac12605f53a18d87acb6124035986bce": " \\rho \\in [0,1] \\, ", "ac12f29267ce448b5deb2a5c56a92fc6": "\\big( \\Delta H_f^\\circ \\big)", "ac13cee3bd1b6fed6c2b0cc358e53432": "\\left( \\frac{\\partial \\xi^{(c)}_i}{\\partial x^k} - \\frac{\\partial \\xi^{(c)}_k}{\\partial x^i} \\right) \\xi^i_{(a)} \\xi^k_{(b)} = C^c_{\\ ab}", "ac13f5960c1a370f60a799e19b3d8b3c": " 5 < M ", "ac145fe968f8678e0d6fe87c79143830": " \\nu (B) = \\mathbb{P} ( X \\in B | Y \\le \\tfrac{1}{3} ) \\mathbb{P} ( Y \\le \\tfrac{1}{3} ) = \\tfrac13 \\mathbb{P} ( X \\in B | Y \\le \\tfrac{1}{3} ). ", "ac14705577906c3e31e0100fca7c8244": "u = q/2 + \\sqrt{(q/2)^2-(p/3)^3}", "ac147d65d71fa92ca4fe7b80bda33fe4": " \\Delta U = - \\frac{W}{m} = - \\frac{1}{m} \\int_{r_1}^{r_2} \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{r} = - \\int_{r_1}^{r_2} \\mathbf{g} \\cdot \\mathrm{d}\\mathbf{r} \\,\\!", "ac14901f6794da57f33dec98a2461e36": "\n \\nabla^2 w \\equiv \\frac{1}{r}\\frac{\\partial }{\\partial r}\\left(r \\frac{\\partial w}{\\partial r}\\right) \\,.\n", "ac149e18819dadf6525e465e895a96d6": "\\delta_{j,j_1}\\delta_{m,m_1}", "ac1544733bbea519816b2055749ea289": "I(z,z) = \\pi/(2z)", "ac1567918cbb3ac224f5b7415c0dc908": "x_1= \\cos \\Omega \\cdot \\cos \\omega - \\sin \\Omega \\cdot \\cos i \\cdot \\sin \\omega", "ac15af2b9683d8910308787bfeb8774c": "Y_{[r:n]}", "ac15ce7c2a7f74efe9f5a9aa072e3a11": " \\displaystyle \\exp(xf(t)) = \\sum_nG_n(x;a,b)t^n/n!", "ac15fde8e619a265850ff05da251c3cb": "\\tilde{X} \\in T_e E", "ac1624df696881d72aaf8dd47a7c8bc3": "a^k x a^m x = a^{k-m+n}", "ac165bcd22f895af74c594e19c49c3bb": "(\\mathbf{a} + \\mathbf{b}) \\mathbf{c} =\\mathbf{a} \\mathbf{c} + \\mathbf{b} \\mathbf{c}", "ac165e2d90aa0d8e736e90792280325e": "d : 3\\, ", "ac1684d65583001fd2355852067cc7ad": "\\phi \\rightarrow [\\alpha]\\psi", "ac16850821ec12f4117dd5c5ef687b58": "\nh(\\mathbf{Y})=\\frac{1}{N}\\sum_{i=1}^M\\sum_{t=1}^N \\ln (1-\\tanh(\\mathbf{w_i^T x^t})^2)+\\ln|\\mathbf{W}|\n", "ac16b190f017e8cc4ff91f8061ac8fe3": "Q < x \\log^{-B} x", "ac16da8d9d2af38a1efe7cc1c526b7b2": "\n\\sum_{i0 ", "ac18f31733d4cde59aa737edfe83069c": " e^x = 1 + x + \\frac{x^2}{2!}+\\cdots ", "ac196575fc3a55f4cf1a9a68d7adc9c4": "\\operatorname{ker} f := \\{(x,x') \\mid f(x) = f(x')\\}", "ac1967226fe35a8a0830d6db6f2bd36a": " A_q(n,d,w) = 1. ", "ac197cb12ae3f52c66fa08317f1dae02": " \\langle H,J,K \\rangle ", "ac1982305f864d8b36e8de51477d87a8": "e^{-\\alpha t} \\cos(\\omega t) \\cdot u(t) \\ ", "ac19e9698092abe64f32476fa29c263c": "d_{n}^{k} = |d_{n+1}^{k-1}-d_{n}^{k-1}|", "ac1a0b0ece79037e565c34afda9c01bb": "(\\mathcal{A},\\mathcal{B}){:}", "ac1a286e8b1b9a31df0fea51113f7698": "\\sigma_A^2\\sigma_B^2 \\geq |\\langle f|g\\rangle|^2.", "ac1aaddc2c112bb89ec31c6a4daf11c3": "F=1", "ac1ae802f224c50b8dce3f51ef5c71b1": "\\mathbf{P}_i := \n\\begin{pmatrix} \nx_i \\\\ \ny_i \\\\\nz_i \n\\end{pmatrix}\n", "ac1b420818c54ccc08a0cf848c1f785c": " Z_\\text{in} = {v \\over i} = -j \\omega C R_1^2 ", "ac1b7c4f7c7b757fccfe341ef0f2c9a0": " P_{\\rm L} = I^2 R \\, .", "ac1b8d789f82d321c40f7aaf557757eb": "X_k = \\sum_{n=0}^{N-1}x_n e^{-i 2 \\pi kn/N} ", "ac1bca1312b038a6c387fed60e0005d1": "= \\operatorname{tr} \\left(\\gamma^\\mu (2 \\eta^{\\nu \\sigma} - \\gamma^\\sigma \\gamma^\\nu ) \\gamma^\\rho \\right) \\,", "ac1bdebca10c25fa7220a205972c9616": "(x_\\alpha)_{\\alpha\\in I}", "ac1bee0a731394b3953b838dc25f97d9": "S_D (\\lambda)", "ac1c194503b9901c5ae8492f14b93d8f": "r(t)= \\left\\{ \\begin{array}{ll} K A e^{2 i \\pi f_0 (t\\,-\\,t_r)} +B(t) &\\mbox{if} \\; t_r \\leq t < t_r+T \\\\ B(t) &\\mbox{otherwise}\\end{array}\\right.", "ac1c3938191ce14d924bbb45b2f11474": "\\omega_{H} = \\sqrt{\\gamma\\frac{A}{m} \\frac{V_n}{L} \\frac{P_0}{V_0}}", "ac1c5e5293b6e9fd483b62fa4e884763": "\\operatorname{ind}(P')\\geq \\operatorname{ind}(P) + \\gamma^5/20", "ac1c6082c4e5a762d3d3248303a25b73": "0\\leq Q\\leq 1", "ac1c8e153a067317abeafef7332c62a8": "x=\\left( a + \\sqrt{a^2-n} \\right)^{(p+1)/2}", "ac1caed87da8e5c74f4fef3ecedf767a": "[f,g]:=\\int_\\Omega f(t)\\operatorname{sgn}(\\overline{g(t)})d\\mu(t),\\ \\ f,g\\in L^1(\\Omega,d\\mu).", "ac1cc51eeed1bec5106e0001daf7794b": "\\frac{}{}\\delta", "ac1d11a9e2cd8530e1ef522c2d4aadf5": "\n\\begin{align}\nu(x=0) &= f_{1}(y) = 0 \\\\\nu(x=x_{l}) &= f_{2}(y) = 0\n\\end{align}\n", "ac1d76c7b0d3d7863e51e27f0b573aa6": "\\tilde{y}_{i+1}", "ac1ded25fb632b2260272865bfb1711b": "r = {(1 - l_B) \\over (l_A a_B)} - 1", "ac1df3c669a7a17df99638716f944351": "\n2\\|\\textbf{u}\\|^2 + 2\\|\\textbf{v}\\|^2 = \\|\\textbf{u}+\\textbf{v}\\|^2 + \\|\\textbf{u}-\\textbf{v}\\|^2.\n", "ac1dfb7471107f4fc9ce9a883d246013": "h\\colon I\\to Y", "ac1e38ad31940aa1e4dbea75699242a6": "x=\\frac{-b \\pm \\sqrt{\\Delta}}{2 a}, ", "ac1e46156116aa6a2363e8e7a1aefb29": "(\\lambda x.K\\ (x\\ x))\\ (\\lambda x.K\\ (x\\ x)) = K\\ ((\\lambda x.K\\ (x\\ x))\\ (\\lambda x.K\\ (x\\ x))))\\ ", "ac1ed26c2d087b5bb7a8dc8e43a6164b": "O(3^k \\cdot mn)", "ac1f2e265bd98238b70646d7f4991f15": " U^*U=I", "ac1f991ea6592a717ed5bf67de648d9f": "\\alpha \\gamma^* = \\mu \\gamma^* \\alpha, ", "ac1fd6f41e91a4eaf03b00d76a71e0ab": "\\mathbf{x_b}", "ac20359d1801996aa06123de0470667f": "\\frac{1}{q}+\\frac{1}{t}=\\frac{1}{y}.", "ac20ae06a6f92400a45835bd4d3bebc2": "t_o = \\frac{(t_a + t_{mr})}{ 2 }", "ac20b0a9f2323ab099161d58790c9850": "(V)", "ac20f78dd266c8e9ab102913870a6734": "f_* : \\pi_1(X, x_0) \\to \\pi_1(Y,y_0).", "ac21c107a31eb75630b86380eb403a6b": "X_t = \\| W_t \\|,", "ac21e0bcdb7b6f0eb20451068ba44271": "\n-\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{jr", "ac23799f06481924eefecbdd21bafda6": "p_e", "ac23e835bd8022dd0e9a551a8a08f2e4": "(p+q+r)^2=p^2 + q^2 + r^2 + 2pq +2pr + 2qr\\,", "ac2418ab117c325a2673377d1db5cacc": "\\mu _b", "ac24e44db943a6c2513ab6271f5e9333": "\nS_{r} (r) = \\sqrt{2m} \\int dr \\sqrt{E_{\\mathrm{tot}} - U(r) - \\frac{L^{2}}{2m r^{2}}}\n", "ac2501a74f507e0eb04b0cac5a1f62b1": "\\kappa(s) = \\frac{1}{R(s)}.", "ac2503d8e54eba0a817b804bdc694734": "J^{\\mu\\nu} = M^{\\mu\\nu} + S^{\\mu\\nu}", "ac254558e043af164b1c8017187126d9": " \\lambda m.\\lambda n.\\lambda f.\\lambda x.(n\\ m)\\ f\\ x ", "ac2585ee2def3cd814a87bf839e54946": " \\alpha\\wedge\\beta\\to\\alpha ", "ac25e982d974436cd636c5c40246ad3a": "\\mathcal{LA}\\,", "ac26130ccd5499ba45403e4d8c2d6abf": "x = L_0 \\left ( 1 - \\frac {1} {2} \\left ( \\frac {k_{B}T}{FP} \\right )^{1/2} + \\frac {F}{K_0}\\right ) ", "ac26d7bacfb8e32bfe71317dd98fbd89": "g_2 = 60 G_4", "ac26e0d487ce75c1af6c4743f5915b56": "A = \\left\\{a_1,a_2,...,a_m\\right\\} \\quad a_1 < a_2 < ... < a_m", "ac26e1b29ab996b2ce2f01582569aaa7": " (h,h(\\hat{k}) , h(\\hat{l})) ", "ac2794014497fda55edc90eaaf8871d8": "x \\in [0,\\ 1]", "ac27ddf8a9c79c93dc3aba8fc61b3ac1": "\\boldsymbol{u}^{(1)}", "ac281fed70809f8b098aeb1bd215133c": "\\sup_{p \\in \\pi_n(\\sigma)} \\|p\\|_\\infty=\\|T_n\\|_\\infty", "ac2849258c53495852bb48c9a7457d2b": "\n\\sigma _z^2 \\approx \\,\\,\\,\\left( {a\\,b\\,e^{b\\mu } } \\right)^2 \\,\\,\\frac{{\\sigma ^2 }}{n}\n", "ac28923d4da40e8060759fde0bc7d64b": "e = \\sqrt{\\frac{2\\alpha h}{\\mu_0 c_0}} = \\sqrt{{2\\alpha h \\epsilon_0 c_0}}.", "ac28ac95afe579e19ed7dd6d9c3733a2": "T_{sample}=T_{clock}", "ac28b7bc1e7e6261eefb477cf1fbe336": "L := - \\delta \\circ \\mathrm{D}.", "ac28d554e0ce26c959f051c323f49ac8": "\\mathbf{q} \\cdot \\mathbf{p} - \\mathbf{Q} \\cdot \\mathbf{P}", "ac2925c06251d509fcaae8206faa5f8e": "\\lambda_{u,d,e}^{ij}", "ac2956d4a2d55e2079038fa225846a5a": "\\begin{bmatrix}0&0\\\\0&1\\end{bmatrix}:\\mathbf a", "ac29a6583fb45490edb52b9ee6dd11ce": "T^*_n(x) = T_n(2x-1)", "ac2a133416225d1b0bfb0f075bd8f677": "\\frac{1}{\\pi} \\cdot \\sqrt{\\frac{3}{e}} \\cdot \\frac{(e n)^n}{n^3-\\frac{3}{5}n^2+\\frac{2}{7}n}", "ac2a17dceb740e3d3e316b89a414f971": "\\ h = z_2 - z_1 = \\frac{R \\cdot \\bar{T}}{g} \\cdot \\ln \\left ( \\frac{p_1}{p_2} \\right )\n", "ac2a3a0dccb508f047e9e43d0b8670f1": "\\Delta\\left[ \\ln (S)\\right]{}_{r_1}^{r_0} = \\rho/\\sigma \\cdot \\Delta\\left[ G \\cdot M/r + \\omega^2 \\cdot r^2/2 \\right]{}_{r_1}^{r_0}", "ac2acd520548339b54cb4ea76613ff85": "\n\\mathbf{AB} = \n\\begin{pmatrix}\n\\mathbf{a}_1 \\\\\n\\mathbf{a}_2 \\\\\n\\vdots \\\\\n\\mathbf{a}_n\n\\end{pmatrix} \\begin{pmatrix} \\mathbf{b}_1 & \\mathbf{b}_2 & \\dots & \\mathbf{b}_p\n\\end{pmatrix} = \\begin{pmatrix}\n(\\mathbf{a}_1 \\cdot \\mathbf{b}_1) & (\\mathbf{a}_1 \\cdot \\mathbf{b}_2) & \\dots & (\\mathbf{a}_1 \\cdot \\mathbf{b}_p) \\\\\n(\\mathbf{a}_2 \\cdot \\mathbf{b}_1) & (\\mathbf{a}_2 \\cdot \\mathbf{b}_2) & \\dots & (\\mathbf{a}_2 \\cdot \\mathbf{b}_p) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n(\\mathbf{a}_n \\cdot \\mathbf{b}_1) & (\\mathbf{a}_n \\cdot \\mathbf{b}_2) & \\dots & (\\mathbf{a}_n \\cdot \\mathbf{b}_p)\n\\end{pmatrix}\n", "ac2b3026e19f4fffb0a1615a1af4fa1b": " ((\\lambda x_1 \\ldots x_{A_1}.\\lambda c_1 \\ldots c_N.c_1\\ x_1 \\ldots x_{A_1})\\ v_1 \\ldots v_{A_1}) ", "ac2bc03d357779516d821b751a488579": "\\mathcal{E}^3", "ac2bc7ba6df87d28f6e4054e7c71cec1": "\\widehat{H} = \\widehat{H}(\\mathbf{r}, t, \\widehat{\\mathbf{p}}, \\widehat{\\mathbf{S}})", "ac2c313f1f139a782bf43c2285a351cb": "e_1 = <1,0>", "ac2c89ae56cd0ccff67fac8a5d74d384": "a=b=\\sqrt{2}", "ac2c8bc4ff4cbf37f3ce2cd6fbc18dc1": "N\\rightarrow\\infty", "ac2d43ef3f26cc74de242202e822ecb0": "10100", "ac2d98c6197937cfcc29f86c6bb960a0": "\\text{Color} = \\frac{\\partial \\Gamma}{\\partial \\tau} = \\frac{\\partial^3 V}{\\partial S^2 \\, \\partial \\tau}", "ac2daafe1e8d2aa3e962b8859d81e291": "u_x, u_{xx}", "ac2dc758c6d91f2614ff52b864463864": "\\Phi_A(\\omega) = \\begin{cases}\n-J\\,\\omega(t_1)\\omega(t_2) & \\mathrm{if\\ } A=\\{t_1,t_2\\} \\mathrm{\\ with\\ } \\|t_2-t_1\\|_1 = 1 \\\\\n-h\\,\\omega(t) & \\mathrm{if\\ } A=\\{t\\}\\\\\n0 & \\mathrm{otherwise}\n\\end{cases}", "ac2dfff35310a48ffb8331bba3819446": "\\scriptstyle{ \\in }", "ac2e2f9511304d2a58f38bb2d8185c97": "\\vdash \\Box A \\rightarrow \\Box \\Box A", "ac2e5ac2ff60eb112a21348ce586f52d": "\\Psi [\\gamma] = \\int [dA] \\Psi [A] s_\\gamma [A]", "ac2e9d0d09761377c13e83ba8dcf1cb3": "\\sin{b\\theta} / {b\\theta}", "ac2ebe619fff28231b7c481f40a7e26c": "\n\\left(\\frac{-1}{n}\\right) \n= (-1)^\\tfrac{n-1}{2} \n= \\begin{cases} \\;\\;\\,1 & \\text{if }n \\equiv 1 \\pmod 4\\\\ -1 &\\text{if }n \\equiv 3 \\pmod 4\\end{cases}\n", "ac2ef055c062f3cf9cc43b58dd0d8f13": "\\Pi(z) = \\Gamma(z+1) \\,.", "ac2fd31c399ab7057a79798d011a469f": "x^{\\underline{m}}=\\overbrace{x(x-1)\\ldots(x-m+1)}^{m~\\mathrm{factors}}\\qquad\\mbox{for integer }m\\ge0;", "ac2fe0f2ae9675d01e42d6c004a7a14e": "\\mathbf{x}_{i+1} = \\mathbf{f}(\\mathbf{x}_i,\\mathbf{u}_i)", "ac2ff8fbffaf11c0e25b8221eeaa6dc5": "G\\xrightarrow{\\;\\eta G\\;}GFG\\xrightarrow{\\;G \\varepsilon\\,}G", "ac3005bff47e8d9bccb4673c9df8a207": "w,h>0", "ac3089d309068d5da87b85c25ba8bd5b": "\\mathrm{[LIV]-[GAED]-X_{2}-[STAV]-X}", "ac308a44607264a36a9af295df50e66b": " \\sigma^1 = \\exp (p) \\, dx, \\; \\; \\sigma^2 = \\exp (p) \\, dy", "ac30af13a7f4631ce20a1f025c7bde8b": "\\big(|0\\rangle\\langle0|, |1\\rangle\\langle1|\\big)", "ac30c2ed56160ba1cf4bf0a539a28a71": "{X \\over \\sigma}={X a \\over \\sigma a} ; a>0", "ac30f9099480963d80562f790ddbf986": "{\\Gamma(x)}", "ac31227896f676fe41fc4750e16f186e": "\n \\alpha := \\cfrac{2D^{\\mathrm{face}}}{D^{\\mathrm{beam}}} ~;~~ \\beta := \\cfrac{2D^{\\mathrm{face}}}{S^{\\mathrm{core}}} ~;~~ W(x) := \\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\n ", "ac3162f5a64ac04ea9aed8e4bfb39b49": "x_l\\,\\!", "ac31663ffdc3b060a4ba7bafcda8ad59": "\\mathbf{r} = x \\mathbf{e}_x + y\\mathbf{e}_y + z\\mathbf{e}_z", "ac3189f08dc4873055badd4172383850": "q(x_0,x_1,\\ldots,x_{n+1}) = -2x_0x_{n+1}+x_1^2+x_2^2+\\cdots+x_n^2.", "ac3195eea5d5fb7c1b8b36c43d6c7e98": "Q(z)|_{z = \\alpha_n} = 0", "ac31e22a1ef21fbcfdf384fa06400ff5": "\\phi(x)=\\phi(Sx)", "ac3200b78c777c7523ef7fa052a801dc": "\\mathbf{v} = c \\cdot \\mathbf{v}_\\textrm{op}", "ac324168309f7ee99190d2a514052328": "\n{\\frac{10^{-2}}{10^5}}+10{\\frac{10^{-2}}{10^6}}\n+10{\\frac{10^{-2}}{10^6}}\n+10^{-2}{\\frac{10^{-2}}{10^4}}\n-{\\frac{10^2+10^2}{10^6}}", "ac329939093b8702bff101704475a24f": " \\frac{-f(x+2 h)+8 f(x+h)-8 f(x-h)+f(x-2h)}{12 h}=f'(x)-\\frac{1}{30} f^{(5)}(x) h^4+O(h^5)", "ac334dacd1e417ef17736db175dae6c6": "\n\\begin{align}\nC_p &= \\displaystyle \\frac{C_{p0}} {\\beta} \\\\\nc_l &= \\displaystyle \\frac{c_{l0}} {\\beta} \\\\\nc_m &= \\displaystyle \\frac{c_{m0}} {\\beta}\n\\end{align}\n", "ac339e412940485ac451af0e77ce3e11": "\\alpha(X)", "ac3415fa7ab00b8fd7cc71f121e16d94": "Scenario \\ II: \\qquad R_B = {\\left ( {\\frac {{0.04333} \\cdot 152.0}{2}} \\right )} = 3.308 AU = 3.3 AU", "ac3425a34834a96af336ad1693a37cd0": "V_a", "ac3436d111a567aca2a5f290f9b4f233": " G = E ", "ac34764d6fb041f6708ca987e897c09a": "\\epsilon_{ij} = \\alpha_{ij}\\Delta T \\,", "ac3480f4df63cbf0308b0d200e6a6234": "A(1,n)=an", "ac34931aae7f3878234cc4a67509d42c": "x = \\sqrt[3]{q/2 + \\sqrt{(q/2)^2-(p/3)^3}} + \\sqrt[3]{q/2 - \\sqrt{(q/2)^2-(p/3)^3}}", "ac351a9b7658bbcf04a9d394d5e6ea34": "{(-1)^F}^2=1", "ac352b5702b3801db6d97f77a8a0ae3c": "\\displaystyle{(\\lambda +{1\\over 2}) \\iint_{\\Omega} |u_x|^2 + |u_y|^2 = (\\lambda -{1\\over 2}) \\iint_{\\Omega^c} |u_x|^2 + |u_y|^2.}", "ac35a9523afa7517a69a0fd457cb51db": "g(E) = g_v m^*/(\\pi \\hbar^2)", "ac35acdd6184a684f5c4a4cfe5dd0891": "A= 250,000 ", "ac35b47fc43531bdf4a62b7419d2e798": "K_2 = C_{e_1} A^T(AC_{e_1}A^T + C_Z)^{-1},", "ac35d6ab0a13aad9f6993271183d5147": " -\\tfrac{1}{2} ", "ac361344f1ddc0d9c90d2b81941d2b68": " \\theta = \\Theta \\sin\\left ( \\omega t + \\phi \\right ) \\,\\!", "ac36250ae0b796544f2e49440e92aab8": " ~|\\alpha\\rangle~", "ac366ce51da53befe152109fae34404a": "C_{\\rm 1} = C_{\\rm 2} = C", "ac36c3286facc71c9527a96892a25657": "\\textrm{Hom}_R(P, - )", "ac37c3534f52b28d7708450145fbc5e6": "\\lim_{n\\to\\infty}R_n(t)=0,\\qquad t\\in I.", "ac382d58b1125bb9aeaf1a83ebf5e9c6": "0.0001", "ac38422ff0a429a41cad0ca1112c4835": " \\mathbf{u} = \\nabla \\Phi\\;", "ac38885ff724da31c9eba254e4905d31": "b_m(y)x^m+b_{m-1}(y)x^{m-1}+\\cdots+b_0(y)=0.", "ac38ca2ba3ef2ff96276fc830a63e2a6": "\\|A\\|\\ge 0", "ac3937bb8833e7ae472560183271fee9": " \\nabla \\times \\mathbf{E} = - {1 \\over c}\\frac{ \\partial \\mathbf{B}} {\\partial t} ", "ac395691ef65ff466b6f796e3f7301c5": "\\,^{244}_{94}\\mathrm{Pu} + \\,^{58}_{26}\\mathrm{Fe} \\to \\,^{302}_{120}\\mathrm{Ubn} ^{*} \\to \\ \\mathit{fission\\ only}", "ac396a955ae3789e42c17ab0e93fd8ea": "\\beta_j = \\max_{x \\in S_j, \\|x\\| = 1} (Bx, x) = \\max_{x \\in S_j, \\|x\\| = 1} (P^*APx, x) \\geq \\min_{S_j} \\max_{x \\in S_j, \\|x\\| = 1} (Ax, x) = \\alpha_j.", "ac397942fcda174f63e2c99e7374eea0": "\\begin{array}{lll}\n&&\\\\[-2ex]\\displaystyle\n {\\tan{\\textstyle\\frac{1}{2}}(A{+}B)} \n=\\frac{\\cos{\\textstyle\\frac{1}{2}}(a{-}b)}\n {\\cos{\\textstyle\\frac{1}{2}}(a{+}b)}\n \\cot\\frac{1}{2}C\n&\\qquad\n&\n {\\tan{\\textstyle\\frac{1}{2}}(a{+}b)} \n=\\frac{\\cos{\\textstyle\\frac{1}{2}}(A{-}B)}\n {\\cos{\\textstyle\\frac{1}{2}}(A{+}B)}\n \\tan\\frac{1}{2}c\n\\\\[2ex]\n {\\tan{\\textstyle\\frac{1}{2}}(A{-}B)} \n=\\frac{\\sin{\\textstyle\\frac{1}{2}}(a{-}b)}\n {\\sin{\\textstyle\\frac{1}{2}}(a{+}b)}\n \\cot\\frac{1}{2}C\n&\\qquad\n& {\\tan{\\textstyle\\frac{1}{2}}(a{-}b)} \n=\\frac{\\sin{\\textstyle\\frac{1}{2}}(A{-}B)}\n {\\sin{\\textstyle\\frac{1}{2}}(A{+}B)}\n \\tan\\frac{1}{2}c\n \\end{array}", "ac39a4db4f36b1137fc5cd26c65602c8": "n,m\\geq N,d(x_n,x_m)<\\epsilon", "ac3a163eef1dad6646346041a3aae0bf": "T_{ant}", "ac3a1c4050ce2dabd89a6982e600078d": " i\\neq j", "ac3a24d482fdb9e1cf27c1f398736969": "f = 2^{3/12} \\times 440 \\,\\text{Hz} \\approx 523.2 \\,\\text{Hz}", "ac3a4eb0762f7cc56c0477f393b1c3fe": "\\left(\\vec \\mu_1, \\Sigma_{y=1}\\right)", "ac3a618f39900f71cf970fe364922de8": "\\coprod \\!\\,", "ac3a63cd83dae29c9b5fc08227e9ff8c": "\\sqrt{\\frac{1}{2}D_{KL}^{(e)}(P\\|Q)} \\ge \\sup \\{ |P(A) - Q(A)| : A\\text{ is an event to which probabilities are assigned.} \\}. ", "ac3a6b1461986df5a4a1c841d7d656ed": "x_3 \\ ", "ac3aa94d096f6d61841278a762a9bae6": "2\\pi k", "ac3af0d234f3a4b6bb2c5a5210b89b90": "s_y = -\\frac{mg}{k}t - \\frac{m}{k}(v_{yo} + \\frac{mg}{k})e^{-\\frac{k}{m}t} + C", "ac3b076f3f9ecc7ae1077b6494fb466a": "dq = \\sigma(\\boldsymbol{r'})\\,dA'.", "ac3b0ef199541495ccbcede67c153b50": " e_i \\le 0 ", "ac3b45a1729f31ef3a5fd3494e7070ed": "\\operatorname{Li}_n(\\tfrac12) = -\\zeta(\\bar1, \\bar1, \\left\\{ 1 \\right\\}^{n-2}) \\,,", "ac3b97adb82c8af89ec6543a7917f988": "\\tfrac{c}{b}", "ac3bb493fabfb2dc21bd360a8525b816": "f(x)=\\sum_n {a_n \\over n!}x^n,", "ac3c1d5f0313a0a9c4d604a3257d5835": "\n w_{\\mathrm{max}} = -\\dfrac{Pb(L^2-b^2)^{3/2}}{9\\sqrt{3}EIL}~.\n ", "ac3c285fe146901075172d2246df81c0": "\nUNC = \\bar{x}(1-\\bar{x})\n", "ac3c472bef5005add55148a7db79fc85": "c_p = \\frac{\\lambda}{T} = \\frac{\\omega}{k} = \\frac{\\Omega(k)}{k},", "ac3c55f29daa17faca3719c580d826d3": "/R", "ac3cb6bb290a63854b9d8794359cd847": " {{e^*}_w} ", "ac3d5562cf7726dbba066259912f1364": "\\begin{align}\nP(X \\to \\widehat{X}) & = Q \\bigg(\\tfrac{||X- \\widehat{X}||^2}{2}.\\sqrt{\\tfrac{2}{N_0||X- \\widehat{X}||^2}}\\bigg) \\\\\n& = Q \\bigg(\\tfrac{||X- \\widehat{X}||}{\\sqrt{2N_0}}\\bigg)\n\\end{align}", "ac3de486287b3fdce2d96fb955f3f188": "\\int_0^\\infty (\\frac {1}{1+x}- e^{-x})\\frac{dx}{x}=\\gamma", "ac3e030d3472a6e27b4a20a550d20e91": "\\frac{2 \\cdot \\pi}{3}", "ac3e0aa038fc3b53136fbae08549d4bc": " C_{(-)} ", "ac3e321efe6a7247a5783df9384f24a1": "t=1+1+\\dots+1 \\ge 1+y+y^2+\\ldots+y^{t-1}=\\frac{1-y^t}{1-y}", "ac3e78c952ad5b2f317ed06ccfed7dce": "\\scriptstyle g_{ij}(q)", "ac3e8c2409ecdf45277f2eb3af8d5f27": "\n\\text{div}\\,\\mathbf{v} = 0\n", "ac3ebf5d1c1200720344e0998baeea1f": "r V_u = f(-E(c)+V_e-V_u)+\\frac {dV_u} {dt}", "ac3eece5d87c831de5e248a5987bf4fd": "X_{N-k} \\equiv X_{-k} = X_k^*,", "ac3f50a75ebc6e5a534eb9c176c096a1": "\\boldsymbol\\omega(t)", "ac3f670ba4341177dd3f75479319ade7": " \\theta \\ ", "ac3f6cd2fb10b95af04ec851bad535bf": "\\varepsilon_1, \\varepsilon_2,\\ldots,\\varepsilon_\\omega, \\varepsilon_{\\omega+1}, \\ldots, \\varepsilon_{\\varepsilon_0}, \\ldots, \\varepsilon_{\\varepsilon_1}, \\ldots, \\varepsilon_{\\varepsilon_{\\varepsilon_{\\cdot_{\\cdot_{\\cdot}}}}},\\ldots", "ac3f7b1163652333e3c461502065073d": "h > 0", "ac3f82f2ea418ba092fbee52c07be25d": "\\langle v_1,\\ldots,v_n \\rangle", "ac3fedd1bd2a76064f596337e1bdc0c1": "G=\\mathrm{Homeo}\\left(F\\right)", "ac4012038492ac9f5b6e97bc72500df8": "P \\and Q", "ac4039ec0acf6beb75bac95e19718e58": " f_{GHz}=\\frac{34}{a\\sqrt{\\varepsilon_r}}\\left( \\frac{a}{L}+3.45\\right) ", "ac405e32b2a205f8e73af43ee7b49a44": "\\tau(H_\\alpha)\\;", "ac406b1b82da4a7724ebbdbf34d3d99d": "\\sigma_{jk} = \\sigma_{kj}", "ac415074e5c80dbb54a06cce8c5428cc": " \\delta = {d \\over n}", "ac419437cc98af938cb2c9ed7650d20b": "A = B^TB", "ac41ef23e72753df6741f85735bb2c71": "N << M", "ac4226515dea1e26f07c7e0463c1662f": " \\lambda=\\frac {1} {2} (.032)^2 = .0005 ", "ac426437d2a41908f0a1e5b81891118a": " t=t_{2} \\ ", "ac42f8b2b87177532296590acb732494": "\\mathrm{Dir}\\left(\\boldsymbol\\alpha+([x=i],\\dots,[x=k])\\right)", "ac431cc9804f6b34a75e251942009a87": "\\mu = (1-\\epsilon + \\mu)\\operatorname{Tr}(Q\\rho)", "ac43222fc3f785ad1b44d3de0f8f3e4e": "\\mathbf{N}\\left( \\mathbf{u}\\right) ", "ac432ef34aa6a855f25078e0eebc6804": " \\hat{K}_G := \\{ z \\in G \\big| \\left| f(z) \\right| \\leq \\sup_{w \\in K} \\left| f(w) \\right| \\mbox{ for all } f \\in {\\mathcal{O}}(G) \\} .", "ac432f7cdd14f99734ab19ca78a2db28": "M=2^N", "ac43410684659398391db593811a7ded": "n_1\\sin\\theta_1 = n_2\\sin\\theta_2\\ ", "ac43424016e00eb059976bea0560c95a": "\\lambda_1/\\lambda_2 =0.15,\\, ", "ac43479171ba8499ef9667797fcbaf45": " g(x) = (K-x)^+ ", "ac43788af9d40e94aa24cc8a859b8cec": " u_y", "ac441a20748de3ef8812a86368ce4430": "g = U(g)\\cdot X(g)", "ac44ae3e20cfce3fb626cd7e2d1c580f": " M \\models \\phi[[a^1], \\ldots, [a^n]] \\iff \\{ i \\in I : M_{i} \\models \\phi[a^1_{i}, \\ldots, a^n_{i} ] \\} \\in U.", "ac44bc4dade959e893a63da25fe47d46": "A= \\left[ \\begin{array}{rrr}\n2 & -1 & 1 \\\\\n0 & 3 & -1 \\\\\n2 & 1 & 3 \\end{array} \\right] ~,", "ac4508c9cea492d2d94e3a0ca8b0925a": "AF_{H} = e^{(Constant*(RH_{s}^n-RH_{o}^n)}", "ac4521857b929a78cede7efc062ff86e": "5400^4 + 1770^4 + (-2634)^4 + 955^4 = (5400 + 1770 - 2634 + 955)^4. \\, ", "ac4547d3da55b3bf59e70a25d99b6995": " B(R) = \\left\\lbrace{ \\left({\\begin{array}{*{20}c} a & b \\\\ 0 & d \\end{array}}\\right) : a,d \\in R^*, ~ b \\in R }\\right\\rbrace. ", "ac45540d54897ea79ab5d8038208b10f": "B_n = (a_1, \\ldots, a_n, b_{m_1}, \\ldots, b_{m_k})", "ac45885f5a7b149456e25c486c3cd44d": "\\rho \\mapsto \\rho", "ac45b7a9b6cb92a4328c3f63d348358b": " U_t=U_{xx} \\, ", "ac45d26e751a516f8cf56081d14f3594": " \\Delta \\phi \\ge 0", "ac4642f0533479d626d4d1c28b113a21": "Q_f \\cdot C_f=C_p \\cdot C_p + C_c \\cdot C_c", "ac46775415632f49ad8cd842413acdf5": "\\, ((x_{min}, y_{min}), (x_{max}, y_{max}))", "ac4685dd95f838f7f096283435392ff1": "z(n;t)=\\Theta(n^{2-1/t})", "ac46b3cc3eb9b30dec9c77babef1645d": "n \\gg 0 ", "ac4715e24a0ff54c118d5a09b082d93e": "Finesse= \\frac{\\pi(R_1R_2)^{1/4} e^{-\\frac{\\alpha d}{2}}} {(1 - \\sqrt{R_1 R_2} e^{- \\alpha_c d})^2} ", "ac475445b1bf193a180c6a3f02970978": " v''_j(x) = \\lambda_j v_j(x), \\, j=1,\\ldots,\\infty.", "ac47d0ae8b4fbe4ec35c4f3a92b133a2": "C_{\\mathrm y} = \\frac{1}{6A}\\sum_{i=0}^{n-1}(y_i+y_{i+1})(x_i\\ y_{i+1} - x_{i+1}\\ y_i)", "ac47d4ed2323aaba412e2677aecc048e": "y(x)=-\\sqrt{2}", "ac482128ccf338c05ed4d499bd3f396d": "r = \\frac{T}{\\tau}.", "ac4853be4fe4d959f92d19dfcc5557b1": "t^- ={1\\over {2}}(t-\\sigma t)\\,", "ac485c5131582bdda2e737d503ccdc0b": "\\begin{bmatrix}\na & -b \\\\\nb & \\,a \\end{bmatrix},", "ac486d1c905461d9e7706fb64a34de97": " P_{t_0}(S\\rightarrow S'|E) > P_{t_0}(S\\rightarrow S') > 0 ", "ac487b8ef9b2765b50619588d7931bc1": "D(s\\otimes s)=D(s\\otimes n)=D(n\\otimes s)=s", "ac49ebec0b635da30198b260ee7c88b7": "g \\left (p \\right )", "ac4a8a9676ee064c111fe5c9c50e5be0": "\\|P\\circ h\\|=\\|P\\|", "ac4b0c2c235e8b2fac33ed0e75ce6018": " G_2", "ac4b7a02d0df6d67aa23bf4532503102": "f(x)-3\\int_{-1}^1(xy+x^2y^2)f(y)dy = h(x)", "ac4b91688ef42a7b65c31e563240cd90": "\\overline{x} = \\frac {\\sum_{i=1}^m x_{i}}{m}", "ac4be95a7df03680877d003b19423cf0": "\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_2)(\\sigma_\\mathrm{n} - \\sigma_3) \\ge 0", "ac4beffac2623ad0743d9da0d71df50e": "i=0,1,\\dots,k-1", "ac4c00824ed96872ea402863ecff5cee": "\\Re(s) \\ge \\frac{1}{2} ", "ac4c4d49f0024e02031f63055485c604": "\\! k ", "ac4c75dac3f3a09545e1f031f7bc5fc5": "a_r:=\\frac{(xp_r,p_r)}{(p_r,p_r)},\\qquad\nb_r:=\\frac{(xp_r,p_{r-1})}{(p_{r-1},p_{r-1})}=\\frac{(p_r,p_r)}{(p_{r-1},p_{r-1})}", "ac4cd20e4c18fe66a4e66b4dbcf58f29": "\\frac{n_{i+1}n_e}{n_i} = \\frac{2}{\\Lambda^{3}}\\frac{g_{i+1}}{g_i}\\exp\\left[-\\frac{(\\epsilon_{i+1}-\\epsilon_i)}{k_BT}\\right]", "ac4d0957d134d13c90c6c8d6cb2ab408": "\\frac{g \\left( t \\right)}{f \\left( t \\right)}", "ac4d1929a51d9cd4bfa45ce98a6eb4bc": "\nJ=\\Psi(x(T))+\\int^T_0 L(x(t),u(t)) \\,dt\n", "ac4d19e65680725604c980215138269a": "(P_i, s_iQ)", "ac4d465faafe210ff81ee7a876a0e511": "\n(XP)_{mn} = \\sum_{k=0}^\\infty X_{mk} P_{kn}\n", "ac4d67efe24eb3db038416763a197001": "\\mathcal S[\\mathbf{A}] = \\int_{\\mathcal{M}} \\left(-\\frac{1}{2}\\mathrm{d}\\mathbf{A} \\wedge *\\mathrm{d}\\mathbf{A} + \\mathbf{A} \\wedge \\mathbf{J}\\right) .", "ac4db2db106ded30bbb8911dc9991ffd": "J_2 \\,", "ac4db8d681232a91538ed96a4ad7c116": "P_n(x)=\\frac{d^n}{dx^n}\\left(x^n(1-x)^n\\right).", "ac4dd38cdea1873c90ae2bf6a6fcba42": "\n{}_2\\phi_1(a,b;c;q,z) = \\frac{-1}{2\\pi i}\\frac{(a,b;q)_\\infty}{(q,c;q)_\\infty}\n\\int_{-i\\infty}^{i\\infty}\\frac{(qq^s,cq^s;q)_\\infty}{(aq^s,bq^s;q)_\\infty}\\frac{\\pi(-z)^s}{\\sin \\pi s}ds\n", "ac4ded85a0dbf2b0c0add82387e0f4f6": "a_j = e^{-i\\pi \\sum_{k=1}^{j-1}a^{\\dagger}_k a_k} \\sigma_j^-", "ac4e6de07a1e76eb0db6be37928b6777": "\\mathcal{B}(X^*_{\\sigma}, Y^*_{\\sigma}; Z)", "ac4ecce9f3667695ae79bd5c1787da81": "A \\otimes_k k'", "ac4f86161efe3e457a9df36e195813dd": " \\dot{\\varphi} = 1 ", "ac4fe1d687bafb543f071dc8049cc7e0": "A + B \\to AB^*", "ac502a0b6b9286244b6e03f9a4c77615": "\\displaystyle \\tilde{u}\\ ", "ac504a8d237d246bf66951560e6ded57": "\\delta^{n}\\lambda\\,(A).", "ac506381a132aa4467abb9d95250f869": "(1+i)^n", "ac50f87ed2278e2f51673dbde14a20d8": "\\nu=\\nu_0+\\nu_1\\, ", "ac50fa74e4a98ca0acd8c059a9d615fd": "P^{-1}AP = D", "ac519be122abae15d34c49756e858854": "R(h)", "ac51e937643df1bb6fd5a32ae29cbf04": "F\\left(n\\right)\n\n= {{\\varphi^n-(1-\\varphi)^n} \\over {\\sqrt 5}}\n\n= {{\\varphi^n-(-\\varphi)^{-n}} \\over {\\sqrt 5}}.", "ac52105a22065176b7d488938a07ff32": "V_{DS,sat} = V_{GS} - V_{th}", "ac526117e42ee3fb665752c0d99e8572": "c_{n} = \\left(n^{\\frac{1}{2}-n}\\Gamma(n+\\frac{1}{2})\\right)^{-\\frac{1}{2}} = \\left(n^{\\frac{1}{2}-n}\\sqrt{\\pi}2^{-n}(2n-1)!!\\right)^{-\\frac{1}{2}}\\quad n\\in\\mathbb{Z}.", "ac52adb6a6fb87dc2d22b8fc08264c9b": " p^{(4)}=m_0\\cdot dx^{(4)}/d\\tau", "ac52bd9e1853e90d6bb5bbd48fa64342": " \\operatorname{Var}(y_{2}) = 2\\sigma^2 ", "ac52edbaa040fd84a02112ff87cb6f73": "\\text{winding number} = \\frac{\\theta(1) - \\theta(0)}{2\\pi}.", "ac5314d09a321345acc0880e07c0d2cf": "~U=\\frac{(\\sigma_{\\rm as}+\\sigma_{\\rm es})\\sigma_{\\rm ap}}{D}~", "ac537233611af2febb063dc76ae196e3": "HME_k(X)=\\pi_k({\\rm Homeo}(X)).", "ac53725cbb7e4bb81ab89f9eab058b3d": "\\frac{\\partial \\ln |a\\mathbf{X}|}{\\partial \\mathbf{X}} =(\\mathbf{X}^{-1})^{\\rm T}", "ac537a9a117addfcf9bc41d087fffee2": "\\Delta V_\\text{acc}/V_\\text{acc}", "ac539147b3927b2e756a7bd304d92b69": " \\textbf{m} = -q\\textbf{f}_p \\cdot K \\pmod p ", "ac53a76a844b9735c5286c34caee12f9": " t^2 g + dt^2,\\,", "ac53a96bee3009768beac40ef37c5246": "\\int f(x)\\,dx", "ac53bb409573a153ca27329fc18a296a": "\\frac{b_i}{n_i}", "ac53d4b886be4431b607c99e63ba19f6": "\\scriptstyle 1\\times 2\\times 4", "ac53e3f8ad4d36243a934e419671b33c": "p_i^2 = m_i^2 \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad (1) \\,", "ac53f743eb1dea66b4da84121c199941": "\\sigma(\\bold{T})", "ac5438ce7168c4cdf514cee52500fc8b": "\n\\mathrm{i} \\hbar \\frac{\\mathrm{d}}{\\mathrm{d} t} \\langle \\hat{\\mathcal{O}} \\rangle\n=\n\\langle [\\hat{\\mathcal{O}}, \\hat{H}_{\\mathrm{System}} ]_{-} \\rangle\\;.\n", "ac54aab5017b337b93d5592ff96d76ea": " E[nX]=n\\frac{\\overline p + \\underline p }{2}, \\sigma_{nX}=\\sqrt{nVar[X]}=\\sqrt{n}\\sigma=\\sqrt{n}\\frac{\\Delta p}{6} ", "ac54b1508587156f39fc3bcdc304ce68": "\\boldsymbol n=(n_1,\\dots,n_K)", "ac54bcb8ea68f80e4063a11664ed3272": "(a,b) = (-\\infty,b) \\cap (a,\\infty)", "ac54d0d316d38a0b5dda3407458b899f": "\nz:\\;\\;\\rho \\left(\\frac{\\partial u_z}{\\partial t} + u_r \\frac{\\partial u_z}{\\partial r} + \\frac{u_{\\phi}}{r} \\frac{\\partial u_z}{\\partial \\phi} + u_z \\frac{\\partial u_z}{\\partial z}\\right) =\n-\\frac{\\partial P}{\\partial z} + \\frac{\\partial {\\tau_{z z}}}{\\partial z} + \\frac{1}{r}\\frac{\\partial {\\tau_{\\phi z}}}{\\partial \\phi} + \\frac{1}{r}\\frac{\\partial {(r{\\tau_{rz})}}}{\\partial r} + \\rho g_z.", "ac54e5357fa98cb1a027353764f51596": "\\displaystyle{\\sum_{i,j} |(T_i v,T_jv)| \\le AB\\|v\\|^2.}", "ac55056f9495835a32455f1d4bb60dc0": "D = 2\\,w_0 \\simeq \\frac{4\\lambda}{\\pi\\, \\Theta_{\\mathrm{div}}}", "ac5511a9cf697d2ab05e295ddf5cb155": "M/N \\,\\to\\, \\lambda \\in (0, +\\infty)", "ac555b25c1cbd13f8c8bf6702289c71a": "\\Delta= \\mathrm{d}\\delta+\\delta\\mathrm{d} = (\\mathrm{d}+\\delta)^2,\\;", "ac5560b0baa5f684a62f34468ace8f71": "\nI \\propto {p^2} \\propto \\dfrac{1}{r^2} \\,\n", "ac557e8e26530ce8e357adde10ec9255": " \\forall \\epsilon > 0, \\exists N \\in \\mathbf{N} \\text{ s.t. } \\forall n\\geq N, |a_n-L|<\\epsilon.", "ac559e2ad90bf05b1b0d6e487b3b4762": "v_{Ar}=0", "ac55d4cd6ea5061a787337e0514fa135": "S \\to \\varepsilon", "ac565d590e2f4a61c4f8b22408dff718": "\\prod_{p\\in A}{1\\over 1-p^{-s}}", "ac567eee0a6fef4b26e9eded313b1e3b": "\\pm 1/p", "ac568d22bb8bd4e355a1bc7367e47484": "\\Phi(\\omega)=\n\\frac{1}{\\sqrt{2\\pi}}\\,\\sum_{n=-\\infty}^\\infty B_n e^{-i\\omega n}\n=\\frac{1}{\\sqrt{2\\pi}}\\,\\left(\\frac{\\sigma_\\varepsilon^2}{1+\\varphi^2-2\\varphi\\cos(\\omega)}\\right).\n", "ac569e3f09962bbb55b38267fae0e567": " \\mathcal{G}", "ac56de678242c592bdba91cb03d2e079": " M_N ", "ac56e3d4b78ac1313dcaec2ac47f4e56": "\n\\sum_i {\\sum_j {T_{ij} C_{ij} = C} } \n", "ac5738b387068353d5b2cde9b67a5f55": " e^B = A. \\, ", "ac5762598fe233904677311da568ef83": "\\frac{x^3 - 12x^2 - 42}{x^2 + x - 3} = x - 13 + \\frac{16x - 81}{x^2 + x - 3}", "ac579035fe50338bb0f3e6d0add8dbaf": "L_n = \\sqrt{9 - {4 \\over {m_n}^2}}.\\,", "ac57f58d60a63aa638ba5b5cf2fc8aad": "(P \\and Q)", "ac581d71e4df95b0bc22173d34db9da8": "\\Omega = \\iint_S \\frac{ \\vec{r} \\cdot \\hat{n} \\,dS }{r^3} = \\iint_S \\sin\\theta\\,d\\theta\\,d\\varphi", "ac58687fedac587bf042ad401814ca40": "|\\mathcal{A}| = |\\mathcal{A}(x)| + |\\mathcal{A}-x|", "ac5872334f458a16f0f784693a162a90": "Sa \\cup \\{a\\}", "ac58733ef195a77b12805d54d1b88872": "+\\alpha", "ac588665121eb148bedcffafb05faee0": "{-x^{n-1}}", "ac58dde9c6edea530fd76fa5827757d7": " {n_3 \\over d_3},", "ac59151262e29901787b899908027d4c": "s=\\sqrt{ng/m}", "ac5939c9e381ad95299e5ca7ba387a32": " \n\\mathbf{H}_1= \\begin{pmatrix}\n\\mathbf{G}_1 \\\\ \\mathbf{Q}_1\n\\end{pmatrix},\n\\mathbf{H}_2= \\begin{pmatrix}\n\\mathbf{G}_2 \\\\ \\mathbf{Q}_2\n\\end{pmatrix},\n\\mathbf{H}_3= \\begin{pmatrix}\n\\mathbf{G}_3 \\\\ \\mathbf{Q}_3\n\\end{pmatrix}\n", "ac594a002e11c4e00f62069afbdf8104": " x\\ne 0", "ac597ebb4c36f141d6de735061c2e04a": "1.9862", "ac59b1181124e1a025f2748614fe6d4c": "R_{ik\\ell m}=-R_{ki\\ell m}=-R_{ikm\\ell}.\\ ", "ac59d5c533a4e45180e8e449c30c5f3e": "J_3(\\mathbb C)", "ac59f3146bafe27f8c02d5e79a2695d6": "h_n=\\langle\\phi_{0,0},\\,\\phi_{-1,n}\\rangle", "ac5a3042e5334a81668c17037deb9dff": " \\frac{x_n}{10} ", "ac5a36beee90794ccf06364188ddef4d": "\\Lambda = \\frac{h}{p}", "ac5aed52e6f6e2fad3f8c20bb11fb74b": " \\frac{dy}{dx} ", "ac5aef16f6c08fc2d4dec47d347e15e1": "\\sqrt{1-\\frac{b^2}{a^2}}", "ac5b34f3f3b27dfe9a61c6c894107810": "ln K=-RT \\ln \\left(\\sum_k {a_k}^{m_k} (solution)\\right)", "ac5b559422794fd336d1b3c294d189a9": "F_r + mr\\Omega^2 = m\\ddot r", "ac5ba95453c64436146b2ef233a4a81c": "o_i=2", "ac5bbb3687d01c127ea2e09c26e79a9f": "-x = - \\{ X_L | X_R \\} = \\{ -X_R | -X_L \\}", "ac5c10d70735ee89ee2c9bcab0ae1d9b": "(x+c)^2 + y^2 - (x-c)^2 - y^2 - 4a^2 = - 4a\\sqrt{(x-c)^2+y^2}", "ac5c193d7df4220226f8742ab033974f": "\n M = R_A x = Pbx/L \n ", "ac5cb70014d349a9423fd3a00cdba45b": "{\\rm Eu}", "ac5d0082777a3ae8fa801aa3df8643b9": "C=P\\otimes Q\\otimes P\\otimes\\dots\\otimes Q.", "ac5d0fdbb2ac7ea13480929e97cdfab6": "E = -\\frac{1}{2}\\int_\\text{magnet} \\mathbf{M}\\cdot\\mathbf{H}_\\text{d} dV", "ac5d130b673b326b265ae014dc5cd2da": "\\Omega^{\\Omega^\\Omega}", "ac5d3fc90f0c69fc32d4409605c0cd84": "b_1=\\frac{10,00}{31}=32\\mbox{ with remainder }8", "ac5d9775b7bf2e332da73818b7960a28": "V = (\\pi r^2)(2\\pi R) = 2\\pi^2 R r^2.\\,", "ac5db7a0ab8919f59ebdba4415f893d6": "x = \\sigma \\tau\\,", "ac5dfe1da6f36f94eee7f96a850adcee": "\\displaystyle M^*F(a^2)=\\int_K F(ga) \\, dg.", "ac5e05e13e026d1a73278ecd1f4919dc": "\nP\\approx P_0 \\left(1 + X + \\frac{1}{3}X^2 \\right)=$333.33 (1+.675+.675^2/3)=$608.96\n", "ac5e17dfe81902e5d26e0133532fc830": "\\begin{align}\n Z \\left(E(K), \\frac{1}{qT} \\right) &= Z(E(K), T)\\\\\n \\left(1 - aT + qT^2 \\right) &= (1 - \\alpha T)(1 - \\beta T)\n\\end{align}", "ac5e5963fb60a759e905dd61d8b20d5f": "\\{\\lambda_k\\}_{k=1}^\\infty", "ac5ed3b29796a5ee75f586553b8f1aa1": "a_2^\\dagger", "ac5f0fdddf7c77ab77ad070214353d7d": "(y\\land x)", "ac5f79345c2917f2ad1782370e993cfe": "{\\mathcal L}_{xx}^2", "ac5f79de62e8ea201e1d31aefc1f6f2c": "Probability \\ of \\ Failure = 1 - Reliability", "ac5f7d0549a68925905bf2502c571d38": "x_{n+1} = x_n - \\frac{f'(x_n)}{f''(x_n)}. \\,\\!", "ac5ffdad1acc6d27c557afa4519dd372": "\\rho(\\lambda)", "ac5fffef3db1632e06fbe5e3abf8a26f": " u_R, c_R, t_R ", "ac6049605c143fb7b50bcb71cf540a93": "P(z)=\\sum_{i=0}^na_iz^{n-i}, \\quad a_0=1,\\quad a_n\\ne 0", "ac6060264b0ae44e991c3f7924414bd8": "F = \\int_{0}^\\infty \\frac{1}{\\Gamma(x)}\\, dx.", "ac607400326ec64b4de422a67d4ad878": " 0 + r_{\\mathrm{A}} V = 0 + \\frac{dn_{\\mathrm{A}}}{dt} ", "ac6095b32baeea48ee84c5d1506f0b88": "f_X(x)>0", "ac6099ca63903f55e5f9a421a8400761": "\\Delta(1)", "ac60b2450430dc2760411cd244c2116d": "D(f) \\leq Q_E(f)^2Q_2(f)^2", "ac60b8262aaa2091a1b9fc4cbe52801d": " (\\mathbf{S}\\times(\\mathbf{S}\\times(\\Delta\\mathbf{r}_i))) \\cdot \\mathbf{S}\\times(\\mathbf{S}\\times(\\Delta\\mathbf{r}_i)) = (\\mathbf{S}\\times(\\mathbf{S}\\times(\\Delta\\mathbf{r}_i)))\\times\\mathbf{S}\\cdot (\\mathbf{S}\\times(\\Delta\\mathbf{r}_i)),", "ac60fa6bb8ec6d421bc75c890219fbae": "(u,w) \\in E(G) \\Rightarrow \\left | c(u) - c(w) \\right | \\notin T", "ac61226ee1912bd294161bba1062da73": "\\gamma=\\textstyle\\frac{1+\\omega}{2+\\omega}\\,", "ac61717620b820d4698bf3ee511f558e": " r \\mathbf{\\hat{r}} \\,\\!", "ac618cdad64183ef7ffb21091168d325": "n\\in\\mathbb N", "ac619b7f876d0c4c3ffa05db1e8f14fe": "\\dot{x}^{a} = \\epsilon \\frac{1}{T}\\int_{0}^{T}f(\\tau,x,0) d\\tau = \\tilde{f}(x^{a}).", "ac620ccc7e9f2f7a612505126d1e4ec8": "g_{ij} \\rightarrow g_{ij} + \\delta g_{ij} \\,,", "ac629b44f26753089f8b98c138e566f8": "OD=Log_{10}(I/I_{0})=\\epsilon\\ *[X]*l*DPF+G", "ac62c02668cff2d75e79d6022b033a49": "m c^2", "ac62e4cff939a466fb5c9edc096576a9": " n<1 ", "ac62f71f2f0d122bd86979ea33b69735": "|\\vec{a}| \\le 1", "ac6309818628bd6565fd7fb9b1738985": "\\begin{bmatrix}\nE'&F'\\\\\nF'&G'\n\\end{bmatrix} =\n\\begin{bmatrix}\n\\frac{\\partial u}{\\partial u'}&\\frac{\\partial u}{\\partial v'}\\\\\n\\frac{\\partial v}{\\partial u'}&\\frac{\\partial v}{\\partial v'}\n\\end{bmatrix}^\\mathrm{T}\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n\\begin{bmatrix}\n\\frac{\\partial u}{\\partial u'}&\\frac{\\partial u}{\\partial v'}\\\\\n\\frac{\\partial v}{\\partial u'}&\\frac{\\partial v}{\\partial v'}\n\\end{bmatrix}\n", "ac63c13e69ea7368800a3e8149352309": "\\{\\,e_\\lambda : \\lambda \\in \\Lambda\\,\\}", "ac63c53ccc35494f09f542f430e59d3a": "X^T \\hat e", "ac64104cf6c2d67f4b460a11a90485ad": "x_2-x_1", "ac64152b4501795453ea978cdc9d240b": "{{P}_{V}}f", "ac645d0f259d900d47e65a167c0dd576": "H_\\mathrm{v} = E_\\mathrm{v} \\cdot t", "ac6463570889371889565663e283dbf4": "f[x_\\nu,\\ldots,x_{\\nu+j}] := \\frac{f[x_{\\nu+1},\\ldots , x_{\\nu+j}] - f[x_\\nu,\\ldots , x_{\\nu+j-1}]}{x_{\\nu+j}-x_\\nu}, \\qquad \\nu\\in\\{0,\\ldots,k-j\\},\\ j\\in\\{1,\\ldots,k\\}.", "ac64e884800c5fd0c48c9ac35cb62c9a": "S(z;x)=x+\\mathrm{mul_c}(z)", "ac64fe2a82a509e5e6c45dffb5e3a014": "\\sigma _b", "ac64ffeed293f17398e36ce96a633821": "f(x) = \\sqrt[3]{x}", "ac650260fe925070f9c7db5294f35c4c": "M\\cdot\\frac{1}{k} = P\\cdot Q", "ac650ff72aed58ebebae303ce1382d7b": "\\sin[\\arctan(x)]=\\frac{x}{\\sqrt{x^2+1}}", "ac651cb13d6b773270e1abc5f36cbc30": "U = VDV^*\\;", "ac6549f04bfc9999d965df64155f1fd7": "\\mathbf{P}=\\{\\mathbf{c}_i=(\\mathbf{x}_i,\\mathbf{y}_i,\\mathbf{z}_i)\\vert^{N}_{i=0} \\subset \\mathbb{R}^3\\}", "ac658bc1190764947ce77af4cc3e3236": "\\mu(A \\cup B) = 0", "ac65d0683f85a6d63c4f6da61e273184": "\\psi_n'(x) = \\sqrt{\\frac{n}{2}}\\psi_{n-1}(x) - \\sqrt{\\frac{n+1}{2}}\\psi_{n+1}(x)", "ac65d90852a77b7a1228091c3ef47a75": "n+g(k)", "ac661a275971c9b20679720fdf9cfb27": "h \\le 4", "ac661a29c48d1fe3fcbd3a93b6f49b66": "u_{l} = \\sum_{q} \\sqrt {\\frac {\\hbar} {2MN\\omega_{q}}} (a_{q} e^{iqal} + a^{\\dagger}_{q} e^{-iqal})", "ac66529482b80579f03a26bb3c3b9cc3": "n^{f(n)}=g(n)", "ac669be63ce65fc35cd80b1289e2862d": "I_1\\subseteq\\cdots \\subseteq I_{k-1}\\subseteq I_{k}\\subseteq I_{k+1}\\subseteq\\cdots", "ac66cfbaf89ece040b12ceb74c922cf2": "D_\\mathrm{X,\\,rest}^{\\mathbf{k},\\mathbf{k'}}", "ac66f5fd2ae9688f313664ab818d1aa3": "\\mathrm{NL} \\left[ \\cdots \\right]", "ac673f3ac2ba6af5bb73b05e6018e478": "=A-\\frac{4\\pi}{k}~\\mathrm{Im}~f(0),", "ac674821b740c5f0ee03783f86eb855b": "\\sum_{j=1}^{n_S} \\sum_{b_j=0}^{a_j} \\sum_{ \\beta_j } x_{b_j} \\ a_j = \\sum_{h=1}^{n_P} \\sum_{ d_h=0 }^{c_h} \\sum_{ \\gamma_h } u_{\\gamma_h} \\ y_{d_h} \\ c_h, ", "ac67a6bbd4e918cf52b6c93b8919e2b1": "B= \\rho_0 \\left(\\frac{\\partial P}{\\partial \\rho}\\right)_{adiabatic}", "ac67bd33955643c8da7df2a8b27239b4": "n(r)=A\\times r^{-q}", "ac67bf448131396b003b86c260d57fb3": " \nx^{\\ast }", "ac67e3986df81b20791e39a9a66e7117": "S_x(\\omega) = \\frac{2 k_\\mathrm{B} T}{\\omega} \\mathrm{Im}\\,\\hat{\\chi}(\\omega).", "ac67fc386d09af4e1aebbb18265aaab9": "\\Delta L = 20\\log \\left( {\\omega_2 \\over \\omega_1} \\right) \\ \\mathrm{dB/interval_{2,1}}", "ac6877a5b1affa393d78c5e21536d1ef": " \\mathbf{a}' = \\mathbf{a} + \\mathbf{A} ", "ac68875869f1c513a9eac93274593a09": "S(Y) \\rightarrow X", "ac68928f385893fe60c26a59abdc6352": "u \\equiv \\sqrt{\\frac{ax + b}{ax}}", "ac68aa85b7ebb48cd1bc2e51983bc29f": "\\sqrt[n]{\\frac{a}{b}} = \\frac{\\sqrt[n]{a}}{\\sqrt[n]{b}} \\,.", "ac68cf8594dce2e13e65414402e3764a": "\\begin{align}\nR &= \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2} \\\\\n&= \\sqrt{\\left[\\tfrac{1}{2}(-10 - 50)\\right]^2 + 40^2} \\\\\n&= 50 \\textrm{ MPa} \\\\\n\\end{align}\n", "ac6a501d84bfa1218ce37b5ebba2f1e9": "Rep(w',\\tilde{P}): ", "ac6ab51682381270751e1d83a9978f24": "E_{abcd} = \\frac{1}{n-2} \\, \\left( g_{ac} \\, S_{bd} - g_{ad} \\, S_{bc} + g_{bd} \\, S_{ac} - g_{bc} \\, S_{ad} \\right) =\n \\frac{2}{n-2} \\, \\left( g_{a[c} \\, S_{d]b} - g_{b[c} \\, S_{d]a} \\right) ", "ac6b05b7902d75131330c9e0f4bc2173": "H \\approx \\frac {f^2} {N c} \\,.", "ac6b090847c1475c1bd561672393be2d": "\n\\Psi(\\mathbf{r}_1,\\mathbf{r}_2,\\dots,\\mathbf{r}_N)=\\psi(\\mathbf{r}_1)\\psi(\\mathbf{r}_2)\\dots\\psi(\\mathbf{r}_N)\n", "ac6b9ceb645fed73f389415d38075131": "\\scriptstyle M_1", "ac6c29d03350956e44d6da252477bc4c": "\\dot\\omega_s", "ac6c4a3c050ea00ceb292f6a5624ec4c": " \\sigma = \\sum_i T_i \\otimes S_i ", "ac6c8d794327dea5d74654298acc72c6": "\\alpha\\left(1 - \\frac{1+2x_3}{K}\\right) < 0, \\text{ or } K < 1 + 2\\frac{\\alpha}{1-\\alpha}", "ac6cb404b55ae5c700e1c72b4bc20f9a": "Q = A_2\\;\\sqrt{\\frac{1}{1-(d_2/d_1)^4}}\\;\\sqrt{2\\;(P_1-P_2)/\\rho}", "ac6cb8272efddf8813cae1caeb2c14b9": "s=(\\ldots,(s_i, t_{ei}),\\ldots) \\in S ", "ac6cf9d25dd26211f3fada1a30ea4b62": "\\int uv = u v_1 - u' v_2 + u'' v_3 - \\cdots + (-1)^{n-1}\\ u^{(n-1)} \\ v_{n} + (-1)^n \\int{u^{(n)}v_{n}}.\\!", "ac6d52854288f0fe148dba2022d24c5b": "\\sigma(x)=x^2", "ac6d8ea88ba58f6ebbea61d35eef2fcb": " Q(t) = Q_\\mathrm{f}(1 - e^{-\\lambda t}) \\, ", "ac6d97dc7e239eb17f1ef109df778bae": " W_{A_{\\rm H_2O}} ", "ac6d9b49f1b2dc815843518138d89445": "P(3) = 3^2-1 = 8, ", "ac6dafaa84518254be6339974a7d0d62": "\\mathbf{A} + \\mathbf{B} = (A_{23} + B_{23})\\mathbf{e}_{23} + (A_{31} + B_{31})\\mathbf{e}_{31} + (A_{12} + B_{12})\\mathbf{e}_{12}. ", "ac6e217f67409af6367f32e279c95bfb": "F = ma_c = \\frac{m v^2}{r}", "ac6e92b9cc1c0fc34afd716878d61e21": "(R, \\cdot)", "ac6ea29b2580cf330194369553c023b7": "yx=\\begin{pmatrix}\n 1 & 1 & 0\\\\\n 0 & 1 & 1\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}. ", "ac6eb5d706652a68ae8bea36c5864675": "\\phi\\colon X \\to Y^{\\text{op}}", "ac6ee0afdb130ebf3cdda8386b5d30e2": "X_2 \\le X_1 ", "ac6ef607418f224973cf10307b27885e": "R=\\left|\\frac{M_{10}}{M_{00}}\\right|^{2}", "ac6f8901feaa1b414f309c624efaba86": "\\mathbf{u}\\oplus_E\\mathbf{v}=2\\otimes\\left({\\frac{1}{2}\\otimes\\mathbf{u}\\oplus_M\\frac{1}{2}\\otimes\\mathbf{v}}\\right)", "ac6fc535f8a17fe3fe3b6d10d2f64bff": " d^2 F_x = \\sum_{i,j}\\frac{de_i de_j'}{r^2} \\left[\\left(1+\\alpha_1 \\frac{u_x^2}{c^2}+\\alpha_2 \\frac{u_r^2}{c^2}\\right) cos(rx) - \\beta_0 \\frac{u_x u_r}{c^2}-\\alpha_0 \\frac{ra'_r}{2c^2} + \\frac{ra'_x}{2c^2}\\right]", "ac6fc5712110f6d886c0e296b9a7bb50": "X^*(s) = \\frac{1}{T}\\sum_{k=-\\infty}^\\infty X\\left(s-j\\tfrac{2\\pi}{T}k\\right)+\\frac{x(0)}{2} = \\sum_{n=0}^\\infty x(nT)\\cdot e^{-nTs}.", "ac6fd8b59b0467b2aa724db812628d76": "Z_n(c) = e^{-c\\beta} q^{n+1}", "ac6fed7873d7b31d1124be438792ca9d": "x_s=0", "ac700f814502ea8fa84c10204b2df7b0": "(a^2 + b^2)^2 =~(a + b)(a - b)^2.", "ac705fffeab2632147a8ac3b5a5ca5d7": "x = (x_1, \\dotsc, x_n)", "ac70cead593142d596009b8bfafee259": " M \\times \\,^{\\prime\\prime} 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\,^{\\prime\\prime} = M \\times (2^5 + 2^4 + 2^3 + 2^2 + 2^1) = M \\times 62 ", "ac7116ffae571a18336f7b0482717cee": "B[u, u] \\geq K \\| u \\|_{H^{1} (\\Omega)}^{2} \\mbox{ for all } u \\in H_{0}^{1} (\\Omega),", "ac714b33b69713c1d167df478f7600a7": "K_j = \\{ z; |z| \\le (1 - 1/j) \\}", "ac71697eaa90a818d0a8b372a4449ddf": "\\textstyle f_{f_{(p)}}(I)=max\\left \\{ 0, \\frac{f(p)^T)f(x)}{\\left \\| f(p) \\right \\|\\left \\| f(x) \\right \\|} \\right \\}", "ac716fe842484ac59f78572e7509d9e3": "p=\\frac{h}{\\lambda}=\\frac{hn}{\\lambda_0}", "ac71a830f77500070e2a5ad571fb3546": "\\{0, 1\\}", "ac71cdec9760eb14f5eb830afcd12079": " \\ = \\lim_{\\epsilon \\to 0} \\frac{1}{(2 \\pi)^4} \\int d^4p \\, \\frac{e^{-ip(x-y)}}{p^2 - m^2 + i\\epsilon} ", "ac727edfb5f1820c5bf856e00f3c0e23": "L/D = Cz / Ci = {{\\pi \\times \\lambda \\times e} \\over Cz } ", "ac72ae7f13022d2482d0f74029073699": "f(x_1, x_2, \\dots, x_n) = \\sum_{i=1}^M \\left(c_i \\prod_{j=1}^n x_j^{a_{ij}}\\right)", "ac72e6ff3b847f81cfd9bad71267da57": " \\frac{|d|}{N^2} \\,", "ac7325c5ae33398cf130fc18419a8402": "\\| T_s\\|_{op} = \\| s \\|_{\\infty}.", "ac738c8e4d5659e2949d3a12fd1cef06": "\\hat{\\phi}\\ ,\\ \\hat{\\lambda}\\ ,\\ \\hat{r}", "ac73a1d41d939b6bb2776ea967baa60c": "G_0 \\;\\xrightarrow{f_1}\\; G_1 \\;\\xrightarrow{f_2}\\; G_2 \\;\\xrightarrow{f_3}\\; \\cdots \\;\\xrightarrow{f_n}\\; G_n", "ac73c9886ffa0bcd16661d73c4e4d731": "\n\\delta^{\\mu_1 \\dots \\mu_p }_{\\nu_1 \\dots \\nu_p} =\n\\begin{vmatrix}\n\\delta^{\\mu_1}_{\\nu_1} & \\cdots & \\delta^{\\mu_1}_{\\nu_p} \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\delta^{\\mu_p}_{\\nu_1} & \\cdots & \\delta^{\\mu_p}_{\\nu_p}\n\\end{vmatrix}.\n", "ac73e376adc7d62f8884dc04f304f3e1": "2\\pi\\over 3", "ac742c2e33bf562745605153583368ae": " % \\mbox{ difference} = \\left ( \\frac{r_{solute} - r_{solvent}}{r_{solvent}} \\right ) \\times 100 \\le 15%.", "ac747d7c43950728212043e2da163e6d": "\\rho_{N_1,N_2}=\\rho_{\\epsilon_1,\\epsilon_2}", "ac748e6222df11a2c035a2bca01edfe1": " \\bar{D}", "ac7509f63413061ed61a6d2d91a54c10": "\\sum a_nb_n ", "ac75293d330037e6af5d9ae332905b93": "x^{64} + x^4 + x^3 + x + 1", "ac75578528c411ec7f86f67913d362b5": "\\, V-E+F=2 ", "ac7562b2c8f3314dbfa6d868b388af66": "{_n\\mathcal{C}^0},{_n\\mathcal{C}^k},{_n\\mathcal{C}^\\infty},{_n\\mathcal{C}^\\omega},{_n\\mathcal{O}},{_n\\mathfrak{V}}\\,", "ac759ef92fd2cf607676766b6404bda3": "1 = A_0 \\leq A_1 \\leq \\cdots \\leq A_n = G.", "ac75c0175415f39b6976f3986c098107": " \\lambda_1\\over \\lambda_2", "ac76223f6d2e18b26ce1d20060be3e0d": "L_g(\\psi)=\\lambda\\psi", "ac766a8fd0a6c11583588b24cb479ee1": "f(x) = \\lim_{h\\to 0}\\frac{A(x+h)-A(x)}{h}", "ac767b1e3ea557add1a188c66f3a4d9f": "G_{T_2}^\\ominus = G_{T_1}^\\ominus-S_{T_1}^\\ominus(T_2-T_1)-T_2 \\int^{T_2}_{T_1} {{C_P^\\ominus} \\over {T}}\\,dT + \\int^{T_2}_{T_1} C_p^\\ominus\\,dT", "ac76c9792578ffc9b5941778167f1cd8": "\\mathcal{F}^{-1}\\colon\\mathcal{S}'(\\mathbb{R}^n)\\to\\mathcal{S}'(\\mathbb{R}^n)", "ac76e9dc9fbdf33348ba0feb50832d6f": "k_d", "ac77513f4153c730b024bf6283dc03af": "\\frac{D}{D t}", "ac776f54906d0e635af42ba47c7d7479": "f_b", "ac77ee2662f1cbf2a55bfe110d3072e1": "K^{-}", "ac77fa5c5e37c8ac6d0218d83ecc426f": "A = \\pi r^2\\,", "ac7851403596aa7a30c934e477cface9": "\\|\\xi\\|=1", "ac786f69ba692f47aa5e320d248c5dd1": "\\to \\;", "ac78bc8772bfefae8b8110fef3887c66": "\n7.000 \\mbox{ metres} = \\frac{L + B + 1/3G +3d + 1/3\\sqrt{S} - F}{2}\n", "ac78d20d44d5fc29946423fcded1f901": "E=CS^r", "ac78f10c275412016fa40d3a1dab747c": "\\dot{Q}_t", "ac790d6d3840cae6edb1a20d04d01690": "\\displaystyle{R_t f(x,y)=f(x,y+t).}", "ac792447ad70c24295c8b982a17c3416": " \\mathcal{C}_{XY} = \\mathbb{E}_{XY} [\\phi(X) \\otimes \\phi(Y)] = \\int_{\\Omega \\times \\Omega} \\phi(x) \\otimes \\phi(y) \\ \\mathrm{d} P(x,y) ", "ac793c01c70d38d2c8e568985fd717f7": "p_5=a_{30} \\omega+a_{21}, ", "ac799c934d9531f0150c0fdf293b31b1": "\\sum \\beta ^t u_t (x)", "ac79c0d1ca12af4b15a896e504bec357": " F_{\\alpha\\beta} = \\partial_{[\\alpha} A_{\\beta]} = \\nabla_{[\\alpha} A_{\\beta]},", "ac79f30984fab7bd68e81b3a506ef97a": "x_{n+1}=x_{n}-\\frac{f(x_n)}{f'(x_n)} \\left({\\frac{1}{1-\\frac{f(x_n)f''(x_n)}{2(f'(x_n))^2}}}\\right).", "ac7a11785204004884f553fd26c36367": "\\lim_{x\\rightarrow -\\infty}f(x)=c", "ac7a323de3b6ee751bd610b514613301": "\\rho(t) = \\rho", "ac7a3959a91fade301dc01e309027f7a": "p_\\varepsilon", "ac7a434fc0b26a03d5718b02a4918c75": " v = \\gamma \\kappa^{-\\frac{1}{3}}, ", "ac7a4fbd0d788d94f43454817e1c208c": "TM", "ac7a57c438fdaed71d97947acdaca557": "W(t=0)", "ac7a7640dae742749d8679b04d33729a": "x = \\mu \\,", "ac7b1f617ec49a445245ac6abe6480fd": "\\mathit{K}", "ac7b98ef3080dc19dac642d2b6f4ae86": "\\mathbf{p}\\rightarrow -\\mathbf{p}", "ac7bcd422dcec803d42fb6c6c5ba0013": "u: V_i \\overset{p_i}\\to U_i \\hookrightarrow X", "ac7bd3eebb32d282c1d8ff30f50c0a94": "\\ p_1'", "ac7c275db7e6590ab19bda8ca44c1f49": "A \\to B\\alpha \\mid C", "ac7c58ae539cc4343ad507088f5171bc": "\\mathcal{F}_{T}=\\mathcal{F}_{0}+\\frac{1}{2}K_1(\\nabla\\cdot\\mathbf{\\hat{n}})^2+\\frac{1}{2}K_2(\\mathbf{\\hat{n}}\\cdot\\nabla\\times\\mathbf{\\hat{n}}+q_0)^2+\\frac{1}{2}K_3(\\mathbf{\\hat{n}}\\times\\nabla\\times\\mathbf{\\hat{n}})^2", "ac7c5cd6c2661cbdc323c77db513b2e6": "n \\in \\mathbb N", "ac7c7b85cf1e98f6e43ddc0fb2ba5e77": "A \\in V", "ac7cae65b4136781e0efd4b11f403e1f": "\n\\begin{bmatrix}\nK_{11} & K_{12} \\\\\nK_{21} & K_{22}\n\\end{bmatrix}\\begin{bmatrix}\nx_{1} \\\\\nx_{2}\n\\end{bmatrix}=\\begin{bmatrix}\nF_{1} \\\\\nF_{2}\n\\end{bmatrix}\n", "ac7cc8d327ff5ec94c3c4c946588fe53": "1+x+x^3", "ac7cef78c641db7f9831a0ab883da5cb": "\n{\\rm disc}_{d}(\\{t_i\\})=\\left(\\sum_{\\emptyset\\neq u\\subseteq D}\n\\int_{[0,1]^{|u|}}{\\rm disc}(x_u,1)^2 dx_u\\right)^{1/2}\n", "ac7d4d3fbb35c342de7cc2864bda4544": "r = f(\\theta)", "ac7d4fea6ff057c7bffad64342f4db58": "\\mu(x) = \\sum\\limits_{y \\in \\{0, 1\\}^n : y \\leq x} \\Pr[y]", "ac7d7159b927004d90aa0f55048bc661": "E[\\gamma]:=\\frac{1}{2}\\int_a^b F^2(\\gamma(t),\\dot{\\gamma}(t))\\, dt", "ac7db01c25d51bc38034f3b1a145613f": "\n\\left(\\frac{a}{2}\\right) = \\begin{cases}\n\\;\\;\\,0&\\mbox{ if } a \\mbox{ is even}\n\\\\(-1)^{\\frac{a^2-1}{8}}&\\mbox{ if }a \\mbox{ is odd. }\n\\end{cases}", "ac7e0729ec6a5836e5e282354c2646ec": "g(w)= \\sum_{n=1}^\\infty g_n w^n \\quad", "ac7e10333dfdfad37efbd24a560486cf": "A=1, B=-1, C=c", "ac7e1c4d65ce6d119cd9415c7ed01dc8": "\\textstyle q^2 = pr", "ac7e2296686feb7aaa54d552f9a764f6": " \\vert\\Delta\\vert \\leq C(\\varepsilon ) \\cdot f^{6+\\varepsilon }. \\, ", "ac7e2d386c26a05b29905e6c7ef0ebaf": "\\mathcal{F}(F)(x)", "ac7e30ae60963d249f43bc20ff862e3a": "i\\, ", "ac7e31aa9efea057446390ff26d6e72a": "\\mathfrak m^{+}\\oplus\\mathfrak l", "ac7e34157d966413043646565553a7f7": "dA_2/dt = \\kappa_2 A_2 + (Q_{21} A_1^2 + Q_{22} A_2^2) A_2", "ac7e49820bc8cba15b55799bbab9bed3": "\\lim_{x\\to\\infty}\\frac{f(x)}{g(x)}", "ac7e5fe9201980ead2f5ee5db12b1b54": "R^{(1/p)} = A[X_1, \\ldots, X_n] / (f_1, \\ldots, f_m) \\otimes_A A_{F^{-1}}.", "ac7e939df1042ccc97c7ff56aa40df4d": "\\omega = \\omega^{-1} = e^{2\\pi i /p}", "ac7ea648636797acef7187a05ff61cfb": "x = I_{\\frac{1}{2}}^{[-1]}(\\alpha,\\beta)", "ac7f0364c26a89a56e7597b41e12638c": "(A_n)", "ac7f0b76f53b3b891f65bd93673da6d9": "\\mu{\\frac{dy}{dx}} + \\mu{p(x)y} = \\mu{q(x)}", "ac7f19d2c6398c89bc22fa25c15dc1dd": "\\frac{dy}{dx}=g(x)h(y).", "ac7f685863bb8c827bf8ca9eb00626dd": "\\gcd(e, q-1)=1", "ac7f86ffbba6688021d318c2b88d1a45": "\\bar{u_i}=\\bar{h_i}-p\\bar{v_i}", "ac7fe526b679cc7c9db7e39560815c27": " g(c_i(\\bold x))=\\min(0,~c_i(\\bold x ))^2. ", "ac806ef41ea15d03d8917acbb8b0b148": "\n\\mathbf{j=}\\frac{1}{\\alpha }\\left( \\psi \\widehat{\\mathbf{v}}\\psi ^{\\ast\n}+\\psi ^{\\ast }\\widehat{\\mathbf{v}}\\psi \\right) ,\\qquad 1<\\alpha \\leq 2.\n", "ac8072fa7e361fec8d254e0815039185": " Cov(e_{nit}, e_{niq}) = \\sigma^2 (X_{nit} X_{niq}) ", "ac80a893933b928b591e8e4bd58363c6": "q = \\left\\lceil y - 0.5 \\right\\rceil = -\\left\\lfloor -y + 0.5 \\right\\rfloor\\,", "ac811a798312eba8bd51841ec37dda49": "\\hat{\\boldsymbol{n}}", "ac8197bfcaba19efa063fd8fa93fe598": "c\\in S", "ac81e26898b284d87a07d969cd5dfe99": "\\Delta:H^2({\\mathbb R}^n)\\to L^2({\\mathbb R}^n) \\,", "ac81f755956cf75bc2d9b4d40b51e8a3": " \\frac{1} {1-RL}", "ac82939575d7e84de714426410d08cd2": "\\Gamma_{S(t)}", "ac829e6e046012abd0461717e3a9010c": "d\\geq N", "ac82ef208010ca8839d4ee1e159078b5": "x=-\\frac{B^\\prime + B^{\\prime \\prime}}{2(B^\\prime-B^{\\prime\\prime})}", "ac8339cea8176f5b87589d794521ec1b": "\\pi_k(X)", "ac835d9bf401a45f950f9ebbf2904d3a": "\\mathrm{Frob}_\\mathfrak{p}", "ac83c8a39fd7f1e7f6aa4b9d99dc680f": "\\begin{matrix}2&2&4\\\\3\\end{matrix}", "ac83cb03a29687ab0aba07cd9ecf374d": " D_{\\mathrm{KL}}(P\\|Q) = \\int_X \\ln\\left(\\frac{{\\rm d}P}{{\\rm d}Q}\\right) \\,{\\rm d}P, \\!", "ac83e098655ec416a72ed72c7873380e": "x_{k_n}<\\lambda\\quad\\forall n.", "ac83fbc2a30dfbd8741e282aaae5c8fc": "\\mu_i^{\\star}", "ac842a99f479ce3820ad4b836a92f3bf": "\\mu = g \\frac{ Q e \\hbar}{4 m}, ", "ac8457b80ff5cffdcc28123ac8a359cf": "\n\\frac{1}{\\Theta\\cos\\theta}\\frac{d}{d\\theta}\\left(\\cos\\theta \\frac{d\\Theta}{d\\theta}\\right) + \\frac{1}{\\Phi\\cos^2\\theta}\\frac{d^2\\Phi}{d\\varphi^2}\\ =\\ -\\lambda\n", "ac8461fbe0884a9e6ace0797dbc5acf0": "P\\times T = P' \\times T' ", "ac846be68077a05bcaeddb7c4ce4883d": "2\\pi f_s = \\omega", "ac8496359d371d3f106ffe12813e418a": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ -\\sqrt{\\frac{3}{2}},\\ \\pm2\\sqrt{3},\\ \\pm2\\right)", "ac84b2ad02982e6b04a13fe2ddbc1d0c": "m\\{x : |I_\\alpha f(x)| > \\lambda\\} \\le C\\left(\\frac{\\|f\\|_1}{\\lambda}\\right)^q", "ac84e541983e44f67d823e8fe5d0c17d": "208_{11} \\ ", "ac84f77bd394e948c2a99b8b25ac4796": "F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'')\n-2y(av'v''-b(u''v'+u'v'')-au'u'')\n+b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'')\n+a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'')", "ac85003e54ae5f4a028c7f2a57e6d97b": "P\\rightarrow P\\and Q", "ac852866316b9d145553e039b9e60ef7": "A = \\overline{B_{R}}", "ac855c4f1c4c3fa6d35cd2bcb7a6763a": "G'=(V',E')", "ac856a002cd01ed937df58dac29c4740": " \\frac{dY}{dt} = B X -X^2Y. ", "ac858d702aeaf70d457f62d3a1b451d9": "\\mathcal{T}(P) =", "ac859d6c693552981a53c02ff801ddba": "\n\\hat z_\\mathtt{RK}(\\mathbf{s}_0 ) = \\mathbf{q}_\\mathbf{0}^\\mathbf{T} \\cdot \\mathbf{\\hat \\beta}_\\mathtt{GLS} + \\mathbf{\\lambda }_\\mathbf{0}^\\mathbf{T} \\cdot (\\mathbf{z}\n- \\mathbf{q} \\cdot \\mathbf{\\hat \\beta }_\\mathtt{GLS} )\n", "ac85b354683c8045f9ac352e4839f573": "[f(x)]^{k/h}=f(x)\\cdot f(x+h)\\cdot f(x+2h)\\cdots f(x+(k-1)h).", "ac85d3d2b24349306d4f4a189f5bc045": "k:=k+1\\!\\,", "ac85f89cd032e312588c29052b47778c": "{\\delta}_1=\\sqrt{{\\mu}\\over {\\rho}{\\omega}}\\,\\!", "ac862aed5625c7cb66b2e82c6219fca7": "\\{0,1\\} = \\partial I", "ac8683163ad2b12626a6de139518bf28": " \\mathbb Q", "ac86caebf28348d596bebbf504f73517": "\\mathcal{L}_y = - y_{u\\, ij} \\epsilon^{ab} \\,h_b^\\dagger\\, \\overline{Q}_{ia} u_j^c - y_{d\\, ij}\\, h\\, \\overline{Q}_i d^c_j - y_{e\\,ij} \\,h\\, \\overline{L}_i e^c_j + h.c.", "ac870dc245178fc8b490a3ddc0995b03": "K_1=K_2=K_3=K", "ac8722535a87a144c18be65eaa533fcb": "\\mathbf{v} = \\dot{r} \\mathbf{\\hat r} + r\\,\\dot\\theta\\,\\boldsymbol{\\hat\\theta } + r\\,\\dot\\varphi\\,\\sin\\theta \\mathbf{\\boldsymbol{\\hat \\varphi}} ", "ac872b407f7694be42f58a709c28af9f": "\\frac{p_n - C_n\\left( \\mathbf{z}\\right) }{p_n}=\\frac{k}{\\varepsilon_n}", "ac8776c97436733e8a9c9efb2ced44ba": "{n_2}/{n_1}", "ac879cac39db2719f97be0c752d1552a": "M_{\\rm Hg}", "ac879d7b5a44cb2d35266e4011fe90eb": "E(k) = \\int_0^{\\pi/2}\\sqrt {1-k^2 \\sin^2\\theta}\\ d\\theta = \\int_0^1 \\frac{\\sqrt{1-k^2 t^2}}{\\sqrt{1-t^2}} dt,", "ac87d970183e542141b0c3a80da00e39": "A = P\\frac{i(1 + i)^n}{(1 + i)^n - 1} = \\frac{P \\times i}{1 - (1 + i)^{-n}} = P\\left(i + \\frac{i} {(1 + i)^n - 1}\\right)", "ac87e908471a3b799f01dcd57d6fe6dc": "H(z)=\\frac{C(z) G(z)}{1+C(z)G(z)}", "ac87e9d456518ef0b0b19238596c810a": "MacD = \\left[ \\frac {(1+y/k)}{y/k} - \\frac {100(1+y/k)+m(c/k-100y/k)}{(c/k)[(1+y/k)^m-1]+100y/k} \\right ] / k", "ac8835dbe6d16c086ee44ad8368348be": " \\log(1+x) = \\sum_{n = 0}^\\infty a_n T_n(x). ", "ac8837d80089f382867183a51a86968b": "|z| = \\sqrt{z \\cdot \\overline{z}}", "ac8881db70b416154780df5a25c067bf": "err(T,S)", "ac88a1ae7e12191ef3ce00508306be52": "t\\rightarrow\\mp\\infty", "ac89395c979b285ab00c9096ace51a71": "w(a,b) = u(a) v(b),", "ac8943837cb52e72d02acb953cec782a": "\\displaystyle \\frac{1}{2\\pi}(\\hat{f} * \\hat{g})(\\nu)\\,", "ac899162d4ae66a5ed23d7db846dbf52": "\n\\dot{r} = \\frac{dr}{dt} = \\sqrt{\\frac{2}{m}} \\sqrt{E_{\\mathrm{tot}} - U(r) - \\frac{m h^{2}}{2 r^{2}}}\n", "ac89e353d01c883dad2415f490a11f26": "x_1 = 3, x_2 = 2, x_3 = 0, x_4 = 5", "ac8a0cc8ff4131b77dde9aa0b3a38a5f": " P = (1-\\dfrac{d}{a})^2 Q \\,", "ac8a3b4b80db2731f61364ace982f1e7": "2^b - 1", "ac8a493b34bbfc621269492a57d3627b": " \\ell \\approx L \\sin \\theta \\approx 0;19,30", "ac8a730bae8c3586fce6d5bc9b00ff12": "\t\\begin{array}{rr|rr} \n 2x & +3 & 8x & \\text{-}4 \n\\end{array}", "ac8ab8fd29f55cdff95deca58b5d5ca8": "H_{ij}(y)", "ac8ac82edf77f8e199be3c309c6f917c": "m_{ij} = 0", "ac8b2928aa5c5d9227c0ed6dd70fde8c": "\n\\begin{align}\n0 & = \\frac{\\partial}{\\partial \\mu} \\log \\left( \\left( \\frac{1}{2\\pi\\sigma^2} \\right)^{n/2} \\exp\\left(-\\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{2\\sigma^2}\\right) \\right) \\\\[6pt]\n& = \\frac{\\partial}{\\partial \\mu} \\left( \\log\\left( \\frac{1}{2\\pi\\sigma^2} \\right)^{n/2} - \\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{2\\sigma^2} \\right) \\\\[6pt]\n& = 0 - \\frac{-2n(\\bar{x}-\\mu)}{2\\sigma^2}\n\\end{align}\n", "ac8b2a77df9cf97ff293587f13320311": "\\Phi(x)=\\begin{pmatrix}y_1(x)&1\\\\y_2(x)&x\\end{pmatrix},\\qquad x\\in I,", "ac8b753bbaa70538d1ce3db3676ffc9d": "\\displaystyle{D(\\varphi)(z)= \\int_{\\partial\\Omega} K(z,w)\\varphi(w)\\, |dw|.}", "ac8c0073a7aa8332a8c95069cb8ad2fa": "-\\sqrt{\\frac{2}{15}}\\!\\,", "ac8c0fe8fb8ce3b87eab4449d32d8cf1": "\\ge R", "ac8c1ce9ee24a6c8e5fbdec180c6edeb": "\\displaystyle \\int_{-\\infty}^{\\infty}f(x) e^{-2\\pi ix\\xi}\\,dx", "ac8ca1f899aa63afcca25775fe7c277b": "\\left|\\mathbf{RPA}\\right\\rangle", "ac8cee8e67641ccd0fe86496b28c52fc": "1/4+\\varepsilon", "ac8d3d548083a7ab6fa669bb6dcaf0a5": "f(x) = \\operatorname{li}(x) - \\sum_\\rho \\operatorname{li}(x^\\rho) -\\log(2) +\\int_x^\\infty\\frac{dt}{t(t^2-1)\\log(t)}", "ac8dcad910a53028ecdce5367f0f0d1a": " \\epsilon \\not \\in Z", "ac8dd016d5e7bb9c81be9c96a1a17a5c": " \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} + 2\\zeta\\omega_0\\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\omega_0^2 x = \\frac{F(t)}{m}. ", "ac8e3ddb9ed9052ad0b2fd370733056a": "g_{11} = g_{22} = g_{33} = 1 \\quad ; \\quad g_{12} = g_{23} = 0 \\quad ; \\quad g_{13} = \\cos\\left(\\frac \\pi 2 - \\phi\\right) = \\sin(\\phi)", "ac8e4491129a32c53f77a63d47fe717a": "\nA_{REX} = \\{\\langle R,w \\rangle \\mid R \\text{ is a regular expression that generates string } w \\}\n", "ac8e53e6234cf7c60b82b22ba566ae24": "\n \\begin{align}\n \\varepsilon_{rr} & = \\cfrac{\\partial u_r}{\\partial r} ~;~~\n \\varepsilon_{\\theta\\theta} = \\cfrac{1}{r}\\left(\\cfrac{\\partial u_\\theta}{\\partial \\theta} + u_r\\right) ~;~~\n \\varepsilon_{zz} = \\cfrac{\\partial u_z}{\\partial z} \\\\\n \\varepsilon_{r\\theta} & = \\cfrac{1}{2}\\left(\\cfrac{1}{r}\\cfrac{\\partial u_r}{\\partial \\theta} + \\cfrac{\\partial u_\\theta}{\\partial r}- \\cfrac{u_\\theta}{r}\\right) ~;~~\n \\varepsilon_{\\theta z} = \\cfrac{1}{2}\\left(\\cfrac{\\partial u_\\theta}{\\partial z} + \\cfrac{1}{r}\\cfrac{\\partial u_z}{\\partial \\theta}\\right) ~;~~\n \\varepsilon_{zr} = \\cfrac{1}{2}\\left(\\cfrac{\\partial u_r}{\\partial z} + \\cfrac{\\partial u_z}{\\partial r}\\right) \n \\end{align}\n ", "ac8e641a998aaa3e1ee290221a6264e2": "E_{n} \\ge E_{g}", "ac8e8de5f6d2fdf09a1777394a4ae6c1": "\n\\begin{array}{lll}\n \\sin\\frac{A}{2}=\\left[\\frac{\\sin(s{-}b)\\sin(s{-}c)}{\\sin b\\sin c}\\right]^{1/2}\n&\\qquad\n&\\sin\\frac{a}{2}=\\left[\\frac{-\\cos S\\cos (S{-}A)}{\\sin B\\sin C}\\right]^{1/2}\\\\[2ex]\n \\cos\\frac{A}{2}=\\left[\\frac{\\sin s\\sin(s{-}a)}{\\sin b\\sin c}\\right]^{1/2}\n&\\qquad\n&\\cos\\frac{a}{2}=\\left[\\frac{\\cos (S{-}B)\\cos (S{-}C)}{\\sin B\\sin C}\\right]^{1/2}\\\\[2ex]\n \\tan\\frac{A}{2}=\\left[\\frac{\\sin(s{-}b)\\sin(s{-}c)}{\\sin s\\sin(s{-}a)}\\right]^{1/2}\n&\\qquad\n&\\tan\\frac{a}{2}=\\left[\\frac{-\\cos S\\cos (S{-}A)}{\\cos (S{-}B)\\cos(S{-}C)}\\right]^{1/2}\n \\end{array}\n", "ac8ed3b3245a06b419e064ea6a9606a7": "H^{n}(G,M) = \\operatorname{Ext}^{n}_{\\mathbf{Z}[G]}(\\mathbf{Z},M),", "ac8eed3cef7c70a14cf4ca5b7ed4f506": "\\epsilon_d={Y_1 + Y_2 \\over Q_1 + Q_2}\\times{\\Delta Q \\over \\Delta Y}", "ac8f0064706c514d8e824e50a5dd844a": "\\{\\gamma^\\mu,\\gamma_5\\}", "ac8f5aa1f827a4711272877aa4d77624": "G=\\langle a_1,a_2,b_1,b_n|[a_1,b_1][a_2,b_2]=1\\rangle", "ac8f647c9a46302c24b421bffb864649": "p(f|e)", "ac8f974dbdf036afd76df0a475d3a490": "\\;C(\\Psi, \\Psi_{id})", "ac8f9f595908d1fb286b5613b028c420": "\\frac{n}{4}", "ac8fe186d10347acec08c5e414a148e5": "(\\sigma_\\mathrm{n}, \\tau_\\mathrm{n})", "ac901a800c439323ac17b4a03d5293b2": "C^{(T)}_p(p,T)=C^{(T)}_V(V,T)-C^{(V)}_T(V,T) \\frac{\\left.\\cfrac{\\partial p}{\\partial T}\\right|_{(V,T)}}{\\left.\\cfrac{\\partial p}{\\partial V}\\right|_{(V,T)}} ", "ac903fc5736f40f289a1dcf9c30a7555": "\\lim_{x \\to \\infty} F(x) := \\lim_{x \\to \\infty} \\int_0^x f(\\xi) d\\xi\\ = \\infty;", "ac909cf8a236d62d4505f095657d1408": " G ", "ac90a3880cb2bfdd1b180d90ac6bfe03": "\\beta_M=1", "ac910d3101640ee4c4e6a573c237ca42": " \\gamma = \\frac{1}{\\sqrt{1-\\beta^2}} = \\frac{2 m_pc^2+ m_\\pi c^2}{2 m_pc^2} ", "ac916bed17656b3fab66d664d727ece3": "{}_2 \\pi_{*}{}^s", "ac91945d4991c1cf89fd4b7b32c58112": "n_2 = |n_2|", "ac91a2111aad280f2a58abae7de5c2bd": "Q\\varphi = T_h \\cdot Q\\varphi", "ac91d6b1fba23ff1b0358ac7f9aed79a": "\\, f(z) = c\\,{\\displaystyle\\prod}_n (z-a_n)", "ac92316008a6e7abe34a913fc2c71e84": "h = \\frac{1}{4} \\mathrm{E}\\left[(d\\log p)^2\\right] \n+ \\mathrm{E}\\left[(d\\alpha)^2\\right]\n- \\left(\\mathrm{E}\\left[d\\alpha\\right]\\right)^2\n- \\frac{i}{2}\\mathrm{E}\\left[d\\log p\\wedge d\\alpha\\right]", "ac9237b6ea0773eb6b5be2ec533323f7": "\\{ u_1, \\ldots, u_m \\}", "ac926d7ccfb426dedd8244d4486f5a47": "\\frac{\\partial S_{ij}}{\\partial t}+\\frac{\\partial (v_kS_{ij})}{\\partial x_k}=P_{ji}-P_{ij}", "ac92973390f12d6f1df4fdf3d77920e3": "Q(x,n) = \\frac{x}{\\sum_{k=1}^\\infty \\frac{1}{k^n}} + O\\left(\\sqrt[n]{x}\\right) = \\frac{x}{\\zeta(n)} + O\\left(\\sqrt[n]{x}\\right).", "ac92ca86699c427e5b68648de90f7748": "\\mathrm{MA} \\equiv \\frac {F_{out}}{F_{in}} = \\frac {2 \\pi r}{l} \\,", "ac92f91e11d5d3418276aee4991e07e8": "(y_{n}^{(i)})_{n \\geqslant 0} ", "ac9337ced4cc03535d9624a64e99a306": "f^l(n) = \\inf \\{m \\in \\mathbb{N} | n \\le f(m)\\}", "ac93aa11750b2daeb7a89597290f3448": "\\pi=p^{\\theta}", "ac93bc8cab05ea83a3a28ee56f9358a4": "n\\ne j", "ac9418a9db6ad7c0efa555d779e1c4cd": " I = \\int w(x,t) g(x)\\, dx ", "ac950286c42020161fbdae0550ca8f99": " \\textbf{H}_{k} = \\left . \\frac{\\partial h}{\\partial \\textbf{x} } \\right \\vert _{\\hat{\\textbf{x}}_{k|k-1}} ", "ac9597db94c04f27094caf6ff1f138e1": "\\mathbb{E}X_k = \\mu_k", "ac95a06bddb26085d451c907c24a1a40": "\\det(A)-{\\operatorname{tr}}(A)\\lambda+\\lambda^2,", "ac95d24d4d5d71ab6fda3dad4c5f088c": "U_n(x)", "ac95f8454e3af8c39ee7cd1e13c41ec0": " \\hat{G}_{-i}\\left(X'_i \\beta\\right),\\, ", "ac961997732555753e4bf95110270d9f": "N-k", "ac96545c5b2728ad7ceb3934ed4bc641": "y = \\pm 1. \\, ", "ac969a128631a169e949bfe3ba6a4eb9": "\\displaystyle A_\\mu(e^X) =\\sum_{s\\in W} \\varepsilon(s) e^{i\\mu(sX)},", "ac96d91385e31fb1d57e3ff0f52706d9": "\\ s_j=\\sum_{k=1}^N{x_k^j}.", "ac96eca19cf18d65b21ea4d5ffad4df1": "\\, 0 < \\theta < 2\\pi \\,", "ac9748290e7c1ca80efd4868e4351670": "\\{ v_1 \\}^{\\perp}", "ac97bc9f822fbd6a46950c89bce64adf": "\\dot{m}=-\\rho\\,", "ac97edb979875bb209761419bb809a76": " \\varsigma_2(\\varepsilon) := \\left(\\psi_0+\\delta_\\psi \\varepsilon(t)\\right)\\left(\\lambda + \\varsigma_1(\\varepsilon)\\right) ", "ac980afd5abbd84a98418d336f2b609f": "\\tau = \\tau(X, X') = \\beta(X, X')", "ac9816fe3f36319e981efcb06fd3bc62": " \\mathbf{E}(z,t) = e^{-\\alpha_{abs} z / 2} \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)})", "ac98299eb2b2b35672426e3b04c22bd0": "\\displaystyle{f_{iI}(x) = e^{-x\\cdot x/2}.}", "ac985f24e2419265b9b99f33c22f2699": " R_1 = Z_0 \\tanh \\left ( \\frac {\\gamma_ \\mathrm T}{2} \\right )", "ac98d661cf3c4c1f0bc710f9aeaed22b": "\nC_n^{(\\alpha)}(z)=\\sum_{k=0}^{\\lfloor n/2\\rfloor} (-1)^k\\frac{\\Gamma(n-k+\\alpha)}{\\Gamma(\\alpha)k!(n-2k)!}(2z)^{n-2k}.\n", "ac98d93faa1fc9f7b2804624c47d3476": "f(x)=x.", "ac99167578e06919715a159902488b11": "d(f(s), f(t)) \\to 0", "ac99434409b985f06e955921848fbfe7": "[\\gamma^{a},\\gamma^{b}]= \\gamma^a\\gamma^b - \\gamma^b\\gamma^a", "ac9985e5426d42cec9c1b1ac47aceb31": "\\langle k_{nn} \\rangle = \\sum_{k'}{k'P(k'|k)}", "ac9a0588c8088117999c4b39b7636297": "\n{\\rm Var}[z]\\,\\,\\,\\, \\approx \\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)^2 \\sigma _1^2 \\,\\,\\, + \\,\\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right)^2 \\sigma _2^2 \\,\\,\\, + \\,\\,\\,\\,2\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right)\\sigma _{1,2}", "ac9ada4c8cf17599b41f557e632b9f48": " \\begin{align}\n&\\lim_{\\alpha\\to 0} \\ln \\,\\operatorname{var_{GX}} = \\lim_{\\beta\\to 0} \\ln \\,\\operatorname{var_{G(1-X)}} =\\infty \\\\\n&\\lim_{\\beta \\to 0} \\ln \\,\\operatorname{var_{GX}} = \\lim_{\\alpha \\to \\infty} \\ln \\,\\operatorname{var_{GX}} = \\lim_{\\alpha \\to 0} \\ln \\,\\operatorname{var_{G(1-X)}} = \\lim_{\\beta\\to \\infty} \\ln \\,\\operatorname{var_{G(1-X)}} = \\lim_{\\alpha\\to \\infty} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}} = \\lim_{\\beta\\to \\infty} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}} = 0\\\\\n&\\lim_{\\beta \\to \\infty} \\ln \\,\\operatorname{var_{GX}} = \\psi_1(\\alpha)\\\\\n&\\lim_{\\alpha\\to \\infty} \\ln \\,\\operatorname{var_{G(1-X)}} = \\psi_1(\\beta)\\\\\n&\\lim_{\\alpha\\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}} = - \\psi_1(\\beta)\\\\\n&\\lim_{\\beta\\to 0} \\ln \\,\\operatorname{cov_{G{X,(1-X)}}} = - \\psi_1(\\alpha)\n\\end{align}", "ac9b02087f136049333738cb14ac9788": "\n \\boldsymbol{F}\\cdot\\mathbf{e}_2 = F_{12}\\mathbf{e}_1 + F_{22}\\mathbf{e}_2 = \\gamma\\mathbf{e}_1 + \\mathbf{e}_2\n \\quad \\implies \\quad\n \\boldsymbol{F}\\cdot(\\mathbf{e}_2\\otimes\\mathbf{e}_2) = \\gamma\\mathbf{e}_1\\otimes\\mathbf{e}_2 + \\mathbf{e}_2\\otimes\\mathbf{e}_2\n ", "ac9b0a2cc08e82e22773dc3892a37285": "\\mathcal{B}\\ne\\{X,\\varnothing\\}", "ac9b6e98b16a4c70210eebb780dc31e7": "d=2^{k-1}", "ac9b957f663e46224b0c2f455fc33984": "v_r", "ac9bc14f1a99a54ed0d997462dbe77a7": "Sym^2 M", "ac9bc9e59da9677620383cbcd267c016": "\\mathbb{N}^\\mathbb{N} \\cong \\mathbb{N}^{\\mathbb{N}\\times\\mathbb{N}})", "ac9be40ec0d2be7aa3471dc91956c295": "f:S\\to \\mathbb{R}", "ac9c130e90586772a4186b8ee7db7e25": "(r, \\theta, \\phi)\\in[0,\\infty)\\times[0,\\pi]\\times[0,2\\pi)", "ac9c686504c7add2709c6555b67a7865": "p(x+r)", "ac9cab8ecd1bf445edd573cc2585046b": "y_i^{(n)} + p_{n-2}\\,y_i^{(n-2)} + \\cdots + p_1\\,y'_i + p_0\\,y_i = -p_{n-1}\\,y_i^{(n-1)}", "ac9cc699ed1c171886cf92f2a6433917": "\\mathbf p", "ac9cebfd7616e94e05eb4f14bc84b53b": "f_{y}", "ac9cfb5964a884cbd482505da49a0826": "1/x = 1 - (x-1) + (x-1)^2 - (x-1)^3 + (x-1)^4 - ...", "ac9d605eda17ff53cd570b600af5ab68": "\n u_i^{\\overline{n+1}} = u_i^n - a \\frac{\\Delta t}{\\Delta x} \\left( u_{i+1}^n - u_i^n \\right)\n", "ac9def5ca3d40428d1ab3019dc8234bc": "\\mathbf{P}(R)", "ac9e0c4c0021e610a67bfb20f44b2b76": "T(v)=\\lambda v", "ac9e20a06038da67aee59205a83bc2e0": " \\phi_{l}\\,= \\phi_{L}", "ac9e404eb2a2d42dd14c58379accea42": " Vg(\\rho_2-\\rho_1)> 2r \\pi\\gamma\\!", "ac9e7174c206fc63ac30e0a0ebb0fb7a": "\\omega=A+B\\,\\sin^2(\\varphi)+C\\,\\sin^4(\\varphi)", "ac9ece2d82b51e64b7417ac65acfcb40": "a \\ge1, b \\ge1, n \\ge3", "ac9efe23bd710abe094c643a0b6e9a39": "V_d", "ac9f0a69aaafff9214979f9461cd6ec4": "\\, H(\\mathbf R(t))", "ac9f0a7583016462a42f855bddca7dc9": "(\\pm 1,\\pm 1, 0, 0, \\dots, 0)", "ac9f30a53ee09902246bec9980f04e78": "f(x+P) = e^{ikP} f(x) \\,\\!", "ac9f58543e10a2d04ba41d0306a58932": ":=n^a\\nabla_a", "ac9f912d0cdab62a7e2af40853435893": "\\mathcal{F}(f)", "ac9fcf492beafeaecc61c15bc7222296": "\\textstyle{\\binom{4}{2} = 6}", "ac9fd838a2880fb8727dec36cbc8d8eb": "\\frac{1}{c_0^2}\\frac{\\partial^2 p}{\\partial t^2}-\\nabla^2p=\\rho_0\\frac{\\partial^2\\hat{T}_{ij}}{\\partial x_i \\partial x_j},\\quad\\text{where}\\quad\\hat{T}_{ij} = v_i v_j.", "aca00bbf1e12e4f40861decb59fa1b6e": "k_{b_1}", "aca0329ce4a70ce9887a65d9c522ded4": " m_j = \\pm {3 \\over 2} ", "aca05ae1d51404d6a28627e96456c5a8": "O(n \\log n \\log^4 |G| + tn \\log |G|)", "aca0a277eec8dff6c564cddfc3ab1c8f": "z,p\\in[0,1]", "aca129096b96c884fb9073adfad13341": "\\mathcal{S}=\\langle{\\rm Fm},\\vdash_{\\mathcal{S}}\n\\rangle", "aca177888c3f5af8d4d74ed768cdc71f": "V=\\left(\\frac{1}{12}\\left(45+17\\sqrt{5}\\right)\\right)a^3\\approx6.91776...a^3", "aca233d18b1fe580f574abbf822d2bf2": "\\cos (A \\pm B) = \\cos A \\ \\cos B \\mp \\sin A \\ \\sin B", "aca25066621498eaa6c0432a663f2f83": "\\textrm{var}(X_t)=\\operatorname{E}(X_t^2)-\\mu^2=\\frac{\\sigma_\\varepsilon^2}{1-\\varphi^2},", "aca27187f4ccf4de3d19c0b0ebfccc96": "\\sigma^2 = \\frac{\\Omega}{2(1+K)}", "aca2731a85306fa9db714fdadede8ae5": "\\ H(s)", "aca2a247249e0e2cc4a1d55d0c7fbb38": " \\psi = \\phi (\\tan\\alpha_2 - \\tan\\alpha_1)\\,", "aca2badad966116bac014ec4f6c0bc7d": "\\lbrace Q_j\\rbrace", "aca32a2d4ede6a4a5babdc499b929bff": "x^5", "aca339bfa7f47149e5b4b65c614f32d3": "ky = k \\int \\frac{dy}{dx} dx. \\quad \\mbox{(1)}", "aca3607ecd212ace5ab76361173b14ba": "c_{6}-c_{8}", "aca36aeda7136ebacf88a7e2f841f5fa": " \\langle x, ya \\rangle = \\langle x, y \\rangle a", "aca37f356ffbc2f5a717639420e88376": "S_2 ", "aca38723251c403b9dca16ad1a45f23d": "M - 2G\\,", "aca3f28bbf7c46a95c0b3a7936ed771b": " t \\mapsto u ", "aca3f55eb06d00c4308c43f9e6a0f38b": "\\hat{T} = \\frac{\\bold{\\hat{\\Pi}}\\cdot\\bold{\\hat{\\Pi}}}{2m} = \\frac{1}{2m} \\left ( \\bold{\\hat{P}} - q\\bold{A} \\right)^2 ", "aca414bfb4ff78047717dd3aec54bc04": " \\|\\boldsymbol{N}_{i=1,\\ldots,k}{(0,1)}\\|_2 \\sim \\chi_k(x) ", "aca41fa6afd4faa3f845e59d55f2e4f5": "h_{Preucil\\ circle} = 60^{\\circ} \\cdot \\left( 4 - \\frac{G - R}{B - R}\\right)", "aca42d640d777de77692b51e0aed5fc1": "\\int_b^a f := - \\int_a^b f.", "aca45839f4cc799f9b078d473117c95a": "\nH(\\lambda(t),x(t),u(t),t)=\\lambda'(t)f(x(t),u(t))+L(x(t),u(t)) \\,\n", "aca48ec64020a4ad0957f5c576267952": "d= \\frac {128}{243}a_0 ", "aca519ff6457e9afedac28f5fe9f021f": "\\scriptstyle\\dot\\ell(\\theta)", "aca5492aee4699b9c63e4f96161257b7": "p(f) {{=}}\\alpha f^{-1-1/s}.", "aca5b2db3e65f2d24e19f0dd19c06f57": "\\overline{S}_p = \\Theta \\backslash S_p", "aca5d69e686f71d0cc7c00ea09fb6957": "A_{1,1}\\subseteq A_{1,2}\\subseteq A_{1,3}\\subseteq\\dots", "aca5dd9ddaa3f5a35bc5393f2016270c": "\nZ' = \\int D \\mathbf{R} \\exp \\left[ - \\beta U_0 (\\mathbf{R}) \\right] \\qquad (7)\n", "aca63a699da5ddf9c9445701a5312253": " \\ln(x) = -\\lim_{\\epsilon \\to 0} \\int_\\epsilon^\\infty \\frac{dt}{t}\\left( e^{-xt} - e^{-t} \\right)", "aca644d1cce2b0d3bf8256ae4a00b616": "h(z) = c_0 + c_1(z-a) + c_2 (z - a)^2 + c_3 (z - a)^3 + \\cdots \\, .", "aca64800e982e3fe0c9456ee00cd6d47": " a = \\frac{g-2}{2} ", "aca6598612497672f0a32d04246244a2": "\\mathcal{SHIF}^\\mathcal{(D)}", "aca66492de8bb086de2248947f06bafb": "\n\\Delta \\phi_i + \\lambda_i \\phi_i = 0. \\,\n", "aca6a99dd02696128f977483b77caa19": "\\Gamma(x_{\\mathrm{min}}) \\approx 0.885603 ", "aca6ad88a322c475d44f5e433833621f": " \\Rightarrow U' = \\int_{R_1} \\left \\{ \\pi_1 \\cdot (U_{11} - U_{21}) \\cdot p(y|H1) - \\pi_2 \\cdot (U_{22} - U_{12}) \\cdot p(y|H2) \\right \\} \\, dy ", "aca6bfdbde74e9790ce3539baa26d3d3": "\\begin{align}\n \\rho(x, y, z) &= \\frac{3B}{r^2 + x^2 + y^2 + z^2} \\\\\n p(x, y, z) &= \\frac{-A^2B}{(r^2 + x^2 + y^2 + z^2)^3} \\\\\n \\mathbf{v}(x, y, z) &= \\frac{A}{(r^2 + x^2 + y^2 + z^2)^2}\\begin{pmatrix} 2(-ry + xz) \\\\ 2(rx + yz) \\\\ r^2 - x^2 - y^2 + z^2 \\end{pmatrix} \\\\\n g &= 0 \\\\\n \\mu &= 0\n\\end{align}", "aca6f3ec7d8cfecaaa27c098ea6c3705": "(x^{l(2t - 1)} - 1)b(x)", "aca7286c5010c024f97be5dc7305b7b8": "\\rho_\\text{crit}=\\frac{3H^2}{8\\pi G}", "aca77169b603ef07a38083179a0d744d": "A_{(i)} \\preceq B_{(i)}", "aca79613be0df9304eceba4de5fdc706": "t' = \\frac{1 - av}{a - v}L", "aca7ca3ebbc6bfdf83da0d07b7393ace": "(r,\\phi,z)", "aca7f53beef597bfcbd4bc7dd42661dc": "(x_1, y_1), \\dotsc, (x_n, y_n) \\in \\mathcal{X} \\times \\R", "aca81481b3193b8883386c4b2e4fe456": "\\mathbf{A}_{\\text{Electric dipole}}(\\mathbf{x},t) =\\frac{-i\\omega\\mu_0}{4 \\pi} \\frac{e^{i k r - i \\omega t}}{r} \\mathbf{p}", "aca81527eb67ba8bcd537ec2e1d44641": " F = \\frac{L}{4\\pi D_L^2}", "aca86e4165538ada58380aede17077a2": "\\mathbb Z[p^\\infty]", "aca8af819b2b7fc1bb8ebd3b7514725f": "\\rho \\, d\\rho \\, d\\phi \\, dz", "aca8e54ccbdbec715b88e376efa47564": "\\frac{\\partial u}{\\partial t}-\\mathcal{L}u\\ge0", "aca9045e75995fefe45f92ac49b73cc2": "(s+1)(s^2+s+1)", "aca91c082000c27977397570352421e1": " \\mathbf{r} = \\mathbf{r}_1 - \\mathbf{r}_2 ", "aca91f4a18befd46de99621a64a01a04": "( A_2 + L_2 A_{12} )", "aca930cdbdf3256b51d9d6112f155c93": "\nu = \\sin x \\cos y \\,F(t), \\qquad \\qquad v = -\\cos x \\sin y \\, F(t),\n", "aca939226111a31d95fd49be7342accf": "c^2 = a^2 + b^2", "acaa29adc42b96d1d09567922defbc96": "\\sum _x \\cos^2 ax = \\frac{x}{2}-\\frac{1}{4} \\csc (a) \\sin (a-2 a x) + C \\,\\,,\\,\\,a\\ne \\frac{n\\pi}2", "acaad114d6cf06e456652577008a3d68": "\\int\\frac{x\\;dx}{s^{2n+1}} = -\\frac{1}{(2n-1)s^{2n-1}} ", "acaaee4d791ce18931dc7de1923710a8": "w_{\\mathrm{abs}}", "acab15104b5d0c4df74a506e60d8c7c6": "i \\neq \\ell(w)", "acab7ee43db0c5cb0cd9448451c40dda": "\n\\frac{d}{dx} RangeVout(x) = 0\n", "acab9143ed837221cb454998d48d55f3": "\\gamma_{jk}", "acac049dcb8be251d092e4ab3b854ed7": "\\mathbf{Z}(p^\\infty)=\\mathbf{Q}_p/\\mathbf{Z}_p.", "acac3b1a1591effd20e30b3e33765e65": "\\mathbf{x}\\in \\Omega \\subset R^N", "acac6b1b4052926e24bc6f0a1218b712": " \\log(t a^p + (1-t) b^q) \\ge t \\log(a^p) + (1-t) \\log(b^q) =\n \\log(a) +\\log(b) = \\log(ab)\n", "acac86a4142ee6875b9387e56fe3d5b8": "\\Box \\Phi=\\frac{\\mathrm{d}V}{\\mathrm{d}\\Phi}", "acacbd7eff5f886e5d13a21b71e25d4d": "\\frac{1}{R} = \\frac{d\\theta}{dL} = 2L", "acacc22825c7b41fc931f88517ee8927": "\\dfrac{J_0}{N}", "acad4dd682f06c96d27816e92061f310": "\\displaystyle{UK_SU^*=A.}", "acad67069c20647b88e7728bbde4578f": "|arg(u_i)| < \\pi - \\frac{\\pi}{n}, i = 1,...,n", "acad883799c945b63088b1cb50b1eefc": " V_r \\, ", "acad965be15367eedc7ae6f26826a960": "D\\nu-\\Delta\\pi=(\\pi+\\bar{\\tau})\\mu+(\\bar{\\pi}+\\tau)\\lambda+(\\gamma-\\bar{\\gamma})\\pi-(3\\varepsilon+\\bar{\\varepsilon})\\nu+\\Psi_3+\\Phi_{21}\\,,", "acadd0c959e9dd6d5841ac342285368f": "\\left\\lceil\\frac{rs-r-s+2}{r-1}\\right\\rceil=s.", "acae4029c7e28c34677cf326de359731": "ds^2 = dx^2 + dy^2", "acae65b6df67b956bf4bb01a810fa0a8": "P_3 = P_0(1+r)^3- c(1+r)^2- c(1+r) - c", "acae8bffc7e9081976644817ca02b97e": "\\operatorname{Var}(aX+bY)=a^2\\operatorname{Var}(X)+b^2\\operatorname{Var}(Y)+2ab\\, \\operatorname{Cov}(X,Y),", "acaee047ef190430b5e7d59a488cc1f5": "Z_P", "acaf2007da34acf9f7dd3ae92df8e896": "w\\Vdash\\Box_i A", "acaf8897122e38e4dc050af3efce3091": "Range = 0.5 C / \\delta t ", "acaf90ade331f42ce1fba4d67553dc8f": "(P-R)", "acb006f815852f5a5467aec61ce3e427": "x^{-1} \\in \\mathfrak{m}_S", "acb02d733693a9864c434df6106a9fd0": " \\sum_{n=1}^{N-1} \\frac{N!}{n!} = \\lfloor (e-1)N! - 1 \\rfloor = \\mathrm{floor}\\left( (e-1)N! - 1 \\right)", "acb05daa0e22e87dd9f5b23b7988c6e6": "e^{ins}", "acb16fc25fd2dbc036e6b2ee864afe58": "\\ell=a(1-e^2),\\,", "acb1b6196997859929bc61b66289d513": "|T|:=\\sqrt{(T^*T)}", "acb1f9f77306b66edae79bd66172d5cd": " \\mathbf{F}_4 \\supset \\mathrm{O}(9) ", "acb20a39e839e353e862d449dfa61552": "L(t, u, \\dot{u}) = T(\\dot{u}) - V(t, u),", "acb22f07ac8327da4fadb78e9b0163e4": " \\cos x = \\sqrt{\\pi} \\; G_{0,2}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} - \\\\ 0,\\frac{1}{2} \\end{matrix} \\; \\right| \\, \\frac{x^2}{4} \\right), \\qquad \\forall x ", "acb248d083ec375ea699ea2c359c902d": "\nq_{k}\\left(i\\right)=\\frac{\\mu_{k}}{v_{i}}E_{i}\\left(k\\right),\n", "acb24ad6e6aeea1a92114871332dcf8a": "V^{-1}AV = \\begin{bmatrix} \\lambda_1 & \\beta_1 & 0 & \\ldots & 0 \\\\ 0 & \\lambda_2 & \\beta_2 & \\ldots & 0 \\\\ 0 & 0 & \\lambda_3 & \\ldots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\ldots & \\lambda_n \\end{bmatrix},", "acb26fcd1f19a8782548ce34bc443e5f": "\\langle (gG)^4\\rangle\\simeq 5:10\\langle (gG)^2\\rangle^2", "acb27b403771f357c1caf25553622aaf": "\\scriptstyle \\|x\\|_p", "acb2da5b59054486f8f8ad151711de2d": "\\frac{i}{y}", "acb35d2f7b8ed6084ab728f9a5d6403a": " f = 1/298.25722\\,21008\\,82711\\,24316\\,28366\\,\\ldots.", "acb37a16cf8b70365b9fd67f1ae91cd4": "\\Delta y = f'(x)\\,\\Delta x + \\varepsilon = df(x) + \\varepsilon\\,", "acb37c8ac9f305432c2b8b7f4f00f41f": "\\frac{d}{dr}\\sum_{k=0}^nr^k = \\sum_{k=1}^n kr^{k-1}=\n\\frac{1-r^{n+1}}{(1-r)^2}-\\frac{(n+1)r^n}{1-r}.", "acb3baee27346e73df858f2f56a6d9d3": "N = \\frac {120f}{P} \\,", "acb3e55fe00e6af9a6d16095f841b683": "S^i", "acb3eaa53997e8e8cd92dfa9a3b25ae0": "x\\rightarrow\\rho_{x}", "acb3eca42d6782c51945e7364c3e69b1": "\\textstyle n-w", "acb420594b11d2582d95b42b01f84372": "\\frac{dy}{dx}\\,.", "acb4cfe56e2e04281e373ad7b37a16ed": "\ny(\\phi)=\\frac{E}{2\\pi}\\psi(\\phi).\n", "acb4dbf1c66a6884575a8fc5521cb661": "I \\,\\ ", "acb4e85201e10edc87f2b80798a1b21f": "F_{5^{2^n}}", "acb4ecfeb936151012086ea6a2557435": "p(n;d) \\approx 1 - \\left( \\frac{d-1}{d} \\right)^{n(n-1)/2}", "acb4f5481f2398093419ccc67addb62c": "( G, \\alpha: E(G) \\rightarrow \\Pi )", "acb4fa0d5bd2de5e08fe2538892c70ee": "A=-\\log( \\frac{V_{dc}}{V_{ex}} )", "acb521c24f182dba1b1ed6e95594dc46": "G_F", "acb5399d4c6d72b5b302d3e06cd65457": "P(G, \n\\mathcal{X}, \\mathcal{Y})", "acb58b43673c051aca748735901d934d": "\\int_W u(x) Q(x, \\partial) \\varphi (x) \\, \\mathrm{d} x=0", "acb5a6d21dc791f205d75de2b11d3693": "x_0 \\in S", "acb5acedb4c877e852d47704109b1944": "u_\\mathrm{em} = \\frac{\\epsilon_0}{2}E^2 + \\frac{1}{2\\mu_0}B^2 \\,", "acb5b43247cc18571f84f9bb9d2598d6": "\\vdash [a]p\\,\\!", "acb6256eec704dfc96682ed762552696": "\\sigma_P(\\mathcal Q)=\\mathcal Q", "acb673d087bda07c5b337b6fd6d88fb1": "[b^{3.14159},b^{3.14160}]", "acb6a83c1d0b73b0dc438c9f913e5a47": "MS_E", "acb6f5ab5881bcfe00f6d21b7f0a4c1d": "A_l \\ (l \\ge 1)", "acb7899fed8cc7d898bb9887daaef698": "a_{\\mathrm{out}}^\\dagger", "acb7b85501d83739e32c3d72c7377344": "Y_{k,l}", "acb7bc85ecad9b0a5ba463144b229605": "r_{2}", "acb7d98e9837a2eefdae10724445044b": "K_0/\\phi=0.00304", "acb811321460397db731e403c57998de": "\\scriptstyle \\mathsf{Boolean} \\rightarrow \\mathsf{Boolean} \\rightarrow \\mathsf{Boolean}", "acb841374c88b0357d8f7b4a7f2b9a34": "-\\tfrac {1}{15} \\pi^2 + \\tfrac {1}{2} \\ln^2 \\phi \\,", "acb8527595db1318d893b069a0e93325": "\\frac{\\partial \\xi}{\\partial x} = \\frac{1}{2 \\sqrt{t}}", "acb863308baadf984e0d07395d8c2ee3": "{}m^2= \\left(\\tfrac{M}{2}\\right)^2 + j^2", "acb87f2eb7e3def60e4d04760bc486fb": "h(I_1,I_2)=m^*exp(-n_1 \\sqrt{I_1-3})+(1-m^*)exp(-n_2 \\sqrt{I_2-3})", "acb883c8af495bb34c13c4a8bdf3c5a1": "10000*2", "acb894f2f85e039114985c601f0637c6": "d \\Phi = \\frac{1}{T}dU+\\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i - \\frac {1} {T} dU - U d \\frac {1} {T}", "acb8ef83a71dd4ef06c0ca6e4567eddb": "p(e|f)", "acb8f4568e712c367577b20881db6d97": "\\scriptstyle B\\, \\in \\,\\mathcal{F}", "acb9073e4a1973884c37d0ea1ab12a64": "\\mathbf{x}=(x,y)", "acb924a66c7c938036f22ca855a914fc": "V_e-V_u=\\frac {rc+ec} {r+e+f}", "acb96b887045c62e8a6a71660cdca056": "s(x) \\equiv p(x)\\cdot x^t - s_r(x) \\equiv s_r(x) - s_r(x) \\equiv 0 \\mod g(x)\\,.", "acb9928515a876bf1a0c89ba2cf7def4": "a_{q_j}^{-}", "acb9acc59e00e6d693c50e59ac126930": " \\lim_{h\\to0} \\max_n |e_n| = 0 ", "acba52518e0e6bca87be396189297b8b": "\\mathcal N=(1,1)", "acba633de4d23ccd556027a352be999d": "L^p(\\partial \\Omega).", "acba73eef02454a834579efcafe24714": "s_{i_1}s_{i_2}\\dots s_{i_m}=s_{j_1}s_{j_2}\\dots s_{j_m}", "acba81db61072f108079d821e24ed6a9": "r\\cos(\\theta-\\theta_0) = a", "acbb312c46a91b79e7f2d6ab3d73e634": "E\\gg V_0", "acbb3f3f330c3c77a01b064ec6b9bf4a": "\\operatorname{P}(Y=r \\mid X=x)", "acbb4812f7b3215eadd5cc8dd1eaf163": "F_{33}", "acbb6211e8cc85af67442674bc9057f0": "\\mathfrak{so}(2r)", "acbb8e3e64181cce97048b5f36c8e27e": "x'_i\\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{n'_i}{\\sum_j n'_j}", "acbbc4c162d250b3b9e7f33dea15c675": "\\text{IOPS} * \\text{TransferSizeInBytes} = \\text{BytesPerSec}", "acbbf1bf32e07cdee43210e25deddd97": "\n\\psi_{p} \\left(\\mathbf{r}\\right) = \\left ( \\chi_{a} \\left(\\mathbf{r}\\right)\ne^{i\\phi_{a} \\left(\\mathbf{r}\\right)}\\right ) \\psi_{0} \\left(\\mathbf{r}\\right)\n", "acbc2c8463e406b7fcb8b781c9df2e74": "\\frac{dR_t}{d L_t}-w_t-\\frac{\\partial w_t}{\\partial L_t}L_t-\\frac{\\partial w_{t+1}}{\\partial L_t}\\frac{L_{t+1}}{1+r}=0\\,\\!", "acbd00719b5c7d3e20ed98da91d602a9": "{{f_n}}/4", "acbd63c0d36152bfc1ca049887071d99": " W_1 \\ne 0 ", "acbd7457473214ac97f71f8e92ecdbc4": "\\Omega^2", "acbd9b1252be80c0c754affa73db761f": "H=\\frac{P^2}{2 M} + V(X) + \\sum_i\\left(\\frac{p_i^2}{2 m_i}+\\frac{1}{2}m_i \\omega_i^2 q_i^2\\right) + X \\sum_i{C_i q_i} + X^2 \\sum_i \\frac{C_i^2}{2m_i \\omega_i^2}", "acbdbab974cdc1cf6b7f78b480b221ec": "3\\pi/2 2\\alpha \\Bigr\\}", "acce110e12ae853a4e99052e93f17cb5": "\\frac{\\partial I(x)}{\\partial x} = -(G + j \\omega C)V(x).", "acce936aad1b1a4e43e52b5a0c9bdfa2": " D, C ", "accee08d2c8a75d61d9019a638f3656e": "\\tan{\\left(\\alpha\\right)}\\equiv S", "accf2ead07f8eddb8d128fb5233ca1fc": "K = h \\cdot S_b S_a P ", "accf5643cccfcf8443587bce78ddcf81": "\\frac{1}{n}\\sum_{i=1}^n |x_i-m(X)|.", "accf5ef38c5df68ba40ce313175278af": "\\begin{align} \n& \\mathbf {{E}_{\\parallel}}' = \\mathbf {{E}_{\\parallel}}\\\\\n& \\mathbf {{B}_{\\parallel}}' = \\mathbf {{B}_{\\parallel}}\\\\\n& \\mathbf {{E}_{\\bot}}'= \\gamma \\left( \\mathbf {E}_{\\bot} + \\mathbf{ v} \\times \\mathbf {B} \\right) \\\\\n& \\mathbf {{B}_{\\bot}}'= \\gamma \\left( \\mathbf {B}_{\\bot} -\\frac{1}{c^2} \\mathbf{ v} \\times \\mathbf {E} \\right) \n\\end{align} ", "accf6382ffbd101930b428b34da80b2e": "\\frac{\\partial J}{\\partial y_m \\Delta t} = F_y\\left(t_m, y_m, \\frac{y_{m + 1} - y_m}{\\Delta t}\\right) - \\frac{1}{\\Delta t}\\left[F_{y'}\\left(t_m, y_m, \\frac{y_{m + 1} - y_m}{\\Delta t}\\right) - F_{y'}\\left(t_{m - 1}, y_{m - 1}, \\frac{y_m - y_{m - 1}}{\\Delta t}\\right)\\right],", "accfa7b9be593f7e5adf6d199ff129e9": "+Fx_1...x_n \\leftrightarrow Fx_2...x_n.", "accfbcf363f3c032f2aa6e0a90e0d221": "|f(x)-f(p)|^\\theta\\le c|\\nabla f(x)|. \\, ", "accfe60aa4398771728b7673a27ea912": "Q= \\begin{pmatrix} {*} & {\\kappa(1-\\pi_{GC})/2} & {(1-\\pi_{GC})/2} & {(1-\\pi_{GC})/2} \\\\ {\\kappa\\pi_{GC}/2} & {*} & {\\pi_{GC}/2} & {\\pi_{GC}/2} \\\\ {(1-\\pi_{GC})/2} & {(1-\\pi_{GC})/2} & {*} & {\\kappa(1-\\pi_{GC})/2} \\\\ {\\pi_{GC}/2} & {\\pi_{GC}/2} & {\\kappa\\pi_{GC}/2} & {*} \\end{pmatrix}", "accfed021f7bdb29414b91e91f1bcd7c": " \\text{EQE} = \\frac{\\text{electrons/sec}}{\\text{photons/sec}}= \\frac{\\text{current}/\\text{(charge of 1 electron)}}{(\\text{total power of photons})/(\\text{energy of one photon})}", "acd05359d339f7526f839063782abaeb": "\\frac{4}{\\pi} A\\,\\!", "acd075c6846eb5446c40d564add2fd0d": "\\scriptstyle\\mathbf{0}.\\mathbf{\\dot{9}}", "acd0b9a53431f92783e0597d54d13fc9": "\\chi", "acd0c90b2ae824ce7a550653329b993c": "G\\left(\\epsilon\\right)=G_0\\,-\\, 685.81 \\left(\\frac{\\epsilon}{\\lambda}\\right)^2", "acd0d443068918e85317149352b60b3d": " \\theta (\\text{radians}) = \\frac{(I_R^{1/2} - I_L^{1/2})}{(I_R^{1/2} + I_L^{1/2})}\\,", "acd16f58ede91fc35dbac9125506782f": " I_2 ", "acd1adc8adfb119b1dad7c1c0260afb9": "VCA(64x^3+192x^2-256x+64,(1,2)) ", "acd1b2832cd9c86565996cfddefa498a": "\\Sigma = LL^T + \\Psi.\\,", "acd1c48cd1546575af6e870b4642b512": "B_1\\supseteq B_2\\supseteq \\cdots \\Rightarrow\\bigcap_{n\\in {\\mathbf N}} B_n\\neq \\empty.", "acd22fd0b585a40eb334c7dcfc69d35a": " F_a(s) := \\sum_{n=1}^{\\infty} \\frac{a(n)}{n^s} .", "acd23e67c1f5e5bdd9639f6e4e1a0d89": "E+\\delta E", "acd250b3d13fd618b2598081858e967f": "(p_1,p_2)", "acd2b09d39705a84bff035c18c9faea9": "dx", "acd2df18fc0dd96a268dcb6d64f0e56b": "\\, \\tilde{t}_\\text{r} + b - t_i", "acd30c7ddea52093aae248255ecf6730": "g(S)=\\phi(f(S))", "acd387491ba572855c1b6fc105654394": "\\scriptstyle\\mu_y", "acd3a0c60e35b1aefb399018cf72d41c": " u,v,x,y ", "acd3f1adc19069be36b72755fa3818a6": "\\frac{V_M}{\\sqrt{2E}} = \\left(\\frac{1+A^{3}}{3(1+A)}+A^{2}\\frac{N}{C}+\\frac{M}{C}\\right)^{-1/2}", "acd3fc5e777f8185296d0fe2239d95e4": "E_{xy,zx} = 3 l^2 m n V_{dd\\sigma} + m n (1 - 4 l^2) V_{dd\\pi} +\nm n (l^2 - 1) V_{dd\\delta}", "acd44493133025ab672042730c117095": " x(n) = d(n)+v(n)", "acd45cda14ff4ebc2131fe1589574fa6": "E_{(X, d, \\mu)}(\\lambda) := \\sup \\left\\{ \\left. \\int_{X} e^{\\lambda f(x)} \\, \\mathrm{d} \\mu(x) \\right| f \\colon X \\to \\mathbb{R} \\text{ is bounded, 1-Lipschitz and has } \\int_{X} f(x) \\, \\mathrm{d} \\mu(x) = 0 \\right\\}.", "acd46438854c0f6040b94de651c8b441": " \\|\\underline{y} - \\underset{=}{A}\\underline{x}\\|_X< \\varepsilon. \\quad (1) ", "acd46c1f258ae858168e9b1ddfd706d8": "\\textstyle 1 + a_1x + a_2x^2 + \\ldots + x^{l_1-1} + x^b(1 + b_1x + b_2x^2 + \\ldots + x^{l_2-1}). ", "acd475c0bad0ed22ee806ca97fc9f7fc": "\\displaystyle{R(R(a,b)c,d) - R(c,R(b,a)d)=- R(R(c,d)a,b) +R(a,R(d,c)b).}", "acd47af7ce87cc1982e41e79407d038f": "\\Phi_{11}:=\\frac{1}{4}R_{ab}(\\,l^a n^b+m^a\\bar{m}^b)=\\frac{1}{2}R_{ab}l^a n^b=\\frac{1}{2}R_{ab}m^a\\bar{m}^a\\,.", "acd4a6ec6db2a0d4717697deacfae8a7": "\\delta\\lambda \\in V\\mathfrak{E}", "acd51996e52559b317ad8f0153993dcc": " \\int_X T f d\\mu = T \\int_X f d\\mu.", "acd542119d1fc6afbb8fd1c5af79c607": "c=m_1 n \\omega", "acd5a2b1c55cb75af3864368e414d25c": " \\sum\\limits_{k=a+1}^{b} f(k)=\\int_a^b f(x)\\,dx \\ + \\sum\\limits_{k=1}^m \\frac{B_k}{k!} \\left(f^{(k-1)}(b)-f^{(k-1)}(a)\\right)+R_+(f,m). ", "acd5ac21cb12941bafcf5f44811ada36": "\\frac{1}{r_l} \\frac{\\partial V}{\\partial x} = -i_l\\ ", "acd5c4dc26887e42fe6b971bce6761a5": " x' = \\frac{x - v\\,t }{\\sqrt{1-v^2/c^2}}\\ ,", "acd61f6f3ecded144c091b95b1d54eb0": " \\operatorname{build-list}[\\lambda o.\\lambda y.o\\ x\\ y, D, V, L_1] \\and D[g] = [x, \\_, \\_]::L_1 ", "acd627ac19ca6949b5118c94fbdc3581": "gz=z", "acd62cb2a01aaf12799bb558ca2d3e9a": " V_1 = L_1 \\frac{dI_1}{dt} - M \\frac{dI_2}{dt} ", "acd6848f1c1924f4180aa57e378f4694": " {dx \\over dt}=P(x,y), \\qquad {dy \\over dt}=Q(x,y) ", "acd68c987c8aa92e0316ef1812d085d5": "\\frac{r_H}{r_e} \\approx 10^{42} \\approx \\frac {R_U}{r_e},", "acd68f339453ca8a2bdf2aa5b1de199e": "R_k^i=\\sum_{j=1}^n \\mathbf{1}(X_k^j\\leq X_k^i)", "acd7092ba67fdd69f1a90b45d0c904e5": "I = I_0 \\frac{8\\pi^4\\alpha^2}{\\lambda^4 R^2}(1+\\cos^2\\theta).", "acd714044d887c541b8ef3c7d3acd701": "a^{N-1} \\equiv 1\\pmod{N}", "acd758f3ce0dc9f4b34790206c90d923": " X^2 - aY^2 = P (T) Z^2. \\, ", "acd766b82e3e8731680b12f585e5723b": "\\mathit{FPR} = \\mathit{FP} / N = \\mathit{FP} / (\\mathit{FP} + \\mathit{TN}) = 1-\\mathit{SPC}", "acd76e9ef783cfaecb157b65a02a3f18": "P=E/T", "acd778c8eeac4e5d31c2e1fddc3afa5d": "\n0.1579 \\left (\\frac{\\mbox{difficult words}}{\\mbox{words}}\\times 100 \\right) + 0.0496 \\left (\\frac{\\mbox{words}}{\\mbox{sentences}} \\right)\n", "acd780ed98d55a9bbac514042bf55f98": "\\mathbf{G} = \\sqrt{ {\\mathbf{G}_x}^2 + {\\mathbf{G}_y}^2 }", "acd7a18d00b8529cb93d2e1d85fc6841": "x^2\\sin(1/x)", "acd7d268161b47d4e57b226df3755299": "\\scriptstyle N' \\;=\\; 2\\;\\uparrow\\uparrow\\;2\\;\\uparrow\\uparrow\\;2\\;\\uparrow\\uparrow\\;9", "acd86dc78eae22556f5e45a3a47647a2": " ", "acd8d96c47b2be537eb958c627d2ceea": "d \\le s", "acd8e3435c8d1acf7d7af9819800ca0a": "G_c = A/(1-AB)", "acd8e43e780f4f799aeb308b324cc8ef": "f \\in L^p([-\\pi, \\pi])", "acd98b3bd24c7cfe5eb067d8e1744b0b": "\nR(x)= 1 - {V(x) \\over c^2}\n\\,", "acd9b23c4ab7052a47c50f5c4267f9f5": "\\Delta U_s+\\Delta U_o=0\\, ,", "acd9d7791db154e7aa0b20891c9b5b83": "A_\\Sigma = \\int_\\Sigma dx^1 dx^2 \\sqrt{det \\; q^{(2)}}", "acd9dd222c02abd953b070ade274ad6b": " \\lambda_\\mathrm{net} = \\sum_j \\lambda_j \\,\\!", "acda16adc68c0b844b6ecd55b5b41450": " \\nu (x) := \\begin{cases}\n{x^4}(35-84x+70{x^2}-20{x^3}) & \\text{if } 0< x < 1, \\\\\n0 & \\text{otherwise}. \\end{cases}", "acda6abc1f60f94736e9958a1965ef75": "\\eta_c = \\pi r^2 N / L^2", "acda736fd1b52ede43b2b9e9b329cb64": "|n \\rangle", "acdaab57946ad81c24e6d07f223ce510": "O(N_{n,k})", "acdab9eb99151f1695dad8ec97cf18bf": "S_{T,L}", "acdaf31e49d0fea5634e575fc059cbea": "a_{v,t} = 0", "acdb358317761ff1104827d3ded1fdd0": "A=K(V)", "acdb4b4b3ea8eb38e732b5cbca546ad8": "H(x)^4 = -\\langle H(x)^2\\rangle^2 + 2\\langle H(x)^2\\rangle H(x)^2 + ( H(x)^2-\\langle H(x)^2\\rangle)^2", "acdb5e36e1885af5cd3a1a444a70b3dd": " h_{AM} = A/A_m \\,\\!", "acdb70a48e459bffb43b4cd328b8c834": "C \\succ 0", "acdbf6851f9ec9b9c613678325ad8f77": "\ny = ( ... ( ( (a_n*x + a_{n-1})*x + a_{n-2})*x + a_{n-3})*x + ... + a_1)*x + a_0\n", "acdbfe105f4d1f01db826ad94c055fb2": " \\int\\limits_\\Omega f\\text{div} \\mathbf\\varphi = - \\int\\limits_\\Omega \\mathbf\\varphi\\cdot\\nabla f \\leq \\left| \\int\\limits_\\Omega \\mathbf\\varphi\\cdot\\nabla f \\right|\\leq \\int\\limits_\\Omega \\left|\\mathbf\\varphi\\right|\\cdot\\left|\\nabla f\\right|\\leq \\int\\limits_\\Omega \\left|\\nabla f\\right| ", "acdc735acf3526a7ec1a0ffbb0e9eb8e": "n_{ }^{ }", "acdc8af3630455cefffb29d2f8b8c28e": "(k_1,\\dots,k_m)", "acdcc635b864c765181db0fe555565af": "\\mathbf a = \\mathbf{u} \\times \\mathbf{v} = \\begin{vmatrix} \\mathbf{e}_1 & \\mathbf{e}_2 & \\mathbf{e}_3\\\\u_1 & u_2 & u_3\\\\v_1 & v_2 & v_3 \\end{vmatrix}\\,,\\quad\\mathbf A = \\mathbf{u} \\wedge \\mathbf{v} = \\begin{vmatrix} \\mathbf{e}_{23} & \\mathbf{e}_{31} & \\mathbf{e}_{12}\\\\u_1 & u_2 & u_3\\\\v_1 & v_2 & v_3 \\end{vmatrix}\\ ,", "acdd147dbc05f2e7814c226ce0f21cc9": "\\sum_{n=0}^m c_n\\,\\mathbb{P}(N=n)=\\sum_{n=0}^m S_n\\,(\\Delta^nc)_0.\\qquad(**)", "acdd417b102de036d6ff77e93538ff41": "\\tilde{N} = 2m - 1", "acdd6f9dbea528fcfbcf9baa262d38b2": "n!_2", "acdd93f884b2e982c6c6396c06ef2c8f": "m^n ", "acdd9632c53d25222d473269c0ee44c4": "g\\colon TM \\to T^*M", "acdde40792830be5957ad526dd192830": " \\mathbf{r}_2 ", "acdde4546e7f6f9ddc14b249bae02cac": "\\bot \\!\\,", "acde1fc85ea3ce4d9764ba03ac623e11": "\\phi =2\\pi\\,", "acde289c30e24e6d3990a6a1f0b360e7": "X \\le 0", "acde4e588502b2739650cbb2590e555c": " K = \\frac{1}{2} HD^2 H\\, ", "acde6aa26d24442e029553b401ea3c50": "\\frac{I}{U_a^\\frac{3}{2}}", "acde80b2942f04a6d497f8ee5e7bf3fb": "2\\pi \\delta(x)\\sim\\sum_{k=-\\infty}^\\infty e^{ikx}=\\left(1 +2\\sum_{k=1}^\\infty\\cos(kx)\\right).", "acde80e93e0641933e0579c84399c033": " \\chi(\\pi(F)) =\\int_G F(g) h(g) \\, dg,", "acdea6717922f989c1bf8359cea6504b": "(\\exists x \\ \\phi ) \\to \\psi ", "acded710f9ec6f5a430cbad23538ec9c": "\\Bbb{Q}", "acdf7a21a954849de15b60d2ef3346a4": "\\lbrace x + y(hi) : x, y \\in R \\rbrace ", "acdfafd8c626074c98a5506762aaab5d": " R_2:= \\sqrt{\\sum_{0\\leq k\\leq n} |a_k|^2 }", "acdfd49e3eb4687d833b44ff1db85d07": "C_n^{(\\alpha)}", "ace02728b31222567e81336576bb1287": "\\forall i \\in \\{1,\\ldots,n\\} \\wedge \\forall j \\in \\{1,\\ldots,n\\}", "ace0747d30266416b57d7283ab23a0d9": "\\displaystyle u_t=3ww_x", "ace117c58aa55ce919df3e6203f70065": "i-j \\mid m", "ace191a75ff2e43a9a4ddf765d10e612": "T \\equiv X^0", "ace1999b1c5f9f268144ee7bd062d899": " \\operatorname{sink-tran}[V, X] = V ", "ace23619ba7616f7fd195cbbe1ea8c68": "( a, b )_{\\text{short}} := \\{ a, \\{a, b\\}\\};", "ace240d4f3d7721ccfa8dc50963e1c78": "\\begin{align}\nE_\\text{k} &= m \\gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\\\\n &= m \\gamma (v^2 + c^2 - v^2) - E_0 \\\\\n &= m \\gamma c^2 - E_0\n\\end{align}", "ace26dfa80cf346a145d085d46224bdf": " \\psi(\\mathbf{\\eta})= \\oint_{\\partial U} \\psi(\\mathbf{y}) {\\partial G(\\mathbf{y},\\mathbf{\\eta}) \\over \\partial n} \\, dS_\\mathbf{y}.", "ace35211f5470ca8816c1d99fb364bf4": " \\prod_{i=1}^n \\mathbf{A}_i = \\mathbf{A}_1\\mathbf{A}_2\\cdots\\mathbf{A}_n \\, . ", "ace3556d8f6a58ddd0051da33cb84213": "x=\\sum_{n=2}^\\infty q_n \\left[\\zeta(n)- \\sum_{k=1}^{m-1} k^{-n}\\right] ", "ace3a85c1ab7a0188af6fe21ac581e0f": "\\begin{align}\n y_{t+h} &= y_t + h \\dot y_t + \\frac{h^2}{2} \\ddot y_t + \\frac{h^3}{6} y^{(3)}_t + \\frac{h^4}{24} y^{(4)}_t + \\mathcal{O}(h^5) = \\\\\n&= y_t + h f(y_t, t) + \\frac{h^2}{2} \\frac{d}{dt}f(y_t, t) + \\frac{h^3}{6} \\frac{d^2}{dt^2}f(y_t, t) + \\frac{h^4}{24} \\frac{d^3}{dt^3}f(y_t, t) \n\\end{align}", "ace3efed7206c540acee0682c3cb667c": "a(m):=\\epsilon(a)m", "ace3f226135c8366d20586549b85fb2d": "T_{cr} = n^{3/2}\\hbar^{2} / km_{\\mathit{eff}}", "ace3fedc66f2ff0ad06fe3172dd52f9a": " F =\\frac{r poisoned}{r unpoisoned} ", "ace51b03cbb1d97a7d0458ce0671282d": "\\scriptstyle\\gamma' \\,=\\, \\gamma - \\pi/2", "ace5530fa08dbba8a9cce75f27775ec7": "t_{2}=t_{1}+\\varepsilon\\left(t_{3}-t_{1}\\right)", "ace574174bfcc43b2ae5ddc1915e28e4": " F=\\{f_1,\\ldots, f_k\\}", "ace611bde2f52927296fb49c4bd94a4a": "-\\otimes B:\\mathcal{C}\\to\\mathcal{C}", "ace687cc08050da0eb3757ad7fa013b0": "|\\psi\\rangle = \\frac{1}{\\pi} \\int |\\alpha\\rangle\\langle\\alpha|\\psi\\rangle \\, d^2\\alpha.", "ace68ddab2d23779ed6d1d907b461a8f": "\n \\varphi\\mathit{\\Delta} = \\frac{h}{\\surd\\pi}\\, e^{-\\mathrm{hh}\\Delta\\Delta},\n", "ace6e16f9c7dae7c6ef7cc0c3d1e37bf": "\\int_{0}^{1} \\sqrt{1 - x^2}\\, dx", "ace72edda51dcd1319b45b82b5e0b13b": "\\bar b_i - \\lfloor \\bar b_i \\rfloor - \\sum ( \\bar a_{i,j} -\\lfloor \\bar a_{i,j} \\rfloor) x_j \\le 0", "ace77091e49e4bb5e297d7e6f5b1efa2": "\\widetilde{R}=e^{-2\\omega}R-2e^{-2\\omega}(n-1)\\nabla^{\\alpha}\\partial_{\\alpha}\\omega-(n-2)(n-1)e^{-2\\omega}\\partial^{\\lambda}\\omega\\partial_{\\lambda}\\omega", "ace80fc773e9ce0608095414dc328a42": "GF^n", "ace82b6a7c4274fd301e4b9f1d8b9927": "\\mathcal D\\subseteq \\mathbb R^2\\,", "ace85fbc78f74230b8b499eb2f97bbe0": "\n \\lim_{n\\to\\infty} \\Pr[ a\\sigma 0", "acfb3ec6c9004fde23246b5bcb1cf446": "Y_t = \\alpha_0+ \\alpha_1 X_{t,1} + \\alpha_2 X_{t,2} + u_t \\,", "acfb509c55a28b79d32766427e1e2ffe": "n = \\frac{(w^{4658j} - w^{-4658j})^2}{(4657)(79072)} \\,", "acfb66dfb2e1cbacd07d60b80dc2c8ba": "\\scriptstyle y_t", "acfb8b20e52306721e93689846c09693": "\\operatorname{cl}(S)", "acfbe0845a735d4be212183fa81835d4": "CE = \\%C + \\frac{\\%Mn}{6} + \\left(\\frac{\\%Cr+%Mo+\\%V}{5} \\right) + \\left(\\frac{\\%Cu+\\%Ni}{15} \\right)", "acfc2d601ea8aef246c0c6e62fe79473": "L(n,k) = \\frac{1}{2}(k^2+k+2)k^{n-k-1}-1", "acfc97a02cb50d33727ffa73ab897a48": "\n\\bar c = \\frac{1}{3} \\left( \\cos (355^\\circ) + \\cos (5^\\circ) + \\cos (15^\\circ) \\right) \n= \\frac{1}{3} \\left( 0.996 + 0.996 + 0.966 \\right) \n\\approx 0.986\n", "acfca7d92f67f0594f4dc635e8981537": "U(\\theta) = aW ~\\mathrm {sinc} \\left [\\frac {\\pi W \\sin \\theta} {\\lambda} \\right]", "acfcbf7ff8e231f0274434e9e81fd028": " \\mathbf{Q} = \\left [ \\hat{ \\mathbf{x}}, \\hat{\\mathbf{y}} \\right ] ", "acfcedecd23f101b8f99faf68f5b0155": "t=\\sum_{i=1}^n\\frac{\\big[(x_i-x_{i-1})^2+(y_i-y_{i-1})^2+(z_i-z_{i-1})^2\\big]^{\\frac{1}{2}}}{v_i}\\!", "acfd149c6690c399644715cd650920a2": "\\mathbb{E}[X_\\tau]=\\lim_{t\\to\\infty}\\mathbb{E}[X_t^\\tau].", "acfd40d4f44093d9b5a489713242bc1c": "Z(x_1), Z(x_2), \\cdots, Z(x_N)", "acfd6a482d40693b670c5c36b4018d24": "\\theta \\simeq \\frac{\\lambda}{\\pi w_0} \\qquad (\\theta \\mathrm{\\ in\\ radians}). ", "acfd9827415a3e5b99960972e6ed49e7": " c'\\ge c+d. ", "acfdf42a789f23a8e0fb548ee4cff02e": "D=2J/(1+J)", "acfe0f1ad57029143d7774958c351150": "G_D = \\{ \\gamma ", "acfe7e5025e2015a5ec007815d09f7a3": " A Q_n = Q_{n+1} \\tilde{H}_n ", "acfe7f8001e5bd6f2b7e2abe522dc28e": "\\sqrt{1/2}", "acfecd93530a94a2f8bc5f3a958020ab": " \\ CVI(ESA) = A\\phi\\frac{\\varepsilon_0\\varepsilon_m\\zeta\\Kappa_s}{\\eta\\Kappa_m}\\frac{\\rho_p-\\rho_s}{\\rho_s}", "acfeec7b28ebea5b196681ad4fa48567": " \\langle S \\cap G^{(i)} \\rangle = G^{(i)} ", "acfefad45aea85d36b980fad3b76df8f": "p, L \\in \\mathbb{R}", "acff1c007e3f2677658dc882b14496b2": "\\frac{1}{\\nu-2}\\!", "acff22f0d2f9b6db2f2ddb4095fc773c": "\n\\begin{align}\nn! & = 2432902008176640000\\\\\n\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n & = 2.422786847e18\\\\\n\\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n\\left(1+\\frac{1}{12n}\\right) & = 2.432881792e18\\\\\n\\sqrt{2\\pi n}\\left(\\frac{n}{e} + \\frac{1}{12 e n}\\right)^n & = 2.43290180e18\\\\\n\\end{align}\n", "acffb3f6ee7ad81b1af5daf7765181fd": "i\\gamma^a e_a^\\mu D_\\mu \\Psi - m \\Psi = 0.", "ad00125dfa85f876d46d35aad76f1fb6": " \\mbox{fall-out}=\\frac{|\\{\\mbox{non-relevant documents}\\}\\cap\\{\\mbox{retrieved documents}\\}|}{|\\{\\mbox{non-relevant documents}\\}|} ", "ad01762918d54d484a3bcf1d0ff92c77": "t > 2", "ad0199277f1fc31588a43f51dcee98a2": " (\\varphi, f) \\mapsto \\varphi \\circ f", "ad01fdffbde44ecbc21ef50f32b96350": "\\textbf{a}", "ad0209c07a0b6289dadb77f3ae18ac7d": "g_s \\approx 2.0023192", "ad022141ea626df9f8414d21038c2c1a": "\n \\sigma_{xx}(x,z) = -z~E(z)~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\n", "ad023b138f710ef380dc234c0ef3df4a": " \\, f=Y_\\nu(x)\\, ", "ad023e6e360b824bd87056bffd38c656": "J_-", "ad0244b7b7e44ae470ffcbfbcd69b08c": "\\{\\Psi_0, \\Psi_1, \\Psi_2,\\Psi_3, \\Psi_4\\}", "ad025b45157a6ac8eb14cffbeff79cc9": "\\tanh x = \\frac{e^{2x} - 1} {e^{2x} + 1}", "ad02a795487ead78e49c799192b71e8f": "L(X, \\mathcal{F})", "ad02b9d8432f6d05cd3a468fa06fea19": "\\begin{align} ds^2 & = \n\\frac{1}{4}\\int_X (\\delta \\log p)^2 \\;pdx\n + \\int_X (\\delta \\alpha)^2 \\;pdx\n - \\left(\\int_X \\delta \\alpha \\;pdx\\right)^2 \\\\\n& -\\frac{i}{2} \\int_X (\\delta \\log p \\delta\\alpha - \\delta\\alpha \\delta \\log p) \\;pdx \n\\end{align} \n", "ad02d88f0bb0daa3c84f67cb96d38132": "\nD_A(\\rho_1,\\rho_2) = \\arccos \\sqrt{F(\\rho_1,\\rho_2)},\n", "ad03254132aa5d91809a8cf56612342b": "[x_1,x_2,\\dots,x_n].", "ad032b49d7dc98a0a141bd3e21aa0442": "\n\\begin{bmatrix}\nc t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\gamma&-\\beta \\gamma&0&0\\\\\n-\\beta \\gamma&\\gamma&0&0\\\\\n0&0&1&0\\\\\n0&0&0&1\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nc\\,t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix} ,\n", "ad034aab5d0e3cc25364fe1ceb6ae5bf": "\\scriptstyle f(n/2W)", "ad036805c82d9efa29f4bb1794ce6d08": "\\mathrm{return}: A \\rarr \\mathrm{M} \\left( A^{?} \\right) = a \\mapsto \\mathrm{return} (\\mathrm{Just}\\,a)", "ad03d54eadb9239e7ca69005efd2705f": "F_X = \\frac{\\sigma_X^2}{\\mu_X}.", "ad04169fa7b5f3f9bc2c40f8eec23763": "(x_i) \\in X", "ad044e2485a09b81239edb647fc1f769": "x_{i,m+j}", "ad047ac3a2f934ed7020451ba5a0e712": "\\!\\mathcal A \\models_Y \\psi", "ad049306ac613841dfb6b7ce533c9278": "\\scriptstyle \\sum_{i \\in I}\\alpha_i u_i \\;=\\; 0", "ad04ec2793be7f4e98d469cad720602c": "e^{-t}/t", "ad054049bfc397395c55958d286d57f8": "\\sum{n_i(n - n_i)} = n^2 - \\sum{n_i^2}", "ad0567a9944f152bfd184f40890ce14f": " D_f(P\\parallel Q) \\equiv \\int_{\\Omega} f\\left(\\frac{dP}{dQ}\\right)\\,dQ.", "ad057c583ebb6d4438bfaee473dd541f": "M_a=\\frac{500000 \\times 12%}{e^{0.12\\cdot 10}-1}=25860.77", "ad05c0b628edadaeaabcaf71ecbefabe": "(8,1)_0", "ad05f3b771692dcfabc7e6820c43902e": "\n \\dot{\\boldsymbol{\\sigma}} = \\mathsf{D}:\\dot{\\boldsymbol{F}} \\,.\n ", "ad065c4ebca745748b8e350eb15fc1e3": "\\sigma \\ll \\omega\\varepsilon", "ad0666f5d6aa0744052b68306d1b3b2c": "\\mathfrak{k}_{\\mathbb{C}}", "ad069d7ed56f5f1f3b5cff0699979da1": "\\mathbf{B} = \\mu \\mathbf{H} = \\mu_0 \\left ( \\mathbf{H} + \\mathbf{M} \\right ) \\,", "ad06c23907b9907e8b361b1cfe04d252": "\\Delta(x_0*d)", "ad07541f27c3268043c88ae708e11fc0": " \\; \\; + \\frac{x^2+y^2}{\\sqrt{2}} \\, \\left( -q^2 \\sin(\\omega u)^2 + q^2 \\tan(q u)^2 \\right) \\partial_v", "ad080ba931fd35a7d0b550f89ac6b396": "x_0^3+x_1^3+x_2^3+x_3^3=0", "ad08354aeedc9913b67a9ab9e974a5b7": "e_C(Tx) = Te_C(x) + \\left(e_C(x)\\right)^q = (T + \\tau)e_C(x), \\, ", "ad0845f3634cea6637a15a94f707489e": "\\theta_\\mathrm{C} ", "ad085c320ae3d2056695ac29a032cdcb": "\\bold a_i^T x \\le b_i, i = 1,\\ldots,m", "ad085fdd6d8372c8d94f397d08123d1e": "\nA_{n-1}B_n - A_nB_{n-1} = (-1)^na_1a_2\\cdots a_n = \\Pi_{i=1}^n (-a_i)\\,\n", "ad08926d5581cdd0d4d3e5cda0fdeaa9": "{O}(n^2\\log n)", "ad08ccd95e52e85c2dcd44a7a3ebae9c": "E_{ABC}=E_A(Z)-0.5[E_B(Z)+E_B(Z+1)]-0.5[E_C(Z)+E_C(Z+1)]", "ad08fa52d58cd932e376fa566f31add9": " \\infty\\ ", "ad090c452c6ba296f47933477b8deb45": "B = \\sqrt{2} - \\sqrt{3}", "ad0949ea83349252abcab854eefe026b": "D = V_{\\beta \\in B}", "ad097788a99fce95854ddf5932af43ef": "\nt_{1/2} = \\frac{\\ln (2)}{\\lambda}\n", "ad098cff3815f700c21be503b6c4b008": " X,Y ", "ad09b123bf990a04adb9a18496695932": "\\frac{\\operatorname{d}I_{\\text{L}}}{\\operatorname{d}t}=\\frac{V_o}{L}", "ad0a2f810c08e439f32c46e2d37b0501": "\\displaystyle{L^2({\\mathbf R}^n)=L^2({\\mathbf R})^{\\otimes n}.}", "ad0a4571b3adfd5d07775181426b480b": "\n\\lambda = \\frac{0.2479 - 0.0000947(7-\\log Re)^{4}}{(\\log(\\frac{\\varepsilon}{3.615D} + \\frac{7.366}{Re^{0.9142}}))^{2}}\n", "ad0ae8547f2d60a46dbf3065989615bc": "-\\frac {a}{b} \\;", "ad0b6f83df7efb8c53cb11753e71908c": "\\zeta(x,y,t)", "ad0b8a125ff5493210f55ad6d374e6f1": " \\det Q = +1 . \\,\\!", "ad0bc67b239e2291a208b837b8cbfd1a": "\\;d\\bar{v}= - \\frac{e \\cdot \\bar{E}}{m}\\;dt", "ad0c59417b072b4141e1e0434351c45c": " \\chi_i \\in \\{ 0,1 \\}^n", "ad0c5b3542cb7a6707764e32aa275ae8": "e_a = \\frac {A_{eff}} {A_{phys}} \\,", "ad0ca7c0e6e61c96379b00af975cb2a2": "\\lambda_i \\ge 0", "ad0cac76ceab27aed84719a59a0aae27": "\\scriptstyle V_2 = DV_1", "ad0ce26b0523de47f1b5177062f92dbf": "\nI_p = \\sqrt{\\frac{\\sum(\\Delta hR_{Fi} - \\Delta hR_{Ft})^2}{n(n+1)}}\n", "ad0ce2a10b6c9e72ba4e24ebd17efc24": "2{K_p}/P_u", "ad0cf9e7df8b45da156ceb46bd1f540a": "G \\exp \\overline{C}", "ad0cfa2bbf19418467d3415cf3e457fe": "\\Delta \\tau = \\sqrt{\\Delta T^2 - (v_x \\Delta T/c)^2 - (v_y \\Delta T/c)^2 - (v_z \\Delta T/c)^2 } = \\Delta T \\sqrt{1 - v^2/c^2}, ", "ad0d2b4b47fb939def45a354ee1bd465": "\\cfrac{V_\\max}{\\cfrac{[I]+K_i}{K_i}} ", "ad0ded57931f809a8d58e646002e0117": "\\mathcal{F}^{-1}g(x):=\\lim_{R\\to\\infty}\\int_{-R}^R e^{2\\pi ix\\xi}\\,g(\\xi)\\,d\\xi.", "ad0e1adb4635fc842fe1f0e1a3e6abb6": "G(z) = c\\, \\exp\\left(\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}\\frac{e^{i\\theta}+z}{e^{i\\theta}-z} \\log\\!\\left(\\varphi\\!\\left(e^{i\\theta} \\right)\\right)\\, \\mathrm{d}\\theta \\right)", "ad0e267647650c75688a43a31b8d9ef6": "x^4+a^4=(x^2+a\\sqrt{2}\\cdot x+a^2)(x^2-a\\sqrt{2}\\cdot x+a^2).\\,", "ad0e7a9071302ad63197194332553921": "I(t)", "ad0ec17c258c02c0a97339ff7b6b5b1e": "{X(t)}, t=0, \\pm 1, \\pm 2, \\dots ", "ad0f8016d509a62519d4eb43432ea03c": "\\alpha_0+n\\alpha,\\, \\beta_0+\\sum_{i=1}^n \\frac{1}{x_i}\\!", "ad0f91f9ac2662bfb373ffba3f94f41a": "\\nu_f^{\\sigma_j} = (\\tau_f^{\\sigma_j}-0.5)c_s^2\\delta_t", "ad0fea7e4abbbd9a25e82b96e04b78a7": "\\operatorname{DG}(a_n;s)=\\zeta(s)^m", "ad10e34c16e66936420140ff29f4ac54": "p_D", "ad116f1e8ffb0fd9190ede397e2943e7": "[IS_{Op} | \\text{pre } Op \\land P]", "ad119f6b80e1bfbffd793099bdef09e6": "H^2", "ad11d51b85b2286ee6f02d88e5577134": " \\tau^* = \\frac{1}{\\omega_0}|\\Delta p|^{-(s+t)} ", "ad11d72eaafa60918ff018956a48542d": " \\Delta t < 2\\frac{\\sigma^2}{(r-q)^2} ", "ad1216189e011f6fa7c861f8dc7d0b7b": "(i, j) \\cdot (k, \\ell) = (i, \\ell) \\, ", "ad123a07f1e5bb8517f0381141f263d1": "R=\\mathbb{Z}", "ad12822cfede0e3c3744ad2b7be7ecd5": "-W^\\mathrm{adiabatic,\\,quasi-static}_{A\\to B}=-W^\\mathrm{adiabatic,\\,quasi-static}_{A\\to O}-W^\\mathrm{adiabatic,\\,quasi-static}_{O\\to B} = W^\\mathrm{adiabatic,\\,quasi-static}_{O\\to A}- W^\\mathrm{adiabatic,\\,quasi-static}_{O\\to B} = -U(A) + U(B) = \\Delta U", "ad1293dfffee4fa039cd66750fe713c4": "B^{(\\alpha)}", "ad12b2b833bd9b517119479a7b4a48dc": " dw_i = \\frac{d\\Gamma}{4\\pi r} \\qquad (6)", "ad12f37be5a9559b1948803c0c72d9ee": " \\triangle DXX'' \\sim \\triangle BYY',\\, ", "ad134f27074ca5a8f5eeec1de6bc53fd": "A_{\\alpha \\alpha}", "ad1389426cf7fbaae0e7dc930d2855b0": " f(x) = \\int_{\\partial D} f(y) \\, d\\nu_x(y).", "ad1438965a28ced49d22f2d6d5b8c850": "{\\mathfrak H}^n", "ad14665003e5517e19f65dd9154d645c": "\n\\operatorname{Li}_2(z) = \\sum_{k=1}^\\infty {z^k \\over k^2}.\n", "ad1470c5d0a3d46e4a6fa3811aa467b0": "\\det\\left(\\partial_y g(x,y,t)\\right)\\ne 0.", "ad14817cc992d3bc938b4dc17e4293aa": " i>0 \\,\\!", "ad148a3ca8bd0ef3b48c52454c493ec5": "mod", "ad154729034445b03f041f46f2025c67": "\\underline d(A)=\\lim_{m \\rightarrow \\infty} \\frac{1+2^2+\\cdots +2^{2m}}{2^{2m+2}-1}\n= \\lim_{m \\rightarrow \\infty} \\frac{2^{2m+2}-1}{3(2^{2m+2}-1)}\n= \\frac 13\\, .", "ad154ef920ac417f1762d3267473b8dc": "f(y; \\,m,\\Omega)", "ad15bf9aa8f1701d291f9c9c9ba1f235": "\\exp_{\\mathbf{R}}\\mathbf{B}(\\hat{\\mathbf{R}})", "ad15c1daa4574176cf5a7f4b6164363e": "\nG^{}_{}(l)=(1-(1-p)^l)\n", "ad15e239915721c0f82c4acdd6998fc8": "\\scriptstyle R=G=0", "ad15f118aa519e6324b3cfb264aebff0": "\\langle \\eta(\\vec x,t) \\rangle = 0", "ad16073491da10b24dc9f70858681c03": "\\alpha_{m1,m2,m3,k}", "ad1626dd38f3ad388f53788be2b95f81": "\\mathcal{L}_f(\\mathcal{E}) = \\{L_f(D) | D \\in \\mathcal{E} \\}", "ad1671e453620443ba0ce1e7f999c088": " x_{1}^{(e)} ", "ad1684c4dabe8f416d5cae29c9e22e13": "\\int\\frac{1}{ax^2+bx+c} dx =\n\\begin{cases}\n \\displaystyle \\frac{2}{\\sqrt{4ac-b^2}}\\arctan\\frac{2ax+b}{\\sqrt{4ac-b^2}} + C & \\text{(for }4ac-b^2>0\\mbox{)} \\\\[12pt]\n\\displaystyle -\\frac{2}{\\sqrt{b^2-4ac}}\\,\\mathrm{arctanh}\\frac{2ax+b}{\\sqrt{b^2-4ac}} + C = \\frac{1}{\\sqrt{b^2-4ac}}\\ln\\left|\\frac{2ax+b-\\sqrt{b^2-4ac}}{2ax+b+\\sqrt{b^2-4ac}}\\right| + C & \\text{(for }4ac-b^2<0\\mbox{)} \\\\[12pt]\n\\displaystyle -\\frac{2}{2ax+b} + C & \\text{(for }4ac-b^2=0\\mbox{)}\n\\end{cases}", "ad1691c78b3ef3ffd684213e12be7f48": "\\displaystyle{ PQ-QP=iI.}", "ad16c9a22c6c7ff5fce4c4ca84d7d155": "\\delta q = \\frac{0.16}{eV}|X_M - X_{SC}| + \\frac{0.035}{eV^2}|X_M - X_{SC}|^2", "ad16d9363867f9730d608ff18c479eac": "\\lambda I ", "ad16fbf2b379191712c5fdc9daea9ff4": "a_0,..., a_k", "ad1741c465b5caff3b12ebeb22a12d06": " w_A \\operatorname{E}(R_A) + w_B \\operatorname{E}(R_B) + w_C \\operatorname{E}(R_C) ", "ad174660af80eb02c99e2f92f67a3ca1": "I(t)=M_at-\\frac{M_a(e^{rt}-1)}{re^{rT}}", "ad174e372007fc59643fe9b60527b77d": "f_{k,i}(x_{\\{k\\}}) = 1", "ad1753d049526dc72fee440f5286a890": "\\begin{bmatrix} U_1 \\\\ U_2 \\end{bmatrix}", "ad17595b4c99261a6890e85ff6454cde": "f(x,y)=x", "ad17752a27b70229afb6af3ccbbdf153": "E\\left(k, \\textstyle \\frac{\\pi}{2}\\right) = \\int_0^{\\frac{\\pi}{2}} \\sqrt{1 - k^2 \\sin^2 \\theta}\\; d\\theta", "ad17dba70511abf12b2fa6728f8d2477": "\\pi_i(\\mathbb{HP}^\\infty) \\otimes \\mathbb{Q} \\cong \\mathbb{Q}", "ad18271b64a6919632429d08da3e0563": "f^e_\\mathbf{k} \\, f^h_\\mathbf{k}", "ad1856a659fcd041a750deb8ded9728d": "\\;\\sum_{\\rho\\in R} \\rho(x) = 1", "ad189fe1bbe368a3aa73dea95ee3cca0": "\\eta_{n}=\\mathrm{Tr}\\lbrace\\Pi_{n}\\rbrace", "ad18be125c83914ced7e1cf2ef403437": "A^T P A -P + Q = 0", "ad19410974e6ac5c6e6555ea0e25b551": "\\{y_1,\\ldots,y_n\\}", "ad19413f59836c5366e005d10ade90eb": "\\displaystyle \\partial_t\\phi + \\partial^3_x \\phi + 6\\, \\phi\\, \\partial_x\\phi=0", "ad19a36414bee03022c5e22fb1e83c46": "\\alpha(s)\\rightarrow 0", "ad19ade3ba12bb98954d3ecba6071293": "\\begin{bmatrix}\nX & XA \\\\ A^HX & X-Q\n\\end{bmatrix}=0", "ad19b199b9bf3769c4d2f72896479bec": "q = \\gamma r^2 \\,\\;", "ad19da162fe57c2b1f00e1dd255e2b9e": "\\mathbb{N}^4", "ad1a8facdd5548365a322675c2e86cab": "\\tau \\propto \\sqrt{\\rho/\\sigma}R^{3/2}", "ad1aff88c54996201da375a8194ac933": " a_i = \\text{Model coefficients}", "ad1b45eb366c8ce6ca5e93ef1e6d3553": "h_f = f_D \\cdot \\frac{L}{D} \\cdot \\frac{\\bar{V}^2}{2g}", "ad1b5e897e1e42b0541f628ba2373316": "\\hookrightarrow \\hookleftarrow \\multimap \\leftrightsquigarrow \\rightsquigarrow \\twoheadrightarrow \\twoheadleftarrow \\!", "ad1b7c62d4a6d0247ec43e55485bbd4f": "u(\\lambda t, t)=0 ", "ad1c2da868dca36dd64722ba2ed689d5": "K^\\text{app}_m", "ad1c7780f8be600335164133778993ba": "\\delta \\mathbf {r}_i\\,", "ad1ca7da6b5e9d12ba1639481ae114a9": "D(n)", "ad1cb7a9ee4c9968ec9ae9fd471cb3f2": "\\lambda\\in\\mathit{\\Omega}", "ad1cf96e860c9d2e0a398ceb04bd059c": "\\det(\\exp(A)) = \\exp(\\operatorname{tr}(A))", "ad1d47517fe03714b004281eb0d34d99": "\\mu=1,2", "ad1df2813134bc1a069fd15e4ef5b7e2": "[x_1,x_2,\\ldots,x_n] \\mapsto \\left[\\frac{x_1}{1-x_n},\\frac{x_2}{1-x_n},\\ldots,\\frac{x_{n-1}}{1-x_n}\\right].", "ad1df793247a0e650d0d7166341b8d97": "1000009", "ad1e360b01c711c93ab17900a331c2ee": "x(k), k=1,2,\\dots,K-1", "ad1e4eba1149b369d9665ea9007714a7": "\\lbrack\\mathbf z\\rbrack = \\begin{bmatrix} R_1 + 2R_2 & 2R_2 \\\\ 2R_2 & 2R_2 \\end{bmatrix}", "ad1e7c00603fbf369bf4cacf66c104bc": "x^{(n)} = \\left( 1, \\frac1{2}, \\dots, \\frac1{n}, 0, 0, \\dots \\right)", "ad1ea0ec32659954b55f34b2a8275482": "F_{Lorentz}=qvB\\,", "ad1f029d9638fa1fc6cad36d5e7a77eb": "\\underline{-2\\mathit{891}}\\, ", "ad1f3e10adf65b602cbec72653ffd864": "\\begin{bmatrix}\n 1&2 \\\\\n 2&4 \\\\\n\\end{bmatrix} \\sim \\begin{bmatrix}\n 1&2 \\\\\n 0&0 \\\\\n\\end{bmatrix}", "ad1fd33daa42c5dc6f24c72c10098205": "\\Delta v_1 \n= \\sqrt{\\frac{\\mu}{r_1}}\n \\left( \\sqrt{\\frac{2 r_2}{r_1+r_2}} - 1 \\right)", "ad1fd8050d7cc74190b140863cfd87ea": " a b = a \\cdot b + a \\wedge b = \\langle a b \\rangle_0 + \\langle a b \\rangle_2", "ad1fe76a40ee2e29af3f9e5afd53f33d": "\\mathrm{Var}(X_\\theta) \\ge \\frac{(d\\mu_\\theta / d\\theta)^2} {\\mathcal I_X(\\theta)}.", "ad1fed8b859be9f584fba5a5346af2b1": "\\frac{d w}{d p} = \\sqrt{\\frac{2}{\\pi }} \\ e^{\\frac{w^2}{2}} ", "ad201a38a6b5fe259ca20ee16298e381": " K_+ \\partial_t^+ u_{n,i} + K_- \\partial_t^- u_{n,i} + L_+ \\partial_x^+ u_{n,i} + L_- \\partial_x^- u_{n,i} = \\nabla{S}(u_{n,i}) ", "ad20572026a258075f94b97c199def12": "\\,\\! \\lambda^* = \\frac{\\partial U/\\partial m}{\\partial E/\\partial m} \\approx \\frac{\\Delta \\mbox{Optimal Utility }}{\\Delta \\mbox{Optimal Expenditure}}", "ad206758377e6aeb5fcc8127b53d8a30": "\\textstyle P_k", "ad20a987c09d236ecc1219b1b0eac2bd": "E_5 \\cong D_5", "ad20f50d2f3a20b9c640b295148a799c": "\\left. \\ln P\\right|_{P=P_1}^{P_2} = -\\frac{L}{R} \\cdot \\left.\\frac{1}{T}\\right|_{T=T_1}^{T_2}", "ad20fe9daba7b82a7ec8a031a622ebba": "\\vec{E} = - \\vec{\\nabla} \\phi \\,", "ad211e30eb156b28954e73841f8323cf": "V_{L2-N}=\\sin \\left(\\theta -\\frac{2}{3} \\pi\\right) * V_P = \\sin \\left(\\theta +\\frac{4}{3} \\pi\\right) * V_P", "ad213685f7f9a5368957d7c14636e2dc": "\\scriptstyle K^{+/-} \\rightarrow \\mu^{+/-} + \\nu", "ad215371919570e0fcda38717b89d31a": "s+t+u = m_1^2 + m_2^2 + m_3^2 + m_4^2 \\,", "ad217abb093ac45b4f470dea89e5a838": "\n\\Gamma = A \\Delta\n", "ad219572d89a2c7c4fb84a68d96f46fc": "\\sum_{k=1}^\\infty 10^{-k!} = 0.1100010000000000000000010000\\ldots", "ad21a96bd2a2f3e4e5ff1f206200b686": "(E_t=0)", "ad21b51cff426a4627886d63623b11f1": "\\left\\langle \\ldots \\right\\rangle", "ad21d285f7a503f23914f20d6a0fb090": "\\sigma_\\bar{C} \\,", "ad21dca2128b28bb14819a6774873c92": " p(X) ", "ad21fdeb1f673b4bc277ad36506384f2": "10^{10}", "ad223d4c5a14badd330b794ba2317e29": "(1-X/2)^A\\,Q_A(X)+(X/2)^A\\,Q_A(2-X)=1", "ad22aac12628edc72863454a00bdfd13": "F_3(a, b) = a^{\\ln(b)} = e^{\\ln(a)\\ln(b)}", "ad22ab935337747f424f19ff51e21602": " \\Pr[C(m_i) \\in B(y, pn)] = \\mathrm{Vol}_q(y, pn)/q^n \\leq q^{-n(1 - H_q(p))}, ", "ad22fecfb5827d7693862909a30f793e": "B_{1,2}^s", "ad233805c929edd552ac82f8ab068b23": "\\xi = \\frac{\\tan \\alpha}{\\sqrt{H/L_0}},", "ad235d999168dee7cf6a3492bd6719a2": "[\\quad]_\\mbox{seq}", "ad23a42c5cf413beee7d906b2e1076c8": "U_\\textrm{eff}(\\tilde{a})\\;", "ad23c0c9ee13d9defaafac6d9d4e01f7": "\\begin{align}\n\\mathbb{E}[\\mathbf{X}] &= \\frac{ \\partial A\\left(\\boldsymbol\\eta_1,\\cdots \\right) }{ \\partial \\boldsymbol\\eta_1 } \\\\\n&= \\frac{ \\partial }{ \\partial \\boldsymbol\\eta_1 } \\left[-\\frac{n}{2}\\ln|-\\boldsymbol\\eta_1| + \\ln\\Gamma_p\\left(\\frac{n}{2}\\right) \\right] \\\\\n&= -\\frac{n}{2}(\\boldsymbol\\eta_1^{-1})^{\\rm T} \\\\\n&= \\frac{n}{2}(-\\boldsymbol\\eta_1^{-1})^{\\rm T} \\\\\n&= n(\\mathbf{V})^{\\rm T} \\\\\n&= n\\mathbf{V}\n\\end{align}", "ad23d4f5835958c256459a9adb173a03": " R = e^{-\\frac{B \\theta}{2}} ", "ad23eeae096ea72e1f35ef98b9809a24": "\\left \\langle \\sigma_y^2(N_2, M_2, T_2, \\tau_2) \\right \\rangle = \\left ( \\frac{\\tau_2}{\\tau_1} \\right )^\\mu \\left [ \\frac{B_3(N_2, M_2, r_2, \\mu)B_1(N_2, r_2, \\mu)B_2(r_2, \\mu)}{B_3(N_1, M_1, r_1, \\mu)B_1(N_1, r_1, \\mu)B_2(r_1, \\mu)} \\right ] \\left \\langle \\sigma_y^2(N_1, M_1, T_1, \\tau_1) \\right \\rangle.", "ad23f8106c8b0dceb80016e0bd1c8108": "\n\\left[\\frac{L}{p}\\right]_3 \\left[\\frac{L}{q}\\right]_3 =1 \\;\\;\\mbox{ if and only if } \\;\\;\\left[\\frac{q}{p}\\right]_3 \\left[\\frac{p}{q}\\right]_3 =1 \n", "ad240428b00641c7f98075d1b5f5ba75": "\\Diamond \\upsilon_1", "ad243d438ae859cd8134570faa4ce9a4": "\n\\frac{\\partial E_f}{\\partial t} \n+ \\overline{u_j} \\frac{\\partial E_f}{\\partial x_j} \n- \\frac{1}{\\rho} \\frac{\\partial \\overline{u_i} \\bar{p} }{ \\partial x_i }\n+ \\frac{\\partial \\overline{u_i} \\tau_{ij}^{r}}{\\partial x_j} \n- 2 \\nu \\frac{ \\partial \\overline{u_j} \\bar{S_{ij}} }{ \\partial x_j }\n= \n- \\epsilon_{f} \n- \\Pi\n", "ad243e98e20cfb6cba3a850e71c9b051": "G_0\\rightarrow G\\rightarrow G/G_0.\\,", "ad245b1eeb7ae17a733feacca4344ff4": "p=1\\,", "ad2498ee355770f3a67da5b8bfe5141b": " b \\in S_{M_a} ", "ad24ccb93fb9fe16aee78196ac7f34f6": " \\left ((-i\\hbar\\mathbf{\\nabla})^2 c^2 + m^2 c^4 \\right ) \\psi = \\left(i \\hbar \\frac{\\partial}{\\partial t} \\right)^2 \\psi ", "ad24f783915561637ab3e4b47c64dbf3": "\\sqrt{\\lambda} = n \\frac{\\pi}{L}.", "ad2550b6103118aacd382279e0cf8d8a": "dx/dt=0", "ad25a64ac87cabcdba13effc6350ca6e": "A_k = \\log{1+r_k \\over 1-r_k}", "ad25b1f62373530aaab97a14f260ba73": "\\forall x_1,\\ldots,x_k .\nL_1,\\ldots,L_m", "ad25b759bdd7b39e660057b8c3a1d6ba": " \\mathbf{r}_\\mathrm{com} = \\frac{1}{M}\\sum_i \\mathbf{r}_\\mathrm{i} m_i = \\frac{1}{M}\\sum_i \\mathbf{m}_\\mathrm{i} \\,\\!", "ad25d52d248e6d3aa26b4727b9af3fce": " \\{ x\\in V: \\|x\\|<1 \\}.", "ad25e702922db38123792ad381b9ce00": " C^\\infty ( [x])=C\\circ C\\circ C\\circ \\cdots \\circ C ([x])", "ad25fb04f7d1081b71f1be0562cd2373": "\\lambda - \\lambda_0\\,", "ad270fd3eb156127488014e055806886": "\\scriptstyle R_0", "ad2744fb23227867b4c76af67eaeea65": "a b c'", "ad2752ae47b77ffe886ce0c58bb998e6": "\\approx \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0.83205 & -0.55470 \\\\\n0 & 0.55470 & 0.83205\n\\end{pmatrix}", "ad2790b1db87855814248592b7af5988": "\n{dV \\over dt} + \\frac{1}{\\tau} V = f(t)\n", "ad27a4ac93f7468705897dd28c49f991": " BT(M)(\\langle\\ \\rangle) = \\perp,", "ad27ce78d6770691a44653316e190830": "\\beta_i(t)=P(Y_{t+1}=y_{t+1},...,Y_{T}=y_{T}|X_t=i,\\theta)", "ad27e0fbe387a580dc01540a0872f569": "q\\frac{(x-a)^{k+1}}{(k+1)!}\\le R_k(x)\\le Q\\frac{(x-a)^{k+1}}{(k+1)!},", "ad27f3ef2730ba24c2d815272ec48312": "\\mathbf{c}_{k_1}(i) = \\mathbf{c}_{k_2}(i) = j", "ad284fb67c16abcdacaf206970e2c252": "\\text{Price}(\\text{complete game})\\times\\text{Price}(\\text{Red Sox win}\\mid\\text{complete game}) \\neq \\text{Price}(\\text{Red Sox win})", "ad285436f3ff10f2045325a317ab4e8c": "\\scriptstyle Y \\;\\sim\\; \\mathrm{Erlang}(k_2,\\, \\lambda)\\,", "ad287c7fee5d4e1299302e6c5c8a534a": "(x_1 + x_2 + \\cdots + x_r)^n.\\ ", "ad288ab4210420840ee21cd181b219fe": "1/r_s^2", "ad288cbcef80162b4b79ba251b8a6e60": "\\left\\{0,...,q-1\\right\\}", "ad289c1214f2fab9373a60b72208144f": "\\sum_{k=1}^\\infty z_k=s \\quad\\iff\\quad \\lim_{n\\to\\infty}\\sum_{k=1}^n\\frac{z_k^n}{\\Pi_k(z_1,\\ldots,z_n)}=s", "ad28b99900bfc7792726ff0661a8aabd": "\\scriptstyle\\boldsymbol{\\varepsilon}=\\boldsymbol{\\varepsilon}(\\boldsymbol{u})=\\left(\\varepsilon_{ik}(\\boldsymbol{u})\\right)=\\left(\\frac{1}{2} \\left( \\frac{\\partial u_i}{\\partial x_k} + \\frac{\\partial u_k}{\\partial x_i} \\right)\\right)", "ad28d3e2b324542ae4b084b9bdda1305": "\\mathbf{K}", "ad28f5ddb8dadd74696e1a639bc20b89": "S\\ = \\gamma_\\text{gw} - \\gamma_\\text{go} - \\gamma_\\text{ow}", "ad29346c7a17385613d5ae979566a866": "K^+", "ad293dbaee54df4d77c2582718c5d63f": "t_{\\operatorname{ev}} = \\frac{5120 \\pi G^2 m_P^3}{\\hbar c^4} = 5120 \\pi t_P = 5120 \\pi \\sqrt{\\frac{\\hbar G}{c^5}} = 8.671 \\times 10^{-40} \\; \\text{s} \\;", "ad295a1ae87715cdf2b7dbb52270b153": "\\det S''_{xx}(x^0) \\neq 0", "ad298dfdb7f1544edd3cf833d9e3b7f9": "{X = X_L - X_C = \\omega L -\\frac {1} {\\omega C}}", "ad2a3c5fd6a158d7f961abb16af4fd88": " \\frac{d[C]}{dt} = k_2 [B]", "ad2a59f94c1abb0d3311a93c9e9fd930": "\\mu + \\frac{\\sigma}{\\xi}\\left[\\left(\\frac{1-\\xi}{1+\\xi}\\right)^\\xi - 1 \\right] ", "ad2a709272cb3da8e9e02c7d3d2b3798": "\\frac{k+1}{k} m - 1,", "ad2a9fbf222d56fde13c1f25c09f02ba": "l_k \\leq f(\\mathbf{x}^*) \\leq f(\\mathbf{x}_k)", "ad2acfcf975cd12d48e304e8431a6514": "spin(6)", "ad2ad00e261c19ce35788f96d741053d": "\\bigl((P_{k-1}+P_k)\\cdot P_k\\bigr)^2 = \\frac{(P_{k-1}+P_k)^2\\cdot\\left((P_{k-1}+P_k)^2-(-1)^k\\right)}{2}.", "ad2aea88247ce8cbe26871fc210ec7e5": "\\lambda [U]", "ad2b07c18b459a3502387cf2ae14a759": " PPV= \\frac{a+b}{2}- 1.5 \\frac{b-a}{6} = 0.25b + 0.75a ", "ad2b494f853797e21a685debd5fb0c37": "d_{\\ell mn}= \\frac {a} { \\sqrt{\\ell ^2 + m^2 + n^2} }", "ad2b4ef8d04ed918232c97d7fb98380e": "\n \\lambda \\approx 0.663\\mu\n", "ad2b7760901ae5661dc14aa75c571fb5": "\\int_0^z\\log\\Gamma(x)\\,dx=-\\int_0^z \\log\\left(\\frac{1}{\\Gamma(x)}\\right)\\,dx=", "ad2b9a68166247fe331e737cfd923379": "\\textstyle 4.\\ Calculate\\ \\theta_{k},\\ i=1, .... q.", "ad2ba8a5cd35bb30d9317d3d15bc7baa": "\\min(c_f(A,D),c_f(D,E),c_f(E,G)) = ", "ad2bb77fa61a041a1fb7c3d3ade8fd40": "\n\\mathbf{A} \\vec x = \\vec y\n", "ad2bda165d6466ec7cbbd520348bb3c3": "\\phi(p+v)=\\alpha(p)+T(v)\\,", "ad2be359ddba2ced6453ab93585818ec": "d \\Xi", "ad2bedb6cb1d239199b748bac75bbbc9": "\\operatorname{Ext}^n_R({}_R M,N) \\cong \\operatorname{Ext}^n_S(M,\\operatorname{Hom}_R(S,N))", "ad2c1986b0573a52539fe06f042fcab1": "\\ y_s ", "ad2c77222aec8fb5befa406d49883c49": "S(\\rho \\| \\sigma) \\geq \\sum_i p_i \\log \\frac{p_i}{r_i} \\geq 0.", "ad2cffba9af0c2623e6e9aa21d0ad81f": "\\tfrac{3}{4}>\\tfrac{2}{4}", "ad2d1263ffac00c6fe0b0a4b4209fc07": "\\mu \\in (-\\infty,+\\infty) \\,", "ad2dc93dcc9cb599d5139c78f73f5ba5": "T_{o-}^{TM}=G sin(\\frac{m\\pi }{a}y)e^{jk_{xo}(x+w)} \\ \\ \\ \\ \\ \\ \\ (24) ", "ad2e02e3fa25aea3a0f775ea734fcc9e": "x_{i+1} = f(x_i)", "ad2e2c2cf5da59dac481bc23a8681f5e": "(A\\cup B)^c = A^c \\cap B^c", "ad2e2e1f456746ade70d262dfa470076": " T_\\mathrm{n} = \\sum_{A} \\sum_{\\alpha=x,y,z} \\frac{P_{A\\alpha} P_{A\\alpha}}{2M_A} \n\\quad\\mathrm{with}\\quad \nP_{A\\alpha} = -i \\nabla_{A\\alpha} \\equiv -i \\frac{\\partial\\quad}{\\partial R_{A\\alpha}}. ", "ad2e376510d670f01d297135068136a0": "\\partial f/\\partial \\vec r^\\prime\\sim 1", "ad2e81721a38fa9910c023f036111fc5": " 00 ", "ad2e9b6a427af10f0517092e816fe4f0": "\\gamma^{n} (A) = \\sup \\{ \\gamma^{n} (K) | K \\subseteq A, K \\mbox{ is compact} \\},", "ad2eaee312d571e05840f9052aa1d204": "y'' + y = 0~", "ad2ec6495865f82aa693e6664390f397": " \\ g_{\\phi}", "ad2f16d20d2425faa75ee0d57cce04d8": "A^{(a)}", "ad2f39995ef86614eb7b48d340e69891": "y_c(x)=(x-c)^2", "ad2f4962edefbf39fd30a266f713b789": "\n(a_3 x_2 + b_3 x_3 + c_3 x_4) (b_2 b_1 - c_1 a_2) -\n((b_2 b_1 - c_1 a_2) x_2 + c_2 b_1 x_3) a_3\n= d_3 (b_2 b_1 - c_1 a_2) - (d_2 b_1 - d_1 a_2) a_3\n\\,", "ad2f600ca9de426a7d297e1743bed644": "\\frac{f(x_0+h) - f(x_0)}{h} \\le 0.", "ad2f8adc6b6964464c859eac1d99bb5c": " \\mathcal{O}(nt) ", "ad2fa75335c6ace2432748221ba887d7": "x=5+2t_1-3t_2,\\;\\;\\;\\; y=-4+t_1+2t_2\\;\\;\\;\\;z=5t_1-3t_2.\\,\\!", "ad2fb4a503e74af83358d7c014ae283b": " \\frac{K \\cdot t}{V} = \\ln \\frac{C_o}{C} \\qquad(4)", "ad2fdd8ee6e6b23d512c21f9be0c1117": "r_m", "ad2feedfb7d9a56ad123af7ea99992bb": "\n\\begin{align}\nM & = \\sum_{j=1}^cl_j(\\mathbf{M}_j-\\mathbf{M}_{*})(\\mathbf{M}_j-\\mathbf{M}_{*})^{\\text{T}} \\\\\nN & = \\sum_{j=1}^c\\mathbf{K}_j(\\mathbf{I}-\\mathbf{1}_{l_j})\\mathbf{K}_j^{\\text{T}}.\n\\end{align}\n", "ad30194f40fc50f75e24fdbc249e719c": "S_X", "ad3081ca1c067b2dc3708b2c02393b4c": " (1-M_\\infty^2) \\phi_{xx} + \\phi_{yy} + \\phi_{zz} = 0 ", "ad3093d2726751b3d4c35d4a51c57db3": "\\hat{n}\\,", "ad30ed7feeb8f21a780b6a686e9e8bfe": "\\;_1\\psi_1 \\left[\\begin{matrix} a \\\\ b \\end{matrix} ; q,z \\right] \n= \\sum_{n=-\\infty}^\\infty \\frac {(a;q)_n} {(b;q)_n} z^n\n= \\frac {(b/a,q,q/az,az;q)_\\infty }\n{(b,b/az,q/a,z;q)_\\infty} ", "ad30fd023e6fcedc28dd9f703366cc73": "\\hat{w}_i = 1 + \\epsilon 0; i = 0..3", "ad3120cd483315d57e6a1a639eb48b88": "\\!\\mathrm{I}(v)= \\langle v,v \\rangle = |v|^2", "ad313d0552ffab1fa55a55b9da5669be": "\\langle A (t)\\rangle = \\langle \\Psi (t)| \\hat{A} | \\Psi(t) \\rangle.", "ad316a5b718454963983bb78ca40a12c": "Gr_n = Gr_n(\\mathbf{R}^\\infty)", "ad319048da73e6b963e76a737dbb3769": "op_A", "ad31cb819fc0db89b65b4a4be935dbfb": "\\max(0,a_i+b_j-N)", "ad31f5e0eceec0b5ba3c8d70a7c3d35e": "\\nabla_{\\dot{\\gamma}(t)}\\dot{\\gamma}(t) = 0", "ad31f686fdaa159fd1e0ff63f511eeb8": " \\pi_1 = (a_D b_S - a_S b_D) /(b_S - b_D) \\, ", "ad31f6bfc6b260a4757db05eb8775c59": "R' = R^e \\pmod n", "ad320126b56683bacf2f017b18158311": "g_-", "ad3228bc35d0fb23895213b0ffa2b283": "\\ \\mathbf{G}_a \\mathbf{G}_a^*", "ad325fc11bdcda9c9c530c3eb01c98ee": "Tr(g^n) \\in GF(p^2)", "ad3283a595ce34980ba891155f83ec2d": "\\scriptstyle \\mathbf{y}=(y_1,y_2)", "ad32b4fecf6367c5494ae457051964d9": "(a_i+b_j-N)^+", "ad3303733486abc36570917187716765": "\\frac{1}{\\Lambda_{(\\mathbf n)}}=\\frac{dX}{dx}.\\,\\!", "ad331c1fda285c215521f2434d74dcc6": "\\textstyle \\lambda_{NN} \\leq \\lambda_{BN} < \\lambda_{PN}", "ad33502b05d52ef733aafe4ebf8b395f": "\\ M_{heel_{max}} > D_{heel} \\times F_{forward} \\times sin( \\pi - \\beta -tan^{-1}(L/D)) ", "ad33512d804a9a2aaf5a5d5eb3b85a57": "V_{2k} = \\frac{\\pi^k}{k!}", "ad33a24916154541dafd25ecbb3eff07": "\n\\begin{align}\n\\arcsin x &{}= -i\\,\\ln\\left(i\\,x+\\sqrt{1-x^2}\\right) &{}= \\arccsc \\frac{1}{x}\\\\[10pt]\n\\arccos x &{}= i\\,\\ln\\left(x-i\\,\\sqrt{1-x^2}\\right) = \\frac{\\pi}{2}\\,+i\\ln\\left(i\\,x+\\sqrt{1-x^2}\\right) = \\frac{\\pi}{2}-\\arcsin x &{}= \\arcsec \\frac{1}{x}\\\\[10pt]\n\\arctan x &{}= \\tfrac{1}{2}i\\left(\\ln\\left(1-i\\,x\\right)-\\ln\\left(1+i\\,x\\right)\\right) &{}= \\arccot \\frac{1}{x}\\\\[10pt]\n\\arccot x &{}= \\tfrac{1}{2}i\\left(\\ln\\left(1-\\frac{i}{x}\\right)-\\ln\\left(1+\\frac{i}{x}\\right)\\right) &{}= \\arctan \\frac{1}{x}\\\\[10pt]\n\\arcsec x &{}= -i\\,\\ln\\left(i\\,\\sqrt{1-\\frac{1}{x^2}}+\\frac{1}{x}\\right) = i\\,\\ln\\left(\\sqrt{1-\\frac{1}{x^2}}+\\frac{i}{x}\\right)+\\frac{\\pi}{2} = \\frac{\\pi}{2}-\\arccsc x &{}= \\arccos \\frac{1}{x}\\\\[10pt]\n\\arccsc x &{}= -i\\,\\ln\\left(\\sqrt{1-\\frac{1}{x^2}}+\\frac{i}{x}\\right) &{}= \\arcsin \\frac{1}{x}\n\\end{align}", "ad33bd1f34fcc3f5f131a843990f834e": "x(t)=0, t \\geq 0", "ad33fafa0a8b623e8e7111f2bdef9165": "c = \\sqrt{gh}\\, \\frac{1}{\\displaystyle 1 - (\\kappa\\,h)^2\\, \\frac{4}{3\\, \\pi^2}\\, K^2(m)\\, \\left( 1 - \\frac12\\, m - \\frac32\\, \\frac{E(m)}{K(m)} \\right)}.", "ad349ab9350969d4e9c6d7362a6d5382": "\\|Ax\\| = \\|A^*x\\|", "ad351d2c93f573f67832782b02165186": "w \\leftarrow w + \\eta(y - \\hat{y})x", "ad3542d391962a7936432142c1defb34": "\\alpha^n +1", "ad357cff2ca1feea62cf6d0e9f0cf562": "\\frac{1-\\theta}{\\exp(t)-\\theta}", "ad35bf0a85bb988f7eac4cc8c001a3d8": " u(ci,x,y) = m_1 + m_2 + m_4 + m_7 = (ci',x',y)+(ci',x,y') + (ci,x',y')+(ci,x,y)", "ad35cb8261928dd5e39845bbebc58cd4": " \\langle X \\mid R \\rangle ", "ad36331174be551ffd48e7ab39d680b1": "K = 1", "ad36550819da5f0eca1e4259cbbc711c": " Z_0 = Z_1 = Z_2 \\, ", "ad36bd90a171655e215afee45fdd02c0": " \\begin{align} r_{k+1} & = \\frac{3}{1 + 2(1-s_k^3)^{1/3}} \\\\\n s_{k+1} & = \\frac{r_{k+1} - 1}{2} \\\\\n a_{k+1} & = r_{k+1}^2 a_k - 3^k(r_{k+1}^2-1)\n \\end{align}\n", "ad36d644abdb9916e9ae59c986242c0c": "f\\circ (g+h)", "ad36df63056a1d422d9ce1e2bc486954": "d_c", "ad371df4ff168f568c61dc3ebf2a6346": "<\\lambda", "ad374494c3a9be600009747d07cd6c9b": "\\chi_i, \\chi_j", "ad375f1dada9d6f2aa7415a75b59dd95": "X = \\mathbb P^n", "ad376131a044f04fda1f2cc17e0a0786": "(Z_1 \\cdots, Z_n) = \\sum_{x \\in \\cap_i Z_i} (Z_1 \\cdots Z_n)_x", "ad377e74dbd841c2f09b9799f3ca3ca4": "\\scriptstyle{n\\frac{\\log(2)}{\\log(3)}}", "ad379321207a2b8124b08b4ecfdfbf20": "H_1(C, \\mathbb{Z}),", "ad387cc259d278535664734466bb624e": " \\frac{ \\sqrt{2 \\pi |\\Sigma_{y=1}|}^{-1} \\exp \\left( -\\frac{1}{2}(x-\\mu_{y=1})^T \\Sigma_{y=1}^{-1} (x-\\mu_{y=1}) \\right) }{ \\sqrt{2 \\pi |\\Sigma_{y=0}|}^{-1} \\exp \\left( -\\frac{1}{2}(x-\\mu_{y=0})^T \\Sigma_{y=0}^{-1} (x-\\mu_{y=0}) \\right)} < t ", "ad38992f0b0e594d4614cda048eb8ad3": "\\scriptstyle \\mu ", "ad389f0d90d8d83b171852432edb0c1c": "A_p(h_c)", "ad38a6cd7bafd5a6a2a15e62aec7126a": " P_{A}V=n_{A}RT", "ad38d8176fc1c5e52e69520de5deebf2": "{L} = {U}.{Ar}.({Tr}-{Ta})", "ad39078a506ef8d691293370d2a2c792": " \\text{vector area of triangle }ABC = \\frac{\\vec{r}(t) \\times \\vec{r}(t + \\Delta t)}{2}. ", "ad3912d11eb0d572ca1a8d1f1f61ab44": "\\frac{1-r\\cos x}{1-2r\\cos x + r^2} = 1 + r\\cos x + r^2\\cos2x+r^3\\cos 3x+\\cdots", "ad397ac1c53080b8cea0f42b776afc3b": "\\vec{\\pi}_E", "ad39ca01e355c4b5f5a1a267986c4a26": "\\mathcal{E}^{\\otimes n}", "ad39d1d0135d81ef41fff005ec10f14d": "fgf", "ad3a27398f9e8e3e73e8217b56922de2": "\\lambda(x)", "ad3a7d4b3110ba04339d9bc3eb17c593": "c_m=\\left(4g\\sigma/\\rho\\right)^{1/2}\\simeq 0.23", "ad3a8fd2674713f2798c4438cb20f16d": " \\textrm{minmod}\\left(z_{1},z_{2},\\cdots\\right):=\\left\\{ \\begin{array}{cc}\n\\min_{j}\\quad & \\textrm{if}\\quad z_{j}>0\\quad \\forall j\\\\\n\\max_{j\\quad} & \\textrm{if}\\quad z_{j}<0\\quad \\forall j\\\\\n0 & \\textrm{otherwise}\\end{array}\\right. . ", "ad3aa6f133bd7a4397cafb33e1e0bc1a": "1^{11} + 2^{11} + 3^{11} + \\cdots + n^{11} = {32a^6 - 64a^5 + 68a^4 - 40a^3 + 10a^2 \\over 6}.", "ad3abd58b31ebe5e82217bfad5bfde5b": "L \\left(\\boldsymbol\\phi, \\partial_\\mu{\\boldsymbol\\phi}, x^\\mu \\right)", "ad3ac08fbcebecf38d53abdd269d2201": "|q(T)(x)|>1", "ad3ad8f4611f0b120ef4f896a7a9cff1": "S_5,", "ad3b74082a8f17b94610efc09b153ec0": "k_i(\\epsilon_i,t,t_i)=m\\left(\\frac{t}{t_i}\\right)^{f(\\epsilon_i)}", "ad3bd3971973276c827b426a20cf4f80": "\\|f*g\\|_r\\le \\|f\\|_p\\|g\\|_q,\\quad f\\in L^p,\\ g\\in L^q,", "ad3bdd714e5a94e3f9caa7e317bc772a": "[H_0]", "ad3c25854515553fa77887e7863584f6": "{\\mathcal F}=\\{h:\\{0,1\\}^d\\to \\{0,1\\}\\mid h(x)=x_i,i =1 ... d\\}", "ad3c3d5c2305e3c7629797020dc1d3e6": "\\|A^k\\|", "ad3ccd0a436ffacd134450bb40fcbd73": "[a_j ,a_0] = i \\lambda a_j ~, \\; [a_j,a_k]=0 \\,", "ad3ce9a0a98c3e47c0c13cdb134f2731": "A_0\\subset A", "ad3cf063bc22654c96547b2161047d05": "(-1)^{n-1}\\left(\\frac 1 n - \\frac{1}{n^3} + \\frac{1}{n^5} - \\cdots\\right) = (-1)^{n-1}\\frac{n}{1+n^2}.", "ad3d0a2b1f8b1a09d277df69d1dc07cc": "(1-q^n) \\cdot B - q^n \\cdot B (2^n - 1) = B (1 - (2q)^n)", "ad3d1a6df6fbfb13e46ed3373028df8f": "\nT_n= \\sum_{k=1}^n k = 1+2+3 \\dotsb +n = \\frac{n(n+1)}{2} = {n+1 \\choose 2}", "ad3d53d34fa2b9d3a564bffb2dea0939": "y_2(x)=x\\,y_1(x)-c_1=\\,c_1x\\ln x+c_2x-c_1,\\qquad x\\in I,", "ad3d6016b77524c0aee789f026f12e8c": "I = 0.004 \\cdot D + 2.1", "ad3d952dcf8d8bbd30d3e023908cc9b0": "\\displaystyle{ \\sum_{n\\ge 0} |b_n(w)|^2 \\le (1-|w|^2)^{-1}.}", "ad3daa882ca1fe1ff7e0eba34a9b3102": "P_1|P_2", "ad3dfe9a6599d33a2be41ab39409737c": "L_{CE}", "ad3e3c2d4aaf97f6b7101a3ee1a1b15a": "(\\exists x (P \\vee Q))", "ad3ef41ec6ed361575ed26a1f5579c0a": "({\\mathbf P_2},F_{\\mathbf P_2})", "ad3ef8ebfb0c4a108989a407dc5720ce": " \\vec{r}_2 ", "ad3f01a7461f90cbd5d1ce22e9aa9cbc": "v=g", "ad3f14e59c02ccd618ffeb0e5cdd36d8": "\n\\hat{M} = \\frac{\\vec{R}_1 \\times \\vec{R}_2}{|| \\vec{R}_1 \\times \\vec{R}_2 ||} \n", "ad3f1bf5bd7ddc2c132c84a7761e0862": " \\phi(x,y)=(f(x,y),c\\pm y)", "ad3f34d3be1af09de7018e22d74589b7": "a_1^k + a_2^k = b^k", "ad3f62bef81bd13bf53a88090b1de76c": "\\mathcal F( \\mathrm{rect}(x) ) = \\mathrm{sinc}(\\xi)", "ad3f97335f03df414419d0baa43d5edd": "\n\\begin{align}\n& \\partial_{t} v = 4\\; \\Re \\left\\{ 4 \\partial_{z}^3v + \\partial_{z} ( v w ) - E \\partial_{z} w \\right\\}, \\\\ \n& \\partial_{\\bar z} w = - 3 \\partial_{ z } v,\n\\end{align}\n", "ad3fe2a63a6326f8813242b69f0bdd00": "\\frac{Y}{2}=\\operatorname{diag}\\left(-1/3, -1/3, -1/3, 1/2, 1/2\\right)", "ad3ff00d7c88f8b663946fe62a413fd0": "\\lim\\limits_{p\\to\\infty} \\frac{x_{n}+\\ldots+x_{n+p-1}}p=L", "ad3ff6d55a7f6d6f97853fc40b704b31": "{\\left\\vert \\mathcal{B}_w \\right\\vert} ^2", "ad400d8832252feb3b7e38e0b4d9d58d": "f_{\\text{a}}=\\frac{1}{{2\\pi}{R_{\\text{F}}}{ C_{\\text{F}}}}", "ad402bbd3783de26526e7171403305c9": "x \\in H_{\\aleph_1}", "ad402d4e5328da57ce7fb5aeff1ad44b": "p(k) = \\tau_0^{-1}", "ad40487f91bc822a805cc68df9037387": "\\,\\!x = 0", "ad404c8e1c0978e4b49bf111c4f406a0": "u \\in \\text{Hom}_R(M,N)", "ad40bfe3593cf804e1235676e66b0a39": "\\sigma_{23} = \\sigma_{32} ", "ad40e2899fdc541d6d1a1fe8dd791257": "x=a(3\\cos t+\\cos3t),\\quad y=a(3\\sin t+\\sin3t).", "ad40ea1df95743f616674c08e9c94de5": "{\\rm add}(I)=\\min\\{|{\\mathcal A}|: {\\mathcal A}\\subseteq I \\wedge \\bigcup{\\mathcal A}\\notin I\\big\\}", "ad40fed19f96f858c793785c66b69988": "\\mathbf\\nabla", "ad4161e4b69b79149e1e7aabb5020fe1": "\\ \\displaystyle s\\in S(d)\\ ", "ad418b504821920ab5945822db6ffeb6": "d\\sigma^2 = dz^2 + dr^2 + \\frac{r^2 \\, d\\phi^2}{1-\\omega^2 \\, r^2}", "ad420b054c202cb316e6ddc34c5e1b10": "\\frac{dh}{dt}=0", "ad421259450f122cb2e6933629afb3b6": " (X,\\Sigma) ", "ad421a969260b60ac5aea21719a0534f": "p^{(l,m)}", "ad4239fbffdb1f252dbcde630436de1a": "HK_\\alpha(x) = \\left\\{\n\\begin{align}\n & \\left(\\sum s_i^\\alpha\\right)^\\frac {1}{\\alpha-1} \\text{ if }\\alpha > 0, \\alpha \\ne 1 \\\\\n & \\prod s_i^{s_i} \\text{ if }\\alpha = 1\\\\\n\\end{align}\\right.", "ad426b67b14ee45934742e8fb438e75f": "S(u) = \\int_{0}^{T} L(t, u(t), \\dot{u}(t)) \\, \\mathrm{d} t,", "ad42d8724086fa8bcfe052fed5f6abbf": "(\\bar{3},2)_{\\frac{5}{6}}", "ad42f37c04c8d5bfa1298ab128b48b20": "\n\\begin{align}\n\\widehat\\varepsilon_i & = Y_i - \\widehat y_i = Y_i - (\\widehat\\alpha + \\widehat\\beta x_i) = \\text{residuals} = \\text{estimated errors}, \\\\\n\\text{SSR} & = \\sum_{i=1}^n \\widehat\\varepsilon_i^{\\;2} = \\text{sum of squares of residuals}.\n\\end{align}\n", "ad42fbc923a930409f09fdd039282cb4": "[0,1,q_0^2,q_1^2,q_2^2,\\ldots]", "ad43ac46b61a23f864d987c43f265064": "(a \\land b) \\lor (a \\land \\lnot c)", "ad442a2f5213849cc035087372913741": "\nTi_s(z) = {1 \\over 2i} \\left[ \\operatorname{Li}_s(i z) - \\operatorname{Li}_s(-i z) \\right] .\n", "ad449fd1279359c43ee3476111d02114": "{\\theta}_{[a,b]}=s(b-a)+a", "ad4547ee131cdba0690f5e2dccfcdf4e": "p_W(w|a,b) = \\frac{ab}{\\pi^2(w^2-a^2b^2)} \\ln \\left(\\frac{w^2}{a^2b^2}\\right).", "ad458211b030a019cc99e6de8734925f": "\\mathrm{Bi} = \\frac{h L_C}{\\ k_b}", "ad45c9b895c71f366af87154ceeeb542": "c^T \\hat{x} \\leq c^T \\hat{x} +\\tilde{\\lambda}^T(b_2-A_2 \\hat{x} ) \\leq c^T \\bar{x} +\\tilde{\\lambda}^T(b_2-A_2 \\bar{x} ) ", "ad468a5041d939757eb3d6e631ecd90f": "\\epsilon_i \\ ", "ad46b8ed8736c6e4d8759a05b2276e30": "x \\stackrel{*}{\\rightarrow} z \\stackrel{*}{\\leftarrow} y", "ad46edbc33c483cafebbd8d1a022823f": "B_{\\lambda,\\text{max}}(T) =\\frac{2k_\\mathrm{B}^5T^5(5+W(-5\\exp(-5))^5}{h^4c^3}\\frac 1{e^{5+W(-5\\exp(-5))} - 1}\\approx 4.09567\\times 10^{-6}\\text{watt m}^{-2}\\text{m}^{-1}\\text{kelvin}^{-5}T^5", "ad47ee0e45d8b56464858852dc8eac9b": "-p_{i,j} \\cdot \\log(p_{i,j})", "ad48061ba38dd8ba69b49262f3171c70": "\\langle c(x,y,z)\\rangle = \\frac{q}{2\\pi\\sigma_y\\sigma_z}exp\\bigg[-\\bigg(\\frac{y^2}{\\sigma_y^2}+\\frac{z^2}{\\sigma_z^2}\\bigg)\\bigg]", "ad480cc795dae1fc1cdbc5ba7a007cf9": "M_{ab}", "ad4831f96f0bde69b58af60a65652c43": "a^d\\equiv 1\\pmod n,\\;", "ad4841da276170344d2ede379eb93b6f": " TSS = ||y - \\bar y||_2^2", "ad48d22365e4e53cf7f90180604f974d": "P_N(r)dr = 4 \\pi r^2 dr\\frac{N}{V}\\left(1 - \\frac{4\\pi}{3}r^3/V \\right)^{N - 1} =\n \\frac{3}{a}\\left(\\frac{r}{a}\\right)^2 dr \\left(1 - \\left(\\frac{r}{a}\\right)^3 \\frac{1}{N} \\right)^{N - 1}\\,,", "ad49033c869cc35527544c95a49dfbc7": "\\ \\frac{S}{C}=4\\times10^{-7}\\frac{\\cos\\psi}{R\\tau\\theta}\\frac{\\sigma}{\\sigma^o}", "ad490a5277a5a88208b130b1d79082b7": "\\alpha_1=\\alpha_2=\\alpha_3=\\zeta_1=\\zeta_2=\\zeta_3=\\zeta_4=\\xi=0", "ad49538f5f27c4d91f5b1199662ef1cd": "\\min(c(A,C)-f(A,C), c(C,B)-f(C,B), c(B,D)-f(B,D))=", "ad496ba534a30f2decf2a01eeb69439e": " {d \\Gamma \\over dt } = k c ", "ad49796e010a3624326450a8aa8a1ea9": "fH_i M", "ad49c33fdff36e127c1a44691ee81f56": "|Z_W| \\approx 240\\pi^2 \\frac r{\\lambda} \\approx 2370 \\frac r{\\lambda}", "ad49e22cc8fba64c6120be1d44073929": "kN", "ad49f2a6c1cf5fac056bd2fd2360e2f4": "\\sum_{i=1}^{\\lceil log_2 n \\rceil} \\frac{i-1}{2^i} < \\sum_{i=1}^{\\infty} \\frac{i-1}{2^i} = 1.", "ad4a2096280228a2ee0ea92c059f7c65": "\\textstyle g(x)", "ad4a28cb675ac57203f26b9d95d7b929": " {a_P}^0 = [ \\frac {a_E + a_W} {2} ]", "ad4a338b457b7c2b4369692b5991a6b4": "\\sigma^2/(2\\theta)", "ad4a510ba07f9ef9dc830fe2cace0410": "\\bar{G}_{ij}(s;L)", "ad4a612defe196ee4e1223f878cfbfa1": "\\,1^3 + 5^3 + 3^3 = 153", "ad4a72b21c6b7c4368eb6c1fe706cd02": "P(x,y) \\vee Q(f(x))", "ad4ae98fd686a76314c00947f8c9c346": " S(\\alpha E,\\alpha\\ell,\\alpha n) = \\alpha\\, S(E,\\ell,n)", "ad4b03c6af9897912a3267ba367f014a": " \\sigma_p^2 = w_A^2 \\sigma_A^2 + w_B^2 \\sigma_B^2 + w_C^2 \\sigma_C^2 + 2w_Aw_B \\sigma_{A} \\sigma_{B} \\rho_{AB}\n+ 2w_Aw_C \\sigma_{A} \\sigma_{C} \\rho_{AC} + 2w_Bw_C \\sigma_{B} \\sigma_{C} \\rho_{BC}", "ad4b169dbd19d7bf34ba4a2d3c25886f": "U_{\\texttt{income}}\\,", "ad4b5108068dba1340473fd53207fe94": "\\zeta = \\frac{\\tau^2}{\\hbar^2}\\left(\\langle 0|\\bar{H}^2|0\\rangle - \\langle 0|\\bar{H}|0\\rangle\\langle 0|\\bar{H}|0\\rangle\\right)", "ad4b6c5471f8b1a67c0de8fd571277cd": "W\\stackrel{\\mathrm{def}}{=}b^{32}", "ad4bd9181a87f73a5d260034ed5d86a8": "\\chi _T", "ad4bfd74d423aa646421e4a2ddbd3ecb": " k = 1 + \\log_2( n ) + \\log_2 \\left( 1 + \\frac { |g_1| }{\\sigma_{g_1}} \\right) ", "ad4c5c7bde0b438a33c5b85084799f40": " = \\lambda (3:0:0:1) + (0:0:1:1) \\ ", "ad4c619397177703f7238f85c12409bc": "\\frac{\\lambda}{4\\Delta t}", "ad4c6e814475fba51a68fc87719e7c13": "S\\subseteq N", "ad4c81b1d6a333ab530d71ae7d99d62d": "\\min(3-0,4-0,1-0,6-2,9-2) = ", "ad4c854806c697cef4bb11cf51fc6345": "\\bar{x}_n", "ad4ce88cba71fb2f58846556dc82aa30": "s_n = 1+n\\frac{2\\pi}{\\ln{2}}i, n\\ne0, n \\in Z ", "ad4d5b1e62e7eddf17f0b22792ccce15": "c_{d_2}\\;", "ad4d928511eba0b6a01709511600eb1b": "v, \\ \\mathrm{and} \\ \\bar{v}\\,", "ad4e4da41e3a76db958e1432354e83d0": "\\ G_R(\\tau)=1+\\frac{(Diff_k(\\tau)+Diff_k(\\tau))}{V_{eff, GR}(+)^2}", "ad4e6dc5001d12eee79d23d812cae3b6": "\\mathbb{Z}^\\omega = \\{n\\in \\mathbb{Z} : X_n(\\omega) = 1 \\}", "ad4e9b75234747169bfee26e7fd55188": "t _4", "ad4ea562a1d69320feb29125494083ee": "\\mu \\approx \\frac{1}{\\chi_a /\\mu_a + \\chi_b/\\mu_b},", "ad4f5b7aabcc5b26463af07f884ba9b2": "\\mathrm{d}U+ \\mathrm{d}(pV) = T\\mathrm{d}S-p\\mathrm{d}V+ \\mathrm{d}(pV)", "ad4f99f9da605f334e13c980f82bf683": "\\overline{\\mathbf{x}}", "ad4faa6bc8324bfa702c17d591496b01": " \\|\\Lambda^k(A)\\|_1 \\le \\|A\\|_1^k/k!.", "ad4ff1fb3dfb1c72fbc5bf33a36132ad": "P \\equiv P'", "ad4ff2a0ac1a79357da764a7ffe4fcb0": "(R[x]/fR[x])_g", "ad4ff4d6922084462b0e1ee6a5b7a46e": "R(M, v_\\max) = \\lambda_\\max", "ad5079f4a2cc21223cc9c1e5f3bae6e2": "\\top \\!\\,", "ad507a32e9cf5a30f307483a1358e621": " T_s = {1 \\over f_s}, ", "ad50bc23b6b087ad38d43395624de0d5": "- \\sigma_{yz} - \\sigma_{xz} + \\sigma_{xy}", "ad50f16c69b7f4cd45d87a7fcb2b7d31": " \\Pr(A=1,B=2,C=3) = \\frac{6!}{1! 2! 3!}(0.2^1) (0.3^2) (0.5^3) = 0.135 ", "ad50f43c76d8a2f4ca3216dd125891b6": " H_{B_1} + H_{B_2} \\geq \\log (d) ", "ad50fdae36081c646aa427b66ed724b1": "\n\\mathbf{I} \\cdot \\dot{\\boldsymbol\\omega} + \\boldsymbol\\omega \\times \\left( \\mathbf{I} \\cdot \\boldsymbol\\omega \\right) = \\mathbf{M}.\n", "ad513e79910ad40a1faae467946b77d9": "H(t):=\\int_0^t h(s)\\,ds", "ad51863e92d6b1e5639a8d60154d97e6": "\\mathrm{Re}(s)>n-\\frac 1 2", "ad5210428e6b44310de9a3940d4d2d48": "I=\\int_{-\\infty}^\\infty e^{-x^2/2}\\,dx ", "ad523ee621fc1190c12849076cd1d653": "r= \\frac{M_2 - (M_1 + C_1)}{M_1 + C_1}", "ad52eedecca04b85487286472df90968": "\n\\frac{1}{2m} \\left( \\frac{\\mathrm{d}S_{z}}{\\mathrm{d}z} \\right)^{2} + U_{z}(z) = \\Gamma_{z}\n", "ad5315bb11856d0c2b70e01c0257401d": "L(a) \\neq \\mathit{in}", "ad5317664156fce6977d0cdbe2c38cd3": "\\textstyle\\frac{\\Gamma \\left(\\frac{\\nu+1}{2} \\right)} {\\sqrt{\\nu\\pi}\\,\\Gamma \\left(\\frac{\\nu}{2} \\right)} \\left(1+\\frac{x^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}}\\!", "ad53499de298c9350dc606565241a8af": "\\begin{align}\n{\\mathbf \\Sigma}\n= \n\\left[\n\\begin{array}{cc}\n{\\mathbf \\sigma} & {\\mathbf 0} \\\\\n{\\mathbf 0} & {\\mathbf \\sigma}\n\\end{array}\n\\right]\\,, \\qquad\n{\\mathbf \\alpha}\n= \n\\left[\n\\begin{array}{cc}\n{\\mathbf 0} & {\\mathbf \\sigma} \\\\\n{\\mathbf \\sigma} & {\\mathbf 0}\n\\end{array}\n\\right]\\,, \\qquad\n{\\mathbf I} \n=\n\\left[\n\\begin{array}{cc}\n{\\mathbf 1} & {\\mathbf 0} \\\\\n{\\mathbf 0} & {\\mathbf 1} \n\\end{array}\n\\right]\\,,\n\\end{align}", "ad538edc8939e30cb08f4a27bde8b693": "m\\ddot x = -k x", "ad53ad4bc214c00147bd566fb7ced4fa": "\\overline{q} = t - \\mathbf{v},", "ad53b34aa1e82fafe714e2ec722016c2": "\\{f_1,f_2\\} (gg') = \n\\{f_1 \\circ L_g, f_2 \\circ L_g\\} (g') + \n\\{f_1 \\circ R_{g^\\prime}, f_2 \\circ R_{g'}\\} (g)", "ad53fbaa3c1f22b70c6526c0853c9770": "\nR_1+R_2+R_1+R_3-R_2-R_3 =\n \\frac{R_c(R_a+R_b)}{R_T}\n+ \\frac{R_b(R_a+R_c)}{R_T}\n- \\frac{R_a(R_b+R_c)}{R_T}\n", "ad5402c8d7b7bf5dbf5da276a7226898": "\n f(x)=\\begin{cases}\n \\frac {1}{2 \\sigma \\sqrt{3}} & \\mbox{for }-\\sigma\\sqrt{3} \\le x-\\mu \\le \\sigma\\sqrt{3} \\\\\n 0 & \\text{otherwise}\n \\end{cases}\n", "ad540f804b5f5864f6059655cc739be0": "\\scriptstyle t_A \\;=\\; 3", "ad5460e91803c5f33707818d28c6af50": "e_m", "ad54ae7e13b20a4fcc0662250672e2f3": " R_\\mathrm{out} = \\frac{v_\\mathrm{x}}{i_\\mathrm{x}} \\ . ", "ad54bcd0857b7c5cec3e974ac8eb399f": "U_\\alpha\\,", "ad54ddc1edf9c0a671f43bcbd3a27f92": "R_\\mathrm{p} =\n\\left|\\frac{n_1\\cos\\theta_{\\mathrm{t}}-n_2\\cos\\theta_{\\mathrm{i}}}{n_1\\cos\\theta_{\\mathrm{t}}+n_2\\cos\\theta_{\\mathrm{i}}}\\right|^2\n=\\left|\\frac{n_1\\sqrt{1-\\left(\\frac{n_1}{n_2} \\sin\\theta_{\\mathrm{i}}\\right)^2}-n_2\\cos\\theta_{\\mathrm{i}}}{n_1\\sqrt{1-\\left(\\frac{n_1}{n_2} \\sin\\theta_{\\mathrm{i}}\\right)^2}+n_2\\cos\\theta_{\\mathrm{i}}}\\right|^2", "ad5509b063da1318dca2b62c641b1729": " \\begin{align} F\n& = m \\cdot a \\\\\n& = m \\cdot \\tfrac {dv} {dt} \\\\\n& = \\dot m \\cdot \\Delta v \\\\\n& = \\rho \\cdot S \\cdot v \\cdot \\left ( v_1 - v_2 \\right ) \\\\\n\\end{align} ", "ad5528b8fad973f13e6df06b144d82f4": "a^n x \\in I_{n+1}", "ad553c00368268ebe2a091f5888355a5": "Z_1 X_2 Z_3 = \\begin{bmatrix}\n c_1 c_3 - c_2 s_1 s_3 & - c_1 s_3 - c_2 c_3 s_1 & s_1 s_2 \\\\\n c_3 s_1 + c_1 c_2 s_3 & c_1 c_2 c_3 - s_1 s_3 & - c_1 s_2 \\\\\n s_2 s_3 & c_3 s_2 & c_2 \n\\end{bmatrix}", "ad55413daba2256bc536432db417e402": "x_{i+1}\\equiv(ax_i^{-1}+ c) \\mod q", "ad554e3174b407e63a3fe5760b190f1c": "\\rho(g)^\\dagger \\rho(g)=\\mathbf{1}\\,", "ad5555b18c7b9f584a51b89a5dc580de": "\\left( \\begin{matrix}\n 1 & j/(n_{r}-jk_{r}) \\\\\n (n_{r}-jk_{r}) & 1 \\\\\n\\end{matrix} \\right)", "ad555e03b17a35ba07e9d41b64d4665d": "\\alpha={ m_{1}^2 \n\\over m_{2} - m_{1}^2}\\,\\!", "ad557178d10274c6b771bdfd146c6c90": "\\varphi (tz)=t\\left( \\varphi (z)+\\varphi (z)^2 \\right)", "ad55a52f243db81ffcb9fbeb2f4d4946": "0 \\leq n \\leq N", "ad55e95e3dbc2406866e1ddccb0b06ec": "A=\\{2n; n\\in\\mathbb{N}\\}", "ad55ee26b2ecc9197be1c81caf00c1d2": "Q_b = C_b (h_b - h)\\,", "ad5636c3fe8d096fdc19fce198e9d821": "\n\\sigma_{j} = p + \\lambda_{j}\\frac{\\partial W}{\\partial \\lambda_{j}}\n", "ad563cc3b3c5d1c0c1c26e7cfbfbd186": "\\gamma(t) = \\exp_p(tv)", "ad56608667c0a0a7fd9a5d2be6a83490": "H(s^\\prime)", "ad56bca08316afa498ad49141d51a970": "d=\\lambda x^{'}.f(x_0,x^{'})", "ad571adc811602e350f5cd64ef6e0522": "f \\colon S \\to T", "ad577afc4259ab7d2b67693b2d3d5dfc": "J \\colon BO \\to BG", "ad578e3e81f63212dfa12755241696d6": " x \\rightarrow 0", "ad57c804a01be18168e02657dc36656a": " Y_i \\,\\!", "ad57d28c0386e08a7d260c949537af6b": "q^\\mu", "ad57dcebee1ecd110f625400861d86a7": "x^2\\in X", "ad57f57ba334b9a31e0433c5b43dfd02": "[a\\otimes t^n+\\alpha c, b\\otimes t^m+\\beta c]=[a,b]\\otimes t^{n+m}+\\langle a|b\\rangle n\\delta_{m+n,0}c", "ad5860354998138e659bd43c1822b4d7": "a_5\\times \\rho^2 \\sin(2\\theta)", "ad58a7c79694a13d0f8d21b73c75872e": "t\\mapsto s", "ad592f64ab01a875375f4b7cc4cacc6e": "J_1^2|j_1m_1\\rangle=j_1(j_1+1)\\hbar^2|j_1m_1\\rangle", "ad597cbdaa96fead3647a52f1d6ee0d1": "\\mathbb C\\otimes\\mathbb O", "ad59b6f2b29abccf7f6b2b84223b0f7f": "\\scriptstyle b=\\partial v/\\partial x=-\\partial u/\\partial y", "ad59e228669694545478e073e08be9d3": "\\vert \\pi^-\\rangle = -\\vert d\\overline {u}\\rangle", "ad59efcd17b79f680b093bcf8caaa363": "ref2", "ad5a4445c3dbbd29e067c23db8f4ce98": "\\{(x,y) \\in \\mathbb{R}^2 \\mid x \\geq 0\\}", "ad5aa69c24d70866599786d06eca6099": "A_\\lambda = C l \\sigma", "ad5aaffe99b2ab9969a75a4dd3b4fe5e": "X_{1..i}", "ad5ae29c09fe08589d89394ace3c2e48": "w_i:X \\longrightarrow \\mathbb{R}", "ad5be5d982221a642ececd360e1981d6": "L_q(x)=L_q^{(0)}(x)=M(-q,1,x)=\\,_1F_1(-q;1;x)", "ad5c10e68c0fc2c2deeebb212c343fc6": "f(n)=2 \\cdot l+1", "ad5c570d43be51d10d689537ede3b94c": "X^2-X-1", "ad5cc1063da84a743e37b64bc3da16f0": "\\mathbf{x}, \\mathbf{y} \\in \\{0,1\\}^n", "ad5ce05406c378852c3817a5f319dc81": "W_X", "ad5d20a25ef06c2c944e6053a2594291": "x \\equiv \\mathbf{e}_1", "ad5dc6f54ef13a220a3770f2fce678dd": "\\partial\\!\\!\\!/ := \\gamma^\\mu \\partial_\\mu", "ad5e03fc16e83e21983542a4d8358e89": "R_x = {{R_3 \\cdot R_2}\\over{R_1}}", "ad5e09f9f7012458cf5c115ae17b7f7c": " f(z)=\\sum_{n=0}^\\infty a_n z^n,", "ad5e7b75605e2609e7b9190d66aa1910": "dx_1 dx_2 dx_3", "ad5f88e9833a575ad05979c522f7f294": " G_2 \\ne G_3", "ad5f8abe2813ff64100f8bb4345fa1b6": " H[(X_{j})_{j\\in S_{i}}]\\leq \\log |P_{i}(A)|", "ad5facd8b6be80a469d8961337da11aa": "F=F_t=0", "ad5faeb470b5752859a80beaee3f1419": "(a_1,a_2,...,a_n)", "ad5fbf513727e5935d7d5c3edfa369e7": "t \\rightarrow t^{\\prime} = t + \\delta t", "ad5fbf78768011dcbfd6c2bb0bb99e51": "\\sup\\limits_n \\{a_n\\}", "ad5fdfad36781cd4265f3f84f5719387": " B_n = \\{x \\in B: f_n(x) \\geq 1 - \\epsilon \\}. \\, ", "ad5fe0c82baaab600a40af8b84a28504": " f(x) = x^3 - x\\,", "ad5ff36592a8abbe847d4749c802c08f": "N_A^{2} \\sigma_{AB} \\sqrt \\frac{8 k_B T}{\\pi \\mu_{AB}}[A][B] =N_A^{2} r^{2}_{AB} \\sqrt \\frac{8 \\pi k_B T}{ \\mu_{AB}}[A][B] = Z [A][B] ", "ad604e2e208707287c3872fb82c33cbb": "\\textstyle {4\\choose 1,3,0} \\ {4\\choose 0,3,1}", "ad60b46ae2e65d6f3278c5ad6200549f": " \\Gamma,\\mathcal A,\\Delta\\vdash\\mathcal C", "ad60bd9104365362629dc3f43255ff14": "\\Lambda_{Roy} = max_i(\\lambda_i)", "ad60bf2b72792dafeb74159fd00e439c": "\\!\\gamma", "ad60d3fa290e2b1e25c954a256dbeaff": " |{\\rm det}(I+A) -{\\rm det}(I+B)| \\le \\|A-B\\|_1 \\exp (\\|A\\|_1 + \\|B\\|_1 +1).", "ad60e6c568fc2a34243cb02bcf58d6f0": " E_{-} 0 ", "ad6199f59028e9d706ea249e7aad4009": "P_b", "ad619b85f2382ae999c4dbb125b659b5": "\\operatorname{gcd}(l',m') = 1", "ad61c25e4bcf37395527c9f30bcfbb84": "\\ E\\{|y_v|^2\\} = E\\{ {y_v}^\\mathrm{H} y_v \\} = E\\{ h^\\mathrm{H} v v^\\mathrm{H} h \\} = h^\\mathrm{H} R_v h.\\,", "ad6244bf99d42febb9a8eb0c7d3ae026": "\\lambda_1+\\lambda_2+\\cdots+\\lambda_r=1", "ad6295bcf1a4d0952d24463e4e2ebd12": "x^2 - y^2 = (x-y)(x+y)", "ad63029c174f4c20559e54cd21c04d00": "y_c=\\sqrt[3]{{(10 ft^2/s)}^2 \\over 32.2 ft/s^2}=1.459 ft", "ad63b32b466634c43f089cbecdfc2bb8": " \\phi(x,k) ", "ad63cf99fc47a0307b6014f764b7126a": "0 \\le \\rho \\le 3a, \\ 0 \\le \\phi \\le 2 \\pi, \\ - \\sqrt{9a^2 - \\rho^2} \\le z \\le \\sqrt{9a^2 - \\rho^2};", "ad63e9c5f5dbc24d3ffcf6dd1e0381e5": "(\\rho-\\sigma)", "ad649d660aa5d76f3e49424cbb12a372": " b + d = 180^\\circ ", "ad64a43e84e30052d77b96fc840f0382": "t=\\frac{1}{1+0.5\\,|x|}", "ad64d564157a892f421976175ad43024": "E_s = K_s(2\\alpha)^2/2 ", "ad6516fbed5fd3ea07272550bafc0962": "(A\\ominus B)\\ominus C = A\\ominus (B\\oplus C)", "ad6519cb890519c550aec41ac301d3b5": "\\tilde{K}^\\prime_\\pm", "ad652d024e1ae1e0ff5a97c8824ca670": "M - M'", "ad65ba39550fd610d1e725bb1264487d": "\\hat\\beta_2=0.556", "ad65d7f50297cfc1cd186cf93f5bc957": "\\langle G,S\\rangle", "ad65d8cf86052db9349ecbbd1ddbb40b": "p(\\mathbf{d}|\\mathbf{x})", "ad66690040bf92bb1f56337579c3c25e": "O(n k^2\\,\\log(n)^2)", "ad667ec54ba3a3cf83c5ca6027dc5833": "x(1-t)+yt-t(1-t)=t^2+(-x+y-1)t+x=0.\\,", "ad667f65f003506f738a6bb75dea8a91": "\\frac{3x + 5}{(1-2x)^2} = \\frac{13}{2(1-2x)^2} - \\frac{3}{2(1-2x)}.", "ad67288ee110d5677553e7aa937800a8": " a=1,2,3 ;~ \\mu,\\nu=1,2,3,4 ;~ \\epsilon_{1 2 3 4}=+1", "ad674c02acc3ffa1e81a4b4eda25fa7d": " \\theta_0 = \\theta_1 + \\theta_2. \\qquad \\qquad (2) ", "ad6755f92b291b930be47d62e57c2db2": "HJD = JD - \\frac{r}{c} \\cdot [sin(\\delta) \\cdot sin(\\delta_{\\odot}) + cos(\\delta) \\cdot cos(\\delta_{\\odot}) \\cdot cos(\\alpha - \\alpha_{\\odot})]", "ad679838da6e57e583716082d9857c88": "L(t)=const", "ad67b3237dcfde19e989725eea76f3ab": "\\hat v_n(x)-K_n(x)=w_n(x)", "ad680677b511c22504d795be77ad8cad": "\\tfrac{N}{df_{t}} ", "ad68349a4d4d8befb2220c3b03b72cc4": "\\mathfrak{e}_6", "ad68356789b262009fda8a5c75f17213": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = -\\left\\{\n \\int_{-h}^h x_3^2~\\begin{bmatrix} C_{11} & C_{12} & 0 \\\\ C_{12} & C_{22} & 0 \\\\\n 0 & 0 & C_{66} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} \\varphi_{1,1} \\\\ \\varphi_{2,2} \\\\ \\frac{1}{2}~(\\varphi_{1,2}+\\varphi_{2,1}) \\end{bmatrix}\n", "ad68893c98b9a98c1f70590a87d16594": "\\sqrt{S} = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3+\\,\\ddots}}} ", "ad6889936aefc25d83027fde133b7ed6": "\\kappa = 8/3", "ad68ebaa2443e34d5c6129353baff7b3": "(A,B,\\Lambda)", "ad690b95a5ff3509f1685d13a5c49c9b": "\\beta_n=\\sum_{r=0}^n\\frac{\\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}", "ad69f527f56cae6fcd050269ac052562": "\\psi(p) = (x, s(x)^{-1}\\circ p)", "ad6a07a012811f8076cf2972af868a64": "\n\\mathcal{J}_3 \\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* =\n m' \\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* ,\n", "ad6a2a44d80265b1ee318558deb5810d": "a = \\frac {p}{1-e^2}", "ad6a84d985e4ec0bf2218d63e52594b6": "\\eta(A,B)=2\\frac{ \\left(\\frac{m_g}{m_i}\\right)_A-\\left(\\frac{m_g}{m_i}\\right)_B }{\\left(\\frac{m_g}{m_i}\\right)_A+\\left(\\frac{m_g}{m_i}\\right)_B}", "ad6a8ad9f91f41df12f30dfddb71d037": "\\lim_{\\lambda \\downarrow 0} \\{ \\lambda^{-1}(z - x) : z \\in E \\} \\to \\{ y \\in \\mathbb{R}^n : y \\cdot \\nu_E(x) > 0 \\}", "ad6ab9f9f04b4da29a9f8072d2558e75": "\n \\sqrt{J_2} = A + B~I_1 + C~I_1^2\n ", "ad6ac1d0686371dd79276fc9e345f497": "F(x\\mid\\mu,\\kappa)=\\Phi(x\\mid\\mu,\\kappa)-\\Phi(x_0\\mid\\mu,\\kappa).\\,", "ad6adeeac19ce5971620ba6f46c877c3": "\\ln y_i = \\beta _0 + \\sum\\limits_n {\\beta _n \\ln x_{ni} + v_i - u_i } ", "ad6b8e0a861a15e2ebc1977fe7f0edf2": "x \\wedge \\bigvee S = \\bigvee \\{\\, x \\wedge s : s \\in S \\,\\}, ", "ad6b929c0e48a4926c29f764d51a39ac": "I_{\\text{C}} = C \\frac{dV_{\\text{c}}}{dt}", "ad6be4ed9f70c8627ed9794a08a841a8": "\n\\sum_{i=1}^3 \\varepsilon_{ijk}\\varepsilon_{imn} = \\delta_{jm}\\delta_{kn} - \\delta_{jn}\\delta_{km}\n", "ad6d0ece2ebb1159cb086b2d15570c4d": "\nJ = \n\\begin{pmatrix}\n-\\mu (-1+y^2) & -2 \\mu y x -1 \\\\\n1 & 0\n\\end{pmatrix}.\n", "ad6d4c05cf4e0ee84d8570429d66401f": "\\begin{matrix}\nx' &=& 2x & - & y & + & z & + & e^{2t} \\\\\ny' &=& & & 3y& - & z & \\\\\nz' &=& 2x & + & y & + & 3z & + & e^{2t} \\end{matrix} ~.", "ad6d590029b59ad659187f122d06ff03": "Y_{3}^{0}(\\theta,\\varphi)\n={1\\over 4}\\sqrt{7\\over \\pi}\\cdot(5\\cos^{3}\\theta-3\\cos\\theta)\\quad\n={1\\over 4}\\sqrt{7\\over \\pi}\\cdot{z(2z^2 - 3x^2 - 3y^2)\\over r^{3}}", "ad6d7c4c92165daac777260bc2f2454a": "\\mathbb T^n", "ad6d8207a2d4bbd54ed8430c63b89739": " T^*_pM ", "ad6dc7b587e2e07454a8168b8c898fe5": "x ^ 3\\,", "ad6dff7a6d865a7a6f46a4e0fdbf7a29": " v(f) = v_x(f(x)) ", "ad6e0610c8dc5c9020e95e363295b765": "\n p_0=\\left(\\frac{E^*F}{\\pi LR}\\right)^{1/2}\n", "ad6e1c467d23be2ae8c8babebc165474": "p_2 = x_1^2 + x_2^2\\,.", "ad6e61ee6fab683e6a9b04c37b937550": "\\displaystyle{u_n=\\partial_{\\overline{z}} f_n,\\,\\, v_n =\\partial_z f_n-1.}", "ad6e7d5c93fd71e6cfbd6372e2a034e0": " \\langle Ax , y \\rangle = \\langle x , A^* y \\rangle \\quad \\mbox{for all } x,y\\in H.", "ad6e8679007e44afc0b7344035b95bd5": "\\Psi\\colon G \\to \\Bbb{R}^+", "ad6e9ad31c7368b32aece5b0d777fbb0": "\\omega=360\\sin\\varphi\\ ^\\circ/\\mathrm{day}", "ad6eaa708d62482fa2ec773ae27e4773": "R_n(\\xi,-x)=R_n(\\xi,x)\\,", "ad6ee5ed441b3c69270490bd3b845890": "M_1 (\\vec X) = \\left( {\\begin{array}{*{20}c}\n {\\bar \\mu _1 } \\\\\n {\\bar \\Sigma _1 } \\\\\n\\end{array}} \\right)\n", "ad6f12e5f235d352e2a30346af6b6e09": "r=a\\frac{\\sin(2\\theta+\\alpha)}{\\sin(\\theta)}", "ad6f5f087f641514eda23accbeba7b85": " IR_{P}(t) = \\dfrac{ 15 \\ (C_2 + C_3) \\text{k}_{3(2)} } { 29 \\ C_1 \\text{k}_{3(1)} + 14 \\ (C_2 + C_3) \\text{k}_{3(2)} }, ", "ad6f6a1deb0bb98b644eb8d32c879421": "iJ_z= \\left[ \\begin{matrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\end{matrix} \\right] ~,", "ad6f6a454a160f940753dcc8165862f2": "\\exist c\\in K: \\mathbf{F}' = c\\mathbf{F}\\text{ (or }\\mathbf{F} = \\frac{1}{c}\\mathbf{F}'\\text{)}", "ad6fadf545aa6ee3da3bea28a7b3726b": " \\mathbf{F} = \\int_V \\mathbf{f}(\\mathbf{r}) = \\int_V\\rho(\\mathbf{r})dV( -g\\vec{k}) = -Mg\\vec{k},", "ad7004f4149e5671d429bc6809fdb90f": "\\frac{1}{d}E(\\alpha)", "ad70063cb71cd9c1596273f830db3954": "D_8", "ad700e485af62ac661d62316f3e04798": "i(jj) = i", "ad70146b431bea9ae74cf8385470c544": "\\mathcal{A}", "ad70725dc49985d7734944f7192e1798": "\\theta(A)", "ad707c98b04fccd8206026b17f495017": "\\operatorname{rank}(AB) = \\operatorname{rank}(A).", "ad70c19648578c6a324927c34ab5959c": "m_1=\\left\\lfloor m/2 \\right\\rfloor", "ad70d8c9174685dc1d03c172d4b4b0fa": "u_{n}= A e^{i q l a - \\omega t}", "ad718443a481232d252fc4c10120ba40": "\\mathbf{G} = \\begin{bmatrix}\n\\sqrt{1-\\rho^2} & 0 & 0 & \\cdots & 0 \\\\\n-\\rho & 1 & 0 & \\cdots & 0 \\\\\n0 & -\\rho & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 1\n\\end{bmatrix}.", "ad71c3ab6129e0e91556e763a5f66c0b": "\\mathbf{\\pi }\\in \\Pi ,u:\\mathbb{R} \\rightarrow \\mathbb{R} ,", "ad72a968ff2ceb1895676ce910084a52": "\\Delta{v_i}\\,", "ad72caf3a28cde539a31670f69d8f66c": "C= B \\log_2(1+\\mathrm{SINR})", "ad72cdcb280c8ae94d5f26a89554cba5": " \\Delta \\rho", "ad72d874e0c18b3428be73e987a32c1e": " \\psi \\in C^q(X;R)", "ad731eb8df7a48b3f35ed7d33dfc717c": "X \\overset{\\underset{\\mathrm{A}}{}}{\\sim} Y", "ad732c257c2d0fb76ae62e77152f17dc": "{d^2\\theta\\over dt^2}+{g\\over \\ell}\\theta=0.", "ad733ded130f28c8c7a86cb3b3d5e00b": "t_0 - D_{heel} \\times drag \\times sin(\\beta) ", "ad7cb0c735dde7c78d7b9867a292dd49": "F_{p,m-p+1}", "ad7ce6fc51be0f1637e9f9229c54a192": "\\Omega^5", "ad7cf7e0d149c60d5cd6eb9b7207e263": "2~\\ln r + 3", "ad7d50cce14de74ca882fb1785c888ba": "|n,+\\rangle= \\cos \\left(\\frac{\\alpha_n}{2}\\right)|\\psi_{1n}\\rangle+\\sin \\left(\\frac{\\alpha_n}{2}\\right)|\\psi_{2n}\\rangle", "ad7d70c7bf9d1fc6c79af907f50a17ee": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}, \\frac{\\partial \\mathbf{v}}{\\partial \\mathbf{x}}", "ad7d8526d7ed3f490c504b4489861268": "R_1,\\ldots,R_n", "ad7db129cc446657f7b8cc4400a2b906": " \\mathbf{e} \\cdot \\tilde{\\mathbf{y}}_{k} = 0 ", "ad7dc384f6b80223e9d8b1e46bbd2822": "\\frac{d\\nu}{dJ}", "ad7dc6fece1a74c91ddec09ea917d51a": "\n\\frac{d^m P_\\ell(u)}{du^m} =\n\\sum_{k=0}^{\\left \\lfloor (\\ell-m)/2\\right \\rfloor} \\gamma^{(m)}_{\\ell k}\\; u^{\\ell-2k-m}\n", "ad7e8b2e2c7f85b3c461818ce09afaaa": "{q} = \\left[\\begin{array}{c}{c}\\\\-{b}\\end{array}\\right]\\,", "ad7ea74375117535c1e209c92ddec28d": " 2^{nR} ", "ad7eaf311c2df00f005b328fcc68ce91": "c_2 = \\sqrt 2", "ad7ec4089cbd13bffa20447cd7ff529a": "\\left\\| {{\\mathbf{x}} - {\\mathbf{w}}_i } \\right\\|", "ad7f03e8f87e3c7454dfeb313fe3f15d": "\\scriptstyle f_2=8.717 \\mathrm {\\ Hz}", "ad7f06d2ea42a1eadd454f208d6ea1cc": "\n \\eta_t(x)\\to 1-\\eta_t(x)\\quad\\text{at rate}\\quad (2d)^{-1}|\\{y:|y-x|=1,\\eta_t(y)\\neq\\eta_t(x)\\}|\n", "ad7f11161540968229f08aa18cf8a171": "!n = \\left\\lfloor\\frac{n!}{e}+\\frac{1}{2}\\right\\rfloor , \\quad n\\geq 1,", "ad7f1825cba67c0cd0f1a5a40353081f": "\\Delta U = Q + W\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{(sign\\,convention\\,of\\,IUPAC)}\\, .", "ad7f4c853d32c94e94029403fda6da08": "I[f] = \\mathbb{E}_z V(f,z)", "ad7f9ba3677e23b433b4b1c3a71af1b7": "\n\\begin{align}\n\\Pr(Y_r \\leq s) & {} = 1 - I_p(s+1, r) \\\\\n& {} = 1 - I_{p}((s+r)-(r-1), (r-1)+1) \\\\\n& {} = 1 - \\Pr(B_{s+r} \\leq r-1) \\\\\n& {} = \\Pr(B_{s+r} \\geq r) \\\\\n& {} = \\Pr(\\text{after } s+r \\text{ trials, there are at least } r \\text{ successes}).\n\\end{align}\n", "ad7fd9b76a65c4b488fd3946e6472b33": "\\sqrt{S} = e^{\\frac{1}{2}\\ln S}.", "ad7fdcd8cb73271ca7bcab45c5637b8e": "x = (x_1, ..., x_n) \\in V", "ad7fe062d6999dee8db781d35205759d": "I = \\frac{c n \\epsilon_0}{2} |E|^2", "ad8060cc0cd025ba497fab99a857442e": " E_\\nu = \\hbar\\omega, ", "ad8064d57c4cefca94f9e89b26c72b0d": "A \\cap C \\leq B \\cap C", "ad80916213002e0467082ac4e1144079": "\\frac{dT}{dx} \\Bigg|_{x=0}=\\frac {\\left(T_L-T_o \\right)A}{e^\\left(AL \\right)-1}", "ad814a55d8d1630d80ac91f02434ce56": "\\sigma = \\langle \\psi |\\cdot \\; \\psi\\rangle", "ad8167152916ac99a39a41e96bdd252a": "M_i = \\sum_{j=1}^i{m_j}.", "ad816e2afd0a0e2cfd8ad1161cff155c": " \\frac{2n}{3} < p \\leq n ", "ad817b973532f01114c86311c3915ed8": "\\xi \\dot \\eta- \\eta \\dot \\xi = h \\cos I ", "ad81a32a4f3750dbe5d92e572fd84446": "A[1] \\oplus B", "ad81e390cda750b7120772848d565022": "\\mathcal{L}_2", "ad8210694a2c702f72b4363e8b0f1cdf": "p\\ n = (n = 0)", "ad82234919ebb82af263a8eee1302900": "\\scriptstyle \\phi ( \\boldsymbol{r}_{\\text{rec}},\\, t_{\\text{rec}} ) \\;=\\; \\sum_{i=1}^n ( \\delta t_{\\text{meas-err},i} / \\sigma_{\\delta t_{\\text{meas-err},i} } )^2 ", "ad827221dbb80be2d7eae35d14161973": "f:\\Omega\\rightarrow\\mathbb{F}", "ad82867a335bfb20f09d87367eec0c62": "(t,y,x).", "ad833bb638bd8d54bdd577c84188da02": "x\\in\\{1,\\dots,n\\}", "ad8382e116b662ffddd378e29566ce84": "B = \\rho_f V g. \\,", "ad83991512c7d5c46c961d8ace1282ef": "I_n := \\inf \\{ X_m : m \\in \\{n, n+1, n+2, \\ldots\\}\\} = \\bigcap_{m=n}^{\\infty} X_m = X_n \\cap X_{n+1} \\cap X_{n+2} \\cap \\cdots.", "ad83a43b5ccb1afe507a059e2e480d8f": "\n\\tan \\beta = \\frac{\\sin \\eta }{\\tan (\\lambda + \\chi)}\n", "ad83f1d01119d7fc6346cec33919e3d4": "=(c_1\\hat {H}|\\psi_1\\rangle+c_2 \\hat{H}|\\psi_2\\rangle)", "ad8434a71e5976298bf148df2422290c": "e^{4\\sigma^2}\\!\\! + 2e^{3\\sigma^2}\\!\\! + 3e^{2\\sigma^2}\\!\\! - 6", "ad846f58ce3010beb42b2fe5bcafd04b": "\\nabla \\cdot \\boldsymbol{\\sigma} = -\\nabla p + \\nabla \\cdot\\mathbb{T}.", "ad84c19a3676ce6b3cf6fe939b316c52": "\\mathcal{Z}(M/\\mathcal{Z}(M))=\\{0\\}\\,", "ad84e5790a081b41df48500ad72da5cd": "\\Box = (-\\star \\mathrm{d} * \\mathrm{d} + \\mathrm{d} \\star \\mathrm{d} \\star) ", "ad8519531ffba3e45853e28b38c5f690": "{x}_{k},{x}_{k-1},\\dots,{x}_{0}", "ad85338df5a2b203b8b2cb3da5065537": "t_1 ", "ad8538da626958c27e41dc42259253bf": "\n \\begin{bmatrix}\n 4 & 3 \\\\\n 6 & 3 \\\\\n \\end{bmatrix} =\n \\begin{bmatrix}\n 1 & 0 \\\\\n 1.5 & 1 \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n 4 & 3 \\\\\n 0 & -1.5 \\\\\n \\end{bmatrix}.\n", "ad85469447dc367b08bc400eda4ce296": "\\ln \\left( {\\frac{{K_2 }}{{K_1 }}} \\right) = \\frac{{\\ - \\Delta H^\\ominus }}{R}\\left( {\\frac{1}{{T_2 }} - \\frac{1}{{T_1 }}} \\right).", "ad85dae0eab1d37dbe532543b5f4c1e8": "\\displaystyle{\\partial_z k(z,w) =h_z(z,F) - ah_w=a_z(z,w).}", "ad85dce51f6ef99ff56d56ec5c4767e9": "\\phi:U\\to\\mathbb{D}", "ad85f734b57b4e682ff22e62ba0d6f8b": "(V^*)^\\mathbb{C} = (V^*)^{+}\\oplus (V^*)^-", "ad8627d4e5d9547cd51c8d35983b1c86": "\\lambda x\\!:\\!\\tau\\to\\tau'\\to\\tau''.\\lambda y\\!:\\!\\tau\\to\\tau'.\\lambda z\\!:\\!\\tau.x z (y z) : (\\tau\\to\\tau'\\to\\tau'')\\to(\\tau\\to\\tau')\\to\\tau\\to\\tau''", "ad86c5b2671ea9c784349e16fb9de4c8": "c_{6}=c_{7}", "ad86ca3df65cd5e4061accfab68c6c04": "\\delta = \\frac{A}{L},", "ad86fe8ce98fe58cb56563c082f890c8": "F_d", "ad88306482b49f8d576e81000697d598": "dS = \\frac{dE+PdV}{T}.", "ad889fd2ffee23bf9b0d1cb75222eb6d": "c^n", "ad88baf9704628dc3e9fbc7a4aee72f1": "\\frac{dy}{dx}=\\tan \\varphi = \\frac{1}{T_0} \\int w\\ ds.\\,", "ad88d63a9374028527e521ebff6fe754": " \\lim_k \\int f_k \\, d\\mu \\geq \\mu(A). ", "ad88f427f0a62d87c4385c998c954f89": " w_A = \\begin{pmatrix}i_1 & i_2 & \\ldots & i_m\\\\j_1 & j_2 & \\ldots & j_m\\end{pmatrix}", "ad8925014e5625378fd2535a9ca5f846": "\n RR_\\mathrm{total}={e V^2 \\frac{\\gamma -1}{\\gamma2R_{sp} }}\n", "ad893ecf1e4b813cca39a2fa7600df96": "f^{-1}(y) = \\sqrt{y} . ", "ad8943555e8bdcf486599e0768ebca70": " \\delta = 1 ", "ad894a5643600c69cbe6c1ce8526a026": "x=XDL(a)", "ad8a0c47ffe493d6e882da85218b803f": "\\vec J", "ad8a19160ac3be779751062c70fd7bbd": "\\frac{3n-m}{m-n}", "ad8a29f055d9509b011b33961772760d": "\n\\mathbf{x}_2 = \\mathbf{x}_1 + \\alpha_1 \\mathbf{p}_1 = \n\\begin{bmatrix} 0.2356 \\\\ 0.3384 \\end{bmatrix} + 0.4122 \\begin{bmatrix} -0.3511 \\\\ 0.7229 \\end{bmatrix} = \\begin{bmatrix} 0.0909 \\\\ 0.6364 \\end{bmatrix}.\n", "ad8a2e615af3f9e5748bdb2869c3a236": "\\beta = \\frac{\\partial f}{\\partial y} = \\frac{1}{a} \\frac{d}{d\\phi} (2 \\omega sin\\phi) = \\frac{2\\omega cos\\phi}{a} ", "ad8a966e68d46f58a0516a7e41bc5b55": "\\frac{(7+2+1+11+6+5+3+4+7+6+10+7+3+11+4)^2}{15} ", "ad8adbbd9eb357770a546733be54f92d": "P \\to (Q \\to R)", "ad8b485cf8d154a334d60bb22ff053d2": "\\gamma=1+\\frac{lnM}{ln(M-1)} ", "ad8b761eae8df248b8effbb1f3b55f53": "(-1)^{r(2)}2q \\cdot (-1)^{r(4)}4q \\cdot \\cdots \\cdot (-1)^{r(p-1)}(p-1)q \\equiv 2 \\cdot 4 \\cdot \\cdots \\cdot (p-1)\\text{ (mod }p).", "ad8bb33a66232186e8e5917a563b2728": "p_i \\left(\\left\\{x_n\\right\\}_n\\right) = \\left|x_i\\right|,", "ad8bb5dd10b7d7ef72d5f852c019028b": "3.2 U_p", "ad8bff56ac03d369f48afbe4936a0a37": " x(t) = x(t + T)\\text{ for all }t \\in \\mathbb{R} ", "ad8c04d4b5dcacbda183de3a114b9244": "a=4+\\alpha", "ad8c245b1489baa088261d07f2dbaf20": "\\frac{dx}{dt}=3x-4y,\\quad\\frac{dy}{dt}=4x-7y.", "ad8c48cbc31907fc833742341efd31e2": "x_1^2=(x_0+1)^2", "ad8c8c21f9cda9eb8779a76341034e41": "\\left( \\frac{2}{3} \\right) ^5 \\times 2^3", "ad8ca40bca5e6a141b9a20fe2250ea50": "\\sum_i \\left | c_i \\right | ^2 = 1.", "ad8cecf6b57a336df34fac21bd3a0a29": "\\beta_n - \\delta_i", "ad8cffa43e54dc4bd90353f1d96591c2": " P(s) = a_2s^2 + a_1s + a_0 = 0 ", "ad8d7ea9f32adcc641d37a3374441f8e": "(D_-, D_0, D_+)", "ad8daec4aef90d732520b061a4a76e9d": "f: M \\to N", "ad8dba3914a17d702ac75870494d8e5e": " L = 2R + S + K, \\!", "ad8e1409cae6c331678964516e9eaf66": "\\mbox {Ein}(z) = \\mbox{E}_1(z) + \\gamma + \\ln z = \\Gamma (0,z) + \\gamma + \\ln z\\,", "ad8e2967c65e9115ae529474ef88110f": "F \\sim \\frac{2\\pi\\mu\\ell u}{\\ln(\\ell/a)}", "ad8e2b8d7fdb9e2c2282f8f6e7c8c100": "\\scriptstyle y \\;=\\; -1/x", "ad8e4a6e719f8a02023764487c67c3f7": "\\mathbb{E}\\left[g\\left(U\\right)\\right]=\\int_0^1 (1+x) \\, \\mathrm{d}x=\\frac{3}{2} ", "ad8e93065d50911a34b4f51d080af16c": "(\\forall) \\frac{\\forall x . \\gamma(x)}{\\gamma(t)}", "ad8ecaf87d655b7f21e6cea3aa1d7600": " k^2 \\gg k_x ^2 + k_y ^2 ", "ad8efa294f5346386e9353264913637b": "\\log\\left|\\psi(t)\\right|", "ad8f2be12384994c3f16f68abe42f911": "\\theta = \\arcsin x ", "ad8f4e56cfa3ec82bdf45608855912fd": "\\log |f(z_0)| = \\sum_{k=1}^n \\log \\left|\\frac{z_0-a_k}{1-\\bar {a}_k z_0} \\right| + \\frac{1}{2\\pi} \\int_0^{2\\pi} P_{r_0}(\\varphi_0-\\theta) \\log |f(e^{i\\theta})| \\, d\\theta.", "ad8f546413922e983aebdd50a93974d9": "\n\\frac{\\text{Annual coupon payments}}{\\text{Clean price}}*100 + \\frac{ (100- \\text{Clean price})/ \\text{Years to maturity} }{\\text{Clean price}}*100.\n", "ad8f68739cb54ede0a0fe501cd78c79a": "\n \\left|A\\right|_{ij} = a_{ij} - \\sum_{l\\neq j} a_{il}\\cdot |A^{ij}|_{kl}^{\\,-1} |A^{il}|_{kj}\n", "ad8f993f4b8c7b52b0e83ea7d91f2edb": "\\Pi \\left(t, f\\right)", "ad8fba0005106869d75f70d40e268415": "\\Pr(A \\and B) \\leq \\Pr(B).", "ad8fd79102f9c0a54cd544c83819e18e": "\\ 1/H(f)", "ad8fdd73974c7abe4e1daadf6743a77b": "u_{\\text{obs}}^{t} = c \\sqrt{\\frac{-1}{g_{t t}}} \\,", "ad8fe0d1f87a9561f4a1e70223d89b32": " \\begin{align} \na_1 + b_1 & = & c_1 \\\\\na_2 + b_2 & = & c_2 \\\\\n & \\vdots & \\\\\na_n + b_n & = & c_n \\\\\n\\end{align}", "ad8ff00578a71ab0daa1fd3e5ff3a1a7": " \\frac{\\mathit P}{\\mathit E} = \\frac{\\mathit ROE - \\mathit g}{\\mathit ROE (\\mathit T - \\mathit g)}\n", "ad9044aa3a4721093d716d8fa4efb893": "\\begin{align}\n {[} A {]} \\{\\Delta X_1\\} &= \\{B_1\\} \\\\\n {[} A {]} \\{\\Delta X_2\\} &= \\{B_2\\}\n\\end{align}", "ad90944800a6ad8153aa5d68c4262394": "e^{\\pi\\sqrt{43}}=12^3(9^2-1)^3+744-2.225\\cdots\\times 10^{-4}\\,", "ad909a6847f720b858df087458cc84ea": "\n \\lim_{n\\to\\infty}X_n(\\omega) = X(\\omega) \\quad\\Rightarrow\\quad \\lim_{n\\to\\infty}g(X_n(\\omega)) = g(X(\\omega))\n ", "ad90df3e4257ddf62c326332eb2838da": "\\frac{\\pi^2}{6ln(2)} ", "ad9124155b3b4c2ec8f22732a155d33c": "B = \\mathrm{clamp}(Y + 1.772 \\times (Cb - 128))", "ad916be79dca78255c6298aa8e924684": "\\delta = 100\\times\\eta = 100\\times\\frac{\\epsilon}{|v|} = 100\\times\\left| \\frac{v-v_\\text{approx}}{v} \\right|.", "ad917999f2546cb05e483cc1107c1534": " \\sum \\frac { A_x } { k + x } = N \\log( 1 + m / k ) ", "ad9184660a5471a37aba8dcbc1c16b01": "\\mathfrak{gc}_n", "ad9185c697c2a966c78f336897a5f25b": " \\nabla \\cdot ( \\mathbf{A} + \\mathbf{B} ) = \\nabla \\cdot \\mathbf{A} + \\nabla \\cdot \\mathbf{B} ", "ad91d0881e68f8a195c857303b4df116": "0\\rarr A_1\\rarr A_2\\rarr A_3\\rarr 0", "ad91e0cd2eff2fe42f09e4fca87c7490": "Z_{IN}\\,", "ad920169f3ff6f0b39025e57550dddee": "x_1 \\in S_1 ,...,x_n \\in S_n", "ad92626fcd8accdf2864b83d1e37ee64": "x\\in S", "ad92d4c383c057eeb1fcdb20cfa39b82": "\\varphi_{j}:W_{j}\\rightarrow U_{j}.\\,", "ad932189e81790ab604e8612236e151d": "\\nabla f = \\nabla \\; y^* T y = \\lambda \\cdot \\nabla \\; y^* y", "ad932e317fc4968b6c8ec54eaa598ac8": "W_{0}^{k, p} (\\Omega)", "ad9332751032b57b83dc05a50d034fba": "+j1.14 = \\frac{j \\omega C_3}{Y_0}\\,", "ad9364d8a15e6e357f1f08234e31fba1": "[\\mathrm{Gr},\\mathrm{Set}]", "ad9372b508af8136ed376989cc89adaa": "B \\to A\\beta \\mid D,", "ad939e25a990a6c4a0803eda4ed46188": " (\\Delta \\otimes \\mathrm{id})T(z)=T_{12}(z)T_{13}(z), \\,\\, (\\varepsilon\\otimes \\mathrm{id})T(z)= I, \\,\\, (s\\otimes \\mathrm{id})T(z)=T(z)^{-1}.", "ad93dc369355a7abcfd7c890422cf6d8": "A_0A_1A_2A_3\\dots", "ad93e079daeee4471486f285f54fee18": "\\tfrac{M + 2\\lambda}{3}", "ad94c7275d5baa05c3fc19c852251611": "l^a\\partial_a", "ad94d43bb9bae680f261cbda601fb44e": "q = q(l)", "ad94da64841cd8c82c3927579e0d999f": " 1=1-0", "ad953e5d19fc4aae35e2c3adf7470be0": "H_3=\\langle a,b, t| [a,t]=[b,t]=1, [a,b]=t^2 \\rangle ", "ad954694aa16c375632f464b84eafedd": " H(\\mathbf{p},\\mathbf{q},t) = \\mathbf{p}\\cdot\\mathbf{\\dot{q}} - L(\\mathbf{q},\\mathbf{\\dot{q}},t) \\,\\!", "ad9562766dbaab6b29e134a7e70e2e21": "p(x)\\cdot x^t", "ad95a2138238d4db05eb8d497f53c322": " ACH_{at 50 Pascal}\\,\\!", "ad9661dd1b2aaadc8471411bba87ce42": "\\{x\\in R:\\forall i\\in I, f_i(x)=0\\}.\\ ", "ad96af06843325feb6155e8ccf4e50ef": "r_1=\\alpha + \\beta", "ad971e01fa335ded772433e4f50d998d": "\\begin{align}\nE_v & = \\frac{d\\phi}{dA}\\\\\n& =\\frac{d\\phi}{d\\Omega} \\times \\frac{d\\Omega}{dA}\\\\\n& =\\frac{d\\phi}{d\\Omega} \\times \\frac{d\\Omega}{dS} \\times \\frac{dS}{dA}\\\\\n& =I_v (\\frac{\\frac{dS}{r^2}}{ds})(\\frac{dA \\cos (\\theta)}{dA})\\\\\n& =\\frac{I_v}{r^2}\\cos(\\theta)\\\\\n& =\\frac{I_v}{h^2}\\cos^3(\\theta)\n\\end{align}", "ad975654c1ebbfd41770bcd64832d16b": "R_{c}", "ad97698774cfd3d98469b3b6b5d5a9ff": "\\int |f_n - f| \\, d\\mu \\to 0", "ad97ae79ce7d1a61111f56dc7b5c4b7b": "\n\\hat{K}(t) = \\lambda^{-1}\\sum_{i\\ne j} I(d_{ij} O_j", "ad98e863e0f137f40830a0f381813e24": "\\mathrm{(SNR)_{O,AM}} = \\frac{A_c^2 k_a^2 P} {2 W N_0}\n", "ad992ba5630ba71a8f10362ccba72863": " \\Lambda^k(V) ", "ad993ef92de1d02f13288dad44bd89e0": "\\exp(X)\\exp(Y) \\approx \\exp(X + Y + \\tfrac{1}{2}[X,Y])", "ad995fd6cab95d1f85b19a4550fb2572": "\\displaystyle x_1=x_0+\\alpha_0 \\Delta x_0", "ad9988c3d6814565a020f4f14cc7ad18": " i=90\\text{ deg}\\,", "ad99c2e999c60beaac228ac96334da47": " Y(s) = \\mathcal{L}\\left \\{ y(t) \\right \\} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty} y(t) e^{-st}\\, dt ", "ad99dfe328b3f096aaefd60ca3eb0f0c": " \\frac{2}{\\pi} .", "ad99e5bcb402576dd5aa4c7f9090968d": "\\Delta y_1^n = y_1(p_1', m) - y_1(p_1',m').", "ad99e9e18fa50f9c886188846b7eee2f": "\n q_{x1} = \\int_{-b/2}^{b/2}q_0\\left(\\frac{1}{2} - \\frac{y}{b}\\right)\\,\\text{d}y = \\frac{bq_0}{2} ~;~~\n q_{x2} = \\int_{-b/2}^{b/2}yq_0\\left(\\frac{1}{2} - \\frac{y}{b}\\right)\\,\\text{d}y = -\\frac{b^2q_0}{12} \\,.\n", "ad9a59472cd36b135d58445d263fdc71": "\\frac{\\zeta^\\prime(s)}{\\zeta(s)} = - s\\int_1^\\infty \\frac{\\psi(x)}{x^{s+1}}\\,dx", "ad9a9ee82ad679bc76b1230978345389": "\n\\begin{align}\n\\operatorname{dCov}^2(X, Y; \\alpha) &:= \\operatorname{E}[\\|X-X'\\|^\\alpha\\,\\|Y-Y'\\|^\\alpha] + \\operatorname{E}[\\|X-X'\\|^\\alpha]\\,\\operatorname{E}[\\|Y-Y'\\|^\\alpha]\\\\\n&\\qquad - 2\\operatorname{E}[\\|X-X'\\|^\\alpha\\,\\|Y-Y''\\|^\\alpha].\n\\end{align}\n", "ad9a9ff06aad121e42f546e8dbadd386": "\n\\begin{align}\n \\frac{\\partial \\eta}{\\partial t}\\, \n & +\\, \\frac{\\partial}{\\partial x}\\, \\left[ \\left( h + \\eta \\right)\\, u_b \\right]\\, \n =\\, \\frac{1}{6}\\, h^3\\, \\frac{\\partial^3 u_b}{\\partial x^3}, \n \\\\\n \\frac{\\partial u_b}{\\partial t}\\, \n & +\\, u_b\\, \\frac{\\partial u_b}{\\partial x}\\, \n +\\, g\\, \\frac{\\partial \\eta}{\\partial x}\\, \n =\\, \\frac{1}{2}\\, h^2\\, \\frac{\\partial^3 u_b}{\\partial t\\, \\partial x^2}.\n\\end{align}\n", "ad9acd84e23eb0efe6b0ff37f052e552": " \\frac{q^2}{4}+\\frac{p^3}{27} >0\\,. ", "ad9b178f059c060175827d59564fef0e": "f(x_1,...x_n)=x_{i_1}\\cup ... \\cup x_{i_k}", "ad9b36a24f984f4206c73d36ad74c889": "\\alpha, \\beta\\in \\Phi^+", "ad9b39b7627e70f1449cb9a7b67532a0": "\\begin{bmatrix}A\\end{bmatrix}=\\begin{bmatrix}2000 & -1000\\\\ -1000 & 2000\\end{bmatrix}.", "ad9b5e88d19271937f539997dffadf91": " \\frac{1}{R_{\\rm eff}^2} = \\left( \\frac{1}{R'_{1}}+\\frac{1}{R'_{2}} \\right) \n\\left( \\frac{1}{R''_{1}}+\\frac{1}{R''_{2}} \\right) + \n\\left( \\frac{1}{R'_{1}}-\\frac{1}{R''_{1}} \\right)\n\\left( \\frac{1}{R'_{2}}-\\frac{1}{R''_{2}} \\right) \\sin^2 \\varphi,", "ad9b72bcd3a849e677d8586b659e4156": " C^S_{v_1} = \\frac{1}{\\varepsilon^{2}_S - \\varepsilon^{1}_S} ", "ad9b8481e31b7600ab8cdf55b75176fd": "\\exists x\\,\\neg P_1(x)\\land\\cdots\\land\\neg P_n(x)\\land P'_1(x)\\land\\cdots\\land P'_m(x),", "ad9bf03f537a1bccac1a489f154b2bcf": "\\delta = \\gamma^s", "ad9bf4de1d22035480ebdd3b5b914bb4": " (\\mathbf{x} \\times \\mathbf{y}) \\times (\\mathbf{x} \\times \\mathbf{z}) = ((\\mathbf{x} \\times \\mathbf{y}) \\times \\mathbf{z}) \\times \\mathbf{x} + ((\\mathbf{y} \\times \\mathbf{z}) \\times \\mathbf{x}) \\times \\mathbf{x} + ((\\mathbf{z} \\times \\mathbf{x}) \\times \\mathbf{x}) \\times \\mathbf{y}", "ad9c92d78fc368dd4a7808646b075e94": "\\sum_{i=k}^{\\infty} a_iT^{i/n},", "ad9ca133409610dd2e02517002be91ba": "\\begin{align}\n&&\\dot u\\,x+\\dot v\\,y+u\\,\\dot x+v\\,\\dot y&=0,\\\\\n\\Rightarrow&& \\lambda(x^2+y^2)-gy+u^2+v^2&=0,\\\\\n\\Rightarrow&& L^2\\,\\lambda-gy+u^2+v^2&=0,\n\\end{align}", "ad9cb145f7951ba79f73b22d675050be": "\\mathrm{d} U = T\\mathrm{d}S-p\\mathrm{d}V.", "ad9d4771c0f2e3c3721606fb29b7415d": "L(x, t) = \\sum_{n=-\\infty}^{\\infty} f(x-n) \\, T(n, t)", "ad9d48ebd7381dba4f8df7220f43b4dd": "O(1,3) ", "ad9d98e55639bdee426d0269ff2465fd": "E(Y) = \\alpha + \\beta x.", "ad9dd8eb9436adeeff5eeee514fee631": "Z_\\mathrm{sun}", "ad9dfe711a779276e2dab9fe1fac92f4": "\\mathbf{\\vartheta} = [\\vartheta^f,\\vartheta^s,\\vartheta^n]", "ad9e19eb66b6772bbdff06b8d5870cc1": "\n\\frac{d^{2}q}{dt^{2}} + \\omega_{n}^{2} \\left[1 + f(t) \\right] q = 0.\n", "ad9e5640cd76257292240ff8386c9121": "X_t=Y \\qquad \\text{ for all } t.", "ad9e5eeb694d30d212a661c575517bf5": "\\phi_{r}\\,= \\frac{\\phi_{R} + \\phi_{N}}{2}", "ad9e646fcc08fbba1631219b1ef23ea0": "\\frac{dT_1}{ds}=[T_2,T_3],\\ \\ \\frac{dT_2}{ds}=[T_3,T_1],\\ \\ \\frac{dT_3}{ds}=[T_1,T_2]", "ad9e7659b6b08612caba9ecf693c9e86": "x_1 < x_2 < x_3 < x_4 < \\cdots < x_{2n-1} < x_{2n}", "ad9e7a46fe6770fcaba75d989b2d248b": "X>X_{C}", "ad9e9fd8e6a9f182378eb93a5444e724": "2^{106}", "ad9ed0be9f20f4973a2a43c46e572baf": "\n\\frac{E[\\Delta[r] + Vp[r] | Q[r]]}{E[T[r] | Q[r]]} \n", "ad9eda6cb6699c556a2a8106f5480f98": "\\{w,u\\}", "ad9ef70969ebc84dd1d0e20238e8bbd7": "l_W = E_B = l_A a_B + l_B < 1", "ad9efab7992d9a7c6f26092b729018e7": "t_l\\le t_1 \\le t_2 \\le \\cdots \\le t_{n-1} \\le t_n \\le t_u", "ad9fb372af0a195312f634caf09fb3d3": " \\tfrac12 + it ", "ada0286d3b1140dfb3823e6a0f866bc1": "p\\equiv 3 \\text{(mod 4)}", "ada02eab128c467389eee956851d5dfe": "\\text{Had}:\\{0,1\\}^k\\to\\{0,1\\}^{2^k}", "ada0817ed81c35403666e75a2a17d877": "bestSubspace := \\min_{s \\in S_k \\wedge s \\subset cand} \\sum_{C_i \\in C^s} |C_i|", "ada090ed68cb81159a7f749210488924": "c'\\not\\in\\mathrm{up}(c)", "ada096ac9312d0444f16fdf81b8595e8": "\\sum_m \\langle j,m | j,m \\rangle = 2j + 1 ", "ada0ab681fd87df114838203fee041ae": "\\begin{pmatrix} \\ \\ & a \\\\ \\ \\nearrow & \\ \\\\ 0 \\longrightarrow & 0 \\end{pmatrix}=\\begin{pmatrix} \\ \\ & 0 \\\\ \\ \\nearrow & \\ \\\\ aR \\longrightarrow & -a \\end{pmatrix}", "ada0b5fb9c7bf97fe9e3a239e1628ddc": "x^N+y^N=k(1+x^N y^N)", "ada130a63f19c1b56887f2a83e42fdd9": "BSC_p", "ada1563042830f461de2cb7cf40738ce": "G/a", "ada15a014f3e2d93a900ef2e649b668a": "\\hat{\\alpha}=\\gamma \\mu_0 m \\frac{\\mathrm{d}\\mathbf{H_{eff}}}{\\mathrm{d}\\mathbf{m}} \\delta{t}", "ada1685ec20f5f232c6221bf39a748de": "\\sum_{j=0}^{N-1} e^{x j} = \\frac{1 - e^{Nx}}{1 - e^x}.", "ada18008b2352e7d0f4eec84a52715a3": "\\Psi(x) = e^{\\Phi(x)}", "ada1839d071959ca9cd25ea85b0c5486": "\\frac{\\partial^2 u}{\\partial t^2} - \\frac{\\partial^2 u}{\\partial x^2} + \\sin u = 0,", "ada1e87eab457a79afcf3ff44a75ec5d": "\\hat{l},\\hat{r}", "ada201aa51e173967b7d6be60b2869dd": "\\varepsilon_{ijk} \\varepsilon^{ijk}=6. ", "ada20c8b4dcfe27324d32778537bcb90": "\\,^{232}_{90}\\mathrm{Th} + \\,^{84}_{36}\\mathrm{Kr} \\to \\,^{316}_{126}\\mathrm{Ubh} ^{*} \\to \\ no \\ atoms", "ada2633de1af63405b436ae8af76bddd": "(a, b).", "ada263a37ff082fce137d898782c52ab": "\\vec{V}=U\\widehat{i}+0\\widehat{j},", "ada268d363b3103aaf1c9517e078ae3e": "R_2//R_f \\ ", "ada284137f1e6d54dfe79cbac6b0ce0b": "H^1(\\mathbb{R}^d)", "ada29722c713d4d8a571d9a4ce353ae6": "\\lambda \\in [0,1]", "ada29e70739458374177a553c092ad35": "\\scriptstyle \\phi\\,", "ada2a65a808570440f16583fca203457": "{\\mathcal O}(n^{11})", "ada2d1d6c0939aa513a16325a8101ee9": "m_a", "ada2f47a74222cdfc9587842e54bef22": "mg = \\rho_f V_\\text{disp} g, \\,", "ada30bd19de55831aed0075075921a39": "Q = Q(x,y,z)", "ada330df81e2994a120c8f14ab27ab31": "Rep(w', P) = R", "ada3b2fb5f5b63f725d24b3d58f37c44": "((x_1,y_1),(x_2,y_2),\\dots,(x_k,y_k))\\,", "ada3b5e0bd9f1cb7513ae4d125399bc6": "S(a_{(1)})a_{(2)}S(a_{(3)}) = S(a) ", "ada42b682c1d5bfb39f36fabf5bed16e": "\\mathbf{A}=\\begin{bmatrix}\n1 & 0& \\cdots & 0 &a(1,n+1)& \\cdots & a(1,m)\\\\\n0 & 1& \\cdots & 0 &a(2,n+2)& \\cdots & a(2,m)\\\\\n & & \\cdots & & & \\cdots & \\\\\n0 & 0& \\cdots & 1 &a(n,n+1)& \\cdots &a(n,m)\\end{bmatrix}\n", "ada4656ee0ed19b96a453cae17b5ed94": "\nu_{12} = {z_1 - z_2 \\over \\sqrt{2} }\n", "ada498fe0ffab8c078cb02eae9165e86": "P(A \\rightarrow B)", "ada4a14156c16478b9260a4c6d636bed": " n_{t} \\times n_{f} ", "ada4a2510f3351cacac0e01613fc8073": "E[X_v] = \\frac{1}{Z} \\left.\\frac{\\partial Z[J]}{\\partial J_v}\\right|_{J_v=0}.", "ada4f6185e490106d643cc39df414e16": "\\sin (2 \\pi f_1 t)\\,", "ada4faac79bf7095475bd8a067297808": "y^2 (y^2 - a^2) = x^2 (x^2 - b^2)", "ada58da664b731438a8992c98f6a9d15": "x \\subseteq y \\land y \\in V \\rightarrow x \\in V.", "ada5fbf16ba60c90661ea0ef5dfa1816": "2 = \\left(\n\\frac{\\left(3-x\\right) \\times 2}{3-x}\n\\right)", "ada6345b651ef2b58b31daa7d83e2d49": "\\sum_i f_i g_i\\,", "ada66824c4435ae482f7fc18dc9318f3": " = \\sum_{i} v_i^{T} x v_j^{T} v_i", "ada66efac7d808ec4c97846ebc68b392": "k(m,n)", "ada6814be5ce9fe76159a6d07f2a4f59": "x^n = E_n (x) + \\frac {1}{2}\n\\sum_{k=0}^{n-1} {n \\choose k} E_k (x).\n", "ada6bb3107630a5037822ff58a498511": "\\left(\\!\\!\\binom n k\\!\\!\\right)", "ada6d921b32dd686be2d413709f2b337": "-\\int_{\\mathbf{R}^n}g(\\mathbf{x})\\,\\frac{\\partial 1_D(\\mathbf{x})}{\\partial n}\\;d\\mathbf{x}=\\int_S\\,g(\\mathbf{s})\\;d\\sigma(\\mathbf{s}),", "ada700af9179217e923483135cb4568c": "\\mathrm{sinc}(x) = \\frac{\\sin(\\pi x)}{\\pi x}.\\,\\!", "ada716276a9edf100419b1341ffa13b1": "U\\ni x\\,", "ada72c7d77fca942f36d345610d9e09f": "A=UTU^H", "ada75fc16b47efb1fe6f0441eb4ae214": " x^5-50x^3-600x^2-2000x-11200", "ada7b42f442e748cb7f242e886f8328e": "\\mu_{\\mathrm{Mulliken}}=-\\chi_{\\mathrm{Mulliken}}=-\\frac{IP+EA}{2}=\\left[\\frac{\\delta E[N]}{\\delta N}\\right]_{N=N_0}", "ada838adeba1adb6961337c649fb8d67": "F_{12}", "ada85f63c6212c3cbf12d6c106125cac": "\n-I =\n\\begin{bmatrix}\n -1 & ~0 & ~0 \\\\\n ~0 & -1 & ~0 \\\\\n ~0 & ~0 & -1\n\\end{bmatrix}.\n", "ada8876c1c9968b0d7ba11c7998f448a": "\\operatorname{kurtosis\\ excess}(X) = \\frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^{2}}-3 . ", "ada8b542fef85299ca3ca141171710de": "A_1, ..., A_n", "ada8d6f3e2079d8f5f1d8c170e9c4060": "F^4,\\ F^2 R^2,\\ R^4", "ada90b12966e8919eaf36747c6707a7c": "\\delta\\alpha-\\bar{\\delta}\\beta=(\\mu\\rho-\\lambda\\sigma)+\\alpha\\bar{\\alpha}+\\beta\\bar{\\beta}-2\\alpha\\beta+\\gamma(\\rho-\\bar{\\rho})+\\varepsilon(\\mu-\\bar{\\mu})-\\Psi_2+\\Phi_{11}+\\Lambda\\,,", "ada96303c2fe815a928cc8e351c91cc5": "V_{\\text{XX}}", "ada9aa25c3eff42b3c01d9dd281354fc": " f(x)", "ada9e6f33ab6780b5367c32c2555c0ab": "E \\oplus E' \\cong M \\times \\mathbb{R}^n. \\, ", "adaae9dc8be1bf972204c32cfc0ebbf2": "\\Delta_2 := (id \\otimes \\tau \\otimes id) \\circ (\\Delta \\otimes \\Delta) : (B \\otimes B) \\to (B \\otimes B) \\otimes (B \\otimes B)", "adab7f43e54ff39e88b3eae3eabb908f": "u, v \\in U", "adab820c298bcd95dc60631d70a2494a": "\\left(1,\\ 1+\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+3\\sqrt{2}\\right)", "adabe5db488beca03e78ab5004735fa9": "k \\Delta x = 2 \\pi", "adac1dda77a34ec578ea55d1ec1c37c2": "\\Delta x=\\Delta y=h", "adac32c1ef3bda0015c98a33e03beda9": " y_2'=0.04y_1-10^4y_2\\cdot y_3-3\\cdot 10^7y_2^2 \\qquad \\qquad \\qquad \\qquad (4) ", "adac33cff83848469b903f970237b14b": "p \\equiv a \\!\\! \\pmod m", "adac83d7541b8a8912222c537931510b": "\\mathcal{P}(\\Lambda \\times \\pi_1)", "adacc9643821abf91c4287d6e6b5dfbe": "T^{\\hat{\\underline{\\gamma}}}_{\\hat{\\underline{\\alpha}}\\hat{\\underline{\\beta}}} = 0", "adacd5089daddab76764085043cb931d": "\\epsilon^{\\text{v}}(p,t) = \\mu^{\\text{v}} E^{\\text{v}}(p,t), \\text{and}", "adad78dc1d0aa8c917af90711bee566c": "\\int_{\\Omega}{}{u(\\partial^{\\alpha}v)}\\,dx = (-1)^{|\\alpha|}\\int_{\\Omega}^{}{(\\partial^{\\alpha}u)v\\,dx}.", "adad89d9253baecc5862b11d197c61e8": "\\text{arc length}/\\text{radius}", "adae50e1af11d53987c3698486c1b8cc": "\\sigma (t) = E \\varepsilon(t) + \\eta \\frac {d\\varepsilon(t)} {dt}", "adaeb1cd6382a196ea5c0373fe82eadd": "\\lambda_{i} = 0", "adaf40fd9dd1a3c7c1fd6eda6513b32b": "\\frac{d\\ \\operatorname{Re} \\{V_c \\cdot e^{i\\omega t}\\}}{dt} + \\frac{1}{RC}\\operatorname{Re} \\{V_c \\cdot e^{i\\omega t}\\} = \\frac{1}{RC}\\operatorname{Re} \\{V_s \\cdot e^{i\\omega t}\\}", "adaf69a920eb0061403acb0e6dea3387": "\\operatorname{sig}(A) = \\frac{KA^2\\log e}{4}\\cdot\\operatorname{sock}.", "adaf7b18dcffa45ae2dc02aadf29ab03": "{a+b\\varepsilon = (x+y\\varepsilon) d\\varepsilon} = {xd\\varepsilon + 0}", "adafa13588ac6efe0dd341edfd992813": "\\mathrm{codim}(U) = \\dim(V/U) = \\dim(V) - \\dim(U).", "adafee1111c076c6a8f3a26fe969abba": "xy \\vee \\bar{x}z \\vee xyz \\vee \\bar{x}yz", "adaff04d21d8cff101de6752c46db927": "\\lim_{n\\to\\infty}\\int_E f_nd\\mu", "adb03071df2fee0b501e927614c231b1": "\\phi < 1.35", "adb0d93a4c67fc920abffc1788d9694d": "\\{(x,L)\\mid x\\in L\\}.", "adb16b2b72bfb43207454e8f0ce25c48": "\\pi(z)", "adb17bf6a4e7b108b51a1ccc8f08fb69": "C^k(\\Omega)", "adb19ff62fbd1367ec055c6e261ed89e": "G(Z) = \\operatorname{Hom}_S (X, Z) \\quad \\text{for } Z \\in \\mathcal{D}", "adb1e64d00743fffaab23c80558f847b": "E[2n+1]\\setminus \\{O\\}", "adb1f8c432527f9659e6e7ae041e870f": "w_i^{(0)} = 1", "adb211bd0cf1b53f0bbe735d2265dcbf": "\\Psi^{}_{}(u_1,u_2,\\dots,u_m,y_1,y_2,\\dots,y_N)=0\\!\\,", "adb21d63c264addcf04969b6163715ae": " \\frac{d}{dx} x^{-1} = (-1)x^{(-1)-1} = -x^{-2} = -\\frac{1}{x^2}.", "adb2412fbe7289b55e52310fb7be9906": " \\begin{align} \n&\\lim_{\\alpha\\to 0}\\text{excess kurtosis} =\\lim_{\\beta \\to 0} \\text{excess kurtosis} = \\lim_{\\mu \\to 0}\\text{excess kurtosis} = \\lim_{\\mu \\to 1}\\text{excess kurtosis} =\\infty\\\\\n&\\lim_{\\alpha \\to \\infty}\\text{excess kurtosis} = \\frac{6}{\\beta},\\text{ } \\lim_{\\beta \\to 0}(\\lim_{\\alpha\\to \\infty} \\text{excess kurtosis}) = \\infty,\\text{ } \\lim_{\\beta \\to \\infty}(\\lim_{\\alpha\\to \\infty} \\text{excess kurtosis}) = 0\\\\\n&\\lim_{\\beta \\to \\infty}\\text{excess kurtosis} = \\frac{6}{\\alpha},\\text{ } \\lim_{\\alpha \\to 0}(\\lim_{\\beta \\to \\infty} \\text{excess kurtosis}) = \\infty,\\text{ } \\lim_{\\alpha \\to \\infty}(\\lim_{\\beta \\to \\infty} \\text{excess kurtosis}) = 0\\\\\n&\\lim_{\\nu \\to 0} \\text{excess kurtosis} = - 6 + \\frac{1}{\\mu (1 - \\mu)},\\text{ } \\lim_{\\mu \\to 0}(\\lim_{\\nu \\to 0} \\text{excess kurtosis}) = \\infty,\\text{ } \\lim_{\\mu \\to 1}(\\lim_{\\nu \\to 0} \\text{excess kurtosis}) = \\infty\n\\end{align}", "adb270d1f97d4f8d68e048c657563e58": "f_x=-Kx\\ ,", "adb2b267be27ff4b58826d555961ce61": "3 \\cdot 2^{402653210} - 1", "adb2dde21431e4bbc5f116c831e5d260": "\\sum_{k=0}^\\infty h_k(X_1,\\ldots,X_n)t^k = \\prod_{i=1}^n\\sum_{j=0}^\\infty(X_it)^j = \\prod_{i=1}^n\\frac1{1-X_it}", "adb2ee58dbc637afc250f95cf2a88991": "x_{\\sigma(1)}\\wedge x_{\\sigma(2)}\\wedge\\dots\\wedge x_{\\sigma(k)} = \\operatorname{sgn}(\\sigma)x_1\\wedge x_2\\wedge\\dots \\wedge x_k,", "adb3720899d038bfbbb8a825c0839c98": "\\begin{align}\nI_0~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T\\cdot\\frac{\\partial I_0}{\\partial \\boldsymbol{A}} & = 0 \\\\\nI_1~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - I_2~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_3}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T\\cdot\\frac{\\partial I_2}{\\partial \\boldsymbol{A}} & = 0 \\\\\nI_3~\\boldsymbol{\\mathit{1}} - \\frac{\\partial I_4}{\\partial \\boldsymbol{A}}~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T\\cdot\\frac{\\partial I_3}{\\partial \\boldsymbol{A}} & = 0 ~.\n \\end{align}", "adb38cf301070d5796952d59e5aab207": "\nH(\\mathbf{Y})=-\\frac{1}{N}\\sum_{t=1}^N \\ln p_{\\mathbf{Y}}(\\mathbf{Y}^t)\n", "adb39a48c317bc064c06bb44a0762ceb": "E=\\frac {1}{2} L I^2", "adb408a5bcb0eb731741c8a6cd2acc5e": "\\|A\\|_F^{2} = \\|P^{\\top} \\cdot B \\cdot P\\|_F^{2} = \\operatorname{trace}\\left( \\left( P^{\\top} \\cdot B \\cdot P \\right)^{\\top} \\cdot \\left( P^{\\top} \\cdot B \\cdot P \\right) \\right) = \\operatorname{trace}(B^{\\top} \\cdot B) = \\|B\\|_F^{2}", "adb40bbb8e80493d44fcf19d1c737bad": "\n {}_t p_x = \\frac{S(x+t)}{S(x)} = \\frac{\\omega-(x+t)}{\\omega-x},\n \\qquad 0 \\leq t < \\omega-x,\n", "adb414ddf55916e9dc656189bc34b8d3": "X=(x_1,x_2,x_3),", "adb41bc7bf68ff83e4b299ad06474de6": "\\rho=\\rho_0e^{-\\frac{mgh}{k_BT}},", "adb421a0022eecf70e2eb5db41ee4e65": "\\scriptstyle{\\zeta \\ll 1}", "adb42ab9a9b1ed8835b0a9a7bc6e6742": "K:[a, b]\\times [c, d]\\to {\\mathbb R} \\,", "adb4356c41cea3c7de4fc29d23af43c3": "\n \\left( B \\or C \\right) , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\vdash \\lnot A\n ", "adb4a708cb27d311ab8e4a1c0c776213": "\\frac{\\partial\\Pi(n,k)}{\\partial k}=\n\\frac{k}{n-k^2}\\left(\\frac{E(k)}{k^2-1}+\\Pi(n,k)\\right)", "adb4d1cb939d4db25b22af251b0aebff": "G = G_{\\infty} \\left( \\frac{T}{T + 1} \\right) + G_0 \\left( \\frac{1}{T + 1} \\right) \\ ,", "adb4d67701bf1a078799812a87f7317f": "\\log{\\left({Bx\\over Bo}\\right)}\\ = \\ -\\left({f\\over r^b}\\right)", "adb5012d2a563cb8b51a0dc93202d0f6": "g(x):=\\sum_{j=0}^\\infty \\,a_j \\sqrt[3]{x-q_j}.", "adb511118fb3d25cfc14741cbe68db41": "\n-I_{+}=I_{-}=I_{ion}^{sat}\n", "adb543e6f1bf92a71afcdbdfbf123a3d": "H_1\\widehat{\\otimes}H_2", "adb54dd3bffda15e4383a0a65cc0ced1": "\\alpha,x,y", "adb58dc0cec8a777932270ff586bf09a": "y' = x \\sin \\theta + y \\cos \\theta\\,", "adb58fd08beefd29d0ed15a7ef973050": "\\max_{s\\in X} I_{S}(s)", "adb5bde8c9dc1ef78acc756b054ef8ef": " \\gamma_3 = - \\frac{1}{6} \\frac{2 m_0}{\\hbar^2} C_0, ", "adb63e5db97ae423fb8a575ee0048a58": "H_{\\min}^{\\epsilon}(A|B)_{\\rho} \\geq H_{\\min}^{\\epsilon}(A|BC)_{\\rho}~.", "adb64966eb28e5e380f28e61ad1947b7": "\\mu = 3", "adb69ea078c9fac283b86b7617588847": "p=\\ln \\frac{\\cos \\beta + \\sqrt{\\cos 2\\beta}}{\\sin \\beta} ", "adb6e021f1146595092244aede38bad4": "F = m_1 m_2 G/r^2", "adb72383bde5461f290788559950faeb": "\\mathfrak{g}_{1}", "adb728f59e90fdb41a14e5f14b2176e2": " \\bar{n}= -1/\\log(1-1/N) \\approx N. ", "adb72bc25510b3364913f226089ee0e8": "\\mathbb{C}_i", "adb73c59b3f652327334a7fe1a972a52": "\\left(1+\\frac1{x^2}\\right)Q(x) =\\int_x^\\infty \\left(1+\\frac1{x^2}\\right)\\varphi(u)\\,du >\\int_x^\\infty \\left(1+\\frac1{u^2}\\right)\\varphi(u)\\,du =-\\biggl.\\frac{\\varphi(u)}u\\biggr|_x^\\infty\n=\\frac{\\varphi(x)}x. ", "adb7dc014e1f2f2c6f56c1343e8612f2": "(x,1)", "adb805636421052fc6695bd3bec4590c": "\\mathcal{L}_i", "adb8498fcd00310f44edb55f331aacca": "I=\\Big\\langle l_1\\equiv\\partial_{xx}-\\frac{1}{x}\\partial_x-\\frac{y}{x(x+y)}\\partial_y,", "adb88cc71e792a50e37b9f7f16b1f1fa": "Z^i_\\alpha", "adb8952dd3a497dc58dd6c876ca97610": " \\mbox{ch}(f_{\\mbox{!}}{\\mathcal F}^\\bull)\\mbox{td}(Y) = f_* (\\mbox{ch}({\\mathcal F}^\\bull) \\mbox{td}(X) ). ", "adb8df18af0bf723c9c5fac3a3ad211c": "f(x)={n \\choose x}p^x (1-p)^{n-x}, \\quad x \\in \\{0, 1, 2, \\ldots, n\\}.", "adb90afc8c7a57e4a78c62407b8c974a": " \\text{(3)} \\qquad U = \\alpha n R T, ", "adb917d5996af355480f6c031e3574ba": "R-H + SO_2Cl_2 \\rightarrow\\ R-Cl + SO_2 + HCl", "adb932011e516c42f31fd6fa2dd3d05c": "\\left\\lfloor \\frac{n^2}{4}\\right\\rfloor + f(k),", "adb9419d0312a85858f900eae31b1457": "N_{W-P} = \\dfrac{\\mathfrak{R} R^2_p}{C_s D_{eff}} \\le 3\\beta", "adb992d3fa81dace746ce7ed74002def": "\\mathbf{E}^{a} [\\sigma_{k}] \\leq \\frac1{n} \\big( R^{2} - | a |^{2} \\big).", "adb9e884b7a7a135bce331d3262e70e0": "V = \\mathbf{E}\\cdot\\mathbf{L} = EL \\cos \\tau = vBL \\cos \\tau", "adb9f4adb7b255f62678d7cfb63f9e83": "d=c_{(p^2-p+1)/q}", "adb9f5a5f4541ed61ffa9a4b0b0edd63": "\\frac{(m - 1)(k - 1)}{k - 2}", "adba45f20d0cbbb824f5d693f04f83b4": " {u_z}_\\mathrm{avg} ", "adba73fab55ec7d9c021287fa25f765b": "\\hat{u} : X/\\ker(u) \\to Y", "adbaf00534500ef474c37f56173d268f": "\n\\pi \\cdot \\cot (\\pi x) = \\frac{1}{x} + \\sum_{n=1}^\\infty \\frac{2x}{x^2-n^2}.\n", "adbb4255a3f716fbc0877274b34f20ac": "U = \\begin{pmatrix}\n 2 & 3 \\\\ 3 &5\\\\\n \\end{pmatrix}", "adbb499a5e214120f2299f8e0ce46447": "%r = \\frac{100r}{f+r}", "adbbc0015f568c23c66c2fdfe720eb96": "f\\in \\mathbb{F}", "adbc3bdbbba8151ffe393513a12c1e1d": "v = \\frac{\\Delta y}{\\Delta x} = \\frac{\\Delta s}{\\Delta t}.", "adbc66dd608f0518210def27880e063b": "\n\\begin{array}{rcl}\n\\overline{ \\overline{ \\phi } } &\\neq& \\overline{\\phi}, \\\\\nG \\star G \\star \\phi = G^2 \\phi &\\neq& G \\star \\phi,\n\\end{array}\n", "adbccc14891dcc63ddc3392a401893a0": " GDOP = \\sqrt{PDOP^2 + TDOP^2}", "adbcf9020b7c341501da8bf3b75b977d": "U = \\begin{pmatrix} \\ \\ \\,a+di & b+ci \\\\ -b+ci & a-di \\end{pmatrix}.", "adbd0e5d7bf9a5158842fc32e1a12935": "\\mu =\\frac{1}{2}\\rho \\bar{c} \\lambda.", "adbd950a1bb1ea5ed075b1eae7003d6e": "Y = \\vec{f}_0", "adbdfecbf00f3727d6dd6821e340b5a3": "\\Longrightarrow u_{\\rm r} = 0.00134 \\Longrightarrow u(c) = u_{\\rm r}c = 0.1\\ {\\rm meq/l}", "adbe30d777a50cb152e21db9e94ace3b": " y' = xy", "adbe4cc30f719295082274f0109c226e": "x\\mapsto \\overline {(x+1)\\cos(\\pi x)}", "adbe79ed34fb480dd69eb53d4f043d4c": "\\Psi_\\omega = \\sum_{\\alpha_1, \\ldots, \\alpha_{G}}A^{G}_{\\omega,\\alpha_{G}}A^{G-1}_{\\alpha_G,\\alpha_{G-1}}\\dotsb A^2_{\\alpha_3,\\alpha_2} A^1_{\\alpha_2,\\alpha_1} x_{\\alpha_1}", "adbedeff4e9d9abd42ce2aa9db68033f": "\\begin{align}\n 0.7e^{-(.1)(\\frac{3}{12})} + 0.7e^{-(.1)(\\frac{5}{12})} = 1.3541\n\\end{align}", "adbf0f0e67f819ebb8e134a7f9771883": "k*t(n)", "adbfab9853a57e78535e96b7fe7b53f6": "\\mathcal{F}_\\alpha[f](u) = \n\\sqrt{1-i\\cot(\\alpha)} e^{i \\pi \\cot(\\alpha) u^2} \n\\int_{-\\infty}^\\infty \ne^{-i2\\pi (\\csc(\\alpha) u x - \\frac{\\cot(\\alpha)}{2} x^2)}\nf(x)\\, \\mathrm{d}x. \n", "adbfe573659ffcaa3a6865f3fea7f3c3": " (m+m_A) \\ddot y =-k_c(y+H \\cos \\omega t) ", "adc002c57c21a0d395beb4b070797530": "X=\\sum_{n=1}^N Y_n", "adc012a9133b1a7f4e114ceedd76e710": "U_{ee} = \\frac{1}{2} \\ e^2 \\int \\frac{n(\\vec{r}) \\ n(\\vec{r} \\, ')} {\\left\\vert \\vec{r} - \\vec{r} \\, ' \\right\\vert } \\ d^3r \\ d^3r' .", "adc0a08f9cef979e09e33513e7146e0d": " \\mathbf{v}_i \\leftarrow \\frac{\\mathbf{v}_i}{\\|\\mathbf{v}_i\\|} ", "adc0a2bcb4dfe33d48ef8197527c2058": "\n A \\begin{bmatrix} 1 \\\\ \\lambda_2 \\\\ \\lambda_3 \\end{bmatrix} =\n \\begin{bmatrix} \\lambda_2\\\\ \\lambda_3 \\\\1 \\end{bmatrix} =\n \\lambda_2 \\cdot \\begin{bmatrix} 1\\\\ \\lambda_2 \\\\ \\lambda_3 \\end{bmatrix}\n \\quad\\quad\n ", "adc0adc5a6909dd54f5ea852f0437957": "\\left \\Vert \\mathbf{p-m_i} \\right \\| = \\sqrt{\\sum_{j=1}^L (p(j)-m_i(j))^2}", "adc1216412ddc22ec1542a723721bb1d": "~E_{\\rm phys}={\\rm Re}\\left( \\vec e E \\exp(ikz-i\\omega t)\\right)~", "adc1257d3246d3100fb3721cc527e081": "\\alpha(x) = 2x, \\quad \\beta = -1,", "adc1259251d35f4ee28a62d11ffcc6fe": "E \\left [ (\\overline{y}_{11} - \\overline{y}_{12}) - (\\overline{y}_{21} - \\overline{y}_{22}) \\right ] ~=~ \\delta (D_{11} - D_{12}) + \\delta(D_{22} - D_{21})", "adc13c282da51a63d0f79411d55aa3f3": " \\mathbf{F} = m \\mathbf{A} = m\\gamma(\\mathbf{u})\\left( \\frac{d{\\gamma}(\\mathbf{u})}{dt} c, \\left(\\frac{d{\\gamma}(\\mathbf{u})}{dt} \\mathbf{u} + \\gamma(\\mathbf{u}) \\mathbf{a}\\right) \\right) ", "adc13c8cded5f7ec4090432c12faef45": "\\sum_{w\\in N^{(t)}(u)\\cup\\{u\\}} \\frac{1}{d(w)+1}", "adc1413da059b9b76bd77db3e8882616": "\\frac{dy}{dx} = -\\frac{\\partial F / \\partial x}{\\partial F / \\partial y} = -\\frac {F_x}{F_y},", "adc145d23077567140b40ccd6b04f9d6": "T(x) = 1 + x\\frac{T(x)^3 + 3 T(x)T(x^2) + 2 T(x^3)}{6}.", "adc16599be758b9a814f2f3f4ba9b5ee": "0\\leq y_3< n_{i,j}", "adc17ef64040ba2aec8a615259dddb3d": "\\mathcal{D}_M", "adc196aeca20f1e5258f350f7e8b9391": "T(\\rho,\\sigma) = \\frac{1}{2}||\\rho-\\sigma||_{1} = \\frac{1}{2}\\mathrm{Tr} \\left[ \\sqrt{(\\rho-\\sigma)^\\dagger (\\rho-\\sigma)} \\right]", "adc1f05a9e275d7cc1eec03a26218b3c": "\\delta(k)", "adc2068334a6803134292b1e37eda6ef": "\n\\mathbf{\\Omega}=\\begin{bmatrix} \\omega_x \\\\ \\omega_y \\\\ \\omega_z \\end{bmatrix}.\n", "adc22fbf6f9b6ceff485012f9a1fd350": "\\tfrac{L}{a}", "adc23a31fdeb40144d82b571fc25572b": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i (\\boldsymbol\\omega^T[\\Delta r_i]^T[\\Delta r_i]\\boldsymbol\\omega) + \\frac{1}{2}(\\sum_{i=1}^n m_i) \\mathbf{V}_C\\cdot\\mathbf{V}_C.", "adc2b689fb675b6745efeb96c76bc0d2": " k\\in \\mathbb{N} ", "adc2c5f4517cadb93a50f6abe87f7dc0": " \\mathbf{F}_i + \\sum_{j=1}^N \\mathbf{F}_{ij} = m_i\\mathbf{a}_i,\\quad i=1, \\ldots, N,", "adc33c3f465116b28d5a61e1b43caccb": "2^8 + 2^{7 \\times 8} = 72057594037928192", "adc386a83ccaa179a8c25524c1d4da6d": " \\mathrm{In} \\begin{bmatrix} H_{11} & H_{12} \\\\ H_{12}^\\ast & H_{22} \\end{bmatrix} = \\mathrm{In}(H_{11}) + \\mathrm{In}(H/H_{11}) ", "adc3a9772809460113ad91ad70033e5a": "\\textstyle \\sum_i \\pi_i=1", "adc3d8ab65bd3c32419c789d2a5ee0b6": "{F_t} = E_t(S_{t + k})", "adc3e5955e3bcf2aa126f1bcc3a8317c": " U \\times U \\subset U \\subset U \\times U. \\, ", "adc41716f35919b3823c36a72c66e005": "a_r(n) c_{n+r} + a_{r-1}(n) c_{n+r-1} + \\ldots + a_0(n) c_n = 0", "adc41b07443501608ada5a2c7006c4b1": "{Z_W} = \\frac{A_W}{\\sqrt{\\omega}}+\\frac{A_W}{j\\sqrt{\\omega}}", "adc4446affa4a5f46bf987658f24b7e2": "\\nu/X", "adc445c3d2ed2f2acf775c372e40361d": "\n\\left(\\frac{\\partial T}{\\partial p}\\right)_{S,\\{N_i\\}} =\n+\\left(\\frac{\\partial V}{\\partial S}\\right)_{p,\\{N_i\\}}\n", "adc44a4c4a465d093a173ea77650d2fa": "\\begin{array}{cccc}\n\\mathbf{e}_\\text{x}\\cdot\\mathbf{e}_\\text{y} & = \\mathbf{e}_\\text{y}\\cdot\\mathbf{e}_\\text{z} & =\\mathbf{e}_\\text{z}\\cdot\\mathbf{e}_\\text{x}\\\\\n\\mathbf{e}_\\text{y}\\cdot\\mathbf{e}_\\text{x} & =\\mathbf{e}_\\text{z}\\cdot\\mathbf{e}_\\text{y} & =\\mathbf{e}_\\text{x}\\cdot\\mathbf{e}_\\text{z} & =0\n\\end{array}\n", "adc456a2a38da7387e04c2b0569a8d39": "n \\not \\in \\mathfrak{p}", "adc46f783da20802d945996ee8e3c92b": "p=-1/2", "adc4a3cdca5be5426e9d24043c0c52a9": "\\sigma: \\tilde{X} \\rightarrow X", "adc4d75a22198d9f167f2cd8e3946748": "\\displaystyle{\\mu(z)=a(z,\\overline{z}),}", "adc503abcd119dc977c6c988d39f304e": "\\left[\\frac{x}{1-z},\\frac{y}{1-z}\\right]", "adc51a36adea24adff169d58e5eb9619": "\\lim_{x\\rightarrow\\infty} \\frac{N(x)\\sqrt{\\ln(x)}}{x}\\approx 0.76422365358922066299069873125.", "adc54b6cac8815a6aa1b2a70a38c8d38": "\\frac{1}{n} \\sum_{x,y}\\frac{(f(x,y) - \\overline{f})(t(x,y) - \\overline{t})}{\\sigma_f \\sigma_t}", "adc5ce08075daf3c79a3920748cb2870": "X = X_1\\rightarrow X_2 \\rightarrow \\cdots \\rightarrow X_n ", "adc5d0fdfe63733a3dfc4bf6560c3029": "\ny=(\\sum^t_{i=j}R^2_{i})-b^T(A+D)^{-1}b\n", "adc5d52493c502c9806d76caf6e1d4b8": "\\pi(z) = \\frac{(1-z)(1-\\rho)g(\\lambda(1-z))}{g(\\lambda(1-z))-z}", "adc5f289396d13696d05c8a264450138": "\n\\mathbf{b_{t-1:T}} = \\mathbf{T}\\mathbf{O_t}\\mathbf{b_{t:T}}\n", "adc60095fc9fa410ec3066d3b293bd7b": "\\frac{x}{\\log_q x},", "adc628513b0611ae63f0153c8fce80da": "\n(x^2+y^2+z^2)^n = f(x,y,z)^2+ g(x,y,z)^2+h(x,y,z)^2\n", "adc654e3daf06f795878f1a28f7974e5": " \\Delta = S_{i} + \\Lambda_1 \\ S_{i-1} + \\cdots + \\Lambda_e \\ S_{i-e}", "adc6be1e13aac292f661b42b2c8f067c": "\\vec{b}\\left(\\vec{r}\\right)", "adc6f588ec47e00141c808b38ac8d900": "n = 0,\\pm 1,\\pm 2,\\pm 3, \\ldots", "adc787907851d418158f8eeb82038394": "R_i\\cap\\{x,y\\}^2=R'_i\\cap \\{x,y\\}^2", "adc863acbcdfb199e99d2dfa00ddbd68": "f \\colon A \\times A \\to B", "adc8663ad59d631c471a910c08079cef": "I_{yz}", "adc92e7a9c69978b0a066c7748e02737": "X = f(X)", "adc94973c43219e80f94c48a68f9e169": "|a|<1", "adc9ce09ab33f641e965ef6f251e5e38": "\\left\\Vert A\\right\\Vert _{1}", "adca507ad6673e6f4f65697b125e41a1": " \\frac{0}{0} = \\Phi ", "adca5bd0c4be3b343baee3c5138bc671": "\\psi(\\Omega^{\\omega^{\\omega^\\omega}})", "adcbb70032e24450c811b3646bae3805": "H(C\\vee f^{-1}C\\vee \\ldots\\vee f^{-n+1}C)", "adcc01c38b3d3eb0aec0095f34cb466f": "\\mathbf{q} \\rightarrow \\mathbf{q}^{\\prime} = \\mathbf{q} + \\delta \\mathbf{q} ~,", "adcc3723e5108b264c27c1f8a6049209": "\\displaystyle{C_\\pm \\circ H = \\pm i C_\\pm.}", "adcc688faeb389ae371e1b55aa698935": "a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_0 = 0\\,\\!", "adcc6924f3dcdf4bea6c0528cbc21ae6": "A_{ij} = 0", "adccd2c5b21d54dfb86b4621cb8db889": " \\langle \\delta, \\psi \\rangle = \\psi(0) = \\frac{1}{(2 \\pi)^n}\\int_{\\mathbb{R}^n} e^{i x \\cdot \\xi} L(\\psi)(x,\\xi)\\, \\mathrm{d} \\xi \\, \\mathrm{d} x, ", "adcd2a0011ecd5f0ba111a45a15da696": " \\kappa = \\exp(m/r) = 1 + m/r + O\\left( \\frac{1}{r^2} \\right) ", "adcda73814cd3ff9aa9712b04003b993": "\\lambda \\le 2\\sqrt{d-1}", "adcdb0d710b84da5595ed65dfb94d6fe": "\n\\hat{I}=\\frac{1}{2}\\left[ \\left( \\frac{\\hat{q}}{\\rho}\\right)\n^{2}+(\\rho\\hat{p}-\\dot{\\rho}\\hat{q})^{2}\\right], ", "adcdbaf35affffc22c079806576a1ff4": "f(T) = e^T = I+T+\\frac{T^2}{2!}+\\frac{T^3}{3!}+\\cdots.", "adcdd51f0a11f8338376e171f9f47bbc": " t_r = - \\tau_0 \\ln(r) ", "adcdd9046d81003e157be92968e290db": "\\bigstar |\\bigstar \\bigstar ||", "adce00ff3cd8ed852a57083a5b519eaa": " \\mathbf{v} = \\mathbf{\\hat{e}}_{\\parallel} \\left ( \\partial A/\\partial t \\right ) \\,\\!", "adce1eccf927fdc70b61b5c2e0214210": "\\{ F(x,y,z) \\leq 0, z\\geq 0, z\\leq h\\},", "adce34bc3546044b9c4c254917f282cb": "\nN(i) = G(U(i)) \n", "adce77c55ab9fee9f16388a3e74811fd": "\\,=(G'WG)^{-1}G'W\\Omega WG(G'WG)^{-1} - (G'\\Omega^{-1}G)^{-1}", "adce79ccc109071bcef2b63c2cadf13d": "\n\\mathrm{var}\\left(\\hat \\theta\\right)\n\\geq\n\\frac{\\sigma^2}{N}.\n", "adceac9e0af0a82ee2548c3391be5b1d": " x^{(k+1)}_i = \\frac{1}{a_{ii}} \\left(b_i - \\sum_{ji}a_{ij}x^{(k)}_j \\right),\\quad i,j=1,2,\\ldots,n. ", "adcec9b8c06ee177d29acde09dc37342": "q\\frac{dN}{dq} = \\frac {LN-M^2}{2}.", "adced53a382406d627baba98f115c671": "f: \\mathbb{R}^n \\to \\mathbb{R}^n", "adcf2341bb6bd27df46b13545331a130": "\n\\begin{align}\np(\\theta|y)\n& = \\int p(\\theta|\\eta, y) p(\\eta | y) \\; d \\eta\n& = \\int \\frac{p(y | \\theta) p(\\theta | \\eta)}{p(y | \\eta)} p(\\eta | y) \\; d \\eta\\,,\n\\end{align}\n", "adcf5e639e12eb6c2281d00f5ee5ab27": "\\mathcal{L}_{EW} = \\mathcal{L}_g + \\mathcal{L}_f + \\mathcal{L}_h + \\mathcal{L}_y.\\,\\!", "adcf603b6660f1d3dd87c7e29b3fce1f": " M(Q) = \\omega_{0} Q. ", "adcf84f2e9c98df6bc5993466aecff8d": "\\pi_n(X) \\overset{\\pi_n(f)}{\\to} \\pi_n(Y) \\to \\pi_{n-1}(Ff).", "add01e8effdb270250da7c081d4ddcec": "\\Sigma=\\{a,b,c\\}", "add08159e2c9e8be1d116da19fec87ec": "\\langle A\\rangle = \\int_{PS} A_{\\vec{r}} e^{-\\beta E_{\\vec{r}}}d\\vec{r}/Z", "add09b550b335e8b35c947fb80fa94de": "\\vdash \\neg(A_1 \\land A_2 \\land \\cdots \\land A_n \\land \\neg B_1 \\land \\neg B_2 \\land\\cdots\\land \\neg B_k)", "add12f52b2e5d53c5f963dcc77b89f57": "G(\\varepsilon_Z)\\circ\\eta_{GZ} :\n\\operatorname{Hom}_S (X, Z) \\to \n\\operatorname{Hom}_S (X, \\operatorname{Hom}_S (X , Z) \\otimes_R X) \\to\n\\operatorname{Hom}_S (X, Z).\n", "add1309884b4582a792c09d34778797b": "\\Rightarrow \\varphi=\\cos^{-1}\\left(\\frac{r_1+r_2}{P}\\right) \\,\\!", "add134f083a3dd1c3d06416cdc2f4179": "\\Phi = \\iota \\circ \\overline\\Phi", "add163969930ebb549512a8e6cd1a5b8": " \\forall p \\exists q \\ne p \\Box (B(x,p) \\rightarrow B(x,q)) ", "add1882d4defc49840c8cf6b4d03854c": "g = |\\mathbf{p}_B - \\mathbf{p}_A| = |\\mathbf{p'}_B - \\mathbf{p'}_A|", "add190f9a3272cf8076b4762fea34b22": "\\,^{z_{16} = x_{16} y_1 - x_{15} y_2 + x_{14} y_3 + x_{13} y_4 - x_{12} y_5 - x_{11} y_6 + x_{10} y_7 + x_9 y_8 + x_8 y_9 - x_7 y_{10} + x_6 y_{11} + x_5 y_{12} - x_4 y_{13} - x_3 y_{14} + x_2 y_{15} + x_1 y_{16}}", "add1eb29c7c2d046d9659acbc6c19096": "I_{SHG FROG}(\\omega,\\tau) = \\left | \\int_{-\\infty}^{\\infty} E(t) E(t-\\tau) e^{-i \\omega t} dt \\right | ^2", "add1f7b3679ad384c651da8cddc5b76c": "\nF_{n\\rightarrow n^\\prime} = \\langle n^\\prime \\vert \\exp(ik_z z) \\vert n\\rangle = \\langle n^\\prime \\vert \\exp(i \\eta (\\hat{a} + \\hat{a}^\\dagger))\\vert n\\rangle.\n", "add205373b194576d21d64500d744f23": "b_{n+1} = b_n Y_n^2", "add22d17a12d295fcfbe317ded46a4c7": "X = \\{ X_1, X_2, \\ldots , X_r \\}", "add2a0eb8901c2600c24ebf5060b27c7": "\\quad A= \\sqrt{2/\\pi} (\\sigma_1+\\sigma_2)^{-1}.", "add33177f15c9b6fa9476f8a6626cee3": "|f(x)-f(a)| <\\epsilon", "add395ce33cf969d7bc47a984a96b078": " M(M-1) d \\leq \\frac{1}{2} n (M^2-1)", "add3bebd8d193580bce7e1f1ce56a98b": "K_+ = span \\{\\phi_+ = a \\cdot e^x \\}", "add3c37d14e3ce7d60a5e403dd582f5e": " N_i\\geq N_{i+d-1}+1,", "add481af47e9668cedddb738ee3c5dbb": "\\textstyle\\vec{M}_S", "add4912efd64faff34f12752a63e92b8": "z\\mapsto\\lambda z(1-z)", "add4a521c7e7679dd7395661d1c3b269": "s_xR\\bar{y}", "add4c40e9c51d8af4f6f489aae525547": "\\frac{2n\\overline{x}}{\\chi^2_{1-\\frac{\\alpha}{2},2n}} < \\frac{1}{\\lambda} < \\frac{2n\\overline{x}}{\\chi^2_{\\frac{\\alpha}{2},2n}}", "add4de9d499f69693cb949759a037de5": "\\big. \nQ=\\frac{1}{V^N}\\int \\prod_i d\\vec{r}_i\n\\prod_{i=1, i 1 ", "ade2745feba6b26f5d7a456d2cb3d714": "\\sigma\\in\\Sigma", "ade281dd1111c4902e4af2dd44a028c9": "=\\frac{1}{2}\\widehat{QO1P1}", "ade28d43714fe207a9b4d976b25e2f05": "\\sum_{S \\in \\mathcal S} x_S", "ade2c1fbf4406b28c4f52dcd3d34ba20": "\\begin{align}\n &N_{bb} = N_{cb} = 0.725 n^{-0.2} \\\\\n &n = Y_b / Y_w\n\\end{align}", "ade30e22a426582fd42754c2898cd4fa": "p \\rarr e^{+} + \\pi^0", "ade318cc7367b7f1b4127eddc5f4899f": "G_2", "ade34c4795450b16bb6db76119013d46": "H\\cdot X_t\\equiv \\mathbf{1}_{\\{t>T\\}}A(X_t-X_T).", "ade35be2e67d3caa09fde879c04c253c": "\\frac{(m_{1}u_{1})^{2}}{2m_1} + \\frac{(m_{1}u_{1})^{2}}{2m_2} = \n\\frac{(m_{1}v_{1})^{2}}{2m_1} + \\frac{(m_{1}v_{1})^{2}}{2m_2}\\,\\!", "ade36de43961a95e5e8cabff50567892": "\n\\bar{\\Phi} \\left[ \\mathbf{r} \\right] = \\frac{N^2}{2} \\sum_{j=1}^n\n\\sum_{k=1}^n \\int_0^1 ds \\int_0^1 ds' \\bar{\\Phi} \n\\left( \\left| \\mathbf{r}_j (s) - \\mathbf{r}_k (s') \\right| \\right) \n- \\frac{1}{2} n N \\bar{\\Phi} (0), \\qquad (2)\n", "ade37f57c6c3b970ef677dc81f02050a": "E_i= C {M_i \\over N_i}", "ade3bc92ed32c8c54dda7a835537a2a2": "\\displaystyle{Q(Q(a)b)= Q(a)Q(b)Q(a),}", "ade40947618f4a0752ea67829148703e": "\\rho:R\\to[0,\\infty]", "ade44f64b37c5fb0db5465333cfb70af": "\\scriptstyle Z_n = \\sqrt{n}(X_n-\\mu)/\\sigma", "ade55f6537a33911bf0721e65913d916": " S(a,b_a)=(b,a^b).\\,", "ade5efee23692306600b878957bf53e9": "\n\\begin{align}\n{} &\\Pr(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots , X_1=x_1) \\\\\n= &\\Pr(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots, X_{n-m}=x_{n-m})\n\\text{ for }n > m\n\\end{align}\n", "ade5f0300689718a3946a5db0e991bab": "\\left[(dx^1)^2+(dx^2)^2+(dx^3)^2-c(dt)^2)\\right],", "ade604c6eb0ffe75b214e2c89cf83f56": "[b]", "ade63dda3b3b03242f58a099a4955113": "\\hat{\\mathbf{\\theta}}", "ade67eae6143b33b45acc4c807df84b2": "\\phi=\\varphi_{2}-\\varphi_{1}", "ade6af44f3ea977d936a5d10e2471f9f": "\\Pi_{a_1, ...,a_n}( R ) = \\{ \\ t[a_1,...,a_n] : \\ t \\in R \\ \\}", "ade6be566d3c16d6b4da3a7b1ed937a2": " \\widehat{\\Omega}^{-1} = \\widehat{\\Omega}^\\dagger ", "ade6d42e5fd4f333b996f577756b852e": "\\begin{align}\n\\gamma(T,R\\!H)&=\\ln\\left(\\frac{R\\!H}{100}\\exp\\left(\\frac{bT}{c+T}\\right)\\right)=\\ln\\left(\\frac{R\\!H}{100}\\right)+\\frac{bT}{c+T};\\\\\nT_{dp}&= \\frac{c\\gamma(T,R\\!H)}{b-\\gamma(T,R\\!H)};\\end{align}\n", "ade6da69c72c4f49bd6087c561d169db": "A[x]A[y]", "ade70117ce02ba078eb67d66f7401e40": "\\mathrm{div}(f) > D ", "ade7148b0bada2f95cb20ab1200b0511": "f(z) = \\int_D f(\\zeta)\\overline{\\eta_z(\\zeta)}\\,d\\mu(\\zeta)", "ade7a9a613adcdfef3575c70047d0d0a": "\n \\begin{matrix}\n a\\uparrow\\uparrow\\uparrow\\uparrow b= &\n \\underbrace{a_{}\\uparrow\\uparrow\\uparrow (a\\uparrow\\uparrow\\uparrow(\\dots\\uparrow\\uparrow\\uparrow a))}\\\\\n & b\\mbox{ multiplied copies of }a\n \\end{matrix}\n ", "ade7da740c6577fc0edc1bfb2f5ea4ea": "t=\\frac{C\\cdot (U_\\text{charge}-U_\\text{min}) }{I}", "ade8000c042a92fa13fabde07e4538eb": "|x-3| \\le 9 ", "ade801bb5bd45ee382e84577156f4906": " \\hat{P}_{MUSIC}(\\theta)=\\frac{1}{V^H(U_n{U_n}^H)V} ......(8) ", "ade810925a7792b05ae5da205b500368": "=m_0 c^2 \\left( \\left( 1 - v^2/c^2 \\right) ^{-\\frac{1}{2}} - 1 \\right) \\ ", "ade82bedfb6c0cc1f2b80455c0ea8eea": "\\det \\left(\\begin{matrix}Z_0&Z_1\\\\Z_2&Z_3\\end{matrix}\\right) \n= Z_0Z_3 - Z_1Z_2.\\ ", "ade9118630eedf98d36830eb8787495f": "\n k_B^2 = {4 \\pi e^2 \\over \\hbar \\omega_c A_M L_B}\n.", "ade9ada08b86de07a836be07bceae8f0": "a*((a+1)*(a+2))", "adea1b76ea2824ade3cf10bde3ed8abd": "J_{X_t} \\leq t \\leq J_{X_t+1}", "adea86af3837400532ddc69ed25dab50": "k_D=2", "adeaa649d1ad7e1f6fecbe62de1811c8": "P_o:\\operatorname{det}(H_q(M))\\stackrel{\\sim}{\\longrightarrow}(\\operatorname{det}(H_{n-q}(M)))^{-1}", "adeaa8ebdb2df9b73e2cb9b020bd9d68": "B(L(T))", "adeab8021eddf44f6e2c632a8cc90f48": "\\mathrm{Operating\\;leverage} = \\frac{\\mathrm{Revenue} - \\mathrm{Variable\\;Cost}}{\\mathrm{Revenue} - \\mathrm{Variable\\;Cost} - \\mathrm{Fixed\\;Cost}} = \\frac{\\mathrm{Revenue} - \\mathrm{Variable\\;Cost}}{\\mathrm{Operating\\;Income}}", "adeb289bd7be7a4dbbe6ebd51045969a": "\\mathfrak{t}_i", "adeb6eae7ccfc21b07c9043b721bcad8": "\\frac{{6 \\choose 4}{42 \\choose 1}}{{49 \\choose 6}}\\approx\\frac{1}{22,196.5}", "adeb9ab79f416ba5986e0bf3d5a58dbb": "\\sigma = 1/\\rho \\,\\!", "adebc776c786e83e7b93d8dc081b12cc": "M_{t,t^{'}} = \\frac{\\sin2\\pi W(t-t^{'})}{\\pi(t-t^{'})}.", "adebd76aaa186af4dd9c0df5253ed8b5": "p_{ij} = \\Pr(X_{n+1}=j\\mid X_n=i). \\,", "adec3f16d38597b244a0e44e049bdc95": "_{interval} \\delta_{13}^2 =~_{interval} \\delta_{24}^2=2^2", "adeca4bf58936639defb63fb56e4873c": "\\color{Mahogany}\\text{Mahogany}", "adecb126c72ccce41f399a98defdc808": "\\theta _s\\,", "adeccf49e50a40930af4326d2bf41f12": "f(t) = 1+t", "adecfa7750515b6b48a16549a9dae78a": "[\\mathcal{L}f](s)=\\int_0^\\infty e^{-st}f(t)dt", "aded119f737a5fddcb4bca983fbf4c5b": "\n\\begin{align}\n X &= a \\cos\\omega\n \\frac{\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}}\n {\\sqrt{a^2 - c^2}}, \\\\\n Y &= b \\cos\\beta \\sin\\omega, \\\\\n Z &= c \\sin\\beta\n \\frac{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}}\n {\\sqrt{a^2 - c^2}}.\n\\end{align}\n", "aded53bb7a30b78471491a0fb62c4252": "\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}\n= \\begin{bmatrix}\n0.41847 & -0.15866 & -0.082835\\\\\n-0.091169 & 0.25243 & 0.015708\\\\\n0.00092090 & -0.0025498 & 0.17860\n\\end{bmatrix} \\cdot \\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix},\n", "adedc64985660bfde0795b304bb669ec": "x \\wedge (y \\vee z ) = (x \\wedge y) \\vee (x \\wedge z )", "adede9781f4fa4d7a4e3dc9cba30dcda": "\\rho(x) := \\rho(A,B; x) :=\\frac{x^T A x}{x^T B x},", "adedfed674a94365bf6d37787a91538e": "\\{gH: g \\in G\\}", "adee31c87561bc55bc97fbd122bf5702": "\\sigma_b", "adee41b39dac61e18a7601473845302e": " f_c(t) = \\frac{1}{2} g_c(t) g_c(-t) ", "adeec3de02d9d5d90f0a801dc997b33a": "(V,B) \\in \\mathcal{G}(V,E,F)", "adeedc6e9f0700c64e8d823847315a38": "\\eta_\\mathrm{rcp}\\approx 0.644", "adef9b4d763ecd516201181c795507c9": "X_j Y_j", "adefa618186933ea28623e8ecc184fb9": "C_*(X)", "adefa6891dd7b5e1a04d931354a40a8d": "\\int_0^\\infty \\frac{x^m \\, dx}{({x^n+a^n)}^r}=\\frac{(-1)^{r-1}\\pi a^{m+1-nr}\\Gamma [(m+1)/n]}{n\\sin[(m+1)\\pi/n](r-1)!\\Gamma[(m+1)/n-r+1]} \\ \\ , n(r-2)\\frac{h}{4 \\pi}", "adfc4ce4b89a89d156008e2f80ed8977": " v = ", "adfc8b73d1e1a2748c17f1ac6e5a6525": " Z = \\begin{bmatrix} X_i & Y_{-i}\\end{bmatrix} ", "adfcbede356bc32f0d777330988cdd8b": "1 \\leq i < j \\leq e", "adfcdcfbd05ebd80c20cbe5be935ca8e": "s_B = \\frac{Q(AC)}{Q(BC)} = \\frac{Q(DC)}{Q(AC)} = \\frac{Q(AD)}{Q(AB)}.", "adfd3f0a80ca9d0986ae5c046ab2e719": "E_\\text{inst,creep}", "adfd6ef912d86df8375ded001762cbe4": "\n\\begin{align}\np(t) &=& tp_1 + (1-t)p_0\\\\\n &=& p_0 + t(p_1-p_0) \n\\end{align}\n", "adfdc72f39503841632d29f0f72cf454": "u_\\varepsilon (y)=\\frac{A}{\\pi D}\\ln\\frac{1}{\\varepsilon}+O(1),", "adfdfe564c87e6e4d7c3bab06a4976ed": " f^{-1}(\\text{Spec }B) ", "adfe459d735d0d9682707f054403d6c5": "FP_{1}", "adfe8792561660c4ecb165da06a9fcd1": "\\neg p \\leftarrow \\mathrm{not}~p", "adfee3591785d81dc5592c7767157fac": "\\alpha < 1 ", "adff8aeea83c829d78cf5c65146b2dd6": "\\Delta t' = \\frac{2 D}{c}.", "adff8f5df1d7435f7597f3d841c356ea": "\\cos\\frac{x}{2}=\\sqrt{\\frac{1+\\cos x}{2}}", "adffd515d0b02621da85f632156ded19": "H_n(X, R)\\ ", "ae0047607c98ee639357cb9d2ca6c639": "\n\\sigma _z^2 \\approx \\left( {{{\\partial z} \\over {\\partial x}}} \\right)^2 \\sigma ^2 \\,\\, = \\,\\,4\\,x^2 \\,\\sigma ^2 \\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,4\\left( {\\mu ^2} \\right)\\sigma ^2 = \\,\\,\\,\\,4\\,\\mu ^2 \\sigma ^2 ", "ae0069d93a8f508446c0a2684c8acbde": "\\int_0^\\infty \\frac {e^{-ax}-e^{-bx}}{x \\csc px}\\ dx=\\tan^{-1}\\frac{b}{p}-\\tan^{-1}\\frac{a}{p}", "ae01176b6cfe4d3ff69e86cfb1f0e9d3": "\\frac{S}{Nk} = \\ln\\left(\\frac{VT^{\\hat{c}_v}}{N\\Phi}\\right).\\,", "ae01724fe287c004f9c8f7c5f83fe808": "x \\mapsto a x, \\quad y \\mapsto y/a, \\quad a \\ne 0,", "ae01cd3560765b24b0e0d643a2e81b43": "\n \\begin{align}\n & bD\\cfrac{d^3 w_x}{d x^3} + n_1(x)\\cfrac{d w_x}{d x} + n_2(x)\\cfrac{d \\theta_x}{d x} + q_{x1} = 0 \\\\\n & \\frac{b^3D}{12}\\cfrac{d^3 \\theta_x}{d x^3} + \\left[n_3(x) -2bD(1-\\nu)\\right]\\cfrac{d \\theta_x}{d x}\n + n_2(x)\\cfrac{d w_x}{d x} + t = 0 \\\\\n & bD\\cfrac{d^2 w_x}{d x^2} + m_1 = 0 \\quad,\\quad \\frac{b^3D}{12}\\cfrac{d^2 \\theta_x}{d x^2} + m_2 = 0\n \\end{align}\n", "ae01f305a6e3bdb1d1891a5c1971dc8c": "w=w'-e", "ae01fb8d278282550edb48ef796a5170": "E_{mass}=\\frac{m R_n + \\rho_a c_p \\left(\\delta e \\right) g_a }{\\lambda_v \\left(m + \\gamma \\right) }\n", "ae02b572d1d17d42fff40a36a6cbc7fb": " E_{-} ", "ae02c4eab6e5298cafbec04dbf81e731": " \\Delta G = -T \\Delta S \\,", "ae031ba5f6837365b3ac78934b3fbf19": "\\scriptstyle{p \\times q}", "ae03268c356670dbba8b79dbfeea6ef6": "{\\mbox{Rate of effusion of gas}_1 \\over \\mbox{Rate of effusion of gas}_2}=\\sqrt{M_2 \\over M_1}", "ae034325adef6e6501e48b911501796f": "\\sqrt{1-z^3}\\!", "ae035d0615a511ccaa5a4492338e1f68": "\\,\\sum_{w \\in V} f(s,w) = d \\text{ and } \\sum_{w \\in V} f(w,t) = d", "ae035e1ee006974bce87c6296b3d368b": "t \\in T", "ae036810a5f608c14e76c6448e6bd756": "S\\cap T", "ae0378bdf7665ea8cce07850c32e4535": "F_n^2 - F_{n+r}F_{n-r} = (-1)^{n-r}F_r^2", "ae038eeb17eda194dfe868f5837fe8ab": "2CO \\rightleftharpoons CO_2 + C", "ae03e6fd7ec13180c1c00ad0dbb0c9a3": "(A_1, \\ldots , A_n)", "ae0414832391f56de3716dd548e26d7d": "n=b-a+1\\,", "ae0415432346a15b811e9dfeeeafcb27": "(p_1,p_0)=((x-a_{0,0}p_0,p_0)=(xp_0,p_0)-a_{0,0}(p_0,p_0)=(xp_0,p_0)-(xp_0,p_0)=0.", "ae04b66f563cdded2152ecfd40db6877": "\\text{Im}(\\rho)=\\text{Im}(-m^a \\bar{m}^b \\nabla_b l_a )\\,\\hat{=}\\,0", "ae04ecf990155597110d55b8097bcbe3": "(3)\\quad L=\\frac{1}{2}\\big(l_+ + l_- \\big)\\,,\\quad l_+ =\\sqrt{\\rho^2+(z+M)^2}\\,,\\quad l_- =\\sqrt{\\rho^2+(z-M)^2}\\,,", "ae051c4bb6460ee43213525a926f5fab": " p\\geq 1/2 ", "ae0552799feee4f82ebf2f80ac541780": "h'(t)", "ae055d59123863e05e4450730d86eace": "2\\mid6", "ae056d46c7bc4d561c5f33bbaf646dad": "x\\leq c_\\kappa x", "ae0572f32f60bf91ec0bbf92e9abde03": " M^2(B)=\\Lambda(B)+\\Lambda(B)^2. ", "ae05effd537106b387d52e088ec0d6da": "\\mathbf F_1", "ae05f2b1843211f27dd7ce7c97cf6fef": "\\arctan _\\eta x=\\arctan x + \\pi\\cdot \\rm{rand}\\frac{\\eta-\\arctan x}{\\pi} .", "ae06012cbd5f1fa6cb2aa39d43b50a88": "K\\subseteq U", "ae06113ce0d044d277f917b8c6eef7bd": "f_1(x)", "ae061a236572fb9fd43913d5ea1c2469": "\\ddot r -r\\dot\\theta^2", "ae0620f054675c50d8d73d018dc0734f": "\\langle\\beta|\\alpha\\rangle=e^{-{1\\over2}(|\\beta|^2+|\\alpha|^2-2\\beta^*\\alpha)}\\neq\\delta(\\alpha-\\beta).", "ae06679c66878c1bae231f1409587a11": "H_1(S)", "ae0741eba16d6d54facd75b00e78502e": "\\begin{bmatrix}0&0\\\\0&1\\end{bmatrix}", "ae0788aaf4519645fe44c766371615b8": " \\bar \\psi D_\\mu \\psi \\mapsto \\bar \\psi D_\\mu \\psi ", "ae07d22a0a4999141e275e5c8794592c": "X\\left(t\\right)=-\\int_t^T f\\left(t,s\\right)\\,ds.", "ae07efea7894a76fe6f1d5f73947605b": "x_3=4(8/9)(1/9)=32/81", "ae0806bc339f805c4b9e5aa133015b3c": "g_\\otimes\\circ\\tau = g_\\otimes", "ae084bd7c450875e503e861caadf424e": "\\exp \\mathfrak m_+", "ae084fc86a3764e72f449c291c227895": "K = \\langle S, s^i, \\mathcal{T}, L \\rangle", "ae0867b993b156dce46f50f9adc5852b": "M = R(1-cos(\\frac {\\Delta}{2}))", "ae087a4cba378a9d3a0df73942c36bb5": "\\phi=0, \\, t=0", "ae089f33494d75ba02fcf25b1f2bdad9": "H_{D}(x,R)", "ae08dfa4d2885e4ddd0752aed1e208d4": "q(x,y) = ax^2 + bxy + cy^2\\quad \\textrm{(binary)} ", "ae08e66242e2555ebba76270e056418b": "M(m,n;\\mathbb{K})", "ae090d40b93b5de83f128ac2d6e66c67": "\n\\begin{align}\n\\sin 0^\\circ & = 0, \\\\[10pt]\n\\sin 30^\\circ & = \\frac 12, \\\\[10pt]\n\\sin 90^\\circ & = 1.\n\\end{align}\n", "ae09320faa20c3a210ee7d3f94ebd40b": "\n\\lambda_r^2 = (t/t_0)^2 = C_2p\n", "ae094769a4ed9b189316833ffac2c89e": " E f( a_i b_i + \\cdots + a_n b_n ) \\le E f( Z ) ", "ae096c3aeb98b14ede366b73643d7222": " k_z = \\sqrt{\\frac{\\omega^2}{c^2}-\\left(k_x^2 + k_y^2\\right)} ", "ae097b5d29385fea6f98b43fb81acff6": "h(0,x) = {1\\over 1+x^2}\\text{ for }t = 0 \\, ", "ae097de0be28a73ceea5824b66f75f40": "V \\approx \\ker T \\oplus \\operatorname{im}\\,T", "ae099bc50f992a7e294ca24135ae4443": "73_{11} \\ ", "ae09d39fb9d77adcbcadba06eefa16a5": "H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k", "ae09e608940294be255c048da5b3980b": "A( x )", "ae0a37b7ef14b72c3e6575db35fff353": "r = \\sqrt {y^2 - x^2} . ", "ae0a6983f5941334d5571f2c7470e3a7": "-1.5788", "ae0a81e277ce33dd808ad1939e0d4243": "a' = a + 1", "ae0a8608c47a17e6a47b9291f1a146ee": "\\sigma_\\mathrm{n} = \\frac{1}{2} ( \\sigma_x + \\sigma_y ) + \\frac{1}{2} ( \\sigma_x - \\sigma_y )\\cos 2\\theta + \\tau_{xy} \\sin 2\\theta", "ae0ab1c404a6ef7ebc35bf3158bd7990": "\\begin{pmatrix} N_t \\\\ D_t \\end{pmatrix} = J^{t} \\begin{pmatrix} N_0 \\\\ D_0 \\end{pmatrix}.", "ae0ad9cefc1ea111e61b78d7488de4d2": "\\Psi_\\ell(X,j(E))", "ae0b2af98abf778a76d96551fc69663e": "{\\tilde{A}}_1", "ae0b38201cc3969be196ce3efccb1875": "\n\\epsilon=10 \\mbox{GeV/fm}^3 = 1.8\\times 10^{16} \\mbox{g cm}^{-3} \n", "ae0bfb3a23809c8cf51bd37bad679cb9": "\\;\\exp\\;[-\\,(z + H)^2/\\,(2\\;\\sigma_z^2\\;)\\;]", "ae0c03f5089b4b4d31fe7f6de55c8b71": "\n\\min_{\\alpha \\in \\mathbb{R}^p} \\|\\alpha\\|_0 \\text{ such that } x = D\\alpha,\n", "ae0c3b8674116a188eae03940e4a0241": "Y \\stackrel{\\Delta}{\\longrightarrow} Y \\times Y", "ae0c3d45350ce9f29b51944121fb6116": "f \\in \\mathcal{H}", "ae0c560bc39962f0d3a7724046823ff4": "\nw_{pj}^{U(new)} = w_{pj}^{U(old)} + \\eta (x_{ij}^p-z_{pj}^{old})\n", "ae0c7a686a2b17a47eec7ce6ddc9d3c9": " {I} ", "ae0c86301913655a7da093a45ab14c85": " \\mbox{Revenue} = PQ_d", "ae0c8938608f8da3c77f87e5e1c9cb75": "\\displaystyle{U=8L(e)^2 -8L(e) +I,}", "ae0c8d4410ffbfea1ed1dfd34404e57d": "\\Psi_g(ab) = \\Psi_g(a)\\Psi_g(b)", "ae0c9267828b126d57f682c10de2fbdf": "\\mathfrak{a}:", "ae0cd1aeff18f7541d888d5821702f0c": "0\\mathbb{Z}", "ae0d2516ba45854f386ff442cee9f9b8": " F=6\\pi \\eta R v ", "ae0d27e115bc241c286df214b6112686": "(\\tfrac\\mathrm{mol}{m^3})", "ae0d8104d9467a208db4b38f04958e5f": "z'=\n\\frac{\\mathit{far}+\\mathit{near}}{\\mathit{far}-\\mathit{near}} +\n\\frac{1}{z} \\left(\\frac{-2 \\cdot \\mathit{far} \\cdot \\mathit{near}}{\\mathit{far}-\\mathit{near}}\\right)\n", "ae0ddb87ee88d6d1681cecd838158f2a": "\\| f \\|' = \\sup_{x \\in \\mathbb{R}} |F(x)| = \\| F \\|_{\\infty}.", "ae0eef8cb3bbd510703902a0f9ea2e22": "N_c", "ae0efd8ef8e094f438c30c5c7e5ca38d": "\\mathbf v = (\\mathbf v \\cdot \\mathbf u)\\frac{1}{\\mathbf u} + (\\mathbf v \\wedge \\mathbf u) \\frac{1}{\\mathbf u}", "ae0f20f574ada5eb85eeca5e7cee4c24": "\n\\frac{d^{\\ell-m}}{dx^{\\ell-m}} (x^2-1)^{\\ell} = c_{lm} (1-x^2)^m \\frac{d^{\\ell+m}}{dx^{\\ell+m}}(x^2-1)^{\\ell},\n", "ae0f2d92206f63e688e9f4280fc07980": " \\bar{x}_n ", "ae0f40d59531db56be70421938472046": "b_2 = V_2^-", "ae0f64edca5a2e9c5b3d863d220bb4b1": " C = \\sqrt{{gL}/{2\\pi}} ", "ae10276778f02098d2f34de9e390323e": "\nm \\ddot x_1 = - k x_1 + k (x_2 - x_1) = - 2 k x_1 + k x_2 \\,\\!\n", "ae10372ba8e34ef8eb80b0dcbbbdec79": "\\sigma_{ik}= - \\frac{\\partial W}{\\partial \\varepsilon_{ik}} \\qquad\\text{for } i,k=1,2,3", "ae10384c95a2476595e4c5e0f18e2431": "\\omega_{c}.", "ae105da046340773eb2b92811e0fe5f6": "F(2)", "ae105df87f29167c63a7751fc83d9304": "I:\\mathcal{C}\\to \\textbf{Set}", "ae109e03c32e10edc5593689a4d84c12": "\\begin{align}\n&\\lfloor x \\rfloor + \\lfloor y \\rfloor &\\leq \\;\\lfloor x + y \\rfloor \\;&\\leq\\; \\lfloor x \\rfloor + \\lfloor y \\rfloor + 1,\\\\\n&\\lceil x \\rceil + \\lceil y \\rceil -1 &\\leq \\;\\lceil x + y \\rceil \\;&\\leq \\;\\lceil x \\rceil + \\lceil y \\rceil.\n\\end{align}", "ae10ab1e4b85571b9a1de76712020a53": "a=1, \\, q=\\frac{1}{5}, \\, \\mu \\approx 1 + 0.0995 i", "ae10c4af42896cc58b325ba9783434e7": "\nr = \\sqrt{\\frac{r_1^2+r_2^2-2c^2}{2}}\n", "ae10dd5112d84cc344d08f13b9058f59": "u=ix", "ae1110eab7ba0e4f1930b5f0c24ad78d": "\\Pi^0_1", "ae112a2312ba101674597a5e4883db14": "g(\\mathbf{x}^\\prime) = \\mathbf{P}_\\beta \\mathbf{P}_\\alpha \\; g(\\mathbf{x}) \\; \\mathbf{P}_\\alpha \\mathbf{P}_\\beta", "ae1144f94b1a440e19b894f9cb9939f5": "\\rho dxdy", "ae119117c4d8619fcac15b31100d3888": "\\int { \\frac{f'(x)}{f(x)}\\, dx} = \\ln |f(x)| + C.", "ae11e1bfb75788e7402f5fb9e9ddc36b": " H^{k}_{\\mathrm{dR}}(M) ", "ae1209f2556a88e5f80cefe95d275ed8": "\\sigma_N^2 = \\int_{-\\infty}^{+\\infty} \\Phi_N (f) \\cdot |H_r (f)|^2 df", "ae1268fdf6e5b7a340acad2989846814": "\\left(-\\sqrt{20/11},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "ae129d4dff4b0f561b458487ac256989": "e^{x} = \\sum^{\\infty}_{n=0} \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots\\quad\\text{ for all } x\\!", "ae12bee4c7dce199dbb3736b27b193fc": "-\\Bigg[\\frac{V_{n k_2}V_{k_2 k_1}V_{k_1 n}+V_{k_2 n}V_{k_1 k_2}V_{n k_1}}{E_{n k_2}^2E_{n k_1}}+\\frac{|V_{n k_1}|^2V_{nn}}{E_{n k_1}^3}\\Bigg]|n^{(0)}\\rangle", "ae131293bb0d8d0df32a71e52fc4078b": "l^n", "ae13af9c17a1c811f9d21c4b01712131": "\\int^T_0 R_N(t-s)k(s)ds = S(t) ", "ae13d2084de8e11866cfa606783d92a3": "{ E = m } \\ ", "ae13d82c2fc547ba46e37338cd4a1f3a": "Z_\\mathrm{nom} = 1.15 \\cdot Z_\\mathrm{min}", "ae13d8bf5257bc28eff6c72a69e92900": "\\Gamma(\\mathbf A,\\ \\mathbf B )=\\begin{vmatrix} \\mathbf{A\\cdot A} & \\mathbf{A\\cdot B} \\\\\n \\mathbf{B\\cdot A} & \\mathbf{B\\cdot B} \\end{vmatrix} \\ . ", "ae13dd3cb22eaa593571c1e8b89b55b0": "H_{\\mathrm{FOH}}(s)\\,", "ae13e1ef14f941f63da46d8de08069ee": "\\frac{i}{c}\\frac{\\partial}{\\partial t}\\psi=-\\frac{1}{2} \\left(\\frac{\\hbar}{m c} \\right) \\nabla^2\\psi - \\frac{\\alpha Z}{r} \\psi", "ae13e5d183dc3701485dbd72e48fe5be": "\\hat{X}=(W^{H}_{L_\\xi}W_{L_\\xi})^{-1}W^H_{L_\\xi}xW^H_{L_\\nu}(W^H_{L_\\nu}W_{L_\\nu})^{-1}\\,", "ae13f935c88f16e07aecb03ccca11bb8": "\\begin{align}\n\\hat{\\operatorname{E}}[\\ln (X)] &= \\psi(\\hat{\\alpha}) - \\psi(\\hat{\\alpha} + \\hat{\\beta})=\\frac{1}{N}\\sum_{i=1}^N \\ln X_i = \\ln \\hat{G}_X \\\\\n\\hat{\\operatorname{E}}[\\ln(1-X)] &= \\psi(\\hat{\\beta}) - \\psi(\\hat{\\alpha} + \\hat{\\beta})=\\frac{1}{N}\\sum_{i=1}^N \\ln (1-X_i)= \\ln \\hat{G}_{(1-X)} \n\\end{align}", "ae141a47ebf980af2be9933a176cacc1": "\\frac{f_0}{1 - v/c} \\qquad \\text{and} \\qquad \\frac{f_0}{1+v/c}. \\,", "ae1434377b9d77654c423b6b600a889e": "\\mu_3=2\\sqrt{2}\\,\\,\\frac{\\Gamma((k\\!+\\!3)/2)}{\\Gamma(k/2)}=(k+1)\\mu_1", "ae14538a4e9ca1603113e058d2b59e89": "\\mathbf{F} = q \\left(\\mathbf{E} + \\mathbf{v}\\times\\mathbf{B}\\right) ", "ae1491938545ebd98a1d4d95e422af90": "\n\\begin{align}\nS &= \\frac{p \\times S_u + (1-p)\\times S_d}{1 + r} \\\\\n&= \\frac{p\\times u\\times S + (1-p)\\times d\\times S}{1 + r} \\\\\n\\Rightarrow p &= \\frac{(1+r) - d}{u-d}\\\\\n\\end{align}\n", "ae14a32bd466191991b49a5b31b2ed65": "\\displaystyle{\\exp Z= \\begin{pmatrix} e^z & 0\\\\ f(z) w & e^{-z} \\end{pmatrix}, \\qquad f(z)={\\sinh z\\over z}.}", "ae14a69360bde0252bc957eb6cdee992": "\n \\mu(p,T) = \\frac{1}{\\mathcal{J}(\\hat{T})}\n \\left[\n \\left(\\mu_0 + \\frac{\\partial \\mu}{\\partial p} \\cfrac{p}{\\eta^{1/3}} \\right)\n (1 - \\hat{T}) + \\frac{\\rho}{Cm}~k_b~T\\right]; \\quad\n C := \\cfrac{(6\\pi^2)^{2/3}}{3} f^2\n ", "ae14c88d6594c555453a872bec830e91": "50i-12", "ae14d2af675afa4b9894d6ffea76e7c2": "i_{X^\\#}", "ae14ec786b44714b1a690882a6f80772": "\\scriptstyle (a,\\, b)", "ae14f2ce8a8f59adfb350596b22005e6": "\\,\\kappa_2", "ae156486fc851684a383f93c23fce823": "b = \\frac{1}{\\tau}", "ae156c8714282f64c6e2fc7dca5c56e8": "|C_\\beta| < \\kappa ", "ae1577141a71d4477f5f8ab50d1fb097": "n_\\eta(i\\omega_m+\\xi)=-n_{-\\eta}(\\xi)", "ae1592643e53bf5d3fc1c3fffba46f84": "PQ ", "ae1599f20901208d2ad639f4294df2eb": "\n2 \\epsilon \\sigma T_a^4 - \\epsilon \\sigma T_s^4 = 0 \n", "ae15b16446df0b4c8258f680f48c5286": "\\hat{H} = \\frac{\\hat{p}^2}{2m_0} + V(r)", "ae15b930d1914e9522ad2f161da4df94": "\\lambda_J=\\frac{2\\pi}{k_J}=c_s\\left(\\frac{\\pi}{G\\rho}\\right)^{1/2},", "ae15c283112b3350b0ac0ca7795ce1f2": "\\mu(B(x,r))\\le r^s", "ae15fab1338d03994f123ebbfe80e65b": "\\boldsymbol{F}_2", "ae16c2ca616293c446913fca87c2ed4f": "T=\\frac{p^{2}}{2m}-\\frac{p^{4}}{8m^{3}c^{2}}+\\cdots.", "ae16d952c6dbf0cdf194a4a0c118ca44": " \\frac{r_{t+\\Delta t}-r_t}{\\sqrt r_t} =\\frac{\\theta\\mu\\Delta t}{\\sqrt r_t}-\\theta \\sqrt r_t\\Delta t + \\sigma\\, \\epsilon_t ", "ae16f80451d0032fb71a01c82621a683": "\n-\\frac{1}{4}S_0(1-\\alpha_p)+\\epsilon \\sigma T_a^4 + (1-\\epsilon) \\sigma T_s^4= 0 \n", "ae17098b33ed2eb1790ffc239382df9b": "[X']=[X]\\,", "ae177a1d0e95f097220e139d3b74c07b": "\\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \\lambda t E(D).", "ae178bf563457bcd14cdfd97125ad6ca": "\n\\begin{align}\n\\vartheta_{00}(w, q)& = \\sum_{n=-\\infty}^\\infty (w^2)^n q^{n^2}\\quad&\n\\vartheta_{01}(w, q)& = \\sum_{n=-\\infty}^\\infty (-1)^n (w^2)^n q^{n^2}\\\\[3pt]\n\\vartheta_{10}(w, q)& = \\sum_{n=-\\infty}^\\infty (w^2)^{\\left(n+1/2\\right)}\nq^{\\left(n + 1/2\\right)^2}\\quad&\n\\vartheta_{11}(w, q)& = i \\sum_{n=-\\infty}^\\infty (-1)^n (w^2)^{\\left(n+1/2\\right)}\nq^{\\left(n + 1/2\\right)^2}.\n\\end{align}\n", "ae17a491daade451114f8cd068ab3d30": "n \\over {w + 1}", "ae17b02badc2fbe87ea0ecf6874b03fc": " \\hat{f} \\, \\hat{f}^\\dagger - : \\hat{f} \\, \\hat{f}^\\dagger : = 1.", "ae17db8dfc64b02c7f238ceea1b06bb9": "V_{3m - 2} \\le v_0", "ae180e5140ee2c56f13742f40509d68a": " \\frac{(RL)^{-1}} {1-(RL)^{-1}} y_t = \\frac{1} {R} L^{-1} y_{t} + \\left(\\frac{1} {R} \\right)^2 L^{-2} y_{t} + \\left(\\frac{1} {R} \\right) ^3 L^{-3} y_{t} - . . . = \\sum_{j=1}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j}y_{t+j}", "ae1835b9b52dcd2b8dfd989cdc48d974": "\\mathbf{C}\\mathbf{P}^1", "ae18442447abb659adeae32665de2dd9": "\\sin(5\\tfrac58 ^\\circ) = \\frac12\\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{2}}}};", "ae1878673b8155997573f00f60bdbcdb": "R_\\text{eff}(\\lambda)=1-\\sigma(\\lambda)nd\\left(\\frac{I_0}{I}-1\\right)^{-1}", "ae188ac20923b140e15497a5acb2e8b9": "\\operatorname{DirMult}(\\mathbb{Z}\\mid\\boldsymbol{\\alpha})", "ae189d45db18dfbd9ec9efd407a98aa1": "\\psi * \\alpha \\models \\mu", "ae18d23603f5d6c8b9ec8c379403647d": "\\mathfrak{e}_{6(6)}", "ae18d394cd255336972c8daf140f6c33": "\\coth\\theta = \\frac{1 + t^2}{2t},", "ae18f520703656d100ae9049e756abff": "x^2 + y^2", "ae1945a83376f9bdb2aadfa0dff99401": "\\mu_i = \\mu_i^\\ominus + RT \\ln a_i", "ae19d29f97b7d23c319637ea86dea481": "\\mathbf D = \\frac{\\mathbf{D \\cdot (B \\times C)}}{[\\mathbf {A,\\ B, \\ C}]}\\ \\mathbf A +\\frac{\\mathbf{D \\cdot (C \\times A)}}{[\\mathbf {A,\\ B, \\ C}]}\\ \\mathbf B + \\frac{\\mathbf{D \\cdot (A \\times B)}}{[\\mathbf {A,\\ B, \\ C}]}\\ \\mathbf C \\ .", "ae19fb2844d3001e7198cf2c3461fb28": " J^{r}\\left(\\pi|_{\\pi^{-1}(W)}\\right) \\cong \\pi^{-1}_{r}(W).\\,", "ae1a02a52660b6faff489905f1406eb7": "|n(t) \\rangle", "ae1a280340c01741a4e5838ecb09edc0": "\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} = 1 + \\frac{W}{m_\\mathrm{0} c^2} + k \\frac{Z e^2}{m_\\mathrm{0} c^2} u", "ae1a5c6ac95733191f50cd0f3578cbd2": "\\phi(\\theta) = -\\tan^{-1} \\theta. ", "ae1a6038ecd0b0c3de71ddc95c04c87f": "R[G]", "ae1aab3b41eb2ffe447a4663c0eef8c4": "P_d = 0.999999999", "ae1bb432ba3ca086f018f5ca04495f84": "\\mathbf{P}(\\omega) = \\varepsilon_0 \\chi_{\\text{e}}(\\omega) \\mathbf{E}(\\omega).", "ae1c05a923377e72b952ef71b559ea23": "x_k|x_{k-1} \\sim p_{x_k|x_{k-1}}(x|x_{k-1})", "ae1c24d53b79e1eb5430165a38c647c3": "\\left[D_\\mu , G^{\\mu\\nu} \\right] = g_s j^\\nu ", "ae1c53133e7292be0590d392c38a1982": "H = \\frac{1}{2m} \\left [ \\boldsymbol{\\sigma}\\cdot \\left ( \\mathbf{p} - \\frac{e}{c}\\mathbf{A} \\right ) \\right ]^2 + e\\phi.", "ae1c861ef8c0cfb7d190630c0199bb45": "\\operatorname{excsc}(\\theta)", "ae1c8d830c1e0756699deae0dab15d7e": "\\mathbf{p} = \\hbar \\mathbf{k}\\;", "ae1cc46015397bebf3ab25231f062d14": "{\\left| G \\right\\rangle}", "ae1cef0d7a64b02031f096aa014d9ffb": " m = \\frac { \\mathrm {log} (F_2) - \\mathrm {log} (F_1)} { \\log(x_2) - \\log(x_1) } = \\frac {\\log (F_2/F_1)}{\\log(x_2/x_1)}, \\,", "ae1cf5863f9bffe0aad0c86639c0c139": "F(y)=0\\,", "ae1d44b34fb91ae147d2042c7d6c572b": " T = -\\frac{1}{r}\\ln\\left(1-\\frac{P_0r}{M_a}\\right) ", "ae1d4ab54458a5757167ad523c6af931": "q\\mathbf{A}(\\mathbf{r},t)\\cdot\\mathbf{\\dot{r}}", "ae1d52491e7803f957f4b825ba0b4db6": "u=\\int_0^\\phi \\frac{\\mathrm d\\theta} {\\sqrt {1-m \\sin^2 \\theta}}. ", "ae1d6ec6dd3dbf432e89ddb6059ad2d3": "\\frac{ \\partial f}{\\partial n} = \\nabla f(\\bold{x}) \\cdot \\bold{n} = \\nabla_{\\bold{n}}{f}(\\bold{x}) = \\frac{\\partial f}{\\partial \\bold{x}}\\cdot\\bold{n} = Df(\\bold{x})[\\bold{n}] ", "ae1e107302713679cedfaa83de8b6fbd": "X = \\operatorname{Proj}\\ k[x,y][z,w]/(xz - yw),", "ae1f01ab654d927291bc83211a418ee4": " \\tilde p(x_i )= p(x_i | x')= \\Kappa \\exp \\Big (-\\lambda f \\big ( \\Big| x_i - x' \\Big | \\big ) \\Big )", "ae1f0354d59f5e1dc6feafa0423dc1c9": "ds^2=dx^2+dy^2", "ae1f1561c38e109c97db6814eac9c202": " R = {\\sum\\limits_{{\\alpha_l}=1}^{{\\chi}_c}{|\\lambda^{[l]}_{{\\alpha}_l}|}^2} = {1 - \\epsilon_l}", "ae1f7bd5410ca26b8d44f4ce2fb13604": " w = d + [2.6m - 2.2] + 5R(Y,4) + 3R(Y,7) \\mod 7 ", "ae1fe7aa5e5466304903d17f0d928716": "O(k\\log n)", "ae1feccb04d649c105639a76b959b011": "\nL^{2} + x_{1}^{2} - r_{1}^{2} = L^{2} + x_{2}^{2} - r_{2}^{2} \n", "ae20749d1b49191b733c4342e74e22f9": "m+0=m", "ae20911ed7f05f8a5953d9d93484b71b": "\\alpha = \\frac{K_s}{K_m} \\approx 1", "ae209850ab35742ca579f429121db3f3": "\\left\\{X_i\\,|\\,i\\in I\\right\\}", "ae20cd37044faebf3f1bc426e5157bc0": "\\sqrt{k_2}(A_\\rightarrow+A_\\leftarrow)=\\sqrt{k_1}(B_\\rightarrow+B_\\leftarrow)", "ae20dd689b98af75e55aabd7c7a77ed1": "f(x) > g(x)\\,\\!", "ae20f1ddd39d0b4454455df9678cae5d": "\\boldsymbol{\\theta} = (\\mu,\\sigma^2)", "ae210622fba74a5c2d5027549b8b9d92": "ln(Y_d)=\\alpha+\\beta^T{X}+d\\ln(D)+\\epsilon\\,\\! ", "ae2122057b782d0081cd44a6565184c9": " \\prod_{k=0}^{n-1} \\cos(2^k \\alpha)=\\frac{\\sin(2^n \\alpha)}{2^n \\sin(\\alpha)}, ", "ae212a5ca170e6d2bc729e5057d55037": "\\displaystyle g=3", "ae214993779abf1b21a7e2c4b9a16c81": "\\phi \\rightarrow (\\exists x \\psi)", "ae214ce1612b31e56788503dc7a566e3": "\nP_{\\mu }(n=0,t)=E_{\\mu }(-\\nu t^{\\mu }). \n", "ae216d29a43b770a2dd03987c958fce2": "\\mathcal{L}_Xf = i_X df", "ae2201d1ad9cdd0fd6f32d925844f54d": "\\left\\{\\omega_1,\\omega_2\\right\\}", "ae220c024f8e48fcb2abb6d40f7d6b0f": " \\begin{bmatrix} v^* & 0 \\end{bmatrix} \\begin{bmatrix} A & B \\\\ B^* & D \\end{bmatrix} \\begin{bmatrix} v \\\\ 0 \\end{bmatrix} = v^* A v \\ge 0. ", "ae226151a2c8531a464444eb5dcf8563": "\\mathfrak{a},", "ae22ad722e54f010340d6b9a457cb963": "({\\mathcal P},{\\mathcal Z};\\parallel_+,\\parallel_-,\\in)", "ae22c14b549aa438f2082f1d743bcf3b": "\\scriptstyle s = 2", "ae22d74db6a49ed650084b282db907fd": "n \\ge 2", "ae2396a962af30bad554e212a19cd81f": "\\{1,1\\} \\uplus \\{1,2\\} = \\{1,1,1,2\\} \\,", "ae2397dfa864124366d79904e721ee90": "S^r\\times S^{q-1}\\rightarrow S^{q-1}", "ae23a73b61e62c334967f5e60f3b4564": "=\\frac{ab^2}{4a^2} -\\frac{b^2}{2a} + c", "ae23ba6bd7256ea959be84f274489451": "\\gamma(s, z) \\asymp z^s \\, \\Gamma(s) \\, \\gamma^*(s, 0) = z^s \\, \\Gamma(s)/\\Gamma(s+1) = z^s/s", "ae23f85a283b7d5858a1f4a28dac5b8b": " x, y \\in \\mathcal{E}", "ae2406f2439ded7254d435c9fb428aaf": "X(t)=\\mu t+ \\sigma W(t),", "ae242f508e74370b1c55d73c3d3ae1b4": "S_{\\alpha \\beta \\gamma} = 0", "ae2441597c6716655b8bdfdd785e2fdb": " \\boldsymbol{\\mathit{1}}", "ae247cf86aafb9eccd886340c5ab17c9": "D \\colon D^b(X)\\to D^b(X) \\,\\!", "ae24ac3f0ecb1bf8d72e7c1f1101f02b": " V_X = (I_X + \\beta I_b) r_O + (I_X - I_b )R_E \\ . ", "ae24b7d11d5da50360099c59979b83e8": "Z^3 - (1-B)\\ Z^2 + (A-2B-3B^2)\\ Z -(AB-B^2-B^3) = 0 \\;,", "ae2531937a44a9e36c56782b63b1fcc9": "e_{(\\mathbf I_1)}=\\mathbf I_1 \\cdot \\boldsymbol \\varepsilon \\cdot \\mathbf I_1=\\varepsilon_{11}\\,\\!", "ae2546d695af4f544574ca3a8fd5cd92": "\\textrm{PIANIST(Jill)} \\vee \\textrm{ORGANIST(Jill)}", "ae25b5ec61ae9ddd6f18ece5b44d3068": "{\\sum_{n=2}^{\\infty}\\varphi^n = 1}", "ae25f7dc2e60302407d34e800f903366": " p \\overset{\\alpha}{\\rightarrow} p' ", "ae26c61c57c9c6cf6cf19fb8efad8fb3": "11\\times 10^{-6}/\\,^\\circ\\text{C}", "ae26ebfa3f71328a1a1f580c883ad234": "(r, \\theta, t).", "ae2708f0b9a5711758322f6e676c4524": "\\lambda, \\xi", "ae27409b6b76b5235d7775d0c6809662": "L^{p_0} + L^{p_1}", "ae275807d7fdbe6bb7b00d5384dbe719": "\nR_{s} = \\frac{2GM}{c^{2}}\n", "ae2776798d919fae669baf113b9445ed": "\\gamma_t", "ae27790d7367b9deb76355518b426460": "c=111 000 111 000 000 111", "ae27a4680c0547613a5200b28d9912a7": "\\{\\land, \\lnot\\}", "ae27c613423ece836012735b4ca7d688": "v = \\theta \\hat{\\mathbf{e}}", "ae27ff01ef8025e03c1303767da3efb0": "PV = C/i", "ae280c91bc7ae811d67ab943417e12ca": "P(\\omega) = - \\frac{ne^2}{m\\omega^2} E(\\omega)", "ae2822dd85d8fbc111090e497ec3fff1": "T_i>t_i", "ae2827d21d71ade904244313479f3e47": "\\langle 1,2,0 \\rangle", "ae282dedc3787fe2bf621682aa181d3c": "\n F_1=F_2 ~;~~ F_4=F_5=F_6=0 ~;~~ F_{11}=F_{22} ~;~~ F_{44}=F_{55} ~;~~ F_{13}=F_{23} ~.\n ", "ae284b2a69dc7609ab896755b0aa01c1": "\\xi\\propto T^{-1}_c", "ae285b83c7928ad3becb83a53bdf0f80": " g \\in \\mathcal{S}(\\mathbb{R}^d) ", "ae289fbe843ad41d8a65a6b06b2c4229": "v_\\parallel \\in T_p N", "ae28d108a659e1c6d9cc9a5a88cdde45": "\\log_b a = n. \\, ", "ae290ec944e8802aaaedf2e3c3ea37c8": "\\sum_{n=1}^\\infty \\frac{1}{2n\\,x^{2n}} = -\\frac{1}{2}\\ln\\left(1-\\frac{1}{x^2}\\right),", "ae2932d61ea95aa51157ce364abe1e58": "p_{{nh}}", "ae299ca8219143ef1ed466df4ed2c02b": "\\!x = 2a \\cot \\theta,\\ y=2a\\sin ^2 \\theta.\\,", "ae299f957f89393efcf32771e77333ce": "F = \\frac{\\mu_0 m_1 m_2}{4\\pi r^2}", "ae29a35cdf57f79f306bf60f57f9cfa5": "6076.12 = feet\\ per\\ nautical\\ mile", "ae29d11157953942ae7ce71d9a909304": "\\mu (n,T) = \\frac{1}{\\beta} \\ln{(e^{\\hbar^2 \\beta \\pi n/m}-1)}", "ae2a0b05b68d126c8dfa21bde862cfee": "\\Phi_{,i i} \\approx \\Gamma^i_{0 0 , i} \\approx R_{0 0} = K \\left(T_{0 0} - {1 \\over 2} T g_{0 0}\\right) \\approx {1 \\over 2} K \\rho c^4 \\,", "ae2a46428fd75c6c8f730b6e40efed8b": "\\begin{align}\n&\\sum_{r=1}^{\\infty }{a_{r-1}(r+c-1)(r+c-2)s^{r+c}}-\\sum_{r=0}^{\\infty }{a_{r}(r+c)(r+c-1)s^{r+c}} +(2-\\gamma )\\sum_{r=1}^{\\infty }{a_{r-1}(r+c-1)s^{r+c}}+ \\\\\n&\\qquad \\qquad + (\\alpha +\\beta -1)\\sum_{r=0}^{\\infty }{a_{r}(r+c)s^{r+c}}-\\alpha \\beta \\sum_{r=0}^{\\infty }{a_{r}s^{r+c}}=0 \n\\end{align}", "ae2a692ecdd06a46626e5ed46bda713c": " \\hat{A} |A\\rangle = A |A \\rangle ", "ae2a737eae3a4ed165ffb3c911057407": " \\Pr(X_t = x) = \\frac{(\\lambda t)^x e^{-\\lambda t}}{x!}, ", "ae2ad0838e7f7243153e7b3e0a8ead2a": "\\displaystyle T=r(2R+r)", "ae2b0208eb35cb9955693ccd120a2ec2": "U(P)= - \\frac{ia \\cos \\beta}{\\lambda r's'} \\int_S e^{ik(r+s)}ds ", "ae2b08c1a26014b04f2a8ca22f91edd3": "(\\pm1,\\pm1,0,0,0,0)", "ae2b6cdf66415619681382295d77189c": "c/n < v_p < c", "ae2bac2e4b4da805d01b2952d7e35ba4": "0011", "ae2bb51fffbccf8d16dc6bedc066bf9b": "\n0_{1,1} = \\begin{bmatrix}\n0 \\end{bmatrix}\n,\\ \n0_{2,2} = \\begin{bmatrix}\n0 & 0 \\\\\n0 & 0 \\end{bmatrix}\n,\\ \n0_{2,3} = \\begin{bmatrix}\n0 & 0 & 0 \\\\\n0 & 0 & 0 \\end{bmatrix}\n.\\ \n", "ae2c05bea1d59f4d22fd9f6ae9b85777": "\n\\left(\\frac{\\mathbf{N}}{\\mathbf{C}}\\right)\n", "ae2c10f2ff55277d996699fcb8cf1fa7": "\\operatorname{ord}(z)", "ae2c20658a455e6ba3e14cc780b13d2a": "\\gamma_2= \\frac{\\mu_4}{\\mu_2^2}-3 = \\frac {a_1+16a_2}{(a_1+4a_2)^2}", "ae2c7ceb11809ce66197dcfb553bd69a": "f(n) \\in O(g(n))", "ae2cdce78cb670fc5f460e89c3f5331c": "V_{i-1}", "ae2cf0da7025212ce4469924f2dd9789": "V(\\phi)= \\mu^2\\phi^2 + \\lambda\\phi^4", "ae2d60a2770dfc3f9b54fcf3f4e56048": "H(Y)", "ae2daa995a16c983d65cfd8ebf6e5097": " \\begin{align}\\hat{H} & = \\frac{\\hat{\\mathbf{p}}\\cdot\\hat{\\mathbf{p}}}{2m} + V(\\mathbf{r}) \\\\\n& = -\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r}) \n\\end{align}", "ae2daef4b0d51b6b5d14636fcecb8108": "p_t^{k_t}", "ae2dc7e2893bc2138568b8baad20d8e5": "\\mathrm{ASPACE}(f(n))=\\bigcup_{c>0}{\\rm DTIME}(2^{cf(n)})={\\rm DTIME}(2^{O(f(n))})", "ae2dfae0e6f2e6824adfeb8335da1e04": " \\{ A, B, C, D \\} ", "ae2e54b3d151c3eb0481c65ff57a0c1a": "\\psi_L\\rightarrow \\psi_L", "ae2ec75b228efe009904168a614ca807": "\\mathcal{L}_H = \\left|\\left(\\partial_\\mu -i g W_\\mu^a \\tau^a -i\\frac{g'}{2} B_\\mu\\right)\\phi\\right|^2 + \\mu^2 \\phi^\\dagger\\phi-\\lambda (\\phi^\\dagger\\phi)^2,", "ae2effcfdbf04059f6d0faddb3357991": "f(x_i)", "ae2f0c342d7720957e5bc64bcef3d10b": "\\nabla:\\Omega^pM\\rightarrow T^*M\\otimes\\Omega^pM", "ae2f1f226ad0efa086af5006770f85dd": "\\scriptstyle |U|", "ae2f8889eebef8457934be5f38e855d5": "\\operatorname{Tr}\\; (\\Phi_x \\otimes I)(\\omega) \\cdot F_y .", "ae301bc66c4ac12d8dda34182275b61b": "\\frac{B_2}{v_0}", "ae301f665024eab26b395b61f1de0f4e": "f''(X\\otimes Y)=\\sum_{g\\in G}\\langle X,\\rho(g)[Y]\\rangle g=\\sum_{g\\in G} \\langle X,Yg^{-1}\\rangle g", "ae30c83515abf64a5c63f00e009d7205": "\\psi(t)= \\sum_{n=- \\infty}^{\\infty} g[n]{\\sqrt 2} \\varphi^{(k)} (2t-n).", "ae318a3a864ee61f7f2a2031c7ac9e3b": " \\Pi^- ", "ae3208c84dbb394ef5908df8be6f7919": " p(t) \\, dt = P[k(t)] \\exp[-\\int^t_0 P[k(t')] \\, dt' ] \\, dt ", "ae32bc393f0a6a8e1c0c2d679403b01d": "\\beta_{m,k}", "ae32c4db9062af37082390d02ee617b0": " \\prod_{p} \\Big(1 - \\frac{1}{p(p-1)}\\Big) = 0.373955... ", "ae32cc891d0e9a1ac1dcd2a7071045df": "(Demand_{p,c} + Backlog_{p-1,c})>0", "ae32d1217c323a3ba368cbeffd768515": "\\det(A^{\\rm T}) = \\det(A).", "ae32fd8f0df3807cc69d27c30da92b84": " \\omega=\\epsilon_{[t, t+dt]}", "ae3302c741f72a121375b717fc54a08c": "X_i \\sim \\operatorname{Beta}\\left (\\alpha_i, \\left(\\sum\\nolimits_{j=1}^K{\\alpha_j}\\right) - \\alpha_i \\right). ", "ae3336e9f1d1f6a2ada98c2551c710a1": "{r_x}", "ae33471194df3609c29698677dbb03c3": "\\scriptstyle H\\psi \\;=\\; E\\psi", "ae337268a115af1167e8d1415b879bf5": "\\Theta_n/bP_{n+1}\\to \\pi_n^S/J\\,", "ae339fc22a7711fa0cdcea163d5afb93": " \\!\\ \\varphi^n - \\varphi^{n-1} = \\varphi^{n-2} . ", "ae33d95a9afc9fa9ef7a42f647c15646": " \\mathcal{N}\\left(\\boldsymbol\\mu_0, \\sigma^2\\mathbf{\\Lambda}_0^{-1}\\right).", "ae34123646d8116408efa0ce7b1bcb0e": "\\ell_j(x_i)\n= \\prod_{m=0,\\, m\\neq j}^{k} \\frac{x_i-x_m}{x_j-x_m}\n", "ae34660b7e43845e725e58cd40ac7b26": "C_\\mathrm h = C\\,\\cdot\\bigg(\\frac{288 - 6.5 h}{288}\\bigg)^{5.2558}", "ae348c16e29f52115316beaf7baf0c70": "Va^{\\dagger}a V^{\\dagger}=a^{\\dagger}a+1", "ae34b5751cd88f95505b5d8f3d1c953b": "\n \\cfrac{\\mathrm{d}}{\\mathrm{d}t}\\int_{\\Omega(t)} \\mathbf{f}~\\text{dV} = \n \\int_{\\Omega(t)} \\frac{\\partial \\mathbf{f}}{\\partial t}~\\text{dV} + \\int_{\\partial \\Omega(t)} (\\mathbf{v}^{b}\\cdot\\mathbf{n})\\mathbf{f}~\\text{dA} ~\n", "ae34c3420c8c20e82c97dfbe9cd58f1c": "\\Sigma g_k", "ae34cee7f40747b5215bfb8d2b1af277": "\\sum_{n=0}^{\\infty} A_{n}", "ae34d9928f475ce27ad1fbe12522bee8": "u \\in L_{\\mathrm{loc}}^{1}(\\Omega)", "ae3586ba4d58f226606311c85cd0cac4": " N = a^{\\dagger}a", "ae35aa6b543588a09c77d30991671251": "\\pi: U\\to L", "ae35c1c4909b9a83b25976246cd7a6b9": "L_v = \\sqrt{L_x^2 + L_y^2}^T", "ae35ded6ec6d7711dfa326b56a93fca0": " EG(b_n;x) = (\\sec x + \\tan x) \\, EG(a_n;x). ", "ae362aa7e035525e0cd17b2a25cab8d2": "V^\\mathbb{B}_\\alpha", "ae363f99f55ba32f8f3885ebd23c8f52": "\\vec{\\mu} = -\\mu_B g \\vec{J}/\\hbar,", "ae3652e065c072411212c421f73cde65": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = \\mathbf{b}\n\\qquad \\Rightarrow \\qquad\n\\rho\\frac{D \\mathbf{v}}{D t} = \\mathbf{b}", "ae36b321359b3981a0bf57c3f5c147ba": "\\ {1\\over x_\\text{eq}} = \\sum_{i=1}^n {1\\over x_i} = {1\\over x_1} + {1\\over x_2} + \\cdots+ {1\\over x_n}.\\!\\,", "ae37742a6df38f5d858ccf78356b3075": " \\ H^{BM}_i(X)=H_i(\\bar{X} , \\bar{X} \\setminus X), ", "ae3801e9709941e5ebddb263af33c118": "\\bigcup_{i\\in I}N_i=M\\,", "ae382ca43c7b15ea16d111b6d9960fdf": "\\mu_0,\\, \\sigma_0^2\\!", "ae38333d19d78fa277a9ac213c64761d": "u(x) = \\sup_{v \\in S_\\varphi} v(x)", "ae387d3c00324151a94df134a57abb31": "\\det(E+(n-i)\\delta_{ij}) = \\sum_{I=(1\\le i_1 0 \\}.", "ae4adc83ac3ff3a0bca232ca88998591": "\\displaystyle{w=H_f(a)}", "ae4adf504e85ca465066cf94a4a82776": "M :", "ae4b0c73741c948631bb41948d21dbb2": "\\mathcal{N}(\\mu_{x_t}, \\sigma_{x_t}^2)", "ae4b63169e690258f98fd789fc19e294": "= P(7,1) + 3P(6,2) + 3P(5,3) + P(4,4)", "ae4ba8c2a72d3cdd4d7abeb028d4f94b": "\\bar{\\Pi}^m_\\ell(z) \\equiv \\left[\\tfrac{(2-\\delta_{m0}) (\\ell-m)!}{(\\ell+m)!}\\right]^{1/2} \\Pi^m_{\\ell}(z) .\n", "ae4bad048cc36318003f843cdbb41bee": "(\\ell_\\nu)_{\\nu=1}^k", "ae4bd59414108a8cc1bd461db43559bc": "\\mathbb{R}^\\omega", "ae4bebd3986bed198274307c46aa5ed3": " BV(\\Omega)", "ae4c0b1f8f0012f8b6d7aca60b53c378": "m^{\\mu} = \\frac{1}{\\sqrt{2}} \\left( \\hat{\\theta} + i\\hat{\\phi} \\right)\\ .", "ae4c0dc7f54f2a2a93e965914d1caf57": "\na^{2^rd} \\not\\equiv -1\\pmod{n}", "ae4c20978506277621ebc28df5cb370e": "b_1,b_2,\\dots,b_n\\in|\\mathcal B|", "ae4c92bb7c933836906d7abdf8ffd5a9": "~a_{s,j}(\\omega)~", "ae4cf0693c9ee75c36c2ec2fd66e4830": "dz_t", "ae4d5f0ad1144210194eef10ca2043be": "\\hat{y}(k)", "ae4dcea2d6cd4ca3fad4a341230d7121": "u(ci, x, y)", "ae4ddbcce55823e0779ee44101de9b57": "U(\\theta)= 2a \\cos {\\frac { \\pi S \\sin \\theta }{\\lambda}} W ~\\mathrm{sinc} \\frac { \\pi W \\sin \\theta}{\\lambda}", "ae4e4a216b3aeda62822354b5c98144b": "\\mathbf{F}' = m \\mathbf{a} \\ ,", "ae4e71dfcfa398305dd359c4c5072360": "E_{TF}=\\frac{3}{10}\\left(3\\pi^2\\right)^{\\frac{2}{3}}\\int{\\left[n\\left(\\vec{r}\\right)\\right]^{\\frac{5}{3}}d^3r}", "ae4ea5a3da5481e2e703fda843745828": "x^2 -2x +1 +1 -x^2 =1 ", "ae4efddf4d745d7121b8b1a65c172152": "V_t \\times F_e = (V_t - V_d) \\times F_a ", "ae4f1d0c9513b2ca4086e6bc154b2262": "k_1(p,n) = k_2(n,p)", "ae4f2ae3e58e54e4c3c6028117ed0aef": "{2p-1 \\choose p-1} \\equiv 1 \\pmod{p^3}", "ae4f583538156fbd547e176754cd8b0a": "\n\\begin{array}{rl}\n\\exp[\\tau (D_T + D_V)] & = \\prod_{i=1}^k \\exp(c_i \\tau D_T)\\exp(d_i \\tau D_V) + O(\\tau^{k+1}) \\\\ \\\\\n&= \\exp(c_1 \\tau D_T)\\exp(d_1 \\tau D_V)\\dots\\exp(c_k \\tau D_T)\\exp(d_k \\tau D_V) + O(\\tau^{k+1})\n\\end{array}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (5)\n", "ae501800b5b962fd73401f68b4134999": "\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}:\\mathbf a", "ae503324ec7ef6f2068b1bdf4f16f262": "M'\\prec_K N'", "ae504089efe2e5db1419e9176003df58": "E_{n_x,n_y,n_z} = \\frac{\\hbar^2\\pi^2}{2m} \\left[ \\left( \\frac{n_x}{L_x} \\right)^2 + \\left( \\frac{n_y}{L_y} \\right)^2 + \\left( \\frac{n_z}{L_z} \\right)^2 \\right] \\quad (23) ", "ae504cfe51ad90b875175e7daf0c0e09": " u_t+(u^2)_x+(u^2)_{xxx}=0 \\, ", "ae5054c1f553e914cc6dba841184d8d1": "\n\\left[ Q_R(\\mathbf{p}^{\\prime}),\nQ_R^\\dagger(\\mathbf{p}) \\right]\n= \\delta( \\mathbf{p}^{\\prime} - \\mathbf{p})\n(1 -{\\overline \\Delta_{12}}(\\mathbf{p},\\mathbf{p})), ", "ae50b307d4a25cdba05a28f1f4312494": "-e^{-aw}", "ae50e65080b4631904c2b880fcedc362": "2NW\\Delta t", "ae512390238b10866a6229149609320e": "\n K_H = \\int\\limits_{-\\infty}^{x'} \\exp(-\\beta u) dx - \\int\\limits_{x'}^{\\infty} \\big [ 1 - \\exp(-\\beta u) \\big ] dx,\n", "ae5147c363f44b5d536cf946a6c36a0d": "\\psi_{ 2m} = \\left ( \\frac { \\psi_{m}}{2y} \\right ) \\cdot ( \\psi_{m+2}\\psi^{ 2}_{m-1} - \\psi_{m-2} \\psi ^{ 2}_{m+1}) \\text{ for } m \\geq 3", "ae517395674ac51312a16f56820d23e9": "K \\,\\ ", "ae5183492a32a81c86f95403602007ab": " |Y - X| ", "ae51a0eb33c86346eab2fa4e2e0fcbe0": " P( k_1 < X < k_2 ) \\ge \\frac{ 4 [ ( \\mu - k_1 )( k_2 - \\mu ) - \\sigma^2 ] }{ ( k_2 - k_1 )^2 } ,", "ae51a3b839976f30fbe69fe0abf3abb6": "\\mathbf{Q} \\oplus O_{K,S} \\otimes_\\mathbf{Z} \\mathbf{Q}", "ae5206ebb3d78ac73edad15ff92ced52": "r\\in {1\\over 2}+{\\Bbb Z}", "ae521f381bddff3319d5c8cc1c9f3876": "q^{\\mu} = (E, q_x, q_y, q_z) \\,", "ae529f8b03f54452d585a0bcde41a901": "\\Omega =\n\\begin{bmatrix}\n0 & I_n \\\\\n-I_n & 0 \\\\\n\\end{bmatrix}", "ae52b372a37ea1c17a0ff6c3f2aea855": "R_F", "ae530819cb1c8640ec456e3056682ca3": "\ng_3 = 4 e_1 e_2 e_3. \\,\n", "ae5313106b3d46867167e2850cc754dd": "\n \\hat{H} |n\\rangle = \\epsilon_n |n\\rangle.\n", "ae531fe69458a420d3e2c453587832bb": "v = - \\frac{\\partial \\psi}{\\partial x}\\,", "ae539319bf4a595e069d48baa96426f0": "\\tfrac{1TeV}{c^{2}}", "ae539dfcc999c28e25a0f3ae65c1de79": "\\gamma", "ae53d9ae5bce5a1daece3b84349f004c": "y^2 = x^2 (x-1)", "ae543151d6e6ddfcdf269f773e757db5": "H = \\frac{f^2}{N c} + f", "ae548e3006648afa9817405847659f28": "X \\ni x \\mapsto f(x)", "ae549a19a4e5c9a24b812cacc9429200": "G(h, F) = \\nu(h, F) G^{\\mathrm{ET}} = \\nu(h, F)b h^{3/2}/F, \\qquad\\qquad (9)", "ae54ae1b74c2e50eacda09dae3b1f26d": "3+\\sqrt{8}\\approx 5.8", "ae54be144d59a94ed163a4b7169b0905": "\\mathit{N-n}", "ae54ff65f8935998308550d3e1a9cc43": "\\ F_{forward} = drag \\times cos(\\beta) ", "ae55b1826c1925191e8c421d98609849": "\\begin{bmatrix}a & \\mathbf v\\\\ \\mathbf w & b\\end{bmatrix} \\begin{bmatrix}a' & \\mathbf v'\\\\ \\mathbf w' & b'\\end{bmatrix} = \\begin{bmatrix}aa' + \\mathbf v\\cdot\\mathbf w' & a\\mathbf v' + b'\\mathbf v + \\mathbf w \\times \\mathbf w'\\\\ a'\\mathbf w + b\\mathbf w' - \\mathbf v\\times\\mathbf v' & bb' + \\mathbf v'\\cdot\\mathbf w \\end{bmatrix}", "ae56252a04d479330273b0305c900726": "\\sqrt{I} \\subseteq I(V)", "ae56652d323fde91d2faf8fd7a90b487": "\\dot{\\vec\\beta}", "ae56c7ee4e8761be19e3cd662de530a3": "b = -\\frac{\\sin2\\theta}{4\\sigma_x^2} + \\frac{\\sin2\\theta}{4\\sigma_y^2}", "ae57300c389f1c004923a8514be21702": "\\alpha^{-1}(J_n)=J_n -{c\\over 6}\\delta_{n,0}", "ae57347b6fd9afdd697dd1fcb1d80318": "y_1=1", "ae574573c0be64cee031c841cff24e1b": "\\mathbf E_{1s} = \\frac{\\zeta^2}{2}-\\zeta \\mathbf Z.", "ae577c9e1ac0eddc1e19ad2d97a479fd": "\\lim_{n\\to\\infty}{\\langle x_{n} - x, x_{n} - x\\rangle} = 0. ", "ae57a8e42a1da41d783afc2018752f75": "y^2 = x(x - a^\\ell)(x + b^\\ell)\\ ", "ae580a69b9ae2fb977bd83fba6824134": "g_i : \\,\\!\\mathbb{R}^n \\rightarrow \\mathbb{R}", "ae58159a9e31d9bad6f2246eb59e6baf": "K_{zz}", "ae5826fdc9666714ad2b1aeb128792a5": " U_n = \\beta z_n + \\varepsilon, \\; \\varepsilon \\sim", "ae589f2a86ab0360c1b87c3428efea0f": "S_{1/T}(f)\\ \\stackrel{\\text{def}}{=}\\ \\underbrace{\\sum_{k=-\\infty}^{\\infty} S\\left(f - \\frac{k}{T}\\right) \\equiv \\overbrace{\\sum_{n=-\\infty}^{\\infty} s[n] \\cdot e^{-i 2\\pi f n T}}^{\\text{Fourier series (DTFT)}}}_{\\text{Poisson summation formula}} = \\mathcal{F} \\left \\{ \\sum_{n=-\\infty}^{\\infty} s[n]\\ \\delta(t-nT)\\right \\},\\,", "ae58ff04457cddbfe34df431579562ab": "I =", "ae5902af9b426543dcbb1b44e6575695": "T_1^2 + T_2^2\\,", "ae59131f0643a3fa48232d8b8ee2ad1b": "\\operatorname{Ch}(E,F,G) = (E \\and F) \\oplus (\\neg E \\and G)", "ae5918998b066673a26babe66195c2f3": "\\frac{bh}{3}", "ae59197fe315396f9f1ea1938bfd9f6e": "\\textstyle {4!\\over 2!\\times 2!\\times 0!} \\ {4!\\over 1!\\times 2!\\times 1!} \\ {4!\\over 0!\\times 2!\\times 2!}", "ae598ffe2b4c6c6453b4a9e533e63c4d": "\\mu_a \\cap (A^T \\mu_b A)", "ae59ac4de8f4711d4efc7b9ed46e7702": "\\sigma_x^2 = \\frac{\\hbar}{m\\omega} \\left( n+\\frac{1}{2}\\right)", "ae59fb7b8d4cfddff10cfafb152d780a": "S(\\mathbf{q}) \\equiv S'(\\mathbf{q})", "ae5a32614ed2734025b1e0f5d6c1b898": "\ny = \\frac{1}{2} \\left( \\tau^{2} - \\sigma^{2} \\right)\n", "ae5a360aae75d157cbdd3a706f26527e": "\\begin{matrix} \\frac{1}{2} \\end{matrix}", "ae5a41477a922e8cd36845185551720c": "f:\\left[-1,1\\right]\\times {\\mathbb R}\\rightarrow{\\mathbb R}", "ae5a69d7f6c2b195ecce380664913592": " \\Delta(x)s_{\\lambda} ", "ae5a6ad79a95b1e022fcd809a1fc1456": "\\vert{\\Psi}\\rangle", "ae5aa09cb62f59c2bd4a28a66f466d8c": "\\textstyle\\tau", "ae5aa7519d9262adb863224c160c9ad4": "\\varepsilon_{HbO2}", "ae5aacda849f569b4b548a575d8b9c88": "a \\times b = *(a \\wedge b) \\,.", "ae5ab52fe37bdd6b2f37158c3516996d": " \\Delta \\mathbf{p} = \\mathbf{F} \\Delta t ", "ae5ad1a3e5bfb8fc359474303e2c161a": "\\nabla\\cdot{\\vec{A}} + \\frac{1}{c}\\frac{\\partial\\varphi}{\\partial t}=0.", "ae5af2e788c4425369df184411f87933": "c^T x+\\lambda^T(b_2-A_2x)", "ae5b379541e258c612c977b14f7dc2a3": "3 \\mid 6", "ae5b49102dacde5698eaba9636b4aee3": "\\rho_1 = \\gamma_1 / \\gamma_0 = \\varphi_1", "ae5b7ec4e29c919fdba15aa5c77f1fdf": "\\frac {B}{C} >1 ", "ae5b87de863e4664ca40cb86d43042c6": "\\mathrm{NF} = 10 \\log \\left(F\\right)", "ae5b9cc8d146efcf9f745d33fad8124d": "\\sum_{n=0}^{N-1}x_n y_n^* = \\sum_{k=0}^{N-1}X_k Y_k^*", "ae5bdcfed58f3c4842375215ff4685f7": "f '' (x)", "ae5c24ac4162f02a4e175ea45e3a3e51": " e^{i\\epsilon S} \\,", "ae5c4c5beb368290f9e4291a26e23619": "\\begin{align}\nf(x;\\alpha,\\beta) & = \\mathrm{constant}\\cdot x^{\\alpha-1}(1-x)^{\\beta-1} \\\\\n& = \\frac{x^{\\alpha-1}(1-x)^{\\beta-1}}{\\int_0^1 u^{\\alpha-1} (1-u)^{\\beta-1}\\, du} \\\\[6pt]\n& = \\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}\\, x^{\\alpha-1}(1-x)^{\\beta-1} \\\\[6pt]\n& = \\frac{1}{\\Beta(\\alpha,\\beta)} x^{\\alpha-1}(1-x)^{\\beta-1}\n\\end{align}", "ae5cc322438f98030ea45b7979912646": "22678^2 \\equiv 3070860^2 \\pmod{15347} ", "ae5d1608ad6245be79e235983a9de6e4": "W_{i,j}=\\begin{cases}\n1 & if\\, X_{i}\\leftrightarrow Y_{j}\\\\\n0 & otherwise\n\\end{cases}", "ae5e4a2a34f8eebd17582d2cfbf36103": "y_{i=2,us} = \\frac{2y_g}{-1+\\sqrt{1+\\frac{8gy_g^3}{q_{i=2}^2}}}=\\frac{(2)(0.5)}{-1+\\sqrt{1+\\frac{8(32.2)(0.50)^3}{7.16^2}}} = 3.63\\text{ ft}", "ae5ef9a7ac291511235d9594150c1d56": "X = \\begin{pmatrix}\n \\;\\;0 & 1 \\\\\n -1 & 0 \n \\end{pmatrix}\n", "ae5f694c1823161d9ada27edd7977339": "\\dot C_{RW} (t)\\,\\,\\, = \\,\\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 \\,{L \\over {2\\,v\\,}}", "ae5f7846142bac8ffe241a75622bc7ec": "\\digamma \\,", "ae5fadbf68be2d234a3428e1ab54422c": "Ts:e^*T^tG \\rightarrow TM ", "ae5fc9b70f02cdc7ef6d997d05d49cc1": " x \\geq 0 \\, .", "ae6001079da25d68d23260634a820b55": "\\sigma_{GB} = \\gamma_{GB} + \\frac{Ad\\gamma_{GB}}{dA}\\,\\!", "ae6017fa084f2a8e560ef1cc1e887a0a": "\\mathbf{y}_j", "ae60211bacf38f925a3c7ebc14b7af40": "p(x) \\in K[x] ", "ae6040b629443755b730bddf1c87de2f": "n(\\mathbf r)", "ae606081f608ca72c2bd61ee6d29a838": "u^k(x)", "ae60705bf823878ee4b73140a1cd0187": "H+\\vec{v}+\\vec{w}", "ae60bb3212f1cbeb042f71e518721f6d": "\n\\alpha(t) = x(t) = (x_1(t), x_2(t), ..., x_N(t)) \n", "ae6135a397f8dcd04464af2cd90da056": "\\scriptstyle{FX}\\,", "ae61bd0b2283c8dcdc72e987a8f6ad21": "\\textstyle x\\in\\textbf{R}^d", "ae620ae9f49060c8d84bd881ab412a06": "E_\\phi = H_r = H_\\theta = 0,", "ae6233dbb4b0c1449fc89b67181b87bf": "\\sigma^2(n)=\\sigma(\\sigma(n))=6n\\,", "ae624af44dc9fe37b52b9dcf7d1adec6": " \\rho = \\frac{M \\cdot P}{R^* \\cdot T}", "ae629045a35388fdab7290b3e8328d05": "J = 0", "ae62bbbffb283490c4696e46c21a3b27": "\\Eta = \\begin{bmatrix} {H_{ii}} & {H_{ij}} \\\\ {H_{ji}} & {H_{jj}} \\end{bmatrix} ", "ae631f3a3d77f0e9090761a1a845a4e7": "_{/}\\!", "ae636e428fa4d899da610f6cc449e9ed": "x^+", "ae63731e5c7bdfe0adec968e5ef2fb80": "\\frac{\\mathrm{d}}{\\mathrm{d}z}\\, \\mathrm{cn}\\,(z) = -\\mathrm{sn}\\,(z)\\, \\mathrm{dn}\\,(z),", "ae63bc9826b9911d0fe9292c251716a5": " E\\in[\\underline E,\\overline E], P\\in[\\underline P,\\overline P] ", "ae63c7eb98cdd41941f4f9d467fb0027": "WHIP=\\frac{BB+H}{IP}", "ae63eb573c620c0bcda6b7b7b7fe07ec": "\\textrm{E}[\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k}] = \\textrm{E}[\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k-1}] = 0", "ae6424c5ce96e7cf700fd854e779a829": "1.5 \\text{m}/\\text{s}^2", "ae645807b4c3f6841199af4312ad7099": "a_H", "ae648a0848cc7ac658fe4370cffc3a7d": " \\int_{t_0}^{t_1} \\bar{u}^*(t) \\bar{u}(t) dt \\ \\geq \\ \\int_{t_0}^{t_1} u^*(t) u(t) dt. ", "ae64b4f6cc01fe809e583a63298b2a37": "K_{G}^{(a)} {\\left| G \\right\\rangle} =(-1)^{k_{a}} {\\left| G \\right\\rangle} ", "ae64cb2fbc0c4d9e0f1af6afd15bdd46": "Q = 1 + \\mathbf{\\hat{i}} + \\mathbf{\\hat{j}}+ \\mathbf{\\hat{k}}", "ae64d78d272ad91f2022aac8010ca251": "P = \\{p_1,p_2,\\ldots\\}", "ae6533f439b0380d741470ea6db2b168": "\\mathrm{d}G = \\sum_{\\alpha,\\beta,S}\\,\\left( V\\mathrm{d}p\\,-S\\mathrm{d}T\\,+\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}n_i\\,+ \\sum_{i=1}^k \\mathrm{n}_i \\,\\mathrm{d}\\mu_i\\,\\right) +A\\mathrm{d}\\gamma\\,,", "ae6560cb18066efd97d55ee3cf6afcbf": "d' \\to d", "ae65757d4aa47b089b5c7af9a5e69c39": "\nw_{k} = \\nu_{k}(\\mathbf{J}) t + \\beta_{k}\n", "ae65beace1997748af688665cbe563cf": "\n\\begin{align}\n \\mathbf{W^T} \\mathbf{F}\n = \\mathbf{(A^{T} w)^T} \\mathbf{F}\n = \\mathbf{(w^{T} A)} \\mathbf{F}\n = \\mathbf{w^{T} A F} = \\mathbf{0}\n\\end{align}\n", "ae6672c0507f295d2c8a02cd956ca15f": "h_1(n), h_2(n), ..., h_i(n)", "ae66b4cac2e2125643c18ef84a6a31a9": "\\theta=\\alpha", "ae672989a676785fb5740faa5bccb00c": " \\langle BC_n \\rangle \\le n\\cos(\\pi/n) ", "ae67524786d9172be9ec02f96ccf711d": "\n\\mathbf{B} = \n\\begin{pmatrix}\nB_{1,1} & B_{1,2} & \\cdots & B_{1,N} \\\\\nB_{2,1} & B_{2,2} & \\cdots & B_{2,N} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\nB_{N,1} & B_{N,2} & \\cdots & B_{N,N} \\\\\n\\end{pmatrix}.\n", "ae67567489bc66b7aca19b182793bbb2": "Q(x) = \\begin{cases}\n \\tfrac{Pb}{L}, & \\mbox{for } 0 \\le x \\le a \\\\\n \\tfrac{Pb}{L}-P, & \\mbox{for } a < x \\le L\n \\end{cases}", "ae67edfe7f2770c0aafef1eaa9ca284d": "[S^n,S^n]=\\pi_n S^n \\to \\mathbf{Z}", "ae681ebdaeee44ae80168c576f73e38f": "Q \\quad = IV", "ae6863873944fa0f7410edff57b96cf2": "\n\\det(-\\mathbf{R}) = - \\det(\\mathbf{R})\n\\quad\\hbox{and}\\quad\\det(\\mathbf{R}^{-1} ) = 1,\n", "ae69617011037a2ead6934ca9030544e": "\\beta-\\alpha", "ae6981198fa9adda167101fd1423f020": " M_B = \\{ B(x,y) : \\mu_B(x,y) = \\mu_B(0,0) \\}. ", "ae69ea7df894abae2b517f1f2942e773": "\\frac{\\lambda }{L}=\\frac{\\mu }{\\rho L}\\sqrt{\\frac{\\pi m}{2 k_BT}}", "ae6a51b48568271c3c623c6bcc475778": "(S,*)", "ae6a731364c9b44cde1eb3f6fdf9e657": "r^2 - 2rv + v^2 = r^2 - (\\frac{L}{2})^2,", "ae6b06cbcc1150a665579832fb9de5f4": "u_{2} '= \\frac{u_2 - v_c }{1- \\frac{u_2 v_c}{c^2}}", "ae6b20e5fd5d0cdfb1680dca574455da": "\\xi= \\ln\\frac{E_0}{E}=1+\\frac{(A-1)^2}{2A}\\ln\\left(\\frac{A-1}{A+1}\\right)", "ae6b4ecc673a1a64ec6fc57f3651d363": "\\vec g - 2 \\vec T - \\vec L", "ae6b565db1a06319ae4b43b1f93d54e4": "y' = (x - y ) / tau", "ae6b5d74b4b96bf566781e468c7572df": "A - B = C", "ae6b606c09e18fd725c7a61a49fed80f": "\\tilde{e}_i L \\subset L", "ae6b79387e624a73dda0396edaf03257": "\\rho_0/U", "ae6bce6bb7ba7f57fa1d664b7492ff01": "\\scriptstyle| f\\rangle", "ae6c2e272cee556d4a2278fe4289f977": "\\frac{A_{n}}{A_{n-1}} =f_{RP}=e^{-k\\Delta t_{p}}", "ae6c622a3c23b83e68db432324ddd8b6": "\\begin{align}\n\\rho &= r\\sin\\theta \\\\\n\\phi &= \\phi\\\\\nz &= r\\cos\\theta \\end{align}", "ae6c8519bfc9f612b9702e5182d24c85": "A^0", "ae6c8d264f06edbdcf3bcc00d01c4fe2": " U_x(x_1, x_2; y_2) = p_1 x_1 + (1 - x_2 - y_2 ) x_2 - (x_1 + x_2)^2/2 - F ", "ae6cd27e20aad9324e4b3f4834856549": "\\rho(m)", "ae6d3e837e2a6f4ce36e66193699a3db": "\\ddot{c}(t)", "ae6d8565bfb7926108ee485e20b71b8a": " F_{X}(x) = P(X \\leq x) = \\frac{1}{(1 + e^{-x})^{\\theta}}", "ae6e14469755f71e3cd42960f3389618": "\\|\\boldsymbol{x}\\| := \\sqrt{\\boldsymbol{x} \\cdot \\boldsymbol{x}}.", "ae6e20b8b37f84f4a0a43803aed617c5": "\\text{2. }\\omega \\notin B : P(\\omega| B) = 0", "ae6e2bd863ccfd96760f9d9372046c3f": "\\nabla(\\rho_i V)", "ae6e53c33182f2c4baa082b092406bff": "T_i ^f = \\left(1-\\frac{2a\\Delta t}{h^2}\\right) T_i ^{f-1} + \\left(\\frac{a \\Delta t}{h^2} + \\frac{\\epsilon u \\Delta t}{2h}\\right) T_{i-1} ^{f-1} + \\left(\\frac{a \\Delta t}{h^2} - \\frac{\\epsilon u \\Delta t}{2h}\\right) T_{i+1} ^{f-1} + \\frac{Q_i ^{f-1}}{c \\rho} \\Delta t", "ae6e72aca4f9e6b6945a053e8905a158": " h_n(t_1, t_2, \\ldots, t_n) ", "ae6ee215cc908e957f89f361f00593c0": "s \\times b^e", "ae6ee87244c7189999acecd8ff19ffda": "\nx = 1 + \\sum_{i=1}^\\infty r_1r_2\\cdots r_i = 1 + \\sum_{i=1}^\\infty \\left( \\prod_{j=1}^i r_j \\right)\\,\n", "ae6ef15c8f716471e48fd523835774a6": "\\mathrm{j}_x, \\mathrm{j}_y,\\mathrm{j}_z,\\mathrm{j}_+,\\mathrm{j}_-", "ae6f01a2a9f1c814067befe84386768a": "AB = BA", "ae6f0b93d805d14752267d9866e4603f": "\\Phi_0 [\\mathbf{r}]", "ae6f145339764b52704cd49b9e1b68d1": " E(c) = \\int_a^b (E\\dot{x}^2 + 2F \\dot{x}\\dot{y} + G \\dot{y}^2)\\, dt. ", "ae6f74d829b005d40a89ddf8a83f8c2e": "\\mathbf{DP}", "ae6f7ad3c2c4bdbac990850452c0ba0d": "\\begin{matrix}\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{f(0+h) - f(0-h)}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{f(h) - f(-h)}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{1/h^2 - 1/(-h)^2}{2h} \\\\\n\\\\ f_s(0)= \\lim_{h \\to 0}\\frac{1/h^2-1/h^2}{2h} \\\\\n\\\\ f_s(0)= 0 \\\\\n\\end{matrix}", "ae6f8a4a17ae021f647c2e1a254523db": "h^{eff}", "ae6f8bd005455a07d55799430b17f957": "\\tau \\sim {1 \\over M_*} \\Bigl( {M_{BH} \\over M_*} \\Bigr) ^{(n+3)/(n+1)} ", "ae702dbb1b0885cf8ec01606f915933f": " \\boldsymbol{\\omega} = \\bold{\\hat{n}}\\frac{{\\rm d} \\theta}{{\\rm d} t}\\,\\!", "ae7038be4929bde1b30b21a25d4b429c": "(x+1)^3 = x^3 + 3x^2 + 3x + 1.", "ae7085071f15da8c6c3ece9e14336049": "r\\in C", "ae70b45cc361765e0e4f6494646b5ba3": "m2 = 10", "ae7145189540eb965b5ca1a731db698e": "p\\to(q\\land\\neg r\\land(s\\lor t)).", "ae71804c2eb1afdfe2e50401c254b0a1": "\\lim_{n \\to \\infty}\\|f_n-f\\|_p =\\lim_{n \\to \\infty}\\left(\\int_\\Omega |f_n-f|^p \\,d\\mu\\right)^{\\frac{1}{p}} = 0.", "ae7189d3c8557417e936739e061dba3d": "\\Psi_\\ell(X,Y)", "ae71a07311d2ceb6eb0952f3b374e445": " F \\rightarrow\n\\begin{pmatrix}\n E_3 & E_1 -i E_2 \\\\ E_1 +i E_2 & -E_3 \n\n \\end{pmatrix} + i \\begin{pmatrix}\n B_3 & B_1 -i B_2 \\\\ B_1 +i B_2 & -B_3 \n\\end{pmatrix}\\,.\n", "ae71b11b7c27cf38e9143c1156d30141": "\\hat{B}(\\xi )", "ae71dcbb016e14d8466396b24f454cba": "p(\\mathbf x)=g(\\|\\mathbf x\\|_\\beta)", "ae7207b01692e2b76bc29b8eb3570e60": "AA^+ b = b", "ae724277ce74cc2e6d8d604aeeb3bb82": "\\lambda_1+\\lambda_2", "ae7277366d5d091ef492ab59b435230c": "\nL_{xx} = \\begin{bmatrix}\n1 & -2 & 1\n\\end{bmatrix} * L\n\\quad \\mbox{and} \\quad\nL_{xy} = \\begin{bmatrix}\n-1/4 & 0 & 1/4 \\\\\n0 & 0 & 0\\\\\n1/4 & 0 & -1/4\n\\end{bmatrix} * L\n\\quad \\mbox{and} \\quad\nL_{yy} = \\begin{bmatrix}\n1 \\\\\n-2 \\\\\n1\n\\end{bmatrix} * L.\n", "ae7277ec5e3d99a92e8c215559f59d7e": "v \\wedge w \\mapsto v^* \\otimes w - w^* \\otimes v,", "ae7294ac8ac7b091405550a6c57150d4": "z=a\\left(\\cos\\left(v\\right)+\\ln\\left(\\tan\\left(\\frac{v}{2}\\right)\\right)\\right)+bu", "ae72c3526a79d7a4d35df5ba899b986c": " \\scriptstyle \\ d ", "ae72e09da1d91b5097d3ae8d5a69e884": " \\mathbf{q} = \n\\begin{bmatrix} \\cos (\\psi /2) \\\\ 0 \\\\ 0 \\\\ sin (\\psi /2) \\\\ \\end{bmatrix}\n\\begin{bmatrix} \\cos (\\theta /2) \\\\ 0 \\\\ sin (\\theta /2) \\\\ 0 \\\\ \\end{bmatrix}\n\\begin{bmatrix} \\cos (\\phi /2) \\\\ sin (\\phi /2) \\\\ 0 \\\\ 0 \\\\ \\end{bmatrix}\n = \\begin{bmatrix}\n\\cos (\\phi /2) \\cos (\\theta /2) \\cos (\\psi /2) + \\sin (\\phi /2) \\sin (\\theta /2) \\sin (\\psi /2) \\\\\n\\sin (\\phi /2) \\cos (\\theta /2) \\cos (\\psi /2) - \\cos (\\phi /2) \\sin (\\theta /2) \\sin (\\psi /2) \\\\\n\\cos (\\phi /2) \\sin (\\theta /2) \\cos (\\psi /2) + \\sin (\\phi /2) \\cos (\\theta /2) \\sin (\\psi /2) \\\\\n\\cos (\\phi /2) \\cos (\\theta /2) \\sin (\\psi /2) - \\sin (\\phi /2) \\sin (\\theta /2) \\cos (\\psi /2) \\\\\n\\end{bmatrix}", "ae72e48bdb47febf50025cb617fd3076": "\\lim_{\\tau \\rightarrow \\infty} y_I = \\lim_{t \\to 0} y_O ,\\,", "ae73c157818fa233ec60b2e7bbe789b3": "\\,u_t\\,", "ae741cf51fea011288eaea0f6a956991": "\\mathbf{G}, \\widetilde{\\mathbf{G}} ", "ae7439782cc306fb54f677f32e56212f": "\\delta v (x) = \\sum_{i = 1}^{n} \\left( x_{i} v^{i} (x) - \\frac{\\partial v^{i}}{\\partial x_{i}} (x) \\right).", "ae743b76e6852c7d1d4ccc0e8115b3f6": "\\theta= \\pi - \\theta^\\prime", "ae74590f81c2f1eb7dac521544cdecae": "MP = M'P = O", "ae7508360c6e738b112608d66c07a2ad": "\\alpha_i-1", "ae75280cde2032160f44943bd80b1862": "\\Phi_{n+1}(z)=z\\Phi_n(z)-\\overline\\alpha_n\\Phi_n^*(z)", "ae756fe66c9abf570ac85ff17d7ba661": "M = {y_1^2 \\over 2} + {q^2 \\over gy_1} = {y_2^2 \\over 2} + {q^2 \\over gy_2}", "ae7589a84a1ca938c3ddd08f2d389058": "T_n \\le T", "ae759bbfe32b567dd5851cc80cd8104b": "\\left(x_0,x_1\\right)", "ae75f22b6373e33264459e14e81e0c53": " n-1,n+1,2n-1,2n+1, \\dots, 2^k n - 1, 2^k n + 1 \\,", "ae76698a7f670211ac7189be69394828": "P'(X_{n+1}=1 \\mid X_1+\\cdots+X_n=s)={s \\over n}.", "ae768ba3648f2292ceb31c708fe7c7f1": "t \\longrightarrow F(t,t_0,x)", "ae76a53bfb1e43ba8b3304dcb8c64e25": "[0 , T]", "ae76c7f25017f84a25e6aecdf15bd266": "\\{ a^m A b^n : m, n \\geq 0 \\}", "ae76dd9d2304517fd7ca6a070f6fd4c3": "u = (300426607914281713365\\sqrt{609}+84129507677858393258\\sqrt{7766})^2.", "ae774bc4ba7630fd6fcc9e5b44bf97e5": "U(t)=\\exp(itA)", "ae779466a3cea06111cc652302c6807d": "V_{in}L \\sim V_{out}\\delta, ", "ae77a08a3766b990e8cdce5fa3cbb742": "{\\rm Ric} (L) \\ge 0.", "ae7833cca77875f1191267fe675af54d": " n = 0, b/a ", "ae7890328a757190537536245733798d": "e^2 = \\frac{2h \\alpha}{\\mu_0 c} = 2h \\alpha \\epsilon_0 c", "ae78f5757049972c2d0dd1aa86544c7c": "\\left( {X}-\\langle {X}\\rangle \\right)\\,|\\alpha \\rangle = -i\\left( {P}-\\langle{P}\\rangle \\right)\\, |\\alpha\\rangle \\text{,}\\qquad \\text{or}\\qquad \\left( {X}+i{P} \\right)\\, \\left|\\alpha\\right\\rangle = \\left\\langle {X}+i{P} \\right\\rangle \\, \\left|\\alpha\\right\\rangle ", "ae790c9ee06832424325c38c26f024d3": "(1-c)(1-ee)=1-c(1+ee'')", "ae7927b40378de7f64ec66a507f4ddea": "p = b/a", "ae796f4819e6c12cbdc707ed5e475daa": "R_p\\simeq0.5", "ae79d2f84a8c42aa2a7c1e53e368d078": "z = ZS\\,\\!", "ae79e53eadfb18d778cdac9db667716c": "X_C = X_L", "ae79f2eaca21df9167600b20f267bf48": "K_3 = -I_1 + 2 \\, R_{ab} \\, R^{ab} - \\frac{2}{3} \\, R^2", "ae7ab2f74409e48fd084aa17cf61e98b": "BBP=\\frac{W}{PA}", "ae7ab6d4189e28b319ebc493de96212a": "f(\\xi) \\stackrel{q}\\longrightarrow \\acute{f}(\\xi) = Tr[\\hat{B}(\\xi )\\hat{U}^{+}\\hat{f}\\hat{U}]", "ae7ac56ba5b515eadbbd7d8a8a2a61bc": "\\frac{\\pi a b}{2}", "ae7ac6b531bd3d47672ef50d03fc862b": "\\begin{align}\n\\theta_{r_i}(x)\\big|_{x=j\\infty} & = \\angle(x-r_i)\\big|_{x=j\\infty} \\\\\n & = \\angle(0-\\mathfrak{Re}[r_i],\\infty-\\mathfrak{Im}[r_i]) \\\\\n & = \\angle(-\\mathfrak{Re}[r_i],\\infty) \\\\\n & = \\lim_{\\phi \\to -\\infty}\\tan^{-1}\\phi=-\\frac{\\pi}{2} \\quad (9)\\\\\n\\end{align}", "ae7addebe3e4b10a75cdee235030b8be": "\\frac{\\sin \\theta}{\\theta} < 1 < \\frac{\\tan \\theta}{\\theta}\\,", "ae7afcb681f17735a4b086229182b14b": "f(r) = R", "ae7b2673ec6face1eeef54b87d2bd960": "(6)\\quad \n\\tilde\\psi\\,=\\,\\psi^{\\langle1\\rangle}+\\psi^{\\langle2\\rangle}\\,,\n", "ae7b42026029e3fb826d5a1f11685b19": "\\Delta^n\\rightarrow Y", "ae7b85146fdd2f7e97869903bbed7ca6": "\n0\\rightarrow \\operatorname{Der}_B(C,L)\\rightarrow \\operatorname{Der}_A(C,L)\\rightarrow \\operatorname{Der}_A(B,L)\n\\rightarrow \\operatorname{Exalcomm}_B(C,L)\\rightarrow \\operatorname{Exalcomm}_A(C,L)\\rightarrow \\operatorname{Exalcomm}_A(B,L),\n", "ae7b86163dbac441bb5144c5e8f5a9af": "\n\\begin{align}\n(P_x, P_y)= \\bigg(&\\frac{(x_1 y_2-y_1 x_2)(x_3-x_4)-(x_1-x_2)(x_3 y_4-y_3 x_4)}{(x_1-x_2)(y_3-y_4)-(y_1-y_2)(x_3-x_4)}, \\\\\n &\\frac{(x_1 y_2-y_1 x_2)(y_3-y_4)-(y_1-y_2)(x_3 y_4-y_3 x_4)}{(x_1-x_2)(y_3-y_4)-(y_1-y_2)(x_3-x_4)}\\bigg)\n\\end{align}\n", "ae7c048832cf1d6be2a5111c839f5763": "Z=1/Y", "ae7c827d150493d019ee32cb361dd5d7": "\\eta^i", "ae7cb935ade7fb4a4ce22341a98c83f8": "\\hbox{not}\\ p", "ae7ceba30f103f87aaf576a606a36dbc": " |\\epsilon \\rang ", "ae7d89a58dcf7b817cced9b0ce3cec7d": "G(t + s) = G(t) G(s)\\,", "ae7dc26febe7792a70337874332d3bab": "B_{1,2} = (B_1 + B_2)/2\\ ", "ae7dd494c676bee39a282283caaebed4": " y=a_{0}\\sum_{r=0}^{\\infty} \\frac{(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}s^{r+c}", "ae7e46a29ca31c5e28348d91492d3d71": "\\Gamma_6", "ae7e60c4e5ca127d95b5438619b96d8f": "\\prod _x x+\\frac {1}{x} = \\frac{C\\, \\Gamma (x-i) \\Gamma (x+i)}{\\Gamma (x)}", "ae7e7efb73d2c4bdea3265a0b926b8e4": "\\rho = \\frac{Z_t - Z_o}{Z_t + Z_o}", "ae7ee17a3b22a1eb22ddc0710fd44d9f": "b(x)=\\left\\{\\begin{array}{ll}0,&x\\in B,\\\\-\\infty,&\\mbox{otherwise}\\end{array}\\right.", "ae7ee713d882359fd9299df9f2023931": "\\frac{\\Gamma\\left(\\frac 1 2 (2+\\nu+s)\\right)}{\\Gamma(\\tfrac 1 2 (\\nu-s))} \\frac{2^{s+1}}{k^{s+2}} \\,", "ae7f1088f0e91a40363fb52fa08e9ff7": "ds^2 = - \\left( k^2r^2 + 1 - \\frac{C}{r} \\right)dt^2 + \\frac{1}{k^2r^2 + 1 - \\frac{C}{r}}dr^2 + r^2 d\\Omega^2", "ae7f20b44c18e9d39359e06d85cd27da": "a_{m} = \\sum^{A}_{n\\neq m} \\frac{U^{A}_{mn} - H_{mn}}{E-H_{mm}} a_{n} ", "ae7f340ade956b9f804e231b073780ae": "\\frac{d}{dt}(e^{-\\mathbf At}\\mathbf x(t)) = e^{-\\mathbf At} \\mathbf B\\mathbf u(t)", "ae7f42600fb21f140bdb24ff09edc7eb": "{\\alpha}", "ae7f5608a869464fdbe638521e94ca09": "1/3 = 0.0\\ 2_!", "ae80a4a206f4c809737476d7d102412c": " \\frac{L}{L_{\\odot}} = (\\frac{d^{2}_{\\odot}}{b})(\\frac{d^{2}}{b_{\\odot}}) ", "ae80be0c50f778d7f68b072017ca8d3b": "L = 0, H = 0.", "ae80ca6070559044a3b3b15a689814b0": "\n \\begin{align}\n g\\, \\frac{\\partial\\zeta}{\\partial{t}} \n &+ \\nabla\\cdot\\left( c_p\\, c_g\\, \\nabla \\varphi \\right) \n + \\left( k^2\\, c_p\\, c_g\\, -\\, \\omega_0^2 \\right)\\, \\varphi\n = 0,\n \\\\\n \\frac{\\partial\\varphi}{\\partial{t}} &+ g \\zeta = 0,\n \\quad \\text{with} \\quad \\omega_0^2\\, =\\, g\\, k\\, \\tanh\\, (kh).\n \\end{align}\n", "ae815b7f7de4588f810a5655a9384639": "\\max_rq(d,r)\\,\\!", "ae815f036fdf54e179337f79b97d8e9d": "(Q_0, \\cdot)", "ae81a5c87cfb63cc1722a80ac77542b8": "z_{1,2}=\\frac{-b \\pm i \\sqrt {-\\Delta}}{2a}=\\frac{-b \\pm i \\sqrt {4ac-b^2}}{2a}.", "ae81c655ec067c4eb522eb126e9bd390": "O(n^{2.3727}).", "ae81d6cf37bf9c45c550aa8787b591da": " \\frac{d F}{d t}=-VRT \\left.\\frac{d \\theta(\\lambda)}{d \\lambda}\\right|_{\\lambda=1}", "ae81d7478798d5463e43941bf65a7548": "\\ f''(x) < 0", "ae82082b25d0dd1e71722375c6c1b753": "d = \\vec a\\cdot \\vec n_0 \\geq 0\\,", "ae821dcef83cfe7092aac4adee1d3d63": "\\text{PR}=\\frac{P_2}{P_1}", "ae82d6d7c53906300b821544b10af3ae": " z(t) = r e^{i \\omega t} \\,", "ae834a625db1ea698d3a376cfde2f6b1": "\\displaystyle\\frac{(-1)^{n-1}}{(n-1)!}\\frac{d^n}{dx^n}\\log |x|", "ae834a818bce13dfe20b10fc2957ef4c": "\\underbrace{1+\\cdots+1}_{n \\text{ summands}} = 0", "ae837e4541406c4172139e4cbaa27c3f": "o(n)", "ae839704175a9040b98df542a35bb908": "x\\ast (x\\ast y)= y\\ast (y\\ast x)", "ae83cd48d8c8afb7546b263917250f29": "h_0 =\\,", "ae842340c8dbee490945399c210f24f7": "OR(\\alpha,\\alpha')=s^TD(u\\otimes v)=\\alpha + \\alpha' - \\alpha\\alpha'", "ae845c34191d9f3b11b61a98b4831f02": "\\hbar \\omega w_{cv} = {1 \\over 2} \\omega \\kappa_2 \\epsilon_0 {E_0}^2", "ae850b771bfe3c9bf7eb443a732b9fc4": "i > 1", "ae85b217407a8c9cb48cf6010ae40356": "y_b = b_0 \\sum_{r = 0}^\\infty \\frac{(c + \\gamma - 1) (c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^{r + c}= b_0 x^c \\sum_{r = 0}^\\infty M_r x^r.", "ae85dfac57c2193d8b64d60aacb5be96": "(\\hbar^2 \\ell(\\ell+1))", "ae85e5d1d9c929a3df267d21443a914d": "\\theta (F,B) = x \\in L_{n+1} (\\pi_1 (X))", "ae860378164d7bec93737eb55692c787": "\\widehat{\\alpha \\wedge \\beta}:= \\hat{\\alpha} \\hat{\\beta}", "ae8611f48b158acec596cd772453d040": "S_{mm}\\,", "ae8659a5e0f945e3cc598ea2eddd6924": "\\partial M=0", "ae866d5222f2ca386f3445c97381e194": "|\\mu|(A)= \\sup\\sum_{n=1}^\\infty |\\mu(A_n)|", "ae868824c3321f6c72f25d4cb6ecddd1": "g^2 \\le \\sum_{{i \\le g }}c_{i} ", "ae8688c95f095c271b20d91fa8995754": "\n0 = - \\frac{r^{2}}{c} \\frac{d\\varphi}{d\\tau} \\delta \\varphi \n- \\int { \\frac{d}{dq} \\left[ - \\frac{r^{2}}{c} \\frac{d\\varphi}{d\\tau} \\right] \\delta \\varphi dq }\n\\,.", "ae868ca14c5554853389300e3f6b23c0": "\\alpha^{x}=\\beta", "ae86959a3bc8028c283231bd2c6b2003": "\\scriptstyle f\\colon [-1,1]^2\\to\\mathbb{R}^2", "ae86a3035ce5b3be164f21aef4eb4690": "\\delta: Q \\times \\Sigma \\times D \\rightarrow 2^{Q^*}", "ae87075b98e97126d880018a79fbc838": "K=ab\\sin{\\frac{B}{2}}\\csc{\\frac{D}{2}}\\sin \\frac{B+D}{2}.", "ae876edef31847ad95fd40ebaf593ba4": "\\phi_n= I_{|n|}(\\kappa)/I_0(\\kappa)", "ae879a8730284180d0756e2c916e2cc3": "C(\\theta)", "ae880de12806272f2616f6801ccb9269": "R_{HOxII,-25} = R'_{HoxII}\\left ( \\frac {1 + \\frac {-25} {1000}} {1 + \\frac {\\delta^{13}C_{HoXII}} {1000}} \\right )^2", "ae884104bd8f6aefc955b646fa6dc8e1": " T = \\sum_{ i = 1 }^K t_i ", "ae88672de91e61566ad2f151f7200296": "J_{ij}=\\frac{\\partial r_i}{\\partial \\beta_j}", "ae886e056934c810dbe9d15c237f374f": " \\mathbb{Z}_{p_{1}},\\dots , \\mathbb{Z}_{p_{r}}", "ae88701cc17c0f46c0e76e4ddeb10641": " y_{k+1} = y_k + h f(t_{k+1}, y_{k+1}). ", "ae88980a5e0aae48e12c8d249b369602": "\\phi_y = \\phi_x + \\pi/2", "ae88abfab61d80b678edcf282ebbb623": " r^{-n}~\\sin(n\\theta) \\,", "ae88f9e768a6e1041e4a8d45d34c17a0": "\\, y_n", "ae891dbd21fc6cfc337d720e8f8d1bba": " 0= q E_r +q v_\\theta B + m {v_\\theta}^2/r ,", "ae89664d26f1223e94ebebb8639c37db": "\ny_{n-1} = y(x_n-h) = y(x_n) - hy'(x_n) + \\frac{h^2}{2!}y''(x_n) - \\frac{h^3}{3!}y'''(x_n) + \\frac{h^4}{4!}y''''(x_n) - \\frac{h^5}{5!}y'''''(x_n) + \\mathcal{O} (h^6)\n", "ae89c5b933e46dcd91ec7afdc427e78b": "\\frac{k}{2}", "ae89f60ba5c7cc542bcc152db846f2f0": "T^{\\alpha\\beta}", "ae89ffcd23c8da87908a88cc2826728c": " F_p = \\underbrace{\\big\\{ 0,1,\\ldots,p-1 \\big\\}}_p ", "ae8a003a2fe94209e73b58be10ef0bc4": "f_n(x) = \\frac1{2^{n-1}}T_n(x) - w_n(x)", "ae8a1fbe271555563971b957b664802f": "d=1,N=0", "ae8a6f9b6d2dd54b6806a01c3ae14eb3": "n \\in P", "ae8a8bb181c358f5d0c7982540ded981": "\\pi r\\left(r + \\sqrt{r^2 + h^2}\\right)", "ae8ad5c7d6c5d5e7a837459543523ee8": "\\nu_p:\\textbf{Z} \\to \\textbf{N}", "ae8ba6297efe7112af0c1e1f47d6dc6c": "\n(u_r, u_\\theta)=\\left( \\frac{1}{r} {\\partial \\psi \\over \\partial \\theta}, - {\\partial \\psi \\over \\partial r} \\right) = \n\\left(-\\frac{A}{r^2}\\cos\\theta, -\\frac{A}{r^2}\\sin\\theta\\right).\n", "ae8beb94cda79be6497b6cee667b0f1c": "T^kV \\otimes T^\\ell V \\to T^{k + \\ell}V", "ae8bebd133e94e2f8fdbc637306c15a1": " a_i \\in F(X)", "ae8c2b3bdf0c2a5167d5cf284953df7a": " V^{1} = x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + (1 + u_{1}u_{1})\\frac{\\partial}{\\partial u_{1}} \\,", "ae8c339812ad36025b7ea503b4c71da0": " \\mathrm{cost}(\\pi,\\sigma) = \\sum_{i=1}^l d(\\pi(i-1),\\pi(i)) + T_i(\\pi(i)).", "ae8cc10dac73f5cc37376b315c8b6fdf": "\\mathbf{K}_j", "ae8cfdfee585f37c3052fd0aac8436a5": "H = pu_t - \\frac{u_t^2}{x_t} - \\lambda_{t+1} u_t", "ae8d233a51b26739dd977f6b24e7ac3e": "x,\\,y \\in K", "ae8d277fe891a6ac970d9162a322950a": "\\displaystyle G= KA_+U,", "ae8decbb3f6d0e0ce6c85b28b5e0d6d8": "\\frac{c^2k^2}{\\omega^2}=1-\\frac{\\omega_p^2/\\omega^2}{1+(\\omega_c/\\omega)}", "ae8e6060a5c2350a971c726914654bcf": "\\scriptstyle\\sum\\limits_{k=1}^\\infty\\frac{\\mu(k)}{k}\\,=\\,0", "ae8e63c994d6e4d73d23cc0d1dcb46bc": "\\underline{P}(A)=\\overline{P}(A)", "ae8ed214cf904f532606d2f5bc5edc4f": "\\,\\!e=(S-a)/2", "ae8f0acd19ff2504e977d4801bdc7a40": "k_\\text{B}", "ae8f5a39b97913f3583fca22787cfe4c": "C'\\supseteq C(\\alpha)", "ae8f780e053b76a441dbb9d59142e198": " \\mu_X[P_X] ", "ae8f7e3fcd6c4387d6a12914daef0fd7": " = \\frac{1}{2}V\\sigma_{ij}\\epsilon_{ij} - ST + \\sum_i \\mu_i N_i\\,", "ae8fe8ded13422a14a609dc0e7342fe5": "I=\\langle l_1,l_2,\\ldots\\rangle", "ae901a53b3b2b6846792905bc9f7d51a": " d_A(z) = \\frac{d_M(z)}{1+z}", "ae901aee6d09f1a2ddc3544979785260": "dx_t = \\theta (\\mu-x_t)\\,dt + \\sigma\\, dW_t", "ae9032ea52a396250372b8cb28024eee": "L_{k}\\ ", "ae9038b18f618fd829649594f3d33486": "T_{M\\;}^{\\;}", "ae9050cdf85fe909d579ba28bb59186c": "\\oint_C \\bold{E} \\cdot {\\rm d} \\boldsymbol{\\ell} = \\frac{\\partial }{\\partial t}", "ae90c351d3aec1e9ca2681381751e2cd": "4s(s-a)(s-b)(s-c) = [s(s-a)+(s-b)(s-c)]^2 - [s(s-a)-(s-b)(s-c)]^2", "ae90c408c0dec02c269ffd3c541f806f": "r=2a\\cos\\theta", "ae9102c16e49824ae844b10909c5d727": "I(k) = \\int g(x) e^{i k f(x)} \\, dx", "ae9153ec0f42ab9b25aed867f102322e": "|\\bigstar |\\bigstar \\bigstar |", "ae920578d835a62a721d526798dcf7a3": " \\mathcal{T}", "ae920a813407fec48feb0583ed7fc70c": "\\sum_{i\\in I}N_i=M\\,", "ae920b837c4993fe76c0215a131430cb": "v_{i,m}\\,", "ae920f54fd12a0b5822bb7cb91d4d975": " {M} ~\\overset{\\underset{\\mathrm{def}}{}}{=}~ \\left(\\left[\\begin{array}{cc}{A} & {0}\\end{array}\\right] \\left[\\begin{array}{cc} {Q} & {A}_{eq}^{T}\\\\ -{A}_{eq} & {0}\\end{array}\\right]^{-1} \\left[\\begin{array}{cc}{A}^{T} \\\\ {0}\\end{array}\\right]\\right)\\,", "ae92296b32c4000d7af31544e31eb2af": "\\ N = \\frac{\\log \\, \\bigg[ \\Big(\\frac{X_d}{1-X_d}\\Big)\\Big(\\frac{1-X_b}{X_b} \\Big) \\bigg]}{\\log \\, \\alpha_{avg}} ", "ae925122246c1cebaaf493db23a97123": " \\Psi^{\\dagger}(\\bold{r})=\\sum_{\\nu} {\\psi^*}_{\\nu} \\left( \\bold{r} \\right) {a^{\\dagger}}_{\\nu}", "ae9285d1829a209c9a23f2151f777a92": "\n\\begin{array}{ll}\n & P\\left(S^{t}|O^{0}\\wedge\\cdots\\wedge O^{t}\\right)\\\\\n= & P\\left(O^{t}|S^{t}\\right)\\times\\sum_{S^{t-1}}\\left[P\\left(S^{t}|S^{t-1}\\right)\\times P\\left(S^{t-1}|O^{0}\\wedge\\cdots\\wedge O^{t-1}\\right)\\right]\\end{array}\n", "ae92862f9c5b0c4c8a55553da98642f3": "\\frac{[n]_q!}{[m]_q![n-m]_q!}", "ae92c1096ea322fdcf271e5633edcdce": "(t_1,t_n)", "ae92c4c1df9d2eef50d948f10277cd1f": "\\sigma_X \\neq 1", "ae92f1fdeddd4c7e29b83fd8e8ad0dbc": "\\pi=3+\\cfrac{1}{7+\\cfrac{1}{15+\\cfrac{1}{1+\\cfrac{1}{292+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{1+\\cfrac{1}{2+\\cfrac{1}{1+\\cfrac{1}{3+\\cfrac{1}{1+\\ddots}}}}}}}}}}}", "ae9323f2d938aa6f08518226d538ea96": "\\{E_i\\}", "ae932984e25a5a5aed4f5aac87e26c5e": "\\begin{pmatrix}e^{\\theta/2} & 0 \\\\ 0 & e^{-\\theta/2}\\end{pmatrix}", "ae932cea9ceed9cde9a7ec0dc7e1db5f": "x,y,z,...", "ae9379a0bed38405ee14c7734e8b6f51": "a_0=0.3635819; \\quad a_1=0.4891775; \\quad a_2=0.1365995; \\quad a_3=0.0106411\\,", "ae937e535876890c855c50948832aed6": "U_{bias}^{LE}(\\mathbf{Q};t)", "ae9390cc3352e70fd19896296154247b": "{\\mathbf v} =v^i {\\mathbf e}_i", "ae93aef978af6e02d2ec11bc27dabb78": "I(2n+1)=\\int_0^\\pi \\sin^{2n+1}xdx=\\frac{2n}{2n+1}I(2n-1)=\\frac{2n}{2n+1} \\cdot \\frac{2n-2}{2n-1}I(2n-3)", "ae940185d1fb9ff8931a519c5ec2ac14": " n", "ae94223db6c419632cec6d0e0b50d694": "\\{p \\lor q, p \\to r, q \\to r\\} \\vdash r", "ae9437cda21a8444e5e3314ca3ea012d": " B_{n}=\\frac{n}{2^{n+1}-2}\\sum_{k=0}^{n-1} (-2)^{-k}\\; W_{n-1,k} \\ . ", "ae948f73bb485b7d014c52b36599ad8b": " \\mathcal{r} F(\\underline{x},\\underline{u}_1) = \\mathcal{r} F(\\underline{x},\\underline{u}_2) = \\underline{0} ", "ae94a79308fba543fc80f5abe60f979a": "H_1 \\tilde P_n", "ae94dff5b55be73d5bd0f151042bde03": "\\mu\\to 0,\\,", "ae9517275ecbc405a4bafc43fef667dd": "c,\\,d,\\,g,\\,h", "ae951ac69a63557c4690c448d64245c7": "y=\\frac{1}{a} \\int_0^{L'} \\sin s^2 \\, ds \\, ", "ae9541230beb0178303001c36945c756": " \\left | \\left \\{x\\in Q: |f-f_{Q}|>\\lambda \\right \\} \\right |\\leq c_{1}\\exp \\left (-c_{2}\\frac{\\lambda}{\\|f\\|_{BMO}} \\right )|Q|.", "ae95450add3c1a6a2d3076947bc2f761": "\\langle B_N\\rangle", "ae955f1b907797881292d48e841b8a63": "y=\\sum_{r=0}^{\\infty }{a_{r}s^{r+c}} ", "ae95920e72f2cd9650009c816946b522": "S_N = a_N B_N - \\sum_{n=0}^{N-1} B_n (a_{n+1} - a_n).", "ae95971e5847f09144dd57064684de6e": "\\mathrm{color2} = \\frac{11}{32} * \\mathrm{color0} +\\frac{21}{32} * \\mathrm{color3} \\approx \\frac{1}{3} * \\mathrm{color0} +\\frac{2}{3} * \\mathrm{color3} ", "ae95c3dfd9d41d7849dd323a73160658": "e^{2\\pi i k / 2N}", "ae96094e1f1be2eff551ee27b7efb902": "\\gamma_n(x) = \\frac{1}{n!} \\cdot x^n", "ae968d4c49248ba4b90ce9931bdd68c8": "V_\\mathrm{LT} = \\frac{R_\\mathrm{E}}{R_\\mathrm{E} + R_\\mathrm{C1}}{V_+}", "ae972be1d0ab5fbf61306cfa604f5649": "\\left \\vert F''(x) \\right \\vert ", "ae975be09d668aef556b7fe547b69968": "0 \\equiv u_{p-1} \\equiv 2 t_r u_r", "ae9768216a248eb33fd8a5799387dc9d": "\n D^{\\mathrm{beam}} \\approx \\cfrac{f[3(2h+f)^2+f^2]}{6}~C_{11}^{\\mathrm{face}} \\approx \\cfrac{f(2h+f)^2}{2}~C_{11}^{\\mathrm{face}}\n ", "ae976c01e16d951131a2c5481dbe0fb3": "f(N) = 1 + 2^{N-2} \\quad \\mbox{for all } N \\geq 3.", "ae977edf26874cd305f70ef3763eb8ea": "\n \\int_0^1 \\frac{h(tx)}{x} \\,(1-x)^{n-1}\\,dx \\leq t^\\alpha\\text{ for all }t\\in[0,+\\infty),", "ae97e37448738bfbafceb1956220be33": "D_{\\mathcal{A}}", "ae986c38762889720075c708aabe0653": "C_{initial} =", "ae9870811c13fc93b7b826c25ae538e0": "\\left|A_{\\text{v}}\\right|\\,", "ae98b6416abfa6816de7b2dcfbfca9e9": " \\sup_{B}\\frac{1}{|B|}\\int_{B}e^{\\phi-\\phi_{B}}dx<\\infty ", "ae990513e01d37961246d0cf682aa9e2": "P_{\\mathfrak{p}}", "ae991ef7abc8293be47d39d5973e84b1": " R_{M,t+1} = {{M_{t+1}-M_t+D_{M,t+1}}\\over M_t}", "ae99248e4442de56367d825443d29943": "\n\\mathbb{E}_{X^{n}}\\left\\{ \\text{Tr}\\left\\{ \\Pi_{\\rho,\\delta}\\ \\rho_{X^{n}\n}\\right\\} \\right\\} =\\text{Tr}\\left\\{ \\Pi_{\\rho,\\delta}\\ \\mathbb{E}\n_{X^{n}}\\left\\{ \\rho_{X^{n}}\\right\\} \\right\\} ", "ae9936702eadb7255154288956ac32ee": " T_{ij} ", "ae993e178a0e8928704d293f8aace441": "\\mbox{EOT} = \\mbox{GHA} - \\mbox{GMHA} ", "ae994359484855d70a77c6fb873460fe": "r=-f_1(\\theta+\\pi),\\ r=-f_1(\\theta-\\pi),\\ r=f_1(\\theta+2\\pi),\\ r=f_1(\\theta-2\\pi),\\ \\dots", "ae995ded33d6f99aa54b3b789c174f61": "j : X \\to \\mathbf P^n", "ae9962443dd3a9d36aa98b937f1e6281": "\n\\phi(r) = \\begin{cases}\n r^{k-1} \\ln(r^r) & \\mbox{for } r < 1 \\\\\n r^k \\ln(r) & \\mbox{for } r \\ge 1\n \\end{cases}\n", "ae9991584f6e5e82d54c60dd30e65775": "\\begin{matrix} {52 \\choose 7} = 133,784,560 \\end{matrix}", "ae9998c9143b13ed68d2f9ead78e069a": "a_{ij}=x_{g_i g_j}", "ae999b7233e3747a700bcd5f446332f7": "\\delta h", "ae99e2b4e62aa37452427d575c3974b7": "\\psi=(u^1,\\dots,u^n,y^1,\\dots,y^{p-n})\\,", "ae9a640824dddd1719b51a56858e935e": "f(t) = \\sum_{i=1}^np_i\\, \\delta(t-x_i),", "ae9a6f2ddbd3c6e5a6b5d2e62f96a297": "\\tilde c_n = \\langle \\tilde f, \\phi_n \\rangle = \\sum_i w_i \\tilde f(x_i) \\overline{\\phi_n(x_i)} = \\sum_i w_i V(x_i) f(x_i) \\overline{\\phi_n(x_i)}", "ae9a9957c624622ac2138dcee3d24344": "\\mathbf \\psi_{1s} = \\left (\\frac{\\zeta^3}{\\pi} \\right ) ^{0.50}e^{-\\zeta r}", "ae9aaf1a55dd88f9a9b9851b9264e0af": "\\frac{a'_\\max}{a_\\max}=\\left[\\frac{p_1(u-1)}{p_1(u)}\\left(1-2|p_1(u)|\\right)\\right]^{\\frac{1}{2}};", "ae9ad243f355f0c1af9754b4d800df0f": " \\gamma\\left(\\cdot\\right) ", "ae9b596f7d228458961100a9633010d3": "m_{em}=\\frac{4}{3} \\cdot \\frac{E_{em}}{c^2}", "ae9b7b219f56685b36afa99db572ff50": "\\beta < 1", "ae9b834189a142978a641a2a75defa37": "d(f_1^{n_1}...f_k^{n_k})=(n_1 , \\ldots , n_k)", "ae9b948f9048215e39daa850985084bc": "\\dots \\rightarrow P^1 \\rightarrow P^0 \\rightarrow A \\rightarrow 0, ", "ae9ba0723d6fd9bf44ac164a08f2af52": "\\operatorname{Der}(\\mathfrak{g})", "ae9c0828c3b91961708b306c4253a96f": " \\operatorname{value} = \\lambda v.(\\lambda h.h\\ v) ", "ae9c25ecdfab52eaeed288fcad91f5bd": "\\sum _x (x)_a = \\frac{(x)_{a+1}}{a+1}+C", "ae9c2712a8c806a08364092eed1ae57d": "G \\cap M", "ae9c9a36f958ae9196d57077af852c38": "dx/y|_D ", "ae9c9f8272b9a8ab81c647dd15d69f88": "V^\\perp\\subseteq U^\\perp", "ae9ca27a75a007b0862d804c8ca3b791": " V^{\\prime }\\left( t\\right) =f_{t}\\left( x,t\\right) \\text{ for each }x\\in\nX^{\\ast }\\left( t\\right) .", "ae9ca687b85044c485a6afea618acc82": "\\begin{align}\n x &= \\lambda - \\lambda_0\\\\\n y &= \\sin \\varphi\n\\end{align}", "ae9ceeaa222478e64cc1b3e4017e13c8": "2\\mathbf{Z}[i]=\\left((1+i)\\mathbf{Z}[i]\\right)^2.", "ae9d4d28f0b8d200dfabbd791c2141da": "{10}^{\\,\\! 4 \\cdot 2^{100}}", "ae9ddf478267d011cd071057920bfabc": "\\displaystyle{\\mathcal{U}f(z)= (\\mathcal{U}f,E_z) = (f,\\mathcal{U}^* E_z) = e^{-|z|^2}(f, \\mathcal{U}^* W_{\\mathcal F}(z)E_0) =e^{-|z|^2}(W(-z)f,H_0),}", "ae9de2ca8a17d5343e7f65b6fa40d88b": "\\chi_\\sigma(\\mathbf{r}) = \\frac{D_\\sigma(\\mathbf{r})}{D^0_\\sigma(\\mathbf{r})},", "ae9de5c1241e96b44a589f1ca3f41261": " \\nabla \\times \\mathbf{H} = \\mathbf{J} + \\frac{\\partial \\mathbf{D}}{\\partial t}. ", "ae9e54cc0a7d7cc879ea105bad907c08": " \\int_{|x|>R} |f(x)|^p dx < \\epsilon^p\\,", "ae9ed7591ea89092eef06ce1ff9b9b06": " PSRR = 20 \\times log \\frac{Ripple_{Input}}{Ripple_{Output}} ", "ae9ef1429a38344c05a13b18e043730e": "(\\tfrac{2}{7}) \n=(\\tfrac{2}{15})\n=(\\tfrac{2}{23})\n=(\\tfrac{2}{31})\n\\dots=1,\n", "ae9f6961fdba246b9e83d074bb63dba5": "P_X(t)=\\sum \\text{rank}(H^n(X))t^n", "ae9fc22900acc0995f0c1fc70cbfe936": "v_{\\alpha}^2(x)/2 m + q_{\\alpha} \\phi(x) ", "aea01230240d8cb594d311cd42ad5391": "t_{2\\alpha'}(\\tilde{x}|\\mu,\\sigma^2 = \\beta'/\\alpha')", "aea03d958973da950ed5e032064dcc80": "\n \\frac{\\partial f}{\\partial {x'}^i} = \\sum_j\n \\frac{\\partial f}{\\partial {x}^j} \\;\n \\frac{\\partial {x}^j}{\\partial {x'}^i}\n", "aea04b144b145e8f1f2ef03d78ba33e9": " \\operatorname{E}(R_p) = \\sum_i w_i \\operatorname{E}(R_i) \\quad ", "aea0ce0fcf96e46b3b452c19ecf3f1af": "x\\vee y", "aea191b41d5ac376730c1efa6f7f03a3": "OOO\\ldots", "aea1a32604221297b1b07bd5daf10eb3": "K-k", "aea1d6515342531759f9331ef8f579e2": "f'(x) = h'(g(x)) \\cdot g'(x). \\,", "aea20b7cafe1d8c62c248b1bfc21ab3e": "\\lambda_i\\neq 0,\\,\\forall i\\in\\mathcal{I}'\\cup\\mathcal{E}", "aea23c010b9f40e29cee2248a64b0a89": "\n\\begin{bmatrix}\n\tZ'_{11} & Z'_{12} \\\\\n\tZ'_{21} & Z'_{22}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\tn_{11} & n_{12} \\\\\n\tn_{21} & n_{22}\n\\end{bmatrix}\n\\begin{bmatrix}\n\tZ_{11} & Z_{12} \\\\\n\tZ_{21} & Z_{22}\n\\end{bmatrix}\n\\begin{bmatrix}\n\tn_{11} & n_{12} \\\\\n\tn_{21} & n_{22}\n\\end{bmatrix}\n", "aea2876a0ecb2bfa347b156b8986e01d": "F_i \\equiv -\\left. \\left(\\frac{\\partial V}{\\partial r_i} \\right)\\right|_{\\mathbf{0}}", "aea2f07a98330b5e92a6b93703fd11ad": "\\left| a \\right| > 1.9143 ", "aea3094f9a691312f1ef68f98a0eb287": " \\left| \\alpha - \\frac{p}{q} \\right | > \\frac{A}{q^n} ", "aea30a4582aca5bae560ef93a40c50a3": " \\|x + y\\|^2 + \\|x - y\\|^2 = 2\\|x\\|^2 + 2\\|y\\|^2. ", "aea3422dfe7fe650b91c3ececfeef34f": " C'(\\beta)\\sim C(0)\\left(1-\\beta\\right)^3.", "aea40d6173a792ba22acd4d69ecb1fe6": " i\\hbar\\frac{d\\sigma_j}{dt}=[\\sigma_j,H]=[\\sigma_j, -\\mu \\sigma_i B_i]=-\\mu\\left(\\sigma_j\\sigma_i B_i - \\sigma_i\\sigma_j B_i\\right)=\\mu[\\sigma_i,\\sigma_j]B_i = 2\\mu i \\varepsilon_{ijk}\\sigma_k B_i ", "aea41ab8e073d91d4ee5a0f930a2ebde": "x : \\mathcal{A} \\to (0,1) ", "aea48456c7ed5276ff5467e079de784c": "f(m)=n", "aea4a5c84f95e486320ea25403d0acd0": "\\ \\displaystyle r_{c} \\le R(q,u) \\ ", "aea4b7931734f5e854bb6bc297f4c77f": "\\hat H \\psi_n = E_n\\, \\psi_n", "aea504859c3f8e2fc881f84385ca8316": "\\mathbf{v}^{b}(\\mathbf{x},t)", "aea5306e965e137ecb60ef101df3509c": "0.5 \\le \\mathrm{Pr} \\le 2000", "aea5366f604c722b48d344a869ffd9d9": "\\sum_i e_i(T) = I,\\,", "aea6087ac85f21226acf0bd4abb3a2d9": "I_{t} = I_{0} + b (C_{t} - C_{t-1})", "aea60d67724b1934ffa0bb2f4f41bd1e": "h_s(t_e)", "aea63352ce212582ec913191ab22ce66": "A_m(3,4) = 1, 4, 18, 88, 455, 2448, 13566, 76912, 444015, 2601300,\\ldots", "aea63d57ddea44046912ec3045e287ea": "A \\oplus B", "aea63efdd651ebfb9ad96e3246fdda92": "Q \\oplus Q' = \\Delta", "aea66fa73a9da048edd7c39e594409a3": "G=E-TS", "aea678fe1ab3d1634895829e4e22f001": "A^T P + P A - P B R^{-1} B^T P + Q = 0 \\,", "aea6b7833910779b6f60a6a1a24c3be7": "S^{j-1} \\times D^{m-j}", "aea71dc0283eae38762cb35539bd2816": "\\begin{align}\nT(\\partial_{\\overline{z}} f) &=\\partial_z Tf, \\\\\nT^*(\\partial_z f) =\\partial_{\\overline{z}} T^*f.\n\\end{align}", "aea757deb53c5bc32db224e9380de6e4": "n_{3}=\\frac{p_{2} V_{t}}{z_{f2} R T_{ref}}", "aea75f414537ce17ecf5f8cdea38ae2c": "\\langle \\mathbf{Ax}, \\mathbf{y}\\rangle = \\langle \\mathbf{x},\\mathbf{A}^* \\mathbf{y} \\rangle", "aea77b7e08bea798e5609d730de031e8": " HETP = A + \\frac{B}{u} + C \\cdot f( \\lambda ) \\cdot u ", "aea780faa086a00774f0a5dff0bab087": "\\frac{d N_i}{d t}=V\\sum_r \\gamma_{ri} w_r", "aea784601123d09b0ec264902610334f": "x < c_1", "aea7b23916878280d6f0d865b093ff5f": " \nA_n=\\{x\\in E |f_k(x)>a~\\forall k\\geq n \\}.\n", "aea7ed499a90fe2b927c665c4dbf82b9": "g_1,g_2,\\dots ,g_n", "aea80585e8900d015113ef3c961557aa": " \\min \\leq H \\leq G \\leq A \\leq Q \\leq \\max ", "aea80c0268a6f6a14b534ca36bc5c231": "\n f(x) = \\begin{cases}\n 4x & \\text{for }0 \\le x < \\frac{1}{2} \\\\\n 4-4x & \\text{for }\\frac{1}{2} \\le x \\le 1\n \\end{cases}\n", "aea824bc5e5471c3fb30e514f83dae1e": "\\omega_{\\mathrm{f}}^2 = \\omega_{\\mathrm{i}}^2 + 2 \\alpha (\\theta_{\\mathrm{f}} - \\theta_{\\mathrm{i}}).", "aea876d9cfdde1f9467f12bec763a382": "f^{-1}(y) = \\pm\\sqrt{y} . ", "aea8c5111742607cf7de72e433ee457e": "t \\times n", "aea91ac64cabb11925dd8963dec6a901": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n (A\\mathbf{x}_n-\\mathbf{b}),\\ n \\ge 0.", "aea965f9a978bbf5389c2ba36283cbf5": "\\begin{align}\n &\\text{If } &g \\circ f = \\mathrm{id}_X\\text{,} \\\\\n &\\text{then } &f \\circ g = \\mathrm{id}_Y\\text{.}\n\\end{align}", "aea99df041688e12bcd5100f06da92e7": "RTS(O_j) > TS(T_i)", "aea99fee29472683fd84dc8cb58f5441": "\\sum_n b_n \\lambda^{-n}", "aea9bd66852635b7c282979d41ef9abb": "L+\\Delta L = \\left ( R+r \\right )\\Theta ", "aea9eb57e724c70a057c85486f8de063": "2^s", "aeaa246407e2cc827c7aa955fb88f2f5": "\\Delta^2(q_1) \\ge 0 ", "aeaa37894efb6b62c31896b7806d68df": "\\textstyle \\Lambda_0(P) \\geq \\Lambda + \\alpha", "aeaac1118bc3926a1d37d4ca74234d8f": "\n\\begin{align}\n& \\sin{\\varphi} + \\sin{(\\varphi + \\alpha)} + \\sin{(\\varphi + 2\\alpha)} + \\cdots {} \\\\[8pt]\n& {} \\qquad\\qquad \\cdots + \\sin{(\\varphi + n\\alpha)} = \\frac{\\sin{\\left(\\frac{(n+1) \\alpha}{2}\\right)} \\cdot \\sin{(\\varphi + \\frac{n \\alpha}{2})}}{\\sin{\\frac{\\alpha}{2}}} \\quad\\hbox{and}\\\\[10pt]\n& \\cos{\\varphi} + \\cos{(\\varphi + \\alpha)} + \\cos{(\\varphi + 2\\alpha)} + \\cdots {} \\\\[8pt]\n& {} \\qquad\\qquad \\cdots + \\cos{(\\varphi + n\\alpha)} = \\frac{\\sin{\\left(\\frac{(n+1) \\alpha}{2}\\right)} \\cdot \\cos{(\\varphi + \\frac{n \\alpha}{2})}}{\\sin{\\frac{\\alpha}{2}}}.\n\\end{align}\n", "aeab120517a102526d9a4812da5adf4f": "\\Pi_e=\\frac{m_\\text{empty}}{m_\\text{initial}}", "aeab2a60e0268f50c685c9ed5b738caf": "\\mathbb{Z}_2", "aeab61761f7fefeb502f3efb3b3418bb": "e^X", "aeab6940aef820bb8d3d023312aaf2ab": "0 < m(x) < 1", "aeab73210d55d2048250dbac8cded996": "X \\!", "aeab7b2dce123bbd58efe9fa613cae8b": "r_kr_{k-1}^{*} = E_se^{j\\left(\\theta_k - \\theta_{k-1}\\right)} + \\sqrt{E_s}e^{j\\theta_k}n_{k-1}^{*} + \\sqrt{E_s}e^{-j\\theta_{k-1}}n_k + n_kn_{k-1}", "aeac1549cc6ba47ce759cd1a5e2d0fdf": "\\Psi(\\bold{r},t)", "aeac2492d1f494d3d86fcaadb8d0c4f1": "Z = \\int DH e^{ \\int d^dx \\left[ A H^2 + Z |\\nabla H|^2 + \\lambda H^4 \\right]}", "aeacaec866f69b1fb901a7abd8cba807": "\\operatorname{Re}", "aead0ce3b61b108bc21db5250bdcea87": " \\phi_{um}(r) = \\max \\left[ 0, \\min \\left(2 r, \\left(0.25 + 0.75 r \\right), \\left(0.75 + 0.25 r \\right), 2 \\right) \\right] ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{um}(r) = 2", "aead3e159de667518a73c5ca7897ce94": "\n\\text{Maximum Profit (upside)} = C_n\\,\n", "aeada2190a142faecca5f5eecf434519": "\\deg_x Q(x,y) = m", "aeadeb18dcc833dd0f4e80c6f419fa57": "J=-1", "aeae7d8cda1003e7eab80896b74cc60a": "\\int_M (f\\mathrm{div}(X) + X(f)) \\omega = \\int_M (f\\mathcal{L}_X + \\mathcal{L}_X(f)) \\omega ", "aeae9a1b03996f6a8629d97b4187b66e": " \\Pi(n; \\,\\mathrm{sn}(u;k); \\,k) = \\int_0^u \\frac{dw} {1 - n \\,\\mathrm{sn}^2 (w;k)}.", "aeaed583aa6296b79a73afa7aafc6d38": " B^2/(2\\mu_0)", "aeaed6446aa8b53ba4a37a7e66b04f99": "V_\\lambda\\subseteq M", "aeaf1784a152be515ad1069e33a74eea": "\\iint \\limits_{R_C} \\left ( c^2 u_{x x}(x,t) - u_{t t}(x,t) \\right ) dx dt = \\iint \\limits_{R_C} s(x,t) dx dt. ", "aeb010ed780a216177210fda33454b82": "{T_v}=T\\frac{w+\\epsilon}{\\epsilon(1+w)}\\, .", "aeb041fd43a6f7248415df7011edd949": "\\quad H^B \\Phi_m^B = E_m^B\\Phi_m^B", "aeb065e020e4e86329c62469a1223987": "I_S[f] = \\frac{1}{n} \\displaystyle \\sum_{i=1}^n V( f(\\vec{x}_i),y_i)", "aeb07379818ba651c9aaa121e7de571c": "p(\\mathbf{y}|m)=\\frac{1}{(2\\pi)^{n/2}}\\sqrt{\\frac{\\det(\\boldsymbol\\Lambda_0)}{\\det(\\boldsymbol\\Lambda_n)}} \\cdot \\frac{b_0^{a_0}}{b_n^{a_n}} \\cdot \\frac{\\Gamma(a_n)}{\\Gamma(a_0)}", "aeb09d486d68c742cb61b8fc76e80891": "n=m^{d_k} + m^{d_{k-1}} + \\dots + m^{d_2} + m^{d_1}", "aeb0c17c9cdeb771105a10af9bed88e9": "w(t_2)", "aeb11284b54f0187c01eb70c16945e4a": " \\delta W_{H,p} = \\rho g\\left( \\left(2bh \\text{cos}\\alpha + b^2 \\text{cos}\\alpha \\text{sin}\\alpha\\right) \\delta h + \\left( \\frac{b^3}{3}\\text{sin}\\alpha + b^2 h\\right) \\delta\\alpha\\right) ", "aeb12fece13ac36fa6d17cc391b86104": "\\{(t,i_O) | t\\in T_O\\}", "aeb13e62a1a680c16824b7f73232f990": "BG(n) \\to BG(n+1)", "aeb1a025917451b1f7e5609cb9394f1a": "10 \\cdot 20 \\cdot 253 = 50,600\\,", "aeb26fb91dc409d3f19315e0c246809e": " K = \\frac{1}{(1-c)} ", "aeb2cfb09bf4183199f25fe1a4c26b67": "P_{m}", "aeb2f8fd636caf4678de018a6e247a45": " {\\psi^*}_{\\nu} \\left( \\bold{r} \\right)", "aeb328bfc33c154e0aff1ba958958993": "\\displaystyle{g(z) = f(z) -z}", "aeb3b74c7d9d73a1816ce0901db11815": "j = i_X \\mu", "aeb41ce04655377a46237bbd4c8c42d3": "\\textstyle \\Gamma(z) = \\int_0^\\infty t^{z-1} e^{-t}\\,\\mathrm{d}t", "aeb457b91cfb46ee4fab5b4edd4f6ed6": "i\\hbar \\frac{\\partial}{\\partial t}\\Psi(\\mathbf{r},\\,t)=-\\frac{\\hbar^2}{2m}\\nabla^2\\Psi(\\mathbf{r},\\,t) + V(\\mathbf{r})\\Psi(\\mathbf{r},\\,t).", "aeb47300ca0bcc6eb10f2923bb743743": " \\int_V \\mathcal D \\phi \\; e^{-\\langle\\phi|S|\\phi\\rangle} \\propto \\frac1{\\sqrt{\\det S}}. ", "aeb4858254faf84606858c464e1f575c": "\\langle B\\alpha^\\mu_z(A)\\rangle", "aeb509d06986c61c1eb549d809836a49": "\\prod_{k=0}^\\infty \\frac{1}{(1-q^kt)}=\\sum_{k=0}^\\infty \n\\frac{t^k}{[k]_q!\\,(1-q)^k} . ", "aeb56018fc66b678da53dfab99832bd7": "\\frac{2}{3}=\\frac{6}{9}=\\frac{12}{18}=\\frac{144}{216}=\\frac{200,000}{300,000}", "aeb5c3f5f9a9bc2e423b70dc84ccd550": "y^{\\ast\n}\\left( t\\right) ", "aeb61e53f99fde00b729e9ee065e1798": "a = x_0 \\leq \\xi_0 \\leq x_1 \\leq \\ldots x_{n-1} \\leq \\xi_{n-1} \\leq x_n = b.", "aeb69818486e973f494057a40bfb188c": "\\sigma_{yy}- \\frac{\\sigma^2_{xy}}{\\sigma_{xx}} +|\\sigma_{yz}-\\frac{\\sigma_{xz}\\sigma_{xy}}{\\sigma_{xx}}|", "aeb6af6f2ca2120ea0e46df13fa223d1": "\n\\ln (z) = 2\\cdot\\operatorname{artanh}\\,\\frac{z-1}{z+1} = 2 \\left ( \\frac{z-1}{z+1} + \\frac{1}{3}{\\left(\\frac{z-1}{z+1}\\right)}^3 + \\frac{1}{5}{\\left(\\frac{z-1}{z+1}\\right)}^5 + \\cdots \\right ),\n", "aeb70562ce8689816a475762020c5ec5": "\\alpha(\\epsilon) = \\sup \\left\\{\\mu( X \\setminus A_\\epsilon) \\, | \\, \\mu(A) \\geq 1/2 \\right\\},", "aeb711d509114823ad66489f2d5cf742": "\\cdots \\rightarrow C_n \\stackrel{ \\partial_n}{\\rightarrow}\\ C_{n-1} \\rightarrow \\cdots ", "aeb7142a2789c49d6f2b49947521d93b": " Q \\rightarrow QM.", "aeb717d721c663285665afce1828fafe": "(s,t)\\,", "aeb75f077446a46031ec527ab86a2ceb": "(\\mathbb{Z}_{mn}, +)", "aeb77a7d96917da189ba5ae4d3bf2d19": "\nPOL = q_1^4 \\left[ 1.15 - 1.15 \\exp \\left( -0.002337 \\left( \\tau_1^T\\right)^3 \\right) \\right] + 1\n", "aeb7accae3d0dd296a7deefb8aef8477": "D_2 \\varphi = 2\\cdot (h * \\varphi)", "aeb7f6c047cc428dbcec7994a5b9467c": "I_1^{(2)}=I_2^{(1)}", "aeb80a43232cb9bb4bb6e6b8bb0392f2": "\\scriptstyle g_i(x)", "aeb84e67c2088eac63592319e744c33c": "\\sup_{P\\in \\mathcal{P}(S,A)}\\|P_n-P\\|_{\\mathcal C} \\sim n^{-1/2}", "aeb86f34b58022b4395d99b5f6aca9af": "X, F, G, U", "aeb8defc1e456472d0e85e94a071691b": "E_{tgu} = 0.5 \\cdot m_{gu} \\cdot v_{gu}^2\\,", "aeb8e9410ca88799c3c6ae935368bdf6": "f\\circ\\varphi \\colon \\Delta \\to {\\mathbb{R}} \\cup \\{ - \\infty \\}", "aeb921cf3acb8a71ea045d1deeee0d9a": "\nh_\\sigma = a\\sqrt{\\frac{\\sigma^2 - \\tau^2}{\\sigma^2 - 1}}\n", "aeb940f3c81492dd2e63c108b2b4aedd": "\\textstyle\\frac{\\Delta{\\rm Principal}}{{\\rm Principal}} \\approx -\\left(1 - \\frac{Nre^{-Nr}}{1-e^{-Nr}} \\right)\\frac{\\Delta r}{r}", "aeba031ed54b6a309c2f321378af60b9": "\nC([x])\\subset [x]\n", "aeba1cd0d42ef0a7345e70bfb71ff1fb": "Y_D(x,s_p)", "aeba205bc5e19f58686451e0b8bdfa78": "R \\in \\{1, \\ldots, n\\}", "aeba5c345c5616a526620985aa4a0c8a": "x\\mapsto x-(x,\\alpha^\\vee)\\alpha", "aebb212e0692d2da6738304bf6c546c8": "r\\alpha}^{M}\\frac{Z_{\\alpha}Z_{\\beta}}{|\\mathbf{R}_{\\alpha}-\\mathbf{R}_{\\beta}|}\\right), \\\\\n&=Z_{\\gamma}\\sum_{i=1}^{N}\\frac{x_{i}-X_{\\gamma}}{|\\mathbf{r}_{i}-\\mathbf{R}_{\\gamma}|^{3}}\n-Z_{\\gamma}\\sum_{\\alpha\\neq\\gamma}^{M}Z_{\\alpha}\\frac{X_{\\alpha}-X_{\\gamma}}{|\\mathbf{R}_{\\alpha}-\\mathbf{R}_{\\gamma}|^{3}}.\n\\end{align}\n", "aebbf8f3914bc8a826c9f99885bdddee": "\\sigma_{xx}-\\frac{\\sigma^2_{xy}}{\\sigma_{yy}} +|\\sigma_{xz}-\\frac{\\sigma_{yz}\\sigma_{xy}}{\\sigma_{yy}}|", "aebc4a1a01ac59d87baeb97858e93594": "\\phi(a,x)", "aebc9f7fb03cec5830183e4b9e5c5c8c": "\\frac{\\rm{AC} \\sin x }{\\sin \\alpha} = \\rm{PC} = \\frac{\\rm{BC} \\sin y}{\\sin \\beta}.", "aebcde1be2abcfe285ff0ae234071062": " \\operatorname{cl}(A \\cup B) = \\operatorname{cl}(A) \\cup \\operatorname{cl}(B) \\! ", "aebce20675e7b553531abc0ef20d8870": "\\Delta_{K_1 \\# K_2}(t) = \\Delta_{K_1}(t) \\Delta_{K_2}(t)", "aebd731dfe2e38bbd7489335c38bde47": " u_p = \\mu_p E \\,", "aebd8a1364ac049fc932fbfa6c50fc71": "\n\\partial \\rho / \\partial t + \\nabla \\cdot( \\rho v) =0", "aebd8b72370be2751660c9130e3bc1d3": "\\begin{align}\n \\left|V_o\\right| &= \\frac{1}{\\frac{2LI_o}{D^2 V_i T}+1}\\\\\n &= \\frac{1}{\\frac{2\\left|I_o\\right|}{D^2}+1}\\\\\n &= \\frac{D^2}{2\\left|I_o\\right|+D^2}\n\\end{align}", "aebd9572546d76116dfec7168633b572": "{\\mathbf x}_1,\\dots,{\\mathbf x}_n\\sim \\mathcal{N}_p(\\boldsymbol{\\mu},{\\mathbf \\Sigma})", "aebdb9a3a47c785d03f3bb18f8482752": "\\mbox{A}^-", "aebdcbebbdfe743438e221e52ba46065": "\\int \\arcsin{x} \\, dx = x \\arcsin{x} + \\sqrt{1 - x^2} + C , \\text{ for } \\vert x \\vert \\le +1 ", "aebe08860787fed6a98f360fa2ba14ab": "e_q(z)", "aebe3f51245726ee72a5982e8334f2c7": "D_k(x)=\\sum_{n\\le x} d_k(n)=\\sum_{mn\\le x} d_{k-1}(n)", "aebe3fd0709a9d419030866f59ceba92": "U_i=F_X(X_i)", "aebe43720845af7b2b9ddb4ea0e01dbf": "f(x) = C\\,", "aebe4574334b918d7c9c22d92aad3bc0": "x_{me}", "aebe4fb3261dfeca3172a5162664e20e": "\\lim_{n\\rightarrow \\infty}\\frac{P(n)}{F(n)}=1.", "aebe6972b1c5ffc08500981db3e7d864": "f=FN=qvB n\\ell A \\sin\\theta = Bi\\ell \\sin\\theta ", "aebe6c3d2d75faba71d68e6d789d33f6": "\\left(a, p, u\\right)\\succsim \\left(b, q, u\\right)", "aebf2e51c8016ae482fb4871477a5840": " L = \\left (\\frac{\\mu_0}{4\\pi} \\oint_{C}\\oint_{C'} \\frac{\\mathbf{dx}\\cdot\\mathbf{dx}'}{|\\mathbf{x} - \\mathbf{x}'|}\\right )_{|\\mathbf{x} - \\mathbf{x}'| > a/2}\n+ \\frac{\\mu_0}{4\\pi}lY + O\\left( \\mu_0 a \\right ).", "aebf4dc5e4a7be473e809f0dff3956bb": "\\exists x \\delta_1(x)", "aebf7c97cb9f5b415ea9f91b5f173429": "dz_{i_1} \\wedge \\cdots \\wedge dz_{i_p} \\wedge d\\bar z_{j_1} \\wedge \\cdots \\wedge d\\bar z_{j_q}", "aebfab71f9190ebca2ea4908dc29eff2": " c \\, ", "aec0091dd3c8a4d02a214ca7f03337af": "c=s+2, \\, ", "aec0b2a619f0dc5247760a577658d708": "\\sum_{i=1}^n \\pi_i p_{ij}= \\pi_j", "aec0d4925fe7422a3ec8218de5ced369": " \\mathcal T : S \\to S. ", "aec11b0fd43e81f5675ef70f94e8c316": "\\langle f(t)f(0)\\rangle", "aec1579af2d4356e9e14b20f219df60b": " -\\frac{1+u^2-v^2}{2} \\, \\partial_u - u v \\, \\partial_v. ", "aec1b50ec48a16ef9c85a54c404ccf03": "c \\subseteq P(a,b) \\iff \\forall R \\in c \\; R \\in P(a,b)", "aec1f3f02148b27d3f579bcab774fc0a": "\\sigma_{zz}-\\frac{\\sigma^2_{yz}}{\\sigma_{yy}} +|\\sigma_{xz}-\\frac{\\sigma_{yz}\\sigma_{xy}}{\\sigma_{yy}}|", "aec1f4cb12a95621ce25cf60237d7ccb": "\n \\Psi_R = \\Psi P_3^{ } \n", "aec1fc89f4616a0ba6448fa94360438b": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{x^{m+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{m+2 n\\,p+1}\\,+\\,\n \\frac{n\\,p\\,x^{m+1} \\left(2 a+b\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}}{(m+2 n\\,p+1) (m+n(2 p-1)+1)}\\,+\\,\n \\frac{2 a\\,n^2 p (2 p-1)}{(m+2 n\\,p+1) (m+n(2 p-1)+1)} \\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}dx\n", "aec20294ef3d796675d514b83ac1cff3": "f_1, f_2, ..., f_n ", "aec21159667b70e7996f990e30c238d4": "\\chi(n)=\\left(\\tfrac an\\right)", "aec2382abd59913b12bd2bbd21645272": "A \\cap B \\subseteq A\\,\\!", "aec23eab074cb29e6cd8c7842c66b7be": "\\omega/k \\equiv c = \\sqrt{gh}", "aec252540365ac09418ac8ec21af0ae0": "\\tilde{\\kappa}_{(l)}l^b:= l^a \\nabla_a l^b", "aec27d37a47e61ac7ed5fad7b2401931": "\\sigma = 2\\sqrt{3}", "aec2b6fdf4d9eef2857704df394b9f68": "\\begin{align}\n {\\mathbf{k}} \\cdot ({\\mathbf{r-r'}}) &= k(z-z') \\quad \\& \\quad |{\\mathbf{r-r'}}| \n \\approx (z-z') + ({\\mathbf{X-X'}})^2/{2(z-z')}\n \\end{align}", "aec2e0c1f5de451a0af9c24f01f88308": "\n\\operatorname{corr}(\\hat{\\beta},\\hat{\\theta})= \\frac{\\operatorname{cov}(\\beta,\\theta)}{\\sqrt{(\\operatorname{var}[\\beta]+\\operatorname{var}[\\epsilon_\\beta])(\\operatorname{var}[\\theta]+\\operatorname{var}[\\epsilon_\\theta])}}\n", "aec30826a03bbef25cb849db7d9e5a42": " \\sum_{n=1}^\\infty \\frac{(-n)^{n-1}}{n!} \n =\\,\\left(\\frac{1}{e}\\right)\n^{\\left(\\frac{1}{e}\\right)\n^{\\cdot^{\\cdot^{\\left(\\frac{1}{e}\\right)}}}}\n= e^{-\\Omega} = e^{-e^{-e^{\\cdot^{\\cdot^{{-e}}}}}} ", "aec30ab1a743bdf25652c2173bc3d5eb": "\\text{Spec }R/pR", "aec323d785ed00e8d5f788b6d7e90664": "x^-=\\frac{t-x}{\\sqrt{2}}", "aec32a986b0da9ed451e293931b0098d": "y_1 + \\frac{5^2}{2(32.2)(y_1^2)} = 3.47", "aec3386f200fec4d214e98961ed3b8d9": " | \\phi \\rangle ", "aec3417846aa9ec3c784cc4fd2c62609": "\\dot{u}_{n+1}=\\dot{u}_{n}+ \\begin{matrix}\\frac{{\\Delta}t}{2}\\end{matrix}~(\\ddot{u}_n + \\ddot{u}_{n+1})", "aec354ed0a3af0ec2bc409a6b1b11578": "\\frac{\\sum_{uv\\in E} d(v)^2}{2|E|}=\\mu + \\frac{\\sigma^2}{\\mu},", "aec3a8c23922f7d9cb66637d01db78d3": "\\textstyle x_1,\\dots,x_N", "aec3ce54f9e0d591f924788f6c55346c": "\\Omega 2", "aec3d3cb7f742121a8f9e516aac02f16": "x \\mapsto \\varepsilon_x", "aec3e9d834421cbd2af392f3d6950872": "[A(t),B(t),C(t)]", "aec4536dff1bd5f47e17166eb5203dee": "\\sigma_s^2", "aec46f7d16aca45f8a05f2aee685c22c": "B_\\delta([z,x])", "aec47c34d6681674e31bf6de7181e2b2": "d = \\frac{A(1)-A(0)}{A(1)} \\,", "aec499fe1849b003884a35aefb6893f6": "(x^n)'=nx^{n-1}:", "aec4b92291e25875125666fc104a6e84": "|z| < 1\\,", "aec54912d09f5f8e0fd725b35957c1b9": "f\\in \\mathbb C \\{ x_1,...,x_m\\}", "aec59fe8a7f07e3182e81cda3df04d20": "f(D^\\prime)", "aec5a2bba9ca3021e990a3f98043dc0c": "\\varphi^*\\nabla", "aec5b0f92247903946ebdc8db1dcf61a": "Y_{4}^{-3}(\\theta,\\varphi)={3\\over 8}\\sqrt{35\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot\\cos\\theta\n= \\frac{3}{8} \\sqrt{\\frac{35}{\\pi}} \\cdot \\frac{(x - i y)^3 z}{r^4}", "aec6a4168e14b1611201211ae96d6426": "\\lim_{t \\rightarrow \\infty} a_t \\geq 0.", "aec6ab701d8584b45b56e798f6505411": "\n (V/W)^* \\cong W^o.\n ", "aec6d0a193a77f8669aa06f9318b457f": "n(x,y,z,t) = \\int f \\,dv_x \\,dv_y \\,dv_z", "aec734fdaf2230c51bdb66fd4ec452ab": "{{\\Delta}V_s}", "aec7c7f9b6447fb73cec1a930a6545f9": "v(w)= -{h(w)\\over h^\\prime(w)}.", "aec7cc23771154b7ba1ad43e3945b3a2": "F = dp/dt = m \\; dv/dt + v \\; dm/dt", "aec7e766d744d75e562aa32bbfeea334": "p_1(x)=p'(x)=4x^3+3x^2-1", "aec829916f544a16d8667c20e3a833f6": "a(n)=a_n", "aec857764b5628d287efcb4c83a083fd": "\\approx 0.44 \\cdot \\frac {\\lambda_0^2} {\\Delta\\lambda} \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (4) ", "aec883e1fb3fa9774692bcb4b2f9ec1f": "\\ \\mathbf u(\\mathbf X,t) = \\mathbf b+\\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad \\text{or}\\qquad u_i = \\alpha_{iJ}b_J + x_i - \\alpha_{iJ}X_J", "aec8f3a5bf91ad4eecdc30c65e565d8b": "M\\left |\\epsilon_0 \\right |<1.\\,", "aec8f5cb8f9bb39605d08301f2571e48": " \\hat{L} = |\\vec{r}_1 - \\vec{r}_2|^{-1} ", "aec9086840ab455386a94e726e4c781b": "/(xy) = /x/y\\ ", "aec919d87970bc5546b513bff81ef2c2": "\\gamma^* = -i\\tilde{k}", "aec9a493908eb92aceb50d1e4cd4ab9b": "F_{thrust-gate} = (62.30)(10)(132-10) ", "aec9e53dccf69f4370fa0563e9b8ea2d": "\\hat f(\\cdot)", "aeca32cf50d343441b43dbeae174e4a9": "_{s.3 \\,}\\!", "aeca3f10827ebab870abb7ab4a88510d": "Z_a=\\frac{d_1}{a}\\,", "aecac0b6d41f2f64227f169bb466e6e1": "dx\\ =\\ (\\lambda x + A)dt\\ +\\ cdW", "aecac8a8d92a3ef9b90f85551c4af754": "\\square ", "aecaf9ba917a397f27b4b2b3fd6e1ea2": "0.89U($0\\text{ M}) + 0.11U($1\\text{ M}) < 0.9U($0\\text{ M}) + 0.1U($5\\text{ M})\\,", "aecb15a50db3b38795f17b0bddf30f13": "\\sigma\\;\\mid\\;(\\tau\\;\\mid\\;\\rho)\\equiv_{b}(\\sigma\\;\\mid\\;\\tau)\\;\\mid\\;\\rho", "aecba22b3c8a0496d38d4358652d55d9": "b_{i}=\\frac{1}{N+1}", "aecbb9ac2eeae056c0d5326b4a04a204": " \\exists x \\, P(x) \\equiv P(a) \\lor P(b) \\lor P(c).\\, ", "aecbc1f3b1c21144a8272210dd8ef30e": "2e^\\psi R_3^0 = \\frac{1}{2}\\varkappa\\psi^\\prime + \\frac{1}{2}\\lambda\\dot{\\psi} - \\varkappa^\\prime -\\frac{1}{2}\\varkappa_a^b\\lambda_b^a = 0,", "aecbd2eee49aa57554b5ca05d9624b7f": " u(\\bold{x},t) = e^{i{(\\bold{k\\cdot x}}-\\omega t)} ~,", "aecc5d1d72106df9112fb63f41764180": "l=-2", "aeccc43f1e2671cfaad89ce12f236eaa": "\\epsilon_0 \\mathbf{E}\\cdot\\frac{\\partial \\mathbf{E}}{\\partial t}", "aecceddcb144373f61bfc7bf9f169a50": "|g(y)| = |g(y) - g(x)| \\le (0) |y - x| = 0", "aecd423d2192fefc568def17954185dc": "f(a) = f(b) \\Rarr a = b.", "aecd47dcd6d37b604690cdf10586e43c": "\\Xi = \\Xi(T,P,\\{N_i\\})", "aecd588fc90ddba761658069545c56d6": "\nf: \\mathcal{X} \\to \\{ -1, 1 \\},\n", "aecdb222738cc0577416faeb98715044": "d(w) < d(u)\\text{ or }d(w) < d(v).", "aecdb7de4b7b17aedbb7d203430824ed": "U = \\langle E \\rangle = \\sum_{i=1}^N p_i \\,E_i\\ .", "aecdc0152ef95b500a243be6642a446f": "V_{TE}\\approx{Z_{TE}V_s}", "aecdca0f6e8433e62897fc95eb9709ca": "(\\mu_{ab}(t))", "aecde193ab3aa4950c1d486becf086e9": " \\mu = \\mu_\\max {S \\over K_s + S} ", "aece4d6d5c7069b478b35db0c1451fa4": "h_B", "aece6c7d0664324840826c0beef31fe5": "\\omega = \\frac{1}{2}(-1 + i\\sqrt 3) = e^{2\\pi i/3}", "aece91ad0e9203be60954fe84f79bdb7": "H=H(\\cdot,\\cdot)", "aece9aa66529f483917ef4bac7c43816": "\n\\begin{Bmatrix}n\\\\k\\end{Bmatrix}\\,\\bmod\\,2 := \\left(n-k\\right)\\!\\!\\And\\!\\!\\left( \\left(k-1\\right)\\,\\mathrm{div}\\,2 \\right) == 0.\n", "aece9e5d02345a638f7356b603e38d95": "f^{(1)}_n(x)", "aeceed2c6215c373413a0af4b23927f3": "\\frac {v^2}{2}+\\left( \\frac {\\gamma}{\\gamma-1}\\right)\\frac {p}{\\rho} = \\left(\\frac {\\gamma}{\\gamma-1}\\right)\\frac {p_0}{\\rho_0}", "aeceff5682a841b3769b665ef8816464": "w(C_1\\mid C_2) \\leq \\min \\{ 2w(C_1) , w(C_2) \\} ", "aecf098be1f5512bc7143e315683a9a6": "A = LL^T\\,", "aecf27c3886e1aa85817a64dbd70d86b": " \\left\\| - {1 \\over z^\\star} \\right\\| > 1 ", "aecf8f00bb3967ccce75d54b30cef9af": " d\\alpha = \\sum_{j=1}^n df_j \\wedge dx^j = \\sum_{i,j=1}^n \\frac{\\partial f_j}{\\partial x^i} dx^i \\wedge dx^j. ", "aecfa437047f198d21df93eca31ff4bb": "\nL_{ji} = {1 \\over {L_{ii} }}\\left( {a_{ij} - \\sum\\limits_{k = 1}^{i - 1} {L_{ik} L_{jk} } } \\right)\n", "aecfa76f00b3bdde3e57b069942ea13e": "i\\ \\Delta", "aecffb77f8d88370fc143c0ccaf0732a": "C_1 + S_1 \\le Y_1", "aed014f67adafb61cfd953699d662899": "AA^\\ast = A^\\ast A = nI", "aed01cfadbc14d78efdfaebb6d240631": "\\forall j", "aed0581f36d4ac99ba11adb789a81a08": "\\sum\\limits_{k=1}^n |x_k| \\le 1\\,,", "aed06f08ee61162387be6bb740c54825": "d(B, A) := \\inf_{b \\in B} d(b, A)", "aed0adb501cef19609a8bf7cb889d2ef": "p=\\gamma(0)", "aed0d53fe012d41663338b5d8f8c72ea": "\\ k<-n", "aed0dc3566949febecafb9be779568c5": "\n\\boldsymbol{F}_{i+1}-\\boldsymbol{F}_{i} = \\tilde{A}(\\boldsymbol{U}_{i+1}-\\boldsymbol{U}_{i})\n", "aed18b28d54d4fd95645899d991e97fb": "\\sum_{n=1}^\\infty\\frac{H_n^{(r)}}{n^m}=\\sum_{n=1}^\\infty H_n^{(r-1)}\\zeta(m,n)\\quad(r\\ge1,m\\ge r+1). ", "aed1ae797761456de6f9378f7e9ccf41": "A_3 \\cong D_3,", "aed1dc823728f46cc3a82ca8c80429bd": "\n 1 - \\frac{3}{\\chi^2} + \\frac{\\chi\\phi(\\chi)}{\\Psi(\\chi)} = \\frac{1}{n}\\sum_{i=1}^n \\frac{x_i^2}{c^2}.\n ", "aed20534cac2132f21e875afd4b2fb7c": "\\Delta \\mathbf M\\,\\!", "aed2280ddb05436650cb8c9c75cdfdb8": "e \\in \\{H,L\\}", "aed231bd38ef70f7b4a3cbf6fa908759": " \\and (S_4 \\implies (\\operatorname{equate}[A_4, p] \\and V[F_4] = A_4)) \\and D[F_4] = K_4 ", "aed23ac143cfcea9afca78d2505732f6": " \\dot{Q}= \\dfrac{\\sigma(T_1^4-T_2^4)}{\\dfrac{1-\\epsilon_1}{A_1\\epsilon_1}+ \\dfrac{1}{A_1F_{1 \\rightarrow 2}}+ \\dfrac{1-\\epsilon_2}{A_2\\epsilon_2}}", "aed25efd3afb8545521c38ace13e95ba": "N_r", "aed27a3c04d9b2c3bba83b3e0a1c15a7": "(2\\pi)^\\infty", "aed2a0aad187d5201e6d539e4310abe7": "M_1+M_2\\,", "aed2fe3ae5d9b9886b7a7bf60d8a39de": "\\mathbf{\\nabla} \\times \\mathbf{B}/\\mu_0 =\\mathbf{J}_{\\text{f}}+ \\mathbf{J}_{\\mathrm{bound}} + \\varepsilon_0 \\frac{\\partial \\mathbf E }{\\partial t}", "aed2ff98c6925317e91f9d3252d33af8": "\nG_F(p,E) = {-i \\over E - {\\vec{p}^2\\over 2m} + i\\epsilon}\n", "aed32ad71bea246946911261baf8285d": "\\beta_{13}[L^{3-}][H^+]^3=K\\beta_{12}[L^{3-}][H^+]^2[H^+]\\,", "aed34f8920e692a5600ac6ea2f19dc20": " \\tau = \\frac{x'''(y'z''-y''z') + y'''(x''z'-x'z'') + z'''(x'y''-x''y')}{(y'z''-y''z')^2 + (x''z'-x'z'')^2 + (x'y''-x''y')^2}.", "aed35bf8836ebadd6bf602f46bed70ee": "\\textbf{S}_{x}(s) = \\frac{2\\sigma^{2}\\beta}{-s^{2} + \\beta^{2}} \n = \\frac{\\sqrt{2\\beta}\\,\\sigma}{(s + \\beta)} \n \\cdot\\frac{\\sqrt{2\\beta}\\,\\sigma}{(-s + \\beta)}. \n", "aed3d0c9eba729def6b10f99ce434a2e": "y=af(b(x-k))+h", "aed3f1239c87e0c1c3e005d53c65253e": "\\sqrt[y]{x} = \\log_y x", "aed430fdf4c64058b58e05bf9ccbbbde": "\\frac{\\sqrt{3}}{2}", "aed4409626921a8d92da49996cc96bbc": "\\vec{e}_2 = \\sqrt{2} \\omega \\, \\partial_y", "aed460e4ac495d30bf3ece6000571956": " \\Delta\\sigma_y = Gb \\sqrt{\\rho} ", "aed4b7bcb9150e1b998efa92af800398": "d\\mu=-\\sin\\theta d\\theta", "aed4bde6049458164db47babfbd24de5": "\nw_B = \\sqrt{\\frac{D}{\\lambda}}\n", "aed51796200e87d7c09a03e9a1e3fe5c": " \\limsup_{t\\to+\\infty} \\frac{ |w(t)| }{ \\sqrt{ 2t \\log\\log t } } = 1, \\quad \\text{almost surely}. ", "aed5407d6db478c4ced31420b298aff4": " (k-p) ", "aed55ef0a94a10162ed6e6452832f740": "= \\operatorname{tr} (\\Gamma^\\dagger)", "aed5a176ca237fe08965022bc5eb4670": "\\lambda^{-\\nu} I_\\nu (\\lambda z) = \\sum_{n=0}^\\infty \\frac{1}{n!} \\left(\\frac{(\\lambda^2-1)z}{2}\\right)^n I_{\\nu+n}(z) ", "aed5b1a2fef554a7b73d58ac16021565": " l=-\\frac{\\frac{\\partial f}{\\partial x}}{\\frac{\\partial f}{\\partial y }}", "aed5e8ea5fcb03c09f569e7509083554": " a = 0 ", "aed5f0f0e09e54cbdc47f64d54d67b1b": " 1 = \\frac{p e^{\\left( \\frac{p}{q} \\right)}}{q}", "aed62d69309a42f8b66d7469127296b7": "N_t", "aed65cec6b386ab92991145660eae850": "\\int_{-\\infty}^\\infty dp\\,P(x,p)=\\langle x|\\hat{\\rho}|x \\rangle.", "aed660e55c892b1daff2c5304ec05796": "\\arcsec (-x) = \\pi - \\arcsec x \\!", "aed6bc44f6b1acf80d24a22eb9db3e13": "v = \\log t", "aed6cafe5fdc2f0b7ee53ea3362d580a": "\\mathbb{Q}(x)", "aed6d8bfe85a12b9fcc53fc2cfe55d2e": " \\mathbf{F}= -\\nabla V,", "aed6f47e10701fd6b264d49853237017": "\\mathbf{\\Phi}_{10}= -\\sqrt{\\frac{3}{4\\pi}}\\sin\\theta\\,\\hat{\\mathbf{\\varphi}}", "aed7012b0d4d59c0a42a0a2a7ba8a4d5": "x\\geq x_\\min", "aed72e5aac543d91818521f0119b09d7": "L^p\\to L^q", "aed739ea0898ff497ebf810b9b78374b": "c_{ij}\\le 0\\ ", "aed7642fd69c1d4b06448629e8d87ab5": "\\psi<0", "aed796204838f402544e07f43dc16e82": "65\\,X^2+125\\,X-245,", "aed81e2c669f58b7b986752df8544c96": "f(x) = F_+(x) - F_-(x)", "aed87dbfc57ce7c84fafe2b013181e7b": " \nC(\\eta)=\\lim_{n\\rightarrow\\infty}\\frac{2n}{\\text{number of clusters in} [-n,n]}\n ", "aed8809b52449ada8c4ab1612bd38aa2": "\\sum_{i=1}^n c^i \\in \\Theta(c^n)", "aed88ac385c99292a54b6a9d8d7816be": "\\Omega_t=\\{x \\in M | (t,x) \\in \\Omega \\} \\subset M", "aed9135e7f4d7411e6a41da37a17a712": "E_p = \\frac{S_p}{p} = \\frac{T_1}{pT_p}", "aed945bcd1f720fd163ff44a24f1f312": " r_{mt} - r_f ", "aed94726442bb21d40c14761a9e77cfd": " {\\vec{J}}_{G}=\\sum_{i=1}^{N}\\,{\\vec{x}}_{i}(t)\\times {\\vec{p}}_{i}(t),\\qquad {\\vec{K}}_{G}=\\vec{P}\\,t-\\sum_{i=1}^{N}\\,m_{i}\\,{\\vec{x}}_{i}(t). ", "aed955d0a153685b950f21c27b83cdfb": " = \\frac{1}{\\eta}P(R_{NP}^c \\cap R_A, \\theta_0) = \\frac{1}{\\eta}\\int_{R_{NP}^c \\cap R_A} L(\\theta_{0}|x)\\,dx \\geq \\int_{R_{NP}^c\\cap R_A} L(\\theta_{1}|x)dx = P(R_{NP}^c \\cap R_A, \\theta_1).", "aed98436585d2b04156168ff159152b3": "Z_v", "aed9fef2ce0f28d04fb53faaf96bce99": "v(t) = \\frac{(\\rho-\\rho_0)Vg}{b}\\left(1-e^{-bt/m}\\right)", "aeda0362a02175380b1f1ca3968b1d58": "P=\\eta\\cdot\\rho\\cdot g\\cdot h\\cdot\\dot q", "aeda2f673c02a333637667e8b60a74c7": "\\{1\\}\\times X", "aeda92e73bb9939fce1fb14bdb953e87": "\\ \\xi", "aedab2cc9650b177d5196f2c9ea5e86c": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\eta + (\\tilde{u}_{1}- E)\\eta + i\\frac{\\hbar^2}{2m}[2 \\mathbf{\\tau}_{12}\\nabla + \\nabla\\mathbf{\\tau}_{12}]\\eta = ({u}_{1} - {u}_{0})\\chi_{0}", "aedaca7e208e1095ebee9bb598e94f7c": "H^{-i}(j_x^*C)\\ne 0 ", "aedae8584d3f89a4fd6d0bf5a2147206": "D_0 \\nabla M + \\rho M = 0", "aedb3fb37b720e4c015b5ee26ca843f5": "\n\\begin{align}\n s &= \\sum_j A_j \\\\\n A &= A / s \\\\\n \\lambda_i &= s \\lambda_i \\text{ for each } i \\\\\n \\phi_i &= \\sqrt{s} \\phi_i \\text{ for each } i \\\\\n\\end{align}\n", "aedb6895ee8d2080fa481fcc310dd65d": "0.7 K_u", "aedb8e41d54a24d26b07cc9367b020cb": "f/", "aedbe4c2ad82ae54f5c5300c61ea6bfc": "\\textstyle |b(x,r)|", "aedbe942957f6802452bcd1f18c1edbf": "\\varphi : G \\rightarrow \\mathbb{Z}_n=\\{0,1,...,n-1 \\}", "aedc5ab33c7dcf8c4c44be47624e93ea": "\n\\Psi_1(a,b,c_1,c_2;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a)_{m+n} (b)_m} {(c_1)_m (c_2)_n \\,m! \\,n!} \\,x^m y^n ~,\n", "aedc7032d2c1109934e8c645de9fc27c": "\\omega=\\frac{|\\mathrm{\\mathbf{v}}|\\sin(\\theta)}{|\\mathrm{\\mathbf{r}}|}.", "aedca8294427c054a3220ef7376b2e18": "c_i+\\alpha_i-1", "aedcac3c35f988cd6c262221a21bfc8e": "L_R = \\sqrt{-g} \\left[R(W)-2\\lambda-\\frac14\\mu^2g^{\\mu\\nu}g_{[\\mu\\nu]}\\right] - \\frac16g^{\\mu\\nu}W_\\mu W_\\nu\\;", "aedcd6f4bae3e37a6b6db0a92a2c7e89": "x\\gcd(a,b) + yc = \\gcd(a,b,c)", "aedd4b9ff7edfa8d2084bff136aab452": "M=x_1^{a_1}\\cdots x_n^{a_n}, ", "aedd4da42244388a9a60a7065877d175": " g_{ab} = g_{\\mu\\nu}e_{(a)}^\\mu e_{(b)}^\\nu ", "aedd5b09e7f99db19ece173df735de60": "\\omega \\in \\Omega^p(M, \\mathfrak g)", "aedd66a2f8390a85f189954877efcef1": "\\gamma = \\alpha + i\\beta = \\frac{1}{2} \\cosh^{-1} \\left(1-\\frac{2m^2}{\\left(\\frac{\\omega_c}{\\omega}\\right)^2 - \\left( \\frac{\\omega_c}{\\omega_{\\infin}}\\right)^2} \\right) +i0", "aedd9cdaa9977c19e9368f10ec61b1b0": "SNR_k", "aedd9dc1e30ac44bfb2de5a289d14678": "\\bold{P} = \\{ T_1, T_2,\\dots, T_m \\}", "aeddc34c7c9f23578defe85c13a359ce": "1.2\\pi^2\\cdot10^{-5}\\frac{\\text{AU}^3}{\\text{y}^2}=3.986\\cdot10^{14}\\frac{\\text{m}^3}{\\text{s}^2}", "aeddda2b3b4b94b4b07cf5043f8132dd": "c(W,t)= \\begin{cases}\\nu \\left(1+(\\nu\\epsilon-1)e^{-\\nu(T-t)}\\right)^{-1} W&\\textrm{if}\\;T<\\infty\\;\\textrm{and}\\;\\nu\\neq0\\\\(T-t+\\epsilon)^{-1}W&\\textrm{if}\\;T<\\infty\\;\\textrm{and}\\;\\nu=0\\\\\\nu W&\\textrm{if}\\; T=\\infty\\end{cases}", "aeddfd15d6271fd03ef2da6066b34ec4": "L^2_{1/2}(\\cdot)", "aede2a17d4924e69e329f299f6ac4240": "\\textit{opendoor}(0)", "aede5ebae46c1c8e5fa942322e39b1ab": "12 / 4", "aede691feb4292ddf93832b54cb7eb37": "L^{X/Y}_1 = J/J^2 = i^*J", "aede92be7395fbdd5842565bcfe711b7": "\\left(\\frac{\\frac{80}{3}}{\\frac{112}{3}},\\frac{\\frac{248}{3}}{\\frac{112}{3}}\\right)=\\left(\\frac{5}{7},\\frac{31}{14}\\right)", "aedea3d9f994342b301191d6f1066ef9": "p(y|x)", "aedeb58717020cce6f58837309266a25": "w = - J (\\mathbf M_1 \\cdot \\mathbf M_2). ", "aededf07bfeb48311a62a8c964612732": " \\mathbb{Z}_{p_{i}}= \\{0,1,\\dots ,p_{i}-1\\}, m_{i}= m / p_{i} ", "aedefb7efa6037ab76409554029bd043": "n(n-1)\\cdots(n-k+1)=\\binom nk k!,", "aedf038af8ed17063a7979b71cd4dee4": "\nX \\sim IG(\\mu,\\lambda) \\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\, tX \\sim IG(t\\mu,t\\lambda).\n", "aedf519fa7ce0017391acb2169d3f84b": " \\Lambda_{0i} = \\Lambda_{i0} = 0 ", "aedf5d904b2835f173cfd98fcbb6e2c9": "\\mathfrak{su}(n)", "aedfa63c1ec122f6c5d43301bca2a5a3": "\n\\beta = \\gamma \\times p \\,\n", "aedfddd728d4ace71de52ff9db4d4136": "p-1 \\geq m(2^m+1)", "aedfe2c74db35c383e0e531ffedd96f3": "| \\psi (\\mathbf{r},t) |^2 .", "aedfe4774d118894012373e93cc4ce02": " \\frac{\\partial w}{\\partial t} = \\tfrac{1}{2} \\frac{\\partial^2 w}{\\partial x^2} - u V(x) w ", "aee015bacc8d588c081b3e8a4fbdb8cd": "f_{\\text{low}} = c/2 \\pi r_2)", "aee039a690732879c0d6742f88c4718d": "z_i,\\;i=1,\\ldots,k", "aee05f0d69e21b12f2c474880cf49835": "\\widehat{n}", "aee06daff3427b63f3b2e7e293db1c92": "a,b \\in V:\\ B(a,b):= -ab - ba.", "aee0b0fc8b67872d6be51a22f93953cd": "-2\\sum_i W_{ii}\\frac{\\partial f(x_i,\\boldsymbol {\\beta})}{\\partial \\beta_j} r_i=0,\\qquad j=1,\\ldots,n", "aee0db605ef466ea91848f902ab09368": " ES \\overset{k_{cat}} {\\longrightarrow} E + P ", "aee183be96e42dac932118850d693e09": " u_z = \\frac{1}{4\\mu} \\frac{\\partial p}{\\partial z}r^2 + c_1 \\ln r + c_2 ", "aee1af1051859ca576aa4980d1f452a4": "\\mathrm{CaCO_3 + CO_2 + H_2O \\ \\rightleftharpoons \\ Ca^{2+} (aq) + 2 \\ HCO_3^- (aq)} ", "aee1c1bcae84de0cf9da78e748c0b8d8": " \\chi = \\chi_{\\mathrm{LIN}} + \\chi_{\\mathrm{NL}} = \\chi^{(1)} + \\frac{3\\chi^{(3)}}{4} |\\mathbf{E}_\\omega|^2,", "aee1fbbee0cc0397e312860d5320c287": "\\left\\{\n\\begin{matrix}\n\\pi &:~n=0 \\\\\n\\pi/2 &:~n\\ne 0\n\\end{matrix}\\right.\n", "aee21cdeffb861d920995497532596a7": "{\\mathbf{}}A^*=A(A^3)^+A.", "aee273e25ba589486a08af8157951229": "x \\wedge y = x", "aee2d81cce2f71c4da628aa16fec0f2b": "\\text{E}(u(c))=\\text{E}[1-e^{-a (c(x)+ \\epsilon)}],", "aee2e13a9f829df0f3da4de02be7340e": "\n z(q):= \\max\\{\\alpha: R(q,u) \\in C, \\forall u \\in \\mathcal{U}(\\alpha,\\tilde{u})\\}\n", "aee2f20e25d996dfa60bd5ef5d62aea9": "t' = M_l t", "aee337b788f09840bb096ada186ef238": "V = \\frac{1}{6} \\left(45+17\\sqrt{5}\\right) a^3 \\approx 13.8355259a^3.", "aee33c821d334cab137de6538bf88b85": "\\sigma \\in \\Sigma ", "aee35340435aab4df857d0a91f9df592": "u(x,t)=g[\\log(t-x)] - g[\\log(x+t)]", "aee38afc13d8a6ea78c432688cc541a2": " \\cos^2(t) + \\sin^2(t) = 1. \\,\\!", "aee4312e64a00e1437b578729577f0f0": "{\\gamma^{\\dagger}(-E)}", "aee45dfdc0ba16dcceaf1c4bd81b715a": " \\epsilon^2 ", "aee46fc92edec9b68188db57ebb1e63e": "\\scriptstyle y_k \\;=\\; y_{2k}", "aee4914cf0567bf2466b57e0cebdc4c9": "-a_6=\\sum_{n}\\frac{7n^5+5n^3}{12}\\times\\frac{q^n}{1-q^n} = q+23q^2+154q^3+\\cdots", "aee59e8e975d5f17eb711c20643c1ff3": "\\textbf{g}=-g\\hat{\\textbf{z}}.\\,", "aee5f8f1fd1bfd90c1d16b17a06cda03": "\nNSD_i = \\left( e_i^t - h_i^t \\right) \\times G\n", "aee64ae5361b3e9139661d533b4f2eaf": "\\delta_{i} = \\begin{cases}\n0, & \\mbox{if } i \\ne 0 \\\\\n1, & \\mbox{if } i=0 \\end{cases}", "aee682f41720ca31235e22f9a5705b19": "s_p^2=\\frac{\\sum_{i=1}^k (n_i - 1)s_i^2}{\\sum_{i=1}^k(n_i - 1)}", "aee6e42dc9045b981da49d9e33e50fdf": "\\left[0,n/\\varepsilon\\right]", "aee74209fcde96d1767c05b426605679": "\\begin{bmatrix} -\\dfrac{\\eta_1}{2\\eta_2} \\\\[15pt] -\\dfrac{1}{2\\eta_2} \\end{bmatrix} ", "aee79ccc01507beba435d2ae0a702acb": " h(t) = \\left(1/4\\right)^{n_1+n_2}\\left(1/4+3/4{e^{-4/3t}}^{n_1}\\right)\\ ", "aee7c3b4e8ebaad994254443e486e45b": "| x_0 \\rangle", "aee7c46f1d2fa46d670ab4fe08652687": "A(D)P(D)", "aee7f5e59876f99aac83cc43a28dcfd2": "Q \\times (\\Sigma \\cup\\{\\varepsilon\\}) \\times \\Gamma \\times Q \\times \\Gamma^* ", "aee8304c83d952c951e620d4cc410d2a": "\\operatorname{Expected Loss} = \\operatorname{E}\\left[\\left(c n S^2 - \\sigma^2\\right)^2\\right] = \\operatorname{E}\\left[\\sigma^4 \\left(c n \\tfrac{S^2}{\\sigma^2} -1 \\right)^2\\right]", "aee84fc92c1020069a0cda221a977b34": "T^{i}_{j}", "aee85d03d645629c397345a0dca991bd": "H^*_G(X) = \\operatorname{ker} d_\\mathfrak{g}/\\operatorname{im} d_\\mathfrak{g}", "aee86c5dad5ad95cb39589566c6d3432": " r \\to \\infty ", "aee8884df41b7dfa9ffc7e72353e66c8": "\\alpha d_{1}(x,y) \\leq d_{2}(x,y) \\leq \\beta d_{1} (x, y).", "aee889714fd1f7e6854608baed5edee2": "V = {1\\over 3}rA.", "aee8ca21d3fb390630f5c6bf98d12f2b": "t_{n+1}-t_n=k\\ \\forall n", "aee9091cff088442abbee629bc9e7135": "\\scriptstyle\\partial_{\\bar{z}}w", "aee9111fe6c6ed26c252cb8af163ac8b": "\\beta_0^{(0)},\\beta_0^{(1)},\\ldots,\\beta_0^{(n)}", "aee96e3b33879c242b9a07a685771b68": "s: \\Lambda_k^n \\to X", "aeea0e7d652cc023f562d4f1e6e8c2ba": "\\Omega=d\\omega +\\omega\\wedge\\omega", "aeea0f626fb362c258ac1654c9e9c476": "Z' = \\frac{1}{i \\omega C'}", "aeea1d8770d357628081b28a4e13c6fb": "H = \\int d^3x N \\Big( {\\tilde{\\pi}^2 \\over \\sqrt{det (q)}} + \\sqrt{det (q)} (q^{ab} \\partial_a \\varphi \\partial_b \\varphi + V (\\varphi)) \\Big) + N^a \\tilde{\\pi} \\partial_a \\varphi", "aeea5a634f3177f54fd63c843308593d": "\\psi(P,x)=\\phi(P,x)\\vee P(x)", "aeea8f92bd7b02008c8638dc4f09e145": " u^* = \\left (\\frac {\\tau_w} {\\rho} \\right )^{1/2} ", "aeeaa7f44b4de0d41eed2d3e559e21a6": "R_H=\\frac{v_{Bullet}^2 \\sin(2\\delta\\theta)}{g}", "aeeb5babc58b24d346bb248d1f799686": " A_\\alpha \\subseteq \\alpha ", "aeeb890ec8f0efb93043ac12eab33cd0": " f_{pm} = 0.39661 + 0.001709p + 0.000010788p^2", "aeec4494b7761b0e1c4889fada1109cb": "\\tau_L", "aeec92ef4a4438bca0f72023daddef4f": "g^x \\equiv h \\pmod{n}", "aeecd87be25dce1cce8ce04778e54da6": "H(X_n | X_0, X_1, \\dots, X_{n-1})", "aeecf319f8e961ad5a9c9e8caf89e3ad": "(x)_k", "aeecfc2ca46fbea15e5f40eaeb3502fe": "\nds^{2} = dx^{2} + dy^{2} + dz^{2} \\,\\!\n", "aeedb5ed378b2d4bb8d61a40cb2c7599": "F_{\\rm rad}", "aeee004da4408003fbf7725961bc9f16": "\\alpha^{-1}W_{\\alpha^2 t}", "aeee899b460fb362ed9627423d9814fd": "Vol_2(d,n) \\geq {n \\choose d} \\geq (\\frac{n}{d})^d", "aeee8a15a4b9de824948f28ef3efc76f": " \\tau_{peak} = \\sigma_n ' \\tan \\phi_{peak} '\\ ", "aeeec2aa7271a8795e0998c4f127ceb3": "1\\to\\{\\pm 1\\}\\to 2T\\to T \\to 1.", "aeeed40490b77356548d3113a93702b6": "f(x) = \\sum_t \\alpha_t h_t(x)\\,\\!", "aeeef176797dc5401472f28182f3beaf": "\\gamma\\in \\Gamma_0", "aeeef262e3d9ee9640dff3aef7d5d70d": " \\mathrm{MTF}(f) = \\frac{M(\\mathrm{image})} {M(\\mathrm{source})}", "aeef00cb48b017fe08fb3a8ec576ad55": "\\ T=T_d", "aeef931d369ef4ce62cfba41c6a19f10": "\nCIQ_t = \\mathcal{A} e^{\\mathcal{B} t} \\mathcal{C}_t\n", "aeef979e7d3065f6d35d63eeae50f2a3": "V_{opt} \\sim 30 \\cdot M^{\\frac 1 6} [m \\cdot s^{-1}]", "aeefc1d9e3ac09095b5f93d4893e305e": "S(n,k) = \\frac{1}{k!}\\sum_{t=0}^{k} (-1)^t \\binom{k}{t} (k-t)^n.", "aeefe4648c0443c8f783f8562b29ae24": "\\frac{\\mathbf{x}_{k+1} - \\mathbf{x}_{k}}{\\Delta t} = \\mathbf{v}_{k+1/2},", "aeeff64c66662563fefb87948db12965": "\n w ( u \\wedge v) = x ( u \\wedge v) + y ( u \\wedge v)\n= \n x \\cdot ( u \\wedge v) + y \\cdot ( u \\wedge v) + y \\wedge u \\wedge v", "aeeffc44b691e1da4d7b910e097357c0": "E_p(a)^b=E_p(ab)", "aef060c03e59c2add29c053e85d83b63": " \\rho (r) ", "aef06d39583fdbb9474814b3c62e589e": "\\mathbb{C}^n = \\mathbb{R}^{2n} = \\left\\{\\left( \\mathbf{x}, \\mathbf{y} \\right) = \\left(x_1,\\ldots,x_n, y_1, \\ldots, y_n\\right) | \\mathbf{x},\\mathbf{y} \\in \\mathbb{R}^n \\right\\}", "aef077a1c7b98f6091af987a5eb78742": "T(x)", "aef09806b200efc47833a510b237c23a": "H_0^1(0,1)", "aef0cebddc745812ca925e9aa9d6395e": " U_i =\\{ (v,v') \\in {\\Bbb R}^{n+d}\\ \\ | \\ \\ v\\in V_i", "aef0dbd1d01568e16c663eb90b1255c7": "R_n \\le \\frac{41}{47} \\ p_{3n}", "aef0fbc632367e448ebc235340f0cbd6": "\\mathbf{B}_{\\theta,\\phi}", "aef187bbae9e06ae82bcf295c366313f": " \\lim_{x \\to a} f(x) = \\infty, \\, ", "aef1a22d87b5379cc6eecc630ae72189": "\\epsilon_{sh}(t)", "aef1e05e58dabb37a1e93da4edd9d90d": "D_{\\mathrm{KL}}(P\\|Q) \\equiv \\sum_{i=1}^n p_i \\log_2 \\frac{p_i}{q_i} \\geq 1\\log\\frac{1}{1} = 0.", "aef2071731671af6bdf5dcb5116927ed": "\\alpha = x_1 + y_1 \\omega", "aef24a08cc5618a6817fd11674099283": "a=n+1", "aef2a2668f54f0e8aea0ecae68dbd94d": "A = U (A^*A)^{\\frac{1}{2}}", "aef37f0d127ab30af8b8abb62c006c72": "\\ J^{a}_{\\mu} = i\\partial_\\mu \\Phi^T T^{a} \\Phi", "aef3802a4eaadfd817c072b496105624": " \\sigma^m(n) = 2n , ", "aef38caaa39e6f4e60925cc63b37bd31": "P(x, y) = \\frac{1}{\\pi}\\frac{y}{x^2 + y^2}", "aef3d64e87e63b113def87d3a541e7e2": "\\frac{\\tfrac{1}{2}}{\\tfrac{1}{3}}", "aef3feeb780b2256c3954a00dcc00e33": " v=v^\\prime(0) {h\\over h^\\prime}", "aef480fa4592522cec2f2161d7bc13a3": "\\frac{5}{12}+\\frac{11}{18}", "aef4af369b156143581b187c3f4d05ff": "\\sum_j \\hat{b}_j^{(\\eta)} = \\sum_j \\hat{b}_j^{(0)}", "aef54d650f19eee6326c3165b2a766f6": "f = \\frac {\\omega}{2\\pi}", "aef5be2f05ed3a5e221def4fd7cf9fd1": "\\,\\rightarrow_M^*", "aef5ce5b61be40d7ccec284b86536173": "\n\\gamma^2+v\\delta\\gamma=1. \\,\n", "aef5e2fa3451d1e408e86356e0cc6e20": " \\mathbf{e}_0 = \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}_1 = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}_2 = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{pmatrix} \\,,\\quad \\mathbf{e}_3 = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix} ", "aef61aad770bee032baeac63f516e127": "V_{max}", "aef6c3b495cb204eb69cd1fc74b9bd72": "1 \\,+\\, r \\,+\\, r^2 \\,+\\, r^3 \\,+\\, \\cdots \\;=\\; \\frac{1}{1-r},", "aef6ed03124b4acc05e378c8054ea706": " y''(t) = f(t, y(t), y'(t)), \\quad y(t_0) = y_0, \\quad y(t_1) = y_1 ", "aef73f12a2f8960d7c534b9d7a4b4eb2": "\\rho R / \\tau^2 \\propto \\sigma/R^2", "aef7687be47784defb13979c1743d82c": "Z \\approx {1\\over 2} {A\\over 1 + A^{2/3} {a_C\\over 4 a_A}}.", "aef79620c9214e5906f0a34f0d3ad7eb": "\\frac{1}{{\\rm M}^{n-1} \\cdot \\rm s}", "aef7acf60dd6869b7d3e913ab0976638": "\\widehat{\\theta}_F (z) = \\sum_{k=0}^\\infty R_F(k) \\exp(2\\pi ikz),", "aef7c6f2255bbb7656eb3538cfef36af": "\\delta=37/20", "aef7de5000b56c95a6d07c8ca6ebc515": "(\\cos \\theta + i\\sin \\theta)^{n} = \\cos n \\theta + i\\sin n \\theta. \\,", "aef7f3ed4cd8fd48b345213112c1a02f": "\\mathrm{mul}_A \\lambda\\,", "aef844086ea01959fac6f36f7097f7ae": "[a,a] = \\{a\\}", "aef880020d0a6c241b3df2e3f78ef13e": "\n f^{*}(\\cdot) = \\sum_{i = 1}^n \\alpha_i k(\\cdot, x_i),\n", "aef88646102820adab607b7f28e7532c": "M=(I,O,S,D,F,U,T)", "aef88ef04b57b73bb95eca06d6c07e14": "L = {H \\over \\log_2 N}", "aef8de7d6c2071c1133ca4dde09973ba": "{q^2 \\rho^2 \\over 2m} A^2.", "aef8e01dca4a9687545d55a2d896c195": "K = \\frac{1}{4} \\sqrt{4p^{2}q^{2}- \\left( a^{2}+c^{2}-b^{2}-d^{2} \\right) ^{2}}.", "aef9952f5eef34d02af9d82bce00a366": "\\Theta^{(c)}", "aefa44b70c0b742d78e4b7cd130b5ab6": " {\\kappa}a < 1", "aefab13583ed4e74b3a831a7925e9635": "\\Delta \\omega_1\\,", "aefad7f2b3391fb368110c8f5d39ba00": "\\mathbb{E}^{-S_{ij}}", "aefb12d7dc769ca427ec2b41b88e3493": "y - X \\hat \\beta = y - X (X^T X)^{-1}X^T y", "aefb307d5b928147885c231518e70f06": " {d \\over d\\tau} \\left( {g_{\\mu \\nu} \\delta^\\mu {}_\\lambda \\dot x^\\nu + g_{\\mu \\nu} \\dot x^\\mu \\delta^\\nu {}_\\lambda \\over 2 \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\right) = {g_{\\mu \\nu , \\lambda} \\dot x^\\mu \\dot x^\\nu \\over 2 \\sqrt{-g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu}} \\qquad \\qquad (2) ", "aefb598412777f7121caa6a1af542034": "(i_P + i_T) = i_{PT} - (c - 2)\\,", "aefb5dac634ba06e34baa5c04958e668": "\\sqrt{10}=3.162277660\\dots .", "aefb8f9e140e54adbc47ad27a839fa41": "|T \\setminus S| \\leq 2(1-\\varepsilon)\\gamma n - \\gamma n = (1-2\\varepsilon)\\gamma n\\,", "aefbb2f78be006ca9e6991e5347d0ca7": "\\rho_{AB}, \\ \\rho_{AC}, \\rho_{BC} ", "aefbd443e566b51dbb15a3eccdf54c15": "\\Box \\bot", "aefc1b938eb9a54ef090c23ce64a0b8f": "\\mathbf{C}(A,B)", "aefc3897f324e53b4f54c589dfeacbdb": " \\frac{1-d}{N} \\mathbf{I} = (1-d)\\mathbf{P} \\mathbf{1}^t", "aefc44fc1f2c7a720cad289f3f3e499e": "\\left(\\cos x + i \\sin x\\right)^k = \\cos\\left(kx\\right) + i \\sin\\left(kx\\right). \\,", "aefc5c478a8033de6d562da90be8da30": "{\\mathbb RP}^2", "aefc90a9a4e1fe1a54d43f65dda57dd1": " \\sqrt{0.75} ", "aefca5aeb586309440b7ad19645765be": " u_c ", "aefda7c37942cef8b283ea323e93edda": "(W,\\partial W)", "aefdc35eeff80588e065006099639553": "\\frac {(20 - 8)}{(3 - 2)} \\; = 12. \\,", "aefde19e62c6f1dd2050861e9ffb5b5a": "\\sqrt{\\frac{\\pi}{ k }}\\,\\frac{\\left ( 2k-1 \\right )!}{2^{2k-1}\\left ( k-1 \\right )!^{2}}", "aefde1ea36ab98cce0ccb9a7a63b9de8": " C = -\\sum_j d_j \\log(p_j) ", "aefec4c723196a4acb8376b691c6a8ed": "P < Q/2", "aefecd063f10e18f3a47456509c8d1be": "H^j = D^j \\times D^{m-j}", "aeff3fc429a38e7e9ba55a650eeb00ef": "\\overline{X}_n=\\frac1n(X_1+\\cdots+X_n) ", "aeff604571a6983017f69133f01ab52f": "U_{Rp}(t)", "aeff93499cc27ab8a1c02788dc26afe2": "\\bar c", "af000f8dec09a251fc65812594de45d6": "\\frac{\\Delta \\varphi}{\\Delta \\alpha}=\\int_a^b\\frac{f(x,\\alpha+\\Delta\\alpha)-f(x,\\alpha)}{\\Delta \\alpha}\\;\\mathrm{d}x = \\int_a^b \\frac{\\partial\\,f(x,\\alpha)}{\\partial \\alpha}\\,\\mathrm{d}x + R", "af003989800dd5db158b00e3ceb324c2": "\\lambda(y)= \\sum_{X \\in \\Phi} h(X,y)", "af00c0ecc420de97d71c9688808d0043": "\\sum_{i \\ge 0} (-1)^i \\operatorname{tr}(f; H^i(X, \\mathbb{Q}_l)) = \\sum_{i \\ge 0} (-1)^i \\operatorname{tr}(f; H^i(\\mathbb{C}P^\\infty)) = 1 + 1 / q + 1/q^2 + \\cdots = {q \\over {q-1}}.", "af00c37efaa4313cd2a00a68e8205c3d": "\\lambda_\\mathrm{A},", "af00ead53b203093af1b9c6d75a500e5": "\\rho_{1} = 1.3569", "af018af5791031c13de056f2688e6379": "\\mathit{C_p} = {1, x, x2,..., x p-I}", "af01dc06bbb68d161b9e46bc105489c6": "(P \\leftrightarrow Q) \\vdash (Q \\to P)", "af01fa9bbc954fefff01e5c29e998d3d": "Pr(U=1|X=0)\\ne Pr(U=1)", "af025a889c05f1e7654296b89e6b01eb": " a'_{\\ell\\ell} = s^2 a_{kk} + c^2 a_{\\ell\\ell} + 2 s c a_{k\\ell}. \\,\\! ", "af02d074b368f06cf2845f9667ab396e": "\\alpha^6 + \\alpha^3 = \\alpha^{6 + Z(-3)} = \\alpha^{6 + Z(4)} = \\alpha^{6 + 6} = \\alpha^{12} = \\alpha^5", "af02e9cb9df03a6e15bf533dabee2cd6": "T_{2B}=R_{2B}(y)", "af031d4a948bc1ecc94cb391f5e7dfeb": " \\hat{L}_z ", "af032d9dcd7e6acd7df61f623de94940": "\\int_0^T|S(t)|^{2k}dt = \\frac{(2k)!}{k!(2\\pi)^{2k}}T(\\log \\log T)^k + O(T(\\log \\log T)^{k-1/2}).", "af03d9aac9906bc6520d99d502902e2e": " w(r) = \\bar{w} + w'(r)", "af041f27d3dfbe13aede103fcec42b95": "\\alpha_6 = 0", "af04baad6149b2e15fe87c4c293312e3": "Y = C \\left({Y}-{T(Y)}\\right) + I \\left({r}\\right) + G + NX(Y),", "af050f91ea86eeaa5238fa135180c263": "\\pi(\\mathbf{x})=P(\\mathbf{X}=\\mathbf{x})", "af052471d64876508e6651ed5da096d4": "N_{\\rm min} = {\\rm min}_i\\{N_i\\}", "af05298bacaca10644a19b0ff4345077": "\\delta\\alpha", "af05677a1515195b4d555dae2bdc9527": "\\ddot{v}_a \\ll \\omega^2 v_a", "af05a1e22f7e84fa4b3a03705fbfb2bd": "\\scriptstyle \\log_e (\\frac {760} {101.325}) -25.99771 \\log_e (T+273.15) - \\frac {14768.57} {T+273.15} + 191.4250 + 2.062331 \\times 10^{-05} (T+273.15)^2", "af05ef44b475874f280cce0e995d1d9d": "b_0 = S", "af06040a5844d93d2967b717481d533a": "f(x) = \\sum_{jk} \\langle \\psi^{jk} \\vert f \\rangle \\psi_{jk}(x)", "af0644441d54bacee762b1e57e155e0e": "0=\\text{max}_u ( r(t,x,u) +\\frac{\\partial V(t,x)}{\\partial x}f(t,x,u)) ", "af0671064e78acdbebce7eb7fb4fd975": "\\operatorname{im}\\, \\sigma", "af0671122076dd4a14bc711efa05587b": "N(A,t) = \\mbox{card} \\{a_j \\in A : a_j \\leq t \\}", "af0690ac0d6b8080a6d1416e18fd67e5": "\\alpha = \\left(1 + \\left(0.480 + 1.574\\,\\omega - 0.176\\,\\omega^2\\right) \\left(1-T_r^{\\,0.5}\\right)\\right)^2", "af06be757e88b733789de838f4988cf7": " d(\\vec x_i,\\vec x_j)+1\\leq d(\\vec x_i,\\vec x_l)+\\xi_{ijl}", "af06c07367ce5a4ccfad8e9adb1cf7ef": "S = {F \\over x} ", "af06f3ba214e10a349242cc1da668c69": "PHM_{1} = \\gamma\\;", "af073691e92723e0b6af56580236481d": "{x} ", "af076425d3c771abfbb9de57b2887cd3": "\\Gamma = \\{ \\gamma_{1}, \\gamma_{2}, \\ldots, \\gamma_{|\\Gamma|} \\}", "af07bc8e5809dffaefe30860455d2084": "\\tau \\in \\mathcal{X}", "af07c655fd82e935c9c731b35c579d57": "\\partial^2\\sigma = \\partial ( ~ \\sum_{j=0}^n \n(-1)^j [v_0,...,v_{j-1},v_{j+1},...,v_n]~ ) =0. ", "af080297731fa4b08bc7d54b5128f43c": "L(1-k,\\chi)=-\\frac{B_{k,\\chi}}{k},", "af084ac6fe1c348535f4687ae4857a78": "f_\\mathrm{M} = \\omega_\\mathrm{m}/2\\pi", "af087dc78647622146ae377d362de7a8": "\\mathcal{\\chi}", "af0884d2a3df78798a4076d914e6a744": "\\tfrac{\\sqrt{1}}{2}=\\tfrac{1}{2}", "af0895c3271b56dbfb924406e71d9cf0": "p(\\mathbf{y}|m)", "af089c1cfcaed71d8199a0bdb2634116": "L\\cdot y_L + F\\cdot y_F + P\\cdot y_P", "af08ed764611f04566155145b38feca7": "F_e(\\phi_E + \\phi_P)/2 + F_w(\\phi_W + \\phi_P)/2 = D_e(\\phi_E - \\phi_P) + D_w(\\phi_P - \\phi_W)", "af094a2e8d985c4dcc1edce41572ddf2": "10^{pH_i}", "af09f0aa2cfe09e07128205508a8cbf7": "g-i", "af0a1ca2f7c52a039bbfe6397530b9c5": "q_1\\ ,\\ q_2\\ ,\\ q_3\\ ,\\ q_4\\ ", "af0a1e2ecd594f1f4cb7261c489ee6ea": "\\sigma_{V,A}^3", "af0a2894d19636c87d0c28f93aa88959": "\n T_m = \\cfrac{m \\nu^2 c^2 a^2}{k_B} .\n ", "af0a8538342e827ac3d05e5f16585a6c": " I = A \\cdot j_0 \\cdot \\left\\{ \\exp \\left[ \\frac { \\alpha_a nF } {RT} (E - E_{eq}) \\right] - \\exp \\left[ - { \\frac { \\alpha_c nF } {RT}} (E - E_{eq}) \\right] \\right\\} ", "af0a9d8dbc9158c803c9bf9efd5988bb": " F(x,u,h(x,u)) = f(x,u)", "af0abba0d1898fd6f710a7e9c614f524": "\\displaystyle \\frac{2 a}{a^2 + 4 \\pi^2 \\xi^2} ", "af0b1015634d3767c108a780a30d48c0": "\\boldsymbol{\\Gamma}=(\\gamma_{ri})", "af0b1e2857db2432ea93223dab9ff608": "(p \\lor (q \\lor r)) \\vdash ((p \\lor q) \\lor r)", "af0b6e97ab6426ca4b1070400a799515": "A = \\{x : \\phi_{x} \\in \\mathcal{A} \\}", "af0b7b11285b366332980be4b8ebd552": " \\frac{V^2}{R} = - \\frac{1}{\\rho}\\frac{\\partial p}{\\partial n} \\pm f V", "af0bb07d22e21feb5e7456c4be886e67": "e^{1/4} 2^{1/12} A^{ - 3} \\approx 0.6450 ", "af0bf959c2b2e5095dbc46b8e4c6caae": "C_n X", "af0c01527b937fc76abd92772b0eef53": "K_{-p-q}A\\otimes Z_\\ell", "af0c3b147733154ff281eedd49e6a7a9": " \\frac {n^k} {k!} \\left({\\frac{w}{d}}\\right)^{k-2}.", "af0c819d75304abe2e6dd79672d00629": "M\\mathbf{u} = a\\mathbf{u},\\qquad M\\mathbf{v}=b\\mathbf{v}.", "af0cd4134166ddb1f16e200095df1040": "{d \\over dt}\\left\\{ X \\right\\} = k_1 \\left\\{ A \\right\\} \\left\\{X \\right\\} - k_{2} \\left\\{X \\right\\}\\left\\{Y \\right\\} \\,", "af0cdcefa2e349c6b1c99b5e3f337822": "a\\mid b", "af0d92f0119e5b4ee67d5e415e3a942c": " P_2 ", "af0d9ca05766460d3623f99b5ddd5ba5": "\n\\begin{align}\n1 = \\sum_{k=1}^{K} \\Pr(Y_i=k) &= \\sum_{k=1}^{K} \\frac{1}{Z} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i} \\\\\n &= \\frac{1}{Z} \\sum_{k=1}^{K} e^{\\boldsymbol\\beta_k \\cdot \\mathbf{X}_i} \\\\\n\\end{align}\n", "af0ecebddb02218a0990700b14b77407": "q=5%", "af0ee5d8fa256de72a0fc70162eab1fc": "Z_{ij} \\partial_i H \\partial_j H ", "af0f42d29f3f823ab3d5ee8abbaee9c3": "T(A-\\lambda_\\star I)x=0", "af0fb85fd8355132c44dd4708f1a0e5a": " \\mathbf{R}_{\\textrm{power}} = \\frac{\\mathbf{R}_{\\textrm{iterative}}}{|\\mathbf{R}_{\\textrm{iterative}}|} =\n\\frac{\\mathbf{R}_{\\textrm{algebraic}}}{|\\mathbf{R}_{\\textrm{algebraic}}|}", "af0ff4dd9186a0e8a79184ed9e15176f": "\\frac{r}{R} = \\frac{4 T^{2}}{sabc} = \\cos \\alpha + \\cos \\beta + \\cos \\gamma -1;", "af102e65b3d44d555569ff21cb485721": "y z = z y \\qquad\\mbox{for all }y,z\\in \\mathbb{R}", "af105717ba6f73bec8076e6e3709b015": "J_{0}", "af105756f1e62528992e00694055af39": "\\left|g^{i}\\left(\\theta_{0}\\right)-\\theta_{i-1}\\right|", "af1089d84639742ab4c561c96f8c8aab": "\\omega'=2\\omega", "af10c0aba71a8c73312d39a2921c658c": "a_7", "af10d34b693c4d88aac0a4198fadf091": "\\tan\\frac{\\pi}{6}=\\tan 30^\\circ=\\tfrac{\\sqrt3}{3}\\,", "af115d9a6059b42f18fdbbb247e691cb": "\nS_W = 1 + 9 + 1 + 0 + 4 + 1 + 1 + 9 + 0 + 4 + 9 + 1 + 9 + 1 + 1 + 4 + 9 + 4 = 68\n", "af115e39322f56c14ed28b9a76c745f5": "y_{min}=0", "af118405f2375558341973e3db0dd938": "M \\times n", "af12ac451a3ecdfd4c7ad2077194e5aa": "W_i(X) = \\frac{1}{M}\\sum_{m=1}^M W_{im}(X)", "af12ba4a0ff8fdb41bd939a0685b948b": "f:{\\mathbb R}^n\\rightarrow {\\mathbb R}^m", "af12f19a4eb846434f8d1bc84ac6499e": "\\begin{matrix}\n\\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot \\varphi} = \\frac{\\partial L}{\\partial \\varphi}\\;,\n\\end{matrix}", "af131f39a3b51256e22555f2a997d92d": "\n \\theta \\neq \\theta_0 \\quad \\Leftrightarrow \\quad f(\\cdot|\\theta)\\neq f(\\cdot|\\theta_0).\n ", "af1392b922d91ed8a3a6b8945942d594": " \nK=\\left[\n\\begin{array}{ccccc}\nK_{11}^{(1)} & K_{12}^{(1)} & 0 & ... & 0 \\\\\nK_{21}^{(1)} & K_{22}^{(1)}+K_{11}^{(2)} & K_{12}^{(2)} & ... & 0 \\\\\n0 & K_{21}^{(2)} & K_{22}^{(2)}+K_{11}^{(3)} & ... & 0 \\\\\n... & ... & ... & ... & ... \\\\\n0 & 0 & ... & K_{22}^{(Ne-1)} + K_{11}^{(Ne)} & K_{11}^{(Ne)} \\\\\n0 & 0 & ... & K_{21}^{(Ne)} & K_{22}^{(Ne)} \n\\end{array}\n\\right]\n", "af139dd9754a3726aeeb007a547a6172": "W\\left(1\\right) = \\Omega=\\frac{1}{\\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{\\,dt}{(e^t-t)^2+\\pi^2}}-1\\approx 0.56714329\\dots\\,", "af13ce531973e6fa35c42fac24339422": "W^-", "af13ff08884c6e470482f9cc18635462": "\\phi\\left(\\bold{x}, \\bold{u}_f, \\rho_p, \\Omega_p, t \\right); ", "af1475338149105be5f2189533779c7d": "X(K)", "af1479155e0de054e6c31bd44d53e20d": "\nE=\\sum_{n=1}^{\\infty}\\frac{1}{2^{n^2}}\\frac{2^n+1}{2^n-1}\n", "af149ea86a14f4322e10771158fd5d72": "\\begin{align} \n&DT + \\delta T = T\\\\ + \n\\Rightarrow\\; &D + \\delta = 1 \n\\end{align}", "af1513ee506545430f1fe3a6e6f62aae": "r_{x'y'}", "af1539ca308854e1de29dbaac67d05d9": "K \\subseteq H", "af156b56cc6377789acd103efa59e2a4": "z'=\n\\frac{\\mathit{far}+\\mathit{near}}{2 \\cdot \\left( \\mathit{far}-\\mathit{near} \\right) } +\n\\frac{1}{z} \\left(\\frac{-\\mathit{far} \\cdot \\mathit{near}}{\\mathit{far}-\\mathit{near}}\\right) + \n\\frac{1}{2}\n", "af15b9e94695e00302f80d1c1979e7d3": "\\mu = G(m_1 + m_2) ", "af15ee95b2016b2e6241b15bc3703fb1": "h(t)=\\sum_{n=1}^\\infty \\kappa_n \\frac{(it)^n}{n!}=\\log(\\operatorname{E} (e^{i t X}))=\\mu it - \\sigma^2 \\frac{ t^2}{2} + \\cdots.\\,", "af15f673af68dc9ca8d3affbc167d20e": "\\tfrac{1}{2}+\\tfrac{1}{3}", "af16192e4c8045d80791a9852466eb7a": "0 \\wedge x = 0 = x \\wedge 0", "af161db0664560f4c127e9cb4fba181c": "U_\\text{outer} = \\frac{P_\\text{Crit}}{2} \\int (w_{x}(x))^2 \\, dx", "af168665ba00c57df2aab7316a5a5ebe": " \\frac{\\operatorname{d}}{\\operatorname{d}t} = U\\frac{\\operatorname{d}}{\\operatorname{d}x}", "af16ba9a1c9234a160f32720752213a4": "d(\\sigma) = \\frac{e(\\sigma)}{2^{(1-s)|\\sigma|}}", "af16da7109bec753342bfb782191da63": "\\beta_0, \\ldots, \\beta_m", "af16e6c5ba292c25e53b1eda3d06f193": "-\\boldsymbol{\\nabla}T", "af17244079a02cfa9d3eb20a18c93872": "\\frac{3}{2}\\left(\\frac{\\pi}{2}\\right)^{4/3}\\left(\\int\\mathrm{d}\\mathbf{r}\\ \\rho^{3}(\\mathbf{r})\\right)^{1/3} \\leq T.", "af1733aee9810a56352c08b5d0943416": "x\\!:\\!\\tau", "af173bd91e46e1df87e2b5fb8f33dd58": "\\ddot x(t)=w^2x(t)", "af1766ba76cd286c656473b7ad5091e6": "I_{n}=I_{n+1}=\\cdots", "af17730957b9bc502282a1fce503c840": "i\\le n", "af1788d04f4bc2045cd70b818948fcb1": "\\begin{matrix}\nF_{\\mathbf{K}} & = & f \\left[ e^{-i\\mathbf{K}\\cdot\\vec{0}} + e^{-i\\mathbf{K}\\cdot(a/2)(\\hat{x} + \\hat{y} + \\hat{z})} \\right] \\\\\n& = & f \\left[ 1 + e^{-i\\mathbf{K}\\cdot(a/2)(\\hat{x} + \\hat{y} + \\hat{z})} \\right] \\\\\n& = & f \\left[ 1 + e^{-i\\pi(h + k + l)} \\right]\\\\\n& = & f \\left[ 1 + (-1)^{h + k + l} \\right] \\\\\n\\end{matrix}", "af17ada00e91fdd8bf86d2d8374f09bf": "u_1\\,", "af17e1bbae3a1e6a0606e5edeccbf878": "\\hat 5", "af17f5ec1e83c7ac9eb8444a179bab03": "\\theta_i > \\theta_f", "af181dd1126dbb8521603d700c674b24": "f(x_0,x_1,x_2,\\dots) = 0 \\, ", "af18532b5434a85f2fdddf3dec14a4cc": "\\overline{A \\cup B}\\equiv\\overline{A} \\cap \\overline{B}", "af1857c6bf45424c72684d81db0a906d": "|V| > \\frac12", "af188644415dea8c0efb08a3ec0ec9ec": "\\vec p = m \\vec v ", "af18b493915ff2879a201af9571d55ae": "\\hat{\\tilde{v}}", "af18c76d7f93c015867a1d6e51461d96": "Q_2=(b_2^2+a_2^2),", "af18d3ebef7d52ba5d51749c60d43932": "R \\equiv \\pm r' \\pmod p", "af18d541596910aaa542820f0f4f7182": "\\boldsymbol{\\Pi}^1_1", "af1905e315fb01d0fa7240900aa47956": "\n\\begin{align}\n\\phi &= \\sin^{-1}\\left[\\tanh(y/R)\\right]\n\n = \\tan^{-1}\\left[\\sinh(y/R)\\right]\n\n = \\sec^{-1}\\left[\\cosh(y/R)\\right] \n = \\mbox{gd}(y/R).\n\\end{align}\n", "af194fc2f10b2517da0d12767cd2fa70": "\\int \\sin ax\\, e^{bx}\\, dx = \\frac{e^{bx}}{a^2+b^2}\\left( b\\sin ax - a\\cos ax \\right) + C", "af19d098e01e2dce39c0e154da8f75a3": "\\mathrm{d}S = \\frac{\\delta Q}{T}\\,", "af1a0be50101fa84b60dfc75fcf26b87": " \\vec{v} = v \\hat{e} _x ", "af1a68f5b118f12f993c04bc3053f661": "\\psi(\\Omega+1+\\alpha) = \\tilde\\psi(\\tilde\\psi(\\Omega)+\\alpha)", "af1a900f00ca83af2135e0b95574b918": "-i\\,", "af1ac4bc67cb253d0d8e5e09dc515d5a": "\\mathbf{Q^Tr=Q^T\\ \\Delta y -R\\ \\Delta\\boldsymbol\\beta}= \\begin{bmatrix}\n\\mathbf{\\left(Q^T\\ \\Delta y -R\\ \\Delta\\boldsymbol\\beta \\right)}_n \\\\\n\\mathbf{\\left(Q^T\\ \\Delta y \\right)}_{m-n}\\end{bmatrix}", "af1ad193d04cd337edeba1051e383046": " S_4 \\implies (A_4 = q \\or A_4 = v[q]) ", "af1af6d276f814629d6ffb4b93ffa3e8": "pd=e^{1-b}", "af1b03a21a3e830dcf317f1053794ca5": "\\Sigma=\\left \\{a,b\\right \\}", "af1b15f193e09af33531da60cf08b7e6": "a(x) = \\sum_{k=0}^\\infty\\frac{(-1)^k(k+1)x^k}{k!} = e^{-x}(1-x).", "af1b43bc665a08a462203cfa53f796e6": " \\prod_{k=0}^n \\mu(k,A) \\leq \\prod_{k=0}^n \\mu(k,B), \\quad n=0,1,2, \\ldots . ", "af1bc558662f3eaa8627b540a1513036": "\\left(\\frac{d}{dt}\\right)^j\\left[v(x)^2+v(x)h(x)-f(x)\\right]_{|_{x=x_i}}=0", "af1bcedca0039a91967dbb97b99156c7": "(23)\\qquad \\bar{\\mu}\\,\\hat{=}\\,\\mu \\,,\\quad \\mathcal{L}_{\\bar{m}}m\n\\,\\hat{=}\\,(\\alpha-\\bar{\\beta})\\delta-(\\bar{\\alpha}-\\beta)\\bar{\\delta}\\,,", "af1c065e7b5d45ffa1ced8c01d835e98": "0.99 \\pm 0.022", "af1c9900155de9530636c5d5193a4e7f": "\\Diamond A := \\neg\\Box\\neg A", "af1c9be501db7e018e740fddb92c3368": "w_i = \\frac {x_i} {(n+1)^2[L_{n+1}(x_i)]^2}.", "af1ce5ba4658c80f2478e1c5248cd0c6": " \\sum_{\\lambda \\vdash n} (f_\\lambda)^2 = n!", "af1d1a79d8e97ddbc1f1e3d63283e803": "L - \\phi' \\frac{\\partial L}{\\partial \\phi'} = \\text{const.}\n", "af1dffb8e7a3d75fea829dd78a15cf9b": "\n \\frac{\\partial \\boldsymbol{\\mathit{1}}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} = \\boldsymbol{\\mathit{0}}\n", "af1e1dbd853909746731867306aca830": "\\scriptstyle \\log_e P_{mmHg} =", "af1e5f33dc524c05d0b301c0c3827713": "\\text{Vol}_q(r,n)", "af1e62413bd3a8ff6c6e9a99eb0c8480": "\\sigma_3(6)", "af1e82c9a3021593281bb610c518b8df": "G \\frac{M m}{r^2}\\ = \\frac{m v^2}{r}\\ ", "af1e9a35e2316f5b8c8d65b517615c3e": "\nx_1(t) = A_1 e^{i \\omega t} \\,\\!\n", "af1eb5593bfee003fee760fd4ebde4a5": "\\Psi = \\frac{1}{\\sqrt{2}}(\\phi_1 \\mp \\phi_2) = \\frac{\\phi_1 \\mp \\phi_2}{\\sqrt{2}} \\,", "af1efda569db68d7cf60103d6a8a1a26": " e^{ibQ} ", "af1f8c5e3bc7a90ee395f614156850e2": "+\\ 0.876793(1-T/T_0) +\\ \\log\\ e^*_{i0}", "af1f903833839556c8f2813fd3e9fdff": "[u\\;\\|\\;v'\\;\\|\\;[v\\;\\|\\;M]_h\\;\\|\\;N]_m\\!\\!\\rightarrow\\!\\![w\\;\\|\\;M]_h \\;\\|\\;[w'\\;\\|\\;N]_m", "af1fa58ab70b5c66ad92f3a6a4ca3cbb": "\\mathrm{I}_{\\mathrm{O},\\mathrm{P}} \\subseteq \\mathrm{I}_\\mathrm{P}\\,", "af1faeacf2ce7c66a29f23c97da6a770": "B=2^{b-1}-1", "af2005b67df95992d6dc7cddbc012414": "(\\overline{w^'\\theta^'_v})_s", "af207efcd250715c5446a9869f45b34e": "m \\ = \\ { 1 \\over 96485 \\ \\mathrm{(C \\cdot mol^{-1})} } \\cdot { Q M \\over n } ", "af2095daecc8d725a8d075f6f8c45c4d": "64 = 8^2", "af20d7f5473c7d04cd6fc699e3a05399": "\\text{Ext}", "af20e2cbfc5b63866f511d3564200b38": "\n\\langle \\phi_m | \\phi_n \\rangle = \\int_{X}\\overline{\\phi_m(x)}\\, \\phi_n(x)\\, d\\mu(x) = \\delta_{mn}\\, .\n", "af2164af21af587f555f9c5d889df58f": "\\begin{align}\n \\left(\n \\frac{1}{\\rho} \\frac{\\partial A_z}{\\partial \\phi}\n - \\frac{\\partial A_\\phi}{\\partial z}\n \\right) &\\hat{\\boldsymbol \\rho} \\\\\n+ \\left(\n \\frac{\\partial A_\\rho}{\\partial z}\n - \\frac{\\partial A_z}{\\partial \\rho}\n \\right) &\\hat{\\boldsymbol \\phi} \\\\\n+ \\frac{1}{\\rho} \\left(\n \\frac{\\partial \\left(\\rho A_\\phi\\right)}{\\partial \\rho}\n - \\frac{\\partial A_\\rho}{\\partial \\phi}\n \\right) &\\hat{\\mathbf z}\n\\end{align}", "af2174f37c76a140f9681334fb751da6": "\\langle E(B) h_1, h_2 \\rangle = \\langle E(B) h_2, h_1 \\rangle", "af2192c1ea39f5a44e7eb497b4a23851": "\\frac {K \\cdot t}{V}", "af21cc9c8ec685320e2b9fe2f564e2c0": "\\lambda_{3} = 1 - \\lambda_{1} - \\lambda_{2}\\,", "af22514052d6f8d9f0c37f986b71dd2e": "i\\frac{d\\Psi(x,t)}{dt}=H\\Psi(x,t),", "af2289df018df185fde3129923495a8e": "\\displaystyle{Q(ga)=gQ(a)g^t.}", "af229d176c4fb1e416aaa9ec5ca215c8": "\\Psi(\\mathbf{r},s_z,t)", "af22a085fabb9eae22f6f5e2cb81ffa5": "\\mathbf{O}(\\mathbf{n}^2)", "af2314afecf0879d55a9d8faacc509a0": "{V}^{\\bullet}_\\mathrm{Cl}", "af2446059e4d16b551d1ae825dbe3eb3": " k = \\frac{2 \\sqrt{2} M \\Gamma \\gamma }{\\pi \\sqrt{M^2+\\gamma}} ", "af2446278bd20f9782a2116a0d8abbc5": " \\Delta t = \\frac{1}{\\Delta f} ", "af2491aa67ccc6773186725de835ac94": "H_n\\left(G,M\\right)", "af24d6e6a43134e267344062697124aa": " \\Delta w_{ij}(t+1) = w_{ij}(t) + \\eta\\frac{\\partial \\log(p(v))}{\\partial w_{ij}} ", "af24df21b628563fd5b45a87bac0baa0": "P!=O", "af24e1b41591af50f91da47e591e5ac4": "\\ f_S + f_L = 1", "af2523b7e4530f351c818c7dadabaaad": "\\,\\!j=1,2,3", "af253c61273f239041c86cbd1539ecb0": " r_\\mathrm{out} = \\frac{v_{out}}{i_{out}}", "af2593b49488f8bc44396795792c2d79": "r_i", "af25a11102575a4b8f2ac1b970eac222": "c_i(0) = \\langle c(0), \\mathbf{v}_i \\rangle ", "af25d53f09e63af655b44ac4e794032d": "p_n: H_n(A) \\rightarrow H_n(B).\\,", "af25d631b81b29fb7babacab1264c8c2": "2\\leq d\\leq n,", "af25dd94489ddd578683ac3d685f4729": "C(n)\\mapsto C(n) e^{-t/t_c}", "af25e53827978215e15d065f348540e4": " X\\in \\mathbf{L}_{M} ", "af262b06e2b0dad47d57c30fe425b960": " \\pi(\\mathbf{x}) \\sum_{i=1}^J [\\alpha p_{0i} +\\mu_i (x_i) (1-p_{ii})] ", "af263c95735d21a38145e95e867e10bb": "\\begin{cases}\\begin{align}4x + 2y &= 12 \\\\\n-2x - y &= -4 \\end{align}\\end{cases}\\,", "af268be6fc4a30fa3fd8cb3033798c23": "\\displaystyle{L(a^{m+1})a^n=2L(a)L(a^m)a^n+ L(a^2)L(a^{m-1})a^n -2L(a)^2L(a^{m-1})a^n\n=a^{m+n+1}.}", "af26c66fa302889554351b2988df9c7f": " -\\frac{\\pi}{2} < \\arctan \\frac{a_1}{b_1} + \\arctan \\frac{a_2}{b_2} < \\frac{\\pi}{2}.", "af26ddd9b30c7296d5abbf70105dff34": "d\\omega/dB=0", "af2700e7f0267cd81b79cb8f1fc6b514": "(a + b)^2 = a^2 + 2ab + b^2", "af270e937afbacdd94c61f410fa6077a": " k = 2 ", "af2723f7f1059abff4aa28170041223e": " x_u(n)=e^{-j\\frac{\\pi un(n+1+2q)}{N_\\text{ZC}}}, \\, ", "af27a3a8d8c0571b0c6f220af086f3b6": "g_{ij}^\\alpha=\\sum{\\partial_i\\ell^{(\\alpha)}\\partial_j\\ell^{(-\\alpha)}}", "af27ca07971e402127529d905fd86dd4": "\\left\\lfloor\\frac{x+m}{n}\\right\\rfloor = \\left\\lfloor\\frac{\\lfloor x\\rfloor +m}{n}\\right\\rfloor,\n", "af2813fa757ea556d92c8a515d9aad01": "h^*_1,h^*_2, \\dots ,h^*_n \\dots \\in G", "af283911b5cea235936a579f45dd62fb": "G(k)\\le k(2\\log k +2\\log\\log k + C\\log\\log\\log k)", "af284c3f4405571f53a4dd19206c76c7": "R = A[X_1, \\ldots, X_n] / (f_1, \\ldots, f_m).", "af29219814f04a3e5905bf1d1b6c5339": "F[y] = \\int_E y \\ln y \\, dy", "af29376001549edb19113cc938b0736c": "{\\rm N}(\\mathfrak p)", "af296833d7fc564cd9a117722a82fb61": "\\beta=(\\beta_1,\\ldots, \\beta_n)", "af299d665ba9942a707257245ef74567": "H(X_1, \\ldots, X_n)", "af29a55892f63c34dcfc7a8ee049b5f8": "\\int \\frac{\\ln x\\,dx}{x^m} = -\\frac{\\ln x}{(m-1)x^{m-1}}-\\frac{1}{(m-1)^2 x^{m-1}} \\qquad\\mbox{(for }m\\neq 1\\mbox{)}", "af2a0bdd67cec3bdbe4813a5275e4b56": "F(\\mathbf{k}) = 0", "af2a29dc9f3a2d4b7d060e55b4d72b41": "(T,\\partial T)", "af2a31c7e77548c2db9983bb625a023c": "\\scriptstyle \\hat{C}=\\frac{1}{N}\\sum_{i=1}^N (f(x_i)-y_i)^2", "af2ab88bacf9db5b1a256a90dd5ca3c2": "\n\\begin{align}\ni&=\\sin \\theta+\\sin (\\theta-\\frac{2\\pi}{3})+\\sin (\\theta+\\frac{2\\pi}{3})\\\\\n &=\\sin \\theta+2 \\sin \\theta \\cos \\frac{2\\pi}{3}\\\\\n &=\\sin \\theta -\\sin \\theta\\\\\n &=0\n\\end{align}\n", "af2af8f6d0880d75af36b3b9b5a27e69": "\\omega = \\sqrt{g k},", "af2b2d9bfb2e51834240234ebcbef056": "\\omega = 2\\pi f = 2\\pi /T ", "af2b710ca100e2fdb7bc4253e615db97": "f(x) = \\sum_{n=1}^\\infty \\int_D\\, \\left( \\varphi_n (x) \\varphi_n^*(\\xi)\\right) f(\\xi) \\, d \\xi.", "af2c2482e5d22714e9854d6a79b815de": "(x, f(x))", "af2c6b8cc4e2a1436b4f94f3783765e0": "s\\le r-1", "af2ce544ae74b48649ca4fc699bf8f8b": "\\sum_{i=0}^x \\mathbb{P}(X = i)", "af2d03fc4d638c2b7ae0fd27cce5a451": "\na^{2} = -\\frac{\\Delta}{\\lambda_{1}D} = -\\frac{\\Delta}{\\lambda_{1}^{2}\\lambda_{2}}\n", "af2d0d0e38aa21f6c5e5dec93254cf87": "\\text{Visibility}(\\text{real})=\\frac{\\max-\\min}{\\max+\\min},", "af2d6fcefd5f28af5a084bf6f4f21974": "c_{n}/c_{n-1}", "af2ddfd45dc7e7508907c4b2d85b5c04": "\\mathcal H_\\mathrm{IN} = \\operatorname{span}\\{ \\left| I, k_1\\ldots k_n \\right\\rangle = a_i^\\dagger (k_1)\\cdots a_i^\\dagger (k_n)\\left| I, 0\\right\\rangle\\},", "af2dee17cf9a3aecbe1535012404f15f": "(2a-R)g", "af2e6382accd190d53804990afd7f957": "h \\in \\mathfrak{h}", "af2e8e279436753e7aa79d91b1be7a53": "a\\circ b=\\sum_{i=0}^k\\sum_{j=0}^lF_{c_i+d_j}.", "af2e978a1e7100601ec491b173a27afd": " \\alpha_{1} ", "af2ebd9ec2727f07384f4e978561db55": "\n \\text{d}\\boldsymbol{\\sigma}:\\text{d}\\boldsymbol{\\varepsilon} \\ge 0 \\,.\n ", "af2fcca8a9618b853a7bd446422a94d2": "x \\in (-\\infty, \\infty)\\!", "af2ffed560a59ceb568ab442f4314411": "r=f(\\theta)", "af3019b7a5f8d6f8760d253db688b29a": "[a , b]", "af30281c8a1eff6803faaa4a05062744": "1\\times k", "af3028bad41c2c20018547071d3ca067": "\\Delta v/c=2\\times10^{-7}", "af30457ddd1b64e4ad027c84c9e7782d": "VO_2 = (CO \\times\\ C_a) - (CO \\times\\ C_v)", "af30541c95d65ddd5487cf40ed20da33": "D_a", "af30978d3ebd8f820c3f5c4b2cfaabc9": "\\left\\{\\tau \\in \\mathcal{H}_g \\ | \\textrm{Im}(\\tau) > \\epsilon I_g \\right\\},", "af30be19eb52ef5f9198607876e8fc18": "\\mu\\left(F\\triangle G\\right)=0", "af3106ab3fcd1d11530ab7799e0f1cce": "\n\\operatorname{Li}_{-n}(z) = \\left( z \\,{\\partial \\over \\partial z} \\right)^n {z \\over {1-z}} =\n", "af31512bd67d3b29989600551e6ef18c": " Q_j = \\sum_{i=1}^n (\\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}_j} + \\mathbf{T}_i\\cdot\\frac{\\partial \\vec{\\omega}_i}{\\partial \\dot{q}_j}),\\quad j=1, \\ldots, m.", "af3187072f109dc13edc079f3381d5c3": "\\upsilon_D\\,", "af31b3ee28c822f0b39dd934ac6af446": "\n\\bar{R}^2=|\\bar{z}|^2= \\left(\\frac{1}{N}\\sum_{n=1}^N \\cos(\\theta_n) \\right)^2 + \\left(\\frac{1}{N}\\sum_{n=1}^N \\sin(\\theta_n) \\right)^2\n", "af31e7b28adcd2ac9e8a1062e4dc5020": "S = -\\lim_{x\\rightarrow 1}\\frac{d}{dx} \\sum_i p_i^x ", "af31ec1ecfb8bc9ce87cb5699f1d430e": "I\\subset \\mathbb{R}", "af3208dd5eef680ff538bfb5048c4b68": "(n-n_A-n_B)+n_A n_B", "af32686d97b437cfca28bce5e1163e5e": "G = \\left\\{ 1, f, f^2, g, gf, gf^2 \\right\\}.", "af3279498c9d7502bd950e444c463906": "|v| = c .", "af3281ff76282a8a48102e75d5923855": "l_a", "af328575e1bdbb44fe0b380876887ad4": "\\begin{align}\n p\\!\\!/ &= \\gamma^\\mu p_\\mu = \\gamma^0 p_0 + \\gamma^i p_i \\\\\n &= \\begin{bmatrix} p_0 & 0 \\\\ 0 & -p_0 \\end{bmatrix} + \\begin{bmatrix} 0 & \\sigma^i p_i \\\\ - \\sigma^i p_i & 0 \\end{bmatrix} \\\\\n &= \\begin{bmatrix} E & - \\sigma \\cdot \\vec p \\\\ \\sigma \\cdot \\vec p & -E \\end{bmatrix} \n\\end{align}", "af328b143cc89c495a4ac2c6a47ba579": "\\textstyle (A - \\lambda_jI)^{\\alpha_j}.", "af329936c4a5eb1d99d4afab448878bf": "\\Pr ( X \\in K ) \\geq \\Pr ( X + Y \\in K )", "af329f290920765163d823fa93c69698": "\\mathbf{f} + \\epsilon_0 \\mu_0 \\frac{\\partial \\mathbf{S}}{\\partial t}\\, = \\nabla \\cdot \\mathbf{\\sigma}", "af3336982b79cc6ce8037449c320b5bd": "A\n\\begin{bmatrix}\ne_x \\\\ e_y \\\\ e_z \\\\ e_t\n\\end{bmatrix} = \n\\begin{bmatrix}\n\\frac {(x_1- x)} {R_1} & \\frac {(y_1-y)} {R_1} & \\frac {(z_1-z)} {R_1} & c \\\\\n\\frac {(x_2- x)} {R_2} & \\frac {(y_2-y)} {R_2} & \\frac {(z_2-z)} {R_2} & c \\\\\n\\frac {(x_3- x)} {R_3} & \\frac {(y_3-y)} {R_3} & \\frac {(z_3-z)} {R_3} & c \\\\\n\\frac {(x_4- x)} {R_4} & \\frac {(y_4-y)} {R_4} & \\frac {(z_4-z)} {R_4} & c\n\\end{bmatrix}\n\\begin{bmatrix}\ne_x \\\\ e_y \\\\ e_z \\\\ e_t\n\\end{bmatrix} = \n\\begin{bmatrix}\ne_1 \\\\ e_2 \\\\ e_3 \\\\ e_4\n\\end{bmatrix}\n\\ (1)", "af336b7754f9db5c7ad57760801ba65c": "\\lambda_{ij}=\\frac{\\partial^2f}{\\partial z_i\\partial\\bar z_j}", "af33769ed9f11e1d120c007bd9a3fe23": " L_{n-\\ell-1}^{2\\ell+1}(\\cdots) ", "af3380b8e8ef9443e58aab805d84bcdb": "f \\colon [a, b] \\to \\mathbb{R},", "af33898e1e2adb7261c88da1fef31252": "x_{n+1} = r x_n (1 - x_n),", "af33a2dbea0cdfe17c954aeb460c844e": "B(t)\\approx \\frac{\\sigma_\\varepsilon^2}{1-\\varphi^2}\\,\\,\\varphi^{|t|}", "af33bf38b97e6077839f77d0687d293a": "\\mathbf{\\hat \\mu} (\\mathbf x)", "af33dfdd08b1ecbe82ac13e55541ee34": "\\Lambda=\\beta_0+\\beta_1\\lambda_1+\\cdots+\\beta_n\\lambda_n", "af33e15a22757ce4b162e0a5ea02a8f3": "\\partial^\\mu F_{\\mu\\nu}^a+gf^{abc}A^{b\\mu}F_{\\mu\\nu}^c=-J_\\nu^a.", "af342eb567646d2870819a9c4ef4df23": "\\mu_{{(Q)}_{[i,\\epsilon]}} = \\frac{P_{[i,\\epsilon]}^Q}{I_{{(Q)}_{[\\epsilon]}}} = ", "af3475959be6ef2e4f70f0ddb3580317": "(s,a,b,d,t)", "af349e9e93b58c52fc0b08ada677d4d9": "X_t-X_s \\,", "af34bac3baaa04c1d9ea3af7d1399134": "\\ell'(b)=\\ell''(b)", "af34c43b576efb841e656ae2220a3185": "_{ordinal} \\delta_{ck}^2 = \\left ( \\sum_{g=c}^{g=k} n_g - \\frac{n_c + n_k}{2} \\right )^2", "af352e1a8041c55a5536a0057fb29f56": "\\operatorname{d}W/{\\operatorname{d}t}", "af352eb902f59229b62cbc55347503e3": "\\rho=\\phi^*\\phi\\,", "af353c28e6126f7998714d0020007e7c": "\n\\operatorname{dVar}^2(X) := \\operatorname{E}[\\|X-X'\\|^2] + \\operatorname{E}^2[\\|X-X'\\|] - 2\\operatorname{E}[\\|X-X'\\|\\,\\|X-X''\\|],\n", "af36bfdf5d776d1515f42f65f776fc35": "2\\pi\\int_{x_1}^{x_2} f(x) \\sqrt{1+f'(x)^2} dx", "af36c2cb1b32575c4465b81c880cb682": "h:A\\to B", "af36c4288773e7c23b5923f3c46f28e2": "d\\sigma = d^2 b", "af36e75c8175c937c065e1fbef2d8884": " \\nabla_h[f](x) = f(x) - f(x-h). \\ ", "af373c654a750e93f50938f4b6b45902": "{\\sqrt{n} \\left[ \\frac{X_n}{n}-p \\right]\\,\\xrightarrow{D}\\,N(0,p (1-p))},", "af379d79f2251fe3f3dfc26999694db3": "\\Pr[X < x] \\le k/q", "af3801b68621d1a6de216218f4f4254b": " C / P_0. ", "af3875e070982a8da158bf60b7b31b10": "H = \\tanh(\\beta J H) = (1+\\varepsilon)H - {(1+\\varepsilon)^3H^3\\over 3}", "af389cf88e44cd791de4ddde7e6e65ad": "\\Box B", "af38aa81989d4ed6d2cc9ecf573b2c60": "h \\approx \\dfrac{G(x)-F(x)}{F'(x)}\\,", "af399a486ca24a50c5f57f63c1e3e281": "M_{M_p}", "af39aa2186df9ced02d33f320d58bf77": "D\\Theta=\\Omega\\wedge\\theta.", "af3a075b7bf2f3639dc75e4f852030cb": "C \\sin(2\\pi ft)", "af3a82eda0af59cc58f6536afc9c8c2c": " l - Q ", "af3a84a65cbb3dc2ff0184cdf435e139": "\\mathbb{P}^N", "af3a9cca32ff2bfbbc8e3547f3fa93a7": "x=0\\,", "af3ab34f15d6ad091faa084932bc8a0b": "(g_\\theta,Z)", "af3ab622577e5d91fe6d79ade47c7a37": "\\sigma_k(t)", "af3af1683d3ff71e60d93a9f22326a2c": "\\frac{\\partial}{\\partial t}", "af3b2158d04fed28fa91e7f926f9ecf4": "Y_{a,b}(\\theta,\\phi)", "af3b47fa824ce47676060f27d041bc82": "\\alpha^{(i)} = \\alpha^{(i)}_1\\wedge\\cdots\\wedge\\alpha^{(i)}_k,\\quad i=1,2,\\dots, s.", "af3c085b71930c88f65c77f9b49eeb39": "\\mathfrak{N}_*=\\sum\\limits_{n \\geq 0}\\mathfrak{N}_n", "af3c09b276d956b9299dcecf0ec7663b": "\n\\mathbf{L} \\ \\stackrel{\\mathrm{def}}{=}\\ \nL_{1}\\mathbf{e}_{1} + L_{2}\\mathbf{e}_{2} + L_{3}\\mathbf{e}_{3} = \nI_{1}\\omega_{1}\\mathbf{e}_{1} + I_{2}\\omega_{2}\\mathbf{e}_{2} + I_{3}\\omega_{3}\\mathbf{e}_{3}\n", "af3c09c3eb1fe5bc60e11b45e6989299": " S \\subseteq G ", "af3c1adca188345d2a91c0d13a891b9f": "x \\in [a,b]", "af3c4b39c4336a55bf3f8ec2ceafb138": "\\sigma^2_{t}(x) = \\int_{C_{t}(x)} h(\\eta,t-s,x) \\tilde{L}(d\\eta,ds)", "af3c603ac56e4d1f497f38638879055f": "F = F^{**}", "af3ca10ea51fcaabc153b2a32cf06776": "\\displaystyle{U_-=v_-, \\,\\,\\,\\,U_+={\\lambda+{1\\over 2}\\over \\lambda - {1\\over 2}}\\cdot v_+.}", "af3cd59f5d9d886aa8b06dcf195cd4c5": " \\mathrm{N}(\\alpha-\\beta)<1 ", "af3d2a838f6dac316f19753449a24e2e": "\\theta'(x)\\,", "af3d98b9ec66e69ae7a1a4ee12824ffc": " \\langle x | \\hat{p} | \\psi \\rangle = - i \\hbar {d \\over dx} \\psi ( x ) ", "af3d9a67bf30bcd9bb42720335e2ebb5": "N_j = \\sum_{i=1}^n \\mathbf{N}_i\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j}", "af3e1cda6b0cc8f9e2f3aa29ac9a82b0": "g: I\\times J \\rightarrow \\mathbb{R}", "af3eb0e2f6a5f9f0d1114908f025eddf": "\\mathrm{We}^*=\\frac{\\mathrm{We}}{48}", "af3ed2d852871195cf3358bf11941a56": "b=237", "af3f350de6ee05ccb95dd4070fd7e561": "\n \\mathbf{u} = u_1\\boldsymbol{e}_1+u_2\\boldsymbol{e}_2+u_3\\boldsymbol{e}_3 \\equiv u_i\\boldsymbol{e}_i\n ", "af3f5eb9542fa70170d8da473d5b605f": "{\\rm SCF} = {\\rm ACF}\\,\\cdot\\,\\left(\\frac{P_{\\rm actual}}{P_{\\rm standard}}\\right)\\,\\left(\\frac{T_{\\rm standard}}{T_{\\rm actual}}\\right)", "af3fa2363525e51962290b0792d9e9d3": "2^{|S|}", "af3fc0bb8d76427c07ca74863a440353": "H B ^{-2} \\rightleftharpoons\\ H ^ + + B ^{-3} \\qquad K_3 = {[H ^ +] \\cdot [ B ^ {-3}] \\over [H B ^ {-2}]} \\qquad pK_3 = - \\log K_3 ", "af40af7531c38caf6087547f3a07b7de": "\\textstyle 6", "af40c4858cfc6c0fe74cecfa90fd3ee0": "V_c", "af40ec3e6925b7b24688a7b2fe93f3b9": "\n \\psi(x,y,z,t) = \\psi(x,y)e^{i \\left(\\omega t - k_{z} z \\right)},\n", "af4108f519890260d177f0738191ed41": " A^T=QR ", "af412f72826130124c3acb945d95f919": "L > 0", "af41455709a56ec5d05bd751c0943016": "\\begin{matrix} {r \\choose 1}{4 \\choose 3}{r - 1 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "af41a5ac08ec0f55d4b1c1652238bb21": "\\text{Cl}_2(\\theta+2m\\pi) = \\text{Cl}_2(\\theta) ", "af42684e33a712eacf2b4f6de50c5efc": " {\\rm X}_{}^{2}\\Sigma_{\\rm g}^{+}", "af42b0b85c9eeace01fa7345fff56929": "(1 - \\theta\\,i\\,t)^{-k}.", "af42c3ef8a744fcd3060cf6105c137e4": " V[\\varphi] = \\iint_D \\left[ \\frac{1}{2} \\nabla \\varphi \\cdot \\nabla \\varphi + f(x,y) \\varphi \\right] \\, dx\\,dy \\, + \\int_C \\left[ \\frac{1}{2} \\sigma(s) \\varphi^2 + g(s) \\varphi \\right] \\, ds.", "af4309bd2d82a6996825b6aff74aa3ba": " \\ K(f) \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\{ z \\in \\mathbb{C} : f^{(k)} (z) \\not\\to \\infty\\ as\\ k \\to \\infty \\} ", "af434dad36f6e110a6b9e7c50e6b95b0": " \\theta = \\arctan{\\left(\\frac{v^2\\pm\\sqrt{v^4-g(gx^2+2yv^2)}}{gx}\\right)} ", "af436c7d41011655fb905f1bf421ed56": "(N)\\;[v]\\;M\\ \\ \\longrightarrow_\\beta\\ \\ M[v := N].", "af436e109d3bdb5003b14aff214cb14a": "f(1)=g(1)=1", "af4386f0b778ca7163acc069af538446": "t\\mapsto \\frac{\\tilde{t}}{\\epsilon}\\,,\\quad r\\mapsto M+\\epsilon\\,\\tilde{r}\\,,\\quad \\phi\\mapsto \\tilde{\\phi}+\\frac{1}{2M\\epsilon}\\tilde{t}\\,,\\quad \\epsilon\\to 0\\,,", "af43a91c46bc152e31f200166c88094d": "\\omega + \\phi_m^\\prime(t).", "af43ae45f1101f0f0f8bcb4dbd65713a": "M_{t}^{D}", "af43bb8f636cccbb1e26aa24f224aefa": "\\,\\,\\sigma_{ij} = H_{ijkl}\\,\\varepsilon_{kl}\\,\\,", "af43d92b4486a506ff512cc7401d3122": "G(\\boldsymbol{x},t)", "af440fd2628b17d9c1a4094053604723": "[[x,y], z ] + [[y,z],x] + [[z,x],y] = 0", "af441d5d4db36ab78d1508dccfd24591": "|i|i", "af444e43bfe7031458ac5378a978f2e6": "PD^{2} = zD^{2} + Aw^{2}.", "af4467566998f59818e37ee69b60aa8f": "\\mathbf{j}", "af4471090f8b9290e8805c7842b67e97": "E= \\frac{GR_L+GR_T}{2}-GBL+1", "af44e70fa977b58e045c90967330bce5": "L_1min=\\frac{(1-D)^2R}{2Df_s}", "af44ea999c698fa93c1454964d8cd33f": "\\sigma_\\mathrm{tot}=\\frac{4\\pi}{k}~\\mathrm{Im}\\,f(0),", "af44ed28bcb4af6073e36805531a714f": " Q(\\alpha)=\\frac{1}{\\pi}\\langle\\alpha|\\hat{\\rho}|\\alpha\\rangle, ", "af458dfd036c3193061eb180a49d8151": "\\ddot x = g", "af45c362c5b56c01ed31beef1c481350": "\\sum_s \\langle X_s(w)X_s(z)\\Phi(v_1,z_1) \\cdots \\Phi(v_n,z_n) \\rangle (w-z)^{-1}", "af45c5905113c7ca7d9ed2d863348d16": "f_0,\\ldots,f_k \\in C^\\infty(M)", "af463fbe5161b4aa6bba5af0b2243e51": " R_x\\left(t ,\\tau \\right) = x(t+\\tau /2)x^{*}(t-\\tau /2).", "af46442dcca6a83c43ceff4a73e65acb": "\\lambda_1\\geq \\lambda_2\\geq\\ldots\\geq\\lambda_\\ell > 0", "af465df4284a125a19f37d39922bdf06": "k+1 \\geq 2", "af46907c3e2c6b2e337dfb3be248d8dc": "|\\mathbf{a}|^2 |\\mathbf{b}|^2 -|\\mathbf{a} \\cdot \\mathbf{b}|^2 = |\\mathbf{a} \\times \\mathbf{b}|^2 \\ ,", "af46adfbb004e32d4130af93b7b0685b": "\\mathfrak{h} \\subset \\mathfrak{g}", "af470e7909712e9e376b0634715ecab2": "\\scriptstyle x^3 + y^3 = z^3", "af474ea77f70bf79a837adabeddaef0d": "g_3=140\\sum_{\\omega\\in\\Lambda\\smallsetminus\\left\\{ 0\\right\\} }\\frac{1}{\\omega^6}.", "af476290cffd76451b16198d76b95e1d": "\\mathbb{T}_{ij} = \\mu\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}+\\delta_{ij}\\frac{\\lambda}{\\mu}\\frac{\\partial u_k}{\\partial x_k}\\right)", "af47e7a3209e07115b78a6d0a5e7d192": "\\hat H^0(G,A)=", "af47f3f72d7d85623b5c23d958a1cdd7": "\\frac{-b + \\sqrt{b^2 - 4ac}}{2a} \\cdot \\frac{-b - \\sqrt{b^2 - 4ac}}{2a} = \\frac{(-b + \\sqrt{b^2 - 4ac})(-b - \\sqrt{b^2 - 4ac})}{4a^2} = \\frac{b^2 - (b^2 - 4ac)}{4a^2} = \\frac{4ac}{4a^2} = \\frac{c}{a}", "af48273cd79dac585caf6cb3de58acf8": "\\sqrt{npq}", "af482845f10d558185f532883644798c": " \\ u(a) ", "af485d5c51e07644c541e908427f911a": "x_n=(7x_{n-1}+c_{n-1})\\,\\bmod\\,10,\\ c_n=\\left\\lfloor\\frac{7x_{n-1}+c_{n-1}}{10}\\right\\rfloor,", "af486e6d98dc73702915ddd695cba09d": "\\sum_{n=0}^\\infty a_n = a_0 + a_1 + a_2 + \\cdots \\,;", "af48a3d5db7210d75f0ce7c3b012a02a": " SubCipher_{n+1}=ENC_{f_n+1}(k_{f_n+1},s_n)", "af48acb6842d1eb3b1bfb46f496f9bcb": " f(i)M^{k+1}(i,j) = \\sum^{N}_{n=0} M^k(n,i)(f(j)M(j,n))", "af49691b43f94e913d8250b6b50a002d": " [e_a, e_b] = e_a e_b - e_b e_a \\ne 0", "af497b0362526d2222483b865eb26389": "\\phi(i)", "af498625853f0fd125bce9ad24b8189a": "p(\\tilde{s},\\tilde{x},\\tilde{u}\\mid m)", "af49a6e1e577b6da2e0e82921e3fcac5": "c=2 r \\sin \\left(\\frac{\\theta }{2}\\right)", "af49ae257e65741ba8f108566422076d": " E = h \\nu = h c/\\lambda ", "af4a1faab9a7846864b44558b5f8821f": "C=\\varepsilon_r\\frac{A}{4\\pi d}", "af4a39bdc6f1a1f192d4599c4041bbff": " P_0", "af4a5555017b9c8ca4ae0e72f9063add": "\\psi_n", "af4a5c0b80df0b49f5ff8c941d8df82e": "\\gamma_\\tau(t)=\\gamma((1+\\tau)t)", "af4a5f5c2fff92a3f9ee48a81db733b2": "E_1=E_2", "af4a619d65e85d8a0d4c2074f977988f": "V_F\\,", "af4a6d456832d52e5cde28a4c31605b0": "\\rho(\\theta, \\omega, t)", "af4b9138e44856ee53ae0faf21377139": "S(\\cdot)", "af4ba2a0ef4e4fd0ce04d2dcca846dbe": "\n\\partial_t P(\\mathbf{x},t\\mid \\mathbf{x_0})=D \\Sigma_{j=-M}^M \\partial^2_{x_j} P(\\mathbf{x},t\\mid \\mathbf{x_0}).", "af4bb97f45e4be6672c0302731ee98bb": "M(x,y) = \\frac{1}{AG(\\frac{1}{x},\\frac{1}{y})}", "af4bd51c532fc021ff3fc260ca8f2c3c": " h_i \\leftarrow principal~eigen~vector(AA^T) ", "af4cee94f17028459b3c30fa8f4b7548": "L_1 : L_2 : L_3 = 2 : 2.5 : 1 = 4 : 5 : 2. \\, ", "af4dc67cceefafa40e03af303f337e08": "X\\vee Y = (X\\amalg Y)\\;/ \\sim,\\,", "af4e9a7286ca8a834244b2f577c0ef3b": "q=\\frac{b}{a}", "af4ed42f3f98877e29e80f1036fc129c": "\\vec{\\sigma}_R = \\frac{-\\,\\vec{S}\\times \\vec{h}}{Mc\\sqrt{1+{\\vec{h}}^2}}", "af4eff7d2ed1e725273d78078c81f969": "{}\\over{\\vdash id_\\tau\\;:\\;\\tau\\to\\tau }", "af4f33f048bdc7163a529795e54acbb9": "\\delta = \\sgn (b) \\sqrt{\\frac{-a + \\sqrt{a^2 + b^2}}{2}},", "af4f5448150c44b2522734384bf22309": " V_0 = BV_0 + \\sum_{t=1}^{\\infty} { RI_t \\over (1+r)^t }", "af4f5bc3b3ac333156d351d72f21f5da": "(C,\\alpha)-\\sum_{j=0}^\\infty a_j=\\lim_{n\\to\\infty}\\frac{A_n^\\alpha}{E_n^\\alpha}", "af4f64393ded7b1189238f8caddfa10a": "\\,\\$ \\in \\Gamma - \\Sigma", "af4f7402baa22c65d278bc1028d95b51": "\\nu.", "af4f87ab4535a91f8dcd3106b4cede86": "\nH + L(\\alpha) \\ge 0,\n", "af4fa08ac43a863a5d65358436206257": "\\frac{Z_2}{E}", "af4fc232156a142a00aa5117ff9b8238": "\\scriptstyle\\ p ", "af4fc4b8af18d5835a56ac29745badc4": "m = \\min(n,2^{2^t})", "af5005466b4a80748cb5de22edf9e4e2": "\\prod_{i \\in I} R_i", "af5025341981763cb6cb4bcf9b245f21": "A_I (t)=e^{i H_{0, S} ~t / \\hbar} A_S e^{-i H_{0, S}~ t / \\hbar} ", "af502cac16692b513bd3b90387f0121e": "p_j\\geq 5", "af507e031f57bd7b15c9246c493bedcb": "(i_1,i_2,...,i_k)", "af5088e331e8058c5424299e7056aad4": "F_x - iF_y = \\frac{i\\rho}{2} \\oint_{C} \\left(\\frac{\\mathrm{d}w}{\\mathrm{d}z}\\right)^2 \\, \\mathrm{d}z", "af50ba77ffc51d0e6caefd72e7458811": "\\bold{j}_m\\cdot\\bold{\\hat{n}}= j_m\\cos\\theta ", "af50e72ac9e6c2ec2e0165edae202f15": "D(E)", "af50ed6d113c99ff7b59d6fe85653850": "F_n = \\sqrt{\\frac{n \\lambda d_1 d_2}{d_1 + d_2}} ", "af50fa0aace51d002f4fe7245e928128": "M_3 = \\frac{1}{16} \\, S^{bc} \\, S_{ef} \\left( C_{abcd} \\, C^{aefd} + {{}^\\star C}_{abcd} \\, {{}^\\star C}^{aefd} \\right)", "af51314d790941e59bce73d32e3a1fec": "4R=(1+u^2)(1+v^2)(1+t^2).", "af514239f28c903a22e5a9a2edfa280d": "f(\\mathbf{v})", "af514f1d4e850822f4fb9f8e7f87a253": "L_{k} = \\left(1 - \\alpha - \\beta\\right) \\left(1 - \\alpha - \\beta - \\gamma\\right)^{k - 1}", "af518d653eac38df25349a8a78542daf": "p,q\\in M", "af518f5c8b1e626c7aed4a2ccfc31535": "c>s", "af51c2085875dbc03bc5a99e01fbc465": "P_n'(1) = \\frac{n(n+1)}{2}. \\, ", "af51f47f0d0018b24a31bdd5c7b59315": "\\left|B\\right|_{ij} = \\left|A\\right|_{ij} \\,\\, (\\forall j; \\forall k\\neq i)", "af520f1702018c4022675dcbb6c11581": "\\frac{\\gamma \\left(m,\\frac{m}{\\Omega} x^2\\right)}{\\Gamma(m)}", "af522263d6b0070903b56923e8cfbfcd": " L(h)=-hp+\\ln(1-p+pe^h)", "af524c014b5deb0fa853790af07116a8": "A^\\mathsf{T} u^\\mathsf{T} = \\lambda u^\\mathsf{T}", "af52ab97868ce69876ab2fc9ab17af07": "L_x=L_y=L_z=0", "af52c15cf7b5411458df44a23b3aba99": "q \\ = C_h (T_{h,i} -T_{h,o})\\ = C_c (T_{c,o} - T_{c,i})", "af53078e1dc2bd672ef03febdfd98595": "\\overline\\theta^+", "af530fe10292371b10c4990dd388fb5f": " \\gamma =\\frac{ \\left( c_p \\right)_{air} * P }{ \\lambda_v * MW_{ratio} } ", "af5315c035e95afd2503300ca6c68d34": "\\sum_{j=1}^N \\epsilon_{ij} \\lambda_{0j} [M_0] \\mathbf{x}_{0j} + [\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} [M_0] \\sum_{j=1}^N \\epsilon_{ij} \\mathbf{x}_{0j} + \\lambda_{0i} [\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i [M_0] \\mathbf{x}_{0i}. ", "af533b0356f3a9efed69c92f704d9f0e": "\n\\delta F_{\\theta} = \\delta L\\sin\\phi - \\delta D\\cos\\phi\n", "af53d7a3676d7e4454e89bce644ff455": "dA_2= \\left(\\mathbf{n} \\cdot \\mathbf{e}_2 \\right)dA = n_2 \\; dA,\\,\\!", "af54128b5251c3f313f6af9341572156": "\\phi^*(x_i)", "af5446ee9627d73cb417a20abe362328": "3. \\; \\; \\mathrm{O}_3 \\; \\xrightarrow{h \\nu} \\; \\mathrm{O} + \\mathrm{O}_2", "af5453349edbc6de3719b5b0423e6da3": "\\mathbf{E}(x,y,z)\\equiv \\!\\frac{\\mathbf{F}_\\text{on q}(x,y,z)}{q}", "af54665a5eb50cf954364d66384ccb2e": "\\lor, \\land", "af54ffa63c915ccee1603320409129c1": "\\int\\limits_a^{a+1}\\log\\Gamma(t)\\,\\mathrm dt = \\tfrac12\\log2\\pi + a\\log a - a,\\quad a\\ge0.", "af5505401b40db2e0d35cff6513b86a3": "n \\in \\lbrace 0,\\ldots,N\\rbrace", "af550f9ef6e071aa35919f8ca4c2e3e2": "y = \\frac{x^3-2x}{2(x^2-5)}", "af55128780c22c8994f40ef39d842648": "\\eta(s) = \\int_{-\\infty}^\\infty \\frac{(1/2 + i t)^{-s}}{e^{\\pi t}+e^{-\\pi t}} \\, dt.\n", "af553dd663e35b19faf98f73e937e8ef": "\\sqcup\\,", "af5584bc9ba07889014f52dfd1845fa6": " \n\\lambda = \\frac {\\mbox{number of new infections}} {\\mbox{number of susceptible persons exposed} \\times \\mbox{average duration of exposure}} \n", "af55c60a971794b6b3009cfb234bebc2": "\\left(x,y\\right)\\to U\\left(x,y\\right)", "af55db0fa43065ea38a4c27f3b414511": "H=\\sum_{\\lambda}\\sum_k \\left[ \\Omega_{+}(k)C_{\\lambda + k}^+C_{\\lambda+ k}+\\Omega_{-}(k)C_{\\lambda - k}^+C_{\\lambda - k}\\right]+const", "af561837b3e9d9869243e401c5d655d6": "W_{A\\to B} = -\\int_{V_A}^{V_B}p\\,dV", "af563036fcb170bd2ae3040167108319": " \\scriptstyle\\theta_1=\\frac{\\mu}{\\sigma^2}", "af568a1afda8fa9ac53b76adca26e282": "f_{xy}(1,0) = p_{xy}(1,0) = a_{11} + 2a_{21} + 3a_{31}", "af56d56ca46476abb399aa2d5fb9b4ea": "\\mathcal{G}= ", "af56f7b6fda3745025f3f4f719e05c6a": "C = span\\big[\\big\\{|j_{k}\\rangle\\big\\}\\big]", "af570d8de92076a82e3178e2aa41d2a4": "\n\\begin{bmatrix}\nL\\\\M\\\\S\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0.7328 & 0.4296 & -0.1624\\\\\n-0.7036 & 1.6975 & 0.0061\\\\\n0.0030 & 0.0136 & 0.9834\n\\end{bmatrix}\n\\begin{bmatrix}\nX\\\\Y\\\\Z\n\\end{bmatrix}\n", "af57aa245c85ecb6c92e3a9075946486": "d(x,y)=0", "af57ada2b56c7fce8501698ff15bbb73": "\\displaystyle{f(e^{i\\theta}) =\\zeta \\overline{f(e^{i\\theta})}}", "af5819c79f3c52c22467c9fb9418cfa3": "\\frac{1}{\\sqrt{2}} \\approx 0.707", "af58359f0e5ba811182dccb1575f99cb": "\\rho\\rightarrow\\lambda^{-1} \\rho,", "af586c0ef606c29bbf4e1d45ab868751": " \\mathit{n} ", "af58d2e96caec842a3731b14193e3125": "O(n^\\epsilon)", "af58e83e955e6d5b1072ef066c7f5301": "\\beta\\gg 1", "af59130f287fcb379e57b8619c0c5c40": "B_2 \\cong C_2", "af592561ede09a7309b94e72f70c26fe": " \\mu(X) < \\infty ", "af595b244e7144c1da1408c33bc6adbb": "(Q,\\le_Q)", "af595f170a3233492d2d044b086e979a": " \\overline{A_{mn\\mu\\gamma}} = \\frac {A_{mn\\mu\\gamma}}{A * I} ", "af59913b4744d262b6b042cdb176ccd9": "\\hat e_x, \\; \\hat e_y", "af59957eaa94df690b4bb6463127cc7e": " a = m^2 - n^2 ,\\ \\, b = 2mn ,\\ \\, c = m^2 + n^2 ", "af59adeb71e82c5a40394cca48321705": "p = 11, p = 12. ", "af59eea08f57f690ff2b77796c75eea0": "\ng(\\lambda A_1 + (1-\\lambda)A_2,\\lambda B_1 + (1-\\lambda)B_2 ) \\geq \\lambda g(A_1, B_1) + (1 -\\lambda)g(A_2, B_2).\n", "af59f96e4d51f1b667ce4f3811fa6325": "d(x, y) = \\inf\\int_0^1 \\sqrt{g(\\dot\\gamma(t),\\dot\\gamma(t))} \\, dt,", "af5a06f376ca2ee185a21a8e4105361f": "FL(KL_i,x)", "af5a1d309c9503c68a6728d50303165d": "\\tilde{E}^a_i \\tilde{E}^{bi} = det (q) q^{ab}", "af5a22dc4fb7daacb50e91d2d2e318cd": "h_f = L \\cdot \\frac{10.67 \\quad Q^{1.85}}{C^{1.85}\\quad d^{4.87}}", "af5a65dd6181403ed93ec3399ec7f085": "\\forall a, b \\in E, (a,b) \\not\\in R", "af5a72a594779d8e948885ec5fad59e5": "L_{\\mathrm{I}}(x_0,\\dots,x_n) = n!\\cdot\\exp[\\ln x_0, \\dots, \\ln x_n]", "af5af0350b55deca7e51e7fcf194bba5": " \\frac{1}{\\xi} = \\frac{\\ln x - \\ln y}{x-y} ", "af5b17387c02d55edd34843f6565a486": "\\forall i\\, [ |R_i| = K ]", "af5b5475a28db6472afccb3631f819fd": "P_T=K(T(R)-T_0)\\,", "af5b931826135983d944ca8b13c74410": " R^\\ell {}_{m;\\ell} = {1 \\over 2} R_{;m}. \\ ", "af5bc5e6142e5b53dce960588756bf2f": "5^2 + (2 \\times 5) + 3", "af5c2fda01072933a38a725400c42217": "V=\\frac{apd}{\\ln(pd) + b}", "af5c31d6b551dc13ba02ed8070bb5c32": "k_r^->0", "af5c484265a1a52d31b88d356dbb2d42": "q_i \\circ q_j^{-1}(u_j,f_j) = (u_i,f_i)", "af5cc561a3b504086658708dbeacfdcd": " \\text{subject to: }\\ ", "af5cec9067fd0a432ec8a74f214608ce": "\n\\leq \n\\left\\| \\sum_{j=1}^{N_\\varepsilon} a_j - A \\right\\|\n+ \\sum_{j= S_{\\sigma,\\varepsilon} }^{ L_{\\sigma,\\varepsilon} } \\| a_j \\|\n\\leq \n\\left\\| \\sum_{j=1}^{N_\\varepsilon} a_j - A \\right\\| + \\sum_{j= N_\\varepsilon + 1}^{\\infty} \\| a_j \\|\n< \\varepsilon\n", "af5cfb02cc7c834feaf13ab3db152d2e": "\\sum_{k = 0}^\\infty a_k x^k", "af5d08c6c2e1254e8fbea6f8cea12018": "\\tau_\\gamma : E_x \\to E_y\\,", "af5d2040ede27ac04a8df6e3192993f0": "(\\forall x \\; \\psi(x) \\to \\exist y \\; \\phi(x,y)) \\to \\exist f (\\forall x \\; \\psi(x) \\to \\exist y,u \\; \\bold{T}(f,x,y,u) \\wedge \\phi(x,y))", "af5dee341893b26c3fadb7a026c2f9c9": "(s_{x_0}, \\ldots, s_{x_n}) \\in V_F", "af5e38d94311bde7c0cff178433c44f9": "\\varphi(3233) = (61 - 1)(53 - 1) = 3120", "af5ed63ff7fd4f368e47fc75c8acbce0": "\\mathbf{B} = \\mathbf{H}+\\mathbf{M}", "af5f2cb6318d9e89f01371454fa6a72c": "\\biggl(\\frac{1-p}{1 - p e^{i\\,t}}\\biggr)^{\\!r} \\text{ with }t\\in\\mathbb{R}", "af5f41ef943226f415d3651c10511e5e": "= -\\frac{\\sigma_n V_{Hn}t}{IB}", "af5f6f5a01c5d72dad34dbcd2d05b3e4": "\\nabla f(x_0+\\Delta x)=\\nabla f(x_0)+B \\Delta x", "af5f8c359108a0074cbf08279cf67d16": "c_b=k_e\\left(1-\\frac{Z/Z_n}{Y/Y_n}\\right)=k_e\\frac{Y/Y_n-Z/Z_n}{Y/Y_n}", "af5f8d4920f35672fa4d638686ef035c": "\\frac{(6+5)-(13+11)}{35}=-0.43", "af5f9d28c29b86501909400687a667b4": "V(\\partial_t,\\nabla_x) := \\frac{\\partial \\varphi}{\\partial t} \\frac{\\partial}{\\partial t} - c^2(x)\\sum_j \\frac{\\partial \\varphi}{\\partial x_j} \\frac{\\partial}{\\partial x_j}", "af5faa8542168fb6d96f81148019676c": "Q * 2 = 24; 24 - 13 = 11 = J", "af5fab9906fdba8c7010a3bee44ad56e": "p^2+q^2=c^2+d^2+2ab.", "af5fc7b19d9e35952d97c070e9dc6cc3": "m\\Omega=n_pm=(m-1)n_s+\\dot\\omega_s", "af5fcef44f2d94e8c932d689633ef74c": "\\!V \\approx \\sum \\pi y^2 \\cdot \\delta x.", "af603996193f8da592fffa36cca27970": "h(z)", "af605906c9c2175a261311affd07fde9": "\\forall i \\in I \\setminus X, \\forall x \\in X, x 3 ", "af60da49eae86743f168a8e1e2d663da": "\n\\begin{align}\nm(\\beta)&=m_p-aE\\left(\\frac{\\pi}{2}-\\beta, \\frac{2\\sqrt n}{1+n}\\right)\\\\\n&=\\frac{a}{1+n}\\sum_{j=0}^\\infty\\left(\\prod_{k=1}^j\\bar{\\varepsilon}_k\\right)^2\\left(\\beta+\\sum_{\\ell=1}^{2j}\\frac{\\sin 2\\ell\\beta}{\\ell}\\prod_{m=1}^\\ell\\bar{\\varepsilon}_{j+(-1)^m\\lfloor m/2\\rfloor}^{(-1)^m}\\right),\n\\end{align}\n", "af60e27e81b78f2d112cd057e7bd5085": "\\varphi:M\\to \\mathbb{R}^k\\ ", "af60f33839c779c2f50a1d96a5cb63b1": "\\beta = \\int^{\\infty}_{G_0} N(\\rho, \\rho)dG = \\Phi [\\sqrt{\\rho} - \\Phi^{-1}(1 - \\alpha)]", "af6155ad4fcd90276970b1d6e97861ee": "\\mathbb Q[\\sqrt[3]2]", "af6165bb0f3705e2c7c55a4b84268f67": "\\displaystyle r_a=s-b", "af618ce60d26d8929c95dbfbf53a1657": "\\int\\limits_{-\\infty}^{\\infty} \\text{sech}(x)dx = \\pi \\!", "af619a75053633c204fdcd9e30dd9330": "\n\\begin{bmatrix}\n u_{1j} , & u_{2j} , & \\ldots, & u_{i-1,j} , & u_{ij} , & u_{i+1,j} , & \\ldots , & u_{mj}\n\\end{bmatrix}^{T}\n\n", "af61a7e3066eda6312a712d61642a518": "F_{1},...,F_{N}\\,", "af61be0a901c6dd41d80eb8dd5bea931": " A \\cap L^p_- = \\{0\\}", "af620b7a410e2ef57195b500eedcfe95": "\\underline{a} \\in U, \\lambda \\in \\R \\Rightarrow \\lambda \\underline{a} \\in U", "af622c1e3ef6fb0966eaa89741385c4d": "\\mathbf{u} \\cdot \\nabla = u_x \\frac{\\partial}{\\partial x} + u_y \\frac{\\partial}{\\partial y} + u_z \\frac{\\partial}{\\partial z}", "af624c2f97e4da308ae462567b7fcfaf": "\\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{F} f\\right\\}(-is).", "af626d1769188eba3d6721a4220e144a": "f(x_0+\\epsilon) < f(x_0) + (K/2)\\epsilon < f(x_0),\\,", "af62748c88042d4dfdb11ce9772000bd": "\\mathit \\Gamma = 1\\,\\!", "af629d68230489996ada0f63f25a87eb": "\nr = \\sqrt{x^{2} + y^{2}}\n", "af62ad26c03e674ed38a160278e1ee9d": " \\mathbf{P} (X \\le a) \\leq \\min_{t>0} e^{ta} \\prod_i E[e^{-tX_i}] . ", "af62f3101f1ec9f17f9ecffb74066a5e": "\\scriptstyle A\\; \\subset \\;B ", "af632437a046e91e1ab982d34a4dbd0c": "s = \\int_a^\\phi \\sqrt { 1 + y'^2 }\\, d\\phi", "af634079dcdcc724a4e8a34c384f188b": "C(X_1, X_2, \\ldots, X_n|Y=y) \\equiv \\operatorname{D_{KL}}\\left[ p(X_1, \\ldots, X_n|Y=y) \\| p(X_1|Y=y)p(X_2|Y=y)\\cdots p(X_n|Y=y)\\right] \\; .", "af635dc1804917640aa92f16aab7e67d": "\\ D_L = \\infty", "af63aa9edf1a785c8bcaa659fdcecaa3": "\\operatorname{codim}(W) = \\dim(V/W) = \\dim \\operatorname{coker} ( W \\to V ) = \\dim(V) - \\dim(W),", "af63c06ed718e5aca1bd2d774f8fb97a": "10^{40}", "af63db8cd5b497ab62b3beb743bb9b9c": "(x, y, z, t)", "af63e6bc9415b4d1c07b935e1cd8ce49": "\\mathbf{J}_i=-D_i[z \\nabla c_i - c_i \\nabla z]\\, . ", "af63fa6d247b18d7a2c3dc3d4c825eb1": "\\phi_\\lambda", "af6409cc82d288d421f94e30a98fee74": "[x:=e] \\Phi(x) \\equiv \\Phi(e)\\,\\!", "af641c6a59282d6d76766e6dfa42d113": "4_0", "af645a1d5e0fbebea7185afb61a60ef8": "T = \\frac{I}{I_{0}} = e^{-x/\\ell}", "af6491afe7eba45333a4f66e0ceb60b5": " a \\otimes_B 1 = \\sum_{g \\in G} t_g g(a)", "af64a28ec3c55f27efa13339ca4fba08": "a^{-1} \\equiv a^{m-2} \\pmod{m}", "af64cf1204edd065a201a664aea1f641": "H = \\frac{(p)^2}{2}+\\frac{{W}^2}{2}+\\frac{W'}{2}(bb^\\dagger-b^\\dagger b)", "af64e839bce022cca9a9ff4507b2db05": "F: C_0\\rightarrow C_1", "af6568d44c4f8bc5827a805e6d58cc3c": "\\frac{\\zeta_\\text{max}}{r} \\ll 1", "af6582c0ddbd72628fe2aa75b0bd7388": "f(x) = \\begin{cases}1 & -1 \\le x < 0 \\\\\n\\frac{1}{2} & x = 0 \\\\ 1 - x^2 & \\text{otherwise}\\end{cases}", "af6583d73c425d6942b524c84f7e75b8": "\\mathcal{I} \\models \\mathcal{K}", "af65a0c64a82f7c77ef3c5aaa3a98d29": "Spec \\oplus_{n \\geq 0} I^n / I^{n+1}", "af65d09f7eeff56eb4bfde5b737875a5": "\\lnot, \\land, \\lor, \\to, \\leftrightarrow", "af6626101d2f33b876450d44885a68d2": "\\sum_{i=1}^n a_i b_i = 1", "af66548439169a67296153c4a0d52aa9": "T(\\rho,\\sigma) = \\frac{1}{2} \\mathrm{Tr} \\left[ \\sqrt{(\\rho-\\sigma)^2} \\right] = \\frac{1}{2} \\sum_i | \\lambda_i | , ", "af6655704d940c7f97ec7165ad8523ce": "(AA^+)^* = AA^+\\,\\!", "af66ffc4647f1403b22facb074940587": "ij=-ji=k", "af673b847ae14b2231328eb2cf4cbde5": "45^\\circ", "af67448c1ee378fbe210db36f5d6ceca": "\\overline{\\mathrm{Nu}}_L \\ = 0.27\\, \\mathrm{Ra}_L^{1/4} \\, \\quad 10^5 \\le \\mathrm{Ra}_L \\le 10^{10}", "af674e69e269100a579027e16f3685c3": "a_2, a_2\\zeta_n, a_2\\zeta_n^2, \\dots, a_2\\zeta_n^{n-1}", "af67777f97c0de945409789275b1f15c": "V_\\max1 - (V_\\max1 - V_\\max2 ) \\cfrac{[I]}{[I]+K_i} ", "af678bb643b99350213529a90b61f722": "A = \\frac{1}{2} \\times \\frac{5t}{1} \\times \\frac{t\\tan(54^\\circ)}{2}", "af6792d6a98b35e8841bfae9b9b6d7bc": "\n\\frac{\\partial^2}{\\partial A^2} \\ln p(\\mathbf{x}; A)\n=\n\\frac{1}{\\sigma^2} (- N)\n=\n\\frac{-N}{\\sigma^2}\n", "af67aa6726412e375494c22baaf5738d": " d_\\lambda f (x) = \\varphi_\\lambda(x) \\mu^{(0)} + \\chi_\\lambda(x) \\mu^{(1)}.", "af67b41f3b111282038bc72417203222": "\\begin{bmatrix}1&0&0\\\\\n0& \\frac{1}{\\sqrt{2}}& \\frac{-1}{\\sqrt{2}}\\\\\n0& \\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}}\\end{bmatrix}\n", "af67ce72e0fbd0c9172d5a555de7714f": "F_n = F_w \\cos \\theta \\,", "af6808282f9861e27270f0d5cf09e87a": "KC(\\mathcal{S} \\cup T(\\mathcal{M}, \\theta)) = KC(\\mathcal{S}) + KC(T(\\mathcal{M}, \\theta)) - 2 \\operatorname{cost}(\\mathcal{S}, \\mathcal{M}, \\theta)", "af6837f60b568c2abf8932e5890a5ad2": "f(x)=\\sum^n_{i=1}x_if(e_i),", "af684073439b4ca1dd02c1b0f8df6c7b": "J_q(n,d,e)\\le qnd", "af687a3f98aa1a1310d2728d82cdae6e": "\\displaystyle{d(x,y)\\ge \\|x-y\\|/R.}", "af689d98166030abe34551468c86759b": "\\Re [S(x)]", "af68d2d31e3b3676aa171f4f4f6a33f7": " \\theta = am^{ b - 1 } ", "af6918be130b7896f8fba57c0968c376": "2000000 \\sin\\frac{\\pi}{1000000}", "af6b414db0d7a8b0b2fd4a91c7e5a3da": "h \\propto Q^{0.4}", "af6bd125dc1548826f2f321ad0fa9519": "KS(x) \\leq \\ell(x) + 4", "af6bd9dfbbc4d19db6e1c6299566839a": "\n\\begin{align}\n\n & \\begin{aligned}\n (\\nu x) \\; & ( \\; \\overline{x} \\langle z \\rangle . \\; 0 \\\\\n & | \\; x(y). \\; \\overline{y}\\langle x \\rangle . \\; x(y). \\; 0 \\; )\n \\end{aligned} \\\\\n| \\; & z(v) . \\; \\overline{v}\\langle v \\rangle . 0\n\n\\end{align}\n", "af6bdbc19b925bb2cd63f58be669ab77": " U(t, t_0) = e^{-iH(t-t_0)/\\hbar},", "af6bef47c78ea4e7592376c963712aba": "\n\\Phi(a) = V \\pi (a) V^* = \\langle V \\pi (a) V^* 1, 1 \\rangle _H = \\langle \\pi (a) V^* 1, V^* 1 \\rangle _K\n= \\langle \\pi (a) \\xi, \\xi \\rangle _K\n", "af6c0ab0cbba5b04d114132cb1c553d1": "\\begin{align} g(x) &= \\frac{1}{\\sqrt{2 \\pi \\hbar}} \\cdot \\int_{-\\infty}^{\\infty} \\tilde{g}(p) \\cdot e^{ipx/\\hbar} \\, dp \\\\\n&= \\frac{1}{\\sqrt{2 \\pi \\hbar}} \\int_{-\\infty}^{\\infty} p \\cdot \\phi(p) \\cdot e^{ipx/\\hbar} \\, dp \\\\\n&= \\frac{1}{2 \\pi \\hbar} \\int_{-\\infty}^{\\infty} \\left[ p \\cdot \\int_{-\\infty}^{\\infty} \\psi(x) e^{-ipx/\\hbar} \\, dx \\right] \\cdot e^{ipx/\\hbar} \\, dp \\\\\n&= \\frac{i}{2 \\pi} \\int_{-\\infty}^{\\infty} \\left[ \\cancel{ \\left. \\psi(x) e^{-ipx/\\hbar} \\right|_{-\\infty}^{\\infty} } - \\int_{-\\infty}^{\\infty} \\frac{d\\psi(x)}{dx} e^{-ipx/\\hbar} \\, dx \\right] \\cdot e^{ipx/\\hbar} \\, dp \\\\\n&= \\frac{-i}{2 \\pi} \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{d\\psi(x)}{dx} e^{-ipx/\\hbar} \\, dx \\, e^{ipx/\\hbar} \\, dp \\\\\n&= \\left( -i \\hbar \\frac{d}{dx} \\right) \\cdot \\psi(x) ,\\end{align}", "af6c4d923c614301f27da634a3488609": "\n S_1 = \\cfrac{I}{c_1} ~;~~ S_2 = \\cfrac{I}{c_2}\n ", "af6c54b42ea45afe997662fce8d96ca3": "\\sigma[N \\setminus(B_\\delta(x))]", "af6c6856fd1ae6101bdd42208c4d2300": "Y_{0,1}", "af6c6fa73b2c53e1e2e943af2d97146a": " \\omega=2.1e^{-\\lambda\\sqrt {-(x^2+y^2)}}", "af6c82bcf4506f6caa6514e7b139ebb9": "K(k) = \\int_0^{\\pi/2} \\frac{d\\theta}{\\sqrt{1-k^2 \\sin^2\\theta}} = \\int_0^1 \\frac{dt}{\\sqrt{(1-t^2)(1-k^2 t^2)}},", "af6c958abc3bf0519fe89b695fa43dd3": "\\textstyle i^{th}", "af6cb909351deb564bb6eaeed94afe24": "\\mathbf{J} = \\epsilon_0 \\int \\mathbf{r}\\times\\left(\\mathbf{E}\\times\\mathbf{B}\\right) d^{3}\\mathbf{r} ,", "af6cde61a3193f47aff7fd3bd58c5f9a": " \\phi: K[X]\\to A, \\quad \\phi(X)=a.", "af6ce256eba8461cd51134b9a9ee63a1": " \\; {}_0F_1(;a-1;z) = (\\frac{\\vartheta}{a-1}+1) \\; {}_0F_1(;a;z)", "af6d24494b6dfb4dd950129471bb52ff": "10\\uparrow\\uparrow\\uparrow\\uparrow 6=(10 \\uparrow \\uparrow\\uparrow)^6 1", "af6d32cc27bb5bfd2f032be3b4b3f3b9": "E/\\mathbb{F}_{q}", "af6d490435569a1fcdc53a9c5adcf0d6": "c^{-1}(x) = 11x^3 + 13x^2 + 9x + 14", "af6dbf74d653d712946cc0ad6ae57d2d": " b_{-n} \\equiv x_n (n>0) ", "af6dd9d337228e8edf1a23505c3cc74f": "1^k, 2^k, 3^k,\\dots", "af6de337f0d23e2f253c1073c624c2f4": " X^k\\, X^l = X^{k+l},", "af6de3ef393b4f48057f5b14b797ac07": "\\pm1", "af6e092132e5d020a811e8e2e64ada81": "\\{\\pm( 1,0),\\pm(0,1),\\pm (1,1)\\}", "af6e1d8cb44f5882e9401d7ac8ba53f9": "(j^{1}_{p}\\sigma)^{*} \\theta \\,", "af6e796d9633e2b9d1c480617f8329e5": "\\omega^{\\beta} (c+c')", "af6ebd20eac9d3ec9fff46026271b11f": "\n\\sum_{\\delta\\mid n}d\\left(\\frac{n}{\\delta}\\right)2^{\\omega(\\delta)}=\nd^2(n).\n", "af6f06f4c4614ce42b8f1344bbf88e16": "\\frac{\\partial \\boldsymbol{\\hat{\\theta}}} {\\partial \\theta} = -\\sin \\theta \\cos \\varphi\\mathbf{\\hat{x}} - \\sin \\theta \\sin \\varphi\\mathbf{\\hat{y}} - \\cos \\theta\\mathbf{\\hat{z}} = -\\mathbf{\\hat{r}}", "af6f41eb05ebf9b60e631003500fcce8": "\\tilde{\\omega}", "af6f421c52246780c9bba295512e272f": "H(X_{i})", "af6f44b4120942cbf6ecdfd6a41afca0": "\\beta = \\frac{\\gamma - 1}{2 \\gamma}", "af6fb038732240438443433f7534f827": "\\oint_{C} \\mathbf{v}\\cdot\\,d\\mathbf{l} = \\frac{\\hbar}{m}\\oint_{C}\\nabla\\phi\\cdot\\,d\\mathbf{l} = \\frac{\\hbar}{m}\\Delta\\phi,", "af6fbfa30c8307f162559808aac1e010": "\n(r, x, y, s) * (t, z, w, u) =\n\\begin{cases}\n(r, x-y + \\max(y , z + 1), \\max(y - 1, z) - z + w, u) & \\text{if } s = 0, t = 1\\\\\n(r, x - y+ \\max(y, z), \\max(y, z) - z + w, u)&\\text{otherwise.}\n\\end{cases}\n", "af6fd9103af97919888cffac59ba9a3a": " f(x)= \\frac1{2X_{max}}", "af6fe13e9a4fada5d1e23f94ecd3a247": "\\varphi_\\alpha(g) = \\int_K \\alpha^\\prime(gk)^{-1}\\, dk ", "af701521c33ff37b8e272f6f41a6f7c0": "2n\\pi", "af70334ae09f014057c91e1056ee7d8b": "\\varphi=\\exists y(y\\cdot y\\equiv x).", "af7054547c72e6aac5bb4e5eaebd91ae": "\\mathrm{2 \\ H_2O + energy \\longrightarrow 4 \\ H + O_2}", "af706df0d8eb3455f45230ee34254ae8": "\\Delta \\ln(A_t)=x \\Delta \\ln(S_t)+(x-x^2)\\sigma^2\\frac{\\Delta t}{2}", "af70b579ee7aa52285df1890c5653311": "k_2 = f(t_2 + \\tfrac23h ,y_2 + \\tfrac23hk_1)", "af710944cde0c57148a35d0e4606bf3e": "|T|\\geq\\gamma q", "af716e3ec1ca078d1404ea1afd6b6cfc": " \\oint {\\chi(\\omega') \\over \\omega'-\\omega}\\,d\\omega' = 0 ", "af7172d38ceebe2ce414b79d5936471c": "\\frac{5\\cdot\\pi}{3(\\sqrt{6}+\\sqrt{2})}", "af71823d5e973dd6e33c1815e33daa6c": "\\scriptstyle \\gamma", "af7184c40ff80aeda8407ca49b4e5aab": "D \\supset D'", "af719c4c11c2fd941dd7f6393c8f75d8": "\\bigcup_{i = 1}^{n} B_{i} \\in B.", "af721080099a1e47c47dd537f3d5800e": "\\varepsilon _{eff}", "af7266ee47b43925c92ef02a73850310": "X\\cup_f Y = (X\\amalg Y) / \\{f(A) \\sim A\\}.", "af7290ef3c4c5fd03c1f6a360587f390": "\\mathcal{E}_\\land(y,z)", "af72a9f7571f17b51791c49fa6e800e9": "P(t) = e^{-t/(\\gamma \\tau)} \\,", "af72e313f94778c817a9b062911530e4": "v_j = \\frac{1}{n}p_f(\\alpha^{-j}).", "af72e5dc8af87a2580b23fbf92c543f6": " q ", "af73024a54a91c6424ec5935b70d6300": "t>1", "af73236f5e9438a12c41f5b64e3e4e6a": " \\frac{7(b-a)^5}{23040}f^{(4)}(\\xi) ", "af7335a77cb118122a387e2f30cd5d72": "j = C*b", "af73499b835f9a9013a43069f2497b25": "T-\\partial T", "af7365870258aa9b25c48c393818ffc1": "p_j(x)", "af73706defd21d1aa5be858aebf20dea": "\\left(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}\\right)\\,\\!", "af73a43b16fe6723b8ca4f37a43fdfee": " \\mathbf{A} \\mathbf{F}= \\mathbf{0} ", "af7433d3216175f8e25d223f96b1307a": "\\bar{x} \\pm t_{0.05,n-1} s\\sqrt{1+\\frac{1}{n}}", "af745907a51ba3c10ff2e82531c8c8a3": "\\frac{\\partial\\Phi}{\\partial z} = \\frac{\\partial\\eta}{\\partial t}", "af74662a2979d1791f911724aaae8808": "\\phi \\mapsto \\psi", "af746df6afa634d006d703f56ffb1943": " B_s=EI\\, ", "af74d136f05e8bd469a19a234ef39ea2": " M u", "af74e033e009b3c6d3c22dc50342b3b0": "J:V \\to \\bar{\\mathbb{R}}", "af74e0d0b9cf89b8d2a67442200ac0af": "\\scriptstyle\\leq(-3.0\\pm2.4)\\times10^{-15}", "af7507208e062ca0f9ad4246febcbb91": "\\frac{b^{p - 1} - 1}{p}", "af75074106bd59bf399a3b64678302af": " \\delta = \\begin{bmatrix} \\beta_i & \\gamma_i\\end{bmatrix} ", "af751e6825b045c1c14a2f42c221212b": "\\sqrt{5} = e^{2 \\pi i / 5} - e^{4 \\pi i / 5} - e^{6 \\pi i / 5} + e^{8 \\pi i / 5}.", "af7534411735f1a62df8ba0f9b974f9c": "N=m^n", "af755e7605c4e2208a7534312a62018b": "\\mathbf{R}=n_1\\mathbf{a_1}+n_2\\mathbf{a_2}+n_3\\mathbf{a_3}", "af757f5cdf9719119441d67284ced643": " \\mu_{app} = \\frac{\\mathrm \\mu_r}{\\mathrm 1+N_z\\times(\\mu_r-1)}", "af75dbde4f6905ba01fa03ab582a6d6b": " x(t)\\,", "af75e668a8eba5333d1c1534487a3df6": "\\theta^j", "af7668ffc92642cc7d948ca137c2e35e": "\\forall a \\, \\forall p \\, [(Ma \\and \\forall x \\, [x \\in p \\leftrightarrow \\forall y \\, (y \\in x \\rightarrow y \\in a)]) \\rightarrow Mp].", "af77189c62259f178f66e159e05c6f6f": "\\left\\lbrace\\Psi_i\\right\\rbrace", "af771bbd76df286b26d5999661d049d1": "R = r/R_E", "af7750d633ea163e48997548be984aa3": "w(\\theta \\circ \\phi)=F_\\circ(w(\\theta),w(\\phi))", "af7779f4945e645a80f51cbca128eee8": "\nY=Y(K',L').\t\t\t\t\t \n", "af778f155d5d93b7af7d70789d25d264": "{{\\mathbf{K}}}", "af77975889b59f536f380d4aeeb649c4": "\\pi: Y\\to \\Sigma\\to X \\qquad\\qquad (1)", "af7837487cd841da77372eba919fda9b": "\\frac1e=e^{-1}=\\sum_{n=0}^\\infty\\frac{(-1)^n}{n!}\\cdot", "af7877e7302cef552749eb8232857bcc": " \\mathbf{W} ", "af78a0f6d92416c14f9caf2b4fd05e13": " \\mathbf{J}_{\\rho} = -D\\,\\nabla\\rho", "af78bb8159f3c25c6ceec3bc3864dea8": " r(d) = R\\,\\tan (d/R)", "af78d9f26a8e3f06d6be213868981e89": "M \\sim N \\iff N \\sim M.", "af790a4306f34da2301cf7eb2ae9085c": "B(\\boldsymbol{u},\\boldsymbol{v}) = -\\int_A \\sigma_{ik}(\\boldsymbol{u})\\varepsilon_{ik}(\\boldsymbol{v})\\mathrm{d}x", "af791e32cf9615390284f8f22f071987": "\n\\mathbf{P} = -\\frac{\\partial G_{1}}{\\partial \\mathbf{Q}}\n", "af793891875d401a6ea4f0957f0ea248": "K[[T^\\Gamma]]", "af7941f6ab7bdb8537b46134a1be77c3": " CS \\, ", "af79596a535b2194f8ce0188a35f9427": "L(n) = 2 F(n+1) - 1, n \\ge 0", "af7961b19a613395ffa273055ff38dda": "P = A_s \\epsilon \\sigma T_H^4 = \\left( \\frac{16 \\pi G^2 M^2}{c^4} \\right) \\left( \\frac{\\pi^2 k_B^4}{60 \\hbar^3 c^2} \\right) \\left( \\frac{\\hbar c^3}{8 \\pi G M k_B} \\right)^4 = \\frac{\\hbar c^6}{15360 \\pi G^2 M^2} \\;", "af79716b282215e75e4aeb49dac652d1": "5_{-2}", "af79e98773c45b1f81acca2600476221": " S = k_B \\ln W \\, ", "af7a3de35a95787aa4e97b346fc8d8cb": "\\gamma \\rightarrow \\sqrt{ZY}", "af7a9538723fc836d5932688d6634260": "\nF(\\rho, \\sigma) = \\sqrt{\\langle \\phi | \\psi \\rangle \\langle \\psi | \\phi \\rangle}\n= | \\langle \\phi | \\psi \\rangle |.\n", "af7b687ef16ffb8ff0aa36b42131bd60": "d_{2} = f \\circ d_{1}", "af7b68a00bd6cea66129f7c61f3d83d2": "\\underline{u} : \\underline{A}", "af7b88d59a4636fc733e38d6c430f047": "\\begin{align}\nx&=r\\sin\\theta\\cos\\phi \\\\\ny&=r\\sin\\theta\\sin\\phi \\\\\nz&=r\\cos\\theta\n\\end{align}", "af7bf029a318eb2c59d4fd429cd009c9": "R = R'.x", "af7c2a0baec1250d0a34075f637d187f": "\\mathbf{v}=\\mathbf{p}_A-\\mathbf{p}_B", "af7c65b5d28160060e0b46267b655d2e": "{O}(n^2 \\log n)", "af7c718155cbd96781f75c11f0f69656": "m_{T}", "af7ca26655e7e3ed9a9763f2eccfc0fc": "\\begin{align}\nP & \\longleftarrow & P\\times EG& \\longrightarrow & EG \\\\\n\\downarrow & & \\downarrow & & \\downarrow\\pi\\\\\nM & \\longleftarrow^{\\!\\!\\!\\!\\!\\!\\!p} & P\\times_G EG & \\longrightarrow & BG.\n\\end{align}", "af7ccd3b4e572676dd46e2caf9344704": "\\tanh \\frac{1}{2} = \\frac{e - 1}{e + 1} = 0 + \\cfrac{1}{2 + \\cfrac{1}{6 + \\cfrac{1}{10 + \\cfrac{1}{14 + \\cfrac{1}{\\ddots}}}}}", "af7d779dd2103235f6bfe6aa887be4f1": "D=\\frac{ct}{2}", "af7d970d5cec1284cc35dbaa994be2f6": "\\mathbf{R}^2 \\times [0,1]", "af7da99a5b8b2f3a9c39843e9d62b250": "P' = (Q_L\\cdot P)\\cdot Q_R = Q_L\\cdot (P\\cdot Q_R),\\,", "af7dab6eea5028da120ff969e5f120ee": "J(\\mathbf{w})", "af7e48399a140a617b755da7e6109b80": "\\mbox{If } \\;(q,r) = 1 \\;\\mbox{ then }\\; c_q(n)c_r(n)=c_{qr}(n).", "af7e4b927403c0c6538319210143d281": "\\chi_p:G\\rightarrow\\mathbf{Z}_p^\\times", "af7e6c402d64a18931f60366a8b7a717": "\\left(\\tfrac{a}{1}\\right) = 1.", "af7e8a85df47250ffab736b7a8807488": "\\partial T", "af7e9fded847803a95399718162eefb5": " z_{t} = \\Delta y_{t} ", "af7eacebcd4cebfbc2a7cc342b0a6946": "! x(y).P", "af7ee0dba9e5f06b87c7e5be8ded9687": " \\begin{align} \\hat{T} & = \\frac{\\mathbf{\\hat{p}}\\cdot\\mathbf{\\hat{p}}}{2m} \\\\\n & = \\frac{(-i \\hbar \\nabla)\\cdot(-i \\hbar \\nabla)}{2m} \\\\\n & = \\frac{-\\hbar^2 }{2m}\\nabla^2\n\\end{align}\\,\\!", "af7f52ecdaee73a352ce2ec761fbec31": "\\frac{1}{\\sqrt 2} = \\prod_{k=0}^\\infty\n\\left(1-\\frac{1}{(4k+2)^2}\\right) =\n\\left(1-\\frac{1}{4}\\right)\n\\left(1-\\frac{1}{36}\\right)\n\\left(1-\\frac{1}{100}\\right) \\cdots", "af7f7f51946e14d2a9c91c6d3ff893a5": "J^a=0", "af7fda3ef2aabe0941b2d05205094a5b": " \\varphi_t^2 = c(X)^2 \\nabla \\varphi \\cdot \\nabla \\varphi. \\,", "af7ff20f71e9b47f3e911cb896b57422": "F(\\mathrm{id}_{X}) = \\mathrm{id}_{F(X)}\\,\\!", "af804f7840a4eecc25d709fbba0760f6": "\n\\begin{align}\nm\\frac{d}{dt} \\langle \\Psi(t) | \\hat{x} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle, \\\\\n\\frac{d}{dt} \\langle \\Psi(t) | \\hat{p} | \\Psi(t) \\rangle &= \\langle \\Psi(t) | -V'(\\hat{x}) | \\Psi(t) \\rangle.\n\\end{align}\n", "af80709026c6dfe82ad5de548a9d934e": "-R_l^l=\\frac{(\\dot a b c)\\dot{ }}{abc}=0,\\ -R_m^m=\\frac{(a \\dot b c)\\dot{ }}{abc}=0,\\ -R_n^n=\\frac{(a b \\dot c)\\dot{ }}{abc}=0", "af808052a7b51876ff529dca329acf14": "\\mu = \\sum x P(x)", "af80bf1447dcebeb9c8b8a8c0cfc8768": "BE \\leftarrow BE \\cup {e}", "af81071231d9959764e1c805fc9f3579": "\\ | \\upsilon_M | > | \\upsilon _D | \\,", "af813a5d288e2afbce3c1cf260e3b5d3": " G(x-y) = \\int d\\tau {1 \\over t^{d\\over 2}} e^{{x^2 \\over 2\\tau} + t \\tau} ", "af8175693d19c6a9556d77d02365bc6c": "\\int_{-\\infty}^{\\infty} x^2 e^{-ax^2}\\,\\mathrm{d}x=\\frac{1}{2} \\sqrt{\\pi \\over a^3} \\quad (a>0)", "af8180a968da4fdd56a4e264652b976a": "\\mu_D", "af81a5b7e3f46c29e5300ec2994219ec": "G(x,s) = G(s, x)", "af81fd7b3dee75ba5dd97bc04cbd6e03": " f'(x) = rx^{r-1},", "af82af7b4e1d594a77b851d1c78d280c": "\\scriptstyle \\mathbb{C}^n\\equiv\\mathbb{R}^{2n}", "af82bead5982eb3167fc1c0c9f6c5ff4": "\\operatorname{B}^\\bullet: A(K) \\stackrel{d}{\\longrightarrow} \\wedge^2 K^*", "af82df5ffe588e7cf6fd41de8edfda27": "(a_k)_{k=1}^{10}, \\qquad a_k = k^2.", "af82e0410600e4120f346bbb9c676294": "-\\triangle_{S}(\\triangle_{S}+2),", "af82ef4d2978c27ddae3aa064c0cb9cc": " p \\in [1,\\infty] ", "af835d5492f88982ddd70f4133c09c7d": " R_{xx}(0) = \\frac{1}{N} \\sum_{i=1}^{N} x(i)^2.\\,", "af8373590e8794c67f3f832504bd2a50": "tI_n-A", "af83df5c3b51c86bfa7860ef718676f5": " \\mathbf{AB} = \\mathbf{A} \\cdot \\mathbf{B} + \\mathbf{A} \\times \\mathbf{B}.", "af84ab2af68a228251d6d33fe3ea40e7": "{}^{0}0 = 1", "af84ab727400dd3e8e8d5a46bc866a28": "{t_{\\alpha/2}}", "af84c69648ff48ae46ea842a3c1d7a59": "\n (a_1, a_2, \\dots, a_n) <^\\text{colex} (b_1,b_2, \\dots, b_n) \\iff", "af856156f0a64dc5bd4a3651639bfda4": "y_1(x)", "af8579de963920fb3c5470a4b69429e5": "\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n -m_1 & -m_2 & -m_3\n\\end{pmatrix}\n=\n(-1)^{j_1+j_2+j_3}\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}.\n", "af85b07518dd1e429a58b20b84f0700a": " \\Delta_3 M_0= M_0(\\Delta_2 + {3\\over 4}).", "af8640dc60754748cf9514a57fc523fd": "W_t = W e^{-\\delta F_t}", "af866a5dd0b6d3d311f564bd359b2815": "\\scriptstyle \\int_\\theta \\tfrac{1}{2} (r(\\theta))^2\\,d\\theta,", "af86a7f303a25ca6cd58f43acf6dfe85": "c_0 = \\ln\\left(\\frac{1-e^{-2\\gamma}}{2\\pi}\\right)", "af86afab5c98cbd62015158f6a0dd62a": "0 \\leq m \\leq 1 ", "af86cece7ba7cc05c52ed824424bf2fb": "q^2+1", "af86e6c6019f4f07d7a231d87cc2c1e0": "X = \\bigcup_{\\alpha\\in A} U_\\alpha,", "af8813d012ddb7761228fc4e357a3ca9": "F_n(x)={1 \\over n}\\sum_{i=1}^n I_{X_i\\leq x}", "af88759582a5b45b289b2f1e42614b2f": "\\mathrm{d}U = T\\mathrm{d}S-p\\mathrm{d}V \\!", "af8883bc30a199aea553696dfc7094a9": "|\\tau^{[3..N]}_{\\alpha_1i_2}\\rangle=\\sum_{\\alpha_2}\\Gamma^{[2]i_2}_{\\alpha_1\\alpha_2}\\lambda^{[2]}_{{\\alpha}_2}|{\\Phi^{[3..N]}_{\\alpha_2}}\\rangle\n", "af88dfc123f812c1b924409cac7cc316": "\nv_w = (1 - 2a)v_{\\infty}\n", "af8926ecb9ff8a2ec59872d13db304b1": " \\mathbf{v} = \\nabla \\varphi.", "af893a6186b4f6d8fd5e57c9d12edef8": "\\pi_1(X).", "af898973322c798414ebb81efe8f7497": "\\boldsymbol x", "af89aff5eb323284af4683ebc2b0ffa9": "m_2 \\,", "af89d69477e63a307ad6a6e4b8d42f75": "\\frac{1}{2}\\left[(\\lambda-\\mu)-\\sqrt{(\\lambda-\\mu)^2 + 4(k-\\mu)}\\right]", "af8a38a7d88cd7bbf6c0f914c2d9cb6d": "T_m(z)=t_{m,0}+t_{m,1} z + \\cdots + t_{m,1} z^{m-1}+t_{m,0} z^m ", "af8a60174012b1473ecac6f1b343360f": "\\limsup_{n\\to\\infty}\\operatorname{Pr}(X_n \\in F) \\leq \\operatorname{Pr}(X\\in F) \\text{ for every closed set } F.", "af8a68d1ebe2e3069e1b3b0d8948a1e1": "c=(v^2+u^2)^2, \\,", "af8a7728946397ee1e19f44e357856ce": "x < a", "af8a7916f39329991344a8a35b3906fd": "0 \\le \\rho \\le 3a, \\ 0 \\le \\phi \\le 2 \\pi, \\ 0 \\le \\theta \\le \\pi.", "af8aa370d7ec13f3e7fb0e0ce5fef862": "\\bar{e}+\\Delta \\bar{e}", "af8ab42080ebaf9a13dab190d9cb68bf": "\\pi_n(X, x) \\cong \\pi_n(Y, y)\\,", "af8ab72d8769b5969dd19599f1781c99": "\\frac{8}{9} \\sqrt{2}", "af8adcfc15af6edfc7614c8305cf4f7e": "\\mathbf{y}=\\mathbf{HWs}+\\mathbf{n}", "af8b2c6529ed19c1c8aa8b509163e19d": "\\begin{align}\n R &= \\begin{bmatrix}\n\\cos \\alpha \\cos \\gamma -\\sin \\alpha \\sin \\gamma & -\\cos \\alpha \\sin \\gamma - \\sin \\alpha \\cos \\gamma & 0 \\\\\n\\sin \\alpha \\cos \\gamma + \\cos \\alpha \\sin \\gamma & -\\sin \\alpha \\sin \\gamma + \\cos \\alpha \\cos \\gamma & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\end{align}\n", "af8b4b7f38f5c58e5a638b4857def7bb": "q^{(p-1)/2} \\equiv (-1)^{r(2) + r(4) + \\cdots + r(p-1)}\\text{ (mod }p).", "af8b7b20e7a90198cc49199da59603ef": " T_+=T_-", "af8bb68b58242c0f1ae197cdd7f2b259": "\\Omega_{n,k}", "af8c060fc30f7769d7cbe86daf9df6a2": "\\mathrm{Hom}(N,\\mathrm{lim}F) \\cong \\mathrm{Cone}(N,F)", "af8cc9f6427f6e6486f16ea5276511ca": "p(x_1,x_2)", "af8cdfecd8d776659ba9449c50dcb8e3": "X = \\sum_{r}P_{r}X_{r}=\\frac{1}{\\beta}\\frac{\\partial \\log Z}{\\partial x}\\,", "af8d376f7f77722ba2eca3218af5ccf6": "\n\\pi(x_k|x_{0:k-1},y_{0:k}) = p(x_k|x_{k-1}). \\,\n", "af8dd265a037f053e453a2acc9766bd4": "\\frac{1}{\\sqrt{2}}|\\uparrow\\rangle + \\frac{1}{\\sqrt{2}}|\\downarrow\\rangle", "af8e27bb4df561f14ad27662799e1b8d": "X^{\\triangledown\\vartriangle}", "af8e48a59c6a8d106ef7b3f9cb89d8a0": "\n\\binom nk_q .\n", "af8ea0491b36d31d4c07acf61cfd2cff": "X_0^2-X_1^2-X_2^2=0", "af8ea2a9df1124da72eef6de0d6f60e9": " (x_j, f(x_j)) \\in \\mathrm{Conv}(\\mathrm{Graph}(f_n)). \\, ", "af8ea48aa2eafc7a3d2bf40ed8ab41ed": "R(\\hat{n},\\phi)", "af8eae6782a8fb006df9d8fdb06f3fca": "A_{FB} = \\frac {A} {1 + { \\beta}_{FB} A} \\ , ", "af8ec36098fa02c9acbb0ab375b41f1a": "\\alpha=e^\\frac{4 \\pi \\kappa}{\\lambda_0}", "af8ed309b9da6108c5b5a282752ac47a": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t}+ ( \\mathbf{v} \\cdot \\nabla ) \\mathbf{v}\\right)=\\rho \\mathbf{f}-\\nabla p +\\mu\\left(\\nabla ^2 \\mathbf{v}\\right)", "af8eecaadfa586cf9300a1c5997d78eb": "\\mathbf{\\Phi}_{l,m} = \\frac{1}{\\sqrt{l(l+1)}}(\\mathbf{r} \\times \\nabla) Y_{l,m}", "af8f078a2fab050305628961ef4b35c5": "X_s\\ll{X_m}", "af8f4f276f8afca7f0e41d50b143dea3": "n! \\simeq \\sqrt{2\\pi n} \\left ( \\frac{n}{e} \\right )^n ", "af8faf538b0f66d2a474a5167ded6f75": "x\\in V(N)", "af901af1e651b2271dc90915b2de84ea": "\\inf \\frac{a_n}{n}", "af903da11c11725cf34d5ae4803cd054": " W = \\gamma \\Delta A \\,\\!", "af91608fc03b1f44a6eedf5cabc20a95": "Q = - \\frac {1}{2m} \\frac {\\quad \\Box \\sqrt \\rho}{\\sqrt \\rho}", "af91db8dee647b5f9583bfbdbc6ed8b3": " (\\kappa+n-1)~r^{-n+1}~\\sin(n\\theta) \\,", "af91f0569bb88e6d77dc39dca085dd48": " \\tan\\left( \\frac{\\alpha \\pm \\beta}{2} \\right) = \\frac{\\sin\\alpha \\pm \\sin\\beta}{\\cos\\alpha + \\cos\\beta} ", "af929a0439812c828ed705cb16985a30": "i: A \\to X", "af92ab3e06622a18648ab3720680325e": "dA = -S dT - P dV", "af9303ff1f5bba174712a161047779a2": "H(z) \n= \n\\frac{z^{-1}-\\overline{z_0}}{1-z_0z^{-1}} \\times \n\\frac{z^{-1}-z_0}{1-\\overline{z_0}z^{-1}}\n=\n\\frac {z^{-2}-2\\Re(z_0)z^{-1}+\\left|{z_0}\\right|^2} {1-2\\Re(z_0)z^{-1}+\\left|z_0\\right|^2z^{-2}}, \\ ", "af9313326bfe4b16145cfc6272ed90de": "\\lang ax, y\\rang = \\lang x, a^t y\\rang,", "af9320eb7f5ca0f509ea23d40b3b5bc0": "\\mathrm{Hi}[\\langle\\hat{N}+1\\rangle]", "af93350a00bbe2f1223686e75933b444": " {IC} = C * ({({\\frac{n_a}{N} * \\frac{n_a - 1}{N - 1}}) + ({\\frac{n_b}{N} * \\frac{n_b - 1}{N - 1}}) + ... + ({\\frac{n_z}{N} * \\frac{n_z - 1}{N - 1}})})", "af934df4e33e631e04c3e61d45d25b57": "g(x) = x^3 + x +1", "af935db07362707196b2fd945c119a75": "\\rho(r) \\sim e^{-2\\sqrt{2\\mathrm{I}}r}\\,.", "af937a0d5a912d6369bf0b3003558d98": "P(s_1), \\; P(s_2)", "af93bd2c9eaec89cfcac0147d28cd795": "a^T (M \\circ N) a > 0", "af93c4b7d64bc0f668f4f95b521f7cf6": " f = f^+ - f^-. \\, ", "af93d27a16206af642cf995f341e8cdf": "\\frac{1}{s_i-1}-\\frac{1}{s_{i+1}-1}=\\frac{1}{s_i},", "af93ed5dfcba46b2961598fdc0a637fc": "\\mathfrak{sp}_{2n}(\\mathbf K)\\text{ or }\\mathfrak{sp}_n", "af94125709afb8274e48a359a3036f2f": "\\alpha = \\arctan u \\,, \\quad \\beta = \\arctan v \\,.", "af94b5f9da3217fc30bed6e0bd53211f": "\\theta_{r_i}(x)\\big|_{x=-j\\infty} = \\angle(-\\mathfrak{Re}[r_i],-\\infty) = \\lim_{\\phi \\to -\\infty}\\tan^{-1}\\phi=-\\frac{\\pi}{2}\\, \\quad (12)", "af955662af73a75a1b74121cab8b101d": "\\mathbf p_i = m_i \\mathbf v_i", "af956437859a7d04b4de0e43dfd12e85": "\\alpha = \\max(\\theta_i, \\theta_r)", "af9570a1bc556cff69496d65944ba388": "\\nu Z. \\phi", "af959f3510d5f2ddbe6ecd07aa5cd0fa": "\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix}:\\mathbf b", "af95d7d45ddb8849334d54973046a6c6": " i \\in K", "af95f7783c9b80c01a350abad653a65d": "\\begin{cases}\n 0 & \\text{for } x < a \\\\\n \\frac{x-a}{b-a} & \\text{for } x \\in [a,b) \\\\\n 1 & \\text{for } x \\ge b\n \\end{cases}", "af96179dd1181cddedf506f48450d492": "H=\\{w=x+iy \\,\\colon\\, y >0\\}", "af963250d7611bfbcec09b4caf03cc58": " P(2\\text{ black}, 2\\text{ white}, 2\\text{ red}) = {{{5 \\choose 2}{10 \\choose 2} {15 \\choose 2}}\\over {30 \\choose 6}} = .079575596816976", "af96333132402547cce199da9accb8f1": "w_i = w_i e^{ -y_i R_j(x_i) }", "af964b461b741a3fbd47e7fc513c70fa": " \\text{Maximize} \\,\\, pQ - wL - rK \\,\\, \\text{with respect to} \\,\\, Q, \\, L, \\, \\text{and} \\, K", "af96650e4fca4a5dbb69c3c09d0aa907": " y\\ F\\ n = (\\operatorname{IsZero}\\ n)\\ 1\\ (\\operatorname{multiply}\\ n\\ ((y\\ F)\\ (\\operatorname{pred}\\ n))) ", "af9668b46080769bbf59d965d95957f0": "X_{1,2}", "af96761ae50cb925ee94d1a0bc002f39": "F = N_{\\rm A} e = 96\\,485.3383(83)\\ {\\rm C\\,mol^{-1}}. \\,", "af967da9294b4f087ac119adc96cdc5b": "(p + q)^2\\,", "af96969c5ffce61a931f6a0bf55ebd5b": "v_\\mathrm N = \\frac {fv} {f - Nc}", "af96f7b284c41028c44b028a566711f3": "z_{\\mathrm{avg}} = \\frac{z_1 + z_2}{2}", "af970a58b53f0495c342914540d9c543": "n_x,n_y=1,2,3...", "af970ac0a71e39b91750b50790f0239e": "[\\hat{\\mathbf{e}}]_{\\times} = \\left[\\begin{array}{ccc} 0 & -e_3 & e_2\\\\ e_3 & 0 & -e_1\\\\ -e_2 & e_1 & 0 \\end{array} \\right]", "af975efde287c292b2efd4c2d120c2cf": "s,t\\in M", "af97a92dba7bd0ac5caf3b6358563936": "\\left(\\frac{8}{21}\\right) = -1\\quad\\textrm{ but }\\quad8^{(21-1)/2} \\equiv 1\\pmod{21}", "af97b447dc470a147d97c8004062d6c9": "(A,B;C,D) = \\frac {AC}{AD}/\\frac {BC}{-DB} , ", "af97bc606e293246f77298f74c18749b": "R_{34,12}", "af97d0522e1ed831000baedd68c22e55": "\\int_0^{2\\pi}e^{\\cos\\theta} \\cos(\\sin\\theta)\\;\\mathrm{d}\\theta.", "af97d054d61924f8175153268040248b": "F=Q^T F", "af98020bc3c39b4b2eeb53742e440461": "\\mu(y,H)\\subseteq H", "af9828188e253bafbf77f5bdd60f8ae2": "\\forall A\\, \\exist B\\, \\forall C\\, (C \\in B \\Leftrightarrow [C \\in A \\and \\theta(C)])", "af9841d6abc7f2f883a2dff242d2c7ea": "\\frac{1}{2}S_{2}^{2}", "af985376ccfa6cbdedb09e23574f90c6": "|\\epsilon_i \\rangle", "af98676085fb502b67ab7824d5d311ab": "n \\times r", "af9906bb3693296449d97dd4525ae08b": "A_1,A_2,\\dots,A_N", "af991a0427319c0cdcd17cb87a538356": " \\gamma <1/2 ", "af99ab5fabbf0d95dc0f03976faea3f4": "Fo = \\dfrac{ \\mbox{diffusive transport rate} }{ \\mbox{storage rate} }", "af99d64e93a874bc1be37107c8f2eecd": "0=(M_X-\\alpha\\cdot id)(H)=((X-\\alpha)\\cdot H) \\bmod P\\,,", "af9a1472ef4e53edbaf25b1c55dfbb08": "1 + {1 \\over 2} + {1 \\over 3} + \\cdots + {1 \\over n} = \\gamma + \\ln n + o(1),", "af9a2c468e49cd4ebf52295750142954": "y_1,\\ldots,y_m", "af9a6c2e33277323fc7e28aa5a148beb": "V_\\text{charging}", "af9acb0a971e14dbf91c684f9e56265a": "L^p(\\omega(x)\\,dx)", "af9af0459bef6dc7fa2fe3ef8a9e9fde": "!n", "af9b090d7100fddbc3026b77c0ad8b51": "s{_{k}^{i}}", "af9b38eef3e6a5767bc3c2bdb4d79ee4": "p'_R(x,y)", "af9b6a5543f2021bc20e5db3df072db3": "x_n \\in U", "af9b7e686b074718e90e52da1a35926e": "+g^{\\alpha \\beta }g^{ \\sigma \\rho}(\\Gamma^{\\mu}_{\\alpha \\sigma } \\Gamma^{\\nu}_{\\beta \\rho } - \\Gamma^{\\mu}_{\\alpha \\beta } \\Gamma^{\\nu}_{ \\sigma \\rho }))", "af9b88d35ef0f93fcb6675bb180e1880": " \\forall x \\,\\exists y\\, (\\text{FARMER} (x) \\and \\text{DONKEY}(y) \\and \\text{OWNS}(x,y) \\rightarrow \\text{BEAT}(x,y)) ", "af9c1b5255b610f825b979e715088f03": "T:W^{1, p}(\\Omega) \\to L^p(\\partial \\Omega)\\,", "af9cc671f27e5791a663b99e3dad7f95": "a = \\frac{1}{\\sqrt{2R_c L_s}}", "af9d17d88dd1d2bba58dae1c1ef9ce8d": "\\sqrt 8", "af9d2817f57d3ae8f438303acad10543": "\\operatorname{H}^i(X; A) = [X, K(A, i)]", "af9d3a4e2bea817b9bf5343969d65431": " y'' + p(x)y' + q(x)\\,y = 0", "af9d7024e5914cdeb5095d3128d41e75": "\\bar \\nu_P = \\bar \\nu _{v'-v''}+B' (J'' -1)J''-B''J''(J''+1) ", "af9dddc485e7304788758bf9d5653c07": "\\displaystyle \\begin{align}E_t &=-J\\overrightarrow{M_A}\\cdot\\overrightarrow{M_B}+K_{1A}\\sin^2\\theta_A+ \nK_{1B}\\sin^2\\theta_B\n\\\\\n&+K_{2A}\\sin^4\\theta_A+ K_{2B}\\sin^4\\theta_B -(\\overrightarrow{M_A}+\\overrightarrow{M_B})\\cdot\\overrightarrow{H} \\end{align} ", "af9dea387500b4803b0d0ad52ad85797": "A^s x_1=\\sum_{r=0}^s\\binom{s}{r}\\lambda^{s-r}A_{\\lambda,r}x_1=\\sum_{r=0}^s\\binom{s}{r}\\lambda^{s-r}x_{r+1}", "af9e417ce1b54f9465073ad013e64255": "w = W(f)", "af9e47190e17ad0538123bb46d9bd071": "log \\hat{e}", "af9ea855579bf5e8f9045bc8b032d5e3": "\\mathrm{RV}(X,Y)=\\frac\n { \\mathrm{COVV}(X,Y) }\n { \\sqrt{ \\mathrm{VAV}(X) \\mathrm{VAV}(Y) } } \\, .", "af9ebcd34e411916aea9d6eb23851cba": "\\theta/\\pi", "af9f4672668bfff3bd69ee9b1e5861c1": " \\frac{GM}{r^2} = \\frac{a^2}{a_0} ", "af9f50ffba2e4ba58532d72e918f76a7": "(u,v)\\longmapsto g^{M_1}_p(T_{(p,q)}\\pi_1(u), T_{(p,q)}\\pi_1(v))+g^{M_2}_q(T_{(p,q)}\\pi_2(u), T_{(p,q)}\\pi_2(v)).", "af9fd7d0955f6bbdacf9ec0632659133": " g_i(x) \\leq 0 , h_j(x) = 0", "af9fdb34b4974a0bf1966b974589a899": "\\{x_1,x_2,\\dots,x_{d+2}\\}\\subset \\mathbf{R}^d", "af9ffec95ee110a7c0380891dff3b068": "\\phi_f=S\\phi_i S^{-1}", "afa0589537ca5004a8feb9a14e3010a7": "i = \\ell-n, \\dots, \\ell", "afa05e4484a28a282089cb78cecfdc3b": " \\frac{-6}{3+2 \\alpha}", "afa0ad9d08299c278954314d342532b4": "\n \\operatorname{Pr}\\!\\left( \\lim_{n\\to\\infty}\\! X_n = X \\right) = 1.\n ", "afa0baada9af7cb3345719433e0c71ed": "G(T)\\geq 0", "afa0c35db6ec542219953fb0a7be935b": "\\|L(v + h) - L v\\| = \\|Lh\\| \\le M\\|h\\|. \\,", "afa0f752349697c9152d19ce695f65cc": "n^2-17", "afa1214f545597f58d1a5b86a3693637": "\\mathrm{SO}(n) \\times \\mathrm{SO}(2)", "afa23eec6f0add4ed51b8feaf2090398": "(\\nabla v)(p,t)(r)", "afa240d750efb5f0414609c2e953f3ef": "\\textrm{pH} = \\textrm{pK}_{a}+ \\log \\left ( \\frac{[\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "afa31b98930809c93c034c78046b0c3a": "V \\cap S \\neq\\emptyset", "afa374ece9cf3e49d489a0144b00ddfe": "\\Delta(y, C(x)) \\leq \\delta n", "afa37d9790e5f7facd041007805508ca": "d \\Delta g", "afa3a0fb8d32693e083adde67bac9b14": "(v,w)\\mapsto (1/v, w/v^{g+1}),", "afa3aac4d74f7edce001d78aa136cfe7": " (a*b)**(c*b)=(a**c)*(b**c) ", "afa3ccd2cc75707de66dc6e1acbee431": "\\ p_{ij \\ldots}", "afa3ddd556046eb23364045545e24df6": "H_{x}=\\left [ \\frac{m\\pi }{a}(Ae^{-jk_{x\\varepsilon }x}+Be^{jk_{x\\varepsilon }x}) + j\\frac{k_{x\\varepsilon }k_{z}}{\\omega \\mu}(Ce^{-jk_{x\\varepsilon }x}-De^{jk_{x\\varepsilon }x})\n \\right ]cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (36) \t", "afa42886472c2363a434c20798a50a6e": "\\tilde{\\mathbf{M}} = \\mathbf{U} \\tilde{\\boldsymbol{\\Sigma}} \\mathbf{V}^*", "afa492d9ac28aeaeea0869294e9767be": "\\displaystyle{A=\\Delta - V^2 - W^2,\\,\\,\\, B=[V,W],}", "afa4a60dee6cf53e0068483aed4bd157": "\n\\begin{align}\n(f*g)(n) \n&= \\sum_{d\\,\\mid \\,n} f(d)g\\left(\\frac{n}{d}\\right) \\\\\n&= \\sum_{ab\\,=\\,n}f(a)g(b)\n\\end{align}\n", "afa505e322c60f2852837b45b6bf0087": "\\{1, e^{\\pm i\\theta} \\} = \\{1,\\ \\cos(\\theta)+i\\sin(\\theta),\\ \\cos(\\theta)-i\\sin(\\theta)\\}, ", "afa50f5d32d629a905a98b7d52618738": " \\tilde{x} ", "afa56b55c9ec4c90f4a6f8f1055dcb5a": "{a^{\\varphi(10)}} \\equiv 1 \\pmod {10}. \\,\\!", "afa56db8578a76412f10e5c959cac759": "\\pi(\\mathrm{M}\\lambda) = \\pi_{\\mathrm{i}} \\pi_{\\mathrm{f}} = (-1)^{\\lambda+1}\\,.", "afa5af7ec51dd47d761c6c2a01889b5c": "\\Leftrightarrow (x+1)^2+(y-0)^2=9(x-0)^2+9(y-2)^2 ", "afa5cd58a50ec51a56219da47025c087": "\\mathbf{O}(\\mathbf{m}\\log{\\mathbf{n}})", "afa62fd6735433f524d3d00bc3683490": "\\cos (\\arccsc x) = \\frac{\\sqrt{x^2-1}}{x}", "afa660e0e7ec38fe304b6e578b3a4e3b": "E = 2 \\hbar \\omega_0 |\\sin(ka/2)| ", "afa68487cf7118336a14e654363be489": "\\left(k,n\\right)\\,\\!", "afa6968ac570381bc9710d108cb24137": "E(m) = m^e \\bmod n", "afa6a9025b646f4392597d22121f573b": "\n \\xi = \\sqrt{3}~p ~;~~ \\rho = \\sqrt{\\cfrac{2}{3}}~q \n ", "afa70db8e1a70ec07a481b554d48162f": "f_{r}:S^{1}\\rightarrow \\mathbb{C}", "afa72b12a08ae960270e310d66aee3bb": "\\eta(1-k) = \\frac{2^k-1}{k} B_k.", "afa79c36aab8a3a2fe78dab8ff85701e": "|P|\\geq (1-\\gamma)\\frac{n}{K+1}", "afa7e97d5bb924160df3645bf600a5b5": "\n\\begin{align}\n\\delta R &= R_{\\mu\\nu} \\delta g^{\\mu\\nu} + g^{\\mu\\nu} \\delta R_{\\mu\\nu}\\\\\n &= R_{\\mu\\nu} \\delta g^{\\mu\\nu} + g^{\\mu\\nu}(\\nabla_\\rho \\delta \\Gamma^\\rho_{\\nu\\mu} - \\nabla_\\nu \\delta \\Gamma^\\rho_{\\rho\\mu})\n\\end{align}\n", "afa820975161ba4ae74ba42b24001805": "x_{static} = \\frac{{F_0}}{{k}}", "afa83ff7ccd4a7e3561fb469116f641a": "r_u = a\\frac{\\sqrt{3}}{4} \\left(1 + \\sqrt{5}\\right) \\approx 1.401258538 \\cdot a", "afa88c06b1420a4ea35996902d982f21": "\\left(\\sum_{k=0}^{2n} a_k\\right) \\cdot \\left(\\sum_{k=0}^{2n} b_k\\right)=\\sum_{k=0}^{2n} \\sum_{i=0}^k a_ib_{k-i} - \\sum_{k=0}^{n-1} \\left(a_k \\sum_{i=n+1}^{2n-k}b_i +b_k \\sum_{i=n+1}^{2n-k} a_i\\right)", "afa89c858f1d2e06eca7c64a5ccf5635": "|K_1|>|K_j|", "afa9033e79d71ecd0dc16b78a57a02d3": "\\frac{1}{2\\sigma^2}", "afa95a9b8510aa0e5c56e3e1046783d5": "\\Re(s) > 0", "afa9ee75ec534f3d2dde227f98fbf627": "t_{nn}", "afaa05df7d221b48e0b619073cd2ed63": "\\gamma > 0 ", "afaa91a3e8253634b6a427f5696d031e": "(x_{k+1},y_{k+1}) = \\mathcal{P}_{F}( \\mathcal{P}_{E}( (x_{k},y_{k}) ) ) = \\mathcal{P}_{F}( (\\mathcal{P}_{C}x_{k},\\mathcal{P}_{D}y_{k}) ) = \\frac{1}{2}( \\mathcal{P}_C(x_k) + \\mathcal{P}_D(y_k) , (\\mathcal{P}_C(x_k) + \\mathcal{P}_D(y_k) ). ", "afaa99c523e981716b3ad73ed0710812": "p_{ij}^{-}", "afaac98ab15d499d98cb96398ade255e": "P(\\boldsymbol{r}) = p_\\mathrm{Boltzmann}(\\boldsymbol{r}) = \\frac{e^{-\\beta F(\\boldsymbol{r})}}{\\int_\\Omega \\, d\\boldsymbol{r} e^{-\\beta F(\\boldsymbol{r})}}", "afaad7642f5759752c61bb6909e655c1": "C_{16}", "afab118fac1b3a2050a326525e0c284b": "c_i \\in F", "afab3a1458204836ca90c21848f8dd0a": "x^ky^n", "afab57a2c49afcb3f8bb6b6fb9c66a8a": "\\sqrt x ", "afab6074518044aa9121a4e6ee8f40b1": "ASA_{unfolded}", "afab68e9720a9ff7cc87134b7c9629fe": "|J_1\\cdots J_n|", "afab6a32c9999257fe43aa6ee975ec35": "B_{c1}", "afaba92e41e1fe40451879623adee13b": "1-\\frac{k}{|E(G_j)|}\\geq 1-\\frac{2}{n-j}=\\frac{n-j-2}{n-j}", "afacf8b7c6733a6234b223486b42a95a": "q>0\\;", "afad1c39ef7027ca506ec28b84116f32": "\n\\phi = \\frac{1}{r^{n+1}}\\ P_{n}^{m}(\\sin\\theta)\\ (a\\ \\cos m\\varphi\\ +\\ b\\ \\sin m\\varphi)", "afad585d79bd1d2cedb493139ad60352": "U(\\mathbf{r},t)= A_oe^{i(\\mathbf{k}\\cdot\\mathbf{r} - \\omega t +\\varphi)}", "afade1f505b2b1a21759d3736dafe6fe": " [\\mathbf{x}_{k}]_{\\times} ", "afae0bfd05f6364d69726d7258245e28": " \\mathbb{R}^{2n} = \\{z = (x_1, \\ldots , x_n, y_1, \\ldots , y_n )\\}, ", "afafaa8f0c72d22bc58cfbd7b5ce8abd": "\\kappa = \\frac{1}{4M}.", "afafdc32cf7a5b2dd950e08f3a36ac7d": "\n\\varepsilon_\\mathrm{ff} = 1.4\\times 10^{-27} T^{1/2} n_{e} n_{i} Z^{2} g_B,\\,\n", "afb0017e5985aac010d802bb1201ceb8": " A \\supseteq \\operatorname{int}(A) \\! ", "afb006d4f819a0b3ea6d12f0eb16ce60": " F^{P}", "afb02460291020c540868f738d7373ef": "C \\sqcup D", "afb031db80764e30170e9633f6fbdf6f": "B^*\\otimes B", "afb04483b68300127e07890c2127fd50": " V_j(a,z) \\Phi(v,w)= \\Phi(v,w) V_i(a,w) = \\Phi(V_k(a,z-w)v,w)\\,", "afb0715d4975c273dd04caa5ddb6cde4": "RC=\\frac{(Hits+Walks)(Total Bases)}{At Bats+Walks}", "afb0905729441cf0f992060c77f9a314": "q=\\int_{t_1}^{t_2}\\iint_S \\mathbf{j}\\cdot\\mathbf{\\hat{n}}{\\rm d}A{\\rm d}t ", "afb0b53c6b56d00c1d2270f8df837d91": "C_P=\\frac {C_{P0}} {\\sqrt{1-M^2}+\\frac{C_{P0}}{2}(M^2/(1+\\sqrt{1-M^2}))}", "afb13eaebd353b594df4682197741c5d": "t = - \\frac{f(r)}{p^k} \\cdot (f'(r)^{-1}).", "afb14adbace6b925eca11034ba19b627": "c_2=hc/k", "afb1f169af4f7ca808fa9c08146fdc17": "\\hat p = i\\sqrt{\\frac{m \\omega\\hbar}{2}}(a^{\\dagger}-a).", "afb21f153788f311a06de646a8d4b300": " W = -\\int_{1}^{2} \\mathit{p} \\cdot \\mathrm{d}\\mathit{V}", "afb2bd16592b3306669aded7029dc528": "A_\\infty = \\bigcup_{p<\\infty}A_p.", "afb2c50cb1ddbbeaa1c632d432991df2": "\\alpha_j \\leq \\beta_j.", "afb2ebd6d9ccc279c88c7084da2d3c75": "V_t = V_1 + (1-2/N^2)V_{t-1}", "afb3081dd6433325b63fb691086af0c2": "-W^{\\mathrm{path}\\,P_1,\\, \\mathrm{irreversible}}_{A\\to B} + Q^{\\mathrm{path}\\,P_1,\\, \\mathrm{irreversible}}_{A\\to B} = \\Delta U\\, .", "afb3121c654d291fcde1148de1da7a0e": "\\Theta = d\\theta + \\omega\\wedge\\theta.", "afb336e0b096cd46514e9c8232688bc5": "T^{\\bar k-1}p", "afb33f5ae228779fa23ebd0e676599a1": "U = \\{U_n\\}_{n \\in \\mathbb{N}}", "afb35c707db1fba548ebc957e6afc89b": "H_{p^3}", "afb36043ad5702b205efb506ad0dd123": "U_2(x,y)=(1-\\alpha)\\left(\\alpha x^\\rho +(1-\\alpha)y^\\rho\\right)^{\\left(1/\\rho\\right)-1} y^{\\rho-1}.", "afb3c1f0529c9f466c316a3b6ce3bf43": "|E(\\mathbb{Z}/N\\mathbb{Z})| = N+1-a", "afb40a3eee24a8d5e57cd766ef847202": " \\Rightarrow v_1-v_2 = u_2-u_1", "afb42b732c44e29f74d8eec948703d9a": "\\prod_{p\\leq X} \\frac{N_p}{p}", "afb496e4a618a6f3edfdc240199bd588": "\\operatorname{Re}\\{x[n]\\}", "afb4a10be4400eec750aa4f0bf377ec8": "(x+1)^2 = -9 \\,", "afb4a2c8ffc8d6b0e05848c4df2d5c4d": "p_a = \\begin{bmatrix} x_a\\\\y_a\\\\1\\end{bmatrix}, p^\\prime_b = \\begin{bmatrix} w^{\\prime}x_b\\\\w^{\\prime}y_{b}\\\\w^{\\prime}\\end{bmatrix}, \\mathbf{H}_{ab} = \\begin{bmatrix} h_{11}&h_{12}&h_{13}\\\\h_{21}&h_{22}&h_{23}\\\\h_{31}&h_{32}&h_{33} \\end{bmatrix} ", "afb508876a48ffbb68ce5d788364eb02": "g(n)=\\frac{1}{1+n}\\,", "afb52925542e78366949ec1e107e3ec8": " \\sup_{f \\in \\operatorname{State}(A)} f(x^*x) = \\sup_{f \\in \\operatorname{PureState}(A)} f(x^*x). ", "afb5986aa3f3b44690d75c98e31a00d5": "(2\\leq N\\leq 112)", "afb5bf774a6fbb08ac8b1f0ef2e9966d": "\nE[F | x^{(t)}]\n~=~\n\\frac{\\sum_{s\\in S^{(t)}} c(s) x'_s \n+ \\sum_{s\\not\\in S^{(t)}} c(s) p_s}{2\\lambda c\\cdot x^*}\n~+~\n\\sum_{e\\in \\mathcal U^{(t)}}\\prod_{s\\not\\in S^{(t)}, s\\ni e} (1-p_s).\n", "afb619275d4c225daf6cdbe7ba63e89a": "\\langle W(C)\\rangle \\propto e^{-\\sigma A}\\;,", "afb61c38f25de6244b57a613f04dfe56": " r \\le f(x) \\ \\ \\longleftrightarrow \\ \\ 1 \\le I(x)\n", "afb6295fadcf439fd7d375e714c524fd": "\\sigma_P(x,\\xi) = \\sum_{|\\alpha|=k} P^\\alpha(x)\\xi_\\alpha", "afb6803c96c3351c85689dcc3e6f3b88": "\\omega \\phi=A(\\phi)\\,", "afb695baf59bea58247f416e7da5980f": "V_{BE}=V_{G0}\\left(1-{\\frac{T}{T_0}}\\right)+V_{BE0}\\left(\\frac{T}{T_0}\\right)+\n \\left(\\frac{nKT}{q}\\right)\\ln\\left(\\frac{T_0}{T}\\right)+\n \\left(\\frac{KT}{q}\\right)\\ln\\left(\\frac{I_C}{I_{C0}}\\right) \\,", "afb71f8bc8983a1ad5fadaac00e9478e": "x = R K \\cos \\phi_1 \\sin (\\lambda-\\lambda_0)", "afb736063831a6d8d95f377f91c670d7": "m_\\mathrm{H}", "afb748f236df25b508be03c1684d8ec1": "\\|f\\|_{L^{p,q}(X,\\mu)}=p^{1/q}\\left(\\int_0^\\infty t^q \\mu\\left\\{x\\mid |f(x)| \\ge t\\right\\}^{q/p}\\,\\frac{dt}{t}\\right)^{1/q}.", "afb7b1200b4d30c8eb8a0170fbd09ecd": "\\frac{d(ux)}{dx} = x\\frac{du}{dx} + u\\frac{dx}{dx} = x\\frac{du}{dx} + u,", "afb7b6633b0d2194a155895a2afe66ae": "P_{i-1}", "afb7bd5c37c6b2d7566fa0dd9d8b55a8": "\\log \\beta_1=pK_{a3}, \\ \\log \\beta_2=pK_{a2}+ pK_{a3},\\ \\log \\beta_3=pK_{a1}+ pK_{a2}+ pK_{a3} ", "afb7d9c6264eee63b9a2ea08e67a5ff6": "2 < r_s < 6", "afb7ee1b55df4b14f85c45637fc6047f": "\\phi(s)=\\frac{1-2^{(s-1)}} {2^{(s-1)}} \\zeta(s)", "afb85b6b6afda08d6a4d44009453cead": "\\begin{matrix} {12 \\choose 2}{4 \\choose 2}^2{40 \\choose 1} \\end{matrix}", "afb89b476f956fca253c86a720a641df": " \\{ 0 , -2, 2, 2, 2, ... \\} \\,", "afb8a480d2be54d1d1cba1bbdd94c5eb": "x\\in \\omega(x)", "afb8b5645a1e73568c4e861b0936490c": "u^{}_0", "afb8c172a228c31d9d813119437b00e6": "(T_\\text{s}/T_\\text{r})", "afb93a8376af5b18be17bd74b00baf18": "\\max_{S_k} \\min_{x \\in S_k, \\|x\\| = 1}(Ax,x) = \\lambda_k ^{\\downarrow},", "afb960b16fccc90c19eea265cfeac083": "(x_n + y_n)", "afb99d3c354369a529292ccd7cac6e66": "\\neg(\\neg Q \\and P)", "afb9a07cea2ccecbc743e9456f5c37e5": "v_{i-1}", "afb9ef78fa91ff24c195acd7cc7c109e": "\n\\begin{array}{lclc}\n\\psi_{x+}=\\displaystyle\\frac{1}{\\sqrt{2}}\\!\\!\\!\\!\\! & \\begin{pmatrix}{1}\\\\{1}\\end{pmatrix}, & \\psi_{x-}=\\displaystyle\\frac{1}{\\sqrt{2}}\\!\\!\\!\\!\\! & \\begin{pmatrix}{1}\\\\{-1}\\end{pmatrix}, \\\\\n\\psi_{y+}=\\displaystyle\\frac{1}{\\sqrt{2}}\\!\\!\\!\\!\\! & \\begin{pmatrix}{1}\\\\{i}\\end{pmatrix}, & \\psi_{y-}=\\displaystyle\\frac{1}{\\sqrt{2}}\\!\\!\\!\\!\\! & \\begin{pmatrix}{1}\\\\{-i}\\end{pmatrix}, \\\\\n\\psi_{z+}= & \\begin{pmatrix}{1}\\\\{0}\\end{pmatrix}, & \\psi_{z-}= & \\begin{pmatrix}{0}\\\\{1}\\end{pmatrix}.\n\\end{array}\n", "afb9f43b4868578f878c72d6051ce3ed": "s[r \\sigma]_p", "afba0265e1e45833686793c3cbd7d315": "(m,w)", "afba1b60dc112f77d0f9144290fe3da9": "f_i=\\frac{\\Delta t}{2}", "afba5eb524f6b8e6e4c05b426d4609aa": "_{s.6.right\\,}\\!", "afbae2f039701d9bc2ce02203fb5ac93": "Adv(\\rho_m)=-\\mathbf{V}\\cdot\\nabla \\rho_m \\!", "afbb142e8c6ec1eec1377097458c6798": "i=1, 2, ..., n", "afbb39cd4de5e00392b67dbf8562c0b7": "\\vec n_0 = {{\\vec n} \\over {| \\vec n |}}\\,", "afbb6fb4c1634eb90098cb044a90573a": " S_3 \\implies (A_3 = x \\or A_3 = v[x]) ", "afbb70f68e460bc368ae1b87d87f139f": " \\kappa: \\operatorname{PureState}(A) \\rightarrow \\hat{A} ", "afbbccde391d5b9165385cae851376fd": "X^{(p)} = \\operatorname{Spec} R \\otimes_A A_F.", "afbc8080bfbef2affc84d559ee2d35c0": " p_{1},\\ldots, p_{n} ", "afbc8e3b9dd0308acf2d6213cc1488fc": "\nf(a) = \\frac{1}{|G|} \\sum_i d_{\\varrho_i} \\text{Tr}\\left(\\varrho_i(a^{-1})\\widehat{f}(\\varrho_i)\\right),\n", "afbccaec2ad2ea83b9f052599d3a5f90": "\\scriptstyle z\\in\\left(-\\infty,\\eta\\right)", "afbd0345bede2eba7571da2656109f7f": "\\scriptstyle\\varphi", "afbd3853293625239d691d5433c42606": " \\Delta E = g_\\mathrm{e} \\mu_\\mathrm{B} B_\\mathrm{0} ", "afbd46540c12efbc980ffdd91cfb5345": "\\{p_4\\}", "afbd65cbee3d1a12c0113eac43784f0c": "g^p \\neq 1 \\mod p^2", "afbd6a4bbc52ce1d64bb3e058fdb21fe": "\\textstyle R = 24/32 ", "afbdad9c220a41e694471b853dba714a": " (u,v)", "afbdfbf7af2861c51f25d69a0c56aaa8": " \\square ABCD ", "afbe2e04dc723e003d43e69111701d0c": "\\frac{n!}{n^{n+1/2}e^{-n}}", "afbe4ab334eaa04e6319ec44292d91ed": "\\chi(G_K,M)=\\frac{\\# H^0(K,M)\\cdot\\# H^0(K,M^\\prime)}{\\# H^1(K,M)}", "afbe5c301c111e5a4f4868700624881d": "x_1=\\frac{p}{q}\\ ,", "afbe65aa9292ee915a6ee17f1edaa02d": "\\Delta\\langle\\left [ B^ {\\dagger}\\right]^J\\,\nB^K\\rangle", "afbe6de3af4a5147014057c1b22521f4": "(S, \\wedge, \\vee)", "afbe94cdbe69a93efabc9f1325fc7dff": "sn", "afbea77b5505e96b8e7d8d0376d1014f": "\n\\begin{align}\nds & = - \\frac{g}{k^2} \\cos \\theta \\,d\\theta \\\\\n\\sec\\theta \\, dx & = - \\frac{g}{k^2} \\cos \\theta\\, d\\theta \\\\\ndx & = - \\frac{g}{k^2} \\cos^2 \\theta \\,d\\theta \\\\\n & = - \\frac{g}{2 k^2} \\left ( \\cos 2 \\theta + 1 \\right ) \\,d\\theta \\\\\nx & = - \\frac{g}{4 k^2} \\left ( \\sin 2 \\theta + 2 \\theta \\right ) + C_x\n\\end{align}\n", "afbeb048dbf9adcaa6b1c4aacb8ddcd3": "\\mathbb{H} \\sim \\mathbb{C},", "afbf42107467e6950069d65e9c011600": "\\varphi I_n-A", "afbf52dd27029fcb9e33431107c7d39e": "(p_1,q_1)", "afbf943326114abd6cfd112bc46382f6": "x_i=(x_{1,i},x_{2,i})", "afbfef43959b3fb5b481d80b164582a9": " E = q, G = q, H = p, Y = (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f)), X = \\{\\} ", "afc016a792acadcf1d7a001e6931bff7": "ZY = U_k(\\omega) + iV_k(\\omega)\\,\\!", "afc0337d9b9405e471cca88fca6de9df": "S \\propto \\nu^{\\alpha}\\,\\!", "afc049f6278cb1d31f39cb05c587f8c2": "a^2 = b^7 = (ab)^{17} = [a,\\, b]^6 = [a,\\, b^3]^5 = [a,\\,babab^{-1}abab] =", "afc08c2e5363eaf5aec75eb35df842af": "\\mathbf{c}=\\mathbf B^{-1}=\\mathbf F^{-T}\\mathbf F^{-1} \\qquad \\text{or} \\qquad c_{ij}=\\frac {\\partial X_K} {\\partial x_i} \\frac {\\partial X_K} {\\partial x_j}\\,\\!", "afc0a7d52fef65e2c77c753fea4ebaed": " R(x): \\mathbb{R} \\rightarrow \\mathbb{R}", "afc0aa32b7e7adc2df63f8231755c802": "B = - \\Delta x", "afc0c3f8dada9aba4813338a0e886383": "\\scriptstyle K_n", "afc0d03dd9a679dbdaaec40bbcd9ac42": "u(t,s)=", "afc102c430ac9a0581ef1cc10f965507": "R_2, \\, W_2, \\, M_1", "afc11585b5362a854d164c7685e5964c": "q_m(X_m) = g_m(X_m,X_m),\\quad X_m\\in T_mM.", "afc12721a21eccf3abaf089a1f33cf1a": "f(x) \\triangleleft g(x) = f'(x) g(x)", "afc174e10956ed42227545d7285ac0c7": "\\gamma(s, x) = \\sum_{k=0}^\\infty \\frac{x^s e^{-x} x^k}{s(s+1)...(s+k)} = x^s \\, \\Gamma(s) \\, e^{-x}\\sum_{k=0}^\\infty\\frac{x^k}{\\Gamma(s+k+1)}", "afc199871a67ec697f4e8eed11cdf12a": "Q(x) = q(L-x), ", "afc1bc4f3a87b7192f8c38b8977e8155": "\\scriptstyle ( |0\\rangle_A \\otimes |1\\rangle_B - |1\\rangle_A \\otimes |0\\rangle_B ) / \\sqrt{2}", "afc226d819917bf775f7d8d1569c2eee": "\\scriptstyle A\\subset\\R", "afc22badbbb70f0cc2eee7a8963b54f3": "\\nabla\\cdot\\left(f(r)\\mathbf{\\Phi}_{lm}\\right) = 0", "afc2e6cd9e340ce372f016732180f11c": "K_{3,4}", "afc33c53edf3066c8b60e52975f0f50d": "\\alpha_2, \\alpha_3", "afc35e32599d8eedde6bc27cb07f6859": "a(x, y)=\\langle Ax, y\\rangle", "afc39fdd91eccb65a7cddc9ea622224c": "{D_S}", "afc46d327d86cb0c086648eba3ded444": "A(S) = \\begin{cases} 1, \\mbox{ if } \\displaystyle\\sum_{k\\in S} x_k \\geq 1\\\\ 0, \\mbox{ otherwise.} \\end{cases}", "afc48b56873694f3d43097841ecc3f4f": "\\frac{1}{x}", "afc48c719724d245859f3558f79414fd": "\\{\\xi < \\kappa : \\xi \\geq \\alpha\\} \\,", "afc4a7bd288555b4036f3db9335d2382": "S:[0, c]\\to\\Sigma", "afc4f2311c39909911e267b9633f8682": "\\hat{\\mu_{j}}=\\frac{1}{n}\\sum_{i=1}^{n} w_{i}^{j}", "afc505cd2bc9ab52c99e9c9b47db9104": "\\frac{1-L^\\alpha x^{-\\alpha}}{1-\\left(\\frac{L}{H}\\right)^\\alpha}", "afc544e9a58f3040c9d51ac65aa9dbf5": " U_q(\\mathfrak{g})", "afc548b868aee21cbcacdd83f1383e5c": "P(\\{2,4\\})/2", "afc5579630315fe079e2a1b5dca8e863": "\\boldsymbol\\Gamma \\, ", "afc5ba20aac1b67bd7e1df4b86af3ab7": "\\langle X_t \\rangle", "afc5de5e11f31e8cf82999eb1a7a854b": "CW=\\sin(A) \\cdot WS", "afc622cbec9bba41fb1e4facb12aef25": "{\\nabla_{2b}}^2", "afc632e523e63e754c75fa02c2796b6d": " (\\lambda p.(\\lambda q.q)\\ \\lambda f.(p\\ f)\\ (p\\ f))\\ \\lambda f.\\lambda x.f\\ (x\\ x) ", "afc650f29e52ac9c285223246c601904": "\\Delta\\,P-\\tfrac{\\rho f}{2\\,D}\\,W^2\\Delta\\,X+\\rho\\left(\\frac {2-\\beta}{2}\\right)\\Delta\\,W^2 = 0", "afc686a373f0c42409fc00347542208b": "\\textstyle \\sum_{n=0}^\\infty b_n", "afc6b3b0fa712285f1752fcc48a3c783": "\\hat{N}_f", "afc7013cec66a473f174e22c77916dc6": "D_s^2 = (\\frac{1+a}{1+2a}) D^2 ", "afc7208923afceffa468310c15684be9": "{}^{\\ast}\\mathbb{R}", "afc73d0e670f37f0b7ac3b75a90779f4": "Va = \\frac{Da . B . Q}{\\tau} ", "afc742ac7e4b919146cd47e318a1170a": "\\left(n,k\\right)", "afc77abf529c5c6b0db9811e307e5b26": "T \\sin \\varphi = \\frac{1}{c}\\int T\\ ds.\\,", "afc7cab3b9da3965e24c393d574412ea": "\\scriptstyle 0 \\;<\\; r_1 \\;<\\; 1", "afc89db38ed9676dd34385fc8381e281": "t_y", "afc8bbbb281f0d8564be71f21ac57e67": "[q,k]_q", "afc8ce286fd0bfc2df36c5cb9e71a60c": "g(e_1), \\ldots , g(e_k)", "afc972a3c03de8771b5e2ca62d17fe93": "i \\rightarrow i\\pm 3", "afc99134529cddc42899cdd6b4ef7509": "\\vec{p}_2, \\; \\vec{p}_3 ", "afc99f082745fcef35be7d256d8b2827": " c_x^2 = \\frac{ s_x^2 }{ m_x^2 } ", "afca2546914ac99a94da862607032030": "\nz(\\theta) \\; = \\; z_{\\mathrm{avg}} + \\frac{z_{\\mathrm{diff}}}{2} \\cos{\\left( 2(\\theta - \\theta_{\\mathrm{ast}}) \\right)} \\;\n = \\; z_1 \\!\\cdot\\! \\cos^2{\\left( \\theta - \\theta_{\\mathrm{ast}} \\right)} \\; + \\; z_2 \\!\\cdot\\! \\sin^2{\\left( \\theta - \\theta_{\\mathrm{ast}} \\right)}\n", "afca429eb4aea40bfe1d68218e9111e3": "\\operatorname{vars}[E] ", "afca91fcc2422a83cd8a6a9965badc52": "\\scriptstyle t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \\quad \\quad p_{n+1} = 2 p_n", "afcac16dc707bb28ab7403ca0c522c20": "\\Delta B_n(x) = B_n(x+1)-B_n(x)=nx^{n-1},\\,", "afcaca2a03573c3007389fd33250d9d5": "(4 \\times 36 \\times 45 \\times 50 \\times 75)^{^1/_5} = \\sqrt[5]{24\\;300\\;000} = 30.", "afcaed3ecbf1ff3a87232473692c055b": "L_+", "afcb2ad15c57011721c6764c49b8d34f": "\\mathfrak{f}_{4}", "afcb4a3f60287755cf845565c61521b2": "F'(c_i) = f(c_i).", "afcb631071913f138e006978be7d14c7": "\\min(x,y), \\max(x,y) \\!", "afcc64e675157a1fd3fd0215551c103f": "u > 0", "afccde5568740d5dcf377a644eb5f7dd": "U = (U_t)_{t \\in [0,T]}", "afcd610810f90236fc01f6c96230110e": "\\mathrm{Hom}(V,W) = V^* \\otimes W", "afcd681feecb2f4b1629b3299a0a577e": "0 \\to A\\times 0", "afcd68b05e92ee5cb605016df78bfe76": "220_5 = 214.\\overline4_5", "afcd8194f845ee00c0e53078a740faa8": "\\frac{\\sqrt {128}}{3}", "afcddde58463b520f448174704122e1b": "I(a) \\,=\\, \\inf \\{ \\mu^+(A) \\, | \\, \\mu(A)\\, =\\, a\\}", "afce25d3d3cf1774ecc01d9b444106cd": "P_r = \\frac{{13 - s \\choose 5} + {13 - s \\choose 4}{35 + s \\choose 1} + {13 - s \\choose 3}{35 + s \\choose 2}}\n {{48 \\choose 5}}", "afce317a79a1e434d7ed6b72f9894b2f": "\\begin{align}\nf_{X_1^n}(x_1^n)\n &= (2\\pi\\sigma^2)^{-n\\over2}\\, e^{ {-1\\over2\\sigma^2} (\\sum_{i=1}^n(x_i-\\overline{x})^2 + n(\\theta-\\overline{x})^2) }\n &= (2\\pi\\sigma^2)^{-n\\over2}\\, e^{ {-1\\over2\\sigma^2} \\sum_{i=1}^n(x_i-\\overline{x})^2}\\, e^{ {-n\\over2\\sigma^2}(\\theta-\\overline{x})^2 }.\n\\end{align}", "afce3b2621ecef372331043856f9a1d7": "C = \\frac{q}{V}.", "afcedbb169c0ac9cc137d121dad41ab3": " v_1 (x) =\\sum_{n=0}^{N} u_{n} T_n (y_1(x)) ", "afcf44958a1f8e851d37e8fd80fccf42": "f(1, f(2, 3)) = 1.75", "afcf483ca2772b920bcc34c037875a5b": "\\mathbf{NC}^i \\subseteq \\mathbf{AC}^i \\subseteq \\mathbf{NC}^{i+1}.", "afcf4d8ebb7740d1b6ba579a4394419b": "\\mathbf{v}_s(t')", "afcf638cb8147295d0c4a6c19fce2a15": " w_j+tw_b \\le w_i", "afcf730c538573a64e325c5dfb30dd11": "\\int_0^1 dy \\int_0^{\\sqrt{y}} (x+y) \\, dx.", "afcf83222df045c8da76f8beb2a2f070": " \\mathfrak{a}_0 ", "afcf9e304436c099eaac4cd9f3546635": "\\begin{pmatrix}i&j\\\\k&l\\end{pmatrix} ", "afd092fa5ac639ea7a26ec8e57f2ccf6": "\\Sigma+\\frac{1}{m}\\cdot K", "afd09f49bd6ee99a218625336f03a486": "\\eta = \\eta(\\xi)\\,", "afd0db8cc1441e48a717b93ecb3fa6c6": " A=e^{ -\\sum_{i,j}b_{ij}D_{ij} } ", "afd116caa3e1faed1a7ce91aded3fd74": "0 \\to d\\Omega^0(S^1) \\to \\Omega^1_c(S^1) \\to H^1_{\\text{dR}}(S^1) \\to 0,", "afd11a6f6b0dbf1f319fe81109bd1710": "\\mathbf{D} = \\varepsilon \\mathbf{E}", "afd132230a84912cfbf37bbf67a5c8c5": "p_2=m_1(1+m_1)\\ ,", "afd13292e8b9aadbf34eba9ebe1ccdc8": "\\theta \\in [0,\\infty)", "afd154afcbe14de295e4542677c1d10a": "w_{i} = \\int_{a}^{b}\\omega(x)\\prod_{\\begin{smallmatrix}1\\leq j\\leq n\\\\j\\neq i\\end{smallmatrix}}\\frac{x-x_{j}}{x_{i}-x_{j}}dx", "afd15d2ac2bcbf7e9107a52fbcbf8776": " N_{t}", "afd17409c5d8d0627ec9c31583016e28": "\\mathbf{x} = \\mathbf{a}", "afd18fe11985815bc7ee7e047f9d3026": "\nJ = \\int_0^T p {d x \\over dt}\\, dt\n", "afd1a0ff109d97cca478477eede98e51": "I_{-1}^1r=0=I_{-\\infty}^{+\\infty}r", "afd1c4d11791e2d97fc6c2eebc5e2b81": "{\\rm{C}}_\\alpha {\\rm{H}}_\\beta {\\rm{O}}_\\gamma {\\rm{N}}_\\delta + \\left( {a{\\rm{O}}_{\\rm{2}} + b{\\rm{N}}_{\\rm{2}} } \\right) \\to \\nu _1 {\\rm{CO}}_{\\rm{2}} + \\nu _2 {\\rm{H}}_{\\rm{2}} {\\rm{O}} + \\nu _3 {\\rm{N}}_{\\rm{2}} + \\nu _4 {\\rm{O}}_{\\rm{2}} ", "afd1c5f72b135bf6d3ffe2d5dfe17cb5": " |\\psi_{2i} \\psi_{2j} \\rangle_\\nu = \\frac{1}{2}(|\\psi_{2i}\\rangle \\otimes|\\psi_{2j}\\rangle + (-1)^\\nu|\\psi_{2j}\\rangle\\otimes|\\psi_{2i}\\rangle) \\in S_\\nu(H \\otimes H)", "afd1d29495bafb53ab8452d7061fc4b6": "O(log(n))", "afd23424c6c632eff930ebdb8ac42fbf": "||f||_{B,p}=\\limsup_{x\\mapsto\\infty}\\left({1\\over 2x}\\int_{-x}^x |f(s)|^p \\, ds\\right)^{1/p}", "afd2ca9721294388cc0ec2aae2effab1": "c \\neq 0 ", "afd2d6f3d20c5eb777b62d7aad82f20f": "\\int_Sg\\,d\\mu < \\infty.", "afd3c77e70508d438a129073c4d09d8b": "\\scriptstyle x/\\delta x", "afd3cea4acc78d548da69746c1a5e643": "\\varphi(r)", "afd3d203841d2aa18591b7c7497cbfef": "\\scriptstyle \\operatorname{P}(\\cdot|\\mathcal{B})(\\omega) ", "afd3de097d167fc7ea10307a61512780": "\n\\begin{align}\n\\frac{\\partial {\\mathcal{L}}}{\\partial \\theta} &= \\frac{d}{dt} \\left(\\frac{\\partial {\\mathcal{L}}}{\\partial \\dot{\\theta}}\\right)\\\\\n-mgr \\sin{\\theta} &= 2mr \\dot{r}\\dot{\\theta} + mr^2 \\ddot{\\theta}\\\\\nr\\ddot{\\theta} + 2\\dot{r}\\dot{\\theta} + g\\sin{\\theta} &= 0\n\\end{align}\n", "afd3fa232ee861d13308e21a7ce816d5": " \\left| \\left| \\widehat{\\mu}_{X^{new}} - \\widehat{\\mu}_{X^{te}} \\right| \\right|_{\\mathcal{H}}^2 = \\left| \\left| \\widehat{\\mathcal{C}}_{(X \\mid Y)^{new}} \\widehat{\\mu}_{Y^{tr}} - \\widehat{\\mu}_{X^{te}} \\right| \\right|_{\\mathcal{H}}^2 ", "afd44cf3c4cd0b09aa1bafd950859218": " \\langle N\\rangle = k_B T \\frac{1}{\\mathcal Z} \\left(\\frac{\\partial \\mathcal Z}{\\partial \\mu}\\right)_{V,T} = \\frac{1}{\\exp((\\epsilon-\\mu)/k_B T)+1} ", "afd4d32f43aded43252abe0348b7da45": "n=\\operatorname{ar}(s)", "afd4fe8942def304d43a0b0c6c90c2cf": "y_{1i}-y_{0i}", "afd53f0cd9885b21b6d51b57e807ce39": "I \\subseteq Q", "afd5453e1907c3bd6049c5b93086b120": "e^{-i\\omega t}", "afd577730a9b67e126d64b3ac2122928": "V_+", "afd62f1241edd941a407c999bcce47a0": "\\left [\\begin{smallmatrix}2&-1\\\\-6&2\\end{smallmatrix}\\right ]", "afd63a7455a93bbb53712112018e287a": "F_{i+1}\\subset F_i", "afd6509b0f0b34345f20dc6989c5b71f": "x_e", "afd6932f58971a6eabd42f45e3009223": "[-1,1]\\times [-1,1]", "afd69b5e60016d9140e423081f9843ba": "2, 2+it", "afd6dad02861a03d4c286897ccc35c9b": " \\widehat{G} ", "afd700837eccc5152adfe9b9d2c4e5d1": "f:x\\mapsto g(h(x))", "afd78ff03251f6f8f5961254807607b1": "\\max(\\deg(P),\\, \\deg(Q))\\geq 2,", "afd7a7c1ea962dc1ad8943b69ad9c6f4": " Q_{Building} = C_{Building}{{\\Delta}P_{Building}}^{n_{Building}}\\,\\!", "afd7b19a27e53b0ed4be10e4fe5414e0": " l_\\text{P}^3 = \\left( \\frac{\\hbar G}{c^3} \\right)^{\\frac{3}{2}} = \\sqrt{\\frac{(\\hbar G)^3}{c^9}}", "afd7f56c5a945c2ef075045bf18cba21": "\\displaystyle{g_1=\\begin{pmatrix} a & 0 \\\\ 0 & a^{-1} \\end{pmatrix},\\,\\, g_2=\\begin{pmatrix} 1 & 0\\\\ b & 1\\end{pmatrix},\\,\\, g_3=\\begin{pmatrix} 0 & 1\\\\ -1 & 0\\end{pmatrix},}", "afd81ba702acf64f145a46ad24f35c6d": "E^c", "afd869c5aba3604cae2af057ae3a31c2": "\n {\n \\rho~\\dot{\\eta} \\ge - \\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right)\n + \\cfrac{\\rho~s}{T}.\n }\n ", "afd88d334feec6179d98b9812314a9fc": "f=f_1=f_2", "afd8d6c9a21e4fd75602659b5b3674ec": "\\left\\langle \\hat{X} \\right\\rangle_\\psi = \\left\\langle \\psi | \\hat{X} \\psi \\right\\rangle", "afd92dc2e39411688df51596598fc50c": "(0,x)\\sim f(x)\\quad\\text{for each }x\\in X.", "afd930f8231f6f6d4e0eff8563cbc26d": "\\left(\\frac{dn_1}{dt}\\right)_{\\mathrm{spontaneous}}=A_{21}n_2\\,.", "afd989480cec5c3f4dadfc6268039a18": " 2 \\sigma_m = 2 \\sigma_1 + \\sigma \\, ", "afd9ad6f62478af42a7f080af97d00cd": "\n\\Psi^\\dagger \\gamma_0 \\gamma_{\\mu} \\Psi.\n", "afd9ff0f91ba357aa07efb5c917a3c6f": "E(Q)\\propto (C_1Q^{4/3}+C_2Q^2)^{2/3}", "afda1cdbbf0854c2a4dccf56a177e953": "f_p(x)=g_p(x)", "afda38e14877df605342a9a56b35fd9e": "\\{|{a_i}\\rangle\\}_{i=1}^n", "afda5b749404dac7e62647c475fc8ffc": "O((d + k) \\log^{2+\\epsilon} q)", "afda68e0d3f422ae88852ccc674c2611": "\\sum_{i=1}^m d_i = m(m-1) + \\sum_{i=m+1}^n d_i.", "afda7bac2c49b1b203b75e1a711139a6": "\n \\frac{\\partial \\mathcal{L}}{\\partial f_i} +\\sum_{j=1}^n (-1)^j \\frac{\\partial^j}{\\partial x_{\\mu_{1}}\\dots \\partial x_{\\mu_{j}}} \\left( \\frac{\\partial \\mathcal{L} }{\\partial f_{i,\\mu_1\\dots\\mu_j}}\\right)=0\n ", "afda963868ac701f9ec2f0588e6a341d": "A \\ ", "afdaa032b2eca1adacb4b81785d946eb": "r \\in R_n\\,", "afdaa09b0c66188dd6011dd9a854ad87": "T(w)=2^{2^n}", "afdab22378e9a25357c4aafb17d9c4de": "\\sqrt{A^*A}", "afdb54b07d7e4cc65d87c747686bb1d5": "\\mathrm{s}^{-2}", "afdb7e5d8072026aef06768cb948910c": "\\sum_{k\\ge 0} \\beta_kz^k = \\exp\\left(\\sum_{k\\ge 1} \\alpha_kz^k\\right)", "afdbbb13c61162f04bd1138360fdbf27": "NNF", "afdbe16eac186c995cc8ab66d61099f9": "\\mathbf{skip} \\equiv v' = v", "afdc190074a7e451cec25ee2fde23fa2": "t_{max}", "afdc19e4310c707e92b582aecd28a644": "S(\\omega) \\ \\stackrel{\\mathrm{def}}{=}\\ \\delta(\\omega - \\omega_0)", "afdc1f597f5da9129cb20528886dee63": " (\\chi^2_k-k)/\\sqrt{2k} ~ \\xrightarrow{d}\\ N(0,1) \\,", "afdc4119083b0ad38a0d1cd797345461": "\\left(\\frac pq\\right) = (-1)^\\nu,", "afdc4644ec429d9465706365864bd97c": "P=\\frac{\\sum (p_{c,t_n}\\cdot q_{c,t_n})}{\\sum (p_{c,t_0}\\cdot q_{c,t_0})}", "afdd0c69120c41b24b2cabf1c174f092": "((1/2)*c_{2}-(1/2)*b_{2}+b_{2})*x^2-(1/2)*(c_{2}-b_{2})b_{2}^2=0", "afdd2fcb6b896c0b43a15dcfa6dff12b": " \\sigma_{p}", "afdd6441e5aa62db383501e01753a2f9": "\\omega_{s2}=\\sqrt{\\frac{2k_2M}{m_A}}", "afdd7e172b5ae1740b832dcc086b71f4": "\\mathrm{Ra}_T = \\frac{\\rho_{0}g\\beta\\Delta T_{sa}D^3 c}{\\eta k}", "afdd81ad51248107c8bc36e81bac6379": "\\,\\!\\tau", "afdd8781e996a65530eb66195282558d": "K_0,K_1,\\ldots,K_n", "afddb92e6a68767389a6b8c77c878c24": " \n\\begin{bmatrix}\n\\mathbf A^T \\\\ \\mathbf B^T \n\\end{bmatrix}\n^{+} = \\left[\\mathbf P_B^\\perp \\mathbf A( \\mathbf A^T \\mathbf P_B^\\perp \\mathbf A)^{-1}, \\quad \\mathbf P_A^\\perp \\mathbf B(\\mathbf B^T \\mathbf P_A^\\perp \\mathbf B)^{-1}\\right],\n", "afddd75c482cba087b3e4af0dba74a8f": " B \\in \\mathcal{B}_C", "afdddaf97c4b77acebb8087f4c492b36": "\\hat{x}[n] = \\sum_{k=0}^{N-1}\\hat{h}_ks[n-k]", "afddde347a49e683ca567853e9e1e1eb": "\\sigma = \n\\begin{bmatrix}\n500\\mathrm{ Pa} & 0 & 0 \\\\\n0 & -4000\\mathrm{ Pa} & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix}\n", "afddf59ae8d636db1911dc1fffccc203": "q_j \\, \\Phi(\\mathbf{r}) \\ll k_B T", "afddf9eba17cc26640ae718ecab61b86": " \\widehat{U}(\\theta) = e^{-i\\theta\\widehat{N}} ", "afde89a53c6ee788e497e5af1e0163ea": "1/\\kappa", "afdf4dd6ca19dee18de6b7d11bd3343e": " \\sum_{\\eta \\le p \\le \\xi} \\frac{w(p) \\ln p}{p} < \\kappa \\ln \\frac{\\xi}{\\eta} + C. ", "afdfb32f3adf86c5a3381b71dad2d82a": "e^{-2\\phi} \\left( - 2\\phi_{,vv} + 4 \\rho_{,v}\\phi_{,v} \\right) + f_{i,v}f_{i,v}/2= 0", "afdfce375c4f20a04076b087075c847a": "E_{kl}={u_{l,k}^2\\hbar^2\\over2m_0r_0^2}", "afdfe142eb72e32b065e4c79d3a70c23": " A > \\sqrt{N}", "afe011f6d9c91aa25a4ea3dbdc927a26": "c_{rs}", "afe020dbda290411bf55239d914b2006": " (C_\\bullet, d_\\bullet)", "afe05d2a96ad163b22a5ab87b8bdce31": "\\mathbf{L} = \\mathbf{r} \\times \\mathbf{p}", "afe0774103c8f49cb9a09d853ab8e8a0": "\\|\\mathbf{\\gamma}'(t_0)\\|,", "afe09df92ef6d5360c2bdabe9ada9032": "\n \\begin{align}\n (1) \\qquad & \\frac{Eh^3}{12(1-\\nu^2)}\\Delta^2 w-h\\frac{\\partial}{\\partial x_\\beta}\\left(\\sigma_{\\alpha\\beta}\\frac{\\partial w}{\\partial x_\\alpha}\\right)=P \\\\\n (2) \\qquad & \\frac{\\partial\\sigma_{\\alpha\\beta}}{\\partial x_\\beta}=0\n \\end{align}\n", "afe11cf106f24f2839c8a6d1fcbf97c2": "\nL_x = \\begin{bmatrix}\n-1 & 0 & +1 \\\\\n-2 & 0 & +2 \\\\\n-1 & 0 & +1\n\\end{bmatrix} * L\n\\quad \\mbox{and} \\quad\nL_y = \\begin{bmatrix}\n+1 & +2 & +1 \\\\\n0 & 0 & 0 \\\\\n-1 & -2 & -1\n\\end{bmatrix} * L.\n", "afe1480f77d9762423ff9298425cf60a": "a = -13.7R^{1/3} + 17.7G^{1/3} - 4B^{1/3}", "afe179de901c076d28827b7d370a73e7": "\\lambda_1,\\ldots,\\lambda_r", "afe1844c59f3013ff484ef86e943c464": " \\sum_{n=1}^k w_n = 1\n", "afe1a98e10ba60c72df9747c66c92aff": "B_{\\lambda k}=e_{\\lambda}(k)\\cdot B_k,", "afe1ddbcb1e6b6015d485848c0aab59a": "a=(a_{-N},\\dots,a_0,\\dots,a_N)", "afe1e18555447b7e6ac89db43224393c": "H = H_0 + V \\,", "afe1fc1e07c36281b750180ca15d274c": " (X_\\alpha - Y_\\alpha) P^\\alpha = 0 ", "afe20ffba4f803d4b5919858efd44cbb": " \\mathrm{ PO_{4}^{3-} > SO_{4}^{2-} > COO^{-} > Cl^{-}}", "afe22e83384c35cf68b27287f95757e1": " \\Pi_2 = ", "afe283315634f8f34a34ddb1eeaece49": "\\chi_v", "afe28b9e7942fc984b7f22cce05c2d1f": "\\lambda\\in\\mathbf{R}^{+}", "afe2e1a5c096f48578b426a05b6e4063": " B_1\\times,\\dots,\\times B_n.", "afe32c7148ada4dffdb9d10e4c892e17": "\\Bbb Z^n", "afe37fa19ca9e3c99e14255bb7ba6957": "P = S = V_\\mathrm{RMS} I_\\mathrm{RMS} = I_\\mathrm{RMS}^2 R = \\frac{V_\\mathrm{RMS}^2} {R}\\,\\!", "afe40a8c76bf0f0805047958fd67f7a9": "\\omega + i\\delta \\to 0", "afe40fb57dc14ab85e9196022e9411bd": "C_J=n^2 (x^2+y^2) + 2 \\left(\\frac{\\mu_1}{r_1}+\\frac{\\mu_2}{r_2}\\right) - \\left(\\dot x^2+\\dot y^2+\\dot z^2\\right)", "afe4276b9c73c38447a55f126178ea01": "\\frac{dE}{ds} = \\frac{dR}{ds}\\mathbf{N}\\left ( s \\right ) ", "afe4824a4b6786d4c33b9dc48e014800": "\\frac{\\partial^2V(x)}{\\partial x^2}+ \\omega^2 LC\\cdot V(x)=0", "afe4a35d0a6261537967336784131574": "{r-1 \\choose 4} \\cdot \\left[ {13-r \\choose 2} \\cdot 16,384 + {13-r \\choose 1} \\cdot 36,864 + 28,160 \\right]", "afe4a5e22fca50adf94d0577162ef65a": "\\prod_{i,j} (1 - x_iy_j)^{-1} = \\sum_{\\lambda} s_{\\lambda}(x)s_{\\lambda}(y)", "afe4d4db4fb1c4cb5b51fd8edbb3288a": "\\langle J,J_z|\\vec \\mu_J|J,J_{z'}\\rangle\\cdot\\langle J,J_{z'}|\\vec J|J,J_z\\rangle = g_J\\mu_B\\langle J,J_z|\\vec J|J,J_{z'}\\rangle\\cdot\\langle J,J_{z'}|\\vec J|J,J_z\\rangle", "afe503561015b8b0d0eaa82d57686ac4": " {\\rm Compression\\;Ratio} = \\frac{\\rm Uncompressed\\;Size}{\\rm Compressed\\;Size}", "afe512117d40d6083b4489be31612309": "\\scriptstyle(0,\\,\\infty)", "afe5622d04523c2d4115f10356714171": " \\frac{s}{s_0}=\\frac{P_a(1-\\theta)}{1-P_{m1}}\\frac{P_a+P_d}{P_a} ", "afe58327e0e7922ce4286c66bba45290": "\n\\begin{alignat}{2}\n\\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v \\right) + \\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v_e \\right) & = \\chi \\left( C_m \\frac{\\partial v}{\\partial t} + I_\\text{ion} \\right) \\\\\n\\nabla \\cdot \\left( \\mathbf\\Sigma_i \\nabla v \\right) + \\nabla \\cdot \\left( \\left( \\mathbf\\Sigma_i + \\mathbf\\Sigma_e \\right) \\nabla v_e \\right) & = 0\n\\end{alignat}\n", "afe5c3592ee8114c274ca27a93f104fd": "\\mathbb P(\\theta_k^1=s|\\theta_{k-1}^1=r)=\\Lambda^1(s|r)", "afe5e0fd369ee417cb70d33df9ad5231": "(Q, I-Q)", "afe5f2c99b3f797e6732ab66963d304d": "j/r", "afe64611637aecca8cbd6f96b702fda2": "f_P \\colon [\\ell] \\mapsto [m]", "afe6554c37b50f1eadd4627b049ed040": "\\scriptstyle \\log_{10} {P_{mmHg}} = 8.04494 - \\frac {1554.3} {222.65+T}", "afe6758df8ee953fa65cf7327450dd89": "N_i^n(x)=\\frac{x-u_i}{u_{i+n}-u_i}N_i^{n-1}(x) + \\frac{u_{i+n+1}-x}{u_{i+n+1}-u_{i+1}}N_{i+1}^{n-1}(x) ,", "afe6d1ab28eef41ae56282851a997bdf": "f^{-1}(\\{y\\})=\\{x \\in X : f(x) = y\\}", "afe6deec500fca0c3400132e6a09cf95": " \\omega = \\frac i2 \\sum_{j,k} h_{jk} dz_j \\wedge d\\bar{z_k} ", "afe6f24c722128a50e8a1afb912cebf2": " t = w + xi + yj + zk, \\ ", "afe74413c7235dc504027b8738dc4a3f": "\\sum_j \\alpha_j v_{\\pi(j)}", "afe75eca7c038d64e113f0a77e2a924f": "\n\\mathbf{A}\\mathbf{u} = \\mathbf{0} \\Rightarrow [\\mathbf{A}_1:\\mathbf{A}_1\\mathbf{B}]\\begin{pmatrix}\n \\mathbf{u}_1 \\\\\n \\mathbf{u}_2 \n\\end{pmatrix} = \\mathbf{0} \\Rightarrow \\mathbf{A}_1(\\mathbf{u}_1 + \\mathbf{B}\\mathbf{u}_2) = \\mathbf{0}\n\\Rightarrow \\mathbf{u}_1 + \\mathbf{B}\\mathbf{u}_2 = \\mathbf{0} \\Rightarrow \\mathbf{u}_1 = -\\mathbf{B}\\mathbf{u}_2 ", "afe796dd3dd2ba89c26cd0eb7ff55544": "t_0\\,\\!", "afe7c3d764df31f273939b42f3ccf777": "\\ A ", "afe7c494638dda399e3e9577a6f7da0c": " s^2 = a p^b ( 1 - p )^c ", "afe7f90dd34deaabbae2c26aab299e8c": "\\hat{Y}", "afe81a3ea48d038b183fda3791857828": "\\eta(x,y)", "afe842f7f239c0f8fabbac2bfc1e75c7": "p_1, \\ldots, p_r", "afe84d5a90a63153a65c7023883c059a": "MRT", "afe88163817b2bb90440ddf605816e24": "(u_r, u_\\theta)", "afe88de18153942f197ff5fc65562337": "d_{\\kappa\\lambda}", "afe8a2094a85cf82781b9eb7a83e0306": "\n \\boldsymbol{S} = S^{ij}\\mathbf{b}_i\\otimes\\mathbf{b}_j = S^i{}_j\\mathbf{b}_i\\otimes\\mathbf{b}^j = S_i{}^j\\mathbf{b}^i\\otimes\\mathbf{b}_j = S_{ij}\\mathbf{b}^i\\otimes\\mathbf{b}^j\n ", "afe8b2e7277d3f141f5a1ecb0727c813": "\\frac{dN}{dt} = -\\lambda N", "afe8cc720c8b27747edda51573599d1e": " F_A = \\mathbf{F}_A \\cdot \\mathbf{e}_A^\\perp, \\quad F_B = \\mathbf{F}_B \\cdot \\mathbf{e}_B^\\perp.", "afe8d014102a3696d954b8f768636aee": " \\text{LSF} = \\frac{d}{dx}\\text{ESF}(x) \\approx \\frac{\\Delta \\text{ESF}}{\\Delta x}", "afe932329be81b4175a10f417e03db77": "H_2 B ^ - \\rightleftharpoons\\ H ^ + + H B ^ {-2} \\qquad K_2 = {[H ^ +] \\cdot [H B ^{-2}] \\over [H_2 B^ -]} \\qquad pK_2 = - \\log K_2 ", "afe99b9bcb72a78f99527ca659a11dd3": "\\delta=\\frac{1}{4\\pi}\\iint_{R_{st}}\\frac{\\begin{vmatrix}x&y&z\\\\\\dfrac{\\partial x}{\\partial s}&\\dfrac{\\partial y}{\\partial s}&\\dfrac{\\partial z}{\\partial s}\\\\\\dfrac{\\partial x}{\\partial t}&\\dfrac{\\partial y}{\\partial t}&\\dfrac{\\partial z}{\\partial t}\\end{vmatrix}}{(x^2+y^2+z^2)\\sqrt{x^2+y^2+z^2}}dsdt.", "afe9ca07e7a626890b8f253f024d1ce5": "\\int_0^\\infty {e^{-ax^{2}}}\\, dx=\\frac {1}{2} \\sqrt{\\frac {\\pi}{a}}", "afe9ddcefaeea53a03575356d2dec5b9": "\\int\\frac{x\\;\\mathrm{d}x}{1+\\sin ax} = \\frac{x}{a}\\tan\\left(\\frac{ax}{2} - \\frac{\\pi}{4}\\right)+\\frac{2}{a^2}\\ln\\left|\\cos\\left(\\frac{ax}{2}-\\frac{\\pi}{4}\\right)\\right|+C", "afea10320fff641049e2a60c5c6e3c19": "\\mathrm e^{-\\gamma t}", "afea2cd4187fdd2ee5ebf07984d4a35a": "H_\\mathrm{v} = E_\\mathrm{v} \\cdot t \\,,", "afea9d6097c5b350e79e5d205471b02b": " \\theta (\\text{degrees}) = \\Delta A \\left( \\frac {\\ln 10}{4} \\right) \\left( \\frac {180}{\\pi} \\right)\\, ", "afeabfae7b1e1b20ce5c3f36142b5ea5": "v,w", "afebb5215686bdbe84b665a08aeefa88": "\\textstyle \\bar{x}_k", "afec38e0fd5f53d24d373c52d1dab7ba": "n=\\lambda", "afec9966f142d82db71c300d6437d2dc": "(A\\downarrow \\mathcal{C})", "afecb9f7500b9d4227abdfdd37b82979": "\\begin{align} 2\\cdot R_*\n & = \\frac{(50.1\\cdot 2.54\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 27.4\\cdot R_{\\bigodot}\n\\end{align}", "afecf589bdfe50566a524f55cc409475": "Supp\\{(p_{i}(t))\\}=(a_{i},b_{i})(\\forall i)", "afecfee4b036fc8f6d29598dd600f491": "f'\\;", "afed042f37256e42dd815513269f1b57": " \\frac{1}{\\sqrt {f}} = {a\\left[ \\ln\\left( \\frac{d}{q} \\right) + D_{CFA} \\right] } ", "afed11449ed0946b5e7337451c138d6f": "\n-\\omega^2 m A_1 e^{i \\omega t} = - 2 k A_1 e^{i \\omega t} + k A_2 e^{i \\omega t} \\,\\!\n", "afed126f109de2e8e6766afaedbee17f": "\\left[\\psi\\frac{\\partial\\varphi}{\\partial n}-\\varphi\\frac{\\partial\\psi}{\\partial n}\\right]dS", "afed82300a064cbf39b00d75d78bf566": "B_i\\subset 3\\,B_{j_k}\\subset X", "afedb0cde602a4affdbab549fecbb728": "\\left(\\frac{\\mbox{Net Income}}{\\mbox{Net Sales}}\\right)\\left(\\frac{\\mbox{Net Sales}}{\\mbox{Total Assets}}\\right)", "afee0388ee68b65d99611a00a29f258d": "\\mathbf{\\bar{x}}", "afee18bcfbe98e4073c1e2ebd549eb0f": "(R,G,B)", "afee5b696933233fbab6627376f84b16": " \\not\\models", "afee62fc420b82a7e1e7f22b1b566e75": "\\{z\\in\\mathbb{C}: \\operatorname{Im}(z)\\in(-\\lambda_-,\\lambda_+)\\}", "afee7b50bfb3df46223a44b956894592": "\\scriptstyle \\mathcal{H}", "afeebcd3436e678ba1f486cbb34182e7": "\\mathbf X^{\\rm T} \\mathbf X", "afeefadf80ee64c619b3807892694b94": "[-\\pi/2,+\\pi/2]", "afef3916048008e4d1416aad7b1d13fc": "\\beta_{S,t} \\,", "afef79e9a2e58551a66fe7a6cb7406d5": "y = \\frac{2v}{2u - 8v + 4}", "afefdbca615c249eccc7c70f939b3159": "\nQ = \\frac{1}{R} \\sqrt{\\frac{L}{C}} = \\frac{\\omega_0 L}{R}\n", "aff00f79ac307d18b4a580c88f911736": "\\mathrm{Ry}=me^{4}/2\\hbar ^{3}c", "aff04715e3115e859636a3caef76dc85": "f_{x(x^2-3y^2)} = N_3^c \\frac{x \\left( x^2 - 3 y^2 \\right)}{2 r^3 \\sqrt{3}} = \\frac{1}{\\sqrt{2}}\\left(Y_3^3 - Y_3^{-3}\\right)", "aff051df65aaac08e1275f435d690820": "r = 8.657 \\sqrt{{D} \\over f}", "aff0dba5c6eba74e63e8dd3b3f21d198": "\\{\\psi_i\\}", "aff1445921876a1dfc47d01fe1c371fc": " (I-Q) ", "aff151781c4668900d67924b8a11c8ed": "m_x\\in[0,1)", "aff1bb96c055033f056d997397b51628": "(1-b_{-1}z)y_{n+1}-\\sum_{j=0}^s (a_j+b_jz)y_{n-j}=0", "aff258936238d58ff1fe8dda35dd0363": "\nH_{\\mathrm{rot}} = \\tfrac{1}{2} ( I_{1} \\omega_{1}^{2} + I_{2} \\omega_{2}^{2} + I_{3} \\omega_{3}^{2} ),\n", "aff2700f964bb01858872bb7f43a5de7": "G\\to G/H", "aff2d63e4d2db1c63e3bc065a447a2ce": "{\\Delta V_w} = V_{w1}-(-V_{w2})", "aff327f09066d09dad830d56066d41d5": "i=\\arccos{h_\\mathrm{z}\\over\\left|\\mathbf{h}\\right|}", "aff35d268d3ea593ea5305792105e758": " (\\hat{c}- \\hat{a}) = \\sqrt{\\text{(sample variance)}}\\sqrt{6+5\\hat{\\nu}+\\frac{(2+\\hat{\\nu})(3+\\hat{\\nu})}{6}\\text{(sample excess kurtosis)}}", "aff384738fe2decf57013490abec5928": " f_{-1}=1, f_d=1. ", "aff3a9d6c254575807ad758e43285b4f": " r \\ = \\ e^i - 1", "aff3cc7e4cc75ecf5bb88695bfeaf164": "f(x) = f(x_0) + f'(x_0)(x-x_0) + \\frac{1}{2} f''(x_0)(x-x_0)^2 + R ", "aff3e1f06c32dd4cd25b4b5f545b8628": "I \\in l(S)", "aff3e765701e98357e13aa352a8bb0b2": "\\|f\\|_{H^\\infty} = \\sup_{|z|< 1} \\left|f(z)\\right|.", "aff3f1bf7f54bdaacefc2e773563eadb": "\\tfrac{1}{6}", "aff42004f3427f3ac03d0721a62c0385": "\\left\\lfloor x \\right\\rfloor", "aff46a7b017e3903b1e0eb1e6599a020": " [ \\sigma_a , \\sigma_b ] = 2i \\hbar \\varepsilon_{abc} \\sigma_c ", "aff5116cb8374fc72950e884fbafc860": " \\angle DVC = \\angle DVE + \\angle EVC. \\, ", "aff515e65fe7b925d19b683a5c517de8": "i_\\max, j_\\max, t_\\max", "aff53a65025654b0bff962a1567e7104": "M_A = PL", "aff54f2a5b832f55c277a7b3dbef0188": "H=A\\setminus\\bigcup_{k=1}^{\\infty} A_k", "aff57e3c0554d5811500e8784e76951e": "\\begin{matrix} {11 \\choose 1}{4 \\choose 2}{10 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "aff5bef244318bec61bc6ec6de959c7c": "Gr(\\Gamma) = \\{(a,b)\\in A\\times B : b\\in\\Gamma(a)\\}", "aff5dfa28167d27593e19861e8d48d42": "\n\\begin{align}\nc_1(n)& = \n1\\\\\nc_2(n) &=\n\\cos n\\pi\\\\\n\nc_3(n)&=\n2\\cos \\tfrac23 n\\pi\\\\\n\nc_4(n)&=\n2\\cos \\tfrac12 n\\pi\\\\\n\nc_5(n)&=\n2\\cos \\tfrac25 n\\pi + \n2\\cos \\tfrac45 n\\pi\\\\\n\nc_6(n)&=\n2\\cos \\tfrac13 n\\pi \\\\\n\nc_7(n)&=\n2\\cos \\tfrac27 n\\pi + \n2\\cos \\tfrac47 n\\pi + \n2\\cos \\tfrac67 n\\pi \\\\\n\nc_8(n)&=\n2\\cos \\tfrac14 n\\pi + \n2\\cos \\tfrac34 n\\pi \\\\\n\nc_9(n)&=\n2\\cos \\tfrac29 n\\pi + \n2\\cos \\tfrac49 n\\pi + \n2\\cos \\tfrac89 n\\pi \\\\\n\nc_{10}(n)&=\n2\\cos \\tfrac15 n\\pi + \n2\\cos \\tfrac35 n\\pi \\\\\n\\end{align}\n", "aff5f1c431e502f35eddb07efc99ca6c": "\n\\begin{align}\n& {} \\quad \\int_0^\\pi \\sin^{n-j-1}(\\phi_j) C_s^{((n-j-1)/2)}(\\cos \\phi_j)C_{s'}^{((n-j-1)/2)}(\\cos\\phi_j) \\, d\\phi_j \\\\[6pt]\n& = \\frac{\\pi 2^{3-n+j}\\Gamma(s+n-j-1)}{s!(2s+n-j-1)\\Gamma^2((n-j-1)/2)}\\delta_{s,s'}\n\\end{align}\n", "aff65a82d805203f10c2846de71cc855": "\\min \\sum_{j = 1}^n \\left ( V_{j}\\prod_{i = 1}^m q_{ij}^{x_{ij}} \\right )", "aff6699dde72b45dade847ebe6484660": " j \\in \\{ 1,\\dots,N_i \\} ", "aff66e6225ed97dfe3370b769175f1dd": "\\frac{d}{dt} \\langle \\psi | Q | \\psi \\rangle = \\frac{-1}{i \\hbar} \\langle \\psi| \\left[ H,Q \\right]|\\psi \\rangle + \\langle \\psi | \\frac{dQ}{dt} | \\psi \\rangle \\,", "aff6a1d3d52fec1872c8ece86a3b68c4": " \\ell(1) = \\ell(x) = \\ell(y) = \\ell(x^2) = \\ell(xy) = \\ell(y^2) = \\ell(xy^2) = \\ell(y^3) = 0, \\ \\text{and} \\ \\ell(y^4) = 3 .", "aff6bd8743c09b461f914b328d7157e1": " kT_D", "aff6c11715a8f41470fe5bfa72092d28": "\\operatorname{var}(X)=\\frac{1}{8}.", "aff7039c606ac6b303f7adbc8bf6d0b4": "{\\rm KILL}[d : y \\leftarrow f(x_1,\\cdots,x_n)] = {\\rm DEFS}[y] - \\{d\\}", "aff7cde03d846864b09d76221a69722c": "\\xi = x - c\\, t.\\,", "aff80297f6793f589fff443a151db376": " \\sigma = \\tfrac{s}{bA} ", "aff8695bbb281ee8e7c9451f58eaa1a8": "\\beta_{max} = \\frac{\\beta_N I}{a B_0}", "aff8ace2fb432df229d6c8be05649c70": "\n \\begin{align}\n \\boldsymbol{\\sigma} \n & = \\cfrac{2}{J}~\\left[\\left(J^{-2/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_1} + \n J^{-2/3}~\\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right)~\\boldsymbol{B} - \n J^{-4/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\boldsymbol{B}\\cdot\\boldsymbol{B}\\right]\n + \\\\\n & \\qquad\n 2~J~\\left[-\\cfrac{1}{3}~J^{-2}~\\left(\\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_1}+\n 2~\\bar{I}_2~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right) + \n \\cfrac{1}{2}~J^{-1}~\\cfrac{\\partial W}{\\partial J}\\right]~\\boldsymbol{\\mathit{1}} \n \\end{align}\n ", "aff8dd966c419974b6cd4950ce967bcd": "\\ \\mathbf{x} = \\mathcal{F}_{n}^{-1} \n\\left [ \n\\left (\n\\frac{(\\mathcal{F}_n(\\mathbf{b}))_{\\nu}}\n{(\\mathcal{F}_n(\\mathbf{c}))_{\\nu}} \n\\right )_{\\nu \\in \\mathbf{Z}}\n\\right ]^T.\n", "aff8de0e28a3b7ef4dcbe9d539fdd75a": "E=b+au\\,\\!", "aff8e35c12fc128b8a7889bdfab87b15": "12-n", "aff8f5af0a09229562296c3473c98ee7": "CS(\\Gamma)=\\frac{1}{2\\pi^2}\\int d^3x\\epsilon^{ijk}\\biggl(\\Gamma^p_{iq}\\partial_j\\Gamma^q_{kp}+\\frac{2}{3}\\Gamma^p_{iq}\\Gamma^q_{jr}\\Gamma^r_{kp}\\biggr).", "aff91ac71f8e502703bfc96ab49950d9": "\\frac{\\operatorname df}{\\operatorname dt} = { \\partial f \\over \\partial x}{\\operatorname dx \\over \\operatorname dt }+{ \\partial f \\over \\partial y}{\\operatorname dy \\over \\operatorname dt }.", "aff96874c6d2fb4aa3c671f56a4dee11": "\\mathit l^{\\prime} \\ge \\mathit l^{ }_{ } ", "aff9c25ec2c8622e769d32e8e3f96929": "\n\\begin{bmatrix}R\\\\G\\\\B\\end{bmatrix}=\n\\begin{bmatrix}\n1.96253&-0.61068&-0.34137\\\\\n-0.97876&1.91615&0.03342\\\\\n0.02869&-0.14067&1.34926\n\\end{bmatrix}\n\\begin{bmatrix}X\\\\Y\\\\Z\\end{bmatrix}\n", "aff9d9b0fc82167682540245922e66f8": "\\arccot (1/x) = \\tfrac{1}{2}\\pi - \\arccot x =\\arctan x,\\text{ if }x > 0 \\,", "affa6dbc3c09b8943cd9959ab4018eb2": " {H_1 \\over H_2} = { \\left ( {N_1 \\over N_2} \\right )^2 }", "affa7a1ac2c378c4080ba7dd58c28471": "\\begin{smallmatrix}\\frac{r_{a}}{r_{p}} = \\frac{2}{1-e}-1 = 2/0.2488-1=7.039.\\end{smallmatrix}", "affa9846bb70e3042cfdec4a23c04404": "\\tilde{S}_{i}", "affaa7a76141cb801102158f622f709c": "U = -\\mathbf{M}\\cdot\\mathbf{B} = -k\\mathbf{B}\\cdot\\mathbf{B} = -k \\left (B_x^2 + B_y^2 + B_z^2 \\right ),", "affaf0f23af9e4d69b3f08c1885e9697": "\\frac{1}{\\tau} = \\sum k_i", "affb0aa2f7561373e5a8aa595be1ccaa": " 0\\leq d_i < b ", "affb2e390a91194fcb31e70227f142a4": "-5 < \\beta \\le -3", "affb5c4d8bb20bda45c1ac50ea593f2f": "\\log 3 ", "affb79599be3b9d3c789a25ae57c54db": "\\|\\mathbf{x}-\\mathbf{x'}\\|_2", "affbabb4dfa253dd63cda734a64afdf6": "\\tfrac{1}{7}\\scriptstyle{\\sqrt{3(2194+1513\\sqrt{2})}}", "affbf749523b86c7464b99cbf614b5c7": "\\sigma_1 = \\sigma_2 = \\tau_2 = \\tau_3 = 0\\Big.", "affc00784827bd77bfb97fce22037412": " C_n^{s_1,\\ldots,s_k} ", "affc298e642d11e10028f1da29285014": "\n \\mathbf{M} = \\mathbf{r}\\times\\mathbf{F} = \\left|\\begin{matrix}\\mathbf{E}_x & \\mathbf{E}_y & \\mathbf{E}_z \\\\ x & 0 & 0 \\\\ 0 & 0 & -F \n \\end{matrix}\\right| = Fx\\,\\mathbf{E}_y \n \\quad \\text{and} \\quad M_y = \\mathbf{E}_y\\cdot\\mathbf{M} = Fx \\,.\n ", "affc7981b9750cefd05a5212016f6332": " { \\mathit l \\over \\mathit l^{*} } = {1 \\over 2} ", "affcdbd83e4ae5ec519060a90a0cfcd7": "B_i (0,3)", "affcf869b8a546e3859337aad4bc3d7e": "\\hbar \\rightarrow 0", "affd18a84c140bedf090b76f84a188be": "\\sigma_i\\sigma_{i+1}\\sigma_i = \\sigma_{i+1}\\sigma_i\\sigma_{i+1}.\\ ", "affd2c09e3a32380c7b54e72ff57f82a": "(A, \\mathfrak{m})", "affe3d9010ad710d08fcb841a9dbe39c": "(2y)^{3} - 3(2y) - 1 = 0", "affe7eb8a9c08d99600141c3947f3f77": "\\scriptstyle m_e\\,", "affec806555703ca808d6c9711057c95": "T f(x) = \\int_0 ^1 K(x,y) f(y) dy.", "affee6a996ae500cb633cbb774a86c57": "dU =C_{V}dT +\\left[T\\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - p\\right]dV.\\,", "affee8090d2e80f4efbbda7dc35b8f4e": "B_>", "afffdbebfc7ca7dfd5bafe7a9d2d3f8b": "L, M", "affff745ed2f95af918553073a22b54c": "||b_k^*||\\ge (\\mathrm{det}(L))^{1/k}\\cdot 2^{(1-k)/4}", "afffff6aa013efeac63fa570a3899bec": "f(x_0), f(x_1), \\ldots, f(x_n)", "b0001fd4675c7008f2f52169719997a2": "N_{++} + N_{+-}+ N_{-+} + N_{--}", "b00060662c8f745eecb9582c31ef93ad": "t \\geq \\tau_{B} (\\omega) \\implies \\varphi (t, \\vartheta_{-t} \\omega) B \\subseteq K(\\omega).", "b0006e9663d585843184bec2b33c05b2": "p^\\mu= \\left(E, |\\vec{p}| \\sin{\\theta} \\cos{\\phi}, |\\vec{p}| \\sin{\\theta} \\sin{\\phi}, |\\vec{p}| \\cos{\\theta} \\right) \\,", "b000985e6015021cf1dd98e1f83c1ce0": "x \\leftarrow a", "b000b15cb551b01fb0219b5d7e94d664": "A_{n-1} = A + \\alpha q^{n-1} \\quad , \\qquad A_n = A + \\alpha q^n \\qquad \\text{and} \\qquad A_{n+1} = A + \\alpha q^{n+1}.", "b000c5445db781f044e39049d358cd21": "46_{11} \\ ", "b001b157212752e9cb7744e56fa6a7d2": "r(\\tau) = r(0) = 0, \\, \\tau > 0", "b0023c4f531bdbf5c77138b57d9633a0": "\\displaystyle{Z_3=\\begin{pmatrix} A_1 -B_1^2(D_1+A_2)^{-1} &-B_1B_2(D_1+A_2)^{-1} \\\\-B_1B_2(D_1+A_2)^{-1} & D_2 -B_2^2(D_1+A_2)^{-1}\\end{pmatrix}.}", "b00260ecfcf3df4bfee482273ac9fcc3": "N = \\frac{MC}{R} = \\frac{10\\times 15}{5} = 30", "b0026bbb574d2cb7b9de5e32219267a9": "\\displaystyle{j(a,\\lambda)=(\\mu -\\mu^2a,\\mu).}", "b00284199b31f31ed91703531b69db8f": "\\,u", "b002aab1af720ff91ad015ea67d2a719": "f(\\hat{x}) = 0", "b002e3e7389da59631da533b49c71de6": " (I-2| \\omega\\rangle \\langle \\omega|)|\\omega\\rang=|\\omega\\rang-2| \\omega\\rangle \\langle \\omega|\\omega\\rang=-|\\omega\\rangle = U_\\omega |\\omega\\rang", "b002f9111128c6302013e9f82eb9b672": " \\| \\cdot \\|", "b003108742a9672754bc4b83ea830337": "x_n \\approx -n + \\frac{1}{\\pi}\\arctan\\left(\\frac{\\pi}{\\ln n}\\right)\\qquad n \\ge 2", "b00333abf1a19d57a26e4426e7d8e26c": "|h|=0,\\frac{1}{2},1", "b00344e5d946804091f7f888d150a257": "p=4m+3", "b0037f1045ad79ee42046a83ffc2d07d": "[0,\\infty]", "b003c7fc3170dc8bad356388a7dbe4ee": "n_i > m", "b003f33a9ce4c9488dfb064afe3e47e4": "\n3Nk_{B} T = -\\biggl\\langle \\sum_{k=1}^{N} \\mathbf{q}_{k} \\cdot \\mathbf{F}_{k} \\biggr\\rangle = 3PV,\n", "b004111fd6f112f5b6b0576b3c5a764e": "s_{(d,\\pm)} ", "b0041d4dea816bf955ba43b293fc34f3": "\n\\frac {\\mathrm d} {\\mathrm dt} \\boldsymbol{H} \\equiv \\nabla \\vec{v} \\cdot \\boldsymbol{H}, \\qquad \\boldsymbol{H} (t=0)=\\boldsymbol{I}\n", "b004297edfcdf9b75ff0a331a0d7bbe5": "x_{n+1}=1", "b004694c650240deecb7bce32acc203a": "\\scriptstyle\\sup_t G_n(t)", "b004791c5701ed55766b9f85970fad77": "[\\![\\sigma_1]\\!] \\times \\; \\dots \\; \\times [\\![\\sigma_n]\\!]", "b004a035d5d920dbdec415378b0c3f50": "I(X;Y;Z;\\cdots) = \\mu(\\tilde X \\cap \\tilde Y \\cap \\tilde Z \\cap \\cdots);", "b004e1767d7881e8b70ad2311a03ee1b": "b(x,t)", "b004efc8e71f975fdaeccf6ccbbad954": "x = ct \\gamma_0 + x^k \\gamma_k", "b005218d3310e485f612b351b24cda5a": "\\mathbf{\\Epsilon = \\left(C^TC\\right)^{-1}C^TA }", "b005347dc23869aa4eafd92ccf80e2f7": "\\scriptstyle{\\boldsymbol{\\hat{a}}}\\in\\mathbb{R}^n", "b005a280e339f9ab9a6317daeb8154f8": "w = \\frac{{t_1}}{{R_1}} + \\frac{{t_2}}{{R_2}}", "b005b22634b1601c4c4fddaea59e6594": "v_1,\\ldots,v_n", "b0060195a36cdadf999b0b73b52086a9": "\\vec{r}_A", "b006290a22399ce40bb5bb0a51aefdbd": "\\vec v_P = \\vec v_C + \\vec \\omega \\times (\\vec r_P-\\vec r_C)", "b0063cda106f340e8ad248c959a90aad": "\\mathcal{C}^2|\\psi\\rangle = \\eta_C \\mathcal{C} |\\bar{\\psi} \\rangle = \\eta_{C}^{2} |\\psi\\rangle = | \\psi \\rangle", "b00642cea31cd9b3d1d5755857dec23b": " \\mathrm{N_2 + 8 \\ H^+ + 8 \\ e^- + 16 \\ ATP + 16 \\ H_2O \\longrightarrow 2 \\ NH_3 + H_2 + 16 \\ ADP + 16 \\ P_i} ", "b0066e761791cae480158b649e5f5a69": "\\textstyle k", "b00675074ace943c92383f0e87dc8b7b": "\\text{abc}", "b00679d443f5f27a021127769f2b8859": "(x_i,y_i)\\quad i=0,1,\\dotsc,n", "b006994ab0e27c08f0fc2e9ba2748b14": "\n\n\\dot C_{CW} \\,\\, = \\,\\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 {{8R} \\over {3\\,\\pi \\,v}}\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\hat Q_0 \\,\\, = \\,\\,{{3\\,\\pi \\,v\\,\\dot C_{CW} } \\over {8R\\,\\varepsilon \\,k\\,F_m \\,\\phi }}", "b006e12213faf27d73498e2bc181fc7d": " \\Gamma_{ij}^k~\\mathbf{g}_k = \\cfrac{\\partial \\mathbf{g}_i}{\\partial \\xi^j} \\quad \\implies \\quad\n \\Gamma_{ij}^k = \\cfrac{\\partial \\mathbf{g}_i}{\\partial \\xi^j}\\cdot\\mathbf{g}_k = -\\mathbf{g}_i\\cdot\\cfrac{\\partial \\mathbf{g}^k}{\\partial \\xi^j}\n ", "b006fab8677f31b7036975c46ab92b36": "0 < z < 1", "b0071619c6ae2488c1ad4d779e2ca17b": "\\Phi: (x,y) \\mapsto (X,Y,Z,T) = (x,y,1,x^2) ", "b0074775711663d90d0bd4fd93d98641": "W = \\int_C \\bold{F} \\cdot \\mathrm{d}\\bold{x} = \\int_{\\mathbf{x}(t_1)}^{\\mathbf{x}(t_2)} \\bold{F} \\cdot \\mathrm{d}\\bold{x} = U(\\mathbf{x}(t_1))-U(\\mathbf{x}(t_2)).\n", "b00754223390cb14caedac2b01de318d": "A_\\text{ellipse} = \\int_\\text{ellipse} dA_\\text{ellipse} = \\iint_\\text{ellipse} abr\\,dr\\,d\\theta = ab \\int_{0}^{2\\pi} \\int_{0}^{1}r\\,dr\\,d\\theta = ab\\pi.", "b00763b17daecc1fabce5ef9d5169e83": " k_{\\rm A} = 1", "b007919465b19dca1570b0b7db79ad6d": "\\bar{x} = x - x_{\\mathrm{eq}} ", "b007b7e594eec03d9eb464a0f0babe39": "u = x^{\\frac{\\alpha-1}{2}}e^{-x/2}L_n^{(\\alpha)}(x)", "b0081e7ab659867d6e17ffc181c29a00": " \\det(M) = \\sum_{(P_1,\\ldots,P_n) \\colon A \\to B} \\mathrm{sign}(\\sigma(P)) \\prod_{i=1}^n \\omega(P_i). ", "b0085c33d2ded3045c17916a6ddc719b": "T = \\kappa \\sqrt{m/k}", "b008971cb69c06e15d5bbc0e9a7bc8c5": "\\mathrm S_2\\mathrm O_8^{2-}\\underset{hv}{\\longrightarrow}2 \\ \\mathrm{SO}_4^{-\\bullet}", "b008dd8e90750a9b97e26f9c8deb3576": "\\begin{cases} X_{t} \\sim \\mathrm{Unif} (\\{X_{t-1} - 1, X_{t-1} + 1\\}), & t \\mbox{ an integer;} \\\\ X_{t} = X_{\\lfloor t \\rfloor}, & t \\mbox{ not an integer;} \\end{cases}", "b0090a2790fdef674bc8d983e5cc9def": "\\tfrac{2A}{r^2}dt = d\\theta,", "b00a2edc6de629791fc9cb56d38018e4": " \\|\\mathbf{A}\\|^2 = - (A^0)^2 + (A^1)^2 + (A^2)^2 + (A^3)^2 ", "b00a74f834a170dbec9c2f6e35593cbf": "t_0 + h", "b00aa29f1629fdf37bd023cd28598ecb": "A \\in \\mathbb{M}_m", "b00acd28de5e0d42095798e5b8481319": "L_B = \\frac{\\mu_0}{\\lambda_0}\\cdot S_B \\ ", "b00ace0e65d8c5a19661e47d8461d9a1": "\\begin{align}\n P &= 3x^2 - 2x + 5xy - 2 \\\\\n Q &= -3x^2 + 3x + 4y^2 + 8\n\\end{align}", "b00ad9e2c97b10d4e72f7cbe8e23ef20": "\\operatorname{det} \\left[ \\mathbf{M} - \\lambda\\mathbf{I} \\right] = 0 ", "b00af9c34835b53f9f4d7ee53567ea94": "\\mathbf{J}_2", "b00b94618e697ab012c5e45044ec8e5a": "C_{\\mathrm y} = \\frac{\\int y S_{\\mathrm x}(y) \\; dy}{A}", "b00be7287468287f01141e036464c792": " \\sum_\\rho F(\\rho) = \\operatorname{Tr}(F(\\widehat T )).\\!", "b00be9a0adb1acb9683cac0a51703da2": "P = \\epsilon_0(\\chi^{(1)}:E + \\chi^{(2)}:EE + \\chi^{(3)}:EEE + \\dots)", "b00bed3e251d962e3e813625c28aac45": "b_i \\equiv b_j \\bmod (m_i,m_j)", "b00c079c0af9d49ad641b915c7b7dcc9": "t=\\infty", "b00c5bd0cdc3e9432babf4fb3512b848": "\\tan", "b00c6dfa7f80b3a6c6f37882a5f22033": "Fe\\;\\rightarrow\\;Fe^{2+}\\;+\\;2\\,e^-", "b00c715f0bb45efce755d6f3bc8697ba": "\\sqrt{\\lang \\psi | \\hat O^2 | \\psi \\rang - (\\lang \\psi | \\hat O | \\psi \\rang)^2}", "b00c7d2695a21ff6cd316dc1d487d182": " O(k^2\\log k) ", "b00c8c0d37a5d4c9d8c8332bd6acb91c": "U_{ix}", "b00c9f50a42f42ea5fda19fc64d872e7": "\\ln(X)", "b00cac1d3940a4191a84a702def969d9": "\\frac{1}{2}\\,\\begin{pmatrix}\\chi& \\eta\\end{pmatrix} \\begin{pmatrix}B' & M \\\\ M& B\\end{pmatrix}\\begin{pmatrix}\\chi\\\\ \\eta\\end{pmatrix}.", "b00cd177b0306289e03a85b67a426cb4": "L, K", "b00d2f9c4307f68760247c15f8dad226": "H|p\\rangle = E_p |p\\rangle", "b00d5ba1e13f2fc4ccaffde24cd83311": "F = C^2 = 1", "b00d67f834955b64552f00fd383285b8": "\\forall x,y: \\neg\\;(x < y \\;\\vee\\; y < x) \\;\\to\\; x = y", "b00db364eb46ed74c05b3b2abc45bc6f": "\n\\textstyle\\frac{1}{2}\\frac{\\partial}{\\partial x_i}\n", "b00dd5b011b4c4ed2f26ebd882c10137": " {D \\over Ds} \\ \\stackrel{\\mathrm{def}}{=}\\ {d \\over ds} + \\Gamma ", "b00df45519331a02d4af287851c965ad": "D^{\\alpha}f(x)=\\frac{1}{\\Gamma(1-\\alpha)}\\frac{d}{dx}\\int_{0}^{x}\\frac{f(t)}{(x-t)^{\\alpha}}dt", "b00e9b7167b85f4943ff3be42d82458c": "\\scriptstyle m_0=\\sigma_\\eta^2=\\overline{(\\eta-\\bar\\eta)^2}=\\frac{1}{2}a^2,", "b00ea3a03ff7fc60731309472994db9d": "\\psi(x)=\\sum_{k=0}^{M-1} b_k\\phi(2x-k)", "b00ea97679c660847e177b5c7214b697": "\\mathbf{A}^{*\\mathrm{T}} \\,\\!", "b00eb436a48bc0494753e5b29b73cefc": "\n(4.1)\\quad\nf(x) = e^{x^2/2}\\int_{-\\infty}^x [h(s)-E h(Y)]e^{-s^2/2}ds.\n", "b00ecc506466aaddd34b17ee484a20ff": "{\\mathcal F}(X,{\\mathbb R}) ", "b00f136caad31158ab2838f7f856bab1": "\\sum_{j=1}^n p_j x_j \\leq W,", "b00f1fb3271ee94e542763932c8691e2": "Px=y", "b00f37e63be4c30d6a056bf96c68bd1d": "j\\in[n]", "b00f9c502b0005213a0f55684e58e552": "dx^\\mu dx^\\nu = dx^\\nu dx^\\mu", "b00fdf63b3a2d8a7e6b27704ac3fc6d8": "\\mathbf{x}_{0k}", "b00ff39df6b11c985944501c666387cc": " [K: \\mathbb{Q}_p]<\\infty ", "b0101260b490efe91e06517526c2fc81": "X _* \\mathfrak P = \\mathfrak P\\big(X^{-1}(\\cdot)\\big).", "b010ae017ec9ab1ff69a64f72bb1fbb0": "\\scriptstyle s\\, _{12,0}. ", "b010ba4a062f96b64cd364bae1f8022f": "a_{11},a_{12},\\ldots,a_{mn}", "b01187dcb2be737fafb41b227cf813db": " \n\\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & -1 \\\\ \n0 & 0 & 1 & 0\n\\end{pmatrix}\n\\quad\n", "b0119fecd5b795cc292bb0d3250686fe": "X = \\frac{x}{x^2+y^2},\\ Y=\\frac{y}{x^2+y^2},", "b011c86549f80860a0502906b333b0af": "(n-1)!\\ \\equiv\\ -1 \\pmod n", "b011ca52c27588510fff7296e52a6893": "H_e = \\alpha\\ M_s", "b0123886b25028b023cd75678bb5df0c": "x = ( x_1, x_2, x_3, \\dots, x_N )^T", "b01270b05c212d3a7b5158a19b4b5860": "\\scriptstyle k \\;=\\; \\frac{1}{ab}\\left(a^2 \\,+\\, b^2 \\,+\\, 1\\right) \\;=\\; \\frac{6}{2} \\;=\\; 3", "b012dee504841fa3b2e68138ebb60fca": "\nE(x,y) = \\max \\{K(x\\mid y), K(y\\mid x)\\},\n", "b01341d0d1795b4dd1e8d26ab44ac482": "\\big / \\Big / \\bigg / \\Bigg / \\dots \\Bigg\\backslash \\bigg\\backslash \\Big\\backslash \\big\\backslash", "b0138e4eb296eee06b27c766e2d1cc76": "\\deg(b(x)) \\le \\deg(a(x)) \\,.", "b013945c092f3ca63a224c8826e503a8": " G_i(t_{ij}, t_{il}) = (Y_{ij} - \\hat{\\mu}(t_{ij})) (Y_{il} - \\hat{\\mu}(t_{il})), j \\neq l, i = 1, \\dots, n. ", "b013a634a60ca0207647c26a20c230c2": "(\\boldsymbol{\\mu}_S)_z=-g_S \\mu_B m_S", "b013add0c4698b9b152ae803114d604e": "\\begin{align}\nI_1 &= \\sigma_{1}+\\sigma_{2}+\\sigma_{3} \\\\\nI_2 &= \\sigma_{1}\\sigma_{2}+\\sigma_{2}\\sigma_{3}+\\sigma_{3}\\sigma_{1} \\\\\nI_3 &= \\sigma_{1}\\sigma_{2}\\sigma_{3} \\\\\n\\end{align}\\,\\!", "b013cbfbb53ecdd1050c6606ed5776ac": "N=10^7 {(1-10^{-7})}^L. \\,", "b014279983b47d5d7d3dc4d332f3db85": "\\mu(n)\\;", "b01431e4770d00fb02eb75cbe5931b67": "\\Phi_E = \\frac{Q_\\text{free}}{\\mathcal{E}}", "b01466d541a80ad76e872b4ab76fde23": "Y^\\sigma \\times_X Z", "b0146824f28676e647960a46dc443129": "\\mathbf{r=y-X\\boldsymbol\\beta}.", "b0146ed529bf5f5ec59a41e136a8b0ed": "T(x_o,y_o) = \\iint T(k_x,k_y) ~ e^{j((k_x/M) (Mx_o) + (k_y/M) (My_o))} ~ dk_x \\, dk_y", "b0146f1d0c37580c2e5da8bcb1f39018": "\\kappa_v", "b0149e8e1c24e9dbbed2eb19d91c131d": "ax^4 + bx^3 + cx^2 + dx + e = 0", "b014ca18e574ca4e51422415fe037adf": "\\begin{align}\nX(u,v) &= (1/3)u^3 - uv^2 + \\frac{u}{u^2+v^2}\\\\\nY(u,v) &= -u^2v + (1/3)v^3 - \\frac{u}{u^2+v^2}\\\\\nZ(u,v) &= 2u\n\\end{align}\n", "b014e307d44fb210ecce00b56f50651b": "a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4", "b014ec6d90a9d2ff0da9d92ac07a0b51": "b\\succeq c,\\;", "b01518e053d26c440d71466b05596b86": "\\mathbf{p} = q\\mathbf{r}", "b015213cacdbf0d505bfe371bd9d19ef": "3\\times 4 = 12", "b01570211a7e73f8b68c358a367a1915": "\\lang \\mathbf x , \\mathbf y \\rang = \\mathbf x \\cdot \\mathbf y = x_1 y_1 + \\cdots + x_n y_n.", "b015b1d6c43f52cb5ed01dc18bc15027": "\\frac{R_1 R_2 R_3}{\\left( R_2 + R_1 \\right) R_3 + R_1 R_2}", "b015bed4bd2be8aab33808f6f13e7ad9": "\n\\log \\sum\\limits_{i = 1}^n {p_i^2 } \\le \\log \\sup _i p_i \\left( {\\sum\\limits_{i = 1}^n {p_i } } \\right) = \\log \\sup p_i \n", "b016262454e7068ffa73a3a34af1a8ef": "\\mathbf{E} = -\\nabla \\varphi", "b0165813698dae0a6a3cb78392cd304c": "\ny[n] = - \\frac{1}{4} y[n-1] + \\frac{3}{8} y[n-2] + x[n] + 2x[n-1] + x[n-2] \n", "b0165cffc655fab4fd47baf87370a314": "v_3 = \\frac{\\partial v_1}{\\partial x_3} = \\frac{\\partial v_2}{\\partial x_3} = 0", "b01668d02e8d1cf9f1caba7b2d229112": "corr[X_2,X_3] = \\left (\\frac{\\mu_2 \\times \\mu_3}{(k_0+\\mu_2)(k_0+\\mu_3)} \\right )^{\\frac{1}{2}}", "b0167fa51333ed7fc99db05ffe147c2b": " \\log_{ 10 } \\left( n + 1 \\right )- \\log_{ 10 } \\left( n \\right ) = \\log_{ 10 } \\left( 1 + \\frac{ 1 }{ n } \\right)", "b016a8798b2f701b557e98d9a7296dec": "F(X) = -\\nabla U(X)=M\\dot{V}(t)", "b017761a5411995328844ac376889f94": "\\overline x=|x|\\hat x", "b017a6b86b9a5816df44b92a96f7faf5": "m_y+\\sigma+e_y=-\\frac{1}{2}m_x-\\frac{1}{2}\\sigma-\\frac{1}{2}e_x", "b017b908e4ed24951bcd88c0eac88eb4": "a \\equiv b \\pmod n\\,", "b018a3ad08e7773b4f058ae152d6f82d": " p-p_0=\\gamma\\left (\\frac{1}{R_1}+\\frac{1}{R_2}\\right)\\!", "b018de0f8575ca1bbbc9f7c84aad0e0d": "G_B(\\tau=0^+)=\\frac{1}{\\beta}\\sum_{i\\omega_n}\\frac{e^{-i\\omega_n 0^+}}{i\\omega_n-\\xi}=-(n_B(\\xi)+1)", "b01944ab4a1f7e31de923da2c53b5464": "O \\left (\\deg(f)^2 (\\log(q)+\\deg(f)) \\right ).", "b01975257f8c352bec4e8bef44a1a0cf": "3+12+14+5=34", "b019c0b7a9ab981c3318cbc7110e8d4e": "(N_{k,p}(u)w_{k} P_{k} + \\sum_{i=0, i\\ne k}^{i=n} N_{i,p}(u)w_{i}P_{i})", "b019ebf6473f22f95e13988af4170b8a": "(c_1, 2 c_2)", "b01a3fff089a9b05443ebd177695f8cf": "U = \\langle E \\rangle = N\\,\\frac{k_B T}{2}", "b01a4d6853762ef4672f34155acae655": "\\phi_n(z)", "b01a5ba687696c8c43c15e34e3bbeaa1": "h'_{x}(\\alpha)", "b01a89bfa3258d61a8f1966d841eb445": "\\delta_{int}(s,t_s)=(s',\\tau(s') ", "b01a8b8a82533f8174d9c26cc1cefd69": "Z^{(-)}", "b01a90dc8146ee1d79a0e27aa1bfa1b1": " (z,t) ", "b01af899a7bf448590245929414cd711": "\\tau_B", "b01b2246ebca2032f36b30352f1e08d1": "\\,\\hat{\\eta}_i", "b01be32afbbafdc65c4f446c2eac8f01": "\\frac{S}{N}=\\frac{S}{\\sqrt{{1\\over n}\\sigma^2}}= \\sqrt{n} \\frac{S}{\\sigma}", "b01bf58a1c780dac6ffceea289aab24f": "\\frac{1}{2}(\\sqrt{6} - \\sqrt{2})", "b01c018931086fcc5acbfd4ddc2f8de1": "\n\\begin{array}\n[c]{cccc}\nZ & X & Z & I\\\\\nZ & Z & I & Z\\\\\nX & Y & X & I\\\\\nX & X & I & X\n\\end{array}\n", "b01c628047a68efef14de27ff1446142": " \\mathbf{B}_\\mathrm{LH} \\ = \\ {\\mathbf{B}_\\mathrm{G} \\over \\sqrt{4\\pi}} ", "b01c79e00dea82369468f5d65b66b111": "f_1g_1+\\cdots +f_sg_s=1,", "b01d584eccbf2c66c6c1dd87891c539c": "N_{\\gamma-norm} = \\left( L_{pp, \\gamma-norm}^2 - L_{qq, \\gamma-norm}^2 \\right)^2 = t^{4 \\gamma} (L_{xx}+L_{yy})^2 \\left( (L_{xx}-L_{yy})^2 + 4 L_{xy}^2 \\right). ", "b01e367d29cd61cdefd50a51c6af0cd1": "\\tilde{g}_{ij}", "b01e4885375d1bacc8b6e748b5cb1e2c": "~g~\\ll 1~", "b01e888e016b5e6ecb7ee2ab2c5a50be": "T\\phi = \\int_\\Omega K(t,x,y)\\phi(y)\\,dy.", "b01f26cc9257f2347f5fcc02e59bd34e": "-2\\ln(LR) ", "b01f8f8d6d33a8c2b25c838953fca293": "\\theta = \\frac{\\nu}{\\sigma}", "b01f9a123ff17c7c5acd3a86de030a5d": "d_{2,-2}^{2} = \\frac{1}{4}\\left(1 -\\cos \\theta\\right)^2", "b01fc7a1b7bf5f6c260ac8ffd2755cb8": "( k + 1 )", "b01fdaa9c36d4807aa99096090ac6caf": "{\\nu}^{\\mu}={\\kappa}", "b01fe390a138c069346647051cbd3cfb": "\\iota_{\\rho(\\xi)} \\omega \\,", "b01feeae0f36566810ab5fc36774d255": "\\, 2\\pi \\,", "b02012b70faf7bc3d271ecf061a36b83": "\\varphi \\lor \\psi\\,\\!", "b0203cf22141cdf80b5ab99f4260d490": "3^{1/\\beta} \\alpha ", "b020772628fce8dbfd9f4b1ed8d855dc": "s\\in\\Bbb{C}", "b0207a4c38182257adff62dfd5b81068": " S = \\varnothing ", "b020852724c2b0d33c40b02a0cd684b9": "\\eta_r", "b020a5b02d20828e36fc03673ae94084": " \\mathbf{v} = \\frac{d \\mathbf{r}}{dt} = \\dot{s}\\mathbf{e}_t = v\\mathbf{e}_t , ", "b020aab9c42d072d658de4e420dbf798": "\\forall x(x\\notin\\emptyset)", "b020cd04527550231fbae754263d6dbf": "3.4_{-0.2}^{+0.3}", "b020e3209106e617c389947a6ddcfbe3": "\\arcsin \\left ( t \\frac {\\sin(\\phi)} {h-t} \\right ) ", "b02105b566c6993c6d020613f7a0dd35": "\n\\| u_{0} \\|_{0} \\leq C \\| u_{1} \\|_{1}^{\\alpha_{1}} \\| u_{2} \\|_{2}^{\\alpha_{2}} \\dots \\| u_{n} \\|_{n}^{\\alpha_{n}}, \\quad n \\geq 2,\n", "b0211e7b5ea43b7ced6f2d4c977bb85b": "N(t) = N_0 e^{-(\\lambda _1 + \\lambda _2) t} = N_0 e^{-(\\lambda _c) t}.", "b0212d655b9a26e2ae99523c2b8591ca": " \\mathcal{A}^t=\\sum \\tilde{a}_{\\alpha} \\partial^{\\alpha},", "b021dab05188a8f80a25e1cce2aa86cf": "f(x)=x^n \\!", "b021e549b2e274fdca12ea9d80e0f3db": "x_{eq} = x_1 + x_2 ", "b022052d43d11f2a5418ff698eccad43": "\\scriptstyle c\\in\\mathbb{R}", "b02228c97a5aba546a19aa7a69270c7b": "v_{Ty}=\\lVert v_T \\rVert \\cdot \\cos(\\theta_{AOB})", "b0222b607c244a4ab3ce8d4289813150": " j \\in [n] ", "b0222f096cc4557e393dae5cd7445677": "f > 100", "b022419bdfcd111ca47d458584094894": "\\scriptstyle s_{\\infty}(x),", "b02280e63f4a6c00898383473860f0c9": "\\lambda \\to \\infty", "b022826c582047d310992f867bb563fb": "\\sqrt{- \\xi^a \\xi_a}", "b022a6937618e08c896352c59c818974": "[y_\\nu,\\ldots,y_{\\nu+j}] := \\frac{[y_{\\nu+1},\\ldots , y_{\\nu+j}] - [y_{\\nu},\\ldots , y_{\\nu+j-1}]}{x_{\\nu+j}-x_\\nu}, \\qquad \\nu\\in\\{0,\\ldots,k-j\\},\\ j\\in\\{1,\\ldots,k\\}.", "b0232872fe7edad08f5e1b511903c3b4": "MSD\\approx log^2(t)", "b023772f0dd74aa9d0c3b786acbb3b77": "(B_E u)(v).", "b0239ba6de62331d0e008807ea23b11e": "F_n=2^{2^n}+1", "b023b6a2bcba95280c04cc6afe78c1ff": "\\bigstar |\\bigstar ||\\bigstar", "b023bc22d33c5d76c1dc2cd8c567147d": "\\scriptstyle \\eta\\equiv 1 ", "b023d2075cac4e9419255d7c42699a62": "\\Rightarrow \\ q_1 = \\frac{a - q_2 - \\frac{\\partial C_1 (q_1)}{\\partial q_1}}{2}", "b023e5721846bcd5d7a712a36e14d9c3": "x = \\frac{-b \\pm \\sqrt {b^2-4ac}}{2a},", "b02432152c4495feb6b3c4fcaf01bfa1": "An-P", "b024c0c6f537b80ddf946f13cdefaba8": " S_{xx}(\\omega) = |\\hat{x}(\\omega)|^2 = \\left|\\int\\limits_{-\\infty}^\\infty x(t)e^{-i\\omega t}\\,{d}t\\right|^2.", "b02522fc26f3033a37a1ae985d6f5515": "\\mathcal{B}A(t) \\equiv \\sum_{k=0}^\\infty \\frac{a_k}{k!}t^k.", "b0257c34dea95d78cc64db7b74d1c7e0": "1-(1 - p)^{k+1}\\!", "b0257fb78728941a6db3a3cb757669d3": "\\chi(\\omega) = \\chi_1(\\omega) + i \\chi_2(\\omega)", "b0259bc91ffdf8a6d6c83c5ff7e24c88": "R_{0}=(R_{1}+R_{2})/2", "b025cfab99e29302ed16163fca00a6f3": "\n\\boldsymbol{\\hat\\beta} = (\\mathbf{X} ^\\mathbf{T}\\mathbf{X})^{-1}\\mathbf{X}^{\\mathbf{T}}\\boldsymbol y .\n", "b02638dcc34980d88f0a42494542e440": "389^{+24}_{-21}", "b0269347b31c7b96fd28b98af644e6b9": "\\Theta = 2 \\theta\\ .", "b026c259aa59b61b46cca91977f03b47": "\\frac{W_x}{W_o}=x(1-p)^2p^{x-1} \\,", "b026fa5688d751968e5f68976c9ea11d": "\\left(\\xi_{i}\\right) ", "b0270982c9f60c95b0b1b04aa1d7dc38": "\\phi(t)=t^\\lambda\\,g(t)", "b0273b5d1ff1864a9d7ada3524f055f5": "\\ln(A'/\\Delta l)", "b0275f1c06fc4d4bf990ef32e8e52820": " df = \\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-m)^2}{2\\sigma^2}}dx ", "b027e4667bb79c2ef4aefea570f6e715": "\\text{with }s^*(v,\\Delta v) = s_0 + v\\,T + \\frac{v\\,\\Delta v}{2\\,\\sqrt{a\\,b}}", "b0282ae9f13ab8f4691d11cedf452779": "| \\psi_3 \\rangle", "b0286ec016b60cde6532dcf8577e9deb": " \\Delta G > 0 \\,", "b02871de5186dfe25ce1ed6bfd7e80bb": "x(1) = A\\textbf{x}(0) + B\\textbf{u}(0) = B\\textbf{u}(0)", "b02875d9775eb54160aa58912dc2cb66": "d(x, A) = \\inf_{a \\in A} d(x, a).", "b028869463f3efe5f70500a5be3d56fd": "X_{i,j}=0", "b028b4ed46186779486159c4b946d9e1": "\\frac{n^2 + 1}{4n^3 + 5n}", "b02907faf58c6415bd3f322462416549": "{{i}_{IN}}-{{i}_{OUT}}=\\left( \\frac{2\\,\\Delta {{V}_{TH}}}{{{V}_{GS1}}-{{V}_{TH1}}}-\\frac{\\Delta \\tfrac{W}{L}}{\\frac{{{W}_{1}}}{{{L}_{1}}}} \\right){{i}_{IN}}", "b0290e3ecdfeca20f7c1454efd0fe622": " \\left( \\log R_e, \\langle I \\rangle_e, \\log \\sigma \\right) ", "b029202511841082dd51cca7971712f4": "b+db", "b02924083300a71ae07fe23ff7bddaa6": "\\int_{b - k}^{b} f(x) \\, dx \\leq \\int_{a}^{b} f(x) g(x) \\, dx \\leq \\int_{a}^{a + k} f(x) \\, dx,", "b029585d42ed445ed65c2b6c24a01a57": "f_1(z) = \\,_2F_1(a+1,b;c+1;z)", "b0296cd33bd3915a736135d5bc3aedd4": "\\sigma>0\\,", "b0296f8afbecb30512ea7e5eb451e233": "\n\\phi_j(q, p) = 0,\n", "b02977a32edf3cd6058a79287df01e39": "\\langle \\langle t, t \\rangle, t \\rangle", "b029f49100607deb243ee8d5ba8b88b8": "U \\subset V", "b029f8dda8764304a330cf9f946c1d8e": "\\varepsilon_{ijk}\\varepsilon_{pqk} = \\delta_{ip}\\delta_{jq} - \\delta_{iq}\\delta_{jp} ", "b02a11e295f1f998a67422fbb6cd9a05": "f_{\\text{flop}} = 1\\,\\mathrm{s}^{-1}\\,\\Leftrightarrow\\, n_{\\text{flops}} = 1", "b02a19bbf1797b237d6954cc66b276d6": "\\begin{cases}\n0; & n \\mbox{ even} \\\\\n\\frac{2}{\\pi n} ; & n \\mbox{ odd}\n\\end{cases}", "b02a4959d4143086dd53005084b70b1c": "N_1\\times N_2", "b02a7a7143e87342951ae05d7270dd42": "\\ddot r '= r ' \\dot {\\theta}' ^2 \\ . ", "b02a9c7851ed38b0bbf84bce84908a97": "a=mn^3, \\, ", "b02aac42c778750e294749bab92d045a": "\\psi(v)", "b02ae6e2be5ddd3e746f631c9a4a069b": "c^2\\left(\\rho_G D\\Psi_G-\\rho_L D\\Psi_L\\right)=g\\Psi\\left(\\rho_G-\\rho_L\\right)-\\sigma\\alpha^2\\Psi.\\,", "b02b2c72f0bbceabc71daef2a13f964c": "n, k_1, k_2, k_3,\\dots ,k_p, p \\in \\mathbb{N}^* \\,\\!", "b02b3678036ec77682064e29c66b53d0": "\\text {R}\\,\\! = \\int\\limits_{}^{} \\big(\\upsilon_n - \\mathit{u_n} \\big)^2 \\text {ds}\\,\\! \\int\\limits_{}^{} \\mathit{u_n} ^2 \\text {ds}\\,\\!", "b02b64762ec0f993e142c6323cc4d625": "t \\ne t_n ", "b02b64daae948bdaa66af4fe62b1441e": "S \\rightarrow abc", "b02b8e9eba33767e3013b3e73f2260a7": "I_x(\\alpha,\\beta)", "b02bd1a027577c06f0ea2445b50da969": "\n\\lambda^{2} - \\left( A_{xx} + A_{yy} \\right)\\lambda + D = 0\n", "b02c0816d5153b8a1d35c085ad9ffac0": "\\dim p, \\deg q, \\det m, \\ker\\phi \\!", "b02c52c2de8814158e6f66ed4fb8930a": "(q^{n+1}(1+(q^{n+1}-1)/(q-1)),q^n(q^{n+1}-1)/(q-1),q^n(q^n-1)(q-1))", "b02c61ae21c5a96d9a2bc02ef963f704": "x^5 + px +q\\,", "b02c654ddf789286816b5cb97b7c59be": "70% * 68.5% = 48.2925%", "b02c7dc3a5c32bc74690ce340ffa65bd": "\\boldsymbol{G}", "b02c9612b62f3e34f0c67d2920cd7eb1": "i = j = 3, 4, \\dots\\,", "b02cba863f5f58ffdaa1902564aac4c1": "\\psi(\\Omega^\\alpha) = \\phi_{1+\\alpha}(0)", "b02cd7ff9eaed417faefff9440af9cdb": "p(o|d)", "b02cedff52845f713b7121aca734a2a0": "\\chi_p", "b02d02edcc59019273802fd11ce00935": "z( x \\pm 1, y) \\rightarrow z( x \\pm 1, y) + 1", "b02d28f3fe68e4223cf50da93f59a2c3": "\\int_{0}^{2 \\pi} e^{x \\cos \\theta + y \\sin \\theta} d \\theta = 2 \\pi I_{0} \\left( \\sqrt{x^2 + y^2} \\right)", "b02d3cb915f6961772ca5b5544719d9b": "x^{lr} = x = x^{rl}", "b02d4af4583dc35a205ae21889d0a405": "\\mu=\\mu^+-\\mu^-\\,", "b02d53d522143362b6b10fa842841142": "P(A \\cup B) = P(A) + P(B)", "b02d6c012576d6e2d0f0e64f2672da67": "B^{\\mu\\nu}", "b02d9a927b32fd8bdacb8b14bcf6dc8b": "\\mathcal{F}=\\ker(F)", "b02db4b201da4012544359bd4ed008f6": " (t/t_{1/2})\\ln \\left(\\frac {1}{2}\\right) = (-t/\\tau)\\ln(e) = (-\\lambda t)\\ln(e)", "b02de1813a76b9a8a0afe915dd9e56fc": "T_a\\ :\\ y^2= x^3 + 3a(x+1)^2", "b02df2ff2aadceba1d822e731057c9c9": " Q_1= \\frac{\\dot{m}}{\\rho_1} ", "b02dfd2acc36968444e735e14f76f85f": "\n\\begin{align}\n\\mbox{total cost} & = \\{\\mbox{cost}((3,4),(2,6)) + \\mbox{cost}((3,4),(3,8))+ \\mbox{cost}((3,4),(4,7))\\} \\\\\n & ~+ \\{\\mbox{cost}((7,4),(6,2)) + \\mbox{cost}((7,4),(6,4)) + \\mbox{cost}((7,4),(7,3)) \\\\\n & ~+ \\mbox{cost}((7,4),(8,5)) + \\mbox{cost}((7,4),(7,6)) \\} \\\\\n & = (3 + 4 + 4) + (3 + 1 + 1 + 2 + 2) \\\\\n & = 20 \\\\\n\\end{align}\n", "b02e0b6f66ec76870124ff85cbb10dd0": "\\Lambda=0.6 \\lambda_1\\sigma_0^4", "b02e15d05c46f6df847d157b043e29bd": "\n\\begin{align}\n\\vdots \\\\\n12 &\\times \\color{blue}{-10} & + \\;\\; 42 &\\times \\color{blue}{3} &= 6 \\\\\n12 &\\times \\color{red}{-3} & + \\;\\;42 &\\times \\color{red}{1} &= 6 \\\\\n12 &\\times \\color{red}{4} & + \\;\\;42 &\\times\\color{red}{-1} &= 6 \\\\\n12 &\\times \\color{blue}{11} & + \\;\\;42 &\\times \\color{blue}{-3} &= 6 \\\\\n12 &\\times \\color{blue}{18} & + \\;\\;42 &\\times \\color{blue}{-5} &= 6 \\\\\n\\vdots\n\\end{align}\n", "b02e485370d5d1b0fe4643aeee7b1ab5": "\\textstyle {n \\choose k}\\, p^k (1-p)^{n-k}", "b02e49f7e1e0a2e7dd2bd3461348cfe6": "T =\n\\begin{pmatrix}\n & - & A & C & A & C & A & C & T & A \\\\\n- & \\color{blue}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nA & 0 & \\color{blue}\\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\leftarrow & \\nwarrow \\\\\nG & 0 & \\color{blue}\\uparrow & \\nwarrow & \\uparrow & \\nwarrow & \\uparrow & \\nwarrow & \\nwarrow & \\uparrow \\\\\nC & 0 & \\uparrow & \\color{blue}\\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\leftarrow \\\\\nA & 0 & \\nwarrow & \\uparrow & \\color{blue}\\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\leftarrow & \\nwarrow \\\\\nC & 0 & \\uparrow & \\nwarrow & \\uparrow & \\color{blue}\\nwarrow & \\leftarrow & \\nwarrow & \\leftarrow & \\leftarrow \\\\\nA & 0 & \\nwarrow & \\uparrow & \\nwarrow & \\uparrow & \\color{blue}\\nwarrow & \\leftarrow & \\leftarrow & \\nwarrow \\\\\nC & 0 & \\uparrow & \\nwarrow & \\uparrow & \\nwarrow & \\uparrow & \\color{blue}\\nwarrow & \\color{blue}\\leftarrow & \\leftarrow \\\\\nA & 0 & \\nwarrow & \\uparrow & \\nwarrow & \\uparrow & \\nwarrow & \\uparrow & \\nwarrow & \\color{blue}\\nwarrow\n\\end{pmatrix}\n", "b02e5eb66a66662f58834b068f9d429f": "\\sigma^o", "b02ea9621c3cab6d35db4ed39a734b81": "\\cos(A + B) = \\cos(A)\\cos(B) - \\sin(A) \\sin(B)\\,", "b02f09cc2fd5a425cc1ae426c183538e": "\n\\begin{align}\n3 & = {2^3+1 \\over 3}, \\\\[5pt]\n11 & = {2^5+1 \\over 3}, \\\\[5pt]\n43 & = {2^7+1 \\over 3}.\n\\end{align}\n", "b02fcb2f2398ef3da73e661f2823bbc4": "M_{i,j} = \\alpha_i^{q^{j-1}}", "b030243ca37bdd8ea6f77cb610749956": " \\qquad \\qquad \\sigma_e = \\frac{1}{\\rho_e}=\\alpha_{ee}, \\ \\ k_e = \\frac{\\alpha_{tt}-\\alpha_{te}\\alpha_{ee}^{-1}\\alpha_{et}}{T^2},\\mathrm{and} \\ \\alpha_\\mathrm{S} = \\frac{\\alpha_{et}\\alpha_{ee}^{-1}}{T^2} \\ \\ (\\alpha_\\mathrm{S} = \\alpha_\\mathrm{P}T). ", "b03032927928a5078cc303ea730c22b8": "\\omega =-\\frac{1}{3} \\omega_0 \\left[ 1 +\\frac{3}{4} \\left( \\frac{V_{ab}}{\\hbar \\omega_0} \\right)^2 \\right].", "b03056b83dcae5234f43c5af67a107f0": "dt_{d} = \\frac{T_{c} - T_{p}}{3S}", "b030f16b1265bcac82a6fb02ec41a331": "q(t) = \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\rho(\\mathbf{r},t) dV.", "b030f6accf8caef7116f768d48f3e952": "u(re^{i\\theta}) = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi P_r(\\theta-t)f(e^{it}) \\, \\mathrm{d}t, \\ \\ \\ 0 \\le r < 1 ", "b03157689c4797bf7ca07a613308f0c5": "x_1, x_2, x_3, \\dots , x_k", "b03165808fcee2d5361c5116b75f759b": "={\\hbar^2\\over 2}(j(j+1) - l(l+1) -s(s+1))", "b0323170a51c7c4828b94aa0a9316329": " {} = x_0 \\left(y_1\\begin{vmatrix}x_2&y_2\\\\x_3&y_3\\end{vmatrix}-\ny_2\\begin{vmatrix}x_1&y_1\\\\x_3&y_3\\end{vmatrix}+\ny_3\\begin{vmatrix}x_1&y_1\\\\x_2&y_2\\end{vmatrix}\\right)\n-y_0 \\left(x_1\\begin{vmatrix}x_2&y_2\\\\x_3&y_3\\end{vmatrix}-\nx_2\\begin{vmatrix}x_1&y_1\\\\x_3&y_3\\end{vmatrix}+\nx_3\\begin{vmatrix}x_1&y_1\\\\x_2&y_2\\end{vmatrix}\\right) \\,\\!", "b032536ddff5d167d1564be84d13cc71": "F(k;n,p) = \\Pr(X \\le k) = I_{1-p}(n-k, k+1) = 1 - I_p(k+1,n-k). ", "b032672621370394a915c2f003bba7ef": " \\mathbf{a} = a_1\\mathbf{e}_1 + a_2\\mathbf{e}_2 + a_3\\mathbf{e}_3 + a_4\\mathbf{e}_4.", "b0327e8a4c28ebcbcead014ce1ac5cf8": "X^{\\omega}\\times Y^{\\omega}", "b032c9a954a90324e229a80b2d4cf61b": "P(M|E)", "b032f6ca5d1d3ed8c6f6a80dead838b2": "\\chi_1(\\omega) = {2 \\over \\pi} \\mathcal{P}\\!\\!\\! \\int \\limits_{0}^{\\infty} {\\omega' \\chi_2(\\omega') \\over \\omega'^2 - \\omega^2}\\,d\\omega'.", "b032ff25d1f50d317b8bebffd08e925c": "\n \\left\\{ g_{1},g_{2}\\right\\} =\\left[ g_{1},g_{3}\\right] =\\left[\ng_{1},g_{4}\\right] =\\left[ g_{2},g_{3}\\right] =\\left[ g_{2},g_{4}\\right] =\\left[\ng_{3},g_{4}\\right] =0.\n", "b033244411ef3f83bbcce25b08789e91": "6s^2", "b0332ee8bdc3b8ac3dfe9cb0d7b98ebf": "\nS=\\begin{pmatrix}\n0 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0\n\\end{pmatrix}; \\quad A=\\begin{pmatrix}\n1 & 1 & 1 & 1 & 1 \\\\\n1 & 2 & 2 & 2 & 1 \\\\\n1 & 2 & 3 & 2 & 1 \\\\\n1 & 2 & 2 & 2 & 1 \\\\\n1 & 1 & 1 & 1 & 1\n\\end{pmatrix}.", "b0333b30c5ff365914b4b15c75625754": " \\delta_{x,n} = \\frac1{2\\pi i} \\oint_{|z|=1} z^{x-n-1} dz=\\frac1{2\\pi} \\int_0^{2\\pi} e^{i(x-n)\\varphi} d\\varphi", "b0335911546487538c5101489a720b10": "u = \\phi(t, x)+v", "b033a984b9e738bc8df16ab860c60b9a": "\\pi : \\mathfrak{g} \\to \\mathfrak{gl}_V", "b033c0225e7b44f6514ae05f38d0b87e": " \\alpha \\,", "b033c79963f39f833c21a2639dc032a0": " x=A^+b + [I-A^+A]w", "b0341478ebb86d65bb8172db39bae3e3": "r^{(g)}", "b034192c16975d5f4f5a3a410c09a82a": "4\\pi/3", "b03419571ce8cf5d7edc6f0386e4ef60": "\\frac{8 \\pi G}{c^4} T_{11} = G_{11} \\;", "b034c771070ce73f556fca3449474580": "\\left [ -\\cos^{-1}\\left(-\\frac{1}{e}\\right) < \\theta < \\cos^{-1}\\left(-\\frac{1}{e}\\right)\\right ]", "b034dda029dfa5cde9d7f365154b0007": "(\\beta)", "b034f0eb25381bafe5e67ae12ecfc8dc": " H(X,Z|W,Y).", "b03531e21637f269eefc077726ff389e": "\\Psi_n(x)", "b0353e2e19550b55405a82c640cefee5": "L(A) \\cap L(B) = \\emptyset ", "b0354908e42865932a11f6f1cf8de5f0": "x_{d+1} = \\sum x_i^2", "b0355f5ee3d15f5d9daa977c7248c4c3": "R=\\frac{GM}{V_k^2}", "b03571b977bfad124ef6d1419a15f8d0": "K=3^d", "b0359bf3d105991b4fb91ffbd6a8ce58": "\n\\begin{align}\n \\mbox{FSPL} &= \\left ( \\frac{4\\pi d}{\\lambda} \\right )^2 \\\\\n &= \\left ( \\frac{4\\pi d f}{c} \\right )^2\n\\end{align}\n", "b035a965587d4f96a2241c0aa6aeed2b": " \\operatorname{build-param-lists}[q\\ q, D, V, T_6] \\and \\operatorname{build-param-lists}[x, D, V, K_6] ", "b035b28927316e68928c645099e25fac": "\\, \\int^N_2 \\frac{1}{\\log(t)} \\, dt.", "b03621ef8954b33bd91c3ea0e3dd7109": "w^{(L)}_k = 1/P.", "b0362adae7661d6fb1bf6c2057c33cfe": "Z^{(\\ell)}_{R\\mathbf{x}}(R\\mathbf{y}) = Z^{(\\ell)}_{\\mathbf{x}}(\\mathbf{y})", "b0368470a6f086c75c6b7bfcba75354b": "\\{ 1,~i_1, \\dots, ~i_7 \\}", "b0374e6e3f8d7910e2cc380cb06e7856": " \\psi(x) = \\ln(x) - \\frac{1}{2x} - \\frac{1}{12x^2} + \\frac{1}{120x^4} - \\frac{1}{252x^6} + \\frac{1}{240x^8} - \\frac{5}{660x^{10}} + \\frac{691}{32760x^{12}} - \\frac{1}{12x^{14}} + O\\left(\\frac{1}{x^{16}}\\right)\n", "b03793e47bcf13611a6b823a06a6a2b8": "\\operatorname{Var}(\\overline{X}) = \\frac {1} {n} + \\frac {n-1} {n} \\rho.", "b037967406947d9c4df2030e348d0c3d": "K=(0,2\\pi)\\times(0,2\\pi).\\,", "b03804dfb02e45ffffdf92b110c9cc08": "S_r(n,k)", "b038777d009c30769574f4c67b796e4e": "(x_1, x_3, \\ldots, x_{N-1})", "b03885e80cf721485b81036e1cda8feb": " s(v) ", "b0395c0bd8ddc8587ff7decddd626ba8": "\\sin x < x < \\tan x.", "b03964054060fc4243178baf038f21aa": "a \\div b = c", "b0397fa32e2c4af972304db2f128bfd7": "\\kappa_p(V)\\geq 1", "b039d36c1c79d1db40ca90d54f40f0d7": "\\mathrm{SU}(n)\\,", "b03a2f67ffae9b34a5098272404ac43b": "\\psi(\\Omega^{\\Omega^2 \\omega^2 6 + \\Omega \\psi(\\Omega^{\\Omega^2 \\omega^2 6 + \\Omega \\psi(0)})})", "b03a61fbe7fab12e47127125429f62b2": "T_\\text{man}", "b03a646eead479cec021c449c9487894": "\\tfrac{a}{b} \\le 3- \\sqrt{2}", "b03ab5bd018275ce6bf15fa888050376": "m\\ll M \\ll m_{0}+t", "b03aca1fbb4c499d9ddf02a167903deb": "\\Pr\\left(\\overline{X}_n-T_a s_n\\sqrt{1+(1/n)}\\leq X_{n+1} \\leq\\overline{X}_n+T_a s_n\\sqrt{1+(1/n)}\\,\\right)=p", "b03acd95ee59e204762d0b13e87919dc": "L_{1/2}(\\cdot)", "b03ae35fdefd0380f5e0d2d1acf17dcf": "S=\\int dt \\left[\\frac{mR^2}{2}(\\dot{\\theta}^2+\\sin^2(\\theta)\\dot{\\phi}^2)+mgR\\cos(\\theta)\\right]", "b03afe8af4af0fa4739135c4258b6cb4": "x_{i-1}", "b03b6b7ef074f16881666df837443c47": "0\\le H(P,Q) \\le 1.", "b03bb69da75d99826fef8c5da6de3861": "Y=AK^\\alpha L^{1-\\alpha}", "b03bd669b67a3b87c242accc5457bef4": "(2^m-1, 2^m-1-m)", "b03be039bceefb46f14fdb659a2829a3": "\\displaystyle{Q(a)L(a)=L(a)Q(a),}", "b03befa028c365c5e6b41544ba1e0ee6": "\\bold{\\hat{\\Pi}} = \\bold{\\hat{P}} - q\\bold{A} ", "b03c1d4bec0bc0efca2fe8e267e9665b": " \\frac{d}{dt}\\int_{\\Omega_{e_{i}}}\\rho_{h}\\phi_{h}d\\boldsymbol{x} + \\int_{\\partial\\Omega_{e_{i}}} \\phi_{h}\\mathbf{J}_{h}\\cdot\\boldsymbol{n} d\\boldsymbol{x} = \\int_{\\Omega_{e_{i}}}\\mathbf{J}_{h}\\cdot\\nabla\\phi_{h}d\\boldsymbol{x}.", "b03c22f9cac43258672942d814a63426": "\\sum_n \\mathbf{p}_n = \\boldsymbol{0}\\,,", "b03c59aa30c8c099e3ba5f40725ca85e": "\\begin{cases}\n 1+\\frac{Log[2]}{a} & \\text{if}\\ b=1 \\\\\n \\left( \\frac{1-b}{a} W\\left(\\frac{ 2^{\\frac{b}{1-b}} a e^{\\frac{a}{1-b}} }{1-b} \\right) \\right)^{\\tfrac{1}{b}} & \\text{otherwise}\\ \\end{cases}", "b03c66025d32dffaa888b50cd05fc38d": "\\{2^k \\ \\mid\\ k \\in \\mathbb{N}\\}", "b03c68587cb4ccc48dc1520d19aaa0fe": "\\{x_{1:t-1}\\}", "b03c6e30c5f7b1b23db6a5d0c881c06f": "6 \\times x", "b03c76fb156ac5bfc8af086e2e02f746": "V^* = V \\times \\{0,1\\}, \\, ", "b03c8392cfa918a5e0567d9c9c32bc34": " \nL = \\frac{c}{\\kappa} \\int_S \\nabla \\Phi \\cdot dS = \\frac{c}{\\kappa} \\int_V \\nabla^2 \\Phi \\, dV = \\frac{4 \\pi G c}{\\kappa} \\int_V \\rho \\, dV = \\frac{4 \\pi G M c}{\\kappa} \n", "b03ccd97321d9cf3e26d1668ceb10c5b": "sin(\\pi/2)=1", "b03ccfc3f0c99bd60af38d16035529ae": "\n2^{bh(v)}-1 = 2^{0}-1 = 1-1 = 0\n", "b03d1740ea1f5507a18d9aefd543196d": "x^{(n)}=x(x+1)(x+2)\\cdots(x+n-1)", "b03d1fbda3dc12054f416faf9ee71689": "\\mathbf{p} = \\frac{\\partial\\mathcal{S}}{\\partial \\mathbf{q}}", "b03d408d00a371ec1e4198b0b8f395c1": "Y' \\to Y", "b03dc6e5fe2f24b41274e305f8aa505c": "\\sin_K x = x - \\frac{K x^3}{3!} + \\frac{K^2 x^5}{5!} - \\frac{K^3 x^7}{7!} + \\cdots.", "b03e1ec9545ad9b73ad0cfc0357fe8c0": "A_{projected} = \\int_{A} \\cos{\\beta} \\, dA", "b03e334ba1c47c88d81ecb0583ed0c91": "\\exists x \\in t\\ (\\phi) \\Leftrightarrow \\exists x ( x \\in t \\land \\phi)", "b03e348e9a32fc602f4c03ffa0d470de": " {\\mathbf{u}}(t)= -L_r(t) \\hat{\\mathbf{x}}_r(t),", "b03e37d22ab5cce498c23d8bf5cada0e": "\\mathbf{F}=m\\mathbf{a}", "b03e4a1623ae326d515b78140419b08c": "x\\in H_p(LM)", "b03e9988ea30b548fa7d1327f50bff1b": "T_\\delta \\quad ", "b03f1c15f67b17175c6c4a8ccdb67b99": "\\frac{-u''(c)}{u'(c)}=a.", "b03f4467c58575605e8f1340c36d25e2": "\n {\\overline P}X = \\{x \\mid [x]_P \\cap X \\neq \\emptyset \\}\n", "b03f467474f21e31ebda062ca3f7d3f8": " F_{\\alpha \\beta} = \\pm \\epsilon_{\\alpha \\beta \\mu \\nu} F^{\\mu \\nu}, \n\\quad F_{\\mu \\nu} = A_{\\mu, \\nu} - A_{\\nu, \\mu }+ [A_\\mu, \\, A_\\nu]\n", "b03f67d67388e1ff9beeeb010b30c590": "a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots,", "b03f7768e96e588d9fc855c9addc1edb": "[(g \\otimes \\alpha) \\wedge (h \\otimes \\beta)] = [g, h] \\otimes (\\alpha \\wedge \\beta)", "b03f7c9ba37c6b32b560bb5bc925e405": " (a, b) = (a+b, -b/a) ", "b03f93d8041232e44b615870e2472b8a": "(\\vec F = I \\vec L \\times \\vec B)", "b04123068d78a64c7715b18710a7dcf8": "M \\approx u a \\cos (\\phi) + \\Omega a^2 \\cos^2(\\phi) ", "b0415467fa7eb1b6feecf8e2c4d438b5": "\\textstyle x_k = A_k (e)", "b041a62b843afa94bacd330bf6f52ad1": "\\tau_E", "b041d4615d2a1e1bf2a4b07c5f817059": " \\int\\limits_{-\\infty}^\\infty |x(t)|^2\\, dt.", "b0424a2a07dd5d94ab96087901e16a71": "r_i(t)", "b0425b32556f84abc2f9c5f8e37e5a6e": " H(s) = \\frac { s C_2 R_1 } {(1 + s C_1 R_1) (s C_2 R_1 / (1 + sC_1 R_1) + s C_2 R_2 + 1 )} ", "b042979db2ae898ff97ac057c969844c": " \\sqrt{S} \\approx 2^n", "b042aeee9ad0cd2ae7a8228834b5b8f0": " V^a_b(f) = \\int _a^b |f'(x)|\\,\\mathrm{d}x.", "b042c8744fab213b9e7394aa3d6a8f23": "\\alpha<\\varepsilon_\\alpha", "b042d603e0522f3b0990f9e4deb12268": "W_s = M_s g", "b042eaf879d8712727f2e445b5492314": "\\lim_{n\\to\\infty} \\frac{a_n} {b_n} = \\frac{ \\lim_{n\\to\\infty} a_n}{ \\lim_{n\\to\\infty} b_n}", "b042f9692274091437b1220bdb14a620": "bs : X \\to X", "b0432d9a7c085967114202d8cb462981": "F_{\\mathrm{Cfgl}} = m \\mathit{\\Omega}^2 r \\ , ", "b043601bcbbe3a7b1ac94fa8b769ca26": "E(z_i u_i) \\neq 0", "b043a5b998f28bf157244be9b7b2c05a": " {\\Pr}_{(x_1,\\ldots,x_k)\\in \\!{F_p^k}} (g_x(m)\\equiv g_x(m')\\mod p) ", "b043dc094f5562580a31cd6953005816": "j: g\\ a", "b043e26fc415857ef6d405d0b1ca5c3f": " SL(3,\\mathbb C)", "b043f53c7c6f18bdf6a61dbe274d8b15": "\n \\begin{align}\n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} + [N_{\\alpha\\beta}~w^0_{,\\beta}]_{,\\alpha} - q & = 0\n \\end{align}\n", "b0463b38427e7541419938e2824cfa40": " \\frac{\\det \\left(-\\frac{d^2}{dx^2} + V_1(x)\\right)}{\\det \\left(-\\frac{d^2}{dx^2} + V_2(x)\\right)} = \\frac{\\psi_1^0(L)}{\\psi_2^0(L)}. ", "b046f800c64eb6ad5def1d6ff0a056ce": "H_1(\\mathrm{S}_n,\\mathbf{Z}) = \\begin{cases} 0 & n < 2\\\\\n\\mathbf{Z}/2 & n \\geq 2.\\end{cases}", "b0472d5a336647230e2a4d79fa562c74": "\\mathcal{L} = \\frac{-1}{4\\mu_0}F^{ab}F_{ab}.", "b0475dda786c30e514e000cc3b6926ea": "{\\tilde{A}}_{2n+1}", "b047763a05ffbe8710682bb2006d8216": "\\ P_* f", "b04793a3bb431aeba63107238546a4eb": "\\cos z = \\prod_{k=0}^{\\infty} \\left(1 - \\frac{z^2}{(k + \\frac{1}{2})^2\\pi^2}\\right).", "b047bcb777651224ab5b4a44e669d237": "\\frac{\\partial}{\\partial x}+i\\frac{\\partial}{\\partial y}", "b047f24e2fa0d6c8825b03766e27b0b5": "v_j", "b048846fcabb7573679a863f8a53bfdc": "\\lim_{h \\to 0}\\frac{f(x+h) - f(x-h)}{2h}.", "b04896d14d7b8a07346aee360536d725": "2N^2/h + \\sqrt{\\pi N^3 h}", "b0489da34d5e5e97e386be8cfe5921a1": "A_2x + B_2y = C_2,\\,", "b048a531a804b5b537aa35179b3c9ceb": "{\\underbrace{\\partial h \\over \\partial t}}_{\n\\begin{smallmatrix}\n \\text{Change}\\\\\n \\text{in mass}\\\\\n \\text{over time}\n\\end{smallmatrix}} \n+ \\underbrace{{\\partial \\overline{hu} \\over \\partial x} + {\\partial \\overline{hv} \\over \\partial y}}_{\n\\begin{smallmatrix}\n \\text{Total spatial}\\\\\n \\text{variation of}\\\\\n \\text{x,y mass fluxes}\n\\end{smallmatrix}}\n = 0", "b0494e2a65b9fdb4893094c8bb5f5943": "\\pm e/A^{7/4}", "b049d45213a36cf71c8cf6c8e7dbafd8": "\\mathbf{\\dot{a}}", "b04a02cd9499822b377063d954e20018": "a_{i,n}s", "b04a59a8d8b346eecadaab6b48ac356f": " \\lim_{x \\nearrow a}\\,f(x)", "b04a6f25f8173bc2935112cf6b808447": "u(x,m)=m+U(x)", "b04a9a071f193b01484a948e7cb0338b": "x^2 + 10x = 39", "b04aab64a7b1c1bdd864936091dd9c06": "\\frac{z}{1-q^{1/2}} \\;_{2}\\phi_1 \\left[\\begin{matrix} \nq \\; q^{1/2} \\\\ \nq^{3/2} \\end{matrix}\\; ; q,z \\right] = \n\\frac{z}{1-q^{1/2}}\n+ \\frac{z^2}{1-q^{3/2}}\n+ \\frac{z^3}{1-q^{5/2}}\n+ \\ldots ", "b04abfe1231e09b85bf15ad633d8604f": "f(x) = \\int_0^\\infty \\frac{1}{\\sqrt{2 \\pi \\sigma^2 v}} \\exp \\left( \\frac{-(x - \\alpha - \\beta v)^2}{2 \\sigma^2 v} \\right) g(v) \\, dv", "b04adf27ce0260051978bebf763e2b07": "|W| \\leq |N_G(W)|.", "b04b32f32002440cb96e80f4ff239674": "\\frac{b}{\\theta} \\simeq \\frac{r}{1}", "b04b37a9ea763c61a6a096f435b13f27": "p=\\frac{\\partial \\mathcal{L}}{\\partial \\dot{x}}=\\frac{\\mathcal{(}m+S(x))\\dot{x}}{\\sqrt{-\\dot{x}^{2}}}+A(x)", "b04ba37465aa0273e4cef2cb5c1532e0": " T = dF", "b04bb1e030821c8a3b01ce148651e453": " I_1=I_2= 2I_3=I", "b04bc2f7422877b9d4e78ef1e3884ba6": "{\\theta}=\\arctan\\frac{r}{h}", "b04bfc079120a2cc6e7709c24f3f4a0f": "\\varepsilon=T\\alpha^2", "b04c112843a0bedac529d4796b1978a6": "\\chi (G) \\le \\Delta (G) ", "b04c85d7240462c1a93ea254b28d2be9": "G'WG", "b04cc0fd48ab214c5864460183ee36d2": "S(0;x)=x ~.\\ ", "b04d1400c87433c8d9ead8c4c69db84e": " v^2 = \\mu \\left( \\frac{2}{r} - \\frac{1}{a} \\right) ", "b04d325ddba6753a1f01c79ee23484d8": "\\frac{\\pi}{4} = 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots ", "b04de31ea0fdcdcd046a82a9bdcab1ac": " y_{n+1} = y_n + hf(t_n,y_n). \\qquad\\qquad (4) ", "b04e100e2c9d8b1d28b613b84d20541b": "dS = \\left[\\left(\\frac{\\partial S}{\\partial T}\\right)_{V}+ \\left(\\frac{\\partial S}{\\partial V}\\right)_{T}\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}\\right]dT+\\left(\\frac{\\partial S}{\\partial V}\\right)_{T}\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}dP", "b04e432e178e7b093d19ad6e48642a58": "\\Delta m^2_\\text{LSND}\\simeq 1\\,\\mbox{eV}^2", "b04e4adbe787059ce0da78c05d12a50a": "\n \\sqrt{n}\\big(\\hat F_n(t) - F(t)\\big)\\ \\ \\xrightarrow{d}\\ \\ \\mathcal{N}\\Big( 0, F(t)\\big(1-F(t)\\big) \\Big).\n ", "b04e678cc504f219559093441fbc0ce3": "\n\\int \\psi_0(x) \\int_{u(0)=x} e^{{\\rm i}S(u+\\epsilon,\\dot{u}+\\dot{\\epsilon})} Du\n\\,", "b04e9d4acb894db196b176c2eb596439": "\\begin{align}\n& \\mathit{CR}_{i,j-\\tfrac{1}{2},k}\\left(h^m_{i,j-1,k}-h^m_{i,j,k}\\right) +\n \\mathit{CR}_{i,j+\\tfrac{1}{2},k}\\left(h^m_{i,j+1,k}-h^m_{i,j,k}\\right) + \\\\\n& \\mathit{CC}_{i-\\tfrac{1}{2},j,k}\\left(h^m_{i-1,j,k}-h^m_{i,j,k}\\right) +\n \\mathit{CC}_{i+\\tfrac{1}{2},j,k}\\left(h^m_{i+1,j,k}-h^m_{i,j,k}\\right) + \\\\\n& \\mathit{CV}_{i,j,k-\\tfrac{1}{2}}\\left(h^m_{i,j,k-1}-h^m_{i,j,k}\\right) +\n \\mathit{CV}_{i,j,k+\\tfrac{1}{2}}\\left(h^m_{i,j,k+1}-h^m_{i,j,k}\\right) + \\\\\n& P_{i,j,k}\\,h^m_{i,j,k} + Q_{i,j,k} = \\mathit{SS}_{i,j,k}\\left(\\Delta r_j \\Delta c_i \\Delta v_k\\right)\n\\frac{h^m_{i,j,k}-h^{m-1}_{i,j,k}}{t^m-t^{m-1}}\n\\end{align}", "b04ec46535b678eaaa052264a684bc03": "b_1,\\ldots,b_{d-1}", "b04ed7f955d142b086939b676ae10d82": "ZZ_3 = E = 4", "b04f65e4058cbe1783f0bda7601c63ec": "n=\\infty", "b04f6f14fdfef861b9bd04361a1b7f51": "f^\\dagger:B\\rightarrow A", "b04f900bb01eb99cbbf08ff85883715b": "\\mathbf{J}^2", "b05008bc7d96edf0a5aadda29af5721c": "\\Delta\\ m", "b05023e89f2d80fbfe5ebcaf1216ee28": "Z_{F^{\\bullet}}(x_1, x_2, \\dots) = x_1 Z_{F'}(x_1, x_2, \\dots)", "b05047cdfcbb090c21e282684a222c53": "f(L_0 \\times \\{1\\}) = L_1 \\times \\{1\\}", "b05068ce9c5dab0896cfbddcf5bfaf2d": "R \\geq 7", "b05099b54ba6833c89e7c45853b96ccc": " 2 \\frac{s_{2n}}{S_{2n}} \\frac{s_n}{s_{2n}} = 2 + 2 \\frac{s_n}{S_n} , ", "b051447f954267c45b1e23c1aecbb791": "r = \\left|p_1 - p_2\\right|", "b051e4169972ac789d7394e94cec7756": "E_{eq}", "b051f223cc606a04995adf2387a0c017": "\\Delta' - \\Delta = A", "b052227c35aed4688ba257bb8edf669e": "U=i\\sigma_y", "b0524d48f9467f57929a3b1defb17eb0": "\\int_r^s f(t) \\Delta(t) = F(s) - F(r).", "b0528d2bd5b0887b836fd3c6dfd6be6d": "(T \\cdot S)", "b052e63b508ca0dab2d6e020026d96a4": "a_{Na^ + } \\cdot a_{Cl^ - } = K_{sp}", "b052fe863fd5963b8384be1f5cb1c469": "R = R_{bd} \\eta^{bd}= h^{a b}_{, a b} - \\square h ", "b0530ef66104b94b3f5b3eea86379316": "\\mathbf{j}_7", "b0530fc36b54468db7e7c7a275555f35": "\\partial_{xx}", "b0532b344a867bff2351af0bfe948bf1": "\\forall x \\exists y . R(x,y)", "b053329c8bd0b3f814ec7000601c2442": " (\\partial G)_T=-(\\partial T)_G=-V", "b0535af217b8c3a92c40a25e07a43d26": "j_{th}", "b053a1ff47b25d911552d8a5062cb72e": " \\frac {p(r)} {P} = 1 - \\frac {P - p(r)} {P} ", "b053b8f225fcc290ba66114eadc9d163": "f(T)", "b053d18e54a1dac30b1a59d05221369a": "T_\\mathrm{i}", "b053fa32cdf4d93f43a23ea19599dd6b": "\\mathcal{X} = \\mathcal{A} \\sqcup \\mathcal{K} \\sqcup \\mathcal{Q}", "b0542ea13489142505baaf6a7d71b0be": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "b05446bf671593ba04da22c1c8037d52": "\\Omega \\leq 1", "b0544a5c71e02d13c2027d1b008b6690": "7^2 + 2^2 + 1^2 = 6^2 + 2(3^2) = 2(5^2) + 2^2 = 54", "b054905292ae021b7589074ae1feb914": "H_{out}", "b0549ae9d31f9e069f897e03a5ddefe8": "V_2 = V_{LN}\\angle -120^\\circ", "b054dd6e8fcd05b7041727d7500a7c94": "f' \\,", "b054ea37bf41dcea1d5c153954cadeb5": "\\nabla_\\ell V^m = \\frac{\\partial V^m}{\\partial x^\\ell} + \\Gamma^m{}_{k\\ell} V^k.\\ ", "b054f343baccb28f6037639617d913fd": "x_n\\sigma_0^2", "b05c4ac68f4195c92270a82893c472fe": "x(t+1) = \\,\\!\\delta(t+1) * x(t)", "b05c5f7b0bf3d2730360c14cef1909c7": " M \\cup_f H^j = \\left( M \\sqcup (D^j \\times D^{m-j}) \\right) / \\sim", "b05c88e78d6cc0f497915f71510cf633": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{4}{\\sqrt{3}},\\ 0\\right)", "b05c8c4c1a9baff88e25fc57db4732d0": "A^{-1}(D)+x_0=A^{-1}(D+Ax_0) \\,\\!", "b05c8eb556758d1ba583fa1cc35ae835": "x'(\\mu - r_f \\cdot k) - \\frac{a}{2}\\sigma^2.", "b05d122cea7858a51fafebb52c16f0a1": "\\frac{\\hbar^2 k_z^2}{2m}", "b05d171f86a561b78190a0574df3eeb4": "\\mathcal{1}", "b05d719866efb1fddef48eecf6cce4be": "\\mathit{Conv}2(R) = \\{(b,(a,c)) \\mid (a,(b,c)) \\in R\\}", "b05d923f2711959266a9b6ed991219ae": "x(t)=\\sin\\left(\\phi(t)\\right)", "b05d9b476aa3ab30b343958155aaf7db": "\\sigma(x) = \\sigma(\\hat x)", "b05dae91491c8763d1674166859401cc": "|1/2,m_1\\rangle|1/2,m_2\\rangle", "b05dba5d84e03ff9af799c5d3e446c96": "\\{v_3,v_4\\}", "b05dc65fe538e4623cc64aa45ff138a5": "\\begin{align}\n e_1(X_1,X_2,X_3) &= X_1 + X_2 + X_3,\\\\ \n e_2(X_1,X_2,X_3) &= X_1X_2 + X_1X_3 + X_2X_3,\\\\\n e_3(X_1,X_2,X_3) &= X_1X_2X_3.\\,\\\\\n\\end{align}", "b05e10a3c7d319ded6cc50daf7d0ff27": "1 -\\text{nth Weighting}", "b05e173e3c80db15b7b7cde2cbcc1684": " S_\\omega = S \\cot \\omega = \\frac {a^2+b^2+c^2} {2}\\,", "b05e17470c0e01b2e97ca745fe31ac87": " P(E) \\leq \\frac{1}{M^\\rho} \\sum_i P(X_1^n(i))^{1-s \\rho} \\left ( \\sum_{i'} P(X_1^n(i'))^s \\right ) ^\\rho \\, .", "b05e731686f6165f35df1806f273216e": "\n\\operatorname{dCor}(X,Y) = \\frac{\\operatorname{dCov}(X,Y)}{\\sqrt{\\operatorname{dVar}(X)\\,\\operatorname{dVar}(Y)}},\n", "b05e756d494034165305df1d0307c136": "1+\\sqrt{2}+\\sqrt{6}", "b05ea1daaf6a15f3f93016d464df5c75": "\\mathsf{n(CH_2CH_2)O\\ \\xrightarrow{SnCl_4}\\ (-\\!CH_2CH_2\\!\\!-\\!\\!O\\!-)_n}", "b05ee475a1f6e7f1af449728c92a85fe": " I(t) = I(t-1)\\times\\frac{\\sum_{i=1}^{35} Cap_{i}(t)\\,}{[\\,\\sum_{i=1}^{35} Cap_{i}(t-1)\\,\\pm J\\,]\\,} ", "b05ee7996fcc96b6761636bd3f111b88": "(x_1,x_2,\\dots)", "b05ef339e52a6cd5509b73427192f00b": "R[A_1, ..., A_n] \\subseteq S[B_1, ..., B_n]", "b05f0c514fd3d46d2a8840c233d82e2b": "\\overline{f} = f", "b05f1360907a3d6931b029c441662b47": "d_G(p,q)", "b05f38111aa635f08d594f45b197586a": "A \\wedge B", "b05f998cf0e4469fbb071bcd08dfde59": "\\lambda \\ll \\rho", "b05f99a49db9cf9651af9d18f8a7255d": " s_{(d,+)} ", "b05fefca49314d60e59f8976261c8afe": " h_j(\\mathbf{x}) \\ge d_j ~\\mathrm{for~} j=1,\\ldots,m ", "b0603860fcffe94e5b8eec59ed813421": "\\beta", "b0605d152905d5d02f6e8e1299020671": "R=\\Z[\\sqrt{-5}], \\quad p=1+\\sqrt{-5}, \\quad a=1-\\sqrt{-5}, \\quad q=2, \\quad b=3.", "b060a7235878f8699dd209616bf50c71": "q_i \\circ q_j^{-1}", "b060ab7a7a95dcb7cc448660d26cbef9": "\\alpha_8", "b060c5b004a312daead8554758222fcd": "\\psi = \\psi_0 + \\delta\\psi ", "b060d24053d8a7a80d83229a653354e3": "n \\equiv a \\mod m", "b060f7ff3d8723e7b70e7c540d021982": "\\lim_{\\theta \\to 0}{\\cos \\theta} = 1\\,", "b061323704dd6cc499d54190686139e1": "q = t e^{ar} \\! .", "b0622752f33a5a40a8bb42072fa75577": "1/z", "b06269a5903808397cf5deb4dfad7c63": "E_n^{(2)}=\\frac{ m a^2}{ 2 \\hbar^2 }\\left ( \\frac{1}{2n-1}+\\frac{1}{-2n-1}\\right )=\\frac{ m a^2}{\\hbar^2 } \\frac{1}{4 n^2-1}", "b0628026ce9a6fd66da663ed22d7ec9d": " {}_{A+1}F_{B+1}\\left[ \n\\begin{array}{c}\na_{1},\\ldots ,a_{A},c \\\\ \nb_{1},\\ldots ,b_{B},d\n\\end{array}\n;z\\right] =\\frac{\\Gamma (d)}{\\Gamma (c)\\Gamma (d-c)}\n\\int_{0}^{1}t^{c-1}(1-t)_{{}}^{d-c-1}\\ {}_{A}F_{B}\\left[ \n\\begin{array}{c}\na_{1},\\ldots ,a_{A} \\\\ \nb_{1},\\ldots ,b_{B}\n\\end{array} ; tz\\right] dt", "b062c2478bc8b9152799df3a2781092d": "n \\geq 1", "b062c83199cffbb8980f964a1883dcd3": " \\mu(S) = \\int_S {1\\over |\\det(X)|^n} \\, dX ", "b062c8ad3d8bfdfe31bf80bf66bd59de": "\\Lambda^\\cdot {\\mathfrak g} \\otimes C^{\\infty}(M) ", "b06306f1cf630f21b568b60c2deb6189": "f^{1}(0) = 6 + 7 \\times 0 = 6 = x", "b063409ffe9e281c74a13dffd1e96d40": "O(|x|^c)", "b063a5fdc1cfbe8ea48b76a926b26a7f": "\\operatorname{E}(X^2)", "b0645e41575d691738535daefc3b804b": "=p_1p_2 + (1-p_1-p_2+p_1p_2)\\ ", "b064647a912302b5132a3e404de5c47d": "\\alpha\\in y\\,", "b0648f48dd04613770f330d581d347a7": " \\text{Assets} = \\text{Liabilities} + \\text{Owner's Equity} ", "b064a73d5f9bef903877a6e6639c1ee2": "q \\nabla \\varphi.", "b064cbd8763f73acc908c4ef1bb76db2": "\\exists \\; x \\Bigg( \\Big( K(x) \\bullet B(x) \\Big) \\bullet \\forall \\; y \\Big( \\big( K(y) \\bullet B(y) \\big) \\rightarrow y=x \\Big) \\bullet \\forall \\; z \\Big( K(z) \\rightarrow B(z) \\Big) \\Bigg)", "b064f8555ec660f2f8bdc927d9636a06": "j \\neq i", "b06505d18d34b54579425426a637d8bc": "(r,\\theta,\\phi)", "b065478b058b21ae35df91a4a86136f6": "\\ \\frac{S}{C} = \\frac{16(\\log2)G\\sigma}{(4\\pi)^3R^2\\theta\\phi c\\tau\\eta}", "b0659d01dc9d12209dd119ca9255e74c": " \\tan \\left({\\pi\\left(X-\\tfrac{1}{2}\\right)}\\right) \\sim \\textrm{Cauchy}(0,1)\\,", "b065c482a016552d8e1c09b60633ef4f": "\\scriptstyle{\\mathbf{N} = \\mathbf{l}-(g_s/2)\\mathbf{s}+3(\\mathbf{s}\\cdot\\hat{\\mathbf{r}})\\hat{\\mathbf{r}}}", "b0660bc2351e3ece0fa9e8ca46b94a9e": "G = (H \\otimes I) \\; C_N", "b0663babbda1b116262508d45bf9fce1": "\\eta_M", "b066446864803a5b294e3c68374ef06b": " \\left|\\Psi\\right\\rang = {1 \\over \\sqrt{2}}\\left|1,n\\right\\rang \\left|2,n\\right\\rang + {1 \\over \\sqrt{2}}\\left|1,n\\bot\\right\\rang \\left|2,n\\bot\\right\\rang ", "b066ca7e5f7003851930d5507d3d9e10": "h\\sim{2 \\times 10^{-13}/\\sqrt{\\mathit{Hz}}} ", "b0676a89388df2892393733934716664": "l_p", "b067c4cc2232606d1c93c0821f86bdaf": "\\lim_{K \\to \\infty} \\sup_{X \\in \\mathcal{C}} E(|X|I_{|X|\\geq K})=0.", "b067c8d663f4b843e6da4dbdae693c13": "\\tilde{\\rho}(r+s)\\cdot v = \\tilde{\\rho}(r)\\cdot v + \\tilde{\\rho}(s)\\cdot v ", "b067f508beb2df611690e2eec879642f": "(M, g_{ab})", "b067f9b548174e85df421fe47b1457c5": "\\vdash_\\vec{s} = \\vdash", "b068078de2ea69e2f96b9c66997b0e72": "G(e_1 - e_2) + i_{V_S} = 0", "b068523367452bcf73f970d2d95b1ec0": "\\mathfrak{J}^\\mu = J^\\mu \\sqrt{-g}", "b068674f0e0eb327f9f147e3d01be9ac": "n_3=\\sqrt{1-n_2^2}=\\pm\\frac{1}{\\sqrt 2}\\,\\!", "b068b32ebbb676a43144f5c40c15e177": "\\nabla\\times\\mathbf{B}=\\mu_{0}\\mathbf{j}", "b068b716789c47fd35816720c88a57bc": "a_k = \\sum_{i=0}^k (-1)^{k-i} \\binom{k}{i} p(i).", "b068d3b565e682411adbf1b9dbc43dff": "\\mathbb T \\cong \\mathbb R/2\\pi\\mathbb Z.\\,", "b068dc65f6bf19622b18f146abc462c4": "\\omega _c = \\frac{qB}{m}", "b0692144c1c41f7f142f5a70f13a1aca": "2\\times f_1-f_2", "b0697dcf76d75d271a93aefdb19e99cb": "\\alpha = \\frac{e^2}{\\hbar c} ", "b069bf920d0d474c64db6bbe81066587": " f_{i} \\leq f_{i-1}^{(i)},\\quad 1\\leq i\\leq d-1.", "b06a8dd1af0d380f8270c58462eb5dcc": "\\Delta(a) = \\langle a,a \\rangle", "b06af4c3881de7096e94502fd672d702": "\\tfrac{dI}{dT} = \\varepsilon E - (\\gamma + \\mu)I", "b06b023b25c9241fb8dffe65c39bdb4c": "W_i \\not \\subseteq \\bar{K}", "b06b06f7a7f8e46d3f00c4599c06a958": "\\begin{matrix} \\frac{3}{50} \\times \\frac{2}{49} \\approx 0.00245 \\end{matrix}", "b06b71f2d255f15a9460e000c47b3e03": "Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\\text{ with }A, B, C\\text{ not all zero.} \\, ", "b06b77764bbbc13165ddb315219529f7": "\\! v_{pec} = a(t) \\dot{\\chi}(t)", "b06bfcbc5bc8346566c8235f152eb2ac": " T = x \\, \\sinh(gt), \\; X = x \\, \\cosh(gt), \\; Y = y, \\; Z = z", "b06c04f31cb64a051a76068f3d96e515": "\\frac{dy}{d\\varphi}=r'(\\varphi)\\sin\\varphi+r(\\varphi)\\cos\\varphi. \\,", "b06c26aed02d969d0f2315ba11b5432b": "B.", "b06c5231a06440d1603c6bfd5daac744": "\nC=\n\\begin{bmatrix}\nc_0 & c_{n-1} & \\dots & c_{2} & c_{1} \\\\\nc_{1} & c_0 & c_{n-1} & & c_{2} \\\\\n\\vdots & c_{1}& c_0 & \\ddots & \\vdots \\\\\nc_{n-2} & & \\ddots & \\ddots & c_{n-1} \\\\\nc_{n-1} & c_{n-2} & \\dots & c_{1} & c_0 \\\\\n\\end{bmatrix}.\n", "b06c74683909f74f8a161078a34a2295": "\\scriptstyle =(6.0\\pm4.0)\\times10^{-43}", "b06c9f2044b3d2bc94be3c7c91c8419f": "1, \\dots, n", "b06ca45c1370b30f4af8137b3923846c": "\\Psi = \\sqrt{\\rho}e^{iS/\\hbar}", "b06cd3422a4a291e99eb2845564ee1c9": " (4) \\frac{\\Delta P}{P} = \\mathit g + \\frac{\\Delta PB}{PB} \\mathit(1 + g)\n", "b06cdcae441fa18f78eefd0f8206bda8": "Y = X^2 \\sim \\chi^2_1.", "b06ce497e066a5754e9285e8831ba03b": "V(S, t) \\geq H(S)", "b06cf05516c536a0fb97b4ee8e521526": "\n\\bar{x} = (0.4\\times80) + (0.6\\times90) = 86.\n", "b06d05dad3a717980f79fc5ebc55a875": "f_{X_1,\\cdots,X_n}(x_1,\\cdots,x_n) = f_{X_1}(x_1)\\cdots f_{X_n}(x_n).", "b06d195f6db07daf641ed951e6a6eb6f": "\\forall s\\in S: (|s^\\bullet|\\leq 1) \\vee ({}^\\bullet (s^\\bullet)=\\{s\\})", "b06d84258ee22339e87f3b4f9b23ba1e": "\\displaystyle{(x,y)_{\\mathbf{R}} =\\Re (x,y).}", "b06da3e472c36ef5f03e2ceb8c5534db": "J_1(z)", "b06dbc91b017817f0fbab0946d087aa8": "\\Omega(\\log^2 n)", "b06e018e0b32477fba4c550bd1533d44": "4t-2n", "b06e5f9a5c96d7b5e3630befa56a53a5": "{{\\Gamma}}", "b06ec3f72c1c923c17b190df7991d516": "\\vec{\\mathbf{r}}", "b06f879dde57ae3ab3822d77abd1ae95": "f=\\coprod_{j \\in J} f_j: \\coprod_{j \\in J} X_j \\to Y", "b06f931578435cc59ef71e1b1e20f473": "\\, Lc(z)\\, ", "b06faad1fdf74ffce3e3c755909e5e63": "X^d", "b06fcae5c26abe00bcf0ad8b69eb6fef": " p = \\frac{L}{r} \\,\\!", "b07001e1331f3c7fd9b5b747962d871e": "\n \\arg\\max_{\\mathbf{p}} p(\\mathbf{p} | \\mathbb{X}) = \\frac{\\alpha_i + c_i - 1}{\\sum_i (\\alpha_i + c_i - 1)}, \\qquad \\forall i \\; \\alpha_i + c_i > 1\n", "b07034ede333cdd392c7f28b0e1485d9": "{\\mathbb Z}\\backslash \\left(\\left[-1,1\\right]\\times{\\mathbb R}\\right)", "b0703ba5c2d65266bb1478f3b48a591c": " v_s = n\\left(\\sum_{i=1}^n (1/v_i)\\right)^{-1}", "b0704cbcbdf00855bc082d0c81f992fc": "w_1=p_{nt,1}*MPL_{nt,1}=p_{nt,1}=p_{t}*MPL_{t,1}", "b070557ff7cfbec04387e07a15035e9a": "f(x+\\delta)-f(x) = a^x(a^{\\delta}-1)\\,", "b07070ab416fb89b14fb13818a30a1f7": "\\mathrm{Pr}(Q_{|x|}(x)=1)\\geq \\tfrac{2}{3}", "b0707cabdf1495faca8c3b7652abee87": " \\Delta : \\mathbb{R}^{n+1} \\times M \\to \\mathbb{R}.\\, ", "b070b665dc76f26dd17283146aea5b6f": "(14.b)\\quad \\nabla^2\\psi =\\,e^{-2\\psi} (\\nabla\\Phi)^2 ", "b070d5a4008c19857ce7fb37f5c64f99": "= | a |^{2} + n \\mathbf{E}^{a} [\\sigma_{k}].", "b070f273e59412103cf0235f3ab533eb": "{\\left | z-\\gamma \\right |} ^2 = r^2", "b07106ca3f9a615cbc0d71092240ceea": "\\pi = 3\\tfrac{1}{8}", "b0719ac7072dc4630a97c8083545463c": "n\\# = (2m + 1)\\# < 4^m \\cdot (m + 1)\\# < 4^m \\cdot 4^{m + 1} = 4^{2m + 1} = 4^n.", "b071b0c123ce3d2e8122e4a022ab64d4": "\n\\begin{align}\n&(\\tfrac{2}{5}) &= &-1, &F_3 &= 2, \\;\\;\\;\\;&F_2 &=1,\\\\ \n&(\\tfrac{3}{5}) &= &-1, &F_4 &= 3,&F_3&=2,\\\\ \n&(\\tfrac{5}{5}) &= &\\;\\;\\;\\;0, &F_5 &= 5,\\\\ \n&(\\tfrac{7}{5}) &= &-1, &F_8 &= 21,&F_7&=13,\\\\ \n&(\\tfrac{11}{5}) &= &\\;\\;\\,1, &F_{10} &= 55, &F_{11}&=89.\n\\end{align}\n", "b071b2787001bfab8088cc441a4199da": "x=\\frac{T_{-1}^{1}-T^1_1}{\\sqrt{2}}\\,,", "b0720857845ec73fbe03ae35133550e5": " p(x) = (x - a) \\sum_{i=0}^{n-1} b_i x^i", "b07242605722771613dc98a3786f899d": "(n_{t}-1)\\times n_{f}", "b0724c93e6fc80de9b80b5c05a06e30a": "V_{CE}>>V_{BE}", "b0726cd60a9e6dfd9ec231ea8d96ca8b": "M_3 = \\{\\, a\\mapsto 0, b\\mapsto 01, c\\mapsto 011\\,\\}", "b0729cbff57df067e63c968b4d2e4ba5": "E_{m}E_{n} \\, ", "b072c2e2d5f791fa563474da8e65ff79": " 1 \\times r = \\int_{V}\\boldsymbol{\\epsilon}^T \\boldsymbol{\\sigma}^* dV ", "b0731b2289a52214e0ea25eab4becc2b": "(\\csc(A+\\theta),\\csc(B+\\theta),\\csc(C+\\theta))", "b0733e0d5d02627ee66fee2e0c37e92c": " a_{i} ", "b07340946e006cf081c2ab5094aa28b6": "\\displaystyle P", "b073731fcc37819eb0743c74015222fd": "\\log_{10}0.012\\approx-2+0.079181=\\bar{2}.079181", "b0739329b07089417f3bff48e4da3158": "\nq_1 + q_2 = 1.\n", "b073afeee849d3822fc4b7a379750ba4": "\n\\begin{vmatrix}\n x & 1 \\\\\n 1 & x \\\\\n \\end{vmatrix} = 0\n", "b074309bd501c868810b9ec87a3d8b21": "\n\\begin{align}\n \\mathbf{F} & = \\alpha\\left[\\frac{1}{2}\\nabla E^2-\\mathbf{E}\\times\\left(\\nabla\\times\\mathbf{E}\\right)+\\frac{d\\mathbf{E}}{dt}\\times\\mathbf{B}\\right] \\\\\n & = \\alpha\\left[\\frac{1}{2}\\nabla E^2-\\mathbf{E}\\times\\left(-\\frac{d\\mathbf{B}}{dt}\\right)+\\frac{d\\mathbf{E}}{dt}\\times\\mathbf{B}\\right] \\\\\n & = \\alpha\\left[\\frac{1}{2}\\nabla E^2+\\frac{d}{dt}\\left(\\mathbf{E}\\times\\mathbf{B}\\right)\\right]. \\\\\n\\end{align}\n", "b0743e3bf51053357bf0aacfc8f784fe": "w_q=p_q/\\rho_q=\\frac{\\frac{1}{2}\\dot{Q}^2-V(Q)}{\\frac{1}{2}\\dot{Q}^2+V(Q)}", "b07444393d99aac2353a66b2ecde829b": "f(u_{in},u_{out})=d_u", "b0744a2f382594ec510335177ba27977": "(-x) \\cdot (-x)=x^2", "b074771be490eddadffd0b136923c8ec": " \\frac{n_1}{n_2} = \\frac{v_2}{v_1} = \\frac{\\lambda_2}{\\lambda_1} = \\sqrt{\\frac{\\epsilon_1 \\mu_1}{\\epsilon_2 \\mu_2}} \\,\\!", "b074fdb33915e3fbb1b5de69de5d91a5": "\\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{bar} = -kA \\frac{\\Delta T_\\mathrm{bar}}{L}", "b07528c7ef5deeb7cccba9ecd62a83f9": "(c_R)^{\\alpha\\beta}_{\\bar a\\bar b}", "b075597544ec9c83c6eecb9a6202cd38": " C_0^* ", "b075c771dbe0309e58cd31c7e69e872b": "f(x)=\\frac{\\operatorname{H}(x-a)-\\operatorname{H}(x-b)}{b-a}, \\,\\!", "b07634519ef6e764b022760c965891cb": "\n\\Gamma(\\mathbf{a_p}) = \\prod_{j=1}^p \\Gamma(a_j).\n", "b0763aa90c17e3026b792e2b8ff441ed": "\\Delta(t)=t^4-4t^3+8t^2-12t-12t^{-1}+8t^{-2}-4t^{-3}+t^{-4}+14, \\, ", "b076af90f38f74d6dab5bf453ef84379": "\\gamma_0 = \\gamma = 0.577\\dots", "b076ebcad3622170db27c7ec08bc6030": "E_x(M)", "b077307a9dba6b8b081870b521516b5c": "\n \\hbar\\oint d\\mathbf{r}\\cdot \\mathbf{k} - e\\oint d\\mathbf{r}\\cdot\\mathbf{A} + \\hbar\\gamma = 2\\pi\\hbar(n+1/2) \n", "b0775f0c06518fb1466e7dc4516fd340": "p_i=1/n", "b0779ee929c658b49ecbb31fec24e08a": "\\epsilon_{abcd}", "b077a0790f898a55907e1e5cbfd9fe53": " \\Delta_i = \\underset{x , y \\in C_i}{\\text{max}} d(x,y) ", "b077b2cd5241144cfc73312f50e345a7": "FGT_2=H \\mu^2 + (1-\\mu^2) C_v^2", "b0782d7122a65808462b7d3ed25e1c78": "j>\\log_p(2n)", "b078321e7fecbd6fac730e8977859a19": "\\{r=r_{ps}\\}", "b07844b1da5f09752e6e2f056906f582": "I_2(6),", "b07861c8ab466eab8dc9f4a9b1da3882": " t_i ", "b078828eae356b38b9c2b775afe76214": "V \\subseteq C^r", "b078a00e66c6fe10c228c1c5b3cccc84": "\n \\lambda^2 = \\omega\\sqrt{\\frac{2\\rho h}{D}} \\,.\n", "b0791cb4b4848de1677f8a86072279ca": "\\Psi_s", "b079715a16df9ae93fd594dc0f99da56": "h^2=pq", "b079bad508a99111d870a3356942b23e": "f\\circ \\gamma", "b07a227977fa7ae3da49839bac932d9e": "\\overline{B}(x_1, r_1) \\subset W \\cap U_1", "b07a343f02c8f9bde67d91e4bf67a2bf": "V_{i,y} = \\sum_{j=1}^n \\frac{\\partial y_j}{\\partial x_i} V_{j,x}.", "b07a3afa21c4946f582070d22a5a551e": "\\rho(z):= \\begin{cases}(-|z-z_0|\\,\\log |z-z_0|)^{-1} & |z-z_0|<1/2,\\\\\n0 & |z-z_0|\\ge 1/2,\\end{cases}", "b07a69bc6e409410acd877acdf9417f6": " \\Theta_+ ", "b07a863687b6c2d485056ceada0135cb": "\\mathbb{E}[z^N]=\\sum_{n=0}^m(z-1)^nS_n,\\qquad z\\in\\mathbb{R}.", "b07a8fc5d559fc7dd33d5e7c4820a394": "B_n(G,T)", "b07a91c215a3e923660d0122d8e69eda": "\\ln(1+x)=\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} x^n = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\cdots \\quad{\\rm for}\\quad \\left|x\\right| \\leq 1,\\quad", "b07b5abd67ee0b72f4136c82c68a0c48": "\\vec{x}", "b07b6f8e269670a73d71005b4a774a54": "A = X \\cup Y", "b07c11dd1fce58dbfff537095e16bcb1": "D \\sigma", "b07c2b0ea530cf411af2184f67aaeab0": "j \\in \\{ 0,0.5,1,1.5,\\ldots \\}", "b07c797a76fc455148631187eaf42682": "\\alpha^{-1}_{1}", "b07cc51b1773cce1ea8ef529bc6f1181": " \\mbox{Exp}_{stock} = 0.5 \\times1500 + 0.3*300 + 0.2\\times(-800) = 680", "b07d3aaefc6fd34e4ed7f63a645731be": "\\mathcal{U}(\\alpha, \\tilde{u})", "b07d7535a2ae6de1583097e80d3aba51": "(m_{ij})", "b07db472c9ec5c96648efcaad07e700c": "\\scriptstyle q_e", "b07e051888acf6be2eda4014b05efef8": "\\scriptstyle 4\\left(1-\\frac13+\\frac15-\\frac17+\\frac19-\\cdots\\right),", "b07e2defa14557f78920e474101d0483": "\\begin{align}\n\\int_0^1 x^{-x}\\,dx &= \\sum_{n=1}^\\infty n^{-n}&&(\\scriptstyle{= 1.29128599706266354040728259059560054149861936827\\dots)} \\\\\n\\int_0^1 x^x \\,dx &= \\sum_{n=1}^\\infty (-1)^{n+1}n^{-n} = - \\sum_{n=1}^\\infty (-n)^{-n} &&(\\scriptstyle{= 0.78343051071213440705926438652697546940768199014\\dots})\n\\end{align}", "b07e61953fa6641297537322b2f2093a": "\\int_{-x}^x f(t-1)\\,dt = \\int_{-x}^1 f(t-1)\\,dt + \\int_1^x f(t-1)\\,dt = -(x+1) + (x-1) = -2", "b07ea54cbc2b65f815311ec81682b386": "q = \\cdots U_\\gamma U_\\beta U_\\alpha q_0.", "b07ed8076fe5fccba58a8d9b4565f250": "W\\left(S\\right)", "b07ef39061da1b95e824cf7ea4a20ade": "P=n^\\searrow.n^\\nwarrow(~)", "b07f4b340ee27bc06700fdb2868c5546": "\\frac{\\beta}{\\alpha-1}\\!", "b07f90147071ed2bc741f01f8f670757": " uPER = \\frac{1}{min} \\times \\left ( 3P + \\left [ \\frac{2}{3} \\times AST \\right ] + \\left [ \\left ( 2 - factor \\times \\frac{tmAST}{tmFG} \\right ) \\times FG \\right ] \n+ \\left [ 0.5 \\times FT \\times \\left ( 2 - \\frac{tmAST}{tmFG} + \\frac{2}{3} \\times \\frac{tmAST}{tmFG} \\right ) \\right ] - \\left [ VOP \\times TO \\right ] \n- \\left [ VOP \\times DRBP \\times \\left ( FGA - FG \\right ) \\right ] - \\left [ VOP \\times 0.44 \\times \\left ( 0.44 + \\left ( 0.56 \\times DRBP \\right ) \\right ) \\times \\left ( FTA - FT \\right ) \\right ] \n+ \\left [ VOP \\times \\left ( 1 - DRBP \\right ) \\times \\left ( TRB - ORB \\right ) \\right ] + \\left [ VOP \\times DRBP \\times ORB \\right ] + \\left [ VOP \\times STL \\right ] + \\left [ VOP \\times DRBP \\times BLK \\right ] \n- \\left [ PF \\times \\left ( \\frac{lgFT}{lgPF} - 0.44 \\times \\frac{lgFTA}{lgPF} \\times VOP \\right ) \\right ] \\right ) ", "b07fc987d7faa3b27f4afdfa28a33679": " \\mu(B(x,2r))\\leq C\\mu(B(x,r))", "b07fe3ffe954a147ede2b9973f781a8d": "\\lceil \\frac{1}{2} |V|\\rceil", "b0801e7cc35cf3ced7d1e7e93d16e238": "G^{\\hat{m}\\hat{n}} = 8 \\pi \\mu \\, \\operatorname{diag}(1,0,0,0) + 8 \\pi p \\, \\operatorname{diag}(0,1,1,1)", "b0802f1639db50ff292d437ff68d9fad": "b_1,...,b_{2g}", "b08042e5d4dbcb7703977dfb1a8c8310": "F^p H= \\bigoplus\\nolimits_{i\\geq p} H^{i,n-i}. ", "b0809e731c18140fa4b8790a05bff824": "\\alpha=(a-\\mu)/\\sigma,\\; \\lambda(\\alpha)=\\phi(\\alpha)/[1-\\Phi(\\alpha)]\\; ", "b08132540e7c3a6261477afcfb3e5025": "\\tan{\\frac{x}{2}}\\tan{\\frac{w}{2}}=\\tan{\\frac{y}{2}}\\tan{\\frac{z}{2}}", "b08150258322942d142cad689c173aa3": "r_2=0.1\\times Do", "b081706ef7fbf2be556d8ba5f07382bd": "\n I_{jk} = \\operatorname{E}_X\\bigg[\\;{-\\frac{\\partial^2\\ln f_{\\theta_0}(X_t)}{\\partial\\theta_j\\,\\partial\\theta_k}}\n \\;\\bigg].\n ", "b0818838e0c67cd14760f45f42ebfecc": "p = p^{\\star}_{\\rm A} x_{\\rm A}", "b0819810b4fd41fa3da5807cec09bb60": "\\eta=\\rho+ \\bar\\rho", "b081b7e2fbb6e424ef6e102545a01e9a": " \nj=1,2,....,n\n", "b0824123acee14aab9341771564f50f4": "\\sigma_2 = \\cos \\psi \\text{d}\\theta + \\sin\\psi \\sin\\theta\\text{d}\\phi", "b0828d5d10626ae9c4f870bfde03c14b": "\\limsup_{\\delta \\downarrow 0} \\delta \\log \\mu_{\\delta} (X \\setminus K_{\\eta}) < - \\eta.", "b082f101d81af99b9debc4583a13cc91": "\\mathit{d}", "b082ff03ae5e29535ae01bc9c0cf286f": " \\mathbf{E}\\left[e^{\\lambda X}\\right] \\leq \\frac{b-EX}{b-a}e^{\\lambda a}+\\frac{EX-a}{b-a}e^{\\lambda b}.", "b08302ec38b2c181e928d905478f5fe1": "m]", "b0833afe97db65cafa5197434f56dc21": "C''+C'^2+C'+\\frac{c}{x}C'+\\frac{c-a}{x}=0\\,", "b083e666dc25b85e8581ca16c6566da8": "\\frac{y_{k+1}-y_k}{\\Delta t} = - y_{k+1}^2", "b08427c38b7d7964c1accc25a2d1308c": "\\frac{\\mathrm{Ma}}{\\mathrm{Re}}=\\frac{U_\\infty / c_s}{\\rho U_\\infty L / \\mu }=\\frac{\\mu }{\\rho L c_s}=\\frac{\\mu }{\\rho L \\sqrt{\\frac{\\gamma k_BT}{m}}}=\\frac{\\mu }{\\rho L }\\sqrt{\\frac{m}{\\gamma k_BT}}", "b0848ce3cc33a68a387c23cb93d5dbe1": "\\omega\\ge \\omega_{max}", "b08589980ea3b0cf90ddb8fc1f852bad": "\\dot k", "b085b7602933fd86e9a781621627bdce": "F_\\mu = qE_{\\mu\\nu}U^\\nu", "b08620f50333be4251e535766f252703": "\\boldsymbol \\nabla T", "b08624b9295db9042280508a197b146f": " \\mathfrak{P}(\\mathfrak{C}(\\mathcal{Z}))", "b086a5dd02389c18b31590889948c7ab": "r_1 |a_{n}|", "b089e9d90c6ed3f35c23b462bb7acb68": "C^\\infty_p(M,N)", "b089eba2bd8a80d3b6bdc0b401441ee0": "B^{\\prime}", "b089f748a7d862acf1b7610771fd35c9": "\\sin \\left(angle \\right) = {perpendicular \\over hypotenuse}", "b08a30b6c684c4e7792edda334dfe928": " \\left\\langle \\!\\! \\left\\langle {0 \\atop m} \\right\\rangle \\!\\! \\right\\rangle = [m=0]. ", "b08adbc793bacf35b1b8397cc2fa390e": " E_d = \\frac{ O - \\frac{ 1 }{ K } - \\frac{ K - 1 }{ N } }{ 1 - \\frac{ 1 }{ K } - \\frac{ K - 1 }{ N } } ", "b08af700f703a669f3fedb0b2ac39f57": "\\scriptstyle \\ h^2 = ab ", "b08b034424b912eca5069ec2a0a59f9d": "\\sin\\frac{\\pi}{30}=\\sin 6^\\circ=\\tfrac{1}{8} \\left[\\sqrt{6(5-\\sqrt5)}-\\sqrt5-1\\right]\\,", "b08b4429be2c06527755a16c325462db": "\\ pV_m = R (T_C+273.15).", "b08b82ae192a2f4780866af050115337": "f=f_{x}=\\frac{R_{M}}{2}\\cdot \\cos \\theta ", "b08bc6280b53e69abcd2992702158481": "\\reals", "b08c10de4deaa2e3319de8e24c99db5a": "a_{j}\\ne 0", "b08c2929757dee806b670132684015cb": "C_i(x,t)", "b08c45dfd54f306086ddceca79adbd57": "\\mathbf{a}-\\mathbf{p}", "b08cb2046bce953b0b1da0815b8eabae": "\\alpha' = \\Omega \\Delta \\tau", "b08cf9ae614d746aaf7c6d32c1e95225": " b_{i,j}=\\begin{vmatrix} a_{i, j} & a_{i, j + 1} \\\\ a_{i + 1, j} & a_{i + 1, j + 1} \\end{vmatrix}. ", "b08cfde7559b0f092f5506122ac3a955": "C = A \\cdot B", "b08d64a5cdeb48bd406c59bb2c46fe5c": "{\\langle F_X \\rangle\\over M_{pl}}", "b08e21b125b369f64ad4303255e25d16": "A\\mathbf{x}=\\mathbf{b}\\;\\;\\;\\;\\;\\;\\text{or}\\;\\;\\;\\;\\;\\;\\begin{alignat}{7}\na_{11} x_1 &&\\; + \\;&& a_{12} x_2 &&\\; + \\cdots + \\;&& a_{1n} x_n &&\\; = \\;&&& b_1 \\\\\na_{21} x_1 &&\\; + \\;&& a_{22} x_2 &&\\; + \\cdots + \\;&& a_{2n} x_n &&\\; = \\;&&& b_2 \\\\\n\\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && &&& \\;\\vdots \\\\\na_{m1} x_1 &&\\; + \\;&& a_{m2} x_2 &&\\; + \\cdots + \\;&& a_{mn} x_n &&\\; = \\;&&& b_m \\\\\n\\end{alignat}", "b08e48cf699800efbe5c794e69875b1e": "\\varepsilon\\colon L\\to\\mathbb{C}", "b08e552224f34a5e027fed8d6f43cb9e": "\\log P = A-\\frac{B}{C+T}", "b08e574bcb124bd838d26c71e37cd15f": "R(\\hat{n},\\phi)|\\psi_0\\rangle", "b08e713328b856fb6e32d087802cb15d": " V_C = V_A (1 + z)^3 ", "b08e7e18235dbfb37ff066a7173097e3": "N^*N = NN^*", "b08f18a6f8745371aa63e07f16295a66": "x \\succ y", "b08f3ba413e46e5f11a131876d4dce14": "\\left (\\sqrt{2}^{\\sqrt{2}}\\right )^{\\sqrt{2}}=\\sqrt{2}^{2}=2", "b0900ce1968e1f6b255bff8e441ad4d7": " F: R_n \\rightarrow R ", "b09023dbae571e79e876e297d3cc32e3": "1 \\geq 1 \\geq 1 \\geq \\cdots", "b0902a9f9a62d64aad259ae03c3ac39c": "2S(n-1,1)=(n+1)\\zeta(n)-\\sum_{k=2}^{n-2}\\zeta(k)\\zeta(n-k).", "b09032be270e5f0f7657a3d47f75b16a": " k-\\epsilon ", "b090de0255faea421987b89c64dac10d": "d\\mathbf{x}=(dx_1, dx_2, \\dots, dx_n)", "b0914f7143110397fe298f6c411969ba": "\\tilde{F}_i = \\begin{bmatrix} \\vdots & \\vdots & & \\vdots \\\\ \\tilde{f}_{i1} & \\tilde{f}_{i2} & \\cdots & \\tilde{f}_{i|J_i|} \\\\ \\vdots & \\vdots & & \\vdots \\\\\\end{bmatrix}_{N \\times |J_i|},", "b091ad31a91dec2141331a8b480b1278": "Y_i = (B_0 + B_1T_i) + \\alpha\\sin(2\\pi\\omega T_i + \\phi) + E_i ", "b091b3894cb744ae473b5082184352fa": "= \\arcsin\\left(\\cos{(\\text{spring angle})} * \\cos{(\\text{wall angle/2})}\\right)", "b09204b14c9df02be450d553c305c573": "M^N", "b0920f366512aa14cb59e6bcc7e068c0": "h:\\tau_3{\\to}\\tau_4", "b092822f51aa21b4f7b081e0449324a5": "S_{\\max}", "b092adacea93dcc76db5aab9d21abeaa": "\nN_K(x) = \\{ p \\in V | \\langle p, x - x^* \\rangle \\geq 0, \\forall x^* \\in K \\}.\n", "b092b0063dd36d0849d9d6703aeae740": "R[t_1, t_2, \\dots]", "b092b81283aa8868c0a9bc664dec79f2": "\\lambda > n+p", "b092e779a98017e72574d21935f3c34c": "S = \\{ (x,y_1, \\dots, y_d) \\in \\R^{1+d}; -N-\\frac{1}{2} \\le x \\le N+\\frac{1}{2}, |\\alpha x - y_i| \\le \\frac{1}{N^{1/d}} \\} ", "b09314b294a763c6e52e92f0a33e29b7": "AM = \\sqrt { ( r+c )^2 \\cos^2 z + ( 2r+1+c ) ( 1-c ) } \\; - \\; ( r+c ) \\cos z \\,", "b09326ee94421277609bbc9f17feb222": "f(n)\\geq\\log_2(n!).", "b093fe86d03e09a5d33c9b234cc6bc90": " \\ S_t \\,", "b0940c2e556df0bfb27f95bb8e291093": "\\Gamma \\vdash \\phi \\leftrightarrow \\psi", "b0940cc4d2af2070bc7fa0ae8cf3844d": " 1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \\ldots\n\\text{ for the signs } +, +, -, -, -, +, -, -, \\ldots.", "b094259c8f3418a59f28b14c0434bf8f": "\\mbox{Lat}(\\Sigma) = \\bigcap_{T \\in \\Sigma} \\mbox{Lat}( T ) \\;.", "b0942823a5e58a98526fbfbc16415163": "h < 2r", "b094292ba067987cb8a5443b637b5b71": " X_1, \\dots, X_n ", "b0944647465b228f4340eef5583df957": "\\tilde{E}_{r}", "b094605cda57feb70c53279f95cf39a1": "G^{(2)}, G^{(3)}, \\ldots", "b09496fd154355c4d49b98305d0ba1a6": " (A f)(y) = \\inf\\{ f(x) : x \\in X , A x = y \\} ", "b094bf11248a56f4646bcc6cacc72bb7": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 680\\cdot 1.77)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 259\\cdot R_{\\bigodot}\n\\end{align}", "b094f013d7dd0056ca9e9466422d00fb": "\n\\rho\\left(\n\\frac{\\partial}{\\partial t}+{\\bold u}\\cdot\\nabla\n\\right){\\bold u}+\\nabla p=0\n", "b094f5c5e123ddf5fb41cfeccd353830": "R^r f_* \\circ g'_* \\circ g'^* \\to R^r(f \\circ g')_* \\circ g'^* = R^r(g \\circ f')_* \\circ g'^* \\to g_* \\circ R^r f'_* \\circ g'^*.", "b0957dbee8fa4e73f290aa71115dd037": "f^{-1}(\\{a\\})", "b0958010b0b988e223cba7bd033ee1f6": " a \\kappa < \\!\\, 1", "b095b75b3b23faa6f98232f54fa8cf46": " C_{\\text{XY}} +1 ~~~~~~~~~~~~~\\text{first sequence pair}", "b095ca0c9d4246c4005b6b5639c6df00": "spin^c", "b095e5865c5d38eeb2912f2aa92c717c": "\\frac{\\sin A}{\\sinh a} = \\frac{\\sin B}{\\sinh b} = \\frac{\\sin C}{\\sinh c},", "b09639675e573ef566df7a52f01c7c4b": "0.2Ur", "b096732058052c145c372be74f137a18": "S^3 = \\left\\{(z_1,z_2)\\in\\mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\\right\\}", "b096f57d7be642b69b8f9fab1fcb6426": " r_{(j)}(\\beta) ", "b09712612275b9adcebc59cc38e19e21": "\\varepsilon_{a_1 a_2 a_3 \\ldots a_n} =\n\\begin{cases}\n+1 & \\text{if }(a_1 , a_2 , a_3 , \\ldots , a_n) \\text{ is an even permutation of } (1,2,3,\\dots,n) \\\\\n-1 & \\text{if }(a_1 , a_2 , a_3 , \\ldots , a_n) \\text{ is an odd permutation of } (1,2,3,\\dots,n) \\\\\n0 & \\text{otherwise}\n\\end{cases}\n", "b0971c2a29a2e76a707593d8cb40579f": "\\max \\big\\{ d(x, m'), d(y, m') \\big\\} \\leq \\frac1{2} d(x, y) + \\delta,", "b097d914131765a722206db7e88fa842": "\\lVert z w \\rVert = \\lVert z \\rVert \\lVert w \\rVert .", "b097e6106ce7c1b5df147910ca1e63a5": "P \\subseteq C", "b098145b26d77efadd349d4250ebc7a2": "\\hat{f}_n \\ \\to \\ 0 .", "b09841d2cce3e4c24e28fd6cbac2433d": " \\ \\displaystyle \\mathcal{U}(\\alpha,\\tilde{u}) \\ ", "b09867c235b83e2a87698f537632c451": "\\Delta I_{\\text{L}_{\\text{On}}}", "b0987ae7f08ea37c08a07e421262d709": "s_{n-1}(t) =\\overset{\\cdot }{s}_{n-2}(t)+\\alpha _{n-1}(t)s_{n-2}^{\\gamma\n_{n-1}}(t)", "b098ae99553cbc9a0245f8970b92c3a7": "\\forall n\\forall x(\\varphi_n(x)\\leq\\varphi_n(y) \\iff S(n,x,y) \\iff T(n,x,y))", "b098f7ee2bbffcf7df1d743ac9085c00": "A_{I}(t) = e^{i H_{0,S} t / \\hbar} A_{S}(t) e^{-i H_{0,S} t / \\hbar}.", "b09955ca3a5c43bb13d1871b457e4118": " \\nabla_l \\left(R^l {}_m - {1 \\over 2} g^l {}_m R\\right) = 0,\\,\\!", "b0996b6d86f4cb78565d379e6cb02133": "J_{F^{-1}}(F(p))", "b099daeebeecf21a03f26e3edb233a19": "k\\leq n", "b099f3aec5c45937c02347b4e0ad87f1": "X_3 = 1\\cdot1\\cdot\\sqrt{2} + \\sqrt{2}\\cdot1\\cdot1 = 2\\sqrt{2}", "b09aa5fcc929e3d648b1bd5ec525f7bc": "H=-\\frac{\\hbar^2}{2 m a^2} \\frac{\\partial^2}{\\partial \\phi^2}-\\lambda \\cos\n\\phi", "b09abd37d6022b8279cc831be0cf7340": "\\ell_{ij} = \\hat{\\mathbf{e}}_i\\cdot\\mathbf{e}_j", "b09b095fd297f6c8544a7712d6f553a3": "O = arg(B_{123})", "b09ba10586d962ac55aaf550945045a2": "M + e^- \\to M^{+\\bullet} + 2e^- ", "b09bccf7122d4243042e3491f20eb826": "E_{tot}", "b09bec5e509227aafa380bb42292aac3": "r(p_i)", "b09c6f646b1383ec048b47d563da6de2": "R=2", "b09cc9965beed1ae5818f28873093de1": "x^2(x-1)", "b09d1080c3a7b893290a4aaa6d590f2e": "Q=f_a(S_a)", "b09d15be6b423f79b4ab4e9e54727ef3": " H_A : \\rho < \\rho_0 ", "b09d91fcc5680fabee5a1a4c1e88e5a5": "g\\sqrt{J(J+1)}", "b09e1bb0ce5e9ec09eb37997943fc2ea": "\\mathbf{c}\\in\\mathbb{C}^d", "b09e73b7d89df839274b362265cdcc0f": "X_{\\text{poly}}=\\frac{m_{\\text{Pol}}}{\\sum_i\\int_0^t\\dot{m}_{i,\\text{in}}(\\tau)d\\tau}", "b09f80a4186bc9855db35fdf09ad8713": "\\mathbb{E}^g[X \\mid \\mathcal{F}_t] := Y_t", "b09fc3dc6d71ccc7a44d6a9e27cd68fe": "\\mathbf{Q}(i\\sqrt{5})", "b09fca95f9ef8665a0fbf54c0528ab8b": "\\psi(\\Omega)^{\\psi(\\Omega)}", "b09ff2913c2e46fbc366417a702a1b2e": "\n\\text{volume} = \\frac{1}{3}\\sum_{\\text{face } i} \\vec x_i \\cdot \\hat n_i A_i\n", "b0a00d94ecf9551014eaa180652b51dd": "E_p=p_p^2/2m_p", "b0a0839b94a22d6b9211e0b9a6c3a569": "x_{p,ni}=\\xi/(x_{ni})", "b0a0fd89b5ad5b86a94a77d23ce141dd": "\\left \\lfloor 1000/(2^{\\frac{2n-1}{4}})+0.2 \\right \\rfloor", "b0a12aa4c7287905815e4d9f63dc9fcb": "\\mathcal D\\,", "b0a1c654d9b36f3a0fc2d87d8b9a77c9": "\\displaystyle w(n,g)", "b0a1dbcc2467e5aa5f6dd877a85b7097": "= \\frac{\\rho kT}{\\mu H}", "b0a1efec958bdcd403a87dceb1582202": "h_e", "b0a1f35e3d362d2575e4b1bd343ea366": "\\mathrm{Tr}_{15} : {{{h^a}_{bc}}^d}_e \\mapsto {{{h^a}_{bc}}^d}_a.", "b0a2491ef753d9d17f23b7ea0bd165ce": " \\rho={1\\over 2} \\sum_{\\alpha\\in \\Sigma^+} m_\\alpha \\alpha.", "b0a29cdc333d1faf656b78194957e812": "e_1(0)=\\dot\\gamma(0)/|\\dot\\gamma(0)|", "b0a2a4033bb8e6fc75d68eb789349973": "z \\approx \\frac{v_{\\parallel}}{c}", "b0a2b8f798909b1f80de23eab07d0e10": "\\frac{\\partial \\boldsymbol{F}}{\\partial \\boldsymbol{S}}:\\boldsymbol{T} = D\\boldsymbol{F}(\\boldsymbol{S})[\\boldsymbol{T}] = \\left[\\frac{d }{d \\alpha}~\\boldsymbol{F}(\\boldsymbol{S} + \\alpha~\\boldsymbol{T})\\right]_{\\alpha = 0}", "b0a2f0cecd20603f49d61da741b934b0": " F(x) = 1 - \\exp\\{ - \\beta( \\log x - \\log \\sigma)^2 \\} = \n1 - \\left( \\frac{x}{\\sigma} \\right)^{-\\beta(\\log x - \\log \\sigma)}.\n", "b0a2f2922576d2ee442510379334b701": "Z_{\\text{rot}}=\\sum\\limits_{J=0}^{\\infty }{g_{J}e^{-{E_{J}}/{k_{B}T}\\;}}", "b0a30ce1a4959b3ade707a759445f452": " \\omega_{pe} = \\sqrt{\\frac{4 \\pi n_e e^{2}}{m^*}},", "b0a31a5cd487644bd5f372ddd360e805": "\\begin{vmatrix}\n & a_4 & a_3 & a_2 & a_1 & a_0 & 0 & 0 \\\\\n & 0 & a_4 & a_3 & a_2 & a_1 & a_0 & 0 \\\\\n & 0 & 0 & a_4 & a_3 & a_2 & a_1 & a_0 \\\\\n & 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 & 0 \\\\\n & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 & 0 & 0 \\\\\n & 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1& 0 \\\\\n & 0 & 0 & 0 & 4a_4 & 3a_3 & 2a_2 & 1a_1 \\\\\n\\end{vmatrix}.", "b0a36e8dd607b70c725038fe531fd0af": "\\tan(\\theta)\\,", "b0a39d661ea4c6fa8fd181a90c75a017": "L'=L_{0}/\\gamma", "b0a3b7c44070c36a994e8a39248f0514": "2y^2(x^2+y^2) - 2by^2(x+y) + (b^2-3a^2)y^2 - a^2x^2 + 2a^2b(x+y) + a^2(a^2-b^2) = 0 . \\,", "b0a3c2109e6956202dd19aca88fa8c02": "t_i = x_i", "b0a3ed2f3e734b387f346f921be4c4a5": "\n\\theta_\\mathrm{c}=\\arccos\\left(\\frac{r_\\mathrm{A}\\cos{\\theta_\\mathrm{A}}+r_\\mathrm{R}\\cos{\\theta_\\mathrm{R}}}{r_\\mathrm{A}+r_\\mathrm{R}}\\right)\n", "b0a4159de5cebc67608f90f4440d6901": " \\begin{align} \n\\boldsymbol{\\partial} & = \\left(\\frac{\\partial }{\\partial x_0}, \\, -\\frac{\\partial }{\\partial x_1}, \\, -\\frac{\\partial }{\\partial x_2}, \\, -\\frac{\\partial }{\\partial x_3} \\right) \\\\\n& = (\\partial^0, \\, - \\partial^1, \\, - \\partial^2, \\, - \\partial^3) \\\\\n& = \\mathbf{e}_0\\partial^0 - \\mathbf{e}_1\\partial^1 - \\mathbf{e}_2\\partial^2 - \\mathbf{e}_3\\partial^3 \\\\\n& = \\mathbf{e}_0\\partial^0 - \\mathbf{e}_i\\partial^i \\\\\n& = \\mathbf{e}_\\alpha \\partial^\\alpha \\\\\n& = \\left(\\frac{1}{c}\\frac{\\partial}{\\partial t} , \\, - \\nabla \\right) \\\\\n& = \\mathbf{e}_0\\frac{1}{c}\\frac{\\partial}{\\partial t} - \\nabla \\\\\n\\end{align}", "b0a45761fd688aad6145c908d4813041": "\\begin{matrix}2&2&4\\\\3&5\\\\6&6\\\\8\\end{matrix}", "b0a463127fe8c970880c8995f29a62aa": " \\sigma^2 = \\sigma_+^2 + \\sigma_-^2. ", "b0a46f8c770478e05b0b6f44b2bc5b84": "f_2(x) = c^x, \\ c \\ne 0, 1", "b0a48359f42798f182d73e0108a116fa": "\\mathcal{I}_{\\alpha, \\alpha}", "b0a4cb7ce577aa2f8ca5df9940854ea5": "\\nabla^2 \\mathbf{E} = \\mu_0 \\epsilon_0 \\frac{\\partial^2 \\mathbf{E}}{\\partial t^2}", "b0a4fd656e2d3fe2f9aed0892c42f982": "i_2 = i_1 + i_s", "b0a4fedf71ce9c71f1a5fc80f65e72fd": "\\hbar{\\mathbf{k}}", "b0a52df617c950265382117ab5c2453a": "\\exp(\\overline{z}) = \\overline{\\exp(z)}\\,\\!", "b0a55e20c11721c2b22c98116ed971b7": "MN/m^{3/2}", "b0a58db1099c5a0ac1433eb1d1f0a18a": " x_1^t \\cdots x_d^t \\not\\in (x_1^{t+1},\\dots,x_d^{t+1}). \\, ", "b0a5ca214bc8cc79598552dc8f9d14fa": " w_i = \\sum_{j=0}^{n} a_j i^j ", "b0a60b3d6b9625e992b6e76260f446d2": " { a \\over {b R T} } = \\sum_i { \\left( {x_i} {a_{ii} \\over {b_i R T}} - { { {g^E_0} \\over {R T} } + \\sum { x_i ln { b \\over {b_i} } } \\over {0.64663} }\\right) } ", "b0a6289567d10a0a779bf4e6078352b5": "S=\\int_M \\left[\\frac{1}{2}\\mathbf{F}\\wedge *\\mathbf{F} - \\mathbf{A} \\wedge *\\mathbf{J}\\right]", "b0a641b71463f9ec4f5af796a6a8ac25": "GL(n,\\mathbf{H})\\cdot Sp(1) < GL(4n,\\mathbf{R})", "b0a680f55c7f281f4e92fcba8f48833a": "\\mathbf{T} = T_{j_1 j_2 \\cdots j_q}^{i_1 i_2 \\cdots i_p} \\mathbf{e}_{i_1 i_2 \\cdots i_p}^{j_1 j_2 \\cdots j_q} ", "b0a726a1b2590f08657b2f8f81b338f5": "\\int d^4 x M^{\\mu\\nu}_0(\\vec{x},t)", "b0a736e17c14e37bfb5be42c7ed82760": "\\mathrm{F}_{12}", "b0a73e4f3e1edae624351e68bb24c55f": "p_{00}/(p_{01}+p_{00})", "b0a756379a4a345703bb0102544099d7": " a_{00} = \\mathcal{L}(p_9)+p_3p_9,\n", "b0a7718f4a5f176de6d21f54f8955bac": "J^a_{-1}J^b_{0}", "b0a7c818eb8972e6e6c4a5ab2ebf5612": "x_0 \\notin\\partial E", "b0a86d53ba85b5636b19239b0cca06c5": "e^{-\\lambda}\\frac{\\lambda^k}{k!} = e^{-5}\\frac{5^k}{k!}.", "b0a8820b1d306649faddbd162eb19860": "h_j \\ (j = 1,\\ldots,l)", "b0a89c71b10c86c9f9b631a4c0b6fda6": "H_k(M;\\mathbb{Z}_2)", "b0a8be8844b4fa1afca3468fa46e779b": " \\frac {d \\vec {J}}{d t} = \\vec{\\mu} \\times \\vec {B} ", "b0a8cf92870c7aea3e21c9d39636b7c9": "s_{ssb}(t)", "b0a8ee21de2ce6fd9addb305b83f3fd0": "\\tau_\\nu = \\min\\left\\{t : \\sum_{i=1}^{t} \\sigma_i^2 \\ge \\nu\\right\\}.", "b0a8ee46b8a8b86c213b91bf3420e932": " SM(t,f) = \\int_{-\\infty}^\\infty ST_x(t, f+\\nu/2) ST_x^*(t, f-\\nu/2) G(\\nu) e^{j2\\pi\\nu\\,t} \\, d\\nu", "b0a9350785c6e7def240390c52a71dd7": "i=1\\ldots m", "b0a9434909b5671ce33add3cb6723f59": "\\begin{pmatrix}1&1\\\\\n2&2\\end{pmatrix}\\begin{pmatrix}1&1\\\\\n-1&-1\\end{pmatrix}=\\begin{pmatrix}-2&1\\\\\n-2&1\\end{pmatrix}\\begin{pmatrix}1&1\\\\\n2&2\\end{pmatrix}=\\begin{pmatrix}0&0\\\\\n0&0\\end{pmatrix}.", "b0a9660c4d6f7599496833934c45ec0a": "A = \\frac{{t^2 \\sqrt {25 + 10\\sqrt 5 } }}{4} = \\frac{5t^2 \\tan(54^\\circ)}{4} \\approx 1.720477401 t^2.", "b0a98c9f991114e92f4a50c9f9dcc64e": "{d^n \\over dx^n},", "b0a98e54153c3d45492aba99cfa69bf6": " \\mathbf{x}_1,\\mathbf{x}_2,\\cdots, \\mathbf{x}_a \\in \\{0,1\\}^n ", "b0a9b336a3ed184f91867928230e2e11": "\nH_1 = \\begin{bmatrix}\n1 \\end{bmatrix},\n", "b0a9d1edf20f41de4b5b5ea8afb99bfc": "\\varepsilon_{12} \\varepsilon^{12}", "b0a9efcc4cc1c1037065bad87326accd": "\\ \\cfrac{1}{15 + \\cfrac{1}{1 + \\cfrac{1}{102}}}", "b0aa112b91bc940e05ee650570f5c9e0": "\\left(\\begin{smallmatrix}1 & \\pm 1 \\\\ & 1\\end{smallmatrix}\\right)", "b0aa2f6c8c4c606f3c587b89de809e95": "{}^IE^0_{p,q} =\nT_n(C_{\\bull,\\bull})^I_p / T_n(C_{\\bull,\\bull})^I_{p+1} =\n\\bigoplus_{i+j=n \\atop i > p-1} C_{i,j} \\Big/\n\\bigoplus_{i+j=n \\atop i > p} C_{i,j} =\nC_{p,q},", "b0aa4106b667feb9fdf231865a596703": "\\begin{align} \n& \\hat{E} = \\hat{H} \\\\\n& \\hat{E}\\Psi = \\hat{H} \\Psi \\\\\n\\end{align}\\,\\!", "b0aa8d9e8f23a9fe8654e458cec2ec15": "0.4 M_\\odot", "b0aac3107a1f145c63359df595af5d0a": "C_i\\equiv0", "b0aaf89036f337adf0823afb23d73477": "|\\vec{\\nabla}E| = \\hbar^2 k/m = \\hbar \\sqrt{ \\frac{2E}{m}}", "b0ab0254bd58eb87eaee3172ba49fefb": "exp", "b0ab090bfd089e0023285d22c7f660af": "\\begin{align}\n & \\mu_1(g_1) = 0, \\\\\n & \\mu_2(g_1) = \\frac{ 6(n-2) }{ (n+1)(n+3) }, \\\\\n & \\gamma_1(g_1) \\equiv \\frac{\\mu_3(g_1)}{\\mu_2(g_1)^{3/2}} = 0, \\\\\n & \\gamma_2(g_1) \\equiv \\frac{\\mu_4(g_1)}{\\mu_2(g_1)^{2}}-3 = \\frac{ 36(n-7)(n^2+2n-5) }{ (n-2)(n+5)(n+7)(n+9) }.\n \\end{align}", "b0ab72dc328ef9c89c3f59178d03c118": "h :\\{1,2, \\ldots,n \\} \\rightarrow \\{1,2, \\ldots,n \\}", "b0ab8a90447399d5ec8ca742435577b8": "\\Gamma_i = - \\frac{1}{RT} \\left( \\frac{\\partial \\gamma}{\\partial \\ln C_i} \\right)_{T,p} \\,.", "b0ab903b05c3437eac72a1856af93473": "P(s,x)=\\frac{\\gamma(s,x)}{\\Gamma(s)},", "b0abc0973221bdcec31073d9fc8c7ffb": "w \\in W,", "b0abfdc45e030b8fcb72390c5587b41b": "\\ell(\\theta | X_1,\\ldots,X_n) = \\sum_{i=1}^n \\log f(X_i| \\theta) ", "b0ac4d4351e388a360ca0aeac283dfc5": "C_3 \\rtimes_\\phi C_2", "b0ac96462e21a5f822d97aad9f36dd61": "P(\\mathbf{x},t)", "b0acb92f8f1051dd230fdecbbbdef0c0": "\\delta \\phi^A = \\epsilon \\Psi^A = \\bar{\\delta} \\phi^A + \\epsilon \\mathcal{L}_X \\phi^A", "b0acbab02bab49c633dec36279271c29": "\\psi(x)=\\sum_{n\\le x} \\Lambda(n)", "b0ad0695e1b5fac3f0e502291f58614f": "\n\\begin{align}\n\\frac{DF(P_0)}{DP} & = F[P_0,P_1]=\\frac{F(P_1)-F(P_0)}{P_1-P_0}=F'(P_0 < P < P_1)=\\sum_{TN=1}^{UT=\\infty}\\frac{F'(P_{(tn)})}{UT}, \\\\[8pt]\n& = \\frac{DF(LB)}{DB}=\\frac{\\Delta F(LB)}{\\Delta B}=\\frac{\\nabla F(UB)}{\\Delta B}, \\\\[8pt]\n& = F[LB,UB]=\\frac{F(UB)-F(LB)}{UB-LB}, \\\\[8pt]\n& =F'(LB < P < UB)=G(LB < P < UB).\n\\end{align}\n", "b0ad3c931b9335dbb027a1ed0d18ddee": "\\textstyle{\\frac {4}{3}}", "b0ada1a2dec8845f3ca70e7ebc391bb6": "\\nu_p(m+n)\\geq \\inf\\{ \\nu_p(m), \\nu_p(n)\\}.", "b0ae0b8077adf7a21dcf08119041323d": "\\begin{align} C(L) &=\\int_0^L\\cos s^2 \\, ds\\\\\n S(L) &= \\int_0^L\\sin s^2 \\, ds\\end{align}", "b0ae187e5574083fe4a8e1c205d93da9": "w_0(n) = \\frac{1}{N} \\sum_{k=0}^{N-1} W_0(k) \\cdot e^{i 2 \\pi k n / N},\\ -N/2 \\le n \\le N/2.", "b0ae3ac4ba5b723d54af7d2bae144325": "K(l, m, P, \\nu) = -\\frac{i e^{ikR}}{\\lambda R}", "b0ae42fab5b15bfde5ccd38520ecdf69": "X_1,X_2,\\ldots,X_k", "b0ae7a20dc117048aa125c0f5c8bb498": "=\\operatorname{st}\\left(\\frac{f(x + dx) - f(x)}{dx}\\right)", "b0ae90a0167de1a10e99defb9abadabd": "P = \\frac{q^2 \\gamma^6}{6 \\pi \\varepsilon_0 c} \n \\left( \\dot{\\beta}^2 - (\\vec{\\beta} \\times \\dot{\\vec{\\beta}})^2) \\right).", "b0aec1d7bafcea512a32b7509aa78fc7": "\\Gamma(z)=\\int_0^\\infty t^{z-1} e^{-t}\\, \\mathrm{d}t. \\!", "b0aec493ec2aec146c91fa82884528ba": "B=\\{b_1,\\ldots,b_n\\}", "b0aed7d3b4cc8bc3c484313b38152ebc": "a_i > 0", "b0aefc1063cb1e358df0311276a43997": "G(x,y,z;x',y',z')", "b0af0457f8ebc581207be94bafb1edc2": "C_L = 2(s/c)(tan\\beta_1 - tan \\beta_2)cos\\beta_m", "b0af76257fd334f5200fa9c2a421688a": "t=2", "b0af9c59b1432f6ebc540600df28c84e": " ...", "b0afbcb7feeff57e1515ec30aa2d7b14": "\\displaystyle{\\varphi_{t,t}(z)=z.}", "b0afc7215c38c3563c283e26aec803f3": "\\,\\!F_0 = 0", "b0afce312c57d7a8b3c0656c527c5202": "\\operatorname{Li}_2(-1)=-\\frac{{\\pi}^2}{12}", "b0b06568d4f17287c12fce9be2f5b56f": "\n\\begin{align}\n\\frac{x_1 + x_2 + \\cdots + x_{2^k}}{2^k} & {} =\\frac{\\frac{x_1 + x_2 + \\cdots + x_{2^{k-1}}}{2^{k-1}} + \\frac{x_{2^{k-1} + 1} + x_{2^{k-1} + 2} + \\cdots + x_{2^k}}{2^{k-1}}}{2} \\\\[7pt]\n& \\ge \\frac{\\sqrt[2^{k-1}]{x_1 x_2 \\cdots x_{2^{k-1}}} + \\sqrt[2^{k-1}]{x_{2^{k-1} + 1} x_{2^{k-1} + 2} \\cdots x_{2^k}}}{2} \\\\[7pt]\n& \\ge \\sqrt{\\sqrt[2^{k-1}]{x_1 x_2 \\cdots x_{2^{k-1}}} \\sqrt[2^{k-1}]{x_{2^{k-1} + 1} x_{2^{k-1} + 2} \\cdots x_{2^k}}} \\\\[7pt]\n& = \\sqrt[2^k]{x_1 x_2 \\cdots x_{2^k}}\n\\end{align}\n", "b0b0766b3bf37cb14022b043af0d9867": " \\mathbf{v} \\mathbf{v} = \\mathbf{v}\\cdot \\mathbf{v} ", "b0b07f74ebf4ec52c2cbb9d0cbb367c5": "\\mathbb{R}^d_+\\operatorname{-}\\min_{x \\in M} f(x)", "b0b08ba2e30d985ecf1f2008fb23b6a2": "\\varphi(r) = r^2 \\log r", "b0b0a01af5a5cd1c6cc1530b5dad854d": "I(1+J) + J(1+I) = 2IJ + I + J \\, ", "b0b105058e826a0724fcd3b3681af682": "c_{t+1} = c_t", "b0b12d9efcd9a407c2e76e97bd32d674": "\\mathbf{B} = \\begin{bmatrix}\na & b \\\\\nc & d \\\\\n\\end{bmatrix}, [a, b, c, d] \\in \\mathbb{R} ", "b0b1702c12c231b9951661e69e495a69": "\\left|x[n]\\right|", "b0b18496fccbc107941f90d42e4f02fc": "|K_C|^*", "b0b19028fa2a74c94730ffef5a0dc9bd": " \\zeta(z) = \\exp\\left({ \\sum_{m\\ge1} \\frac{z^m}{m} \\left|{\\mathrm{Fix}(f^m)}\\right|}\\right) ", "b0b2068fd3cdddf2606f6e13ccefc794": "J_{\\rm c} = A_{\\rm c} T_{\\rm e}^2 e^{-E_{\\rm barrier}/kT_{\\rm e}} ", "b0b221d0c616d0d728408692051fa3e4": "f: X_i \\rightarrow X_i^+", "b0b2ad6d392ce683a24b008f6c504d71": " (a[0], a[1], \\dots, a[n]) \\, ", "b0b2d9a9897fb514f3e2cc6f8b094f9d": "N(a+bi) = a^2 + b^2 \\text{ and }N(c+di) = c^2 + d^2, \\,", "b0b2dd85fe1e54bf05874ad039773e91": "\\R \\mathbb{P}^2 \\times S^1", "b0b2e0c647f3ef01f13696528bac5f29": " T_3(z)=26+(K-22)z +(K-22)z^2 +26 z^3", "b0b35dd991c33f87e257f69989531f7e": "a \\in S ", "b0b37f4a6a7b8248d2e03ff6877bd73e": "h_n{\\rightarrow}0.", "b0b3ca599c6c326fe0222508085ebd2f": "\\rho_{12}", "b0b3ec6536a94c87c132594e05e2757e": " 0< y < \\infty ", "b0b3fb087b45cb8020e2f9f067cefe24": "S_z|s,m\\rangle=m\\hbar|s,m\\rangle ", "b0b40c1df51b85f1221b7a7e9929c53a": "S(mag/arcsec^2)=M_{\\odot}+21.572-2.5\\log_{10} S (L_{\\odot}/pc^2),", "b0b44d52127dc10ec06d3f2b8841808e": "G^u H/H = (G/H)^u", "b0b4ae8d7123d56f8ce4cfdf7fa41172": "\\mathbf{l}_a=(x_a, y_a, z_a)", "b0b512acc55543f6e203e2b5795faca5": "\\rho.", "b0b64cb6125c98087e97f0e8ca4b5e34": "\\scriptstyle c_t", "b0b657d76e20ef959678874393e6f925": "\n \\boldsymbol{\\nabla}\\times\\boldsymbol{F} = \\boldsymbol{0}\n ", "b0b69b347fb86ac0368df955a2a74233": " X+Y \\sim \\textrm{Cauchy}(x_0+x_1,\\gamma_0+\\gamma_1)\\,", "b0b74a547d62e4409d891b270e8eaec0": "\\theta = \\sum_i p_i dq^i", "b0b750a9767559ef800b58323351a794": "(y^{q^{2}} - y_{\\bar{q}})^{2}=y^{2}(y^{q^{2}-1}-y_{\\bar{q}}/y)^{2}", "b0b78e4eca40a2cb991a38d3b6022a64": "(z - \\omega_N^k)", "b0b7903dc8b617bda1aa240f300fa8d6": "5\\pm0.1", "b0b813eb87b99e8d5ee0f3c5ea234b55": "C(u_1,u_2,\\dots,u_d)=\\mathbb{P}[U_1\\leq u_1,U_2\\leq u_2,\\dots,U_d\\leq u_d] .", "b0b81485e7fc31be7c5a38fac7e38be6": " P =\\int_K \\rho(k) \\, dk", "b0b81843f8dd3c8d9a0f2630d8345a41": "R(w^*,w).\\,\\!", "b0b890fc6654a8f3a7e65d8db0f69129": "0 0", "b0d0bd53fe2a96f20fef1deea5a21a20": "\\int_{\\mathbf{R}^n}\\hat{f}(x)g(x)\\,dx=\\int_{\\mathbf{R}^n}f(x)\\hat{g}(x)\\,dx.", "b0d133cd46f3a4e096e6b59e517132be": "\\beta^{2} \\left( 1 - \\beta^{2} \\right) \\left( 4 - \\beta^{2} \\right) = 0", "b0d149d0999db5d5fab5aa8dd531fda6": "O(\\ln ^2 n)", "b0d1854fd6ce1d9e698288168af72a3a": "\\mathbf{\\hat s}", "b0d1a4bc88aec39c9b1100adca0b9bfd": "\\! \\alpha", "b0d1d6f407f84acc917ce32af6edf78b": "2 \\times n", "b0d1d800bef357b4ca31aab7ebdc87cb": "\\begin{bmatrix}\n\\dot{x} \\\\\n\\dot{e} \\\\\n\\end{bmatrix} = \n\\begin{bmatrix}\nA - B K & L C \\\\\n0 & A - L C \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nx \\\\\ne \\\\\n\\end{bmatrix}", "b0d24d0672298c811f4502a66123cf49": "(\\phi_{u,v})(x,y)", "b0d2803d6314e2603065efaf29f279c0": " \\Delta V_{tc} = \\left[\\frac{\\eta \\cdot I_t}{\\rho}\\right]^{2/3}", "b0d29b039fd3a8c1566e974994f334b0": "f_i(\\vec{x}+\\vec{e}_i\\delta_t,t+\\delta_t) = f_i(\\vec{x},t) + \\frac{1}{\\tau_f} (f_i^{eq}-f_i)", "b0d2ede138d931ac4f33aaacb98b577d": "b(R)(x,y,z,w) = R(x,y,z,w) + R(y,z,x,w) + R(z,x,y,w).\\,", "b0d2ee14b6a8ef8b87f6b140391484fa": " ~\\epsilon_t = ~\\sigma_t z_t ", "b0d30f8c4127aab80c38f5ed5cf6ceee": "Z \\mathbf{y} = \\mathbf x", "b0d32aa4352e9337e3ebff3b76ea3ba7": "S_\\eta=\\frac{1}{\\beta}\\sum_{i\\omega} g(i\\omega)=\\frac{1}{2\\pi i\\beta}\\oint g(z) h_\\eta(z) dz", "b0d3416af47ddecba00dfcb2cf850439": "S \\subseteq S^{\\prime \\prime} = S^{\\prime \\prime \\prime \\prime} = S^{\\prime \\prime \\prime \\prime \\prime \\prime} = \\ldots = S^{2n} = \\ldots", "b0d45e1dbfb1b9552ec9e30d48369647": " E[X] = \\int_{-\\infty}^\\infty x f(x) \\, dx= \\sum_{i=1}^n p_i\\int_0^\\infty x\\lambda_i e^{-\\lambda_ix} \\, dx = \\sum_{i=1}^n \\frac{p_i}{\\lambda_i}", "b0d47a5d4685af1084c0b4a5ef6bd841": "1+X", "b0d494aa680a86edfea0642db2fdf4bb": "k = n^{\\frac{1}{2}+\\gamma}", "b0d49b865cddd6cbe6dc1a307b5b5f2a": "\\langle f_i | e_j\\rangle = \\delta_{ij} . ", "b0d503a05840435c001e719e40a08e7f": "V=\\frac{4}{3}\\,x_m\\,y_m\\,z_m", "b0d58eb92dc859a1af188a7f157277d3": "f=50 \\text{ mm}", "b0d5beeffba1dc3fb668921815a66cbb": "S(\\vec R) = k_B \\log (P(\\vec R)) + C_{st}", "b0d62106c4e5cde13ca78490c3bf4da2": "\\tfrac{4}{2}", "b0d63763dd0d590130deae735347416b": "\nSS_E \\equiv SS_T - SS_A - SS_B - SS_{AB}\n", "b0d64b0d1b713c19222734005f9105c6": "\\rho(X) = D(X) - \\mathbb{E}[X]", "b0d65ac79e37b2f37bc9c0b639afe599": " h_s^{(j)}(t) = \\frac{w_s^{(j)} p_s(x^{(t)}; \\mu_s^{(j)},\\Sigma_s^{(j)}) }{ \\sum_{i = 1}^n w_i^{(j)} p_i(x^{(t)}; \\mu_i^{(j)}, \\Sigma_i^{(j)})}. ", "b0d661faa8eb5de509339bef78286c3c": "S_{xx}(\\omega)=\\frac{(\\Delta t)^2}{T}\\left|\\sum_{n=1}^N x_n e^{-i\\omega n}\\right|^2", "b0d6893b5705762672b3eb637094e7b1": "\\displaystyle{D(X) = TX -XT}", "b0d6aa37903652c314cd25aa75cd69f8": "x_10~ ", "b0e035e2f02cbcd829a72a584e823c93": "\\varphi(\\lambda_1 x_1+\\lambda_2 x_2)\\leq \\lambda_1\\,\\varphi(x_1)+\\lambda_2\\,\\varphi(x_2)\\text{ for any }x_1,\\,x_2.", "b0e0666351e371beddbeb82f48417b4e": " \\frac{dA}{dz}=[A,B]. ", "b0e1326365729673e417839e572d1ced": "\\frac{8}{5}", "b0e13f2c3bc7369dc6be11acc6e4e0d1": "x=l\\|r", "b0e14caf7e6f0dd705bfcae76d664ccb": "E^2 = c^2\\mathbf{p}\\cdot\\mathbf{p} + (mc^2)^2\\,.", "b0e1572e1cd3093886eebd7ece9d1313": "S^2\\to M", "b0e1938d94db062e04ed33c5076f8fc1": "\\scriptstyle{\\vec{F}=m\\vec{a}}", "b0e1972c8bac46c6c9a2016d8d3db0f9": "\\alpha(xy)c=\\alpha(x)(\\alpha(y)c)", "b0e19c48c587537c4eaac6f0b85ea629": "\n\\frac{1}{\\sqrt{p}}\\sum_{n=0}^{p-1}\\exp\\left(\\frac{2\\pi in^2q}{p}\\right)=\n\\frac{e^{\\pi i/4}}{\\sqrt{2q}}\\sum_{n=0}^{2q-1}\\exp\\left(-\\frac{\\pi in^2p}{2q}\\right).\n", "b0e1a79e18a8a15a2d7e6b1c6eea881d": "\\epsilon_r = 1", "b0e1cd3497c281427e87021794e6b6c8": "D_{ij} = D_{ji} \\geq 0", "b0e1d1725a26e1c05d2b5563c9ff90a6": "t_i > t_{\\mathrm{min}}", "b0e239eac7e1bfa9138ba47ec12cf4db": "\\frac{a}{b} = q_0 + \\cfrac{1}{q_1 + \\cfrac{1}{q_2 + \\cfrac{1}{\\ddots + \\cfrac{1}{q_N}}}} = [ q_0; q_1, q_2, \\ldots , q_N ] ", "b0e25f4cc1f2ec664644367146873931": " \\cos \\theta\\!", "b0e26a322ea30f4234cda5df41664fe9": "A \\subseteq \\mathbb{N}^{k+1}", "b0e26c4430d7cdebfa3e16de6ed4e138": "\\neg P\\or(P\\and Q)", "b0e286520da5b7a2e0a0f16252e0064d": "N_J", "b0e326067ac39376910e919d0e2ac74b": "(a,\\sigma^2, \\Pi)", "b0e334b8ffa613c7b463bc17e61bb9a1": "\\begin{align}\\Pr(Y \\leq x) &= \\Pr(\\{|X| \\leq c\\text{ and }X \\leq x\\}\\text{ or }\\{|X|>c\\text{ and }-X \\leq x\\})\\\\\n&= \\Pr(|X| \\leq c\\text{ and }X \\leq x) + \\Pr(|X|>c\\text{ and }-X \\leq x)\\\\\n&= \\Pr(|X| \\leq c\\text{ and }X \\leq x) + \\Pr(|X|>c\\text{ and }X \\leq x) \\\\\n&= \\Pr(X \\leq x). \\end{align}\\,", "b0e3b809ba9c4e4d42b58bf74a394cb0": " q = x_1+\\bold{i}x_2+\\bold{j}x_3+\\bold{k}x_4.\\,\\!", "b0e3fe9ac49b3dfea8186ae5fd127f5a": "F(x_1, x_2, x_3)", "b0e3fea20b60b73a1b0f45b08c2fdb0c": "x = x", "b0e3feaf9d087ce01821d8033e3b2501": "\\mathop{c.h.}", "b0e411378656915b0f107274278d4bd7": "{\\rm 1~Pa = 1~\\frac{N}{m^2} = 1~\\frac{kg}{m \\cdot s^2}}", "b0e459861a9874560d25c047e67f9ab7": "\\; i(E_{jk}-E_{kj})", "b0e46b56e9bdc6f6af43b141b28d596d": "|s_N(x) - f(x)|\\ ", "b0e497198fa4bad3fb4d19a92eee5971": " \\omega = \\frac{-1 + \\sqrt{1+\\delta^2}}{\\delta \\, r}", "b0e4aba163070b44cb702d029ec957ae": "P_{\\leq \\beta} := \\{X \\in 2^\\omega : X\\ \\mathrm{has\\ effective\\ packing\\ dimension\\ } \\leq \\beta \\}", "b0e4e3e939361e0a06c636e13fc22a82": "\\mathbb{E}(\\log_2 (1+|h|^2 SNR))", "b0e4ef51b55ea0700e5fc82830bb7966": "J = \\frac f {\\sin \\theta}.", "b0e573cb06c155e7e718b2e2586d122b": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor", "b0e5985a06483cb10764f9c88cf621ab": "\n \\varphi_\\mathbf{x}(\\mathbf{u}) = \\exp\\Big( i\\mathbf{u}'\\boldsymbol\\mu - \\tfrac{1}{2} \\mathbf{u}'\\boldsymbol\\Sigma \\mathbf{u} \\Big).\n ", "b0e5be5cdad1dfe1c2b9018a38a25666": "\\dot{v}_3", "b0e5c2ad9c02d4a9b3153e5c2977d007": "t'_0 = \\max (t',t_0)", "b0e5cee7a8fd83a42e71d63dbc30fb49": "x_n=x(t_n)", "b0e5dba0d9dd7414d79bbc5f5b189eb3": "0 = (g^{\\mu \\nu} \\sqrt {-g})_{, \\nu} + g^{\\sigma \\nu} \\Gamma^{\\mu}_{\\sigma \\nu} \\sqrt {-g} + g^{\\mu \\sigma} \\Gamma^{\\nu}_{\\sigma \\nu} \\sqrt {-g} - g^{\\mu \\nu} \\Gamma^{\\sigma}_{\\sigma \\nu} \\sqrt {-g} \\,", "b0e5fd33c496329a1b9584db65bca124": "\\vec{b}=\\vec{B}/B", "b0e631a7a3f36d3395c5fb976cbc01b3": "\\boldsymbol{\\nabla} \\times (\\boldsymbol{\\nabla} \\times\\boldsymbol{\\epsilon}) = \\boldsymbol{0}", "b0e673a80ebdb03274573967922b5db9": "{{i}_{D1}}", "b0e67ea1dd7486aa420dc4ecd139407e": "N_{\\text{S}} = \\frac { V_c } {a * L_c}", "b0e69d4d08b39719c4df7964410712a1": "\\omega^2 = -U^2 \\left(\\frac{2k}{rd}\\right) \\theta^2", "b0e6b03253cd3701fe7b71fadfdcfa26": " \\text{ if } [ E( X )> k \\text{ and } E( X^2 ) \\ge kE( X ) + ME( X ) - kM ] \\text{ or } [ E( X ) \\le k \\text{ and } E( X^2 ) \\ge kE( X ) ]", "b0e71aa0aa333762a6fc21501ecdc385": " \\mu_a = {{\\mu_0} {{(1-0.5kH)}^{-2}}} ", "b0e74ad29fb8f740b6fe903e3d05d428": "-P = \\left(\\frac{\\partial F}{\\partial V}\\right)_{T}\\,", "b0e75a3b6dc3407f0f5ab88f91964323": "S(T,aV,aN)=a S(T,V,N).\\,", "b0e78f4cfb6e1a15f27275fbd9a0b0a6": "\\sum_i \\phi_i(x) = 1.\\,", "b0e7a386c3bc9554d58b4d928100ba01": "\\omega_0", "b0e7dccdaaed10455d0ec8f5a55edf9d": "{t_{ab}}^b", "b0e7dcdb380bc3d7d53bcf56e801db3e": "\\mathfrak{so}(32)", "b0e810ceaf566c84b25408c4a50e6f7d": "t\\begin{Bmatrix} p \\\\ 2 \\end{Bmatrix}", "b0e81b08b0beabf8f09e42380fdef6d5": "\\bold{F}", "b0e84870ae23e84ef5c1bf0fa9d56211": " \nM = \\begin{bmatrix} \n0 & 1 \\\\\n0 & 0 \n\\end{bmatrix}\n", "b0e898969d193da7b5ffa471ab916a12": "g(y)=\\int_{-\\infty}^\\infty f(x) e^{-2 \\pi i x y}\\,dx,", "b0e8995628d1a112ca922409de992510": "x\\in\\mathbb{R}^n", "b0e8b86cb1ffbe35e1efdddb565e9828": " x = \\int dx \\,", "b0e8bc18825717f3fd6233317c25f136": "{625 \\pi}", "b0e8cfac4e9ad9f286844ddd81f8b9c2": "R(z)=z^{-1}(I_n-z^{-1}A)^{-1}=z^{-1}\\sum_{k=0}^{\\infty}\\frac{1}{z^k}A^k\\cdot", "b0e90dd1fbeaa51070f6447c7772ff52": "\\mathbf{B} = \\mathbf{B}_0 \\sin(kx-\\omega t)\\,\\!", "b0e90df4adba620ae3e040aa3e9e7a44": "c(x) = \\tanh\\left(\\frac{x}{\\sqrt{2\\gamma}}\\right),", "b0e91f12c1c6d587f6a25aec3e67336e": "{v}' b - \\frac{1}{n}l", "b0edce0aa0c7a6c53d2df4a4443fc767": "\\eta_i(\\theta)=\\partial_i\\psi(\\theta)=\\sum F_i(v)p(v;\\theta)=E[F_i]", "b0ee8c037f92234d07c5374fe67cc0ec": "2v_h v_v/g", "b0eea9be63df29ee7491457d2ada9510": " \\vec{\\nabla}\\cdot", "b0eead36bdc4dcfe5d0d810d6de87f47": "S = \\{ z \\in D \\; | \\; f^{(k)}(z) = g^{(k)}(z) \\quad \\mbox{for all} \\; k \\geq 0\\}.", "b0eeedb282de29dc04bc37b58b15be1b": "p({\\rm label}|\\boldsymbol{x},\\boldsymbol\\theta) = \\frac{p({\\boldsymbol{x}|\\rm label}) p({\\rm label|\\boldsymbol\\theta})}{\\int_{L \\in \\text{all labels}} p(\\boldsymbol{x}|L) p(L|\\boldsymbol\\theta) \\operatorname{d}L}.", "b0ef37ab6a1cbfcaed956356fea24825": "a \\cdot W_0^a \\cdot \\text{E} [R_1^a \\cdot R_2^a \\cdots R_T^a].", "b0ef5909d2a60a6c7ea9fdcd98f578dd": "\nQ = \\frac{2\\pi f_o \\mathcal{E}}{P}\n", "b0efbf0dda7a34cd76a823bc0edb63a0": "(V, 2^V)", "b0f0565982aff48d319ef259b6fabb59": "M_{2^n} \\hookrightarrow M_{2^{n+1}}.", "b0f05f9ac53de35eb88b481262b1c8b9": "\\log_{\\sqrt{2}}3=\\frac{\\log_2 3}{\\log_2 \\sqrt{2}}=\\frac{\\log_2 3}{1/2} = 2\\log_2 3", "b0f0f7c3463ea729675a13565b834ec2": "X = \\bigoplus_{j \\in J} X_j.", "b0f0f90da81c2aa35a86685912bb66ff": "E(\\epsilon)=0", "b0f107df7cf8b7a3f8c44fb502256ee5": "B = {\\frac{Q_h}{Q_e}}", "b0f14ad30fd7b3cde54ba7cba7b0d9ed": "\n{\\rm E}\\left[ {{1 \\over 2}{{\\partial ^2 z} \\over {\\partial x_1 ^2 }}\\left( {x_1 - \\,\\,\\bar x_1 } \\right)^2 } \\right]\\,\\,\\, = \\,\\,\\,{1 \\over 2}\\,{{\\partial ^2 z} \\over {\\partial x_1 ^2 }}\\,{\\rm E}\\left[ {\\left( {x_1 - \\,\\,\\bar x_1 } \\right)^2 } \\right]\\,\\,\\, = \\,\\,\\,{1 \\over 2}\\,{{\\partial ^2 z} \\over {\\partial x_1 ^2 }}\\sigma _1^2", "b0f19c5714fe9f9891ed26ff783cf639": "\\varepsilon >0", "b0f1af0f252a0236ed24d6f4fed597a7": "b = \\frac{m^b(x)k_B}{m^b(x)(k_R+k_G+k_B)} = \\frac{k_B}{k_R+k_G+k_B}", "b0f1e6e5325b9f7ebc6b5096dd14b5ee": " \\langle (\\Delta N)^2 \\rangle = k_B T \\left(\\frac{d\\langle N\\rangle}{d\\mu}\\right)_{V,T} = \\langle N\\rangle (1 - \\langle N\\rangle) ", "b0f21feba15b271553b66baf0b4f03d6": "s \\in E^p", "b0f251c932ad905f812c60ffece1b1dc": "\n\\text{P} = \\frac{ TP + FP } { N }\n", "b0f2a73d9e68ac9591a9e2b108945946": "\\lim_{x\\to 0}\\varphi(x)", "b0f2a8cceaacf5c229a5b930ddd11de5": "f = p_i \\circ g", "b0f2fb4e42fe0a6f39e5f88dc8db314a": " \\sum_{x\\in {N}}f(x) ", "b0f374580528936fefb599854a63d26f": " (x,y) = (x_0,y_0)+(x_1-x_0,y_1-y_0)s+(x_2-x_0,y_2-y_0)t\\,", "b0f3b48644cd0b034ed2d0d80efe9f78": "\\phi(z)/|\\phi'(z)|", "b0f3ed60e0f117aeb7e652f174de079d": "\n\\left(\\frac{\\zeta_m}{a }\\right)_m =\n\\zeta_m^{\\frac{a^{m-1}-1}{m}}.\n", "b0f3efadab9319dd6f2c855bb6030f2b": "{\\rm WF}(f) = \\{ (x,\\xi)\\in \\mathbb{R}^n\\times\\mathbb{R}^n \\mid \\xi\\in\\Sigma_x(f)\\}", "b0f45016f7759ff0919b993051be422f": "Ax=\\lambda Bx", "b0f48abb14306386eafba7d159d2ba44": "\\zeta(-1) = -\\frac{1}{12}", "b0f4efef8fcfd8ad61a3fc9c72639a10": "f(\\alpha \\mathbf{x}) = \\alpha f(\\mathbf{x}) \\!", "b0f51255c168d1a8b8bed12cbd211e6b": "\\textstyle (n-l)q^{l-2} \\leq |B(\\mathbf{c})|", "b0f55fdb0cb061803d4b48ad6cd0cfa5": "\\Lambda_m=\\frac{\\kappa}{c}", "b0f59b2c132ac3350c95273172a7f91a": "h(t) = \\frac{1}{\\sqrt{2^j}} \\psi \\left( \\frac{-t}{2^j} \\right) ", "b0f5aaa20e02fbc0b863c6335d45cdf4": "\\hat{\\beta_{\\tau}}=\\underset{\\beta\\in R^{k}}{\\mbox{arg min}}\\sum_{i=1}^{n}(\\rho_{\\tau}(Y_{i}-X\\beta)) .", "b0f5d579adbd66734a4f36bca82afa8e": "\\sigma\\ge 0", "b0f6387fbe35f7786dfb103349e46386": "\\displaystyle a(t)", "b0f63886c28538180948dfe1168fa1e8": "\\sum_{i=0}^{k} p_{ii}^0 = |X|", "b0f677d2752ffd22cad9d23ad45e672e": "x^3 - x^2 - x - 1 = 0", "b0f6a163f0009bc3c1934715430f1117": " d{[}ES{]}/{dt} \\; \\overset{!} = \\;0 ", "b0f6fe81e00ed72c169a53640ed19362": " S_4 \\implies A_4 = p ", "b0f760815f18068f2d8353e1f1781cdd": " X(u,v) = r \\, (1 + \\cos v) \\, \\cos u, ", "b0f7cdc68850e12daeb7553a69b3ad97": "x\\in B_{r}", "b0f7f1c3d6ea96ffd5ff297793d7ffb0": "\nP(Q) \\, dQ = \\frac{e^{-Q/2}}{(2\\pi)^{k/2}}A\\,dR= \\frac{1}{2^{k/2}\\Gamma(k/2)}Q^{k/2-1}e^{-Q/2}\\,dQ\n", "b0f83f6ccf1948fb6e83175e669f19e9": "\\boldsymbol{\\mu}_s = -e\\mathbf{S}/m_e = g_s \\frac{\\mu_B}{\\hbar} \\mathbf{S}\\,\\!", "b0f85b65cd02c938b3140833623aa308": "\\mathbb{RP}^3", "b0f8c70c78ee0ca7ca552296137f8676": "\\Delta^1_n,", "b0f8cd041c0b3db16698b7675019f959": "E_D(X) = \\sum^k_{i=0} h_iD_i .", "b0f8db8264be52f6501f82f169685986": "B_{\\mu\\nu}-B^{(0)}_{\\mu\\nu}\\,", "b0f8ef5b05040014d2e15c6d446acb3f": "p(x, y) = p(y, x) \\propto \\exp\\left[-\\frac{1}{2(1-\\rho^2)}(x^2+y^2-2\\rho xy)\\right].", "b0f8fed8e67315098071267f1e657aee": "\\sigma^2(n)=\\sigma(\\sigma(n))=2n\\, ,", "b0f955b87baf7377b98b47ed9c723949": "x[n]\\,", "b0f96b97d80e0abc5546f3b1cb09bcc6": " (X,Y) \\mapsto h(X,(I- \\Delta S)Y) ", "b0f9e169814a058e3f268d80d28ad516": "p(\\mathbf{y},\\boldsymbol\\beta,\\sigma|\\mathbf{X})", "b0fa0aa63119ef1dd9a5554251d6afc0": "\\sum_{k=0}^\\infty ar^{2k} = \\frac{a}{1-r^2}", "b0fa1d3fa14cc6e5fcafef8d2c8db8a7": "1.6<{\\frac{L}{m}}", "b0fac8f5a806875f752147ea6d076124": " dG = -SdT + VdP + \\sum_{i=1}^n \\mu_i dN_i \\,", "b0fb8545d62c4f294bb8882b59fd0416": "\\chi_-^z = \\begin{bmatrix}\n 0\\\\\n 1\\\\\n \\end{bmatrix} \n", "b0fb8c302cd141ea8a1daf6f2fad52e4": "\\langle B\\rangle_{\\mathcal A}", "b0fbb8d1e24a78bd48ec0d224d25039f": " \\partial_\\mu\\partial_\\nu = \\partial_\\nu\\partial_\\mu ", "b0fc2ed75dd8444021d1ecee459213b9": "\\delta_{skin}", "b0fc3733b365ec8c3ffd1eba00a6f3c8": " U = W \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & d \\end{pmatrix} V^T ", "b0fc6bfba8c6fc56b5b21b217985bf78": "T_s N=Tk", "b0fc6ff34964072b6eec27e9ac640565": "4m+1", "b0fce080d13742b31089e75623ddb598": " \\mid \\psi_R \\mid^2 ", "b0fd3b3e82872188b11b0dfa0b8a3158": "\\langle m|\\partial_\\mu H|n\\rangle + E_m\\langle m|\\partial_\\mu n\\rangle=\\partial_\\mu E_n\\langle m|n\\rangle+E_n\\langle m|\\partial_\\mu n\\rangle.", "b0fdcaa638b469f3aaa5a4d84bc34019": "\\frac{\\mathrm{d}\\,T_1}{\\mathrm{d}\\,z}=k_a (T_1(z)-T_2(z))=-k_a\\,\\Delta T(z)", "b0fe139f2cfc7f266b0cbbaa25c4ea8c": "x \\ge 4", "b0fe13c015ea3e274c7d50aefe0aecf1": "E_{p-1}", "b0fe27776eac7bb5ceb37d40c994cdf8": "c=1+\\epsilon", "b0fe32a437b536d43779bc2a3a263879": "CV^{-1}\\,", "b0fe3b000d3d709261799b0ec6afeb64": "=\\pi \\rho(1-2/3+1/5)R^5= \\frac{2}{5}mR^2,\n", "b0fe7b31bbd5910bf1df10ebbbb65290": "\\mathrm{Sc} = \\frac{\\nu}{D}", "b0fe904737f4c9448c7df0a6f7751fbc": "F(\\psi)=RB_{\\phi}", "b0fe9835c14da0c730840a639330493f": " \\int_{t}^T \\gamma (s)H(t-s,s)ds =\\frac{1}{2 \\pi }\\int_{0}^\\infty K(0,iw) exp(iwt)dw+ \\frac{1}{2 \\pi } \\int_{t}^T \\gamma (s) \\int_{0}^\\infty K^2 exp(iwt) exp(-\\phi(s,iw)) ds \\quad(2.13)", "b0fea1650b2be7e894a441b5d13961e8": "\\mathrm {DOF} = \\frac\n{2 N c \\left ( m + 1 \\right )}\n{m^2 - \\left ( \\frac {N c} {f} \\right )^2} \\,.\n", "b0feb12682b2ab51ad2e575dcce4c32a": "a\\sqrt 2", "b0fed3a5af25604d68e181049d25f14a": "\\left(\\frac{a}{x}\\right)^n+\\left(\\frac{b}{y}\\right)^n=\\left(\\frac{c}{z}\\right)^n", "b0ff9637e5889942f1f1a7488fa6ffa9": "g = 1", "b0ffacf21ba5624fab781c58530dbed0": "\\mathcal{L}_{N} = e J_\\mu^{em} A^\\mu + \\frac g{\\cos\\theta_W}(J_\\mu^3-\\sin^2\\theta_WJ_\\mu^{em})Z^\\mu", "b0fff5de9fb5028e165f5757885f4b4e": "(\\sigma,\\omega)", "b10016870c0908c1526830f1694f270d": "P(Y \\in S)", "b100a996f565d63c867e77c31dd0fd34": " (u^2 \\mp 1)^2 - d(uv)^2 = 1 \\, ", "b100c5a3aeaa49cc4ab83f934af1b51b": "n_2 < n_1\\!", "b1011b266a080a0f5552d63d1ce8b3f2": "dx_2 \\ ", "b10139e65040adfefa195a2958f23a17": "32M=72n^3-108n^2+90n-27+{5\\over2n-1}", "b1019c205ada8bc54e2af7dfa6005c3f": "(F,+,\\cdot,<)", "b101b0d9d219f02b3201d6a0fd58d173": "dt, \\, dr, \\, d\\theta, d\\phi", "b101be3347c24febf120e72c40d142e8": "i^2t", "b10214fe1a3e764e23853466da9b7882": "(A,\\mu,\\eta)\\,", "b102296d38f0da96072c5a6351d3820e": "M_{0}=\\begin{bmatrix} 1 & 0 & 2 & 1 \\end{bmatrix}", "b1031fd5faed6c2f30969be7f8f17b80": " \\mathbf{a}_i = \\alpha\\times(\\mathbf{R}_i-\\mathbf{R}) + \\omega\\times\\omega\\times(\\mathbf{R}_i-\\mathbf{R}) + \\mathbf{a}.", "b103313889895630a9a8e8575659a04a": "f(x) \\;=\\; \\begin{cases} x^2\\sin (1/x) & \\text{if }x \\ne 0 \\\\ 0 & \\text{if }x=0\\end{cases}", "b103530936a3b627585a440f8201764b": "I_s", "b103594657ee76b9b94b3b549bc7d2e5": "\\mathbf u", "b10382cd8363b852d50677844e9a67a7": "\\mathrm{d}\\theta", "b1039685676961c3c962d3ef2df475ed": "\\rho_{XB} = \\sigma_X \\otimes \\tau_B", "b103a4d942ef51e2f38189841c5cddb1": " g(\\overline{D}) \\subset D.", "b103d43db689f3950d83d16293e3dc02": " F_G = F_T \\;", "b103de7979b4b024e7409a0cbfb3599d": "S^1 \\times S^1 = \\{ ( e^{i\\theta}, e^{i\\phi} ) \\, | \\, 0 \\leq \\theta < 2\\pi, 0 \\leq \\phi < 2\\pi \\}.", "b103ea433e0f7581cd1f861c8c9f12be": " \\mathcal{H}^{12} ", "b1043c81f01412b466d3d82e4b458bfd": " \\le \\int (|f| + |g|)|f + g|^{p-1} \\, \\mathrm{d}\\mu", "b10461c36355ef500a1015e1f817e490": "\n\\begin{align}\n\\mu'_6 & =\n\\kappa_6+6\\kappa_5\\kappa_1+15\\kappa_4\\kappa_2+15\\kappa_4\\kappa_1^2\n+10\\kappa_3^2+60\\kappa_3\\kappa_2\\kappa_1 \\\\[6pt]\n& {}\\quad + 20\\kappa_3\\kappa_1^3+15\\kappa_2^3\n+45\\kappa_2^2\\kappa_1^2+15\\kappa_2\\kappa_1^4+\\kappa_1^6\n\\end{align}", "b10481adc941987ad06fe59dd6474fec": " \nINT\\_MIN \\le x + y \\le INT\\_MAX \n", "b104d8a3f67a98a44243527ccc5ccb82": "\\mathbf{f} = \\frac{1}{\\sigma} \\mathbf{J}", "b104e6d9634887dd6e5216773ec731c4": "\\nabla \\times \\mathbf{F} =0\\boldsymbol{\\hat{x}}+0\\boldsymbol{\\hat{y}}+ \\left[{\\frac{\\partial}{\\partial x}}(-x) -{\\frac{\\partial}{\\partial y}} y\\right]\\boldsymbol{\\hat{z}}=-2\\boldsymbol{\\hat{z}}\n", "b104f213c5dd706b3cb9c3d95143e561": "v_1,v_2, \\ldots, v_r", "b104fe97276a090ca0fd05947558a49a": "\\lambda_0 \\geq \\lambda_1 \\geq \\ldots \\geq \\lambda_{n-1}", "b1058b7fac1582bcabb79b62cff0a92f": "V_{BE} = V_{BE}(I_B)", "b1060d05722b9eac8df539cdd63d8b9d": "\\begin{align} \\gamma = \\lim_{n \\to \\infty} \\frac{1}{n}\\, \\sum_{k=1}^n \\left ( \\left \\lceil \\frac{n}{k} \\right \\rceil - \\frac{n}{k} \\right ).\\end{align}", "b1064ad00b0e9f65262afcbca6d12cac": "\\scriptstyle{R_0^0}", "b1067f1f690ce02f6bf1c0d3b0bf090f": "(0) = 0", "b106acef78b29da1b6dbb89851b8c2e3": "O(1/n^\\alpha)", "b1073e39978e9d85c8fce44219d20086": "p_{2H}", "b1077787fa6476e9d12cb56d84e40ed0": "49s_2^2", "b1079b9c010fde2726ec6200c278c711": "k = \\frac Nn", "b107aeca1a6b6bc3ba713f678443dc2f": "\nx_{1,2} = -\\frac{b}{4a} - S \\pm \\frac12\\sqrt{-4S^2 - 2p + \\frac{q}{S}}\n", "b107c9c617b0c253066ebe62dcdfd9e7": " \\|x\\| = \\|(x-y) + y\\| \\leq \\|x-y\\| + \\|y\\| \\Rightarrow \\|x\\| - \\|y\\| \\leq \\|x-y\\|, ", "b107d35b03fe3c14f9950f69c1425fe8": " \\begin{Vmatrix} R^1_1 & R^2_1 & R^3_1\\\\ R^1_2 & R^2_2 & R^3_2\\\\ R^1_3 & R^2_3 & R^3_3 \\end{Vmatrix} = \\begin{Vmatrix} R_\\perp & 0 & 0\\\\ 0 & R_\\perp & 0\\\\ 0 & 0 & R_\\| \\end{Vmatrix} ", "b10819aa0ee1c21e28e060eaba29ddbb": "\\delta^2 E\\ge 0\\,", "b10848b6683d3839d06c042374f47e8f": " \\mu(g^{-1} A) = \\mu(A) \\quad ", "b10875bc434a705765885f5f0886c065": "\\mathcal{Z}=(\\mathbb{Z},<)", "b1088c41c2f9148b66ae1402671cb9d6": "L^M", "b108c2b766e8430c6171c227a84cb56b": "p-p_\\infty = - \\frac{3}{2} \\rho U^2.", "b109145162caa9783dde1aaab9e833f8": " H = \\sum_{g=1}^{G} \\frac{(O_g - E_g)^2}{N_g \\pi_g (1-\\pi_g)} .\\,\\!", "b10914a26219d2bb9b98bb4e71aea776": "(\\Sigma _{11} )^{ - 1} \\Sigma _{12} ", "b1099658f007b968720ccae0040857e3": "\n\\begin{pmatrix}\n X_1 \\\\\n X_2\n\\end{pmatrix} \\sim \\mathcal{N} \\left( \\begin{pmatrix}\n \\mu_1 \\\\\n \\mu_2\n\\end{pmatrix} , \\begin{pmatrix}\n \\sigma^2_1 & \\sigma_{12} \\\\\n \\sigma_{12} & \\sigma^2_2\n\\end{pmatrix} \\right)\n", "b109d99798651e71ada9772280c0b166": "{1,2,...,m}\\,\\!", "b10a4d445480e94e92f16acc934f3474": "(A_\\bullet, d_{A,\\bullet})", "b10a6543100ecca098e7e3130b907475": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 1}{10 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "b10a84a37d3ba937bb679b7552f77961": "\\ \\sin(a+b)=\\sin(a) \\cos(b) + \\sin(b) \\cos(a)", "b10aae38fb52f69ed306dc09a1d39464": "\\alpha \\left ( x - \\beta \\right )^2 ", "b10ac5ddd8e73a46e6025883f190f593": "{EL}_{oil}", "b10b01dfc1a84f952a5365976b8d097a": " X^\\alpha = \\prod_{i=1}^n X_i^{\\alpha_i} =\nX_1^{\\alpha_1}\\ldots X_n^{\\alpha_n}, \\quad\np_\\alpha = p_{\\alpha_1\\ldots\\alpha_n}\\in\\mathbb{K}.\\ ", "b10b36f6c916a969f912883de5e5247a": "n \\Vdash \\phi", "b10b5bd1ef3ff5b0352b3fb3f38daa27": "I+\\varepsilon A", "b10b63dac9e13dd52845ef1a55bea6d7": " A \\subseteq \\kappa", "b10b82d2c4bf554894dd2937664d1398": "\\{i:i+1", "b10c267aef806f1863555266a8d1e695": "v < {\\sqrt{r\\mu g}}.", "b10c2b771ce21ca2f80a5160a89a4408": "R_{aft} = R_{bef} + K (S_{act} - S_{exp})", "b10c34d76368e8b9912dec0e25763231": "u \\text{ and } y", "b10c5a4391d008a320120dc962ea08fa": "n_s=n\\sqrt{P}/H^{5/4}", "b10c6cabcb6feb3b12bd85e98c95a456": " k^2 = \\mathbf k \\cdot \\mathbf k .", "b10cae44982aa946131e4ccabfe7443f": "y_2 = \\frac{2y_1}{-1+\\sqrt{1+\\frac{8gy_1^3}{q^2}}}= \\frac{2(5.0ft)}{-1+\\sqrt{1+\\frac{8(32.2\\frac{ft}{s^2})(5.0ft)^3}{\\left(10\\frac{ft^2}{s}\\right)^2}}}=0.59ft", "b10d23add155e9b4b303f93f026c8740": "(\\nabla \\cdot \\mathbf v) f \\ne (\\mathbf v \\cdot \\nabla) f", "b10d847c083dd9e227702542e84c11bf": "\\begin{matrix} (\\frac{1}{2})^2 = \\frac{1}{4} \\end{matrix}", "b10df83df07fc30201182970b565906d": "\\frac{\\mathrm{d} \\sin\\theta}{\\mathrm{d}s} = \\cos \\theta \\frac {\\mathrm{d}\\theta}{\\mathrm{d}s} = \\frac{1}{\\rho} \\cos \\theta \\ = \\frac{1}{\\rho} x'(s)\\ . ", "b10e76532570f1c2a3a190476521a79d": "x = w^2 \\cdot r_0/g_0.", "b10e97c32e5addd1a76c3fa77face990": "\\mu = 398600.440\\text{ km}^3/s^2 \\, ", "b10e9c50b6aa38ad5aeb37517e91d003": "\\frac{p_t}{p} = \\left(1 + \\frac{\\gamma -1}{2} M^2\\right)^{\\frac{\\gamma}{\\gamma-1}}\\, ", "b10eb74ee6acee236aff04e628e8dd49": "c_0=\\frac{\\rho\\,b_0}{1+ \\sum_{j=1}^{n}{b_j M_j}}=\\frac{\\rho-\\sum_{j=1}^{n}{c_i M_i}}{M_0}.", "b10eb788aa7493fb6f71d84abe675ad5": "X_i= {\\rm grad} \\frac {\\partial s(n)}{\\partial n_i}\\ ,", "b10ef86a9df086613794000c6d1eeff0": "e^{-(\\frac{x-m}{s})^{-\\alpha}}", "b10efaa647c7de32900edfed61dacbe2": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\cos\\theta = \\lim_{\\delta \\to 0} \\left( \\frac{\\cos(\\theta+\\delta)-\\cos\\theta}{\\delta} \\right) . ", "b10f16c4540352f1b9a0b1f5c442f6c6": "J_n=J_n(z), \\,", "b10f513824eb543bc124098f72c0eac4": "M_E = \\int_\\Sigma d^3x {H (x)^2 - q^{ab} V_a (x) V_b (x) \\over \\sqrt{det (q)}}", "b10fbcd57b7ea5f1fc7a7deb31a8d4ad": "\\frac{1}{\\sqrt{5}\\, q^2}", "b10fc51a57589581ad5ad4d1ff470933": "\\begin{align}\n0 &= \\int_S \\varepsilon_{ijk}x_j\\sigma_{mk}n_m\\, dS + \\int_V\\varepsilon_{ijk}x_jF_k\\, dV \\\\\n&= \\int_V (\\varepsilon_{ijk}x_j\\sigma_{mk})_{,m} dV + \\int_V\\varepsilon_{ijk}x_jF_k\\, dV \\\\\n&= \\int_V (\\varepsilon_{ijk}x_{j,m}\\sigma_{mk}+\\varepsilon_{ijk}x_j\\sigma_{mk,m}) dV + \\int_V\\varepsilon_{ijk}x_jF_k\\, dV \\\\\n&= \\int_V (\\varepsilon_{ijk}x_{j,m}\\sigma_{mk}) dV+ \\int_V \\varepsilon_{ijk}x_j(\\sigma_{mk,m}+F_k)dV \\\\\n\\end{align}\n\\,\\!", "b10fe62b94a43f62eefdb8b856aa169a": "s \\ne 0", "b10ff078f8fc2235af7f6ed9da680596": "Lk = Wr + Tw", "b110016fb948d64894a17091792aa223": " \\sum_{i=1\\atop (i,n)=1}^{n-1} \\frac{1}{i} \\equiv 0\\pmod{n^2}.", "b11037a7c92d84bc62f27efd18b7705b": "-\\sqrt{2}", "b1105fdf3c3524189e3ba095f3d1f60d": "\\sin(78\\tfrac34 ^\\circ) = \\frac12\\sqrt{2+\\sqrt{2+\\sqrt{2}}};", "b1108e1f80940a655f711d0c6726ec6e": "\nP=\\frac{\\rho k_{B}T}{\\mu m_{H}},\n", "b1110e042878e13020c4cb60f289138d": "F(a_1, a_2,\\ldots, a_n) \\ne 1", "b11121131b3fb9cc51a07f43518ecde5": "\n\\|D-\\widehat D^*\\|_{\\text{F}} = \\min_{\\operatorname{rank}(\\widehat D) \\leq r} \\|D-\\widehat D\\|_{\\text{F}} = \\sqrt{\\sigma^2_{r+1} + \\cdots + \\sigma^2_m}.\n", "b11143d2bae63168fe8bf6e8aec5d104": "\\displaystyle{f(e^{i\\theta})=e^{ig(\\theta)},}", "b1114bf21f95f3fb19a2fa64d6e4fea6": "\\alpha_1(\\vec z),\\dots,\\alpha_n(\\vec z)", "b1117971e2d16bfbc514f610758170aa": "\n\\begin{align}\nR^\\dagger_{pq} & = R^{-1}_{pq} \\\\[6pt]\n\\Rightarrow\\ R^{\\dagger^\\dagger}_{pq} & = R^{-1^\\dagger}_{pq} = R^{-1^{-1}}_{pq} = R_{pq}.\n\\end{align}\n", "b111d39fd95625e851427ae01f1e1bfb": "u^T a(x) u > \\alpha u^T u \\;\\;\\; \\forall u \\in \\mathbb{R}^n", "b1122ef2eab9d999603a695f0c222114": "{}_1 Y_{1\\pm 1}(\\theta,\\phi) = -\\sqrt{\\frac{3}{16\\pi}}(1 \\mp \\cos\\theta)\\,e^{\\pm i\\phi}", "b112304e26c16cb02e857599f0ba7bc2": "f(x)=\\|x\\|_p", "b11234e39ca98ec36f5677d901b059e2": "x^3, x^5, ... , x^{2^k-1}", "b1129819ab10dec9902c160d66d8d9d3": "\\sigma_{j}^{+} =( \\sigma_{j}^{x}+i\\sigma_{j}^{y})/2 = f_{j}^{\\dagger}", "b112cb6b08c291dd1b9aff0be4d205f1": " R^2_4(\\rho) = 4\\rho^4 - 3\\rho^2 \\,", "b112de5d0f367bb1274d7197647d7a34": " R^{ab} = \\left( R/2 - \\Lambda \\right) \\, g^{ab}", "b1130034ceb2d895989e5318cec819c5": "\\boldsymbol{J}_2 = diag[J_{2xx},J_{2yy},J_{2zz}] = diag[0,J_{2},J_{2}]", "b113088acd6277093167d9d2d204fef0": "\\overline{X}_n={X_1+\\cdots+X_n \\over n}", "b1139a7e700b8817e6b3994099e0816e": "du(X_s,s)", "b113df9d7a8ce5298d5b439635fec270": "\\hat{H} = \\frac{1}{2m} \\left ( -i\\hbar\\nabla - q\\bold{A} \\right)^2 + q\\phi ", "b1144199298c05fe7c6acdd53900d47d": "\\alpha(d) > c^d", "b114420560cb3ebedb7211b6cc865fd3": "\\frac{dC}{dx} = 0 \\text{ at } x= L", "b114c7d0c0c6ba653801b799dd18354c": "a^p\n = \\sum_{k_1,k_2,\\ldots,k_a} {p \\choose k_1, k_2, \\ldots, k_a} ", "b114cacee591f0deb7038dd2c5d3f4d9": "\\mbox{BI2} = \\sum_{k=1}^m \\frac{n_k + \\sqrt{n_k}}{2}", "b114da6017704896bf94d739ad1a4200": "P={D \\over {k_e}} + {(r/k_e) (E-D) \\over {k_e}} \\,", "b11508129408fb97c84dabd0932bfe4f": "\\textstyle{R=log_2 (a / b) }", "b11535fc3773205f19891280b9443e91": "\\ P_0", "b1153e1eabd872eba57fe50893d48d57": " E = \\gamma mc^2 = \\hbar \\omega = \\gamma \\hbar \\omega_0 ", "b1156e1c096fcbe3902a52c5bb935dd0": "l=(1-\\zeta_l)(1-\\zeta_l^2)\\dots(1-\\zeta_l^{l-1}).", "b115d6e6b4315f8d86f66c0d7cdb2493": "\\Pi = \\frac {n R T i}{V} = c R T i", "b115dd0eaf0cb2b58ab69cf16ec6c245": "A = \\pi r^{2}", "b1161f6fd674bed84c273f4792a18b8f": "d > r_1 + r_2", "b116a0f3db53cd612ac3a0844ef4a95e": "\\nu = \\frac{\\mu}{\\rho} = \\frac{(3 \\sim 4) \\cdot 10^{-3}}{1.06\\cdot 10^{3}} = (2.8 \\sim 3.8) \\cdot 10^{-6} \\, \\frac{m^2}{s}", "b117860c5e3976483d0abc15fd5731ba": "(x + p)^2 \\,=\\, x^2 + 2px + p^2.\\,\\!", "b118049106534c1ea5023db1d4098990": "u(T) =\\frac{8\\pi (k_\\mathrm{B}T)^{4}}{(hc)^{3}} J,", "b11851c54a76356fd0473144e7369dad": "S_{\\text{RST}} = -\\frac{\\kappa}{\\pi} \\int dx^+\\,dx^- \\left[ \\partial_+ \\rho \\partial_- \\rho + \\phi \\partial_+ \\partial_- \\rho \\right]", "b118b14c5bc54bba023d0807847187c5": "\\mathrm{Nu} = \\frac{h L}{k}", "b118b34f00aa8b99f551678224108d47": "X^*\\,", "b118e5102d931f9add1ace88a1e6bf4b": "{\\rm conv}(\\{(1,1),(2,0),(0,5),(0,0)\\}).\\ ", "b11947d0762e32879113b93e65358bf2": " x^3+(D+G)x^2+\\left(DG +\\frac{D^2}{3}\\right)x=P-\\frac{D^2G}{3}", "b11964267b358a00c282799c4cc0d419": "\\pi_{ii} = 1", "b11995826184c9fd576412690627368e": "A_0:=(A^* A+ \\delta I)^{-1} A^*", "b119ad98b3fe36f8d25938c900f715b9": "\\scriptstyle\\ p > \\alpha ", "b119bd686fa4386a8de05969f8761eb6": "f_t\\colon S^2\\to \\R^3", "b11a635d6db634adb7a46dfbc0faa57d": "\\text{var}[Z_i^{(m)}]=m^2 \\text{var}[Y^{(m)}]=(\\hat{\\sigma}^2 /\\hat{\\mu}^{2-d})\\text{E}[Z_i^{(m)}]^{2-d}", "b11a7ba13b817dc67902c3faaceaba84": "S=\\frac{1}{2\\kappa^2}\\int d^Dx\\sqrt{-g}\\left[-\\frac{2(D-26)}{3\\alpha'}e^{\\frac{4\\tilde\\Phi}{D-2}}+\\tilde R-\\frac{4}{D-2}\\partial_\\mu\\tilde\\Phi\\partial^\\mu\\tilde\\Phi+O(\\alpha')\\right],", "b11aaca003db13d6a28942c088662228": "c\\!", "b11b95bf2939a330ce7e0b719bb9286f": "\\varprojlim G/G_n", "b11c09acdac4cb58f8d3640ca55db0d7": "\\Delta{E} = E_2-E_1=h\\nu \\ ,", "b11c34751cf11439cacc59670b1f0471": "(r,r+\\sqrt{3}r/3,r+\\sqrt{6}r2/3),\\ (3r,r+\\sqrt{3}r/3,r+\\sqrt{6}r2/3),\\ (5r,r+\\sqrt{3}r/3,r+\\sqrt{6}r2/3),\\ (7r,r+\\sqrt{3}r/3,r+\\sqrt{6}r2/3),\\dots. ", "b11c6e1bc5531c3ec5dd91efd290969c": "s \\equiv \\mathrm{false}", "b11c94227a9893460a57da0f215b9a91": "\\mathcal{G}(\\Omega)=\\mbox{Volume}(\\Omega)-\\mbox{const.}", "b11cb7f6b013fc6f6033b4529d708f94": "(x,\\alpha) \\sim (y,\\beta) \\Leftrightarrow x = y", "b11d1ef61711356579a14faa6a0b7afd": "\\phi_{\\Gamma}\\left(\\omega\\right)", "b11d23ff2c1bed912935f31995b25360": "f^e_{\\mathbf{k}}", "b11d40946feb1563a7226983b191bd30": "\\mathfrak{H}_l", "b11d9e9960ba4486bc9a7c30b685c67f": "Q\\mbox{rej}", "b11dc7873d25937fecc529e5508103be": "\\eta(6) = {{31\\pi^6} \\over 30240} \\approx 0.98555109", "b11e74bb67307c1024177df41c940fcb": "\\frac{{\\rm d}\\mathbf{p}}{{\\rm d}t} = q\\left(\\mathbf{E} + \\frac{\\mathbf{p} \\times \\mathbf{B}}{m}\\right) \\,\\! ", "b11ee64ef3660acebc3eb7089e69e808": "|\\psi_{1n}\\rangle := |n,e\\rangle", "b11f091028002066f34babf59d804f19": "\\{\\neg,\\lor\\}", "b11f21930059fad55269a506608293ec": " I(x) ", "b11f746f6fe208397a79f54b312215e1": "\\int_u^v t^{s-1}\\,e^{-t}\\,{\\rm d}t = \\gamma(s,v) - \\gamma(s,u)", "b11f827f514cb0955967cae989b85586": "E(v,h) = -\\sum_i a_i v_i - \\sum_j b_j h_j -\\sum_i \\sum_j h_j w_{i,j} v_i", "b11f82e24c770eed4fa7bf95d197ad17": "V(t)=\\text{Re}(V_0 e^{j\\omega t}), \\quad I(t)=\\text{Re}(I_0 e^{j\\omega t}), \\quad Z=\\frac{V_0}{I_0}, \\quad Y=\\frac{I_0}{V_0}", "b11f847f6ff11e32c1c1205517b4152a": "(A*A)x_0=A*y", "b11f96ff7267aeaf786722c192cb657e": " k=\\frac{EA}{L} ", "b11fb689c2bac211cf4b53a8e1a2d24c": "C_p - C_V = -T \\left(\\frac{\\partial p}{\\partial V}\\right)_{T} \\left(\\frac{\\partial V}{\\partial T}\\right)_{p}^2 ", "b11fc66f351c39c3a0b0ec4d444eaeb9": " S(a) = \\sum_{x_i-y_j \\in a} \\sigma(x_i,y_j) + \\text{gap cost}", "b120497571a1899a292ea1b1117d7607": "\\vec{W}=\\vec{U}+\\left( -\\vec{\\omega }\\times\\vec{R} \\right)", "b120822dcac3f98882e11888d7db2ae6": "\\omega(n)", "b120c443afe60bde62d51e451c120db9": " \\tan \\delta = DF = \\frac {1} {Q} ", "b1211e0856de45355909dd99b7253cb6": "\\scriptstyle\\phi x", "b121a2814d60010ef1123579b3734056": "F_X(x) = F_Y(x) \\, ", "b122ead0ff85f3776528926ef773f213": "\\gamma_1, \\gamma_2", "b1231e378ca215f1e89b4afba1cb2c2a": "\\left[f\\right]_{\\lambda,p}^p = \\sup_{0 < r< \\operatorname{diam} (\\Omega), x_0 \\in \\Omega} \\frac{1}{r^\\lambda} \\int_{B_r(x_0) \\cap \\Omega} | u(y) - u_{r,x_0} |^p dy ", "b1232ac0f982e4220c20c6b90b8c899f": "\\Gamma = \\{ 0, 1 \\}", "b1235edf019dc37e4934afe242fa613d": "\\frac{1\\cdot1/3}{1/2}=2/3.", "b12363d26ae358d6609812e891828c3e": "A_1, \\ldots A_n", "b12398ab8fc7bc0007d74659c919cadc": "F=\\left[\\frac {B_0^2 A^2 \\left( L^2+R^2 \\right)} {\\pi\\mu_0L^2}\\right] \\left[{\\frac 1 {x^2}} + {\\frac 1 {(x+2L)^2}} - {\\frac 2 {(x+L)^2}} \\right]", "b123c11e6d25a25ae5268c3b4107273b": " [a_j,a_k^*] = \\delta_{j,k} ", "b124995aa595d6f733256b1622dc083c": "\\beta=\\frac{2\\pi}{\\lambda}", "b1249dd08ec1585bf8eccd2123f3b813": "A - \\lambda B", "b124c4703da151b2792f6c677b69d342": "J^\\mu(x)\\equiv\\frac{\\delta}{\\delta A_\\mu(x)}S_\\mathrm{matter}", "b124c7528cf7aa76ecd3fa2628fa7bc9": "\n\\begin{align}\ng(v) & = \\sum_{n=0}^\\infty \\left(\\frac{\\partial}{\\partial x}\\right)^n \\left[ \\frac{(y f(x))^n}{n!} g(x) (1-y f'(x))\\right] \\\\[10pt]\n& =\\sum_{n=0}^\\infty \\left(\\frac{\\partial}{\\partial x}\\right)^n \\left[ \n \\frac{y^n f(x)^n g(x)}{n!} - \\frac{y^{n+1}}{(n+1)!}\\left\\{ (g(x) f(x)^{n+1})' - g'(x) f(x)^{n+1}\\right\\} \\right]\n\\end{align}\n", "b124ceb21167638740f3e805c5302cf7": "A^{(n)}", "b1250fdae7bab2d70d00986328690f8d": "\\scriptstyle\\mu_x", "b125159f013262e1b5fcbcd7e02e6085": "\\theta = (m_0,\\mu,\\bar{\\sigma},b,\\gamma_1)", "b125907d7e613932cbdc452a784c2fcf": "\\frac{31}{69}=.4492753623188405797101\\,4492753623\\ldots", "b125a5ecbaa37b26e79651d1017a63fd": "\\frac{1}{C_\\mathrm{total}} = \\frac{1}{C_1} + \\frac{1}{C_2} + \\cdots + \\frac{1}{C_n}", "b125e2f2c61878e4c60e8c028089bf17": "s = a \\tan \\varphi\\,", "b126676c87d3ec1c3820c0e9ca4ee5fa": "m_{ii}=1", "b1266aaa09e07c262e8e5e52550c5285": "U_d", "b1267e02eb75e5d645063bb06323b499": "\\Delta_3 Ef = E(\\Delta_2 - R_2)f. \\, ", "b126b6b3f742c5dede9e2f69e1c79059": "f(x_0).", "b126d201e71b5bb897919fda8152fc3a": "(1/n!)\\le f(n)", "b1270f1434d3ef15fc50ccb4575e7364": "\\displaystyle \\|x + y\\| \\leq \\|x\\| + \\|y\\| \\quad \\forall \\, x, y \\in V", "b1273bc760e93100dd330ee55a13173b": "x : [n] \\rightarrow \\{-1,+1\\}", "b1277f19b8710735cc46a2f6857cbd41": "\\bigl[ \\begin{smallmatrix} 1 \\\\ -1 \\end{smallmatrix} \\bigr]", "b1277f8575d35f67dabf7019e79fcd06": "\\mathbf{p} \\cdot d \\mathbf{s}=nds", "b1279e6bda234b74110b8f5f4e3fcc14": "\\eta(x,t) = H\\, \\operatorname{sech}^2 \\left( \\frac{x - c\\, t}{\\Delta} \\right).", "b127bc6e2af2c66432516913828003fe": "C^\\infty", "b127e11e723cd038ba3a6bbcc16048bd": "f^{-1}(x)", "b128ea924e9725e672fb02ba383c246d": "F_n = \\frac{\\varphi^n-(-\\varphi)^{-n}}{\\sqrt 5}.", "b128eade957fa281154b62a27f62fef5": "\\eta(\\mathbf{q} | \\Gamma)", "b12973224bb4e64f40c5f1ed06440baa": "\\ll 1 ", "b1298c1df62184e6f11b687200e0cdce": "(a_1,a_2,\\dots, a_m)", "b129a55e4eb24c45aa99010845fc03f5": "\nE^{(1)}_\\mathrm{electrostatic} = \\int\\int \\rho^A_\\mathrm{tot}(\\mathbf{r}_1)\\frac{1}{r_{12}} \\rho^B_\\mathrm{tot}(\\mathbf{r}_2) d\\mathbf{r}_1 d\\mathbf{r}_2,\n", "b12a03de674f19bf9237a2486df50955": "= \\bar \\nu _{v'-v''} + (B '+B'')J' + (B'-B''){J'}^2 ", "b12a0772c82f9e498936b9b7040298dd": "\\boldsymbol{\\sigma}_\\mathbf{r}(s)", "b12a1b5cb9ae96a7ee51f5abf812bf66": "\\{S,S'\\}", "b12a3f6f033f80051f96c7c005b34f17": "\\mathbf{e}_{k}", "b12a455a4571568c8ef0e3a92214a13b": "c\\log n", "b12a4c6387dbc58c0641f9ca0f946846": "\\frac {\\partial }{\\partial \\mathbf{q}}=\\left(\\frac{\\partial }{\\partial q_1},\\frac{\\partial }{\\partial q_2},\\cdots \\frac{\\partial }{\\partial q_N}\\right)", "b12a80f7fee3782080387b607348e189": "i \\in \\bar{K}", "b12a846b29db1753d98839ef41ad01f4": " = n_0(v_0 | \\sum_{j=1}^{l}{v_j} )\n\\cdot \\prod_{j=1}^{l} n(v_j | e_j)v_j! \n\\cdot \\prod_{j=1}^{m} t(f_j | e_{a_j}) \n\\cdot \\prod_{j:a_j\\not =0}^{m} d(j | a_j,l,m). \\,", "b12b1e3665ded56ea8ef6c2d1e082fac": "\\mathbf{n} - \\boldsymbol{\\beta}", "b12b4369c9e5d022be74182b7e6cbab2": "\\vartheta = z\\frac{{\\rm{d}}}{{\\rm{d}}z}.", "b12b5cb9d63e2c5c68a8cbc013a39953": " \\ell \\left( \\left[ J_F \\right] \\right) > 0 . ", "b12bb0a35647ae55827341a71958d603": "\\mathcal{L}=\\bar \\psi_a \\left(i\\partial\\!\\!\\!/-m \\right) \\psi^a + \\frac{g^2}{2N}\\left[\\bar \\psi_a \\psi^a\\right]^2", "b12bcbc167b55255e47462c21ddc6127": "{\\bar{N}}_5", "b12bcebfb89ad6149ac63ce1ef753560": " u(r,t) = \\frac{F(t)}{r^2} ", "b12be649b1b4543851695fe8292ef1c4": "\\{z: h(z) < +\\infty\\}", "b12bf6f7cc0375df4f1b3227e38acc4b": "G,", "b12bfacbde24bd2f343369de68a6950e": "Attr_i(U)^{j+1} := Attr_i(U)^j \\cup \\{v \\in V_i \\mid \\exists (v,w) \\in E: w \\in Attr_i(U)^j \\} \\cup \\{v \\in V_{1-i} \\mid \\forall (v,w) \\in E: w \\in Attr_i(U)^j \\}", "b12cd3edc6c275ef31648ba5fe6629f9": "\nL = \\int dt\\sum_{i,j}\\left\\{ \\tilde{A}_{i}\\lambda_{i,j}\\tilde{A}_{j}-\\widetilde{A}_{i}\\left[\\delta_{i,j}\\frac{dA_{j}}{dt}-k_{B}T\\left[A_{i},A_{j}\\right]\\frac{d\\mathcal{H}}{dA_{j}}+\\lambda_{i,j}\\frac{d\\mathcal{H}}{dA_{j}}-\\frac{d\\lambda_{i,j}}{dA_{j}}\\right]\\right\\},", "b12cde645632df9bf1bf7e4ea8267e31": "\\tilde{M}.", "b12ce78fa06152dffd57d53b9c43405b": "P^{-1}~T_0~P", "b12d10c60ac7543d7820704dc755a66c": "f^{\\mathrm{AA}}_p(x) = (-\\log x)^p.", "b12d44ec1a9da95bf72159365c1172b3": "\\mathcal{L}(\\sigma^2|D,\\mu) = \\frac{1}{\\left(\\sqrt{2\\pi}\\sigma\\right)^n} \\; \\exp \\left[ -\\frac{\\sum_i^n(x_i-\\mu)^2}{2\\sigma^2} \\right]", "b12d71f7799a6e757424e2ad0ca4de64": "G_X \\approx \\frac{\\alpha \\, - \\frac{1}{2}}{\\alpha +\\beta - \\frac{1}{2}}\\text{ if } \\alpha, \\beta > 1.", "b12d871e51c4353b35e22a74dfa08922": "\\Gamma_n^*", "b12de912e832cad9ecf86a73c6780f96": " {\\epsilon_0 \\over e}\\nabla\\cdot\\vec E = [\\sum_i n_{i0}Z_i - n_{ne0}] + [\\sum_in_{i1}Z_i - n_{e1}] ", "b12deaa864a976b5b24afbb43645280d": "\\scriptstyle\\vec S", "b12e0164e5067a721239fed30632a03f": " xyxyxy^3 ", "b12e048ac0500233bc7d865a45be7575": " \\log_b(x^c y^d) = \\log_b(x^c) + \\log_b(y^d) \\!\\, ", "b12e5a95f5b2792a16918c9116ea9afb": "\\begin{align}S^2_a &= \\frac{n-1}{a}S^2_{n-1}\\\\\n&= \\frac{1}{a}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2\\end{align}", "b12ec47452774173386ca0c30404136a": "c\\cdot V\\subset V", "b12ec5d483d671f1ea99b42688529b73": "xDy", "b12ef35c62ee4a2731dc8141ec9d4f6a": "\\chi(\\nu)=i^{(b^2-a+2)c+(a^2-b+2)d+ad}.\n", "b12f51f1445b9311dc5068c12295b89f": "M_X(t)=\\operatorname{E}\\bigl[t^{X}\\bigr]", "b12f6cef784e5ae89436538a62551f8f": "\n\\omega_{3}/2 = i \\int_{-e_{3}}^{\\infty} \\frac{dz}{\\sqrt{4z^{3} - g_{2}z - g_{3}}}.\n", "b12fba4deb2258f82ee6646345d4c1cd": "\\phi_1 = \\arctan \\left({R_1 \\over L_1}\\right)", "b12fbd45b5ca4afa92b05316360ec270": "m(x)=\\mathbb{E}^x[e^{-\\delta\\tau}w(X_{\\tau-},X_{\\tau})\\mathbb{I}(\\tau<\\infty)]", "b12fec5abc8f668f77495a8e7a1e8dba": "P_{R50}", "b130465db5d494301da588769cc2cff7": "K^ {top}_n(C^r_*(G))", "b130cea41c1390db2561fff3fede2475": "A\\backslash (B\\backslash C)", "b130f78916d8f16879591af21b7a6b8f": " \\{ |i\\rangle \\}", "b1310865c869b569d6f2c501486646b1": "= -\\operatorname{tr} \\left( \\gamma^5 \\gamma^5 \\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\right) \\,", "b1313fad99e7f8a885ba7ec69c01ca7e": " e_k(t) = \\sqrt{2} \\sin \\left( \\left(k - \\textstyle\\frac{1}{2}\\right) \\pi t \\right)", "b1319866f827a2bb06c49953ecb4e324": "\\,\\! p_1,p_2,m", "b131ada07ed8d85169a9865f9876f260": "\\begin{align}\n \\phi_L &\\to 0\\\\\n \\phi_R &\\to -90^{\\circ} = -\\frac{\\pi}{2}^{c}\n\\end{align}", "b131cef2b14f3d020ddca58ec3abf026": " \\Xi = \\Phi - pV/T \\,\\!", "b131ef6d7e3202c5979f70fb5ef8d675": "b = 2i", "b131f01006bc1f7ac1fd66c714067a88": "(WA,W_R B)", "b131f258ee417fcc87ebe39631f072b6": "\\hbar = m=1", "b1322c2ec8e0a4cc5a6ec5fb691409ff": "e_i =", "b132537fb46ab641a4d2859835bb058d": " \\left(\\sum_{k=1}^n a_k b_k\\right)^2 = \n\\sum_{k=1}^n a_k^2 b_k^2 + 2\\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i b_i a_j b_j \\ ,", "b132798668cd58f559b09b1a65e582d6": "B_i^\\star = (-1)^i B_i", "b1328a1ad76e9e164e7bfac820a998d4": "SaO_2", "b132e7d166b5c04b862f27032cf7cd1d": "Q'(p; \\gamma) = \\gamma\\,\\pi\\,{\\sec}^2\\left[\\pi\\left(p-\\tfrac{1}{2}\\right)\\right].\\!", "b1330158b1fb6b6827c2c69f26757cf7": "\\,A = i\\phi + A_1\\mathbf{i}+ A_2\\mathbf{j}+ A_3\\mathbf{k}\\quad ", "b13335e7fef939f7a3032ba1a899c320": "\\mathcal{A}^n_\\alpha", "b1335980d2af58c76da0ea0b3b7ffb8a": "\\begin{pmatrix}\\frac{1}{3}&0&0&0\\\\0&\\frac{1}{3}&0&0\\\\0&0&\\frac{1}{3}&0\\\\0&0&0&-1\\end{pmatrix}", "b133bba9e960b1f74e1872e1914aae07": " (\\forall x) A(x)\\ \\equiv \\ \\neg (\\tau_x(\\neg A)|x)\\neg A\\ \\equiv \\ (\\tau_x(\\neg A)|x)A", "b133c73cbf6b46ddb652d1809636c873": "p({\\rm label}|\\boldsymbol{x},\\boldsymbol\\theta) = \\frac{p({\\boldsymbol{x}|\\rm label}) p({\\rm label|\\boldsymbol\\theta})}{\\sum_{L \\in \\text{all labels}} p(\\boldsymbol{x}|L) p(L|\\boldsymbol\\theta)}.", "b133ce6aa41209b4ad485ffdb8e740df": "\\mathbf{x}_i", "b133fd04dba42e3f0869f042d167133d": "W = (L/L_1)^b\\!\\,", "b13430a841c79bb952d9b315eee1b1b6": "\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n\\equiv \\frac{(-1)^{j_1-j_2-m_3}}{\\sqrt{2j_3+1}} \\langle j_1 m_1 j_2 m_2 | j_3 \\, {-m_3} \\rangle.\n", "b1345f58ece1b9bfa5eaa77035bd84ec": " \\oint dS = \\oint {dQ \\over T} = 0", "b134a6f2de21299320dec4b98fe8cf50": " R_z(90^\\circ) \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ \\end{bmatrix} =\n\\begin{bmatrix} \\cos 90^\\circ & -\\sin 90^\\circ & 0 \\\\ \\sin 90^\\circ & \\cos 90^\\circ & 0\\\\ 0 & 0 & 1\\\\ \\end{bmatrix} \n\\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ \\end{bmatrix} =\n\\begin{bmatrix} 0 & -1 & 0 \\\\ 1 & 0 & 0\\\\ 0 & 0 & 1\\\\ \\end{bmatrix} \n\\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ \\end{bmatrix}\n", "b1350543e150091904b6b29d100e0257": "a\\wedge \\lnot a = 0", "b1350ab6dd4465d84e507c21a7a2752e": " \\sinh(\\alpha t) \\cdot u(t) \\ ", "b13589b173cbefbb4cee76b25488f23f": "p\\in (0,1)", "b135bbdbbdd37f80657be59e6420e4e7": "f(t) = t^{2^i}", "b135c6d410c2eaed3d730250bcdb1ebb": "K_i(x) = \\frac{x_i}{|x|^{n+1}}", "b135ed10c90fe2b12e99e39c12164b35": "\\frac{16384}{19683} \\sqrt[3]{2}", "b135f99c269b229445248885a4c9d43d": "S^*(t)", "b13639bdae38d62049d530d130868116": "\\tau_{\\text{max}}:=\\begin{cases}N&\\text{if }A_N=0,\\\\\\min\\{n\\in\\{0,\\dots,N-1\\}\\mid A_{n+1}<0\\}&\\text{if } A_N<0.\\end{cases}", "b1367bfdaa78c662ca2b99ead06757fb": "y_{\\text{ave}}(a,b) = \\frac{ \\int\\limits_{a}^{b} \\! f(x)\\,dx\\, }{ b - a }", "b1368c92e82c22dc21428d64ea304b45": "E[Z]_{ab}", "b13691451dda317e500c89009c459905": " 2^{596,000,000} -1", "b1369e626512bac8ab3b12aec8dc79cb": "z \\to 1", "b136ae4c61d0607d4b6dbfe56aa1f7a0": "a\\succ_Wb \\wedge b\\succ_Wa", "b136b1142e272d316c82d3acb1fb36e9": "\\frac{adsorptive\\;capacity\\;before\\;adsorption}{adsorptive\\;capacity\\;after\\;adsorption\\;and\\;electrochemical\\;regeneration} \\times 100", "b13739115575a44e4fc526b0c5b351b8": "1 \\times 3", "b13794c8a8099c400a39af5423f9b6b1": "\\Rightarrow \\ln x = kt + \\text{constant}\\, .", "b137cc07b996634c65993f4094ee9eeb": "\\scriptstyle\\mathfrak{P}", "b137ec322b39715a061eb8b4e28fee4a": "g_i^*=h_i^*-Ts_i^*", "b1385f6df6fdc6962977da1ab4933cf5": "\\lim_{M \\to \\pm\\infty}G(n,M) = 0. \\,", "b138dcac14a4db156fe0d651556b28f6": "\\mathbf{w}\\cdot\\mathbf{x}_j + b < -\\gamma ", "b1393d502fbf86d740e6f3a913d9dc95": "\\cos \\theta =\\frac{x'(s)}{\\sqrt{x'(s)^2+y'(s)^2}} = x'(s) \\ .", "b1396e446a28cf7a11438fd56d906c61": "P=(x_1,x_2,\\ldots,x_n)\\text{ and }Q=(y_1,y_2,\\ldots,y_n) \\in \\mathbb{R}^n", "b139704c73c33d92256f392b9e119d7b": "\nH(\\lambda,k) = \\gamma\\left(1\\!-\\!\\frac{1}{k}\\right) + \\ln\\left(\\frac{\\lambda}{k}\\right) + 1\n", "b13983116770ed7824c102bcc757ae73": "\\textit{carnivore} \\subseteq \\textit{animal}", "b139ac1a1835371e6c8b110533582021": "0 < a < 1:", "b139b7eaf8327fea84aae3bc097f7589": "~\\hat{U}~", "b139d6cad83a231254ba764ceca73d1e": "\\int_a^b f(x)\\,dx\\ = f^{(-1)}(b) - f^{(-1)}(a).", "b139e46209a45282571db44bba4e8bb7": "\\lim_{n \\to \\infty} f/g = 1 ", "b139f613f109cdef85fd98b795a50f03": "P_{ji}\\!\\left(\\frac{x}{z}, Q^2\\right)", "b13a517bf0a297b03e4cf0ca1cdb55b9": "ST_x(\\bot) \\equiv \\bot", "b13a5799bb58928e7cd8908345495625": "[[M]_{S(n^{\\nwarrow}.\\sigma\\mid\\sigma_0)}\\;\\|\\;N\n]_{S(\\overline{n}^{\\nwarrow}.\\tau\\mid\\tau_0)} \\rightarrow M\\;\\|\\;[N]_{S(\\sigma\\mid\\sigma_0\\mid\\tau\\mid\\tau_0)}", "b13a86d75bd61d727499af1bb9eca8fb": "\\frac{1-\\epsilon}{2n^2}.", "b13aee2e4ae7ec57f8fa3275b5e14374": "\\mathrm{st}(C, x_1, \\cdots, x_n) \\in \\overline{\\mathcal{M}}_{g, n} (X, A)", "b13af6ebac5826cc2885003511328bd4": "\\propto w_t^{[i]}", "b13b1bb7caccf9b5e15bacebff63acc8": "g:R^n\\rightarrow R^n", "b13b8e411eb1977abd48009da0008c05": " 0 = \\frac{-gx^2}{2v^2} p^2 + xp - \\frac{gx^2}{2v^2} - y", "b13be40341fd44a1f4b363313147aeb7": "F\\,\\!", "b13c21f58c2aef4965c54083e2321e72": "C = \\sum\\frac{1}{s_{2i}}=\\frac12+\\frac17+\\frac1{1807}+\\frac1{10650056950807}+\\cdots", "b13cbdda562bbe5a72bc1bafcb24ba1b": "\n\\Beta(x,y)=\n\\dfrac{\\Gamma(x)\\,\\Gamma(y)}{\\Gamma(x+y)}\n", "b13cd5ca80926a73bdb2718cdf423e20": "\ne = \\sqrt{1 + \\frac{2 \\epsilon h^{2}}{\\mu^2}}\n", "b13ce01327ca6b09a10667ad600ee7e4": "T =(t_{ij})_{m \\times n} = ( w_jr_{ij} )_{m \\times n}, i = 1, 2, . . ., m ", "b13d7223d7a16b7f2c4a9a0e3d69bab4": "\\mathit{b}_0\\mathit{b}_1\\mathit{b}_2...\\mathit{b}_{2n-1}", "b13e027706dd366acb2e99bdb6df1834": "\\{y \\in V_{\\alpha} \\mid \\phi_3\\}", "b13e2e0f647dd7ec48eac033713139a3": "\\textstyle \\mathbb{C}", "b13e58b215c7795fa9c047efc17931d7": "\\Delta V_{0}", "b13e691e0e3843724babf70eead88a12": "= 90^\\circ", "b13e99739f3178dee0a33000bb369ba9": "|\\mathbf{AXB}|\\mathbf{X}^{-1}", "b13eed1c676bee61448123b74d2da123": "\n\\tan\\sigma_{01} = \\frac{\\tan\\phi_1}{\\cos\\alpha_1}\n\\qquad", "b13f3b7ce769eeb98828869c7959edd6": "\\mathbb EQ_n=\\mathbb E\\|X-X'\\|^\\alpha", "b13f44af7891ec1651a5e352f330b943": "\\ell_a \\sim 10^{-10}", "b13f465f0343e740c6c549025d5d7a87": "f_{yz^2}", "b13f60ba22b37e12cced44db7e597ef5": " \\left\\lfloor \\, \\right\\rfloor", "b1401339c97c28d07a8abad3ef4b9aa4": "\\phi(k)", "b140497bd7d32d6d3baa98bd1e424819": "F = -\\frac{dW}{dt}", "b1404fe340b0f89e8296c198c2a91261": "\\ell^{\\alpha}(v;\\xi)=E(v)+\\xi^iF_i(v)", "b140692065454d046e25c6b04164f7e9": " W_1=X_1\\,", "b1407f3a2efd30adbee483ec2e4b1517": " j^{\\star}", "b1409d82620d8cab9a17858c3f812d93": "\\sum_{n = -\\infty}^{\\infty}{\\left|h_{inv}(n)\\right|} = \\| h_{inv} \\|_{1} < \\infty", "b1411ea1831fe024758d07f7e2342eca": "\\| e \\| =\\min_{y \\in e} \\| y \\|, \\quad e \\in E, ", "b1418ceb1628f4fc5b10655f043253a2": "k\\geq 1.", "b141c13a13f8d6158127ee607cb626d4": "\\left[\\frac{\\Delta \\lambda_B}{\\lambda_B}\\right] = (1-p_e)\\epsilon + (\\alpha_\\Lambda + \\alpha_n)\\Delta T", "b141c6c76a3ed026c4197baec8791f8d": "x-5 = -\\sqrt{7} \\quad\\text{or}\\quad x-5 = \\sqrt{7},\\,", "b141cbd5fbf9d72a2cab9c21b3c8f81f": " \\frac{b-a}{2} (f_0 + f_1) ", "b141d3ac3448174f99ae5caa0b61a9bd": "\n\\mathbf{R} \\sim\n\\begin{pmatrix}\n\\cos\\phi & -\\sin\\phi & 0 \\\\\n\\sin\\phi & \\cos\\phi & 0 \\\\\n0 & 0 & 1\\\\\n\\end{pmatrix}, \\qquad 0\\le \\phi \\le 2\\pi.\n", "b141dac78114fbc5a535e0dd21893b4a": "\\displaystyle{L(a)L(a^2)=L(a^2)L(a).}", "b141ddadf5e0495f3f7eb71a19fc86b3": "\\left[\\begin{matrix}\n & a_n & a_{n-1} & a_{n-2} & \\ldots & a_1 & a_0 & 0 \\ldots & \\ldots & 0 \\\\\n & 0 & a_n & a_{n-1} & a_{n-2} & \\ldots & a_1 & a_0 & 0 \\ldots & 0 \\\\\n & \\vdots\\ &&&&&&&&\\vdots\\\\\n & 0 & \\ldots\\ & 0 & a_n & a_{n-1} & a_{n-2} & \\ldots & a_1 & a_0 \\\\\n & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \\ldots\\ & a_1 & 0 & \\ldots &\\ldots & 0 \\\\\n & 0 & na_n & (n-1)a_{n-1} & (n-2)a_{n-2} & \\ldots\\ & a_1 & 0 & \\ldots & 0 \\\\\n & \\vdots\\ &&&&&&&&\\vdots\\\\\n & 0 & 0 & \\ldots & 0 & na_n & (n-1)a_{n-1} & (n-2)a_{n-2}& \\ldots\\ & a_1 \\\\\n\\end{matrix}\\right].", "b14207852307601f3e6a885e6bbca7ab": "z = {x- \\mu \\over \\sigma}", "b14210cabedb11adf96edf9a88b2220d": " \\frac{dP}{P} = - \\frac{M g\\,dz}{R^*T}", "b14225ff140aee1525bedf830fce8890": "R_{\\nu}^+=\\{r|\\alpha_r=\\nu \\}; \\;\\;\\; R_{\\nu}^-=\\{r|\\beta_r=\\nu \\}", "b142902445bf7e190b784764876fa888": " x^{20}-210 x^{19}+20615 x^{18}-1256850\n x^{17}+53327946 x^{16} \\,\\!", "b14396eef551234e560a5a7f55bf4ea2": "A = R[x_1, \\ldots, x_n]", "b143d8b39a838dc0725700cd1917b5a1": "\\operatorname{Hom}(-,Q)", "b143ed4c4e432d364d121a581763311a": "\\rho(X,t)=|\\psi(X,t)|^2", "b143f322bb9741a42452fbc372145d2b": "\\mathbb{N}^* = \\mathbb{N}^+ = \\mathbb{N}_1 = \\mathbb{N}_{>0}= \\{ 1, 2, \\ldots \\}. ", "b14435371b0fb5972013cb9b0c5652f5": "\\Omega^r(E) = \\Gamma(E\\otimes\\textstyle\\bigwedge^rT^*M).", "b144b5a7d78974d74fe8ad1462363df0": "e(t) \\rightarrow 0", "b144b825de930d53a341601c97e3717e": "\\tau(\\omega) = \\left[1 + \\Gamma(\\omega)\\right]", "b14597656469abaafa9ea55af43eb7b1": "Y = \\{y_1,\\dots,y_{\\ell}\\}", "b146306ad9831e0c9da46d54ff2c2966": "\\mathbb{C}^m", "b14642604806870949312d14309c3467": "v(t) = \\mathrm{Re} \\left\\{ V(t) \\right\\} = A e^{\\sigma t} \\cos(\\omega t + \\phi)", "b146454a953faf542ea75cd7c963ca92": "\\mathbf{k_1}, \\mathbf{k_2}, \\ldots, \\mathbf{k_n}", "b1467e69911887adc3b0a5e0da980140": "(~ | z_k| = 1 ~)", "b14681ecb137124b5ddac444c8649aef": "A' = 1, B = 1, C = c+1", "b146af453fcf665b355c4c06f748b614": " MW_{ratio} = ", "b1471bb1af9d123329d87c64646d9ece": "(\\gamma^5)^2 = I_4. \\,", "b1473bdbccdecad53cf9edfd42827f29": "\\Omega_L", "b147741062234ce6da4e4fd164bec907": "\\scriptstyle i,j", "b14794f45342b0b277fbeceec309a453": "\\rho(\\theta) = \\rho^{\\pi_\\theta}.", "b148331980abe0a6464957ec0eb4eca0": "1/S", "b148748249a2f454a9ca8ac94b1c6b98": "V=\\frac{1}{12}(45+17\\sqrt{5}+30\\sqrt{5+2\\sqrt{5}})a^3\\approx14.612...a^3", "b148b40085b2e970d334e4334797e6e1": "\\frac{dL}{dt}= -k_r L ", "b148ead747289e1ccb199fa937a8cf37": "\\Psi_p", "b149a98bbebadbb15a4952fc0ad4920e": "f(a_0)=a_1", "b149c0b42aa87f2c4ec14dcf84b4d354": "w_0=(1+\\textstyle\\sum_{j=1}^{n}{b_j M_j})^{-1}=1-\\sum_{j=1}^{n}{w_j}.", "b149d639912f4b25fad4065e48eee707": "A \\or \\neg B \\or \\neg C \\or D \\or \\neg E.", "b149e786e385fe0cb5baab1ad3e6dfcf": "\n\\begin{matrix}\n\\mbox{If } a \\geq 1: & \\\\\n& x_{t+1} = x',\n\\end{matrix}\n", "b14a12caee0410ffe3f7f0821bbf91d6": "m_{s}", "b14a42a49a21c924871f86ba89c7a20b": "\\displaystyle{T_K\\varphi -{1\\over 2}\\varphi = 1}", "b14a6888ad1a5bef40b886dd30304e20": "P(R,G|do(S=T)) = P(R)P(G|R,S=T)", "b14a757b6bca1c60d7042e8acc0c670d": "\\hat{E}=i \\hbar \\dfrac{\\partial}{\\partial t}", "b14a869a727eb41fc927c1303bce1933": "t^{(k+1)} \\gets X^{(k+1)}w^{(k+1)}", "b14a8ae401c01d66739b99eca422a459": "\\omega_{0}", "b14af0a0ce80fe986b6851f42d0c6a1a": "k \\ge j \\ge i", "b14b4c262a5551811ec0d36a4b49edec": "\\Delta\\lambda/\\lambda \\approx \\Lambda/L", "b14b50602c5942b664a1bb30e54a2db5": "V(t)<=V(0)", "b14b5bc70c7f68f6c23e958fc3d490e9": "Map(8,M) \\subset LM\\times LM", "b14b8a802bad010c113ff05dc7667f74": " \\int_0^t X_{s-} \\circ d Y_s : = \\int_0^t X_{s-} d Y_s + \\frac{1}{2} \\left [ X, Y\\right]_t^c,", "b14bade8deddd923a059807f6b0f14d0": "E_n = \\frac{n^2\\hbar^2 \\pi ^2}{2mL^2} = \\frac{n^2 h^2}{8mL^2}", "b14bbad84e3491274965607189be00a4": "\n{\\hat{\\alpha}}(q) = \\max \\{ \\alpha: \\ \\mbox{minimal requirements are satisfied for all } u \\in \\mathcal{U}(\\alpha,\\tilde u)\\}\n", "b14c017a5b88bc70b922737f0ffba7f1": "\\mathbf{A}x", "b14c0e488ab4562b0d7285a1576dacac": "a_{k+1}/a_k", "b14c6073990824870aa92d96a06ce144": " N_c, N_q, and N_y", "b14c60f49c1f76be84756b34c71d8faa": "\\,\nC_v N \\log T + N \\log V\n", "b14ca0d13af1171efedfa6cb5c69dcac": "\\,\\!\\gamma(x) = \\gamma(y)", "b14ca5dcb7a6009760c67d9edb3c8177": "\n\\operatorname{E}(s^2) = \\frac{1}{n-1}\\left[\\sum_{i=1}^n \\sigma^2 - n(\\sigma^2/n)\\right] = \\frac{1}{n-1}(n\\sigma^2-\\sigma^2) = \\sigma^2. \\,\n", "b14cad9903dc2ad385128810b00a1478": " \\nabla\\cdot\\mathbf{F}=\\nabla_\\mu F^\\mu ", "b14cec9142c33e54427f0c1bac863fcd": "r_x(\\tau) = E[(x(t+\\tau)-m_x(t+\\tau)) (x(t)-m_x(t))]", "b14d0254d62d89793aca2b1afb873ed9": "\\operatorname{Pr}(a_{j-1}\\le X < a_j) = p_j \\quad \\mbox{ for } j=1,\\ldots,k", "b14d0e30c9b15d076c4d07af11af03a5": "S \\cup \\{ \\neg P \\} \\vdash \\mathbb{F}", "b14d327607693cd09d15ec60bfb3bc32": "\n\\lim_{k\\to 0} k\\cot\\delta(k) =- \\frac{1}{a}\\;,\n", "b14d3d5395ded5c83cbc25439364a767": "S^E=S^E(t)", "b14d41b4926c299b70ad2b55dc576393": " n \\operatorname{inc}\\ \\operatorname{init} = \\operatorname{value}\\ (f^n\\ x) = \\operatorname{value}\\ (n\\ f\\ x) ", "b14d4c6559495b811cf51871d140c3bf": "\\omega: S \\rightarrow \\Gamma", "b14d58dc31f3e065e7ccefa121e9fd34": "\\Omega , i, \\omega", "b14d6bc5ff7318032f94d9dd79684f18": "\\hat{x}(k)", "b14d820635533a420def0a81c5899e29": "\\text{Pr}(X\\geq a)\\leq e^{-za}M_X(z),\\quad \\forall z>0.\\,", "b14db62755e9934e48f2482c55333150": "\\phi_{sl,m}", "b14e1803c6b1ffeda5a90f69bb5a9296": "|\\Psi^+\\rangle", "b14e790a71581b32bf212e899c1617f2": "\\log 2", "b14e98b51be569c77c32944f31a6d303": "x_0^2=n \\in \\mathbf{F}_p", "b14eb5e636889e876c4065a0983b247c": "H = \\sqrt{8\\pi G \\rho_\\mathrm{full} / 3} = \\sqrt{\\Lambda / 3}.", "b14ef532a91b6a826c93e81adefa66cd": "[0,\\infty )", "b14f32ac0005b994af3a6ddb53ec89d7": "h_{\\epsilon}^{*}(T_{\\epsilon}) = T_{0} + \\epsilon \\, h_{\\epsilon}^{*}(\\mathcal{L}_{X}T_{\\epsilon}) + O(\\epsilon^{2})", "b14f732d561982c308b65e08d1582157": "Q_D\\left(n\\right)\\,\\!", "b14fac1b6cb11c9242084185b9b19b11": "\nf^n\\left(x\\right) = a + (x-a) f'(a)^{n} + \\frac{(x-a)^2}{2!}(f''(a)f'(a)^{n-1})\\frac{f'(a)^n-1}{f'(a)-1}+\\cdots\n", "b1501faeb0df363afe65c276472321cb": "\\nabla^2 u_{\\bold{k}}(\\bold{r}) \\ll k^2.", "b1501feaa533fa4f1c794af3da7efb30": "f = X_{i1}\\beta_1 + X_{i2}\\beta_2 +\\cdots", "b1502aa9ae0a6622ec2e043bf8188c60": "\\|X^\\alpha\\|^2 = \\frac{\\alpha!}{|\\alpha|!}.", "b1503ab93c61e79cc190afaade410a60": "\\alpha \\mapsto \\phi_\\alpha(0)", "b150aeddf14a3e945c19cafafc6b08d9": "{z}={h} \\,", "b150dacc28d2f7dec3b710499e844376": "S_a = x + \\bar{X} a", "b151059ec427062f01e3c66bf001a78d": "\\Delta(G)=2", "b15132214159ed654378904b189a3b50": " P= \\sum_{t=1}^{\\infty} D_1\\times\\frac{(1+g)^{t-1}}{(1+r)^t}", "b151345dbb7026783424ac89cbb1cd1d": "\\mathfrak{P}(\\mathfrak{V})", "b1517494e1323100c7bbd01aabe8d829": "e = E_0/N", "b151f652b3f021ff2b644356609adb7d": "l_1\\equiv\\partial_x+1", "b151fa626f3536a661497f975cd116cb": " \\frac{\\part L}{\\part u} =\\frac{d}{dx} \\frac{\\part L}{\\part u'} \\, . ", "b152076e9dd48472122ecabcc072f527": " var( H ) = \\frac{ \\sum p_i [ log( p_i ) ]^2 - [ \\sum p_i log( p_i ) ]^2 } { N } + \\frac{ K - 1 }{ 2N^2 } + \\frac{ -1 + \\sum p_i^2 - \\sum p_i^{ -1 } log( p_i ) + \\sum p_i^{ -1 }\\sum p_i log( p_i ) }{ 6N^3 } ", "b1520790a8af94fea41b3bff915a0d30": "d\\vec{\\ell}_1=(dx_1,0,0)", "b15228e82aede9520b1c411698e36704": " \\boldsymbol{\\sigma}^* = \\mathcal{G}(\\boldsymbol{F}^*) ", "b1522c00cb1f9d50e250df3ef818af06": "\\scriptstyle t_f\\,", "b1522dbd040b9055a1a10f42395c1d04": "u_1 \\ll c", "b152977745c263cdd73f76d07cac8965": "\\theta_{(\\ell)}=0\\;, \\theta_{(n)}<0", "b152ae0a68127d3342197967da8dded9": "* \\mathbf{F}\\,,", "b152d3d32200a21b0ddd01de4e2bf2e6": "K_a = \\frac{x(x+y)}{C_0 - x}", "b152dfc478e041c2f8ca8c8c170977b2": "\\alpha<\\delta", "b1533707aff6038a2ae0ba3d48b9d0ba": "\\lim_{x \\to a} g(x) = \\lim_{x \\to a} h(x) = L. \\, ", "b15375f2ec3123a66379a505fd21d00d": "\\left| \\left\\langle f,{{g}_{{{m}_{k}}}} \\right\\rangle \\right|\\ge \\left| \\left\\langle f,{{g}_{{{m}_{k+1}}}} \\right\\rangle \\right|", "b1551abc1c365bea7fa082798f6e46db": "\\ddot{f}(t) \\ge 0.", "b1555bf6f74ba9d2e553f243674dbb13": "\\alpha \\mathbf{A} + \\beta\\mathbf{B} \\in M_{mn}(F) ", "b155ae980d1680deca6d8ce5b2d59146": " \\Delta^k x^n = \\sum_{j=0}^{k}(-1)^{k-j}{k \\choose j} (x+j)^n.", "b156070f49739b185c9a5400c0dd3953": "\nLCS\\left(X_{i},Y_{j}\\right) =\n\\begin{cases}\n \\empty\n& \\mbox{ if }\\ i = 0 \\mbox{ or } j = 0 \\\\\n \\textrm{ } LCS\\left(X_{i-1},Y_{j-1}\\right) + 1\n& \\mbox{ if } x_i = y_j \\\\\n \\mbox{longest}\\left(LCS\\left(X_{i},Y_{j-1}\\right),LCS\\left(X_{i-1},Y_{j}\\right)\\right)\n& \\mbox{ if } x_i \\ne y_j \\\\\n\\end{cases}\n", "b1561183abca1fc087b691d7112f9eb7": "\\inf_n x_n \\leq \\liminf_{n \\to \\infty} x_n \\leq \\limsup_{n \\to \\infty} x_n \\leq \\sup_n x_n", "b15627153df9cd85d0d6e191ea8adb5e": "\\mathcal{A}(M)", "b156273df52bf06b4a1642a839f8c687": "\\beta_{max}=\\kappa_3+\\kappa_4 d_3", "b1562ec09cfad3b898b701625e4173c0": "M_i \\setminus V", "b156ac9408f4ee2537199f040c16ec23": "C=\\bigcup_{n=0}^\\infty C_n.", "b156c9c9c101f3366a46bcd4410e6a63": "C_p = {p - p_\\infty \\over \\frac{1}{2} \\rho_\\infty V_{\\infty}^2 }", "b156dd8154dbf3fe6ccdacba0c5517fa": "\\text{octal } 756", "b157177b4a7bc1d93ee812fc12ed636c": "H = H_{\\mathrm{ac}} \\oplus H_{\\mathrm{sc}} \\oplus H_{\\mathrm{pp}}.", "b157241ba4c33969aded643826dfe94f": "(M_i \\otimes I)", "b1576c8aecc847836fd21d01fe0fc5f8": "n>9\\,\\hat{\\sigma}_D^2.", "b1576f91985f709a5b05288637e11972": " \\Delta_i = \\dfrac{\\underset{x \\in C_i}{\\sum} d(x,\\mu)}{|C_i|} , \\mu = \\dfrac{\\underset{x \\in C_i}{\\sum} x}{|C_i|} ", "b15780357b31ca065b579d6a2db8c3be": "X=\\left(X_1,\\ldots,X_k\\right)\\sim GD_k\\left(\\alpha_1,\\ldots,\\alpha_k;\\beta_1,\\ldots,\\beta_k\\right)", "b157d842424e4488850d4c05893b96a6": " \\psi_t = K_{it} * K_a = K_{a+it} \\, .", "b157e1f32b5233ac6434a30cba3d04a3": "\\mathbf{F} = \\mathbf{\\nabla} \\left(\\mathbf{m}\\cdot\\mathbf{B}\\right),", "b157fa5cabfbf8e6208656f2344f5ac1": "\\mathcal{L}_g = -\\frac{1}{4}W^{a\\mu\\nu}W_{\\mu\\nu}^a - \\frac{1}{4}B^{\\mu\\nu}B_{\\mu\\nu}", "b158158ff854c08f9062aa56ee2fd88e": "s = |\\mathbf{a}_1| ", "b15897a46a4144bf83c5e128480b9499": "LQ _l = K_l + E_{l,k} \\times LP^k", "b158bc6958dcc94aba0eef593806c5f3": "\\mathbf{r}\\,\\!", "b158d383c90f68da1ca0cb990b313807": "\\theta=\\frac12 dz+\\sum_i y_i\\,dx_i", "b15909b9fdeef7da41408ce2e885eb3e": "S'^2\\sim S'\\,", "b1595d1f59dd13c08063dd82a250715d": " \\frac{\\partial \\psi}{\\partial t}=i\\left(\\omega_1 \\sigma_x + \\left(w_0+\\frac{\\omega_r}{2}\\right)\\sigma_z\\right)\\psi ", "b15996a0a0598d3153cb6b5502fca013": "N=\\{1, \\ldots, n\\}", "b159de4a1a12e16da0be9f99372a7eb6": "b_{1},b_{2},...,b_{n}", "b15a112a15f0e9ffe29d914e232b0083": " \\sum_{i=0}^{j} a_{i} n_{i}(x_j) = y_j \\qquad j = 0,\\dots,k.", "b15a26665522d698b7051b1bb873a3d3": "\\lfloor year/100 \\rfloor", "b15a3315a69ce382beea50cbaa779f80": "\\ell_p", "b15a83ab884dba5503520bc6a469bd68": "(s,s')", "b15abd6812f3d6c6ddf6c9949a2b4e0e": "\\mathbf{F}_\\nu", "b15b8ac91905ed86004cfa56d42f07a5": "\\mathbf{b_2} = 2\\pi \\frac{(\\hat{y} \\otimes \\hat{x} - \\hat{x} \\otimes \\hat{y}) \\mathbf{a_1}}{\\mathbf{a_2} \\cdot (\\hat{y} \\otimes \\hat{x} - \\hat{x} \\otimes \\hat{y}) \\mathbf{a_1}} ", "b15bba76d4c80be3f7219f1a4756a7bb": "|f + g|^p \\le 2^{p-1}(|f|^p + |g|^p).", "b15be8d30ad34b32dd32fa9845b143f9": "[y]^{<\\omega}", "b15c18104ff9f540eeae0fb865bfa925": "f:\\ A\\to A", "b15ca00571e83c98a3ccf47b11e38487": "O(2^nn^r)", "b15cc96c58a06c0e9a295eb478ab9a3a": "B \\land C", "b15d63984dc6f9cbb31c7fa272fd3708": "S_{wn}", "b15d7da847e6298e7c96a2c8e84b1512": "\\frac{P_\\mbox{out}}{P_\\mbox{in}} = \\left| \\Gamma_1 \\right|^2", "b15dd04eb5b4b56f09a8714358143b42": "\\epsilon = 0.005,", "b15e233fefd088b7879098ce266f52e4": "\\scriptstyle M_{bol_{\\odot}}=4.73", "b15e44a5a383828f866ab23a559956dd": "W=\\overrightarrow{F}\\cdot \\Delta \\overrightarrow{r}", "b15e74b6e84b0f970597a523dd19166f": "\\frac{\\partial}{\\partial g_i}(g_j) = \\delta_{ij}", "b15e884300982c50ab41f68964bf96b2": "(q_1+q_2, p_1+p_2)", "b15eb051873f6aa38b25107acb6b1aa8": "\\tau_\\mathrm{min}\\,\\!", "b15eee5e90f63fae3ea9bbcf5d096dd5": "C_{\\mathrm{p}}= T\\left(\\frac{\\partial S}{\\partial T}\\right)_{P},", "b15f093c3a05442a5b87d417e3392d90": " 1+\\sqrt 2", "b15f4a67a47efe4dad6e21fd2b362007": "\\epsilon = \\gamma x^2 + 2\\alpha x x' + \\beta x'^2", "b15f86ff0ab26e1e323bec2e592178bc": "h \\approx 5\\times 10^{-22}", "b15f9b99a71f9578a4111c1497b0568b": "\nI = \\sum_{k=1}^{N} m_{k} |\\mathbf{r}_{k}|^{2} = \\sum_{k=1}^{N} m_{k} r_{k}^{2}\n", "b16010a8b08f3ddaa6613ae2b509cb3a": " Y = f(x + b (\\alpha - a)) \\ ", "b16026bb85144da269f849ecf167fe00": "\\gcd(a, a) = a.\\;", "b160d9f3f2761414090754d82fe16180": "\\begin{smallmatrix}m\\ =\\ M_v - 5(\\log_{10} \\pi + 1)\\ =\\ 4.3.\\end{smallmatrix}", "b1614c55006f29ae6b033f6895b345b0": "S^2\\rightarrow S^2", "b161b3c0eab21de57e28e209d958df51": "\\sum_{k \\geq 3} (k-2) (f_k-g_k) = 0.", "b16247057a5fd42c41e1cd05334a143f": "c_k=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(t)e^{-ikt}dt,", "b1626dddb74e28f6706919be72bbc408": "\\mathbf{A}\\colon\\mathbf{B}=\\left(\\mathbf{A}\\cdot\\mathbf{B}^\\mathrm{T}\\right)\\colon \\mathbf{I}\n=\\left(\\mathbf{B}\\cdot\\mathbf{A}^\\mathrm{T}\\right)\\colon \\mathbf{I} ", "b162769cbbfa5541e07a33ce587b199b": "[K] = [K_0]+[\\delta K] \\, ", "b162c0697c63c877325b0c12669cfe59": "\\mathfrak{e}_{6(-26)}", "b162e4214c4455a1242a3630991f3112": "S \\subseteq R", "b162f269e431fe6b9e4039220424204f": "\\beta=2, \\alpha=\\frac{n}{2}.", "b163d8c0a4dd0ac876f918bbb2939ede": "\\mathbf{z}_{\\rm{l}} \\vec{v} \\mathbf{z}_{\\rm{r}} = \\begin{pmatrix}\n1 &-dt_{ab}&-dt_{ac}&-dt_{ad}\\\\\ndt_{ab}&1 &-dt_{bc}&-dt_{bd}\\\\\ndt_{ac}& dt_{bc}&1 &-dt_{cd}\\\\\ndt_{ad}& dt_{bd}& dt_{cd}&1\n\\end{pmatrix}\\begin{pmatrix}\nw\\\\\nx\\\\\ny\\\\\nz\n\\end{pmatrix}\n", "b164349668fbd7c75fde749e56aea849": " \\delta p ", "b1649231c35fd9bc9b2fb6790969b935": "\\delta_F(a) = \\delta(Fa) \\colon MF(a) \\to RF(a)", "b164931473edbbe531153e39ee705073": "\\nabla\\Lambda = 0", "b164d47306605b60358d498325e89c60": "\\mathbf A_+=\\langle F,\\le,V\\rangle", "b164d735a1442441757a25939687bb1e": " \\sum_{j=0}^n \\frac{f_j(x) f_j(y)}{h_j} = \\frac{k_n}{h_n k_{n+1}} \\frac{f_n(y) f_{n+1}(x) - f_{n+1}(y) f_n(x)}{x - y}", "b164ff32e945cf3bb77d47c27ddae628": " Y_\\ell^m (\\theta, \\varphi ) = N \\, e^{i m \\varphi } \\, P_\\ell^m (\\cos{\\theta} )", "b1653e8d0f28b4d62ff97b5bd4510c46": "k^s\\zeta(s)=\\sum_{n=1}^k \\zeta\\left(s,\\frac{n}{k}\\right),", "b1654339ecbdc1b668aed07af70f0d79": " X_C = \\frac{V_\\text{0}}{I_\\text{0}} = \\frac{V_\\text{0}}{{\\omega C}}{V_\\text{0}} = \\frac{1}{\\omega C} ", "b1658d474b359f87e80dd3d941ee75db": "S = (1C1)^{-1} C", "b1659e57b68d26a8c0f5568ee2e56ce2": "D=\\left\\{ \\frac{\\partial\\theta_{i}}{\\partial\\xi_{j}},\\ i,j=1,...,n\\right\\},", "b1664b60f6215e77ee1c57c7efa7eea1": "K=10", "b1668a01ba656e95f581b0a5a25979ad": " ds^2 = -dX_0^2 + dX_1^2 + dX_2^2 + dX_3^2, ", "b166a6bfe90131963aedb357b5e6b6e7": "E(B)=E(A)=\\infty", "b166aea9432d4d27b0e1096acc1b65d3": "c^2 = a^2 + b^2 - 2ab\\cos90^\\circ = a^2 + b^2 \\therefore c = \\sqrt{a^2 + b^2}", "b166c28e9f68f410fa4ab964e880433f": "\\sum_{r=0}^\\infty N_r \\frac{z^r}{r!} = \\frac{e^{-2z}}{(1-z)^2}.", "b1672b2348002d3b8ef1c34bd90dd389": " \n \\pi_T: L(\\mathbb{R},\\mathbb{C}) \\rightarrow \\mathcal{B}(\\mathcal{H})\\quad f \\mapsto f(T)\n ", "b167324015aee9df7eba30f5eba4bc44": "(x_1-x_2)^T(F(t,x_1)-F(t,x_2))\\leq C\\Vert x_1-x_2\\Vert^2", "b167448eacafa32b14d5f1b739efd059": "\\begin{align}\n \\nabla_a (X^b + Y^b) &= \\nabla_a X^b + \\nabla_a Y^b \\\\\n \\nabla_a (X^b Y^c) &= Y^c (\\nabla_a X^b) + X^b (\\nabla_a Y^c) \\\\\n \\nabla_a (f(x) X^b) &= f \\nabla_a X^b + X^b \\nabla_a f = f \\nabla_a X^b + X^b {\\partial f \\over \\partial x^a} \\\\\n \\nabla_a (c X^b) &= c \\nabla_a X^b, \\quad c \\text{ is constant}\n\\end{align}", "b167ceed566ff7d78915601708e6f074": "|{\\psi^{\\bot}_{Tr}}\\rangle", "b167f697831bc5116be8c5691c67b757": " \\lim_{t \\rightarrow \\infty} e^{(z-1)t}\\sin \\left( e^{zt} \\right) = 0 ", "b167f996ed4b04c0542f8ea01aa1217d": "E_{2}(\\mathbf{R})", "b167f9e6329cd83b6793f45e7fd86f0c": "\\frac{1}{R_\\mathrm{net}} = \\sum_{i=1}^{N} \\frac{1}{R_i}\\,\\!", "b168d28fa1998064815f6f08e29d1070": "\\frac{\\sigma_f}{f} \\approx b\\ln(a)\\sigma_A", "b1693548182443931c0b4e28e9c10d43": "L_n=n!(n-1)!R_n", "b16962166310e3eb53b01b2d9deb3721": " \\text{FNBW} \\simeq \\frac{115 \\lambda^{3/2}}{C \\sqrt{NS}}\\, \\text{degrees}", "b169957b9849662d6f9e5731422d5295": "d_\\alpha \\overline{X}=0", "b169a8e7abb135f654e3e5485ab797f2": " \\quad I - I_0", "b16a31e6b747b70a02a185a6039b5d72": "\\operatorname{ch}(V)={\\sum_{w\\in W} (-1)^{\\ell(w)}w(e^{\\lambda+\\rho}) \\over \\sum_{w\\in W} (-1)^{\\ell(w)}w(e^{\\rho})}", "b16a5825fd95e6ea10c1cf08dfefa20b": "r_1 = \\frac{c}{R_0 H_0} \\frac{q_0z+(q_0-1)(-1+\\sqrt{1+2q_0z})}{q_0^2(1+z)}", "b16a8c7bcdc3994d9b90e1951222b43c": " {\\mathbb I} ", "b16a915c5e2f2c50839e8b709f463fd1": " \\frac{te^{nt}}{e^t-1}=\\sum_{m=0}^\\infty B_m(n)\\frac{t^m}{m!} \\ . ", "b16a95dda5502d49ee4b6adce9559374": "\\bar{y}_i = \\frac{x_{i+1}-x_i}{\\tau}. ", "b16aa452ea04f91469fb3b501f2f9433": "\\xrightarrow{\\mathcal{D}}", "b16ac9c81b58361ecc29fe5e7fa55ea9": "n_P", "b16aef03417cb331500b3d31f17b7a67": "k_c = \\frac{\\dot{n}_A}{A \\Delta c_A}", "b16b4bd8d156c6c2268ca5138460c3b8": "K_i+w_i \\cdot 1/w_i = \\sum_j P_{ij}K_j + 1", "b16b80b01aff35c4b281f35509216c62": "T=\\frac{1}{4}\\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}", "b16b899eaf50ac59e78959704c9fb9c0": "x_2\\in F_2^k", "b16b89e9bf765ff86e21c59186595b66": "T_{eff} = (2600 K)\\mu^{13/51}(\\frac{M}{M_{\\odot}})^{7/51}(\\frac{L}{L_{\\odot}})^{1/102}", "b16ba0cfb7a097751a9b82ec8fe54973": " \\left|\\sum_{1\\le m,n \\le N} c_{mn}\\lambda_m \\lambda_n \\right|\\le \\sum_{1\\le n\\le N} |\\lambda_n|^2/n ", "b16bb29edf8a8a6d38c021362b9a93aa": "Z=0.9152", "b16bc5195b896d86b5a8e3aa7a533384": "W_n, K_n,\nN_n,", "b16bd71cb53f9ecb5ea777ba398f3c49": "v_s^2 = {\\gamma_eZ_iT_e \\over m_i}[{1 \\over 1+\\gamma_e(k\\lambda_{De})^2} + {\\gamma_iT_{i0} \\over \\gamma_eZ_iT_{e0}}]\n", "b16bd865d240572559c3582edef29809": "u_{1,1}^{0} = \\delta_{1}^{0}(1) = 2", "b16beec766046c34ff66572d524ac954": " y(t) = A_{c} \\cos \\left( 2 \\pi f_{c} t + \\frac{f_{\\Delta}}{f_{m}} \\cos \\left( 2 \\pi f_{m} t \\right) \\right)\\,", "b16bf8f93ec3cc62ad9634567990c0d7": "\n(\\star\\leftarrow) \\quad {X\\leftarrow \\Gamma \\qquad Y \\leftarrow \\Gamma'\n \\over\nX \\star Y \\leftarrow \\Gamma\\Gamma'}\n", "b16bf9ed30a4813b65ce791e6027bc58": "\\left\\{1,2,3,4\\right\\}", "b16c193c45df2569aeb55986622c757d": "\\mathbf{A}^{-1} = \\begin{bmatrix}\na & b & c\\\\ d & e & f \\\\ g & h & i\\\\\n\\end{bmatrix}^{-1} =\n\\frac{1}{\\det(\\mathbf{A})} \\begin{bmatrix}\n\\, A & \\, B & \\,C \\\\ \\, D & \\, E & \\, F \\\\ \\, G & \\, H & \\, I\\\\\n\\end{bmatrix}^T =\n\\frac{1}{\\det(\\mathbf{A})} \\begin{bmatrix}\n\\, A & \\, D & \\,G \\\\ \\, B & \\, E & \\,H \\\\ \\, C & \\,F & \\, I\\\\\n\\end{bmatrix}", "b16c8e38625d676cbfe1ea2cbc8466ab": "\\{ \\sqrt{\\pi}, 2\\pi+1 \\}", "b16ca40b1c70490c34e4278285f503b8": "\\Delta {{V}_{TH}}={{V}_{TH2}}-{{V}_{TH1}}", "b16cbd132e0a460af40df02b80fc55df": "\\hbox{Z}^2/\\hbox{A}\\ge47.", "b16cbda417e99213421b86f8d6d1b075": "\\tilde{\\kappa}(x, y;t) = L_x^2 L_{yy} + L_y^2 L_{xx} - 2 L_x L_y L_{xy}", "b16d559add8bb6f204db09f1623c3c86": "V \\propto T\\,", "b16d8a3d4340323330e21b336648a9b1": "i^{4n+2} = -1\\,", "b16d987ec08683620216c9a9342fbd5c": "f_\\#(\\pi_1(Z,z))\\subset p_\\#(\\pi_1(C,c)). ", "b16dcf208ebd8f0b62b5e3860cdd5428": "0 = f(a) = f(x_n) + f'(x_n) (a - x_n) + \\frac{f''(x_n)}{2} (a - x_n)^2 + \\frac{f'''(\\xi)}{6} (a - x_n)^3", "b16dcfae4e53399309b7add6b1d63786": "(x_{n-1},y_{n-1})", "b16dd0f39b6d5d979d98168dab6bf750": "y=+\\infty", "b16dda52c2bd26572bb648d09389a37f": "\\text{Let}~ r_{\\text{XY}}= \\text{mutation of base X to base Y} = K~C_{\\text{XY}}/P_{x}P_{s}", "b16e214f9c41320505766391b98b9a0a": "x^{ 16 }+x^{ 14 }+x^{ 13 }+x^{ 11 }+1", "b16e7ee3164806b1fc7ee11273e1367d": "x_i =\\gamma^i ", "b16ec18ec93db63598f346d041945d48": " \\alpha_2 = -\\frac{3(1-k)}{4} ", "b16ed84ee4f55df3a1f5aa6678334a81": "\\scriptstyle\\vec{P}\\cdot\\vec{k}<0", "b16f0ff1a711580c1b3c5ccc3db6b38c": " \\frac{\\partial D}{\\partial t} = k_1 L_t -k_2 D ", "b16f28e553cca2c5ab953a8487c13d0b": "\\|x\\|_p:=\\biggl(\\sum_{j=1}^p |x_j|^p\\biggr)^{1/p}", "b16f29e221a1f6e8fb9c73e995ee4014": "v_i = R_i v_{i-1}", "b16f71465fc529157df6b6e5d94cfaed": "Uf_w(x)= \\frac{1}{\\sqrt{\\pi}} \\frac{1}{(1-w)(x-\\overline{z})}", "b16f8ab4f582463313727bf4ddf30af5": "\\begin{align}\n \\int_{0}^{\\infty} \\frac{dx}{(x+1)\\sqrt{x}} &{} = \\lim_{s \\to 0} \\int_{s}^{1} \\frac{dx}{(x+1)\\sqrt{x}}\n + \\lim_{t \\to \\infty} \\int_{1}^{t} \\frac{dx}{(x+1)\\sqrt{x}} \\\\\n &{} = \\lim_{s \\to 0} \\left(\\frac{\\pi}{2} - 2 \\arctan{\\sqrt{s}} \\right)\n + \\lim_{t \\to \\infty} \\left(2 \\arctan{\\sqrt{t}} - \\frac{\\pi}{2} \\right) \\\\\n &{} = \\frac{\\pi}{2} + \\left(\\pi - \\frac{\\pi}{2} \\right) \\\\\n &{} = \\frac{\\pi}{2} + \\frac{\\pi}{2} \\\\\n &{} = \\pi .\n\\end{align}", "b16faffaaf3b3800901cb29da601ca01": "\\stackrel{}{} \\int_B\\lambda(x) \\, dx", "b16ff8539963c116ae77852e03b9e387": "\\rightarrowtail", "b1701cceadcf4e836ef162c973d68be4": " x \\succ y ", "b17029f18ec742eb53a4811940efd62b": "\\operatorname{Log}(z_1) + \\operatorname{Log}(z_2) = \\operatorname{Log}(z_1 z_2) \\pmod {2 \\pi i}", "b17035d78e5215c342a29fe6699c3861": "\n \\nabla^2\\nabla^2 w = w_{,1111} + 2w_{,1212} + w_{,2222}\n = \\left[\\frac{\\partial^4 W}{\\partial x_1^4} + 2\\frac{\\partial^4 W}{\\partial x_1^2 \\partial x_2^2} + \\frac{\\partial^4W}{\\partial x_2^4}\\right] F(t) \n", "b17049ab4478d3ca3b31a8572205f9e5": "\\begin{bmatrix}\n 1 & \\cdots & 0 \\\\\n \\vdots & \\ddots & \\vdots \\\\ \n 0 & \\cdots & -1\n\\end{bmatrix}\n", "b1706c978324aa6773ff6a3ee55eb311": "Z_{c}", "b171407dc0b06299778a5b732bebc8e0": "x = \\rho \\sin\\theta ", "b17141f97bb07454ddc086569c5c85fc": "\\left(\\frac{105V}{16\\pi^3}\\right)^{1/7}", "b172360604bb105e7af2a630c3c0c314": "~m_{w}", "b172ee204e8015140161b7aac42a5d77": " p^\\mu = m v^\\mu\\,", "b1731843bef1cad1513de1755be2fea4": " u^n du \\wedge dv_1 \\wedge dv_2 ... \\wedge dv_n = du_0 \\wedge du_1 ... \\wedge du_n \\,.", "b1736036e91c5cca6744e8917c9f4581": "|{\\Phi}\\rangle = {(}\\frac{1}{\\sqrt{3}}|{0_{A}}\\rangle-\\frac{i}{\\sqrt{3}}|{1_{A}}\\rangle{)} \\otimes {(}|{0_{B}}\\rangle+\\frac{1}{\\sqrt{2}}|{1_{B}}\\rangle{)}", "b173824c3c343f6077422d007300b34d": "\\mathcal{K} = (\\mathcal{T}, \\mathcal{A})", "b173ef878bdcfc228de8b27ad1dc4a97": "\\Omega_3(t) =\\frac{1}{6} \\int_0^t dt_1 \\int_0^{t_{1}}d t_2 \\int_0^{t_{2}} dt_3 \\ (\\left[ A(t_1),\\left[\nA(t_2),A(t_3)\\right] \\right] +\\left[ A(t_3),\\left[ A(t_2),A(t_{1})\\right] \\right] )", "b17401050c4bd629f544d8cfd6365f60": "(a)_k=\\Gamma(a+k)/\\Gamma(a)", "b1740c4f88c37803aded4e5573ed72e4": "w_2 = 5.0 \\text{ ft and } w_min = 3.30 \\text{ ft}", "b1741345e80a40e510c636e14e70a24e": "\n\\text{Risk}_i = C \\cdot \\frac{IR_i \\cdot EF_i \\cdot ED_i}{BW_i \\cdot AT} \\cdot SF \\cdot ADAF_i\n", "b1741dec7cb03db3517ceed8cf5cbe72": "-7.90298 \\left( \\frac{373.16}{T}-1 \\right) + 5.02808 \\log_{10} \\frac{373.16}{T} ", "b174261228db91d2a6e35048ed859933": "\\textstyle\\int_A^C I(y)\\,dR_t", "b174392c5c097908ae2bb4bf74152ebd": "\\Delta\nx_k=\\Delta x", "b174997ab492bef349c12f93cb220ed2": "\\begin{matrix}\nA_n & = & \\#\\{ (x,y,z) \\in \\mathbb{Z}^3 | n = 2x^2 + y^2 + 32z^2 \\} \\\\\nB_n & = & \\#\\{ (x,y,z) \\in \\mathbb{Z}^3 | n = 2x^2 + y^2 + 8z^2 \\} \\quad \\\\\nC_n & = & \\#\\{ (x,y,z) \\in \\mathbb{Z}^3 | n = 8x^2 + 2y^2 + 64z^2 \\} \\\\\nD_n & = & \\#\\{ (x,y,z) \\in \\mathbb{Z}^3 | n = 8x^2 + 2y^2 + 16z^2 \\}.\n\\end{matrix}", "b1749f1a0fde559c96586aff7a7fc35f": "R = nk\\,", "b174ac530e9aff32d525a1c78c9dfb7a": "\\mathbf{Y} = \\sum_k \\mathbf{X}_k", "b175cd96d67b4517cafe768fa4ff9250": "r(S)", "b175e061916db22dc1d6c4b54452c90b": "I \\, ", "b17633b1ac40f43b75cb4ddf4b9fd09c": "I(X; Y) = D_{\\mathrm{KL}}(p(X,Y) \\| p(X)p(Y)).", "b1764f3edba93784b38d9f4d266a50af": "\\lambda \\mathbf{I\\gg{}J^TWJ}, \\ \\mathbf{\\Delta \\boldsymbol \\beta} \\approx 1/\\lambda \\mathbf{J^TW\\ \\Delta y}.", "b1769d3e6d455896e99a8b9e887c8a2b": "b=\\log_2(M)", "b1769f7c4def3bc1d9b55748c35b5dd1": "{{\\Delta z} \\over z}\\,\\,\\, \\approx \\,\\,\\,r\\,\\,{{\\Delta x} \\over \\mu }", "b176ab2d25e031d43fd95c476e075be6": "S^{r+q}\\rightarrow S^{q}", "b17731820841f7be975c7b0e5cf333d6": "D_1^{\\psi} = \\{d^{\\psi}\\colon d \\in D_1 \\} = g D_2", "b1777f17fc24178afaff323dc0a00a70": "p_U(\\lambda) = (-1)^n\\lambda^n.", "b17792df221a159b336e853c3f76ad5c": "\\Sigma_{(x:A)} B(x)", "b177ad71ce74008730c2ff17f891a6c9": "V=\\frac{1}{6}\\sqrt{4(c^2+d^2)s(s-a)(s-b)(s-c)-a^2b^2c^2}", "b177b59dc9eeb7ff7cec9b90d5ab72ef": "-mgl \\sin\\theta = m l^2 {d^2\\theta \\over dt^2} ", "b177d8b36872d3c8b674343b938cfa83": "p_n\\ge 0", "b177e430d23ba511ffe058e25b18c240": "\\chi(S') = N\\cdot\\chi(S) \\,", "b17818d0da2c4a69cbd2328309f54f56": "|4| \\ge |-1| + |2|", "b17846e4d90b88e80f850625fe88d21e": "\\frac{\\partial}{\\partial \\tau} c^{i}(\\xi,\\tau) = \\{\\zeta^{i}, \\mathcal{H}(\\zeta)\\}|_{\\zeta = c(\\xi,\\tau)}", "b17894a84a359c9f3aa9d467b9f79594": "K \\subset X", "b178cd882e3759630a58f025abf49f53": "F^+=B_\\nu J(J+1) +2B_\\nu\\zeta_r (J+1)", "b178d5a3a196182f21ed1048a445ff32": "xRy", "b1794edc54c438b4dcd8f1ba9741552a": "\\tilde{I}[f] = \\sum_{i=1}^{N}\\ w_i f(\\theta_i,\\varphi_i),", "b1796b2c7ceccc2dd25b75743dd67962": "E \\, \\exp(-i\\omega t)", "b1796f4c59718f87103d4b5064494709": "\\mathcal U.", "b179a501223973af8490f2e1a8964190": " \\Sigma", "b179cc6c27f08ef780a42277f3279df7": "(1-1) + (1-1) + \\cdots = 0", "b179f461399fc6ccf97642a454134831": "e_{rs}=\\frac{1}{2}\\left(\\frac{\\partial u_r}{\\partial x_s} +\\frac{\\partial u_s}{\\partial x_r}-\\frac{\\partial u_k}{\\partial x_r}\\frac{\\partial u_k}{\\partial x_s}\\right)\\approx \\frac{1}{2}\\left(\\frac{\\partial u_r}{\\partial x_s} +\\frac{\\partial u_s}{\\partial x_r}\\right)\\,\\!", "b179f507fbdaeeea3d7bd1e664cb446b": "\\alpha_r=\\alpha_{ri}", "b17ae088de3cc166035562ff27ffcc16": " S(n,j) \\!", "b17aee1253456a2feeb779ebb848d063": "k_z = \\mathbf{k}\\cdot \\hat{z} = |\\mathbf{k}|\\cos\\theta = \\cos\\theta (\\frac{2\\pi}{\\lambda})", "b17b06339eb32cb39467fe3c0a666da6": " \\textstyle \\beta ", "b17b12a68c0360288c360b66c7d164e5": "4 \\| \\frac{y_n + y_m}2 -x \\|^2 = \\|y_n -x\\|^2 + \\|y_m -x\\|^2 + 2 \\langle y_n - x \\, , \\, y_m - x\\rangle", "b17b19b1105908110708ff7cba45c0cf": "\\phi(s,t)", "b17b1b72dfe9ed6d65e2e1f5e7d59db9": "\\omega = \\begin{bmatrix} \\phi \\\\ \\chi \\end{bmatrix} \\,", "b17b3d7fbfb44342e7da1ce0d73ef033": "\\Pr \\left [\\tilde f_1(r_1) = f_1(r_1) \\right ] < \\tfrac{1}{n^2}.", "b17b42857dcc6f0c6b642f323f19403d": "AO = \\frac{R^2}{R^2/r}= r", "b17b60c997392d36b82cd0fe54c89e88": ". \\ ", "b17bf8f47f42cc1f7819de21b75ca920": "\\mathrm{M} \\, A", "b17c145a9aea39b86d65f5f5b1f7f271": "\n\\left(\\frac{a}{p}\\right) \\equiv a^{(p-1)/2} \\pmod p.\n", "b17c2c41c2f623ed6c132ddbcc77f1d2": "\\ddots", "b17c571cb944b89bdf5448e0d83d22fa": "\\gamma_\\tau(t)=\\gamma(\\tau+t)", "b17cb2916f80d354489966c0856a2539": "E_{\\alpha \\beta \\gamma \\delta} E_{\\rho \\sigma \\mu \\nu} = - g_{\\alpha \\zeta} g_{\\beta \\eta} g_{\\gamma \\theta} g_{\\delta \\iota} \\delta^{\\zeta \\eta \\theta \\iota}_{\\rho \\sigma \\mu \\nu} \\,", "b17d30899a9047efcf74e562297dea22": "v_n", "b17d3ee6cb549cfd276eccdddc631074": "\\mid z \\mid = q^{-\\operatorname{ord}(z)}", "b17d5cca8288affbc01133d6a73e7f00": "\\overline{0} = 1.", "b17d709fe0b8e035e278b0b723751af9": "\\scriptstyle \\frac1TX'\\!X", "b17d99387062c2a69705c953b180c1a4": " \\frac{\\Delta m^2\\, c^3\\, L}{4 \\hbar E} = \\frac{{\\rm GeV}\\, {\\rm fm}}{4 \\hbar c} \\times \\frac{\\Delta m^2}{{\\rm eV}^2} \\frac{L}{\\rm km} \\frac{\\rm GeV}{E} \\approx 1.267 \\times \\frac{\\Delta m^2}{{\\rm eV}^2} \\frac{L}{\\rm km} \\frac{\\rm GeV}{E},", "b17d9f05f375f9b95c8f2bb235bea02d": "\\text{accuracy}=\\frac{\\text{number of true positives}+\\text{number of true negatives}}{\\text{number of true positives}+\\text{false positives} + \\text{false negatives} + \\text{true negatives}}", "b17dc478c2455348c03fe53d08ee6ad4": "n=6.", "b17dc52b0e420515e1cd6fcb5b2ea9b7": "K_i \\varphi", "b17dfdc0e369defa5f6618d67e7184f6": "V=\\Gamma(TM)", "b17e6cd4f37e7a8bd801171dd77644b0": "\\mathbf{J} = \\rho \\mathbf{v}", "b17ebfb4bc8fc72ad2f2825be0417645": "(1 \\pm i)/\\sqrt{2}", "b17eede807822336f4d0aff50cd9a929": "\\textstyle f : \\Omega_1 \\to \\Omega_2 ", "b17ef3dfa9b1eec4f176361535d30d5a": "a,b\\in\\mathbb{C}", "b17f0b246e59bf25b9a47f5804c17e32": "0\\leq\nk\\leq n", "b17f2bf5f090df49db18f9b6338080e4": "\\forall a_n \\in A", "b180555618c1ac211c609f4a5ca071d7": "\np(h_i,x_i^m|X_i^o,\\Theta) = \\frac{p(h_i,x_i^m,X_i^o|\\Theta)}{\\textstyle \\sum_{h_i \\in H_b} \\int p(h_i,x_i^m,X_i^o|\\Theta) dx_i^m}\n", "b180a8dfb2a28fdd78be990ca01881d6": "\\boldsymbol{\\Omega \\times v} = \\begin{vmatrix} \\boldsymbol{i}&\\boldsymbol{j}&\\boldsymbol{k} \\\\ \\Omega_x & \\Omega_y & \\Omega_z \\\\ v_x & v_y & v_z \\end{vmatrix}\\ = \\begin{pmatrix} \\Omega_y v_z - \\Omega_z v_y \\\\ \\Omega_z v_x - \\Omega_x v_z \\\\ \\Omega_x v_y - \\Omega_y v_x \\end{pmatrix}\\ ,", "b18147d9388828931cd268dccff53503": "(s\\mathbf{I} - A)\\mathbf{X}(s) = B\\mathbf{U}(s), \\,", "b1814e2514bf0725ed90ba8b891a17fd": " S = \\sum_{j=1}^n \\sum_{k=1}^n (a_j - a_k) (b_j - b_k).", "b1817ffd9084ba1f6a38fa6b3dbf1031": "\\operatorname{Ext}^n_R(N, {}_R M) \\cong \\operatorname{Ext}^n_S(S \\otimes_R N, M)", "b181909e58757be28dc84d40fb69dc0c": " \\int dk |k\\rangle\\langle k|\\, ", "b181b0930814406a40420ff02f971193": "|\\Phi_n(q)| > q-1", "b181bae3873060969aa4b12d8fcb6587": "\\operatorname{tr} (\\gamma^{\\mu 1}\\dots\\gamma^{\\mu n}) = \\operatorname{tr} (\\gamma^{\\mu n}\\dots\\gamma^{\\mu 1})", "b181c712123b5936569cb0f42e826513": " -k\\frac{\\partial T}{\\partial n}\\bigg|_A + k_e \\frac{\\partial T_e}{\\partial n}\\bigg|_A = q \\, ", "b181edbfe651a6f42f75225984fab4c6": "X(\\omega)= \\frac{1}{|det(V)|}\\sum_k\\! X_a(\\hat\\Omega-Uk )", "b182162dd23057637250533fb53fd3e3": "\\sum_{j=1}^N ", "b1821b7e099e46e11506aa24a714c135": "a^2=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]=", "b18228c626adff7546b864649b242f55": "\\sum_{k=\\alpha}^n {n \\choose k} (-1)^{n-k} f(k) = \n\\frac{n!}{2\\pi i}\n\\oint_\\gamma \\frac{f(z)}{z(z-1)(z-2)\\cdots(z-n)}\\, \\mathrm{d}z", "b182509af021daf39b884a16d57d6fdd": "h=1.0", "b182a03eba72b533286ca6593327c68a": "\\left\\vert 0.5 - \\epsilon_{t}\\right\\vert \\leq \\beta", "b182a9307422a979ee3775a50f23c135": "\\operatorname{E}[c\\chi^2(k')] = ck' ,", "b182abf14e13e1012eed7e26297b744a": "x = x_\\star", "b18378e7fd7b760abcd2a7579d2b41ad": "{\\rm SPACE}(f)", "b183b1339b385c7916b6ad5fe35275b3": "{\\mathbf d}_{cv}", "b183ee26373b3ae7fdaec6ddcb96bbe8": "2P=(X:Y:Z)", "b1841747f5ff6b77832bb3f9df8d83c9": "a_{10}+b_{10}+c_{10}=b_{1}+c_{1}-a_{1}", "b1846956b55eab7c86bbf6944852e601": "D: M_n(K) \\to K\\, ", "b1849f750e3bd428cde7462d6d2c07e3": "(x, y, z) = \\left(\\sqrt{1 - \\frac{X^2 + Y^2}{4}} X, \\sqrt{1 - \\frac{X^2 + Y^2}{4}} Y, -1 + \\frac{X^2 + Y^2}{2}\\right).", "b184a877710fa8a3e92581deb144ab59": "\\begin{align}\\sigma_{\\mathrm{effective}}^2 & = \\tfrac{1}{2} \\left [ \\left (\\sigma_{11} - \\sigma_{22} \\right )^2 + \\left (\\sigma_{22} - \\sigma_{33} \\right )^2 \n + \\left (\\sigma_{11} - \\sigma_{33} \\right)^2 \\right ] \\\\\n & \\qquad + 6 \\left(\\sigma_{12}^2 + \\sigma_{13}^2 + \\sigma_{23}^2 \\right )\\end{align}", "b184b935cbc37f728b49a2326670a3c3": "\\psi_{\\bold{k}}(\\bold{r}) = \\frac{1}{\\sqrt{\\Omega_r}} e^{i\\bold{k}\\cdot\\bold{r}}", "b184cc07cf4d0d0b630877328cd4996a": "(2\\pi\\alpha)^{-1/2}e^{-x^2/(2\\alpha)}.\\,\\!", "b1851df0764671f49513e8a7a8eacd16": "\nREL = E(p-\\pi(p))^2\n", "b1855d965bc476c7273362bb03ebe591": "f: X \\to X", "b1856aae62785885c4ad61a80b5a50fc": "\\partial f \\pitchfork Z", "b185ffb47b695dba9d93358b8cee6bb9": "R(\\hat{n},360^\\circ) = -1", "b1862d209ef023d22c04705d79ebea52": "Z^*_n", "b1864cad967d07dbab71116cf4cf155a": " 0 \\le a_i < n ", "b186642415a69ad75058e243873dc35d": "B_{ik}", "b1869c384bb8c0064c0dea010b858abd": "z=\\pm\\infty.\\,", "b1877823d350816dc8a4835b69d71892": "G_X(e^{t}) = M_X(t)", "b187fb9c8cc2db55204b36e24382c4d5": "P(m)=e^{-E_m/(k_BT)}/Z=e^{xm/J}/Z", "b1880934d9d465c0bc7c7ea232104633": " x_{n+k+1} = x_{n+k} - \\frac{f(x_{n+k})}{f'(x_{n+k})} ", "b18813d82177ff5a5e8ba1feb40a4c04": "f = {nv \\over 4(L+0.4d)}", "b1881b22e26c9d0446b8addfa41624b7": "\\bar{y}=\\frac{1}{A}\\int_a^b \\left[\\frac{f(x) + g(x)}{2}\\right][f(x) - g(x)]\\;dx,", "b1881be603cc44c1b4e7ac98635f483d": "x_2(b)=0", "b1884371ba66eb7cb9f7661802b6dddc": "\\ \\kappa_0 (\\mathcal B)", "b1886af18aa1330acb3e6bdcc7ab8192": "\\alpha \\to \\gamma_n^{-1}\\cdot \\alpha\\cdot\\gamma_n", "b18884355467022ae3c3cb6e42439569": "\\nabla\\cdot\\mathbf{F}= X \\rho", "b188958c51be734d0b4b77e423775e5f": "QC' = \\left(\\frac{1}{2} \\times \\frac{M_3}{E_2 I_2} \\times L_2\\right)\\times L_2\\times\\frac{1}{3} + \\left(\\frac{1}{2} \\times \\frac{M_2}{E_2 I_2} \\times L_2\\right)\\times L_2\\times\\frac{2}{3}+ \\frac{A_2 X_2}{E_2 I_2}", "b1889a624eb834ddc2c20e5c7a7b62b7": "\\omega_{\\pm}^2 = K\\left(\\frac{1}{m_1} +\\frac{1}{m_2}\\right) \\pm K \\sqrt{\\left(\\frac{1}{m_1} +\\frac{1}{m_2}\\right)^2-\\frac{4\\sin^2(ka/2)}{m_1 m_2}} \\ , ", "b188da9d9bd3477bc5e517fd5a38535d": " | 0 0 \\rangle \\mapsto | 0 0 \\rangle ", "b1891ba8f86acb0cafc7a02bb69a6998": "(\\lambda - \\lambda_i)^k", "b18920911be41b547a6ca99dbb9cb7d1": "\\left(\\frac{\\partial P}{\\partial T}\\right)_{V}=-\\frac{\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}}{\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}}=\\frac{\\alpha}{\\beta_{T}}\\,", "b18996e3a2623598d8e64e7c6cb9397f": " \\prod_{p} (1+2p^{-s}+2p^{-2s}+\\cdots) = \\sum_{n=1}^{\\infty}2^{\\omega(n)} n^{-s} = \\frac{\\zeta(s)^2}{\\zeta(2s)} ", "b18a70503c0fa9d8b0f952920fe1658d": "c_0, c_1", "b18ade54a6889d8f386cf8cdecaeb640": "A_x = 2\\pi\\int_a^b y \\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2} \\, dx = 2\\pi\\int_a^bf(x)\\sqrt{1+\\left(f'(x)\\right)^2} \\, dx", "b18b2b21e57cd14cf9cceca839ca014d": "\\big\\{0, 1, \\ldots, 2^{32}-1\\big\\}", "b18bba563f296ca27c8fa6b0df8afdf6": "[f]:[\\mathbb{R}]^n \\rightarrow [\\mathbb{R}]", "b18bc101c14333355db0a5f22fdd11ad": "\\Phi_{00}=\\Phi_{10}=\\Phi_{20}=\\Phi_{12}=\\Phi_{22}=\\Lambda=0 \\,,\\quad \\Phi_{11}=-\\frac{M^2}{2r^4} \\,.", "b18c292d9cd9804034a97b770f4a769b": " \\begin{pmatrix} \\hat{a} \\\\ \\hat{b} \\end{pmatrix} \\rightarrow \\frac{1}{\\sqrt{2}} \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix} \\begin{pmatrix} \\hat{c} \\\\ \\hat{d} \\end{pmatrix}. ", "b18c63f340c16736f8cf43cc2138d908": "x \\frac{d}{dx}", "b18c988d7cd3f1be09e04d09bb128d45": "\nL_v = 20\\, \\log_{10}\\left(\\frac{v_1}{v_0}\\right) \\mathrm{dB}\n", "b18d0fcb6b750fbc1ec4a79a7ed4d18b": "a_ny^{(n)} + a_{(n-1)}y^{(n-1)}+...+a_1y' + a_0y = g(x).", "b18d4f3b563c111b31daaf6dda0e4d53": "x = y^n + r", "b18d7466c94c17d756eb00d951debe7a": "I_yI_x^+=I_uI_z^+", "b18d8e288b6dc0a68582abbca9429611": " \\omega = \\frac{\\sigma \\left (1 - k^2 a^2 \\right )}{2a \\mu_B} \\frac{1}{k^2 a^2 + 1 - k^2 a^2 K_0^2 \\left ( ka \\right ) / K_1^2 \\left ( ka \\right )}", "b18dc455039eb0701d07f0174321f0b6": "u(x,t)=(\\phi(x)+r(x,t))e^{-i\\omega t}", "b18dd9969fdd8cb84b9895b8aa7154e2": " S^2 ", "b18e625f3cd453a00985a4eaff5b97f6": "dp/ds > 0", "b18e66a49acefe6efd677a1d1f695cb3": " \\varepsilon^{ij} = {h^{ij} \\over r_{0}^{ij}} = { \\left( q^{ij} + q^{ji} \\right) - \\left( d^{i} + d^{j} \\right) \\big / 2 \\over \\left( d^{i} + d^{j} \\right) \\big / 2 }", "b18ec7aa4c51c961b79d47ec4f42198a": " \\forall x_n A(x_1, \\ldots , x_n) ", "b18f3f93aad26c89c9de88cd20088fb0": " |f(z)|\\le M(r) |z|", "b18f9ac0dc710d313e8b6f6699216737": " \\frac{{\\rm d}\\mathbf{T}'}{{\\rm d}t} = \\frac{{\\rm d}\\mathbf{T}}{{\\rm d}t} - \\boldsymbol{\\Omega} \\times \\mathbf{T} ", "b190439ba05b805a5773a847b3e6ddee": "\\Delta E_\\mathrm{Lamb}=\\alpha^5 m_e c^2 \\frac{k(n,0)}{4n^3}\\ \\mathrm{for}\\ \\ell=0\\, ", "b1905e0428e3f7f578a414e97ee7f284": "\\limsup_{x\\to x_{0}} f(x)\\le f(x_0)", "b1906bf6805a1fdb21fa9bec7f6cfa51": "\\Delta p_\\text{B}(x) = -\\frac{1}{W}\\left[\\Delta p_\\text{B}(0) - \\Delta p_\\text{B}(W)\\right]x + \\Delta p_\\text{B}(0)", "b19091543f0047d39232fb5f932fb0cd": "\\left( u_{external}\\,\\,\\Delta M \\right)_{in} = \\Delta U_{system}", "b190edffa413da7afbc4a300ace6f8c2": "f(x) \\geq g(x)", "b19102176c702b6c2502e54b92f34820": "F = G", "b1914a0bdd47c521cd4b3ebcc6839ff6": "dl^2 = dx^2 + dy^2 + dz^2 + \\frac{(xdx+ydy+zdz)^2}{\\kappa^{-1}R^2 - x^2 - y^2 - z^2}", "b191641f43764be5d670c3e761b20da0": "\\sigma_m = \\sigma_w * \\left( 1. - 0.411 * log \\left( 1 + \\frac{x}{a} \\right) \\right)", "b191a6f14c8cfe93d6a7824fe7514662": "1 = \\frac{1}{3} + \\frac{1}{7} + \\frac{1}{8}+ \\frac{1}{15} + \\frac{1}{24} + \\frac{1}{26}+ \\frac{1}{31} + \\cdots", "b191d659fe8712c6546d12d2d0c2b32a": "\\{P|K(P)>c\\}", "b191d88108bd35e3157684fbea90d1ca": "\\left(\\frac{\\Delta Q}{\\Delta t}\\right)_\\mathrm{water} = C_\\mathrm{w} \\frac{\\Delta m}{\\Delta t} \\Delta T_\\mathrm{water}", "b191fef56561fd383de177e0ba50a4a6": "\\rho_n(u)=\\rho(u)", "b1921063346b829a17cef5d3b6314b98": "\\mathcal G = \\mathcal C \\, e^{(i \\pi I_2)}", "b192326f3446487c06b16e95605fe4ec": "\nFRC = \\frac{SSNR}{SSNR + 1}\n", "b1924a75439298a7cee8516735ad477d": "\\sin\\left(\\,\\theta(t)\\,\\right)=0", "b1924db398b2a674ebf874aab99e02a2": "\\begin{align}\n &\\Pr(N=n\\mid M=m,K=k \\ge 2) = (n\\mid m,k) \\\\\n = {} &\\frac{\\mathcal{L}(n)}{\\sum_n \\mathcal{L}(n)} \\\\\n = {} &[n\\ge m]\\frac{k-1}{k} \\cdot \\frac{\\binom{m - 1}{k - 1}}{\\binom n k} \\\\\n = {} &[n\\ge m]\\frac{m-1}{n} \\cdot \\frac{\\binom{m - 2}{k - 2}}{\\binom{n - 1}{k - 1}} \\\\\n = {} &[n\\ge m]\\frac{m-1}{n} \\cdot \\frac{m - 2}{n - 1} \\cdot \\frac{k - 1}{k - 2} \\cdot \\frac{\\binom{m - 3}{k - 3}}{\\binom{n-2}{k-2}}\n\\end{align}", "b192779452ddb9e56ecd18fe69cbc5d0": "d = \\sqrt{h(D+h)} =\\sqrt{h(2R+h)}\\,,", "b192b444ec4e44acf1b1e5681715a839": "\\sum_{i=1}^n(X_i - \\bar X)^2 \\sim \\sigma^2 \\chi^2_{n-1}", "b1931523cbb540b76c31e308705b362c": "= g(b f - c e : c d - a f : a e - b d)", "b19331ebd3e5a176466f2ec7cae6c74b": "f(t) = \\int_0^\\infty e^{-tx} \\,dg(x),", "b1936cb66c18c108fd8589f449c40521": "R+2 \\pi i", "b19374bcd90f9fb6dc6f9c117a5ed615": "V_{in} \\sim \\frac{\\eta}{\\mu_0\\delta},", "b1937aa13654ffda3498f863f3339a3d": "M = \\alpha h \\coth (\\alpha h) \\,", "b193877a39ee52ab1c39a32ebb037fc0": "U\\subset\\mathbb{R}", "b193cb1a1a6f3ced260798c1214141c7": " d\\Omega = 2\\pi\\sin{\\Theta}d\\Theta", "b1940cf6ba7bb50eba76b65a78d3710e": "f^* R^i g_* \\mathcal F \\rightarrow R^i g'_* f'^* \\mathcal F.", "b19421c0ee1d369ed3a1c0fbc414ffc2": "f(\\pi) = 0 \\, ", "b194c15036a18dfc1b7fe299fbf7f779": "\\mathfrak{su}_4 \\cong \\mathfrak{so}_6", "b194e6e0e0ce4829ee8ad905aac5508a": " P_i = ABC ", "b19560cceb72af41e63fed9e56fed827": "T_p^{(e)}", "b19595a7c32e7662345df13f3b91bd1d": "\\textstyle 1.\\ Find\\ W_{K}\\ to\\ minimize:", "b195a0e10669ea79fa13b210056935d1": "AS'B = ACB - \\pi/3", "b195c7fbe0154829d402ccd4b347b79d": "\\displaystyle{T={1\\over 4} (H +JHJ)={1\\over 4}(H+UH^*U^*),}", "b195cbf223072974b26cb4128e032c7a": "u^{}_1", "b195e42a68e175040a364466e746eff0": "\\epsilon := \\textstyle{\\frac{1}{n}} \\operatorname{tr}", "b1961580cf8067b379c44d0778a271e3": "\\frac{dS}{dt} + \\frac{dI}{dt} = 0 \\Rightarrow S(t)+I(t) = N", "b1964db885f5fc758435be991ccbb24e": "\nP(Q) \\, dQ = \\int_\\mathcal{V} \\prod_{i=1}^k (N(x_i)\\,dx_i) = \\int_\\mathcal{V} \\frac{e^{-(x_1^2 + x_2^2 + \\cdots +x_k^2)/2}}{(2\\pi)^{k/2}}\\,dx_1\\,dx_2 \\cdots dx_k\n", "b196e7c875c9162a01df9c5247342db9": "c\\in {\\Bbb R}", "b19732d25f1a9438168e0bddb9f56061": "A_1 \\times A_1", "b197a4debe2058db9267ab3223c1db76": "P_\\ell(\\cos\\gamma)", "b197c95b57705130c6785b4b120ac43f": "y_x = \\frac f {v'} \\frac {u'} u_\\mathrm{h} J \\,,", "b1983ced6fccda24caf3231309473611": "\\zeta'(s)", "b1989712bf96269bffed2662309d7093": "Y^n", "b198c2aa42c82f16f9a4b55dc651e694": "a^2+b^2-c^2=0.", "b19930487b77bb2b470dafd74c4e60fa": "y_i\\,", "b19952c82a18485ec10add490fd34895": "2^{2^n-n-1}\\prod_{k=2}^n k^{{n\\choose k}}", "b199b7f7bf286536d54fb0d225401862": "{\\mathbf{}}\\Psi^2_{i+1}=\\left(A_i-A_iP_iC'_i(C_iP_{i}C'_i+W_i)^{-1}C_i\\right)'\\hat{S}_{i+1}\n\\left(A_i-A_iP_iC'_i(C_iP_{i}C'_i+W_i)^{-1}C_i\\right)", "b199eca953bb29571273393bd52f9788": "t_{OX}", "b19a26c9e683f5d146582841583f55c7": "F_{rest}", "b19a4f8b3b78965c6146726ed6222d4b": "I_{\\sigma,\\varepsilon}:=\\left\\{ 1,\\ldots,N \\right\\}\\setminus \\sigma^{-1}\\left(\\{ 1,\\dots,N_\\varepsilon \\}\\right)", "b19a697c21a02c54dc8929347ad0e1bf": "X_2^{(4)}k = \\frac{1}{2}[H,X 1]^{(4)}k = \\frac{1}{2}[H^{(4)},X 1^{(4)}]k = 0.", "b19accce41f42a61898b9b603ed6e5c7": "g^{y}", "b19ae2540c87083b363440c562f58f70": " H(k[\\Delta]; t,\\ldots,t) = \n\\frac{h_0 + h_1 t + \\cdots + h_d t^d}{(1-t)^d} ", "b19af5d7fde0cbd375c58078fd04965d": "A^{op}", "b19b687c436fd8e9c814ff7ec2b1aa75": " \\left( {P \\over P_0} \\right) = \\left( {V \\over V_0} \\right)^{-{\\gamma}}, ", "b19b7f2a72eeab9913f53d0ce662693b": "\\boldsymbol\\chi", "b19bda5fa08db29ff7b0c2afb1f02f72": "H_n(M,\\partial M)\\cong \\mathbf{Z}", "b19c342386dd724597e1a37f9ef041f7": " F^*(-s^*)\\cdot G(s) ", "b19cadeefda1fd6c2e48e158dcb1c55f": "(\\tfrac{a}{b})", "b19cd07d814dd2b66bd17ce09d625204": "1+\\epsilon^2R_n^2(-js,\\xi)=0\\,", "b19cd90da1b8990119ad33e315100f2b": "1~\\text{rad/s} = 60/2\\pi~~\\text{rpm} = c\\times9.55~\\text{rpm} = 1/2\\pi~\\text{Hz}", "b19ce4d91cca43dfbd9ea833084b9da2": "S[A[i]+H[i+1],A[i]+d(w)-1]", "b19cff9a97aaa83e75d4cecbbf463032": "(\\bar{3},1,3)\\rightarrow2\\,(\\bar{3},1)_{\\frac{1}{3}}\\oplus(\\bar{3},1)_{-\\frac{2}{3}}", "b19d26f7ba7e8d35a9ebfd9c835b91fc": "y^2 \\equiv \\pm\\, x v_1^{a_1}v_2^{a_2} \\cdots v_k^{a_k}\\pmod{n}.", "b19d57d5d52be56427044a9bdaec9b6b": "M_n=2^n-1", "b19dde892a5f307cfbd7adbb2e4e5db1": "V(r) = \\frac{Z e}{r} exp[{-q r}]", "b19dfd822e2517400335d53c1556df72": "x\\ ", "b19e0830c32a5d133685ea4a5fc6c280": "D[\\partial_i\\partial_j||]=D[||\\partial_i\\partial_j]=-D[\\partial_i||\\partial_j]", "b19e18b0951c9af472c1315cf52d32f2": "\\mathbf{x}_{k}", "b19e1c09f77b9d202eba1ec635db53a4": "(x_i^1, x_i^2, y_i)", "b19e322806d98b4edc3651a12af4607a": " \n\\begin{cases}\n3x_1-\\cos(x_2x_3)-\\tfrac{3}{2}=0 \\\\\n4x_1^2-625x_2^2+2x_2-1=0 \\\\\n\\exp(-x_1x_2)+20x_3+\\tfrac{10\\pi-3}{3}=0\n\\end{cases}\n", "b19e4cac8a740f5a856e8d7329790da9": "\\mu'_5=\\kappa_5+5\\kappa_4\\kappa_1+10\\kappa_3\\kappa_2\n+10\\kappa_3\\kappa_1^2+15\\kappa_2^2\\kappa_1\n+10\\kappa_2\\kappa_1^3+\\kappa_1^5\\,", "b19e5e53592e0305f3d4b94124471866": "\\begin{pmatrix}\\alpha_1 \\\\ \\alpha_2 \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} \\omega_1 \\\\ \\omega_2 \\end{pmatrix}", "b19e6a211a3088122c690281010ba54e": "s + 1", "b19e6e878c894cc7a44999c50d96dfae": "K(x\\mid y) \\leq K(y\\mid x)", "b19ee900abf5f74bc85b6b62f72b9699": " | a+bi | | c+di | = | (ac-bd)+i(ad+bc) |,\\,", "b19f0ffca7d1b2e5b9bfc622880eb4b6": "P =h \\nu \\ g(\\nu) B_{21} \\ \\rho (\\nu) \\ \\Delta N ", "b19f1e652762b49cec89cdf2190bbe68": "\\Chi^2(k_0) = \\sum_{i}{\\frac{(x_i-\\hat{\\mu_i})^2}{\\hat{\\mu_i} \\left (1+ \\frac{\\hat{\\mu_i}}{k_0} \\right )}}", "b19f40bfb48ffec62620ccd9115c7e0b": "L(t_1,\\ldots,t_n)", "b19f4b1717a03b04704b328cea5e5c51": "\\mathbb{Q}(\\sqrt{-m})", "b19fa91b34346b628b4a8d7209356136": "\\phi_x(x) = N_x e^{\\kappa x}", "b19fcd1a310caf459087237656ac120f": "3\\frac{\\ddot{a}}{a} = \\Lambda - 4 \\pi G (\\rho + 3p)", "b19fce4d9c61ca37120418acae635e2c": " \\dot{P}(t) = A(t)P(t)+P(t)A'(t)-P(t)C'(t)W^{-1}(t)\nC(t)P(t)+V(t)", "b19fea06cde535ffefd2d49680430d6d": "a_1 < a_2,", "b1a03a1c7de12ce6f25f40802ced3791": "\\Psi\\left(\\pm\\infty\\right)\\,", "b1a08cafc37b99e80e4553ff18b27ab6": "c_0, c_1, \\ldots", "b1a11054963b70ca807c6784eb40b471": " \\sin{\\frac{D}{2}}=\\sqrt{\\frac{efg + fgh + ghe + hef}{(h + e)(h + f)(h + g)}}.", "b1a14893e88ebddf1bdbd3fca0e3e397": "T \\ \\stackrel{\\mathrm{def}}{=}\\ { 1 \\over f_s } ", "b1a1604620f7c3a3f2b8f8c1af7d3bd3": "\\frac{d(uv)}{dx}\\,", "b1a16377cfe2043a593711c2f8341594": "ax \\in \\left( n\\pi, n\\pi + \\pi \\right) \\,", "b1a165a2c94ec60d99fd3a058f7e9a1d": "g <^* f \\text{ if } B\\subseteq A \\text{ and } g(X) <' f(X) \\text{ for every } X\\in[B]^{\\omega}", "b1a1bb65c0078e26975036f6f18ab72d": "|F,m_f \\rangle", "b1a1e099dc6974042f7c2494ae353f49": "e_0", "b1a20e93ab964f8bfacb1603e492fd84": "x' = x \\cos \\theta - y \\sin \\theta\\,", "b1a21faed624e5e41a30df18a4aae3e9": " \\textstyle \\delta ", "b1a27dbf6bcf45f350e6a397db2635ad": "y_2(x)", "b1a2cb0ca2b9f4c9079504ddcea808c8": "\\begin{align}\n{\\rm E}_{\\lambda_0} \\left[ \\Delta(\\lambda_0\\mid\\mid p_{\\rm ML}) \\right] &= \\psi(n) + \\frac{1}{n-1} - \\log(n) \\\\\n{\\rm E}_{\\lambda_0} \\left[ \\Delta(\\lambda_0\\mid\\mid p_{\\rm CNML}) \\right] &= \\psi(n) + \\frac{1}{n} - \\log(n)\n\\end{align}", "b1a2cb1b566c6bfb7ff3749480c06c27": "y \\mapsto \\left( \\frac{y}{2} \\right)^2 - \\frac{1}{8} ", "b1a3145bb19172c1357f017daa87ba26": " HH^n(A,M) = \\text{Ext}^n_{A^e}(A, M)", "b1a32cf77751105d02bf97e93875bfc8": "C_p = \\frac{2} {\\gamma M_\\infty^2} \\left( \\left(1+\\frac{\\gamma-1} {2} M_\\infty^2 \\left[\\frac{1-|\\vec{V}|^2}{|\\vec{V_\\infty}|^2}\\right]\\right)^{ \\frac{\\gamma}{\\gamma-1} } -1 \\right) ", "b1a32e328541068bec3cde20e51c552d": "R_\\text{p} = \\max_{i} y_i", "b1a3dd93335f66dc3c5adb09d6107db2": "(0.1 \\cdot 2 \\cdot 1 + 0.14154 \\cdot 0.05 \\cdot 2 \\cdot 1) (1+0.1030) = 0.14154 \\cdot 1", "b1a4189b12d0335d7bcaf5beadf63f89": "v = -K_m { v \\over [S] } + V_\\max", "b1a467ad97fd2b1180aa130f3a6b40cc": "\\alpha .", "b1a495f9d647d9d44f914a64b0f60f1d": "f_c(f_c(z)) = (z^2+c)^2+c = z^4 + 2z^2c + c^2 + c.\\,", "b1a49c109378eca79558bfbec7b49f6c": "F_n = \\hat{f}(\\Delta_t n) = \\sum_{m=1}^{M} \\Beta_m e^{\\lambda_m \\Delta_t n}.", "b1a4d3cbd4b3f295a0aee501f3cfdae3": "pa^n=r^{n+1}.", "b1a520b5a2f7e21d3b9aa31a99275424": "c\\delta(x)", "b1a52f02c88b5457603ffe330131ddd3": "G(x) = \\frac{-1}{2c}\\left(-cg(x)+\\left(\\int_{-\\infty}^x h(\\xi) d\\xi +c_1 \\right)\\right).\\,", "b1a534b2e2fd29534bee221f52da0a2f": "\\int_X\\left(\\int_Y |f(x,y)|\\,\\text{d}y\\right)\\,\\text{d}x", "b1a568d57a7b776425ab5411a9b71c6d": "y = \\frac 1 2\\mathrm{laea}_y\\left(\\frac\\lambda 2, \\phi\\right)", "b1a57296f23b9f86943f10e1426d8e59": "\\min J_1 (w,\\xi )=\\frac{1}{2}w^T w + c\\sum\\limits_{i = 1}^N {\\xi _i } ,", "b1a5745f1bff63ad3df9bdc031437d81": "(\\mathbf{g},\\mathbf{f}) = \\frac{\\mathbf{g}+\\mathbf{f}-\\mathbf{f}\\times\\mathbf{g}}{1-\\mathbf{g}\\cdot\\mathbf{f}}", "b1a5b28aa744883c7b2825e4414595f4": "~\\delta(x)^2=0~", "b1a5c19a13c6916a922c962deb2eecc5": "i(t) - \\ ", "b1a5d251fa4fe598cb947ffc42b9cbed": "ut", "b1a5d739c560afa811c98886bd10bbd6": "L_\\mathrm{A}", "b1a5f756b8ea5d4ffc7a90e109549b4a": "\\,R^4", "b1a63622330234db8ec748a3c16b74fc": " \\left(1 + \\beta \\frac{R_E}{r_\\pi + r_E + 2R_E} \\right) r_O + \\frac{r_\\pi + r_E + R_E}{r_\\pi + r_E + 2R_E} R_E", "b1a6476a3ae50596f1af668de0720d82": "c_{v \\times 1}", "b1a65899c5095279ba479f793addb001": "(\\mathcal{T},\\Theta,G)", "b1a687638370b93e4f97815e19384a2d": "r_i = y_{i+31}", "b1a6a9c402678b8a451a7a697992a50d": "\n\\sum_x \\psi(x) |x\\rangle\n", "b1a75d9f1d86b2ab1abfb7928633a3e0": "\\sum_{n=1}^\\infty\\operatorname{E}\\!\\bigl[|X_n|1_{\\{N\\ge n\\}}\\bigr]\n=\\sum_{n=1}^\\infty 2^n\\,\\operatorname{P}(N\\ge n)\n=\\sum_{n=1}^\\infty 2=\\infty.", "b1a773f3976ad93dabd87998f18b05ed": "m \\ge 0", "b1a79e74b39d6c6e37f85fcb40caec25": " \\frac{2 m}{L_1 + L_2} ", "b1a7de7cedad1716c4e9c026738d0b29": "\\psi_s", "b1a85e6f7f1b2a19e9c32e5c9b0c2363": "p \\ll p_0", "b1a88331eee388f7f7df4cb373fd799c": "\\boldsymbol{\\beta} = \\frac{\\bold{v}}{c} \n\\equiv \\begin{bmatrix}\n\\beta_x \\\\ \\beta_y \\\\ \\beta_z\n\\end{bmatrix} \n= \\frac{1}{c}\\begin{bmatrix}\nv_x \\\\ v_y \\\\ v_z\n\\end{bmatrix}\n\\equiv \\begin{bmatrix}\n\\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\n\\end{bmatrix} \n= \\frac{1}{c}\\begin{bmatrix}\nv_1 \\\\ v_2 \\\\ v_3\n\\end{bmatrix}", "b1a8d8f34533660e651d8a0922ab715d": "\\bigcup_{\\alpha < \\omega_1} \\mathbf{\\Sigma}^0_\\alpha = \\bigcup_{\\alpha < \\omega_1} \\mathbf{\\Pi}^0_\\alpha = \\bigcup_{\\alpha < \\omega_1} \\mathbf{\\Delta}^0_\\alpha", "b1a8ffba4724f3f27da85a48a4f8e5b6": "I^m_{\\ell}(\\mathbf{r})", "b1a9243f1a94eaa10dce873ed5ba14c3": "N N^* = \\begin{bmatrix} UP^2U^* + D_{U^*} P^2 D_{U^*} & -D_{U^*}P^2 U \\\\ -U^* P^2 D_{U^*} & U^* P^2 U \\end{bmatrix}.", "b1a93df6dab6b289d3b4cd4d6c8a6923": " f(x,\\lambda)= (E(\\lambda)f)(x). ", "b1a951e7ce4ca4252c6e9a71d82080e7": "\n\\begin{align}\nD^{-\\frac12} p(t) &= D^{-\\frac12} A D^{-1} p(t-1) \\\\\n& = \\left[ D^{-\\frac12} A D^{-\\frac12} \\right] D^{-\\frac12} p(t-1).\n\\end{align}\n", "b1a959fb7c9f23edb3d4afd1e0e54a2d": "\\mathbf E_J\\,\\!", "b1aa26743b420c61f3a8330229630f5a": "S+O", "b1aa369da7bedce498e67b4c27858cf7": "\nJ_{\\alpha}^{\\prime\\prime} = \n\\int_{0}^{\\infty} \\frac{x\\ dx}{\\left( x + b^{2} \\right)^{3} \\sqrt{\\left( x + a^{2} \\right)}}\n", "b1aa58facc6fbe9fc76b3069a30acc67": " u(\\vec x) = \\vec p \\cdot (\\vec x - \\vec{x_0}), \\,", "b1aa757cdd0c8f2b918706b388762359": "(N, M_0)", "b1aa8a36646975e9230970173707609e": " v = -{cu \\over \\left( b - \\lambda_{\\pm} \\right)}.", "b1aac9e37f16ea21532c66304ebaad3c": "\nE = A_{xy} \\xi + A_{yy} \\eta + B_{y}\\,\n", "b1aadd76d51f2eefe550b9856338063b": "\\int_0^\\infty f(x) dx", "b1ab02dfd0dba66a92b2c5e81996b6e8": " \\alpha^{-1} = 137.035\\,999\\,074(44).", "b1ab0e3579efa0e6927077c54dd264d7": "\\displaystyle{[L_m,L_n]=(m-n)L_{m+n} + {m^3-m\\over 12} \\delta_{m+n,0}.}", "b1ac0099b267222ef2b4fc9ca6118247": "\\delta_{ij}\\,\\!", "b1ac9c8a76a5e9f7175deba6624fc13d": "f(x_{1}, \\dots, x_{n}) = f(0, \\dots, 0) = 0", "b1ace37a0c8f06a27ba4daad5f38542a": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\n \\end{Bmatrix}\n = (-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6).\n", "b1acfc5798a5e9a85d28a257fda7bda1": "|f(\\boldsymbol{x}) - L| < \\varepsilon ", "b1ad0419726a7d274e4e77823c27a5d0": "\\mathcal{L}_K", "b1ad0c0bcc81598721f32f9538f33f5e": "f_\\zeta(v;\\,\\gamma) = \\phi(v)\\,\\textstyle\\sum_{j=1}^J \\!\\gamma_j v^j", "b1ad1f51c299253aa4d7a86250da5f50": "G = \\frac{1}{4} \\int_{-\\pi/2}^{\\pi/2} \\frac{t}{\\sin t} \\;dt \\!", "b1ad687f1dd83f2608ca501db365b628": "_{q\\rightarrow p\\,}\\!", "b1adf878ce5d72f1f2150d6372f8c1c6": "c_f(v,u) = f(u,v)", "b1ae56f57321218e3f458b33f99eaa6e": "P(b \\mid N) = \\frac{N}{c}", "b1ae5826936b2c6baf4494b9bd71162d": "\\Phi(t) = \\int_{0}^{t} \\varphi(s) \\, \\mathrm{d} s,", "b1ae80fe8c8453ce0b715e5e821f2e42": "\\begin{matrix} {2 \\choose 1}{3 \\choose 1}{10 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "b1ae887c73ac02747fe976bc7f86d5b8": "\\hat\\Gamma", "b1ae98b2fe50a6be48196f0f4e0b8225": "\\bold{j}_{{\\rm m}, \\, i} = \\rho_i \\mathbf{u}_i ", "b1af54ea3ef72df054afea31e354951b": "g: S^1 \\times S^1 \\rightarrow S^2", "b1af85c6ffd87c7f7d84abc0be6ad967": " \\int_{-\\infty}^{\\infty} w(\\tau) \\, d\\tau = 1.", "b1af92c2a74b3c58d6043e8504a16c34": "\\mu=\\frac{t}{n}", "b1af98886f6d3465484c12754eee03a3": "3^4+4^4+5^4+6^4=7^4-143", "b1aff8dff519b1ffe92325b670345894": "j^\\mu", "b1b04c543903cd729297dbf3998341e1": "\\scriptstyle\\subset", "b1b0968508cc0740bff08a94b947fef5": "(E,d)", "b1b0d3e4f55784eb2ad63dadadcae77e": "L_2(x)", "b1b11f811a64939e7e10514448918206": " \\mathbf{k}_i=\\mathbf{k}_{i\\parallel}+\\mathbf{k}_{i\\perp} ", "b1b1277d751f94f04953ffa751786613": "z=\\frac{(\\hat{p}_1 - \\hat{p}_2) - d_0}{\\sqrt{\\frac{\\hat{p}_1(1 - \\hat{p}_1)}{n_1} + \\frac{\\hat{p}_2(1 - \\hat{p}_2)}{n_2}}}", "b1b1534515bc3ce47c02921a7300905f": " \\hbar \\omega \\left[\\frac{1}{\\sqrt{2}} \\left(-\\frac{d}{dq}+q \\right)\\frac{1}{\\sqrt{2}} \\left(\\frac{d}{dq}+ q \\right) + \\frac{1}{2} \\right] \\psi(q) = E \\psi(q)", "b1b1897fb17253b9d2a5720ba6ed6a1d": " F= \\frac{\\partial F_i}{\\partial x_j}(x_0) ", "b1b2295029f4f847241d8537bcd97955": "\\mathsf{plus}\\ x\\ y \\Leftarrow \\underline{\\mathrm{case}}\\ x\\ \\{", "b1b2344262ad98a70ca709f0819f55e0": "\\mathcal{T}_\\mathrm{k}(\\mathbf{R})", "b1b25d67623182faebd3e9c42707465c": "b(v_1) \\cdot v_1 + 0", "b1b289fddfc0cb5a192dbc3bcff14d19": "\\ -20 \\log_{10} \\left( \\sqrt{ 1 + (f/f_1)^2 }\\right). \\ ", "b1b30639481e128953f623feb63881b6": "\\frac{\\partial \\mathbf{A}\\mathbf{u}}{\\partial \\mathbf{x}} =", "b1b3365bceedc0aa4495a1093d8cbf9c": "g_{\\mathrm{n}} \\,\\!", "b1b395d13deae6851a003a23dd326ac5": "z-y-z", "b1b3cadc8dc4792b9218328b9e8c7a8b": "(r_1,\\ r_2)", "b1b4236ff0cd09962d508f2249ad5d6d": "1 \\leq \\sum_{k=1}^\\infty \\lambda(V_k) \\leq 3.", "b1b42f8e786b15b0000a7139f641ae9c": "W_\\Theta(r) = \\frac{\\Gamma}{2\\pi r} \\left ( 1-e^{-r^2/R_c^2} \\right )", "b1b4b0d655c5efac4d0088f28227a4c2": "b \\simeq h \\theta", "b1b4d01b8fbf482f178bec6bce84970a": " = (v_j^{T} x ) v_j^{T} v_j", "b1b5303246152ee78f4d518f2c5a3632": "\\Gamma(z) = \\sqrt{\\frac{2 \\pi}{z}}~{\\left( \\frac{z}{e} \\right)}^z \\left( 1 + O \\left( \\frac{1}{z} \\right) \\right).", "b1b5af87952746714df72e54367057ab": "x_1^{t_1}\\ldots x_r^{t_r}y_1^{s_1}\\ldots y_c^{s_c}", "b1b5bff69f79a08fd448b0d7fc65465b": "E_u[Y_t(u)-Y_c(u)]", "b1b5f2822b0d963f207cff3518b1ba96": " V_{\\lambda} ", "b1b63bf636dd92b09560a16dc420ec8f": "Q_{CA} = cG \\,", "b1b63ee26e583aa26ae80261b0871624": "f_\\mathrm{A}\\ =\\ f_\\mathrm{la} \\sin \\theta", "b1b6642d37bea2ace820e4167235a4e3": " I(x):= \\begin{cases}\n1 &, \\ \\ r \\le f(x) \\\\\n0 &,\\ \\ r > f(x)\n\\end{cases}\\ , \\ x\\in X\n", "b1b66a9db54174594f3302f880cc37f2": "{\\rm Shi}(x)=\\sum_{n=0}^\\infty \\frac{x^{2n+1}}{(2n+1)^2(2n)!}=x+\\frac{x^3}{3!\\cdot3}+\\frac{x^5}{5!\\cdot5}+\\frac{x^7}{7! \\cdot7}+\\cdots.", "b1b679012e3ce779dab2e8c06cc237a4": "\nE_{U}=E_{L|T>0}+D-DT\n", "b1b69ab6637864a286b2b4688839119e": " {\\left( 1+{\\left(\\frac{x}{b}\\right)}^{-a} \\right)}^{-p} ", "b1b6b2584a2dec2d281251af995467da": "\\sigma_\\text{tot} = \\frac\\pi4 \\;(r+R)^2\\ .", "b1b6f92188ded61b6bce01f6af6555f8": "\n\\frac{R}{i}- \\left( 1+i \\right) ^n \\left( \\frac{R}{i} - P \\right)\n", "b1b6fcae547db1506cc1f4f3cbc5ecb3": "\\varphi_{S_n}(t)=\\varphi_{X_1}(a_1t)\\varphi_{X_2}(a_2t)\\cdots \\varphi_{X_n}(a_nt) \\,\\!", "b1b71528103d2d1e59831a0a38ad6a7f": "\n\\ H(\\omega) = \\frac{1}{1 - \\alpha e^{-j \\omega \\tau}} \\,\n", "b1b71c119db57ced2c2592444b0563d3": "\\ k_\\mathrm{B} ", "b1b7552cd5d78d26d1bcbdbb54133adc": "\ndW = \\sum_{r=1}^{D} Q_{r} dq_{r} = \\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\sum_{r=1}^{D} dq_{r} \\left( \\frac{\\partial \\mathbf{r}_{k}}{\\partial q_{r}} \\right) = \n\\sum_{r=1}^{D} dq_{r} \\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\left( \\frac{\\partial \\mathbf{r}_{k}}{\\partial q_{r}} \\right)\n\n", "b1b797ea90a2e273ac403f3d0f4c4d8d": "A I_n", "b1b7c6ae5e5cdd17b43e5d7b8f62c3bf": "S_1S_3+S_2S_4=\\overline{AC} \\cdot\\overline{BD} =2R^2[\\cos(\\theta_2-\\theta_4)+\\cos(\\theta_1-\\theta_3)]\\;", "b1b7e1a001bfd7885286ed5bdcac04e5": "\\frac{1}{2} + \\ln \\frac{b-a}{2}", "b1b7e9834940ea63cc1123d4d8e0fd83": "\n\\mathbf{Q} \\mathbf{F} \\mathbf{Q}^\\mathrm{T} = \\boldsymbol{\\Phi} \\quad \\mathrm{with}\\quad\n\\boldsymbol{\\Phi} = \\operatorname{diag}(f_1, \\dots, f_{3N-6}, 0,\\ldots,0).\n", "b1b83c023651b6e7383adc9ca89adc6c": "\\frac{\\mathrm{d}U}{\\mathrm{d}t} = \\Sigma_k \\dot Q_k + \\Sigma_k \\dot H_k - \\Sigma_k p_k\\frac{\\mathrm{d}V_k}{\\mathrm{d}t}+P,", "b1b8489509db92e2a7ad833bb4c295b9": "\\tilde{K}_0\\left(A\\right)", "b1b87ce6840ce6d48fea729dfe6aa7d9": "\\frac{6[(\\alpha - \\beta)^2 (\\alpha +\\beta + 1) - \\alpha \\beta (\\alpha + \\beta + 2)]}{\\alpha \\beta (\\alpha + \\beta + 2) (\\alpha + \\beta + 3)}", "b1b89fd07bebe5910dd61f9884b9422c": "E_L = \\{(u,v)\\in E_f : \\operatorname{dist}(v) = \\operatorname{dist}(u) + 1\\}", "b1b8f0bdd9f8ca3166c303c80382d81c": " \\frac{d\\vec{P}}{dt}=\\mathbf{A}\\vec{P},", "b1b8fb840e495bed58f382ca24013e23": "\\left([\\operatorname{ad}_x,\\operatorname{ad}_y]\\right)(z) = \\left(\\operatorname{ad}_{[x,y]}\\right)(z)", "b1b91fdb29cc302357e11517a2b1455e": "\\text{cl}(A)\\,", "b1b9ff762c44a081e3efff2bbc55e149": "F(x-1)=(x-1)^3+6(x-1)^2+12(x-1)+8=x^3+3x^2+3x+1.", "b1ba4274b474f3d046c81ce8c1e74987": "APCS{-}M \\equiv 1-ub_1-ub_2.", "b1ba71af061dc6fd6f03b78cd6a4f1f9": " \\mathbb{F}", "b1baab9dc00f9b9800dbd8022cb79308": "\\ker(\\alpha)", "b1bab2698439ee7dc34302d5ceb4ca16": "C = \\frac{Q}{V} \\approx \\frac{Q}{|\\mathbf{E}| d} = \\frac{A}{d} \\varepsilon", "b1bac8a96474b1892627a2eba5836804": "h=z \\quad", "b1bad49c6be23482617bed7a6f8e7472": "\\Pi(1; \\tfrac \\pi 2 \\,|\\,m)\\,\\!", "b1bb2fdbfac74e4b9815f42c2a1d434b": "\n\\nabla\\cdot\\vec{E}=4\\pi\\rho,\\quad \\nabla\\cdot\\vec{B}=0\n", "b1bb8c7afea66d1a513279d3ee3907a1": "a_1, a_2, a_3, \\dots ", "b1bb96487bf8a34ff7650f44ebb81ebd": "\\scriptstyle M_\\text{A}(H)", "b1bb9a6d25dcfb94e1f3cf72c2778165": "\n\\begin{align}\n\\textrm{ATM}(K_0) &= \\frac12 \\left(\\textrm{Call}(K_0,\\sigma_0) + \\textrm{Put}(K_0,\\sigma_0)\\right) \\\\\n\\textrm{RR}(K_c,K_p) &= \\frac{}{}\\textrm{Call}(K_c,\\sigma(K_c))-\\textrm{Put}(K_p,\\sigma(K_p)) \\\\\n\\textrm{BF}(K_c,K_p) &= \\frac12 \\left(\\textrm{Call}(K_c,\\sigma(K_c)) + \\textrm{Put}(K_p,\\sigma(K_p))\\right)- \\textrm{ATM}(K_0)\n\\end{align}\n", "b1bbab6ffa5ec9035cc625d016c17b50": "Q_B l_A a_B", "b1bc0b52530604995f6c2db544e83685": "M = v_1 v_1 ^* + T", "b1bc12761ea2cf77920bd330ed7d1340": "{E}_0", "b1bc2b7d6f34b77eab6f115457e257df": "y_n(x)=\\sqrt{\\frac{2}{\\pi x}}\\,e^{1/x}K_{n+\\frac 1 2}(1/x)", "b1bc37c1b33a9e35818613624bf5e602": "\\sqrt{a^2+b^2+c^2}=1", "b1bc9ac7b02ae98f1aade6c6204b8ded": "\\eta_G = (-1)^{S + L + I}\\,", "b1bd76ec62ac5a5dd2c4cd2b06b01d27": "\\,\\,\\dot{\\boldsymbol{\\sigma}} = \\mathsf{H}(\\boldsymbol{\\sigma}):\\dot{\\boldsymbol{\\varepsilon}}\\,\\,", "b1bd8fd4b422bd47efbbab1c340be468": "\\scriptstyle \\left|x\\right|^0", "b1bd9c3806421d407d026bd55052eb26": "\\bigcup{}_{i \\in I}", "b1bdc43eae72efb1824daa856fb425ef": "x_2=0.5878", "b1bde21760bf1e5775e69219e72badac": "\n\\begin{align}\n\\lim_{|x| \\to \\infty} x \\sin \\frac{1}{x}\n& = \\lim_{|x| \\to \\infty} x \\left( \\frac{1}{x} - \\frac{1}{3!\\, x^3} + \\frac{1}{5!\\, x^5} - \\cdots \\right) \\\\\n& = \\lim_{|x| \\to \\infty} 1 - \\frac{1}{3!\\, x^2} + \\frac{1}{5!\\, x^4} - \\cdots \\\\\n& = 1 + \\lim_{|x| \\to \\infty} \\frac{1}{x}\\left(-\\frac{1}{3!\\, x} + \\frac{1}{5!\\, x^3} - \\cdots \\right).\n\\end{align}\n", "b1bde8cabdb309329c5faf1204e85585": "\\log y = a \\log x + \\log k\\,\\!", "b1be3afc890ab20e94cc2d6c2d5f0eb7": "\n \\quad (2) \\qquad \\mathbf{u}^{n+1} = \\mathbf{u}^* - \\frac {\\Delta t}{\\rho} \\, \\nabla p ^{n+1}\n", "b1beaca82efb3420477f27dce0dea938": "\\, t=T, \\, X=x(T), \\, Y=y(T)", "b1bebe11ba0f20e8301061fe18822e43": "w \\in \\{0,1\\}^N", "b1bec884cf920d9cf06ae79930387919": "-0.000739 - (235 \\times 10^{-12})\\times N^2", "b1bf42a0a37f7318b9bbf7a4d13ca942": " x^2 + y^2 + {A y \\over \\psi} = 0, ", "b1bf82dfc0253aaa25e20f35b6fabf9e": "y' = k y", "b1bfab5573a2755db0837678e64a06fb": " {d^2 x^\\mu \\over ds^2} =- \\Gamma^\\mu {}_{\\alpha \\beta}{d x^\\alpha \\over ds}{d x^\\beta \\over ds}\\ .", "b1bfc5183fc374f2d30fb0aa4689034b": "3.1415926535897932382\\dots\\!", "b1c00939df742be3abef95d817194cc3": "\\gamma L=F/2", "b1c056020f47f57100dfd8dceaa37097": "\\, A_{(a_0,\\;a_1,\\ldots,\\;a_n)}(x)\\,", "b1c095b78602d0d0c8ab94fb75f6ee2b": "\nq(1)=1 ", "b1c0c2df5f88bd63a8f697078cf320e3": "S(T,X)=S(0,X) + \\int_0^T \\frac {C(T^\\prime,X)}{T^\\prime}\\mathrm{d}T^\\prime.", "b1c118ded2bf848527d7737a5b2e0fd6": "(b,\\infty), (\\infty,c)", "b1c11fd4a529f7f696cf17dbbc96b390": "S^1 \\subset M \\subset S^m", "b1c12e60a109db429ab19e7d791af4b0": "\\mathcal{O}\\left(\\frac{M^2_{\\mathrm{bare}}}{M^2_{\\mathrm{physical}}}\\right)", "b1c1b168b6213c57e5727cf3f209025b": "\\operatorname{Cl}_2\\left(\\frac{\\pi}{4}\\right)=\n2\\pi\\log \\left( \\frac{G\\left(\\frac{7}{8}\\right)}{G\\left(\\frac{1}{8}\\right)} \\right) -2\\pi \n\\log \\Gamma\\left(\\frac{1}{8}\\right)+\\frac{\\pi}{4}\\log \\left( \\frac{2\\pi}{\\sqrt{2-\\sqrt{2}}} \n\\right)", "b1c1c1599603cec8c02e24f56cc29a86": " Fr_2 = \\frac{30}{5.55 \\times \\sqrt{(32.2)(5.55)}} = 0.40", "b1c1c639be8669c894c8d84c2b1fe80e": "\\eta\\ = f_8(\\phi,\\psi)= f_8\\!\\left({Q\\over{ND^3}},{gH\\over{N^2D^2}}\\right).\\,", "b1c1c6676a593daefc7bf9101562b2dc": " \\left.\\otimes\\right. ", "b1c1eb35922f1a033cf63e8dec66624f": "\\bar{\\beta}", "b1c1ec0b0523093f517126fd8d8fb3bb": "\\langle 0|\\varphi(0)|p\\rangle", "b1c2025bba6e4b0b35c85065f2f7c03e": "\\textstyle\\deg(P)<\\deg(Q)=\\sum_{j=1}^{r}\\nu_j,", "b1c245002f3bf083a2a3067e750eb7b9": "\\sqrt[-p]{\\sum_{i=1}^nw_ix_i^{-p}}=\\sqrt[p]{\\frac{1}{\\sum_{i=1}^nw_i\\frac{1}{x_i^p}}}\\geq \\sqrt[q]{\\frac{1}{\\sum_{i=1}^nw_i\\frac{1}{x_i^q}}}=\\sqrt[-q]{\\sum_{i=1}^nw_ix_i^{-q}}", "b1c297496ac92769de2feda7e315508b": "\\sum_{k=1}^n A_{jk} = \\sum_{k=1}^n f_j(g_k) = 0", "b1c344e019b72a7f291900209f5a0b71": "F = k(a_1, ..., a_r)", "b1c361cabc3938a2002a2cbd9c761193": "f^\\prime = 0", "b1c373194092151c4c38fb7b35cf6d30": "\n\\mathcal{A}_n(\\mathbf k)=i\\langle n(\\mathbf k)|\\nabla_{\\mathbf k}|n(\\mathbf k)\\rangle.\n", "b1c38d091014e7e763f99573794e4f29": "f(x,y,z,w) = x^2+y^2+z^2+w^2", "b1c3aaa135ed50aa6b4720bb5758852d": "R = \\mathbb{F}_{q}[x, y]/ (y^2-x^3-Ax-B, \\psi_l)", "b1c3ace081166024a4d1ebc17138f378": "l_c", "b1c3b4adb15ad59fc4d7c568849cbd43": "\\lambda f.(p\\ f)\\ (p\\ f)", "b1c3c6275500b8c6a15ad85c3925734e": "S(A|B)_\\rho", "b1c3d433a015ce18964d108cc1ad9233": "\\scriptstyle U\\left(x,y\\right)=x^\\alpha y^{1-\\alpha }, 0 \\leq \\alpha \\leq 1", "b1c3f7e410221c9315edf4e3c5a2a850": "\\scriptstyle{6\\sqrt{29-2\\sqrt{2}}}", "b1c3fc3468a5805a2a049982ef726f5e": "C'=\\{e'_1,\\ldots,e'_n\\}", "b1c40b3a3ae9d790ba73bb6d82a9d32f": "\\alpha_1 = 0 ", "b1c4535fe21fc33ecca8eb3cd3e527bd": " v = \\sqrt{\\gamma\\frac{P_0}{\\rho}} ", "b1c4708891baa61f8bf60a8d334c4668": "(S ; \\vee, \\wedge, /, 0)", "b1c4aa4f0d644dc2c1096b99ba236d57": "\\cfrac{\\cfrac{stC \\qquad \\overline{s} tD}{tCD} \\, \\operatorname{var}(s) \\qquad \\overline{t} E}{CDE} \\, \\operatorname{var}(t) \\Leftarrow \n\\cfrac{\\cfrac{stC \\qquad \\overline{t} E}{sCE}\\, \\operatorname{var}(t) \\qquad \\cfrac{\\overline{t} E \\qquad \\overline{s} tD}{\\overline{s}DE}\\, \\operatorname{var}(t)}{CDE} \\, \\operatorname{var}(s)", "b1c4c6403237d5b1b5a38d12763bd28d": " Y_k = \\sum_{n=0}^{N-1} y_n \\ e^{-i 2 \\pi \\frac{nk}{N}} \\, ", "b1c4e13b2f2414eff8512d27925aa086": " \\sigma \\frown \\psi = \\psi(\\sigma|_{[v_0, \\ldots, v_q]}) \\sigma|_{[v_q, \\ldots, v_p]}.", "b1c516512fcdb2a02eae610cf6ab07f5": " \\sum_{i=1}^n \\alpha_i y_i = 0.", "b1c527cd52f5c9829f42f0b2914849a2": " d(\\vec{x},\\vec{y})=\\sqrt{(\\vec{x}-\\vec{y})^T S^{-1} (\\vec{x}-\\vec{y})}.\\,\n", "b1c54450fc5b14966cf71182283cbe13": "x=a(1-\\cos\\psi),\\,y=a\\frac{(1-\\cos\\psi)^2}{\\sin\\psi}", "b1c5660b1392ecb094b31a0e42253ff9": "G'", "b1c56f8b7f603e2120e1e63981341887": "a = -1(1-2) = 1, b = -(-1)(2)=2, d = 2.", "b1c59d032657cc7c4acd8823a381fcb1": " \\mathrm{ ICS } = s^2 / m - 1 ", "b1c5bfd4fc10fc2ddaf93e5096524f06": "f(x_1,\\ldots,x_k) \\simeq g(x_1,\\ldots,x_l)", "b1c5e2f374acbfcf6a10df8f2be7eb7a": "f^{(k)}(t)", "b1c5f5e3ac8412d3325d9c7cfb88221c": "NP/N\\,\\!", "b1c632a0a619406fb90cd97206852fc8": " P(E) = P(E|I) \\cdot P(I) + P(E|\\sim I) \\cdot [1 - P(I)] ", "b1c64ced58d03cd29d1d4fa6077ca9f2": " \\ln \\, \\mu_g ", "b1c6628cfd275b340399cc6d7f25bb98": "A - (y^T A x)^{-1} A x y^T A", "b1c6b3b6f46c3f83cf80fa00ef8a453a": "\\{\\boldsymbol{v}_1,\\boldsymbol{v}_2,\\ldots,\\boldsymbol{v}_{i-1}\\}", "b1c73331f15ce9513163ccf4c7fce2ed": "[D_P]", "b1c7d3bad56148cdc8cdaec1e2183d59": "\\tilde{A}=\\mu A^{-1}", "b1c7d5c9fde1b06bf4a7b2b92ed98347": "f(x_1;\\theta)\\cdots f(x_n;\\theta)", "b1c7dc28b06a14f73d3a61879ea20d93": "\\sigma_F(a) = \\sigma(Fa) \\colon LF(a) \\to MF(a)", "b1c828f579f38d53fae593f9a7d2e3a2": "\\delta_p^v: u \\mapsto u(p) + \\epsilon v(u), \\quad v \\in \\mathcal{O}_p^*.", "b1c87eb4290ebda1773b62beab19b89c": "n!! = \\frac{(n+2)!!}{n+2}.", "b1c8c45effe675814282df4c37172565": "\n\\begin{align}\nY_{3,-3} & = f_{y(3x^2-y^2)} = i \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 3} + Y_3^3 \\right) = \\frac{1}{4} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{\\left( 3 x^2 - y^2 \\right) y}{r^3} \\\\\nY_{3,-2} & = f_{xyz} = i \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 2} - Y_3^2 \\right) = \\frac{1}{2} \\sqrt{\\frac{105}{\\pi}} \\cdot \\frac{xy z}{r^3} \\\\\nY_{3,-1} & = f_{yz^2} = i \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 1} + Y_3^1 \\right) = \\frac{1}{4} \\sqrt{\\frac{21}{2 \\pi}} \\cdot \\frac{y (4 z^2 - x^2 - y^2)}{r^3} \\\\\nY_{30} & = f_{z^3} = Y_3^0 = \\frac{1}{4} \\sqrt{\\frac{7}{\\pi}} \\cdot \\frac{z (2 z^2 - 3 x^2 - 3 y^2)}{r^3} \\\\\nY_{31} & = f_{xz^2} = \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 1} - Y_3^1 \\right) = \\frac{1}{4} \\sqrt{\\frac{21}{2 \\pi}} \\cdot \\frac{x (4 z^2 - x^2 - y^2)}{r^3} \\\\\nY_{32} & = f_{z(x^2-y^2)} = \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 2} + Y_3^2 \\right) = \\frac{1}{4} \\sqrt{\\frac{105}{\\pi}} \\cdot \\frac{\\left( x^2 - y^2 \\right) z}{r^3} \\\\\nY_{33} & = f_{x(x^2-3y^2)} = \\sqrt{\\frac{1}{2}} \\left( Y_3^{- 3} - Y_3^3 \\right) = \\frac{1}{4} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{\\left( x^2 - 3 y^2 \\right) x}{r^3}\n\\end{align}\n", "b1c8e3ad8be402c537d94c1109c1d8e3": "\\varphi^2=\\varphi", "b1c94ca32b405f28a2e6dc508c90ec73": "{\\mathit l \\over n} ={1\\over 3}, {2\\over 5}, {3\\over 7}, \\mbox{etc.,} ", "b1c993087edc47449dd061baf0bfd1fa": "\\mathcal{L}_X R_{ab}= -3 \\psi_{a;b}", "b1c9f2bd6b9cb58cfff10193cc59dee6": "V=\\{(a^n,b^mc^n)\\mid n,m\\in\\mathbb N\\}", "b1c9fea3a4f3ca8fcc6e4bc6efddc506": "\\cos(2j+1)\\frac{\\pi y}{2} \\cos(2k+1)\\frac{\\pi y}{2}", "b1ca38fc476fedb2f78a5ec735a2072c": "\\hat{S}_a", "b1ca84ceac83ee2b4bb113842083d004": "\nl_\\phi = m \\hbar\n\\,", "b1caaecc752132b80cefe69464f18d84": "\\Phi(a,b)= \\frac{1}{\\sqrt{2\\pi}}\\int_a^b e^{-t^2/2} \\, dt. ", "b1caeb9a239844ce9f5011b54da5e2b8": "\\forall u\\,(w\\;R_i\\;u\\Rightarrow u\\Vdash A).", "b1cb067acfce0f1fbce32093bb279b16": "\\mathfrak{n}.", "b1cb0a5214d82471743ed0310b972082": "\\frac{d}{dt}\\mathbf{u}_1 = \\mathbf{\\Omega \\times u_1}= \\omega\\mathbf{u}_2\\ ;", "b1cb28fc827ea62c88b4a15da2f25aff": " k_1 \\omega_1 + k_2 \\omega_2 + \\cdots + k_n \\omega_n = 0. ", "b1cb80cd1e3f517f8680fe3eeaad1b0f": "p(x_t=i,x_{t+1}=j) = \\,p(x_t=j,x_{t+1}=i)", "b1cbade00c5dfbba0f578b744adb2fcc": "P_{gen}= \\frac{\\eta_{gen} mv^2}{2 \\Delta t} ", "b1cbeeb5bfe1ef5883c210578b1e67af": "X^\\ast", "b1cc187fef332dcea3a4e71f1756f8a9": "\\text{call/cc} \\colon \\left( \\left( T \\rarr \\left( T' \\rarr R \\right) \\rarr R \\right) \\rarr \\left( T \\rarr R \\right) \\rarr R \\right) \\rarr \\left( T \\rarr R \\right) \\rarr R", "b1cd1826ab12cb4d4708a1ac1b1c6743": "Z_0 \\ \\overset{\\underset{\\mathrm{def}}{}}{=}\\ \\mu_0 c_0 = \\sqrt{\\frac{\\mu_0}{\\varepsilon_0}} = \\frac{1}{\\varepsilon_0 c_0}", "b1cd439998d6699a5c3df293a79eda6d": "\\ell\\cdot \\mathrm{Ann}(I)=\\{0\\}\\,", "b1cd45001d084b076e994bf6ec675821": "\\boldsymbol{Y}", "b1cd9f8cfa5a19a16a3c9277ab483fb5": "\\mathbf{p}\\rightarrow \\mathbf{p}", "b1ce07750f944eec631a70e6afb8b900": "H_1=\\{A^n|n\\in \\mathbb Z\\}=\\left\\{\\begin{pmatrix}1 & 2n\\\\ 0 & 1 \\end{pmatrix} : n\\in\\mathbb Z\\right\\}", "b1ce58773dc119c7b75a60372f6718a9": "\\left\\langle u,v\\right\\rangle = \\sum_{k=1}^{n}u_{k} \\cdot v_{k}", "b1ce984ac839926872a0c02d55ece68c": "\\Delta V^o = -RT \\left(\\frac{\\partial \\ln K}{\\partial P} \\right)_T ", "b1cea662ac2068dd62b80ab81eaeff70": "f^{(m*n)}(x) = (f^n)^m(x)", "b1cea9b3a520fced7a27c00d25eee19c": "\\Delta \\varepsilon_{\\mathbf k}", "b1cef7010465a3cc0cbf95ada60cfb71": "\\frac{1}{2}\\rho\\phi", "b1cf745e93356fb9fc8c26ac88b79d79": "\\sigma_{3c}", "b1d01ba576fc0c4beb7bb683dde67e85": "\\frac{\\sqrt2}2 \\cdot \\frac{\\sqrt{2+\\sqrt2}}2 \\cdot \\frac{\\sqrt{2+\\sqrt{2+\\sqrt2}}}2 \\cdot \\cdots = \\frac2\\pi\\!", "b1d05f297496103569c8043d420f721c": "c=(m+n)(mn-k^{2}) \\, ", "b1d083037d71e27111494c9e36913418": "\\{(x,y,z) \\in \\mathbb R^3 \\mid x^2 + y^2 = z^2 \\mbox{ and } 0\\leq z\\leq 1\\}.", "b1d10db2016c2f83c13b25fcb170cdeb": "b-1", "b1d15bea99b4b68f62de3d1029daadb3": "\n\\mathbf{\\hat{b}_{1:5}} = \\alpha\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix}0.3763 \\\\ 0.6237 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.2745 \\\\ 0.1889\\end{pmatrix}=\\begin{pmatrix}0.5923 \\\\ 0.4077 \\end{pmatrix}\n", "b1d17d25adc487b9336039cad57ed7de": "\\varphi = \\varphi_f + \\varphi_b \\ . ", "b1d1862de3416e7cf6acb5248cbde79f": " \\varepsilon_{2} ", "b1d1c126f163d5bbd85851822009ba2c": "\\operatorname E(X|Y)", "b1d1de19605ee31fd39d6a0f5fe75f00": "\\textrm{fuel} + \\textrm{oxygen} \\to \\textrm{water} + \\textrm{carbon\\ dioxide}", "b1d250b72a38495fccda4ba6650ecab6": " g(z, u) = \\exp\\left(uz - z + u^2 \\frac{z^2}{2} - \\frac{z^2}{2} +\nu^3 \\frac{z^3}{3} - \\frac{z^3}{3} + u^6 \\frac{z^6}{6} - \\frac{z^6}{6} \n+ \\log \\frac{1}{1-z}\\right)", "b1d25e8d29ade4d7f4d887b892dae717": "m < \\alpha -1", "b1d2687bcaebeec4b499952de9652396": "\\ln (1+x) \\;=\\; \\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n} x^n.", "b1d2cbb56725869232ffb5db7c424059": "|E| = \\sup_B \\|E(B) \\| < \\infty", "b1d3ae599bb4845aea585aaf24ad11f6": "\\mathbb{Z}_n \\times \\mathbb{Z}_m", "b1d3eb918e7b2ca67f8124f6c7c1069e": "En=\\{0,1,2,...,n\\}", "b1d41617df1f169032b499595799d3d9": "\\frac{V_{\\,alveolar\\,dead\\,space}}{V_t} = \\frac {P_{\\,a\\,CO_2} - P_{\\,end\\ tidal\\,CO_2}} {P_{\\,a\\,CO_2}}", "b1d41c87b8d83286b88d2780e6149591": " e^{r_{1}x} \\, ", "b1d421949205639ceda258f53b942e76": "Res_{L/k}\\mathbb{A}^1", "b1d42495efc90cd6935b07a4ed8c19cd": "\\delta = k - L", "b1d4523480d88bb55dd62d5bedbafc70": "\\scriptstyle a \\;=\\; b \\;=\\; 1", "b1d47ef227676fca1066fc9abe3b8088": "1 - \\exp(- \\lambda A)", "b1d4bc8df65b79b8888186af3a39ca71": "\\frac{\\partial R}{\\partial y}\\ne0", "b1d4e1a432264c56d2e4125dda85425f": "I_{\\mathrm{RMS}} = \\sqrt {{1 \\over {T_2-T_1}} {\\int_{T_1}^{T_2} {(I_\\mathrm{p}\\sin(\\omega t)}\\, })^2 dt}.\\,\\!", "b1d4f8037d5e03ce4baab18ef3fb07ac": " Q = -\\Delta H_{ad}*N", "b1d521904feceb2cbffa1a1b9fb38c9a": "n_{\\rm Io} - 2\\cdot n_{\\rm Eu} + \\dot\\omega_{\\rm Io} = 0 ", "b1d53402e037aa97818e246df06664d4": "F(\\vec{k},t) \\equiv \\int G(\\vec{r},t)\\exp (-i\\vec{k}\\cdot\\vec{r}) d\\vec{r}", "b1d539c1cb8274f98212423df4aabef2": " \\Delta W = \\oint_\\mathrm{cycle} p \\mathrm{d}V \\,\\!", "b1d5465369a625ade20ad81425213000": "x:[a,b]\\to\\R^n", "b1d54b1d1eae225599d636eb9cb7bb61": "|H(i\\omega)|=1 \\quad \\text{and} \\quad \\angle H(i\\omega) = 180^{\\circ} - 2 \\arctan(\\omega RC). \\,", "b1d5a0b838536e095a7a206a19eb83e8": " \\ \\frac{d}{dt}\\mathbf{u}_2 = \\mathbf{\\Omega \\times u_2} = -\\omega\\mathbf{u}_1\\ \\ .", "b1d5a7e74b4b2a1ad7ae453e66648b33": "\\frac{v_1^2}{2g}+y_1+\\frac{P_1}{\\gamma}-h_f = \\frac{v_2^2}{2g}+y_2+\\frac{P_2}{\\gamma}", "b1d5d3cb85dc72c6198e6d824143a356": " \\zeta(t) ", "b1d604b819bfb3ae6571d113599b0f16": "r(x) = 0 = p(x) - q(x) \\implies p(x) = q(x)", "b1d6484254cd0baa67cc4435e15c264d": " A_n^\\epsilon =\\; \\left\\{(x_1, \\ldots, x_n) : \\left|-\\frac{1}{n} \\log p(x_1, \\ldots, x_n) - H_n(X)\\right|<\\epsilon \\right\\}.", "b1d669f65d56cc854e5c05b487749ed0": "\\beta = {\\omega \\over v}", "b1d68f262b5f40c924c811f8bacb4949": "\n\\begin{matrix}\n\\mathrm d\\pi &= \\theta^1\\varepsilon_1+\\cdots+\\theta^n\\varepsilon_n\\\\\n\\mathrm d\\varepsilon_i &= \\omega^1_i\\varepsilon_1+\\cdots+\\omega^n_i\\varepsilon_n\n\\end{matrix}\n", "b1d6cded94848df015cd6354e1680328": "T_{\\mathrm{h}} = \\{ (x, y) : ", "b1d6d1d5b541f3830a566b368411d9d6": "E_{x,x} = l^2 V_{pp\\sigma} + (1 - l^2) V_{pp\\pi}", "b1d73c7108e08dd26e8a010517e892f3": "I(t) = I(0) e^{-bt}", "b1d7622d3b8eb48d16ece89a991d5132": "\\int_a^b f(x) \\,dx \\approx \\sum_{i=0}^n w_i\\, f(x_i)", "b1d81ba3854b035a687241284dceb885": " TE_{10n} ", "b1d81bbb16ac047a271a9b9bb535184b": "Q(x) = 0", "b1d8273be8b19b70951e4103502540e7": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\frac{\\partial F}{\\partial y'}=\\frac{\\partial F}{\\partial y}", "b1d8967bbd431b3a5feb66d644b15393": "V_{TBG} = V_c - V_m", "b1d99688d5fc7f3ecd699a893793a139": " Y_{\\ell}^{m}(\\vartheta, \\varphi ) \\,", "b1d9c16294a048f52072b4f39cda732b": "[T_A^1]=T_B^1", "b1da05f04364f14254bc9d549d5bca00": "p \\vert N", "b1dae32d79326c3284015c8cb417f348": "(X^\\ast, \\Phi, X_\\ast, \\Phi^\\vee)", "b1db01b3f5c4c8b6e2b5366d23b4ac07": "p = \\nabla \\cdot \\mathbf{\\Phi}", "b1db2f32e42812f002f9f316494318d3": "A_m, B_m, C_m, D_m", "b1db4bf4082746e8219343f94a97ef2d": "\\theta_{m+1} = \\theta_{m} + \\mathcal{I}^{-1}(\\theta_{m})V(\\theta_{m})", "b1dbda9de3b63b631355fc95f7742081": "G^{\\mathrm{A}}", "b1dbea0f47e01a637dfde85ecf36b1df": " \\gamma = 1 - \\sum_{k=2}^{\\infty}(-1)^k\\frac{\\lfloor\\log_2 k\\rfloor}{k+1}.", "b1dbfa575cbf69875c4237ca53ac2875": "T^n", "b1dc21bfe298d9c5408dced55789670b": "\\scriptstyle L", "b1dc2ec890a84c779c2d91939fad28cc": " \\begin{align}\n\\mathbb{P}(\\sup_{0\\leq s\\leq t}W(s) \\geq a, X(t-\\tau_a) < 0) &= \n\\mathbb{E}[\\mathbb{P}(\\sup_{0\\leq s\\leq t}W(s) \\geq a, X(t-\\tau_a) < 0| \\mathcal{F}^W_{\\tau_a})]\\\\\n& = \\mathbb{E}[\\chi_{\\sup_{0\\leq s\\leq t}W(s) \\geq a} \\mathbb{P}(X(t-\\tau_a) < 0| \\mathcal{F}^W_{\\tau_a})]\\\\\n& = \\frac{1}{2}\\mathbb{P}(\\sup_{0\\leq s\\leq t}W(s) \\geq a)\n\\end{align} ", "b1dc3321d750f1cf207aa208a7542f6e": "2p^2\\,", "b1dc939aedeb8602fbd8e6f2175e4f9f": "\\forall x \\in I: s(x) = s^\\prime", "b1dca1392ecd95181c33278dd8fdc9ae": "f(C_{ij})", "b1dcf3b095edd58cbe4fcd30a6058e04": "{D}_{7}^{(2)}", "b1dd3c8b978976444978fabd46948ae7": "N_r \\times 1", "b1dda9664c3ceed011b477bb18ea0966": "\\! d(t) = a(t) \\chi", "b1ddeae984d7c294737d9b66480377ea": "[ES]", "b1de0281f2acc89a3847d2500cef0a69": "U \\mapsto ", "b1de3c70ac298557eb752cb1f5fa96b5": "\\Phi_{1}(x)=x-1", "b1de52403850af04fba4f52c49d20d9f": "i = R", "b1de5a728e24a03d288ae603f3c7f6e4": " \\rho = r \\sin \\theta\\,", "b1de67ee15389fd841bdaae5b6603f79": "d_\\mathfrak{g} \\alpha", "b1dea217b8fa1e7c6c7f94243b3ca4e3": "\\alpha(t):=vt\\ ", "b1dec047a59105f20cd077aead558edb": "Ba \\rightarrow aB", "b1decc163a355b267e647cf6a9ad59d4": "U(X)", "b1dede758ffefa90100a765450b08b25": "P(x)=ax^2 + bx + c", "b1def1385303ae9dac39ddf434a440d2": "(\\hbar^2 s(s+1))", "b1df2b9ae737f1b6a0e5e2d2e7601957": "\\lim_{n\\to\\infty} \\frac{n}{\\sqrt[n]{n!}}=e", "b1df55110838626fd19b80855b5924e7": "i = C \\frac{dv}{dt} \\,", "b1dff6f2f6e8d28fe4efec1c55986123": "2^{\\omega} = \\{0,1\\} \\times \\{0,1\\} \\times \\cdots ", "b1e00023907ec552c9ec4b56da93c296": "ds^2 = -e^{2\\nu}dt^2 + \\rho^2 B^2 e^{-2\\nu}(d\\phi - \\omega dt)^2 + e^{2(\\lambda - \\nu)}(d\\rho^2 + dz^2)", "b1e05a583e87318d114119b928ef15f4": "S = \\sum_{i=1}^{n} f(x_i^*)(x_{i}-x_{i-1}), \\quad x_{i-1}\\le x_i^* \\le x_i.", "b1e06ace954c35d8373e85d3c7450c90": "(b^2 + {{a^2}\\over 2})(\\pi - 2\\arccos {b \\over a}) + 3 b \\sqrt {{a^2} - {b^2}}.", "b1e08169e22cd6c911360ee60097c685": "T:M\\rightarrow N", "b1e0c36f50853cf247546f6defd9e6b3": "|\\Phi(p,t)|^2 \\sim \\mathcal{N}\\left( -m x_0 \\omega \\sin{(\\omega t)} , \\frac{\\hbar m \\Omega}{2} \\left( \\cos^2{(\\omega t)} + \\frac{\\omega^2}{\\Omega^2} \\sin^2{(\\omega t)} \\right)\\right),", "b1e0c735fdf2695538bdad8c5d874b0d": "\\lambda ^3 -3 \\lambda ^2 + 3 \\lambda -1 = 0. \\, ", "b1e0ea191f5778f09c1b9c096623f263": "\\delta =\\delta _T= \\delta _c= {5.0*x\\over\\sqrt{Re}}", "b1e0f9e9e1bb994cb695462313b929c1": "F = rN^{r}_{rr}=rN^{r}_{\\theta\\theta}", "b1e10e2df39cb10dc30822e8ae8a8f17": "*S^{IJ} = i S^{IJ} .", "b1e1175f3eeb4db44866e9af298a7fc1": "EM_{kinetic} = {k_{intramolecular} \\over k_{intermolecular}}", "b1e135fe7949486aa6490eaac952bfc5": "~\\sgn~", "b1e14447c95c0a097f59aaf67db4032b": "x = \\gamma \\left ( \\gamma \\left ( x - v t \\right ) + v t' \\right )", "b1e146d1af1aef980db5537485fb4df3": "\\left(\\frac{2n}{\\pi}\\right)^{1/4}e^{-\\sqrt{4\\pi n}}\n\\cos\\left(\\sqrt{4\\pi n}-\\frac{5\\pi}{8}\\right) +\n\\mathcal{O} \\left(\\frac{e^{-\\sqrt{4\\pi n}}}{n^{1/4}}\\right).", "b1e18f0fe26004a1ed8cf8dcedb56110": " R=\\frac{\\ln(D_e / D_i)}{2 \\pi \\lambda}", "b1e1a82fe971c0f59dd0499b217044f2": "(x-3) (x-2) (x-1)^8 x^2 (x+1)^8 (x+2) (x+3) (x^2-6)^2 (x^2-5)^4 (x^2-2)^2 (x^4-6 x^2+3)^8", "b1e1b14138dbeeb0329165de07428922": "E_{f}", "b1e1b4bdfbe507872d7b1cdbe91e2c0a": "\\operatorname{E}\\left( \\sum\\left( X ^ 2\\right) \\right) = n\\sigma^2 + n\\mu^2", "b1e1e505392279deaaec7376894a0abb": "=b_{7}", "b1e235746bb67e2176280cef08e9bcc4": "m_1 = \\Delta_1 \\quad \\text{and} \\quad m_n = \\Delta_{n-1}", "b1e29598daf893ef05a0e423524eff09": "V(z)=D \\big\\{exp \\left[-2\\alpha (z-z_m)\\right] - 2 exp \\left[-\\alpha(z-z_m>)\\right]\\big\\} + 2\\beta D exp \\left[2 \\alpha (z-z_m) \\right] \\xi(x,y)", "b1e2d01f2a1704adf693860b594f016b": "I(x,y,t)", "b1e304dae60a3ac493eb9d805ef40bcf": "\\overrightarrow{b_2}", "b1e3557158da1b8bd2442cf4d3af8213": "f(u)\\sin v ", "b1e3cce286fc6f598268163e147f311b": "\\underline{\\underline{\\mathbf{A}}}", "b1e3d93bf335b7f3ee77d0014a4e4238": "R = k \\cdot r", "b1e3dddec4f846c9df5c9170c4cc97c2": " (X : Y : Z : 0) ", "b1e3e87c5dd2ea421ab9ac031f0e27fc": " \\lambda = \\frac{2 \\pi}{k} = \\frac{2 \\pi v}{\\omega} = \\frac{v}{f}.", "b1e41d22580a408594c599252cbaa71c": "\\mathbf{E}_2 \\times \\mathbf{H}_1 = \\hat{\\mathbf{r}} (\\mathbf{E}_2 \\cdot \\mathbf{E}_1) / Z", "b1e4d1495e1260e02705644429b77ccb": " \\eta(\\varepsilon) = \\eta_0 + \\delta_\\eta \\varepsilon(t) ", "b1e4dfcaa47ef45d74a72541f534dc7d": "C = \\cos^2 \\phi_1 + 2 n \\sin \\phi_1 ", "b1e4e90d7eec86101a8dcf71fa2a10c0": " \\mathbf{p} = m\\mathbf{v},\\quad \\mathbf{L} = \\sum_{i=1}^n m_i (\\mathbf{r}_i-\\mathbf{R})\\times \\frac{d}{dt}(\\mathbf{r}_i - \\mathbf{R}),", "b1e5752bf1f2ab884c5d55ee9969f8cf": "\\nu > 1", "b1e60da4a6cb9d279d19cef8d9bf317a": "2\\pi \\log\\left( \\frac{G(1-z)}{G(1+z)} \\right)= 2\\pi z\\log\\left(\\frac{\\sin\\pi z}{\\pi} \\right)+\\text{Cl}_2(2\\pi z)", "b1e66d475d9974a93e6ddbd0cec60420": "\\overline{A_{1} A_{2} A_{3}}", "b1e698450bf04a347db3ed9dfbe61e10": "g_{ij} = \\rho(|| \\mathbf x_j - \\mathbf x_i ||)", "b1e6ef73ebfa7ee485f32dc580647859": "b+c+\\dotsb = 1", "b1e6ef928fdf08248de14c989f50e783": "\\operatorname{mwnchypg}\\left((x_1,\\ldots,x_{c-1}+x_c); n,(m_1,\\ldots,m_{c-1}+m_c), (\\omega_1,\\ldots,\\omega_{c-1})\\right)\\, \\cdot", "b1e6f16620e2faaaf21b39dca1a9d163": "Pr(X n", "b1e9e7f9951f537496698cefd55de63e": "\\{\\mathbf{x} | \\mathbf{d}^T\\mathbf{x} + \\beta > 0\\}", "b1e9fc74c575e610dbb5dfff059ade61": "\\mathcal{L}(\\phi_1, \\phi_2, \\ldots, \\phi_n) = \\sum_{i=1}^n \\sum_{j=1}^n g_{ij} \\; \\mathrm{d}\\phi_i \\wedge {*\\mathrm{d}\\phi_j}", "b1ea44ae4f7defb2c277035f8624b2cc": "P=\\Delta^\\dagger\\begin{pmatrix}f&0\\\\0&f\\end{pmatrix}\\Delta.", "b1ea751d8e5969d16820ed614d40b152": "\\sqrt{X}", "b1ea8af760925dbc2f44d6421cf3343d": "\\boldsymbol{v}_g", "b1ea9003a01ef9b73653b820ae79c8af": " X = \\{\\,(x_1,\\cdots,x_p) \\in G^p : x_1x_2...x_p = e\\, \\} ", "b1eaa22cd41542bc9290f4c16b3d92f0": "\n \\sigma_{11} = -p + 2~\\lambda^2~\\cfrac{\\partial W}{\\partial I_1} = \\sigma_{22} ~;~~\n \\sigma_{33} = -p + \\cfrac{2}{\\lambda^4}~\\cfrac{\\partial W}{\\partial I_1} ~.\n ", "b1eb3882c9acde2a91faf366c6dfa6ca": "\\frac{V-(k+\\lambda)}{\\sqrt{2(k+2\\lambda)}}\\to N(0,1)", "b1eb5211d6032eed5816f7f80e972751": "f_\\mathrm{obs} = f_\\mathrm{rest}\\sqrt{\\left({1 - v/c}\\right)/\\left({1 + v/c}\\right)}", "b1eb6c159179669cf674c8a61b57c356": "\\frac{\\mathrm{d}^2 x^a}{\\mathrm{d}\\tau^2} = 0", "b1ebf2e8036f8c123707148f4d861b62": "\n\\begin{align}\n\\biggl( \\sum_{k=1}^n a_k^2\\biggr) \\biggl(\\sum_{k=1}^n b_k^2\\biggr) - \\biggl(\\sum_{k=1}^n a_k b_k\\biggr)^2 & = \\sum_{i=1}^{n-1} \\sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 \\\\\n& \\biggl(= \\frac{1}{2} \\sum_{i=1}^n \\sum_{j=1,j\\neq i}^n (a_i b_j - a_j b_i)^2\\biggr),\n\\end{align}\n", "b1ec5bb4a4e7660c7ac21b7cfb839973": "\\delta Q = \\sum_{i=1}^N E_i\\,dp_i", "b1ecbefbe45745f00635340a1a1efce7": "r_{yy}", "b1eceabd39c6939e5d3e3993cc22cbbd": "\\Phi_E = \\iint_S \\mathbf{E} \\cdot d\\mathbf{S}", "b1ed9d604c1dcce653fc963e0d98eb6c": "\\lim_{n \\to \\infty} x_n = x", "b1edb8fa2644df0ed0e07500d13daf49": "\\mu\\;", "b1edc9b4b586fdfc381ad4c3b73f3d26": "\\bar{\\mathbf{y}}_j", "b1edcd07f10b0791c3a972b97365c201": "\\varphi_1\\neq\\varphi_2", "b1ee2c791a1dc32d3165194928996c6b": "e^{j2\\pi f_0\\cdot t}\\,", "b1eedb98b9ee8b1200498441d7773c79": "y(t) = \\lambda f(t)", "b1eee0229f3c7f7bbbcffdd75ff3d790": "I_{\\rm sp} \\,", "b1ef14547f9a5a6b2e766c10a31fe0f0": " 2u^2y+2\\alpha y + y^2 ", "b1ef1f3d41d005af7726bb2ec5272e08": "a = t_0 < t_1 < \\cdots < t_{k-1} < t_k = b", "b1ef865938acae145379723fa0e85ab3": "\\frac{\\Delta y}{\\Delta x}=f'(x)+\\varepsilon ", "b1efd729aece47dede501a528db58ede": "a_0,\\ldots, a_{N-1}", "b1efdc12bb1749dfe6c952d79ff76d6c": "f'(x) = 1 + \\frac{4}{3}x^\\frac{1}{3}.\\!", "b1eff30fb2ae3bd07db91a92e83f296e": "\\lambda_i = \\frac{2L}{n_{i}},", "b1f00291547da5086e09333e5822cc47": " I_1(a,b) = 1 \\, ", "b1f04df7494c3d50652e562ccdbcc9aa": "H: G \\sim N(0,\\frac{N_{0}E}{2})", "b1f0b2d37ccfdfa7344a1d3a0c90e5f5": "\\mathrm{eFG\\%} = (\\mathrm{FGM} + \\mathrm{.5*3FGM}) / \\mathrm{FGA}", "b1f0ba72c1b12cf59c528bedb61f5a7d": "T = \\sum_{i=1}^r \\lambda_i \\, v_i\\otimes v_i.", "b1f1298ffb25acf6de2a38d9063886c1": "2xy", "b1f1b1897e779ca7f6c1bae69f3c3749": "M_{biomass}(t) = M_{biomass} (0) e^{-kt}\\,", "b1f1bcb6a3e691cc3e86f1451f86f849": "|\\psi\\rangle = \\int \\psi(\\mathbf{x},t) |\\mathbf{x},t\\rangle \\mathrm{d}^3\\mathbf{x} ", "b1f1f8583be8908ffaa737a0df011cca": "\\text{Changing ohms from one kva base to another:}", "b1f20f45bc2900a378757926256231f9": " \\psi_2 = \\angle EVC, ", "b1f22668cf08f8070391d898e41f6a26": "s(n,k) = (-1)^{n-k} \\left[{n \\atop k}\\right] .", "b1f235014f9731dffc8286217bd7776b": "|x+y| \\le \\max(|x|,|y|)", "b1f2581825b726570e0cd345b0dec2ed": " \\Pr(T < T_i|\\theta) = F(T_i|\\theta) = 1 - S(T_i|\\theta) .", "b1f25d5ac356789e6b2397b5e5c8a446": "h_{t} \\,", "b1f288d01dcb8084da070b3db9bdd8df": "\\sum_{n=N}^{M} \\frac{x^n}{n!}", "b1f29830b25cd1b87c4846bf0d6afc84": "r = k [A][B] = k'[A] \\, ", "b1f3716a870a37d36d7e853f60364c2c": " \\mbox{pefsu} = \\frac{\\mbox{number loaves of bread (or jugs of beer)}}{\\mbox{number of heqats of grain}}", "b1f39f4cec5f8d4c568a19de1672d1e3": " \\kappa = \\sqrt{ \\frac{ \\sqrt{ \\epsilon_1^2+ \\epsilon_2^2}- \\epsilon_1}{2}}.", "b1f3b8f45ac4dcc3b3c01af2feac6f4d": "\\nabla \\times (f \\vec v) = (\\nabla f) \\times \\vec v + f (\\nabla \\times \\vec v", "b1f3c3d711c6e4f1a53ecc26fe374c3d": "\\text{factorial} = [\\![(1,\\times),(g, p)]\\!]", "b1f418855a4271604c8a9d397615df0e": "\\begin{align}\n\\operatorname{var}(X) &\\equiv \\sigma^2_X\\\\\n\\operatorname{var}(X_1+X_2) &\\equiv \\operatorname{var}(X_1) + \\operatorname{var}(X_2)\\\\\n\\operatorname{var}(cX_1) &\\equiv c^2 \\, \\operatorname{var}(X_1)\n\\end{align}", "b1f450a630478345ee58a042ab294aeb": "(gate8)", "b1f48ef7e2aad94f81a71f382341fcc1": "\n\\sigma_t =\\frac{8 \\pi}{3} \\left(\\frac{\\alpha \\hbar c}{m c^2}\\right)^2 = 0.66524574 \\ldots\\times 10^{-24}~\\textrm{cm}^2=0.66524574 \\ldots~\\textrm{barn}\n", "b1f4ae3617fa709a2defea574b76b1dd": "\n\\alpha_{ij}\\equiv -2\\sum_{k>0} \\frac{\\langle \\psi^0_0 | \\mu_i | \\psi^0_k \\rangle \\langle \\psi^0_k | \\mu_j | \\psi^0_0\\rangle}{E^{(0)}_0 - E^{(0)}_k}.\n", "b1f4eb6cb3d66c6e73532534df74ba3e": "\\scriptstyle{\\varepsilon_1(t)}", "b1f5408e7336c92f550164d35832ea71": "3^{27}", "b1f54b6fc512fa446f17dd790683d9b3": "\nL_2=\\frac{1+t}{1-t},\\qquad \nL_4=\\frac{1+t_2}{1-t_2},\\qquad \nL_8=\\frac{1+t_4}{1-t_4}\n", "b1f58065e719ead02b97fd16c9e3e954": "\\frac{(x - X_c)^2}{a^2}+\\frac{(y - Y_c)^2}{b^2}=1", "b1f6640317aea7a3341c2a6745b691e6": "{\\tilde{B}}_{n-1}", "b1f6646554b6b837c39855f0781eea38": " u(n\\Delta{t}, i\\Delta{x}) ", "b1f699ff7d694eafcf154fa86ff894ac": "\\omega = \\sqrt{\\frac{g}{L}}\\,\\!", "b1f6accaf73cf3442bd72b00315c9433": " \\mu_k(p^a) = (-1)^a \\binom{k}{a} \\ ", "b1f6bd8037d80086cabd3f7746b9ba5d": "U_{\\texttt{age}}\\,", "b1f6bd814b709b6f61220c0e3b211fa0": " O(h^{p+1}) ", "b1f7437ef0ee5d3d5cb11d3e554360e9": "\\langle c, 0 \\rangle", "b1f75fb0884331aec1d974b44e9c1381": "f(x) = \\|x\\|\\varphi(x).\\,", "b1f7c5ab9e2a6bd63e6e15655fdaca29": "x_i \\gets y_j+y_k", "b1f7d274d75d55f60cbbb27ab974d006": "(a{\\uparrow})^n(x)", "b1f83af4f8d7121ce44b4b7cc219a794": "100\\frac {\\bar{X}_m - \\bar{X}_f}{\\bar{X}_f} ,", "b1f885bb117d384b4281ceabf2c528eb": "\\sum_{d|n}\\mu(d)=i(n)", "b1f897c26cc33fc7caebf27e62480ad6": "\\pi_{\\mathbf R}\\colon{\\mathbf R}\\to M\\,", "b1f92e93e26ebed1e52dec9f85b3e5a2": "O = \\frac {A}{4\\pi(\\frac{3V}{4\\pi})^{\\frac 2 3}}", "b1f932f5fa7af9a41b172e75de92fc7e": "\\frac{v^2}{2}", "b1f93d12f4cd740025c75bfa6025d43d": "\n\\mathbf{A}(\\mathbf{r}, t) = \\frac{\\mu_0}{4\\pi} \\int q\\mathbf{v}_s(t') \\frac{\\delta(t' - t_r')}{|\\mathbf{r} - \\mathbf{r}_s(t')|} \\, dt'\n", "b1f9dc931ec1d2272c2754c60014e5aa": "\\Psi_{mn} = \\mathrm{exp} -\\left[ \\frac{U_{mn} - U_{nm} }{RT} \\right]\\mathrm{\\,;\\,\\,}", "b1fa24da2f834e39ecc6c42ec7ba753c": " \\tilde D", "b1fa4cf8624d0c78e088abc9ddc9c461": "\\epsilon \\, \\gamma_{ab}", "b1fa555384f65a24b230c0a236d35e60": " \\diamondsuit\\left(\\sum{_{\\exists\\exists}}(\\mathbf{A},\\mathbf{b})\\right)=[\\underline x_1,\\overline x_1]\\times \n[\\underline x_2,\\overline x_2]\\times ... \\times [\\underline x_n,\\overline x_n] ", "b1fa7c43f3d4a606961577d38aeba405": "(S, \\Lambda, \\rightarrow)", "b1fa8227a5e4842457d24514a0cd63f9": "\\phi_x(t) = M_X(it)", "b1fa983ad95ffd2e9ddb787d91e0d351": "\\frac {\\theta_c} 2", "b1faae42f19015a24b341eb75a33c61a": "x^{\\underline n}=\\frac{x!}{(x-n)!}=x(x-1)(x-2)\\cdots(x-n+1)", "b1fadb71b95b34e306e9f3860d9aa478": "\\Delta m = \\Delta E/c^2.", "b1fae97c8f8415c96a91e034eea14c12": "(17)\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;\\; \\rho_1 u_1^2 + p_1 = \\rho_2 u_2^2 + p_2 ", "b1faeaa1797dd4acc12d9e88a591e309": "\n\\nabla \\cdot \\mathbf F =\n\\frac{1}{h_1 h_2 h_3}\n\\left[\n\\frac{\\partial}{\\partial q^1} \\left( F_1 h_2 h_3 \\right) +\n\\frac{\\partial}{\\partial q^2} \\left( F_2 h_3 h_1 \\right) +\n\\frac{\\partial}{\\partial q^3} \\left( F_3 h_1 h_2 \\right)\n\\right]\n", "b1faec8989142cab1ca8f92586a5b502": "\\tfrac{385}{1539}", "b1fb42cd865768cdad76bda300f4d495": " M_3 = \\left\\{\\frac{1}{2}(1,-i,-i, -1),\\frac{1}{2}(1,-i,i,1),\\frac{1}{2}(1,i,i,-1),\\frac{1}{2}(1,i,-i,1)\\right\\} ", "b1fb821f353d9fea7e3ca838d3595c9f": "I = I_0 \\sin \\left(\\frac{4\\pi (x_0 + v \\Delta t)}{\\lambda}\\right) = I_0 \\sin \\left(\\Theta_0 + \\Delta\\Theta\\right)", "b1fbc451252848e20abc1cc93fce6aa2": "a=rs^m", "b1fbda6e68ca9bdc5e6f598b0a304c24": "a_1:=(-1)^{i-1}t_i;\\quad b_1:=(-1)^i s_i;", "b1fc00b2ad5520e423fd6afa1d759206": "\\phi_{i,1 \\dots N}", "b1fc24c28e6377697e945ca807ad904f": "z^2=x^2+y^2", "b1fc36b4ff6be3cf1362e30c96b0e59a": "P_9", "b1fc620fbe597b6ac8ac9d1d79c42f9d": "-H(\\sigma)-H(\\sigma')", "b1fc8dfc4e5c858e86f1e3398153d305": "F_k = \\frac{\\partial}{\\partial \\lambda_k} \\log Z(\\lambda_1,\\dotsc, \\lambda_m).", "b1fc93fc4e5fa16d5ddbe2b6fdb629bf": " \n \\begin{align} \n \\hat\\alpha_\\mathrm{mle} & = 34.09558\\\\\n \\hat\\beta_\\mathrm{mle} & = 31.5715\n \\end{align}\n", "b1fcab8bdfebf3cfa947d55d1af391ce": "\n\\mathbf{x}_{m+1} = \\mathbf{A} \\cdot \\mathbf{x}_{m}\n", "b1fcd737282cb84aa5367d915f807299": "\\dot{x}=\\kappa\\left(-2Cp+y-(i\\Theta+1)x\\right) ", "b1fcd777cdd594dbe5f39dd36d7115d5": "\\frac{\\partial \\ln \\Beta(\\alpha,\\beta)}{\\partial \\beta}= - \\frac{\\partial \\ln \\Gamma(\\alpha+\\beta)}{\\partial \\beta}+ \\frac{\\partial \\ln \\Gamma(\\alpha)}{\\partial \\beta}+ \\frac{\\partial \\ln \\Gamma(\\beta)}{\\partial \\beta}=-\\psi(\\alpha + \\beta) + 0 + \\psi(\\beta)", "b1fd03e2504877df90bad3a13332663d": "\\rho_{XB}", "b1fd4bd93e8643f5e116cb9863c31241": "L(x) = \\frac{\\sum_{j=0}^k \\frac{w_j}{x-x_j}y_j}{\\sum_{j=0}^k \\frac{w_j}{x-x_j}}", "b1fd4fb76313c6aba3a05b03811d5bfa": "n \\ge2", "b1fd6744d0fd3617860b368904d167ee": " \\frac{2\\omega\\kappa_\\omega}{u_{ph}} = (\\frac{2n_{e,c}e_c^2\\langle\\langle\\tau_e\\rangle\\rangle\\omega}{\\epsilon_\\mathrm{o}\\epsilon_em_eu_{ph}^2})^{1/2} ", "b1fd941bab0603984169f8f36d51e8bb": "\\psi = \\frac{P}{\\gamma} = \\frac{P}{\\rho g}", "b1fdb508c74de5bdd32369959f61039c": "\\int \\frac{\\ln x}{x^2}\\,dx.\\!", "b1fdfef99e5c6ef51fed9b3498ff7760": " A_{\\mathrm{G}} = \\lambda_{\\mathrm{B}} (1-r_{\\mathrm{av}}) A_0 ", "b1fdffdf4d2af63633a9859e2ee49505": "H_\\lambda(i,j)", "b1fe34d978f668189f7cae96ddcccdbe": "T_J", "b1fe39510205f2b487442a5f6ccc27f9": "\\mathbf{(mn)(nm)} = \\mathbf{mnnm} = \\mathbf{mm} = 1", "b1fe3e27835112e8f6c41dbc3e4402c8": "\\mathrm{length}(c_k)", "b1fe5415bec5f5f55fd5f1b184d64cc2": "\\mathcal{N}\\,", "b1fe79806fcbe3689272ab9fba138a95": "\\rho(\\mathbf{r}, t)", "b1feacdec6350958e80d9ebf162f9de4": "\\Pr\\left(\\bar{X} - \\frac{cS}{\\sqrt{n}} \\le \\mu \\le \\bar{X} + \\frac{cS}{\\sqrt{n}} \\right)=0.95\\,", "b1febd412fac8fde3ae610f13e8d2d61": "\\mathbb{C}\\backslash\\{0\\}", "b1fec2bc4b826a3c8e96a2bbee131b48": "\n\\begin{align}\n\\mu'_x & = \\sin\\theta\\cos\\varphi \\\\\n\\mu'_y & = -\\sin\\theta\\sin\\varphi \\\\\n\\mu'_z & = -\\cos\\theta \\\\\n\\end{align}\n", "b1feef7a1f1cc65c020ea6a0f79e4b4d": "g(\\eta,t,x) = q(\\eta,t,x) = 0 ", "b1ff20c432d59f26609b3bc470da1757": "e^t<1", "b1ff4527c0ed6231114895fcea0a8fdc": "\\scriptstyle P'_E", "b1ff56da9f5c8286aa017516e9904a59": "\\textstyle{\\left(\\sum_i V_i\\right)^\\perp = \\bigcap_i V_i^\\perp}", "b1ff602a4b91ff11580a306ddc8ff327": "\\lim\\limits_{x \\to 0^+} \\frac{1}{x} = +\\infty\\text{ and }\\lim\\limits_{x \\to 0^-} \\frac{1}{x} = -\\infty.", "b1ff92f901f2028f6cbe9efcd0eeeec5": "\\ \\Delta\\ K=I- \\delta\\ K ", "b1ffc5901f7eb768f7f634fb2afd6895": "O(x^n)", "b1ffcfb867890c608537015015e371aa": "0<\\lambda(T,W)<1", "b2000f4eb8c429f1d7c8a3e49ecc4b7b": "\n-w_k\\prod_{j\\ne k}(z_k-z_j)=-w_kg_{\\vec z}'(z_k)=f(z_k)\n", "b2003e3d15e7e1bc137de4befed7c280": "\\qquad \\qquad k_p = k_{p,S} = \\frac{3.1\\times10^{12}\\langle M\\rangle V_a^{1/3}T_{D,\\infty}^3}{T\\langle\\gamma_G^2\\rangle N_o^{2/3}}\\qquad \\mathrm{\\ high\\ temperatures}\\ ( T > 0.2 T_D,\\mathrm{\\ phonon\\mbox{-} phonon\\ scattering\\ only)},", "b20052d5de37f4dd63c45a542f4ea0ae": "2\\times 2^n", "b2009a6e77cc5365348b95642e0da135": " {2n \\choose n} \\sim \\frac{4^n}{\\sqrt{\\pi n}}\\text{ as }n\\rightarrow\\infty.", "b200dae6e79fc0362ab95748e05f7213": "\\textbf{z}_{k} = \\textbf{H x}_{k} + \\textbf{v}_{k}", "b200f725a7040fe60683d6075c2a0ae6": "q_{t+1} - 3q_{t} + q_{t-1} = 0 \\mod N", "b20107c122924833da223e401c1c288b": "b-a.", "b2011e1e2e6a7918776035da9ac57af7": " \\langle B,E,G,C,E,B \\rangle ", "b201419564b9265d2dabf0eefd1cdff6": "x_n = \\frac{1}{N} \\sum_{k=0}^{N-1} X_k \\cdot e^{i 2 \\pi k n / N},", "b20151dab6478e35d387a51ea61c97a0": " \\mathbb{E} |X_t| < \\infty ", "b201ddd787b17688df70cf50b5778960": "a^2 = 2 b^2", "b2020e5cce821e04213ecc278c47a5fc": "\\delta:\\mathfrak{g} \\to \\mathfrak{g} \\otimes \\mathfrak{g}", "b2023e0bd2b7956c9aed41785ce5e034": " r^{\\frac{3}{4}} \\exp(\\tfrac{3 \\pi i}{4}) (3+r)^{\\frac{1}{4}} \\exp(\\tfrac{2 \\pi i}{4}) = r^{\\frac{3}{4}} (3+r)^{\\frac{1}{4}} \\exp(\\tfrac{5 \\pi i}{4}).", "b20258f4fb410232643efd47a1dabf3b": "F(x)=x^2\\sin\\left(\\frac{1}{x^2}\\right)", "b20271d7c6e068ef9bef6d5bb7957dc6": " {\\partial x^\\lambda \\over \\partial X^\\mu}", "b202788231f9b0e58d6e4dd971a84c88": " m = {f_1 \\over S_1} \\,.", "b202a2317b546d7549e4729e334c2af2": "h = C_p T", "b202b0d8bb6f41aaaa460bbf83924153": "\\left|\\alpha - {p_i\\over q_i}\\right| < {1\\over \\sqrt5 q_i^2}, \\qquad\n\\left|\\alpha - {p_{i+1}\\over q_{i+1}}\\right| < {1\\over \\sqrt5 q_{i+1}^2}, \\qquad\n\\left|\\alpha - {p_{i+2}\\over q_{i+2}}\\right| < {1\\over \\sqrt5 q_{i+2}^2}.\n", "b202fdf1fe5f27f110e35672c0982fc2": "+2370''\\sin(2D)", "b20319359c1431deff0a2bbe3eafdcaa": "a_n \\sim \\frac{B(r)}{r^{\\alpha} \\Gamma(\\beta)} \\, n^{\\beta-1}(1/r)^{n}\n= \\frac{-1/2}{(1/4)^1 \\Gamma(-1/2)} \\, n^{-1/2-1} \\left(\\frac{1}{1/4}\\right)^n\n= \\frac{n^{-3/2} \\, 4^n}{\\sqrt{\\pi}} \\,.", "b203240f3e9896c821211dedf0849f67": "R/r", "b20342605d831fd595e704906e7ae7ad": " m_e \\omega^2>>{e^2\\over 16\\pi\\varepsilon_0 Z^3},", "b2036acd90655104e017d3541e3e5bdc": " | \\Psi (t) \\rangle ", "b203eee69fd72eae59e0a64109e8774e": "p_{i_{j}} = (L, R)", "b2043bb379e1e4b7c4561419eb3d0dd1": "q = \\left\\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\\right\\rangle \\ . ", "b204bc307b1d966e397beb097d1e0259": " [ai + k + 1 - l - i]^2 - ", "b204fac838376fec1922434afe97ae54": "\\nabla \\vec X = X^a{}_{;b} \\frac {\\partial} {\\partial x^a} \\otimes dx^b = (X^a{}_{,b} + \\Gamma ^a _{bc}X^c) \\frac {\\partial} {\\partial x^a} \\otimes dx^b", "b2050205fbfed08fd910138dcfb3f0d2": "\\mathfrak{T}^{\\mu \\dots}_{\\nu \\dots}", "b20513a448e84e26c8111ae1955609da": "\\Psi_M:\\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}\\to\\mathbb{\\hat{C}}\\setminus M", "b20613f10658c6386d37b6c4fcf0f304": " B_{2p} = -\\frac{(2p)!}{2^{2p} - 2} \\begin{vmatrix}\n1 & 0 & 0 & \\cdots & 0 & 1 \\\\\n\\frac{1}{3!} & 1 & 0 & \\cdots & 0 & 0 \\\\\n\\frac{1}{5!} & \\frac{1}{3!} & 1 & & 0 & 0 \\\\\n\\vdots & & \\ddots & & & \\vdots \\\\\n\\vdots & & & \\ddots & & \\vdots \\\\\n\\frac{1}{(2p+1)!} & \\frac{1}{(2p-1)!} & \\frac{1}{(2p-3)!} &\\cdots & \\frac{1}{3!} & 0\n\\end{vmatrix}", "b20656fa3a475aa605fd62b4e73b95f8": " \\langle E^2 \\rangle - \\langle E \\rangle^2 = k T^2 \\frac{\\partial \\langle E \\rangle} {\\partial T} + k T \\mu_1 \\frac{\\partial \\langle E \\rangle} {\\partial \\mu_1} + k T \\mu_2 \\frac{\\partial \\langle E \\rangle} {\\partial \\mu_2} + \\ldots ,", "b2067a22573d99ae5954bf3d70115de4": "R_{\\mathrm{K}} = h / e^2 \\,", "b20683b8a9c62bac5bc089cac1247e00": " b = ( \\frac{1}{2Q})", "b20697f2fb554b2ce2722b222d5574ea": "m_{od} \\;", "b206b941551da550a3697202dbf0afac": "I_{k} \\equiv \\frac{q}{a^{k+1}}", "b206d8f14e7f754906a0faf0800670c3": "\n\\frac{S(T_2)}{S(T_1)}=\\frac{S_b(T_2) \\left (1-1/\\rho_i \\right)}{S_b(T_1)}\\frac{\\rho_b(T_2)}{\\rho_b(T_1)}\n\\exp \\left \\lbrace \\frac{c}{\\rho_i \\left[S_b(T_1) - S_b(T_2)\\right ]} \\right \\rbrace\n", "b206da1b6347e05a6b0dd59f8aa1e8a3": "{ \\dfrac{p_{11}/(p_{11}+p_{10})}{p_{10}/(p_{11}+p_{10})} \\bigg / \\dfrac{p_{01}/(p_{01}+p_{00})}{p_{00}/(p_{01}+p_{00})}} = \\dfrac{p_{11}p_{00}}{p_{10}p_{01}}.", "b20705db889eeb63a290eb51f43d32dd": "x = 10 + 2 \\epsilon_3 - 5 \\epsilon_8", "b2070fdd2cda831dc3a53cdd6ca282aa": " \\sum_{i=1}^{n}P_i.D = \\sum_{i=1}^{n}P_i.S = 0. ", "b2073596eaf5fc725d3b4594ec4c3bf4": "\\overline{R}^2 = 1 - \\tfrac{n-1}{n-p}(1-R^2)", "b2076343f79d0798a4310f9a2a9b1010": "P(B)\\cdot P(A\\mid B)", "b207aa939095073192d74a4607e30fb6": "(++++)", "b207b7c153b62dbaa84693ed99a01a3c": " | \\psi \\rangle = {1 \\over \\sqrt{2}} \\bigg( | 0 \\rangle - | 1 \\rangle \\bigg) ", "b208340165fd32ccb6e077aaa4de2cfc": "\\Delta S_{mix} = -nR(x_1\\ln x_1 + x_2\\ln x_2)\\,", "b208f6d3357690031fc6f919ea455c68": "k_s \\approx \\delta_\\nu\\,", "b2097887f8d828c8d8ac2df2d0a88993": "y_2 = \\frac{2y_1}{-1+\\sqrt{1+\\frac{8gy_1^3}{q^2}}}", "b209c950944a4e587b257275962a91dc": "q = a + bi + cj + dk,", "b20a05c79ef6217c0046068231c1bc29": " P\\rho(f)P = \\int_G f(g) (P \\rho(g)P) \\, dg", "b20a6b0f89d3b10ab2c2da917e4f28f9": "\\scriptstyle{\\chi_{i,j} = \\chi_{j,i}^*}", "b20aaad0861fb14ec7563e9d0c28e862": " R \\setminus \\{ 0 \\} ", "b20aee2bbc353e6df4146fd0a7c03a35": "{{Tonnage}} = \\frac {{Length}\\times\\ {Beam} \\times \\frac {Beam}{2}} {94} ", "b20b39a3c3965c4ecd315a175a6a4344": "TEE", "b20b7170f3549a378d62619b11b4cc9d": "\\sum_i w_i \\log \\left( m^{-1} q_i \\right) = 0", "b20b780ef47582aac5c479b87254cd51": "n_1,\\,n_2", "b20b7b16a4549cdf8c5678e2ec2abb08": " s = s(t) \\ . ", "b20b9b3015569b84eb7a8f4bec6215a1": "H^q(X,\\mathcal F)", "b20b9c95777ca8efbcd37b72add06018": "1_{FY}\\in\\mathrm{hom}_C(FY,FY)", "b20bbefee863c351bcf07689868ecb32": "u(t) = \\sqrt{u_{min}^2 + \\left ( \\frac{t-t_0}{t_E} \\right )^2}", "b20c0122cf206c84698d743b19f8ba55": "\nT_{\\pi,\\lambda}(x) = x - a_i + a'_i\n", "b20c21d03514156099a28ad2d125699f": " {\\partial D \\over\\partial y_k} = 0 \\Rightarrow y_k = { \\int_{b_{k-1}}^{b_k} x f(x) dx \\over \\int_{b_{k-1}}^{b_k} f(x)dx } = \\frac1{p_k} \\int_{b_{k-1}}^{b_k} x f(x) dx ", "b20c65ce46ed7e1a7bd4f5aafb9258f4": "x\\in P", "b20c834ff8e69befcd4fb3bc196a4e31": "\\mathbf{x}_1,\\ldots,\\mathbf{x}_N", "b20cbb7ac7113949c027dd07ebde2157": "L_n^{(\\alpha+m)}(x) = (-1)^m L_{n+m}^{(\\alpha)[m]}(x).", "b20d06226bc0444f1b75e70769ba1f6a": "\\lambda_{n-1}<0", "b20d183679ef26674d3562af5247a0d4": " \\rm{rand}", "b20d2735b038c894e4cb4f25863d8ea4": "\\displaystyle \\frac{2 (-i)^n T_n (2 \\pi \\xi) \\operatorname{rect}(\\pi \\xi)}{\\sqrt{1 - 4 \\pi^2 \\xi^2}} ", "b20dc76099cd35de6217ef94d2dc1c62": "r' = B^n x + \\alpha - (B y + \\beta)^n.", "b20dfc8aecb4a17bab2050ad4ea848e6": "\\frac{\\partial g(u)}{\\partial u} \\frac{\\partial u}{\\partial \\mathbf{x}} ", "b20e054aca26275d419c518e03249c8d": "|\\lambda_i| < d", "b20e1c03df2a61582a3f0402c3f890c2": "(Q_1 x_1, phi_1)...(Q_k x_k, phi_k)", "b20e57a9bc0bc985f0fb4aad53d4bf56": "r_i = \\frac{n_i}{n_{\\rm mix}- n_i} ", "b20e591cc01e2a02165395ccac8b8cfa": "\\nu_\\mu\\,", "b20e61b8871171d6e294e0ab7bbc46de": "\\mathfrak{P}^{55}", "b20f230f6ab87238b6f03a0e0fe86d99": "(\\mathbb{N}, [\\mathrm{zero},\\mathrm{succ}])", "b20f91b34a598af41f4330fa0f9e2df7": "k \\geq 9", "b20f94ef263b6ebc40f36163981f1d85": "Z_1=1", "b20f96435e630751bd82b89867944a46": " \\nabla _\\mu T_{M\\;\\nu }^{.\\;\\mu } =4\\pi\nf_\\nu \\left( \\phi \\right) T_{M\\;}^{\\;}=f_\\nu \\left( \\phi \\right) \\Box \\phi", "b20fa74827e027ca6292b771765ca7f4": "\nQ_\\text{BP} = n \\sum_{k=1}^h \\hat{\\rho}^2_k,\n", "b20fb0b62ef5ba81415e25a5df7f3f1c": "\\Gamma(\\omega)\\phi(\\omega)", "b2100c1f880372ac491582d41fc0b296": "K_1 = \\frac{2\\mu\\sqrt{2\\pi}(4V_B-V_C)}{(k+1)\\sqrt{L}}", "b2100e0b0a770000ba9ce9c14956a990": "f(x,y)=e^x\\log(1+y).\\,", "b210194abacd83044f9e8696b8d31cd3": "S = (S_1,S_2,\\dots,S_k)", "b21041d60e8c933b2b7cafe39826efe2": "C_{16} = G_{12} + G_8 \\cdot P_{12} + G_4 \\cdot P_8 \\cdot P_{12} + G_0 \\cdot P_4 \\cdot P_8 \\cdot P_{12} + C_0 \\cdot P_0 \\cdot P_4 \\cdot P_8 \\cdot P_{12}", "b2105398a08fa66d3d57226a429dc911": "(0,0,0),", "b210625d08f5f266c08307c384a72e94": "p = \\lfloor 10^nx\\rfloor", "b21089caf00b179019196b5aa46f1e2f": "I_{\\text{S}}", "b210b83a37b79089e2ceabd3c2f787bd": "N(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{x} e^{-\\frac{1}{2}z^2}\\, dz", "b210c8e620d3a207cae3eec34e608045": "s= h(g_\\theta(z_1),\\ldots, g_\\theta(z_m))= \\rho(\\theta;z_1,\\ldots,z_m)", "b21110ea434f7c31293f606d1b7c0f9e": "\\eta_{Receiver}", "b2119e600bcd51107bc1a081b8c9224d": "(\\beta=v/c)", "b211e75f0ce7a52b9b0039d73e900e6a": "f=\\frac{100}{2\\pi} \\sqrt{\\frac{2p\\cdot M_h}{J_r} }", "b21211e24fd00411d9b68b493b24be04": "K_{I}", "b2123fa349f965e11612dd74cbcd10cf": " \\left(\\frac{1}{2}+o(1)\\right) n^{3/2}", "b212555f2264191f6c69b4fdd8c14175": "\\int \\mathbf{J}\\cdot d\\mathbf{A} = \\oint \\mathbf{H}\\cdot d\\mathbf{l}", "b2129932d6760fcf09fd4d722c70a691": "x_k[n] \\ \\stackrel{\\mathrm{def}}{=} \n\\begin{cases}\nx[n+kL] & n=1,2,\\ldots,L\\\\\n0 & \\textrm{otherwise},\n\\end{cases}\n", "b212c1efc7215e7926546ed7ef9d6b8f": "|\\bar{r}|=\\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}", "b2131c0195ed3425a71ed9b3df795a9a": "|\\Pr\\{A^{r}(f(r)) = 1\\} - \\Pr\\{A^{r}(R) = 1\\}| \\leq \\epsilon(n)", "b2131ee8f5de9aa3f7ef57b553852ca2": "SS_{tot}", "b2134cdfad79a5f2ebd1a9b81403f3df": "{1\\over U_{o}} = {R_{fo}} + {R_{fi}}\\cdot \\frac{D_{o}}{D_{i}} + \\frac{D_{o}}{2k_{w}}\\cdot \\ln {\\frac{D_{o}}{D_{i}}} + {1\\over h_{o}} + {1\\over h_{i}}\\cdot \\frac{D_{o}}{D_{i}}", "b213b23490cd9b3004a46503d415b35b": "\\beta(\\varphi,\\psi) = B(\\varphi)(\\psi).", "b213f2579119ee72fde52aaef9577fc0": "\\sum_{\\ell,\\ell'}", "b2140ec20466e975e1b1ae57cbfad9af": "-1>0", "b2142506231a217404725257845a00d7": "u^{*}_{z} = \\frac{e_s - e_i}{e_i} \\eta N_i \\bar{r_i} \\,", "b2145aac704ce76dbe1ac7adac535b23": "var", "b21463b7bf495a414913585c1a0d37cd": "dRA = d\\psi*cos(\\epsilon)", "b21480ca995fe8716a1d282fe0645cda": "W_i^* = 14 \\left(1 - \\frac{0.894}{\\phi^{0.5}}\\right)^{4.5}", "b2149e9abd7900e7222ad3a1eae87d29": "N(r)=1+4\\sum_{i=0}^\\infty \\left(\\left\\lfloor\\frac{r^2}{4i+1}\\right\\rfloor-\\left\\lfloor\\frac{r^2}{4i+3}\\right\\rfloor\\right).", "b214d15cf7204fcc59d3a8f02e97e4df": "\\begin{matrix}{4 \\choose 1}^2{32 \\choose 2}\\end{matrix}", "b2152103751168ca9e318ba71b86cec1": "B_qe^{j\\phi_q}", "b2152cb2095e216ce109776f7bbe0b1c": "\\exp(-\\lambda_i t)", "b21564df8fb015c5b57cb0b7b6e2d1c4": "\\begin{align}\nTu &= u|_{\\partial\\Omega} && u\\in W^{1,p}(\\Omega)\\cap C(\\overline{\\Omega}) \\\\\n\\left\\|Tu\\right\\|_{L^p(\\partial\\Omega)}&\\leq c(p,\\Omega)\\|u\\|_{W^{1,p}(\\Omega)} && u\\in W^{1,p}(\\Omega).\n\\end{align}", "b21569633aaf568b07a5359bd138235a": " W_1", "b2156b9f335324beccf1ab70ee909b5c": "2v_1 + 3v_2 -5v_3 + 0v_4 + \\cdots", "b215721e69df92b9ba1960fd25eec227": "r+s-1", "b21597bf63359679da5c540620dce28c": "\n\\begin{align}\na_0 & = & m - q_0 k , & & q_0 & = & f\\left(\\frac m k \\right) & \\\\\na_1 & = & q_0 - q_1 k , & & q_1 & = & f\\left(\\frac {q_0} k \\right) & \\\\\na_2 & = & q_1 - q_2 k , & & q_2 & = & f\\left(\\frac {q_1} k \\right) & \\\\\n & \\vdots & & & & \\vdots & & \\\\\na_n & = & q_{n-1} - 0 k , & & q_n & = & f\\left(\\frac {q_{n-1}} k \\right) & = 0\n\\end{align}\n", "b215acfd5900e3efa5be2c6f9258d964": " \\textbf{m} = c\\textbf{f}_p \\cdot \\textbf{g}+c\\textbf{f}_p \\cdot \\textbf{f}-q\\textbf{f}_p \\cdot K \\pmod p ", "b215c0686d2ea5feb08b88778c09c173": "z_0=(9.0, 0.0)", "b2162024070024f4d5f1a0367527f73b": "\\scriptstyle \\delta t_{\\text{clock,rec}} \\;=\\; \\tilde{t}_{\\text{rec}} \\,-\\, t_{\\text{rec}}", "b216483921d32c23e44ffca5081fbcfb": "C/N", "b2164b7e33e4ef00732070a0b76908dd": "{\\Gamma (\\mu (k+n)+1)}", "b2166b22eb366c80950a66d879fb4c50": " J(x)=\\sum_{n=1}^{\\infty}\\frac{\\pi(x^{1/n})}{n}", "b2168391c4d41fb6cb18ee257fbc857f": "f: X' \\rightarrow Y", "b2168860562f7c26755bf402bb0b8aa6": "G\\times G'", "b216d0e9c6deb03a9ea73b770d64b394": "\\,C_0 = \\sqrt{2C_p\\,T_{01}\\,(1 - \\frac{p_3}{p_{01}})^{\\frac{\\gamma - 1}{\\gamma}}}", "b2176dc8bf55f8f189b6e79f02501082": " {\\hat p}", "b217a57fa7676efeba0a44c6eb92245b": "(p_x, p_\\lambda)", "b217e9804667f547d370052f9b605567": "\n\\Bigg(\\frac{q}{p}\\Bigg)_4 \\Bigg(\\frac{p}{q}\\Bigg)_4 =\\Bigg(\\frac{ac-bd}{q}\\Bigg).\n", "b2180b02b7c329f6492554c77e28afb6": "E\\left[ x_i^2 x_j^2\\right] = \\Sigma _{ii}\\Sigma_{jj}+2\\left( \\Sigma _{ij}\\right) ^2", "b218de4a60594ad8624171994d30c5ab": "r_i = \\displaystyle\\sum_{k=1}^{n} \\nu_k R_k \\mathrm{\\,\\,;\\,\\,} q_i = \\displaystyle\\sum_{k=1}^{n}\\nu_k Q_k ", "b218fabcfac4887d034666cfc477c0af": "x = (x_1,x_2,\\ldots,x_n)", "b2199848c9d5d3b03c071c785c4b6bc2": "\\dot{Q}''=\\kappa \\nabla T", "b219baabaaca7d9246089b1df361b7ab": "0 \\to M' \\to M \\to M'' \\to 0", "b21a4239849e2467517e97afb4197757": "2^n, n \\leq 6", "b21a6056ecfbbd926380d83199f5a621": "\\exp \\left\\{ {v_i } \\right\\}", "b21a96683e78df0a0494bc14cdc267e9": "\\frac{\\partial V}{\\partial p}\\ ", "b21a98c8319c874c691297dafff3c2fb": " \\eta = 0", "b21abfa62ce96369e7bc01628cbcf545": "[-1, 1]^2 + [-1, 1] = [0,1] + [-1,1] = [-1,2]", "b21add5c533e1a77236dc30f127629bc": "Ext^1", "b21b00f59a1570ff319ede55e4858e87": "10{,}36 \\%", "b21b01bc014573c9f4fb803cb577e95a": "\\nabla_uR(v,w)+\\nabla_vR(w,u)+\\nabla_w R(u,v)=0", "b21b2679b6dce21ef5b841dc0da4b624": "\\nabla \\cdot \\mathbf{E} = \\rho", "b21b2be1e14c9c561e89f52870860b89": "\\sum_{j=1}^n p_j x_j", "b21c3d901f618b503a36c7cbe21dc896": "\\mathbf C_i", "b21c440b5d60bcb5a9dc5972374c4b02": "t = \\langle \\alpha | M | \\alpha \\rangle", "b226276ac951b9ed159feeb9ce3bdcad": "E_{\\text{int}}^{\\text{ion}} = E_{\\text{int}}^{\\text{neutral}} + h \\nu - IE_{\\text{ad}}", "b226307d34927441fbfa72eccd0a5e51": "X\\to \\text{Spec}(k)", "b226d54e76eace18783a3a0448d11840": "\\scriptstyle \\int a \\,+\\, \\int b \\,=\\, \\int (a \\,+\\, b)", "b227a3d6f756b4fa6709610768e77429": "[z,x^{-1},y]=1", "b227e25a491edbca1893db95baa4d645": "\\mathrm{-C(=O)H}", "b227e9a1c083c04c10dce6d88de81ecc": "\\,\\Omega=d\\omega +\\omega\\wedge \\omega, ", "b227fd4b46a2e0d68301989814779099": "B \\wedge (t + \\alpha v - q) = 0", "b228628da5e8a985db45fc37321cef4f": "E(R_i)", "b2288c0ea50a746c7314e7b37e3ea3eb": "\\varrho(T_h) \\le a", "b228a029aabaf1a01df223283075acc1": "f\\left(a+\\frac{k^2(X-a)}{(X-a)^2+(Y-b)^2},\\ b+\\frac{k^2(Y-b)}{(X-a)^2+(Y-b)^2}\\right)=0.", "b228ab4861641cad1932b9006bc40a16": "i = 1, \\ldots, k", "b2291ad12bfe6b7fac6ff782505ee4fa": "r_2 = \\pm \\sqrt{p^2 - 4q}.\\!", "b2291be636b18bffe7133f77731f6377": "I_\\mathrm{min} \\propto ||C_A|-|C_B||^2 ", "b2294a0f3421d0a6380a504d6fb419d6": " \\mu_{e}=\\frac{\\mu_p}{[1-(1-\\frac{\\delta}{R})^4(1-\\frac{\\mu_p}{\\mu_c})]} ", "b2295e235ff62838a30a24bb3a0e0da8": "\\frac{\\int f(x_1)\\cdots f(x_{2N}) e^{-\\iint \\frac{1}{2}A(x_{2N+1},x_{2N+2}) f(x_{2N+1}) f(x_{2N+2}) d^dx_{2N+1} d^dx_{2N+2}} \\mathcal{D}f}{\\int e^{-\\iint \\frac{1}{2} A(x_{2N+1}, x_{2N+2}) f(x_{2N+1}) f(x_{2N+2}) d^dx_{2N+1} d^dx_{2N+2}} \\mathcal{D}f} =\\frac{1}{2^N N!}\\sum_{\\sigma \\in S_{2N}}A^{-1}(x_{\\sigma(1)},x_{\\sigma(2)})\\cdots A^{-1}(x_{\\sigma(2N-1)},x_{\\sigma(2N)}).", "b229738ccd293880d734dab039d14fb2": "\\pi/2 = 90^\\circ", "b2297d72ae9829fddf4284604c883b1a": "I = \\prod_{i=1}^k [0, \\ldots, n_i]", "b229a076bc4156bed0705fca44534300": " \\eta_{gen}=\\frac{W_{out}}{W_{in}}", "b229a3cb1adfc546f6288dad025941f0": "1-f", "b22a19e157e836d1087546cd4333f621": "\n\\begin{align}\nx(t) * h(t) &= O_t\\left\\{\\int_{-\\infty}^{\\infty} x(\\tau)\\cdot \\delta(u-\\tau) \\, \\operatorname{d}\\tau;\\ u \\right\\}\\\\ \n&= O_t\\left\\{x(u);\\ u \\right\\}\\\\ \n&\\ \\stackrel{\\text{def}}{=}\\ y(t).\\,\n\\end{align}\n", "b22a20ce4b43efb51fc1c1cd446015ae": "135 = 1^1 + 3^2 + 5^3", "b22a43a6f58d37ebc827de0bd5282307": "\n\\begin{align}\n\\frac{d\\mathbf{x}}{dt} & = \\mathbf{f}(\\mathbf{x},\\mathbf{z},t), \\\\\n\\mu\\frac{d\\mathbf{z}}{dt} & = \\mathbf{g}(\\mathbf{x},\\mathbf{z},t).\n\\end{align}\n", "b22a9d5fac34e3dd40942c5a81f04bfe": "\\begin{align}\n &A\\cos(\\omega_1t+k_1)\\cos(\\omega_2t + k_2) \\\\\n = {} &\\frac{A}{2}\\cos\\left[\\left(\\omega_1t + k_1\\right) + \\left(\\omega_2t + k_2\\right)\\right] + \\frac{A}{2}\\cos\\left[\\left(\\omega_1t + k_1\\right) - \\left(\\omega_2t + k_2\\right)\\right] \\\\\n = {} &\\frac{A}{2}\\cos\\left[\\left(\\omega_1 + \\omega_2\\right)t + k_1 + k_2\\right] + \\frac{A}{2}\\cos\\left[\\left(\\omega_1 - \\omega_2\\right)t + k_1 - k_2\\right]\n\\end{align}", "b22ab481e90b308b206231448dd6e583": "x_m=-\\tfrac{2^k\\,y_m}{\\dot y_m}.", "b22b0ec60c917058687b9691b9999e3f": "B (\\cdots, a_{-1}, {\\hat a_0}, a_1, \\cdots) = (\\cdots, {\\hat a_{-1}}, a_0, a_1, \\cdots),", "b22b247d849cad3d8818d6662cb099bc": "\n\\begin{pmatrix}\n\\cos(\\theta) & -\\sin(\\theta) \\\\\n\\sin(\\theta) & \\cos(\\theta)\n\\end{pmatrix}\n\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 0\n\\end{pmatrix}\n\n\\begin{pmatrix}\n\\cos(-\\theta) & -\\sin(-\\theta) \\\\\n\\sin(-\\theta) & \\cos(-\\theta)\n\\end{pmatrix}", "b22b6774075fb79240ee821fe3760834": "~|{\\rm final}\\rangle~", "b22b9c91758d368074ec2acd327add2a": "f_N(n) y^{(N)}(n) + a_{N-1} f_{N-1}(n) y^{(N-1)}(n) + \\cdots + a_0 y(n) = 0,", "b22bc706b123ba369ff9fcefc9985780": "\\lim_{n\\to \\infty} \\prod_{k=1}^n \\left( 1+\\frac{2k}{n^2}x \\right)=e^x, \\qquad x \\in \\mathbf{R}.", "b22bdbabf8ccfa2ebb14198179cc1c59": "x^\\top y", "b22be762dcefab82b7eebd7e18fd4c56": "\\Delta H_{fus} \\, = \\, -0.88 + \\sum H_{fus,i}", "b22bed7dcd15fb6c161a5585291a35c0": "z_1 z_2 = r_1 r_2 (\\cos(\\varphi_1 + \\varphi_2) + i \\sin(\\varphi_1 + \\varphi_2)).\\,", "b22c0074a7268a77038b113780b7b487": " \\zeta_K(s)", "b22c2100f1ddee997f0bba7ccdb8f382": "\\ell(x,t) = \\lim_{\\varepsilon\\to 0^+}\\frac{1}{2\\varepsilon}\\int_0^t \\mathbf{1}_{[x-\\varepsilon,x+\\varepsilon]}(B(s))\\,ds", "b22cb97b2e484856d8e539d34deb04c8": "N_j = N_{1j} + N_{2j}", "b22cba4fdccc573a7a04645c8db18567": "/4m", "b22ccb8f67480069ee142c9d085fb8c7": "r = \\cos(q\\phi)+4", "b22cd99bc4ae29980fc013577097f2b3": "\\begin{bmatrix} \\dfrac{-b_{22}}{b_{21}} & \\dfrac{-1}{b_{21}} \\\\ \\dfrac{- \\Delta \\mathbf{[b]}}{b_{21}} & \\dfrac{-b_{11}}{b_{21}} \\end{bmatrix}", "b22d2e66eafc6bab338579d6b28a24a8": "\\frac{\\Delta V}{V_0} = \\frac{\\left ( 1 + \\varepsilon_{11} + \\varepsilon_{22} + \\varepsilon_{33} + \\varepsilon_{11} \\cdot \\varepsilon_{22} + \\varepsilon_{11} \\cdot \\varepsilon_{33}+ \\varepsilon_{22} \\cdot \\varepsilon_{33} + \\varepsilon_{11} \\cdot \\varepsilon_{22} \\cdot \\varepsilon_{33} \\right ) \\cdot a^3 - a^3}{a^3}\\,\\!", "b22d648beef1849cc0df1cb210d0881b": "\\int (ax + b)^n \\, dx= \\frac{(ax + b)^{n+1}}{a(n + 1)} + C \\qquad\\text{(for } n\\neq -1\\text{)}\\,\\!", "b22d7f9c32f843eb3506a7173285ccb3": "\n \\Gamma(x)\\,\\Gamma(y)=\\Gamma(x+y)\\Beta(x,y) .\n", "b22d88cb94c5e2ff76e474084613366a": "PV = nRT \\rightarrow P.P^{-1/ \\gamma} \\sim T \\rightarrow P \\sim T^{\\frac{\\gamma}{\\gamma-1}}.", "b22db6865ff75755d6628a4ae6d0fec0": "\\hbar= h/2\\pi", "b22dcbc847d48a3d3c67b6d4940fa0e1": "\\tfrac{OC}{OF} = \\tfrac{21}{10 + 9} = \\tfrac{21}{19}", "b22e17b8c289f50801ebfc6e38151501": "E(Q)/M_{Pl}^2R(Q)<1", "b22e2a8a3fff4bab0fd5a7b9b1ba421a": " \\operatorname{Tr}(AB)=\\operatorname{Tr}(BA)", "b22e53f4844ad42cba603e3f5517437b": "n \\ge 3", "b22e8896d51004edd1e00d02095ef271": "\\rho(x)\\ddot u(x,t)=\\int_R f(u(x',t)-u(x,t),x'-x,x)dV_{x'} + b(x,t)", "b22e9bd10effecca9cefac6199d7baf4": "A_{\\emptyset} = S", "b22ed21a2808e5cf9e0a13299ecc173f": "\\vec{R}_0: \\Omega \\rightarrow P", "b22f0191164f7abee4e51374c30ece01": " \\frac{dc}{dt}=-8\\pi\\,DRc^2 ", "b22f1db8b53545b8cff0c07c16facae0": " \\hat{H} = \\frac{\\hat{p}^2}{2m} + V(x,t) = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2} + V(x,t)", "b22f22c737688a7630a4b9dd796dc39a": " 1/\\sqrt{N}", "b22f655b41aa9e3a65b98367a787494d": "\\int x^3 r^3 \\; dx = \\frac{r^7}{7}-\\frac{a^2r^5}{5} ", "b22f9f07466b620c0ed5eb98b195419b": "G=C_{p^2}", "b23039afe5a1623f57f22332f997b8a0": " C : S \\rightarrow \\mathbb{R} ", "b230b0e5581b6f2a9738938c70d5d0e8": "s_1 \\to s_2", "b231050be0040fb29c9235da3a1bc85c": " A(x) = {\\sum_{n\\le x}}^{\\star} a(n)\n=\\frac{1}{2\\pi i}\\int_{c-i\\infty}^{c+i\\infty} g(z)\\frac{x^{z}}{z} dz.\\; ", "b23123224af129cc4e439277fc073026": "U_2=R_2 - {n_2(n_2+1) \\over 2}. \\,\\!", "b2313794b9e9847256ff800baaf834b3": " y\\ F = \\operatorname{fact} ", "b231627ec439760b67e104509faa2476": " \n\\psi_0(x) \n= \\frac12\\left( \\sum_{n \\leq x} \\Lambda(n)+\\sum_{n < x} \\Lambda(n)\\right)\n=\\begin{cases} \\psi(x) - \\frac{1}{2} \\Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\\dots \\\\ \n\\psi(x) & \\mbox{otherwise.} \\end{cases}\n", "b23195767bec9aeeb214709fb40211cb": "U(t) = \\frac{\\hbar}{2 e} \\frac{\\partial \\phi}{\\partial t}", "b231b896e8a76755b71423a8f69ef61d": "L_\\mathrm{v} = \\frac{\\mathrm{d}^2 \\Phi_\\mathrm{v}}{\\mathrm{d}A\\,\\mathrm{d}{\\Omega} \\cos \\theta}", "b231d54eff297383f18a6ed996851494": "\\tfrac{C_{A}}{T_{A}} + \\tfrac{C_{B}}{T_{B}} + \\tfrac{C_{Z}}{T_{Z}} = \\tfrac{100}{T_{M}} ", "b23203c5f3d0669a6e6b4c3162198882": " \\langle \\Psi_1 , \\Psi_2 \\rangle = \\sum_{\\mathrm{all\\, }s_z} \\int\\limits_{\\mathrm{ all \\, space}} \\, d^3\\mathbf{r} \\Psi^{*}_1(\\mathbf{r},t,s_z)\\Psi_2(\\mathbf{r},t,s_z) ", "b23228959598aafd91be99d849f7ede1": "alive(1)", "b2324bc4d52d31f92e87159b9fe83662": "\\dot{V}_1", "b23293e799dbcfe1fffa244d284c85c4": "f(n)=10 \\uparrow^{n} 10", "b232a9762eebb5e0856ed1ad52356a2b": "P(A|X=2/3).", "b232d03cc135f7bd4854c06d92883334": "k_2 > 0", "b23390035b6f8377441c3edc8a3074c3": "\nE\\tau = \\frac{|\\Omega|}{\\alpha D} \\left[\\log \\frac{1}{\\varepsilon} +O(1)\\right].\n", "b233b1de4d99022d07f3c0d60e55be88": "\\nabla V(\\langle w_t, x_t \\rangle, y_t)", "b23487a271d4a44aaf4e35486504b32a": "A \\lor A", "b234e7b64400d5a3c2d383e40d2fa247": "5x+3x\\,\\!", "b234f631a36ffdb8b9eff0453cacabb1": "y(\\mathbf x_i) = b_i, i=1, \\ldots, N", "b23515c34e0830e5421feda4640a2f1c": "\\log(y)", "b236208a7dd35b9dfce5f1b323b10eee": "H^{1}(B(x,r)\\cap\\Gamma)\\leq Cr", "b2366c85c3432909168eba1b5ca34cff": " p _ {n+1} ^ x - p_ n ^ x < 1 ", "b23673c5daafff80daa3bdb28410c64c": "f'(x^*)(y-x^*) \\geq 0\\qquad\\forall y \\in I", "b23720f163156874b54003a622d8f042": " x = b (X - v T) \\,", "b2373a75a6beaf851e191fcef92e2877": " \\gamma_V", "b2374b104405ee67a89c1d3fdcab93ce": "p\\geq 3", "b2379e0db097369ad8246463d20e35ff": " \\lambda_u = \\lim_{q\\rightarrow 1} \\operatorname{P}(X_2 > F_2^{\\leftarrow}(q) \\mid X_1 > F_1^{\\leftarrow}(q)). ", "b237f8eabc8572b1b05b95bd69caca90": "\\operatorname{GL}(V)", "b238260444871d21939a7bdf44afb4fd": "\\left\\{ XX,ZZ\\right\\} ", "b238370374fe08e7460ce34b5d3909fc": "|S(f)| \\le 6.", "b2385b72ce8109c9a7872dba1168e1be": "{d\\eta_b\\over d\\rho} = 0 ", "b238f88c25045a0bcd9e04bbc9488339": "P_1 = P_0(1+r\\Delta t)-M_N\\Delta t\\;", "b23908dcb4df542372450cb69d4207cb": "m_{\\rm p}", "b23911e46bf6dd5fb90f7a74bdb02945": "\\lambda_+^0", "b2391cde056c30c5ad5528694def220b": "{\\mathbf{c}}_0", "b23934a062492374a6a6b1cc5ac8fb04": "1-2H", "b2395511da658c0dde285e938a7fb915": "a_1, a_2, \\ldots, a_T", "b23965c2df24111baac6244689881cf4": "<3\\times10^{-8}", "b23988d6fe20113406e7fb6e4136108b": "\\Delta H_\\text{SO} =-\\boldsymbol{\\mu}\\cdot\\boldsymbol{B}.", "b239e176fd6cbe5c9663783123edd193": "{\\mathbb P}\\biggl( \\bigcup_{i=1}^n A_i \\biggr) \\ge \\sum_{j=1}^k (-1)^{j-1} S_j.", "b23a315c109152076598d1863a7fec24": "d\\boldsymbol{\\ell}", "b23a582833d29b97ec5c620574963677": "x=1,2,3,4,5,6", "b23a9d0420d49a41a5ce4568b13664ae": "{\\partial \\over \\partial H } \\left( t H^2 + \\lambda H^4 \\right ) = 2t H + 4\\lambda H^3 = 0", "b23b3bafb22287e5833a84c1cc7f9af0": "\nY = R \\tan \\alpha \\tan \\delta \\,\n", "b23b40b44ca41ba8f66cac7ead73a2b4": "\\Lambda u = f.\\ ", "b23b4fb4075eed5c1cfd786f8c9f1ba9": "\\int_{t-\\Delta t}^{t+\\Delta t}\\int_{x-\\Delta x}^{x+\\Delta x} [c_p\\rho u_\\tau - k u_{\\xi\\xi}]\\, d\\xi \\, d\\tau = 0.", "b23bb843aef389d9e4009055b32538b7": " r + s + \\dots + u ", "b23bf954624725d78f5bff70ce36294a": "T^3\\cdot e_1 = -4T^2\\cdot e_1 -T\\cdot e_1 + e_1", "b23c1c68e0927a49279b00c0ddd8de4f": "\nL = p A + (1-p) B\n\\,", "b23c30843775a53ac79c87253b7aef19": "m^{\\varepsilon}", "b23c960c11468dfb2c69ff021c072c81": "\\scriptstyle d_n d_{n-1} \\ldots d_1 d_0", "b23cbd6091e090d635d1cfcacdc95178": "\\mathcal{F}_{\\mathsf{Sec}}", "b23cd70f8ad4eb0823296457aae4fcfe": "b_{i}(\\sigma_{-i})", "b23d2a50a67a711424cbfa93a1f2a7e0": "C^{(V)}_T(V,T)\\ ", "b23d4f21c7dab04a73625d14a3d37222": "\\ (1+\\epsilon)*m+\\epsilon^{-1}*ln(n)", "b23d8bcdb490736c53d5b677455a8cd2": "f(z)", "b23d8d3987a26b17a32a762026a0834e": "a\\le x\\le b", "b23ddb5372a93ab5a70bef450d6c9102": " \\frac{d}{d t}N_2=-c_1 N_1", "b23e589fe6c12931d7d6fbd0e0d092f6": "g^{-1}", "b23e70b95a6e25239f6fb27745749477": "\\sum_{n\\ge 1} \\frac{\\varphi(n)}{n^s} = \\frac{\\zeta(s-1)}{\\zeta(s)}", "b23f082f391a8810745330f464d501fa": "D_{\\mathrm N} = \\frac {s f^2} {f^2 + N c ( s - f ) }", "b23f815810637d1c4b8a8922992033dd": "\\sum_{i=0}^n {X_i}", "b2402455f1613f9905162ce4eca80099": "w^{j}_{\\sigma_{i}}-w^{j}_{\\sigma_{m+i}}\\,\\!", "b2405424d4d6f61ae7dcc185a4abc5cf": "$100*1.43/3.9 = 36.67", "b240589ec1fc132498ae619c468b9625": "\n\\omega_{\\varphi}^{2} \\approx \\frac{GM}{r_{\\mathrm{outer}}^{3}} = \\left( \\frac{r_{s} c^{2}}{2r_{\\mathrm{outer}}^{3}} \\right) = \\left( \\frac{r_{s} c^{2}}{2} \\right) \\left( \\frac{r_{s}^{3}}{8a^{6}}\\right) = \\frac{c^{2} r_{s}^{4}}{16 a^{6}}\n", "b2406f8a5da83fb0752595a017affea0": "\\displaystyle \\|u\\|_{L^\\infty(\\Omega)}\\leq C \\|u\\|_{H^1(\\Omega)}^{1/2} \\|u\\|_{H^2(\\Omega)}^{1/2},", "b2409f58d03e597e86a9a17a0e000d64": "= \\frac{d}{dt}\\left[\\frac{ds}{dt} \\left( x'(s), \\ y'(s) \\right) \\right]\\ ", "b241106cb64c9d29047ba3decbdbe766": "\n F(x;\\,k) = \\frac{\\gamma(\\frac{k}{2},\\,\\frac{x}{2})}{\\Gamma(\\frac{k}{2})} = P\\left(\\frac{k}{2},\\,\\frac{x}{2}\\right),\n ", "b24124aa75288df8a65be3dd8dd63256": "\\hat{\\gamma}(h)", "b2412c37ae749271a3e0a1cd7a4c80d3": " \\subseteq ", "b2418e3671b0875372fae9bed02fa0a3": "\\scriptstyle R_k", "b2419bb5da225eb1774f0534174a5700": "R = (x,y)", "b241ba217976de2679a6fdd0e99350c9": "\\notin \\!\\,", "b2422afa9d6035d44a29762f3c9cb8f1": "P^{\\ast} F^2 < \\infty", "b24289dd4f3597fa6f898304c3570b09": " x\\in[0,L^{(e)}]. ", "b242bb2bf1de1bfa0ee0179c0f9ec760": " SL(2,R) ", "b243559b8df66b802e7af1a5fa41e020": "\\mathbf{w}_i", "b243a2f5b069c34fa3dcbc69b17168b4": "M_0 \\subset M", "b243ad50ab0d8b9e6af750e40459f4e6": "x^{-1} \\cdot 2^1 = \\frac{2 \\sqrt{5}x}{5}", "b243e96d5c8b64335ad3add497c4f67b": "\n \\frac{1}{2}\\int_{0}^{2\\pi}(a\\cos (k\\theta))^2\\,d\\theta = \\frac {a^2}{2} \\left(\\pi + \\frac{\\sin(4k\\pi)}{4k}\\right) = \\frac{\\pi a^2}{2}\n", "b243ff6241850369065865096efdadc7": "c \\equiv 4^{13} \\pmod{497}", "b2447f5497bdd35b265e31f15d359b7c": "P\\big(A\\cap T^{-1}(B)\\big) = \\int_B \\nu(x,A) \\,P\\big(T^{-1}(d x)\\big).", "b244b39e680169d27fc966281af6ee90": "R_\\text{sk}", "b24515d3c97caba18db0f452aaa34f7d": "H^q_{\\mathrm c}(X)", "b2452064e7d93399db7fd0cb7859b290": " \\mu(E) = \\inf \\{\\mu(U): E \\subseteq U, U \\text{ open}\\}.", "b24534cee79b99f23a2ed9bedd1bb32a": "I_1,I_2", "b2453c9b3fa9a43176934e4d82bb7919": "E(m) = (\\frac{0}{b}+\\frac{c}{\\sqrt{b}(\\sqrt{a}+\\sqrt{1-a}-1)})\\cdot m - \\frac{c}{\\sqrt{a}+\\sqrt{1-a}-1}\\cdot\\sqrt{m}.", "b24545a9e0ae42f059b5da0cc5ed44c2": "A \\setminus B \\in \\mathcal{R}", "b2459349c02dfaf999171de19adec3e7": "\\mathbf F_B", "b245cbec1641e05ddc97e91445a03a4e": "m(\\theta) = C_0\\,\\theta + C_1\\sin \\theta + C_2\\sin 2\\theta + \\cdots + C_n\\sin n\\theta.", "b245ed291ff9ca24caf6ddcf45aea7db": "\\inf \\left\\{ \\left. \\int_{X} c(x, T(x)) \\, \\mathrm{d} \\mu (x) \\;\\right| \\; T_{*} (\\mu) = \\nu \\right\\},", "b24629b630e9db2b8f1505b22c9af539": "O(\\sqrt p)", "b246d2df444781a5924919110d540f53": "\\mathcal{M}(P) = \\{Q \\ll P\\}", "b246e763015159591e671978660de481": "F\\cup\\{1-tf\\},", "b24705f623fd3e634a0d29fc0434de28": "\\,{(1+i)}^{n}", "b24706b0ccdfef2e717e271b7a9aaeef": "H^*", "b2470acdda43d4bba7256b91430e296f": "\\left(\\wp,\\wp'\\right)", "b247137b482d35f1faf753f5ab5534f6": "\\left(1 + \\frac{n_2}{2}\\right)\\psi\\left(\\frac{n_2}{2}\\right) + \\frac{n_1 + n_2}{2} \\psi\\left(\\frac{n_1\\!+\\!n_2}{2}\\right)", "b2474b6ce3dc04cb7bc6f8c12d30dabe": "\\sqrt{ax^2 + b} \\ (a \\ne 1)", "b247985c61d89d0f79e37281e9c4b344": "x_{\\min}, x_{\\max}, y_{\\min}, y_{\\max}\\,\\!", "b247e30586f89117f34208f169cfcc7e": "\\forall x (x \\in A \\leftrightarrow x \\in B)\\rightarrow A=B.", "b247e3b7599155a01f1ce7d00a5f89d3": "K\\otimes_{\\mathbb Q}\\mathbb Q_p,", "b247ed7ca9a044e762e9a7eeaa56616a": "\\operatorname{Var}[Y]=\\operatorname{E}(\\operatorname{Var}[Y\\mid X_1 , X_2])+ \\operatorname{E}(\\operatorname{Var}[\\operatorname{E}[Y\\mid X_1,X_2]\\mid X_1]) + \\operatorname{Var}(\\operatorname{E}[Y\\mid X_1]),\\,", "b2489a547dd650ea6989ec153d35636a": " \\alpha_{k} \\, \\mathbf{x}_{k} = \\mathbf{A} \\, \\mathbf{y}_{k} ", "b2489cd82e70b90de0e108e56401492f": "r_{\\text{s}}\\,", "b248da97426ec52b04a39ecaee857094": "n \\sin \\theta", "b248ea13e86a172c47e7572cf5a4fabf": "\\mathbf{a}-\\mathbf{b}\n=(a_1-b_1)\\mathbf{e}_1\n+(a_2-b_2)\\mathbf{e}_2\n+(a_3-b_3)\\mathbf{e}_3.", "b24912e94c5945beca4b5ac18dca0fd0": "M(P) \\ge M(x^3 -x - 1) \\approx 1.3247 ", "b2491e6ebc936d451bb330f8b4aa0db7": "u_t = \\alpha \\nabla^2 u = \\alpha \\Delta u, \\quad \\,\\!", "b24a712d89ff2d14914698f28f4875f7": "\\mathbf{\\Delta k}=\\mathbf{G}", "b24a9f2357dd9ded95e9cc2342f94149": "A_{(\\alpha}B^{\\beta}{}_{\\gamma)} = \\dfrac{1}{2!} \\left(A_{\\alpha}B^{\\beta}{}_{\\gamma} + A_{\\gamma}B^{\\beta}{}_{\\alpha} \\right)", "b24ac1454ab244cc589a945221614e4d": " Com.3 ", "b24b2ca972bab94d38967a97fdbf690b": "22\\pi^4\\approx 2143;", "b24b54908f39298f54a3a89fe3bddc83": ":U", "b24b8f15d968c1335ade60af01a0ba3b": "z=a", "b24bb32883f303522231cf556249b517": "\\eta\\ne 0, \\gamma \\ne 0", "b24be3b2b15ff7dc4903711fe311eb2b": " \\tan x + \\sec x = 1 + 1x + \\frac{1}{2}x^2 + \\frac{1}{3}x^3 + \\frac{5}{24}x^4 + \\frac{2}{15}x^5 + \\frac{61}{720}x^6 + \\cdots ", "b24c37f364bf4559f9a4e6404dd37c5d": "f(x|\\mu) = \\frac{e^{-(x - \\mu)^2 / 2\\sigma^2}}{\\sqrt{2 \\pi \\sigma^2}}", "b24c6bc81d2633fc3fb898f13caee939": "G = 10^\\frac{3}{10} \\times 1\\ = 1.99526... \\approx 2 \\,", "b24c739e0aee77a89b32aaea55de8e34": "{\\tilde{B}}_{n}", "b24c9e6dc1326f75032ec252346efa7b": "s_i\\colon T_i \\rightarrow A_i", "b24cad4296db34332ed04c332cc4c46c": "\\mathbb{CP}^3", "b24ce0cd392a5b0b8dedc66c25213594": "Free", "b24d762204a589d01f1cd44d0900051b": "H_{\\omega^{\\omega + 1}}(1) - 1", "b24d8c9a79678bea79fc5550b1fd9818": "\\sqrt[3]{\\sqrt[3]{2} - 1} = \\frac{1 - \\sqrt[3]{2} + \\sqrt[3]{4}}{\\sqrt[3]{9}} \\,.", "b24d8e45ac8f5880c07aa50dbc00f180": " L(n,n-1) = n(n-1)", "b24de13033be62c0f54f5108cdb0d6e3": "V_{\\text{out}} = -\\int_0^t \\frac{ V_{\\text{in}} }{RC} \\, \\operatorname{d}t + V_{\\text{initial}}\\,", "b24e7ded29aa654c8d5f86a7d39b78f9": "\\operatorname{E}(w_i z'_i) = \\frac{1}{n}\\sum_i\\frac{n'_i}{n_i}z'_in_i = \\frac{1}{n}\\sum_i n'_i z'_i = \\frac{n'}{n}~\\frac{1}{n'}\\sum_i z'_i n'_i", "b24e96a58c506109141e12bebc16b08e": "(\\sigma,d^i t) \\sim (d_i \\sigma, t) \\quad \\text{ for all} \\quad \\sigma \\in S_n, t \\in \\Delta^{n-1}.", "b24e97f775c1d590d27a09d32e5a070a": "i = \\operatorname{arg}(\\ z_3\\ ,\\ \\sqrt{{z_1}^2 + {z_2}^2}\\ )", "b24ebd773af28c56a6b9e4a87b5359f3": "\\Vert J \\Vert", "b24ed48c74475247bc54451c0b353e2e": " |v|^2= |u|^2 + 2 \\, a \\cdot s ", "b24ef0077c518ded1952af65df4e3d69": "H(z) = \\frac{B(z)}{A(z)} = \\frac{1}{1 - a z^{-1}}", "b24f1213058775408741f5b167a46165": "\\ln(\\phi(q))=-\\sum_{n=1}^\\infty \\frac{q^n}{n} \\sum_{d|n} d", "b24f35e151db110b7965af878f2c9b9b": "y(e) = \\sum_{f:\\mathrm{out}(f)=v}(m_e(f)y(f))", "b24f5cbded864c573918739ce83e8757": "\n\\dot{r} = \\frac{dr}{dt}\n", "b25035e6a08b702624f2c05055706ad8": " H_0 |n^{(1)}\\rang + V |n^{(0)}\\rang = E_n^{(0)} |n^{(1)}\\rang + E_n^{(1)} |n^{(0)}\\rang ", "b250b9f82d9c0cae90cd6200dd946cb8": "K e^{-rT}", "b250dc2b6221d08097fe33d63e334666": "K^\\mu(x)", "b250dd36d4cd78a734f90dade09c2d06": "\\alpha\\in\\mathbb{R}\\backslash\\mathbb{Q}", "b2513c3178f19bae2ffb57ef4dec44c2": "\n\\begin{array}{c|c}\n1 & 1 \\\\\n\\hline\n & 1 \\\\\n\\end{array}\n", "b251455100fe9be47ea386c4383e0142": "\\frac{200-8}{100}=1.92", "b25174129646e13a5ea684fb8c805619": "\\begin{cases}\n\\Delta u + \\lambda u = 0& \\rm{in\\ }\\Omega\\\\\nu|_{\\partial\\Omega} =0.&\n\\end{cases}\n", "b251ba52faf079f748fede470d64f777": "\\cos a \\cos b = \\frac{\\cos(a+b) + \\cos(a-b)}{2}", "b251cf0fcfd9192d2baddb57fac542d4": "u\\in\\mathcal{U}^N", "b251e2668ded2ba1b87fae39624bbebc": "\nh_\\phi = \\frac{a \\sinh \\tau}{\\cosh \\tau - \\cos\\sigma}\n", "b251e477c2411df23a54a4f05ef65b31": "t_1= T_1", "b2835812f85495a13c03cfe81dbcefec": "Av_j=\\lambda_jv_j", "b28367b88a69b32ffd19721127a6ac2f": "= | a |^{2} + \\mathbf{E}^{a} \\left[ \\int_{0}^{\\sigma_{k}} n \\, \\mathrm{d} s \\right]", "b283813a9b50dccfca8be8a16dfc16b5": "-785\\pm 7", "b2838d2526296c923b604c53e4c7e615": "\\sqrt{E^2 - (m_0c^2)^2}", "b283a6b9d62c9beb1c887a376469249c": "\\dfrac {b+a}{c} = \\dfrac{\\cos \\left( \\alpha/2-\\beta/2 \\right) }{\\sin \\left( \\gamma/2 \\right) }", "b283ac9f2678a2c2ff7dd3e3f845d8fd": "\\color{Plum}\\text{Plum}", "b283c3393bec056fda0aab1ac8aefd3e": "x_{.,.} = \\sum_{i=0}^{2}\\sum_{j=0}^{3}{{x_{i,j}}}", "b283e6e314ab7ff62b019819c1e6cc20": "\n\\phi(q)=\n\\sum_{d\\,\\mid\\,q}\\mu\\left(\\frac{q}{d}\\right) d\n.", "b2842e28f2977818c0c0b4f6ac45ad5c": "\\boldsymbol \\xi", "b284487b833f18819c47f746ae45e265": "P_A~dA = \\frac{dN_A}{N} = \\frac{dg_A}{N\\Phi_A}", "b284a8607ea25d985ccf5d1f0b1e0072": " \\Box \\varphi = - \\dfrac{\\rho}{\\epsilon_0} \\,,\\quad \\Box \\mathbf{A} = -\\mu_0\\mathbf{J}", "b284dd8a44eeffd13ad2022e1534dcc7": "B_k = \\mathfrak{so}_{2k+1}, D_k = \\mathfrak{so}_{2k}.", "b2851c4ec03702787da602afed881b08": "v_e=v_0 [OH^-]_{0^{ }}/[H^+]_0", "b2854ab43cf6a5420b2199e932e1097e": "N = \\sum_{\\{Sb\\}} \\mbox{Sb.z} \\times \\mbox{Sb.n}", "b285709d398b3076b9a297f37421b45a": "(x = y) = \\lnot(x \\oplus y) = \\lnot x \\oplus y = x \\oplus \\lnot y = (x \\land y) \\lor (\\lnot x \\land \\lnot y) = (\\lnot x \\lor y) \\land (x \\lor \\lnot y)", "b28586e9b2b4b6a956dbc5756a0f1c0e": "f^{64}(1) < f^{64}(4) < f^{64}(27)\\, ", "b2861664fefde20b465bfcb95531a880": "F=\\frac{s_1^2}{s_2^2}", "b2862a03051e0005b21a675d87f3eb42": "\\,x\\in(0,1]\\,", "b286853b86de6c28bfff2e12756fbe50": " y^4-x^4 = xy,\\, ", "b286b8c4bc57830d232fbb479d0e41fe": "y(t) = I(t) \\cdot \\cos(\\omega_c t)", "b286d5b617f827a047837ff0b7a07224": " \\sqrt{2E_i}.\\delta(\\mathbf{r}) ", "b286f52c6cf49daef86b801b4f24d451": "\n\\exp \\left (-{a } (q^2+p^2)\\right ) ~ \\star ~\n\\exp \\left (-{b} (q^2+p^2)\\right ) = {1\\over 1+\\hbar^2 ab}\n\\exp \\left (-{a+b\\over 1+\\hbar^2 ab} (q^2+p^2)\\right ) ,\n", "b2875a3fa82e9c5f5fcf7bfb6e3bc308": "\\color{Black}\\tfrac{6}{m}", "b2876eb1ab53c60666414de34a85c998": "811228\\mbox{-}987x", "b287d3430bd7ee394ec8098fcf3dea28": "n=1,2,3,\\dots", "b287f03727f2e4ba703a259321b90eb1": "\\mathrm{ymid}", "b287fa827e859a779ff79e86f2c46bf0": "\\delta(\\alpha x) = \\frac{\\delta(x)}{|\\alpha|}.", "b288166f9c09da22dec38f4d7579af1e": "w_i = c_i \\cdot \\frac{M_i}{\\rho}", "b2889a5a432ba0504a2d4a43a1d3487d": "\\xi_C = \\frac{W_C}{W_0} = 2\\pi\\frac{\\rho_q}{R_H} = \\begin{cases} 4\\pi \\alpha, & \\mbox{at }\\rho_w \\\\ 2\\pi, & \\mbox{at }\\rho_{DOS} \\end{cases}", "b288cf1331d4185bea6fd71899e2a59e": "\\frac{1}{(1-z)^{\\beta+1}} = \\sum_{k=0}^{\\infty}{k+\\beta \\choose k}z^k.", "b288d8c85109267bfff41f4e0c0b87ac": "g(\\mathbf{X})=1", "b288f5c9a3a4b58ce6cd40c7ad3f1d9c": "S(q)", "b2896b287b1872f36ee26816070649e1": "\\displaystyle{H_{\\partial\\Omega}^\\varepsilon g(s) -H^\\varepsilon g(s) = {1\\over \\pi i} \\int_{ |t-s|\\ge \\varepsilon} K(s,t)\\cdot g(t)\\, dt,}", "b2899ffd74be22232b4bc09604d1ee3f": "\\int x^2\\,\\operatorname{arcoth}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{arcoth}(a\\,x)}{3}+\n \\frac{\\ln\\left(a^2\\,x^2-1\\right)}{6\\,a^3}+\\frac{x^2}{6\\,a}+C", "b28a0ae5bc0bde91a95fd2bacbc3cb57": "\\|x\\|^2 = \\|x - y\\|^2 + \\|y\\|^2 \\ge \\|y\\|^2 = \\sum_{j=1}^n|\\langle x, f_j \\rangle|^2.", "b28a421e90e41dec6ffde85eac31dbbd": "\\mathrm{CmCl_3\\ +\\ 3\\ NH_4I\\ \\longrightarrow \\ CmI_3\\ +\\ 3\\ NH_4Cl}", "b28a4f15ceb966a97124fed6a89197b3": "\\frac {c}{s}= \\frac {10} {(D_h/D_t)(N_s/1000)^{1.5}}", "b28aa3d09ec610171b94f8e781fab524": "M=(M_{i,j})\\in\\mathbb{R}^{(n)}", "b28ac3d2ce2a75cf4c8239b7cd1b8545": "{x}_{k}+sh", "b28b3de6ae143ffd5ba1a5aaad7a7959": " \\begin{bmatrix} T_{1 1} & T_{1 2} & \\cdots & T_{1 n} & \\cdots \\\\ T_{2 1} & T_{2 2} & \\cdots & T_{2 n} & \\cdots \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\cdots \\\\\nT_{n 1} & T_{n 2} & \\cdots & T_{n n} & \\cdots \\\\\n\\vdots & \\vdots & \\cdots & \\vdots & \\ddots \n\\end{bmatrix}. ", "b28b802b982da5c4ba4f696a649dc4ec": "\\boldsymbol\\Sigma={\\boldsymbol\\Sigma_1+\\boldsymbol\\Sigma_2 \\over 2}", "b28b870288338b43a984749c8b5e36db": "d\\beta", "b28bb898fe1080d7acaeb12d2d2c3826": " E_1 + \\sum_{n=0}^\\infty \\sum_{k=0}^n \\binom n k E_k E_{n-k} \\frac{x^n}{n!} = 1 + \\sum_{n=0}^\\infty \\sum_{k=0}^n \\left(E_k \\frac{x^k}{k!}\\right) \\left( E_{n-k} \\frac{x^{n-k}}{(n-k)!}\\right). ", "b28be720b8ce2237a2978edb64fbb03c": "M_{s}", "b28c3ddf23fff5fde64283aac99f03ef": "\\tfrac{1}{16}", "b28c91a7e2ba8e9896629fdc9efaf2e3": "\\beta_1 \\beta_2 = c + 1", "b28cec1e4bccf72c556bd49690e1343d": "20\\log_{10}\\left(\\tfrac{1}{\\sqrt{2}}\\right) \\approx -3.0103\\, \\mathrm{dB}", "b28d562b320d99956f26203ceb6cb6d1": "\\ddot u=c^2\\nabla^2u", "b28d586e1ae4d912b2e9ad2911e1cb3b": " J[n]\\times J[n] \\longrightarrow \\mu_n", "b28d603d04eae96cbd00c7b6b7a1a602": " 760 000= 750000 +10000= 19\\times 4\\times 10000 ", "b28d8d4995a89e71b056fc94f12775d5": " u(0,x)=f(x), \\quad u_t(0,x)=g(x).", "b28d8e22da7bc87a650ad40c40ab9179": "\\frac{\\partial \\Lambda}{\\partial \\lambda}\\approx\\frac{\\Lambda(x,\\lambda+\\epsilon)-\\Lambda(x,\\lambda)}{\\epsilon}", "b28e14d0aa8a972ec4664069c72678c0": "\\{X_t, \\mathcal{F}_t\\}_{-\\infty}^{\\infty}", "b28e3696d881f699fad4d7640cdce015": "p_1 \\ldots p_m", "b28e5c541f1a4c5aa0c6c321b9fa6c4e": "F^0 B^n = B^n", "b28e5d61d18e16b164ba6378efb123eb": "\\rho(x, \\theta)=\\frac{(x - \\theta)^2}{2},\\,\\!", "b28e7714a0790030292169a1fc89a7c3": "g \\circ f = h \\circ f", "b28eb62a7d1d39fc7ca131828db95079": "f(x) = 1+2x+4x^2+8x^3+\\cdots+2^n{}x^n+\\cdots = \\frac{1}{1-2x}", "b28ebc0055a1bb2b27a7ba9ab7ee3718": "\nH_{mn} = i{d \\over dt} U_{mn}\n", "b28ee39d8a0053aefd7b2e7b58b2d781": "M_{l}", "b28f16a6b48f12614b3e9f750bc5dcfc": " \\chi(X,E) = \\int_X \\operatorname{ch}(E) \\operatorname{td}(X)", "b28f3ec1ecb3c7cbbcba3db5a895f1ef": "\\left(\\frac{\\cdot}{p}\\right)\\,", "b28f66f9f1e99f37ac125e36bb011544": " \\det(\\cdot)", "b28f8f46dce074bad6e74e819e8f881d": "s=\\sigma+j\\omega", "b28fa72b2cae14d443e980b5bcf146ed": "\n\\Bigl\\langle \\mathbf{r} \\cdot \\frac{d\\mathbf{v}}{dt} \\Bigr\\rangle + \n\\frac{1}{\\tau} \\langle \\mathbf{r} \\cdot \\mathbf{v} \\rangle = 0\n", "b28ff9b2da02841dc70a9b1d28102aad": "E[\\hat{Z}(x_i)]=E[Z(x_i)]", "b29034c89b3880d4d73184a19a603eae": "V = \\int_{\\Omega}d\\overline{\\mathbf{x}}", "b2903f7769064feb062908d823fbcbc7": "\n\\mathbf{u} =\n\\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{S}^{-1} \\mathbf{e}_{k} \\rangle \\mathbf{e}_{k} =\n\\sum_{k} \\langle \\mathbf{S}^{-1} \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle \\mathbf{e}_{k} =\n\\mathbf{S} (\\mathbf{S}^{-1} \\mathbf{v}) = \\mathbf{v}\n", "b2906b28ab75b60b6e31fec575db6417": " K_i ", "b2906efc593fad1360c0e095116264fb": "\\mathbf{E} \\big[ X \\big| F_{k} \\big] \\to \\mathbf{E} \\big[ X \\big| F_{\\infty} \\big] \\mbox{ as } k \\to \\infty", "b2916345a264458c40cfadb785bd415d": "\\tilde{\\chi}(\\omega) = \\frac{\\tilde{x}(\\omega)}{\\tilde{h}(\\omega)} = \\frac{1}{\\omega_0^2-\\omega^2+i\\gamma\\omega}. \\, ", "b2916c6b00e1ec5d92802ef293fefce8": "\\int_0^{2\\pi} \\cos(m\\varphi)\\cos(m'\\varphi)d\\varphi=\\epsilon_m\\pi\\delta_{|m|,|m'|}", "b29179b236c5fd36473b1c88fd60d32a": "\\hat\\beta = (\\overline{x y}) / (\\overline{x^2})", "b2917c32c52d7d43f2f697a7bd04fa60": " \\mathrm{DCG_{p}} = rel_{1} + \\sum_{i=2}^{p} \\frac{rel_{i}}{\\log_{2}(i)} ", "b291a578a5f5b482d8a1e09acc7e6ded": " e^{i\\phi}=W/|W| ", "b2929243e8ef76d486458050f7f1328c": "(x)_{n}=x(x-1)(x-2)\\cdots(x-n+1)", "b293304b1a559f4a54aa55fb919436b1": "Z_0 \\approx \\sqrt \\frac{R}{i\\omega C} = \\sqrt \\frac{R}{2\\omega C} - i \\sqrt \\frac{R}{2\\omega C}", "b29343932405739f797d987848c26c33": "c_0 \\le c_1", "b293b00d30b86fd2b58b9edfd508cc38": "|M_x| \\le 2^k\\sqrt{x} \\qquad(2)", "b2944ed09ca4f6a2f9ca6b359417f1b9": "u=(x,y,z)", "b29499b89ad95d265078b6d72a665547": " BI = \\delta \\sqrt{ p( 1 - p ) } ", "b294a140646a15380d91952b3a7b31ac": " G(T) = H(T) - T \\times S(T) ", "b29512829e54e5d529a53e147e5d13fb": "\\mathrm{R}_\\mathrm{m} \\ll 1", "b2955769bfaa74d83761376efc65bac8": "\\sum_{k\\in A} e^{-\\sqrt{k}}k^n", "b295647d304d987e315b1f1835790f9d": "\\Delta \\omega=\\omega-\\omega_0", "b29572f86288b0c022b9b823dbdc5ea4": "c_z", "b295a54293b91685fafc341cc1727423": " r = a \\pm bi \\, ", "b296020a44114e0cade1b082da2be8e7": "S(\\rho) = - \\hbox{Tr} \\left( \\rho \\log_2 {\\rho} \\right),", "b29636aec8fd5ce27a853b536be88e61": "(\\nu x)(\\nu y)P \\equiv (\\nu y)(\\nu x)P", "b29697436110e6fc47686bca2c393da6": "\\frac{\\delta T}{\\delta q_j} = \\sum_{i=1}^n \\mathbf{F}_i\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j}\n=\\sum_{i=1}^n (-\\nabla V_i + \\mathbf{N}_i)\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j}\n=-\\sum_{i=1}^n\\nabla V_i\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j}+\\sum_{i=1}^n \\mathbf{N}_i\\cdot\\frac{\\partial \\mathbf{r}_i}{\\partial q_j}\n=-\\frac{\\partial V}{\\partial q_j} + N_j\n", "b296b049c2b28670a0cf9d40d0067d89": "\\scriptstyle \\left(u_i\\right)_{i \\in I}", "b296cfc2ae7884dc45b0a1797be35d4f": "\\ p_{i} =P_{\\text{total}}y_i ", "b296d5c1da62458b8ed74014faf0214b": "G = \\mathbb{R}^n", "b29745e3aef96a15dccae5a677d4b461": "\\Theta_0\\subset\\Theta_1", "b2977b4d7e89bb3c93d7b8c022711886": "Z = X \\otimes Y", "b2979065afd7fd9bfdb47276d6c5f43e": "{\\rm trig}(M)=(0,g_2,g_3)\\quad \\mbox{or}\\quad (1,g_2,g_3)", "b297a5990c9e2442f206c11ab6bce3e2": " \\Sigma^2 + \\Omega^2", "b297c2ff2e7ccda20218d0e616495701": "\n Q \\approx \\cfrac{\\mathrm{d} M}{\\mathrm{d} x} ~;~~ q \\approx \\cfrac{\\mathrm{d} Q}{\\mathrm{d} x}\n ", "b297d9114cd7bb1f41ae6bfe17d3a994": "X=P^{-1}AP", "b297e35185210c852bc91b0ac07cf7d3": "D = \\{(\\mathbf{x}_1,d_1),\\dots,(\\mathbf{x}_s,d_s)\\} \\,", "b297f33907be235ac2c834731ee4dec9": " X_1 ", "b298349d4e28e8e08c8e0a8e69c8ac1a": "(f_k(x))", "b29845044d103daeb520f4d257b2d622": " \\int |M(f)(x)|^p \\, \\omega(x) dx \\leq C \\int |f|^p \\, \\omega(x)\\, dx,", "b29888291bf45c70effe8c8852f77618": "\\tilde{B}_n(x) = \\frac{(-1)^{n + 1}}{n!} \\left( \\delta^{(n - 1)}(x - 1) - \\delta^{(n - 1)}(x) \\right)", "b298a498943e93e2b0e2948de8caf304": "Clipped(t_1,f,t_2) \\equiv\n\\exists a,t \n[Happens(a,t) \\wedge (t_1 \\leq t < t_2) \\wedge Terminates(a,f,t)]", "b2993ad9836fb15cd08949c4d99222c8": "\\textrm{Bl}", "b2999c995fbf73003cadbe42738169e7": "\\Omega = \\lbrace z: |z - 4/3| \\leq 2/3 \\rbrace.\\,", "b2999eebf3d614f76007714e70bded3f": "\\Omega(k)=ck,", "b299e0a82ae4cc195a8de36433f8e723": "f\\upharpoonright N=\\sum_{U\\in S}f_{U}\\upharpoonright N\\,", "b29a4164c5146dc8f5652ee4e17ed9c9": "\\textstyle\\binom{n-k}{k}", "b29a7dac584357f7cfa081fad5d04ec7": "k=\\sec80^{\\circ}=5.76", "b29abc1bd9ba25cae99c111bebc2c98d": "\\{|a \\rangle \\otimes |b \\rangle\\}, 1 \\leq a,b \\leq n ", "b29ae459ca90adf1b934f56b9d702f38": "\\tilde{\\mathbf{C}^n} \\setminus E", "b29afec38e7d7590942ba17cc56813f5": "\\begin{align}X(z) &= \\sum_{n=-\\infty}^{\\infty} (a_1x_1(n)+a_2x_2(n))z^{-n} \\\\\n &= a_1\\sum_{n=-\\infty}^{\\infty} x_1(n)z^{-n} + a_2\\sum_{n=-\\infty}^{\\infty}x_2(n)z^{-n} \\\\\n &= a_1X_1(z) + a_2X_2(z) \\end{align} ", "b29b4a6c4b54adb39719a98e8219d108": "- 7 x^9 + 2 x^5 - 5 x + 3", "b29c11eddd8cbd0774986b45fe0366d5": "0 = m_{i}^{2} + m_{f}^{2} - 2 m_{i}m_{f}\\gamma", "b29c2a0c63f92e6868da058a4e097d99": "\\begin{smallmatrix}\\left[\\frac{m}{H}\\right]\\ =\\ 0.00\\end{smallmatrix}", "b29c4cea1e004f8bef55ca442910d66f": "[v_1, v_2] = (\\phi(v_1), \\phi(v_2))\\;", "b29c6b4ba7a0795d59d0fea7b520c763": "F(x) = 1 - \\frac{\\mu_2}{\\mu_2-\\mu_1}e^{-\\mu_1x} + \\frac{\\mu_1}{\\mu_2-\\mu_1}e^{-\\mu_2x}", "b29ca6ad521aaccd188a7a943bd6aeff": "\\left|\\mathrm{RPA}\\right\\rangle=\\mathcal{N}\\mathbf{e}^{Z_{ij}\\mathbf{a}_{i}^{\\dagger}\\mathbf{a}_{j}^{\\dagger}/2}\\left|\\mathrm{MFT}\\right\\rangle", "b29ce83792d3bff525e258e597dcd21f": "[\\phi^i(x),\\phi^j(y)]=[\\chi^i(x),\\chi^j(y)]=\\{\\phi^i(x),\\chi^j(y)\\}=0.", "b29cf34da2933b28666fbf79909869f0": "\\chi_\\lambda=\\chi_\\mu", "b29cf6c5102d5ed326b043a3112f86c5": " \\frac {P - p(r)} {P} \\ll 1 ", "b29cffba3d429722b44ac3d0363b3e88": "E_{kin}=E_{\\text{Core State}}-E_B-E_{C}'", "b29d03b09c21e5fd21a525f60af542cc": "\\log(w^z) \\equiv z \\cdot \\log(w) \\pmod{2 \\pi i}", "b29d212b27c4d2e39c12ea065984420e": "\\displaystyle |\\langle \\psi_i | \\phi_j \\rangle|^2 = \\frac{1}{d}, \\quad \\forall i,j ", "b29d47595d32440b88c12202ede52bd2": "f(tx_1+(1-t)x_2) < t f(x_1)+(1-t)f(x_2)\\,", "b29d684733101977ee3de8018401e310": "x\\delta(x) = 0.", "b29d8900c2508e13a4393715afaae0fa": " 1 + \\sum_{i=0}^\\infty \\sum_{j=0}^\\infty E_i\\frac{x^i}{i!}\\cdot E_j \\frac{x^j}{j!} ", "b29dd5a1c8063277a31006f300d9931f": "\\chi^{(3)}|\\mathbf{E}_0|^2 \\mathbf{E}_\\omega", "b29dda0b7b963fc105978c711a5814c6": "\n \\xi(\\omega) =\n \\begin{cases}\n -r, &\\omega \\ne \\omega_0\\\\\n 35 \\cdot r, &\\omega = \\omega_0\n \\end{cases}.\n", "b29e1cf6937cef562973339a8484d4a3": "T > T_\\textrm{inv}", "b29e574144cd232991b2d8a19cd0cedb": "\\sigma_{xz} (x,0)=0 ~;~~ \\sigma_z (x,z)=-P\\delta (x,z)", "b29e7914bdafd062db067959521706a3": "\\Delta x = 0 \\ .", "b29e7a84b9dcebee8409bb2754ef9ae1": "|\\mathbf{J}|=g'(\\mathbf{y})", "b29e9b339056646c1c6beeb737595263": "r_\\infty=R_0 e^{-{B/T_0}}", "b29ea1a8edcd01846ca43ed66e0b7cd0": "\\sum_i \\rho_i \\cdot \\bar{v_i} = 1", "b29ecac498c3e2ba75ee0043c44effac": "\\Gamma\\backslash G/K", "b29ff5a3c2db4c5a01e421e825ed768f": "q''(x_1)=2\\frac {b-2a}{{(x_2-x_1)}^2}", "b29ffffc4be33026a90aba250172a31d": "\\sum_{k=0}^\\infty \\frac{(-1)^kz^{2k+1}}{2k+1}=\\arctan z, |z|<1\\,\\!", "b2a0b55a42b86b5fa63a0665fab3215e": "f(x)={{x^3}-9x} \\!\\ ", "b2a0d52330184175247d24748a5920dd": "p+q\\ge 2\\sqrt{ac+bd}.", "b2a131827df5229f2fb8c4150befce5d": "p'_l=\\frac{|p_1|}{1-2|p_1|}, p'_m=-\\frac{2|p_1|-p_2}{1-2|p_1|}, p'_n=\\frac{p_3-2|p_1|}{1-2|p_1|}.", "b2a15c24d72e4ad4e39d290c24abe3eb": "\n\\begin{align}\nL_{1/2}(x) &=\\,_1F_1\\left( -\\frac{1}{2};1;x\\right) \\\\\n&= e^{x/2} \\left[\\left(1-x\\right)I_0\\left(\\frac{-x}{2}\\right) -xI_1\\left(\\frac{-x}{2}\\right) \\right].\n\\end{align}\n", "b2a178b444ac510d2180ad476efae4c5": "\\det(\\mathbf{M}) = AD - BC = { n_1 \\over n_2 }. ", "b2a1874a3fb143565a94a391ce1487b4": "\n\\Omega = \\begin{bmatrix}\n0 & -z\\theta & y\\theta \\\\\nz\\theta & 0 & -x\\theta \\\\\n-y\\theta & x\\theta & 0\n\\end{bmatrix} .", "b2a19898941ab3e2716cb7677e8e5fe3": "(\\Delta t, \\Delta x, \\Delta y, \\Delta z) = (t_B-t_A, x_B-x_A, y_B-y_A, z_B-z_A)\\ ,", "b2a19cc11389bdd6c56398ce32fa57c0": "g_m/I_D=1/(nV_T)", "b2a1d748a8c5afebbb34299fd52444f4": "S \\subset \\Delta \\mathcal A", "b2a1f17ca8a2c60be1919f0bff6bf5fa": "\\omega^\\alpha=\\alpha", "b2a2045673899839342e2a4cd33ff1d4": "\\cfrac{0.30}{\\sqrt{0.47 * 0.52}} = 0.607", "b2a20c2e1e3ba2643f41fdeda7bf5d97": " T(F)f(x)=\\int F(x,y)f(y) \\, dy.", "b2a224d517eee472ce1ff6ccb6aefed4": "Z_{A,mono}=\\frac{1}{2}Z_{A,Dipole}", "b2a25c2df7f82a411f9ac132bec89c37": "[T^i_j,\\overline{S}_k]= - \\delta^i_k \\overline{S}_j", "b2a2b16625b9cd09c36a0ebd9f1775cd": "w=\\frac{2(\\rho_p-\\rho_f)gr^2}{9\\mu}", "b2a3916abbf2b36cfd8fe93358b0a23b": "\\mu \\ne 3", "b2a392f872bb2b00eeee9c13b7fb095c": "\\frac{2-a}{\\sqrt{2-2 a}}", "b2a3a5b3bd9ed80c0671949213a7971e": "\\forall (x_n)_{n\\in\\mathbb{N}} \\subset I:\\lim_{n\\to\\infty} x_n=c \\Rightarrow \\lim_{n\\to\\infty} f(x_n)=f(c)\\,.", "b2a3b6c2ddce0d2a310eed1d04732d24": "\\langle (\\delta \\vec{r} )^2\\rangle _{vac}=\\sum_{\\vec{k}} \\left(\\frac{e}{mc^2k^2} \\right)^2\\langle 0|(E_{\\vec{k}})^2|0\\rangle =\\sum_{\\vec{k}} \\left(\\frac{e}{mc^2k^2} \\right)^2\\left(\\frac{\\hbar ck}{2\\epsilon _0 \\Omega} \\right)", "b2a3d5da467afbfe4b1bc4a4cfbd66d5": "L u = \\lambda u.", "b2a3dcd6b7346707d18a04dc1436081d": " \\mathbf{\\hat{n}} \\cdot \\mathbf{\\hat{e}}_{\\angle} \\mathrm{d}A = \\mathbf{\\hat{e}}_{\\angle} \\cdot \\mathrm{d}\\mathbf{A} = \\cos \\theta \\mathrm{d}A \\,\\!", "b2a47061cb5cc01f39ad1757ad3bf6e3": "h(t) = 0, \\quad \\forall \\ t <0 ", "b2a4ead0f0aff36e3865323c22f4274d": "D(G) \\ge M(G) = 1-r + \\sum_i d_i \\ . ", "b2a4f9a3281b86366555030cd6936969": "{ix,iy,iz}", "b2a51110e0a7d8e04aeb8ab827f1878f": "u_n\\in V_n", "b2a57d7b4c8887d47e9845cdef9f1b7b": "w - 1 + n", "b2a5b920d6cfaf414bc3a828d55c7245": "\\mathbf{V}_{B} = \\mathbf{V}_{A} + \\mathbf{V}_{B/A} \\,\\! .", "b2a5f64c75587230505ee8d1a706edd2": "\nL_z = l_1n_1+l_2n_2+l_3n_3,\n", "b2a5fdfcbbe24ac058a8a9141bc9ae98": "\n(\\mathbf{\\hat{f}_{0:4}})^T =\nc_4^{-1}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.1907 \\\\ 0.8093 \\end{pmatrix}=\nc_4^{-1}\\begin{pmatrix}0.3386 \\\\ 0.1247\\end{pmatrix}=\n\\begin{pmatrix}0.7308 \\\\ 0.2692 \\end{pmatrix}\n", "b2a618c32c403560e84428aeedfaca44": "dH\\left(S,p,n_{i}\\right) = TdS + Vdp + \\sum_{i} \\mu_{i} dN_{i}", "b2a6370b9aea642d539b57d35328217c": "\\nabla^2 L = L_{xx} + L_{yy} \\, ", "b2a675fa3596a2718e5dc0151594bce4": "\\tfrac{kilometer}{hr}", "b2a6cfa5d751948f9694b02e351a57bc": "h(y) = {\\rm res}_x(g_1,g_2) \\in \\mathbb {Z}_N[y]", "b2a710a58dc7ef2253065d0520ca6326": "\\sigma=\\sum_k\\lambda_k\\sigma_k", "b2a7686369af6fc146d91ca84a898228": "P_2-P_1=m\\Delta V-v_e\\Delta m\\,", "b2a79e265659333176263a376ef46d7a": "S:=S\\cup\\{c_{i,j}|c_{i,j}\\neq 0\\}", "b2a7e3cbdc0bd343ad40ffc6752589b0": "\\displaystyle{W(x_1,y_1)W(x_2,y_2)=e^{i(x_1y_2-y_1x_2)} W(x_1+x_2,y_1+y_2),}", "b2a7ee071b9b15ff67ad7058633e8344": "E_{\\text{bit}}", "b2a80d18b15b0c456722e84ed705d895": "S_0=2\\qquad S_{n+1}=2\\pi V_n", "b2a834060291d284eab0fbe9bfe42e82": "P(Y\\leq x+1/2)", "b2a84ad8cb3bde2600e13698ea618681": "n \\le \\sqrt{p}/\\lambda", "b2a875f1ffbd326141c3d1645de03199": " P(| X | \\ge 1) \\le \\min(1, \\sigma^2) ", "b2a8d267d1a123f063087bbbf73c9aee": "(P \\downarrow Q)", "b2a9197a5681a40cb0d4698ced2449dd": "\\sigma(r)=0", "b2a92e83693925a96004a35a3ad1a2aa": "\n\\begin{align}\n &\\frac{\\partial}{\\partial t}\\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\right] \n + \\nabla \\cdot \\boldsymbol{M} = 0,\n \\\\\n &\\frac{\\partial \\boldsymbol{M}}{\\partial t} \n + \\nabla \\cdot \\left[ \\overline{\\boldsymbol{u}} \\otimes \\boldsymbol{M} + \\mathbf{S} \n + \\frac12 \\rho g (h+\\overline{\\eta})^2\\, \\mathbf{I} \\right] \n = \\rho g \\left( h + \\overline{\\eta} \\right) \\nabla h \n + \\boldsymbol{\\tau}_w - \\boldsymbol{\\tau}_b.\n\\end{align}\n", "b2a9328a2b2b98dbe22c648dac189bee": "\\text{side }a : \\text{side }b = \\left({a \\over 2}\\right)^2 - 1 : \\text{side }c = \\left({a \\over 2}\\right)^2 + 1", "b2a948987cd25ecd90da801860d664d6": "\\textstyle \\underset{i\\in e}{\\sum }w_{i}^{t}", "b2a9799adac892f2dbf1eb54ecd72d2f": "r_n= \\sin i \\ \\sin u\\,", "b2aa3321eabdff001134ecaaea56bdf9": "C^\\prime", "b2aa5c440762961047ee1d4318e034fa": "\\sigma_1=\\sigma_2=\\sigma_3=0\\!", "b2aa77efcd83437aa06e359c930b5835": "%C* = 5%C \\mbox { for } %C \\le 0.30% ", "b2aad55be1ed8d18584ebd05936a74d4": "\\tfrac{d}{dx} \\, T_n(x) = n U_{n-1}(x) \\mbox{ , } n=1,\\ldots", "b2aaf51e188540d74591d71ca5ad7fdc": "Z\\times Z_2", "b2aaf5b659ea03cdf984cba8439b1c89": "\\mathrm{CINT}", "b2ab00255db7033e0eea0640cfa8c9c8": "\\tbinom{c_k}k\\leq N", "b2ab713b421d618102204d6b2896e6ca": " \\ q_{max}", "b2abc6ea7394f2c9ef889f5422bb0fd0": " \\hat{b} \\, \\hat{b}^\\dagger = \\hat{b}^\\dagger \\, \\hat{b} + 1 =\\, : \\hat{b} \\, \\hat{b}^\\dagger : + 1.", "b2abca8acad0203d40179d4fab9e0088": " 1 = \\mid a_R \\mid^2 + \\mid a_L \\mid^2 ", "b2ac27eca36ba4337e353255c001e987": "F : \\begin{array}{rcl}\nSmProj(k) & \\longrightarrow & Corr(k) \\\\\nX & \\longmapsto & X \\\\\nf & \\longmapsto & \\Gamma_f\n\\end{array}", "b2ac34e4412c10b84740f93222826646": "\\langle Ax, y\\rangle =\\langle x, Ay\\rangle", "b2ac4223f077afe02a125d12ee79a239": "\\Delta\\Delta G_{i, j}^{stat} = \\Delta G_{i | \\delta j}^{stat} - \\Delta G_i^{stat}", "b2ac55f34e929c4a57b2c5c78e1b48ef": "\\int \\arccos{x} \\, dx = x \\arccos{x} - \\sqrt{1 - x^2} + C , \\text{ for } \\vert x \\vert \\le +1 ", "b2ac7e0eca494a427265c904af9577c0": "r \\leq s ", "b2ace1bdc6e897d910330be41683095c": "\\rho_1 = \\frac{H(x)Q(x)(G(x)Q(x)+x^2H(x^9)Q(x^9))}{Q(x^3)^2}", "b2acf1bccaef91741702e922d0c532b2": "\\mu_A.", "b2ad3d92fdb22063d1eb100609ae29bd": "m>2", "b2ad748a14102441e38a60a94c8ab2f6": "\\tfrac{1}{X} \\sim \\operatorname{Log-\\mathcal{N}}(-\\mu,\\ \\sigma^2).", "b2ae0ff86a05502e4057edca71b55a31": "\\mathbf{F}_5", "b2ae64b55442c7c9684a111472756d19": "\\ p_1", "b2af43dae40a7ef20394cbab68225135": "T_{sample} \\ge T_m + T_a ", "b2af5829edee88975e0de0c8a0e8f115": "A_n \\equiv {\\rm as}_n(\\mathbf{x})", "b2afa3d597d85d6407f6bd6f0509a957": "f(x)=x\\cdot\\left[1-\\left(\\frac{x}{c}\\right)^2\\right]^p\\exp\\left\\{-\\chi\\cdot\\left(1-\\left(\\frac{x}{c}\\right)^2\\right)\\right\\}", "b2afbf48e4423b72000a68ec5c7512c7": "T_{12}=\\eta_0 \\dot \\gamma \\left(1-\\exp\\left(-\\frac t \\lambda\\right)\\right)", "b2b019062e4461b70cd23c4ed33e88ec": "\\hat{H}_{2} = - \\sum_{i>j} \\frac{q_iq_j}{2r_{ij}m_im_jc^2} \\left[ \\mathbf{\\hat{p}}_i\\cdot\\mathbf{\\hat{p}}_j + \\frac{r_{ij}(r_{ij}\\mathbf{\\hat{p}}_i)\\cdot\\mathbf{\\hat{p}}_j}{r_{ij}^2} \\right]", "b2b08bdb59a69aa76c327da93b0e4533": " \\mathrm{sys}(X). \\, ", "b2b0d00c3e9c5c209a997deec6ae942c": "\\scriptstyle \\mathbf{F}=\\nabla f", "b2b0d0ee07ee3563fa17144760ef82fb": "\\gamma \\in \\Gamma^* \\}", "b2b0d1d95370a822d4c506acd497423b": "\\begin{matrix}{52 \\choose 3} = 22,100\\end{matrix}", "b2b0f05294163681518decab691a9901": " \\tfrac{1}{2\\pi} ", "b2b0fcf75948ad1f8e7787bfa1e55597": " \\!\\ K_n = \\frac{1}{2\\sqrt{2}} {(\\delta_S^{n+1} - {(2-\\delta_S)}^{n+1})} ", "b2b11f660c622f85a891a91b8b0f8198": "\\operatorname{hacovercosin}(\\theta)", "b2b1299c2b422e258a497fe265ebc152": " \\mathbf{T} = \\mathbf{Z} \\cdot \\mathbf{W} = \\mathbb{KLT} \\{ \\mathbf{X} \\}.", "b2b165fabc7672fbb131ea9642e94981": " A,B, \\epsilon \\in \\R, A > 0", "b2b199eca231239ffb412d81d635a2d8": "\nw_{k} \\equiv \\frac{\\partial W}{\\partial J_{k}}\n", "b2b1d5b148e392b958aeffa8fc78570a": "S(x) = a_0 + x b_1(x) = b_0(x)", "b2b2a7567ce5ab3bca737e1b48270c8f": "X = \\left \\{ a, b \\right \\} \\,\\!", "b2b2ea39924d1562fd02adf7f2d097dc": " \\begin{bmatrix} \\ln x_1 \\\\ \\vdots \\\\ \\ln x_k \\end{bmatrix} ", "b2b31feb84e9f5250e22d8070fad1af3": "U_1 \\setminus \\{x_1\\}", "b2b3531a3baf9951f5dbd3d826e11358": "\\scriptstyle \\{|i \\rangle_A\\}", "b2b36bab8a62c3e52467eff868fe1a71": "x_{n+1}=2x_n\\ (\\textrm{mod}~1)\\,", "b2b3994d117ceb314790fa224d9d95a1": " \\begin{bmatrix} U \\end{bmatrix} ", "b2b3a7e02b6911ef179f5553ab56075b": " \\frac{(1+\\sigma)^r|Df|}{\\Xi}", "b2b3c26a2bc59d18a5361483bcacd59a": "x>e", "b2b3cc72284dac7840a83981bd5a9294": "I=\\{r\\in M\\,:\\,a s_{V_2}(\\mathcal{R}') ", "b2dab0a1ba9bf4e5197be178c68adbd4": "c_n=\\sum_{k=0}^n a_k b_{n-k}.", "b2dab1d88c0ffc15b608c845ba04d817": "p_1(x), p_2(x), \\ldots, p_s(x)", "b2db06b7cf1fffdbc4e2f18b10bb855f": "P-R=\\sigma_{n,n}\\,Q", "b2db69eca084ea38bb96cde2b807f3f6": "P = -VI \\,", "b2db85aed1ee20d39134f5e8eb33be0f": "\n\\frac{1}{2}\n\\exp \\left( \\frac{1}{2} \\log \\frac{1}{(1-z)^2} \\right)\n+\n\\frac{1}{2}\n\\exp \\left( \\frac{1}{2} \\log (1+z)^2 \\right)\n", "b2dbc0e345c4f2f1eff0be5cd4d7eb88": "\\textbf I_n\\;=\\;\\int_0^{\\frac{\\pi}{2}}\\,\\frac{1}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^n}\\;\\mathrm{d}x,\\,", "b2dbe9a0d5cbefb70c64fdce9cac0bab": "\\boldsymbol{p} .", "b2dc5b9ec8ec71b297573b65ca0ac56a": "\\frac{dL}{dt}=LA-AL", "b2dce6ada2c5d8fe8660ccdc92084e1c": "\\scriptstyle i(t) \\;=\\; C \\frac{\\operatorname{d}v_{\\text{out}}}{\\operatorname{d}t}", "b2dd28b7041ed7f4e91147debbfa5655": "(\\alpha^k \\smile \\beta^\\ell) = (-1)^{k\\ell}(\\beta^\\ell \\smile \\alpha^k).", "b2dd502a6fe4cf33bf9dec4637bd05b3": "C_0 = F(C_1)", "b2dd828f3bbd3b9a139d97c3b292a7dc": "f_i(r_1,\\dots,0)", "b2dd8c12339542d4b86fe519288e4811": "T_{\\mathrm f}", "b2dda93bc6c53abfac92cdc9f8052f31": "-\\Delta\\,H_s = \\gamma\\,_0^s - \\gamma\\,_1^s - \\frac{2H_m}{ZX_c(1-X_c)}", "b2ddb046ae3f72d3a624b321af4c51f0": "|K(x)| \\geq a |x|^{-n}", "b2ddd76f0fbb14334d0c58ee5f8abbc6": "\\mathcal{Q}^1_{Hur}(I) = \\{x \\in \\mathcal{Q}_{Hur}^1 : x \\equiv 1 (", "b2de265cbed4fe2a2b00d733ef445151": "\\frac{E^2}{c^2} - p^2 = m^2c^2", "b2de7d60c0a1dcef693df2bb4f2d96af": "A(n)\\simeq\\R^n", "b2df7aed5d4b9ae45a6f8daeb0c97d59": "P_n(unknot)=q^{n-1}+q^{n-3}+\\cdots+q^{1-n}", "b2dfc0bed9e638d3714c96febfe347a1": "K_{k+1}", "b2dfd62d121e33a2fcb5cc1b1d7b91a8": "1^{st}\\,\\!", "b2e000f5b490a1c71738edbbbb286827": " \\left[ \\bar{x} - \\frac{cs}{\\sqrt{n}}, \\bar{x} + \\frac{cs}{\\sqrt{n}} \\right], \\,", "b2e0353a73b30def3cbb203a3898d73e": " q^ \\ast \\equiv \\ (c/z) \\bmod \\ (d/z) ", "b2e049a0417e1965cb465b354db5c035": "z = R_C + R_D + R_L + R_G + R_R + j \\omega L - \\frac {1}{j \\omega C} \\, ", "b2e0ceac6bd18da00a1e23684fe47f6e": "x,n \\in \\mathbf{F}_{p}", "b2e0d54f64466804fed9e95428bdc0ab": "\n D\\,\\left(\\frac{\\partial^4 w^0}{\\partial x_1^4} + 2\\frac{\\partial^4 w^0}{\\partial x_1^2\\partial x_2^2} + \\frac{\\partial^4 w^0}{\\partial x_2^4}\\right) = -q(x_1, x_2, t) - 2\\rho h\\, \\frac{\\partial^2 w^0}{\\partial t^2} \\,.\n ", "b2e0ff135c9c01d43edc18f679bd90ec": "P|Q \\equiv Q|P", "b2e1010a52a5cae426542daada810cf9": "U_i = \\{[x_0:\\cdots: x_n], x_i \\neq 0\\},\\quad i=0, \\dots,n.", "b2e144ade832e9ce66fe984019dac672": " \\langle Tx \\mid x \\rangle \\ge 0 ", "b2e153c9cd0b3250e5f10919ca76bfba": " {\\rm det}\\, XTX^{-1} ={\\rm det} \\, T.", "b2e1862f056ce431613ebf06e044950d": "\\mathbb{E}[z^N]=\\sum_{n=0}^m\\frac{(z-1)^n}{n!},\\qquad z\\in\\mathbb{R}.", "b2e2363f3e560706fa957ad3e0137f59": "M_{1} \\equiv \\int d\\zeta \\ \\lambda(\\zeta) \\ \\zeta", "b2e274a038406731e8821a099c845286": "_M^7", "b2e27bc9b8a78206347fb5622e1eaf69": "\\frac {B^{1/3} \\cdot \\textrm{Size}}{ \\textrm{Productivity} } = \\textrm{Effort}^{1/3} \\cdot \\textrm{Time}^{4/3}", "b2e2fb58b8fec5c6b987fdc0cbefe470": "(H_1,\\{O^i_1\\})", "b2e2fb6d5a4309812c9841600f9c3884": "\\! e^{\\lambda(e^{it}-1)}", "b2e309e74e0ac25b253392e685614ebb": "BS(1,-1)", "b2e30ac010c1ba2afc6549c66554e4c8": "\\begin{align}\nr_1 &= x_1 + x_2 + x_3\\\\\nr_2 &= x_1 + \\zeta x_2 + \\zeta^2 x_3\\\\\nr_3 &= x_1 + \\zeta^2 x_2 + \\zeta x_3\n\\end{align}", "b2e326c865572138d1885c88af745bd7": "\\Gamma(U,-)", "b2e35a7c3c1b400bc4f39ff7b5887002": "[f,\\Delta](s)=\\Delta(f\\cdot s)-f\\cdot \\Delta(s).", "b2e3acac70abbfedae49ec0e8f444c2a": " = \\frac{1}{4\\pi}\\left(\\iint_{\\vec{r}' \\; \\text{is at infinity}} \\frac{\\vec{E}(\\vec{r}')}{\\|\\vec{r}-\\vec{r}'\\|} \\bullet d\\vec{A}' - \\iiint_{\\vec{r}'} \\left(\\vec{E}(\\vec{r}') \\bullet -\\frac{\\vec{r}' - \\vec{r}}{\\|\\vec{r} - \\vec{r}'\\|^3}\\right)d\\tau'\\right) ", "b2e3e180ad364034821d8bf7ec82cb50": "e^{\\varphi(p)/p_1} \\mod p", "b2e3ebf684f01631e7752bdd0a4bdc94": "\\left\\langle\\mu\\right\\rangle = \\mu \\tanh\\left(\\mu B\\beta\\right).", "b2e3fd46a3de8bba5c68b1b50bcadeb2": "\\operatorname{Li}_2(\\tfrac12) = \\tfrac1{12} \\pi^2 - \\tfrac12 (\\ln 2)^2", "b2e4185c41cfe3d8d09f3ab47b19065f": " \\bar \\nu_{P,R} = \\bar \\nu _{v^\\prime,v^{\\prime\\prime}}+(B ^\\prime+B^{\\prime\\prime})m +(B^\\prime-B^{\\prime\\prime})m^2,\\quad m=\\pm 1, \\pm 2 \\ etc. ", "b2e455b7a658dfdd358ca73816832dde": "r={{h^2}\\over{\\mu}}", "b2e4cb285567a8e5904a6fa8b97754d3": "ax + b \\leq \\varphi(x),", "b2e4fd2fc5b03e0195b5552778c2327e": "\\displaystyle v", "b2e5084b2328ec8cf93a641391f84186": "(1 - z)^{-1} = 1 + z + z^2 + z^3 + \\cdots", "b2e52c90e7a5aa1958669aec5199452e": "\\Delta V = \\omega PAr\\Delta T\\left(1+\\omega^2\\tau_E^2\\right)^{-1/2}", "b2e5b279ab92b8478c292cceec504993": "\\alpha_{i}(t) = \\gamma_i(t) + \\frac{\\sigma_{ii}(t)}{2} ", "b2e5c93aaaebb9668f06a33103e8703e": "\\mathbf{g}(\\mathbf{x})\\leq\\mathbf{0} ", "b2e6312fdcfe80d572b1cdae39997d81": " \\frac{2}{3n} = \\frac{1}{2n} + \\frac{1}{6n} ", "b2e63854758885c274da75aa5c35ea20": "g=G \\frac {m_1}{r^2}", "b2e65f286b63bb822bb86a88c42c2396": " \\eta_{ab} = \\eta(f_a, f_b) = {\\rm diag}(1,\\ldots 1, -1, \\ldots, -1)", "b2e6721d25a3bd1609ad0ae71723a537": "v_{\\text{c}}={2\\pi DcdN_{\\text{A}}}", "b2e6984038e1e4beff84a141ed052086": " \n\\mu_{\\alpha}=\\lim_{Q\\to P}\\frac{P'Q'}{PQ}\n= \\lim_{Q\\to P}\\frac{\\sqrt{\\delta x^2 +\\delta y^2}}\n{\\sqrt{ a^2\\, \\delta\\phi^2+a^2\\cos^2\\!\\phi\\, \\delta\\lambda^2}}.\n", "b2e6b4b500a2061ad279556a1bbb6982": "D \\left( x, y, \\sigma \\right) = L \\left( x, y, k_i\\sigma \\right) - L \\left( x, y, k_j\\sigma \\right)", "b2e6dbb29dfbfbcfb16a00c824118ee4": "P^m", "b2e7507cfe51436a016df9936ff919e2": "\\Lambda = \\begin{pmatrix}\n\\gamma&-\\beta \\gamma&0&0 \\\\\n-\\beta \\gamma&\\gamma&0&0 \\\\\n0&0&1&0 \\\\\n0&0&0&1\n\\end{pmatrix}\n", "b2e787a96969004634d09a91b45b491c": "e_j \\in S_i", "b2e7e732d0690b0a49e17c326cb9fd60": "\\mathrm{A\\ m^{-1}}", "b2e7ed348aa73743e400a026ac0577a0": "\\frac{dv}{dt}=-\\frac{dp}{dt}z+\\frac{dr}{dt}x", "b2e7f628b8992f939cf0337f00edbd84": "\\alpha^{\\frac{N\\pi - 1}{4}}", "b2e82fbb4c4cfbeaf7dc39b02eb31b1d": "V_{eff}(q) - V_{ind}(q)", "b2e85d8886d7472d9f74819fc03716be": "p_{1}p_{2}\\cdots p_{n}", "b2e8dc0479156a2925ea1ca702a7c3e5": "\\frac{\\rm d}{{\\rm d}t}x(t)=f\\left(t,x(t),\\int_{-\\infty}^0x(t+\\tau)\\,{\\rm d}\\mu(\\tau)\\right)", "b2e8e5cf0b7748c712e621941398c5e7": " \\hat{p}_x = -i\\hbar \\frac{\\partial}{\\partial x}, \\quad \\hat{p}_y = -i\\hbar \\frac{\\partial}{\\partial y} , \\quad \\hat{p}_z = -i\\hbar \\frac{\\partial}{\\partial z} \\,\\!", "b2ea640611d80f2a795fea9c07bd454a": "\\text{Headwind} = \\cos[60^\\circ] \\cdot 15 \\mathsf{knots} \\approx 7.5 \\mathsf{knots} ", "b2ea88139d5f7f9a9c00fade0f7173d3": "M_{\\lambda}", "b2ea9af34c1ffde31984b205e1f9a79b": "\\textbf{P}_{k\\mid k}^a = (\\textbf{I} - \\textbf{K}_k\\textbf{H}_k)\\textbf{P}_{k\\mid k-1}^a(\\textbf{I} - \\textbf{K}_k\\textbf{H}_k)^T + \\textbf{K}_k\\textbf{R}_k^a\\textbf{K}_k^T ", "b2eaa0b506519cf669dd0639da2b8af5": "f\\colon[0,1]^n\\to\\mathbb{C},\\quad\\hat f\\colon\\mathbb{Z}^n\\to\\mathbb{C},", "b2ead5b98b521d128682fccb511112a2": "L_2 (\\mathbb{R} , dx)", "b2eaf702d925b88c3ec665e6a336ff88": "\n\\phi:\\;\\;\\rho \\left(\\frac{\\partial u_{\\phi}}{\\partial t} + u_r \\frac{\\partial u_{\\phi}}{\\partial r} + \\frac{u_{\\phi}}{r} \\frac{\\partial u_{\\phi}}{\\partial \\phi} + u_z \\frac{\\partial u_{\\phi}}{\\partial z} + \\frac{u_r u_{\\phi}}{r}\\right) =\n-\\frac{1}{r}\\frac{\\partial P}{\\partial \\phi} +\\frac{1}{r}\\frac{\\partial {\\tau_{\\phi \\phi}}}{\\partial \\phi} +\n\\frac{1}{r^2}\\frac{\\partial {(r^2{\\tau_{r \\phi})}}}{\\partial r} + \\frac{\\partial {\\tau_{z r}}}{\\partial z} + \\rho g_{\\phi}", "b2eafb8e367a3cc13a58e20a05d477c2": "Target_{mod} = \\min \\left (\\max \\left( \\mathrm{round} \\left( \\frac{ \\max \\left( ATEE - 1000, 1000 \\right) }{35} \\right) - 7 - 4, 26 \\right), 71 \\right) ", "b2eb03b3fb8b8f0e66b1d0e4a0f9f15e": " \\psi(x) = e^{-\\bold{r}\\cdot\\bold{r}/ 2a}~ ,", "b2eb095329eabea5d4a99dd07f7c5c65": "1+\\left({3\\over 8}\\right) N_R.", "b2eb544e1b59658d6ce4acc496b69d5f": "\\mbox{eGFR} = \\mbox{144}\\ \\times \\ \\mbox{(SCr/0.7)}^{-1.209} \\ \\times \\ \\mbox{0.993}^{Age} \\ ", "b2eb6e84218664816010c3988fc69423": "K(\\mathbf{p}) = \\det(S(\\mathbf{p})),", "b2ebdf2d768b21e435c3a0ae34ba369e": "a_i \\in \\mathbb{R}", "b2ec5784b737f9fa5bf00d12b8ff4c31": "u_{X \\otimes Y} = (1 \\otimes u_Y)(u_X \\otimes 1)", "b2ec74744cae58cc57975f303c887315": "\n\\sum_{i=1}^n r^{-\\ell_i} = \\sum_{i=1}^n r^{-|s_i|} = F(r) \\le 1 \\; .\n", "b2eca40648477dfae9f18801936d7c5d": "\n \\boldsymbol{B} = \\lambda^2~\\mathbf{n}_1\\otimes\\mathbf{n}_1 + \\cfrac{1}{\\lambda}~(\\mathbf{n}_2\\otimes\\mathbf{n}_2+\\mathbf{n}_3\\otimes\\mathbf{n}_3) ~.\n ", "b2ecc83d04e940653d9eff95f9dc8820": " S_0 = \\frac{P_0}{\\pi R^2}.\\qquad(9)", "b2ed2baa67c2fc85d5c297aa3eed542f": "m_1,...,m_k \\geqslant 2", "b2ed3c45c7e5f719ea8e2afb23ce2305": "\\Delta P = - \\rho g \\Delta h ", "b2edc7312fd620f931aa1dd1e50fcf1f": "\\left(0,\\ \\pm1,\\ \\pm2,\\ \\pm2,\\ \\pm2,\\ \\pm2 \\right)", "b2edd231f40e36803d2f240d7bcef490": "\\left\\lfloor\\frac{K}{4}\\right\\rfloor", "b2ee0be1a1c704c9ba5e8eac1f1c3d93": "\n (\\mathrm{j}_i \\otimes 1)|j_1 m_1\\rangle|j_2 m_2\\rangle \\equiv (\\mathrm{j}_i|j_1m_1\\rangle) \\otimes |j_2m_2\\rangle\n", "b2ee68cef3f3af61a701db53f3647234": "\\langle H_1,\\dots, H_k\\rangle=H_1\\ast\\dots \\ast H_k.", "b2ee7a07d4c4fe8a5606f346ce368931": "\\hat \\sigma", "b2ee7e0381a59bfa625a007bd3676937": "n^{\\underline k}=n(n-1)(n-2)\\cdots(n-k+1)", "b2ee7e8f0ee751b3f0cd519dd0738a4d": "(\\Gamma \\vDash A)", "b2eeb7362ef83deff5c7813a67e14f0a": "596", "b2eebdcd905f27b3b455eb936bf4ce89": "(\\nabla\\cdot\\nabla) \\Phi = (\\nabla_i \\nabla_i) \\Phi ", "b2eebe3a241a72cbec1699b9bfe9d7d4": "\\ G^{*}(f)=G^{'}(1 + jh) ", "b2eeee520a7cd68c51cdfe3205deaf1a": "\\omega_0 = \\sqrt{\\frac{1}{LC}}", "b2ef0c95e03503e9a3a50a98cb64634d": "\\lceil x - 0.5\\rceil.", "b2ef1288b47bfaf1802c2eb74a0f80a5": "I_{lm}", "b2ef7b90bc1c9ee2211b9327f3fd450b": "W = cL^b \\,", "b2eff04897fc670e738ece90af718fe3": "U = \\{1, 2, 3, 4, 5\\}", "b2f00e3023a4f99d9ca4ee5e3d368f29": "{_uM_u}=\\frac{q^2}{gy_1}+ \\frac{y_1^2}{2}", "b2f08186a452656e6f3f79091acb3545": " \\left \\| A \\right \\| _{\\alpha,\\beta} = \\max \\limits _{x \\ne 0} \\frac{\\left \\| A x\\right \\|_{\\beta}}{\\left \\| x\\right \\|_{\\alpha}}. ", "b2f0ab81fe4336470a2c03dddfda88e7": " e^{-q \\tau} \\Phi(d_1) \\, ", "b2f0dd5b0f2a9affd64d997a97221b96": "d^2_i", "b2f0eb04cee6ad98a70f95240f0e5210": "\\mu_1:=-\\sum_{j=2}^k \\mu_j", "b2f117a0da071372814db6f6503b5912": "\\scriptstyle \\delta t_{\\text{orbit-relativ},i} (\\boldsymbol{r}_i,\\, \\dot{\\boldsymbol{r}}_i)", "b2f11b285846c67f583bae66a39ef4fa": "l(l+1)", "b2f146c7defd3eec04ee0e85d0976284": "|v_{\\alpha}(\\phi_{\\alpha})| = \\sqrt{v_{\\alpha,0}^2 + \\frac{2 e \\phi_{\\alpha}}{m_{\\alpha}}},", "b2f1738b69736644232e7f3f8baad0db": "g(\\epsilon)\\sim \\epsilon^{(d-2)/2}", "b2f193fcbedd4789a019c94c8f7b23cc": "\nF L ( \\sin \\theta_1 + \\sin \\theta_2 ) = k_\\theta \\theta_1\n", "b2f19ee47984c2463249d5eac26ba658": "G_{2k}(\\tau) = \\sum_{ (m,n)\\in\\mathbf{Z}^2\\backslash(0,0)} \\frac{1}{(m+n\\tau )^{2k}}.", "b2f1fc0bd0352b69869deb3dfaf7b708": "p(t) = 0\\,", "b2f2942efce4451ac9b652c7b0ca32b8": "\\sum_{k=0}^{n} z^k = \\frac{1-z^{n+1}}{1-z}\\,\\!", "b2f3133f811e3de65dc09b272f39bd07": "% Right-to-left shunt = ((Total body counts - Total lung counts)/Total body counts)*100", "b2f36239a2ad48a493a52dafd6ada9c9": "\\epsilon = \\phi_y - \\phi_x", "b2f43ecbbb4ffab23f024c2552e700f4": " \\underset{h}{\\operatorname{arg\\,min}} \\frac{ 2 \\bar{m} - v } {h^2} ", "b2f454442c9c270eb99771d760b26704": "L_w=qS_w\\frac{\\partial C_L}{\\partial \\alpha} (\\alpha-\\alpha_0)", "b2f4600889cf0fd31291cf2056daf45b": "M_{AB} = \\frac{EI}{L} \\left( 4 \\theta_A + 2 \\theta_B \\right) = 0.4EI \\theta_A + 0.2EI \\theta_B", "b2f4719d592f69b002e7e02fceba00c8": "\nx^2 = c\\qquad(c>0)\\,\n", "b2f4dc99edb7623881fa8e36e0703e1e": "p' = 1 - \\frac{33}{20}n \\cos 2 \\theta \\qquad q' = \\frac{33}{20} n \\sin 2 \\theta", "b2f547e61c2795dbf015b14ae0144d24": "s(A) = \\max \\{|\\lambda_i - \\lambda_j| : i,j=1,\\ldots n\\}.", "b2f5824096f4800f33f2f72a9c2c8161": "\\Bbb{Z}[y]", "b2f599ddbe130d56b9442b1a6116408e": "y \\in \\mathbb{Z}_q", "b2f5e3fefa176986e02a2a436617df45": "t_1 \\leq t_2", "b2f5f4b90fcc45031eb3beea74bc3ab6": "\n N_U=\\min_{t_Q}\\{N_Q+q_0(t_P-t_Q)-k_0(x_M-x_U)\\} \\qquad (6)\n", "b2f5ff47436671b6e533d8dc3614845d": "g", "b2f603ad23395f7e5658146ee8dfded9": "\\langle m|H(0)|l\\rangle=0", "b2f627a7f1cf87fdf6a1b7bba9c3a57c": "\\approx 1.29/\\sqrt{n}\\,", "b2f670056e1be0a795c8422a8871da1f": "\\epsilon_{ij}^{\\text{s}} = \\epsilon_{ij} - \\tfrac{1}{3} \\delta_{ij} \\sum_k \\epsilon_{kk}", "b2f69b142fc2b480f7dc019c7222f80a": "\\rho =\\frac{-g_{0k} v^{k} \\pm \\sqrt{(g_{0k} g_{0q} -g_{00} g_{kq}\n)v^{k} v^{q} } }{g_{00} v^{0} },", "b2f6e729347446a74200c3682fff825b": "T = T + \\alpha \\, ", "b2f6f615bc97af533b59ecd718536983": "O(R^2)\\,\\!", "b2f709e9eb8b0ac70d8863157fba36e0": "\n{\\sigma_1 \\dots \\sigma_n} = {k^{1}\\sigma_0 \\dots k^{n}\\sigma_0}\n", "b2f758012ecfcc655aac22f5b5d218e7": "\\widetilde{K}^n", "b2f784b358c52293aaa5de8dc695e289": "{\\mathcal M}_p", "b2f78f4a7f0ac769984e77779d28acf9": " s=1 \\, ", "b2f8941efa1f1bf607daa2f44d82154a": "B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\\frac{182}{3}x^{12}+\\frac{572}{3}x^{10}-429x^8+\\frac{1820}{3}x^6\n-\\frac{1382}{3}x^4+140x^2-\\frac{3617}{510}", "b2f8fac895ae3321ac21105f79bd9924": "s\\le D", "b2f96b60fb479ed55ee6ddd695241e68": " g \\in X' ", "b2f98901702434983310d0737eacc0a8": "\\scriptstyle{R}", "b2f98aeaa0d2cf14e5a9542d752b63a3": "\n \\varepsilon_{ij} = \\left(\\tfrac{1}{3}\\varepsilon_{kk}\\delta_{ij}\\right) +\n \\left(\\varepsilon_{ij}-\\tfrac{1}{3}\\varepsilon_{kk}\\delta_{ij}\\right)\n", "b2f992829fb7acded0075c616b48eb1b": "CO + DO + EO + FO + \\text {arc} CD + \\text {arc} EF \\,\\!", "b2f9b7797b78dd8aceb2b926c63fb373": "a(u,v) \\cdot f(u,v)", "b2fa7603ed3ff550f3d5aa08790ce620": "x=x(u_1,u_2,u_3), y=y(u_1,u_2,u_3), z=z(u_1,u_2,u_3)", "b2fa95084cd1d4e7d24eeda52c75e012": "23 = 12", "b2fa98e9248be0ac3d8f22b78a46b3b6": " P_n = \\frac{E^2}{R}\\,\\!", "b2faa9bd06c141b7e73fffb5bf5c01db": "O(pn^{\\frac{1}{2}}\\log^{\\frac{3}{2}}n)", "b2fb74518ea302a224d437f9394191f6": "\\frac{dy}{dx} = F(x)\\,\\!", "b2fb953b9957a2af96592bd740ee3937": "pV = NRT", "b2fb98d0dc742c236e19302fa2799d83": "\n\\Delta \\hat{z}\\ =\\ \\int\\limits_{0}^{2\\pi}\\frac{f_z }{V_t} (\\hat{g} \\cos u + \\hat{h} \\sin u)\\frac{r^2}{\\sqrt{\\mu p}}du \\quad \\times \\ \\hat{z}\n=\\ \\frac{1}{\\mu p}\\left[\\hat{g}\\int\\limits_{0}^{2\\pi}f_z r^3 \\cos u \\ du\n+\\ \\hat{h}\\int\\limits_{0}^{2\\pi}f_z r^3 \\sin u \\ du \\right]\\quad \\times \\ \\hat{z}\n", "b2fbfa1a687362042e2a958aed18e212": "H^\\ast(X; \\mathbf{Z}/2\\mathbf{Z}) = \\bigoplus_{i\\geq0} H^i(X; \\mathbf{Z}/2\\mathbf{Z})", "b2fc22965700de03d2b96a7f9de832d9": "P(x)=2(x+1)(x-2)(x-\\frac{1+i\\sqrt{31}}{4})(x-\\frac{1-i\\sqrt{31}}{4}).\\,\\!", "b2fc41af400b39dbbd7498243ad5ffad": "(1+x)^n=\\sum_{k=0}^n{n \\choose k}\\cdot x^k.", "b2fc4e3792aa9246c0fc56cec5321f73": "\\{a \\vee b\\}", "b2fcb9284254e6c9550462067ff17916": "\\begin{align}\n\\mathbf{F}&=q_{\\mathrm e}\\left(\\mathbf{E}+\\mathbf{v}\\times\\mathbf{B}\\right)\\\\\n & + q_{\\mathrm m}\\left(\\mathbf{B}-\\mathbf{v}\\times\\frac{\\mathbf{E}}{c^2}\\right)\n\\end{align}", "b2fcfa587253cd4b756f09aea29e68c7": "\\ \\mathbf{F}_{jk} = \\langle \\chi_j\\mid \\nabla \\chi_k\\rangle;\\qquad j,k=1,2,\\ldots,M ", "b2fd80a7662a2a0212f7fd2a57ab3e23": "x'_{m+j,i}=1", "b2fdcdf4cb3ecf4dcc1f523f0c6f37ec": "\\frac{GVD(\\lambda_0)}{A}L \\le \\tau_p \\le A GVM(\\lambda_0)L ", "b2fe7b064a31b70c828db758ae6374b8": "\\mathrm{d}S", "b2fec01db153f4efdb5354fa740f252d": "\n\\vec{R}_A^0 \\equiv \\vec{\\mathbf{F}} \\cdot \\mathbf{R}_A^0\n=\\sum_{i=1}^3 \\vec{f}_i\\, R^0_{Ai},\\quad A=1,\\ldots,N\n", "b2feffe0e4cc095e51cd7cfd0f072d1c": " \\langle \\beta, \\alpha \\rangle", "b2ff05b986b4b9b38933dbdda149c689": "\\scriptstyle f \\;=\\; t_A / (t_A \\,+\\, t_B) \\;=\\; 0.75", "b2ff665e53fd029b9e15d9d06fa5e9f3": "f(x^*)", "b2ff77986c0f00b77ca693c3ef8f0b2a": "\\eta_2 := (\\eta \\otimes \\eta) : K \\otimes K \\equiv K \\to (B \\otimes B) ", "b2ffac8b267e74089a92fef2bf7f1c64": "w =A\\exp(S/k)", "b3000d9be909c6a79da301e48aae025c": " \\log\\left( \\frac{G(1-z)}{G(z)} \\right)= z\\log\\left(\\frac{\\sin\\pi z}{\\pi} \n\\right)+\\log\\Gamma(z)+\\frac{1}{2\\pi}\\text{Cl}_2(2\\pi z) ", "b3003516b6ffffcfaaf49dfa36042e27": "i_{\\text{B}}", "b3003578382fe20fffc884128088df72": "(i-1,j-1,k-1,y_4)", "b300423d9c46046302a6e86a172d1c30": "g(\\mathbf{a}, \\mathbf{b}) = a_1b_1 E + a_1b_2 F + b_1a_2 F + a_2b_2G.", "b3004a3ab4fe299ccddb3bb6e07206d6": " f(x) \\approx f(x_0) + (\\nabla f)_{x_0}\\cdot(x-x_0) ", "b300cb8a2728cf364877cac6eae4df58": "(s\\circ t)(q)=s(t(q))", "b300f7e8b563e1a506acb5a5479bfa56": "\\Psi\\propto\\begin{pmatrix}\n(1+\\gamma)r^{\\gamma-1}e^{-Cr}\\\\\n(1+\\gamma)r^{\\gamma-1}e^{-Cr}\\\\\niZ\\alpha r^{\\gamma-1}e^{-Cr}(x-iy+z)/r\\\\\niZ\\alpha r^{\\gamma-1}e^{-Cr}(x+iy-z)/r\n\\end{pmatrix}", "b302362eb6571956b12353e6b259c295": "C = \\frac{1}{B} \\frac{d^2\\left(B(r)\\right)}{dr^2}. ", "b30245c9c5bf00cd984c6f92e0f57633": "2^\\kappa>\\kappa^+\\,", "b302a520c037429b65c5ca6a9ad7da16": "Q(D_2 \\varphi) = Q(2\\cdot (h * \\varphi)) = 2\\cdot (h * Q\\varphi)", "b302a5b8a882603d1c5707882a06c11f": "L=K[a]", "b302d8ceedb346012a3c9e687d6cf504": " \\ x^2 -Ny^2 = 1 ", "b3049c97377f3ffa2c7e68aa271549df": "(U,m)", "b304ac50e37f4acc0c5663163739c917": "y_i^{j} = k_{i}^j- k_{i-1}^j", "b304e16efd77abf5c37023ac5a2f6675": "a_1=a_2=\\ldots=a_s=a\\text{ and }a\\notin\\N\\text{ and }s\\in\\N^+.", "b304f396b1f3ae96386badc95727d6db": "L_{n-l-1}^{2l+1}", "b30528dbb7987ca34dd8f2da55f9bb55": "T(p)=\\Box p", "b30556631747b6e17035fcb3b9f65678": "\\det\\begin{bmatrix} 3-\\lambda & -4\\\\4 & -7-\\lambda \\end{bmatrix}", "b305b1a674914c8aa741db076fda10c7": "\\textbf{X}_0 = \\textbf{H} \\textbf{X}", "b306099f62e689346834ff72bf6e5318": "2^{rel_{i}} - 1", "b30644b0401c972293b9c7c0e1b4fa22": " p \\le \\tfrac32 k(k-1) + 1 \\quad\\text{and}\\quad n \\le \\tfrac32 k(k-1). ", "b30749630f0d843a7816660092fe95e7": "\\dot R =\\ddot R =0 \\ ,", "b3078514e84bda9c4c8e0a8ccdca6339": " E \\subseteq X ", "b3079ab7855a3c3cd3dc307ec3c053e1": "\\mathcal{S}: \\mathcal{P} \\rightarrow V^*", "b308789ef3a773624318da2fc3104a37": " \\sum_{k=1}^{n-1} \\frac{k}{n-k} =\n\\sum_{k=1}^{n-1} \\left( \\frac{k}{n-k} - \\frac{n}{n-k} \\right) + n H_{n-1} =\nn H_{n-1} - (n-1)", "b3087fa76022e975b69541b859f17657": "P_\\mathit{SW} = \\frac{VI_o (t_\\mathit{rise} + t_\\mathit{fall})}{2T}", "b308b466147c3c531fa9fac2bf00ad3b": " \\mathbf{p} \\cdot \\mathbf{p} \\equiv p^2 \\equiv p_1^2 + p_2^2 + p_3^2 \\,\\!", "b308d2a7bd88d40bca9c5abf77113d16": " B^*=B-2p \\rho \\phi_0,", "b3090eb4b3cddbe31fa59b035d2bb197": " = e ^ {V1 / V_e} e ^ {V2 / V_e}", "b309279814f54a31b68340d6a07c997e": " x^2 + y^2 = c^2(1 + dx^2 y^2) \\, ", "b30935a8734afb58faf9e57e5fad45ae": "\\displaystyle x^2 = 4x.", "b309484499cbcf2457f5e9c07d868315": "L[F[r]] = r", "b30993644036ddabf4131e14f00712ce": " U(s) \\geq U^*-\\epsilon", "b30a620bc89d8ab28e86be994b3adf07": "(4/3)\\pi p_f^3(\\vec{r}).\\ ", "b30a7accf23abebd0c6c167b0b931851": "J_n = \\sum_r : e_r^*e_{n+r} : ", "b30a7e6ab32107c2cca3cb56bf5ba2a2": "1S_{1/2}", "b30aa35982fe7f21ba3df0bbfbea4b37": "Y_n(x) = \\left(\\frac{x}{2}\\right)^n {\\mathcal K}_n\\left(-\\frac{x^2}{4}\\right).", "b30aa944b03b526ccbd484016048a5af": "\\lceil m/(n-1)\\rceil", "b30acb6105c9d299c61d235e593c9b57": "\\theta^{\\alpha} = du^{\\alpha} - u_{i}^{\\alpha}dx^{i}\\,", "b30aee2e1984443816de3ac728770261": "u \\ge 2", "b30af5818e9af7a574d215f8ae7c2abf": "S = \\cfrac{B^2(H-h)}{6} + \\cfrac{(B-b)^3h}{6B}", "b30b1664ca55b8cb511cbecdb731421f": "\\Delta Y_t", "b30b3d45ab6c61113f5d53ea192238f0": "\\oplus{}", "b30b90ca6f6b3cd7007c9bb2e4dde996": "[Q_B, \\mathcal{H}] = 0", "b30b9412c731ad29e422b5f87be502d1": "g(x,y) = x^2+y^5", "b30bf69c74f25f95f9ebd9b057eb22c0": " B = {\\left ( \\frac{4}{5} \\right )}^{3/2} \\frac{\\mu_0 n I}{R}", "b30bfb7e993aa1297e6433fdf03ae6d4": "i_\\mathrm{D} = I_\\mathrm{R} \\left(e^{v_\\mathrm{D}/V_\\mathrm{th}}-1\\right), ", "b30cd799ba51f154bbafd54f5fc8b6f4": "R_{\\text{on}}=R_1 || R_f || R_L", "b30cdfe8a035d4672a88872c1cdae03d": "\\sum_{r=0}^n a_r\\zeta^r = 0.", "b30cee410b41b4ef81903cd4040429bc": " E = {1 \\over 2} m v^2 - {{e_M}^2 \\over r} = -{1\\over 2} {{e_M}^2\\over r} ", "b30d0c93987469d25336926968dee626": " \\chi_{tdt}^2 = \\frac{4(i-j)^2}{h}. ", "b30d45c0d2cc6584ed17fe999311f124": " \\ell + x = t + (t-h) = 2t-h \\Rightarrow x\\approx 2 - \\left(\\frac{\\varphi}{\\theta}+1\\right) L \\sin \\theta ", "b30d4c59fbeec6dbb27b475b89f22bf6": " R = P + Q = (x_3'(x_1-x_2):y_3':(x_1-x_2)^3)", "b30da265c153eef9c3bd2cd7d77e4822": "F(x(A+\\Delta A)+(h+\\Delta h))", "b30da86868c737218153a938d2adc5a7": "\\|V\\|=n", "b30db2758e539f86bf1280f0b356c1bf": "{\\mathcal O}^\\star(M)", "b30e0678c6c7fb33704461c7310f31f2": " 1 + 0\\omega", "b30e2c450a95ff948877a9270c17c326": "\\forall u_1u_2\\dots u_n(t=t')", "b30e5fabd719836e82d496163995b1f9": " I_D \\approx I_{D0}e^{\\begin{matrix}\\frac{\\kappa(V_{G}-V_{th})-V_{S}}{V_{T}} \\end{matrix}}, ", "b30ea4a70758f8616a64f318963d15c1": "y^3 = x^4 - z", "b30ecdafcb792b2280917c47250b1b20": " \\Delta_D ", "b30f48a69117eb1a1eb7222300635a81": " T_{2n+1}\\left(\\sin\\theta\\right) = (-1)^n \\sin((2n+1)\\theta) ", "b30f693afecb91c1b700c1fab118fe56": "a< b \\in [0,1]", "b30fcbe9bdbe6d85743092a3101f1df3": "\\begin{align}\nX(s) & = \\frac{s \\sin \\phi}{s^2 + \\omega^2} + \\frac{\\omega \\cos \\phi}{s^2 + \\omega^2} \\\\\n& = (\\sin \\phi) \\left(\\frac{s}{s^2 + \\omega^2} \\right) + (\\cos \\phi) \\left(\\frac{\\omega}{s^2 + \\omega^2} \\right).\n\\end{align}", "b30fea55217da33035791cde8a882c36": " s^{signed}_{ij}=0.5+0.5 cor(x_i,x_j)", "b3100106408ffdba59dfd464f97fba8c": "p_{X}(x) = \\int_y p_{X|Y}(x|y) \\, p_Y(y) \\, \\operatorname{d}\\!y = \\mathbb{E}_{Y} [p_{X|Y}(x|y)]", "b3101a8ecf881f3cdc70591f51aef163": "[\\mathcal{F}_n(f)](x) = \\frac{\\sqrt{n}}{n\\sqrt{c\\pi}} \\sum_{k=-\\infty}^\\infty {\\exp{\\left({\\frac{-n}{c} {\\left({\\frac{k}{n}-x}\\right)}^2 }\\right)} f\\left(\\frac{k}{n}\\right)}", "b310378952bc98552e1b65a78ef63bc2": "\\scriptstyle L \\;=\\; L^{-} \\;=\\; L^{+}", "b3105b3ae9c60714dbd807ab2889468e": "[X,Y]:M\\to\\mathbb{R}^n", "b31067fd75038af8a3e3a9d8df83136b": "E_\\lambda : z \\to e^z + ln(\\lambda)", "b31080c386310d0751938d89585f883f": "\\operatorname{OE}[a](t) = 1 + \\int_0^t a(t') \\operatorname{OE}[a](t') \\, dt'. ", "b3108c627ec5bd07fc9beae5c9c84dd0": "\n\\omega = 2 \\hat{k} \\cdot \\vec{\\Omega} = 2 \\Omega \\cos{\\theta},\n", "b310993a88f98558edd11107ae0b86d3": "a_{11} = l_{11} u_{11}", "b310cddca095488132c7573da68c5946": "\\ell_j(x) := \\prod_{\\begin{smallmatrix}0\\le m\\le k\\\\ m\\neq j\\end{smallmatrix}} \\frac{x-x_m}{x_j-x_m} = \\frac{(x-x_0)}{(x_j-x_0)} \\cdots \\frac{(x-x_{j-1})}{(x_j-x_{j-1})} \\frac{(x-x_{j+1})}{(x_j-x_{j+1})} \\cdots \\frac{(x-x_k)}{(x_j-x_k)},", "b3112a7ae75e5f47d5c1a0f22eabf414": " \\mathbf{J}_{\\rho} = L_{\\rho u}\\, \\nabla (1/T) - L_{\\rho\\rho}\\, \\nabla (-\\mu/T)", "b3117136921a56b346e406acba6d16d9": "z'", "b3117b9798edb910ce6bd181d9a04a6e": "A\\mathbf{x} = \\begin{bmatrix} \\mathbf{r}_1 \\cdot \\mathbf{x} \\\\ \\mathbf{r}_2 \\cdot \\mathbf{x} \\\\ \\vdots \\\\ \\mathbf{r}_m \\cdot \\mathbf{x} \\end{bmatrix},", "b3118c3a5c1eef915dca0ec74405a1d7": "\n M = -\\cfrac{f(2h+f)^2}{2}~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} - \\cfrac{f^3}{6}~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\n ", "b311a1d738b40d575b131d9f32224949": "du_o/ds (s) < 0 ", "b311b4b85ff453db25712409b1bb0820": "Q_G(u,v) = u^{k(G)} \\, R_G(u,v).", "b312064cb4116badf7c325488c4b4346": "K=\\frac{3\\mu_0^2}{160\\pi^2}\\frac{\\hbar^2\\gamma^4}{r^6}", "b3121755893394c8d5d26ed07d1e29be": "x=x_\\mathrm{med}", "b3121a40c0a0646f6750ee5192dd29ff": "\\Pr(2\\text{ heads}) = f(2) = \\Pr(X = 2) = {6\\choose 2}0.3^2 (1-0.3)^{6-2} \\approx 0.3241 ", "b312658b0f3f6ff981c9fffd45646c0a": "P_\\sigma", "b3128cc5b9d0c30bb4264e5d04439dd9": "B_{mn}", "b312bda716894a19696c86bea86a990c": "\\mathbf{\\hat{S}} = {\\hbar \\over 2} \\boldsymbol{\\sigma} \\,\\!", "b312db50682b283d93891800813612bc": "j'=\\tfrac{1}{2}(1+\\eta i + \\tau j)", "b312fd02300054c5337f72bede2113ef": " \\delta W = \\mathbf{F}\\cdot\\delta \\mathbf{r}.", "b312fdfec89ba8e8105e162ca2201fcf": "\\#(n)", "b31350ca70b6562aa3fed27c2d10704a": "\\Delta(s)=I-sA+s^2(D-I)", "b31389c17026fbd0d868973dd3b941a0": "\\alpha^{{\\rm N}(\\mathfrak a)} \\equiv \\alpha \\bmod {\\mathfrak a}", "b3138fb853182c3e592f223cc922b216": "\\textstyle p=1", "b313e9a2561d7053a4671a323a856196": "\\lim_{\\Omega\\to\\infty} P(N <= xn) = 0", "b3141da828891903b8f23b9e5433351d": "\\mathcal{A} f (x) = \\lim_{U \\downarrow x} \\frac{\\mathbf{E}^{x} \\left [ f(X_{\\tau_{U}}) \\right ] - f(x)}{\\mathbf{E}^{x} [\\tau_{U}]},", "b3142375086cdfb4aba891363486fc82": " - a^n = c_1(E). a^{n-1}+ \\ldots c_{n-1}(E) .a + c_{n}(E) .", "b31464ffea35b9d3addd4517e3178f40": "\\text{Saybolt Furol viscosity} = \\frac{\\text{Saybolt universal viscosity}}{10}", "b314971dbcdb40fa195389e703f464e1": " P_\\lambda \\, ", "b3149ecea4628efd23d2f86e5a723472": "-3", "b314bde45ace44e762eed4135b031001": "87\\pm 2", "b314fabe05ff014a1466d871bc64b117": "S \\in \\{0 \\text{ for background}, 1 \\text{ for foreground/object to be detected}\\}", "b31509052fe9979febb271ffc4adca94": "\\begin{align}\nL^{4k}(\\mathbf{Z}) &= \\mathbf{Z} && \\text{signature}\\\\\nL^{4k+1}(\\mathbf{Z}) &= \\mathbf{Z}/2 && \\text{de Rham invariant}\\\\\nL^{4k+2}(\\mathbf{Z}) &= 0\\\\\nL^{4k+3}(\\mathbf{Z}) &= 0.\n\\end{align}", "b3150d04a1bd76ffd34fa53bfea53823": "H_y", "b31529a9c8e387f647fd8a3c88335512": "p = 1-\\prod_m(1-p_{m}).", "b3152f0a4ba958e31fe774b1fe2d7847": "v_i=0", "b3157397b7c9184739fafcd0552b9e6c": "a(x) \\in G ", "b315756b0da325edc77c00bdba2f2390": "\\zeta(-n)=-\\frac{B_{n+1}}{n+1}.", "b315af5e591cf5ee380c9e5bea76969c": "{\\mathbf{}}P_i,S_i,\\hat{P}_i,\\hat{S}_i", "b315d2fab08915c5908789c5def916e0": "T(z)", "b31686e4b7c7b015181b11a8e7af7866": "C \\subset N", "b316bf2a47c3c1b73ca67287287059b8": " z= \\sum_{i=1}^n { \\lambda_i x_i }", "b3170c05fa6ad6ade832911adddf47ac": "\\mu_P - b \\sigma_P ", "b3172ae3ec0f17928fb978bc224f4694": "O(\\log^{2} n)", "b317835ba394db0e29300958826d1e1b": "\\scriptstyle{\\dot G \\neq 0}", "b317b3ce8ecbc2b595b8a0547e04c6c3": "F(B,C,D) = (B\\wedge{C}) \\vee (\\neg{B} \\wedge{D})", "b318034ca7d342511903de7f00c4a647": "D_A*D_A\\Phi=0", "b31849f9bb2ccd821fef9373a90fb41e": "\\scriptstyle t \\in \\cup_{k=1}^\\infty [U_k,V_k] ", "b3184c475c28fa3b1e9cfa70808557d9": "\\alpha_i, \\alpha_j, \\beta,", "b3184f185af8acd73a9eb21bad0164aa": "\\deg(Q(X)) \\le {e + K - 1}", "b3187382f055309c56c798a60c43cc12": "P_{(a,b,c)} = \\frac{1}{2}\\left(1 + \\sigma_{(a,b,c)}\\right)", "b31881e86c55958dcd13c9e857c510c4": "dp(x,t) = p(x,t + dt) - p(x,t)", "b318ca600711ba79f277ce06cf6dea1a": "\\bar J(t,t',t_0)", "b31944feb2985a3e4537ce33e3773154": "-3.8\\times10^{-6}<\\frac{v-c}{c}<3.1\\times10^{-6}", "b3194771005d40112302e6e90c6e83fb": "Y,", "b319506040e6eed21c5974bc8a026736": "I_p(P + QR,Q) = I_p(P,Q)", "b319533e116db21161e7a185c279aca0": "E_{\\text{half-cell}}= E^0 - \\frac{0.05918 V}{n} \\log_{10} [ M^{n+}]", "b3196f76cd4858dd9f16fe8ba50303d2": "\\mathbf{U} \\otimes \\frac{\\partial \\mathbf{V}}{\\partial x} + \\frac{\\partial \\mathbf{U}}{\\partial x} \\otimes \\mathbf{V}", "b31a28e5d09fd0b38b8d0d541ed68389": "F = \\mathbb{C}", "b31a532df2bbd20c8e731e326e9b4d8b": "\\bar{U_{j}^2}", "b31afbff0c4b02f22a5544c31d308636": "S=\\int L\\,\\mathrm{d}t ,", "b31b01e02215d61157b561ab12cf731f": " \\hat{S} = \\cos\\frac{\\hat{\\phi}}{2} + \\sin\\frac{\\hat{\\phi}}{2} \\mathsf{S}. ", "b31b8ded308e4640393b9cf0a3bd7041": "T^{ab} = \\mu \\, u^a \\, u^b ", "b31bb0874ce73cdcefbe1310f8739ccd": "\\sigma_{zz} + \\sigma_{yz} + \\sigma_{xz}", "b31c1f0afaa7c6e5ba165f0830de5b8d": "\\theta_0={\\frac{14.1MeV}{\\rho \\beta}}\\sqrt{\\frac{l}{X_0}}", "b31c25a2b8f4123e762bc010f3be3511": "\\displaystyle{\\|u_n\\|_p,\\,\\, \\|v_n\\|_p \\le {\\|\\mu_n\\|_p\\over 1 - \\|\\mu_n\\|_\\infty}.}", "b31c3ddf18a9a937849a46a2f4693b57": "V_{Y0},", "b31c629eaf5e4795120e1a0d712fc85b": "\\mathbf{y} \\sim \\mathcal{N} \\left(\\mathbf{c} + \\mathbf{B} \\boldsymbol\\mu, \\mathbf{B} \\boldsymbol\\Sigma \\mathbf{B}^{\\rm T}\\right)", "b31c6ccf6a6e02cc981d847f1a0a0b84": " N_D = \\frac {4\\pi}{3} n \\lambda_D^3 ", "b31c9601ded98fbf5d257b7ec22a75b5": "Y_\\alpha(x)", "b31c9921732aafa1b9b4b49d8e6afa77": "| \\alpha |^2 + | \\beta |^2 = 1 \\,", "b31ca2303486f163f40932ed95e05904": "\\delta_u(t) := \\delta(t-u).", "b31cfca20db51c47d413a12a780951bd": "X_i | Z_{i-k}, \\ldots, Z_{i-1}, Z_{i+1}, \\ldots, Z_{i+k}", "b31d4535817b7bdec96e8a69ff294d12": "4n+2", "b31d95b3244309dfa0abf15a9074aeea": "d = \\sqrt{2Rh} \\,.", "b31dc509c423e0fb67def2013ef13f6b": "F,G\\in\\mathcal{A}", "b31e11c4fe034d6c22e16b98187b0bcc": "F(-\\mathbf{q}) = \\left|F(-\\mathbf{q}) \\right|\\mathrm{e}^{\\mathrm{i}\\phi(-\\mathbf{q})} = F^{*}(\\mathbf{q}) = \\left| F(\\mathbf{q}) \\right|\\mathrm{e}^{-\\mathrm{i}\\phi(\\mathbf{q})} ", "b31e25f36a1ce68650ec2bd1831ad488": "2L = (N + 1/2)\\lambda/n_2\\,\\!", "b31e6bba30984fb845b22f4066ce237d": "{\\vec{u}} = \\begin{bmatrix}u^1 \\\\ u^2 \\\\ u^3\\end{bmatrix} = {d \\vec{x} \\over dt} = {dx^i \\over dt} =\n\\begin{bmatrix}\\tfrac{dx^1}{dt} \\\\ \\tfrac{dx^2}{dt} \\\\ \\tfrac{dx^3}{dt}\\end{bmatrix}.", "b31ea7c177c487cbd5ed7c8fe6865e20": "\\dot\\gamma(0) = V", "b31f4c451709226268b712d41cb6f291": "\\hat{p}_0=0, \\hat{p}_n=n.", "b31f73db14e189372965962dbc0a6b76": "{}^T", "b31f79ba4fc080b7edcaa94b5f8ce29d": "\\pi r^2 ", "b31ff4a0f2f4f59d51881eff549d85e4": "\n \\underline{\\underline{\\boldsymbol{K}}} = \\underline{\\underline{\\boldsymbol{A}^T}}~\\underline{\\underline{\\boldsymbol{K}}}~\\underline{\\underline{\\boldsymbol{A}}}\n ", "b320382e5bb840835e286c46dde1516e": "\\mu-\\beta\\log \\left(\\tfrac{e^{-X}}{1-e^{-X}}\\right) \\sim \\mathrm{Logistic}(\\mu,\\beta) ", "b3207a621a400a0520e5c64aaf14d75e": " \\left({{o^p}_w}\\right) ", "b3209099a28ec585f44496856a5c97bd": "\\pi(x) > \\operatorname{li}(x),", "b320921360d2d5b5082a89b84ad98d82": "d = {{2D^2}\\over{\\lambda}}", "b320a438d85fa52a48eadd6f45e12cc8": "E_b = \\frac{1}{1 + 10^{\\tfrac{R_a - R_b}{400}}}", "b320dd96e0efaaabd7e13eddf120ed1a": "n^\\text{th}", "b32145d6e95e6810497b83ed3d630a2f": "\\frac1{p} + \\frac1{q} = 1.", "b321ca2c38c25d7605560e7b5c0ce988": "+(n-i)\\delta_{ij} ", "b321dced0e4448ff0eeb40d54a488d2c": "P_{12}=P_{21}", "b321ddba22f01c0aeb93b8c1f951e1a2": "\\frac{1}{64} \\sum_{x=0}^7 \\sum_{y=0}^7 |e(x,y)| = 4.9197", "b3225327d2c7f28fd9d02381725ffc37": "\\mathrm{Cd + 2OH^- \\rightarrow Cd(OH)_2 + 2e^-}", "b32286714953424f7724f8910dea2367": "L_*(R)", "b322ddec5b38908a592e0b68309d38bc": "\\Phi(\\infty) = 0", "b322f846fa2db6bccf378cec979fdc6f": "\n \\frac{z-n}{z+n} = \\left( \\frac{\\zeta-1}{\\zeta+1} \\right)^n, \n", "b3230a50b29f56162a54af630acb7da9": "B_{p,r}(z) = [1+zB_{p,r}(z)^{p/r}]^r", "b32331c9d3542a9cc697069b46ea868c": "\\langle \\ ,\\ \\rangle_k", "b323329972e337616e0fa27a835b0427": "\\dot{v}_4 = {1 \\over C_4} ({v_6 \\over R_6} - {v_2 \\over R_2}) ", "b3233a2c65e055da223ee3c86fdbac31": "dp_t / d \\tau = dp_t / d \\tau \\big/ dp_t / d \\tau = 0", "b3233c3759c3ef3cad9f4c317caddc96": "A\\vdash_L B\\leftrightarrow\\sigma B", "b3236ae2d4b6a357535b4a96d9dad8c7": "\n S(\\boldsymbol{\\varphi}(w)) = \\frac 12 \\sum_{j=1}^n \\mu_j w_j^2.\n", "b32421ebcab57c70f24366e27d6105bd": "H^{2} [G_\\Psi, U(1)]", "b32426a79a04e6b0e9fdf840f2121c5e": "n=\\sqrt{\\frac{\\mu } {a^3}}", "b3245f9ec6840c5302882cf02817c259": " IE = \\sum_{ i = 1 }^K \\frac{ A_i }{ A } \\frac{ B_i }{ t_i } ", "b324785b22639471552a9e94eb3fc14b": "\\Re(s)>-1", "b3252e35021b416593f66d7381274ebf": "a = \\frac{-d}{D} \\begin{vmatrix}\n1 & y_1 & z_1 \\\\\n1 & y_2 & z_2 \\\\\n1 & y_3 & z_3\n\\end{vmatrix}", "b3252ec46f025d45d6385f8e1ced9bf6": " T(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(x_0)}{n!} (x-x_0)^{n}", "b3262abb3cdda0a9e2445e12bfec11c3": "\n\\Delta S_\\mathrm{rxn} = [yS(Y) + zS(Z)] - [aS(A) + bS(B)] \n= \\Delta S^0_\\mathrm{rxn} - k \\ln \\frac{[Y]^y [Z]^z}{[A]^a [B]^b}.\n", "b326553b37a5795d96e162bdcd7ba3a3": "Z_{N}=\\sum_{all\\atop spins}\\prod_{all\\atop faces}w\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right),", "b3266ea17e4ca3b960780c69415f6ab9": " c_2 = -\\frac{1}{4\\mu} \\frac{\\partial p}{\\partial z}R^2.", "b32687ae9c17829711c1b9d8a202b754": "x_c=1/ \\sqrt{2+\\sqrt{2}}", "b326b5062b2f0e69046810717534cb09": "true", "b326e89a91bcd8405d5c344da7d41b95": " \\begin{bmatrix} \\theta_1, & \\theta_2, & \\dots & \\theta_N \\end{bmatrix}\n", "b3271ab515cb4bd333b5eee184a689d5": "T>1\\ ", "b3272004eb76882c27889566afd23b0a": "R_G(p)", "b32740e101b4e92f1abff60f50de070a": "y = C x + D u \\,", "b32741c45e23e7aa30645d835f7abe41": "P_c^{ n+1}(c) = P_c^n(c)^2 + c", "b327a899bb419b25561e0f26a5c9ffe1": "\\sum c_n z^n", "b328126984640e5176f136cdf26111cb": "\\Theta(u)", "b3284a45880400170745efcd00270276": "\\int_0^\\pi f(\\cos \\theta) \\sin(\\theta)\\, d\\theta \\approx a_0 + \\sum_{k=1}^{N/2-1} \\frac{2 a_{2k}}{1 - (2k)^2} + \\frac{a_{N}}{1 - N^2}.", "b32853341973dfad9cb9bd75b2e621ca": "J(U)\\supseteq V\\cap J(X)", "b3288769a036705e209b00c386f5863d": " c_{\\mathrm{s}} = \\sqrt {\\frac{G}{\\rho}} ", "b3288a64e76ff5e5842a772bf8a1e895": "\\displaystyle f(s,t) = (s e^{{\\rm{i}} 2 \\pi t}, t). ", "b328c4df2535645212a4bd50a4223ab2": "p_1,\\ldots,p_r", "b328d7644fea9ec9d61617be81ebe1e4": "|n-a^2/4n|", "b329170f783fc89f7dc93accecca88db": "\\varepsilon > \\max_{x,y} d(x,y)", "b329725f8f77913b47ad9d25f9c35dff": "S^0>S^1>S^2>\\cdots", "b32a0da9bca051430aa4aa1e53925d3d": "\\mathrm{diam}(E_\\lambda)=o(1)", "b32a32f2f1abaf677b2d6920345fb3a8": " \\Gamma(V,L). ", "b32a51b0ba083ef43143959adad058bc": "s = w_{2}", "b32a7f14629938a51c890bcef09d8884": "\\nabla\\times\\mathbf{B}=\\alpha\\mathbf{B} ", "b32a946afccd7015d8e6061b06c35ec1": "\\displaystyle{\\|u\\|_{(k+2)} \\le C(\\|Vu\\|_{(k+1)} + \\|Wu\\|_{(k+1)})\n \\le C^\\prime \\|\\Delta_1 u\\|_{(k)} + C^\\prime \\|u\\|_{(k+1)}.}", "b32ab703d0cbc24c1f669ca7c89db915": " { {d^2 \\acute{x}^{\\mu}} \\over {d\\tau^2}} + \\acute{{R}^{\\mu} }_{\\alpha \\nu \\beta } \\acute{u}^{\\alpha} \\acute{x}^{\\nu} \\acute{u}^{\\beta} = 0 ", "b32add837fecb55b6b5369cac41db771": "f_j^{(p)} = \\sum_\\beta f_{j\\beta}^p X^\\beta.", "b32afa304b6d4f9c2de74286af75cc40": "CP + DQ = 1 \\, ", "b32b12944ec7cb7711916d23d01ba14e": "\n\\xi^{Pickands}_{(k(n),n)} =\\frac{1}{\\ln 2} \\ln \\left( \\frac{X_{(n-k(n)+1,n)} - X_{(n-2k(n)+1,n)}}{X_{(n-2k(n)+1,n)} - X_{(n-4k(n)+1,n)}}\\right)\n", "b32b2a24aa7d6c3ec1ed3acc01a7d2fa": "0 \\le F_{\\mathrm{sf}} \\le \\mu_{\\mathrm{sf}} F_\\mathrm{N}", "b32b4e36c9cbd23e672d531ba0bf2425": "\\ln \\prod_a^b p(x)^{dx} = \\int_a^b \\ln p(x) \\, dx", "b32baea0f7f01917a0a1b74f94095bdf": "O(E + V \\log V)", "b32bf4e20c042e6d911013989a90986f": "2 \\binom{n}{j}", "b32c3dcaf69168cbebf752ae92ba79b1": "\\beth_{d-1}(|\\alpha+\\omega|+\\beth_2)", "b32c408e0dde5e669097fb7409ba1f8a": "(m-1,k)", "b32c5669b1250bf336774ea8c6324817": "x'= \\frac{1}{x-a}", "b32cc5f8bb408d3a3fed5d24b2bb1e5b": " S= AB\\cap M,", "b32cd45992e1909f88beda04183cf3c5": "g(l)", "b32cf808dc3ea0eec3be6e7cb2666a6a": "\\hat{w}=w \\iff (\\exists w)( (x_1^n(w),y_1^n)\\in A_{\\varepsilon}^n(X,Y)) ", "b32d297ddb7570d449d31c0920b0e0da": "({\\mathbf v},{\\mathbf w}) \\mapsto e^{2\\pi{i}{\\mathbf v}\\cdot {\\mathbf w}}", "b32d3c1e8ccc17f86f37d21b07021f72": "{\\varphi \\;=\\; \\forall u\\forall v(\\exists w (x\\times w=u\\times v)\\rightarrow(\\exists w(x\\times w=u)\\lor\\exists w(x\\times w=v)))\\land x\\ne 0\\land x\\ne1,}", "b32d3f4dbd0c3082b48de0578c74ed2b": "y(w)=F_{\\nu}(-w)=e^{-iw \\nu} P(-w) \\,", "b32d75a8bdcd055e83037b39e53d55cc": "\\chi_{0}", "b32da3fe98fd8d27f929efe7d5351ff8": "\\begin{bmatrix} 1 & 0 & 0 \\\\\n 0 & c & -s \\\\\n 0 & s & c \\\\\n \\end{bmatrix}\n \\begin{bmatrix} 7.8102 & 4.4813 & 2.5607 \\\\\n 0 & -2.4327 & 3.0729 \\\\\n 0 & 4 & 3 \\\\\n \\end{bmatrix}", "b32da413bb5de013d3ae3af802b60100": "P_{ex}", "b32da9ee7a0d49f2fe98c93e82d0f57a": "\\phi_B=(\\mu_0\\,I_c\\,N)(\\pi\\,r_0^2)\\quad(1)", "b32e2999e2be729a09396e09c795e093": "S=\\sum e(x_n)", "b32e5082a0368de493b979eb74cae645": " z = r(\\cos \\varphi + i\\sin \\varphi )\\,", "b32ef36c594908179e8a1ee8cf7f5844": "\\bigotimes", "b32f3b852e3ce6fc744c9b8320658db2": "\\sin n\\theta = \\sum_{k=0}^n \\binom{n}{k} \\cos^k \\theta\\,\\sin^{n-k} \\theta\\,\\sin\\left(\\frac{1}{2}(n-k)\\pi\\right)", "b32f8af374ae02160584405099f78bab": "x = a \\cdot (\\cos E -e)", "b32fd428658d9c329d705ffe1cd69546": "f(t) = a + (b-a)t", "b32fdb28dc2cb2ccc57256c4d3a1ecd3": "\n\\frac{d \\mathbf{r}}{d\\theta} = r \\hat{ u} {i} e^{{i} \\theta} = \\mathbf{r} {i}\n", "b32ff2b6c199ae545415e4e3f68734e9": "\\left(-i \\vec{\\alpha} \\cdot \\vec{\\nabla} + \\beta m \\right) \\psi = i \\frac{\\partial \\psi}{\\partial t} \\,", "b3301173e3873b3cf6376855ac759057": "\\sin 2\\ell\\beta", "b3304241c6b2ecd18f3a84b13671b3a0": "\\frac{\\partial \\mathbf{B}}{\\partial t} = - \\nabla \\times \\mathbf{E}. ", "b3306031b7d1e33c51e02af8be120a45": "\n\\begin{align}\nY_{i,1}^{\\ast} &= \\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i + \\varepsilon_1 \\, \\\\\nY_{i,2}^{\\ast} &= \\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i + \\varepsilon_2 \\, \\\\\n\\cdots & \\\\\nY_{i,K}^{\\ast} &= \\boldsymbol\\beta_K \\cdot \\mathbf{X}_i + \\varepsilon_K \\, \\\\\n\\end{align}\n", "b330ac1093e45568fb78964edf401aae": "I\\in C^1(H,\\mathbb{R})", "b330eb13198a151ececa2c4480d66b3a": "\\ell\\approx{\\frac{T_0^2}{4}},", "b3315157c523b05d5e33ab539c39cac4": "A_{nr}=\\frac{4(2-\\delta_{n0})}{a^2}\\,\\,\\frac{\\sinh k_{nr}(L-z_0)}{\\sinh 2k_{nr}L}\\,\\,\\frac{J_n(k_{nr}\\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}\\,", "b3316cb05fb29bf839331b3777d798d6": " PSF(r,z)= I_0 e^{- 2 r^2/\\omega_{xy}^2} e^{-2 z^2/\\omega_{z}^2} ", "b331e927c8c03cfef9165178f84ab284": "g^{\\pm 1}(x)\\equiv\\exp\\{\\pm i\\alpha^j(x)\\frac{\\sigma^j}{2}\\}", "b33247df6cbcd6f310600c2ac52c4397": "P \\ast R", "b332f207622fad4b0730a9f806ae81d1": "M_r\\approx\\left[\\frac{a^{3/2}+b^{3/2}}{2}\\right]^{2/3}\\,", "b333184ed4da45b161e98bb0c3ae1e7d": " p_6 ", "b33322b97da820c9b0ce0bbd49e0b789": "y=M(x)+\\frac{ax+b}{cx+d}", "b333749b4586bdf2399c7e4ed97ab78a": " c = 4 a E(e). ", "b3339762fae67db0039b6813b6433b37": " \\vec{R} ", "b333cbc03a1fb219a75e3a5262df8c2b": "q = \\sqrt{2}^{\\sqrt2}", "b333ec3c90e57cb6ec74c5dcc55d2b8f": "f_{\\varphi}: M^n\\to M", "b33472739444deafea370a1bad523f11": "\\gamma = i \\omega \\sqrt {LC}", "b334aa039552810ed5b6bdbab477c823": "x_{k+1} = A x_k + B u_k \\,", "b334c19d8f51116276b3e0c3cbeb60cb": " \\displaystyle{Pf(x)=if^\\prime(x),\\qquad Qf(x)=xf(x).}", "b3350d6c8c68388034ec69d244914265": "q(X)", "b335530431e90353f44f9b0df9b7d26e": "a(\\vec{k})=\\left(E\\tilde{\\phi}(\\vec{k})+i\\tilde{\\pi}(\\vec{k})\\right),", "b3359d18a9fc3088284c1a31ad9c804b": "c \\equiv (a \\cdot b) \\pmod{m}", "b3361350b1f3af592fe33db674f07026": "\\bar {u} \\frac{\\partial \\bar{T}}{\\partial x} + \\bar {v} \\frac{\\partial \\bar{T}}{\\partial y} = \\frac{\\partial}{\\partial y} \\left [(\\alpha + \\frac{\\epsilon_M}{\\mathrm{Pr}_\\mathrm{t}}) \\frac{\\partial \\bar{T}}{\\partial y}\\right].", "b33680be67cd21fc8762ccd8a90ee6b3": " {{\\Delta K} \\over K} = {{{\\Delta K} \\over Y} \\over {K \\over Y}} = {s \\over k }", "b336cf833ae5e4cfaaac2ecd295bd8ed": " \\gamma \\,", "b336eb25d5467291568ec5160a659649": " \\det(A_i) = \\det\\begin{bmatrix}a_1, & \\ldots, & b, & \\ldots, & a_n\\end{bmatrix} = \\sum_{j=1}^n x_j\\det\\begin{bmatrix}a_1, & \\ldots, a_{i-1}, & a_j, & a_{i+1}, & \\ldots, & a_n \\end{bmatrix} = x_i \\det(A)", "b337327b592d4f7c920a96ee55073076": "A(\\omega) = \\frac{R_0}{i \\omega L + R_0}", "b3378b4e7f20fa396e458bce56568953": "Q=\\int_R d^3 \\mathbf{r} \\sum_{i=1}^N\\ q_i\\delta(\\mathbf{r} - \\mathbf{r}_i) = \\sum_{i=1}^N\\ q_i \\int_R d^3 \\mathbf{r} \\delta(\\mathbf{r} - \\mathbf{r}_i) = \\sum_{i=1}^N\\ q_i ", "b33800f651c044cb34d124043561eca6": "\\displaystyle{\\Delta(z)=(1-z/4)\\cdot \\prod_{n\\ge 1} (1-z\\lambda_n^2).}", "b3380e338b49c4f4eb5cf2282e117001": "\n\\operatorname{cov}_{\\mathrm{id}}(X,Y) = \\left\\vert\\operatorname{cov}(X,Y)\\right\\vert.\n", "b3384ad941a998a799993c4b09e42f09": "\\mathrm{diam}\\;U", "b3384c2ea470d1c79e0591ccf988f553": "Y \\to \\frac{R'}{R} \\,Y", "b3385e9f6a0c391c02d9b23bad88918f": "\\scriptstyle d\\,=\\,2t(\\cos{54^\\circ}\\,+\\,\\cos{18^\\circ})", "b33868f1ec7c96058510cf9d3f084896": "(d_i)=0", "b338e99c971ef409a8b7f5d58d0d28bb": "\n\\psi^\\dagger(k)|..,n_k,\\ldots\\rangle = \\sqrt{n_k+1}\\, |...,n_k+1,\\ldots\\rangle\n", "b33944bd1792ef0433e44921b3690860": "H(z) = T \\sum_{k=1}^N{\\frac{A_k}{1-e^{s_kT}z^{-1}} - \\frac{T}{2} \\sum_{k=1}^N A_k}.", "b3395ff62538e03ecc132268803d6858": "n_s={120\\times{f}\\over{p}}", "b3398e572dd0f3721e17de32fe309a26": "DH(\\gamma^x,\\ \\gamma^y)=\\gamma^{xy}", "b339bce2296fb2d252cb2a4283cec9c7": "\\lambda^{\\mathrm{state 2}}_{\\mathrm{observed}} > \\lambda^{\\mathrm{state 1}}_{\\mathrm{observed}}", "b33a002ae5293de29d823ba042768d43": "\\left\\{ x_i y_j : i = 1 , \\ldots , n , j = 1 , \\ldots , m \\right\\}", "b33a271c0cd3f4f08a6b9921498fc24b": "{13 \\choose 2}{4 \\choose 2}^2{11 \\choose 1}{4 \\choose 1}", "b33a53fcbdf659bad9c5a7a559f47ab5": "Z_n = R_n", "b33a9d817bdc32669029c1aaeb5e0277": "\\rho^{(n)}", "b33abcff3b5678553bd93749ce21f59c": "y^n b // c > d", "b33d406c77937b0081669cbdbb0638e8": "\\Pi_1 = \\bigg(a - b\\bigg(q_1+\\frac{a - bq_1 - \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2b}\\bigg)\\bigg) \\cdot q_1 - C_1(q_1),", "b33d80a056b4824710e254ae09fc85b6": "V_S\\,", "b33d843d3a4413c1a1bab3f3ec09bfea": "H_5(x)=32x^5-160x^3+120x\\,", "b33ddbe9da8a132cc7e738b2ffa9769b": "\\frac{\\epsilon^2}{\\delta^2}S_0'^2 \\sim Q(x).", "b33def2e4c569cf404c57c927b3195cc": "\nd_{\\pm} = q [1 \\pm e^{-d_{\\pm} R}]\n", "b33e355a9d218d30d0bb99cf38884d60": "f(a)=r\\,.", "b33e709d574598ba81b7159064dc82a1": " \\mathcal{M}_k ", "b33edba9ee8b74b308047a1267014f6c": " VK ", "b33f1f616d0996319a45528b4efc3152": "1 + i = (1 + r)(1 + \\pi).", "b33f43c8b0c90c824ac9eccb03991d05": "P = \\left( -1 \\right)^{L+1}", "b33f82eee659a8d579ed693235e664de": " i' \\circ g =f \\circ i ", "b33faa14b1478c5147e6e34c0629f3cd": "d_a=d_e\\left(\\frac{\\rho_p}{\\rho_0 \\chi}\\right)^{\\frac{1}{2}} ", "b33fc13fecddb4683a4cd09bb6bad3fc": "((v_1,f_1),(v_2,f_2)) \\mapsto f_2(v_1)", "b3401fd131e8473a8f3542a3cf33a521": "b+d=1\\,\\!", "b3403ed07f9f0d30a3f1a1dbc3c2b2cc": "I_a\\,", "b340547a9b6eafa6b14137eec39f6e94": "\\{z_1,\\dots,z_{i-1}\\}", "b340970cac551a50508324e5e1d42ad8": "d(x,y) = d,\\,", "b340d3e11a01b97cc9a572c939977fa5": "2^{k}", "b340ed6d0c9b8c51c37ee9d22033710b": "D_e = \\frac{1}{n(n-1)} \\sum_{u=1,w=1}^{N} \\sum_{i=1,j=1}^{m}{_{metric}} \\delta_{c_{iu} k_{ju}}^2", "b340f8f2387f1f8a3c6161e673780970": "A^\\mathsf{T}", "b341269eefa76742e89ce1292fff8f39": "\\Zeta(s)", "b3415aa7c82f88356c4d59306f30927e": "\n[\\phi(x),\\partial_t \\phi(y) ] = {\\rm i} \\delta^3(x-y) \\,\n", "b3418d3b3583947f265eead65473b8ae": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \\mathbf{\\hat k} \\mathbf{\\hat k}\n { \\exp \\left ( i\\mathbf k \\cdot \\mathbf r \\right ) \\over k^2 +m^2 } \n=\n\\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } \n\\left[\n\\left( \\mathbf{\\hat k}\\cdot \\mathbf{\\hat r}\\right)^2\\mathbf{\\hat r} \\mathbf{\\hat r}\n+ \\left( \\mathbf{\\hat k}\\cdot \\mathbf{\\hat \\theta}\\right)^2\\mathbf{\\hat \\theta} \\mathbf{\\hat \\theta}\n+ \\left( \\mathbf{\\hat k}\\cdot \\mathbf{\\hat \\phi}\\right)^2\\mathbf{\\hat \\phi} \\mathbf{\\hat \\phi}\n\\right]\n { \\exp \\left ( i\\mathbf k \\cdot \\mathbf r \\right ) \\over k^2 +m^2 } \n\n", "b341b3edbd7853a8ae8dda83e6391dc4": "\\scriptstyle{\\lambda_1(M)},", "b3424e502226e35b25a534c3b81c4d6a": "t_1 \\longrightarrow_R^+ t_n", "b342550f3773a09372a3bf3612edfaef": "{{V}_{GS}}-{{V}_{TH}}", "b3427f6be335d120941a28ca1519a531": "\\nu(W)=[n/(n-q)]", "b34291a96ca4ca9bb8d9dd964eb9757e": "E(\\sqrt{2})=a^\\sqrt{2}", "b342dbcef67ad5d51ba8cb6f248f9c83": "L = \\mathbf{D_{xx}}\\oplus\\mathbf{D_{yy}}=\\mathbf{D_{xx}}\\otimes\\mathbf{I}+\\mathbf{I}\\otimes\\mathbf{D_{yy}}, \\,", "b342ee7058e06c723ae8f60310de3401": "n = 30 \\text{ years} \\times 12 \\text{ months/year} = 360\\text{ months}", "b34333d9f2ce02b48ead7a30ff5fe378": "\n \\bar{E} = \\frac{1}{2}\\, \\left[ (\\rho-\\rho')\\, g + \\sigma k^2 \\right]\\, a^2.\n", "b3435b89f7db0aaf4bcc99599b034af2": " \\frac{N_i}{N_{total}} \n= \n\\frac \n {e^{-E_{rel}/RT}}\n {\\sum_{k=1}^{N_{total}} e^{-E_k/RT} }\n\n", "b343d4e41380ba25343ee5056b984b12": "M \\ge 2", "b343d6bcd575348855f4a7e940c6830c": "p_n = 1-\\frac{1}{1!}+\\frac{1}{2!}-\\frac{1}{3!}+\\cdots+(-1)^n\\frac{1}{n!}", "b343e460483272f7784a5021e874388d": "e^k\\subseteq Y", "b343ee5121a249580aae8b68d278dbb6": "E_3=-d/a", "b3443fad7d3e7ea54caee660eff3209d": " f_{GHz} ", "b3447cb579d4c6208f05e638d051d6c4": "f=\\epsilon s + \\delta n", "b34498f8e980b65963ac6d7957f3ad97": "\\mathrm{RuO_2 + xH^+ + xe^- \\leftrightarrow RuO_{2-x}(OH)_x}", "b344c1de54cdc774e1816e8f04de606c": "\n\\gamma(s, z) = \\cfrac{z^s e^{-z}}{s - \\cfrac{s z}{s+1 + \\cfrac{z}{s+2 - \\cfrac{(s+1)z}\n{s+3 + \\cfrac{2z}{s+4 - \\cfrac{(s+2)z}{s+5 + \\cfrac{3z}{s+6 - \\ddots}}}}}}}.\n", "b344c3b6b25358e7f46eb7c776d158cf": "\\frac{q}{A}={{\\mu }_{L}}{{h}_{fg}}{{\\left[ \\frac{g\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)}{\\sigma } \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{2}\\;}}{{\\left[ \\frac{{{C}_{pL}}\\left( {{T}_{s}}-{{T}_{sat}} \\right)}{{{C}_{sf}}{{h}_{fg}}Pr_{L}^{1.7}} \\right]}^{3}}", "b344ce67247eb6013ef79f527ae4b2db": "\\mathcal{E}(k)= A \\cos{ak} ", "b344dd2aebaa87f6c22bfb94ec3830a7": "D_{3}", "b344dddb18d239ca2e94294d48ddfef4": "\n\\begin{align}\n\\iota^2 & = 1, \\\\\n\\kappa^3 & = 1, \\\\\n\\lambda^5 & = 1.\n\\end{align}\n", "b344e8ca8288f0d9da8b0a9ee55fe485": "T(X_1^n)=\\overline{X}=\\frac1n\\sum_{i=1}^nX_i,", "b345472ff7e8e9153c11ad96dd36f3ad": "a b=0", "b345bbcfd75efd10bfde14dfce019a22": "\n \\Longrightarrow S''_{yy} (\\boldsymbol{\\phi}(0)) = \\boldsymbol{\\phi}'_y(0)^T S''_{zz}(0) \\boldsymbol{\\phi}'_y(0), \\qquad \\det \\boldsymbol{\\phi}'_y(0) \\neq 0; \n", "b345c32572eb8b7f81a19143ca688fe1": "\\scriptstyle n\\equiv a_2 \\pmod m_2", "b345da2b90491540c9a3ccc83bd5e21f": " 1- \\sqrt{R} ", "b345ddcc812207b8e53ba7761ab5dcef": "\\int_{x_0}^x \\int_{x_0}^{x_1} \\dots \\int_{x_0}^{x_{n-1}} f(x_n) \\,\\mathrm{d}x_n \\dots \\, \\mathrm{d} x_2\\, \\mathrm{d} x_1= \\int_{x_0}^x f(t) \\frac{(x-t)^{n-1}}{(n-1)!}\\,\\mathrm{d}t .", "b345e1dc09f20fdefdea469f09167892": "a,b", "b345e2c7d9a3709952a119c7040c96a4": "\\mathbf{e}_{i_1 i_2 \\cdots i_p}^{j_1 j_2 \\cdots j_q} = \\mathbf{e}_{i_1}\\otimes\\mathbf{e}_{i_2}\\otimes\\cdots\\mathbf{e}_{i_p}\\otimes\\mathbf{e}^{j_1}\\otimes\\mathbf{e}^{j_2}\\otimes\\cdots\\mathbf{e}^{j_q}", "b345ec9f776ec99ee5bf634b25d3fe96": "\\lim_{N\\to \\infty } \\, P^N=\n\\begin{bmatrix}\n 0.625 & 0.3125 & 0.0625 \\\\\n 0.625 & 0.3125 & 0.0625 \\\\\n 0.625 & 0.3125 & 0.0625 \\\\\n\\end{bmatrix}", "b3460fdfc0b774b0f7339530a46f3027": "A \\to abc", "b3463b124517193a0a4d229ac943c801": " x \\ge 0 ", "b34660069c8ab7b486601906583a503f": "\\mathrm{\\delta}W = \\mathrm{d}(p_{out}V_{out}) - \\mathrm{d}(p_{in}V_{in}) + \\mathrm{\\delta}W_{shaft}", "b3466edc3160797aa1190884a753d549": " h\\in H,v\\in V", "b3467597e96791bba2e1f887c47db761": "\\det \\begin{bmatrix}\na_2 & a_3 \\\\\nb_2 & b_3\\end{bmatrix}\\mathbf{i} - \\det \\begin{bmatrix}\na_1 & a_3 \\\\\nb_1 & b_3\\end{bmatrix}\\mathbf{j}+ \\det \\begin{bmatrix}\na_1 & a_2 \\\\\nb_1 & b_2 \\end{bmatrix} \\mathbf{k}", "b3469dc9ed71762bdd9e79b538ff0d63": "(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \\ldots)", "b346e54c96599b1832520537fb2831ee": "|\\alpha;t\\rangle=e^{-{\\rm i}Ht / \\hbar}|\\alpha;0\\rangle", "b347065361874a0d878895cda4d74721": "\\scriptstyle \\beta l < \\pi /2", "b3473849992273069fe21b5e3254d102": "\\lambda^8 = \\frac{1}{\\sqrt{3}} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2 \\end{pmatrix}.", "b3473ae03a86552468eb45cca3455646": "R_R", "b34755e6b67935ea37b38cbf96da9444": "b_1=0", "b34792eabe56e43c15a8bea87b8521ac": "4 a^2 x^2 + 4abx = -4ac", "b347c0a2f23c8113bc13cb6d40101286": "L(s,\\chi) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}.", "b347f001f8e8cc0924c930fcd9b79306": "2(1-\\varepsilon)\\gamma n > \\gamma n\\,", "b3482fc38719d81f2e089940131db687": " {\\rm R}^i F (A) = 0 \\,\\!", "b34857d1a418c5e231b34d5d7051a878": "S\\approx 4\\pi\\!\\left(\\frac{ a^p b^p+a^p c^p+b^p c^p }{3}\\right)^{1/p}.\\,\\!", "b34888b4ddc928a126adc47a04cc14ec": "\\operatorname{Ber}(X) = \\det(A-BD^{-1}C)\\det(D)^{-1}", "b348b487e5f6e90ecfe8c615bfa22613": "x_{pi}", "b348bba098d1943b5f2355f745bcc15d": " U_n(r) = A_nJ_0 \\left( \\alpha \\frac{r}{R} n^{1/2}i^{3/2} \\right) + B_nY_0 \\left( \\alpha \\frac{r}{R} n^{1/2}i^{3/2} \\right) + \\frac{iC_n}{\\rho n \\omega}\\, ,", "b348c06960c023702d6d5365752b7c5f": "p \\cdot z \\geq r", "b348eb4a8fd0bd1e3615822e8eefb38d": "S = \\left \\{ (x, y) \\, : \\ a \\leq x \\leq b, 0 < y < f(x) \\right \\}", "b3490d104fb2099d4d4b8eab62bc6b69": "A \\subseteq X", "b34918ac8aa42e3560af00c21d31c158": "\\underset{x\\in(-\\infty,-1]}{\\operatorname{arg\\,min}} \\; x^2 + 1,", "b3494a1e6a2202410f1391875a02756a": "3 \\uparrow\\uparrow X \\ = \\ 3 \\uparrow (3 \\uparrow (3 \\uparrow \\dots (3 \\uparrow 3) \\dots )) \\ = \\ 3^{3^{\\cdot^{\\cdot^{\\cdot^{3}}}}} \\quad \\text{where there are X 3s}.", "b349b6f05bf0b4580fc7a71f362d6795": "{{E^*}_{1/2}}^{ox} = {E_{1/2}}^{ox} - E_{0,0} + w_r ", "b34a2d749412fa939b6081335ebb6d47": "(\\bar{4},1,2)", "b34a5f1b36d8d28ee3bb600cefcd2bf1": " C_{p} = \\partial H/\\partial T\\,\\!", "b34aaec4657b093456dba586562624e0": "\\mathrm{Wea} = \\frac{w}{w_\\mathrm{H}} 100", "b34abc0f6c68ff3b47701159f93d9d07": "S_i = \\Sigma_{j=1}^i \\; y_j\\,", "b34b01a0880d496617c074e3607592ba": "C_\\beta", "b34b53e3091df4ac7431705bad995f93": "\\mathit{g(x)h(x)} = \\mathit{x^N} - 1", "b34ba421687f4eecdfb35688516e3256": "\\begin{align}\n|n(x^\\mu)\\rangle = & |n\\rangle +\\sum _{m\\neq n} \\frac{\\langle m|\\partial_\\mu H|n\\rangle }{E_n-E_m}|m\\rangle x^\\mu \\\\\n&+\\left(\\sum _{m\\neq n} \\sum _{l\\neq n} \\frac{\\langle m|\\partial_\\mu H|l\\rangle \\langle l|\\partial_\\nu H|n\\rangle }{(E_n-E_m)(E_n-E_l)}|m\\rangle -\\sum _{m\\neq n} \\frac{\\langle m|\\partial_\\mu H|n\\rangle \\langle n|\\partial_\\nu H|n\\rangle }{(E_n-E_m)^2}|m\\rangle \\right.\\\\\n&\\qquad\\left.-\\frac{1}{2}\\sum _{m\\neq n} \\frac{\\langle n|\\partial_\\mu H|m\\rangle \\langle m|\\partial_\\nu H|n\\rangle }{(E_n-E_m)^2}|m\\rangle \\right)x^\\mu x^\\nu+\\cdots.\n\\end{align}", "b34bac6aa097d3c3038f3be7c1f6339e": "a_i \\ne 0", "b34bccbb95dc0e00e0713abfe979810a": "\\scriptstyle \\lbrace x \\isin V : r(x) = 1 \\rbrace ", "b34bd672d88459b10a9c59a6cce41c37": " \\binom nk = \n\\begin{cases}\n n^{\\underline{k}}/k!, & \\text{if }\\ k \\le \\frac{n}{2} \\\\\n n^{\\underline{n-k}}/(n-k)!, & \\text{if }\\ k > \\frac{n}{2}\n\\end{cases}.\n", "b34c29833f70bb980f404f0c2c94a15d": "\\Phi_n(x)=\\prod_{d\\mid n}(x^d-1)^{\\mu(n/d)} ", "b34cbdef44efec100b3f0c1310097541": " \\gamma \\in \\left[\\frac{1}{6},\\frac{1}{2}\\right]", "b34cc51545e8c6b4329336cab2c87a99": "f(r) = - \\frac{k}{r^{3-\\beta^{2}}}", "b34cc8b74ee6bad420198e637176ef00": "\\varepsilon _{1} =\\frac{w^{2}+m_{1}^{2}-m_{2}^{2}}{2w},", "b34d205d8dcffa8bb7cf827a10f5e5b2": "V_T^n", "b34d2e1641d69277fcbb01ec7af8b3c3": "\\mathrm{INL} = \\max_{0 \\le c \\le c_{\\max}} \\left| V_{\\mathrm{out}}[c] - V_{\\mathrm{out}}[0] - c \\cdot m \\right|", "b34dc97c727eb9b7f419b43e76169d06": "E_c\\epsilon_c = fE_f\\epsilon_f + \\left(1-f\\right)E_m\\epsilon_m.", "b34e0f95baa48a3058826e247b7c96a5": "-2^{31}", "b34ea53ef0dda813197dfd0885610d89": "t \\in [0,t_x]", "b34ebd59522899ddec37dca0a9c2227b": "\np = \\frac{nh}{2L}", "b34eff8e8c279be5abae18e43d276c99": " \\oplus \\! ", "b34f15d534acf8451066759f529abe5c": "\\int^T_0 R_N(t,s)k(s)ds = S(t)", "b34f246dfb92585bd76bc38bba96f4f7": "\\lbrack\\mathbf y\\rbrack = \\lbrack\\mathbf z\\rbrack^{-1}", "b34f4bf13148160dbfda56307a96f59d": "P_n=\\begin{cases}0&\\mbox{if }n=0;\\\\1&\\mbox{if }n=1;\\\\2P_{n-1}+P_{n-2}&\\mbox{otherwise.}\\end{cases}", "b34ff8d858a330c8427f57642cacd249": "H(s) \\, H_{inv}(s) = 1", "b3505ca0a05b8e4e2191b44771990d0f": "\\sum_{j=0}^N R_w[j-i] a_j = R_{sw}[i] \\quad i = 0,\\, \\ldots,\\, N ,", "b350cc9e7850ab88f5ff0cafcaebee52": "\\sigma^2 = \\frac{\\sum_{i=1}^N w_i (x_i - \\overline{x}^{\\,*})^2}{\\sum_{i=1}^N w_i}", "b3512911fb98b3b6e1d00f891c396b3c": "O(n^{1+\\rho})", "b35158eae6bb2a77be02cb9c5b91723c": "B=\\frac{-\\Delta\\mu_1^{excess}}{{v_s}{c^2}}", "b35162bfcc36257fe605567b154dd1a3": "\\dot Q_L=\\dot Q_a", "b3518bb9d1de78faea0aa1786d1e105b": "\\sigma(E) = \\{ \\emptyset, E, \\Omega \\setminus E, \\Omega \\}.", "b3519ad352b40f2022d40e539aa198eb": "J = J_S \\left(\\exp\\left(\\frac{qV}{kT}\\right) - 1\\right)", "b351bb9b0af6e4fc678749675c53ad67": "mg", "b351c64955303b25eeefae451967a25a": "\n\\begin{array}{lcl}\n x' & = & \\frac{dx}{dA} \\\\\n & = & -r\\sin A + \\frac{(\\frac{1}{2}).(-2). r^2 \\sin A \\cos A}{\\sqrt{l^2-r^2\\sin^2 A}} \\\\\n & = & -r\\sin A - \\frac{r^2\\sin A \\cos A}{\\sqrt{l^2-r^2\\sin^2 A}} \n\\end{array}\n", "b35267b2afd635325dd53d65ace01932": " \\operatorname{D}_A(U) =\n\\operatorname{Tr}(\\operatorname{E}_A(U) S). ", "b352c1829f7f1599d3629f0ddda63baa": "u\\in H^2(\\Omega)\\cap H^1_0(\\Omega)", "b352c4ef2b0b4597e9daaedcc59efb36": "i = \\frac{d}{1-d}\\approx d+d^2", "b352d6b73674c5e7b6d21a1e5de27a21": "L_n^{(k)}(x)=(-1)^k\\frac{d^kL_{n+k}(x)}{dx^k}\\,", "b352f8fc501767ac358d0e0b8881320b": "\n\\tilde{V}(\\mathbf{k}) = \\int d\\mathbf{r} \\ v(\\mathbf{r}) \\ e^{-i\\mathbf{k} \\cdot \\mathbf{r}}\n", "b352f9959b806dba523e3b10925f7a55": "National Savings = Y - C - G = I", "b35307d2f9666c6266e56b0d74bd3337": "f(x)\\cdot g(x) = \\sum_{k=0}^\\infty \\left(\\sum_{i=0}^k a_i b_{k-i}\\right) x^k.", "b35310cd1a45eb09fc8b8c82524ff239": "{V_s}/X", "b353364570282e70ced477bc59280428": "(M_1,M_2,M_3,M_4,M_5,M_6)", "b35363bdea65a08bf3c0c4d3d6921960": "R_{\\alpha \\beta \\gamma}^{\\;\\;\\;\\;\\;\\; \\delta}", "b3537b3fabc090f5b5eb03b61602413f": "\\forall i, (a_i,a_{i+1}) \\in R", "b35388ba43c80d6ea937659a7f381feb": "h[n] \\ \\stackrel{\\text{def}}{=}\\ O_n\\{\\delta[m];\\ m\\}.\\,", "b353b7625a68c190a26507f668237bc0": "\\scriptstyle 1 - \\frac{1}{2}\\alpha", "b353cc24a470cb31d366004a0b9c9395": "\\cos \\varphi\\mathbf{\\hat{x}} + \\sin \\varphi\\mathbf{\\hat{y}}", "b353dee241385457391a347d246efaba": " b_s \\ne 0 ", "b353eb79ceda6569345ae9a7094e7b42": " W = - E T ", "b355248e5271fdbb48442c341f1c708f": " \\forall\\varepsilon > 0\\ , \\exists M_{\\sigma,\\varepsilon}\\, \\forall N > M_{\\sigma,\\varepsilon} \\,\\, , \\left\\|\\sum\\limits_{i=1}^N a_{\\sigma(i)}-A \\right\\|< \\varepsilon ", "b3552d62afe58cd377cd3993962a22ec": "\\mathbf{x} \\in \\mathcal{X}", "b3553c94b1d70ba1c68eb15a2967887b": "\\|\\delta_h w\\|_{(1)} \\le \\|L \\delta_h w\\|_{(-1)} + \\|\\delta_h w\\|_{(0)} \\le \\|[L, \\delta_h] w\\|_{(-1)} +\\|\\delta_h Lw\\|_{(-1)} + \\| \\delta_h w\\|_{(0)}\\le\\|[L, \\delta_h] w\\|_{(-1)} +C\\|Lw\\|_{(0)} + C\\| w\\|_{(1)}.", "b3555732fdc356cffd422f9a2b079404": "\\sqrt{-1} = (-1)^\\frac{2}{4} = ((-1)^2)^\\frac{1}{4} = 1^\\frac{1}{4} = 1", "b35613fbe41dd8a19ce67bd05e38fbe5": "\\bold{F} = - \\bold{\\nabla} V. \\, ", "b3563887df0309f64156a9e49392cdfa": "(F_{q})^{n}", "b356dd682b757a7a63522888bf22f7f1": "X\\odot Y", "b356ef7125071eb8f3d99073e7b4f70d": "y(t+\\Delta t) = y(t) \\varepsilon \\sin \\theta \\cos \\Phi", "b357202f9d1b377632f93cda1aef7ef9": "x\\sin x - \\int 1\\sin x \\, dx\\!", "b3572b565ad8cfa6b45136938be50df5": "\\lambda\\colon\\ A \\to R", "b35733c204115fcf8512dcaee3f5acf8": "F^{B}", "b35734e57fc04837260a305006ac2adf": "a \\odot b", "b35792b8fb57f437acca0210cc6e4923": "\\alpha = \\alpha_{\\mathrm{max}}", "b357c8c8e593e8bb1bc78788ec2444bd": "1 / \\gamma", "b35805702bd1f1257445260ec1637f2b": "\\phi_{i=1 \\dots N, j=1 \\dots N}", "b35900311b4f724019c442a43e03c7e6": "\\,i^{(12)}=0.1139", "b3596cfd4e7fc7c5a9ad677696fd9351": "x \\cdot 0 = 0\\ ", "b359934d2e5450223427a582752152fc": "\\displaystyle{\\Delta v= Lv=L(\\psi u) =\\psi Lu +[L,\\psi]u=\\psi(f-u) +[\\Delta,\\psi]u.}", "b35a3936acfe224203ec57ba6ec72a68": " y'(t) = f(t, y(t)), \\quad y(t_a) = y_a, \\quad y(t_b) = y_b, ", "b35a68c01fd529f6dfa5bfaca6ff7094": " M^1(\\mathbb{R}^d)", "b35a99837a88dbb0dab5aa5c154df209": "\n f(\\mathbf{x}) = f[\\boldsymbol{\\varphi}(q^1,q^2,q^3)] = f_\\varphi(q^1,q^2,q^3)\n ", "b35abce555d5a8bf34422672181b1947": "\\; (A - 4 I) p_3 = 0 ", "b35acd1ae3d77c915f9cd31707748596": " T = \\inf\\{t \\geq 0 \\colon B_t = 0\\}", "b35b3328c700ab86ee2b61ab205f82dc": "\n \\mathrm{length}(AB) = dx \\,\n ", "b35be7b06e97445fb790f3974a158298": " \\mathbf{\\bar Y} ", "b35c02eafb6c7aa5c583bcf6214d97a9": "t_{12} \\cdot t_{34}+t_{41}\\cdot t_{23}-t_{13}\\cdot t_{24}=0 ", "b35c0806776670e5e6f68b8d8061ceb4": "\\sum_{i=1}^n \\log(i)^c \\cdot i^d \\cdot b^i \\in \\Theta (n^d \\cdot \\log(n)^c \\cdot b^n)", "b35c0c98b5e2434367a8ab1871e9dddd": "a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{ \\ddots + \\cfrac{1}{a_n} }}}", "b35c60f0e907175c309bc870851eccae": "\\frac{\\text{(votes cast)}}{\\text{(available seats)}}", "b35c6cc06d406e42284e5fbd483b9564": "\\le50", "b35c72ef245ba891620272cc8a8adf48": "\\underline{A}=A_{z}\\underline{z_{o}}=L^{TM}(z)T^{TM}(x,y)\\underline{z_{o}} \\ \\ \\ \\ \\ \\ \\ (10) ", "b35c88cd484f719bfa43a22ee92ceaa8": " {1\\over 2} {1 \\over 4 \\pi r } \\left[ \\mathbf 1 - \\mathbf{\\hat r} \\mathbf{\\hat r} \\right] . \n", "b35c98a019d8553c49a8bb8e608ff40f": "{\\delta}_1\\,\\!", "b35cfaf2796e71504b71bd7006af232c": "[M+25H]^{25+}", "b35d70d7cae923b540645ea83a126407": "\\int \\sinh ax \\sinh bx\\,dx = \\frac{1}{a^2-b^2} (a\\sinh bx \\cosh ax - b\\cosh bx \\sinh ax)+C \\qquad\\mbox{(for }a^2\\neq b^2\\mbox{)}\\,", "b35d72029cdb4c6c437ff98c81ec2b7e": "\\tfrac{1}{2}\\left(\\tfrac{a}{c} + \\tfrac{b}{d} \\right )+\\tfrac{i}{2}\\left (\\tfrac{b}{d} - \\tfrac{a}{c} \\right )", "b35da90c70b9ce58ea277c5a2ee048ce": "x, y \\in A", "b35dbbd8c26593b79ae8c4e6249b29cc": "(x-X)^2", "b35de87b64267319e75b8768d9f16f4b": "\\varepsilon \\approx \\left|Q - \\int_a^bf(x)\\,\\mbox{d}x\\right|", "b35df029ce82b4812f7727816ec1427e": "\\left(\n\\mathbf{a}\\odot\\mathbf{b}\\right)_i = a_i b_i", "b35e04a53eee7cedcb2439460cdf2118": "\nF(x_1, x_2) = 1 -\\sum_{i=1}^2\\left(\\frac{x_i}{\\theta_i}\\right)^{-a}+ \\left(\\sum_{i=1}^2 \\frac{x_i}{\\theta_i} - 1\\right)^{-a}, \\qquad x_i > \\theta_i > 0, i=1,2; a>0,\n", "b35e37a7549556fd19791f74efec5156": " \\partial_x^+ ", "b35e3ac0ad832edc4fd0415803dccbd4": " = k\\int_R^\\infty \\rho dr", "b35e3b1aacabe629382caaa90d804e39": "2 ^ x\\,", "b35e7c8b4e0be57a65dad8a09170fd3a": "x^\\mu = (t,x,y,z)\\,", "b35e9d869fae7ac18e7a3de4388bce3d": "\\begin{align}\nx & {} = -u+\\frac{2\\left(1-a^2\\right)\\cosh(au)\\sinh(au)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)} \\\\ \\\\\ny & {} = \\frac{2\\sqrt{1-a^2}\\cosh(au)\\left(-\\sqrt{1-a^2}\\cos(v)\\cos\\left(\\sqrt{1-a^2}v\\right)-\\sin(v)\\sin\\left(\\sqrt{1-a^2}v\\right)\\right)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)} \\\\ \\\\\nz & {} = \\frac{2\\sqrt{1-a^2}\\cosh(au)\\left(-\\sqrt{1-a^2}\\sin(v)\\cos\\left(\\sqrt{1-a^2}v\\right)+\\cos(v)\\sin\\left(\\sqrt{1-a^2}v\\right)\\right)}{a\\left(\\left(1-a^2\\right)\\cosh^2(au)+a^2\\,\\sin^2\\left(\\sqrt{1-a^2}v\\right)\\right)}\n\\end{align}", "b35ee83d50b95cb8a0a1286afc9e6ac4": "V_q(\\mathbf{R})=G \\frac{1}{2} \\frac{1}{|\\mathbf{R}|^3} \\sum_{i,j} Q_{ij}\\, n_i n_j\\ .", "b35f00b531f3a450be3c7ec28b3495f5": " \\varphi_X(t) = \\operatorname{E} \\left [ e^{itX} \\right ]", "b35f1a60bae8833c8b350d05a34eb42f": "N_G(W)", "b35fd07936e094c30419400cbbd2a3c2": "\\frac{L}{L}", "b35fd32eeee83fad1dab0ec8b5e06b20": "n,m \\ge 0", "b36005b87d4229290c4e83f019f7e176": "A_{k+2}={(2k-1)\\over (k+2)(k+1)}A_k\\;\\!", "b3601ffb1d02c3a239d595b13a50deee": "A^{*2}", "b36029777e8aedd8430e73250b87665c": " \\alpha = \\gamma _1 + {5 \\over 2} \\gamma _2 ", "b3604604335332b2da44677432a69008": " d = 2\\,", "b36085e1357776dbdf3cd35aaf2fee45": "\\scriptstyle{\\phi^l_i = \\hat{l}_{z_i}/L_z}", "b3608a8fee792aeaf9265fd104077e75": "\\sigma_{pH}", "b360bc82a37a1a9fe361d6caff8150c6": "\\zeta(2n) = (-1)^{n+1}\\frac{B_{2n}(2\\pi)^{2n}}{2(2n)!} \\!", "b36109bd208450a6bae83ffcef737c3d": "3N = 6 = 3 + 2 + 1.", "b3613bb4d21e1509a1af858c5f67e353": "\\scriptstyle 0.8(0.6)\\times10^{-16}", "b361972ec1c94e300d839838dd2efacd": " x_{t+1} = rx_t(1-x_t),", "b361cb71d682e3a905945f2ec8762a23": "\\frac{dy}{dx} + P(x)y=Q(x)\\,\\!", "b361e2a3ad90ce53d965f6e8691a495d": "|j,m_{max} \\rangle", "b361ed3c558e1bba8823eceb4b533dc9": "T\\colon S^l \\to S", "b362988f7da2e144a902ae9c272ce2a2": "[r_i, d_i] \\in I", "b362ab6c823a35fc13ed7e85a8e99b4b": "\\left\\{X_t ; t\\in T\\right\\}", "b362ae44f4e6f4b321d39c6d0a3b80a0": " \\subseteq", "b362eca1fe8d5953a40b69da01bbb88d": "g(\\tau)=\\delta(\\tau),", "b363b18fd791f221ce237fadbd61257b": "\\xi_1,\\dots,\\xi_{t-1}", "b363b64da6ac1c009f5597f3f936a8a0": "0 = x^T A^T A x = |A x|^2", "b363fd7ffbafd275da11c56429c48a9f": "C^\\infty_c(\\mathbf{R}^n)", "b36438ae2f76559ea476c4aea673e362": "q(x)=E[Q(x,\\xi)]", "b3644fb71e7a528937ba95b6a506e85a": "\\chi_A(x) =\n\\begin{cases}\n1 & \\mbox{if } x \\in A, \\\\\n0 & \\mbox{if } x \\notin A. \\\\\n\\end{cases}\n", "b36457de496e6ed6464075cdfcb41614": "\\sum_{i=1}^{n}\\sum_{k=1}^{m} J_{ij}J_{ik}\\Delta \\beta_k=\\sum_{i=1}^{n} J_{ij}\\Delta y_i \\qquad (j=1,\\ldots,m)\\,", "b364619857ce96409bca705f33f791d3": "2^{-\\delta^{\\prime}n}", "b3646e9eb64dba305087a3aa5eb58f40": "S_{NNR(n)} = 2\\cdot S_{RRB(n-1)} + 1 + S_{RBN(n-1)} + 1 + S_{RNN(n-1)}", "b36474aa07e3af4b7285e72a90e5dc83": " \\hat{\\mathbf{t}} ", "b3649440aa030932f3312f0ac32ef05c": " p b > q a . \\! ", "b3649b26e958de545e7c6f6a8173e3bd": "Nm + n = P(Mn + m) \\,", "b3656b361df183cfea80ab9c709772e2": "M = \\mathbf{r}_1\\times \\mathbf{F}_1 + \\mathbf{r}_2\\times \\mathbf{F}_2 + \\cdots", "b3658b394f30495e6bce78ab7e301924": "=u_1 \\|\\nabla q_1 \\| \\frac{\\nabla q_1}{\\|\\nabla q_1 \\|}\n+u_2 \\|\\nabla q_2 \\| \\frac{\\nabla q_2}{\\|\\nabla q_2 \\|}\n+u_3 \\|\\nabla q_3 \\| \\frac{\\nabla q_3}{\\|\\nabla q_3 \\|}", "b36612089e1f0bd0cde0f798e7724c4c": " f(\\Gamma (t),0) ", "b36676580adc58e12ca7ad3da565aba3": "I(t) = I_0 \\cdot e^{- {R_2 \\over L} t}", "b3667a128c90a96573f505a7dd5bcb12": "x_{vc}", "b366de74df58874c71900ea68f46934f": "c_A", "b3672bbc2630193a91569f9672ff61d1": "S(C) = \\{ n \\in \\mathbb Z_+^R : An \\leq C\\}", "b36815ef789ae148d310904a64ef840e": "\n\\psi=2Axy.\\,\n", "b3682b5f84f8d5e9ce6cbad13b13ac15": "\n d^D = -\\left(\\cfrac{2\\sigma_0 a}{E^*}\\right)\\sqrt{m^2-1}\n ", "b3682c6ee2f4fac99c559d583a0378d1": "\\sqrt2 = a / b,", "b368341ef7d01484ee11adc693a1aa91": "X \\in A", "b3683d9ea49140537eb356f5a835fdea": "e_b(-1,i) = 0\\,\\!", "b36857ff350e259eaa517f02245ae92a": "\\scriptstyle {\\sqrt{2}a^3}", "b368759510d307c1f2cde90254103af1": "E_t^{\\ast}", "b368c3bfa5eb6a5573a03d19c50ddd8c": "L^\\infty([0,1]) ", "b368fe83f35f2c17ea337faf4ec82cb4": "\\displaystyle (f * g)(x)\\,", "b369dfc5c333bfb04888f44fd42f1aa3": "\\sigma \\circ (\\tau_1, \\dots, \\tau_n)", "b369f3914c582967d1e03631005bd439": "m = (f^{0}(2) - f^{0}(1)) / (2 - 1) = (x_{2}^{0} - x_{1}^{0}) / (2 - 1) = (10 - 8) / (2 - 1) = 2 / 1 = 2", "b36a07ef51423c416b6650b9fc4be85b": "Z_b(s,t)=\\frac{\\zeta(s,t)}{2^{(s-1)}}", "b36a0863000d8c601829ef0c51e34e62": "\\displaystyle \\hat{f}(\\boldsymbol \\nu)=", "b36a11265e1d5004cb32797e429636f6": " Lu = \\sum_{ij} a_{ij}(x)\\frac{\\partial^2 u}{\\partial x_i\\partial x_j} + \n\\sum_i b_i\\frac{\\partial u}{\\partial x_i} \\geq 0 ", "b36a348ad393bf30fb1d1e8954d32879": "\\{0, \\ldots, q-1\\}", "b36a6e999cc64c671c5d09d9262e4bd6": "\n\\Psi_2(a,c_1,c_2;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a)_{m+n}} {(c_1)_m (c_2)_n \\,m! \\,n!} \\,x^m y^n ~,\n", "b36aa4a601c6fccd04511f4243506187": " i = 1, \\dots , n ", "b36b40ee4f475d990d279d40a8f03dc4": "h \\geq 2b", "b36b68d80f1a60376372eba84ffb24b9": "s_1\\left(t\\right) =s_0\\left(t\\right)-B_0 h_0\\left(t\\right),", "b36baa627ad650ba563eec933f8fc247": "\\overline{P}(A)=1", "b36bde39b728e5886f1952b4df78eb50": "var\\{ \\omega \\} = \\frac{2}{N} \\left( 1 - \\frac{|R_N(1)|}{R_N(0)} \\right). ", "b36c04e4a6db3012f7bc9c7a148f7cbf": "\\mathrm{Cout}_{i+n-1}", "b36c4842c4c457e2a89e7b3f261cebdb": "\\mu_m=E_\\pi[\\mu_f(\\theta)] \\,\\!,", "b36c5f2a6d17f607113c431b791c903c": "0,1,i", "b36c7c583683f741fbf2f214650a8cc2": "\\pi_1 (X) =1", "b36d22ff30b19946fbce0871725af2ec": "\\psi(\\Omega(1+\\alpha)) = \\phi_2(\\alpha)", "b36d403260838aaab9f27ffbdc7ca90a": "\\ \\sigma_2'=\\sigma_3'", "b36d6aacc05d916cb4ef6b901378aa60": "h(x, y)= x^2+y", "b36d700973b6d0c0f7a4bba3bc01aa04": "A \\ang \\theta =A \\ang \\delta - \\beta", "b36d7366d83b0907ccc7468675c136b1": "\\pi_1(V,w) = \\langle v_1,...,v_m | \\beta_1,...,\\beta_n\\rangle", "b36d8f246067068ec06a80d0e25fc844": " \\textbf{y}(t) = \\begin{bmatrix} 1& 2\\end{bmatrix}\\textbf{x}(t) + \\begin{bmatrix} 1\\end{bmatrix}\\textbf{u}(t)", "b36d97a1a7e6c30c7c37076a36bf72fc": "P(D|M2)/P(D|M1)", "b36db1eb0211ec40ac28f73c9cecb8d5": "\\boldsymbol p", "b36e24227a1d000cf7f26648326ba6c7": " {} = x_0 \\begin{vmatrix}x_1&y_1&y_1\\\\x_2&y_2&y_2\\\\x_3&y_3&y_3\\end{vmatrix}\n-y_0 \\begin{vmatrix}x_1&x_1&y_1\\\\x_2&x_2&y_2\\\\x_3&x_3&y_3\\end{vmatrix} \\,\\!", "b36e327ab3919a88a50a8b665160ed22": "\\frac{\\partial\\rho}{\\partial t}=D\\frac{\\partial^2\\rho}{\\partial x^2},", "b36e4544c145826e0e6a017898dd403e": "(4) \\ \\Lambda \\sin (\\theta_m) = m\\lambda,\\,", "b36eac3d201a1a25dda928990a79d751": "\\mathbb C", "b36ec7652a58d22bce4bbbe61e82a861": "q = \\begin{pmatrix} a & b \\\\ b^* & a^* \\end{pmatrix}", "b36ece1067bbfd55b1b27e04f291aa89": "\\sum_{g\\in G}g", "b36efcbf626df764cc2503645e0806bf": "\\varepsilon^{-\\frac{1}{3}}\\operatorname{Ai}\\left (x\\varepsilon^{-\\frac{1}{3}} \\right). ", "b36f11d568b44daf17e024c19201f421": "Q, Q \\notin \\mathcal{P}", "b36fb91255979fed4b060317066e6afd": "VAG(x^3 -7x + 7,(0,1)) \\cup VAG(x^3 -7x + 7,(1,2)", "b3707129cafd2b4490b23f0448584c01": "\\sqrt{2} = \\scriptstyle\\frac{\\gamma^3-9\\gamma}2", "b37077f82287da6bfecc19762e13f555": "n\\geq10^{m-1}>81m", "b37136c6e2e1745e2e3d2f01fd79c9d0": "-\\infin \\le x \\le \\infin", "b371446eca42545722d7065e07de6f16": " \\mathbf{J}_1 \\cdot \\mathbf{E}_2 - \\mathbf{E}_1 \\cdot \\mathbf{J}_2 = \\nabla \\cdot \\left[ \\mathbf{E}_1 \\times \\mathbf{H}_2 - \\mathbf{E}_2 \\times \\mathbf{H}_1 \\right] .", "b37156de5b9223bf800c43f63bd6a995": "\\operatorname{ord}_P(f)=\\max\\{d=0,1,2,\\ldots: f\\in m^d_P\\};", "b371971db602b547b769b4e5d267bce5": "\\vec{F}_{12}", "b371d27fd2d6d207ddea1bb840e69060": "V=U\\oplus W", "b372000806dc1678fdabfa038f704823": "R^\\omega", "b3722522b1475d6efb288f72531f913e": "d\\,f(r(t)) = \\left [\\theta(t) + u - \\alpha(t)\\,f(r(t))\\right ]dt + \\sigma_1(t)\\, dW_1(t)\\!", "b3722c9a801cb823c3a331ee76dc256d": "0<\\alpha<2", "b3724742141abc278ec80258e9f6359c": "\\hat{\\mathcal{T}}_R", "b3726ad5b574f39002feef05d034cd66": " \\mathbf{p} = p_1\\vec{r}_u + p_2\\vec{r}_v", "b3732fec900e19059f6b5d7ccfe64a0d": "K_{\\text{cond}}=\\frac{[\\mbox{Total Fe bound to EDTA}]}{[\\mbox{Total Fe not bound to EDTA}]\\times [\\mbox{Total EDTA not bound to Fe}] }", "b373671248a0e0bc5aa3a22ea421caf1": " G=(W \\cup V, E)", "b3737811fad340bb9b5430de0081be61": "(P_1, \\ldots , P_r)", "b373a0b62e2c6a525f05747553a453e8": "(10a+b) \\cdot (10c+d)", "b373c9aaf6e3d64b82b5212883729798": "W(\\cdot) \\equiv \\frac{f(\\cdot)}{f_*(\\cdot)} ", "b373eca5bbb843d644c7154f844f6ff9": "I_S^\\prime=I_S/a", "b3740755d048945985fe129d70bb7657": "V_\\mathrm{out} ", "b3742fbe3d571f543dd5212555db08f0": "f(t)=\\sum_{i=0}^n a_it^i", "b37434f4790b79ce3ce332abc01d0be5": "\\varepsilon(e_i) = 1\\ ", "b3748f4c1fbe6213346f653cc81c2937": "\\varphi_Y(t) = 1 - {t^2 \\over 2} + o(t^2), \\quad t \\rightarrow 0", "b37498a6f70e4f65d3a8f57c1dcd78a1": "h(n) = \\max\\{h_1(n), h_2(n), ..., h_i(n)\\}", "b374a3a801d4f978e1bda6c21b02a610": "(\\mu-\\nu)(X)=0", "b374b10a005dc8c9ce872e04caf212d7": "\\eta_1", "b375b646d9b14ba907e5f6375e546ecf": "\\sigma_n(x_0) \\to \\frac{1}{2}\\left(f(x_0+0)+f(x_0-0)\\right).", "b375cc228f11f5ae0e381b38b88b1557": "f_1,f_2,\\ldots", "b3761bb70837c899c53474662995c320": "\\eta_{ij}={\\mathrm {diag}}(-1,-1,-1,1)\\,", "b37636ee3368838e4bf28a77372582bb": " Ehr_P(z) = \\sum_{t\\ge 0} L(P, t)z^t ", "b376521ee33af236d55cbe38ab1c4edf": "H^1_0\\subset H^1", "b3766082a2199d7eb7dd2676911b8d70": "10^{10,000,000,000}", "b376c0c0c1e2799185778697bb1ffae9": " EROEI = \\frac{\\hbox{Usable Acquired Energy}}{\\hbox{Energy Expended}}", "b37702735ecfb7e4faf1e5830cd52481": "\\frac{Z_i}{N_i} = \\frac{Z_{i+1}Z_e}{N_{i+1}N_e}", "b37729121173dd9c19fdd33f2e299cbf": "m(A,Z) = Z m_p + N m_n - a_{V} A + a_{S} A^{2/3} + a_{C} \\frac{Z^2}{A^{1/3}} + a_{A} \\frac{(N - Z)^{2}}{A} - \\delta(A,Z)", "b3772e8c8d295d4c4bb9c718fdfdbb2c": "\\mathcal{M} = \\mathcal{O}_X \\cap j_* \\mathcal{O}_U^\\times", "b37745adfab9189dc10e8f8040fce875": " w(i,j)", "b37765e97e48972bfa859d9dcff9609c": "w_i^{(t)} = \\big|y_i - X_i \\boldsymbol \\beta ^{(t)} \\big|^{p-2}.", "b37787c457be8dc8099e7661799b6dfc": " ^n\\sqrt{\\,\\,\\,}", "b377c0c20988bde2e65ac3de417a6416": "L_J(0)\\,", "b377cfb1f5190a3826638d226ca6e697": "(a_0 : \\cdots : a_n)", "b3780f6ac9cce3a027250a414358e9f9": "\\left(\\!\\!{n\\choose k - 1}\\!\\!\\right)", "b37882b4804a6ea437529681774cc9bc": "\\operatorname{End}_R(U)", "b378d9656f01f277d044907fe5bc5d53": "\\mathbf{u} = (p_2-p_1) / \\left|(p_2-p_1)\\right|", "b3794e291f18d28d4e7c3f545df80810": " \\sigma = \\sigma_0 \\, \\left(\\frac{T_0}{T} \\right)^\\frac{1}{2} ", "b379a906e71a3677585e2eecef4238fa": "\\overset{\\sim}{\\to}", "b379b95e703764163b8c9e014b44ebfd": "c_1,\\ldots,c_k", "b379ccbc3f1df0fbd02ec573aa97e3f8": "\\scriptstyle (u_T,v_T)", "b379d18b8a4daef0d77193765f8122f2": "\\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} =\\frac{(w_1-w_3)(w_2-w_4)}{(w_2-w_3)(w_1-w_4)}. ", "b379db2cb7fcdd7062903d961af59e1c": "1/2 + i \\hat H", "b379e294d8931c6844b3307dac427c69": "R \\!\\, ", "b379ffd61c53f58edcdb2e9602c0dcc7": "\\textstyle{EI\\frac{\\partial^2 u}{\\partial x^2}}\\,", "b37a14f502010d17637ed5dd77546d22": "\\Pr(M|D) = \\frac{\\Pr(D|M)\\Pr(M)}{\\Pr(D)}.", "b37a22dfb32a75941b3cd2456f39d0ba": "K_{\\mathrm{sp}} = \\left[\\mbox{Ca}^{2+}(aq)\\right]\\left[\\mbox{SO}_4^{2-}(aq)\\right].\\,", "b37a56f788e443f00f5e5424263f113d": "W = XY", "b37a62f6c2480e696d2899a19fbd65cd": " \\min \\Phi_k (\\bold x) = f (\\bold x) + \\sigma_k ~ \\sum_{i\\in I} ~ g(c_i(\\bold x)) ", "b37a7ee75d37935770ebb1949fc5a6ff": "A_x,A_z", "b37ae6172d17349fb2b23726eaa1800b": "M_{member}", "b37b3fab9981fe9c399ab07cb0f32994": "l_a=(-1,0,0,0)\\,,\\quad n_a=(-\\frac{G}{2},-1,0,0)\\,,\\quad m_a=\\frac{r}{\\sqrt{2}}(0,0,1,\\sin\\theta)\\,.", "b37b608fcbc020f7af48eb0e66caa3c8": "\\mu^{-1}(\\epsilon)", "b37b69605d183ba16f9c11dd872bcf50": "^2", "b37b6d620656a554d3507923f675e54f": "a^n \\sin(\\omega_0 n) u[n]", "b37b93483657a9c705e05712b28e668f": "t_n=\\frac{H_{2n}-1}{2}", "b37bc44f9ae6f10b620792736b4e47eb": "\\partial s", "b37bd73369b035569566df1f1e0fea67": " (\\arctan x)' = { 1 \\over 1 + x^2} \\,", "b37bf130683f747eb2cd2ba7125a1774": "\\frac{\\operatorname{d}i_{\\text{L}}(t)}{\\operatorname{d}t} = \\omega I_p \\cos \\left( \\omega t \\right)", "b37c223d06332ae94c57536f377d3b35": "S^* M (S^{-1})^* = S^{-1} M S \\quad \\Rightarrow \\quad SS^* M (SS^*)^{-1} = M.", "b37c5244c9e1d25a73a1e423cb3b10b8": " m_1 \\mbox{sinh}(s_1)+m_2 \\mbox{sinh}(s_2)=m_1 \\mbox{sinh}(s_3)+m_2 \\mbox{sinh}(s_4) ", "b37c63ca2b850257f9c767423738fb95": " t \\to \\pm \\infty ", "b37cab6dd322f926f9fe13fe13038aba": "||F_\\ell(\\mathbf{P},\\mathbf{K})||_\\infty = \\sup_\\omega \\bar{\\sigma}(F_\\ell(\\mathbf{P},\\mathbf{K})(j\\omega))", "b37cdef9afc80698567023bc700f8074": "S_{\\rho}(z)=\\sum_{k=0}^{k=n}\\frac{m_k}{z^{k+1}}+o\\left(\\frac{1}{z^{n+1}}\\right).", "b37d7222860df5b7dd68a79a3c84cbce": " FV(A_4) \\not\\subset V ", "b37d9342efae67c23aba7006462ea570": "w'_L=w'_G\\,", "b37e337ff7bbf858ca4fabfb32a22a99": "\\frac{(b-a)^3}{24}\\,f^{(2)}(\\xi)", "b37e3cb3b8d0d610a28c266b418d5737": "e \\leftarrow e||enc||I_{W^{\\ast}}^{B^{\\ast}}(mac \\oplus mask_m)", "b37e4da811b7d6daa3c5361357fa0104": "H^?_*", "b37e6c07e23bbf201907eb35271e6222": "\\mathbb{Q}\\cap[0,1)", "b37e80cb7b6752ac1e20f7e81c0c1d33": " P(\\partial_t,\\xi) ", "b37e8f6cf48d9fd42eac64bf31350234": "\\mu(X) = 1", "b37e946806135df659e5c6eb7189c3cb": "Q_c(t)", "b37ec5cdeb911cf15079a8d2686f0f74": " \\sin x = \\cos\\left(\\frac{\\pi} {2} - x \\right) ", "b37ee86c3d8a1e52642740408a15c12f": "\\,\n\\mathbf{S} = { \\mathbf{V} + \\mathbf{U} \\over 1 + \\mathbf{V} \\cdot \\mathbf{U} }\n", "b37f7f2003fda8b85fa802c0714a76f7": " |\\psi|^2 = - \\frac{\\alpha} {\\beta}.", "b37f8680e372f1187285ab703e280162": "\\xi_\\mathrm{cutoff}", "b37f86d1c1848292a8157aa249791d67": "f(S)=\\min t", "b37f9f97bbec8ec16901ae9625d9409f": "\\{G_n\\}_{n \\in \\mathbb{N}}.", "b37fc9a40d91499accf6036b90cd956c": "(\\phi \\to (\\psi \\to \\chi)) \\to ((\\phi \\to \\psi) \\to (\\phi \\to \\chi))", "b37fefe850564f437494b78e6d96c1c5": "\\partial \\bar{\\Omega}", "b37ff0c92ffd413cf3fce6a590f390f1": "\nI_\\circ = 2\\pi\\varepsilon_\\circ c \\, r\\vert E_{\\pi/2}(r) \\vert \\, .\n", "b37ff8977709d2c73a49456ec854bbcd": " \\mu(x+t) = \\frac{q_x}{1-tq_x}", "b3801bdeb21de63ecb516fab522e8d9e": "\n \\infty \\text{ for } {5 \\over 3} \\le q < 2 ", "b38073024b171a3951dfa3347a7a3791": "p(\\theta|\\varphi)", "b380752fd7e896d832790bf7725b3cf1": "{ P }_{ in }={ L }_{ \\bigodot }\\left( 1-a \\right) \\left( \\frac { \\pi { { R }_{ p } }^{ 2 } }{ 4\\pi { D }^{ 2 } } \\right) ", "b38099561056c01c992acc10daa1119f": "{T} = \\frac{30m}{sdl(1+l^2)}", "b380bb023bfc7f1582c8ef85923356f8": " \\psi_1 \\left(\\tfrac14\\right) = \\pi^2 + 8G", "b380e7b2e7bb0590eabcc6aacf6376f1": " p = (p^0+p^3)P_3 + (p^0 - p^3)\\bar{P}_3 + (p^1+ip^2)\\mathbf{e}_1 P_3 + (p^1-ip^2)P_3 \\mathbf{e}_1", "b380f933c93fc298f869759a1974f4c4": "+g^{\\nu \\alpha }g^{\\beta \\sigma }(\\Gamma^{\\mu}_{\\alpha \\rho }\\Gamma^{\\rho }_{\\beta \\sigma }+\\Gamma^{\\mu}_{\\beta \\sigma } \\Gamma^{\\rho }_{\\alpha \\rho } - \\Gamma^{\\mu}_{ \\sigma \\rho } \\Gamma^{\\rho }_{\\alpha \\beta } - \\Gamma^{\\mu}_{\\alpha \\beta } \\Gamma^{\\rho }_{ \\sigma \\rho })+", "b3810fd6e47114e4a71bdf168b56cce1": "\\textstyle A", "b381378e501b74ce040733b2c11e93df": "\\ u'", "b381a1aa59913f90570ca05a3a396076": "D_\\alpha (P \\| Q) = \\frac{1}{\\alpha-1}\\log\\Bigg(\\sum_{i=1}^n \\frac{p_i^\\alpha}{q_i^{\\alpha-1}}\\Bigg) = \\frac{1}{\\alpha-1}\\log \\sum_{i=1}^n p_i^\\alpha q_i^{1-\\alpha}.\\,", "b381bdecec33b5dbae5ec863d77b7f61": "v_o", "b381e19964cb4720ae0bada81d798897": "\\operatorname{M}_n(k) = B", "b381f27d583bb4d89d258ff2f2cbb34f": "\\ln\\bigl( 2 \\sqrt{r} \\bigr),", "b3820c51ee6b05023aee24c49466057b": "h(n) = 0 \\,\\, \\forall \\, n < 0", "b382290ae1c3d3bf6d81e0af683723d0": "\\scriptstyle \\, p", "b38239354a684ca16363ba43122f119e": " u(0) = e^{ikL} u(L)=e^{ikL} u(N a) \\rightarrow e^{ikL} = 1 \\,\\! ", "b3823d48eb5ec1575a8f250837e283ae": "\\frac{P \\land Q}{\\therefore Q}", "b382491d6095c4f1ce3ec785f2f87306": "\\frac{d}{dt}\\!\\left(\\frac{\\partial \\mathcal{L}_{1}}{\\partial\\dot{\\alpha}}\\right)=\\frac{\\partial \\mathcal{L}_{1}}{\\partial\\alpha}\\,,", "b3826fac186d74aa9455f482fe041301": " \\frac {70 \\text{ miles}}{\\sin D} = \\frac {\\text{ritorno}}{\\sin 45} = \\frac {\\text{avanzo}}{\\sin 90} ", "b3828df35423515fe3ace06511209f09": "t = a \\cdot \\sqrt{\\frac{a} {\\mu}} ( E - e \\cdot \\sin E)", "b38293c8ae24b1c4ab76dab720ea6131": "\\partial_t \\left( \\frac{1}{2} \\partial_x \\psi \\right) = \\frac{1}{2} \\partial_t \\phi \\,", "b3831e7b5cc3ee40d01e05cb8d275846": "\\Gamma_\\infty", "b38337820d7bea18eee0affb6e964911": "k=2\\pi/\\lambda", "b383399758251160bf32256a8775ef40": "\\epsilon^\\ell", "b383c2ac5820fb9972e87ff44b14a1eb": "\\begin{align}\n\\mathbf{P}^{NL}= \\varepsilon_0 \\chi^{(2)} \\mathbf{E}^2(t)\n&= \\varepsilon_0 \\chi^{(2)} [\n|E_1|^2e^{-i2\\omega_1t}+|E_2|^2e^{-i2\\omega_2t}\\\\\n&\\qquad+2E_1E_2e^{-i(\\omega_1+\\omega_2)t}\\\\\n&\\qquad+2E_1E_2^*e^{-i(\\omega_1-\\omega_2)t}\\\\\n&\\qquad+2\\left(|E_1|+|E_2|\\right)e^{0}],\n\\end{align}", "b383daf7aeef95779a202ed6ef3add5e": "(a,b) := (-1)^{\\left|a\\right|}\\Delta(ab) - (-1)^{\\left|a\\right|}\\Delta(a)b - a\\Delta(b)+a\\Delta(1)b .", "b384432c84f0a465088caec1bec364bb": "\\textstyle\\sum x_i", "b384af1185ca42f15fa01bac03666243": "\\theta=\\frac{1}{n}\\arccos\\left(\\frac{\\pm j}{\\varepsilon}\\right)+\\frac{m\\pi}{n}", "b384c82d779cb1b7fc6df5796f629a46": " L = \\frac{\\hbar}{2 e} \\frac{1}{I_0 \\cos \\delta_0}.", "b384d40749b8036bef6e0f961ac520b0": "x^{\\underline{n\\!}}", "b384df06bb5c67230c97fbbdba9367bc": "k_2/k_1=0.5", "b38531db306867a742f831403cdc324e": "\\color{Turquoise}\\text{Turquoise}", "b385ea2c0b4fc9b394414da1791682cf": "3 \\times 2 = 6", "b38600633036962f873b2fd9ec257438": "a^p\\equiv a\\pmod{p}.", "b38622de509edd6e7542eeda95e3bca8": "O(2^{{(\\log N)^{1/3} (\\log \\log N})^{2/3}})", "b38640769d4a75803896a9ef6f346063": "dH = nC_pdT\\,\\!", "b3865b63db6bc011a8d44a1f4aa8bfa2": "\\mathrm{2\\ O_2 ^-\\ +\\ 2\\ H^+\\ \\xrightarrow {SOD}\\ \\ H_2O_2 +\\ O_2}", "b3868ba70a8f4cf55464dd9926381db1": "-p", "b386b479f620d744966f02f5d2dbf279": " d = \\frac{a}{\\pi} \\sqrt{(n^2 + nm + m^2)}=78.3 \\sqrt{((n+m)^2-nm)} \\rm pm,", "b386eb6ae61bf3e373507ab0ab2a781c": "Q = ~~\\frac{\\partial F_2}{\\partial P} \\,\\!", "b387177f43aa8a211fd49ca8610b4a44": "\\mathbf{B}''(t) = 6(1-t)(\\mathbf{P}_2 - 2 \\mathbf{P}_1 + \\mathbf{P}_0) + 6t(\\mathbf{P}_3 - 2 \\mathbf{P}_2 + \\mathbf{P}_1) \\,.", "b387631b29aaa92b444083dd56f63e05": "\\langle \\epsilon_i|\\epsilon_j \\rangle = \\delta_{ij}", "b38783c82b15a70659b4d7e307e05f2b": "J_a", "b387b386b36d6db4a13b9f26e1c4fe11": "\\mathbf{a}, \\mathbf{b} ", "b387c75f47adfb21b6bb09c66932d14f": "\\mathcal{P}_\\mathrm{s}(k) \\propto k^{n_\\mathrm{s} - 1}.", "b38835d235e064308d951041dac58027": " H_2^{+}", "b388546f7265be0102e65b0dc69fb619": "\\lim_{n \\rarr \\infty} p_{ii}^{(kn+r)}", "b38871c5ec94bc4ee4c7091fd44cd199": "\\oint_C \\bold{B} \\cdot {\\rm d} \\boldsymbol{\\ell} = \\mu_0 ", "b3888789e551c6fb4be8949470727115": "r (m + n) = rm + rn", "b388d52865e7a0dcb4f115521fafe9cb": "FWER", "b388db4877b92469aa0cd2416e4023b2": " 0<\\mu \\leq 1", "b388e5bdcbc976ae05d4348d57c81ea2": "\\alpha^B", "b3891a8196e1187e20de1649ad562421": "|g| = 1", "b3892a8dd641d8c4c67b888b7986b1a8": " A^{(0)} := A", "b389f34bc8cb94f69543459905f8e6e8": " \\mathbf{g}(\\mathbf{r}) = \\frac{\\mathbf{F}(\\mathbf{r})}{m}.", "b38a3cc9739bed58fa4fbe967a480961": "\\operatorname{Ext}_R^n(A,B)=(R^nT)(B).", "b38a60453f9f58140239c63cddc7882b": "\nf(H_{\\text{eff}})= \\int dE | \\Psi_{E}\\rangle f(E) \\langle \\Psi_{E}^* |\n", "b38b062594ba46f7b421b41687746257": "\\exists f\\in C,g\\in I,n\\in\\mathbb N\\;f+g+\\left(\\prod H\\right)^{2n}=0.", "b38b330daebcffca87abf9f1791b8dad": "ANH = \\frac {BL_s}{450}", "b38b39a84deeef3e5236c4a2fbc9942d": "\\Sigma _{i} =\\gamma _{5i}\\beta _{i}\\gamma _{\\perp i}, ", "b38bd6a9dcc764cde7305542b22b2674": "r_1 = r_2 \\approx 1 \\,", "b38c6811f506ff1cd035862b05203a0c": "\\{B_k\\}", "b38c8fbc96120899130bc7601fa70ec8": "\ns^{(k+1)}_n:=\\sum_{k=-N}^N a_k s^{(k)}_{2n-k}+\\sum_{k=-N}^N b_k d^{(k)}_{2n-k}\n", "b38cfbb72ddcb359898996f96b2b96d9": "\n \\begin{align}\n M_{11} & = D\\left[\\mathcal{A}\\left(\\frac{\\partial \\varphi_1}{\\partial x_1}+\\nu\\frac{\\partial \\varphi_2}{\\partial x_2}\\right)\n - (1-\\mathcal{A})\\left(\\frac{\\partial^2 w^0}{\\partial x_1^2} + \\nu\\frac{\\partial^2 w^0}{\\partial x_2^2}\\right)\\right] \n + \\frac{q}{1-\\nu}\\,\\mathcal{B}\\\\\n M_{22} & = D\\left[\\mathcal{A}\\left(\\frac{\\partial \\varphi_2}{\\partial x_2}+\\nu\\frac{\\partial \\varphi_1}{\\partial x_1}\\right)\n - (1-\\mathcal{A})\\left(\\frac{\\partial^2 w^0}{\\partial x_2^2} + \\nu\\frac{\\partial^2 w^0}{\\partial x_1^2}\\right)\\right] \n + \\frac{q}{1-\\nu}\\,\\mathcal{B}\\\\\n M_{12} & = \\frac{D(1-\\nu)}{2}\\left[\\mathcal{A}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2}+\\frac{\\partial \\varphi_2}{\\partial x_1}\\right)\n - 2(1-\\mathcal{A})\\,\\frac{\\partial^2 w^0}{\\partial x_1 \\partial x_2}\\right] \\\\\n Q_1 & = \\mathcal{A} \\kappa G h\\left(\\varphi_1 + \\frac{\\partial w^0}{\\partial x_1}\\right) \\\\\n Q_2 & = \\mathcal{A} \\kappa G h\\left(\\varphi_2 + \\frac{\\partial w^0}{\\partial x_2}\\right) \\,.\n \\end{align}\n", "b38d2aa331233fde8dcfdb96fdd45309": "\\mathbf{H}(\\mathbf{x},t)=\\mathbf{H}(\\mathbf{x})e^{-i \\omega t}", "b38d4974b3df137b6a6a8ad612c42ade": " \\sqrt{x^2+y^2} - L=0\\,\\!", "b38d890e96d2a15ad28f375ac559c2eb": "\\forall A: \\varnothing \\subseteq A\\, .", "b38d8e34c51cf27957c6278e56eba275": "\\mu=-(\\mathbf{X}\\boldsymbol{\\beta})^{-1}\\,\\!", "b38d90f7e51a6ad43644de01f3e85423": "\\{0,1\\}", "b38dab56401444c630f3e2eacabfbdbf": "{q^2 \\over g} \\left({1 \\over {y_1y_2}} \\right) = {1 \\over 2}({y_2 + y_1})", "b38e3ad00511bc4140c288dacf2dfbce": " p \\times 1", "b38e494949f1cd22a892052c39edcee8": "(i)^k-(i + 1)^k", "b38e584e95858f729f5cc4399f3b995e": "D=d", "b38e7328e89228bc97f00d286e67f050": "W_0'', W_1'' = solve(G \\setminus B)", "b38e9c3d807d739ff9d07527f6ecb7fc": "H = H(q_1,q_2\\cdots q_N;p_1,p_2\\cdots p_N;t).", "b38f210942590a3b0406aeb629dc1a9d": "{\\bar{S}}_7", "b38f58c2612b7bf4d4b4b49adb012540": "h=(b-a)/n", "b38ff6af9b33e93bc39bd1813be241f2": "0\\le i < n", "b390aa14d8dde0171bca2032f615970b": "T \\equiv -L_{-1}=-{1\\over 2} \\sum_{n \\in \\mathbf{Z}} :b_{1-n}b_n: \\equiv - \\sum_{n>0} n x_{n-1} \\frac{\\partial}{\\partial x_n}", "b390c95094083219e808a1bc0ef4c1a2": "\nb=\\sin (a) \n", "b3911243ed11006ffffe65100ceb9d4f": "SP_4(2)", "b392094b2201328bc152fcf2666bffd4": "\\gamma/\\alpha", "b3925a9fb323b83be61d3dc70b0118a8": "\\mathcal{L}\\left(f\\right) = \\frac{f_{osc}^3 \\sigma^2_c}{f^2}", "b3928e5bd12f21bfc94ba211d1b358f7": "y(t) = A\\cdot \\sin(\\omega_c t) + \\begin{matrix}\\frac{AM}{2} \\end{matrix} \\left[\\sin((\\omega_c + \\omega_m) t + \\phi) + \\sin((\\omega_c - \\omega_m) t - \\phi)\\right].\\,", "b392e21323720230c1d408b3e5b07533": " B = \\sum_{i=1}^n y_i", "b39312e08f2f13570205529ce7a7c38b": "\\operatorname{E}u(w)=\\int_{- \\infty} ^ \\infty \\! u(\\mu_w+ \\sigma _w x)f(x) \\, dx \\equiv v(\\mu_w, \\sigma_w),", "b39325c04b95ae5fac370e2080a686f2": " 3 \\times 2", "b39335b6584e8455ab4de3c86b439e21": "l_i", "b393ada365a2128453e9be55253ab7f0": "V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \\, ", "b3941db2f83d80bde12d22290207e688": "G_{\\mu \\nu} + g_{\\mu \\nu} \\Lambda = {8 \\pi G \\over c^4} T_{\\mu \\nu}.", "b3947d1f29478a80e2d08fd9df087c8c": "\\scriptstyle \\bar{E}", "b394842da8b4177367dfebdcf72eb8e1": " w = \\frac{1}{ x_0 } - \\frac{ v_0^2 }{ 2 \\mu } ", "b394d55ba74f325a73c180b0dc7d80d8": " \\gamma ", "b394ee85321f0af76e12702282e15c16": "\\vec{v} = (b,c,d)", "b3951bdc2799efacc710b5fcc17dd9db": "\\zeta_n^{s-r} a_i \\equiv a_j\\pmod{\\mathfrak{p}},", "b395256ce832a253cf407f6b27705f26": "\\Delta\\rho \\simeq \\frac{3\\rho}{2E_F} \\Delta E_F", "b395db2fd8497ce4609e9f72d6e9798a": "r = a\\cdot{1 - e^2 \\over 1 + e \\cdot \\cos\\nu}\\,\\!", "b395eaa21fda78530038895582b7decc": "\\omega e_l = e_l \\omega \\,", "b395f49a26833b8b26a81b934a2fab83": "\\textstyle S^n \\rightarrow Y^n", "b3962043ce9cbb6c800fe58006ff1e3a": "z^N-1", "b3962fa8f1ad4dba5a6dedf03a5cf598": " \\theta + p\\theta\\alpha = p\\alpha", "b39647df33473d968498e3e2f33f8c98": "\\frac{\\Delta u_n}{\\Delta u_{n-1}} = \\frac{u_{n+1}-u_n}{u_n-u_{n-1}}", "b3964f255295d14da82c7ff2a8ea0c59": "b_1,...,b_k \\in \\mathbb{Z}", "b396790d1105a4317ed94f4cb86b6aa7": " \\mu_{j,k} = \\operatorname{E} \\left[ ( X - \\operatorname{E}[X] )^j ( Y - \\operatorname{E}[Y] )^k \\right] = \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} (x - \\mu_X)^j (y - \\mu_Y)^k f(x,y )\\,dx \\,dy. ", "b39681ee380b6e9d2e8378344a4a440f": "\\nabla^2 E + k_0^2 n^2 (I) E = 0", "b396e4d0706ca316bd0766e5c68bf592": " \\delta^l {}_m = g^l {}_m,\\,\\!", "b397612d537f1ddaa62263f8eeaa189a": "A = {X_1}^2", "b39768a953359c5d5b382c9ae1640262": "n^a\\partial_a=-\\partial_r \\,,", "b39787fcfd4c7e9ca21f07703304defb": "1 + \\frac{1}{3} + \\frac{1}{3 \\cdot 4} - \\frac{1}{3 \\cdot4 \\cdot 34}", "b397d810bf083201c7d6c530a752230b": "e^{\\int f(x)\\,dx}", "b397e11c9a9c57cb3a9347462da89a6b": "\n\\begin{array}{rcl}\n t & = & t_+(A,t) - (R/C) \\sin( 2 \\pi F_n t_+(A,t) + A) \\\\\n t & = & t_-(A,t) + (R/C) \\sin( 2 \\pi F_n t_-(A,t) + A) \\\\\ne(A,t) & = & \\cos( 2 \\pi F_c t ) ( 1 + c(t) ) \\\\\n & + & g(A,t) \\\\\n c(t) & = & M_i \\cos ( 2 \\pi F_i t ) ~ i(t) \\\\\n & + & M_a ~ a(t) \\\\\n & + & M_n \\cos ( 2 \\pi F_n t ) \\\\\ng(A,t) & = & ( M_d / 2 ) \\cos( 2 \\pi (F_c + F_s) t_+(A,t) ) \\\\\n & + & ( M_d / 2 ) \\cos( 2 \\pi (F_c - F_s) t_-(A,t) ) \\\\\n\\end{array}\n", "b39803d248f298e99fc15a08b59fff6b": "w = u+iv \\in \\Omega", "b398535a4d9fb56bd99dcd87cf64bdd8": "\\scriptstyle{\\mathrm{R}^- \\notin \\mathrm{R}}", "b3985ae8019991ee009dd32e713340ef": "\nP\\left( m;\\;0,\\;z_1, \\;z_2, \\;z_3,\\ldots,\\;z_{n-1}\\right)=0\n", "b39885fa283d70d4c334fb327faa0af0": "\\frac{\\partial \\theta_f}{\\partial t} + \\nabla \\cdot ( \\theta_f \\bold{u}_f ) = 0,", "b398ba53b9abc587f794956242d299a0": "\\sum_{i=0}^k {k\\choose{i}}^2\\lambda^i", "b398f9d0eabb493faec006adf15a9056": "\\scriptstyle+\\pi/2", "b3991e6fe3e116d93aa13c892dc8e23b": "x\\in\\mathfrak{X}", "b399b644255a84697ed3b22ee7e41132": "\n \\begin{bmatrix}\n \\cdot & \\cdot & \\cdot \\\\\n \\cdot & \\cdot & \\cdot \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\cdot & \\cdot \\\\\n \\cdot & \\cdot \\\\\n \\cdot & \\cdot \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\cdot \\\\\n \\cdot \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n \\cdot & \\cdot & \\cdot & \\cdot \\\\\n \\end{bmatrix}\n =\n \\begin{bmatrix}\n \\cdot & \\cdot & \\cdot & \\cdot \\\\\n \\cdot & \\cdot & \\cdot & \\ \\\\\n \\end{bmatrix}\n", "b399d9a7128d8bb2be48ba82639487e7": "\n[d(\\rho, \\rho+d\\rho)]^2 = \\frac{1}{2} \\sum_{j,k=1}^{n} \\frac{|\\langle j| d\\rho | k\\rangle |^2}{\\lambda_j+\\lambda_k}.\n", "b399fc886c47d6e3ef41c013f594825c": "\\mu_F\\,", "b39a429c3296a78be133c17569058621": " \\delta (v)=v_{(-1)}\\otimes v_{(0)} ", "b39a791ce556a5edc1d3291b998140d8": "v_j(N)", "b39b486ab0e05c15da9fcc485c6e4226": "f(x) = x^n - A", "b39bd1665971691a6b356df32ab21eab": " \\iint_{|z|<1}|F(z)-f(z)|^2 \\, dS ", "b39c009f737a310a830d919fbf6e88f2": "\nm=2,~x^{\\{m\\}}=\\left(\\begin{array}{c}x_1^2\\\\x_1x_2\\\\x_2^2\\end{array}\\right),\n~H+L(\\alpha)=\\left(\\begin{array}{ccc}\n1&0&-\\alpha_1\\\\0&-1+2\\alpha_1&0\\\\-\\alpha_1&0&1\n\\end{array}\\right).\n", "b39c472a9bd5285df46a1c4a205aa1ce": " z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.", "b39c6ae9a2906c98e980928da0628e21": "{{{2}}}", "b39c86ab0a38d7fdb51bc957ad398842": "H_\\text{norm}", "b39cbc9dfa6017ac5e90c926518c54df": "\n\\begin{align}\n- \\ln [(n/2)! ] &\\approx -\\frac n 2 \\ln \\frac n 2 + \\frac n 2 -\\ln \\sqrt{n\\pi}\\\\\n&= \\underbrace{-\\frac n 2 \\ln n}_{keep} + \\underbrace{\\frac n 2 \\ln 2 +\\frac n 2 -\\ln \\sqrt{n\\pi}}_{drop} \\\\\n\\end{align}\n", "b39cccb7c469a2ae5b93f067d86abde2": "\\begin{align}\n h_1(X_1,\\ldots,X_n)&=e_1(X_1,\\ldots,X_n),\\\\\n h_2(X_1,\\ldots,X_n)&=h_1(X_1,\\ldots,X_n)e_1(X_1,\\ldots,X_n)-e_2(X_1,\\ldots,X_n),\\\\\n h_3(X_1,\\ldots,X_n)&=h_2(X_1,\\ldots,X_n)e_1(X_1,\\ldots,X_n)-h_1(X_1,\\ldots,X_n)e_2(X_1,\\ldots,X_n)+e_3(X_1,\\ldots,X_n),\\\\\n\\end{align}", "b39d0b2b164c8c43c53a7d376f5a5d93": "\\Delta y \\approx dy", "b39d1cca30eaf3a36ac1da1f2f09f5d2": "\\frac{\\partial U}{\\partial V}\\ ", "b39d4c8e85558822ffc346decc43e8a4": " \nT_t^x=\\int_0^t \\eta^\\rho_s(x)\\mathrm{d}s.\n ", "b39d522071c16e0669969513193819e4": "\nE[J,B] = C \\int d \\vec x (I(\\vec x) - J(\\vec x))^2 + A\n\\int _{D/B} \\vec \\nabla J(\\vec x) \\cdot \\vec \\nabla J(\\vec x) d \\vec x + B \\int _B\nds\n", "b39dbae8093555878a2742b698f35583": "I \\otimes \\Lambda (\\rho) ", "b39df834231ec9abf2e9875e03093bb1": "x \\times S(y) = (x \\times y) + x", "b39e28f8f1bf6ed199cc61565ade41a3": "\\tilde{\\mathbf{e}}_{k}", "b39ef54e9236caff9980be39ae4ef785": " \\int_a^b\\! e^{M f(x)} \\, dx ", "b39f14f00721ed0224f5808142679467": "\\tan\\sigma_{12}=\\frac{\\sqrt{(\\cos\\phi_1\\sin\\phi_2-\\sin\\phi_1\\cos\\phi_2\\cos\\lambda_{12})^2 + (\\cos\\phi_2\\sin\\lambda_{12})^2}}{\\sin\\phi_1\\sin\\phi_2+\\cos\\phi_1\\cos\\phi_2\\cos\\lambda_{12}}.", "b39f3e74e31db56a89e7baa14725bf7d": "H(X_1, X_2),", "b39f49056a60e48104392bc4ab7593c7": "z^2f''-zf'+(1-z)f=0\\,", "b39f8e90133824606d848a45d4db0be4": "\\rightarrow H^1(X, \\mu_n)\\rightarrow H^1(X, \\mathbf{G}_m)\\rightarrow H^1(X, \\mathbf{G}_m)\\rightarrow H^2(X, \\mu_n)\\rightarrow H^2(X, \\mathbf{G}_m)", "b39f906430806cf3424faaa09e8e1fa8": "\\scriptstyle k \\;=\\; 3", "b39feb04c5e1fbd98665e1d5351269f8": "\\displaystyle{(\\pi(S)\\pi(R))^3 =\\pi(J)}", "b39feb5ff547d50a9038925208efba96": "\n(a_p - a_1) \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1-1, a_2, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right) +\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_{p-1}, a_p-1 \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right), \\quad 1 \\leq n < p,\n", "b3a009daee86850e9d26b4ff536abd06": "\\liminf_{n\\to\\infty}\\frac{g_n}{\\sqrt{\\log p_n}(\\log\\log p_n)^2}<\\infty.", "b3a0541656fed02d4a3472f96dbee070": "\\frac{M}{r^3}", "b3a0af66e0a94103bdb6dee06bf272e7": "O(\\frac{1}{(1-p_f)^2}(1+\\frac{1}{n}))", "b3a0d33c255af8581b2e7c0791b5757e": "T\\in\\mathcal{F}", "b3a0e31764bc939c66eba960765eca10": "\\epsilon_{t}\\sim^{iid}WN(0;1)\\,", "b3a17f7c339aaaad2c632e168123c7bc": "Lu", "b3a19ae9e2897d910b7eb89cf2d51af8": "T_{em}\\approx\\frac{1}{\\omega_s}.\\frac{3V_{TE}^{2}}{R_r^{'}}.s", "b3a1d6a8e0ee9ae64a71ecfd3a39ce01": "i_n\\,\\!", "b3a235acf1abcaafc27ec1b9ac9261cd": "w_2=p_{nt,2}*MPL_{nt,2}=p_{nt,2}=p_{t}*MPL_{t,2}", "b3a2654aef16b4900309f66e4afe32ff": " PSF(r)=\\frac{1}{\\pi (1+\\eta)} \\left[\\frac{1}{\\alpha^2} e^{-\\frac{r^2}{\\alpha^2}} + \\frac{\\eta}{\\beta^2} e^{-\\frac{r^2}{\\beta^2}}\\right] ", "b3a26b4ea9749e0b6fc4804cd4436a9a": "T_A^1, T_A^2", "b3a27cd4a69443c0a0120768f9184dcd": "(\\mathfrak{m})", "b3a291bf19f9c818145637387e56185d": "Y_i = S(\\alpha, \\beta_{y_i}, \\gamma_{y_i}, \\delta_{y_i})", "b3a29d3554abdf78e3881e428c519449": " \\omega_k", "b3a2a995912df97e66bbfd7c6d52d31c": "f(x,q) \\in V_q", "b3a2aefa71b94924549fafde68478b76": "[n,n-t\\alpha,2t+1]_{2}", "b3a2bb8314802b4ac9a06ee630c549e8": "x \\in D = \\{x_1 = 0\\}", "b3a2eb9f5d8877ecd8d5ab5d711d577f": "{\\color{white}-}\\nabla \\left( \\nabla \\cdot \\mathbf{H} \\right) - \\nabla^2\\mathbf{H} = \\nabla \\times \\mathbf{J}.", "b3a2fb8ef7f08828d72283e1080bbb60": "\\pi(x_1,x_2,\\ldots,x_m) = C \\pi_1(x_1) \\pi_2(x_2) \\cdots \\pi_m(x_m),", "b3a3488cbcc121528a4d54727e82d686": "\\forall a,b \\in F, x,y \\in \\mathfrak{g}, v,w \\in V", "b3a378da4ebe900db368cd100dffee2a": "N_{B}", "b3a3ba605e89a0ab34429674317a5a8d": " \\mathrm{MSE} \\,", "b3a3bf5a76f4d500c49ba444b91d6d43": "{\\tilde{C}}_{n}", "b3a3c26c853baf4d5f2d12cb862275da": "\\operatorname{prox}_{R}(x)", "b3a42b71b26793a5b8f3b0f1d0682f9f": "\\;\\alpha=\\beta^n", "b3a43536c7dd3a7a15689a5044f30c49": " F(x) = y ", "b3a44b0fde4ea6d2342f98feb669f20d": " d = r \\left( \\frac{ 2 \\rho_M R^3 }{ \\rho_m r^3 } \\right)^{1/3} ", "b3a45381adee15c9d50d0d9638721119": "\\mathit{Y_{i}}\\,", "b3a487f015b42efe8ccfb9e5721f38a7": "\\Delta u = \\frac{\\partial^2u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2}.", "b3a4c7c78a2207a4d6b0d8d604403c0e": "G_\\eta(\\tau)", "b3a4cb9709ba1df56a251673d769563b": "q^d-1", "b3a4da7f488a3d3557fced48fc762ad5": "\\frac{dX}{dt} = \\alpha . X - \\beta . X . Y", "b3a4e1897a83d548944c49412476032c": "\\displaystyle \n\\begin{align}\n{\\partial A_i \\over \\partial t_j} &= {\\left[ A_i, \\ A_j \\right] \\over t_i - t_j}, \\quad i\\neq j \\\\\n{\\partial A_i \\over \\partial t_i} &=- \\sum_{j=1 \\atop j\\neq i}^n {\\left[ A_i, \\ A_j \\right] \\over t_i - t_j}, \\quad 1\\leq i, j \\leq n\n\\end{align}\n", "b3a4f1dc43a008216070e00b4f495e91": "\\varphi^{-1}", "b3a51289e6b2b788da1f219db23f1d8c": "5^2", "b3a52140be3915a93fe870426fd3db44": "\\textstyle \\left(\\sum_{i\\in\\N} a_i X^i\\right)+\\left(\\sum_{i\\in\\N} b_i X^i\\right) = \\sum_{i\\in\\N}(a_i+b_i) X^i", "b3a550b4b5c513d5071d0a70ff319b93": " \\frac {\\dot{W}}{\\dot{m}}=h_3-h_4 ", "b3a6aa279e57d777bb4712ee91ee76e2": "\\gamma = \\chi^r.", "b3a6f2e99be80bcc93dbeca17627920d": "\\psi\\,\\!", "b3a710a752fc5abab9234a6d52c8a8f7": "u(n_\\mathrm{A})^2 \\propto \\left ({\\frac{\\partial{n_\\mathrm{A}}}{\\partial R_\\mathrm{AB}}} \\right)^2 u(R_\\mathrm{AB})^2 = n_\\mathrm{A}^2 \\frac{(R_\\mathrm{A}-R_\\mathrm{B})^2}{(R_\\mathrm{A}-R_\\mathrm{AB})^2(R_\\mathrm{AB}-R_\\mathrm{B})^2} u(R_\\mathrm{AB})^2", "b3a72c4a564fadaa0415339cbf83fbb4": "\\mathbf{v} = \\langle r, \\angle \\theta, h \\rangle", "b3a7609bc347ad3436edf7764dd98f76": "\\mathit{r}", "b3a7623a349d8a2f4d3059917b6b6b2b": "X(f) = H(f) \\cdot X_s(f),\\,", "b3a79691da2d23edc28c85b135b26201": "\\gamma_{GB}", "b3a7ab87953ff7ed9729308c28b6b0e2": "(\\varphi ,p)", "b3a7bf6fd35c4dd60da3bee7b597b018": "F_\\text{thrust} = I_\\text{sp} \\cdot \\dot m \\cdot g_0,", "b3a7dd982652063aaedb9b17a973babd": "[X]_t=\\lim_{\\Vert P\\Vert\\rightarrow 0}\\sum_{k=1}^n(X_{t_k}-X_{t_{k-1}})^2", "b3a803180a6f785fcdf9f211d9dca4fa": "\\sum_{k=1}^\\infty (-1)^k \\frac{\\zeta(k)}{k} = 0", "b3a85cb43f33055ed304e7398d792a89": "\\mathrm{Ra}_{x} = \\frac{g \\beta} {\\nu \\alpha} (T_s - T_\\infin) x^3 ", "b3a85dc95e28843a8080680bdcf83976": "f^{147} = -f^{156} = f^{246} = f^{257} = f^{345} = -f^{367} = \\frac{1}{2} \\,", "b3a9a8d975e8102ee63142cc457774e5": "k = A e^{{-E_a}/{RT}}", "b3a9bd0b98286cc550fb3352e7ade6a4": "pE = \\frac{E_{H}}{0.0591 V}", "b3a9d12a15dffeb61fa9e35a43e46a0e": "\\phi: M_1 \\to M_2", "b3aa3f714afb39e1a9ce8ff67032ca05": "ds^{2} = -\\frac{32G^3M^3}{r}e^{-r/2GM}(d\\tilde{U} d\\tilde{V}) + r^2 d\\Omega^2,", "b3aa69183b685b6ddf4bdd3775378f38": "f^1(\\theta)", "b3aa8435013ec596c1854e0cdd3121aa": " \\sqrt[4]{49 + 20\\sqrt{6}} + \\sqrt[4]{49 - 20\\sqrt{6}} = 2\\sqrt{3},", "b3aabbf4f196efac1ad8d6a66470de8d": "\\eta:1_D \\to G\\circ F", "b3aada04a5c6b23f428a8c3179d6a000": "\\pi = 0", "b3ab0a10b9866e0e0d720e135a12c5df": "\\phi\\otimes X", "b3ab37dc878e7e652c0c2d4f873d5d3b": "V_\\mu ^\\prime =\\left( A_\\tau ^\\nu V_\\nu +B_\\tau \\right) \\left( C_{\\tau \\mu\n}^\\nu V_\\nu +D_{\\tau \\mu }\\right) ^{-1}.", "b3ab4ca469ac063d953202d45decbf58": "{x}={r} \\,\\cos\\theta", "b3ab6f2b5bf20aa1f88d6e6ba7bca74d": " \\varphi(q,t) = 0 ", "b3abc3002a91b757662a1731e617e5d5": "\\,E[Y|X=x]", "b3abf56eced42e42a03ed8d22ad4f0af": "e^{-x^2}\\,", "b3ac2dce3c8dc26697e37e46e847ea36": "\\tilde{\\sigma}_{ij} ", "b3ac48eb1cd6efd3e3977851bc222c1d": "f\\star g = fg + \\sum_{n=1}^{\\infty} \\hbar^{n} C_{n}(f,g)", "b3ac61bc2a7926ddb5786b2723806228": "q_i,p_i", "b3ac96feedba78abe953f4985639064f": "A_{\\alpha}=0", "b3aca807d7daa2e4b90547c456901206": "\n\\| u \\|_{L^{4}} \\leq \n\\begin{cases}\nC \\| u \\|_{L^{2,\\infty}}^{1/2} \\| \\nabla u \\|_{L^{2}}^{1/2}, & n = 2, \\\\\nC \\| u \\|_{L^{2,\\infty}}^{1/4} \\| \\nabla u \\|_{L^{2}}^{3/4}, & n = 3.\n\\end{cases}\n", "b3adabf97cd8b92afe0f9dbf08d7fefa": "x[n] = \\left \\{\\cdots, 0, 0, 0, 1, 0.5, 0.5^2, 0.5^3, \\cdots \\right \\}.", "b3adc544110a800d578aae8e960e0a5c": "W^{1-\\frac{1}{p},p}(\\partial\\Omega)", "b3adde09dc59c8e4a538e3b828f4cb6c": "F : C_{0} \\to \\mathbb{R}", "b3ae4510ff61ea912717bb178e043956": "L_n = F_{n+1} + F_{n-1}=2F_{n+1}-F_n \\,", "b3ae58a45038b5c809b20b7544987ec4": "\\lambda \\in A, x \\in I", "b3ae6d88ceb4b908afe01f7a012dda55": "G(B,C,D) = (B\\wedge{D}) \\vee (C \\wedge \\neg{D})", "b3ae710aaefa771a9fdddae8f6641799": "E[\\vec{X}]_{ab} = \\frac{2}{3} \\, \\theta \\, \\sigma_{ab} - \\sigma_{am} \\, {\\sigma^m}_b -\\omega_{am} \\, {\\omega^m}_b", "b3aeb4acba0a8df23bef03696723a1c1": " \\mu_{H}(A) = \\frac{1}{n} \\, \\# \\left\\{ \\text{eigenvalues of }H\\text{ in }A \\right\\} = N_{1_A, H}, \\quad A \\subset \\mathbb{R}. ", "b3aec1950705f48365cc763e00724fd0": "f(x) = \\begin{cases}\n1/x&x\\not=0\\\\\n0&x=0\n\\end{cases}", "b3aec70c4bb216d19e40e111a9b13b72": "\\displaystyle{\\int_{-\\infty}^\\infty f(x)\\widehat{g}(x)\\, dx= {1\\over \\sqrt{2\\pi}}\\iint f(x)g(\\xi)e^{-ix\\xi} \\,dxd\\xi=\\int_{-\\infty}^\\infty \\widehat{f}(\\xi) g(\\xi)\\, d\\xi.}", "b3af37ea8abe538bf728346b6d8966c3": "(x_1+x_2+\\cdots+x_m+x_{m+1})^n = (x_1+x_2+\\cdots+(x_m+x_{m+1}))^n ", "b3af40e3fb137f641a28f1db8c2c14fe": "\n(u,v)=\\left( {\\partial \\psi \\over \\partial y}, - {\\partial \\psi \\over \\partial x} \\right) = \n\\left(A\\frac{y^2-x^2}{(x^2+y^2)^2},-A\\frac{2xy}{(x^2+y^2)^2}\\right).\n", "b3af46ccd0660f494fe83f656ccbcaed": "\n2 \\cos(\\frac{k \\pi}{n+1}) = h^2 \\lambda_k + 2\n\\,\\!", "b3af847984ab4e22c016e88ccfc1f1e1": "S \\propto \\nu^{\\alpha} T.", "b3afa72af562c27146b17c3625065fa0": "x_{11}", "b3afcb0d18ca24fd0c7a41df30c1b9d6": "\\frac{L(tx)}{L(x)}\\to 1,\\quad x\\to\\infty", "b3afd55bc0c07f6bedbd190e8f66eba3": "\\begin{alignat}{1}\n Q(1) & = 4*E(1) \\\\\n Q(2) & = 3*E(2) \\\\\n Q(3) & = 4*E(3) \\\\\n Q(4) & = 1*E(4) \\\\\n\\end{alignat}", "b3afdc4dc7644ad636e084509b04ddab": "v'", "b3afebc9edba83e666e66d3982cff1ca": "- \\int_V \\left[ \\mathbf{J}_1^* \\cdot \\mathbf{E}_2 + \\mathbf{E}_1^* \\cdot \\mathbf{J}_2 \\right] dV = \\oint_S \\left[ \\mathbf{E}_1^* \\times \\mathbf{H}_2 + \\mathbf{E}_2 \\times \\mathbf{H}_1^* \\right] \\cdot \\mathbf{dA}", "b3b05ba48a598018046500f43ee1b8f0": "\\int\\!\\!\\!\\!\\int_{\\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\;\\;\\;\\bigcirc\\,\\,\\mathbf D\\cdot\\mathrm{d}\\mathbf A", "b3b090ee8ddff83e3902e680b0599473": "b(t)", "b3b0ab307caacddf6fc8ac5c5af0cb21": "F^{*2}", "b3b0ac9390df50a5c92903e634ed77ae": " \\alpha' + 2\\beta + \\gamma' \\geq 2. ", "b3b0aeb783045f4084554f724fa28c75": " E [ w | X, D=1 ] = X\\beta + \\rho\\sigma_u \\lambda(Z\\gamma),\\, ", "b3b0e7a8d5614f56a81a3957ba3f1e19": "p e = \\ell \\, ", "b3b0f6d78ec0b12211e0e8bf96f36375": "b_{2}=1", "b3b117b2245457456c6de148ed3ef1cd": "\\textstyle \\delta^{(n)}(\\xi)", "b3b153f6a289d2b6f5b0f60039ecb154": "\\frac{1}{2}\\int \\left(\\sqrt{f(x)} - \\sqrt{g(x)}\\right)^2 dx = 1 - \\int \\sqrt{f(x) g(x)} \\, dx,", "b3b1eff27d394f97f93d245d09f8c77a": "\\lambda(z)", "b3b1ff55fbc27187371b82217326516f": "s = r + tp^k", "b3b22bf389d8eb8d8113730a6c4cfb1a": " \\zeta^2(s) =\\sum_{n=1}^{\\infty}\\frac{d(n)}{n^s}", "b3b234d2326e0a90386b283e94277f24": " L^1 ", "b3b243e9eee09ca1f444fd281cfabfb4": "\\cos(\\theta-\\alpha)/\\cos(\\theta-\\alpha)", "b3b2ddd1222c2d8cacfeaa23412a0b2b": "\\tau(n)", "b3b2fc69392c221055e54ef8762b7de3": "H_{\\text{D}3}=b_{41}^{8\\text{v}8\\text{v}}[(k_\\text{x}k_\\text{y}^2-k_\\text{x}k_\\text{z}^2)J_\\text{x}+(k_\\text{y}k_\\text{z}^2-k_\\text{y}k_\\text{x}^2)J_\\text{y}+(k_\\text{z}k_\\text{x}^2-k_\\text{z}k_\\text{y}^2)J_\\text{z}]\n", "b3b343da1414f298a86825083bd698c4": "S = \\frac{IP}{I+E}", "b3b3866f96243f1d23c99fa8e585e3b5": "[\\![(x_1, x_2, \\ldots, x_n)]\\!] = ([\\![x_1]\\!], [\\![x_2]\\!], \\ldots, [\\![x_n]\\!])", "b3b395997bd96898f9b65aa69d227ad8": " P\\times_H \\mathfrak g/\\mathfrak h", "b3b3a9cf8678098c9c7d06b0cd0e9391": " \\wedge : \\mathrm{Con}(\\mathcal{A}) \\times \\mathrm{Con}(\\mathcal{A}) \\to \\mathrm{Con}(\\mathcal{A})", "b3b3b9698bd8774e61ffd2d9a007e71d": "T_E = \\{1_{E0}, 1_{E1}, 1_{E2}, \\ldots \\} \\cong E.", "b3b40c23f5616214368f8605b1021dc8": "\\lim_na_n+\\lim_nb_n = \\lim_n(a_n+b_n).", "b3b44cea36b3f5c3690e89f5159e978c": "\\pi_0 A", "b3b4c9349d687b4c81ac9ebcffd6b500": " P( | Y | < k ) \\ge \\frac{ ( k - \\mu_Y )^ 2 }{ ( k - \\mu_Y )^2 + \\sigma_Y^2 } \\quad\\text{ if }\\quad \\sigma_Y^2 \\le \\mu_Y ( k - \\mu_Y ) ", "b3b50ec7666ab0bf9ac804c7c393043f": " c_{2n} = \\sqrt{2+c_n} . \\,\\!", "b3b559952df80f3ab3b39d9d6433423d": "F=\\Re\\{W\\}", "b3b5752f30ad142315669d281f32a3bc": "d\\theta/dt", "b3b57c8795ce3a78acbbab74e88b4790": "\\mathbf{e}_1", "b3b5ac58f1f5844f04cbeb07d1a19ee5": "op \\circ \\overline{op}", "b3b61c6df83363e52df1c5a4c666362a": "\n\\begin{array}{ll}\n\\min & \\max_{i=1,\\ldots,k} \\left[ \\frac{f_i(x)-\\bar z_i}{z^{\\text{nad}}_i-z_i^{\\text{utopia}}}\\right] + \\rho\\sum_{i=1}^k\\frac{f_i(x)}{z_i^{nad}-z_i^{\\text{utopian}}}\\\\\n\\text{subject to }& x\\in S,\n\\end{array}\n", "b3b6276389187ff4c5214d66f87532e9": "{c \\over v}", "b3b630407f134ab421c641ccf5a3d7f5": "\\beta _0", "b3b6887182188a0fd9be3b90a4a1ed20": "\\displaystyle{F_n(x)= \\left(x-{d\\over dx}\\right)^n e^{-x^2/2} = (-1)^ne^{x^2/2} {d^ne^{-x^2}\\over dx^n} =(2^nx^n + \\cdots )e^{-x^2/2}.}", "b3b6be59dd1b37733e8a035e9abc5c15": "m^{(k)}", "b3b7148a89cd39f73598dc8277f48cb1": "c_{ii}>0\\ ", "b3b7562cd5e7f896a951d5d2b027480c": "a(S) = \\sum_{i \\in S} \\sum_{j \\in V} a_{ij}", "b3b77b8b84c6393c0fcce178da5d4491": "\n \\mathbf{F}_1 = \\int_{-t/2}^{t/2} (\\sigma_{11} \\mathbf{e}_1 + \\sigma_{12} \\mathbf{e}_2 + \\sigma_{13} \\mathbf{e}_3)\\, dx_3\n \\quad \\text{and} \\quad\n \\mathbf{F}_2 = \\int_{-t/2}^{t/2} (\\sigma_{12} \\mathbf{e}_1 + \\sigma_{22} \\mathbf{e}_2 + \\sigma_{23} \\mathbf{e}_3)\\, dx_3\n ", "b3b798fac75b7194288d2d09dc47376f": "\\varphi(n)", "b3b8706d79ba378fee61bcfdb14be587": "\\displaystyle{S(\\varphi)(z)=\\int_{\\partial\\Omega} N(z-w)\\varphi(w)\\,|dw|,}", "b3b894fc88c79ab58a5886a8865d35f2": " \\!\\ \\eta(1) = \\ln2 ", "b3b897d8bf421d0606280573a28c60b4": "\n\\frac{c_\\text{transit} }\n{c_\\text{auto} } = R\n", "b3b8b0a6b3ef3ef6ef3be41b15ab5f61": " z_{k+1} = z_k + hf((z_{k+1}+z_k)/2) \\, ", "b3b92ef11679db9309d134c545aabdb4": "x,y,", "b3b93124feacf4e89e3e5f450af915ff": "\\hat{\\Delta}", "b3b93be208fdcf41aaf020d13642ef6d": "\\exp(-i\\omega t)", "b3b95b45b8c6691da097cf68d274001c": "( +, \\ \\times, \\ \\uparrow, \\ \\uparrow\\uparrow, \\ \\uparrow\\uparrow\\uparrow, \\ \\dots)", "b3b98d4deafdfcf56ebb06c88b8347f7": "q=(kt)iW", "b3b99f91f0bf592e7e2640ac52d4f974": "=\\operatorname E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right )^2\\right ]+\\operatorname E\\left [ \\left ( m(\\vartheta)-\\Pi\\right )^2\\right ]+2E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi\\right ) \\left ( m(\\vartheta)-\\Pi\\right )\\right ]", "b3b9a75bd787d05f6c16f5d86cdaf98e": "S^\\cdot M", "b3b9aac83312be933d13aaad49af0028": "\\displaystyle{a(s,t)={|v(s)-v(t)|^2\\over |e^{is/L} -e^{it/L}|^2}}", "b3b9b7ce491af7850cc272b65cf94921": "k^{k^3}", "b3b9e98b51e7d6b38811a376bada397f": " D_L = 10^{\\frac{(m - M)}{5}+1}", "b3b9f109e20a30ff8e7bfb118b768a47": "\\int ( T dS - p dV + \\sum_i \\mu_i dN_i )", "b3ba464c1cdb4483e2d2a06de60e33d8": "f_{ik}(x_i) = f_{jk}(x_j)\\,", "b3ba64f3a3dc3177a64ba3badb2390a7": "k=1\\ldots K=\\infty ", "b3ba73aa66f39d666ca150c823e92f78": "\\Delta N_{ph} = \\frac{2}{h^3} \\ \\Delta V_{ph} = \\frac{8\\pi}{3h^3}p_{f}^3(\\vec{r}) \\ \\Delta V .", "b3bb083b8a5f99531ab092dca3016db3": "(\\ ,\\ ] \\!\\,", "b3bb18fad60a888eac05a33b7c1b8b68": "F: C \\to D", "b3bb497fe9dd38820d99eaa0163854a9": "w(u,x_0,r) = \\sup_{B_r(x_0)} u - \\inf_{B_r(x_0)} u", "b3bb4df14c22be68c643aebcdcb714d3": "\\! \\phi", "b3bb612ab648c23852e4a6344f78bcda": "m_{max} = j", "b3bb9fe98d289e9e4b1d6cbd233c04c6": "P_v\\ ,", "b3bbdf9d76964764f1368688e747254a": "\\scriptstyle f : E \\to \\mathbb{R}^+", "b3bbf13dc13cfe7441748aec337c711e": "\n \\frac{\\partial x^i}{\\partial X^\\alpha}\\frac{\\partial x^i}{\\partial X^\\beta} = C_{\\alpha\\beta}\n", "b3bd18879d1356b28facc6c7d1c3efb7": "v(\\cdot,\\cdot)", "b3bd19a60fba14265dc78caa62f8ebaf": " A_N = \\int D\\mu \\int D[X] \\exp \\left( -\\frac{1}{2\\tau\\alpha} \\int \\partial_z X_\\mu(z,\\overline{z}) \\partial_{\\overline{z}} X^\\mu(z,\\overline{z}) \\, dz^2 + i \\sum_{i=1}^N k_{i \\mu} X^\\mu (z_i,\\overline{z}_i) \\right) ", "b3bd1af06991dd1963ccff8464e09176": "\\frac{12}{11}", "b3bda0b5e86aea92bab5764d678e94e9": "\\gcd\\left(\\sum_{a=1}^{p-1} a^{p-1}, p\\right)=1", "b3bddeff63dbd0886c1d29e4a4a506b4": "\\mathit{d_H}(a \\bar{b}, d^{RC}c^R) \\geq 2 \\lceil \\mathit{d_{min}} /2 \\rceil \\geq \\mathit{d_{min}}", "b3be13c4c99ae88af3c8e3ecaecd5c96": "9 * 2 = 18; 18 - 13 = 5", "b3be31c37afc0ba9363cf8a0658ffcae": " (\\beta,\\beta-2\\rho)c_\\beta = \\sum_{\\gamma+\\delta=\\beta} (\\gamma,\\delta)c_\\gamma c_\\delta \\, ", "b3be6dafce285ec472ab0f3ca2f4c98e": "\\Delta'(x-y;\\mu^2)=\\int\\frac{d^4p}{(2\\pi)^3}e^{-ip\\cdot(x-y)}\\theta(p_0)\\delta(p^2-\\mu^2)", "b3bea076aeac19d6986042b9322e0f94": "\\rho^{[1..{k-1}]}=\\sum_{\\alpha}{(\\lambda^{[k-1]}_{\\alpha})}^2|{\\Phi^{[1..{k-1}]}_{\\alpha}}\\rangle\\langle{\\Phi^{[1..{k-1}]}_{\\alpha}}|=\\sum_{\\alpha}{(\\lambda^{[k-1]}_{\\alpha})^2}|{\\alpha}\\rangle\\langle{\\alpha}|.", "b3beb4cefcfa14e55177c4a51d67ac12": "\\mathcal{L}\\{f(t)\\}.", "b3bec36514bc77e891fd3a3a42c6faf1": "\nF_3(a_1,a_2,b_1,b_2,c;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n} {(c)_{m+n} \\,m! \\,n!} \\,x^m y^n ~,\n", "b3becc94dfee9c0c28916bc7602c5807": "\\Phi(m,0)=\\lfloor m\\rfloor", "b3bedc7a5cca8532c31e0ce57623fc6a": "\\aleph_\\alpha", "b3bef2728b38f398d231942e6db1b268": " \\mathrm{hom}_{\\mathcal{C}}(FY,X) ", "b3bf335cd5b2c4b0cab047c38072d465": " \\alpha^*F", "b3bf48373eeac2443f97206ec5378052": "\\ U(h)", "b3bfb2e674294303f455cfcc5ee71b1d": "\\hat{Y}(X_{0})", "b3bfd590747d61b5e23c4299fdfdb67b": "end\\;date - start\\;date", "b3bff26ff4c629f205fd7b2f0c4193c7": "FOV_C= \\frac{57.3d}{f_T}", "b3c051740480543aeabfc9bbe384859f": "z = W(z)e^{W(z)}", "b3c05b61064af74abc7fdb6aff95608e": "Q^2+\\left ( \\tfrac{J}{M} \\right )^2\\le M^2\\, ", "b3c09d54dbadd407ab5ba9e5b8bf7834": "P_{gap}", "b3c0eca779cd4f71d6b6ddb2444c5ecf": " dF = \\sum_{i=1} ^n \\frac {\\partial F} {\\partial \\rho_i} \\ d\\rho_i \\ ,", "b3c11c5a8c0d72587fb61a56d7f5128c": "\\left(\\frac{a}{n}\\right) = 1", "b3c1575fed74ea99b89769178a559c61": "\\displaystyle x^{2} + c = (x + \\sqrt{c})^{2}.", "b3c178941564e6dfb17be3c11a517753": "\\partial_\\mu", "b3c1956526e3792621a47f16b8c4ee88": "(p,\\epsilon,Z,q,Z)", "b3c1d87d981186d9f40fe847201a0b2b": "C=C^\\ast;\\ C_i = C_i^\\ast;\\ C_{ij} = C_{ij}^\\ast;\\ C_{ijk} = C_{ijk}^\\ast;\\ \\ldots", "b3c1f4d2d7f3e04b44dfd93f9f7cf73a": "B = \\{(x,y) : x > 0, y \\geq 1/x \\}.\\ ", "b3c28f388bb590e49e14f61bcd0de061": "\\left(\\sigma^2+\\tfrac{2\\sigma^2}{\\sqrt{2\\beta}} H_{-1}\\left(\\tfrac{-2+\\alpha}{\\sqrt{2\\beta}}\\right)\\right)-\\mu^2 ", "b3c2b9e9bac532a349664ef9d040dbbd": "f_2 \\in L^2(\\mathbb{R}^d)", "b3c2dca9cbf41b46609f41fafadb1a37": "2\\ln(2|\\mathcal U|)", "b3c2e8a276b659af1638c6244e618e3c": "I^+[S] = I^+[I^+[S]] \\subset J^+[S] = J^+[J^+[S]]", "b3c37035fb7d49bd4560071829ad3be7": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}}(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{A}):\\boldsymbol{T} = \n \\left(\\frac{\\partial \\boldsymbol{A}^{-1}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T}\\right)\\cdot\\boldsymbol{A} + \n \\boldsymbol{A}^{-1}\\cdot\\left(\\frac{\\partial \\boldsymbol{A}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T}\\right)\n = \\boldsymbol{\\mathit{0}}\n", "b3c375d68954e9f7b99ec57775794cab": "P_{n+1}(z) = P_n(z) + z^{2^n} Q_n(z) ; ", "b3c3a5cf6210f2b65a670fe5394bb557": "\\frac{8}{\\tfrac{1}{3}}=8\\times\\tfrac{3}{1}=24", "b3c3a6c2157a94e7d5a8d50718d40eb0": "\n (*) \\qquad P(A) = \\sum_{\\omega\\in A} p(\\omega) \\quad \\text{for all } A \\subseteq \\Omega \\, .\n", "b3c445123d4ea9c2125ad7ba63f0dcc9": "0<\\lambda_1\\le\\lambda_2\\le\\cdots,\\qquad\\lambda_n\\to\\infty.", "b3c449847c5253894761a20f2f69ce17": "T(x,y)=\\lambda \\delta(x-y) - K(x,y)", "b3c457bc544830e45d785f87ddd52d95": "\\mathbf{Z_0} = \\mathbf{R_0} + \\mathbf{j} \\mathbf{X_0} ", "b3c46e3ff69fedaa248b0051d55ab140": "C^n = 1", "b3c49ce0fd8ba156756ddffa9751029a": "\\,(s,x\\$)", "b3c4c451bc90a3086c0ed2c149cad691": "\\delta Q = C dT\\,", "b3c532cb648648558195532b123d7a3b": "r = a \\frac {\\sin n \\theta}{\\sin (n-1) \\theta}\\!", "b3c53bb91864c72f844dd460b092923c": "R^{\\alpha \\beta} - {1 \\over 2}R g^{\\alpha \\beta} + g^{\\alpha \\beta} \\Lambda = \\frac{8 \\pi G}{c^4 \\mu_0} \\left( F^{\\alpha}{}^{\\psi} F_{\\psi}{}^{\\beta} + {1 \\over 4} g^{\\alpha \\beta} F_{\\psi\\tau} F^{\\psi\\tau}\\right).", "b3c5e424cb998bcc16e063922ee52974": " = \\frac{-1}{(n-2)A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} \\left(\\nabla^2_{\\vec{r}}\\frac{1}{|\\vec{r}-\\vec{r}'|^{n-2}}\\right)\\vec{F}(\\vec{r}')d\\tau'", "b3c5e90fb32b517fd0a2b8b5a64e6f01": "\\tfrac{1}{d} A", "b3c64c8ee611632152795bf71f12cd45": "x - x = x + (-x) = +0\\,\\!", "b3c67f3eaa504cf0d55f6b0309833f36": "\\mbox{MAIFI} = \\frac{\\mbox{total number of customer interruptions less than the defined time}}{\\mbox{total number of customers served}}", "b3c687f1651c4169fd7717200f40474a": "(x_2\\lor\\lnot x_4)\\land{}(x_0\\lor \\lnot x_5)\\land (x_1\\lor\\lnot x_5)\\land (x_2\\lor\\lnot x_5)\\land ", "b3c6a13815fc2f549a513a98dd7ee5ed": "\\sum_{k=0}^\\infty p(5k+4)q^k=5\\frac{(q^5)_\\infty^5}{(q)_\\infty^6}", "b3c6c268881f63bc1e65e427f3a91937": "\\mathbf{R} \\smallsetminus (x_0+ 1/y_0)", "b3c71947e2b9c4c7db41d0ebd21c01be": "\\psi = -\\frac{1}{\\varphi}", "b3c757bf50cf13d08e08609c98d669bc": "\\textstyle \\alpha \\mathbf{x} \\cdot \\nabla f(\\alpha \\mathbf{x})= k f(\\alpha \\mathbf{x})", "b3c780e42aa19529e114467e765827f8": " q_{ij}x^i y^j", "b3c8184e7b59c39b5234e3b409bf40e3": "(f,g)", "b3c8fddf3a9677c20bac0377bce6a289": "\\ln{a \\over {a-x}} = \\lambda \\ln{ b \\over {b-y}}", "b3c9417edac4788902dfc0419fb34f31": " t^3+y^3+1=d\\cdot x\\cdot y ", "b3ca0c4688ccd2a1ce4b86fc21c09ef0": "y(x_0) = y_0\\,", "b3ca4d0a4d5895975416dc1ef5489517": "\\mathbf{E}=-\\mathbf{\\nabla}\\varphi", "b3caac68124ca7e6009b016c7662e92d": "\\Psi(k, n, m) = \\frac{1}{\\sqrt{c_0^n}}\\cdot\\Psi\\left[\\frac{k - m c_0^n}{c_0^n}T\\right] = \\frac{1}{\\sqrt{c_0^n}}\\cdot\\Psi\\left[\\left(\\frac{k}{c_0^n} - m\\right)T\\right]", "b3cab44ccb70c7cf1da41e3542770823": "\\psi(x-\\epsilon) = \\psi(x) - \\epsilon {d \\psi \\over dx} ", "b3cab663e30306aa161ab729793cf389": "M(256,256,3)\\approx(256\\uparrow)^{256}257", "b3caceb0430a012a7c7cd227be4d361c": "\\scriptstyle\\boldsymbol{x}|_t\\in M", "b3cae7ad8f2e53185c44657cf85b5303": "\\rho(\\psi(P_B))^a) \\in \\mathbb{F}_q^m", "b3caeb008f2d81c0db9792af597daaa5": "y_i \\in X", "b3caf492db153e0a5cd735d09e35d8da": "\\beta^G_{\\mu\\nu}=R_{\\mu\\nu}+2\\alpha'\\nabla_\\mu\\Phi\\nabla_\\nu\\Phi+O(\\alpha'^2),", "b3cb373d7e2623f9bb28014e79ee03be": "x \\in\\mathbb{R}_j", "b3cb40f42240992366d62d63b96c3c7d": "\nW = T \\cos \\theta ,\\quad F = T \\sin \\theta\n", "b3cbb43b2a200c1d0410ab7819b0e263": "1\\otimes v_\\lambda", "b3cbba34886af60f53946a216c584ef1": "\\vec{x}, \\vec{y}", "b3cbd728c9ed527d32dad4b621b586a4": "E_{\\hat{m}\\hat{n}} = a^2 \\, \\exp(a^2 r^2) \\, \\operatorname{diag}(0,1,1)", "b3cbe00bded6d903ce22576f701df12b": "2T", "b3cc52e84cae595805eab2ef677ca00d": " C( v ) = \\left\\{ x \\in \\mathbb{R}^N: \\sum_{ i \\in N } x_i = v(N); \\quad \\sum_{ i \\in S } x_i \\geq v(S), \\forall~ S \\subseteq N \\right\\}.\\, ", "b3cc771848410cf41e36cfbd5e5ea463": "\\{s_c,\\ldots,s_{c+d-2}\\}", "b3cc86cb8d6e751560751bd13037f029": "K_{p}", "b3cc889d471c2ba6e65be9a70ba2cca7": "\\vec\\theta", "b3cccc6f101e4bc729a16f81c779c4fd": "\\{1,2,4,5,7,8\\}", "b3cce15296d36494c71c00a80e7a0408": "\\alpha<\\beta\\,", "b3ccf884da30a65dbd76743e06acf402": " a \\neq 0", "b3cd06d186206970f2d4c23518df6a06": "\\Phi_E = E 4\\pi r^2 ", "b3cd432801cbb1b2b597002250d1f7e7": "|F,m_F \\rangle ", "b3cd7c12a3ddf8473116d4220b0e91b7": "\\int_0^a x^{n-1}\\,dx = \\tfrac{1}{n} a^n \\qquad n \\geq 1.", "b3cdbebc991eab324f08223c8f2732f4": " x_1*x_1 = x_2*x_2 = \\mathfrak{0} ", "b3ce227d491e220b120d66d061f384d5": " ~\\epsilon_{t-1}^{+} = ~\\epsilon_{t-1} ", "b3ce52be4840b282733271462f6c8a64": "U=D+U_E+U_G.", "b3ce78d7668e13811f6a571e2ecf3a16": "\\hat{f}(\\xi) = \\lim_{R\\to\\infty}\\int_{|x|\\le R} f(x) e^{-2\\pi i x\\cdot\\xi}\\,dx", "b3ce840291438de3cf32ea48176e7c68": "FC = 200", "b3cf0e2b804e3921aab10e96b38eb596": "m_q = |\\mathcal{A}_q|", "b3cf2ae4668acb5b76983f9bd0d7f0e7": "f <^* g", "b3cf2dd5537058291bdbcba55ca950b9": "r = \\lfloor 0.5 + \\mu_{3,2} \\rfloor =1", "b3cf2df098a2e88d08e4ebed8e23e50a": "C_{0}^{*}", "b3cf7cadee13c0309f267d835d399ee9": "\\begin{align}\n\\left(\\frac{\\mathrm{d}s}{\\mathrm{d}t}\\right)^2 &= c^2 - v^2 + \\varepsilon\\gamma_{\\mu\\nu} \\frac{\\mathrm{d}x^{\\mu}}{\\mathrm{d}t} \\frac{\\mathrm{d}x^{\\nu}}{\\mathrm{d}t}\\\\\n&= c^2 \\left(1 - \\beta^2 + \\varepsilon\\gamma_{\\mu\\nu} \\frac{\\mathrm{d}x^{\\mu}}{\\mathrm{d}x^0} \\frac{\\mathrm{d}x^{\\nu}}{\\mathrm{d}x^0}\\right)\n\\end{align}", "b3cffd2c588e15263223473874dbef6a": "TkN \\equiv -T \\pmod{R}", "b3d023901f623e68333dcec327bf9899": "s:A\\times K \\rightarrow A, \\ \\left(a,k\\right)\\mapsto ka", "b3d06c70cba3a1038a6da9b893ef9752": "x_n = p\\ -\\ p_n", "b3d087898af0b50af7869de61030b2b7": "\\textstyle \\textbf{R}^{ n}", "b3d08887814e4d507114bbd9fe05dded": "\nz \\frac{d}{dz} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1 -1, a_2, \\dots, a_p \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) +\n(a_1 - 1) \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n \\geq 1.\n", "b3d0d6e6eae8b9bf995831188ec66554": "\\bigl(G_nf\\bigr)_{f\\in\\mathcal{F}}", "b3d1051c1f716edacf7452cd570fb4b2": "\n\\omega(n,2m,r)=\\frac{1}{2}r\\left(\\sigma(n,m)\\left(r\\sigma(n,m)+1\\right)-(r+1)\\sigma(n,2m)\\right).\n", "b3d1090a278119f24cdd0075f775c846": "s_q(x) = \\begin{cases} 0 & \\text{if } x < -\\nu \\\\ \\frac{1}{c(q)}E_{q^2}^{\\frac{-q^2x^2}{[2]_q}} & \\text{if } -\\nu \\leq x \\leq \\nu \\\\ 0 & \\mbox{if } x >\\nu. \\end{cases} ", "b3d1173a2711680dece9ef31d67a07e6": "\ns's^{-1}t't^{-1}u'u^{-1} = 1 \\Rightarrow s' = s, t' = t, u' = u\n", "b3d13b827d7cfd34fb26b282a880f1ac": "\\|A\\|_{\\ell_\\infty \\to \\ell_\\infty} \\leq M", "b3d1565cfb8b9d6b7f005f9997d0f665": " C_1 \\leqslant \\max_{p\\in[0,1]} \\Big\\{ H_2 \\left(\\eta \\, p \\right)- H_2 \\left(\\frac{1 + \\sqrt{1- 4 \\,\\eta\\,(1-\\eta) \\,p ^2}}{2} \\right) \\Big\\} \\;", "b3d1668cd792b0d8ea3ba2e802c3641c": "\\mathcal{O}_K", "b3d19cb30cfadaa3d55be207cec6a3ca": "R_{\\mathrm{sen}}", "b3d1c41eee036ee05e7d81c18ec40572": " \\left(\\frac{\\mathrm{d} y}{\\mathrm{d}x}\\right)^2 = (1-y^2) (1-k^2 y^2)", "b3d1cd2c61f4cf36c98bb766a324a4b9": "\\scriptstyle t=l/\\kappa", "b3d1cf971c6b1afd63dfb0d5d9675c12": "n^{\\underline k} = \\frac {n!}{(n-k)!}", "b3d1dd0f50eeb9254dbae0c6453e4673": "(t,x)=(0,0)\\in(\\R\\times\\R^n)\\,", "b3d1e494a173d3549e2e41c721e62624": "R(x,y_1,\\dots,y_k)", "b3d221eb1cf3038507f74c8d6ae08184": "m\\cdot \\widehat{Q_{s}}(h)\\,\\!", "b3d233bae58a7640fb560da1196dabab": "|{\\psi_{D}}\\rangle = \\frac{1}{\\sqrt{1-\\epsilon_n}}\\sum\\limits_{{{\\alpha}_n}=1}^{{\\chi}_c}\\lambda^{[n]}_{{\\alpha}_n}|{\\Phi^{[1..n]}_{\\alpha_n}}\\rangle|{ \\Phi^{[n+1..N]}_{\\alpha_n}}\\rangle", "b3d24475b8862471e5f38d085864ac8d": "E[X(t)] = E[X(0)],", "b3d298fa1a4b433d6a6f06e05e7ced14": "\\lim_{x_{i} \\to +\\infty} \\partial f(x)/\\partial x_i =0", "b3d299e85ac0c329843d0d130a45861f": "\\eta(x)", "b3d2bf2fd06cbd82e92540f73175b3b2": " \\frac{ \\left\\Vert A^{-1} e \\right\\Vert / \\left\\Vert A^{-1} b \\right\\Vert }{ \\left\\Vert e \\right\\Vert / \\left\\Vert b \\right\\Vert } .", "b3d2d7d75ea7f8ce18e9bbb9f89d1745": "X_k =\n \\sum_{n=0}^{N-1} x_n \\cos \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right) \\right] \\quad \\quad k = 0, \\dots, N-1.", "b3d2d990c6c9a6be94ecc51066881c23": "\\lambda_i \\geq 0", "b3d2df3d656465520bc82ba9cc16572c": "f_{W}:X\\to[0,1]\\,", "b3d349b69ea80c0865f0125ac5126862": "\n b_2=\\frac{3}{2},\\qquad\n b_4=\\frac{15}{8},\\qquad\n b_6=\\frac{35}{16},\\qquad\n b_8=\\frac{315}{128}.\n", "b3d34ab01c52bf842b4a1d747c527630": "\\rho \\frac{D \\mathbf{v}}{D t} = \\nabla \\cdot \\boldsymbol{\\sigma} + \\mathbf{f}", "b3d3ba6220d174609f538afa38e9ff9c": "E_{1},E_{2}", "b3d3e195e9dbdfc4886917feea60bbbc": "\n\\begin{align}\n\\frac{ds}{(a^2-b^2)\\sin^2\\omega + (b^2-c^2)\\cos^2\\beta}\n&= \\frac\n{\\sqrt{b^2\\sin^2\\beta + c^2\\cos^2\\beta}\\,d\\beta}\n{\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}\n \\sqrt{(b^2-c^2)\\cos^2\\beta - \\gamma}}\\\\\n&= \\frac\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega}\\,d\\omega}\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}\n \\sqrt{(a^2-b^2)\\sin^2\\omega + \\gamma}}.\n\\end{align}\n", "b3d420e0223260ca38aa0dc3147b6a2f": "\\,\\!10^{10^{100}}", "b3d425b19c11909e4d43bda63900981a": "O(N (\\log \\log N)^{-c_k})", "b3d428065a200a5fc31f1fe88b2de8c1": "D([a,b]) = [a,D(b)] + [D(a),b]", "b3d42b7ad778c12efe37fbbbc3a4d611": "\\log (X_j)", "b3d440d3937f2555da8c64003399ad76": "\\ln(\\gamma_i)", "b3d492de2b4af67b460f0f2406972716": " {\\rm det} \\,( I - \\mu(D - \\lambda)^{-1}) = \\prod \\left( 1 - {\\mu \\over \\lambda_n -\\lambda}\\right) = \\prod {1 - (\\lambda+\\mu)/\\lambda_n \\over 1-\\lambda/\\lambda_n} = {\\omega(\\lambda+\\mu)\\over \\omega(\\lambda)} .", "b3d4a2c5b40d7af881f9d19730b6ea7c": "\\frac{n}{n-1}", "b3d4d63b2e58c14a1a1338fade533819": "\\mathbf{X} = (x_1, y_1, \\ldots, x_k, y_k)", "b3d4f52d241a1831b815b5568d7101f5": "E_j", "b3d546c5815bb11c379066c3cba92861": "\\mu f", "b3d566e48447230dedb291970bd5fca5": "i_{\\text{L}}(t) = I_p \\sin(\\omega t)", "b3d56c78cb67fe031c7b0bb05b55c90e": "TWA \\subsetneq DPA", "b3d600e5ebe36e29f4a306cac5bf2004": "\\displaystyle{\\mu(z)={z^2\\over \\overline{z}^2}\\overline{\\mu(\\overline{z}^{-1})},}", "b3d63b5fc2695753842a63f54776f759": "\\tilde{u} (\\vec e_j) = u_j.", "b3d68d9b3d5c81dec3c2cf54fc141872": "f(t)\\,dt", "b3d693bd3fb59325d096762d8a22409c": "\\mathbb{H} \\, \\tilde{x}_r = \\tilde{y}_r", "b3d6b4dcbd87131eccfa1bb09b6088b3": "i\\bar{\\psi}\\gamma_{\\mu}\\partial^{\\mu}\\psi", "b3d6da6d5b550669e9acf54fb349bdad": "p(t_1),", "b3d6e4d3fd6a3965f0bdd166e844c7e1": "\\rho = \\rho(t_0) ", "b3d7060a714b8b612c35c9ac6a06bbe8": "\\hat h(P)=0", "b3d70ac0681e7ecac848fd0be1deb3b4": "i = \\hat e_x \\cdot ( P3 - P1 ) ", "b3d748e3c305bd25f3d31688c4351358": "\\delta_{in} \\ge H_q ^{-1}(1 - r) - \\varepsilon, \\varepsilon", "b3d7fa428a35c4da250553c3942cd746": "G(z_{t}, \\zeta, c)x= (1+exp(-\\zeta(z_{t}-c_{1})(z_{t}-c_{2}))^{-1} \\zeta>0 ", "b3d814d8e2fc1c96b83740ffc166bf2d": "i_L = \\frac {R_{out}} {R_{out}+R_{L}} A_i i_{i} \\ . ", "b3d872aacfbed35f71d9c939b77ca8bf": "\\deg(r) \\geq d ", "b3d8761a6c22d8f5350a7c060582c757": "d V_R = 0.", "b3d87af37d061fafe21b7f9bcce0251d": "\\ S^* = (F^*)^{-1}P^* = (QF)^{-1}QP = F^{-1}Q^{-1}QP = F^{-1}P = S. ", "b3d8866d4d045928d3d12dcba43ee2b0": "K' = K - \\mathbf{1_N} K - K \\mathbf{1_N} + \\mathbf{1_N} K \\mathbf{1_N}", "b3d899b5de1e2843bf8f1e0266445e33": "f = v / \\lambda\\,", "b3d8e7af1795dd90e0e03b4ede9b1480": "x = \\Phi(y)", "b3d8faa0b596f36a712288013522967f": "\\nabla^{2} \\textbf{A} + k^{2}\\textbf{A} = -\\textbf{J}\\,", "b3d96fb571406467917956c8b95ca05e": "-T \\leq i \\leq T", "b3d97ef3dd062da43a9839603bdcb427": " I = p \\vee q", "b3d97fb80e18ca855349f24f2ebb96c7": "Force(N)\\!", "b3d9d07a09519e0efd66b2fa9d89c413": " \\Diamond_2 P", "b3d9f06bb651033d1bbbd25436b30235": "\\hat{\\mathbf{x}}, \\hat{\\mathbf{y}}, \\hat{\\mathbf{z}}", "b3da0277fabe45aa55188316b3589b05": "(1 + x)^m (1 + x)^n", "b3da4d245e10bef3c52b629eea4e49ed": " \\ A/\\Phi \\ ", "b3da502c4e486c75373bf2f5fba2eda5": "s_{k+2} = \\frac{ s_{k+1} + \\left[ k(k+2) \\frac{1}{4R^2} - E \\right] s_k }{(k+2)^2}", "b3da5d2e30a49290903f4a87895d0ce3": " b\\in\\R ", "b3da91fb2a15bdb73358caf791d422ef": "\\mathcal{H}^{12}", "b3dac1a41e24e20818296ea6683033f5": "dz'd\\overline{z}' = \\frac{dz\\,d\\overline{z}}{|cz+d|^4}", "b3dad0c5f43ebbbd73c67c8c6dd29f0c": "f: C \\to D", "b3daf701c9df42b8e201f260e9d64c53": "I = cN - P ", "b3daf88ddf2262fdd877b0b91c933d6a": " \\{ \\langle \\mathbf{e}_\\mu \\bar{\\mathbf{e}}_\\nu \\mathbf{e}_{\\lambda} \\rangle_I \\},", "b3db0f4d3cfb1e151a1c7b128e9aae50": "Z \\to \\frac{R}{R'}\\,Z", "b3db5189f09a6a8011825784e668dffe": "T_C D", "b3db841d1eaa70b4943a81b1c86c3ae8": "<3\\times10^{-9}", "b3db9bf1334ea5084424c0b310528fcf": "\n{\\hat{\\beta}}(q, {r_{c}}) = \\min \\left \\{ \\alpha: \\ {r_{c}} \\le \\max_{u \\in \\mathcal{U}(\\alpha, \\tilde{u})} R(q,u) \\right \\} = \\min_{\\alpha \\ge 0} \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})} \\psi(q,\\alpha,u)\n", "b3db9f971d73876a57de1084b379c534": "1/\\sqrt{1-v^{2}/c^{2}}", "b3dc791f086de956c9585035ce0fa24d": "d =_{def} \\lfloor \\frac{w_\\min - 1}{a_\\max} \\rfloor", "b3dcba7548fb84b24a987160adae3ff9": " (2 - \\lambda) \\bigl[ (3 - \\lambda) (9 - \\lambda) - 16 \\bigr] = -\\lambda^3 + 14\\lambda^2 - 35\\lambda + 22", "b3dcbadb9ff7d6a7d667b3d57d0f4684": "\\scriptstyle i \\;=\\; \\hat x \\hat y \\hat z", "b3dcd2c8dad362baf1a969176570ca33": "\\sqrt{s^2-v^2}=2a\\sinh \\tfrac{h}{2a}.\\,", "b3dd1563020c9e655cb71451282fdb58": "H=mv^2/2=p^2/2m", "b3dd1fdd47e881a7225527d1c38a5f02": "y\\in M", "b3dd91f02d77318b0b5438928c0fae66": "M', M''", "b3ddec5d673a632217318f109276dbe8": " \n\\text{(Eq. 4)} \\qquad 0 \\leq \\mu_{ab}^{(c)}(t) \\qquad \\forall a, b, c, \\forall t \n", "b3de035d5bb03b32a4e850a3ebb62240": "s(t) = a_{I}(t)\\cos{\\left(\\frac{{\\pi}t}{2T}\\right)}\\cos{(2{\\pi}f_{c}t)}-a_{Q}(t)\\sin{\\left(\\frac{{\\pi}t}{2T}\\right)}\\sin{\\left(2{\\pi}f_{c}t\\right)}", "b3de1f7f3433320e8a41ca58d6e83477": "\\frac{u(t)}{v(t)}\\le \\frac{u(a)}{v(a)}=u(a),\\qquad t\\in I,", "b3de2049c76972e1b1e957772f908c1d": "2x+1", "b3de3bd42e6030a8f85ef1f328d7718d": "\\beth_{\\beth_{\\omega}}", "b3de4acf4668c2067091c166b31a1270": "e'", "b3de59619a2a120e26ff15abecbaa5bc": "\\forall a \\in A, L(a) = \\mathit{in}", "b3de6e992503c9658197399597ff3380": "\n\\mathfrak{P}(\\mathfrak{C}_\\operatorname{odd}(\\mathcal{Z}))\n\\left(\n\\mathfrak{P}_3(\\mathfrak{C}_2(\\mathcal{Z})) +\n\\mathfrak{C}_2(\\mathcal{Z})\\mathfrak{C}_4(\\mathcal{Z}) +\n\\mathfrak{C}_6(\\mathcal{Z})\n\\right)", "b3deaeb4b9253415748caf31bc912dca": "T^0_1(V) \\cong L(V;\\mathbb{R}) = V^*", "b3df24f360f2df5a72c996bb1c142c5e": "= 0", "b3df289a8474e68ef635027c58b17d0e": "440 \\rm{ Hz}\\cdot (\\sqrt[12]{2})^{-29} \\approx ", "b3df8a165b8f741f126f0de01dbb82a4": "l^2 = \\{ (a_n)_{n \\geq 1}: \\; a_n \\in \\mathbb{C}, \\; \\sum_n |a_n|^2 < \\infty \\}.", "b3df9ff495b9d1f05374368314a6b9d5": "m_1,s_1,\\ldots,m_t,s_t,\\ldots", "b3dfaeed011f0f4cecb6d1467ce29eed": "L_g", "b3dfcac668be0e614fa8bc21c8a033b8": "\\Sigma_2 \\in \\mathbb{F}^{p \\times r}", "b3e0082d92cf933ec249a2badbf9cb07": "\\tfrac{\\varepsilon}{2}", "b3e0484d1eb4c3cee531476393094ada": "r = \\frac{[Fe(CN)_6]^{2-}}{k_\\alpha + k_\\beta[Fe(CN)_6]^{2-}}", "b3e08326ab3cca71236c8a6a9db851f4": "\n\\overline{z} = \\overline{R}e^{i\\overline{\\theta}}\n", "b3e087c2c260c9ea1a5b65ddcce7087f": "X \\subseteq \\mathbb{R}^{n}", "b3e095751b1bdbac8454886c03423526": "\\lim_{x \\to n^\\pm} \\tan \\left(\\pi x + \\frac{\\pi}{2}\\right) = \\mp\\infty \\qquad \\text{for any integer } n", "b3e0a83c151255895da5edfa032be385": "\\mathbf{F}_\\mathrm{rad} = -\\frac{\\mu_0 q^2 R}{24 \\pi c^3} \\frac {\\mathrm{d}^3 \\vec a} {\\mathrm{d}t^3} ", "b3e1373b2d3d154ceda7cbea3a45c45d": "\n\\mathbf{P}(\\tau, \\mu | \\mathbf{X}) = \\text{NormalGamma}\\left(\\frac{\\lambda_0 \\mu_0 + n \\bar{x}}{\\lambda_0 + n}, \\lambda_0 + n, \\alpha_0+\\frac{n}{2}, \\beta_0+ \\frac{1}{2}\\left(n s + \\frac{\\lambda_0 n (\\bar{x} - \\mu_0 )^2}{\\lambda_0 +n} \\right) \\right)\n", "b3e27b649f6fd56cc00529bf18234126": "\n\\mathbf{\\dot{q}}=\\mathbf{Kq}+\\mathbf{u}", "b3e2a4e5448529fb9bfea798079a15b1": "\\bar{S}", "b3e355a9d18170d4adfca6dd8a002439": "r^2 \\equiv nt \\pmod p", "b3e36eac359f563e62eedfe601862ead": "\\frac{\\pi}{4} = 2 \\arctan\\frac{1}{2} - \\arctan\\frac{1}{7}\\!", "b3e37f6567760c6c25d6600d50e42e27": " \\sigma(x) = x^p \\mod pR ", "b3e3b3b47782da95ee4c1f1935943e01": "\\displaystyle{G(r,\\theta)=(x(\\theta)-(1-r)y^\\prime(\\theta),y(\\theta) +(1-r)x^\\prime(\\theta)).}", "b3e3d5ae345a151e480bcbd40bef4e78": "\\ \\{f,\\{g,h\\}\\} + \\{g,\\{h,f\\}\\} + \\{h,\\{f,g\\}\\} = 0", "b3e5505e14a51ab95afec169410a508e": "\\displaystyle r_a^2+r_b^2+r_c^2+r^2=a^2+b^2+c^2", "b3e588a61bf40babf508717913600811": "f_r=\\sqrt{\\frac{1-\\dfrac{2GM}{(R+h)c^2}}{1-\\dfrac{2GM}{Rc^2}}}f_e.", "b3e5cc44b7782482691bd73a32bbd7e4": "\\{ 1, i, j, k\\}", "b3e5f3e687b926acb5b7c06955c7f27c": "B = X - R - N \\cdot x \\,", "b3e611b5b275e9e229aed24f35c8cf27": "TSV(X,t) = \\left(\\mathbb{E}[(X - t)^2 1_{\\{X \\leq t\\}}]\\right)^{\\frac{1}{2}}", "b3e6c07f851a4dcb68098b129a6ef136": "x \\not\\in B^c", "b3e6cf702fd2bab4b523954265379e27": "\\ \\sum_i(X_i,f_i,\\alpha_i)=0", "b3e73e4cd676213bf7f551a3fdfc812c": "a_n=\\sum_{k=0}^n (-1)^k {n\\choose k} s_k.", "b3e7543ca2b01ccf5435d8cf576c119b": "\\Delta E=2\\beta (a_{r}+a_{s})=\\beta N_i", "b3e77fa0668e74b333523b097a06e4a6": "\\left.|0,0\\rangle=(\\uparrow\\downarrow - \\downarrow\\uparrow)/\\sqrt2\\;\\right\\}\\quad s=0\\quad\\mathrm{(singlet)}", "b3e79fb25e5f5c241e5f1b6a9c15ed7d": "\\omega\\sqrt{\\frac{t'}{\\xi}}=e^{ik}, \\qquad \\frac{d\\omega}{\\omega}=idk, \\qquad \\frac{d\\omega}{\\omega^2}=i\\sqrt{\\frac{t'}{\\xi}}e^{-ik} dk", "b3e7d3a218a8ac2437e1dda9d2250500": "\\phi(\\mathbf{x})", "b3e8942b91a72fdf31f675ab0c084809": " \\lambda_1\\ge 0,\\,\\lambda_2 \\ge 0, \\,\\lambda_1+\\lambda_2 \\le 1. \\, ", "b3e8fdefec52e8dc5953ab8bcf7a8dce": "\\langle\\psi'_r|P_0|\\psi'_r\\rangle = 0", "b3e91d1053230755791f73a5579f3558": "\\theta\\ = G \\cdot\\ \\left( \\frac{V_1 - V_2}{V_1 + V_2} \\right)", "b3e946ea9bd13f0cf59f31f6ccdb4ad1": "\\frac{V_B}{V_A} = \\frac{p_A}{p_B} ", "b3e99b5990d9665f9416ab1572cef77d": "\\hat n_i \\hat n_i^\\top", "b3e9cb33118de10acd6f482df469b8a5": "\\left(\\sqrt{\\frac{2}{5}},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{5}{\\sqrt{3}},\\ \\pm1\\right)", "b3e9cc6953a0977af3730a295649d553": "f_1,f_2 : X \\to Y\\,", "b3e9e808c9a4b5ce3441755f5dad2d19": "S^2 \\times S^1", "b3e9fcb5c7b51ffc29b7cbc958ebddbd": "x^\\alpha (\\tau)", "b3ea218f49843e02fc62e35247296172": "\\phi S = \\phi, \\, ", "b3ea31f017035b77df54877e0f3ae677": "G \\propto 1/t\\,", "b3ea46eb5b8877f092065c7f125c20a4": "1-A", "b3ea57d5ade457d6ec38d82e9b4e42f5": " q = \\frac12 \\times \\rho V ^ 2 ", "b3ea6480b9eaee4a45d3d6508b028a46": " E = m c^2 ", "b3ea78edea872df8cec402d0eafaa228": "\\Gamma(r) := \\left\\{\\begin{bmatrix} a&b\\\\c&d \\end{bmatrix} \\in \\Gamma : a\\equiv d\\equiv \\pm 1,~b\\equiv c\\equiv 0\\mod r\\right\\}.", "b3eaa369f7d928d04f799d81568ce476": "\\frac{\\partial \\mathbf{A}\\mathbf{x}}{\\partial \\mathbf{x}} =", "b3eaad9d1d606f1c71d31571409d066e": "QP_n^{[r+1]} = (ax+b)P_n^{[r]} + cP_{n-1}^{[r]}", "b3eaadd2e7a988a3992f3cb515afbea1": " \\mathrm{ran} f_x (x_0,\\lambda_0)=Y_1 ", "b3eaf2de90500de110cd1cd74e0048e2": " | a+bi |^2 | c+di |^2 = | (ac-bd)+i(ad+bc) |^2,\\,", "b3eb288736cda51f5c8160bf6a2e978f": " \\int_\\Omega uD^\\alpha\\varphi\\;dx=(-1)^{|\\alpha|}\\int_\\Omega \\varphi D^\\alpha u\\;dx", "b3eb33725287a446eeec70446c2848a8": "Z_t=\\mathcal{E} (X)_t,\\,", "b3eb3c93b876cbcc73e2a6d7cf35117c": "\\lambda_{CW}(M^\\prime)=\\lambda_{CW}(M)+\\frac{\\langle m,\\mu\\rangle}{\\langle m,\\nu\\rangle\\langle \\mu,\\nu\\rangle}\\Delta_{W}^{\\prime\\prime}(M-K)(1)+\\tau_{W}(m,\\mu;\\nu)", "b3ebb415a32c1f95046c1f63b8d21303": "p(t)=Pu(t)", "b3ebf6672bfc41423f04e0b1e34b4fed": "\\frac{d^2\\beta}{dt^2}+(\\frac{4k}{MV}+\\frac{2k(a^2+b^2)}{VI})\\frac{d\\beta}{dt}+(\\frac{4k^2(a+b)^2}{MV^2I}+\\frac{2k(b-a)}{I})\\beta=-\\frac{2k}{MV}\\frac{d\\eta}{dt}+(\\frac{2ka}{I}-\\frac{4k^2 b(a+b)}{IMV^2})\\eta", "b3ec317e5d643805cce5e982fd4484a6": " \\mathrm{De} = \\frac{t_\\mathrm{c}}{t_\\mathrm{p}}", "b3ec63fd10a18f7cc72241fcd7fde999": "= \\mathcal{L} \\{ h_{\\mathrm{FOH}}(t) \\} \\ ", "b3ec8ea76a383ee408da10d0c19d0af9": "h(x)=g(x)e^{-x^2/4}.\\,\\!", "b3ece0b94018b9ae13c986d80f911021": "f\\colon I \\rightarrow J (\\subset \\mathbf R), g\\colon J \\rightarrow \\mathbf R,", "b3ed0f6c963cdc28311c17a8179e783b": "\\theta_A = \\frac{[A_{ad}]}{[S_0]} ", "b3ed271416ff6059e88909b137014b6f": "V = x^5 + ax^3 + bx^2 + cx,", "b3ed3873a6f87baa50e86877d4e1192b": "\\mathrm{HA1} = \\mathrm{MD5}\\Big(\\mathrm{A1}\\Big) = \\mathrm{MD5}\\Big( \\mathrm{username} : \\mathrm{realm} : \\mathrm{password} \\Big)", "b3ed4251b9f719e41d71545df31094ec": " \\delta_{\\mu\\nu} = \\begin{cases}0 & \\mu \\ne \\nu \\\\ 1 & \\mu = \\nu \\ \\end{cases} ", "b3ed6bb9c1b2052f1127a36eac2e3581": " \\widehat{A} ", "b3eda10ef04d977700292f2a3a40fdf1": "\n\\begin{align}\nc^2 &= m^2 + d^2 - 2dm\\cos\\theta \\\\\nb^2 &= n^2 + d^2 - 2dn\\cos\\theta' \\\\\n&= n^2 + d^2 + 2dn\\cos\\theta.\\, \\end{align}\n", "b3edff0458c3b467c5488de4cfcd77b3": "\\sigma_a=0.02[\\text{barn}]\\left (\\frac{k_0}{k}\\right )^3Z^4", "b3ee05f439034348c077877162fd587f": "\\mathrm{N}S := \\coprod_{p \\in S} \\mathrm{N}_p S", "b3ee14dca969ebea8e02e6e2d146abba": "P-\\epsilon", "b3eea734fb1e98dcebe3d0e0f1929fc5": "\\sigma_x^2 = \\int_{-\\infty}^\\infty x^2 \\cdot |\\psi(x)|^2 \\, dx - \\left( \\int_{-\\infty}^\\infty x \\cdot |\\psi(x)|^2 \\, dx \\right)^2", "b3eec1f49c488ba828f6b82a4ab1e5f8": "f\\circ p\\neq g\\circ p", "b3eec6f2a3d5a2ef3b8862d122eafe43": "\n\\sum_{a \\in G} f(a^{-1}) g(a) = \\frac{1}{|G|} \\sum_i d_{\\varrho_i} \\text{Tr}\\left(\\widehat{f}(\\varrho_i)\\widehat{g}(\\varrho_i)\\right),\n", "b3ef705cbe065533e4adcea916eeb73f": "\\left[\\begin{array}{c} L \\\\ M \\\\ S \\end{array}\\right]=\\left[\\begin{array}{ccc}1/L'_w & 0 & 0 \\\\ 0 & 1/M'_w & 0 \\\\ 0 & 0 & 1/S'_w\\end{array}\\right]\\left[\\begin{array}{c}L' \\\\ M' \\\\ S' \\end{array}\\right]", "b3ef72ebcf42e08a76728163571ed3a0": "\\scriptstyle \\dot m_{01} \\,0\\, p_{01} \\,", "b3efa631a484f7a7d7168fceea9694b7": "\n\\langle v_e \\rangle = \n\\sqrt{\\frac{k_BT_e}{2\\pi m_e}}\\,\ne^{-e\\Phi_{sh}/k_BT_e}\n", "b3f086ae433fbc1e03a09581679a42cd": "C_{0} = C_{\\infty} = 8", "b3f09b8c1b54b8905b29265e8dfd1b64": "\\mathbf n_a", "b3f0e2dd99ec1ff379ef84cc729a4f3a": "\\{ V_i\\vert i\\in I\\}", "b3f12f512212d208f000c458ccb22fb9": "\\int_0^\\infty x^m \\exp(ix^n)\\mathrm{d}x=\\frac{1}{n}\\Gamma(\\frac{m+1}{n})e^{i\\pi(m+1)/(2n)}", "b3f1519622f3a6a580491acc126f5513": "I \\subseteq \\{1,2,\\ldots,n\\}", "b3f16aba6fce262c871f55dc737d6a50": "y=a^2-b^2,", "b3f1879fce37d7507481070069a5ca54": "\\epsilon = |v-v_\\text{approx}|\\ ,", "b3f1dad522fa79bb97bb554e5a13d3e2": "\\mathrm{Mo} = \\frac{g \\mu_c^4 \\, \\Delta \\rho}{\\rho_c^2 \\sigma^3}, ", "b3f1f363e4dfcfd077dc09a26d612348": " R(n) = \\begin{cases} \\{2n\\} & \\text{if } n\\equiv 0,1 \\\\ \\{2n,(2n-1)/3\\} & \\text{if } n\\equiv 2 \\end{cases} \\pmod{3}. ", "b3f25f118b37a6ce41742c923a8f2e2b": "S^{m-1}", "b3f28893006d66a92b714614024e3d19": "T_a", "b3f2a8d36c0d4d41fd75c544a9271b4d": " D(\\alpha\\wedge\\beta) = (D\\alpha)\\wedge\\beta + (-1)^{|\\alpha|}\\alpha\\wedge(D\\beta),", "b3f2d298e7d24c841720591c6fa3cf28": "T_{em} = \\frac{7.04P_{gap}}{n_s}", "b3f2db4285db1ed8f1b9744ace51eb12": "a_n a_{n-k} = a_{n-1} a_{n-k+1} + a_{n-2} a_{n-k+2} + \\cdots + (a_{n-k/2})^2", "b3f2f0a849db86404aa17f0cb4aacdf7": "\\tan \\frac{\\varphi}{2} = -\\frac{X}{R_0}", "b3f30005bf4e53b7f839a623f00cffe6": "A\\in", "b3f3094020a04ffe2b70e6e2c8147c3d": "(p - p')(y-y')", "b3f31370b22d855f4c67f86584fea39a": "\\frac{T''(t)}{c^2T(t)} = \\frac{1}{R(r)}\\left(R''(r) + \\frac{1}{r}R'(r)\\right).", "b3f3261d85a720477d8c616c16b7faa4": "Q_r \\neq K_{eq}~", "b3f33e0c1fb4320f52d6264bee4ff616": "Q(\\alpha,\\beta) = \\sum_{i=1}^n\\hat{\\varepsilon}_i^{\\,2} \n= \\sum_{i=1}^n (y_i - \\alpha - \\beta x_i)^2\\ ", "b3f3eef36077645c8768db713e6f6621": "M_{2} \\equiv q a^{2}", "b3f434111c8385e713a66b411a3bf716": " W_f( | x_1, x_2, \\cdots,x_n \\rangle) = |f(x_1, x_2, \\cdots, x_n) \\rangle. ", "b3f438536e151a0e49ecd98f3d637aa1": "V\\textbf{y}", "b3f46ddfd0947bc49cc0770ccefe1eac": "\\phi \\,(t)", "b3f4a250358ba04c85283bc2ee67db35": "\\scriptstyle dt", "b3f5000b218c300cfc1dcfeb003216fb": "\\tau\\ =\\ \\frac{16h}{\\pi} \\frac{\\sigma}{\\Delta G_v^2a^4} \\exp\\left( \\frac{\\Delta G}{k_BT} \\right)", "b3f556daf943d4a86d013d77df6d3320": "(n,m)", "b3f56f95c4276796be87ca1762226c2b": "e^{-z/2}>0", "b3f57ab1224569072d83cd7460f1fa25": " y = Zr/a_0\\,", "b3f57bca7e6d2f5d9e2550654850e7e7": "\\ Ra = [1, 0, ... , 0]^{\\mathrm{T}}", "b3f5ac845d19b6fe999149e9fdce352f": "\\tilde{\\tau}", "b3f5ba84c9c5785b2dac2ddc4a4137c2": "\\psi_\\alpha |n \\rangle", "b3f610065895a852fcdd86dbab7c3521": " g_\\varepsilon ", "b3f69243d14facc10d0a06bb37b129da": "S_1 = 1.", "b3f6b31becfecb50f94df899ca0749c9": "v( \\sigma ) = M \\operatorname{sgn}(\\sigma)", "b3f6c8f1701b360e7a97ee82b481dab8": " a_{20}-\\mathcal{L} a_{30} = p_3 a_{30} +p_7,", "b3f70e7c005368b43884b220f476119e": "[xG_i, yG_j] = (x,y)G_{i+j}\\ ", "b3f80543047d31043c2c1948b7b53c5c": " t = \\frac{ \\arccos \\Big( \\sqrt{ \\frac{x}{r} }\\Big) + \\sqrt{ \\frac{x}{r} \\ ( 1 - \\frac{x}{r} ) } }{ \\sqrt{ 2 \\mu } } \\, r^{3/2}", "b3f8385244601e46e1a38a02f9350aa9": "\\varphi(z) = \\lim_{k\\to\\infty} \\frac{\\log|z_k|}{2^k}. ", "b3f8a8f23a887c015e701792bcde4bf5": "\n\\operatorname{E}(X_1 | X_2 < z) = \\rho (X_2 | X_2 < z)", "b3f8bca6f98a1c1edd6da6d810c3ec83": " \\lim_{t \\rightarrow \\infty} e^{-t}\\mathcal B A(zt) = 0, ", "b3f936e8530b351c9ee59e6e84bc900a": "\\{2\\pi+1\\}", "b3f94b2e3500ec96f38ac36c2bcb7950": "\\{|t_i\\rangle\\}", "b3f96f7daf1c948be68bcd4244390e26": "\n\\begin{align}\n\\mbox{E}(x) & = (m-1)(x+1)\\mod{m} \\\\\n & = -(x+1)\\mod{m} \\\\\n\\end{align}\n", "b3f9fb8869bb62e9af47677a184680f8": "a_3= \\lfloor 11^\\frac{3}{2} \\rfloor = \\lfloor 36.482\\dots \\rfloor = 36, ", "b3fa454c7d3fd661b31a8635f91db006": "\\sqrt{V}", "b3fa5b8ab3c9689b357f56b321043b35": "d1=sin(trajpar \\cdot 8\\pi)", "b3fa6a433c371d3b87f98c6fff134f61": "=\\frac{a_2\\cos (\\theta-\\alpha)-a_1\\cos (\\theta+\\alpha)}{\\cos (\\theta+\\alpha)\\cos (\\theta-\\alpha)}", "b3fa7427906619e46d0de73fbed2a496": "\\operatorname{VaR}_\\alpha(L)=\\inf\\{l \\in \\mathbb{R}:P(L>l)\\le 1-\\alpha\\}=\\inf\\{l\\in \\mathbb{R}:F_L(l) \\ge \\alpha\\}.", "b3fa8a34afff7de21f38da8a87e15ab4": " \\mathbf{F}_A = -(\\mathbf{i}_w D + \\mathbf{j}_w Q + \\mathbf{k}_w L) ", "b3fab08523b4448fa57ba593811b3fb8": "1 - \\omega(x, D)(E) = \\omega(x, D)(\\partial D \\setminus E);", "b3fac9c03c45ab1eca6a4bfabacad7d4": "\\int_0^x 1\\,dt = x.", "b3fafeb29497bce970b35ca06ea053d1": "\\textstyle \\mathbb{P} (A|B) = \\mathbb{P} ( A \\cap B ) / \\mathbb{P} (B) ", "b3fb7841f5d61c2df34d98d160d18b42": "g_i = \\left ( f_i\\left ( x \\right ) \\right )^{e_i}-C_i \\bmod N_i", "b3fb8e1ceafb87e6c147431dbf5029f7": " \\delta \\theta = \\epsilon ~,", "b3fba11893a9432b3dede6d9150d98cf": "\\forall x. \\phi(x, D_n(x, z), z)", "b3fbaabad879b60c6866c27b1f815e0f": "Pr_{U_{m}}[A] > 1 - \\epsilon", "b3fbd5d7018ef18bf725509b01bb8542": "E(g(T)) = 0", "b3fc9c062f9c7235f14a2cc7b9c89a99": "{\\chi}^2", "b3fca307b5d7ebb95f6ed0d0f8f9d5cf": "(y',x')\\sim(y,x)", "b3fd0efc78eba0926cc12ebd2310c35a": "M=P/G", "b3fd10264ec402fdcd9629a74170bf7f": "\\displaystyle{Q(1-x)Q(C(x)+C(y))Q(1-y)=-4B(x,y).}", "b3fd7249f267cb83ed71313ac4945671": "u(t_0),u(t_1),...,u(t_n)", "b3fdf1683f1320aebf08d2c374f93a2b": "\\alpha = 0.5", "b3fe33dc3abf4031bfb5d34a00226afb": " S \\to LS (80%) |L (20%)", "b3fe3a4ab83e11c8e51545a00b03d9ae": "\\scriptstyle \\dot e \\;=\\; 0", "b3fece7fa595a0680eb82282500ac9da": "\\scriptstyle\\Pi_n^2([-1,1]^2)", "b3ff2432f060d962bf9bcb7d828940e9": " B_{ij} ", "b3ff3893997c4b69a49c0684c5111680": " StDev_2 = 1 - \\sqrt{ \\frac{ \\sum^{ K - 1 }_{ i = 1 } \\sum^K_{j = i + 1 } ( f_i - f_j ) }{ N^2 ( K - 1 )} }", "b3ff3d84fd8eed4dcac51793d52d1131": "\\begin{bmatrix}\n3/2 & 0 \\\\\n0 & 2/3 \\end{bmatrix}", "b3ff94e844afdaf524beda403eb9cb89": "F=\\Pi_1(Z_1)\\ast\\Pi_1(Z_2)", "b3ffcabe77329fb06d378be57e1f2995": "= \\sgn( \\sec ( \\frac{4 \\theta - \\pi}{2})) \\frac{\\sqrt{\\sec^2 \\theta - 1}}{\\sec \\theta} ", "b3ffe9324c448a373ecbaa1bee59bffd": "k\\ge n\\!", "b3fff766c44be9d65de796897350eebd": "\\forall x\\forall y( (f(x)=f(y)) \\to (x=y) )", "b400411c2d0067c87cea807a33126912": "Z_1,\\ldots,Z_n", "b40084e401024a3c947c14562c72334d": "x = f_0 \\log_g(-1) + f_1 \\log_g2 + \\cdots + f_r \\log_g p_r - s.", "b40135b43459f64749fea496651027d9": "\\,p_x = 1-q_x", "b401865541e90b020522ff543215b80f": "R_{i+1}", "b401a41e2af3fbe3336a41e75f1b3f9c": "F''(x) \\approx \\dfrac{G'(x)-F'(x)}{h}.", "b401c39904773d8aeabe5f92e212e484": " \\sum_{n=1}^\\infty\\frac{n^2-\\frac12}{\\left(n^2+\\frac12\\right)^2}\\left[\\psi_1\\left(n-\\frac{i}{\\sqrt{2}}\\right)+\\psi_1\\left(n+\\frac{i}{\\sqrt{2}}\\right)\\right]=\n-1+\\frac{\\sqrt{2}}{4}\\pi\\coth\\left(\\frac{\\pi}{\\sqrt{2}}\\right)-\\frac{3\\pi^2}{4\\sinh^2\\left(\\frac{\\pi}{\\sqrt{2}}\\right)}+\\frac{\\pi^4}{12\\sinh^4\\left(\\frac{\\pi}{\\sqrt{2}}\\right)}\\left(5+\\cosh\\left(\\pi\\sqrt{2}\\right)\\right).\n", "b401fd9b7bbd61fe859300e6f05de5f4": "A, B, C", "b40252feb5f6825834f87169d1e2bab8": "f_k:\\{0,1\\}^{{n \\choose 2}}\\to\\{0,1\\}", "b402679e957a6b7b1eb981bb1146cf19": "H_n(A_\\bullet) \\to H_n(B_\\bullet)\\ (\\text{respectively, } H^n(A^\\bullet) \\to H^n(B^\\bullet))\\ ", "b40295ddc3f9266edb4689c3ca2016de": " Q = T_3 + \\frac{1}{2} Y_W \\, ,", "b4034f6b24a696e4147cb887d5f2a0f3": "b=c>a", "b40366e0a38593f9044ae2565dccf117": "\\int x^{2k+2} \\phi(x) \\, dx = -\\phi(x)\\sum_{j=0}^k\\frac{(2k+1)!!}{(2j+1)!!}x^{2j+1} + (2k+1)!!\\,\\Phi(x) + C", "b40387f817b754da00e5460c3e8e58e4": " (x^{q^{2}}, y^{q^{2}}) + \\bar{q}(x, y) \\equiv \\bar{t}(x^{q}, y^{q}),", "b4038ee0c7e0361c3881d75debc1a86d": "\\lambda=6341\\AA", "b40394a7b465104c70bc9767b77848fa": "\\mathfrak{R}_x", "b403bb2c0078f0598ee4e7cc80101596": "n=1,2,\\ldots", "b404034f043d49294a8df73b46c8a1c9": "\\mathbf{\\tau}_{ext}", "b40408b786089b80afdb26bffe1b232f": "L^2 \\equiv L_x^2 + L_y^2 + L_z^2", "b4043fda6efe16881dc37743934c8bc9": "x_{n+1} = e^{\\lambda (x_n - 1)},\\, ", "b4046041fdcc8806e31609db02ec0e4e": "= 2\\,y\\, \\left( \\frac{1}{1} + y^{2} \\, \\left( \\frac{1}{3} + y^{2} \\, \\left( \\frac{1}{5} + y^{2} \\, \\left( \\frac{1}{7} + y^{2} \\, \\left( \\frac{1}{9} + \\cdots \\right) \\right) \\right)\\right) \\right) ", "b404e5ba06e2f1ce5044b7639cc12f4e": "E_{h}", "b4054fd679b34e3ab8bdd7ec6d863ed6": "\\text{Step 4}", "b406060768f5b2bd91780808b3aa76c9": " \\begin{align}\n\\tilde{y}_{i+1} &= y_i + h f(t_i,\\tilde{y}_i), \\\\\ny_{i+1} &= y_i + \\tfrac12 h \\bigl( f(t_i, \\tilde{y}_i) + f(t_{i+1},\\tilde{y}_{i+1}) \\bigr). \n\\end{align} ", "b4063324cff37d1a18ab823a7aa09ba2": "\\|x\\| \\leq 1+\\delta", "b40658c3a27f45bdcbba002e7761fd8b": "0.4383 + 0.3606i", "b40669fa7371edf874815830e563b2ce": "j\\,\\!", "b406795f956dcd69658da2c820a42dc6": "\\langle x,y \\mid xyx=yxy \\rangle. \\, ", "b40679a7e11d3c22d167968c7af91a66": "\\beth_{\\alpha} \\!", "b406d9d0ff1f74e5d98729a2d9814892": "H_{c1}", "b406e104006db334f6e330cf8ccccb2f": "\\scriptstyle\\| \\; \\|_{L^1}", "b406eb8d3ec54db681f8428f5cb5914a": "-h_i(x) \\leq 0", "b40718bd8dbf6917cfe2524c67d8fd23": "\\mathbf{CP}^2\\#\\overline{\\mathbf{CP}^2}", "b4075fc02c134cd1743ea1798bb61d9a": "U_n(P,Q)\\, ", "b4076f291516fa6c4077be7ad7941cd3": " +\\left(\\frac{\\partial S}{\\partial V}\\right)_T = \\left(\\frac{\\partial P}{\\partial T}\\right)_V = - \\frac{\\partial^2 F }{\\partial T \\partial V} ", "b4082bb8cccfc975d1b8c0592749b384": "\\langle P_2\\rangle=\\eta\\mathbf{H}^2", "b408349fc963c30f9d03b5325d18da86": " {\\vec x}_{(nr)} = - \\frac{{\\vec K}_G}{M}, M = \\sum_{i=1}^N m_i ", "b408773b3b4bda819d3ac8244a7a6131": "EIS=-\\frac{\\partial(\\dot{c}_{t}/c_t)}{\\partial(\\dot{u}'(c_t)/u'(c_t))}=-\\frac{\\partial(\\dot{c}_{t}/c_t)}{\\partial(u''(c_t)\\dot{c}_{t}/u'(c_t))}=\\frac{\\partial(\\dot{c}_{t}/c_t)}{\\partial(RRA\\cdot(\\dot{c}_{t}/c_t))}=\\frac{1}{RRA}=-\\frac{u'(c_t)}{u''(c_t)\\cdot c_t}", "b409b63d32faf9ae549e1eb08ce8b46f": "t_8", "b409c5d7a79f05a84571dd51a6162666": "\\langle v_i \\rangle = 1.389\\times10^6 \\sqrt{\\frac{Z_i}{M_A}}\\int\\limits_0^\\infty i^\\prime _i (U_g)dU_g \\left ( \\int\\limits_0^\\infty \\frac{i^\\prime _i}{\\sqrt{U_g}}dU_g \\right )^{-1}", "b409f29e0e4a117e30f2d87cf65fb1df": "H(f) = 1 + j \\frac{w}{B}", "b40a5b82ed1b9686eced5a9fd13dbc95": "ab\\cdot cd", "b40a98411ed2e9d127f5775375675cbb": " \\langle A,B,D \\rangle ", "b40ad2027edfb9236ed48fd9f9a90627": "f(x_1, ..., x_n) = ...", "b40ad8aabd5f6d3178634c5efa6fef69": "\\scriptstyle\\chi", "b40ae2a7b156e7cef81f590f0ee155d7": "R_g^*\\theta = g^{-1}\\theta", "b40b084ea81480577d4f787e49fd501c": "\\,y_t\\,", "b40b0a1c46e8a526cb97f195df486821": "h_{ii} \\,", "b40b250574793392e383376257e7f594": "\n\\begin{bmatrix}\n \\frac{\\alpha - E}{\\beta} & 1 \\\\\n 1 & \\frac{\\alpha - E}{\\beta} \\\\\n \\end{bmatrix} \\times\n \n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n \\end{bmatrix}= 0\n", "b40b35046686a79469b1d0345bcb8d51": " A=dx^i\\otimes(\\partial_i + A^j_i(x^k)\\widetilde\\partial_j)", "b40b8917c7c49d16568cbf928b2300a5": "\\Delta{V} = -\\int_{t_0}^{t_1} {V_{exh}\\ \\frac{\\dot{m}}{m}}\\, dt", "b40ba20edf1e293b6b099253c085917b": " \\varepsilon = \\varepsilon_0 \\sin(t\\omega)", "b40c4b4bbd04b95c34e10ba60be30413": "\\, c wz -az+dw -b=0 ", "b40c7799199c93ab5abac71d241be0f0": "x_F", "b40c9116624d5ca3934e6df17a7f1f06": " x\\rightarrow x+\\varepsilon ", "b40cabbdb80121b9a9eb24c639e95d56": "g(\\boldsymbol{x}) > 0", "b40d226176be678b0d2703e46048034e": "\\partial E_1(x) / \\partial x_1 + \\partial\nE_2(x) / \\partial x_2 +\n\\partial E_3(x) / \\partial x_3", "b40d49683a705e20a9f1c17c744d18c6": "\\sum_{n=0}^{\\infty} a_n (z - c)^n.", "b40da596e258b27e43aa76e22b8bc2e7": "n_i/n_\\mathrm{tot}", "b40db63d9f1ee474e1380c6d147c86e1": "\\psi_W(t)\\approx \\psi(t)", "b40dd751cb7bb262ac66946207a6a2ba": "H={\\sqrt{6}\\over3}a=\\sqrt{2\\over3}\\,a\\,", "b40dd7fd7a25c817890b03418be26473": "2^A", "b40dfc306f8d079032a0d81cb959ed19": "\\frac{1}{16}\\left[\\begin{array}{ccccc}- & \\# & 7 \\\\ 3 & 5 & 1 \\end{array}\\right]", "b40e15eb39c33582ef078acef2167460": "h \\in H\\,\\!", "b40e8ee778b5405f4857b03c6375b962": " P = KD", "b40e9b3b41b3a6af89c4633e0a768717": "z_{n}=\\sin^{2}(2 \\pi x_{n})", "b40eace757bf9557eb125f3ff71c4b8c": "V_{2k}=V_k^2-2Q^k", "b40ece258841cb68ddecc67c417ff93b": "\\scriptstyle\\text{partial}(f) \\colon (Y \\times Z) \\to N", "b40edb65518356463bf114337d04d5c4": "\\lim_{x\\to c}\\frac{f'(x)}{g'(x)} \\text{ exists, then}", "b40efdf0f485282c15b218743c1d5383": "\\left [\\begin{smallmatrix}2&-1\\\\-4&2\\end{smallmatrix}\\right ]", "b40f2434ee99e5318c636cdfa25ebde5": "{\\bar{BV}}_3", "b40f67c341af0fa793f733f53d974b3c": "{F_{drag}= A_{cross} \\cdot C_d \\cdot \\frac {v_{air}^2 \\rho_{air}} {2} }", "b40fb81d96f402ca1f2b652d08438749": "f=f_a + f^a,", "b40fbd957f5970efeed776fa2de69c73": "\\,\\!V[n]", "b40fdbd2fa5ccd89ebf1266c6105b457": "\nH_{2}O_{(l)} + \\Delta H \\rightarrow H_2+ \\ ^1\\!/_2 O_2\n", "b40fe106e7aed76d2a82a1f99bd52fc8": "\nR(i,j,p)=\\mathop{\\sum^p_{k=1}}_{k\\neq j}r_k\\left(\\lambda_j-\\lambda_k\\right)^{-i}~~~(i=1,\\ldots,r_j-1)\\,.\n", "b40fec8f5fc6175c6f06b6c8c71ddaa0": " X\\frac{d}{dX} ", "b4101d9449547636fe4cba6c48a47c98": "\\scriptstyle \\|\\,\\cdot\\,\\|", "b4104cf5653be840cd0f4e61709bfba1": "|t|^2+|r|^2=1", "b4107cb8bd412cc417542e71ddd9a9a1": "\\sqrt [12]{2}=1.059463094359295264561825", "b41085e6b469331604d68321d911d18f": "\\operatorname{csch}\\,x = \\left(\\sinh x\\right)^{-1} = \\frac {2} {e^x - e^{-x}} = \\frac{2e^x} {e^{2x} - 1} = \\frac{2e^{-x}} {1 - e^{-2x}}", "b410bd40302e72c97302ec70cdbb3d25": "E[F \\,|\\, x^{(t)}]", "b410d540fb1425307f3619e56cf7b948": "\\langle u+v,w\\rangle= \\langle u,w\\rangle+ \\langle v,w\\rangle.", "b410e2512bc96c4b7e05861e942b6882": " \\frac {1}{R_T} = \\frac {1} {R_1} + \\frac {1} {R_2} + \\frac {1}{R_3} + ... \\ . ", "b410ffc4b3f966c104be7343ac0cd98a": "\\mathrm{Coupling}(C) = 1 - \\frac{1}{d_{i} + 2\\times c_{i} + d_{o} + 2\\times c_{o} + g_{d} + 2\\times g_{c} + w + r}", "b41110283ad02ef6316474a308115c6d": "=\\int (\\mathbf{\\theta - x})^T(\\mathbf{\\theta - x}) \\left( \\frac{1}{2\\pi} \\right)^{n/2} e^{(-1/2) (\\mathbf{\\theta - x})^T (\\mathbf{\\theta - x}) } m(dx)", "b4111782f9942b695c7694c5a2aa7097": "\\mathbb{Z}^\\omega", "b41152e27af121a898790bc4e54bf730": "\\mathfrak{m}_\\pm", "b411b30fc040e7739d4b591e1cd542d0": "f(z) = \\sum_{n=0}^\\infty a_nz^n", "b411e3494b2e394deda0994ae3d97a8f": "\\pm 1 \\,", "b41213f350c2f546db302333d8aac352": "s_1 = 01", "b4122480209c68407485d24fa87f648e": "\n(\\mu,\\tau) \\sim \\text{NormalGamma}(\\mu_0,\\lambda_0,\\alpha_0,\\beta_0) ,\n", "b412322401dde09df54091c733fbcf12": "\\{ \\emptyset, \\{ \\circ \\}, X \\}", "b412845d6762f3e4d61fca7b0d1af2fb": "\\frac{M(tx,ty)}{N(tx,ty)} = \\frac{M(x,y)}{N(x,y)}", "b4128c6dbaa35f740083111588a002f8": "M_{ax}", "b412aabf724a0fc1b19d43a1b520a6b6": "1 / e^2", "b413867189781f95eb6ad3cac23fa47f": " p = (n, m, h)", "b413be5c0a3afc99652051fac49c5071": "|B|=(s t+1)(t+1)", "b413c3da292d1d0eed198b418207ef77": "r_0,", "b413e8f3e9cc62f1764dfde2a3d37381": "\\epsilon_F\\equiv {\\epsilon_F}_p = {\\epsilon_F}_n", "b4140ea0e2ec06de591abe315b982a25": "E^{(+)}(\\mathbf{x}, 0)", "b4143e82fca0dad8c3f89debcb47f79a": "\\scriptstyle \\omega \\;\\to\\; 0", "b414689c3d6fd4e7b0367aaed28c5aaf": "\\textstyle B_1 \\supset B_2 \\supset \\dots ", "b4148647de0a31cd452bb3e65a41a68e": "\\rho_X", "b414bdcaa0dc57b0e0c96fae524e8885": "\\textstyle s>-1", "b414ce48b49b2ca6eaebe665d7e10baf": "\\mathfrak m_+", "b414e9c17ff4f40d81835eb8724d89ab": "f_1=f_0+\\Delta f", "b41543f48905ce7f8c3bf6e8a250e1c8": "\\int_{-\\infty}^{\\infty} e^{a x^4+b x^3+c x^2+d x+f}\\,dx =\\frac12 e^f \\ \\sum_{\\begin{smallmatrix}n,m,p=0 \\\\ n+p=0 \\mod 2\\end{smallmatrix}}^{\\infty} \\ \\frac{b^n}{n!} \\frac{c^m}{m!} \\frac{d^p}{p!} \\frac{\\Gamma \\left (\\frac{3n+2m+p+1}{4} \\right)}{(-a)^{\\frac{3n+2m+p+1}4}}.", "b4154b47112b5bc96fbab1550df05574": "V(\\mathfrak{f})", "b415784152d3478d7a770bcb0543eb53": "e^*", "b415814cc17d8ff75c7e0399465892c4": "|\\alpha_{x}|^{2}\\,\\!", "b4162d1243d16c59a994033a8dd6f3d6": "St << 1", "b4163e93ff1e6ae9fbe63df70634bd8e": "\nX = \\frac{1}{n}\\sum_{k=1}^n U_k.\n", "b4164d4e58f7d929dc5139f9e49a4fd9": "\\mathbb{C}[H_n \\backslash S_{2n} / H_n]", "b416cb4f20811d78bee9be2d2af0c33c": "\\frac{dZ(t)}{dt}", "b416fde3561b910dcd1332699f6a03cb": " H=p^2 + f(x) ", "b4172575243f198d6219bce1d3b9fda8": "\\Phi = Gm \\left[\\frac{1}{a} - \\frac{1}{a+z} \\right]", "b41769c8a9d52e53cbcb47ee308af850": "\\scriptstyle\\ell,", "b417b48199b3d78bcc164abf759cf631": "\\scriptstyle dx\\int{dx\\int{dx\\int{Vdx}}}", "b417e5836449a695fdf89fce38c8efe8": "\\log (f(x_1,\\dots, x_{K-1}; \\alpha_1,\\dots, \\alpha_K)) = \\log(\\frac{1}{\\mathrm{B}(\\alpha)} \\prod_{i=1}^K x_i^{\\alpha_i - 1}) ", "b417ec45ef46b0be3382a13653b7bbcf": "k_d = P k_t", "b418054fb5ab38d9691069f46119d452": "(E+H\\wedge T) {\\rm d}^4\\Omega", "b4180564eb9903d92c4259beb25600c5": "S_n = {X_1 + X_2 + \\cdots + X_n \\over \\sqrt{\\sigma_1^2+\\sigma_2^2+\\cdots+\\sigma_n^2} }", "b41833e1c0d90a2d190ac6a35f4a1bf5": " = \\left( X : Y : Z : {1 \\over 2^n} \\right). ", "b418c3a788c05a5b77bb65a4c9170d76": "\\mathrm{J}", "b418e2e413f0e04d6b762d1f08b400ad": "\\mu_{0}", "b4194b9dcfccaee380fece4273eb4bde": " DX = 0 ", "b41952e9dfed8e1ed562fddafeca7c70": "x^n", "b419f3e5dad9fdd027402d8e2980bf56": " \\textstyle I", "b41a2c96c8053b13a5cc94f26bb6f182": "x_1=011", "b41a39d4c56d885b285aaca3499c7a87": "\\underline{x}^k", "b41ab39ee9d8c817cf377cc034e82dd0": "\\frac{\\dot{Q}_{in}}{\\dot{m}}=h_3-h_2", "b41b5ff0b5306cddf49873420de7ae0b": "{d X\\over dt} = {P\\over m},\\quad {d P \\over dt} = - \\nabla V ", "b41b690fad87753207ebf0a73894985e": "\\mathbf {\\beta_2}", "b41b71f56baca0590bdb140a520c4dbc": "{\\mathbf{\\delta }}_{\\mathbf{0}}", "b41b8cf8d8310a19bb5a3e50f6feee26": "\n\\nu_p(n)=\n\\begin{cases}\n\\mathrm{max}\\{v\\in\\mathbb{N}:p^v | n\\} & \\text{if } n \\neq 0\\\\\n\\infty & \\text{if } n=0 \n\\end{cases}\n", "b41b9dd23fcbbf9d222a2b66fd285d85": "\\Lambda ", "b41bc82a4129f817ff3f69ee740f9bf4": " t [ x := s ] ", "b41c3ad079cebf2d708fc7aa7b000c8f": "A_{ij} = 2\\frac{(\\alpha_i,\\alpha_j)}{(\\alpha_i,\\alpha_i)}", "b41c74d88003d07c3a33c0ba86c4dd40": "C: \\mathbb{R}^d \\to \\{1,2,\\dots,K\\}", "b41c7a728ea72728831e3747fcbe4893": "J = \\det\\frac{\\partial(x,y)}{\\partial(r,\\varphi)}\n=\\begin{vmatrix}\n \\frac{\\partial x}{\\partial r} & \\frac{\\partial x}{\\partial \\varphi} \\\\[8pt]\n \\frac{\\partial y}{\\partial r} & \\frac{\\partial y}{\\partial \\varphi}\n\\end{vmatrix}\n=\\begin{vmatrix}\n \\cos\\varphi & -r\\sin\\varphi \\\\\n \\sin\\varphi & r\\cos\\varphi\n\\end{vmatrix}\n=r\\cos^2\\varphi + r\\sin^2\\varphi = r.", "b41c9ae08a20934d4a11c12cbb31c1f8": "\\,H_{1,m}=(tan\\phi_{1,m})/n_m", "b41ca7ca01349e145dca22d17019c28e": " F_y +F_u q + F_p q_x + F_q q_y =0,\\,", "b41ccb59f19a2a801d73af4a9f2e39d4": "y_n\\to x", "b41d42f08b0c12a7376bb05a6f0aabb6": "\\alpha (1-\\sqrt{\\lambda})^2,\\alpha (1+\\sqrt{\\lambda})^2]", "b41de50a0b96d03ba3763dac390c31f5": "\\mathbf M = \\mathbf r_{\\rm cm} \\times \\mathbf a_{\\rm cm} m + I \\boldsymbol{\\alpha}", "b41dff6be59799b5508d65a4c85c4c5f": "f_c^n(z) = f_c^1(f_c^{n-1}(z)) \\,", "b41e11c071abba0e9c5c3a64bece7c25": "G_{\\rm yz}", "b41e561e3a312b33dc1ffc0d571fe432": "\\left[\\frac{1}{\\cos B + \\cos C}, \\frac{1}{\\cos C + \\cos A}, \\frac{1}{\\cos A + \\cos B}\\right]", "b41e69585ee66a3ecb6d72089e9003f8": "\\textstyle{2}", "b41f1ca6c1a5628a473b36410309cab6": " \\frac{\\mbox{energy}}{\\mbox{volume}} = \\int_{0}^{\\epsilon_f} \\sigma\\, d\\epsilon ", "b41f4c0a1edc94bd1df67625e97f0db8": "\\gamma_{0}, \\gamma_{1},\\gamma_{2},..., \\gamma_{p}\\,", "b41f8b39d530442a8673e4fbc4b3bd5c": "a_G=\\sum_\\chi f(\\chi)\\chi", "b41fb7b8d11404c411e6924448113b29": "\\bold{r} = r\\bold{u}", "b41fbdf14f9a2b3e3faf557c339761a8": "\\left(1-n\\mp\\Delta_n,4+4n\\right)", "b4200c075166c47f6027af975f0092fb": "|B \\cup \\beta|=|(B \\cup \\beta) \\times (B \\cup \\beta)|", "b4200ef65e617ab25280439076018368": "\\displaystyle{{1\\over \\sqrt{2\\pi}}\\left|\\int_a^b { 2 \\sin t \\over t} \\, dt\\right|.}", "b420164c11abab4393442816046d7bed": "\\sum_{n=0}^\\infty \\varepsilon_n a_n", "b420375fa419bdc1a03c6a92c9622832": "P \\in S_i", "b420785bf5617fcb3546f4a337d0f12d": "\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1", "b420859365d1ce113bee2cbfaf4e1ee2": "{{\\partial (8b) \\over \\partial x} - {\\partial (8a) \\over \\partial y}}", "b420d230e9393a1df0102d86a286ee34": "\\,\\!\\sum_{i} m_i v_i", "b4212a723fb78759d206dbdb7689a78d": "\\mathrm 1.9 \\times 10^{-5} \\ll .20", "b4217238a094325ed947b236c36de83e": "\\,\\delta", "b421c6a07c8ec3b72222603a69f78739": "|S_i|0}\\left\\lbrace\nt\\ln\\left(t\\frac{e^{t^{-1}b}-e^{t^{-1}a}}{b-a}\\right)-t\\ln\\alpha\n\\right\\rbrace.\n\\,", "b429e1bf86abbcebcd1881e21d0cfb7a": "R_{\\alpha,\\beta}", "b429e69a723cd6825cf0329ef7e7e029": "\\mathcal{C}^{i+1}", "b429f5c79da3dabda25f15c25fd26dc5": " f({\\mathbf v}+{\\mathbf w}) = f({\\mathbf v}) + f({\\mathbf w}) ", "b42a3fa7c073518e0937006628d2c5ff": " {S_4 \\over S_1} = {{27\\over25} \\div {25\\over24}} ", "b42ab20d6b1a9418cb16f27e30b3c52c": "X_t \\,", "b42aefbc872f861ae135d4eee1b25367": "N_\\beta", "b42b2230798776f7337df357e928e5db": "\\frac{s}{t} \\leq \\frac{\\log_2 b}{\\log_2 a} \\leq \\frac{s+1}{t}.", "b42b2a7f481df8d65d265f46c51a0930": "O(n(\\log n)^2)", "b42b52d48821a5a2193d8b3fd16b5a13": "l^\\infty(R)/I_\\omega", "b42b9f04ed1ab62cc0ec3d24ae21b9d4": "\\begin{align}\n\\mathbf{C}_x &= E[(\\mathbf{x} - \\mathbf{m}_x)(\\mathbf{x} - \\mathbf{m}_x)^H]\\\\\n &= \\mathbf{R}_x - \\mathbf{m}_x\\mathbf{m}_x^H\\\\\n \n\\end{align}\n", "b42ba69f7527f754818886b21d6a879d": "b_k = r_k \\sin(\\phi_k) = -2 f_0 \\int_{0}^P y(t) \\sin(2 \\pi k f_0 t)\\, dt, \\quad k \\ge 1\\,", "b42bc71a982ce1a8d338700480fcfd01": "[h,e_i] = \\alpha_i(h)e_i", "b42bdb6c4118d6d8a87cb5530f9b5384": "e^{-r(T-t)}[F(t) - F(0)]", "b42c514effdc97dbe7da381362c5cab2": " \n\\int_o^{\\infty} {k\\; dk \\over k^2 +m^2} \\mathcal J_1^2 \\left( kr \\right)\n=\nI_1 \\left( mr \\right)K_1 \\left( mr \\right)\n . ", "b42c5ae0273f882a2cda1c704694c002": "\\lfloor 9.5n\\rfloor-1", "b42c99526e116327a5aa83272c51f9b9": "F = \\{\\emptyset, \\{H\\}, \\{T\\}, \\{H,T\\}\\}", "b42c9c76bdd9c3156fa800bc8cc4d397": "F(\\dot x(t),\\, x(t),\\,t)=0", "b42e308d377d5599c0ff36acc3e4953f": "({X\\cup X^{-1}})^*", "b42e845a3fdccb4272820c68bbf29aa1": " \\kappa(\\vec{\\theta}) = {\\Sigma \\over \\Sigma_{cr} }", "b42edc50ccb0e0815cb57194f6981840": "[0,\\epsilon)", "b42ee43cccf14ac7746455b75a974be6": " A^{-1}[i] < A^{-1}[j] ", "b42f251139b1224c2127ab15bfb18d11": "-b/A^{1/3}", "b42f56fd53401a31180a39c5edaf5b8e": "U_e = \\int {k x}\\, dx = \\frac {1} {2} k x^2.", "b42fc3bfa9443e4198d4eb57d610cb39": "\\sum_{n} a(n)n^{-s}\\,", "b42fdafeaec91ae720ec2b509dfcc361": "\\approx 3.8631 \\,", "b4306ba7de637b5f326c7a612576930a": " I = -\\frac{Ze^2B}{4 \\pi m}.", "b4306d2dce25904c2e05f66ec51ec6d7": "\\begin{align}\nq_+&=(1-\\tfrac1{24}(wh)^3+\\mathcal O(h^5))e^{wh}\\\\[.3em]\n&=e^{-\\frac{1}{24}(wh)^3+\\mathcal O(h^5)}\\,e^{wh}.\n\\end{align}", "b4306f405d2a4b7aa99067373b20fd80": "{x^2 \\over a^2} + {y^2 \\over a^2} = 1 \\,", "b430895f6ac205c6a39291a0280cefe9": "F(x)=\\int_0^x f(t)\\,\\mathrm{d}t", "b430c5c99751f143b9b0700f926a0963": "\\scriptstyle\\mathbb{Z}^2", "b430d03801784674db571a65cb1cca4d": "a < x < L", "b430feac417ef28453e6827fc73d3f0b": "\\frac{32\\pi^4}{945} R^9", "b4316a8b5ff6bb65d89fd89f71281552": "\\left|\\frac{(a^{m-1}_{\\;\\ell+i})^2}{a^{m}_{\\;\\ell+i}}\\right|", "b4319e527a74a2aab0a540c448095ca4": "\\frac{\\partial \\phi(x,t)}{\\partial t}=\\nabla\\cdot (D(x) \\nabla \\phi(x,t))=\\sum_{i,j=1}^3\\left(D_{ij}(x) \\frac{\\partial^2 \\phi(x,t)}{\\partial x_i \\partial x_j}+ \\frac{\\partial D_{ij}(x)}{\\partial x_i } \\frac{\\partial \\phi(x,t)}{\\partial x_j}\\right)\\ . ", "b431ed872884c2829589ed841d5f1335": "\\zeta(x)", "b4321c2a8acbf8ee9b743ba11756ca7b": "\n s = \\frac{2n+1}{n+1} \n ", "b432312e9aa93ca14fdf504f6c930f0e": "\\scriptstyle T_1, T_2, \\ldots", "b432364211db87fd2775672e6b7d1357": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\;\\mathrm{gd}\\,x=\\mathrm{sech}\\, x;\n\\quad \\frac{\\mathrm{d}}{\\mathrm{d}x}\\;\\operatorname{gd}^{-1}\\,x=\\sec x.", "b4333a01979ed7c715d5ada0ff44f9b2": "\n\\partial_t u = \\Delta u + \\xi\\;,\n", "b433a5e7ad028cbf282699928501e7f0": "\\Theta_{m,n}(\\tau,z)=(W(z)f_\\tau,\\Psi_{b(n)}).", "b433cf90974c094655169d07d747a03c": "\\mathcal{S} \\ ", "b4340833ecd2858a4fffce45d7622843": "\\pi_{S_1}(R) \\bowtie \\pi_{S_2}(R) \\bowtie ... \\bowtie \\pi_{S_k}(R)", "b4342042ece210c0443dd2a9ab50fb21": " Z = i \\left ( \\frac{\\omega L}{1 - \\omega^2 LC} \\right ) ", "b434312799c08de6925fc732edbba0dc": " \\nabla^2 E_u + k^2E_u = 0 ~~~~~~~~~~~~~~(2.0) ", "b4345a0c58f694c88dd229458a23c866": "A=A_0+A_1g+A_2g^2+A_3g^3+\\dots", "b43475967d1e10bfecc27a40968be9ac": " P_0 = \\sum_{t=1}^ T C(t) \\times P(t)", "b4349ad339c9a4cea7b8a0457532447b": "\\mu(n) \\,n^\\alpha", "b434dd49c6a402f6234aa1f89c2c37dd": "\\frac{d}{dt} \\langle P\\rangle =\\frac{d}{dt} \\langle \\psi|P|\\psi \\rangle = \\langle \\psi | (-\\nabla V) |\\psi\\rangle = -\\langle\\nabla V\\rangle \\, .", "b434e3eed7c0a3e0406f66acfc8985f4": "w_A = \\begin{pmatrix}1 & 1 & 1 & 2 & 2 & 3 & 3\\\\\n 1 & 3 & 3 & 2 & 2 & 1 & 2\\end{pmatrix}", "b4357ef60445333323b9822363df5388": "x_3=\\tfrac{3}{4}", "b4359481fa18d442c313e5b8ab515afb": "x_2 + \\cos(x_1)", "b435e227d5dd201e1768b2bcb2e0aa81": "height", "b43607836a20abbef6a9d3b1a93dbc67": " B\\left(\\omega\\right)", "b43613569ed5fd0e51b0f05bf30e0ea2": "x,y \\in B'", "b4364ce7cf07c554e4b38429bb24c463": "d \\left( \\gamma(s), \\gamma(t) \\right) \\leq \\int_{s}^{t} m(\\tau) \\, \\mathrm{d} \\tau \\mbox{ for all } [s, t] \\subseteq I", "b43659f1b6e46577ea986ff86d9c28dc": " z = C \\mathrm{ln}( r ) ", "b436770591d8c0bd26fcd6420e758374": "A = X_2ZZ_1", "b436bee9d22bc4a22d7a01bce9a45174": "(\\mathcal{C}).", "b436c841ca5eb0ababe36c382f4b98ed": "\\tfrac{bc}{bd}", "b436e7a6454082a2cc65c26ed05c3a1d": "L_x = L_y", "b4372f2dd42c65e8d511e685abc65c27": "-\\varepsilon", "b4373d2efd47f88b4bc9a6d69b17894f": "F^a_{X/S} : X^{(p)} \\to X \\times_S S \\cong X", "b437409256b1cef1fcc7b697e5ee8c3e": "F(y)< -\\frac{K}{\\sqrt{y}}", "b43752cc712c9030eb7e14b4cf7e1b48": "\\Phi: A \\rightarrow B", "b437678dcdb3b04fe0f02b9882fe9a15": "R_{13}(w) = \\phi_{13}(R(w))", "b43798d42498cccf7946770cc3ddb2a3": " \\hat{\\sigma}^2 > 0; ", "b437d463f6d5607effce8623ca95a3e7": "f*K(x) := \\int_{R^n} f(y) K(x-y)\\, dy", "b4382640baf53452112286c961c2e517": "\\Phi (\\rho \\otimes \\omega) = \\sum_i (Id \\otimes \\Psi_i)(M_i \\otimes I)(\\rho \\otimes \\omega)(M_i \\otimes I)", "b438337428286740a07825d9398b7c12": "G_{mn}=\\sum_{k=0}^n (-1)^k {n \\choose k} {k+m+1 \\choose m} \\left[ \\zeta (k+m+2)- 1\\right].", "b4383ba8c3e8e5fd38ef91094ea59a8b": "\\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k}}{(2k)!}=\\cos z\\,\\!", "b4384281fd8464ec4dfb34d6ee23a733": "\\Omega_k \\equiv \\frac{-kc^2}{(a_0H_0)^2}", "b4384b48cede149e0859e2c52e0d2b26": "{\\kappa_p-\\kappa_r}\\over{\\kappa_o-\\kappa_r}", "b4385ff0fefddaffb613f04ef4971462": "t\\downarrow 0", "b4387ef76b8e099fc6a875ae013d2f26": "\\beta=2, \\alpha=\\frac{n}{2}", "b438d17304e25de33eeaa8c099f6f6dd": "2^{n-1}\\,", "b438dea668d65ee3da0d3d8136693f38": "r=2a\\sin^2\\theta/\\cos\\theta=2a\\sin\\theta\\tan\\theta", "b438e026526d51abb2a082926792d1a5": "w^+_r=w^{\\rm eq}_r \\left(1+\\sum_i \\frac{\\alpha_{ri}(\\mu_i-\\mu^{\\rm eq}_i)}{RT}\\right); \\;\\; w^-_r=w^{\\rm eq}_r \\left(1+ \\sum_i \\frac{\\beta_{ri}(\\mu_i-\\mu^{\\rm eq}_i)}{RT}\\right);", "b43943c21cee89e2a9628e2970bf83e5": "t_k", "b439fe2996cd1a4e0ad1020864b44f6a": " z = -2\\,", "b43a58225d95cac4316d2a91a337bfac": "\\mathbf{v} \\cdot \\nabla \\mathbf{v} = \\nabla \\left( \\frac{\\|\\mathbf{v}\\|^2}{2} \\right) + \\left( \\nabla \\times \\mathbf{v} \\right) \\times \\mathbf{v}.", "b43a7f2e65800ecac00195148f39d829": "\\int\\arcsin(a\\,x)^n\\,dx=\n \\frac{x\\arcsin(a\\,x)^{n+2}}{(n+1)\\,(n+2)}\\,+\\,\n \\frac{\\sqrt{1-a^2\\,x^2}\\arcsin(a\\,x)^{n+1}}{a\\,(n+1)}\\,-\\,\n \\frac{1}{(n+1)\\,(n+2)}\\int\\arcsin(a\\,x)^{n+2}\\,dx\\quad(n\\ne-1,-2)", "b43a8b6c71ec7d7ace48f1c0d38f4fdf": "f_t = g_{t,1}e_1 + \\dots + g_{t,s}e_s", "b43ad6507e69f853d1be4ad19bca49e4": "\\ k_af_a + k_bf_b + k_cf_c", "b43ada50edbae22900bf3460cbe98f80": "\\mathrm{K}^{\\mathrm{M}}_n(K)", "b43af77ff040b1718c956500bbe5e831": " d\\mathbf{F} = E \\ \\frac {e} {\\rho} \\ dA \\ \\mathbf{e_x} \\ .", "b43b617c91c012e483ed1766e4518cfd": " \\mathbf{a}_1 = a_1\\mathbf{\\hat b}", "b43b866a28319f25e8857c5086475152": "h_{t_1}(t)", "b43ba514e4d9d67e792bfa82f8308f4b": "D ", "b43bc6d5141a70d3215a2b502597ce08": "n_x=2", "b43be11d478a301e30c0f7e0a138b340": " x_V = \\frac {x_1 + x_2} {2} = -\\frac{b}{2a}.", "b43c4d791589b450acacff8eb52f42c7": " \\operatorname{E}_{GB}(Y^{h})=\\frac{b^{h}B(p+h/a,q)}{B(p,q)}{}_{2}F_{1} \\begin{bmatrix}\n p + h/a,h/a;c \\\\\n p + q +h/a;\n\\end{bmatrix},\n", "b43ccd1f9ed242caabe84e106f0996c7": "{1 \\over \\sqrt{N-3}},", "b43cf38d897d640c8ef143e03cfcc116": "W(\\mathbf{x})", "b43d63dbb1e9fb09ecf991ec280b949c": "\\gcd(62264\\, +\\, 2^{2^4}\\times 20449,\\, F_{5}) = 641.\\!", "b43d659a3f93c7efcffab75b59ee32cc": "p(u, y)>0", "b43d8b063d5cbc14e99b377670753285": "W_L^+ W_L^-", "b43d97c56d425e5e0bb87727da9aec61": "\\{\\theta^{}_i\\}", "b43dfd8d50bfad4ef868b65bc83ecda2": "\\Pi (x:A),B(x)", "b43e1afea15e7932e8186fe889b78372": "x^2 \\in I", "b43e53ac7ef423eae3780123762ac1b9": "2p_1\\operatorname dp_1=p_3 \\operatorname d p_2+p_2\\operatorname d p_3", "b43eda57220236bcf6065b0deeb02f44": " \\varepsilon^v_S = \\frac{\\partial \\ln v}{\\partial \\ln S} ", "b43ee769ee2719ce8eab553d50a7312c": "\\text{sign}(x_1,\\dots,x_k)", "b43f748cb308ff11cbb14950a19ebe3b": "O(r^{-2})", "b43f9f9ebfb541f8ed7146fcaac57508": "S(n)", "b440515e8a254d5c1c5cc54a7a84056f": " h_i(t) ", "b44068d871f70758eb564490dbf0b2cb": " F_m ", "b44081922a9a63015d84b1c2a238377e": "M_{A,B}: f(x,y)=0", "b440cb7c5b59e1b562fa197c1a754bbc": "P(t) = \\frac{dW}{dt}=\\bold{F}\\cdot \\bold{v}=-\\frac{dU}{dt}.", "b44159616dfef62a8b4c1aa88f205441": "h(k)=\\begin{cases}\n 1, & \\mbox{if } k \\ge 0 \\\\\n 0, & \\mbox{if } k < 0\n\\end{cases}", "b4418dced2533e7c6bc1b54c015de44f": "\\mathcal{M}_2(\\mathbb{R}) = \\left\\{ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\bigg|\\ a,b,c,d \\in \\mathbb{R} \\right\\}. ", "b441ba3f5beb9fefdce70e1822ebacdd": "\\mathrm{K}_{x}\\mathrm{Na}_{1-x}\\mathrm{Cl}", "b441c131bd5d04fbd84c282b2a02d606": " \\Gamma^l{}_{jk} = \\Gamma^l{}_{kj}. ", "b4428ffafe0fd452728516a73ee9df7c": "\\frac{d k}{d\\omega} = \\frac{t}{x}", "b44297828d0a0afc507ff3d808a30d91": "\\beta=2^{O(n(\\log \\log n)^2/\\log n)}", "b442b1b14e59c7520913303dfec564bd": "\n\\chi \\approx \\chi_s+2g \\approx \\chi_s+2\\gamma\n", "b442e3a3a6f9ce3f9822b1c4d04ce21e": "p = 1/T = {m\\over n}", "b44319d78378ef677add34223f090879": "s-t=s'-t'=0", "b44381ce89c4fcc59d7ec098e1ce22d2": " T_m(0) \\neq 0 , \\quad m=n-1, \\ldots ,1 ", "b443d3e0622ee21499fee1300ca3c1e4": " {{documentation}}\n\n[[Category:Wikipedia formatting and function templates]]\n[[Category:Templates with minimal expansion depth]]\n", "b443da8e7567757f4e5fe18e5621c99c": "(1-r^2) \\sum_{k=0}^{n} ar^{2k+1} = ar-ar^{2n+3}", "b4441129c553528a24ad99afc810876f": "x_p(t) = \\frac{\\int{\\bar{p(t)}.e^{At}dt}.e^{-At}-\\int{\\bar{p(t)}.e^{Bt}dt}.e^{-Bt}}{P}", "b44425b2f65654074ab594aab8440cb1": "\\mathbf{e}_i\\wedge \\mathbf{e}_j = - \\mathbf{e}_j\\wedge \\mathbf{e}_i,", "b4446a108c36b87faef092585dd123d2": "(f(t))(x):=g(t,x)", "b444ddd304b9ff0b6f24c06df64f37cc": " Re(\\lambda_t) < 0, \\qquad\n\t t = 1, 2, \\ldots , n, \\qquad \\quad (7) ", "b4450c28def79135453c45607c60c8ff": "M(x)=\\frac{ax+b}{cx+d}", "b4454057d5dc2ed41fb5f4230498bfb8": " Q=0 ", "b445529f8bac657b0ea8a332869a6557": "T_{ref}", "b4455dcc2b84b4d0639b86841d4a3c23": "C_{4,n} = {(2n-1)^2 + 1 \\over 2};", "b4458b50c0b2840c80a4ddd872e60bd2": "V=\\mathfrak{g}", "b445b1711121fa792a474c95e25ba519": "j-th", "b445da84ad46d5b53cf91debd3ae9031": "(1,0,0)", "b445eca80799c56b0ff5057e58173b76": " a^* \\cdot a = 1 ", "b445f44ea90a363d9cd4d308c7acced5": " q_n = q_{n-1} - \\frac{f(q_{n-1})}{ (q_{n-1}-p_n)(q_{n-1}-r_{n-1})(q_{n-1}-s_{n-1}) }; ", "b445f537ccb118df96d69619ce43232c": "B(\\mu ,\\frac 12)", "b445ff61ce5ad48da2672c1af4a37e6d": "v = -\\frac{\\partial\\psi}{\\partial x}", "b4461085e8d7797a635cd066f57eb1ba": "a, b \\in V", "b4466a57e929c5539798af3c3c8a846b": "\\lambda_i^ {1/2}", "b44685792715718c064d2551e5fa13e7": " \\mathcal{A}\\phi + L \\le 0 ", "b446a28a9c9371ede32daff111b157f3": "\\displaystyle{T_c(\\partial_z S(\\psi)) = -\\partial_z S(\\psi).}", "b446a9c0004aebffec88c3b44744cad8": " u^*_{i - \\frac{1}{2}} ", "b446c120b6768d1fb770d4d7210abfed": " Q(X,Y) ", "b446c480b478ddfea3eccea8c8fd2dd2": "W_{i+1} = \\exp(\\frac{\\ldots}{\\ldots})", "b446fc6feedcd4a6135a14efe01e8473": "P\\in\\mathbb{C}[X]", "b4473b4292bdb35816395e64f18b0b55": "A^{k-1}", "b447ae6504a55c09b8d9db97d81c8e23": "\\psi_1, \\psi_2, \\dots", "b447b5ff6bf716ce9c2fddeba99007a8": " \\frac{\\partial u}{\\partial t}=u(1-u)+\\frac{\\partial^2 u}{\\partial x^2}.\\, ", "b448431baa5636b9b3d879498dbf5b33": "\\mathcal{L}[\\vec{x},t] = - \\rho [\\vec{x},t] \\phi [\\vec{x},t] + \\vec{j} [\\vec{x},t] \\cdot \\vec{A} [\\vec{x},t] + {\\epsilon_0 \\over 2} {E}^2 [\\vec{x},t] - {1 \\over {2 \\mu_0}} {B}^2 [\\vec{x},t] .", "b4487866137b9112e1db74926f48c1be": "\\cos^n\\theta = \\frac{2}{2^n} \\sum_{k=0}^{\\frac{n-1}{2}} \\binom{n}{k} \\cos{((n-2k)\\theta)}", "b448adfddf99931baa50c05b76d7039c": " s=(\\ldots, (s_{i}, t_{ei}),\\ldots)", "b4497d4d3ca9c7496c62657779ffc6c9": "y = x^2", "b4497ded69853df5048550e838a2d44e": "s=n", "b4499ad723d35f7f8dc19e175d32db6e": "G=k_{i}-k_{0}", "b449c4eb7bbcae83e03fc9118b65540c": " r^2 = \\mathbf r \\cdot \\mathbf r ", "b44a6dadf27ce0f7964c16f7228db6ec": "G.x", "b44a80875737cd69113399f6a4b36d19": "X_4,X_5,X_6", "b44ae9037e3b805e9faaeb3cc92f9c80": " {x}^{ }_{ }(t+1) = 4 x(t) [1-x(t)] +c[x(t),t] ", "b44b03a8575a95b0b56ee55f663e461d": "\n\\rho_{ij} =\n\\begin{pmatrix}\ni\\\\\n2&0.27\\\\\t\n3&0.03&0.27\\\\\n4&-0.06&0.03&0.27\\\\\n5&0.01&-0.06&0.03&0.27\\\\\nj&1&2&3&4\\\\\n\\end{pmatrix}\n", "b44b5c2ee407d180d52a34b9877151c2": "(w,b')=\\mathrm{Rot}_{G}(v,a')", "b44b69d1b43e78ed1338be3fbe2730bf": "\\oint_{C} L\\, dx = \\iint_{D} \\left(- \\frac{\\partial L}{\\partial y}\\right)\\, dA\\qquad\\mathrm{(1)}", "b44b7281aa33c73e84372f595bbd112e": "A_{ij} = X_{ij}/x_j", "b44bd3c13387c64590c6d95ad2d6c1e1": "c_{0}", "b44bff1eb08dfd91e56a59d825877f65": "u_{e} < 0", "b44c03ccc80c92d9b8f4c2c9bc4cfae5": "\nE + S \\, \\overset{k_f}{\\underset{k_r} \\rightleftharpoons} \\, ES \\, \\overset{k_\\mathrm{cat}} {\\longrightarrow} \\, E + P\n", "b44c2b967168cd60a01a85f65094fca2": "Y_l^m\\, \\left ( \\theta , \\phi \\right )", "b44c664113a1d76f3e587ba0c12f0d70": "x=10^{k-1}", "b44ca220d7a934beb1ce1d31e76a5bc9": " k^{th} ", "b44cd7ce23a4f987025e2a288971ad24": "2\\sin\\frac{\\theta}{2}", "b44ce005bf0ac9b57ff80c54e8fe8fb2": "a=n/n^{\\ominus} + o(n/n^{\\ominus})", "b44d0007b3f77528ac461c92d6cfa6e3": " \\sum_{i=1}^m (a_i \\cdot x_i) ", "b44d24420e85b98a54f77fffca2c7b24": " q_t(V)= - \\int_{\\partial V} \\mathbf{H}(x) \\cdot \\mathbf{n}(x) \\, dS ", "b44d3de79c86db3cbebbb0e127010c98": "\\mathbf{T}^{(\\mathbf n)}= \\mathbf n \\cdot\\boldsymbol{\\sigma}\\quad \\text{or} \\quad T_j^{(n)}= \\sigma_{ij}n_i.\\,\\!", "b44d59b3aed8c0a7f292f2cabdaacad2": "\\frac{568\\mbox{ ml} \\times 4}{1000} {{=}} 2.3\\mbox{ units}", "b44d83228c3f975ed8579c62c73a0e3d": "p_i=\\mathrm{Pr}(X=i),\\quad i=0,1,2", "b44d85c38803e0aa3e03ad312a7b3555": "k=|\\mathbf{k}|", "b44d92f139319f60a10ee65946907778": "x \\leftarrow \\frac{1}{1+x}", "b44dba8c56faecb9d0d5cd3eafed3e16": "c_k(V \\oplus W) = \\sum_{i = 0}^k c_i(V) \\smile c_{k - i}(W).", "b44e1c2cbcc2f527446c5e9de79857d3": "M(p)", "b44e3fbb292172e3f12702ee2cec9927": " \\int u\\mathrm{div}\\phi = -\\int \\phi\\, d\\mu = -\\int \\phi \\nabla u \\qquad \\forall \\phi\\in C_c^1 ", "b44e6104bf410bb73d7d0f41d664e1af": "\\alpha(gX) = g\\alpha(X)", "b44ecc18c4f5193fb60bca8fe88c8aec": "\\theta+2\\pi n-\\mu \\le 0", "b44ed3606f9b2387a9f43341b2638551": "E(X)", "b44edfb63bffbc1972f93c246bbf6226": "\\sum_{i=1}^n \\mathrm{Binomial}(n_i,p) \\sim \\mathrm{Binomial}\\left(\\sum_{i=1}^n n_i,p\\right) \\qquad 0 2", "b44ff88c805eaaf3af21ba38cf5454a8": "\\frac{\\pi^4}{384} \\cong 0.25367.", "b45039891f4e63085f13348559ae6dc7": "\\mathbb{Q}(x)\\supseteq \\mathbb{Q}", "b450750d9b997b4241acdd04ca94fc3d": "\\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^*", "b45083c4ec51548da9d19034f0fb5034": "y_0=\\sqrt2-1,\\ a_0=6-4\\sqrt2", "b450dcec5c98c56c6f40dd957d2f5e64": "\n\\begin{bmatrix} w_1 \\\\ w_2 \\\\ \\vdots \\\\ w_n \\end{bmatrix} \\;=\\;\n \\begin{bmatrix} A_{1,1} & A_{1,2} & \\ldots & A_{1,n} \\\\\nA_{2,1} & A_{2,2} & \\ldots & A_{2,n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n,1} & A_{n,2} & \\ldots & A_{n,n} \\\\\n\\end{bmatrix}\n\\begin{bmatrix} v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_n \\end{bmatrix}\n", "b450eb225ea98ebcea170ff906196b95": "\\{|u_m\\rangle\\}", "b45127fa83bcf38fe2340296bcd769ca": " \\det(A-\\lambda I) = (\\lambda_1 - \\lambda )(\\lambda_2 - \\lambda)\\cdots(\\lambda_n - \\lambda)", "b45168862d2ede3b4a90c5356685d06f": "\\Box \\phi = \\phi^{;a}_{\\;\\; ;a}", "b451b111e69ed271d9838c6396d40d47": "\\chi_m^2", "b451c43cb0c64229f84a5d047f7193bf": "b_n=\\sum_{p_i}{\\prod_{i}{{b_i+p_i-1}\\choose{p_i}}}", "b451e94bde39552d0298fc7233ee0a1b": "\\frac{N_e}t = \\Phi_{\\xi}\\frac{\\lambda}{hc}", "b4520b8864da82fc92311d8d1c871e02": "\\frac{\\mathrm{d}q}{\\mathrm{d}t}=u(t)-kq", "b4524dcb5b509691b2c63353eef05abb": "\n\\mathcal{L}(\\overline{x},\\Sigma)\\propto \\det(\\Sigma)^{-n/2} \\exp\\left(-{1 \\over 2} \\sum_{i=1}^n \\operatorname{tr}((x_i-\\overline{x})^\\mathrm{T} \\Sigma^{-1} (x_i-\\overline{x})) \\right)", "b4526bc114c359f022230d0167b857bb": "\\dot{Q} = 2 k \\pi \\ell \\frac{T_1 - T_2}{\\ln (r_2 /r_1)}", "b4527e0091094c13aa10ddd6a0ab36be": "s_{t+1}", "b4528db2856b17a72cb53e0ae64716f3": "R = \\sqrt{\\sqrt{I} + \\sqrt{J} } = \\sqrt{I + J}", "b452a4592c7c7199281040f521c6683c": "f(x)=RxR^\\dagger \\, ", "b4539de0f16a5694ce32204796025294": "dL=\\delta L + d_H \\Theta_L,", "b453a7e25e69acfd060d38e945870b08": "\\frac{75}{64}", "b453b7a6c0dc109ee6f630965eacc50b": " {[}E{]}_\\text{tot} = {[}E{]} + {[}ES{]} \\; \\overset{!} = \\; \\text{const} ", "b453c2588c4a3c9af293219ed1519ae5": "N(t)=0", "b453d7e889bb434cd6221827ee25c53d": " \\operatorname{let} p : \\operatorname{de-lambda}[p]\\ \\operatorname{de-lambda}[f] = \\operatorname{let} x : \\operatorname{de-lambda}[x]\\ \\operatorname{de-lambda}[x] = \\operatorname{de-lambda}[f]\\ (\\operatorname{de-lambda}[x]\\ \\operatorname{de-lambda}[x]) \\operatorname{in} f\\ (x\\ x)] \\operatorname{in} p ", "b453e7b271e5a20897db79faf9f9e0a8": "x=\\frac{p+q}{2}.", "b453fbc0af49072637dd9e2080d13773": "\n df = \\frac{\\partial f}{\\partial \\boldsymbol{\\sigma}}:d\\boldsymbol{\\sigma} + \\frac{\\partial f}{\\partial \\boldsymbol{\\varepsilon}_p}:d\\boldsymbol{\\varepsilon}_p = 0 \\,.\n ", "b454025860634b889bbc65c2f7e51d2b": " \\frac{K \\cdot t}{V} = \\frac{7/3 \\cdot K_D}{V_D} \\qquad(7a)", "b4544bcdf210df45b564ca713da6be35": "E_n\\equiv E_n(0)", "b4551da5510e8ab4b0972a36285834ea": "f(x) = \\frac{(1 + x^2/\\nu)^{-\\frac{\\nu+1}{2}}}{\\sqrt{\\nu}B(\\frac{1}{2},\\frac{\\nu}{2})}", "b4552446a272a59fbfcfe74af2267840": " v_1, v_2, \\cdots, v_n \\in \\mathbb{R}^d ", "b455332ec3d21f3e01f66f091c41c42e": "\n\\begin{Bmatrix}CS:\\\\DS:\\\\SS:\\\\ES:\\\\FS:\\\\GS:\\end{Bmatrix}\n\\begin{bmatrix}\\begin{Bmatrix}EAX\\\\EBX\\\\ECX\\\\EDX\\\\ESP\\\\EBP\\\\ESI\\\\EDI\\end{Bmatrix}\\end{bmatrix} +\n\\begin{bmatrix}\\begin{Bmatrix}EAX\\\\EBX\\\\ECX\\\\EDX\\\\EBP\\\\ESI\\\\EDI\\end{Bmatrix}*\\begin{Bmatrix}1\\\\2\\\\4\\\\8\\end{Bmatrix}\\end{bmatrix} +\n\\rm [displacement]\n", "b4553a2effe26ae4a69dfebaa759e400": "\ns(1+e^{i \\Delta k \\Lambda})=1-(-1)^N e^{i \\Delta k \\Lambda N}.\n", "b455eda3674a49ab83301aeed8d8bba8": "\n u_2(z,t) = U_0\\, \\Bigl[\\, \\cos\\left( \\Omega\\, t \\right)\\, -\\, \\text{e}^{-\\kappa\\, z}\\, \\cos\\left( \\Omega\\, t\\, -\\, \\kappa\\, z \\right)\\, \\Bigr],\n", "b455fe05c1fa9a8bbbabbb1e6c6df5c8": "D<0", "b4568d5775f04355b3acc95596e63a3d": "x \\equiv 11 \\pmod{60}", "b456a79df4fbdb3229883fa8baf8f16c": "0 \\in T_pM", "b456d76f520ebf947aa6c877ade6377b": "p(x, y, z)", "b456f6664ade6c043c384dd658b72b78": "k < 2\\times10^{-9}", "b457388c580843a32353e48d678ccc78": "f:m_+ \\rightarrow n_+", "b45741412b2aadba21e8bad6f02bcad1": "F: \\mathbf{Set} \\to\\mathbf{Set}", "b457d9cd937cec99623cc42d6b4324e8": "x > 0\\,\\text{ or }\\,y \\neq 0.", "b457e8c172d357c9577709d19373b4c9": "S(A,P,z) \\le XV(z) \\left( F_1 \\left(\\frac{\\log y}{\\log z} \\right) + O\\left(\\frac{(\\log \\log y)^{3/4}}{(\\log y)^{1/4}}\\right) \\right) + \\sum_{m|P(z), m < y} 4^{\\nu(m)} \\left| r_A(m) \\right|", "b45814117f9ab3179afe8157042cb076": "\\eta : 1_{\\mathcal{D}} \\to T", "b458468a82de49393c3e1d6dc4b233da": "E_{edge}=-\\left | G_{\\sigma} * \\nabla ^2 I \\right \\vert ^2", "b45894d123bb8944a9bdfc2f7d67b505": "\\tilde{P}(q,\\omega) \\ \\ = \\ \\ \n\\frac{d}{d\\omega} \\left[\\frac{\\omega}{1-e^{-\\omega/T}}\\right]\n\\times \\frac{2Dq^2}{\\omega^2+(Dq^2)^2}\n", "b458bc178a8aa0217eae8b30adb0dd29": "h(x_1), \\dots, h(x_k)", "b4591c2c17b2b965497bd850f60b4a32": "{\\partial u \\over \\partial y} = -{\\partial v \\over \\partial x} = e^x \\cos y.\\,", "b4591fcd7df669d5a70687fd23b8357b": "(e_k)", "b45926724b06b8bc4e039e4a67864611": "A = \\vec{r}(0)", "b4593c174e6b1f25bede48f72a8ee85a": "\\delta F = F \\, dr", "b459bde00acd873d596c285b72441b02": "\\mu \\pm 1.96\\sigma", "b459c62edd5151704cb60b6c2dd0dc60": "x_1=6,\\ \\ x_2=9,\\ \\ x_3=3,\\ \\ x_4=8,\\,", "b459cfbae2f04854d6530730ab2aadf8": "\\vdash \\Psi \\leftrightarrow F(\\Box \\Psi)", "b459e5b981985978bbb00ce550152f76": " d\\tilde{S}_t = \\sigma \\tilde{S}_t \\, d\\tilde{W}_t.", "b459f9e8b5454970c178c9ac41b8fa25": "N(\\sigma)", "b45a362a9980dfddd51e5a5240ae3417": "\\tilde{S}:=\\varphi\\circ{\\theta}_{D}[{I}^{2}]", "b45a8997532bedeb0adec8164ed316cb": "s\\ge 4r", "b45aede5e3730b2bb35b155af4b603d8": "\\{ 1, ~\\epsilon \\}", "b45b187b7d1dcfc489e60a81b68611e6": "\\mathbf{x}_{0}", "b45b39b7dcafd646e7ac1a29d102c0fc": "\\textstyle [\\mathbf{x}] = \\bigcup_{i=1}^k [\\mathbf{x}_i]\\mbox{,}", "b45b8425664315935a99b51f62f4393b": " Y = g(X, \\varepsilon) ", "b45b8dd13cca3d26e0ad6c2407e5b2d9": "\\sigma_1,\\sigma_2,\\sigma_3", "b45ba8e7ab04820201a9134dda03904c": " \\lambda = \\frac{Q[u]}{R[u]}. \\,", "b45bd609bfd875764e53b26cc3e1ea8d": "1+\\frac{z}{1!}+\\frac{z^2}{2!}+\\frac{z^3}{3!}+\\dots", "b45bfa929f973c525d3dccffae844854": "\\ln S=\\ln T_{max}. \\ ", "b45c24bbf1541163fd9f08c6cc1a71ac": "MRT = T_1 F_{p-1} + T_2 F_{p-2} + ... + T_n F_{p-n}", "b45c5d45de29c581fd82b96bbc42a422": "M=-J,\\ldots,J", "b45c71f1cff651f3b8165fc8e747f04a": "\\log(z) = \\operatorname{Log}(z) + 2 \\pi i k", "b45c8afea9bf3ec0db0fb0bc67fb09bf": "\\alpha t \\in (0, \\pi)", "b45ccc58111c53de6568e65e94ba125a": " \\Delta G^{\\mathrm{prot}}(\\mathrm{pH}) = -\\mathrm{RT}\\ln\\left[ \\frac{}{1-} \\right] ", "b45ce36c5912b5de4cda2ca77247dd53": "\\min_A(x)\\geq\\min_B(x)", "b45cf4d6fb986448578e215b1c6ccffc": "S_3 \\overset{\\sim}{\\to} \\operatorname{Inn}(S_3) \\cong S_3.", "b45d083db191e371584f8c513dfa5a84": "1\\,\\text{slug} =1\\,\\frac{\\text{lb}_F\\cdot\\text{s}^2}{\\text{ft}}\n\\qquad\\Longleftrightarrow\\qquad\n1\\,\\text{lb}_F = 1\\,\\frac{\\text{slug}\\cdot\\text{ft}}{\\text{s}^2}", "b45d18f4febd154bb67798c3a259533c": "p = 2^{61}-1", "b45d4875924b97b30f8f0365c8dbc2f7": "(\\tfrac{p + r}{2}) (\\tfrac{q + s}{2})", "b45d5fe2db01e15206f52f55e133fa60": "\\left[-\\ -\\right] : V^{op} \\times V \\to V", "b45ed4c9165d03125c964c0d7b999e50": " G_{\\gamma\\delta}F^{\\gamma\\delta}=\\frac{1}{2}\\epsilon_{\\alpha\\beta\\gamma\\delta}F^{\\alpha\\beta} F^{\\gamma\\delta} = -\\frac{4}{c} \\left( \\bold B \\cdot \\bold E \\right) \\,", "b45f0216deed1f42d1ffe316d21babb3": "Q^{-1/2} \\sum_x \\left|x\\right\\rangle \\left|f(x)\\right\\rangle.", "b45f0dd1aaf96a3673b689a460780e97": " A + B \\rightleftharpoons 2B", "b45f2438d78d8ee4fde495ebb5930f2f": "\\sigma_x \\sigma_p = \\frac{\\hbar}{2} \\sqrt{\\frac{\\pi^2}{3}-2} \\approx 0.568 \\hbar > \\frac{\\hbar}{2}.", "b45f2b06fe14797f9c10e4a8d3152e89": "K_2 = -1 - \\frac{2}{(M - 1)^3}\\left[M(2M - 1) + \\frac{B}{D}\\right]", "b45f5c702b11483b774e94ddc09336b1": "|R_k(x)| \\le M\\frac{|x-a|^{k+1}}{(k+1)!}\\le M\\frac{r^{k+1}}{(k+1)!}", "b45f6ba89dc73a6355d87d66a325bd16": "{\\Phi (x,y)}", "b45fb66d4e0bcdaa5aab4687b34f58da": "D_{i}=\\epsilon_{ij}^{0}E_{j}+\\delta_{ijk}\\frac{\\partial E_{j}}{\\partial x_{k}}=\\epsilon_{ij}^{0}E_{j}+(ie_{ijl}{g}_{lk}k_k)E_{j}\\, ", "b45ffda60604a2b58c3eaf27a070a7f4": "U_n", "b46035a209ea24f73b5f651f0bf69f35": " 0^y ", "b46063eeeb03bf19b61f7a266ba47af7": "\\mathbf{\\hat{z}}", "b4606d6b572a35e658b331cdfeb81984": " \\Delta(q, \\omega)=((\\ldots,(s_i, t_{ei}+dt),\\ldots), t_e+dt).", "b4607652036b9817488e6ca46a90dacd": " \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left ( {m\\dot{\\mathbf{r}} \\over \\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2}} \\right ) = e\\left ( \\mathbf{E} + \\mathbf{v} \\times \\mathbf{B} \\right ) . \\,\\!", "b46087c3e7f96bd3b5666db23f1058de": "\\chi_{r,n} = \\chi_{r-1,n-1} + (-1)^r \\chi_{r, n-1}\\,", "b460cd0817ce69e677e491c0cf6db39a": "A=(a_{i,j})_{i,j=1}^n", "b460e6fa421d7420403c88406e40962b": " \\mbox{if }s \\in S. \\,", "b460fbb14d07b0f87fd86ae7b451666e": "\\begin{align}\n {} & {} \\log(n+1) \\\\\n < &\\sum_{i=1}^n\\frac{1}{i} \\\\\n \\le &\\prod_{p \\le n}{\\left(1 + \\frac{1}{p}\\right)}\\sum_{k=1}^n{\\frac{1}{k^2}} \\\\\n < &\\frac53\\prod_{p \\le n}{\\exp\\left(\\frac{1}{p}\\right)} \\\\\n = &\\frac53\\exp\\left(\\sum_{p \\le n}{\\frac{1}{p}}\\right)\n\\end{align}", "b4610c3d18223aa8668d18c3bbf09eeb": "Fo(x) = \\frac{\\varphi^x + \\varphi^{-x}}{\\sqrt{5}}", "b46182ab13eb2bdea118f5d711cddf45": " 0 = \\left( 2 h_{1} h_{0} + h_{1} h_{2} \\right) \\frac{J^{\\prime\\prime}(u_{0})}{2} + h_{1}^{3} \\frac{J^{\\prime\\prime\\prime}(u_{0})}{8} = \\frac{h_{1}^3}{24 \\beta^2} (3 \\beta^2 J^{\\prime\\prime\\prime}(u_{0}) + 5 J^{\\prime\\prime}(u_{0})^2) ", "b46185aa2fbbef1aa8485eeddac877aa": " \\mathbb{A}_\\mathbb{Z} = \\mathbb{R} \\times \\widehat{\\mathbb{Z}}.", "b4619f36d5baf962e394e25532080565": "b+c=9", "b461d6cce888232b3f5a9b2effe59ebb": "S_{2,t}", "b461e624280b70a5185dce080fab600b": " dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2. ", "b4621ae7a201d85cc07297e550e409c3": "a = A(P_{a}, n)", "b4623ba64c59d890a957a90228c462f4": "\\,(-p^2 + m^2)G(p)=-1.", "b4623d525e4c5153ed0a80672be47ec7": "0.\\dot{8}\\dot{1}", "b46280f54e83e584213d5f9ba97e93ea": "Bxy", "b462f451c083af2c752e662b05e577fb": "\\sum_{n=0}^{\\infty} \\frac{1}{n!} = \\frac{1}{1} + \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{6} + \\frac{1}{24} + \\frac{1}{120} + \\ldots = e\\,.", "b4632806a55f4e5bee48e898f168d911": " s^2 = \\frac{1}{n - 1}\\sum_{i=1}^n (x_i - \\overline{x})^2, ", "b46332089f518b670403e11063162975": "P^{\\mu}W_{\\mu}=0,", "b4636032426a2fef0fd1de55cf6e779f": " \\forall \\mathbf{c}^1 \\neq \\mathbf{c}^2 \\in C^* \\text{, } |\\{i | \\mathbf{c}^1_i = \\mathbf{c}^2_i = 1\\}| \\leq a_\\max ", "b463a84d7bf58235dfafce8763ab5f51": "\\nu_{lo}", "b463bdc97643aff802bf41f512c1a91c": " A = VSW^T \\, ", "b463fd4218d02f80097506c8dd56696a": "\\hat{f}(\\mathbf{x}')", "b46461a00a69a73a9c3385e5ad388abd": "\\scriptstyle H^2(M,\\mathbb{Z})", "b46470bb313e5bc4274b0bea39d9e166": "\\color{red}\\exists y", "b464d11cc23cafe83a245e56d0feaa0b": " {v_x - v_S \\over v_\\infty - v_S} = {v_x \\over v_\\infty} = {v_y\\over v_\\infty}= 0", "b464d412cac55dd51d713ec5d25bbe97": "\\frac{\\partial u}{\\partial\\tau} = \\frac{1}{2}\\sigma^{2}\\frac{\\partial^2 u}{\\partial x^2}", "b464e24729f0a8273e24a741b14e45d8": "\\displaystyle p=q+t", "b464f53189af11722808f7e044924c34": "\\frac{\\operatorname{d} u}{\\operatorname{d} x} + u^2=0\\,", "b465413ec20d3f268f01ac020464a3bc": "x_1 h = h", "b465c239d2a3d8bc724e958da83a9ee1": " \\mbox{density of body} = \\mbox {density of water} \\frac { \\mbox{weight of body}} { \\mbox{weight of body} - \\mbox{weight of immersed body}}\\,", "b465fc5142fa1418926096c25da9e996": "C(m,n) > 0 ", "b465fc9b2e1544bda5b3d70c0abc47a0": "I(\\omega)(c) = \\oint_c \\omega", "b4667aa067433427b3050ec509396893": "\\frac{d}{dt}\\langle \\hat{A}\\rangle = \\frac{1}{i\\hbar}\\langle [\\hat{A},\\hat{H}] \\rangle+ \\left\\langle \\frac{\\partial \\hat{A}}{\\partial t}\\right\\rangle", "b466b7e7f170fea7e20793823df51ee4": "PSU(n,q^2)", "b46772aaa7aa31b2d0e7427592354623": "df(X_t)= \\sum_{i=1}^d f_{i}(X_t)\\,dX^i_t + \\frac{1}{2}\\sum_{i,j=1}^d f_{i,j}(X_{t})\\,d[X^i,X^j]_t.", "b467aaaf6069c732234b7395b6a64292": " \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} + 2\\zeta\\omega_0\\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\omega_0^{\\,2} x = 0, ", "b46824dbe142795dce9550ef2f17e90b": "X(f)\\,", "b468734497376c8595eace2486c50332": " \\frac{\\partial \\rho}{\\partial t} = - \\nabla \\cdot (\\rho \\mathbf{u}) ", "b4688185c39aa7f432b22e74a0af529a": "\n\\begin{bmatrix}M\\end{bmatrix}\\begin{Bmatrix}\\ddot{x}\\end{Bmatrix}+\\begin{bmatrix}C\\end{bmatrix}\\begin{Bmatrix}\\dot{x}\\end{Bmatrix}+\\begin{bmatrix}K\\end{bmatrix}\\begin{Bmatrix} x\\end{Bmatrix}=\\begin{Bmatrix} f \\end{Bmatrix}\n", "b4688aaaaf17fad03225929fe56ad458": "dr", "b468c5ac8aea59279face1fb343ebee2": "\\,\\eta_ts\\, =\\,\\frac{(h_{01} - h_{03})}{(h_{01} - h_{3ss})}", "b468eb894b0b2306cad0a1d8b979ab5a": "G_{\\mu \\nu} = \\frac{8\\pi G_N}{c^4} T_{\\mu \\nu},", "b468f400051e51e42856bffa4c7a166d": "X_n=\\frac{1}{1-p}=\\frac{N_o}{N}", "b469783410f2f5a50bb63c666eadf8a0": "C^0\\ ", "b469af9974f8d9eb8843d39720fc7075": "I^-(E)", "b469b2e00ec70b8959488c9c9a726b6c": "\\sum_{r=0}^{k-1} f(x_r)(\\rho(x_{r+1})-\\rho(x_r))", "b469c23e7f02db2fd7fe8e730c87902f": " \\int_{-\\infty}^\\infty \\Phi(bx)^2 \\phi(x) \\, dx = \\pi^{-1}\\arctan \\sqrt{1+2b^2} ", "b469e36fd31bbdd794013b7d0b9a1e7d": "\n\\mu = \\frac{1}{\\det A} = \\frac{1}{[(1-\\kappa)^2-\\gamma^2]}\n", "b46a050229ef828f125219740a0376ff": "L(t)=U(t,s) L(s) U(t,s)^{-1}", "b46a18e9890d22402ae78d8944c6696a": "E={\\pm}mc^2.", "b46a5e533fa9d48c25c74bc2e5467a30": "\\langle B|a \\rangle \\langle A|b \\rangle", "b46a8ed7354c34976b40a2f02d3e2312": "u:A \\otimes - \\rightarrow - \\otimes A", "b46a9b45f4789315551e44d8a2cd99ef": "(-\\ln \\hat{G}_X) >0", "b46adb44f5cad8cf8556f732b7d41094": "\\rho = \\frac{i\\hbar}{2mc^2}\\left(\\psi^{*}\\frac{\\partial \\psi}{\\partial t} - \\psi \\frac{\\partial \\psi^*}{\\partial t}\\right)\\, ,", "b46b001ed8d2448265c899634d88a8fd": "\\begin{align}\n\\sin z & = \\sum_{n=0}^\\infty \\frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \\\\\n& = \\frac{e^{i z} - e^{-i z}}{2i}\\, \\\\\n& = \\frac{\\sinh \\left( i z\\right) }{i}\n\\end{align}", "b46b1562fd94ed61ac8bf3b19a876370": "f\\circ g=f\\circ h", "b46bbfd9b4fe358e5d86069a93fba21e": "0 \\le r < 2^{160}", "b46bccf1d6601fa2dfc61f96ac36e7fa": "\\forall_{i,j \\in N_i,l, y_l\\neq y_i} d(\\vec x_i,\\vec x_j)+1\\leq d(\\vec x_i,\\vec x_l)", "b46c5d302861054c6520f1d0e2b2c26f": " \\mathrm{USp}(2n) \\supset \\mathrm{O}(n) ", "b46c6db0a403506f3c3c164af0d19aa1": "\\Delta \\phi = \\phi - 2.", "b46c736be8ef852989ec337e46887519": "\n\\boldsymbol{P}=-p~\\boldsymbol{F}^{-T}+\\frac{\\partial W}{\\partial \\boldsymbol{F}}\n = -p~\\boldsymbol{F}^{-T} + \\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{E}}\n = -p~\\boldsymbol{F}^{-T} + 2~\\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{C}} ~.\n", "b46cdb6d2b154c0975cfac765468c45e": "V_{12}\\equiv V_1\\otimes V_2", "b46d315b6b46c421eeb8bd38f455740f": "u(x),", "b46d8a0836f37e79bbbae45e4da4c9c6": "V:=\\bigcup_{n \\in \\mathbb{N}} B\\left(n\\,;\\,1/n \\right),", "b46d9fd32c09c402c1f76fc52c46d07b": "\\mathbf{D}^2_{xy}=\\begin{bmatrix}1 & 1 & 1\\\\1 & -8 & 1\\\\1 & 1 & 1\\end{bmatrix}.", "b46dac031aa91c29202b1e3c199839fa": "\\Delta[2] \\to C", "b46dbbacdc78af7cbc001ca1135a723f": "y^*_j", "b46dc2c5045b85c8cdac04c678815bad": "\\|\\vec{r_V}\\|=1", "b46dd8aedc4a0c40b1cf5f1995b8114b": "m_0\\rightarrow q_0", "b46df97853c3812f55660d479e0d88b5": "J_m = I_m -3", "b46e20fdb71852921d92098c7253fb87": "A_{0}=0", "b46e4c4d65410bdb3678f32a555127ff": "\\bar R^2", "b46e6ba7708d84d491c0599e519a62d4": "\\ \\eta", "b46f1443ff9c3cb4f577bb897a2de277": "{\\color{Blue}~5.4}", "b46f3b513e40dc5fd5178d486f9635ce": "u_{o}= L/t_M", "b46f52dda33aec7dfecce5e4ed6ba9cd": "(x_n)_{n\\in\\mathbf{N}} \\mapsto \\left(\\sum_{i=0}^n x_i\\right)_{n\\in\\mathbf{N}}.", "b46ff5b69a0be3f6dcf978051c0c9197": "e_i=\\sum_j[\\mathbf{O}]_{ij}a_j", "b470530f47f92bf24ceff5e439da9016": " d\\Omega ", "b4705b69397942be01f29bc0a46ca8d2": "e^{i H_0 t_2/\\hbar}", "b47087833632969d05305bd06511a056": " S_{uvw}", "b47091324076c5a0eaa8857a9e3a12fe": "B \\geq D", "b470d5f2a2524b706d9d8efbbfc5a0c4": "\n\\varphi(2m) = \n\\begin{cases}\n2\\varphi(m) &\\text{ if } m \\text{ is even} \\\\\n\\varphi(m) &\\text{ if } m \\text{ is odd}\n\\end{cases}\n", "b470f825c32bc8405263a7602ec06300": "\\!\\mu_4(v_3)", "b471443e4771511d5c9cac903de9f89d": "\n S_x = A \\bar y = \\sum_{i=1}^n {y_i \\,dA_i} = \\int_A y dA\n", "b4714516723ddd97a003ef855c775e17": "\n\\rho_{\\,\\mathrm{humid~air}} = \\frac{p_{d}}{R_{d} T} + \\frac{p_{v}}{R_{v} T} = \\frac{p_{d}M_{d}+p_{v}M_{v}}{R T} \\,\n", "b4715d62e8f972de36630c27da901992": "\\sum_{k=0}^\\infty k\\frac{z^k}{k!} = z e^z\\,\\!", "b471c28a8f89b550ef4df41183a4db7e": "\n x(nP) = \\frac{A_n}{D_n^2} \\quad \\text{with}~\\gcd(A_n,D_n)=1~\\text{and}~D_n \\ge 1.\n", "b471e6669b87577f07cf0c4a72c6d3ef": "\n\\begin{pmatrix} a \\\\ b \\end{pmatrix} =\n\\begin{pmatrix} q_0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} b \\\\ r_{0} \\end{pmatrix} =\n\\begin{pmatrix} q_0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} q_1 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} r_0 \\\\ r_1 \\end{pmatrix} =\n\\cdots =\n\\prod_{i=0}^N \\begin{pmatrix} q_i & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} r_{N-1} \\\\ 0 \\end{pmatrix}\n", "b472393cb5f32c5a211ade0f6a14ea29": "{}^{239}\\mbox{Pu} + \\mbox{neutron} \\rightarrow \\mbox{fission fragments} + 2.9\\mbox{ neutrons} + 198.5\\mbox{ MeV}", "b472771ca5b773401ef71415e3ee1d84": "\\scriptstyle \\lfloor{k}\\rfloor", "b472cff555464972f1ad58df41fb10c9": "m^{*}", "b4731272cc56c1945e08d1133c68c135": "a = E(Y) - t_{a/2,n-1}S/\\sqrt{n} \\qquad and \\qquad b = E(Y) + t_{a/2,n-1}S/\\sqrt{n}", "b4731b868a2b181869ecfaaf23995388": "\nK_{ij} \\leftarrow K_{ij} + (1-K_\\max) \\,\n", "b47322e86d8492a5ebae7379b6199fed": "0 \\leq r", "b473a856f2605059cb8e1d26b4bf6b8a": " R_\\mathrm{E} \\parallel \\left( {r_\\pi + R_\\mathrm{source} \\over \\beta_0 + 1} \\right) ", "b473f8442413aed4cc75735d7a712425": "\n{{\\sigma _{\\hat g}^2 \\,} \\over {\\hat g^2 }}\\,\\,\\, \\approx \\,\\,\\,{{\\sigma _L^2 \\,} \\over {L^2 }}\\,\\,\\, + \\,\\,\\,\\,4{{\\sigma _T^2 } \\over {T^2 }}\\,\\,\\, + \\,\\,\\,\\,\\left( {{\\theta \\over 2}} \\right)^4 {{\\sigma _\\theta ^2 } \\over {\\theta ^2 }}", "b4745fb790abc9b75d8ec80c25f0ad2c": "u= \\frac {\\langle u, v \\rangle} {\\langle v, v \\rangle} v+z,", "b47497e42a930a7696eb4476f7b9d453": "\\Gamma(t;\\gamma_1,\\lambda) + \\Gamma(t;\\gamma_2,\\lambda) = \\Gamma(t;\\gamma_1+\\gamma_2,\\lambda)\\,", "b474d19e83df6e92b500868f67c08320": "|\\Phi^+\\rangle = \\frac{1}{\\sqrt{2}} (|+\\rangle_A \\otimes |+\\rangle_B + |-\\rangle_A \\otimes |-\\rangle_B).", "b4754c509fa4da2ba20a3221ed819ebe": "m = \\frac{\\eta_1 - \\eta_2}{\\eta_1 - \\eta_3},", "b47550fedd0a4952c5eabda411e6794a": "n^{2/7}", "b475ab443d758b40d57525fa56c1bdd3": "\\tfrac{\\sqrt{4}}{2}=1", "b475f3deb6b3ccca69f77963c1687937": "\\tbinom n k p^k (1-p)^{n-k} \\!.", "b47620c7a43a5343a6f255206a7c79fd": "\\begin{align} 2\\cdot R_*\n & = \\frac{(23.97\\cdot 4.24\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 21.9\\cdot R_{\\bigodot}\n\\end{align}", "b47696f0bc27bd2fd2d87e4d629c3363": "(x_k, y_k) = d_A Q_B", "b476df18dee8856651415fb3f269d99e": "[a(\\vec{k}_1),a(\\vec{k}_2)]=[a^\\dagger(\\vec{k}_1),a^\\dagger(\\vec{k}_2)]=0,", "b4772376be95bee4a6d735e9c554aa89": "\\pmod{\\mathfrak{p}}.", "b4773be040cad2c6eb5b1485caab1316": "A_{(\\alpha\\beta\\gamma)\\delta\\cdots} = \\dfrac{1}{3!} \\left(A_{\\alpha\\beta\\gamma\\delta\\cdots}\n+ A_{\\gamma\\alpha\\beta\\delta\\cdots} \n+ A_{\\beta\\gamma\\alpha\\delta\\cdots} \n+ A_{\\alpha\\gamma\\beta\\delta\\cdots}\n+ A_{\\gamma\\beta\\alpha\\delta\\cdots}\n+ A_{\\beta\\alpha\\gamma\\delta\\cdots}\n\\right)", "b477e15b312155554c106011fa84c8ef": "b^x = y \\,", "b47826e42eb80268d4ea0c3a19e13830": "f^{''}:Z\\coprod \\partial D^2/\\!\\sim", "b478875e767e2e4e8d0a11782d3f301e": "2\\,H^+\\;+\\;2\\,e^-\\rightarrow\\;H_2\\;", "b478a53738d66bd5e531bd8893c8f00d": "\\rho_0\\Pi", "b478ab01fc211597afddbc4a7035698b": " \\frac{1}{4\\pi}\\int_{T^2} \\frac{|(f'(s) \\times f'(t)) \\cdot (f(s) - f(t))|}{|(f(s) -f(t))|^3} \\, ds\\, dt.", "b479eb9db178ce9cd1c3dd5a9d08c3e0": "\n z_f^{\\mathrm{topface}} := z - h - \\tfrac{f}{2} ~;~~ z_f^{\\mathrm{botface}} := z + h + \\tfrac{f}{2}\n ", "b47a172c867945390c2b35f7e49c7013": " |\\epsilon_i\\rang", "b47b05f767b9b7c170f441a0b298973d": "-\\sqrt{\\frac{9}{20}}\\!\\,", "b47b3f5531fbf62a3a6db5074de7d5d4": "s = \\frac {m + 1} {m} f", "b47b68c629bcd1b61fad554f2adb0586": "\\Gamma_i", "b47bb4915bb607854a7dac5dc3a24be4": "F_n+G", "b47bb9d90548a8df5469e2251d75d182": "\n= \\sum_{ij} \\psi^*_i \\psi_j \\varphi^*_j\\varphi_i= \\sum_{i} |\\psi_i|^2|\\varphi_i|^2 + \\sum_{ij;i \\ne j} \\psi^*_i \\psi_j \\varphi^*_j\\varphi_i\n", "b47bccde348ea8dd0b7824eeb3c681e7": "\\mu_x", "b47bddc8e3df384ef2ea67e83472ee4c": "(k,\\,k) \\ \\stackrel{\\mathrm{def}}{=}\\ \\langle k,\\,Jk \\rangle = \\pm \\langle k,\\,k \\rangle = 0", "b47c02257f87c90967ee9530a9af0f2f": "g_3(p) = g_2^p(1)", "b47c0d70ab3ff9847f3dfdf34ffc6cfb": " d_L(z) = (1+z) d_M(z)", "b47c3e6593ddc14aaba20983acc81f21": "\\psi(\\alpha) = \\zeta_0", "b47c43c367d5c83b506e3d0d42802e0c": " i = \\sqrt{-1} ", "b47c4801c54838dc2a916c2f255d1310": " \\mathbf{k}", "b47cebf461715ee4b7baf5fd41395415": "S = \\int \\frac{\\mathcal K}{T}\\, dT. ", "b47cfb84a40079e0a7675225ea1c24cf": "n - r", "b47d021a8ef6accd20db5d38632ce4bd": " p = {\\partial L \\over \\partial \\dot{q} } \\,", "b47d560f6ce84ba6ff1a978aa6aa75ad": "\\sqrt{A^2+D^2}", "b47da8ee0401fe06ba3217c6a1d172f9": " \\hat H (a\\psi_1 + b \\psi_2 ) = a \\hat H \\psi_1 + b \\hat H \\psi_2 = E (a \\psi_1 + b\\psi_2). ", "b47de9a757fbc98f0956aaae44a12bdf": "Cl^-\\big(4M\\big) \\big| Hg_2Cl_2 \\big(s\\big) \\big| Hg\\big(l\\big) \\big| Pt", "b47df78feb9149d5a9ace342fe7eeccc": "c_n = \\frac{1}{a_0}\\left(b_n - \\sum_{k=1}^n a_k c_{n-k}\\right).", "b47dff7010b2f031f41d3f8056d48af3": "\\left(\\frac{P}{S}\\right)_\\text{ant} = \\frac{1}{2}c \\varepsilon_0 E_\\theta^{\\,2} \\simeq \\frac{E_\\theta^{\\,2}}{240\\pi},", "b47e774584945983245c337da77aef48": "H^2 \\cdot \\left ( \\frac {d^2s} {d\\theta^2} + s \\right ) = \\mu", "b47e7c4e2b16e9217a007cb2984149b6": "M_{10}", "b47f1a763f96fd2c0e4a982b62270ef1": " \\mathbf{u}_k^{(i)} ", "b47f4a52c63cc9ff6f51576116e93ca8": "\\frac{|x|}{|C(x)|}", "b47f4d013503c82f6201cc955b22652a": "k0} \\tfrac{1}{h} (K-x)}.", "b49dd1566f5caed0973c19c86dd93deb": " (I-2| \\omega\\rangle \\langle \\omega|)|x\\rang=|x\\rang-2| \\omega\\rangle \\langle \\omega|x\\rang=|x\\rangle = U_\\omega |x\\rang ", "b49dd5fbba77c04434d9f7e1ac42e8e5": "x_{P+1}=x_1", "b49e2259d284cb10aeb04610d2e409cd": "w^{||}{}_i=\\nabla_iA", "b49e6490e5fe841a61eec12e18d9adc4": " m_1 \\ne m'_1 ", "b49e6b2ce02c6cb5dc3b31cd3e1e2649": "\\hat{l}\\,", "b49e97d4c8857505eb99f070178cb19a": "FWER=Pr\\left\\{ k\\geq i_{0}\\right\\}", "b49e988a23052db6efb40a4ad197b1c0": "Y_t=X_{t}-m \\text{ for } t=1,2, \\dots ,n \\, ", "b49ec529ea50f42ac059de127b31ec15": "Range \\ Limit = 0.5 \\ c' \\ t_{radar} ", "b49ef929564fea3bef02acbb33e3b49a": "\\Pr(S;r_1,r_2,\\ldots,r_s,0,\\ldots,0)", "b49f4cb6e5f31d15cb98a87814e051aa": " {1\\over p^2+m^2} = \\int_0^\\infty e^{-\\tau(p^2 + m^2)} d\\tau ", "b49f4ddfe70dcb566cbacc9ff5a08dbb": "\\Phi(x_1, x_2, \\dots, x_n) = \\Psi(x_1)\\Psi(x_2)\\cdots\\Psi(x_n).", "b49f82b760b828a761dccdccd723bff3": "(i+j)/2", "b4a00e509a2ad349a2ad3e495bf616a2": "X_2\\rightarrow 2X", "b4a03fc5908b5802be7486ba506c0a71": " \\lambda =\\begin{matrix} \\frac {1}{V_E L} \\end{matrix} ", "b4a0d3adc8189adce651bb05fb689cd0": "S_\\mathrm{BET} = \\frac{S_\\mathrm{total}}{a}, \\qquad (6)", "b4a0f5b8d0f12406406e8a8f901213c6": "P(Q-Q')", "b4a112666c0f2f19e0cb65880aefe351": "\\ln (2)", "b4a1a121ee7fe86368be661bba118368": "R_{ext}", "b4a1cd0c10fd19ebce223337eca4f397": "\\pi(x)P(x\\rightarrow x') = \\pi(x')P(x'\\rightarrow x)", "b4a1e93ecd8b7f30fafdcab7bd436a91": "S_k(I)", "b4a202118e0b2ca20ba87569ce6ce92e": "|{\\psi}\\rangle=\\sum\\limits_{\\alpha,\\beta,\\gamma=1}^{\\chi}\\sum\\limits_{i,j=1}^{M}\\lambda^{[C-1]}_{\\alpha}\\Gamma^{[C]i}_{\\alpha\\beta}\\lambda^{[C]}_{\\beta}\\Gamma^{[D]j}_{\\beta\\gamma}\\lambda^{[D]}_{\\gamma}|{{\\alpha}ij{\\gamma}}\\rangle", "b4a20407d8d225c2e9dffb16b943f9cb": "\\mu_r = \\mu / \\mu_0 \\approx 2000 - 6000\\,", "b4a23df6a56e7f4fe98c93924d9b43a8": "L \\ln{(1-q)} \\approx -Lq > \\ln{(1-s)} \\approx -s", "b4a273be56587f67fd99de24b44cf7f9": "S(\\mathfrak{g})", "b4a2752b15ecebbdf24f20abf187a3d2": " \\frac{f''_x(x)}{f_x(x)}+ \\frac{f''_y(y)}{f_y(y)} + \\frac{f''_z(z)}{f_z(z)} + k^2=0 ", "b4a2b427ccec3c3902f82892f67942d4": "(u|v)=\\int_a^b\\! u(x)v(x)\\,dx,", "b4a2bd4fb97d5a33297635ea8bcae917": "\n\\langle r^2 (t)\\rangle \\sim \\langle r^2 (t) \\rangle_\\text{nrml}^\\alpha. \t\t\t\t\t\t\n", "b4a43e7e0d50a0ebd610e2dff5d50b33": "Z[-1] \\xrightarrow{-w[-1]} X \\xrightarrow{u} Y \\xrightarrow{v} Z.\\ ", "b4a441e696b2e02ed4b9be8691e7ba3f": "M(X) = I - H(X)", "b4a471bcfee1881e424a8d65561d60ce": "c_n \\,\\!", "b4a4a97f3eaff15415cfa155eebd6bcd": "\\dim(\\ker(T)) > 0", "b4a4b43da4cbd76817bbf8b40572da82": " \\nabla^2 \\Psi = {n_{0} e \\over \\varepsilon \\varepsilon_{0}}\n \\left( e^{e\\Psi (x,y,z)/k_{B}T} -\n e^{-e\\Psi (x,y,z)/ k_{B}T} \\right), \\; ", "b4a4b79a5edb1ddc487acf6f8b86cc57": "f(y) = f(x) f(y - x) \\neq 0", "b4a4b952dc983fdda39db7716dafe21e": "|H\\rangle, |D\\rangle, |V\\rangle, |A\\rangle", "b4a51c44a1d63949b1c4fa05b5a0de7e": "\\mathcal{R}_\\mathcal{R}(C^{wnn}_{n}) - \\mathcal{R}_{\\mathcal{R}}(C^{Bayes}) = \\left(B_1 s_n^2 + B_2 t_n^2\\right) \\{1+o(1)\\},", "b4a55b1fd55706b7c7fd191d56e1f8a2": "\\,f(x+1)=xf(x)\\,", "b4a560ce93dfddf9f0e9f27ba10db3df": "s_y(t) = -\\frac{mg}{k}t - \\frac{m}{k}(v_{yo} + \\frac{mg}{k})e^{-\\frac{k}{m}t} + \\frac{m}{k}(v_{yo} + \\frac{mg}{k})", "b4a56b04dc317b73411509704819a078": "\\Phi_{15015}(x)", "b4a5c8a9fb5300d4131fb372f44ab18f": "1/2^{n-1}", "b4a6665159a5c9d953cf58a02aeccaa1": "\\sin{\\left(\\alpha\\right)}\\approx\\tan{\\left(\\alpha\\right)}", "b4a68a1a0f7da433f52c3d9a254c9fca": "G=ve^2N/LE", "b4a6d3ee3558c5fb543b94814ac7b330": "\\partial\\Phi/\\partial{t}", "b4a6df6990e642311027aedadf4fa65c": "\\nabla \\cdot (u,v) = \\rho,", "b4a709d281d7b03866dcbdc5ee767144": "A(\\mathbf{D}) = H^\\infty(\\mathbf{D})\\cap C(\\overline{\\mathbf{D}}),", "b4a75d9f396d3ee58b74efd210a99cfa": "w = \\tfrac12 \\Big(1 + \\tfrac32 z \\pm\\sqrt{1 + z + \\tfrac94 z^2}\\Big), ", "b4a774c4bb6485ff89c3ec1d714546ab": "\\left( R = \\frac{b^2}{a} \\right) ", "b4a795b0c431898253a6e979f606c35b": "= \\int_0^L{\\frac{Pl^2}{EI}dl}", "b4a807930c85239e73e47a9410159bdd": "72=\\tbinom85+\\tbinom64+\\tbinom33", "b4a81b4c5a20b5f2db0d322e8d1bd10b": "\\begin{bmatrix} 1 & 1 \\\\ F_1 & F_2 \\\\ P_1 & P_2 \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} \\le \\begin{bmatrix} L \\\\ F \\\\ P \\end{bmatrix}, \\, \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} \\ge \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}. ", "b4a88417b3d0170d754c647c30b7216a": "result", "b4a885f48e9437f5c594e9fd2def86bd": " a_1,\\ldots,a_k,\\quad 1 \\leq k \\leq n", "b4a89d0d7d530a97218ae50547894393": "F = m a = m \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} = -k x. ", "b4a97854cd363edeba28316f68c7ace4": "G(x, y, u, v) = 0\\,", "b4a9ccf993bec2f5e96a58cb46f8c456": " \\scriptstyle \\sqrt{3} ", "b4a9f5825b7e4f93e297f816c63580a2": "\\sgn(x)", "b4a9f9b41a298f1eb63cf9b81ceb09ce": "\\begin{align}\\varphi(s)&=\\sum_{k=m}^\\infty f(k;m,r,p)s^k\\\\\n&=\\frac{(1-ps)^{-r}-\\sum_{j=0}^{m-1}\\binom{j+r-1}j (ps)^j}\n{(1-p)^{-r}-\\sum_{j=0}^{m-1}\\binom{j+r-1}j p^j}\n\\qquad\\text{for } |s|\\le\\frac1p.\\end{align}", "b4aaafb753b05f6907754259d3666085": "\\tan(\\alpha) = tdw/dx ", "b4aac4a510f9f1a9a31400b0793caf16": "\n \\mathbf{M} = -(\\mathbf{r}_X-\\mathbf{r}_A)\\times\\mathbf{F} - \\mathbf{r}_X\\times\\mathbf{R}_O = \n \\left[(x_A-x)\\mathbf{e}_x\\right]\\times\\left(-F\\mathbf{e}_y\\right)\n - \\left(x\\mathbf{e}_x\\right)\\times\\left(R_O\\mathbf{e}_y\\right) \\,.\n", "b4aaed0cad77922de5d336efdc463624": "\\Phi_{bh} = \\frac{1}{2}[\\bar{E_C} - \\bar{E_V}] + \\delta_m = \\frac{1}{2} [\\bar{E_C} - E_V - \\frac{\\Delta_{so}}{3}] + \\delta_m", "b4aaff74c5dc6b809d721353f18b3c23": " C_{pq}^{mn} = \\frac{1}{2}g^{ma}g^{nb} \\epsilon_{abpq} \\sqrt{-g} ", "b4ab1e7588b3c05e6a6b30f0d106948b": "10x^3+y^3+1=15xy", "b4ab215883fb505586253857e655fd49": " x_5=\\frac{1+x_4}{x_3}=\\frac{1+x_1}{x_2},\nx_6=\\frac{1+x_5}{x_4}=x_1,x_7=\\frac{1+x_6}{x_5}=x_2,\\ldots", "b4abbcbfe681b2f893c425c4d061c880": "O(\\ell!2^\\ell n)", "b4abd1a533f11b97c8a1f1899369e886": " f(r)= 1 - b r^2 \\,", "b4ac0fcc7f393c76e3d0178f3a0e66d4": "J_-|j\\,m\\rangle = \\beta|j\\,m-1\\rangle.\\quad", "b4accbdefaef0b8b9af349b348592644": " X_\\alpha \\preceq X_\\beta ", "b4acd643874b29aa12d1de7fb9d03127": " 0 = \\int_\\gamma g(z) dz = \\int_{-R}^R \\frac{e^{ix}}{x +i\\epsilon} dx + \\int_{0}^{\\pi} \\frac{e^{i(Re^{i\\theta} + \\theta)}}{Re^{i\\theta} +i\\epsilon} iR d\\theta ", "b4ad087ef85dfbf2f21e92dad4a792eb": "x=e(a(x,t),b(x,t))", "b4ad0bcd2a9cba90cf40f1a225ef6765": "\n E(\\mu) = \\sum_{j=1}^N \\sum_{i=0}^M \\mu_{ij} Q_{ij}\n", "b4ad485cd57b2677023a2900a075b27b": "m_N - m_{N-1} + \\ldots \\pm m_0 \\geq \\beta_N - \\beta_{N-1} + \\ldots \\pm \\beta_0", "b4ad80684ed94b4380f0d33ba54b91da": "N_{i}", "b4ad964a1d5d7773bbd709668cc04728": "2=\\{H,T\\}", "b4ae0274471448ed7212169454f485ea": "K_n(x)", "b4ae732b84ab369c4310e928eceee49d": "x^x", "b4ae75602cf68f8c16539b1a5acc19d7": "E_{tot} = E_{inc} + E_{scatt} \\ ", "b4ae82e9c3f32fc8353d353a8099de92": "A^f", "b4aea502775ddf89174c79907aae6d04": "\\sum_{i}^{}{\\Delta H^0_i} = \\Delta H^0 ", "b4aec90a452193e7e21f8a7501708c6f": "MPI = H \\times A", "b4aeedb3f9d2f3e8878f7ea4045696b3": "x'^2 + y'^2 + z'^2 = c^2t'^2", "b4af28d95eb82c8d3e2530f220ab935d": "dy=\\dot { y } dt", "b4af7bdd73fad574f6542ab293e7b885": "8119-5741\\sqrt{2}=-0.00006\\ldots", "b4af9695779e4224862e5ed9065bb4d9": "\\phi _0\\,", "b4afa1d2b2488f73695d60302a0a616b": "t \\in \\{ 0, 1, \\dots, N \\}, \\mathbb{N}_{0}, [0, T] \\mbox{ or } [0, + \\infty).", "b4afba90a745393ed04564c6f556e87e": "t_{diffusion} = \\frac{^2\\pi}{4D}", "b4afbdece947d0310b7921f8a6ffe54b": "(f):=\\sum_{z_\\nu \\in R(f)} s_\\nu z_\\nu", "b4afbe61d4833bef257ca285e3f37a28": " u_i(s^\\prime_i,s_{-i})> u_i(s_i,s_{-i})", "b4afd08834664281d6b74a3bf35e8a9c": "\\begin{pmatrix} x_a \\\\ x_b \\end{pmatrix} = \\begin{pmatrix} \\cos \\theta & \\sin \\theta \\\\ -\\sin \\theta & \\cos \\theta \\end{pmatrix} \\begin{pmatrix} x_1 \\\\ x_2 \\end{pmatrix}", "b4afe193b2dfbead2372207b5bfcfd22": " \\!\\ f(x) = x^{x^n} ", "b4b00ac692e6bffe6d3feba0d00e160b": "\\operatorname{E}\\left[ X^n\\right] = i^{-n}\\, \\varphi_X^{(n)}(0) = i^{-n}\\, \\left[\\frac{d^n}{dt^n} \\varphi_X(t)\\right]_{t=0} \\,\\!", "b4b069ed276abf905a43b084a6dedef7": "\\frac{1,340,000\\ \\mathrm{N}}{(1,686\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=81.04", "b4b0bcb0abcb405b279851a0d26a6a2b": "y = (K-W)/2.", "b4b0c53769991646580d0a68dbd6b5b1": "\\left( {}^1\\!\\!/\\!{}_2 + {n}\\right) \\cdot \\pi", "b4b0ef4dbd8411e6c18b097e1b26acf2": "y_2=\\frac{1}{x^2}", "b4b109b194892a78dcf5c500ab233275": " \\overline T = T^{**} ", "b4b147bc522828731f1a016bfa72c073": "00", "b4b19f2b8e1145203692973c75278b3f": "\\ln x = \\ln a + \\int_a^x \\frac{1}{u}\\,du", "b4b1d228434c31ff5b98e3b213c8c126": "PL=10log (I_s/I_r)", "b4b1e54761c254a3b9f8f5d4b33ad79d": " \\gamma_p'(t) = X_{\\gamma_p(t)}, \\qquad \\gamma_p(0) = p, ", "b4b21a61ef310a97feb59f77124820b1": "\\sigma \\in B^{\\ast}\\,", "b4b23e2166ee0237d543332886296dea": "\\dot{\\textbf{x}} = A\\textbf{x}, \\qquad \\text{where } A = \\frac{df}{dx}(x^*). ", "b4b2a18ed632e7f3434cfa4f2097db7a": "\\, t_2>t_1\\!", "b4b2ae7315e82a6690adff3b61c3a6e4": "\\Gamma\\,|\\!\\!\\!\\sim B", "b4b35c52ea1d178b6cf6171f902e072f": "O(1/n)", "b4b3868777f2fb150e27c7599a7dce01": "H_{X_k}", "b4b3879cb35fa7f1ecd46bd580676cdb": "(x)_r", "b4b3887e499a6f95cc9c316586382f07": "\\gimel(\\kappa)", "b4b39e4620aa52f75e31a22fc3119c55": "|z_1|=|z_2|=|\\overline{P}|+1", "b4b3a1c8b50d7d1a59f3f5afde4d4e04": "\\Omega = (G M)^{1/2} r^{-3/2}", "b4b3cfdb9c6ac55693f1484c939b1ca8": "1-\\frac{V(t)}{V_0}=e^{-\\frac{t}{\\tau}}", "b4b3d6156f258511e0e10215bc79ebf7": "{{B}_{x}}(v)", "b4b40bafacd66ea6b116de530a0deae0": "f(x):=(-1)\\cdot g(x)", "b4b42a3e11a6efb4ee53e3a098e7b1fc": "\\frac{\\mathrm{d} x_i}{\\mathrm{d}t}= F(x)-V(x)", "b4b4ad5a543e85af49d736b14290d808": "\\operatorname{let} p = \\lambda f.\\lambda x.f\\ (x\\ x) \\and q = \\lambda p.\\lambda f.(p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p ", "b4b4add87d4c6e5748cada1eb82b4b6f": "R_\\theta", "b4b4ec070037b907e6e3a5c1dbd2fe66": " \\lambda^2 \\,-\\, \\mathrm{tr}(A)\\,\\lambda \\,+\\, 1 \\,=\\, 0", "b4b4f11522008be3a0d73351d6969a22": "g \\circ f = \\mathrm{id}_X . ", "b4b506f464b5ebe379d3b59013a98eef": "\n \\begin{cases}\n \\displaystyle \\frac{(2\\pi)^{n/2}\\,r^n}{2 \\cdot 4 \\cdots n}, & \\text{if } n \\text{ is even}; \\\\ \\\\\n \\displaystyle \\frac{2(2\\pi)^{(n-1)/2}\\,r^n}{1 \\cdot 3 \\cdots n}, & \\text{if } n \\text{ is odd}.\n \\end{cases}", "b4b5172740464f53e47b365012cbb515": "\\lbrace\\Phi_j\\rbrace", "b4b53f761cdc87e12f9d05fd3e3fe286": " \n\\Delta(t) \\leq B \\,\n", "b4b58d5fa01cad57f8d9551623f86423": "(M_p \\times F)/Homeo(F) = M", "b4b5c0ca9b2271d664e36a80a316fd4e": "\\frac{{\\mathcal C}_{n+1}(x)}{{\\mathcal C}_n(x)} = \\cfrac{1}{n+1 + \\cfrac{x}{n+2+\\cfrac{x}{n+3+ \\cfrac{x}{\\ddots}}}}.", "b4b5c2fd239b1f4cf161620566f94813": " \\theta_1, \\theta_2 > \\theta_{ 12 } > 0 \\, ", "b4b5e90b0c27ca848caacf5c155b3a05": "0\\rightarrow K(Z,2)\\rightarrow \\text{String}(n)\\rightarrow \\text{Spin}(n)\\rightarrow 0", "b4b61d69cb6c7e0fd8c5f5e29ff5be08": "C^\\infty(X,Y)=\\bigcap_k C^k(X,Y)\\subset \\mbox{Hom}(X,Y)\\,", "b4b6697d24f51347740aaf3918ae47a9": "\\angle PBN + \\angle ACP = \\angle PBA + \\angle ACP = 180^\\circ", "b4b6ab053f977fff88b0f6d49ef93417": "f(x,t) = \\frac{1}{2\\pi} \\int_{\\mathbb R} F(\\omega) e^{i [k(\\omega) x - \\omega t]} \\, d\\omega", "b4b6e894569917b9ca2210bae107c888": "LX", "b4b6f1bbb8e00c9db240414a3b0ccc65": "R(v) - r(v)", "b4b7215932f559fc4f1efe4da1342d5f": "C_\\bullet", "b4b72658ce06f123f8a2a77bc5a40f6b": "f \\mapsto \\hat{f}\\,", "b4b7891ccbdca5478a5c38022a5df3de": "\n\\int (A+B\\,x) (a+b\\,x)^m (c+d\\,x)^n (e+f\\,x)^p dx=\n \\frac{B(a+b\\,x)^m (c+d\\,x)^{n+1}(e+f\\,x)^{p+1}}{d\\,f(m+n+p+2)}\\,+\\,\n \\frac{1}{d\\,f(m+n+p+2)}\\,\\cdot\n", "b4b7ac5ecdad1a5fdabbc8597c0b0ee8": "\\sum_{t=1}^{\\infty}\\delta^t 35=\\frac{\\delta}{1-\\delta}35", "b4b7cdc339bf8e28873b3f6573706dc8": "M_A(\\alpha)+M_A(\\beta)=0 ,", "b4b7fa3207aa5d9fc11207a446c9ad8d": "\\gamma={\\dfrac{Mt+1}{MN}} \\approx {\\dfrac{t}{N}}", "b4b7ff67cdbc3c18717f019066c8a02f": "\\frac{\\omega^2}{k^2}=c^2\\,\\frac{v_s^2+v_A^2}{c^2+v_A^2}", "b4b814b5cf195681eaf0889a1a51bc0a": " m(x) ", "b4b8325d5a582e2cfaa553ff14168399": "{n \\choose k} = (-1)^n n! \\cfrac{\\sin (\\pi k)}{\\pi \\prod_{i=0}^n (k-i)}.", "b4b83dba59ae4c71b9e85c8053a18eaa": "x \\wedge y \\wedge x = x ", "b4b8e24d57fd7a9fcca45bc7bfc1a9d7": "10^n \\cdot y + z,", "b4b8e9385d05b35b7f5222688f84dd5d": "V=\\frac{5}{12}(11+5\\sqrt{5}+6\\sqrt{5+2\\sqrt{5}})a^3\\approx16.936...a^3", "b4b913ade688bde49cdbbcd2274c157b": "{\\vec c}, y", "b4b944bf64fa264153ed2aa121306f06": "x^4 + 1", "b4b9ac99b5673095cf9139c7539fb576": " K_1(R)", "b4b9bda428763a72645b8e5caa867d02": " \\sigma_y = \\frac{1}{\\alpha} \\sinh^{-1} \\left [ \\frac{Z}{A} \\right ]^{(1/n)} \\,\\! ", "b4b9c33a024467163d9b5646289be203": "z\\mapsto \\lambda z(1-z),\\quad \\lambda\\in[1,4].\\,", "b4b9d3a927b2647eae824a687ded8892": "L' = L \\, \\sqrt{1-v^2/c^2}", "b4b9e56c16cecd265ee99e8f8894b5ed": "\n|\\psi(t)\\rangle = \\exp\\left({- {i \\over \\hbar } \\hat H t}\\right) |q_0\\rangle \\equiv \\exp\\left({- {i \\over \\hbar } \\hat H t}\\right) |0\\rangle\n", "b4b9fce855ee66e7860f797e4197e262": "2X(X^2 + Y^2) = a \\sqrt{2}(3X^2-Y^2)", "b4ba913d69272ea8829b381ba60e07dc": "(A,C)\\rightarrow (X,B) \\, ", "b4ba935f896f3cd01c616de37f8c931a": "-\\infty < t,z < \\infty, \\; 0 < r < \\infty, \\; -\\pi < \\phi < \\pi", "b4baa64b4124d4b122bf353a83f053ab": "\\sum_{j=1}^r(r-j)\\delta(\\alpha_j).", "b4babaffb3671ceaaa08cec89bdd93f1": "i=0, \\dots , \\infty ", "b4bb178f16b2b0689a3aa7bef70fc988": "\\!\\,\\gamma_2 : J \\rightarrow X", "b4bb2ff1e8bf4071d9f268ab7a1f04de": "(\\widetilde{c}_i)_{i\\in I}", "b4bb8b6d596628c43e0d1c626494773f": "\\mu_3(X)=E(\\mu_3(X\\mid Y))+\\mu_3(E(X\\mid Y))\n+3\\,\\operatorname{cov}(E(X\\mid Y),\\operatorname{var}(X\\mid Y)).\\,", "b4bb8f9ab1f431c06161d7ba9db5d4ea": "\\,\\rho = 0", "b4bb93de26e26a007a7ae5f4fde1360c": "x = \\sum_{k \\in B} \\, \\langle x, e_k \\rangle \\, e_k. ", "b4bb9b3d210eefdbd2080df54f6ffa52": "\n\\dot{\\mathbf{p}} = -\\frac{\\partial H}{\\partial \\mathbf{q}}\n", "b4bbb4d83d34a2b4d18e88ffd16846a6": "\\textstyle\\ f_{max} ", "b4bbb7efbfd459db800e5e959fcf968d": " 10 ", "b4bbc4ffd3fb9462f6e814d960e6c211": "\n\\alpha^4 = \\alpha^3 \\alpha = (\\alpha^2 + 1)\\alpha = \\alpha^3 + \\alpha = \\alpha^2 + \\alpha + 1\n", "b4bbdfee0e587eb58d7a4bb30c7e2777": "\n\\sigma_{-s}(1)+\n\\sigma_{-s}(2)+\n\\dots+\n\\sigma_{-s}(n)\n", "b4bbe80d2f76ce3b68e380e663c55ca0": "\\gamma_{xy} = \\Delta x/l = \\tan \\theta \\,", "b4bc3dd9d72ccf5bee765c74a6d694a5": "D^2F(u)\\{h,k\\} = \\lim_{\\tau\\to 0} \\frac{DF(u+\\tau k)h - DF(u)h}{\\tau} = \\left.\\frac{\\partial^2}{\\partial\\tau\\partial\\sigma}F(u+\\sigma h + \\tau k)\\right|_{\\tau=\\sigma=0}", "b4bc52e27e5b729bc89d6921e32f4657": "\\frac{dz}{dt}=-\\beta y", "b4bcc41e051915534afd8389de6d4adc": " r_1,r_2 \\in Z_{q}^*", "b4bd3191ac7c399c255b7f03d908558b": "C = \\mathbb{N}", "b4bd5799f3e431d35abbee93c3bf0d05": "\\kappa = \\frac{2\\pi e^2}{\\epsilon} \\frac{\\partial n}{\\partial \\mu}", "b4bdac4d98dea99a6997ac385e355a48": "\\dot{x_2}(t) = - \\frac{g}{l}\\sin{x_1}(t) - \\frac{k}{ml}{x_2}(t)", "b4bdbc86cca534e405be9bc583978096": "\nf(\\vec{x}) = \\sum_{i = 1}^{m} \\tfrac{1}{c_{i} + \\sum\\limits_{j = 1}^{n} (x_{j} - a_{ji})^2 }\n", "b4be199cf758f80d944244c3ff0d565e": "\\tbinom{p^r}{s}", "b4be4aef5bdb4e33a59f531920bebedb": " e^{-\\beta(H_\\nu-H_\\mu)}.", "b4be690b796ff090e1c1ed31a0531be1": "\\mathbf{d}_{nm} = \\int \\Psi^*_n \\,\\, \\mathbf{\\hat{d}} \\,\\, \\Psi_m \\, d^3r = \\langle\\Psi_n|\\,\\mathbf{\\hat{d}}\\,|\\Psi_m\\rangle", "b4be6a44d849cab4127c2f96a1bcf7b4": "e=p_0(p_1p_2p_3)^*(p_4p_5p_6)", "b4be765182da3f69bf00ce524e99893c": "\\mathbf{B}(\\mathbf{A} \\vec{x} ) = (\\mathbf{BA}) \\vec{x}", "b4bebc8b1f4ef9057c77e2c860b00123": "b= d\\,\\left(\\frac{\\nu}{\\mu}\\right)^{\\frac{1}{3}}\\,\\!", "b4bec1a163ae8e218e5570f4872dfc23": "p_n(z)= {z \\choose n} = \\frac{z(z-1)\\cdots(z-n+1)}{n!}.", "b4bec3b5059077915adfef789780d33e": "\\nabla\\times\\hat{\\mathbf{B}} = \\mu_0 \\hat{\\mathbf{J}} + \\mathrm{i}\\mu_0\\varepsilon_0\\omega \\hat{\\mathbf{E}} \\quad\\Rightarrow\\quad -\\frac{B^r}{r}+\\frac{\\mathrm{d}B^{(1)}}{\\mathrm{d}r}+\\frac{B^{(1)}}{r} = \\mu_0J+\\mathrm{i}\\omega\\mu_0\\varepsilon_0E", "b4bee4aa2931acb9477864624d7624ef": "Q_{ij}=\\sum_l q_l(3r_{il} \\cdot r_{jl}-r_l^2\\delta_{ij})", "b4bf318b4a05dcd44090fe31622ae8bd": "\\phi_s\\geq0 ", "b4bf50f41725a8db242752933606f4f7": "\\operatorname{rev}(X)", "b4bf6d4856e5f8a169930a86ac673104": "S\\circ R\\subseteq X\\times Z", "b4bf8f88b6ed3da20c8d535d4200c502": "\\langle x,cy\\rangle=\\bar{c}\\langle x,y\\rangle.", "b4bfb5d4c668da442ea77900f788c78c": "A\\times B=\\{a\\times b|a\\in A, b\\in B\\}", "b4bfc25d2b91f9b257af483cf5669406": "\n\\begin{align}\n H(\\nu) = \\mathcal{F}\\{h\\} &= \\int_{\\mathbb{R}^n} h(z) e^{-2 \\pi i z\\cdot\\nu}\\, dz \\\\\n &= \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} f(x) g(z-x)\\, dx\\, e^{-2 \\pi i z\\cdot \\nu}\\, dz.\n\\end{align}\n", "b4bfd89dba39af360e9880b46abedcca": "I_3\\,\\!", "b4bffb219415530412ba360274f620fa": "C_L(1-k) \\ df_S = (1-f_S) \\ dC_L", "b4c0269d7fb1f53867dd3ab8b285ac90": "\n (2.3)\\quad\n(\\mathcal{A}f)(x) = f'(x) - x f(x)\n", "b4c0a437a0b8204de141f1d962bcdc2b": " = \\int_0^\\infty G(\\tau,\\xi)F(t-\\tau,\\xi)\\,d\\tau ", "b4c0b472ea4160b12d176e86faa8f62b": "\\left(\\!\\!{n\\choose k}\\!\\!\\right)=(-1)^k{-n \\choose k}.", "b4c0b9132c3575f65b9d566e1abfeb83": " \\quad 0 < 4 - 2\\alpha - \\beta ", "b4c10a8b89678141e0c5da38bed6ab67": "P_1=(X_1:Z_1)", "b4c11ec7e320e0ddfacbdbe90442e2e2": "TN^* = (NT)^* = (TN)^* = N^*T.\\,", "b4c12e9fdfb0f756a266c23c64790051": "\\begin{align}\nt' &= \\gamma \\left( t - \\frac{vx}{c^2} \\right) \\\\ \nx' &= \\gamma \\left( x - v t \\right)\\\\\n\\end{align}", "b4c130adabaa9389bbc3098f3dab6ce8": " NL\\,\\!", "b4c1317909a771b1b257de320ec85517": "\\mathbf{H} = \\sum_{i=1}^N {p_i^2 \\over 2m} + {1\\over 2} m \\omega^2 \\sum_{\\{ij\\} (nn)} (x_i - x_j)^2\\ ", "b4c1a0f6e1e0545cf711b07cf83d2c57": " \\mathcal{O}_v=\\begin{bmatrix} C \\\\ CA \\\\ CA^2 \\\\ \\vdots \\\\ CA^{v-1} \\end{bmatrix}.", "b4c1a9f7db63fc5e8f8e0bda6f7e40ab": " \\alpha \\preceq \\beta ", "b4c1aa7346dff3987f2b0dceb89321c0": "r, s, t, \\ldots", "b4c21a07bd71871beacf81a2f6816a46": "P(x_{1}+x_{2})", "b4c26c530ffbe7d04dbbeca331112a2d": "\n\\{S\\} = \\left [s^E \\right ]\\{T\\}+[d]\\{E\\}\n", "b4c289759094992509c0bef9b74cd280": "t+\\tau", "b4c28c023dde554acbfd1aabf9dd43b8": "B(\\omega) \\subseteq X", "b4c28fb206c3c1cd16f3222a1ef998e5": "\\times 2,\\times 3", "b4c290f26d038a9c6fa3be0b472128c6": "T_c = \\sqrt{\\frac{9}{16 \\pi}} \\frac{1}{f_d} \\simeq \\frac{0.423}{f_d}", "b4c2a6bb6a0ec0d9099caa312a70ebf7": " \\operatorname{E}[\\,\\text{gain from }$1\\text{ bet}\\,] = -$1 \\cdot \\frac{37}{38}\\ +\\ $35 \\cdot \\frac{1}{38} = -$0.0526.", "b4c2bce11c1744628b30207aa02acd17": "\\scriptstyle \\{x\\},", "b4c2e12c4724a6386b11a5b8235585cf": "\\sigma=\\tfrac{1}{2}(uvx+vxy+xyu+yuv)", "b4c3146c9bf2990f8578c1dbe09cfcb4": "\\mathbf{J}_\\mathrm{b}=\\mathbf{J}_\\mathrm{P}+\\mathbf{J}_\\mathrm{M} ", "b4c3a337202d604b8effee1d4d5bbb63": "X_n \\ \\xrightarrow{\\!\\!as\\!\\!}\\ X \\quad\\Rightarrow\\quad g(X_n)\\ \\xrightarrow{\\!\\!as\\!\\!}\\ g(X).", "b4c3d7f1eb9f6396041c05db1df6c35f": "\\bar{X}_n", "b4c3ebc7168383aae0242e43e78c581e": "\\ C=\\frac{c}{2^7\\pi^3}\\frac{P_tG^2\\lambda^2}{R^3}\\tau\\theta\\sec\\psi\\sigma^o", "b4c42f5f85bc08b39aed5b8c9bc0a7ac": "\\int_{-\\infty }^{+\\infty }p_{i}(t)dt=1", "b4c45bd5a39a4c4fb58ea25350b4923f": "\\frac{dC_{B}}{dt}=(k_{1}C_{WD})-(k_{2}+k_{E}+k_{M}+k_{G})C_{B}", "b4c484c8301f24fdd320f088a06dd8b3": " h^0(X,L)", "b4c4967e1de1e8cdba27501a2fd4c0ad": "H_c=-\\int p(x)\\log\\frac{p(x)}{m(x)}\\,dx.", "b4c497bb27e462db431e3806037b36fc": "GF(4)=\\{0,1,\\omega,\\omega^2\\}", "b4c4a5f8d12104b9a64943f225d5d02f": "G=(V=(X,Y),E)", "b4c4d53110bd987dfd8223a8629ac4da": "\n\\operatorname{Li}_s(z) = \\sum_{k=1}^\\infty {z^k \\over k^s}.\n", "b4c4e73d31b676c507cc86ab3f011b33": "f_4", "b4c4feecef97770f88bec2a39d4f9c45": " |V\\rangle ", "b4c52d813dcdc0237be7cf4d5f7b08c2": "W=W_{exp}-\\Delta V \\rho_{b}(p,T)", "b4c53db33bedeea127fd6d22b09514c4": "z=e^{i\\omega}.", "b4c54c9031781ad93b6e91b767e68773": "L:V(T)\\to\\{+1,-1,+2,-2,...,+n,-n\\}", "b4c5530a77b8f8c60cc1caaf1adc49e5": " \\mathfrak{p} ", "b4c58556f25b2f351400c956c48c1404": "CE = \\%C + \\left(\\frac{\\%Mn+\\%Si}{6} \\right) + \\left(\\frac{\\%Cr+%Mo+\\%V}{5} \\right) + \\left(\\frac{\\%Cu+\\%Ni}{15} \\right)", "b4c5951739d8b6a1fcfce67d15fea2fc": "\\mathrm{ber}(x) \\sim \\frac{e^{\\frac{x}{\\sqrt{2}}}}{\\sqrt{2 \\pi x}} [f_1(x) \\cos \\alpha + g_1(x) \\sin \\alpha] - \\frac{\\mathrm{kei}(x)}{\\pi}", "b4c5a06a970c265ffda68b5c43b64834": "P_1, P_2, \\ldots, P_8", "b4c5bf322323a071f0d2319176c70613": "\\cos\\theta=1/(n\\beta)", "b4c5e190e54726cb04170994a1818d52": "T = n \\omega^2 = \\frac{1}{12n} + \\sum_{i=1}^n \\left[ \\frac{2i-1}{2n}-F(x_i) \\right]^2. ", "b4c5e47b75fbe8576df9ddf184c0fd09": "\\chi_\\lambda, \\,\\chi_\\mu", "b4c5e501212ef8cf15aa28ea188a86aa": "\\scriptstyle s_iN/n_i", "b4c5f021f797d4f160ec61628d9aec3b": "(f \\otimes 1)(a \\otimes z) = b (g \\otimes 1)(a \\otimes z) b^{-1}", "b4c683bb3734fecce048da578b3aab93": " i^2 = (\\mathbf{e}_2 \\mathbf{e}_3)^2 = \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_2 \\mathbf{e}_3 = - \\mathbf{e}_2 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_3 = -1,\\!", "b4c6935eaf06fb7910ac5b9510e81900": "\\begin{align} \\mathbb P [ R \\leq \\rho, \\Theta \\leq \\theta] &= \\int_{R \\leq \\rho, \\, \\Theta \\leq \\theta} \\frac{1}{2\\pi}\\exp\\left({-\\frac12(z_1^2 + z_2^2)}\\right) dz_1dz_2 \\\\\n& = \\int_0^\\theta \\int_0^\\rho \\frac{1}{2\\pi}e^{-\\frac{r^2}{2}}r dr d\\tilde\\theta \\\\\n& = \\left( \\int_0^\\theta \\frac{1}{2\\pi}d\\tilde \\theta \\right) \\left( \\int_0^\\rho e^{-\\frac{r^2}{2}}r dr\\right) \\\\\n& = \\mathbb P[\\Theta \\leq \\theta]\\mathbb P[R \\leq \\rho] \\end{align}\n", "b4c6ddb06c63b2a5fdfbf9124727cd1e": "G \\subseteq SL(m,\\mathbb C)", "b4c6e0b50d44ab12eab12ef4ba567bfc": "k=rn", "b4c7435ef87862f11bc2c631951796d6": "k\\!", "b4c7a92d32c5278369554b7979877403": "\\Phi^{t}(x)\\,", "b4c84630b0e5b771b2c074a7b534767e": "2B \\log_2(M) = B \\log_2 \\left( 1+\\frac{S}{N} \\right) ", "b4c871bae409e79da77feeb8b885abd7": "\\kappa_2(V)=1", "b4c88899c55da40d2bf6b3cd60621cde": "A / \\operatorname{nil}(A)", "b4c8fc7ce034afb8018ced9be94f72ee": "\\scriptscriptstyle O(E\\sqrt{V})", "b4c94aa127c0a7298b73cc42303e3563": " V = \\sum_{j=1}^n V_j + \\sum_{j=1}^{n-1} \\sum_{k=j+1}^n V_{jk} + \\cdots + V_{12\\dots n}.", "b4c94e91f99ef1ef677e64a08ea14923": " (\\lambda x.x\\ x)\\ \\operatorname{get-lambda}[x, x\\ x = \\lambda f.f\\ (x\\ x\\ f)] ", "b4c98b428be4a4ec579bfd73cb4c1d9d": "X \\circ Y", "b4c9c3ea7b4b685ebefc011afd52dc78": "f^{64}(1)\\, ", "b4c9ebb84280e57487a4b48370c48187": "\\omega(x)", "b4ca1dafe9018cfa437f7a10e076a9bf": "\\zeta(3) = 1 + \\frac{1}{2^3} + \\frac{1}{3^3} + \\cdots = 1.20205\\dots\\!", "b4ca21873f3bc106e616cd58e1b2fc05": "p(x) = \\frac{e^{-x/V}}{V}.", "b4ca2f8dc780db0841a1b2e00f4da394": " s F(s) - f(0) \\ ", "b4ca59ffc853526a95d58a9a0c426898": "\\textit{VERB}", "b4cb17a85a61568050c140675c023f8f": " 2^{d}=O(n^{\\lfloor \\ell/2 \\rfloor})", "b4cb3ec0807bd7ea5025378f99428f71": "\n\\mbox{Capitalization Rate} = \\frac{\\mbox{annual net operating income}}{\\mbox{cost (or value)}}\n", "b4cbcd59edfc57d013fcf8629dc5eff0": "v_{k+1}=\\sum_{j=1}^k \\mu_j v_j+\\sum_{j=k+1}^n \\mu_j w_j.", "b4cbfeac895b884b90b588e7b0a78e30": " o_i ", "b4cc24ef0d6deb7c83cd873062ab9694": "f(x)=2x+3\\,", "b4cc29197668ab0a2e423c5e79dd0aea": "C = {\\theta_2 \\over x_1 } \\bigg|_{\\theta_1 = 0} \\qquad D = {\\theta_2 \\over \\theta_1 } \\bigg|_{x_1 = 0}.", "b4cce309cb6c90b9859dcc3795594935": "\n\\begin{align}\nD_{\\mathrm{KL}}(P\\|Q)\n& = \\int_{x_a}^{x_b}P(x)\\log\\left(\\frac{P(x)}{Q(x)}\\right)\\,dx\n= \\int_{y_a}^{y_b}P(y)\\log\\left(\\frac{P(y)dy/dx}{Q(y)dy/dx}\\right)\\,dy \\\\\n& = \\int_{y_a}^{y_b}P(y)\\log\\left(\\frac{P(y)}{Q(y)}\\right)\\,dy\n\\end{align}\n", "b4ccf441aa3cda5dbfc77fd82dca7bdf": "\\begin{matrix}\n\\left(H_0 + \\lambda V \\right) \\left(|n^{(0)}\\rang + \\lambda |n^{(1)}\\rang + \\cdots \\right) \\qquad\\qquad\\qquad\\qquad\\\\\n\\qquad\\qquad\\qquad= \\left(E_n^{(0)} + \\lambda E_n^{(1)} + \\lambda^2 E_n^{(2)} + \\cdots \\right) \\left(|n^{(0)}\\rang + \\lambda |n^{(1)}\\rang + \\cdots \\right)\n\\end{matrix}", "b4ccfc7a2a110889cda16b0c3c4a1370": "\\varepsilon_{ijk} = \\mathbf{e}_i\\cdot \\mathbf{e}_j\\times\\mathbf{e}_k", "b4cd49f9eea1eed89681537213fc1864": "TPI = TM \\times\\sqrt{count}", "b4cdd9bed1b384e14510209d97d39fbd": "p \\neq 2", "b4cddc5b73d75b560a023245d529dc11": "A^3", "b4cdfcaa940bb9d437c89136c297e459": "\ny = a \\ \\sinh \\mu \\ \\sin \\nu\n", "b4ce2dbbf136bc07c36afbe686369614": "RR=\\frac {a/(a+b)}{c/(c+d)} = \\frac {20/100}{1/100} = 20.", "b4ce37da897651014626943caa4432ce": "P_H = 760\\cdot e^{65.3319-(7245.2/T_A)-(8.22\\;\\ln T_a) + (0.0061557\\;T_A)}", "b4ce4572939c6e6cc95d9136f8daade3": "T_{01}", "b4ce49c5253feb65515c895bd99dbc08": "\n\\begin{align}\n\\mathbf e_1 & = (1, 0, \\ldots, 0) \\\\\n\\mathbf e_2 & = (0, 1, \\ldots, 0) \\\\\n& {}\\ \\vdots \\\\\n\\mathbf e_n & = (0, 0, \\ldots, 1)\n\\end{align}\n", "b4ceec2c4656f5c1e7fc76c59c4f80f3": "\\beta_1", "b4cf2626f0fec73e8abdc92a45385634": "U_2 = v_j-\\left(V^{M}_{N \\setminus \\{b_1\\}}-V^{M \\setminus \\{t_j\\}}_{N \\setminus \\{b_1\\}}\\right)", "b4cf73322aa5485ba6daa8405da4538f": "\n f(k;\\rho) = \\frac{\\rho\\,\\Gamma(\\rho+1)}{(k+\\rho)^{\\underline{\\rho+1}}}\n ,\n\\,", "b4cf77f26f3aa93e5fb7e37a30aa904e": "B_i, 1 \\le i \\le n", "b4cfbc20b094dcac91bf5b9cb18dcd60": "S \\to ABC", "b4cfcb063606167450721758ecab6754": "\\mu =\\sin^{-1}(1/M)", "b4cfd53d067e2dbe295a64a2fcca99f0": "O(n^{d})", "b4d0090c603b49992bb93d812feb3e29": "xy = 0", "b4d0632945d7d7559f90aff9acbd1fc6": "k \\approx \\frac{3 - s + \\sqrt{(s - 3)^2 + 24s}}{12s}", "b4d0c42b30868aacde119075349a349a": "\\delta g_F = \\frac{2g}{R} \\times h ", "b4d10483b1572b5a7487bd1325690db5": "G^*_d", "b4d104f48fcd973ff4a600586c0100cb": "\\bigl\\{ [x, +\\infty) \\bigr\\} \\cup \\Bigl\\{ \\bigl(-\\infty, x - \\tfrac{1}{n} \\bigr) \\,\\Big|\\, n \\in \\mathbb{N} \\Bigr\\}.", "b4d10ce22a49c5d8957c338169967588": "J_0,", "b4d150f213b923c3c70de5aefa4eeb6b": "(K,\\, +)", "b4d1825fe6110f1ba66315c83510095c": " \\gamma = \\frac{q}{2m} ", "b4d1aae080c61c0653a8ec5f47c77acd": "(\\tfrac{2}{221})", "b4d1dd84c5ddc1cc4278b4961dc12cf4": "L=100\\sqrt{Y / Y_n}", "b4d26e2a6ed2d6ea7ab4d1f03413762a": "R\\ ", "b4d2cd6cdd2344e69bb11e284d5ee1e1": "\\overline \\rho = \\frac{\\rho}{\\rho^{\\text{*}}}, \\text{ and } ", "b4d3705332ff642b54762b5651c5c599": "T(n) = O(n) + T(0) + T(n-1) = O(n) + T(n-1).", "b4d40400da36c0ff0642c22fd25f35aa": "\\sigma_x^2=\\frac{L^2}{12}\\left(1-\\frac{6}{n^2\\pi^2}\\right)", "b4d414045ddc319a7da99472065d3b35": "->", "b4d428a21b801e24074b3bb62352e09c": "R_{NP}", "b4d44476eb54f26f00b897aa42d9a5b3": "a_k \\cos (2\\pi \\nu_k t)", "b4d44c11068b631881c56a4451933b44": "K\\circ L", "b4d4aaa6e8ad0c4787b1ba2225d4396b": "\\mathbf v \\wedge \\mathbf u = - (\\mathbf u \\wedge \\mathbf v)", "b4d4b6f79e8f5bf0c293f217bc6e6419": " R(x) = 2 \\sum_{k=1}^{\\infty}\n\\frac{k^{\\overline{k}} x^{k}}{(2\\pi)^{2k}\\left(B_{2k}/(2k)\\right)}\n= 2\\sum_{k=1}^{\\infty}\\frac{k^{\\overline{k}}x^{k}}{(2\\pi)^{2k}\\beta_{2k}}. \\ ", "b4d4f58bbdfe43fffb98064b9eec040c": "E_v = A_v + T_v = I_v + S_v \\,,", "b4d510403435638d7132b8a694c692a9": "\\rho\\colon {\\mathrm {Mp}}(n,{\\mathbb R})\\to {\\mathrm {Sp}}(n,{\\mathbb R}).\\,", "b4d5a3b88f65ca786e2117df57d257b7": "\\mathbf{H}_\\mathrm{eff} = - \\frac{1}{\\mu_0 M_s} \\frac{\\mathrm{d}^2 E}{\\mathrm{d}\\mathbf{m}\\mathrm{d}V}", "b4d5ef7fd52361e20859305b2481a1fa": "{m,d,e}", "b4d5f26cb6d29eca898208b70948226f": "v, v'\\in V", "b4d61beeeb444c0058e4657d91b6b48b": "H=\\mathcal{L}+\\boldsymbol{\\lambda}^{\\text{T}}\\textbf{a}-\\boldsymbol{\\mu}^{\\text{T}}\\textbf{b}", "b4d6576fb136a8c29d6093c5540d11d7": "9.\\sigma_{1}(p_{2}) = \\alpha_{1}(p_{2}).\\mu_{2,1}(p_{2}).\\mu_{3,1}(p_{2})", "b4d66375940a034a87c1355eff23cadb": "q_e", "b4d6c813adb7b1313189353be9038ae7": "\\scriptstyle f(t)", "b4d718cc4e0dba192796ff21413bbc45": "\\begin{align}\n\\sin \\pi z = \\pi z \\prod_{n = 1}^\\infty \\Bigl( 1- \\frac{z^2}{n^2} \\Bigr).\n\\end{align}", "b4d721b564a6d36e8c9afa6aeccbbb12": "\\mathit{d}_H^{RC}(\\mathbb{D}) \\geqslant \\mathit{3}^{k-1}", "b4d73714fd423a5194ebe46a1856c11a": "\\gamma (g) = \\left \\lfloor \\frac{7 + \\sqrt{1 + 48g}}{2} \\right \\rfloor,", "b4d737d6afa0193a8bffdc96a0c59ffb": "Fr_{\\left(q=\\sqrt{g}\\right)} = \\frac{v}{\\sqrt{gy}} = \\frac{q}{\\sqrt{gy^3}} = \\frac{\\sqrt{g}}{\\sqrt{gy^3}} = \\frac{1}{\\sqrt{y^3}}", "b4d7496eba3dcf53ce37a7ff52ca087e": "\\frac{\\partial {\\rm tr}(\\mathbf{X^{\\rm T}AX})}{\\partial \\mathbf{X}} =", "b4d81a160c83e0c370bcdc0ebfbebead": " \\tilde{g}_n = \\begin{bmatrix} g_n \\\\ \\gamma_n \\end{bmatrix} ", "b4d8331f6c7e3a9e6cb2bc2366c265ac": "\\operatorname{sup}_N:(f:N\\to FX)", "b4d88fd381c4c1a4e800afc1f90153a3": " u v = (x+y)(x-y) = x^2 - y^2 .", "b4d89ca36ccd30048439f9fa4cd8e3ff": "\\phi(nT) \\ = \\ \\phi((n-1)T) + 2\\pi T f(nT) \\ = \\ \\phi((n-1)T) + \\underbrace{\\arg(s_a(nT)) - \\arg(s_a((n-1)T))}_{\\Delta \\phi(nT)}.\\,", "b4d8cfdfcb5d518b13733fdcbaec18cd": "Q(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)})", "b4d8ee22d89f54e05d9b76bcd152e33e": "3 k_b T", "b4d91d16c122ab01d8192af0b08db5b4": "n^c", "b4d92b6a95b9d8e96de89158f352e245": "\\dot{\\boldsymbol{\\beta}}", "b4d931cd5cae476e92b6adb652814453": "(x,x^2,x^3)", "b4d98061c1f53ba1f1795001017194ae": "(I_{\\alpha}f) (x)= \\frac{1}{c_\\alpha} \\int_{{\\mathbb{R}}^n} \\frac{f(y)}{| x - y |^{n-\\alpha}} \\, \\mathrm{d}y", "b4d9c9396bb87fa191e618fef0cad3b9": "\\phi/ 3", "b4d9f22bfe287cc04c80661ca8700b76": "\\pi_{\\mathbf P}:{\\mathbf P}\\to M\\,", "b4da2d15689435c0da75f75c6a003dff": "\n\\begin{align}\n\\mbox{total cost} & = 3 + 4 + 4 + 2 + 2 + 1 + 3 + 3 \\\\\n & = 22 \\\\\n\\end{align}\n", "b4da5ce72c70980028d7ceaa8acf75d1": "\\mathrm{abs}(\\cdot)", "b4dadd3fa66225a21cb09b64ee3a7a00": "(1-\\gamma)n", "b4daf118210e07a580a0f97bc40a4a66": "\n \\begin{align}\n F_1 & + 2\\int_{-\\pi}^{0} \n (C_1\\cos\\theta + C_3\\sin\\theta)~\\cos\\theta~ d\\theta = 0 \\qquad \\implies F_1 + C_1\\pi = 0\\\\\n F_2 & + 2\\int_{-\\pi}^{0} \n (C_1\\cos\\theta + C_3\\sin\\theta)~\\sin\\theta~ d\\theta = 0 \\qquad \\implies F_2 + C_3\\pi = 0\n \\end{align}\n ", "b4daf451b964aa346904ace7d4101414": "\nW\\left( \\lambda_1,\\lambda_2,\\lambda_3 \\right) = \\sum_{p=1}^N \\frac{\\mu_p}{\\alpha_p}\\left( \\lambda_1^{\\alpha_p} + \\lambda_2^{\\alpha_p} + \\lambda_3^{\\alpha_p} -3 \\right)\n", "b4daf80bbfffbbabc659de25b22c46f8": "f_x = m \\gamma^3 a_x = m_L a_x, \\,", "b4daf9b85a7eb68ad7329b5b0aea9686": "Q_\\mbox{acc} \\subset Q", "b4db429d4409b9fdd98f957b603c9246": " w_\\min \\geq \\frac{t}{2d} ", "b4dc35d35f72e04399232e931b815ae5": "\\begin{align}\np(\\mu|D, I) &\\propto \\int_0^{\\infty} \\frac{1}{\\sqrt{\\sigma^2}} \\exp \\left(-\\frac{1}{2\\sigma^2} n(\\mu - \\bar{x})^2\\right) \\;\\cdot\\; \\sigma^{-\\nu-2}\\exp(-\\nu s^2/2 \\sigma^2) \\; d\\sigma^2 \\\\\n&\\propto \\int_0^{\\infty} \\sigma^{-\\nu-3} \\exp \\left(-\\frac{1}{2 \\sigma^2} \\left(n(\\mu - \\bar{x})^2 + \\nu s^2\\right) \\right) \\; d\\sigma^2\n\\end{align}", "b4dc73ce07ee6864c72ac3922677040b": "\\begin{align}\n \\mu_{X \\cup Y} &= \\frac{1}{N_{X \\cup Y}}\\left(N_X\\mu_X + N_Y\\mu_Y\\right)\\\\\n \\sigma_{X \\cup Y} &= \\sqrt{\\frac{1}{N_{X \\cup Y} - 1}\\left([N_X - 1]\\sigma_X^2 + N_X\\mu_X^2 + [N_Y - 1]\\sigma_Y^2 + N_Y\\mu _Y^2 - [N_X + N_Y]\\mu_{X \\cup Y}^2\\right) }\n\\end{align}", "b4dcdada429a28bc7ce8bef44299ef3b": "\\mathbf{x}_h+\\mathbf{x}^*", "b4dd4d4e5fd4e415d8966f38b81c383f": "p(x) = \\frac{1}{Z} f(x)", "b4ddc890a47a33948f4293f9aefda613": "\\int \\delta_\\epsilon \\left(\\mathcal{F} e^{iS}\\right) \\mathcal{D}\\phi = \\int \\epsilon \\lambda \\mathcal{F} e^{iS} \\mathrm{d}^dx ", "b4ddcc4fd358d0c77e1fbbf31c365628": "y = -x/m \\, ,", "b4dddd12bd53aff93fc3afbbe8227c8b": "\\int_E f_n\\,d\\mu \\to \\int_E f\\,d\\mu", "b4ddfe545f28bbfa160dd51646848ff6": "I\\otimes A \\approx A", "b4de65839a534830b1cf99c2a4be0c84": "\\begin{align} \np_\\varphi &= a \\wedge b \\vee \\neg c \\\\\n&= a \\wedge \\left (b * (1-c) \\right ) \\\\\n&= a \\wedge \\left ( 1 - (1-b)(1 - (1-c)) \\right ) \\\\\n&= a \\left ( 1 - (1-b)(1 - (1-c)) \\right ) \\\\\n&= a - (ac-abc) \n\\end{align}", "b4de759501f7737b2743a16f32a58992": "Q = d ", "b4debeaf92802a04d40a877302516216": "\\gamma = \\frac{1}{\\sqrt{1 - v^2/c^2}}", "b4ded27bb1df5a56174ab90e78c9d4ef": "|\\psi^\\prime\\rangle =U_\\alpha |\\psi\\rangle", "b4df2338eaba50964870071419908ee4": "|E(2\\omega)|_{z=l}=E_0\\tanh{\\left(\\frac{-iE_0l\\omega d_{\\text{eff}}}{n_\\omega c}e^{2i\\phi(\\omega) - i\\phi(2\\omega)}\\right)}", "b4df24c45b38fcc3a5789ea7eab594d7": "w'_1 = 0", "b4df2c9d040d3323dd62a55fb17d096f": "-c_1(E_1)", "b4dfceb38634a59959942197ee14d8a6": " t_\\mathrm{W} \\equiv \\frac{1}{8} \\frac{\\nabla\\rho \\cdot \\nabla\\rho}{ \\rho } \\qquad \\text{and} \\ \\ \\rho = \\rho(\\boldsymbol{r}) \\ . ", "b4dfef62a2262de39d198b56e58e6b86": "i[Z := T]", "b4e017ac97c88935b86d551220f9f91f": "f(x)=\\tfrac{x}{\\sqrt{1+x^2}}", "b4e0584cada4d87dcacb8c2b56e3ea52": "f_t(z) = \\sqrt{z^2-4t}", "b4e0812b32b31146c425bbe5c74612f1": "\\eta_2 = 0.\\,", "b4e0aecee31455dea4ee1b150b24f669": "\\Rightarrow \\lambda_i = \\frac{\\left\\| A v_i \\right\\|^2}{\\left\\| v_i \\right\\|^2} \\geq 0.", "b4e0d70db0680a7213c96d88c9535e0c": "f_* : H_n(X) \\rightarrow H_n(Y).", "b4e0fdc3742e9b3bb933f60f27c4abc7": " \\int_{0}^{2 \\pi} \\left| \\psi ( \\theta ) \\right|^2 \\, d\\theta = 1\\ ", "b4e1177f7c72c8bda868a96416feb0b5": " \\text{Relative change}(x, x_{reference}) = \\frac{\\text{Actual change}}{x_{reference}} = \\frac{\\Delta}{x_{reference}} = \\frac{x - x_{reference}}{x_{reference}}.", "b4e148a029c38e6ce18aa899069d6542": " p_0 + \\tfrac12\\, \\rho\\, w^2 =\\, p_0 + \\frac {T}{A}.", "b4e19ab196fd5458a543d06547330d6f": " w(x) = \\prod_{i=1}^{20} (x - i) = (x-1)(x-2) \\ldots (x-20). ", "b4e1a40f6e4e92c618a0388b4e2aba95": "\\tau = e^L", "b4e21a1d957f7b5274983241056944a1": " \\sigma_A^2\\sigma_B^2 = \\langle f|f\\rangle\\langle g|g\\rangle. ", "b4e23b008dab6a1bbe52112bf8e4668f": "\\scriptstyle y'_1", "b4e2450c0105722d1d6252fd0b1ae4a5": " \\iota_X^2 = 0 ", "b4e249e089a5d74d0c9a60badd2aad94": "z\\mapsto 1/z^2", "b4e25e0c2475a06faf3ce075a42cce55": " (\\bar{x}-0.98,\\, \\bar{x}+0.98). ", "b4e2c6553091235edca58fb4a84e6c92": "I_0+\\delta I = I_c \\sin(\\phi_0+\\delta\\phi)\\,", "b4e2e5b4b0276586a8614f3e65da282e": "\\alpha = [\\alpha_1,\\ldots,\\alpha_p]^T", "b4e3130295f4fbc325738a67a88d779f": " f^{*} = \\frac{b_2 - 1}{2b_2} - \\frac{b_1 - 1}{4}\\left(\\frac{1}{f_1} - \\frac{1}{f_2}\\right)\\frac{f_2-1}{f_2+1} \\! ", "b4e355d934b62e4798f806c1467711ce": " G \\sim \\widehat{\\widehat{G}} {\\rightarrow} \\widehat{H} ", "b4e3590a7ead24b5faf21e42d48f6a9a": "g(x,u)\\le b,", "b4e3a0b26b915ffd756cbbd331698a27": "(2, 7, 4, 1, 7)", "b4e3b2e1644e79f214fca35e0b175fc3": "w\\Vdash B", "b4e3c34cc97f2bc10c509472ce1f8673": " \\hat T ", "b4e43e2f7dd690a32e56dcd9efa08030": "\\infty_2", "b4e455900e8f51c11fd31be5ef878520": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {S(A)} \\right) =i\\,\\mathrm{Im}\\ \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( {A} \\right).\n ", "b4e472b03d55443b44d09fdcdfc802b9": "Z(X_0, F_0, t) = \\prod_{x\\in |X_0|}\\det(1-F^*_xt^{deg(x)}|F_0)^{-1}", "b4e4938ef564a6dd0de495b56c80fff2": "K*P \\models Q", "b4e4a782d960a2a06111ebc9e8dfbe7e": "A \\cap A = A\\,\\!", "b4e4ce1fe0597a065f0922c4c6b15740": "T_{l}(u) = u^{m}\\sum_{k=0}^{\\infty}\\ a_{k}u^{k}.", "b4e4d55933eeefec9f91ae76fab94b86": " |a \\rangle = \\frac{1}{\\sqrt{2}} (|0 \\rangle + |1 \\rangle) ", "b4e4db51d394d583b59e9236a9b4b11b": "\\hat{N}", "b4e57b4f5ad2e1ef7957dcff769f0156": "\\cos(\\omega_0 t) = \\begin{matrix} \\frac{1}{2} \\end{matrix}(e^{j \\omega_0 t}+e^{-j \\omega_0 t}).\\,", "b4e608ee136f764187d9d00ff399f785": "\\begin{align}b^2 - b & \\not\\equiv b\\\\\n3b - b & \\equiv 2b\\\\\nb^2 - b & \\equiv b(b-1)\\end{align}", "b4e657e33c1c47b37a657e156fe534dd": "|V_{ub}|", "b4e679694431b4c46be8b68e72370934": " \\frac{\\partial \\phi}{\\partial x}= \\frac{\\delta \\phi}{\\delta x}", "b4e731c8239f77c8cb80ad0b8c8dbae8": "P(|X_N - X_0| \\geq t) \\leq 2\\exp\\left ({-t^2 \\over 2 \\sum_{k=1}^N c_k^2} \\right). ", "b4e736f3d710e3776a29b349a8248af9": "c_1 = 2,\\text{ } c_2 = \\pi,\\text{ } c_3 = \\frac{4 \\pi}{3}", "b4e78ed4d089bcd1ae56e1ff7ac669ce": "NEG = Energy_{\\hbox{Consumable}} - Energy_{\\hbox{Expended}}.", "b4e7b06e2ea912b9675bec14d93a78bb": "p(x,y) = 1\\cdot x^2 + (2y^2+3) \\cdot x + (-y^2+5y-8)", "b4e7b57c5011b7bc76483ebcfdce2eb4": "a_0 + a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n < 0", "b4e801c5e802c8bba2394b8dcb68ce3d": "T_{\\rm H} = \\frac{\\hbar c^3}{8\\pi GM k_B} \\;\\quad(\\approx {1.227 \\times 10^{23}\\; kg \\over M}\\; K)", "b4e902be33edd8dc30d6704a064a3626": " E = \\int \\frac{1}{2} \\nabla\\phi\\cdot\\nabla\\phi d^2 x", "b4e903c4108aae76d15dbf9b4ad46ea2": " g\\ q\\ n ", "b4e963d52864ffe4dad957ccfcfd8e6c": "y_1=y_2", "b4e9b923eccb2e0ca041dbda2951b186": "Q_D l_A a_D", "b4e9c50e0f754586c416d3652e1eeda3": "\\frac{g} {\\cos\\theta} = \\frac {v^2} {r\\sin \\theta}", "b4ea43395ff2296f7aec63a071ffe6fe": " (x_0\\lor x_2)\\land(x_0\\lor\\lnot x_3)\\land(x_1\\lor\\lnot x_3)\\land(x_1\\lor\\lnot x_4)\\land", "b4ea45e298f8d9d4ef4f3c50468cd266": "T(n) = a \\; T\\left(\\frac{n}{b}\\right) + f(n)", "b4ea693c42bc6c9114db459e71f69e11": "\\displaystyle{\\pi(T) W(x) \\pi(T)^*= W(Tx).}", "b4ea755da20ceac7b6be84bc7f586db1": "\\delta(x)\\,\\!", "b4ea8b6c6851fbf6c731702fc0388b47": "E = K A^m S^n", "b4eac423e3217c36b9ee2a05c7da8a0c": "\\pi_i(S^8)= \\pi_i(S^{15})\\oplus \\pi_{i-1}(S^7) . \\,\\!", "b4eacdad5298e9ce9c683f3a47539384": " \\varphi(\\mathbf{r},t) = -\\mathbf{r}\\cdot\\int\\limits_{0}^{1}\\mathbf{E}(u \\mathbf{r},t) du.", "b4eb1f8a77e3f221cd01fffe38556710": "\\dot{\\vec{j}} = 0 \\Rightarrow {\\vec{P}}-M \\dot{\\vec{x}}_{CM} = 0.", "b4eb37063db94d4eefa0e8232dea8729": "w-w'", "b4eb4f2148b1098ed40aeaa55b229914": "\\phi(z_0)=0", "b4eb64a9617960307250c0a0a88d010e": " \\epsilon = 1 ", "b4ebb31955a90e03b0637414c2b7380e": "f=\\pi_{\\mathbb{Z}}", "b4ebdd421104d52de0a62322927957c5": "\\Omega_j = \\left(\\int_{\\gamma_j} \\omega_1, \\dots, \\int_{\\gamma_j} \\omega_g\\right) \\in \\mathbb{C}^g.", "b4ec2035260ca2ee78ae3f7f4a0f0b2c": "\n\\operatorname{P}( \\left| \\overline{X}_n-\\mu \\right| \\geq \\varepsilon) \\leq \\frac{\\sigma^2}{n\\varepsilon^2}.\n", "b4ec2a6d7a9bd3a7cc23c3e96ee5becd": "1+x^{-18}+x^{-23}", "b4ecadab6f46354e5ce47377ca70f1bc": "\\delta_x = \\frac{q x} {24 E I} (L^3 - 2L x^2 + x^3)", "b4ece36c7273c13c7c7b911d5263cb62": "\\Omega(k)\\, =\\, k\\, \\sqrt{g\\, h}\\,", "b4ed15068e8603001f12e096368c625e": "A(x)=\\sum_{p^\\nu \\le x} f(p^\\nu) p^{-\\nu}(1-p^{-1})", "b4ed3595c01ff80603704c588764a20d": "\\scriptstyle n_1", "b4ed497e71088f7bc2a827a932b72741": "V_{base}", "b4edcfe28a927b547bee6d7bfce4b33a": "B_n'(x)=n(b+x)^{n-1}=nB_{n-1}(x).\\, ", "b4edf0561e883fcc8600b70c46b79501": " \\text{CSS}(C_1,C_2)", "b4ee79d0544a3ce0935580d36d0d0197": "\\displaystyle{S(\\varphi)(z)={1\\over 2\\pi}\\int_{\\partial} (\\log |z-w| - \\log |z|) \\varphi(w)\\,|dw| + {\\log |z| \\over 2\\pi} \\int \\varphi(w) \\, |dw|,}", "b4eeb6983de4a1b477029584a602b52e": "\\omega =\\ 90\\ \\text{deg}\\,", "b4ef01c5a4e87e7d2a053e0f6ba11057": "x_\\perp =(0,\\vec{x}=\\vec{x}_1 -\\vec{x}_2)", "b4ef15cd4fc7cda682565ca2d7ae2fbd": "s=P_{\\rm s}/P_{\\rm d}", "b4ef23f4dc285f81f000f54125d89f80": "\ng(z) = \\int_{0}^{\\infty} k(z,y) \\, f(y) \\; dy, \\quad\nf(z) = \\int_{0}^{\\infty} h(z,y) \\, g(y) \\; dy.\n", "b4ef3f2519815e10c3e3abbbef6e09fe": " u_i\\,\\!", "b4ef8fe4cf351baec62ab8f742488ed3": "g(S)=\\sum_{T\\subseteq S}f(T)", "b4ef98c111a47d5c00e7465ed5cb64a9": "S = \\sum_{i=1}^N X_i\\,", "b4efa421f97d4f436838b1a0e811c20a": "\\Omega=\\Omega(G)", "b4f01ce6f096eb5d9970addb7bc2ccc3": "a^2 x^3 + a x y^4 + a^2 y^2 = 0", "b4f04016fd134f5b3a53532934713c07": "\n\\begin{align}\nt \n&= q_1(q_2 \\cdots q_n) - p_1(q_2 \\cdots q_n) \\\\\n&= s - p_1(q_2 \\cdots q_n) \\\\\n&= p_1((p_2 \\cdots p_m) - (q_2 \\cdots q_n)).\n\\end{align}\n", "b4f057222f6f344fe6d5a3da6db164e8": "\\begin{align}f_1^\\prime &= P_1(x,f_1)\\\\\nf_2^\\prime &= P_2(x,f_1,f_2)\\\\\nf_3^\\prime &= P_3(x,f_1,f_2,f_3),\\end{align}", "b4f091268b9f97fa9cf1d024715d3985": " \\mathfrak{n}_0 ", "b4f0e1fe299a172a7b3ffa5c7f2e3c82": "\nd_{\\pm} = q ~+~ W (\\pm q R e^{-q R} )/R \n", "b4f123de8a92ab2b0bfdba0096ca3033": "\\frac{A}{\\pi r^2}=\\frac{\\theta}{2\\pi}.", "b4f180ab6b4d296ad1079649b2368ba3": "\n2y = \\frac{x^{2}}{\\sigma^{2}} - \\sigma^{2}\n", "b4f184140e0a127daeac2e7110003086": "\\sigma = 25/3", "b4f1d66c65ee666e2a28d56b450d0004": "[g|\\partial D^2]\\notin N. \\, ", "b4f240cb80f7406ea802b1df89e89944": "\nA = \\begin{pmatrix}\n0 & 1 & -1 \\\\\n-1 & 0 & 1 \\\\\n1 & -1 & 0\n\\end{pmatrix}.\n", "b4f24871d473e147533bd647fed7420c": "\nR_\\mathrm{eq} = R_1 \\| R_2 = {R_1 R_2 \\over R_1 + R_2}\n", "b4f25116db97e7c1394935c114bcdbeb": "\\left| M(n) \\right| < \\sqrt { n }.\\,", "b4f31befa4e3ef15d06078a10af0c3e7": "\\begin{align}\n\\mathbf{L} & = \\begin{pmatrix}\nL^{11} & L^{12} & L^{13} \\\\\nL^{21} & L^{22} & L^{23} \\\\\nL^{31} & L^{32} & L^{33} \\\\\n\\end{pmatrix} = \\begin{pmatrix}\n0 & L_{xy} & L_{xz} \\\\\nL_{yx} & 0 & L_{yz} \\\\\nL_{zx} & L_{zy} & 0\n\\end{pmatrix} = \\begin{pmatrix}\n0 & L_{xy} & -L_{zx} \\\\\n-L_{xy} & 0 & L_{yz} \\\\\nL_{zx} & -L_{yz} & 0\n\\end{pmatrix} \\\\\n& =\\begin{pmatrix}\n0 & xp_y - yp_x & -(zp_x - xp_z) \\\\\n-(xp_y - yp_x) & 0 & yp_z - zp_y \\\\\nzp_x - xp_z & -(yp_z - zp_y) & 0\n\\end{pmatrix}\n\\end{align}", "b4f341229d5002a5fb4bc989f4f64031": " n! (-1)^n \\frac{1}{n(n-1)} = (-1)^n (n-2)!", "b4f374f6cf12734d9efb2f321e66c14f": "\\begin{align}(1-x^2)^\\alpha=& -\\frac 1 {\\sqrt \\pi}\\frac{\\Gamma(\\frac 1 2+\\alpha)}{\\Gamma(\\alpha+1)}+ 2^{1-2\\alpha} \\sum_{n=0} (-1)^n {2\\alpha \\choose \\alpha-n} T_{2n}(x)\\\\=& 2^{-2\\alpha}\\sum_{n=0} (-1)^n {2\\alpha+1 \\choose \\alpha-n} U_{2n}(x).\\end{align}", "b4f3a448c2bbf59f4ee619567d478c6d": "\\mathbf{B}=\\mu_0\\mathbf{(H + M)}", "b4f3e65da9f66501cc2c03a4ff6f5097": " F_S = - m g \\ \\sin \\theta ", "b4f47c07a64ffeda969641197fba63ec": "\\forall \\lambda, \\mu \\in E, \\,\\, \\lambda > \\mu \\iff \\lambda - \\mu > 0 \\iff \\,\\, \\, \\exists k_1, k_2, ..., k_n \\in \\mathbb{Z}^+, \\, \\alpha_1, \\alpha_2, ..., \\alpha_n \\in \\Delta, \\,\\, \\lambda - \\mu = \\sum_i k_i \\alpha_i ", "b4f491d44007493678b29e338c136c1d": "\\vec r=\\vec r(u,v),", "b4f4af47028b9c3671ee55e7e507ab8b": "G_{\\rm Ic}", "b4f4b833df5244ef26d8cd236cfc4911": "_{q\\,}\\!", "b4f4ca5cf277db349470aae173359506": " \\boldsymbol{\\sigma}=(\\sigma_x,\\sigma_y,\\sigma_z) ", "b4f54b06b18b41a0cf7f39409c3bdd25": "|E_n|/|S_n|", "b4f551363b0014f9b6854ea56d9d9288": "\n\\varphi(n) =n \\prod_{p\\mid n} \\left(1-\\frac{1}{p}\\right),\n", "b4f56356a68f1ca1b6ce93728c8a8501": "\\textstyle\\frac {2}{2-1}=7", "b4f58132acdde0c16e48f71c4bda51da": " IR_{P}(t)= \\cfrac{ \\displaystyle\\sum_h \\sum_{d_h\\neq 0} \\sum_{\\gamma_h} \\cfrac{d_h q}{^{d_h}M_{P_h}} \\ \\cfrac{\\text{d} [{^{d_h}_{c_h}}P^{\\gamma_h}_h (t)] }{\\text{d} t} }{\\displaystyle \\sum_h \\sum_{d_h\\neq c_h} \\sum_{\\gamma_h} \\cfrac{(c_h-d_h) p }{^{d_h}M_{P_h}} \\ \\cfrac{\\text{d} [{^{d_h}_{c_h}}P^{\\gamma_h}_h (t)] }{\\text{d} t} } \\qquad \\qquad (5) ", "b4f583cddb817f9a0d749a49b40ee379": "\nU = \\sigma_y k =\n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix} K,\n", "b4f5fd6f59befe3d10d0a9473b86e615": "2k^2 = b^2", "b4f60ddb6d2d6914ecbd9dc3476f59a6": "e^{\\pi\\sqrt{163}}\\,", "b4f73df1667ec204362afd062f4b2d43": "K = \\frac{1}{2}\\overline{\\upsilon_i' \\upsilon_i'}", "b4f77a07895589a7ee4dc8b33b544bbd": "\\Phi_{12}=\\delta\\gamma-\\Delta\\beta-(\\tau-\\bar{\\alpha}-\\beta)\\gamma-\\mu\\tau+\\sigma\\nu+\\varepsilon\\bar{\\nu}+(\\gamma-\\bar{\\gamma}-\\mu)\\beta-\\alpha\\bar{\\lambda}\\,,", "b4f7edacfc23fbfa7f57b82d4c757c6c": "E[Q(x,\\xi)]", "b4f8302965c44d6aa5350742aea8410e": "{\\kappa(\\omega, E)} = {{\\pi^2} \\over{\\epsilon_0 m^2 \\omega^2}}|e \\cdot p_cv|^2 {{e|E_x|} \\over {\\hbar\\theta_x^2}}\\int_{-\\infty}^{\\infty}{J^{2D}}_{cv}(\\hbar\\omega - \\epsilon_G - \\epsilon_x) \\cdot|Ai(-{{\\epsilon_x} \\over {\\hbar\\theta}})^2 |d\\epsilon_x.", "b4f852dd635d8ca7dbee71a7a0e317dc": "\\mathbf{h}(\\mathbf{x}) = \\mathbf{0} ", "b4f8d13d0136f4ca485840779df1588c": "= \\arctan \\frac {1*5 + 1*5}{5*5 - 1*1}", "b4f90c1b56afd1ab99428ef95ef2e46e": "G^{a}_{\\mu\\nu}", "b4f92e1b27491a579508f67a9eececea": "\\color{blue}\\mathcal{M} \\rightarrow \\mathcal{S} \\rightarrow \\mathcal{E} \\rightarrow \\mathcal{I} \\rightarrow \\mathcal{R} \\rightarrow \\mathcal{S}", "b4f93bc54bcde8d5fbe028a9a5c705d3": "\\langle,\\rangle:TS\\times TS^*\\rightarrow\\mathbb{R}", "b4f94ac3f8b75d998ca2ac9c51da44da": "I = \\sum I_j", "b4f96c4a6cea2c181556edafdc8b7ad7": "Z=-Z_f\\cos(\\theta-\\alpha)+X_f\\sin(\\theta-\\alpha)", "b4f9fbac5ff208700d03cc2984d72257": "\n\\mathbf{AB} =\n\\begin{pmatrix} \\mathbf{\\bar a}_1 & \\mathbf{\\bar a}_2 & \\cdots & \\mathbf{\\bar a}_m \\end{pmatrix}\n\\begin{pmatrix} \\mathbf{\\bar b}_1 \\\\ \\mathbf{\\bar b}_2 \\\\ \\vdots \\\\ \\mathbf{\\bar b}_m \\end{pmatrix}\n= \\mathbf{\\bar a}_1 \\otimes \\mathbf{\\bar b}_1 + \\mathbf{\\bar a}_2 \\otimes \\mathbf{\\bar b}_2 + \\cdots + \\mathbf{\\bar a}_m \\otimes \\mathbf{\\bar b}_m = \\sum_{i=1}^m \\mathbf{\\bar a}_i \\otimes \\mathbf{\\bar b}_i\n", "b4fa6ec8d5abb531ac054e0bfceb543c": "\\vec a_n=A(\\vec x_n)", "b4fa98f28ac66b8f384055fbb4c2b12a": "S: (x(u,v),y(u,v),z(u,v))\\ ,", "b4fad4cf4ddd2861774df34d7075f473": "\\vec{p'}", "b4fadef9fa71bb336e92176b88b9538e": "\\|\\mathcal F\\|_{q,p} = \\sup_{f\\in L^p(\\mathbb R^n)} \\frac{\\|\\mathcal Ff\\|_q}{\\|f\\|_p},\\text{ where }1 < p \\le 2,\\text{ and }\\frac 1 p + \\frac 1 q = 1.", "b4fae4f0a8d53a32ced764327b0e0f69": "Y^- = X^+", "b4fb0f5e25dfacb6f6e4e87ab0bfedf3": "dof\\,", "b4fb34bcd339799557d8040b27c8337f": "H_{\\mathrm{ZOH}}(f)\\, = \\mathcal{F} \\{ h_{\\mathrm{ZOH}}(t) \\} \\,= \\frac{1 - e^{-i 2 \\pi fT}}{i 2 \\pi fT} = e^{-i \\pi fT} \\mathrm{sinc}(fT) \\ ", "b4fb42d5b32229bad85ed392f86baa6a": "(1-(1/2)v^2/c^2)", "b4fb80d8a73238c381c870f8c3e6191e": "\\beta_{w=1 \\dots V}", "b4fbaefd262975a5f0bd60af684e94ca": "PA_{n} \\subsetneq PA_{n+1}", "b4fc249b6271bfbe123cc85e886c85e7": " \\displaystyle\n u_i := - \\log_b (p(x_i))\n", "b4fc3dd4350e1ed94f736d4c7eae8033": "{\\scriptstyle\\frac{1}{2}} (x^2-4x+2) \\,", "b4fcd0f172bc49358ec33eab4bef416e": "z_b=-\\sin i\\,", "b4fcf3d2fdc88fc4df4183a385f2f919": "\\gamma _j", "b4fd4871902898e1b784f825eeda5128": "\\phi(z)={1+{1\\over 2}z \\over 1-{1\\over 2}z}", "b4fd5a492dd5b9ba0a48521130799f98": "C = \\frac{F}{F+M} (1-e^{-(F+M)T}) N_0", "b4fd9d3e3da91c9751facf6c23c6c898": " \\frac{\\partial u}{\\partial t} = \\hat{H} u +f(x,t) u+g(x,t)", "b4fdfe9e16ab428401ad08ff432292ae": "\\sim r\\leftarrow \\hbox{not }p", "b4fe7fca5d5c33a01bea4f5b63e917f5": "(A \\to w, k)", "b4fe821773d54d7079ad2871fc8fc380": "\\epsilon_H = -I ", "b4ff29aba58a0db4b26380d1c1c0b20f": "A(m-1,A(m,n-1))", "b4fffcf44a922e4459c9d460a87d5db4": "\\arctan z = \\sum_{n=0}^\\infty \\frac{2^{\\,2n}\\,(n!)^2}{\\left(2n+1\\right)!} \\; \\frac{z^{\\,2n+1}}{\\left(1+z^2\\right)^{n+1}}", "b500549e9a7413c599d98cee7e5ede2a": "Q=\\frac{-kA}{\\mu} \\frac{(P_b - P_a)}{L}", "b500be3a4df641aa3feb53c126e050a9": "\\hat{y} = \\sin \\theta \\cdot \\hat{r} + \\cos \\theta \\cdot \\hat{t}", "b5013af88c7eeb9ef254b80f5fe9441d": "\\sum_{n=0}^\\infty\\frac {\\alpha^n}{\\sqrt{\\varepsilon_n !}}", "b5013dce15f82535e5b3c3c370b69d2b": "\\frac{x - x^*}{2} = x_1\\,e_1 + x_2\\,e_2 + x_3\\,e_3 + x_4\\,e_4 + x_5\\,e_5 + x_6\\,e_6 + x_7\\,e_7.", "b50175f624d6b1ba60f8987a99e47587": "\\textstyle\\frac{1+\\omega}{2+\\omega}", "b501826417a079bbd72666082748ca8d": "\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k) < 0 ", "b50185b61ba0bb946e21f3b71260692c": "e_i=s_i\\cdot M_i/m_i", "b501eb212483345f81fc36b6464c40a2": "\\mathrm{D}^{\\alpha} u = \\frac{\\partial^{| \\alpha |} u}{\\partial_{x_{1}}^{\\alpha_{1}} \\cdots \\partial_{x_{n}}^{\\alpha_{n}} }.", "b502038b84efb1de5bc22da79d01ee38": " \\dot{R}^2 = E^2 - P^2 - \\frac{L^2}{R^2} ", "b50227f0be274c25b99746337f0af662": "\\left \\|\\mu- \\nu \\right \\|_{TV} = 2\\cdot\\sup_{A\\in \\mathcal{F}} | \\mu (A) - \\nu (A) |.", "b50263cb508beb0b97588e7ec26fc196": "\\text{Asymmetrical short-circuit kva = symmetrical kva * X/R factor }", "b50266b0ec343f01cfb183db6a32dfb5": " \\mu_Y = \\alpha \\mu_X + \\beta ", "b502c18d71a9e1f5bf2403da6a3bd9d1": "\\ d={gt^2\\over 2}", "b502f02d4a41d40cc71ba98c01380cb7": " \\hat{J}_z \\,\\!", "b5032ecb5dc0a7767e7c22d5a9b1010a": "\\begin{pmatrix}\\omega &0\\\\0&\\omega^q\\end{pmatrix}.", "b503308f11a5cc7ef47345999a9a897d": "\n A_{\\alpha\\beta} := \\int_{-h}^h C_{\\alpha\\beta}~dx_3\n", "b503367ef3a352abe4f7a842f4dc2697": " {x}(t) \\approx \\varphi(t-1) ", "b503873ccab92b17de2796c57143d51c": " x \\and y \\to y = 1 ,", "b503a169c74f983f5d1138a223c91cb5": "C_{t,i} \\,", "b503e122497372a8fd1c95112f788a31": " (K,\\omega) ", "b503e1eb0698b04c29cb00d397b6622b": " A_{\\mathrm{G}} = \\; \\lambda_{\\mathrm{R}} A_0 ", "b5042e27fd97a8abd2988eb6b49d22e8": "ax^3+bx^2+cx+d\\,", "b504ab8b363ce810f9b9dc22d5a3adec": "k_{-2}", "b504ae44bf31924cef5289d0414eff19": "(\\operatorname{row } M)^\\bot = \\ker M.", "b505027b880e2d13220d16ad4b198616": "R=\\frac{1}{2}r\\left(1-r\\right).", "b50509cc0040a4648ecea110b458e72e": " \\begin{align}\n{\\tilde x}^{\\mu}(\\tau ) & = \\left({\\tilde x}^0(\\tau ); {\\tilde {\\vec x}}(\\tau)\\right) = \\left(\\sqrt{1 + {\\vec h}^2} (\\tau + {\\frac{{\\vec h \\cdot \\vec z}}{{Mc}}}); {\\frac{{\\vec z}}{{Mc}}} + (\\tau + {\\frac{{\\vec h \\cdot \\vec z}}{{Mc}}\n}) \\vec h\\right) \\\\\n& = z^{\\mu}_W(\\tau ,{\\tilde {\\vec \\sigma}})= Y^{\\mu}(\\tau ) + \\left(0,\n{\\frac{{- \\vec S \\times \\vec h}}{{Mc (1 + \\sqrt{1 + {\\vec h}^2})}}}\\right) \\\\ \\end{align} ", "b5051a221a63db88b24424d07c724de4": "\\left[T^a, T^b \\right] = i f^{abc} T^c \\,", "b5051da41a711bcdf458f7645692ff2b": "\\kappa \\, ", "b50569b66221d5a9b91e3257ff9591e2": "a_1\\dots a_n+a_{n+1}\\dots a_{2n}=10^n-1. \\, ", "b5057ea575f9f4849850d07a0e5d2116": "L.L=1", "b505bcb804c714bd47cb27fff108576f": "A B A B^{-1}", "b505d107dd4e6913240b9780bc4d8a64": "{x_{n+1}}/{\\overline{x}}", "b505e3a7e7c1458191ac7bda20b0bece": "\\sum_{k=0}^{n-1} 2^k = 2^{n} - 1", "b506570107affe7e87140fa82fd1d3bd": "\n\\underline{P}(Cl_t^{\\leq}) \\subseteq Cl_t^{\\leq} \\subseteq \\overline{P}(Cl_t^{\\leq})\n", "b507333cf90b398ad4affba5b06da850": "V_{DS}", "b50753d104b8bbf8181f8fc400067fbc": "I_mA = AI_n = A. \\,", "b507939335cbb800f7d05719c815c57e": " g(aA+bB,C)=ag(A,C)+bg(B,C) ", "b5079439139710163df771d22bada048": " f = C (m + 2n)^p \\ ", "b507cd5b4054ac3b44dc42af45918153": "\\scriptstyle n \\, > \\, 1 ", "b507d4c6cd11397973f7d288bf2cbb3b": " {\\mathbb R}^3", "b507fda1585fd50fbd201331484dda3c": "\\vec \\mu_J= \\vec \\mu_L + \\vec \\mu_S", "b50860aa80041a11e9d4cecb97cde9e1": "\\Delta \\cap DS", "b5086d47635520198193d3add00b0aa1": " X \\lessdot Y \\iff \\begin{cases} A \\to \\alpha X B \\beta \\in P \\\\ B \\Rightarrow^+ Y \\gamma \\\\ A, B \\in V_n \\\\ \\alpha , \\beta, \\gamma \\in (V_n \\cup V_t)^* \\\\ X, Y \\in (V_n \\cup V_t) \\end{cases} ", "b5089922b03890078a6925a3a4888d50": "\\mathrm{Hom}(k^n, \\mathcal{R})", "b508a346371d7c7eb0ea705ab81dac3b": "P(A) = \\dfrac{365!}{365^N(365-N)!}", "b508e49dcdcf4291a38a3d775b9bd021": " \\square ", "b50902bab2351b7e4bf180012c49364c": "P=a^{n+1} r^{1+2+3+ \\cdots +(n-1)+(n)}", "b5092d379ce6662734109ffe625396af": "n^\\frac{1}{2}", "b50969fbc26f9ffc93c2e1160388595e": "Z'", "b509e1adbda1a4cf95c7a01676cce1c7": "{}^{x}a \\approx \\begin{cases}\n\\log_a(^{x+1}a) & x \\le -1 \\\\\n1 + x & -1 < x \\le 0 \\\\\na^{\\left(^{x-1}a\\right)} & 0 < x\n\\end{cases}", "b50a2bc0fd041cc3709467d44da4b30c": "\\frac{\\partial \\Pi_i }{\\partial q_i} = \\frac{\\partial P(q_1+q_2) }{\\partial q_i} \\cdot q_i + P(q_1+q_2) - \\frac{\\partial C_i (q_i)}{\\partial q_i}", "b50a3d782a3344338bfe874ce09fb0a1": "\\sum_{n=0}^\\infty (-1)^n a_n = \\sum_{n=0}^\\infty (-1)^n \n\\frac {\\Delta^n a_0} {2^{n+1}}", "b50a6e2a0bf89cdd6b0b40b1e2b95475": "b_i = \\frac{n_i}{m_\\mathrm{solvent}}.", "b50aea7f81aa83f65b33ceb720fa3ed4": " r_{m} ", "b50af84492fdbecca5543caf6426920f": "A\\in \\mathcal{S}", "b50b4a9aae036f392b7a4e7eecfc7fe9": "QC_G(Q)", "b50b661f3f01034c37558990520a0679": " \nE[\\Delta(t)] + VE[p(t)] \\leq B + Vp^* - \\epsilon \\sum_{i=1}^NE[Q_i(t)] \n", "b50b8b64964286fa9c9ddc80b7855a20": "2\\hat\\gamma(h)", "b50b8c5268f146268dcaa92124807f19": " X_{p} + X_{o^{N}}+ E = X_{po^{N}}", "b50be05e366ee8ef4288ed95159f7edf": "x=y+m=y-\\frac{b}{2a}", "b50c0c59ddaf7a238f98209524f4e750": " \\scriptstyle\\mathfrak{G} ", "b50c0e68cf1ec94a39940c5ef3e441e9": "\\, B", "b50c7519bcccecb145825b267bfda020": " \\dot{p} = - 2x ", "b50cff473c8235d5323c88f9fd30505d": " \\ln(S) + pS = X - aC \\,", "b50d45f654fd97b93056672d2aba1077": "\\sum_{n_x=0}^n (n-n_x+1)=(n+1)(n+2)/2", "b50d856ed0fc2a65da97296264657f7e": "\\textstyle (t-1)m+1", "b50da6d94b883c42627719c5d8aa2fc2": "\\alpha \\mapsto \\phi(1,\\alpha,0)", "b50e6266cfe73b3ba93b2f5c15a4eb31": " u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \\Rightarrow\n \\Phi(a'_{i},a_{-i})-\\Phi(a''_{i},a_{-i}) >0 ", "b50efb1d787ccea115f12d70b5491abc": "z = r\\cdot(\\cos\\varphi+i\\sin\\varphi)", "b50f28fd03eb2dfbf596b99684b2f120": "\\bold X = [x(t_{1}), ......, x(t_{M})]", "b50f373c43a62b7b0460a861c475bfb5": "t : A \\rightarrow V", "b50f41573472c246b711bff087dc63f1": "\\ddot\\mathbf{P} = \\mathbf{\\hat r} (\\ddot r - r \\dot\\theta^2)\n + \\boldsymbol{\\hat\\theta} (r \\ddot\\theta + 2 \\dot r \\dot\\theta)\n + \\mathbf{\\hat z} \\ddot z", "b50fc5179237853f806560b9b4aa1906": "S(f-f_0)", "b50fe02422c20cbf10181503fcaeacd9": "\\frac{\\partial I}{\\partial x}\\Delta x+\\frac{\\partial I}{\\partial y}\\Delta y+\\frac{\\partial I}{\\partial t}\\Delta t = 0", "b50fed9ce0a969928b903a0ee554859a": "f(x;\\beta ) = \\frac{1}{\\beta} e^{-x/\\beta} ,\\; x \\ge 0 ", "b50ff0f4a9eaff400abb72525d74107a": "\\sum_{i=1}^{n} {\\alpha_{i}}=1. ", "b510195a3db47ab46fd7c2eef96c1512": "z_k = \\frac{\\lambda + \\mu y_k}{\\frac{y_k^2}{C}-3}\\,", "b5104dd3538b71ce33cd1b84d5b8779e": "\\mathrm{Hol}_q(\\omega) = g^{-1} \\mathrm{Hol}_p(\\omega) g. \\, ", "b510a73de7ba6e052827c54225551729": "\\sigma^2_u", "b510b1a9cf17c9e236625127708256a5": "\\sigma\\sqrt{c_4^{-2}-1} .", "b510c0dfe1fdd7905edff67189a24948": "B8^{-}", "b510c3fca0a1f79d6494f455f9f34d5f": "rab", "b510fb5161af0934e976612dc512b587": "O( n^2 / 2^j + 2^j n )", "b510fdd431f6ea5b6870a87fb9a3d698": " \\frac{(c + \\gamma - 1)(c + \\alpha)_{\\gamma - 1} (c + \\beta)_{\\gamma - 1}}{(c + 1)_{\\gamma - 1}(c + \\gamma)_{\\gamma - 1}} x^{\\gamma - 1}", "b51119fac5a2fe2168898584eb2afa1e": "\\textstyle \\mathrm{d}\\mathbf{r}", "b5111c5bf72878ab558bad80660685fd": " \\scriptstyle \\rangle ", "b5116f2d1ff6eab11a0de33f117f4064": "\\cos a = \\cos b \\,\\cos c + \\sin b \\, \\sin c \\,\\cos A.", "b5122d8e3bf584ecbc5357f3f2f10425": "\\tfrac{j}{i+j} > \\rho", "b51230518591d83381e158cab213b376": " I = \\frac{V}{R} ", "b512551b1858f9e82551003314fc13b2": "t(t-1)(t-2)\\left(t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352\\right).", "b512a36ab7216dfb855b771ed74dc5a7": " f(z) = \\frac{1}{z}\\left(\\sum_{k=0}^{\\infty} \\frac{(-1)^kz^{2k+1}}{(2k+1)!} \\right) = \\sum_{k=0}^{\\infty} \\frac{(-1)^kz^{2k}}{(2k+1)!} = 1 - \\frac{z^2}{3!} + \\frac{z^4}{5!} - \\frac{z^6}{7!} + \\cdots. ", "b513bc1e66c0a81bcbe6ee98e05a12fa": "\\textstyle\\int_a^b", "b513bef155311274861913d6d5af8120": "K_a^i = \\{ A_a^i , K \\}", "b513fb1f0d12a240135ce169665b7615": "e^{-\\lambda}", "b5140cb2eb8c260a46931e47570bba2f": "((p \\to (q \\to r)) \\to ((p \\to q) \\to (p \\to r)))", "b5143e66f3af5cdb929c6366d79bb745": "p \\mid m_i - m_j", "b5144e36aad39f6c785b5ba1dea79845": "\\tfrac{1}{C} = \\tfrac{P(1)\\times S(1)}{Q(1)} = \\tfrac{P(2)\\times S(2)}{Q(2)}", "b514b22ead8e943bae05286ae2de1790": "\\frac{d\\mu}{dn} < 0", "b514df83f88619a2796a8c0c0a2f1c98": "\\boldsymbol{\\rho}", "b5152870a4226795d7ca9b84ce48d01a": "\\int_0^\\infty\\frac{\\mathrm{d}x}{1+x^2}", "b51537ecf21f35ec145934a73f70dd9e": "c / c_0 = 1 + (u_{0} / u -1) \\left(\\frac{1}{1 + \\mathrm{Stk} (2 + 0.617 u / u_{0})} - 1\\right)", "b515a59901f3838c0722e89247108882": "\\eta\\colon K\\to Z(A),", "b515e04edfab040d018ee443d7c4590a": "\\tau_{-} ", "b515e20913f49df2d711f2fee27798c2": "X^{(p/S)} \\times_S S' \\cong (X \\times_S S')^{(p/S')}.", "b515e809d7e02954a916fab298fcbe12": "\\ \\displaystyle u\\ ", "b51613de586175f237f4bcd8c6e69cce": "L = 0.01595 N^2 \\left(D - \\sqrt{D^2 - d^2}\\right)", "b5167ebc5e76b15a9654168c644c8143": " q^1 = \\mathbf{v} \\cdot \\mathbf{e}^1; \\qquad q^2 = \\mathbf{v} \\cdot \\mathbf{e}^2; \\qquad q^3 = \\mathbf{v} \\cdot \\mathbf{e}^3. \\, ", "b5169c4cb306ca804bab52cb9ef06bc4": "\\lambda=\\mu", "b516c96d270a0211c80d25a0e05c6b2e": "f_H = \\frac{f_0}{\\cosh \\left(\\frac{1}{n} \\cosh^{-1}\\frac{1}{\\varepsilon}\\right)}.", "b516f2b183cb993d6594c64bc3953b46": " c-\\varepsilon ", "b51718a3e4891fa5a12a6615b7a4bd0d": "C_{ijk} = - C_{ikj} \\, ", "b51719c877f26f3c8f26a22cbbb8d542": " a = a^{\\perp \\perp} ", "b51733143a013ed937cdb52428b08f66": "Q_\\mathrm{lost} = A \\epsilon \\sigma T_H^4 ", "b5179f8c65147b0fcd4f2f2beda5bcd0": "\\mathbf{p}={\\hbar\\over i}\\nabla=-i\\hbar\\nabla\\,,", "b5184ba8ee8f0d1ab43ccb368537c5cf": "s/Q", "b51860a1f1bab55af1f80e818ee81b27": "\\,a_0 \\in \\mathbb{C}", "b518757d9afda57deb135ba9006e4f3a": "\n\n\\hat \\sigma _{\\bar x} \\,\\,\\, \\approx \\,\\,\\,\\,{{s\\,} \\over {\\sqrt {\\,n} }}{{\\sqrt {\\,\\gamma _2 } } \\over {\\sqrt {\\,\\gamma _1 } }}\\,\\,\\,\\, = \\,\\,\\,\\,{{s\\,} \\over {\\sqrt {\\,n} }}\\sqrt {{{n\\,\\, - \\,\\,1} \\over {{n \\over {\\gamma _2 }}\\,\\, - \\,\\,1}}}\n\n", "b518d1426b703776d952b4d7aee1fe34": "\\forall X,Y \\in \\mathbb{B}, X \\not\\subset Y", "b518d90ffc58451224c736564d92d568": "\\partial^\\mu \\partial_\\mu A=0", "b5191b0ca42178fab674a9a7aa9e21af": "20\\,0\\,", "b532cced6fa2b1b7bd9440e8575aede8": "g_j^{t-1}", "b532f2f79a54fff709a7be69698ca81a": "[H f(\\mathbf{x}_n)] \\mathbf{p}_{n} = \\nabla f(\\mathbf{x}_n)", "b53312f47914b19a03d32c44885a0730": "S^{2k+1} \\times S^{2k+1}", "b53368dd662ec84c035908f83e89eee8": "k=0,1,2", "b533a0717892c46be101013c79a30c3b": "I_{3,2} = -10 \\log{\\left( \\frac{P_3}{P_2} \\right)} \\quad \\rm{dB}", "b5340357aa5967b0c0b47caf0fb7c098": " a_{L} ", "b534772dff96961555b796a358abbcc9": "\\cdots \\oplus \\!", "b5348a7338c9b8c7d771f6158925c65f": "\\Phi(m,n)", "b534ae4f11b6fc8405311a63a2654c12": "F_{\\mathrm{g}} = mg \\ , ", "b534e6e54c7c6562172786f12745e58c": "\\dfrac{1}{5}=2.01210121\\dots_3\\mbox{ or }\\dfrac{1}{15}=20.1210121\\dots_3.", "b534fc70be91b44ee3c72495f6442557": "(-1)^{p_3 r_3}", "b5350fe416a7e82c4d2e45a457ba4027": "B= \\{0=b_1 1", "b5487907d6956b4d1bd7fb7dc0f883a1": "y_1''(t)+p(t)y_1'(t)+q(t)y_1(t)=0", "b548a5132f6409ee8a9cad55d9f0f64b": " B ", "b548c2d3eb71eb151f98b6acc41f2836": "P(x_{1})", "b548c83048b324ade5094af73cc13e15": "\\nabla^2 \\psi + k^2 \\psi = \\delta(\\bold r)", "b549269f6eb1d3ccffde4da15738ac51": "R_{down}", "b5493015a5af23b1343f6adb44f55a92": "\\Omega = \\{0, \\ldots, 36\\}", "b5493af0f4e07e88d96498e744c86602": "\\gamma = \\frac{0.236R}{\\omega L}", "b549426a4d4e826070f5ee41df9cd333": "\\bar{\\ }", "b54956f3b3bb7253a1d7496c2682f814": "\\mathcal{S}_n(f)", "b5496e7120868a4564a4ca507f5c5250": "S^2(A_n)=S(S(A_n)),", "b54a1e6dc48022d650ff39c7c876cb42": "E(R_i)~~", "b54a22676a0a1bb415fca022218f9d60": "y^2 = x^3+x", "b54a25b238beeae6ee215f64608fe898": "S_R^\\delta f(x) = \\int_{|\\xi| \\leq R} \\left(1 - \\frac{|\\xi|^2}{R^2} \\right)_+^\\delta \\hat{f}(\\xi) e^{2 \\pi i x \\cdot \\xi}\\,d\\xi.", "b54a304d5eb56b87420f9e518326beb5": "\\mathrm{Sp} = (\\mathrm{Sp} \\times \\mathrm{Sp})/\\mathrm{Sp}", "b54aee514ec2dda08e97780cd7fe2898": " g(E) = 1 ", "b54af01ced003f86510cf34474d77417": "\\delta(b)", "b54b1ae58a2406186e0d6aae52dcdf22": "(R, \\Theta) = \\left(\\sqrt{2(1 + z)}, \\theta\\right),", "b54b1e50a84bd1c17d87167c42e4c42a": "\\ P_v", "b54b237fc9cc692d97d16af8525af325": "i\\frac{k}{i}", "b54b4f00cd8a5ea223d3b0e96caa8c64": "\\sigma^2 = \\frac{1}{2} \\sigma_{mn} \\, \\sigma^{mn}, \\; \\omega^2 = \\frac{1}{2} \\omega_{mn} \\, \\omega^{mn}", "b54b5dbf5d84023eb89bea1c5076265b": "\\mathcal{G}_1", "b54b916164eb2b5c43e92821210fff7c": "\\ \\xi ", "b54b950bbd1d8b68cac4d0fd192f25d3": "\\textstyle \\Pi", "b54bf112ce613354c8b98c3760dd1482": "\\langle A\\rangle \\simeq \\frac{1}{N}\\sum_{i=1}^N A_{\\vec{r}_i} e^{-\\beta E_{\\vec{r}_i}}/Z", "b54c20a1791d643a2313438d36296302": "\\tfrac{\\lambda_\\min(U)}{\\lambda_\\max(U)}", "b54c34fe7f1b03ec3eddee0e3e1a924d": "2 \\max \\left( \\left|\\frac{a_{n-1}}{a_n}\\right|,\\left|\\frac{a_{n-2}}{a_{n-1}}\\right|, \\dots \\left|\\frac{a_1}{a_2}\\right|, \\left|\\frac{a_0}{2a_1}\\right|\\right)", "b54c46d54f638653c6ee8ba7b42cdda2": "Y,Z\\in\\mathcal{H}", "b54c820a305418085ee91336364d2829": " \\frac{\\partial F}{\\partial n_1}=0\\,\\!", "b54cd2204027b0b14c3886f30775208f": "\\begin{align}ds &= \\sqrt{\\rho^2\\phi'^2 + R^2}\\,d\\lambda \\\\\n&\\equiv L(\\phi,\\phi')\\,d\\lambda,\n\\end{align}\n", "b54d16333fb82ab8fa85d2d88a5cb37a": "-\\frac{\\hbar^2}{2m}\\frac{d^2 \\psi}{dx^2}(x) +V(x)\\psi(x) = E\\psi(x)", "b54d185c1ff233c7a7d8f247505a3c68": "y\\approx Y+YZ+\\dfrac1{42}Y^3+\\dfrac12YZ^2", "b54d2ab08d3eea1141af1fb0c7e305bd": "\\omega = -\\frac{3}{2}", "b54d347535629f8f3742e2a88d0af4c1": "\\det \\begin{pmatrix} zw & z & w & 1 \\\\ z_1w_1 & z_1 & w_1 & 1 \\\\ z_2w_2 & z_2 & w_2 & 1 \\\\ z_3w_3 & z_3 & w_3 & 1\\end{pmatrix}\\, ", "b54d543e5e32131f240f8cf63abfad01": " \\frac{\\partial f}{\\partial y} - \\frac{d}{dx} \\frac{\\partial f}{\\partial y'} = 0", "b54d7481d0ebddb9e6264ebeb132061b": "\\scriptstyle \\log_{10} P_{mmHg} = 6.83029 - \\frac {945.90} {240.0+T}", "b54e0455ab77ce6a7a22e5f82fd0b8d7": "p\\in P\\;", "b54e07b13feee5c2cfc4dd0ae1b04866": "\\hat P_{EV}(e^{j \\omega}) = \\frac{1}{\\sum_{i=p+1}^{M}\\frac{1}{\\lambda_i} |\\mathbf{e}^H \\mathbf{v}_i|^2}", "b54e1fc81f9b0bc0a2d60e48f61f1529": "\\nu_{\\rm xz}", "b54e8e32c5ee71b93d0b366aba8d3ddf": "2.4 M_\\odot", "b54e9644666eda03df33de248fec46e8": "f^{\\prime\\prime}(x).", "b54ef194d7d81725f4c99a315e9cf79e": "R'_w", "b54f1682614b39af5acd7706e88d54a2": "f(a) \\equiv 0 \\, \\bmod{\\mathfrak m}", "b54fa61736bd2d5c11e5dbc348f669e9": "\nL(\\chi,s) = \\sum_{n=1}^\\infty \\frac{\\chi(n)}{n^s}\n", "b5504dfd07db78004d570e40181e156c": "\\left(\\frac{\\alpha}{\\lambda}\\right)_3 = \\left(\\frac{\\alpha}{\\pi_1}\\right)_3^{\\alpha_1} \\left(\\frac{\\alpha}{\\pi_2}\\right)_3^{\\alpha_2} \\dots", "b55056b60a379ca95cd29604f07b4568": "\\begin{cases}\\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} & \\text{if}\\ \\xi\\neq0,\\\\ \\frac{12 \\sqrt{6} \\zeta(3)}{\\pi^3} & \\text{if}\\ \\xi=0.\\end{cases}", "b5505e6bb2a9b20c19666b053716f3f3": "X_t = e^{-t} W_{e^{2t}}", "b5508248c4f113bb733b0c5653359cb5": " i\\rho n\\omega U_n = -C_n +\\mu \\left(\\frac{\\partial^2 U_n}{\\partial r^2} + \\frac{1}{r} \\frac{\\partial U_n}{\\partial r}\\right) \\, .", "b550aecfe5039e3b6d00053df822ad64": " \\sum_{n=0}^{\\infty} 2^n(1/2^{2n + 2}) = \\frac{1}{4} + \\frac{1}{8} + \\frac{1}{16} + \\cdots = \\frac{1}{2} \\,", "b5517476f2d3fa86fca4d4e9c1994abe": "\\beta + \\bar{\\nu}_\\text{e}", "b55197d655cf582483061cd10efc6457": "[\\overline{C_1},\\overline{S_1}] \\xrightarrow{d} \\mathcal{N}([C_1,S_1],\\Sigma/N)", "b55199e17a654fa4f49bea75fe962569": "L\\ :=\\ L\\ +\\ 256", "b551f22c2340c2759698ddff70ea06b3": "\\rho(p,q;t)", "b5520eb3b46e58f0c2a923292cfa0fd7": "\\cot\\frac{\\pi}{60}=\\cot 3^\\circ=\\tfrac{1}{4} \\left[(2+\\sqrt3)(3+\\sqrt5)-2\\right]\\left[2+\\sqrt{2(5-\\sqrt5)}\\right]\\,", "b55282257ddfcf307334d6f44b16072d": "\\gamma(1) = y\\,", "b55326243a4f048c5acc9d47fe0f1153": "\\nu_{g}(\\lambda)", "b5536a3dd03f39ca1313a2991ac89818": " \\mathbf{F}_{RR} = \\mathbf{B}_R^T \\mathbf{f} \\mathbf{B}_R ", "b5537a450bbdcb90c7b806a283613507": "M_p(x_1, \\dots, x_{n \\cdot k}) = M_p(M_p(x_1, \\dots, x_{k}), M_p(x_{k + 1}, \\dots, x_{2 \\cdot k}), \\dots, M_p(x_{(n - 1) \\cdot k + 1}, \\dots, x_{n \\cdot k}))", "b553d32b903db4fec7f7041cd0aa3f2a": "1_{c}", "b553e5e50285981e9ae45f98b0b59005": " \\mathbf{n} = \\mathbf{k} \\times \\mathbf{h} = (-h_y, h_x, 0)", "b553e6fcd2b96d7db8988e930735a787": "u=\\mathrm{tanh(x)}", "b55407894f2e361fdfae1a364bdeed61": "[Ca^{2+}]", "b55433dc79ce75599a4f4ee28e4fe5f3": "B_1 = \\begin{bmatrix}1 \\\\\n0 \\\\\n0\\\\\n0\\\\\n\\vdots \\\\\n0 \\\\\n0 \\\\\n\\end{bmatrix}", "b5543eac9c3854311e9674fd59047896": "F_{s,t}", "b55497653615abb12afe2b02bfd6b1c0": "\n\\mathrm{Gyr}[\\mathbf{u},\\mathbf{v}]=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & \\mathrm{gyr}[\\mathbf{u},\\mathbf{v}]\n\\end{pmatrix}\n", "b554b2f1422fbb44548bdf8d4719e2c5": "E_{fs}=-(mc^2\\alpha^4/(2n^3))[1/(j+1/2)-3/4n]", "b554be435bd00946e7e894be2c9e2463": "\\pi _T > 0 ", "b554e8a10e609483ef3ea3d14f4b3fb0": " S(\\mathcal{N} \\rho)", "b55524184371aa07d12bad1953a4218d": "\\varprojlim{}^1A_i=\\mathbf{Z}_p/\\mathbf{Z}", "b55564d54baeba580f041520c442c061": "\\eta_g()", "b555782b59f3ee643b8eb3a7b97b19fa": " A_o = \\frac {T_m}{T_m + T_d} \\begin{cases}A_o = Operational \\ Availability \\\\ T_m = Mission \\ Duration \\\\ T_d = Observed \\ Down \\ Time \\end{cases}", "b5559c76e1d288c05115a5cd638869ca": "p(\\mathbf{E}|\\mathbf{\\theta},\\mathbf{\\alpha}) = \\prod_k p(e_k|\\mathbf{\\theta})", "b556305f7f04af67acb7bc7c6ca1360a": " \\gamma = \\gamma' = \\beta(\\delta -1)=\\nu(2-\\eta) .", "b55687725a5cdeb864cc8738979c239a": "\\int x(ax + b)^n \\, dx= \\frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} + C \\qquad\\text{(for }n \\not\\in \\{-1, -2\\}\\mbox{)}", "b556e8bb6e9e62a57749b3b312bba994": "ns+at=1", "b556fdc077fb1b476e794a91cab1b254": "s_{\\Lambda}=\\sum_{i=1}^m x_i", "b557025233da6a9fe18e5fb6c49c0223": "dl^2 = \\gamma_{\\alpha \\beta} \\,dx^\\alpha\\, dx^\\beta,\\,", "b557292c5b9f56146b7f2d1c630bfca2": "\\mathbf{N}(t) = \\frac{\\mathbf{T}'(t)}{\\|\\mathbf{T}'(t)\\|} = \\frac{\\mathbf{r}'(t) \\times \\left(\\mathbf{r}''(t) \\times \\mathbf{r}'(t) \\right)}{\\left\\|\\mathbf{r}'(t)\\right\\| \\, \\left\\|\\mathbf{r}''(t) \\times \\mathbf{r}'(t)\\right\\|}.", "b55801e55d8b2a8dbe619f515a07626d": " a \\in \\mathbb{R}", "b5586f36c7d5416f4c7f3eb1adab90a8": " (g_m R_S \\gg 1)", "b558a385bb1efbc390701bb82874a578": "v=v_{O'|O}=-v_{O|O'}", "b558cd29fec8b0881bc42476972eebfb": "\\psi(t)=0\\,", "b558cf7e148e39ee538ecb77a32404e9": "c=1/4", "b558ea8ccf34dd8a099a5f6bd47872a0": "d=3", "b558fd167b47a0c3e5ddd120fea78030": " \n\\begin{cases}\n(q, \\omega, (s', ta(s'), 0))\\in \\Delta& \\textrm{if } ~\\delta_{ext}(s,t_s,t_e,x)=(s',1),\\\\\n(q, \\omega, (s', t_s, t_e))\\in \\Delta& \\textrm{otherwise, i.e. } ~\\delta_{ext}(s,t_s,t_e,x)=(s',0). \n\\end{cases}\n", "b5592c8f6d66c20ea40c3b56c9a2adf7": "\\|d\\varphi\\|^2", "b5593c9e9294bbcf67535541ebe12236": "\\sqrt{\\varphi} = \\sqrt{\\varphi^2 - 1}", "b55948cfddfd9327698141d3e34a9526": "(x_p,\\, x_m) \\times (y_p,\\, y_m) = (x_p \\times y_p + x_m \\times y_m,\\; x_p \\times y_m + x_m \\times y_p)", "b55955b209563603fa10745017411764": "ds = 0", "b55962c92133939f596a96e3429e2a30": "G/\\Gamma", "b55990f82b8f3281c4da0faa8cd9a2aa": "\\operatorname{Li}_{-3}(z) = {z \\,(1+4z+z^2) \\over (1-z)^4}", "b55a2591359f0c7f1d65f016d6a36209": "\\sqrt{3}=1.7320508075689...", "b55a331282202e89499af372c0cabc3a": "b,", "b55a68c56c465af21d8cc7c82db9f943": "\\mathbf{v}(t) = \\frac{dR}{dt} \\begin{bmatrix} \\cos (\\omega t + \\pi/4) \\\\ \\sin (\\omega t + \\pi/4) \\end{bmatrix} + \\omega R(t) \\begin{bmatrix} -\\sin(\\omega t + \\pi/4) \\\\ \\cos (\\omega t + \\pi/4) \\end{bmatrix}", "b55a897031cd98fc7e559721ab334151": "a*(1-e)", "b55b21896c012dfc06e3e1d22c41a157": "I^*=\\{x^*:\\sup_{x\\in I}(x^*x-f(x))<\\infty\\}", "b55b85c6c6381a2612e9b5414096aebf": "\\begin{align}\nd\\mathbf{X}^2&=d\\mathbf X \\cdot d\\mathbf X \\\\\n&= \\mathbf F^{-1} \\cdot d\\mathbf x \\cdot \\mathbf F^{-1} \\cdot d\\mathbf x \\\\\n&= d\\mathbf x \\cdot \\mathbf F^{-T}\\mathbf F^{-1} \\cdot d\\mathbf x \\\\\n&= d\\mathbf x\\cdot\\mathbf c\\cdot d\\mathbf x\n\\end{align}\n\\qquad \\text{or} \\qquad\n\\begin{align}\n(dX)^2&=dX_M\\,dX_M \\\\\n&= \\frac{\\partial X_M}{\\partial x_r}\\frac{\\partial X_M}{\\partial x_s}\\,dx_r\\,dx_s \\\\\n&= c_{rs}\\,dx_r\\,dx_s \\\\\n\\end{align}\\,\\!", "b55b9211149200f23cec9b1cc71c941d": "\\frac{\\partial M_r}{\\partial c}= \\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r^2}\\sum_{k=0}^{r-1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta + k} - \\frac{2}{c + 1 + k}\\right).", "b55bdbd0e413382285a3e68cae3f06d4": "\n \\begin{bmatrix}\n 1 & -\\mathbf{c}^T_B & -\\mathbf{c}^T_D & 0 \\\\\n 0 & I & \\mathbf{D} & \\mathbf{b}\n \\end{bmatrix}\n", "b55beff7729de6fad042a51f5b45cf13": "(w,d)", "b55c2713f905a9358d09d2ff92607500": "\\sqrt{2 \\log p}", "b55c49a8c1e3987b296af9639926156d": "\\Delta T = Q \\times R_{\\theta}\\,", "b55c4a3034263c412a87f556d44d7e45": " \\|x - z\\|^2 = \\|x\\|^2 - 2x\\cdot z + \\|z\\|^2,\\, ", "b55c99d189c3eeb760a22fc9b00eb655": "L(\\mathbf{q},\\dot{\\mathbf{q}},t)", "b55cef16fb1acc2e1cf64f7d87271d05": "Happens(a, t)", "b55d0ede02aca1f78c09a7e9b737fbe8": "\\! M.", "b55d23ea86203094d3a57ec2dbacd5b8": "\\scriptstyle \\forall f \\in F", "b55d67cff8e9e4a13c1a024ebb6caecb": "|f_j(x) - f_j(z)| < \\epsilon / 3 \\,", "b55d7bd389b3c7db3354d4bde0dc90d5": " O_{O}^{x} \\leftrightarrow V_{O}^{\\bullet\\bullet} + 0.5O_{2} (g) + 2e'", "b55dfe709f37544d8223c8fac431ea3d": "\\lVert\\boldsymbol{\\omega}\\rVert \\ll 1 ", "b55e0184b6bf3374a2c0bd54a95d4002": "\\Pr(1\\text{ head }) = f(1) = \\Pr(X = 1) = {6\\choose 1}0.3^1 (1-0.3)^{6-1} \\approx 0.3025 ", "b55ee80477b930cb4efc0c8358cbe8bc": "\n\\widehat{\\mathcal{R}}_S(\\mathcal{H}) \n= \n\\frac{2}{m}\n \\mathbb{E} \\left[\n \\sup_{h \\in \\mathcal{H}}\n \\left|\n \\sum_{i=1}^m \\sigma_i h(z_i)\n \\right| \\ \\bigg| \\ S\n\\right]\n", "b55ef753868952d12da8e1fe7f128e91": "r={\\ell\\over{1-e}}\\,\\!", "b55f019b53c7b64f82042c744d7aa790": "A_y(\\mathbf{r},t) = \\frac{\\mu_0}{4\\pi} \\int_\\Omega\\frac{j_y(\\mathbf{r}',t_r)}{|\\mathbf r - \\mathbf r'|}\\,{\\rm d}^3\\mathbf{r}'", "b55f0d5f2caa23c2b230ba754a160eb3": " M = \\prod_{ i\\in I }M_i/U.", "b55f1e2562d04efe88ca89d17a7f2ce4": "\\gamma_{\\mathrm{la}}\\ >\\ \\gamma_{\\mathrm{ls}} - \\gamma_{\\mathrm{sa}}\\ >\\ 0", "b55f5c009e4319c4857333557bd5a2c7": "P=|\\mathbf{v}+\\mathbf{w}|^2", "b55f6a1cc49fe913cf382a41c4782ec8": "t = \\log\\left(\\frac{(1-p)q}{(1-q)p}\\right).", "b55f6e2fcc43bc66174535284aeda1d0": "B = \\gamma B_0", "b55f7f2aa87e3969147cee9aef64b1f6": "g^{(2)}", "b55fcf79eed30434755558266245edc6": "T: W^{1,p}(\\Omega)\\to L^p(\\partial\\Omega)", "b56029b66e7a8f9f347604b35a9359b3": "\\ p'=\\frac{1}{3}(\\sigma_1'+2\\sigma_3')", "b5608a0848bd171bb644b4ac3a2ff9b6": "2^{(-1)^s\\,\\times\\,E}\\,\\times\\,M\\ ", "b56096894d618d9cbbdb36e9b93ac573": "\nP=\\left(\\frac{EV}{X}\\right)\\sin(\\delta)\n", "b560ab73d8e81a27a95ebe3820d4a77d": "\\begin{cases}\n \\frac{1}{b - a} & \\text{for } x \\in [a,b] \\\\\n 0 & \\text{otherwise}\n \\end{cases}", "b560e3f99f391ef60f87d7a5ff7fd652": "P_{diss}=T_a \\dot S_i", "b560ea169d42024b739cdc4dd38551e1": "F_2 ={x}^{2}-sx+ \\frac{b}{2} + \\frac{s^2}{2} + \\frac{c}{2p}", "b561256f6ad6aa6af1aee444574b704e": "\\mathrm{III}_T(t/a)=|a|\\,\\mathrm{III}_{aT}(t)", "b5617d37a902ff804217410ec54ae0c9": "X^\\mathrm{opt} = \\frac{(\\mu - Wr_f)}{(r-1r_f)^TV^{-1}(r-1r_f)}V^{-1}(r-1r_f).", "b561ebfeace709dc294814a0acbb06ec": "\\delta_H = \\frac{\\Delta R}{R}=\\frac{R_{\\uparrow\\downarrow}-R_{\\uparrow\\uparrow}}{R_{\\uparrow\\uparrow}}=\\frac{(\\rho_{F+}-\\rho_{F-})^2}{(2\\rho_{F+}+\\chi\\rho_N)(2\\rho_{F-}+\\chi\\rho_N)}.", "b5620eb67498d4cee040829f1bc1d29b": " n_r=dim(\\hat{\\mathbf{x}}_r(t)) ", "b5625820dca72ea7fb78b753064e1555": "s, f: P\\rightarrow 2^{P}", "b5629da22d0111b860dd528e61f4edd2": "\n\\begin{pmatrix}\n\\frac{\\cos^2\\gamma}{I_1}+\\frac{\\sin^2\\gamma}{I_2} &\n\\left(\\frac{1}{I_2}-\\frac{1}{I_1}\\right){\\scriptstyle \\sin\\beta\\sin\\gamma\\cos\\gamma}&\n-\\frac{\\cos\\beta\\cos^2\\gamma}{I_1}-\\frac{\\cos\\beta\\sin^2\\gamma}{I_2} \\\\\n\\left(\\frac{1}{I_2}-\\frac{1}{I_1}\\right){\\scriptstyle \\sin\\beta\\sin\\gamma\\cos\\gamma}&\n\\frac{\\sin^2\\beta\\sin^2\\gamma}{I_1}+\\frac{\\sin^2\\beta\\cos^2\\gamma}{I_2} &\n\\left(\\frac{1}{I_1}-\\frac{1}{I_2}\\right){\\scriptstyle \\sin\\beta\\cos\\beta\\sin\\gamma\\cos\\gamma}\\\\\n-\\frac{\\cos\\beta\\cos^2\\gamma}{I_1}-\\frac{\\cos\\beta\\sin^2\\gamma}{I_2} &\n\\left(\\frac{1}{I_1}-\\frac{1}{I_2}\\right){\\scriptstyle \\sin\\beta\\cos\\beta\\sin\\gamma\\cos\\gamma} &\n\\frac{\\cos^2\\beta\\cos^2\\gamma}{I_1}+ \\frac{\\cos^2\\beta\\sin^2\\gamma}{I_2}+\\frac{\\sin^2\\beta}{I_3} \\\\\n\\end{pmatrix}.\n", "b562ab9f9905d559d12c5623a0480bb0": "\n\\frac{H^2}{M} = 0.01\\,\\text{eV}.\n\\,", "b5633c502df6b9c64af8e715675e9925": "\\bar{v_i}=\\left( \\frac{\\partial V}{\\partial m_i} \\right)_{T,P,n_{j\\neq i}}.", "b56345a50292f47f654a3eb76a93bf96": "y(\\hat\\theta): \\Theta \\rightarrow Y", "b56353d4c23d056b3982a56ef5933725": " -v'(0) + a v(0) = 0, \\quad v'(L) + b v(L)=0.\\,", "b5636cc667397f17d07fcf4e9610219d": "\\bigl( \\begin{smallmatrix}\\\\ 0&1\\\\ 0&0\\end{smallmatrix} \\bigr)", "b5636e9023aa770fd7b0d6b208363ee6": "r/f^n, \\, r \\in R , \\, n \\ge 0", "b5638bf42985093933f1966b00bff7a1": "S_{CHSH}", "b5639cb63fe5f4ffc78fda1ffcb31602": "\nH = \\begin{bmatrix}\n 1 & 0 & 0 & 1 & 0 & 1 & 1\\\\\n 0 & 1 & 0 & 1 & 1 & 0 & 1\\\\\n 0 & 0 & 1 & 0 & 1 & 1 & 1\n \\end{bmatrix}.\n", "b563b4b2a05fa5debbe6a1e605195ccc": "m_i \\mathbf {a}_i", "b563cba5d8b737e50b9e733a03226435": " \\Delta_i = \\dfrac{1}{|C_i| |C_i - 1|} \\underset{x , y \\in C_i, x \\neq y}{\\sum} d(x,y) ", "b563cef45b7d9dd48cc279303a3341b3": "f = 0 \\text{ on } S", "b5644e7081ae879fca36699a5f990337": "p[Y,Z] > p[Z,Y]", "b564adbbb566f444252c9489a394b7d5": "\\mathrm{SO}(p+4)\\,", "b565238e2e7e7ea81782e382d69b6c80": "\\tfrac{1}{\\sqrt{2}}", "b565317a0e4f61771df1683cfd611332": "\\phi = \\forall x \\exists y \\; \\psi(x, y)", "b56546a86ab832a9b2a5b15f96519319": "\\Delta x", "b56573c0ea1866096bcda94811c3840a": "\\left. a\\right\\rangle", "b5663ad6b80031ac72445514da7824e0": "\\ a_j ", "b566499503ea4af4acd1687d8d445a12": "H_{\\text{cv}}=\\sum _{i\\neq j} V_{\\text{cv}}^{\\text{ij}}a_{c_i}{}^{\\dagger }a_{v_j}{}^{\\dagger }a_{v_j}a_{c_i}", "b5667b46e4cc293d8c130e4c70bb0da0": "Z^{-2}_4", "b566d682735aa82795857debbaa33e1a": "\n\\frac{d}{dt} \\left( r^{2} \\dot{\\varphi} \\right) = r (2 \\dot{r} \\dot{\\varphi} + r \\ddot{\\varphi}) = r a_{\\varphi} = 0\n", "b566e2db91de64240d9a02bf8442b226": "x_{(1)}=3,\\ \\ x_{(2)}=6,\\ \\ x_{(3)}=8,\\ \\ x_{(4)}=9,\\,", "b5673211f9fdd77156adf5b733f9a59f": "-\\mathrm{i}\\eta", "b5673e6f96292e6b27f1d52892227a7a": "E \\left (\\sum_{g \\in G} n_g g \\right ) = \\sum_{h \\in H} n_h h \\ \\ \\ \\text{ for } n_g \\in k ", "b567722bea2f0928d4a0a68d9bf58dc7": "= \\omega^2 v t \\left(\\cos\\alpha, \\sin\\alpha\\right )=\\omega^2 \\mathbf{r_B}(t) \\ .", "b567ae9adc29314111b84a9578d4695e": "\\| u_j \\| \\leq 1 .", "b567af445f78514da30cd41c5103e444": "\\mathrm{Mo} = \\frac{g\\mu_c^4}{\\rho_c \\sigma^3}.", "b567cb2a054af0bde90a9d024cb088a5": " \\sigma ^2 = \\langle R_x^2 \\rangle - \\langle R_x \\rangle ^2 =\\langle R_x^2 \\rangle -0 ", "b567ccfd2bd6802177ed27366aa085ae": "r(t) = ce^{i(a/c)t}", "b567e6c34ca15a11d3b7eb6b67fca596": "N= {t \\choose e} = {t! \\over {e!\\,(t-e)!}}.", "b5681d2305e8a29bb01ccd0536a767f0": "[\\omega]_\\times", "b56829338d2afdfa6ed68d5e1ca419e4": "e^{j(\\pi+2k\\pi)}=K\\frac{A_1 A_2 \\cdots A_ne^{j(\\theta_1+\\theta_2+\\cdots+\\theta_n)}}{B_1 B_2 \\cdots B_m e^{j(\\phi_1+\\phi_2+\\cdots+\\phi_m)}}=K\\frac{A_1 A_2 \\cdots A_n}{B_1 B_2 \\cdots B_m}e^{j(\\theta_1+\\theta_2+\\cdots+\\theta_n-(\\phi_1+\\phi_2+\\cdots+\\phi_m))}", "b56878d58249eb3e838fb50a79e5bf22": "E_{\\rm ext}= E_{\\rm sc} \\cdot \\left(1+0.033412 \\cdot \\cos\\left(2\\pi\\frac{{\\rm dn}-3}{365}\\right)\\right),", "b56886db3c9d33fe297e8fc6f110cc08": "(B f)(z) = \\int_{D} \\frac{(1 - | z |^{2})^{2}}{| 1 - z \\bar{w} |^{4}} f(w) \\, \\mathrm{dA} (w),", "b568c0c3b9b778f5d2de848448f056e8": " \\and (S_3 \\implies (\\operatorname{equate}[A_3, f] \\and V[F_3] = A_3)) \\and D[F_3] = K_3 ", "b5692e090382eae290cbfe7cf04a2f16": "H^*(M,\\mathbb{R})", "b5697c684aadbdb56993973b0ffba1fa": "\\operatorname{mult} \\equiv \\lambda m.\\lambda n.\\lambda f. m\\ (n\\ f)", "b569ea09e6888720fdc34972c54761a9": "\\omega_{\\text{C}}=2\\pi f_{\\text{C}}", "b56a00802fe0784956ad63475c138994": "\\mathrm{MA} = \\frac {F_w}{F_i} = \\frac {\\cos \\phi} { \\sin (\\theta - \\phi ) } \\,", "b56a008a68cf34511ac8e913dce8ab8f": "\\ \\ln{[A]} = -kt + \\ln{[A]_0}", "b56a08ea423db9a934600049c0ec5345": "O(h^4)", "b56a0b128cb79cea58479c5253709e23": "p_m", "b56a36812ed1b8fe38b62c81a7e82f94": " \\sqrt{1 - v^2/c^2} ", "b56a5cecf0e777246615610b764ee92b": "dq = \\lambda(\\boldsymbol{r'})dl'", "b56a819305ab7f9e2f3ef399bfd11482": "N(xy) = N(x)N(y)", "b56a85cc63dd584acf92bb267a16c2c1": " 2 \\, \\int e^x \\cos x \\,dx = e^x\\sin x + e^x\\cos x, ", "b56ab2e02c53a275f69437e0ce33afbc": "W \\to TW", "b56af68bc0881292eaa0c63d3ca76fb5": "H_0 : \\theta \\le \\theta_0", "b56b21f3225c9429433774a8c1eee307": "Trips = a + b * Area", "b56b5b5573fe084234787c9d22620790": "n>2\\;", "b56b6c076ce34bb6cb06a9e42ce9d645": "C^{(\\beta)}_D(\\{x\\}) = \\{d|x \\epsilon A_X^{(\\beta)}(\\{d\\})\\}", "b56b85bb53c97eed82f7b37da8e57797": " = c*D + K (D/EOQ) + h (EOQ/2) ", "b56b8c9a33e4814bf35edc1ea73bbc58": "\\frac{{6 \\choose 2}{42 \\choose 3}}{{49 \\choose 6}}\\approx\\frac{1}{81.207}", "b56b9a870c6796c0add9f0e4bdceb535": "f(\\phi(t_1), \\cdots, \\phi(t_n)) = \\phi(f(t_1, \\cdots, t_n)) = 0", "b56bbce22b62c41037e72aa0a92266f4": "k = A (T/T_0)^n e^{{-E_a}/{(RT)}}", "b56c13c44064324aa8b6b56295e8f731": "g(x)=\\begin{cases}\nx\\ln x & x>0 \\\\ \n0 & x=0 \\\\ \n+\\infty & x<0, \n\\end{cases} \n\\,", "b56c24adeec880bb838545ad2ce3809f": "\\exp(i\\omega_nt)", "b56c8ed2a6b4933d547a2e7b56b3dd9c": "[2^r, r, 2^{r-1}]_2 ", "b56cd64a67234ca76e122a5e4fea87eb": "\nV(x)=\\infty ,\\qquad x<-a\\qquad \\qquad (\\mathrm{i})\n", "b56d3e2a8cc0795757960db1ec2ab900": "\\Delta G_{reaction}^\\ominus = \\sum \\Delta G_{\\mathrm f \\,(products)}^{\\ominus} - \\sum \\Delta G_{\\mathrm f \\,(reactants)}^{\\ominus}.", "b56d4a19a38c5889b8e591c4c327b9cf": " \\mathrm{sat}(T \\cap k[x_1, \\ldots , x_i]) = \\mathrm{sat}(T) \\cap k[x_1,\\ldots , x_i] ", "b56d77399aac837cb601949bd6ff72a6": " x^{(1)} = \n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 1.0 \\\\\n 1.0\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.5000 \\\\\n -0.8636\n \\end{bmatrix}. ", "b56da18a0da8c58d13080a401a3850cd": "\\mathbf{x}\\,", "b56dad9e7a00cd4202979add2bb4c3dd": "RSS = \\sum_{i=1}^n (y_i - f(x_i))^2, ", "b56dea67d4fe0a1937c2d467b9354feb": "\\frac {\\log 3}{\\log 2} = \\frac{\\sum_{n=0}^\\infty \\frac{1}{2^{2n+1}(2n+1)}}{\\sum_{n=0}^\\infty \\frac{1}{3^{2n+1}(2n+1)}} = \\frac{\\frac{1}{2}+\\frac{1}{24}+\\frac{1}{160}+\\cdots}{\\frac{1}{3}+\\frac{1}{81}+\\frac{1}{1215}+\\cdots} ", "b56e264b3f650350da9622a88e9325d3": "g_0^-", "b56e703d9a5e889c9ef2138f3e45a35b": "\\mathbf{e} = (e_1, e_2, \\dots, e_n) \\in \\mathbb{R}^n", "b56e8d865984fcd82f75e056885ad9df": "g^{x_1 + x_3 + x_4}", "b56e9b6357bff15d91503e7dc28f76d0": "\\neg(R \\and \\neg S)", "b56f1e986804834a7d33b263c071cd00": " \nPSRR(dB) = 20 \\log_{10}\\left({\\Delta V_\\mathrm{supply} \\over {\\Delta V_\\mathrm{out}}} \\cdot A_v\\right)\\mbox{dB}\n", "b56f23e6b7c9de5b867d65e6a971fb81": "K\\left(\\pi\\left(x,y\\right)\\right) = x", "b56f9675d930c935bc564029c006c3d6": "\\mathrm{NA} = \\sqrt{n_\\text{core}^2 - n_\\text{clad}^2},", "b56fc4b20f6979a169a498565be5bb7f": "x^2+3x-2", "b56fcdd407b69b7c0fde2740d581e1ce": " \\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_z) = I - \\frac{i}{\\hbar}\\Delta\\theta \\widehat{L}_z", "b5705206106078f4060031f5d23a89b7": "\n\\begin{align}\n\\overline{X_t} &\\leq \\sum_{i=1}^{[at]} \\mathrm{Geometric}(p) \\\\\n\\mathbb{E}\\left[\\,\\overline{X_t}\\,\\right]^2 &\\leq C_1 t + C_2 t^2 \\\\\nP\\left(\\frac{X_t}{t} > x\\right) &\\leq \\frac{E\\left[X_t^2\\right]}{t^2x^2} \\leq \\frac{E\\left[\\overline{X_t}^2\\right]}{t^2x^2} \\leq \\frac{C}{x^2}.\n\\end{align}\n", "b5708a94d6f531c82f5acd306bf67daf": " E (b x^a) = a \\ ", "b5708f9dc17c864800c6ba5664e668a8": "\\left[ C_{j'} \\right]", "b570cd8d264c728274987f95a6f16b35": "\\, e^{ta}", "b570ef901c5add3aceb857788f676fb8": " f'(\\infty):= \\lim_{z\\to\\infty}z\\left(f(z)-f(\\infty)\\right) ", "b5710837e6bf2a13b1aba5708297c2ac": "\\sigma^2_x", "b5711a2c87bb7d493510fb0937a32e83": "{\\mathrm{Pic}}^0", "b571277c47689cbc9d18c5ad188a6810": "\\beta_S=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_S\n\\quad = -\\frac{1}{V}\\,\\frac{\\partial^2 H}{\\partial P^2}", "b5713ce152cc921666a3d543cbc566e0": "\\sum_{i=1}^n \\log(i)^c \\cdot i^d \\in \\Theta(n^{d+1} \\cdot \\log(n)^{c})", "b571a38a82cf057d9ece9fb7f357a292": "\\forall n\\in \\mathbb N^* \\quad \\forall u \\in\\mathbb R_+ \\qquad e^{-u} \\leqslant (1+u/n)^{-n} ", "b571afaca3e7b33fbad4e46dbbec3955": "\\theta = 0.90{\\pi\\over 4 }", "b571bfd5956b988de94aa3e5c0757fcd": " i\\hbar\\frac{\\partial }{\\partial t}\\Psi = \\hat{H} \\Psi ", "b571dba34a3bcbcea537a1f2073491b0": "\\cos(A \\pm B) = \\cos(A) \\cos(B) \\mp \\sin(A) \\sin(B) ", "b571ea0216ca9ea4f93f840b04ecdf41": "n_1, n_2", "b571ef1516fed518679621831ae47423": "\\alpha_{n-1}", "b572607bac523af2dbec9529328b93e7": "I_W = \\frac {V_C} {\\sqrt{G_S}}.", "b5727b63e2216b7b9b13e57cca562853": "\n \\Gamma_{ijk} = \\frac{1}{2}(g_{ik,j} + g_{jk,i} - g_{ij,k})\n = \\frac{1}{2}[(\\mathbf{b}_i\\cdot\\mathbf{b}_k)_{,j} + (\\mathbf{b}_j\\cdot\\mathbf{b}_k)_{,i} - (\\mathbf{b}_i\\cdot\\mathbf{b}_j)_{,k}]\n", "b57282d86f86286698465d5b0452a0f5": "Q - W\\;", "b572890fa5547318444e6b929fb1dd7d": " \\left(\\rho,R,L\\right) ", "b5734b93904d340f8771c8c26115d4ef": "\\partial W = i(M) \\sqcup j(N)~.", "b57370ef42c5eeb2c3f6e4a96e89034d": "i_\\alpha : C_\\alpha \\to X\\qquad \\alpha \\in A.", "b573762a6864e22bd17685294fb15416": "p'_x(a,b)=p'_y(a,b)=0,", "b5738acdf98625ce3e7f4be0774509d7": "f(v) = s(e^v)", "b57456ea79d674c253d523219159a39b": "\\frac{\\partial \\mathbf{f(g)}}{\\partial \\mathbf{g}}", "b5745b3a657715b4f37e9536e88a94bd": "a(t) = (22\\cdot 4^t-6(-4)^t-4+3(-1)^t)/15", "b574a848490754a0e717730eb9acc00a": " n_i ", "b57595842e1238fa7d2a834dc886b21e": "E\\left({\\mathbf{x}}(0){\\mathbf{x}}^\\mathrm T(0) \\right)", "b57597b6992c1fdf164d2cd2fd22e897": "\n\\, \\cos \\phi_\\mathrm{s} = \\frac{\\sin \\delta \\cos \\Phi - \\cos h \\cos \\delta \\sin \\Phi}\n {\\cos \\theta_\\mathrm{s}}\n", "b5759e863eee0afa9c5e8a42a9b9868a": "F = dp/dt = 0", "b575a5547b7f2e7892513a3487eb8cc4": "x = x_1 - f(x_1)\\frac{x_1-x_0}{f(x_1)-f(x_0)}", "b575cc9060e978a486efe6c32b8c3e41": "\\overline Y'", "b5760b3219193c188bd1aa5c92f3a880": " \\mathbf N = \\mathbf R \\times \\mathbf P ", "b5764cb805fbfe65b4d37e052f428722": "[J_f](\\mathbf{[x]})^{-1}\\cdot f(\\mathbf{y}))", "b5766f640fbe108c229c4309b21d7fda": "\nx_{\\mathrm{rms}} =\n\\sqrt{ \\frac{1}{n} \\left( x_1^2 + x_2^2 + \\cdots + x_n^2 \\right) }.\n", "b576c354f80be0fd47b85185682f30e5": "-z \\left[ \\log z+\\gamma z +\\sum_{k=1}^{\\infty} \\Bigg\\{ \\log\\left(1+\\frac{z}{k} \\right) -\\frac{z}{k} \\Bigg\\} \\right]", "b576e479e45cb450b6d5d400ea9235b0": "\n\\begin{align}\n\\alpha & = \\frac{x_1 + x_2 + \\cdots + x_n}{n} \\\\[6pt]\n& = \\frac{\\frac{m}{n} \\left( x_1 + x_2 + \\cdots + x_n \\right)}{m} \\\\[6pt]\n& = \\frac{x_1 + x_2 + \\cdots + x_n + \\frac{m-n}{n} \\left( x_1 + x_2 + \\cdots + x_n \\right)}{m} \\\\[6pt]\n& = \\frac{x_1 + x_2 + \\cdots + x_n + \\left( m-n \\right) \\alpha}{m} \\\\[6pt]\n& = \\frac{x_1 + x_2 + \\cdots + x_n + x_{n+1} + \\cdots + x_m}{m} \\\\[6pt]\n& > \\sqrt[m]{x_1 x_2 \\cdots x_n x_{n+1} \\cdots x_m} \\\\[6pt]\n& = \\sqrt[m]{x_1 x_2 \\cdots x_n \\alpha^{m-n}}\\,,\n\\end{align}\n", "b577351c874393262ef832bbc025cd20": "\\mathbf{F}(\\mathbf{r},\\mathbf{\\dot{r}},t,q) = q[\\mathbf{E}(\\mathbf{r},t) + \\mathbf{\\dot{r}} \\times \\mathbf{B}(\\mathbf{r},t)]", "b5781455f1c247290d030b4597c75165": " g = \\operatorname{value}\\ v", "b578268d31442a2a442b648ee816caaf": "\\begin{bmatrix} \\boldsymbol\\Sigma^{-1}\\boldsymbol\\mu \\\\[5pt] -\\frac12\\boldsymbol\\Sigma^{-1} \\end{bmatrix}", "b5782e836bd96005b28e0bda3c5df127": "Q\\to \\mathbb R", "b5787cd88f7886418474f65bd605b2b1": "\\!1/\\alpha \\approx 137", "b5787daf7424256fbc05ab74e69c84a6": "\\phi\\rightarrow 0", "b5789375e21eb131d3c4ac46a1d383ac": "\\psi=\\frac{X}{c+vt}\\,\\!", "b5789e71da522c4d24c38db916f53b58": "n \\lambda = 2 \\pi r.\\,", "b5790b115afbcab048e7d2fe00d7485c": "\\mu = \\mu ^\\circ + RT\\ln \\frac{P}\n{{P^\\circ }} + \\int_{P^\\circ }^P {\\Phi dP}", "b5790e70feb338c83d60b66339c268d9": "\n\\begin{align}\n\\frac{\\mathrm{d}t}{\\mathrm{d}x}&=\\frac{1}{2}\\sec^2\\frac{x}{2}\\\\[8 pt]\n&=\\frac{1+t^2}{2}\\\\[8 pt]\n\\Rightarrow\\mathrm{d}x&=\\frac{2\\,\\mathrm{d}t}{1+t^2}.\n\\end{align}\n", "b5790f6afb09c06e900201f7ea62cad8": " \\mathbf{v} = \\sum_i v^i {\\mathbf e}_i = \\sum_j {v'\\,}^j \\mathbf{e}'_j.", "b579368e7711aea24182878e63eb11ce": "l_1 \\,", "b579753dcf8b884680c3f967cbcab30d": "\\frac {1}{2} \\omega^2 = -\\frac{1}{2} U^2 \\left(\\frac{2k}{rd}\\right) \\theta^2", "b579848db719a2a8763cd8117e12538f": "\nX_{1/T}(f) = X_{2\\pi}(2\\pi f T)\\ \\stackrel{\\mathrm{def}}{=} \n\\sum_{n=-\\infty}^{\\infty} \\underbrace{T\\cdot x(nT)}_{x[n]}\\ e^{-i 2\\pi f T n}\\;\n\\stackrel{\\mathrm{Poisson\\;f.}}{=} \\;\n \\sum_{k=-\\infty}^{\\infty} X\\left(f - k/T\\right).\n", "b579d1768cd704ae726700fbed449821": "\\operatorname{Perf}(f,r)", "b57a02e8bff23b13dbea87b024ef5270": "\\scriptstyle s_{c'}", "b57a485b7e970689185f7664b58b9d61": "\\widehat{E}^2 \\psi = c^2\\widehat{\\mathbf{p}}\\cdot\\widehat{\\mathbf{p}}\\psi + (mc^2)^2\\psi \\,,", "b57a521502b8b9db257a8c2bf234d305": "\\tfrac{5}{1}", "b57a89f71c4f8f8d50cfc67dd2566075": "\\mathbf{I}_A(x) = [x\\in A].", "b57ab3c0a708bbdb6eee0ff010e893ab": "\\hat{\\beta}_{\\tau}", "b57ab7ddbc16f10ac195536ccad8daaa": " \\partial A", "b57ad96dc74b69cb78daf4cc31d961d9": "C_p", "b57b32918873851971d261344bc539d9": "x \\mapsto x^2,", "b57bf1c8aa4527eb79e84c2761a6bda0": "y-z", "b57c06412d2bb792633cb8ebe122e17c": "\\mathbb{A}=\\mathbb{R},\\mathbb{C},\\mathbb{H},\\mathbb{O}", "b57c14ef058c9ced1cf63e1f8210feab": "Cost(x=a) = \\sum_{i=1,\\ldots,n} \\min_{y_i=b} ( Cost(y_i=b) + Violates(x=a, y_i=b) ) ", "b57c1eacfa9177838e8bc41c107abe09": "S_{BPA} = \\frac{y}{xy+y+1}", "b57c2f34f6f7df53a82bbcefaa2831d2": "\\vec{N} (x)", "b57c719719847ab68b6c3c34c367a390": "\\langle k_{nn} \\rangle", "b57c858ff6b5191fd7bdcddb146a4f6e": "\\hat{H}_\\text{D} = \\hat{A}\\mathbf{I}\\cdot\\mathbf{J}", "b57c87d46b4026f482b6dde151ab378a": " U_n(r) = \\frac{iC_n}{\\rho n \\omega} \\left[ 1 - \\frac{J_0(\\alpha \\frac{r}{R} n^{1/2}i^{3/2})}{J_0(\\alpha n^{1/2}i^{3/2})} \\right] \\, ,", "b57c887ab27876807fdc91a99b456096": " d = \\left( p_{01}, p_{02}, p_{03} \\right) ", "b57c933b10b80ad4ae8b7e59146a59e0": "\\mathbb{H}^3 / \\Gamma", "b57c949c85f3313aff49254a11b397e4": "\\mathbf{D}=\\varepsilon_0 \\mathbf{E}+\\mathbf{P}", "b57cc9df53abb24ce8ac0ee2d74988cc": "\\forall x_v\\in Dom(v),\\; \\mu_{v \\to u} (x_v) = \\prod_{u^* \\in N(v)\\setminus\\{u\\} } \\mu_{u^* \\to v} (x_v).", "b57d2ec19bad7631c5924c970eadbac1": "\\mathrm{MA}=\\frac{\\tan \\varepsilon}{\\tan \\varepsilon_0}", "b57d6ce52daa3edeb526178c22b41bf2": "\\sigma_y(\\tau) = \\frac{\\pi\\sqrt{2\\tau}}{\\sqrt{3}}\\sqrt{h_{-2}}", "b57dd8855290e9233cb1bb5282da95ed": "y'' + y = t \\cos {t}. \\!", "b57dddc7cafee5e12e59c8bf4f243a74": "\\overline{I}", "b57de2b58a6a31f5d20853038e8d558b": "d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2}", "b57defdb0e0747a438482f89c69e5f88": "d\\sigma(Z)", "b57e195c9ad2d2036ecd4ff213d5d742": "\n\\begin{align}\n u(y=-y_{l}) &= g_{1}(y) \\\\\n u(y=y_{l}) &= g_{2}(y) \n\\end{align}\n", "b57e3e573977846f5261894a491f03e0": "E_m(h) \\oplus h = h", "b57e47c5db4ee8c18d62b6e01135a3c2": "p_0(x) + p_1(x) \\cdot f + p_2(x)\\cdot f^2 + \\cdots + p_n(x)\\cdot f^n = 0.", "b57e5691ade0c06aeb8470d6594f92c2": "y_n^* = \\underset{y \\in \\mathcal{Y}}{\\textrm{argmax}} \\left(\\Delta(y_n,y) + \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y)\\right)", "b57e6329a2d87473ce9c7b475c565f4c": "\\langle a,\\;b \\mid a^2 = e,\\; b^n = e,\\; aba^{-1}=b^{-1}\\rangle.", "b57eabb87a5f1593f89417a93bbb9021": " i = A_{\\mathrm{r}} J_{\\mathrm{r}} = \\int J \\mathrm{d} A, ..........(32) ", "b57eb533b3d9c18d104bb0b9a5a80bbe": "G_1", "b57eb536d6b1082f81e04e27db867e82": "b = f^{0}(1) - m = x_{1}^{0} - 2 = 8 - 2 = 6", "b57ef8a8f51e13145770c1b1f49a815d": "\\frac {\\partial \\Pi_1}{\\partial q_1} = \\bigg(\\frac{a - 2bq_1 + \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2}\\bigg) - \\frac{\\partial C_1 (q_1)}{\\partial q_1}=0.", "b57f1e5d633ec21a86d607cfe88d9d8c": "s=\\pm \\sqrt{{{\\left( {{R}_{\\text{E}}}+{{y}_{\\text{obs}}} \\right)}^{2}}{{\\cos }^{2}}z+2{{R}_{\\text{E}}}\\left( {{y}_{\\text{atm }}}-{{y}_{\\text{obs}}} \\right)+y_{\\text{atm}}^{2}-y_{\\text{obs}}^{2}}-({{R}_{\\text{E}}}+{{y}_{\\text{obs}}})\\cos z \\,.", "b57f645e15c4418cf51449d7571132e9": "v_2 = \\left( {1 \\over 2} , {-\\sqrt 3 \\over 2} \\right) \\,", "b57f8edd17f362c86b4af9c13e020c7e": " \\dot{z}(t) = \\dot{u}(t) \\left\\{A - \\left[\\gamma+\\beta\\operatorname{sign}(z(t)\\dot{u}(t)) + \\phi(\\operatorname{sign}(\\dot{u}(t))+\\operatorname{sign}(z(t))) \\right]|z(t)|^n \\right\\} ", "b57fd8be0576c706aba43eba09cca614": "\n \\overline{X}_n\\ \\xrightarrow{P}\\ \\mu \\qquad\\textrm{when}\\ n \\to \\infty.\n ", "b57ffecfbd2e3b1b06528d4388435972": " a = 1.5 ", "b58007d92aea212e5c109ba3e84faab7": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi \\epsilon_0}\\int \\frac{\\rho(\\mathbf{r}', t_r')}{|\\mathbf{r} - \\mathbf{r}'|} d^3\\mathbf{r}'\n", "b5804196f83592d3521e8e37fe5d9554": "\\operatorname{dCov}^2(X, Y; \\alpha) = 0", "b5808cdf7992ff6006d3c96dd0fea2d8": " \\langle \\psi \\mid \\operatorname{E}_A \\psi \\rangle ", "b581178eba64b79066aeee141afc5aca": "R + N = b \\ ", "b581407905ddf237cb241d145403d02f": "\\text{var}(\\hat{\\beta}_j)", "b581e00c6d20de2e8785a2d829c4861b": "\n\\tan^{-1}\\cfrac{x}{y}=\\cfrac{xy} {1y^2+\\cfrac{(1xy)^2} {3y^2-1x^2+\\cfrac{(3xy)^2} {5y^2-3x^2+\\cfrac{(5xy)^2} {7y^2-5x^2+\\ddots}}}}\n=\\cfrac{x} {1y+\\cfrac{(1x)^2} {3y+\\cfrac{(2x)^2} {5y+\\cfrac{(3x)^2} {7y+\\ddots}}}}\n", "b5825a6794db70d759fd6fa7eb747fd7": "Y_2", "b58262ba5a4433930c01382efd78fe93": "(I^2-C^1-P^2-M^3) / (I_2-C_1-P_2-M_3) \\times 2 =32.", "b5826ccea2352f6a3083ccfe47de072a": " 48842^2 - 67 \\cdot 5967^2 = 1.", "b582d37aace0ac0b2564d89428405d2a": "\\textstyle \\langle e_i, e_j\\rangle=0", "b584338cb789b1a2f45279a208429c67": "\\sqrt{\\sum_1^k \\left(\\frac{X_i-\\mu_i}{\\sigma_i}\\right)^2}", "b5844c56537f625cbb22ee72ec6d4cae": "\\textit{par}: \\textit{parent}", "b584926a1eecbe09088f2eda8fde1314": "(*) \\,", "b584f16652ca59f6dee92cdda222c857": "{\\mathcal G}=\\{\\{\\langle\\vec x\\rangle \\in {\\mathcal P}\\mid \\vec x \\in U\\} \\mid U \\text{ 2-dimensional subspace of } V_{n+1}(K)\\}", "b5851291f285b0a7ad8cdbc950be0bdd": "\\partial_\\alpha F^{\\alpha\\beta} = \\mu_0 J^\\beta_{\\mathrm e}", "b585525d0667dcb1eecd04099b4dcf69": "f: \\mathcal{X} \\to \\{-1,1\\}", "b5857704d060bca981babedbb505a0e9": "\\forall x ( \\exists y ((\\lnot x = y) \\land x R y ) ) \\land ( \\exists z ((\\lnot x = z) \\land z R x ) ) ", "b585f05c124e93ada14d5620deb1c83f": "p_n = 0", "b586353ccfd59a0c2a2e241eed01e2a6": "3\\, ", "b586ca0679ff19a636ae766ccbb094be": "D_{T}", "b587126b0e9bb6e95fee80cf2ca56df6": " \\frac{\\partial ^2 f}{\\partial z_i\\, \\partial z_j} \\geq 0 \\mbox{ for all } i \\neq j.", "b5876fd631e5ea015841e0469bb900ab": "\\|T\\|_1\\leq 1", "b5881797840eb3cb4e614eb5644011fd": "\\phi: U_0 \\to X', \\quad (1:x_1:...:x_n) \\mapsto (x_1, \\dots, x_n)", "b58840ff6674127d89f9ef77711288bb": "g: [0, \\infty)\\rightarrow R", "b5884e98f283bf5d372805bb4f502968": "\\tbinom{n+x-1}{x-1}", "b58892e40b2b9b7ceba79f0fc78a4e3d": " \\mu r \\dot \\theta ^2 -\\frac {dU}{dr} = \\mu \\ddot r. \\, ", "b588c544fe9cb8a2567b682bfc8f6a5e": "\ng(\\theta) = \\frac{1}{\\cos[\\beta \\frac{\\pi}{6} - \\frac{1}{3} \\cos^{-1}(\\gamma \\cos 3\\theta)]},\n", "b588cb269da092dd6fc8b99c55bdce01": "(ab,c)", "b588edb0f01efc5ddcb938d4e7faeaa2": "\\textstyle \\frac{\\partial C_1}{\\partial x} = -\\frac{\\partial C_2}{\\partial x}", "b5893f576fbe7e6dc93633f4e005dd76": "\\mathbb{E}[Y_n] = \\Pr(Y_n = 1),", "b5897e74f63d9c2811f4424b45c2e52c": "L_i(x)", "b5899b7aca806eb5132335f3ff0a1b8d": "p_6(x)=-248832x+103680x^2-17280x^3+1440x^4-60x^5+x^6;", "b589ff39912e1563c93f3f69aa8706c9": "\\displaystyle r_a=r_b=r_c.", "b58a266a343b78ad6e531167d68e9b5a": "\\lim_{i\\rightarrow\\infty} \\frac{f(i)}{i}=\\frac{1}{2}.", "b58a27003a8246bca74c5aa0c8d1a994": "\\beta(\\alpha^{-m})^i=\\alpha^j\\,.", "b58a3df70156589da21b6ee1a66f8a6c": "\\left(\\sum_k m_{i,k} n_{j,k} a_k\\right)^2 > 0", "b58a4f671ec26253527788d422892f25": "\nR^M_L(\\mathbf{r}_{Ai}-\\mathbf{r}_{Bj}) = \\sum_{\\ell_A=0}^L (-1)^{L-\\ell_A} \\binom{2L}{2\\ell_A}^{1/2}\n", "b58ae777dd3d3173f96e20b83f217277": "\\mathrm{Tr}^U_{U,U}(\\gamma_{U,U})=U", "b58aee0d4e57de666d62059f7a228032": "w=2^{2^t} \\pmod n", "b58b2cd6ed234211790e5abe31aadd93": "P(z) = A(z) + z^{-(p+1)}A(z^{-1})", "b58b86dba39a5f3078258e9f4bc0cf7c": "U_{--}", "b58c3695f840246334d7105036ba482d": "\\langle \\ldots \\rangle", "b58c3efc9a2ff8059d63b876132f42f9": " D\\cdot F = \\mu_0 J ", "b58c4bfe2a46b39ea6b8bcf342cc92e1": "{}^d", "b58c6cad449c2446930b3136fa5c1c83": "\n \\sigma_{11} - \\sigma_{33} = 2C_1\\left(\\lambda^2 - \\cfrac{1}{\\lambda}\\right) ~;~~\n \\sigma_{22} - \\sigma_{33} = 0\n ", "b58c8a51ae8379aace5d9d25a6d6f4bd": "k,n \\in \\mathbb{Z}, k \\ge 2", "b58cd1f13da97654764c90df7fcd49ec": "VP/G\\to M", "b58ce5d1d26a24b13bf8c1eec5fe5e4c": "-5q_p(5) \\equiv 4\\sum_{k=1}^{\\lfloor\\frac{p}{5}\\rfloor} \\frac{1}{k} + 2\\sum_{k=\\lfloor\\frac{p}{5}\\rfloor+1}^{\\lfloor\\frac{2p}{5}\\rfloor} \\frac{1}{k} \\pmod{p}.", "b58d00a792c61d33909998fcc6815a26": "\\gamma^0 = \\gamma_0", "b58d4e0df51ff5cde9ac0559c61e6960": "C^{p+q}", "b58d95462c716905858f1312422d3f39": "\\alpha = (\\alpha_1,\\dots,\\alpha_N) \\in \\mathbb{N}^N", "b58ddbf329a199641958b55696809080": "\\frac{\\partial \\ell(r,p)}{\\partial p} = - Nr\\frac{1}{1-p} + \\sum_{i=1}^N k_i \\frac{1}{p} = 0", "b58de7a77d2f47af67682e4fe74c751f": "\\scriptstyle u\\in {S\\!BV}(\\Omega) ", "b58deb8d900253c3359de971eb589a0c": " A = -\\log_{10} \\left( \\frac{I}{I_0} \\right)", "b58e7310a300fd63e121ea262e447e21": "y_{g}-y_{f}", "b58e7d33948064e898e56582fad07fab": "\\gamma \\approx 0.577", "b58ee2a4cf50298fdd56bc3f87f647da": "\\operatorname{var}(X)=\\sum_{i=1}^N \\sum_{j=1}^N w_i C(x_i,x_j) w_j.", "b58f09d94b3bc41909f99900c199833c": "{-\\hbar^2 \\over 2m} \\psi_{xx} = (E+V_0)\\psi \\,\\! ", "b58f2b59b309ed7e6b4627a60ef62bea": "\\frac{W}{m^2K}", "b58f4cd361d4a7f680f6d56491fd4eae": " V = \\frac{256}{81} r^2\\ h ", "b58fc384d67537ed01e7ede10445d683": " \\left(D + \\frac{b}{2m} - \\sqrt{\\frac{b^2}{4 m^2} - \\omega_0^2} \\right) y = 0 ", "b58fd48da47e76171af231455ad47284": "\\operatorname{Ext}^n_G( M, N\\uparrow_H^G) \\cong \\operatorname{Ext}^n_H( M\\downarrow_H^G, N)", "b58fde817df50216754662f0cd75644d": "\\scriptstyle \\hat{\\mathbf{v}}", "b5902c078230a89414303185251ec9c5": "\\ell_{(N,\\psi)}(\\bar x,\\bar y)", "b59034871cb4c9af89847678b88f09ad": "\\frac{1}{4}(1 + 4z - (1 + 2z)\\cos(\\pi z))", "b59036ea76bbbe7704169f41085a72ae": "\\frac{\\partial f}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\frac{\\partial f_1}{\\partial f_2}~\\frac{\\partial f_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u}", "b590381175fc73aa925d33f3d23bb5f7": "M_2/M_1 \\to 0", "b59069fa7cfd9003cf15e76a2a92235f": "\\isin ", "b59087f1bc422632979b352e5fccf173": "E_r = E_\\phi = H_r = H_\\theta = 0.", "b590b6f4cc4d7c2dcf1c3afc89afe591": "\\ \\Delta^r(a\\ \\alpha_{i,j,k} + b\\ \\beta_{l,m,n}) = a \\ \\Delta^r(\\alpha_{i,j,k}) + b \\ \\Delta^r(\\beta_{l,m,n})", "b5910a2ba63cf60546758737af872b68": "\\binom{N}{n} = |\\mathcal{P}_n(S)| \\ge R(\\underbrace{ m, m, \\ldots , m }_{k}).", "b59130a3e198b926c546d0aca11b97dd": "s \\leftarrow p,\\ \\hbox{not } q.", "b5913b3c185e37512c34e9116dd61d9e": "\\Gamma(\\tfrac14) = \\left(4 \\pi^3 e^{2 \\gamma -\\mathrm{\\delta}+1}\\right)^{1/4}", "b591625f552038b180cd114ca7230937": "\\; s_1", "b5916b9d8af6e55e1f434d975323a665": "\\mathbf{V} = -\\frac12{\\boldsymbol\\eta_1}^{-1},", "b59170a6e97c1165418a1e79d028131c": "\nc_n \\ \\stackrel{\\mathrm{def}}{=} \\ \\begin{cases}\n\\frac{A_n}{2i} e^{i\\phi_n} = \\frac{1}{2}(a_n - i b_n) & \\text{for } n > 0 \\\\\n\\frac{1}{2}a_0 & \\text{for }n = 0\\\\\nc_{|n|}^* & \\text{for } n < 0.\n\\end{cases}\n", "b5917133d9991ddb2f42645e5d2127e6": "\nl_{ch} = E G_f /{f'_t}^2\n", "b591aa501f4b5135c98b2eee02a0a8fa": "\nP_2(\\overline{R})=\\frac{2}{\\pi \\sqrt{1-\\overline{R}^2}}.\n", "b5925307569ff125c378d66130f22420": "I_C=M(\\alpha I_E +I_{CBO})\\,", "b5925f6ba6d30a039d222a38afadda8d": "I : J = (I : (g_1)) \\cap (I : (g_2)) = \\left(\\frac{1}{g_1}(I \\cap (g_1))\\right) \\cap \\left(\\frac{1}{g_2}(I \\cap (g_2))\\right)", "b5929daff203ba192c1343956ba40352": "x^{32} + x^{31} + x^{29} + x^{27} + x^{24} + x^{23} + x^{22} + x^{21} + x^{19} + x^{17} + x^{13} + x^{12} + x^{10} + x^9 + x^7 + x^4 + x + 1", "b592ace5923b566556b80f56e74f0a4e": "R = \\sigma\\sqrt{X}.", "b592ff845626595c1904eeefe1104665": "\\operatorname{Cl}_{6}(2\\theta) = 32\\operatorname{Cl}_{6}(\\theta) - 32 \\operatorname{Cl}_{6}(\\pi-\\theta) ", "b593702d63d2a809173537382805bc5a": "R_\\text{t} = R_\\text{p} - R_\\text{v}", "b593cc3389cbe2e6d3e2e7e265302aa9": " f_{a}\\;(x)", "b593e21b21dee802d341bd00adc87908": " a = m^2+n^2-p^2-q^2,\\,", "b593e94e4aac7597e79e63b0cdee90b4": "\\langle x, y\\rangle = \\frac {1}{2}[q (x + y) - q(x) - q(y)] = \\left(x_1y_1+\\cdots + x_ky_k\\right)-\\left(x_{k+1}y_{k+1}+\\cdots + x_ny_n\\right).", "b593ee29d4114e08e21e6a0b3a1d8933": "3x \\equiv 2 \\pmod 6\\ ", "b594bbdf48989e647ddcb16b60de8229": "E\\Psi=\\hat H \\Psi", "b5951ef16f6e76573401925bdb7ca76f": "u_{x_i x_j}", "b5955f81bd64162adbb2c30273313b99": "\\mathbf{Hr} = \\mathbf{H} \\left( \\mathbf{x}+\\mathbf{e}_i \\right) = \\mathbf{Hx} + \\mathbf{He}_i", "b5957d28bab6b3b897fe10a3525032c2": "\\frac{dx}{2y+a_1x+a_3}", "b595d22b3e4cf1a5f9287928c4b5c344": " U^{\\dagger}(t,t_0)U(t,t_0)=I.", "b595dab97c6dc0e3c396e7fa19101565": " \\forall A ", "b595dd587456b9f05eb14ea2e1d65149": " f(x) \\!", "b595e06452c4f30fa2bcaa73a00520ba": "A_1B_1,\\ A_3B_3", "b5960c986bdd55fdb39dccf68171c5cf": "V_-=V_{out}+Be^{\\frac{-1}{RC}t}", "b596216b8c374d4820c2a8f2a009f6c6": "F(9)", "b5972127f2a9e102201a5a8e9a478dfe": "S = \\begin{pmatrix}\n 0 & 0 & 1\\\\\n 1 & 0 & 0 \\\\\n 0 & 1 & 0\n\\end{pmatrix}", "b5977838b50e003ef76318e1349f662e": "\\mathbf{F} = \\mathbf{R}\\mathbf{U} = \\mathbf{V} \\mathbf{R}\\,\\!", "b59793a7331fa02f8c6cff7c57a69d33": " \\tilde{n} = 1 + .01 j ", "b597ac40d874352b2143c16ef0a37016": " \\vec{s}(C_{-1}^{(2)}) = [-1,+1,-1,+1], ", "b597b209c4fad652f5105576fc29d78b": "c_1:=\\det\\Phi(x)=x\\,y_1(x)-y_2(x),\\qquad x\\in I,", "b597d293eb2d368778634c99e52ed6c7": "\\displaystyle{\\|f|_{\\partial \\Omega_s}\\| \\le e^{Ms/2} \\|f|_{\\partial \\Omega}\\|.}", "b597eb951f099baa98018819815bbeb6": " R_{\\mu\\nu} - {\\textstyle 1 \\over 2}R\\,g_{\\mu\\nu} + \\Lambda\\ g_{\\mu\\nu} = \\frac{8\\pi G}{c^{4}}\\, T_{\\mu\\nu} ", "b598032a1eef1079f3e0969a24cba3d8": "\\mathbf{P}\\!\\!\\!\\!/ = P_\\alpha \\gamma^\\alpha = P_0 \\gamma^0 + P_1 \\gamma^1 + P_2 \\gamma^2 + P_3 \\gamma^3 = \\dfrac{E}{c} \\gamma^0 + p_x \\gamma^1 + p_y \\gamma^2 + p_z \\gamma^3 ", "b59851a0d20b597a87817c63850ac3be": "\n \\operatorname{Var} (x) = \\frac{(b-a)^2 (3-2\\theta)}{36} ,\n", "b5986c8c82229d7fe9ea3c1dd36546cc": "\n\\hat{\\bar{w}}_{1L}(s, 2z+\\gamma_{1L};L) = \\sum_{n=0}^\\infty \\bar{w}_{1L}(s, 2n+\\gamma_{1L};L)z^n = \\bar{\\Gamma}_{1L}(s)\\big[1-\\sum_{c=1}^{[L/2]} (-1)^{c+1}\\bar{h}(s,c;L)z^i\\big]^{-1}. \n", "b598813a8ce79d1c385908aa798f0899": "\nS_{xy}(\\omega)=\\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty R_{xy}(n)e^{-j\\omega n}\n", "b59887381ff15fcf883404555adb6e6d": " a^2 - b^2 = (a+b)(a-b),\\,\\!", "b59893f0c314e91cc1ee2fbe02dae701": "1/d_{xy}", "b5990ebf0079e910f2dea3ad6d867e45": "\\sigma _{\\bar x} \\,\\,\\,\\, = \\,\\,\\,{\\sigma \\over {\\sqrt {\\,n} }}\\,\\,\\sqrt {\\,\\gamma _2 } ", "b59943c454ce5cca4e85db89847b286f": "\nB_{\\max} \\le \\sup_{\\tau \\ge 0} \\{E(\\tau) - S(\\tau) \\} = (E \\oslash S)(0).\n", "b59952a19854e7cff05d55a4ac247dd7": "\\Re(\\mathit \\Gamma) < 1\\ ,", "b5996e56a2b75faf8f5e41fc04aad484": "(p,0) \\mapsto (0:-1:1)", "b59971a4abd50132818532e6cfb957d3": "||.||_\\infty", "b59a0cc893165ee3725126f8913d51b1": "\\{ \\rho_1^k \\}", "b59a9bcacc5ab1c3432f602d6d169a26": "R = 1/\\limsup_{n \\rightarrow \\infty}{\\sqrt[n]{|c_n|}}.", "b59ad048146da050e21276678f916d71": " \\Delta P = \\frac{128 \\mu L Q}{ \\pi d^4}", "b59af95b206ece4d4c99bfa09e5be6f2": "M_f=[f(x,y)]_{x,y\\in \\{0,1\\}^n}", "b59b15da2e9712d821385c9f0cbbba55": "x\\in M", "b59b6f8a89cc6febe351ac0454b72c37": " TL_i = 1 + \\sum_j (TL_j \\cdot DC_{ij})\\! ", "b59bee72c2b965aad6dc8266006b67a2": "L^x(t) =\\int_0^t \\delta(x-b(s))\\,ds,", "b59c4b52a381623c6142feff7e79d532": "d_{i+1}", "b59c66d367817424ec5d03dffd4cdddb": "\np_2 = p_{1} \\cdot \\dfrac{r_1}{r_2} \\,\n", "b59c940ea403e349a4a777c87c8d0d03": "\\{x,y\\} \\in R_i", "b59cebf62c72b513d86a92235550b15c": "15.9\\,\\text{kHz}\\,", "b59cec9e95964de5a6781efa220fddc7": " \\bigcap_{i\\in J} C_i \\neq \\emptyset ", "b59cfde620b3982bada3f8355a3f0361": "\\sum_{j = 1}^e V_{ij} U_{jk}", "b59d0d15cdabf0e1bb04d27ed379ed1c": "J^\\alpha{}_{;\\alpha} \\, = 0", "b59daa2bfba847df45648fd9e6e9952a": "M^{*}", "b59eada559a73fbc21d4a6380134f8cc": " \\left| \\log {\\zeta g^\\prime(\\zeta) \\over g(\\zeta)}\\right| \\le {|\\zeta|^2 + 1\\over |\\zeta|^2 -1}.", "b59ed35f45f798ed10b9c41a91419aa8": "z_{xx}-z_{yy}+\\frac{4}{x+y}z_x=0", "b59ed4489e42356035ee636651e0c12d": " z : \\mathbb{R}^2 \\rightarrow \\mathbb{R} ", "b59ede180266dd9cb7e420706be50968": "S_{x}(n\\Delta_{T}\\, ,m\\Delta_{F}) = \\sum_{p=0}^{N-1} X[(p+m)\\,\\Delta_{F}]\\,e^{-\\pi\\frac{p^2}{m^2}}\\,e^{\\frac{j2pn}{N}}", "b59ee23005d35eb3359bfe3d5631f2bd": "\\frac{(s+1)(s t+\\alpha)}{\\alpha}", "b59ee502e09945baf7db84342c10bc90": "1 + \\cos (\\theta)", "b59f07b7f2025bb7f7af90a78cff7285": " E_\\textrm{h} = m_\\textrm{e} \\left( \\frac{e^2}{4 \\pi \\varepsilon_0 \\hbar}\\right)^2 \\quad\\hbox{and}\\quad \na_{0} = {{4\\pi\\varepsilon_0\\hbar^2}\\over{m_\\textrm{e} e^2}}. ", "b59f5c6e66e2d6fedf78b8bc9565556b": "\\mathbb{Q} \\big(\\sqrt{- d} \\big)", "b59fbcfc9b6b8a5b4e057c7665bb5579": "\\sin(\\theta\\,1_z+(\\pi/2)\\,1_z)=1_z\\cos(\\theta\\,1_z)", "b59febdc0ca94d82211fc617d7363786": "K_{X'}", "b5a00b1c13c9254d2a4454642c31a0ba": " m=3", "b5a0186ff622e0012de7b6930248bf4c": "N=mg\\,", "b5a07cf06917d525235bf00dd2d21e48": "R^{\\dagger}= (a_1a_2....a_r)^{\\dagger} = a_r....a_2a_1", "b5a0a6cbd06c7dd5d3c9cd21b7fc6e3b": "\\Delta d = \\frac{\\lambda}{2n\\cdot\\sin\\alpha\\cdot\\sqrt{1+\\frac{I}{Isat}}}", "b5a0f6174964b6839ddaca7bdfade926": "L(\\theta,\\delta)= \\|\\theta-\\delta\\|^2 \\,\\!", "b5a11cad2b00d27709dcfcd63c88f758": "f_c(x) \\,", "b5a17428027a2c7d21eb2bcc4c0785a6": "M=3N\\langle x-0\\rangle^1\\ -\\ 3Nm^{-1}\\langle x-2m\\rangle^2\\ +\\ 9N\\langle x-4m\\rangle^1\\,", "b5a17afb75924794fba93b97b1432a5f": "I,J,", "b5a19f87dca8b0072a84e25e51887bec": "h\\in \\mathbb C \\{ x_{n+1},...,x_m\\}", "b5a1a802b7b4daaa1d4b163f338ea39e": "\\begin{align}\n & S = \\frac{ \\hat{\\mu}_3 }{ \\hat{\\sigma}^3 } \n = \\frac{\\frac1n \\sum_{i=1}^n (x_i-\\bar{x})^3} {\\left(\\frac1n \\sum_{i=1}^n (x_i-\\bar{x})^2 \\right)^{3/2}} \\\\\n & K = \\frac{ \\hat{\\mu}_4 }{ \\hat{\\sigma}^4 } \n = \\frac{\\frac1n \\sum_{i=1}^n (x_i-\\bar{x})^4} {\\left(\\frac1n \\sum_{i=1}^n (x_i-\\bar{x})^2 \\right)^2} ,\n \\end{align}", "b5a1bc5fe021b6a01be1890a3959e8a6": "L_{\\lambda}(A) = \\bigcup_{\\alpha < \\lambda} L_{\\alpha}(A) \\! ", "b5a1bf2c6cc28459c06b8b2081e690fb": "\\scriptstyle \\beta=v/c ", "b5a20b0571d91b8e4b710342fe949b01": "b: V_g \\to V_f", "b5a25142bb574f7c6e0a57ba2fd3b644": "(x_1,D,0)", "b5a278cbda0511c6106c3ac2dee3f640": "\\mu_2(x\\wedge y)\\mu_1(x\\vee y) \\ge \\mu_1(x)\\mu_2(y)", "b5a2b9eadabef119a8ceeaf466a76832": "\\phi=-\\pi, \\, \\pi", "b5a37557355ce42adf225546586c3b1d": "\\frac{d^3 W}{d\\Omega d\\omega}=\\frac{e^2\\gamma^2N^2}{4\\pi\\varepsilon_0 c}L\\left ( N\\frac{\\Delta \\omega_n}{\\omega_\\text{res}(0)} \\right )F_n(K,0,0)", "b5a39ad91c7838537580c5c266fdd223": "\\lnot \\varphi\\,\\!", "b5a39f3ef3d54c80aff98acc8b8abe67": "R_{m,n} = \\lim_{n \\to \\infty} R_{m,n} + O \\left( \\frac{1}{n^m} \\right),", "b5a3a8e75fb47e4f0f6a2416230960f1": "\\scriptstyle \\uparrow\\uparrow", "b5a3b3a3ccdf440d73d9c20c9ce04b49": "A = -R_{\\text{f}}/R_{\\text{in}}", "b5a3ee6aca18dcfbd2a5a02f252a3601": "A +_{\\mathrm{e}} B = \\{ z \\in \\mathbb{R}^{n} \\,|\\, \\mu \\left[A \\cap (\\{z\\} - B)\\right] > 0 \\},", "b5a3f741cc8dd365804afa8464e28fd6": "\n \\begin{align}\n \\frac{dy}{dt} \n &= \\left( \\frac{\\partial Y_0}{\\partial t} + \\frac{dt_1}{dt} \\frac{\\partial Y_0}{\\partial t_1} \\right) \n + \\varepsilon \\left( \\frac{\\partial Y_1}{\\partial t} + \\frac{dt_1}{dt} \\frac{\\partial Y_1}{\\partial t_1} \\right)\n + \\cdots\n \\\\\n &= \\frac{\\partial Y_0}{\\partial t} \n + \\varepsilon \\left( \\frac{\\partial Y_0}{\\partial t_1} + \\frac{\\partial Y_1}{\\partial t} \\right)\n + \\mathcal{O}(\\varepsilon^2),\n \\end{align}\n", "b5a41fea50b4a2453c11866691ff4d0b": " r_1 \\,\\!", "b5a4205e65c59048fc1dbf7d6660f5b2": "n = \\Pi_{i \\left(\\frac{6}{29}\\right)^3 \n\\end{cases}\\\\\nu^* &= 13 L^*\\cdot (u^\\prime - u_n^\\prime) \\\\\nv^* &= 13 L^*\\cdot (v^\\prime - v_n^\\prime)\n\\end{align}", "b5aa6ae34ff1ca83d2d6c57d80ff0da7": "r_2 = (1 + r_{1})^p - 1 \\, ", "b5aa86e1f469633fedcf91cc673e8a16": "[G:G_1],\\ldots,[G:G_k]", "b5aad0e7ad636810d8b9747084d17bbc": "B_I R \\simeq R[It]", "b5aadcf4cb867681b41ea1233226eff2": "E_\\nu=h\\nu_0(n+\\begin{matrix} \\frac{1}{2} \\end{matrix}),", "b5aae3ee3cc278a2a10774a13df46f3e": "\\sqrt{2} \\cdot \\sqrt{5}", "b5ab1808eab29fb02f966490ca07c5a2": "S=\\frac{1}{16\\pi G_N}\\int d^4x\\sqrt{-g}\\mathcal L", "b5ab2af5326372cab90e4c10a5b3a106": "f_{W_t}(x) = \\frac{1}{\\sqrt{2 \\pi t}} e^{-\\frac{x^2}{2t}}.", "b5abd342f012c319cd267d474136012f": "m<20%", "b5ac1e83c72f0148b60bf27e48df8c4e": "~z=L~", "b5ac309851c714925167a08ca1cf3232": " \\sigma = 1-\\frac\\pi{z}\\frac{\\sin\\beta_2}{(1-\\phi_2\\cot\\beta_2)} ", "b5acaf9bad34b4f50a893612dbd905f1": "v_1\\in V(\\Gamma)", "b5acb32277a0db65312eb8dde7a7eaba": " \\alpha = \\frac{(1-m^* / m_0)^2}{E_g} ", "b5ad141afc644f037547926e87ceca41": " V = W_0 \\oplus Z(v_1;T) \\oplus Z(v_2;T) \\oplus \\cdots \\oplus Z(v_r;T)", "b5ad186d19b5fd678d5dc23a0c6fb7b8": "\\zeta^a", "b5ad24dfb5ecf3114fa2cca08b81d7de": " xyxyx = y^2 ", "b5adcebb06257d386e562ff0ea0e8a15": "\\begin{align}\\underline{\\mathsf{f}}(x \\wedge y) &= R(x \\wedge y)R^{\\dagger} \\\\\n&= R(xy - x \\cdot y)R^\\dagger \\\\\n&= RxyR^\\dagger - R(x \\cdot y)R^{\\dagger} \\\\\n&= RxR^\\dagger RyR^{\\dagger} - x \\cdot y \\\\\n&= (RxR^\\dagger) \\wedge (RyR^{\\dagger}) + \\cancel{(RxR^{\\dagger}) \\cdot (RyR^{\\dagger})} - \\cancel{x \\cdot y} \\\\\n&= f(x) \\wedge f(y) \\end{align}", "b5ade5d5393fd7727bf77fa44ec8b564": "a\\;b", "b5ae773dbd5065334943f8881e62384d": "[B]=\\frac{[A]_0+[B]_0}{1+\\frac{[A]_0}{[B]_0}e^{-([A]_0+[B]_0)kt}}", "b5ae78ef9722e7dcfcc3857e34e9dfed": " \\mu = m - M \\!\\,", "b5af07121ceb57dfd87f810c123087ca": "\\Omega_+", "b5af411569af41a493883bdb1b3b6b9c": "\\vdash p", "b5af8f227db697298b55dca327b7d956": "\n\\overline{R}=|\\overline{\\mathbf{\\rho}}|\n", "b5af9393b0065cfb45bb8e1d19358238": "\\Delta{P}_t \\over P-\\Delta{P}_t", "b5af96f3d66f30c6d1e2e4b31f10b0e9": " P_{if} = 1-\\exp[-\\frac{4\\pi^2 {H_{if}^2}}{hv \\mid(s_i - s_f)\\mid}] ", "b5afb0097352cf4b2bd9d2db48010754": "A = U \\begin{pmatrix}I_r &0\\\\\n0 &0\\end{pmatrix} V", "b5afd576f8017adbc34ae5bac5d3b5ff": "w^2=w^iw^j\\delta_{ij}", "b5afe8c2a3265ca6b3288c50a580f6de": " H^s_p, \\ \\ s \\in \\mathbf{R}, \\ 1 \\le p \\le \\infty \\, ; \\quad B^s_{p, q}, \\ \\ s \\in \\mathbf{R}, \\ 1 \\le p, q \\le \\infty.", "b5b0301f69cc0285cbe8cf5f98b9151a": "Q=\\alpha K + (1-\\alpha)N.\\,", "b5b053917195bde623ade564a79535a2": "r \\geq s\\,\\!", "b5b05439a9d4bf77785feb3ab02e920e": " E_{hb} (\\sigma)=c_{hb}(T)\\max[0,\\sigma_{acc}-\\sigma_{hb}] \\min[0,\\sigma_{don}+\\sigma_{hb}]", "b5b09e1b0079b20c91e84fe3387e5eb1": " \\Beta(x;\\,a,b) = \\int_0^x t^{a-1}\\,(1-t)^{b-1}\\,\\mathrm{d}t. \\!", "b5b0be4761784d8366d704ce4b835d7e": "x\\tan(x)", "b5b0d72e23e62a925f148000871eb710": "\\neg \\phi\\,", "b5b0f60a15d847d70995ccbf9c091a0b": "U(n+1)/(U(1) \\times U(n)) \\cong SU(n+1)/S(U(1) \\times U(n)).", "b5b10e790e49e8f656e4de6cf20b7535": "\\tan \\frac{A}{2} = \\sqrt{\\frac{(s-a)(s-d)}{(s-b)(s-c)}}.", "b5b1300599ad520f117012448bd7b8f7": " \\mathbf{E}(z,t) = \\mathrm{Re} (\\mathbf{E}_0 e^{i(\\tilde{k} z - \\omega t)}) = \\mathrm{Re} (\\mathbf{E}_0 e^{i(2\\pi(n+i\\kappa)z/\\lambda_0 - \\omega t)}) = e^{-2\\pi\\kappa z/\\lambda_0} \\mathrm{Re} (\\mathbf{E}_0 e^{i(kz - \\omega t)}).", "b5b1318b83d3dfa1e5879a3be246a16e": "f(y_1, z_2 + y_1^r, z_3 + y_1^{r^2}, ..., z_m + y_1^{r^{m-1}})", "b5b14d1e0a7187f596eb7e85268ab502": "s_\\Lambda=6.36", "b5b169eabf2664fbd08c717a67deef6e": " \\sigma_{\\theta\\theta} = 0", "b5b194f680b5465058e2a13c111e3cda": "Z_r^{p,q} = \\ker d_0^{p,q} : F^p C^{p+q} \\rightarrow C^{p+q+1}/F^{p+r} C^{p+q+1}", "b5b1a40ac4768ad9bc9746c1621dfda8": "c_n = \\prod_{i=1}^{n-1} i^{n-i}=\\prod_{i=1}^{n-1} i!.\\,", "b5b1f0fbf4c1516d349902d401407670": "\\mathbf{F}_q = q (\\mathbf{E} + \\mathbf{v}_q \\times \\mathbf{B}) \\,", "b5b24586dd675564aadc1f6f5bfcb094": "dS = \\frac{\\delta\\langle q_{rev} \\rangle}{T}", "b5b2a0d828b35dc653031f51e350c3cd": " \\scriptstyle f > f_0 ", "b5b2ca5c0650649f209c2d07d4c82028": "ACG_*", "b5b2cfb52b9c47e273c36d3ccd7148a0": "\\left(\\bar\\psi (\\partial^\\mu + ieA^\\mu)\\psi\\right)_B = Z_1 \\, \\bar\\psi(\\partial^\\mu + ieA^\\mu)\\psi", "b5b2de50634dcd8daa628fbb52dd0ede": "\\operatorname{\\Gamma L}(V) = \\operatorname{GL}(V) \\rtimes \\operatorname{Gal}(K/k)", "b5b37e2b08c3cfa84f550994f0646fd0": "\\scriptstyle \\frac{1}{\\sqrt{p(1-p)}}", "b5b3e047196a3dc3db7366d51887f440": "R''", "b5b3f036ccef31c770b270ee155b776a": "h(G) = 0.\\,", "b5b43c9cc839e27aef4e4e568cc3e236": "R^{\\beta}{}_{\\nu \\rho \\sigma}\\,", "b5b46268e61e89ad442e530d5e03b24c": " \\mathcal L_X\\omega = d (\\iota_X \\omega) + \\iota_X d\\omega", "b5b46f2966b214539d0035fc9fb1289b": "\\frac{\\partial^2 \\epsilon_y}{\\partial z^2} + \\frac{\\partial^2 \\epsilon_z}{\\partial y^2} = 2 \\frac{\\partial^2 \\epsilon_{yz}}{\\partial y \\partial z}\\,\\!", "b5b4b3bad247ae87f337e7696bf10ea6": "F(z) = z^{n} + A_{1}z^{n-1} + \\cdots + A_n = 0.\\,", "b5b5205b8ca5c87ccf026bc2c564f3ed": "\\begin{bmatrix} 0 & 1 & -3 \\\\ 3 & -13 & 23 \\\\ -4 & 19 & -36 \\end{bmatrix}\n +4\\begin{bmatrix} 0 & 0 & 1 \\\\ -1 & 4 & -6 \\\\ 1 & -5 & 10 \\end{bmatrix}\n +\\begin{bmatrix} 1 & -1 & -1 \\\\ 1 & -2 & 1 \\\\ 0 & 1 & -3 \\end{bmatrix}\n +\\begin{bmatrix} -1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & -1 \\end{bmatrix}\n =\\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}\n", "b5b55b0f2beaf37a0321cb5c7524ec12": "f(\\lambda x)", "b5b55e0b425ef0b5307d0fa448b6f7a0": "R * (x,\\,y) = y", "b5b5b11dbc6e1ae42f46671411c9291e": "\\text{incremental capital output ratio} = \\frac{\\Delta K}{\\Delta Y} = \\frac{\\frac{\\Delta K}{Y}}{\\frac{\\Delta Y}{Y}}= \\frac{\\frac{I}{Y}}{\\frac{\\Delta Y}{Y}}\n", "b5b5ceb42f5177e102503e48789ca642": "U(a,1)\\begin{pmatrix}v&0\\\\0&u\\end{pmatrix} = U(av,u) \\thicksim U(u^{-1}av,1).", "b5b5d5266a80712fec5f4c0e88b4d345": "L_{xy}(x, y) = (L(x-1, y-1) - L(x-1, y+1) - L(x+1, y-1) + L(x+1, y+1))/4, \\,", "b5b5e44c83a9a5e07104e9de1d43d070": "v(3) = 0", "b5b5ec40a51d4338595d153052472a7a": "\\beta_j^-", "b5b5ef9dc79ba028fff5b35318f7d08c": "P(\\nabla) f(x)", "b5b669c9ae93de03aba3cd63144f8b13": "p(w,b|\\log \\mu ,\\log \\zeta ,\\mathbb{M}) = p(w|\\log \\mu ,\\mathbb{M})p(b|\\log \\sigma _b ,\\mathbb{M}) .\n", "b5b6c7bfd135d675e7aaa69b14028f49": "\nAccessibility_i = \\sum_j {Opportunities_j } \\times f\\left( {C_{ij} } \\right)\n", "b5b6cb8534c02028704c6bb5d5feda71": "\\widehat{\\rm BAC}.", "b5b6ec4aed1b9fd68e43067816cdd207": " \\frac{\\Delta L}{L} ", "b5b73597fd8d1782dfe9a8d6feceb083": "g^y", "b5b748371d9eff1ef46a8feaa869ebe1": "\\sum_{k\\in\\mathbb{N}} g(n,k) = f(n,\\lambda(n)) = \\varphi(n)", "b5b751f95cfac4ed7feb8597ddf88b5d": "y,y^{\\prime}\\,", "b5b763ac95e519a955cfe3a8c52db0da": "\\dot \\gamma ", "b5b77e5108b9009196818647b67ee80e": "\\frac11+\\frac12+\\frac13+\\cdots+\\frac1n", "b5b7c1492582190761bdd2c5c9a5afc5": "(\\overline{Y}-Y)/\\overline{Y} = 1-Y/\\overline{Y} = c(u-\\overline{u})", "b5b7c59b9f44de1b10f57923d094bb59": "k-\\epsilon", "b5b7c75c82edf42828cf226d9bef5309": "(1-H_q(\\delta)-\\varepsilon)", "b5b7e35de4dd018ea02f006a7a5d35c3": "x_3^2g_2x_4g_1=1", "b5b8304b110b66a5c81e76624db144c4": "c=a^2=b^3", "b5b835e4b3b47a46a432e3379e46b0ff": "\\bar x = \\alpha(x^t) = \\alpha(x)^t.", "b5b86b3edb5fb4cfd75641b2671e4686": " L = \\Q x_1 + \\Q x_2 + \\Q x_3 ", "b5b890af9b874f24f249c109a54d4ad3": " \\mathrm{Area} = 4 a b \\frac{\\left(\\Gamma \\left(1+\\tfrac{1}{n}\\right)\\right)^2}{\\Gamma \\left(1+\\tfrac{2}{n}\\right)} . ", "b5b89f34bb3b0233a835a112365dae01": "P^{eqb}=\\frac{a-c}{g-b}.", "b5b8d6d103ecf28ff17d5cb1e99ac973": " H = -t \\sum_{\\langle i,j \\rangle,\\sigma}( c^{\\dagger}_{i,\\sigma} c^{}_{j,\\sigma}+ h.c.)", "b5b8e083af2aa7e2959009464d35dc2d": "H_{0} (x|q) =1 ", "b5b8f30465a4c96aa83a57f040bbeb47": "I_k(z) = \\frac{(z-z_0)(z-z_1)\\cdots(z-z_{k-1})(z-z_{k+1})\\cdots(z-z_n)}{(z_k-z_0)(z_k-z_1)\\cdots(z_k-z_{k-1})(z_k-z_{k+1})\\cdots(z_k-z_n)}", "b5b96987cd60ee4adc10df4592792ffb": "\\ w", "b5b988aeee964df7e0acf6f6c906dd46": "\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda q.f\\ (q\\ q)) ", "b5b9a888a219f26a1b4da5c9d08e11dd": "c=\\sqrt{\\frac{g}{k}+\\frac{\\sigma k}{\\rho}},", "b5b9d300b33cedf87f836f8a71e03afc": "\\operatorname{tr} (\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma) = 8 \\eta^{\\rho \\sigma} \\eta^{\\mu \\nu} - 8 \\eta^{\\nu \\sigma} \\eta^{\\mu \\rho} + 8 \\eta^{\\mu \\sigma} \\eta^{\\nu \\rho} \\,", "b5b9db4989b533e8418b7e68de6f70d0": "C\\cap D", "b5b9f3bc4fda3d42a4a0b77532480ffc": "\\frown\\ : H_p(X;R)\\times H^q(X;R) \\rightarrow H_{p-q}(X;R).", "b5b9ff87b41362d6434ff77869a92e3e": "J = \\frac{\\mathrm d \\Phi}{\\mathrm d A} = \\int_\\omega{L_{e+r}(\\theta,\\phi)\\cos\\theta \\, \\mathrm d\\omega}", "b5ba02a830e3da1c6c0fc7bd9b07bef4": "\\mid \\!\\,", "b5ba59556fd78279e185d74b42a731e3": "\\ \\{\\{x\\},\\{x,y\\}\\}", "b5ba7a6c4e2621eed19065abd8363b1c": " \\gamma' ", "b5bb4b3875a5b81f8305d5a3fe135d3b": " D_x(r)=1-e^{-\\Lambda(b(x,r))}, ", "b5bbead3a5fd6a8aebb7f9128f46006a": "B_{21}=\\frac{4\\pi^2 e^2}{m_e h\\nu c}~\\frac{g_1}{g_2}~f_{12}", "b5bc1e0b62d042bdc31e6cae8778c1c3": "\\mathcal{K} \\subseteq \\mathcal{P} (M)", "b5bc514d4ea99dcb379b88d3651d73d5": "\\mathbf{\\hat{t}} \\in \\mathbb{R}^3", "b5bcbe2b836cde4e8d562a514e4442d3": "\nY = X' \\beta + U, \\,\n", "b5bce437117ad38017aaeff29de5e955": "\\max|X_j|\\le \\left(D^{-1}\\sqrt{\\det(AA^T)}\\right)^{1/(N-M)}", "b5bd301d40782786ce800c6ff28f087e": "\\frac{E_1}{E_2} = \\frac{\\frac{1}{2} k_1 x_1^2}{\\frac{1}{2}k_2 x_2^2}, \\,", "b5bd34dde5951165d737d45d43aca121": "\n\\begin{align}\nJ(\\sigma) &=\n\\int_0^\\sigma \\frac{k^2\\sin^2\\sigma'}{\\sqrt{1 + k^2\\sin^2\\sigma'}}\\,d\\sigma'\\\\\n&= E(\\sigma, ik) - F(\\sigma, ik).\n\\end{align}\n", "b5bd45ead9aaade606c58defe83cb3b0": "\n= \n{2\\over r} \\int_0^{\\infty} {k dk \\over \\left ( 2 \\pi \\right )^2 } {\\sin\\left( kr \\right) \\over k^2 + m^2} =\n{1\\over i r} \\int_{-\\infty}^{\\infty} {k dk \\over \\left ( 2 \\pi \\right )^2 } {\\exp\\left( ikr \\right) \\over k^2 + m^2}\n", "b5bd4f3fe06a2b0d9d9375e6f2e58868": "UH=HU", "b5bd913391528370d091679316fc99f4": " m= p^k ", "b5bdb99aaa96fa491096b71ef146ce0f": " G( A(a,b), C(a,b) )=G\\left({{a+b}\\over 2}, {{a^2+b^2}\\over {a+b}}\\right) ", "b5bdd193671cbaffd33383d261e1e572": "\\Gamma^{\\rho}_{\\mu\\nu} G^{\\mu\\nu} = 0", "b5bde8bddb001d60a75126d5cb3803d1": "w_{ij}=A_j q_{ij}", "b5bdf35ebf4c82aa3a1537c2a7e629ee": "Z\\to0", "b5be4fab1d623da19867f993b13d8698": "c = \\lfloor year/100 \\rfloor.", "b5bed9540a8a9b967311dc761f5ee7a7": "p_g", "b5bf2ff201296f322a56bb40941b887f": "\\displaystyle \\min_{x \\in X}P(x) \\geq \\beta", "b5bf7c8dd11bb7d851314c970a9a0f9d": "\\displaystyle{P_s(x)= {s\\over 2\\pi(|x|^2 + s^2)^{3/2} }.}", "b5bfa2758b0c8cbf8bca9c13f427a7eb": "j\\colon V\\to V\\,", "b5bffabbfddd4b274281bb37445286e3": "2^1 \\times 0.0001_2", "b5c0173f43393be29caf7a3d1411997f": "B=\\begin{pmatrix}\n\\lambda & 1 & 0 & 0 & \\cdots & 0 \\\\\n0 & \\lambda & 1 & 0 & \\cdots & 0 \\\\\n0 & 0 & \\lambda & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & 0 & \\lambda & 1 \\\\\n0 & 0 & 0 & 0 & 0 & \\lambda \\\\\\end{pmatrix}\n=\n\\lambda \\begin{pmatrix}\n1 & \\lambda^{-1} & 0 & 0 & \\cdots & 0 \\\\\n0 & 1 & \\lambda^{-1} & 0 & \\cdots & 0 \\\\\n0 & 0 & 1 & \\lambda^{-1} & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & 0 & 1 & \\lambda^{-1} \\\\\n0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{pmatrix}=\\lambda(I+K)", "b5c028dc722bb5cda1493aa19a923e81": "f(t_1,\\dots,t_k)=f(Ad_g t_1,\\dots, Ad_g t_k) \\, ", "b5c06e60657057bca1290497d70eaf8d": "AB=\\frac{1}{2}\\pi\\left(2x+y\\right)^2", "b5c0abb83ef6ede3d5bf41b1a15d48a8": " v_f = v_i + at ", "b5c151dea7bbf35a163d77d7ee7d1e82": " x \\mapsto \\langle \\xi, \\pi(x)\\xi\\rangle ", "b5c17436e585356f7c7603efbf85f482": "X = \\mu + b \\sqrt{2 V}Z \\sim \\mathrm{Laplace}(\\mu,b)", "b5c1a0c623fa8d6fa1c5247ab098bc62": " \\mathbf{u}_j = \\frac{1}{D_{j-1} } \\begin{vmatrix}\n\\langle \\mathbf{v}_1, \\mathbf{v}_1 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_1 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_1 \\rangle \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_2 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_2 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_2 \\rangle \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_{j-1} \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_{j-1} \\rangle & \\dots &\n\\langle \\mathbf{v}_j, \\mathbf{v}_{j-1} \\rangle \\\\\n\\mathbf{v}_1 & \\mathbf{v}_2 & \\dots & \\mathbf{v}_j \\end{vmatrix} ", "b5c2260f7b4deaa6a9bd9f0873fcac16": "M_k(n)={1\\over n}\\sum_{d\\mid n}\\mu(d)k^{n/d}", "b5c22f08f82037e761ab59411be574c1": "\\Delta\\sigma= \\sigma_\\mathrm{max} - \\sigma_\\mathrm{min}", "b5c23fce9714645dfb9bbe0fac056eae": "v(x_1,x_2)=u(e(x_1,x_2),f(x_1,x_2))", "b5c270e072eadfe12e60cc09004a6c71": " s_\\lambda(x_1, \\ldots, x_n) = \\det \\left ( (h_{\\lambda_i +j-i} )_{i,j}^{r \\times r} \\right ), ", "b5c2922a4fb9ca4c5d627388f9d29cdd": "\n\\begin{align}\n\\Phi(z) & = P(Z \\le z) = 1 - \\tfrac{\\alpha}2 = 0.975,\\\\[6pt]\nz & = \\Phi^{-1}(\\Phi(z)) = \\Phi^{-1}(0.975) = 1.96,\n\\end{align}\n", "b5c2a23af4cff7cdd7561b1bf5f957be": "\\Omega(x)=S(x)\\Xi(x)\\bmod x^{d-1}=r(x)", "b5c2dd7d8dee91e52c2e147a7914117d": "|j_1m_1;j_2m_2\\rangle", "b5c30ebb6129c18661378d7f7f5fda43": "\\begin{align}\n\\frac{d}{d\\alpha} \\textbf{I}(\\alpha) &= \\int_0^{\\frac{\\pi}{2}} \\frac{\\partial}{\\partial\\alpha} \\left(\\frac{\\ln(1 + \\cos\\alpha \\cos x)}{\\cos x}\\right)\\,\\mathrm{d}x \\\\\n&=-\\int_0^{\\frac{\\pi}{2}}\\frac{\\sin \\alpha}{1+\\cos \\alpha \\cos x}\\,\\mathrm{d}x \\\\\n&=-\\int_0^{\\frac{\\pi}{2}}\\frac{\\sin \\alpha}{\\left(\\cos^2 \\frac{x}{2}+\\sin^2 \\frac{x}{2}\\right)+\\cos \\alpha\\,\\left(\\cos^2\\,\\frac{x}{2}-\\sin^2 \\frac{x}{2}\\right)}\\,\\mathrm{d}x \\\\\n&=-\\frac{\\sin\\alpha}{1-\\cos\\alpha} \\int_0^{\\frac{\\pi}{2}} \\frac{1}{\\cos^2\\frac{x}{2}}\\frac{1}{\\left[\\left(\\frac{1+\\cos \\alpha}{1-\\cos \\alpha}\\right) +\\tan^2 \\frac{x}{2} \\right]}\\,\\mathrm{d}x \\\\\n&=-\\frac{2\\,\\sin\\alpha}{1-\\cos\\alpha} \\int_0^{\\frac{\\pi}{2}}\\,\\frac{\\frac{1}{2}\\,\\sec^2\\,\\frac{x}{2}}{\\left[\\,\\left(\\frac{2\\,\\cos^2\\,\\frac{\\alpha}{2}}{2\\,\\sin^2\\,\\frac{\\alpha}{2}}\\right) + \\tan^2\\,\\frac{x}{2} \\right]} \\,\\mathrm{d}x \\\\\n&=-\\frac{2\\left(2\\,\\sin\\,\\frac{\\alpha}{2}\\,\\cos\\,\\frac{\\alpha}{2}\\right)}{2\\,\\sin^2\\,\\frac{\\alpha}{2}}\\,\\int_0^{\\frac{\\pi}{2}}\\,\\frac{1}{\\left[\\left(\\frac{\\cos \\frac{\\alpha}{2}}{\\sin\\,\\frac{\\alpha}{2}}\\right)^2\\,+\\,\\tan^2\\,\\frac{x}{2}\\,\\right]}\\,\\mathrm{d}\\left(\\tan\\,\\frac{x}{2}\\right)\\\\\n&=-2\\cot \\frac{\\alpha}{2}\\,\\int_0^{\\frac{\\pi}{2}}\\,\\frac{1}{\\left[\\,\\cot^2\\,\\frac{\\alpha}{2} + \\tan^2\\,\\frac{x}{2}\\,\\right]}\\,\\mathrm{d}\\left(\\tan \\frac{x}{2}\\right)\\,\\\\\n&=-2\\left(\\tan^{-1} \\left(\\tan \\frac{\\alpha}{2} \\tan \\frac{x}{2} \\right)\\right) \\bigg|_0^{\\frac{\\pi}{2}}\\\\\n&=-\\alpha\n\\end{align}", "b5c3131115f52172acbfe9df1fd51313": " O_T = \\sqrt { \\frac { n - 1 } { 2 } } \\frac { s^2 - m } { m } ", "b5c33fcffc729e69649c601a9a515dca": " A\\to B, A \\vdash B", "b5c422551f5c5caf36d6b9391aa06eb0": "\\nabla f \\cdot \\nabla (r - R) = 0", "b5c452b7c15ef5c83aba8ee5700b10fb": "0 \\le \\delta \\le 1-{1\\over q}, 0\\le \\epsilon \\le 1- H_q(\\delta)", "b5c468a5d387562f191efe36c812d6f2": "|b|_\\ast>1", "b5c4746b87123f00f1fd9ee176958b57": " \n c_\\text{kdv}(k) = \\sqrt{gh} \\left( 1 - \\frac{1}{6} k^2 h^2 \\right),\n", "b5c481e177fe4bc6149c694d536adc89": "\\mathop{\\rm Char}P\\cap N^*\\Sigma=\\emptyset", "b5c4b99616dd98ab7127c1a6d126da98": "\\int p\\,dq", "b5c51019c721c4b5f3fb146a9b9afee9": "R = \\sqrt{\\sqrt{\\rho}\\tilde{\\rho}\\sqrt{\\rho}} ", "b5c51033694be213443a71a684ad1dcd": "\\varphi(t) = \\frac{e^{i2\\pi t} - 1}{i 2\\pi t}.", "b5c5c2980e3d08eccb95b57800a754e8": "\\mathcal R = \\frac{\\mathcal F}{\\Phi}", "b5c5ebf6bc4b6e1043d81473528cc82d": "t_f = \\frac {2 A_m\\sqrt {h_t-(h_t-h_m)}} {A_g \\sqrt {2g}} ", "b5c60558a3bac5db7129adfc0e74b6f7": "f(x) = \\sum_{n=0}^N a_n \\varphi_{n}(x) + O(\\varphi_{N+1}(x)) \\ (x \\rightarrow L).", "b5c619c5be87f9d5e81f3fa4ad2f9366": "\\int \\frac{dx}{a+bx+cx^2} = \\frac{1}{c} \\int \\frac{dx}{\\left( x+ \\frac{b}{2c} \\right)^2 + \\left( \\frac{a}{c} - \\frac{b^2}{4c^2} \\right)}. ", "b5c61d7c58fd37ffd94f35d642c724ec": "\\lim_{x\\rightarrow\\infty}\\frac{\\pi(x)\\ln(x)}{x}", "b5c629bba4344a90fff99133498789bf": "(A \\leftrightarrow C)", "b5c62fad7a632050daab4978c8bbf6f2": "1/R_c = 2L_s", "b5c64346d5cdef3b47c8124936921179": " M = \\tfrac{1}{2}(P+Q). \\, ", "b5c65ad7a070557473d3f68a403664d6": "\\{U_i : i\\in I\\} ", "b5c6ae7f4724b4278d0b35ea0d3e8e01": "F\\colon \\mathbf{R}^{2N} \\to \\mathbf{R}^N", "b5c6fb64412e52016d22cffa76a31f5c": "\\Phi-\\Phi=\\Phi", "b5c74117da5d318679b33b3cd107b53f": "u=0 \\text{ on }\\partial \\Omega", "b5c779f34ffff5e68f7daffdd1a3fc12": "\\prec ", "b5c77f8bbf18dd132ad969f9542265ac": "\\vartheta_{01}(z|q) = \\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 - 2 \\cos(2 \\pi z)q^{2m-1}+q^{4m-2}\\right).", "b5c7d38f05ec52ca6923cb6621d49f13": "\\lim_{n\\to\\infty} diam(C_n)=sup\\{d(x,y): x,y\\in X\\}\\rightarrow 0", "b5c7dcb4df9eaf4eb9ea8c0659fd329b": "\\{\\alpha | a_\\alpha\\neq0\\}", "b5c7ed640115eb9d71c091fab7bb1a54": "a_n x^n + a_{n-1}x^{n-1} + \\dotsb + a_2 x^2 + a_1 x + a_0,", "b5c81131a3946539a5be94b5dba30370": "R_{\\mathrm{min}} = \\frac{2V_{l}}{RTln(s)}\\sigma_{lv}\\,", "b5c88ff4e8253b5ee9dda7b3c571911c": " \nL_n \\subseteq \\overline{\\Gamma_n}^* \\subseteq \\Gamma_n\n", "b5c8afcfcaad4a8fd4986cea56f01d32": "I\\ nat\\ (2+1)\\ (2+1)", "b5c92d730e17bbcdb277be89fade0736": "F_a(H)", "b5c9601b14021f65fa067e0c89ccfbbe": "\\frac{\\bigl( 4n(n+1) \\bigr) \\bigl( 4n(n+1)+1 \\bigr)}{2} = 2^2 \\, \\frac{n(n+1)}{2} \\,(2n+1)^2.", "b5c979175f35d96619427dd5704ddce6": "m = a_k n^k + a_{k-1} n^{k-1} + \\cdots + a_0,", "b5c9b7f3f76f3d5bc7f6034bcc6a3648": " F_i(x)", "b5c9d59b43d51211b097198261f68eac": "a^2 / b^2 = 2", "b5c9fb29c75232cfd72cc063da0f4a84": "{W_{I}(\\mathbf {r}, t)} =\\langle \\psi | {E^{(-)}(\\mathbf {r}, t)} \\cdot\n{E^{(+)}(\\mathbf\n{r}, t)}| \\psi\\rangle", "b5ca22ed8865f8077d61b4485e7884ae": "R_{\\text{c}}", "b5ca24cdfc1f4897195c13630b9737c2": "{{x}_{i}}", "b5ca4406a7e0b4f20a0ffa5ffdd8f0a2": "Ba", "b5ca64c3f92d337ddf375ee16124c869": "G > R \\ge B", "b5caa48bf98162e649a24ed43e47a7f4": " \\lambda =1", "b5caab59b5e04076f9f99cab087b5437": "c(\\zeta, \\tau=0)", "b5cadfcd67a1a1668b3d990d57e43cf3": "{g-k+r \\choose r}.", "b5cafb18e382e5b38417d5f056103d44": "-\\pi < t < \\pi", "b5cb3eca74cfc71186b5cd6bc7702c70": "\\scriptstyle \\langle f|H'|i \\rangle", "b5cb978249c4494ce29817d23a1c378b": "d_i = 0", "b5cbc56d126ee84790a37a6e59ab5c87": "\\forall (h:c\\rightarrow c^\\prime)\\in \\mathrm{Mor}\\, C", "b5cc0bc0c0b1a5e1525c338b712cea40": "\\frac{\\alpha^2}{\\kappa^2} e^{\\kappa^2} \\left( e^{\\kappa^2} - 1 \\right)", "b5cc250d5f99ad1bd9e77149431a75bd": "\\mathbf{BA} = \\begin{pmatrix} \nx \\\\\ny \\\\\n\\end{pmatrix}\\begin{pmatrix} \na & b\n\\end{pmatrix} = \\begin{pmatrix} \nxa & xb \\\\\nya & yb \n\\end{pmatrix} \\,.\n", "b5cc6f02d99f0ab58efa925a8e6d5ddf": "\\begin{cases} G\\times H\\to H \\\\ (gf)(x)=f(g^{-1}x) \\end{cases}", "b5cca1cd2bc2171cb2f87a5c4d920e5a": "u(t)\\in K", "b5ccc64842dfd475fa53cb039cb30ad4": "\\frac{D}{ds}\\left(m u^\\lambda + u_\\mu \\frac{DS^{\\lambda\\mu}}{ds} \\right) = -\\frac{1}{2}u^\\pi S^{\\rho\\sigma} R^\\lambda{}_{\\pi\\rho\\sigma}", "b5ccc76bd4961eae3fa65dd66c56a832": "-u_{l}^{\\alpha} \\left[ \\frac{\\partial \\rho^{l}}{\\partial x^{i}}\\, dx^{i} + \\frac{\\partial \\rho^{l}}{\\partial u^{k}}\\, (\\theta^{k} + u_{i}^{k}dx^{i}) + \\frac{\\partial \\rho^{l}}{\\partial u^{k}_{i}}\\, du^{k}_{i} \\right ] \\,", "b5cd1fa4c671ce9e9227c64b21a4024e": "\\ Z_{\\text{eq}} = R + jX = (R_1 + R_2 + \\cdots + R_n) + j(X_1 + X_2 + \\cdots + X_n) \\quad", "b5cd47e29142b917076a33178d4f2138": " 1/U = 1/h_1 + dx_w /k + 1/h_2 ", "b5cd7bf819263bfaea1a737ebe4a3bc5": "a\\in\\Omega_x", "b5cdf14b93ce889b192e14e0e2ecd177": "[z^{n+1}] H (g(z)) = \\frac{1}{(n+1)} [w^n] (H' (w) \\phi(w)^{n+1})", "b5ce65163c12a67c8d0aa92dfad451c3": "\n\\begin{array}{lcl}\n\td_x = c_y (s_z \\mathbf{y}+c_z \\mathbf{x})-s_y \\mathbf{z} \\\\\n\td_y = s_x (c_y \\mathbf{z}+s_y (s_z \\mathbf{y}+c_z \\mathbf{x}))+c_x (c_z \\mathbf{y}-s_z \\mathbf{x}) \\\\\n\td_z = c_x (c_y \\mathbf{z}+s_y (s_z \\mathbf{y}+c_z \\mathbf{x}))-s_x (c_z \\mathbf{y}-s_z \\mathbf{x}) \\\\\n\\end{array}\n", "b5ce653633f03eceb128269469da6550": " \\Pr \\{X_{ni}=1\\} ", "b5ce6f1c1f0bbe099c7ff8e4ef13143f": "S_x(t,f) \\approx S_y(t,f-at) \\, ", "b5ce78e25c3c273ea461c23b5c238459": "t/2 \\pi=100000000", "b5ce9cd67a3c4e62df1c3fcd734fde6d": "\\frac{L} {T^2}", "b5cebe6e2f51755bfcba7afea96326ed": "\\frac {F_w}{F_i} = \\frac {1} {\\sin \\theta - \\mu \\cos \\theta} \\,", "b5cebf1e428c34fb8d9f71b9233aff6e": "E=\\frac{1}{2}mv^2=\\frac{3}{2}k_B T", "b5ced3be9af09ca40df4e83ccaa3e1ee": "\\alpha\\le \\frac{1}{m(Q_k)} \\int_{Q_k} |f(x)| \\, dx \\leq 2^d \\alpha.", "b5cf0fd687ac0f6f3a526facca001db4": "\nQ = \\frac{\\sqrt{M k}}{D}, \\,\n", "b5cf2c441a1c397f4d10eae65346a9f4": "\\log_5(x - 3) = 2\\,", "b5cf8d1d8e6ab88f2f69c769bb6bda23": "BC_{Bullets} = \\frac{SD}{i} = \\frac{M}{i \\cdot d^2}", "b5cfb27e6840828dd5cae98d07e97fa1": "\\gamma_{ij} := 2\\varepsilon_{ij}", "b5cfde056e4d6e824338becaf3b397ab": "\nPoss(drop(o),s)\\leftrightarrow is\\_ carrying(o,s)\n", "b5cffd525243f6bbc277d7abc49a7528": "\n \\begin{bmatrix} \n 1 & 2 \\\\ \n 3 & 4 \\\\ \n \\end{bmatrix}\n\\otimes\n \\begin{bmatrix} \n 0 & 5 \\\\ \n 6 & 7 \\\\ \n \\end{bmatrix}\n=\n \\begin{bmatrix} \n 1\\cdot 0 & 1\\cdot 5 & 2\\cdot 0 & 2\\cdot 5 \\\\ \n 1\\cdot 6 & 1\\cdot 7 & 2\\cdot 6 & 2\\cdot 7 \\\\ \n 3\\cdot 0 & 3\\cdot 5 & 4\\cdot 0 & 4\\cdot 5 \\\\ \n 3\\cdot 6 & 3\\cdot 7 & 4\\cdot 6 & 4\\cdot 7 \\\\ \n \\end{bmatrix}\n\n=\n \\begin{bmatrix} \n 0 & 5 & 0 & 10 \\\\ \n 6 & 7 & 12 & 14 \\\\\n 0 & 15 & 0 & 20 \\\\\n 18 & 21 & 24 & 28\n \\end{bmatrix}.\n", "b5d0004c40debcacb69ebb187bdeca0b": "\\omega^{2^{p-1}} \\neq 1", "b5d01f28c8e00d1d814179db8e835844": "\\sigma^*", "b5d0231875e239cf3638f4faa4ffac4a": "\\sum_{\\Omega(n)=2} \\frac{\\ln n}{n^2} \\approx 0.28360", "b5d059b83b3f0bd0477b41b53e433919": "\\ell+1", "b5d05ce684aa761e8e47929fbf6dc3bf": "L_{PE}", "b5d066e317d19fb7deef66b7fcf3b700": "\\sigma\\geq 0,\\;\\gamma\\geq 0", "b5d0de27d29299844b2b5f2976ff363f": "-y'", "b5d11692b46623ea806f4f4df8fa2d1e": "{{T}_{\\infty }}", "b5d12b88cda85cf18d87c2dffae314b3": "\\pi\\colon P\\to M", "b5d166e25a3d8380eae15cf283630174": "\\{g_1, g_2, \\dots g_m\\}", "b5d17546de90c54e5370cea890ed865a": "\\sqrt{\\mathrm{1T}}=\\mathrm{1.11T1TT00T00T01T0T00T00T01TT...}", "b5d18f6466ac6ab01d58241355de658e": "\\ R(\\theta) = \\frac{I_A(\\theta) n_0^2}{I_T(\\theta) n_T^2} \\frac{R_T}{N(\\theta)} ", "b5d1a371670070244753ef5f1dee8727": "\n \\begin{align}\n \\hat{X}_{Bayes}(\\mathbf{v}) =\n \\text{argmin}_{\\hat{x}\\in\\mathcal{X}}\\lambda_{\\hat{x}}^\\top\\mathbf{v}\\,.\n \\end{align}\n ", "b5d20d3126a51c0d36af90a7517830d2": "=m_0 c^2 \\left( \\frac{1} {\\sqrt{1 - v^2/c^2}} - 1 \\right) \\ ", "b5d233ed26be49cd86442e013dd03e5a": "(id_\\mathcal{C} \\downarrow G)", "b5d28865d4c812be7993552d33ba3ba2": "\\scriptstyle \\dot{M} < 10^{-6}", "b5d33129ecab7058f6fc7096d94c89b5": "\\mathrm{Cov}(x, y) = \\int_{B} \\langle x, z \\rangle \\langle y, z \\rangle \\, \\mathrm{d} \\mathbf{P} (z)", "b5d35792e8b738afb5fda57440fc8a86": "10^{-3}1, \\,", "b5e9ca1f3b9cabac91a746e38bc2d56a": "p - t \\times p", "b5e9f9299c1b859e16f93295ddd04f40": "nFE = nFE^\\circ - R T \\ln Q_r \\, \\,", "b5ea47054fd51edcaef459254f9457f7": "\n~\\sigma_t^2 = K + ~\\delta ~\\sigma_{t-1}^2 + ~\\alpha ~\\epsilon_{t-1}^2 + ~\\phi ~\\epsilon_{t-1}^2 I_{t-1}\n", "b5ea5e35834a952e82adfd750939a4dc": "e\\ne 1", "b5eae28d6e16da95d82f8eee56e386b9": "m = kq", "b5eb0e55eaae364940f347834720e79b": "\\Sigma_{g,1}", "b5eb3e54a631391e5239136bf56006e7": "w=\\frac{az+b}{cz+d}", "b5eb82cd477f9d284a3a599f132e03e6": "\\sqrt{\\frac{1}{21}}\\!\\,", "b5eb976f24772df2f35d12bc042b3ca1": "x^2 \\equiv 13 \\pmod {17}", "b5ebc15c41c4e0a083233089587e167d": "A A_{\\mathrm{right}}^{-1} = I_m", "b5ebdd892cc15c1284a266f78793ec4f": "R_2 =2", "b5ebde760a90c550ac92f154cb8b6d50": "= [1 + m(t)]\\cdot c(t) \\,", "b5ebfe906f163ed61531b93943f6fe23": "(A\\cap B)^c = A^c \\cup B^c", "b5ec4a0a9559038d13be51d22661f143": "\\log(\\log|1/\\Gamma(z)|)", "b5ec5b8e36f8407249764a55efdf3035": "E=\\frac{e^2}{4\\pi\\varepsilon_0 R}", "b5ec8733fba5fc1d57816dd81a4567ee": "\\Gamma(x)\\Gamma(y)=\\left(\\int_{\\R}f(u)\\mathrm{d}u\\right)\\left(\\int_{\\R}g(u)\\mathrm{d}u\\right)=\\int_{\\R}(f*g)(u)\\mathrm{d}u=\\Beta(x, y)\\,\\Gamma(x+y)", "b5ed9970a044e6019d606720a540ad99": "\\neg(P\\lor Q)\\iff(\\neg P)\\land(\\neg Q)", "b5edac89c1fffa443bb5bb8543a94d53": "\\rho = \\frac{m}{\\Delta x}", "b5edd7958b76532b3b6957775521be99": " \\operatorname{fact}\\ n = \\operatorname{if} n = 0 \\operatorname{then} 1 \\operatorname{else} n * \\operatorname{fact}\\ (n - 1) ", "b5edf44587d20078ae3bc7e3cb9797c9": "n \\sum_{k=1}^n \\frac{(-1)^{k-1}}{k} \\sum_{n_1+\\cdots+n_k=n} \\frac{n!}{n_1! \\cdots n_k!} \\binom{\\binom{n_1}{2}+\\cdots +\\binom{n_k}{2}}{n}.", "b5ee1b175fa73d31f94dd1bd8c9d72a0": "f_N(x) = \\sum_{n=-N}^N \\hat{f}(n) e^{inx}.", "b5ee3131eda42a6da67aa7f58298d69f": "bp_1", "b5ee5261417e3ef128457d1c0213c467": "\\lim_{(x,y)\\to(a,a)}\\frac{f(x)-f(y)}{x-y}", "b5ee536114972786ee4546a290252f07": "\\frac{V_{\\,physiologic\\,dead\\,space}}{V_t} = \\frac {P_{\\,a\\,CO_2} - P_{\\,mixed\\,expired\\,CO_2}} {P_{\\,a\\,CO_2}}", "b5ee6c10617ae952a5b5d18e6d993de2": "\n\\begin{align}\n\\frac{\\pi}{4} =& 83\\arctan\\frac{1}{107} + 17\\arctan\\frac{1}{1710} - 22\\arctan\\frac{1}{103697}\\\\\n& - 24\\arctan\\frac{1}{2513489} - 44\\arctan\\frac{1}{18280007883}\\\\\n& + 12\\arctan\\frac{1}{7939642926390344818}\\\\\n& + 22\\arctan\\frac{1}{3054211727257704725384731479018}\\\\\n\\end{align}\n", "b5ee71c389673e71530b1bfbd735a50f": "t=pq^{-1}", "b5ee8171de5562f0d236b021857d8ea9": "\\Psi : X \\rarr C_b(X)", "b5eec292b0e1dac4fe2ae0d7956f0dcc": " G(x_1, \\ldots, x_n) = \\sqrt[n]{x_1 \\cdots x_n} ", "b5eef7bc2b038d2afc6f6b0a04d88796": "k=2,3,\\ldots", "b5ef746f23d3d5a6266a70d37ca09bda": " F_i = \\sum_{j=1}^m \\left(\\mathbf{F}_j \\cdot \\frac{\\partial\\mathbf{v}_j}{\\partial\\dot{q}_i} + \\mathbf{M}_j\\cdot \\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_i}\\right), \\quad i=1,\\ldots, n.", "b5efbfb9f9870cee381c2105777ea9c0": "b = \\frac{(a_1 - a_2)}{2}", "b5f017f69fb727f11631379e320f16d2": " y_3' = \\lambda'(x_1(x_1-x_2)^2-x_3') - y_1(x_1-x_2)^3", "b5f02a62ae65973ddf77a7db0bfe5348": "t^{(0)} \\gets X w^{(0)}", "b5f08bec6985157be16dff8ba2f0946c": "\n(x,y) \\mapsto (u,v) = \\left(\\frac{1+y}{1-y},\\frac{1+y}{(1-y)x}\\right)\n", "b5f11caebe7992081eb0fe87d98aa1c2": "\\ q = \\frac{4\\pi n_0}{\\lambda}\\sin\\left(\\frac{\\theta}{2}\\right)", "b5f13881ba52040ff1d2cb63faa31d38": "q(z)=p(z)\\overline{p(\\overline z)}", "b5f1cc6065e64f56f9ec2b3fbc49ffa0": "(S_{T}-K)\\vee 0", "b5f1d57df2bc01da4f1647fb873ef886": "k = \\frac{{2\\pi n_0}}{{\\lambda _0}}", "b5f1dbde8494d1482120db59a0c2c9b6": "\\mathcal\n+\\sum_{(\\alpha,\\beta),(\\gamma,\\delta)}\\overline{\\phi(x_\\alpha)\\phi(x_\\beta)}\\;\\overline{\\phi(x_\\gamma)\\phi(x_\\delta)}\\mathopen{:}\\Pi_{k\\not=\\alpha,\\beta,\\gamma,\\delta}\\phi_i(x_k)\\mathclose{:}+\\cdots.\n", "b5f22eac2c1ee76af29ad9f037417cc9": "\\chi(1_G) = \\operatorname{tr}\\ I_V = \\dim V.", "b5f248d87d62f958ad03728a433cc9a7": "F_4(a, b) = a^{(a^{(b-1)})}", "b5f288ffb0dae77db8367195dc2a99f8": " g^2 > 1", "b5f2b060370b637602d2fa5cf746f8b4": "\\mathbf{E}^{x}[M_t | F_s] = \\mathbf{E}^{x} \\left[ \\mathbf{E}^{x} \\big[ M_{t} \\big| \\Sigma_{s} \\big] \\big| F_{s} \\right] = \\mathbf{E}^{x} \\big[ M_{s} \\big| F_{s} \\big] = M_{s},", "b5f31254115067e90bb4c68e07239e8d": " {\\text{External Memory Fragmentation} = 1 - } \\frac{\\text{Largest Block Of Free Memory}}{\\text{Total Free Memory}} ", "b5f36e113b7e7bea8714be7cedd69df7": "\\nabla^2 \\Phi = - 4 \\pi G \\rho", "b5f37dc1f1cfcd287a937be7c3a9c6ba": " \\begin{bmatrix} S_{11} & \\cdots & S_{1 n}\\\\ \\vdots & \\ddots & \\vdots \\\\ S_{n 1} & \\cdots & S_{n n}\\end{bmatrix} ", "b5f39a38ad472176c3b3262f6fa17cbb": " x^{\\mu }=(x^0,\\vec{x})", "b5f4044efb6ad9d48e0cc465a9e73f14": "\\scriptstyle\\{w_1,\\dots,w_m\\}", "b5f4b098d2d147926be7e70e4bd89187": "=a^n+\\sum_{k=1}^{n-1}a^kb^{n-k}-\\sum_{k=1}^{n-1}a^kb^{n-k}-b^n", "b5f4bb4e9a2e0379662d12ac93597014": "d_n(D):=\\sup_{z_1,\\ldots,z_n\\in D} d(z_1,\\ldots,z_n)^{\\frac{1}{\\binom{n}{2}}}", "b5f4d4dbc9d77eb93ed154ced39f2b57": "\\{0, 2, 7, 13, 16, 17, 25\\}", "b5f532c1f4af9c9b9293390e67df97b6": "\\left(\\pm1,\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2}),\\ \\pm(1+\\sqrt{2})\\right)", "b5f5bda5b135931831c4d5ee274b719d": "S_n^0", "b5f5c56bb9b0949d5da5c61f3393cef1": "(e_r)", "b5f5d0b0ae2fa4dfd85d3e832b513a65": " (\\partial H)_S=-(\\partial S)_H=-\\frac{VC_P}{T}", "b5f5d9cbfb40edf495ee267b11947bd9": "\\scriptstyle \\text{GF}\\left(2^{128}\\right)", "b5f5e12cd5d40f92e7433b3384a5814d": "H_n = \n\\begin{cases} \n 0, & \\mbox{if }n\\mbox{ is odd} \\\\\n (-1)^{n/2} 2^{n/2} (n-1)!! , & \\mbox{if }n\\mbox{ is even} \n\\end{cases}\n", "b5f5ecaac73faacf3a8f2739ab46168e": "D^3 = \\{x\\in\\R^3\\ |\\ |x|\\leq 1\\}. ", "b5f6746e456d24d9796d448c9e2a87ea": "\\omega_{x\\land (y\\cup z)} = \\omega_{(x \\land y)\\cup (x\\land z)}\\,\\!", "b5f6771b9577e973e0ac604e59a54472": "\\arg\\min", "b5f6a3bfe3db82d56c91ba1018e81a32": " \\frac{\\partial \\bar{u_i}}{\\partial t} + \\bar{u_j} \\frac{\\partial \\bar{u_i}}{\\partial x_j}\n= - \\frac{1}{\\rho} \\frac{\\partial \\bar{p}}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 \\bar{u_i}}{\\partial x_j \\partial x_j}\n- \\frac{\\partial\\tau_{ij}}{\\partial x_j}.\n", "b5f6d453f012edf61ad2e3b07ba447ed": "2/11 = 0.0\\ 1\\ 0\\ 1\\ 4\\ 6\\ 2\\ 8\\ 1\\ 9_! = 0.0\\ 1\\ 0\\ 1\\ 4\\ 6\\ 2\\ 8\\ 1\\ 8\\ 11\\ 12\\ 13 ..._!", "b5f71b48f5e9ed532dadaaadddf19f14": "= \\mathrm{Var}(\\epsilon) + (\\mathrm{bias})^2 + \\mathrm{Var}(g_i)", "b5f730a3a1fcfdf53a3c30fa1d61f68b": "t \\rightarrow f(X_t)", "b5f731bc9930456a038f9cf1dd0e3233": "_{0}\\!", "b5f7603f638cdcd8ed84e0eb651490e1": "xy\\cdot uv = xu\\cdot yv", "b5f7b43c8814c10589c0dd10f0bcb5f6": "0 \\le\\mu\\leq\\tfrac12", "b5f7c06288222fcfc1b7994b827f467d": "I = [-3.8117, 1.3117]", "b5f7c2f96f17e3cdb2734f2180072ad4": "\\mathcal{K}(X,K)", "b5f7e60e340c9674ec2f7559eb9505d5": "e\\,", "b5f805dc42cd2a08cf6d146db48974c3": "\\displaystyle{x^3y^4z^7}", "b5f84ff248b2da54309e64cd81dd6f0f": "P_e = I V", "b5f8528efa7bf791235eae3ccf08cb67": "\\mathbf{F} = \\mathbf{E} + ic \\mathbf{B}", "b5f891ce584fb02342601b37f6710a9d": "B_\\tilde{\\nu}(T) =2 hc^2\\tilde{\\nu}^3 \\frac{1}{e^{hc\\tilde{\\nu}/(k_\\mathrm{B}T)} - 1 }", "b5f8aaa88ffb741b52d1e6005f830f94": "N_n(t) = \\prod_{j=1}^{n-1} \\lambda_j \\sum_{i=1}^n \\sum_{j=i}^n \n\\left ( \\frac{N_i(0)e^{-\\lambda_j t}}{\\prod_{p=i, p\\neq j}^n (\\lambda_p-\\lambda_j)} \\right )", "b5f8b69e61734e17ee8ead9bf1077ce2": "r \\in \\overline I", "b5f8f9e9596481b15b9cf180da09b2a2": "\\sup_{\\theta\\in\\Theta_0}\\; \\operatorname{E}_\\theta\\phi'(X)=\\alpha'\\leq\\alpha=\\sup_{\\theta\\in\\Theta_0}\\; \\operatorname{E}_\\theta\\phi(X)\\,", "b5f93cc533cdaba5b51e732aa1cb2328": "F = AP", "b5f960b51c53fc20261c019ccaef829f": " \\tau = -\\kappa\\theta\\,", "b5f98007f5983b371d9981cdf91b306d": "k\\in N(h)", "b5f9809b7704a2856f05da24ba24edf5": "f: \\operatorname{Spec}(S^{-1}R) \\to \\operatorname{Spec}(R)", "b5f980c77f880b983d9f221da4326cdc": " f(g)=\\int_{\\widehat{G}} \\xi(g) d\\mu(\\xi),", "b5f9b9ddf37b31222a24af7e9574e672": "v \\in A, B", "b5f9dec433d4a6d357da901e9796582d": "KR_n^G(\\{\\cdot\\})\\cong K_n(R[G])", "b5f9e9cc4add1503f6f1177ec8eb0dd4": "F(L)", "b5fa19e43e23bedb92690d71458e68c0": "R/R_1, \\; R/R_2", "b5fa2f3185f2c7c632b61655792315a7": " C_l \\, ", "b5fa2f464d58456e6372592624194499": "r \\equiv n^{\\frac{Q+1}{2}}\\pmod p", "b5fa680e8e949d824ffcb14359f5b20d": " g^*(x)=\\frac{1}{\\alpha}\\max\\{0,x\\} ", "b5fa8929f8b0f3bac9706232140c6640": "\\sum_{k=0}^n \\; {\\alpha\\choose k} \\; (-1)^k = {\\alpha-1 \\choose n} \\;(-1)^n, ", "b5fab4a924ae3047ee331abb6d554d7b": "D=P^2-4Q,", "b5fabb1d12ac7cadc800765285cd00c3": "\\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix}", "b5fb230f69758c90675cd819926b7e84": "p=9", "b5fbb7fa8551b12a2e81ac599526a59d": "\n D \\partial^2_{\\xi}\\hat{u}(\\xi)+ c\\partial_{\\xi} \\hat{u}(\\xi)+R(\\hat{u}(\\xi))=0.\n", "b5fc7d94e3679f0161b72d97cdd2e6b0": "\\nu(x\\cdot y)=\\nu(x)+\\nu(y)", "b5fc81e0c4effd9ea321fddfb6319f0d": "F_i,F_j", "b5fc8b877f02c72981ab48f4e3be8152": "\\bar{K}_{i,j}(\\tau_1,\\tau_2)", "b5fc8ec656510459eef48116f0969560": "\\begin{align}\n\\text{sensitivity} & = \\frac{\\text{number of true positives}}{\\text{number of true positives} + \\text{number of false negatives}} \\\\ \\\\\n& = \\frac{\\text{number of true positives}}{\\text{total number of sick individuals in population}} \\\\ \\\\\n& = \\text{probability of a positive test, given that the patient is ill}\n\\end{align}", "b5fc93a18a72a23e3ec51540d61dd267": "\\scriptstyle{f}", "b5fc97dcd5cf85e2eaa4d6ffc8a006c6": "n_{\\alpha}(\\phi_{\\alpha}) = n_0 v_{\\alpha,0} v_{\\alpha}^{-1}(\\phi_{\\alpha}),", "b5fcc53ff6ee1c79c39011ccafcec582": "\\Theta(g n)", "b5fd0863242c3d9d3648bf7029e39bc3": "{1 \\over 1}+{1 \\over 4}+{1 \\over 9}+{1 \\over 16}+{1 \\over 25}+{1 \\over 36}+\\cdots = {\\pi^2 \\over 6}.", "b5fd373294232842fa2484cc27a86359": "\n \\begin{align}\n \\boldsymbol{\\sigma}\n & = \\cfrac{2}{J}~\\left[\\left(\\cfrac{\\partial W}{\\partial \\bar{I}_1} + \n \\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right)~\\bar{\\boldsymbol{B}} - \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\bar{\\boldsymbol{B}}\\cdot\\bar{\\boldsymbol{B}}\\right] \\\\\n & \\qquad + \\left[\\cfrac{\\partial W}{\\partial J} - \n \\cfrac{2}{3J}\\left(\\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_1}+\n 2~\\bar{I}_2~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right)\\right]\\boldsymbol{\\mathit{1}} \n \\end{align}\n ", "b5fd4e16b4b306ce5afc0bd82f92b582": "\\mathcal{L}\\{f(t)\\}=\\frac{\\mathcal{L}\\{\\phi(t)\\}+\\sum_{i=1}^{n}\\sum_{j=1}^{i}a_is^{i-j}c_{j-1}}{\\sum_{i=0}^{n}a_is^i}", "b5fd5fa8e336bf1f4dde623c11cfd0bf": " j^{\\star} = \\sigma T^{4}", "b5fd706005db1ed0da7820f99b5a1120": "\\Pi^0_n", "b5fd7f24810677aab406512dae7a3a50": "\n d\\mathbf{a} = J~\\mathbf{F}^{-T} \\cdot d\\mathbf{A}\n\\,\\!", "b5fde5dc95f09efaec8f78c00e5649d7": " R_{w}, G_{w}, ", "b5fe27dc0c9387d83477924336f279eb": "g_{nm} (t) = s(t-m \\tau_0 ) \\cdot e^{j\\Omega nt}", "b5fe35be1a9dcb523d16898e8d7a7d25": "{\\mu}\\,", "b5fe656a33b35558cfec4b4b4c52299b": "\\begin{align}\n\\int x^m (\\ln x)^n\\; dx\n& = \\frac{x^{m+1}(\\ln x)^n}{m+1} - \\frac{n}{m+1}\\int x^{m+1} \\frac{(\\ln x)^{n-1}}{x} dx \\qquad\\mbox{(for }m\\neq -1\\mbox{)}\\\\\n& = \\frac{x^{m+1}}{m+1}(\\ln x)^n - \\frac{n}{m+1}\\int x^m (\\ln x)^{n-1} dx \\qquad\\mbox{(for }m\\neq -1\\mbox{)}\n\\end{align}\n", "b5ff2555dbb611296ff01465f17c6750": "\n G = K_{\\rm I}^2\\left(\\frac{1-\\nu^2}{E}\\right)\n ", "b5ff701deb782c6ef54fd1ffedd82d9d": "\\frac{25}{24}", "b5ff71ca3617985476347b2954da2eda": "\\delta^n: C_{n-1}^* \\rightarrow C_{n}^*", "b5ff8cb0b995e57524ed166083b08dc8": "\\frac{d^2\\Phi}{d\\phi^2}=-m^2\\Phi", "b5ffa09e2dcdf336331f92a176c726b3": "\n\\mathrm{THD_F} = \\frac{ \\sqrt{V_2^2 + V_3^2 + V_4^2 + \\cdots + V_n^2} }{V_1}\n", "b5ffe0dca537fb03fe2b6b246aba5e5b": "X_{\\epsilon(X, X^*)}", "b6009f7a7cd2b3bce4e229b6ac7e1f99": "Lu_G=A(x)Lu_1(x)+B(x)Lu_2(x)+A'(x)u_1'(x)+B'(x)u_2'(x).\\,", "b600a4558d43f47614382063a055462e": "[WE_1]", "b6010e55e500135d34b5a98e08d0d306": "\\exp_a b = a^b, \\exp b = e^b, 10^m \\!", "b601705266abd4ce473a6892320c52dc": "\\left(\\begin{array}{ccccc} 0 & 2 & 6 & \\frac{1}{3} & 4\\\\2 & 0 & 4 & 8 & 0\\\\ 9 & 4 & 0 & 2 & 933\\\\\n1 & 4 & 4 & 0 & 6\\\\ 7 & 9 & 23 & 8 & 0\\end{array}\\right)", "b60189e71e2347f7fe47c29825a5e6b0": "f(x; \\boldsymbol{\\theta})", "b601a6be75a9cbc7dc7a28fa154f1cb7": "\\Delta R=k\\Delta T \\,", "b601d1ffb40cd41011217487fe40901c": " f(x) \\approx f(0) + f'(0)(x - 0).", "b601daf67e7beb498303e5be9a2b9ea4": "F(b)", "b601e2162effc84a925b1ccd20799f7f": "\\scriptstyle{1}", "b60239728f23ac093dbada1358b2c944": "X^\\infty", "b602b96aedf5560677023878af48f261": "\\sigma^x_i.", "b602e3660f38ffcc08be88e8b5460523": "\\cos (\\alpha + \\beta) = \\sin\\left( \\pi/2-(\\alpha + \\beta)\\right) = \\sin\\left( (\\pi/2-\\alpha) - \\beta\\right)\\,", "b602fe0c1c73a4a77e3f8eb489d6384e": "\\left.\\begin{array}{l}\n\\mathrm{d}\\theta^a = -\\sum_{b=1}^p\\theta_b^a\\wedge\\theta^b\\\\\n\\\\\n0=\\mathrm{d}\\theta^\\mu = -\\sum_{b=1}^p \\theta_b^\\mu\\wedge\\theta^b\n\\end{array}\\right\\}\\,\\,\\, (1)\n", "b60350edc53cb11a3d71acf8bf0354b8": "F(\\theta)", "b60357309b038076dca368f72f193939": " {f_2 \\over f_1} = \\exp \\left ({1 \\over RT} \\int_{G_1}^{G_2} dG \\right) = \\exp \\left ({1 \\over RT} \\int_{P_1}^{P_2} \\bar V\\,dP \\right) \\,", "b603629934a490be25bfc47d48fdba8a": " i_B = -v_{ \\pi} \\frac {1+R_2/R_1+R_f/R_1} {(\\beta +1) R_2} \\ . ", "b603934c48c20ec905190d6b31048439": "\\mathbf{f}_0", "b60399dae71453c6dd79b46fd277cbe2": "F\\left(x,w\\right)=w^2-P\\left(x\\right), \\, ", "b603b3c0a14bb2de5ba69ce6023c5b30": "\\Gamma(\\gamma)_s^s = Id", "b60563836876273ca6693f89e062e424": "i \\in \\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left({A}\\right)\n", "b60578ea9286326c00bce2d00bd2cc7f": " \\Phi_B = \\int_S \\mathbf{B} \\cdot \\mathrm{d}\\mathbf{A} .\\,\\!", "b6059572028eb3337b1980b35cbc12e3": "CCA = tdUCC - \\frac{1}{2}td\\left ( a - b - c \\right )", "b605cef3cd3a461c32450b656da8d53f": "c \\to 1", "b605f22144cb7b29fc68e465c26d0717": "f(\\boldsymbol{x})=\\boldsymbol{x}^\\mathrm{T}\\boldsymbol{A}\\boldsymbol{x}-2\\boldsymbol{b}^\\mathrm{T}\\boldsymbol{x}\\text{.}", "b6063aecc8ee46d35205b0d97e91ab91": "\n\\mathrm{LEC} = \\frac{\\sum_{t=1}^{n} \\frac{ I_t + M_t + F_t}{\\left({1+r}\\right)^t} }{\\sum_{t=1}^{n} \\frac{E_t}{\\left({1+r}\\right)^{t}} }\n", "b6065bb97aa73920366cba01dc4dbd8c": "s_n:=\\inf\\{\\mu(D):D\\in\\Sigma,\\, D\\subset X\\setminus N_n\\}.", "b606686363620a63377067cae470a818": " \\beta = \\frac{1}{\\sqrt{\\sum_{{\\mu}=1}^k \\frac {c^2} {v^2_{\\mu}}}};", "b6068a77fa5bcfc074e4a19384703cef": " M = m - 5 (\\log_{10}{D_L} - 1)\\!\\,", "b606a400873dad5625d810e0ff16eaa5": "n=\\frac{pq}{{\\sigma_P}^2}", "b60761c70639e1803e0a2edc8c5c1e4a": "F_\\mathrm{DEP} = \\frac{\\pi r^2 l}{3}\\varepsilon_m \\textrm{Re}\\left\\{\\frac{\\varepsilon^*_p - \\varepsilon^*_m}{\\varepsilon^*_m}\\right\\}\\nabla \\left|\\vec{E}\\right|^2 ", "b6083ad79ceaa342cd2e53f1d4e16385": "\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1", "b608ae6e9987ec8255ee2bce5c963aa0": "\\Im(z)-F(w,w)\\in V \\, ", "b608c2688809ea931e2d08767fc17b15": "ac^*b,", "b608e9db129c774a2458eef05ef9fabb": "a_k^\\dagger\\,a_k|\\dots, N_k, \\dots \\rangle=N_k| \\dots, N_k, \\dots \\rangle.", "b6091496fb783209519bc7a58f0be172": "a^{s \\dagger}_{\\textbf{p}}", "b60930b60e5c04fba5691e251b886c4a": "\n\\rho_i = \\rho(\\mathbf{r}_i) = \\sum_j m_j \\frac{\\rho_j}{\\rho_j} W(| \\mathbf{r}_i-\\mathbf{r}_j |,h) = \\sum_j m_j W(\\mathbf{r}_i-\\mathbf{r}_j,h),\n", "b60995eccc4701b5d366618048e2ef96": "2T \\cong \\widehat{A_4}.", "b60998b034ebcf39a9f3b7125bb6a9e1": "V=\\{\\vec{u}\\in (H^1_0(\\Omega))^n|\\operatorname{div}\\,\\vec{u}=0\\}", "b609a24b7be4d2ad849b4c49ebccbc0e": " f(z)={1\\over2i\\pi} \\int_\\Gamma{\\phi(t)-\\phi(z)\\over{t-z}}\\, dt\n+ {1\\over2} \\phi(z) ", "b609b5dbf535b21bd0a57154820a04b2": " \\hat{X_a} ", "b609dc7efa41c8635fd50aac2ab879e9": "\nw\\left( {T_{ij} } \\right) = \\frac{{7!}}\n{{2!1!1!0!2!1!}} = 1260\n", "b60a20ba93f2377eee1394b934a9d38a": "dx = f(x,t) dt + dw", "b60a727bae3021dd4fce1c9c3f0f7943": "(L \\otimes_A L',B \\otimes B')", "b60a7f20247e3fb6e52a65d8846d3380": "VA^{-1}(I-UY) = C^{-1}Y", "b60b24b3129215c63d80400c6e1e9c03": "log_2(K)", "b60b36645369aed26adac65533077b63": "\\partial u/\\partial z<-\\gamma_0", "b60b512048732a7f4a297a3da09d40c7": "a_p^{\\dagger}", "b60b8f210d5bf91d7936fe8a320e00f1": "(2, \\delta, \\frac{1}{2}-2\\delta)", "b60b92366c36210afd86b1477704a635": "E[\\psi,\\lambda]=\\frac{\\langle\\psi|\\hat{H}_{\\lambda}|\\psi\\rangle}{\\langle\\psi|\\psi\\rangle}.", "b60bc2530ddf9c351b97d0cdd2dc3258": "\\Gamma_x \\subset \\Gamma'_x", "b60bcc14c5fb3ee3ccab3ec91911e78d": "X\\in Ob(C), Y\\in Ob(D)", "b60bf52c9b56205956b1928e2907f850": " P( X > 0 ) \\ge \\frac{ C_0 }{ \\psi } ", "b60c1a3f0aab20da0ffe2668b311599d": " A_\\mu(p) ", "b60c35ec85bdde32495c13fef0ce7b33": "\\sigma_{n,d}=0, \\; d = 1 \\ldots D", "b60c5bddd33cf816bcb6c6c68bf6a821": "\\epsilon\\ll 1", "b60cc3473a00994267b6d68fc2e2213d": "\\psi \\in Q_\\epsilon(F)", "b60cc95e3cbd67da95533fb5d97256cc": "T\\left(eV,eV,r\\right)", "b60d2312008aea67872e839acc27a0e1": "u_n(\\mathbf{r}) = u_n(\\mathbf{r-R}) ", "b60d47beebc133bf395a3f49401287ce": "u_t+u_x=0", "b60d6af9bdc84a7793c79498faf92b29": "0<\\epsilon<1/2", "b60eaf692beab6b7728fe1bceba9ff50": "|f^{(k+1)}(x)|\\leq M", "b60f30276a32adf49b5b6e5b8d6bfb9e": "u, v \\,", "b60f4400a8d6f91685758b286d999b4f": "\\qquad \\qquad k_p = \\frac{1}{6\\pi^3}\\sum_{\\alpha}\\int c_{v,p}\\tau_p {u}_{p,g}^2\\kappa^2d\\kappa \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm{for \\ isotropic \\ conductivity}.", "b60f8885bfe001b2323c3701ffa403df": "\\frac{1}{9+1} = 0.1", "b60fea8bbcda86c78dfb8fa626250337": "R_n(V) = \\frac{\\Gamma(\\frac{n}{2} + 1)^{1/n}}{\\sqrt{\\pi}}V^{1/n}.", "b60ff03e8aebb32b3fb8d47da662cc3a": "\\Gamma\\vdash e\\!:\\!\\sigma", "b610110c1217b14be4d5d4aef26bd9db": "f(x,y)=x^2 + y^2", "b6107ac5363e951c88160c06ca95a7ab": "\\mathbf{W} = \\{ W[j, k] \\} ", "b610823c2043aad9a078c09e64338317": "p_n(x)=x(x-an)^{n-1}. \\,", "b610a6be2280368b199dac98ac99d2f8": "\\frac{1}{\\psi}", "b61140d008e61c59e46f1e5f19268f95": "\n R_k(x) = \\frac{f^{(k+1)}(\\xi)}{k!}(x-\\xi)^k \\frac{G(x)-G(a)}{G'(\\xi)}.\n", "b611c66792087b71041987a2fe2fb5a3": "\\langle K\\rangle_N", "b611d3c842d44661de59757af134d178": "(\\tfrac{\\mathrm m^2}{\\mathrm s})", "b612955441f59aa8098f4df8349e91e4": " j = \\{ a \\in l_\\infty : {\\rm diag}(\\mu(a)) \\in J \\} . ", "b612c94332b3a4403c9df46c15952253": "\\mathbf{v'}=R\\mathbf{v}=\\begin{pmatrix}\n\\cos \\theta & -\\sin \\theta\\\\\n\\sin \\theta & \\cos \\theta\n\\end{pmatrix}\\begin{pmatrix}\nv^1 \\\\\nv^2\n\\end{pmatrix}.", "b612e87c6ff8fbfbd5bf02b49cdf27e4": "\\zeta(s,n/k)=\\frac{k^s}{\\varphi(k)}\\sum_\\chi\\overline{\\chi}(n)L(s,\\chi),", "b61310e800f19b4bd59aa3c5ad2c4fba": "\\mathrm{adj}(\\mathbf{A}^\\mathsf{T}) = \\mathrm{adj}(\\mathbf{A})^\\mathsf{T}.", "b61340057e8dc4df64776e4b28a873a5": "\\sum_{n=2}^\\infty \\Lambda(n) e^{-ny} \\sim \\frac{1}{y},", "b613f697434176a930d2aea7b7095e5a": "\\bar{S}_{2k+1}(c)", "b6144b558fcc0a5505b21d6b05456ab6": "o \\in O", "b6148e05adce7e60adaa7ae79512344c": "a_{123}\\,dx\\wedge dy\\wedge dz.", "b615415d6a15ca54d6d21737c6056e27": "dY=\\lambda \\delta X", "b6155309baf501dba761eeac25b9f8e6": "E_\\lambda : z \\to \\lambda * e^z", "b61560cc582fdd6edd8fde165d5fafa1": "N \\cup O", "b615704df7838928bbe5e52ebaf79792": "y'(t) = -y^2", "b6159a70e4a00a8d99f2d671827aafda": "\\{(x,y) \\mid x R y\\}", "b6159d91b09ffb836190d1219d700bc8": " \\sin x = \\tfrac{1}{2} (\\sin(nx) + i\\cos(nx))^{1/n} + \\tfrac{1}{2}(\\sin(nx) - i\\cos(nx))^{1/n} ", "b615b17230a88d57227ead6f77e85f06": "\\operatorname{perm}(J - I - I') = \\operatorname{perm}\\left (\\begin{matrix} 0 & 0 & 1 & \\dots & 1 \\\\ 1 & 0 & 0 & \\dots & 1 \\\\ 1 & 1 & 0 & \\dots & 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 1 & 1 & \\dots & 0 \\end{matrix} \\right) = 2 \\cdot n! \\sum_{k=0}^n (-1)^k \\frac{2n}{2n-k} {2n-k\\choose k} (n-k)!,", "b6160aceebee1fc058815af24736b4c0": "\nP_{l}(\\cos \\theta) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{4\\pi}{2l+1}} Y_{l0}(\\theta, \\phi)\n", "b6166425b0ed13ba7fe1b59c6994ddac": " n_1 \\over d_1. ", "b61675e0d4931e0edac7031280f2bc67": "\nn=2^{n_2}3^{n_3}5^{n_5}7^{n_7}\\cdots=\\prod p_i^{n_{p_i}}.\n", "b6167b6afd511906373a961fc9d81944": "\\lim_{n \\to \\infty}{ P\\left( \\left| \\frac{K}{n} - x \\right|>\\delta \\right) } = 0", "b616a193642642f7941278aa624fab9c": "\\textstyle x \\notin Z ", "b61719e4483d24b6b51917d6c1d2bb14": "\\varrho", "b617300c883fc8793d71ff280af48e87": "\nPV = Nk_{B} T = nRT,\\,\n", "b6173aa5cf83e30d76ac19f12b69ee51": "\\bar{n}_i ", "b617a71d30d11e7acf3ef74f25e3bb00": " \\mathrm{Hom} (-,-)", "b617fd80cc98614bae4b2ffb1eb95161": "y \\equiv_{pc} x", "b6182f8f2256775999b7fcb33d9af43c": " 1\\,\\mathrm{lbf} = 1\\,\\mathrm{slug} \\times 1\\,\\mathrm{\\tfrac{ft}{s^2}} ", "b6184b6403377c7a3e0977d56ef5b04a": " = |(a_1 - b_1) + i(a_2 - b_2)|", "b6188ef2a499e2b5f84645d17c4420cd": "\\boldsymbol{k}=k\\boldsymbol{N}", "b61938be4d7ab7d3b11bd662f3a36bf2": "\\Delta^n_{kh} (f, x) = \\sum\\limits_{i_1=0}^{k-1} \\sum\\limits_{i_2=0}^{k-1} ... \\sum\\limits_{i_n=0}^{k-1} \\Delta^n_h (f, x+i_1h+i_2h+...+i_nh).", "b619515b8506dc2970977dda63aeff99": " g_1(x) = \\lim_{\\varepsilon\\to0+} \\mathbb{P} ( Y \\le 1/3 | x-\\varepsilon \\le X \\le x+\\varepsilon ) \\, , ", "b619612161d3f27b6d678c9585ba27c5": "E^\\ominus {\\rm (H^+/H_2)(abs)}", "b61964640fb3a5af848e0d8ea7d9ea9d": "L(x)^{-1} = L(x^{\\lambda})", "b6196ac10489e856109c6b6cdc48c200": "\\Omega = \\sqrt{|\\chi_{g,e}|^2 + \\delta^2}", "b619a665eea2c405c560ebaae572fbbf": "S \\setminus \\{ e \\} \\!", "b61a6f4a5ad27ce0ecba71108b9d8b55": "\\tilde u_i", "b61affb0d13f266ae4eb0587ae96c0c7": "\\int_0^{\\frac{\\pi}{2}} \\frac{dx}{a+b\\cos x}=\\frac{\\cos^{-1}(b/a)}{\\sqrt{a^2-b^2}}", "b61b1300e834e1c6a5b11d7d2a3a77b8": "Y_{5}^{3}(\\theta,\\varphi)={-1\\over 32}\\sqrt{385\\over \\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(9\\cos^{2}\\theta-1)", "b61b77220fc105c422a2426ea0f44977": "A= \\frac{n}{4}\\left[(a_1^2+a_2^2)\\cot \\frac{\\pi}{n} + \\sqrt{(a_1^2-a_2^2)^2\\sec^2 \\frac{\\pi}{n}+4 h^2(a_1+a_2)^2} \\right]", "b61bc46f22550f19681f90494eb058ac": "\\begin{align} P(Hypercalcemia~WHOIFPI~by~PH) = P(PH~WHOIFPI) * r_{PH \\rightarrow hypercalcemia} = \\\\\n 0.00125 * 1 = 0.00125 \\end{align}", "b61be5c647c498a135c0d23432353aca": " \\sum_{i=1}^k a_i^{\\downarrow} \\geq \\sum_{i=1}^k b_i^{\\downarrow} \\quad \\text{for } k=1,\\dots,d,", "b61be62e4461f7026449741fedd9106d": " 2^2 ", "b61c45d9da25d4fa897bf62a932f6013": " R=(R_k)_{k\\in K} ", "b61c717336a888013511a106676b8297": "t(c) + t(d) = 0", "b61c9bcb991ec4a2006b896f1f339a0e": "\\sigma(M) = 4\\pi \\chi(M).", "b61c9d858e39944ebd01ac6766b9d790": "\\frac{1}{50} + \\frac{1}{30} + \\frac{1}{150} + \\frac{1}{400} = \\frac{1}{16}", "b61cf542a810b924c8604d4401fa58da": "\\int \\tan^2{x} \\, \\mathrm{d}x = \\tan{x} - x +C", "b61d0ddd2b530b34a1004aa7d358348e": "\n F = -\\cfrac{3}{2}\\Delta\\gamma\\pi R\\,\n ", "b61d404c266bf6df70b5e4008ef38b74": "\\textstyle \\forall a,b \\in \\mathbb{Z}_p^*:\\ e\\left(P^a, Q^b\\right) = e\\left(P, Q\\right)^{ab}", "b61da6081735ca4e2585bbcc2f5bac66": "\\cos \\theta = \\frac{ \\mathbf{A \\cdot B}}{\\|\\mathbf A \\| \\|\\mathbf B \\|} \\ \\ ( -\\pi < \\theta \\le \\pi )", "b61de492e114fdea590c98ae3e6a69cd": " \\nabla(\\psi+\\phi)=\\nabla\\psi+\\nabla\\phi ", "b61dfd183394411dec2dde1f699604f0": "\\left(\\mathcal{F}_i\\right)_{i \\in I}", "b61e3e119f0c82ac140a6035ebde1adc": "y = x^{-1}(xy).", "b61e3f18cb3956c9f140a97ab14fc019": "\\mathfrak c ^{\\mathfrak c} = 2^{\\mathfrak c}", "b61e4311384fc4e2776d32d3792df273": "3^2=2^3+1", "b61e4c9b29a2fb83dbb1badc0b00a4f0": "M(x)= e^{2\\int \\frac{1}{x}dx} = e^{2\\ln x} = x^2.", "b61ea8c787ce313f529effda346778d0": "\\Pr[\\vec{P} = 0] \\leq \\Pr[p_i = 0] \\leq \\frac{1}{2}.", "b61ec2d76acbcba46737b1b7e5206295": "\\propto r", "b61ecd66c5584991e31dae11c601c7ea": "\\phi \\colon TM \\longrightarrow M\\times {\\mathbb R^n}\\,", "b61f2797fc12b10ed5c51dba473f4118": "b=\\frac{8\\pi^5 k^4}{15c^3h^3}", "b61f44a7e8ec129ea32abe414c2c08f1": "\\lambda_D \\ll\\lambda_{Te}", "b61f870fe50364a8c4721035916c8355": "\\sin(\\alpha_R) = 1.22 \\frac{\\lambda}{D}", "b61fa43ff71c6c5931c3612d13564fbc": "l_\\text{A} = \\frac{\\hbar^2 (4 \\pi \\epsilon_0)}{m_\\text{e} e^2}", "b61fc4fa7f4945d850411fd956319354": "\\scriptstyle\\Psi(x,B)", "b6201992f4c59679d538891b928453d7": "Xy'=g\\,", "b62057487de2f4ff559d6e405bf048da": "(a_1,a_2,...,a_n) \\rightarrow (|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\\, .", "b6209cb13840b8f9092904da80842ac3": " a + y > b + x ", "b620a81dd3ece010422b26a18d14f485": " f(X \\to Y) = X \\to Y ", "b620ec647f64441164349ca2e067fa38": "S=\\int d^4x\\sqrt{-g} \\; \n\\left(\\frac{\\phi R - \\omega\\frac{\\partial_a\\phi\\partial^a\\phi}{\\phi}}{16\\pi} + \\mathcal{L}_\\mathrm{M}\\right)", "b620fc620006dfbe43c20173814f5654": "\\left \\lfloor qu/p \\right \\rfloor", "b6212b218375d5b5e82dd21cc96293ab": " D(N) = \\sup_{a\\le c\\le d\\le b} \\left\\vert \\frac{\\left|\\{\\,s_1,\\dots,s_N \\,\\} \\cap [c,d] \\right|}{N} - \\frac{d-c}{b-a} \\right\\vert . \\,", "b6212c9d2d6221ae8f6edcfc39e85cec": " \\tfrac{6}{\\pi^2} ", "b62161aeccf6a03f53d132aec1ae3c69": "\\rho = \\frac{MP}{kT}", "b621c0e82db4fe3bc62666acc32421c0": "\\mathbf{y} = \\sum_{k=1}^{K} \\mathbf{h}_k \\sqrt{q_k} s_k + \\mathbf{n}", "b6221638128486d49dce5e43e94a4fea": "\\tan (A \\pm B) = \\frac{ \\tan A \\pm \\tan B }{ 1 \\mp \\tan A \\ \\tan B}", "b6221cf1b2e5db7d42a32afd7591f4c0": "\n\\begin{bmatrix} x\\prime \\\\\ny\\prime \\end{bmatrix}\n= \\begin{bmatrix} A & B & C\\\\\nD & E & F\\end{bmatrix}\n\\begin{bmatrix} x \\\\\ny \\\\\n1 \\end{bmatrix}", "b6229106f49ec7deacf8531e8d1631c1": "S_a \\circ T_b = \\exp (2\\pi iab) \\; T_b \\circ S_a.", "b622c552b6b1086d1a01ca374f43f5b9": "M=\\mathcal{T}(A)", "b622e878a39805053c37ce0bd981f575": "\\scriptstyle x\\,=\\,\\mathbb{E}\\{X|\\mathfrak{G}\\},\\,y=X-\\mathbb{E}\\{X|\\mathfrak{G}\\}", "b62300b5a36934f95edd44ac74feaa18": " H = - {1\\over2} {d^2 \\over dx^2 } + {1 \\over 2} x^2 ,", "b6231cba3180037e6378398375fe0239": "\nP\\approx P_0 \\left(1 + X + \\frac{X^2}{3} - \\frac{1}{45} X^4 + ...\\right)\n", "b62379de90e5e7ed935bb1321a066c9e": "K:=\\R", "b623b17a9f7bd178d1848ccf71c12246": "\\tilde{u}_{t+1}", "b6244fb778ddf76b17d0476b6c033c80": "\\textstyle S=3N-6 ", "b624c16b84d2290a6c1f0a1d0a827d11": "\\mathrm{(CH_3)_2C}", "b62521b04c1403ec150c915a10ca729d": "w(X)\\leq\\kappa\\,", "b6252c47c600d72b4b0f484c229f580d": "\\wedge", "b62568e907a52e4499a530490090d5b8": "U = {3\\pi\\over2} kT \\left({2LkT\\over hc_s}\\right)^3\\int_0^{hc_sR/2LkT} {x^3\\over e^x-1}\\, dx", "b6258554e480f5ea71258543f9e9bf48": " N_{0}", "b6258cd164c7cf4f47c570f7b80f8e70": "G_{k} =\\frac{ 4 A_{k-1} A_{k} }{ B_{k-1}G_{k-1} },\\qquad k = 2,3,4,\\dots,n ", "b625fdaa13b0d27f635a55acc01a54ef": "P(x) = \\Pr(X > x) = C \\int_x^\\infty p(X)\\,\\mathrm{d}X = \\frac{\\alpha-1}{x_\\min^{-\\alpha+1}} \\int_x^\\infty X^{-\\alpha}\\,\\mathrm{d}X = \\left(\\frac{x}{x_\\min} \\right)^{-\\alpha+1}.", "b6265014adfe1cee3336aa6865eee898": "\nF(r) = Ar^{-3} + Br + Cr^{3} + Dr^{5}\n", "b626b2c0d15115fcfb684d2b0e853091": " (1 + x)^\\alpha \\approx 1 + \\alpha x.", "b626e12170402a3377d307693f4d46fc": " [AB,CD] = A[B,CD] +[A,CD]B = A[B,C]D + AC[B,D] +[A,C]DB + C[A,D]B", "b6271d4cd1696b9073f318278ae53304": "(x,\\,y) \\ \\stackrel{\\mathrm{def}}{=}\\ \\langle x,\\,Jy \\rangle,", "b6272461f4a2a97db67e9ec1fcd60b51": "\nm_0^2 c^4 = E^2 - c^2p^2 = \\hbar^2 \\omega^2 - c^2 \\frac{\\hbar^2 \\omega^2}{c^2} = 0,\n", "b62752675ac6d21b4fe23b787d16c18f": " M_2 ", "b627e7bb16c49a3cff64c4e132280f73": " M = \\frac{D N_v}{f''} ", "b627f49b8abdc28c1bc54f0c6dbae06b": "g_i \\sim \\mathcal{N} \\left( 0, 1 \\right)", "b62818a470ccd48807e36dfcac35cfbb": "n=N", "b62843f564ee90c0a0db6acd1a9dd032": " 1<\\alpha <2 ", "b6287e27ae37f971e61370f5a63e074e": "2|U_K|", "b628b982d7282524edc91803451374c8": "f_{ij}=f_{ji}", "b628fa976c7f9e03e61feb9c50975e84": "\\pi_A(\\{ \\langle A=a, B=b \\rangle \\}) \\cap \\pi_A(\\{ \\langle A=a, B=b' \\rangle \\}) = \\{ \\langle A=a \\rangle \\}", "b62935b764e9eb443cbfcf56130e5503": "\\nabla F(x)\\cdot(y-x) \\ge 0\\text{ for all }y\\in C. \\, ", "b6295da69df7f9b08e227e98997f6a1a": "s \\,", "b62a0171d6050a21e9e11f5df93dc1a8": "\\mathrm{^{243}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{244}_{\\ 95}Am\\ \\xrightarrow [10.1 \\ h]{\\beta^-} \\ ^{244}_{\\ 96}Cm}", "b62a5b5760bcde2a77528f950527f26c": "\\varepsilon_{ij} = \\frac{1}{2} \\left( \\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i} \\right)", "b62ad37ac87dff1d7ffe4d2df02d1a97": "{a \\over R_1} \\cdot {m_2 \\over m_1}", "b62ae3289cfa5cdd7f71a143c37b22a0": "{i=1,2,...,N}", "b62ae37c4127a7163ab0fb2bd47664a5": "2|U_{\\bold{G}}|", "b62b594be33424b382ba19c162855f2f": "\ne_i^{t+n} - e_i^t = NS_i + IM_i + RS_i\n", "b62b8016ce0dd2f0856bd0bd6c960c09": "\\frac{h_n}{k_n}-\\frac{h_{n-1}}{k_{n-1}} = \\frac{h_nk_{n-1}-k_nh_{n-1}}{k_nk_{n-1}}= \\frac{-(-1)^n}{k_nk_{n-1}}.", "b62bb57f5eb7cf503a25cdc33b2635d1": "\\alpha,\\alpha^2,\\dots,\\alpha^{n-k}", "b62c02c3d22092006f4306ea73ec7be1": "x \\to a^{-n}xa^{n} ", "b62c84921249d98121f43ef273f07f6c": "\\displaystyle{Mp(2,\\mathbf R)=\\{(g,G): \\,G(z)^2=m(g,z)\\}.}", "b62ca46b4e1e0ec0b6df4f012ecabd44": " (\\overline T)^* = T^*. ", "b62ce09fb8e35fb39fb82756b1a0b406": "q(D,\\widehat{D})\\leq-\\alpha\\,\\!", "b62ce1b42a86d3b781d62418bc90e05b": "\\epsilon>0", "b62cf19167e84df21cbdb9db87fb708d": "\\forall x,y [x\\neq y\\rightarrow[xRy\\vee yRx]]", "b62cf57b8a1f72804dac059593c6f465": "\ny= \\ln \\tan\\!\\left(\\dfrac{\\phi}{2} + \\dfrac{\\pi}{4}\\right).\n", "b62d30f88785fa46e313b8d0c3696c6f": "U = -\\boldsymbol{\\mu}_L\\cdot\\mathbf{B}", "b62d3a6ab32f22c940e77c7795fe328e": "d(p, q) = \\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2+(p_3 - q_3)^2}.", "b62d451d80868bfc6c30ebe203c9d62c": "b \\in \\{0,1\\}", "b62d5a93e55145b80caafca2cc1ff327": "\\mathbf{A}(\\mathbf{x},t)=\\frac{\\mu_0}{4 \\pi}\\int d^3\\mathbf{x'}\\int dt'\\frac{\\mathbf{J}(\\mathbf{x'},t')}{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}\\delta\\left(t'-(t-\\frac{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}{c})\\right)", "b62d661ae2e04de5fdef7d4cd54dffa4": "A \\oplus A = 0", "b62e4a1d299e14473fc6b8628dd5ce70": "E = mc^2 \\oplus pc.", "b62e69d0d6e3aadf0deccba88965e6bf": "\\frac{x^3 - 12x^2 - 42}{x^2 + x - 3}", "b62e726c08f2bdb4891bc2a5bd66e497": "F_\\text{Poisson}(k;\\lambda) = 1-F_{\\chi^2}(2\\lambda;2(k+1)) \\quad\\quad \\text{ integer } k,", "b62e7bb9f6b49bd8f5482289ee207615": "T^\\mu = \\frac{\\partial x^\\mu(s, t)}{\\partial t}.", "b62e8dabe822649c5173bb6dda8564a6": "2^k \\leq s(F) < 2^{k+1}", "b62e8f790d30244f96fbcd522df71965": " d \\leq 2 ", "b62ea5126e7b75c575ff17f06d12b17c": "Z = {E_0^-(x) \\over H_0^-(x)} ", "b62eb75d9c07ee61affacf9958060def": "T_{is}", "b62ee571e757703a39b83dc350098df6": "\\begin{vmatrix} a & b\\\\c & d \\end{vmatrix}=ad - bc .", "b62f2fdd3d492ccb999d9ccbd4824a55": "\\rceil", "b62f5ce620c4e3e4bbc0f2999f2d2f44": "\\rho (\\mathbf{r},t)=\\psi ^{\\ast }(\\mathbf{r},t)\\psi (\\mathbf{r},t)", "b62f67898064a320f25adc285a3884e5": "\\phi^M", "b62f853c2ff6e52496f35b881eded54f": "\\approx 0.82/\\sqrt{n}\\,", "b62f9e439088c520fec2f873bf99d779": "mG", "b62faab0eb98f03d8b559e0db28d760e": "22 = 11+11 = 5+17 = 3+19", "b6302350f794fa9befd42eeb64366e41": "\\phi = |\\mathbf{X}|/|\\mathbf{Y}|", "b630284b3456dfe296af12c179843354": "\\gamma_\\mathrm{SV} ", "b630b323bae7d5ee0ec0cacf9a88a273": " \\int x\\Phi(a+bx) \\, dx = \\tfrac{1}{2b^2}\\left((b^2x^2 - a^2 - 1)\\Phi(a+bx) + (bx-a)\\phi(a+bx)\\right) + C ", "b630d235b87d8cbad90764fe221f3c12": "\\ \\xi'", "b6314e10219744f982ae898b2f685a3c": " h A_{combined} {(T_{\\infty} - T_{0} )}+k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 \\, ", "b631851e34dc2cd400e63b3252d6618a": "\\phi(t) = t^\\lambda g(t)", "b631c15e91b6cfa8a4142f2ab2d965d9": "F(x)\\ge\\limsup_{n\\to\\infty} F_n(x_n)", "b631d80f20466b7da8866cd04ef8a4b9": " \\gamma_V ", "b632190fce8188f49b99ba1fd1332a0b": "S(n) = \\tfrac{2E}{N}", "b642776ceb7bcc3d9ebeab48fb8d020f": "\\psi =[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2, \\dots ] \\!\\, . ", "b64291c7ba1da657ec2cf6f6b38cb565": "\\omega =\\ 87\\ \\text{deg}\\,", "b642de9303588ac375ef295c545e0fbb": "Q_{in:k}", "b642e9c580b69e7ac27dd2e211bf889e": "f_B = 0.03E^{-0.8} \\;", "b642ecb28cf6920f5edcd6f580b065ac": "r_\\pi = \\int R(\\theta,\\delta_{\\pi}) \\, d\\pi(\\theta) \\, ", "b642efa543e4478b1401d0c447525b1d": "Z(\\cdot),", "b64311ce56643ddea9e2d5967e8e0734": "s = g(t)", "b6432531b257c1701b7f60be70f3b414": " C_i = f_i(M)^{e} ", "b6433f97229adc351b130a94cc92f38d": "Cu_{2}0", "b643a386e4669efdcfedf1f6f5aef259": "\\psi: U \\to X \\times_S P.", "b64423086bbdc0d1d06113d3745de4eb": "\\mathrm{M} \\left( A^{?} \\right)", "b644874ec93c6b34a0dcc17fc02abd37": "\\Delta H = - R * slope,", "b644dc4f5d956615ab1a51470684e917": "\\scriptstyle\\mathfrak{f}(L/K)", "b644e07a575e165ea9860c27c416d4ff": "s_{n}=\\sum_{k=1}^r a_k s_{n-k}", "b644e1be852e79cb7a022763c2645d07": "\\tau+1", "b6452f6dcd3906eb93584997794a7605": "\\ a/L", "b64580297758acc7b262382c6d5ee72a": "\\textstyle R(a_B\\mid[x]) = \\lambda_{BP}P(A\\mid[x]) + \\lambda_{BN}P(A^c\\mid[x]),", "b645a5eae231c6c3bfa505a7587994e1": "\n\\lfloor E \\rfloor (F) \\rightarrow^\\sigma F\n", "b645a747cce869984b9dbaa3f78f1388": "v^{\\alpha} \\!", "b645cf0fea128dc5b6a0b7491599bdaa": "z \\in U", "b645f5f4390239658cdda3c4ef77368d": "t_n =\n\\begin{cases} \n1 & \\text{if } m = 1 \\mod 4 \\\\\n0 & \\text{if } m = 3 \\mod 4\n\\end{cases}", "b6460a8e0c5c41c0d4af00303776e2a0": "P'(s) \\equiv \\frac{d}{ds} P(s) = - \\sum_p \\frac{\\log p}{p^s}", "b6460ec0d170c6bcad114994de0b8f35": "(2^3)^4", "b64647fd6da85f674b3f3cffc7195711": "f(\\alpha)=0", "b64670ebf2b954e8acab7543cf4ddcf1": "\\int_0^\\infty\\|T(t)x\\|^p\\,dt<\\infty.", "b64682e3783325aacefa2c8fa98131b8": " \\langle v | w \\rangle = \\langle v | \\sum_{i \\in \\mathbb{N}} | e_i \\rangle \\langle e_i | w \\rangle = \\langle v | \\sum_{i \\in \\mathbb{N}} | e_i \\rangle \\langle e_i | \\sum_{j \\in \\mathbb{N}} | e_j \\rangle \\langle e_j | w \\rangle = \\langle v | e_i \\rangle \\langle e_i | e_j \\rangle \\langle e_j | w \\rangle ", "b646a0caab874a1ed07d518c14a7f399": " \\lim_{k \\to \\infty}\\tfrac{\\chi_k(x)-\\mu_k}{\\sigma_k} \\xrightarrow{d}\\ N(0,1) \\,", "b646b7e49416328096f4accf008cdbfc": "f \\circ P_G = P_F", "b646d9d40fad993b93f494d64cc7b653": "\\epsilon_{x,p}<0", "b6472a3f67d525b24f10876787f6b76b": "2 \\pi \\left (1 - \\cos {\\theta \\over 2} \\right).", "b64793d84e854222a9a448c2b8629db0": "A''", "b647bf9862cd7401cb8b18b703400eb4": "\\left. y_{i+1}=\\left( \\sum_{j=1}^m \\frac{x_j}{\\| x_j - y_i \\|} \\right) \\right/ \\left( \\sum_{j=1}^m \\frac{1}{\\| x_j - y_i \\|} \\right).", "b6487269fc440ec0ed4e4852ca9801d2": "kL = n\\pi - \\tan^{-1} kx", "b648fdd6a5e4eea65339c16a4f99c3d9": "k-3", "b6490d18b58112f41ee45b5bb8b219e6": "\\psi(z+1)= -\\gamma +\\sum_{n=1}^\\infty \\frac{z}{n(n+z)} \\qquad z \\neq -1, -2, -3, \\ldots", "b64935de754a12bdb4697430f70c4464": "\\varepsilon=\\begin{pmatrix}0 & 1 \\\\0 & 0 \\end{pmatrix}\\quad\\text{and}\\quad a + b\\varepsilon = \\begin{pmatrix}a & b \\\\ 0 & a \\end{pmatrix}", "b6496a36ad044a1737e24ada592a3f0f": "m_{es}=E_{em}/c^2", "b6496ed903887d000dcaf4c0400a6936": "{R\\omega_0/c}", "b6497980d493702276b50780cb4431fa": "\\lfloor p/m \\rfloor / (p-1)", "b649b410ba183a925265d1428378897b": "c_4=b_2^2-24b_4", "b649ba36bc8146f86ca06da4a2b1735e": "\nD = \\sqrt{\\sum^n_{i=1}\\left(R_{Fi} - \\frac{i-1}{n-1}\\right)}\n", "b649f49cbbc0ad75cd358ecb0273f5f3": "H^d", "b64a0eaacf43445957bd7be1f0cb54d0": "\\mathrm{d} X_t = a(X_t) \\, \\mathrm{d} t + b(X_t) \\, \\mathrm{d} W_t,", "b64a6949eff02aba09d74620661abb02": "f(-10,1) = 0", "b64a6dcf23e38df1d0d7a1b8f7fb6551": "(\\delta_{init}, P')", "b64ab42a350067da01eb3391962f2e1d": "x_{i_1 \\cdots i_N}", "b64afa041d14a08f5892de29bce81e0d": "\\, G \\,", "b64affcb49a011663e89db6b20e3811b": "\\mathbf{v}\\ \\stackrel{\\mathrm{def}}{=}\\ \\mathbf{s}-\\mathbf{c}", "b64b167a372abdc9e4303d0bda6507dc": "\\log n+2\\gamma", "b64b7538e8eb0e0110333aaf5e43869d": "\\text{EoT}=720\\times (C-\\text{nint}(C))", "b64bcac64cfd3d1389fc5ae85eb88244": "{\\mathit{H}}_n(\\gamma x)=\\sum_{i=0}^{\\lfloor n/2 \\rfloor} \\gamma^{n-2i}(\\gamma^2-1)^i {n \\choose 2i} \\frac{(2i)!}{i!}{\\mathit{H}}_{n-2i}(x).", "b64c5f80b875220df1b9a36bbf7cf63f": " \\neg Q \\rightarrow \\neg P ", "b64d7cfb9fd9d66669cfb3a4c3dcfe21": "h_{B\\mathrm{,max}}={P_\\mathrm{atm} \\over \\rho g}", "b64dafb94036f29780e6e5988c430724": "\n\\frac{d\\left[ \\mathrm{A}\\right]}{-\\frac{j_{\\mathrm{A}}}{D_{\\mathrm{A}}} + \\frac{n_{\\mathrm{A}}FE_{m}}{RTL} \\left[ \\mathrm{A}\\right]} = dz\n", "b64dbf25233ace904116789cafe46af0": "\\left(\\left\\{a, b\\right\\}, \\left\\{\\left(a, 2\\right), \\left(b, 1\\right)\\right\\}\\right)", "b64dd22742c425f3f425548bb6783c91": "\\Pi_j n_j", "b64dd42af859ee5cd23185709f5d3f80": "\\begin{align}\n & T_n + S_n \\ \\xrightarrow{p}\\ \\alpha+\\beta, \\\\\n & T_n S_n \\ \\xrightarrow{p}\\ \\alpha \\beta, \\\\\n & T_n / S_n \\ \\xrightarrow{p}\\ \\alpha/\\beta, \\text{ provided that }\\beta\\neq0\n \\end{align}", "b64dfc230e8341e0cc1148ba690c8052": "O(n(d_f^2 +d_g^2))", "b64e2824e2fc56e37e418df00206bb9e": "r_{\\pm}= \\mu \\pm (\\mu^{2}-q^{2})^{1/2}", "b64e83482a0678487c1a9a5a45afa779": "(10x+y)^{10}-100x ^ {10} =y^{10} +100xy=\n\\begin{cases}\n0, & y=0\\\\\n100x+1, & y=1\n\\end{cases}\n", "b64ea1b421fa75b09004eeeac5926761": "~E^i/c = -F^{0 i} \\,", "b64f20fc185268beaa7eb53e3d313897": "\\scriptstyle A\\,\\subseteq\\,\\mathbb{R}", "b64f5526db17b257ec1034defaa83aba": "\\varepsilon = E_0", "b64f5b8b30b6722decc9babd0bd9c9f7": "f(x) =\\frac{1}{x^2+2x-3} =\\frac{1}{4}\\left(\\frac{-1}{x+3}+\\frac{1}{x-1}\\right)", "b64f97e9733979d716c30f69f719d092": "\n\\hat{\\Theta}_{m}=D\\mathcal{P}_{m}\\hat{\\rho}_{m},\n", "b64fc1a0b986f1db8152439ed2a80b6a": "\\vec{C} = {\\vec{W}\\times\\vec{P} \\over P^2} - \\vec{P}t", "b64fd69c7485da9a498ccf114183c8fd": "\\begin{array}{rcl}q&=&\\frac{1}{e}\\\\&=&4\\frac{1}{\\sqrt{10+2\\sqrt{5}}}\\\\&\\approx&1.05146\\end{array}", "b65082b6e25c581436688ef0855431e6": "p(0)=1", "b6509b9d910f9724e94457efc8f11fda": " \\operatorname{E} \\circ \\operatorname{tr} = \\operatorname{tr} \\circ \\operatorname{E} ", "b650a6be693852e0a2aa75e84f50d894": "\\Phi _E : C(X) \\rightarrow L(H)", "b650f81eea343d589dda3861d0d3dc67": " \\sum_{i=1}^n V_{i,j} = \\sum_{i=1}^n \\sum_{k=1}^m \\frac{B_{i,j,k}-C_{i,j,k}}{(1+r_k)^{t_k}} ", "b65113d1ec808520e75d35850031c3d9": " A_{ij} + B_{ij} = C_{ij} ", "b651b808e10cfe6c5ffa9f5a57305e9d": "1,1,t_1^2, t_1^{-2}", "b651e9ce2bc905d10cd0507e5fb08c7e": "P = \\frac{k_B}{m} \\rho T", "b65223f7bc09d870f23c05efeb3cf62c": "v_r = \\frac{1}{r} \\frac{\\partial \\psi}{\\partial \\theta},\\,", "b6528d2b48cf609f9df3903e419f2dc1": "\\frac{\\zeta(s)\\zeta(s-k)(1-2^{k-s})}{\\zeta(2s-k)(1-2^{k-2s})} = \\sum_{n\\ge 1}\\frac{\\sigma_k^{(o)*}(n)}{n^s}.", "b652903a8bff28e6b1fadda1d2606dd0": "\\mathbf{p}_B", "b652d5aa78b40e9d276fdb8421f6aa3c": "\\sum_{\\sigma : E \\to \\{0,1\\}} \\prod_{v \\in V} f_v(\\sigma|_{E(v)}),~~~~~~~~~~(1)", "b652f96e7b10d06086755c656aecba95": "i_n i_m = i_m i_n", "b653120ab1fe8c7e43d01b7a32b562ac": "\\scriptstyle \\partial S", "b6536ae61b09dbab60a7fa0d9ebad29e": " [q, p] = i \\,", "b6538872bcb1caaeeb04c11c7393300b": "P(x_1,\\ldots,x_n)=1", "b653b59198dfbabdd0f786e38b2229f6": "f^{\\mathrm{h}}", "b653c7ae728050acfd4345e0b390e260": "b_{1}-a_{1}", "b653e11806af554fa151494ff81052c6": " \\nu_\\pm = \\nu \\pm \\eta ", "b65427f0d47d47735b1d9a5c1561b584": " \\begin{align}\nx_0=x-2h,\\quad x_1=x-h,\\quad x_2=x,\\quad x_3=x+h,\\quad x_4=x+2h.\n\\end{align} ", "b6549c04d85aec4fcdf6c0e74914e528": " |\\psi'\\rang = |x\\rang.", "b6556d68800ca8de590ba7c5b4bb1a31": "A(w)", "b655a12bf59af99597d20a4072217748": "\\rho_{a / b}(\\rho_{b / c}(R)) = \\rho_{a / c}(R)\\,\\!", "b655c15b3935eb478a2b84966a68c8cb": "2 \\ln(\\Gamma(z)) \\approx \\ln(2 \\pi) - \\ln(z) + z \\left(2 \\ln(z) + \\ln \\left( z \\sinh \\frac{1}{z} + \\frac{1}{810z^6} \\right) - 2 \\right),", "b655f9998f2613650eefddaa7c7cc1e6": "\\left\\{\\exp\\left(\\lambda W_t-\\frac{1}{2}\\lambda^2t\\right),t\\geq 0\\right\\}", "b6560d3a4bcc2903d447ec24c42b7446": "\\frac{\\sigma^2}\\mu = \\frac{m - k + 1}{(k - 3)(k - 2)}", "b6564a23ff08387af74dd596b1a2452f": "f(x) = \\int_{-\\infty}^\\infty \\hat f(\\xi)\\ e^{2 \\pi i \\xi x}\\,d\\xi", "b6568aea9ab38e416418357c9611c5cd": "\\cos(\\varphi) = \\sum_{n=0}^\\infty \\frac{(-\\varphi ^2)^n}{(2n)!},", "b6568ef8d9731efa94454aa366750546": "\nV = \\frac{W}{Y}.\n", "b656a8820b1dd231ab2b7578f0e9998d": "r (\\cos(\\theta) + i \\sin(\\theta))", "b65707bc1abece673856df5f634eea57": " e = - N\\frac{\\mathrm d \\Phi}{\\mathrm dt}", "b6571cc8cf5bc281cc0deb82860eeeaf": "\\frac{ \\Delta \\phi}{\\omega_{\\rm orb}} \\approx \\frac{6 \\pi m}{r}", "b6574aac9bbfc4590470c212eeddb38f": "0 < v < \\pi/2", "b6585048e55a3d9165cae2853ef3ee52": " V_L = V_S \\frac {T} {2}\\, ", "b6588c9cd67a7c2391504fc1af9f64de": "\\Sigma^1_n \\subset \\Pi^1_{n+1}", "b65895675bf7f4e12e2fdb24adb8a239": "\\tilde{f}_i B \\subset B \\cup \\{0\\}\\text{ for all } i, ", "b658a74fc45ebcdf76b57106c905c020": " \\omega_m \\approx \\omega'_0 \\sqrt{1-\\frac {1} {2Q^4_L} } ", "b658c2c4e5077339c1d74d7ffbb8ac62": "0.80 + j1.40\\,", "b658fcd40309b4d3f06d5e7c8a28d7a2": "=\\,\\dot{m}\\,v_e + (p_e - p_o)\\,A_e", "b659131e701ba032b4be3951b3db1abb": "\\gcd(a_1,a_2,\\dots,a_n) =\\gcd(a_1,\\, \\gcd(a_2,\\, \\gcd(a_3,\\dots, \\gcd(a_{n-1}\\,,a_n)))\\dots),", "b65928a130f935417c22aae725bd2d01": "\\sum_{i= 1}^\\infty (1-p)^i=\\frac{1}{p}-1 ", "b6592fae1d948065cf58d0e8dcca8ce3": "z=\\prod_{p_i\\in P} p_i^{a_i}", "b659766f76f3e493cf3a93f159f34642": "1- 1/e", "b6598bc64482b686c6915029fd5839a7": "\\scriptstyle \\Delta \\lambda_B/\\lambda_B", "b659b176c63693fd8c1c2c2d1b369955": " |x|\\leq 1, |y|\\leq 1", "b65a01e13ee3105a3722dc612ddda1e1": "u = 0 \\text { on }\\partial \\Omega;", "b65a185307c6a26dba37873cfc84b472": "\\log_{10} K = \\frac{nE^o}{59.2\\text{ mV}} \\quad\\text{at }T = 298 \\text{ K}.", "b65a4171c4e10f5a8603fa50a2cb67c7": "dx^i \\wedge dx^j = - dx^j \\wedge dx^i. \\, ", "b65a625a63e7e29b339eefcb5e2c2bfe": "\\frac{\\partial F(x)}{\\partial x}=\\frac{\\partial y}{\\partial x} \\frac{\\partial F(x)}{\\partial y}", "b65a7b88be425ddfbeaabb815738d0a9": "(a_{2k-1})_{k=1}^\\infty, \\qquad a_k = k^2", "b65aaf90bdd1af36e2d61a2507dd4b06": "\\mbox{log}_{10}\\gamma_\\pm = -Az_i^2 \\frac{\\sqrt I}{1 +Ba_0\\sqrt I}", "b65b0d763cb8cec54727e4b446ec40c1": "T_{64}", "b65b23bcb178465b6866e890306707b1": "z_4", "b65b659984ffcaec52f97371b9bbac5e": " \\bar{P} = I_{rms} V_{rms} = I_{rms}^2 R = \\frac{V_{rms}^2}{R} ", "b65bf014160ecbedf4846b0bfc7446f4": "S_n.", "b65c1eb9941ed29912f721e63ac83266": "\\delta = 3\\,", "b65c1fcf6eabd6bafa89f5ec1ccf8e64": " J(\\omega) = \\eta \\omega ", "b65c9354e2318142e9c3bc56edb56327": "\nf_X(x)= \\begin{cases}\n\\frac{1}{2}x^2 & 0\\le x \\le 1\\\\\n\\frac{1}{2}\\left(-2x^2 + 6x - 3 \\right)& 1\\le x \\le 2\\\\\n\\frac{1}{2}\\left(x^2 - 6x +9 \\right) & 2\\le x \\le 3\n\\end{cases}\n", "b65caf3f79c9f0def5813fb51be44dc1": "f(z_0,\\dots,z_n)", "b65d1d0df375a7e3e536ec2a4cfccf87": "bestSubspace", "b65d7e59f858547c1ae3b3b0e24c9a2a": "\\alpha_t(s,a) = 1", "b65de8c46904da74fb7df853a50d4e60": "P^{T}QP = \\begin{bmatrix}\n\\begin{matrix}R_1 & & \\\\ & \\ddots & \\\\ & & R_k\\end{matrix} & 0 \\\\\n0 & \\begin{matrix}\\pm 1 & & \\\\ & \\ddots & \\\\ & & \\pm 1\\end{matrix} \\\\\n\\end{bmatrix},", "b65e14ccf9950f20fc42160724bba72c": "a \\in N_O", "b65e2f30edb62ae80fbc62810e25b240": " \\int \\rho \\, Q \\, dx = \\frac {\\hbar^2}{8m} \\mathcal{I}", "b65e3b75d7c17bdafe6f1585e6dba3f7": "H_A=\\left(A, \\lbrace e_i \\cap A |\ne_i \\cap A \\neq \\varnothing \\rbrace \\right).", "b65e45ba095a47b580dcf3eb70103bd0": "A_{[\\alpha}B^{\\beta}{}_{\\gamma]} = \\dfrac{1}{2!} \\left(A_{\\alpha}B^{\\beta}{}_{\\gamma} - A_{\\gamma}B^{\\beta}{}_{\\alpha} \\right)", "b65e640c029f3e18db0d46c47eb0058b": "\\mathit{Var}(f; N) = E(f^2; N) - (E(f; N))^2", "b65e760c0ea282c246c5f715e7fd359f": "J_i = \n\\begin{bmatrix}\n\\lambda_i & 1 & \\; & \\; \\\\\n\\; & \\lambda_i & \\ddots & \\; \\\\\n\\; & \\; & \\ddots & 1 \\\\\n\\; & \\; & \\; & \\lambda_i \n\\end{bmatrix}.", "b65e9787783be9fae0903239eb6d00ca": "\\frac{dy}{dx}+p(x)y=q(x)", "b65efd6b8116022fcb2cecb9ca9b4ad8": "\n\\int x^m\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{x^{m+1} \\left(b+2c\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{b (m+1)}\\,-\\,\n \\frac{2c (m+n(2 p+1)+1)}{b (m+1)} \\int x^{m+n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx\n", "b65f4cb3d484c73c17f35aa97bd39fd7": "\\alpha (\\text{dB}/(\\text{MHz}\\cdot\\text{cm}))", "b65f63fc2e5889921c0a04a7083d7c04": "f\\left( q\\right) =F\\left( A\\left( q\\right) \\right) +R\\left( q\\right),", "b65f9585cc50e45718bf591c9f52bfad": "x \\approx 1/\\varepsilon\\,", "b65fc35882cb985db36e71a7cb842f03": "10^{33} - 10^{34}", "b6602d0d147f3f730cba696b1e927890": "S_{N}(f) = \\frac{N_{0}}{2} \\text{ for } |f|w", "b6604e27b1de01f95e5eaf85a659ca20": " d = a = \\frac{K M}{r^2} ", "b660610f78d66a8c12171cd9fe641d25": "r\\leq n", "b6609a28b7c6f29a328e55e41c7d8b99": "W_{Ce} = \\frac{C_eV_e^2}{2} = \\frac{Q_e^2}{2C_e} = 2\\pi \\alpha W_0, \\ ", "b660bc77fc0cb14625b4db8e9700dea3": "S_c\\,", "b66110554ae79882e14faac4f78cc53c": "\\lambda _1 \\geq \\lambda _2 \\geq ... \\geq \\lambda _n", "b661377449259af7129e62ad6f0d892a": "\\frac{dt}{d\\tau} = \\frac{1}{ \\sqrt{1 - \\left( \\frac{2GM}{rc^2} \\right)}}. ", "b6613a68b5cc331709c4826ec3e22249": "E_p =\\frac{\\frac{Q_2-Q_1}{(Q_1+Q_2)/2}}{\\frac{P_2-P_1}{(P_1+P_2)/2}}.", "b662d8ecda1daeb9f728cd938d6caf76": "\\scriptstyle\\sqrt x/\\ln x", "b6633e4b0da04e49ee59e70173a66d6c": "L \\rightarrow R", "b6635e70379babe7a49fd53135ddc3a3": "\\theta(t) = \\theta_0\\cos\\left(\\sqrt{g\\over \\ell}t\\right) \\quad\\quad\\quad\\quad |\\theta_0| \\ll 1", "b664b2871ea148309e1b79c49bfa1985": "X=\\{ x=(\\ldots,x_{-1},x_0,x_1,\\ldots) : \nx_k \\in \\{1,\\ldots,N\\} \\; \\forall k \\in \\mathbb{Z} \\}.", "b6652ce6eca61dea5213c527f6442f3f": "R[X_{0},X_{1},\\ldots,X_{n-1}]\\,", "b665c23089b00afd961f7e8e870941bf": "M_A(B)=M_A(\\beta)", "b665cc1a1886ebc2f71b68ff55ec8335": "\\ A +\\ B \\rightleftharpoons \\ AB", "b6663903b8b76cab879396575c264b09": "\\frac{n!}{k!(n-k)!}", "b6664f5a659d01dfc55c1efafc70cfed": "v=yq_2", "b6667ad5364f3f2edb91ae4ca2410f84": " E_2 = E_1 + Z", "b666acc49d61fdc481ab5ecd749b92d1": "M \\to L_C M", "b666ade6c484cd866cd38e1b18067428": "\\operatorname{sign}(\\det(UV^*))", "b666e8903395d3e539c9108e21e6b03f": "d(y,z)", "b666fe5f7a0b83c3c2f9cdcc2a3a9634": "\\scriptstyle\\varphi\\,", "b66740a9da8aac794e611e96d56a9527": "E \\{ (\\hat{x}_{\\mathrm{MMSE}}-x) g(y) \\} = 0", "b6677cca304c2c1d4df6971d4a36505b": "\n\\begin{matrix}\nH(A : B | \\Lambda)&=& H(A|\\Lambda)+ H(B | \\Lambda)- H(A, B | \\Lambda)\\\\\n&=&H(A, \\Lambda)+H(B, \\Lambda)-H(\\Lambda)-H(A, B, \\Lambda)\n\\end{matrix}\n", "b667d414585c34426165c2944a20663a": " A\\setminus B = \\bigsqcup_{i = 1}^n K_i", "b667e9d71f962ff35357af0a36832842": "\\exp\\left[-\\mu_ac(t-t')\\right]", "b66806fc09f086c13ff5a81d97ef6c65": " h' ", "b668564cd6e7084f3df4b22c3d9f485d": "\nB =\n\\begin{pmatrix}\n1&0\\\\\n0&-1\n\\end{pmatrix}\n", "b668e5099c41e69ab4e1c67d78ee06aa": "Q^{n+1}(c) = Q^{n}(c)^{2} + c", "b6690288c49f0e82126d8f77fc48541e": "\\lambda_1+\\lambda_2=2 \\alpha = A,", "b66955e7b8b0aae2ad2a7fe39b3fe54c": "a=\\sum_j u_j a^{(k)}_j", "b6695eed71d38bca8b6f59578f0e11fa": "P_{[i,\\epsilon]} = \\frac{m_{[i,\\epsilon]}}{M_\\epsilon} = ", "b6698255344e7ccd631bbbc9e05d2d0d": "\\mathbf{C} = \\mathbf{A} \\mathbf{B} \\qquad \\mathbf{A},\\mathbf{B},\\mathbf{C} \\in R^{2^n \\times 2^n}", "b66992f272799cb9ad851757b5e66a0d": "T,~T_e", "b66a1fb8c1a76a44af26eeee6ffb7506": "\\mathrm{Factor} = \\frac{\\mathrm{Days}(\\mathrm{Date1}, \\mathrm{Date2})}{365}", "b66a36b8c8bb857d358999683afc444b": "\n(5.1)\\quad\n||D^k f|| \\leq C_{k,l} ||D^l h||, \n", "b66a521bc144929e9fcbd777d25a5685": "2/\\sqrt{3}", "b66b0de1de4915381fd5e0e38990abdb": "k_{int}=k_s-\\frac{(R_{12} (k_s+1)-R_{12} (k_s-1))}{2(R_{12} (k_s+1)-2R_{12} (k_s )+R_{12} (k_s-1))} ", "b66b0e21404e0f99b544713b64fe535d": "\\mathcal O(\\Delta t^3)", "b66b28f00e374b2f3b6f8319643866da": "A_{1,m}=\\left\\{x\\in A_0\\left| d(f_m(x),f(x)) \\le \\frac12 \\ \\forall n\\geq m\\right.\\right\\}", "b66b30a8157e5dc4999171b231e59204": "G(x, H) = \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau_{D}} \\chi_{H} (X_{s}) \\, \\mathrm{d} s \\right],", "b66b682bd67a09bc0cbb0c58e1cf191f": "\\scriptstyle \\lambda", "b66b8a11464a4aacdf5b0a8cf04b2fdc": "y=\\sqrt{1-r}", "b66b99fa0df635cf2df3286f51433312": "\\vec{f}_1 = \\partial_r - \\frac{\\sqrt{2m/r}}{1-2m/r} \\, \\partial_t ", "b66be95d560ade889c052960d5dd8993": "L = \\{p_1, \\ldots , p_n\\}", "b66c6cb035badd4a5e09f884163a08f5": "\\prod_{j0\\\\\n0 & x=0\\\\\nk_2 e^{1/x^2} &x<0\n\\end{cases},\n", "b67364fa49d54e29e13e1a1455646a5c": "gcd(a, n) = 1", "b6737df269d6289b1be82f0ed7962754": "f(n,m) = n^2 + m^3 + g(n,m).\\,", "b6748ffa10d280bff701f37fd3477f0f": "\\overline{n^2}_\\mu ", "b674c2692956b940c2ed473f36066d23": "\\theta_0=\\frac{\\pi}{2}", "b674ca3b68ad0d282b4f35cc013482d8": "\\operatorname{gd} \\,u = i^{-1}\\ln\\tan\\left(\\tfrac14\\pi+\\tfrac12ui\\right)", "b675067d29587c8c58d43c07fdede7b1": " x_{k-1} ", "b6750dcc04d293eee4ac89a2b8e9d0eb": " \\scriptstyle{\\Phi(\\cdot)} \\ ", "b675312965a066842c3a63e14cb021d3": "(p+\\varepsilon) u^k \\left \\{ \\frac{\\partial u_i}{\\partial x^k}-\\frac{1}{2} u^l \\frac{\\partial g_{kl}}{\\partial x^i} \\right \\rbrace =-\\frac{\\partial p}{\\partial x^i}-u_i u^k \\frac{\\partial p}{\\partial x^k},", "b6753ca65c71b9d338fad9974391d6c1": "\\operatorname{Var}(X+Y)=\\operatorname{Var}(X) + \\operatorname{Var}(Y)", "b675c699383bacf31087d83e83aec89b": "\\mathrm{NA_i} = n \\sin \\theta = n \\sin \\left[ \\arctan \\left( \\frac{D}{2f} \\right) \\right] \\approx n \\frac {D}{2f}", "b675e6e2f88bea0b1fc3549a2b328b4f": "0\\leq\\sigma_a-\\sigma_c\\leq L:=\\limsup_{n\\to\\infty}\\frac{\\log n}{\\lambda_n}.", "b6761918a56406e74c680ddbeafa681a": "\\{p,q,r,s\\}", "b676265f2c8667386598687e8d249ebe": " \\vec{m} = m \\hat{e}_z ", "b67637770070b35459053543643eb6ee": "\\pm\\sqrt{\\Big( 3 - 2\\sqrt{6/5} \\Big)/7}", "b6766916061cf181cde659b8513ec651": " \\mathbf{I}\n\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\times \n\\end{array}\\!\\!\n\\mathbf{A}=(\\mathbf{A}\n\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\times \n\\end{array}\\!\\!\n\\mathbf{I})\\mathbf{I}-\\mathbf{A}^\\mathrm{T}", "b676a3053b7e7ba00dd7338b4d686103": "\\eta_X", "b676a4890add4a8b24aa7d8ebed5d323": "\\forall p", "b67764c25190a17a5ca928712d086e7e": "C_H \\; = \\; 0.8 \\; + \\; (\\; 1.1 \\; \\log_{10} f \\; - \\; 0.7 \\; ) \\; h_M \\; - \\; 1.56 \\; \\log_{10} f ", "b677819a0cfef0707d486ca49139d253": "\\!\\, = x.", "b6778912e44f21229ff6b20302b2f6ea": "\\forall S\\in Z^m, \\forall i\\in\\{1,...,m\\}, \\sup_{z\\in Z}|V(f_S,z)-V(f_{S^i},z)|\\leq\\beta", "b677a3c472026bb26371d850252cfa06": "L= \\frac{\\lambda}{i} = N\\frac{\\Phi}{i}", "b677b860ba2645ca70741d1b5d6e443f": "\n\\rho(A_1 A_2 \\ldots A_n) \\leq \\rho(A_1) \\rho(A_2)\\ldots \\rho(A_n).\n", "b677c14df4a8faeffeb721dd87ce1376": "\n \\frac{\\partial f}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = Df(\\mathbf{v})[\\mathbf{u}] \n = \\left[\\frac{d }{d \\alpha}~f(\\mathbf{v} + \\alpha~\\mathbf{u})\\right]_{\\alpha = 0}\n", "b677cae71bd1938e8369e545d9328f6d": "F_{20}", "b677d10abaad74aea2aed32face46461": "\\frac{}{}I_0", "b677de8ef6ce15ffb2866723339e7d41": " C_a \\gg C_0 , ", "b677f56a1beeb7ae1bfbd1e0257c455a": "\\tilde f_1", "b67831192703e79cff5c4ad2c1a34992": "\n \\begin{Bmatrix}\n i & j & \\ell\\\\\n k & m & n\n \\end{Bmatrix}\n= (\\Phi_{i,j}^{k,m})_{\\ell,n}", "b6783fe6b996528d1b36c863940222c0": " \\Rightarrow(y_1 + \\frac{q^2}{2gy_1^2} = 6.20)", "b6786375db94fcb50161093eaa48282c": "\\chi(G_K,M)=p^{-[K:\\mathbf{Q}_p]v_p(m)}", "b6787f7f9a7cad0ca9fb72d61bec672d": "A =\\left ( \\frac{1329\\times10^{-H/5}}{D} \\right ) ^2", "b67901b21404de63647cfae5066c43d6": "(A\\to\\neg A)\\to\\neg A", "b679a18fb2bd13045febaab5972b4f51": "\\beta(s,i) = \\mathrm{rem}\\left(K\\left(s\\right),\\left(i+1\\right)\\cdot L\\left(s\\right)+1\\right)", "b679abe0ce49cbae1d660d78ad62e489": " \\tilde{U} ", "b67a433599d3052e425f90d23fc344ba": "f(x, y) \\rightarrow [r, \\theta]", "b67a44c3710f8f974b1ab52515265cd0": "\\Omega(v) = D(D v) = D^2v\\, ", "b67a653040de73d1fe260e345033ca19": "z = (T_0/L)((P/P_0)^{-LR/g} - 1)", "b67a80bdea9e07ed4f91a7249c66be05": " x = \\frac{1}{ u}( u \\wedge v)", "b67abe28ebff3b23affb192fdc031100": "P\\left( \\frac{d-c}{a-b} ; a\\frac{d-c}{a-b}+c \\right)", "b67ac0d388323cec5c29589ea2c3a792": "h^a,h^b,h^\\alpha", "b67aea742321119a7d616ca58f066e67": "V = \\frac{n}{4}hs^2 \\cot\\frac{\\pi}{n}.", "b67b7b6cea2a72931e6891b5f8a19b95": " B = {1 \\over 4} \\left[ \\begin{array}{rrrr} {1 \\over a} & {1 \\over b} & {1 \\over b} & {1 \\over a} \\\\[6pt] {1 \\over b} & -{1 \\over c} & {1 \\over c} & -{1 \\over b} \\\\[6pt] {1 \\over b} & {1 \\over c} & -{1 \\over c} & -{1 \\over b} \\\\[6pt] {1 \\over a} & -{1 \\over b} & -{1 \\over b} & {1 \\over a} \\end{array} \\right],", "b67bc639d5ff9ce7a4296b90c5eafcc8": "L = L_0 \\; + \\; \\gamma g\\log d \\; - 10 (\\log {F_A} - 2 \\log (\\frac{H_{ET}}{30}))", "b67bd43cf6886d3f3ccfa17ef2056470": "\nA_{xx} \\xi^{2} + 2A_{xy} \\xi\\eta + A_{yy} \\eta^{2} + \\frac{\\Delta}{D} = 0\n", "b67bfd87696daa8f3fc1ab36f07a66ed": "x = \\pm\\sqrt{r}", "b67c02f56b464eea7c5d5af02ae49eab": "\\varepsilon=r+\\gamma P\\hat{v}-\\hat{v}=r+\\gamma P\\Phi\\theta-\\Phi\\theta", "b67c0503b7f67f0b82a87b614da8b6f3": "\n \\frac{\\partial^4 w}{\\partial x_1^4} + 2 \\frac{\\partial^4 w}{\\partial x_1^2 \\partial x_2^2} + \\frac{\\partial^4 w}{\\partial x_2^4} = 0 \\quad \\text{where} \\quad w := w^0\\,.\n ", "b67c2f234467703a7f051dc2f92bd186": "\nV(x,T) = D(x)\\,\\!.\n", "b67c6c58067aee24446144fae966125c": "\\,\\alpha_1 ", "b67c7583857e403c3385d4590d7798b2": "\\scriptstyle f'(x^*) = 0;", "b67c991443973c3987ae359e6025193a": "T_c=\\left(\\frac{N}{Vf\\zeta(3/2)}\\right)^{2/3}\\frac{h^2}{2\\pi m k}", "b67d6ce22b98a2a997793aaf73dce9a5": "L_n(x)", "b67d94d03b5b02adb27d65fc9ddc71f0": "\\frac{\\Gamma(\\lfloor k+1\\rfloor, \\lambda)}{\\lfloor k\\rfloor !}", "b67dbbdf4ca7749f440f29dfefc379bd": "{n\\choose 2}+1", "b67dbff8a71a7974902cd7e1f017af58": "\\frac{\\sin x}{\\sin y}=\\tan \\phi.", "b67dd82c4491032baa6a6039d44a843f": "VCA(64x^3+384x^2-1024x+512,(1,\\tfrac{3}{2})) \\cup VCA(64x^3+576x^2-64x-64,(\\tfrac{3}{2},2))", "b67def68f7b971494744f6e78005008d": "\\sum_{k=0}^\\infty \\frac{z^{2k}}{(2k)!}=\\cosh z\\,\\!", "b67dfa2cd9bc45caeb135634d8e30e17": "\\overline{a}_n \\overline{b}_n = \\overline{(ab)}_n.", "b67e21eb64907bf8ac8a9597c4a08c09": "\\mathbf{w}\\cdot\\mathbf{x} - b=0,\\,", "b67e2498ca535b375f644f29d1c6052c": "A(r)", "b67e3d0a2f2ef20da04c415d94a5e438": " \\neg q, \\neg p, \\mathit{false} ", "b67e67743863e4cec701eff334d88280": "X = a \\to b", "b67e8dc9b01b4c7ebb2a860a921894bf": "\\langle t_m \\rangle = \\int_\\tau^\\infty tP_m(t)dt = \\langle t \\rangle+\\tau", "b67e9954a9673b5b06b08d0f50a5cc04": " \\operatorname{M}_E(S) = E S E + (I - E) S (I - E). \\, ", "b67ed2bad70fbd0e7be1eb911fd13cff": "w(n)=a_0 - a_1 \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right) + a_2 \\cos \\left ( \\frac{4 \\pi n}{N-1} \\right)", "b67fabc466a922c92f9e768291ab730b": "\\scriptstyle\\mathcal{A}", "b67fabcf73384011c463e62bd48eabbb": "\\frac{\\partial \\mathbf{f(g)}}{\\partial \\mathbf{g}} \\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}", "b680702aa533b29153b410a5925ee9bc": "T_{22}-T_{33}", "b680915d3365ac4e9039800023c9f31c": "a_{p}", "b680a292551a2a8591cef4ca5cf37a90": "\\phi = v", "b680ca5da7822594d38cfde54d763a25": "\\scriptstyle -\\colon\\, F \\;\\to\\; F", "b681237b09e92cb165a8f787b2a9387f": "\\vartheta(z; \\tau) = \\sum_{n=-\\infty}^\\infty \\exp (\\pi i n^2 \\tau + 2 \\pi i n z).", "b6816de3f96512ea8efe48e8b38b1d5f": "E \\cap F = \\emptyset", "b681b2a31b2da520722163ca63e99267": "S\\in G, H(S(q,p))=H(q,p)", "b681b8d487bfc6b7c46b98b350f20a23": "1/{x^2}", "b681d19f9cc8d8f0c05a23e2b3d22dbd": "\\int_L \\frac{\\delta Q}{T}", "b6827453b96c950065634ca2ab86aed2": "\n \\dot{J} = J~\\text{tr}(\\boldsymbol{d}) = J~\\text{tr}(\\boldsymbol{l})\n", "b68318ccb719e44c7ec1d68cd22ed787": "\n\\overline{\\mathbf{GP}} \\cdot \\overline{\\mathbf{GQ}} = \n\\overline{\\mathbf{GT}_{1}} \\cdot \\overline{\\mathbf{GT}_{1}} =\n\\overline{\\mathbf{GT}_{2}} \\cdot \\overline{\\mathbf{GT}_{2}}\n", "b6831ce6df1d484222b5e375dff31bd1": "=3 * \\frac{\\text{Voltage}_{L-N}^2}{\\text{ohms reactance}\\text{ * 1000}}", "b683f558d32f8c7930ad91cdce350394": "\\rho_c = K_n\\rho_{avg}", "b68405770f86eafce0c68df7fad0e7ab": "(A\\to B)\\to(\\neg B\\to\\neg A)", "b6840667baeed0cede98f9fff1fa1e60": "z_j=0", "b6840c8a722605fa92c1a0d3bb47bfdb": " \\left(\\frac{N_1}{N_2}\\right)^2", "b68484df16e4563ada91d58be278f732": "\n\\left( \\frac{\\partial P_{m}}{\\partial p_{n}}\\right)_{\\mathbf{q}, \\mathbf{p}} = \\left( \\frac{\\partial q_{n}}{\\partial Q_{m}}\\right)_{\\mathbf{Q}, \\mathbf{P}}\n", "b684a1c21d0748816800bef19ea9ed23": " \\ \\psi (0) = \\psi (\\pi) = 0 \\quad (6) ", "b684a3016f4f6ff157d42c0b61eae2f1": "\\scriptstyle e(t)", "b684c7ad1ccc6e601765336a93264132": "b\\in\\{0,1\\}", "b6850e1f451890933375fdcd04bb82f4": " G_\\mathbb{Q} ", "b6851faf3de16cd9f2aa8d07bbc04b7f": " (1+X)^\\alpha = \\sum_{k=0}^\\infty {\\alpha \\choose k} X^k.", "b685ae28e9168659333a5006495be1fa": "b_0={\\partial f\\over\\partial x},\\,b_1={\\partial f\\over\\partial y},", "b685b6754f9861b835672be85d3d8599": " K = \\mathbb{Q}(\\sqrt{-15}) ", "b685ee5572d63318f8fac40d09b4fda0": "11^{14}\\ \\equiv\\ 54\\ \\not\\equiv\\ 1 \\pmod {71}", "b686041ba1d128ca0614ff54d8fa5af7": "K^n \\to K", "b686145bc4c810532476d279e70713ab": "X_1, X_3", "b6861b031adf8082e37dc181bd68d6ac": "f\\,''(x)=\\frac{2}{x^3}", "b68655fc04819eabf029a9f006604e77": "\\Lambda(s,\\chi) = \\left(\\frac{\\pi}{k}\\right)^{-(s+a)/2}\n\\Gamma\\left(\\frac{s+a}{2}\\right) L(s,\\chi),", "b686a4801cbd3a3ef8a58e1905928a5c": " \\frac {\\partial}{\\partial n} \\left( \\frac {e^{iks}}{s} \\right)=\\frac{e^{iks}}{s} \\left[ik - \\frac{1}{s} \\right] \\cos {(n,s)} ", "b686da6225c24dedbe9f33cf9a2c217b": " \nf_2(x; \\nu) =\n\\frac{(\\nu/2)^{\\nu/2}}{\\Gamma(\\nu/2)} x^{-\\nu/2-1} e^{-\\nu/(2 x)} .\n", "b68717e3be7604f8de30c23b89a3a49c": " \\|\\omega\\| := \\sup\\{|\\langle \\omega,\\xi\\rangle|\\colon\\xi \\mbox{ is a unit, simple, }m\\mbox{-vector}\\}.", "b6874ea82b91ddf82b268f5f3bbd8744": "x^{\\ell} \\equiv 1 \\pmod{n}", "b687583a1866359f604a4a435d3843c6": " g = v + \\sum_{i0} \\left (e^{\\frac{\\alpha}{2}}-e^{-\\frac{\\alpha}{2}} \\right )}", "b691da3755f175d6879d3cd876305109": "\\mathrm{-Im}(\\gamma)x \\,", "b691dd84f69c70b9780b1c5d0a9cbc30": "\\lambda^2", "b6920e4d7523855085a3d4c7a975c139": "\\lim_{N\\rightarrow\\infty} \\overline{Q}_N = \\langle Q\\rangle.", "b692cce0e6425c2647164a709b00e5d9": "\\mathfrak{P}^{35}", "b6934d2278c70718177d0a49585fed80": "\\nabla : \\Omega^0(M,E) \\to \\Omega^1(M,E).", "b6936283cf5a6793a38534242fa265a8": "X \\subseteq U(d)", "b69376cc7a977cfad74c0eb1eabd41c0": " \\alpha_{\\rm 1} = 90\\deg ", "b6938a6be190a230c12bb3631885e842": " \nk=1,2,....,n_c\n", "b693c49f6bb90839e4bc36c3a45748c0": "Y_i=\\phi_i^T X", "b693f475b35beebe3dbb3b93205afa8c": "\\dot x \\sin\\alpha=\\dot y\\cos \\alpha.\\,", "b69466b536f8ce43b6356ec1332e05a4": "pd", "b6946cac8739e46329f033a1a5f289fb": "O\\left(\\sqrt{\\log y}\\right)", "b694a96c7ffb9edd1e45d331fbde48d5": "\\lambda, n_1, n_2,\\,\\!", "b694c539ea5655cde31cfac456a3c99e": " \\mathcal{I}_R, \\mathcal{I}_\\Theta", "b694e5e1ac76b6c55d764147bc1b7e70": "m\\{x:\\, |Tf(x)| \\ge \\lambda\\} \\le \\left (2\\mu\\|T\\|^2 +2\\mu^{-1} + 2A \\right )\\lambda^{-1}\\|f\\|_1.", "b694f835fd8af4252c2a78c80cd03822": "\\frac{1}{\\frac{1}{N}\\sum_{i=1}^N \\frac{\\hat{c} - \\hat{a}}{\\hat{c} - Y_i}} = \\frac{\\hat{\\beta}- 1}{\\hat{\\alpha}+\\hat{\\beta} - 1} = \\hat{H}_{1-X}", "b6951164891b0d700390e088a85086b1": "a_1x_1 + a_2x_2 + \\cdots + a_nx_n > b.\\ ", "b69517746e5f7304d30baf9c8a3cebd3": " \\gamma = \\frac{1}{\\sqrt{1 - v^2/c^2}}.", "b695258b63afe86614bd2e3fa157061d": "2S+1", "b6953509adc36893ee3e5abc3de2d96c": "(\\kappa)\\,", "b695592b78ede38ba112e182bf323803": "\\omega_J = \\sqrt{\\frac{1}{L_JC_J}}. \\ ", "b6958be5bc33b38c5cc2e3295914c304": "\\tau_S = \\lbrace S \\cap U \\mid U \\in \\tau \\rbrace.", "b695bca0d4c8463e002d36182f7bddd1": " {n \\choose k} = {n-1 \\choose k-1} + {n-1 \\choose k}", "b695f45b2bd557050ac7a4ff3df1ea90": "\\sin a = \\sin\\phi_o \\sin\\delta + \\cos\\phi_o \\cos\\delta \\cos h", "b6962f3ba2b6db751b1ce894422c7dd4": "P(0, T)", "b69633479f4d8e7bbf0ca24e6b69affc": "n_F^\\prime(\\xi)=-\\frac{\\beta}{4}\\mathrm{sech}^2\\frac{\\beta \\xi}{2}", "b69633dea882a89d84d9f240b69214db": "Q=[I_V]_{\\beta'}^\\beta", "b6967ec70f20b3f5bded7b56b61ece04": " |z| > |e^{-\\alpha}| \\,", "b6976baebb1c995920678da24af70ecc": "\n\\begin{align}\n \\mathbf{u}\\times\\mathbf{v} & = \\varepsilon_{ijk}~\\hat{u}_j~\\hat{v}_k~\\mathbf{e}_i\n = \\varepsilon_{ijk}~\\frac{\\partial x_j}{\\partial q^m}~\\frac{\\partial x_k}{\\partial q^n}~\\frac{\\partial x_i}{\\partial q^s}~ u^m~v^n~\\mathbf{b}^s \\\\[8pt]\n & = [(\\mathbf{b}_m\\times\\mathbf{b}_n)\\cdot\\mathbf{b}_s]~u^m~v^n~\\mathbf{b}^s\n = \\mathcal{E}_{smn}~u^m~v^n~\\mathbf{b}^s\n\\end{align}\n", "b697c8ee21cc8aeabc53f54726d01304": "\\sum n_j\\Delta_j", "b697ed26e426f950acac968d464c6ce6": "v_\\mathrm F = \\frac {fv} {f + Nc}\\,.", "b69808f6aaa7bd103419be28dd866913": "T_n=\\sum^{n+1}_{i=1}X_i,", "b6983df235f9655e49c0bf87308aa7e5": "\\rho \\ell", "b698452a65010ffc9dcc3ffc8c45bcaa": "V(K)", "b6987318502ad058a4cf78ca9d5dcc97": "x\\cdot y=x_0 y_0\\oplus x_1 y_1\\oplus\\cdots\\oplus x_{n-1} y_{n-1} ", "b6988bc27bfc910183102a2199093536": "\\log 5", "b698afc77fc2cd0035ef31be7562587f": "k_{ij}=0", "b69909c779ece687f91eaaf1065f7a8a": "\\mathrm{PI} = \\frac{mass}{height^3}", "b6991133ff9e47d74affe1374eef6ed2": "\\vartriangle^2_n", "b699172573481512250ebeea57430023": "\\hat{p}_k = \\frac{n_k}{n}", "b699ae15b453ad6213635a763e704519": "r_1, r_2 \\in R", "b699e1fcf455797aa6f3b8f6ea125b40": "n\\in\\Z", "b699f74124afc1785c9ed707ed1a98d1": " C_1 , C_2^\\perp", "b69a1c937fbeee4a59cc97e9b222f29b": "v = \\frac{d [P]}{d t} = \\frac{V_\\max {[S]}}{K_d + [S]}", "b69af4ab26b2973a3a6cc98f927fd7af": "0 \\equiv (z + tf'(r))p^k \\pmod{p^{k+m}}", "b69afcfc81ea462d910bb0d05abb3a0e": "\\mathfrak{P}^{105}", "b69b1b7f27f20e3b8475b4b7fa30a4e9": " ~ \\omega + \\langle T^2 - 3 \\rangle ", "b69b31fe811d9e52fc629fcf42fbabc8": "2^7\\ln(2) =\\sum_{k=1}^{7}\\frac{2^{7-k}}{k}+\\sum_{k=8}^{\\infty}\\frac{1}{k2^{k-7}}\\, .", "b69b8f85bbd62fbd210ba29353d6ac58": "O(\\sqrt{t})", "b69b983e085d60965c7338aa7f160480": " \\mathbf{d^2F} = -\\frac{k I I'} {r^3} \\left[ 2 \\mathbf{r} (\\mathbf{dsds'}) - 3\\mathbf{r} (\\mathbf{r ds}) (\\mathbf{r ds'}) \\right] ", "b69c76f1ffcec715085ed98e8f0bf3c0": "P(t) = \\nu e^{-\\nu t},", "b69cc55182ee927b1cfedd7998987790": " s \\in R_q ", "b69ce9678655b4a586d1132116c53ec9": " P(x,s;y,t)", "b69cf130348b5d53a7aaeb7f79eea645": "(t,x)", "b69d195fc5694495e83532110f163518": "j(q) = \\frac{1}{q} + 744 + 196884 q + \\cdots.", "b69d3d9da215903181229b10b164eb71": "\\eta(4) = {{7\\pi^4} \\over 720} \\approx 0.94703283", "b69d4c36dabcc9ab2a75b79068a41c13": "\n T^{*}_{11} = 2C_1 + 2C_2\\beta ~.\n ", "b69d5e303dde8d36c7f13927b19fd20a": "X_1, X_2,\\dots, X_n\\,", "b69d5ff7e3568bc82ad28545877b0813": "(k+t)=(i+r)= N, \\, ", "b69d73dc693024a0511ae10a02532f53": "B\\subset U", "b69dca48a439cee8ee89a79cddf7b6a3": "\\omega_{X} = (\\vec{b},u,\\vec{a})\\,\\!", "b69de7fe175517870f2a9468b30047fb": "c_p = \\left(\\frac{\\partial C}{\\partial m}\\right)_p,", "b69e1ea7d4a97762d41738adddddc922": "\\Psi_0(q) = -1 + \\sum_{n \\ge 0} { q^{5n^2}\\over(1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^{5n+1})}", "b69e55ed03d4ece8e14e6d89f8b29a1e": "2^{d}", "b69e81bc9714378dccfe2313431340e6": "\\,\\ \\cos x", "b69e9a6ff9b2de6a812dca4153b75c66": "U+pV", "b69e9c8e927b92655a9220f06379400b": " (-\\pi , \\pi ) ", "b69ebd9fd620cd431896a5d60d0974eb": " (X, cl) ", "b69ec53af9cde848716f7081240133ae": "\\langle x,y,z \\mid (xy)^2=x^2=y^2, zxz^{-1}=y,zyz^{-1}=xy, z^{3m}=1\\rangle", "b69f3155dbb5aa9f0365032732ca52be": "g:d\\to x", "b69f82d27ae5532c51356f933efe0959": "\\overline{z}^n", "b69f93397446d59ccd2ab18dca7c6824": "\\mu = \\mu_o + \\frac{\\mu_1}{1+(\\frac{N}{N_\\text{ref}})^\\alpha}", "b69ff6f07f4d77ff31df03cfbd5d4e9d": "\\mathrm{E}(e_t e_{t-k}') = 0\\,", "b6a014aa7766fb9a16c387be09ae176b": "F_\\nu = \\frac{d P_{\\nu}}{d \\tau} = m A_\\nu ", "b6a050aa002bbf28bf75c0278d2e8a46": "\\psi_{L/R}", "b6a055e85a546c40c0954f635041e4c4": "\\frac{n-p_f(n-c-1)}{n}", "b6a061ac73651760e4f93eaf2a787788": " a_0 + a_1 x + \\cdots + a_{n-1} x^{(n-1)} ", "b6a09c4593ee4c0bfaaf2b1221a2aff7": "\\alpha -b/a\n\\end{cases}", "b6d308839cb85814f6e8dace283de0cc": "{U}_{\\mathrm{Cfgl}} = -\\frac{1}{2} m \\Omega^2 r^2 \\ ,", "b6d3436fdd4103441c47d099f09e429b": "U = I^{*a}N^{-1}", "b6d3a69028bc43d299002fb98e49a008": "P_P=\\frac{\\sum (p_{c,t_n}\\cdot q_{c,t_n})}{\\sum (p_{c,t_0}\\cdot q_{c,t_n})}", "b6d3ed45d408d4c5144ef8f378205ac3": "\\ \\Delta^s(a\\ \\alpha_{i,j,k} + b\\ \\beta_{l,m,n} )= a \\ \\Delta^s(\\alpha_{i,j,k}) + b \\ \\Delta^s(\\beta_{l,m,n})", "b6d414bbe0a55ffee0abf59999a366de": "\\prod_{t=0}^{T}\\frac{\\widehat{P}_{\\eta}(z^{t+1})}{\\widehat{P}_{\\eta}(z^{t})}", "b6d478a5a30fcef39bd4004d82450c86": "\\alpha>0 ", "b6d483abd46a6f0fe3ffcc854e94b62f": "2^m-m-1", "b6d4d33b4a7a2e788804a7d66dcfe114": "K(x,y)", "b6d4d6108047eecabd1d28c5211cd781": "\\sqrt[2]{n}_s", "b6d559316680dbab3dd9a96c9c77f5e5": "\\cos\\theta_W = \\frac{g}{\\sqrt{g^2+g'^2}}", "b6d55fa9927655cb65fa8a704796b8cf": "F(mode)=\\exp \\left( -\\frac{\\alpha+1}{\\alpha} \\right).", "b6d577044f6d9c582475336bc1c63f81": "\\int_D f\\, \\Delta g = 0", "b6d59a6a6a9a896479d8c25683b4f5f2": "4^n + 2^{n + 1} - 1", "b6d64150f605b9e12639dbf03fc83cc3": "f^{-1}\\left(\\bigcap_{s\\in S}A_s\\right) = \\bigcap_{s\\in S} f^{-1}(A_s)", "b6d6497e215bcffc9a9b2ab0f18b2265": " \\textstyle [1, 0, 8, 4, 1, 0, 1, 5, 1, 2, 2, 3, 1, 1, 1, 3, 6,...]", "b6d661675d69663dbf9e20b7a3d40c4d": "Length = \\frac {A}{W} ", "b6d6cb9c1f2d4850cd57eca2bbd5cb49": "\\gamma_1(t)=[-\\cos((n+1)t),-\\cos(nt)],\\quad t\\in [0,\\pi].", "b6d6e3398d8aa769b047837ff02c2a8f": "\\mu=1.665", "b6d6e346150500233ab7398ed87e54a6": "S_{11} = \\frac{T_{12}}{T_{22}}\\,", "b6d731bd3cff1dc9fe55868b2ba73399": "S = \\lim_{n \\to \\infty} S_n . ", "b6d767d2f8ed5d21a44b0e5886680cb9": "22", "b6d7b30853f6e0687d7c6ff453146b2d": "X=A \\cup B", "b6d808515c9c07e08f1ca5cadb0d03fb": "T^1_1(V) \\cong L(V;V).", "b6d815135af1aadc05347f3d67537fc6": " 3^1 + 1 - 1 = 3^1 ", "b6d817a7de2d320cf24924769bb70eda": "\\lambda_0 = 0", "b6d81f2c3bda69f1f50b9f231cd8fd31": "V(x)=\\begin{cases}V_0 & |x|B>0", "b6dbf81b750e9317026b593887c8f097": "c \\geq 0", "b6dbfa0a896dd9e35465b3d70416264d": "\ndu = \\frac{d\\xi}{\\sqrt{E \\cosh^2 \\xi + \\left( \\frac{\\mu_1 + \\mu_2}{a} \\right) \\cosh \\xi - \\gamma}} = \n\\frac{d\\eta}{\\sqrt{-E \\cos^2 \\eta + \\left( \\frac{\\mu_1 - \\mu_2}{a} \\right) \\cos \\eta + \\gamma}},\n", "b6dc261f366b0c10cb56b600ae675f0c": "C_{2n}(x) = \\frac{(-1)^n}{4(2n-1)!}\n\\pi^{2n} E_{2n-1} (x)", "b6dc5f05487e98bedc01b1c45a9bdb95": " \\|x - x_c\\|^2", "b6dcba07cdf219bd49c3f3fd1ba6038a": "E \\alpha \\Delta T", "b6dcce81573ff19909cc933ab18ac347": "MI(U,V)=\\sum_{i=1}^R \\sum_{j=1}^C P(i,j)\\log \\frac{P(i,j)}{P(i)P'(j)}", "b6dce60a1fbb8a791e4a194cadc0a243": "C_V = C_P - nR", "b6dd4fbbf36f1ffd2e409c02dcda8f1e": "2\\binom{n+2}2 - 3\\binom{n+1}1 + \\binom{n}0= 2\\frac{(n+1)(n+2)}2 -3(n+1) + 1 = n^2\\,,", "b6dd6c0cdf5f1663fd8665e19833964d": "g_{\\rm total}(r)", "b6dda12d483ec68ce36096c9b1301499": "\\widehat{\\rho}", "b6ddd84a9cc636257258701ca934e763": "wp", "b6de19c09ffda5b5946c15601572a681": "\\mu=1,\\sigma=1", "b6ded21c24eec4c5c6ed8a4269795251": "\\operatorname{pd}_R M_1 = 1 + \\operatorname{pd}_R M", "b6df1ca955358221c80c622cfdbe6912": "\\lambda \\rightarrow 0", "b6df368921b202c45a82b9bcc4a36f1f": "\\left\\{x_1,\\frac{x_1+x_3}{x_2},\\frac{(1+x_2)x_1+x_3}{x_2x_3} \\right\\},", "b6df59463b8976b3a2f19c66f7dde533": "R_S=R_H \\left(1+\\frac{\\cos(\\theta)-\\cos(\\alpha)\\left(\\cos(\\alpha)\\cos(\\theta)+\\sin(\\alpha)\\sin(\\theta)\\right)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "b6df61f9e628de42f72903a60d685329": "K^* ", "b6df672c19bdc530001f479c2b545272": "T= T_u", "b6dfe13686b4064de88996320015290a": "H(2,N)", "b6dfe9b2cb6f2015f14e9cdf355b45de": "\\zeta(s)\\zeta(s-1)", "b6e02fff4248a448d170e9db0ee8591a": "s(\\mathbf{x},t)", "b6e043dc80cae926b36c888de4044918": "\n\\zeta_{G, p}(s) = \\sum_{a\\geq 0}\\sum_{b\\geq 0}\\sum_{c=0}^{a+b} p^{-as-b(s-1)-c(s-2)} = \\frac{1-p^{3-3s}}{(1-p^{-s})(1-p^{1-s})(1-p^{2-2s})(1-p^{2-3s})}\n", "b6e04960fc717fa80bcbb17ffe52df75": "\\frac{1}{G}", "b6e081076094b9d162cf2d83fa02effb": "\\bar{x} = \\frac{(20\\times80) + (30\\times90)}{20 + 30} = 86.", "b6e08d9eb167b20c2528790237e2ab3c": "F(\\omega) = \\frac{H(\\omega) + H(-\\omega)}{2} - i \\frac{H(\\omega) - H(-\\omega)}{2}.", "b6e099a7694b00e8255f7d7f8d27cc10": "\\exp (-t^2) = \\sum_{n=0}^\\infty H_n \\frac {t^n}{n!}\\,\\!", "b6e0f7c3a55c55fbd547e30acac480da": "=R \\left( \\frac {d \\omega}{dt}\\ \\hat u_\\theta + \\omega \\ \\frac {d \\hat u_\\theta}{dt} \\right) \\ . ", "b6e0fb7ce3da205f28974dc3b323c989": "D_0", "b6e126c839ebd82964eedadaf88e5c62": "1+2a=b+c+d+e-m.", "b6e13a599c4829769d5b67580f770764": "\\sum_{k=1}^n k z^k = z\\frac{1-(n+1)z^n+nz^{n+1}}{(1-z)^2}\\,\\!", "b6e14f0b7fd37d42a58bad2889b94f1a": "\\Delta C = \\bar{C_i} - \\sum_{i\\neq r} q_{ir}\\bar{C_r} \\, ", "b6e19724204b26861523de89ac4da149": "t_{n-1}", "b6e19a1fe80ca9cfafeafff9c250426c": "z\\mapsto 1/z", "b6e1bf3cd18042ddbb9b753f9b82528e": "a_{ijk}", "b6e1e7b3683b714d8b7c11808aa56274": "\\mathbb{HP}^{\\infty}_{\\mathbb{Q}} \\simeq K(\\mathbb{Z}, 4)_{\\mathbb{Q}}", "b6e1f60f68221fa1e7c2b0aa55c6264a": "I_0 = \\frac{\\pi}{4} \\left({r_2}^4-{r_1}^4\\right)", "b6e21e3748f6cfbfd4583f6fde5811da": "\\mathbf{f}=0", "b6e2417d703646c5dadff78d8d59ecc2": "\\vec{\\textbf v}=\\frac{\\partial\\vec{\\textbf x}}{\\partial t}\\,", "b6e24af854f119fbcd58cd16ad9d10b9": "2^{-52.2}", "b6e2568139f5fabbb474d1ad1a71a26c": "\\sqrt g_{00}", "b6e2f7e0b8e5609540e413f421d42457": "\\le'", "b6e311f9e6bac042f20023d6287b3604": "|\\nabla L| = \\sqrt{ L_x^2 + L_y^2}", "b6e363fbd2526d269185b693bbda8557": "g_1, \\ldots, g_m", "b6e38b85ed79ba9d3cad8cbcbd669a74": "\\arg\\max_{k}{(T_1[k,i-1]\\cdot A_{kj}\\cdot B_{jy_i})} ", "b6e44014f2d190e7d122255b127b1952": "\\scriptstyle {(\\bull \\downarrow \\mathbf{Top})}", "b6e49a615ad6f2860442f6cd76411a94": "\\begin{align} \\tan \\frac{\\theta}{2} &= \\csc \\theta - \\cot \\theta \\\\ &= \\pm\\, \\sqrt{1 - \\cos \\theta \\over 1 + \\cos \\theta} \\\\[8pt] &= \\frac{\\sin \\theta}{1 + \\cos \\theta} \\\\[8pt] &= \\frac{1-\\cos \\theta}{\\sin \\theta} \\\\[10pt]\n\\tan\\frac{\\eta+\\theta}{2} & = \\frac{\\sin\\eta+\\sin\\theta}{\\cos\\eta+\\cos\\theta} \\\\[8pt]\n\\tan\\left(\\frac{\\theta}{2} + \\frac{\\pi}{4}\\right) & = \\sec\\theta + \\tan\\theta \\\\[8pt]\n\\sqrt{\\frac{1 - \\sin\\theta}{1 + \\sin\\theta}} & = \\frac{1 - \\tan(\\theta/2)}{1 + \\tan(\\theta/2)} \\\\[8pt]\n\\tan\\tfrac{1}{2}\\theta & = \\frac{\\tan\\theta}{1 + \\sqrt{1+\\tan^2\\theta}} \\\\ &\\mbox{for}\\quad \\theta \\in \\left(-\\tfrac{\\pi}{2},\\tfrac{\\pi}{2} \\right)\n\\end{align}", "b6e50dc52a61bb3ffebc19f1983346f7": "F_0(x)=1", "b6e51091f8406fbdf9afa339cae46d25": "M_{dd} + \\frac{c_{l,design}}{10} + \\frac{t}{c} = K", "b6e52059d0858b24b49090de0315bd70": "M_0,\\ldots,M_{n-1}", "b6e5258eb8b3bed4595891cac44884d6": "g=\\langle h|a,b\\rangle", "b6e5bfc600837b6a9fd76d2d76e1191a": " P(\\boldsymbol{W}, \\boldsymbol{Z}, \\boldsymbol{\\theta},\n\\boldsymbol{\\varphi};\\alpha,\\beta) = \\prod_{i=1}^K\nP(\\varphi_i;\\beta) \\prod_{j=1}^M P(\\theta_j;\\alpha) \\prod_{t=1}^N\nP(Z_{j,t}|\\theta_j)P(W_{j,t}|\\varphi_{Z_{j,t}}) ,", "b6e5d8b11bbe27410cc67867fe72809c": "^hp(x,y,1)=p(x,y)", "b6e604e37671241f5db80c4ae38ebd8d": "\\,_S\\sim \\;= \\{(s,t)\\in M\\times M \\,\\vert\\; s\\setminus S = t \\setminus S \\}.", "b6e619e9f02d1d0a9cd46fc3ad6d681a": "x_n=q_-^{\\;n}", "b6e652ab96f9d945925ec4a919d759c0": "\\left(\\tfrac{2n}{3}, n\\right)", "b6e66caa5b1e6f21d922d05aa4381a1a": " \n \\mathbf{u} \\mathbf{w} + \\mathbf{w} \\mathbf{u} = 0\n", "b6e6ac2ec634386daf1ea99a76e04657": "S_w=1", "b6e6bc83bcef34501cb482ddedf41585": "\\tilde{L}(\\mathbf{\\alpha})=\\sum_{i=1}^n \\alpha_i - \\frac{1}{2}\\sum_{i, j} \\alpha_i \\alpha_j y_i y_j \\mathbf{x}_i^T \\mathbf{x}_j=\\sum_{i=1}^n \\alpha_i - \\frac{1}{2}\\sum_{i, j} \\alpha_i \\alpha_j y_i y_j k(\\mathbf{x}_i, \\mathbf{x}_j)", "b6e6e21d654c8153978a21d1aef12599": "\\beta_\\mathrm{Darlington} \\approx \\beta_1 \\cdot \\beta_2", "b6e6ed3b7444d66c7d8d1609742a260a": "\\scriptstyle{\\xi^2_0}", "b6e6eefafa2dae37d8c4029cd9049024": " \\int z^n d\\mu(z) = 0, n=1,2,3\\cdots,\\, ", "b6e73fecf812de26d98c11f7c3793349": " \\left\\vert A_p \\right\\vert = \\frac{1}{f(p)} X + R_p ", "b6e740a833e78eb18b9d6aa92b104010": "\\mathit{d_{min}}", "b6e76c7fa3d8d36af56090191cc91945": "\\frac{[A]_{f}}{[A]_{i}}=e^{-k(\\frac{1}{k})}=e^{-1}=\\frac{1}{e}\\approx 0.37", "b6e79f0e9d83fc9e5247da5937e79995": "x = \\cos \\varphi,\\ y = \\sin\\varphi\\,\\cos\\varphi = \\sin(2\\varphi)/2", "b6e7cb6212a04bce916fbca99d16a03f": " \\frac{e^{1+\\gamma /2}}{\\sqrt{2\\,\\pi}} = \\prod_{n=1}^\\infty e^{-1+1/(2\\,n)}\\,\\left (1+\\frac{1}{n} \\right )^n ", "b6e8418ca6b4d11a44b0159af33033ed": "\\mathbf{(X^\\top X )\\hat{\\boldsymbol{\\beta}}= {}X^\\top Y},\\,", "b6e855e277f8a9d224a5f6ad559be35d": "\\displaystyle{\\Omega(g,h)^2 =b(gh)b(g)^{-1}b(h)^{-1}.}", "b6e88d0dc78da15dde1432eb53943ceb": "1 = \\int_x p(x) dx = \\int_x g(\\eta) h(x) e^{\\eta T(x)} dx = g(\\eta) \\int_x h(x) e^{\\eta T(x)} dx .", "b6e89371cc1ce24ac26578f460ef98b9": "x \\sim_j y", "b6e8b013207d20278c24c27baef7174b": " \\mu < \\mu_0 ", "b6e8c09f11be550dee03d40ce521b9bc": "\\Gamma = c + jd\\,", "b6e8d0fc534320a1b1d3bb04f4a2dd65": "\\rho^\\pi = E[ V^\\pi(S) ]", "b6e8d7de934866e4316eb3a88f20a64a": "\\partial \\,\\sum\\nolimits_c M_c", "b6e8e221f3912172eb4054b526ef1d65": "r_x(\\tau)", "b6e939531f37dcbbd94a67323f019c5c": "dU = \\left(\\frac{\\partial U}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial U}{\\partial y}\\right)_x\\!dy", "b6e97f0fe1e72654fca4ea6062e6c6fe": " (\\partial A)_V=-(\\partial V)_A=-S\\left(\\frac{\\partial V}{\\partial P}\\right)_T", "b6ea137ae8d2ec0de253bcd60116f279": "\\Phi_{\\rm B}^{(n)} + \\Phi_{\\rm B}^{(p)} = E_{\\rm g}", "b6ea3a37f15b635d4e2834dd67807fa8": "(h*g)*f", "b6ea641ae2b83b7291553982df2a763f": "\\text{srg}\\left(v, \\tfrac{1}{2}(v-1), \\tfrac{1}{4}(v-5), \\tfrac{1}{4}(v-1)\\right).", "b6eb0ced14427ecb7a16117a9b3075e4": "r_{Y,i} = y_i - \\langle\\mathbf{w}_Y^*,\\mathbf{z}_i \\rangle", "b6eb3ea391a1864a320bfcc1b5969a8f": "[v,u]\\in E", "b6eb59bd3ad2ed8da27c17d146298806": "dS = \\frac{1}{T} dU + \\frac{P}{T} dV - \\sum_i^n \\frac{\\mu_i}{T} dN_i - \\frac{\\Phi}{T} dQ - \\frac{v}{T} dp", "b6eb881adad04eaeb7db5e55cf24d176": "\\alpha \\alpha^* + \\mu^2 \\beta^* \\beta = \\alpha^* \\alpha + \\beta^* \\beta = I,", "b6eb8fd623429edf6b5936f380192ac1": "E[S]=\\frac{1}{\\mu}", "b6ebe1f8dd5c3600685dca82a1cd4066": "p(x) = x^{3} - 3x - 1", "b6ec3006539c8d54609bf7620f43d5d4": "I_n(w)", "b6ec4a80bcfec214d625381b9e1a7420": " V_{\\text{out}} = V_2 - V_1.\\, ", "b6ec4bd5baa0f9ad4f59bf8ace0d9aa6": "k = A / \\mathfrak{m}", "b6ec5bfd8aafca620a378565923d8a12": "\nQ_i(t+1) \\geq Q_i(t) + y_i(t)\n", "b6ec5ebd85b56e08222d95e97f6a70b6": "\\mathbf{\\hat{n}}", "b6ec9b872abe5069cb6a8793af9b38a8": "\\int x^n e^{ax} dx . \\,\\!", "b6ecc071ea9c9cfa64ee1a7be0bae906": "H_n(D^n,S^{n-1})\\cong H_n(S^n)\\cong \\mathbb{Z}.", "b6eccb7749a25a41b26c05a27386aa85": "S(x)=\\alpha^{-7}+\\alpha^{1}x+\\alpha^{4}x^2+\\alpha^{2}x^3+\\alpha^{5}x^4+\\alpha^{-7}x^5,", "b6eced81a8546457d9e79aea97e9797e": "\\det A_Q \\ne 0", "b6ed361a01d2d76ff50fd4a9e5d079ef": "\\operatorname{prox}_{\\gamma R}(x)", "b6edf500dbd452dc7f442e52f1f8d1ee": "\\,n_t=L_t-u_t\\,", "b6ee2c9d72ed588fad8ab26d81a35c8e": "G = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\nW(A) & \\\\\n & R(A) \\\\\n & W(A) \\\\\n & Com. \\\\\nAbort & \\\\\n &\\end{bmatrix}", "b6ee5a169a243759913a274deb41eea9": "l^\\mu = \\left(\\frac{r^2 + a^2}{\\Delta},1,0,\\frac{a}{\\Delta}\\right)", "b6ee85fab65380532f44d9e6be2e7921": "V(x) = V(x_0) + (x-x_0) V'(x_0) + \\frac{1}{2} (x-x_0)^2 V^{(2)}(x_0) + O(x-x_0)^3", "b6eef11e290a26eb3ff833dbf1a3e162": "\\nabla J ", "b6efbeb3e561301d165346ee013e3946": "B_i = \\frac{8\\pi^2}{3}< \\Delta R_{i} \\cdot \\Delta R_{i} > = \\frac{8\\pi^2 k_B T}{\\gamma}(\\Gamma^{-1})_{ii}", "b6efc039f1c18527b52f122ceebdb2c0": "u\\in C^\\infty(\\Omega)", "b6efc33675866ef05f6f7c367bf5fcbc": "\\mu_p(N_D) = 130 + \\frac{370}{1+(\\frac{N_D}{8\\times10^{17}})^{1.25}}", "b6efda8c757c3691117d15f81cd2cb60": "\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_z)\\psi(x,y,z,t) = \\psi(x - \\Delta\\theta y, \\Delta\\theta x + y, z, t)", "b6f008cdc901ba1d39345e1131dc618b": "\\Delta=\\sqrt{\\frac{4}{3}\\frac{m}{H}}.", "b6f024ee1f418e14143182ee1cdfe4a6": "A = U\\Sigma V^*", "b6f09dc3e9c4892b30ff7dac63469d41": "1/2^{M-1}=2/m", "b6f0b640ad0c96c278735a4a6ce9dbbe": "K_6", "b6f0f330ad2c032fbcffb72d43599714": "\\alpha^{-i_j}", "b6f15eed19fa629bd7d72843c5ee2c80": "\n\\Pr[|X| \\geq k\\sigma]\\leq 2e^{-k^2/4n}\n ", "b6f167f3c6348f1f472215d98418181e": "\\quad (2) \\qquad \\qquad \\bar{\\rho}_i \\left( t_1 \\right) = \\frac{1}{ x_{i+\\frac{1}{2}} - x_{i-\\frac{1}{2}}} \\int_{x_{i-\\frac{1}{2}}}^{x_{i+\\frac{1}{2}}} \\rho \\left(x,t_1 \\right)\\, dx ,", "b6f193b0d5b785142c7abeb26253c872": "m = m(l)", "b6f19bd9cacf5cb9b2c03d5615325e07": "e_j", "b6f1bff246572498bf3711e092395bf5": " \\{ i \\mathbf{e}_{1}, i \\mathbf{e}_{2},\n i \\mathbf{e}_{3} \\}", "b6f22330e49d2c8c5a22e44868f61ad9": "\\epsilon \\circ \\eta = \\eta_0 \\circ \\epsilon_0 : K \\to K", "b6f228ba706c088b635c2cbc089d81c2": "\\mathcal{D}_b", "b6f239901e57cdbc844e1c1ed8524fcd": " Tf(x) = \\int_c^x K(x,y)f(y) \\, dy,", "b6f249ce50e15f472af6d072f3ad569f": "\\ \\displaystyle g(d,s)", "b6f271ee1439404e25319ff9b6ab8cfc": "E:X\\rightarrow \\mathbb{R}", "b6f2c2a9bf6ef452c92b237c041b12cd": "\\scriptstyle N\\, = \\,5x\\,", "b6f2f9d5c604372acd54304776ba0551": " L_{\\alpha + 1} := \\text{Def}(L_\\alpha). ", "b6f2fa3caaf30c4673006bba49fa1aff": "\nm(A)=\n\\begin{cases}\n1 & \\text{if }A\\in F \\\\\n0 & \\text{if }S\\setminus A\\in F \\\\\n\\text{undefined} & \\text{otherwise}\n\\end{cases}\n", "b6f352daef08740b75294e0f52d1a135": " (\\pi [s,t] \\psi)(x) = e^{i t y_0} e^{i s x} \\psi (x). \\quad ", "b6f378d534e71b8b40b1c9c534c4212b": "G = \\left ( \\frac{\\pi d}{\\lambda} \\right )^2 e_A ", "b6f380ad171c42a39b177d48108d6fdb": "\\mathrm{Ste} = \\frac{c_p \\Delta T}{L}", "b6f3f002b01f7b6adc82ebe15de58095": " \\Delta P =\\rho g (\\Delta h)\\,", "b6f40f5650d2c8d4659f702f4b97feee": " = \\mathbb{E}_\\theta\\left[ |\\mathbf{\\theta - X}|^2 + 2(\\mathbf{\\theta - X})^T\\frac{\\alpha}{|\\mathbf{X}|^2}\\mathbf{X} + \\frac{\\alpha^2}{|\\mathbf{X}|^4}|\\mathbf{X}|^2 \\right] ", "b6f45974b5bd0b31604f155245fed82b": " \\phi : X \\rightarrow \\{ 0 \\} ", "b6f46a787f6d75265beb761c5e02de7f": "\nb^2 = \\frac{q_{xx}+q_{yy} - \\sqrt{(q_{xx}-q_{yy})^2 + 4q_{xy}^2}}{2}\n", "b6f49426301faafd013269bc725e95d9": "=a\\left(\\frac{1-2b+b^2}{4a^2}\\right)+\\left(\\frac{b-b^2}{2a}\\right)+c", "b6f49c9984be4224621517564c44f76a": " \\eta =\\frac{\\tanh(hp)}{hp} ", "b6f4c121cf603125d2989b8856279f3b": "p_{t}(x,p)", "b6f500b5cbd2230e54b1a4fc9ae7e65e": "\\scriptstyle y= \\frac{(ax+b)}{(cx+d)}", "b6f50bf7d5997fc496f230a4bf5c4503": "0\\le \\Delta C/\\Delta Y <1", "b6f50eec2b1c884e2e4cece11426237c": "\\lim_{r\\rarr\\infty}\\|A^r\\|^{1/r}=\\rho(A). ", "b6f5a4658d29442f6b2e589aa3f4c8c9": " T_\\text{sun} = 15.7 \\times 10^6 \\; \\text{K} \\; \\pm 1% ", "b6f5e9c1bfe12039af724f6d3fd74103": "\\tilde{t} = \\sqrt{\\frac{g}{h}}\\, t.", "b6f612d14ef2832798251d7c631bce2c": "\n\\begin{matrix}\ny_0 & & & \\\\\n & \\triangle y_0 & & \\\\\ny_1 & & \\triangle^{2} y_0 & \\\\\n & \\triangle y_1 & & \\triangle^{3} y_0\\\\\ny_2 & & \\triangle^{2} y_1 & \\\\\n & \\triangle y_2 & & \\\\\ny_3 & & & \\\\\n\\end{matrix}\n", "b6f66897271b7c5bbbb760f7242071b2": "(V\\otimes W)_0 = (V_0\\otimes W_0)\\oplus(V_1\\otimes W_1),", "b6f668fc3c85346daf47fa97c37162fe": "\\scriptstyle k_{\\mathrm{Alice / Bob}} = g_{\\mathrm{Alice}}^t I_{\\mathrm{Bob}}", "b6f6a3cf1f00b6beff68fbc7e9b19697": " X_v\\, ) = P(X_v=x_v \\mid X_j=x_j ", "b6f6bf1bbd9c5cd226e3c4203413ca34": "I_x = I_y = \\frac{1}{12} m\\left[3\\left({r_2}^2 + {r_1}^2\\right)+h^2\\right]", "b6f6da6a02bc6cfe48b256fe69a5c60b": "\\textstyle \\frac{1}{2} = \\frac{1}{3} + \\frac{1}{9} + \\frac{1}{27} + \\frac{1}{81} + \\cdots", "b6f6e45dd40e6c3e292b3efaedc83e4c": "0.5349N\\sqrt{N}-0.4387N-0.097\\sqrt{N}+O(1)", "b6f70c125f8356ec6572622b8e534a29": "\\lambda_{max}(U^{(k)}) = 1", "b6f732ad503ad02dd50fc191ab12f07e": "\\phi - \\ ", "b6f73a80440ba66cf28744ea730169cc": "\\nabla B ", "b6f754083d466a75d830a58892808b26": "A(z) = \\sum_{k=1}^{\\infty} \\frac{\\phi(k)}{k} \\ln \\frac{1}{1 - B(z^{k})}", "b6f776834e64cf409f03e4db112e41f9": " \\hat{H} |\\{n_{\\mathbf{k}}\\}\\rangle = \\bigg( \\sum \\hbar \\omega \\big(n_{\\mathbf{k}_l} + \\frac{1}{2} \\big)\\bigg) |\\{n_{\\mathbf{k}}\\}\\rangle ", "b6f7bbdcf1c3f4f0369a552daffe1081": "\\mathbf{P}[ A ] = \\mathbf{1}_A ", "b6f7e8f3c106b837c1b840b8f30198ee": "\\phi_{A,B}(na,b)=\\phi_{A,B}(a,nb)", "b6f814e5374e4f13a495a7dce107a984": "c_{T-j}(k) \\, = \\, \\frac{1}{\\sum_{i=0}^j a^ib^i} Ak^a", "b6f848990e08649f2d7862c36b0de02b": "f_* \\colon A(X) \\to A(Y),\\,", "b6f86515477dcb5234788f79d1b73db7": "\\ p'", "b6f874a2cc2566f9b6283bf014136393": "\\dot Q(t)\\ ", "b6f8f98f5df93aa4a854a9616318232a": "M^w(x)=1", "b6f90f01e9c5c23353d23fb45bfbfb59": " \\textstyle T ", "b6f918b172d5ea50f266bef374dbdcfd": "f(z + \\omega) = g(z,f(z))", "b6f955eea05c3fbfdf450ee8d7f170b7": " \\hbar \\rightarrow \\hbar{{k}_{0}}^{2}T/m ", "b6f9667bb00b0f068e8bec3e5f6a6a34": "\\frac{d E}{d {\\lambda}}=\\int{\\psi^{*}(\\lambda)\\frac{d{\\hat{H}_{\\lambda}}}{d{\\lambda}}\\psi(\\lambda)\\ d\\tau},", "b6f9898c121c415ae9a1cbdf6f4585b1": " |I(f)| \\leq M_K \\sup_{x\\in X} |f(x)|. ", "b6f99dd2dba1958451a25f7141e07658": "\\,J^-(x)", "b6f9b31124654a7e509d6415a71c30e1": "P(A|B) = \\frac{P(B | A)\\, P(A)}{P(B)}\\cdot", "b6fa6f2861fdc4fd175f2bf5cae15b78": " = \\int_{-\\infty}^{\\infty}{\\left|h(t) (e^{\\sigma + j \\omega})^{- t} \\right| dt}", "b6fa76c3812f4238c5f51c4aea9f6d16": " g(s)=s\\int_0^\\infty K(st) f(t)\\,dt ", "b6fb51d0fcd3b494a43b7e4609d00c16": "\n\\widehat D^* = U_1 \\Sigma_1 V_1^{\\top},\n", "b6fb5b048cd2a44207b4ba5bc6b2a849": "G \\times_{G_x} T_x M / T_x(G \\cdot x)", "b6fbddbd6c4529ae0d5a6dd66de3ad66": "h(x)=f(x-x_0),", "b6fbf0e3f2f8e145382cfa033977b3c6": "\\mathrm{MV(t)}=K_p\\left(\\,{PV(t)} + \\frac{1}{T_i}\\int_{0}^{t}{e(\\tau)}\\,{d\\tau} + T_d\\frac{d}{dt}PV(t)\\right)", "b6fc0f44c837b3fbdd0506021c29e483": "u(0)=u(1)=0", "b6fc4c3d38796d052ec06444b0919b0e": "\\left(L_2, \\langle\\cdot, \\cdot\\rangle_2\\right)", "b6fcb4e5874b0e46e621cd95970dcf7d": " \\cdots \\, ", "b6fcf8047972f167cff752ed5da2e188": "\\hat{U}(t,t') ", "b6fd0610efb6839ab8e412d591444acd": "(\\varepsilon_1^T, \\mu_1^T)", "b6fd0cd8bb4fc48eb2f3b1bdb2381323": "\\textstyle\\sum_j a_j < \\infty.", "b6fd5d75f8ce31050022214853d1247c": "(x,x^2).", "b6fd690f14c928129b6cf69de12b70df": "(\\tilde{U}_1^i,\\tilde{U}_2^i,\\dots,\\tilde{U}_d^i)=\\left(F_1^n(X_1^i),F_2^n(X_2^i),\\dots,F_d^n(X_d^i)\\right), \\, i=1,\\dots,n.", "b6fd761f5955a58d0ca1351db6863a95": "y=0,", "b6fd810683654ba952403f748a6b2f0d": "U>>1", "b6fda545e2a5d648a6722552373f93ec": "R~", "b6fdd38ae4f2b5d17ce5a1ba02a13cd4": " \\int_{0}^\\infty e^t \\sin(e^{2t})dt = \\int_{1}^\\infty \\sin(u^2)du = \\frac{\\sqrt{\\pi}}{8} - S(1) < \\infty,\n", "b6fdf05e32c392d349ae90a49d668ec6": "\\begin{array}{cc}\n \\begin{array}{rr} \\\\ &3 \\\\ \\text{-}1& \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & \\text{-}12 & 0 & \\text{-}42 \\\\\n & & 3 & \\text{-}39 \\\\\n & \\text{-}1 & 13 & \\\\\n \\hline \n 1 & \\text{-}13 & 16 & \\text{-}81 \\\\ \n \\end{array}\n\\end{array}", "b6fe69e1e5755764eb061bd395eabfd6": "x_A = H \\cdot x_B ", "b6fe7b4a22e2ce1a41258ebe4cf85e22": "\n \\rho_0 c^2 = \\boldsymbol{Q}(\\boldsymbol{N}) \\boldsymbol{m}\\cdot\\boldsymbol{m} = B_{ijkl} N_j N_l m_i m_k,\n ", "b6feb6a09203c54946ad88c1852e8fb9": "\\mathbb{Q}(\\sqrt{-d})", "b6ff82ee2d26313a82825b9d64ebf9e0": "\\langle\\cdot, \\cdot\\rangle_W", "b7000dd9eb57a731c41f8c85a55ff36a": "R_1", "b700875d6c3fddddeaacac673ca2ada7": "z_8 = x_8 y_1 + x_7 y_2 - x_6 y_3 + x_5 y_4 + x_4 y_5 + x_3 y_6 - x_2 y_7 + x_1 y_8", "b70092a599ac47df27dbaf46c2f8c000": "f(v_1,\\ldots,v_n)", "b700d79a31284d76d065bf9ed1d198fa": "i\\frac{\\partial}{\\partial t}\\rho_{2} =\\left(1-P\\right) L\\rho_{2}+\\left(1-P\\right)L\\rho_{1}.", "b7012ad3adb00f3601145d679d464d49": "-\\phi_0 < \\phi < +\\phi_0", "b70130c1d12a295824f82d854bc7fdde": "{1 \\over {1-e}} > {r_1 \\over R_1} > {1 \\over {1+e}}", "b70139d86c17b5cd249cf2bc02a06d56": "w_i(t) \\,", "b701d484050ffe1c8cd528e613153102": "\\mathbf{a} \\cdot \\mathbf{b} = \\begin{bmatrix}\n a_1 & a_2 & a_3\n\\end{bmatrix}\\begin{bmatrix} \n b_1 \\\\ b_2 \\\\ b_3\n\\end{bmatrix}.", "b701dd50eefa4c3f6566dde196c161d2": "\\mathbf{v} \\cdot \\nabla \\mathbf{v} = \\nabla \\left( \\frac{\\|\\mathbf{v}\\|^2}{2} \\right).", "b701e2ca975d792d3c5b5fa7eefbffd9": "W^{\\mathbb C} \\cong W\\oplus \\overline{W}.", "b701ec9736fce67fb38d767ee2e0632e": "\\pi / 4", "b702433c5df70078f71bcb0ab39e985e": " \\mathit T = \\mathit g + \\frac{\\mathit ROE - \\mathit g} {1} = ROE ", "b702477cbe61c0ce557079bf1dbe3630": " \\phi(x) = \\infty", "b70249d6bb132fc46c5c673323126231": "l = ", "b70281fe485dd30f89119c04c16daf48": "\\scriptstyle <7.8\\times10^{-12}", "b702dcd6abdfcc347f4b09c3cb0822ad": "\\left\\lfloor\\frac{J}{4}\\right\\rfloor + 5J", "b702f32b67f9030008cccd793cfa7251": "P=2^{-U}", "b702faf51aa59a307d5db0fbbc277dc9": "T_{q'}M", "b7030f4a7787255971b9a34a4abe635d": "H^\\infty", "b7031fb93d19bf83a63d60a032c46a8c": "\\pi_1(\\mathbf{RP}^n)", "b703c4c1813b414656e13efe313a14d0": " \\mathbf{b}' = -\\tau\\mathbf{n}. ", "b7041ecec08137a4b6e518729927bb93": "\\mathbf{D} = \\varepsilon_0\\mathbf{E}, \\;\\;\\; \\mathbf{H} = \\mathbf{B}/\\mu_0", "b7042343c6a2dc1a35132132ada079b6": "H_n(x)=(-1)^n e^{x^2}\\frac{d^n}{dx^n}e^{-x^2}= (-1)^n e^{x^2}{n! \\over 2\\pi i} \\oint_\\gamma {e^{-z^2} \\over (z-x)^{n+1}}\\, dz.\\,\\!", "b7049455cbf26ced4b530ca689125412": "C->M->C", "b704be109e5c20a5708a6244349d93bc": "H \\approx \\frac {f^2} {N c}\\,.", "b704c6ad803fe71423959f624378048b": "\\mathbf{a}_r", "b704ca997a0aa43986e2db1da19d2da5": "r = r_1 \\left(1+\\frac{Gm}{2c^2 r_1}\\right)^{2}", "b704d560ca15b68216d1411a19005b78": "d(x,y) \\ge 0", "b70510e3f4e5aea1c5b474c9c7abfb15": "\\textstyle E", "b70540876ce2c01efdcd3f12813bbe52": "d_\\mathfrak{g}", "b7054f9f87003b41d29617e8e685b217": "\nc_1(E) = -s_1(E), \\quad c_2(E) = s_1(E)^2 - s_2(E), \\quad \\dots, \\quad c_n(E) = -s_1(E)c_{n-1}(E) - s_2(E) c_{n-2}(E) - \\cdots - s_n(E) \n", "b70583f3a015c1875d0e207f5f3fe783": "\\mathbf{C} \\!\\,", "b705f2e462333a01b22347ccc32cdfca": "t=0,\\, \\pm\\tfrac{2\\pi}{3}", "b7060f195bc25bdfcf21243b9bd7e304": "f(n) = g(n) + o(g(n))\\quad(\\text{as }n\\to\\infty),\\,", "b70649418dc8f8c1bb34fe7ed3fe6ce6": "b_i=\\frac{r_{i-1},c_{i+1} - \\textstyle \\sum_{j=2}^i b_{i-j+1}\\times a_j}{a_1}\\mbox{ with remainder }r_i", "b7071dd8b0abc1e3f749079fa043c9a1": "N \\delta(t_1-t_2)", "b7072c4b6d9ab6f15d48d7b3b675ca33": "~I_{\\rm p}~", "b7073127ccadb9fd1034fcb6e186c01a": "\\mathcal{G}= ", "b707ba750ce12a00c44697bce2aecf97": "\\mathbf{\\hat \\beta}_\\mathtt{GLS}", "b707cadc05082564bb511ece68b23ff6": "\\bigtriangleup\\xi = \\bigtriangleup\\eta = 1 ", "b707d765022d77421731a95a69ea2fd7": "\\mathcal{L}=\\frac{1}{2}\\partial^\\mu \\phi \\partial_\\mu \\phi -\\frac{m^2}{2}\\phi^2 -\\frac{\\lambda}{4!}\\phi^4.", "b70838b84e4c483e59f5774173358cda": " \\langle X,Y\\rangle_{\\beta H}= \\int\\limits_0^1 \\langle{\\rm e}^{x\\beta H} X^\\dagger{\\rm e}^{-x\\beta H}Y\\rangle dx", "b7085b515908cd51f60e49823154819d": "F_{d+2} \\ge \\varphi^d", "b708bebdd71d20ee2823e71671a21717": "Z_{base} = \\frac{V_{base}}{I_{base} \\times \\sqrt{3}} = \\frac{{V_{base}^2}}{S_{base}} = 1 pu", "b708e6b70a87e363c0d29bda205d8abc": "|Z|\\leq 1", "b7096a6454bc1195e86186bd80dc88f8": "\\inf \\{ \\|S x\\| : x \\in X_0, \\, \\|x\\| = 1 \\} = 0. \\,", "b70979a357cfbe7c61e4c686ea2beb93": "\\perp", "b709afff82db4608f377c5b328f31cd6": "x_2(t)\\,", "b709be2d63ae9a0e7ab33643a4a7330e": "|f|_{0,\\alpha;\\Omega} = |f|_{0;\\Omega} + [f]_{0,\\alpha;\\Omega} = \\sup_{x\\in\\Omega} |f(x)| + \\sup_{x,y\\in \\Omega} \\frac{|f(x)-f(y)|}{|x-y|^\\alpha}.", "b709c49a56ec3eeb5c46897368c090f3": "F = \\frac{1}{2} \\frac{\\partial C}{\\partial dx_{drive}} V^2 ", "b709c9de900b88eda72a282a81060ca4": "a_r = \\ddot{r}-r\\dot{\\theta }^2 \\, ", "b709fcaeeba09ee82f38bf54b8841e1d": "D : \\Gamma(E) \\rightarrow \\Gamma(E\\otimes\\Omega^1M)", "b70a2439386ddca4aa95baf4affa981d": "g(y^p) = f(x)", "b70a49a65c0e3d8453415cc954d3464f": "x_1y_2 + y_1x_2 = 0", "b70a89e03bf391c1918ed6fc1b9aa147": "n\\equiv m\\mod\\begin{cases}4|a|,&a\\equiv2\\pmod 4,\\\\|a|&\\text{otherwise.}\\end{cases}", "b70a99a6ca5ef90011fd5a983483f6bd": "\n\\tau(n) = \\frac{65}{756}\\sigma_{11}(n) + \\frac{691}{756}\\sigma_{5}(n) - \\frac{691}{3}\\sum_{0\\sigma_j", "b70cb49f619d6c081c95fd317e776a72": "R = \\frac{\\gamma l}{g \\omega} \\big( 0.01 \\frac{v}{d}+ \\frac{0.09}{d} \\sqrt{ \\frac{v_{eff}}{d}} \\big)", "b70d16981fc62e54bf7bab0749e16b1e": "P(X)\\in K[X]", "b70d19481861cb4a7cc978497ee4ac9b": " Q = T_z + \\frac{Y}{2}", "b70d88a22d54efd83965a691422d56d9": "\\text{floor}(L/2)", "b70d8d61e2504a493a7bbdddd12ae97f": " W_f \\,\\!", "b70d8fb18274dea964226af93e926975": "\\psi'(\\theta)", "b70dcb6f493a301d38e443b1725032e6": " v_2=\\begin{bmatrix}* \\\\1 \\end{bmatrix}", "b70e9bb62a290a2679ca6112867dec04": "T(n) = \\sum_{k=0}^{n}S(n,k)\\,T_0(k)", "b70eada2724beaf367c7b5eeb70ffa97": "{\\rm d}P = |\\psi|^2 {\\rm d}^3\\mathbf{r}. \\,", "b70ed2c76e4da83ec195c322dc5f1f31": "\\alpha = 5", "b70ed2cf76e6573c9505c28601b1f88b": "\\bigcup X = \\{y | (\\exists x \\in X) y \\in x\\}", "b70f413207c200b3c32fd6bc291c47b4": " \\delta A[f_0,f_1] = f_1(0)\\left[ n_-\\frac{f_0'(0_-)}{\\sqrt{1 + f_0'(0_-)^2}} -n_+\\frac{f_0'(0_+)}{\\sqrt{1 + f_0'(0_+)^2}} \\right].\\,", "b70f51f953e24916512b0914f9782ecf": "P(q_1+q_2)", "b70f6f8339b908f7fadf27c4c16db1a4": "\\langle a \\mid a^8 = 1, a^4 \\neq 1\\rangle.", "b70f849db0a7d272eb98ed71f021b8fc": "\\omega(t)=\\inf\\big\\{at+b\\, :\\, \\forall x\\in X,\\, \\forall x'\\in X\\,\\, |f(x)-f(x')|\\leq a|x-x'|+b\\big\\}.", "b70f8b6dc105a4f5a38ac446d2d870d0": " T \\leq S ", "b70fbc7e1ed18cafdaa1104119dabb01": " |A_0\\rangle ", "b70fca9cc917f0febd9bc5a9740b0469": "{n-1}", "b70fdd3b8c8c417ddbfcec546ca06196": "\\frac{{}_1F_1(a;b+1;z)}{{}_1F_1(a;b;z)} = \\cfrac{1}{1 + \\cfrac{\\frac{a}{b(b+1)} z}{1 + \\cfrac{\\frac{a-b-1}{(b+1)(b+2)} z}{1 + \\cfrac{\\frac{a+1}{(b+2)(b+3)} z}{1 + \\cfrac{\\frac{a-b-2}{(b+3)(b+4)} z}{1 + {}\\ddots}}}}}", "b7100f8517ce594a0e3b8ab5b2e028ed": "P[ s(t) | \\{ t_i \\} ] = P[ \\{ t_i \\} | s(t) ] * (P[s(t)]/P[\\{t_i\\}] )", "b71037ae905865aaba0f3bfc125bac58": "x_1+ x_2 i + x_3 j + x_4 k \\mapsto \\begin{pmatrix}\\;\\;\\,x_1 + i x_2 & x_3 + i x_4 \\\\ -x_3 + i x_4 & x_1 - i x_2\\end{pmatrix}.", "b710740e0f1178528f7b4ed05dfff1cb": "\\mathcal{A} = (A, (f^{\\mathcal{A}}_i)_{i \\in I})", "b7107d0fd54b56a387c3224c25d98e01": "G \\to \\mbox{Aut}(G)", "b710a4112dce2b3f7856bba2a9263074": "\\varphi(q-1)", "b7110e4d7f4ac116a4b99264d0fc7710": "x^{11} \\pm 1", "b71136c2cda4a8b0615224d4a4c425bc": "D_i = \\frac{ \\sum_{j=1}^n (\\hat Y_j\\ - \\hat Y_{j(i)})^2 }{p \\ \\mathrm{MSE}} .", "b7113fea4e3404a780e9b027bad65ca1": "\\langle T_C \\rangle = \\frac{1}{\\Delta S} \\int_{Q_{out}} TdS ", "b711a0deb74c93592fc17de2d0160232": "\\mu(A)\\,\\!", "b712420b56855b39d4548e570c650004": "(I, I+1)", "b7128774951504f57261cb61ef0e0b10": "r(v) - R(V)", "b712a61e9e79e8db42d8c8fcf4eb495f": " T=-D^{-1}(L+U) ", "b712eff16e12e6873dab9ac2141b8339": "WT_\\psi\\{x\\}(a,b)=\\langle x,\\psi_{a,b}\\rangle=\\int_\\R x(t){\\psi_{a,b}(t)}\\,dt.", "b7134155f3265c49d67db82de8b09ade": "p(s|i) = \\frac{1}{(2\\pi \\sigma^2)^{D/2)}} \\exp{\\left(-\\frac{\\lVert s - m_i \\rVert^2}{2\\sigma^2}\\right)}", "b7136a45d284b590ed25adc2fb004580": "<_y", "b713f7bf79b2939fb7a0a9451ea07493": "p<1", "b713fdbbac4bb8ead3d5d28126e9a392": "\\mathrm{2S_2F_2 + 6NaOH \\ \\xrightarrow{}\\ Na_2SO_3 + 3S + 4NaF + 3H_2O }", "b71428441cd0287813287ecb58210eff": " (a_1 + b_1x) + (a_2 + b_2x) = (a_1 + a_2) + (b_1 + b_2)x, ", "b714698d390c731fbd748505bc3fcd2b": "H_k g_k = z\\,\\!", "b715116e28695ccea3792d287e70cb52": "\\mathbf{x} = (x,y)", "b71533d42f31ccbf072658e9a0a9d1dd": "\n\\ln\\Gamma(z)\n", "b71566754973f9a89e6e8496b65a59a5": "\\sum_{p|n} \\frac{1}{p} - \\prod_{p|n} \\frac{1}{p} \\in \\mathbb{N}.", "b71567c7178dc7766034699509500165": "\\lambda^{(1,2)}, \\dots, \\lambda^{(n-1, n)}.", "b715a0527e93f5bdf5d348679cde84a0": "{T_v} \\approx T(1+0.61w)\\, .", "b715d9d253089a286ab7ae3d6711e76e": "\\{N(t),t\\geq0\\}", "b716216856be1c8fccda8bc181fe7253": "\\Big( \\pi \\models F\\phi \\Big) \\Leftrightarrow \\Big( \\exists n\\geqslant 0: \\pi[n] \\models \\phi \\Big)", "b716679f73ec2a26d01720cefbe2abad": " (\\tilde{\\mathbf{x}}')^{T} \\, \\mathbf{E} \\, \\tilde{\\mathbf{x}} \\, \\stackrel{(1)}{=} \\,(\\tilde{\\mathbf{x}} - \\mathbf{t})^{T} \\, \\mathbf{R}^{T} \\, \\mathbf{R} \\, [\\mathbf{t}]_{\\times} \\, \\tilde{\\mathbf{x}} \\, \\stackrel{(2)}{=} \\, (\\tilde{\\mathbf{x}} - \\mathbf{t})^{T} \\, [\\mathbf{t}]_{\\times} \\, \\tilde{\\mathbf{x}} \\, \\stackrel{(3)}{=} \\, 0\n", "b716b0adc2e9496db885b50c1caa9214": " \\theta ", "b716bee89287e8c9e136c38b18bed223": " \\operatorname{k}: \\hat{A} \\rightarrow \\operatorname{Prim}(A). ", "b716d1cedea9acee32e6b23115f97cb1": "\\textbf{P}_{k\\mid n} = \\textbf{P}_{k\\mid k-1} - \\textbf{P}_{k\\mid k-1}\\tilde{\\Lambda}_k\\textbf{P}_{k\\mid k-1}", "b7175eaa1ab0177a5cbb63ca45f24862": "\\approx 6 m^{3}", "b7176c7de5276623f52f0ed15e54ecf9": "v_1 \\mathbf e_1 + v_2 \\mathbf e_2 = \n\\frac{\\mathbf{e}_{1}}{h_{2} h_{3}} \\frac{\\partial}{\\partial x_{2}} \\left( h_{3} \\psi_{3} \\right)\n- \\frac{\\mathbf{e}_{2}}{h_{3} h_{1}} \\frac{\\partial}{\\partial x_{1}} \\left( h_{3} \\psi_{3} \\right)\n", "b7178fa3d9f8403f02aa7e943e286b09": "\\alpha (\\mathrm{cm}^3) = \\frac{10^{6}}{ 4 \\pi \\varepsilon_0 }\\alpha (\\mathrm{C} \\cdot \\mathrm{m}^2 \\cdot \\mathrm{V}^{-1})", "b717d214ce0ab853da42f1c8abb8dbee": "\\lambda^2+1=0", "b717f837526f07405b9800ce93093033": " \\vec{\\psi}_{P} = \\vec{\\psi}_{C}+\\vec{\\alpha}\\times(\\vec{r}_{P}-\\vec{r}_{C}) ", "b7182a40aa88eafb5e9ee6e8b104c54e": "\\omega=\\frac{\\hbar k^2}{2m}.", "b71833f3f5f11723da0b818346bcd5ce": " \\mathbb{Q}(\\sqrt{3}) ", "b71857a807646a46b503f94d02750565": "g = 266", "b718adec73e04ce3ec720dd11a06a308": "ID", "b7198f1c96a413d82c06eec9e5f174e2": "\\operatorname{P_i(S_x) =P(s_x/S_{x,j}) = f(S_{x,j})/\\sum_{b e {[acgt]}}\\operatorname{f(S_{x,i},b)}}", "b719ee6b5fdde9c019605f2905d1a3f0": "(m, r_\\alpha)", "b719f187556bec7ff266644f9c83af64": "p(F_i \\vert C, F_j) = p(F_i \\vert C)\\,", "b719f4a7450516398394be47c084f748": "[0,\\lambda) = \\{\\alpha \\mid \\alpha < \\lambda\\}\\,", "b71a40aa9818bdd6d86d29e13063ae0d": "|n|^k\\hat{f}(n)", "b71a7b7f99f98a29d62751fab4ed5d59": "f(v)", "b71a8f41306207d3a8bcbd7eb83d092e": "\\mathrm{Ca} = \\frac{\\mu V}{\\gamma} ", "b71ab1368f7a45daf474a0a6c235b46f": "\\nexists \\mathbb{A}: \\aleph_0 < |\\mathbb{A}| < 2^{\\aleph_0}.", "b71ab2e674f64393001d115ddb18d797": "\\min_{\\boldsymbol{w}\\in\\boldsymbol{W}, t>0}\\{t\\ln M_{G(\\boldsymbol{w},\\boldsymbol{\\psi})}(t^{-1})-t\\ln\\alpha\\}.\\,", "b71ac9e048a0d5a9b6093426bfff9dd9": " N_{MSY}", "b71afe59bc8e8e7a8216215075ed6a5f": "\\lim_{n\\to\\infty} \\left( 1 + \\frac{1}{n} \\right)^n", "b71b05c43288664471b251106e9bb6b5": "T_i \\downarrow", "b71b0c9abcfe420e111b2e640fc826af": "f(x)=ax+b", "b71b51656a17aef177b8e01a29f35b28": "= \\tan^2\\left(\\frac{\\alpha}{2}\\right)", "b71b8b9192ec2c95b40933a2b31dde4c": "p(\\lambda)", "b71bb38f7710ac70019012651874209a": "\\rho^{(k)} :(\\mathbb{R}^d)^k \\to [0,\\infty) ", "b71bda47d1fff20e9d74e86c61cef209": "f(x):= \\ln x-\\ln(1-x) = \\ln\\frac x{1-x},\\qquad x\\in(0,\\tfrac12].", "b71bdf4ba72f9af42fe853bf57e818f9": "1729 = 7 \\cdot 13 \\cdot 19 \\qquad (6 \\mid 1728;\\quad 12 \\mid 1728;\\quad 18 \\mid 1728)", "b71c45d07337b82ea264ff4e76f311d2": "\\begin{align}\n{\\mathfrak{T}_{\\mu}^{\\nu}}_{; \\nu} \\, + \\, f_{\\mu} \\, & = \\, - \\frac{1}{\\mu_0} ( F_{\\mu \\alpha ; \\nu} g^{\\alpha \\beta} F_{\\beta \\gamma} g^{\\gamma \\nu} \\, + \\, F_{\\mu \\alpha} g^{\\alpha \\beta} F_{\\beta \\gamma ; \\nu} g^{\\gamma \\nu} \\, - \\, \\frac12 \\delta_{\\mu}^{\\nu} \\, F_{\\sigma \\alpha ; \\nu} g^{\\alpha \\beta} F_{\\beta \\rho} g^{\\rho \\sigma} ) \\sqrt{- g} \\\\\n& + \\frac{1}{\\mu_{0}} \\, F_{\\mu \\alpha} \\, g^{\\alpha \\beta} \\, F_{\\beta \\gamma ; \\nu} \\, g^{\\gamma \\nu} \\, \\sqrt{-g} \\\\\n& = \\, - \\frac{1}{\\mu_0} ( F_{\\mu \\alpha ; \\nu} F^{\\alpha \\nu} \\, - \\, \\frac12 F_{\\sigma \\alpha ; \\mu} F^{\\alpha \\sigma} ) \\sqrt{- g}\\\\\n& = \\, - \\frac{1}{\\mu_0} ( (- F_{\\nu \\mu ; \\alpha} - F_{\\alpha \\nu ; \\mu}) F^{\\alpha \\nu} \\, - \\, \\frac12 F_{\\sigma \\alpha ; \\mu} F^{\\alpha \\sigma} ) \\sqrt{- g} \\\\\n& = \\, - \\frac{1}{\\mu_0} ( F_{\\mu \\nu ; \\alpha} F^{\\alpha \\nu} - F_{\\alpha \\nu ; \\mu} F^{\\alpha \\nu} \\, + \\, \\frac12 F_{\\sigma \\alpha ; \\mu} F^{\\sigma \\alpha} ) \\sqrt{- g} \\\\\n& = \\, - \\frac{1}{\\mu_0} ( F_{\\mu \\alpha ; \\nu} F^{\\nu \\alpha} - \\frac12 F_{\\alpha \\nu ; \\mu} F^{\\alpha \\nu} ) \\sqrt{- g} \\\\\n& = \\, - \\frac{1}{\\mu_0} (- F_{\\mu \\alpha ; \\nu} F^{\\alpha \\nu} \\, + \\, \\frac12 F_{\\sigma \\alpha ; \\mu} F^{\\alpha \\sigma} ) \\sqrt{- g} \\ ,\n\\end{align}", "b71c4fcfa773cb137f0b4a75eb25926a": " \\operatorname{ker} f := \\{(m,m') \\in M \\times M : f(m) = f(m')\\}\\mbox{.} \\! ", "b71c79d41fa0322512bb89172bbf4074": "N=30*12=360", "b71ca241e19e8f4f887c67be97704b0c": "g_j(q_1,... , q_n, \\dot{q}_1,... , \\dot{q}_n, t) = 0, j=1,.... , k.", "b71cb44d2d56df29547726103cba3eca": "Pc = \\tfrac{3}{5} \\tfrac{2}{5} + \\tfrac{2}{5} \\tfrac{3}{5}", "b71cb4edef0ab1120b64437aeed2ae74": "\\delta S = \\frac{\\delta Q}{T}.", "b71cd5bd7478f766b5d09c8edfc3ac76": "\\mathrm{sys}^2 \\le \\frac{\\pi}{2}\\cdot\\mathrm{area}", "b71d0cdf4bffb48011728f5b706a2566": "\n \\sigma_{11} = 2C_1\\left(\\lambda^2 - \\cfrac{1}{\\lambda^2}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right] ~;~~ \\sigma_{22} = 0 ~;~~ \\sigma_{33} = 2C_1\\left(1 - \\cfrac{1}{\\lambda^2}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right]~.\n ", "b71d15cfd6c3626ebd6e68931993b5b4": " x \\mapsto e^{ix} , ", "b71d1fd4eaedb49ce92134a432d9e93b": "B_n(X)=\\operatorname{im} (\\partial_{n+1})", "b71d93df6fd82aa52133bb049b6e246b": " \\hbar ", "b71dbd5cfef4ff47b25b2b5de7031983": "\\sigma \\sim \\exp (-\\overline{\\mathcal{R}}_{nn})", "b71df392b5551e4809e8d8a1d1ee491e": "g\\in U_G\\subset G", "b71e7ee6baf044a8312fbb869ae447f1": " \\left(\\frac{L^3}{T}\\right)", "b71e82328343553d3aa52fbdfa4f3fe3": "y_n(x)=\\,x^{n}\\theta_n(1/x)\\,", "b71e90b6174a396ddf8cb6fb308d81c1": "\\mu\\,", "b71eb63c887ec22cbc52f8260c394d12": "\\pi_* F \\to \\pi_* G", "b71eb7b75a86d7a398cbea6e17345abe": "\\tilde{g}", "b71f06ac1df0b9fb998db4f18a819d65": "A_D(x; y)", "b72019e25aaa0c74f562f993ec65ee80": "\\mu = \\frac{B}{H} = \\frac{ B_0 e^{j\\left(\\omega t - \\delta \\right) }}{H_0 e^{j \\omega t}} = \\frac{B_0}{H_0}e^{-j\\delta},", "b7201d4299f25dc171f446e414f32044": "\\frac{dx}{dt}=L(x)\\cdot\\frac{\\delta E}{\\delta x}(x)+M(x)\\cdot\\frac{\\delta S}{\\delta x}(x).", "b72032b1a5937004d3a2b2c502829f00": "\\alpha, \\ \\beta", "b720b3cfe9461d6667673c0aadf83770": "1/(2\\lambda)", "b720c5b49807bded2f98e9fad2725953": "\\gamma_k = \\begin{bmatrix}0&\\sigma^k\\\\ -\\sigma^k& 0\\end{bmatrix}", "b720cf7e5332303e663b782e6a220f0a": " E(\\log \\sigma_{k+1} \\,|\\, \\sigma_k) = \\log \\sigma_k ", "b7212f63fcdb7e2774768cc7dbd675b6": " 1,1,0,1,1,0,1,1,0,1,\\ldots.", "b72188cdd8f585a00c0ff9658eb5bfa9": "T=37 \\ ^\\circ \\mathrm{C}=310 \\ \\mathrm{K}", "b7218bee342893da26184ff4615787cb": "\\widehat{\\mathbf{G}_m}", "b722b90cb6462b165d9679060468f7e3": "W_\\theta", "b723472c22ef1e77e0c6916988c57109": " {1\\over \\eta}\n \\begin{bmatrix}\n { {1\\over 2} + {\\sqrt{ 5} \\over 2 } } \\\\ 1 \n\\end{bmatrix} \n ", "b723ff4c56b767b84cd2bfcc1f3b6bd2": "\\tau_w = \\dot\\gamma_x \\mu", "b7240066b3a44e5aa21fe402eab73bce": "{\\part A_N \\over \\part z} = i \\gamma | A |^2 A = \\hat N A. ", "b7245f14c8e44535aa04e5be98acb788": "L_{\\kappa, \\omega}", "b724963291df0df7ae33f9da72fac8eb": " \\hat{f}_{+}(k,0) = -\\frac{G^{+}(k)}{K^{+}(k)}, ", "b724bb16b81305234f11d40062150690": "I_1,\\ldots,I_m", "b7258031702ba169464fd04280758f19": " Z[J] ", "b72594ebd2ca0a329d98c9af456e01fb": "P=P_f^{a}P_m^{1-a} ", "b725c8a3d9e23c3271b76b0a38db5dfe": "\\ H^{BM}_i(X)= H^{m-i}(M,M\\setminus X),", "b72614cb60a8b7ae16f7468c89d9d167": "s=[s(x)]", "b72638902951e86e34c5deaee68312d5": "1 \\cdot 2 \\cdot 3 \\cdot \\cdots \\cdot \\frac{p-1}2", "b7268300b994278950d4bcfa020b30e9": "{x^i}", "b72774d31577caded518da7b47e87844": "\\hat{x}_1 = x_1", "b7277a3bac777f05d8acc80a07ed0e19": "0\\,", "b7277dac5f51b36159d72733d307c79b": " (y(x),z(x))=\\int_a^b y(x) z(x) w(x) \\mathrm{d}x ", "b7279bc489d8cc728e1272a51856efc0": "F_d\\, =\\, \\tfrac12\\, \\rho\\, v^2\\, c_d\\, A", "b727a8c635e22f761f8c2f2860382285": "z = x + iy", "b727c8d32a0ffffc01f2d0be815fa8f4": "G = \\lbrace \nf_1 = \\lbrace \\alpha,\\beta \\rbrace,\nf_2 = \\lbrace \\beta,\\gamma \\rbrace,\nf_3 = \\lbrace \\gamma,\\delta \\rbrace,\nf_4 = \\lbrace \\delta,\\alpha \\rbrace,\nf_5 = \\lbrace \\alpha,\\gamma \\rbrace, \nf_6 = \\lbrace \\beta,\\delta \\rbrace\n\\rbrace", "b727f656afa0e0b7571f358cacd852e4": " \\operatorname{Tr}(S E) = \\langle \\psi | E | \\psi \\rangle .", "b7284ce6e294fe69458cebca16517548": "\\text{maximum speedup } \\le \\frac{5}{1 + 0.75 \\cdot (5 - 1)} = 1.25", "b72876f25f0d187508e9b29062584264": "T_W[n]=\\frac{1}{8}\\frac{\\hbar^2}{m}\\int\\frac{|\\nabla n(\\vec{r})|^2}{n(\\vec{r})}d^3r.\\ ", "b728784a4800d2ad560fa4a5e3f9b84b": "\\pi=\\pi(x_1,x_n)=\\{x_1,x_2,...,x_n\\}", "b728786dc3d5aa549c373aa07176d4e3": "b(s)", "b728811f76420a5fb05c289be1e35261": "\\eta_k(s) = \\frac{k \\mu}{k \\mu + s + \\lambda-\\lambda \\eta_{k+1}(s)}.", "b728941c7f7d1c1b9a4062a67e772bb5": "\\mathbf{\\hat u_\\theta}", "b728c7e0603f2df0840946aea0bcd9fa": " n(\\vec r)", "b728d097f878c2ed00d20d675a57692d": "\\scriptstyle \\frac{1}{P} \\hat s\\left(\\frac{k}{P}\\right).", "b728ef24813aefab19c9da23433ec533": "x_n \\in \\mathbb{R}", "b7294e6f9054b856ac4d0f7744053b00": "T_{\\rm Kep}=2\\pi\\sqrt{a^3/GM}", "b72957dfc2d6c2b26b5db9240577efa5": "\n\\operatorname{dCov}^2_n(X, Y; \\alpha):= \\frac{1}{n^2}\\sum_{k,\\ell}A_{k,\\ell}\\,B_{k,\\ell}.\n", "b72983103ecab73d3a85f4bdee25707b": "G_{c,v,w}(p) = F_{c,v}(p + w);\\,", "b729e39538482b6f45945cca3cb90246": "\\left[A_i ,A_j\\right] = \\varepsilon_{ijk}A_k\\,,\\quad \\left[B_i ,B_j\\right] = \\varepsilon_{ijk}B_k\\,,\\quad \\left[A_i ,B_j\\right] = 0\\,,", "b729ef86edcc88f21d4386199efaa425": "P = \\left ( \\frac {10.0 m} {1 ~ \\mbox {kg}} + \\frac {6.25 h} {1 ~ \\mbox {cm}} - \\frac {5.0 a} {1 ~ \\mbox {year}} + s \\right ) \\frac {\\mbox {kcal}} {\\mbox {day}}", "b72a0307b92aca8a2ae3a7cc16377f24": " G'' = \\frac {\\sigma_0} {\\varepsilon_0} \\sin \\phi ", "b72a2389eed9c6013a6224b8a5ab5a46": "N = \\tau R_{ext} / (-\\alpha)", "b72a44322f7f1f0ff4b306d4ae1bc080": "\\prod _x ax+b = C\\, a^x \\Gamma \\left(x+\\frac{b}{a}\\right) \\,", "b72a8eea71d8d942c27dc2976dc87741": "B = \\frac{u \\rho}{\\mu ( 1-\\epsilon) D}", "b72aea9f6f6668f8117f4547cf09e24a": "\\left.\\frac{\\partial}{\\partial x_1} f(\\boldsymbol{x})\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}\\,,\\quad \\left.\\frac{\\partial}{\\partial x_2} f(\\boldsymbol{x})\\right|_{\\boldsymbol{x}=\\boldsymbol{a}}\\,,\\ldots, \\left.\\frac{\\partial}{\\partial x_n} f(\\boldsymbol{x})\\right|_{\\boldsymbol{x}=\\boldsymbol{a}} ", "b72b0fb9cb61833669b42cc49a667434": "l(b - 1)^2 \\ge (b / (b - 1))^{l - 1}", "b72b2674ebb6be0e6445bf811693b115": "{Q(X) \\over E(X)} = P (X)", "b72b444aff4d04735e0f59d65848223b": "V_y < V_x", "b72b60c2096a9c73b059b403e18f7fa9": "\\beta \\approx \\sqrt \\frac{\\omega CR}{2}", "b72b930c9ba7c3d97833a6529a3505bf": "(n-1)!\\ \\bmod\\ n", "b72b9733fee316a48709941d483b7fd4": "\nH_D \\left( {C\\cap \\left( {a,b} \\right)} \\right)+H_D \\left( {C\\cap \\left( \n{b,c} \\right)} \\right)=H_D \\left( {C\\cap \\left( {a,c} \\right)} \\right), ", "b72bb56ac811da450bd58b2be5fe3b10": "\\displaystyle{AF=T\\mu F,\\,\\,\\,\\, BF =\\mu TF}", "b72bb92668acc30b4474caff40274044": "\\mu ", "b72bfa47bf19f5433d3645bdcf77d039": "e\\approx \\gamma\n-4\\gamma^{3/2}/\\pi", "b72c0f9fe37953fe7433edf622c2eb8e": "x\\in O", "b72c3ba1914157cf7e988597d49d7181": "\\frac{\\partial u}{\\partial t}=\\frac{\\partial u}{\\partial x} \\frac{\\partial x}{\\partial t}+\\frac{\\partial u}{\\partial y} \\frac{\\partial y}{\\partial t} = \\left(2x\\right) \\cdot \\left(r \\cos(t)\\right)+\\left(2\\right) \\left(2\\sin(t) \\cos(t)\\right)", "b72c5f6d7870b9bed62c86bd79e30acb": "f_*", "b72c7a2e21307c6027efe236c89d0fd5": "d.f. \\cong \\frac{(N+1)(N-2n)}{2(N-n)}", "b72cac1453b7b00006e2b072179ba301": " Q = - \\frac{(1+k)} {2r} ", "b72cbad2e091cd742eebd82128199388": " y\\,'(x) = \\frac{dy}{dx} \\, , \\ \\ y_1=f(x_1) \\, , \\ \\ y_2=f(x_2) \\, . ", "b72cc7a41abc7de384e5745289176ac4": "\\widehat{\\theta}(X)=\\frac{n\\overline{X}+a}{n+\\frac{1}{b}}.", "b72d5458dee2dce176eea36016ee8001": "p(n,t)", "b72e155ce74e0c0af2af9ca0491cbb8c": " P_A O_2 = F_I O_2(PB-P H_2 0) - P_A CO_2 (F_I O_2 + \\frac{1-FIO2}{R})", "b72e84e6af2a4e193e3f11d4d2c1c56b": "\\ \\displaystyle \\hat{\\alpha} \\ ", "b72ebeb95f4bb3e84afd94141a57b13b": "z_{i}", "b72ed2122ce1c7093864cd52faec5548": "\\tau_{\\{c\\}}((a+b)\\cdot c) = (a + b) \\cdot \\tau", "b72f04b9556c1fc82cb2c9cbfe13a84c": "X=\\mu\\otimes1_{1\\times N}+LF+\\epsilon", "b72f04dc7652f905c471e9a2d38b1695": " \\{a_1,\\ldots,a_n\\} \\mapsto \\langle\\!\\langle a_1, a_2, ... , a_n \\rangle\\!\\rangle ,", "b72f68beddd78a277cbdd2d974e2fc00": "\\frac{1}{12n}", "b72ff1865a1c435377cc456a7d19201b": "\\frac{1}{R} = \\limsup_{n \\to \\infty} \\big( | c_{n} |^{1/n} \\big)", "b72ffbc8d1ea730b63e2c15237aa7dca": "\n \\boldsymbol{\\mathsf{I}}^{(s)} = \\frac{1}{2}~(\\delta_{ik}~\\delta_{jl} + \\delta_{il}~\\delta_{jk})\n ~\\mathbf{e}_i\\otimes\\mathbf{e}_j\\otimes\\mathbf{e}_k\\otimes\\mathbf{e}_l\n", "b73003a67a9aaebf48ca825d031767c3": "A I_n-A", "b73043195bf477a064e5db88cbc00fba": "B_0 \\,=\\, \\frac{\\mu_0}{2}M", "b730592e311d7960a9b65120a07221c7": "\n \\gamma=\n \\begin{cases}\n 1 & \\mbox{if }p\\ne 0 \\\\\n 1/2 & \\mbox{if }p=0\n \\end{cases}\n ", "b7306b4a63cb774969d761dfb6b1f38d": "\\left(x',y'\\right)", "b7307dd6be83caee53dcefe74c204455": "\\overline{W}", "b730895dd0f031e2bbdccef16cd5432e": "B \\to K \\to K_i", "b7312144c3abb3480e6a46ae8ffc4cab": " \\vec{y}", "b7313cb06f4e36f2fd97b7f74013ecd6": "R_{4,1} = 84 r^4-168 r^3+105 r^2-20 r", "b73149ba47df47463f4674816df0122c": "S = M_S * x + v_S", "b73156ede6517270f8229a4ffe7da24d": "\\lim_{x \\rightarrow \\infty} \\left ( \\frac {1}{x} \\cdot \\#\\left\\{ n \\leq x : a \\le \\frac{\\omega(n) - \\log \\log n}{\\sqrt{\\log \\log n}} \\le b \\right\\} \\right ) = \\Phi(a,b) ", "b731647d0db1474aadeaca9c50c53abd": "x_{m+1}(z) = x_{m-1}(z) + a_m\\cdot(2\\cdot m+1)\\cdot(z+z^{-1}) \\cdot z^{(-1)^m} \\cdot x_{m}(z)", "b7317a8504f7a16c22f1eddbca314404": " {\\rm Tr}(A) = \\sum_{n=0}^\\infty \\langle A e_n , e_n \\rangle = \\sum (\\{ \\langle A e_n , e_n \\rangle \\}_{n=0}^\\infty )", "b7317b5cb95ffcb1bc74dd5624afd461": "(a_n) \\,", "b731ef3659d797633eb61fdd20c387d1": "\n{\\hat{\\beta}}(q) = \\min \\{ \\alpha: \\ \\mbox{sweeping success is possible}\\}\n", "b732129839d9486ee52896d389adaa49": "J(y)=\\int_a^b yy' dx.", "b732143e7dcffcd5507ff74d79f6dfed": "|x| = \\begin{cases}\n -x, & \\mbox{if } x < 0 \\\\\n x, & \\mbox{if } x \\ge 0 \n\\end{cases}\n", "b732241bde1398a427d44c562abb8296": "\\epsilon_v=\\pi n\n\\frac{\\hbar^2}{m}\\ln\\left(1.464\\frac{b}{\\xi}\\right)", "b73298ed357c4076e95d6848e01f3f3b": "f(\\boldsymbol\\mu,\\boldsymbol\\Sigma|\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi,\\nu) = \\mathcal{N}(\\boldsymbol\\mu|\\boldsymbol\\mu_0,\\tfrac{1}{\\lambda}\\boldsymbol\\Sigma)\\ \\mathcal{W}^{-1}(\\boldsymbol\\Sigma|\\boldsymbol\\Psi,\\nu)", "b7329efb5304cda3e6fc4c595e23c083": "s_{n-1} = \\sqrt{\\frac{1}{n-1} \\sum_{i=1}^n (X_i - \\overline{X})^2}.", "b732c3021be10c52ecd557d3384dfb4f": "\nx = k_x \\cdot \\frac{I_b - I_d}{I_b + I_d} \n", "b732d58bed04059f3855a888dd72667f": " SubCipher_1=ENC_{f_1}(k_{f_1},P)", "b732eb1d255985e8e456541c7978c214": "\\scriptstyle -\\frac{df(E)}{dE} \\approx \\tfrac{1}{kT} e^{-(E-\\mu)/(kT)}", "b732eee7963f88bd0b2630d9f291291d": "g(\\operatorname{E}(Y))=\\beta_0 + f_1(x_1) + f_2(x_2)+ \\cdots + f_m(x_m).\\,\\!", "b7332cacd745f6cbb3f5bb6ca9009da9": "E(S, T)", "b73336f10fe3c5af96c4b2afc11b84ac": "\\begin{matrix} {12 \\choose 1}{4 \\choose 4}{44 \\choose 1} \\end{matrix}", "b7334379d9d9cae4fc2f7e18f5ab2a0f": "R\\equiv7 \\pmod {13}.", "b73371cf71ad52dbe7533f3e98c3ee85": "v(t) =\\frac{gt + v_0\\gamma_0}{\\sqrt{1 + \\frac{ \\left(gt + v_0\\gamma_0\\right)^2}{c^2}}}.", "b733f0076f2768cebd2b54579a77c0d5": "\nU = \\frac{-Q_{1}\\ln \\rho^{\\prime}}{2\\pi\\epsilon} \\int \\rho d\\rho \\ \\lambda(\\rho, \\theta) \n+ \\left( \\frac{1}{2\\pi\\epsilon} \\right) \\sum_{k=1}^{\\infty} k \\left( C_{2k} I_{1k} + S_{2k} J_{1k} \\right)\n", "b7342158273bb3538120848e898d907b": "\\begin{alignat}{3}\n\\mathbf{a} & = \\frac{\\mathrm{d} \\mathbf{v}}{\\mathrm{d}t} \\\\\n & = \\frac{\\mathrm{d}v }{\\mathrm{d}t} \\mathbf{u}_\\mathrm{t} +v(t)\\frac{d \\mathbf{u}_\\mathrm{t}}{dt} \\\\\n & = \\frac{\\mathrm{d}v }{\\mathrm{d}t} \\mathbf{u}_\\mathrm{t}+ \\frac{v^2}{r}\\mathbf{u}_\\mathrm{n}\\ , \\\\\n\\end{alignat}", "b73436c83fbc627bad6e0938bf559e72": " D = Y-X ", "b73449e8c260fc11c456f2256efb5e2c": "W = ", "b7346d5afda9c24f9d1c530fc51d0c72": "i+j", "b734b80ff93e407a3953ad7303f93cdf": "\\bold{X}", "b734ca571082fd69b834b64e364ab194": "ln K=-RT \\ln \\left(\\frac{\\sum_k {a_k}^{m_k} (solution)}{a (solid)}\\right)", "b734f44b8e7fc9cc95254827bcec3488": "ABV = (Starting SG - Final SG)/7.36", "b73508f56c37c003a2803625513a9111": "s(x) \\equiv 0", "b735cc5b0975b697cc2f85de2faf96fc": "\\begin{align}\n\\delta\\mathcal{L} & = \\partial_\\mu\\left[\\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)}\\right]\\delta\\phi + \\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)} (\\partial_\\mu \\delta \\phi)\\\\\n& = \\partial_\\mu\\left[\\frac{\\partial \\mathcal{L}}{\\partial (\\partial_\\mu \\phi)}\\delta\\phi\\right] \\\\\n& = 0\n\\end{align}", "b735e0a957a11bf8ff3f8f16b9d1cb8a": "\\operatorname{RMSD}(\\hat{\\theta}) = \\sqrt{\\operatorname{MSE}(\\hat{\\theta})} = \\sqrt{\\operatorname{E}((\\hat{\\theta}-\\theta)^2)}.", "b735f21818ebe03d9a1f376c65c96756": " B=-1 ", "b7371bdea1e1da12ac7bf20ecb3ce940": "\\pi - \\arctan{\\left(\\frac{Z_o\\Omega}{(1+R)}\\right)}", "b73727739d8874f04d7325d7d6e4bfc6": "\\det(A) = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) A_{\\sigma(1),1}A_{\\sigma(2),2}\\cdots A_{\\sigma(n),n},", "b7372f1fb70474f7233efe8f7a92fa6d": "MA = \\frac{M_{2}}{M_{1}} = \\frac{a}{b}.\\!", "b737679c7fa91766b1cff05b6ece61e6": "s<1", "b737840eb757d695a9a0a4ba0d83a27c": "|b-a| < \\delta", "b7378d95b623bf6259d186043315af1e": "f\\in L^p", "b737ae12bc87ecb2514594d9ab0f4951": " \\left[ \\begin{matrix} 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & \\cos(\\theta) & -\\sin(\\theta) \\\\\n 0 & 0 & \\sin(\\theta) & \\cos(\\theta) \\end{matrix} \\right] ", "b737b79143b2a605a59b766cac00f682": "B_m(x)=\n\\sum_{n=0}^m \\frac{1}{n+1}\n\\sum_{k=0}^n (-1)^k {n \\choose k} (x+k)^m.", "b73888927dce68abd96c23864ae9db79": "Coverage > Readiness", "b73895d1388919eb1718ad4722b7283d": "e\\cdot U = \\frac{1}{2}\\cdot m\\cdot v^2", "b738e5afd28bd269038d5f77ba3bcd08": "h_{\\text{in}}^2", "b738e7f3a7e38cde7618aa3f6f3cd09f": "B \\rightarrow A: \\{N_A, N_B\\}_{K_{PA}}", "b738fb4e0d8e58506249303b6723a878": "\\sum_{j=1}^n w_{ij} x_{ij} \\le w_i \\qquad i=1, \\ldots, m", "b738fdd36c372c425656120a55360015": "(\\mathbf{C}, \\otimes, I)", "b7394fe68a1b8d395eebf06e9becf1ae": "\\tfrac{1}{2}\\left(\\tfrac{a}{c} + \\tfrac{b}{d} \\right ) -\\tfrac{3 i}{2\\sqrt{3}} \\left (\\tfrac{b}{d} - \\tfrac{a}{c} \\right )", "b739750c4075307803b83ccdf9ca8d0e": "g_{k\\times A}(t) = k\\cdot g_{A}(t).", "b73980c5c4b201a342ea02702357870e": "\\begin{align}\n\\chi(\\phi)&=2\\tan^{-1}\\left[\n\\left(\\frac{1+\\sin\\phi}{1-\\sin\\phi}\\right)\n\\left(\\frac{1-e\\sin\\phi}{1+e\\sin\\phi}\\right)^{\\!\\textit{e}}\n\\;\\right]^{1/2}\n-\\frac{\\pi}{2}\\\\[2ex]\n&=2\\tan^{-1}\\left[\n\\tan\\left(\\frac{\\phi}{2}+\\frac{\\pi}{4}\\right)\n\\left(\\frac{1-e\\sin\\phi}{1+e\\sin\\phi}\\right)^{\\!\\textit{e}/2}\n\\;\\right]\n-\\frac{\\pi}{2}\\\\\n&=\\sin^{-1}\\left[\\tanh\\left(\\tanh^{-1}(\\sin\\phi) -e\\tanh^{-1}(e\\sin\\phi)\\right)\\right]\\\\\n&=\\mathrm{gd}\\left[\\mathrm{gd}^{-1}(\\phi)-e\\tanh^{-1}(e\\sin\\phi)\\right].\n\n\\;\\!\\end{align}", "b73a40f6f4400a50f18d20fa9bf8170c": "\\sqrt{x}=1+\\cfrac{x-1}{2 + \\cfrac{x-1}{2 + \\cfrac{x-1}{2+{\\ddots}}}}", "b73b3f5e95ca478727d6475fd0f8e582": "E=A\\oplus A^*", "b73b42665dec4ea9a47e90cf6eeec8d3": "T_\\max = \\frac{ { \\tau }_\\max J_{zz} }{r}", "b73b539f0e5814395b409a2654cb692f": "\\mathbf{J}\\cdot\\mathbf{E}", "b73b7f5df4eaaa09ad8f9dc93909e7c1": "e_g", "b73b99ca8f07ad5fa9218decf912e3d2": "A = Q_2 x_2 \\cdots Q_n x_n \\phi(0, x_2, \\dots, x_n),", "b73bbbba27cb4dc47632c5f3e2e0a210": "\\text{ where }z^\\star = [3 + \\eta - (\\eta^2 + 2\\eta + 5)^{1/2}]/(2\\eta)", "b73bbd75e8042393528f0a4f58accef2": "\\Diamond p \\rightarrow \\Diamond \\Diamond p", "b73bf34b66c7e1be5ba8848482fda0f3": " \\varrho(z) ", "b73c3280b6f85a6ac520af103083f535": "t=1", "b73c48c15ff51e2d2ede12ba616344cf": "t=b", "b73d6e2880555cdd41bd45dcae9e1dbd": "u_f", "b73d6f9a40e17771bddb987c60b955e3": "x ^ {10}\\,", "b73d7c47421efc8fa77c53f98179b711": "\\in\\lceil A", "b73d8a0a08742e015561b75f06a6120e": " FV^{-1} ", "b73db75ec9c85e450950c6e7e72d4881": "\\exp_{10}^4(1)", "b73df68ddda6a05486b79fe45b490a7b": "R\\left(X,Y\\right) = \\left[X_H,Y_H\\right]_V", "b73e0d3df83d0d0be91a1bdf7a7705a0": "\\alpha_{t}|\\alpha_{t-1} \\sim N(\\alpha_{t-1},\\delta^2 I)", "b73e2d23fd3e793fb3748def89e736c6": "\\displaystyle ((a^2+b^2-c^2)(a^2-b^2+c^2) : (a^2+b^2-c^2)(-a^2+b^2+c^2) : (a^2-b^2+c^2)(-a^2+b^2+c^2)).", "b73e8d0e56cb94ca0948f5d134a3c521": "\\mathrm{Volume% \\ O_2 \\ in \\ propane \\ combustion \\ gas} = -0.1208(\\mathrm{% \\ excess \\ O_2})^2 + 0.186(\\mathrm{% \\ excess \\ O_2}) ", "b73ec8a7ba13ba6e500eb4e292703b97": "2G(1+\\nu)\\,", "b73efa24bcd9b42693aa5fd36cf84194": "j=1,..., n", "b73f24e6c15ca11ecef3aee820e02186": "\\Delta E = \\frac{E_{n}(Z\\alpha)^{2}}{n}\\left( \\frac{1}{j + 1/2} - \\frac{3}{4n} \\right)\\,,", "b73f2cc8b1f234f512bfa68fcedcd569": "n=0\\,", "b73f381364977f102ba6e685517e26e9": "-\\hbar k_{max}/m", "b73f4c8f6e8498ade871c984d59b62d4": "g \\circ t", "b73f56bb40d07add30b93d39c1369f44": "Ax=-x',~~~D(A)=\\left\\{x\\in X: x\\text{ absolutely continuous }, x'\\in L^2(0,1)\\text{ and }x(0)=0\\right\\}.", "b73f6935a955b8d459ba0e35b06e69c5": "\\int_{s_1}^{s_2}\\left|\\frac{dR}{ds}\\right| ds.", "b73fbefe34c93dac39a1da2141cf7a63": "|V_i|=\\lfloor n/b\\rfloor", "b73fcf47c7e3902e303c52abfa954ccd": "(1 - X)^{-1} = \\sum_{n=0}^\\infty X^n.", "b7402da51686f2cfa8d6d28927e966cd": "\\stackrel{\\circ}{S}", "b7402de0bffac15010302e42486bdcf7": "\\lnot \\phi \\in \\Phi", "b74093923941d33ee19becc5f4b48b25": "[0,T]", "b74099ac29f6340841ebacb79702ffcb": "0.09091 \\times 1,000 = 90.91", "b740d224af0df8454cc9c5d50e5eb93a": "\nG = \\frac{\\bar{Y}-Y_\\min}{s}\n", "b7414ac71042f4f99b0bd4dba57f624b": "\\mathrm{CINT}_x", "b74200cc012602fbf74dd969d36fd7e8": "\\frac{16}{5}\\,\\sqrt{\\frac{2}{7\\pi}}", "b7421fe6ee1ced892d47268225e02c0d": "\\mathbf y'(x) = A(x)\\mathbf y(x)", "b742409bb408cc4bdfa7db9d4386ee14": "\\,\\!\\nu", "b742409f5163b83a3cc2903aa7030e14": "Nil\\ M\\ N = N", "b74273098d76aecf6d1bae2220d22e51": "f\\in\\mathbb K(\\mathfrak g^*)^{Ad(G)}. \\, ", "b742781c7807e2c86d3de001bfc33e49": "R_{in} = -g_m \\omega ^ 2 L_1 L_2", "b7429252ea18065a8bc4dae2e48901b9": "A_1 = 1", "b742b4b16842311b0be58bff47f19fd4": " r_2 = r_m+A \\ ", "b742cb32a5cb3d1ec3579ac659469715": "z=c\\,", "b742d361c0f1764ef8b628e38c9babb7": "Q^{\\mathrm{path}\\,P_0,\\, \\mathrm{reversible}}_{A\\to B}", "b7437009fef1da3857dacbc1b61ec79f": " C:\\Lambda^2\\ni\\bold{F}\\mapsto \\bold{G}\\in\\Lambda^{(4-2)}", "b743c44ba997bde8305c1aa937e58dc5": "\\operatorname{P}\\!\\left( A| \\{\\emptyset,\\Omega\\} \\right) \\equiv\\operatorname{P}(A).", "b7446546dd11ad75bf320f60e546bba1": "|B:A|", "b744ae1c3eaa1d70cc905799a540117b": "H(\\omega_1)", "b74514f71318ae36a0d0df15daf8da2e": " \\nabla^2 \\varphi =-\\dfrac{\\rho}{\\epsilon_0}", "b7453f3a1f5b0d2af2b956f83c7718b4": " S_1, S_2, \\ldots, S_m \\subseteq [n]", "b745b0aa00289d4b946077f926eb4fdd": "\\tilde{\\boldsymbol{a}} = \\sum_{k=1}^{d} \\dot v_k\\boldsymbol{e_k} \\ .", "b746438cbfcbc1d8fbc373654aa2e1f2": "Z_T = 17.5 + j32.7 \\ \\Omega\\,", "b7465d1c23024768d3df2fbdb60cd44f": "H(f)\\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{cases}1 & |f| < B \\\\ 0 & |f| > f_s - B. \\end{cases}", "b7470ad6919bcca25914e399934e9904": "v_{min} = 2\\pi a^2 \\frac{\\sqrt{1-e^2}}{T a (1+e)}", "b747491a0c06001ba8283559e87247d0": "\n P_q = \\int_0^a \\int_{-b/2}^{b/2}q(x,y)\\, w(x,y)\\,\\text{d}x\\text{d}y \\,.\n", "b747939038108b08cd7b34797107cbda": " \\left[ \\text{d}_t^2 - (u\\Delta t/\\Delta x)^2 \\text{d}_x^2 \\right] \\Psi(x,t) =0. ", "b747c78f38c764a29282a0d4b1904d3c": "\\vert\\Phi_{i_{1}\\ldots i_{n}}^{a_{1}\\ldots a_{n}}\\rangle", "b747f30dd525fbe8c7e4c3daf135993c": " = \\frac{32e^4}{4(k-k')^4} \\left( (k' \\cdot p') (k \\cdot p) + (k' \\cdot p) (k \\cdot p') \\right) \\,", "b7480c66f6e62effe269d485e3cc7996": " \\rho(x) = \\rho_+(x) - \\rho_-(x),", "b748875ae0725b3a1806722577fdcd07": " R_{ab}-\\frac{R}{2} g_{ab}=0", "b748aa8cccd5e37a9687f7550d0d06f8": "\\forall t\\in T: |t^\\bullet|=|{}^\\bullet t|=1", "b748bfc651a1c153253ade48f583f52c": " \\int_{\\Gamma} \\mathbf{v} \\cdot \\hat{\\nu}\\, d\\Gamma = \\int_\\Omega \\nabla\\cdot\\mathbf{v}\\, d\\Omega.", "b7493d04f879d8d6f518d826869fc944": "H=G=\\left|2A-1 \\right|=\\left|1-2B \\right| \\, ", "b749733a27896c2e23e84b729583444d": "\\begin{cases}\n\\dot{\\mathbf{x}} = f_x(\\underbrace{\\mathbf{0}}_{\\mathbf{x}}) + ( g_x(\\underbrace{\\mathbf{0}}_{\\mathbf{x}}) )(\\underbrace{0}_{z_1}) = 0 + ( g_x(\\mathbf{0}) )(0) = \\mathbf{0} & \\quad \\text{ (i.e., } \\mathbf{x} = \\mathbf{0} \\text{ is stationary)}\\\\\n\\dot{z}_1 = \\overbrace{0}^{u_1} & \\quad \\text{ (i.e., } z_1 = 0 \\text{ is stationary)}\n\\end{cases}", "b74adb4033c8efbb6964af11715749c0": "U^\\mu \\,", "b74ae2e5325187384712c448c0b81b89": "O(K/N)", "b74b19343e85a50417844d5c0c087e73": "d = 2\\pi / |\\mathbf{g}_{\\ell m n}|", "b74b19baede16d1062a997509b052e2f": "f_{\\pi^{\\pm}} = 130.41 \\pm 0.03 \\pm 0.20~\\mbox{MeV}", "b74b1f8d50605ddf8e5b53e46a684ea6": "[0,1] \\cup [2,3]", "b74be3deed0453f0de3fe0293a2c2623": " F_0", "b74be84a4ad10414203cd3737b482d56": "W(k) \\cdot Y(k)", "b74beb6c46fe5d8da69909283cbe4eb2": "\\zeta(3) = \\sum_{k=0}^\\infty (-1)^k \\frac{P(k)}{24}\n\\frac{((2k+1)!(2k)!k!)^3}{(3k+2)!(4k+3)!^3}", "b74bfd3a1ffe715e39bc8e076dbef617": "H=E=const.", "b74cca13275d27e5578ee05ef6a5a6be": "\\textstyle |\\psi_{+} \\rangle ", "b74d00f8653f4ff4b4457652d0162146": "V(\\theta^*) = 0", "b74d5165c419544d101e2ae2b3c83de7": "\\scriptstyle L sin\\theta", "b74d5b9690449ae221829c5e182dd071": "\\varepsilon_3 = \\frac{1}{E}(\\sigma_3-\\nu(\\sigma_1+\\sigma_2))", "b74dd0149907ca969bd2722dd409e3a0": "a = 0.3", "b74e0ed6426ee76d92c00953f0776f55": "A P + U\\, ", "b74eadeba42c003d8fab5cfc8deb6e24": "NP \\to John ~|~ Susan ~|~ ...", "b74f0bc7ddb189760ea36d7b0d4bdf60": " \\frac{1}{c^2}\\frac{\\partial\\phi}{\\partial t} ", "b74f4ddd9198cb5f3ad4845acaa39f90": "{B}_j= \\sum_{n (\\ne n_0)} \\frac{2\\hbar \\ {\\rm Im}[\\mathcal{I}_{n_0n} \\ \\mathcal{F}^j_{nn_0}]}{(E_n-E_{n_0})^2} ", "b74f9e63dccf24519e653cc6613532aa": "\\mathfrak{P}^{70}", "b7504fce1dddfae4389f5b11c760b5fa": "\\sum_{m=0}^n \\tbinom m k = \\tbinom {n+1}{k+1}\\,.", "b7509c7d3c658df14df71ae3ab93c0bf": " | \\psi_{S}(t) \\rang = e^{-i H_{ S} ~t / \\hbar} | \\psi_{S}(0) \\rang ", "b75104fa8d57db818147f3e20937cb21": "85\\frac{1}{3}", "b7518302a60dadc4beff26128fab8bb9": "\\hat{\\mathbf{L}}_\\mathrm{GR}=p^\\alpha\\frac{\\partial}{\\partial x^\\alpha}-\\Gamma^\\alpha{}_{\\beta\\gamma}p^\\beta p^\\gamma\\frac{\\partial}{\\partial p^\\alpha},", "b7518db9190998212d2a5c3bd08ac24a": "\\pi_1(U_1\\cap U_2, x_0)", "b751a4d9ab4423916c5b4443262750cb": "n_{2, t} = \\frac{s_1}{\\lambda}n_{1, t} = \\frac{s_0s_1}{\\lambda^2}n_{0, t}. ", "b751c8ce619dc6e565cb921859939593": "V_{\\text{EB}}", "b751ca8d9ecd6da46521d8e2a1fda6f3": "\\min_{1 \\leq i \\leq k} \\frac{\\sum_{w \\in V} f_i(s_i,w)}{d_i}", "b751d0fd8d2020bea3840b401adcdf14": "\n \\begin{align}\n R'_c = \\tfrac{3}{\\sqrt{6}}\\mbox{m} \\\\\n L'_s = \\tfrac{1}{\\sqrt{6}}\\mbox{m} \\\\\n \\\\\n \\end{align}\n", "b75256fc7ca348e6a7ee60a1fb23b398": "\\{|{i\\alpha}\\rangle\\}", "b752abe6bedbf4873ab314a686f9c73a": "\\frac{\\partial}{\\partial x} \\left[ T_{xx} \\frac{\\partial h}{\\partial x} \\right] + \\frac{\\partial}{\\partial y} \\left[ T_{yy} \\frac{\\partial h}{\\partial y} \\right] = S_{c} \\frac{\\partial h}{\\partial t} - q - L_\\mathrm{above} + L_\\mathrm{below}", "b753404f3901f0920868e553b66de80a": "i = 1...N", "b75357e348ca95f84bdd7601de6ace35": "Q_{-\\frac12}(z)=\\sqrt{\\frac{2}{1+z}}K\\left(\\sqrt{\\frac{2}{1+z}}\\right)", "b7539d8f48588ea602543f6ef3ae75ce": "\\prod_{k=0}^{n-1} (1+q^kt)=\\sum_{k=0}^n q^{k(k-1)/2} \n{n \\choose k}_q t^k ", "b753c7afcf3edb2d098b1e79ce94ede2": "\\quad P(x,D) u (x) = \n\\frac{1}{(2 \\pi)^n} \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} e^{i (x - y) \\xi} P(x,\\xi) u(y) \\, dy \\, d\\xi ", "b75424e2fe5cf22853fae8a5fdc10e5f": "\\tau = \\frac{{N - 1}}{L} \\cdot \\sum\\limits_{i = 1}^N {\\left( {\\frac{{L_i }}{{S_i }} - 1} \\right)}", "b7546a0c1eccddb863c5ffe950fb3191": "T = T_m", "b7546f4a65b346102f3c004e5ccb6a3f": "m=\\frac{4}{3} \\cdot \\frac{h \\, \\varepsilon_0}{c^2}", "b7548848db6772a8861b4eb838354657": "V_{\\rm P-P}", "b754e158a268d9af71619b9cbec22494": " \\overline \\sigma = \\sigma (\\overline \\varepsilon) = E\\overline \\varepsilon = \\frac{\\overline P}{A} ", "b754eb39a857ecdf410e517c538b4d28": " \\text{IQE} = \n\\frac{\\text{EQE}}{\\text{Total Absorption}}= \n\\frac{\\text{EQE}}{\\text{1-Reflection-Transmission}}\n", "b755302272b91203da4c3344de99babc": "| \\alpha \\rangle \\langle \\beta|", "b75539b1a15a036ac0080778d6e9d040": "p_0 = S\\rightarrow ABC", "b75554f3af7d5be9e89c4731ef146610": "\n \\begin{align}\n & \\frac{\\partial \\sigma_{rr}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{r\\theta}}{\\partial \\theta} + \\frac{\\partial \\sigma_{rz}}{\\partial z} + \\cfrac{1}{r}(\\sigma_{rr}-\\sigma_{\\theta\\theta}) + F_r = \\rho~\\frac{\\partial^2 u_r}{\\partial t^2} \\\\\n & \\frac{\\partial \\sigma_{r\\theta}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{\\theta\\theta}}{\\partial \\theta} + \\frac{\\partial \\sigma_{\\theta z}}{\\partial z} + \\cfrac{2}{r}\\sigma_{r\\theta} + F_\\theta = \\rho~\\frac{\\partial^2 u_\\theta}{\\partial t^2} \\\\\n & \\frac{\\partial \\sigma_{rz}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{\\theta z}}{\\partial \\theta} + \\frac{\\partial \\sigma_{zz}}{\\partial z} + \\cfrac{1}{r}\\sigma_{rz} + F_z = \\rho~\\frac{\\partial^2 u_z}{\\partial t^2}\n \\end{align}\n", "b7556138f3d570744182637d619e0b96": "y_{t+1},...,y_{T}", "b7558d3231540219f900afeb2549b197": "F_{max} = \\mu_s F_{n}\\,", "b755b17af66a4f8f1a46b127aa567679": "\\int\\frac{\\sin^2 ax\\;\\mathrm{d}x}{\\cos^n ax} = \\frac{\\sin ax}{a(n-1)\\cos^{n-1}ax}-\\frac{1}{n-1}\\int\\frac{\\mathrm{d}x}{\\cos^{n-2}ax} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,\\!", "b755ef73b5d0b973a1c43e85cfeccb0e": "\n{\\rm Enc}(x) = (f(r), x \\oplus h(r))\n", "b7562a0ba55d51f24f25cc095b9d22ce": "\\text{Discount Yield} (%) = \\frac{\\text{Face Value} - \\text{Purchase Price}}{\\text{Face Value}} \\times \\frac{\\text{360}}{\\text{Days Till Maturity}} \\times 100[%]", "b756454b42f81793b752a2d0649b8ae8": "m(\\varphi_1)-m(\\varphi_2)", "b75648dd9eac0213faf3d0fe575301ed": "(\\mathbb Z^2,d)", "b756759cee2116ec9a6f430ad45d1281": "\\Sigma_{1,2}", "b756ad4d90a80bad1b1e43f1505bc4a2": "\\Pi_D = - {1 \\over A} \\left( \\frac{\\partial G}{\\partial x} \\right)_{T,V, A} ", "b756c45011bf4832a0dfbb3c9a0b0bb5": "R < I(X;Y) - 3\\epsilon", "b756eae735a30f3105ba58fc453b18b3": " \\frac{v_o}{a_e*l_o}", "b756f0e59190228047847567caa5efbd": "R \\leq ", "b7570c5d0ac799299396a957cc92659e": "k^a\\partial_a = \\frac{\\partial}{\\partial t}+\\Omega\\frac{\\partial}{\\partial\\phi}", "b7578e1adf02c327997583d89d8325c9": "A_{i,j} = A_{i+1,j+1} = a_{i-j}.\\ ", "b757f45717eedf2afb77a6a3557ed9cf": "\\ln \\gamma_\\pm =|z^+z^-|f^\\gamma+m\\left(\\frac{2pq}{p+q}\\right)B^\\gamma_{MX}\n+m^2\\left[2\\frac{(pq)^{3/2}}{p+q}\\right]C^\\gamma_{MX}\n", "b758017a2948b6bd97cf8eacc80f0b5e": "\\overrightarrow{C}", "b75808adcaae2efa080903b79af93dd1": "\\int_{0}^{\\infty} \\frac{1-\\cos px}{x^{2}}\\ dx=\\frac{\\pi p}{2}", "b759113b8251f2d79900a8bce517869d": "\\gamma \\in\\left(0, 1\\right)\\;", "b75a0622ce936a9b43378002a7cd549c": "\\frac{e^{-k^2/2a^2}}{a^2}", "b75a1b92802b7cf27e5b1d3343e9e61f": "v_C=\\sqrt{2gh_C}", "b75a1e0a5c83768879b096c022f493e0": " f(\\epsilon) = \\frac{1}{e^{(\\epsilon-\\mu) / (k T)} + 1} ", "b75a45535bedf1759984fee5b936c8eb": " \\hat{f}(a) = \\sum_{x \\in \\mathbb{Z}_m^n} e^{\\frac{2 \\pi i}{m} (f(x) - a \\cdot x)} ", "b75a83da5f96838941399e025d53ed55": "t_{ij}", "b75ac6208a384af96ed4d6c6d9fa36d3": "\\ [A]_t", "b75ae5009006961f01d4017ec8bb8683": "(q_i)\\neq R", "b75babe7843cacf6eb17a422073c1a6a": "(\\boldsymbol{\\varepsilon}_2 \\le \\boldsymbol{\\varepsilon} \\le \\boldsymbol{\\varepsilon}_R)", "b75c75d4f899aa30588e300171fc226d": "= \\frac{\\ln(\\textrm{GNIpc}) - \\ln(100)}{\\ln(107,721) - \\ln(100)}", "b75c8c04ea6237adbdf3899bc64c97c4": "h^{\\sigma \\sigma_j}\\,\\!", "b75c913f9ba692b2ce2603fe6f259fd4": "(ab/R)\\cdot R^2/R = ab", "b75cad744dc1bd73a837172a20ca7f09": "0 = \\left (dr+\\left ( 1+\\sqrt{\\frac{2M}{r}}\\ \\right ) \\, dt_r \\right)\\left (dr-\\left (1-\\sqrt{\\frac{2M}{r}}\\ \\right ) \\, dt_r\\right ), \\, ", "b75cb77f2f5f6d4f9f2f8ccdec97524d": "\\Theta = \\frac{L}{R} = \\left ( \\frac{d_{31}L}{t_{r}} \\right )U_x", "b75ccaa32bcaaf3c761257439988af75": "\\scriptstyle X_0 \\,=\\, \\mathbb{E}\\{ X \\}", "b75d841651a5eb18e2a974a39772dad3": " \\sum_{l=1}^n \\lambda^{l} x_l = \\sum_{k=1}^n (1-\\lambda)^{k-1} \\lambda^{n-k+1} \\sum_\\mathrm{samp} \\max ( x_{\\alpha_1}, x_{\\alpha_2}, \\dots, x_{\\alpha_k} ) ", "b75da5329299b4e2bdb9e8424d9ec23d": "{ {-q' \\text{ ln}_{q'}({1 \\over 2})} \\over {\\lambda}} \\text{ where } q' = {1 \\over {2-q}}", "b75df081dba8e17135028fd6bd00ac3b": "\\mathbb R\\times TQ", "b75e2f6cea9ebe81310b28c4e6d586a1": "y=\\sqrt{a^2+p^2}+\\frac{T_0}{2Ea}p^2+\\beta.", "b75e40e255191d74eb6fe4252a9df75a": "S(M) = M-2 ", "b75e47a600c3dd2ab1b59b7a29d24a86": "\\,K_{1A},\\ K_{2A}", "b75e86d06edb5538b61eac441c1ebba4": " u(z) = \\lim_{n\\to\\infty}u_n(z)", "b75e8eb7e3638c8facd532bac47a6c07": " k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 ", "b75eb3aa77434f4c16440e5af4170f65": "\\dot sU ", "b75ec38e3802479a2c1b1ee13a205add": "\\tau = t_1 - t_2", "b75ef0b356e92862e74603fbba6f2e5e": "\\operatorname{Spin}(3) \\cong \\operatorname{Sp}(1)", "b75f4361c1507712e443293b4aa9f979": "n!\\approx\\left(\\frac{n}{e}\n\\right)^n\\sqrt{2\\pi n},", "b75f52f90437f88652c2b04f68960e16": "\nrn+sa = 1.\n", "b75f834179259928cadcf3799eddfa3d": "f(\\mathbf{x})=\\mathcal{A}\\boxtimes_{n=1}^N \\left(\\mathbf{w}_n(x_n)\\mathbf{U}_n\\right) = \\mathcal{A}\\boxtimes_{n=1}^N \\mathbf{w'}_n(x_n),", "b75fcd4cc4f92dae11f5b9830c7eda6c": "\\dot {\\bar{r}} = \\bar{v}", "b75fd3b43e30c4ac7b032ce248f41b1b": "n^a\\partial_a=\\partial_u +U\\partial_r +X\\partial_\\varsigma+\\bar{X} \\partial_{\\bar \\varsigma}:=\\Delta\\,,", "b76035d5c2fd54a73d7398c2e0d6d904": "T^{\\nu}_{\\ \\alpha}", "b76077f766f1b01fefe8c7441e148eff": "Zn_{(s)} \\leftrightarrow Zn^{2+}_{(aq)} + 2e^-", "b7608e943795b81346c02d066ab45fbc": "S - \\sum_{i \\in A} a_i \\in V, \\quad A \\supset A_0.", "b761061fb8b5471c5d339e457730bb85": "a^2+i^2=2e^2", "b761ee46d0ce80a41707764df46d2507": "p_1\\sqcup p_2", "b761ffd32949be8a693e449aba2535c0": "b = (b_1, \\; b_2)", "b7621b374e969f02799569a4cea55976": "\\mathbb F_q", "b76225acfb742d19d9afc0670f031a49": "\\sigma_i=\\theta_N(\\sum_{j=1}^{N}w_{ij}\\oplus x_{ij})", "b7622b686c654306a5d85879c9acd15b": "A \\rightarrow S: A,\\{T_A, B, K_{AB}\\}_{K_{AS}}", "b762449c9355f2002dbbc189528ff78a": " \\mathcal Z:=\\{\\{(x,y)\\in \\R^2 \\mid y=ax^2+bx+c\\}\\cup\\{(\\infty,a)\\} \\mid a,b,c \\in \\R\\}", "b76254bceeaf0179a8c1835ba27866b2": "\\mathcal V_i=\\{V_{i,\\alpha}\\}_{\\alpha\\in\\mathcal A}", "b76270672dc3afbabd12335b076080d5": "x^2 + 3x + 1", "b762720c7d7f3d5eb21a8c911fdb85d6": "V_{s} = V_{c} + \\frac{ V_{r}} {1-\\frac{P_{1}}{P_{2}}}", "b762cd368a68a32a6e68d95be86d7247": "dv/dt", "b76357dcb234222b2d96924ba9613bdf": "\nc_{k} = \n\\int_{\\zeta_{a}}^{\\zeta_{b}} d\\zeta \\ \nc(\\zeta, \\tau=0) e^{\\zeta/2} P_{k}(\\zeta)\n", "b76419de9ed9c13cc66af389483c9ad9": "\\scriptstyle L^p(M)", "b7643fd11a7f05d1122597aea9a3270b": "R_{\\mu\\nu} = \\frac{8 \\pi G}{c^4} \\left( T_{\\mu \\nu} - \\frac {1}{2} g_{\\mu \\nu}T \\right) \\,", "b76451de6540df3062e59b59530f7d0d": "\\alpha \\approx 4.8", "b764a689149b33c8f42cbd38251ad5cb": "\\Sigma_{D,b}", "b764e05a8a02c6bb416e07c29edc87c1": "\\hat{x} = C_e A^T C_Z^{-1}(y-A\\bar{x}) + \\bar{x}.", "b76530f37a5cbc3d17ebe8df6fed402f": "m_{1}", "b76540089ab3627d47c3b420480d300c": "\\tfrac{W}{kg}\\;", "b7659385a300e99d13b2be86085c861b": "\\overline{\\theta}\\ \\stackrel{\\mathrm{def}}{=}\\ i\\theta^\\dagger\\gamma^0=-\\theta^\\perp C", "b765d94ed0ac192454c5eded90aee7d8": " ModD = \\frac{MacD}{(1+y_k/k)} ", "b76605589b0ac1078d2a797f40bc29d9": "\n p_1\\, +\\, {\\scriptstyle \\frac12}\\,\\rho\\,v_1^2\\, +\\, \\rho\\,g\\,z_1\\,\n =\\,\n p_2\\, +\\, {\\scriptstyle \\frac12}\\,\\rho\\,v_2^2\\, +\\, \\rho\\,g\\,z_2\\,\n +\\, \\Delta E,\n", "b766c5cb07662c545c79455796b6366a": "(\\textbf{A},\\textbf{B})", "b766eb43028405e2a2fcbe84d8ea45b4": "RT=\\left(P+\\frac{a}{TV_m(V_m-b)}-\\frac{c}{T^2V_m^3}\\right)(V_m-b)", "b7674808d08baa4a0b7b7c5ede627a33": "u\\left(z\\right)=\\begin{cases}\n\\frac{n}{n+1}\\frac{1}{\\pi_0}\\left[\\left(\\pi_0\\left(z-z_1\\right)+\\gamma_0^n\\right)^{1+\\left(1/n\\right)}-\\left(-\\pi_0z_1+\\gamma_0^n\\right)^{1+\\left(1/n\\right)}\\right],&z\\in\\left[0,z_1\\right]\\\\\n\\frac{\\pi_0}{2\\mu_0}\\left(z^2-z\\right)+k,&z\\in\\left[z_1,z_2\\right],\\\\\n\\frac{n}{n+1}\\frac{1}{\\pi_0}\\left[\\left(-\\pi_0\\left(z-z_2\\right)+\\gamma_0^n\\right)^{1+\\left(1/n\\right)}-\\left(-\\pi_0\\left(1-z_2\\right)+\\gamma_0^n\\right)^{1+\\left(1/n\\right)}\\right],&z\\in\\left[z_2,1\\right]\\\\\n\\end{cases}", "b767590c778a62f0dc5457f7cc53c4da": "\\sin(z), \\cos(z), \\exp(z), \\text{ and }\\exp(-z^2).", "b767b69298f626cb57f5d822017e656d": "B:= \\begin{pmatrix} a_0 & b_0 & 0 & \\cdots & 0 & 0 \\\\\nc_1 & a_1 & b_1 & \\cdots & 0 & 0 \\\\\n0 & c_2 & a_2 & \\cdots & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & a_{d-1} & b_{d-1} \\\\\n0 & 0 & 0 & \\cdots & c_d & a_d \\end{pmatrix} ,", "b767edc252dc472ebff3e08f2d9ecd35": "\\sigma_1,\\sigma_2", "b768114d72f2f9aee72720925722b83a": "\n\\Delta \\omega\\ =\\Delta \\omega_1\\ +\\ \\Delta \\omega_2\\ =\\ \\ -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ 3 \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)", "b768a3a29d00fe4c91fc0f8eda50b33b": "=1.", "b768bc23df6eaf0bf2dba78edc44dc89": "\\displaystyle{K(\\mathbf{u},\\mathbf{v}(t))=-{1\\over 2\\pi} {(\\mathbf{u}-\\mathbf{v}(t))\\cdot\\mathbf{n}(t)\\over |\\mathbf{u}-\\mathbf{v}|^2}.}", "b769048e91cf95da0825929e439101c7": "\\begin{matrix}\\frac18\\end{matrix} (35x^4-30x^2+3)\\,", "b76939a99122a1d7abf713807bb152de": "\\int_{R^n} m(\\xi) \\hat f(\\xi) e^{2\\pi i x \\cdot \\xi} d\\xi", "b7693c042f9e5697e620f9bbb0e047ce": "x_n^1=x_{n}\\left(\\alpha\\right)", "b769e9b8714eef54abb8535d5ee65151": "\\frac{x^{-\\nu-\\alpha+1}}{\\Gamma(\\alpha)}\\int_0^x (t-x)^{\\alpha-1}t^{-\\alpha-\\nu}f(t) dt ", "b76a45f3533739ddfdcf52463a9d429d": "\\sqrt{(x-c)^2+y^2} = -{1 \\over 4a} (4xc - 4a^2)", "b76ab9d91d67741f0efc49ca473ace9e": "c=e^{a+bx+cx^2+dx^3}", "b76ac747ba9bf2014a6f6cb2c6a49c11": " U =\\sqrt{ \\frac{ 2neV_1}{m}}", "b76af6019c297461898fa12f2850ca87": "P,Q", "b76bae539d137197e947b243c819831e": "X=\\{1,2,3\\}", "b76bc731b702e5b1199cdf61df983cba": "r=\\frac{D}{\\sqrt{p_1p_2q_1q_2}}", "b76c161c8af2e1802886de3cf3ad3758": "\\frac{d}{dt} \\langle \\psi | Q | \\psi \\rangle = \\frac{-1}{i \\hbar} \\langle \\psi|\\left[ H,Q \\right]|\\psi \\rangle + \\langle \\psi | \\frac{dQ}{dt} | \\psi \\rangle \\,", "b76c1b1b365888cd69210c89ffa96781": "\\ell_1 \\leq \\ell_2 \\leq \\dots \\leq \\ell_n", "b76c3bc2f3acdd2aeceda78602e8e0d1": "15\\oplus \\bar{6}\\oplus \\bar{6}", "b76c5b55fa2e2d978a7b8acfdc5a4129": " \\langle \\chi \\rangle = 2\\langle \\gamma \\rangle ", "b76c6189df4a454d0b3ae042747034e8": "\\frac{\\partial f(x,y)}{\\partial x}=0", "b76c655fc38d3166b5db8bdb7f6c1119": "F = U - H = \\sum_{\\mathbf{X}} P(\\mathbf{X}) E(\\mathbf{X}) + \\sum_{\\mathbf{X}} P(\\mathbf{X}) \\log P(\\mathbf{X}).", "b76c91fdd098eba7af96f1d44836c169": "\\{J_1^2, J_{1z}, J_2^2, J_{2z}\\}", "b76cabbf35531be4206a333f3b69b9b6": "1/(\\cos\\theta - \\sin\\theta)", "b76cac4b8f057fbb76690435f3f86d59": "\\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\\\\n&\\, \\, \\, \\, \\, + (H^e{}_{(ac);e} + H_{(a|e|}{}^e{}_{;c)})g_{bd} + (H^e{}_{(bd);e} + H_{(b|e|}{}^e{}_{;d)})g_{ac} \\\\\n&\\, \\, \\, \\, \\, - (H^e{}_{(ad);e} + H_{(a|e|}{}^e{}_{;d)})g_{bc} - (H^e{}_{(bc);e} + H_{(b|e|}{}^e{}_{;c)})g_{ad} \\\\\n&\\, \\, \\, \\, \\, -\\frac{2}{3} H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})\\end{align}", "b76cc2549eae3243eb4dc854655af38d": "E=\\frac{1}{2}mv^2", "b76ce2721ef566d2c0b3b0713ae690cc": "\\sigma_3 \\otimes \\sigma_2 ", "b76ce741ebdca551d10d1a130f0ecb5f": "g(x_1, x_2, x_3)=\\sum_{m_1=-\\infty}^\\infty h^\\mathrm{one}(m_1, x_2, x_3) \\cdot e^{i 2\\pi \\frac{m_1}{a_1} x_1}", "b76ce76ab370f2ada6af3c3e64b6d828": " \\left[ \\mathbf{M} - \\lambda\\mathbf{I} \\right] {x_1 \\choose \\theta_1} = 0 ", "b76ced1c8b28bbeaed301c8b20b5eb18": "\\theta_5", "b76cffd853c713f28192272303a2dd59": "\\scriptstyle a_{n+1} = \\frac{a_n+b_n}{2} \\quad \\quad b_{n+1} = \\sqrt{a_n b_n}", "b76d08cc3d2d6ee705362bf7acebe479": "\\mathbf{T} = \\mathbf{X} \\mathbf{W}", "b76da0fbfc5e2818b86530eddbc21fbc": " \\textstyle P+W=C_W+C_P+I \\,\\ ", "b76da133c3b0491d29fe72a4e3372ff1": "R_z(\\varphi) = \\begin{bmatrix}\\cos\\varphi & -\\sin\\varphi & 0 \\\\ \\sin\\varphi & \\cos\\varphi & 0 \\\\ 0 & 0 & 1\\end{bmatrix}.", "b76da9f6e936faecfa97483f8f7744f9": "x = a_0 + \\underset{i=1}{\\overset{3}{\\mathrm K}} ~ \\frac{1}{a_i} \\;", "b76e0dde52c1d34cc026f8dc6d1d6ee3": "F(u)", "b76efb34cc3bb3a1eb50983b97e453c0": "\\Phi = \\mu \\left(2\\left(\\frac{\\partial u}{\\partial x}\\right)^2 + 2\\left(\\frac{\\partial v}{\\partial y}\\right)^2 + 2\\left(\\frac{\\partial w}{\\partial z}\\right)^2 + \\left(\\frac{\\partial v}{\\partial x} + \\frac{\\partial u}{\\partial y}\\right)^2 + \\left(\\frac{\\partial w}{\\partial y} + \\frac{\\partial v}{\\partial z}\\right)^2 + \\left(\\frac{\\partial u}{\\partial z} + \\frac{\\partial w}{\\partial x}\\right)^2\\right) + \\lambda (\\nabla \\cdot \\mathbf{v})^2", "b76f0b6f903e78c08b499fb3020f4cc5": "v=\\sqrt{2gh}", "b76f49177d62b3bb5a1fa679b5bdcd11": "\\Omega_\\infty", "b76fb4a3c70dfc89f5f95b2bb901fd8b": "\\varphi(v, t)", "b77018b9555e5e71604dcdb6d2890492": " \\mu = 0 ~,~ \\xi = {{q-1} \\over {2-q}} ~,~ \\sigma = {1 \\over {\\lambda (2-q)}}", "b77088438db6cd390f148352fbd3628b": "f_\\theta(x) = \\underset{y}{\\operatorname{argmax}}\\ p(y|x,\\theta)", "b770b0f3dcc21e4030c2d6c6e57aac65": "R = \\left \\{(x,y)\\in\\mathbb{N} \\times\\mathbb{N} | 10.", "b77235ff9a27e24b6b7d268facb3ab29": "\\tfrac{\\lambda}{3K-\\lambda}", "b77252cbaf1fa0cfe7d9c8c5f5b750f1": "(a, b\\, and\\, c)", "b7725b58b41317e6c25a0a135f4d7113": "k_B\\,", "b77265736ac463b1cd4bd7615c742bb0": "\\frac{\\sqrt{X}}{ln X}", "b772c93f01654dfbe6bccd09897090a5": "b_n=b_n^{(k)}, k=1,2,...g", "b772fa113f545507aaa7f2729dda91e7": "\\mathsf{T}", "b7730e68c273310c006be79539bbd2f1": "B_1,\\ldots,B_k ", "b7732db01537da5f6da1ee6a977c839c": " \ny^\\prime(t) + y^{2}(t) = -1,\n", "b7733942a1f32da349dc714ebf419c87": " bc-ad=1", "b77348eeaea2f34c39d1473d57efaa8a": "\n(D X - X D) |\\psi\\rangle = \\int_x \\left[ \\left(x \\psi(x)\\right)' - x \\psi'(x) \\right] |x\\rangle = \\int_x \\psi(x) |x\\rangle = |\\psi\\rangle\n\\,", "b77357311677938e2b887b4eefeb905c": "\\tfrac{1}{2}N(N-1)", "b773d60e5f6f08d6b3ff73230b863ac4": "|W| > \\frac12", "b773e187c20e6e83b676023b2fc59638": "\\mu \\setminus \\left\\{c\\right\\}", "b773eda0ec3883f1b64c5d318a598808": "\\phi^{(Sha)}(t)= \\frac {\\sin \\pi t} {\\pi t} = \\operatorname{sinc}(t).", "b773f71275e431f6cc55ea896624c049": "z=x+yi.", "b774008ede478f4701f8a8efa2df896d": "\\left(\\operatorname{prox}_{\\gamma R}(x)\\right)_i = \\begin{cases}\nx_i-\\gamma,&x_i>\\gamma\\\\\n0,&|x_i|\\leq\\gamma\\\\\nx_i+\\gamma,&x_i<-\\gamma,\n\\end{cases}", "b7740e100f1cebfb292f2a7da1096473": "\\frac{1}{\\phi(q)}=\\sum_{k=0}^\\infty p(k) q^k", "b77444bdd3650c29c240013c1c1f4bc8": "\\,\\! m_{1}\\vec{u}_{1}+m_{2}\\vec{u}_{2}=m_{1}\\vec{v}_{1} + m_{2}\\vec{v}_{2}.", "b774a872d1446ef483f72759429edcdf": "\\zeta(n)-1", "b774bf10cc82867b13fd386c66758a7e": "\\frac{d A}{A} = (M^2 - 1) \\frac{d V}{V}", "b774cc8af8bc7db75e593ab21c44fbd2": "100\\uparrow\\uparrow 3=10^ {2 \\times 10^ {200}}", "b774f667cc193c2c20b0364183ea6a55": "X_{k-\\ell}\\,", "b775517335ea365f861ccfaa06de48bc": "u_s(x) = 1_B ", "b775b49467be5b1590a6a68557888dea": "\\displaystyle \\mu_{\\frac{1}{\\sqrt{n}} X_n}(x,y)", "b775fa00fca0b673d6aa384d1849662d": "\\gamma=\\det\\gamma_{ij}", "b77634f2e9c56b7e1140ce3dee924ce4": " SM(t,f) = \\int_{-\\infty}^\\infty \\tilde{ST}_x(t,f+\\nu) \\tilde{ST}_x^*(t,f-\\nu) P(\\nu)\\, d\\nu ", "b776b65cbe248db1aacb6a49b88002a1": " \\vec{R}_{A}^0 \\mapsto \\vec{R}_{A}^0 + \\vec{t} ", "b776c31cd8be4da5eff1621bd1239f4d": "p=4n+3", "b776e142a7d578565c9608f84ae94038": "K_{AB}", "b776e1e3e8d8ea77a06508d290c80f5e": "\nf = \\begin{bmatrix} f_1 \\\\ \\vdots \\\\ f_n \\end{bmatrix} \\in C(X).\n", "b776efbd9a31b72d991a19aca334a028": "\n p = \\frac{1}{V_0}\\left[2C\\chi \\left(1-\\tfrac{\\Gamma_0}{2}\\chi\\right) + 3D\\chi^2\\left(1 -\\tfrac{\\Gamma_0}{3}\\chi\\right) + \\dots\\right] + \\Gamma_0 E\n ", "b777aad006891c8c439b4f1361b4ab87": "I\\ nat\\ (2+1)\\ 3", "b777cc9f5ab1431ae82461b38f2ef329": "\\left\\{3,{3\\atop5}\\right\\}", "b777fe6e01a85f416ea0ed212e1f7b8d": "\\lambda_c\\,\\!", "b77866205d31791c11358bd81d2bb952": " OSIN2_i=max(INDEXA_{i},OSIN3_i) ", "b778aa74646f1b3bcf1c80c07bbf4459": "T(n) = 0.5T\\left (\\frac{n}{2}\\right )+n", "b778f61aec32802dd8dd14467ac8e173": "X_{1,2}, X_{1,3}, X_{1,4}, X_{2,3}, X_{2,4}, X_{3,4}", "b7791b2ece4c55226260092bb614d2b6": "X_0 = 10", "b7795686d2d3b9a166c0fff0b011eaef": "F_d = C_dbc \\left(\\frac {1} {2}\\rho W_m^2\\right)", "b77979b64761e74681ce4e82def6e49d": "\\phi \\equiv \\psi", "b77998926ddaee48260b460438ca2659": "\\Omega(k)\\,", "b779b8d68c22fb7c45d881df469f2fbc": "x_{n}\\left(\\alpha\\right)=\\prod_{k=0}^{n-1}\\frac{\\sin\\left(\\pi k/N+\\alpha/2N\\right)}{\\sin\\left[\\pi\\left(k+1\\right)/N-\\alpha/2N\\right]}", "b779e74d58fc27e7937acfc318c852a3": "f(z) = {1 \\over (z-1)(z-2i)}.", "b77a19aa325864d73ff45f72c4b229db": "\n \\frac{\\partial\\tilde{\\rho}}{\\partial t} +\n \\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\nabla \\cdot\\langle\\mathbf{v}\\rangle+\n \\langle\\rho\\rangle\\nabla\\cdot\\tilde{\\mathbf{v}} +\n \\nabla\\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\cdot\\langle\\mathbf{v}\\rangle+\n \\nabla\\langle\\rho\\rangle\\cdot\\tilde{\\mathbf{v}}= 0\n ", "b77a85919c3fea556130a3fe90cfdb91": "\\delta W = \\bold{F} \\cdot \\delta \\bold{r}", "b77aadc394c8bd5ece3b3a44d131f75b": " \\frac{\\lfloor k \\rfloor -a+1}{n} ", "b77ad743f8b6e07684131ba3293fdbea": "\\sum_{j=-\\infty}^{\\infty}\\left| W(2^{-j}r) \\right|^2=1, r \\in (0,\\infty).", "b77b10f92e69c6bcac1f16a13a890feb": "\\pi_0(C)", "b77b12bdba88cf791055456eb9e6e090": "\\sum_{i=1}^m\\sum_{j=1}^n p_{ij} x_{ij}.", "b77b47427f75c1c25ea44999903e8bba": "8^2 - 67\\cdot1^2 = -3", "b77b71413d8cfa002edde168c56ac77e": "f:D \\rightarrow C", "b77b9d5a3451ca811ab357dbff97a764": "\\Phi_{2n}(x) = \\Phi_n(-x)", "b77bacbae170dc5f92e2e7ce3d20694f": "\\lim_{n\\rightarrow\\infty}\\mathbb{P}(\\sup_{0\\leq t\\leq T}|H_n(t)-H(t)|>\\varepsilon)=0.", "b77bba84f5bbecbe0599ab20c917b942": " \\psi_1\\left(\\frac{1}{4}\\right) = \\pi^2 + 8K", "b77c08c715bf14dde84546b31b3d241d": "\\scriptstyle U\\left(x,y\\right)= \\min \\{ \\alpha x, \\beta y \\}", "b77c0bf8e2e7c1a83b427dd2efb60ca5": "B_l \\ (l \\ge 1)", "b77c1ee7a52e04b2b8744ef618da8229": "\\rho \\le 2", "b77c347d0475a53e31a6670f0b2032c2": "P_{k-1} = Q + A^T \\left( P_k - P_k B \\left( R + B^T P_k B \\right)^{-1} B^T P_k \\right) A", "b77c45da156e0a128c696e132e315596": " 1/3 ", "b77c5eb6fa42011ed99de67f5bc3355d": "\\partial{C}", "b77c921a5cbc5fbbf6d65cf4ff2fbbab": "{\\rm ci}(x)={\\rm Ci}(x)\\,", "b77cf59c4bca9f6e3fb4be5408511ed5": "\\lim_{\\Delta x \\to 0} x_1 = x_1", "b77d00a3f8b0b0fb6bf5194beef0b2ac": "\\begin{align}\nx[n] &= T \\int_{\\frac{1}{T}} X_{1/T}(f)\\cdot e^{i 2 \\pi f nT} df \\quad \\scriptstyle {(integral\\ over\\ any\\ interval\\ of\\ length\\ 1/T)} \\\\\n\\displaystyle &= \\frac{1}{2 \\pi}\\int_{2\\pi} X_{2\\pi}(\\omega)\\cdot e^{i \\omega n} d\\omega. \\quad \\scriptstyle {(integral\\ over\\ any\\ interval\\ of\\ length\\ 2\\pi)}\n\\end{align}", "b77d36c54af6474ea86d7134f57770b9": " \\exp\\left(- \\frac{q_j \\, \\Phi(\\mathbf{r})}{k_B T} \\right) \\approx \n1 - \\frac{q_j \\, \\Phi(\\mathbf{r})}{k_B T}", "b77d5fd2ff0d1914e30cdeec70d093a7": "\\frac{\\mathrm{A_{1}}(\\theta,\\Phi)}{\\mathrm{G_{1}}(\\theta,\\Phi)} = \\frac{A_{2}}{G_{2}}", "b77d7b187738837dec2dc3b49cd96315": "\\text{Gal}(\\mathbf{K}/\\mathbf{Q}_{p})", "b77dae16635412d3e6bd3c0b21083c55": " i,j ", "b77dd1db2d1c400523c50c7a0c65a606": "\\alpha_\\lambda\\in\\mathbb{Q}", "b77dfd3fe7aa76a515f6242d4f1860e3": "p_{i0}=1-\\sum_{j=1}^J p_{ij}", "b77e415eb0aedb34f37b4a7d861053ad": " e^{-2\\pi i m} = 1 ", "b77e8abd504f20c6bdeb7dc60fd39c2b": "g(hU)=(gh)U.", "b77ef9215f68cac632b7fb9fb85eeed9": "f_x \\triangleq \\nabla f ", "b77f864c47beeac5fd7b5fd7add0fa7e": "\\mathrm{Ca}^{\\bullet\\bullet}_\\mathrm{i}", "b77f8dd486452d072e1128ed1e3b5363": "\\hat {a} = \\hat {a_1} + j\\hat {a_2} = \\frac{\\hat {q}}{q_0} + j\\frac{\\hat {p}}{p_0} \\ ", "b77feb319bad6e57c15d82a6f8ed1669": "T_n(x) = U_n(x) - x \\, U_{n-1}(x), ", "b78011dd0059087572ef32f4b8c90463": "\\Box_i \\alpha", "b7803023ae299db155cd0355001cdcf4": "d = \\frac { \\lambda } { 2 NA }", "b780fe4052f5d9e748d22734d03a3676": "\\rho = | \\Gamma | ", "b78166506211b1c4d78912520a3db665": "\\sum_{i=1}^n d_i \\leq \\sum_{i=1}^n \\lambda_i \\qquad n=1,2,\\ldots,N", "b781a01147ad14bd52a9bddd65715c44": "P(n) \\sim F(n)\\,", "b781ba17c33ff79f408d85b817ff5ca8": "I(p_{t_n},p_{t_m},q_{t_n},q_{t_m})=\\frac{1}{I(p_{t_m},p_{t_n},q_{t_m},q_{t_n})}", "b781d1387ea6735db16b31ca99f30d46": " s(y_2-y_1)-t(y_4-y_3)=y_3-y_1 \\ .", "b781f69978bd70b3f8f984e1c3d2d38f": "f(t) = Ae^{\\lambda t}.\\ ", "b7822365e57cdadf21b84fb98e4011ea": " \\rho \\overline{ u'_i u'_j} ", "b7824a6baf48fc36762383445a974005": "\\scriptstyle a,\\,", "b78288e15cc528faca9b9323393a570a": "p_i=\\frac{(1-p_1)F}{(F-1)i(i-1)}", "b7829056827045069cc90546017a7f1c": "\\beta_T=\\beta_S+\\frac{TV\\alpha^2}{Nc_P}", "b78291a2b533bbfb143fcfd48d12dd04": "\\alpha_3 = {{5\\alpha_0 + 2\\alpha_1} \\over 7}", "b7832b5cfbd02c0fd2c9454c39a46b71": "\\begin{align}\n\\frac{d}{dr}\\left(\\frac{1}{\\rho}\\frac{dP}{dr}\\right) &= \\frac{2Gm}{r^3}-\\frac{G}{r^2}\\frac{dm}{dr} \\\\\n&=-\\frac{2}{\\rho r}\\frac{dP}{dr}-4\\pi G\\rho\n\\end{align}", "b7832f37dc0cf91f19012c608fb48fa7": " \\mathbf{X} = \\langle X, \\le\\rangle", "b783639913d35b45dfb3f0ce342324af": "u\\to 0 \\quad r\\sin\\theta\\to b \\quad(\\theta\\to\\pi).", "b7838911065da606c708efbe5bff62f7": "ij=k", "b783909eb0bcee927f07374557fcdc1c": "\n\\mathrm{Gyr}[\\mathbf{u},\\mathbf{v}]=\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & \\mathrm{gyr}[\\mathbf{u},\\mathbf{v}]\n\\end{pmatrix}\\,,\n", "b783ac6dc483d3ad522fe4626f8679f6": "L,H", "b7841bfccb62fb5c035691dcfa92277c": "s/V_k", "b78425c3e997f550f20c30da7d9122d6": "\\textstyle x^b", "b784e9964ebefa5a04f9896e7cbd04f3": "{\\Rightarrow_G}^*", "b784eee16964b199883877873459f259": "\\omega, \\tau", "b7851790cf5e4739895ceda6db72c500": "\\prod_{i 0", "b78f4bc851c931b45a007c8110531514": "\\gamma = k' \\left( \\frac{M}{\\rho N_A} \\right)^{-2/3}(T_c - 6 \\ \\mathrm{K} - T) = k' \\left( \\frac{N_A}{V} \\right)^{2/3}(T_c - 6 \\ \\mathrm{K} - T)", "b78fbac69f6254390f84945e68e11ce3": "w = (n,1)", "b78fc48bef6a346a4699d9142ee83e63": "\\ell\\to\\infty", "b78fdb5669ca1c7f63886386bd63d673": "\\Omega_2", "b7909b5361363957711757c0d73b169f": "Y[x,y]=\\frac{(x^2-y^2)x'+2xyy'}{xy'-yx'}", "b790a42a315e0dcc5521f3d906d7f991": "\\forall a \\in \\mathrm{Q}", "b790d19ef6d508598ba427446c12f566": "\\Delta{H} \\,", "b790e9c4aee69bf438b491aceb9b59a2": "x^8 + x^2 + x + 1", "b790ef7beb4828e979db87cae8ec2332": "\\chi = 0", "b790f54c9da268b0549c4fa62623ac77": "\\widehat{\\theta}(X)", "b790fc27357d649aa9ef3fd4612368d7": "T = \\begin{bmatrix} Q_{1} & \\\\ & Q_{2} \\end{bmatrix} \\left( \\begin{bmatrix} D_{1} & \\\\ & D_{2} \\end{bmatrix} + \\beta z z^{T} \\right) \\begin{bmatrix} Q_{1}^{T} & \\\\ & Q_{2}^{T} \\end{bmatrix}", "b79107dba3754abdfbf71a6de07c70d9": " = (2^n X : 2^n Y : 2^n Z : 1), \\ ", "b7914b43d3d1b8f5942df72afc2764ac": "(g,e')N=N", "b791c0d2799dbd2932e43c44b89e624d": "90^\\circy)] = f(a)[E(u(X)) - u(a)] ", "b7920206b31488ee52e6f55fb81e9ee6": "\\mathbf{a} = \\begin{pmatrix}\na_1 \\\\\na_2 \\\\\n\\vdots \\\\\na_n \n\\end{pmatrix}, \\quad \\mathbf{a} = \\begin{pmatrix}\na_1 & a_2 & \\cdots & a_n \\\\ \n\\end{pmatrix}", "b79221ae70f21629a59c13da90909765": "H(\\ln T)^{1-\\varepsilon_{1}}", "b7924311bbe4367ad8f1a21dac329b37": "RRR = {\\rho_{300K}\\over \\rho_{0K}}", "b79247f47ead44b96ca6ad6039614b35": "a_q, b_q, m_q, d_q", "b7926576753f22add8d093ff49d64a58": "\\,\\theta=\\theta_0,", "b792880d272f5f0af02fec1c3e927032": "\\operatorname{tr}\\exp(A+H) \\leq \\operatorname{tr}(\\exp(A)\\exp(H)). ", "b792b1164423d03ca17802333e697b3e": "Y_t = a \\cdot t + c \\cdot t^2 + b + e_t.", "b792b79837bab8604ea280cdcc14606d": "m \\geq 0 ", "b7937e0b357095313295ca9bafbac180": "\\textstyle (\\mu-\\alpha,\\mu+\\alpha)", "b7938d74e16a5a5bfcda365c70076b6b": "r_{HS} = \\frac{c}{H_0} \\ . ", "b79429e27902447707de19ef35b23cf3": "\n D_1 = \\frac{x_{(1)}-a}{b-a}, \\ \\ \n D_i = \\frac{x_{(i)}-x_{(i-1)}}{b-a}\\ \\text{for } i = 2, \\ldots, n, \\ \\ \n D_{n+1} = \\frac{b-x_{(n)}}{b-a} \\ \\ \n ", "b7945d725ffb953b74666001713257a6": " G(T) \\simeq \\sum_{\\alpha}{\\int{ \\frac{\\lambda_a(k_z)}{L} \\frac{\\hbar \\omega_{\\alpha}(k_z)}{2\\pi} \\frac{df_b}{dT}v_z(\\alpha,k_z)dk_z}} ", "b79461fb4c84b6d095f53e417e7fceef": "(ar,b)-(a,rb)", "b794640c6fc523797116ab1b67e36b1d": "{\\alpha\\,\\pi/\\beta \\over \\sin(\\pi/\\beta)}", "b7946ea248cb8d49667d213b7687f519": "J_3 \\simeq J_2 \\times J_3 / J_2 \\simeq J_1 \\times J_2/J_1 \\times J_3 / J_2.", "b79473cac494ce2e7292f083ac59ae16": " \\nabla_{\\vec{\\theta}} = D_d \\nabla. ", "b794953137f4f96e47a80417a66b50f8": " T(X,Y)= \\nabla_X Y - \\nabla_Y X - [X,Y]", "b794d1205f89607944ea64b81414c3d9": "v_i,v_j,v_k", "b794df480ca8781f03ad48e01c21203d": "(1-x^2)y'' + ( \\beta-\\alpha - (\\alpha + \\beta + 2)x )y' + n(n+\\alpha+\\beta+1) y = 0.", "b7955eca36a386ce4fdca25dd32553b0": "R_t-R_0", "b795bf837b378bedf824d91780638900": "\n\\nabla_i \\phi=\\phi_{;i}=\\phi_{,i}=\\frac{\\partial \\phi}{\\partial x^i}\n", "b7964470b43a86739aae2d02263d30f2": "\\displaystyle{{f^\\prime(z)f^\\prime(w)\\over (f(z)-f(w))^2} -{1\\over(z-w)^2},}", "b796498a3b0726423061b0887e814b5f": "f_{uv} \\le c_{uv}", "b79653feb383377db13554d33305b5bb": "\\nabla p", "b79668eec38e11f73714eceabee0a2a3": " \\frac {d\\varepsilon} {dt} = \\frac { \\frac {E_2} {\\eta} \\left ( \\frac {\\eta} {E_2}\\frac {d\\sigma} {dt} + \\sigma - E_1 \\varepsilon \\right )} {E_1 + E_2}", "b796799761752e1c60bb6272153ce23b": "= \\sin\\left( \\pi/2-\\alpha\\right) \\cos \\beta - \\cos\\left( \\pi/2-\\alpha\\right) \\sin \\beta\\,", "b7967f28fd06e6955acc9271228f7001": "Q_{i,j,k}<0.0\\,", "b796beb2da2e65c1aa9b6a0935bbac1c": "\\underset{\\longleftarrow}{\\mathrm{Lim}} (F/E) = \\mathrm{Cart}_E(E,F).", "b796df2faf22509236a5b60a19a4c12b": "x \\in (-\\infty, \\infty)\\,\\!, \\; \\tau \\in (0,\\infty)", "b797137e1313bc2f6057743e99a69e3f": "\n\\begin{align}\nX & = X_0 \\exp(x) \\\\\nY & = Y_0+X_0 y \\\\\nt & = z/w\n\\end{align}\n", "b797821d5855122e2c2ae2f8df737011": "\\gamma_{x} \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\gamma_{x}^{-} \\cup \\gamma_{x}^{+} \\,", "b7979923815f2b669b3ae50d2de8f224": "r = E\\left(r\\right) = \\rho - \\frac{\\rho \\left(1 - \\rho^2\\right)}{2 \\left(n - 1\\right)}", "b797b40eaa3f27329b3fa86b9bd5a6a4": " \\mu_B = \\frac{e \\hbar}{2 m c} ", "b797cc89cb8f41a16ffd148ae8f8cc47": "p \\in P^N", "b797e5b6eddafce4bc819c1feeee5fba": "V=\\mathcal{O}_{[g]}^\\chi=\\mathcal{O}_{[g]}^{X}\\qquad V=\\bigoplus_{h\\in[g]}V_{h}=\\bigoplus_{h\\in[g]}X", "b7980d04369cb3c0edb2f0c0fc1fb5fb": "\\frac{4a}{3\\pi}", "b79874a8ccff5c615e9665f1c363d91c": "\\bar H = HN.", "b7988e9474ad4e1a346c164fe08f161b": " lift = \\frac12 \\times \\rho_{air} \\times S \\times C \\times V^2", "b79890000b5e5d807575259917dec4ec": "d_\\lambda = \\sum_{\\mu\\uparrow\\lambda}d_\\mu, ", "b7989003046eaa38ec7d7d0858710c0e": "\\big(-G\\gamma _{2}\\cdot \\mathcal{P}_{1}-E_{2}\\beta _{2}+M_{2}+G\\frac{i}{2}\n\\Sigma _{1}\\cdot \\partial (\\mathcal{L}\\beta _{1}\\mathcal{-G}\\beta\n_{2})\\gamma _{51}\\big)\\psi =0, \n", "b7989b947b67e808333e8d6bdec1b546": "Q = \\begin{pmatrix}\n\\cos(2\\theta) & \\sin(2\\theta)\\\\\n-\\sin(2\\theta) & \\cos(2\\theta)\n\\end{pmatrix}", "b798a81736b39547f3b6253b7467f1bb": "1,\\ldots,D", "b798fc560f4bc716ec0cb41066fcb9a3": "X=(X_1,\\ldots,X_k)", "b7990f89c3817cf3f1ee23fd24f28f1b": "\\vec{\\nabla} \\cdot", "b7991ba449e98ca71d115635041c94eb": "\\frac{\\sigma^2}{N}", "b7994662826f7519f6cf27f13a52488a": "\\displaystyle{J(f)=(1-|\\mu|^2)|e^{2 k}|}", "b799647f1dbea78a6dabda7ba4e11b58": "4r^2 \\le IA\\cdot IC+IB\\cdot ID \\le 2R^2", "b7996e760c2857d9277ce6e5cc07f23c": " \\det \\left( F \\right) = \\frac{1}{c^2} \\left( \\bold B \\cdot \\bold E \\right) ^{2} ", "b799ce95f4d5ca25af231ea9ef35ca4e": " f(x; a_1, \\ldots , a_n) = \\sum_{i=1}^n \\, w_i \\, p(x;a_i) ", "b799eb2a509e3cbfd330f4538054ff65": " d \\ln {f \\over f_0} = {dG \\over RT} = {{\\bar V dP} \\over RT} \\,", "b79a061362e71cdfe458a526cc1f61aa": "[x,p]=1/2", "b79a29731558948b032c87079370a753": "(\\delta_{ij})_{i,j=1}^n\\,", "b79a32c19d4648dc1c0a3e091b02b65f": "\\int\\sec^2x\\ln|\\sin x|dx.", "b79a8e461e9bf30fc1f40a2464480100": "x_1=1", "b79ac722e9a4dc3a861f30c3d02a5b1f": "\\dot{\\varphi}", "b79b13386bbc27969daa20e33cc39993": " E = -\\nabla V ", "b79b47e14056624916071bd4bae5ffb1": "\\gamma \\in \\Gamma - \\{e\\}", "b79b5451bc9de0cce30f30098dabf9d0": "(v)", "b79b7164d9422832cb0d75f520cbd621": "\\rho(\\mathbf{r}',t_r)=[\\rho(\\mathbf{r}',t)],", "b79bfdb88013c2f0c194fc84754692a2": "n=\\frac{N_{A}\\cdot Z\\cdot\\rho}{M_{u}}\\,,", "b79c3a7338d0af0cfbc77a8361959a14": "a\\le a+b", "b79c6aad39de736e3c71e4c95a213b97": "I_n = \\int \\cos^n x dx . \\,\\!", "b79cc2e7d5138385da531f79ff5301a8": "K= \\frac{[S]^\\sigma[T]^\\tau}{[A]^\\alpha[B]^\\beta} \\times \\frac{\\gamma_S^\\sigma \\gamma_T^\\tau}{\\gamma_A^\\alpha \\gamma_B^\\beta}", "b79cc7f72b31d0d1982eef9bf0ed17a9": "\\beta(4)\\;=\\;\\frac{1}{768}(\\psi_3(\\frac{1}{4})-8\\pi^4),", "b79ccd742593b70f3af0278a099edeca": "P_{i,i-1} = P_{i,i+1} = q_i", "b79cfcb790ca485671dc83411d53bfaf": "= [\\,1 + u_{1}u_{1} - \\rho(x,u,u_{1})\\,]dx + u_{1}\\theta \\,", "b79d0b2488369d86fc6b18e734bdf3f1": "b )\\overline{~a~}", "b79d3087872bae18873f7677979a91ed": "L(\\theta; x) = f(x; \\theta)", "b79d3cf247fe41b7c878bbb75e58e21c": " y_d \\pm t_{\\frac{\\alpha }{2},m - n - 1} \\sqrt{\\text {Var}} ", "b79d749d3b15389cda7b0d6d566b1b38": "\\mu'_3=\\kappa_3+3\\kappa_2\\kappa_1+\\kappa_1^3\\,", "b79d83cec9e5296c59ac21e0c129880f": "( 3^{53} / 2^{84} = 19383245667680019896796723/19342813113834066795298816)", "b79e46a819f58a1f2f30485a350ab86c": "p_\\mathbf{A}(\\lambda) = \\det(\\mathbf{A} - \\lambda \\mathbf{I}) = 0", "b79eaf6d743eafd2ad6f84d6e4fd2200": "\\,x = \\ln{y}", "b79eff431a40095f21f524be473d0ed3": "d_S", "b79f044affa1835e069b621adae218ea": " x=1.7 ", "b79f09064b24b1a50d9c70eeb89986a4": "x, k \\in \\mathbb{Z}_q^\\times", "b79f62c91fe182070e4f41422fc0a661": "\\varphi (x) = \\int_{\\partial G} \\varphi (y) \\, \\mathrm{d} \\mu_{G}^{x} (y).", "b79f64f923ca8a52982ac3959c6c85c5": "\\textstyle 10^{-3}", "b79fb824f314b736ccb58393913ef3b4": "\\sqrt{n}\\bigg(\\bigg(\\frac{1}{n}\\sum_{i=1}^n X_i\\bigg) - \\mu\\bigg)\\ \\xrightarrow{d}\\ N(0,\\;\\sigma^2).", "b7a020c9b82a337d7e1451b9ba3dffd6": "\\mathbf{L}_{\\mathrm{total}} = \\mathbf{L}_{\\mathrm{spin}} + \\mathbf{L}_{\\mathrm{orbit}}\n", "b7a045b747e93d3017494c97a6829db2": "\n\\nabla^{2} \\Phi = \n\\frac{1}{a^{2} \\left( \\sigma^{2} - \\tau^{2} \\right) }\n\\left[\n\\sqrt{\\sigma^{2} - 1} \\frac{\\partial}{\\partial \\sigma} \n\\left( \\sqrt{\\sigma^{2} - 1} \\frac{\\partial \\Phi}{\\partial \\sigma} \\right) + \n\\sqrt{1 - \\tau^{2}} \\frac{\\partial}{\\partial \\tau} \n\\left( \\sqrt{1 - \\tau^{2}} \\frac{\\partial \\Phi}{\\partial \\tau} \\right)\n\\right].\n", "b7a06ec3b6aaee9a6f00c730595d6966": "\\phi : R \\to S", "b7a0c3760f24a66fa381963f36bac361": "u(\\lambda,T) = \\frac{2\\pi c^2}{\\lambda^5}~\\frac{h}{e^\\frac{hc}{\\lambda kT}-1} \\approx \\frac{2\\pi ckT}{\\lambda^4}", "b7a13d1e0b257580b5e7518b53acb172": "a_{8}", "b7a187d8c0331d89373399519166149d": " \\sum_{k=0}^n\\sigma_3(k)\\sigma_3(n-k)=\\frac1{120}\\sigma_7(n)", "b7a20f9ce2b35634b0e139a4bc5c8ba1": "\\mathcal{J}(\\boldsymbol\\beta^{(t)})", "b7a29088014d09dda63feb154289cee5": "\\operatorname{perm} \\left ( \\begin{matrix} 1 & 1 & 1 & 1\\\\2 & 1 & 0 & 0\\\\3 & 0 & 1 & 0\\\\4 & 0 & 0 & 1 \\end{matrix} \\right ) = 4 \\cdot \\operatorname{perm} \\left(\\begin{matrix}1&1&1\\\\1&0&0\\\\0&1&0\\end{matrix}\\right) + 0\\cdot \\operatorname{perm} \\left(\\begin{matrix}1&1&1\\\\2&0&0\\\\3&1&0\\end{matrix}\\right) +0\\cdot \\operatorname{perm} \\left(\\begin{matrix}1&1&1\\\\2&1&0\\\\3&0&0\\end{matrix}\\right) + 1 \\cdot \\operatorname{perm} \\left(\\begin{matrix}1&1&1\\\\2&1&0\\\\3&0&1\\end{matrix}\\right)= 4(1) + 0 + 0 + 1(6) = 10. ", "b7a2dd356798da53bb75a97aa00338cd": "\\mathit{Fo}_h = \\frac{\\alpha t}{L^2}", "b7a3618bcda69250715977870b1d09f0": "\\varphi\\left(\\frac{\\tau + 16m}{n}\\right)\\,", "b7a3ed9ee23ea08385040181382645ad": "X/G", "b7a41bc98c347d420698e71f251f61c5": "\nx = \\int \\cos \\left[\\int \\kappa(s) \\,ds\\right] ds\n", "b7a47071a3b9e1997ab22a6fa2e2c9be": "\\eta : 1_{\\mathcal{D}} \\to GF", "b7a4b773f6098561481c4ea1d002e4f8": " 1/k = am^{ b - 2 } - 1 / m", "b7a4c68225b375505f2622072f66c455": "s\\triangleleft t", "b7a4cf3183b22fb93371b74eb2d5478e": "e_b = i_b r_b + { {d \\varphi_b} \\over {dt}}", "b7a559e3346957bbaafb56a96b9f9dbf": " (\\mathbf{A}-\\lambda \\mathbf{I}) = \\mathbf{0} ", "b7a55a10c79e672f1ed9b3b2894a7c72": "F^{*}_{A}", "b7a58213d45b7eb8c24f8bf35502500a": "\\mathbf{E} ( \\vert Y_n \\vert )< \\infty ", "b7a5b7421bcf521053ece507676ea7b9": "\\int \\frac{\\sqrt{ax+b}}{(px+q)^n}dx = -\\frac{\\sqrt{ax+b}}{p(n-1)(px+q)^{n-1}}+\\frac{a}{2p(n-1)}I_n\\,\\!", "b7a5e7317809cddbc42bba826a2fef41": "\\displaystyle{f_{\\overline{z}}=g_{\\overline{z}}.}", "b7a64102c3dc75babf0d85a5b9eec2b2": "\\delta=\\pi/\\sqrt{\\sup K}", "b7a65dd08bf9af0a353482901179db12": "0.0123123123\\cdots = \\frac{123}{10000} \\sum_{k=0}^\\infty 0.001^k = \\frac{123}{10000}\\ \\frac{1}{1-0.001} = \\frac{123}{9990} = \\frac{41}{3330}", "b7a6635c3d4097f3718e654ee989adb6": " I= I(\\theta, \\lambda, t)", "b7a69ecbc06f094649626a1df816f614": " GeneralFormula.truckload.Emissions:Truckloads.Nec.Intake(scalar)*distance(miles)*emissions(emissions/miles)=Emissions ", "b7a6a0da5a85238a6adf04f68b470a9c": "\\langle a \\cup b \\rangle p \\equiv \\langle a \\rangle p \\lor \\langle b \\rangle p\\,\\!", "b7a6d7337b214fe14bb139c20b5be210": " {b_2(\\rho_m-\\rho_c)} = {h_w(\\rho_c-\\rho_w)} ", "b7a73ca1e611888a87190a9b1b74bb08": "F_{n-2} = F_n - F_{n-1}, \\, ", "b7a75d1128d4db36134861d43d3f7204": "\\!\\mu_1 \\ldots \\mu_4", "b7a77087e82a0c7cf768211bbccd8428": "\nJ(u)=\\Psi(x(T))+\\int^T_0 L(x,u,t) dt\n", "b7a7774cd632d8b68d923b55e5fa56cd": "[B]^\\Phi", "b7a82f9dd676bd47ba44e35b543f2cdc": "\\frac{E_P}{E_{ann}} = \\frac{1}{r}", "b7a855339e079aa1a58a324c41586d63": "\\omega _+", "b7a879b4432dccb74950d3de90d0e076": "E_1(\\omega) = \\frac {E\\eta^2 \\omega^2 } {\\eta^2 \\omega^2 + E^2} ", "b7a8a252f29279e8aedd1d27ec01bf12": "\\hat \\theta_0", "b7a9115ee22351f5c9429453280324f1": " h_d(n) = \\min_i \\left \\{H(n,o_i) | o_i \\in OPEN_{d'} \\right \\} ", "b7a91dd13db8e1ec2bd5bb6279cdc07a": "\\scriptstyle\\tilde{\\nu}", "b7a95045c941d5a36aded1b309a55ba0": "\\psi_\\tau", "b7a961256d22f45d0455ac3aabe0ff59": "w_n\\sim\\mathcal{N}(0,\\sigma^2)", "b7a9bd6d713710903b9402861dd6f1d1": " f_c^{(k-1)}(z_{cr}) \\neq f_c^{(k+n-1)}(z_{cr})\\,", "b7a9d6dbe30dab60cdeef52c3bbc473d": "S_5 \\cong \\operatorname{PGL}(2,5),", "b7aa3661c591d5afcaf9bccf30806c02": "\n\\mathbf{E} = \\rho \\mathbf{J}\n", "b7aa37a77522d09dd67ae29e86c67356": "H_\\alpha(X)=\\frac{\\alpha}{1-\\alpha} \\log \\left(\\|X\\|_\\alpha\\right)", "b7aa6ab57a1acb55cc30c3c48207cec0": "\\{Y_i", "b7aa85d8dc37049f95ca0c719bea5df3": "B=\\lim_{t\\to 0} \\frac{1}{2}\\sum_n \\sgn(\\omega_n) \\exp -t|\\omega_n| ", "b7aa8cb8feef8ea465eec5f95cbe124c": "x^2 \\equiv 18 \\pmod {23}.", "b7ab334c197ae6ea06259315b162a5a6": "\\ Z_R = R", "b7ab3b4c7fff78272e66d17e30e362e8": "1 \\le n \\le \\lfloor 5.4 \\rfloor = \\left\\lfloor { 108 \\ \\mathrm{MHz} \\over 20 \\ \\mathrm{MHz} } \\right\\rfloor", "b7ab8345268eac99dd68fceaed15773f": "c\\sqrt{\\frac{R_s}{2r}} = c\\sqrt{1 - \\frac{R_s}{r}}", "b7abaf6b63107b53800877b27a93e10b": "\\varepsilon^*", "b7abb54a474d5ab0afa65a85afd55192": " n \\frac{R}{R+W} ", "b7abf5b8acb15524057224fc5a8641a7": "T(n) = O(n) + 2T\\left(\\frac{n}{2}\\right).", "b7ac16d18db6ab1417b68e2224f8db5e": "M(a,b,z)\\text{ and }U(a,b,z)", "b7ac3f79e5e78b28ddfb288d3061b9dc": "\\lambda_r \\propto \\frac{\\lambda_u}{2 \\gamma^2}", "b7aca8b6420901ef1f6e08ba3f00f520": "C_{pi}", "b7ace99a0680ddaa75813a2ff2e1f93a": "\\phi(\\theta)", "b7ad0927f20a74c7dbaa78521b2609d5": "2\\mathbb{Z} ", "b7ad1cb0656036eb2062243a18a1077b": "\\,\\Pi(x_i, x_j)\\,", "b7ad3fe5626d2aa1092a9fbdeeb0eed4": "\\left[{n\\atop 2}\\right] = (n-1)!\\; H_{n-1},", "b7ad796a189e76f856d731f927a25ef6": "\n \\begin{align}\n K_{\\rm I} & = \\lim_{r\\rightarrow 0} \\sqrt{2\\pi r}\\,\\sigma_{yy}(r,0) \\\\\n K_{\\rm II} & = \\lim_{r\\rightarrow 0} \\sqrt{2\\pi r}\\,\\sigma_{yx}(r,0) \\\\\n K_{\\rm III} & = \\lim_{r\\rightarrow 0} \\sqrt{2\\pi r}\\,\\sigma_{yz}(r,0) \\,.\n \\end{align} \n", "b7ada57e7eac15ced7fb6cdd4b00dcae": "\\begin{bmatrix}w^1 \\\\ \\vdots \\\\ w^k\\end{bmatrix}", "b7adb18b53a3fe3759ffb44ac437fbb8": " \\frac{S}{L} = \\frac{1}{\\cos \\varphi} = \\sec \\varphi. ", "b7ae3108b5d0b410c2b6ca3c86a55336": "R_{m,\\nu} = \\sum_{n=0}^{[m/2]}\\frac{(-1)^m(m-n)!\\Gamma(\\nu+m-n)}{n!(m-2n)!\\Gamma(\\nu+n)}(z/2)^{2n-m}.", "b7ae643f5e09c96a6e85312645d0841e": "\\Sigma = \\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V \\sigma \\, dV", "b7aea0aeeeb43d16a2e3fe28eac97558": "\\displaystyle 3.14=\\pi", "b7aec960198717151f1f6fd3e5daf83b": "\\Psi_0=\\Psi_1=\\Psi_2=\\Psi_3=\\Psi_4=0", "b7af53ee24c5cdc13cc7280ef036c4b7": "B^2/8\\pi n_im_ic^2 = 26.5\\,\\mu^{-1}n_i^{-1}B^2", "b7b021f2a04a520229ceac20d373084a": "\\widehat{A}", "b7b02a7ef04199ab9fa81672dcacb71d": "\\alpha_1:=T", "b7b0740f2c6f3767bc16c2074ba43163": " I(J \\cap K)=IJ \\cap IK. ", "b7b0eee0cac0f44201236a3e54cd6dbb": "\\Delta(x) < -Kx^{1/4}", "b7b1467e0bc6c521df4ca91cca51e05b": "IA\\cdot IC+IB\\cdot ID=\\sqrt{AB\\cdot BC\\cdot CD\\cdot DA}.", "b7b1a79ac01329e7299a22af1983f443": "\\frac{\\partial (n/n^{\\ominus})}{\\partial t}= \\nabla \\cdot [\\mathfrak{m} a (\\nabla \\mu - (\\mbox{external force per gram particle}))]", "b7b22eb6304e2e9f34da515cd344daa8": " ~\\varrho^{(m)}_{G,1...m} ", "b7b23267c9a5a677b0fb254a436b3608": "2^{f(n)}\\geq n!", "b7b25cf5d6d1492bbef0913f1ad91976": "F_2 = \\frac{{(q/2)}^2}{4 \\pi \\epsilon_0 L_2^2}=\\frac{q^2/4}{4 \\pi \\epsilon_0 L_2^2} \\,\\!", "b7b2783dfc2fc44e01fd91646c6b615d": "\\frac{d[\\mbox{A}]}{dt}=-k[\\mbox{A}].", "b7b3079ac6d8af2b6dacc8c3e6b42eb4": "\\dots \\rightarrow \\operatorname{H}^k(B) \\overset{\\delta}\\to \\operatorname{H}^{k+m+1}(B) \\overset{\\pi^*} \\rightarrow \\operatorname{H}^{k+m+1}(E) \\overset{\\pi_*} \\rightarrow \\operatorname{H}^{k+1}(B) \\rightarrow \\dots", "b7b317b473364c48a9ef93ac73a8d853": "f_{ij}{}^k", "b7b3554b5574d921cff587042e7f69f8": "LG = \\Omega G \\rtimes G", "b7b3bb91478672c26dd3b9b9212029f5": "\n\\left\\{ D_{i}, D_{j}\\right\\} = \\sum_{s=1}^{3} \\epsilon_{ijs} L_{s} ~;\n", "b7b3fce75d60e04a7743c46ff7e0defb": "\n\\begin{align}\nx + \\frac{ab}{x} & = u^2 + v^2 = \\underbrace{\\left(u^2 - 2uv + v^2\\right)}_\\text{a perfect square} + 2uv \\\\\n& = (u - v)^2 + 2uv = \\left( \\sqrt{x} - \\sqrt\\frac{ab}{x}\\, \\right)^2 + 2\\sqrt{ab\\,{}}.\n\\end{align}\n", "b7b3fd87a5454294a47d052d6ee781bd": "\\omega(g\\cdot v,g\\cdot w)= \\omega(v,w)", "b7b463a712104818b3743b09768e77d7": " p(t_0) = y_0, \\, ", "b7b4ece98fb05dce623ce268369d0fc0": "(x^2 + y^2)^2 = ax^2y.\\,", "b7b505ae93c6d3ff541f212d00b5a9ee": "f^{1,2}(\\theta(t))", "b7b50b22ec17ecc1fbd33068f51595d5": "\\,J^-(x) = \\{ y \\in M | y \\prec x\\}", "b7b54c3d7db1743114428605e0177658": "\\sqrt{\\frac{1}{35}}\\!\\,", "b7b551345fe6353796a7d459971b249a": "I = \\frac{dQ}{dt}.", "b7b56ded25b7108bbebe19cf76450d0b": "|\\mathbf{r} - \\mathbf{r}_s(t_1)| = c(t - t_1)", "b7b5bcce4e933b9c89b7be462b0e64d1": "L_+, L_-, L_0", "b7b5ed7572a24ac00d5ebc7961a1c6b3": "\n[P,X^n] = - i n~ X^{n-1}\n\\, ,", "b7b623bd9fc244aa94c8e12e2a4adb68": "\\left(\\begin{smallmatrix} 0 & 1\\\\ -1 & 0\\end{smallmatrix}\\right).", "b7b6c22afc15196ccf5158f03ec4ba7c": "n \\!\\, ", "b7b6d3b9b400bd4da798e27bdde46369": " n := [n-0] ", "b7b6dc77fa522a1126d0fe10f8bedfe7": "R_{sk} = \\frac{1}{n R_q^3} \\sum_{i=1}^{n} y_i^3 ", "b7b716918bf85c93a5ce2674a5341716": "\\mathbf{b_1} = 2\\pi \\frac{(\\hat{x} \\otimes \\hat{y} - \\hat{y} \\otimes \\hat{x}) \\mathbf{a_2}}{\\mathbf{a_1} \\cdot (\\hat{x} \\otimes \\hat{y} - \\hat{y} \\otimes \\hat{x}) \\mathbf{a_2}} ", "b7b7345126f5771e028799ff5ccda48b": " r_i ", "b7b73c594d81415b4b43b6053f8899a7": "\\textstyle I(\\mathbf{q})", "b7b7459bdbde68592519fb31b7a50b11": "d(f, g) \\equiv \\sup\\left\\{d[f(x), g(x)]: x \\in X \\right\\}", "b7b76d3b46f6ac3787d7bc7aba5f161b": "n = 0, 1, 2, 3, \\dots \\, ", "b7b7f6f266747ed39bc8fc431bc55a35": "\\ Y=A[\\alpha K+ (1-\\alpha) L]", "b7b861fb229333b75fd5bff6cea72e1e": "A\\ -\\ C\\neq\\ 0", "b7b8d9ae13fa97702d12da78f6cc3b53": "d = \\lambda_0 > \\lambda_1 ", "b7b90a10f235bdc8863e14916883eb36": "n \\nmid q - 1", "b7b90c0f97414aa72b5c001a38c4e185": "k_1+\\cdots+k_n", "b7b9299b5c2fb9ba654b66a649be09c3": " \\ln 2 = \\frac{1}{1} -\\frac{1}{1\\cdot 3}+\\frac{1}{1\\cdot 3\\cdot 12} -\\cdots. ", "b7b937c28de597d804eaef48f6f45bba": "\\,y(t)=(C+m(t))\\sin^2(\\omega t)\\cos^2(\\omega t)", "b7b9614009c8952b4a32cb232adabac4": "(x^{-1})^{-1} ", "b7b97ea61f067e8e567e52776281c0ec": "\\frac{1500}{1538} = 97.53%", "b7ba04b284ecbc991794b53d478ba2fe": "P_i\\,\\!", "b7ba708e9a4a876aabf4559b29d9ca64": " (V, F) ", "b7baae6b46b38404223045b901e968d8": "\ng(r)=\\frac{P(r)}{Q_{R}(r)}\n", "b7bb09fbd197486190e64b90a98ba492": "(-1)^{| \\alpha |} = (-1)^{\\alpha_1+\\alpha_2+\\cdots+\\alpha_n}", "b7bb15757ede6e8c9254572abffa7fbb": "V+V^\\top", "b7bb6183dffe8956be88436614c30821": "P(x_1,\\ldots,x_n)", "b7bb80c55b650ca92c99904a6cbab641": "\nM = \\begin{pmatrix}\n1 & 2 & 0 & 6 \\\\\n0 & 2 & 9 & 4 \\\\\n0 & 0 & 0 & 4 \\\\\n0 & 0 & 0 & 0\n\\end{pmatrix}.\n", "b7bbc1e89d249bab81c39db8b89547f9": "{\\widehat{AR}}_5", "b7bc25228c03b496dc5904e7d6e6da0e": "\\overline{D}", "b7bc9429486bd00e86561530a52563d5": "\n\\varphi_{r} = \\int dq_{r} \\sqrt{v_{r}(q_{r})},\n", "b7bd0d234924167e94091d3d5cd3b3d6": "\n\\Psi_t(x)=(\\lambda_1^t\\psi_1(x),\\lambda_2^t\\psi_2(x),\\ldots,\\lambda_n^t\\psi_n(x))\n", "b7bd3329a8336c7b575a79c1ea24c0e3": "\nA*B = {1\\over 2}(AB+BA) ~\n", "b7bd50d2b0cf7ee36492988227f6a3d0": " m , m'", "b7bda529db8d0b44669b1c829ec15f67": " \\phi_{ab} \\, ", "b7bdc88e8dbb3f5092e7ad80b75024eb": "\\scriptstyle GF(p)", "b7bdde1d24353a4001e0bb4a2f610131": "T_G(x,x) = m_{\\vec{G}_m}(x).", "b7bdf4eacc2efb11e58c48a735b6ad88": "\\frac{q_0}{w}+k_0=k_j", "b7be0eec5ee3bf54abfa289a0dc5f842": "\n d^H = \\cfrac{a^2}{R}\n ", "b7be253286360d0f5f8631af991445f1": "1 + max(MD(\\Box p), MD(p)) =", "b7bf2fea9594ae8ce75b3ebb2b83034e": "\\det(H)={{c_n^{\\;4}}\\over {c_{2n}}}", "b7bf34a097286c6ad7faa770c58854b5": "u^2=a_0+a_1u", "b7bfb8c1055bc61b8e237c8dff9b6b4f": "d(x_n, y) \\to 0", "b7bfc2163ab12aecf3c27d2c472bee9b": " t_1 = \\frac {2 \\pi R + \\Delta L}{c} ", "b7c00b834142d6583530cdfd40310fa9": "n_o", "b7c01691fa3ef67549426aca0cd235ff": "\\frac{AQ}{OQ} = \\sin \\alpha\\,", "b7c02df40b0327091bc5349530ec324b": " \\ \\displaystyle g(d,s) \\ ", "b7c053e082bb79d93dae2a4d1df71bd6": "\\textstyle \\{(1,2),(1,3),(3,2)\\}", "b7c09d9c75a32aacf421121bb19340ec": "\n2 T dt = \\mathbf{p} \\cdot d\\mathbf{q} = \\sqrt{2 T} \\ ds.\n", "b7c0aa43f8ee4af765158e8c9f6b793d": "-2.4\\times10^{-5}<\\frac{v-c}{c}<12.6\\times10^{-5}", "b7c0af8996eaef0cfe89b22fa8a9498f": "\\alpha_2 = \\frac{1}{2}+ m", "b7c0ba461ba6b440e441ca09474eb0cf": "E\\propto 1/a^2", "b7c0c53edf03af27adf74d717c5904cf": "Q(z) = \\frac{2\\sqrt z}{\\sinh 2\\sqrt z} \\ ", "b7c12f2fe442f8ba5cef553885b4887c": "\n\\begin{matrix}\nI(W;X;Y;Z)& = & H(W)+H(X)+H(Y)+H(Z) \\\\\n\\ & - & H(W,X)-H(W,Y)-H(W,Z)-H(X,Y)-H(X,Z)-H(Y,Z) \\\\\n\\ & + & H(W,X,Y)+H(W,X,Z)+H(W,Y,Z)+H(X,Y,Z)-H(W,X,Y,Z) \n\\end{matrix}\n", "b7c1832e8960a9244d06097cbfe1f75d": "\\pi_p(EU(n))", "b7c1cb8a8cbd1df89cb7a23c0fae743c": "\\Delta > 0,", "b7c1dd0af3bf62fb51e696bc1060700c": "\\frac {1}{4 \\pi \\varepsilon_0}\\int \\frac { \\bold{p} ( \\bold{ r}_0 )\\bold{\\cdot (r - r_0)}} {| \\bold{ r}- \\bold{r}_0 |^3 } d^3 \\bold{ r}_0 =\\frac {1}{4 \\pi \\varepsilon_0}\\int \\bold{p} ( \\bold{ r}_0 )\\bold{\\cdot \\nabla}_{\\bold {r}_0} \\frac {1}{|\\bold r - \\bold{r}_0|} d^3 \\bold{ r}_0 , ", "b7c2114a11b4856d9e831e440782b751": "\\mathrm{Re}_p*", "b7c2a05d0e6220189f3a032840d30b44": "", "b7c336fee7d111216583b70995236493": "\\sqrt[n]1", "b7c33bf97610fa6e59edc6a44f148751": "H(q)", "b7c33ea221ff9de847467d7550b90b24": "K_\\mathrm{fluid}^{(1)}", "b7c4092e843abf3b303121807fd13df1": "\n\\begin{bmatrix}\n1 & 2 \\\\\n3 & 4 \\\\\n5 & 6 \\end{bmatrix}^{\\mathrm{T}}\n=\n\\begin{bmatrix}\n1 & 3 & 5\\\\\n2 & 4 & 6 \\end{bmatrix}\n", "b7c44bf187171e4ff94d1d6aa26d786c": "l_D", "b7c46e686b220fa636a32c919efd813c": "f(x_0 + 1, y_0 + 1/2)", "b7c49a49035b2f0e02a8eae9c417b07e": "{\\rm det} \\left( \\alpha I - A \\right) = \\alpha^3 - \\alpha^2 {\\rm tr}(A) - \\alpha \\frac{1}{2}\\left( {\\rm tr}(A^2) - {\\rm tr}^2(A) \\right) - {\\rm det}(A) = 0.", "b7c49f2a7f3ff1b7903b2791df2c3e37": "\\hat{\\textbf{x}}_{0\\mid 0} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} ", "b7c4be617a2758f2225d0a2e6dbff239": "(\\Delta E \\sim 1/r^3)", "b7c4e7bc52aba676865a3ee10a381418": "\\sigma^2_{Y_i}", "b7c541ddc48b39ad88e31c37681a3093": " \\{ z_k \\} \\leftarrow \\{ x_k \\} ", "b7c54c00a8a78d4b30e1565764f0f0e9": "q = Ze", "b7c5e624b3a2994fc3ff999c8518e120": "f_{kl} := p_k\\circ f \\circ i_l\\colon A_l\\to B_k.", "b7c659fa009f1d54db51ce9579783e42": "H_- = {\\rm p}K_{\\rm a} + \\log {{c_{\\rm B^-}}\\over{c_{\\rm BH}}}", "b7c65f6d3230142d3429d5e1916df77e": "L(\\omega)\\neq 0", "b7c6a4f4490187091cf7392cbc2e6c6d": "\n \\displaystyle \n S(i,g-1) \\ , \\ {\\rm for} \\ i = 0, \\dots , n\n", "b7c6e3d5b75627b0bb1a76e67366762a": "\\tilde{G}^{liq} = \\tilde{G}^{vap}", "b7c77b0576a7afb4ab800e1af0ee660c": "R_0+x\\ ,", "b7c7bbd395e093af41e0408ca62c5b29": "\\mathcal{T}(A) \\Rightarrow_{mem} \\mathcal{T}(B)", "b7c7eb7771f344e3956c3d02fd538636": "C_{V,m}=\\frac{3R}{2}+R+R=\\frac{7R}{2}=3.5 R", "b7c804b499d6390f797cf160766584e3": "{\\frac{T_y}{T_x}=\\frac{\\left(1+\\frac{\\gamma-1}{2}M^2_x\\right)\\left(\\frac{2\\gamma}{\\gamma-1}M^2_x-1\\right)}{\\frac{1}{2}\\frac{\\left(\\gamma+1\\right)^2}{\\left(\\gamma-1\\right)}M^2_x}}", "b7c863804f5039c55ee14f6a2c538613": "\\mathbb{Z}_{(p)}[v_1,v_2,\\dots]", "b7c8b183a159bf2c54d8d56c775de89e": "R_{abcd}", "b7c8ddac43e631fa531d5a99d255a2f2": " G P=P", "b7c919cd1744c8625358b82ae84e80c1": "\\phi_U\\colon \\pi^{-1}(U) \\to U\\times\\mathbb C^k", "b7c93deb1a7b0a1c116192530d664aa8": "\\frac{m^2}{sr}", "b7c961ccb6c2a9ae7b28bc0e5f733b98": "I_{sp} =\\, \\frac{F}{\\dot{m}\\,g_o}\\,=\\, \\frac{\\dot{m}\\,v_{e}}{\\dot{m}\\,g_o}\\,=\\,\\frac{v_{e}}{g_o}", "b7c9b914349311de06d16d335de7dda5": "r \\approx \\frac{\\alpha \\hbar c}{10^6 GeV} \\approx 10^{-24} m", "b7ca1f09ce6f41c9fc3b03a37373fb16": "\\displaystyle n \\cdot a = \\underbrace{a + \\cdots + a}_{n},", "b7ca6aa557d6769b0d760135d27f1fc2": " \\mathbf{J} \\cdot \\mathbf{\\hat{n}} A = I ,\\,\\!", "b7ca7c018266a087ba1ad761fe7caef0": " -\\frac{1}{2m}\\nabla^2 \\psi(\\vec{r}) + \\frac{-Ze^2}{r} \\psi(\\vec{r}) = E\\psi(\\vec{r})", "b7ca8f4064eb805b11c18a2679cd1329": "\\frac{d}{dx}(f * g) = \\frac{df}{dx} * g = f * \\frac{dg}{dx} \\,", "b7cacc8a75b41bd0be9c116cf711a285": " \n\\Delta(t) = E\\left[L(t+1) - L(t) | \\boldsymbol{Q}(t)\\right]\n", "b7cae5804f5be3c8b6eb610c73118a57": "\\displaystyle{(x_1^2 + \\cdots +x_N^2)(y_1^2 + \\cdots + y_N^2) =z_1^2 + \\cdots + z_N^2,}", "b7cb1995536b8e47615cd8d941b22fdf": "q = \\left\\lfloor \\frac{m a}{4^k} \\right\\rfloor ", "b7cb6b182a6ee0c8693d35bba5c79ff8": "V_{\\lambda} = \\bigcup\\{V_{\\beta} \\mid \\beta<\\lambda\\}", "b7cbef4654837135cedfdb4272b6c5c8": "B[u, v] = \\langle f, v \\rangle \\mbox{ for all } v \\in H_{0}^{1} (\\Omega),", "b7cc4b6b2b8c0f37377b5cc259385de0": "-t", "b7ccc44d94dac448e361a36537a5a498": "|B_n-B|\\le\\frac{\\varepsilon/3}{\\sum_{k\\in{\\mathbb N}} |a_k|+1}", "b7ccc4e37fd8dcb4fef364dea7533ef6": "\\begin{matrix}\nF(a,b,b') & \\xrightarrow{F(1,1,g)} & F(a,b,b) \\\\\n_{F(1,g,1)}|\\qquad & & _{\\eta(a,b,c)}|\\qquad \\\\\nF(a,b',b') & \\xrightarrow{\\eta(a,b',c)} & G(a,c,c)\n\\end{matrix}", "b7cd21b7ff89d4b4679dc666fa10c1c6": "\\mbox{E}(x)=\\mbox{D}(x)=((m-1)x+(m-1))\\mod{m}", "b7cd3d94d13c3b3659b01e610b1d38d7": " x\\in A_i", "b7cdc3a6313fc337d042bb5e9e7c4d7d": "\\displaystyle{\\pi_\\pm(g^{-1})F_\\pm(z)= (\\overline{\\beta} z + \\overline{\\alpha})^{-1\\pm 1/2} F_\\pm(gz)}", "b7cdc76cf13e423c8757db43affc5f5e": "E\\{\\hat{x}\\} = \\bar{x},", "b7ce7e29c1cfce3bd400000312e3427f": "\\pi_{XZ}(R)", "b7ce8fafde4741ce5ddd93d65bf4c3b2": "\\mathbf y - X \\hat{\\boldsymbol{\\beta}}", "b7cec77da70a912096a767df09607536": "\\nabla f = \\nabla \\cdot f + \\nabla \\wedge f", "b7cf4fa748a9ab6e267e8dad901cefee": " \\operatorname{sys}^2 \\leq \\frac{2}{\\sqrt{3}} \\;\\operatorname{area}(\\mathbb T^2),", "b7cf916bbb2426224b3d416449d9f6ea": "\\mathbf{v}_\\mathrm{B} = s \\mathbf{u}_R, ", "b7cfe7f1257dc26db0faf9d6973b2a67": "F^*(x) = a_0\\delta_0 + \\sum_{n=1}^\\infty a_n x^{*n}.", "b7cfe8c90a3b52b3c9c52edeaae61467": " \\; \\; \\; + \\frac{\\omega}{2} \\, \\frac{C^\\prime \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u \\right)}{C \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u \\right)} \\, \\left( x \\, \\partial_x + y \\, \\partial_y \\right) ", "b7d00a071eaeaabbd80481aaa3f89d60": "a_{r}=-\\frac{\\left ( (r+c-1)(r+c-2)+(2-\\gamma )(r+c-1) \\right )}{\\left ( -(r+c)(r+c-1)+(\\alpha +\\beta -1)(r+c)-\\alpha \\beta \\right )}a_{r-1} =\\frac{\\left ((r+c-1)(r+c-\\gamma ) \\right )}{\\left ( (r+c)(r+c-\\alpha -\\beta )+\\alpha \\beta \\right )}a_{r-1}", "b7d068eb9b2314ba593eb0f06241efe3": "|000\\rangle", "b7d079fad30c8a8fa9b24aa052006a44": "(-1)^{2+3}\\;\\operatorname{det}\\begin{pmatrix}\\!-3&\\,2\\\\ \\,3&\\!-4\\end{pmatrix}=-((-3)(-4)-(3)(2))=-6.", "b7d0866c7fecc54443c6254c1dd3b6d7": "A=\\pi a b\\,.", "b7d0cc0433f57da36836ae4180c58d89": "v = in(e)", "b7d0cf3c80b1fb1c261557640eeed9a9": "\\forall b_1,...,b_n \\notin T(s) ", "b7d0deb1644263dc9d2fd1c310ef5088": "Y_{6}^{-5}(\\theta,\\varphi)={3\\over 32}\\sqrt{1001\\over \\pi}\\cdot e^{-5i\\varphi}\\cdot\\sin^{5}\\theta\\cdot\\cos\\theta", "b7d134a35dd9df91c9f2d6e2e72d6437": "\\lambda\\,/2", "b7d156780296bd277b8eab0d56582322": "\\chi\\left(\\Psi\\otimes\\Psi\\right)=\\chi\\left(\\Psi\\right)+\\chi\\left(\\Psi\\right)", "b7d1819984ee35c19e52759bd5a58855": "\\operatorname{Var}\\left(\\sum_{i=1}^n X_i\\right) = \\sum_{i=1}^n \\sum_{j=1}^n \\operatorname{Cov}(X_i, X_j) = \\sum_{i=1}^n \\operatorname{Var}(X_i) + 2\\sum_{1\\le i}\\sum_{0", "b7d801a9c1e90aa803768a7d9562c217": "a\\ne b \\ne c \\ne d, \\alpha \\ne \\beta \\ne \\gamma \\ne \\delta \\ne \\epsilon \\ne \\zeta \\ne 90 ^\\circ", "b7d85ff32d83964f95c873a135e1e459": " \\tilde{h}^{ab} \\frac{\\partial}{\\partial \\tilde{\\sigma}^a} X^\\mu \\frac{\\partial}{\\partial \\tilde{\\sigma}^b} X^\\nu = h^{cd} \\frac{\\partial \\tilde{\\sigma}^a}{\\partial \\sigma^c} \\frac{\\partial \\tilde{\\sigma}^b}{\\partial \\sigma^d} \\frac{\\partial}{\\partial \\tilde{\\sigma}^a} X^\\mu \\frac{\\partial}{\\partial \\tilde{\\sigma}^b} X^\\nu = h^{ab} \\partial_a X^\\mu \\partial_b X^\\nu ", "b7d8e4ef8572a7e71d8656288e2fecc9": "\\int_{-\\infty}^\\infty \\delta(\\tau)\\, g(t - \\tau)\\, d\\tau = g(t)", "b7d8e882d4749e1724d81a99f81a9a6c": "h:\\mathbb{R}\\to \\mathbb{R}", "b7d9c577c7283a4292285ff31882c615": "k/n=1-H\\left( \\mathbf{p}\\right)\n-4\\delta", "b7d9fc34b02eb0f2427532d4d5a3cbbb": "X = j \\frac {10^p-1}{F}", "b7da4fb56a7f83df3b9d2a051cc669a8": "|x\\rang", "b7db3f24c875752b001a68c006786406": "\\varepsilon_2''' = -\\frac{\\nu}{E}\\sigma_3", "b7db800dc5214a95515b82140c024c08": "N(C)(?) = \\mathrm{Fun}(i(?),C)", "b7dc20bc80b3af65cf4ea7af73b386f2": "\n(Eq. 1) \\text{ } \\Delta(t) \\leq B(t) + \\sum_{i=1}^N Q_i(t)(a_i(t) - b_i(t)) \n", "b7dc75a17b589ae3e07431c88ab40db1": "\\hat{D}(\\alpha) = e^{ -\\frac{1}{2} | \\alpha |^2 } e^{+\\alpha \\hat{a}^{\\dagger}} e^{-\\alpha^{*} \\hat{a} } ", "b7dd61c8920269118e459753218b9572": " \\mathbf Z=1", "b7dd78815adf0fcb4bd88b8c4fe1e6ac": "\\scriptstyle \\boldsymbol \\nabla T", "b7dd8b53cbfd22e6ac39a832e7c111b5": "q_1, ..., q_n", "b7dda7ddfce9692420b16565b407df3c": "\\mathbf{P}_0 = \\mathbf{I}", "b7ddb9d956541e77a2e8884e8cebc626": "y=-1, y=1;", "b7ddbb24750bc66fcde74204a365cbfa": "[1, \\infty]", "b7ddc771da0452a49d50a0528acc2510": "N_I", "b7ddcbc92226dbadd8ea3d1bcf972e44": " t \\le t_n )", "b7de165bfe7930d84a508e8711f89265": " a|\\uparrow\\rangle + b|\\downarrow\\rangle", "b7de17f8cf0e396ddda031f3aba016f7": "|\\langle Tx,Ty\\rangle|^2=|\\langle x,y\\rangle|^2", "b7de2cd7725f9ca514440bea498f0475": "\n\\begin{pmatrix}\ne^{i\\phi_x} & 0 \\\\ 0 & e^{i\\phi_y}\n\\end{pmatrix} ", "b7dec04f11da5aa06afb48ffadc8ae89": "\\langle W',R'\\rangle", "b7df0075b53b1156771529a2282b0abd": "\\sigma\\Gamma\\vdash\\sigma A", "b7df11cc71dbdc4e68aa00d75f0eef7c": "p(\\theta|\\eta)\\,", "b7df2a267dfd6a7706d640af01158f61": "\\nabla\\cdot\\left(\\frac{\\mathbf{q}}{\\theta}\\right)~+~\\rho~\\frac{\\partial s}{\\partial t}~=~\\sigma,", "b7df2f71f0b736213e1d652048b246bd": "E(\\overline{K})", "b7df3569a957776aa0b102635468f72b": "v(x) = a(x)g(x) + e(x)", "b7df3ac0bb54daa668d0c4198c1b7359": "\\mathcal{B}(p,q) = \\begin{cases} \\{ U_{\\epsilon}(p,q):= \\{(x,y): (x-p)^2+(y-q)^2 < \\epsilon^2 \\} \\mid \\epsilon > 0\\}, & \\mbox{if } q > 0; \\\\ \\{ V_{\\epsilon}(p):= \\{(p,0)\\} \\cup \\{(x,y): (x-p)^2+(y-\\epsilon)^2 < \\epsilon^2 \\} \\mid \\epsilon > 0\\}, & \\mbox{if } q = 0. \\end{cases} ", "b7df617d338f95fb95807d746658b70d": "\\begin{align}\n\\mathbf{A'}_x &= \\frac{4R - \\mathbf{A}_x}{3} \\\\\n\\mathbf{A'}_y &= \\frac{(R - \\mathbf{A}_x)(3R - \\mathbf{A}_x)}{3\\mathbf{A}_y} \\\\\n\\mathbf{B'}_x &= \\mathbf{A'}_x \\\\\n\\mathbf{B'}_y &= -\\mathbf{A'}_y\n\\end{align}", "b7e023d108958f7a9cd1b8df1daa047d": "\\left.p(\\vec{r},t)=\\frac{\\beta}{4 \\pi C_p} \\int \\frac{d \\vec{r'}}{|\\vec{r}-\\vec{r'}|} \\frac{\\partial H(\\vec{r'},t')}{\\partial t'} \\right|_{t'=t-|\\vec{r}-\\vec{r'}|/v_s} \\qquad \\quad \\,\\,\\,\\,(2). ", "b7e0662a74a00d483fae168777618454": " = \\vec{\\nabla}_{\\vec{r}}\\bigg(\\frac{-1}{(n-2)A_n}\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\vec{\\nabla}_{\\vec{r}} \\bullet{} \\bigg(\\frac{1}{|\\vec{r}-\\vec{r}'|^{n-2}}\\vec{F}(\\vec{r}')\\bigg)d\\tau'}\\bigg)", "b7e0f7d14f90bda506e48eb7a851b1fd": "P=(x,y,z,t)", "b7e110dc4cd9572d475843c01feab57d": " Y_1 \\supseteq Y_2 \\supseteq \\cdots ", "b7e153016a41e53d159526baf3172126": "\\textstyle \\frac{d^2\\psi}{dx^2} = -\\frac{2mE}{\\hbar^2} \\psi;", "b7e1a0fe2f4389acfb090c4095691126": "\\alpha k_{B}", "b7e1e96d5cb91fbf8242282ff4954985": "r_0 = 2GM/c^2", "b7e1e98677fa65a1345432a6809c73cc": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{r}} \\right) - \\frac{\\partial L}{\\partial r} = 0 \\qquad\n\\Rightarrow \\qquad \\ddot{r} - r\\dot{\\varphi}^2 = 0 ", "b7e22083726f51daff311007a364723b": "P(X) \\, ", "b7e22564eab60311f2b790a07c3a4fa2": "\\langle \\phi| A | \\psi \\rangle^* = \\langle \\psi | A^\\dagger |\\phi \\rangle", "b7e26c14d3652afbb66cf6cbe5056b85": "\\Vert\\Phi(a^*b)\\Vert^2 \\leq \\Vert\\Phi(a^*a)\\Vert \\cdot \\Vert\\Phi(b^*b)\\Vert.", "b7e287165bbddc489804720c9f39aa9e": " a^2 = \\frac{\\hbar^2}{12m^*k_BT} ", "b7e29356c80eac1ceb44596af3519b2c": " O\\left(n^{-1/4} \\right) ", "b7e2da2c8f82a0ddac841904f8d272b2": "223/71=3.140845", "b7e3d5666931535292f8485ef752ae1f": "\\frac{1}{R} = \\frac{2}{M^2} \\sum_{i,j \\ne i}^{N} \\frac{m_i m_j}{\\left| \\vec{r_j}-\\vec{r_i} \\right| }", "b7e3e77e66b69abdfb8637e71203376d": "n_r(n - n_r) = 4", "b7e3ef02df75984bbdc00e000035a3aa": "[w : x : y : z]", "b7e421440ca56d6d328c56355fa8a886": " \\cos(X) := \\sum_{n \\ge 0} \\frac{(-1)^n} {(2n)!} X^{2n} ", "b7e424558f485d695fecacabb4515a74": " \\left|\\Psi\\right\\rang = \\left|1,V\\right\\rang \\left|2,H\\right\\rang ", "b7e479b31166f04c7a288a77ab9ead5a": "\\!(t_1 \\ldots t_n)", "b7e4dc7d6febe3bdbff80882ee5fa066": "(D^+)_{ij} := (D_{ij})^+", "b7e4e5820da8d002ec816eff06cadb73": " 0 \\rightarrow H_{2}(X) \\rightarrow\\, \\mathbb{Z}\\ \\xrightarrow{\\alpha} \\ \\mathbb{Z} \\oplus \\mathbb{Z} \\rightarrow \\, H_1(X) \\rightarrow 0 \\! ", "b7e4e7da68e69666554c8d803e03ad5c": "R(M,x) = \\frac{x' A' A x}{x' x}", "b7e51e5a0eb8af538fd4b97c276a829a": "\\sum_{k=1}^m k=\\frac{m(m+1)}{2}\\,\\!", "b7e52fc2aacf9ab9ef03a52157fb78a9": "d(x,y) = f(x) + f(y)", "b7e5564a667dfb196a740a7e29ad1576": " \\gamma = 5/3 \\,", "b7e5addb2d8ddc61f82c1d392694a2a8": "A[x-\\ell] = Ax - \\ell", "b7e5cfed52175cfc3dbf5eec9c582477": "{ 1-e_{q'}^{-\\lambda x \\over q'}} \\text{ where } q' = {1 \\over {2-q}}", "b7e60e606a4ee251adbca6b2d81ee8dd": "\\, B_n(x)\\,", "b7e659a8db62e16be10033d0e57576b5": " y(t_0+kh) ", "b7e6809cf9d3359abbe6e49d6f3ebcf7": "\nN=T/\\gamma\\,,\n", "b7e6b3b50615673bf8efb54d5e9f90ac": "\n \\underline{\\underline{\\mathbf{f}}} = \\underline{\\underline{\\boldsymbol{K}}}~\\underline{\\underline{\\mathbf{d}}}\n \\implies \\begin{bmatrix} f_1\\\\f_2\\\\f_3 \\end{bmatrix} = \\begin{bmatrix} K_{11} & K_{12} & K_{13} \\\\ K_{21} & K_{22} & K_{23} \\\\\n K_{31} & K_{32} & K_{33} \\end{bmatrix} \\begin{bmatrix} d_1\\\\d_2\\\\d_3 \\end{bmatrix}\n ", "b7e6c23ee7a7ff2622dc04340b013584": " \\mathrm{Be} = {{32 Re L^3} \\over {d^3}}", "b7e706872bae2f1902485bb157e89a31": "A(x) = \\frac{1}{\\hbar} \\sum_{k=0}^\\infty \\hbar^k A_k(x)", "b7e72a533bf9bb637abcc45e153979ae": "x^2 - Ny^2 = k_1", "b7e72d1e5edad2ac2c3d0419ee69d192": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = \n -\\cfrac{2h^3E}{3(1-\\nu^2)}~\\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} w^0_{,11} \\\\ w^0_{,22} \\\\ w^0_{,12} \\end{bmatrix}\n", "b7e7400b3c69ed331dc7a83bdb9d587a": "E\\!\\left[e^{tx}\\right] = \\int_{-\\infty}^\\infty e^{tx} f(x) \\, dx= \\sum_{i=1}^n p_i \\int_0^\\infty e^{tx}\\lambda_i e^{-\\lambda_i x} \\, dx = \\sum_{i=1}^n \\frac{\\lambda_i}{\\lambda_i - t}p_i.", "b7e7485c807a5613cd4c93c6b7a02464": " (y'_{1}, y'_{2}) ", "b7e74959be53b9287428f2de47b49a45": "{\\rm unif}(I)", "b7e7526478b4d29e9aa642107ab6ffea": "h:\\mathbb{R}^n\\to \\mathbb{R}", "b7e75fb93ffe60420e042ed9eb360b44": "\\approx 10^{-14}", "b7e793ad72ea0e9877791b4d2a0e050c": "\\sin x = \\operatorname{Im}(e^{i x}). \\,", "b7e7aa4054047885b9306a7fa12490ff": "\\neg\\exists N \\in V: N \\stackrel{*}{\\Rightarrow} N", "b7e7eae2f2276316efcbe4200535881e": "\\delta\\Gamma^{\\lambda}{}_{\\mu\\nu}", "b7e7f1401949b28b4206290cc446d9be": "\\left|\\psi\\right\\rang = \\frac{1}{\\sqrt{2}}\\bigg(\\left|\\uparrow\\downarrow\\right\\rang - \\left|\\downarrow\\uparrow\\right\\rang \\bigg)", "b7e80289721f52b48e65f081e42bc00b": "B_{ab}^{IJ} = {1 \\over 2} (E^I_a E^J_b - E^I_b E^J_a)", "b7e8220bf182eadf440a08badbe88ade": " \\begin{align}\nA_1 &= c_1 r_1 = \\begin{bmatrix} 3 \\\\ 4 \\end{bmatrix} \\begin{bmatrix} 1/7 & 1/7 \\end{bmatrix} = \\begin{bmatrix} 3/7 & 3/7 \\\\ 4/7 & 4/7 \\end{bmatrix} = A_1^2\\\\\nA_2 &= c_2 r_2 = \\begin{bmatrix} 1/7 \\\\ -1/7 \\end{bmatrix} \\begin{bmatrix} 4 & -3 \\end{bmatrix} = \\begin{bmatrix} 4/7 & -3/7 \\\\ -4/7 & 3/7 \\end{bmatrix}=A_2^2 ~,\n\\end{align} ", "b7e8a6c911bd4d847a47aaa611f89ff8": "\\overline{R}^2=\\overline{z}\\,\\overline{z^*}=\\left(\\frac{1}{N}\\sum_{n=1}^N \\cos\\theta_n\\right)^2+\\left(\\frac{1}{N}\\sum_{n=1}^N \\sin\\theta_n\\right)^2", "b7e8b4e8d42361d1ca16bba0324a622b": "\\sum^{\\infty}_{n = 1} \\Pr(E_n) = \\infty", "b7e8b51d46ac30532e7d892e99644e1f": "\n \\begin{bmatrix}\n 1 & \\mbox{coat}\\\\\n 10 & \\mbox{lb. of tea}\\\\\n 40 & \\mbox{lb. of coffee}\\\\\n 1 & \\mbox{quarter of corn}\\\\\n 20 & \\mbox{yards of linen}\\\\\n 1/2 & \\mbox{ton of iron}\\\\\n x & \\mbox{commodity A, etc.}\\\\\n \\end{bmatrix}\n=\n2 \\mbox{ ounces of gold}\n", "b7e8bd3e4461dd6a830c9a155bbf53c7": "y = \\pm \\pi \\tan {\\theta / 2 }", "b7e8bf86dc6cc717e7edd808270314b9": "R\\mathcal S: \\mathcal F \\in D(X) \\mapsto R\\hat p_\\ast (p^\\ast \\mathcal F \\otimes \\mathcal P) \\in D(\\hat X)", "b7e9290399296bb30621bbfbe7ec3e18": " x_0=1, x_{n+1}=\\frac{x_n+\\frac{2}{x_n}}{2}", "b7e93a604eb38cfbc18104d65d417e21": "\\Delta (x)\\geq 0", "b7e94a9c08b7f1b9c5a4a82dd9c9bf28": "A : [0,1] \\rightarrow S", "b7e9834a5532ae8ea8eeebeaa9d0d355": "[\\cdot,\\cdot]\\circ (id+\\tau_{A,A})=0", "b7e99ce99291c6f8753c162d86c7cd2e": "\\left( \\left| 0 \\right\\rangle - \\left| 1 \\right\\rangle \\right)/\\sqrt{2}", "b7e9c6e337a051345deb2291cae155df": "U^{(n)}=1+\\mathfrak{m}^n=\\left\\{u\\in\\mathcal{O}^\\times:u\\equiv1\\, (\\mathrm{mod}\\,\\mathfrak{m}_K^n)\\right\\}", "b7e9c9ed0f864e3402b7ac232e8e7731": "H_\\max^\\prime", "b7e9e4a9b9823b38a3b6bb4c2a18b891": " 2p ", "b7ea0287279bb9b529bc86c732bfb3f4": "c_1^2+c_2 \\equiv 0 \\pmod{12},", "b7ea5c45a7582fbde3a2e3e9c125af6e": "|{\\psi'}\\rangle=\\sum\\limits_{\\alpha,\\gamma=1}^{\\chi}\\sum\\limits_{i,j=1}^{M}\\lambda_{\\alpha}\\Theta^{ij}_{\\alpha\\gamma}\\lambda_{\\gamma}|{{\\alpha}ij\\gamma}\\rangle", "b7ea997ad3ddff4e9291f7688651d354": "r= a \\frac{\\sin 3\\theta}{\\sin 2\\theta} = {a \\over 2} \\frac{4 \\cos^2 \\theta - 1} {\\cos \\theta} = {a \\over 2} (4 \\cos \\theta - \\sec \\theta)", "b7eaae8bf128dbe48592f7599922d3c7": " \n\\Delta E = {} - 2 \\, \\frac{\\int_M \\phi_A^{} \\mathbf{\\nabla} \\phi_A^{} \\bullet d{\\mathbf{S}} }{1-2 \\int_R \\phi_A^2 ~dV}\n", "b7eabfb7e0998079116eb6d174cac099": "f = \\left( 1 - \\frac{v}{u^\\prime} \\right) f^\\prime.", "b7eafeeb9235299d8b11bf80b4a7bd10": " \\cot \\theta = \\frac {\\cos \\theta}{\\sin \\theta}.", "b7eb05ab811c4d9205d554ea13a192fb": "\\textstyle \\mu = np = n\\theta = 98,451 \\times 0.5 = 49,225.5", "b7eb208c782703c2bd0111433660631f": "P(A|b) = \\frac{P(b|A)P(A)}{P(b|A)P(A)+P(b|B)P(B)+P(b|C)P(C)} =", "b7eb8ae8b7fc7f755e200afd8959c9ee": "t_{0}\\in \\left( 0,1\\right) ", "b7eba324cbc0e7f9978f84e9d484bc8c": "\\lambda C_1 + \\mu C_2.", "b7ebb2994de557723de6594e7033c988": "\n\\frac{j}{m} \\leq \\alpha \\leq 1.\n", "b7ec0b1eef5c262672261158f01e0f05": "\\psi(\\Omega\\psi(0))", "b7ec0f8e6af32acf00209dd063fe46ab": "~q_{s}~", "b7ec7f6cdbb33b87ead9ad8009e20c6b": "{\\mathbf{e}_k\\cdot\\mathbf{e}_i\\times\\mathbf{e}_j}=\\left[ \\begin{array}{cc}\n+1 & \\text{cyclic permutations: } (i,j,k) = (1,2,3), (2,3,1), (3,1,2) \\\\\n-1 & \\text{anticyclic permutations: } (i,j,k) = (2,1,3), (3,2,1), (1,3,2) \\\\\n0 & i=j\\text{ or }j=k\\text{ or }k=i\n\\end{array}\\right.\n", "b7ec8392054ac095c5e773dbca467e3c": "\\Theta_n/bP_{n+1}", "b7eceedb8619384a52fee99fbd200fb1": "\\mathbf{B}({\\mathbf{r}})=\\nabla\\times{\\mathbf{A}}=\\frac{\\mu_{0}}{4\\pi}\\left(\\frac{3\\mathbf{r}(\\mathbf{m}\\cdot\\mathbf{r})}{r^{5}}-\\frac{{\\mathbf{m}}}{r^{3}}\\right).", "b7ed0c227ec9e1436e0eeea79150e432": "\\epsilon\\!", "b7ed6729825aa00b8872311700f4517d": "\ny = a \\ \\frac{\\sinh \\tau}{\\cosh \\tau - \\cos \\sigma} \\sin \\phi\n", "b7ed7c825f245170177326a8bc2ed32a": "[n] ", "b7ed9adcd3e5051a15329f6452928557": "\\frac{\\pi}{k_2}", "b7edd2e5786caf2c4d3cd52307a994f8": "\\sin(\\phi) = 0\\ ", "b7edfa468af86640068ed256263fd8aa": "p \\to q\\,\\!", "b7ee6e2ef79ecd1caa723c168a7d278e": "f(p)={(n+1)! \\over s!(n-s)!}p^s(1-p)^{n-s}.", "b7ee8e0f2cd53807afee015cec5e7da1": "V_\\ell", "b7ee9d43439dd698ddf71e48abf84a7a": "gj}\\frac{1}{r_{ij}} - \\sum_{i>j}\\hat{B}_{ij} \\right) \\Psi ", "b7f92950658e373ddce5e0b79a208d13": "\nc \\int_{t_\\mathrm{then}+\\lambda_\\mathrm{then}/c}^{t_\\mathrm{now}+\\lambda_\\mathrm{now}/c} \\frac{dt}{a}\\; =\n \\int_{R}^{0} \\frac{dr}{\\sqrt{1-kr^2}}\\,.\n", "b7f9e14b5e868ab2044ccdd3c5675d0c": "p={S \\over N}", "b7f9eb8b0c14c82445b8f69fb2c48afc": "xy = a^2", "b7fa641e3b224dc9e8c686819bf9f6da": "x = 2 \\quad (L_1)", "b7fa660074967ceab080a1c709b414f7": "Z_{i,j} Z_{k,l} - Z_{i,l} Z_{k,j}.\\ ", "b7fab0de84b7adb7b2c29c2dc92bcdb4": "\\kappa = 5/6", "b7faea0340c55ba978e664ba848a514a": "P_i^R\\,\\!", "b7fb618072508c8101d47b0e581006be": "\\lim_{k \\to \\infty} \\bigg( \\sum_{i=1}^n \\left| p_i - q_i \\right|^k \\bigg)^{1/k},", "b7fb672c544d8d27d8beee73a62f31a2": "V_t(k)", "b7fb6ad459383cd018bc45ee244f79ea": "a=-k", "b7fb7a491b6c4c8c08ba6cf69ad6eb24": "\\frac{B_k}{k(k-1)}", "b7fb892098b677ebb84364f689555527": "ac_0/c_1", "b7fb8fdc5f38572a231460173384c165": "\\begin{matrix} {52 \\choose 7} \\end{matrix}", "b7fbc54261487977503b9f79c6e3b90e": "\\omega = {2 \\pi f} \\!", "b7fbc830885f6fdd8cd00cd8aa528285": "T_M(d)=\\frac{2T_{MB}}{H_fd}(\\sigma\\,_{sl}-\\sigma\\,_{lv}3(\\sigma\\,_{sv}-\\sigma\\,_{lv}\\frac{\\rho\\,_s}{\\rho\\,_l}))", "b7fbe33834453942126375e78e02a296": "\\vartheta_{00}(0; \\tau)", "b7fbf7955f26999a6acc3e996835072a": "k \\leq g", "b7fc055ca24d5d1e6b3f8e0810bf58c3": "\\mathbf{G_x}", "b7fc1fdacb92023398c74a344f7c22ad": "f(e_{ij}) = 0", "b7fc2e699d36c258ae4ef27235daabab": "<\\overline{16}_H><\\overline{16}_H>16_f 16_f", "b7fc30fef6dd2b6706b281ae9d7a5848": " h_1,\\dots,h_p ", "b7fc3f646461b51e4056cddcf5c5e38e": "n>k", "b7fc663831c2a43ff7b8e638afbb8a2e": "2 \\pi M_i {L_{cm}}/N_i", "b7fd1fff0c8ae7c71eeb6f611588d4dd": "\\ell(w_0w) = \\ell(w_0) - \\ell(w).", "b7fd3ffc8855d0bd89e501489d587156": "\n B_\\mu = A_\\mu + \\partial_\\mu \\omega.\n", "b7fd7223347aaf8dbc22fc2fcd70072d": "\\angle", "b7fdb37abe4e2f06d60b4c033cf9f541": "S = \\{v_1, \\ldots, v_n\\}", "b7fdc1b211b5c305f9c00018c6ed4fcc": "r_{i+1}(x_j) = r_i(x_j) + \\alpha h(x_j) y_j", "b7fddd7bf463639d4ee15a688e0e746e": "\n\t{\\nabla^2 u -\\dfrac 1{c_0^2}\\frac{\\partial^2 u}{\\partial t^2} + \\tau_\\sigma^\\alpha \\dfrac{\\partial^\\alpha}{\\partial t^\\alpha}\\nabla^2 u\t- \\dfrac {\\tau_\\epsilon^\\beta}{c_0^2} \\dfrac{\\partial^{\\beta+2} u}{\\partial t^{\\beta+2}} = 0.}\n", "b7fde4b6aeb47bda09d17dd86fd25012": " \\alpha,\\beta \\in \\mathbb{N}^N", "b7fdf8b134c2d946858b3f7424be8108": "K \\subset L", "b7fe3be5922782d74eacff43604b59ec": "\\left| \\psi \\right|^2", "b7fe76b5602277ffbcff6363fef1ed6a": "(1-|z|^2)|f^\\prime(z)|", "b7fef59c720e1097bab325454999193b": "code(x)", "b7ff01d5e1765edbfa5322aebe921384": "\\frac{d[A]}{[A]}=-k dt", "b7ff051ae1ec21a1cd2bd77c2a9d9054": "\\Theta_1, \\cdots, \\Theta_s", "b7ff21f9f6c4f22354621c151c1755d1": "\\textstyle \\underline{X}", "b7ff2f896c3cc38db0fa80314d115e92": "X=\\sum_{i=0}^{m-1} a_i n^i.", "b7ff486cacb0a8c573c1f9cc87e701ce": "\\frac{56}{60} = 0.9\\bar{3}", "b7ff735e0ca4418cec97d416aad39cb7": "F_X(x)(f) = f(x), \\quad x \\in X, \\ \\ f \\in X',", "b7ffc0e38fd8b09ddd7e0d00eaeb6919": "\\Pr(\\mathbf{x)_A} = N! \\prod_{i=1}^k \\frac{p_{i}^{x_i}}{x_i!}.", "b80054a17a801f558add29c75e80bcaf": "\\!\\,\\gamma : I \\rightarrow X.", "b800a72afc92adc9af78439c92965bca": " k_x = k ~ \\sin \\theta ~ \\cos \\phi ", "b800d2314816d68d9970ae4b93cc4c67": "\\frac{x^{T} M x}{x^{T} x}", "b80145380e89514839fe3b0fb573daf3": "J_i = \n\\begin{bmatrix}\nC_i & I & \\; & \\; \\\\\n\\; & C_i & \\ddots & \\; \\\\ \n\\; & \\; & \\ddots & I \\\\\n\\; & \\; & \\; & C_i \\\\\n\\end{bmatrix}.", "b8014864b3a8cf343991ac4b219686b4": "\n 1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4) \n + p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 = 0\n", "b80154d3af5b253acc6b0d44562fb7b4": "\\wedge^{m+1}", "b8018d79411ff313d1546892816ca065": "Q(s_t,a_t) \\leftarrow Q(s_t,a_t) + \\alpha [r_{t} + \\gamma Q(s_{t+1}, a_{t+1})-Q(s_t,a_t)]", "b8019917054b350b99e6b86e092537a5": "\\mathcal{H}^m \\llcorner M", "b802546f482b4dcf1597133d932e888f": "M/Q", "b802b95df66ebd6faa827cb240a35df3": "\\text{Total costs} = \\text{fixed costs} + (\\text{unit variable cost} \\times \\text{number of units}", "b802c98ba1917880b477e7217330a97f": " y_n = p(x_n) = \\frac{1}{N} \\sum_{k=0}^{N-1} Y_k \\ e^{i 2 \\pi \\frac{nk}{N}} \\, ", "b802d16d3619892c1f8e62df79f704e2": "M(B) = \\inf\\{ m(V) \\mid V \\text{ is an open set with } B \\subseteq V \\subseteq X \\} .", "b8031d5131b6a0cc03a646dd10d6de4b": "\\overline{P_1P_3}\\cdot \\overline{P_2P_4}=\\overline{P_1P_2}\\cdot \n\\overline{P_3P_4}+\\overline{P_1P_4}\\cdot \\overline{P_2P_3}", "b803be0c70d543b51833076d04b64cf2": "Z^Y", "b803e28e43222a0bd42dd1e9cb1d60a0": "i_t = \\pi_t + r_t^* + a_\\pi ( \\pi_t - \\pi_t^* ) + a_y ( y_t - \\bar y_t ).", "b80449a31f661fca7353453d7bf6899c": "\n\\begin{align}\nP(X_{(k)}\\leq x)& =P(\\text{there are at most }n-k\\text{ observations greater than }x) ,\\\\\n& =\\sum_{j=0}^{n-k}{n\\choose j}p_3^j(p_1+p_2)^{n-j} .\n\\end{align}\n", "b80450c901f5bbda210d513c9118020a": " y = \\rho \\sin (\\alpha \\theta), \\ ", "b804b198db9dc940133270a74935447c": "D_i = \\frac{L^2}{\\frac{1}{2} \\rho_0 V_e^2 S \\pi e AR} ", "b804b1acac63098f1141a1920bbd8452": " i \\hbar {d \\over dt} U(t) = H U(t).", "b804d0ecfc09846662d8a139d1008c43": "\n\\Psi_{\\theta\\theta}+\n\\frac{v^2}{1-\\frac{v^2}{c^2}}\\Psi_{vv}+v\\Psi_v=0.", "b804f77ee0ecd149f48b12066658e545": "X_1, X_2,\\dots ", "b805a5c4f0315b99770bba237275e251": "H^n(X, F) \\rightarrow \\check H^n(X, F)", "b805aa9baed73948950fab8041fd235f": " \\int \\mathbf{v}_i dm ", "b80705cae0394601a62c164e8abf590c": "U=3PV", "b807bba33c04236e8fe26ef83de1ca46": "1 \\leq q_0 \\leq q_1 \\leq \\infty", "b807ea01665cfa0b6ac2c7136a7624a4": "(A_2,\\le_2)", "b807f7efdefd9dfe222e51972863e2c2": "\\text{Min} =\n\\begin{cases}\n f\\left(8.05502, 9.66459\\right) & = -19.2085 \\\\\n f\\left(-8.05502, 9.66459\\right) & = -19.2085 \\\\\n f\\left(8.05502,-9.66459\\right) & = -19.2085 \\\\\n f\\left(-8.05502,-9.66459\\right) & = -19.2085\n\\end{cases}\n", "b8080657a5db65775cb2d5e4c60b88b6": "e^{(\\varepsilon_{\\rm min}-\\mu)/kT} \\gg 1, ", "b808bf72b850187d20ab6ca71b2467f1": "\n \\bigg(\\frac{1}{T}\\sum_{t=1}^T \\frac{\\partial g}{\\partial\\theta'}(Y_t,\\hat\\theta_{(i)})\\bigg)' \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\hat\\theta_{(i)})g(Y_t,\\hat\\theta_{(i)})'\\bigg)^{\\!-1} \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\hat\\theta_{(i)})\\bigg) = 0\n ", "b808f684e6601ac93328e346f41cf077": "A^{n}\\,", "b809b034e642d8b8b70287202e30c9fe": "n=2\\mod 4", "b80a3c347d61a79d455168927351ec90": "\\ V^2 = 0.10875 \\times \\frac{4 G}{3 D_3}", "b80a8f76ac79791c54019485d23bc7f5": "f_{tsc} = {{1 \\over{2 \\pi f C_{tsc} L_{tsc}}}}", "b80a9a2652690f18b3cf0a45fe5dfc3d": "\n \\boldsymbol{\\nabla}\\mathbf{u} = \\boldsymbol{\\epsilon} + \\boldsymbol{\\omega}\n", "b80aa2fbaf064390802304d6f4eb3ca1": " \\displaystyle \tf(\\boldsymbol{x})=-2x_1+x_2-x_3+4x_1x_2+4x_1x_3-2x_2x_3-2x_1x_2x_3. ", "b80aa416f81ff0e16ae33f68ccc48d24": "M = \\frac {f}{f_{e}} = \\frac {1200}{3} = 400", "b80afa3b3aefaea1241a4149171eefc4": "(E,\\,M)", "b80b8b1209aab7d17f973a594f30ff17": " + \\left( {\\partial v_x \\over \\partial z} - {\\partial v_z \\over \\partial x} \\right) \\mathbf{j} + \\left( {\\partial v_y \\over \\partial x} - {\\partial v_x \\over \\partial y} \\right) \\mathbf{k} ", "b80b8fca2cc90974040645b8993a4351": "u(D) = \\sum_i n_i u(p_i)\\ ", "b80ba298a5a61a021ae1f14eacd9b18a": "\\rho_1,", "b80bae73ffd445a2111bd3dac5b21f91": "\\mathfrak{o}_{2n+1},", "b80bb7740288fda1f201890375a60c8f": "id", "b80bdc42db24cbd3caf5e911c1175094": "\\sum_{p^k} f(p) = f(2) + f(3) + f(4) +f(5) +f(7)+f(8)+f(9)+\\cdots", "b80bf9f6699f85e916569ba4502154f5": "{A}_{7}^{(1)}", "b80c0eb4daa4390f48d13a09f84c9eec": " s > 0 \\;", "b80c132c276e9c7dc780a0b7864a656d": "x^p = y^q", "b80c66751d461e1977c00ba46ac29b9e": "B^+ = \\Omega BQ", "b80ccf3527672b23893d6850b10a5924": "\\sum_{n\\in S}\\frac{1}{n} = 1.", "b80d2105be69c7299c9b32677fc6d765": "(\\Re)", "b80d3743a94f68ef56ac6e8de7341bdc": "\\left(R\\left(x,u,u_x,\\ldots,\\frac{d^n u}{dx^n}\\right),w_i\\right)=0 ", "b80d49cef94af4573ad492189cf1bd9e": "X_C = \\frac{1}{\\omega_\\mathrm{d} C} \\,\\!", "b80dbceef3938176350492ba59c1ce69": "\\operatorname{perm}(A + B) = \\sum_{s,t} \\operatorname{perm} (a_{ij})_{i \\in s, j \\in t} \\operatorname{perm} (b_{ij})_{i \\in \\bar{s}, j \\in \\bar{t}},", "b80dd089111078f55e4b1f3e91d5a0b8": "(R,\\mathfrak{m}_R)", "b80df78e21075311116ed1cceb9ad242": "2 h f f' m_e c", "b80df81a5faf8b9892bef0691286dbe7": " \\Lambda (n) ", "b80e1799da326ae676f176849ac128b9": "2\\lambda", "b80e329b92db0b3801b1a7f7ecaac250": " | 2(p-n)-1 | \\le 3k^2 - 3k + 1, ", "b80e62b23e9280fc62fa8cc38d47c7f1": "x^2+16x", "b80ea96507544cc137f47a4adab446aa": "\\delta(\\alpha\\mathbf{x}) = |\\alpha|^{-n}\\delta(\\mathbf{x})", "b80f60b2240d539d24c4e944a59ff85f": " \\boldsymbol{v} ", "b80fcc547cdd29ca4ba79c13f638c5a7": "6\\times \\left[ 1^2+2^2+\\cdots+\\left( \\tfrac{a-1}{2} \\right)^2 \\right]", "b80fe26f9922bb3d4c6881149ea822fd": "\\phi(9)=6", "b81045dcb6b722093ac0cb8273e23f43": "T_e = \\frac{(1-R)^2}{1+R^2-2R\\cos\\delta}=\\frac{1}{1+F\\sin^2(\\delta/2)},", "b810639022fb634222a351cf6a360c5e": "\\ ASA_{native}", "b81070f0d46c41f7bd201235dd5a1c6b": "k= 2", "b8107e269a90b9b511302b4fcbd47e26": "r_M=\\sqrt{rR}={a\\over\\sqrt{8}}\\,", "b810cd46a205e24425d6e8f23e6a11be": "f\\lambda = f^\\prime\\lambda^\\prime = c,", "b810d1dedc8fd896d253b051dbe5e8f4": "\\,f_1'(x)=-\\sin (x)", "b810f069a1b55d774ecfcb05ea587b84": "\\Delta G = RT \\log \\frac{C_{inside}}{C_{outside}}+\\Delta G^b", "b81131fa69e5aa46d91728222b9f62b3": "\\lim_{t\\to\\infty} \\frac{N(A,t,k,q)}{N(A,t)} = \\frac {1}{q}", "b81178954303281124ec5dbe41a2a694": " {u_2}=p-c", "b811ad605a77c1640898116e6ca222e8": "(x - \\alpha) ^r", "b811c19e856929dbb7d2a1f8ce6613cc": "\\tau_{a b ;c}\\equiv \\partial_c \\tau_{a b}-\\Gamma^d{}_{c a}\\tau_{d b}-\\Gamma^d{}_{c b}\\tau_{a d}", "b8121ae7c3f6b7b3f1fc9683ad4814ae": "\n\\Delta E = \\sum E_{products} - \\sum E_{reactants}\n", "b8123d40e9f602d6d6447eea79969c94": "x = 10 - 0 = 10", "b8125447d7ad29baf3a1254a66ecacc7": "(K_b(message))", "b81285463678bc5a76db8cdf1065fdac": "\nR_n(D) = \\frac{1}{n} \\inf_{Q_{Y^n|X^n} \\in \\mathcal{Q}} I(Y^n, X^n)\n", "b812fc909cd2045960ccb93b939aaf85": "\\{\\ :\\ \\} \\!\\,", "b813540bd9163909f71955a6d0c46466": "R(\\Psi_b) = e^{\\pi i m b^2} \\Psi_b.", "b81356a91208e2860157f4dbc108819a": "\\cot (2 \\theta) = \\frac{\\cot^2 \\theta - 1}{2 \\cot \\theta} = \\frac{\\cot \\theta - \\tan \\theta}{2}\\,", "b813f9c78257b57925d6d472b66ea8e6": "2^{\\aleph_0} = \\aleph_n", "b814591d75089ea3f1d491e610c8eb31": "\\mathrm{CH_3OH + H_2O \\to 6\\ H^+ + 6\\ e^- + CO_2}", "b814a1ab6d418b7af545ccbcefbbe66d": "\\gamma = -i", "b814edf85cd6c858ce819e0d97167107": "\\mathrm{Inv}^1 \\langle X | T\\rangle.", "b814fd22cd53f108fe452fe5090a68f3": "K+B*n", "b8151f6178433af03a399b1971d6448c": "\\Delta=0", "b8157129df45d665c4148d55e1adeb0d": "= (x_1*y_1)(1+\\delta_1)(1+\\delta_3)+(x_2*y_2)(1+\\delta_2)(1+\\delta_3)", "b8159f8b10c7e9a7d4daf9c560f93ab4": "w(\\Gamma)", "b815ae449bac5c9f4b8a16ed24586776": "\\begin{cases}\nF:\\mathbf{R}^2\\to\\mathbf{R},\\\\ \nF(x,y)=-50(y-cos(x))\n\\end{cases}", "b81603b86fcb2802ebe4710de6b801fe": "\\alpha\\colon A \\to C, \\beta\\colon B \\to C \\in \\mathsf{Hom}(\\mathbf{CRing}) ", "b81609ee8cb7e92f44be27248a70cef7": "\\Sigma F_y=-T_{2y}-T_{1y}=-T_2 \\sin(\\beta)-T_1 \\sin(\\alpha)=\\Delta m a\\approx\\mu\\Delta x \\frac{\\partial^2 y}{\\partial t^2}.", "b81621b44ab1144158d50e5549e5b3a7": "LWE", "b8162735544391d07f4bfdcc71247992": "\\operatorname{MSE} = (c (n-1) - 1)^2\\sigma^4 + 2c^2(n-1)\\sigma^4", "b816c033c294a32f4d34a40bd09a1c4f": "p(y) \\propto \\exp\\left[ -\\frac{1}{2\\,b_2} \\ln\\!\\left(1+\\frac{y^2}{\\alpha^2}\\right) -\\frac{\\ln\\alpha}{b_2} +\\frac{2\\,b_2\\,a + b_1}{2\\,b_2^2\\,\\alpha} \\arctan\\left(\\frac{y}{\\alpha}\\right) + C_1 \\right]", "b816d12c3c71d1bc2a5df2a5b4188e20": "\\boldsymbol{u}^{(1)}(\\boldsymbol{x}, t)", "b816f57af9249104ca2d5f62050491e6": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\binom{t}{k} = \\binom{t}{k} \\sum_{i=0}^{k-1} \\frac{1}{t-i}\\,.", "b816fe110ef11ad51e27f35a2e833840": "I = \\lim_{\\Delta t \\rightarrow 0} \\sum_{i=1}^n \\mathbf{F}(\\mathbf{r}(t_i)) \\cdot \\mathbf{r}'(t_i)\\Delta t", "b8170cb7d3efb46b90d1382f83803d58": "r_y", "b8176f1c20e0fe79b8d090564a1a8524": "y\\in H_q(LM)", "b8179104036b155358217a078e53097a": "P_1(m,n)=\\pi(m)-n", "b817f1c34c5298355697bc680c79e933": "B_{z/2}=2", "b817fc306a2ef387d2d80dcce5815bb4": "a_{2n+1} = \\frac{T_{2n+1}}{(2n+1)!} = \\frac{2^{2n+3}}{\\pi^{2n+2}} \\sum_{k=0}^{\\infty} \\frac{1}{(2k + 1)^{2n+2}}", "b8188de67cd4e56b94fedc1253e82ae1": "\\frac{A t_c}{a x_c} = \\frac{A}{b x_c} = 1 \\Rightarrow x_c = \\frac{A}{b}.", "b818a00353f60a8085181e899340c3a4": "\nf \\sim \\mathcal{GP}(m,k).\n", "b818b9e18078b8cb3dac3a825385ce77": "({P}_{3}/{P}_{2})", "b818bf403266637a1a17c64f181d54c9": "(\\Box p \\lor \\Box q) \\rightarrow \\Box (p \\lor q)", "b818ebeaed41ff1adf7aefe86154e804": "v_S = 0", "b81921016641270de75dce10647054e9": " h_{ab} \\rightarrow \\tilde{h}_{ab} = \\Lambda(\\sigma) h_{ab} ", "b81a1bc2f4b65265e783b541e1e8abed": " i \\partial \\bar{\\Psi}^\\dagger \\mathbf{e}_3 + e A \\bar{\\Psi}^\\dagger = m \\Psi ", "b81a1c8d6b35c25f32a1c5a7b718680b": "N_e = N + \\begin{matrix} \\frac{D}{2} \\end{matrix}", "b81a2307cf9dfafbaeb3f69952db5dc9": "B = Attr_{1-i}(W_{1-i}')", "b81a2ecf1a506afffd4cbb25cc4ce4bc": "{\\ \\mathrm{d}U = \\delta Q + \\delta W }.", "b81a47dc5e6dbddc318eae52b59f9e48": "F(2) + F(6)", "b81ae774fb9a9772fdb84cf877eb3c03": "f=P(D)y_p=P(D)(u_1y_1)+P(D)(u_2y_2)+\\cdots+P(D)(u_ny_n).", "b81af82a9bf3af2cf172f87cf029d8a5": "\\lim_{x \\to c} \\, [f(x) \\pm g(x)] = L_1 \\pm L_2", "b81b432dc9e045d57db3f9ee2d12eb74": "\na |n \\rangle = |n-1 \\rangle \\sqrt{n} \\quad\\hbox{in particular}\\quad\na |0 \\rangle \\propto 0, \n", "b81b62f7f5a9caeaf5d7d5868cc9bd65": "\\mathbf{F}_{\\mathrm{net}} = \\frac{\\mathrm{d}\\mathbf{p}}{\\mathrm{d}t} = \\frac{m\\mathrm{d}\\mathbf{v} - (\\mathbf{u} - \\mathbf{v})\\mathrm{d}m}{\\mathrm{d}t} = m\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t} - (\\mathbf{u} - \\mathbf{v})\\frac{\\mathrm{d}m}{\\mathrm{d}t}", "b81bb5f29240bb60a3280e000a6163ab": "z=g^{ab}", "b81c424a641eef222f9055b01c436df7": "Math \\geq good", "b81d029719f7f31a33dbe2eff214a7fc": "\\left( p,f\\left( p \\right) \\right)", "b81d6e4c57aeed0348950759cce8f6a8": " (a,b) \\mapsto (a):", "b81dc6f35f108a053d40693836998778": " \\and (S_5 \\implies (\\operatorname{equate}[A_5, m] \\and V[F_5] = m)) \\and D[F_5] = D[m] ", "b81ec1fb5edac998ae644f70b1c2315c": " \\sigma_\\theta = \\dfrac{Pr}{t} \\ ", "b81ecf3b3dcec3a2f3e76c067c95023b": "\\psi(r, \\theta, \\phi) = R(r)\\Theta(\\theta)\\Phi(\\phi) \\,\\;", "b81ee109551bc7a3dae3f9e6f707f687": "\\scriptstyle\\bold{S}", "b81f15acff7574a7b55c62e2ad516548": "\\frac{x ( 180 - x )}{ 90 \\times 90} = \\frac{x ( 180 - x )}{ 8100}.", "b81f1d152b34fd793ba0ed270c7abd5b": "ATR = {1 \\over n} \\sum_{i=1}^n TR_i", "b81f6a0e3d64ca6463a122054e3ea3e4": "1.6521", "b82044cbe3094dc03e35dec3de4c0f0d": "\\scriptstyle\\square^2", "b820475bf60ed0c263730d061ad40590": "\\vec{b}(x_i),u,\\vec{a}(x_i) \\in [0,1]\\,\\!", "b8204f63f0bfdfbbd50cbd1239bced1b": "(\\xi \\vee \\eta)_{inf}(\\alpha)=\\xi_{inf}(\\alpha)\\vee\\eta_{inf}{\\alpha}", "b8205c8a9ccbdff2945e12fb6754f5ee": "O(P_{r}+P_{s}+P_{r}\\log(P_{r})+ P_{s}\\log(P_{s}))", "b820a9dd1d18bfe1c906624633d98afc": "6+5+5+3+6 = 25", "b820f0f036441fd093b0a48f39f6cd9f": "\\|A\\|_F=\\sqrt{\\sum_{i=1}^m\\sum_{j=1}^n |a_{ij}|^2}=\\sqrt{\\operatorname{trace}(A^{{}^*}A)}=\\sqrt{\\sum_{i=1}^{\\min\\{m,\\,n\\}} \\sigma_{i}^2}", "b8218ed94ba96d430f6737eff255e57e": "y_0, \\ldots, y_m \\,\\!", "b821c59a6940bd76ded585a5b4dcfe57": "d = a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n,", "b821ccbeb808b349c1a37bc63b33d15d": "Ra=\\frac{g\\alpha T D^3}{\\nu \\kappa} , E=\\frac{\\nu}{\\Omega D^2} , Pr=\\frac{\\nu}{\\kappa} , Pm=\\frac{\\nu}{\\eta} ", "b8221c3fbf010127b9a27e0370da40d2": "[g_{ij}] = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & r^2 & 0 \\\\\n0 & 0 & r^2\\sin^2\\theta \\\\\n\\end{pmatrix}", "b8222346a3e827efe6165fc9c286188e": "x = (a - b) \\cos(t)\\ + b \\cos(t ((a / b) - 1))", "b82244d7bf94239987fb5b799043e1bc": "\\displaystyle{H_f(a)=f^{-1}(a).}", "b822563bee1367c98ac46728ce5da588": " P_B,2=p_0+g\\rho_2z\\!", "b82294f5183775732257c5f9da84be66": "h\\leq i < j", "b822e378e7b2840e2cad7184a22da916": "(10\\uparrow\\uparrow)^{2} (10\\uparrow)^{497}(9.73\\times 10^{32})", "b8234ddeec5ab2c1ef081869ecf12529": "\\beta \\ \\stackrel{\\text{def}}{=}\\ \\pi\\alpha.", "b8236ec2fa05659e0394f9e062339e16": "\n\\beta = { F_{\\rm r} \\over F_{\\rm g} } \n= { 3L Q_{\\rm PR} \\over { 16 \\pi GMc \\rho s } }\n", "b823ba11deac57a95afaf50ae898671f": "Z_\\mathrm S = Z_\\mathrm L \\, ", "b824284121fe0cb8822d6116f429d7f4": "x^8 + x^7 + x^6 + x^4 + x^2 + 1", "b824540ec3486856ae4f8bb71f20cbd8": "R, \\, R_1, \\, R_2, \\, R_3, \\, \\Re (W_1), \\, \\Re (M_1), \\, \\Re (M_2)", "b824573201ea26f26e4a5f81fc58b368": "u_x(\\mathbf{x})", "b8248ff40ad49638e66e5e4060c4aa2d": " N+O ", "b824c87d57cb94c0bb654d488c4c0db8": "d(1)+d(2)+ \\cdots +d(n) = n \\ln n + (2 \\gamma -1) n + O(\\sqrt{n}).", "b824f68fb07f81a2296aa3619c9a3feb": "x \\cdot (x^{-1} \\cdot y) ", "b8254dd3c8126609a0bfb7b6143aa2ce": "\\scriptstyle P(x)", "b8257326220b9167e2803b08c55ed5e4": "\\sum_{x\\in X}\\left|f(x)\\right|^p<+\\infty", "b82586551a45082f52e32203d6d680c9": "\\mathbf g(\\mathbf r) =\n- G {m_1 \\over {{\\vert \\mathbf{r} \\vert}^2}}\n\\, \\mathbf{\\hat{r}}\n", "b825b8a7f0d566903cdfd22376b461ae": "F = IdB_{\\text{avg}} = \\frac{\\mu_0 I^2}{ 2\\pi}\\ln \\frac{d}{r} ", "b825f2ef7386c322b740f1bf147b441a": "\\lambda_T = 0\\!", "b8266ecf0c4a61c66f9e2df8ae3d5543": "s_i = \\left \\lceil \\frac{f_{\\mathrm{max}}\\cdot(t_i - t_{\\mathrm{min}})}{t_{\\mathrm{max}}-t_{\\mathrm{min}}} \\right \\rceil", "b82675bb398d5dfe002a65bd169c69f6": "Q = \\{P_4, P_5\\}", "b8267b04c8fd091bf4ffefd514ddd4bf": " f(t) ", "b826926a2f39071df8465411c0c25bf8": " \\min\\limits_{x \\in X}\\;\\; f(x) ", "b826b1fa375ad9fda751f0598cc14304": "\\displaystyle \\tan{\\frac{\\theta}{2}}=\\sqrt{\\frac{bd}{ac}}.", "b826ffe1e8f9a455d68fb566d0e256ac": " u = \\frac{1}{4\\pi r}.", "b82731c4dc50304f71eabd7777583d9f": "\np = \\rho R_s T\\,\\!\n\\,\\!", "b827d64a41806c50ed774a4e71693e23": "R \\to S", "b827ddaebd7b07bd4d4caf964e9ccdf3": "D/l_0 \\to \\infty", "b827e8f7e0c79093c044da645ab6c07b": " U = 0", "b827f5b1d826250f6221152c4cdaa45f": "S_F(x-y) = \\int{{d^4 p\\over (2\\pi)^4} \\, e^{-i p \\cdot (x-y)} }\\, {(\\gamma^\\mu p_\\mu + m) \\over p^2 - m^2 + i \\epsilon} \n\n= \\left({\\gamma^\\mu (x-y)_\\mu \\over |x-y|^5} \n+ { m \\over |x-y|^3} \\right) J_1(m |x-y|).\n", "b82813d60777838a0bd49b5ed7c01cac": "U=C-\\lambda \\frac{\\delta C}{\\delta \\lambda}", "b828269912d295cde899a40067033807": "d=3:", "b82831376352e9b2a605a954fb8f95c3": "\\displaystyle{|\\mathbf{v}(t)-\\mathbf{u}|^2 = \\lambda^2 + t^2(1-\\lambda\\kappa(0)) + O(t^3)\\ge (\\lambda^2 + t^2)/2,\\,\\,\\, \\,\\,\\,|(\\mathbf{v}(t)-\\mathbf{u})\\cdot \\mathbf{n}(0)| =|-\\lambda + {t^2 \\over 2}\\kappa(0) + O(t^3)|\\le |\\lambda| + C_1t^2}", "b8284f13bbdfab44dbb820065b67a338": "\\mu_j", "b82860435e4b39b977533a2f87ca0b9a": "\\omega\\{n: d(x_n,x)\\le \\epsilon \\}=1.", "b8289d17f97ba52d49c1b0a69f0fa28e": "\\mathfrak{g}_0 = \\mathfrak{k}_0 + \\mathfrak{a}_0 + \\mathfrak{n}_0", "b828f3e76ffa893672ffdf0c2216d2b2": " d\\Omega^p_X(\\log D)(U)\\subset \\Omega^{p+1}_X(\\log D)(U) ", "b829523742460585ba70a24be27211a5": " W = P_1 V_1^\\gamma \\frac{V_2^{1-\\gamma}-V_1^{1-\\gamma}}{1-\\gamma} ", "b829b07a4f7dbe5903a7c0c85deded88": "\\ y = a_0 + a_{1}r + a_{2}q + \\nu ,", "b829da9fcb62a395452955350f668cd0": "\n\\Omega_{k,l} = \\Big\\{ [i_1,\\ldots,i_m]\\in \\mathbb{Z}^m;\n\\sum_{j=1}^M i_j \\!= l-1, i_k=0, i_j\\geq 0 \\,\\, \\text{, for all } j\n\\Big\\}.", "b829f02a39c49f5574f79b57eb55f9fb": "F[x,y]= \\int_a^t \\sqrt { x'^2 + y'^2 }\\, dt", "b829f2c9998dfb9456883bfa6ae414c4": "pIC_{50} = -log_{10} (IC_{50}) ", "b82a13156c4dd1727e1fe9e6fcb1b663": "n > n(\\sqrt{2} - 1) = m - n > 0", "b82a32a661ce9d3204abeb3b41a249e8": "b > 2a", "b82aa7bb6835aa988cca044034af8edb": "A, B\\in K", "b82ad0f4c05ef06a455c2d80e9cd8ff9": "4 + (11 \\cdot z ), z \\in \\mathbb{Z}", "b82af737d80fbbc821493725d28876f1": "P_m \\times P_n", "b82b22a4c1aab2f92879c851dceda92e": "c \\colon U \\to \\mathbb{R}^n\\;", "b82b303b7b7f8a40d4c5a0fd1c992932": "{{r}_{i}}={{\\varepsilon }_{i}}/(cycle\\text{ }count)", "b82b53226cf3a845627d2dd0a1f60220": "\\boldsymbol{U}_i,\\boldsymbol{U}_{i+1} \\rightarrow \\boldsymbol{U}", "b82b5d705ff42045e393ad7a5517446b": "g^{-1}(s)", "b82bb955c089797d74812bd63db2fcd5": "\\delta \\cdot d_e(x_e)", "b82bfb74798194aec4fe9651ab5b0b69": "r_>", "b82cbc85f7c6a7500d47aaf7cc29f6f3": "\\scriptstyle\\vec\\varepsilon", "b82d6bdc7c931e0d0169f884aeb75ca4": "f(x_i|\\theta)", "b82d6dbcf20b18e9db0cfe8cc7ff4c25": "P(A,B)", "b82d6ede334f248454bf3e5df32bd92c": "\\int_M df(X) \\;\\omega = - \\int_M f \\, \\operatorname{div} X \\;\\omega ", "b82d9736580a6c2c30a08f87054fb71d": "\\frac{\\partial }{\\partial \\tau }W(\\xi ,\\tau ) = - W(\\xi ,\\tau ) \\wedge H(\\xi )", "b82dde5e24ea4e44c222d1e867c11a8d": "screen\\ x\\ coordinate\\ (Bx)\\ =\\ model\\ x\\ coordinate\\ (Ax) \\times \\frac{distance\\ from\\ eye\\ to\\ screen\\ (Bz)}{distance\\ from\\ eye\\ to\\ point\\ (Az)}", "b82e0c100f6cafd57cc7f7053614c721": "\\mathfrak l=\\mathfrak k\\otimes\\mathbb C", "b82e4fc2a3dba2d2142c79f4af704ae1": "\nf \\propto \\sqrt{T}\n", "b82e6d5d0e451d9c4495018a82e22bba": "\\chi^2=\\sum_n \\frac{\\left[p_1(R_n)-p_2(R_n)\\right]^2}{\\sqrt{p_1^2(R_n)+p_2^2(R_n)}}", "b82e873c804cb3989c463901ddeee322": "\\langle\\mathbf{p}\\rangle", "b82e992d49de1b14203535f4c7fe5b24": "n^2/4", "b82ead4fc1ebd49ccdd8df2992266f26": "\\ll 200", "b82eb111fc2c3cf3691d2cafc782acee": " |\\psi \\rangle := (2^T-s)|0 \\rangle + s|1 \\rangle. ", "b82ed60eb1e2d8bc5b82ffe4c1cae427": "\\sum_{v \\in V} \\deg(v) = 2|E|\\, .", "b82f22cffb52cf43c269f562ed0220cf": "\n W_n = \\psi_n(P)\\qquad\\text{for all}~n \\ge 1.\n", "b82f6aa248bfe451b37194b9b7af428e": "E = E_{\\rm{electronic}}+E_{\\rm{vibrational}}+E_{\\rm{rotational}}+E_{\\rm{nuclear}}+E_{\\rm{translational}}\\,", "b82fd7c1600f413fb50ee0429cba5e06": " \\mathbf{f}_n=\\begin{bmatrix}\n\\mathbf{x}_n \\\\\n\\mathbf{z}_n\\end{bmatrix} ", "b83015a9c976474a0cfb14180fda179e": "F(x) = \\sum_{k=1}^\\infty \\frac{x^k}{c_{2k}(k-1)!} = 12x - 720x^2 + 15120x^3 - \\cdots", "b830690ab2d6e3a64083e9d2d6cdf052": "T_{d} \\le B(log_{B}(M) - 1)", "b83090ebdd36c12daa183f389518bed7": "\\begin{matrix} t_1 & \\rightarrow & t_2 \\\\\n \\alpha |\\uparrow\\rangle + \\beta |\\downarrow\\rangle & \\rightarrow & |\\uparrow\\rangle.\n \\end{matrix}", "b830bb0441cdeca49c8a7c3111facd53": "r_n=r_{n+ik}", "b830c8cca62b66614c6fcffe0403afa6": "\\frac{T}{W} > 0.15", "b831479ce4fd15624d48f471ca282de4": " \\lim_i M(x_i) \\leqslant \\lim_i \\frac{f'(y_i)}{g'(y_i)} + \\epsilon_i ", "b83184d27fc82343101f1039b170b34b": "\\begin{align}\n \\mathbf{A} &\\rightarrow \\mathbf{A} + \\nabla f\\\\\n V &\\rightarrow V - \\frac{\\partial f}{\\partial t}\n\\end{align}", "b8318c01377a8e682622ab674926965b": "C_1 := C_1 \\cup C^{\\{a\\}}", "b831cbd57bb91a779c9756db4a5c93d5": " [\\textrm{HCO}_3^-]_{eq} = \\frac{K_1[\\textrm{H}^+]_{eq}}{[\\textrm{H}^+]_{eq}^2 + K_1[\\textrm{H}^+]_{eq} + K_1K_2} \\times \\textrm{DIC}, ", "b8320004139d8cd13a02dd72be3fc127": "\n \\boldsymbol{\\nabla} \\varphi = \\frac{\\partial \\varphi}{\\partial q^l}~\\mathbf{b}^l = g^{li}~\\frac{\\partial \\varphi}{\\partial q^l}~\\mathbf{b}_i\n \\quad \\Rightarrow \\quad\n [\\boldsymbol{\\nabla} \\varphi]^i = g^{li}~\\frac{\\partial \\varphi}{\\partial q^l}\n", "b832012cc23cc15f9892c6d5349c7571": "G_0", "b832846503e99cc427cb9948f5ad4e47": "\\mathbf{X_0}", "b832939e6e64aeac4edd8ea28e3bd655": " E_{2ss} = E_{1ss} + Z ", "b832e86768a630832e6d71ed85e361f0": "iDa>0,iDb,iDc<0", "b8331ffc95806516017969bb2709d2a5": " \\frac{d[C]}{dt}=k_2[A]", "b8332320e1835e0f8e6767945a389899": "\\lim_{n\\to\\infty} \\Pr[\\sqrt{n}(S_n-\\mu) \\le z] = \\Phi(z/\\sigma),", "b8332ce5142709b827fb92d5348a3c7b": "\\theta(b)", "b8334b3383767a68160aaaa6dbffab80": "X_{s'}(z)=X(z)", "b8341ffa2db4b65fa2cd64053595a7bf": "E_{Fn}", "b8346a0547f779cad0ec4c28e6366c3c": " \\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N ", "b834f516f9faaf72a347d1d26aa5f385": "(A\\times B)\\times C \\neq A \\times (B \\times C)", "b8351348a464132a2d01e4fa15e70824": "W_x(t,f) = W_y(t,f-at) \\, ", "b8355cb9cc3ea4c93fe57f29cc2e4238": " \\hbar c / \\sqrt{2V} ", "b83575dc1ded8d81994d0a6891faba6e": "M(n)=\\min_{A,B}\\max_k M_k \\,\\!", "b835b3f4863bfef85741ed2a29b5e5f8": "\\sum_j \\sum_i (y_{ij})^2", "b835eae8895f630834f193fe58e0e95c": "=4*x^2+644*x+25921-", "b8360c6bd1ab6340b14d0b14e1b1eaab": "\n\\mathbf{B}^{\\text{cgs}} \\ = \\ \\mathbf{H}^{\\text{cgs}} + 4\\pi\\mathbf{M}^{\\text{cgs}} \\ = \\ (1+4\\pi\\chi_{v}^{\\text{cgs}}) \\mathbf{H}^{\\text{cgs}}\n", "b8364333833f6246231e5463b3372841": " \\frac{d\\theta}{d\\xi}=-\\frac{\\phi}{\\xi^2} ", "b83659408f1c36dca6ed81810c5ce540": "k_{i,i+1}.", "b8367e84204b541d1d69b9a8c150b001": "\\vec x_1", "b836b42372534bffca8de53c80cccca3": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0\\right)", "b83717b56bedc9a78a21c4ddfab1b1cb": "M|S", "b8371ee786e89c3c88450d1e59252c3a": "\\beta\\mathbb{N}", "b83781994c5cb591e05faf2489ea545b": "N_{k} = \\rho^{k-1} N_1", "b837b9a9ecd8df527042223be8e7c118": " G = G_m ", "b837de4664310acfca4899d0b0aad1dd": " C^J_{v_1} = 1 ", "b837f40ee100bf25f753e5a6805475bc": "\\scriptstyle\\mathbf{\\hat{z}}", "b83804837650ca714da2d3f774c9a90a": "x^2+y^2+z^2-1=0,\\,", "b8381ec81dcc05fafff780f2972d28c6": "\\operatorname{DP}_1", "b838f2bb7b90186a93d87f52c485427a": "\\subset A \\to B", "b838f57acb7c0836c7ddc8ac58e99c21": "\\left( \\mathbf{I}\\otimes\\mathbf{A}\\right) \\left\\vert \\Phi_{n}\n^{+}\\right\\rangle", "b8396a36215a06b21a08452788b1d661": "0 \\leq i \\leq 31", "b839af431c1fe0acc0486066b85216d4": "\\frac {dA(t)} {dt} = W\\cdot A(t)", "b839c0a45fd15dac69be08a4dbb0b7e3": "[0,4]", "b839cdf1d4482f1950560f0b5bbef417": "\\bigg( \\pi \\models \\phi_1 \\Leftrightarrow \\phi_2 \\bigg) \\Leftrightarrow \\bigg( \\Big( \\big(\\pi \\models \\phi_1 \\big) \\land \\big(\\pi \\models \\phi_2 \\big) \\Big) \\lor \\Big( \\neg \\big(\\pi \\models \\phi_1 \\big) \\land \\neg \\big(\\pi \\models \\phi_2 \\big) \\Big) \\bigg)", "b839d55ad8d5333784bf000c99786611": "U_\\epsilon C \\subset \\mathbb{R}^2.", "b83a32da9f4f84a9a9838f0ba789f2a8": "R_2 - R_1 = \\frac{2 \\pi c}{\\omega}(ul + vm)", "b83a38709a22473708f2acbebb8dc505": "\\rho_\\text{e} = 2\\rho_\\text{bulk} \\ \\frac{ Z}{A}", "b83a3a0764cbc248893b450a923d8e74": " \\Delta \\epsilon", "b83a49bfa027b956443911f4401ae167": "I_f^0", "b83a6f892ed17fc2dfcffb78e20ec6b4": "P(s)=\\log\\sum_{n=1}^\\infty \\frac{a_n}{n^s}.", "b83a8210155f7c0165144c96b00d1652": "\\lambda g\\colon X\\to Z^Y", "b83aaa62f1888d91508f72a4ad65b88c": "X_{a+1}", "b83b3607f39d5ccd59a62d407f5bac0e": "U(K)\\times U(M)", "b83b55f69d5649c1e308e14c7d237b9f": "F = \\langle\\psi|p|\\psi\\rangle", "b83b6d5dfde907c605ad06675cd89ad7": "\\frac{d\\theta}{dt}=1-\\cos\\theta+(1+\\cos\\theta)I(t), \\;\\;\\; \\theta \\in S^1", "b83b9c7644d89565a6fa36c4f528870c": "g=1-\\frac {b}{a}=1-\\sqrt{1-e^2},", "b83bd19dbb6901233364f3b901373617": "\\mathord{\\underbrace{D^+ \\Bigl( \\mathord{\\underbrace{2 \\mathord{\\overbrace{\\sqrt{V}}^{ {} \\propto \\|\\sigma\\|_2}}}_{W}} \\Bigr)}_{D^+ W \\, \\triangleq \\, \\mathord{\\text{Upper right-hand } \\dot{W}}}} = \\frac{ 1 }{ \\sqrt{V} } \\frac{\\operatorname{d}V}{\\operatorname{d}t} \\leq -\\mu", "b83bfe93a5e34ed6d07f82c3e811ff2c": "\\frac{}{}U(\\delta)", "b83c012a0d238a1990b3d7b776423e4b": "\\frac{\\mathrm{d}^2\\gamma(s)}{\\mathrm{d}s^2}", "b83c1341878b1c7215f574a19eeeaba5": "R(N) \\ \\stackrel{D}{=}\\ \\{ M' \\mid M_0 {\\to_{(S,T,W)}}^* M' \\} ", "b83c7e0a61b824d027734d00c40d74d0": "\\mathbb{P}(B\\mid E) = \\frac{0.1% \\times 5 \\cdot 10^{12} \\times 99%}{0.1%\\times 5\\cdot 10^{12} \\times 99% +10% \\times 50\\cdot 10^{9} \\times 1%} = \\frac{99%}{99% + 1%} =99%=\\mathbb{P}(B) ", "b83c8966f239ff418c5ca21def72155b": "A^{\\mu}=(\\varphi, \\mathbf{A} c)", "b83c92fb3f6f02ef031af394e756fdbb": "x,y \\in A", "b83cc84e76d32ec4f35f596c4e21dba4": "\\delta R/R=2\\rm{Re}(\\it{\\delta r/r})", "b83d038bc28c69dc846f11317360b148": "\\ H(f)", "b83d048f91eb4cc730a23b4c381013bf": "r_o = \\frac{2\\sqrt{3}}{3}a \\approx 1.154700538a,", "b83d1f54c8da81d6a2c3a390fcb72005": "\\frac{dx}{dt}(t)\\in F(t,x(t)), ", "b83d69f97a03cbabeec84bebe9a7461f": "(S\\rightarrow S')", "b83dad6e39c6bcbe308f1b4e10a0ea8c": " \\mathrm{d}S =\\sum_i \\frac{\\partial S}{\\partial q_i} \\mathrm{d}q_i + \\frac{\\partial S}{\\partial t}\\mathrm{d}t ", "b83dd1d9a78148b5bcf48e5f481b2e96": "\\int \\left[\\frac{\\partial}{\\partial\\theta}f(\\theta)\\right]\\, d\\theta = 0. ", "b83de01a63fb21bdd9972d81a074aa79": "f\\left(x;\\eta, b\\right)=b\\eta e^{bx}e^{\\eta}\\exp\\left(-\\eta e^{bx} \\right)\\text{for }x \\geq 0, \\,", "b83dff54b515c5245927108fae2ace42": "x^2=2", "b83e1270c8ddaaf6ea4ad61ffa96641d": "\\frac {d\\theta}{ds} = \\frac{1}{\\rho} = y''(s)x'(s) - y'(s)x''(s)\\ ", "b83e141129f65a3843ba401d0ae4832f": "0.5\\leq a\\leq 0.8", "b83e198f8e82aae8a683cbd855052ede": "Margin % * (Sales / Avg Inventory Cost)", "b83e78dbd48bc230257a4052e1d5c83b": "\\mu \\ddot r = -\\frac{dU}{dr} + \\frac{\\ell^2}{\\mu r^3}. \\, ", "b83e9e2f83ba5231f63ab67c194ef949": "D = \\{t_1, t_2, \\ldots, t_m\\}", "b83ebe80ab639c666c4503528d5ca2ba": " \\frac{k_1}{k_{-1}}=\\frac{[\\mathrm{C}]^c[\\mathrm{D}]^d}{[\\mathrm{A}]^a[\\mathrm{B}]^b}=K_{eq}", "b83ef8eea106fd5294feff091bbd7692": "\\lambda(n) = (-1)^{\\Omega(n)},\\,\\! ", "b83f0a478e1ae6a7e2d4027b78200c89": " A = \\begin{bmatrix}\na_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\\\\na_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\\\\na_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\\\\na_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\\\\na_{51} & a_{41} & a_{31} & a_{21} & a_{11}\n\\end{bmatrix}. ", "b83f23cd1b808027ac47716efacf17c6": "(H, \\mu, \\eta, \\Delta, \\varepsilon, S)", "b83f352d46df6703f46c09bfcc0913af": "|X_k|/N = \\sqrt{\\operatorname{Re}(X_k)^2 + \\operatorname{Im}(X_k)^2}/N", "b83f4e8812d667d312fe5a55247687c0": "O(n^5)", "b83f59a742fee766dffe89a0933a65fd": "M = \\begin{pmatrix}I_p & -A \\\\ B & I_n \\end{pmatrix} ", "b83f960b00fc8ec633e70b85952290c7": "M\\cdot V_T =\\sum_{i} (p_i\\cdot q_i)=\\mathbf{p}^\\mathrm{T}\\cdot\\mathbf{q}", "b8400422f522e01f7ae103721923b09c": "\\sum_i (\\dot{x}_i dp_i - \\dot{p}_i dx_i)= 0.", "b84022ebaeea7ac342e0a9209248ca2d": "\\sqrt {-\\gamma}d^{4}x", "b8407f4f379757d45ce6cf19f5f7dc76": "\\mathcal{L}_{[K, L]} = [\\mathcal{L}_K, \\mathcal{L}_L].", "b84091d00e54d3449525f2493f2df59d": "P_{0,0}=1", "b84127588be1e217c659a44d45c96161": "E_s", "b841647b32dee1399ba830a821e607aa": "\nKZP(t,m,k,\\nu_{0}) = 2|\\frac{1}{2S\\rho_{0}} \\sum\\limits_{\\tau=-S\\rho_{0}}^{S\\rho_{0}}2Re[KZFT_{m,k,\\nu+{0}}[X(\\tau+t)]]^{2} |\n", "b841919b9636810e60139e020bf94d13": "\\overrightarrow{\\mathbf{B}}", "b841a0b80e4426cfc9a2a7c475d4df05": "\\sum_{i =1}^m \\alpha_i f_i", "b841a1d483c900a402c9002c70351da3": "a_{0}=j", "b841bc548698d50c8f27b1c72bacf767": "ds^2 = g_{ij}dq^idq^j = \\frac{\\partial \\bold{r}}{\\partial q^i}\\cdot\\frac{\\partial \\bold{r}}{\\partial q^j}dq^idq^j", "b841fa6244c8ca9e7c460714e2633787": " {g_m R_\\mathrm{E} \\over g_m R_\\mathrm{E} + 1} ", "b8421c93619d2634595fd3452b3b0dd3": "(t_2-t_1)", "b8427e1b5c4d5850ac075306e2b64534": "\\lim_{n\\to\\infty} \\frac{N_S(a,n)}{n} = \\frac{1}{b}", "b842b3e648a0edd697bfc08f6a678ecc": "\\textstyle x_{i+1} = x_i + h", "b8431d1354b6c27607fe41a98cfe4365": "\\hat \\beta _k", "b8437aa084a9f025956ac04aa3588bd3": "P\\subset Q", "b8437ab03db3e8ba01c7afe79067f866": "\\sum_{u=u_{1}}^{u_{2}}\\sum_{v=v_{1}}^{v_{2}}b_{uv}=p_{u_{2}v_{2}}+p_{u_{1}v_{1}}-p_{u_{1}v_{2}}-p_{u_{2}v_{1}}", "b843e855c26fd03a77cfaa554b17b86b": " \\psi)", "b8441fce120a835919f704e62ea1834b": "g(x)' \\in G", "b84458419d588e888416b44ba129f31a": "\n\\begin{array}{lcll}\nd\\theta / dt&= &-(I_2\\,/\\,M)\\,(K\\,/\\,I_2)\\,+\\,(\\omega_0\\,+\\,K\\,/\\,M)\\,e^{-(M/I_2)t} &\\cdots(Eq.9)\\\\\n\\theta &= &-(K/M)t\\,+\\,(I_2/M)\\,(\\omega_0\\,+\\,K/M)\\,(1\\,-\\,e^{-(M/I_2)t}) &\\cdots(Eq.10)\n\\end{array}\n", "b8445fa2ace71c9f38cc1f78791cfe07": "\\langle \\phi(k_1) \\phi(k_2) \\phi(k_3) \\phi(k_4) \\rangle = {\\int e^{-S} \\phi(k_1)\\phi(k_2)\\phi(k_3)\\phi(k_4) D\\phi \\over Z}", "b84472bcdf85eed3cebe9e373ae330e0": "\\frac{255}{219}", "b8447accaf27cd956a35a8d4e5d4b60b": "\nK_i = \\frac{1}\n{{\\sum_j {K_j T_j f(C_{ij} )} }},K_j = \\frac{1}\n{{\\sum_i {K_i T_i f(C_{ij} )} }}\n", "b844884860796a0432da0e32092a2d45": "\\sqrt2 W + X", "b844988d451f6206dd8f4ae858640c38": "\\forall w\\,\\exists v\\,(w\\,R\\,v)", "b8449f75f5233e2c215d1d79b6326152": "c_{n}", "b844b9623cf92ff28fa26a79418d3cb6": " {k+r-1 \\choose k} = \\frac{\\Gamma(k+r)}{k!\\,\\Gamma(r)} = (-1)^k\\,{-r \\choose k}\\qquad\\qquad(1)", "b844eec8df154c6dbc16de69f24e29c4": "{\\mathcal M}^3", "b845322b63d37e2be8d379b596847621": "1 = 1\\cdot 221 - 20\\cdot 11 = (-5)\\cdot 187 + 72\\cdot 13 = 5\\cdot 143 + (-42)\\cdot 17", "b845469fbf49f2d64aaa49ed7c48de29": "\\scriptstyle\\pi^0", "b8455e7e24b28da1636437ed1ff50349": "F_j(x,y) = \\sum_{V_k \\in Variations} w_k \\cdot V_k(a_j x + b_j y +c_j,d_j x + e_j y +f_j)", "b845a767b417b5e226da760e9d8870f9": "\\mathrm{\\theta}", "b845cf537f179670d16877122b364bb0": "\n \\begin{pmatrix}x^\\prime \\\\y^\\prime \\end{pmatrix} =\n \\begin{pmatrix}x + m y \\\\y \\end{pmatrix} =\n \\begin{pmatrix}1 & m\\\\0 & 1\\end{pmatrix} \n \\begin{pmatrix}x \\\\y \\end{pmatrix}.\n", "b846164cab1ec6d6524f2d87fc326fd0": " \\mathbf{m} ", "b846a247a7d8ef31a739baca0bb5c14c": "N = \\frac{g_0\\,z}{1-z}+N~\\frac{\\textrm{Li}_\\alpha(z)}{\\zeta(\\alpha)}~\\tau^\\alpha", "b84702f319c2a6edf3767a4951139f3d": " l(x) = f_{\\theta_1}(x) / f_{\\theta_0}(x)", "b8471cf60ef536cbe9070142a69d0d58": "L(u) = \\frac{u-1}{n}", "b8472c188c37a0b3fecc10bb11f5aa1b": "3(b-1)^2", "b8474c8be230b3ea199fe19accb22f45": "S \\cap T \\neq \\phi", "b84757a15f5829f9cc0c6825c111782e": "\\displaystyle{TH_0=H_0}", "b847640f683e17db3188c7217af63921": "E(\\ln(X))=\\psi\\left(k/2\\right)+log(2)", "b8476b95ecc1750ad8fbb2baa56e8705": "f_{0} (U) \\subseteq U'", "b847b9d4d0fd1991d31d2dd924e84545": "\n \\sum_{n=1}^\\infty \\frac{1}{n} = \n 1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots\n", "b847edea4ded8e64711639b1df06244c": "\\forall m\\,\\exists C\\,\\exists M\\,\\forall n\\dots", "b8482c4ff438808900a3f8933e48777b": "\\mathrm{ad}_x[y,z] = [\\mathrm{ad}_x y ,z] + [y,\\mathrm{ad}_x z]", "b84858d89b9ce9b7e4e8674af46ff8f6": "\\!\\mu_3(v_1)", "b8487f56f4766bc315c8664e3ca67c55": "\\mathbb{P}[X=x]", "b8489c94e4f713942f159aaa84d3805b": " e^{K(N-2s) +L(N-2r)} ", "b848c9f3487a61f3eacb26d7e1af7e40": "C=\\prod_{p>2} \\frac{p(p-2)}{(p-1)^2}\\approx 0.660161", "b8490566e3034fbd9e024910036ee895": "A(t)L(t)", "b8491cdb583ee764e18d1267558cfddd": "a(x) \\neq 0, \\pm 1 ", "b84922ec9338555d0b13639708c81c97": "\\|u + v\\|^2 = \\langle u + v, u + v \\rangle = \\langle u, u \\rangle + 2 \\, \\mathrm{Re} \\langle u, v \\rangle + \\langle v, v \\rangle= \\|u\\|^2 + \\|v\\|^2.", "b8492e8e03b9236a2c10c73ddb46bbed": "x_a\\in vals(a)", "b8499d12c57188e9b69c080ffa41ac32": "0\\rightarrow U\\rightarrow V\\rightarrow W\\rightarrow 0", "b849f1916c453390ddaa075acc6d00ae": "\\quad\\int_{4\\pi}d\\Omega^\\prime\\int^{\\infty}_{0}dE^\\prime\\,\\Sigma_s(\\mathbf{r},E^\\prime\\rightarrow E,\\mathbf{\\hat{\\Omega}}^\\prime\\rightarrow \\mathbf{\\hat{\\Omega}},t)\\psi(\\mathbf{r},E^\\prime,\\mathbf{\\hat{\\Omega}^\\prime},t)+s(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)", "b84a26ff3e49bd6033824f03c7d8c44c": "(\\gamma=5/3)", "b84aff51d22648e22f887cf45976149c": "v \\not= 0", "b84b37fab5bf71504f77336981f55e67": "M \\setminus L", "b84b4522323fc4952510d43fe0eba9ea": "r_s\\!", "b84b996034dac0e6823fe4e7ec3f6c56": "\\mathrm{R} \\left( s \\right) = \\mathrm{P_c} + \n\\cos \\left( \\frac{\\mathrm{s}}{\\mathrm{r}} \\right) \\left( P_0 - P_c \\right) + \n\\sin \\left( \\frac{\\mathrm{s}}{\\mathrm{r}} \\right) \n\\left[ \\hat{n} \\times \\left( P_0 - P_c \\right) \\right].", "b84bdd460960e61123a76bae6187f8de": "b \\to d", "b84c2714e0e6be5a0b72cebb8ac54750": "s\\models\\Phi", "b84c3fba79c7a511446287e9c98ef602": "E^\\ominus, E^\\ominus\\left( \\mathrm{X} \\right) ", "b84c6428ba259ca12217fa1159eed052": "\n\\operatorname{NGD}(x,y) = \\frac{\\max\\{\\log f(x), \\log f(y)\\} - \\log f(x,y)}\n{\\log M - \\min\\{\\log f(x), \\log f(y)\\}}\n", "b84d26db70e09655e3c3807491e9783f": "\\bigcap_{X\\in G} X=\\varnothing,", "b84dbe9c34c676593314b39951d9a495": "\\mathbf{p} = m\\mathbf{v} + q\\mathbf{A} \\,\\!", "b84dd9db8a8adbade7e22916726d0e63": "[x_0,\\ldots,x_n]f,", "b84dde4da00f8d3d068fa870ef60da4f": "\\vartheta_j", "b84e37f34131a2e918e9e99b37358e6e": "\\alpha_k = m_k/\\Delta_k", "b84e3b977e66d880d989aee7030e3136": "\\mathbf{L} = \\mathbf{r} \\wedge \\mathbf{p} ", "b84e64e0369cf2a9b72dc4be1ed968e2": "F_1 = AB + AC + AD,\\,", "b84eac179e4982ee661fa17aaa64fa84": " \\int_{-\\infty}^\\infty \\ln(x^2) \\tfrac{1}{\\sigma}\\phi\\left(\\tfrac{x}{\\sigma}\\right) \\, dx = \\ln(\\sigma^2) - \\gamma - \\ln 2 \\approx \\ln(\\sigma^2) - 1.27036 ", "b84ed0f6f042e31272baa9896fdfa4e5": "F = m a = P A = \\rho L A {dv \\over dt}.", "b84eecf6276ff8eaaea000216444e885": " \\overline{u}{\\partial \\overline{v} \\over \\partial x}+\\overline{v}{\\partial \\overline{v} \\over \\partial y}=-{1\\over \\rho} {\\partial \\overline{p} \\over \\partial y}+\\nu \\left({\\partial^2 \\overline{v}\\over \\partial x^2}+{\\partial^2 \\overline{v}\\over \\partial y^2}\\right)-\\frac{\\partial}{\\partial x}(\\overline{u'v'})-\\frac{\\partial}{\\partial y}(\\overline{v'^2}) ", "b84ef86ab09f5023314400a3fb0d7a13": "\\phi_2 = \\arctan \\left({R_2 - R_1 \\over L_2}\\right)", "b84efc51ae2eaf6573546ffeec303838": " {\\partial^2 u(x,t) \\over \\partial t^2}={KL^2 \\over M}{ \\partial^2 u(x,t) \\over \\partial x^2 } ", "b84f05651df861ccac3774ae39d126cb": " \\tan \\beta = (1 - f) \\tan \\phi,", "b84f2abdf594a2e85899651523f8e691": "'E'", "b84f3e680935950935579fe18b0f8860": "\n\\int_a^b \\omega(x) \\, x^k p_n(x) \\, dx = 0, \\quad \\text{for all }k=0,1,\\ldots,n-1.\n", "b84fa0190dc4f5e2fa8496d7ebfeaa78": "v(n,d)\\leq2^n, \\, ", "b84fac8449ee71c486517423cae70491": " 2 \\pi g(x)= \\int_{-\\infty}^{\\infty}h(u)e^{iux} \\, du. ", "b84fce20ffbb727266f3f6ef3bbbd05a": " \\theta H^2(\\mathbb{D},\\mathbb{C}), ", "b84fd397a041fe2e49410838c79c3e3b": "E_{a^{n}}^{\\dagger}E_{b^{n}}", "b84fd4a6af376668d364dfd918463c80": "\\begin{align}\n&\\underset{x, u}{\\operatorname{maximize}}& & f(x) + \\sum_{j=1}^m u_j g_j(x) \\\\\n&\\operatorname{subject\\;to}\n& & \\nabla f(x) + \\sum_{j=1}^m u_j \\nabla g_j(x) = 0 \\\\\n&&&u_i \\geq 0, \\quad i = 1,\\dots,m\n\\end{align}", "b84fe9e210a7d118be23493d4cc5cb16": "H_* (M, R) \\to H^{n-*}(M, R)", "b8500dbc13734fc3e5e6795a45932538": "c_n = \\frac{1}{n} \\sum_{i=1}^{n} a_i", "b85046de574d7022fcbdfc05223c1ec3": " \\Im=\\{Q\\ll P:H_g(P,Q)\\leq\\beta\\} ", "b85074cd5b8a6bb3479c8d44b5c971f0": "L_c = \\frac{\\sigma}{\\rho g}", "b85107f85f717b2c67b01eeaa11db276": "\\max(W(n-1,x-1),W(n,k-x))", "b8514a49fc24ec843e8ccea0d5ae862c": "(rk_0,\\ rk_1,\\ \\ldots,\\ rk_{31})", "b851680de2440595c0a83cced43932cb": "\\chi_{XXZ}=\\frac {1}{2}N_s[\\langle\\cos \\theta \\sin^2 \\theta\\rangle \\beta_{Z'Z'X'} + \\langle\\cos \\theta\\rangle\\beta_{X'X'Z'} - \\langle\\cos \\theta \\sin^2 \\theta \\sin^2 \\Psi\\rangle(\\beta_{Z'X'X'} + \\beta_{X'X'Z'})]", "b851ccae0726dbc4844a5187509ee861": "\\sigma(Y)", "b851f60778740913aa39de7812aeaa50": "\\sum_{n=0}^{N}\\left | c_{n}\\right |^{2}\\left [1-\\cos\\left(E_{n}T\\right)\\right ]", "b851fdfbb4d84e00932feab3ccc66bb1": "w= w(r)", "b8526a6c02b161d23d3e5d74fe332639": " y \\rightarrow 0", "b8529247a122be84e2689e53eeefa3db": "\n\\begin{align}\n\\phi&=\\mu+D_2\\sin 2\\mu+D_4\\sin4\\mu+D_6\\sin6\\mu+D_8\\sin8\\mu+\\cdots,\\\\\n\\end{align}\n", "b852ad3222a8d5cd8ba50edd327017f4": "p(drunk|D)", "b852c3ecc842544ba6028d834c15f6f3": "{\\sigma}= \\sqrt {\\sigma_i^2 + \\sigma_j^2}.\\,", "b852ca18b86701a48600366c50bc1cbc": "/n", "b853000eecac22f06dc4349059e05c70": "L \\to \\,\\frac{\\omega_c'}{\\omega_c}\\,\\frac{R}{R'} \\,L", "b8532871f6f605a67d19482f890884b0": "x \\cdot y = xy + x + y", "b85335298b3cad6f0de68b8b95832a16": "\\mathcal{S}_n(f)\\geq f", "b853576fe0748244f812196e889f17d0": " \\sum_{j=0}^k \\tbinom m j \\tbinom{n-m}{k-j} = \\tbinom n k", "b853d8f8bb997d6e8ef60ce45bf51ab9": "=\\left(\\frac{1-2b+b^2}{4a}\\right)+\\left(\\frac{2b-2b^2}{4a}\\right)+c", "b853def8685897e230a331d157e40140": "x_\\text{red} = {1 \\over {1 \\over x_1} + {1 \\over x_2}} = {x_1 x_2 \\over x_1 + x_2}\\!\\,", "b85415604eb84e3ab18b02c6ed10171d": "\\forall \\alpha", "b8541f48c0ec118ae3108e9dd75b6799": "{\\zeta \\over \\mathit{\\Delta}} = \\sin \\delta", "b85421983777748289ebdb271e158a4a": "\\Delta(x-y)", "b854353541dff10da003ca9f47c8270c": " \\pi (p^*) - p^* y^*(p) ", "b85440ed5f218dd4337a85cbcc9d1e49": " \\int \\, K(x,y)\\varphi(y)\\mathrm{d}y = \\lambda \\varphi(x)", "b8544b3bb10b55b70f788ddb08dfaffe": "S^{2n-1}\\hookrightarrow {\\Bbb R}^{2n, *}={\\Bbb C}^{n, *}", "b8544f3f00befc81eee074e3f60b27fc": "X_{in}", "b8545d0411c7461585fddc77297d278a": "\\Re(s)>0", "b8545e53a042928f50ba71b4fe1bbd47": "C_{3} = T_{3} + T_{1}T_{2} + \\frac{1}{6}(T_{1})^{3}", "b854786b93e6351bff411807455f6940": "\\,M[f^n] = M[f]^n", "b8547b7524e757bab800bb4bbd3c33e4": "\\displaystyle{f_\\varepsilon(h)=\\min_{z} 2h\\cdot(x- z),}", "b854899362530a91dd45e68abc6b9eec": "Var(X) = V(\\mu) = \\nu_0 + \\nu_1 \\mu + \\nu_2 \\mu^2.", "b854b164e0d34f20afc88c9b014a15c7": "\\Pr(X_n=1|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots) = \\bar{X} + \\sum_{r=1}^{m} F(r) (x_{n-r}-\\bar{X}),", "b854ce4a07d5e88e9d4ca2c47c44af77": "[M_{\\mu\\nu},K_\\rho]=\\eta_{\\nu\\rho}K_\\mu-\\eta_{\\mu\\rho}K_\\nu", "b854e9b60e8f6e70280d526849d7b9a6": "\\,\\mbox{T}(a) \\mbox{T}(da) = \\mbox{T}(a + da)", "b854edf8184d99b8b14f56f3d50ba910": "[ \\Delta , \\Delta ] \\subset \\Delta.", "b855017e48827729e83ed27e0154bdbf": "O(p^{N})", "b8552c3955b6c6763620d874cf7dc725": "R_\\star", "b8555b8b81034ecbb1977bcc9e29043c": "D={N_s \\over N_t}", "b855b4d7940cb719873a0ba7fa313e45": " x_0 \\, \\log \\left(1 + \\frac{h}{x_0} \\right) = h - \\frac{h^2}{2 \\, x_0} + O \\left( h^3 \\right) ", "b8564e26c025cc0e3555531fdf97c7ab": "\\begin{align}\nR^{\\mu}(\\tau ) & = \\left({\\tilde x}^0(\\tau ); \\vec R(\\tau )\\right) = \\left(\\sqrt{\n1 + {\\vec h}^2} (\\tau + \\frac{\\vec h \\cdot \\vec z}{Mc}); \\frac{\\vec z}{Mc} + (\\tau + \\frac{\\vec h \\cdot \\vec z}{Mc})\n\\vec h - \\frac{\\ \\vec S \\times \\vec h}{Mc \\sqrt{1 + {\\vec h}^2}} (1 + \n\\sqrt{1 + {\\vec h}^2}) \\right) \\\\\n\n&= z^{\\mu}_W(\\tau ,{\\vec \\sigma}_R) = Y^{\\mu}(\\tau ) + \\left(0; {\\frac{{-\n\\vec S \\times \\vec h}}{{Mc \\sqrt{1 + {\\vec h}^2}}}}\\right) \\end{align} ", "b8567a18dbeb8af216ab3774c16ea2a4": "\\boldsymbol{\\hat \\phi}", "b856a49b1090bf3cc58abc4f84af41ec": "w = z^k", "b856e883d6bb2473af6f4c9ca0891977": "(x, y) = x", "b85703357a3e33246be93fb1ac56c005": "f_{t}\\left(\nX^{\\ast }\\left( t\\right) ,t_{0}\\right) ", "b8574bcf99565a1414c57b66585e5a5f": "(n_0,n_1) = (12,3)", "b8575d36c811f53b01cbc1f0a1624135": " C_L = \\frac{2 \\pi \\alpha}{\\beta + 2/AR} ", "b8578226c9ab7c7fa43b3e1214fe49e4": " E = \\frac{ N - \\sqrt{ \\sum_{ i = 1 }^K n_i } }{ N - \\frac{ N }{ \\sqrt{ K } } } ", "b857aeb906b3a136b4073ac8098226c8": "\\mathcal L", "b857b9fd29fead3060def94530a93cde": "\n \\left(\\frac{\\partial \\boldsymbol{A}^{-1}}{\\partial \\boldsymbol{A}}:\\boldsymbol{T}\\right)\\cdot\\boldsymbol{A} = - \n \\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T}\n", "b857f70d1af41cd4868e2d86c3719df8": " \\det(M_k)/\\det(M_{k-1}) ", "b858329f340fe2a24f4be716b5b290e9": "K^{\\mu\\nu}=2\\Sigma\\ l^{(\\mu}n^{\\nu)} + r^2 g^{\\mu\\nu}", "b8583c4914cd6e5740f8397a1012422c": " \\varphi(x)", "b8584febca9c3579f3be86a723146888": "q\\equiv7\\ \\text{mod}\\ 12", "b858573f90759afde9928d26ed5f5994": "\\frac{\\operatorname{d}i}{\\operatorname{d}t}", "b858609ffecf018131877e1bb7e6124b": " \\frac{\\partial \\det(A)}{\\partial A_{ij}}= \\operatorname{adj}(A)_{ji}= \\det(A)(A^{-1})_{ji}.", "b85875c36a327ceefcb495846ddb344c": "\\,\n N = V \\int {d^3k \\over (2\\pi)^3} {p(k)\\over 1-p(k)} = V \\int {d^3k \\over (2\\pi)^3} {1 \\over e^{k^2\\over 2mT}-1} ", "b8587cf0aef3305986da72fddfd4c6d4": "\\scriptstyle H^d_\\delta(S)", "b85889fb6bb63cf04014f38a284f2d61": "{C_k}", "b8589b3ae082bec42381770421ceec85": " \\sigma = 1-\\frac\\pi{z} ", "b859326f038f3e9db3841d3acde0b951": "A_1,A_2\\subset A", "b85a13835d9dedd28827b9598ce6cb23": "A(\\alpha)", "b85a7b79c8036dd1f97ec37c42ca687d": "V \\to V", "b85ace437b35af9b3e7a2720ae5ba7fd": "V \\mapsto V^\\mathfrak{g}", "b85b22b73899fb60c9a94ad32b3b8917": "f_D=f_T=1/2", "b85b44845177209bd7b1d898fe15ed8b": "\\rho=\\sqrt{x^2+y^2}", "b85b695aa5e6926ed8c875e883141310": "v \\gamma_0 = \\gamma (1 + \\mathbf{v})", "b85ba3a7e24cdcd09f2249a6ad36f6b4": "\\Omega > T^2(\\Omega) > T^4(\\Omega)\\ldots", "b85bcc97d4f94096885959e9af909d10": "[y_1, y_2] \\subseteq [x_1, x_2]", "b85c608b832c33cdd200d152c51462da": "M^2 = \\begin{bmatrix}a^2 & b^2-a^2 \\\\ 0 &b^2 \\end{bmatrix},\\quad\nM^3 = \\begin{bmatrix}a^3 & b^3-a^3 \\\\ 0 &b^3 \\end{bmatrix},\\quad\nM^4 = \\begin{bmatrix}a^4 & b^4-a^4 \\\\ 0 &b^4 \\end{bmatrix},\\quad \\ldots", "b85d0f710b362f9c50c0bbc642d9ea6a": "t = \\left\\lfloor\\frac{d-1}{2}\\right\\rfloor", "b85d13e3e82ead731ca364add40e1cca": " \\sin^2 x + \\cos^2 x = 1 \\;\\!", "b85d477cea482709ae01579a7353a4ed": " \\theta = \\frac {1}{2}", "b85d8ab9158dc8ba71f365673dcd69bd": "\\hat{a}\\ ,\\ \\hat{b}\\,", "b85de4edb33de81c25b595eeef343b2e": "\\rho_{\\rm in} \\in \\mathcal{S}(\\mathcal{H}_{\\rm in})", "b85e3cad0e14ba918ab369cf955e394d": "p=1,2,", "b85e91bb4163b3f53246fa67de69f91e": "C=\\Delta_{i<\\kappa} C_i", "b85e945f8461d1e32329187aeadd078a": "0 \\log 0 \\; = 0", "b85eab975c26d6819d437878389696e8": "\\left[ \\max_{i:[Av]_i\\leq 0}\\frac{[b-Ac]_i}{[Av]_i}, \\min_{i:[Av]_i\\geq 0}\\frac{[b-Ac]_i}{[Av]_i}\\right],", "b85ed23fae9e21174601545934aabe54": "Cb = Cb - 128;", "b85efe812005c13ef6cbc374834a7bf6": "\\{ C_{in}^\\alpha \\} _{\\alpha \\in \\mathbb{F}_{q^k }^* }", "b85f29b76957b36a3518c23be8823604": " \\begin{align} \n\\Omega & = \\det\\begin{bmatrix}\\mathbf{v}_1 & \\mathbf{v}_2 & \\cdots & \\mathbf{v}_i & \\cdots & \\mathbf{v}_j & \\cdots & \\mathbf{v}_n \\end{bmatrix} \\\\\n& = - \\det\\begin{bmatrix}\\mathbf{v}_1 & \\mathbf{v}_2 & \\cdots & \\mathbf{v}_j & \\cdots & \\mathbf{v}_i & \\cdots & \\mathbf{v}_n \\end{bmatrix} \\\\\n& = - \\Omega\n\\end{align}", "b85f2a2c197f3ef9f95e92c282cebdb0": "\\frac{3^1 2^0 + 3^0 2^1}{2^4 - 3^2} = \\frac{5}{7}.", "b85f564c1c7d3234d4ca4d9dfddada67": "\\ j\\omega C_C (V _i - V _O ) = j \\omega C_M V _i, ", "b85ff6d5f8f512d64796be342791fc97": "\\omega\\!", "b860b4573af3b733e0d0d6620df576a0": " F_{\\theta} = -mgL\\sin\\theta.", "b860d0acb01b8cabe3b013939b4adfff": "\\scriptstyle (f_n)_n", "b860e196543eac3961dd61c6101972a6": " S_{mi}=R\\ln 2.", "b860ee2c5c8a1ee35ec51ee6145e6f47": " \\frac{\\pi^4}{15} ", "b860fa70bed1ed0b9800daaa1c839eaa": "AICc = AIC + \\frac{2k(k + 1)}{n - k - 1}", "b86152b2d6dfa035242ef70c30778eb5": "\\frac{1}{(\\beta E_c)^\\alpha}=\\frac{Vf}{\\Lambda^3}", "b861a992a449a386942968a35a4d03ee": "f^*\\omega'=\\omega,", "b86258713f510e04419eece800fe893c": "F\\left(k\\right) \\equiv 3\\cdot \\left(8^k - 4^k\\right) \\pmod{11}.", "b8626014b9ad4fd87b3a16834c0a644d": " \n \\hat{\\beta} = \\big(\\hat{\\operatorname{E}}[\\,\\xi_t\\xi_t'\\,]\\big)^{-1} \\hat{\\operatorname{E}}[\\,\\xi_t y_t\\,],\n ", "b86285f242e2b8bec1ad5f8bc532d1fd": "I(Y_m)=I(Y_{m+1})=I(Y_{m + 2})=\\cdots.", "b862cdc19adae1aae1d00a31d9ff3853": "\\frac{p V_m}{RT}=Z=1 + Z^{\\rm{rep}} + Z^{\\rm{att}}", "b86414f1eef70449e842e8a3d211c89c": "(\\alpha\\, , \\, \\beta).", "b86484ecfdec71f1ab69bc5b4fbf9d71": "\\frac{x_1+x_2+\\ldots+x_n}{n}=\\alpha=\\sqrt[n]{\\alpha\\alpha \\ldots \\alpha}\\ge\\sqrt[n]{x_1x_2 \\ldots x_n}.", "b864c9800529ccbe5e190fedcb7fd0b6": "y = Cz, ", "b864eb68d02d14fecb85ce91a10f619b": "\n P\\left(X\\in\\left [ x,x+d \\right ]\\right) \n = \\int_{x}^{x+d} \\frac{\\mathrm{d}y}{b-a}\\,\n = \\frac{d}{b-a} \\,\\!\n", "b864fa55c2cd993aa4d5eea34b10f081": " P_i = BC ", "b86578d7b264b4b363c6d397fefd2166": "\\langle W,\\le,\\Vdash\\rangle", "b865a619532e01d189d9ed4db46310d7": " Q_B \\Psi + \\Psi * \\Psi = 0 \\left. \\right. \\ .", "b865ad7936afa56569f3822c221a9e77": "\\mu^* (S) = \\inf \\left\\{ \\left. \\sum_{i = 1}^{\\infty} \\mu_0(A_{i}) \\right| A_{i} \\in R, S \\subseteq \\bigcup_{i = 1}^{\\infty} A_{i} \\right\\}", "b865c30a9dc10c2cc3254774da79c29d": "p \\in N", "b865cadcbdbc2d9afa53a8c224e435a3": " \\Lambda^\\alpha {}_\\beta ", "b865cbc1106e464bf530ae2d0a709b71": "\\mu = \\mu_b^\\ominus + RT\\ln{\\left( \\frac{\\gamma_b b}{b^\\ominus}\\right)}\\,", "b8665cc0e16b2761fcd016bcfbfcb79f": "w_{i_{max}}\\cdots w_{i_1}\\alpha\\omega = a_{i_1\\dots i_{max}}\\omega", "b8669e407850fb7bd8084a6e3ff11696": "\\nabla _{e_i} e_j = \\Gamma ^k _{ji} e_k", "b866b51e8d83324762c882309a2a4484": "G = \\frac{(OA-IB)}{S}", "b8670442949f3b9e606da7bb16e258ea": "1\\le i< j\\le n", "b867077b43b6d5e232b474f09ce5e536": "\\displaystyle{p_t(z)={1+\\kappa(t) z\\over 1-\\kappa(t) z}}", "b86757d785714bd45fd1652dc170d18c": "\\scriptstyle{\\alpha}", "b867b19c85e2f20c589d858c4ded0919": " \\{ z_k \\} ", "b867d9a393228cfe609b9e484cc4d310": "F_n = \\dot m \\cdot (V_{jfe} - V_a)", "b867e268edebfc3093c6e8ac7ed75d98": "n^{\\log_b a}", "b8685bbcf08e616e092e975c6677a8b3": "A/J(A)\\cong B/J(B)", "b8688eae87e677799ad469012b9f0174": "\\sum a_n x^n \\to s, ", "b868b595d0eda4dfdf99e9ac1ebf8a37": "\n p(r) = p_0\\left(1 - \\cfrac{r^2}{a^2}\\right)^{1/2} + p_0'\\left(1 - \\cfrac{r^2}{a^2}\\right)^{-1/2}\n ", "b868c8846248949b3359a7acf25421fa": "VP \\to ITV ~|~ TV \\ \\_ \\ NP_{obj}", "b868ee258b2579a725718dd627bbfca3": "\\tfrac 12\\chi^{2}(\\alpha/2; 2k) \\le \\mu \\le \\tfrac 12 \\chi^{2}(1-\\alpha/2; 2k+2), ", "b868ef08a68316a6c8128bcaf43049b1": " 0 = y_0 + v t_g \\sin \\theta - \\frac{1} {2} g t^2 ", "b8698ca51fa9649301ebece2e3063d90": "P \\lor (\\forall x Q(x))", "b86a2b6970b206f5f3581c9d3174eb14": "\\scriptstyle \\mathbf{Q}=1/N_t \\mathbf{I}", "b86a3983ca5646d02080a01d371c2882": "\n\\frac{dS_{\\varphi}}{d\\varphi} = p_{\\varphi} = L\n", "b86a4e360a575bd5bb637ff8f78b4998": "f(x+y) = f(x)g(y)+f(y)g(x)\\,\\!", "b86a832781d2a7d353b774d81ca52704": "\n\\Phi(\\rho, \\theta) =\n\\frac{-\\lambda}{2\\pi\\epsilon} \\left\\{\\ln \\rho^{\\prime} -\n\\sum_{k=1}^{\\infty} \\left( \\frac{1}{k} \\right) \\left( \\frac{\\rho}{\\rho^{\\prime}} \\right)^{k}\n\\left[ \\cos k\\theta \\cos k\\theta^{\\prime} + \\sin k\\theta \\sin k\\theta^{\\prime} \\right] \\right\\}\n", "b86aa6189291a696d4652c2a809f3ee7": "s = \\tfrac{1}{2}(1+i+j+k) \\qquad t = \\tfrac{1}{2}(\\varphi+\\varphi^{-1}i+j).", "b86b0594d40397704d608adb70701c37": "h'", "b86b1dc650b88dbb696e790ec07e3b1d": "H(\\omega)=|K(j \\omega)|=\\frac{1}{\\sqrt{1+\\left(\\frac{\\omega}{\\omega_0}\\right)^2}}.", "b86b30bd3400189fd3813a82abf61165": "(8.c)\\quad \\gamma_{,\\,\\rho}=\\rho\\,\\Big(\\psi^2_{,\\,\\rho}-\\psi^2_{,\\,z} \\Big) ", "b86ba1363dd6bcd44e8b5d1944a9e582": "\n\\begin{alignat}{2}\n\\lambda_{k} & = \\frac{4}{h^2}(\\cos(\\frac{\\pi (k - 0.5)}{2n + 1})^2 - 1) \\\\\n& = -\\frac{4}{h^2}\\sin(\\frac{\\pi (k - 0.5)}{2n + 1})^2.\n\\end{alignat}\n", "b86bd1b59766d968de08cdf357527a09": "(AB)^\\dagger = B^\\dagger A^\\dagger", "b86be4624f298b12dbd1d397cde1168c": "\\sigma_c(T) = \\sigma_{ap}(T) \\setminus (\\sigma_r(T) \\cup \\sigma_p(T)) ", "b86beba0f80bfe1b6c1f2d07d2e7cb88": " E_n = \\hbar \\omega \\left(n + {1\\over 2}\\right)", "b86c2f91d45b743ddaa3e843e6d83ae7": "1 / \\sqrt{\\mu(\\varepsilon_1 \\pm g_z)}", "b86c47fcc148e7a92a9889f11ce4cbbe": "f(x)f''(x) \\leq (f'(x))^2", "b86c5249e645e8d330b585a2232bd91e": "co(ci, x, y)", "b86cddfddcd82d12867b1540aa2afff2": " \\epsilon = e_1 \\wedge \\cdots \\wedge e_n", "b86ce46dec7b12a7580e4d62a607d1b0": "\\pi\\colon \\tilde{X} \\rightarrow X", "b86d5133a073ea6f0e1db510dd344975": "h[n] = \\frac{1}{\\sqrt{2}} [-1, 1]", "b86d765a1128c9465c1aed778a4bf97f": "U = \\frac{1 - L^\\alpha x^{-\\alpha}}{1 - (\\frac{L}{H})^\\alpha}", "b86d9bef614fda3ba417a5a1db289c56": "\\int \\exp\\left[\\theta^TA\\eta\\right] \\,d\\theta\\,d\\eta = \\det A ", "b86da085d2cf7314761019454867b038": "\\mathit{c} \\in \\mathcal{B}", "b86dcab6c3ac700b3c90f0df2c3868e1": "\\Gamma^{[{C}]}", "b86ddcf6a50f204bde05a2637b16b903": "\\displaystyle{D=\\pi_z(a)+\\pi_z(a^{-1}) + \\pi_z(b) +\\pi_z(b^{-1})}", "b86e1e84ee5bf78cdb6dc5cdc41ceb1f": "\\sigma_x^2 = \\int_{-\\infty}^{\\infty} |f(x)|^2 \\, dx = \\langle f | f \\rangle.", "b86e29d296818481de9780303db4051c": "H(X) = -\\int_\\mathbb{X} f(x)\\log f(x)\\,dx", "b86e763975fccf203dcdebdc48c08cc0": "\\frac{\\ln(1 - p\\,\\exp(t))}{\\ln(1-p)}\\text{ for }t<-\\ln p\\,", "b86e9a8d808943ba8aa6a31fa1a59e4d": "\\frac{d^2 y}{d t^2} = A y^{2/3}", "b86ea39c86cf8459f07257ca5a345961": "-x(x^2-x-3)(x^2+x-1)", "b86ec03508adbfb449d3715270294153": "\\textbf J = \\int_0^{\\frac{\\pi}{2}} \\frac{1}{a\\cos^2 (x) + b \\sin^2 (x)}\\;\\mathrm{d}x.", "b86f1ff0e3c8d1008a3de1558f35b8e1": "\\left\\{x_1,\\frac{(1+x_2)x_1 + x_3}{x_2 x_3},\\frac{1+x_2}{x_3} \\right\\},", "b86f2184dae7c1876066944ec0f48991": "d S(q) = p_i(q) dq^i", "b86f2e589fbc70df18e95acc3232913a": "E_1 > E_0", "b86f30598d919e522ec43786889820ef": " x_{il} ", "b86f3a3297f012dc8e9836e7fe9fe93d": "(0:1:0)", "b86f46920a412e8ed6b2e4785acf48fe": " - \\varepsilon < f(x) - L < \\varepsilon ", "b86f4f59d71006869ef690e1e2b6ac21": "r_m = \\frac{a \\varphi}{2} = \\frac{1}{4} \\left(1+\\sqrt{5}\\right) a = a\\cos\\frac{\\pi}{5} \\approx 0.80901699\\cdot a", "b86f5720afac102c13276da808c45e73": "S_1,S_2,S_3,S_4\\;", "b86fbebeba347ad988940e8be0d9c983": "K[A_1,\\ldots,A_k]", "b86fc6b051f63d73de262d4c34e3a0a9": "AB", "b8700f134280f8ff7ce4f9b75e92d92f": "n=0,1,2,\\ldots,\\quad s=n+\\alpha", "b8704984486bcd1e703af272fb529e50": "\n\\mathrm{d}s^2 = \\left[ 1 + \\left(\\frac{\\mathrm{d}w}{\\mathrm{d}r}\\right)^2 \\right] \\mathrm{d}r^2 + r^2\\mathrm{d}\\phi^2,\n", "b8704a6eb6cd579fbe6accd25c37ac42": "[1.7,4] ", "b8707b8338538c522d300b60cd285519": "\\scriptstyle \\rho \\,\\in\\, [0,\\, 1] ", "b870cdd568c46fba7f22dabf23d9bb1c": "\\forall x ( \\phi \\land \\psi)", "b871efe87c7f438ba6ba9dde00168e4d": "f_{\\Delta E} = \\frac{10^{(\\frac{3}{2}(m_1 + 10.7))}}{10^{(\\frac{3}{2}(m_2 + 10.7))}} = 10^{\\frac{3}{2}(m_1 - m_2)}.", "b8722c861e5ab48371a035da6a20cfbd": "\\alpha_1 = \\frac{1}{2}-m", "b8728c96149bf7a4887ff37591fdd6d7": "\\scriptstyle k_B", "b872a8afd266d4c706cee9170e01fa18": "\nK=\\frac{\\operatorname{E}[(\\mathbf{y}-\\mathbf{\\overline{y}})^4]}{(\\operatorname{E}[(\\mathbf{y}-\\mathbf{\\overline{y}})^2])^2}-3 \n", "b872bcc1cd58d9ad16996e4a36be611c": " 1-p ", "b872c3abb88b5848d1c7b2f40917af18": "O(n\\log^{4k-8} n)", "b872df4ec31ea877f8cf1dd2bfa91a58": "|n_{{\\mathbf{k}}_{1}}\\rangle |n_{{\\mathbf{k}}_{2}}\\rangle |n_{{\\mathbf{k}}_{3}}\\rangle... \\equiv |n_{{\\mathbf{k}}_{1}}, n_{{\\mathbf{k}}_{2}} ,n_{{\\mathbf{k}}_{3}}...n_{{\\mathbf{k}}_{l}}...\\rangle \\equiv |\\{n_{\\mathbf{k}}\\}\\rangle ", "b87320cbeefe93392900ca113edf306f": "\\pi_j = \\sum_{i \\in S} \\pi_i \\, \\Pr(X_{n+1} = j \\mid X_{n} = i)", "b8732250f44bd33f732db5c055f8f434": "n_{r1}", "b87338e37185999ddd2ff0c4bb24d604": "\\alpha_i \\geq 0,\\, ", "b873b93485fd4aebf0de9fae695b3e3b": "HS_{A/f}(t)=(1-t)\\,HS_A(t)", "b873bc07b241a0553ecc145520382add": " \\left(\\frac{D_1}{D_2}\\right)^2", "b87426e8d181f7ffad3d42a4a069fa54": "\\mathcal{F}_t = \\mathcal{F}_{t+} := \\bigcap_{s > t} \\mathcal{F}_s", "b8745f46d66d09df91fc7208ace9b614": "\\mathrm{Tor}^{\\mathbf{Z}}_1(h_1, h_1) \\cong \\mathrm{Tor}^{\\mathbf{Z}}_1(\\mathbf{Z}/(2),\\mathbf{Z}/(2))\\cong \\mathbf{Z}/(2) .", "b8748e5e5d9922e95bea4b1ab02024d4": "\\frac{\\partial f}{\\partial t} = - \\frac{1}{i \\hbar} \\left(f \\star H - H \\star f \\right),", "b874e35a0f8f4e2d259e33f0aa223f76": " z_{2}\\, +\\, \\frac{p_{2}}{\\rho g}\\, +\\, \\frac{V_{2}^2}{2\\,g}\\, =\\, z_{3}\\, +\\, \\frac{p_{3}}{\\rho g}\\, +\\, \\frac{V_{3}^2}{2\\,g},", "b874f142584d9b5db146f35375642971": "\\scriptstyle{\\dot G = G^\\prime = 0}", "b8751bd2be0443a6a41f59fb31761be4": " \\mathbf{b}_1,\\mathbf{b}_2, \\dots, \\mathbf{b}_n \\in Z^{m}", "b8755ec95024debee43dba01a0dd381c": "0\\le(x_{\\sigma(j)}-x_j)(y_k-y_j). \\quad(2)", "b87596adcd88a55af62e73bd57f832f2": "h_n = t_{n+1} - t_n", "b875cbec7390c6753db97ef017a8a717": "(\\pm 1, \\pm 1),", "b875e9737f4b03a0bb633855891ef2c5": "\\forall k \\in n(\\bot)", "b87617b4396726988bd30f00e396c44d": " \\frac{ e^{-x} } { 1 + e^{-x} } \\exp\\left( -\\theta \\log\\left(1 + e^{-x} \\right) + \\log(\\theta)\\right) ", "b8762602e2c648ed23225e7705cd3f87": "e^s", "b87659b0ccc7cd23002826acbd404c4a": "\\textstyle A_{p}", "b8766065fc0e85ffbcba39860bcfa3fb": " y_{n+1} = r(h\\lambda) \\, y_n ", "b8768af5777eb852ed8413169485439f": "A = N + P\\,\\!", "b876bc071ade36ea2c2fb3ce0b3add4d": "C_1=\\cup_{i=0}^{\\infty}D_{i}", "b876c33bdf7943bb16811d7a474cd5fb": " r=\\int_0^s\\frac{dt}{\\sqrt{1-t^4}}", "b876c9885d35bf683137ab531f1fc130": "\\scriptstyle I \\,\\cap\\, J \\;=\\; (I \\,\\cap\\, J) (I \\,+\\, J) \\;\\subset\\; IJ \\,+\\, JI", "b87731acbe6a326d0424c5b150811065": "n_3^2\\sigma_3^2-2\\sigma_1n_3^2\\sigma_\\mathrm{n}+n_3^2\\lambda=0\\,\\!", "b877422e14fc4a8eb3b55e9d73761774": "(u_i,f_i)\\in U_i \\times F", "b877f6a4668e896b3f6d09dcbdc2d795": "\\displaystyle u_{tt} - u_{xx} - u_{xxxx} - 3(u^2)_{xx} = 0", "b8783fb7e048bea2f3021ee3c9f48ff7": "m_2 = m_1/10", "b878944996de71ec1815a6f512a2be11": "f : A \\longrightarrow A", "b878d23bcaddaa53cf8dab89ee272c50": "\n\\ln(\\gamma_i)=\\frac{\\displaystyle\\sum_{j=1}^{n}{x_{j}\\tau_{ji}G_{ji}}}{\\displaystyle\\sum_{k=1}^{n}{x_{k}G_{ki}}}+\\sum_{j=1}^{n}{\\frac{x_{j}G_{ij}}{\\displaystyle\\sum_{k=1}^{n}{x_{k}G_{kj}}}}{\\left ({\\tau_{ij}-\\frac{\\displaystyle\\sum_{m=1}^{n}{x_{m}\\tau_{mj}G_{mj}}}{\\displaystyle\\sum_{k=1}^{n}{x_{k}G_{kj}}}}\\right )}\n", "b8790351417af4d21a9817b6683443b7": "\\int_\\R\\int_\\R [f(x)-f(y)][g(x)-g(y)]\\,d\\mu(x)\\,d\\mu(y) \\geq 0.", "b8794b518b88666f2e75dad6979380c2": "-2{mht \\over u} \\equiv \\sum_{0 < k < d} {\\chi(k) \\over k}\\lfloor {k/p} \\rfloor \\pmod {p}", "b879509e3604348bb5375d44279aabff": "\n= \\int \\psi_e'^* \\psi_v'^* \\psi_s'^* \\boldsymbol{\\mu}_e \\psi_e \\psi_v \\psi_s \\,d\\tau + \\int \\psi_e'^* \\psi_v'^* \\psi_s'^* \\boldsymbol{\\mu}_N \\psi_e \\psi_v \\psi_s \\,d\\tau \n", "b879bd1e6db5e255989f36bcf36d8919": "\\nu(x, z) = \\mu(x) - |z|^2.", "b879c30a501714179774990b54953d8c": " \\limsup_{n \\rightarrow \\infty} r_B(n) = \\infty ", "b879ce3d707fd4033f2dfa698127812f": "\\vec{\\jmath}", "b879d262c08c8e279734c0b03620c57c": "F_{Y|X=\\frac{3}{4}} (y) = \\mathbb{P} \\left ( Y \\le y | X =\\tfrac{3}{4} \\right ) = \\lim_{\\varepsilon\\to0^+} \\mathbb{P} \\left ( Y \\le y | \\tfrac{3}{4}-\\varepsilon \\le X \\le \\tfrac{3}{4}+\\varepsilon \\right ) = \\begin{cases}\n 0 &\\text{for } -\\infty < y < \\tfrac{1}{4},\\\\\n\\tfrac{1}{6} &\\text{for } y = \\tfrac{1}{4},\\\\\n \\tfrac{1}{3} &\\text{for } \\tfrac{1}{4} < y < \\tfrac{1}{2},\\\\\n\\tfrac{2}{3} &\\text{for } y = \\tfrac{1}{2},\\\\\n 1 &\\text{for } \\tfrac{1}{2} < y < \\infty,\n\\end{cases}", "b87a1e4f24caa9af697450eb67b890d3": "\\,s \\in Q", "b87abb5a9809312679634898e0329551": "p = (A_1 \\to w_1, ..., A_n \\to w_n)", "b87aed890b6f3382845061a26851d12b": " C_{int} ", "b87af2adf7660d102c494dbba397790e": " K_1^M (F)=F -\\{0\\} ", "b87b04b7c4967b6b3b4f7a12a91d6e43": " \\mathrm{Re}_x = \\rho u_0 x/\\mu", "b87b6f3b2205286a2b6a3060c6c007b1": "\\psi_0(x)", "b87b93932f1029a5f12b9f479bd10629": " (a_0, a_1, a_2, \\ldots) \\leq (b_0, b_1, b_2, \\ldots) \\iff a_0 \\leq b_0 \\wedge a_1 \\leq b_1 \\wedge a_2 \\leq b_2 \\ldots ", "b87bc906326bc1e00bd35b095eb13ea4": " x = a_1 \\cdot a_2 \\cdots a_k \\mbox{ where } a_i\\in T. ", "b87bcf1d429f5dfdef241e504ad88cd7": "P(x)=(x-1)(x+1)(x+2).\\,\\!", "b87bd1ee1a0be4e1d783db7189296850": "ds^2 = -c^2 dt^2 + dr^2 + r^2 d\\Omega^2 \\,", "b87be097cd1aafba8b57620f1b6f7f8f": " V^a_b(f)=\\sup_{P \\in \\mathcal{P}} \\sum_{i=0}^{n_{P}-1} | f(x_{i+1})-f(x_i) |. \\,", "b87c367b8c3cd666f7aab0413637487b": "\\frac{d}{d\\varphi}{\\mathbf{s}}_u = -\\cos\\varphi\\cdot{\\mathbf{x}}_u -\\sin\\varphi\\cdot{\\mathbf{y}}_u=-{\\mathbf{n}}_u", "b87c7644190fc9f57a891bef36bd5343": "Q[z/y]", "b87d2660eb1eaef389ed7bb98f55522a": "\\nu_m \\propto t^{-3/2}", "b87d99b77cb30bc8425412a8cd411e81": "\\langle f,g \\rangle = \\int_0^{2\\pi} f(x)\\cdot g(x)\\,dx,", "b87dad47f694d3ffabf5a3c80f5e2fdb": "C=\\varepsilon \\cdot { {n \\cdot A} \\over {d} }", "b87e884e2cd7d28af67d8d3392194572": "FX/SO(4) ", "b87eba667a5fa094be60b61b0cfc2f12": "T^{*}_{11}", "b87eec3b12f547e37800a3ef8c344413": " \\delta = 1 - R ", "b87f07d4750c59a6ca98e2ccec1d6822": " \\Delta = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. \\,", "b87f6806dfaff0baca547a0818e5867b": "Y(s) = P(s) U(s)\\,\\!", "b87fc5ab6613cff2c6227e76ed214631": "\n\\begin{align}\n\\lim_{t\\to 1/2} \\operatorname{sinc}(t) \\frac{\\cos \\pi t}{1 - (2t)^2}\n& = \\operatorname{sinc}(1/2) \\lim_{t\\to 1/2} \\frac{\\cos \\pi t}{1 - (2 t)^2} \\\\\n& = \\frac{2}{\\pi} \\lim_{t\\to 1/2} \\frac{-\\pi \\sin \\pi t}{-8 t} \\\\\n& = \\frac{2}{\\pi} \\cdot \\frac{\\pi}{4} \\\\\n& = \\frac{1}{2}.\n\\end{align}\n", "b87ff58eebe2547d8c7de5a500440bb5": "\\overline{\\mathcal{R}}_{nn} = \\frac{\\Gamma(\\frac{d+2}{d+1})}{K^{\\frac{1}{d+1}}}", "b87fff35cf5e31794f8da932ca0aece6": "\\sqrt{\\hbar/2}", "b880087f34c93cb314f2f21705a30239": "\\tfrac 1 {12n}", "b8800c974087202fc14b553501b78e8f": " x \\in I_k \\Rightarrow y = y_k ", "b88023d62a62bfb03f7b5fdbc6cd0048": "\nIII = LL - LA\n\n", "b880252cdf3650d09d56d32599069846": " H_1 = \\sum\\limits_{k\\sigma ,k'\\sigma '} {B_{k'\\sigma ',k\\sigma } c_{k'\\sigma '}^* } c_{k'\\sigma '} ", "b88077033ee418194ce6a6c98c131866": "\\mathbb{A}^n;", "b88077f4f430adc303b3a8067fdee07e": "(-2)^n\\,\\frac{\\Gamma(n+1/2)}{\\sqrt{\\pi}}\\,", "b880884db10b0c0f52919b0487b0aa8f": "R_G=\\prod_{i=1}^g (Y-P_i)", "b880b97dcd70418e6d554eb63ae5e040": "O(N^2P)\\,", "b880e16250f062744c564d21a575bd27": "\\scriptstyle F^{\\ast n}_l(x)", "b8815c0742c901c9c013a87813afe77c": "W = \\int \\ldots \\int \\frac{1}{h^n C} f(\\tfrac{H-E}{\\omega}) \\, dp_1 \\ldots dq_n ", "b8816a9347da383cde69cf90b20349ea": "y = k a^x.\\,", "b881877a78d5e10b4313d905a8e85851": "d=b", "b881cb76e4d8544e1bf119fbde20b063": "\\sum_{n=1}^N |\\lambda_n(A)|\\leq \\sum_{m=1}^M s_m(A)", "b88230e7a92dc09f0ecace84fe206c3e": "\\pi_k:\\Sigma^*\\to\\Sigma_k^*", "b8826d7df9e770ff83dc09e2a532d940": " \\mu_n ", "b882bd1bbec35b1acd184a0f772b0a5a": "x(p_1,p_2,w) = \\left(\\frac{aw}{p_1}, \\frac{(1-a)w}{p_2}\\right).", "b882ef4516cd840a2409536792bf96fe": "\\frac{r}{s}", "b8832dde9e183a2f1355e4f08ea899f2": "\\frac{\\partial}{\\partial t}\\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\right] + \\nabla \\cdot \\left[ \\rho \\left( h + \\overline{\\eta} \\right) \\overline{\\boldsymbol{u}} \\right] = 0,", "b8832ee351fbcbfef5e9d8246229300f": "\nI_\\mathrm{No} = {1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega \\over (1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega)} \\cdot I_\\mathrm{total} \n", "b8836f4a6dd18e1da0f780b31c892525": " \\mathbf{d}_A \\equiv \\mathbf{R}_A -\\mathbf{R}_A^0 ", "b88376e34f6e524ef0f2102f2fde7e71": "1 \\over 11", "b883c0380e92a5ea6c5ed3c42945ea54": "\nMgr - mgr \\cos{\\theta}=Mgr_0 - mgr_0 \\cos{\\theta_0}\n", "b883c3e887c1fd9de1b1fbe1649bf970": "\\frac{P \\to Q}{\\therefore \\neg Q \\to \\neg P}", "b884cf991c936b14639d3a5016054e5e": "L = n^2 A_L", "b8856e134afcc03f0221e2994b0559f6": " [0,1] ", "b885a473c00c3276bb78ee97eb83708a": "\\sum_j L_{ij}X_j", "b885a8e0f4203def40dc124c3cdb5d61": "\\mbox{Seq}(\\mathcal{F}) = \\epsilon\\ \\cup\\ \\mathcal{F} \\ \\cup\\ \\mathcal{F} \\times \\mathcal{F} \\ \\cup\\ \\mathcal{F} \\times \\mathcal{F} \\times \\mathcal{F}\\ \\cup \\cdots", "b885ac8554d3ee614aa6b20b1a27eba5": "R_0 = \\left(\\frac{n_1-n_2}{n_1+n_2}\\right)^2", "b885af2c4c1b313e5aeff8050582f90d": "B_\\nu", "b8860772568e78732b1d08045ef1a102": "\n\\left(-\\frac{\\partial ^2}{\\partial x^2}+V(x)\\right)\\Psi _n(x)=\\varepsilon _n\\Psi _n(x),\n", "b8863e9144aaa266de49127f2e39dd84": "m\\cdot Q_{P}(h)(1-Q_{P}(h))\\,\\!", "b886918c4e92b8b6e7da3de17d1fea94": "C = [c_1:c_2:\\ldots:c_r]", "b886cc3a6f1edc02707033e9714d52af": " \\mbox{End}\\, {}_BA_B ", "b887096006fe99e58b906715db2004fe": "I = \\frac{1}{2} mr^2", "b887865b76bf88ef699a653e18fbbde5": "\\mathrm{Mass% \\ O_2 \\ in \\ propane \\ combustion \\ gas} = -0.1433(\\mathrm{% \\ excess \\ O_2})^2 + 0.214(\\mathrm{% \\ excess \\ O_2}) ", "b887b3aa8584c4760216f76b888e33d3": "\\int\\,[a f(\\theta) + b g(\\theta) ]\\, d\\theta = a \\int\\,f(\\theta)\\, d\\theta + b \\int\\,g(\\theta)\\, d\\theta ", "b887edff57a9f799631baa3fead9ee4e": "\\lnot \\exist \\lnot \\alpha", "b887f583a5b331b8420ebbdf0f43eff8": "\\alpha_{ij} = \\begin{bmatrix}1 & \\alpha_1 & 0 & 0 & 0 \\\\ \\alpha_{-1} & 1 & \\alpha_1 & 0 & 0 \\\\ 0 & \\alpha_{-1} & 1 & \\alpha_1 & 0 \\\\ 0 & 0 & \\alpha_{-1} & 1 & \\alpha_1 \\\\ 0 & 0 & 0 & \\alpha_{-1} & 1 \\end{bmatrix}", "b888a165bf8eb750bf740e0556b19e5f": "\\begin{align}\n\\tau_\\mathrm{oct} &=\\sqrt{T_i^{(n)}T_i^{(n)}-\\sigma_\\mathrm{n}^2} \\\\\n&=\\left[\\tfrac{1}{3}(\\sigma_1^2+\\sigma_2^2+\\sigma_3^2)-\\tfrac{1}{9}(\\sigma_1+\\sigma_2+\\sigma_3)^2\\right]^{1/2} \\\\\n&=\\tfrac{1}{3}\\left[(\\sigma_1-\\sigma_2)^2+(\\sigma_2-\\sigma_3)^2+(\\sigma_3-\\sigma_1)^2\\right]^{1/2} = \\tfrac{1}{3}\\sqrt{2I_1^2-6I_2} = \\sqrt{\\tfrac{2}{3}J_2}\n\\end{align}\n\\,\\!", "b889a9f1d0145264605acee08c55cb0a": "\\widehat{p}|p\\rangle = p |p\\rangle", "b889aa5cd7208d0ca05e0bb93d3e0db4": "{s}^{6}+2b\\,{s}^{4}+({b}^{2}-4c)\\,{s}^{2}-{c}^{2}\\qquad(3)", "b889ca901749452f189a0647c0166cf9": "A=A_0e^{-kt}\\,", "b889d15a0696e3ebb492b8e4f4d8e2b9": "B=2p\\rho\\phi_0", "b88a066dfed9c874f0216f5ac58d75f9": "F \\circ G", "b88a23bee9a5fa27020aa994d8545539": "\\scriptstyle{R_{\\alpha}^3}", "b88a3bd7d0304017cd3da6c6347b1e7b": "10 \\uparrow ^{10^{10}} 10 \\!", "b88a5ffa43a50eee1a9c9c40cf38bcd9": "C_{uv}^* = \\sqrt{(u^*)^2 + (v^*)^2}", "b88a6a8d27cad215a9166dbf2ceabdfa": " G = \\langle X \\mid R \\rangle ", "b88a7567ac3a39a434d530c67fe94edb": "(x_3,y_3),(x_4,y_4)", "b88a7fd35477f0d725d81fff4a38612f": "r_i = \\left\\| \\left( x,y \\right)-\\left( x_i,y_i \\right) \\right\\|_2", "b88afac170d7fac5d8e09a152b0e9c8f": " K_{t+t'} = K_{t}*K_{t'} \\, ,", "b88afbf38271ca99768716c2a740a37e": "H= \\sum_i p_i \\ln p_i, \\,", "b88b66545ef6f8bca4965ca63e1d188b": "x \\land y = x", "b88babc37fa9d6df6ed4c47f33029e40": "(-\\infty,b)=\\{x\\,|\\,x 0,", "b898527a8c7829c3ad640446f2a4a206": "\\begin{matrix} {13 \\choose 1} \\end{matrix} ", "b898ad35d34383e22735efcce089c40b": "{n} \\cdot \\pi", "b898b881f599f37913814ad42c6348cf": " \n\\left. \\psi_{-} \\right|_M = \\mathbf{z} \\cdot \\left. \\mathbf{\\nabla} \\psi_{+} \\right|_M = 0 \\; .\n", "b898c724600078a5bcea5a68d6130cd8": " \\bar{X} \\sim \\textrm{IG}(\\mu,n \\lambda)\\,", "b898cb2d3ee8be3d1ae0b9fd7c54f93c": "p | q^k - 1", "b898cf6fe1d8abcd05ec424e28d4da52": " ds^2 = dr^2 + \\frac{r^2}{4} \\left({d\\psi\\over n} + \\cos \\theta \\, d\\phi\\right)^2 + \\frac{r^2}{4} [(\\sigma_1^L)^2 + (\\sigma_2^L)^2]. ", "b898e78dc14e8bac3d0244f5e25db974": " x_k=\\cos\\left(\\frac{\\pi\\left(k+\\frac{1}{2}\\right)}{N}\\right) .", "b8990ee3a38f071fa5a4fdf71dfcba36": "x \\gg |\\alpha^2 - 1/4|", "b89916cc26e19e24f53476729b65d222": " |\\mathbf{a \\times b}|^2 = |\\mathbf{a}|^2 |\\mathbf{b}|^2 \\left(1-\\cos^2 \\theta \\right) .", "b8998437ec2e471ec626e23ce9082c31": "D^\\mathrm{op}", "b8999d506517abc298147641a340bb6b": " \\beta_0^{(P)} ", "b89a166fca759ed4774f75c0d6e09f6d": "\\lim_{k \\to \\infty} f^*(\\theta_{n_k}) = \\lim_{k \\to \\infty} f(x_{n_k},\\theta_{n_k}) = f(x,\\theta) = f^*(\\theta)", "b89acf6eb18efdf1b0c95cf3dfa80691": "\\Psi:~\\sum\\limits_{g\\in G}\\mu_g\\lambda_g\\mapsto\\sum\\limits_{g\\in G}\\psi(g)\\mu_g\\lambda_g.", "b89ad38724bd3e6843f3ad323edef061": " G(z) = 1 + \\sum_{n\\ge 1} \\left(\\frac{1}{|E_n|}\\right) g(z)^n = \n\\frac{1}{1-g(z)}.", "b89ad8f87bb5a656a9e64ecb4bc52c8a": "z\\prod_{n=1}^{p}\\left(z\\frac{{\\rm{d}}}{{\\rm{d}}z} + a_n\\right)w = z\\frac{{\\rm{d}}}{{\\rm{d}}z}\\prod_{n=1}^{q}\\left(z\\frac{{\\rm{d}}}{{\\rm{d}}z} + b_n-1\\right)w", "b89af3b244f8bb244ba26b184f477d3f": "pb+aq", "b89b3f88e44018ee6ca79c81b5495824": "E_{k}=\\tfrac{1}{2}mv^{2} ,\\quad p=mv . \\,", "b89b998f6ae262c645370bdd2d52fa1e": "T_L(x)=T_T\\left(\\frac{1}{x} \\right)", "b89bbaa324bc34a01b3a8464173bd4b1": " q_m = \\frac{\\mathrm{d} q}{\\mathrm{d} m} \\,\\!", "b89bcb869f04a07c85548ed4ab181b98": "(1/RR),", "b89be1c446d8657b8e91f2dab8ec5a07": "\\tfrac{1}{6} \\left ( (\\operatorname{tr}A)^3-3\\operatorname{tr}(A^2)(\\operatorname{tr}A)+2\\operatorname{tr}(A^3) \\right )", "b89bf077af082ecb188ffa6b716e7fbf": "S_{spo}", "b89c0948ab0bc319ae1544a910f09b4d": "\\le R|\\arg u|\\,R^{\\Re s - 1}\\,e^{\\Im s\\,|\\arg u|}\\,e^{-R\\cos\\arg u} \\le \\delta\\,R^{\\Re s}\\,e^{\\Im s\\,\\delta}\\,e^{-R\\cos\\delta} = M\\,(R\\,\\cos\\delta)^{\\Re s}\\,e^{-R\\cos\\delta}", "b89c394a7ffa42e7b1df542cbfa8bbdc": " N^a ", "b89c6015a02f4ce6e7e66dca2b910faf": "\\begin{Bmatrix} 4 \\\\ 4 \\end{Bmatrix}", "b89c9d03f4dd41369dad742dbb15d452": "y_{2c} = \\left ( \\frac{q_\\mathrm{max}^2}{g} \\right )^\\frac{1}{3}", "b89ca6d0c050f0a414b5a1679298bfd2": "N_{\\mathrm{sectors}}", "b89cbe73e93e541422a4bff3a3bb916c": "\\ell_1(x) = {x - x_0 \\over x_1 - x_0}\\cdot{x - x_2 \\over x_1 - x_2}\\cdot{x - x_3 \\over x_1 - x_3}\\cdot{x - x_4 \\over x_1 - x_4}\n = {} -{8\\over 243} x (2x-3)(2x+3)(4x-3)", "b89ccb62f143d578bc098eebb9141978": "\\mu = E[X]", "b89ce2f1b41707a5da3749f4f7993755": "R = 1/\\kappa", "b89d0423634bb189c4137797d33a79cd": "\\cosh^2 x - \\sinh^2 x = 1\\,", "b89d8b55ccf078b8702df32820ed20c4": "d(p,q) \\ge cR", "b89d9587f350a9ab37826dd3059ec4ca": "1+\\frac{1}{4}+\\frac{1}{4^2}+\\cdots+\\frac{1}{4^n}=\\frac{1-\\left(\\frac14\\right)^{n+1}}{1-\\frac14}.", "b89e57bfe860691e546011b992f089d8": "\n\\forall y\\in Y\\qquad \\sup_{x\\in A}|\\langle x,y\\rangle|<\\infty.\n", "b89f49363491c1fd8d5def9f17008acf": "\\int_{0}^\\infty \\frac{1}{\\Gamma(x)}\\, dx \\approx 2.80777024,", "b8a09a40cd74d61afad3aed8a57c9c98": "\\operatorname{deg}(p)", "b8a0cb9ce72ea102c6e765a1467e93d0": "\\{\\text{A}, \\text{B}, \\text{C}\\} \\,", "b8a162e7bc884f5c8365fe1c7ba7b747": "\\frac{c(c + \\alpha)_{1 - \\gamma}(c + \\beta)_{1 - \\gamma}}{(c + 1)_{1 - \\gamma}(c + \\gamma)_{1 - \\gamma}} x^{1 - \\gamma} ", "b8a2a8bd6200b323cb5b47225b8d06ff": "\\nabla^2 f = \\frac{1}{{c_0}^2} \\frac{\\partial^2 f}{\\partial t^2} \\,", "b8a2d4470b74c63c49dbaa725b738a25": "\\forall x \\forall y [ \\forall z (z \\in x \\Leftrightarrow z \\in y) \\Rightarrow \\forall w (x \\in w \\Leftrightarrow y \\in w) ],", "b8a36f43d57f942e036244f278dc145c": "|\\psi_k\\rangle = U_k|\\psi\\rangle", "b8a3974423fae23ee4795685d7908bcd": "I=-\\omega q_0 \\sin(\\omega t + \\phi)\\,\\!", "b8a3a72cb1c04d8c00ccad532fd1ca98": "g = \\tfrac{G}{R+G+B}", "b8a3e636ff7b23b72a14999b9dca80a7": "\nEQ_{DFA} = \\{\\langle A, B \\rangle \\mid A \\text { and } B \\text{ are DFA's and } L \\left( A \\right)= L\\left( B \\right) \\}\n", "b8a3f6735ea7e153d5c05f6efd65ddcf": "R = \\frac {\\pi \\times 12.5} {330} \\approx 11.9%", "b8a414a22a3eaaaebe4b6cb0ba0cc1ef": "n^T t = (M_l^{-1T} n)^T t'", "b8a4727d14ee7f43c3c8d9501e8d39fc": "c_1, c_2,\\ldots,c_n", "b8a4bbb7fb49e4a36f191d1fcd5550c8": "g(x) = \\sin\\left(\\frac{1}{x}\\right)", "b8a514ecf545799f8ac25834f5c6ec6f": "\\overline{1} = 0", "b8a521de83da2b17a2024c8b7db2877e": "f(f(x)) = 1-(1-x) = x \\, .", "b8a57fc266b5e80f80201a27daa5b1e4": "f(x)< M g(x)", "b8a58b224152f9fd9e18340723f7337a": "G_{vv}", "b8a5a6f3a54da36307e6eabaa55ea8e8": "w \\in \\reals_{++}^N", "b8a5d65a9e8fb41b76b53ace1c255449": "V \\propto L^3", "b8a5f4a94c70b68a20f80f4b0138270a": "A_q(n,d)\\ge q^k", "b8a5f6b21dee452848597fec7b3bb5d0": "L_2\\colon U_2 \\to \\mathbb{C}", "b8a6094f07db0d32940d7889436d4f36": "C_1\\,\\!", "b8a61362f670646db691a7424555b7b0": "h(n) = a^{n} u(n)", "b8a634038a70d816164289c48b4b05df": "I_Q\\!", "b8a74bd4a57bf1d1068d3cb240498581": "i e ^{ix}", "b8a793aeb1bd78e9d9f6e01115062994": "\\vec{f}", "b8a7b586630bdaa7035df355d6c56bc8": "y_{(2)}(t)", "b8a7c839046accfa720be5fac1adb14d": "P = I^2 R = \\frac{V^2}{R},", "b8a803c54482e67d07dcd7dcdd356206": " \\langle x, x \\rangle = 0 \\iff x = 0.", "b8a86e50eb27fd624a623bfc4d6a3596": "\\mathrm{2 \\ FeS_2 + 7 \\ O_2 + 2 \\ H_2O \\longrightarrow 2 \\ Fe^{\\,2+} + 4 \\ SO_4^{\\,2-} + 4 \\ H^+}", "b8a8913df88de1f6d2474aee7bee1e67": "\n \\mathbf{v} = v_k~\\mathbf{b}^k\n ", "b8a8a019ca9770d65559651a4d99330c": "\\max(u+v-1,0) \\leq C(u,v) \\leq \\min\\{u,v\\}", "b8a904a576b5d779486bd8bd67a62ca5": "\\left(A^{c}\\right)^{c}=A.", "b8a90bb2f6cb3c807a8318c9a17f87ac": "f(x|0,\\lambda) = \\frac{1}{2\\lambda} \\exp \\left( -\\frac{|x|}{\\lambda} \\right) \\,\\!", "b8a9eabe79fe64cf742780b71414c552": "\n\\cfrac{\\mathrm{d}}{\\mathrm{d}t}\\int_{\\Omega(t)} \\mathbf{f}~\\text{dV} ~.\n", "b8aa0dc923aa67b1de31179dfa4e84b6": "\\scriptstyle{t}", "b8aa37385c748d0b8da95eb6ee11b0c5": " R_b = \\frac {Z_2} {\\sqrt {1 - Z_2/Z_1 } } \\, ", "b8aa46b097b892bcd2503ec49b4eade1": "A, B, C\\in\\mathbf{H}_n^+", "b8aaaed68aed33ee2b0ecf9a9593a717": "f : \\alpha \\rightarrow A", "b8ab2254eeaeb5b93687d3e0375d9cb0": " s_c ", "b8ab68e98451ed29beedf917ae082599": "\\textstyle l_2", "b8aba318727aee243c401328aa5601bb": "\\psi\\ = f_1\\!\\left({Q\\over{ND^3}}\\right),\\,", "b8abe49beb0fec9e69ed763b6fabc079": "\\int k \\frac{dy}{dx} dx = k \\int \\frac{dy}{dx} dx. \\quad \\mbox{(3)}", "b8ac038a0060449c342de82881d0b074": "dim\\,G \\le 2\\,\\chi(G)", "b8ac1b11d2eaed10f3a1f8cb1f2ad2f9": "\\frac{dQ}{dt}", "b8ac1d27aabf73233e2fc83a209654ef": "\nh(v) =\n \\mathrm{round}\n \\left(\n \\frac {cdf(v) - 1} {63}\n \\times 255\n \\right)\n", "b8ad26640244956a617ff0b9a6dca70f": "\\mathfrak{P}^{44b}", "b8ad338cbc4f91289deeddbb0f1294fa": "\\int (F_A,F_A)+(D_A \\phi,D_A \\phi) - \\lambda(1 - \\| \\phi \\|^2 )^2.", "b8ad43e1f47eae9eb58210df285b742f": "Z_n^{m}(\\rho,\\varphi)=(-1)^m Z_n^{m}(\\rho,-\\varphi)", "b8ad6b4b404bb5e3f7371aa8fc920c1c": "R_n^{(b)}={b^n-1\\over{b-1}}\\qquad\\mbox{for }b\\ge2, n\\ge1.", "b8ad8d87ecae08cb9f678b8bebcdffeb": "\\vec x_1=\\vec x_0+\\vec v_0\\,\\Delta t+\\frac12 A(\\vec x_0)\\,\\Delta t^2", "b8ae408e67e5e1c0cf6692f72c224ed2": "\\Delta\\;h = u\\cdot \\Delta\\;v_w", "b8ae956639f3013f4188dd92ca760760": "\\chi_\\text{e}^\\text{SI} = \\chi_\\text{e}^\\text{LH} = 4\\pi \\chi_\\text{e}^\\text{G}", "b8aeaa721511472d40ab24447aa4fc63": "\\displaystyle f(x,y)", "b8aecd02bed058c8186d57773120d73c": "s_n > 1", "b8aee4e4681fb34cb675c7a3f6661335": "S = \\left\\lbrace\\mathbf{v}_1=\\begin{pmatrix} 3 \\\\ 1\\end{pmatrix}, \\mathbf{v}_2=\\begin{pmatrix}2 \\\\2\\end{pmatrix}\\right\\rbrace.", "b8aeec221bddf80461166eb0994c5da3": " [z + pl(a - p) + t(2ap - p^2 - 1) - pm]^2) ", "b8aeec89ec825951da5c2353deec77a7": "\\exists K >0, \\forall x,y: \\quad |f(x)-f(y)|\\leq K|x-y|.", "b8af1c8555ccf5a2218bed6d7c33ed71": "S(x) = f(x/|x|)|x|^\\lambda\\,", "b8af37fffd70ae4cb72a2dc54e5ca40a": "\\displaystyle{\\{G_r^+,G_s^-\\}=L_{r+s} +{1\\over 2}(r-s)J_{r+s} +{c\\over 6} (r^2-{1\\over 4}) \\delta_{r+s,0} }", "b8af38f3bf6df72f6f8a351b6a97e9e5": "E:y^2=x^3+Ax+B", "b8af81c0e3a8a513f397ea40a3ba522b": "(\\lambda v.\\ M)\\;N\\ \\ \\longrightarrow_\\beta\\ \\ M[v := N]", "b8afc7f218ef8ad469d071094fc0491d": "s\\ne s'", "b8afd874782496d59c182e5395d31c2c": " \\mathfrak{H} ", "b8afe267018201cf468f96cbfba03882": "(y + 1)^2", "b8affe3bfb0b2fa9593c1608e83458ed": "\n{\\left( \\frac{dr}{d\\varphi} \\right)}^{2} = \\frac{E^2 r^4}{L^2 c^2} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{m^2 c^{2} r^4}{L^2} + r^2 \\right)\n\\,", "b8b016e7605567c365328696ae2a8c18": "V_n(\\mathbf{R}^n)", "b8b0222aa3e3baf952adb1207d1cf057": "\\displaystyle \\hat{f}(\\nu - 2\\pi a)\\,", "b8b02cdacd901f7863d5a818b41ca3e8": "\\phi(zv) = \\overline{z} \\phi(v)", "b8b05bc60bff6aecdba0dd2141ee2a48": "x = \\textstyle\\sqrt{\\frac{b}{3}}", "b8b0682996dfb7f881d43fba5491a89d": "D\\colon H^1(M;{\\mathbb{Z}}_2)\\to H_2(M;{\\mathbb{Z}}_2), ", "b8b08c56acdf4d880c5a14affadae5b5": " F = \\R ", "b8b0a651ff9d02b67e99a8506c25729f": "a_{i1}=a_{i2}=...=a_{im}=\\frac{v^2}{\\sigma^2+mv^2}", "b8b0a85adc2932b8868f3b7b7766d9a8": "A_1=\\{n | (X_n,d_n)=(X,d_X)\\}\\,", "b8b0fe84eb67cdfddd266e8fb51c7c46": "2^{\\ell/3}", "b8b10cbfbe5701bf4a62648fc6423296": " G = \\gcd (p-1, q-1) ", "b8b10fd1bbebed59341134a12e42d051": "\\nabla \\lang\\mathbf{r}|\\Psi\\rang", "b8b15c368aae3f06195e985f4167915f": "i^2=-\\!1=j^2=k^2", "b8b174d56467e23951b7672df481a8e2": "X^K", "b8b1ab099a8bc07578abdf1d3037af2b": "(x'_1,x'_2,x'_3,t')", "b8b1ada879bc1925069081ed49d5e818": "\\text{ Skewness }=\\frac{\\text{ optimal cell size - cell size }}{\\text{optimal cell size}}", "b8b1b55df143d778ab22f2fc15026742": "c(V \\oplus 1) = c(V)", "b8b1db3250c69bb5840eb25681e8d39a": "(\\sigma^2)^2", "b8b221a858263358b5acb42285af1743": "p(f_i|c_j)\\ ", "b8b24952ca8c58c8368b8c960169360a": "\\kappa_n(W_1+\\cdots+W_m)=\\kappa_n(W_1)+\\cdots+\\kappa_n(W_m).\\,", "b8b2512e6c854fcefb7bff4e3664b43a": "\\ D_\\mu = \\partial_\\mu - i \\frac{e}{\\hbar} A_\\mu", "b8b28691a5a198514d96a4cc282977c7": "U\\to L(X,Y) \\,", "b8b28b504014ff8c587be4deeee04791": "X_i := \\exp(-i/N);", "b8b2f2e244a8edf372dd17734fad41c5": "\nu(y=-y_{l}) = g_{1}(x)\n\\quad\\text{and}\\quad \nu(y=y_{l}) = g_{2}(x) \\qquad \\text{(1-b)}\n", "b8b3bf912a01e331e1a4193eaf27a7f4": " EBAC = (0.806 \\cdot 2.5 \\cdot 1.2)/(0.49 \\cdot 70) - (0.017 \\cdot 2) = 0.036495627 \\approx 0.037 g/dL", "b8b414fb745fcda3625b8ce3836529ea": "\n \\begin{align} \n & \\boldsymbol{\\sigma}^* &=& \\mathcal{G}(\\boldsymbol{F}^*) \\\\\n \\Rightarrow & \\boldsymbol{R}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{R}^T &=& \\mathcal{G}(\\boldsymbol{R}\\cdot\\boldsymbol{F}) \\\\\n \\Rightarrow & \\boldsymbol{R}\\cdot\\mathcal{G}(\\boldsymbol{F})\\cdot\\boldsymbol{R}^T &=& \\mathcal{G}(\\boldsymbol{R}\\cdot\\boldsymbol{F})\n\\end{align}\n", "b8b4326ebb88870f8cc97ab3f59a0867": "[0;1]", "b8b4336a693a503aecd17a6d875852c7": "\\sigma_\\text{tot} = \\oint d\\Omega \\;\\frac{d\\sigma}{d\\Omega}", "b8b43a6a19626dedac465d47b83d19f6": "\\widehat{f'}(n)", "b8b48e584cbd8a35c600b116f05407cb": " \\frac{\\Delta Y}{Y} ", "b8b4e27fa432d23118897720a0d9b538": "f \\circ g \\sim Id_B", "b8b5089c70ae62534970e2d8aa67e96d": "FX", "b8b51ea823fd3e69ab50a57ec9c316e3": " \\langle E\\rangle=\\left\\langle \\psi \\left|i\\hbar\\frac{\\partial}{\\partial t}\\right|\\psi\\right\\rangle = \\int_\\mathrm{all\\,space} \\psi^*(\\mathbf{r},t)\\left(i\\hbar\\frac{\\partial}{\\partial t}\\right)\\psi(\\mathbf{r},t) d^3 \\mathbf{r} = \\hbar\\omega ", "b8b52ef19625b9857849c5648a9ab2ff": "F(\\mu) - L(F(\\mu)) = \\frac{\\text{mean deviation}}{2\\,\\mu}", "b8b5331c19c0a68648f99c16842742f8": "\\mathrm{dim}\\pi", "b8b5565b7d78d328f9bd7f8f41517156": "\\begin{align}\n\\mathbf{E}^{x} \\left [ f(X_{t+h}) \\big| F_{t} \\right ] &= \\mathbf{E}^{x} \\left [ \\mathbf{E}^{x} \\left [ f(X_{t+h}) \\big| \\Sigma_{t} \\right] \\big| F_{t} \\right] \\\\\n&= \\mathbf{E}^{x} \\left [ \\mathbf{E}^{X_{t}} \\left [ f(X_{h}) \\right] \\big| F_{t} \\right] \\\\\n&= \\mathbf{E}^{X_{t}} \\left [ f(X_{h}) \\right ].\n\\end{align}", "b8b56aa84f66a5cdf5bb46d8b0c36a6c": "\\rho_{\\mathrm{bound}} = \\nabla\\cdot (-\\mathbf{P})", "b8b56e19ded5afaf332c6c09f34e3363": "R \\subseteq R^{+}", "b8b57d2dfd6dd09f3ba94f6612ab5f8f": " \\frac{d}{dx}\\left( x^x \\right) = x^x(1+\\ln x).", "b8b5d407f904f7eaa00b2f90f71d6114": "\n\\alpha^T = \\alpha \\left( \\frac{293 K}{T} \\right)^{0.8}\n", "b8b6287baf54ad22ce8257c674acca4b": "r^{\\prime}", "b8b6314b2684338a1e6eaced2d0ae3db": "W_{1/2} \\,", "b8b696352011641e48e77ea29ca425d0": "D = A \\times B - C = (d_{ij})", "b8b703afc2ec1bb172ad654e315bd270": "{1 \\over 2} \\oint_{t_0}^{t_1} \\vec u \\times \\dot{\\vec u} \\, dt.", "b8b778fdc1d4e258734273493ce6821a": "\\scriptstyle |V_\\mathrm T|", "b8b7988d77e46705ce64b7e12006589e": "{\\tilde f}: {\\tilde X} \\to {\\tilde Y}", "b8b7b0b49e954884fc8b2343a741a88d": "\\overline{x} = \\frac{\\sum_{i=1}^N x_i}{N}", "b8b7b69dd3b0a2fb034737ca01fa64aa": "\\frac{\\partial p}{\\partial z}+\\rho g=0.", "b8b80c7b788c66734337e7b2b3ebfc11": " X^{\\star\\star}", "b8b85b3218d54bebe6ec3752d4050cc7": "m_\\mathrm{h}", "b8b891599266ca164f75a2484a2e7088": "\\Gamma_{r}", "b8b8e4b2341c8c749bbf7e07ec69bcb3": "t^3-t^2", "b8b9dba56c4a984f334962a6864c5e8c": "v_{i}=\\binom{n}{i}", "b8b9e363bef496a20ace5e6ae20710c1": "1 \\to \\operatorname{Homeo}_0(\\mathbf{T}^n) \\to \\operatorname{Homeo}(\\mathbf{T}^n) \\to \\operatorname{MCG}(\\mathbf{T}^n) \\to 1,", "b8ba62cb85f0c34b3b2d160b77db0eb3": "|x=S\\rang = \\sum_n | \\psi_n \\rangle \\left\\langle \\psi_n | x=S \\right\\rangle = \\sum_n | \\psi_n \\rangle \\sqrt{ \\frac{2}{L} }~{\\rm sin}\\left(\\frac{n \\pi S}{L}\\right)", "b8bab924acdac9218f63254f0405c3f9": "\\frac{\\rho_2}{\\rho_1} \\approx\n \\frac{\\gamma+1}{\\gamma-1}.", "b8bac7ec102b3e1a265123b044789f14": "\\tau: \\mathbb{N} \\to \\mathbb{N}", "b8bb965b9c9f9e7a9fac4cb6db651fac": "\\mathrm{lim}\\,\\mathrm{Hom}(N,F-) = \\mathrm{Cone}(N,F).", "b8bb9c82a7dddc5d5c2a02629328b558": "\\{1,2,\\ldots,N\\}", "b8bbafec952a72c34efacaeb248f8725": "\\alpha^*(t), \\omega(t)", "b8bbd0b01e42ea0bd9613f94d832a6cf": "\\pm 1\\text{ to }\\pm 10", "b8bc089b0efdc2502bf402c5f5f42893": "(\\ell_g \\varphi)(g')=\\varphi(g^{-1}g').", "b8bc392eac89235d45634ba09bdc7122": "\\mathfrak{R}=\\operatorname{Hom}(\\pi,H)", "b8bc676e18856ab99264e412cd746aa4": " g(a;p)=\\left(\\frac{a}{p}\\right)g(1;p). ", "b8bc998a3a8fb403c931cd97646b7e0c": " L_c^+(f) = \\left\\{ (x_1, \\cdots, x_n) \\, \\mid \\, f(x_1, \\cdots, x_n) \\geq c \\right\\} ", "b8bc9dd7df553c39dd656c4e79fa0ad2": "AX + U(C^{-1}+VA^{-1}U)^{-1}VA^{-1} = I", "b8bcc58f10b6407dfb37e21ea16d7e28": "= - \\frac {A}{T}", "b8bcd3cade794b8fe0ac7b178b644d70": " \\sigma_{ij}(\\lambda) = \\lim_{\\delta\\downarrow 0} \\lim_{\\varepsilon \\downarrow 0} \\int_\\delta^{\\lambda +\\delta}{\\rm Im}\\, M_{ij}(t +i\\varepsilon)\\, dt.", "b8bd1f94ba7fff77c0b46488f3eafc55": "4 \\pi \\alpha \\,", "b8bd398aa7d53f30f584447aa52adf2d": "\\scriptstyle\\nabla", "b8bd64655075b083226ec23a9ba00625": "\\varphi(n) = \\sum_{k=1}^n \\frac{1}{k - m_1}.", "b8bdb83a7ea12ab1788f3429aa7f2aaa": "\\|Mf\\|_p^p = \\int \\int_0^{Mf(x)} pt^{p-1} dt dx = p \\int_0^\\infty t^{p-1} |\\{ Mf > t \\}| dt", "b8bdffbd8ca48dfef02f31204c0c9335": "\\, E(r)/r^2 \\, \\to 0 \\,", "b8be0571a618a68b6309c170d3b04207": "(G,+)", "b8be07936ed033bf1711a67213850909": "n_{pas} = {3}\\times \\frac{3600}{0.2}", "b8be53fc91a7e328809afdadd5e47448": "x(t-t_0) \\rightarrow W_x(t-2,f)\\,", "b8be6f99dc879c9ee26d4ab41996313c": "\\mathcal{S} = \\int \\left( - \\frac{\\hbar^2}{m} \\eta^{\\mu \\nu} \\partial_{\\mu}\\bar\\psi \\partial_{\\nu}\\psi - m c^2 \\bar\\psi \\psi \\right) \\mathrm{d}^4 x ", "b8bec58337a36d66981c3c2c8bd52ee8": "\\binom{n}{d}", "b8bed6415701390d318a028cb3856cee": "\\{1,1,1,3\\} \\cap \\{1,1,2\\} = \\{1,1\\} \\,", "b8bedda91f4d222aa38fdc933118cb18": "\\Delta(c_1,c_2)", "b8bee02962d38439bed5ecc2fe0563eb": "f^{-1}[A]\\in V", "b8bef56ec0fe5a44206e942434b8efc9": "\\psi(\\mathbf{r_1},\\mathbf{r_2},\\,t)", "b8bf00456be21c8a3e36090c569857ae": " J\\cdot (K\\cdot X) = (JK)\\cdot X", "b8bf1b353991830947fe347499bbc436": " \\chi(2,5) = q_3 + q_2 q_3 - q_3", "b8bf5fdbc25b7e9513fa710fbd9e856b": "n \\wr \\pm 1", "b8bf897ab08afec17ac7b54cbc93c05a": "\\hat{\\nabla}", "b8bf9ec2857409070721a067209a2932": "\\Gamma_{21} = \\frac{\\eta_2-\\eta_1}{\\eta_2+\\eta_1}", "b8bfa16badb3a9727306d18fedc5c19a": "T = (t_1,\\ldots,t_n) \\,\\!", "b8bfbf83d35dcab315dadfea4c446dbc": "a^k", "b8bfcbcf93d124fb19a235413b77877e": " \\Omega(\\langle E\\rangle)=\\frac{\\langle E\\rangle^{m-1}m}{\\sqrt{2\\pi m}m^me^{-m}}", "b8bff3608421623acb56df270957d4d2": "\\sqrt{m}", "b8bff5e58215db4c1196838907485685": "A[y/x]", "b8c05e674d257ef0a9727bf317fcb0de": " G(.,\\boldsymbol{s}) ", "b8c05f64d379f12b69d5d3a712cd91fe": "X^w=\\{X_i^w\\}", "b8c0a17c9b74cb68a7434c45d8130b5a": "T:X\\to \\mathcal{N}(X)", "b8c0a2786aea1a238e2fe83086da76aa": "s_1 \\dot{x}_1 + s_2 \\dot{x}_2 + \\cdots + s_{n-1} \\dot{x}_{n-1} + s_n \\dot{x}_n = 0", "b8c0d3b6102b7c9a276b6d2157f4ff73": "(K + U_g)_i = (K + U_g)_f \\,", "b8c12a259d80a755209777a806776028": "E_{kin}", "b8c13c95ded9efa205d2f3b4d0c73240": "I_N", "b8c153bc1156d055d3661f462ffc6951": "L=(L_{ij})", "b8c176b39791f34b31fe7d5f652c215d": "G_i = (G_0)_i", "b8c1791f9f7576cd2d0242549e203790": "v = \\operatorname{Re}\\{v\\}+i\\,\\operatorname{Im}\\{v\\}", "b8c1c7f867fa85502f166e8cba1bcda0": "\\sigma'_n", "b8c1d3ae0c6ca9971093a40313949636": "\\psi_n(x,y,z;t) = e^{-i\\omega_nt} e^{ik_xx+ik_yy} \\sin \\left(k_n z \\right)", "b8c27e905b917bdf4b30a958201d2860": "\\frac{d}{dt}", "b8c29f30ca02c1ac596565ef1a550746": "f = {1 \\over 2 \\pi \\sqrt {LC}} \\,", "b8c2afb23dd94c997007e9e54ae96717": "\\sigma = \\cup^{\\infty}_{i=1} \\sigma_i", "b8c32666f8ac8f928d12371ee0552199": "d = 2^{m-r}", "b8c37f722bd3586e7a9b26421c14a12b": "\\mathbb{N}^n_0", "b8c39ce03437652ee00aacb2610dee47": "\n\\Psi(1,2, \\ldots, N) = \\psi_{n_1}(1) \\psi_{n_2}(2) \\cdots \\psi_{n_N}(N)\n", "b8c3a49211477ad0f22b2df5df64d696": "\\mathrm{d} X_{t} = b(X_{t}) \\, \\mathrm{d} t + \\sigma (X_{t}) \\, \\mathrm{d} B_{t}.\\ ", "b8c40550c2b8a4e4e6d7731a27727b94": "s>1", "b8c40f9a56ca9394d51ab29eb43fb95d": "\\mathbf{\\begin{bmatrix} \\hat{a} \\\\ \\hat{b} \\\\ \\hat{c} \\\\ \\end{bmatrix} =\n\\begin{bmatrix}\n \\frac{1}{a} & -\\frac{\\cos(\\gamma)} {a\\sin(\\gamma)} & \\frac{\\cos(\\alpha)\\cos(\\gamma)-\\cos(\\beta)}{av\\sin(\\gamma)} \\\\\n 0 & \\frac{1}{b\\sin(\\gamma)} & \\frac{\\cos(\\beta)\\cos(\\gamma)-\\cos(\\alpha)}{bv\\sin(\\gamma)} \\\\\n 0 & 0 & \\frac {\\sin(\\gamma)} {cv} \\\\\n\\end{bmatrix}}\n\\begin{bmatrix} x \\\\ y \\\\ z \\\\ \\end{bmatrix}\n", "b8c428dd641001a0b6294e9bf9beaae0": "\\textstyle \\mathrm{\\Beta}(a,b)", "b8c42eb4d42f50322b55b182b8a749a5": "x_n\\to x", "b8c460643608d59ace5819d3971b4261": "m^{2}", "b8c498cf0232b664377dcc76a33c4ead": "S^2 \\subset \\R^3 \\,\\!", "b8c4a19be70635d845b266b71448a556": "\\frac{da}{dt}=-2a^2b \\quad\\text{and } \\frac{db}{dt}=0.", "b8c4fa5f860a81bbad87f88179c3b21d": "b = \\frac {fm_\\mathrm s} {N} \\frac { x_\\mathrm d } { s \\pm x_\\mathrm d } ,", "b8c56a334a113610975f7ee16d4f88f7": " (1)\\quad Y = \\exp(-aX) ", "b8c571de06aaa367f173cbf514606ab9": "j = 1 \\dots T", "b8c5b66a6b0f220aa3d2e1c4b973f2c4": "c(\\theta)", "b8c607b4701524574858cfc7de17d637": "\\phi \\to (\\psi \\to \\phi)", "b8c60893a57e3050eed7850b181959f5": "\\frac {(s - 23.5)}{4.2} 2 + 5.5", "b8c6f88c26ba671b21f514301841797a": "\\sum_j a_j ^* a_j = 1.", "b8c7762661b0203c7a0a0f46b1b4aa62": "u(c,l) = U\\left(c - G(l)\\right), U'>0, U''<0, G'>0, G''>0. ", "b8c7ebadd2dde0c933d95bfe090f98c9": "\\mathbf u' = T\\mathbf F(\\mathbf u,\\lambda),\\, \\mathbf u(0)=\\mathbf u(1) + N.\\Omega", "b8c7eefde7b7b427b7dbd3ca78414d1b": "x_{i+} > 0,\\ x_{+j} > 0", "b8c80e8ba283a2534f9843764d0faf3f": "(3, 3),\\quad (4, 3),\\quad (3, 4),\\quad (5, 3),\\quad (3,5).", "b8c85270cb1476bf1c340c23e0db2f4f": "\\tan^{-1}", "b8c85bb33048b386a35e94bde61735a7": "\\cosh x = 1 + \\frac {x^2} {2!} + \\frac {x^4} {4!} + \\frac {x^6} {6!} + \\cdots = \\sum_{n=0}^\\infty \\frac{x^{2n}}{(2n)!}", "b8c8735bc2f2a6061d0e839ac017632b": "\\omega_N^{m n k} = \\omega_{N/m}^{n k}", "b8c891d47240a6a3cae65c73d971d011": "T(X_1^n) = \\left(\\min_{1 \\leq i \\leq n}X_i,\\max_{1 \\leq i \\leq n}X_i\\right)\\,", "b8c8d73789f94b50e8d9e1f449e8fa72": "(x;q)_\\infty = \\sum_{n=0}^\\infty \\frac{(-1)^n q^{n(n-1)/2}}{(q;q)_n} x^n", "b8c8dd339acb5f2afbd591a04d67777d": "C : y^2 + h(x) y = f(x)", "b8c8e818d33671bb9e0d7c8fbf26d4f2": "\n(B.3)\\quad \\nabla^2\\Phi\\,=\\,2\\psi_{,\\,\\Phi}\\cdot (\\nabla\\Phi)^2,\n", "b8c95200b78eb1605732fce4580e9ef2": "v_j = \\sum_{i=1}^t c_i Y_i", "b8c992c0f7aed18b9b5ac3ebef3825f5": "{\\bold \\ f}", "b8c9d0574c70663a333eba54c9f22c9d": "\\varepsilon_m", "b8c9f9605af106d4f89f771083873dea": "\\eta_{th} ", "b8ca2ecb289dd9949f207b597483765e": "W_k= \\frac{1}{\\sigma^2_E+\\left( \\frac{\\partial E}{\\partial v} \\right)^2_k\\sigma^2_v} ", "b8cab0aa8fa1e318c2df3e81fbc9c615": "i V_- \\\\\n0, & \\mbox{if }V_+ < V_- \n\\end{cases}", "b8f4584ea698debb88f1516bbc97821e": "\\mathbb{Z}_+", "b8f4bd032232b3745a66cfd8a29f6b58": "\\nu=\\tau=\\gamma=0\\,,\\quad \\mu=\\bar\\mu\\,,\\quad \\pi=\\alpha+\\bar\\beta\\,.", "b8f4d69a05fe63143ca663861582939d": "\\int\\limits_{-\\infty}^\\infty |f(t)|\\,\\mathrm{d}\\,t \\;<\\;\\infty\\quad\\mbox{und } \n \\int\\limits_{-\\infty}^\\infty |g(t)|\\,\\mathrm{d}\\,t \\;<\\; \\infty", "b8f55348be790e3b3610fbfca38e39ba": "p(H1|y) = \\frac{p(y|H1) \\cdot \\pi_1}{p(y)} ", "b8f568c048f69b664340cd088005c8e1": "k = \\sqrt{1- v^2 / c^2}", "b8f5984150495c1bf05a55706c291719": "F:M \\to N", "b8f64155886e69e5e0cb7a1f1dc1bfb0": "-\\frac{1}{2}e^{\\xi(\\beta-\\tau)}n_\\eta(\\xi)\\left(\\tau^2+\\eta\\beta(\\beta+2\\tau) n_\\eta(\\xi)+2\\beta^2n^2_\\eta(\\xi)\\right)", "b8f6a8e633fd38db84eaec533e2d7e57": "\\sigma_{ij}\\,", "b8f6be62fd7198eff2326362a2d01ed0": "\\pi_1(B(S^{-1}S)^0) = \\pi_1(B(S^{-1}S)^0)^\\text{ab} = H_1(B(S^{-1}S)^0) = H_1(BGL(R)) = H_1(GL(R)) = GL(R)^{\\text{ab}} = K_1(R).", "b8f6f29dc1f5be1c98ba23bb455cc655": "\\tfrac{3250433}{479001600}", "b8f6f7d2d03f0f1776b1dde049db694d": "\\lim_{n\\to\\infty} k(n) = \\infty ", "b8f727d0b6cc9ba7ea8bbde5f1642d7e": "c_{i+1}:=\\frac{f(x_{i+1})-f(y_{i+1})}{x_{i+1}-y_{i+1}}\\le\\frac{f(x_i)-f(y_i)}{x_i-y_i}=:c_i", "b8f72ae30b7ea69075e0762214a5adde": "\na_{1,2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{x-x_0}{u_x} = \\frac{y-y_0}{u_y}\n", "b8f72f293239adf1bf0810b30d228cea": "a \\mapsto a^\\#_u: \\mathfrak{g} \\to T_u P", "b8f76e28bafc572673efaa24cb16fd5a": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta = (1\\times\\cos\\theta) + (0\\times\\sin\\theta) = \\cos\\theta \\, . ", "b8f78abf35fb328cd78b7a508ffacf50": " \\and T_9 = [F_9, S_9, A_9]::L ", "b8f7c9998f5c6f6e6906d6fa04b77fc2": "v: B \\to C", "b8f812088cd6806b9c3fffab69c4d96c": "\\sum_{s\\ni e} x^*_s \\ge 1", "b8f81a0d3ac34e81e171a7a82b62ceaf": "G[Y]", "b8f8967311d06be27429bbb16f908e45": "dr_t = (\\theta-\\alpha r_t)\\,dt + \\sigma \\, dW_t", "b8f8fb440d35f10d9730dc7227d366aa": "O|\\psi_t\\rangle=O\\,T(t)\\,|\\psi_0\\rangle ", "b8f8fd1addd90fba817479ec7f24acd8": "u_1 > u_2", "b8f915ead929c7e5e2a0263359cc76ca": "u(0,t) = 0 = u(L,t) \\quad \\forall t > 0 ", "b8f91f01f2674fc98918734b5c6e0988": "dx = \\varepsilon y\\,dt \\quad\\text{and}\\quad dy=-y\\,dt+dW \\,. ", "b8f924708653f5fbb8ddc13626740b9a": "+\\, \\exp\\;[-\\,(z + H + 2mL)^2/\\,(2\\;\\sigma_z^2\\;)\\;]", "b8f94065029949847785048768ce9438": "E_t(S_{t + k})", "b8f98a7842122c822acab181361e37c0": "\\begin{align}\ns&=(Y-x) \\left (Y-x^q \\right )\\cdots \\left (Y-x^{q^{d-1}} \\right ) \\\\\n&=s_0+\\cdots+s_{d-1}Y^{d-1}+Y^d\n\\end{align}", "b8f9f0d8e28d9823d00aedaf41172a9a": "\\displaystyle{T_{K,0}=P_0 T_K P_0.}", "b8fa1d9e2d5c13188cc7bf3d03103bb9": "\\left( k/n,c/n\\right)", "b8fa393a2917364ab6c81dece669152a": " \\phi_> = \\left( {-r}+\\frac {\\kappa-1}{\\kappa+2}\\frac {{R^3}}{r^2} \\right)E_{\\infty} \\cos \\theta \\ ,", "b8fa7f5c084b58130e3144eb9aed7f57": "y = (1 + x) \\left(1 - x/K \\right)", "b8fa9472929a82aa484b61f944a2799e": "\\mathbf{P} = m_0\\mathbf{U}\\,,", "b8fabe6c058bc8848a6d19c3246258fe": "k_{\\rm C} = 1", "b8fb0ac928e1704de53ae8b4afacfe86": "6500\\ \\text{K} \\times \\frac{1.4388}{1.438} = 6503.6\\ \\text{K}", "b8fb5569f7a270b79832eb8e36e9a5ef": "\\langle g_{\\Omega}(\\hat{a},\\hat{a}^{\\dagger}) \\rangle = \\langle g_{\\Omega}(\\alpha,\\alpha^*) \\rangle.", "b8fb62f274d10516d752d34a491cbb81": "x=\\sum_{i=\\frac{x}{y} - \\frac{y-1}{2}}^{\\frac{x}{y} + \\frac{y-1}{2}}i.", "b8fb6888bfbd6fc0fb3a22a8f91979b3": " P\\to M ", "b8fbb50e5e5e2d989d20a78076665e18": " x_2 ", "b8fbd856bd6154ff191b28b864faa798": "C=\\frac{8L^3}{E^*wt^3}", "b8fc365b2d7b0b8c9f7ca6c9abf2d91c": "\\textstyle r(i,j)", "b8fc6c4d6682b4ee832ef7f9ff059f8d": " \\mu(A) \\leq \\|A\\| ", "b8fc829c59d349e94d79ee4f4d298562": "\\mu = \\frac{1-e^{-r}}{r}", "b8fcc9e75a4c93044689078912ec932d": "1+\\frac{1}{\\sin(\\frac{\\pi}{7})}", "b8fcecc7e5cf92c0b394842acfd2c601": "\\boldsymbol{M} = \\frac{C}{T- \\theta}\\boldsymbol{H}", "b8fcef7402b93755d04bac587a7c9a78": " ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2 ", "b8fcf44cb01112b85a7a6da1fe6fb9ec": "\\left( { e^2 \\over L_B}\\right)", "b8fd5c93b90bf57d095183cb10af9442": "1_A =\\sum_{k=1}^n (-1)^{k-1}\\sum_{\\scriptstyle I\\subset\\{1,\\ldots,n\\}\\atop\\scriptstyle|I|=k} 1_{A_I}\\qquad(*)", "b8fd80228025d9c272ea4c642abad45e": "\\vec x' = \\begin{pmatrix} a^2+b^2-c^2-d^2 & 2(bc+ad) & 2(bd - ac) \\\\\n 2(bc-ad) & a^2+c^2-b^2-d^2 & 2(cd + ab) \\\\\n 2(bd+ac) & 2(cd-ab) & a^2+d^2-b^2-c^2 \\end{pmatrix}\\vec x.", "b8fd8d94458afec95f678e96cee5ba8d": "C = c + v", "b8fdd509044f396c8d4d4dd9190538c0": "\\begin{cases}\\dot{\\mathbf{x}}_f & = \\mathbf{A}_f\\mathbf{x}_f + \\mathbf{B}_f\\mathbf{u}\\\\\n\\mathbf{y}_f & = \\mathbf{C}_f\\mathbf{x}_f\\end{cases}", "b8fdd66b94d710c9b668e3b88b0fed95": " \\bar D ", "b8ff88560f39d94a49696d7dad859757": "(A_n \\; | \\; n<\\omega)", "b8ff8a62e8c6034e456103df97a9619e": "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx", "b8ffe5c6d3f8e5161c9972930275237e": "F=f-f^{(2)}+f^{(4)}\\mp\\cdots,", "b8fff79b68dfe278089119b566c90e8f": "\\frac{b_i}{b}\\geq 0", "b90009dbe61d4426321b152b808c5439": "\n\\begin{array}{lccccccccccccc}\n\\text{Minimize} & f^T x & + & g^T y & + & p_1h_1^Tz_1 & + & p_2h_2^Tz_2 & + & \\cdots & + & p_Kh_K^Tz_K & & \\\\ \n\\text{subject to} & Tx & + & Uy & & & & & & & & & = & r \\\\ \n & & & V_1 y & + & W_1z_1 & & & & & & & = & s_1 \\\\ \n & & & V_2 y & & & + & W_2z_2 & & & & & = & s_2 \\\\ \n & & & \\vdots & & & & & & \\ddots & & & & \\vdots \\\\ \n & & & V_Ky & & & & & & & + & W_Kz_K & = & s_K \\\\ \n & x & , & y & , & z_1 & , & z_2 & , & \\ldots & , & z_K & \\geq & 0 \\\\ \n\\end{array}\n", "b9003cb490ff737d481f7f9dd047545c": "\\mathrm{NA} = n \\sin \\alpha_0\\;", "b90045124f688480477618351250c39d": "\\chi_c(G)", "b9005d988380e3a74f51d5070dff5f63": "t_f = \\frac {2 A_m(\\sqrt {h_t}-\\sqrt {h_t-h_m})} {A_g \\sqrt {2g}} ", "b900c2cf9c743226167919b726928be2": "\n D^j_{m k}(\\alpha,\\beta,\\gamma) D^{j'}_{m' k'}(\\alpha,\\beta,\\gamma) =\n \\sum_{J=|j-j'|}^{j+j'} \\sum_{M=-J}^J \\sum_{K=-J}^J \\langle j m j' m' | J M \\rangle\n \\langle j k j' k' | J K \\rangle\n D^J_{M K}(\\alpha,\\beta,\\gamma)\n", "b900e99be47fc9a6a02e34ee11eee442": "y=\\pm\\sqrt{L^2-x^2}", "b9010a13bf9fbe83e93f7926ab119e61": "\\displaystyle \\hat{f}(\\nu)=", "b901439144359f9dd77057c018153180": "\\mathbf A \\mathbf u = \\mathbf f,", "b901bbd4673a98ef5ea882db5bfc0346": "0\\leq |\\operatorname{tr}|/2 < 1,", "b901e8d37852c2c4f5c9d10d29d12dba": "\\ T_q{\\mathcal O}_{q_0}=\\mathrm{Lie}_q\\,\\mathcal{F}", "b901f0f07d9c5d049ad380ad4489ec84": "\np = \\frac{NkT}{V-Nb'} - \\frac{N^2 a'}{V^2} \\Rightarrow\n\\left(p + \\frac{N^2 a'}{V^2} \\right)(V-Nb') = NkT \\Rightarrow \\left(p + \\frac{n^2 a}{V^2} \\right)(V-nb) = nRT.\n", "b90280cedb3f5a3a1038b4d143124398": "I(p)", "b9028363f5ed74877f999968c8bcded9": "\\frac{4 \\times 10^{-18}}{(25812.807) (483597.9)^2} \\ ", "b9029b128d06779d39cca7e5aedba928": "W\\leq -\\Delta A\\,", "b903bb1911963c4fae36794da6f92b06": "e = \\sum_{n = 0}^{\\infty} \\frac{1}{n!}\\cdot", "b903beeadf66057b1304053a9b33d2fb": "(y_1,\\dots,y_m)", "b903e52192fa0c2d68c4b76162463227": "\nf(x; a, d, p) = \\frac{(p/a^d) x^{d-1} e^{-(x/a)^p}}{\\Gamma(d/p)},\n", "b9040b8f90d994c213ff123d47bf457b": "E(x,S,C,\\lambda)=E_{\\rm color} + E_{\\rm coherence}", "b9046a2cae8c5d8cd88bbc551b692060": "u_3 = u_2a_2-1=0.175\\cdot6-1=0.05, a_3=\\left\\lceil\\frac{1}{0.05}\\right\\rceil=20 \\, ", "b904adfaf2dd6df97d614ecf79bbe395": "{\\mathbf e}_r", "b904e6c3213161406f89e33ba5c5af8b": "\\lambda^{(k+1)} = \\lambda^{(k)} - \\omega(\\theta_1\\psi_1^{(k)}|_\\Gamma - \\theta_2\\psi_2^{(k)}|_\\Gamma)", "b90509a57312494627db1598b498aae2": "t > 5", "b9052201c1faae1c07ad99848af0435d": "\\xi_\\mathrm{cutoff}=\\frac{1}{\\lambda N}", "b90538fd6e9acde717c05aa935632276": "(v_x,v_y,v_z)", "b905712bfa8b4c42fbd10ae93954df2c": "\\nabla_a e_b^I = 0", "b905c4eeff6a7969c330b11edd1aa416": "\\left \\lfloor \\frac{32}{5} \\right \\rfloor + \\left \\lfloor \\frac{32}{5^2} \\right \\rfloor = 6 + 1 = 7\\,", "b90681d267e372ce8ea3c28bdb03701a": "\\,l_{x+1}", "b90691978d08cb603b904476132ac936": "T_r = 0.7", "b9069bf2b3c8855933836d4be255ae33": "|z-a| < r\\,", "b906d84dec62cdb87a11efe33bb6e1a0": "\\pi^{ij}e^i-e^j, 1 \\leq i,j \\leq d", "b9071299813b2e3226a5d6d4d493b208": " \\int_{A} d\\omega = \\int_{\\part A} \\omega , \\,\\!", "b90794a9b2468bc8d4712997d27b8c0e": "F(A,B)", "b9081a527812dd6cd61914ec1220215b": " \\frac{(\\mu_2-\\mu_1)-(\\bar X_2 - \\bar X_1)}{\\displaystyle\\sqrt{\\frac{S^2_1}{n_1} + \\frac{S^2_2}{n_2} }}. ", "b9081d208114a6c507fcd01e1f29200b": " k(x,y) \\geq 0\\qquad \\forall x,y, k ", "b9085f8317feb54c75c57b10fd57b5a1": "T_p = \\frac{4\\pi^2 I_s}{\\ mgrT_s}", "b908a9de5097803d84bc1b940cd41b52": " h = t_{n+1} - t_n ", "b90917ba2f043766873da72d523abe80": "\\widehat{R} = \\widehat{R}(\\theta,\\hat{\\mathbf{n}})", "b90918f8950b61072610a4727bd2e528": "P(x)=0", "b90934e94dcb4c0efe0d4926675d8cc7": "(x+y\\omega)^2 = \\left(x^2 + y^2 \\omega^2 \\right) + \\left(\\left(x+y\\right)^2-x^2-y^2\\right)\\omega,", "b909359e5fb15b2437e2bddbe9c93a25": " \\min_{{\\hat x},r} \\left\\{ r:\\left\\| {\\hat x} - x \\right\\|^2 \\leq r, \\forall x \\in Q \\right\\} ", "b9094b64eb48d8a1d02131436031e117": " V_i = \\sum_k D_{i k}p_k + \\sum_j C_{i j}V_j - k_iI_i", "b9097d715b19b3008260cfafcd2829f1": "\\begin{align}\n a &= \\frac{1}{\\frac{4}{C_n\\sigma}\\sin^2\\phi + 1}\\\\\n a'&= \\frac{1}{\\frac{4}{C_t\\sigma}\\sin\\phi \\cos\\phi - 1}\n\\end{align}", "b909bac8f7fe07a9f672c3c76e7cb3fe": "\n\\hat{\\beta} = \\beta + \\epsilon_{\\beta} , \\quad\\quad \\hat{\\theta} = \\theta + \\epsilon_\\theta,\n", "b909bfee018dfafb2874d43dc6c4e156": "\nI(q,\\alpha,u):= \\begin{cases}\n\\quad \\alpha &, \\ \\ R(q,u) \\in C\\\\\n-\\infty &, \\ \\ R(q,u) \\notin C\n\\end{cases} \\ , \\ q\\in \\mathcal{Q}, u\\in \\mathcal{U}(\\alpha,\\tilde{u}) \n", "b90a2cf28b25d7e71147ee2ed146ea16": "2 / 6", "b90a4d894f15f2e0469861dc264e7cc9": "f(x)= \n\\begin{cases}\n d(x, \\Omega^c) & \\mbox{ if } x\\in\\Omega \\\\\n -d(x, \\Omega)& \\mbox{ if } x\\in\\Omega^c\n\\end{cases}\n", "b90a555c7bc03349d92ea7f8bbc7a2e1": "\\Psi_Q = \\log M_Q", "b90a669e1d34005f8f0dce450eee21d1": " |\\, S\\, \\rangle ", "b90aaf7cb592305c0789977ab6b4fc12": "\\left(A \\land \\left( A \\rightarrow B\\right)\\right) \\rightarrow B", "b90ab47bbc86995eda8af64ca79d461b": "e(P,cP)", "b90aea2e0e12931faf3d2df36aaee37a": "Q_{2 \\times 2} = \\begin{bmatrix}\\cos \\theta & \\sin \\theta \\\\ -\\sin \\theta & \\cos \\theta\\end{bmatrix} , ", "b90b15f6c7ee24a546aca82fbd4e30e5": "c_0\\;=\\;{d_0^{\\gamma}}10^{\\frac{-L_0}{10}}", "b90b193ae15bc8cb0ec4873947b14205": " K_{0(OC)} = K_{0(NC)} * OCR^{(\\sin \\phi ')} \\ ", "b90ba633b1042bb72be385e0e4ab8efd": "P(e|X)\\le\\sum_{\\widehat{X}\\neq X} P(X \\to \\widehat{X})", "b90be3704ef844286db9634653fb3890": "\n \\boldsymbol{S} = J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T} = \\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\tau}\\cdot\\boldsymbol{F}^{-T}\n", "b90c3cd5392558407dde73abebe0b6db": "n=\\text{poly}(\\log k)*k", "b90c7cc8c953964d0a000b2642a0b4b9": "F(\\delta)", "b90cbeab6ccaaa9f4274fd5fb1d75614": "\n \\begin{align}\n \\sigma_{xx}^{\\mathrm{f}} & \\approx \\pm \\cfrac{M_x}{2fh} ~;~~&\n \\sigma_{xx}^{\\mathrm{c}} & \\approx 0 \\\\\n \\tau_{xz}^{\\mathrm{f}} & \\approx 0 ~;~~ &\n \\tau_{xz}^{\\mathrm{c}} & \\approx \\cfrac{Q_x}{2h}\n \\end{align}\n", "b90cc52f2a6835cae6a24b87da728a25": "\\dot{d}(t) = \\frac{d(t) \\dot{a}(t)}{a(t)}", "b90d0989f07f1238730d6c96e20fe821": "a=0.3+i0.3", "b90d109dc3d8120ea1a2fa80c332bfee": "\\forall\\alpha.(\\mathtt{Set}\\ \\alpha)\\rightarrow \\mathtt{int}", "b90d2a100e6c5b13b8bcb074a1c8227d": "\\textstyle f, \\quad f(T) = \\sum_{i=1}^N f(\\lambda_i ) E_i .", "b90d910974c7e30d4658ae306bd863a1": "\\displaystyle{R(y,Q(x)y)=R(y,x)^2 - 2Q(y)Q(x).}", "b90d9832ea4fb9678417a79b7b53c006": " \\Delta H_{DF} \\,", "b90dde30d773c05a12a2bd1d22f6b41b": " \\hat{\\mathbf{B}} = (\\mathbf{X}^{\\rm T}\\mathbf{X})^{-1}\\mathbf{X}^{\\rm T}\\mathbf{Y}", "b90ec7be51640136151c729ada9b058c": " \\omega _0 ", "b90efdf0fe93ca5b5a085bffc9a867d8": "v_{\\mathrm{g}}", "b90f05a8544cb73776ff59f830c4d7d2": "\\ \\lambda=\\frac{\\ell}{L}", "b90f1456a938509ed6f775282cfd34d2": "p^{rs}", "b90f5c5ce7877b8e869f658439585671": "(G,\\,M)", "b90ff92766cbdf83844f36c31df2e886": "\\begin{cases}\nx' - ct' = \\lambda \\left( x - ct \\right) \\\\\nx' + ct' = \\mu \\left( x + ct \\right) \\,\n\\end{cases}", "b910681d600f86dbd1636fcfe236b391": "\\Sigma^{1,Y}_n", "b910c111ac8440bf4f4863bb5fc83aa8": "A \\cup B", "b910c1db6be4ee8fcfc1d447382008e7": "\\begin{matrix}\\frac1{128}\\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\\,", "b911b5612a6bca38ab90de129660b1f9": "q \\to \\infty", "b911d64428b2e6ddff4e24fa94350dc3": "{L_{cm}}/ N_i", "b91200a83c8bd2d78783bb5a8e21fd44": "y(k) = C x(k) + D u(k)", "b912134d3c76d86f4e0cfef56c4874cc": "\\frac{V_1}{n_1}=\\frac{V_2}{n_2} \\,", "b9125d9c3d750408f348b56a66549547": " {\\partial R(r)\\over \\partial r} = \\left[\\frac{(n - 1)}{r} - \\zeta\\right] R(r) ", "b912cbe641859f749e578750f452ceb0": "6\\times6", "b912fca18e1915fc242bd28839cb345c": "\\ \\mathcal{L}_\\mathrm{loc} = \\frac{1}{2} (D_\\mu \\Phi)^T D^\\mu \\Phi -\\frac{1}{2}m^2 \\Phi^T \\Phi", "b9135a4821a5a5aba42691e6def31d96": "\\frac{1}{T_{1/2}} = \\frac{1}{t_1} + \\frac{1}{t_2}", "b9135e27f55be74199b4b6c9acc7a650": "f_i(S_1,...,S_n) = \\{ x \\in X :", "b913af3def72a4f34a2a20132bed1798": "|a|<1;\\Re(s)<0 ;z\\notin (-\\infty,0) ", "b913b8e2f20eee6d17b9aaab24f8b8a5": "\\tfrac{\\hbar^2 j(j+1)}{2I}", "b914291761a7fddb21c4ef1d2cdb9ac8": "(a * b)(h) := \\sum_{fg=h} a(f)b(g).", "b91434287ab62d915ef5acb4a6082a24": " 0.5 = \\operatorname{E}[X_1] ", "b914af7dbd35f18612c7b7a2e64d4172": "c_i = a_i\\cdot b_i \\mod m_i", "b914dbb44bebb6311735971de9dc8f00": "\\tau_\\sigma", "b9150322187bb3f5ed09c90af95e44c6": "\\rho = \\sqrt{x^{2}+y^{2}}", "b91513276f11131f13bab655e6365f05": "M_{t}(\\omega)", "b915379db65cf229e37c46551608f0b7": "G_{S_L}", "b915837d28f2dd7e59c0a7e1d82f7c69": "X = (X_1,X_2,\\ldots,X_n) ", "b915a44009946c4d28baf54e13bed534": " h(X) = -\\int_X f(x) \\log f(x) \\,dx ", "b9162a968d41efc5513bd438448af51d": "\\left.\\begin{matrix} \\gamma & = & \\cos(\\theta/2) - \\{a_1 \\sigma_2\\sigma_3 + a_2 \\sigma_3\\sigma_1 + a_3 \\sigma_1\\sigma_2\\} \\sin(\\theta/2) \\\\\n& = & \\cos(\\theta/2) - i \\{a_1 \\sigma_1 + a_2 \\sigma_2 + a_3 \\sigma_3\\} \\sin(\\theta/2) \\\\\n& = & \\cos(\\theta/2) - i v \\sin(\\theta/2) \\end{matrix}\\right\\}", "b91665717a6c1ed792690c7d0a5e125c": "\n\\langle \\pi_1 (a^*)k, l \\rangle = \\langle \\pi (a^*)k, l \\rangle \n== \\langle \\pi(a) ^* k, l \\rangle == \\langle k, \\pi (a) l \\rangle \n== \\langle k, \\pi_1 (a) l \\rangle ==\\langle \\pi_1 (a) ^* k, l \\rangle .\n", "b916c79c91e7a3e6cb38927abce54041": "{\\vec m} = {\\vec c} \\oplus {\\vec b}", "b91700bb7194efc3c035f1f467cbdb8e": " K_n = K_n(A,b) = \\operatorname{span} \\, \\{ b, Ab, A^2b, \\ldots, A^{n-1}b \\}. \\, ", "b91732a5bb8ca52279670398840e4019": "\\; a_{-m} = -(\\lambda - T)^{m-1} e_{\\lambda} (T)", "b9173fe17a82cd9655ccb8a4814f8787": "s_{uv} = \\frac{C^*}{L^*} = 13 \\sqrt{(u^\\prime - u^\\prime_n)^2 + (v^\\prime - v^\\prime_n)^2}", "b9177e3fa0776d122feace67da9f9cd7": " x = \\mu_g {\\sigma_g}^z. ", "b917975a7d05c4497d5453d9ad11270d": "\\Phi_l(X,j(E))", "b91858a81333df456565c3ad32c31c7f": "ef^k v= (-k^2+(\\lambda+1)k)f^{k-1} v", "b9186c58860ce3528aaf67aa2656948c": "\n\\forall j \\in \\mathrm{N} \\; \\quad \\nu\\ _j (a) \\le\\ \\nu\\ _j (b) \n", "b9187a0720713323643a8316856d475d": "E_r^{p,q} \\Rightarrow_p H^{p+q}(C^\\bull)", "b918f94e33beef9bd8225cdb96325060": "\\displaystyle I_i = V_i / Z_i ", "b91902652a46b22e848b8a69e4265f92": "a_{11}", "b91956c9e61796a0499173b75b1e9603": "\n\\begin{array}{c|cc}\n0 & 1/2 & -1/2\\\\\n1 & 1/2 & 1/2 \\\\\n\\hline\n & 1/2 & 1/2 \\\\\n\\end{array}\n", "b9195b17a8c47054df536a3b64077669": "a_{11}=0.85", "b9195ba4f565cd37732b7a04d49f209d": "\\frac{1-(-1)^m}{2} \\le c \\le \\frac{(m-1)(m-2)}{2}+1.\\ ", "b9199c3ba33ff14b241563024c80f724": "\\mbox{f(t)} =sin(\\omega t+\\theta-\\beta) ", "b919a76e7234c0a68a05a6ab162f2aac": "(\\phi \\vee \\psi) \\to \\neg (\\neg \\phi \\wedge \\neg \\psi)", "b919b061a40cb07328cacaf0acd4c0fc": "h_{\\tau} = \\sqrt{\\sigma^2+\\tau^2}", "b919be7840494b711e6232f3d1ab47a6": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{T}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "b919ddfd02be81bcb9894a4ac5bdb2f2": "\\mathbb{C}_p", "b919dfc2ec008fdef4d67726625fc657": "f^2 = \\frac{(ac+bd)(ad+bc)}{(ab+cd)}. \\,", "b919ed0fcdb0ef78fb7cdd4d5f82b163": "[f,g]_\\varepsilon", "b91a0d7f6f1e59874dfbabe0eb8e9cca": "R_2 =1.6303", "b91a0ff479e43b3fa0ad600eb66a8ae5": "B' = U B = \\begin{pmatrix}\n 14 & 9 \\\\ 21 & 15 \\\\\n \\end{pmatrix}", "b91a1729631876d6354d9e0d922e90f8": "U = \\rho_\\mu (x-x')_\\mu. \\, ", "b91a1ae5a2907163b090d20c859e4b1f": "\\textstyle I_{1-p}(n - k, 1 + k)", "b91a3a44fbb0d969e9759d9eb8b73ecf": "2\\nu", "b91a98f2d075e6d4a51345cd1d7b9d35": "\n\\begin{bmatrix}\n -415 & -33 & -58 & 35 & 58 & -51 & -15 & -12 \\\\\n 5 & -34 & 49 & 18 & 27 & 1 & -5 & 3 \\\\\n -46 & 14 & 80 & -35 & -50 & 19 & 7 & -18 \\\\\n -53 & 21 & 34 & -20 & 2 & 34 & 36 & 12 \\\\\n 9 & -2 & 9 & -5 & -32 & -15 & 45 & 37 \\\\\n -8 & 15 & -16 & 7 & -8 & 11 & 4 & 7 \\\\\n 19 & -28 & -2 & -26 & -2 & 7 & -44 & -21 \\\\\n 18 & 25 & -12 & -44 & 35 & 48 & -37 & -3\n\\end{bmatrix}\n", "b91b8170baad9da75fd07e174dc76147": "R=A-\\arctan\\left({p \\over h}\\right)", "b91b94a12237afc0aa3dbbebc15a5300": "\n K_{\\rm I}= \\cfrac{6P}{BW}\\,a^{1/2}\\,Y\n", "b91bf10d1291ad15226d30bd775bde08": "K \\big( \\tfrac{\\sqrt{2}}{2} \\big) = \\tfrac{1}{4 \\sqrt{\\pi}} \\;\\Gamma \\big(\\tfrac{1}{4} \\big)^2", "b91bfe4e5d7244204f84bf3bf47d104b": "\nm_2 = \\frac{F r^2}{G m_1}\n=\\frac{0.032[N] \\times (200[m])^2}{6.674 \\times 10^{-11} [N m^2 kg^{-2}] \\times 10^9 [kg]}\n\\approx 19200 kg\n", "b91c612c442206fb82c143a82c18acb5": " \\operatorname{E}(X\\cdot Y) ", "b91c7be9bcaa9282d1225bd8b5ca3481": "\\{0, 1\\}^k", "b91cb7e014659c1044b2077dc258e346": "d \\bmod\\ \\varphi (N) ", "b91cefdc52e83ebd4febb714bed8c73e": "\\scriptstyle V_S=0", "b91ecd42da51f059a48fe5ccc85ccf0c": "Z_\\alpha(n) = \\log_\\alpha(1 + \\alpha^n),", "b91ef6e35ecc9d446de03cd4b708445b": "\\textstyle -\\frac{\\pi^4}{90}", "b91f441250c9c08c3409647b52c712d3": "(n-m) \\times m)", "b91f79d15e895f4dbd6d763e08f00852": "\\sigma_\\mathrm{true} = (1 + \\varepsilon_\\mathrm e)(\\sigma_\\mathrm e)\\,\\!", "b91f8b384aa497eb6f3db123eda07d84": "\nx(t) = \\sum_{n} A_{n}(t) e^{j \\theta_{n}(t)}\n", "b91fa79f5df502a40aa15de0c67a4ce4": "Z_T", "b9201cddb8d356d15ab508a9d02d11de": "{ap \\choose bp} \\equiv {a \\choose b} + {a \\choose 2}\\left({2p \\choose p} - 2\\right){a -2 \\choose b-1} \\pmod{p^3}.", "b9202baac42d0c8bb147f0992e180440": " Q_1 \\rightarrow Q_2\\,", "b92047d25384be01a6692c83ef59150b": "e_n = \\frac{\\pi^{\\frac{n}{2}}}{\\Gamma(1+\\frac{n}{2})} ", "b9204ee88c045724c5d289a0c4088973": "F(t) = \\exp\\{ -8 [1-(1-h_e)S(t)] \\}", "b92060adfa818e5fdaf6ce649113d25c": "\\infty = [1 : 0].", "b920969a68b9bc978a87ceefd55a56f3": " x_1,y_1,x_2,y_2,\\ldots\\,. ", "b920f3ac7d175a98559c6d40dcc79ded": "m=-j,-j+1,\\dots, j ", "b9212d2e30251f85d89bf73f517ec668": "\\hat{\\mu}_3", "b92196b297154c741332c38dc6dd5334": "\\sigma\\, a\\, \\frac{z\\, +\\, h}{h}\\, \\sin\\, \\theta\\,", "b921a58a2f0cb81ef7b97a1bf5fea631": "Z[J]=\\int \\mathcal{D}\\phi e^{-\\int d^4x \\left({1\\over 2}(\\nabla\\phi)^2+{m^2 \\over 2}\\phi^2+{\\lambda\\over 4!}\\phi^4+J\\phi\\right)}.", "b921abd1fb297f889b1330991d096fa5": " {1\\over 2} \\left [ D\\left ( x-y \\right ) + D\\left ( y-x \\right )\\right ] \n", "b921e17d9650779d62b4649934b2369a": "E(\\bar{\\mathbf{Q}})", "b92205e8807d4c55bc949dd19cc940c2": "U = U_k + U_e", "b9220c0b4bd067820f6e892855ef712e": "\\mu_B={e\\hbar}/2m", "b9223ffd274864028d5bd115a690ab13": "\\frac{4 \\pi}{3\\sqrt{3}},", "b9226bb4ed4d95b5c918239a869e85e6": "a \\sim t^{p'_l}, b \\sim t^{p'_m}, c \\sim t^{p'_n}", "b922baf609bf9de9b9820e71512b53e3": "F_i \\propto f(\\mathbf{x}_i)", "b922d097c2c988fa6e309120d77b108f": " \\left(- \\frac{\\hbar^2}{2m x_c^2} \\frac{d^2}{d \\chi^2} + \\frac{m \\omega^2 x_c^2}{2} \\chi^2 \\right) \\psi(\\chi) = E \\psi(\\chi) \\Rightarrow \\left( -\\frac{d^2}{d \\chi^2} + \\frac{m^2 \\omega^2 x_c^4}{\\hbar^2} \\chi^2 \\right ) \\psi(\\chi) = \\frac{2 E m x_c^2}{\\hbar^2} \\psi(\\chi).", "b92304f5d825a884b1fb29a015df3ad6": "x_i = \\alpha^{a_i} \\beta^{b_i}", "b92344113fdfd3cfe5f51f207b45d967": "T^T T = I_r \\quad D^T D = I_r ", "b92345c1866ec9047df6ee6e9f3abb7a": " A = |N,0\\rangle\\langle 0,N| + |0,N\\rangle\\langle N,0|. \\, ", "b92358f8adab006dbcda7a3da7e29ce0": "\\overline{\\mu}_j = \\mu_j + z_j F \\phi", "b923be5f5a01dcecbf71a9cb720b3d76": " \\Delta E \\Delta t ", "b923c2cf28c1279128e6606a77532c8b": "2. \\; \\; {\\mathrm{Cl} \\cdot} + \\mathrm{O}_3 \\; \\xrightarrow{} \\; {\\mathrm{ClO} \\cdot} + \\mathrm{O}_2 \\; ", "b92400fd745f044490fa3a7bf8832ed2": "|\\Psi_{mn}^{pq}\\rangle = \\mathcal{A}(\\phi_{1}(\\mathbf{r}_{1}\\sigma_{1})\\phi_{2}(\\mathbf{r}_{2}\\sigma_{2})\\cdots\\phi_{p}(\\mathbf{r}_{m}\\sigma_{m})\\phi_{q}(\\mathbf{r}_{n}\\sigma_{n})\\cdots\\phi_{N}(\\mathbf{r}_{N}\\sigma_{N})).", "b9245f80d2d981672547d4c6bec01bbe": " \\dot{V}_E ", "b924817bcf921d224d4d54afdfc8e999": "DR_{T}^{S}", "b92481eddde8c0a762bc2eab35a80a37": "\\subseteq", "b9250d8504392fc9989d69e51b30b5b2": "(x_1,y_1) , (x_2,y_2)", "b9251576b0572468c770245b64d32626": "\\ln \\zeta(s) = s \\int_0^\\infty \\Pi_0(x) x^{-s-1}\\,dx", "b9252478fb34a5cba141ab845fa3894f": "HS_{A/f}(t)=(1-t^d)\\,HS_A(t)\\,.", "b9253df50d7ba1062b240cb34576d067": "G+G'", "b925570f8f29956b817a048704780419": "p(x) = a(y)p(x|y) +\na(\\overline{y})p(x|\\overline{y})", "b9256cbf05352d345ec34289d57ca42a": " M _\\infty (A) = \\varinjlim \\cdots \\rightarrow M_n(A) \\rightarrow M_{n+1}(A) \\rightarrow \\cdots .", "b9258cef5fb9377aa23fecd3e6d4b49f": "u(s) = u(x_1(s),\\dots,x_n(s)).", "b925b32ee7085c3527cf2f2e8029efaf": "M^*", "b925cba2058fa7d1057a9c35016e1119": "\\lim_{n\\to\\infty} \\frac 1n\\sum_{k=0}^{n-1}(f\\circ T^k)(x)=\\int_I f d\\mu\\quad\\text{for }\\mu\\text{-almost all }x\\in I.", "b925d871a60443249ff7a55c936ae42a": "L(f) = f(0.3)\\text{.}\\,", "b925ed5267a000c017231d1097cfecc8": "\\gamma_{\\epsilon}", "b926137669ab2509e0b53251e4ce2eff": "\\boldsymbol{U}_f:", "b9263e4f2641095df60c159852cba310": "Q = \\iiint_\\Omega \\rho \\, \\mathrm{d}V\\,,", "b9265d12a5ea211457b8618389c2ec94": "\\psi_n(x)\\equiv \\left\\langle x | n \\right\\rangle = {1 \\over \\sqrt{2^n n!}} \\pi^{-1/4} \\hbox{exp} (-x^2 / 2) H_n(x),", "b9265d9eae4e2b9514bce5e6b4a04046": "\n \\boldsymbol{L}^p := \\dot{\\boldsymbol{F}}^p\\cdot\n (\\boldsymbol{F}^p)^{-1}\n ", "b926657b48de569547ba819ae969b604": "c_{3,0}(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta, \\zeta \\widehat{z} \\eta) = \\alpha \\widehat{x} \\beta \\gamma y \\delta \\zeta z \\eta", "b9266b82ce201847aeff3c64afede0ad": "\\sinh(z) = z\\,_0F_1({\\tfrac{3}{2}};{\\tfrac{z^2}{4}}),", "b92696456c623463672bfc24331c57b8": "t={18\\eta \\over Egwr_0}", "b926a67b2cf907dee9f2fdc2b99d26a7": "\\overline Y", "b926b59ce3616d3ff06a43ced8a978aa": "(\\cos(\\phi)r,\\sin(\\phi)r) ", "b926fd1401bf550c7d483747876b0c15": "W_{F} = F \\cdot |\\phi^{-}\\rangle\\langle\\phi^{-}| + \\frac{1-F}{3}|\\phi^{+}\\rangle\\langle\\phi^{+}| + \\frac{1-F}{3}|\\psi^{+}\\rangle\\langle\\psi^{+}| \\frac{1-F}{3}|\\phi^{-}\\rangle\\langle\\phi^{-}|", "b92715e7fdcb7fb77da93af22bfb40d6": "\\mathrm{SO}(p) \\times \\mathrm{SO}(q)\\,", "b9271f5a26c974861efbb8b29606d3d6": "\n\\operatorname{Li}_{-n}(z) = (-1)^{n+1} \\sum_{k=0}^n k! \\,S(n\\!+\\!1, \\,k\\!+\\!1) \\left({{-1} \\over {1-z}} \\right)^{k+1} \\qquad (n=1,2,3,\\ldots) \\,,\n", "b927458b056f9e625af0357915368028": "\\lim_{n\\to\\infty} R_n", "b92778d8fc08064c3b54874e5565a860": "\\mathrm{Nu}_L = \\frac{\\mbox{Convective heat transfer }}{\\mbox{Conductive heat transfer }} = \\frac{hL}{k}", "b927878584460c3716e8b25afd83a752": "T(\\phi,\\psi,\\chi)=\\frac13\\langle [\\phi,\\psi],\\chi\\rangle +\\text{cycl.}", "b927ba818b5a11cd03c35e93650ad0f4": "1/\\overline{z_0}", "b927df09d52481777c4ac38c40fb934b": "\\hat{q}", "b927ec028bb5319a42a64a1ee72a64ce": "\\mathcal S_1", "b92856f40c0f6b126473f730b0c51f45": "\\hat{x}_j=s_i ", "b928d90eb312541e8a465adb53adf9e6": " \\hat{M} ", "b929006398a7f5878c9b71d0ee948cfc": "l(b - 1)^{l + 1} \\ge n \\ge b^{l - 1}", "b92919a8507b1381fd9fa7272279f50e": "VCA(64x^3-448x+448,(0,1)) ", "b92a16fa4ed9254a4bd558b2d6af9389": "C_{exp} \\approx 4\\times 10^{-3}\\;\\mathrm{F/m^2} \\ ", "b92ad8f9e33a38bdcc9a232cd38830dc": "\\frac{\\partial \\ln \\Beta(\\alpha,\\beta)}{\\partial \\alpha} = -\\frac{\\partial \\ln \\Gamma(\\alpha+\\beta)}{\\partial \\alpha}+ \\frac{\\partial \\ln \\Gamma(\\alpha)}{\\partial \\alpha}+ \\frac{\\partial \\ln \\Gamma(\\beta)}{\\partial \\alpha}=-\\psi(\\alpha + \\beta) + \\psi(\\alpha) + 0", "b92bcf6cc5bf38eeb396162b2544ee07": "1 \\leq n \\leq N", "b92c0e7449108fefe2406bc174fda978": "B(z) = \\sum_{n=0}^\\infty B_n z^n", "b92c240c8a338d7582a5bf40ecc87a1c": "0 \\leq N", "b92c284ef3cc7e7f7b96264cff1269a3": "D: \\Omega_+(M) \\rightarrow \\Omega_-(M) ", "b92c751f642e771716a4ca666faca163": "y_0 \\approx 1/\\sqrt{S}", "b92c7c27a7f6cc48a42631d4437c66e8": "J = (7^{e - 1/e} - 9) \\cdot \\pi^2 = 867.5309...", "b92c963a7269efd559c84541bf79086f": " \\Sigma=\\frac{1}{\\sum_{i=1}^{N}w_i - 1}\\sum_{i=1}^N w_i \\left(x_i - \\mu^*\\right)^T\\left(x_i - \\mu^*\\right). ", "b92cd01af2eeeb166ec6dc7aedf303dc": " S - S_0 = k_B \\ln \\, \\Omega \\ ", "b92d2164f6da715600b4fc72d08cbea7": "a,b \\in A", "b92d58e39ba8227e0fc0cd147d195be7": " {\\nabla \\cdot \\mathbf u = 0} ", "b92df035eb31b1d5a2d2f7929f29556a": "\\left[ x'(s)^2 + y'(s)^2 \\right] = 1 \\ . ", "b92e091636fd440c41c22fdab3f73f2b": " C_h f = a_0 + h C_hU^*f.", "b92e522d9306bb8e2a3b4b8f4cc5eb36": " l_2\\equiv\\partial_{xy}+\\frac{1}{x+y}\\partial_y,", "b92e88ea5ab81d9e0f2a7a4f979b0107": "\\mathrm{lcm}(m,n)\\cdot \\mathrm{gcd}(m,n) = m \\cdot n.", "b92f230079a017886723c3816cbea9a4": "\\hat{\\boldsymbol{\\beta}}", "b92f3d91f4b148d492944c3122256ae4": "y=Q(x)", "b92f47c52da1c94a583e890a4cb47eb2": "k_f = k_{f_{frs}}\\sqrt{2}", "b92f9a5cf425d8d1eae1c55595ae6426": "c = E_k(m)\\!", "b92f9ba7978aaba7dbc7fb6e2225b1ce": "ds^2 < 0", "b9303378aace20ed5fa78d6be9f21649": "\\Delta B^\\prime = \\plusmn 1", "b930558dd9f6e07de4388d55b2ae5a1f": "SL(2,\\mathbb{C})^{\\otimes 2}", "b930c0f5ac424cb0ced45bb1926709db": "\\mathrm{Cov}(F)=I", "b930c9663825350baac9038373f3bdfd": "H(\\mathbf{x})", "b93121c5f83511d99d1c9377565b1c8c": " S_F(z) = \\sum_{a \\in F} p_a(z)", "b9315b68f55b2e5d02eed4974bb83cc0": " \\frac {2}{nm}\\sum_{i,j=1}^{n,m} \\|a_i- b_j\\|_2 - \\frac {1}{n^2}\\sum_{i,j=1}^{n} \\|a_i-a_j\\|_2 - \\frac{1}{m^2}\\sum_{i,j=1}^{m} \\|b_i-b_j\\|_2 ", "b9316a1ef8ab85d4d6304328aebd0077": "(1 - \\epsilon)n", "b9317e3ab08276ed67a93cad7750526b": "\\overline{X} \\to e^{q\\overline{\\Lambda}}X", "b93185c292f590f7451d13372cb9c633": "Df(x)=A_x", "b93192c8feb6edb240d69db59bf129f5": "A_{\\alpha ;\\beta} = A_{\\alpha,\\beta} - \\Gamma^{\\gamma} {}_{\\alpha\\beta}A_\\gamma \\,.", "b931ac2b3c613fd9104a79ab5894be19": "\\begin{align}\n\\Omega & = -kT \\ln \\Big( \\sum_{N=0}^{\\infty} \\frac{1}{N!} e^{\\frac{N\\mu - N\\epsilon}{k T}}\\Big) \\\\\n & = -kT \\ln \\Big( e^{e^{\\frac{\\mu - \\epsilon}{k T}}}\\Big) \\\\\n & = - kT e^{\\frac{\\mu - \\epsilon}{k T}},\n\\end{align}", "b931bda0e6873f73a61315a156785442": "\\sum_i p_i\\,dq^i", "b931feaf69c1e53a7a93e0d0c705a283": "\\scriptstyle (A+j\\omega B)^2", "b9320aeac3a88f53405054e1cef91d32": "S_n^{\\;2} = \\frac{1}{n-1}\\sum_{i=1}^n\\left(X_i-\\overline{X}_n\\right)^2", "b932123c869cecce9017223fff8b57c0": " \\sigma_P: S^{k} (T^*X) \\otimes E \\to F ", "b932bb7019d1ce32f5e6aaa67db2bf1c": "D_{1,2}", "b932cab20a3e98285d9f37acafcfcef1": "\\text{Short-Circuit Calculation Formulas}", "b932cc0dd03b60e73ded0f5f8d6461ee": "A_o", "b932f229793eb59392b04a901c51954c": "\\Gamma_\\mathrm{res}", "b933bd59c33616b544cb1f66878475d5": "1/4\\pi", "b933c22ef70e25c19a0138498a84d38e": "{\\mathbf{S}}({\\mathbf{p}}(t)) ", "b933d68005b8ec42986af370c15d34ea": "J(f,x)=\\mbox{det}(Df(x))", "b9342755128d00bd6a34fccc672aac98": "{K_{p}}", "b934864d651db6fd2a5726920858e61c": "\nu = \\int \\frac{d\\xi}{\\sqrt{E \\cosh^{2} \\xi + \\left( \\frac{\\mu_{1} + \\mu_{2}}{a} \\right) \\cosh \\xi - \\gamma}} = \n\\int \\frac{d\\eta}{\\sqrt{-E \\cos^{2} \\eta + \\left( \\frac{\\mu_{1} - \\mu_{2}}{a} \\right) \\cos \\eta + \\gamma}}.\n", "b9349ec7c60e95034d755144d904b9a5": "{\\sigma}", "b934ec3206a8f4e13049c726f5a5e8a0": "g(t^-)", "b935012777bd266fc3e4a0ea10f56b23": "\\epsilon_{2}(p) = E_{2}(p)-E_0", "b9350da5c05d387b2e4a501adb2021f0": "\\frac{1}{2}(1+\\pi_{1}-\\pi_{0})>1/2", "b93516dbccf7a15e3766064fa6b41471": "A (x) = \\lfloor x \\rfloor \\,", "b935372daabab8b68114be6465f1306d": "Z = \\sqrt {\\mu \\over \\varepsilon }.", "b93549a27db51b8875f2b1c63ea5305f": "C \\frac{d^2 \\phi}{dt^2} + \\frac{1}{L} \\phi \\, =0", "b9354f14725abba21a627ced3cf71642": "G_N(z)=\\exp(\\lambda(z-1)),\\qquad z\\in\\mathbb{R},", "b93609597653c86039ddc61b0c3133bf": "V^m", "b93648ec0b4fea0a0a0a4e751706b4ac": "1-\\epsilon(1-\\beta)\\delta n = 1-\\epsilon e^{-\\lambda \\delta n}", "b936857484bfcbe1af4f82ac98076e01": "\\displaystyle \\varepsilon\\gamma_{\\mu\\nu}", "b936b76e27bfe0ba03994a616b01e268": " \\alpha := p_2^* r_2 / p_2^* a_2 ", "b936f8cfc0f708a91cb68a4ed88ac83b": "E_2 = E_1 + Z", "b937097fc08510e49539b5e6078c410a": "s_n=\\max_i x_i", "b93720faeb9c508960aad35c127c35d8": "\\sin 2\\theta=1", "b937440153f2b6514321f4fc4501b57e": " \\Leftrightarrow 8(x^2+y^2)-2x-36y+35 =0 ", "b93744585cb8785b650b688d1d659b4c": "1 \\,\\text{Ry} \\equiv h c R_\\infty = 13.605\\;692\\;53(30) \\,\\text{eV}.", "b9375d07918418c1152ff1a6125b0f58": "\\alpha\\in S", "b937660f99cae84bacb98a5fb200de90": "f_{PAD}= f_{0} \\frac{v}{c_{a}} cos \\theta = \\frac{v}{\\lambda} cos \\theta ", "b938355277958aaeb941d01b4297ae10": "\\varphi(x_1)=0, \\quad \\varphi(x_2)=0. \\,", "b938b073798fe33d80885a1816be84f5": "P(X_1^n(2)) > P(X_1^n(1))", "b93950628f1d7ff598967f325af724df": "y_0-y_P", "b9396b5e36a071fc70a778ade10dc370": "4N", "b9397c8bef6e61ca7269a5a45977189d": "\\ \\log x", "b93992717c6068a7043aa8ece4c84cfe": "\\ln \\left( {\\ln \\left( {\\nu + 0.7 + e^{ - \\nu } K_0 \\left( {\\nu + 1.244067} \\right)} \\right)} \\right) = A - B*\\ln \\left( T \\right)", "b939c90a3e294d1a3c8b6e675004896e": "\\left \\{ 1, 2, 3, 4 \\right \\}", "b939d4f27f8d32e5e61b00517e1d3254": "\nR_{\\text{ct}} = \\frac{1}{f\\,j_0}\n", "b939f0ab3bc50adfb138070886bdf916": "\n \\sigma_i = 2C_1J^{-5/3}\\left[ \\lambda_i^2 -\\cfrac{I_1}{3} \\right] + 2D_1(J-1)\n", "b939fb6d81c8639ee3675de3c1fcd2db": "\\mu = E_0 + E_F \\left[ 1- \\frac{\\pi ^2}{12} \\left(\\frac{kT}{E_F}\\right) ^2 - \\frac{\\pi^4}{80} \\left(\\frac{kT}{E_F}\\right)^4 + \\cdots \\right] ", "b93a018029326a9c13735a2eb9f7fed1": "S\\ni x^2=-1-y^2\\in-1-S", "b93a2aa5febedd10d4f137148490c6c7": "\\left\\lceil \\frac{n+1}{3} \\right\\rceil", "b93a5a30d4e3f3bbcb5f90944463b57d": "{{i}_{C}}=\\left( \\frac{\\beta +1}{\\beta +2} \\right){{i}_{C3}}", "b93a71639dde67d6b03cc183869b35d0": "u_2 \\leftarrow g_2^r rem P", "b93a850917cb1c7cac37a7560e2dfcb4": "t = R\\ {\\sqrt { \\frac {5-\\sqrt{5}}{2}} } = 2R\\sin 36^\\circ = 2R\\sin\\frac{\\pi}{5} \\approx 1.17557050458 R.", "b93a9b6f1a6515c50fa7ef0735fba061": "\\theta = \\frac{1}{b} \\ln(r/a).", "b93aa8caf5a9290ec76b585a09229e84": "1/\\phi(q)", "b93afcf1b21aed9f4f18f192d42f2b07": "Y_{9}^{4}(\\theta,\\varphi)={3\\over 128}\\sqrt{95095\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(17\\cos^{5}\\theta-10\\cos^{3}\\theta+\\cos\\theta)", "b93b105d538065eb81ebdc5594af2dba": "\\qquad Y_W = 2(Q - T_3)", "b93b2d58591e77cd05faf5899efd7e25": "L = c \\Delta P = \\rho V v c =-\\rho V\\Gamma.\\,", "b93b6a66a839dc69ebaefd1e224f3de9": "\\bar{\\mathbf{x}}=\\boldsymbol{\\mathsf{L}}\\mathbf{x}", "b93b8a1a6c03f499a76f31aa44bdada6": "T=PA\\cdot PB,", "b93b8bf786ec65f88415fd0006eba52f": "w = (\\left|\\cos \\theta\\right|^r + \\left|\\sin\\theta\\right|^r)^{-1/r}.", "b93b98c75aa2620021dafbcd5bf3d191": "w_i \\ge 0", "b93ba875ab06bb9c758c4c62b13557bd": " y (\\partial_t + \\partial_z) + (t-z) \\partial_y \\,\\!", "b93bad48d5b4afe24ce92638f5f5565b": "a = A_y - B_y\\,", "b93bd5cacbfe7c277d4ac092f7f16b18": "LN[i]", "b93c5775963a676b1703d31d20f60965": "\n\\begin{align}\nr(x) & {} = 3084841486175176 \\\\\n & {} \\quad + 6740415721237444x \\\\\n & {} \\quad + 3422416581971852x^2 \\\\\n & {} \\quad + 13128433387466x^3 \\\\\n & {} \\quad + 12193131840x^4.\n\\end{align}\n", "b93c88c2f0b71ae0759707362c0b5a73": "Y_{10}^{-2}(\\theta,\\varphi)={3\\over 512}\\sqrt{385\\over 2\\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(4199\\cos^{8}\\theta-6188\\cos^{6}\\theta+2730\\cos^{4}\\theta-364\\cos^{2}\\theta+7)", "b93c9bb6a3b43550d5693faf072691f2": "R/\\mathfrak{p}", "b93ca6a652b6ae7f9644152055cad9a9": "\\hat {\\textbf{q}} = \\textbf{q} + \\varepsilon \\textbf{q}^0", "b93d1aa17b59b6fedbe519dff6ca0584": "f(\\pi)=0", "b93d1f29d4f44d7bdd1e080d28078896": "\\mathbb{Q} = K_0 \\subseteq K_1 \\subseteq \\dots \\subseteq K_n", "b93e05c3d19557cdf6d533e747b05d97": "R=\\frac{aM}{c_T}\\frac{C_L}{C_D}ln\\frac{W_1}{W_2}", "b93e55f041afea432bc4512574304035": " Y_p = X_i \\cdot S_p ", "b93e841276c84b79011d521461ee8879": "\\sigma \\equiv_{b} \\tau \\mbox{implies } a.\\sigma\n\\equiv_{b} a.\\tau", "b93ead3a68f59f81c398d3aa909bec28": "HS=SH", "b93f0362b9803f783c2978c33e531b76": "f\\bot g", "b93f4f8f77089f6a756b9d33148937f6": "C\\cap D=\\emptyset", "b93f52316ffff92a9c85cd679320025b": "C_{p^{\\lambda_i}},", "b93f95b48e4b10e1c52423507987cdaf": "\\overline{\\mathrm{Nu}}_D", "b93fa409d09c6a3975d3b05a28d32aa1": " (Uf)(\\lambda) = \\int_0^\\infty f(x) \\Phi(x,\\lambda) \\, dx,", "b93fb539b136fa2c26579b30941703c0": "y_c = \\biggl(\\frac{q^2}{g}\\biggr)^\\frac{1}{3} = \\biggl(\\frac{(4.00)^2}{9.81}\\biggr)^\\frac{1}{3} = 1.18m", "b940adca161bbb9e7c0bb648f451fd79": "\\sqrt {e}", "b941032b60a3cc9f2e063ccfaec60283": "\\varnothing\\times\\varnothing = \\varnothing", "b9416a75117ebac2d3888f17912c1cd5": "D(P)(x)(h) = \\lim_{t\\to 0} \\,\\frac{1}{t}\\Big(P(x+th)-P(x)\\Big)", "b942425e7b6f75c904ff0d5606392a5d": "d(\\mathrm{Ad})_x: T_x(G) \\to T_{Ad(x)}(\\mathrm{Aut}(\\mathfrak g))", "b942bb400ea83797bf47f945861230ca": "(P \\and (Q \\or R)) \\Leftrightarrow ((P \\and Q) \\or (P \\and R))", "b943155b01a851ab453f9386060cd0f7": " \\frac{dp}{dz} =- \\rho w^2 W \\quad (2.3)", "b9436a5559a5b3899f02c0c4ef884afa": "f(A) = B", "b943d150ecf217fc0c38bad3262c0243": "{\\bar{BH}}_4", "b943ed66810170df7939f608c9b6061d": "\\Lambda_m^0", "b94431f67d51b6156e048ef9c219beb4": "\n G(t) = \\int_a^t \\frac{f^{(k+1)}(s)}{k!} (x-s)^k \\, ds,\n", "b94448994236e00d8e41c9b6b51dd141": " 1\\leq a<100", "b9445a7d45f12f7b2b525b88e5bbf7ba": " y\\pm\\frac{2}{3}x^{3/2}=C,", "b94469d70fdeaff76d49ffee4329433a": "\\frac{\\mathrm{d}}{\\mathrm{d}x} \\left (\\int_{a(x)}^{b(x)}f(x,t)\\,\\mathrm{d}t \\right) = f(x,b(x))\\,b'(x) - f(x,a(x))\\,a'(x) + \\int_{a(x)}^{b(x)} f_x(x,t)\\; \\mathrm{d}t.", "b944733efc4f1a74a5c498075efdf4c3": "h_\\eta^{(1)}(z)", "b944a84220329d7125beac3e5c323957": "\\sigma_k^*(n) = \\sum_{d\\mid n \\atop \\gcd(d,n/d)=1} \\!\\! d^k.", "b944aa4b58233ece791ba6f97b8764d1": "\\{(x, y) \\ :\\ x > 0,\\ y > 0\\ \\} = Q\\ \\!", "b9459997f199fe3cf85174a325685c07": "\\dot{m} =\\int\\rho c_{x}dy", "b945b864ba27839a5c052aae347a0153": "\\lfloor \\quad \\rfloor", "b945d1191545f30972c6e56dcec9d492": "a(x)=\\frac{a_0x^0}{\\mu_0}+\\frac{a_1x^1}{\\mu_1}+\\cdots", "b945d4e3b36dfd75e9aaf47a8e3fdec8": "b=\\bar x + \\sigma \\sqrt{\\cfrac {m-q}{q}}", "b945f7ba59226a24c5f334c285b174ef": "\\Phi(z_0,\\ldots,z_{n-1}) = \\sum_{\\lambda_0=0}^L\\cdots \\sum_{\\lambda_n=0}^Lp(\\lambda_0\\ldots,\\lambda_n)z_0^{\\lambda_0}e^{\\lambda_n\\beta_0z_0}\\alpha_1^{(\\lambda_1+\\lambda_n\\beta_1)z_1}\\cdots\\alpha_{n-1}^{(\\lambda_{n-1}+\\lambda_n\\beta_{n-1})z_{n-1}}", "b9461fa47b97b9a8b2e0e9c7f55b7bf0": "\\; \\mathcal{H}_{A_1}", "b94644a7da84ec539258c4af66228197": "\\omega_{+}/\\omega_{-}=8", "b94660acfcbabd4c668962d645006de3": "t := (T + mN)/R", "b9467e87395a68096fd3fb6558bbf840": "\\mathbf A^{-1} = \\sum_{n = 0}^\\infty (\\mathbf I - \\mathbf A)^n.", "b94681533c5d80c4b857078f4ebeaf1b": "\\frac{X_b(t) - X_b(0)}{X_b(\\infin\\,) - X_b(0)} = \\frac {2}{\\beta\\,} \\frac {b^2}{a^2} \\sqrt{\\frac {FDt}{\\pi\\,}}", "b9469af1c64f172e2fc3f6ba7040b2e4": "(a^2 + b^2)\\,", "b946d9febc558947b5bca1952b7594af": "C \\subseteq P", "b946f99c26d7de207fe8ef33a75d2df2": "\\hat\\sigma", "b9470cd33df2483201103d02a624ab7d": "\\begin{align}\n\\Omega & = -kT \\ln \\Big( \\sum_{N=0}^{\\infty} e^{\\frac{N\\mu - N\\epsilon}{k T}}\\Big) \\\\\n & = + kT\\ln\\Big(1 - e^{\\frac{\\mu - \\epsilon}{k T}}\\Big).\n\\end{align}", "b9478a48803909a0babe0592b16e003b": "\\omega_{1}", "b947f812dc91d77355771d8059b381d6": "\\displaystyle \\frac{1}{2}\\left(\\frac{1}{i \\pi \\xi} + \\delta(\\xi)\\right)", "b947f8db3816a66d35a6ac2197b06a33": "v_{\\lambda}", "b9480c8b6e26347aa38d2782f69a0628": "\\mathcal{EL}", "b94838a844b2cb766084e7ce51d12191": "\\delta_H = \\frac{R(0)-R(H)}{R(H)},", "b948510b5f866759a20677ff1ba706ea": "(Q,\\Sigma,P, v, F)", "b9487bfd3cb2fd4d3a1f6eb8e1f8adbf": "\\omega_j", "b948b1b8c3cf4b769673fe317ce9a494": " 3 (x + 35)^2 =3x^2+ 210x + 3675", "b948d02a09c3318e3fa1680e9fc439a5": "\\nabla_a {C^a}_{bcd} = 2(n-3)\\nabla_{[c}S_{d]b}", "b9496d94322251c806ea78aad1598bd7": "\\mathrm{n}", "b949c87a7c8dc0c72d589e1d46bb8465": " \\sum{c_i(e_j) \\cdot y_{ij}} + \\sum{c(S_i) \\cdot x_i} \\leq B ", "b949d0d25f7eeab4f7af7bc455cf5e2a": "\\begin{align}\nx & = \\left(r + \\cos\\frac{\\theta}{2}\\sin v - \\sin\\frac{\\theta}{2}\\sin 2v\\right) \\cos \\theta\\\\\ny & = \\left(r + \\cos\\frac{\\theta}{2}\\sin v - \\sin\\frac{\\theta}{2}\\sin 2v\\right) \\sin \\theta\\\\\nz & = \\sin\\frac{\\theta}{2}\\sin v + \\cos\\frac{\\theta}{2}\\sin 2v\n\\end{align}", "b949f2c33392aea22245de3460e15f05": "(t-1)^5(t+2)^4(t-3)", "b949fcc832c2ea997346d644870e0047": "\\alpha, \\beta", "b94a63370565f3d52f78789ca25f5c82": "f^k", "b94ae7515f142f04bcd71114651a49c0": "\\begin{align}\n \\mathrm{Ai}(-x) &{}= \\sqrt{\\frac{x}{9}} \\left(J_{\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right) + J_{-\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right)\\right), \\\\\n \\mathrm{Bi}(-x) &{}= \\sqrt{\\frac{x}{3}} \\left(J_{-\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right) - J_{\\frac{1}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right)\\right). \n\\end{align}", "b94aef1d75439b969e5b20e8d736ccfd": "I_\\mathrm{1}, I_\\mathrm{2}", "b94b0b6b321bf68a2dd423caad9ecfe2": "E = \\frac {Z}{ 2 \\, \\sqrt{n} }", "b94b4b30620550d568c0dad160a9ce7e": "\\epsilon_b/\\epsilon_a \\to 1", "b94b72d6f6f4953ee5808ae0783abe70": "\\lim_{p\\rightarrow 1}\\frac{1}{p-1}({\\rm Tr}(B^{1-p}A^p)-{\\rm Tr}\\, A)=R(A\\|B),", "b94b7384d8e7063e90ab37d62e0f5618": "f_{e}", "b94b8c4f500711383088a388d2f31dec": " n_1=4, n_2=2, n_3=2, n_4=1", "b94b997ca3146ae81be69568590c9e8c": "V=\\begin{bmatrix}\n 1 & 0 & \\ddot{\\rho}(0) & \\cdots \\\\\n 0 & -\\ddot{\\rho}(0) & 0 \\ & \\ \\\\\n \\ddot{\\rho}(0) \\ & 0 \\ & \\ddot{\\ddot{\\rho}}(0) \\ & \\ \\\\\n \\vdots \\ & \\ & \\ & \\ddots \\ \\\\\n\\end{bmatrix}", "b94b9ee395975bccc91fb0a274a5a0ea": " K \\in L^2_{\\mu \\otimes \\mu}(X \\times X). ", "b94c4f2b808f0528badc4ee6a7141fbf": "\\mathcal P:=\\{A_1,A_2,B_1,B_2,C_1,C_2\\} \\ ,", "b94c774c0ac40ac0969778c538bb0b82": "\\rho = \\left| z \\right| = \\sqrt {z \\overline{z}}", "b94cc201ab648ef0acf74c1ba9eb0020": "\\varepsilon=r+\\gamma P \\Phi_{i}\\theta_{i}-\\Phi_{i}\\theta_{i}", "b94d3bfe2d62b773d5550ed9333a7f2b": " V {\\mathrm{d} t} = - N \\mathrm{d}\\Phi_B , \\,\\!", "b94eb989b1c42ee020ba861ef9fdf993": "\\text {E}=\\text {mgh}=0.1\\times980\\times10^{-2}=0.98\\text {erg}\\,\\!", "b94f4ff472eb246b688240a6d0250716": "\t\nx_{ij}\t\\in \t\\{0,1\\} (i, j = 1,2,\\dots,n)\n", "b94f8ec8ea416c6594033b85fd4a28be": " \\operatorname{de-let}[\\lambda V.E] \\equiv \\lambda V.\\operatorname{de-let}[E] ", "b95049cc9dace9125f15c14a2d45617e": "T''_n(1) = \\lim_{x \\to 1} n \\frac{n T_n - x U_{n - 1}}{x^2 - 1}", "b9507732f60b55f6ae1454ae3185f7dc": "\\textrm{dom}(\\eta) \\subseteq \\textrm{dom}(\\tau)", "b95082e3db0006ec1fdd1de9645dfcdf": "t_0 = \\frac{\\mu_0 q^2}{6 \\pi m c}.", "b950dfc2d7082d4d890110699a6e70de": "c = x_c^2 + y_c^2 + x_1^2 + y_1^2 - 2(x_c x_1 + y_c y_1)-r^2.\\,", "b9512c5c2e6c80e8104ff76ee9f46d65": "\\mathbf{X} = \\{\\mathbf{x}_1, \\dots, \\mathbf{x}_N\\}", "b951693deaaccba2c592e2e63fd8f36c": " dr_t = (\\theta_t-\\alpha_t)\\,dt + \\sqrt{r_t}\\,\\sigma_t\\, dW_t,", "b9516b8d352a2e5049b1c2ac6f81f73b": "\n \\frac{\\partial\\eta}{\\partial t}\\, +\\, \\boldsymbol{\\nabla}\\Phi \\cdot \\boldsymbol{\\nabla} \\eta\\, -\\, \\frac{\\partial\\Phi}{\\partial z}\\, =\\, 0.\n \\qquad \\text{ at } z\\, =\\, \\eta(\\boldsymbol{x},t).\n", "b951741b288c5f66a2332f538e5b954f": "C_{4,4} = 1 + 4 \\times 6.", "b951976a6f3b38f517d5496c7c4bd971": "\\; s", "b951e05fe7fa09870554dc46e1190b61": "\\sqrt{a^2 + b^2} = c. \\,", "b951f494d416997bca8bf7571eefb746": "\\mathbb{F}^k", "b9520b4340f8db4e4203db8f9d9b9af7": "\\vec x(i)=[x_1(i),x_2(i),\\ldots,x_m(i)], \\qquad i=1,2,\\ldots N", "b95213bdeb26c6596f1f3b1cfb9bc44a": "M_{k,t+1}", "b9522ec4cf751017a898967e394f5c0c": "F^\\chi=\\prod_{i=1}^mF_i^\\chi", "b9522ecc1538acfae68946c21b760eb9": "E(x)=1/\\lambda\\,", "b95239a8305de6ec0b3ca809118e3030": "M_\\lambda", "b95242033c1eb117ea788f79c85832a1": "\np_\\text{total} = p_1+p_2+\\cdots+p_n = p_\\text{total} = \\sum_{i=1}^n p_i.\n", "b9524ae615f21c59215e167b14a4582d": " \\log m + \\frac{7}{5})^s", "b953c6c459c33a5f132071d7326a72fc": " \\mathbf{e}(z+\\Delta z,t+\\Delta t) = \\mathbf{e}(z, t) e^{i k (c\\Delta t - \\Delta z)},", "b953e4967adc37888dcd6eb729e4c024": "M=\\frac{A}{S-A\\cdot\\left(1-P(b)\\right)}.", "b95444f5544b11bb988c9251af11682b": "\\Theta(x)=0,\\; x<0;\\; \\Theta(x)=1,\\; x>0", "b954df5da4d4c2a91144038f55b4788a": " R_{ ik } = \\frac{ H_{ max } - H_{ obs } }{ H_{ max } - H_{ min } }", "b95521f38077efc71cf7c7434fae9a43": "\\mathit{MPC}=\\Delta C/\\Delta Y", "b95548e4c063029489c933b9ea7acf66": "\\mathbf J =\\mathbf{{\\partial Y\\over \\partial a}}", "b95575a6c2a227da6e8dcec1f483f647": "k = k_{\\mathrm{B}} = R / N_{\\mathrm{A}} \\,", "b955b6e68976d2beb4ac8f262902c8cf": "x_4\\mapsto g_8", "b955e6bbd91d7ea1a35f7b8cd0dbf852": "\\det(V) = \\prod_{1\\le i2 \\\\\n \\infty & \\text{otherwise}\\ \\end{cases}", "b96161628ac73d83ecb5d92e093e95f1": "\\vec k\\perp\\vec B", "b9616e931cbeaf8e1543f4300fa26066": "\\delta l^a =(\\bar{\\alpha}+\\beta)l^a-\\bar{\\rho}m^a-\\sigma\\bar{m}^a\\,,", "b961a5c36e26c4991d685cf4a9515e04": "\\psi_1(\\Omega_2)", "b961cdfcb88fc6e13c1f3432f1aec31b": "U = -\\int\\vec{F}\\cdot d\\vec{x}=-\\int {-k x}\\, dx = \\frac {1} {2} k x^2.", "b9621b8b0f23cc967b4287db8a198890": "C \\equiv P \\and Q", "b96257595b03b3246cae68dc0ebdd19f": "\\tilde \\nu: M \\to X_k", "b9626f46a7da6a24bedf1a056f5f00dc": "S = S_1 \\times ... \\times S_n", "b962a7fd741217fe4cd9f8db4da6d7dd": "\\ M_{heel} = D_{heel} \\times (lift \\times cos(\\beta) +drag \\times sin(\\beta) ) = D_{heel} \\times lift \\times (cos(\\beta) +{(L/D)_{\\alpha}} ^{-1} \\times sin(\\beta)) ", "b962e2d174cb42060204285f35ca2fad": "C : y^2 = x^5 -4 x^4 -14x^3 +36x^2 + 45 x = x (x + 1) (x - 3) (x + 3) (x - 5)", "b962e3b189b0041d78401d6ebf5f2a15": "\\left|[f,g]\\right|\\le [f,f]^{1/2}[g,g]^{1/2}\\quad \\forall f,g\\in V.", "b962ea3f2d135b03d86099b354a7621b": "\\scriptstyle \\sqrt{1/2} \\ \\approx \\ ", "b9637075ca09c210eefd213e7affbb0e": "\\frac{\\partial^2 C}{\\partial x^2}= \\frac{1}{2 (\\Delta x)^2}\\left(\n(C_{i + 1}^{j + 1} - 2 C_{i}^{j + 1} + C_{i - 1}^{j + 1}) + \n(C_{i + 1}^{j} - 2 C_{i}^{j} + C_{i - 1}^{j})\n\\right)", "b9637c5888bbcde544169d353b7f8375": "\\scriptstyle (x[n])_{n\\in\\Z}", "b9638e8e3d6b8387c304a4d1a6e7f9ce": "\\det(A) = \\sum_{\\sigma \\in S_n} \\sgn(\\sigma) \\prod_{i=1}^n A_{i,\\sigma_i}.\\ ", "b96398abecce3e62cecc63ca0f1ef61c": "{}_1F_1(a,b,x) = \\frac{\\Gamma(b)}{\\Gamma(a) \\Gamma(b-a)}\\int_0^1 e^{xt} t^{a-1} (1-t)^{b-a-1}\\,dt,", "b963fde6416aac853a4a14e2f269f23e": "{}_3Q_1 = \\left( {U_1 + p_1 V_1 } \\right) - \\left( {U_3 + p_3 V_3 } \\right) = H_1 - H_3 ", "b96418893166a80def26dd77daed74a7": "\\varphi(-1)=-1", "b9646fd95c052cac2f772aa489549dc4": "H^{p,q}(M)\\cong H^q(M,\\Omega^p)", "b9647962c91189afec881ebc1143e49f": "%=((M-(S*P))/(S*P))*100", "b964c540f618e10ad210023507757cf4": "D\\colon {\\tilde M}\\to{\\Bbb R}^n", "b964cd02ca13b2d49adec910f4908eeb": "\\bar{\\delta} \\phi^A = \\epsilon \\Psi^A - \\epsilon \\mathcal{L}_X \\phi^A\\,.", "b964ea812d5905cef85daaccc7dfba17": "{{\\Delta}{\\rho}}", "b96547401192de04e4b93a745e598797": "I(t)=C_\\mathrm{m} \\frac{d V_\\mathrm{m}}{d t}", "b96573a898970a16e9b34d32228e7087": "I_2 \\otimes T", "b965785d0b240702c61687fb99d54f12": "D\\not\\subseteq A", "b965bc1e410e35af3cf6b26aef411019": "2 l^2 + l", "b965ebff676c48d93e0bfc559f2325da": "\\displaystyle{\\begin{pmatrix} \\alpha & \\beta \\\\ \\overline{\\beta} & \\overline{\\alpha}\\end{pmatrix}.}", "b965f3572a11c11b42ac0c97a261dd5e": "m\\times N", "b96609ff7571c63c0789736df39eaab5": "Z_{in} = \\frac{v_1}{i_1}", "b9665ea2d6c57056bd2b43ce0d6dbfe7": "{\\textbf C}_ X", "b966976ade5c91461449a97a2cd9c48b": "G^{ab}= 2 \\, \\left( F^{a}{}_{j}F^{jb}-\\frac{1}{4}g^{ab} \\, F^{mn} \\, F_{mn} \\right )", "b966ba91375d4f9a1e700f5b200383d8": "\\scriptstyle \\hat v \\;=\\; \\frac{1}{\\sqrt{3}}(\\hat x \\,+\\, \\hat y \\,+\\, \\hat z)", "b966e17c3a484185b3729b0db60546a5": "\\frac{1}{v} = \\frac{K_{M}}{V_\\max [\\mbox{S}]} + \\frac{1}{V_\\max}", "b96715ea8d29a384744d34e6414859a6": "t')", "b9675b5d61a63b01419a46d9a48e3c7b": " S ", "b9676153013660b7f8e92709283324e0": "R_\\oplus", "b967f855ebe5375fccc126bdcf975404": "I={c}{i_x}", "b9682475432c1a2833a2ffe001f722a2": "h(i)", "b9683a9fdab7e5d1575da13b0c5abce3": "\\varphi_i\\,R\\,\\varphi_{i'}\\iff\\forall j\\le k\\,(\\Box p_j\\in\\varphi_i\\Rightarrow\\{p_j,\\Box p_j\\}\\subseteq\\varphi_{i'}).", "b96844a8772c1c0f774e89971e8916c4": " \\sum_{i=1}^m r_i^s = 1. ", "b96882ea96f14f0f7ce3d2ac1290f456": "B \\lor \\lnot B", "b968bfbe5ebfbc1c35c66c569e6f8707": " \\begin{align}\nY^\\mu (\\tau ) &=\\left({\\tilde{x}}^{0}(\\tau);\\vec{Y}(\\tau)\\right)=\\left(\\sqrt{1+{\\vec{h}}^{2}}(\\tau+\\frac{\\vec{h}\\cdot\\vec{z}}{Mc});\\frac{\\vec{z}}{Mc}+(\\tau+\\frac{\\vec{h}\\cdot\\vec{z}}{Mc})\\vec{h}+\\frac{\\vec{S}\\times\\vec{h}}{Mc(1+\\sqrt{1+{\\vec{h}}^{2}})}\\right) \\\\\n& =z_W^{\\mu}(\\tau ,\\vec 0) \\end{align},", "b9690ccee1a31cc3e265dc160784e73f": "\\mu \\circ T\\mu = \\mu \\circ \\mu T", "b9691f47a6204ac04362116373c1c90d": " 1 - [-0.95 \\log_2(0.95) + -0.05 \\log_2(0.05)] \\approx 71.4%.", "b96927aedf1d26a8f82ba6143a3ba2bd": "A_\\text{c}", "b96929734b76a9c25502de0ad38b1ccb": "\\pm \\frac{R_1}{R_2}V_{\\text{sat}}", "b9693f762b7f415ddd14bf6f955c5de4": " \\mathbf V(x)=-10x+75 (kN) ", "b969de8bcf61e4d1ee59381c86c90fdc": "y^2=x^3+3x^2.", "b96a37b1e0e6638149365ff82483bfd1": " \\mu(q,p) = \\begin{cases} (-1)^{|p| - |q|} & \\text{if the skew diagram }p/q\\text{ is a disconnected union of squares} \\\\\n& \\text{(no common edges);} \\\\[10pt]\n0 & \\text{otherwise}. \\end{cases} ", "b96a884bdb7eba0f0dd86d2cdee1496e": "(1-p)\\gamma_{\\delta^{(p)}}^*(t)", "b96acbef7740318f1c241a96c42f3cd0": " R_{S}(t) = \\frac{ 165 \\ {_2^1}S }{ 154 \\ {_2^1}S + 315 \\ {_2^0}S }", "b96b29ae5fa989fefc3e947e4f755b6d": "A=\\frac{h}{3} ", "b96b3d27230f39849c2035128fa7642b": " H(a,p)=(a)\\log\\frac{a}{p}+(1-a)\\log\\frac{1-a}{1-p}. \\!", "b96b5d9dc2ee4d2caca9199b60852326": "C_{144}=B", "b96b902c70e2a3ed5ec85897fd39ddfd": "\\left[\\tilde\\delta\\omega\\right](v_1 , v_2 , \\ldots , v_{p-1}) = e^{-2\\varphi}\\left[ \\delta\\omega - (n-2p)\\omega\\left(\\nabla\\varphi, v_1, v_2, \\ldots , v_{p-1}\\right) \\right]", "b96be2207cbca2749077375d0f5fe94e": "\\epsilon^p_{\\mathrm{z}}", "b96c0851dc265a48c4b429a9669a328d": " g_D = \\frac{\\hbar c}{2e} = 68.5e", "b96c25fdb5cd4b92c8d7699e1e17bfb2": "[R]", "b96cc28dc9a28a4740de93b0a33f8346": "a_1, a_2, \\ldots, a_n", "b96d35b98aaa5cb7b88b5edc384108e3": "8|n", "b96d3cb2a38a0cc4d59d3f343f4d1495": "k[t]_{(0)} \\cong k(t)", "b96d8c41855d78894532240d6aef43f6": "\\log | f |", "b96d8d807897b16f48702f34c0724d70": "I_{T} = np - B_{0}", "b96dd3916287a5484fd50d1627a16763": "\\{\\mathrm{milk, bread, butter}\\}", "b96df954c928d541e2255f118a688b5a": " v_\\text{lab} = \\frac{u_\\text{cm} + v_\\text{cm}}{1+u_\\text{cm}v_\\text{cm}/c^2} \\approx 0.64 c", "b96e0218b58f8445b5e2d407356b93d0": "{\\tilde{A}}_{11}", "b96ec497b55f6f25e072c22101112414": "\\tau_{ij}^{r}", "b96eeba5bc87a1624d01be9c183947f9": "\\frac{\\partial}{\\partial y}", "b96f4a50eb44544f19ec64be82af191f": " i A ", "b96f54e599159010c49fb586280dd79d": "\\hat{d}\\ ,\\ \\hat{e}\\ ,\\ \\hat{f}", "b96fac8da19b01ee19c92db0dcac14cf": "\\scriptstyle{E_1}", "b9701884be7018b325b9840061e36766": " \\vec{s}_a ", "b9701ef26ef48799f7219b4f056aa3a0": "f(x) = (2\\pi)^{-n}(-1)^{n/2}\\int_{-\\infty}^\\infty \\frac{1}{q}\\int_{S^{n-1}}\\frac{\\partial^{n-1}}{\\partial s^{n-1}}Rf(\\alpha,\\alpha\\cdot x + q)\\,d\\alpha\\,dq", "b97029bb76ac9d922fe5f2150ae5600c": "P_{\\rm rad} = 4 \\pi r^2 \\sigma T^4", "b970b96e0f4186fac7ef26097e7917e2": "F_F\\;=\\;\\big( \\frac{f}{900} \\big)^{-n} \\mbox{ for } 2 < n < 3", "b970c956d9aee45ffd4886ef6b264941": "F_l[X]/\\langle X^p-1\\rangle", "b970e02b6f255db0bcf7180cd231df6b": "(f^n)^\\prime", "b9715386b9c73d78bad986b02af38b59": "RMD = \\frac{MD}{\\mbox{arithmetic mean}}.", "b971714a78ddc5ce03211134d635617c": "-\\sqrt{\\frac{27}{70}}\\!\\,", "b971bef60b58a493ee19800eb7b7623d": "{}D_{192} = 314\\frac{64}{625} - 313\\frac{584}{625} = \\frac{105}{625}", "b971f906855c5b31fb2ffed6e9b3416a": "\\textbf{M}_{O}=\\textbf{r} \\times \\textbf{F}", "b97257b9ed4d3cd928884e129b3ff9ba": "c\\mathbb{E} \\left [ [M]_t^{\\frac{p}{2}} \\right ] \\le \\mathbb{E}\\left [(M^*_t)^p \\right ]\\le C\\mathbb{E}\\left [ [M]_t^{\\frac{p}{2}} \\right ]", "b972bd2b3c27a0f72e7b6271bdcb2e50": "V_0\\subsetneq V_1\\subsetneq \\ldots \\subsetneq V_d", "b972f2f44c9d1069173d7c7320d95d40": " X_1^n(i') ", "b9734b2f80d39cf54e5be70c42dd5f62": "n = 1.4 \\pm 0.15", "b97351c111c1fbea3a56675260b4fc63": "\\mathrm{Cov}(x, y) = \\langle Cx, y \\rangle", "b973a4bb1c91719f54b72c2df2d5a314": "x = \\Psi(t, y)", "b973dd37ec5474f5276f2f142bc9b0cd": "\n\\begin{align}\n\\exp(B)\n& = P \\exp(J) P^{-1}\n= P \\begin{bmatrix} e^4 & 0 & 0 \\\\ 0 & e^{16} & e^{16} \\\\ 0 & 0 & e^{16} \\end{bmatrix} P^{-1} \\\\[6pt]\n& = {1\\over 4} \\begin{bmatrix}\n 13e^{16} - e^4 & 13e^{16} - 5e^4 & 2e^{16} - 2e^4 \\\\\n -9e^{16} + e^4 & -9e^{16} + 5e^4 & -2e^{16} + 2e^4 \\\\\n 16e^{16} & 16e^{16} & 4e^{16}\n\\end{bmatrix}.\n\\end{align}\n", "b97490fccf847a07db5f3568813c9a29": "\\hat m_ +", "b974d2c57d06f492920f8d819419339a": " H = G^2 - \\frac{p''(x_k)}{p(x_k)}", "b974eaedff79e43bd664664c87e6e858": " \\mathbf{F}=\\frac{1}{2}\\alpha\\nabla E^2. ", "b974fab4d589248cc77f788d0824417e": "\\widehat{\\mathbf{r}} \\psi = \\mathbf{r} \\psi ", "b9755d3c79287a7966dc0a97e5ad41bf": "T(\\mathbf{x})=\\sum_{i=1}^nX_i\\sim Po(n\\lambda)", "b9758953de341f297b056f0822280ee7": "b = \\frac{(b+d)+(b-d)}{2}\\, ,", "b9759fa06220d9f786393fa5f1b5873a": "_4^6", "b975aa939cdf6288a254885fbfcee8c2": "T^{\\mathrm{D}}_p (x,y) = \\begin{cases}\n 0 & \\text{if } x = 0 \\text{ or } y = 0 \\\\\n T_{\\mathrm{D}}(x,y) & \\text{if } p = 0 \\\\\n T_{\\mathrm{min}}(x,y) & \\text{if } p = +\\infty \\\\\n \\frac{1}{1 + \\left(\n \\left(\\frac{1 - x}{x}\\right)^p + \\left(\\frac{1 - y}{y}\\right)^p\n \\right)^{1/p}} & \\text{otherwise.} \\\\\n\\end{cases}\n", "b9763895bc43f4b115029a8f123ff327": "w_{\\mathrm{max}} = \\tfrac{qL^4}{185EI} \\mbox{ at } x=0.5785L", "b976516d0f6fb6fdc9455d1d8b4bbb0d": "D^{\\epsilon}(\\rho||\\sigma) \\geq \\log \\frac{\\epsilon}{\\epsilon - (1-\\epsilon)\\delta} ~.", "b976bc6c13ef83c214ccdcc37c9d085c": "\\begin{pmatrix}0 & -1 & -1 \\\\ 1 & a & 0 \\\\ z & 0 & x-c\\\\\\end{pmatrix}", "b9775a5d315f0d1b6a931805177e7947": "A = 8 \\tan \\frac{\\pi}{8} r^2 = 8(\\sqrt{2}-1)r^2 \\simeq 3.3137085\\,r^2.", "b9779cbb81654eb586a577617dc819ba": " \\phi, ", "b977ce9c8c404a2970562ad5d26b504f": "m \\le \\left \\lfloor \\frac{d}{2} \\right \\rfloor", "b977d77f8f911ac4a5dc9ad40e9e5ea3": "0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A", "b977f5b92456036524518471f99f166d": "\\Lambda_K(s)=\\Lambda_K(1-s).\\;", "b9780b42dd9f04a07d96b05c3a20cef3": "x^n - 1 = \\prod_{d\\,\\mid\\,n}^n \\Phi_d(x).\\;", "b9781d2b945133445a129163c583e282": "\\left|G\\right| = \\left[G : H\\right] \\cdot \\left|H\\right|\\mbox{,}", "b978b09bd055bd2d83bffed2408fe866": "\\zeta_m=e^{2 \\pi i\\frac{1}{m}}", "b978dcb25dfd853d3a10d7fb2757dcdc": "g_3=-x;", "b9795958e9b801c7b2095befacd3ae7f": "Z_C(t,t,t,t) = \\frac{t^6+ 3t^4 + 12 t^3 + 8 t^2}{24}", "b97988dc6b568f34bf9963b8c6ab2b9f": "\n\\begin{align} \n& V_{\\text{obs, r}}=-R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}d \\cdot \\cos\\left(l\\right)\\sin\\left(l\\right) \\\\\n& V_{\\text{obs, t}}=-R_{0}\\frac{d\\Omega}{dr}|_{R_{0}}d \\cdot \\cos^{2}\\left(l\\right)-\\Omega d \\\\\n\\end{align}\n", "b979cf6c48569dc6dd40d29e5a17992d": "\nu^{'''}\\left(x\\right)>0\n", "b979e87c35910c4a0ccbafa50867e57d": " m_1\\bold{u}_1^\\prime + m_2\\bold{u}_2^\\prime = \\boldsymbol{0} ", "b97a0f22e8342e4c8e8be1d021d1b43a": "O(2^k) = O(2^{\\log\\log n}) = O(\\log n)", "b97a188f779af7f41ad9789d711afb9b": "N=2345678917", "b97a64d0aa53b8787334510202931f11": "(K,\\Lambda)", "b97aedb4cd2ddbbc2d9ec2709066e56e": "\\frac{(d.f.)s^2}{\\chi^2(0.95)} \\le \\sigma^2 \\le \\frac{(d.f.)s^2}{\\chi^2(0.05)}", "b97af27a4b21cb93ef15b9f34f5f79b7": " 0\\to V_\\Sigma Y\\to VY\\to Y\\times_\\Sigma V\\Sigma\\to 0, \\qquad\\qquad (2)", "b97b28aafdeffaff55ecbcb34906fd98": "r:B \\times A \\to \\mathbb{R}", "b97b40227f91f1565bccf3af1f3afac7": "p^{d_i} (1-p)^{n_ib-d_i} 2^{-(N-n_i)b}", "b97b92a360403e58e31082f873ba3746": " P=\\frac{1}{2}C_{p}\\rho A\\nu^{3} ", "b97be21bdc1ea3a582f0d28d54ff950c": "\\Theta(n_i + n_j)", "b97c0971f30ca54c963bd46635154a3c": "a=\\sqrt{L_P/L_S}", "b97c128cd3ceab343c7f48d74717e2e6": "\np\\,\\textrm I\\,g\\iff\\begin{cases}\nx=b&\\text{if }m=\\infty\\\\\ny=\\frac{1}{2}mx+b&\\text{if }m\\leq 0, x\\leq 0\\\\\ny=mx+b&\\text{if }m\\geq 0 \\text{ or } x\\geq 0.\n\\end{cases}\n", "b97c2927858c54d8a86d21b5fba0ca9d": "T_m M \\approx \\mathbf{R}^n", "b97c2edd78f5b76334bf072e57827524": "\\, |k\\rangle", "b97c4f9b2410e61d1bc994857acc6242": " M = n \\, t = \\sqrt{\\frac{ G( M_\\star \\! + \\!m ) } {a^3}} \\,t ", "b97c6750d395dddadf1df4b954ab9099": "\\zeta(s)=\\prod_p(1-p^{-s})^{-1} ", "b97cb7d9980edfcc7d394cecc0defcb3": "\\inf(S) = -\\sup(-S)", "b97cc6402bb032c2199270813199b231": "{\\scriptstyle \\Gamma(z)}", "b97e0c4e94a232c598cb33740a0ff9cf": "2\\ell+1", "b97e775a12308facc150f35c45a4336d": "\\log(f(z)/z)=2 \\sum^\\infty_{n=1}\\gamma_nz^n.", "b97e7e2689efc5d8f0645d45467c0536": "\\tau_Y = \\left\\{ U \\subseteq Y : \\bigcup U = \\left( \\bigcup_{ [a] \\in U} [a] \\right) \\in \\tau_X \\right\\}.", "b97ea367e4a7e02d8343aaf278dd0f3c": " \\ln \\Gamma_p(a)= \\ln \\left(\\pi^{\\frac{p(p-1)}{4}}\\prod_{j=1}^p \\Gamma\\left(a+\\frac{1-j}{2}\\right)\\right) = \\frac{p(p-1)}{4} \\ln \\pi + \\sum_{j=1}^p \\ln \\Gamma\\left[ a+\\frac{1-j}{2}\\right] ", "b97ed23283e478267be2d01504f5da02": " x \\in B", "b97f238ff58fc8a7e7201f459aa9be73": " \\mathbf{A} = \\begin{pmatrix} a_{11} & a_{12} & a_{13}\\\\ a_{21} & a_{22} & a_{23} \\end{pmatrix} ", "b97f272cdce1751105a03caa193a7392": "\\,y_0\\,", "b97f623b28f8a3d1523cc44be398e9e2": "\n G(k)=\\exp(ikx_0-k^2Dt).\n", "b97f7152b1579bb48fa5f6ce4ca6a3b2": "\\zeta(s)=2(2\\pi)^{s-1}\\Gamma(1-s)\\sin(\\pi s/2)\\zeta(1-s).", "b97f83f20024f3ddfb3ea7fcaa0f668d": " \\operatorname{lambda-process}[\\operatorname{none}, L] = L", "b97f9b4ad79d2772c9c437b1982b65ab": "id_X - T", "b9803c5d3caa408f20ec7494b1bdd3c5": "\\sin\\beta", "b9807aa0b387a138de03cf2d09294e04": "H_4 = \\tfrac{1}{2}\\left[\\begin{smallmatrix}\n1 & 1 & 1 & 1\\\\\n1 & -1 & 1 & -1\\\\\n1 & 1 & -1 & -1\\\\\n1 & -1 & -1 & 1\\\\\n\\end{smallmatrix}\\right].", "b980a56c8e2341539e2d11efcf2f10ea": "e^{-\\frac{x^2}{2}}\\cdot H_n(x) \\sim \\frac{2^n}{\\sqrt \\pi}\\Gamma\\left(\\frac{n+1}2\\right) \\cos \\left(x \\sqrt{2 n}- n\\frac \\pi 2 \\right)\\left(1-\\frac{x^2}{2n}\\right)^{-\\frac{1}{4}}=\\frac{2 \\Gamma\\left(n\\right)}{\\Gamma\\left(\\frac{n}2\\right)} \\cos \\left(x \\sqrt{2 n}- n\\frac \\pi 2 \\right)\\left(1-\\frac{x^2}{2n}\\right)^{-\\frac{1}{4}}", "b980a93ab5a68069c3f3a55122d7ec7b": " I(1) ", "b981609d3776538ab51b81c9f35d5827": "\\begin{align}\nE_{KL}&=\\frac{1}{2}\\left( \\frac{\\partial x_j}{\\partial X_K}\\frac{\\partial x_j}{\\partial X_L}-\\delta_{KL}\\right) \\\\\n&=\\frac{1}{2}\\left[\\delta_{jM}\\left(\\frac{\\partial U_M}{\\partial X_K}+\\delta_{MK}\\right)\\delta_{jN}\\left(\\frac{\\partial U_N}{\\partial X_L}+\\delta_{NL}\\right)-\\delta_{KL}\\right] \\\\\n&=\\frac{1}{2}\\left[\\delta_{MN}\\left(\\frac{\\partial U_M}{\\partial X_K}+\\delta_{MK}\\right)\\left(\\frac{\\partial U_N}{\\partial X_L}+\\delta_{NL}\\right)-\\delta_{KL}\\right] \\\\\n&=\\frac{1}{2}\\left[\\left(\\frac{\\partial U_M}{\\partial X_K}+\\delta_{MK}\\right)\\left(\\frac{\\partial U_M}{\\partial X_L}+\\delta_{ML}\\right)-\\delta_{KL}\\right] \\\\\n&=\\frac{1}{2}\\left(\\frac{\\partial U_K}{\\partial X_L}+\\frac{\\partial U_L}{\\partial X_K}+\\frac{\\partial U_M}{\\partial X_K}\\frac{\\partial U_M}{\\partial X_L}\\right)\n\\end{align}\\,\\!", "b981619657184ce1c90ebba21fcdc0e3": "\\lambda_{k} = \\lambda", "b98166bd0a654548447542fddaa46805": " S|e_i \\rangle = |e_{i+1 \\mod{d}} \\rangle ", "b9818200eb104fef2b9bd2473d41166a": "p_k(n)", "b9823787b877610bdb31bc151511580f": " K_{\\lambda \\nu}", "b982725718a95a3bf61d06b5a2c45d07": "\\sum |a_n|", "b982c3341186506ddc5ef6c16db140f1": "\\operatorname{VIF}(\\hat \\beta_i) > 5", "b982dece9edf001590c097a355a7dd21": "{\\Gamma\\vdash e \\Rightarrow \\tau}\\over{\\Gamma\\vdash e\\Leftarrow \\tau}", "b982e10688c603a42fe1386ce7842f0a": "T_s = {1 \\over f_s}", "b9836cb62532a526b41184f038124e92": "\\gamma > 1", "b983e1c22f8028be985e4e51a08457d6": "(\\Gamma_r, k)\\not \\in \\Pi", "b98407b2f376a38bd0077db4031d66bc": "\n\\lim_{n \\to +\\infty} \\left( \\frac{\\int_a^b e^{nf(x)} \\, dx}{\\left( e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}} \\right)} \\right) =1\n", "b984336fc9f83ca4aa87368a7f0593c2": "\n f(t)\\equiv\\frac{|x_c-x_{0}|}{\\sqrt{4\\pi Dt^3}} \\exp\\left(- \\frac{(x_c-x_{0})^2}{4Dt}\\right).\n", "b984431eec201bc8dec92e7be5aca234": "X_{i,j}", "b9846310b55b19a141239ce7ce829c4b": " M \\times \\,^{\\prime\\prime} 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\,^{\\prime\\prime} = M \\times (2^5 + 2^4 + 2^3 + 2^1) = M \\times 58 ", "b9849a381df28f9362d25f801a276f4c": "\\Delta w_{ji} (n) = -\\eta\\frac{\\partial\\mathcal{E}(n)}{\\partial v_j(n)} y_i(n)", "b984a455700a031c819e30638d5de5b1": "\\mathbf{y}(t) = C(t) \\mathbf{x}(t). \\, ", "b984e14b4f20798137490fb8a436ed55": "\\sum_{n=2}^{\\infty} \\zeta(n,k) = \\zeta(k+1)", "b9851d886d2f1b2687b54d3883c06e71": "\\beta = \\arccos{(\\vec{\\tau_1} \\cdot \\hat{k})}", "b98541a405c6677e4930edc7abc37dae": "S'=S", "b9856e2c53938e3a83be45e6930f5bd1": " f_{Y_{[r:n]}}(y) = \\int_{-\\infty}^\\infty f_{Y \\mid X}(y|x) f_{X_{r:n}} (x) \\, \\mathrm{d} x", "b98582386a77a81b56b8aafa5bfb8b76": " IL = -20\\log_{10}\\left|S_{21}\\right| \\,\\text{dB}", "b98597d396de6787a152c619b1a6683c": "\\left | \\mathbf{x} \\right |.", "b985cc1b662cb11d7df47d57584c3d94": "\\textstyle\\text{rate(initiation)} = k_i[\\text{I}^+] [\\text{M}]", "b9863932b82bb2f540aeb1f4d93d5c9a": "\n\\begin{cases}\nu_t - Lu = 0 & (x,t)\\in D\\times (s,\\infty)\\\\\nu|_{\\partial D} = 0 &\\\\\nu(x,s) = f(x,s) & x\\in D.\n\\end{cases}\n", "b98668f929644e01cb59b3263337cdcc": "E(m, \\Theta_i)", "b98675139bce859e1f429a25760a0faf": "\\left| k\\right\\rangle", "b986d30af12fb91db908e703c74f803c": "p(\\hat{x}): \\Re^m \\rightarrow \\Re^3", "b98776166a550cde051c9b3a82e9c9dd": "{\\partial u \\over \\partial y} = e^x \\cos y, {\\partial^2 u \\over \\partial y^2} = - e^x \\sin y,", "b9879c38c282eabf57577de876736321": "\n\\varepsilon^v_{P} = \\frac{-\\Gamma/K_{eq}}{1 - \\Gamma/K_{eq}} - \\frac{P/K_{m2}}{1 + S/K_{m1} + P/K_{m2}} \n", "b988016ec00d9d5e7f46fcbd4fcc6740": "C(i,1) = -C(i-2, 1)\\,", "b9884cf80c3c261a056bf045b5bfc69b": "r\\in {\\Bbb Z}", "b9887bd8e3c08217a487b9e7ba73a863": "\\text{Sl}_{2m}(\\theta) = \\frac{(-1)^{m-1}(2\\pi)^{2m}}{2(2m)!} B_{2m}\\left(\\frac{\\theta}{2\\pi}\\right)", "b988ae2610dd8a7871f5ea887cc1299d": "z_-\\,", "b988f0fab5be344e15d2160d52fb792d": "k^{O(k)}\\log |V|", "b9899c96a2e78424ce1820504a622632": "\\mathbf{a}^{\\rm T}", "b989b19581ec78801e176255e105070c": "\\mathrm{Hom}_{\\mathbf{X}}(F(-), G(-)) : \\mathbf{C}^{op} \\times \\mathbf{C} \\to \\mathbf{Set}", "b98ad416d82e363e9ddda8b08ac4cb75": "\\Phi_{2}\\left(\\mathrm{R}_{i+1}\\right)", "b98adbb153610f4a7f705254445ff20c": "\n\\tau_i = \\exp \\left (- \\frac{\\alpha_i \\, \\Delta z_i}{\\cos \\theta_i} \\right )\n", "b98aff60242bc0631d04c462b03425ad": "(\\delta, n)", "b98b2f5d2598ee9fba2b7db6743a2259": "\nC_m = \\frac {A_m}{B \\cdot T}\n", "b98b6d9b00fae0b36e4f8c2a91231e2f": "\\int_{-1}^1 f(x)\\, dx \\approx \\sum_{n=0}^{N/2} w_n \\left\\{ f(\\cos[n\\pi/N]) + f(-\\cos[n\\pi/N]) \\right\\} .", "b98bc74ffe450c40440cf8bea71e69b8": "\n\t\\forall i:g_j^t(\\boldsymbol{x_{j,i}}) = g_j^{t-1}(\\boldsymbol{x_{j,i}})+\\alpha_th_t^j(\\boldsymbol{x_{j,i}})\n", "b98c1e6b6c387d4d51c47c543856932f": "\\mathbb{C}P^N", "b98c4465dca3e88120be2e3983ea51a7": "\\frac{n}{\\frac{1}{x_1} + \\frac{1}{x_2} + \\cdots + \\frac{1}{x_n}}", "b98c78badaf657e0ac0dbbd685be2ec4": " f=f_1 + H f_2 + \\alpha ", "b98c7ca1e8959293bb5a7f9abf42f43c": " (A,B,C,D) ", "b98ca956afcbf3b3a54407bb3e146962": "\\beta = (\\beta_1,\\beta_2,\\ldots,\\beta_n)", "b98cf70afb8bf09a0a93f8dc4bb13e0c": "\\frac{am+Nb}{|k|}, \\frac{a+bm}{|k|}, \\text{ and }\\frac{m^2-N}{k}", "b98d5cbde65d355e482b7e7e305f484b": "-2\\sqrt{3}", "b98d5e2753690ff027e65af2f5071e8a": " b_{k, i} = \\sum_{j=1}^n a_{k j, i j}. ", "b98da12a6fffabb3b4e8969305296972": "f=g", "b98e364b9e09b7a7aa4c9a35b3380596": "\\delta_a(x) = \\langle x,x \\rangle", "b98e3985e414bbeb2a5905e14a1796de": "S_{yy}(\\omega)", "b98e55edcce7765524865c9db8f062c0": "1\\to A^G\\to B^G\\to C^G\\to H^1(G,A) \\to H^1(G,B) \\to H^1(G,C).\\,", "b98ed0efdb19db1edfab1caa029e226c": "(x_1,\\ldots,x_n)\\,", "b98f1a2c352ba6306d6d596bb63827b1": "k = k_0\\;-\\;10\\;\\log {[A_0\\;(1\\;-\\;e^{\\frac{-A^i}{A_0}})(1-e^{R_ff})]}", "b98f31187a40b0f798a9430ad1fbc237": " q", "b98f5d4b8da7fed35f521f995c92640f": "\n \\begin{align}\n w(x,y) & = \\frac{2M_0 a^2}{\\pi^3 D}\\sum_{m=1}^\\infty\n \\frac{1}{(2m-1)^3\\cosh\\alpha_m}\\sin\\frac{(2m-1)\\pi x}{a} \\times\\\\\n & \\qquad \\left[\n \\alpha_m\\,\\tanh\\alpha_m\\cosh\\frac{(2m-1)\\pi y}{a} -\\frac{(2m-1)\\pi y}{a}\n \\sinh\\frac{(2m-1)\\pi y}{a}\\right] \n \\end{align}\n", "b98f83032f6e8ca0c8f5a38bca1e3d75": "DF", "b98fbc08b339fbe15c1064d712c3fb97": "\\begin{matrix}3&4&5\\\\4&7\\\\6\\end{matrix}", "b98ff324f95688bec8c08176e6518157": " t = t^{n+1} ", "b98ffcfd5f59b9691a62eadb5e084168": "\\frac{\\partial L}{\\partial q}", "b98fffd08a7cc9efa78d14e66713aef8": "\\sum_{n=1}^{\\infty} n = - \\frac {1}{12}", "b99052646d7cc70f4a545ffd96aa01c7": "I_{r,A,i}", "b99052caaf9ecf2fb2ab1f19fdf1f6c2": " \\varphi=\\varphi_0+\\xi", "b9906f6f9c389d6e609c1cc4a3c4bdd9": "\\mathbf{r} = \\mathbf{a}_0 + \\alpha_1 \\mathbf{a}_1 + \\cdots + \\alpha_n \\mathbf{a}_n", "b9908ceebeaf01f742bff6d31f9178c9": " V_f = \\frac{v_f}{v_c}\\! ", "b990b4c4c71628916fc45ea6e3485632": "\\frac{n^2\\pi^2}{L_e^2}", "b990e86fe337d996ce8619bb7d0df239": "\\det A_{33} > 0 ", "b9915ff5b5bad7ed3090cb436a6b2902": "0.1123\\pm0.0035", "b9916d82db00911b797a6eae76d06e12": "2.2) \\ \\text{Stock B}\\ += \\text{Flow}\\ ", "b9919ad61ae723116a99d8454fb2b241": "-\\eta n_\\eta^\\prime(\\xi)=\\beta n_\\eta(\\xi)(\\eta+n_\\eta(\\xi))", "b9919dd2c370090188c4f5da3dc4c965": " \\varphi \\left(\\bigcap_{j=1}^\\infty A_j\\right) = \\lim_{j \\to \\infty} \\varphi(A_j)", "b991a4874b21d0bb91a33b840e3e6d69": " \\widehat{\\boldsymbol{\\beta}}_L = \\underset{\\boldsymbol{\\beta}_{*} \\in \\mathbb{R}^{p}}{\\text{arg} \\; \\text{min}} \\; \\|\\mathbf{Y} - \\mathbf{X}\\boldsymbol{\\beta}_*\\|^2 ", "b991ad0f34b7f2dacf8a57b29e4e9257": "\nED_{ik}=\\sum\\limits_{j=0}^n \\left( \\frac{|x_{ij}^p-z_{kj}|-\\frac{w_{kj}^U-w_{kj}^L}{2}}{|\\frac{w_{kj}^U-w_{kj}^L}{2}|}+1 \\right)\n", "b991b909308d7fbbf216d862ab4658f6": " v = \\sum_{n=0}^\\infty \\alpha_n b_n,", "b99217f54e7280d24fa0891a12dde3b2": "\n L_J(\\phi_0) = \\frac{\\Phi_0}{2\\pi I_c \\cos(\\phi_0)}\n = \\frac{L_J(0)}{\\cos(\\phi_0)}.\n", "b992f3280ae4ccc4667868c6493c12ca": "S = \\left ( \\begin{array}{cc}0 & -1 \\\\ 1 & 0 \\end{array} \\right )", "b9936a99467b36af27e26e0e437c7011": "\\phi_{23}(a \\otimes b) = 1 \\otimes a \\otimes b.", "b993850b8585f987f7e998ce8d32ba7d": "g:\\mathcal{C}\\to\\mathbb{R}", "b994075bf8982474c8ba3912f4ede4b7": " \\mathbf{w}_{n+1} = \\mathbf{w}_{n}+\\Delta\\mathbf{w}_{n}", "b9940daa148fdb0fcb7aaa6d09c3d10c": "\\xi \\rightarrow \\infty", "b99448623b0641cd3e77eed8fd0f111e": " w(x) = \\,\\!", "b9944f852776634eb830d0e6b21d0a74": "S = - k_\\mathrm{B}\\mathrm{Tr} ( \\rho \\log \\rho ) \\!", "b994d9bac37f362349ccf622a1e51e77": "\\mu'_{20} = \\mu_{20} / \\mu_{00} = M_{20}/M_{00} - \\bar{x}^2", "b994e3915cb9d9e2d912a6b76dbaa069": "|f(x)-(S_Nf)(x)|\\le K {\\ln N\\over N^\\alpha}", "b995409e4e6203551e94b1a36599fcfd": "\\left|\\sum_{i=0}^{m-1} f(s_i) (y_{i+1}-y_i) - s\\right| < \\varepsilon.", "b995753c5b55ad1e757ae2f8e4942227": "a_n = \\frac{1}{2\\pi i} \\int^{}_{\\gamma} \\frac{f(z)}{z^{n+1}} \\, \\mathrm{d}\\ z ", "b995c74b9e55e8634d92474af98fd843": "[Fe/H] = -0.04\\pm0.04", "b995ca772e3f09f371a34411dd98d87b": "\\begin{align}\n & \\hat{\\beta }(X_{0})=\\underset{\\beta (X_{0})}{\\mathop{\\arg \\min }}\\,\\sum\\limits_{i=1}^{N}{K_{h_{\\lambda }}(X_{0},X_{i})\\left( Y(X_{i})-b(X_{i})^{T}\\beta (X_{0}) \\right)}^{2} \\\\ \n & b(X)=\\left( \\begin{matrix}\n 1, & X_{1}, & X_{2},... & X_{1}^{2}, & X_{2}^{2},... & X_{1}X_{2}\\,\\,\\,... \\\\\n\\end{matrix} \\right) \\\\ \n & \\hat{Y}(X_{0})=b(X_{0})^{T}\\hat{\\beta }(X_{0}) \\\\ \n\\end{align}", "b9963a530d99a95f87030470573e4058": "x \\equiv \\pm n^{\\frac{p+1}{4}}", "b99677b6098e3d7bb659818814ed5a22": " R_k(x) = \\frac{f^{(k+1)}(\\xi)}{k!}(x-\\xi)^k \\frac{G(x)-G(a)}{G'(\\xi)} ", "b9968a7e4e6d2fa6ef2abc9e61da0b09": " (u) = R ", "b9973d4a3ffe9eb7ebb1dba9eb65d97f": "V \\ne W \\to \\operatorname{sink}[(\\lambda V.\\lambda W.E)\\ Y, X] = \\lambda W.\\operatorname{sink-test}[(\\lambda V.E)\\ Y, X] ", "b9974285610b7a82c94b6a504726df8c": "h_n(ix;q^{-1}) = i^n\\hat h_n(x;q)", "b9977dbee9f3c9a6e5d03f8875f43b2a": "\\mathcal{C}_x^\\omega\\,", "b997a1e4b5dd1d89819c37df2a68acfe": "\\frac {{\\dot{m}} {\\sqrt {RT_{01}}}} {{D^2} {p_{01}} }\\ ", "b998305fbc201dfeea95f6615660ad7a": "\\kappa > 0 ", "b99834bc19bbad24580b3adfa04fb947": "|", "b998909b5187196e437d8720ae319cc3": "{dx \\over dt} = rx\\left(1-{x \\over K}\\right).", "b9989476e71416e0f1e3829cef76cfaf": "p_b", "b998f64dddb0a23ee73dcb48051a99fc": "\\zeta=\\frac{z}{d}", "b9990144634f3cfb62b3305b8836ff39": "\n \\mathbf{R} \\mathbf{U} = \\mathbf{U}\n\\begin{pmatrix}\ne^{i\\phi} & 0 & 0 \\\\\n0 & e^{-i\\phi} & 0 \\\\\n0 & 0 & \\pm 1 \\\\\n\\end{pmatrix}\n", "b99906faea74951ef7c7ae09fc6ca661": "\\vec x - \\vec{x_0}", "b99954e05a394732bd6d4abaad4a7b84": "n+1, f \\times (n+1)", "b9997f155ec5f9e3d8737239d8b579e7": " \\sigma_1, \\sigma_2", "b999c907848214c4407592d24153b83e": "b = b' - a' \\, \\pmod{2^n}", "b999ceecea2cb6bc102df14831d2e54c": "\\hat{\\mathbf{R}}", "b99a31f00d8eff828fb2b1657efe2f4b": "1\\times 1", "b99a978ca6daff29217155ab52235464": "\\sin^2\\theta=\\sin^2(-\\theta)\\text{ and }\\cos^2\\theta=\\cos^2(-\\theta).\\,", "b99aa2ff86df5e07cfc6fdad91e5d031": "1 = \\frac1{1\\cdot 2} + \\frac1{2 \\cdot 3} + \\frac1{3 \\cdot 4} + \\dots + \\frac1{(n-1)\\cdot n} + \\frac1n", "b99b0ec9c6f3dc63ace67040a29ab23c": "\\pi:I\\times S^1\\rightarrow S^1,\\qquad(x,z)\\mapsto z.", "b99b20bb1bf2e887720f7fd4f5124f8e": "CSO(1,1)=\\left\\{\\left.\n\\begin{pmatrix}\ne^a&0\\\\\n0&e^b\n\\end{pmatrix}\\right|\na,b \\in \\mathbb{R}\n\\right\\}\n", "b99b54951a07c1b878ccdd8a26563e0d": "[T_A^1]", "b99bcb3567d0c5e96ca2b631fa7c6df7": "{BC}^{2}={AB}\\times {BH} \\text{ and }{AC}^{2}={AB}\\times {AH}. \\,", "b99bddedafab5b1c558b0af55bfc7240": "\\tfrac{(humerus+radius)}{(femur+tibia)} \\times 100", "b99bdee6ca7ad9e8d4696709cfa1f038": " \\langle k \\rangle = \\frac{k_1 + k_2}{2} \\,\\!", "b99c326f6c41bc0fecb4241a0a0cbf8f": "F(x) = \\begin{cases}\n0 &:\\ x < 0\\\\\n1/2 &:\\ 0 \\le x < 1\\\\\n1 &:\\ 1 \\le x.\n\\end{cases}", "b99c8bb0c7891e48050346bca3cf4e85": "\\delta_{\\vec{\\xi}}h_{\\mu\\nu}\\approx \\left(\\mathcal{L}_{\\vec{\\xi}}\\eta\\right)_{\\mu\\nu}=\\xi_{\\nu;\\mu} + \\xi_{\\mu;\\nu}", "b99c8e371e3f6ec9f2f5f2e7040e5171": "Y = G - \\left\\lceil \\frac{U}{2} \\right\\rceil", "b99c956111da1e6c8ccea318c7c9e997": "\\partial_\\mu\\partial_\\nu E_n=\\sum_m\\left (\\langle \\partial_\\mu n|m\\rangle\\langle m|\\partial_\\nu H|n\\rangle + \\langle n|\\partial_\\nu H|m\\rangle\\langle m|\\partial_\\mu n\\rangle\\right),", "b99cfe5f8671d9c38c435a7f83cf64af": "\\textstyle \\bar{m}", "b99d9edf6c05afa64cbddaabfb01e378": "\\mathcal{B}_{X,D} = (\\mathcal{D}, \\mathcal{X},\\mathcal{P}) ", "b99daeaed303a0643b2a3e32fe1d7d06": "P = r_M I_{M}^2", "b99dc1815b22b7a4f2d89a522551132a": "\\Delta(v) = v \\otimes 1 + 1 \\otimes v", "b99e1515ecf21a120a60dd1de4c74f55": " \\sum_{j=0}^n b_j \\frac{d^j y}{dx^j} = r(x)\\,\\!", "b99e2e974a3671d8e927fb520c58d963": "i \\in m", "b99ec51f9a92b4c4ebd93b7eef8ace66": "\\mathbf{v}_{\\mathrm{reflect}}=\\{0.707107, 0.707107\\}\n,~\\mathbf{v}_{\\mathrm{refract}}=\\{0.636396, -0.771362\\}", "b99ee89a34d8299ee5843b06a3ff9c14": " - V (m z - p_z t) = V_z N_x - V_x N_z = \\left(\\mathbf{V}\\times\\mathbf{N}\\right)_y ", "b99f077b460c4982de558c95a8045477": "\\lambda = \\sqrt{\\frac{2E}{mR^2}}", "b9a06227ee6d9f7055fb4dfc12fdca75": "\\mathcal L (D)", "b9a085e4724285c1ec37bc5bf06b5381": "\\rho(x)=\\rho(x;\\alpha;\\beta,N)=\\binom{\\alpha+x}{x}\\binom{\\beta+N-1-x}{N-1-x}/\\binom{N+\\alpha+\\beta}{N-1}", "b9a09fc91c189866ddda195e61c679e0": "P'_x(x,y,z)=P'_y(x,y,z)=P'_z(x,y,z)=0.", "b9a0d1673d97e866d033f3eeeafdeabf": "O(M) = O(2^m) ", "b9a0de7c9a98267021e4a113a8e5ac8c": "L_{aa} = L_{bb} = L_{cc} = L_{lr} + L_{mr}", "b9a10457e87f759a56c8d4183ad4e81b": "~ \\sin \\theta_n = n \\lambda /S, n = 0, \\pm 1, \\pm 2 ...... ", "b9a137fb19237c41e9dffde79f287722": " r_\\mathrm{in} = \\frac{v_{in}}{i_{in}}", "b9a13db700f39922c1499e08898bb4b1": "\\varepsilon_{st}", "b9a164f87773ae33c5cdb36ecda4c37e": " \\mathrm{FWER} = \\Pr(V \\ge 1), \\,", "b9a1df853dec067d4bd6835b8a3c8bf2": "y\\in \\varphi(x)", "b9a1f4068e7ea8d16eca3ce74803494e": "\\ E_\\text{noncovalent} = E_\\text{electrostatic} + E_\\text{van der Waals} ", "b9a2c20bbebe879bbb3cc097dc50ae9a": " w_k^{\\top} f_k (x_{ \\{ k \\}}) = \\sum_{i=1}^{N_k} w_{k,i} \\cdot f_{k,i}(x_{\\{k\\}})", "b9a2f879198dfcb574d1369cefc9851b": "dn/dz", "b9a2fca69f54df7caa2f9a2210a69fb6": "S_k = M_{2,k}", "b9a372b7185bd7e4cdafdd4b9e488a9b": " \\iint_D \\nabla \\cdot (v \\nabla u) \\,dx\\,dy = \n\\iint_D \\nabla u \\cdot \\nabla v + v \\nabla \\cdot \\nabla u \\,dx\\,dy = \\iint_C v \\frac{\\part u}{\\part n} ds, \\,", "b9a3ada043aeaa0740ea989ed9f20460": " \\frac{\\zeta(s)}{\\zeta(2s) } =\\prod_p \\frac{(1-p^{-2s})}{(1-p^{-s})}=\\prod_p (1+p^{-s}). ", "b9a3dd1d1942f98df096ff7012963092": "A x = 0 \\Longleftrightarrow x^\\mathsf{T} A = 0 .", "b9a423d03870b9395eaf7400a04dc6e3": "\\scriptstyle \\sum {x_i \\cdot y_i} \\;=\\; 0", "b9a455246ea08eb147c1c309733e38bd": "{r_{\\rm w}}", "b9a4a6572dadf50c5ac415790b7ac3f7": "\\sup_j \\|x_j\\| \\le \\|x''\\|, \\ \\ x''(f) = \\lim_j f(x_j), \\quad f \\in X'.", "b9a4b8196b32c89cd4288481555c3178": "d_{118} = d_{228} = d_{338} = -d_{888} = \\frac{1}{\\sqrt{3}} \\,", "b9a4ee16e97954b83abd8c1b9bc5a1c5": "\\sum _x \\frac1x = \\psi(1-x) + C ", "b9a5029049b784d0fd6bb87db84fec7f": "\\dot{y} = x (3z + 1 - x^2) + \\gamma y \\, ", "b9a572b420f4354e767ab704b864fb03": "\n\\big[ A, B\\big] \\equiv AB - BA\n", "b9a5a34976eeacd48a07d5c90a70eb72": "\\mathfrak{gl}(V)", "b9a5be4bab84ccfb46c2d8e956e2b5f7": " I_1 > I_2 > I_3 ", "b9a5d7bd6ec9f150d5e31322c3840c9e": " \\prod_{k=1}^n \\left(k!\\right)^{n/k}\\geq L(n)\\geq\\frac{\\left(n!\\right)^{2n}}{n^{n^2}}", "b9a63607c785eca6ba85a5ab7c00d44d": "\\mathcal{P}^*", "b9a687c5d3a35b4e8a1f544b2e419027": "D_+(x) = \\frac12 \\int_0^\\infty e^{-t^2/4}\\,\\sin{(xt)}\\,dt.", "b9a6c0f48487fa8a521453c1c352895a": " M = \\operatorname{E}[X X^\\top]", "b9a6c1cbc3d90d1c911dffc171ef8c0b": " \\Delta S = S_{final} - S _{initial} = \\frac{q_{rev}}{T}", "b9a7003c4b3bfc976739dd56b6a7443b": "\\mathbf{W} = {\\mathbf{I_1}}^2 \\mathbf{R_{01}}", "b9a7220bb5c37d41b2d11d54401e304e": "x_1 \\le \\cdots \\le x_k", "b9a741940bca80cc21c7cb7890f2139c": "\\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{\\partial \\alpha^2}= -\\operatorname{var}[\\ln X]", "b9a7999cedb8e7b89224418f38613c9d": " \\Lambda^3_0\\mathbb C^6\\cong\\mathbb C^{14} ", "b9a7c6efd2f01cc9a003634521a25eb8": "\\chi=X^8-X^7-5X^6+2X^5+10X^4+2X^3-7X^2-5X-1", "b9a81338ec85349318c261baa38cd65c": "x\\in \\mathbb{F}^N", "b9a815899d5dd32551a6f939216b7e68": "\\sum_{i=1}^n \\alpha_i =1", "b9a88728cd5c4cedf96a41aff96ad322": "[y_0:\\cdots: y_{i-1}: 1: y_{i+1}: \\cdots : y_n] \\leftarrow \\left (y_0, \\dots, \\widehat{y_i}, \\dots y_n \\right )", "b9a8ac9853421bef1d567769c1995677": "f(a,b,x) = a \\cdot x + b", "b9a93d5b5a6452e781f7fece7a7ba807": "H=(b_{1}-b_{0})'\\big(\\operatorname{Var}(b_{0})-\\operatorname{Var}(b_{1})\\big)^\\dagger(b_{1}-b_{0}),", "b9a94d1d7b4aa026eb099b92cec36a68": "v_o \\approx \\sqrt{(m_2)^2 G \\over (m_1 + m_2) r}", "b9a9a489cec6b5037c226d04c96b32fe": "B^0 \\to K^- \\pi^+", "b9aa657cda79f009ea61bf87638ec037": "w_2w_{4k-1}", "b9aa7696f9327fae051ef89737a6f16f": "s_0=\\tfrac{3}{11}, t_0=\\tfrac{6}{11}", "b9aa7b9c6194d9ebe2323ce19cbc1914": "P(t) = \\boldsymbol{\\tau} \\cdot \\boldsymbol{\\omega}, \\,", "b9aa9bdc5288dc634da1df5bad7bc967": "k_{3,n}(x) = \\sum(x_i - \\bar x_n)^3 n/((n-1)(n-2))", "b9aaafeedbd8fb1c43a7eaace65f38a9": "d\\cdot2^s", "b9ab0e6db048dce86b6d40435e7f7fc1": "z \\mapsto \\begin{pmatrix}x & y \\\\ y & x\\end{pmatrix}.", "b9ab1a65d099ad6c6fbcbbc92bb279f8": " {\\Bbb C} P^n", "b9ab51329782387f83b8c00a44280bfc": "\\,\\ \\sec^2 x = 1+\\tan^2 x", "b9ab567e6c94ce93582c30dcad2fa42d": "\\beta=2, \\alpha=\\frac{\\nu+p-1}{2}.", "b9ab99f46a6985aae69dfcf6c227b8ea": "\n \\rho_0 U_s = \\rho (U_s - U_p) ~~, \\quad p_H - p_{H0} = \\rho_0 U_s U_p \\quad \\text{and} \\quad\n p_H U_p = \\rho_0 U_s \\left(\\frac{U_p^2}{2} + E_H - E_{H0}\\right)\n ", "b9abfd80497e5a91bc8c9dc0269719ce": " C_L = \\pi A\\!R A_1 ", "b9ac1a3811f8eb2bca398caf566011e3": "T \\square \\square F = 0 ", "b9ac1f207c04be2fc9dda2284aa6560a": "\\mathit{\\bar{c}} \\in \\mathcal{B}", "b9ac6a57dfc4b13b6f2ec5fa83824a48": "\\vec{P} = D \\times \\vec{r} = (p_1, p_2, \\dots, p_n)^T", "b9ac7e2878a6028ef83cdd38af2a1577": "\\operatorname{dom}(\\mathcal A)", "b9ac8a4a81a0edfb908ec32cc7c1f4f3": "\\textstyle\\{ -(1,2),(1,3),(3,2)\\}", "b9aca72a99b0eea3d4b4e0ab6a0dc445": " H_n = \\frac{1}{n!}\\left[{n+1 \\atop 2}\\right]. ", "b9acad20bc803c565d548b8ee39b57a9": "-\\sum_i \\frac{z_i q n^{0}_i}{\\varepsilon_r \\varepsilon_0} e^{-\\frac{z_i q \\varphi}{k_B T}} \\approx -\\sum_i \\frac{z_i q n^{0}_i}{\\varepsilon_r \\varepsilon_0} (1-\\frac{z_i q \\varphi}{k_B T})=-(\\sum_i \\frac{z_i q n^{0}_i}{\\varepsilon_r \\varepsilon_0}-\\sum_i \\frac{z_i^2 q^2 n^{0}_i \\varphi}{\\varepsilon_r \\varepsilon_0 k_B T})", "b9accdb61a2bcc736e0bd89b6ef347b2": "y=\\frac{2x+2}{x+2}\\,", "b9ad22349a216d55add97ecb5f6fbbe2": " |\\mathbf{a} \\times \\mathbf{b}|^2 = |\\mathbf{a}|^2 |\\mathbf{b}|^2 - (\\mathbf{a} \\cdot \\mathbf{b})^2 .", "b9ad2606b43062f9c7bd48797ba3b3e2": "p \\,", "b9ad281c4116c04f58b98edc8ef495f5": "\\tilde{y} \\in \\left\\{1,...,N-1\\right\\}", "b9ad5fbc6cc9290d0d1c2a66374cf4e5": "\\hat H^{\\text{core}}(1)=-\\frac{1}{2}\\nabla^2_1 - \\sum_{\\alpha} \\frac{Z_\\alpha}{r_{1\\alpha}}", "b9ad66ff822e00497667f0a726ae956f": "f(n)=1", "b9ad7cf1d6fbc654012641e94ff34b8e": "b^2 = 2k^2", "b9adc6aebdb0ddd6546d9fd723091ae5": "\\omega_1\\mid\\omega_2", "b9ae2bd322ab9b4fec7fbc196cbf83f0": "l_i(A_{ij}X'-a_{ij}T')=0", "b9ae88e78673718ef56d80215ae76971": "v*=\\frac {\\gamma + \\alpha} {\\rho}", "b9ae9974805e130a5f43c8fda36ea968": "\\tau^{00} = \\rho", "b9aed3e843eab936af2ccc5741dd3c80": "\\oint_C(v_x\\,dy-v_y\\,dx)=\\oint_C\\left(\\frac{\\partial \\psi}{\\partial y}dy+\\frac{\\partial\\psi}{\\partial x}dx\\right)=\\oint_C d\\psi=0.", "b9aedda31653a7d4e7f102d797166249": "\n v = \\lambda \\sigma\n", "b9aee135287df7e0e2ed02228fad32c5": "a = (r_a + r_p)/2\\,", "b9aef63fbeffc58957c847a261567deb": "\\sigma_n(A_\\epsilon) \\geq 1 - C \\exp(- c n \\epsilon^2) ", "b9af0912e5bf67359d36c3552172f790": " t", "b9af761b10f673bb403ebf06c049638d": "X= \\{X_1, X_2, \\dots \\}\\,", "b9afc70a4d618f9e2db0198231d72534": "10^{-5}", "b9b01461b5deb8d4e6cf5b4a969ba040": "\\pi : \\operatorname{Bl}_\\mathcal{I} X \\to X", "b9b0327da519113dc17e439cd93eabad": "\nF(x) = \\sum_{j=0}^\\infty P(j) I_x(\\alpha+j,\\beta), \n", "b9b058c1b74dd2b54642e998b7d0ba59": "H(M)", "b9b0ac3de2a84a16c13acc469f925c76": ".\\qquad \\qquad\\underbrace{NP,\\; \\qquad (NP\\backslash S)}", "b9b0b23644365a8ef85a31cb67048514": "\\mathbf{F} = -k\\mathbf{x}\\,\\!", "b9b0e0af2e124c97b0e2cf35cb7b7da3": " D = \\frac{ 1 }{ 2 } \\sum_{ i = 1 }^K \\left| \\frac{ A_i }{ A } - \\frac{ B_i }{ B } \\right| ", "b9b1307bd6285288213da5aed3c94bce": "\\mathbb{E}\\bigl[X_n^-1_{\\{X_n^->c\\}}\\,|\\,\\mathcal G\\bigr]<\\varepsilon, \n\\qquad\\text{for all }n\\in\\mathbb{N},\\,\\text{almost surely}", "b9b13ed799970c996af5a9d93a4b28d2": "p(x)-\\tilde p(x) = (p(x) - f(x)) - (\\tilde p(x) - f(x))", "b9b16f1d465f090f4d669bc3b448b76f": " E_{pot}^p = \\beta h", "b9b186eecd710603ea40c72485106652": "(1)\\quad ds^2=-e^{2\\psi(\\rho,z)}dt^2+e^{2\\gamma(\\rho,z)-2\\psi(\\rho,z)}(d\\rho^2+dz^2)+e^{-2\\psi(\\rho,z)}\\rho^2 d\\phi^2\\,,\n", "b9b196b047743edaccef1bb6080548bc": "\n\\vec{\\alpha}(\\vec{\\theta}) = \\vec{\\nabla} \\psi(\\vec{\\theta})\n", "b9b1aa6c7903804e4343d1fb6797e1fe": "\\dot{\\gamma}_{ph,e,sp}", "b9b246445097a44296f754ad7e0700a2": "\\chi_{[a,b)}(x) = (b-x)_+^0 - (a-x)_+^0", "b9b28ed18d4ce5f5e6a07b70e9f86140": "d [x,y] = [d x,y] + (-1)^{|x|}[x, d y]", "b9b2a6df9c4c8ff2e472aa82e329b8b2": "S_n \\to \\operatorname{Aut}(S_n)", "b9b2be5035e4689198fdeba3f5169f3d": " \\mathfrak g", "b9b2bf843b5f25ff70445f6646a6e378": "J_\\alpha(x) = \\frac{1}{\\pi} \\int_0^\\pi \\cos(\\alpha\\tau- x \\sin\\tau)\\,d\\tau - \\frac{\\sin(\\alpha\\pi)}{\\pi} \\int_0^\\infty e^{-x \\sinh(t) - \\alpha t} \\, dt. ", "b9b326ece5284d055de96098000d55fe": "\\langle Tx, Ty \\rangle = \\langle x, y \\rangle", "b9b32f21a2ceecbaa32bba992698af3b": "\n\\begin{bmatrix}\n -26 & -3 & -6 & 2 & 2 & -1 & 0 & 0 \\\\\n 0 & -2 & -4 & 1 & 1 & 0 & 0 & 0 \\\\\n -3 & 1 & 5 & -1 & -1 & 0 & 0 & 0 \\\\\n -4 & 1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}\n", "b9b3538e209bc0930e5032e03ba3e244": "\\beta = 1,8392", "b9b355d9face217e8494531db3a6fc25": "R_{\\mathbf{B}} = e^{\\frac{\\mathbf{B}}{2}}.", "b9b35f4d7e06031254176294e7dde71a": "Z_{0}", "b9b3647aea01b66f3cf91943c2f34628": "\\tau''", "b9b3e9b4dde507b600d825a4800d6c7f": "\\mathbb{P}_{\\boldsymbol{\\theta}}(S(X)\\geq\\gamma)", "b9b3ec02d24a866506ef46e8cc0b85a2": "\\left(1+x+1/x\\right)^n=\\sum_{k=-n}^{n}{n\\choose k}_2 x^k", "b9b471da6656429a641f5a112b5db22c": "0.\\overline{428571}", "b9b47e8027c547ca5769e19a0725019a": "\n\\dot{\\lambda}(t)=-\\frac{\\partial H}{\\partial x}\n", "b9b5464fb915dcedca1c5acc36838a47": "(m',n_m')", "b9b54978b6079dc758fe869b211ee240": "S_i=n_aR\\ln\\frac{V_t}{V_a}+n_bR\\ln\\frac{V_t}{V_b}.", "b9b56e07e2d8464d1653af6cde083bd9": "s = ut + \\frac{1}{2}a.t^2 + \\frac{1}{6}jt^3 ", "b9b583bf45b3ed4f9bc001eba7a8f126": "+-", "b9b602783394854b5580a5730729bb7f": "1 \\not \\in n\\mathbb{Z}", "b9b604233f5a9438478f2b87373fff86": "x(t)=\\int_{0}^{\\infty}\\int_{-\\infty}^{\\infty} \\frac{1}{a^2}X_w(a,b)\\frac{1}{\\sqrt{|a|}} \\tilde\\psi\\left(\\frac{t-b}{a}\\right)\\, db\\ da", "b9b64c4322fac490ce3c22b1b6b01fb1": "\\alpha_{A,B,C} \\colon (A\\otimes B)\\otimes C \\cong A\\otimes(B\\otimes C)", "b9b64e322a9a73d7681bd0a4a1b2bb70": "\\scriptstyle{T = \\{ t_1, t_2, \\ldots \\}}", "b9b6751318dbf69706e7b461ae7415ae": "\\begin{cases}\n\\overbrace{ \\begin{bmatrix} \\dot{\\mathbf{x}}\\\\ \\dot{z}_1\\\\ \\dot{z}_2 \\\\ \\vdots \\\\ \\dot{z}_{i-2} \\\\ \\dot{z}_{i-1} \\end{bmatrix} }^{\\triangleq \\, \\dot{\\mathbf{x}}_{i-1}}\n= \n\\overbrace{ \\begin{bmatrix} f_{i-2}(\\mathbf{x}_{i-2}) + g_{i-2}(\\mathbf{x}_{i-2}) z_{i-2} \\\\ f_{i-1}(\\mathbf{x}_i) \\end{bmatrix} }^{\\triangleq \\, f_{i-1}(\\mathbf{x}_{i-1})}\n+\n\\overbrace{ \\begin{bmatrix} \\mathbf{0}\\\\ g_{i-1}(\\mathbf{x}_i)\\end{bmatrix} }^{\\triangleq \\, g_{i-1}(\\mathbf{x}_{i-1})} z_i &\\quad \\text{ ( by Lyap. func. } V_{i-1}, \\text{ subsystem stabilized by } u_{i-1}(\\textbf{x}_{i-1}) \\text{ )}\\\\\n\\dot{z}_i = f_i(\\mathbf{x}_i) + g_i(\\mathbf{x}_i) u_i\n\\end{cases}", "b9b68521710db71141358b678e612544": "\\cos(m\\varphi)", "b9b69a70ac85e110c1c5f3c33b5cbd4a": "x = z - e ", "b9b6cbbec8b28fa5dbd71ee6965b091c": "\\lambda_N,", "b9b73025a3da75dc46d6096ac1617963": " v_S(T) = v(T), \\forall~ T \\subseteq S.\\, ", "b9b76ae7b4be1257acd4d2fb52ad4223": "\\tau_0", "b9b778a4db518d496cfc987928a7b24e": "v'(s) = \\biggl(\\underbrace{u(s)-\\int_a^s\\beta(r)u(r)\\,\\mathrm{d}r}_{\\le\\,\\alpha(s)}\\biggr)\\beta(s)\\exp\\biggl({-}\\int_a^s\\beta(r)\\mathrm{d}r\\biggr),\n\\qquad s\\in I,", "b9b785a9b5754916ed9ebf9f382ce063": "\nx = k_x \\cdot \\frac{I_4 - I_3}{I_4+ I_3} \n", "b9b7c722e67b3c72487933418c01eed1": "N_{\\rm C}(T) = 2(2\\pi m_e^* kT/h^2)^{3/2} \\quad N_{\\rm V}(T) = 2(2\\pi m_v^* kT/h^2)^{3/2}.", "b9b86e448627ff2d115e41f9541379ca": "\\frac {f{(x)}+f{(-x)}}{2} + \\frac {f{(x)}-f{(-x)}}{2} ", "b9b8b1c0d2280f1c4ff9bdb570a191e5": "B(\\rho,z) = \\frac{\\mu_0 m}{4 \\pi (z^2+\\rho^2)^{3/2}} \\sqrt{1+\\frac{3 z^2}{z^2 + \\rho^2}}", "b9b8e22689d1975a092a87ea00870fad": "a\\mathbb{P}(X \\geq a) \\leq \\mathbb{E}(X)\\,", "b9b92c1571fae89040982523167bdcae": "r \\in R", "b9b994bd0e61af129449c255e05f9de8": " e^x = G_{0,1}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} - \\\\ 0 \\end{matrix} \\; \\right| \\, -x \\right), \\qquad \\forall x ", "b9ba30c942c821f254169f7699f744c1": "g^f", "b9ba33e2b0c432fe18480ba4b7fb7459": "Q(1,1)=1", "b9ba3dd66ab518da657a791da866a82c": " \\tilde{F} ", "b9ba83bd1f78308ec076e403e1a4357f": " f(x+h) \\approx f(x) + f'(x)h + \\tfrac{1}{2} f''(x) h^2", "b9baba85a74cf924e5cf3969bfdce992": "\\left\\| \\frac{\\mathrm{d} f}{\\mathrm{d} t} \\right\\|_{L^{1} (W)} \\leq \\liminf_{k \\to \\infty} \\left\\| \\frac{\\mathrm{d} f_{n_{k}}}{\\mathrm{d} t} \\right\\|_{L^{1} (W)}. ", "b9bb5158d865b18629530bd452202448": "(a \\uparrow b) \\uparrow c", "b9bb8769ad3d3d253723eec9bbede67b": "\\tfrac{a-b}{a} \\sim \\tfrac{1}{\\sqrt{\\xi}}", "b9bb962193d7d23210bf6582c7698b7a": "(r*S)/1000", "b9bb99f8d50decb4d51c66ce85b9be03": "\\displaystyle{\\|u\\|_{(k+1)} \\le C\\|\\Delta_1 u\\|_{(k-1)} + C \\|u\\|_{(k)},}", "b9bbbac81a8ee78d623ff4736ccd9d99": "J_{oxide} = D_{ox} \\frac{C_s- C_i}{x}", "b9bc0392891c623c4552070f141e322f": "\\Lambda^2 V ", "b9bc237195dd2d0273f1654cb32d6e6d": "x^q", "b9bc6c5a230c4fad6d7b00cf7604830d": " \\scriptstyle {\\mathbf r}_2(t) ", "b9bca147d694233bd569a14e94c3a2a7": "\\mathbb{P}[\\omega = T] = 1 - p", "b9bca61eb5ad241470fc2756a819bf31": "G (\\mu_1+\\mu_2) \\approx 1 \\approx G (\\mu_1+\\mu_3)", "b9bcc215ce6de0f16a457ef6f8f00ff0": " f(r) \\equiv 0 \\,\\bmod{p^{k+1}},", "b9bced20dfb7b01ee9381882e7523366": "|\\psi\\rangle = |\\phi\\rangle_A\\otimes|\\gamma\\rangle_B", "b9bcff7239a673511519919daa16ba2d": "v_\\text{p} = \\frac{\\nu}{k}", "b9bd0c3bcb6b2ca29a2e3800f457eca0": "1+\\sqrt[3]{2}+\\sqrt[3]{2^2}", "b9bd6ff6c3d2e46b1e5a06974ab3c314": "\\mathbb{\\hat{C}}", "b9bd9b9a55a05f3e7bc3cd856db48a71": " Y^{(n+1)}(t) = f^{(n+1)}(t) - \\frac{R_n(x)}{W(x)} \\ (n+1)! ", "b9bda630e24b56a97d6093b26aec2f0a": "z_n' = \\frac{d}{dc} f_c^n(z_0).", "b9bdf5aa2d8aa85c340a23aa15dede27": " \\begin{bmatrix}\n 1/4 & 0 \\\\\n 0 & 1/4 \\\\\n \\end{bmatrix} ", "b9bdf8e07297cb3e699d2b92333294ef": "\\xi = \\left ( f(x_0), f(x_1), \\ldots, f(x_k) \\right )", "b9be3179a87de24fd2091ab6a8b2d0eb": "M = \\lbrace q : q^* = q^{\\star} \\rbrace = \\lbrace t + x(hi) + y(hj) + z(hk) : t, x, y, z \\in R \\rbrace .", "b9be43622f4256b4fb6f1248e3d0c485": "f(x,y)=ax^2+by^2-r^2", "b9be9b051b07b9ff1f69bc988c17d42f": "\\left(\\frac{\\partial H}{\\partial T}\\right)_P=\\left(\\frac{\\partial Q}{\\partial T}\\right)_P=C_P.", "b9bea5ba520464f5879d5d1ba8026ba9": " \\begin{align} \\mathbf{ab} &= (a_1\\mathbf{e}_1 + a_2\\mathbf{e}_2 + a_3\\mathbf{e}_3)(b_1\\mathbf{e}_1 + b_2\\mathbf{e}_2 + b_3\\mathbf{e}_3) \\\\ &= a_1 b_1{\\mathbf{e}_1}^2 + a_2 b_2{\\mathbf{e}_2}^2 + a_3 b_3{\\mathbf{e}_3}^2 + (a_2 b_3 - a_3 b_2)\\mathbf{e}_2\\mathbf{e}_3 + (a_3 b_1 - a_1 b_3)\\mathbf{e}_3\\mathbf{e}_1 + (a_1 b_2 - a_2 b_1)\\mathbf{e}_1\\mathbf{e}_2. \\end{align}", "b9beab72f0feee620392e3e442c824af": "G_S(z) = G_{X_1}(z)G_{X_2}(1/z).", "b9beb85a05f6f5d04e6714aeb012e8ce": "\n\\begin{Bmatrix} \\mathbf{Q}-\\mathbf{P} \\\\ \\mathbf{P}\\times\\mathbf{Q} \\end{Bmatrix}\n= \\begin{bmatrix} A & 0 \\\\ DA & A \\end{bmatrix}\n\\begin{Bmatrix} \\mathbf{q}-\\mathbf{p} \\\\ \\mathbf{p}\\times\\mathbf{q} \\end{Bmatrix}.\n", "b9beed9af843086956bf0bcc5df807cf": "\nr = \\sqrt{x^{2} + y^{2}} = \\frac{1}{2} \\left( \\sigma^{2} + \\tau^{2} \\right)\n", "b9bf452e9d0fd2173814b6a9fa7a02f8": "f:\\mathcal A^2\\rightarrow\\mathcal A", "b9bf803b2cb737254622dbc1cb0823ee": "\n\\overline{X_{3}A_{1}} \\cdot \\overline{X_{3}A_{2}} = \\overline{X_{3}B_{1}} \\cdot \\overline{X_{3}B_{2}}\n", "b9bf93a4c3c27af89ff0f6a20ddb9b41": "V_N", "b9bf9e9bb3b1e282188f8fd6328d037d": "\\langle f, \\varphi\\rangle \\le C\\sum\\nolimits_{|\\alpha|\\le N, |\\beta|\\le M}\\sup_{x\\in\\mathbf{R}^n} \\left |x^\\alpha D^\\beta \\varphi(x) \\right |=C\\sum\\nolimits_{|\\alpha|\\le N, |\\beta|\\le M}p_{\\alpha,\\beta}(\\varphi).", "b9bfc05d744cd7d809107cb567db9dd5": "R_j", "b9c01d2e251d8242278fe18f4e745171": "\\tau:\\Omega^1(\\mathrm{Ad}_Q)\\to \\Omega^2(TM)\\,", "b9c0e7233c4cede09f33451598d6e50e": " E^2 \\approx \\mathbf{p} \\cdot \\mathbf{p} ", "b9c162b7a62b1d6c971709473afea6ff": "x' = x", "b9c1797f6d94b37d4822c8659a0d953b": "\\cos_k(\\angle\\zeta^i)=(2^{-1}\\bmod{p})\\cdot(\\zeta^{ik}+\\zeta^{-ik}),", "b9c1c4ae19059a83c9af42a7f183caa8": "S\\subset[X]^\\lambda", "b9c272336b3fa656c898b916c4122089": "\\int_0^1\\int_0^1 \\frac{1-x}{1-xy}(-\\log(xy))^s\\,dx\\,dy=\\Gamma(s+2)\\left(\\zeta(s+2)-\\frac{1}{s+1}\\right).", "b9c285b5f0316db797e7e9cb93e7cc7a": "(s-1)\\,\\zeta(s)", "b9c28c2b90732cc80c74232bc339b4b4": " \\kappa(A) = \\left\\Vert A^{-1} \\right\\Vert \\cdot \\left\\Vert A \\right\\Vert .", "b9c2ccaacfbf7f833ece124233c8f626": "\\mathbf{F}_{\\mathrm{Coriolis}} = \n-2(\\mathbf{\\omega} \\times \\mathbf{p}) = -2(\\mathbf{\\omega} \\times) \\mathbf{p} = \\begin{bmatrix}\\,0&\\!-2\\omega_3&\\,\\,2\\omega_2\\\\ \\,\\,2\\omega_3&0&\\!-2\\omega_1\\\\-2\\omega_2&\\,\\,2\\omega_1&\\,0\\end{bmatrix}\\begin{bmatrix}p_1\\\\p_2\\\\p_3\\end{bmatrix}", "b9c2db92e2991398b51278fe6151c246": " \\begin{bmatrix} \\ln x \\\\ \\ln (1-x) \\end{bmatrix} ", "b9c30a6f5a6ae60d74699919cb051322": "\\frac{\\delta F}{\\delta t}=\\frac{\\partial F\\left( t ,S \\right)}{\\partial t }-v^{i}Z^{\\alpha }_{i}\\nabla _{\\alpha }F", "b9c325478e9faef67735b13b533d5d9f": "\\Delta E_{int} = E(A,B) - \\left( E(A) + E(B) \\right)", "b9c340bcacd78496692d0c1d71b9d583": "I\\left(\\theta\\right) = I_0 \\left[ \\operatorname{sinc} \\left( \\frac{\\pi a}{\\lambda} \\sin \\theta \\right) \\right]^2 \\cdot \\left[\\frac{\\sin\\left(\\frac{N\\pi d}{\\lambda}\\sin\\theta\\right)}{\\sin\\left(\\frac{\\pi d}{\\lambda}\\sin\\theta\\right)}\\right]^2", "b9c34eca0acf3b79a94bc239dbffef61": "\\Delta G_\\mathrm{mix}=RT\\sum_i{x_i\\ln x_i}", "b9c36e3f67b27fde8bbae8f95a3c8371": "\\rho_{Ref}\\,\\!", "b9c38508cf9ec4bdd7d09eb8681f76d0": "\nH = \\frac{p_1^2}{2m} + \\frac{p_2^2}{2m} + \\cdots + \\frac{p_N^2}{2m} + V(x_1,\\dots,x_N).\n\\,", "b9c38daa8b302ae7052b4077c9e448d1": "P(t)=-t\\frac d{dt}\\log(E(-t))= t\\frac d{dt}\\log(H(t)),", "b9c4386e071aa15713d0a504ff819176": "\\tan{\\frac{A}{2}}=\\sqrt{\\frac{bc}{ad}}=\\cot{\\frac{C}{2}},", "b9c4aa1c09db93df6495ceb4dc0cfa91": "J(E)", "b9c4d6f7d3431318568f0af9eba45ac4": "\\exists a \\forall c \\in b \\forall y\\in c\\, (y \\in a)", "b9c4e7cd220c11276c7ab524f6dd554b": "f_i : X_i \\to X_{i+1}", "b9c511cc405274c251b1c4f42d0f820e": "W_h\\approx\\eta\\beta^{1.6}_{max}", "b9c5576f3360bc915210a556eb4331ce": "\n\\begin{bmatrix}\n1 & p & u\\\\\n0 & I_n & q\\\\\n0 & 0 & 1\n\\end{bmatrix}", "b9c5829eadb06283fbf2ecb300bd937e": "\\mathcal F \\to \\mathcal G", "b9c5a7b8eafb9bba272983ee2676c82d": "f(\\bar{u})", "b9c5b726fd8ce90fe42ecc8ed5306dc9": "e^{-i\\omega t}\\phi\\,", "b9c5db1f4a3bfd57b92bb859623c2c02": "C_{k,l}", "b9c5fbc71813ea66ccf2ff7213c41f9a": "\\ L_j", "b9c631a7f2d494314b2c03471b58fa96": "\n\\; _{p}F_{q} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right)\n= \\frac {\\Gamma(\\mathbf{b_q})} {\\Gamma(\\mathbf{a_p})} \\; G_{p,\\,q+1}^{\\,1,\\,p} \\!\\left( \\left. \\begin{matrix} 1-\\mathbf{a_p} \\\\ 0,1 - \\mathbf{b_q} \\end{matrix} \\; \\right| \\, -z \\right)\n= \\frac {\\Gamma(\\mathbf{b_q})} {\\Gamma(\\mathbf{a_p})} \\; G_{q+1,\\,p}^{\\,p,\\,1} \\!\\left( \\left. \\begin{matrix} 1,\\mathbf{b_q} \\\\ \\mathbf{a_p} \\end{matrix} \\; \\right| \\, -z^{-1} \\right),\n", "b9c65b1af4e3bf5d43d4485e8d1dd086": " C_r \\ = 1 ", "b9c6c1f484dec227223f056e2056b1e9": "g_\\mathrm{n}", "b9c6e2526564fb7bcc9ed337e2d3e2c9": "P_B = 1.4 \\times 10^{-34} \\times N^2 \\times T^{1/2} \\frac{\\mathrm{watts}}{\\mathrm{cm}^3}", "b9c72009dddfec492b25e6598d50d76b": " \\left( L_{-1} - \\frac{1}{2(k+h)} \\sum_{k \\in \\mathbf{Z}} \\sum_{a,b} \\eta_{ab} J^a_{-k}J^b_{k-1} \\right)v = 0,", "b9c73e8cc0e5987b4dbd492ce3cf93d4": " h(-1) = 1, h(0) = 2, h(1) = 1 ", "b9c7927032fde756ab5854625aa7dd6e": " \\left (\\frac{T_1}{T_2} \\right )^\\frac {1}{\\gamma-1}", "b9c7f8bc9617d3242f9d0dfb46440624": "\\displaystyle{Q(a)R(b,a)=2[L(a),L(b)]Q(a)+2L(ab)Q(a)=R(a,b)Q(a).}", "b9c7fb058dcd08a0589310fe82a0e6fc": "(\\lambda x.(\\lambda x.x)) x", "b9c890d2a1bdac8371b67363638b7d55": "\\mathbf{p} = \\gamma(\\mathbf{u}) m \\mathbf{u} ", "b9c8da5b5f64da0e2263c572095f18b9": "\\mathbf{P(s)=\\tilde{D}^{-1}(s)\\tilde{N}(s)}", "b9c917d062a77de04f2391dc9dbbdfc8": "A \\wedge (B \\vee (D \\wedge E)).", "b9c9b5d6966ea9cebf8741d8a5e3e253": "t_{n+1} = t_n + h", "b9c9b7cb70d85246c8b9e95eb59ff410": "\n e_{ikr}~e_{jls}~\\varepsilon_{ij,kl} = 0\n ", "b9ca17d6d5d7c8bc2724a2f00ae01073": "O(log log N)", "b9cab46de2314348b05127d55852e93e": " G(t{m+1}) - G(t_m) = \\sum_{n=0}^N \\nu_n(t_m) [Y_n(t_{m+1}) - Y_n(t_m)] , \\quad m = 0 \\ldots M-1, ", "b9cab4fadb5fdd08d96118b828c85610": "h_{0} = hash(A_{x}\\; ||\\; L\\; ||\\; ts\\; ||\\; K_{s})", "b9caf87c8317f14da6ed9bc1e5b4c906": "\n-\\Bigl\\langle \\frac{dP_{b}}{dt} \\Bigr\\rangle = \n\\frac{192G^{5/3}m_{1}m_{2}\\left(m_{1} + m_{2}\\right)^{-1/3}}{5c^{5} \\left( 1 - e^{2} \\right)^{7/2}} \n\\left( 1 + \\frac{73}{24} e^{2} + \\frac{37}{96} e^{4} \\right) (\\frac{P_{b}}{2 \\pi})^{-5/3}\n", "b9cb1a4417d507db5b25542cfbc91d92": "R_{(n)}^t=\\max{\\{\\frac{S_1^t}{S_1^0},\\frac{S_2^t}{S_2^0},...,\\frac{S_n^t}{S_n^0}\\}}, ", "b9cb25dbfaa2475f222b59b4c329300a": "\\;= \\sum_i p_i (\\log p_i - \\sum_j (\\log q_j )P_{ij}),", "b9cb5f3d3d0f7211bd76efc57d7b7f10": "\\kappa = \\sqrt{ \\frac{4\\pi e^2 n \\beta}{\\epsilon} }", "b9cc16a19e8ed20adda366107c7fa535": "f_L(x)=x/3", "b9cc659f0b202ffd5a1a55e2eb7e4d72": "\\lambda x + (1 - \\lambda)y \\in C", "b9cc676cdcbcbfcefc36730acbf56bc3": " \\Delta_k^\\alpha = \\{Q+\\alpha: Q\\in \\Delta_k \\}.", "b9cd538606450e3755de5b7176a11dc8": " c_{t+1} - c_t = (1-R^{-1}) \\left[A_{t+1} -A_t + \\sum_{j=1}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j-1} E_{t+1} y_{t+j} - \\sum_{j=0}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j} E_{t} y_{t+j} \\right] ", "b9cd75cafc268116fc32417321bea630": "{}^{\\bullet}t = \\{ s \\in S \\mid W(s,t) > 0 \\}", "b9cd8944970495808a4c0681172e1eb5": "\\zeta_3", "b9cd91b300863e086fdfbd00e8e6895b": "k_{\\rm C}=\\frac{1}{4\\pi\\epsilon_0}=\\frac{\\mu_0 (c/100)^2}{4\\pi}=10^{-7}\\cdot 10^{-4}\\cdot c^2 = 10^{-11}\\cdot c^2 .", "b9cdb4af3783e8cb4ccd5c8355e6a2d7": "\n\\frac{\\partial \\tilde{\\nu}}{\\partial t} + \\frac{\\partial}{\\partial x_j} (\\tilde{\\nu} u_j)= \\mbox{RHS}\n", "b9cdef1f4e396fa23d5eba13be178a48": "H_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\mu}(Ce^{-jk_{x\\varepsilon }x}+De^{jk_{x\\varepsilon }x})cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (38) ", "b9ce40bcbb171ca1aa015d52607b4e92": "b_1 = a_1a_3 - 4a_0 \\,", "b9ce66f29ca8494702df8f090aa21683": "f(x)=\\Omega_R(g(x))", "b9ce6e5aea35c7181602e1df355632c9": "\\text{input} x\\to x=(x_1,...,x_n)\\overset{\\text{secret}: S}{\\to}x'\\overset{\\text{secret}: P}{\\to}y'\\overset{\\text{secret}: T}{\\to}\\text{output} y", "b9ceb717935301286e58700550f64603": "dE = \n\\frac{\\partial E}{\\partial k_x}dk_x +\n\\frac{\\partial E}{\\partial k_y}dk_y +\n\\frac{\\partial E}{\\partial k_z}dk_z =\n\\vec{\\nabla}E \\cdot d\\vec{k}", "b9cf0ed16e363f88c392e65e2fef7e50": "\\frac{E}{\\eta} t", "b9cf2597e4d646e0906ef5b51281b963": "\\begin{align} 7_{10} = 010303_{2i} = 10303_{2i}.\\end{align}", "b9cfa756a9c8854d7fc52774e960a721": "(c,\\infty)", "b9cfe1b67969f25dd148bc6200ec8943": "S(X)\\leq S(Y)", "b9d0054d22815e8c7aafcecf95513376": "P(\\text{ill}\\cap\\text{positive}) = P(\\text{ill})\\times P(\\text{positive}|\\text{ill}) = 1%\\times99% = 0.99%.", "b9d011645bfa9400ecf9add3ff284bc5": "\\scriptstyle\\mathcal{U}_\\Sigma", "b9d05c0e5f48a18b8da10e57d7d6a105": "C(f,h)(x)=\\sum_{k=-\\infty}^\\infty \\textrm{sinc}(\\frac{x}{h-k})", "b9d06dec758e474e919f0da02a08b61d": "\\{f_i:X\\to X|i=1,2,\\dots,N\\},\\ N\\in\\mathbb{N}", "b9d0aa4f828b718c72bc050c28aeb6cb": "e^{it\\mu}\\frac{\\pi st}{\\sinh(\\pi st)}", "b9d0c8e580a03e2df95ade4bb4230e47": "\n!P = P \\vert !P\n\n", "b9d0f77e58d861485dbc7a9b8abc41dc": "E[z]\\,", "b9d10cef6be83275ad40916a8fb9868a": " T_0(x) = 1 \\,", "b9d137027731ca1ef79c63a0930fe23c": " M \\subseteq M' ", "b9d140f8840370b4cf2cb8af1abcdefd": "\\tilde A_N(\\omega, z) = \\int_{-\\infty}^\\infty A_N(t,z) \\exp[i(\\omega-\\omega_0)t] dt ", "b9d19ffbe256928ba9cfc33fca0a0fe0": "\\textstyle \\dot{u}(t)", "b9d1d1ae1d635d6dc703e6c584ea86e8": "y_b = b_0 \\sum_{r = 0}^\\infty \\frac{c(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^{r + c} = b_0 x^c \\sum_{r = 0}^\\infty M_r x^r.", "b9d248fba6ef2835243be38aed9d7881": "-w_{\\rm max} = \\sqrt{2\\times\\hbox{NAPE}}", "b9d28274daab68977a00b2f6d93fdfd6": "\\epsilon_k(p)=q", "b9d2bf4ce514c3bb250e1cf930f4255a": "\\boldsymbol{\\mu}_L = -g_L\\mu_B \\frac{\\mathbf{L}}{\\hbar}.", "b9d2deed6ef920cac731bbe9b1b9f0e7": "\\frac{\\partial}{\\partial x_i} \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_j(\\vec{r}')d\\tau'} - \\frac{\\partial}{\\partial x_j} \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_i(\\vec{r}')d\\tau'} = 0", "b9d3088c5b91ea81fc32975130464045": "\\rm \\ C_6F_5XeF + Me_3SiCN \\rightarrow C_6F_5XeCN + Me_3SiF", "b9d34f6560d095c61bf917828cb68a67": "H_k = \\left\\{\n\\begin{array}{lr}\n\\mathbb Z & k \\in \\{0,n\\} \\\\\n0 & k \\notin \\{0,n\\}\n\\end{array}\n\\right.", "b9d36383b92e6b088c03d85e92294503": "\\displaystyle{ T(iu_n)=-\\lambda_n iu_n}", "b9d379c4dc0c7719d35252466c66e661": "H^*(X; R) = R [w] / \\langle w^{n+1} \\rangle ", "b9d386732886ff57844aebda690a1b5f": "f = \\sum_{n=1}^\\infty \\varphi_n \\left ( \\varphi_n^\\dagger f \\right). ", "b9d3c8119d94a4085d113e2b7d297e9d": "\\{(x_1,x_2,\\ldots,x_{n}) \\in \\mathbf R^{n} \\vert x_{n} \\geq 0\\}.", "b9d3df73b352098d63380f5dc6e3f4c8": "\\alpha = \\frac{a + \\sqrt D}{2}\\text{ to }\\alpha' = \\frac{a - \\sqrt D}{2}.\\,", "b9d3f88c1af4d625351377114ede6ec9": "\\textstyle x^i", "b9d41b7bc0c706fdc5dcb941a5df5e50": "(1-x^2)\\,y'' - x\\,y' + {\\lambda}\\,y = 0\\qquad \\mathrm{with}\\qquad\\lambda = n^2.\\,", "b9d446ca4a007887193ff3b69a0c6edc": "p(x_1,\\dots,x_n)", "b9d47b00d2beb7b77a10daa418ed21a1": "S_{r_1} (r_2) > 0 ", "b9d4b71d8f13d861df9a38d560181d60": "d_\\sigma", "b9d4c80aa30d44f35197be54fdd3028e": " W(C;1,-1) = \\sum_{w=0}^{n}A_{w}(-1)^{n-w} = A_{n}+(-1)^{1}A_{n-1}+\\ldots+(-1)^{n-1}A_{1}+(-1)^{n}A_{0} ", "b9d5158105d9dda2e8f52092b01a5f66": " e = \\sum_{k = 0}^n \\frac{1}{k!} \\approx 2.71828 ", "b9d53fe9e078f6eab8696575a2523c5e": "\\textrm{volume\\ percent} = \\frac{\\textrm{volume\\ of\\ solute}}{\\textrm{volume\\ of\\ solution}}*100\\%", "b9d5990f0cd832576abcda79afa8fc05": " g(X_1,X_2,\\dots,X_n)", "b9d5bc2d26ae23d9488fda95a40e2378": " P(z) = P\\left( -{1 \\over z^\\star} \\right) ", "b9d5cb9349eed45f6ee53a00cbefa79d": "[E_{2T}] = [E_2] + [W'E_2] ", "b9d5ce15d23afb8a202dda58cec0ede0": "\\left(-3\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ 0\\right) \\pm \\left(0,\\ \\sqrt{\\frac{2}{3}},\\ \\frac{2}{\\sqrt{3}},\\ 0\\right)", "b9d5d48ebf3183379edc7b1343a18db6": "\\Omega X \\times \\Omega X \\to \\Omega X ", "b9d606eec26e53f4d2d853bc8eeaa1ae": "\\operatorname{dCov}^2(X, Y)= 0", "b9d6402881ca240f8f24dc1e3505e6f0": "\\ F_{u,v}", "b9d6a53f6f846487330e8360ec0038c9": "\\mathrm{li}(x)", "b9d6aeb74899c71808f83e5be98712b0": "-\\psi -L", "b9d6c69752013a0b1cb9de46cd9b8d01": "MM(p) = 2^{2^p-1}-1", "b9d6e037dafca6126c8e263dfe8093f4": "\\alpha_7 = {{1\\alpha_0 + 6\\alpha_1} \\over 7}", "b9d6e17586b69681f29ca11ceaad8467": "2^n (a n + 1)", "b9d6fd50fbfd55575ce027042f1397c8": "Q =", "b9d7185eac4d1a4f4824d1c4ea6ea104": " T_q^{(k)} = Y_{\\ell = k }^{m = q} (\\mathbf{a}) = \\langle \\mathbf{a}|k,q\\rangle", "b9d733706fb2a5297f5959723edaa694": " [\\exists \\text{ internal } A\\subseteq{^*\\mathbb{R}}\\dots]\\ .", "b9d735dc0ee160b5f3a3af78e3332990": " 2^{(1-\\alpha)} ", "b9d74e7d39b0762ebed690f7a986e4ff": "\\nabla \\times \\mathbf{B} = 4 \\pi \\mathbf{J} + \\frac{\\partial \\mathbf{E}} {\\partial t}", "b9d77c3c22e56c9bda0407017863ef8d": "\\mathbf{m} = m(\\mathbf{X}) = [m(\\mathbf{x}_1),\\ldots,m(\\mathbf{x}_N)]^\\top", "b9d808c74a1f944c4bf2862a43888ff3": "a_1 \\cdots a_n \\mapsto \\frac{1}{n!} \\sum_{\\sigma \\in S_n} a_{\\sigma(1)} \\otimes \\cdots \\otimes a_{\\sigma(n)}.", "b9d81759864fec9f1ee4f46f286740ca": " D[p] = [q, \\_, \\_]::[x, \\_, \\_]::L_3", "b9d8c69e01ecbee9540aac83049f937f": "C=N_\\mathrm{A}/V_\\mathrm{m}", "b9d8cd78ba80059e01a45849772c4d01": "c_0 = \\sum_k q_k\\times c_k", "b9d8e38b1af8543c75c1f614a44d883e": "\\lambda, z", "b9d8ee5c3ec6efb5542e4eba54ea8406": "\\varepsilon_i=\\gamma z_i + u_i", "b9d9113e41e0a528097a016d4c0e449f": "f:R \\to S", "b9d94d0b84bc6915e94fbedeb89f820c": "R_{\\rm S} = 6.96 \\times 10^8 \\ \\mathrm{m},", "b9d9652962ca11b31af01298c9d8e54a": "\\overline{\\tau} = \\tau", "b9d982b54ba3cfcc61a9b63f7966d20e": "\\subseteq, \\nsubseteq, \\subsetneq, \\varsubsetneq, \\sqsubseteq \\!", "b9d99db8626de63193c7fe96273a6cae": "\\Omega ", "b9d9f37c043a7028a00da7e4cd9cd8ee": "\\textbf{K}(s)", "b9da150f3140ab78061f34537b70118a": "1_{d_{1}}", "b9da8405725ad9899272143927be455f": "\\nabla_i", "b9da84c1cf62dce9b7abc6f515950695": "\n \\boldsymbol{\\sigma}^T\\cdot (J~\\boldsymbol{F}^{-T}\\cdot\\mathbf{n}_0~d\\Gamma_0) = \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0\n", "b9dac10249a4a0fc0416f7dc05c60589": " a \\approx \\; 1.541434 \\times 10^{-6} \\; \\mathrm{A \\; eV} \\; {\\mathrm{V}}^{-2};\\;\\;\\;\\;\\; b \\approx 6.830890 \\; {\\mathrm{eV}}^{-3/2} \\; \\mathrm{V} \\; {\\mathrm{nm}}^{-1}, ..........(30b)", "b9dad82d242d9a7ab1288cd6794473a3": "U = -\\boldsymbol{\\mu}\\cdot\\mathbf{B} = -\\mu_z B\\,\\!", "b9db39b4a2fe5fab16e6eb185c32e915": "2-2g", "b9db508e5ba06a66635cc653bc8d7a19": "\\varphi:\\mathcal H_\\Delta^k(M)\\rightarrow H^k(M)", "b9db510a1ff3f88c33aed6d57fac8e50": " \\mathbf{ \\hat n }", "b9db5f176bc7ba26f9cfec54eb2d5c27": " t = \\frac{w(w + 1)}{2} = \\frac{w^2 + w}{2} ", "b9db6ff56842decdd7444f220dc37c77": "f(x,y) = P(x,y)", "b9dbf3ccac0c8a4d7f15add827784767": " g(1;p) =\\sum_{n=0}^{p-1}e^{2{\\pi}in^2/p}=\n\\begin{cases} \n\\sqrt{p} & p\\equiv 1\\mod 4 \\\\ i\\sqrt{p} & p\\equiv 3\\mod 4 \n\\end{cases}.", "b9dc09df912fc3cc250187f627f006d1": "\\{x[m-k];\\ \\mbox{for all integer values of m}\\}", "b9dc09fad1f8e102f3a5ce14a3f828dc": "\\overline{c_{i,j}}", "b9dc139da6c57a999f610d2929284d66": "\\ \\Box(A^* \\rightarrow B^*)", "b9dc24009d7340ca9cdbc6b238074351": "\\cdots \\to H_n(A) \\to H_n(X) \\to H_n (X,A) \\stackrel{\\delta}{\\to} H_{n-1}(A) \\to \\cdots .", "b9dc54fc0b0620ae9ef39f187fbd066f": "g^{(1)}( \\mathbf{r}_1,t_1;\\mathbf{r}_2,t_2)= \\frac{\\left \\langle E^*(\\mathbf{r}_1,t_1)E(\\mathbf{r}_2,t_2) \\right \\rangle}{\\left [ \\left \\langle\\left | E(\\mathbf{r}_1,t_1)\\right |^2 \\right \\rangle \\left \\langle \\left |E(\\mathbf{r}_2,t_2)\\right |^2 \\right \\rangle \\right ]^{1/2}}", "b9dce96eb3d5a71b28f9f198c28d2d1b": "\\Theta", "b9dd05be16380c4a5b6e4c2518ef7ead": "\n\\mathbf{X}_k \\succeq \\mathbf{0} \\quad \\text{and} \\quad \\lambda_{\\text{max}}(\\mathbf{X}_k) \\leq 1\n", "b9dd08cc3484cd9a981a09ff91ab6868": "y=\\arccos x\\,\\!", "b9dd19c284f3f8130c2d56b1c6ea4de9": "(\\ell+1)^2", "b9dd523da765ee84d8b5cf79177a07a7": " j_{\\text{t}}\\; vs.\\; E ", "b9dd767427bafdde55e14dc1058bdc71": "\\frac{v_r}{c} = \\cos \\varphi \\cdot 1,54\\cdot10^{-6}", "b9ddc3d27a579004ac1fcaab875bfb61": "\\{(a_i, \\lambda_i)\\}_{i\\in I}", "b9ddc796513f9a7b2e8c83532047d140": "y(x) = A x^{\\frac{1}{2} + \\sqrt{a}} + B x^{\\frac{1}{2} - \\sqrt{a}}", "b9ddda4489c7f4be52a6974d2b62482b": "S_N f\\left(\\frac{2\\pi}{2N}\\right) = \\sin\\left(\\frac{\\pi}{N}\\right) + \\frac{1}{3} \\sin\\left(\\frac{3\\pi}{N}\\right)\n+ \\cdots + \\frac{1}{N-1} \\sin\\left( \\frac{(N-1)\\pi}{N} \\right).", "b9ddf81067de6b598b22bad47af47328": "\\left[P_G\\left(\\eta\\right)+p'_G\\left(0\\right)\\right]-\\left[P_L\\left(\\eta\\right)+p'_L\\left(0\\right)\\right]=\\sigma\\eta_{xx}.\\,\n", "b9de3b19a18f502d1d53a3d4bd8db097": " \\rho=\\limsup_{n\\to\\infty}\\frac{n\\ln n}{-\\ln|a_n|},", "b9de3d5dab9ead77662b0af1ae87b23a": "x(t) = \\frac {c^2}{g} \\left( \\sqrt{1 + \\frac{\\left(gt + v_0\\gamma_0\\right)^2}{c^2}} -\\gamma_0 \\right).", "b9de43224f0cca3054be38773886de96": "x = \\cos(n_x t + \\phi_x),\\qquad y = \\cos(n_y t + \\phi_y), \\qquad z = \\cos(n_z t + \\phi_z),", "b9de7620d2eda2792e8e067044047811": "\\forall i", "b9def9c6092310c2b38e2c1bf6f45336": "S^{(j)'}", "b9dfd714d582737988d8d16f27d6aadf": "S(A,B) = S(A) + S(B),\\,", "b9dfe473344a2bf44d9e321d694b2d64": "f(\\delta)", "b9e04879a9104941c2edb08abfc9b969": "\\left(\\frac{\\gamma}{\\mathfrak{p} }\\right)_n \\equiv \\zeta_n^{b(1)+b(2)+\\dots+b(m)} \n\\pmod{\\mathfrak{p}},\n", "b9e055e62da74d74dc78f590586c659c": "Q_{W|X}(\\tau)=\\exp(X\\beta_{\\tau})", "b9e05bba1f9e728b551685c2312d2dab": "W\\ ", "b9e0cf7c46782344ea43345a6f1ca57b": "\\begin{pmatrix} 0 & -1 \\\\ 1 & 0\\end{pmatrix}", "b9e103e0d1a0aed0487e66e5f12217b8": "\\phi(z)\\,", "b9e10f00ff12a08bf338bfa7027e1b1f": " R(x) = [m/n]_{f}(x)\\,", "b9e16eac2b15f6f31da906acc73d654c": "c_k=\\sum_{m=0}^{k-1}\\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \\left\\{1,1,\\frac{7}{6},\\frac{127}{90},\\frac{4369}{2520},\\ldots\\right\\}.", "b9e1e197f301c905e0a70e8b6253e170": "1 / G", "b9e219db066a6b7ee9229a2c86df2b52": "d=3\\,", "b9e2984827f26e7e259340cf2b16e007": "\\mathbf{u}' = ( u - v ) \\mathbf{d} \\, .", "b9e2adb6ba3cea9de90b242ab097a1b9": " \\oint dZ = 0 ", "b9e2b65f1237a23735bf7d54745f7147": " \\sigma (l_A a_B + l_B) (Q_B l_B + Q_D l_D) ", "b9e2edfb86610de94b1da2e77b37abc2": "\\mu_\\alpha(\\phi) = \\sec\\phi \\left[\\frac{\\sin\\alpha}{\\sin\\beta}\\right].", "b9e31bb31262a5b47d4c9af8d9a3267a": "\\int_0^x \\, \\frac{du}{1+u^2} = \\arctan x = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\frac{x^7}{7} + \\cdots ", "b9e32aad5fcebc8b53a15ea565772d5d": "P_{\\ell +1}^{\\ell}(x) = x (2\\ell+1) P_{\\ell}^{\\ell}(x)", "b9e339a33416d37cd602158e7bc49cee": "\\epsilon\\tbinom n2", "b9e4114d1fedbf8c4eb702422cdeadaf": "\\psi(x)-x \\ne o\\left(\\sqrt{x}\\right).", "b9e45da104a403ee45932d721ea85f91": " \\frac{\\mu_0 l}{\\pi }\\operatorname{arcosh}\\left( \\frac{d}{2a}\\right) = \\frac{\\mu_0 l}{\\pi }\\ln \\left( \\frac{d}{2a}+\\sqrt{\\frac{d^{2}}{4a^{2}}-1}\\right)", "b9e467164dfd41e020ad709667fa1985": "D=(c,f)", "b9e469c6becaa5888d53ecb9f1566f0f": "\\tau\\in S_n", "b9e4700dbf37f251500418f7e1e1643c": "\n\\begin{align}\nU(\\rho)\n&= \\frac {2 \\pi J_1(\\pi W \\rho / \\lambda z)}{2 \\pi W \\rho/\\lambda z}\\\\\n&= \\frac { 2 \\pi J_1(\\pi W \\sin \\theta /\\lambda)}{W \\sin \\theta/\\lambda}\\\\\n&=\\frac { 2 \\pi J_1(k W \\sin \\theta /2)}{ kW \\sin \\theta/2}\n\n\\end{align}\n", "b9e4914805203ca09487b655377c74e2": "\\frac {d\\mathbf{v}_\\mathrm{B}}{dt} =\\sum_{j=1}^3 \\frac{d v_j}{dt} \\mathbf{u}_j+ \\sum_{j=1}^3 v_j \\frac{d \\mathbf{u}_j}{dt} =\\mathbf{a}_\\mathrm{B} + \\sum_{j=1}^3 v_j \\frac{d \\mathbf{u}_j}{dt}. ", "b9e492e4b87b15e0549294d8555ffe17": " \\sum_{n=0}^\\infty (x)_n ~\\frac{t^n}{n!} = (1+t)^x ~, ", "b9e4a8c817ff4dc3009d5a78aec250e2": " y\\ ", "b9e53640fe36b5f6d5911fe071f2f1c9": "\\begin{cases}4n,&n\\equiv2\\pmod 4,\\\\n,&\\text{otherwise.}\\end{cases}", "b9e56bb14e6ebd52d24eabdc4374ecf9": "R(z)=\\frac{\\lambda \\alpha}{1-\\alpha z}. ", "b9e57dc3472906d461a4e09cda11c827": "\\frac {v^2}{2}+ \\int_{p_1}^p \\frac {d\\tilde{p}}{\\rho(\\tilde{p})}\\ + \\Psi = \\text{constant}", "b9e58906604bf0b02e6373431a1eec33": "y=\\wp(u).", "b9e593718e45d5f9b63c929f9c856e29": " \\widehat{K} \\left(X, Z\\right) = \\frac{1}{n m} \\sum_{i=1}^n \\sum_{j=1}^m k(x_i, z_j) ", "b9e5c1f6d80b2e44e1387675901bedf0": "H = c_1 a^{k-1} + c_2 a^{k-2} + c_3 a^{k-3} + ... + c_k a^{0}", "b9e5f900bd482c2fae222bdf5d4fb8b5": " f(i)M^{k+1}(i,j) = \\sum^{N}_{n=0} (f(n)M^k(n,i))M(n,j)", "b9e604ef5f51aef7874e92e2f4511f24": "\\dot{r}=\\frac{\\tilde{F}}{2}\\dot{v}", "b9e6231a6bb976777fe2c1783cb612be": "\n\\mathbf{F} = q_0 \\mathbf{E}\n", "b9e6acb521bc120e81be095c97327582": "(\\pi_n(t); \\; n = 1 \\ldots N)", "b9e6b3768802ca3c844eb529ad00654c": "P_4(x)=1 \\,", "b9e6b473dcb33e428aaf475bc03c8069": "B_{j_1}", "b9e6c2e0dacf187e292c68a742e44eb2": "\\frac{new\\ tempo}{old\\ tempo} = \\frac{number\\ of\\ pivot\\ note\\ values\\ in\\ new\\ measure}{number\\ of\\ pivot\\ note\\ values\\ in\\ old\\ measure}", "b9e6f4d225b1896c2662c08eb0ad71fe": "c_1= \\left[\\frac{i}{2\\pi} \\mathrm{tr} \\ \\Omega\\right] .", "b9e6fd523be59930e8543ce542d7f35b": " H = 1 - \\sum_{ i = 1}^K \\sum_{ j = 1 }^i \\sqrt{ p_i p_j } ", "b9e78ade22ce06124590960d49061e11": " \\frac{1}{q} = \\frac{1}{R} - \\frac{i\\lambda_0}{\\pi n w^2} ", "b9e79ddc223ce2eae7ec37d734c016fb": "t\\mapsto [\\mathbf{x}(t),\\dot{\\mathbf{x}}(t),\\dots,\\mathbf{x}^{(n)}(t)]", "b9e83869c795cf66e6d9d0a894e17be5": "(C)", "b9e83ca4c733a84c1a17b963941c77a9": "\\left\\{ v_{C} \\right\\}", "b9e853702db2a875c295e56409af25d5": "\n\\int\\left(\\nabla^2\\psi\\right)^2\\,dV = \\int\\left(\\nabla\\times \\mathbf{ v}\\right)^2\\,dV.\n", "b9e86052bf0db1c4a455b88d63f43311": "m_e^*", "b9e87a2ccf7871b6ae7f89d2b02565c0": "N ", "b9e8b95daba513d9b4a8fbaedd55afe6": "*F", "b9e8dfade75870725dac7001428a6fb5": "{\\rm d}A = V\\sum_{ij}\\sigma_{ij}\\,{\\rm d}\\varepsilon_{ij} - S{\\rm d}T + \\sum_i \\mu_i \\,{\\rm d}N_i\\,", "b9e939be6d862b2bac1be409fa7ab4b6": "\\sigma^{\\otimes n}", "b9e94569390ed3a2f4a4be3157ecdc3d": "\\frac{\\partial u}{\\partial\\bar{z}}-iz\\frac{\\partial u}{\\partial t} = \\varphi^\\prime(t) ", "b9e95fd5bb336253e70c271ebd42d9b3": "z = r_1 e^{i \\omega_1 t} + r_2 e^{i \\omega_2 t},\\,", "b9e9877cdbb006209f1d4797dc8c487d": "\\rho_A = \\frac{M\\;P_A}{R \\;T_A}", "b9e994cb4d7023c725385744d836b962": "C_2 \\approx", "b9e9ae8f952d4ffd75017c14f9043b08": "a(=\\infty) \\frac a{\\ln a}", "b9e9af7c7713fd0e8a98f2285c285a2d": "T(\\alpha v) = \\alpha T(v) = \\alpha(\\lambda v) = \\lambda(\\alpha v)", "b9e9cbfba44e0cd5bae7694ab5d79b90": "p_i (k)p_\\ell (k)", "b9e9d119fb05a9127bb284f8c5eca949": " a \\triangleright (b \\triangleleft a) = b", "b9ea76d06a2c8c3e7f609f0b883fa0c4": "{(1-10^{-7})}^{10^7} \\approx \\frac{1}{e}. \\,", "b9ea9ada867405e843fa07e4d8d8cf7d": "\n\\delta_d + \\delta_c = \\ln \\left (\\frac{I_{1d}}{I_{2d}} \\right )\n+ \\ln \\left (\\frac{I_{1c}}{I_{2c}} \\right ) \n=\\sum \\left (\\beta_i^* +\\alpha_i \\right )\n\\left (\\sigma_{i2} - \\sigma_{i1} \\right )\n= \\sum_i \\beta_i^* \\left (\\sigma_{i2} - \\sigma_{i1} \\right)\n+\\sum_i \\alpha_i \\left(\\sigma_{i2} - \\sigma_{i1} \\right)\n", "b9eade921d6b57ef1190199aa5a1bffc": "\\mathrm{mul}_c ", "b9eb0bde49ead948ede78e1d5b4fc49d": " \\{ v_0, v_1, v_2, v_3, v_1 \\wedge v_2, v_1 \\wedge v_3, v_2 \\wedge v_3 \\} ", "b9eb2112fe156a61170bdb1fef3f3e0e": " \\psi^{-}=\\phi", "b9eb34d4f0b8f6ed99e4b6f61ce48670": "P_0 \\, ", "b9ebb9a7309f6a1efad7a1b232bc28b4": "_1^1\\text{S}_1 + _2^0\\text{S}_2 \\rightarrow {_1^0}\\text{P}_1 + {_2^1}\\text{P}_2", "b9ebfa4c9632a82653130075e6c7a37e": "(\\mathit{LR})\\qquad\\frac{\\Box p\\to p}p", "b9ec6d34d662b4a7a1c7a7879ca864b2": " (1 + x)^{\\alpha} \\approx 1 + \\alpha x.", "b9ecd49b3245fff5c8a6ccb8e4a645be": "[x_i, x_{i+1}]", "b9ece18c950afbfa6b0fdbfa4ff731d3": "T", "b9ed25240b5e9e247c863467c6c72920": " J_1 =\na_1 \\delta^3\\left ( \\vec x - \\vec x_1 \\right ) \n\n", "b9ed436a02dbf642f176d45ccf486f7e": "\\bigl|\\!\\operatorname{E}[X_n]\\bigr|\\operatorname{P}(N\\ge n)\n=\\bigl|\\!\\operatorname{E}[X_n1_{\\{N\\ge n\\}}]\\bigr|\n\\le\\operatorname{E}[|X_n|1_{\\{N\\ge n\\}}],\\quad n\\in{\\mathbb N}.", "b9ed47966f26bba6280ecb8f87861542": "\\frac{E_p}{E_f}=\\frac{m_f}{m_p}", "b9ed6d47d6f8ced44cf67dea5a664328": " {\\Delta}_{\\rho} ", "b9ed795e8a2f07d853bf50e8dab08f6a": "Q(x) = P ", "b9eda55e1abb4ebb36d2ed32d78673c9": "T:I\\to \\mathbb{N}", "b9edc9fdf9219ea792c0123fa31a1781": "\\frac{\\Delta PE}{\\Delta t} = m g \\frac{\\Delta z}{\\Delta t}", "b9ee4c7f0b92bc756ea3c1061499262e": "\\mathbb{Z}_p \\!\\,", "b9ee64f3fbd5bb177eb14aa7d03cf3c2": " \\mbox{DL} = (y\\bmod 4\\times 2 + y\\bmod 7\\times 4 + c \\bmod 4\\times 2) \\bmod 7, ", "b9ee891073166295b76fbd6817423f2c": " {\\tau}^{0.5} = {{a}{|\\gamma|}^{0.5} + b^ {0.5}} ", "b9ee8b12053d6449dc19ddd808c51844": "\\textstyle p_k / q_k", "b9eec145b65de6bf7a11add7d60a9882": "b(s)=(s+1)\\left(s+\\frac{n}{2}\\right).", "b9eee2e7106d66fadaf15e55bec28f48": "\\mathbf{v} = \\left[ \\begin{matrix} \\rho & \\angle \\theta & \\angle \\phi \\end{matrix} \\right]", "b9ef4de40a6f3b663cf5868726351097": " {\\mathbf E}({\\mathbf r},t)=E_0 {\\hat{\\mathbf z}} \\cos(ky-\\omega t), \\ \\ \\ {\\mathbf B}({\\mathbf r},t)=B_0{\\hat{\\mathbf x}} \\cos(ky-\\omega t). ", "b9ef52bd85089a8885b434ddc6d689d8": "\\gamma(E)", "b9efa03dee28eeab1564acaed3f0e627": "WHIS = \\frac{\\alpha_i}{\\beta_i} ", "b9eff7e0d36f577f46ab154b7a5ef6ee": " a_1=\\frac{1}{2\\pi i} \\oint_C w'\\, dz. ", "b9effa7b6f409f5a21df7cd2f5a5801d": "\\|A\\|_2", "b9f0f55977a0da757719ad68ade88196": "x' = x-x_0", "b9f1029503e19706e0328165c4f9ec65": "\\alpha + \\beta = 1", "b9f113081a424b059ae1922886519977": "e^{i (\\alpha + \\beta)} = \\cos (\\alpha +\\beta) + i \\sin(\\alpha +\\beta)", "b9f1276034ffa4171c37839db1ba85ea": "\\hat B", "b9f1700a4afa63d337c7e5d244ed85b0": "\n\\begin{bmatrix}\n\\boldsymbol{I}_m & \\boldsymbol{V}_1^{(b)}\\\\\n\\boldsymbol{W}_2^{(t)} & \\boldsymbol{I}_m\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1^{(b)}\\\\\n\\boldsymbol{X}_2^{(t)}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{G}_1^{(b)}\\\\\n\\boldsymbol{G}_2^{(t)}\n\\end{bmatrix}\\text{,}\n", "b9f1713d8f0d758373389dbda71f979a": "\\tbinom55", "b9f182e5cd7aa4ed9146f5bf56eaf9f1": "|F (k)| ", "b9f1a1e885f2fff2c3658d12d01c4756": "x^* = \\operatorname{prox}_{\\gamma R}\\left(x^*-\\gamma\\nabla F(x^*)\\right).", "b9f1a920e6f59cd03bbc4084848925cf": "\\mbox{IF}=(1+f(t))\\cdot \\sin(\\omega_{I} t)", "b9f1b7817dc53359d3099af5f31253ed": " n_{j+1} - n_j \\leq n_j - n_{j-1}, \\qquad \\mbox{for all } j = 1,\\ldots,q-1. ", "b9f1d681ab0fd22acc70125a6a3d4c35": "f_2(f_1(x)) \\rightarrow g(x)", "b9f20153eb1493733dd41821ad454918": "\\varphi \\in V^*", "b9f233e3d7bf6cc398e4a1c458bfde98": "R_b = {1 \\over T_b},", "b9f244deaed1ceefca1838fc80fb12d5": "\\displaystyle\\frac{\\deg(V)}{\\hbox{rank}(V)} < \\frac{\\deg(W)}{\\hbox{rank}(W)}", "b9f2b42c0b3441c46df2397518164e3b": "I_\\mathrm{e} = c w,", "b9f2fb09fc41226eb720ce0df4ce760d": "f\\colon X \\to\\mathbb{R}^{+}", "b9f38000287caafda39194cd93dc47e2": " l_i ", "b9f3b86436b81f16b5f81ce791f44b4c": "0.\\overline{72}", "b9f4673bed0bba054e8c635722b08bbd": "\\, m, \\, \\, m \\ge 1 ", "b9f46ee8b891d1407d20f6ede9ce4bed": "ax^2+bx+c\\,\\!", "b9f4ced174aa86d67e8017a75e92b520": "f(a-)", "b9f51e88b56191174ee1ed5140d8b9d0": "Aa+Bb+Cc=0", "b9f5302d907d3e217c15c089389b21fb": "(7)\\; h_j=2.4\\;m", "b9f5405f61ed356725963d91c5755cd0": "\\textstyle \\{ (1,2),(1,3),(3,2),(3,4),(4,3)\\}", "b9f5a1748162dc797e590c9286cbb67f": "e^{-\\frac{z_i q \\varphi}{k_B T}}", "b9f5fe890a1339290fecb527094d9837": "n_{\\rm air} < n_{\\rm water} < n_{\\rm oil}", "b9f6400617f1a6ee27428c6640211bda": "s_{A}=\\sum_{i=1}^m \\log x_i; s_K=\\min_i x_i", "b9f6d459ddfe35a633fa3863855869b9": "y_{\\bar{q}}/y", "b9f70f95f10ee9b762f731390cf2c569": "h\\rightarrow 0", "b9f72ab3554d2d4630ec2a9ac86d7290": "disc(\\mathcal{H}) < 2t", "b9f74f31a6dcadb7af390894f688d992": "\\mathfrak{h} < \\mathfrak{g}", "b9f75335491983e9512097aea2ae548c": "\\scriptstyle{p(\\sigma^2) \\;\\propto\\; 1 / \\sigma^2}", "b9f75b5d87bbaec04b948e176e693ed6": "\\vec{a}(x,y,z) = a_x \\mathbf{\\hat{x}} + a_y \\mathbf{\\hat{y}} + a_z \\mathbf{\\hat{z}} ", "b9f76805020cbe44e6aa871f657bbdb6": "\\tfrac{k-1}{2}", "b9f7700a3a18704836b9238e0564c22b": "\\Delta\\mathbf{w}_{n-1}=\\mathbf{g}(n)\\alpha(n)", "b9f7d1dabef4ae33a6cdfe7928b07bcd": " \\alpha(p_1, \\, p_4) = \\displaystyle\\sum\\limits_{p_2 \\in A_2,\\, p_3 \\in A_3, \\, p_5 \\in A_5 } f(p_1,\\, p_2,\\, p_5 ) \\cdot g(p_2, \\, p_4)", "b9f811960de58b613159fea9313caa0e": "z=2ae^{it}+ae^{-2it}", "b9f82c822ee60a2cc3b9d8b12e04125e": "I(\\omega_n) = \\log \\left(\\frac{1}{P(\\omega_n)} \\right) = - \\log(P(\\omega_n)) ", "b9f83c1e059749e6b1dd302c99222e27": "\n\\int_0^1 \\cdots \\int_0^1 \\left(\\prod_{i=1}^k t_i\\right)\\prod_{i=1}^n t_i^{\\alpha-1}(1-t_i)^{\\beta-1}\n\\prod_{1 \\le i < j \\le n} |t_i - t_j |^{2 \\gamma}\\,dt_1 \\cdots dt_n \n", "b9f849fd5a982ebbfc046dab77beedd5": "X=\\mathbb{R}", "b9f883999a50d85c710d3610092eea3a": " n = child_1 + 1, if child_1 = child_2 ", "b9f8f5a47b5ad432d2fdd181c94a495d": "\\mathbf{x}\\in \\mathcal{Z}_{+}^J ", "b9f9030a48b0b01a2e459f0a2d5be784": "\\phi^*h=g", "b9f9052cc6177bc659a9685b02beefc8": "\\mathbf P'_k=\\tfrac{k}{n+1}\\mathbf P_{k-1}+\\left(1-\\tfrac{k}{n+1}\\right)\\mathbf P_k", "b9f90d7df71228bf749c5088d422964b": "\\mathbf{x} = (x^1,x^2,x^3) ", "b9f9673072fdabf06909378b2ff01462": "\\{x|\\Lambda \\le c\\}", "b9f98f47e1371ed8b195cd8ff6237b9f": "A = 2A_0\\!\\left(\\frac{h}{2}\\right) - A_0(h) + O(h^2) .", "b9fa0b07f90055c2e160f15a880d945b": "l_\\phi", "b9fa1fb3a4d24e2fd62e8c19cc0cb429": " t_p = \\mathrm{thickness \\ of \\ plate} ", "b9fac11b0014a523c125cafca25c3dd6": "P_D = 2\\rho\\eta\\bar{J}^2", "b9fb557e9420c3b5356a86acf1110829": "x'_{s'}", "b9fb863e555166c3f3a1b11e5287c5d6": "\\mathcal{Z}(M)=\\{0\\}\\,", "b9fb8df4a6edc847fbf6f694fd3ff292": "\\mathcal{C}\\in\\mathbb{N}[\\mathfrak{A}]", "b9fb9509fe5af60d15f119556c1285da": "\n\\mu_{i}=\\frac{\\ell-2i+1}{\\ell-i+1}\\binom{\\ell}{i},\n", "b9fc176742c04df91dcdd89eb6680692": "\\Delta=\\frac{D-2}{2}", "b9fc19dbb19db42216a39c582c497345": "g_b = g - \\mu\\sum_{i=1}^m \\frac{1}{c_i(x)} \\nabla c_i(x)~~~~~~(3)", "b9fc27ec87e7c52eeff4237b36f0daba": "y_2=x_1+z", "b9fc7f671f31d6dd71c977e8914145fb": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\int_{D(t)} F(\\vec{\\textbf x}, t) \\,\\mathrm{d}V = \\int_{D(t)} \\frac{\\partial}{\\partial t} \\,F(\\vec{\\textbf x}, t)\\,\\mathrm{d}V + \\int_{\\partial D(t)} \\,F(\\vec{\\textbf x}, t)\\, \\vec{\\textbf v}_b \\cdot \\mathrm{d}\\mathbf{\\Sigma}", "b9fd2910da8296ee7012a189bd112343": "t_P \\equiv \\sqrt{\\frac{\\hbar G}{c^5}}", "b9fd3635fbfb0140d06cb6485d8cfbe6": "A_v =\\begin{matrix}\\frac {v_{out}}{v_{S}} = \\frac {R_L}{r_E} \\approx g_m R_L\\end{matrix} ", "b9fde6e1a0a2715c53c8c511b4b39610": "\\otimes(A,B,A)", "b9fe3a0e0c6df6354539c0de59f28d1d": "R_H=\\frac{(p-nb^2)}{e(p+nb)^2}", "b9fe3b30add7abc9d6250e8ef982f1c5": " \\mathrm{atm} \\,", "b9fe5c99e9e249d0e066f4781cd7cc34": "n(z,\\rho)=\\frac{2N}{\\pi}\\arcsin\\left(\\frac{z}{\\rho}\\right)", "b9fe787382e7dd0bc05fbcaef4277d40": " m = m \\cdot U \\cdot U^{-1} ", "b9feed48ea508903a1fc6d3bb247944c": "y(0)=2.", "b9ff383f9f73fbbe1c5a1a7b4ac57754": "\n\\nabla \\cdot \\left( \\mathbf\\Sigma_i \\nabla v_i \\right) = \\chi \\left( C_m \\frac{\\partial v}{\\partial t} + I_{ion} \\right)\n,", "b9ff9540826918b3a5656774fd03ad1b": "\\scriptstyle Y\\,=\\,\\varphi(X)", "ba0028f56b496ebede399e396c60d4fb": " E^{\\pm} ", "ba005bd281844e1892f1f9e1df6670d6": "xy \\leq \\frac{1}{2},", "ba009552f99a8ba57dbe7d7414df44e0": " \\operatorname{E}(R_p) = w_A \\operatorname{E}(R_A) +\nw_B \\operatorname{E}(R_B) = w_A \\operatorname{E}(R_A) + (1 - w_A) \\operatorname{E}(R_B). ", "ba00ff13d8beee9db2785c5003ff2d25": "k_i = f\\left(y_t + h \\cdot \\sum_{j = 1}^s \\beta_{ij} k_j, t_n + \\alpha_i h \\right)", "ba013b3b37124c8e0eadfcb925a797a2": " R = \\frac{ A_r }{ A_l } ", "ba013be6b3bd2b8052436a39b501c861": "\\tau = \\frac{\\lambda}{c},", "ba014edfbac4c12564ce978b4b147e42": "\\psi=\\psi(\\zeta,\\eta,\\lambda)", "ba0188b2f77b1c1533ec9aa77b0ebca6": "\\Theta \\subset \\mathbb{R}^{k}", "ba0284b4ba78980af8411a7d96e1f435": " \\frac{\\mathrm{d}N_B}{\\mathrm{d}t} = - \\lambda_B N_B + \\lambda_A N_{A0} e^{-\\lambda_A t} ", "ba02b3ca1a848212b89332197d07f672": "p_3 = C\\rightarrow cC", "ba033bd31d8d967f0e5200b97b0dff29": "\n\\begin{align}\nI(d) = I_0 * e^{-d/\\Lambda(E)}\n\\end{align}\n", "ba0385d91e6786b776b57151a02b2c57": "C(s\\otimes n)=C(n\\otimes s)=C(n\\otimes n)=n.", "ba03fc54fb1d0add9285288f35feee7e": "Q^2", "ba0435236656b24c084dce6efe6ca7d8": "x \\cap y = \\varnothing.", "ba046557a0be81dea60516f4bce86f12": "\\ P'(q) = \\sum_{x=q}^n C(n,x) \\cdot (1/6)^x \\cdot (5/6)^{n-x}", "ba04b0f0cd224cf41613acf66f6a6a48": "g_2 = \\frac{n M_4}{M_2^2}-3.", "ba04edde761bb8001c30a5bb0c35016a": "T_1,\\ldots, T_e", "ba04f6865e9830cc88fc8a05abf933a6": "df = \\nabla f \\cdot d\\mathbf r \\quad \\Rightarrow \\quad df = \\nabla f \\cdot \\sum_i \\mathbf e_i \\, dq^i", "ba04f72440c2ebf0eb7361b79eb3f198": "\\epsilon^{-1}Q(x)", "ba0593b3db2fa8535b077516f4b0d70b": "\\textstyle l", "ba05f1c33890f84fb2b70f4a267a1414": "\\boldsymbol\\omega_p = \\frac{\\ mgr}{I_s\\boldsymbol\\omega_s}", "ba0618f368b0295dbd7a5218d4dbd13a": " Y = |Y|e^{j\\arg(Y)}. ", "ba0685fe6069237f8d60a1b600a9c1cf": "\\hat{\\mathbf{h}}(n+1)=\\hat{\\mathbf{h}}(n)+\\mu \\mathbf{x}(n) \\, e^{*}(n)", "ba06b75bbf9629def26a44342e34d797": "\\log_b(\\sqrt[y]x) = \\log_b(x^{\\frac{1}{y}}) = \\frac{1}{y}\\log_b(x)", "ba070f1111fecd4e69888509378c300a": "h(y) dy = {g(x)dx}", "ba075dff8e28a6851ba220f95abfd929": "\\varphi\\sim", "ba07bae672c94490b800b7108dcb0f2b": " s^2 = n p ( 1 - p ) ", "ba07bd079d8a9fc8e9536c8fe5692506": "h_{a,b}(x) = ((ax + b) \\bmod 2^w)\\, \\mathrm{div}\\, 2^{w-M}", "ba07f14e4a861871a3e4228826f7179f": "\\Bigg(\\frac{2}{p}\\Bigg)_4 =\\left(-1\\right)^\\frac{b}{4} =\\Bigg(\\frac{2}{c}\\Bigg) =\\left(-1\\right)^{n+\\frac{d}{2}} =\\Bigg(\\frac{-2}{e}\\Bigg), ", "ba080d615fd9c99a4a1025b42a894c58": "x_i = \\frac{y_i}{1 + \\sum_{j=1}^{n-1}{y_j}} \\quad i=1, \\ldots,n-1 ", "ba081c66c293fd790d4ebbddd16c5965": "f_i(q_1,\\ q_2,\\ q_3,\\ \\dots,\\ q_N,\\ t)=0\\,\\!", "ba083cb38352865a6242d0e5ff5507cd": "q(t) = Cv(t) \\ ", "ba08b339c7f0b53c99e51fb97143a970": " - \\varepsilon < \\frac{a_n}{b_n} - c < \\varepsilon ", "ba08fa5a4e5c3ef76385b1ecd26f509f": "n \\le M - 1", "ba09284e48396dad2af91976129d330a": "\\textstyle l_1-1", "ba0965330d709297a9d8b07f305169cd": "1/3", "ba09723afc7dc13bdf1f9d4db20d4f26": "i=0,1", "ba09824b15f1c4485c61c44c34b3bc8d": "\\textrm{Rad}(\\mathfrak{g})", "ba09f421a64355b94e3cfe736e501f3f": " K =\n\\frac{\\Gamma_p\\left(\\frac{\\nu+n+p-1}{2}\\right)}{(\\nu\\pi)^\\frac{np}{2} \\Gamma_p\\left(\\frac{\\nu+p-1}{2}\\right)} |\\boldsymbol\\Omega|^{-\\frac{n}{2}} |\\boldsymbol\\Sigma|^{-\\frac{p}{2}}.", "ba0a067c29c2d17025e2663753e2d7bf": "t(n) = 3 t(\\lceil n/2\\rceil) + cn + d", "ba0a13356ae56f1aed63dce9ef8260db": "\\operatorname{L}(\\theta;\\mathbf{X}) = p(\\mathbf{X}|\\theta)", "ba0a6bac3505cf5a6986ca3efaa363a0": "\\langle\\mathbf{A}\\rangle=\\sum_{jk}{\\epsilon_i}^{jk}A_{jk}.", "ba0a7ff0a200e867c60df7bff9d72894": "\\begin{bmatrix} \\dfrac{1}{C}e^{\\eta_1} \\\\ \\vdots \\\\ \\dfrac{1}{C}e^{\\eta_k} \\end{bmatrix} =", "ba0b7f56ec4d67f2973890cf97682870": "\\Phi_\\epsilon(f)(x)=\\int_{\\mathbb{R}^n}\\varphi_\\epsilon(x-y) f(y)\\mathrm{d}y", "ba0b8f8c6a2a940b6eccae1efc9334be": " L(n!)^3\\equiv 1 \\pmod{p},", "ba0ba844fd2bb7a2da5a649311a104b7": "\\; \\sum _i E_i = I", "ba0c0d65bab991cfa5ea1d743d4b0c5d": "\n \\begin{align}\n \\cfrac{d}{dt}\\left( \\int_{\\Omega(t)} \\mathbf{f}(\\mathbf{x},t)~\\text{dV}\\right) & = \n \\int_{\\Omega_0} \\left[\\lim_{\\Delta t \\rightarrow 0} \\cfrac{ \n \\hat{\\mathbf{f}}(\\mathbf{X},t+\\Delta t)~J(\\mathbf{X},t+\\Delta t) - \n \\hat{\\mathbf{f}}(\\mathbf{X},t)~J(\\mathbf{X},t)}{\\Delta t} \\right]~\\text{dV}_0 \\\\\n & = \\int_{\\Omega_0} \\frac{\\partial }{\\partial t}[\\hat{\\mathbf{f}}(\\mathbf{X},t)~J(\\mathbf{X},t)]~\\text{dV}_0 \\\\\n & = \\int_{\\Omega_0} \\left(\n \\frac{\\partial }{\\partial t}[\\hat{\\mathbf{f}}(\\mathbf{X},t)]~J(\\mathbf{X},t)+\n \\hat{\\mathbf{f}}(\\mathbf{X},t)~\\frac{\\partial }{\\partial t}[J(\\mathbf{X},t)]\\right) ~\\text{dV}_0 \n \\end{align}\n", "ba0c349b78141e809860adb506f4a441": "d_{1,1}^{2} = \\frac{1}{2}\\left(2\\cos^2\\theta + \\cos \\theta-1 \\right)", "ba0c637c3412d6c20ccc779f993342b1": "(1-f_v(x))-t_v(x)", "ba0c828dc2dce02eb62bc52aac41aa8f": " d(h_{1}) \\cdot h_{2} = h_{1}h_{2}h_{1}^{-1} \\! ", "ba0ca4ef54810b50be010ad192094bfc": "\\zeta_K (s) = \\sum_{I \\subseteq \\mathcal{O}_K} \\frac{1}{(N_{K/\\mathbf{Q}} (I))^{s}}", "ba0cae8cd75cf18ea1490755fe18b04e": "_SM = S \\otimes_R M", "ba0cc1294aea939170683255b6d89565": "(u_{\\omega})_{\\omega \\in \\Omega}", "ba0d324edec72d5902607bd47820d189": "h : V \\longrightarrow E[p]", "ba0d329c7fd08886fbb37b41d2b50cb5": "r_i= y_i - f(x_i, \\boldsymbol \\beta) ", "ba0d3e525aa243331668487233a94a1a": "1/\\lambda^5", "ba0d501129e621abc42185dc86f414a9": "\\Pi_2", "ba0d6773c9cf604f95f3195ae6894358": "\\langle y, G(\\lambda) z\\rangle = \\left \\langle y, (O-\\lambda I)^{-1} z \\right \\rangle = G(y, z; \\lambda),", "ba0d75ad021ab7de55b9b760fcd276de": "(a + b) \\cdot c = a \\cdot c + b \\cdot c", "ba0d8bb3a7b9d14aa7496d59f463efb7": "\\mathbf{p}", "ba0d9b4fcc1920dfd664e420ad0b7c8b": "(i,j) \\in [q] \\times [q]", "ba0dae8e2b4820981b8484e766b1ae80": " \\{ 3,3,5 \\} ", "ba0dffbf721ff09756465187f3aca372": "3\\times 3", "ba0e082710581d68da83a215fac32d80": "Nn^2", "ba0e825588550edfdae8b934514c97ad": "\n\\vert E_{\\pi/2}(r)\\vert \\, = \\, {1 \\over \\pi\\varepsilon_\\circ c \\, r}\n\\sqrt{{ P_{avg} \\over 160}} \\, = \\, \n{9.48 \\over r} \\sqrt{ P_{avg} } \\quad (L \\ll \\lambda /2)\\, .\n", "ba0ee9f88a765b4718d9fedf9d3640ab": "\\zeta=r \\cdot\\epsilon^\\theta", "ba0f138a28b843f3d89e8f9fb3d66628": "\\lambda(y) = \\lambda^{\\star}/(1+\\exp(-X(y)))", "ba0f412e72d4285c9176e0570ff7a0bd": "\n\\langle \\Psi(t) | \\hat{A} |\\Psi(t) \\rangle = \\sum_{\\alpha, \\beta=1}^{N} c_{\\alpha}^{*} c_{\\beta} A_{\\alpha \\beta} e^{-i \\left ( E_{\\beta} - E_{\\alpha} \\right )t / \\hbar }.\n", "ba0fc2aeb51a2ff250da5d9d63b223ea": " S^3\\cong\\operatorname{Spin}(3)\\cong\\operatorname{SU}(2)\\cong\\operatorname{Sp}(1) ", "ba106b32b6a8c37852f0f2aaac5d9d59": " \\log S = B - KI \\,", "ba10a8a75c092a8f5506bc0840904b2d": "\\mu_{x} \\left( Y \\setminus \\pi^{-1} (x) \\right) = 0,", "ba10b7ea103fd308b117a65e7d08f791": "u_n = u_0 = W_0\\, W_1 = \\frac{\\pi}{2}", "ba10d752289f8b043860dac430270bb0": "x_{N+2}", "ba10d7872db5c3b28760bd959f892f90": "\\|T(t)\\|\\leq M{\\rm e}^{\\omega t}", "ba110e2e35ad8f4913fde324a5148c18": "\\beta_e", "ba1129f61370c1d00d6d847005afd618": "\\theta = \\tan^{-1} \\left( \\frac{1}{\\mu} \\right) \\,", "ba1150bd532505bdf83a3e92a1df9f84": " \\frac{1}{4} \\oint_{|z|=1} \\frac{4 i z}{z^4 - 6z^2 + 1}\\, dz = \\oint_{|z|=1} \\frac{i z}{z^4 - 6z^2 + 1}\\, dz.", "ba115b5bac0594a010ddd35135f50e55": "\\displaystyle{|([L,\\delta_h]g,h)|\\le A\\|g\\|_{(1)}\\|h\\|_{(1)},}", "ba11e03d47e48dbf58df9243292fb93e": "H_{3}=H_{g} \\,", "ba12029129953be50d2a67ca7c39dfd1": "|k|\\leq N", "ba1209a74dfc5caf0fb6d72dc39e2cbe": "\\left(\n\\frac{16}{27}\n\\right)^{\\frac{1}{3}} \\approx 0.840", "ba126040f3526ec46d9f5d55b1d2ef7a": "CE = \\%C + \\frac{\\%Mn}{6} + 0.05", "ba12910406a84332d3fcd9f1f1ee3682": " SNR_{transform} = \\frac{w^TX^TCXw}{w^TX^TNXw},", "ba12d76598447362ce6c90e184c3b696": "l^a\\partial_a\\,=\\, \\Big(1\\,,\\frac{F}{2}\\,,0\\,,0 \\Big)\\,,\\quad n^a\\partial_a\\,=\\,\\Big(0\\,,-1\\,,0\\,,0 \\Big)\\,, ", "ba12fb03e8057bad75518fcbee8f7bf6": "A_{\\mathfrak{p}} \\to k(\\mathfrak{p})", "ba1353aafd82f719ad44575be75f5de0": "A = b h ", "ba1366c23927c5ce3380316f2aac4deb": "C_{M} = \\frac{\\Phi}{\\phi_{M1}-\\phi_{M2}}", "ba1369f86d338ff2b361ff54ae77749f": "a_n\\leq a_m", "ba13e11c8cfda15db18cace103b0b78b": "a = \\arctan\\left\\{\\frac{2\\sin\\alpha}{\\cot(c/2) \\sin(\\beta+\\alpha) + \\tan(c/2) \\sin(\\beta-\\alpha)}\\right\\},", "ba1406065eb0834926b2731d1ba1cb89": "\\ell_{(M,\\varphi)}", "ba1430cdb25f20380bec6ae909417ba7": "\\mathrm{d}{\\omega} = \\sum_{i=1}^n \\frac{\\partial f_I}{\\partial x^i} \\mathrm{d}x^i \\wedge \\mathrm{d}x^I.", "ba1446196eb5b2dcba0347b0680b6911": "B = B_+ + B_- + \\delta_+ \\delta_-", "ba14478ad11baa5f1887036ddb73f2da": "\\mathcal{H}il", "ba14596cbd44ac3d0152d2a58b6610c6": "\\mathrm{tr}(\\lambda_i \\lambda_j) = 2\\delta_{ij}", "ba147e769d1fa5e8fea3fa3716ad43c2": " P(x,y) = 3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y", "ba1564cbe2a3a58562b0711273e723cf": "x_i^*", "ba161ea992dec5fa221a50de2a82b67d": " Cxy \\rightarrow Cyx.", "ba16948cd108422ee3b4755d263b5c8c": "\\displaystyle{E=mc^2}", "ba169516a48bea9e759c9586b827d9ff": "x \\,", "ba16baa1ff18ebd5a731db7eed1b9820": "\\varepsilon = \\left[23.4393-0.013\\left(\\frac{D}{36525}\\right)-2\\cdot 10^{-7}\\left(\\frac{D}{36525}\\right)^2+5\\cdot 10^{-7}\\left(\\frac{D}{36525}\\right)^3\\right]\\mbox{ deg}", "ba16c12fe13857ce8d08abd6c5433f8e": "v_\\phi=v_\\phi(r),", "ba170ad56198bc8b54f87215d0c2b36e": " S_{ab} = R_{ab} - \\frac{1}{4} \\, g_{ab} \\, R = G_{ab} - \\frac{1}{4} \\, g_{ab} \\, G", "ba17557091f2c392f3d2853d672f868a": "a^2 - b^2 = (a + b)(a - b)", "ba175a66d5dacdb8fc85fa29fa9281f1": "m = t_1+t_8+t_9\\ ", "ba1783f6c03278e39cce9fbdcc4944ca": "\\mbox{BFD} = f \\left( 1 - \\frac{ (n-1) d}{n R_1} \\right). ", "ba1787377e133906dd0c2735f0a1cd1f": "\\widehat{f_p}(a)", "ba17e5404c00e4ebaacba6e90c90361a": "x + 3y + 2z = 0 \\;\\;\\;\\;\\text{and}\\;\\;\\;\\; 2x - 4y + 5z = 0", "ba1876099732f9763debc3f370e08a01": "\n\\begin{array}{lcl}\nminimize: V(\\vec w, \\vec \\xi) = {1 \\over 2} \\vec w \\cdot \\vec w + CF \\sum{\\xi_i^\\sigma} \\\\\ns.t. \\\\ \n\\begin{array}{lcl}\n \\sigma \\geqq 0;\\\\\n \n \\forall y_i(\\vec w \\vec x_i +b) \\geqq 1-\\xi_i^\\sigma;\n \\end{array}\n\n\\\\\n\nwhere\\\\\n \\begin{array}{lcl}\n b\\ is\\ a\\ scalar;\\\\\n \\forall y_i \\in \\left \\{ -1,1 \\right \\};\\\\\n \\forall \\xi_i \\geqq 0;\\\\\n \\end{array}\n\\end{array}\n\n", "ba18cabbd8fabd5d0f1af73fa3666d54": "\nM_{i_1} \\oplus M_{i_2} \\oplus \\cdots \\oplus M_{i_d}\\,\n", "ba191367582f630bd68a2bec58ab6edf": "f(\\vec x,\\vec x)=0", "ba19168854afbf8161781c080087a7a0": "\\frac{e^x}{\\cos x}=1 + x + x^2 + {2x^3 \\over 3} + {x^4 \\over 2} + \\cdots.\\!", "ba19189f7cb45e02e726761c340ad8d6": "S_{n-1}", "ba19d312896ccd189b444da7c031bdca": "\\int_a^b\\alpha f(x)+\\beta g(x)\\,dx=\\alpha\\int_a^bf(x)\\,dx+\\beta\\int_a^bg(x)\\,dx.", "ba19f7c5a1eda0919e470c57ea1eab19": "\\mathbb{R}^2 = \\mathbb{R}\\times\\mathbb{R}", "ba1a159999ac59295921d2ce13953e1e": "\\bigcup_{s\\in S} f_s(K)=K. \\,", "ba1a1cc77f82f898b8eab374dc422c30": "C_{p,m}=C_{V,m} + R=\\frac{5}{2}R", "ba1a8f0fa12af7350b307e63adc2a6f6": "H(f)_{ij}(\\mathbf x) = D_i D_j f(\\mathbf x)\\,\\!", "ba1ac9170a37743433ec1aac5e77e797": "\n\\rho \\ \\stackrel{\\mathrm{def}}{=}\\ u^{2} + v^{2}\n", "ba1affd65bf79ce189a6157d1699978d": "E_J = {\\Phi_0 I_c}/{2\\pi}", "ba1b25f8bf4176fb3f4a48978f516e52": "\\begin{matrix}\\Delta L = 0, \\pm 1 \\\\ (L = 0 \\not \\leftrightarrow 0)\\end{matrix}", "ba1b28e01ad39a82ae3913415143b9db": "\\mathrm{d} U = \\delta Q -p\\mathrm{d}V-\\delta W^\\prime", "ba1b9bb5fdce5ef7a645ec7628bbcbdb": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "ba1ba0ac3c4b60d4685d9ec9ea540ba4": "4.49\\approx\\frac{3\\pi}{2}", "ba1c5e25522bb6546176a5e8e9a90fde": " |T^{-1}| \\le \\tfrac{1}{1-q} |S^{-1}| \\quad\\text{where}\\quad q = |S-T| \\, |S^{-1}|. ", "ba1c61b92889dea3952d604081b46c30": "a \\in V, u,v,x,y \\in V^*,\nux,vy \\in V^+", "ba1cdfe91b56f482c68bfce0a28b7746": "u \\in \\mathbb{R}^d", "ba1d2adeb5aa0dc3608d7ad0456f0586": " \\limsup_{\\varepsilon\\to0+} \\frac{ |w(\\varepsilon)| }{ \\sqrt{ 2\\varepsilon \\log\\log(1/\\varepsilon) } } = 1, \\qquad \\text{almost surely}. ", "ba1d4700d066e215a9ee4446977c6b5e": " \\quad Q(\\mathbf{r_1},\\mathbf{r_2},\\,t) = - \\frac{\\hbar^2}{2m} \\frac{(\\nabla_1^2 + \\nabla_2^2) R(\\mathbf{r_1},\\mathbf{r_2},\\,t)}{R(\\mathbf{r_1},\\mathbf{r_2},\\,t)} ", "ba1d4a8b565027c7e786eafac2d18499": "\\scriptstyle S ", "ba1d53582ea22b31d461fdeda49e219c": "\\mathrm{Hom}_M(Q, B)", "ba1d61dcec5080b41ac446ee961f709c": "V = \\mathbb R^n", "ba1d749363c96983e4742a0e2923e732": "\\scriptstyle f_i", "ba1daa95f69a9ea73a59a31b4718045c": " \\operatorname{de-lambda}[V] \\equiv V ", "ba1dc5576cf82405452f474a919083dd": "i_c(U_g)", "ba1e31a591edcdfce264baa6fd8221c0": "\\epsilon_{t}\\stackrel{\\mathit{iid}}{\\sim}WN(0;1)\\,", "ba1e3938e58337806e860777e4ce3ad0": " \\beta=0 ", "ba1ec05cece0590164b6d0f26459cabd": "P.Y", "ba1ef213db9e66f4ab829fc64114a1cc": "\\frac{a}{b}.", "ba1f3051fe01b8b97d244dfb7c3411bc": "0\\leq\\sigma", "ba1f4ae1697f8d0a884e867962902adb": "corr[X_2,X_3] = \\left (\\frac{2 \\times 4}{(10+2)(10+4)} \\right )^{\\frac{1}{2}} = 0.21821789023599242. ", "ba1fe7a35972211dd6a472511c2baa54": "[f,P]:\\Gamma(E)\\rightarrow \\Gamma(F)", "ba20d0d7bc7eb8fd2693660190209256": "dU = \\delta W + \\delta Q\\,\\!", "ba20d85f7fcc992ec26534b024215072": "N \\approx \\frac {f^2} {c}\n\\frac {D_{\\mathrm F} - D_{\\mathrm N} } {2 D_{\\mathrm N} D_{\\mathrm F} } \\,.\n", "ba20ee00ffaa162b49a7c55b825cc243": "\\scriptstyle h\\in V^*\\otimes V^*", "ba21355c1c59f317f2414ed3da0709b4": "{\\mu_i}'_r=\\left[\\frac{(\\lambda_i/\\delta)^{t_i/\\mu_i}}{\\Gamma(\\nu)}\\sum_{k=0}^r \\binom{r}{k}\\left[\\frac{\\ln(\\lambda_i/\\delta)}{\\mu_i}\\right]^{r-k}\n\\frac{\\partial^k\\Gamma(\\nu+t_i/\\mu_i)}{\\partial t_i^k}\\right]_{t_i=0}.", "ba21515035e4e9e8ba1aba08c16d5f99": "q = \\frac{4\\pi \\sin(\\theta)}{\\lambda}", "ba217160da8743c7569884fe2a4c7f59": "\\mathbf{y}\\in [\\mathbf{x}]", "ba21a099021b04b1e437bd35aa4e7f2d": "x\\sim y \\iff f(x) = f(y)", "ba21fa9082638cd1d9c6979a46ec49c8": " x_1^2+x_2^4. ", "ba2223d4bfe144a42c757bc707efb9cb": "P(\\text{reject }H_0 | H_0 \\text{ is valid}) = P(X = 25|p=\\tfrac 14)=\\left(\\tfrac 14\\right)^{25}\\approx10^{-15},", "ba223361d6f89bbdce8c7a133cc7893f": "\\frac{\\Delta I_L}{\\Delta t}=\\frac{V_i}{L}", "ba2240f5fa4aae09820e5dba97add443": "\n\\bar{Y}=\\frac{[\\text{bound sites}]}{[\\text{bound sites}]+[\\text{unbound sites}]} = \\frac{[\\text{bound sites}]}{[\\text{total sites}]}\n", "ba22feea876eacf152c81b5cd20255c4": "y=\\arccot x\\,\\!", "ba233a31701f26b34429e360addd16cc": " (x^\\mu,\\dot x^\\mu)", "ba2344328841f78ca48bb634f784b332": "\\mathcal{B}\\,", "ba235b5880a9b549a5e5504ab6d2a3ab": "y_i \\ne 0", "ba23677a3c31dfe435e394896607c876": "\nI_P \\times V_P = I_S \\times V_S\n", "ba236d83572d76144157a730ac25b46c": "\\phi(z)=-1/\\bar z", "ba23818df7b58f41dae937355bc5413f": "1 / \\Omega", "ba23b2afdfbe1d1d87cb7bbf4f3f3c33": "Y_1 Z_2 X_3 = \\begin{bmatrix}\n c_1 c_2 & s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 + c_1 s_2 s_3 \\\\\n s_2 & c_2 c_3 & - c_2 s_3 \\\\\n - c_2 s_1 & c_1 s_3 + c_3 s_1 s_2 & c_1 c_3 - s_1 s_2 s_3 \n\\end{bmatrix}", "ba23e7fcf51c212a0d1165d71c665652": "A \\leftrightarrow \\neg \\neg A.", "ba243fa3aa61eb6397c997897e8cee01": " \\Gamma^{}_{} ", "ba24416021fb4cea3ff4ab4ebe461e55": " \\mathbf{J}_{\\mathrm{M}} = \\nabla\\times\\mathbf{M} \\ , ", "ba249332a557244c3fddd4b5dcd00491": " \\log_{10}(\\gamma_{0}) = b I", "ba249e4e56e7a9f411959020bb6b063c": "\nU \\frac{(V_{\\rm w2}-V_{\\rm w1})} {g}\n", "ba24c00f8314a4a7270cca5dc62d3517": "(h_{\\mathrm{e}})_k = h_{2k}", "ba251020052ec8585b37bac295d1c036": "\\scriptstyle\\gamma", "ba255eac608b4881c519abb75ab96ed1": "\\pi^2/(12\\ln2)", "ba25d1c49421b65af620ec7e6acec5ef": " (A,B) \\mapsto {\\rm Tr}(K^*A^qKB^{-r})", "ba26147bdc025123da95a7c7e01b7faf": "g(X,Y)(p) = g_p(X_p,Y_p)\\,", "ba261e3953ec1a5b1fa783959d9210ad": " (1-z)^{c-a-b} \\;_2F_1(c-a,c-b;1+c-a-b;1-z)", "ba26231ab8c7266178877e0f98363b29": "p(\\sigma[1] \\sigma[2] \\ldots \\sigma[L] \\gamma[1] \\gamma[2] \\ldots \\gamma[L]) = \\prod_{t=1}^{t=L} p(\\gamma[t] C_{(-)}[t] \\sigma[t] C_{(+)}[t])", "ba264a01a874104fa292b4ddc8fa8e63": "\\begin{align}\n e_0 (X_1, X_2, \\dots,X_n) &= 1,\\\\\n e_1 (X_1, X_2, \\dots,X_n) &= \\textstyle\\sum_{1 \\leq j \\leq n} X_j,\\\\\n e_2 (X_1, X_2, \\dots,X_n) &= \\textstyle\\sum_{1 \\leq j < k \\leq n} X_j X_k,\\\\\n e_3 (X_1, X_2, \\dots,X_n) &= \\textstyle\\sum_{1 \\leq j < k < l \\leq n} X_j X_k X_l,\\\\\n\\end{align}", "ba26850b49861060d9be13fa0b99ee90": "\\text{d} C / \\text{d} Q", "ba269a30783923a2a05dce1426acdb0a": "\\; S(p(x)) + S(p(y))", "ba269b3f01b90a7cfb962da8d1d07a99": "\\sigma > \\beta+1", "ba26a39e885e7a0356dceabcbd3a2c44": "\\mathcal{F} f(\\xi)", "ba26d38529fc409ea72148cf70fdd670": "f\\in C", "ba26e7e6b9cd1320494d7436a9a415ff": "t=(9, 13, > 13, 18, 12, 23, 31, 34, > 45, 48, > 161),\\, ", "ba271e6fd4cbd3546dfe8175c28bcd1a": "\\{U, X, \\omega, \\xi^3, \\xi^4\\}", "ba2726db47605e68f0421eac5b358394": "y=mx^a", "ba2760b31ddde9d9d923e1db7346f2a7": "NpH", "ba27680944e2cfa348635000bd6acb97": "\\pi=-g_y+g_m \\,", "ba284473b8399c585472e1372cd5bd37": "h^{999}(password)", "ba2873051912ceedc11325c6965cb961": "\\text{right}", "ba28d379ba2164703aa2f32750ec9330": "\n\\rho_\\text{TOT}(\\mathbf{r}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{n_1, n_2, n_3} \\sum_{\\mathrm{charges}\\ k} \nq_k \\delta(\\mathbf{r} - \\mathbf{r}_k - n_1 \\mathbf{a}_1 - n_2 \\mathbf{a}_2 - n_3 \\mathbf{a}_3)\n", "ba29022e4d808423320010459e45e1c6": "\\psi(1)", "ba29c163d35ef5539a7dcd6306337a64": "h,m \\in\n\\mathcal{N}, u, \\overline{u}\\in V^+, v,v',w,w'\\!\\! \\in V^*", "ba29c190bf535fc8d7681efc965c430d": "\\mathbf{v} = \\left[ \\begin{matrix} r \\\\ \\angle \\theta \\\\ h \\end{matrix} \\right]", "ba29de4c27c5c2085aae68d010477b07": "C_{\\mathrm{D}}", "ba29e0bb433b67bb9c1a09c2f3726b9a": "ds^2_{us}", "ba2a034f4d913f87fe07cad29368d114": "MG", "ba2a14a4496d6d3aac71ac292941b5c0": "\\bar f:V\\to\\bar W", "ba2a73f94e9094d86ef936d0280ccfa3": "\\frac{\\mu }{\\rho L }\\sqrt{\\frac{m}{\\gamma k_BT}}\\sqrt{\\frac{\\gamma \\pi }{2}}=\\frac{\\mu }{\\rho L }\\sqrt{\\frac{\\pi m}{2k_BT}}=\\frac{1}{\\nu L }\\sqrt{\\frac{\\pi m}{2k_BT}} = \\mathrm{Kn}", "ba2b1bee30d4577fb7f6374ca4551515": "a \\!\\, ", "ba2b247157fe937b2f51ec3b1cc85a07": "X: I \\times \\Omega \\to S", "ba2b3f442782cd1e181750e57c502913": "\\{O_1,O_2,O_3,O_7,O_{10}\\}", "ba2b62d255927ce5d3d692e255529d80": "C_p = 1-|\\vec{V}|^2 + M_\\infty^2 u^2 ", "ba2bacf73891dcd89456dd325f3bc996": " {\\mbox{d} T \\over \\mbox{d} r} = - {3 \\kappa \\rho l \\over 64 \\pi r^2 \\sigma T^3},", "ba2becbaed2a09fce6efd50ee5142ce3": "(\\dot u_0,\\dot\\lambda_0)\\,", "ba2c30ae90da5264dcc5f1641fba719f": "\n \\begin{align}\n q_1(x) & = \\int_{-b/2}^{b/2}q(x,y)\\,\\text{d}y ~,~~ q_2(x) = \\int_{-b/2}^{b/2}y\\,q(x,y)\\,\\text{d}y~,~~\n n_1(x) = \\int_{-b/2}^{b/2}n_x(x,y)\\,\\text{d}y \\\\\n n_2(x) & = \\int_{-b/2}^{b/2}y\\,n_x(x,y)\\,\\text{d}y ~,~~ n_3(x) = \\int_{-b/2}^{b/2}y^2\\,n_x(x,y)\\,\\text{d}y \\,.\n \\end{align}\n", "ba2c8064da3e4988de3b819605de6f71": "\\text{graph discrepancy index}=100 \\left(\\frac{a}{b} - 1\\right)", "ba2c8ced0d082ed3e59ddd08ad383cc0": "\\mathbb E[\\bar v_N]", "ba2d0676365a777286c8fbc5356e496a": "R \\in N_R", "ba2d261ba80c873c805df465e4b44636": "\\Phi^{-1}(z,\\mu,\\sigma)", "ba2d7032007d6740f559790ee9e9a083": "\nG(k_1,\\dots,k_m) \\, = \\, \\bigl[x_1^{k_1}\\cdots x_m^{k_m}\\bigr] \\,\n\\prod_{i=1}^m \\bigl(a_{i1}x_1 + \\dots + a_{im}x_m \\bigl)^{k_i}.\n", "ba2d776cefaed4e8103a92655210741f": "\\begin{align}\n \\tau_s & = \\max\\left\\{s_0 - (s_0 - s_{\\infty})\n \\rm{erf}\\left[\\kappa\n \\hat{T}\\ln\\left(\\cfrac{\\gamma\\dot{\\xi}}{\\dot{\\varepsilon_{\\rm{p}}}}\\right)\\right],\n s_0\\left(\\cfrac{\\dot{\\varepsilon_{\\rm{p}}}}{\\gamma\\dot{\\xi}}\\right)^{s_1}\\right\\} \\\\\n \\tau_y & = \\max\\left\\{y_0 - (y_0 - y_{\\infty})\n \\rm{erf}\\left[\\kappa\n \\hat{T}\\ln\\left(\\cfrac{\\gamma\\dot{\\xi}}{\\dot{\\varepsilon_{\\rm{p}}}}\\right)\\right],\n \\min\\left\\{\n y_1\\left(\\cfrac{\\dot{\\varepsilon_{\\rm{p}}}}{\\gamma\\dot{\\xi}}\\right)^{y_2}, \n s_0\\left(\\cfrac{\\dot{\\varepsilon_{\\rm{p}}}}{\\gamma\\dot{\\xi}}\\right)^{s_1}\\right\\}\\right\\} \n\\end{align}", "ba2dbdce2434fe6c7ba3476b0dc42604": "d_i = s_{0i} - S", "ba2dca7ca06c3defe0ffa1b94cfc005e": "0 \\dots 2r", "ba2de48cb11da4a8649b13e25a3ef7ba": " B=H-T_RS \\qquad \\mbox{(5)} ", "ba2e23e2fec504fc83fae3c81b34db39": "g_J\\langle J,J_z|\\vec J\\cdot\\vec J|J,J_z \\rangle = g_L {{\\vec L}\\cdot {\\vec J}}+g_S {{\\vec S} \\cdot {\\vec J}} ", "ba2ef2a94eaf85823264f64780c5274a": " d_H(C) \\le 1 + k", "ba2ef7a41145290f56556aef840d0e3e": "\\eta:\\operatorname{id}_\\mathbf{Set}\\to UF", "ba2f058851723feab0fd77f6e72ce8ad": "C(\\varepsilon) = \\frac{1}{N^2} \\sum_{\\stackrel{i,j=1}{i \\neq j}}^N \\Theta(\\varepsilon - || \\vec{x}(i) - \\vec{x}(j)||), \\quad \\vec{x}(i) \\in \\Bbb{R}^m.", "ba2f3deb9c797b107313837ea4e2ff43": "7^2 = 49 \\equiv 10 \\pmod {13} ", "ba2f4b7d31d5382c3d5f9f23ad323d5f": " A_1 + A_2 ", "ba2fbd561e16ee872177d658ef91cde0": "({\\rm sech}^{-1}(y)-\\sqrt{1-y^2},y)", "ba2fcd70f4179c09fb40d9c10eea8727": " a > 0 ", "ba309b7bf4807e440c0f08aff086d74f": "\\frac{d^2}{dx^2}f_x(x) + k_x^2 f_x(x)=0", "ba30f8bd19dfe3883d3ae752b9b4d98e": "\\text{Area} = \\left|\\det\\begin{bmatrix}{\\mathbf v}& {\\mathbf w}\\end{bmatrix}\\right| = \\left|\\det\\begin{bmatrix} a & c\\\\b & d \\end{bmatrix}\\right| = \\left| ad - bc \\right|.", "ba30ff66532ff5c093a06e1a5f7555b7": "\\zeta(2)", "ba31df305456593b06ec17c00ed7530c": "\\mbox{lim}_{i\\rightarrow\\infty}M\\{|\\xi_i-\\xi|\\leq \\varepsilon \\}=0", "ba320bd13f9b22e3b99c91f36dd4a826": "H^1(\\partial D)<\\infty", "ba321e3cd38516713a050c261066f6d9": " (i, \\sigma_i) ", "ba323dc814191392965d1fa1728664fc": "\\sigma_y", "ba324831fa19f09d2d1681983bb80bcf": "\\ell, m, n", "ba327876babb3f6eb784f7d09cae9a18": "f(x) = c x_1^{a_1} \\cdots x_n^{a_n} \\mapsto e^{a^T y +b}", "ba32aa91a51a8e2ecbf55c490728f8b5": " \\delta^4 \\left ( x - y \\right ) ", "ba32b392193cfaf546e7c9189053bb04": "\\log N(\\varepsilon||F||_{Q,2}, \\operatorname{sconv}\\mathcal{F}, L_2(Q)) \\leq K \\left(\\dfrac{1}{\\varepsilon}\\right)^{\\frac{2V}{V + 2}}", "ba32c3260ad0a202843a39e58162e074": "j = 1, 2, \\ldots, m\\ ", "ba330b361c6066f1ff27333d69a2cd42": "d=\\frac{a \\mp \\sqrt {a^2-b^2c}}{2}.", "ba3390f40401fac9c9639c6940c1d034": " S_\\mathbf{r} = \\bigcup_{j \\in [n], \\mathbf{x}_j = 1} S_{M_j} ", "ba33c1a3e3b8326a5695430b97a07f7a": "x^nQ(t/x)", "ba3404c3e28be030ea56cff76fbe1eaa": "\\log{g(\\zeta) - f(z)\\over \\zeta} -\\log {g(\\zeta)\\over \\zeta}= -\\sum_{m,n\\ge 1} c_{-m,n} z^m\\zeta^{-n}", "ba3414d2700859174c2c39b7ddbd65e5": "\n\\begin{align}\nw_1=w_2&=a\\\\\nw_3=w_4&=b\\\\\nw_5=w_6&=c\\\\\nw_7=w_8&=d.\n\\end{align}\n", "ba3485479be7ea44a6eaf6243fd5f1e6": " y = u + v \\,", "ba34c31144ea53ed49d2635ddfd08286": "0.50{K_u}", "ba35215f7902f446e81c62b96c466aaa": " \\nabla^2 \\mathbf{B} \\ - \\ { 1 \\over c^2 } {\\partial^2 \\mathbf{B} \\over \\partial t^2} \\ \\ = \\ \\ 0", "ba3543c161b4529993023a59aee09b20": "FP(\\mu, \\sigma, 1, 1, \\alpha) = P(II)(\\mu, \\sigma, \\alpha)", "ba358c645b8ee3489088628f0db5d767": "|\\psi (t)\\rangle", "ba35d7bbb2d6dcd8d10fad741123373b": "ExprRest \\rightarrow \\epsilon\\,|\\,+\\,Expr", "ba365b95f8f3f3a058b182a3963dae26": " C(Y) = \\sum_i |{\\mathbf{Y}_i - \\sum_j {\\mathbf{W}_{ij}\\mathbf{Y}_j}|}^\\mathsf{2} ", "ba3680cd4e215e6942c77787fc738543": "s(3)=1", "ba369f668147aba92b18387a892b3605": "\n\\mathrm{J}_{i, i+k} = \\frac{1}{\\Gamma} \\sum_{q=1}^{\\Gamma} \\left[\\Phi_{q}\\left(\\mathrm{R}_{i+k}\\right) - \\Phi_{q}\\left(\\mathrm{R}_{i}\\right ) \\right]^2\n\\qquad \\text{(6)}\n", "ba36c6f2b6aed3942865278c5b801c13": "mP", "ba36f5e063abcdb42f0d413e7f74cc4b": "Y_3 = T_1Y_2Y_1T_2 - Z_1X_2X_1Z_2", "ba36fff7f97db291c2e020b1c77e39d5": "\\partial_tH=0 ", "ba37160ce1b14e34a6f860c444bd019d": "- s", "ba37bc0c4d67ac9dc725b6a9daa13c48": "ds^2 = d\\tau^2 + dx^2 + dy^2 + dz^2", "ba37c4e2464ee14f46b7a2dc7a2e20b6": "\nc_q(n)=\n\\sum_{d\\,\\mid\\,(q,n)}\\mu\\left(\\frac{q}{d}\\right) d\n,\n", "ba37d0a984f019d52549cde23f93fd24": "f(x_1)+f(x_2)+ \\cdots +f(x_n) \\ge f(a)+f(a)+\\cdots+f(a) = nf(a).", "ba3810e3beccb1fe6400fee870b21333": "\\lambda_3=1/\\lambda^2\\,", "ba38380d4c4f91f86d21c1dd310e420d": "\\mathbf{b}", "ba383a7e2514a8b4ff547567f905b900": "\\epsilon(X, X^*)", "ba3846a8fcc5c3b76886357208059892": "t(x^{q}, y^{q})= \\bar{t}(x^{q}, y^{q})", "ba386b43c4ea65aab9731b49b0f3649b": "X(e^{i (\\omega-a)}) \\!", "ba38db0cb854061a963e6c426eac1b55": "\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}", "ba3900b69edf3c1787048beac182e687": " \\overline u_i = \\max \\{u_i : K(p)u = Q(p), p\\in {\\mathbf p}\\} ", "ba3942f11fd1fed5d5ca3fbae1162f28": "m_0^2 c^2=\\left(\\frac{E}{c}\\right)^2-\\|\\mathbf{p}\\|^2\\,", "ba39e48c889fd46dfb0baae2261d4d16": "\n\\langle v_e \\rangle = \\frac\n{\\int_{v_{e0}}^\\infty f(v_x)\\,v_x\\,dv_x}\n{\\int_{-\\infty}^\\infty f(v_x)\\,dv_x}\n", "ba3a185629593f098a0a000f9a9b039c": "\\lim_{x\\rightarrow c}\\left(\\limsup_{y\\in S_x} \\frac{f(y)}{g(y)}\\right)=\\limsup_{x\\rightarrow c}\\frac{f(x)}{g(x)}", "ba3a4c0341bfa2e67057e9a04a598eff": "B \\times B", "ba3aba578f82932dead58375fb5e70e3": "\\nexists{x}{\\in}\\mathbf{X}\\, P(x) \\equiv \\lnot\\ \\exists{x}{\\in}\\mathbf{X}\\, P(x)", "ba3ad2f84db2a88128e4751d61296f1e": "c_i \\sim \\chi^2_{n-i+1}", "ba3ae596369a83c351d076ebf486d357": "\\sin \\alpha \\approx \\alpha", "ba3b35aa613f49d99f567e0903810c52": "\n\\begin{alignat}{4}\n\\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v \\right) + \\nabla \\cdot \\left(\\mathbf\\Sigma_i \\nabla v_e \\right) & = \\chi \\left( C_m \\frac{\\partial v}{\\partial t} + I_{ion} \\right) & \\,\\,\\,\\,\\,\\,\\, & \\mathbf x \\in \\mathbb H \\\\\n\\nabla \\cdot \\left( \\mathbf\\Sigma_i \\nabla v \\right) + \\nabla \\cdot \\left( \\left( \\mathbf\\Sigma_i + \\mathbf\\Sigma_e \\right) \\nabla v_e \\right) & = 0 && \\mathbf x \\in \\mathbb H \\\\\n\\nabla \\cdot \\left( \\mathbf\\Sigma_0 \\nabla v_0 \\right) & = 0 && \\mathbf x \\in \\mathbb T \\\\\n\\vec n \\cdot \\left( \\mathbf\\Sigma_0 \\nabla v_0 \\right) & = 0 && \\mathbf x \\in \\partial \\mathbb T \\\\\n\\vec n \\cdot \\left( \\mathbf\\Sigma_0 \\nabla v_0 \\right) - \\vec n \\cdot \\left( \\mathbf\\Sigma_e \\nabla v_e \\right) & = 0 && \\mathbf x \\in \\partial \\mathbb H \\\\\n\\vec n \\cdot \\left( \\mathbf\\Sigma_i \\nabla v \\right) + \\vec n \\cdot \\left( \\mathbf\\Sigma_i \\nabla v_e \\right) & = 0 && \\mathbf x \\in \\partial \\mathbb H\n\\end{alignat}\n", "ba3b373335983f485d79c93ec80842e0": "0 \\leq \\mathcal{I}(\\theta) < \\infty", "ba3c6c4abd3ba37a47c7b2c1eb924a36": "C_B=\\dot{m}/K. \\qquad(6a)", "ba3ccae03457d2c3d32f997a8b4ed3dd": "L_0 + \\frac{{L_0(T-T_0)}}{{EA}} = \\sqrt{S^2 + 4\\left(\\frac{{W(L_0+\\frac{{L_0(T-T_0)}}{{EA}})}}{{4T}}\\right)^2}", "ba3ccd4b13bec4773e53be23da20c2f1": " {\\int e^{iS} \\phi(x_1) ... \\phi(x_n) D\\phi \\over \\int e^{iS} D\\phi } = \\langle 0 | \\phi(x_1) .... \\phi(x_n) |0\\rangle \\,.", "ba3cce9011cc8060756bb7b17b85a1c9": "\n \\ln\\mathcal{L}(\\theta\\,|\\,x_1,\\ldots,x_n) = \\sum_{i=1}^n \\ln f(x_i|\\theta),\n ", "ba3d5d7f05b6edff9242390eb4d8cd14": "KK(A,B) = [qA, K(H) \\otimes B]", "ba3d5fbc299b55f3390e666a517877c7": "\\operatorname{var}(X) = \\frac{1}{4 (1 + \\nu)} \\text{ if } \\mu = \\tfrac{1}{2}", "ba3d7b18ea7388857c06410f4fc0effe": "10^{500}", "ba3d82e84f351f5d273dac734168cfe8": "\\frac{\\pi}{4} = 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots + \\frac{(-1)^n}{2n + 1} + \\cdots", "ba3d83828507892bc0a8c58936722a83": "\\beta \\subseteq R", "ba3d99b016f93884d822f9f310fd79a5": "\\delta_1<0", "ba3dc499874588257e0496d32ae6ef52": "\n\\hat{\\textbf{x}}_{k} \\leftarrow \\hat{\\textbf{x}}_{k} + (\\alpha)\\ \\hat{\\textbf{r}}_{k}\n", "ba3e20fb16c2cbec605cda80dbd15368": " O \\rightarrow P_1\\,", "ba3e6033b05510f1390d2ac3ce8111de": "\nx=z/H\n", "ba3e90eb9b9d5e7789e348e54720e794": "\\kappa = \\frac{\\sqrt{\\det\\left( (\\gamma',\\gamma'')^t(\\gamma',\\gamma'') \\right)} }{\\|\\gamma'\\|^3}.", "ba3f45251d617895a4c9f5b7f93f64b0": " i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t} ", "ba3f530ab26e57d66c604690f243940b": "v_a=-C_R u_a", "ba3fcbc389692dbe1eb822eea9f7abd6": "{A}_j", "ba4006ac4fce67e8735477009b951dca": "1, 1, 3, 10, 41, 196, 1057, 6322, 41393, \\dots", "ba404f144da76afb0ad9ee47351ed6bf": "F_{ij}", "ba4093d0a84ff9eccfba0e930c79b78c": "K(X)\\to K(x_0)\\cong\\mathbb{Z}", "ba411274bb57e9a8cd3c81b6002e99dc": "(\\varphi I_n-A^\\mathrm{tr})\\cdot E=0\\in V^n", "ba41374bbe52b231e8958c296f89cafe": "J^1C", "ba415662af540c0e23acfbce5e165c54": " z_k=\\cos\\left({{2\\pi k}\\over 5}\\right)+i \\sin\\left({{2\\pi k}\\over 5}\\right), \\, k=1, \\ldots, 4 \\ .", "ba417875d9c031d56b94117fa694f8cb": "\n\\mathbf{M}_{\\rm orb}=\\frac{1}{2V}\\int_V d^3\\mathbf{r} \\, \\mathbf{r}\\times\\mathbf{J}(\\mathbf{r}) \n", "ba41b9335277139fea5b442e12cd82da": " D (d_\\lambda f) = \\lambda \\cdot d_\\lambda f,", "ba41dd284ac15dfcc5613ac586c7f35e": " p_b^{'} ", "ba41f5d1ccca9adc9370f5c4d9f091a1": "f_j(x^1) < f_j(x^2)", "ba422aec1ee8fc7f4247221494319917": "U.", "ba425f8c5f881b879c63c2632c48ffbe": " A(x,p) ", "ba429d55f5cf6012eaf30faed8ec4507": "\\pi(x)<\\operatorname{Li}(x) -\\frac13\\frac{\\sqrt x}{\\log x}\\log\\log\\log x.", "ba42a52e8d7be8d6c7339193056a35d6": "(t_k,\\boldsymbol{p}_k)", "ba42bd4eeca9ad56736681b43f8023ec": "\\mathcal{L}_{\\mathrm{matter}}", "ba42caf042989747e6e3aab9e240d845": "F\\ ", "ba42d2ba86c56e3bf9e863f1a08c70d7": "I_x = \\Omega_x \\cap \\mathbb{R}^n", "ba42fff18d19511fbb0729338977a0a2": "P_{\\mbox{NL}} = \\epsilon_0 \\chi^{(3)} (\\Xi_1 + \\Xi_2 + \\Xi_3)^3\\ ", "ba432d1d368e727b0c08e93be7e7e82e": "\\operatorname{Tr}(\\beta ) = \\sum_{i=0}^{m-1} \\beta^{p^i}", "ba4360756ce2bde23dcdcf563eed9ec8": "\n \\frac{E_{\\mbox{steel}}}{E_{\\mbox{brass}}} = 2.06\n ", "ba4365913db635d06bea26e48fc91566": " \\psi \\mapsto e^{i\\Lambda} \\psi ", "ba440087fb6fced6ea6ba49f9d3c5ad1": " \\begin{align}\nL(x) &= {1}\\cdot{x - 2 \\over 1 - 2}\\cdot{x - 3 \\over 1 - 3}+{4}\\cdot{x - 1 \\over 2 - 1}\\cdot{x - 3 \\over 2 - 3}+{9}\\cdot{x - 1 \\over 3 - 1}\\cdot{x - 2 \\over 3 - 2} \\\\[10pt]\n&= x^2.\n\\end{align} ", "ba44161db276575d10cb07814c41737d": "\\phi(F_1)=G_1", "ba4426fd467b498d040a2fff8d71ab58": "(X,Y,Z)", "ba44ca17883ac6306c798b5568608c9b": "H = \\cos(30^\\circ) \\times P = \\frac{ {\\sqrt 3}}{2} \\times P \\approx 0.866 \\times P", "ba4521e3ff18d5746c6b1fb417180b04": "\\eta \\geq 0", "ba4539fd08ac66ab36d4782cd4d08c36": "\\left(\\frac{\\alpha}{\\pi}\\right)_3 = \\omega^k \\equiv \\alpha^{\\frac{\\mathrm{N} \\pi - 1}{3}} \\pmod{\\pi}.", "ba456174cc989dc565d6a33a1ceb0e55": " t^q \\cdot u(t) ", "ba45d4f9648f3cfd73a2905a49941fad": "\\phi(v_j,v_k)", "ba45fbb750e96a66c8608b8c539bb8c6": "\\delta\\subseteq\\delta^*", "ba4665eb3a16996494c37092e12ca213": "\\ y = X \\beta + Zu + \\epsilon\\,\\!", "ba4669f2cb710dbf1cf447f701a16339": "Z[\\eta ,\\eta ^{+}]", "ba4679dc22fc4f6c3e610219bf0149ac": "sk(t)^{\\alpha} = (n + g + \\delta)k(t)", "ba4693816a43d6e0a249142db4e5e203": "= \\omega t v \\left(-\\sin\\alpha, \\cos\\alpha\\right )\\ ,", "ba469a3a216882b342618a29b0f62b29": "X_1+ \\cdots + X_n \\sim N(0, \\sigma_1^2 + \\cdots + \\sigma_n^2).", "ba46b2f7575f2aa7eb22a2b41fbea1a4": "cf(\\prod A/D)", "ba46f7b8eda6fc591c3a755dcbb2af13": "\n\\begin{align}\n\\sin(-\\theta) &= -\\sin \\theta \\\\\n\\cos(-\\theta) &= +\\cos \\theta \\\\\n\\tan(-\\theta) &= -\\tan \\theta \\\\\n\\csc(-\\theta) &= -\\csc \\theta \\\\\n\\sec(-\\theta) &= +\\sec \\theta \\\\\n\\cot(-\\theta) &= -\\cot \\theta \\\\\n\\end{align}\n", "ba472bdfe7d3d0fc82003bc3ae988293": "\\text{with E}_{t/b}\\left(\\eta\\right)=\\int_1^\\infin e^{-\\eta v} v^{-t/b}dv,\\ t>0", "ba47327d99cfb5bd75074ff9d698e6c4": " \\mathfrak{G}^k = \\{i^k\\}_{i=1}^\\infty", "ba47655ba4edfcc24b5746ed76579c11": "F_i>F_j ", "ba4768a794ee09bc702a5b3b78affb83": "F\\supseteq F", "ba47abcff1696fb541c9d198ee09924f": "\\Delta_{2\\pi}(\\theta)=\\sum_{k=-\\infty}^{\\infty}{\\delta(\\theta+2\\pi k)}.", "ba488e09707f4ad4aa15ff1368ab9425": "\\displaystyle \\sin{2A}=\\sin{2B}=2\\sin{A}\\sin{B}.", "ba48a3d54271cb3defe0bb2d1cb9d105": "\\partial_t u + \\partial_x^3 u + \\partial_x f(u) = 0.\\,", "ba48bda3e8fdcb942ca0c64263752690": "U = \\frac{H\\, \\lambda^2}{h^3} = \\frac{H}{h}\\, \\left( \\frac{\\lambda}{h} \\right)^2.", "ba48d7188dc8f8812339501c74f6f059": "h_t(x,y) = h_t(T(x),T(y))", "ba48fe807fbb3bd9b876a71539ea38e7": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = 1 - \\exp \\left(-4x_{1}\\right)\\sin^{6}\\left(6 \\pi x_{1} \\right) \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = g\\left(\\boldsymbol{x}\\right) h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) \\\\\n g\\left(\\boldsymbol{x}\\right) & = 1 + 9 \\left[\\frac{\\sum_{i=2}^{10} x_{i}}{9}\\right]^{0.25} \\\\\n h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) & = 1 - \\left(\\frac{f_{1}\\left(\\boldsymbol{x}\\right)}{g\\left(\\boldsymbol{x} \\right)}\\right)^{2} \\\\\n\\end{cases}\n", "ba4915b964297cc38cc2543a0c43dfed": "\\ p(\\lambda):=\\det (\\mathbf{A}-\\lambda \\mathbf{E})", "ba49a142af0f09ff78a93024195754a2": "m_i\\in \\mathbb{R}", "ba49da1c7aa3ad9082411df54aaa913d": " r(z) = 1 + z b^T (I-zA)^{-1} e = \\frac{\\det(I-zA+zeb^T)}{\\det(I-zA)}, ", "ba49f3c9cedf22e18aced41764cf6bb5": "\\mathcal L_{\\mathrm{int}}", "ba4a102d160646d2d780f70d65f7c67f": "j_i : V \\hookrightarrow M_i,", "ba4a31bc0b72ea90eb1f59d2072bb57c": "\\psi (s) \\ ", "ba4a33598cd18866cbe1def374f76f21": "P^{i} = 0", "ba4aa6c197c1960035d0e957f2847628": " \\sigma_3 \\otimes \\sigma_1 \\otimes \\sigma_0 ", "ba4b9b9ac9801c2a22225a706af752ee": "\\omega^A", "ba4bec8ff246524f8a9f8bab01151dab": "H_0:\\beta_i=\\beta_0", "ba4c02666b11f39f1a1e8a3d3d0efe94": "x\\in\\R\\ ", "ba4c026d656e78eab1c54d1498a7b195": "\n\\begin{matrix}\n\\overline{N}_{\\Delta f}(f) & = & \\int_{f-\\Delta f/2}^{f+\\Delta f/2} \\overline{n}(f) df \\\\\n\\ & = & \\overline{N}(f+\\Delta f/2) - \\overline{N}(f-\\Delta f/2)\\\\\n\\ & \\simeq & \\frac{8\\pi lwh}{c^3} \\cdot f^2 \\cdot \\Delta f\n\\end{matrix}\n", "ba4c9b912833eb332ef2b90a84a96178": "\\overline{\\omega} = \\frac{\\Delta \\theta}{\\Delta t} = \\frac{\\theta_2 - \\theta_1}{t_2 - t_1}.", "ba4d2594d07ea453624646404987092b": "V = N \\times \\log_2 \\eta ", "ba4d29f6c00f4c6ab8bde4861d450f63": " \\sum_{l=1}^n \\lambda^{l} x_l = \\sum_{k=1}^n (1-\\lambda)^{k-1} \\lambda^{n-k+1} \\sum_\\mathrm{samp} \\min ( x_{\\alpha_1}, x_{\\alpha_2}, \\dots, x_{\\alpha_k} ) ", "ba4d61db4bfe2d3e03b7a3d439fc6f26": "r \\in \\mathbb{W}", "ba4d8b3ac1cef5a6acef962426e12a72": "10) \\ \\mbox{Valve New adopters}\\ = \\mbox{New adopters} \\cdot TimeStep ", "ba4df2f28bc4478e30618ae6c8ca21f8": "x \\le_S y", "ba4e3c00811bf821a366166d534cd3fd": "p_{\\psi}\\,\\!", "ba4eaefd518848ee1a667c3d141d1f7f": "x = \\chi x_c, \\ t = \\tau t_c", "ba4f1a1bce8656d1e7ee84cbc854c52d": " [\\theta] = 3298.2\\,\\Delta \\varepsilon.\\, ", "ba4f1f85a59e6cb6e1042cb343763829": "\\frac{\\partial u}{\\partial t}=\\nu_c\\,\\frac{\\partial^2u}{\\partial y^2},", "ba4f71984f9419b38c3eb0cd44d16c5d": "\n\\begin{align}\n& (a_1+a_2+a_3+ \\dotsb +a_n)^2 \\\\\n=& \\, a_1^2 + 2a_1a_2 + a_2^2 + 2(a_1+a_2) a_3 + a_3^2 + \\dotsb + a_{n-1}^2 + 2 \\left(\\sum_{i=1}^{n-1} a_i\\right) a_n + a_n^2 \\\\\n=& \\, a_1^2 + [2a_1 + a_2] a_2 + [2(a_1+a_2) + a_3] a_3 + \\dotsb + \\left[2 \\left(\\sum_{i=1}^{n-1} a_i\\right) + a_n\\right] a_n.\n\\end{align}\n", "ba4fa0f781b41c64926772379536d350": "a_i.", "ba4fd30f3123bb9215e8c00634cb67bb": "A \\Rightarrow B", "ba4fdbe1b32fbb78fe2365bbd2b0bf90": "\\Delta T - \\Delta G = \\alpha - \\alpha = 0 \\, ", "ba506659db6f8a67bcef375f72584dbf": "\n \\begin{align}\n \\hat{X}_i(z^n) := g\\left( l^k(z^n,i) \\,,\\, z_i \\,,\\, r^k(z^n,i)\\right)\\,.\\end{align}\n ", "ba510d567e85c4fd5175606c9abd270f": "\\left (x-r\\right )\\left (ax^2+(b+ar)x+c+br+ar^2 \\right ) = ax^3+bx^2+cx+d\\,.", "ba511e55ddd9c4b8a42da7b110e181ca": "\\{\\lambda_{n-i}\\}_{i=0,\\cdots,n-2}", "ba514e0d52bd0176b98f42b72885916a": "k = \\left\\lfloor\\frac{b}{2}\\right\\rfloor", "ba517eba22506860137404b2b8fd777f": "\\inf\\, \\{1, 2, 3\\} = 1.", "ba51973adec6c86ba75fe6f16d9994c7": "d(z_1,\\ldots,z_k):=\\prod_{1\\le i 2b/2D", "ba5441ad572bf8c84b5dbf28d433e676": "R: A \\times S \\to \\mathbb{R}", "ba546961eb270d3e8128821dcea790bc": " n\\in\\N _ + ", "ba546f6fe2c5eca547a946a426af5af8": " b\\in[1,4]=\\mathbf{b} ", "ba548651910eb62ce8faac0b37516b3b": "\n\\bar{h}^{\\alpha \\beta} (t,\\vec{x}) \\approx\n-\\frac{4}{r}\\, \\int\\, \\tau^{\\alpha \\beta}(t-r,\\vec{x}')\\, \\mathrm{d}^3x'\n", "ba552b911b50b9d353e299f202584035": "{\\scriptstyle[0;\\overline{1,M}]-[0;\\overline{M,1}]\\ge\\frac{1}{2}}", "ba558491b41a622c80f6f05186d98b85": "x, z \\in C", "ba55a83c2e4d158bd9ad69b03415b9b5": "L_3=\\partial_b^2 +\\partial_x^2 +\\partial_y^2,\\,\\, \\, R_3=b\\partial_b + x\\partial_x + y\\partial_y. \\, ", "ba55b4ecaf92fc759b7d7a57ac640fb1": "p_i \\equiv 2^{i-1} - 1 \\pmod{p_1}", "ba5619f1c4531bcdf2d845cb81bd8983": "(\\log 31)^5~\\dot{=}~7.37", "ba566485e055321975e1d73398caaa07": "\\; (f/g)^* = f^*/g^*", "ba5676323b24b2f899ba3b39c9214c42": "(T-T^*_C)^{-1}", "ba567cd886670efd7dfa357175a29b15": "\n\\left[-{\\hbar^2 \\over 2m_0} {d^2 \\over dr^2} + {\\hbar^2l(l+1) \\over 2m_0r^2}+\\frac{1}{2} m_0 \\omega^2 r^2 - \\hbar\\omega(n+\\frac{3}{2}) \\right] u(r) = 0.\n", "ba56835ad996bf0d4f4a6fcad2ecbcca": "{}^{n - 1}T_n\n = \\operatorname{Trans}_{z_{n - 1}}(d_n) \\cdot\n \\operatorname{Rot}_{z_{n - 1}}(\\theta_n) \\cdot\n \\operatorname{Trans}_{x_n}(r_n) \\cdot\n \\operatorname{Rot}_{x_n}(\\alpha_n)", "ba56972d7dae9ed9d6b66c5f4a85c43a": "c = 1 + c_1 + c_2 + \\dots.", "ba56aa0f7d67a760e4c0061168353275": "\\lambda_1 = \\lambda_2 = \\lambda", "ba571f314b629ffc9e16bee2bc900e38": "(1/4!)\\pi^4 = (1/24)\\pi^4 ", "ba572198fe7d16fb6f8333b0bef5fe83": " \\operatorname{sink}[(\\lambda q.q\\ p)\\ (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f))] ", "ba5737073bb703dd3ed96db4c922d559": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} =\n \\cfrac{2Eh}{1-\\nu^2} \\begin{bmatrix} 1 & \\nu & 0 \\\\ \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix} \\,,\n", "ba575e1a0af463c6d0480a9917e5aec1": "\\Delta H_\\mathrm{ads}=\\Delta H_\\mathrm{liq}-RT\\ln c,", "ba57bb7079200174aa89934614f51bef": " \\ln x_2 = - \\frac {\\Delta H^\\circ_{fus}} {R}\\left(\\frac{1}{T}- \\frac{1}{T_{fus}}\\right)", "ba57c1ac98d95a3fe0ac2a10ce6c47d5": "\\overline{\\lambda}_t ~=~ \\frac{1}{n}\\sum_{i=1}^{n}\\lambda_t ~=~ \\lambda_t", "ba57fa3ccf2c87378112d88feb3ede7e": "Y \\cup \\{A\\}", "ba58622e816d5a9fe535e168641f4ca4": "\n\\begin{align}\n\\Pr(Y_i=1\\mid\\mathbf{X}_i) &= \\Pr(Y_i^\\ast > 0\\mid\\mathbf{X}_i) & \\\\\n&= \\Pr(\\boldsymbol\\beta \\cdot \\mathbf{X}_i + \\varepsilon > 0) & \\\\\n&= \\Pr(\\varepsilon > -\\boldsymbol\\beta \\cdot \\mathbf{X}_i) &\\\\\n&= \\Pr(\\varepsilon < \\boldsymbol\\beta \\cdot \\mathbf{X}_i) & \\text{(because the logistic distribution is symmetric)} \\\\\n&= \\operatorname{logit}^{-1}(\\boldsymbol\\beta \\cdot \\mathbf{X}_i) & \\\\\n&= p_i & \\text{(see above)}\n\\end{align}\n", "ba5883f7b8125749557b914936115ae3": "\\|A \\hat x-b\\|", "ba5886250360b2fb2084ffe1f1ee6cd5": "\n \\sigma_{jk}~\\cfrac{\\partial\\epsilon_{jk}}{\\partial x_1} = \\cfrac{\\partial W}{\\partial\\epsilon_{jk}}~\\cfrac{\\partial\\epsilon_{jk}}{\\partial x_1} = \\cfrac{\\partial W}{\\partial x_1}\n ", "ba58aa53a7d8ded8435e6615a0f2b56f": " D(n) = \\sum_{k=0}^n \\binom{n}{k} \\binom{n+k}{k} . ", "ba58f6384c6fd6fb3f5ecaad8ecfa003": "i\\in\\{1,2,\\dots,k\\}", "ba5a547c5c5956feb6e69ead91eba700": "\\scriptstyle\\sqrt{X_1^2\\,+\\,X_2^2}", "ba5a6282642e572e8602f75d61225c88": "\\{x\\} = x -\\lfloor x\\rfloor.", "ba5a6e0fc8e7ba50baa4a6158ff6258d": "H_2 = \\begin{bmatrix} 4 & 2 & 2 & 1 \\\\ 2 & 2 & 1 & 1 \\\\ 2 & 1 & 2 & 1 \\\\ 1 & 1 & 1 & 1 \\end{bmatrix}", "ba5b2deb11a254a6f0b0c2cf6deffa05": "(\\theta_1, \\theta_2, ..., \\theta_{10})", "ba5b42084395788c801145a09e3b4de2": "r_j \\, s_k = s_{(j+k) \\text{ mod }n}", "ba5b44d2ff79907e9b86a4b0d53a7fe9": "\\textit{Coeff} = \\frac{500}{a + b \\cdot x + c \\cdot x^2 + d \\cdot x^3 + e \\cdot x^4 + f \\cdot x^5}", "ba5b5b6dda0017f40dd5d4ce77aabe9a": "\\frac{p_{j-1}(x)}{g_j(x^2)}\\approx \\frac{f_{j-1}(x)}{g_{j-1}(-x)}", "ba5ba5d0c240ea55c9caa93c495de9ab": " \\frac{dD}{dt} = k_{N} N -k_2D ", "ba5c2b20590f6a0247cef4ef93bf9733": "a^2 + b^2 + c^2 + d^3 + e^{6k-1} = 0\\ ", "ba5c3bc0401c59fba8d5f87a1c59bc5f": "K_i(s_i,t_i,d_i)", "ba5c5ffd2b4ad28d284cd5d651add311": "\\mathcal{H=}p_{\\perp }^{2}+\\Phi -b^{2}\\approx 0\\,. ", "ba5c64ef1f71170e1be751ba8b73893e": "Z= \\cup_i Z_i", "ba5ca0751856c4ac7c762a37944e7912": "\\begin{align}{\\rm{gd}}\\,x&=\\int_0^x\\frac{\\mathrm{d}t}{\\cosh t} \\\\[8pt]\n&=\\arcsin\\left(\\tanh x \\right)\n\n=\\mathrm{arctan}\\left(\\sinh x \\right) \\\\[8pt]\n&=2\\arctan\\left[\\tanh\\left(\\tfrac12x\\right)\\right]\n\n=2\\arctan(e^x)-\\tfrac12\\pi.\n\\end{align}\\,\\!", "ba5cce3dbd307aa1405fdf9e3c71af6b": "d = 2 r \\arcsin\\left(\\sqrt{\\operatorname{haversin}(\\phi_2 - \\phi_1) + \\cos(\\phi_1) \\cos(\\phi_2)\\operatorname{haversin}(\\lambda_2-\\lambda_1)}\\right)", "ba5dcf24fddde71317ce1ba89bca52bd": "E_{211}=E_{121}=E_{112}", "ba5deb368211ded2864396835632f5e9": "p_{\\theta} = \\frac{dL}{d\\dot {\\theta}} = mr^2\\dot {\\theta}", "ba5eb492e111b77d8be11cbf73c1d7b4": "X_2/(1-X_1)", "ba5eead710a1a9f0eb10a6d434365f96": "\\frac {1} {2} \\dot m v_e^2", "ba5efcf3f87c93f80af9ef4f9b9671a5": "d<4\\,", "ba5f18aa0aa1179d9a09e6c9786a1091": "a=2\\sin\\tfrac{\\pi}{10}=\\tfrac{-1+\\sqrt{5}}{2}=\\tfrac{1}{\\varphi}", "ba5f2237404c01253d310e7b278b0122": "\\ u=s", "ba5f3abb5e5dcb138bb137236b020ed2": "0 \\;\\rightarrow\\; A\\;\\rightarrow\\; B\\;\\rightarrow\\; C \\;\\rightarrow\\; 0", "ba5f923cb32d02556aab145e711f9673": "\\kappa_0(\\mathcal B)\\,\\!", "ba5fb8f74cfe5daa2fd92924e0ef52b4": "v = \\sum_{i=1}^n x_i a_i", "ba5fcfa2c4587192b275a66f722aa8dd": "x \\mapsto 2x \\mod 1.", "ba600654b293bfb5f283826f6f7360e5": "p^2=-\\frac{1}{8}(r^2-a^2)", "ba605c13bf8f27616ececc02a36770f0": "\\gamma(\\lambda_1) = x", "ba6113ea76ed82edf883f8909a32f671": "\\scriptstyle-\\sqrt{3}", "ba618b922f37e05b095b9b530c1b6786": "\\partial A = \\{ (\\lfloor N / 2 \\rfloor, \\lfloor N / 2 \\rfloor + 1), (N, 1) \\},", "ba61cd2977b86c1784f8b5a085cb7392": "\n0\\leq [D_{k}]_{i,j}<\\infty\\;\\;\\;\\; 1\\leq k\n", "ba61e76d1f2c4e3d9a9d3e02957c5cda": "t_k\\leq t \\leq t_k+\\delta t", "ba623634b25d2314bb519710e20764a1": "F = \\frac{MS_B}{MS_W} \\approx 42/4.5 \\approx 9.3", "ba624c11dba77ff64591930c5bd7a239": "U(b)=w(b)(v-b)=2b(v-b)=\\tfrac{1}{2}[{{v}^{2}}-{{(v-2b)}^{2}}]", "ba6275457a491e6b480434106e4cce9b": "Z^{-m}_n(\\rho,\\phi) = R^m_n(\\rho)\\,\\sin(m\\,\\phi), \\!", "ba628311e7365111cf96643ea6884d98": "a \\land (a \\lor b) = a", "ba62d12df342fbcc2088311fc5e11d15": "Q_1,.., Q_{i-1}, Q_i,...", "ba62d4671b2565aa119b5d37a7c6df86": "\\lambda \\in \\mathbb{C}", "ba62d4c51b5ba5228ea6c8ec53535a8f": " R = I + [\\mathbf{k}]_\\times \\sin\\theta + (1 - \\cos\\theta) [\\mathbf{k}]_\\times^2 ", "ba633efbd4ad97774de7849226761aa5": " {\\Delta}_{\\rho} := \\Delta-\\Delta(1) . ", "ba63465fd8cd9aba567ab72c2defb674": "\n\\begin{align}\nCLI & = 0.0588 \\times 537 - 0.296 \\times 4.20 - 15.8 \\\\\n& = 14.5 \\\\\n\\end{align}\\,\\!", "ba637b9540e93abfe249a9e186c47a76": "f(x_n)", "ba638bf390954c59fb0fcba6d466c97f": "MPS=\\frac{\\text{Change in Savings}}{\\text{Change in Income}}", "ba639def679e2abab7282b0c8809ffbf": "(X_t^\\tau)_{t>0}", "ba63ec10815c5b6fea1b6c22d86072f6": "L_n, \\bar{L}_n", "ba63ed057d80a7b531d8f110643cae14": "\\scriptstyle Z_\\mathrm i", "ba64b04df6ce1d1834ed807e6f94149c": "GG = G_3 + G_2 \\cdot P_3 + G_1 \\cdot P_3 \\cdot P_2 + G_0 \\cdot P_3 \\cdot P_2 \\cdot P_1", "ba653c4eff6fc38c2de653a4224ea23d": "p = \\,", "ba66b2f7fcbce2e88676efd960f4ea0a": "\\left|x\\right|^2 = x_1^2 + x^2_2 + x_3^2", "ba673c572e184b4315b1322665054a1d": "(P \\and Q) \\vee R", "ba6748ca0f8a8ca517da7bba036bc06a": "d_1 \\cdots d_n,", "ba6752595ca0cf2fddf08410836663da": "(e_0,e_1,e_2,\\ldots,e_r,k)", "ba676ddcb676c458b25bafa281569c96": "\\mathbf{a} = \\left( \\ddot{r} - r\\,\\dot\\theta^2 - r\\,\\dot\\varphi^2\\sin^2\\theta \\right)\\mathbf{\\hat r} ", "ba684fb46ede184a5c61516794c354f0": " u_1 \\otimes \\dots \\otimes u_n ", "ba6890ea39e502f388d92f96a579ecd3": "8.85 x 10^{-12}", "ba68e6232ffc3673e6925625c4fc1e4a": "\\sin(Rt\\cos(\\theta))", "ba6907859d2afe9d2d06499f55987e94": " \\lim_{n\\to\\infty} \\int_S |f_n-f|\\,d\\mu = 0", "ba6945bc4c319bcff1261b0fd97721c8": "x \\equiv a_i \\pmod{u_i} \\quad\\mathrm{for}\\; i = 1, \\ldots, k", "ba6996774a267e98d4ccd064d128c77a": "S = S_k(q_k) + S_{rem}(q_1\\cdots q_{k-1}, q_{k+1} \\cdots q_N, t). ", "ba6998ecf85565b342aa7bc02b755630": "\\underset{x}{\\operatorname{arg\\,max}} \\, f(x) = \\{ x\\ |\\ f(x) = M \\} = f^{-1}(M) = f^{-1}(\\underset{x}{\\max} \\, f(x) )", "ba6a9ce23df390159173b44ba1ee4cf2": "\\Im(\\mathit \\Gamma)=0", "ba6af5992af901e90141a1aa64c59c65": " x^5+\\frac{20}{17}x+\\frac{21}{17}", "ba6b198381e4258c71aef5367fec0861": "\\frac a b \\nwarrow \\frac c d \\quad \\frac a b \\nearrow \\frac c d.", "ba6b38ee4c37b188fe47cf832e962ac3": "t = \\frac{3 \\times 10^{-4} \\, \\mathrm{Pa} \\times \\mathrm{s}}{P}", "ba6b8b03c6a27cc75e372b1c396e0e03": "f(x \\pm 2h) = f(x) \\pm 2h f'(x) + 2h^2 f''(x) \\pm \\frac{4h^3}{3} f^{(3)}(x) + O_{2\\pm}(h^4). \\qquad (E_{2\\pm})", "ba6baa5339a8cdc0c7a5c208e27fd024": "\\alpha \\leq \\beta", "ba6bc6ea0362fffde01971ff21a65461": "\\vec{y}:=(y_1,\\ldots,y_n)", "ba6bd390708104418d43ff01f62b72f5": "e^{i\\gamma_n}", "ba6c0cc44bb78aaae0ad170d5235734a": "\\scriptstyle k>\\frac{10^{10}-1}2", "ba6c4ed3ebaa97e8ee3a21f56f58d1f4": " C_1\\triangleleft C_2\\triangleleft C_4\\triangleleft C_{12}", "ba6c71b4c4802031cbc5f4645e9c0c73": "A\\to C,(A\\to B)\\to C\\vdash(C\\to B)\\to C", "ba6caaf9685183ba17192abb048ea5fb": "V_{\\rm bin}", "ba6cb1732ef6e9ee5028f24e77c67eb6": "n_i \\in \\{-1,0,1\\}", "ba6cf07fbee10c109805ba5e43fe463f": "R^4", "ba6d2cabeea380a98787286ac1640416": " K\\backslash G/H ", "ba6d51b782c9b5c9d34408410c3a215b": " f_+(z) = 1 + wz", "ba6db82f500e93df8f61b2cef58eeb4d": "i! = \\Gamma(1+i) \\approx 0.4980 - 0.1549i.", "ba6dec84c931271e495c61c977fb2233": "(z^{k+1})", "ba6e1e8136d73aeb7cae24c1261c76eb": "2|V|-1", "ba6e227bcbbeb70d1849993294ba37f9": "\\forall z (z \\in x \\Rightarrow z \\in y) ", "ba6e3e62f3f9ba4259908d2a49de8004": " \\frac{d^2 \\chi}{d \\tau^2} + t_c \\frac{b}{a} \\frac{d \\chi}{d \\tau} + t_c^2 \\frac{c}{a} \\chi = \\frac{A t_c^2}{a x_c} F(\\tau).", "ba6e5da8c689a733b14da0891a031abc": " \\gamma (I) \\subset \\mathbb{R}^2", "ba6e8e634061dd39ef520f228b6ca2f9": "DICE=3.00 + \\frac{13HR + 3(BB + HBP) - 2K}{IP}", "ba6eaf195a0ec25791b7ff440c8bc828": "\n\\omega_\\chi (x,y) = \\chi ( [x,y] ).\n", "ba6ecc351317ce59c181259c29654395": " c_2 ", "ba6f58df99239ec14bd035b047f61c08": "\\gamma(i+1)", "ba6f8f5dfd4ef1085f8cc0c7cc7e1df9": "\\ [\\omega,\\eta]_{gr} := \\omega\\eta - (-1)^{\\deg \\omega \\deg \\eta} \\eta\\omega.", "ba6fe80a614d62aed14c8238419c68ef": "H_n(X;\\mathbb Z)", "ba703426d09ac4ac9e5d39ffa77e4544": "P\\#+2, P\\#+3,\\ldots,P\\#+(Q-1)", "ba70a624bbc3040baccbd4f629efc78d": "q^i", "ba70fcd2238f5f30d0a65e0b9eb37792": " [Q_i, Q_j]_{p,q}=0, \\quad [P_i,P_j]_{p,q}=0,\\quad [Q_i, P_j]_{p,q}=-[P_j, Q_i]_{p,q}=\\delta_{ij}. ", "ba710d0e205091ead19c9960523ef9c6": "\\nu(d)\\leq2d+\\lceil\\frac{d+1}{2}\\rceil", "ba71c468fa084b815ab848f0579b691b": "R_\\mathrm{total} = R_1 + R_2 + \\cdots + R_n", "ba71e09081b78a0c5120fa56ec959933": "|q\\rangle", "ba71e713784013dfcd6b3fd45b556a32": "f''(x)\\!", "ba72cc37557d8ab20fe4415695c67dca": "\\hat{s}_j", "ba731eebd50406b078890a9efb13f8ea": "\ne_1=\\tfrac{1}{2},\\qquad\ne_2=0,\\qquad\ne_3=-\\tfrac{1}{2}.\n", "ba7359f159cb8b2e460cb0400173545f": "x = b", "ba73f757afdcdd7e9f3b0aac112f1258": "Z_o\\left(\\omega_o\\right)", "ba7443e545fc82bc79774ea00149c418": "\\hat{S}_{i}, \\hat{B}_{i}", "ba7457b824914ad4fb9ed475de4e0c6a": "\\mathbf{k}=(k_x,k_y,k_z)", "ba74714fcca1c050fe8e4fe55e40c44f": "|df_p(v)\\times df_p(w)|=|v\\times w|\\,", "ba74a0f0e9e818659e5956ecc17a37de": "\\displaystyle e=|(x'+u')-(x+u)|-|x'-x|.", "ba74fbdb6007db3e58353861390c705b": "F_t = \\frac{1}{N}\\left(\\frac{1+F_{t-2}}{2}\\right)+\\left(1-\\frac{1}{N}\\right)F_{t-1}.", "ba750841907d3b131bd49309922615f8": " f(g(x)) = g(f(x))\n= \\frac{(\\cos(\\alpha+\\beta))x - \\sin(\\alpha+\\beta)}{(\\sin(\\alpha+\\beta))x + \\cos(\\alpha+\\beta)}. ", "ba7555e544bf7c1b4d71d20e33df6112": "\\tan \\theta = S/D\\,", "ba755afaa3734b28f2e2258bc69bdb8a": "~G_3", "ba759ebf0c3a2f351f8016bf7971384d": "p_{n+1}(x)", "ba75ce071d3c3cd4257251cd3f7edccc": "f: \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}", "ba75cfb697a3081b9bb4e7ee136f52d5": "\\displaystyle (i\\nu)^n \\hat{f}(\\nu)\\,", "ba75d44678c80cd5b66f29b217d001c8": "2\\uparrow^{m-2} (n+3) - 3.", "ba760789275a80840870e04ec9096216": "\\scriptstyle i", "ba760d52d4ebf6f6b1dfd16e8b12fdff": "(\\mathbb{F}_p) \\cong \\mathbb{Z}_{2^kn}", "ba761bbf587a98a7109de6b2f692aeb7": "0 < \\left| \\sqrt{2} q_1 - p_1 \\right| < \\frac{1}{2}", "ba76293da4d2e4ca17e2f3e326f15da2": "e_{i_1}e_{i_2}\\cdots e_{i_k}", "ba766097028c03d2acbae7b25bc25e64": " A_{ij} ", "ba7663494aed6a9b024557ecb9aabab9": "\n(R_a)(10) + (R_b)(25) + (R_c)(50) - (1)(15)(17.5) -50 + M_c= 0 \\,.\n ", "ba76cb2a4b1502ee518d59e7eb045da9": " O(1/n) ", "ba76cd2c38fe90916e050987d8600099": "=\\Pr[A\\cap B]+\\Pr[A\\cap B^c]", "ba7766c5f4906e385ac618e0f127419f": " m = (X_1-a_1, \\ldots, X_n-a_n), \n\\quad a = (a_1, \\ldots, a_n) \\in \\mathbb{K}^n. ", "ba7794282933eccfae4a96e645437f78": "H_1 : Y(t) = N(t) + X(t), 0 10^{-3}", "ba86615d96decf833f69cb0708e629da": "-I \\not\\in SO(2k+1)", "ba8667b570943f74cc2ecee0cb8a65ac": " \\frac{d^2u}{d\\xi^2}+[a_u-2q_u\\cos (2\\xi) ]u=0 \\qquad\\qquad (1) \\!", "ba8667ea3145b03c28740a0400bc7bf9": " \\Tau = \\frac{1}{\\text{PRF}}", "ba866c91316768938b3c9b059bba8ee4": "\\hat w", "ba866d719b99ee5bad737a5a016f6801": "CA\\parallel BD", "ba86961edb3ad2da01ad27adf1e08c80": "C_{2n}", "ba86b59827b2c2579197f96540ef4ece": "\\scriptstyle J = MR^2 \\dot\\theta", "ba86b65773914366b19f85367caff6a1": "L = n^{\\rho}", "ba86c63d6e0487a0bd96882528610a7c": "g_*(T) = \\sum_\\mathrm{i=bosons} g_i{\\left(\\frac{T_i}{T}\\right)}^3 + \\frac{7}{8}\\sum_\\mathrm{j=fermions} g_j{\\left(\\frac{T_j}{T}\\right)}^3", "ba872a9667617dd7e91f42aabf13db20": "R_{k}=\\left(I+\\frac{1}{n+2k-2}uD_{u}\\right)D_{x}.", "ba878e26494800f709cd3d17905801fa": "p\\geq0.5", "ba879f2e6d7fe7b966964872ee261d5f": "|N| = \\delta + 1", "ba87a6cb7c689d7b53e364e63da2b817": "\\frac{D_F X}{d s}=0,", "ba87fc5513baabe2a2c39ee57a49a29d": "y = x^2 \\iff x = \\sqrt{y} ", "ba8891e2862e270086aaa0135161d437": " a \\left( v'' y_1 + 2 v' y_1' + v y_1'' \\right) + b \\left( v' y_1 + v y_1' \\right) + c v y_1 = 0.", "ba88bfc60064cdbb45de9262e7f57469": "\\Delta U=Q - W=Q - P\\, \\Delta V \\text{ and }\\Delta (PV) = P\\, \\Delta V \\,.", "ba88feb7c05148ab497abab66426a297": "\\left(\\dfrac{dn}{2}\\right) \\left(\\gamma^2 + \\left(\\dfrac{\\lambda}{d}\\right)\\gamma \\left(1-\\gamma\\right)\\right)", "ba89010414d81f15838fff18cd52d6f3": "\nK_0 \\left( k_{Ds} r_{12} \\right) \\rightarrow -\\ln \\left( {k_{Ds} r_{12} \\over 2}\\right) + 0.5772\n.", "ba89015ace40a4e3b800bdb2cb283df6": " y = 2\\pi t \\sin(2\\pi t)+ 2\\pi [(\\cos^3 2\\pi nt)+\\sin^3 2\\pi nt)\\sin(2\\pi nt)- \\sin(2\\pi t)] ", "ba89610087348952873fd9e6e67d7277": " S(x,y)=\\left( \\frac{S_1+S_2}{2},(S_1-S_2) \\right). ", "ba898392182e6f3d11a7388959706750": "c = 26", "ba898539c0796da27c99d173011744ff": " \\langle s \\rangle=\\frac{q\\lambda}{1+q\\lambda}", "ba898c1a07e51127a0f62688194c5e07": "f=g/u", "ba89c8c210964d6117c40eb28aaac4ab": " \\frac{dv} {dr} = 0 ", "ba89da6701e9e7114792a23ffc391e0c": "u(\\lambda,T)=\\frac{\\beta}{\\lambda^5}\\cdot\\frac1{e^{hc/k_BT\\lambda}-1}", "ba89e9bcbe1e7627579fec3497e622bf": "(~x_1 \\and ... \\and x_n~)~\\or~(\\neg x_1 \\and ... \\and \\neg x_n)", "ba89f18a5287475c4faa37fdd564a555": "\n\\begin{array}\n[c]{c}\nX=[\\underset{TN\\times K1}{X_{1it}}\\vdots\\underset{TN\\times K2}{X_{2it}}]\\\\\nZ=[\\underset{TN\\times G1}{Z_{1it}}\\vdots\\underset{TN\\times G2}{Z_{2it}}]\n\\end{array}\n", "ba89f9c03b4821159728bee995502231": "w_{\\alpha \\beta}( n_\\mathbf{p}+1 \\leftarrow n_\\mathbf{p} ) e^{-E_{\\beta} /kT} = w_{ \\beta \\alpha}( n_\\mathbf{p} - 1 \\leftarrow n_\\mathbf{p} ) e^{-E_{\\alpha} /kT}, ", "ba8a6fd6baf6471c0447a46bce1faea2": "\n X = \\begin{cases}\n -1, & \\text{with probability }\\frac{1}{2k^2} \\\\\n 0, & \\text{with probability }1 - \\frac{1}{k^2} \\\\\n 1, & \\text{with probability }\\frac{1}{2k^2}\n \\end{cases}\n ", "ba8a8118087112b83ee22e8be0d68a72": "\\Gamma_\\mathbf{C}(s)=2(2\\pi)^{-s}\\Gamma(s)", "ba8abf8d4d1d4f95e7075cfdda38086d": " \\textstyle \\forall j \\neq i: \\beta z_{ni} - \\beta z_{nj}, \\; ", "ba8acd598e496039cfb4b027915b4553": "X(X^TX)^{-1}X^T", "ba8b13b333207ada467690c5f36a965c": "\n\\begin{align}\n\\frac{d\\beta}{ds} &=\n\\frac1{\\sqrt{(a^2-b^2)\\sin^2\\omega + (b^2-c^2)\\cos^2\\beta}}\n\\frac\n{\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}}\n{\\sqrt{b^2 \\sin^2\\beta + c^2 \\cos^2\\beta}} \\cos\\alpha,\\\\\n\\frac{d\\omega}{ds} &=\n\\frac1{\\sqrt{(a^2-b^2)\\sin^2\\omega + (b^2-c^2)\\cos^2\\beta}}\n\\frac\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}}\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega}} \\sin\\alpha,\\\\\n\\frac{d\\alpha}{ds} &=\n\\frac1{((a^2-b^2)\\sin^2\\omega + (b^2-c^2)\\cos^2\\beta)^{3/2}}\\times\\\\\n&\\quad\\biggl(\\frac\n{(a^2-b^2) \\cos\\omega\\sin\\omega\n\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega - c^2}}\n{\\sqrt{a^2\\sin^2\\omega + b^2\\cos^2\\omega}} \\cos\\alpha \\\\\n&\\qquad+\\frac\n{(b^2-c^2) \\cos\\beta \\sin\\beta\n\\sqrt{a^2 - b^2\\sin^2\\beta - c^2\\cos^2\\beta}}\n{\\sqrt{b^2\\sin^2\\beta + c^2\\cos^2\\beta}} \\sin\\alpha\\biggr).\n\\end{align}\n", "ba8b22a1e13f3062e76590f29ccff38a": "d_H(\\mathbf{x}, \\mathbf{y})\\leq 1", "ba8b44d129e4ccbb898c74b423153f27": " x\\mapsto |x|", "ba8b6cb18e41c14e93462b052a3af380": "\n\\partial_t P(\\mathbf{x},t\\mid \\mathbf{x_0})=\\Sigma_{j=-M}^M D_j \\partial_{x_j}^2 \\int_0^t k_{\\alpha}(t-u) P(\\mathbf{x},u\\mid \\mathbf{x_0}) \\, du. \n", "ba8bbb96aff0950e76a0cca58779718d": "\\bar{J}^{\\mu} \\, = \\, \\frac{\\partial \\bar{x}^{\\mu}}{\\partial x^{\\alpha}} \\, J^{\\alpha} \\, \\det \\left[ \\frac{\\partial x^{\\sigma}}{\\partial \\bar{x}^{\\rho}} \\right] \\,.", "ba8bbeb315889843b03e21c8017f73db": "dM", "ba8c320d2b91ff31667e4191fd507a33": "N/\\sqrt{T}= (N/\\sqrt{\\theta}) / \\sqrt{288.15}", "ba8c4af6d4f694ab81f9a728208db763": "x_i:S^1 \\subset M", "ba8c4e76ff5f89a174063f9344d5d7b2": "N(3,\\delta) \\leq C^{\\delta^{-2}\\log(1/\\delta)},", "ba8c7ed7f2ab9b734cf406a5da35584d": "\\displaystyle \t-x_1x_2x_3=\\min_{z\\in\\mathbf{B}}z(2-x_1-x_2-x_3)", "ba8cd6886a27297c2c2f2ae70d6cbcd9": "\\|\\ \\|", "ba8d2562016dfdde9498264d6a432e36": "u_x = \\frac{\\partial u}{\\partial x} \\,", "ba8e44ca4fdff057b0a7c1f460d913d4": "\n\\begin{align}\nP(G|D) &= \\frac{P(D|G) P(G)}{P(D)}\\\\[8pt]\n&= \\frac{P(D|G)\\,P(G)}{\\sum\\limits_{i=1}^{n} P(D|G_i)\\,P(G_i)}\\\\[8pt]\n\\end{align}", "ba8e8e9ac4613641f1b0d6e64e205bf7": "\\iint_{\\Sigma}\\left\\{\\left(\\frac{\\partial R}{\\partial y}-\\frac{\\partial Q}{\\partial z}\\right)\\mathrm{d}y\\mathrm{d}z +\\left(\\frac{\\partial P}{\\partial z}-\\frac{\\partial R}{\\partial x}\\right)\\mathrm{d}z\\mathrm{d}x +\\left (\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\\right)\\mathrm{d}x\\mathrm{d}y\\right\\}", "ba8ead7e6ef626f5e3781012b8e95e1d": "n_1-n_4", "ba8ee02596b55dce5401924b35eddc79": " L_{ij} = \\frac{1}{D_j} \\left( A_{ij} - \\sum_{k=1}^{j-1} L_{ik} L_{jk}^* D_k \\right), \\qquad\\text{for } i>j. ", "ba8f2d5bdd297506e9e4e31a22f1cc28": "e_1e_2-e_2e_1 = 2e_3, \\quad e_2e_3-e_3e_2 = 2e_1, \\quad e_3e_1-e_1e_3=2e_2.", "ba8f42d2c4d5f382a018fa5c25f6ee3f": "f \\mathop{\\,$\\,} g \\mathop{\\,$\\,} h \\mathop{\\,$\\,} j \\mathop{\\,$\\,} x", "ba8f8566ab1d6993661748b4e31ff937": "\\sqrt {n\\sigma^2 + 1/n} ", "ba9013bfaaf42aa0e58576a5fe5ca9b2": "\\Lambda=h\\sqrt{\\frac{\\beta}{2\\pi m}}", "ba90213c719e873fcce42232c2b4606f": "(1+{0.1299 \\over 365})^{31}-1", "ba90a50bca98db034cd3cdfea43c2903": " G = (\\mathbb{Z}/n\\mathbb{Z})^\\times ", "ba90ac20b4971b47c6c697db32702e49": "\\phi \\in C^1_c(A, \\mathbb{R}^n)", "ba90b43808c43913a60e095b042f3bec": "\\frac{B_7}{{v_0}^6}", "ba90cd4e7d44593f09a89b5360622b2f": "\\{ \\ldots \\}_{\\text{eq}}", "ba914b44b61c7951020ddc57b906a931": "\\exp^*(x) = \\delta_0 + \\sum_{n=1}^\\infty \\frac{x^{*n}}{n!}.", "ba916a5b4b18e07c09d1e412f8e9083b": "\\mathcal P(x)", "ba917b260f2bac43b832289d150b98a5": "{2^{-n(H(X,Y) + \\varepsilon)}}\\le p(X_1^n, Y_1^n) \\le 2^{-n(H(X,Y) -\\varepsilon)} \\}", "ba91e8dd256b456f3713e5b59d78ab89": "s_A, s_B", "ba92151a4f842421069eb3d129e3546e": "\\frac{dA_{i}}{dt}=k_{B}T\\sum\\limits_{j}{\\left[ {A_{i},A_{j}}\\right] \\frac{{d}\\mathcal{H}}{{dA_{j}}}}-\\sum\\limits_{j}{\\lambda _{i,j}\\left( A\\right) \\frac{d\\mathcal{H}}{{dA_{j}}}+}\\sum\\limits_{j}{\\frac{d{\\lambda _{i,j}\\left(A\\right) }}{{dA_{j}}}}+\\eta _{i}\\left( t\\right).", "ba92210da772d601a61805a7aa76ddf3": "\\hat{\\beta}(\\tau;Y,XA)=A^{-1}\\hat{\\beta}(\\tau;Y,X) .", "ba924c5619ed6d9c072a0a4d11f6ce02": "s = \\frac{2 n_t}{n_x + n_y}", "ba9265bec2734b9ede16b282c25bea46": "\\delta U", "ba92b5cc71cfd8af5382924cfcd87a12": " \\rightarrow (\\lambda x . z) ((\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w))", "ba92c0307bf7f0c7172353e88303a316": " \\rightleftharpoons ", "ba93ae628c254175e10798a20fabb3eb": "M_T", "ba93c47786d5646bb7f6ae0f3b5f2a9a": "\n \\nabla_{e_j} {\\mathbf v} \\ \\stackrel{\\mathrm{def}}{=}\\ v^s {}_{;j}e_s \\;\\;\\;\\;\\;\\;\n v^i {}_{;j} =\n v^i {}_{,j} + v^k\\Gamma^i {}_{k j}\n", "ba942d3a88514f57368c60bb5c9b2452": "S(q)-1", "ba943e739116b527f79ab67270760299": " \\sum_{i=1}^n \\frac{1}{z-a_i}=0\\ ", "ba94918e231bde39c89a3bb2218fce37": "t_{1/2}", "ba94a1bfa9145cc88348e5c2560f59ba": "\\frac{\\partial K}{\\partial t}(t,x,y) = \\Delta K(t,x,y) \\rm{\\ \\ for\\ all\\ } t>0 \\rm{\\ and\\ } x,y\\in\\Omega", "ba94c45ae8b60041c875e6261507e81f": " X_i = \\mu_i + (W_i / Z)^{\\gamma_i}, \\qquad i=1,\\dots,k, \\qquad (5)\n", "ba94d870a039daf0b37fa8cebc1f4a0b": "P \\to Q \\Leftrightarrow \\neg P \\or Q", "ba94eacb55a3dd2cf57a267c32841a73": "\\|\\left(\\Phi^{(n)}_\\lambda\\otimes\\Psi\\right)(\\rho)\\|_p\\le v_p(\\Delta_\\lambda)v_p(\\Psi)", "ba9509c10f9c8d57e176311bc72aef3c": "\\left(\\Lambda/Q\\right)^p", "ba951a6e9a213ed3d1bb64fdb1ce1ada": " \\begin{align} A & = 212175710912 \\sqrt{61} + 1657145277365 \\\\\n B & = 13773980892672 \\sqrt{61} + 107578229802750 \\\\\n C & = (5280(236674+30303\\sqrt{61}))^3\n \\end{align}\n", "ba953073531a1de2f02b1fffb83d726d": "k_{eq} = k_1 + k_2 ", "ba953d381c3cac1c5d3a09195bd582cd": "g(x)=c_1 \\varphi_\\lambda + c_2 \\theta_\\lambda - \\int_c^x (\\varphi_\\lambda(x) \\theta_\\lambda(y) - \\theta_\\lambda(x)\\varphi_\\lambda(y))r(y)g(y)\\, dy.", "ba9541d28a993df60975d22557e96b98": "~ f_b", "ba954b47ab8a54201d5763d876945d81": "E^k", "ba95a086c3d80a5ad9571480efcd5a55": "(-1)\\cdot x = (-x)", "ba95c31172d81280b9751673345220dd": "y_2 = e^{(2-i)x}", "ba961e99548adc7c986617e942dc3156": "\n\\text{cov}\\left(\\log \\frac{\\hat p_{x-1}}{\\hat p_x}, \\log \\frac{\\hat p_x}{\\hat p_{x+1}} \\right) \\approx - \\frac{1}{np_x}\n", "ba9624ae6d7b06a81d0c9e82a6420b42": "\\begin{align}\n\\sigma_2^2-\\sigma_3^2-2\\sigma_2\\sigma_n+2\\sigma_2\\sigma_n&=0 \\\\\n\\sigma_2^2-\\sigma_3^2-2\\sigma_n\\left(\\sigma_2-\\sigma_3\\right)&=0 \\\\\n\\sigma_2+\\sigma_3&=2\\sigma_n\n\\end{align}\\,\\!", "ba962cc8baf3d61c25958242bbcdb98a": "\\boldsymbol{\\mathsf{T}}", "ba9632ed951b71776f6be11677e0911d": "TSHI = LN(TSH) + 0.1345 * FT4", "ba9681b16bfa747ed782862321543e2b": "\n\\left(\\part^2+m^2\\right)\\varphi(x)=j_0(x)+\\left(m^2-m_0^2\\right)\\varphi(x)=j(x)\n", "ba968a334c5e09c054babfd3a775aca8": "r> 9", "ba969f947d179f6b00d4eb46d05fd47e": "\\mathbf{x}^{(1)}=\\mathbf{x}^{(0)}-\\gamma_0 \\nabla F(x^{(0)})", "ba96ad1529cb296068d2b6a960d6de00": "{\\mathbb{Q}}", "ba96dfcf0566d488bdb6a4ac4aa424b0": "V_{\\omega+\\alpha} \\!", "ba96f58f16e488d79d13c7213b076924": " \\pi_F\\circ\\varphi = f\\circ\\pi_E ", "ba97033e8461393c30b72ca4450b947f": "p_i(t) = t - \\lambda_i", "ba974d87268f4dbfab2c05698ee36269": " V_{\\alpha i} \\,\\!", "ba978e664924758f5c3ab9d11d3a8279": " \\alpha_x = \\alpha_y \\ \\stackrel{\\mathrm{def}}{=}\\ \\alpha. ", "ba97bda50c74a00d88a6373c1116cd45": "\\sigma_{xx} - \\sigma_{xz} + \\sigma_{xy}", "ba97f7dab4891ca41acf0567b5be0da1": "h'_{x}(\\alpha) = \\min_P \\{-\\log P(x) : P(x)>0, K(P) \\leq \\alpha \\} ", "ba98060c528e0901d24501e41f3b064c": "c_{6}*c_{8} ", "ba982605b93234413f19e8439669fadf": "\\tau_{Ace}=r_{Ace}C_{Ace}\\,", "ba982d45b139e0a106b917fbc16fb8d5": "\\phi(.)", "ba9844fcba8e49068265d9204d3a7513": "\n\\begin{align}\nr^2 &= \\left(a^{\\frac{n+5-p-q}{8}}\\right)^2 \\\\\n &= \\left(a^{\\frac{n+5-p-q - \\Phi\\left(n\\right)}{8}}\\right)^2 \\\\\n &= \\left(a^{\\frac{n+5-p-q - (p-1)(q-1)}{8}}\\right)^2 \\\\\n &= \\left(a^{\\frac{n+5-p-q - n+p+q-1}{8}}\\right)^2 \\\\\n &= \\left(a^{\\frac{4}{8}}\\right)^2 \\\\\n &= \\pm a \\\\\n\\end{align}\n", "ba98773206a5bda9aa37dedfda2e0d57": "\\frac{\\partial G}{\\partial x} \\frac{\\partial x}{\\partial u} +\\frac{\\partial G}{\\partial y} \\frac{\\partial y}{\\partial u} = -\\frac{\\partial G}{\\partial u}", "ba98fca7493b9da7adc4af5a537d4074": "dy = r \\omega \\cos(\\omega T) \\; dT", "ba995023b7f1dca6f1eb2ca13d156e9f": "e_\\alpha^I e_\\gamma^J", "ba9a6d7ebd57920366a001de0cd0e7b6": "\\{v_i\\}", "ba9a8cf37db813e40756a6eb88c41d0e": "\\eta = \\frac{\\mathit{W}_{1-2}+\\mathit{W}_{3-4} }{-\\mathit{Q}_{2-3}} = \\frac{\\left(\\mathit{u}_1 - \\mathit{u}_2\\right) + \\left(\\mathit{u}_3 - \\mathit{u}_4\\right)}{ - \\left(\\mathit{u}_2 - \\mathit{u}_3\\right)} ", "ba9a9d1380f9de8816819d6af737a0cb": "\\frac{1}{r^3} P^2_2(\\sin\\theta) \\sin2\\varphi = \\frac{1}{r^3} 3 \\cos^2 \\theta \\sin 2\\varphi", "ba9aaebe38451c53e773ab92d3b6f221": "f^p(z_0)=z_0", "ba9ac2115ad4df35f38c1ec0ee90c601": "f(x)=\\frac{\\nu^{\\frac{\\nu}{2}}\\Gamma(\\nu+1)\\exp \\left (-\\frac{\\mu^2}{2} \\right )}{2^\\nu(\\nu+x^2)^{\\frac{\\nu}{2}}\\Gamma(\\frac{\\nu}{2})} \\left \\{\\sqrt{2}\\mu x\\frac{{}_1F_1\\left(\\frac{\\nu}{2}+1;\\, \\frac{3}{2};\\, \\frac{\\mu^2x^2}{2(\\nu+x^2)} \\right )}{(\\nu+x^2)\\Gamma(\\frac{\\nu+1}{2})} + \\frac{{}_1F_1\\left(\\frac{\\nu+1}{2};\\, \\frac{1}{2};\\, \\frac{\\mu^2x^2}{2(\\nu+x^2)} \\right )}{\\sqrt{\\nu+x^2}\\Gamma(\\frac{\\nu}{2}+1)}\\right \\}", "ba9ada8495306ee37f553c37d2854f3a": " \\frac{dI}{dt} = \\beta S I - \\gamma I ", "ba9b0865bcccc864fd38ecf267c9d58f": "x, \\, y", "ba9b168cf3979ac498c61e10ae756ce3": "K = \\frac{|\\tan \\gamma|}{2} \\cdot \\left| B^2 - C^2 \\right|.", "ba9b653031d15d32c54f0f68b0229221": "A=P^{-1} B P", "ba9b869e9d30565168d58300615c1b15": " \\mid \\langle \\psi_{x+} \\mid \\psi_{y-} \\rangle \\mid ^ 2", "ba9b94fa87203a8ba2de1e07a615297e": "\\mathbf{\\bar{5}}\\oplus\\mathbf{10}\\oplus\\mathbf{1}", "ba9b9a828d49a6e0faa2e46f0c64360b": "M^*((Mf)\\cdot F) =f \\cdot (M^*F),", "ba9bf43ad118aadf5af90daf8adbe32e": "a, b, c, d", "ba9c12974f273482904ffea372cf942e": "\\mathbf{\\hat{e}}_i = \\sum_j R^j_i \\mathbf{e}_j = R^j_i \\mathbf{e}_j,", "ba9c159a57992f38d430025575ff9e82": " d = m^2+n^2+p^2+q^2,\\,", "ba9ccf6340002e6d509a5d405f121819": "\\left (\\lim_{x \\to p^{-}}{f(x)} = L\\right )", "ba9ce2ab196b647ae276015f8e827efe": "\\displaystyle{f_{\\delta,y}(z)=f(z) +\\delta (f(z)-f(y))}", "ba9ced8acac51040b5d7cd1f22fbfa6d": "\\sqrt{K}=\\sqrt{S}+\\sqrt{T},", "ba9cef05a14eca7b29fc5f8b437e8a66": "V_{\\text{in}}", "ba9d12c0ccecfe5f2f551f71be184bf3": "{x \\in R^{N}}", "ba9d900c1981afef7b87ebd8aa261914": "\\gamma(p) = \\sqrt{1+\\left(\\frac{p}{mc}\\right)^2}", "ba9d95b1469598bb40243556e8daa2ef": "\nf_X(S) = \\tfrac{\\min(S-1, 13-S)}{36}, \\text{for } S \\in \\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\}\n", "ba9e5e3b215000085ba42de879b9c13d": " \\varphi_{tt}- \\varphi_{xx} + \\varphi\\ = 0.\\,", "ba9e675095ed796f337ce5c9bcd782f8": "\\begin{align} \\mathcal{Z} \\{x^*(n)\\} &= \\sum_{n=-\\infty}^{\\infty} x^*(n)z^{-n}\\\\\n&= \\sum_{n=-\\infty}^{\\infty} \\left [x(n)(z^*)^{-n} \\right ]^*\\\\\n&= \\left [ \\sum_{n=-\\infty}^{\\infty} x(n)(z^*)^{-n}\\right ]^*\\\\\n&= X^*(z^*)\n\\end{align} ", "ba9e8b76836212e141e84052b358d111": " y_{n+k} = \\Psi(t_{n+k}; y_n, y_{n+1}, \\dots, y_{n+k-1}; h). \\, ", "ba9e9f46818c15f83a7a504b9fc217d0": "\\{ x_n \\}", "ba9ece35e50daae9607a6d8b1f8854d8": "\\det A_{33}", "ba9ed23090ed544513a9dd726edd7bd4": "\n {n \\choose k} = \\frac{n!}{k!\\,(n-k)!} = \\frac{\\Gamma(n+1)}{\\Gamma(k+1)\\,\\Gamma(n-k+1)} = \\exp(\\ln\\Gamma(n+1)-\\ln\\Gamma(k+1)-\\ln\\Gamma(n-k+1)),", "ba9fbd2891b5c03df1962332091e0299": "\\prod_{j = 1}^{d} f_{j} (\\pi_{j} (x))^{1 / (d - 1)} = 1.", "baa068b8cb5f55c4ababe7d22941b453": "\\frac{\\partial J_{ij}}{\\partial t}+\\frac{\\partial (v_kJ_{ij})}{\\partial x_k}=\\left(\nx_j\\frac{\\partial P_{ki}}{\\partial x_k}-x_i\\frac{\\partial P_{kj}}{\\partial x_k }\\right) +(P_{ji}-P_{ij})", "baa0ca82ad6b144dfc4f571a15416e5d": "\\beta_1,\\beta_2,\\ldots,\\beta_n", "baa0faec7624f2c00654ba88c9987a55": "\\alpha = 13.3", "baa0faf2e6857dd138517bda00e3f097": "\\theta = \\arg z = \\dfrac{1}{i}\\ln \\sqrt{\\frac{z}{\\overline z}} = \\dfrac{\\ln z - \\ln \\overline z}{2i}", "baa12815d5532cbb2af63416e1613cdd": " \\mathcal{S} = \\int_{t_1}^{t_2} L \\mathrm{d}t \\,\\!", "baa137bef5be13d041506d04c44055e2": "(E_f \\mathbf{\\hat f} + i E_s \\mathbf{\\hat s})\\mathrm{e}^{i(kz-\\omega t)}.", "baa18aba5c66b0b5dca9138c81df18a6": "cons : A\\times List(A)\\longrightarrow List(A)", "baa1a26954eab123acfd1d12504d7e07": "xP'_x(a,b,c)+yP'_y(a,b,c)+zP'_z(a,b,c)=0,", "baa1a7ff11e2771a872953087cda913e": " {\\partial w\\over \\partial \\overline{z}} = \\mu {\\partial w\\over \\partial z}", "baa1bf4a23550d345923ab5f65309625": "\n\\rho^{2} = x^{2} + y^{2}\n", "baa1d93f7c32cc734ae77facfc6e13db": "\\displaystyle{F_n(z) = F(z) e^{z^2/n} e^{-1/n}}", "baa257dcec16aef54b89884d89dfb26c": "1, 4, 10, 23, 57, 132, 301, 701", "baa25c0373e841a749f6eb1868659764": "k/n", "baa2956e0b83dc196644d5e603c77597": "\\partial^{2}B/\\partial x^{2}", "baa29773cb7705d67eefc8b96b27697d": "\nG=\\langle x, y, z\\rangle\\triangleright\\langle y, z\\rangle\\triangleright\\langle z\\rangle\\triangleright 1\n", "baa2a0767ff6c982f816043ad79f0d5f": " \\ f^{(k)} (z) ", "baa2e7f05759669d3896c2af82f53e89": "X=\\lambda \\frac{\\partial f}{\\partial x}(p, q, r),\\,Y=\\lambda \\frac{\\partial f}{\\partial y}(p, q, r),\\,Z=\\lambda \\frac{\\partial f}{\\partial z}(p, q, r).", "baa322c5b1cb29f7b10d4fb8b6c53f28": "z_6=\\chi_{\\psi_{6,6}}(z_6,\\rho_{\\psi_{3,6}}(z_3))=\\chi_0(z_6,\\rho_{3}(z_3))=0+(-x_2)=-x_2", "baa38c023150bbe9db3057e7f62f360b": "S_\\theta \\cap s_i \\subseteq S_\\phi", "baa38e3055f0669d15ccfb70164249ae": "\\mathbf{A} = -\\mathbf{K}^{-1} \\mathbf{Q} C_p(t)", "baa39104ec8e9751f398ec784e10dfc4": "\n\\frac{{x}^{2}}{a^{2}} - \\frac{{y}^{2}}{b^{2}} = 0\n", "baa391107779807079c64fa015c2b8e1": "A_L=A_G=A.\\,", "baa39f91ab24c52c9f2ae1afc54ffe87": " \\ \\frac{\\pi}{2} \\ ", "baa3d5416a77f2342f08e6e445a31e57": "H \\left(\\mathbf{q}, \\frac{\\partial S}{\\partial\\mathbf{q}}, t\\right) = -\\frac{\\partial S}{\\partial t}", "baa3e042d1a6d08b2bb0b68a099656bb": "e=2.71828182845905...", "baa3eebb5a8e3d8eceb12df7f5c60785": "\n C_0^N=\\mathrm{tr}[(\\bigotimes_{j=1}^{N}\\rho_j){[S_N-K]}^+]\n ", "baa40258f8212af9547e48577b377902": "\\mathbf{q} = [q_1\\ q_2\\ q_3\\ q_4]^\\mathrm{T}", "baa489ce969586af081a7434213d759a": "\\mathrm{core}_t(p^e) = p^{e\\mod t}.", "baa4ce9c3eb41eeae2d2df3caa27f814": "P_{x}(\\Delta x)=P(xx)=\\frac{F_X(x+\\Delta\\;x)-F_X(x)}{(1-F_X(x))}", "baa4d2af06996adc04d003dcfe01b86c": "|\\alpha_0\\rangle", "baa52b85c066dbd5eeff3c078a69205b": "n\\,\\!", "baa5401ed00ebd45207ba156e7115463": "\\tau \\in \\mathbb{R}", "baa5cca94b703744ca84bf8cc0f3a542": "\\beta(x)=\\sum_{k=0}^{\\infty}\\frac{(-1)^k}{(2k+1)^x}", "baa5d4f70910cb72cc4d416ffb3e6348": "\\pi_0\\,\\text{Diff}^+(D^n)", "baa604484d742402e9c220a099bf3e1a": "\n \\begin{cases}\n a_1 = \\mu (2-d) \\\\\n a_2 = \\frac{\\mu(d-1)}{2}\n \\end{cases}\n ", "baa64d109209aed8010d5a1cd8d8400a": "\\alpha_{mk}, \\ \\beta_{mk}", "baa65bc4a5f97e0cef812a7721a37ec4": " f_n - \\chi_{[1,2)} ", "baa6622640ecab75cd238a153f65150c": " e(x)=x \\, ", "baa6dbd23c686a57f7e189c911395b09": "V= \\frac{a}{\\rho}\\,\\,S_\\nu(\\sigma)T_{\\mu\\nu}(\\tau)\\Phi_\\mu(\\phi)\\,", "baa701a9413cf33c494f6e11b3f86cc4": "\\Pr\\{h_a(x) = h_a(y)\\} \\le 2/m", "baa70d4e3d4da1bbf342a98fe0c39273": "\\gamma_\\pm", "baa71180b5ff97bbb2d86a9e26a107d9": "t[n]", "baa7ac8ad153d36b886a4518466f10d4": "\\text{Points Earned} = \\frac{\\text{Points earned in 2008}}{100} \\times 40 + \\frac{\\text{Points earned in 2009}}{100} \\times 70 + \\text{Points earned in 2010}", "baa845df6ae9753b7cb09c114ff8f7f0": "\\phi(\\bar{n})", "baa859cfd5304c05ad074ca16514a635": "A_d = ([A_P(0)\\frac{\\lambda_d}{\\lambda_d-\\lambda_P} \\times (e^{-\\lambda_Pt}-e^{-\\lambda_dt})] \\times BR ) + A_d(0)e^{-\\lambda_dt}", "baa8794a95e010ab23fe5bcc71c234b0": "b_n^2=(c_{n+1}c_{n-1}-c_n^2)/(c_nc_{n-2}-c_{n-1}^2)", "baa89a9c78b7f1e3d5714858a2df6cd0": "\\, A \\mapsto NAN^{-1} .", "baa8b18bcf0f26e6cbb3c303b40d768f": "f \\bigg( \\sum_{k=1}^\\infty a_k 3^{-k} \\bigg) = \\sum_{k=1}^\\infty \\frac{a_k}{2} 2^{-k}.", "baa8d956eb32dd20ebbaf6b991aff34c": "\\sum_{b=0}^{a} x_b {_{a}^{b}} \\ \\text{S} \\rightarrow \\sum_{d=0}^c y_d {_c^d} \\ \\text{P}.", "baa909e99da564c44b2cef48346c7df4": "u(r_f) = Eu(r_f + \\pi + x).", "baa914baba914f160ca34a682c433f4d": "\\mathcal O_X", "baa922952cf9bd66f04a5a29991d78f0": "B_\\nu(T) = \\frac{2 \\nu^2 k T}{c^2}.", "baa93c172f43421c3476a0a8725b7598": "Q_c^{(c)}(t) =0", "baa94765bb86470eae4dff66e1ef4e15": "X = \\sec\\,z \\,-\\, 0.0018167 \\,(\\sec\\,z \\,-\\, 1) \\,-\\, 0.002875 \\,(\\sec\\,z \\,-\\, 1)^2\n\t \\,-\\, 0.0008083 \\,(\\sec\\,z \\,-\\, 1)^3 \\,,\n", "baa9824eab5670bccf4316038cb0dda0": "{{P \\over Q} + 1} = {{X + Y + \\sigma (100 + \\alpha + \\beta)} \\over {Y + \\sigma (100 - l_1 + \\beta)}}", "baa99ca70610391881b6589666178fba": "\nT(TM\\setminus 0) = H(TM\\setminus 0) \\oplus V(TM\\setminus 0)\n", "baa9eb1ad696761df860689931a35827": "x = (p+q) \\cos\\theta\\,", "baaa0f7b2fe75e7b0ad7d0cb69d93286": "f'(x_n) \\simeq \\frac {f(x_n)-f(x_{n-1})}{x_n-x_{n-1} }, ", "baaa4f3867c6afb16fce31f0c1377ae2": " L_0 = - \\frac {\\beta} {sM} \\, ", "baab41a5d4ac1b8e4c953f5808d09236": "P=p_k", "baab4b64dc889852439c95b9dd3bbc26": " \\delta(t) \\, ", "baab98e8a30363c3072fbb644951cf3b": "2 b^2", "baaba1c869f1a7eb85b06302d83578b5": "GL(n,\\mathbf{C}) < GL(2n,\\mathbf{R})", "baabb199ba3b1566da6945bd31e14def": "\n\\,y = \\,a \\sinh \\xi \\sin \\eta,\n", "baac0f0729151e67b45adfda61f21846": "\\ u^*=Q(t)u, ", "baac2e6aa5b6100bf82d373171cc9afd": "\\beta_{\\gamma+1}\\setminus\\beta_\\gamma", "baac3574f6a9a97b1f84b25c63e62a58": "(X,x_0)", "baac480cfa2532ab62a4eae83da5a234": "L(\\vec{r},\\hat{s},t) \\approx\\ \\sum_{n=0}^{1} \\sum_{m=-n}^{n}L_{n,m}(\\vec{r},t)Y_{n,m}(\\hat{s})", "baac4caf11c8e58ed14b5b2c14df3612": "E_{s}", "baacbe90a611155ee8de5fa5e833e134": "b_{1}", "baad638b44f1b3c7d797dace412d1253": "v_j, j = 1, 2, . . ., n.", "baad724ef903dd1825688d08bf49df0f": "b=\\frac{1-\\mid{\\frac{\\partial}{\\partial{z}}P_c^p(z_0)}\\mid^2}\n {\\mid{\\frac{\\partial}{\\partial{c}}\\frac{\\partial}{\\partial{z}}P_c^p(z_0) +\n \\frac{\\partial}{\\partial{z}}\\frac{\\partial}{\\partial{z}}P_c^p(z_0)\n \\frac{\\frac{\\partial}{\\partial{c}}P_c^p(z_0)}\n {1-\\frac{\\partial}{\\partial{z}}P_c^p(z_0)}}\\mid}", "baada5b81dd5ed15dcef0792df8f5b0e": "\\sup_{x\\in D}f(x) =f(x_0)", "baae909d89bb66e1e8dd868caedb43d9": "\\frac{du_2}{dt}=J_2 \\frac{dT_2}{dx}", "baae9c37ff11d55f95670ee11fc4dd94": " A f = f'\\,", "baae9c442fc628d0a7e752b4bf0e2e17": "M'= \\frac{M}{y_c^2}", "baaed71801434ae6fdae6489b28d2e05": "X \\rightarrow Y \\in S^+ \\land Z \\subseteq H~\\Rightarrow~(X \\cup Z) \\rightarrow (Y \\cup Z) \\in S^+", "baaf048c66e6487ae6fa23b2c7997b44": "\\forall x \\exists y\\, \\phi(x,y)", "baaf263c5593fa5f44fb1a5859996e72": "f*F_n \\rightarrow f", "baaf2ff5ab9695bc7670534ef13612c7": "\\mathbf{G}_2", "baaf44b4d1d7d66d31e42be8a8f5185d": "d = 1 / p", "baaf610d2fd731ae81ed2ace55f4d37e": "\\begin{align}\n \\zeta(s) &= \\sum_{n=1}^\\infty \\frac{1}{n^s}, \\Re(s)>1 \\\\\n \\eta(s) &= (1-2^{1-s})\\zeta(s) \\\\\n &= \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s}, \\Re(s)>0\n\\end{align}", "baaf69d74d1ce5987be0a0e29650c461": "\\alpha = \\arctan{\\frac{1}{n}}", "baaf9bc7c15871fa8a7ddfd6f91feb7c": "k_2\\,", "baafef37984789b810eac3694d638240": " \\gamma^2 x^2 + \\gamma^2 v^2 t^2 - 2 \\gamma^2 v t x + y^2 + z^2 = c^2 {\\gamma^2} t^2 + \\frac{ \\left( 1 - {\\gamma^2} \\right)^2 c^2 x^2}{ {\\gamma^2} v^2} + 2 \\frac{ \\left( 1 - {\\gamma^2} \\right) t x c^2}{ v}", "bab01555432a0d2ea0f04a0cfa2ad3ed": "f_1 \\otimes f_2", "bab0647dc460bb805dde529575ea032c": " \\tan \\theta = b/a.", "bab07a8d585ac51c785590b1f383de3a": "(A, B)", "bab0de6c0d4aa8dba0d31a4bfb739e12": " ZY^2=X^3+aZX^2+16aXZ^2, ", "bab0e1f271310a07fe05f14291dcd73f": "b\\in B\\setminus A", "bab110509cf9e28060a059bdde863bbb": "\\langle A |^\\dagger = |A \\rangle, \\quad |A \\rangle^\\dagger = \\langle A |", "bab13da8f971f7af2dda91dedb3a6760": "\nJ_K(t,t^\\prime) = 3(1-2\\nu) J(t,t^\\prime),~~~J_G(t,t^\\prime) = 2(1+\\nu) J(t,t^\\prime)\n", "bab1685168134a1d536e73e478e1a7cd": "\\vec{\\mathbf{F}} = \\{ \\vec{f}_1, \\vec{f}_2, \\vec{f}_3\\}", "bab2268911a9107df58fd017091cb88a": " 2~r~\\sin\\theta \\,", "bab227d364d28cfd877cbc3fcb89caf2": "2n-3j-3>0", "bab26f258476c3b3c67abd4c7e32c33e": " \\oint_{\\partial \\Sigma(t)} \\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{F}/q(\\mathbf{r},\\ t) = \n\\oint_{\\partial \\Sigma(t)} \\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{E}(\\mathbf{r},\\ t) +\n\\oint_{\\partial \\Sigma(t)}\\!\\!\\!\\! \\mathbf{v} \\times \\mathbf{B}(\\mathbf{r},\\ t)\\, \\mathrm{d} \\boldsymbol{\\ell}\n", "bab2df46c18ba7c3859f29a8664e0274": "\\left( H_m \\right)_{k,n} = \\frac{1}{2^\\frac{m}{2}} (-1)^{\\sum_j k_j n_j}", "bab2f779451c3263f2e436c7da62662f": "\\begin{bmatrix} \\mathbf{x} \\\\[5pt] \\mathbf{x}\\mathbf{x}^\\mathrm{T} \\end{bmatrix}", "bab36981ac0c2be165df4d57fa1198cc": "T_1 / T_2 = - Q_1 / Q_2 \\,\\,\\,.\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)", "bab37e123b1c2c97a4bafc18cf14ddb1": "\\,(1+\\frac{0.1139}{12})^{12}=1.12", "bab37e7085f7c0f63629b74eebdb84aa": "{\\varphi^2 +\\varphi -1 = 0}", "bab385b8634317f5a2506b96dd990db2": "f_{out}(x) = f_{in}(x) + \\alpha f_{out}(x-1)", "bab437bc8e19e5b989342d93d86fe5d6": " \\vec{R}_{cm} = h\\mathbf{\\hat e}^3", "bab462b87c333de7bfb2e058243071f7": " c_{\\frac{p}{q}} = \\frac{e^{2\\pi i\\frac pq}}2\\left(1-\\frac{e^{2\\pi i\\frac pq}}2\\right).", "bab470a732ce4e5819399525494645ea": "\\Phi(\\omega)=\n\\frac{1}{\\sqrt{2\\pi}}\\,\\frac{\\sigma_\\varepsilon^2}{1-\\varphi^2}\\,\\frac{\\gamma}{\\pi(\\gamma^2+\\omega^2)}", "bab4bfc910e3da844bc4728a29a8a400": "\\langle \\Phi _E (f) h_1 , h_2 \\rangle = \\int _X f d \\langle E(B) h_1, h_2 \\rangle", "bab4dc7e3ec4538d8376a5b0af6cc9a4": " \\tfrac{\\text{baseline} \\cdot \\text{altitude}}{2}", "bab502ec6cc805b203df2f76889e4686": " Y \\colon(\\tilde\\Omega,\\tilde{ \\mathcal F}, \\tilde{\\mathbb P}) \\rightarrow \\mathbb R ", "bab5583a97aae5ad9fee3a156584c2f7": "(\\kappa x.f)\\circ \\operatorname{lift}_\\tau(c) = f[c/x]", "bab560eccaa0f6ac28000fa7149ec620": " \\operatorname{drop-formal}[D, \\lambda x.\\lambda o.\\lambda y.o\\ x\\ y, F] ", "bab57b9b0c9a3cf0a648e8dc76f7b2c1": " \\begin{cases}\n\\text{Mesh 1: } I_1 = I_s\\\\\n\\text{Mesh 2: } -V_s + R_1(I_2-I_1) + \\frac{1}{sc}(I_2-I_3)=0\\\\\n\\text{Mesh 3: } \\frac{1}{sc}(I_3-I_2) + R_2(I_3-I_1) + LsI_3=0\\\\\n\\end{cases} \\, ", "bab5b53c98e031a83ef5fd09fcbfa5f7": " ds^2 = {4(du^2 + dv^2)\\over (1+u^2+v^2)^2}.", "bab5e8f9ccb3a2bb04f0b1bd8e9d8da4": "~ n_1+n_2=1 ~", "bab5fdfec445af09c9bd493363f6890c": "\\delta_{ijk}\\, ", "bab600a18dae568479eb615c58c7573b": "2m-1", "bab60175392463ff66d48083422156c9": "\\displaystyle H(e^X) = A_\\rho(e^X)", "bab6339bec35c71863f7e263cb606116": "\\beta_x(\\alpha)", "bab65e3465fe96a3595c671c9db551f1": "F_{ab}", "bab66dd001e1687b60e803a688133097": "=\\,\\dot{m}\\,v_{eq}", "bab6a6b00cd74919c40a0c5539d9f6d5": "G=K\\rtimes H", "bab6eb70ec7f480b9f5b175a49173dea": "\\sin \\,0 = 0", "bab706cbe46a08fb4d7afecac650de34": "\\ln J_2 - \\ln J_1 = \\ln A_G - \\ln A_G + 2\\ln T_2 - 2\\ln T_1 + {\\frac{W}{{kT_1 }} - \\frac{W}{{kT_2 }}} ", "bab7107646e81b58d810cce54a7ff590": "\\scriptstyle h\\left(E F\\right)\\,=\\, h\\left(E\\, \\mid\\, F\\right)\\,=\\,\\max \\left(\\, h(E), h(F)\\,\\right)", "bab72a1a2b33eb2807b1ea9a8cb24df1": "3.14159261864 < \\pi <3.141592706934", "bab74aa43cb7a272691d93dd1eb082fc": "\\int^{\\infty}_{0} f_k e^{-x}\\,dx,", "bab7635cfb583681be215462161e878c": "\\rarr\\!\\larr", "bab7a758463f6b2a927ae4840e3decc1": "\\sigma\\circ\\sigma=id_{V}\\,", "bab7b15bc7ca37be5b3070feae64f1b7": "M_{ax} = \\left | {d \\over d(s_o)} {s_i \\over s_o} \\right | = \\left | {d \\over d(s_o)} {f \\over (s_o-f)} \\right | = \\left | {-f \\over (s_o-f)^2} \\right | = {M^2 \\over f} ", "bab7dc5258abd0b325f21e10e8bb712d": "a=0.6+i0.45", "bab7e6f7b31abd2365859f615069f04d": "K-\\tfrac{2G}{3}", "bab7f8a600b04fcd059a62b58a7163e4": "A\\subseteq X'", "bab823a7af6de5e8101b30f3652953cb": "\\sum_{k=-\\infty}^\\infty [F(n+1,k)-F(n,k)] = 0 ", "bab850aa25261a9a52edb46c81adb230": "\\varphi_\\lambda(g)=\\int_K \\alpha^\\prime(kg) \\, dk,", "bab8934edf3459496de15161170914a1": " \\vec{F} = -\\ C \\vec{r} + m \\Omega^2 \\vec{r} + 2 m (\\vec{\\Omega} \\times \\vec{v}) ", "bab8a60bed9a6153bd7168dcf1c9d02e": "g^{ac}", "bab90fe1c208d169454ccf49d9ed3672": "d_{jk}", "bab91b68b91b07abd08b769f8b6332a9": "Z(s) = { V(s) \\over I(s) } \\bigg|_{V_o = 0}.", "bab91e2fba314a76b28fa106e304e729": " \\sup_{B}\\frac{1}{|B|}\\int_{B}e^{-\\frac{\\phi-\\phi_{B}}{p-1}}dx<\\infty. ", "bab944c919598d3b2b72cda7428fac4b": "\\text{b. If system reactance is given in short-circuit symmetrical rms kva or current, convert to per-unit as follows:}", "bab949c5fc82553497e8189f0138cef3": "S_1 = Y_1", "bab94b4b8c17fc20459a49d962119945": " |\\overline{PR} | = \\frac{|y_0 - m||B|}{\\sqrt{A^2 + B^2}}.", "baba22b9a5810fe16b869cba8615b105": "\\Delta \\mu _{Xm+} (kJ\\cdot mol^{-1}) = \\Delta G(kJ\\cdot mol^{-1})", "baba46f75c4d54e1f76697c72e1edaf9": "\n\\xi_{++}(\\Delta\\theta) = \\langle \\gamma_+(\\vec{\\theta}) \\gamma_+(\\vec{\\theta}+\\vec{\\Delta\\theta}) \\rangle\n", "baba7a8ae0082067bb908536d770c0a5": "\\{g, k\\}", "babaad776533a325e5a5c2ac716ce1a4": "f(t) = \\int_{-\\sigma}^\\sigma e^{jxt}g(x)\\, dx \\qquad (t\\in \\mathbb{R}), \\qquad \\forall g\\in L^2(-\\sigma,\\sigma),", "babac06303aae8ed0cf57f1d78bbdf41": "\\mathcal{O}^+(z_0)", "babbdc3168e738712438ea16f8d80aa6": "\nx = \\cfrac{1}{1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cfrac{a_4}{b_4 + \\ddots}}}}\\,\n", "babc7d8472e85733f1d2a23bff21f56d": "\n\\to\n\\begin{pmatrix}\n2 & 0 & 0 \\\\\n0 & 6 & 12 \\\\\n0 & 0 & -12\n\\end{pmatrix}\n\\to\n\\begin{pmatrix}\n2 & 0 & 0 \\\\\n0 & 6 & 0 \\\\\n0 & 0 & 12\n\\end{pmatrix}\n", "babd191d2227f2fbd1360e9256d94a95": "(\\vec r -\\vec a)\\cdot \\vec n_0 = 0.\\,", "babd84e7489bf81cc58271754dd6d5ca": "\\eth\\eta = - (\\sin{\\theta})^s \\left\\{ \\frac{\\partial}{\\partial \\theta} + \\frac{i}{\\sin{\\theta}} \\frac{\\partial} {\\partial \\phi} \\right\\} \\left[ (\\sin{\\theta})^{-s} \\eta \\right].", "babd85cc8ad08e7b82b236349b390483": "u(x, \\tau) = \\frac{1}{\\sigma\\sqrt{2\\pi\\tau}}\\int_{-\\infty}^{\\infty}{u_0 [y]\\exp{\\left[-\\frac{(x - y)^2}{2\\sigma^2 \\tau}\\right]}}\\,dy", "babda774fc3921c9ea706c28f2454abe": "\\frac12+\\frac13+\\frac17+\\frac1{43}=\\frac{1805}{1806}", "babdfd10865b9b81304ab37cb153beb4": "{\\Delta} G={\\Delta} H-T{\\Delta} S", "babe5557d034cacc8df486b4f1a50433": "x(=\\{x_i\\})", "babe6946e09d56c12c94fa1956855b07": "(x,y,z)\\mapsto(-x,-y,-z)", "babe82b9009bad9cd5f4bc4c20328c40": "P(S|N,n,s=0) \\propto {(N-S-1)! \\over S(N-S-n)!} = {\\prod_{j=1}^{n-1}(N-S-j) \\over S}\n", "babe9d07a41a01686dfe6dd2ef6de286": "x=S_2\\cos(\\theta_2+\\theta_3)\\;", "babea9f91c976d82e65085802b30759b": "(x^2-1)\\frac{d}{dx}(x^2-1)^n = 2nx(x^2-1)^n", "babed71eb5b11a1271d72412612bbc91": "R=R_s \\cos \\alpha \\ ", "babf4717bb0fde997db926db0668f13e": "\\sqrt{3} \\times \\sqrt{5}", "babf66b60565b31785ebb0c6b1753a74": "x^{\\omega}/{\\omega}", "babf7dff69ec1d43855103aa8e94a6b0": "0< t_i < T.", "babfce4f1b99f82fea6858e19a634305": "\nR=P(2 | \\vec{y})-P(1 | \\vec{y}) = \\mathrm{erf} \\left [ \\frac{q_0 - \\bar q (\\vec y)}{\\sqrt 2 \\sigma_q} \\right ]\n", "babfce8c64fc41ca6895d7f429fc5b8c": "\\kappa(\\beta) = - \\mu^{-1} \\beta", "bac04aa589eda23dfeb9dc457485de2f": "v_{\\infty}", "bac04c238ea62df1128650c54bb78639": "s_0\\le s\\le s_1", "bac05dad8a5db4e98f071bdd4249b2d7": "120 \\pi", "bac0bac98b88cd8a13cfb766a2b0ebbe": "\n \\theta_n^{(h)}= \\frac{1}{I}{\\sum_{k=1}^{K} \\Big(x_k-m_1^{(h)}\\Big)^n \\, h(x_k)}\n", "bac1059b618b2183b941a2b949e93051": "\\mathbf{S}(0) = (S_0(0),\\ldots S_N(0))'", "bac106869f266b1fda2801b27a99b20e": "i \\mapsto \\begin{pmatrix}\n 1 & 1 \\\\\n 1 & -1\n\\end{pmatrix}", "bac1123299fbfa6c819117534ca12eac": "\\frac{dx(t)}{dt}= \\alpha*(y(t)-h)", "bac1ab1b8ec3a251cabf813431c52f19": "N+1", "bac1f656d7e2f8b7b4c5d50e2f4526c5": "(\\hat{Z}(x)-Z(x))", "bac21673a2e7bc2a60284a3e95fd7dcc": " \\{ a^n b^n : n \\ge 1 \\} ", "bac2280b4bbf1995e442c384d848d31c": "\\alpha=\\frac {K_i}K_j=\\frac {(y_i/x_i)}{(y_j/x_j)}", "bac2697eb974d759245f21185b1130ee": "X\\le 1 ", "bac2c4e48a6108c7d596fe48f9161c81": "-\\overline{\\rho v_i v_j}", "bac35b1be7ddf96fc78a0f2fea9bfa2f": "(r_E + R_E) \\| (r_\\pi + R_E) ", "bac38a549df5e2eb6faad6c126389e24": " 4.40 = y_{1ss} + \\frac{15.0^2}{2(32.2)y_{1ss}^2} ", "bac3c9c1f3a17be78f7c430ab959585c": "\nP = - \\frac{h^2}{2\\mu u_0} \\left(\\frac{dp}{dx}\\right).\n", "bac3d6a280036a5ac4a57747367d60ad": "y^{2}=4a\\left(x+a\\right)", "bac3dfa0d459b91e7b5f6a47c07eaa5e": " \\lambda(s)=\n\\begin{cases} \n\\phi &\\text{if } \\lambda_{i^*}(s_{i^*})=\\phi \\\\\nC_{yy}(\\lambda_{i^*}(s_{i^*})) &\\text{otherwise}.\n\\end{cases}\n", "bac3f79fd08a26dafa320462e141e7fc": "\\log(k) = 0.6s_EN + 0.6s_EE", "bac4581ce84ae2fdeff73c92d3552999": "T=T_c", "bac4a7fe320df1d892715ede7817a9f9": "b = ", "bac4f3cfdf892783a2b29fb5efd1fbac": "\\nabla\\cdot\\mathbf{B}=\\mu_{0}\\nabla\\cdot(\\mathbf{H+M})=0,", "bac5404ed49654f8dce8ac687de3f2e9": "f \\colon V \\to W ", "bac590c0e60c9e13fba55ee465e9032a": "\\sum_{N=1}^\\infty P(n|N) = \\sum_{N} \\frac{[N \\ge n]}{N} = \\sum_{N=n}^\\infty \\frac{1}{N}", "bac5ea322cb0af15bcf1e60e1efd41f9": "{R_\\mathrm{total}} = \\frac{R}{N}", "bac604b221ec2a4b7853f7e5b7d0caed": "\\mu_A(x)", "bac626a9370d3087eec9bb766eb8bfd8": "\\ Y(s)", "bac6fc1614afb248b94164f556a31b22": "\n\\exp\\left\\{\n-\\beta \\sum_{i=1, i-axis and the major axis of the ellipse.\n\n==== Parametric form in canonical position ====\n[[File:Parametric ellipse.gif|thumb|right|200px|Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly ''t'' is the angle of the blue line with the X-axis.]]\nFor an ellipse in canonical position (center at origin, major axis along the ''X''-axis), the equation simplifies to \n:X(t)=a\\,\\cos t", "bade67761084cb5b1601e405143fb5db": "\\textstyle{\\frac {\\log(4)} {\\log(2(1+\\cos(85^\\circ)))}}", "bade6c43bc660edb01c9f71388966539": "|f(z)| \\leq C e^{\\tau|z|}, \\quad z \\in \\mathbb{C}", "bade72e8cda3786d020f3d10b9114167": "X_0 = N - a_0^2.", "bade944662ce53cb4f358d50f414c8d5": "(k=1\\dots d)", "bade98231f65e53147ff990f621a9d34": "\\log x", "badea6429d8322bee59b555f933f6acd": "R / \\cap \\mathfrak{a}_i \\simeq \\prod R/ \\mathfrak{a}_i, \\quad x \\mapsto (x \\text{ mod } \\mathfrak{a}_1, \\ldots , x \\text{ mod } \\mathfrak{a}_n)", "badecb6f6a9700c69e5abfcac2202294": "\\sum_{j = 1}^\\infty \\left\\lfloor \\frac{n}{p^j} \\right\\rfloor,\\ ", "badf1138d41ffa4a6aa8d332bf3aa87a": " \na_{30}\\ne 0,\n", "badf81deee879e5aea507378d09dcc3d": "g_3 = g_3(x_6)", "badfb077b831e846fd08ad710475943b": "OTP = MAP - (AMP/3)", "badfbdfaf954893efa0e9a44b223d2ff": "H_1(M)=\\mathbb{Z}^{2g}", "bae03d86f30b1a46bdbc5c453c751d82": "\\scriptstyle R = \\sqrt{s}", "bae0a3c23be572e582d563a9e987cb06": "\\frac{1}{2}\\left[1\\!+\\!\\frac{x\\!-\\!\\mu}{s}\n\\!+\\!\\frac{1}{\\pi}\\sin\\left(\\frac{x\\!-\\!\\mu}{s}\\,\\pi\\right)\\right]", "bae0d3cdd4a9088d17091786388f7ed1": "L(X) + h(x) \\le L(X) + d(x,y) + h(y) = L(Y) + h(y)", "bae10214f07acb6fe2fba117ca64b299": "C_{p_1,p_2}^0,C_{p_2,p_3}^0,...C_{p_{m-1},p_m}^0", "bae1037ce5b22dae3814b60ed7ebb354": "(z_{1k},z_{2k})", "bae155438877126b42cbddee193c048a": "f_t", "bae1702c310b0e688d8ba226b8531709": "A(D)f(x) = 0", "bae1a1195e7864425df3267cfe4dbd45": "R_{c,\\theta}(p) = c + R_{0,\\theta}(p - c).\\,\\!", "bae1b27e402251662d0addc28fc95b54": "x(0)=\\phi(0)", "bae1bd6b42ef40c389767a39e8b81762": "~T(\\gamma)=2\\pi\n\\left(\n 1 + \\frac{\\gamma^2} {24} + O(\\gamma^4)\n\\right)\n~", "bae1d6c42845f1a2f0ee483e89fda10b": "(nN)^{O(1)}", "bae332353ff0b2e9afcb21c3ecc5888d": "\\sin a\\sin b", "bae33b2138b42213df38823b2cb646f3": "(x^2+y^2-ax)^2=b^2(x^2+y^2). \\,", "bae35cc8970cf4b5799cf1750875fee8": "f(n)=(3n+1)/2", "bae39854ee9cb514ff5221ef7ec93d07": "\\mathbf{r}\\rightarrow \\mathbf{r}(-t)", "bae3bd8c501bff82621a0b8329334739": "\\sigma_{\\mathrm B}", "bae3d6023e15e7afb492a84c828371b3": "\\scriptstyle{R_l^m}", "bae3ee37212197763e7935c586954c55": "\\scriptstyle \\check{\\mathbf{q}} \\;=\\; [q_1\\ q_2\\ q_3]^\\mathrm{T}", "bae407c6f2a426eaa43856f268b11d1c": " \\pi(\\mathbf{x})=\\prod _{i=1}^J P(Y_i=x_i).", "bae42440374e6e03e384da68a898a7d8": "[a]_\\sim = \\{x\\in S\\vert\\; x\\sim a\\}", "bae457fa50b4453414e59d840f292f24": "\\|AB\\|\\leq \\|A\\|\\cdot\\|B\\|", "bae47dd8842c5eb498ba02d3190bf6bb": "\\frac{x_1}{x_0}, \\ldots, \\frac{x_n}{x_0}, \\frac{y_1}{y_0}, \\ldots, \\frac{y_n}{y_0} ", "bae4840189ced03034d2ebbb9f139216": "u_i=\\varepsilon_i - \\beta\\nu_i", "bae4a0257e695ef4f0a4ef8489a25172": "\\lim_{x \\rightarrow c}m(x)=\\lim_{x \\rightarrow c}M(x)=\\lim_{x \\rightarrow c}\\frac{f'(x)}{g'(x)}=L", "bae4a4a3c25370a16eb78ebdf150428a": "\\mathbf \\theta (x)= \\int \\frac{M(x)}{EI}= -\\frac{5}{3} x^3 + \\frac{75}{2} x^2 + C_3(\\frac{m}{m})", "bae5010d3a3e6c1fd5c4a4cb2a1e6988": "I_{sat}", "bae5223a1b5cc633ce59c914a993bf24": "A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix},", "bae571387ef11f57c6d4b0d2cfb81db7": "\\,W = -\\Delta U", "bae5ae20b85f0c5250c1217197530a07": "C(b)", "bae6151366c748a8f3135f4bfd918c31": "t=\\tau,\\,\\tau+1,\\,\\tau+2,...", "bae6227fc0a5295d2dc6ca499c4727f7": "X_{\\hat{\\phi}}\\,,Y_{\\bar{\\lambda}}\\,,Z_{\\tilde{\\eta}}\\,,T_{\\mu'} \\cdots ", "bae64d80dd1b3c8cd705f63eedcf72b3": "G(z) z^{-k}", "bae69c89b03bf2b1e1fb98f985d0c043": "r_0^2 - a^2 = 1. \\, ", "bae6c0da55fe9fa6e90aa865b298c900": "z = A_{z} \\cos \\left(\\omega_{0} t + \\phi_{z} \\right)", "bae6c50ed8fad779e343e97f98930582": "({\\mathbf v},{\\mathbf w}) \\mapsto e^{i{\\mathbf v}\\cdot {\\mathbf w}}", "bae6d076b0af6d766f00ea312018b575": "p_{t-1}", "bae6db451c1e4c34bdf29590745370ff": "\\{A_i\\}_{i \\in I}", "bae7119f1b2c3660f88464dad8a7bfae": " v \\mapsto q v q^{-1} ", "bae757619ec763cb2dc31e3a62659a28": "f(n) = \\left[ \\frac{1}{2} (n^2 + n) \\right] T_6 + \\left[ \\frac{1}{2} (n^2 + 3n) \\right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7\n", "bae79f29df0aff6d0bf7fbd05be2b5cc": "k_0=\\sqrt{2m E/\\hbar^{2}}\\quad\\quad\\quad\\quad x<0\\quad or\\quad x>a ", "bae7a7ec39642bc006216e7786faf231": "\\frac{1}{\\sqrt{1-4x}} = 1 + 2x + 6x^2 + 20x^3 + 70x^4 + 252x^5 + \\cdots. ", "bae7ea023996f4436232194bdf34b3af": "c_R(x)=c(x)", "bae80e1ed605d486dffa57793c1bd9ef": "\\mathbb{Z}_4 \\times \\mathbb{Z}_4", "bae8278af81bad8f9e05d9514cbcdc85": "z = u(x,y)\\,", "bae84c968e61e660a485ef119b63a208": "Q_{next}", "bae8a8c45b14c9c09cb4b775a7fa3b55": " \\hat{U}^{\\dagger} \\approx I - i\\hat{H}^{\\dagger}. ", "bae8d89d8e9b037b4871191134f1cb1f": "[{\\mathcal L}_m,{\\mathcal L}_n]=(m-n){\\mathcal L}_{m+n}", "bae90bbf07b6bae749053ba35d9e876b": "2 \\alpha", "bae93c90b9ca0a57b5f20d93ed0ae52c": "\n\\vec{\\alpha}(\\vec{\\theta}) = \\frac{1}{\\pi}\\int d^2 \\theta^{\\prime} \\frac{(\\vec{\\theta}-\\vec{\\theta}^{\\prime})\\kappa(\\vec{\\theta}^{\\prime})}{|\\vec{\\theta}-\\vec{\\theta}^{\\prime}|^2}\n", "bae9c6ffd8a59511914ee5ecc34dba5e": "p_{e'}^{\\, 2}c^2 = (h f)^2 + (h f')^2 - 2(hf)(h f')\\cos{\\theta}. \\qquad\\qquad (2) ", "bae9e6fc40bae491dc669bd36a45ad26": "p^*", "baea3aed6334fc6836fd96e2ba1f481b": "\n{\\varphi}_{{\\lambda}_{1}}\\circ\\delta_{[1,{j}_{1},{c}_{1}]}[I]\\cap\n{\\varphi}_{{\\lambda}_{2}}\\circ\\delta_{[1,{j}_{2},{c}_{2}]}[I]\\neq \\varnothing\n", "baea842a53d22ca7669e25caa7183568": "\n\\Delta E_n= { h\\over T(E_n) }.\n", "baea939835c7286a270227db063eb725": "x=a\\ ", "baea994ff2ceb1050bc45e4bd1310553": "a(x)", "baeaec63e19d225c09956138c519dfc4": "\\int \\tan (x) \\,dx = \\ln{\\left| \\sec (x) \\right|} + C", "baeb020211e4b16960873059cf342d5e": "f_1(x) = f(x) \\ \\ \\text{if} \\ \\ |f(x)| < s, \\ \\ \\text{and} \\ \\ f_1(x) = s f(x) / |f(x)| \\ \\ \\text{otherwise}.", "baeb16871e6d72e16baab7a4c14f4c54": "\\psi = \\tilde{\\psi}", "baeb190e1b1ffc516aa28e6b1cdbb80a": "R_Y \\geq 5", "baeb24a6b69ff0541d84c9aed01467d0": "\n\\Phi(z,s,a+x)=\\sum_{k=0}^\\infty \\Phi(z,s+k,a)(s)_{k}\\frac{(-x)^k}{k!};|x|<\\Re(a),\n", "baeb4ac3dd056379e7a4f7dbdc4f4589": "\\phi_{c_1,c_2,t}", "baeb977ede52407099dae4050e8c2fc5": "|f(s)-f(t)| < \\varepsilon/3\\,", "baebda531a2ca8eeeb920632d65767b1": "C_V = \\frac{ \\pi^2}{2}R \\frac{T}{T_F}", "baec90b7fc5679b03165033a3c962196": " A f(x) = xf(x) ", "baecf9b3b8c8fc6b832abee091aa7a95": " \n\\int \\exp\\left( - \\frac 1 2 x^T A x \\right) d^2x\n= \\int \\exp\\left( - \\frac 1 2 \\sum_{j=1}^2 \\lambda_{j} y_j^2 \\right) d^2y\n= \\prod_{j=1}^2 \\left( { 2\\pi \\over \\lambda_j } \\right)^{1\\over 2}\n= \\left( { ( 2\\pi )^2 \\over \\prod_{j=1}^2 \\lambda_j } \\right)^{1\\over 2}\n= \\left( { ( 2\\pi )^2 \\over \\det{ \\left( O^{-1}AO \\right)} } \\right)^{1\\over 2}\n= \\left( { ( 2\\pi )^2 \\over \\det{ \\left( A \\right)} } \\right)^{1\\over 2}\n ", "baed24671e637f3d719adffba6c5c7de": "={u}\\frac{dv}{dx} + {v}\\frac{du}{dx} ", "baedb64646b8243d9b631e787dc24566": "S_m - S_n < a_{m}", "baedec314f9f3343464cfb1c50541ebf": "a_1 < 0 < a_2", "baee23a79d84293517b16436294f9d73": "\\mu^n = 4", "baee690a2cb18da629e83692a3350d58": "\nI(u+x,v+y) \\approx I(u,v) + I_x(u,v)x+I_y(u,v)y\n", "baee7954015e2064f783597b5b46b98e": "\\infty.", "baeea2cdff8986b808f26e2baeeab0ec": "0 \\in B' ", "baeed5a86ed74ced1b1874334fc3d116": "\n \\int_{\\Omega} F_{ij}\\,G_{pj,p}\\,{\\rm d}\\Omega = \\int_{\\Gamma} n_p\\,F_{ij}\\,G_{pj}\\,{\\rm d}\\Gamma - \\int_{\\Omega} G_{pj}\\,F_{ij,p}\\,{\\rm d}\\Omega \\,.\n ", "baef6868daeada2c5d3e97dea209ea1b": "\\pi_0=\\frac{H}{P_0}\\frac{\\partial p}{\\partial x},", "baf039f424070cd3e5775d77e3e4dc9c": "\\mathfrak{P}^{33}", "baf0f8f3ab6a378cf226442d28839203": "\\tilde{k}", "baf17c2542c7e2354aecf6b37d4a4693": " \\rho' = \\frac{P_n \\rho P_n}{\\mathrm{Tr}(P_n \\rho)}", "baf219e28a6bbd5effc88e033f194c3f": "y(N-1)", "baf225af2519cac48f9ac7cb40cbb328": "\\scriptstyle \\vec{l}", "baf22bb8f11d966ae64f9c362235d1a8": "R_{0} = R \\setminus \\{0\\} \\,", "baf23a76b6190d4961cb2a667a4fce31": "\\mathbf{a}(t) = \\frac{d^2 R}{dt^2} \\begin{bmatrix} \\cos (\\omega t + \\pi/4) \\\\ \\sin (\\omega t + \\pi/4) \\end{bmatrix} + 2 \\frac {dR}{dt} \\omega \\begin{bmatrix} -\\sin(\\omega t + \\pi/4) \\\\ \\cos (\\omega t + \\pi/4) \\end{bmatrix} - \\omega^2 R(t) \\begin{bmatrix} \\cos (\\omega t + \\pi/4) \\\\ \\sin (\\omega t + \\pi/4) \\end{bmatrix}", "baf248137587c51b6c0a05708c0f1478": " \nL(\\mathcal{G},t)=\\{\\omega \\in \\Omega_{Z,[0,t]}: \\exists (q_0, \\omega, q) \\in\n\\Delta, q_0 \\in Q_0, q \\in Q_A\\}. \n", "baf26bc73fb503fdeee70d4b4098c42b": "PV = {Nm\\overline{v^2} \\over 3} ", "baf2830ab9df98e367c65db81727f2b3": "\n P[V^B, V^S] = (1 - \\alpha) P[V^B,\\epsilon] P[V^S,\\epsilon] + \\alpha(\\delta P[V^B,\\epsilon] P[V^S,\\mu + \\epsilon] + (1 - \\delta) P[V^B,\\mu + \\epsilon] P[V^S,\\epsilon]) \\;.\n ", "baf28c0ab9cebbb57061437b387dcf98": "\\,\\Upsilon_m", "baf3090423c9ec582648ee210ffcdd2f": "g(\\mu)\\to g*", "baf34000863ff61152fa0623f1c0d8fe": "\\hat{\\boldsymbol{z}}", "baf3795e9447238c7f24eb584bffbf59": "\\wedge_F", "baf3c00cf96323f154f35fe4c4242b5a": " \\tfrac{\\lambda}{10} ", "baf3e82dfe09986ed28955db3d3db07f": "\\lceil\\log{n}\\rceil +1", "baf49397984663078f6c984821516402": "= \\frac{1}{T} \\int_{-T/2}^{T/2} \\delta(t) e^{-i 2 \\pi n t/T}\\, dt \\ ", "baf4a981e01b377c1722c39fb73e55d6": "\\textstyle O\\left(\\frac{n}{p} \\log\\frac{n}{p}\\right)", "baf4b92702c036cdea2915725420f1dd": "t(x^{q},y^{q})", "baf4d65124ffce24ea7a6ecb05b19634": "\\Pr[X \\le x] \\ge k/q", "baf50091a2fd982a2a0b18a451101b8c": "Q = \\lambda ,", "baf526b31545752ff6902608666559f9": "\\sum_i T^i T^i = J (J+1) 1", "baf54e53c0dba46064a120d625b469c9": "|\\vec{p}_i| = p_i \\gg m_i", "baf55b54f7e3be741b037b7ee72f61c4": "\\partial_y=\\frac{\\partial}{\\partial y}", "baf5611db6001b7b16291a0ff77f8b78": "\\displaystyle K = \\sqrt{abcd} \\sin{\\frac{B+D}{2}}.", "baf5b838be5dad94d944c12e45f70775": "{\\mathbf u}, {\\mathbf v}", "baf5eeda080a17ca4f3f4197dc6a2d2e": " B_n \\subseteq Z_n \\subseteq C_n. ", "baf606802b02a2603db47ea634c06429": "r_1", "baf60f9a0cabd5c9c6df59c5482d8bde": "{\\rm st}(f(x_{i_0}))= f({\\rm st} (x_{i_0}))=f(c)", "baf68a13eac2bfcfc3c57cd481bdc89c": "\n\\sqrt{\\frac{\\pi\\sqrt{e^\\pi}}{2}}\\frac{1}{\\Gamma^2\\left(\\frac34\\right)}=i\\sum_{k=-\\infty}^\\infty e^{\\pi(k-2k^2)}\\vartheta_1\\left(\\frac{i\\pi}{2}(2k-1),e^{-\\pi}\\right),\n", "baf6f4c81740cb51bf4f6c244ef29e0c": "\\mathrm{d}G =V\\mathrm{d}p-S\\mathrm{d}T+\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}N_i - \\sum_{i=1}^n X_i \\,\\mathrm{d}a_i + \\cdots", "baf6fe567d7bb558d4dc7d9f8cdd3660": "\\mu_{12} = M_{12} - 2 \\bar{y} M_{11} - \\bar{x} M_{02} + 2 \\bar{y}^2 M_{10}, ", "baf7018ce0de62ae6cbf3e3144559f43": "C \\setminus U", "baf70e8fe8a270a7acfceb8c5a755a26": "\\left.R_d(r)=D\\frac{\\partial \\Phi}{\\partial z}\\right|_{z=0}= \\frac{a 'z '(1+\\mu_{\\mathrm{eff}}\\rho_1)\\exp(-\\mu_{eff}\\rho_1)}{4\\pi \\rho_1^3} + \\frac{a '(z '+4D)(1+\\mu_{\\mathrm{eff}}\\rho_2)\\exp(-\\mu_{\\mathrm{eff}}\\rho_2)}{4\\pi \\rho_2^3}", "baf72936aff4d27838f145ac1a9f4d26": "h_{\\sigma} = \\sqrt{\\sigma^2+\\tau^2}", "baf72ad4f78f7b4fb65ef4c94366c84f": " H= H_S + H_B + H_{BS} \\, ", "baf79daaf3e9ff810091dc146d171988": "\\begin{align}\nf_n(z)&=a_n z + c_{n,2}z^2+c_{n,3} z^3+\\cdots \\\\\n\\rho_n &= \\sup_r \\left\\{ \\left| c_{n,r} \\right|^{\\frac{1}{r-1}} \\right\\}\n\\end{align}", "baf7fdf955b2926774d12f9e93052d0a": "n=4 \\,\\!", "baf83f5d27e35e8bed59808e502590df": "\\phi^*\\gamma^i(y)=g^i_j(x)\\theta^j(x),\\ (g^i_j)\\in G", "baf84e13e4745f517c06300b3f542e7a": " \\frac {1} {s^2 +s \\left( \\frac {1} {\\tau_1} + \\frac {1} {\\tau_2} \\right) + \\frac {1+ \\beta A_0} {\\tau_1 \\tau_2}} ", "baf85669da3e40da73acfad1063add6b": "\\lim_{\\xi=\\rightarrow\\,\\infty}R_n(\\xi,x)=T_n(x)\\,", "baf8662ec55d0a5fe18be23f272227b6": " z f^\\prime(z) = g(z) f(z)", "baf8ae7c464c8646b3362714cddc13f4": "g(22) = 3", "baf9300209f85bb107d59da81d77d66c": "\n u = a(y) + b(y) x + L_{x}^{-1} \\rho(x, y) - L_{x}^{-1} L_{y} u - L_{x}^{-1} N u \\qquad (3)\n", "baf9b2f265e61a4b57a56cb1a299da67": "f=f(x)", "bafac51b20b0f05e890ad952c78956ca": "\\begin{align}\n & \\int_0^1 f(x)\\,dx + \\dotsb + \\int_{n-1}^n f(x)\\,dx \\\\\n &= \\int_0^n f(x)\\, dx \\\\\n &= \\frac{f(0)}{2}+ f(1) + \\dotsb + f(n-1) + {f(n) \\over 2} - \\int_0^n f'(x) P_1(x)\\,dx\n\\end{align}", "bafad3c14ea75d115e4f399b1a102aa9": "R_c = \\frac{\\Delta T}{\\dot{Q}}= \\frac{\\ln (r_2 /r_1)}{2 \\pi k \\ell}", "bafae80c2cb57bbcfd2caa4935f6201c": "x\\subseteq{\\mathcal A}\\,", "bafb4216f285a15a6a45871c530351b6": "{{\\mu }_{\\text{turb}}}", "bafb486a45b6e9523ef7ed8dbf306b59": "\\epsilon^{-p}", "bafb7cdb3c110a4227627daf06ee9d81": "T_n(x)=\\frac{n!}{2\\pi i}\\oint\\frac{e^{x({e^t}-1)}}{t^{n+1}}\\,dt", "bafb8a0a567b336fbf3921d53a0d0265": "e^{i\\varphi} = \\cos \\varphi + i\\sin \\varphi", "bafbd97f4cdfae7cb458c6e351e8f950": "\n\\sqrt{r} = [a_0;\\overline{a_1,a_2,\\dots,a_2,a_1,2a_0}].\\,\n", "bafbfe0b3dc76e9abbcd1caf385938b0": "\\;\\deg(GH)=\\deg(G)+\\deg(H)", "bafc8853bb7cdfe1222547bc7aaefd5d": "\\begin{bmatrix}\\begin{bmatrix}K\\end{bmatrix}-\\omega^2\\begin{bmatrix}M\\end{bmatrix}\\end{bmatrix}\\begin{Bmatrix}X\\end{Bmatrix}=0.", "bafc88bde477f6409499cc88a2b3ab7e": "\\textit{even} \\subseteq \\textit{int}", "bafce582ad46ebd3592650f13c7e83de": "Z=\\sum_{n=0}^{\\infty } \\frac{(-1)^n(4n)!(260n+23)}{(n!)^44^{4n}18^{2n}}\\!", "bafd35a908d3d9708a8ab4fa5b4e93c6": " s=P_a(1-\\theta)\\sum_{n=0}^{\\infin} P_{m1}^n ", "bafd6f9af0e019c754781b9ca57dd4ec": "{z \\choose n}", "bafdaa8db207498cecd960b4860136c6": "\n\\frac{d}{dt} \\left( \\mathbf{r} \\cdot \\mathbf{r} \\right) = 2 \\mathbf{r} \\cdot \\frac{d\\mathbf{r}}{dt} = \\frac{d}{dt} \\left( r^{2} \\right) = 2r\\frac{dr}{dt}\n", "bafdabbebe66fb4b221d650ce3d734fb": "X\\cap (\\cap_{n<\\omega} J_n) = \\emptyset", "bafde36da130d6fd86e39133492cb185": " \\mu(T) = |T| ", "bafde8c442fcc6319776027e13683915": "\n \\begin{align}\n \\nabla p & = \\cfrac{\\partial p}{\\partial r}~\\mathbf{e}_r + \\cfrac{1}{r}~\\cfrac{\\partial p}{\\partial \\theta}~\\mathbf{e}_\\theta + \\cfrac{\\partial p}{\\partial z}~\\mathbf{e}_z \\\\\n \\nabla\\cdot\\mathbf{v} & = \\cfrac{\\partial v_r}{\\partial r} + \\cfrac{1}{r}\\left(\\cfrac{\\partial v_\\theta}{\\partial \\theta} + v_r\\right) + \\cfrac{\\partial v_z}{\\partial z}\n \\end{align}\n ", "bafdeef47a01f45f7f6d988f81171ef3": " H^1(X, O_X)", "bafdf18634fb4173831bf31a73cd6529": "\\theta(\\Omega^\\Omega)", "bafdf5ea37a19f860601c2c2ad691980": "\\begin{align}\n& \\mathbf{T} = \\begin{pmatrix}\n\\frac{1}{4} & \\frac{1}{2} \\\\[4pt]\n0 & \\frac{1}{4}\n\\end{pmatrix}.\n\\end{align}", "bafe85a42129d01dbd3982a3e84292c9": "-x_0", "bafeda928a7c52a780cbe0ec07f23128": "V = \\frac{1}{3} B H", "baff064a34052749a7bcb80620eba3d7": "\n \\min_{B,R} \\|X - BR\\|_{F}^{2}\n", "baff14eb0b28df47a4a1370ff40d5e05": "\\, F_{ab;c} + F_{bc;a} + F_{ca;b} = 0", "baff74252389d78cdf9052ed059e8074": " \\quad \\psi = R \\exp i S / \\hbar", "baff7b8908ee67020b730938d31889b4": "\\mathbf{w}_{new}=\\frac{\\mathbf{w}_{new}}{|\\mathbf{w}_{new}|}", "baff9212422ee2c21c5965d7653ce76d": "\\{ f[x] : x \\in B \\}", "baff9a04007759960fc84e5947724bbd": "\\ \\displaystyle \\alpha \\ge 0 \\ ", "bb00705b3ffb03499cd0b82d376c977c": "{\\mathbf u}", "bb01244e5030c7c1c7cf188f41ec4085": "T_V=\\int{pd\\Theta}=\\cfrac{1}{2K}p^2=\\cfrac{1}{2}K{\\Theta}^2", "bb01c6d95c0ef817e72f01c96c0e9458": "T_{m,n,k}", "bb01c9060a1bc36a09e1d2744326b155": " = \\frac{P(x_{\\sigma(1)},\\ldots,x_{\\sigma(n)})}{P(x_1,\\ldots,x_n)} \\cdot \\frac{P(x_{\\sigma(\\tau(1))},\\ldots, x_{\\sigma(\\tau(n))})}{P(x_{\\sigma(1)},\\ldots,x_{\\sigma(n)})}", "bb01e6736023e376a4b05b8136ff74e3": "\\scriptstyle L_g", "bb0238ebc3aaffab6105f66878528235": "\\theta = \\arctan (m)", "bb0255a8649ea26dcc7e43b4cfdf5931": "u(x,t,\\theta) = V(x,\\theta) - t", "bb0263526d051dc417417f5e486c0511": " \\ X_1,\\dots, X_n", "bb0274fb00f17bae9dd962bea4a1036a": " k \\ge 7 ", "bb02c578b9c4780fb23000723d825703": "uu'=-f_0(x)u^3-f_1(x)u^2-f_2(x)u-f_3(x). \\, ", "bb02e0f869f4a2cfbd1c8ba072e31948": "Q(x) \\, f_n^{\\prime\\prime} + L(x)\\,f_n^{\\prime} + \\lambda_n f_n = 0", "bb02eaa2eeee27a666c7dd52f72106b7": "(x'(s),\\ y'(s)) = (\\cos \\varphi,\\ \\sin \\varphi)", "bb0389bbd391f58036794e3cc7035147": "w = z = 0.", "bb03b7925dafac83fda0da287dc2fc91": "b' = b\\sqrt{\\tfrac{3}{2}}", "bb03ddfae4c8be482114ac8e3654d47e": "\\mathbf{e}^k=\\mathbf{e}_k=\\mathbf{i}_k", "bb04f4d9ce355c917d872e53d1ab80d8": "(m+2n,n)", "bb057b6f1739112f8b861b440f437f4c": "\\mathbf r_x\\,", "bb0595afc164c113f1fc5b9bfbba79e7": "\\sum_{k=1}^n {I}_k = 0", "bb05a2405c7094392d816aaacdff8917": "n = \\frac{V_v}{V_t} = \\frac{V_v}{V_s + V_v}= \\frac{e}{1 + e}", "bb05b4328ae6bfc5c6e74c8dbcc30944": "\ny = \\mathbf{w}^H \\mathbf{r}\n", "bb05c38b47d6ec5d44d73c4f271173f7": "\n\\int \\exp\\left( - \\frac 1 2 x \\cdot A \\cdot x +J \\cdot x \\right) d^nx\n=\n\\sqrt{\\frac{(2\\pi)^n}{\\det A}} \\exp \\left( {1\\over 2} J \\cdot A^{-1} \\cdot J \\right)\n", "bb060aedd64dcd62201b067321e1d635": "A = \\bigcup_{t \\in T} A_{t}.", "bb0618f0e7771d2efffcf3e0e8e53086": "f \\sim g \\,\\!", "bb061b5643b5ef629be1dfa92a28e994": "\\text{DOR} = \\frac{\\text{PPV}\\times\\text{NPV}}{\\left(1-\\text{PPV}\\right)\\times\\left(1-\\text{NPV}\\right)}", "bb066dafa0adcbe0f7d96211da09ca12": "\\displaystyle\\begin{matrix}\n|x_1| \\le 1 \\\\\n\\vdots \\\\\n|x_n| \\le 1\n\\end{matrix}\n", "bb06d9dc2e28a049db0533b05e0af8ac": "A\\ominus B\\subseteq A\\circ B\\subseteq A\\subseteq A\\bullet B\\subseteq A\\oplus B", "bb0731a3bca224a3977b9a22337d4faa": "\\rm \\ 2 KSO_2F + S_2Cl_2 \\rightarrow S=SF_2 + 2 KCl + 2 SO_2", "bb07857a9042b22dfad8e38987fce5c1": "\\tbinom{2n}{n} = \\sum_{i=0}^n \\tbinom{n}{i}^2,", "bb0793f27e7186e29ca8f9ce8bfadd68": "c^k", "bb07a0afe5a382e336945b30a6796116": "g\\geq R(i,a)+\\sum_{j\\in S}q(j|i,a)h(j) \\quad \\forall i \\in S \\,\\, and \\,\\, \na\\in A(i)", "bb07d4f8242d71333ace466725cec195": "\\limsup_{n\\to\\infty}t_n \\le e^x \\le \\liminf_{n\\to\\infty}t_n \\, ", "bb07f64a91af4ee429f37d98761c5760": "P_i=(x_i.y_i)", "bb087a1d5df2d7ce3d29881335624bf6": " \\begin{align} \\textrm{ad} : & \\mathfrak{g} \\to \\textrm{gl}_{\\mathfrak{g}} \\\\ & x \\mapsto \\textrm{ad}_x \\end{align}", "bb08d9dd364ae4375f4c485f77244a34": "\\int\\limits_0^1 \\!\\frac{\\,\\ln\\ln\\frac{1}{x}\\,}{1+x^2}\\,dx\\, =\n\\,\\int\\limits_1^\\infty \\!\\frac{\\,\\ln\\ln{x}\\,}{1+x^2}\\,dx\\, =\n\\,\\frac{\\pi}{\\,2\\,}\\ln\\left\\{ \\frac{\\Gamma{(3/4)}}{\\Gamma{(1/4)}}\\sqrt{2\\pi\\,}\\right\\}", "bb08dd704db8da80ca9f92f93699ac96": "{\\rm E}_n(x)", "bb08eb05ee52e0ef44314778e6fe1cf0": "\n\\begin{align}\n\\begin{bmatrix}\n\\underline{2} & \\underline 3 & \\underline 4 \\\\\n1 & 0 & 0 \\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\n0 & \\underline{1000} \\\\\n1 & \\underline{100} \\\\\n0 & \\underline{10} \\\\\n\\end{bmatrix}\n&=\n\\begin{bmatrix}\n3 & \\underline{2340} \\\\\n0 & 1000 \\\\\n\\end{bmatrix}.\n\\end{align}\n", "bb08f416d77e57081c0d154405a14d95": "|\\mathrm{cat}_o\\rangle \\propto 2e^{-{|\\alpha|^2\\over2}}\\left({\\alpha^1\\over\\sqrt{1!}}|1\\rangle+{\\alpha^3\\over\\sqrt{3!}}|3\\rangle+{\\alpha^5\\over\\sqrt{5!}}|5\\rangle+\\dots\\right)\n", "bb09069342a80f22277976a945403e5c": "G(\\omega)=G'(\\omega)+i G''(\\omega)\\,", "bb09197b611c23a56dd528d401734508": "d_e(t) = a(t) \\int_{t}^{t_{max}} \\frac{cdt'}{a(t')}", "bb0954899597dcfc4dd752729a6b6f40": "e_0 ^2 = e_1 ^2 = -1", "bb09e44c153541a8c11572df235ac098": "\\qquad \\sum_{i=1}^n v_ix_i", "bb09e5f5719ce9d64b9fdcb066c153be": "\\,q(x)", "bb09f8ad19447181390384383df3839c": "\\int_\\gamma g(z)\\,dz=\\int_a^b \\frac{d}{dt}f\\left(\\gamma(t)\\right)\\,dt=f\\left(\\gamma(b)\\right)-f\\left(\\gamma(a)\\right).", "bb0a3296098c3e8b7157ae86f831e4a9": "\\tfrac{9K(M-K)}{3K+M}", "bb0a54404ff03f7c68ddb30ec5b6d06f": "\\int\\arcsin(a\\,x)^n\\,dx=\n x\\arcsin(a\\,x)^n\\,+\\,\n \\frac{n\\sqrt{1-a^2\\,x^2}\\arcsin(a\\,x)^{n-1}}{a}\\,-\\,\n n\\,(n-1)\\int\\arcsin(a\\,x)^{n-2}\\,dx", "bb0a7c6063d3bfd0913ad40c50dd8b3b": "b \\cdot (1 \\cdot a)", "bb0ab0a954afd7e4fa0603afa54ad8ba": "s = \\tfrac{1}{2}(a + b + c + d)", "bb0b50f15548b4738cfcb79f58ba06ef": "1/T_{\\rm s} = dS_{\\rm s}/dE = dS_{\\rm B}/dE", "bb0b78218f2d37f43f0b91ebbcd7b824": "\n\\mathbf{H}_{SB}=-J\\sum_{}\\mathbf{S}_{x_i}\\mathbf{S}_{x_j}+\\mathbf{S}_{y_i}\\mathbf{S}_{y_j}+\\mathbf{S}_{z_i}\\mathbf{S}_{z_j}\n", "bb0b9fcdf6dde8c8b42483ae2c8fb6cd": "\n a_{33} = - \\frac{2\\lambda(\\lambda + 2\\mu) + \\lambda A + 2(\\lambda - \\mu)B - 2\\mu C}{\\lambda + \\mu}\n ", "bb0ba3b6080070c74905ef02df196890": "\\displaystyle{k_{\\overline{z}} = \\mu k_{z} + \\mu_{z}.}", "bb0bef8c4f114ebddaff86b9db84154a": "x \\in \\mathbb{R}^n,\\quad b \\in \\mathbb{R}^n, \\quad c \\in\n\\mathbb{R}^n, \\quad", "bb0c18a311d4277afb73fce7bbaf20b6": "d_{G}(v,x)\\leq p", "bb0c70907cacbd246e930bd43263cda7": "\\scriptstyle \\frac{\\pi-\\theta}{2}", "bb0cd4d4275b79502d35c361055ac51b": "0=g(0),", "bb0ceb932d836b9056c7d7b6629c4542": "\\cfrac{\\qquad}{\\vdash (\\lambda x. t) x = t}\n", "bb0d108d37ca7355749480c17a3abb25": "L_n(x)=\\frac{e^x}{n!}\\frac{d^n}{dx^n}\\left(x^n e^{-x}\\right), \\text{ for } x \\ge 0", "bb0d1495b7bc35c69a63a0936775a25d": "N_x", "bb0d5e324e2251a05af53ae349c7020a": "[t_0,t_f]", "bb0d7736b3b54f6a564a0a8104a81e18": "\\text{Numerator}_{M+1} = \\text{Numerator}_M + n p_{M+1} - Total_M \\,", "bb0e1da13dab6a65d1c8bfc8d99d4d47": "Q(x'|x_t)", "bb0e29aec416bbc3fbbe42a5a70e5321": " r_{0} ", "bb0e5b655a6f8377863d92505060d41a": " S \\equiv \\frac{\\mu_0 L V_A}{\\eta}. ", "bb0e6cc8084bdd766e31682c691bbd84": "V_i \\not\\subseteq V_j", "bb0eecff6f9fe278eb45fc6f22ce6250": "r_{x_1,x_2}=x_1^*[-n] * x_2[n]", "bb0ef5c00e84a5dbbe260314d723e935": "\\,^{z_3 = x_3 y_1 - x_4 y_2 + x_1 y_3 + x_2 y_4 + x_7 y_5 + x_8 y_6 - x_5 y_7 - x_6 y_8 + u_3 y_9 - u_4 y_{10} + u_1 y_{11} + u_2 y_{12} + u_7 y_{13} + u_8 y_{14} - u_5 y_{15} - u_6 y_{16}}", "bb0ef6a7dab4274633746a73df17abcd": "\\textrm{dom}\\, \\phi_e", "bb0f002b656f0be19eb64294f7bc75dd": "E[X(t)|X(0) = i] = i.", "bb0f2c04905717785eae8f15594e613e": "c_1\\leq x\\leq c_2", "bb0f4b133c9326482b502fa8c71c8ed3": "\\displaystyle K=rs.", "bb0f5ea3b10061bcf032be8028976ce9": "\\delta W = \\sum_{i=1}^N p_i\\,dE_i", "bb0f6f521801074c8e9747deb60eee0a": "\\vert \\phi \\rangle = W_\\alpha", "bb0f786c74ff5e31d949b1d88f1697c3": "\\textstyle P\\sim \\exp (-\\mathcal{R})", "bb0f9981d99c966810da8916a10c3e14": "C_{p} = C_{V} + V T\\frac{\\alpha^{2}}{\\beta_{T}}\\,", "bb0fb6debc8daaf2d0bb0ab6806cb9c2": "\\frac{1}{\\pi}\\log \\left \\vert \\frac{t - a}{t - b}\\right \\vert ", "bb0ff29a4bc364bedefe30db1e159080": "(\\textrm{ad}_x)^N y = [x [x \\ldots [x [x, y] \\ldots] = 0\\ \\forall y \\in L", "bb1025ff77079b9ecff97b446127e810": "\\setminus", "bb107148298ef8c0a51802983932d17f": "\\mathcal{E}(u) = \\iint_{\\mathbb{R}^n \\times \\mathbb{R}^n} (u(y)-u(x))^2 k(x,y)\\, \\mathrm{d}x \\mathrm{d} y ", "bb10f0006222256ac59257efa192481e": "\\bar{Z}_1^{p,q} = \\ker d_0^{p,q} : E_0^{p,q} \\rightarrow E_0^{p,q+1} = \\ker d_0^{p,q} : F^p C^{p+q}/F^{p+1} C^{p+q} \\rightarrow F^p C^{p+q+1}/F^{p+1} C^{p+q+1}", "bb11019dbf0bb779bd6788eb81561b14": "\\pi(n)", "bb110602838b3eddfceb3d002bafcd71": "\\mathcal K", "bb115f24911e8f19bfa131897d55e39b": "f(x,\\sigma^2|\\mu,\\lambda,\\alpha,\\beta) = \\frac {\\sqrt{\\lambda}} {\\sigma\\sqrt{2\\pi} } \\, \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} \\, \\left( \\frac{1}{\\sigma^2} \\right)^{\\alpha + 1} \\exp \\left( -\\frac { 2\\beta + \\lambda(x - \\mu)^2} {2\\sigma^2} \\right) ", "bb118841d15312f5c1c57cac13f69ec2": "d\\leq e", "bb11a3db60cdd8d4eb77289b87dc43d6": "y=s(\\sum w_i x_i)", "bb11afdcca0dd78a85aaf1b85519a11f": "8 \\over 15 ", "bb11cbec87d1e0ae16983956dafc5b56": "P =0.07 V L/t +P_1", "bb135eeadfc5744c3de4e5593cbb2f55": "\\mathrm{E} \\left (X\\ - s | X\\ > s \\right ) = \\frac{s^{1-b}}{a}, \\qquad \\forall s \\ge 1. ", "bb136453ec6e1305266816ba036b4a51": "\\left|E(S, T) - \\frac{d \\cdot |S| \\cdot |T|}{n}\\right| \\leq d\\lambda \\sqrt{|S| \\cdot |T|},", "bb13757e14aff823fbee0d1646c74c15": "\\nabla^2\\psi(\\theta, \\phi) + \\lambda\\psi(\\theta, \\phi) = 0,", "bb137e2c56797ad1e41bd80b4855866e": " \\hat a \\ \\bar b \\ \\vec c", "bb137e7f6b31bd41256ae3b13f838355": "\\frac{1}{w}=\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}+\\frac{1}{d+1}", "bb138448d22e6abe610a422106324601": " \\mathbf x \\to \\infty ", "bb13c4e2b28bbaad996960486f0a002c": "{}^{65}{\\rm Zn}", "bb13cb15c070b07dc7b14ebf2c17226b": "dr_t = \\theta_t\\, dt + \\sigma\\, dW_t", "bb13dec01f41b622adfbbc41e2fe25fe": "\\omega_{nl}", "bb1427b184255e775aa9858ba32a33bb": "B(v)-{{B}_{x}}(v)", "bb143c69f8ef3221ca8c601260061c06": " \\frac {\\partial p}{\\partial x} = \\frac{ P_p - P_w} { \\partial x_u} ", "bb149b83fb83c10f54e241b71fa71877": " \\begin{align}\n - 1 = C^2 - A^2 & \\Rightarrow & A^2 - C^2 = 1 \\\\\nc^2 = (Dc)^2 - (Bc)^2 & \\Rightarrow & D^2 - B^2 = 1 \\\\\n2CDc - 2ABc = 0 & \\Rightarrow & AB = CD\n\\end{align}", "bb14b0d3a22c8dfdb4dd2b9108bc5414": "\\eta \\colon RF \\to X", "bb1628584c88a140924f49543eea126e": "x^6 + x + 1", "bb1647cc848965ace8fa9da03b1b7076": "\\mathbf{\\Psi}(\\mathbf{x})= \\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell \\frac{i k}{\\sqrt{\\ell(\\ell+1)}} h_\\ell^{(1)}(kr) \\mathbf{X}_{\\ell m}(\\theta,\\phi) \\int d^3\\mathbf{x'} j_\\ell(kr') Y_{\\ell m}^*(\\theta', \\phi') \\mathbf{L'}\\cdot\\mathbf{V}(\\mathbf{x'})", "bb1699fcc992ba1c60a597b758e80cd7": "\\vec{x}(i) = (u(i), u(i+\\tau), \\ldots, u(i+\\tau(m-1)),", "bb17f7a45302e42d0e694b38d8ec1c77": "| \\beta_1 | = | \\beta_2 | = 1", "bb17f96ed4feec551de62b0653ebc068": "V(\\mathbf{x}) = \\sum_i \\frac{m_i}{|\\mathbf{x}_i - \\mathbf{x}|}.", "bb17fc8ecf947279911e814c8e48f8a7": "Pr", "bb184d7df222e0c3b0076be1c0ff7d24": "\\operatorname{E}[\\boldsymbol Y]_i=e^{\\mu_i+\\frac{1}{2}\\Sigma_{ii}} ,", "bb1850496ac61845d16db13eb22eabfa": " \\langle y \\rangle = \\frac{1}{P} \\int{I(x,y) y dx dy}", "bb194054af8fc2f1876dd0c77c985597": "\\sqrt{z} = \\sqrt{r} \\exp(i \\phi/2)", "bb1a1e69b7aa139f09ebd51af0941bd4": " x_0\\pm R ", "bb1a93786ebf5dae06012fd81de73167": "i\\mathbf{k}= \\nabla\\psi /\\psi", "bb1adcf74c0b12a88571bbface9b98a4": "\\begin{align}\n(\\nabla x) \\times (\\nabla y) &= \\left (\\mathbf{\\hat x} \\frac{\\part x}{\\part x}+\\mathbf{\\hat y} \\frac{\\part x}{\\part y}+\\mathbf{\\hat z} \\frac{\\part x}{\\part z} \\right ) \\times \\left (\\mathbf{\\hat x} \\frac{\\part y}{\\part x}+\\mathbf{\\hat y} \\frac{\\part y}{\\part y}+\\mathbf{\\hat z} \\frac{\\part y}{\\part z} \\right ) \\\\\n& = (\\mathbf{\\hat x} \\cdot 1 +\\mathbf{\\hat y} \\cdot 0+\\mathbf{\\hat z} \\cdot 0) \\times (\\mathbf{\\hat x} \\cdot 0+\\mathbf{\\hat y} \\cdot 1+\\mathbf{\\hat z} \\cdot 0) \\\\\n& = \\mathbf{\\hat x} \\times \\mathbf{\\hat y} \\\\\n& = \\mathbf{\\hat z} \\\\\n\\end{align}", "bb1b74ce65dbb78a06b465ae9c8068fb": " \\omega/k = v ", "bb1ba35bb2ff22641f8a89b5b849e34c": "a_{i0}", "bb1bd1fbdfe6c2903e81f5efa905d161": "\\begin{align}\n\\frac{\\partial P_1}{\\partial v} &= \\left\\langle \\frac{\\partial (\\mathbf{F}\\circ \\psi)}{\\partial v} \\bigg | \\frac{\\partial \\psi}{\\partial u} \\right\\rangle + \\left\\langle \\mathbf{F}\\circ \\psi \\bigg | \\frac{\\partial^2 \\psi}{ \\partial v \\partial u} \\right\\rangle \\\\\n\\frac{\\partial P_2}{\\partial u} &= \\left\\langle \\frac{\\partial (\\mathbf{F}\\circ \\psi)}{\\partial u} \\bigg | \\frac{\\partial \\psi}{\\partial v} \\right\\rangle + \\left\\langle \\mathbf{F}\\circ \\psi \\bigg | \\frac{\\partial^2 \\psi}{\\partial u \\partial v} \\right\\rangle \n\\end{align}", "bb1c1c59fab5451a091524f865007d7a": "\\delta : Q \\times \\Sigma \\rightarrow Q", "bb1c2d93029e2f8c3a4cec866aa8bcff": "\\cdots \\to H^i(X) \\to H^i(Y) \\to H^i(Z) \\to H^{i + 1}(X) \\to \\cdots.\\ ", "bb1c60291e95aa783f9b0dc3b47d75de": "\\qquad \\qquad c_{v,p} = \\frac{dE_p}{dT}|_v = \\frac{9k_\\mathrm{B}}{m}(\\frac{T}{T_D})^3n\\int_0^{T_D/T}\\frac{x^4e^x}{(e^x-1)^2}dx\\qquad (x = \\frac{\\hbar\\omega}{k_\\mathrm{B}T}),", "bb1c64ea94aedb439295260a56c24d98": "\\Leftrightarrow, \\nLeftrightarrow, \\Longleftrightarrow \\iff \\!", "bb1cfced548747823630c4d014fce819": "\\frac{33705\\,\\frac{\\mathrm{W \\cdot h}}{\\mathrm{gal_{ge}}}}\n {135\\,\\frac{\\mathrm{W \\cdot h}}{\\mathrm{km}} \\times \\frac{1.6\\, \\mathrm{km}}{\\mathrm{mi}}}\n \\times 77.6 \\% {\\mathrm{_{charging\\ eff.}}}= 120 \\,\\mathrm{mpg_{ge}} = 1.95 \\frac{\\mathrm{L_{ge}}}{100\\, \\mathrm{km}}", "bb1d79195b819f53957d96d626a649fa": "N_0(T+H)-N_0(T) \\geq cH", "bb1dc756d7468a85f6f476c164a1aea2": " B U \\xi = U A \\xi ,\\quad \\xi \\in \\operatorname{dom}A. ", "bb1de1d611164deb1ea9c49d3fe77ee4": "T(\\mathbf{x}) = f(\\mathbf{a}) + \\mathrm{D} f(\\mathbf{a})^T (\\mathbf{x} - \\mathbf{a}) + \\frac{1}{2!} (\\mathbf{x} - \\mathbf{a})^T \\,\\{\\mathrm{D}^2 f(\\mathbf{a})\\}\\,(\\mathbf{x} - \\mathbf{a}) + \\cdots\\!\n\\,,", "bb1de668c4ff89a780ac6f19a2cbd11f": "i = \\frac{h}{L}", "bb1e3b93a174f61c335531730b835bdb": "s = k^{-1}(z + r d_A)\\,\\bmod\\,n", "bb1e4aa0b6ea5d371fe93b713ac37d48": "\nf(n)\\sim\\dbinom{n}{m} (n-m)! 1/n!,\n", "bb1e60ecd6fbf7a34500dac31bff93bf": "f(x) = e^{kx}", "bb1e96ddfa4f71b0b8196b0903cc7fa6": "\\mathcal{O}_Y \\simeq (\\pi_*\\mathcal{O}_X)^G", "bb1f1be4221af98ce40787f409cfefce": "Y = y^{q} _ {\\bar{t}}", "bb1f58727d9a4d07982d84a628dffb2a": "\\scriptstyle \\lfloor\\frac{37+1}{4}\\rfloor=9", "bb1f7fb72d58a8f0189b04f452b2085f": " \\gamma_0 = \\sum_{i=1}^p \\varphi_i \\gamma_{-i} + \\sigma_\\varepsilon^2 ", "bb1f83aaaa27e21d4657e7f8488d69cd": "\\int_S{ \\mathbf{J} \\cdot \\mathrm{d}\\mathbf{A}} = -\\frac{\\mathrm{d}}{\\mathrm{d}t} \\int_V{\\rho \\; \\mathrm{d}V} = - \\int_V{ \\frac{\\partial \\rho}{\\partial t}\\;\\mathrm{d}V}", "bb1fbb45b8cfdadc3c47dad1662d3f84": "N=\\mathcal{T}(B)", "bb1fc8b45e69eb5ee9dc2916f93343ad": "(3^{\\alpha\\beta})", "bb1fcde8df83e01858e18f2d9fc5657c": "\\mathbf{ \\tau } = \\mathbf{ l \\times F_g },", "bb1fd1491735adaf94e0e2f49fa1ec62": "a(x)\\leq t\\leq b(x),", "bb1fde7b959051c5118b3bf923e67973": "\\dot{z} = A(u(t)) z+ \\phi(y,u(t) ), ", "bb2012a188317789580d975e8fea91ec": "\nML_\\mathrm{dB} = - 10 \\log_{10} \\bigg(1-\\rho^2\\bigg) \\,\n", "bb2055e5ce917077f06d80ef5c8715d3": "X_1(\\tfrac{1}{z^*})", "bb210cc84bdfd665ee1d3412c642dd85": "x^n+y^n=z^n", "bb21149af5d9768e14069a1e305c3054": "~m_{max} = 2", "bb2123359f1ccc9c13ef31c6b0d6c230": "\\begin{align}\n I(0) &= \\int_0^\\pi dx = x|_0^\\pi = \\pi \\\\\n I(1) &= \\int_0^\\pi \\sin xdx = -\\cos x|_0^\\pi = (-\\cos \\pi)-(-\\cos 0) = -(-1)-(-1) = 2 \\\\\n I(2n) &= \\int_0^\\pi \\sin^{2n}xdx = \\frac{2n-1}{2n}I(2n-2) = \\frac{2n-1}{2n} \\cdot \\frac{2n-3}{2n-2}I(2n-4)\n\\end{align}", "bb2144e32f668a2220516ab1a51da90f": "\\scriptstyle{|\\phi_1(t)\\rangle}", "bb21532dfc292770f9145dba3ffd28ad": "\\int d^2 \\theta\\; \\kappa\\; L H_u", "bb218c60a24e424d39011f8f5995ba37": "\\begin{array} {l}\nf'(x_0)=-\\frac{f\\left(x_0 - h\\right) - f(x_0)}{h} + \\frac{f^{(2)}(x_0)}{2!}h - \\frac{f^{(3)}(x_0)}{3!}h^2 + \\frac{f^{(4)}(x_0)}{4!}h^3 + \\cdots \n\\end{array}", "bb21bd0bbba2968010cff52c8241ea5c": "1 \\leq i,j \\leq d", "bb2264d0a69bc1158847b3be2d312129": "\\begin{align}J_\\nu(z) &= \\frac{ (\\frac{z}{2})^\\nu }{ \\Gamma(\\nu + \\frac{1}{2} ) \\sqrt{\\pi} } \\int_{-1}^{1} e^{izs}(1 - s^2)^{\\nu - \\frac{1}{2} } \\,ds, \\\\ &=\\frac 2{{\\left(\\frac z 2\\right)}^\\nu\\cdot \\sqrt{\\pi} \\cdot \\Gamma\\left(\\frac 1 2-\\nu\\right)} \\int_1^\\infty \\frac{\\sin(z u)}{(u^2-1)^{\\nu+\\frac 1 2}} \\,du,\\end{align}", "bb22ed8158cad8879cb143ffd3bb30f6": " | \\psi^{(-)} \\rangle \\,", "bb231b1f9878c425c8693956dbe8aaba": "\\begin{align}\n&f(x_1,x_2)=0\\\\\n&g_0(x_1,x_2)\\neq 0\\\\\nx_3&=\\frac{g_3(x_1,x_2)}{g_0(x_1,x_2)}\\\\\n\\vdots &\\\\\nx_n&=\\frac{g_n(x_1,x_2)}{g_0(x_1,x_2)}\n\\end{align}", "bb2327846da7f38a0b0abe5c68a85bfd": "\\displaystyle{\\pi_\\sigma(g^{-1}) f(z)=|\\overline{\\beta}z+\\overline{\\alpha}|^{1-2\\sigma}f\\left({\\alpha z +\\beta\\over\\overline{\\beta}z +\\overline{\\alpha}}\\right).}", "bb236679c34583a061196b76fce347b8": "\\mathrm E(|X|)\\leqslant K", "bb23f7cebf5cd71322a35674c62c9c39": "\\neg (\\neg \\phi \\land \\neg \\psi)", "bb24a8db95aef161a252f268eee6f67d": "\\mathrm{d}U = \\delta Q - \\delta W+\\sum_i \\mu_i\\,\\mathrm{d}N_i", "bb24f744eb581c5153b20d3d11f978fe": "f(x_1,\\dots,x_n)=\\bigwedge_{i\\in I}x_i", "bb251c95a829beb8d2750b69f5a010c7": "\\sigma = 0.1714 \\times 10^{-8}\\ \\textrm{BTU}\\,\\textrm{hr}^{-1}\\,\\textrm{ft}^{-2}\\,\\textrm{R}^{-4}.", "bb257f34141f5175ca81910f47418daf": "\\neg \\neg A \\vdash A", "bb25bca8667d4b5e58e8112d7f0d93b3": "Y_{(n)}= \\sum_{i=1}^n X_{(i)}", "bb2607526951cec0b17344e5f1775d33": "|\\alpha,\\beta\\rangle = |\\beta, \\alpha\\rangle", "bb2626de5dc0ebe8a664e94d6d93e47f": " \\frac{1}{w(x)} \\left(\\frac{d}{dx}\\left(p(x) \\frac{d}{dx}\\right)+q(x)\\right) ", "bb26e3e0ce2d75deb6d81d0ce4afe213": "s_r=\\exp{\\frac{2\\pi i q}{N}}", "bb26e62e386ae96a6e4ddfd1fe137f1d": "\\begin{align}\nF_1(X,Y,Z) &:= Y^2 + pYZ + qXZ + rZ^2,\\\\\nF_2(X,Y,Z) &:= YZ - X^2\n\\end{align}", "bb271550a79c1310ed4435abc0499436": "(3) \\ \\Delta n = - \\frac{1}{2}n^3p_{ij} a_j,", "bb274b43a6bff408e8735567fb97bb12": " G \\left(T,P,d\\right)= \\left( n_{l}\\mu_{l}\\left(T,P\\right) \\right)d + \\gamma_{total}\\left(d\\right) ", "bb27a66f482c8fb252675532d81d4c38": "2_0", "bb27c6a181570f95963697acf058e099": "A_J:=\\bigcap_{j\\in J} A_j", "bb27d2ec5b7b691d36d370e52c177906": "F_n\\Bbb R", "bb280019c0ad7546db3b3fbc57cbb3f3": "(1)~~ ~~ A= \\frac{\\alpha}{1+I/I_0}", "bb282cf43d8bc5ccc38d4bf4590a5d7c": "N_{\\hat{B}}=J+K", "bb28a28153614a4a85cb33ebe2196090": "t_{2\\alpha'}\\left(\\tilde{x}|\\mu',\\frac{\\beta'(\\nu'+1)}{\\alpha'\\nu'}\\right)", "bb28a4861b2ef01da002b41cf2def260": " HC_n(A)\\simeq \\Omega^n\\!A/d\\Omega^{n-1}\\!A\\oplus \\bigoplus_{i\\geq 1}H^{n-2i}_{DR}(V).", "bb28df4cfac62f9b2e3af9823192aad7": "\\frac{1}{N}\\sum_{i=1}^N \\ln \\frac{Y_i - \\hat{a}}{\\hat{c}-\\hat{a}} = \\psi(\\hat{\\alpha})-\\psi(\\hat{\\alpha} +\\hat{\\beta} )= \\ln \\hat{G}_X", "bb290fa0914dcb156f3b97015faa7ac5": "\\!X[F/x] = \\{ s[F(s)/x] : s \\in X\\}", "bb2a5ab3050d72dc72bea3a9015cb855": "Rb = 1", "bb2a783775769adcc2e956b8d71dd4c4": "\\varrho_B^\\lambda", "bb2ac97b4922a3eee2cb3197bb793622": "\\displaystyle K = \\sqrt{abcd} \\sin \\frac{A+C}{2} = \\sqrt{abcd} \\sin \\frac{B+D}{2}.", "bb2ae79f0684f13541c939152a8949f8": "\\pi_0 = \\left[\\sum_{k=0}^{c-1}\\frac{(c\\rho)^k}{k!} + \\frac{(c\\rho)^c}{c!}\\frac{1}{1-\\rho}\\right]^{-1}", "bb2b3235aa857163b2e31dae75d0e4dd": "P(G, 4)>0", "bb2b3740c0be4f58a6e6a5b484610e41": "=-0.00005-1.4775x-0.00001x^2+4.83484x^3", "bb2b56f7c3ebb4d5dd20ed17a29d7df0": "\\sqrt{10} \\rho^4 \\cos 4 \\theta", "bb2b5d99c3b9a41ce401310ee80aff66": "a_1 \\geq a_2 \\geq \\cdots \\geq a_n", "bb2be9f78116308428dcc3cfb35490fd": "\\overline{A \\cap B}=\\overline{A} \\cup \\overline{B} \\iff \\overline{A \\cup\n B}=\\overline{A} \\cap \\overline{B}", "bb2c36514a92eb549fcd2221e7c9549f": "C_\\beta(s) = \\begin{bmatrix}1&0\\\\s&1\\end{bmatrix}", "bb2c5239abfc7447c65626e8557d4ff0": "(\\Omega, \\mathcal F^*)", "bb2c78023ceafe51132e9bbe2a0bb636": "\\pi(i)\\,\\!", "bb2c8c1270113e54050bffec982662eb": " v_n = \\frac{c}{n} ", "bb2ca6bdb21deccecfdfce3cedecb0b5": "(C_*,\\partial_*)", "bb2cbfb656d3f396d44463d809ade4b9": "\\iiint_T \\rho^2 \\rho \\ d \\rho d \\phi dz", "bb2cdd0145e04e6bfd0f2f9a0cb2534a": "\\mu = \\alpha\\beta", "bb2d11e6b23dbf2f929536c1c4191af9": "\\hat{t}=-\\frac{R-r}{r}t.", "bb2d620b4ad201bbeea33d1d4122f2c0": "R=10^2 \\equiv -4, \\; t\\equiv 10^3 \\equiv -1 \\pmod {13}, M = 2.", "bb2d6e38eaad9a7084bb1f9f6b7c88b1": "\\prod \\left( 1- d_i \\right)", "bb2dc7b2cc97b516bc154679eb945381": "M[f] = \\left(\\begin{array}{cccc}\n1&0&0& \\cdots \\\\\n0&f_1&f_2& \\cdots \\\\\n0&0&f_1^2& \\cdots \\\\\n\\vdots&\\vdots&\\vdots&\\ddots\n\\end{array}\\right)", "bb2e2079868aaab1999aeb60c486d24b": "\\{a(f),a^\\dagger(g)\\}=\\langle f|g \\rangle", "bb2e262e459a2afd80d9f0729019f5bc": "\\mathbb{F}_{Q}^* ", "bb2e3e52a6dddcb4315c33df60b28e71": "[D,K_\\rho]=+K_\\rho", "bb2ea3e51c0a42e9a32535d33af75461": "\\mu_f(\\theta)", "bb2ec0ad1e0648206de578c58dad0277": "\\omega\\;", "bb2ef54fde1a29b82cbac859b51609b1": "\\begin{align} S(L) &= \\int_0^L \\sin s^2 \\, ds\\\\\n &= \\int_0^L (s^2 - \\frac{s^6}{3!} + \\frac{s^{10}}{5!} - \\frac{s^{14}}{7!} + \\cdots) \\, ds\\\\\n &= \\frac{L^3}{3} - \\frac{L^7}{7 \\times 3!} + \\frac{L^{11}}{11 \\times 5!} - \\frac{L^{15}}{15 \\times 7!} +\\cdots\\end{align}", "bb2f50d61db1bcff49fc3d37a2d232e6": "\ns_{\\overline{n}|i} = (1+i)^n \\times a_{\\overline{n}|i}\n", "bb2f6ad2e0a95e19532a5bfd62e8b097": "O^1(Z)", "bb2f6c44fec5f8cd09ec2719c252cc2a": "\\textstyle D_1=kTB_1(1+N_1\\frac{dln\\gamma_1}{dlnN_1})", "bb2ff7eed766a2334e113f8bce3e3e94": "(x,y) \\in \\overline{K} \\times \\overline{K}", "bb300f909d905a9f58568dd6baa83271": "x = f_1(p),\\ y = px", "bb302d839fa6778207b0b0213c665a0a": "\\int_0^\\infty \\frac {e^{-ax}-e^{-bx}}{x \\sec px}\\ dx=\\frac{1}{2} \\ln\\frac{b^2+p^2}{a^2+p^2}", "bb30479355600a9a55492ad89bed7114": "\\binom{a}{0}=1", "bb30604f882d01410edeab7acf1fbbda": "O(\\alpha(V))", "bb307f8bb7ecec8aeb9bab7d909c27e6": "M= \\left ( \\frac {CY} {2i} \\right )^{\\frac {1} {2}}", "bb3092ed94db3592822b08ac853da21a": "\\scriptstyle{1/\\sqrt{1-{v^2}/{c^2}}}", "bb30d6bb39e3acc4d2fc7eeabd4bd2b4": " \\|x\\| ", "bb30fda650554a35a9578f879f8f8ba2": "A \\in \\R", "bb31eefef27994ed715fae5bad92434a": "F_\\text{seq}(2)=\\big\\{1,5,12,6\\big\\},v_2=0", "bb320cc2b38097530605847a5b03cb36": "\\tilde{f}_i", "bb323c390fbfdfe4017c74d29cf95506": " s_{ij}^v(x) = \\max \\left\\{ v(S) - \\sum_{ k \\in S } x_k : S \\subseteq N \\setminus \\{ j \\}, i \\in S \\right\\}, ", "bb325b2a393b7e81107715d1e02a831e": "P_t = \\prod_{i=1}^{n}\\left(\\frac{p_{it}}{p_{i0}}\\right)^{\\frac{1}{2} \\left[\\frac{p_{i0}q_{i0}}{\\sum_{i=1}^{m}\\left(p_{i0}q_{i0}\\right)}+ \\frac{p_{it}q_{it}}{\\sum_{i=1}^{m}\\left(p_{it}q_{it}\\right)}\\right]}", "bb328dd8b9a9aec267808b73ca7c97a4": "S[\\phi,\\psi]=\\int d^dx \n\\left[\\frac{1}{2}\\partial^\\mu \\phi \\partial_\\mu \\phi -V(\\phi) +\n\\bar{\\psi}(i\\partial\\!\\!\\!/-m)\\psi \n-g \\bar{\\psi}\\phi\\psi \\right].", "bb32b80ae48f215fe27dcbb78a35d0b4": "\\boldsymbol{r}_0=\\boldsymbol{b}-\\boldsymbol{Ax}_0", "bb32bb1b206b73964b4b933685f9bd73": "\\mathrm{population} \\propto (2J+1)e^{-\\frac{E_J}{kT}} ", "bb333f0401d6c144f8b1a82d268ccc2f": "1*1=\\sigma_0=\\operatorname{d}=\\tau", "bb33608dd6a33d53a868b56d9580206b": "\np\\left( m\\right) =\\text{Tr}\\left\\{ \\Lambda_{m}\\rho\\right\\} ,\n", "bb338d245a5ff3598919b54ff24c74fc": " H(X|y) = \\mathbb{E}_{{X|Y}} [-\\log p(x|y)] = -\\sum_{x \\in X} p(x|y) \\log p(x|y)", "bb343e445cdaac0b9cbba74cd8635b6d": "C_n = \\frac{1}{n+1}{2n\\choose n} = \\frac{(2n)!}{(n+1)!\\,n!} = \\prod\\limits_{k=2}^{n}\\frac{n+k}{k} \\qquad\\mbox{ for }n\\ge 0.", "bb346a4b70961486eeb6e209d840dd22": "\n\\dot{p}_x = -\\frac{1}{2}\\frac{\\partial V}{\\partial x}\n", "bb349127590b517e26a8079d7e6dd75d": "H_\\text{KL}(k_\\text{x},k_\\text{y},k_\\text{z})=-\\frac{\\hbar^2}{2m}\\left[(\\gamma_1+{\\textstyle\\frac52 \\gamma_2}) k^2 - \n2\\gamma_2(J_\\text{x}^2k_\\text{x}^2+J_\\text{y}^2k_\\text{y}^2\n+J_\\text{z}^2k_\\text{z}^2) -2\\gamma_3 \\sum_{m \\ne n}J_mJ_nk_mk_n\\right]", "bb34925bfefb3a3d1fbff1e3e1fed6ff": "\\mathbf{L}\\!{}_\\mathbf{\\displaystyle Y}\\!\\mathbf{X}", "bb34bbe4e3c674bcdc73e9970657db23": "\n \\nabla^2 \\varphi = \\cfrac{1}{h_1 h_2 h_3}~\\sum_i\\frac{\\partial }{\\partial q^i}\\left(\\cfrac{h_1 h_2 h_3}{h_i^2}~\\frac{\\partial \\varphi}{\\partial q^i}\\right)\n", "bb34dac0a0f191f825f126dca87e2f9f": "jk_{x\\varepsilon }tan(k_{x\\varepsilon }w)+k_{xo}=0 \\ \\ \\ (6) ", "bb35282671b724c79ba513be1012efe3": "AD = cum\\frac{(C - L) - (H - C)}{(H - L)}\\times V\\!\\,", "bb35b3dd6c53e3260f7befb8a3c09043": "\\frac{1-x^n}{1-x}=1+x+\\cdots +x^{n-1}.", "bb35b551a893d6d6bb1ef6a5a14eba17": "b \\mapsto [(f(b), 0)] = [(0, f'(b))]", "bb35ccf88dd64f8b606b91e884cf55dd": "\\beta F = \\int d^dx \\left[ A H^2 + \\sum_{i=1}^{d} Z_i (\\partial_i H)^2 + \\lambda H^4 ... \\right].", "bb36185edfb7e6337a1f9d810a7cc6b9": "\\theta_{1},...,\\theta_{n}", "bb3630a9fa87b407e064fb58cf811f78": "\\mathbf{F}_{\\mathrm{ext}} + \\mathbf{v}_{\\mathrm{rel}}\\frac{\\mathrm{d}m}{\\mathrm{d}t} = m\\mathbf{a}_{\\mathrm{cm}}", "bb3640700986ca417655be2f03d39389": "K^M_2 (K)", "bb37340d07af7c1b63c9fc4e133e8ac9": "q = \\frac{ln(10)}{400} = 0.00575646273", "bb3744e1c3ce9087a0863323e0e192f9": "(G_0)^{-1}(i\\omega_n)=i\\omega_n+\\mu-\\Delta(i\\omega_n)", "bb3769a96e8cf59015a0ccec884e94a9": "F_i - \\mu F_w \\cos \\theta - F_w \\sin \\theta = 0 \\, ", "bb376dcf8d52be15743d33b938047387": "E = \\lambda Q\\mathcal{E}\\qquad\\qquad(13)", "bb376ebbb657f82c00312a3a96c09709": "\\scriptstyle 2\\times 3", "bb379bc87d3a2021507a57c65b3118b4": " j \\ne n ", "bb37ad79f67fefc1d728a92a35eb2718": "\\int d^4p/(2\\pi)^4", "bb37d2344e6991d1a187ed13b64b7869": "\\lim_{V_{m}\\rightarrow0} \\Phi_{S} = P_{S}\\frac{z_{S}^2F^{2}}{RT}\\frac{[V_{m}([\\mbox{S}]_{i} - [\\mbox{S}]_{o}\\exp(-z_{S}V_{m}F/RT))]'}{[1 - \\exp(-z_{S}V_{m}F/RT)]'} ", "bb380c3f99a2b96c4f1c9165041693ce": "{\\sqrt{3}-\\sqrt{5}}", "bb384a8f21ca5e7cb9e58dc8fc4b696b": "M_{M_7} = M_{127} = 170141183460469231731687303715884105727 ", "bb39291841787f3b7b78a16dbbfc40c3": "\\underline{1} \\ ", "bb392c67803f4d0b5858d5dcd9e2126a": "\\sum_{n=0}^\\infty (-1)^n {s \\choose 2n+1} = 2^{s/2} \\sin \\frac{\\pi s}{4}", "bb39641f762f54e397d7acaddd2562a1": "z_R", "bb396bee47a8cf8736f061c05a181881": " M = 1 + { A \\over \\Delta V }. ", "bb39981ded66a53f49c9f5044cb6210d": " \\nabla\\left(\\mathbf{A}\\cdot\\mathbf{B}\\right)=\\left(\\mathbf{A}\\cdot\\nabla\\right)\\mathbf{B}+\\left(\\mathbf{B}\\cdot\\nabla\\right)\\mathbf{A}+\\mathbf{A}\\times\\left(\\nabla\\times\\mathbf{B}\\right)+\\mathbf{B}\\times\\left(\\nabla\\times\\mathbf{A}\\right) ", "bb39ee23b894cd961787dde0ead0780e": "Q=\\frac{\\partial}{\\partial \\theta}-i\\Theta^*\\frac{\\partial}{\\partial t}\\quad \\text{and} \\quad Q^\\dagger=\\frac{\\partial}{\\partial \\theta^*}+i\\Theta\\frac{\\partial}{\\partial t}", "bb3a6d15be568af98af5236badd2257b": "\\displaystyle{H^\\varepsilon f \\rightarrow Hf}", "bb3a7512b585ca5753629156fe548c43": "f: S \\rightarrow R", "bb3afff8239e7bfd8c3093e2cb2d78c2": "x_0=(x_{10},\\dots,x_{n0})", "bb3b48a74c72835d9eb179df94fdc745": "L_{i2,1} = - 10 \\log{\\left(\\frac{P_2}{P_1} \\right)} \\quad \\rm{dB}", "bb3b6ac1a7cbac341aca9d433ba2901f": "A_{ry}", "bb3b7cf4fbc40f050e4aa57029d58d2c": "Z=\\frac{\\partial}{\\partial z},", "bb3b8de25cf43bd422e3090838328c93": "L_\\mathrm {L2}=\\sqrt{Z_\\mathrm {i \\Pi} Y_\\mathrm {i T}} \\ e^{\\gamma_\\mathrm L}", "bb3b9778cfcf171b25f67bfc93f93269": "\\frac{D} {Q} ", "bb3bc4f0cfb5927d37a12caf30479ab6": " J_n = \\sqrt { \\frac { n } { 2 } } \\frac { s^2 - m } { m } ", "bb3bef10bccf9896fa6a70fbfc1530dc": "\\vec{Y} = \\, f \\, \\vec{X}", "bb3c416a0cc945eff9c7abb9144468cc": "\\phi^2=\\phi\\circ\\phi", "bb3c9a147b7f957bff11272c95d0955b": "g(hu) = (hg)u", "bb3cb31e25c44f196b21a16be92bd9f9": "\n I_1 = \\mathrm{tr}(\\boldsymbol{B}) = 3 + \\gamma^2 \n ", "bb3cfbc1ec6e7df3e71815fe50244199": " \\mathbf{\\mu}=\\frac{1}{2}\\, q\\, \\mathbf{r}\\times\\mathbf{v}", "bb3d08be83cafaafdd7afcecc749f7fd": "(\\mathbf{h}P_\\sigma)P_\\pi = \\mathbf{h}P_{\\pi\\circ\\sigma}", "bb3d1664a324bd750c4f5cb7811e6715": "{\\nabla}^2 \\varphi = -\\frac{\\rho}{\\varepsilon_r \\varepsilon_0} = - \\sum_i \\frac {z_i q n^{0}_i}{\\varepsilon_r \\varepsilon_0} e^{-\\frac{z_i q \\varphi}{k_B T}}", "bb3d1cdcf8ccd7205148e74cdf302109": "G^* + M \\to MG^{+\\bullet} + e^-", "bb3d30e69ceec4ce2c20c4fc1e2e4515": " \\begin{align} \nz = \\frac{[Z]}{[Z]_0 } ; \\ \nv_1 = k_1 [X] ; \\ \nv_2 &= k_2 [Y] ; \\ \nJ_1 = \\frac{K_{M1}}{[Z]_0 } ; \\ \nJ_2 = \\frac{K_{M2}}{[Z]_0 }; \\ \\qquad \\qquad (2)\n\\end{align}", "bb3d8ddeb18f7c5667d0d1f1f5509ba5": "t<-\\ln(1-p)\\!", "bb3d9f4c90636c86982be3a11b2a0d41": " \\, \\frac{|SC|}{|CD|}=\\frac{|SA|}{|AB|}", "bb3da18fdcee82129bed460f66341f75": "f_r(x) = \\frac{1}{|B(x, r)|} \\int_{B(x, r)} f(y) dy.", "bb3dd2340362b59d281fb5d7cab6787b": "\\text{beta}=1/3", "bb3de43d139820ecaae2f4bf7f3e7417": "\\boldsymbol{\\iota\\kappa\\lambda\\mu\\nu\\xi\\pi\\rho} \\!", "bb3e03e69cf74eec3409c14ddc71e852": "\\begin{smallmatrix}T\\ =\\ \\frac{T_{\\textrm{eff}}(1-qp_{\\nu})^{1/4}}{\\sqrt{2}}\\sqrt{52/r},\\end{smallmatrix}", "bb3e8d1b13d0b38532aa8c2f3bf629b7": "NA_{obj} + NA_{cond} = 2 NA_{obj} ", "bb3e9905681ca2b302f091a5e7b8c339": "P_\\mu^*(n)=(1-\\mu)\\frac{e^{-\\mu n}(\\mu n)^{n-1}}{(n-1)!}.", "bb3ea69cd58f6221cc3b62033c72469c": "\\begin{align}\n(\\Pi V)_0 &= V_1 \\\\\n(\\Pi V)_1 &= V_0.\\end{align}", "bb3eb0bbab4b305aac9ef34c4676de93": "HoldsAt", "bb3f00cac9bbe7a2970d5af61210f809": "r_i\\in \\mathbb{R}", "bb3f31d110a2cdebb60aca0be290d3a8": "\\left[ - \\frac{\\hbar^2}{2m} \\nabla^2 + V(\\mathbf{r}) \\right] \\psi(\\mathbf{r}) = E \\psi (\\mathbf{r}),", "bb3f51a3f24324e0336abf47ae80567e": "\\sigma=e(n\\mu_e+p\\mu_h).", "bb3f6929ee0839c8992f749196f44bce": "\\alpha_{\\tau\\tau}=-\\frac{1}{2}\\lambda^2e^{4\\alpha},\\ \\beta_{\\tau\\tau}=\\gamma_{\\tau\\tau}=\\frac{1}{2}\\lambda^2e^{4\\alpha}", "bb3f70a2a1f82293915d24cae2cc8d6b": "\\nu(W)=4", "bb401d3b80f8198f6eac74e83376f488": "k \\ge n ", "bb403142ee5600e5ca48aa5b927dd14d": "\\mathbf{P} \\!\\,", "bb40385852aa907b6063b8361e0d562d": "\\textstyle \\R", "bb40389c5c64fd72a97eb9cf718e176d": "ds^2~=~dx^2~+~dy^2~+~dz^2 ,", "bb403abfb2b73931c3a5ba684a8bb83d": " \\delta_x: S \\times X \\rightarrow S \\times \\{0,1\\}", "bb4046b64b7f1554a8afae934d0a275e": "X=A\\cup B", "bb409045a914e2e1c7b8faae95ab084c": "\n\\operatorname{Li}_s(e^{2 \\pi i m/p}) = p^{-s} \\sum_{k=1}^p \ne^{2 \\pi i m k/p} \\,\\zeta(s, \\tfrac {k}{p}) \n\\qquad (m = 1, 2, \\dots, p-1) \\,,\n", "bb40b204c002d9ac2615a89d90bf297b": "c, \\; q", "bb40c4bbda5ad91fd85704aa2539a5fc": "A\\ang \\!\\ \\theta, ", "bb41932265810a9a7a04962f682616fb": "\\Sigma^{0,C}_{n+1}", "bb41f950c259bff1a5cb7f3b20126ddf": "\\theta_{e}\\,", "bb4213ebdd4d53ad5b73e9c420f08437": "dx = dx_\\infty\\,dx_2\\,dx_3\\,dx_5 \\cdots \\text{ and } dx^{*} = {dx^{*}}_\\infty {dx^{*}}_2 {dx^{*}}_3 {dx^{*}}_5 \\cdots", "bb421ad4c9c39bd685922e4c24447f17": "\\{\\Phi_{10},\\Phi_{20},\\Phi_{21} \\}", "bb422483470763d536fb8cae35efcb70": "r \\approx R \\sqrt[3]{\\frac{M_2}{3 M_1}}", "bb42a27a35a9700d552bf00412e6b774": "H_i = \\sum_{j=1}^{N} {F_{ij} J_j}", "bb42a95247994ef62f79cb24fd513da4": "\\Psi_{gh} = \\Psi_g\\Psi_h", "bb4347346900555955a70beb2197d99e": " \\int P_n(x)^2 W(x)\\,dx = 1~.", "bb43b3c110f9b5c5eb2954bcbf63877c": "ds = \\left( {{{\\partial s} \\over {\\partial T}}} \\right)_P dT + \\left( {{{\\partial s} \\over {\\partial P}}} \\right)_T dP", "bb43b95c72e0a8cc3a0bc14c0c8f4f98": "Z_i := D_i([ZZ_{i,1} [,ZZ_{i,2} [,ZZ_{i,3} [,ZZ_{i,4} [,ZZ_{i,5} [,ZZ_{i,6}]]]]])", "bb43e597c30e69225d41b84d14137def": "K_I", "bb4404f60344e37a78f537b2d93b71f6": "(\\ \\cos E\\ ,\\ \\sin E\\ )", "bb4417a55139328a12d5673eca148247": "I_{max} = (x_-,x_+), x_\\pm \\in \\mathbb{R}, x_0 \\in I_{max}", "bb447d9e32369180155160a32374003d": "\\frac{11}{7}", "bb44d742c4e023b7007bd52fd787bf33": "\\frac{1}{R}\\frac{d}{dr}\\left(r^2\\frac{dR}{dr}\\right)", "bb450e425f67273287b08b73995369c3": "wp(S,\\mathbf{true})", "bb453b8f91c47deeaf9bd5d7849872ec": "n \\ge N", "bb45eaa761d4952b84fe1bce502b7521": "w\\Vdash \\Box p", "bb460611c640064de26a3a2b322d453f": "f^\\#: \\operatorname{Spec} B \\to \\operatorname{Spec} A, \\quad p \\mapsto f^{-1}(p)", "bb460ee743c6ae88c46d3fe79471ed40": "\n\\mu_p := \\frac{1}{2}\\sqrt{\\theta^2 + \\frac{2\\sigma^2}{\\nu}} + \\frac{\\theta}{2}\n\\quad\\quad\\text{and}\\quad\\quad\n\\mu_q := \\frac{1}{2}\\sqrt{\\theta^2 + \\frac{2\\sigma^2}{\\nu}} - \\frac{\\theta}{2}\n\\quad.\n", "bb46ac516f4e48113c27db52a52d061b": "\\phi^{A,\\bar{x},\\bar{a}}", "bb46be17c770c663efc2e1ee55ec56d1": " \\ -\\gamma = \\Gamma'(1) = \\Psi(1). ", "bb46d799963434e4a6487d3079981cbe": " B_{i}=\\frac{k}{\\kappa_{s}}\\int_{\\Omega_{1}}N_{i}p(x,y)d\\Omega+\\frac{k}{\\kappa_{s}}\\int_{v}N_{i}gdV-kT_{o}\\sum_{j=0}^{N_{D}}\\int_{v}\\nabla N_{j}^{D}.\\nabla N_{i}dV", "bb46e1c9d0c09489c403fb5d40cead7a": " T = T_1 + ... + T_n ", "bb46e581b6e0fc97c8ae1673c7913b46": "-\\exp(-(1/2)x-(9/4)\\tau) u(x,\\tau) ", "bb47339cac1047068698bb6d178294d2": "\\lim_{n\\rightarrow\\infty}\\int_0^{2\\pi}\\biggl|a_0+\\sum_{k=1}^n \\bigl(a_k\\cos(kx)+b_k\\sin(kx)\\bigr)-f(x)\\biggr|^2\\,dx=0", "bb473d57e7677ff6449732f677f430c6": " \\{I_\\alpha, I_\\beta\\} \\in E \\iff I_\\alpha \\cap I_\\beta \\neq \\varnothing. ", "bb4754d838d3b6f4fadc79e812888d7a": "\\hat{r}\\ ,\\ \\hat{t}\\ ,\\ \\hat{z}\\,", "bb4789b6191521c2770dee45fe95ac7b": "\\aleph_0 = |N|", "bb486aff7ef20b440eaf4dd495df3d92": "x'\\,\\!", "bb486e51ce82fc24b7172171327b1bb4": "|z_1 - z_2| < \\frac{1}{\\left \\|(z_1 - T)^{-1} \\right \\| } .", "bb487b0587d84f9952423c2f7c9cfda5": "\n\\lim_{N\\to\\infty}a_N = \\lim_{N\\to\\infty}b_N = M(a,b), \\,\n", "bb4896d8e7bd57dac43ea00156d3d2ee": " \\psi_E (x)", "bb48da6da473c6ad3fd0cb3de3c0cadc": "L_t=L_{t+1}", "bb48ef27310d3f74e6e0616dd8465539": "\\sin^2(x) = \\sin(\\sin x)", "bb49f9e210fdf673a7ab33c443f3707f": "j = k", "bb4a49cb79c606a6caf17bc2e3c8a76f": "c_{ij}^k", "bb4a4b2e0386dc0d440cf6ecfbaa794c": "Categories and interwikis go on the /doc subpage.\n", "bb4a8e905a167c2da46fc52f024820f5": "\\theta_1=36.53^\\circ", "bb4acfce94eb58ddc5cc8657cd69b942": " \\mathrm{d}y = \\frac{\\mathrm{d}y}{\\mathrm{d}x} \\,\\mathrm{d}x, \\,", "bb4acfe9f46e3223425eae204bf8a691": "0 \\in S", "bb4aec551c4db7e91d2b96eb8f0bc27e": "a_s^{m,k}=\\frac{c_s^{k,m}}{m^k},s=\\frac{-k(m-1)}{2},\\dots,\\frac{k(m-1)}{2}", "bb4b4be258a81ca5d7c1de20a7416e3c": "(x \\cdot y) \\cdot (u \\cdot v) = (x \\cdot u) \\cdot (y \\cdot v)", "bb4b59795d899a664f6f5a8f03f12422": " \\frac{1 \\text{ lbf}} {(1 \\text{ in})^2} = \\frac{4.4482216152605 \\text{ N}} {(0.0254 \\text{ m})^2} ", "bb4b6e55de9b4b3d244af9b6dd704808": "P(A|B) = \\frac{P(A \\cap B)}{P(B)}", "bb4b753330b45b39e2023d4dfedaaa30": "J^{\\mu} = \\left(c \\rho, \\mathbf{J} \\right)", "bb4ba496bfef2409e50ed88bffe07faa": "\\chi_{[a,b]}(t) \\,", "bb4c1318779a4be7196322a08e4738ce": "\\scriptstyle -\\colon F \\,\\times\\, F \\;\\to\\; F,\\,", "bb4c2a6c7d44d33d31fd237ab5ff9bf7": "\\Delta=\\nabla^*\\nabla", "bb4c3792c712c062987042b3892a4f65": "\ng(n) = \\sum_{d|n}f(d).\\;\n", "bb4c3b08ee01dab851365c5196a5c9ba": "|p_0\\rangle", "bb4c5f0294e16e513b6f45466c717ea9": "\\sigma~=~\\frac{k}{\\theta^2}~\\left[ {\\left(\\frac{\\partial\\theta}{\\partial x}\\right)}^2~+~{\\left(\\frac{\\partial\\theta}{\\partial y}\\right)}^2~+~{\\left(\\frac{\\partial\\theta}{\\partial z}\\right)}^2\\right] ,", "bb4ca3d3dafece5ebe7a57fa24dfb71d": "\\mathcal{O}_X", "bb4cb9a9f2c4bb86055249c2c3e10045": "E(\\bar{K})", "bb4ceaf9614980e207d86df8263138cc": "\\tfrac{19}{720}", "bb4d063adc39665a9e509992420a99aa": "B_6(x)=x^6-3x^5+\\frac{5}{2}x^4-\\frac{1}{2}x^2+\\frac{1}{42}.\\,", "bb4d598686f39b8d501d855cf4c927c7": "\\Omega(n^2)\\,\\!", "bb4dc6a52ccdc56d62744a1d7bad6daf": "\\mathbf{x}_0 \\in U", "bb4e0330ef7fb649101b1c067d3b7fe3": "\\frac{d\\theta_\\mathrm{(t)}}{dt}=\\R'(1-\\theta)(1+k_\\mathrm{E}\\theta).", "bb4e107e647170db46a868203ef2fee6": "\\langle a,b,j \\mid aba=bab, (aba)^4,j^2,(ja)^2,(jb)^2 \\rangle\\,\\!", "bb4e66bc6e3f24ad2f9d45501a56b340": "n=1,\\dots,N", "bb4e7c5b5ffdea6ae84cd0ade212e0ac": "q,", "bb4e9117605045e19a6ee1e1e9c77540": "\\left\\vert S_n - \\ell \\right\\vert \\le \\ \\epsilon.", "bb4efac64f5310f4bdcbcaacc63c5a4e": "1/(k+1)", "bb4f625d80b70e47da4b2b92fbfee156": "E = eV = \\frac{Ze^2}{a} \\,\\!", "bb4fbc77f4d57566e0e86d39d0aa375c": "\\sigma_B \\geq 0~.", "bb4fc39d3c5f7655ce59241a92f04fc0": "\\vec V", "bb4fc6592b12acbede41441fa153a119": "open,open'", "bb4fd54e8d877a9ad04c3f697894342e": "f:X\\to TY", "bb4fe7ebc922bf8a7e2f6415bb7e08af": "a \\times 3a \\times 7a \\times 9a \\equiv 1 \\times 3 \\times 7 \\times 9 \\pmod {10}. \\,\\!", "bb50504bc0912800122bba6eff9b469d": "\\tanh(1/n) = [0; n, 3n, 5n, 7n, 9n, 11n, 13n, 15n, 17n, 19n, \\dots] \\,\\!", "bb50abfdae3fa0074565bb32bb202dee": " d_1\\quad d_2\\quad d_3\\quad d_4\\quad d_5\\quad d_6\\quad d_7\\quad d_8 ", "bb50c42d82e381a0c2c2bed89f3f2dbe": "\\langle\\Psi,\\Phi\\rangle", "bb50c43bb8e92b4bc4ea25e0a744036f": "\n\\begin{align}\n\\cos(2 \\alpha) & = \\frac{\\sin(4 \\alpha)}{2 \\sin(2 \\alpha)} \\\\[6pt]\n\\cos(4 \\alpha) & = \\frac{\\sin(8 \\alpha)}{2 \\sin(4 \\alpha)} \\\\\n& {}\\,\\,\\, \\vdots \\\\\n\\cos(2^{n-1} \\alpha) & = \\frac{\\sin(2^{n} \\alpha)}{2 \\sin(2^{n-1} \\alpha)}.\n\\end{align}\n", "bb50cef24bf13793015bf5862be32c18": "F(u+h) - F(u) = \\int_0^1 dF(u+th;h)\\,dt", "bb50d21e1723e9cd71fa5e7509b020fa": "\\varphi_m(x)={\\rm E}_{-m}(x).\\,", "bb50dd2c7402223d135fc631d1a195a8": "(((\\dotsb((a_n x + a_{n-1})x + a_{n-2})x + \\dotsb + a_3)x + a_2)x + a_1)x + a_0.", "bb50f41437089454551b069f420460cc": "\\tau _{ij}^{\\mu }=\\left\\langle \\psi _{i} | \\partial ^{\\mu }\\psi _{j} \\right\\rangle ", "bb5146245f5d44c39d4ac9d84f873870": " \\mathbf{y}_{k}. ", "bb51591592a9338e4ad19bd19634216c": "V = -Y =X", "bb517908bbca75499541e8a4c20fafec": "(V_\\alpha,\\in,U\\cap V_\\alpha)", "bb519c9957fe4645a1152856df434749": "C,D", "bb51a2d3e3053e904ee44b13a33b7dfb": "\\alpha = \\frac{k_\\mathrm{e} e^2}{\\hbar c},", "bb51bbb759ffea94d0cba82d3e2e6eca": "x_1 = x_0 - \\frac{f(x_0)}{f'(x_0)} = 0 - \\frac{1}{0}.", "bb51cf70b3a2d83e0f6b41f7465184be": "C_n^k=C_n", "bb523011662ed1392b96ccf6c3628624": "R' = R^e\\pmod n", "bb523ed9704bc3475d054b1374d25884": "L_P^\\sigma", "bb525f0e3122df3f2726b5dc601c3c15": "\\sum_{j=1}^n f_jdz^j+g_jd\\bar{z}^j.", "bb526e8731d9dc4dd44698844a3458fb": "\\eta\\ = f_2\\!\\left({Q\\over {ND^3}}\\right),\\,", "bb5294ead8fa07af980f9033e7005696": "\\displaystyle{S_y(a,T,b)=(a,T-R(a,y),b-T^ty-Q(y)a).}", "bb52b1c9ed1ad0cc3728a0edb0134a37": "[x,y] \\subseteq B_{\\delta}([y,z]\\cup[z,x]),", "bb52b273c3930ae6b05beb5515026d55": "(1+n)^x \\equiv 1+nx\\pmod{n^2}", "bb52cce3181a7d2e315fdb0791de2cdf": "\\mathfrak{a}_+^*", "bb5376582042c0b0c244e2ffbba7bac2": "j=0,1,\\ldots,a_1-1", "bb53892132035a371093545cbacf3590": "R_1R_2", "bb538c43f4c498e1dc92e7b6931588e9": "U(r_0) = \\frac {ae^{ikr_0}}{r_0}", "bb54269594be1bed03736c3575765956": "\\lim_{n\\to \\infty}g_n = g", "bb548b9148094dcf18e6b98b518c4a61": " \\ln|u-1| ", "bb54a6c0fd3c050dd63419fd0aaf67e9": "\\log G(1+z)= \\frac{z}{2}\\log 2\\pi -\\left( \\frac{z+(1+\\gamma)z^2}{2} \\right) + \\sum_{k=2}^{\\infty}(-1)^k\\frac{\\zeta(k)}{k+1}z^{k+1}", "bb54df709bc82561427975c78890bee3": "x^{-1} \\cdot x ", "bb54f2258a434346acc333a848f96707": "\n\\begin{align}\nP & = \\left(\\frac{3}{50} \\times \\frac{2}{49}\\right) + \\left(\\frac{3}{50} \\times \\frac{(13 - x) \\times 4}{49} \\times 2\\right)\\\\\n& = \\frac{3}{1225} + \\frac{12 \\times (13 - x)}{1225}\\\\\n& = \\frac{159 - 12x}{1225}.\\\\\n\\end{align}\n", "bb556908c5cfe41bfe22be042e788a79": "e_y-f_x=e\\Gamma_{12}^1 + f(\\Gamma_{12}^2-\\Gamma_{11}^1) - g\\Gamma_{11}^2", "bb558c843b1be96491bdcc41ddd411c3": " = \\frac{1}{2} \\eta_{\\mu \\nu} \\{\\gamma^\\mu, \\gamma^\\nu \\} \\,", "bb55a487b346b85686dc9e5f5650ea5e": "\\Psi_2 \\left( \\alpha; \\gamma, \\gamma'; x, y \\right) = \\sum_{n=0}^{\\infty}\\sum_{m=0}^\\infty \\frac{(\\alpha)_{m+n}}{(\\gamma)_m(\\gamma')_n} \\frac{x^m y^n}{m!n!},", "bb55bf48b946db3f7987acb7e19a87c5": " E: W^{1,p}(\\Omega)\\rightarrow W^{1,p}(\\mathbb{R}^n),", "bb55d1bc515040ea37826f050dba8f74": "d(f \\cdot x,x)<\\epsilon", "bb55d3c9cda2e13bb1b243aa6363d643": "= b \\frac{A}{W}", "bb55edc48a98a09e546cf81e665bab52": "\\begin{align}\nv_x & = u_x + 2a \\exp \\Bigl( \\frac{u+v}{2} \\Bigr) \\\\\nv_y & = -u_y - \\frac{1}{a} \\exp \\Bigl( \\frac{u-v}{2} \\Bigr)\n\\end{align} \\,\\!", "bb560ca55f4d1e7da560cf4f7fe8cb01": "\\frac{dQ}{dP} \\leq \\alpha^{-1}", "bb5640e1665b1052c15fa57c6cccaf48": "r_{i-1}", "bb565f37f54e864870aeee9b411ef631": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx = \n \\frac{x^{m-n+1} \\left(a+b\\,x^n\\right)^{p+1}}{b\\,n (p+1)}\\,-\\,\n \\frac{m-n+1}{b\\,n (p+1)}\\int x^{m-n} \\left(a+b\\,x^n\\right)^{p+1}dx\n", "bb56b20e372084127d386c1c96a8d4a2": " \\hat{\\mathbf{h}}(n+1) = \\hat{\\mathbf{h}}(n)+\\frac{\\mu\\,e^{*}(n)\\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)}", "bb5708d1312e1c40aeeb4d4be0484471": " i = E_v q v ", "bb57a411a8bbaee17c112295f97aa993": "\\Sigma_k=\\mathbb{P}^n", "bb57b91d05be7469a42e5fe99fea6ddc": " \\ddot{\\mathbf{r}} = - \\frac{\\mu}{r^2} \\mathbf{\\hat{r}}", "bb581215d6b6cadf52fcdf124cb33426": "T(n)=n", "bb585e88fcdc940edd266a713af8c03e": "\\tan(x) = \\cfrac{x}{1 - \\cfrac{x^2}{3 - \\cfrac{x^2}{5 - \\cfrac{x^2}{7 - {}\\ddots}}}}.", "bb587534cfb943244916df210b453dc4": "l(\\Phi)\\equiv\\partial_x+a\\partial_y+b(\\Phi)", "bb58ea210dfe7e658969e8e2b7c1bc5a": " a = 3 \\ m ", "bb58f08b453dca9777fae6d41037c08c": "\\textstyle{\\left(\\!\\!{n + 1 \\choose d}\\!\\!\\right)},", "bb590bfd61b20098b1d5378a8a336a21": "\\int xe^{-c x^2 }\\; \\mathrm{d}x=-\\frac{1}{2c}e^{-cx^2} ", "bb5965b64ee7c611c5e0cb814507e437": "r = 2kn", "bb59c1ec46c813f0077056337d44f53b": " \\theta=(0:1:0) ", "bb5a3c0aa78e47c056574c2afd9e11f7": "\\begin{align}\n1_{FY} &= \\varepsilon_{FY}\\circ F(\\eta_Y) \\\\\n1_{GX} &= G(\\varepsilon_X)\\circ\\eta_{GX}\n\\end{align}", "bb5a4c6aec76388261617c7f67a1add9": " G = G", "bb5a65518109d93861558396615ab9ab": "\\operatorname{rank}\\begin{bmatrix}B& AB& A^{2}B& ...& A^{n-1}B\\end{bmatrix} = n. \\,", "bb5a935ddeef07f93a577f062b7eba9c": "{\\mathbf{}}P_i", "bb5b00f58e00f972d23476b0225b4fb9": "\\left( \\phi_k, \\phi_i\\right)_d=0", "bb5b3bcb8c23bfd7cd9570310e44c4f2": "T_n=0", "bb5b4edf4497b75351b60ba54ef5a76b": "q \\to \\infty ", "bb5b8140e191bad59e43f5fe2a57a101": "\\; Z = \\sum _s e^{- E(s) / kT}, ", "bb5c24c1fe0cc9ee39825e58542babf0": "c = 2\\arctan \\left\\{ \\tan\\left(\\frac12(a-b)\\right) \\frac{\\sin\\left(\\frac12(\\alpha+\\beta)\\right)}{\\sin\\left(\\frac12(\\alpha-\\beta)\\right)}\\right\\},", "bb5c4529134e162387104653415ccac2": "\\bold {n}_1", "bb5cde8e82b10b6ca4ae9c7bf36a9f26": "|a(u,v)| \\le \\|\\nabla u\\|\\,\\|\\nabla v\\|", "bb5d022158c0a3ac882bafb55f7b2d4c": "G=\\mathrm{SL}(n,\\mathbb{C})", "bb5d152d30870304df9bdfe10f5b132d": "\\mathcal{S} (X)", "bb5d7fb4d13a34272c256026c7fea8e7": "F(y) = \\int_0^\\infty F(x+y-D)\\frac{\\lambda^c x^{c-1}}{(c-1)!} e^{-\\lambda x} \\text{d} x, \\quad y \\geq 0 \\quad c \\in \\mathbb N.", "bb5d988f67ab54a6c4613358f2e2e774": "\\nabla \\cdot \\textbf{A} = - j \\omega \\epsilon \\Phi \\,", "bb5db1787afa082bb98ba6cec22c2e58": "\nZberm=0.125Hb^{5/8}(gT^2)^{3/8},\n", "bb5ddaf9bf85383b86b722146fe3313e": "\\frac1x \\times \\frac1y = \\frac1{xy}.", "bb5de6ffd7598d54855921826d9644a5": "(Bu|v)=-\\int_a^b\\! u''(x)v(x)\\, dx=\\int_a^b u'(x)v'(x) = (u|Bv) ", "bb5e6a3f1b453fda389759b44410fed5": "b_1,b_2,\\dots,b_n\\in B", "bb5ea476a1ca8aca4860be553828512f": "\\text{ln}_{q'}", "bb5eca882a383c03a2be0d1827a3a68a": "g_S \\approx 2", "bb5f212dbe5a7555fe5df8ab4d9d459d": "\\pm E", "bb5f82aed81603f825f820d581680bb0": "S' = S(E(\\boldsymbol{r}'))", "bb5f8444d6eacf31cd695b4466e20637": " \\tau = \\tau_0 + G \\alpha b \\rho_\\perp^{1/2}\\ ", "bb5f9ae663258cf4fb3c6e544fcc346c": "\\begin{pmatrix}\n\\cos^2(\\theta) & \\cos(\\theta)\\sin(\\theta) \\\\\n\\sin(\\theta)\\cos(\\theta) & \\sin^2(\\theta)\n\\end{pmatrix} =\n", "bb5fad98edb4f2b4ef9a42feeb59c39f": "f_{k(m,n)}", "bb5faee9517fd0bd3b20a694017ee4d7": "\n\\frac{P_{n1}Q_{n2}}{Q_{n1}Q_{n2}+P_{n1}Q_{n2}+P_{n1}P_{n2}} .\n", "bb5fe849e0f5798893f4a540e47e6784": "G \\times X \\to X, \\quad (g, x) \\mapsto g \\cdot x", "bb6018fda9755072b0c4e5955a103c7b": "a,b,\\dots", "bb606e6d6d1a7156fbd9a7c12c24ef42": " y_4 = x_3 \\sin(x_1) \\, ", "bb608b2cee08276c52a6e5844e30d71e": "n = \\frac{40}{100}(5+1)=2.4.", "bb60db793fae525d84a4856ee205185c": " A = A^\\dagger\\,.", "bb611d8f13acd6d0bde73b2efe583202": " {dL \\over dx} = {\\part L \\over \\part u}u' + {\\part L \\over \\part u'}u'' + {\\part L \\over \\part x} \\, ,", "bb6153fa085abef335b14dda0e0e3eea": " r_{\\mathrm{A}} V = V \\frac{dC_{\\mathrm{A}}}{dt} ", "bb615d99e539f61fc0f53640faa25d3d": "\\left| x - a \\right|^n + \\left| y - b \\right|^n = |r|^n.\\,", "bb6178040b9acfff3af0d57ba1ade967": "\\left(X^\\top X \\right)^{-1} X^\\top", "bb618ccc054e4d0f1bfcd40b4b7c15ef": "(W_t)_{t \\geq 0}", "bb61eddb7756a517d02994dee20f1e11": "\\psi(\\Omega^2 3 + \\psi(\\Omega^2 3 + \\psi(0)))", "bb62415aef959869a2e2f21f43e7212d": " \\cos \\boldsymbol{\\Phi}_r \\rightarrow \\sqrt{\\frac{r}{2M}}\\,\\!", "bb624936ad0d034447ae027850d18cca": " \\ c_0 = c_{00}(1 - y_d) + c_{10}y_d", "bb628534a4104598f27b6a965f331829": "b_{ip}", "bb629bd419ef06478b40ed143990f3ed": "\\mathfrak{b}\\leq\\mathfrak{a}", "bb62b8f8776ba87b10babbab70453bf7": "\\mathbf{EE}^\\mathrm{T}=\\mathbf{M}", "bb62bd969ce847be17bf807850f6d4e3": "f=r-\\sqrt{r^2-\\left({p \\over 2}\\right)^2}", "bb62e071bb094a467c255539c0dd44ff": " f^{\\mu} = - 8\\pi { G \\over { 3 c^4 } } \\left ( {A \\over 2} T_{\\alpha \\beta} + {B \\over 2} T \\eta_{\\alpha \\beta} \\right )\\delta^{\\mu}_{\\nu} u^{\\alpha} x^{\\nu} u^{\\beta} ", "bb62f7f7f2945e757b92863dcfccdc47": "\\bar{e}\\ \\,", "bb635c596d79e37fc323a003c9635eb8": "\\lim_{n\\rightarrow\\infty} x_n - a_n = 0.", "bb635f27b336f2271c640f9c865bfbd1": "\\pi:[0,1]\\cap \\mathbf{Q} \\rightarrow P\\otimes_{\\mathbf{Z}}\\mathbf{Q}", "bb637eaa99062c285bfd7718dfcfeb38": "\\frac{\\operatorname{d}E}{\\operatorname{d}y}=1-\\frac{q^2}{gy_c^3}=0", "bb6395dbe3fc33fa78b9c4166fcf6f94": "f(x)=|x|^a,\\quad 0 < a < \\tfrac{1}{2}", "bb63fa04e9784e5103d6fee5dc266e14": "SiO_{2} + Si \\rightleftharpoons 2 SiO", "bb640085d65cd57a23fb737876a2a662": "\\mathbf{r}_{k+1} := \\mathbf{r}_k - \\alpha_k \\mathbf{A p}_k", "bb64023e4ce51a5c70d23327409d4375": "\\hat \\beta \\pm se_{\\beta}", "bb644acfdbac4d1d37a215a107fd3de5": "H={P^2\\over 2m}+mgz+V_0 e^{-kz}\\quad (1)", "bb64cc5d7984d551811cb0fccb5a7c42": "\np_x = \\frac{\\partial L}{\\partial \\dot{x}} = -\\frac{q B}{2c}y\n", "bb651aa4e020374f3ccf593c78c4be04": "\\bar{S}_k(c) = S_3(c)", "bb65d63592442c0bb8959f587aa0411c": "\\prod_{s=1}^l \\zeta (\\sum_{j \\in P_s} i_j)", "bb65dd135dc8ac915e7a3928e6f9fcbb": "\\ \\sup_{T \\in F} \\|T\\| < \\infty. ", "bb65f89e096d769fd401ecc6b7a21057": "f\\geq 0", "bb6685ff3d9e63eb84e8f2eb8c2b51a2": " y' = y ", "bb672a4e6d7b7f6497d0a3d0f6b6ac22": "D = - \\frac{\\lambda}{c} \\, \\frac{d^2 n}{d \\lambda^2}.", "bb679bcac5aa7986adc87631aa5fb2ac": "(M,\\varphi)\\ ", "bb679c46b14db5681ecc318725143e76": " \\mathcal{L}_Y ", "bb67ff73be588d8a11275296b569c219": "\\sqrt 6 -\\sqrt 2\\approx 1.03527", "bb6884f964e93eb6fec8c3997af47bc8": "\\sqrt{T}\\big(\\hat\\theta - \\theta_0\\big)\\ \\xrightarrow{d}\\ \\mathcal{N}\\big[0, (G'WG)^{-1}G'W\\Omega WG(G'WG)^{-1}\\big]", "bb68966eed03a8bfe91d7b1ed65aa865": "\\chi^2_{k-1}", "bb68e1228bc0c4daec545ed25641410b": "\\operatorname{pf}\\begin{bmatrix}\n\\begin{matrix} 0 & \\lambda_1\\\\ -\\lambda_1 & 0\\end{matrix} & 0 & \\cdots & 0 \\\\\n0 & \\begin{matrix}0 & \\lambda_2\\\\ -\\lambda_2 & 0\\end{matrix} & & 0 \\\\\n\\vdots & & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\begin{matrix}0 & \\lambda_n\\\\ -\\lambda_n & 0\\end{matrix}\n\\end{bmatrix} = \\lambda_1\\lambda_2\\cdots\\lambda_n.", "bb68e5a8458e6cffee5192b303de3e68": "\\dfrac{f}{|f|}: \\left(S^{2n+1}_{\\varepsilon} -V_f \\right) \\rightarrow S^1", "bb6908689df3cab257c093f83fe3a984": "R_{ik}", "bb6979b49d1c2178d16c5b8db37b73e7": "i = 1, 2, 3,\\ldots, r\\,", "bb697ecf46d863dbbb0a1738a33a71ac": "\\beta(T,\\mathcal{A})\\le \\mbox{dist}(T,\\mathcal{A})", "bb6a4f7f1c24506698331395c4c878e2": " X_{\\delta} ", "bb6a9fea6ae6df0e147ee1fbf852e7e5": " [V^*] = \\chi(V^*)[k] ", "bb6b0986664b6824b935c05a9371dfb3": "\\scriptstyle{\\left(|\\psi(t)|^2 = |\\psi(0)|^2\\right)}", "bb6b0a1643726df326d9589bf33f7288": "y \\in \\{0,1\\}^n\\,", "bb6b360f5cd4959dfd4054c1cdb7c7d8": "\\operatorname{cov}(X,Y)=\\operatorname{E}(\\operatorname{cov}(X,Y \\mid Z))+\\operatorname{cov}(\\operatorname{E}(X\\mid Z),\\operatorname{E}(Y\\mid Z)).\\,", "bb6b39f5ac329f1b3812be645a1ac101": " \\frac{BC}{\\sin{A}} = \\frac{AC}{\\sin{B}} = \\frac{AB}{\\sin{C}} = \\frac{HA}{|\\cos{A}|} = \\frac{HB}{|\\cos{B}|} = \\frac{HC}{|\\cos{C}|} = 2R. ", "bb6bb3d48f027e7218628a5a4dc0eb3a": "\\sup_{t>0}|f \\ast \\Phi_t(x)| \\leq (Mf)(x) \\int_{\\mathbf{R}^n} \\Phi", "bb6c68e5ed2f7f24129bdef34ff55277": "FVA \\ = \\frac{ C ( 1 - (1+i)^n )}{1 - (1+i)} \\ = \\frac{ C ( 1 - (1+i)^n )}{-i} ", "bb6c8e35b4ab3a003b9f2dca824811e5": "\\max\\{|z_i|\\}", "bb6cd28f89b0c2991f3c28c1b26da8f0": "\\vec B=\\nabla\\times \\vec A\\,.", "bb6d2017090b409176f64685875542a8": "a \\rightarrow P", "bb6d365afa442a4db163875ff3c2cfbd": " P_i = Q_i ", "bb6d5dca92835ce8efbfd63cbedfb16e": "f(z) = \\sum_{k=0}^{\\infty} \\operatorname{PP}(f(z); z = \\lambda_k),", "bb6dfea7a28bffb63f4f27c9ae3f3497": "\\left (p + a\\left (\\frac{n}{\\tilde{V}}\\right )^2\\right ) (\\tilde{V} - nb) = nRT", "bb6e39738c1edab96ea600848c3e983b": "e^I_\\mu e_J^\\mu = \\delta^I_J", "bb6e55a45ffc57038a31186f0c0d5b2f": "\\hat{\\sigma}_e^2", "bb6e765d9054c6361239677c44e90bd4": "e^{-\\rho t}S_t", "bb6ea3e57db2823c1117291c70b5f0d7": "\\displaystyle x^0 = ct", "bb6f84370e08c5e04347907d2e14c5ee": "\\frac{1}{\\operatorname{pf}(A)}\\frac{\\partial\\operatorname{pf}(A)}{\\partial x_i}=\\frac{1}{2}\\operatorname{tr}\\left(A^{-1}\\frac{\\partial A}{\\partial x_i}\\right),", "bb6fa012750ced4774de71294850c238": "X'VX", "bb7077f66fa4b072be5648f122cf3ec2": " S'", "bb70f2c932611ea7ea6d87f19521ece0": "e^{\\theta' x}", "bb71339752ab32bc4403493c1184f0e7": "M_{l}=\\frac{M_{s}}{\\phi_{sl}}-M_{s}", "bb71500ae6948878a65a9a1ccf133d20": "\n\\begin{align}\nO_t\\{x(u-\\tau);\\ u\\}\\ &\\stackrel{\\quad}{=}\\ y(t-\\tau)\\\\\n&\\stackrel{\\text{def}}{=}\\ O_{t-\\tau}\\{x\\}.\\,\n\\end{align}\n", "bb7201e607352ac81dd187084f449f80": "x\\mapsto \\lfloor x\\rfloor", "bb7221e73746dd2961702ea776876349": "\n\\bar{\\Pi}^m_\\ell(z)\n= \\left[\\frac{(\\ell-m)!}{(\\ell+m)!}\\right]^{1/2}\n\\sum_{k=0}^{\\left \\lfloor (\\ell-m)/2\\right \\rfloor} \n (-1)^k 2^{-\\ell} \\binom{\\ell}{k}\\binom{2\\ell-2k}{\\ell} \\frac{(\\ell-2k)!}{(\\ell-2k-m)!}\n\\; r^{2k}\\; z^{\\ell-2k-m}.\n", "bb7253f800686ccd2734630b80ce1263": "\\mathbf{p}(u, v) = \n \\sum_{i=0}^n \\sum_{j=0}^m \n B_i^n(u) \\; B_j^m(v) \\; \\mathbf{k}_{i,j}\n", "bb72b4fdd5151cbd19cfae9192376b2d": "= I_0 {\\left[ \\operatorname{sinc} \\left( \\frac{\\pi a}{\\lambda} \\sin \\theta \\right) \\right] }^2 ", "bb73a87f91e041f5707c005772d58680": " L = \\beta J* ", "bb73c44b65a7fe49b93c999e18983002": "\\Gamma_{12} (u,v,0) = \\iint I(l,m) e^{-2\\pi i(ul+vm)} \\, dl \\, dm", "bb73c9bb2e8526110e93a5430af6830e": "f(x) = (2x + 8)^3 ", "bb73ef3821c4fedf147fbce7011b7c3a": "\\rho_c / \\rho_f \n= \\frac{c^3}{8\\pi n^3 \\nu^2}\\frac{Q}{\\nu V}\n= \\frac{1}{8 \\pi} \\left ( \\frac{\\lambda}{n} \\right )^3 \\left ( \\frac{Q}{V} \\right )", "bb745769155fb0530205911ce0a6625f": "v_n \\ne 0", "bb746f7137ccd7122cf9d79adfc5394d": "\\{ \\dots, -11, -7, -5, -3, -2, 2, 3, 5, 7, 11, \\dots \\}\\, .", "bb74b38d796fb5314d36de69d08781f6": "\\,A' = P A P^{-1}", "bb74f9a727588d9168cad1c36c83316a": " X_n = X_{-n}^* ", "bb751181bddf990b27e26801032cfc12": "n_i^2 = n_h n_e = N_{\\rm V}(T) N_{\\rm C}(T) \\exp((E_{\\rm V}-E_{\\rm C})/kT),", "bb75321bc17551664c78e2a6a9232549": "{\\it exocytosis}", "bb75812c8d1a2397ab9ca0698f29f0e1": "c,j \\in \\mathbb{Z}", "bb758a63cd2d23420d9a5a5aa95d0bf6": "\\alpha_4 = {{4\\alpha_0 + 3\\alpha_1} \\over 7}", "bb75e9ad16ead32d8198b3d88b529d27": "0.20 + j0.50\\,", "bb75f263867ff5e6a7a3d8a3a89e9bac": "\n \\delta (q) ~ \\star ~ \\delta(p) = {2\\over h}\n\\exp \\left (2i{qp\\over\\hbar}\\right ) ,\n", "bb7600a69ec4809141c422f19ad76b98": " c = 2^3 \\equiv 8 \\pmod {13}. ", "bb761bde88e0ca09579ae129d047ce5f": "\\sigma:R \\to R", "bb76348d66d3f7ebe60415c5aa85f51a": " AC ", "bb764e0275f38f05d6b32fcadf742969": "x=a(3\\cos t-\\cos3t),\\quad y=a(3\\sin t-\\sin3t).", "bb76959c4affe088d89122647737e6c9": "(2,1)\\prec (3,0)", "bb769e3dfb2db3eb9b852f5f2e3c9b3d": "g=\\{g_n\\}_{n\\in\\Z}", "bb76b7d43a28608d1d1be434cef89b65": "r(1+e\\cos\\theta)=\\ell.\\,", "bb7742dcaed696f357b7f5331e8e0842": "\n r^k \\geq 0 \n", "bb7752b61a9599fe22c5c1d719443af6": "\\hat{p}_j^{\\mathrm{after}}", "bb7763c9bc129207910eb846dad6ea38": "\\textstyle S_{12} = S_{21}", "bb77c251ac792e757864a0dd3c9e8fef": "H^{\\alpha\\beta}_{a\\bar b}", "bb77d1268a515929cc099056e1a16b41": "K \\not\\models c", "bb77e4b4dd3163acaaa3785cc665dff5": "E\\geq \\left\\|\\int_{S^2} \\operatorname{Tr}\\left[\\varphi \\vec{B}\\cdot d\\vec{S}\\right]\\right \\|.", "bb77fe9588a0846ef0f08db59b3afff8": "{AAOV}=2 \\times arctan \\frac{0.5d}{f_E}", "bb78072cfd8f838d192a74c9d0a15d49": "(|e\\rangle|0\\rangle+|g\\rangle|1\\rangle)/\\sqrt{2}", "bb78137297463c3d408b9ed85d778a0d": "\\langle S_i \\rangle = 0", "bb7877cc0a32c67514bace3840febe09": "h(x)=x^{\\{m\\}'}\\left(H+L(\\alpha)\\right)x^{\\{m\\}}", "bb788e62f4c4d7a924311b3ad0e4460d": "\\displaystyle \\nabla^4 u = E(w_{xy}^2-w_{xx}w_{yy})", "bb78d66f75a21867cc70ee71fb6ca346": "abs(\\lambda) > 1 \\,", "bb79713c22ccb6ec06f7ee8c2ab1cea4": "\n (1)\\quad\\left(\\frac{\\partial \\mathbf{x}}{\\partial \\mathbf{X}}\\right)^T\\cdot\\left(\\frac{\\partial \\mathbf{x}}{\\partial \\mathbf{X}}\\right) = \\boldsymbol{C}\n", "bb79f26e414f4b8500c8b32391f9f7f6": "D = \\{ x^2 + y^2 \\le 9, \\ x^2 + y^2 \\ge 4, \\ y \\ge 0 \\}", "bb7a0f687202c4060328fb9a0c01fd7c": "X\\rightarrow X", "bb7a277df8f0ea1e29fb705f72e18296": " -y \\partial_x + x \\partial_y \\,\\!", "bb7a3c7fe5ade579ec8e99f0a3b73291": "w(n)=1 - \\left|\\frac{n-\\frac{N-1}{2}}{\\frac{L}{2}}\\right|,", "bb7a7f6f296d39f3021cbb907578e469": "\\overline{k}", "bb7a8e51850a3197f75c11f5f2fa8ba3": "A\\subset K^n", "bb7aa5f754128087c7052a62a2387abd": "1 \\ \\mathrm{ Jy} = 10^{-26} \\frac{ \\mathrm{W} }{ \\mathrm{m^2} \\cdot \\mathrm{Hz} }", "bb7af94640de9a627cd17587b363f477": "\\nabla f = \\left(\\frac{\\partial f}{\\partial x_1},\\dots,\\frac{\\partial f}{\\partial x_n}\\right).", "bb7b0e1f82d5915120e52f52f18dbac5": "V = {e^2\\over 4\\pi\\varepsilon_0}\\left(\\frac{-1}{|2\\mathbf R|}+\\frac{-1}{|2\\mathbf R+\\mathbf r-\\mathbf r^'|}+\\frac{1}{|2\\mathbf R-\\mathbf r^'|}+\\frac{1}{|2\\mathbf R+\\mathbf r|}\\right).", "bb7b11d5f32bf42903b51f28afc3579c": "[h,e]=2e", "bb7b2c08afd890adf0c07f15c0e930ae": "h \\circ g", "bb7b70e1360a8f3cce7214b700c7d85b": "p_0(x)\\subseteq p(x)", "bb7baac8e56b583265fa47070ff442b8": " P + Q_x = 0 \\implies Q_x = -P\\,.\n ", "bb7bcd90aa26e8dfa7072e6a99614b02": "E_{3} = \\Delta z \\Delta p^{2} + \\Delta x \\Delta p^{2} + 2 \\Delta x \\Delta y \\Delta p + \\Delta x \\Delta y \\Delta z + 2 \\Delta y \\Delta z \\Delta p + \\Delta y \\Delta p^{2} + 2 \\Delta x \\Delta z \\Delta p", "bb7bd5c0e742510e3af2c22dae240614": " 0= \\mathrm{P.V.} \\int \\frac{e^{ix}}{x} dx - \\pi i \\int_{-\\infty}^{\\infty}\\delta(x) e^{ix} dx ", "bb7c0e24976efc47dec57ca0a800d6f6": "\\mu(v_1,\\ldots,v_n) := \\mu_1(v_1,\\ldots,v_n)\\mu_2(v_1,\\ldots,v_n).", "bb7c6658ccd40e86bf4253c408cd0837": " \\psi_0(\\vec{r}_1,\\, \\vec{r}_2) = \\frac{8}{\\pi a^3} e^{-2(r_1+r_2)/a} ", "bb7cdba13e62bb68e776dc27f7a1efae": "M(x)= \\frac{ax+b}{cx+d}=x, \\qquad a, b, c, d \\in \\mathbb{N}.", "bb7ce425b411ce5884679a36aac991b0": "{\\rm E}_1(x)", "bb7d04f1dafe1a2cb0919f2b863f5286": "\\xi\\in C", "bb7d18803da7e67f5866c36d41ed933b": "ID(x,y) = E(x,y)+O(\\log \\cdot \\max \\{K(x\\mid y), K(y\\mid x)\\} )", "bb7d9b93f22d3d3382fb59b1502ab792": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 2}{3 \\choose 1} \\end{matrix}", "bb7da6d0da67cf849f01ec7deb383f3b": " \\min_{\\hat x} \\max_x \\left\\{ \\left\\| {\\hat x} - x \\right\\|^2 :f_i (x) \\le 0,0 \\le i \\le k \\right\\} ", "bb7dd51591abbc641356a4d483705142": "\\epsilon \\over d", "bb7def67d9d1b967409d73534d483e8e": "v_{\\text{approx}}", "bb7e02fd109e5326862d00c6b33ae5f6": "H_f = (\\kappa_1 + \\kappa_2) \\,", "bb7e089fb7805a39ba8aba8ce63e4d30": "Zsummary", "bb7e14d2f8f6e270be13a48157e194ad": " \\epsilon > \\mu \\ ", "bb7e4c5cdd93a7e073c885b3b7b838c5": "\\widehat{x_a}", "bb7f5a571f6ae540301ba57d48f77935": "\\bigcup_{\\alpha<\\pi}(0,\\alpha).", "bb7fc203e3965b4bb3a17d8eab46b509": "Z_{iT}=\\sqrt{1-\\omega^2}", "bb8007be06b82d355ecf36f3c472c54d": "\\bar {\\delta} =\\frac{\\sum c_i \\delta_i}{\\sum c_i}", "bb801fb27fa09e35a5172ed0d4427b25": "\\operatorname{mod}\\sigma_y^2(n\\tau_0) = \\frac{1}{2n^4\\tau_0^2(N-3n+1)}\\sum_{j=0}^{N-3n} \\left\\{ \\sum_{i=j}^{j+n-1} x_{i+2n}-2x_{i+n}+x_i\\right\\}^2", "bb803b54e6e01944dda31b1d719eca33": "\n\\mathcal M=i\\sqrt{\\frac{2\\omega_p}{Z}}\n\\int \\mathrm{d}^4 x f_p(x)\\left(\\part_0^2-\\Delta+m^2\\right)\\eta(x)\n", "bb808f740af1836f934c5cbef27d1f31": " \\nabla \\cdot (k \\nabla h^2) = - 2N. ", "bb80b12da84a98f0fba104673c2c9014": "r=p_1p_2\\cdots p_n", "bb80b611a1eb600af85b631eefe7cb56": "m \\frac{\\mathrm{d}^2 \\xi}{\\mathrm{d} t^2} + k \\xi = I \\delta(t),", "bb814d3ce84b6f06229cd61bb6d92fe1": "\\displaystyle{ c_n= {1\\over n} \\sum m^2 |a_m|^2 c_{n-m}}", "bb81b08f3cbee00945625bc0c16ec282": "\\omega_l", "bb8243e71726632c27dac33f5ae61808": "\\langle x,y \\mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\\rangle.\\,\\!", "bb824d685cf56934ebbe8cd4498640c2": "f(x)=\\sum_{\\alpha\\in\\mathbb{N}^r} a_\\alpha x^\\alpha, \\alpha=(i_1,\\ldots,i_r), a_\\alpha\\in \\mathbb{R}.", "bb82523146cf07afb8f5460c1d01326f": "\\textstyle [mn,mk]_2", "bb82531c2c4eb6a67995527a7366b579": " \\mathbf{F} + \\mathbf{R} =m\\ddot{\\mathbf{X}}, ", "bb8256124b5618445fb4ade41d440594": "gW = gW^{T} - f(U) + \\lambda.gP^{ex}.", "bb828b7b624b6f952901465f45e03f73": "l_i \\leq x_i \\leq u_i", "bb82bdec8e532667416bdcd76be70c6b": "\\frac {1}{T^2} ", "bb82d3578611a6172b9bbed38cfc9a99": "N=2^l", "bb82e5be46deb0719b3262eda7d657f9": "\\tfrac{2^{10}-1}{11}", "bb833cb0999aa4349120e585bd8f92d2": "y(t) = k(t)^{\\alpha}", "bb83472937132b2fa7927e3c3ab53fd7": "\\scriptstyle I_\\text{ext}", "bb83cb72fddd4f2db0661213fc620cbf": "A \\wedge b \\wedge A =", "bb8435bf707e7824fde61fe4b5af967c": " n_\\mathrm{osc} = U_{p} / \\omega ", "bb845bbd71f6ad9773c748892a3b7a81": "f(n,k)", "bb84718fd353a04048472973b19aa2f8": "M=p.\\overline{c}.M+p.\\overline{t}.M+p.(\\overline{c}.M+\\overline{t}.M)", "bb852917a1e6836cd67d0ea56175e2e6": "q2(q1)=(5000-q1-c2)/2", "bb855ffff01523dca25cd305a9015380": "v=\\dot z = \\frac {d (R e^{i \\theta})}{d t} = R \\frac {d \\theta}{d t} \\frac {d (e^{i \\theta})}{d \\theta} = iR\\dot \\theta e^{i\\theta} = i\\omega \\cdot Re^{i\\theta}= i\\omega z", "bb85766da3d55dcd107aad22a7b4195f": "AX_1=\\lambda X_1", "bb859a4d37bb682ebc6141fa2ff75080": "\\scriptstyle{\\mathfrak K}(w,x,s) = \\sum_{k=0}^\\infty {e^{2k\\pi ix} \\over (w+k)^s}", "bb85b410c856a6c0fe40efd41df461fb": "(D\\varphi)(x)\\cdot y:=\\lim_{\\theta \\downarrow 0} \\frac{\\varphi(x+\\theta\\,y)-\\varphi(x)}{\\theta}=\\inf_{\\theta \\neq 0} \\frac{\\varphi(x+\\theta\\,y)-\\varphi(x)}{\\theta}.", "bb85d3858f56aace31a39106a0ac3f32": "a = (1+\\sqrt d)/2", "bb85e0aefc52cbd74c99e98ba45ace88": "n_{Fan}\\,\\!", "bb8609dc510279af12408e99b5e77ec9": " u_i^{n+1} = \\frac{1}{2}(u_{i+1}^n + u_{i-1}^n) - a\\frac{\\Delta t}{2\\,\\Delta x}(u_{i+1}^n - u_{i-1}^n)\\,", "bb860ba07ddc39a5b05c44e4bfbabb10": "UR = a + b(\\text{ern}-u) + e", "bb864935e72b0395953fc9b88e131652": " \\delta = \\arcsin(\\sin \\epsilon \\sin \\lambda)", "bb864a15dfd581ebb0a44e047d17f17e": "z_i = z_{i + 1}\\implies f[z_i, z_{i+1}] = \\frac{f(z_{i+1})-f(z_{i})}{z_{i+1}-z_{i}} = \\frac{0}{0}", "bb864d05a4f5107fa6f1da44f9d55e46": "\\begin{bmatrix}\nu_{11} & 0 & . & 0\\\\\n0 & u_{22} & . & 0 \\\\\n. & . & . & . \\\\\n0 & 0 & . & u_{nn} \\end{bmatrix}", "bb86a2da172ff054b5dcf21c59bc3e39": "\\varepsilon(x) = \\sup \\{ |(a'\\otimes b')(x)| : a' \\in A', b' \\in B', \\|a'\\| = \\|b'\\| = 1 \\}", "bb86d13f5f8b1cc5f9e37d6c20fdc838": "E=-E_1 \\sum_{j,k} S_{j,k}S_{j,k+1} - E_2\\sum_{j,k} S_{j,k} S_{j+1,k}", "bb86efce4fcf02b73cbc005a673ea85a": "\\sqrt{n/2}", "bb8778734f4675e8b10e257e02e8bce2": "\n \\begin{align}\n N_{11,1} & = \\cfrac{2hE}{(1-\\nu^2)}\\left(u^0_{1,11} + \\nu~u^0_{2,21}\\right) ~;~~\n N_{22,2} = \\cfrac{2hE}{(1-\\nu^2)}\\left(\\nu~u^0_{1,12} + u^0_{2,22}\\right) \\\\\n N_{12,1} & = \\cfrac{hE(1-\\nu)}{(1-\\nu^2)}\\left(u^0_{1,21}+u^0_{2,11}\\right) ~;~~\n N_{12,2} = \\cfrac{hE(1-\\nu)}{(1-\\nu^2)}\\left(u^0_{1,22}+u^0_{2,12}\\right) \n \\end{align}\n ", "bb877cd4504ea5d32c04b108589daaed": " X\\subset\\mathbb{R}^n, \\mu ", "bb878ae4ece4f8eea4b1828a989b0e29": "\\theta=44.537^\\circ ", "bb87ae4a20a7f9ced74cdd9d2a0bf4d1": "a\\neq b,", "bb87e2c025fe16e290327e3043541911": "\\mathcal{F} = \\mathsf{id}^{\\otimes N}", "bb885c567559f727bb24a04a5592ae52": "00\\ 00\\ 00\\ 01\\ 01\\ 10\\ 1110\\ 110", "bb88a6ad6d3d72b527950d6c6439cff5": "F_{\\mathbf P}\\colon{\\mathbf P}\\to {\\mathbf R}\\,", "bb89594893f2806847ed7e56e7233191": "~\\beta~", "bb8a0a401cd53fae70f5b1f71d2b083f": "\\left( \\frac{\\partial \\frac{g}{T}}{\\partial T}\\right)_P=-\\frac{h}{T^2}", "bb8a234a1fa4d88c5f15b8c9efef912a": "A \\;\\overset{f}{\\hookrightarrow}\\; B \\;\\overset{g}{\\twoheadrightarrow}\\; C", "bb8a539cc55bc5fa8f233280dd33bd92": "M:= \\exp\\left(|\\alpha|^2 +\\mathrm{Re}\\, \\alpha \\right)", "bb8a66c1dce7f48aa8d08fb3ae198d60": "\\scriptstyle w[N/2-1]\\ =\\ w[N/2]\\ <\\ 1.", "bb8a7737cfe99e20b960cde6697b948d": "dG=n^2 dS\\int\\cos\\theta\\,d\\Omega = n^2 \\int_{0}^{2\\pi}\\int_{0}^{\\alpha}\\cos \\theta \\sin \\theta \\, d\\theta\\, d\\varphi", "bb8ae9ad50381b3f2acb35f1c6576c6b": "\\langle Sx, y \\rangle = \\int_{H} \\langle x, z \\rangle \\langle y, z \\rangle \\, \\mathrm{d} \\gamma(z),", "bb8afc1919990a746b42b10464553801": "\\scriptstyle k\\left[M\\right] / \\mathrm{Ker} F_i \\;\\cong\\; F_i\\left(k\\left[M\\right]\\right) \\;=\\; k", "bb8b16386d45f596098afc423b3c2a10": " f_{X,Y}=f_{X,Z}=f_{Y,Z}, ", "bb8b3cf36b6dbd1966a58cc5dd832fad": "p_{\\mathbf k}", "bb8b56a03834ee067dce91b7391efd73": " r_D = \\left . \\frac {\\Delta v_D}{\\Delta i_D} \\right| _{v_D=V_{BIAS}} \\ , ", "bb8c00db24673e5545225fafeb403c44": "B_{n}^{(-k)}=\\sum_{j=0}^{\\min(n,k)} (j!)^{2}S(n+1,j+1)S(k+1,j+1),", "bb8c511ee4c7aed5bb61ed4814a4798d": "L^2(\\mathbb{T})", "bb8cc68bcbf9b7fb2898d7e3586b3725": "L_{m}", "bb8ce32e64235a7ab76671174e82c207": "L/W", "bb8d15823c04894cc8b61d8acbe36c80": " = 6.1 + 1.3 ", "bb8d2463e54dae1d4fad47caed2ca000": "\\scriptstyle m \\;\\to\\; \\infty", "bb8d32b106cb3969290746fd88dd4b71": "\\nabla\\cdot\\mathbf{B}=0 ", "bb8d6211517738e62edd3e997cfb79e1": "I_{n,m}= \\int \\frac{x^m}{(x^2+a^2)^n} dx\\,\\!", "bb8d628a61a7818e219d4749ede74e80": "\\frac{1}{\\sqrt{p(1-p)}}", "bb8da7a5ff9c6982267414aad053a166": "\\frac{\\mathrm{m}^2}{\\mathrm{kg}}", "bb8db3aa7d25f5165a5550190504d0f1": "f_1(x)=4x^4-6x^3+3x-5\\,", "bb8dc2849644ee73f14771e0d4b7c164": "f:{\\mathbb R}^m\\rightarrow{\\mathbb R}^\\ell", "bb8e1779103bd208154a000a79db2187": "\\mathbf{A}^{\\mathrm{T}} = \\mathbf{A} .", "bb8e44b4d641bdb8904d86693c9b3159": "\\mathbf{\\bar a}_i = \\begin{pmatrix}A_{1i} \\\\ A_{2i} \\\\ \\vdots \\\\ A_{ni} \\end{pmatrix}\\,,\\quad \\mathbf{\\bar b}_i = \\begin{pmatrix}B_{i1} & B_{i2} & \\cdots & B_{ip}\\end{pmatrix}\\,.", "bb8f59f7a664514dab9e3705b46ea990": "p_a=\\tfrac{2aT}{a^2+b^2-c^2},", "bb8ff3194d0e023b0c249c7a29b6138a": "J_d = \\frac{f}{2} = \\frac{Sh}{Re\\, Sc^{\\frac{1}{3}}} = J_h = \\frac{f}{2} = \\frac{Nu}{Re\\, Pr^{\\frac{1}{3}}}", "bb8ff621ca2ded258a384b40ca9fd2be": "\\mathbb{Z}_4", "bb907b9894c9c07fe23fd9add593b891": "I_B \\geq 0~.", "bb91070c190e17a88a31f74e77654640": "(x^3+x^2-6 x-2) (x^4-x^3-6 x^2+4 x+4)^4 (x^5+x^4-8 x^3-6 x^2+12 x+4)^8", "bb916b2ffb24cdd6029f50becd94ccde": "\\mathbb{R}_+", "bb9175caa5b41b177c47d81ea62a300f": "\\mathbf{{\\Sigma}}_y", "bb91812e128124554e908744f598c046": "\n M^{\\mathrm{face}}_{xx} := \\int_{-f/2}^{f/2} z_f~\\sigma^{\\mathrm{face}}_{xx}~\\mathrm{d}z_f \n ", "bb91e68fe7541e66f6951661abac8587": " y_1'=y_1+2y_2+t", "bb9233e04d20656c0d14ce3ddd9acc73": "\n\\tilde{\\mu} = \\frac{1}{I} \\sum_{i=1}^I E[z_i]\n", "bb9244d96de98dd6663239637d5b026d": "H(z) z^{-k}", "bb924e1fa13ba59fe284918cc196abc7": "x^0=ct", "bb92d3d16c031b7ce2989b3b6fe598bd": "\\sum_{n=0}^t \\sum_{i=0}^{z-1} f(z\\cdot n+i) = \\sum_{n=0}^{z\\cdot t+z-1} f(n)", "bb93c0da3b5e6d9b1fbe2f071237780a": " p_1 (x_1, x_2, \\dots,x_n) = x_1 + x_2 + \\cdots + x_n \\, ,", "bb93f9f3ff7dca86bda22a161f00dd0b": " PPV= \\mu-t\\sigma ", "bb94027bb57843eee634a196d8a5dae2": "\\frac{M - L}{H - L}", "bb94774690863b1fa1803121f25b670e": " \nf_{r, \\textrm{sphere}} = 8 \\pi \\eta R^3 \\!\n", "bb94bb01fdb7a2a9798eca6c287a6f5e": "\\exists p \\in \\mathrm{Prime}", "bb94c4b6d2f68a18629d309e835786ba": "S^n \\,", "bb94c693b55dab63cb99f287eadf0536": "\n\\text{Execution time}(T) = \\text{CPI} \\times \\text{Instruction count} \\times \\text{clock time} = \\frac{\\text{CPI} \\times \\text{Instruction count}}{\\text{frequency}} = \\frac{1.55 \\times 100000}{400 \\times 1000000} = \\frac{1.55}{4000} = 0.387 \\, \\text{ms}\n", "bb952e5bfd1cb8137abfee99179f33b8": "(\\mathbf{F}\\cdot\\mathbf{n})\\,dS .", "bb9549c7c55cd046f7baef8e2d3c2c0e": " \\underline{d}(A) = \\liminf_{n \\rightarrow \\infty} \\frac{ a(n) }{n} ", "bb9564a8f18d2f0d40e5eb2ea36be15c": "\n\\begin{bmatrix}\n1 & 1 & 0 & 0 & 0 \\\\\n0 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 \\\\\n0 & 0 & 0 & 1 & 1 \\\\\n1 & 0 & 0 & 0 & 1\n\\end{bmatrix}.\n", "bb95fb1f675d78c7aa77a03eedede129": "L=D^2+p(x)D+q(x)\\,", "bb960183fbbf8415c33216aa8cc898a8": "x^m\\,", "bb964b9e144986e5b5c7995a6a784bc4": "\\psi(\\psi_1(0)+1)", "bb969720ec61e7759be914b74462bcf1": "\\mathbb{F} = \\mathbb{R}", "bb96c977980f721099a4c5b19edf4d6c": "\\textstyle\\prod_{s\\in S}(1+X_s);", "bb96f585983a5fe11f63c4e300305743": "\\operatorname{rank}\\left(\\bigoplus_{j\\in J}A_j\\right) = \\sum_{j\\in J}\\operatorname{rank}(A_j),", "bb970fc05f6d3fb62d40d87a6a19635d": "\\sigma_j=\\sigma", "bb97325572ea32f46bcca467a1594c76": "p_{n+1}(x)=x^2p_n'(x)-(2nx-1)p_n(x),\\qquad n\\in\\mathbb{N}.", "bb9790fa5d0c014c1a0716e9e5dd212d": "\\frac{dC}{dt}=0", "bb97af20fb12b8029db3fffb3b60b7de": "\\tfrac{11}{4}", "bb97b8296ff2125a5e4b18ce63b82016": "\\gamma\\in S", "bb9812b3f80262983a6126aae2c460b0": "F_{O_2loop}=\\frac{(P_{amb}*K_{bellows}*K_E*V_{O_2}+V_{O_2})F_{O_2feed}-V_{O_2}}{P_{amb}*K_{bellows}*K_E*V_{O_2}}", "bb9815ce80b9ec9781788d456f62d300": "Q_1(A) \\le Q_2(A)", "bb985d83f595e88bd833c92a89e13d2c": "\\mu=A+Br^{1/4}", "bb989cf5b9ccc538bc9923ff2f2e20ac": "\\scriptstyle \\cos \\theta = \\cos 2 \\pi U_2", "bb98b7c205ca223cd8381eb62424a726": "{\\mathrm{d} \\over \\mathrm{d}t}{\\partial r_j \\over \\partial q_i} = {\\partial \\dot{r_j} \\over \\partial q_i}, \\quad {\\partial r_j \\over \\partial q_i} = {\\partial \\dot{r_j} \\over \\partial \\dot{q_i}},", "bb9919380a0314f55de4d4372853bab2": "F_n(x)= \\begin{cases}\n0, & \\mbox{if } n = 0\\\\\n1, & \\mbox{if } n = 1\\\\\nx F_{n - 1}(x) + F_{n - 2}(x),& \\mbox{if } n \\geq 2\n\\end{cases}", "bb99b76a3c6e9b2f36a692e60fbc4115": "Y_{i} = A_{i}.K_{i}^{a}.L_{i}^{1-a}", "bb99c816f2611dc1cc7895e33df9abb2": " Y\\ni y\\to y-s(\\pi(y))\\in \\overline Y, \\qquad \n\\overline Y\\ni \\overline y\\to s(\\pi(y))+\\overline y\\in Y. ", "bb99ff6e47242e0e278d3d151dc4ffaf": "[f_k,[f_{k-1},[\\cdots[f_0,P]\\cdots]]=0.", "bb9a2f1e8e2dd5d06f32cb38529fcb0b": "\\scriptstyle Z_{\\mathrm i mm'}", "bb9a3198c8198d54bb725544bd514cc2": "P(Y)=\\sum_{X}P(Y\\mid X)P(X),\\,", "bb9ada68920b05b8869cdb12d86883c0": "\nP(\\overline{C},\\overline{S}) \\, d\\overline{C} \\, d\\overline{S} =\nP(\\overline{R},\\overline{\\theta}) \\, d\\overline{R} \\, d\\overline{\\theta} = \n\\int_\\Gamma ... \\int_\\Gamma \\prod_{n=1}^N \\left[ P(\\theta_n) \\, d\\theta_n \\right]\n", "bb9b0011343f974d591555f3ca392a2f": "\\frac{\\partial}{\\partial t}\\mathcal{A} + \\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\mathcal{B}} = 0,", "bb9b190dad5a3adfc26f0e6c0f990e64": "I \\subset \\mathbb{R}", "bb9b2e9c3d35c4d8eb97a2ae04b8247f": "\n(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1}\\mathbf{BD}^{-1} = \\mathbf{A}^{-1}\\mathbf{B}(\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\,\n", "bb9b46a16179477cafb63f159aed98f8": "u_1,\\dots,u_n", "bb9b494393996f62620d8b38659767ff": "D_{DC} = 0 ", "bb9b9795bbf2dd4e2134c46f2a0aabfe": "0 \\vee x = x = x \\vee 0", "bb9ba39c7f5c9db46084467411e9725e": "Var(\\epsilon_j)=\\sigma_\\epsilon^2", "bb9c2ca510dde1f011c018bfd037bd2b": "x \\in B(x,r).", "bb9c7946daff0c3d88d5acae42b5eb02": "\n\\begin{align}\n\\operatorname{E}[(X - \\mu)^j] & = \\sum_{i = 1}^n w_i \\operatorname{E}[(X_i - \\mu_i + \\mu_i - \\mu)^j] \\\\\n& = \\sum_{i=1}^n \\sum_{k=0}^j \\left( \\begin{array}{c} j \\\\ k \\end{array} \\right) (\\mu_i - \\mu)^{j-k} w_i \\operatorname{E}[(X_i- \\mu_i)^k],\n\\end{align}\n", "bb9d14e27a639c3abef205ed8323c07f": "W_{\\kappa,\\mu}\\left(z\\right) = \\exp\\left(-z/2\\right)z^{\\mu+\\tfrac{1}{2}}U\\left(\\mu-\\kappa+\\frac{1}{2}, 1+2\\mu; z\\right)", "bb9d1fe1ca7e034e27c1710cb8e6019e": "U_c", "bb9d382722fa73f397b2402238e3a47f": "(x - 1)(x - 2) = 0", "bb9d5006ba314da4e92b164695899ef5": "\\mathbf{S} = \\left[\n\\begin{array}{ccc}\n0& \\mathbf{S}_0& \\mathbf{S}^T_0\\\\\n\\mathbf{S}^T_0& 0& \\mathbf{S}_0\\\\\n\\mathbf{S}_0& \\mathbf{S}^T_0& 0\n\\end{array}\\right]", "bb9d5efe03c39f5cbe5a30fa35e66e89": "d\\mathrm{vol}_M", "bb9d620e0438b2ea30bc77cf5d291980": " g(z) = z + b_0 + b_1z^{-1} + b_2 z^{-2} + \\cdots", "bb9d68a60f64e75e6701904b6f4fe533": "\\ell j\\,", "bb9d7a0d1e1731a883d3f4171ca284cb": "\\begin{bmatrix} \\dfrac{-b_{12}}{b_{11}} & \\dfrac{1}{b_{11}} \\\\ \\dfrac{-\\Delta \\mathbf{[b]}}{b_{11}} & \\dfrac{-b_{21}}{b_{11}} \\end{bmatrix}", "bb9daf6b79e806209e9a67f5e36154ba": "\\phi(a)\\in \\operatorname{Aut}(G)", "bb9db992fc0f5caa2113ab10938b7ced": "R_{\\text{p}i}", "bb9e545154d1da7a31c8234345efc0e2": " I = 2 \\pi i \\; 2 \\left( - \\frac{\\sqrt{2}}{16} i \\right) = \\pi \\frac{\\sqrt{2}}{4}.", "bb9e79260d825c7d06aab9ca790e89d6": "y \\in [y_1, y_2]", "bb9e8c22fbec72de581ea6f395558172": "\\textstyle N_v=2\\left(\\frac{2\\pi m_h^*kT}{h^2}\\right)^{3/2}", "bb9e8ed4959cea54b6c910c06004d0f1": "\\,mg = \\pi d \\gamma \\sin \\alpha", "bb9ecd1d0d94b953e7c0b95863a4c8bd": " V = I = 0 ", "bb9ef5d61f2163ff1f07eca8f846c210": "d_{n+1} \\circ \\gamma_n + \\gamma_{n-1} \\circ d_n = id_{D_n}", "bb9f7df553e76a141b00331357ed90ec": "(y^1,\\dots,y^n)\\ ", "bb9f8f2b6b538c66939f3472fc2194ab": "C_{Ace}", "bb9f96d521cd5f30a49b4465dc1e1679": "\n \\Phi(u, 0, v) - e^{-u + v^2/2+ \\log(\\Phi(u, v^2, v))\\; }\n", "bb9f9712ea81bb0440acb5b2ac7e6d8c": " z_{k+1} = z_k + hf(z_k) \\, ", "bb9faeae578b8f447e83a31a7d0c1961": "P(k)-i-1", "bb9fd77684530503f3765280b54c2413": "(0, 0)", "bb9ff3d6a5d9ca601e8b82eca3c2e3db": "\\begin{align}\nk_e &= \\frac{1}{4\\pi\\varepsilon_0}=\\frac{c_0^2\\mu_0}{4\\pi}=c_0^2\\times 10^{-7}\\ \\mathrm{H\\cdot m}^{-1}\\\\\n &= 8.987\\,551\\,787\\,368\\,176\\,4\\times 10^9\\ \\mathrm{N\\cdot m^2\\cdot C}^{-2}\n\\end{align}", "bba00d2beb9289cf212bba871ea239c3": "[P\\rightarrow Fred(\\mathcal H)]_{PU(\\mathcal H)}", "bba03df87f96a2bccd68d9c53bc8f849": "\n0 = \\frac{dI}{ds} = \\left(\\frac{dQ}{ds}\\right)Q^T + Q\\left(\\frac{dQ}{ds}\\right)^T \n \\implies \\left(\\frac{dQ}{ds}\\right)Q^T = -\\left(\\left(\\frac{dQ}{ds}\\right)Q^T\\right)^T\n", "bba040c19ecd19b757ad020e44e59a63": "\n=\\frac{\\operatorname{cov}(\\beta+\\epsilon_{\\beta}, \\theta+\\epsilon_\\theta)}{\\sqrt{\\operatorname{var}[\\beta+\\epsilon_{\\beta}]\\operatorname{var}[\\theta+\\epsilon_\\theta]}},\n", "bba0493d5a21e6641d795298c769d921": "a/e", "bba08f8558259dc5d6781d1cc4df6e5f": " V(r) = kr^d, ", "bba0bb612073ca81417f1e40b2a29e55": "m^{e - 1}\\text{ (mod }q\\text{)} = 1", "bba141f54da4bbd4ada9d9837c85c0b1": " q_1 = \\frac{Q}{w_1} = \\frac{150}{10.0} = 15.0 \\text{ ft}^2/s", "bba16ae4ae3f2348e89aff374155f327": "1\\leq i,j\\leq n", "bba16f0338de26ed885d18450857485c": "d(x, A) = \\inf \\{ d(x, a) | a \\in A \\}", "bba172e28fbf36bb9bbfeb7be6c87b43": "\\sin\\varphi", "bba177e0a6805406181fcfb736188e2d": "\\mathit{L(Q)}", "bba1916242e7ad4b8d4f1ae9d47004e3": "U_{}^{}", "bba19fea927b71d74e753f2487e107fd": "2e", "bba20f9b26e5231cb0239d32ba387b40": "(X-1)^n \\equiv X^n - 1 \\pmod{n, X^r - 1} \\,", "bba2508f090afb0b4f583d952676fe2c": "\n= \\sum_{k\\ne n} |k^{(0)}\\rangle\\langle k^{(0)}| V|n^{(0)}\\rangle + E_n^{(1)} |n^{(0)}\\rangle,\n", "bba25b4122161102db82d6af465ca252": "\\Gamma_3(a)=\\pi^{3/2}\\Gamma(a)\\Gamma(a-1/2)\\Gamma(a-1)", "bba28db82c681e0ef69a54bfbdfe5a0d": "z_{t}", "bba2d09cd2772b89b8a21f01423b20bb": "\\frac{m}{2}(p\\ln 2 - \\ln|\\boldsymbol\\Psi|) + \\ln\\Gamma_p\\left(\\frac{m}{2}\\right)", "bba303605bcbd86b8a19b11c942b17be": "\\{ p(a^n b^n c^n) : n \\geq 1 \\}", "bba323204a8777a1de68627fc96c626f": "\\beta_c = \\sqrt{ \\frac{f''}{2 \\kappa} }", "bba339ebb24b24ee5450ae0a96196694": "91_{11} \\ ", "bba36344dc1eebedec17d8453567e25c": "C_{QL} = \\xi C_{Q0} \\approx 2.187\\times 10^{-3}\\;\\mathrm{F/m^2} \\ ", "bba37c8587c80798a3dced0357076e1d": "\\frac{V_D}{V_T} = \\frac{P_A CO_2 - P_E CO_2}{P_A CO_2}", "bba392ec52e55dd0955793cc1f599cd1": "F(r) = m(\\ddot r - r \\dot\\theta^2) ", "bba3ed1954dd4a088f786b03834abe13": "P = \\frac{|\\beta_2| A_\\mathit{eff}}{T_0^2 n_2 k_0}", "bba44ecc6a5fca28f87b1081ccfc83bb": "P_0 = \\frac{1}{1 + R_f} \\left[E(P_T) - \\frac{\\mathrm{Cov}(P_T,R_M)(E(R_M) - R_f)}{\\mathrm{Var}(R_M)}\\right]", "bba45ce2254e53a300e910df97bd2f21": "\\mathbf{v}_\\parallel = -\\hat{n}\\times(\\hat{n}\\times\\mathbf{v}),", "bba52f5098a36b5a76ede1529d3a8e82": "\\hat{w}_i = R_i^j w_j.", "bba57c5e8cc8ff53ca863de886b7254f": "\\scriptstyle \\rangle ", "bba5e80e329425379d30f7458eee15ee": "\\mathbf{s} = 0", "bba5f1c890840e4b160b42893d14215d": " f(\\mathbf{X}) = f_0 + \\sum_{i=1}^d f_i(X_i) + \\sum_{i k_\\theta", "bbbebe81181dd24a10b04d6b62f81fbd": "\n\\begin{bmatrix}\nt' \\\\ x'\n\\end{bmatrix} =\n\\frac{1}{\\sqrt{1 - {v^2 \\over c^2}}}\n\\begin{bmatrix}\n1 & {- v \\over c^2} \\\\\n-v & 1\n\\end{bmatrix}\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix}\\;,\n", "bbbf102df1c0ffec3720ec82bde4fdc8": "z_k\\rightarrow w.", "bbbf2029f4dca65771204368fc09a790": "T_{eq} = T_{1} + 4 T_{2} + \\cdots", "bbbf6d0c90ac5ca24fc5d28e4a2fdfc7": "(f(x),g(y))", "bbbfb32e1b83017c858afe3dec69eacd": "f(v_x)\\,dv_x \\propto e^{-\\frac{1}{2}m_ev_x^2/k_BT_e}", "bbc01fffff9b6c6427b386b48552a227": "\\displaystyle 1", "bbc0ad4e01dc12c1a2a5f7f5e5eed3fc": " \\Delta V = \\frac{\\hbar}{2 e} \\frac{d \\Delta \\delta}{dt} = \\frac{\\hbar}{2 e} \\frac{1}{I_0 \\cos \\delta_0} \\frac{d \\Delta I}{dt} = L \\frac{d \\Delta I}{dt} ", "bbc0b8a6ecde4e4a3b04fd76b5a4cdcd": "u_1 = zw\\,\\bmod\\,n", "bbc0c6836f0577b78af41061f363101d": "x_1,\\ldots, x_n\\in\\mathbb C", "bbc0fa53e7c3dcf1a46aefb474460604": "\\Delta Q_{i} = -Q_{i} + \\sum_{k=1}^N |V_i||V_k|(G_{ik}\\sin\\theta_{ik}-B_{ik}\\cos\\theta_{ik})", "bbc12ce901dadc8bc3188cd5a917f680": "S^1 \\wedge \\dots \\wedge S^1 \\wedge X_n \\to S^1 \\wedge \\dots \\wedge S^1 \\wedge X_{n+1} \\to \\dots \\to S^1 \\wedge X_{n+p-1} \\to X_{n+p}", "bbc195522988fd33ee013539fe596ba9": " \\mathbb{H} \\mathbf{c} = \\mathbf{e}\\mathbb{S}\\mathbf{c}, ", "bbc30d5aca3137b587ea1bffbcec25df": "\\scriptstyle\\frac{1}{300}", "bbc370ba0746f10af60f700ec234059c": "\\scriptstyle{\\mathbf C}", "bbc376ea04b3d6f5b9ef906fbe573e54": "\n\\int \\bar{\\psi} \\gamma^\\mu \\partial_\\mu \\psi + {1\\over 4}F^{\\mu\\nu}F_{\\mu\\nu} +\\bar{\\psi} e\\gamma^\\mu A_\\mu\\psi\n\\,", "bbc378b5ebbd819b08e487922d402546": " C_0^\\infty(M) \\rightarrow C_0^\\infty(M). \\quad ", "bbc3b8599c244230429ad0410f84d1c0": "\\Delta\\rho=0", "bbc3c8f4ef55127df878f6e416d73bc2": "x,y,z,w", "bbc3d4143fae91e6632b45c41d2a4bb0": "\\tau(\\mathcal{H})", "bbc3e95e340f1edf4b456b6d21c35876": "I(x, y)\\, \\mathrm{d}x + J(x, y)\\, \\mathrm{d}y = 0, \\,\\!", "bbc43d6d2d90ca0dffe542028b37cce4": "A\\dagger B", "bbc43d6ed972c6e2de680faea6b148d7": "e_p", "bbc481a811ebb23f5b9909b164345115": " Q = \\Delta U + W.", "bbc4a582c9969876728b420ce6404d2f": "_{nominal}\\alpha = 1 - \\textstyle\\frac{n-1}{n} (1-\\pi) \\ge \\pi", "bbc4f56098bff0fae6b8e237ac236215": " j_*\\Omega^{\\bullet}_{X-D} ", "bbc50c394fb22e95138798f18228a15c": "N\\to \\infty", "bbc52643a1b35116a3feeb38846929be": " \\mathfrak{so}(S_+)\\oplus\\mathfrak{so}(S_-) ", "bbc554cbb3376b5af65ec1912690cc11": "\\begin{matrix} 64 \\times {4 \\choose 1}{3 \\choose 2}{3 \\choose 1}^2 = 6,912 \\end{matrix}", "bbc55db45842d3fa36ebf5ff90f5ba31": "\\Box F", "bbc56328d415f957848ce11cec53bb66": " \\frac{1}{v} = \\frac{1}{V_\\max} + \\frac{K_m}{V_{\\max}[S]}", "bbc57ebca49610110f89e055905b451f": "= F_1 + F_2 \\,", "bbc63b2e536e3588d1b2bd5173a8a9ec": "\\Delta_{(x, y, z)}^2 f(a, b, c)=x^2{\\partial^2 f\\over\\partial x^2}(a, b, c)+2xy{\\partial^2 f\\over\\partial x\\partial y}(a, b, c)+\\dots=0.", "bbc68214daa23745a8fa8da1da46a3cf": "\\log{n \\choose k} \\approx k \\ln(n/k-0.5) + k - 0.5 \\ln(2 \\pi k)", "bbc6a62551f8b4aef565bf82622ef6e7": "\\mathrm{\\frac{Q}{L t T}}", "bbc6cd3a96f8b1af88cda8f48f225cc7": " \\frac{a_j}{\\lVert a_j \\rVert^2}. ", "bbc6de0a6767f0f10d761c32d2162edc": "0.\\overline{42857}\\overline{1}", "bbc6f2e55dfd227ffcd3395e725c5217": "C\\bar C=\\Delta_0", "bbc70ccfd6acd55c6c3bff3a703f165c": "\\displaystyle{e^{-2Ct}D(c(t))}", "bbc736568c6094d9f5d03d56f5d67967": "\\begin{bmatrix}\na & b & c & d & e \\\\\nb & c & d & e & f \\\\\nc & d & e & f & g \\\\\nd & e & f & g & h \\\\\ne & f & g & h & i\n\\end{bmatrix}.", "bbc7564cba8f40aacdf6f175c9993131": "L = \\{a + t b : t \\in \\mathbf{R} \\}, \\quad a, b \\in \\mathbf{C}, \\ b \\ne 0,", "bbc756bac22d2485778a6a5eee1b5970": "ka \\sin{\\theta} = 3.8317...", "bbc806b4bada43fcb0c758f4b9fe1644": "\\frac{}{\\Gamma \\vdash (\\alpha\\!\\rightarrow\\!(\\beta\\!\\rightarrow\\!\\gamma))\\!\\rightarrow\\!((\\alpha\\!\\rightarrow\\!\\beta)\\!\\rightarrow\\!(\\alpha\\!\\rightarrow\\!\\gamma))}\\;\\text{Ax}_S", "bbc84b44d97f490e1e9d2edd3c97ae77": "\\mathbf{g}\\cdot d\\mathbf{A} = -4 \\pi GM", "bbc84b927458e4b01c11afe4db614aa3": "\\overline{A \\cdot B} \\equiv \\overline {A} + \\overline {B}", "bbc856f279e35641a6712d14eed36a6a": "a = d \\ne b = c, \\alpha = \\zeta = 120 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ, ", "bbc8815e85ab6707a3f80db202d5b705": "\\mu_{n_k}", "bbc89b6ac6f0936b52f6b21bcea3f261": "even(0)\\ ", "bbc89d1f9d11043894ddc5ae07a48523": "X_j\\supset A_2", "bbc8e1d1ab403d62d22ed75d7885c74a": " C = \n \\begin{bmatrix}\n 0.0625 & 0.0000 \\\\\n 0.0398 & -0.0909\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 11 \\\\\n 13\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7439\n \\end{bmatrix}. ", "bbc9627bd6ac33b7b24b38365e09dd05": "\\alpha_1 = g_{1}^s h_{1}^c ", "bbc9908828c4521e53e2b121a82966e0": "\\tilde P", "bbc9aeb2a5c6aed7034daddefaef2982": "[G\\to Y]", "bbc9d28df39f5e2ee980d44deb131e47": "\\tbinom{n}{1}", "bbc9e23fd8cd18c70defa1588e8b6f57": "a^b = b^a, a\\neq b", "bbca6e1b938d9d3f73e5b25c095fea7c": " (p_1,...,p_n) ", "bbca87e8ce1fe4cc9670f0d72c56968f": "\n\\Delta \\bar{e}\\ =\\frac {1}{\\mu}\\ \\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ f_r\\ + \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ f_t\\right)\\ r^2\\ du\n", "bbcaa7c1af4bb8e91232f90dd77a7e64": "Q(z) = z/\\sigma_L(z) = \\exp\\left(\\sum_{k\\ge 2} {2G_{2k}(\\tau)z^{2k}\\over(2k)!}\\right)", "bbcad09fb6fecfd981cf7a4f1139a649": "\\frac{\\sqrt D}{3}", "bbcb787e58679a5149ca28d90880e29e": "(r)", "bbcbccac08b4e749ab6332a7e94980ae": "\\Gamma \\left ( \\tau^- \\rarr e^- + \\bar{\\nu_e} +\\nu_\\tau \\right ) = K_2G_F^2m_\\tau^5,", "bbcbef96f26ab546a9ab2c7c7883667c": "2^{-57.7}", "bbcbf530f71adcca1bcf9e9490eb90fc": "\\frac{\\hbox{apotome}}{\\hbox{limma}}\n=\\frac{3^7/2^{11}}{2^8/3^5}\n= \\frac{3^{12}}{2^{19}}\n= \\frac{531441}{524288}\n= 1.0136432647705078125\n\\!", "bbcc47760d186ed7a33c98c4e6626cd0": " \\sum_{j=0}^k (-1)^j\\tbinom n j = (-1)^k\\tbinom {n-1}k", "bbcc9441aae04a669af42b88b1875d63": "\\mathcal{F}_{\\tau^+}=\\{A \\in \\mathcal{F}: \\{\\tau=t\\} \\cap A \\in \\mathcal{F}_{t+} ,\\, \\forall t \\geq 0\\}", "bbcd3b46d7fb4cdcfde575aef1003af0": "\\mathrm{crd}\\ \\frac{\\theta}{2} = \\pm \\sqrt{2-\\mathrm{crd}(180^\\circ - \\theta)} \\,", "bbcd4f437ead2fd88398fe0b5929212e": " j+k \\le p", "bbcd78d4e35782ca1566de945409e118": "V_D=", "bbce1233ac70a8862dd5d46c09484017": "0\\to C_\\bullet(A) \\to C_\\bullet(X) \\to C_\\bullet(X)/C_\\bullet(A) \\to 0.", "bbce1ae5723b9f72bda44ae694e19299": "Q_i\\supseteq P\\;", "bbce47e32e4bdbe17ba09ae3b7092568": " \\mathcal{I}\\mathcal{G} (n) ", "bbce78e75e26e07812aa78e3037df021": "\\sigma_i \\sigma_j = - \\sigma_j \\sigma_i \\qquad (j \\neq i).", "bbcefe2f915d1decc8497d7dd71f871c": "\n\\mathbf{S} = \\begin{bmatrix}\n\\lambda_{x}^{2} & 0 & 0 \\\\\n0 & \\lambda_{y}^{2} & 0 \\\\\n0 & 0 & \\lambda_{z}^{2}\n\\end{bmatrix}\n", "bbceffd092edca2a2be0d4b62cb67cf7": "C = \\{c,\\bar{c}\\}", "bbcf352387f97cfa99113e343bd21981": "a_i = \\sum_{j\\in J} \\mu_{i,j} b_j ", "bbcf64fd9b0426d1e1871ec734d4c046": "r = \\limsup_{n\\rightarrow\\infty}\\sqrt[n]{|a_n|},", "bbcf6c6382b9662aaab8ebb4e95ad938": "\n \\frac {\\mathbf{u}^{n+1} - \\mathbf{u}^*} {\\Delta t} = - \\frac {1}{\\rho} \\, \\nabla p ^{n+1}\n", "bbcf7f5a5ec99a4a8c85d97eae9c6ffa": "0 + 1 + 4 + 1 + 0 = 6", "bbcfbeaa3d1dce81352b55609b13c904": " \\frac{Kc}{\\Delta R(\\theta, c)}=\\frac{1}{M_wP(\\theta)}+2A_2c+O(c^2) = \\frac{1}{M_w}\\left(1+ \\frac{q^2 R_g^2}{3}+O(q^4) \\right)+2A_2c+O(c^2)", "bbd11292be1f51c96d3ca087a72b63ee": "dS/dz > 0", "bbd130db3bbc3d4f067b57e6e06491bb": "q^* = q - 1", "bbd14a48451cb7e9bf390a10cee50658": "D(EQ) \\leq n-1", "bbd15ae8fd710c21c7767b91ff77ff5a": " q_{0} A+ k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 ", "bbd16d6efcf1df35e364ec3a2db9d2bf": "\\mathrm{Cov}[\\varphi(x),\\varphi(y)]", "bbd183b6fca2bb24bc70e8b7ef591a17": "/2) = n", "bbd20b4d599e8cc21573f79cdf29853b": "RBF = \\frac{RPF}{1 - HCT}", "bbd2304a7352c6ce817428c5a2dd2ce9": "f: \\pi \\to \\pi^{ab}", "bbd27eb869a0a40dfba51d4c374a77bb": "2\\tfrac{2}{3}", "bbd2d784e12fadbdb5619e73d5944553": "[X,Y] \\in \\mathfrak{h}", "bbd2f83e89660e6f6131b174391f4bc4": "\\tilde{H}_{\\mathrm{\\infty}}(W|SS(W),E) \\geq \\tilde{m} ", "bbd33eafb33a8471bb3badb54772ebce": "P_i = r_iQ", "bbd3533f20ae455fddf43fc4e66853f6": "\\neg (x < x)", "bbd38276a0a774e1c0b1f143d46c2406": " W_{NS}(A^{n+})=\\sum_{i=1}^{n-1}\\alpha_n(\\lambda)W_{ADK}(A^{i+})", "bbd4428b8e11c67402062afa6c1941ef": "\\left\\{ X_{k_{1}},X_{k_{2}},\\cdots\\right\\}", "bbd45151eefa33fe8c5129499e7669bb": "1-\\alpha ", "bbd4531a90d1f34a4684e8e5d8983e58": "P=(p_{ij})", "bbd480440b413f47a818466f3d171741": "\\underline{\\varphi(\\beta / \\alpha) \\vdash \\psi}\\,\\!", "bbd4b1590ce85c7bad3f1bba07f1c795": " \\mathbf{L} = \\mathbf{L_{M}} + \\mathbf{L_{n}} \\, ", "bbd4bc053a47bad0dcbbd53f45fc1927": " \\mathcal{C}_{Y \\mid X} \\phi(x) ", "bbd4c22201c7690dba6a9b18255fb740": " 4 s V_\\infty \\sum_{n=1}^\\infty A_n \\sin(n \\theta) = \\frac{1}{2} V_\\infty c C_{l_\\alpha}\\left[\\alpha_\\infty + \\alpha_{geo} - \\alpha_0 - \\sum_{n=1}^{\\infty} \\frac{n A_n \\sin(n \\theta)}{\\sin(\\theta)} \\right] \\qquad (10) ", "bbd4ee465d17c754389769a00955830b": " x = r \\cos(t) \\, ", "bbd51a148ef49db528f7f34812eb882d": "H_2 \\overline{C_2 P_n}", "bbd53983c2afc17431f20d7703971622": "\\gamma_{ab} = \\lambda_a \\lambda_b \\xi^{2 s_1} + \\mu_a \\mu_b \\xi^{2 s_2}, \\,", "bbd5a1352773ca82c685affb33bdcdd4": "|L_1(x)\\cdots L_n(x)|<|x|^{-\\epsilon}", "bbd61434f9d46b084ea72c2eccc166e9": "\\sigma=1+\\sum_{j=1}^{3}2j\\alpha_j\\cos\\left(2j\\xi'\\right)\\cosh\\left(2j\\eta'\\right),\\,\\,\\,\\tau=\\sum_{j=1}^{3}2j\\alpha_j\\sin\\left(2j\\xi'\\right)\\sinh\\left(2j\\eta'\\right).", "bbd66338b92605931d1c9add526ed91c": "O \\left ( \\log n + { \\left ( \\frac{1}{ \\epsilon\\ } \\right ) }^d \\right )", "bbd6aaca31cb1d74b12047516d0cffb3": "(\\log 5 - \\log 2).", "bbd6c14eb843868655fc2682d04cee3a": "e=10^{-k},\\;k\\in\\mathbb{N}_+", "bbd774c2aa00da58571ce6b588027046": " H^k(X; \\mathbf{Z}) ", "bbd8077ae7976ba53e3c38b53f30f939": "\\hat{L}_m", "bbd8140c23f7eccd5d0dd8943604da3a": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=18,\\cdots ,20", "bbd843b156d2311f8fac4ff728594d58": "G^\\circ", "bbd87e394c9a2e7a45fd63ed40ba0722": "a^b", "bbd8816b29095b55baf191c88f4a9b11": "\\tfrac{4}{3}", "bbd8aa84f5500f868e100cd680b6440d": "c_{\\lambda,\\mu}^\\nu=2", "bbd8dfc9d695dfd147ded7ef2949c611": "{{{7}}}", "bbd8ff55788947c1ebe71725ae8b9d70": "\\forall x \\in A \\ : \\ \\left|f(x) - \\sum_{n=1}^{k} f_n(x)\\right| = \\left|\\sum_{n=k+1}^{\\infty} f_n(x)\\right| \\le \\sum_{n=k+1}^{\\infty} \\left|f_n(x)\\right| \\le \\sum_{n=k+1}^{\\infty} M_n = \\left|M - \\sum_{n=1}^{k} M_n\\right| < \\varepsilon.", "bbd930b2a5050aa23d2377431d25998a": " D'= \\left({d \\over {dx}}\\right)' = \\operatorname{Id}_{\\mathbb K [x]} = 1.", "bbd97b00c539801e32317ab550867ec4": "B2", "bbd9ab8e0698b1599cfd1ae40464e427": "\\varnothing \\in y", "bbd9e557aa6e5e3094482c8fbf21bba9": "\\sigma_{gt}=\\frac{\\epsilon_{graphene}} {-lg( \\frac{I} {I_0}) \\rho_{sample}} ", "bbda5d1d4630916bc3b757997eb9534e": "D_n = \\mu_n k_B T/q, \\quad D_p = \\mu_p k_B T/q,", "bbda6bb08d3ca2b1b674500ed8ec1633": "\\begin{matrix} \\frac{3}{2} \\end{matrix}", "bbda6e267347f48942b5667506f8aeec": "\\alpha_{\\sigma(i)}", "bbdaa9edbc4936d95cc339051366eb52": " \\epsilon_{i} \\cdot A_{i} \\cdot B_{ij} = \\epsilon_{j} \\cdot A_{j} \\cdot B_{ji}", "bbdaf44eabc92afaeea6c8491b28a0e0": "S \\upharpoonright n", "bbdb459c18c538281ac37849d2609907": " \\boldsymbol{\\Rho} = \\frac{{\\rm d}\\mathbf{\\tau}}{{\\rm d}t} = \\mathbf{r}\\times\\mathbf{Y} = \\frac{{\\rm d}(\\mathbf{I} \\cdot \\boldsymbol{\\alpha})}{{\\rm d}t} \\,\\!", "bbdb6736d0263cf66fcf64fbdeaaeab5": "F_n\\left(\\frac{d}{dx}\\right)\\cosh^{-1}(x) = \\cosh^{-1}(x)P_n(\\tanh(x))\n={}_3F_2(-n,n+1,(x+1)/2 ; 1,1; 1)\n", "bbdb69970ef2b2cd4707261724a10aed": " {d|\\nu|\\over d\\mu} = \\left|{d\\nu\\over d\\mu}\\right|. ", "bbdb775a30b98d2ccee1892d70871479": "\nB = \\sqrt[3]{A + \\sqrt{A^{2}+1}}\n", "bbdbeceeb36eca6ec12971bb804bcf98": "U_0(P,Q)=0, \\,", "bbdc139e0ac120f57a6bf5284e104d4e": "\\ ax^3 + bx^2 = c.", "bbdc52880341d0aeb3b3006af0a6583a": "p_{2}", "bbdc7b61bac3bea8b5b74395d1c5861a": "\\pi_0(t) := 1 - \\sum_{i=1}^n \\pi_i(t)", "bbdc94ad9bda12e8621fe212f86051c9": "P(n \\mid N) = \\frac{P(b \\mid N)}{N} ", "bbdca3fe884005b02475aabf6f87a568": "\\left ( \\tfrac{\\Delta Q}{\\Delta P} \\right ) \\times \\tfrac{P}{Q}", "bbdcfa12335dbd27ae238941482e0693": "P(R_A,\\theta_1)", "bbdd0039565b43ad7412b5d09f968433": "w\\in\\mathbb{R}", "bbdd476fdd52bed4f824a53412df91cb": "\\mathcal{H}_{C} = \\oplus_{j=1}^{N}(\\otimes_{i=1}^{l_{N}}\\mathcal{H}_{ji}).", "bbdd7361d44e8036783ec2b5ab6c63fc": " | N_1, N_2, N_3, \\dots, N_j, \\dots \\rang .", "bbdeac3e68b831b324e53785d1e96022": "2.9901", "bbdefebfd8eddafa82d4ef9b92c40709": " 1034 cm^{-2} s^{-1}", "bbdf45dbe9c5b10b9c97a13fd313cd29": "x>0.", "bbdfa573f0b35f1c65a668f7d0433635": "v_n(t+\\tau) \\le v_n(t)+2.5 a_n \\tau( 1-v_n /V_n) ( 0.025+v_n( t)/V_n)^{1/2}", "bbdfbbf285451356a0276bbfbd5ec312": " \\int \\delta(E^2-k^2 - m^2) |E,k\\rangle\\langle E,k| dE dk = \\int {dk \\over 2 E} |k\\rangle\\langle k|", "bbdfd6aee2306bea29cebf2b9d61478d": " \\nabla \\cdot \\mathbf{B} = \\nabla \\cdot (\\nabla \\times \\mathbf{A}) = 0", "bbdff003438ca083d138c6816699ba78": "e=\\sqrt{1 - \\left(\\frac{b}{a}\\right)^2 } \\ . ", "bbdffccbec47019eb3d2e1c7fdf226f0": " \\gamma^{\\mu_1 \\mu_2 \\cdots \\mu_{2j}}", "bbe039f0f3c3e173234ec873956eda51": " \\mathbf{x}_{k} \\in \\mathbb{R}^{2} ", "bbe043315c4c54e96c2da85ef6359432": "<...>", "bbe04fe1701cb9e4343d43eccc45e295": "(1-\\tfrac1{2\\alpha})", "bbe0a63c8cc61f5ec579343a9769c0e9": "\\textrm{pK}_{a} = - \\log_{10} (K_{a}) = - \\log_{10} \\left ( \\frac{[\\mbox{H}_{3}\\mbox{O}^+][\\mbox{A}^-]}{[\\mbox{HA}]} \\right )", "bbe0de7b422ab71871b989aea8f36512": "A_N(z)", "bbe12d4f0b1c259e78df8b7956711779": " \\dot{e} = (A - LC) e", "bbe19515821409ecc6d00cae04ad178f": "t = t^{i_1i_2\\dots i_r}\\, {\\mathbf e}_{i_1}\\otimes {\\mathbf e}_{i_2}\\otimes\\dots\\otimes {\\mathbf e}_{i_r}", "bbe198a0485954e1c9aa77c34f45ea24": " d_i^7 = \\frac{80 Q^3 \\rho_{\\text{fluid}} f }{ \\pi^3 k [ \\rho_{\\text{tube}} (c^2+2c) + \\rho_{\\text{fluid}} ] } ", "bbe2039951b2f5c3408f393fff27cece": " \\Phi (r)dr = \\frac{1}{4 \\eta} \\frac{|\\Delta P|}{\\Delta x} (R^2 - r^2) 2 \\pi rdr = \\frac{\\pi}{2 \\eta} \\frac{|\\Delta P|}{\\Delta x} (rR^2 - r^3)dr ", "bbe25f29b798313701cb1352afe2e7c3": " \\int\\limits_{-\\infty}^{\\infty} dx \\, |\\Psi|^2 = 1 . ", "bbe29cb0240eed8d68f0697916b53591": "f:\\omega\\to\\omega", "bbe2d6d9adaa5810d57d2129753cfa9b": "\\scriptstyle |f|", "bbe30e4771ecb9b793bd887f7a8cb3c1": " SD (( Ext(W;X),SS(W),X),(U_l,SS(W),X)) = SD((R,P),(U_l,P)) \\leq \\epsilon. ", "bbe332975b4098c59e7f6c13223a6171": "\\scriptstyle A\\geq B", "bbe34742e238671d511c07270c57a7da": "F(b)-F(a) = \\operatorname{P}(a< X\\leq b) = \\int_a^b f(x)\\,dx", "bbe355cf591ff62b88be5c2a87dc321f": " \\tfrac12 - \\tfrac1{10} \\sqrt{15} ", "bbe35ea2722c88b84c616667779768bc": "10 * 0% = 0", "bbe37222a830030962410339a2b87d2a": "\\cos(3x) = \\cos^3 x - 3 \\cos x \\sin^2 x \\quad\\text{and}\\quad \\sin(3x) = 3\\cos^2 x \\sin x - \\sin^3 x.", "bbe37c7a6dd0aee54390d8de181b79ab": "\\int_0^\\pi \\frac{1}{\\delta}\\omega_f(\\delta;t)\\,d\\delta < \\infty.", "bbe382ef9f3f40c528604217544252ef": "\\mathrm{Sper} A = \\cup_i\\tilde{P}_i", "bbe392de856d4bc1920740a662bb39b9": "H(A_n)", "bbe3a4e8d6c05e27fa2029698208087d": "a_n = \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} x e^{-inx} \\, dx = \\frac{n\\pi \\cos(n\\pi)-\\sin(n\\pi)}{\\pi n^2} i = \\frac{\\cos(n\\pi)}{n} i - \\frac{\\sin(n\\pi)}{\\pi n^2} i = \\frac{(-1)^n}{n} i", "bbe3ac368bbba428d818c0ad2365e3bb": " \\dot{x_i} = \\frac{dx_i}{dt} = f_i(x_1, \\ldots, x_n) ", "bbe3b2a7420a900e48c8d31606284e95": "R\\mathcal S \\circ R\\widehat{\\mathcal S} = (-1)^\\ast [-g]", "bbe3e5b47ab282dac1fc63d158afa8f2": "ln(2) \\approx 0.693147", "bbe4289dbff1196916b26cae15c5605b": "e'>N^{ \\frac{3}{2}} ", "bbe4997106b7852cc9b218a9b9e1a08b": "\\text{WAL} \\times r \\times P", "bbe4bde9e982f9f6f61c3aadab003331": "\\max_{s}U(s,p)", "bbe4c2378655154c2cf4da7638a37cc9": " D_+(x) = {\\sqrt{\\pi} \\over 2} e^{-x^2} \\mathrm{erfi} (x) = - {i \\sqrt{\\pi} \\over 2} e^{-x^2} \\mathrm{erf} (ix) ", "bbe4f670ac607eb171a5dd96fb2bed82": "\\partial_{\\alpha}(\\tfrac{1}{2}\\epsilon^{\\alpha\\beta\\gamma\\delta}F_{\\gamma\\delta}) = 0 ", "bbe536d178bdd0ee80b1afb4a7bc1719": "SL(2,13)", "bbe536e90cf7432e2071ea1ca0b1f598": "\\bar{R}_P(X,Y) = (\\operatorname{Id} - P)[PX,PY]", "bbe55c6ace53648be1021d9a095be9ed": "\\Delta E = E_{out} - E_{in} = 0", "bbe583184bd3d6ef462f4b0f8fc55823": "\\mathbf{F} = q(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B})", "bbe5f31f9329179a432c3e95b993f82d": "\n\\begin{align}\ny_p &= t [F_1 (t) e^{\\alpha t} \\cos{\\beta t} + G_1 (t) e^{\\alpha t} \\sin{\\beta t}] \\\\\n&= t [F_1 (t) \\cos{t} + G_1 (t) \\sin{t}]\\\\\n&= t [(A_0 t + A_1) \\cos{t} + (B_0 t + B_1) \\sin{t}] \\\\\n&= (A_0 t^2 + A_1 t) \\cos{t} + (B_0 t^2 + B_1 t) \\sin{t} .\\\\\n\\end{align}\n", "bbe642f926d0728d1fb7201f401f2234": "\\left [\\begin{smallmatrix}2&-1\\\\-3&2\\end{smallmatrix}\\right ]", "bbe643421677a81aa77de89bacbe7856": " f(x) = c x(1-x) ", "bbe643ecaa441a3d5fd1286d2e5dbe47": "\\lim_{n\\to\\infty}\\frac{F_{n+1}}{F_n}=\\varphi", "bbe66a4f47e7adb96fa1f6f412caa9d0": " \\bold B \\bold x = \\bold g, \\quad (3) ", "bbe673e19382ec69f355a313766f9980": "\\begin{align}\\mathbb{E}[M]\n&=\\mathbb{E}[|X_0|]+\\sum_{s=0}^\\infty \\mathbb{E}\\bigl[\\underbrace{\\mathbb{E}\\bigl[|X_{s+1}-X_s|\\big|{\\mathcal F}_s\\bigr]\\cdot\\mathbf{1}_{\\{\\tau>s\\}}}_{\\le\\,c\\,\\mathbf{1}_{\\{\\tau>s\\}}\\text{ a.s. by (b)}}\\bigr]\\\\\n&\\le\\mathbb{E}[|X_0|]+c\\sum_{s=0}^\\infty\\mathbb{P}(\\tau>s)\\\\\n&=\\mathbb{E}[|X_0|]+c\\,\\mathbb{E}[\\tau]<\\infty,\\\\\n\\end{align}", "bbe68f1be57781002a4b0a6104e6fd2c": "(a;q)_n = \\prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\\cdots(1-aq^{n-1}).", "bbe6a0b14913c4e18db25d19834b8a63": "Z(C_6) = \\frac{1}{6} \n\\left( a_1^6 + a_2^3 + 2 a_3^2 + 2 a_6 \\right).", "bbe6ec700bc50dc3f97437fbeebc5958": "{\\sqrt{n}[g(X_n)-g(\\theta)]\\,\\xrightarrow{D}\\,\\mathcal{N}(0,\\sigma^2[g'(\\theta)]^2)}", "bbe6f4d0b7ade2b36ec685144b9881f7": " \\forall a_m \\in A, \\, a_m \\rarr a, \\forall b \\in \\Gamma(a), \\exists a_{m_k}", "bbe757b9675d43b4fe1304f2229683dd": " \\epsilon L = \\epsilon p \\dot{q} - \\epsilon H \\,", "bbe7758ed0186b728c060a753ef8fbfa": "R(w^*,w)\\,\\!", "bbe7cc099585dd36c51ec3941c6b10af": "H_1(X;\\mathbb Q) \\times H_1(X;\\mathbb Q) \\to [\\mathbb Q[\\mathbb Z]]/\\mathbb Q[\\mathbb Z]", "bbe7fa13d596f8cb4f8b9c013352996f": " \\,a'=\\lambda_1 a+\\lambda_2 b ", "bbe840aba4b466ab41ac2be6f03a3b10": "(x_1 + x_2 + x_3)^n = \\sum_{k_1+k_2+k_3=n} {n \\choose k_1, k_2, k_3} x_1^{k_1} x_2^{k_2} x_3^{k_3};\\ \\ k_1, k_2, k_3, n \\in \\mathbb{N}_0", "bbe854490eab49a73d2528591f7b6a34": "Q_1A = \\begin{bmatrix}\n \\alpha_1&\\star&\\dots&\\star\\\\\n 0 & & & \\\\\n \\vdots & & A' & \\\\\n 0 & & & \\end{bmatrix}", "bbe85a3ddbde90b4cc660a58531aba74": "x_0,y_0,a\\in\\mathbf{Z}/n\\mathbf{Z}", "bbe8642f43b20f888085a572f6aa47fd": "z\\in[-M,M]", "bbe87a6993999dad9dd197255f1eb656": "(A \\circ B)_{i,j} = (A)_{i,j} \\cdot (B)_{i,j}", "bbe89a0387c0948e921fd276ada3155a": "SP_x", "bbe8aaf748c17f4c5337b2ba0556b328": "2f_{xxx} + f\\,f_{xx}=0", "bbe8b5c17ac8fc8ae1690a99df745e3d": "P(t)V(t)", "bbe8e8d4fa9cae9decea01a86bc11956": "k = (S \\to a, a^{*}S^{+})", "bbe913eff774e504f23c9b6c16e3ea9c": "\\omega^{<\\omega}\\,", "bbe91787441e804af2313a6c683c83f4": "\\mbox{C}_6", "bbe921eb05f6c72433864f8b1010e1c3": "\n\\mathbf{AB} = \\begin{pmatrix}\n\\mathbf{A}\\mathbf{b}_1 & \\mathbf{A}\\mathbf{b}_2 & \\dots & \\mathbf{A}\\mathbf{b}_p\n\\end{pmatrix} = \\begin{pmatrix}\n\\mathbf{a}_1\\mathbf{B} \\\\\n\\mathbf{a}_2\\mathbf{B}\\\\\n\\vdots\\\\\n\\mathbf{a}_n\\mathbf{B}\n\\end{pmatrix}\n", "bbe9662f2e882111673e90991b8dcbbe": "h_i > 0\\,", "bbe98c823c90a25ca09adef9f4045413": " p_X(x)=1 ", "bbea53549a6650c79074c93931418e8a": "f(x,t)", "bbeab42de558bdcf9e58fa46f0996ab5": "N_{i,p}(t)", "bbeb20f15d8f11194f61da691a098a75": "\\mathrm{A}_4 \\twoheadrightarrow \\mathrm{C}_3.", "bbeb37aacd9af57e74ee73c834e0556e": "\nI_i(x|k,t) = \\sum_{m=i}^j (t_{m+k+1}-t_m)M_m(x|k+1,t)/(k+1).\n", "bbeb8a78de6c8d229210a0a1e655e7a1": "-\\mu(-A) \\leq \\mu(A)", "bbebccafcd22020fbb01ebdb608e9c78": "\nc^2=\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_s", "bbec02837f914b5145ee322bb602ef19": "Y(g) \\geq -\\left(\\textstyle\\int_M |R_g|^{n/2} \\,dV_g\\right)^{2/n}", "bbec0af3385f98ce6d2a7f568eb68b3d": " \n{1\\over 2} m \\omega^2 \\sum_{j} (x_j - x_{j+1})^2= {1\\over 2}\\omega^2\\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\\over 2} \\sum_{k}{\\omega_k}^2Q_k Q_{-k} ,", "bbec269d4698698d793774a16735a1b4": "s_1^2, s_2^2", "bbec41208f865ccb1c8108e1618dfc63": "\\beta=1/k_B T \\, ", "bbec418fa4d191017b6f5b67c33be633": "\\zeta = x^0 + x^1,\\eta = x^0 - x^1", "bbec604361b557e2150bfb5de18d3714": "\\bar{A_i}=\\bar{U_i}-T\\bar{S_i},", "bbec9d375fa7cd6b5f6fe0f5e2cf4ac2": "h_L", "bbeca9097e0ab80e98236eed28a36a69": "k_{int}", "bbed5368cd1ecd6dc57e4d1b2aef3f6a": " x_i, y_i", "bbed9c9db735071cfd59cf2508737f7a": "p(x) = \\inf \\left\\{r > 0: x \\in r K \\right\\}. ", "bbeda0f262db432b3b8afbd872ccb976": "\\mu= \\sqrt{2+\\sqrt{2}}", "bbee722a3544d31beb6bd8b6b36237bc": " \\displaystyle \\alpha", "bbeebef16eb63685fb3c41a1dfcb33fd": "c_g = \\beta/k^2", "bbeef22793970f6e3683f0eaa14c071b": "\nH_{DM}=\\mathbf{D}_{AB} \\cdot ( \\mathbf{S}_A \\times \\mathbf{S}_B )\n", "bbef8a0ca55663170261e1e91d667c3e": "M_\\mu\\to M_{\\mathfrak{p}}(\\mu)", "bbef919d2a24106e2d6bff7830e5b0be": "\\pi (a,b)", "bbefa2f30c7b7c6c9d81e23db637ccd3": " G = F+PV. \\;\\, ", "bbefcec2ce50358bf2aef415243c8ac7": "\\sum{\\sum{ \\left | L(r,c) - R(r,c-d) \\right \\vert }}", "bbeffce2850319a829cb61f4e3379023": " Pr(X2", "bbf613a005f865d52057a190b42ae82a": "\\{\\varphi_i\\mid i\\le n\\}", "bbf6226c62867be81941ad439d343b33": "Z^{-m}_n(\\rho,\\varphi) = R^m_n(\\rho)\\,\\sin(m\\,\\varphi), \\!", "bbf64655f9bbfa4a3a3f51098e691de4": "F(u)- F(v)=L(u,v)\\ (u-v).", "bbf64a40daca55845a55c7fdac63a45a": "\\sum z^4 = {m(m^2-1)(3m^2-7) \\over 240} ", "bbf66b670ce44eb92efa1f849b774a73": "{5 \\choose 5}{8 \\choose 0}{4 \\choose 1} = 4", "bbf6a384c25418fe79b5c6e6a349245a": "f^{-1}(y) = \\dfrac{\\sqrt[3]{y} - 8}{2} .", "bbf6d2f7372b3649a6b93068e0f67761": "S(x_1\\otimes\\dots\\otimes x_m) = S(x_m)S(x_{m-1})\\cdots S(x_2)S(x_1).", "bbf6fbd83baa2ef1ed4556294cec0f25": "M_z(t) = M_0 = \\text{const} \\,", "bbf7ea1d373e03d16d1418909b05eaf6": "ctrl", "bbf7f5302ecba92bccf41d29bfbf01f4": "-\\sin(kz)\\ .", "bbf83720bae2fc907e65b01eff489ba2": " \\scriptstyle (X,\\mathcal{T})", "bbf853ace98aa090f4d80500e3bf1a9d": " \\Delta \\left(\\frac{1}{\\gamma }\\right) \\approx \\frac{G M_{\\text{earth}}}{R_{\\text{earth}} c^2}-\\frac{G M_{\\text{earth}}}{R_{\\text{gps}} c^2} ", "bbf8614c1674a7ad6004de521f5554c2": "W_k = W_{k+1} + \\rho K N_k", "bbf8619a9375454117b431af3f0afc8a": "XE", "bbf8848cc1975d7994370e78cbd026bb": "e=\\frac{-( 3p^2+4a)}{16}, \\ \\ d=\\frac{3p}{4}", "bbf8b6b5d6776ae3a085e899b0cd450f": "\\log q_\\mu^2", "bbf8dec14d4c5d4f06a331c6260f81d7": "(S \\to x, a^{*}Sb^{*})", "bbf9825953caf18cb203d04f255cf4fa": "l^*", "bbf9836c365791083242909bac026ce8": "\\textstyle{m(\\mathbf{s})}", "bbf98b2ba1c0b7a43e9bf3ac4f6f9da5": "S = \\left\\{p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}\\mid e_i\\in\\{0,1,2,\\ldots\\}\\right\\}", "bbf9b8d8763f19000470d3fe9e8f660c": " L = n{h \\over 2\\pi} = n\\hbar.", "bbf9ba3b0f2e177d6c52b25d2f3ae601": " \\Delta J = J^{\\prime} - J^{\\prime\\prime} = \\pm 1 ", "bbf9bb72bda8e52158d274155382df2b": "A_{\\text{obs}} \\triangleq A", "bbf9ebba660d845d186c603769990d94": "\\left(D^{(0)} +\\epsilon D^{(1)}\\right)\n\\left( f^{(0)}_n(x) + \\epsilon f^{(1)}_n(x) \\right) = \n\\left( \\lambda^{(0)}_n + \\epsilon \\lambda^{(1)}_n \\right)\n\\left( f^{(0)}_n(x) + \\epsilon f^{(1)}_n(x) \\right)\n", "bbfa1a20f3909a5da90133f362911417": "_{s.9 \\,}\\!", "bbfa67c8461f7c63cf2809f5534dc939": " f_k ", "bbfa9a7497d35f1af1576075f80f2b96": "j_\\nu", "bbfabef1607eca628d6c8d68e27bb543": " \\vartheta(C_5) = \\sqrt{5} ", "bbfb01d9c5795c94f31bf2f0b42dbd44": "(x,v) \\mapsto (x,f(x)(v)) \\in D^n_- \\times F", "bbfb849afa1c5e7748fe88e6997d49bd": "\\left( A-UC^{-1}V \\right) ^{-1} = A^{-1}+A^{-1}U(C-VA^{-1}U)^{-1}VA^{-1}.", "bbfbb016dcd9c0cfb5b0996e501f4d43": "w(x\\Delta)=wx", "bbfc2919ae656c630a2d1facdcdc1f75": "i\\hbar\\frac{\\partial \\psi(\\vec{r})}{\\partial t} = \\left(-\\frac{\\hbar^2\\nabla^2}{2m}+V(\\vec{r})+U_0|\\psi(\\vec{r})|^2\\right)\\psi(\\vec{r})", "bbfc4725f29e67385e6ddae500b3531d": "\\operatorname{nassoc}(A, B) = \\frac{w(A, A)}{w(A, V)} + \\frac{w(B, B)}{w(B, V)}", "bbfcd669d9e1a00961bb4150722c82ce": " \\Delta_{ti} ", "bbfce40239f06e8c70729c1fdaf79f6c": "\\Phi(\\mathbf{p},t) = \\frac{1}{\\sqrt{\\left(2\\pi\\hbar\\right)^3}}\\int\\limits_{\\mathrm{ all \\, space}} d ^3\\mathbf{r} \\, e^{-i \\mathbf{r}\\cdot \\mathbf{p} /\\hbar} \\Psi(\\mathbf{r},t) ", "bbfd6c479a806b8b8dcfdef41e56f96e": "\\mathbf{Z}^2.", "bbfd9f0bbbbc12fb3a46a6e4293badb9": "CM=2r.", "bbfe086e7b9ef1138508e76aac0d51f1": " \\operatorname{cr}(G) \\geq e^3 / 64 n^2.\\,", "bbfe1873f099ae470add3f89cea3e894": " \\left(\\rho + e \\right ) \\ d \\theta ", "bbfe273f0212848aaca7eb5fb3bb271e": "\\gamma_n = \\frac{(-1)^n n!}{2\\pi} \\int_0^{2\\pi} e^{-nix} \\zeta\\left(e^{ix}+1\\right) dx.", "bbfe27c59446a116003d36c6d58e9ccc": "V_{D}=3.75\\mbox{ L per exchange or }15\\mbox{ L/day}", "bbfe41b005b86a7a538fda9bc4d78df0": "|z - p| < 1/\\limsup_{n \\rightarrow \\infty}{\\sqrt[n]{|c_n|}},", "bbfe52ca8532b13fbf98bb912e26d39d": "P_{start}", "bbfef0452ddce9237fbffa54100282f0": "\\omega_{X|\\overline{Y}}", "bbff16cc802afa4373c89d130e19f13e": "D_\\infty(P \\| Q) = \\log \\sup_i \\frac{p_i}{q_i} ", "bbff1faabecc3672a8362010bf9a709b": "m\\rightarrow-\\infty", "bbff3e97dd1ed63466315aa30fdcd150": "0 \\leq s < 1", "bbff57e14b083fd521590c3662ef6826": " a_{x,y,z} : (xy)z \\mapsto x(yz).", "bbff5d9106dd26d7528e19a83904bc04": "\\theta y + (1-\\theta) z \\succ x ", "bbffa7165772e4348476296f743d4024": "(\\nabla v)_{ij} = \\frac{\\partial v_j}{\\partial x_i}", "bbffc389c944cc9169465759ff1a5e3e": "\\Omega^k(P,V)", "bc00047abd8af760d0dc5ac3ff14af5c": "\\frac{16}{3} r^3", "bc000bc6629ca0bf912ff477e7e9a6da": " \\Delta f = \\frac{1}{\\rho^2}\\frac{\\partial}{\\partial \\rho} \\left(\\rho^2 \\frac{\\partial f}{\\partial \\rho}\\right) + \\frac{1}{\\rho^2 \\sin\\theta} \\frac{\\partial}{\\partial \\theta} \\left(\\sin\\theta \\frac{\\partial f}{\\partial \\theta}\\right) + \\frac{1}{\\rho^2 \\sin^2\\theta} \\frac{\\partial^2 f}{\\partial \\varphi^2} =0.", "bc000d2954770adb3d3b7a5674279a2d": "N = N_1 + N_2 \\,", "bc003bed59d4065c54eecbe15942eeb8": "m \\times g \\times h", "bc003f9a25a1a214fbf2d49b7a7f0763": "\\Delta U_i(x)", "bc00b7036d34c1ef83acdf2c4d140f97": "ed = \\frac {k}{g} (p-1) (q-1)+1", "bc00e60b1f7b501d192c147505c6407f": "\\text{True negative rate}=\\frac{tn}{tn+fp} \\, ", "bc0120424da6b0da7afef11d565a36b5": "\\psi_X\\colon N \\to F(X)\\,", "bc012ba7b7841ef58df960c90d63fbf5": " \\mathbf{R} = \\begin{pmatrix} - \\mathbf{r}_{1} - \\\\ - \\mathbf{r}_{2} - \\\\ - \\mathbf{r}_{3} - \\end{pmatrix} ", "bc0137aab72ea049ff1a379d3e56117d": "G_n(z)=z\\cdot \\prod_{k=1}^{n-1}\\left( 1+\\frac{G_k(z)^2}{n^3}\\right)", "bc0151f7949c4251c994ff2e6773b372": "f(-2.805118, 3.131312) = 0.0, \\quad", "bc0231e454606f334ddfb84973611372": "E\\langle C \\rangle = 1 - e^{-R},", "bc0262adcd445d20c86e1a2fd12265b2": " I_n = \\mathrm{diag}(1,1,...,1). \\,", "bc02bc4bd22043160ab520d3f875edc3": "a+b>c, b+c>a, c+a>b.", "bc03579d2e3fce1fd8465e381368dbd4": " \\min_R \\max_D T(A,D) = \\max_D \\min_A T(A,D) \\, ", "bc03be75e52772eb6dcc44dc47635e3c": "x_0 = \\tfrac13(s_0 + s_1 + s_2),\\,", "bc03d4c96aca58e29428684525db5b8e": "\\scriptstyle T_p(\\cdot)=\\cos(p\\arccos(\\cdot))", "bc03f60b440322e9fbfb2f4ba04740cf": "\\pi_{j} : x = (x_{1}, \\dots, x_{d}) \\mapsto \\hat{x}_{j} = (x_{1}, \\dots, x_{j - 1}, x_{j + 1}, \\dots, x_{d}).", "bc040000007492ef2a2dbaf5014fa95a": "\\sum_{i=0}^{n-1}t(i) + 1", "bc048564bf5e3207ac785c6c6b48f28c": "\\bot_{\\mathrm{H}_2}(a, b) = \\frac{a+b}{1+ab}", "bc048ad7c156f91b078f9f8cc86fc899": "~ f ~", "bc04a87f058e29762303e1e77261b43c": "\\sum_{x\\in\\mathbb{F}}x^i=0", "bc051712095464cc0a09ade2f0eca137": "\\mu_c", "bc053f9ca58072737ded11c3ab0df3c5": "(Tf)(x) =\\int_0^\\infty G_0(x,y)f(y) \\, dy. ", "bc0544a9083698bb9f13ad408b76666f": "\\frac{\\partial\\varphi}{\\partial t} = -\\boldsymbol{v} \\cdot \\boldsymbol{\\nabla} \\varphi + R(t) = -\\boldsymbol{v} \\cdot \\boldsymbol{u} + R(t). ", "bc0552fada300baf97062e6ff5474847": "B^{32}\\,", "bc0572983b967465400efb7ff51e5336": "\\varphi = \\exp_{p'} \\circ \\psi \\circ \\exp_p^{-1}", "bc057f2dd49255725afbfc097a9339a9": "x_0=1\\,", "bc05c3521018898849a05b47f0bd463e": "a_{n+1}/a_n", "bc05f816927117dfa4e104b3788e88cf": "\\mathbf{z} = \\mathbf{U}", "bc060f191dd74933a83f24fdc1a9f09f": "\\lnot \\varphi \\vdash \\psi\\,\\!", "bc063d1424c43a5d244c7efbce3be95b": "dN_E= \\frac{dg_E}{\\Phi(E)} ", "bc063ee2b40cee00ccb06c1db5354650": "\\mathfrak{M} = \\frac{u_\\mathrm{o}}{(k_\\mathrm{B}T_\\mathrm{e}/m_\\mathrm{i})^{1/2}}", "bc06522994770da8ed6c238f9afa1d7e": "\\mathbf{y}_p = e^{tA}\\int_0^t\n\\begin{bmatrix}\ne^{2u}( 2e^u - 2ue^{2u}) \\\\ \\\\\n e^{2u}(-2e^u + 2(1 + u)e^{2u}) \\\\ \\\\\n 2e^{3u} + 2ue^{4u}\\end{bmatrix}+e^{tA}\\mathbf{c}", "bc068a80c63e3e19b92726d778a35b9b": "q_\\theta^*", "bc06d38fc5ce779bda9ad725cbd02822": "\\dot{v} = \\delta u - \\frac{v}{T} + \\kappa E_0", "bc0705e0aa69d9767a90b2ede0f3219b": "\\Phi_n^*(z)=z^n\\overline{\\Phi_n(\\overline{z}^{-1})}", "bc0769829fa8a0dacbd5d6ee031cf6ea": "I_{j}\\left( \\Gamma ,N \\right)\\to e^{i\\beta _{j}}", "bc07d6b2a439b7af6904d22c5cf6cb7c": "f(p_i,q_i,t)", "bc07fd6e7d69477df7c49251642f171d": "F:{\\mathbb Z}[x]\\rightarrow{\\mathbb Z}[x]", "bc0829a77e1a2edd7f863389aabf7a04": " \\omega_a = 0 \\ ", "bc088f37c4935b8446186ec93d974428": "m_i < n_i \\!", "bc08a7752f904c00d0db9d5e418df569": "\\, \\Phi", "bc08a92151e3307ab9b8bfc5a17b7420": "\n\\chi_\\nu(z) = \\sum_{k=0}^\\infty \\frac{z^{2k+1}}{(2k+1)^\\nu}.\n", "bc08d429016ce706ce6d49da8a21859e": "\\min(normal) = 0", "bc08d71229a7dd1f0215045c52e8499d": "\\forall a,b \\; \\exists c \\subseteq P(a,b) \\; \\forall R \\in P(a,b) \\; \\exists S \\in c \\; S \\subseteq R", "bc0935b4829945f700cb1b6d7d0e011e": "1 - \\sum_{w\\in N^{(t)}(u)\\cup\\{u\\}} \\frac{1}{d(w)+1},", "bc0955788eaabafd7212d6c47f1db37c": "\\operatorname{E}[xy] \\approx \\operatorname{E}[x]\\operatorname{E}[y]", "bc095cdf506a10288692a3a3ff4b81ec": "s_1 + s_2 + \\cdots + s_i \\ge {i \\choose 2}, \\mbox{for }i = 1, 2, \\cdots, n - 1", "bc09a31ba549793e35967c6a91f954dc": "\\big| \\mu (x, t) - \\mu (y, t) \\big| + \\big| \\sigma (x, t) - \\sigma (y, t) \\big| \\leq D | x - y |;", "bc09aa63b26db1758652355222eb6c81": " \\sum_{m=1}^{p-1}{(-1)^m{p-1\\choose m} m^{2n}}\\equiv \\sum_{m=1}^{p-1}{(-1)^m{p-1\\choose m}}\\pmod p\\!", "bc09b9a9867b5d64e303c330c019f22a": "\\dim V_\\lambda", "bc09df16a00c8339246327ea8a5f364b": "\\mathcal C = (C, \\sigma_f, I_{\\mathcal C})", "bc09ede002daa473b5d0e118e220d2a6": "\n W = \\rho_0 u \\;.\n ", "bc09fe19e1165de9c3bdd48f49ab36a1": "W_s", "bc0a110afce8a7385771e562075f6d5b": "L[\\gamma] = \\int_a^b |\\gamma'(t)|\\,dt", "bc0a661ac3fd0fbf2affdbf9eebbd767": "\\ Z_C = \\frac{1}{j\\omega C}", "bc0aa6d4658d7a4d472f9bcf685cb462": "\\|c\\cdot x\\| = |c| \\|x\\|", "bc0aecf31732321638ec398b2fdbaac8": "\\tilde{M_k}", "bc0b010a170079637bac3ea68658060b": "w\\left(1/2\\right) = 0,", "bc0b379b5413c1c8dd6c74f171fcfd74": "10^{-35} \\text{m}", "bc0bb7bc3cf030fbc60830d4f184702a": "\nR = O\n\\left(\\frac{HU}{b-a} + HT_a + HT_b +\nH\\log\\left(f'(b)-f'(a)+2\\right)\\right);\n", "bc0bd54b84771683a2773a032fb50c5f": "1f, 3f, 5f, \\dots \\ ", "bc0be51a52d7aba60419205095c07d38": "\\mathrm{D}_5 \\cong \\mathrm{E}_5", "bc0be8817b32d29489a0efdf9a765139": "\\scriptstyle{J_\\pm|j\\,m\\rangle}", "bc0c0da5e802469c7686c5797bb052ed": "F = \\frac{G m M}{r^2}\\,", "bc0c5cae94a7c39f27fc866748dbb6d6": "C \\approx \\pi \\left[3(a+b) - \\sqrt{(3a+b)(a+3b)}\\right]= \\pi \\left[3(a+b)-\\sqrt{10ab+3(a^2+b^2)}\\right]", "bc0c72a2df5a1593dfacb259f9720ffb": "P_i = \\sum_{n=0}^\\infty l_{in} w {(1+r)^n}", "bc0ca427e35c436c687a25ffb70b8db4": "(2P-H)^2-2(H-P)^2=-(H^2-2P^2) \\,", "bc0cc04dcc1fb93fcd4f1222a0bcf8ef": "U_\\tau(\\lambda,a,b)\\in H", "bc0d14627195b3fc9609c4201104b31e": "~i>0", "bc0d497741a7b4780c3369919f1f38e5": "Prob_z[Accept] = 1/2", "bc0d980155deb6854a454140fcaa7748": "\n\\mathbf{j}^2 = \\mathrm{j}_x^2+\\mathrm{j}_y^2+\\mathrm{j}_z^2. \\, \n", "bc0da0409e746e69196b64bc1e00f38e": "\\pi_{\\alpha,\\mu} = e^{-\\alpha}\\sum_{n=0}^\\infty \\frac{\\alpha^n}{n!}\\mu^{*n}.", "bc0dc5d1176cb8349010b43d4438ddf2": "\\scriptstyle c(E) = e^2 D(E) \\nu(E)", "bc0e1d179aa267124e7b83035a7d5550": "\\scriptstyle{\\theta = \\arctan\\left( \\frac{x}{b}\\right)}", "bc0f340bf40d15ccc279eda66ac2942c": "(R_1, R_2)", "bc0f3af731f48b4c88ba7cbf610022af": "S(E,V) = \\log \\int_{H(\\mathbf{p},\\mathbf{q};V) \\leq E} d^s\\mathbf{p}d^s \\mathbf{q}. ", "bc0fa76142427e4500b7ab1c7d9c3d4c": "d^dk", "bc0fc24907c3b4394528e7fdce11c7d0": "\\mathbf M_x", "bc0fd2946dad180d149d6869a3bc452c": "H_n(x) = (-1)^n\\,e^{x^2} \\ \\frac{d^n}{dx^n}\\left(e^{-x^2}\\right).", "bc0fde6ddc733d40ed2fcbb1214a5b91": "V_{\\text{interior}} = { n \\choose 4 },", "bc0febaf397a46b80540b93d178825e1": "\\bigg. J = -\\frac{P(p_2 - p_1)}{\\delta} \\bigg. ", "bc0ff388d4dce09880c859241491b9e7": " \\mathrm{E}(k) = \\sum_{d=1}^\\infty d^{1-k} \\zeta(k)^{-1} = \\frac{\\zeta(k-1)}{\\zeta(k)}. ", "bc102beb1486bb194ce3b2f441cf8f75": " \\langle \\psi '| = \\langle \\psi | \\hat{U}^{\\dagger} ", "bc10317efd8652a077105bdf67ceee04": "E_\\mathrm{p,g} = -\\Phi m \\,\\!", "bc10334e8f0b09b18f36bcdf1023f1c4": " G_0 ", "bc10725e1adfb13af350c0edbf870701": "f_{pe} \\approx 8980 \\sqrt{n_e} ", "bc10b7d0cb068a7d79c9f1d398441652": " \\text{(2)} \\qquad\n \\frac{b^3D}{12}\\,\\frac{\\mathrm{d}^4\\theta_x}{\\mathrm{d}x^4} - 2bD(1-\\nu)\\cfrac{d^2 \\theta_x}{d x^2}\n = q_2(x) - n_3(x)\\cfrac{d^2 \\theta_x}{d x^2} - \\cfrac{d n_3}{d x}\\,\\cfrac{d \\theta_x}{d x}\n - \\frac{n_2(x)}{2}\\,\\cfrac{d^2 w_x}{d x^2} - \\frac{1}{2}\\cfrac{d n_2}{d x}\\,\\cfrac{d w_x}{d x}\n", "bc10d5589c741fca643ed406ebed7ec2": "R_{emp}(g) = \\frac{1}{N} \\sum_i L(y_i, g(x_i))", "bc10eb58fa811c9d101532e6d59b7586": "\\frac {dm} {dt} = A \\frac {D} {d} (C_s-C_b)", "bc111f0238a83b749116100f167f3ce2": "\\left \\{ (0, P(G, 0)), (1, P(G, 1)), \\cdots, (n, P(G, n)) \\right \\}.", "bc1137d9f5d0fb1128d18b3a1ae635b4": "I = G(0,0 \\cdots )", "bc1139d5dc7581ed9a97aa27c21a1eed": "X = X^l + X^r", "bc1160de50eba24ab3c3bfcb3386460d": "O( n k^2 )", "bc11cf658715d130a37ac60ac17afb52": "\\R^n", "bc11dcdd275d3161fc8a94f7f3fff74c": "k\\xi \\ll 1", "bc121177e3daa3829a416db7427915c3": "C\\sqcap D", "bc12179320d2471d5c8626c4ff58c9c8": " A[X_1,\\ldots,X_n]^{S_n} ", "bc123b0077e4bdc8923a519d0e548fc2": "\\delta \\bar{c} = i \\delta\\lambda B", "bc12579b9b75cff5ebd99c37c58edc10": "k(\\omega) = \\frac{\\omega}{c} \\sqrt{1+\\frac{a^2 \\omega_0^2}{\\omega_0^2-\\omega^2}}", "bc125909eac9ce70cb572040555a9bfe": "\\left( \\gamma^\\mu \\right)^\\dagger = \\gamma^0 \\gamma^\\mu \\gamma^0. \\,", "bc128c5b7593dc238ee77251a7a2ca1c": "-r u_{i + 1}^{n + 1} + (1 + 2 r)u_{i}^{n + 1} - r u_{i - 1}^{n + 1} = r u_{i + 1}^{n} + (1 - 2 r)u_{i}^{n} + r u_{i - 1}^{n}\\,", "bc12ad5d2c0eeb90d0addb2c85b29873": "\\hat \\rho = \\exp\\big(\\tfrac{1}{kT}(\\Omega + \\mu_1 \\hat N_1 + \\ldots + \\mu_s \\hat N_s - \\hat H)\\big),", "bc12be465d8896183f27ea852a16fdcb": "f : \\widehat{\\mathbb{R}} \\to \\widehat{\\mathbb{R}}, p \\in \\widehat{\\mathbb{R}}, L \\in \\widehat{\\mathbb{R}}", "bc12c38dae955b2d041c6f6b1e36b3eb": "\n | \\arg\\sqrt{-\\mu_j}| < \\pi/4.\n", "bc12f7980e796e3f4238db1976f0a954": "\\frac{ \\partial f }{ \\partial x }(P)=\\frac{ \\partial f }{ \\partial y }(P)=\\frac{ \\partial f }{ \\partial z }(P)=0.", "bc131eae75f3a8522cfa4e50471ced1d": "i\\neq j .", "bc1327d120ef71e8c011b034f4fdac1e": "\\ddot{\\mathbf{r}} \n= (\\ddot{r} \\hat{\\mathbf{r}} +\\dot{r} \\dot{\\hat{\\mathbf{r}}} )\n+ (\\dot{r}\\dot{\\theta} \\hat{\\boldsymbol{\\theta}} + r\\ddot{\\theta} \\hat{\\boldsymbol{\\theta}}\n+ r\\dot{\\theta} \\dot{\\hat{\\boldsymbol{\\theta}}})\n= (\\ddot{r} - r\\dot{\\theta}^2) \\hat{\\mathbf{r}} + (r\\ddot{\\theta} + 2\\dot{r} \\dot{\\theta}) \\hat{\\boldsymbol{\\theta}}.", "bc1383df44d4d7828629c10ec47ff031": "NI = H_{\\mathrm{core}} L_{\\mathrm{core}} + H_{\\mathrm{gap}} L_{\\mathrm{gap}}\\,", "bc13b3b4702a0fdc44b97bcea37f833f": "\\alpha=0.9, 0.1", "bc13cd279c7047534073c787fa05e9fe": "\\mathbf{-3} \\,\\,\\mathsf{nat}", "bc141e04e7fda27d5767ce7eb142d2b0": "\\scriptstyle\\nabla^2\\psi=0.\\,", "bc14a86f9330901d65de4c178d329f0d": "x_{text} = {x_{marker\\_orig} \\over x_{orig}} \\cdot x_{scaled} + {marker\\_size \\over 2} + x_{adjust} + x_{textadjust}", "bc14cda5f9a72f7a64963c0019b79037": " \\delta_S^3 = 5\\delta_S + 2 = [14;14,14,14,\\dots] \\approx 14.07107", "bc14d0b36245a70db0774c92687278f2": "t\\in R^m", "bc14d6db123fb08760de06222323bb66": "\\left(x+\\frac{b}{2a}\\right)^2=\\frac{b^2-4ac}{4a^2}.", "bc14d9da2c2d7264bfea99213e58f097": "\\, (1-t\\lambda^{-1})^{-1}", "bc14fab2843c1d35801694eb81933f8a": "1-e^{-n^2/(2 \\times 365)}", "bc1517e856cba306878493383113e096": " \\langle A,k \\rangle ", "bc15215600c407c790835d94e393a1d2": "\\scriptstyle f_0\\,=\\,10", "bc153e2cad94b24ae8c1a5dd1c4b02a7": "a_0, \\ldots, a_n", "bc165bc8e3201dc07052e2a8b654d4b2": " (1-z^{-1})X(z)", "bc16f5b0df2879fbfdd65f8c306eee09": "z_1 = \\sqrt{-2 \\ln U_1} \\sin(2 \\pi U_2) = \\sqrt{-2 \\ln s}\\left( \\frac{v}{\\sqrt{s}}\\right) = v \\cdot \\sqrt{\\frac{-2 \\ln s}{s}}.", "bc17243f2ecc0d3951f5fcfbc57ad28d": " \\lambda_n = \\frac{n^2\\pi^2}{L^2} + A \\qquad (n \\in \\mathbb N_0). ", "bc172ded4c85824ef74607c6652a54db": "C_z(y)=\\sum_{k=1}^\\infty b_k z^{k}.", "bc178b5020aad313b74971b43c230a49": "\\operatorname{ch}(E) = \\sum \\exp(x_{i}) ", "bc1791bbe94b9ac8e61e2e840cf0977c": "\\nabla\\cdot\\mathbf{E}=\\frac{\\rho}{\\varepsilon_0}", "bc17aa4f59026708d0568f27322c9ebf": " \\mathrm{SNR} = \\frac{\\mu_\\mathrm{sig}}{\\sigma_\\mathrm{sig}}", "bc17ab29814ac5ab6d345beb66736ea1": "\\lambda_1 = \\lambda\\,", "bc181336362fd0ac89a727ab4d6d0a5e": " R_{\\perp} = {4 \\pi G \\over {3 c^2 } }\\rho (r) ", "bc18a2086f496a252e1044819a3f13d1": "\\mathcal M(Q)\\to Q.\\,", "bc18c8f7ce1b588adb728e0f4d465597": "A = \\alpha_1 + \\alpha_2 + \\beta_1 + \\beta_2.\\,", "bc18f68fe5bd9c523f5d75bd0351d971": "R = R_\\alpha R_\\beta R_\\gamma", "bc193cd78126c75f5cabdf834e3c8757": " W[\\varphi] = \\int_{-1}^{1} (x\\varphi')^2 \\, dx\\,", "bc194dbb18bc13b7e0a04ff39cf7b27c": "\\langle \\Phi \\left(x\\right), \\Phi\\left(y\\right) \\rangle_{\\ell^2(B)} = \\langle x, y \\rangle_H", "bc19778dfce344e071c44e5aa291c99a": "C \\to \\frac{\\omega_c'}{\\omega_c}\\,C", "bc19d332a13b1879543bfa256adde188": "{\\mathfrak c} = 2^{\\aleph_0}", "bc19d71084467fd838fe4f06e484f520": "p_2(x)=-2x+x^2;", "bc1a2631ba3a3097c98d88de5082e8f4": "(A \\lor A) \\land A", "bc1a651d4083d415fdac11098d94cc5c": "X \\to M", "bc1a7bae78627701da7ace74ea14bfe3": "f^{\\star}(x)=\\exp[(\\ln\\circ f)'(x)]", "bc1abb0851943f643257edd7e501e5ac": "\\langle \\text{ß} \\rangle", "bc1ad411a6a46e2cad7fa02fcbf9515e": "(t,x) = (t,x_1,\\dots,x_n).", "bc1ad98bf630bf90c55268bc3f84d1b1": "R_x(t;\\tau)", "bc1b1561ecbc8fbfe8ffe2ccf86ee3b9": "(B,\\pi)", "bc1b261413007bb08dd83de049faceda": " p = \\frac{m v}{\\sqrt{1 - (v/c)^2}} ", "bc1b87aaaccb3fd2fb365e86a1963de2": "\\tfrac{135}{11}", "bc1b984c6ef3eba3fd79e48c1834150c": "\\,\\Theta(N \\log N)", "bc1bef420d05d8005139138f82089446": "h(x) = (f*g)(x) = \\int_{-\\infty}^\\infty f(y)g(x - y)\\,dy,", "bc1c03531ff65097ddc100cde90d85d5": "\\alpha = \\gamma", "bc1c13f727ebeeb42b752265bfaedba0": "C_{ijk} = \\nabla_{k} R_{ij} - \\nabla_{j} R_{ik} + \\frac{1}{2(n-1)}\\left( \\nabla_{j}Rg_{ik} - \\nabla_{k}Rg_{ij}\\right).", "bc1c4d4cd88af4bb763e86836dfe8fdc": " \\beta_0 = \\frac{1+k}{2} ", "bc1cdf97b0b0241bfc1e9bc6373fcbaf": " d\\sigma^\\hat{m} = -{\\omega^\\hat{m}}_\\hat{n} \\, \\wedge \\sigma^\\hat{n} ", "bc1ce9bb24e9e246ea480a88bac6be20": "\\phi(\\mathbf{r}) = \\frac{Q}{r} e^{-k_0r}", "bc1d17016982419c1dd6c64fddb05fa8": "\\zeta_f(t)=\\prod_{i=0}^{n}\\det(1-t f_\\ast|H_i(X,\\mathbf{Q}))^{(-1)^{i+1}}.", "bc1dc366ec4060492626385d1641d853": "\\gamma_{5}", "bc1dd1f726b27fa55d0382c0b17c4ceb": "\\Sigma = \\{0, 1\\}", "bc1ddf4ca0178bd9b1dbb7a20a1bcb07": " Q(t) ", "bc1de67a73fd1f016632a8ce42045490": "L = ac^*b+ac^*dc^*a+bc^*a.", "bc1e38467607e03ff44cd976a94a123e": "(b - 1)^l", "bc1e39d2d5ec98a34bc33cf1ba1bf6af": "H^n = \\{ x | q(x) = 1, x_1>1\\}. \\, ", "bc1e884949371f13085abd79e3e685a6": "n_x,n_y", "bc1ee39ebf2247bd3b2d4d845e33ba87": "\\beta_c", "bc1f5811871acc3c7b06c7953a110acd": "\n\\begin{bmatrix}\nt \\\\ x\n\\end{bmatrix} =\n\\frac{1}{\\gamma^2+v\\delta\\gamma}\n\\begin{bmatrix}\n\\gamma & -\\delta \\\\\nv\\gamma & \\gamma\n\\end{bmatrix}\n\\begin{bmatrix}\nt' \\\\ x'\n\\end{bmatrix}.\n", "bc1f5b479adff82a5d49b234e2e36bd3": "~\\phi", "bc1fa49848dfa3d3ea653565e21b600a": "\\langle f,g\\rangle=\\int_X f(t) \\overline{g(t)} \\ d \\mu(t).", "bc1fc3e551b4e691344cb7bc2e92cba2": "\\left\\{a, b\\right\\}", "bc1fee9e180c999f1414ea3d6ab2ee2f": "F \\colon \\mathcal{A} \\to \\mathcal{B}", "bc202c7d528188450c136883c308c4e3": "\\mathbb P(\\mathcal E(x))", "bc203b474a6178dcee7c53c7595bdf53": "V=I*Z", "bc20434c8e63d048a2db3905ba3f3407": " \\dot{V}_A ", "bc208dc1c1fd9bc014d8ca8ed0318273": "d=(n-1)/2-", "bc20950ac9a1a1701abe106fb69bc71a": "y_2 = \\frac{2y_1}{-1+\\sqrt{1+\\frac{8gy_1^3}{q^2}}}=\\frac{(2)(2)}{-1+\\sqrt{1+\\frac{8(9.8)(2)^3}{4^2}}} = 0.75m", "bc20dd9a7900de549b14609e55d81271": "|x^3|", "bc211085df31bad71349cd0c372fbc80": " f(gb)=\\Delta(b)^{1/2} \\lambda(b) f(g)", "bc2145775fa07d048a39d1846a4f4148": "\\widehat{y_i}", "bc2147edbf19acf1d33d250174c3a73a": "I(R) = \\frac{I_0}{(1+R/R_H)^2}", "bc215e938eabcd0783992868d9584993": "\nE = E_0 + K_0 \\left( V_0 - V + V \\ln(V/V_0) \\right). \\,\n", "bc21828ab0e7d27bfc6acaa871537f7c": "C \\rightarrow df \\mid eg", "bc219a452a316e7c7efa84706b425dfa": "(x^{q^{2}}, y^{q^{2}} ) + q(x, y) = t(x^{q}, y^{q})", "bc21a82cfefe9b012b93a1f551070108": "H(X|Y=y)", "bc21c7387ec7fb674888c194676f5e7a": "\\sqrt{\\frac{9}{35}}\\!\\,", "bc21cb9de0b59b7c96c5ee6d17ccaf57": "\n\\frac{1}{\\omega_{ci}}\\frac{\\partial}{\\partial t} \\ll 1\n", "bc21f1480afe766aa2e661d62dbddb0d": "\\sqrt{I^n} = \\sqrt{I}", "bc2210896c3f51ab1317c0ee1e7e331d": "~g_\\mathrm{e} = 2.0023193043617(15).", "bc222b3a8824dae4326cdee9c7c59d30": "(1+x)^{3}p(\\frac{2x}{1+x}) = x^3-7x^2+7x+7", "bc22303b3c0595de1119f36144f65baf": "t=0\\,\\!", "bc2247e0c342ef41018509b57621e81e": "k\\int_{t-\\Delta t}^{t+\\Delta t}\\left[\\frac{\\partial u}{\\partial x}(x+\\Delta x,\\tau)-\\frac{\\partial u}{\\partial x}(x-\\Delta x,\\tau)\\right]\\,d\\tau = k\\int_{t-\\Delta t}^{t+\\Delta t}\\int_{x-\\Delta x}^{x+\\Delta x}\\frac{\\partial^2u}{\\partial\\xi^2}\\,d\\xi\\, d\\tau", "bc226f89f3e1c1202b0e7c8a20e89cb0": "\\left|\\varepsilon(a,b,c,d)\\right|=1", "bc235f320fd96b113ffdcf20a7fd1149": "\nRangeVout(x) := V_{outsensmax}(x)-V_{outsensmin}(x)=\\left(\\frac{R_{sensmax}}{R_{sensmax}+x}-\\frac{R_{sensmin}}{R_{sensmin}+x}\\right) V_{in}\n", "bc237e1a91a4b7b7384aca53f7c29185": " \\int \\sec^3 x \\, dx = \\int u\\,dv ", "bc238e5f552d1ed2ff05e4b483901727": " \\operatorname{sys}^2 \\leq \\frac{\\pi}{2} \\operatorname{area}(\\mathbb{RP}^2),", "bc23a9efaa7a0f2260293026db76e501": "+ \\frac{\\partial}{\\partial x} E_{ext} (\\bar v_i) \\Bigg\\} ", "bc23da86e20de8a8f30dd268198f1ea8": "\\operatorname{erf}(x)=\\begin{cases}\n1-\\tau & \\mathrm{for\\;}x\\ge 0\\\\\n\\tau-1 & \\mathrm{for\\;}x < 0\n\\end{cases}", "bc23e52745eaf4b47cb95067826327e3": "\nM_x =\n\\begin{bmatrix}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \n\\end{bmatrix}\n= - \\beta \\Omega\\,,\n", "bc2430d80ba58e7537eff41c199bbfe5": "2\\cdot \\log_2 n", "bc245eb7ee4218e02ce01900b6267b8a": " \\int_S K \\, dA = 2 \\pi \\chi(S), ", "bc2487a8b798eb9690933ea00f364883": "E_\\alpha (z) := \\sum_{n=0}^\\infty \\frac{z^n}{\\Gamma(1+\\alpha n)}", "bc248bd4bdbc4af05c1879cd8abfb9b3": " \\begin{align} E & = T + U \\\\ \n& = -\\frac{G m M}{\\left | \\mathbf{r} \\right |} + \\frac{1}{2} m \\left | \\mathbf{v} \\right |^2 \\\\\n& = m \\left ( - \\frac{GM}{\\left | \\mathbf{r} \\right |} + \\frac{\\left | \\boldsymbol{\\omega} \\times \\mathbf{r} \\right |^2}{2} \\right ) \\\\\n& = - \\frac{GmM}{2 \\left | \\mathbf{r} \\right |} \n\\end{align} \\,\\!", "bc24f6e33886e509d0c2ae2b103a1b13": "\n\\mathbf\\Omega_n(\\mathbf R)=\\nabla_{\\mathbf R} \\times\\mathcal{A}_n(\\mathbf R).\n", "bc251cb205e3a152b139d107c0f953ec": "\\hat{t}, \\hat{\\omega}", "bc2556e18207936c5f69aee7dba1a583": "\\frac{dv}{dr}\\cdot v = -\\frac{GM}{r^2}\\,", "bc2581ec27e4519065919f679a57add4": "a = \\frac{d^2 s}{dt^2}.", "bc25d2644ecddc216fdb0e3b9b43524d": "|G| = 24\\,", "bc25d7bd862ba133a3803a2729663d43": "c\\in\\mathbb{R^{+}}", "bc261473b054c1b0befe06c6d640c1f5": "\n \\nabla^2 \\mathcal{M} = -q \\,.\n", "bc26320b42d68655a5a6c69bd2d6f83c": " \\mathbf{Q}^{fe} = -\\int_{V^e} \\mathbf{N}^T \\mathbf{f}^e \\, dV^e \\qquad \\mathrm{(18b)}", "bc267733e1020bfcbf6989997f574414": " \\lambda = 1 ", "bc26fb21dd6bb264b363b873d224a462": "SG_A= {{\\rho_s \\over \\rho_w}-{\\rho_a \\over \\rho_w} \\over 1 - {\\rho_a \\over \\rho_w}} ={SG_V-{\\rho_a \\over \\rho_w} \\over 1 - {\\rho_a \\over \\rho_w}}", "bc27acef93cded9cf91f091b3f8a309b": "\\eta = 1 - \\frac{T_C}{T_H} \\qquad \\mbox{(1)}", "bc27ebfeed8e79a9208c48e8f95a545d": "s_a = \\frac{1}{n} \\sum_{i=1}^n \\sin \\theta_i,", "bc2843a626ba8a29324a09a370199a8c": "u_{2}", "bc284d32b7b330f0d15f67dce889ab1b": "\\omega_m^2 = \\left(\\frac{\\Delta \\omega}{2}\\right)^2 + \\omega_0^2", "bc285e9783160bdf450893a7aa9310ca": "(e\\!:\\!\\tau)", "bc2885855be0d87cca3e723814054c68": "\\mathcal U^{(m)}= \\Big\\{ e : \\prod_{s\\ni e} (1-x'_s) = 1\\Big\\}", "bc2898404504912281df6f3cd3f5a57c": "L_{ij}=\\rho (x_i v_j-x_j v_i)", "bc28dc7a62fbe0e6005afee6efedfcde": " \\nabla^2 \\Phi(\\mathbf{r}) = -\\frac{1}{\\varepsilon_r \\varepsilon_0} \\, \\sum_{j = 1}^N q_j \\, n_j(\\mathbf{r})", "bc2900523d29b29eb252969f134c8697": "Z_\\mathrm{C}\\,", "bc297ffbee6b089ef4a5917f873f3ea2": "f(x,y)=5x^3y^3+xy^9-2y^{12}", "bc29eb8b168eae87802c952fd87904b5": " \\Gamma^t_a = \\frac{\\sqrt{\\sum_{i=1}^N |b_i|^2}}{\\sqrt{\\sum_{i=1}^N |a_i|^2}}. ", "bc2a0c5d5a3a6f727117f0f2aec03021": "\\begin{align}\n 2\\phi(-q) - f(q) &= f(q) + \\psi(-q) = \\theta_4(q)\\prod_{r > 0}\\left(1 + q^r\\right)^{-1} \\\\\n 4\\chi(q) - f(q) &= 3\\theta_4^2\\left(0q^3\\right)\\prod_{r > 0}\\left(1 - q^r\\right)^{-1} \\\\\n 2\\rho(q) + \\omega(q) &= 3\\left(q^{-\\frac{1}{2}\\frac{3}{8}}\\theta_2\\left[0, q^\\frac{3}{2}\\right]\\right)^2\\prod_{r > 0}\\left(1 - q^{2r}\\right)^{-1} \\\\\n v(\\pm q) \\pm q\\omega\\left(q^2\\right) &= \\frac{1}{2}q^{-\\frac{1}{4}}\\theta_2(0, q)\\prod_{r > 0}\\left(1 + q^{2r}\\right) \\\\\n f\\left(q^8\\right) \\pm 2q\\omega(\\pm q) \\pm 2q^3\\omega\\left(-q^4\\right) &= \\theta_3(0, \\pm q)\\theta_3\\left(0, q^2\\right)^2\\prod_{r > 0}\\left(1 - q^{4r}\\right)^{-2}\n\\end{align}", "bc2a55b249912558e096f1131d4d0844": "\\int_0^\\infty x^{n}e^{-a\\,x^2}\\,dx = \\frac{\\Gamma(\\frac{(n+1)}{2})}{2\\,a^{\\frac{(n+1)}{2}}}", "bc2b148c3f78e4d0305854438602fe5d": "X w = \\{ \\lambda \\in X : s ( \\lambda ) = w \\}", "bc2b493270665f283e4a251c4e0d48e1": "\\rho < \\rho_c", "bc2b97a7f52b3ad19595ffbdf6a5d329": "u(C) = Eu(r);", "bc2bf5cc8f05a62c37c1efa547b9fc52": "\\tilde c_n", "bc2c108805ca687d2fd9627967f5c825": "x=p^{n}\\dfrac{a}{b}", "bc2c2a918131d94dd699949b6a12abc3": "\\frac{\\left|f'(z)\\right|}{1-\\left|f(z)\\right|^2} \\le \\frac{1}{1-\\left|z\\right|^2}.", "bc2c375202b98d65f888d4adc6c815b3": "\\Delta S = P - ET - Q - D \\,\\!", "bc2c4742f8b89d12f22ffd1817b81ca0": " \\langle x, y \\rangle = \\langle y, x \\rangle^*,", "bc2c65de7ab086704b2a886e54e17e57": "\\ (u_n)", "bc2c859f120ddb15bf4054020d6917d2": " \\scriptstyle \\mu = G(M\\!+\\!m)\\,", "bc2c9f94d4206d09fef01fa877f5b516": " u ", "bc2d72621ddcf0a44494f5c48f7ce9db": " \\mathcal{S}_n = \n \\{ \\alpha \\in \\mathbb{R}^n: \\alpha_i \\in [0,1], \\sum_{i=1}^n \\alpha_i = 1 \\}\n", "bc2e9774e37f572e3e072a3a9e6aa499": "b^\\dagger", "bc2ebd049d7870c9a5e378accdc7cff0": "e_2 = \\{e_1\\}", "bc2eccd9769c218e373342cf72d88999": "B = a^{-1}M \\cup a^{-2}M \\cup\\dots", "bc2f3fa6c5f02020bc972603cafa4b75": "S_m", "bc2f53b04f54cc5374110231d134eb9d": "{p = \\frac{R\\,T}{V_m-b} - \\frac{a}{\\sqrt{T}\\,V_m\\left(V_m+b\\right)}}", "bc2fd7448c4e5bf0c0d0a335acfdefdc": "E_\\text{k} = \\int \\frac{v^2}{2} dm = \\int \\frac{v_i^2}{2} dm + \\mathbf{V} \\cdot \\int \\mathbf{v}_i dm + \\frac{V^2}{2} \\int dm. ", "bc2fdb9bba0c639b194e53a9ea1c1add": "(\\beta, s)", "bc2fe0cc60d4174a117461994771959c": " \\begin{align}\n \\frac{\\mathrm{d} p^1}{\\mathrm{d} \\tau} & = q \\gamma \\left[-c \\left(\\frac{-E_x}{c} \\right) + u_y B_z + u_z (-B_y) \\right] \\\\\n &= q \\gamma \\left(E_x + u_y B_z - u_z B_y \\right) \\\\\n & = q \\gamma \\left[ E_x + \\left( \\mathbf{u} \\times \\mathbf{B} \\right)_x \\right] \\, .\n\\end{align} ", "bc3043135215831f4a2b5c32f42ba1fe": "D_{cl}\\approx K^2/2", "bc30841159c3e4804f6c8764c267c347": "\\tau\\epsilon", "bc30887c2bd3a74f72e3de8dbd64d7b0": "O(2^d* \\log ^d n)", "bc308b7a6807a591ad7a40d53a5e0a4c": "\\phi_1(q) = \\sum_{n\\ge 0} {q^{(n+1)^2}(-q;q^2)_{n}}", "bc308d779d7d72bef0691d3a72fc07c4": "\\textit{mother}(\\textit{my\\_favorite\\_oak})", "bc310d4c21b54ab7dd388d94b5453083": "\\{ X \\leq x \\}", "bc31148a0f8a4e298d832b01f2275dff": "t_c = \\frac{a}{b}, \\ x_c = \\frac{A}{b}.", "bc311def35b635b464049e459f4600e4": " \\alpha = \\alpha' = 2-\\nu d,", "bc31472a96cc26b2902cf51568541d0d": "\ng(\\varepsilon) = g_0 \\sqrt{\\varepsilon}\\, [f^{\\mathrm{e}}(\\varepsilon) + f^{\\mathrm{h}}(\\varepsilon) - 1] ~,\n", "bc31b08d41db3b3218c5fe97f7aab804": "R \\ge 1", "bc31b36d46dd90d5f24cbcf813835308": " P = \\tfrac{F \\times d}{t}", "bc31cb910c8900e0bc846755baeb3a74": "\\text{Min} =\n\\begin{cases}\nn=2 & \\rightarrow \\quad f(1,1) = 0, \\\\\nn=3 & \\rightarrow \\quad f(1,1,1) = 0, \\\\\nn>3 & \\rightarrow \\quad f\\left(-1,\\underbrace{1,\\dots,1}_{(n-1) \\text{ times}}\\right) = 0. \\\\\n\\end{cases}\n", "bc31d271dbe356f68662eaf9e7c1c90d": "\\mathbf{x} = x^k \\sigma_k", "bc323eb523023ee2b6f956881d4dcb2d": "\\frac{\\partial\\ell}{\\partial w_i} = \\begin{cases} -t \\cdot x_i & \\text{if } t \\cdot y < 1 \\\\ 0 & \\text{otherwise} \\end{cases}", "bc32463e38763b1e1ff46d9fe38cd0a8": " m_1\\bold{v}_1^\\prime + m_2\\bold{v}_2^\\prime = \\boldsymbol{0} ", "bc32927aee9c0a0f166176f4241ecc6b": "\nJ_{\\beta} = \n\\int_{0}^{\\infty} \\frac{dx}{\\left( x + b^{2} \\right)^{2} \\sqrt{\\left( x + a^{2} \\right)}}\n", "bc32a332a7f241f4ed46fbdce76fb127": "\\mathbf{A}=\\begin{pmatrix}k & 0\\\\ 0 & 1/k\\end{pmatrix}", "bc3365f2a4f0a748894baecea96f0b3c": "\\Box A^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ \\partial_{\\beta} \\partial^{\\beta} A^{\\mu} \\ \\stackrel{\\mathrm{def}}{=}\\ {A^{\\mu , \\beta}}_{\\beta} = - \\mu_0 J^{\\mu}", "bc33930868619371af02545a50b07460": "(\\mathbf{z}_{\\rm{l}} \\vec{v}) \\mathbf{z}_{\\rm{r}} = \\mathbf{z}_{\\rm{l}} (\\vec{v} \\mathbf{z}_{\\rm{r}})", "bc33a0a8b83e6a8ea9f75cb06dc62d08": " \\Box", "bc341300286f8ea840c0cf242f71ba1b": "\\vec{t_1} \\mid \\vec{t_2}", "bc3415740f316d9c9c6799371d9bf2ad": "|2 m_2\\rangle", "bc342e40f4e0f3452e7d84bf7687fb09": "\\star \\mathrm{d}t \\wedge\\mathrm{d}z = - \\mathrm{d}x\\wedge \\mathrm{d}y", "bc3438bc8fb7a111ed9f9512705c362d": "f(w) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ w\\neq1\\ \\mbox{in}\\ H \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ w=1\\ \\mbox{in}\\ H.\n\\end{matrix}\\right.", "bc344d1e52738b6ed3543455f952d208": "f_r(x, \\vec w', \\vec w) = \\frac{\\mathrm{d}L_r(x, \\vec w)}{L_i(x, \\vec w')(\\vec w' \\cdot \\vec n) \\mathrm{d}\\vec w'}", "bc345a40dc068de41cab1d739cc81a0b": " B_{ij}=|U_{ij}|^2 \\text{ for } i,j=1,\\dots,n. \\, ", "bc34984bfac77e2a1478eb27c71d1649": "\nim [\\hat{H}, \\hat{x}] = \\hbar \\hat{p}, \\qquad i [\\hat{H}, \\hat{p}] = -\\hbar U'(\\hat{x}).\n", "bc352fc10ca296a872b51d91a1132127": "F\\,", "bc3557d0edb7a135c6ff5692cee705f4": "\\sigma < a_1", "bc358a9b132558b073565ae0d4fe6489": " v_c = \\eta \\lambda_c\\,\\!", "bc35a274ec8c4e053abbcc57693fa710": "\\mu(k)=\\min\\{w \\mid w\\leq\\nu(w-k-1)\\} \\, ", "bc360221d7bbaa656bdc3513cb22ba7c": "f: f^{-1}(V_i) \\to V_i", "bc36138c33bb67212933edcb86655858": "g\\colon S^2\\to M", "bc36844ed447210e7af661a40bfa5e80": "\\rho_\\text{P} = \\frac{m_\\text{P}}{l_\\text{P}^3} = \\frac{\\hbar t_\\text{P}}{l_\\text{P}^5} = \\frac{c^5}{\\hbar G^2} ", "bc36f13b7075cee29d03de2894e70c74": "\\sigma \\in C_{0}", "bc372d62bf19798de03010be07a1f92b": "C(u) = \\frac {\\sum_{i=1}^k {N_{i,n}w_i P}_i} {\\sum_{i=1}^k {N_{i,n}w_i}}", "bc37698563e17375c50f88c93e413575": "Gm/r^2", "bc37998a169faef6e8595708cbe077f3": "k =1/\\sum_{u^\\prime \\in U}|simil(u,u^\\prime)| ", "bc37b93ead8b18b6f643c2d3cf06ae87": "E \\approx m_0c^2\\left[1 + \\frac{1}{2}\\left(\\frac{m_0u}{m_0c}\\right)^2 \\right]\\,,", "bc37bc38dc176b3f5b228856f02f15de": "\\sum{n_i^2}", "bc3831215b33413be6699445f9af779f": " Qx[\\lnot\\alpha (x)] \\leftrightarrow \\lnot Q'x[\\alpha (x)].", "bc385952ef9e6e6c27b8c1c5ed84aa56": "\\mathcal{P}(gg') = L_{g \\ast}(\\mathcal{P}(g')) + R_{g' \\ast}(\\mathcal{P}(g))", "bc3866a1c16ebdeda4597c9f6cf20cd5": "5\\sqrt{3}\\,s^2", "bc38c7464c20f3c94ab5881739346c9c": " m=1,2,...,p-1 \\!", "bc38cf690a8274ab273346d35d076f29": "\n\\text{If }a \\equiv 1 \\pmod4 \\text{ is prime there exists a prime } \\beta \\text{ such that }\\left(\\frac{a}{\\beta}\\right)=-1, \\,\n", "bc38d522f8c024ecf03188ae85b06f04": "r e^{i\\varphi}", "bc38e61fa1c3071dac07d09d05cd5520": "b_i = 1\\,", "bc38fd9a40f0a2e5844a605e78952a35": " \\Delta G_{micelle} ", "bc3929b7b631756cf25d5e5041589104": "F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1}]} = F_\\nu{[\\text{erg} \\text{ cm}^{-2} \\text{ s}^{-1} \\AA^{-1}]} \\times w", "bc3950901ead22069ca59359641d1ff0": "b=2(m^2+3n^2)/g, \\,", "bc399a67f07a1013b59b0d622e2ebd72": " N > M_{\\sigma,\\varepsilon}\\ \\ (N \\in \\mathbb{N})", "bc39eb928ed1019f696d2fe7c8e88e4a": "\\exp_p: T_p(S) \\longrightarrow S", "bc3a26422f9086f068d2fc42f3c7e8e5": "|\\!\\!\\!\\sim_L", "bc3a5ff0833d04d37b7b04541050b2d8": " B(t,T) = e^{-r(T-t)}. \\,", "bc3a73d960b5262a1163c928c45c5b7f": " V(h) = \\int_h^{\\infty} W(h') \\, dh'.", "bc3b0824c6c71f705a86cb9fba06a368": "\\tau \\simeq \\left[\\left(\\frac{2a}{9g}\\right)^3 \\frac{2\\pi\\mu a}{M}\\right]^\\frac{1}{5}", "bc3b0fdd739142e392fee425338efd15": "q_{t-1} = q_{t} - p_{t} \\mod N", "bc3b3ddfae5ae298698f4e363d6e75ce": " \\mathbf{g}(\\mathbf{x},\\mathbf{z},t) = 0. \\, ", "bc3b62d8ce67f7387ebff8dfda619056": "V = \\bigoplus_{n \\in \\mathbb{N}} V_n", "bc3b9ae730a46cae85b2b87bb6c38c0d": "\\scriptstyle M_{\\text{A}\\cup\\text{B}}", "bc3bc0f4e607ec211eb408c516d17e9e": "\\theta_{\\text{i}}", "bc3bdc9724f93314682698c9ae8bb22f": "\\Delta(t) = t^2 - t + 1 - t^{-1} + t^{-2}", "bc3c1084478b441badd251b7bd22368b": "\\|A\\|^2_{HS}=\\sum_{i,j} |A_{i,j}|^2 = \\|A\\|^2_2", "bc3c287222c0938aef30b8dd36014c98": "\n \\cfrac{\\partial{W}}{\\partial \\bar{I}_1} = C_1 ~;~~ \\cfrac{\\partial{W}}{\\partial \\bar{I}_2} = 0 ~;~~ \\cfrac{\\partial{W}}{\\partial J} = 2D_1(J-1) - \\cfrac{2C_1}{J}\n ", "bc3c9ce6c03a87c29700092569f8e77e": "{\\mathbf{}}G'_iH_i=\\tau_i, G_iH'_i=I_{n_r}", "bc3ca3bed92eaa47a8bc2a380896120c": "D<1", "bc3d1fdffdc1fc2d0eadcb83e439c07b": "x \\cdot q", "bc3d5395316181aad8cc7b756ac8d69d": "\\delta(k_m, i_x)", "bc3d8c5e889f2e89e640b7f12a2910ac": "X \\lll 1=\\left\\{\\begin{matrix} 2X, & \\mbox{if } X < 2^{31} \\\\ 2X + 1 - 2^{32}, & \\mbox{if } X \\geq 2^{31}\\end{matrix}\\right.", "bc3dd50665943cd7b81c4d51a84570da": "TTF=A_0 (RH)^{-2.7} f(V) \\exp\\left(\\frac{E_a}{K_B T}\\right)", "bc3ed96951d451bae798d9e760624aec": "P'_x(a,b,c)=P'_y(a,b,c)=P'_z(a,b,c)=0.", "bc3f3f596bd097f1a7fc7e7b54a02032": " \\operatorname{cov} (X_i,X_j) \\ge - \\frac{\\sigma^2}{n-1} \\quad\\text{for }i \\ne j.", "bc3f9b2f5d2a8c0b854bba333e342aee": "a_0, a_1, \\cdots, a_{d-1} \\in \\mathbf{F}", "bc3fa1581c996183673f424b9501c4aa": "\\chi_{\\text{2}}", "bc3fc416ad91eff0baf3b1a4c3b9dd75": "\\left\\langle M_B\\left(l_B\\right)\\right\\rangle \\sim l_B^{d_B}", "bc3fcb43ebb4129e201538a169affeab": "M(256,3,3)\\approx10^{\\,\\!10^{1.99\\times 10^{619}}}", "bc405cd059f4906d424c7bfa038b199b": "\\int_{-\\infty}^\\infty f(x)\\, \\mathrm{d}x", "bc407cf977cbedd3c65f7635b1034926": "\\pm i\\beta", "bc4087958df85c8deeb9f32a4788bace": "R_S=\\frac{v_{Bullet}^2}{g}\\, 2\\frac{\\cos(\\theta)}{\\cos(\\alpha)}\\left(\\sin(\\theta)\\cos(\\alpha)-\\sin(\\alpha)\\cos(\\theta)\\right)\\sec(\\alpha)\\,", "bc40bc190bd3fafa9696cd433b79e67c": "\\left\\{\\frac{1+x_2}{x_1}, \\frac{x_1 +(1+x_2)x_3}{x_1 x_2},x_3 \\right\\},", "bc40d69b6b19d10339e77a9898793e99": "\\textstyle V_s ", "bc412094321e628f9d8068f982c8f2ff": "\\mathbf{v} = v_i \\mathbf{e}^i .", "bc413f1edcc15eb1af54a676d29329b7": "\\,(n+1)", "bc41a23f464ca929a6077e0a08392774": "\\ell(s_1s_2)=\\ell(s_1)+\\ell(s_2).", "bc41f77f02e7a81fcfb2aacad05e5e7f": "U(0)=u[X(0),0]=u(\\eta,0)", "bc41fb9219219b40ed71b8cb9bd30806": "(i+1)^{k-1}", "bc427e4e32980c0ebdc0b0af903161f0": "MX=0", "bc42e8a1afaf1497ef727f7866095bab": " R: M_f \\times I \\rightarrow M_f", "bc43232425cf5bc6044db6617a8c4338": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta = \\lim_{\\delta \\to 0} \\left( \\frac{\\sin(\\theta + \\delta) - \\sin \\theta}{\\delta} \\right) . ", "bc43bc4366a9a825ecf8ddf3aa25422e": "f'(x) = 2x \\!", "bc44059088ca72319b6526e22654edbb": "~-2 \\le y \\le 2~", "bc44429cd7a78f322c04621b0f7cfa61": "|\\psi\\rangle", "bc44471844597d981a8533d54625c5b5": "1=\\sum_{n=1}^NZ_n+\\int_0^\\infty d\\mu^2\\rho_c(\\mu^2)", "bc44fc192689b6c18eb5ff8a421487c0": " \\mu _2 - \\mu _1 (\\Sigma _{11} )^{ - 1} \\Sigma _{12} ", "bc450cb7462671fb43eea532ee53bc9d": "|\\tan(z)|^2 < \\frac{\\cosh^2(\\pi k)(\\sin^2(t) + \\cos^2(t))}{\\sinh^2(\\pi k)(\\cos^2(t) + \\sin^2(t))} = \\coth^2(\\pi k)", "bc45170b65450ce70e49d02283e6e8e1": "X = Y_1 + Y_2 + \\cdots + Y_K", "bc4517f2b615362810942c20c7782ac1": "~a[\\delta {(x-S/2)}+ \\delta {(x+S/2)}]", "bc453d6996835b57b15f980e8eb279bd": " d_n(m\\otimes a_1 \\otimes \\cdots \\otimes a_n) = a_n m\\otimes a_1 \\otimes \\cdots \\otimes a_{n-1} ", "bc45509c13e98408c47d68bc810da20f": "\\frac{d}{dt}\\log f_W(t)=-\\sum_{m\\ge 1}\\frac{X^{(m)}Y^{(m)}t^m}{m}", "bc4562e9578f2b37515b9870201ded41": "Z \\!\\,", "bc45a0b9de3f96976c8d65365720eac9": " R_{\\zeta} = \\left( \\zeta I - T \\right)^{-1}.", "bc45e6e45137f2864c725cd726adccfe": "\\scriptstyle y \\,=\\, 1 ", "bc46305d1c61903bb8b090b48681f77a": "\n\\boldsymbol{H^\\prime}=\\boldsymbol{H}^{-1}\n", "bc46879be9eb6c808402a06590c84147": "\\mu_1(t)", "bc468ce982e873f59f928820178b6c08": "n-mt", "bc4693537d04815fbd01779aa775eaf5": "C_0 \\subset \\mathbb R^d", "bc46e4a0435fc3ea476cc2bb93f41c50": "a \\in S.", "bc471f5ac7120d0c0a2de62661764c60": "F=-\\frac{1}{4\\pi\\epsilon_0}\\frac{3p^2}{16a^4}(1+\\cos^2\\theta)", "bc475c7904a55ec99258b52d62a62a97": "\\tau(L) = \\inf \\tau(x).", "bc476e991d97675f6ae157be98ddd201": "\\tfrac{12}{1+n}", "bc47997b7fd20376673f595b7c49a970": "h(a) = 0", "bc47de9920be052640cf06c958935378": "\n\\langle \\psi |x,x\\rangle + \\langle \\psi |x,y\\rangle + \\langle \\psi |y,x\\rangle + \\langle \\psi | y,y \\rangle\n\\,", "bc47f620f44417e1d6b029879c923167": " G_1, G_2,\\dots, G_n", "bc490949c334c8be665e6d711130078b": "\\mathbf{\\hat{n}}\\,\\!", "bc490fe55576d629d21d55868a20b0d7": "\\mathcal{R}(a \\wedge b)=\\mathsf{R}(\\mathsf{h}(a \\wedge b))", "bc491afce997f3e6e2e056f95ee25764": "-T_L = J_m W_m^r + D_mW_m^r", "bc49307af40cd91d43146cef48cad4dd": "\\begin{align}\n \\mu &= \\ln(\\operatorname{E}[X]) - \\frac12 \\ln\\!\\left(1 + \\frac{\\mathrm{Var}[X]}{(\\operatorname{E}[X])^2}\\right) = \\ln(\\operatorname{E}[X]) - \\frac12 \\sigma^2, \\\\\n \\sigma^2 &= \\ln\\!\\left(1 + \\frac{\\operatorname{Var}[X]}{(\\operatorname{E}[X])^2}\\right).\n \\end{align}", "bc49a3b68d37164b5a12ff2a602b215e": "F(x_1 + \\Delta x) = \\int_a^{x_1 + \\Delta x} f(t) \\,dt.", "bc4a573673bd90f51804c4aafd780630": " (\\exists x)(P(x))", "bc4a652d5f7b2960341f4ef18524aee9": "V_{avg} ", "bc4a92061d5ccfd42eca09af64b8aa9d": "x_2 s", "bc4ab9a83404585e00fae38f287afd50": "\\sum\\nolimits_{n=0}^{N-1}{f\\left[ n \\right]=0}", "bc4acceffb16f151dbbc5a5fd1752640": "\\tau = (1+\\sqrt{-163})/2", "bc4b975c1e34b8014f42ddb532255e9d": "f(x,y) = y x", "bc4bbe5ef5c97727c165fe2bdbde5f41": "\nR(q,u) \\in C \\ \\ \\ \\longleftrightarrow \\ \\ \\ \\alpha \\le I(q,\\alpha,u)\n", "bc4c002e62f13a0aaf14f467217cdd68": " \\operatorname{tr}(A\\rho)", "bc4c09ebd430dcea6fcea61c0b46dc54": " \\left(x\\right)_v\\in \\mathbb{F}^d", "bc4d1781c5c3bfbaff592d4ea8c1f4b2": "\\begin{align}\ns_0 &= 0\\\\\ns_1 &= s_0 + t_0 = t_0\\\\\ns_2 &= s_1 + t_1 = t_0 + t_1\\\\\ns_3 &= s_2 + t_2 = t_0 + t_1 + t_2\\\\\n&\\dots\\\\\ns_n &= s_{n-1} + t_{n-1} = t_0 + t_1 + \\dots + t_{n-1}\\\\\ns_{n+1} &= s_n + t_n = t_0 + t_1 + \\dots + t_n = 1\n\\end{align}\n", "bc4d2eae46f1b5561228fe4d6903f297": " H^q(X,\\mathcal F) \\cong H^q(\\mathcal F^\\bullet(X)), ", "bc4d3d6eb8fe10756ab2cb7b57b85eea": "\\,\\!y", "bc4d796280c69d02978a7303e20cee77": "T = \\rho_f V g - m g . \\,", "bc4e7f0b23ee25838724df93ce8a42e2": "-\\pi^2a/4", "bc4ec7d397ec587b5d144c471b0f8304": "Q |\\psi_1\\rangle = S_\\psi |\\psi_1\\rangle = -2 \\sin(\\theta) \\cos(\\theta) |\\psi_0\\rangle +(1-2\\sin^2(\\theta))|\\psi_1\\rangle", "bc4ec8af887727920e5e277db4ada6a3": "\\operatorname{\\bar{F}}(\\bar{r},\\dot {\\bar{r}},t)", "bc4ef3482cf04f0146d0167c19d7c72c": " -\\frac{1}{n}\\log p(x^{(n)}=(1,1,\\ldots,1)) = -\\frac{1}{n}\\log (0.9)^n = 0.152", "bc4f03f018abdbcfd2b0ec354a90eef8": "\\sum_{i_1,i_2,\\dots}\\varepsilon_{i_1\\cdots i_n} a_{i_1 \\, j_1} \\cdots a_{i_n \\, j_n} = \\det(\\mathbf{A}) \\varepsilon_{j_1\\cdots j_n}", "bc4f0a1932ddb448ee08ab3513537226": "\\le 2^{k +H(p)n-n}", "bc4f6701143ab3355414753fea5b7e1b": "\\eta_V = v_1 \\; \\mathrm{d}x^1 + v_2 \\; \\mathrm{d}x^2 + \\cdots + v_n \\; \\mathrm{d}x^n.", "bc4f87efe2261a77adc88761fc22bdd6": "\\lim_{a\\rightarrow 0}\\int_{-\\infty}^\\infty \\frac{1}{a}\\textrm{sinc}(x/a)\\varphi(x)\\,dx\n = \\varphi(0),\n", "bc4fa69f41e6a7ee3b4b1e284f03bcb5": "(t,x,y,z)", "bc4faf0bbb2680669d5d3fdcd8ec145a": "\\int_{-\\pi}^\\pi \\sin(\\alpha x) \\cos^n(\\beta x) dx = 0", "bc4fe36089d889f229dbd3709f5b289b": "a \\frac{d^2 x}{dt^2} + b\\frac{dx}{dt} + cx = A f(t).", "bc500a1295a3c89a416c0ef4801e719e": "C(E/K)>0", "bc503500e3f9b58bf1dc6e965fb6ec24": " \\dot{\\varphi} = -H_x \\cdot \\dot{y}.", "bc50725aa150e566e3f8380c2bbd8903": " J_n( \\omega t) \\cdot u(t)", "bc50ff076cc6152244a17feb235db9ff": "p_n\\# = \\prod_{k=1}^n p_k", "bc515bbcef70c8f951820af14c395c5d": " K_0\\backslash G_0/K_0 = A_+", "bc518bd42e0cfa424357a26712828988": "\\ x", "bc5199cf5a2b5835aa8f32337152fbca": " f(x,\\cdot): N \\to \\mathbf{R} ", "bc51cf7abf6225ab950b5de03c5b60e7": "\\varepsilon = \\begin{pmatrix}\n\\varepsilon_1 & + i g_z & 0 \\\\\n - i g_z & \\varepsilon_1 & 0 \\\\\n0 & 0 & \\varepsilon_2 \\\\\n\\end{pmatrix}", "bc51e1f3922b3e4965f6672d5af0b03a": "u^3_i = 192", "bc51f0873b4193986f8d2a333b552d4e": " n = p_1^{\\nu_1} \\, p_2^{\\nu_2} \\cdots p_k^{\\nu_k} ", "bc5205b988d05080b56161e1cef640b0": "q_k(i)", "bc521818a9e52ca69f66ab4e6eeb00d9": "\nu(x=0) = f_{1}(y)\n\\quad\\text{and}\\quad \nu(x=x_{l}) = f_{2}(y) \\qquad \\text{(1-a)}\n", "bc52536749f2709af553cad17e0b3e85": "(wRv \\land wRu) \\implies vRu ", "bc525bfe96d932b73299eb0f16f09a52": " a]=\\{x\\leq a\\}", "bc52603d99602b162611b5936bf37ba6": "w_n\\Vdash p", "bc52bf85905fb98b27d07233c530fd28": "\\mathcal{O}_m.", "bc52ec9bd67dee8138d1353d9abb7788": "\\partial \\Omega_D \\cup \\partial \\Omega_N =\\partial \\Omega ", "bc532877f339de93e618971e4caad87a": "\\sqrt{(-3)^2+4^2} = 5.", "bc53326d5d40f45e0b1fd5ce724d38c2": "P\\to X ", "bc5352560d0298aaa2514feb58d152dc": "\n(\\mathbf{\\hat{f}_{0:5}})^T =\nc_5^{-1}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.7308 \\\\ 0.2692 \\end{pmatrix}=\nc_5^{-1}\\begin{pmatrix}0.5331 \\\\ 0.0815\\end{pmatrix}=\n\\begin{pmatrix}0.8673 \\\\ 0.1327 \\end{pmatrix}\n", "bc5356f585f962701cb5b478f18eb43c": " i = \\left( \\frac {FV} {PV} \\right)^\\frac {1} {n}- 1", "bc535c5c19d72d7d86716528555f1de8": " \\begin{array}{lll}\n\\phi_1&=& 2.8669x - 4.0225x^3 - 0.5092x^5 \\\\\n\\phi_2&=&-4.4622x + 20.6505x^3 - 18.4582x^5 \\\\\n\\phi_3&=&-0.7745 + 7.9272x^2 - 9.2789x^4 \\\\\n\\phi_4&=&-0.8960 + 2.4523x^2 - 0.1239x^4 \\\\\n\\end{array}\n", "bc53972d9cc7d8bdbe299a2887a8ee44": "\\begin{align}[][x_1, x_2] \\,\\langle\\!\\mathrm{op}\\!\\rangle\\, [y_1, y_2] & = \\left[ \\min(x_1 {\\langle\\!\\mathrm{op}\\!\\rangle} y_1, x_1 \\langle\\!\\mathrm{op}\\!\\rangle y_2, x_2 \\langle\\!\\mathrm{op}\\!\\rangle y_1, x_2 \\langle\\!\\mathrm{op}\\!\\rangle y_2),\n\\right.\\\\\n&{}\\qquad \\left.\n\\;\\max(x_1 {\\langle\\!\\mathrm{op}\\!\\rangle}y_1, x_1 {\\langle\\!\\mathrm{op}\\!\\rangle} y_2, x_2\n{\\langle\\!\\mathrm{op}\\!\\rangle} y_1, x_2 {\\langle\\!\\mathrm{op}\\!\\rangle} y_2) \\right]\n\\,\\mathrm{,}\n\\end{align}\n", "bc540fc77989c7da05c666933c7fda8f": " \\frac{\\partial \\varepsilon}{\\partial u} = \\frac{1}{L} > 0", "bc5431c9bd9ec2abc5304e1f863dad9d": "V_L = - {d\\Phi_B \\over dt} = - L {dI \\over dt}", "bc543d7a7613fdba8b43f631679de283": "V_{rrm}", "bc545a94872a4630d6d4cc2e3bdf8337": "f(t)\\star g(t) = f^*(-t)*g(t),", "bc54994e76197e6ef25bce08717c8a37": "TR=P \\cdot Q", "bc54cd8b9c2d489305fe7514367aa7e9": " m(r_n) ", "bc54d70ef5846deb2cc17c31d68db40e": "P(\\textstyle H_0 \\mid k) \\approx 0.95", "bc54ef40f1e0ddbff0ec143bfab57205": "\\beta(s) = \\sum_{n=0}^\\infty \\frac{(-1)^n} {(2n+1)^s},", "bc54f4d60f1cec0f9a6cb70e13f2127a": "pc", "bc5519999d1da98bfea622ca28e9e7e8": "{{\\log_{\\epsilon}{N}={-D}=\\frac{\\log{N}}{\\log{\\epsilon}}}}", "bc5572af307e74986365042fc172ba87": "\\sum_{g\\in G}f(g) g,", "bc5598abb2a5459f0f6bd248046d88ce": "X\\neq Y", "bc559999da4b1b5baba315dfc9678cdb": "\\sum_{k=0}^\\infty k^3 \\frac{z^k}{k!} = (z + 3z^2 + z^3) e^z\\,\\!", "bc55d7473bb8614df896fa1d7f9befd3": "(x,t)", "bc55f47d89fb9010fa11ad9d73089c64": "Z' \\to Y' \\to X' \\to", "bc57299cb99f66fcd562fe574bd867d3": "k=2\\pi n/\\lambda", "bc57358cf71cdeb82d5b3f225b67e9cf": "X_j \\sim S(\\alpha, \\beta_j, \\gamma_j, \\delta_j)", "bc575696d76bc0e7bf83e933fd6ea78e": "\\operatorname{exsec}(\\theta) = \\sec(\\theta) - 1. \\,", "bc57654ade2ec46f0a8e2f4e9eac429e": "\\{|\\psi_n\\rangle\\}", "bc57943a94ffe2e7851277da8bbd2c36": "\\mathcal{Z}(a_n) \\subseteq \\{ 1 \\le n \\le 30 \\}", "bc57a7590080f64b0f5ea302307c7ca9": "\\mathbf{X}\\boldsymbol{\\beta}=-\\mu^{-2}\\,\\!", "bc57a9d8056475e64a8acdd85b2d043e": "\\phi_v = (u_1,u_2)\\mapsto\\langle L(u_1,u_2), v\\rangle", "bc57c814c1fd601a8e5b5f83e53ae7ea": "0.8814", "bc57dab2af9ad0fe9ee03ecdbaf1eda7": "s \\begin{Bmatrix} q , p \\end{Bmatrix}", "bc58c3e463b34f556ccdf6be69a4835f": "S_{ab} = 36 m/r", "bc58d9b8470563baf485420cb67efae4": "\\frac{1}{(n-1)!}\\sum_{k=0}^{\\lfloor x\\rfloor}(-1)^k\\binom{n}{k}(x-k)^{n-1}", "bc595ba79358f9d6b1c97bd59c6c0eb9": "\\log |S|", "bc598253010051ebaf09a8186aca7fe6": " = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v \\cdot \\left(v_1^2 - v_2^2\\right) ", "bc59b78aa3072aa46e892826a72747aa": " T(A \\vee B) = T(A) \\vee T(B) ", "bc59f2e7cc53852d55ed60f996b8f59d": "p(x_{0},t={0} \\mid x_{0})=\\delta(x-x_{0})", "bc5a558b95b5afffc72a004cefdcc30f": "\\varphi_v(g) = \\pi(g)v", "bc5a5acee3eaeb545a242695e84f4b9d": "U^\\prime ", "bc5a60a883624ac0a079372d782174bb": "\ny_{n-1} + y_{n+1} = 2y_n + {h^2}y''_n + \\frac{h^4}{12}y''''_n + \\mathcal{O} (h^6)\n", "bc5aab2827583b39566ee3fea362e16f": "\\ \\pi[n]", "bc5b3e565679d0addd7a0987aea6bd67": "\n\\begin{array}{lcl}\n v & = & \\frac{dx}{dt} \\\\\n & = & \\frac{dx}{dA} \\cdot \\frac{dA}{dt} \\\\\n & = & \\frac{dx}{dA} \\cdot\\ \\omega \\\\\n & = & x' \\cdot \\omega \\\\\n\\end{array}\n", "bc5b4713e8628414f35f8aa00d26ec6b": "K_{k} (\\tilde M)", "bc5b6f3317ebf6632decbc2fd0ad4f45": "nB_z\\subseteq mB_y", "bc5bba065d63063402eca740ffb9bfcb": "q \\equiv w^{2} \\pmod l", "bc5c11c2f36342bea7deebef7f0e8c6b": " \\sum_{i = 1}^n (y_i - \\bar{y})^2 ", "bc5ca44354481ae5c6c5a4b4f25f7548": " A_v = k(p-m)\\, ", "bc5cedbb76acb4cf87db6d24c5afe4e5": "A\\in\\mathcal F", "bc5cf935a2dcdd920efde61f4f36ede3": "{\\mathbf e}_j\\,", "bc5d01096369aff54596f5d8ba1a3436": "e_1 \\wedge e_2 \\wedge\\cdots\\wedge e_n", "bc5d0d1e2e286e0dafef5f8f0ddcb629": "PV = \\left[\\sum_{k=1}^{n} Fr(1+i)^{-k}\\right]", "bc5d39ca37592e9136fe019a88898d7a": " H_{\\mathrm{Darwinian}}=\\frac{\\hbar^{2}}{8m_{e}^{2}c^{2}}\\,4\\pi\\left(\\frac{Ze^2}{4\\pi \\epsilon_{0}}\\right)\\delta^{3}\\left(\\vec r\\right)", "bc5dabf91fd4731126b527b1ea9d4f1e": "\\displaystyle \\int_{\\mathbf{R}^n}f(\\mathbf x) e^{-2\\pi i \\mathbf x \\cdot \\boldsymbol \\xi }\\, d^n \\mathbf x ", "bc5e1551c5e01006b41c631651385606": "\\scriptstyle 0.5/L", "bc5e1992589d794f9797f8ba3755d14b": "x \\odot y = \\sum\\limits_{i=1}^{k} x_{i}y_i", "bc5e265a68f4db730eee81d42bb060aa": "R=\\frac{t}{2} \\csc \\frac {180}{19}", "bc5f222141b1ccc1f5214bf979153df0": "\\frac 1{x}", "bc5f3b2dbffc1c229e73472f00be8b11": "\np(n)=\\frac{1}{n}\\sum_{1\\le k\\le n}\\sigma(k)p(n-k).\n", "bc5f4e89155c592e8b6935693d47aa6e": "\\theta_i \\theta_i = -\\theta_i \\theta_i.", "bc5f90d8db2fc440069599979635521d": " \\widehat{B}_x, \\widehat{B}_y, \\widehat{B}_z", "bc5fd71a22ad8a8ae3e69ea8abf61932": "\n\\begin{align}\ns_{lsb}(t) &= Re\\big\\{s_a^*(t)\\cdot e^{j2\\pi f_0 t}\\big\\}\\\\\n&= s(t)\\cdot \\cos(2\\pi f_0 t) + \\widehat s(t)\\cdot \\sin(2\\pi f_0 t)\n\\end{align}\n", "bc5ff4b67ac656b22daefb40988f3b33": "B = \\begin{bmatrix}-2&2&-3\\\\\n0 & 0 & 4.5\\\\\n2 &0 &-1\\end{bmatrix},\n\\quad\nC = \\begin{bmatrix}-2&2&-3\\\\\n0 & 0 & 4.5\\\\\n0 & 2 &-4\\end{bmatrix},\n\\quad\nD = \\begin{bmatrix}-2&2&-3\\\\\n0 & 2 &-4\\\\\n0 & 0 & 4.5\n\\end{bmatrix}.\n", "bc6076dd1dcda72512fc527f341d5fa0": "(7)\\quad ds^2=-\\Big( 1-\\frac{2M(v)}{r} \\Big) dv^2+2dvdr+r^2(d\\theta^2+\\sin^2\\theta\\,d\\phi^2)\\;.", "bc6078d31598b25f3972d3f16142fabe": "A_\\alpha^{\\;\\;\\; IJ}", "bc60f62efc2afe9a54ea8bfd1445bad6": "\\phi^{A,x,\\bar{a}}", "bc6105fffb7d0e58021ac2f334b9af80": "\\frac{1}{2} \\begin{pmatrix}\n1 & -i \\\\ i & 1\n\\end{pmatrix}", "bc61081b76835ef0e76ed73b52a8b5da": " U = m g h ", "bc614838e36bcdf7f50d651d0a67f2b8": "=b_1 b_2 + a_2 b_1 i + a_1 b_2 i - a_1 a_2", "bc6163fbca663bf79098f7db68e64769": " d/d\\theta (\\widehat{U}^{\\dagger}\\widehat{a}\\widehat{U}) = i\\widehat{n}\\widehat{U}^{\\dagger}\\widehat{a}\\widehat{U} - i\\widehat{U}^{\\dagger}\\widehat{a}\\widehat{U}\\widehat{N} = \\widehat{U}^{\\dagger}i[\\widehat{N},\\widehat{a}]\\widehat{U}", "bc61a568c1831f9d88756c506581b8ba": "4x^2-7x-686=0", "bc61f32c4b2e60cc53f778918990bc7f": "E_s[n] = \\left\\langle \\Psi_s[n] \\left| \\hat T + \\hat V_s \\right| \\Psi_s[n] \\right\\rangle", "bc61f78b1218adc843f2708971761e52": " \\operatorname{lift-choice}[\\lambda f.(p\\ f)\\ (p\\ f)] ", "bc62a12fa55d5c10d5fe731a9c076a87": "V = \\frac{2 \\pi r^3}{3}(1-\\cos\\phi),", "bc62e9e73384945b8bb4852c245438a1": "G^{3j}=\\frac{1}{2\\pi i}\\mathrm{trace} \\left(P_A\\frac{\\partial S}{\\partial X_j} S^\\dagger \\right)", "bc635f263b520c6e0e7fb46fec96c445": "\\dot \\theta'", "bc636accf423165274b4ed38c36fd78f": "y \\to y f^{-1}(y) -F(f^{-1}(y))", "bc636cba0328cee03e6c3702b40c124d": "\\textit{open}", "bc6395080d354ba8fcd5faa92d6a9ae9": "\\Delta \\hat{z}\\,", "bc63e1214ee96586fa65b76090d1b0b5": "u' = {u / V}\\,\\!", "bc640fda19ba1355decf50da5066374b": "0 < \\left\\langle e_1\\right\\rangle < \\left\\langle e_1,e_2\\right\\rangle < \\cdots < \\left\\langle e_1,\\ldots,e_n \\right\\rangle = K^n.", "bc64da524e498792465b500f8e0e885c": "i: N\\hookrightarrow M", "bc6570d55ddbde2cae6602fc9e669ca4": "E_{a,d}\\ :\\ ax^2 + y^2 = 1 + dx^2y^2, \\,", "bc65be1977b28c469751b02dd6bae407": " F = \\mathrm{Frob}_k", "bc65c2c97b39a6c5d84c5c2549ed4ff4": "A_1(\\omega)= \\frac{{{H_0}}}{{\\sqrt {1 + {{\\left( {\\frac{\\omega }{{{\\omega_c}}}} \\right)}^2}} }}", "bc65e8d65befc4549d9f98fb84cb643b": "\\textstyle \\phi(\\mathbf{q})", "bc66063c38c978fc661e85560a58244a": " \\tau, \\sigma \\in K", "bc66897cefcaa1845e38f48d1e3fcdb2": "f_c(z) = z^2 +c.\\,", "bc66b608e65a590f96c466cd0a50995c": "T(n; t)", "bc66ee573b1edc7e19d015ad93c3faad": "2^{2 + 1} - 1", "bc6728710dfcabff6643ec7981ff2204": "\\displaystyle \\{U_n\\}", "bc6728da065e55da79e7d2146216b39f": "\\begin{align}\n L(x)L(x)^{-1} &= 1\\qquad&\\text{corresponding to}\\qquad x(x\\backslash y) &= y \\\\\n L(x)^{-1}L(x) &= 1\\qquad&\\text{corresponding to}\\qquad x\\backslash(xy) &= y \\\\\n R(x)R(x)^{-1} &= 1\\qquad&\\text{corresponding to}\\qquad (y/x)x &= y \\\\\n R(x)^{-1}R(x) &= 1\\qquad&\\text{corresponding to}\\qquad (yx)/x &= y\n\\end{align}", "bc677e81bb15b26ad403ad3d3c75a186": "5x-3", "bc67a9f19bf6f888c97232df42bdb370": "\nu(t,x)=f(x-ct) \\,\n", "bc681f5a3820994f9b02047b00e5203e": "g^{\\alpha \\beta}", "bc6829892f1efb0083551b250dbfaeff": "\\langle \\rangle", "bc687091d1a0073e1bd0cd3a480c4b62": "\\mathbf{C}_i", "bc687bc49a850d62c9dd055dbe05e6a9": "U(1)", "bc68a4b2f977e4d5cd0baa3e270503b5": "\n\\begin{bmatrix}\nK_{11}-K_{12}K_{22}^{-1}K_{21}\n\\end{bmatrix}\\begin{bmatrix}\nx_{1}\n\\end{bmatrix}=\\begin{bmatrix}\nF_{1} \n\\end{bmatrix}\n", "bc68a8604737dc3333f1695b6d77c15f": "f_i^{neq}= f_i^{(1)}+K f_i^{(2)}+O(K^2) ", "bc69156d64a65851cbbe6e0fd9980845": " k_0 \\text{ or } k_1 ", "bc69195f580841c8f53b0c5a8bafd0d2": " \\mathbf{E} ( \\mathbf{r} , t ) = \\begin{pmatrix} E_x^0 \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\\\ E_y^0 \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\\\ 0 \\end{pmatrix} = E_x^0 \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\hat {\\mathbf{x}} \\; + \\; E_y^0 \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\hat {\\mathbf{y}} ", "bc69355ed3ebe2169c4a6118b7526eac": "n_1^2\\sigma_1^2-2\\sigma_1n_1^2\\sigma_\\mathrm{n}+n_1^2\\lambda=0\\,\\!", "bc6953a31e25a5a2b21d58dd9e704b92": "\\rho_\\perp\\ ", "bc69671423d881f6e8718199da88f390": "-\\pi < \\phi \\le \\pi", "bc69c71d3708a3d204abed9d803db3b0": "PSL_2(\\mathbb{Z})", "bc6a17fb61b022aa34dedea8fd31ba65": "\\lim_{t \\to \\infty} g(t) = \\lim_{s \\to 0} \\frac{s}{s} \\frac{6}{s+2} = \\frac{6}{2} = 3", "bc6a430b477fa6c4fb7c5f8c054278e2": "T_u\\mathrm FM=T_u \\mathrm F_{SO}(M)\\oplus \\mathcal M_u\\,,", "bc6a69dd5a0af965c5e62b1b758c7aed": "\\alpha \\approx A^*\\sqrt{h\\nu - E_{\\text{g}}}", "bc6ab990f036543ea7edb1c21111801f": "\\hat{\\textbf{y}}_{k\\mid k} = \\textbf{P}_{k\\mid k}^{-1}\\hat{\\textbf{x}}_{k\\mid k} ", "bc6b00b9f6f181e9e28770def9e484ae": "\\epsilon=\\phi_{1,3}", "bc6b0efd3bed4dfabe15757cf4089d87": "F_1", "bc6b17338aa160a88fbc732ad5d07a53": "h(x) = \\frac{g(x)}{a}", "bc6b839904c41c27c097cdfb37a5d0b5": "\\le 40", "bc6ba1ae6a8bcf145ae99e52ee0ddbbd": "E_{n}=-\\frac{mc^{2}\\alpha_{g}^{2}}{4n^{2}}", "bc6c0d9bfde025ae802e6c6dd44eb17f": "t_u - t_l \\rightarrow \\infty", "bc6c25654c4ae73e7fff01ed0a434b7e": "y \\ll z", "bc6c994d8d52f5e470f841d080f79700": "n_\\mathbf{p} = {1 \\over e^{\\omega_p /kT} \\left( 1 + {1 \\over \\Omega} \\right) - 1}.", "bc6d0de85e84afdaf232791d9aafa398": "I_k", "bc6d6d765a1e3372c80deacaab1b5d31": "n_s\\,", "bc6d862edf5fa53f79e86911a6191a1a": "\\,\\Phi(\\exp (2\\pi i\\lambda), s,\\alpha)=L(\\lambda, \\alpha,s).", "bc6db3a1839dd7c394635a3dc41126c0": "\\left(\\phi\\right)", "bc6db8ce68690d52a482c20c56f5724c": "f^n=f\\circ\\stackrel{\\left(n\\right)}{\\cdots}\\circ f", "bc6e380143f5e6ab6b7fc25e7aeeb0d2": "b_2 = S_{21}a_1 +S_{22}a_2\\,", "bc6e6d8623b942cb4ae44568a95f5d6e": "\\mathbf{S}V = \\mathbb{R} g {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} g", "bc6ed08df5a25b4798e1b71860209143": "\\langle \\psi|", "bc6eeeedc5cf9143a5a74a8873159e9b": "\\begin{align}\n X(K) X'(K) & = y(N)\\ y'(N) = y(N-1)\\ y'(N-1) \\\\\n & = s(N-1)^2 + s(N-2)^2 - 2 cos(2 \\pi \\frac{K}{N})\\ s(N-1)\\ s(N-2) \n\\end{align} ", "bc6f033d77ea9f26dccd3fad2e57830b": "z^2 = 0.638169999974373 - (0.239864000011495)i{\\;}{\\;}{\\mathrm {(green)}},", "bc6f6c4c10dbfd3205d3a877afeaf752": "\\,kp+p\\, ", "bc6fce76d375244bf6137252e884f118": "G_1\\in[0,255]", "bc6ffd6f05ce5f7a3975cd9a2b6fceae": " ASA_{i}", "bc708bb71903c8ff24b1b1794bb42f01": "\\scriptstyle\\gamma^\\sigma \\partial_\\sigma\\!", "bc70c1bed7eb6f03ba08d29cb671d9bd": "(4) \\,", "bc70f4c63d3ddbed127485e7bf260ac2": "\\frac{dx}{dt}=\\alpha [y-x-f(x)]", "bc70f52508d6ed214afba9abf47d911d": "{\\Bbb C}^n\\backslash 0 \\times {\\Bbb C}^m\\backslash 0", "bc70fdbc03ca4a9de34f1c49a6091ac3": " T = \\frac{m}{2}\\mathbf{v}\\cdot\\mathbf{v} \\nrightarrow T = \\frac{\\gamma m_0}{2}\\mathbf{v}\\cdot\\mathbf{v} ", "bc71a2736a4414502642a91b0fb13348": "P(v)=v^n+A_1v^{n-1}+\\cdots+A_n.", "bc72484cc49cdb99907f163cd306bbb5": "\\frac{\\partial }{\\partial T}\\epsilon", "bc72ea83ed6812be2d14a6b6fd5bafbd": " \\textbf{P}(t) = R\\textbf{e}_r + Z(t)\\vec{k},", "bc73000ca1f0550960b3eb9414d2faf3": "G=(V,A)", "bc734bcf2983d232758ff394827b39a1": "\\,v_f = c\\left(\\frac{f}{f_0} - 1\\right)", "bc73b81ec481d77097c4702a999ebd03": "\\tau (y) = \\mu \\frac{\\partial u}{\\partial y}", "bc73ba0748f3006c75b7555f26a859bb": "\\sigma_{\\varphi \\and \\psi}(R) = \\sigma_\\varphi(R) \\cap \\sigma_\\psi(R)", "bc73c77b8d08081505dffb94d0287767": "y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6.\\ ", "bc73e6a9a1820229b13204dd1c24e285": "E_{a,d}", "bc74238a96a7420dbd138441156d9732": "Y(g)", "bc744126d5ab96db07248619437f0b5f": "\\frac{\\partial h(\\vec x,t)}{\\partial t} = \\nu \\nabla^2 h + \\frac{\\lambda}{2} \\left(\\nabla h\\right)^2 + \\eta(\\vec x,t) \\; ,", "bc744196dd90c347f3828b689d66a238": "Y ", "bc746292c52f55ac306d6302da450d74": " n_h = \\frac{K' W_h S_h}{\\sqrt{C_h}}. ", "bc746644ba31816fed06dcaa187ebfc0": "\\cos \\frac{\\pi}{h+k} < \\frac{p}{q} < 1", "bc74cf2e4e5a9b8fd1b8643a1e94936f": "\\displaystyle{\\psi(a)f(x) =\\int K(x,y)f(y)\\, dy,}", "bc7544fd7e8fa6477b5ffd697b59eeb4": "k \\in \\{0, \\dots, 2^n-1\\}", "bc755a84295f1dede06b3dce79fada96": "\\varphi = \\frac{1+\\sqrt{5}}{2}\\approx 1.61803", "bc756c3b7de2b3ee5823e4f405058ae8": "\\mu \\in \\mathbb{R}, A_t (x), B_t (x)", "bc75bb356eb1451bef3aed12359becf1": "[A]^{\\omega}\\to Q", "bc75f3d0e99ebda58deb84378e3d3210": "\\scriptstyle Q \\,=\\, -\\gamma^0", "bc760a6e3e27a0a5cb8574a64b91a685": "I_{[0,x]}", "bc768f4d7a1414ee48897f957d43c244": "V = 8\\sqrt{2} a^3 \\approx 11.3137085a^3.", "bc769422220f70508a64a013903aac9e": "3\\uparrow\\uparrow\\uparrow2 = 3\\uparrow\\uparrow3 = 3^{3^3} = 3^{27}=7,625,597,484,987", "bc7697052aabe7c4db2dfe9b14a30dd4": "\\operatorname{lcm}(a,b)=\\left({|a|\\over\\operatorname{gcd}(a,b)}\\right)\\cdot |b|=\\left({|b|\\over\\operatorname{gcd}(a,b)}\\right)\\cdot |a|.", "bc769c816b81cab99af9c864f74d86d8": " G = ", "bc769ff8d2d5cfb3a2d4005dda94a8b5": "\\sigma_2=I_1-\\sigma_1-\\sigma_3\\,\\!", "bc76c1348f4fee15cf4e440c2ed38e05": " R^0_0(\\rho) = 1 \\,", "bc76ffef16390811594b267b8bf983e5": "x=e^{i\\pi \\tau}", "bc774582457c9c10ac4e0dabb6486cd6": " \\{ f,g \\}(x) = {(\\mathrm{d} f)_{x}}((X_{g})_{x}) ", "bc77816299bf3ccb82444662e6b2ddde": " 1 = \\nu^2 + \\sin(\\arccos(\\nu))^2 ", "bc77844c890d771c7b767a80a8c9b1b2": "\n\\langle \\psi|x,y\\rangle + \\langle\\psi |y,x\\rangle = 0\n\\,", "bc7784762541d448d321eb227991ba52": " u_{i} ", "bc779e730d9f2ecfac6a90765983c941": "+\\sum_{j=1} ^{N-1} \\sum_{i=j+1} ^N f_{ij}\\biggl\\{\\epsilon_{ij}\\biggl[\\left(\\frac{r_{0ij}}{r_{ij}} \\right)^{12} - 2\\left(\\frac{r_{0ij}}{r_{ij}} \\right)^{6} \\biggr]+ \\frac{q_iq_j}{4\\pi \\epsilon_0 r_{ij}}\\biggr\\}\n", "bc77a4e6431a916ab74e48d4dff8b4da": "p(\\infty ) \\,", "bc77acb55a8a5094528829a80ab522cf": "x^2 y^2 = U^2,\\,", "bc77b3910a530a89a1052f7e512e2fe7": "b^n-1 = b^n-1^n = (b-1)(b^{n-1}+b^{n-2}+...+b+1)=(b-1)b^{n-1}+...+(b-1)", "bc77dfc3e57b6d5d8352e5eb8d898980": " A^x +B^y = C^z,", "bc77ff635aa24c404f96a98e64749147": "C_\\text{out} \\circ C_\\text{in} (m) = (C_\\text{in} (C_\\text{out} (m)_1), \\ldots, C_\\text{in} (C_\\text{out} (m)_N ))", "bc781f7e79776d3e82b32d57cb0c6e39": " \\tan \\theta = \\frac{E_R - E_L}{E_R + E_L} \\,", "bc785f7473a43eff28b96536ad69fcb6": "\n\\begin{align}\n \\mathcal{I}(v) &= v \\\\\n \\mathcal{I}(\\lambda v.\\ M) &= [v]\\;\\mathcal{I}(M) \\\\\n \\mathcal{I}(M\\;N) &= (\\mathcal{I}(N))\\mathcal{I}(M).\n\\end{align}\n", "bc7883dd9ef864e3434c4038633eb4b1": "\\rho(T)=\\rho(0)+A\\left(\\frac{T}{\\Theta_R}\\right)^n\\int_0^{\\frac{\\Theta_R}{T}}\\frac{x^n}{(e^x-1)(1-e^{-x})}dx", "bc79052a3f1515cb9df3d2092d031fe0": "A b", "bc79e1d3d08a23bbccb0abb23ca56ccc": "\\phi(\\omega)=-\\arg(H(j\\omega))=\n-\\arctan\\left(\\frac{15\\omega-\\omega^3}{15-6\\omega^2}\\right). \\, ", "bc7a5f6f4af5264f79cdb2429ddec471": "y_{2}=2-1", "bc7a7f80c06a9d21f2f7bbeeae861898": "\\textbf{b}^*", "bc7a800c9ee52a586850bf8e8049ea53": "\\mathfrak{sl}_3(\\mathbf H)", "bc7ada52c0f323ff0c6d36809f3441ca": "PP'", "bc7af4f454c2c9aaee5c583062af8f4a": "\\textstyle\\tau_{(Q)}", "bc7b36fe4d2924e49800d9b3dc4a325c": "fs", "bc7bc940b6a7703ea15e51dd86aa34d5": "N^*\\Sigma\\,", "bc7c0c4d362bd401dd2cf509563ac10f": "T \\to w(S, \\widehat{b}c)", "bc7c0dfa509ab2be2540361b6fcf71d2": "\\tau*", "bc7c5631fcbd2ca1d2c78b85d2018bab": " = x_k - \\frac{x_k}{n}+\\frac{A}{n x_k^{n-1}}", "bc7ca6107acb97f702fb59c8436317b6": " R ", "bc7ce353d9937cd8f6b7fbf2161043ab": "\\dot{\\textbf{x}}(t) = \\begin{bmatrix}\n -2& -1\\\\\n 1& 0\\\\\n \\end{bmatrix}\\textbf{x}(t) + \n \\begin{bmatrix} 1\\\\ 0\\end{bmatrix}\\textbf{u}(t)", "bc7d1b54e28c3718e6d17bd665273527": "n^{\\log_23}", "bc7d7533b380140af484b1840cd5f022": "\\left[\\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix}\\right]", "bc7d85d19951ebba6919b1bb0581923d": "k_\\beta = \\frac{\\omega}{\\beta c} ", "bc7dc02a98095717581006df42f273dd": " X_t = \\sum^{\\infty}_{n=1} \\mathbb{I}_{\\{J_n \\leq t\\}}=\\sup \\left\\{\\, n: J_n \\leq t\\, \\right\\}", "bc7dde2c6447f941c1bfc454a1661979": "sk_n(K_*)", "bc7dfae1c9847390ea5ddf4969a33453": "\\mu,\\lambda", "bc7e1abaf9a619923eff9ce20745d491": " \\psi_t(x) = {1\\over \\sqrt{2\\pi (a+it)} } e^{- {x^2\\over 2(a+it)} } \\, .", "bc7e206c04a5bac40b3e33d5612fa58a": "\\left(1- 2 X Z + Z^2 \\right) ^{- \\frac{1}{2}} \\ = \\sum_{n=0}^\\infty Z^n P_n(X)", "bc7e3cee019e8fd3d1ee510a13e6cdcb": "y^4+4y^3+\\frac{4}{5}y^2-\\frac{8}{5^3}y-\\frac{1}{5^5}=0\\,.", "bc7e5224c4955e6b907420bd0e7a57a8": "I^*I \\subset R", "bc7e6f1d4dc690456b97914400de91e6": "n-k", "bc7e721ff80bb19a310d0cdcaa3edefd": "\\displaystyle{D(f,g)=(u,g)}", "bc7e8e5c904b50aac12d0a16f82653f0": "y>-1/2", "bc7ebab691ca84a54d3af9338d89175e": "f(r_1)/p^1=7/7=1", "bc7ebcdaaf2c6d154e0e3c3765710d1f": "p =\\text{cont} (p)\\,\\text{primpart} (p),", "bc7ebfd12e10995b8b7d512ea85b39d1": "\n\\mathbf{\\hat{b}_{3:5}} = \\alpha\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix}0.6273 \\\\ 0.3727 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.4175 \\\\ 0.2215\\end{pmatrix}=\\begin{pmatrix}0.6533 \\\\ 0.3467 \\end{pmatrix}\n", "bc7ecca6c6ce762e0341cdc33590bb08": "\\Phi \\colon F(X)\\times \\cdots \\times F(X) \\longrightarrow F(X)", "bc7f34e784f2912e5c69e777ba458912": "\\displaystyle{\\sum_{m=0}^k (-m-1)^{-k} a_m =1.}", "bc7f6af5d946c46e8b6c49da798d7b4d": "n + m = 3n - m + 2(3n - m) + p", "bc7f93c534cb0ca5d6bcf06e11e8924c": "s = H = \\frac {f^2} {N c} + f,", "bc7f9c0d0a37b3edc5c729b32385700d": "\n\\begin{array}{rcl}\n2 H_2 + O_2 & \\rightarrow & 2 H_2 O \\\\\nC + O_2 & \\rightarrow & C O_2\n\\end{array}\n", "bc7ff488480d3a90be7d5e2e3edbf4e2": " \\phi=\\mathit{I}\\int_{r_0}^{\\infty}\\frac{dr}{r^2 \\ \\cos \\boldsymbol{\\Phi}_s},\\,\\!", "bc80026e0281558be651ee627f031e73": "fWAR = wRAA + UZR + Position + \\tfrac{20}{600}*PA", "bc803af6c83e7ec45445969ded111387": "c_V = \\left(\\frac{\\partial C}{\\partial m}\\right)_V.", "bc803ffeccf66823def1f06b7b18aa50": "B^n x + B^n \\le B^n (y+1)^n\\,", "bc8064645cbd0836a2c3b79465d7219a": "\n\\forall 1 \\leq i \\leq N, ~ \\forall a \\in A_i, ~ \\text{Gain}_i(\\sigma^*,a) = 0 \\text{.}\n", "bc809869ed301eea6e81a3d925371e83": "Y \\sim \\mathrm{Herm}(a_1+b_1,a_2+b_2)", "bc80b1407e8a501e1b161c3f63a75e5e": "r = \\sin(2\\theta) \\,", "bc80ca02d51d1856b205cc3f4eb555a3": "-v_{\\mathrm{rel}}\\mathrm{d}m = m\\mathrm{d}v \\,", "bc81052dbde220b0a33098d6a7b3928f": "\\kappa_3=\\mu'_3-3\\mu'_2\\mu'_1+2{\\mu'_1}^3\\,", "bc8154c0b6d222ee5080f88170a5da14": " \\beta \\in [1,3] ", "bc8175d214316e711521fd45392e2721": "Q_1 Q_2 BC", "bc82008506f1d15fdcafcaa91673c393": " \\forall", "bc82354ed842663d5dd32ff2edba6959": "\\displaystyle{\\|u\\|_{(1)} \\le \\|L u\\|_{(-1)} + \\|u\\|_{(0)},}", "bc824ed47ec758b465e28d3f390d28c8": "\nI_2=\\frac{4 A_1^2 \\chi_0^2 L^2}{\\pi^2}.\n", "bc82dfdf178b4a25026eafa1f0ee500b": "\\mathit{Assoc}1(R) = \\{((a,b),c) \\mid (a,(b,c)) \\in R\\}", "bc82e16f0b0b78d8f4a864b6a7bb4c12": "\\rho_A \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_j \\langle j|_B \\left( |\\Psi\\rangle \\langle\\Psi| \\right) |j\\rangle_B = \\hbox{Tr}_B \\; \\rho_T ", "bc82f6800ea9645e50ff1b80103002b6": "\\left(\\frac{n}{p}\\right)", "bc833cdbfe24a6563135542bac06f994": "Q_1=(b_1^2+a_1^2),", "bc838c2d36e46b1ad4967a4b8e9761b2": "\\vec{e}_i = c\\times\n\\begin{cases} \n (0,0) & i = 0 \\\\\n (1,0),(0,1),(-1,0),(0,-1) & i = 1,2,3,4 \\\\\n (1,1),(-1,1),(-1,-1),(1,-1) & i = 5,6,7,8 \\\\\n\\end{cases}", "bc839a4e91950f2dae989e4d278b46b8": "N_{t+1}=N_{t}+B-D+I-E,", "bc83a359a9add2518eb86b82908e0204": "\\frac{1}{T}\\cdot{dT}=-\\mu\\cdot{d}\\varphi", "bc83bdcb7c9356c7ca03f8affdb68be9": "c_{1,p},\\ldots,c_{m_p,p} \\in H^p(E)", "bc841768b250d76eee3e41f4f9cdfe04": "f_1 \\cdot V_1 + f_2 \\cdot V_2 = 0", "bc84247a121d7fc96029c3544358a4b1": "\\; Tr(W\\varrho_{A_1\\ldots A_m}) \\geq 0", "bc8445ce4b43fe1a90b9f7f2327f6ab6": "x^g \\cdot x^h = x^{g+h}.", "bc847cc3f7ea0af6cc911b5b9cbfc624": "\\vec{J}", "bc84bd14471febc2854d1d2aebdc3f8d": " \\frac{n_h}{N_h} = \\frac{K S_h}{\\sqrt{C_h}}, ", "bc84f1c761b8125f38ddea4b10aecaa5": " \\int x^2\\phi(a+bx) \\, dx = b^{-3} \\left ((a^2+1)\\Phi(a+bx) + (a-bx)\\phi(a+bx) \\right ) + C ", "bc84f9efb2fc42633f27a814f1720ad9": "A \\cap B \\in \\mathcal{R}", "bc8511864e923d73236afc7a906d66b3": "\\dot{X}^a = {X^a}_{;b} X^b", "bc853f49a5900f73a83357f97843fe36": "\\frac {\\mathrm{d}^3 \\vec a} {\\mathrm{d}t^3}", "bc85747909205d14a2b8a54be03a75ba": "T(t,\\sigma) = C(t)\\phi(t,\\sigma)B(\\sigma)", "bc859f5d27e888bd63ba8875af901623": " \\mathbf{a}\\cdot(\\mathbf{b\\times c}) = \\varepsilon_{ijk} a^i b^j c^k.", "bc85b1dec4728c2b7a5e0f5aa5a8c02c": "\n\\Theta_{\\ell}^m (\\cos\\theta) \\equiv \\left[\\frac{(\\ell-m)!}{(\\ell+m)!}\\right]^{1/2} \\,\\sin^m\\theta\\, \\frac{d^m P_\\ell(\\cos\\theta)}{d\\cos^m\\theta}, \\qquad m\\ge 0,\n", "bc85e57a059a864e6eaf1e5998e28e7c": "f(n) = L(n/50)^{1/2}", "bc863b98c95e7a376688b90f7a4a5a14": "\\beta = \\beta_{c}", "bc8685752bcc734c62647f4751b48f2f": "M = p + 1", "bc86a9b1fe83aec5576e67ed479d17be": "v(S\\cup \\{i\\}) = v(S)", "bc871bb7b12d4b027d6a8cf9f08b996c": "F[x,y,z]=\\int_a^t\\sqrt[3]{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}", "bc87303b6d4516a53e376e91237f9ac0": " \\forall r \\forall x [Fxr \\leftrightarrow [\\forall y (y \\in x \\rightarrow Byr) \\and \\lnot Bxr] ] \\,.", "bc874042160933dccb96ffce7270e07c": "[X_0:\\ldots:X_n]", "bc875f581737a02dcf22f67d4e1e3639": "S_\\ast(X)\\xrightarrow{\\triangle} S_\\ast(X)\\otimes S_\\ast(X)\\xrightarrow{f_\\ast\\otimes 1} S_\\ast(B)\\otimes S_\\ast(X)", "bc878da0be138a488c40872a698f7635": "(a^{ij})", "bc887b255284f4682bd75c55dc7a94ea": "R_0=1 \\,,\\, \\omega_c=1", "bc88c0b89410feeab4775637a8df2ac8": "\\vec{r}=\\vec{r}(u,v),", "bc88cc93ece5abc3451be37c21a570fb": "K \\rightarrow L", "bc894e9744def722a83ce0daa2324c5d": "|m_I,m_J \\rangle", "bc898c6ea042886e6764d3bc8a0282bc": "L(N)", "bc89e90c0921301ef04de8208a6bd152": "f_2(\\omega)\\,", "bc8a00a1dff29cd897997025985b2e02": "{U\\over V} = \\frac{8\\pi^5(kT)^4}{15 (hc)^3},", "bc8a475845e35987e73b70972e913267": "(0.5)^2 = 0.25", "bc8acedaab77511884465aca3bd119c8": "\nPoss(pickup(o),s)\\rightarrow is\\_ carrying(o,do(pickup(o),s))\n", "bc8ae75196252e9d27604b88101f8c4d": "\\theta=dx^\\mu\\otimes \\partial_\\mu", "bc8afe0839b33befa45da475e6ee5249": "Z_\\mathrm L=R_\\mathrm L\\,\\!", "bc8b44b66b445e615960d2f1542052fb": "X=[x_1,...,x_n]^T \\in \\mathcal{R}^{n \\times m}", "bc8b6400cff7648b83005806b8fb161e": "\\mathbf{g}_i", "bc8ba71df089aa9fbfb50cde95500a24": "Y=\\frac{m_1y_1+m_2y_2}{m_1+m_2}", "bc8be85562ff0367b0e146d121ca9d7e": "|z - p| < 1/\\sqrt[n]{|c_n|}", "bc8c04be5869ef28417507c27bd69ced": " |f(0) - f(y)| = 0 \\; \\forall y \\in G", "bc8c2dbfdc300148acc89b3132c01757": "=\\sum{n! \\over j_1!j_2!\\cdots j_{n-k+1}!}\n\\left({x_1\\over 1!}\\right)^{j_1}\\left({x_2\\over 2!}\\right)^{j_2}\\cdots\\left({x_{n-k+1} \\over (n-k+1)!}\\right)^{j_{n-k+1}},", "bc8c5999d9c5e668c634c1b74811e882": "D\\colon \\Omega^\\bullet(A,E)\\to\\Omega^{\\bullet+1}(A,E),", "bc8c786df5b91b7a7e20b76549d1a560": "\n \\vartheta(G) = \\max_{d,V} \\sum_{i \\in V} (d^\\mathrm{T} v_i)^2.\n", "bc8ce9965ab1721f8219539001d9ab7a": "\\widehat{Pf}=-Q\\widehat{f},\\qquad \\widehat{Qf} =P\\widehat{f}.", "bc8d2de0161fa94eb3a6122eaa265b46": "\\ntriangleleft", "bc8d56f921f6c4bcb46abe382b3203a3": "\\mathcal{X} \\subseteq {\\mathbb R}^n", "bc8da3d4deab6fe2e5279ab567f5d326": "\\frac{W}{(n + a)} \\left\\{ \\cos [ \\pi W (n+a)] - \\operatorname{sinc} [ W (n+a)] \\right\\}", "bc8ddb20ee766c4d14b22f062d3a6b0f": "Q(i,j)=(r-2)d(i,j)-\\sum_{k=1}^r d(i,k) - \\sum_{k=1}^r d(j,k)", "bc8e03dc71b9c2bdb2e27bdad46d7986": "622614630 = \\frac{90!}{6! \\, 84!}", "bc8e5f26d15916cb9e1f7989ae066e7f": "\\mathbf{C}(t)=(1-t)^3\\mathbf{A} + 3(1-t)^2t\\mathbf{A'}+3(1-t)t^2\\mathbf{B'}+t^3\\mathbf{B}", "bc8ee86a15b3effab73a4766cc099a94": "(x_1,\\ldots,x_n)+(y_1,\\ldots,y_n):=(x_1 + y_1, \\ldots, x_n + y_n)", "bc8f35e7dfe1706543f16b5357080033": "W_q^{(n)}", "bc8f8e4039100a8aa47ea60a21e536e0": "f(V_i) \\sub W_{1-i}", "bc8fbf36014d0c6e67388a19a4a67248": " \\rho = \\frac {AV} {Il} ", "bc9001a48de9800d413c7d0559c82e0a": "K=\\mathbb{Q}(x_1,\\ldots,x_n,e^{x_1},\\ldots,e^{x_n})", "bc902318aadcc6ec5ec0451d34afb328": "\n\\frac{dG}{dt} = 2 T + \n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = 2 T - \n\\sum_{k=1}^N \\sum_{j0\\mbox{)}\\,\\!", "bc9e7ac65ee0907f90ba42645611677a": "h \\colon T \\to A \\times B", "bc9ebba36f3790d136e7b3399301deb3": "\nX(t)^i=tv^i-\\frac{t^2}{2}\\Gamma^i{}_{jk}v^jv^k+O(t^3)\n", "bc9ec1a7a70dddc0cb706b52c6df4bb9": "y = \\mathop{\\mathrm{Im}}(z^2+c) = 2xy + y_0.\\ ", "bc9ec66e619ed52cdbc1b2d5ad1e5487": "L_i:\\Gamma(E_i)\\rightarrow\\Gamma(E_{i+1})", "bc9ecbad427a9115b02513afcdef9284": " \\psi_j = \\sum_{i=1}^{n} c_{ij} \\chi_i.", "bc9f0932417e0988364e729f3418debf": "1727636", "bc9f2422656ce1ca9b5e6900582c9d98": "\\alpha , \\beta", "bc9f2f43829bb4bbac0e872ef42071f9": " H(f) g = \\{f,g\\} ", "bc9fcabf792355cb6ad48217b20e3fbc": "\\Lambda_1", "bca02b7c9fb5c395fb49c37ff5a0f102": "f^!", "bca02e795910850d9061641510c81b3d": "\\textstyle m-1", "bca0301fb6a6f31a9eef11d509a50b64": "E\\oplus\\epsilon_1\\cong F\\oplus\\epsilon_2", "bca0f19b47cdf656a003b6da716ee890": "~\\langle \\hat a^\\dagger \\hat a\\rangle - \\langle \\hat a^\\dagger \\rangle \\langle \\hat a\\rangle~", "bca10719f29c571bcdc01592690180d2": "\nF_\\nu(k) = \\int_0^\\infty f(r)J_\\nu(kr)\\,r\\,dr\n", "bca128e90191065b1dfdbc1cbf14cd02": " \\frac{ds}{dt}= \\left[\n\\frac{1+2i}{10}\\alpha s\n-\\frac{3+16i}{15}|s|^2s \n\\right] +{ O}(\\alpha^2+|s|^4)", "bca2025da0f5379503bf3bad7f1736da": "\\frac{dx}{dt} = 8t", "bca219c9a046a8013a69acaf1617fc58": "V_0(k)", "bca223625418f8d701584210e950861e": "\n\\widehat{R}_y \\equiv \\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_y) = \\begin{pmatrix}\n\\cos\\Delta\\theta & 0 & \\sin\\Delta\\theta \\\\\n0 & 1 & 0 \\\\\n-\\sin\\Delta\\theta & 0 & \\cos\\Delta\\theta \\\\\n\\end{pmatrix} \\,,\n", "bca22ad09771b2ed3118e02065d00414": "R_i(x_1,\\ldots,x_n)", "bca22c70330df273def8dd23d4bfb30d": "P|R \\rightarrow Q|R", "bca288ecd74bd3b00bc3816825893103": "\\gcd(a,b,c) = \\gcd(\\gcd(a,b),c)", "bca2969ab46de0424785a236c9f12414": "\\scriptstyle(+0.5\\pm3\\pm0.7)\\times10^{-10}", "bca2c38c70fadbb8a6f9e8295298bf72": " F \\left( u_{i - \\frac{1}{2}} \\right) \\ ", "bca2cfe6c0d167cf4ba84f1da7e1cd84": "\\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}, \\quad \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}. ", "bca2e70efd4b0ae8862d3b89f32ae8aa": "1/(1-ay)", "bca32cc8d2edac0002718d9ebab85f12": "\\boldsymbol{F} = m \\boldsymbol{a} = m(\\ddot r - r\\dot\\theta^2)\\hat{\\boldsymbol{r}} + m(r\\ddot\\theta + 2\\dot r \\dot\\theta)\\hat{\\boldsymbol\\theta} \\ , ", "bca377027bb6483ab4ab40dfb803ec5f": " {\\omega \\over {\\omega_\\mathrm{c}}}", "bca3ca5925faa7346c2d3dbd310c19ee": "A = \\left(6+2\\sqrt{3}\\right)a^2 \\approx 9.4641016a^2", "bca4187fb83cbf49510ce58933264595": "\\mathfrak{gl}_n(\\mathbf C)", "bca41cec3f5df6eaaa71ea5bfb8c5a15": "\\scriptstyle x,y\\in Z^d ", "bca48e09c894629d597e5de54f5b868e": " \\vec y ", "bca495e0887a3525111ca9d2db556ec0": "\\alpha \\beta = \\mu \\beta \\alpha,", "bca4a0ad769782402de00bc6718e9f37": "\\begin{align}\nA\n&{} = \\iint_T \\left\\|\\left(1, 0, {\\partial f \\over \\partial x}\\right)\\times \\left(0, 1, {\\partial f \\over \\partial y}\\right)\\right\\| dx\\, dy \\\\\n&{} = \\iint_T \\left\\|\\left(-{\\partial f \\over \\partial x}, -{\\partial f \\over \\partial y}, 1\\right)\\right\\| dx\\, dy \\\\\n&{} = \\iint_T \\sqrt{\\left({\\partial f \\over \\partial x}\\right)^2+\\left({\\partial f \\over \\partial y}\\right)^2+1}\\, \\, dx\\, dy\n\\end{align}", "bca5084a4f3011c5c5bc5f990b27a00b": "\n\\begin{align}\nx^2-p^2& = c-p^2\\\\\n(x+p)(x-p)& = c-p^2\\\\\nx-p& = \\frac{c-p^2}{p+x}\\\\\nx& = p + \\frac{c-p^2}{p+x}\\\\\n& = p+\\cfrac{c-p^2} {p+\\left(p+\\cfrac{c-p^2} {p+x}\\right)}& = p+\\cfrac{c-p^2} {2p+\\cfrac{c-p^2} {2p+\\cfrac{c-p^2} {2p+\\ddots\\,}}}\\,\n\\end{align}\n", "bca5631af729a175b488d07afa758c25": "k = \\eta f p \\epsilon P_{FNL} P_{TNL}", "bca58132a0418a80715f6b26a61dc341": "|y|", "bca58b6e88d85e0d8024c19ecb0daae7": "T=p_1\\cdots p_r", "bca5992e8c51d2b19fee2d2c81878e2e": "\\frac{dN}{N} = -\\lambda dt.", "bca5aec9edf7b5111c10ab572d1811f0": " (\\nabla^2 + k^2)\\mathbf{E} = 0,\\, \\mathbf{B} = -\\frac{i}{k} \\nabla \\times \\mathbf{E},", "bca5d9824a653048c4928a992a3e038a": "\\min_{x\\in\\mathbb R}\\; (x^2 + 1)", "bca683a4723d7c6815ebf42cb0d3c3ad": " S_{\\phi} ", "bca694d253d0810c9574d0ef15d85b52": " (g_m R_{\\text{S}} \\gg 1)", "bca7616fc459f9f8ca21656ab34ec98e": " 9\\leq n", "bca7663e4a6679bf4e683449913d4020": " u{\\partial u \\over \\partial x}+\\upsilon{\\partial u \\over \\partial y}={\\nu}{\\partial^2 u\\over \\partial y^2} ", "bca79a61d1b3d6d1e6bfbb9e1aca2a8a": "\n A_{ij} = A_{ij}^{(s)} + A_{ij}^{(a)}, \\qquad A_{ij}^{(s)} = \\left( A_{ij} + A_{ji} \\right)/2, \n \\qquad A_{ij}^{(a)} = \\left( A_{ij} - A_{ji} \\right)/2.\n", "bca81de698242d0f5c4a5d87ac1b57a6": "|c_n|\\geq (t-\\epsilon)^n", "bca8b9f63a2106f69c647e416a4658f2": "Ae^{i\\omega t + i\\beta\\sin(\\Omega t)} = Ae^{i\\omega t}\\left( J_0(\\beta) + \\sum_{k=1}^{\\infty}J_k(\\beta)e^{ik\\Omega t} + \\sum_{k=1}^{\\infty}(-1)^k J_k(\\beta)e^{-ik\\Omega t}\\right) , ", "bca8e3e6db7bb96d41a7f025708732ab": "X \\rarr f_1, \\, X \\rarr f_2 \\, \\ldots \\, X \\rarr f_n", "bca8f94ac631dc7ba8765ea9476097e9": "j^* = R \\left( \\frac{\\omega \\rho}{\\mu} \\right)^\\frac{1}{2}", "bca95bd8a25374a5d4553e56b8e4b450": "n= \\left\\lfloor \\frac{n}{2}\\right \\rfloor + \\left\\lceil\\frac{n}{2}\\right \\rceil.", "bca96548ba79e60bbc3fe3ab0b6cc94b": "\\int^{\\infty}_{-\\infty} \\varphi(s)/s\\,\\mathrm{d}s.", "bcaa2a173bdaf8a10598a6fec16dd165": "\\sqrt{9.2345}", "bcaa465bd04b837ad9737eabc5d22252": " L = \\chi^2 + \\alpha S ", "bcaa51a5ff271a3a824894a38b401f18": "t'=T_2", "bcaa59ec0e4fed3df2064564e6d3d959": "y^{(k)}\\quad\\quad(k = 1, 2, \\dots, n).", "bcaa8381151a356cf8209cfe6056c349": "\\mu_{\\rm lattice}", "bcaaa8f517fe21dff26b6c35166665b2": "f(x) + \\frac{h(x)^2}{4}", "bcaab054284fceadafaee5ef90df270b": "f(x) = b_2\\,(x-a_1)(x-a_2), \\!", "bcab194a78e203782846497342276023": "\\ell(\\gamma)", "bcab32c2c9bfa56a862761e6a6f9167c": "\\int_0^\\frac{\\pi}{2}\\sin^n{x}\\,dx = \\int_0^\\frac{\\pi}{2}\\cos^n{x}\\,dx = \\frac{2 \\cdot 4 \\cdot 6 \\cdot \\cdots \\cdot (n-1)}{3 \\cdot 5 \\cdot 7 \\cdot \\cdots \\cdot n}", "bcab6f6d9443a46de1cf6694c7ee1b21": "\\ f_1(x)= (x_1-x_2)^2 \\, ", "bcab79e46346557a8afb9aab2a1f913b": " \\min_y \\; E(x,y) + \\lambda V(y) ", "bcab9e216d144060a3b8d0a444ddda68": "1 - r / (2^r-1)", "bcabbd362f974801208fbb729de0afc4": "\\operatorname{csgn}(z) = \\frac{z}{\\sqrt{z^2}} = \\frac{\\sqrt{z^2}}{z}. ", "bcabd7569840f52d5dec09707050b860": "d_{ij}=d(\\{X_i\\}, \\{X_j\\}) = { \\|X_i - X_j\\|^2}.", "bcabdc681c06fd924ed0c867561b1ad1": " G(\\mathbf{k_\\perp}) = \\iint d^2 r~G(\\mathbf{r}_\\perp)e^{-i\\mathbf{k}_\\perp\\cdot\\mathbf{r}_\\perp}. ", "bcac33a6f8d6c2957821296250803de3": "v_i^j", "bcac97f5877fd269dae9bb236b77c2d5": " (\\mu_{X1}*, \\mu_{X2}*, \\mu_{Y1}*, \\mu_{Y2}*) = (1/2, 1/2, 1/2, 1/4)\\, ", "bcacdf7cd18956a7172bc0bd08f13dce": "\\eta = 1 - \\frac{q_C}{q_H} = 1 - \\frac{T_C}{T_H}", "bcad639e78e2079f66ad00ed44a1e978": " F = \\{ (x,y) \\mid W(x,y) > 0 \\}", "bcad6dedd3c7ce414de47773d0b2f0fe": "\\hat{U}(t_1,t_0) = 1 + {1 \\over i\\hbar}\\int_{t_0}^{t_1}\\hat{H}(t)dt + {1 \\over (i\\hbar)^2}\\int_{t_0}^{t_1}dt^\\prime\\int_{t_0}^{t^\\prime}dt^{\\prime\\prime}\\hat{H}(t^\\prime)\\hat{H}(t^{\\prime\\prime}) + \\ldots", "bcad85b0b28f0663ac0da7c9e7da2029": "P_\\alpha = \\frac{\\partial S}{\\partial x^\\alpha}", "bcadd70ee20d41564b9ac5d75a1f916e": "\\begin{align}\n\\Delta \\tilde{\\mu} & =(\\exp (\\Delta t\\cdot J)-I)J^{-1}\\dot{\\tilde{\\mu}} \\\\\nJ & =\\partial_{\\tilde{\\mu}} \\dot{\\tilde{\\mu}} =D-\\partial _{\\tilde{\\mu}\\tilde{\\mu}} F(\\tilde{s},\\tilde{\\mu})\n\\end{align}", "bcadd7a6d09397d07d66bf008355d128": "*(z)", "bcaddcd9a503f5fe39d4f8c4e5888206": "\\pi^{ij}", "bcadf8a467e0fb7efda919479ac5033e": " { 1 \\over P/E } \\ = \\ i ", "bcae0b62ad70fbbfbc69698d446611ac": "f_Y(y) = \\left| \\frac{d}{dy} (g^{-1}(y)) \\right| \\cdot f_X(g^{-1}(y)).", "bcae15cb74009d8c32f7e70dbea69640": "\\lim_{x_{i} \\to 0} \\partial f(x)/\\partial x_i =+\\infty", "bcae3862df19a6e4e637f4106681b5f6": "|0 \\rangle , |1 \\rangle ", "bcae6502020386760e56b17bf900437d": " \\left( \\frac{\\mathrm{d}S_{\\theta}}{\\mathrm{d}\\theta} \\right)^{2} + 2m U_{\\theta}(\\theta) + \\frac{\\Gamma_{\\phi}}{\\sin^{2}\\theta} = \\Gamma_{\\theta} ", "bcaefdfeac974cd593c4bd8a7b7c1fe6": "\\pi= 2\\times\\frac{2}{\\sqrt{2}}\\times\n\\frac{2}{\\sqrt{2+\\sqrt{2}}}\\times\n\\frac{2}{\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}\\times\\frac{2}{\\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}}\\times\\cdots", "bcaf1d084317663843ea1c6c147c6c63": "\\rho = \\delta\\left(\\theta - \\psi - \\arcsin\\left(\\frac{\\omega}{K r}\\right)\\right)", "bcaf504e02c523953e13e8567682620b": "z_{3,4} = -\\frac{1}{2} \\pm (-\\frac{3}{4} - c)^\\frac{1}{2}. \\,", "bcaf8a7922074e2315e45bb4d044d2cc": "\\{(t_1,t_1^{-a(1,2)}t_2,\\dots,t_1^{-a(1,i)}\\dots t_{i-1}^{-a(i-1,i)}t_i,\\dots, t_1^{-a(1,n)}\\dots t_{n-1}^{-a(n-1,n)}t_n, t_1^{-1}, \\dots, t_n^{-1}) : t_i \\in T, 1\\leq i\\leq n \\} < T^{2n}", "bcaf9adc715e202413ab30519c8e5270": "u_1,u_2,\\ldots,u_r", "bcaf9b2f40df6c793ffc6299da01abea": "a_{\\ell m}^{(E)}=\\frac{-ik^2}{\\sqrt{\\ell(\\ell+1)}} \\int d^3\\mathbf{x'} j_\\ell(kr') Y_{\\ell m}^*(\\theta', \\phi') \\left[-ik\\mathbf{\\nabla}\\cdot(\\mathbf{x'}\\times\\mathbf{M}(\\mathbf{x'}))-\\frac{i}{k}\\nabla^2(\\mathbf{x'}\\cdot\\mathbf{J}(\\mathbf{x'}))-\\frac{c\\partial}{r'\\partial r'}(r'^2\\rho(\\mathbf{x'}))\\right]", "bcaf9fc9d7c748f345aac27f87d9c424": "\\textstyle (E_{1}-E_{2})", "bcafea3556ed9d0b57a8a609b035aa7d": "x \\in (\\xi+\\alpha/\\kappa; +\\infty) \\text{ if } \\kappa<0", "bcb012494b2e45b2dcc62f6a0ec28d5c": "f(x + y) = f(x)f(y), \\,\\!", "bcb040110dbde5fc4d2c95424591e06f": "| \\mathbf p | = m c \\, \\sinh \\varphi ", "bcb0530f3acc96f91c315bba97cc7950": "k \\, t", "bcb0623f65460bcd9e2297b77acd8ede": "r = L \\sin \\theta \\,", "bcb0c2200e10fdd1ff9ed781242f6c54": " \\lambda f.f\\ ((x\\ f)\\ (x\\ f)) ", "bcb110af8328250ee2657a2c174bcd9d": " P(Y)", "bcb1552937fb353819b59171412b4bfc": "\\sum_{m=b+1}^{\\infty} \\frac{b}{m^2 - b^2} = \\frac{1}{2} H_{2b}", "bcb1e5d69ade173f16a72a335ea0f1d5": "v_2=v_1\\left(\\frac{\\ell_2}{\\ell_1}\\right)^3", "bcb205a241b4146ac080f49086591892": " TEE_f = 387 - (7.31 \\cdot age) + 1.14 \\cdot (10.9 \\cdot weight + 660.7 \\cdot height)", "bcb208eda3c20cf9c11a824868a53579": "(1-\\beta(n_i, \\tilde{n}_i))\\frac{w_i}{n_i} + \\beta(n_i, \\tilde{n}_i)\\frac{\\tilde{w}_i}{\\tilde{n}_i} + c\\sqrt{\\frac{\\ln t}{n_i}}", "bcb23a05189ca6a8087b5c0424ce7ab1": "\\tau(n)\\equiv 1217 \\sigma_{11}(n)\\ \\bmod\\ 2^{13}\\text{ for } n\\equiv 3\\ \\bmod\\ 8", "bcb266e899a44f2088286560046a55ec": "\n\\tan 2\\theta = \\frac{2q_{xy}}{q_{xx}-q_{yy}}\n", "bcb316079e347ad0a46c5ef9fdce919d": "I(\\rho, \\mathcal{N})\\ \\stackrel{\\mathrm{def}}{=}\\ S(\\mathcal{N} \\rho) - S(\\mathcal{N},\\rho)", "bcb3197149eb7730839be24919520f1c": "{\\Delta h_m} = \\frac{V_{r2}^2 - V_{r1}^2}{2}", "bcb3403d7c62d324fd1797f959f4178f": "V_\\mathrm{swap} = B_\\mathrm{domestic} - S_0 B_\\mathrm{foreign}", "bcb370c359c2e87e40757e80ba507ba8": " \\mathcal{C}_{YY}^\\pi = \\mathbb{E}_Y [\\phi(Y) \\otimes \\phi(Y)] ", "bcb387e456bba5813ec52564a62d70d2": "n > u", "bcb38eb1a3895571c2bdd23cb1ecdc55": "\\begin{vmatrix}\nA_x & A_y & A_x^2 + A_y^2 & 1\\\\\nB_x & B_y & B_x^2 + B_y^2 & 1\\\\\nC_x & C_y & C_x^2 + C_y^2 & 1\\\\\nD_x & D_y & D_x^2 + D_y^2 & 1\n\\end{vmatrix} = \\begin{vmatrix}\nA_x - D_x & A_y - D_y & (A_x^2 - D_x^2) + (A_y^2 - D_y^2) \\\\\nB_x - D_x & B_y - D_y & (B_x^2 - D_x^2) + (B_y^2 - D_y^2) \\\\\nC_x - D_x & C_y - D_y & (C_x^2 - D_x^2) + (C_y^2 - D_y^2)\n\\end{vmatrix} > 0\n", "bcb3ce3101b7faeacb61014a0f2e1807": "s(x) = \\sin^2(x)", "bcb3cf322e2ecf491e5c8fcd2bd66386": "a_{14}+b_{15}+c_{13}", "bcb406a82b873292b902980cfbe6d496": "R^{-1}\\ ;\\ \\overset{\\alpha}{\\rightarrow}\\quad {\\subseteq}\\quad \\overset{\\alpha}{\\rightarrow}\\ ;\\ R^{-1}", "bcb40bfbf0a26d2f7f4f961c13a75cde": " \\mathbf{\\ddot{n}}", "bcb4ac5418dc96b2920a3692433d0424": "Yield=\\frac{mTBV}{mREFILLING}", "bcb4d1c89410acc1fafa3258fbaa28d7": "(1-x^2){d^2 f \\over dx^2} -2x {df \\over dx } + n(n+1)f = 0.", "bcb561269867d9487e18e111af013f51": "\\big(\\forall b\\in B\\setminus \\{0\\}\\big)\\big(\\exists a\\in A\\big)\\big(a\\leq b\\big)\\big\\}", "bcb5a71d98c357d4c0c703b90b39ba9d": "\\Lambda_{TC} \\ll F_{EW}", "bcb5c56fdb85fc4ce67ef170b8f625b5": "a \\prod_{i=1}^n x_i,\\, b + n,\\, c + n\\!", "bcb5ee900c93bdbd38299834c30cdb04": " TE ", "bcb64d191dc658c5b012b5bd140e21de": "D+C(sI-A)^{-1}B", "bcb674421ee89025768a522da98323fe": "r \\leq s", "bcb70b306359b869c26fec7096d4a27b": "\\upsilon(I)=1", "bcb7124ee162b402a6e67ecf71c82f90": "V'_y=\\frac{ V_y }{ \\gamma \\left ( 1 - \\frac{V_x v}{c^2} \\right ) }", "bcb78214f183b68635b51525df88dbf0": "\\psi_{6,7}=6", "bcb7d90ff075c9e9270326b558f7fd62": "\\psi^{jk}", "bcb7f071b02c7bc22e8c2a6c2f6d9b4a": "e^{i \\phi}+c.c.", "bcb7f294e0067cee37c00a71b3a154a0": "T_F = \\frac{1}{8 \\pi^2}\\frac{N_A^2h^2}{MR}\\left( \\frac{3\\pi^2N_A}{V_m}\\right)^{2/3}. ", "bcb80b9025f70a90569628905965e2fc": "\n\\begin{align}\n& {} \\quad \\left(1-\\frac{1}{2}\\right)-\\frac{1}{4}+\\left(\\frac{1}{3}-\\frac{1}{6}\\right)-\\frac{1}{8}+\\left(\\frac{1}{5}-\\frac{1}{10}\\right)-\\frac{1}{12}+\\cdots \\\\[8pt]\n& = \\frac{1}{2}-\\frac{1}{4}+\\frac{1}{6}-\\frac{1}{8}+\\frac{1}{10}-\\frac{1}{12}+\\cdots \\\\[8pt]\n& = \\frac{1}{2}\\left(1-\\frac{1}{2}+\\frac{1}{3}-\\frac{1}{4}+\\frac{1}{5}-\\frac{1}{6}+\\cdots\\right)= \\frac{1}{2} \\ln(2).\n\\end{align}\n", "bcb891853a0cb755e95e53a65cdffe4c": "\n\\Pr(n_1,n_2,\\ldots,n_S| \\theta, J)=\n\\frac{J!\\theta^S}\n{\n 1^{\\phi_1}2^{\\phi_2}\\cdots J^{\\phi_J}\n \\phi_1!\\phi_2!\\cdots\\phi_J!\n \\Pi_{k=1}^J(\\theta+k-1)\n}\n", "bcb8b3f1d2ca33c28d189c4bacf3aa1c": "(n^4 + n^2)/2", "bcb8bf7f82ecd5610271353f249620f9": "u = v \\circ w", "bcb96febd8881d1aa209c9d1bf6123d8": "P=(X:Y:Z:Z^2)", "bcb9cd07da3b7b3253347af8869cd78f": "\\displaystyle{A={1\\over\\sqrt{2}}Y={1\\over\\sqrt{2}}\\left(-{d\\over dx} +x\\right),\\,\\,\\,A^*={1\\over\\sqrt{2}}X={1\\over\\sqrt{2}}\\left({d\\over dx} +x\\right).}", "bcba34aae2c2dba1e298fa3ae5b5ef4f": "\n \\Delta x_{\\mathrm{meas}} = \\frac{\\Delta\\phi}{2 k_p} \\,.\n", "bcbaa02a2e886c3a96fdd25d74201a6d": "kT_0 = 4.00 \\times 10^{-21}\\,[Ws]", "bcbab0d6e2b8a8c560b143b4f3990c93": "H_{\\aleph_2}", "bcbb10d03bd0006d1f35a126960ab499": "\\vec{f}_1 = \\partial_r ", "bcbb222c164071f6e93a53c176ba6178": " L = (( \\Sigma \\cup { \\theta } ) \\times \\Gamma) \\cup (\\Sigma \\times ( \\Gamma \\cup { \\theta } )) ", "bcbbbf52cc654457a84c5b7d6685a149": "f^!,", "bcbbc268ed86ba66f6e1f2188617e08b": "\\mathbf{E} = \\sum_{i=1}^N \\mathbf{E}_i = \\frac{1}{4\\pi\\varepsilon_0} \\sum_{i=1}^N \\frac{Q_i}{r_i^2} \\mathbf{\\hat{r}}_i. ", "bcbbf0316b9f6a75864a35f4dc85aae9": "\n\\begin{bmatrix} \\mathbf{A} & \\mathbf{B} \\\\ \\mathbf{C} & \\mathbf{D} \\end{bmatrix}^{-1} = \\begin{bmatrix} (\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1} & -(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1}\\mathbf{BD}^{-1} \\\\ -\\mathbf{D}^{-1}\\mathbf{C}(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1} & \\quad \\mathbf{D}^{-1}+\\mathbf{D}^{-1}\\mathbf{C}(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1}\\mathbf{BD}^{-1}\\end{bmatrix}.\n", "bcbca0214006afb96120a0975d3e5056": "P\\left [ {Z}_i^r \\right ] \\le e^{(-\\beta n \\log n ) / n} = n^{-\\beta}", "bcbcaa9f681a5d4c101689424c4d038d": "\\frac{b-a}{4}", "bcbcc5b1de5c42e7ee3349b34fcf23ae": "\nz\\,\\,\\, = \\,\\,\\,f\\left( {x_1 \\,\\,\\,x_2 \\,\\,\\,x_3 \\,\\,...\\,\\,\\,x_p } \\right)", "bcbd000f380aaff405b00ae8422be924": "Df \\;", "bcbd2b4f60deaeae6ac539ee1b9b3364": "\\sum F_x=-F_{CD}=-0=0 \\Rightarrow verified", "bcbd9e03c91245dfecc35f3336e92377": "\\frac {1} {\\theta}", "bcbda6ba123832a052e6aa6c561230bc": " I_{PPT}=(2(2E_i)^{\\frac{3}{2}}/F)^{n^{*2}} ", "bcbe236e2a511d2fa7829c5bb3eb7e5b": "r^2=b^2-(a^2\\sin^2\\theta\\ + c^2-a^2\\cos^2\\theta \\pm 2a\\sin\\theta\\sqrt{c^2-a^2\\cos^2\\theta}),\\,", "bcbe4438d5ef70feb360e3aac3ec752a": "\\mathbf{k}\\times\\tilde{\\mathbf{E}}=\\omega\\tilde{\\mathbf{B}}", "bcbe8fc1f612272091981ceb3c69db05": "\\sum_{k=0}^{\\lfloor\\epsilon n\\rfloor} {n \\choose k} \\leq 2^{H(\\epsilon) \\cdot n},", "bcbecc32aa767cbd35756086eaf3fffa": "T = \\frac{1}{4} \\sqrt{2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}", "bcbedc1f676303589d8c4252a2b72d79": " \\gamma_m = \\sum_{i=1}^p \\varphi_i \\gamma_{m-i} + \\sigma_\\varepsilon^2\\delta_m ", "bcbf3a3eb23782d41a9a1bda583050f2": " E_h = \\alpha^2 m_e c^2 \\,", "bcbf3b9dc614281fdb19a163119e0b31": "x \\leq x \\land y \\leq x \\land", "bcbf3be76227db979dbac07aa75d0afe": "=(\\vec{N} \\cdot (\\{ 0.133; \\; 0.65; \\; 0 \\}))^3=(0\\cdot 0.133 + 1\\cdot 0.65 + 0 )^3=0.65^3=0.274625.", "bcbfa0ea91b8455ab4a90c16a36cc0b9": "(t-a_1)(t-a_2)(t-a_3)\\cdots.\\,", "bcbff1bd77712373ebe8811bbf3a8b8a": "L\\left(n\\right)=F\\left(n-1\\right)+F\\left(n+1\\right).\\,", "bcc000a1ddebbfc0ba140ca290222b4c": "\nE_W [\\rho] = \\int dr \\rho \\hbar^2 [\\nabla (\\ln \\rho)]^2 / 8m = (\\hbar^2 / 8m) \\int dr (\\nabla \\rho)^2 / \\rho = \\int dr \\rho \\, Q\n", "bcc0498f553f75e44a5b58710331ffd1": "H^n (X, F) = \\varinjlim_{K_*} H^n(F(K_*)),", "bcc04f06bdfab4d40561757e2dc04b2d": " \\theta_s = \\frac {\\theta_a + \\theta_b + \\theta_c}{2} ", "bcc05fb3e9ab0ef56c071860be59ec77": "f(n)\\sim e^{-n/n_0}", "bcc0610b7d5cc08ec2270dcadf102319": "\n \\forall A\\in {\\mathcal A}\\qquad \\forall\\lambda\\in{\\mathbb F}\\qquad \\lambda\\cdot A\\in {\\mathcal A},\n ", "bcc068b66fe1930e7f3c9fb5e1c51597": "z = 1", "bcc0b7f83f4aebb3e519581cfb7f5a22": "\\sigma_{xz}", "bcc127cdb75514816603fe7e0bc5d8c1": " \\sigma \\in S_n ", "bcc146f673783f7045b159b41b4fa7a1": "\\|\\Psi_nx\\|\\geq k_n\\|x\\|", "bcc14f1fe55db0f62485521bb4efb19e": "\\delta^2", "bcc19c67fd0b99acf0be5b2319ac367b": "R_{eq} = \\frac{1}{fC_s}", "bcc1d46a50a92f30674bee2846e108e5": " f'(\\infty)\\neq \\lim_{z\\to\\infty} f'(z) ", "bcc21004617f492e4bd02aeb15530605": "\\int_a^b K(x,y) \\phi(y) \\,dy = \\lambda \\phi(x)", "bcc228ceda745ac530ef7aa443104658": " F_1 = \\frac{q^2}{4 \\pi \\epsilon_0 L_1^2}", "bcc27c35b938075beae36044ab481818": " \\frac{b_k^\\top A b_k}{b_k^\\top b_k} = \\frac{b_{k+1}^\\top b_k}{b_k^\\top b_k}. ", "bcc299b8101bd5245028b5d821d8a92b": "even(s(X))\\leftarrow \\hbox{not } even(X)", "bcc2aa107a151656e7f6703735ab866c": "\nK_n = s\\lambda + \\alpha_n x - \\sigma t = 0.\n", "bcc2d525cb4e5a3142c2431d054218e9": "w^T \\Sigma w", "bcc2e7a6ed1080cf267ca6bf53321c68": " 0\\leq p \\leq 1 ", "bcc38f953cfa0b216f4ed3aa418306eb": "P_{a_j}=f_0 f_1 \\cdots f_j. \\, ", "bcc3bceb26bb8501c59f7252839ba9d8": " 2 \\log (-x) = \\log ((-x)^2) = \\log (x^2) = 2 \\log (x) ", "bcc41342fa2d1ba887adc9f9c161b163": "f(\\lambda x + (1 - \\lambda)y) \\leq \\lambda f(x) + (1-\\lambda) f(y)", "bcc449e4a4d25c399c4113f158198ba0": "\\begin{align}\nP(A\\cap B) & = P(A|B)P(B) = P(B|A)P(A)\\\\\nP(A\\cap B) & = P(A)P(B) \\qquad\\mbox{if A and B are independent}\\\\\n\\end{align}", "bcc44acebec7a40a22cf42361ee783a4": "\nD\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n-{1 \\over \\vec k^2 + m^2}\n", "bcc48a1d7a7c1d7a8fd7ee0e0a458a54": "\\begin{align}\nK_1&=111.13209-0.56605\\cos(2\\phi_m)+0.00120\\cos(4\\phi_m);\\\\\nK_2&=111.41513\\cos(\\phi_m)-0.09455\\cos(3\\phi_m)+0.00012\\cos(5\\phi_m).\\end{align}\\,\\!", "bcc4c2946dde03546d09caea0345f30d": "\\lang \\psi_1|\\psi_2\\rang", "bcc4e95887936b915e769c0e1c2d1a1b": "f_{y} ", "bcc5370abad6a8f063ba5131948d1a30": "4{\\pi}^{2}{\\nu}_i^{2}", "bcc55c40efc8ca0b2b061fccf4f8b831": "g(x) = 3 \\left(2 + x^2\\right)^{-5/2}, \\!", "bcc60e8361f18d1db7ad9f197114e6ea": "E(\\vec{r})=E_{\\vec{r}}", "bcc625a2d7d30ab71a187cdbd7b970be": "\\sigma_B-\\sigma'_B", "bcc63a61119d3928e0457034f077bc20": "P_{m+1,\\mu+1}=P_{m,\\mu}+P_{m+1,\\mu},", "bcc64cfb3e300a9b59abed68a481fc02": "\\det(V(\\lambda_3,\\ldots,\\lambda_n))=\\prod_{4\\le j\\le n}(\\lambda_j-\\lambda_3)^3\\prod_{4\\le i\\frac{\\alpha}{m_{0}},\\forall i\\in I_{0}\\right\\}", "bcd2f538774585f1c084b5cde96142c6": "\\delta ( f \\nabla g) = - \\nabla f \\cdot \\nabla g - f L g.", "bcd31318629828d0fd10b0f3f62c2387": " \\sum_{r=0}^{k-1} |\\rho(x_{r+1}) - \\rho(x_r)|", "bcd3805a4c70e2a9f7e03ccd5346373c": "x \\in V\\cup\\Sigma", "bcd399cdc2129775c8abacf6c4b7e0b5": "dA = \\frac{4}{(1 + X^2 + Y^2)^2} \\; dX \\; dY.", "bcd3b6388adb4066bbdc4aa0def15de5": "\\Phi^X_t", "bcd452adbd609a7a0f02aeb21b8c1e02": "W(A)", "bcd47188603676ec49db2f4d83c5ae99": "\\hat{a}=\\bar{y}-\\hat{b}\\bar{x}", "bcd471b00d0811f9c486a14983ca686c": "q>p", "bcd475909f5ea43b8a3a863d76b47fb0": "PG(F)\\subset PG(F^2)\\subset PG(F^4)\\subset PG(F^8)\\cdots", "bcd4cf81d4c6b19d179362be3a64a82f": "\n\\vec{X_1}=\\vec{X_{\\alpha}}-\\vec{a_1}.(\\vec{D_{\\alpha}})\n", "bcd5036734178323b14fc400abb57796": "u_t = \\alpha u_{xx}", "bcd51b3a35691db9e3050a99bf1f5756": "I_{gt}", "bcd5396b5cdb97494a37d4bc99aaca2d": "\n\\sigma_t^2 = \\frac{1}{E}\\int |t-u|^2|\\psi(t)|^2dt\n", "bcd58eddcec78f940f9012263ccdb16c": " \\lim_{T\\to\\infty} \\frac{1}{T} \\underset{free-action} {\\underbrace{\\int_0^T F(s(t),\\mu (t))dt}} \\ge\n\\lim_{T\\to\\infty} \\frac{1}{T} \\int_0^T \\underset{surprise}{\\underbrace{-\\log p(s(t)|m)}} dt = H[p(s|m)] ", "bcd5937de0ffaf5c44095ac8d3e6368e": "q_d", "bcd5c16ceda8a689e223d741f045f3ef": "O(M)", "bcd5e55af524202eb49f91259357eb36": "Jf = y", "bcd61d66149607a4ba4b1c2af97db3cb": " H^d(S):=\\sup_{\\delta>0} H^d_\\delta(S)=\\lim_{\\delta\\to 0}H^d_\\delta(S).", "bcd67bdb3b262e737c243930d59c3455": " \\sigma_d = i C \\omega ", "bcd691169782d5401abe8a4f36bc7484": "\\Delta L = \\frac{F}{E A} L", "bcd691dc2d5b500f0c86879ac531ca2e": "\\widehat{\\sigma}^2=\\frac{\\sum \\left(X_i-\\bar{X}\\right)^2}{n-1},\\,", "bcd6941ab2be888375a1bbcb79dbcca6": "\\sum\\limits_{n=1}^\\infty {(-1)^{n+1} \\over n} = 1 - {1 \\over 2} + {1 \\over 3} - {1 \\over 4} + {1 \\over 5} - \\cdots", "bcd6e43a9ed7a9a86a06afdf9a2bd512": "\\scriptstyle f_s/2.", "bcd728235988ab80a5e5caf8ef1ddbb6": "\\mathcal{S}_{i0}", "bcd7a55e0f843d3a84797b8288d5ddc5": "C_i = {2e_i\\over k_i{(k_i - 1)}}\\,,", "bcd7bbaae58d0751844462f2d7b75fdb": "\nx = [a_0; a_1, a_2, a_3, \\dots] = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3 + \\cfrac{1}{\\ddots}}}},\n", "bcd80215b2c368081508d57a04e89f0b": " a=12^2-1^2 =143 \\,,\\ b = 24 \\,,\\ c = (12^2 + 1^2)=145", "bcd8b0c2eb1fce714eab6cef0d771acc": "()", "bcd8c18ec733d2835ac03d839738b225": "\\frac{\\partial A(c)}{\\partial c} = -\\frac{u'(c)u'''(c) - [u''(c)]^2}{[u'(c)]^2} < 0", "bcd9041349934b16130e5d29490cb2dc": "P_j[i]=p_{ij}-p_{ik}", "bcd910e539450fd304aa201f0bb78dde": "\nv(tI + ds)=f(t\\mu(I))+f'(t\\mu(I))\\mu(ds)\n.", "bcd976de311ee902b3fa85399c29fd16": "\\langle V,\\cap,\\cup,-,\\Box\\rangle", "bcd99b53532f6446b360b8fb2b86864b": " \nP(E,\\Omega) = V\\left(\\chi_E,\\Omega\\right):=\\sup\\left\\{\\int_\\Omega \\chi_E(x) \\mathrm{div}\\boldsymbol{\\phi}(x) \\, \\mathrm{d}x\\colon \\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n),\\ \\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\\le 1\\right\\}\n", "bcd9cda32974706ca010c76904be5197": " \\pi({ab\\choose 01})f(x) = \\tau(bx)f(ax). ", "bcda10a70a47f357c59f5e74f7a7f372": "\\operatorname{dCov}^2(X,Y)", "bcda1c9d5d9484f4527e949826ada94d": "X \\sim NC\\chi_k(0)", "bcda1cde590d1cab7a58543b7796d8b6": "-{\\mathrm{ord}}_P (G)", "bcda36ff87925e9e56ca078e68cee4dc": "\\textbf{x}_{i}", "bcda3d91b803292f63f09e0cdf16dfd9": "\\begin{align} & Pr(y_i=1) = 1\\text{ if }\\operatorname{sign}(u_iv_i)=1\\, \\\\\n& Pr(y_i=-1) = 1\\text{ if }\\operatorname{sign}(u_iv_i)=-1\\,\n\\end{align}", "bcdab4b7d57f101af5e6a81bf476f457": "\\delta_{\\lambda\\mu}", "bcdaea39c18ac89691ced2247a6d9c14": "e_{ij} v = \\frac1{2} ( \\partial_{i} v^{j} + \\partial_{j} v^{i} ).", "bcdb091b660b612f72f7e9a48c8ed3e9": "p_{\\mathrm{sat}} =", "bcdb45ba812dc7bcbf352f062c81b9f5": "\\left|1,0,0\\right\\rangle", "bcdbb4e3002fef7ac7555c59f2186c48": " p=a\\cos\\beta, \\qquad z=b\\sin\\beta; ", "bcdc098ade4e290c363bae972eee9925": "\nX_i=(x_{i},\\ldots,x_{i+L-1})^\\mathrm{T} \\; \\quad (1\\leq i\\leq K)\n", "bcdc6f4ad46817cf7ec21c9a52eca378": " r(x) = R_{max}[4x(1-x)]^{3/4} ", "bcdc7b73b2d2c60b10c1feead2469957": "\\tilde{g} = F^T g F ", "bcdc97eebf7de1d04b88a079ac859f4a": " \\mathbf{f} = \\rho\\mathbf{E}+\\mathbf{J}\\times\\mathbf{B} \\ \\rightarrow \\ \\int_V \\mathbf{f} dV = \\mathbf{F} = \\int_V (\\rho\\mathbf{E}+\\mathbf{J}\\times\\mathbf{B} )dV ,", "bcdcc22cc8aef5c12667f84eaf15668a": "\\equiv 10 \\pmod{84}", "bcdd098f25c4f86c7caa11d3dd7b258d": "\\sin \\alpha = \\frac{4}{A^2}\\,", "bcdd4c0456c6b800958afc0b4f9c3d76": "\\{ z \\in {\\mathbb{C}} \\mid a + b z \\in G \\}.", "bcdd5b4f31db9578f9ff28edaaf74da0": "r_2=\\alpha - \\beta", "bcdd824109921bdbda929506401a3de4": "C_H^d(S)", "bcdd856c3cc9ab7e4b520d41ae9e21e7": "q(\\xi) = (\\xi-x_1)\\cdot \\dots \\cdot(\\xi-x_n)", "bcdd90cc86ab66383999cfaa25bc0872": "16 - 6\\;", "bcdde35e703bbf839ad4ecb927674063": "A=\\sum_{i\\in I} a_i=-\\sum_{j\\in J} a_j.", "bcddedfc512b62576e1fec49fc5a641a": "\\operatorname{E}(e^{t X})=\\sum_{n=0}^\\infty \\frac{t^n}{n!}\\operatorname{E}(X^n)", "bcde283ff0ea2852104722d0001b196f": " d_0(m\\otimes a_1 \\otimes \\cdots \\otimes a_n) = ma_1 \\otimes a_2 \\cdots \\otimes a_n ", "bcde49ea86fc665329977d51e6a52068": "\\rho = \\left | \\Psi \\right |^2 = \\Psi^* \\Psi ", "bcde4c72532b7fdf71809d269585114f": " L(\\alpha,\\beta,\\gamma) \\leftrightarrow P(\\alpha,\\beta) \\land I(\\beta,\\gamma) ", "bcde56a327ae5162a036acd1fd94dcd7": "{}\\qquad\n\\left(\\begin{array}{rrc}\n0 & -1 & 2 \\\\\n-2 & 3 & 1 \\\\\n1 & 0 & 2\n\\end{array}\\right)\n\\quad\n\\begin{array}{c}\n\\stackrel{\\eta}{\\longmapsto}\\\\\n\\stackrel{\\varrho}{\\longleftarrow\\!\\!{}^{{}_{\\!{}_\\mathsf{l}}}}\n\\end{array}\n\\quad\n\\left\\{\n\\begin{array}{rcrccr}\nx'_0 & = & & -x_1 &+& 2x_2 \\\\\nx'_1 & = &-2x_0&+3x_1&+& x_2\\\\\nx'_2 & = & x_0 & & +&2x_2\n\\end{array}\\right.", "bcde579718db782b7c4dcf08c197aba0": "\\mathcal{L}_2 = \\mathcal{L}(\\Alpha, \\Omega, \\Zeta, \\Iota)", "bcde655710f0abb657af22f5629872a7": " \\rho(t')\\rightarrow \\rho(t)", "bcde97762bc91e41f0d2563c2a09fe81": "E \\approx m_0c^2 + \\frac{1}{2}mu_0^2 \\,,", "bcded2b63f3093d26a7df346919e1131": "I[{{f}_{1}},{{f}_{2}}]={{P}_{V}}[{{f}_{1}},{{f}_{2}}]+{{P}_{V}}[{{f}_{2}},{{f}_{1}}]", "bcdee530286a4f606771b37d1c42070d": "\\exists x (x = y)", "bcdf3c17ac3e3a0fc7f076a4835e09c9": "\\eta = 1/(1-2a/g)", "bcdfdc3f8f46e6f068471ba809eb5c92": "\\hat{l} = (\\hat{x}+i\\hat{y})/\\sqrt{2}", "bce02343af40ab17c04110085e31f2d7": " \\partial_t |f_t(z)|^2 =2\\Re\\, \\overline{f_t(z)} \\partial_t f_t(z) = 2 \\Re\\, \\overline{w} v(w),", "bce03186f0c68c4bf4aeb7545f37ee48": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda}e^{i\\pi\\tau\\|x\\|^2}\\qquad\\mathrm{Im}\\,\\tau > 0.", "bce05dfd8fb486dd98f2f139cb15495f": "(G_2, P_2) = (G_1 \\or G_2 P_1, P_2 P_1) ", "bce08a8bec955b47528a27539215ef9b": "a^{1+n}=a a^n", "bce0bb4bfe72fa4ff9c5120bc738795b": " F = /^2\n", "bce0c263e559598999b8f9a74c5a11f6": "\\,{}^{n}(a, x)", "bce0d270638fcee0eed73554b9c4bb2e": "M' = C U^2 (\\theta + \\alpha_0) ", "bce0f248eceed72f357aa8695f4b2739": "L_\\mathrm{e} = \\frac{c}{4\\pi} w.", "bce10be70680dac8f823c023594d4803": "\\omega_0 = \\gamma \\mu_0 H_0 \\ ", "bce10dd4d9f4c50b21a5642cb106bbd7": "\\dot \\epsilon_{sh}", "bce1251c44dea90a422c2a73f9312c8a": "X\\subset\\mathbb R", "bce12d79d4b71ead792175cc1f7261c2": "A_t = \\{X \\in L^{\\infty}_T: \\rho(X) \\leq 0 \\text{ a.s.}\\}", "bce19ad78b8d3ef0c4aaa06de6c3c0ea": "s-t", "bce1cc2d959256d75e3854f54c195c36": "X_u \\perp\\!\\!\\!\\perp X_v | X_{V \\setminus \\{u,v\\}} \\quad \\text{if } \\{u,v\\} \\notin E.", "bce20f17ce36aaafcb57a5f8b0514d8e": " u(e^{i\\theta}) = \\frac{a_0}{2} + \\sum_{k \\ge 1} a_k \\cos(k \\theta) + b_k \\sin(k \\theta) \\longrightarrow v(e^{i\\theta}) = \\sum_{k \\ge 1} a_k \\sin(k \\theta) - b_k \\cos(k \\theta). ", "bce2591a703c1a115aa37658e3fafb33": "l(t,s)=u(t,s)=m", "bce2929b6f41cc656173cc4af317b649": "\\scriptstyle \\mu=4\\pi \\times 10^{-7} ", "bce2ab55e47614c3257e6d70d1e88099": "r(u) - r(v)", "bce2e62f9b6f9dcbc5f68338c6dac671": "\\Delta p_{2} = \\sqrt{\\frac{\\sigma^2(1+\\epsilon^2/\\Omega^2)+\n \\hbar^2/16\\Omega^2}{1+4\\epsilon^2(\\sigma^2/\\hbar^2+1/16\\Omega^2)}}.", "bce3943be36b04aea7b989cb0b78d92c": "J_X", "bce3ca9fc218918cacd51abc56d295eb": "r(\\theta)", "bce409f9e2283fad368d2d345fb340fc": " f(K,L) = \\lim_{N \\rightarrow \\infty} f_N(K,L) = -kT \\lim_{N\\rightarrow \\infty} \\frac{1}{N} \\log Z_N(K,L) ", "bce4934c92a9b675bf55222455985d66": " \\tan U_1 = (1 - f)\\tan \\phi_1 \\, ", "bce51ecbacf481280497d6c5cae158cd": "\\displaystyle{E(z) =-{1\\over 2\\pi} \\log |z|.}", "bce55d68812b80914bcf077bbe55170c": "A \\rightarrow B: A, \\{T_A, A\\}K_{AB}, \\{L, A, P'_A\\}S_A, \\{\\{K_{AB}\\}P_B\\}S'_A", "bce569ada457cf9977152e10e8034f50": "\\hat{\\textbf{x}}_{k \\mid n}", "bce596646bc4c506ed3760fd1b86e028": "\\omega_2=\\tau", "bce5a32882a06453c21697f71717a40d": "n_\\Sigma^2 \\cos\\theta_\\Sigma d \\Omega_\\Sigma=n_S^2 \\cos\\theta_S d \\Omega_S", "bce5d8a9b32ec72c0f4ebb5f2db958c4": "\n\\begin{align}\n\\mu & \\sim \\mathcal{N}(\\mu_0, (\\lambda_0 \\tau)^{-1}) \\\\\n\\tau & \\sim \\operatorname{Gamma}(a_0, b_0) \\\\\n\\{x_1, \\dots, x_N\\} & \\sim \\mathcal{N}(\\mu, \\tau^{-1}) \\\\\nN &= \\text{number of data points}\n\\end{align}\n", "bce6824562e2254f3d5f331e5547f07b": "(Ff)a = b", "bce72578767922eaf8cc2a4dc390b143": "NPV = PV_{1}+PV_{2}+PV_{3} = \\frac{100}{(1.05)^{1}} + \\frac{-50}{(1.05)^{2}} + \\frac{35}{(1.05)^{3}} = 95.24 - 45.35 + 30.23 = 80.12, ", "bce7348acd2b5a8c2c71599252b0ada7": "\\frac{4\\pi emk^2}{h^3} \\approx ", "bce73ac63af509f2043cc7490660b1e4": "\\mathbf x\\cdot\\mathbf U \\cdot \\mathbf x\\ge 0 \\,\\!", "bce772c43922c4bb71cd6bfeb893179e": "x^\\mu = (x^1,x^2,\\cdots)", "bce7cf9d1fdbf8a5b8abcdafa3a7b671": "r(t) = e^{s t}", "bce833798a9c0b5191d6088f77c675a3": " \\int_0^{R_g} N(r) 4 \\pi r^2\\,dr + \\int_{R_g}^\\infty N(r) 4 \\pi r^2 \\,dr,", "bce83ab90d8d6309afb816d62a2be762": "T_x \\mu = P", "bce8c25241244464efc6a6364f0a1a09": "R\\in\\mathbb{R}^{d\\times d}", "bce8c60547c71aa66683418d2cd8ee15": "k_0=2", "bce8ca5d66de267a610b7b659491452f": "\\begin{align}\n\\boldsymbol{P}_i^\\mathrm{T}\\boldsymbol{AP}_i&=\\boldsymbol{U}_i^{-\\mathrm{T}}\\boldsymbol{V}_i^\\mathrm{T}\\boldsymbol{AV}_i\\boldsymbol{U}_i^{-1}\\\\\n&=\\boldsymbol{U}_i^{-\\mathrm{T}}\\boldsymbol{H}_i\\boldsymbol{U}_i^{-1}\\\\\n&=\\boldsymbol{U}_i^{-\\mathrm{T}}\\boldsymbol{L}_i\\boldsymbol{U}_i\\boldsymbol{U}_i^{-1}\\\\\n&=\\boldsymbol{U}_i^{-\\mathrm{T}}\\boldsymbol{L}_i\n\\end{align}", "bce954dc4b061daf857c7801f8c03d08": " ds^2 = -dT^2 + \\left( dr + \\sqrt{2m/r} \\, dT \\right)^2 + r^2 \\left( d\\theta^2 + \\sin(\\theta)^2 \\, d\\phi^2 \\right) ", "bce9974ce956b43e5d08505c7980f252": "x' + y'", "bce9b1b27a894cf6d210a94160f932f1": "N_1 \\setminus V \\cong N_{M_1} V \\setminus V \\to N_{M_2} V \\setminus V \\cong N_2 \\setminus V,", "bce9b7bf9c7a34c973b10aaf8762d973": "c = \\frac 1 {\\sqrt{(L/C) \\cdot C^2}}", "bce9cdbc2db46ad3077705e9aa7a3fd8": "\\int_{M} d(x, x_{0})^{p} \\, \\mathrm{d} \\mu (x) < +\\infty.", "bcea0d218379849440499338a155f525": "E_0^{p,q} = \\frac{Z_0^{p,q}}{B_0^{p,q} + Z_{-1}^{p+1,q-1}} = \\frac{F^p C^{p+q}}{F^{p+1} C^{p+q}}", "bcea17d83a7cfc41bd7134d753bfc051": "\\psi\\left(x,z,t\\right)=e^{ik\\left(x-ct\\right)}\\Psi\\left(z\\right),\\,", "bceaa74a66e006b87e97fab1929b8a0c": "G \\times X \\to X,\\ \\ (g,x)\\mapsto g.x", "bceabac53495b74029d32265dc429e16": "\\frac{\\partial N_2}{\\partial t} = \n-\\frac{\\partial N_1}{\\partial t} =\n - B_{21} \\ \\rho (\\nu) N_2 ", "bceac2d8a1b80f8ed4256e1bc6cf5413": "p= \\sum_{i\\in I}\\frac{a_i}{A} x_i=\\sum_{j\\in J}\\frac{-a_j}{A}x_j,", "bceb01d12447ebdbff64b56e15c730dd": "A(\\omega)=\\left|{\\frac{V_o}{V_i}}\\right|", "bceb64350855d3f86ce7eb275502472e": "\\Big [ \\big[\\mbox{un-}\\big] \\big[\\mbox{easi}\\big] \\Big ] \\Big[\\mbox{-er}\\Big] ", "bcebb51378c578704e052209aed7caee": "a_{j} = e^{-i\\pi \\sum_{k=1}^{j-1}f^{\\dagger}_k f_k} f_j", "bcebe5adc40058198e140aa576d48c09": "E_i =", "bcec0a7910250155d820a827aa82e80a": " r\\frac{\\partial}{\\partial r}\\left(r\\frac{\\partial u}{\\partial r}\\right) + \\frac{\\partial^2 u}{\\partial \\theta^2} = 0", "bcec2223190838bfafbb26fb658ea0df": "+(1-x)", "bcec4b14faeffbacccb5a3c0e962859c": "0=L_-L_+^k Y = (\\lambda - (m+k)^2-(m+k))Y.", "bcecfb7cf693ddf6097f1215bef3057f": "\\scriptstyle\\mathfrak{{S}}\\ \\equiv \\ S", "bced091a5cc12ff1e9ca707da97b17b9": "\\scriptstyle f_\\text{curried}(1)(2)(3)", "bced10a1088e54adb192e22d31dc48c3": "\\eta(-,b,c)", "bced89b08dae75b09769e9d01479fd06": "u = \\frac{x}{x^2+y^2+z^2},\\quad v = \\frac{y}{x^2+y^2+z^2},\\quad w = \\frac{z}{x^2+y^2+z^2}.", "bcee0b90ed0cfab3e9620fdc8ddce4fe": "\\sum_n^\\infty t^n L_n(x)= \\frac{1}{1-t} ~ e^{\\frac{-tx}{1-t}} ~ .", "bcee0d78f5b7b3562c2a028178a49e90": "\\mathcal{R}\\mathcal{R}^t", "bcee4381759e591e4e9153b2ca9d957e": "W=\\int Fdx\\approx \\frac{1}{2}F_{max}\\Delta x", "bceea2409a51c09473a928031fd38797": "x^n + t_1 x^{n-1} + \\cdots + t_n", "bceebd840b6d04544b3c1da173f14c8d": " S_{xx}(f) = F'(f) ", "bceec6343d715cd616294827ec0a4af9": "f^{*}(x)=e^{f'(x)\\over f(x)}", "bceeefdf10860348ca77cb9ba55afdd3": "(A \\land (B \\lor C))", "bcef00974854bf888a85289fd794808d": "\n(\\mathbf{\\hat{f}_{0:2}})^T =\nc_2^{-1}\\begin{pmatrix}0.9 & 0.0 \\\\ 0.0 & 0.2 \\end{pmatrix}\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.8182 \\\\ 0.1818 \\end{pmatrix}=\nc_2^{-1}\\begin{pmatrix}0.5645 \\\\ 0.0745\\end{pmatrix}=\n\\begin{pmatrix}0.8834 \\\\ 0.1166 \\end{pmatrix}\n", "bcef8e07c5a6580a225b851bd520b245": "\n 0 \\to R \\xrightarrow{\\ d_2\\ } R^2 \\xrightarrow{\\ d_1\\ } R\\to 0,\n", "bcf0143668c6e3e5074d4cc7cdb2dbb9": "(-1)^{\\epsilon(\\hbox{deg}\\ x)\\epsilon(\\hbox{deg}\\ z)}[x,[y,z]] + (-1)^{\\epsilon(\\hbox{deg}\\ y)\\epsilon(\\hbox{deg}\\ x)}[y,[z,x]] + (-1)^{\\epsilon(\\hbox{deg}\\ z)\\epsilon(\\hbox{deg}\\ y)}[z,[x,y]]=0.", "bcf016b550d0e02a5527d6f52560167b": " f(x) = \\int e^{i \\phi(x,\\xi)}\\, a(x,\\xi) \\, \\mathrm{d} \\xi ", "bcf07d51c7582ff0478437982d94b7aa": "L^2(\\Gamma \\backslash G)", "bcf0ad541d90abfbc91844946acd3102": "\\theta\\mapsto f(x\\mid\\theta), \\!", "bcf0bcd1aa381721056ef55f5d4d4e57": "\\frac{\\partial v}{\\partial t} + u \\beta y = -\\frac{\\partial \\phi}{\\partial y}", "bcf11833de545f1a6b1c6f7c4e29fd10": "P(\\mathbb{N})=\\{\\varnothing,\\{1, 2\\}, \\{1, 2, 3\\}, \\{4\\}, \\{1, 5\\}, \\{3, 4, 6\\}, \\{2, 4, 6,\\dots\\},\\dots\\}.", "bcf14dcc23af906ddc8bbf6cfdd52a93": "W=\\frac{12\\sum_{i=1}^n (R_i^2)-3m^2n(n+1)^2}{m^2n(n^2-1)-m\\sum_{j=1}^m (T_j)},", "bcf1b2fb51509e906acbba4c199a725d": "A\\sum_{n=0}^\\infty(-1)^n = \\lim_{x\\rightarrow 1}\\sum_{n=0}^\\infty(-x)^n = \\lim_{x\\rightarrow 1}\\frac{1}{1+x}=\\frac12.", "bcf1b6cf0dc31051842744f31cd16bf3": "\\scriptstyle N ", "bcf268fa81d33445b1e761a09dbe40e2": "\\ P_1= x_1P^*_1f_{1,M}\\,", "bcf2c94681c322a08d242f6166da3124": "R_{\\mathrm{H}}", "bcf31249fbb834b75b31c1dc15511a69": " \\lambda_1 ", "bcf38d6030036e7096768a453fea24f1": " Compliance = \\frac{\\Delta V}{\\Delta P} = \\frac{.5\\;L}{(-5\\;cmH_2O - (-10\\;cmH_2O))} = \\frac{.5\\;L}{5\\;cmH_2O} = 0.1\\;L\\;\\times\\;cmH_2O^{-1}", "bcf3ee5f51327cdfae78cacdc1bbf290": "\\mathbb{R}^{n\\times n}", "bcf4056c7fc25042c197068b565c919b": "\n{\\bold f_x}=\\begin{pmatrix}\\rho u\\\\p+\\rho u^2\\\\ \\rho uv \\\\ \\rho uw\\\\u(E+p)\\end{pmatrix};\\qquad\n{\\bold f_y}=\\begin{pmatrix}\\rho v\\\\ \\rho uv \\\\p+\\rho v^2\\\\ \\rho vw \\\\v(E+p)\\end{pmatrix};\\qquad\n{\\bold f_z}=\\begin{pmatrix}\\rho w\\\\ \\rho uw \\\\ \\rho vw \\\\p+\\rho w^2\\\\w(E+p)\\end{pmatrix}.\n", "bcf46c546f6db1d39918363359670334": " a,b \\in \\{0, 1\\}", "bcf493cd687a87ef50bc679dadf7325d": "\\ \\bar{u}=\\dot{u}-wu, ", "bcf4bf802559cac168b2e8ff83862928": "0.07 + {LWL\\over150}", "bcf4d67c21e47132fdc01a0ddc5abd9d": "\\beta(u)", "bcf4ec199e34869368df30fd0b0c6262": "x \\in [0; \\infty)\\!", "bcf553d201ed80a2def89535154f0d7d": "e^\\varepsilon", "bcf576a010b44a755fc9b16e0ab59abd": "\\varphi^*(x, y) = \\tfrac{1}{4}\\left(\\varphi(x{+}h,y)+\\varphi(x,y{+}h)+\\varphi(x{-}h,y)+\\varphi(x,y{-}h)\n\\,-\\,h^2f(x,y)\\right)\\,,", "bcf58b3a053f555331826fde0708e924": "\\int r^3 \\;dx = \\frac{1}{4}xr^3+\\frac{3}{8}a^2xr+\\frac{3}{8}a^4\\ln\\left(x+r\\right)", "bcf5c93180d4b7914b29c1d3b86051a3": "\\dot{v}_\\alpha(t) = f(x_\\alpha(t), v_\\alpha(t), x_{\\alpha-1}(t), v_{\\alpha-1}(t), \\ldots, x_{\\alpha-n_a}(t), v_{\\alpha-n_a}(t))", "bcf5d1ec95c448c01f326fb39846d993": "\n\\begin{array}{rcl}\nf(x) \n&=& \\int_0^\\infty \\frac{sin(t)}{t+x} dt = \\int_0^\\infty \\frac{e^{-x t}}{t^2 + 1} dt \n~=~ {\\rm Ci}(x) \\sin(x) + \\left(\\frac{\\pi}{2} - {\\rm Si}(x) \\right) \\cos(x) \\\\\ng(x)\n&=& \\int_0^\\infty \\frac{cos(t)}{t+x} dt = \\int_0^\\infty \\frac{t e^{-x t}}{t^2 + 1} dt \n~=~ -{\\rm Ci}(x) \\cos(x) + \\left(\\frac{\\pi}{2} - {\\rm Si}(x) \\right) \\sin(x) \\\\\n\\end{array}\n", "bcf5fbc63b9ad933511df2186023a791": "P(k,x)", "bcf5fe804451b6dd5a36b317e0584765": "Pj", "bcf61157ec3b14e4ed941446d11c36fd": " \\textstyle \\mathbf{q}_r = \\mathbf{q}_{ph} = \\int_0^\\infty\\int_{4\\pi}\\mathbf{s} I_{ph,\\omega}d \\Omega d\\omega.", "bcf63c6e3d34300aeb43a46a74545c3e": "D_{LS}=\\sqrt{\\frac{1}{2\\pi} \\int_{-\\pi}^\\pi \\left[ 10\\log_{10} \\frac{P(\\omega)}{\\hat{P}(\\omega)} \\right]^2 \\,d\\omega },", "bcf6400a3dbf60d9739573a2d5235814": "[\\hat H, a ] = -\\hbar \\omega \\, a.", "bcf64f991733ac711a4fad66b739223e": "\\gamma \\circ \\eta", "bcf6b871c3f427cc43098bb51fb57acc": "d(w,v)", "bcf6ca5bb5866e2c9bfb9bc118b673d3": "\\|A\\mathbf{x}-\\mathbf{b}\\|^2+ \\|\\Gamma \\mathbf{x}\\|^2", "bcf7a30f6d844909ffb7cc24037bab96": "R_k^i", "bcf80a79f61b2ea27594eb672a8ccef6": "S(X, Y) = \\frac{| X \\cap Y |}{| X \\cap Y | + \\alpha | X - Y | + \\beta | Y - X |} ", "bcf82f8b7c618c08973fcdf0688ed227": "\\textstyle g=\\frac{GM}{R^2}", "bcf859f911f97a29890875198420b6e1": "\\nu = \\exp\\left( \\exp\\left( \\frac{\\text{VBN}_\\text{Blend} - 10.975}{14.534} \\right) \\right) - 0.8,", "bcf869bf7ac01f42aed2975f4ec9ee8f": "\\lim_{T \\rightarrow 0}\\alpha_V=0.", "bcf869c3ce3657ae35397bba5cd957b7": "\\frac{\\partial u}{\\partial t}(x,t) + \\mu(x,t) \\frac{\\partial u}{\\partial x}(x,t) + \\tfrac{1}{2} \\sigma^2(x,t) \\frac{\\partial^2 u}{\\partial x^2}(x,t) -V(x,t) u(x,t) + f(x,t) = 0 ", "bcf8742a4ba5a45f78b3844de44c8291": "E(\\tilde{R}) - R_f = -R_f cov (\\tilde{m}, \\tilde{R}).", "bcf8b45c50c2d6581bccda3abcef4540": "\\epsilon_{\\downarrow}", "bcf8e42130c4d4d28ed8cd0875a5fc01": "a_{\\mathit{vf}}=\\left\\{ \\begin{array}{ll}\n1, & \\mathrm{for}\\ \\Delta_{0,50}\\leq0\\\\\n\\\\\\frac{1}{1+4\\Delta_{0,50}^{2}}, & \\mathrm{for}\\ \\Delta_{0,50}>0\\end{array}\\right.", "bcf9487b0158387f206b36cb24493042": "var[G|K] = E[G^2|K] - (E[G|K])^2 = \\rho + \\rho^2 -\\rho^2 = \\rho. ", "bcf9798dc8df3db3ad0b17e773b5070a": "\\ F", "bcf9f264dda3484c6ab1cb0b09ece399": "\n\\operatorname{cs}^2(u)+m_1=\\operatorname{ds}^2(u)=\\operatorname{ns}^2(u)-m\n", "bcfa30ab2fa8c595e734fb72fe62c87e": "z_{nk}", "bcfaa71258857f954c54f90d8b8cbac8": "E_\\theta= {AI \\over r}", "bcfaf7fbfae6af68b35915f745b83960": " \\underset{free-energy} {\\underbrace{ F(s,\\mu)}} = \\underset{complexity} {\\underbrace{ D_\\mathrm{KL}[q(\\psi|\\mu)\\| p(\\psi|m)]}} - \\underset{accuracy} {\\underbrace{E_q[\\log p(s|\\psi,m)]}}", "bcfb586aafee37599aa9838429f56b92": "15\\rightarrow 10\\oplus 5", "bcfb6a451744147f33faf4cc8cc32dfa": "C_{GDj}\\left(V_{GD}\\right)=A_{GD}\\sqrt{\\frac{q\\epsilon_{Si}N}{2V_{GD}}}", "bcfb744a0bc9b79910de87013f7f2793": " g^{-1}P = g^{-1}(g^{-1}P_1 \\cap g^{-1}P_2) = g(g^{-1}P_1).g(g^{-1}P_2)", "bcfbce6a52229a30d8e7d862231848ad": "U(L) = T(L)/I", "bcfbd2a039d5cdf3dd7515fe4f6752e0": "\\scriptstyle \\vec F_R ", "bcfc3fbe26b17743608c4ea194798dcb": "ab \\leq cd", "bcfc5a770fe0c9e0751e33f814b83185": "\\frac{X_b}{X_c} =\\frac {\\exp\\left (\\frac{-\\Delta\\,G'}{RT}\\right )}{X_c^0}", "bcfc6401bb3203bcd178d796d5f830e2": "u^{\\tau}", "bcfd17775e5eecfacbd477817b838d0c": "{\\bar{T}}_9", "bcfd353bed73b997058ec114e01321c0": "f = 1.5 \\cdot F", "bcfd59954e3d2040cabb16c7bdfff62c": "\n\\sum(X_i-\\mu)^2=\n\\sum(X_i-\\overline{X}+\\overline{X}-\\mu)^2\n", "bcfd5ea0d0d21242d7e42a4b07f55ca1": "Re(s) > 0", "bcfd6773beedec8f6a89729c47dde947": " c_H \\geq \\frac{T}{\\chi_T} \\left( \\frac{\\partial M}{\\partial T} \\right)_H^2 ", "bcfd96eadd19d87cd8164f9008065502": "\n\\mathbf{S}_m (\\mathbf{r},t) = \\sum_{\\alpha} \\frac{m_{\\alpha}}{2} \\dot{r}^2_{\\alpha}\\dot{\\mathbf{r}}_{\\alpha} \\delta(\\mathbf{r}-\\mathbf{r}_{\\alpha}(t)).\n", "bcfe37e5b15ed1a462cc886785e51095": "\\!c", "bcfe49b867308a41c0e25c8fad0959ac": " ((a,b),[\\infty])\\notin I", "bcfe7d492dfb1fce573366c2258a1028": "F = e + \\int_0^\\infty \\frac{e^{-x}}{\\pi^2 + \\ln^2 x}\\, dx", "bcfe8fcd3a933ac9e1c29ed540815191": "\\det \\boldsymbol{\\varphi}'_w(0) = -1", "bcfec23cdb06d5cad044ce5272b84738": "E \\left [X_n^2 \\right ] \\le c_1 E[X_n]^2", "bcfedb6d79d4ec2887afe739408e2063": "F=\\frac{3\\pi R^2}{2h (t)^3} \\frac{d h}{d t} \\eta", "bcff4e889139f8149c0802ab2e54c38a": "\n\\dot{y} = \\frac{c}{q B} \\frac{\\partial V}{\\partial x}\n", "bcff9f30cbc9dcbd1d42999c169a8e66": "(\\nabla^2+k^2)\\mathbf{H}(\\mathbf{x})=-\\left[k^2\\mathbf{M}(\\mathbf{x})+\\mathbf{\\nabla}\\times\\mathbf{J}(\\mathbf{x})+\\mathbf{\\nabla}(\\mathbf{\\nabla}\\cdot\\mathbf{M}(\\mathbf{x}))\\right]", "bcffce19824166e20ee0fdf099e74d77": "P(a)=P(b)=true", "bcfff60732910e98f453767c6d5dc68d": " m=2 ", "bd00176a53ca8945ba2cd15d6bbb163a": "\n\\ R_0 \\cdot S = 1.\n", "bd00a4d1ec40cebbb9bc8722c5a574df": "\\mathop{\\textbf{y}}\\downarrow-: \\hat C \\to \\textbf{Cat}", "bd00ab8e53c766360eb6094d5f5a22e4": "s(\\alpha^i) = 0, \\ i=1,2,\\ldots,n-k", "bd01088e71ee8b066ae45f9bd2995bc7": "(-f)(x) = -(f(x)), \\forall x\\in E", "bd016483b1db39dcdb1bb7750a899309": "\\scriptstyle \\log_{10}P_{mmHg}=7.18807 - \\frac {1416.7} {211+T}", "bd01e55ec81dc87cff9090442b6c4cbe": "ds^2=-\\frac{r^2}{M^2}\\,dt^2+\\frac{M^2}{r^2}\\,dr^2+M^2\\,\\big(d\\theta^2+\\sin^2\\theta\\,d\\phi^2 \\big)", "bd020867cf1ebd806e24eadf6bf89d27": "\\Sigma^{0,Y}_n", "bd02308055b3ff96d5e8c526b49f3944": "\\alpha\\leq\\beta", "bd02517d89d1a7f3c319d08433e4d3f0": " (-1,0, . . . ,0) ", "bd029141958f74343d250fdee92fff5d": "\\sqrt{p \\over n-p} \\approx \\sqrt{p \\over n}", "bd029f2da202af0e6bc064a6c98aba25": "\\partial_\\mu \\left( \\bar{\\psi}\\gamma^\\mu\\psi \\right) = 0.", "bd02afc6c10d8e2bd5bf7952fc04a616": " 2f'(x) = 2E_1 + 2\\sum_{n=1}^\\infty E_{n+1}\\frac{x^n}{n!} = 2E_1 + \\sum_{n=1}^\\infty \\sum_{k=0}^n \\binom n k E_k E_{n-k} \\frac{x^n}{n!}. ", "bd02b92a1e334174f07121fa1082ad02": "\\vec{\\nabla} f", "bd02e45e2a62135a1be48df6a89025e9": "(q, a, q_1 \\vee q_2)", "bd033180a8269eb5adaef9aa45be8767": " a_{r} + a_{l} + (F_{r} - F_{l})", "bd035dfcc3777056656ffa2d13a2005b": "\\displaystyle{2\\pi|K(z_n,w)+K(w,z_n)| ={|(z_n-w)\\cdot (\\mathbf{n}_{\\zeta_n} -\\mathbf{n}_w)|\\over |z_n-w|^2}\\le {|\\mathbf{n}_{\\zeta_n} -\\mathbf{n}_w|\\over |z_n-w|}={|\\mathbf{n}_{\\zeta_n} -\\mathbf{n}_w|\\over |\\zeta_n-w|}\\cdot {|\\zeta_n -w|\\over |z_n-w|}.}", "bd03d28f1070c25de6a83bd318e798d2": "stdKt/V = \\frac { \\frac {10080 \\cdot (1 - e^{-eKt/V})}{t} }{ \\frac {1 - e^{-eKtV}}{spKt/V} + \\frac{10080}{N \\cdot t} - 1} ", "bd03f0d5c103f027067c2a86b51cc087": "f \\mapsto (\\mathbb{P}_n - P)f = \\dfrac{1}{n} \\sum_{i = 1}^n (f(X_i) - Pf) ", "bd0419eeefd68a1cf8b3ada02be1c607": "\\mathrm{core}_t(n)", "bd0471cc6559e5b0af3980d5d6c7405a": "SFRatio_{i}=\\frac{E(R_{i})-\\underline{R}}{\\sqrt{Var(R_{i})}}", "bd04726287c64a0b06661a839c5301ba": "\\begin{align}\n a_1 &= (h\\cdot a_0) \\downarrow 2 \\\\\n d_1 &= (g\\cdot a_0) \\downarrow 2\n\\end{align}", "bd04a7981785bfb04e17e90e088ea172": "\\mathbb{C}/\\langle 1, \\tau \\rangle", "bd04f90dd3ab3a6e3443b9c396f4d95c": "H_{\\nu}(\\omega)=-e^{j\\nu [ \\frac {\\omega} {2}]}. \\frac {ce_{\\nu}( \\frac {\\omega} {2},q)} {{ce_{\\nu}(0,q)}}.", "bd0531079a09b3a0859784fef4ba9c5b": "\\frac{dL}{dt}v+L\\frac{dv}{dt}=\\frac{d\\lambda}{dt}v+\\lambda \\frac{dv}{dt}.", "bd0531680cfd4f8f1c1021d658182fd4": "\\frac{11}{9}", "bd053d458ae7dbe7f3f5798494c8593d": "{{v}_{GS1}}+{{v}_{GS4}}", "bd053f912dc245c409502219017166c4": "y^T A", "bd054851bb1a55b53cac05d43d08b805": "\\sigma_3 \\otimes \\sigma_3 ", "bd059cffd2e39d311f3881c6aab7b7b6": "\\mathcal D_k,\\ 1\\leq k\\leq n", "bd05cfff1609ae74b72f1fe2bb65966b": "G=SO_{0}(n+1,1)", "bd05d83f07a577e2de7abec242d9707c": "U = n S^2 / \\sigma^2 \\sim \\chi^2 _ {n-1}", "bd062db9735bb663f4e12f37bff99baf": "l_i\\in{\\mathcal D}", "bd063840e15d583ef9aa8f06cb178bc8": "R|f(z)| \\leq r|f(z) - 2A| \\leq r|f(z)| + 2Ar", "bd06606115e6517f6636b798d8a3f348": "p\\notin f(\\partial\\Omega)", "bd06653fa389fd124a423b1eb2238ee7": "\n{u}_n(x,z) = \\left(\\frac{2}{\\pi}\\right)^{1/4} \\left(\\frac{1}{2^n n! w_0}\\right)^{1/2} \\left( \\frac{{q}_0}{{q}(z)}\\right)^{1/2} \\left[\\frac{{q}_0}{{q}_0^\\ast} \\frac{{q}^\\ast(z)}{{q}(z)}\\right]^{n/2} H_n\\left(\\frac{\\sqrt{2}x}{w(z)}\\right) \\exp\\left[-i \\frac{k x^2}{2 {q}(z)}\\right]\n", "bd0670e8dc8763903325216e94b6372b": "F = \\frac{MS_\\text{Treatments}}{MS_\\text{Error}} = {{SS_\\text{Treatments} / (I-1)} \\over {SS_\\text{Error} / (n_T-I)}}", "bd0686284f74e2670a8777f124d1a9b9": "I(v)=\\log n", "bd06c801293949d1fe0913f22f48907d": "M^{[n]}", "bd074ee2443ffa68afbba9f9f7437f10": "H_1 : \\sigma_{\\text{Tx A}-\\text{Tx B} }^2 \\ne \\sigma_{\\text{Tx A}-\\text{Tx C} }^2 \\ne \\sigma_{\\text{Tx B}-\\text{Tx C} }^2", "bd0776f7dc00e0c60ffb11de6ca23f61": "[y_1 \\; \\cdots \\; y_n]^{\\rm T}", "bd077d2e4853d57f74610cbe18ed3606": "\\liminf_{x\\to a} f(x) = \\sup_{\\varepsilon > 0}(\\inf \\{ f(x) : x \\in E \\cap B(a;\\varepsilon) - \\{a\\} \\}).", "bd077e57a116458876550ba95d0d0b12": "|\\psi_I(t)\\rangle=\\left[1-\\frac{i\\lambda}{\\hbar}\\int_{t_0}^t dt_1 e^{\\frac{i}{\\hbar}H_0(t_1-t_0)}V(t_1)e^{-\\frac{i}{\\hbar}H_0(t_1-t_0)}\\right.", "bd080369c9e8175dab30edc7d6153f6f": "|\\overline{P}_+\\cap z| = 1 = |\\overline{P}_-\\cap z|", "bd081457e3547537ad0981269aae8557": "\\frac{dx}{dt} = x(\\alpha - \\beta y)", "bd08543acf3e90a310b808bc160bdb17": "p(\\vec{r}) = \\frac{ e^{-\\beta E_\\vec{r}}}{Z}", "bd088289e1ce9095bcc65527929eebc8": "\\frac{4}{3}\\pi R^3", "bd088b0b39439367329a12245def7fb7": " P^2 = M^2", "bd08a29c8c7eff1628798c1df749569d": "{x^n}", "bd08dce1872c5fa8499d6190546e6e5d": "\\sqrt{21\\over{8}}\\sin(\\theta)(5\\sin^2(\\phi)-1)\\cos(\\phi)", "bd0927513f01847d8c5f49491c4b69ae": "a(n)= \\sum_{k=0}^n k! \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}=\\sum_{k=0}^n \\sum_{j=0}^k (-1)^{k-j} \\binom{k}{j}j^n=\\frac12\\sum_{m=0}^\\infty\\frac{m^n}{2^{m}}.", "bd09278b82e1b1e7f2ed671f86c1afba": "\\log(\\Gamma(z\\!+\\!1))=P(z)", "bd096abbba638438bdde3d60d6e5248c": "\\langle E\\rangle=\\left\\langle \\psi \\left|i\\hbar\\frac{\\partial}{\\partial t}\\right|\\psi\\right\\rangle = \\hbar\\omega.", "bd099ba2f570d75ac214c9985d8aeebf": "\n\\begin{align}\nI_1\\dot{\\omega}_{1}+(I_3-I_2)\\omega_2\\omega_3 &= M_{1}\\\\\nI_2\\dot{\\omega}_{2}+(I_1-I_3)\\omega_3\\omega_1 &= M_{2}\\\\\nI_3\\dot{\\omega}_{3}+(I_2-I_1)\\omega_1\\omega_2 &= M_{3}\n\\end{align}\n", "bd09a84ddf69efcccf159217dbafa8b4": "{1\\over 3}, {2\\over 5}, {3\\over 7}, \\mbox{etc.,} ", "bd09adeac843fa573703871e1c456ea6": "2^P", "bd09d756fce73011a37a48787d107a6a": "D\\subset\\mathbb{R}^n", "bd0a5a0b6f1522694386136d5446db5b": " 49 = 7 \\times 7", "bd0a7d3f8f4c08adfe4d25a38edbdf1f": "b_t=t_i-t_a\\;", "bd0ab067d560553bc2b0da98ba431f00": "A = \\frac{x_1 + x_2}{2}", "bd0b2d00ed4b9f68b48f60f9f7f713a1": "7 * 2 = 14; 14 - 13 = 1 = A", "bd0b50409e847836a49c051fe29b01f9": "\\frac{300\\,\\rm Hz}{300\\times10^6\\,\\rm Hz}=1\\times10^{-6}= 1\\,\\rm ppm \\,", "bd0b585967a59ef1ffc929f1df518f95": " n \\ge m \\, ", "bd0b82d0a78cf71df61a4b5de07979bc": "\\scriptstyle z \\mapsto z e^{-mj}", "bd0bfa874d9262d292412c07b109635e": "\\phi,\\lambda", "bd0c05f07fdd90250d7b42c8848f76cd": "y_{\\bar{t}}^{q}", "bd0c894f93eeb1c36aacefa4763743c9": " \\frac{\\partial \\Phi(\\vec{r},t)}{c\\partial t} + \\mu_a\\Phi(\\vec{r},t) - \\nabla \\cdot [D\\nabla\\Phi(\\vec{r},t)] = S(\\vec{r},t)", "bd0cbab3489b12ae7b00400a72fd084e": "\\mathbf{R}^{n+1}_+ := \\left \\{(x,t) \\ : \\ x \\in \\mathbf{R}^n, t>0 \\right \\}", "bd0d085610a6aa9f283073a74edba162": "\\mathcal{C}(X)", "bd0d5092e4fe4d15a3ec438e3c021b10": "\\mathcal{L}_C", "bd0d6e65c8e5bd1b2c15a36fc5b48e23": "\\displaystyle{e^{2a} =Q(e^{a})1= e^{2L(a)}1 = e^X\\cdot 1,}", "bd0d8259b12ecd27d62915dfbb80f012": "Z(S_n)", "bd0da827380e8448a68be3bed2c15512": "\\lambda=\\sum_{i=1}^n \\lambda_i", "bd0dade194e906f9e6c53267b6826186": "P_\\mathit{BODYDIODE} = V_F I_o t_\\mathit{no} f_\\mathit{SW}", "bd0dce5879d433bbcc72fde585209a69": "\n W = \\mu~\\mathrm{tr}(\\boldsymbol{\\varepsilon}^2)\n ", "bd0e17786a5e689f5ae14ee6376392a4": "p_{5} \\in A_{5}", "bd0e2b55bfa5e05bc806eec07b5a6fd7": "\nQ_1^{(\\text{blue})}(t) - Q_2^{(blue)}(t) = -1\n", "bd0e36747fc87a21705299deba8ee0e5": "\\mathbf{P}_{1}, \\mathbf{P}_{2},\\mathbf{P}_{3},\\ldots,\\mathbf{P}_{N}", "bd0e3c9cebc05152c913156404d74099": " \\mu_2 = \\xi{\\left(\\theta\\right)} \\sigma^2.\\, ", "bd0e4304f37e6c280fa9ec3b7ef44a0e": "\n\\operatorname{Var}(x_{ij}) = \\frac{(\\nu-p+1)\\psi_{ij}^2 + (\\nu-p-1)\\psi_{ii}\\psi_{jj}}\n{(\\nu-p)(\\nu-p-1)^2(\\nu-p-3)}", "bd0ebfd87b5d01d4e0011d336a8dc5f9": "\\mathcal{E} = -{{d\\Phi_B} \\over dt}\\quad", "bd0ec27e3c4f6193fde001dd48eeab39": "\\mathbb{P}(Z_\\pi (T) > Z_\\rho (T)) = 1.", "bd0ee688485c16c9ddc1d3c28fb719b2": "\nD_{\\alpha }(-\\hbar ^{2}\\Delta )^{\\alpha /2}\\phi (\\mathbf{r},t)+q^{2}|\\mathbf{\nr}|^{\\beta }\\phi (\\mathbf{r},t)=E\\phi (\\mathbf{r},t). \n", "bd0eeeba0c2d7c5010e0c52c131e7e26": "\\int_a^b w(x) p(x) dx = \\sum_{i=0}^N w_i p(x_i)", "bd0f1a4bb4587d0ae1f9d6aa2a5e401c": "{G}_{2}^{(1)}", "bd0f249c738bdc1a9449579b81ec092a": "f(y)=f(z)", "bd0f473ae258c92a711cf228c1ade769": "\\left ( \\sin \\phi_N - \\sin \\phi_S \\right ) \\left ( \\theta_E - \\theta_W \\,\\! \\right)", "bd0f5bd521e3eccb26f604f88866ba98": "y^2 = x^3 + ax + b, \\, ", "bd0f6b337992b40a7e1f69b617164ed9": " c_1^2 = 3 c_2", "bd0f969dda459f857a866efa5c4c93e9": "1 - \\frac{1}{e}", "bd0faa462fe976ea959a16bd506016be": "\\hat A_{\\psi}^{\\dagger}=\\int d\\mathbf{k}\\psi(\\mathbf{k})\\hat a(\\mathbf{k})^{\\dagger}", "bd0ff8a0d0a4a68149ea99a360d42370": "\\sum_{r} \\tilde{P}_{r}\\log\\left(\\tilde{P}_{r}\\right)\\geq \\sum_{r} \\tilde{P}_{r}\\log\\left(P_{r}\\right) \\,", "bd0ffcbe956f74e928abe1f48432e14f": "k \\geq x_{max}", "bd1018eae1a19f0ff01ce38a6d0ade88": "\\sum_{a\\in A}C_{a,f(a)}", "bd104681629da4d85545ccd1cd083b79": "V_R\\,", "bd10741b49458d951a8800d77cf18f34": "e=\\sqrt{g(2-g)}.", "bd10a049115504bd1d220ef654de1c6a": "d_{pd}=\\frac{\\sum_{i=1}^N \\sum_{t=2}^T (e_{i,t} - e_{i,t-1})^2} {\\sum_{i=1}^N \\sum_{t=1}^T e_{i,t}^2}.", "bd10aa65a223ea197c6d876014a4f57f": "dE_L = L I \\delta I\\,", "bd1170bfacacc290617643f0ff37ae3d": "t_{ij}(x) = t_{ji}(x)^{-1}\\,", "bd11fe6ad1fb02b346efc16399aaaf31": " A_b = \\{ \\langle min(t_{ij} | i = 1,2,...,m)| j \\in J_- \\rangle, \\langle max(t_{ij} | i = 1,2,...,m)| j \\in J_+ \\rangle \\rbrace \\equiv \\{ t_{bj} | j= 1,2,...,n \\rbrace, ", "bd124309c4f48c25cf73b27600ba7376": "f_i = \\langle f,e_i \\rangle ", "bd127a6eb496673b468dfb0f5cb3d30f": "\\ \\Delta H(T_d)", "bd12c697bc8cb343034a2055cd4452ca": "\\eta_c = \\pi x N / (4 L^2 (x-2))", "bd12d0730f99ad55b084d29f2266f7ea": "V( \\mathbf x) > 0 ", "bd132a992034386bcb6b9f340d4e040c": "a + b\\mathbf{i} + c\\mathbf{j} + d\\mathbf{k} = a + \\vec{v}", "bd1339fa21a1f050ebdc6281765926cb": "\\mu_3 = k_3 = a_1 +8a_2", "bd135da798150347a1e6aff190f23d7d": "c^\\prime(g,h) = f(gh)f(g)^{-1} f(h)^{-1} c(g,h)", "bd13928335641fe2e045ee521ecce40a": "[T(t)] = e^{[S]t}.", "bd13af805829957f36b4d69737832dd2": " T = \\int_{-\\infty}^{+\\infty} \\lambda d \\operatorname{E}_T(\\lambda). ", "bd1423899541389e8b59915afec0ece5": "\\displaystyle Z(P,P,s)\\sim\\frac{T^{N/2}}{(2\\sqrt{\\pi})^N\\Gamma(N/2+1)} ", "bd143560ee47f0a97b53ff039e1d0a00": "a^{-3}", "bd1442f8c0440700dc85edfb1da199ba": " \\tilde{\\nu} = \\frac{1}{\\lambda} ", "bd1491719a1629b149ae87453455ab02": "C_{\\mathrm{Katz}}(i)", "bd151e0b103592ead1d70f28401f3979": " \\begin{align}\\Psi(\\bold{k}, t) &= (2\\pi a)^{3/2} e^{- a \\bold{k}\\cdot\\bold{k}/2 }e^{-iEt/\\hbar} \\\\\n&= (2\\pi a)^{3/2} e^{- a \\bold{k}\\cdot\\bold{k}/2 - i(\\hbar^2 \\bold{k}\\cdot\\bold{k}/2m)t/\\hbar} \\\\\n&= (2\\pi a)^{3/2} e^{-(a+i\\hbar t/m)\\bold{k}\\cdot\\bold{k}/2} .\\end{align}", "bd15aee66d3e277a76faaddd54d4b391": "\\ell '", "bd15cc5e7a92033aa354208804e5c4fb": "\\det S''_{ww} (\\boldsymbol{\\varphi}(0)) = \\mu_1 \\cdots \\mu_n", "bd15f38fa61a84145f61c9ddcd2c4246": "\\left | f^\\prime (p) \\right | \\in (0,1)", "bd15f495fc1079c7477ff513b9b6dd68": "v_p = \\sqrt{M/\\rho}.\\ ", "bd1607d7d8fe8f662a816d093c5f44c6": "\\;\\;\\;\\;\\; = 2 \\int d^4 x \\; e \\; \\Big( e^\\beta_J \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} - {1 \\over 2} e_M^\\gamma e_N^\\delta e_\\alpha^I \\Omega_{\\gamma \\delta}^{\\;\\;\\;\\; MN} \\Big) (\\delta e_I^\\alpha)", "bd160f8f22d190848971709bc6cd4bd4": "(0, e^{-x^2/2}/\\sqrt{2\\pi}]", "bd16574a5db14d9998738286a03a80e5": " \\Delta (v)=1\\otimes v+v\\otimes 1 \\quad \\text{for all}\\quad v\\in V.", "bd166201ceb89be2bfdd8eae3c3d63f7": "k(v_1, v_2, \\dots, v_n)", "bd166ec7bd5ab9a99f722115ca6f9485": "\\phi\\in\\mathbb{R}^S", "bd1681d3912e4b7310ae67be4edcc0c0": "\\det(t+E+(n-i)\\delta_{ij}) = t^{[n]}+\\sum_{k=n-1,\\dots,0}t^{[k]} \\sum_{I,J} \\det(X_{IJ}) \\det(D^t_{JI}),", "bd16ae62bc10dfc3637a34fc74abd1d5": "g = g^{\\star}", "bd16c2233bbbbfd85d5045b568fe76ed": "F_b \\,", "bd1775dfa0e36f59812acbf909fe8cf5": "E = X + D", "bd17ac345974b4fb9bb73d3cefa6878e": "p_t(x)", "bd17fe5dd0e614f21e8370f72a34ea2a": " \\log(x_1/x_2) = -\\log(x_2/x_1). \\, ", "bd18434a3a9d2cf410cdbdd6ae7c0487": "x_{12}", "bd18772c2eac404eb837c21e32fe1aa1": "a \\bold{v}", "bd190fb4e339ff29449314cc89298a33": "\\sum_L L^2 4^{-2L} e^{-4\\beta L}", "bd1947997969b2a7179e6b7f399f5a52": "\\widehat{D}\\in(\\{0,1\\}^{k})^{m}\\,\\!", "bd198e8f0a93c0fa884094402a755a53": "\\beta^-=\\frac{cov(r_i,r_m |r_m0", "bd25d754aaa9a7d1a0190e0068686a33": "X_{r} = -\\frac{\\partial E_{r}}{\\partial x}\\,", "bd260652abe9c6e6ba188c5c5bc6b3b2": " g_i(x) \\le a_i , h_j(x) = 0.", "bd263586485b9182e1cd261cdd860fee": "(\\tfrac{47}{221})", "bd2638d8b06489b9e278d82202c20687": "\\bar{q}q^\\prime", "bd266f7e84bbfeb60e58410e2c391886": "|\\Psi_{A_1\\ldots A_m}\\rangle", "bd2699dea9ff1bb527f69efb09c6cd50": "{R_{ox}=-C_R\\rho \\frac{m_{ox}}{s}\\frac{\\varepsilon}{k}}", "bd26a7993bbab7c4c02dce65ef0f06d2": "m=\\int_{t_1}^{t_2}\\iint_S \\bold{j}_m\\cdot\\bold{\\hat{n}}{\\rm d}A{\\rm d}t ", "bd26b6485d66ceeeb23c98de6dbf2cfd": "V_n \\subset V", "bd26c744fd9cd71a9d640887d1c93936": "\\Pi(x)\\,\\!", "bd26cc626311ceac9778d9c36868fe2a": "v(z):=\\sum_{k\\geq 1}v_k z^k", "bd278da126d6e201ecd800d174ca6671": "\\frac{{}_{(s)p}\\partial f(x)}{\\partial x}, \\ p=0,1\\,\\!", "bd27c32d6b46ca79778985fb4dd8c6d9": "\\text{Protocol overhead} = \\frac{\\text{Frame size} - \\text{Payload size}}{\\text{Frame size}}", "bd27c9dc0758e10ed9f4a31497f7bb6c": "K_d", "bd27f512b6b84e7124ff6260b048f2d8": "3\\zeta(2)\\zeta(3)-\\tfrac{11}{2}\\zeta(5)", "bd285dd6f5e1a0aae14065e3214c6f51": "\\begin{align}\n\\mathrm O(k) &\\to V_k(\\mathbb R^n) \\to G_k(\\mathbb R^n)\\\\\n\\mathrm U(k) &\\to V_k(\\mathbb C^n) \\to G_k(\\mathbb C^n)\\\\\n\\mathrm{Sp}(k) &\\to V_k(\\mathbb H^n) \\to G_k(\\mathbb H^n).\n\\end{align}", "bd28763d683cd60f69aced0d5f70cfc7": " t^2 - \\operatorname{tr}(A) t + \\operatorname{det}(A) ", "bd2885c4d565130382e345b8794c00d8": "\n\n\\begin{pmatrix}\nI & P_1 & \\cdots & P_1 \\\\\nP_2 & I & \\cdots & P_2 \\\\\n\\vdots & & \\ddots & \\vdots \\\\\nP_p & \\cdots & P_p & I \n\\end{pmatrix}\n\n\\begin{pmatrix}\nf_1(X_1)\\\\\nf_2(X_2)\\\\\n\\vdots \\\\\nf_p(X_p)\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nP_1 Y\\\\\nP_2 Y\\\\\n\\vdots \\\\\nP_p Y\n\\end{pmatrix}\n", "bd2896abeafb0d0827da75e6f3d74158": "\n\\mathcal{G}(\\mathbf{k},\\omega_n) = \\frac{1}{-\\mathrm{i}\\omega_n + \\xi_\\mathbf{k}},\n", "bd28ef5c2a7e43263a157c9acde6602c": "\n\\Delta G = -RT \\ln K_{eq}\n", "bd2918239b11f927f561f72a61e4bff9": "ASC = ABC + \\pi/3", "bd291c5280e347fa36a84b04f8c54b25": "n = {1 \\over F} = 4N_e u + 1", "bd29410546a722ddfcee276596f5514f": "d(\\Lambda)", "bd295d7127b2460023430f00e76baf1f": "\\sum_{s\\in S} (r_s)^D=1 \\, ", "bd296873fe93374f42e5d71037770281": " SK_2 = \\frac{ Q_3 + Q_1 - 2 Q_2 }{ Q_3 - Q_1 } ", "bd296e0c0505fb0753097dcaefebbb10": "p_n>0", "bd299d9abdecd7fa58803910511dc5da": "\\frac{\\partial x_j}{\\partial\\bar{x}_k}=\\frac{\\partial}{\\partial\\bar{x}_k}(\\bar{x}_i(\\boldsymbol{\\mathsf{L}}^{-1})_{ij})=\\frac{\\partial\\bar{x}_i}{\\partial\\bar{x}_k}(\\boldsymbol{\\mathsf{L}}^{-1})_{ij}=\\delta_{ki} (\\boldsymbol{\\mathsf{L}}^{-1})_{ij}=(\\boldsymbol{\\mathsf{L}}^{-1})_{kj}", "bd29d576e103d5fed67d5d728d43aa08": "K(\\Pi) \\leq \\widetilde{K}(\\widetilde{\\Pi})", "bd2a1def24d2102a95e99b2f64356a55": "\n \\sigma_{r\\theta} = \\sigma_{\\theta\\theta} = 0 \\qquad \\text{at}~~\\theta=\\alpha, \\theta=\\beta \n ", "bd2a1eae6578bf7f7085d5381fa3f2e6": "\\ell_1,\\ell_2,\\ldots,\\ell_n\\,", "bd2a36455ca59a0cef6d59e9cb8eee06": "T_{i,r}", "bd2a51ef9691cf734f6a4ef47555ad8d": " \\tau_{r} = \\frac {-\\mu b} {2 \\pi r} ", "bd2a58b5cb7f42552c3296fbcf9e24b5": "\\det(\\mathbf{B})=\\operatorname{pf}^2(\\mathbf{B}) \\, ", "bd2a6ae84256d09023472376cb2d0114": " \\mathcal{S}[\\phi]\\,=\\,\\int_M \\mathcal{L}[\\phi(x),\\partial_\\mu\\phi(x),x] \\mathrm{d}^nx.", "bd2aadec076ac659d03c5fa1c50679e1": "2^{14}", "bd2b01910a5c8fcd520fbdaa60971129": "X\\sim\\mbox{Beta}\\left(\\alpha,\\beta,\\lambda\\right)", "bd2b566ffb8ae6e70ac6a1e5a8927e17": "10^{10^{100}} = (10 \\uparrow)^3 2", "bd2bef56d579bf5c88d1a976522f502a": "\\delta t ", "bd2bf71091cb66b8138fff375d4e0b3a": "\\Gamma^*_n \\subseteq \\Gamma_n.", "bd2c186a1af1220aae6b6f328d9da3e0": "\\begin{align}\nh_\\sigma(x) &= \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{x^2}{2\\sigma^2}} \\\\\nT_\\sigma(x) &= \\frac{x}{\\sigma} \\\\\nA_\\sigma(\\mu) &= \\frac{\\mu^2}{2\\sigma^2}\\\\\n\\eta_\\sigma(\\mu) &= \\frac{\\mu}{\\sigma}.\n\\end{align}", "bd2c307756d9c6ef773cb84e6b2d85d6": "4x^2+y^2+6x+2=0:\\ \\hbox{an intersection of multiplicity 4}", "bd2c3d7a6b80d7fcbbe79ca4dca837b5": "\\begin{align}\nW^{+} \n& =\n\\left(\\frac{1}{\\sqrt{2 \\epsilon}}\\right)\n\\left[\n\\begin{array}{c}\n- J_x + {\\rm i} J_y \\\\\nJ_z - v \\rho \\\\\nJ_z + v \\rho \\\\\nJ_x + {\\rm i} J_y \n\\end{array}\n\\right]\\,, \\quad \nW^{-} \n=\n\\left(\\frac{1}{\\sqrt{2 \\epsilon}}\\right)\n\\left[\n\\begin{array}{c}\n- J_x - {\\rm i} J_y \\\\\nJ_z - v \\rho \\\\\nJ_z + v \\rho \\\\\nJ_x - {\\rm i} J_y \n\\end{array}\n\\right]\\,. \n\\end{align}", "bd2c502dcbdb7a02c1e00dea36335429": "\\sum_{i=1}^n (x_i r_i) j_i = \\sum_{i=1}^n x_i", "bd2c67d124865581cec5fbf9bf9225b2": "\n\\left\\langle \\bar{\\partial} J \\right\\rangle_{S}=0\n", "bd2c992b3525cb2322e78b0efdb6e03a": "\\mathbf{E_T}=\\mathbf{E_0}e^{i(xk_T\\sin(\\theta_T)+zk_Ti\\sqrt{\\sin^2(\\theta_T)-1}-\\omega t)}", "bd2c9f3bd598b16df096bb3088465253": " U = - \\frac{W_{\\infty r}}{m} = - \\frac{1}{m} \\int_\\infty^{r} \\mathbf{F} \\cdot \\mathrm{d}\\mathbf{r} = - \\int_\\infty^{r} \\mathbf{g} \\cdot \\mathrm{d}\\mathbf{r} \\,\\!", "bd2cbbf2c113b351d534515d6366877b": " z = r \\, \\sin \\theta. ", "bd2cded85b4ccf4696c66652f70487a2": "\\lang -,- \\rang", "bd2d818ae16cdb1206ab9f554b45bad8": "A\\to B", "bd2d84e2d3ad3874b9965c1ddfbbfde3": "Q_M (a,b) = \\int_{b}^{\\infty} x \\left( \\frac{x}{a}\\right)^{M-1} \\exp \\left( -\\frac{x^2 + a^2}{2} \\right) I_{M-1} \\left( a x \\right) dx ", "bd2d97589dd4282976dc67ddb16c434c": "\\omega =\\omega_\\mathrm{c}", "bd2da5815de58f5c37c7ff14b7d93684": "r_1+r_3=r_2+r_4.", "bd2e476762f97870096920e2d52d1cad": "\\boldsymbol\\beta \\cdot \\mathbf{X}_i", "bd2e505886b6a6b2988caed30eccc20b": "{\\omega^2 \\tau^2}", "bd2f7e5ef15f8f3246661487ab2e085f": "V_c\\,", "bd2f936acd0a2cc2f63419c8631c5524": "\\Omega^\\prime_0 - \\Omega_0 \\equiv \\Omega_k - \\Omega_0 = k \\left ( k + x + \\frac{1}{x} \\right ) \\delta_0\\Omega_0", "bd3015188b778a441dff9d8da10f06c4": "\nQ_{\\nu-\\frac12}^\\mu(z)=\\frac{e^{i\\mu\\pi}\\Gamma(\\mu-\\nu+\\frac12)(\\pi/2)^{1/2}}{(z^2-1)^{1/4}}\\biggl[\nP_{\\mu-\\frac12}^\\nu\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)-\\frac{2}{\\pi}e^{-i\\nu\\pi}\\sin\\nu\\pi Q_{\\mu-\\frac12}^\\nu\\biggl(\\frac{z}{\\sqrt{z^2-1}}\\biggr)\\biggr].\n", "bd3062b6bd0dfca37763e0adc4020042": "j_d = {\\mu}_d j_r", "bd306a66198e624522d25092b801a7c6": "a_0\\,\\!", "bd30b127dae3f4a62a50a33bfb44082b": "{LOE}", "bd315c5885e3f70d255ba8b93f81c1e8": "\\frac{x^2+x+1}{x}, y=x+1", "bd31b7b2c882cbc141fa101cb505b543": "2\\sin(x/2)\\!", "bd31cdac54ce6924745808a18e10deaf": "\\rho \\left[ \\frac{\\partial \\overline{u_i}}{\\partial t} + \\overline{u_j} \\frac{\\partial \\overline{u_i}}{\\partial x_j} \\right] = - \\frac{\\partial \\bar{p}}{\\partial x_i} + \\frac{\\partial}{\\partial x_j} \\left( \\mu \\frac{\\partial \\bar{u_i}}{\\partial x_j} - \\rho \\overline{u_i' u_j'} \\right),", "bd324c1d75e77afd3d0e70ffe2b6cc59": "v(f_h) \\approx 1 - f_h + \\tfrac{1}{6} f_h\\ln f_h...........\\rm{(11)} ", "bd32cbb2f31d45a38bade2e09ee1a3b0": "\\psi(x,t)=\\theta(x)\\,", "bd332d2c6fca58705c899a3bc02ea5c0": "y''(x)+y(x)=0 \\, ", "bd3330f18aaa9373ae9e3d809ba1bf5b": "9 \\times 6", "bd3338dad64d3ef7d71ea7a131c0d1aa": "{\\delta_{ij}}\\ ", "bd33ee20cd6f4d9df61207ee2e7d26d4": "x_{n+1} = 4 x_n(1-x_n) \\,", "bd3431741fa758e82dfab9353096f05b": "g_p(aU_p+bV_p,Y_p) = ag_p(U_p,Y_p)+bg_p(V_p,Y_p),\\ \\ \\text{and}", "bd343a9f1a0564e1450937c5476f3175": "1947 = [37, 23]_{52}", "bd3441e3852d503fd4bd2d48b4ca837c": "\\tan\\lambda = {\\sin\\alpha \\cos\\epsilon + \\tan\\delta \\sin\\epsilon \\over \\cos\\alpha}; \\qquad\\qquad \\begin{cases}\n \\cos\\beta \\sin\\lambda = \\cos\\delta \\sin\\alpha \\cos\\epsilon + \\sin\\delta \\sin\\epsilon; \\\\\n \\cos\\beta \\cos\\lambda = \\cos\\delta \\cos\\alpha.\n\\end{cases}", "bd34624719644c1f7f0d03904c2947e3": " R(T^{n}) ", "bd3475e17ce5c8285f36cb9ec6563388": "R_{3,1} = 21 r^3-30 r^2+10 r", "bd34890db4b8e417b7943a4db2eeaf56": "\\mathbb{P}(y \\mbox{ sent})", "bd34f8113bf1c89a4a4edf927a054ba2": "\\nabla^2_{X,Y} T = -\\left(\\nabla_X \\nabla_Y T - \\nabla_{\\nabla_X Y} T\\right).", "bd3515d0dee40ec4f19bdaab368174d0": "[\\text{OH}]=K_\\text{W}[\\text{H}]^{-1}\\,", "bd35195f6b2c230ea7bbafe0badd22fd": "\\mathrm{u}(1)_{B-L} \\,", "bd354159915f4458d2f13e4445a42036": "\\scriptstyle C_{-1}", "bd3560da33792097cba8005721bbf746": "E_{+}=[(E_1+E_2)/2+|W_{12}|]", "bd3569ad9730755d9daa9dd01eb7a267": " V[x] = A_3 ", "bd35710ca447aa78305e60773e1e5e44": "(s, \\omega) \\mapsto X_{s} (\\omega)", "bd3595a57a4111be9b6c0ddabd893a98": "\\sqrt{n}(t_n - \\theta) \\xrightarrow{D} N(0,V),", "bd35987b7222ce28633ec21b4f3b8faa": "\\frac{C_1}{C_2} = \\frac{{G_1}/{G_2}}{{G_3}/{G_4}} = \\frac{G_1 G_4}{G_2 G_3} = \\frac{G_1}{G_2} \\times \\frac{G_4}{G_3} = C_1 \\times C_3,", "bd35a26674b979881b7b23d1025b2eb4": "S=\\sqrt[5]{K_{sp}\\over 108}", "bd35a620b265709408a56297a9fa2ed6": "U=\\int_0^\\infty\\epsilon(\\rho)\\,{\\rm d}\\rho\\,\\int_0^N\\,{\\rm d}n\\,\\frac{kT}{2Na^2}\\left(\\frac{\\partial z(n,\\rho)}{\\partial n}\\right)^2", "bd35c8c102677ad8535dce0787561436": " \\frac{v^2}{2} - \\frac{GM}{r} = - \\frac{GM}{2a} ", "bd35f42304dd99af31ee411c72353aa0": "x(0)=0,\\, x(1)=1.\\,", "bd3631f05264d04c59f62aa820c45839": " \\Phi = n \\, v ", "bd3663d4de8ae4c5b04260a33874267d": "\\nu_\\max = { \\alpha \\over h} kT \\approx (5.879 \\times 10^{10} \\ \\mathrm{Hz/K}) \\cdot T ", "bd370a78f22c1b4ddb48d3782b96e6e1": "N=\\frac{-4.61}{-6.7\\times10^{-6}}", "bd3784ae0f4827723a54753b4a7ef633": "\\frac{1}{\\tau} = \\frac{1}{\\tau_B} + \\frac{1}{\\tau_N} ", "bd37c39c0cb729c7d05bb8169e741d25": "\\mathrm{2\\ CmO_2\\ +\\ H_2\\ \\longrightarrow \\ Cm_2O_3\\ +\\ H_2O}", "bd37e8ac5e18834acc89ca31c4b4d9f4": "\\tau\\to 0^+", "bd37fb1a13fc2809b3a8935d5785ae23": "A \\| \\mathbf{v} \\|^{2} \\leq\n\\langle \\mathbf{S} \\mathbf{v} \\mid \\mathbf{v} \\rangle \\leq B \\| \\mathbf{v} \\|^{2}\n\\text{ for all }\\mathbf{v} \\in V", "bd3878423d602dd56d30b3c7efac7b88": "\\int_{0}^{\\infty} \\cos t^2\\,\\mathrm{d}t = \\int_{0}^{\\infty} \\sin t^2\\,\\mathrm{d}t = \\frac{\\sqrt{2\\pi}}{4} = \\sqrt{\\frac{\\pi}{8}}.", "bd38b54acc747367b32084fb43ca1d83": "\\tfrac{0}{0} = 0", "bd38b944fb3e7eaec77fa3659083c295": "O_{p}", "bd38f9b3fa5d5e3d919bd0aa002a47ad": "(P_1,P_2,\\ldots,P_d)", "bd3907f4bb738cb6d018da631026a17d": "((A\\equiv B)\\equiv C)\\equiv(B\\equiv(C\\equiv A))", "bd3946372d5fadf876d0779779b394b5": "+\\frac{1}{2}\\left[\\frac{V_{nk_1}V_{k_1k_2}}{E_{nk_1}E_{k_2 n}^2}\\left(\\frac{V_{k_2 n}V_{nn}}{E_{k_2 n}}-\\frac{V_{k_2k_3}V_{k_3 n}}{E_{nk_3}}\\right)-\\frac{V_{k_1 n}V_{k_2 k_1}}{E_{k_1 n}^2E_{nk_2}}\\left(\\frac{V_{k_3k_2}V_{nk_3}}{E_{nk_3}}+\\frac{V_{nn}V_{nk_2}}{E_{nk_2}}\\right)\\right.", "bd397ab261cc85876142768e23f383f2": "G \\ltimes X", "bd397cee584dbe511c0b5daae9f43138": "\\widehat T = t_1 \\dots t_{m}", "bd3980a98a6bb323ccd2925af533bff2": "8_{S+} \\times 8_{S+} \\,", "bd39cd6a9a352f415a754ee725a679c8": "c_1,c_2,\\ldots", "bd39e01a01dad6ecb8afb9160061ae2b": "\\Delta_n^r", "bd39e8748f4a43690e58f92e8482ed77": " r=\\int_0^s\\frac{dt}{\\sqrt{1-t^4}}.", "bd3a26f037bd712733a95e06b5729ef5": " \\gamma_y ", "bd3b2ff6295e24e83d4af00b22668fa7": " [[x,a,a],b,b] = [[x,b,b],a,a] ", "bd3b50e4330e1f3a8d5ba057b5b445e5": "\\mbox{curl}\\,(\\mbox{curl}\\,\\vec v ) = \\nabla \\times (\\nabla \\times \\vec v)", "bd3bf5120f9aa2d3436dbc61cebc9b6d": "P(t) = P_0e^{rt} - \\frac{M_a}{r}(e^{rt}-1)", "bd3c086b13efcb804ebc0526514b2a49": "N_k = \\frac{1}{k}\\sum_{d|m}\\phi(d)\\left(m^{k/d}\\right),", "bd3c32d1b9d47448b7b64e44649fb0dd": "g:\\mathbb{C}\\rightarrow{}\\mathbb{C}", "bd3c49261ce84eefdaa8e38434b420f4": "\n \\sum_{k=1}^n{\\frac{1}{k^2}}\n < 1 + \\sum_{k=2}^n\\underbrace{\\left(\\frac1{k - \\frac{1}{2}} - \\frac1{k + \\frac{1}{2}}\\right)}_{=\\, 1/(k^2 - 1/4) \\,>\\, 1/k^2}\n = 1 + \\frac23 - \\frac1{n + \\frac{1}{2}} < \\frac53\n", "bd3c55ee6cf19fa7766109ceee894f73": "1^3+2^3+3^3+4^3+5^3 = 15^2 \\,", "bd3c747f738dd2116ffa8144895e9500": "f(x,y) = y \\oplus x (y \\oplus 1)", "bd3ce927343fbb645ceadd8eb0a7c20f": "\\forall x\\in\\mathbb{R}\\cup\\{-\\infty\\}", "bd3d025ef6bd138ad6e9e3e922b85f42": "Z = -2 x - 3 y - 4 z\\,", "bd3d0d66f10c17b73f89073a0ad989ea": "h = \\frac{g r}{\\cos \\theta} = g r \\frac{1}{\\cos \\theta} = g r \\sec \\theta", "bd3dc07238985f3c97e3ad85641105fc": "~\\cos^{-1}(x)~", "bd3e16c775e6e4bbf81f68c84bfbeb21": "H_4(x)=16x^4-48x^2+12\\,", "bd3e290ef7048a4aa74a6ca89bcd669a": "u(c)", "bd3e4b64fa4df0513e2fa8cbcecd0425": "x^2+ax+b=0, \\ a,b \\in K", "bd3e7d1a0e374794c258bca0633b0bb8": "\\scriptstyle h\\left(\\emptyset\\right)\\,=\\,0", "bd3e86cbaabce5bea6ca48f07c78f682": "(S(f \\circ g))(z) = (Sf)(g(z)) \\cdot g'(z)^2.", "bd3e9131f46d27d2a4d38320e69fce2d": "|1 \\rang, |2 \\rang, |3 \\rang,...", "bd3f2fefb7c12056c575d661a06ccaf4": " \\delta_y: S \\rightarrow Y^\\phi \\times S", "bd3f5e5761f53f746629fcb780486145": "{ (Qg)(x) = (Qf)(x + a)}.\\,", "bd3fdf8f3a036e1785c378af74707e43": "X \\sim \\mbox{Scaled Inv-}\\chi^2(\\alpha,\\tfrac{1}{\\alpha})\\,", "bd401fb84ca6d8a3f41f9953747e878e": " \\mathbb{P} (Y=0|X) = \\begin{cases}\n \\binom 7 X / \\binom{10}X &\\text{for } X \\le 7,\\\\\n 0 &\\text{for } X > 7.\n\\end{cases} ", "bd40205f6a3818788f1b5458a3c3f26a": "y^0=1", "bd4039209dd60d6004b3b61616272a1c": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & -1 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2\n\\end{smallmatrix}\\right ]", "bd4059d83b7c2ddb037f9d01a1202f4b": "I_{MAX}", "bd40da48bf138f80d70ff03892f834ac": "\\mathbf{B}_{dR}", "bd40ec5485793302610e9b3c7a2fd0ad": "\\operatorname{pair} \\equiv \\lambda x.\\lambda y.\\lambda z.z\\ x\\ y ", "bd4139e482f28f770e89c5ebc376bb6b": "\\frac{1}{x_n}=(\\frac{1}{x_n})_o+\\frac{k_{tr}[solvent]}{k_p[monomer]}", "bd41b9476222197dd320c95bfeea335f": "\\bar{\\sigma}=\\bar{\\sigma}(\\bar{\\epsilon})", "bd41da9271335d863969ecbcec2a7219": "H \\otimes C", "bd420004d6db45ecc9bf1fedca5cc37f": "C\\colon V \\to \\overline V", "bd4258324ddbabdea1bff3d0372a47ac": "\nW = \\prod_i w(n_i,g_i) = \\prod_i \\frac{g_i!}{n_i!(g_i-n_i)!}.\n", "bd42baab38ef2da123e8eff3eb5d2f44": " x = S^+ b = E^T \\mbox{Diag} (e^+) E b ", "bd42cbe48bb362c79c2ee01cb926d140": "z = \\sqrt{\\frac{1+\\beta}{1-\\beta}} - 1.", "bd434c2188e9adadeb869477eaceda96": "var[(1/n)x_{1}+(1/n)x_{2}+...+(1/n)x_{n}]", "bd43628486820699aeed0516b1456b9a": "~\\sqrt{!~}~~", "bd43ca22529dbfd7bff6c43a35ccfdff": "E_{FIN}(\\Z)", "bd43f2e86e14e323b94a9bcd1028e4fd": "\nJ_i^2 =J_i, i=0,\\ldots,n, \\qquad (3)\n", "bd441169490cabfd9e8955cf47b8dfce": "R_{j}^2", "bd44986cc04ea0c5eec625d6da314869": " \\gamma(s,x) \\rightarrow \\Gamma(s)", "bd44b48341f734cdd579323d60db74dc": "MP = \\frac{\\Delta Y}{\\Delta X}", "bd44bede20e23408a0c4843e077e7e3f": "1/n^{1/3}", "bd44d6fdec2d90a529ccba09e99d5bfd": "k=9=3^2=a^2", "bd45179857ba68d6d941afd2bb61d0bc": " \\frac{dp}{p} = {\\frac{c_p}{R}\\frac{dT}{T}}, ", "bd451eb73308564e96bd82e28e5d9f73": "\\mathcal{F}_{\\mathfrak{H}}= \\{|x\\rangle\\, , \\, x \\in X \\}", "bd45214b82f3aa1ea4361c247febd9ed": "\\mathrm{P}(E_i | \\rho) = \\mathrm{Trace}(E_i \\rho)", "bd4543c3163156a6de837a1600d2277c": "\\{sa \\mid s \\in S\\} \\cup \\{a\\}", "bd4563c4ff55da948dcdffae4a9def73": "\\tan{\\alpha} = \\frac{\\Delta h}{d}", "bd456961c34e969d8a2c2195d0d5a7e3": "\\mathbf{v}_g = \\frac{1}{f} \\mathbf{k} \\times \\nabla_p \\Phi", "bd45db4d22a04c5d8137095a7d55a042": " FPR(T)=\\int_{T}^\\infty P_0 (T) dT ", "bd461d09b0849c7735ac314f5e02bd04": "S \\to c_{1,0}(\\widehat{\\epsilon})", "bd4657b38cd392f7feba2fed368ef7ed": "A = R \\oplus V,", "bd468161bd61d2b5f553d0a44cf88f66": "\\dot{x} = 0", "bd4688fb8dbdce142f35136eb707bdac": "\\varphi^3=1+2\\varphi\\!", "bd46af57380b35fe2da6590ea5c1d099": "\\begin{align}\n x &= \\ell_{BP} = \\ell_{PD} \\\\\n y &= \\ell_{OB} \\\\\n h &= \\ell_{AP}\n\\end{align}", "bd46c485e2bb3d7cfc9c8d75c295a27f": "w\\cdot \\lambda:=w(\\lambda+\\delta)-\\delta", "bd46c6da1e315db96c063309877482be": "\\frac{1/k^s}{\\zeta(s)}", "bd46cdd913ff99e068b9e791aa377565": "\\lambda_1, \\lambda_2 = \\alpha \\pm \\beta i.", "bd46d41510d36b3b0e1df545a55a9351": " \\frac{\\mathcal{O}(x,y)}{\\langle 2x, 5y^4 \\rangle} = \\{y,y^2,y^3\\} \\ . ", "bd4743185977ba9514e8eeee21202c90": "\\overline{GM_{T}}", "bd479d7d56e37ebb70dd10cceb276726": "\\scriptstyle \\eta \\in S ", "bd47aba73a616c8cfc7f66678dd68de4": " [F_rL, F_sL]\\subset F_{r+s}L ", "bd481086c87ef3ce5d097fddaef8d2cb": "x_{ij} = 0, \\forall j", "bd4812c256fb3fca124ce17cc428c201": "\\mathcal{R}^K _{\\theta} ", "bd48580351c5725ada2d2af124c9f777": "n \\neq 3", "bd48605b4e9c17b93effa549fbea8fb5": "a = 1.40 \\times 10^{-3}", "bd488e50221add6a4452ec2046bb4197": "V_D = const", "bd48ed85b0ecd95464ae89e983d224f1": "(-,+,+,+)", "bd4916323432426aea693ea75e9e6898": " W^{-}=\\begin{bmatrix} * & t1 & t2 \\\\ p1 & 1 & 0 \\\\ p2 & 0 & 1 \\\\ p3 & 0 & 1 \\\\ p4 & 0 & 0 \\end{bmatrix} ", "bd491cdc6f4ca1d38a6a0154c94ae89d": "e^{xf(t)}=\\sum_{n=0}^\\infty {p_n(x) \\over n!}t^n", "bd4935c76044f41835aa2971788209bb": "{\\omega'}_i\\in \\mathbb{R}^3", "bd495edd4dd32f3de851f8dfee4442ad": "6\\zeta_4=3\\alpha_3+2\\zeta_1-3\\zeta_3", "bd4a042e6302df5ecf2fb615b6fe5894": "C_p =\\,", "bd4a09211d877a753f4ee03dc2c6ee81": "a \\propto t^{2/3}", "bd4a2a94d64cafb5da0bbadbcf932b98": " R_e", "bd4ad156631022f3e76290fa3c82a6b6": " \\lambda X^{\\tfrac{1}{k}} \\sim \\mathrm{Weibull}(\\lambda,k)", "bd4b18c2f8f2fee0c1c922b489aa986a": "3\\uparrow^{3 \\uparrow\\uparrow\\uparrow\\uparrow 3}3", "bd4b1c547645a029f8e41917e5f409ea": "Z(G) = Z(G; a_1, a_2, \\ldots, a_n).", "bd4b235338810c8818fe2d9c733cf988": "^{rd}", "bd4b41210f7b8af526ca5bd69ea9322b": "\\langle \\text{U},\\text{L}\\rangle", "bd4bda948310a484170ab9d96ccfe637": "|P-Q|_\\pi = |P\\setminus P'| + |Q\\setminus Q'|", "bd4bef862017fd6ddc414c8bbb96a356": "({v_0+v_i})10^{-pH_{i}} \\text{ vs. } v_{i^{ }}", "bd4c35e5e710824b8141f9dcc1a57fb4": " \\sum_{i=0}^{n} {n \\choose i}^2 = {2n \\choose n} ", "bd4c6147969648b4fc0bc1cc57b6d93d": "\n\\psi=U(x)\\delta(x)f(\\eta) = y \\sqrt{\\frac{2{\\nu} U_{0}L}{m+1}}\\left(\\frac{x}{L}\\right)^\\frac{m+1}{2}f(\\eta)\n", "bd4c9fa175c594f39cdd08df09ca03fb": "f_i \\in \\mathbb{C}", "bd4cade77412b2abbc581988053e3abb": "20^2+21^2 = 29^2", "bd4d036e482070cd5ddc758986d61758": "\\mathsf{JKLMNOPQR} \\!", "bd4d0b71de405fc441d46e4204d8b894": "q^*(\\mathbf{Z})", "bd4d1a3285f03fd309e38e9357945060": "\\sqrt[3]{n}", "bd4d4d6be6a7ba505ecade650ef99f0c": "\\{v_1,v_2,\\cdots,v_{|V|}\\}", "bd4dd431a0e73bb599c38718770f1726": "\\mathsf{(CH_2CH_2)O+CO_2}\\rightarrow\\mathsf{(O\\!\\!-\\!\\!CH_2CH_2\\!\\!-\\!\\!O)C\\!\\!=\\!\\!O\\ \\xrightarrow[-CO_2]{+H_2O}\\ HOCH_2CH_2OH}", "bd4dd72bf0d2e1e7f42736bff2a725c3": "\\Delta\\langle \\hat{B}^\\dagger \\hat{B} \\rangle", "bd4decbae003cac3ffa6031b7d907715": "K = W \\,", "bd4e0bca500fb42678be86b2b563aad5": "s= \\frac{v^2 - u^2}{-2g}.", "bd4e286dbdab0699e9ea309e26b00e5c": " \\theta = \\omega _0 t + \\frac{1}{2} \\alpha t^2", "bd4e4fc1ad2f8cc69389943436ff46a1": "\\langle u_1,v_1,...u_g,v_g,q_1,...q_r,h|u_ih=h^{\\epsilon}u_i, v_ih=h^{\\epsilon}v_i,q_ih=hq_i, q_j^{a_j}h^{b_j}=1, q_1...q_r[u_1,v_1]...[u_g,v_g]=h^b\\rangle", "bd4e64a70eea297ec8b1ac283c4bf619": "\nf_{\\mathrm G} = 2.31 \\, \\lambda^{-6/5} \\left[ \\sec \\zeta \\int_{\\mathrm{Path}} C_n^2(z) \\, v_{\\mathrm{Wind}}(z)^{5/3} \\, dz \\right]^{3/5}\n", "bd4e71b896a9ebe8829fb19a52a8f8fc": "\n\\vec{y} = A \\vec{x} + \\vec{b}.\n", "bd4ed1166d48e6752d4767bd77d0c739": "{F(0) = G(0) \\quad F(S(x)) = H(x,F(x)) \\quad G(S(x)) = H(x,G(x)) \\over F(x) = G(x)}.", "bd4f723890a0f277509822e3dc08ea50": " \\mathrm{Bd}\\; K ", "bd4fc13e5d5d7de22ed9d6601f2f7053": "g(x) = \\prod_i h(x_i)", "bd4fd3c1f8807bea66d0eb0db2e6f260": "\\{\\lor, \\lnot\\}", "bd4feba6e0f935f9e86dd8a696df9dc7": "G(\\alpha)+\\int_0^t\\,f(s)\\,ds\\in \\text{Dom}(G^{-1}),\\qquad \\forall \\, t \\in [0,T].", "bd500bbe2943fbef5db6d12ee0d2424e": " \\lim_{\\underline x \\to \\underline \\alpha} \\sum_{i=1}^n \\frac{F^{(n-1)}(x_i)}{\\Pi_i(x_1,\\ldots,x_n)}=\\lim_{\\underline x \\to \\underline \\alpha}\\frac{1}{n!}\\sum_{i=1}^n f(x_i)=\n\\frac{f(\\alpha)}{(n-1)!}", "bd501cbd2db5e6ed8c7fcbbbec316690": "\\frac{ar + 1}{b(r + 1)}", "bd506560570958eb3f3eae5345d593dd": "L - \\ ", "bd50799c638954f003dfd9eed321786c": " \\sqrt{g_{tt}} u^b\\,", "bd50aaa2f347aa0e51198ecd2811cfb3": "\n \\mu \\Delta\\Delta w + \\rho w_{tt}= 0\\,.\n ", "bd50ce465c232b9179bfb07a0d5c6387": "h_t(x) = \\Psi^{-1} ( s^t \\Psi (x))~,", "bd50d50e85997f176e9b74d73105bc7a": "\\frac{y'x^3 - 2x^2y}{x^5} = 0", "bd50f41235ca8e1f6b62f443fe87712b": " s = 0 \\text{ and } |s| \\rightarrow \\infty ", "bd51592f6c031a5489e790b2084289f0": "\\frac {d M_y(t)} {d t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _y", "bd516ee95e4b9402fdc51bb8709ee1d5": "K(z) = \\frac{1+|\\mu(z)|}{1-|\\mu(z)|}.", "bd51750857ee2ecaec5d2b4d3761884d": "D\\in(\\tfrac{1}{2},1]", "bd51986a9b55df9e9b248641d4d2cdcd": "R(\\theta,\\delta) \\,\\!", "bd51cc8b190640140318d4b9f79a3322": "\n\\begin{align}\n\\|x + y\\|^2 & = \\langle x + y, x + y \\rangle \\\\\n& = \\|x\\|^2 + \\langle x, y \\rangle + \\langle y, x \\rangle + \\|y\\|^2 \\\\\n& = \\|x\\|^2 + 2 \\text{ Re} \\langle x, y \\rangle + \\|y\\|^2\\\\\n& \\le \\|x\\|^2 + 2|\\langle x, y \\rangle| + \\|y\\|^2 \\\\\n& \\le \\|x\\|^2 + 2\\|x\\|\\|y\\| + \\|y\\|^2 \\\\\n& = \\left (\\|x\\| + \\|y\\|\\right)^2.\n\\end{align}\n", "bd51e8022bf7af338d0fe40d8dd6a023": "\\ \\gamma_0, \\ \\chi_0, \\ \\tau", "bd51fd5b3f0b28416a603cf12909b43e": "A_0(x)=\\sin(x);\\,", "bd5233e9b0887c21dba785159a0e3869": " (x)=x-[x] ", "bd5235aa8911a9ccfb0c2f60d8e02549": "k_f = \\frac{RMS}{ARV}", "bd524f7229d76de1dd43719f0069d201": "\\frac{dN}{dt} = rN \\left( 1 - \\frac{N}{K} \\right),", "bd52b93a7ddd38dce7110b7b5def6175": "T\\ |p,\\sigma,n\\rangle \\ \\propto \\ |-p,-\\sigma,n\\rangle ,", "bd52d5ea685b18f168c337038f540a08": "\\ N = \\frac{\\log \\, \\bigg[ \\Big(\\frac{LK_d}{HK_d}\\Big)\\Big(\\frac{HK_b}{LK_b} \\Big) \\bigg]}{\\log \\, \\alpha_{avg}} ", "bd53619d96943376be1955d0f71d43ea": " dE + \\delta w_u \\le 0 ", "bd546bac80c7ef2c903eded88455fb6e": " \\mathbf{F}(\\mathbf{r}) = - \\mathbf{\\nabla} V(\\mathbf{r})\\text{, where }V(\\mathbf{r}) = \\int_{|\\mathbf{r}|}^{+\\infin} F(r)\\,\\mathrm{d}r", "bd547b95e10313f96218833f98699ba2": "\\Omega^r(M)^\\mathbf{C}=\\bigoplus_{p+q=r} \\Omega^{(p,q)}(M). \\, ", "bd54b3920cf6eeb40dfc2affef6520bc": "\n s-\\frac{P(s)}{\\bar H^{(\\lambda)}(s)}\n =\\alpha_1+O\\left(\\ldots\\cdot|\\alpha_1-s|\\right).", "bd54da1b0da1344d45992d3c83f61d90": "\\lim_{\\tau \\rightarrow 0}\\hat{U}(t_1,t_0) = 1", "bd54e9eeac4f7960eaaa02ce776df503": " l_+ = l / \\sqrt{1-v^2/c^2} ", "bd54eec9ac0fea3cc2810f0c7b0cdf93": "A|\\psi\\rangle", "bd54ff6ab40adb2d84718c348a79e3c3": " {d \\over dx}\\left(\\frac{x}{x^2 + ab}\\right) = \\frac{ab - x^2}{(x^2 + ab)^2} \\qquad \\begin{cases} {} > 0 & \\text{if } 0 \\le x < \\sqrt{ab\\,{}}, \\\\ {} = 0 & \\text{if } x = \\sqrt{ab\\,{}}, \\\\ {} < 0 & \\text{if } x > \\sqrt{ab\\,{}}. \\end{cases} ", "bd55296f80f126e8e4172dee7b7bd55c": " h(g) = (\\sigma(g) v,v).", "bd555850ec8f76c4db96806869cdf5ed": " \\hat{X}\\hat{Z} = \\omega\\hat{Z}\\hat{X} ", "bd55b4c89e02b4c7bff331d32f5d352b": "\\scriptstyle\\pi^\\pm", "bd5653685be88aa535c4595fe9e90bfd": " H(S)=\\left\\lfloor\\frac{7+\\sqrt{49-24 e(S)}}{2}\\right\\rfloor", "bd56eb0e7fb8f7a4330af8b1bb119463": "p \\bar{p}", "bd570616db9c57a9c26e4a9f07493072": "\\det(A)", "bd570a268e26baa140665ec18c5efc54": "S_2^P", "bd5724aa311ed9c987c71d73ddb0f7a1": "K_{\\rm w} = \\frac{a_{\\rm{H_3O^+}} \\cdot a_{\\rm{OH^-}}}{a_{\\rm{H_2O}}^2}", "bd57cac6f0e265b31b4eec1b2687276d": "y_1 \\alpha_1 + y_2 \\alpha_2 = k,", "bd5842fca64557de0f80ff6b94b18d10": "\\mathbf{Z}\\times O/(O \\times O)", "bd586db7e0e2a9e4b95ad4ff181c4eac": "\\Delta^\\text{op}\\text{Shv}_\\text{Nis}(\\text{Sm}/S)", "bd58a033faae68294909ecd644c5bbf3": "\\ \\dot{Q} = w^*Q - Qw \\quad \\text{and} \\quad \\dot{Q}^T = -Q^Tw^* + wQ^T ", "bd58bcb39533454de05a8e431beeff3d": "R_x = (R_g + R_y) \\cdot \\frac{R_{B1}}{R_{B2}}", "bd58bcb684f1aed87164adbd9b1c3a18": " \n\\|e^{(k+1)}\\| = \\|(I-\\omega A) e^{(k)}\\|\\leq \\|I-\\omega A\\| \\|e^{(k)}\\|, \n", "bd58be1c34899f71e289d7705a405332": " \\begin{align}\nE [ X(t) | X(t-1) = i ] &= p s \\dfrac{1-p}{p s + 1} + i \\\\\nVar( X(t+1) | X(t)=i) &=p(1-p)\\dfrac{ (s+1) + (p s + 1)^2 }{(p s +1)^2}\n\\end{align}", "bd58d4f8876efd7cad6f4b67fdc43778": " \\frac{v^2-u^2-1}{2} \\partial_u - u v \\, \\partial_v ", "bd58dbdec956752e435c81ec311ea2c5": "\\delta = d / D", "bd58e634d33926015151895e5610672a": "S(\\mathbf{r},i) = \\frac{r_i}{\\lVert \\mathbf{r} \\rVert} = \\frac{r_i}{\\sqrt{r_1^2 + \\cdots + r_c^2}} ", "bd58f95b40eece9aeb69393d43e03382": "B^2=B", "bd594cffe0fa501ec1d1352c5485c33f": "(E, [\\cdot,\\cdot], \\rho)", "bd595c752884bd8e15e234f1197f1482": "\nP(u,v) = \\left(\\begin{array}{c}\n X(u,v)\\\\\n Y(u,v)\\\\\n Z(u,v)\n\\end{array}\\right)\n= \n\\left(\\begin{array}{c}\n (a + C_{u}^{s}) C_{v}^{t}\\\\\n (b + C_{u}^{s}) S_{v}^{t}\\\\\n S_{u}^{s}\n\\end{array}\\right)\n", "bd595d95e10b1d932e7a49d36dfd8473": "\nm_n(x_n)=\n\\begin{cases}\n 1 & x\\in S_n \\\\\n 0 & x \\notin S_n\n\\end{cases}\n", "bd5a1a13c2d442bab8209bc70cd4bc77": "\\tilde v_i = \\tilde u_i", "bd5a6b2cfd0c7b2f5d8b3c9b88e43bd5": "\\scriptstyle \\frac{1}{\\Gamma(k)} \\gamma\\left(k,\\, \\frac{x}{\\theta}\\right)", "bd5acb099e60303f9e98bb2d5e02ac4a": "\\cosh x + \\sinh x = e^x", "bd5af63a7293ee9cb2b777e320635bb6": "P \\phi_r=\\frac{1}{2}(P+|P|)[f_r^+\\phi_R+(1-f_r^+)\\phi_L]+\\frac{1}{2}(P-|P|)[f_r^-\\phi_P+(1-f_r^-)\\phi_{RR}]", "bd5b55e66371e2852af85136ec81cef4": " = \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {-\\left(\\frac{\\partial}{{\\partial x_i'}}\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\right)F_j(\\vec{r}')d\\tau'} - \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {-\\left(\\frac{\\partial}{{\\partial x_j'}}\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\right)F_i(\\vec{r}')d\\tau'}", "bd5b7e23b10a38192deb36370e260a01": "\\textstyle\\frac{1}{x}", "bd5c019ac70d95cd739605b96f5622fe": "\n\\tau = -p 1 + C_1 A + C_2 A^2 + C_5 B,\n", "bd5c370d2490afc14ef3959c646d1520": " \\mathbf{E}_6 \\supset \\mathrm{O}(10) ", "bd5c51809d573267dca5495b5cfe41ff": "\\scriptstyle F_{\\mu \\nu}\\!", "bd5c5269ec6f2aca2948864f3d488b15": "|f(x)-f(x_0)|<\\varepsilon.", "bd5cbd8d59bcd5d3b3cc28397f48e2cb": "V=\\{x\\in X^{m}:|Q_{P}(h)-\\widehat{Q_{x}}(h)|\\geq\\epsilon\\,\\!", "bd5d556ba63e2589c07492c9ae893172": "r\\cdot(xy) = (r\\cdot x)y = x(r\\cdot y)", "bd5d6fea40d31fb61632878b32a4dd6a": "\\tau_3\\Big.", "bd5e1c66965b4f905bd7883e614caf68": "\\delta \\left( x^{\\prime }-x\\right) =\\int_{-i\\infty }^{i\\infty} \\frac{d \\tilde{x}}{2\\pi i }e^{\\tilde{x}\\left( x-x^{\\prime}\\right)}", "bd5e3a78bd902ae6ac9c8e787bc8191e": "\\frac{\\phi(t) -\\phi(0)}{t}", "bd5e4eb6e88f52da47fc6af5bdec04bb": "f_1,\\dots,f_k\\in{\\mathcal F}", "bd5e5c0c7a6fd08c0d4681736ff57aa5": "\nu \\rightarrow u^\\prime = L u L^\\dagger.\n", "bd5e69b719d9722da9755f6cfa69edb7": "B_x, B_y, B_z", "bd5e9071b80f0469a610b0526207c4b7": "\\int\\frac{\\sin^n ax\\;\\mathrm{d}x}{\\cos^m ax} = -\\frac{\\sin^{n-1} ax}{a(n-m)\\cos^{m-1} ax}+\\frac{n-1}{n-m}\\int\\frac{\\sin^{n-2} ax\\;\\mathrm{d}x}{\\cos^m ax} \\qquad\\mbox{(for }m\\neq n\\mbox{)}\\,\\!", "bd5ece1b4efaa7b8869a8424c294c0ad": "2\\times 2\\times 2\\times 2\\times 2 = 32", "bd5efe560473039d1f0bbfd74c346475": "\\Psi _{BETA}(\\omega)", "bd5f120afa9bfe2ad95251a2870746fe": "D^n(x^a) = (a)_n\\,\\, x^{a-n}.", "bd5f5113953dd1ee740ce74f9766a8e9": " \\mbox{SMAPE} = \\frac{\\sum_{t=1}^n \\left|F_t-A_t\\right|}{\\sum_{t=1}^n (A_t+F_t)}", "bd5f566f38d3613b3649503c4c606673": "a_{i,j} b_{f'}", "bd5fe94e1cc1d832b86b5f929b6f007a": " H = \\int_X^\\oplus H_x d \\mu(x), \\quad K = \\int_Y^\\oplus K_y d \\nu(y) ", "bd5ff661aac2d93e3537a1056bf3e57e": "{d \\over dx} \\left[ (1-x^2) {d \\over dx} f(x) \\right] + n(n+1)f(x) = 0", "bd602c27d8c77ed60673fe50cf7ce9d2": "M_\\text{bol,star}", "bd6032849eca57998455c01e14f6103a": "\n\\Phi(z,s,a)=\\frac{1}{a^s}\n+\\sum_{m=0}^\\infty (1-m-s)_m \\operatorname{Li}_{s+m}(z)\\frac{a^m}{m!}; |a|<1,\n", "bd603e36fdcc12541a39ac5b56ad6997": "a=2(1-1/2) = 1, c = -2(1/2) = -1, d = -1", "bd6050c2618cfa91378c68d57b6caa3d": "{O}(\\log n)", "bd605f3fc80714eacaa71d812c794491": "P \\land 1 \\le Q", "bd6091fcb2999804b41eafea5de7b99a": "\\frac{\\Delta \\phi}{\\omega_{\\rm orb}} = -\\frac{\\pi \\, m}{R}", "bd609dc09119a8e9e2c35632abdb1f09": "0.5 \\ge \\lambda > 0.25", "bd60a95e073622c196e1b7ab14f9d02a": "\\scriptstyle \\Rightarrow", "bd60d710ce19420dade12257b132cbda": "\\mathcal{X}", "bd6169a6da726904f27c5c8bbe2dffb5": "P(A \\mid B)", "bd6183562998ad9a5c0a609ac1ffa9ed": "\\int_{X} f(y) \\, \\mathrm{d} \\delta_{x} (y) = f(x),", "bd6229c4aabe16ba01e2d57229a3d058": "S=\\{\\infty_1,\\infty_2 \\}", "bd62af9d26f00d725548c98e6b75b715": "A + B + C + D + \\cdots + Z + \\frac13 Z = \\frac43 A.", "bd62ecf1a43719e7541c07d6ee2e587d": "\\Sigma_k \\hat{\\textbf{d}}_q ", "bd6360ea4245ba2e296be41a122a0ebd": " H^p(X) \\times H^q(X) \\to H^{p+q}(X). ", "bd637c73cd24c4bf2531f97d784b648d": "J(f) = \\left [\\frac {\\partial f_i}{\\partial x_j} \\right ]_{1 \\leq i \\leq m, 1 \\leq j \\leq n}.", "bd63c51c18793fca3cf0aa6fc25558de": "s(F) \\le 2", "bd63e7c1fd76f4fd3de69162a8c9139a": " x^2+y^2=1+dx^2y^2", "bd63eede1b9ac8becf61651405919cef": "|H_{\\nu} (\\omega)|", "bd6408cba8d23bc00962412e47c3c1bd": "\\approx .4\\mu m", "bd649d73aa7ae383158d3c80d6e6c680": "\\mu y_{y0\n\\end{array}\\right.,\n\\end{align}\n", "bd919c1f503521b56916153ab98bfc3c": "p = (p_1, \\ldots p_d)", "bd919d0e75e8a1b38e38b425b7431449": " u=\\frac{1}{\\psi} \\frac{\\partial^2 \\psi}{\\partial x^2} - \\lambda.", "bd91a2e001fc549904264085f604110b": "\\vec{F} = m \\vec{g}", "bd91c77e97e8e180e4eaf2ee8abc44c6": "\\left| S \\right| = \\Delta(C_{out}(m_1),C_{out}(m_2)) \\ge (1-R)N", "bd91cc4f6aee2d3ce8c11b35c3cc4576": "\n\\psi(y;t+\\epsilon) = \\int_{-\\infty}^\\infty \\;\\;\\psi(x;t)\\int_{x(t)=x}^{x(t+\\epsilon)=y} e^{{\\rm i}\\int_t^{t+\\epsilon} (\\frac{\\dot{x}^2}{2} - V(x)) dt} Dx(t) \\, dx\n\\qquad (1)", "bd926bbd1e7c6eb221f70e4345e5e527": "\n \\begin{align}\n u_\\alpha(\\mathbf{x}) & = u^0_\\alpha(x_1,x_2) - x_3~\\frac{\\partial w^0}{\\partial x_\\alpha} \n \\equiv u^0_\\alpha - x_3~w^0_{,\\alpha} ~;~~\\alpha=1,2 \\\\\n u_3(\\mathbf{x}) & = w^0(x_1, x_2)\n \\end{align}\n", "bd9298f77556bfdf395905c64d3e80f1": "\n\\begin{align}\n D(a, b) &= f_{xx}(a,b)f_{yy}(a,b) - \\left( f_{xy}(a,b) \\right)^2 \\\\ \n &= 2b(b+1) \\cdot 2a(a + 3b + 1) - (2a + 2b + 4ab + 3b^2)^2.\n\\end{align}\n", "bd92e54911c55c7f15d147063e76869a": "P\\left(k\\right)\\sim k^{-\\gamma}", "bd92f8bbf2797705d566305589808b44": "(\\mathbf{w}^{\\text{T}}\\mathbf{S}_B\\mathbf{w})", "bd931d8cc9f9fbe9e58200e8e059a652": " A_x = 2 \\pi \\int_a^b y(t) \\ \\sqrt{\\left({dx \\over dt}\\right)^2 + \\left({dy \\over dt}\\right)^2} \\, dt. ", "bd9357dd600c7477920dd0220b931af0": "\\sum_{n=1}^{\\infty} f_n(x)", "bd9379709121646fd1c1ad927c585330": "f(x_0 + h) = f(x_0) + f'(x_0)h + R_1(x),", "bd93a1e5ca2a7c0fdfda82a075b47ff4": "\\frac{3}{4} 0", "bd983f8895c65658c3954db9c16e4892": "\n\\begin{align}\nim \\langle \\Psi(t) | [\\hat{L}, \\hat{x} ] | \\Psi(t) \\rangle &= \\langle \\Psi(t)| \\hat{p} |\\Psi(t)\\rangle, \\\\\ni \\langle \\Psi(t) | [\\hat{L}, \\hat{p}] | \\Psi(t)\\rangle &= - \\langle \\Psi(t)| U'(\\hat{x}) |\\Psi(t)\\rangle.\n\\end{align}\n", "bd9841a11a78574aad8602b07c855e4b": "f^{(n)}(x)", "bd9882eee6ae24c6d2d66d0cf9150df0": "\n\\begin{align}\nU(x,z)\n&\\propto \\frac {\\sin{ \\frac {\\pi Wx} {\\lambda z}}}{ \\frac {\\pi Wx} {\\lambda z}}\\\\\n&\\propto W \\mathrm{sinc} { \\frac {\\pi Wx} {\\lambda z}}\\\\\n& \\propto W \\mathrm{sinc} { \\frac { \\pi W \\sin \\theta} {\\lambda }} \\\\\n& \\propto W \\mathrm{sinc} (kW \\sin \\theta /2)\n\\end{align}\n", "bd98bd88cb212b51f814f32acfc26cf8": "\\forall x_1,x_2\\in X: x_1\\leq x_2\\Leftrightarrow F(x_1)\\leq F(x_2)", "bd98f92750aa4a039c112954bb1e5ff1": "k_\\mathrm{spec} = \\|R\\|\\|V\\|\\cos ^n\\beta = (\\hat{R} \\cdot \\hat{V})^n", "bd992149bfb23cd3dad401846a6eabfb": "\\,2(b^2 + c^2) = 4d^2 + a^2", "bd9930b0328a4549e815323d6e358814": "\\left [\\hat{a}_i^\\dagger, \\hat{a}_j^\\dagger \\right] = 0 ", "bd99f1d9cb677df79ade058e005b84a8": "R^3", "bd9a078cf0ed656ddd3adfc591577c25": "u=v=\\frac{\\partial p}{\\partial x}=\\frac{\\partial p}{\\partial y}=0.", "bd9a79b08fe176021508b377a48a0c81": "\\nabla \\cdot (f \\vec v) = f (\\nabla \\cdot \\vec v) + \\vec v \\cdot (\\nabla f)", "bd9ac0252e42c9aa1d41fa34ec8e2180": "\\frac{1 - \\cos \\theta}{\\theta^2} = \\frac{\\sin \\theta}{\\theta} \\times \\frac{\\sin \\theta}{\\theta} \\times \\frac{1}{1 + \\cos \\theta}.\\,", "bd9ac0f7d73785a9ed4de850db52f11d": "M_{i'}", "bd9b97a8990cfd0769957c8422ee36f2": "p(5k+4)\\equiv 0 \\pmod 5\\,", "bd9bb8279ded447b5cd7e946461af692": "\\langle X \\rangle_s = \\frac {a\\mu+b\\lambda}{\\mu+\\lambda}.", "bd9bd3ae3235ccab3d57ae407678dbfa": "y(x)=2\\sin(x). \\,", "bd9bdddceec87e842249541b3e8c5fe8": " J = 1", "bd9c2d4428513d122e7e0a237dbc31f6": "q^{O(n)}", "bd9cbe8959c89af631d52eb92b64eb14": "\\frac{\\partial f}{\\partial x} = \\frac{\\partial \\phi(x,\\overline{z})}{\\partial x}.", "bd9cc145972976ab672cd0ab2159b2e6": "\\mathbf{p}_2", "bd9d0515bcef0c71104187b18a73b87f": "(f\\ominus b)(x)=\\inf_{y\\in E}[f(y)-b(y-x)]", "bd9d145600421e0ebf09c4f8f30f4379": "\\psi(X,t)", "bd9d66ac9cc994663535b93d2eb6b983": "L_x =L_0\\cdot 2^\\frac{T_0-T_x}{10}", "bd9d6d16b6439dfdd3359c2439d7daab": "\\psi:T_p(H) \\rightarrow T_{p'}(S')", "bd9d72ed6d7a33a5526f3bfb50a661da": "P(\\boldsymbol{Z}, \\boldsymbol{W};\\alpha,\\beta)", "bd9d908ecf869226c9e58b6ef821404a": "a + bi", "bd9e37c54e4d3f0ba3bc8684fee78a86": "\\frac{g(z)\\,dz_1 \\wedge \\dotsb \\wedge dz_n}{f(z)} \\mapsto (-1)^{i-1}\\frac{g(z)\\,dz_1 \\wedge \\dotsb \\wedge \\widehat{dz_i} \\wedge \\dotsb \\wedge dz_n}{\\partial f/\\partial z_i}\\bigg|_{f = 0}.", "bd9e6534fbbe8b65bec4096e92653ded": " \\xi^1, \\xi^2, \\xi^3 ", "bd9ea68ebb779682861ce8c2b9866675": "\\textstyle \\left\\Vert \\ell\\right\\Vert _{W_{p}^{m}(\\Omega )^{^{\\prime}}}", "bd9ebb46730f2360292b1e32b71ec308": "-\\frac{1}{\\theta} \\log\\!\\left( 1+\\frac{(\\exp(-\\theta u)-1)(\\exp(-\\theta v)-1)}{\\exp(-\\theta)-1} \\right)", "bd9eeb94b009b7f25bede0870ae751f9": "%\\text{ ohms reactance on kva base}_2=\\frac{\\text{kva base}_2}{\\text{kva base}_1} * %\\text{ ohms reactance on base}_1", "bd9f292da2c3cfb33fb71852e20fcded": "|\\Psi_0\\rangle", "bd9f53048c6dba2a64fbcdc51b799ad0": " \\nabla ^2 p - {1 \\over c^2} { \\partial^2 p \\over \\partial t ^2 } = 0 ", "bd9f74baa3f39a7e7c2a95013876fcb5": " d_X(f(x),f(y)) - MD(f,z)(x-y) = o(|x-z|+|y-z|). \\, ", "bd9f7b1860a9acb194c60ab76da21a26": "\n d\\mathbf{a}^{T} \\cdot d\\mathbf{l}= dv = J~dV = J~d\\mathbf{A}^{T}\\cdot d\\mathbf{L}\n\\,\\!", "bd9f7beebaa97b2aa39b3d804b64cdc9": "G = \\begin{bmatrix}\nT1 & T2 \\\\\nR(A) & \\\\\n & R(A) \\\\\nW(B) & \\\\\n \\end{bmatrix}", "bd9f8f411cfdf7655a854eb717a68c0d": "p_B (\\lambda) = \\lambda^3 - a_1 \\, \\lambda^2 + a_2 \\, \\lambda - a_3 \\, ", "bd9fb11d90b56e5e82ecf6de6eeef474": "\\frac{W(-\\ln(z))}{-\\ln(z)}", "bd9fb545195373e0a3cb1678f4e6dfa6": "M_1,M_2,\\ldots", "bd9fcc39363c10bce0de2d88d1e24233": "\\Gamma \\vdash \\Delta", "bd9fe3b3817cbd84fc9c6150f04c222c": "\\mathbf{x}_{k+1} = \\mathbf{x}_k + \\alpha_k\\mathbf{p}_k.", "bd9ffd7249f0a439ea5899d16d51b59a": "d(\\varphi)", "bda005f68d3436f92f148db482f8e766": "\\left | \\frac{a}{b} \\right \\vert \\quad \\left \\Vert \\frac{c}{d} \\right \\|", "bda0358298638ddb3f41194585b15f8d": "\\begin{matrix}a_{11}x_1+a_{12}x_2&=b_1\\\\a_{21}x_1+a_{22}x_2&=b_2\\end{matrix}", "bda0411e4b6129d514dcbfa5810fb14a": "b_1", "bda0836123cd8b14a399490e8d435e9e": "f^{(i)}(0)=c_i", "bda12d72722f3828fdf818c1583ea3f2": "Ff", "bda143f49b32d0c405426b84523b8794": "a_{ij} \\ne 0", "bda1596eb7e6f87dd50dee743295fb8e": "\\left[{n\\atop 0}\\right] = 0", "bda1a62d0146cc558dabbfd3b8d78768": " \\log s_i^2 = \\log a + b\\log m_i ", "bda2169ed5990eeca2bf2a6f2a9edb96": "r \\le 9", "bda27f813d1af78afabc736abf2bc4f5": " (T_\\Omega u)(z) = \\lim_{\\varepsilon\\rightarrow 0} {1\\over \\pi} \\iint_{|z-w|\\ge \\varepsilon} {\\overline{u(z)}\\over (z-w)^2} \\,\\, dx dy", "bda297ac247b5b5d5b6f459b261e2676": " \\log R = \\left\\{ \\begin{matrix}\n0 & \\mathrm{if} \\; \\theta = 0 \\\\\n\\frac{\\theta}{2 \\sin(\\theta)} (R - R^\\top) & \\mathrm{if} \\; \\theta \\ne 0 \\; \\mathrm{and} \\; \\theta \\in (-\\pi, \\pi)\n \\end{matrix}\\right.", "bda2babb056c39851445ec528657c27b": "\\delta^{\\ast}A", "bda31969d0d647f666563687c6a36673": "H = \\frac{n}{\\frac{1}{x_1} + \\frac{1}{x_2} + \\cdots + \\frac{1}{x_n}} = \\frac{n}{\\sum_{i=1}^n \\frac{1}{x_i}} = \\frac{n \\cdot \\prod_{j=1}^n x_j }{ \\sum_{i=1}^n \\frac{\\prod_{j=1}^n x_j}{x_i}}.", "bda33c68e41c985887f87c8cb5d968af": "a^{\\,}_i", "bda396866533a2e89853cfd0b4e6c802": "\\hat a^\\dagger_{i\\sigma} , \\hat a_{j\\sigma}", "bda39d4f033613c137d77e2f97d4bb78": "\\ell_P=\\sqrt{\\frac{\\hbar}{m c}\\cdot\\frac{G m}{c^2}}", "bda3a15b6b4f064fdf2eacd2024c60e4": "L u =\\sum_{i=1}^n\\sum_{j=1}^n a_{i,j} \\frac{\\part^2 u}{\\partial x_i \\partial x_j} \\quad \\text{ + (lower-order terms)} =0 \\,", "bda3c606e1ce64e3bda86c1fe6232693": "G_4", "bda3ce9585d53bb9acf1f7dcf2f392bc": "(7)\\qquad \\delta\\rho-\\bar{\\delta}\\sigma=\\rho(\\bar{\\alpha}+\\beta)-\\sigma(3\\alpha-\\bar{\\beta})+(\\rho-\\bar{\\rho})\\tau+(\\mu-\\bar{\\mu})\\kappa-\\Psi_1+\\Phi_{01}\\,\\hat{=}\\,0", "bda3d69fca2f1f318af61dd5d461a0d8": "f^* \\colon A^k(X') \\to A^k(X) \\,\\!", "bda425da0f98798184bd677b48a0ca17": "S_{n-1} = \\frac{n}{2 \\pi} S_{n-1+2}", "bda454128c44a14fdd8e0c33af35cf38": " p(r) \\approx P + \\frac {2 \\gamma V_{molecule} P} {k_B T r} = P + \\frac {R_{critical} P} {r} ", "bda4942193611a75b2c2ed9831d10e17": "\n \\boldsymbol{\\sigma} = -p~\\boldsymbol{\\mathit{1}} + 2C_1\\boldsymbol{B} \n ", "bda4b64d59ca093f441f90b7131e0b59": "\\mathrm{ISZERO} = \\lambda n^{\\forall \\alpha. (\\alpha \\rightarrow \\alpha) \\rightarrow \\alpha \\rightarrow \\alpha}{.}n\\, \\mathsf{Boolean}\\, (\\lambda x^\\mathsf{Boolean}{.}\\mathbf{F})\\, \\mathbf{T}", "bda4f87e2188bd6a8e3c96cedc37441f": "1 : \\sqrt{2}", "bda588be4d9518e7bd776d85084c2e84": " \\nu = cos(\\theta/2) ", "bda59d5aa34fd0b3aec260f7b04b5d2e": "\\chi = \\frac{C}{T-T_c}", "bda5b753c1cc44ecf01f6f2446ba2a77": "{dC_m\\over dC_L} =0", "bda748fd895afb66313c9caa771ce2fa": " S_n=\\frac{n}{2}[ 2a_1 + (n-1)d].", "bda7584ce450e3c1ab0ce4425333e50f": "[J^{\\mu\\nu},J^{\\rho\\sigma}] = i(\\eta^{\\sigma\\mu}J^{\\rho\\nu} + \\eta^{\\nu\\sigma}J^{\\mu\\rho} - \\eta^{\\rho\\mu}J^{\\sigma\\nu} -\\eta^{\\nu\\rho}J^{\\mu\\sigma}).", "bda75ca2eac71c3dc245d1c00c22e58a": "\\boldsymbol{r}_{i+1}^\\mathrm{T}\\boldsymbol{r}_i=(\\boldsymbol{r}_i-\\alpha_i\\boldsymbol{Ap}_i)^\\mathrm{T}\\boldsymbol{r}_i=0", "bda77bfddeacf6bcff4a67b0d2c7e16a": "\\mathit{3}^k - 1", "bda7e0f20ef3faa740f9e27fcdc0974f": "b_n=0", "bda7f291b1161db5ec22e779f68a27a2": "a_3b_0", "bda8c3e530a22f7d724189e2595edaf8": "(A + C) \\det A_Q > 0", "bda8c749814f8799e86a632603d38df7": "x_{k_n}", "bda8d29ea12cb629bb58d3c25c22a8ca": "y = px + b~.", "bda95678f32b004b846fa356a0edcbf7": "S_\\mathbf{r} = \\bigcup_{j \\in T} S_{M_j} ", "bda95869d8d03343d8bc5a33acd4b435": "L_1/L_2", "bda9643ac6601722a28f238714274da4": "red", "bda9bfdba4a1d531e6107dcee35f5aa6": "d(R) \\ge \\operatorname{dim}R", "bda9d2d3b0437dff45fe9115d6ba8f05": " P = P_1 \\left(\\frac{V_1}{V} \\right)^\\gamma ", "bdaa38af471931fb1226c2093491aad3": "\nP_\\mathrm{avg} = \\frac{1}{T} \\int_{0}^{T}p(t) \\mathrm{d}t = \\frac{\\epsilon_\\mathrm{pulse}}{T} \\,\n", "bdab5462b12b7848c3385c68b67721c0": "\\underline{x}^{-k} = \\frac{(-1)^{k-1}}{(k-1)!}\\frac{d^k}{dx^k}\\log |x|.", "bdab89f96441cfced554cac5e90ba5dc": "\\ \\gamma = \\frac{1}{ \\sqrt{1 - { \\beta^2}}}", "bdabbe5f8a56d11e76b093ef464176f8": "\\mathcal{C}_{2}", "bdabd973c47be470c6552383a19bc210": "\n2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\sin i\\ \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right) \n\\left((1-{e_g}^2+4\\ {e_h}^2)\\ \\hat{g}\\ -\\ 5\\ e_g\\ e_h\\ \\hat{h}\\right)\n", "bdac35e948ee89a9d2c4be37f7664b9c": "EPR_{AB}\\otimes EPR_{CD}", "bdacc26d650e11c3c9a824b6c503968a": " E_{+}=\\frac{1}{2}(E_{1}+E_{2})+\\frac{1}{2}\\sqrt{(E_{1}-E_{2})^{2}+4|W|^{2}} ", "bdad15ea9884c44eef2f958b7d3d5307": "\\mathcal O_L = \\mathcal O_K[\\alpha]", "bdad2afd43ea9c3915a25f26962fd9f1": "b<\\infty", "bdad3d2a4a009fa582125a15028cdde0": "\\begin{matrix}\n\\frac{\\partial \\phi(t, \\omega)}{\\partial t} & \\approx\n\t \\frac{1}{\\Delta t} \\left[ \\phi(t + \\frac{\\Delta t}{2}, \\omega) - \\phi(t - \\frac{\\Delta t}{2}, \\omega) \\right] \\\\\n\\frac{\\partial \\phi(t, \\omega)}{\\partial \\omega} & \\approx\n\t \\frac{1}{\\Delta \\omega} \n\t \\left[ \\phi(t, \\omega+ \\frac{\\Delta \\omega}{2}) - \\phi(t, \\omega-\\frac{\\Delta \\omega}{2}) \\right] \n\\end{matrix}", "bdad9592897b698de731dd710aa946ab": "x_2 = 2.054360090947453 \\times 10^{-8} / 1.786737589984535 = 1.149782767465722 \\times 10^{-8}", "bdadb83f2b86250838d67e9183a1a38d": "\\Delta x_i, i = 1, ..., n", "bdadbc49b7aaeed484e111bd1ca0c7ce": "(\\emptyset,\\mathsf{i})", "bdadfb708863a8b471c50c06f51cda5c": " C(x-t) ", "bdae2189d01268c1e0cecd6945446d3b": "\\frac{1}{f} \\approx \\left(n-1\\right)\\left[ \\frac{1}{R_1} - \\frac{1}{R_2} \\right].", "bdae303660216c7c4e2b7d6af056c54e": "V_{ACDA} \\approx \\sqrt{1372.3+ 177.8 d_{ACDA} } - 37.0", "bdae525a20269dff89cece818b68ef62": " \nf(x)=x\\left(\\sqrt{x+1}-\\sqrt{x}\\right)\n\\text{ and } g(x)=\\frac{x}{\\sqrt{x+1}+\\sqrt{x}}.\n", "bdae850b3b1ea9d05ad8922a9b030af0": "c\\ = n_m + n_{m-1} + \\cdots + n_0.", "bdae909dd640386186258b09459bbd31": "y(t) = (h*x)(t)\\ \\stackrel{\\text{def}}{=}\\ \\int_{-\\infty}^\\infty h(t - \\tau) x(\\tau) \\, \\operatorname{d} \\tau\\ \\stackrel{\\text{def}}{=}\\ \\mathcal{L}^{-1}\\{H(s)X(s)\\}.", "bdaf0957ecaef89ee58191c989fb514a": "\\Psi_4 = \\frac{1}{2}\\left( \\ddot{h}_{\\hat{\\theta} \\hat{\\theta}} - \\ddot{h}_{\\hat{\\phi} \\hat{\\phi}} \\right) + i \\ddot{h}_{\\hat{\\theta}\\hat{\\phi}} = -\\ddot{h}_+ + i \\ddot{h}_\\times\\ .", "bdaf724a60d1a26fc270c207aa43ec96": "K=\\sqrt{rr_ar_br_c}.", "bdafa1dc5172f4a53d28ee27ad965cf9": "C_{P} - C_{V} = n R\\,", "bdafd259ff486ffeee49b5585144360f": "\\Delta\\mathcal{W}", "bdafe832701324189955973f39046804": "\\tau\\rightarrow \\infty", "bdb017da72ffd1138398e269ea27289f": "\nf(z) = \\sum_{n=1}^\\infty \\left(z^2 + n\\right)^{-2}.\\,\n", "bdb028359a95d4904cd2e1959665f1aa": "\\frac{I}{2}", "bdb0947ef98d216d2886e00e6e6331e5": "\\displaystyle{C(x)+C(y)=-2i(1-a^{-1} -b^{-1}),}", "bdb0eb3306c020a5a4bd3d3c2ef3cb02": "\\frac {1}{kT}", "bdb0f74f71db4777b8e795c0f0cc1d4c": "A^\\mu\\to A^\\mu+\\partial^\\mu f", "bdb10fb1ed0dffe46848b30f214e5a58": "\\|\\gamma'\\|^2 = x'(t)^2 + y'(t)^2 \\not= 0", "bdb11f9384571f2674a92fa545403f41": "E_{ij}^\\text{dual} ", "bdb13bb68e673ced95749883d3ea6506": "r = \\frac{4 + \\lambda - \\sqrt{8\\lambda + \\lambda^2}}{4}", "bdb1d86b9155e1c5d0eb51ab67b6442f": "(\\sigma_1\\ge\\sigma_2\\ge\\sigma_3)\\,\\!", "bdb1f8328b8f4b7d0e4e88138cabf13e": "u \\equiv \\sqrt{ax+b}", "bdb20bf8bb020967b21d45edabfd652e": "t/\\tau", "bdb246cabc729f5a8c86c7f4cf23295c": "\\#", "bdb2bddbb10ec1ab97b0430eebca4d9d": "_{metric}\\alpha = 1 - \\frac{1_{metric} \\delta_{13}^2 + 2_{metric} \\delta_{34}^2}{\\frac{1}{26-1}(4\\cdot7_{metric} \\delta_{12}^2 + 10\\cdot7_{metric} \\delta_{13}^2 + 5\\cdot7_{metric} \\delta_{14}^2 + 10\\cdot4_{metric} \\delta_{23}^2 +5\\cdot4_{metric} \\delta_{24}^2 + 5\\cdot10_{metric} \\delta_{34}^2)}", "bdb30e7398dbbd84a2ca4367634238f5": " E=\\langle E\\rangle", "bdb33ad8d3f706a0d5e00dad3b24edb2": "\n D_f(p\\parallel q) = \\int p(x)f\\bigg(\\frac{q(x)}{p(x)}\\bigg) dx\n ", "bdb3ce735283612b6df231db12abeb61": "\\left(A_{i}\\right)_{x,y}=\\left\\{\\begin{matrix} \n1, & \\mbox{if } \\left(x,y\\right)\\in R_{i},\\\\ \n0, & \\mbox{otherwise.} \\end{matrix}\\right. \\qquad(1)", "bdb3f3601de72ca395d9158a1acbe580": "W_\\delta(t)({ b})", "bdb425114b55ee24c3fb78a496ea77f0": "\ns_1 = y_1 + y_2 + y_3 + y_4 + y_5\n", "bdb43d7c9303438d13dbaad3077c148a": "\n \\begin{bmatrix}\n a_{11} & 0 & 0 \\\\\n 0 & a_{22} & 0 \\\\\n 0 & 0 & a_{33} \\\\\n \\end{bmatrix}\n ", "bdb465280b2d551fad938e02ed1c2ee4": "|\\phi_1\\rang, |\\phi_2\\rang,\\dots,|\\phi_j\\rang,\\dots", "bdb4665b9fc4e64b840ec849b815b08b": "d \\vec{\\ell}_n", "bdb4668d8bf05db7b723def6acaa221d": "N^{O(\\log\\log N)}", "bdb4722c2da8a7add6e1f63fbbed4ea0": "F=qvB\\,", "bdb4a8e82a5247b221b8f30d74a4be56": "\\begin{array}{rcl}\n \\dot x &= &Ax + b \\varphi(\\theta_r(t) -\n\\theta_c(t)), \\\\\n G(t) &= &c^{*}x,\n\\end{array}\n\\quad\nx(0) = x_0,\n", "bdb4c8a7ae8ba3644c4fa7b11163959c": "O(G)", "bdb4ff0e35a86ce2fd8e31eb4bdf347d": "m'_0 = m_0 + k_0", "bdb501c1f92e0ed11ab6e9e8f9c97452": "\\int d^Dx \\sqrt{-g}\\, f\\left( G \\right)", "bdb517cfac450c8771bf96b2b4dac96d": "\\gamma:I\\to D", "bdb5196464b599cc36a7bad6870e16e1": "p \\equiv 1 \\pmod{4}.", "bdb534533456946426af7279df6234e4": "\n\\left( \\frac{\\partial S}{\\partial E}\\right)_{N,V} = \\frac{1}{T} \n", "bdb569fb853cd0ef54f6c092a71d6eab": "\\varepsilon\\to 0", "bdb579fc4c09876cea00e538fc4e1422": "\\mathrm{SO}(12)\\cdot\\mathrm{SU}(2)\\,", "bdb5995b5c77012f77703a11f9cbd30f": "\\sum_{i,j} a_{i,j}(x) D_i D_j u(x) + \\sum_i b_i(x) D_i u(x) + c(x) u(x) = f(x)", "bdb5c6fa2818532735752d9cee5cf0b2": "\\Pr[\\mbox{there is a cycle of length}> n] = \\sum_{k=n+1}^{2n} \\frac1k = H_{2n}-H_n.", "bdb5cdf1abc14a738a252d7ee22bff82": "\\hat d", "bdb5ea74631d8c7730411c6223ec31da": "\\ln \\Gamma(z) \\sim z \\ln (z) - z - \\tfrac{1}{2} \\ln \\left (\\frac{z}{2\\pi} \\right ) + \\frac{1}{12z} - \\frac{1}{360z^3} +\\frac{1}{1260 z^5}\\qquad \\qquad \\text{as }|z|\\to\\infty\\text{ at constant}\\quad |\\arg(z)| < \\pi", "bdb63ff200d1edf2515fcf6b4de4044d": "\\{c_1\\}\\cup\\mathrm{up}(c_1)-\\{c_2\\}", "bdb63ff3065ca01e55f8085b9ffc30e9": "a<_sce(ab)<_s b", "bdb670793ffda937abbf04ddd56bf8b6": "a \\sim e^{-\\Lambda p_1\\tau},\\ b \\sim e^{\\Lambda(p_2+2p_1)\\tau},\\ c \\sim e^{\\Lambda(p_3+2p_1)\\tau},\\ t \\sim e^{\\Lambda(1+2p_1)\\tau}.", "bdb67599a52f9822f9009fd50e3c7570": "\\frac{j}{2^n}", "bdb7071ef55230028e224bedfd7ecf2b": "\\varphi : H_0(F) \\to H_0(V)", "bdb725204de40bf0d87df58bf34d76bb": "\\Omega_p ", "bdb73538f3ea1854a5d69843d6d94970": " \\displaystyle{K_S(a,z)=C \\cdot \\exp\\,{1\\over 2\\alpha}(\\overline{\\beta} z^2 + 2az - \\beta a^2)}", "bdb7d16783b9608667cb5755b905c86f": "\nX |\\psi\\rangle = \\int_x x \\psi(x) |x\\rangle ~ .", "bdb7d69fcaa984a12a3bde4205b94202": "A\\times B\\times \\{1\\}", "bdb7fd4ddc610a781bec5f9a649f9ded": "\n \\boldsymbol{\\sigma} = \\cfrac{2}{J}~\\boldsymbol{F}\\cdot\\cfrac{\\partial W}{\\partial \\boldsymbol{C}}\\cdot\\boldsymbol{F}^T \\qquad \\text{or} \\qquad\n \\sigma_{ij} = \\cfrac{2}{J}~F_{iK}~\\cfrac{\\partial W}{\\partial C_{KL}}~F_{jL} ~.\n ", "bdb812d241c286581d237b4d33ede512": "(1\\,n)(2\\,n-1)\\cdots,\\text{ or }\\sum_{k=1}^{n-1} k = \\frac{n(n-1)}{2}\\text{ adjacent transpositions: }", "bdb8898ed76853769c90e96bb7a183d1": "217 \\cdot 15,540 = 3,372,180\\,", "bdb89f8123c82ba589f1a588c8414560": "\\overrightarrow{D\\varphi}\\mp\\frac{1}{g}\\vec{B} = 0", "bdb91627e4227c3dceaa9e47744c6d1b": "\\mathrm{Iso}(\\mathcal{A})", "bdb918167ddc06a1160bfbdf0b283c8a": " T_H = [H] + \\beta_1[A][H] + 2\\beta_2[A][H]^2 - K_w[H]^{-1} \\,", "bdb92424d687d9adf7836ba22e5c3fa0": "(Q,T)", "bdb92e783064112720cf3b66cc2e323a": "\\,\\zeta(s)=\\Phi (1,s,1).", "bdb92f698427fcfc6a3bdde77e230e88": "\n\\mathbf{A} = \\begin{pmatrix}\na_{11} & a_{12} & a_{13} \\\\\na_{21} & a_{22} & a_{23} \\\\\na_{31} & a_{32} & a_{33}\n\\end{pmatrix}\n= \\begin{pmatrix}\n1 & 2 & 3 \\\\\n4 & 5 & 6 \\\\\n7 & 8 & 9\n\\end{pmatrix}", "bdb9766a955ad77c51691d2a1ecf7c27": "L_{m\\cdot n}(\\xi)=L_m(L_n(\\xi))", "bdb9e0eff76affcd54ae23d00c4a1292": " \\exp \\left( \\frac{ i \\theta}{2} \\, \\sigma_2 \\right) =\n\\left[ \\begin{matrix} \\cos(\\theta/2) & -\\sin(\\theta/2) \\\\ \\sin(\\theta/2) & \\cos(\\theta/2) \\end{matrix} \\right]. ", "bdba0233d7fcd8d8d399cbc00c697b32": " P_2 : [x_2 : y_2 : z_2]. ", "bdba5650480b8ceb007a3fb0612b515a": "\\Phi^{(n)}_\\lambda", "bdba81b651876aac88c546a1e995a01d": " \\int_0^\\infty \\! \\mathrm{P}(X\\ge x)\\;\\mathrm{d}x =\\int_0^\\infty \\int_x^\\infty f_X(t)\\;\\mathrm{d}t\\;\\mathrm{d}x = \\int_0^\\infty \\int_0^t f_X(t)\\;\\mathrm{d}x\\;\\mathrm{d}t = \\int_0^\\infty t f_X(t)\\;\\mathrm{d}t = \\operatorname{E}[X]", "bdba9cc156ecd7a8db38b206d6cc492e": " \\forall x \\in A, \\Phi(x, \\alpha_1, \\ldots, \\alpha_n) ", "bdbad48f91246e4472d60f58cfc4c124": "a=\\frac17(34\\pm 6\\sqrt{21}).", "bdbb03f6ca4fb296f8ec516358cac4ea": "w(r) = -2 \\pi \\, C \\rho _1\\, \\int_{z=D}^{z= \\infty \\,}dz \\int_{x=0}^{x=\\infty \\,}\\frac{xdx}{(z^2+x^2)^3} = \\frac{2 \\pi C \\rho _1}{4}\\int_D^{\\infty }\\frac{dz}{z^4} = -\\pi C \\rho _1/ 6 D^3", "bdbb15c7eb6f84cce253a4ba00700dad": "{\\bar{S}}_5", "bdbb20df53356ef9ab737609779999b2": "\\frac{\\prod_{i=1}^{c} \\binom{K_i}{k_i}}{\\binom{N}{n}}", "bdbb8aceb0f5c8bdaf497fd6493b6e50": " M(i,j)=\\frac{a_{ij}}{\\sum_{j=1}^n a_{i,j}}", "bdbbf0133763eea5dd2c3bbf69abbb6e": "r_1 = 2, r_2 = -1, \\omega_1 = 1, \\omega_2 = 2", "bdbc424a69449035ac4dd646112327ca": " 0 \\leq i \\leq L ", "bdbc429f9f120e8f7046846e044af62e": "\n\\sum_{n=0}^\\infty \\frac{2 a_n}{3^{n+1}}.\n", "bdbc4bb248d2fa28ddef56daf6768df0": "\\beta_{pq...}^\\ominus=\\mathrm{\\frac{[M_pL_q...] } {[M]^p [L]^q ... }\\times \\Gamma}", "bdbcce19944eedc1a5665e3ff83e7c40": "\\,\\mathfrak{Re}\\left(\\text{Fourier} \\left[ \\frac{\\sin(x)^1}{x}\\right] \\right)", "bdbd822b7151e63cd6cc3bc6d522ec24": "_2F_0(\\cdot, \\cdot; ;-1/x)", "bdbda826c4a255ae0a87301d017b1962": "\\sum_{o\\in O}J_o^T = \\sum_{\\chi\\in X}J_\\chi^T.", "bdbdb2fede62d05353d5ad56ca90654d": "F(x) + C", "bdbdd45dd2551d68448e4e5200169bbb": "\\lambda(n),", "bdbdf0d6a3fa520f36cdbae052820b0b": "\\Theta \\subset \\mathbb{R}^k", "bdbe0b6a2746e2ea2d40ac535ae2e869": "\\vec r_1,\\ldots ,\\vec r_N", "bdbeb5666d280c9c98765905805cb078": " J^2 = J_x^2 + J_y^2 + J_z^2 ", "bdbf617b20313e6707d538f6bfc0e8a5": "(\\text{coin},\\text{black area})", "bdbf889589c70fd4f95e334068d5e538": "\n\\begin{bmatrix}\nx' \\\\\ny' \\\\\n\\end{bmatrix} = \\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\\begin{bmatrix}\nx \\\\\ny \\\\\n\\end{bmatrix}", "bdbfc3695d6644472effb0bccc67511e": "x_{k+2}=6x_{k+1}-9x_{k}\\,", "bdc00c75ad00927574dfb0dc8e4934a2": "y = (x + 5)(x - 2)", "bdc027bbaabb563ce819663d131c4cff": " -\\ln[-\\ln(F)] = (x-\\mu)/\\beta ", "bdc03e2c22f50574abb6a7ca190cef92": " r=L_c \\,", "bdc05d7dc025bdb8744a41c81997f6b2": "(\\sqrt[5]{100})^{15.2-8}\\approx 758", "bdc07b655170a58f742105b8b0595b8c": "L=\\frac {AS^2}{100(\\sqrt{2h_1}+\\sqrt{2h_2})^2}", "bdc09e2a80a459cdce64c523eb49289e": " \\Delta\\tau = Gb\\sqrt{c} \\epsilon^{3/2} ", "bdc09e307e822ea0c8a87a329e6f16af": "0, 1, 2, 3, \\ldots, n, \\ldots ; \\aleph_0, \\aleph_1, \\aleph_2, \\ldots, \\aleph_{\\alpha}, \\ldots.\\ ", "bdc0bfbdc0cdd02efdca174a81cfe67e": "\\nu=1/(2p+1)", "bdc0f476a70f430b015db9ab167b2704": "\n\\begin{align}\n\\Box & = \\partial^\\mu \\partial_\\mu = g^{\\mu\\nu} \\partial_\\nu \\partial_\\mu = \\frac{1}{c^2} {\\partial^2 \\over \\partial t^2} - \\nabla^2 = \\frac{1}{c^{2}} \\frac{\\partial^2}{\\partial t^2} - \\frac{\\partial^2}{\\partial x^2} - \\frac{\\partial^2}{\\partial y^2} - \\frac{\\partial^2}{\\partial z^2}.\n\\end{align}\n", "bdc1456d10dae8739acec38aa0110e89": "\\delta^n(\\varphi) = \\varphi \\circ \\partial_n", "bdc233f6983b0d7b79e1558c8edc099a": "- \\left(\\frac{d t}{d x}\\right)^{-3} \\frac{d^2 t}{d x^2} = \\left(\\frac{d t}{d x}\\right)^{-n} f(x)", "bdc28031cb4a94b320d0dfc87aa1e9df": "dm_e", "bdc29947292c779b768dca47063c5c05": "E_1 \\subseteq \\mathcal{D}", "bdc2bb340897e34504b1041306124496": "\\mathcal A \\stackrel{F}{\\rightarrow} \\mathcal B \\stackrel{G}{\\rightarrow} \\mathcal C, \\,", "bdc2dd71faf26f89aab7bccb56e70820": "\\sum_{n=1}^{\\infty} M_n < \\infty", "bdc2f992f59cfd68656264e93f2f63c3": "\\frac{d_1 X/d_2}{1+d_1 X/d_2} \\sim \\operatorname{Beta}(d_1/2,d_2/2)", "bdc3472578e4cf2bdfe0a4d3c8ee6b61": "d_{1,-1}^{2} = \\frac{1}{2}\\left(- 2\\cos^2\\theta + \\cos \\theta +1 \\right)", "bdc3bc196d9f93ccac41390ce5793f53": "p:\\mathbf{R}^k \\to E_x", "bdc3c190b2bedddb58faf4697ff6e23e": "w_i=a", "bdc3c7dd5cf86455e220fea7dbf2abc0": "\\beta_T(X)\\subset \\beta_G(X)", "bdc53ada38114e72507714d7cc50a62c": "\\lambda_{2}= 0.0971028 - 0.995786i \\,", "bdc547e36c7691ec97e5d3d31ede9fed": "x_1,x_2,x_3,x_4,x_5 \\ge 0", "bdc5bd650581cc477a9a07d097797de6": "\\lambda m.\\lambda n.\\lambda f.m\\ (n\\ f) ", "bdc5d25a342fb54a19f75e97a3486838": "M\\{\\,a\\,\\}=\n\\begin{bmatrix}\n1&0&0&0&1&1&1&1 \\\\\n1&1&0&0&0&1&1&1 \\\\\n1&1&1&0&0&0&1&1 \\\\\n1&1&1&1&0&0&0&1 \\\\\n1&1&1&1&1&0&0&0 \\\\\n0&1&1&1&1&1&0&0 \\\\\n0&0&1&1&1&1&1&0 \\\\\n0&0&0&1&1&1&1&1\n\\end{bmatrix}\n", "bdc617e548f3cf2c3eb231a202d75f5f": " B(\\Gamma) = 1-\\Gamma/\\Gamma_0 ", "bdc634820cb035ab2586a6e6605453fc": "\n\\epsilon_S = 1 - \\frac{\\mu_a \\cap (H^T \\mu_b H)}{\\mu_a \\cup (H^T \\mu_b H)}\n", "bdc6582b8426b699257dc63f66c7b041": "\\phi_i(v+w) = \\phi_i(v) + \\phi_i(w)", "bdc6641e7429f770187cf8c3d3395800": "w = 1 + 0 + 4 + 1 \\ \\bmod\\ 7 = 6 = Saturday", "bdc67a6c178f09dd8716def1da948ece": "(\\partial\\phi_{2,m}/\\partial\\lambda) = H_{2,m} (\\partial n_m/\\partial\\lambda) + (k_{1,m}k_{2,m})^{-1}\\bigg(H_{1,m}(\\partial n_m/\\partial\\lambda) + (\\partial\\phi_{2,(m-1)}/\\partial\\lambda)\\bigg)", "bdc6c0a05e2d0dfcd3d1dfd85efdab80": "\\tilde R_{ijkl} = e^{2\\varphi}\\left( R_{ijkl} - \\left[ g {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} \\left( \\nabla\\partial\\varphi - \\partial\\varphi\\partial\\varphi + \\frac{1}{2}\\|\\nabla\\varphi\\|^2g \\right)\\right]_{ijkl} \\right)", "bdc7070471d64842d62b4d3fbb232824": "a^{ij} = L_{p_ip_j}(Dw)", "bdc734b371906f96638ae82e99903c80": "\\lnot [\\forall z (z \\not \\in A)].", "bdc7862e131f5d5df5f4a752ee22cc5a": "z \\in [2,4)", "bdc7c0f82fad8f1d58a7e535b3529da3": "\\ T_{ij}", "bdc7f07114a2f127369290817d6178f1": "\\sigma_{i\\,j}", "bdc7f8496348d633593fe43a4ecf3eaa": "C=C (Y-T(Y), i-E( \\pi )) \\, ", "bdc81afe3a1ec95c284d35ede0a3319b": "\\mathbf{Z}/p\\mathbf{Z}.", "bdc81bec57f2dd901205ffefee15897d": "R = \\begin{bmatrix}\n\\star & \\star & \\star \\\\\n0 & \\star & \\star \\\\\n0 & 0 & \\star \\\\\n0 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix}.", "bdc84ce56ab6793e71359305ef69a2c3": "\\boldsymbol a ' = \\left( \\ddot {r } ' -r ' \\dot {\\theta } ' ^2 \\right) \\hat { \\boldsymbol r } ' + \\left( r ' \\ddot \\theta ' +2 \\dot r ' \\dot \\theta ' \\right) \\hat {\\boldsymbol \\theta } ' \\ ", "bdc85221b7a070ef34e3caeb5a866194": "{1 \\over 2} (2A) + {1 \\over 2} \\left({A \\over 2}\\right) = {5 \\over 4}A", "bdc89a08402b70149b14fd75ec3fc837": " F_i = \\sum_{j=1}^m \\mathbf{F}_j\\cdot \\frac{\\partial \\mathbf{r}_j}{\\partial q_i},\\quad i=1,\\ldots, n,", "bdc8a6ff587fdf331ef5021472751ff2": "\\epsilon_i \\in \\{\\pm 1\\}", "bdc8f68e0ec07e4e67299a2f4399068d": "Z_{11} = {Y_{22} \\over \\Delta_Y} \\,", "bdc90190232cd065a18bf21a301727d5": "F=qE\\,", "bdc921126e95ca33059e0f6b162df48f": " y_\\mathrm{net} = \\sum_{i=1}^N y_i \\,\\!", "bdc9617d1b95f54f0a4690d650a7725d": "a=\\frac{r_{a}+r_{p}}{2}", "bdc986e12d436e05719eb2dd93751951": " \\int_{-\\infty}^\\infty \\frac{\\sin^3(\\theta)}{\\theta^3}\\,d\\theta = \\frac{3\\pi}{4} \\,\\!", "bdc9eff6fca120dc051d3e5dc90e0dec": "(p-k)^2\\equiv k^2 \\pmod{p}", "bdc9f24bd89807a974ccf192f5a0221b": "{d^2 f \\over dx^2} - {2x \\over (1-x^2)} {df \\over dx } + {n(n+1) \\over (1-x^2)} f = 0.", "bdca204b3bd59c1ee1c0dfcc1e394d7b": "\\Omega={\\mathbf e}\\Omega(\\mathbf e){\\mathbf e}^*", "bdca743629332992fd0aa0a8f8444087": " D^{creep} = \\Phi^{creep} \\int_0^t Ae^{(-\\Delta H/RT(t))} \\left(\\frac {\\alpha_1 \\bar{\\sigma} + \\alpha_2 \\sigma_H} {K} \\right)^m dt ", "bdca7cb8b0af95a8f2ecb38a46fb48e2": "x_2 = \\tfrac13(s_0 + \\zeta s_1 + \\zeta^2 s_2).\\,", "bdca971c2e545ac7b208c6a7e5dd61a9": "C: y^2 = f(x)", "bdcaf44edae22c80c658b8b3bff015b7": "N_X \\mathcal U ", "bdcb2df7e4fd9a307784c02c4461e47c": "\\sigma(x)>\\sigma(y)", "bdcb6c6365f0667dffe20fe80bdba882": "\n\\sigma_3(n) = \\frac{1}{5}\\left\\{6n\\sigma_1(n)-\\sigma_1(n) + 12\\sum_{00\\\\\n \\beta & {} = (1 - \\mu) \\nu , \\text{ where }\\nu =(\\alpha + \\beta) >0.\n\\end{align}", "bded6cd26d0a012bb8cfe7e993bcf6f3": "E_n^{(+)}=a_nexp(-i\\nu_nt)", "bded7640d8880d1640f6bf8b29e33733": "B'=B^\\frac{1}{2.19921875}", "bdedb3ac145d23c7543d57173977cae8": "\\mu^{\\text{v}}", "bdeddf8a34fc1e60917063b6899c50d6": "C_k(s)=1", "bdedfea755e3953c08c0c021499438b4": "A \\cup A^C = U\\,\\!", "bdedfecbb1c2c4733514a9278593e3a0": "2b\\sqrt{npq}", "bdee03534d2aca7a5890dbf7cec2d5b2": "y_{\\mathrm{low}} = (x*g)\\downarrow 2 ", "bdee25c6213bc1694681bfee53c9c777": "\nh_1 = \\frac{L_1}{m} = r v_1 = r^2 \\frac{d\\theta_1}{dt} = 2 \\frac{dA_1}{dt}\n", "bdee612aa542dfcd2f377c13541a5b24": "\\cfrac{\\partial x_i}{\\partial q^k} \\mathbf{e}_i = \\mathbf{b}_k, \\quad \\cfrac{\\partial q^i}{\\partial x_k} \\mathbf{b}_i = \\mathbf{e}_k", "bdee6c26ce956535b88bf71e3bbeeee4": "\\sigma'\\ = {\\sigma\\over\\sqrt{1 - v^2/c^2}} ", "bdef057f6811885befdf33a53c0b3686": " \\bar{x} = \\frac{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y) x\\, dx\\, dy} {\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y) \\,dx \\,dy} ", "bdef1801c4b417df01d7ef091692229b": " \\Lambda_D ", "bdef1ba72b1b2fd97b2d9d8ec8b37e2b": "\\mathrm{Be} = \\frac{\\Delta P L^2} {\\rho \\delta^2}", "bdef5e458dc2d2cec7d8406f0dbfc069": "\\epsilon_{n+1} = \\text{constant} \\times \\epsilon_n^m, \\ m=1 .", "bdef799f6f8ecf660179854b20dad334": "\n a^3 = \\cfrac{3R}{4E^*}\\left(F + 3\\Delta\\gamma\\pi R + \\sqrt{6\\Delta\\gamma\\pi R F + (3\\Delta\\gamma\\pi R)^2}\\right)\n ", "bdefb2a50505bfe41a93dca4330e9dbf": "z^2+w^3", "bdefd406814a16c46631eeb6eaec7b8c": " \\sum_{n=1}^\\infty \\pm \\frac{1}{\\sqrt{n}},", "bdf0192b4884ffc5d8952b151eb9529f": "\\alpha_{mk} \\in D_R", "bdf09c6e9eb113e30482ca7b953833ad": "k=\\frac {AE} {L} ", "bdf0c01e226cd1a74123369691890ea5": "\\hat{F}_{\\phi}=\\frac{1-|\\psi\\rang\\lang\\psi|}{1+|\\lang\\phi|\\psi\\rang|} ", "bdf0cc8bbcc9d976f3675b85fb744541": "\\omega = \\frac{K}{N}", "bdf1392d13f945aff9eeae6e7a39d42d": " r^{-2}~\\sin\\theta \\,", "bdf190fe9c3aa1ec48505bda6e582964": "m=\\frac{-b}{2a}", "bdf1bb6d8b8f9ab1171f0100a5b62e50": "Q_{Actual}\\,\\!", "bdf1f4686923731229742043787a3176": " g(x) = \\frac{(\\cos\\beta)x - \\sin\\beta}{(\\sin\\beta)x + \\cos\\beta}, ", "bdf21431bd396e93a3ff7103856063ad": " g_j \\in X' ", "bdf24cb32f74b97200209b3cc2dcf146": "\\mathcal{}MU_*MU", "bdf2c244ac338b0f16fbd30c22360ba6": "_{(qp')'\\,}\\!", "bdf3259f701b44c696d58676b11b8512": "{\\rm TIME}(C)", "bdf32b1e7f2d63fd9503e54324ec00b8": "{q_c}", "bdf34453d03e1ccbfa8792d0c6591801": " q+\\lambda ", "bdf35164fa9030e26234376efab4cd6e": "n \\log n\\,2^{O(\\log^* n)}", "bdf3b3ed73dc67b5f7dad028664d43b8": "x^2 + 6x + 2", "bdf3fcf5ee93515e1e417e5c101216c0": "\nD_{i_1, \\dots, i_m}(\\mathbf{x},s) = s^m L_{i_1, \\dots, i_m}(\\mathbf{x},s)\n", "bdf409c50f6910d31a3e0ad499d85080": " \\scriptstyle e^{i \\omega t} \\,=\\, \\cos(\\omega t) + i \\sin(\\omega t) \\, ", "bdf4974ee7a4eb686d65b641cc7cabe5": "p(x) \\propto \\exp\\left( -\\frac{1}{b_2} \\int\\!\\!\\frac{x-a}{(x - a_1) (x - a_2)} \\,\\mathrm{d}x \\right). \\!", "bdf4b507084a6ba89bcd8c0bd00b64b8": "i_\\text{i} = C{d \\over dt}(v_\\text{i} - v_\\text{o}) \\,", "bdf4ec9a20363474995369686037a346": "Q_{A} \\,", "bdf58568ffd21d3ceea5689399f6aa35": "\\sqrt{1.5} \\approx 1.22", "bdf58f5704ccb300f82da0b9824467e0": " \\quad 0 < \\beta \\leq 2 ", "bdf5b51c254c0b92eefcaf7832b75e42": "\\text{DC Fan-out} = \\operatorname{min}\\left ( \\left\\lfloor\\frac{I_{\\text{out high}}}{I_{\\text{in high}}}\\right\\rfloor ,\\left\\lfloor\\frac{I_{\\text{out low}}}{I_{\\text{in low}}}\\right\\rfloor \\right ) ", "bdf5bb1ff26e73061b7b4e66eef439d7": "\\frac{1}{\\sigma^3}\\sum\\limits_{i=1}^n {\\left( 1-2{p_i} \\right)\\left( 1-{{p}_{i}} \\right){{p}_{i}}}", "bdf5d4b1ce055ed3182a4c2b144fb183": "x\\mapsto mx\\ \\bmod\\ 1", "bdf5daf6a969fe5c88830b36f7c0c069": " r_2 \\,\\!", "bdf5e0bc4c65258ef69a960cf673c1c6": "\n\\ H(z) = \\frac{z^K}{z^K - \\alpha} \\,\n", "bdf60a2beb1f5d6d147194ea09933479": "\n\\mathbf{Ta} \\cdot (\\mathbf{Tb} \\times \\mathbf{Tc}) =\n\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c}),\n", "bdf60f26645d19d2f18ca3ae2a5b8b80": "Expr' \\rightarrow {} + Term\\ Expr'\\,|\\,\\epsilon", "bdf6841c119d7e35d66abeb09bacf4f7": " \\int_{a}^{b} P(x) \\, dx =\\frac{b-a}{6}\\left[f(a) + 4f\\left(\\frac{a+b}{2}\\right)+f(b)\\right].", "bdf68d445859b0f87aaca621c3edb316": "S(x)=2x^2-x+4.\\,\\!", "bdf68f92f66b3e02cac0a53dbb21d8d7": "(B \\setminus A)^C = A \\cup B^C\\,\\!", "bdf6997060b4c4e4a77b953d5bb92713": "\np \\rightarrow p+dp = p -{\\partial H \\over \\partial x} dt ~.\n", "bdf6bbfa906dbc2d729e9363f5e4c98b": "R-r", "bdf6c3662836fe5683a083246c07d749": "z_n=e^{i\\theta_n}", "bdf6ca28f2f91a481dc163e2c5b89368": "\\scriptstyle \\mathbf{n}_{A}", "bdf6e60242d313a62cd1d8d89c4e04e8": "\\mathbf{F} \\cdot {\\rm d}\\mathbf{A}", "bdf724cc943f6f9d572480684400dc3c": " \\max_{0 \\leq x \\leq 2\\pi} | f(x) - P_n(x) | \\leq \\frac{C(f)}{n^\\alpha}, ", "bdf73d504e145a975cecf17f5af3aca6": "g_{mk,\\ell}\\equiv\\eta_{mk,\\ell}=0", "bdf7b8b03eb7addab1bfc180f7bbfead": " u(t,0)=0, \\quad u(t,L)=0,", "bdf8466e259107339874eea9181557d3": "T := \\{t_1,t_2,\\ldots,t_n\\}", "bdf891fe7d51bba75de6a8b8ee74495f": "\\int\\sin a_1x\\cos a_2x\\;\\mathrm{d}x = -\\frac{\\cos((a_1-a_2)x)}{2(a_1-a_2)} -\\frac{\\cos((a_1+a_2)x)}{2(a_1+a_2)} +C\\qquad\\mbox{(for }|a_1|\\neq|a_2|\\mbox{)}\\,\\!", "bdf8d185dbce8b2b93fab2b65fb0d99f": "\\eta = \\sinh^{-1}[\\frac{w}{c}] = \\tanh^{-1}[\\frac{v}{c}] = \\pm \\cosh^{-1}[\\gamma]\n", "bdf956f35e199be3037efdd9c5099e47": " \nI^m_{\\ell}(\\mathbf{r}) \\equiv \\sqrt{\\frac{4\\pi}{2\\ell+1}} \\; \\frac{ Y^m_{\\ell}(\\theta,\\varphi)}{r^{\\ell+1}}\n", "bdf98549953e24f477a24fced1a3e753": "w(t),w", "bdf9dda2ee50bf86baaa88e04c90d886": "\\hat{\\Theta} ", "bdf9f3e689b685d4587030bc1c2d75ce": "U=w(v-B(x))=F(x)(v-B(x))", "bdf9f794e718b37ee0331e867499054b": "t>T", "bdfa0d8a277bcb67862b81fb03629433": "\\neg \\neg P \\to P", "bdfa0dea7b916a174675f27ac04beb43": "\\frac{d}{dz}\\frac{S_0}{\\hbar}= \\sqrt{k^2 - 2mV/{\\hbar}^2}", "bdfa146669860f3362a5237f35716ce4": "\\mu\\left(\\bigcup_{\\alpha\\in\\lambda} X_\\alpha\\right)=\\sum_{\\alpha\\in\\lambda}\\mu\\left(X_\\alpha\\right).", "bdfa2e69f2a18b75df271311a44dcae2": "R_0^0=-\\frac{1}{2}\\frac{\\partial \\varkappa_{\\alpha}^{\\alpha}}{\\partial t}-\\frac{1}{4} \\varkappa_{\\alpha}^{\\beta} \\varkappa_{\\beta}^{\\alpha}=0,", "bdfa3c0340d9ca72f759ef5b843f6c67": " \\gamma ", "bdfa4ed2dea9fd83dcbf8413ea3c774a": " \\prod_{k=0}^n \\mu(k,A) \\leq \\prod_{k=0}^n \\mu(k,B) ", "bdfaa36ba3cd25abf8cf37e2de7bbbf6": " X_1,X_2", "bdfb008bfbe589aee9bc497655f7b31f": "\\sigma(T) = {\\bar \\sigma_{\\mathrm{pp}}(T)} \\cup \\sigma_{\\mathrm{ac}}(T) \\cup \\sigma_{\\mathrm{sc}}(T).", "bdfb00e89ee2174b94092b830da41edf": "T = 2\\pi \\sqrt\\frac{I}{mgR},", "bdfb0fe0737e83edc5c8cc930e38b43c": "p_1, \\, p_2, \\, \\cdots,\\, p_{20} ", "bdfb764ba3335b78ea5881e31c850c54": "\\|w\\|_0", "bdfb974277fb00fdece3043eacfce258": "\\psi(r,\\theta)", "bdfbba598e5b241566a6424fb9edb41f": "S^1 \\times D^2 \\subset S^3", "bdfc3e5877f8c57d44bd9b402db99b31": "\\frac{\\partial y}{\\partial \\mathbf{x}}", "bdfc939103dbb704506cea8f8140aaed": " S^1\\cong\\operatorname{SO}(2)\\cong\\operatorname{U}(1)\\cong\\operatorname{Spin}(2) ", "bdfcab164d08c3271b2843d0981fccd0": "\n\\begin{align}\n\\big[ a^{(\\mu)}(\\mathbf{k}),\\, a^{(\\mu')}(\\mathbf{k}') \\big] & = 0 \\\\\n\\big[{a^\\dagger}^{(\\mu)}(\\mathbf{k}),\\, {a^\\dagger}^{(\\mu')}(\\mathbf{k}')\\big] &=0 \\\\\n\\big[a^{(\\mu)}(\\mathbf{k}),\\,{a^\\dagger}^{(\\mu')}(\\mathbf{k}')\\big]&= \\delta_{\\mathbf{k},\\mathbf{k}'} \\delta_{\\mu,\\mu'}.\n\\end{align}\n", "bdfcdaae512b793e751cac8743366531": "\\textstyle \\Omega_2 ", "bdfce85d85798d3936171af9bff8d902": "b=f(0)=0.", "bdfd00eda0d09ba32b2e7946257dfab8": "\\nu = \\tfrac{\\mu}{\\rho}", "bdfd08eb77e9325f5dfc5b1e211b1056": "\\mathbf{r}_{X} = \\mathbf{B}_X^T \\mathbf{q} = \\mathbf{B}_X^T \\Big[ \\mathbf{f}\n\\Big( \\mathbf{B}_R \\mathbf{R} + \\mathbf{B}_X \\mathbf{X} + \\mathbf{Q}_v \\Big) + \\mathbf{q}^{o} \\Big] = 0 \\qquad \\qquad \\qquad \\mathrm{(7a)} ", "bdfd1759f0e74571412b077c85cf69a7": "\\nabla x^\\mu=-\\omega^\\mu_\\nu x^\\nu\\;", "bdfd4e69cc88555e49fc5dc1cb6e37c1": "{(m - M)}_{0}", "bdfd9b20ff80c1c0c4e58771866b27d4": "\\begin{align}\ne^{i \\theta} z &= (\\cos \\theta + i \\sin \\theta) (x + i y) \\\\\n &= x \\cos \\theta + i y \\cos \\theta + i x \\sin \\theta - y \\sin \\theta \\\\\n &= (x \\cos \\theta - y \\sin \\theta) + i ( x \\sin \\theta + y \\cos \\theta) \\\\\n &= x' + i y' ,\n\\end{align}", "bdfdb30d82d96edd76fd87476a2a0de5": "\\scriptstyle \\partial V", "bdfdc51f291e5595cabcf599ee1f3cfe": "\\lang O_i|O_j\\rang = \\delta_{ij}", "bdfdd9855e0e91724b2e0f669e144133": "\\psi_2(t)=C_{\\psi_2}\\cos(\\omega_0 t){\\rm sech}(t)", "bdfdf62e3b235f909ebc80b2ce388cd7": "R = \\begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\\end{bmatrix}", "bdfe65565546b202d1cee271a28c441c": "\\begin{align}\n \\langle w_i^2z_i \\rangle &= \\frac{\\langle w_i z_i \\rangle}{z} \\\\\n \\langle w_i^2 \\rangle &= \\frac{\n - b z^2 w^2\n + 2 b z^2 w \\langle w_i z_i \\rangle\n + b z \\langle w_i z_i \\rangle^2\n - b z^2 \\langle w_i z_i \\rangle^2\n - 4 \\langle w_i z_i \\rangle^3\n }\n {b z^2 - 4z \\langle w_i z_i \\rangle}\n\\end{align}", "bdfe6587745eba1c893bfb877d0e6031": "0.7\\overline{8}5714\\overline{2}", "bdfe65f7dda54c68271cd83af0a28302": "{d(m_{rel}\\mathbf{v})}\\!", "bdfe6c93fa90825a0e87cb850719caae": "P_C = 6.3872304\\,d", "bdfe7416faed1eff14823ec146689d06": "f_0(E,E_F,T)=\\frac{1}{1+e^{(E-E_F)/(k_BT)}}", "bdfebd9ac8dfe2be56e07820135e95cf": "\\displaystyle f(x) = \\frac{1}{(2 \\pi)^{n/2}} \\int_{\\mathbf{R}^n} \\hat{f}_3(\\omega)e^{i \\omega\\cdot x}\\, d \\omega \\ ", "bdff02c25c202c3443714db4e0a7a195": "\\ SS = 1- \\frac{MSE_{forecast}}{MSE_{ref}} ", "bdff1eeccb2f53c78f7ca40f345bcff5": " \\hat{A}^* = \\mbox{Sc}(\\hat{A}) - \\mbox{Vec}(\\hat{A}).\\!", "bdff5906728187bb804afacdaacbb8ba": "\\sqrt{{1\\over m}} \\sigma", "bdff6259d5205c5bfebfb43565c15dd7": "p(b)", "bdffcbaa8ded7e9d55ba2b96d6f72279": "[x]_{\\mathrm{RED}}", "be00f1d8e6b6fa3ae45e7baea2f04eb5": "K_{\\mbox{k}}", "be010b885dabd6541540908720a7f917": "\\lambda_{\\text{e}} = h/m_\\text{e} c", "be013af3ce833433142e2eeeaaaf4a3b": "\\phi + \\omega \\, t - \\frac{t}{\\omega \\, r^2} = 0, \\; \\; -\\pi < \\phi < \\pi ", "be013b2eb716d9fbad3c6fe685f01612": " P(W_i) = {\\frac{K}{N}} .", "be017bc8773fd646db762e6cf391d6a6": " \\chi(ab) = \\chi(a)\\chi(b)", "be01cfaac2fa87280bdb95a8edeb2fdd": "\\beta(1)\\;=\\;\\tan^{-1}(1)\\;=\\;\\frac{\\pi}{4}, ", "be01daffd685ce17c3427f48e803868e": "Log(K_{eq}) = \\frac{9141}{T} + 0.000224T - 9.595", "be033dc7f45139c969cf8ea40fc1ae5b": "k=\\sqrt{1-v^2/c^2},", "be034638ef0b98ceab11b7234eefc124": "\\Delta I = \\mu I\\Delta t", "be03652d2bb4d1f854c52819df9e9e8c": "\\left\\{\\begin{matrix}\n &\\frac12 \\exp \\left( \\frac{x-\\mu}{b} \\right) & \\mbox{if }x < \\mu\n \\\\[8pt]\n 1-\\!\\!\\!\\!&\\frac12 \\exp \\left( -\\frac{x-\\mu}{b} \\right) & \\mbox{if }x \\geq \\mu\n \\end{matrix}\\right.\n", "be03797eb6d055b0d24a01ce43959510": "\\mathbf{p}(t)\\in \\Omega \\subset R^N", "be038ba0d271f59232ab6b29b936c293": "{D\\rho \\over Dt} = 0", "be03b6c4f45519e696c0ba49bd003c90": "|\\boldsymbol\\Sigma|", "be040ba5b5a3805240f1700fa42c995c": "m = {\\hat m}S^{-1}", "be044c4b9e4e6b78c2bba207256e229a": "5^3 = 125 \\approx 128 = 2^7 ", "be04533a8b6f90f43cfbd31056bb5c11": "-1.645 < \\beta \\le -1.28", "be046422d72781015f8b9f3de436755d": "z=ae^{\\pm(\\frac{2j\\pi}{3})}", "be04cfde6046808f210a0c2006167f67": "k\\cdot \\epsilon=0", "be04e39816cb5b09db11676b5a022dc9": "\\mathrm{freq} = \\frac{1}{\\mathrm{mass}^{1/3}}", "be04f53ec12730b31e1e7d86f79ff9ba": "T_\\mathrm{sol-air}", "be04fb2381852402e9d1c52056ecb957": "(S (\\delta), P')", "be04ff143d94738aa3094b5275bfb090": "AF_{T} = e^{(E_{a}/k)*(1/T_{o}-1/T_{s})}", "be053207749bf65f1c5166580f0fa630": " Q = \\frac{U Ar \\int^{\\Delta T(B)}_{\\Delta T(A)} \\Delta T \\frac{\\mathrm{d}\\,z}{\\mathrm{d}\\,\\Delta T}\\,d(\\Delta T)}{\\int^{\\Delta T(B)}_{\\Delta T(A)} \\frac{\\mathrm{d}\\,z}{\\mathrm{d}\\,\\Delta T}\\,d(\\Delta T)} ", "be0570246313150f43f5e2ac25a81fc4": " u(x(\\theta),t(\\theta),\\theta) \\geq u(x(\\theta'),t(\\theta'),\\theta) \\ \\forall \\theta,\\theta' ", "be0629a348e6a004a4718229cffe4981": "u_k(x)=\\sum_{K} \\tilde{u}_k(K)e^{i K x}", "be06417e0bfdbf969a99ceb55fe6fa27": "v(t) = I \\frac{dl(t)}{dt} ", "be068e111c9f00276b22affcc996b8c3": "\n(\\mathbf{\\gamma_3})^T = \\alpha\\begin{pmatrix}0.1907 \\\\ 0.8093 \\end{pmatrix}\\circ \\begin{pmatrix}0.6533 \\\\ 0.3467 \\end{pmatrix}=\\alpha\\begin{pmatrix}0.1246 \\\\ 0.2806\\end{pmatrix}=\\begin{pmatrix}0.3075 \\\\ 0.6925 \\end{pmatrix}\n", "be0722ea92d0d999165835f7d930e0af": " \\scriptstyle L ", "be073d7ae95ab54b8396bfc14d03b3ce": "\\|f+g\\|_p^p + \\|f-g\\|_p^p \\geq \\big( \\|f\\|_p + \\|g\\|_p \\big)^p + \\big| \\|f\\|_p-\\|g\\|_p \\big|^p.", "be073f575c9ab6e9081043279d3b8692": "\n \\begin{align}\n \\varphi_1 = - \\frac{\\partial w^K}{\\partial x_1} \n - \\frac{1}{\\kappa G h}\\left(1 - \\frac{1}{\\mathcal{A}} - \\frac{\\mathcal{B} c^2}{2}\\right)Q_1^K\n + \\frac{\\partial }{\\partial x_1}\\left(\\frac{D}{\\kappa G h \\mathcal{A}}\\nabla^2 \\Phi + \\Phi - \\Psi\\right)\n + \\frac{1}{c^2}\\frac{\\partial \\Omega}{\\partial x_2} \\\\\n \\varphi_2 = - \\frac{\\partial w^K}{\\partial x_2} \n - \\frac{1}{\\kappa G h}\\left(1 - \\frac{1}{\\mathcal{A}} - \\frac{\\mathcal{B} c^2}{2}\\right)Q_2^K\n + \\frac{\\partial }{\\partial x_2}\\left(\\frac{D}{\\kappa G h \\mathcal{A}}\\nabla^2 \\Phi + \\Phi - \\Psi\\right)\n + \\frac{1}{c^2}\\frac{\\partial \\Omega}{\\partial x_1} \n \\end{align}\n", "be07cf879fc7ed490eface5f74b368f8": "\\mathbf{r} = \\left[ r_\\textrm{x}, r_\\textrm{y}, r_\\textrm{z} \\right]", "be0815236bfc7242207bb075397b535b": "\nC = {{\\varepsilon A } \\over d} \\propto L\n", "be087534e2a4cfd3ecdcf43e8258c68c": "d=2 \\arctan \\left[ \\frac{k-1}{k+1}\\tan(s/2)\\right]", "be08a1a5834381cc0f5c8a1733c3a96b": "A_{sn} = A_s \\left ( \\frac {\\left (1 - \\frac {25}{1000} \\right )\\left ( \\frac {^{13}C}{^{12}C} \\right )_{PDB}}{\\left (1 + \\frac {\\delta^{13}C}{1000} \\right )\\left ( \\frac {^{13}C}{^{12}C} \\right )_{PDB}} \\right )^2", "be091dee5590410dd458295ef6c9d7cc": "\\scriptstyle \\emptyset^c a", "be0922af898c54c77ae79eaccd2b11e3": "\\varepsilon_i\\,", "be09291862fcb3f4aaab9708b0958507": "-\\frac{\\partial}{\\partial t} \\int_V u dV=", "be0937a4a15cb840e5bffd930f61adb1": "A^T A x = 0", "be0962205f2f030ee982503fdc47922b": "\\hat{x} = \\sqrt{6}\\, x", "be09b3b3eb4d4207d853d299fd74d74a": "\\tilde v=dx^{(4)}/d\\tau", "be09db7ecac6e971afd35d1f5ccef868": "r=2a\\sec\\theta", "be0a4019d0f85628977a6bfd1012e9c9": "f_{\\omega^\\omega}(f_1(f_0(3))) - 2", "be0a68ab470de9be845e681e60230e80": "V \\approx {cw \\times cd \\times wl \\over 3 \\pi}", "be0b0bb1a6fa94d0ef5f2ee893796034": "P\\mapsto P(\\mathbf{x})", "be0b10d7a5483e3e69e1edcfdd0a5eeb": "e^\\eta", "be0b512a4993d12cf01b4185d114e8ca": "\\textstyle H_3", "be0c568c8ef3008351b5fd8ac22a5e4b": "\\frac{dx}{dt}=r+x^2", "be0d2be4d03d65669c42f4ed8bcb7490": "E(m)=\\int_0^{\\pi/2}{\\sqrt{1-m \\sin^2 \\theta} } d\\theta,", "be0d3746aa8ea624f95d30c05ea2e57c": "\\frac{\\mathrm{d} P}{\\mathrm{d} T} = \\frac {L}{T \\Delta v}.", "be0d556cd4738b55c9cc85186fb484df": "u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x = \\frac16 (6u_{xxxx}+60uu_{xx}+45u_x^2+40u^3)_x. ", "be0d9ad89acdf99174b2de3cb77a63f9": "x_1 = \\cdots = x_k = 0", "be0e03c8daf2ad6156109023ab21991b": "\\left [\\begin{smallmatrix}2&-2\\\\-3&2\\end{smallmatrix}\\right ]", "be0e21a53d044912466e8b6663cdb405": "\\arctan x", "be0e3c80092a2bf754ef94b8e23a0775": "\\frac{\\Gamma, A, A \\vdash \\Sigma}{\\Gamma, A \\vdash \\Sigma}", "be0e78eb82eeee4fea72c84390b22cb1": "\\lambda \\le \\Lambda", "be0e8922ab3569d2776e284169a1786f": "H_n (\\omega)", "be0e92c973281a7fd0a135ac46c0e220": " 8\\pi W= \\frac{D^2_1}{\\varepsilon_1} + \\frac{D^2_2}{\\varepsilon_2} + \\frac{D^2_3}{\\varepsilon_3}. ", "be0eaf7ffbe0810f97ff11712901977a": "\\scriptstyle \\tilde{t}_{\\text{rec}}", "be0eb8d04654c02153173935041f3236": "H_h=-\\sum_i hf(ih)\\log (f(ih)) - \\sum hf(ih)\\log(h).", "be0ebc75f57256b2927a7f6d5a447839": "E/N", "be0ec5636289125b1dcdb51bc54645ba": "i > n", "be0ee399cef24c410a435fc287571628": "\\operatorname{Li}_s(z) = \\operatorname{Li}_s(0,z) \\,.", "be0ef7381b90f14641253182df9f7faa": " \\begin{align} & \\psi_1^0(x) = \\frac1{\\sqrt A} \\sinh x\\sqrt A, \\\\ & \\psi_2^0(x) = x. \\end{align} ", "be0eff9f51cf2679c4d89293ddee681c": "\\eta_{\\mu\\nu} = \\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n0 & -1 & 0 & 0\\\\\n0 & 0 & -1 & 0\\\\\n0 & 0 & 0 & -1\n\\end{pmatrix}\n.", "be0f13570ad89ff3e8aaded606301931": "2/v", "be0f99c039f39f6ab123731a4d7d6878": "y= y_p + y_c", "be0f9edb82c3c7112eb22621cc872e77": "a_1 = 2", "be10803bf1eb1433805ca6063458a59d": "\\begin{align}\n P_D &= e^{-2\\pi\\Gamma}\\\\\n\\Gamma &= {a^2/\\hbar \\over \\left|\\frac{\\partial}{\\partial t}(E_2 - E_1)\\right|} = {a^2/\\hbar \\over \\left|\\frac{dq}{dt}\\frac{\\partial}{\\partial q}(E_2 - E_1)\\right|}\\\\\n &= {a^2 \\over \\hbar|\\alpha|}\\\\\n\\end{align}", "be1081cebd7d953a43a33720b76d828b": "\n\\begin{array}{r c l}\n\\mathbf{r}_1 &=& \\mathbf{r}_1(q_1, q_2, \\cdots, q_m, t) \\\\\n\\mathbf{r}_2 &=& \\mathbf{r}_2(q_1, q_2, \\cdots, q_m, t) \\\\\n & \\vdots & \\\\\n\\mathbf{r}_n &=& \\mathbf{r}_n(q_1, q_2, \\cdots, q_m, t)\n\\end{array}", "be10e2a871d2c2c81543e3818d7b2b6f": "\\sup_{y \\neq 0} \\int_{|x|>2|y|} |K(x-y) - K(x)| \\, dx \\leq C.", "be110c2b0ea2c8aec31d5b0670abc71d": "r z = C, C \\rightarrow \\mbox{constant}", "be110c611a2dabdf886900237a9e0bda": "V=\\{1,\\dots,n\\}", "be11492b73e6c23703fba4073526f3bd": "V_{id} = V_{in+} - V_{in-}", "be1194e622f6cf4ac4cb874ff5eaa472": " \n\\Phi'(\\varphi, \\lambda, z, t) = \\hat{\\Phi}(\\varphi,z) \\, e^{i(s\\lambda - \\sigma t)}\n", "be11da5507b0711a089861015e776ddc": "I(t) = I_0 \\exp \\left (- t / \\tau \\right)", "be1209e79e38657935a2719b89df1b2e": "J_+J_- = (J_x + iJ_y)(J_x - iJ_y) = J_x^2 + J_y^2 - i[J_x,J_y] = J^2 - J_z^2 + \\hbar J_z.", "be123a652db3df7faea653b66c7ce1f0": "g_iB_{ij}=g_jB_{ji}", "be126e8296829fe8d8a09ea7d759102d": "\\frac{\\dim \\mathbf{H}_\\ell}{\\omega_{n-1}} = \\sum_{j=1}^{\\dim(\\mathbf{H}_\\ell)}|Y_j({\\mathbf{x}})|^2", "be1276cd2ac1bb57aa68b15ea701c48b": "O(f)\\colon O(X)\\to O(Y)", "be12b9d21df37cb3d98f0babe2b51cf9": "x_{n+k}", "be12c342118f714b3942320690f7c5d3": "a_n, b_n > 0", "be12f1a8a9586ebecc48e397f40f52be": "R_\\mathrm E", "be132b67a052cfb4485459a33f3b48dd": "(1-yf(x))_+", "be13b97a480cc06bd57c488562e97b53": "\\textstyle m ", "be13bcec7006b0c34dc6ef402071892a": "\\mathcal{R}(X)", "be13ed787879a19dcc1cacf259f60ddd": " V^a_b(f) = \\int _a^b |f'(x)|\\mathrm{d}x", "be149617ce84f1b65d9eab27b44d3538": " \\lim_{\\varepsilon\\to 0^+} \\int_{-\\infty}^{\\infty}\\eta_\\varepsilon(x)f(x) \\, dx = f(0) \\ ", "be14ed2634f66264cb0e3b6b6511fa38": "K_q(\\mathcal{O}_X)", "be14f4f60dc55a60e88632f3f554748b": " MPGe = \\frac { E_G} {E_M*E_E} = \\frac{ 32,600 } {E_M}", "be1530545b696e4a2458d69c85374e26": "\\, (X,Y,Z) ", "be154364f5d1c0f946ce8e672f414f10": "\\frac{\\mathrm{d}}{\\mathrm{d}\\alpha}\\,\\varphi(\\alpha)\\,=\\frac{2\\pi}{\\alpha}\\,", "be15536d8ea2aef75e0b2e7d5f7c6c1b": "c \\in {\\{R,G,B\\}}", "be1578dd6d01c7215a83942e77f3bdfb": "A=A(a, \\lambda)", "be157b8edd64c3a38eb08d0856251188": "f_i(x) < 0, i = 1,\\ldots,m", "be16c1240c46fe2f8d4c8f48055a9b6a": " \\langle \\psi_{1}|W|\\psi_{2}\\rangle \\neq \\langle \\psi_{2}|W|\\psi_{1}\\rangle ", "be16c56b5703b6c1c5cf85c8db4a6dc6": "\\begin{bmatrix}\n\\frac{\\alpha_1^\\mu}{\\alpha_1-w_1} & \\cdots & \\frac{\\alpha_n^\\mu}{\\alpha_n-w_1} \\\\ \n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\alpha_1^\\mu}{\\alpha_1-w_s} & \\cdots & \\frac{\\alpha_n^\\mu}{\\alpha_n-w_s} \\\\ \n\\end{bmatrix} \n", "be1784cf9894427176f69aad4ef16006": " H = - \\sum_i p_i \\ln p_i \\!\\,", "be17df0289d848ce5096d5a5de3365e4": "1 \\leq i, j \\leq n", "be180ed26c983d9d42c60501e1732afd": "\\text{fmap} \\, \\text{extract} \\circ \\text{duplicate} = \\text{id}", "be183cdd1e68ce30a59b96233609b08f": "\\sqrt{n}", "be18497d740eea3ffa4c8a6dda23e115": "\\sqrt{a^2+r} = a + \\frac{r}{2a} - \\frac{(r/2a)^2}{2(a+\\frac{r}{2a})}", "be1857f25045cff35f35b455cee588d7": "\\tilde{u} = \\sum_{i=1}^n u_i \\, \\tilde{\\omega}^i. ", "be1866bdcff2957e40808261436a8bb1": "c\\cdot\\ln{n}", "be18de1ab5ac957030fb9a29dcf4704f": "y=\\operatorname{sign}(uv) \\,", "be195822eb37f96a2b36b7143a2013f9": "\\frac{1}{2}\\int \\frac{6 + e^{2ix} + e^{-2ix} }{e^{ix} + e^{-ix} + e^{3ix} + e^{-3ix}} \\, dx.", "be195a956147f1c887862ac4ecdca9ba": "Z = X\\cup Y", "be196b3c6645e6e2fe15bd1ff63bf850": "\\scriptstyle X_0=x ", "be19849edbbac70381e97ff43639c709": "K\\subset \\mathcal{P}(S)", "be19f385ad30c71f1fd9008e5deb26eb": " S_2 = U_2 ", "be19f8a2eb641270d6947b77e9c1c15a": "A_{\\ell m}^{(2)}", "be1a31750d826e7f66b28f0150aece3c": "(xy')'+ \\left (x-\\frac{\\nu^2}{x}\\right )y=0.", "be1b0537b193b62b72defb49c43376c3": "\\mathbf{r} = \\lambda_{1} \\mathbf{r}_{1} + \\lambda_{2} \\mathbf{r}_{2} + \\lambda_{3} \\mathbf{r}_{3},", "be1b10a900ea4d9eb37acea9137b9ce1": "F(0) = a = G(0)", "be1b56e769fc0fcd1e1fde86b3d4fedd": "\\det(AB) = \\det (A) \\det (B).\\ ", "be1b8f4847a85b3d57958ee6a9cd0b96": "1-p_1\\,", "be1bd074038fe4b002cd8eb674f4cb49": "\n \\int x^m\\left(c ((A\\,b-a\\,B) (1+m)+A\\,b\\,n (1+p+q))+(d(A\\,b-a\\,B) (1+m)+B\\,n\\,q(b\\,c-a\\,d)+A\\,b\\,d\\,n (1+p+q))\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^{q-1}dx\n", "be1be1c6963b0b557359e981f6fdc2c8": "n^{\\mathcal{O}(d)}", "be1c23065840576391bc4616ebe59f09": "\\tau_C = RC\\,\\!", "be1d29f6c4a4a08b16e1a00e406dc8b2": "P(T|G)", "be1d469325c7a2ff7eebc98db90a20e9": "\\frac{f(x')-f(x)}{x'-x}", "be1e5025ca37e166f989201ea10d4466": " E_2^{p,q} = H^p(B, H^q(F)) \\Rightarrow H^{p+q}(X).", "be1e53563c9742e710b79720575bdf76": "f(A)=\\sum_{S\\subseteq A}\\mu(A - S)g(S)\\qquad(***)", "be1f1c34d83eab89cabced4554cc0e6b": "\\mbox{SAIFI} = \\frac{\\mbox{total number of customer interruptions}}{\\mbox{total number of customers served}}", "be1f2aa667c053c4fc22170355e2e820": "S/k \\approx \\ln{\\left(q+N^{\\prime}\\right)!\\over q! N^{\\prime}!}", "be1fa6840eb86e20d92199baae90bfce": "U_n(P,Q)", "be1fc80d965e57435ca91104e17a8b6d": "+2/3", "be1fd56e612ccf10c14f287aac66467a": "\\scriptstyle D_F(7\\rightarrow 3)= 4(1)-2+2-3=1", "be211b25b08fda776dc0e9914e8940d6": "\\scriptstyle r(i) \\le j \\le r(i+1)-1", "be214b421d966199065db8848f5581e9": "\\mathfrak{P}^{90}", "be214cfc8de580112d1831d8c7de08d1": "p({\\rm label}|\\boldsymbol{x}) = \\int p({\\rm label}|\\boldsymbol{x},\\boldsymbol\\theta)p(\\boldsymbol{\\theta}|\\mathbf{D}) \\operatorname{d}\\boldsymbol{\\theta}.", "be21561e15167d146cf99280ef36c228": "-j0.23\\,", "be21690ce1efe9316b8688017b209cab": "\\begin{align}\n\\begin{pmatrix}\\pi_0 & \\pi_1 \\end{pmatrix}\n\\begin{pmatrix}B_{00} & B_{01} \\\\ B_{10} & A_1 + RA_0 \\end{pmatrix}\n= \\begin{pmatrix} 0 & 0 \\end{pmatrix}\n\\end{align}", "be21777fb5c9299bddcdc61a33a21c8d": "\\rho_1,\\rho_2,\\ldots", "be218c5daa1ccf72ee5a887157188ca6": "\\lbrace v_\\beta:\\beta<\\alpha\\rbrace", "be218c62d8aaff669ca73f0c083f828b": "h(w) = \n\\left\\{\\begin{matrix}\n0 &\\mbox{if}\\ w\\ne 1\\ \\mbox{in}\\ S\\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ w=1\\ \\mbox{in}\\ S.\n\\end{matrix}\\right.", "be226a01238f1b61df76b5db2c0d84ff": "\\ f=\\rho \\exp({\\rm i}\\varphi)=(x+{\\rm i}y)!=\\Gamma(x+{\\rm i}y+1) ", "be22bcadbaac86cae38467a9c7e39a2e": "P(n)=\\sum_{k=1}^n\\gcd(k,n)", "be22cf7cb5f78e040dd5798edcf68190": "m_1, m_2 \\in \\mathcal{M}_{+} (X)", "be2300f96eadd6a23e74f4f50909a4e9": " e = \\sqrt{1 - b^2/a^2} ", "be230db0cdc1a224435b61e2a32d735f": "\\prod _x f(x)^a = \\left(\\prod _x f(x)\\right)^a \\,", "be2310c2b79e2f01ca1e0480c0039098": " e = ", "be2391e4617ee7a5f605850aa5b3aae3": "D = D_0\\exp(\\frac{\\Omega H - E_A}{kT})", "be239bf9205494afcfe34bec838c2b7d": "m = -2.5 \\log_{10}L(d) ", "be23bc49eb7808ff319e5250c075da4c": "O=2^I", "be23bfa2055641d91bd44330ec9e61d7": "cov[X_1,X_3] = \\frac{10 \\times 0.2 \\times 0.2}{0.5^2}=1.6", "be23f361681eec545ea4f524956d0a59": "\\{ G(\\lambda) , A_a^i \\} = \\partial_a \\lambda^i + g \\epsilon^{ijk} A_a^j \\lambda^k = (D_a \\lambda)^i.", "be240fc6228ee81b631c00c7600a42a0": "{ dI \\over dz} = { \\gamma_0(\\nu) \\over 1 + \\bar{g}(\\nu) { I(z) \\over I_S } } \\cdot I(z) ", "be244a186d19a342f52026670d2ea7ec": "\\sum_{i=1}^n \\lambda_i = 1.", "be2475e65fb4d1440fafc5686ac62e56": "H_1:\\sigma^2=\\sigma_1^2", "be248e4d759472af3b1a8cad19cb471b": "\\sqrt{s(s-a)(s-b)(s-c)}\\,\\!", "be24ceaca9a45cdff59e1a4f7f137a91": "\\sum_{x=1}^\\infty \\mathrm{sinc}(x) = \\mathrm{sinc}(1) + \\mathrm{sinc}(2) + \\mathrm{sinc}(3) + \\mathrm{sinc}(4) +\\cdots = \\frac{\\pi-1}{2}", "be24d279e0258dc564d544ec66339e58": "{a_{A}^{t_{1}}} | {a_{A}^{t_{0}}}\\; \\sim N\\left( a_{A}^{t_{0}},\\ \\frac{t_{1}-t_{0}}{\\tau }\\sigma _{a,A}^{2} \\right)", "be2536c4a5b27dee51f27ac14d5c5fe4": " Q\\,\\!", "be255060a36472cfc84b58b0cd20c392": "f(x) = f(-x)", "be25a0d91d73b9470a8d954ff4e0e59f": "\\ln M! \\approx M\\ln M", "be25a0f3144424e5ad1adb7890982f7a": "(\\mathbb{R}^{n})^{k}", "be25a1184eedef1abda3948f2f822eb8": "A_m = fA_x + (1-f)A_s", "be25b5c2942c59eb386a67a02fa1eea8": "\\gamma_1\\,", "be25bc87ac7e58845a01429ca38353bd": "\\mathbf{M}_{5} := (\\mathbf{A}_{1,1} + \\mathbf{A}_{1,2}) \\mathbf{B}_{2,2}", "be25c15a665fda4ef8c0bae23d64f8d9": "\\mathrm{zero} : 1 \\to \\mathbb{N}", "be2617e8d3b7474be297244690bb1638": "[D,K_\\mu]=-K_\\mu, ", "be2633409178b341c3b69d30974b48f0": "N(p)", "be269cf96a0e9d60afb8d4b813223b03": "(2n+1)\\times{(5n^2+5n+3) \\over 3}", "be26a738841ad0bb163b71ba6ef44af7": "A \\cdot (\\neg B \\cdot \\neg C + B \\cdot C) \\,", "be26b723d0038bc45a0892bad30aa602": "R(i,a)", "be26d65e3168fad96d603a54f17f4207": "S \\in \\mathbb{R}^n", "be2713aad08825429302308bea1e8f9a": "W(A+B)\\subseteq W(A)+W(B)", "be27440276630872fe041776e6e19f71": " g_\\alpha(n) = g_{\\alpha[n]}(n)", "be277e52e4e0f40a74aaeb4febb9439a": "C^+ = \\mathcal{F}\\cdot\\Sigma^+\\cdot\\mathcal{F}^*\\,\\!", "be27a0e6b3e7f18ccf9d3f4f31c94943": "\\ R_1 = r", "be27ba3bdd8297ce1419b45228c5d006": "h_{-2}=0\\,", "be27db9892e7f2008eeb03ddd92cce53": "A_q(n,d) \\leq q^{n-d+1}.", "be27fdfc5353842e876ffdcef7b9ab47": "\\lnot\\lnot (x \\vee y) = x \\vee y \\mbox{ for all regular } x, y \\in H,", "be28119b7cf11f18a1d0f37b3b79ebe8": " |\\psi\\rangle = \\cos\\left(\\tfrac{\\theta}{2}\\right) |0 \\rangle \\, + \\, e^{i \\phi} \\sin\\left(\\tfrac{\\theta}{2}\\right) |1 \\rangle =\n\\cos\\left(\\tfrac{\\theta}{2}\\right) |0 \\rangle \\, + \\, ( \\cos \\phi + i \\sin \\phi) \\, \\sin\\left(\\tfrac{\\theta}{2}\\right) |1 \\rangle ", "be283d0ab3865f43b981c3d4e9db248d": "M = \\frac{1}{2}fr^2 + vr", "be284b51c031ef25ef80b778085eb0f2": "J_\\nu=\\frac{1}{2}\\int^1_{-1}I_\\nu d\\mu = a", "be286ce5d27ac78bd6183288f73d187e": "I_n = \\int \\frac{dx}{x^n\\sqrt{ax+b}}\\,\\!", "be289404d12ada278e56ffc62f95d952": " X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = g(\\nabla_X Y + \\nabla_Y X, Z) + g(\\nabla_X Z - \\nabla_Z X, Y) + g(\\nabla_Y Z - \\nabla_Z Y, X). ", "be28b95ec255031a1ff43f7f0472133f": " f_{2} ", "be29d84e89261618a862f6ea7740d010": " Amino~acid + ATP \\rightleftharpoons{} aminoacyl~AMP + PP_i", "be29f307d4cd6b51cdeb1fb17e29e095": " v = u e^{-2 \\alpha t} ", "be2a29c487b1e2ae222792bc40de296c": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}", "be2a810f0c6dc72012f84edae7fa723c": "\\frac{\\partial^2 u}{\\partial \\xi \\partial \\eta} = 0", "be2a8371d0531e3b8a63cf417a7e1c97": "E[X_i^2]=E[Y_i^2], i=1,\\dots,n, \\ E[X_iX_j] \\le E[Y_i Y_j] ", "be2abbd01485a8f98e24f66f66537657": "\\mathrm{Bi}(z)\\sim \\frac{e^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt\\pi\\,z^{\\frac{1}{4}}}.", "be2aca6c673ba2d9f6d91ae8f1baae7c": "h[n] = \\begin{cases} 1; & n = 0 \\\\ 0; & n \\neq 0 \\end{cases} ", "be2b12571a964fe83cb0dff9293ab3a8": "\\begin{align}\n\\left|j_1 , m_1 , j_2 , m_2 , ... j_n , m_n \\right\\rangle & \\equiv \\left|j_1,m_1\\right\\rangle\\otimes\\left|j_2,m_2\\right\\rangle\\otimes\\cdots\\otimes\\left|j_n,m_n\\right\\rangle \\\\\n & \\equiv \\left|j_1,m_1\\right\\rangle \\left|j_2,m_2\\right\\rangle \\cdots \\left|j_n,m_n\\right\\rangle\n\\end{align}", "be2b2986a418a3780abcb16612b19d53": "\n\\cfrac{\n \\cfrac{\n (1)\\cfrac{C_1 (1,3)\\qquad (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{C_8 (-1)}}{C_3 (3)}\n }\n {C_7 (3)}\n}\n{\nC_9 {\\color{red}(3)}\n}\n", "be2b620a3398f4ae29ea9a112173bc52": "H_Z", "be2bb09559df9e71b17b8c000f69079d": "\\mathfrak{p}_nR[x]", "be2bc03b2ef80c9a0a1acb5d62aa5724": "g_2(\\tau)-1=\\exp\\left(-2 \\gamma \\sqrt{\\langle\\Delta r^2(\\tau)\\rangle k_0^2}\\right)", "be2bcede94f51a333178769ff656a5a5": "\\tan^2(x) + 1\\ = \\sec^2(x) ", "be2c28f2856ed70592f88a28b9852a2b": "A \\nrightarrow B", "be2c698ecf110e629e49dc642dbad38b": "R_F\\left(\\kappa x,\\kappa y,\\kappa z\\right)=\\kappa^{-1/2}R_F(x,y,z)", "be2c8ecd6fd465d6d47131e7130a5aef": "B^{2} = 4 AC,\\,", "be2cb5e6dee16e6c6686ebfa0f7e03e6": "\\chi_{\\text{e}}(0)", "be2cb6dcfb495aebc304232961b8bd28": "HPI(0,T)=\\frac{\\widehat{P}_{T}(z^{\\tau})}{\\widehat{P}_{0}(z^{\\tau})}.", "be2d287b020ffb0cdc527cb85568ef45": "\n\\max_{C_i} \\sum_{r=1}^k w_r \\sum_{x_i,x_j \\in C_r} k(x_i,x_j).\n", "be2d40890622342279b7755354d866f7": "W_0,W_1,W_2,... ", "be2d4a0b2130d6bc88220ab0eb3d4b8d": " p_{X_v} (x_v) \\propto \\prod_{u \\in N(v)} \\mu_{u \\to v} (x_v). ", "be2d5f7f2ace2463d6afb36a6a4ce5a5": " \\delta_0 \\approx \\sqrt{1-(I/I_0)^2} ", "be2dab4fb5e0aa7054e6e73e656b2c98": "\n0 < r < \\infty ~, ~~ 0 < \\theta < 2\\pi ~,~~ -\\infty < z < \\infty\n ", "be2df553a1b45b139ba822f3f5743747": "\\begin{align}\nw_0\\left(n-\\frac{N-1}{2}\\right) \n= \\frac{1}{N} \\sum_{k=0}^{N-1} W_0(k) \\cdot e^{i 2 \\pi k (n-\\frac{N-1}{2}) / N}\n=\\frac{1}{N} \\sum_{k=0}^{N-1} \\left[(-e^{\\frac{i\\pi}{N}})^k\\cdot W_0(k)\\right] e^{i 2 \\pi k n / N},\n\\end{align}", "be2df626f0ab5dcf5214b8fb64cdd50f": "\\max", "be2e0b6ef53b4ca93046102a71046453": "SL(2,\\mathbb{F}_5)", "be2e1b70c5046a4d4059b377fba214c6": "\\mathrm{MAE} = \\frac{1}{n}\\sum_{i=1}^n \\left| f_i-y_i\\right| =\\frac{1}{n}\\sum_{i=1}^n \\left| e_i \\right|.", "be2e64bf81bdd5b763e34032688a451d": "\n\\; \\Phi (E_{kl}) = \\sum _{i = 1} ^{nm} P_k v_i ( P_l v_i )^* = \\sum _{i = 1} ^{nm} V_i e_k e_l ^* V_i ^*\n= \\sum _{i = 1} ^{nm} V_i E_{kl} V_i ^*.\n", "be2e88cf6edffbc5019ccbf506dbc1dd": "r_i=Rank(A_i)", "be2ead4212a1478dd2595f27ea3488dd": " U (Q, \\mathbf{x} ) = \\begin{matrix} \\frac{1}{2} \\end{matrix} QV = \\begin{matrix} \\frac{1}{2} \\end{matrix} \\frac{Q^2}{C(\\mathbf{x})}~", "be2ee47d51ebc1ed985603e7a7c950b0": "\\delta \\beta_j\\,", "be2eebcb4baddc03698274ac919963d6": "a_{21}=0.1", "be2f58e383cf21649a4febe0a112f985": "\\boldsymbol{\\alpha\\beta\\gamma\\delta\\epsilon\\zeta\\eta\\theta} \\!", "be2f8982694877dad701051ab2ed410b": " r_g = \\frac{\\left(\\gamma - 1\\right)}{2\\gamma}\\frac{GM\\mu}{k_B T}\n\\approx 2.15 \\frac{\\left(M/M_\\odot\\right)}{\\left(T/10^4 \\ {\\rm K} \\right)} \\ {\\rm AU},\\!", "be2f8a10ac15fe1b4c585a4b7247bf69": "q_s = A \\mathrm{e}^{\\mathrm{i} ( \\omega \\tau + \\phi ) }, \\ \\frac{\\mathrm{d}q_s}{\\mathrm{d} \\tau} = \\mathrm{i} \\omega A \\mathrm{e}^{\\mathrm{i} ( \\omega \\tau + \\phi ) }, \\ \\frac{\\mathrm{d}^2 q_s}{\\mathrm{d} \\tau^2} = - \\omega^2 A \\mathrm{e}^{\\mathrm{i} ( \\omega \\tau + \\phi ) } .", "be2f8dfcc8988afd0d165621cf9b16ec": "e^{i(2h+1)\\theta}", "be2f902cec0c8ccbd8dfd6a1d219230b": "\\scriptstyle u \\;=", "be2fce1f6c14f2afc746c89a0ef632e1": " L_2=\\partial_t+(\\lambda^2+u)\\partial_x+(-\\lambda u_x+u_y)\\partial_{\\lambda},\\qquad (3b)", "be2fe6ac34fa5997367353bd3d5528c6": "\\begin{matrix}\\mathrm{Cabtaxi}(7)&=&11302198488&=&1926^3 + 1608^3 \\\\&&&=&1939^3 + 1589^3 \\\\&&&=&2268^3 - 714^3 \\\\&&&=&2310^3 - 1008^3 \\\\&&&=&2492^3 - 1610^3 \\\\&&&=&4230^3 - 4008^3 \\\\&&&=&9492^3 - 9450^3\\end{matrix}", "be3015321efef81e22339d5568321d96": "\n\\sigma_j=\\tau_j+\\tau_j'~,\n\\qquad\n\\sigma'_j=\\tau_j-\\tau_j'~.\n", "be30377db9e70a92dd723ec0d6439a95": "\\text{Area}(R_1) < \\text{Area}(R_2) < \\text{Area}(R_3) \\iff \n\\frac{1}{2}r^2\\sin\\theta < \\frac{1}{2}r^2\\theta < \\frac{1}{2}r^2\\tan\\theta \\, . ", "be3037adbef7ba0303ffe86e78b20564": "\\begin{matrix} {10 \\choose 1}{4 \\choose 2}{9 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "be30567e7c74bce181624bbbc08f5bc2": "\\boldsymbol{\\omega}", "be307efda1cc91158a5b6c183aa658e9": "\nk_{B} T = \\Bigl\\langle x_{n} \\frac{\\partial H}{\\partial x_{n}}\\Bigr\\rangle = 2\\langle a_n x_n^2 \\rangle,\n", "be3082fb02fc4c854b43a1bae14aad06": "\\mathbf{a}(t) = \\frac {d\\, \\mathbf{v}(t)}{dt} = [-x(t), -y(t)] = -\\mathbf{r}(t)\\, .", "be309c37fffedb9700956fb8898e9fc2": "\\cos(\\theta_1)=1/\\sqrt{1+a^2}", "be30d18eafc3752b159677fea48a241b": "\\inf_{x \\in X} f(x). \\,", "be30d362da3ea1f43b03bb8077197b6b": "s_n = \\sum_{k=0}^n a_n", "be30e2aabc8adc09c105ce4341f81610": "H_{rot}=B\\mathbf R^2=B(\\mathbf J-\\mathbf L-\\mathbf S)^2", "be3126c40535b19838705c0c0552680d": "K=\\frac{2MN\\cdot EI\\cdot FI}{EF}", "be317e4b3d2d6d45c0d36a5ae990e09e": " K_n(R) = \\pi_n(BGL(R)^+\\times K_0(R)) ", "be31a4e99421d806bc7db4668b21ef0f": "\n\\mathrm{Fr} = \\frac{v}{\\sqrt{\\displaystyle g \\frac{A}{B}}}.\n", "be31ac6fcb84bdcdf7452ade504a0c1b": "ka << 1", "be31c00a6dcd5369eb274cd92519d8c6": "X_1^n(2)", "be31f27508f2c40a95f95752eb29cf67": "\nA_\\mu(x) = \\sum_\\mathbf{p} {-1 \\over 2 \\sqrt{V p_0}}\\left\\{\n\\left[Q_R(\\mathbf{p}) u^{-1}_{-1}(\\mathbf{p})^\\dagger \\gamma_0 \\gamma_{\\mu} u^{+1}_{+1}(\\mathbf{p})\n+ Q_L(\\mathbf{p}) u^{+1}_{+1}(\\mathbf{p})^\\dagger \\gamma_0 \\gamma_{\\mu}\nu^{-1}_{-1}(\\mathbf{p}) \\right]e^{i p x}\n\\right. ", "be3219f10fd61718a6abaeec9d7b9369": "\\mathbf{r'}", "be321ca26f71d086b8e58bf9e3480648": "\\ m0.\n", "be4799f3025bd3b64c2193e9a7554192": " (2)\\quad g_1\\cdot (g_2 \\cdot v) = (g_1g_2) \\cdot v ", "be47a88a385860f34393cb1072c99c20": "(i \\pm L^{d-1}) \\mod N", "be47ce0451493b94ae92f7acd41d327e": "\n J_{\\Gamma_1} = \\pi\\,(\\sigma_{\\text{far}})^2 \\,.\n ", "be47f1a6cda726c53c7b0b2f0104ea92": "\\textstyle \\sum_{i=1}^{N}w_i=N^*", "be4884419a84ab11aec5ec9522009c10": "h_{z}=1\\,", "be48c078f8bec0604c3e7ee0d43540b2": " z \\rightarrow \\tfrac{1}{\\overline{z}}", "be49211c98cc7f446546bff5f0640aef": "\\mathbb P(n_i = N) = \\frac{X_i^N}{G(N)}[G(0)].", "be4966f6d70c8195397e7a6ab94733de": "Z-n", "be4a3d7289b163da3d090f4ca91e76fc": "\\Delta u + f = 0\\,", "be4a4d7b45cd058b7e175c5fd4d5fe47": "\\scriptstyle EIRP", "be4ad86b82a218bbf0737e00cef0334a": "[\\gamma_1]=p, [\\gamma_2]=q", "be4aec3ff0f60dfdd9f20276740306a6": "\\int_a^b \\! f(x)\\,dx", "be4b15a4aeb6035db1caf39c9e0c7a63": "\\ C_L", "be4b1e8a26c6ae286b4adbae1e37a84f": "n_{r}=n_{o}\\left ( 1-\\frac{A r^2}{2} \\right )", "be4b1fa2eb1efa31d98ea15bee2789f6": "\\mathfrak{su}_3\\oplus\\mathfrak{su}_3", "be4b4c508dc31a88a166c4264598dcf7": " g(z^2)^{-1/2}= z^{-1} -{1\\over 2} a_2 z + \\cdots. ", "be4b54911f62a0f74d3aac2bb40d855f": "dF(u;\\alpha\\psi)=\\alpha dF(u;\\psi).\\,", "be4b7fb4af1557fa6f557bc40cb61bf6": "\\mathcal{O}_{\\mathbf{C}_p}\\to \\mathcal{O}_{\\mathbf{C}_p}/(p)", "be4c02f886b962acf7ad63f8a95902f3": "x - y", "be4c189597393a529a1fb685b5c78ffb": "E_{VCYC}(G)", "be4c549cb79966b4568c38ae78ac090f": "\\psi_x = v, \\quad \\psi_y=-u,", "be4c623245866f6a489782352e49be6f": "(a^2 + b^2)^2(1 + (\\lambda\\ - 1)^2 + \\lambda^2)^2 = 2(a^2 + b^2)^2(1 + ((\\lambda\\ - 1)^2)^2 + (\\lambda^2)^2)\\,", "be4cec24ca8743531ca7d6e36fdd564f": "\\theta << \\lambda / D", "be4d618ab64fb0d5627605ab02b48431": "\\,\\Gamma(x+1)=x\\Gamma(x)\\,", "be4d72b82d0bf10137711ed413eef391": "\\lfloor \\log_2(x) \\rfloor + 2 \\lfloor \\log_2 (\\lfloor \\log_2(x) \\rfloor +1) \\rfloor + 1", "be4e17e2f0d86a738dad3992b51626c5": "\\|x\\| \\ne \\|y\\|", "be4e37e9a4a1b7bea67ac7cde17d8fff": "W_c \\approx {2 \\pi \\over \\ D}", "be4e86366eaf8967b09d60f66048f98d": "\\sum_{n=a}^{b} \\nabla f(n) = f(b)-f(a-1)", "be4e9b8b32e6dd5bb3a98961f3f68d7a": "\\mathbf{r}'", "be4ec263d10e772aa024daa01488bd78": " y( t ) = \\sum_{n=1}^{ \\infty }\n\\left[\n \\lim_{ r \\to 0 } \\left(\n {\\frac{ x^{ n }}{ n! }}\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{ d } r ^{\\,n-1}} \\left[\n r^n \\left( \\frac{ 7 }{ 2 } ( \\arcsin( \\sqrt{ r } ) - \\sqrt{ r - r^2 } ) \n \\right)^{ - \\frac{2}{3} n }\n \\right] \\right)\n \\right]\n", "be4ec64c5316077d35560036c197747f": "A(z) = B(C(z))\\,", "be4ed87b1a3076fe860dac8d15787477": "\\sum\\nolimits_{m_{j+1}} \\Pr\\nolimits_r \\left [V(w,r,M_j)=m_{j+1} \\right ] * \\Pr \\left [V\\text{ accepts }w\\text{ starting at }M_{j+1} \\right ].", "be4f0f0bd7a2a7c8490de1bf521ef86d": "Z_2^T", "be4f9b0809e5843e74da872a11c47c9e": "U_s = 2 \\left|s\\right\\rangle \\left\\langle s\\right| - I", "be4fa808001eee81c8844b50d59bf0da": "\\chi = V - E + F = 0. \\,\\!", "be4fddc36d12509820b4f061e7654cb6": "\\iota \\jmath x \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\hat{y}(y = x)", "be4ffa129169d0472fa76830ff47d3f5": "\\hat T = -\\frac{\\hbar^2}{2m}\\nabla^2.", "be506a249238a91b3f310382e0e5cefd": "Q = Q(x,p,t) \\,", "be50717efc5b86679f64d47c190e4486": "|f^n(z)|\\le \\delta(r)^n", "be50a0672df40516db72cbea17e37c82": "c (q, t) dq", "be50a66a89409d8fc01f546ddcc6c696": "dZ=(R+i \\omega L)dx=Zdx\\, ,", "be50b4896ad6abd2b6bd857cc97f8ded": "S1\\ \\delta^o\\ S2", "be50cf42652835bac20fc2ea120051a4": "\\varepsilon : H \\rightarrow k", "be513f9e89f8f198cd80e4f720721a34": "(m,M)", "be5194cb253d4820db4c525a07de78bc": "f\\colon S^k \\to X \\, ", "be51b0c28ef20dfd09f05dc296af3f0e": "\\overline{\\mathbf{A}}", "be51e5b78105c9173d182b9f5c75f6ec": "d \\doteq 2r \\quad \\Rightarrow \\quad r = \\frac{d}{2}.", "be51e87157af0fdee9585b4c4efe937e": "1 + \\frac{1}{3} + \\frac{1}{5} + \\cdots + \\frac{1}{2n-1} = \\sum_{k=1}^n \\frac{1}{2k-1}=H_{2n-1}-\\frac{1}{2}H_{n-1}", "be51fb144dd08fb5d30d82b40dffb687": " e = \\sum_{i=1}^n x_i \\otimes y_i = \\sum_{i=1}^n y_i \\otimes x_i", "be52a170013d131f9875e416924553ab": "{U}", "be52fbf49174370905414e46e7c4e7ca": "a_{m} = F/m = (eE/m) \\, \\exp(-i\\omega t)", "be5323f252de6435ca391ca07bf774d3": "\n \\frac{\\partial x^i}{\\partial X^\\alpha}~g^{\\alpha\\beta}~\\frac{\\partial x^j}{\\partial X^\\beta} = \\delta^{ij}\n", "be53536edd0f8ad106ede3628feac182": "a_{P}\\,= [(D_{w} + F_{w}) + D_{e} + (F_{e} - F_{w})]", "be53bf42f38e11660ede3efef0b4c14b": "\\begin{array}{lcl}\nS &\\to& NP \\;\\; VP\\\\\nVP &\\to& VP \\;\\; PP\\\\\nVP &\\to& V \\;\\; NP\\\\\nVP &\\to& \\textit{eats}\\\\\nPP &\\to& P \\;\\; NP\\\\\nNP &\\to& Det \\;\\; N\\\\\nNP &\\to& \\textit{she}\\\\\nV &\\to& \\textit{eats}\\\\\nP &\\to& \\textit{with}\\\\\nN &\\to& \\textit{fish}\\\\\nN &\\to& \\textit{fork}\\\\\nDet &\\to& a\n\\end{array}", "be53d48a1b0706e83ffed093708357e9": "f(z) = z\\sum_{k=1}^\\infty \\frac{1}{k(z-k)} ", "be53eb2f0a4401e6a90238845e32f0a2": "\\Delta d", "be53fbf88694876d65b98c3d386d9d2e": "(1,3,1)\\rightarrow (1,3)_0", "be547ccde7b43bef62c075c4bd08dfcf": "\\epsilon = 1 - |\\langle{\\psi}|\\psi'_{D}\\rangle|^2 = 1 - (1-\\epsilon_{n+1})|\\langle{\\psi}|\\psi_{D}\\rangle|^2 = 1 - (1-\\epsilon_{n+1})(1-\\epsilon_{n})", "be54e5cf3c18b44328ef1b664b066837": "= \\lim_{N \\to \\infty} \\sum_{i=1}^N b \\frac{s_{i+1} - s_i}{W(s_i)}", "be553aa705453d5a7fc177284fc3528b": "c = {g}_{1}^{x_1} g_{2}^{x_2}, d = {g}_{1}^{y_1} g_{2}^{y_2}, h = {g}_{1}^{z}", "be56c148025bf561319d0dc6bdbdec7e": "\\text{span}\\{\\pi(k)v : k\\in K\\}", "be575ab900d275505c595e870db23f9a": "p \\Downarrow a", "be5774fb5e7d9621e415c167840865ad": "\\operatorname{erf} (\\overline{z}) = \\overline{\\operatorname{erf}(z)} ", "be57a7904f86c7f028c9cf9169d59272": "k/2^k", "be57a92d9ecc76c03302b4c680610ede": "\\displaystyle -i\\pi\\sgn(\\xi)", "be57ddfba778eae49b52b6e108ed35c6": "\\, \\mu", "be57f8c7a66ba7b9075ba5cb134de42a": "\\begin{align}\nJ_1 &= s_{kk}=0,\\, \\\\\nJ_2 &= \\textstyle{\\frac{1}{2}}s_{ij}s_{ji} \\\\\n&= \\tfrac{1}{2}(s_1^2 + s_2^2 + s_3^2) \\\\\n&= \\tfrac{1}{6}\\left[(\\sigma_{11} - \\sigma_{22})^2 + (\\sigma_{22} - \\sigma_{33})^2 + (\\sigma_{33} - \\sigma_{11})^2 \\right ] + \\sigma_{12}^2 + \\sigma_{23}^2 + \\sigma_{31}^2 \\\\\n&= \\tfrac{1}{6}\\left[(\\sigma_1 - \\sigma_2)^2 + (\\sigma_2 - \\sigma_3)^2 + (\\sigma_3 - \\sigma_1)^2 \\right ] \\\\\n&= \\tfrac{1}{3}I_1^2-I_2,\\,\\\\\nJ_3 &= \\det(s_{ij}) \\\\\n&= \\tfrac{1}{3}s_{ij}s_{jk}s_{ki} \\\\\n&= s_1s_2s_3 \\\\\n&= \\tfrac{2}{27}I_1^3 - \\tfrac{1}{3}I_1 I_2 + I_3.\\,\n\\end{align}\n", "be58311b4aafa09d78749539695cb27d": "N (x_n)", "be5858e4690cd2d4caf14552b7572b21": "(b, c) \\times (0, d)", "be5896fba115b3c40c601bf76316b0ed": "(-\\gamma^\\mu \\hat{P}_\\mu + mc)_{\\alpha_r \\alpha'_r}\\psi_{\\alpha_1 \\cdots \\alpha'_r \\cdots \\alpha_{2j}} = 0 ", "be58a6efcf08220e495d0f7e0b45ca42": "\\hat u_i", "be58ca80bb8afcfac6c4ab5f220c9428": "\\mathbf{v}_0=[1,1,\\dots,1]", "be58dc1585ae8f0e93f4b97754bafac4": "\\phi *\\psi (\\operatorname{id})(1)=\\operatorname{id}(1)=1", "be58eebf55fea7451c1e6d0666a8caf3": "b_1=\\frac{c_1,c_2}{a_1}\\mbox{ with remainder }r_1", "be5a245cb71b53e792e42810fbf1c461": "x^0\\rightarrow-\\infty", "be5a40095b24ef15398a4284653fff57": "O(1)\\hookrightarrow O(2)\\hookrightarrow\\ldots\\hookrightarrow O(k)\\hookrightarrow\\ldots", "be5a43ca7039b973bec5b00d02ec8b46": "\\mathbf{v}=(u(x,y,z),v(x,y,z),w(x,y,z))", "be5a51c028fd4767b4690e685f544025": "q^4 + s^4 =t^4", "be5a97203be4dedb236e8a0ee158f94f": "g \\in \\mathbb{R}^p", "be5b15163e0a56fac340bfcd8a6c3891": "U''", "be5ba9455a61f9bf2ae36909dca694bc": " \\binom{x+n-1}n = \\frac{(x+n-1)(x+n-2)\\cdots(x+1)x}{n(n-1)\\cdots2\\cdot1} = \\frac{(x+n-1)^{\\underline n}}{n!}.", "be5c3b466fdc788381e91d2051e1ae96": "V_F = A \\exp(\\gamma l)\\,", "be5cb85055fcd4574e8317e22810d873": "v \\colon [-1,1]\\to [-1,1] \\colon t \\mapsto v(t) = \\begin{cases} 1, & \\mbox{if } t > 0; \\\\ 0, & \\mbox{if } t = 0; \\\\ -1, & \\mbox{if } t < 0. \\end{cases} ", "be5d020681555f686020b08c9fac768f": "x_t, x_{t+1}, x_{t+2}, \\ldots, \\, ", "be5dfa7e103b620ca21bd0697cf6ccab": "V = x^3 + y^3 + axy + bx +cy \\, ", "be5e54f5bcbbf992f6fc6492fdee7802": " Z =\\underset{s \\in \\mathbf{R}}{\\operatorname{argmax}}\\ (W(s) - s^2), ", "be5e5dee1e2256ead8d910f1efcc01bc": "R\\tilde{R}=\\tilde{R}R=1 .", "be5e7a9ab80ef1f79a62cfca26c9e40d": "\n W = C_1(\\lambda_1^2 + \\lambda_2 ^2+ \\lambda_3 ^2 -3) + C_2(\\lambda_1^2 \\lambda_2^2 + \\lambda_2^2 \\lambda_3^2 + \\lambda_3^2 \\lambda_1^2 -3) ~;~~ \\lambda_1\\lambda_2\\lambda_3 = 1\n ", "be5e7c8d12b2a28601fc1d0605eb44f8": "L_2=\\ln\\left(R_2\\right)", "be5ed23ca7b81c50755559191c52e886": " \\delta_x", "be5ed5fc358769008adb58f90396e6ff": "\\xi_\\mathbf{k} = \\epsilon_\\mathbf{k} - \\mu", "be5efc3b10eb1987ca0a385763767c5a": "I=I_{0}\\frac{f}f_0\\left[1+\\frac{\\lambda(f_0-f)}{2c}\\right]\\cos^2\\theta_i\\; ", "be5f727ecb4a3920890171d9d8aafdb0": "(M,\\varphi)", "be5f7d0028c1c35dbeca4e329b1611cc": " N \\hbar \\omega ", "be5fa506bb254607a01a8eee4ba7cb18": "\\frac{9}{8} \\sqrt[4]{2}", "be5fb7b9013a13e68b56cbc6d3b4af2b": "(P_1^2 \\or P_3^0) \\and (P_2^0) \\and (P_1^2 \\or P_3^0) \\and (P_1^2 \\or P_2^0 \\or P_3^0) \\and (P_2^0)", "be600049f00bf88bd403b6d0e359c43f": " \\mathbf{C}_{0} = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} = \\left ( \\begin{array}{c|c} \\mathbf{I} & \\mathbf{0} \\end{array} \\right ) ", "be603123f8badd99a54e17b6ed4b7cfd": "0 + 0", "be604c96277dc408355856206d317883": "\\frac{\\partial{n_i}}{\\partial{t}} = -\\nabla \\cdot n_i \\mathbf{q} +\\nabla \\cdot D_p\\nabla_i + \\left(\\frac{\\partial{n_i}}{\\partial{t}}\\right)_{growth} + \\left(\\frac{\\partial{n_i}}{\\partial{t}}\\right)_{coag} -\\nabla \\cdot \\mathbf{q}_F n_i", "be606ce58050fcff8d1a485f8664335f": "{\\Delta}E=E_A-E_I+J\\,", "be606f26b1eadc6c83f1f0169e61db95": "\\mathrm{SO}(n+2)\\,", "be60f053e7784c841c915164ecb115b7": " k_0 \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\frac{\\rho e^2}{\\epsilon_0 k_B T}} ", "be6129e89af53c2444e83aec27b4345b": "\ne =\nx \\mid\nid_\\tau \\mid\n!_\\tau \\mid\n\\operatorname{lift}_\\tau(e) \\mid\ne \\circ e \\mid\n\\kappa x:1{\\to}\\tau . e\n", "be614c67fc2ddf9efdbe87d1a827b45f": "F_x = -T \\frac{x}{L}", "be616fcf9fc20351b58a3923a918c800": "i=nFk \\exp \\left( \\pm \\alpha F \\frac {\\Delta V} {RT} \\right)", "be6194deb29e4ae12a0329c6bbd3ae69": " b=\n \\begin{bmatrix}\n 11 \\\\\n 13\n \\end{bmatrix}.\n", "be61da5241b7ec72990cee637f75dd38": "\\frac{1}{\\lambda}=R \\left( \\frac{1}{n_{f}^2} - \\frac{1}{n_{i}^2} \\right). \\,", "be624289f0831cb79b3c25a4a69268a1": "K = 1/r_0^2", "be624dd96c8626fcdc627abaa5532679": "(x_i, x_i^+) ", "be62c8336be5155ac133372592462e81": "r_{0}, r_{1},\\cdots r_{m}\\in Q_{1}", "be63735e514dd584684d3ca01e453dad": "x+2=0", "be63ea2913ad7d604603d804b6a83ed8": " \\psi = \\begin{pmatrix}\n\\psi_1 \\\\\n\\psi_2 \\\\ \n\\end{pmatrix} = \\chi e^{-i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)}= \\chi e^{-i(\\mathbf{p}\\cdot\\mathbf{r}-Et)/\\hbar}", "be640f9f75929001f3a18d6bd2c5cb46": "\\varphi = (1+\\sqrt 5)/2 \\doteq 1.618", "be646b656bd25cb91b31cf419ed3ac3b": "\\frac{\\partial }{\\partial \\tau} f(\\xi,\\tau) = f(\\xi,\\tau) \\wedge H(\\xi ).", "be646fb209244e6216cd89ed5fb530ea": " B = \\begin{bmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{bmatrix}. ", "be647d1bf838d10a7b48862003a278e5": " \\nabla^2 \\mathbf A - {1\\over c^2} {\\partial^2 \\mathbf A \\over \\partial t^2} = -{4\\pi \\over c} \\mathbf J_t", "be6498a3b2fee56f7c8f8c2fa908cd3c": "\\theta\\in[-1,1)", "be64b12e7585735191d3b51f9980ec20": "\\varepsilon_r", "be64c0e9fff1e4247e6073159ee6d3a9": "= 4 \\arctan \\frac{1}{5} + \\arctan \\frac{-1}{1}", "be6537307fb7a54236855c4695023cfd": "T_{1/2} = \\frac{t_1 t_2}{t_1 + t_2} ", "be654227d95864c1676d3c7869e459b9": "\\mathbb{Z}/n \\mathbb{Z}", "be65790de965e5d4d1fdb314b48e3c76": "\\theta_{i}", "be65c62f6fbd909279a0c1cd4155cbd2": "e_\\nu^I", "be6609700ae0cc3b09b66ad497646925": "\\emptyset \\neq [T] = \\bigcap_{t \\in T} [t] \\subseteq X.", "be66693fa55aac932aa7190ebb637557": "\\delta Q=T{\\rm d}S", "be666f6aee3247c865e04ca6dfc8f2a8": "{\\overline{a}}", "be6678c6914668ba2fe3c7ee5312b83a": "f \\in L^{p_\\theta}", "be667c321b0d8f3768bd4a7403343823": "r\\to \\int\\limits_{\\partial B(x, r)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\, u(y) \\,\\mathrm{d}S(y)", "be66a2efb8bc69d4818fba9948677e89": "\\left(\\sqrt{\\frac{2}{5}},\\ \\pm\\sqrt{6},\\ 0,\\ \\pm2\\right)", "be6739a7c6ea1be5b20b36574cd02d97": "r_{min} = -\\frac{\\sum_i{a_i b_i}}{1 - \\sum_i{a_i b_i}}", "be67504b1b8ffe3c174cb320b8f78f05": "\\alpha=\\frac{2}{\\beta}.", "be676c86809ee03e5e98882409aa88e9": "\\begin{align}\n AA^{T} &= \\begin{bmatrix}\n 1 & 2 & 3 \\\\\n 4 & 5 & 6\n \\end{bmatrix}\\cdot\n \\begin{bmatrix}\n 1 & 4\\\\\n 2 & 5\\\\\n 3 & 6\n \\end{bmatrix} = \n \\begin{bmatrix}\n 14 & 32\\\\\n 32 & 77\n \\end{bmatrix} \\\\\n (AA^{T})^{-1} &= \\begin{bmatrix}\n 14 & 32\\\\\n 32 & 77\n \\end{bmatrix}^{-1} = \\frac{1}{54}\n \\begin{bmatrix}\n 77 & -32\\\\\n -32 & 14\n \\end{bmatrix} \\\\\n A^{T}(AA^{T})^{-1} &= \\frac{1}{54}\n \\begin{bmatrix}\n 1 & 4\\\\\n 2 & 5\\\\\n 3 & 6\n \\end{bmatrix}\\cdot\n \\begin{bmatrix}\n 77 & -32\\\\\n -32 & 14\n \\end{bmatrix} = \\frac{1}{18}\n \\begin{bmatrix}\n -17 & 8\\\\\n -2 & 2\\\\\n 13 & -4\n \\end{bmatrix} = A^{-1}_\\text{right}\n\\end{align}", "be677411b24311fdc5aadf8c7dce68af": " \\mathfrak P ", "be67862c04bca400cf335dd356482b35": " S(1 \\lor 2) \\to 3 \\lor 4 \\qquad S(3 \\lor 4) \\to 1 \\lor 2", "be67cb79ad716c2539c006a541e9dde4": " \\forall x, y \\in A : \\|x \\, y\\| \\ \\leq \\|x \\| \\, \\| y\\| ", "be67deb841d7381ef4029435603deef2": "z_+\\,", "be682859672c2d226c7d11e40a655cd6": "k_{max}=\\pi/a", "be687a043d2d4b5e2a280b625ab8c72b": "\\mathsf{S}(a) \\times \\mathsf{S}(b)", "be6890a42d976569ffcde25a2f5d9a55": "\\displaystyle W", "be68b6fcbe8335eb559ad37e5c8e344f": "\n \\mathrm{d}\\mathbf{x} = \\sum_{i=1}^3 \\sum_{j=1}^3 \\left(\\cfrac{\\partial x_i}{\\partial q^j}~\\mathbf{e}_i\\right)\\mathrm{d}q^j\n ", "be6926df17518422447e9e8a87d60cc0": "2\\#", "be69568abbd883c64964157ee9054fd3": "\\mathrm{COP}_{\\mathrm{heating}} \\le \\frac{T_H}{T_H - T_C}\\,", "be695d6064139ce547407417a025d05b": "\\mathcal{M}=", "be69b9a48077e93682919420d1db509c": "\\lambda m.\\lambda n.\\lambda f.\\lambda x.m\\ (n\\ f) \\ x ", "be69bd666fce3ddb22af5e27e3228c6f": " \\eta_0= \\eta_s + \\eta_p ", "be69bd952d906a1caa6c6828f9b5e59b": " D_o \\simeq 15 N \\frac{C^2 S}{\\lambda^3} ", "be69d933e508ca21f8c4281191eed0e1": "\n\\Vert \\mathbf{\\varphi} \\Vert = \\left( \\int_\\mathcal{T} \\varphi(t)^2 dt \\right)^{\\frac{1}{2}}. ", "be69daf2ca15c80df4b5db81ead421d5": " \\sigma_y = C (\\dot{\\epsilon})^m \\,\\! ", "be69dfca0e6d31fe81ec145cdee57cb2": "g(k) = z^k", "be6a1d8f3145ce3810396d767454af40": "\n\\begin{align}\nd\\Omega_i&=\\sin\\Theta_i\\ d\\Theta_i,\\\\\nd\\Omega_f&=\\sin\\Theta_f\\ d\\Theta_f.\n\\end{align}\n", "be6b540fadd2ffb4944a2f33a3b24d19": "Z=-{{1}\\over{2\\pi i}} \\oint_{\\Gamma_s} {D'(s) \\over D(s)}\\, ds + P", "be6be6fc0cd587760523c34fddd815ba": "y(t)=C x(t) + D u(t)\\,", "be6c217c61c877e655980977308565bc": " g", "be6c36da517f067449ee3d1d1524868b": "\\qquad", "be6c3b4fef68de84e22d95de14d914a0": "\\mathrm{Euc}(n)\\simeq \\mathrm{O}(n)\\rtimes \\R^n", "be6c95526ed0f02413e1beb5bd60a6a0": "\\frac{\\sigma(n)}{n^{1+\\varepsilon}}", "be6ca27d1dcd6c2da927f868044f4df9": "W_\\lambda\\chi(x)", "be6cec2578c42c1d5c18d74d937f7c0c": "\\Pr(x_i\\mid I) = \\frac{1}{Z(\\lambda_1,\\ldots, \\lambda_m)} \\exp\\left[\\lambda_1 f_1(x_i) + \\cdots + \\lambda_m f_m(x_i)\\right].", "be6d7a1e8f35685ca5bcbbffcf114efa": " W_d ", "be6d9b1ac0170b98428b34ad56a3246d": "\\mu\\left(\\bigcup \\mathcal{G}\\right)=\\sup_{G\\in\\mathcal{G}}\\mu(G)", "be6e10730a347dc80247452c87fa2e86": "\\Sigma(x - \\overline{x})^k", "be6e1ad51c4a4786ad71a72b2f5cfc39": "\\langle\\vec{k}_a;\\epsilon_\\mu|\\vec{k}_b;\\epsilon_\\nu\\rangle=(-\\eta_{\\mu\\nu}){1\\over 2|\\vec{k}_a|}\\delta(\\vec{k}_a-\\vec{k}_b)", "be6e7af9eef11093fd14b11d099dc460": "P = {Nm\\overline{v^2}\\over 3V} ", "be6e8160d9c733251bb6e51eed8e385f": "U(\\mathbf{P}_\\phi)=e^{i\\phi}", "be6e965b2ae421179af9643280249400": "\\mathcal{O}(\\epsilon^2)", "be6ebf88846bad4a11efe2971118939e": "\n\\left(\n 1 - \\sum_{i=1}^p \\phi_i L^i\n\\right)\n\\left(\n 1-L\n\\right)^d\nX_t\n=\n\\left(\n 1 + \\sum_{i=1}^q \\theta_i L^i\n\\right) \\varepsilon_t \\,\n", "be6f15a803a280348b7edb0a873e4ef4": "\\sqrt{1 - x^4}", "be6f16868b0e3fb3ef20dd92f9aa2888": "\\scriptstyle A^{B\\times C}", "be6f46e2cdd574a5bd6d6e9ea87658b9": "\\mathbf{L}_X \\, d\\bar{z} = f(\\bar{z}) \\, d\\bar{z}", "be6f4aaeec71201d095d0c270b968f68": "\\forall\\varepsilon > 0\\;\\exists \\delta >0 \\;\\forall x \\in I \\;(0 < x - a < \\delta \\Rightarrow |f(x) - L|<\\varepsilon)", "be6f4c67faf880bf06dcdd5a581a5700": "\\mathbf{Z}/\\ell\\mathbf{Z}", "be6f5b811fad685e25fdc3bfec286f43": "c={\\partial C \\over \\partial m},", "be6fbb5ad32d692edbed96e19fd8d8b7": "\\mathbf{R} ", "be7008b526e1b9a326ecafba64e8d5a4": "k_{-2}=1\\,", "be70ff3f5df65d4fe3bbfe7998e8667a": "a_1 = \\frac{a + b}{2},\\qquad b_1 = \\sqrt{a b}.\\,", "be711036c4b423816af4fd40c16db0de": "\\langle k_1, k_2 \\rangle \\,.", "be71b65d0636c3842621ee53c4e82244": "\\Phi=e^{\\beta(E-\\mu)}\\,", "be71e9d0419232e0a072127a3b425c7b": "Lclm", "be720732726bae13b7bcad579ad3099e": " (p'_x,p'_y) ", "be720a79367232ddce8ae74ab4187b22": "E = m_0c^2\\left[1 + \\frac{1}{2}\\left(\\frac{p}{m_0c}\\right)^2 - \\frac{1}{8}\\left(\\frac{p}{m_0c}\\right)^4 + \\cdots \\right]\\,,", "be720c0e4be7419950ecdb57d51353fc": "(B \\or C)", "be7213bbbf93fe30cf104998059cb869": " \\vec a = \\frac{\\sum \\vec F}{m}", "be727d4572cc6139abf5410f4fb58a9a": "(x,r, \\theta)", "be7280c2e77f45082653907bb03dcdfd": "1-2^{2k}+3^{2k}-\\cdots = 0.", "be72949b5996be265f3e211acc28829f": "\\det X = c^2 - a^2 - b^2\\,", "be72a2352b7a779163d0a85afd4f424f": "\\oint_C ", "be72b70393551c7409f9aecef442ec3a": "B\\le qdn", "be72fb7e571b48ad667eaabafef48d12": "\\tfrac{m}{m+1}\\tbinom{2m}{m}", "be7332df733c4976eb4317d2defe66e0": "{\\eta_{stage}} = {\\eta_b}*{\\eta_N}", "be738a0bc0411efd0d66472baf4225dd": "1305184", "be73b8534106a0723d4eaea449c8e767": " \\sum_{i=1}^\\infty r^i=\\frac{r}{1-r}, ", "be741fe55ae08f617413928c580e4e24": "P(\\text{well}\\cap\\text{positive})=P(\\text{well})\\times P(\\text{positive}|\\text{well})=99%\\times1%=0.99%.", "be742521f222f7a71b42feae30a33c28": "\n\\begin{align}\n(1) ~~~e > e_s > e_i \\\\\n(2) ~~~e_s > e > e_i \\\\\n(3) ~~~e_s > e_i > e \\\\\n\\end{align}\n", "be747de70c1827ab24295172129a8b60": "\\vec x|n\\in T", "be74db548621dd1bd0569772d1d2ee2f": "\\Delta = \\delta \\cos \\left (\\phi \\right )", "be74deb8a220372eb1fe925badbd0802": "u\\Vdash B", "be75133fa05b17ecfb36d5984d7fe56a": "f^{-1}(\\mathcal{F})", "be75163cf103c0cce5ddf3cebae53dd3": "\nd\\mathbf{r}_{k} = \\sum_{r=1}^{D} dq_{r} \\frac{\\partial \\mathbf{r}_{k}}{\\partial q_{r}}\n", "be751b9685929d4fad95e83b6fdc41cb": "\n-\\frac{1}{2}\\psi''(x)+ U(x)\\psi(x)=E\\psi(x)\n", "be7520317c6b8f8c604acfa150adb4d6": "P = T^2", "be755a14c3c93e5e6004bce4940df5dd": "\\square~=~\\frac{\\partial}{\\partial \\tau}~\\mathbf{o}~+~\\frac{\\partial}{\\partial x}~\\mathbf{i}~+~\\frac{\\partial}{\\partial y}~\\mathbf{j}~+~\\frac{\\partial}{\\partial z}~\\mathbf{k}~=~\\frac{\\partial}{\\partial \\tau}~\\mathbf{o}~+~\\nabla~=~\\frac{-i}{C}~\\frac{\\partial}{\\partial t}~\\mathbf{o}~+~\\nabla .", "be757116cf3edd8952f6afe84c8a4031": "\\textstyle \\tfrac{1}{p}", "be757864df6d03f9f7782a6a5620e33f": " \\log_{10}(F(x)) = mx + b ", "be75d40fbc57657321d2f9bbb2ae4c38": "\\omega_c = \\frac{1}{\\sqrt{L_{\\frac{1}{2}} C_{\\frac{1}{2}}}}", "be76b2a9b69e5eec389ddfc0ad9f74d5": "\\mathbf{r}_0 = (x_{10},x_{20},\\dots,x_{N0})", "be76eca2b73f3edf7863870c43b56c4d": "\\epsilon_{graphene}", "be77357cc48202a8fd4cbdb6119617c3": "P(F, \\tau) = D \\left[ N(-d_-) K - N(-d_+) F \\right]", "be776571819ba7e3de83e69ef184e1ae": "T=\\frac{\\Delta s}{\\sqrt{1-\\omega^2 \\, R_0^2}}, \\; Z=0 ", "be778389cb1425ba081953d3ed0734e6": "x_i, p_j", "be77a060005aa645fa899660696dbc31": " \\left | q \\alpha -p \\right | \\le \\frac{1}{N+1} ", "be77c7ea100d36fdd298f8b0baf4291f": "\\lambda(L) \\leq 1", "be77ee028017cb26fbf040c12a7ded08": "(\\frac{1 + \\gamma}{2})G_M, ", "be78009eacd0f9948f7d5e210f1ec354": "\\color{Purple}\\text{Purple}", "be784a93354fdcbe9d4b83b148d8c92d": "x^+ = \\left\\{\\begin{matrix} 0^T, & \\mbox{if }x = 0;\n \\\\ {x^* \\over x^* x}, & \\mbox{otherwise}. \\end{matrix}\\right. ", "be784f2ac607d516cc86484df2eb94ed": "p(x)=0.", "be78811c6b5a2671681f6a350bf12859": "\\displaystyle{H_i=\\{\\xi:(T-\\lambda_i I)^{m_i} \\xi=0\\}.}", "be78dd31229c162b39a0bc12869fcc27": "O_n\\,", "be79026d2695f7ac58c0407e0869dde6": "\\ \\Delta_0 = 0,", "be7962944255c3eda89ddf57607fb7a9": "\\mathrm{Ref}_l(v) = 2\\frac{v\\cdot l}{l\\cdot l}l - v", "be798990949f1fd430dd3ae3d3f9dac4": "A(\\omega) = \\frac{R_0}{Z_{in} + R_0}", "be79a7c743e196ac62850c0800363371": "p\\neq q", "be79b5c0726688c4c4761661f94783cd": "H^1 = W^{1,2}", "be79bef15867ae77963aa6afd3fda415": "\\rho>\\rho_0", "be79c5d803240f269e5c1d725f3341d4": "\n\\mathbf{\\hat{n}}=(\\cos \\psi \\sin \\mu,~\\sin \\psi \\cos \\mu,~\\cos \\mu),\n", "be79f7f945df09954a5c59a467a713c3": "~J_n(x)~", "be7a87b45fa292446698a5f397255873": " \\Delta f = 0", "be7a8c9a92b2e3faebbcf17155162414": "\\rho = \\frac{1}{qn_m\\mu_m}", "be7b096889f20b704a0cc20de6146be7": "\\Gamma_{ij,k}^{(0)}", "be7b58dfe827c4ea0ba5bcdbe4e47c7b": "\\mathbb N^{(n)}", "be7ba347eb65edc5d58b0bbecff3e055": "1\\to A_5 \\to S_5 \\to Z_2 \\to 1", "be7bae39a858bc3f40aadaaed913054f": " \\arccos z = \\frac {\\pi} {2} - \\arcsin z\n= \\frac {\\pi} {2} - \\left( z + \\left( \\frac {1} {2} \\right) \\frac {z^3} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {z^5} {5} + \\cdots\\ \\right) \n= \\frac {\\pi} {2} - \\sum_{n=0}^\\infty \\frac {\\binom{2n} n z^{2n+1}} {4^n (2n+1)}; \\qquad | z | \\le 1 ", "be7bc047602f883532174b038ab60d52": "dU=-pdV+TdS", "be7bdda608f4dece44b2d3ef60b58168": "v_{t}", "be7bef3084af6f04ab139859aeb30119": " \\| \\cdot \\|_{\\operatorname{C}^*} ", "be7c388f2091c1dfb0c9e1da709bb736": "ddded", "be7c5958a4223379b79bf3047327a0c0": "P_n(x) = {1 \\over 2^n n!} {d^n \\over dx^n } \\left[ (x^2 -1)^n \\right]. ", "be7d88949bcb95bc145c6889a108bcf8": "\\mathbf{n} = \\frac{\\mathbf{r} - \\mathbf{r}_s}{|\\mathbf{r} - \\mathbf{r}_s|}", "be7d976bdd18c902f894b85190337527": "(x^\\mu,a^m_\\mu) ", "be7d9b8593073e55bed0acdcd954804a": "2^{\\aleph_0}= \\aleph_1", "be7ddaaf2c62feeef8ba7e34b19ea83e": "\\frac{|AP|}{|BP|} = \\frac{|AC|}{|BC|}", "be7df41863c2be643001fcf33a0b4770": "2\\sqrt 2", "be7e4dc9420caffd105e2c1726e175c2": "P'_D", "be7e71dc7e4af510bbe81fc0c37022fd": " R_{0} < \\frac{N}{S(0)} ,", "be7f116e5c225353c1b70eff8f58f49d": "q_1q_2=1", "be7f25eaf5466ecff2e414b724c286d5": "\\left \\langle \\frac{a}{b} \\right \\rangle", "be7f759dd13dd5697ce6a8917df6b932": "\\int \\arcsec{x} \\, dx = x \\arcsec{x} - \\ln \\vert x \\, ( 1 + \\sqrt{ 1 - x^{-2} } \\, ) \\vert + C , \\text{ for } \\vert x \\vert \\ge +1 ", "be8004936cefab400798858bce1d2b54": "x\\rightarrow \\infty", "be8015bba62c968bbe17ffc867044584": "\\alpha > -1", "be8036e31fad5ddff11ce6a39009a3fa": " \nK^{(0)}=\\begin{bmatrix}\n 1\n \\end{bmatrix}\n\\qquad\nK^{(1)}=\\left [ \\begin{array}{rr}\n 1&1\\\\\n 1&-1\n\\end{array}\\right ] \n\\qquad\nK^{(2)}=\\left [ \\begin{array}{rrr}\n 1&1&1\\\\\n 2&0&-2\\\\\n 1&-1&1\n\\end{array}\\right ] \n\\qquad\nK^{(3)}=\\left [ \\begin{array}{rrrr}\n 1&1&1&1\\\\\n 3&1&-1&-3\\\\\n 3&-1&-1&3\\\\\n 1&-1&1&-1\n\\end{array}\\right ] \n", "be80513999c1cc08af82a484ce2d6324": "x, y, z\\in A", "be808b39d3bc00ab017bc010c72c8ede": "T = \\frac{1}{2} \\sqrt{abh_ah_b},", "be80da424bfb2d1164e95e4f27c9c404": "\\inf \\{s : \\mathrm{some}\\ s\\mathrm{-gale\\ succeeds\\ on\\ all\\ elements\\ of\\ } Z \\}", "be815b03094ed8c1507e53ef3bb442f9": "(\\tfrac{a}{m}) = -1,", "be81922317267247d263ba29a94ee925": "f_1,\\dots,f_k", "be81b031738dc714184b7b8b5853c083": "{C_{ab}}^{cd} = {R_{ab}}^{cd} - 4S_{[a}^{[c}\\delta_{b]}^{d]}", "be81b80a5098bd824e7f663b7d9a6645": "C_{t+dt}", "be820da0fd356ad5467f36d60a8a6c96": "\\displaystyle K_f(x,y) = \\sum_{\\gamma\\in \\Gamma}f(x^{-1}\\gamma y).", "be823ef17f25d28008b3a5b2a38be9fb": "\\begin{align}\n S'_{ii} &= c^2\\, S_{ii} - 2\\, s c \\,S_{ij} + s^2\\, S_{jj} \\\\\n S'_{jj} &= s^2 \\,S_{ii} + 2 s c\\, S_{ij} + c^2 \\, S_{jj} \\\\\n S'_{ij} &= S'_{ji} = (c^2 - s^2 ) \\, S_{ij} + s c \\, (S_{ii} - S_{jj} ) \\\\\n S'_{ik} &= S'_{ki} = c \\, S_{ik} - s \\, S_{jk} & k \\ne i,j \\\\\n S'_{jk} &= S'_{kj} = s \\, S_{ik} + c \\, S_{jk} & k \\ne i,j \\\\\n S'_{kl} &= S_{kl} &k,l \\ne i,j\n\\end{align}", "be828375a09ee8b1e12d4ced32566b4a": "\\{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \\}", "be82ab786bd8f8246be89e0f093f3196": "F(x) = \\tfrac{1}{2}+\\frac{x}{2\\sqrt{2+x^2}}.", "be82f968ba5d0c8adfa33588cf8232d0": "\\log(n)", "be8376b63e4b85240153c58e9fdb101c": "z_\\mathrm{res}", "be840e962af2a41f9a733bb2b1f33d78": "0 \\le i \\le k - 1", "be8434141c1d744549c6bc9cbcfb9e14": "\\rho(x)=\\frac{8}{\\pi}\\sqrt{x(1-x)}", "be84437aefc062ec45eecbe5956d7571": "X(x) = B \\sin(\\sqrt{\\lambda} \\, x) + C \\cos(\\sqrt{\\lambda} \\, x).", "be846fbc8bce5bdaa1d318b6632fa1a5": "\n\\begin{array}{rccl}\n\\text{Decision Space} & D & = & (0,\\infty)\\\\ \n\\text{State Spaces} & S(d) & = & \\mathcal{U}(d,\\tilde{u})\\\\\n\\text{Outcomes} & g(d,s) & = & \\varphi(q,d,s) \n\\end{array}\n", "be84797495015839a64708a8e38ebc5c": " \\eta \\otimes \\eta(1) : k\\to R\\otimes R", "be848abd3cc43327d24461a96aa53c90": "{\\rm si}(x) = -\\int_x^\\infty\\frac{\\sin t}{t}\\,dt", "be84b5b1e85514254a19253e019ad4ea": " H\\psi(\\vec{r}_1,\\, \\vec{r}_2) = \\Bigg[-\\frac{1}{2}\\nabla^2_{r_1} - \\frac{1}{2}\\nabla^2_{r_2} - \\frac{Z}{r_1} - \\frac{Z}{r_2} + \\frac{1}{r_{12}}\\Bigg]\\psi(\\vec{r}_1,\\, \\vec{r}_2). ", "be84d426e33608e6002b31e6e2aff90e": " \\mu = \\frac {1} {2} \\sqrt { \\frac {4 \\beta A_0} { \\tau_1 \\tau_2} - \\left( \\frac {1} {\\tau_1} - \\frac {1} {\\tau_2} \\right)^2 }. ", "be850bda5b9b9b8d9942a69fb4ac2a3d": "{{v}_{i}}=\\phi ({{x}_{i}})", "be855e669301412ac3f3b83755234fb6": " \\frac{1}{x(1-x)} ", "be857dae69368927e9217c941e997159": "A_{\\Sigma} = 8\\pi \\ell_\\text{PL}^2\\gamma\n\\sum_i \\sqrt{j_i(j_i+1)}", "be85dafdab3661e98553d386053519fa": "g\\colon\\,x\\mapsto x^2.", "be85e6288f05f084f9258a0b9fd70963": "\n I_{\\nu\\eta}(\\theta) = \\operatorname{E}[\\, z_\\nu z_\\eta' \\,] = 0 \\quad \\text{for all }\\theta,\n ", "be85f0b79b611da0bffabbfaa5d409c6": "\\Phi\\or\\Phi", "be86fd4cd4ef851245e3bd4efb51adb1": "\\frac{\\Gamma(3/2)^{2n}4^n}{\\Gamma(\\frac{n}{2}+1)^2}.", "be877926cd55f3f21e638eed0bfb7739": " \\mathrm{Spec}(R) ", "be878688d6c5004a90173a01a1523800": "\\prod_{k=1}^n \\left({x_k \\over x_k + y_k}\\right)^{1/n} \\le {1 \\over n} \\sum_{k=1}^n {x_k \\over x_k + y_k},", "be87e9be40417ea133dbe331b30dccc1": "(B / N K)", "be885a3c468d7a69e997453e0ee60459": "F_{0}", "be885e873a17f79a275aacad678716dc": "\nq(t) \\ \\stackrel{\\mathrm{def}}{=}\\ e^{D(t)} x(t)\n", "be887c63ba9ee034f232d4b8ef81cd24": "E_n = - h c R_{\\infty} \\frac{Z^2}{n^2} ", "be88da065bc68261b6a360b3ef965867": " \\mathcal{H}_A ", "be89bff36f58d6dff9fd4070749f6df7": "Z=0\\,\\!", "be8a3658d1b18d6a36e91e2be6605ef0": " \\psi(x,\\ t=0) = A\\ \\exp \\left( -\\frac{x^2}{2\\sigma^2} + i k_0 x \\right) \\ , ", "be8a967cfadcf008fdbb423ec0bd9b7f": "\\mathrm{z}^{-N}", "be8aeaedb748c499c41301debbcb2c33": "\n\\hat \\sigma _z \\,\\, \\approx \\,\\,\\bar z\\,\\,\\sqrt {\\,\\,\\left( {{{2\\hat \\sigma _x } \\over {\\bar x}}} \\right)^2 \\,\\, + \\,\\,\\,\\,\\left( {{{\\hat \\sigma _y } \\over {\\bar y}}} \\right)^2 \\, + \\,\\,\\,4\\left( {{{\\hat \\sigma _{x,y} } \\over {\\bar x\\,\\bar y}}} \\right)}", "be8b2c2230c2c01e811e06fb15932853": "\\mathbf{\\left(J^TWJ\\right)\\Delta \\boldsymbol \\beta=J^TW\\ \\Delta y}.", "be8b4a43eb741f8a1b6b7e0f2f696fa5": "\\text{(mean deviaton around mean)}(Y)=(\\text{(mean deviaton around mean)}(X))(c-a) =\\frac{2 \\alpha^{\\alpha} \\beta^{\\beta}}{\\Beta(\\alpha,\\beta)(\\alpha + \\beta)^{\\alpha + \\beta + 1}}(c-a)", "be8ba7735738722966476d26232dadd7": "\n\\begin{align}\n\\frac{g,L^{norm}_{g}}{g, g} & = \\frac{g, D^{-1/2} L D^{-1/2} g}{g,g} \\\\\n& =\\frac{f, Lf}{D^{1/2} f, D^{1/2} f} \\\\\n& =\\frac{\\sum_{u~v}(f(u) - f(v) )^2}{\\sum_{v} f(v)^2 d_{v}},\n\\end{align}\n", "be8bca388e52c65579c1c4d04b795ad6": "v \\cdot dv = -\\frac{GM}{r^2}\\,dr\\,", "be8bd8da29c0ba185b3152d482446a8d": " M_q\\equiv C^{d_q} \\bmod\\ q ", "be8c6c62ecfe7b3edcb037c37d32f053": "370_{-150}^{+260}", "be8c969010055a73ce3ab81c594a6233": "E > \\operatorname{min}\\{ V( r \\to - \\infty ) , V( r \\to + \\infty ) \\}", "be8ce83c4012411934119c24512dcc54": "!_\\tau:\\tau{\\to}1", "be8d15e89f4e45eba5a34e1b1a927135": " u_3(X_1,X_2,X_3)=E(X_1X_2X_3)-E(X_1)E(X_2X_3)-E(X_2)E(X_3X_1)-E(X_3)E(X_1X_2)+2E(X_1)E(X_2)E(X_3)", "be8d42199aeb9532c4f8f002ecb135e1": "1 - k\\,\\mathrm{B}(k, \\rho+1)\\,", "be8d6880f782946e4e9eed3f4b4188dd": "x\\cdot\\left(\\frac{1}{x}\\right) = 1", "be8d90b9e60c43e9f3c0c9d122a80d6d": "T(E)=\\frac{2\\epsilon}{k_B}\\left[\\ln \\left( \\frac{(N+1)\\epsilon - E}{(N+1)\\epsilon + E} \\right)\\right]^{-1}.", "be8dae2a8c3a12c072ca75beb4396f30": "L_x = L_y = L", "be8de749306572fd4cda8ccbfcdd4f47": "\n\\sum_k \\psi^\\dagger(k)\\psi(k) = \\int_x \\psi^\\dagger(x)\\psi(x)\n", "be8e7a92acaeb659aa26819e796e08c8": "m^q / 1", "be8ee8d7cbad25374e39bd48aca87b41": "\\Pr(X=n)=\\frac{1}{n}\\Pr(S_n=n-1)", "be8f05a57a81fdafc8f2f05860d1ce2b": "H_{x}", "be8f15e62ddaf5495535dbe7b95dfb4c": "\n\\begin{align}\n\\int_0^{2\\pi}\\frac{\\mathrm{d}x}{2+\\cos x}&=\\int_{x=0}^{x=\\pi}\\frac{\\mathrm{d}x}{2+\\cos x}+\\int_{x=\\pi}^{x=2\\pi}\\frac{\\mathrm{d}x}{2+\\cos x}&&\\\\\n&=\\int_{t=0}^{t=\\infty}\\frac{\\mathrm{d}x}{2+\\cos x}+\\int_{t=-\\infty}^{t=0}\\frac{\\mathrm{d}x}{2+\\cos x}&t&=\\tan\\frac{x}{2}\\\\\n&=\\int_{t=-\\infty}^{t=\\infty}\\frac{\\mathrm{d}x}{2+\\cos x}&&\\\\\n&=\\int_{-\\infty}^{\\infty}\\frac{2\\,\\mathrm{d}t}{3+t^2}&&\\\\\n&=\\frac{2}{\\sqrt 3}\\int_{-\\infty}^{\\infty}\\frac{\\mathrm{d}u}{1+u^2}&t&=u\\sqrt 3\\\\\n&=\\frac{2\\pi}{\\sqrt 3}.&&\n\\end{align}\n", "be8f2af4fb50dbe564da4c9a84d4a73a": "S_{n+1}=\\max(0, S_n+x_n-\\omega_n)", "be8f31f215a0f44ac908ed05d01acfd3": "\\mathbb{I}_S(T)=\\{s\\in S \\mid sT\\subseteq T \\text{ and } Ts\\subseteq T\\}", "be8f3521d7afe06c82239aad4f60151c": "R_1 R_2 < 1", "be8f678f6c72cc54cce3b19f921bae19": "DC_G(D)", "be900306f4701280920e643e71ec2066": "r = z \\left(1 + \\frac{\\left(x - x^\\prime\\right)^2 + y^{\\prime2}}{z^2}\\right)^\\frac{1}{2}", "be9013786e48fd4efad8fc6670ad31c8": "J_{\\nu}(x)", "be906b108c4bf6a97c620fc37344e83c": "C(y_1=a_1,\\ldots,y_n=a_n) = \\max_a \\sum_i C_i(x=a,y_1=a_1,\\ldots,y_n=a_n)", "be9098f4d0b79e0b882184f16b163e90": "l~", "be90bbf96fde13297f30c2edac95f114": "\\frac{\\partial u}{\\partial x}+ix\\frac{\\partial u}{\\partial y} = F(x,y)", "be90f8541fc61ac8810ff435d99dafc6": "L_{mnl} = \\mu k_{mnl}^2V\\,", "be911ca1e9424c00fff3704a15425bb9": "\n\\begin{align}\nB_{1_1} =& 128 A^3 - 366 A^2 + 288 A - 80 I \\\\\nB_{1_2} =& 16 A^3 - 44 A^2 + 40 A - 12 I \\\\\nB_{2_1} =&-128 A^3 + 366 A^2 - 288 A + 80 I\\\\\nB_{2_2} =& 16 A^3 - 40 A^2 + 33 A - 9 I\n\\end{align} \n", "be912c520dc89293138e5e4c5421ae2a": "\\mathcal{Z}(M)\\cdot \\mathrm{soc}(M)=\\{0\\}\\,", "be91a566f7dcc511921f756653ec3fbd": "x'(t)=V(x(t))", "be91ece6bd0d613d5b06356b56121a92": "S^\\ell = \\cup_i S_i^\\ell", "be91f496e2796a758538e14bbb67f596": "P_{4}^{3}(x)= - 105x(1-x^2)^{3/2}", "be91f876dd0f1403a8674aa7e5612fbd": "{\\left(\\frac{\\partial \\left(n\\right)}{\\partial t}\\right)}_{diff\\text{.}}=-{v}_{x}\\frac{\\partial {\\left(n\\right)}^{0}}{\\partial T}\\frac{\\partial T}{\\partial x}\\text{.}", "be921caed3f8e40e0a4627837cf319a5": " D_j = \\begin{vmatrix}\n\\langle \\mathbf{v}_1, \\mathbf{v}_1 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_1 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_1 \\rangle \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_2 \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_2 \\rangle & \\dots & \\langle \\mathbf{v}_j, \\mathbf{v}_2 \\rangle \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\langle \\mathbf{v}_1, \\mathbf{v}_j \\rangle & \\langle \\mathbf{v}_2, \\mathbf{v}_j\\rangle & \\dots &\n\\langle \\mathbf{v}_j, \\mathbf{v}_j \\rangle \\end{vmatrix}. ", "be922f9b0d80ebf5a02cc8c9d810c3fb": "\\begin{align}\ne_{ij}(\\lambda) e_{ij}(\\mu) &= e_{ij}(\\lambda+\\mu) \\\\\n\\left[ e_{ij}(\\lambda),e_{jk}(\\mu) \\right] &= e_{ik}(\\lambda \\mu) && \\mbox{for } i \\neq k\\\\\n\\left[ e_{ij}(\\lambda),e_{kl}(\\mu) \\right] &= \\mathbf{1} && \\mbox{for } i \\neq l, j \\neq k\n\\end{align}", "be929a55272c47898f885dca10d3487a": " \\begin{align}\nf(x_1,\\ldots,x_k;n,p_1,\\ldots,p_k) & {} = \\Pr(X_1 = x_1\\mbox{ and }\\dots\\mbox{ and }X_k = x_k) \\\\ \\\\\n& {} = \\begin{cases} { \\displaystyle {n! \\over x_1!\\cdots x_k!}p_1^{x_1}\\cdots p_k^{x_k}}, \\quad &\n\\mbox{when } \\sum_{i=1}^k x_i=n \\\\ \\\\\n0 & \\mbox{otherwise,} \\end{cases}\n\\end{align}\n", "be92ab67016e908b3d58350ff6ca73d4": "I_{yes}", "be92e65cbd9596640eab62efaf04ccc6": " b_n=0\\ ", "be9305b5eeedc1d76dc33d5fda39da78": "\\int_{a}^{b} dx \\nabla f = \\int_{a}^{b} dx \\cdot \\nabla f = \\int_{a}^{b} df = f(b) -f(a)", "be933ceac1686bd11ff758d12975f698": "O_T", "be93d8913a062285278dd19862176525": "IOP =\\frac {F}{C} +PV", "be941ada323e76aed5d4968cac2b8f2c": "\\int_0^1 e^{-xs} (1-s)^{a-1} s^{b-a-1}\\,ds \\sim \\Gamma(b-a) x^{a-b} \\quad \\text{as } x \\to \\infty \\text{ with } x > 0,", "be94211f5066cb6b4a4adced2b2a68e7": "(2^{\\aleph_0})", "be945d7021012c7b505d7e350eb217a7": "\\scriptstyle{6(1/3)^s+5{(1/3\\sqrt{3})}^s=1}", "be94908305228dcb6b393cc3f0a2ae18": " \\begin{pmatrix} A & 0\\\\ 0 & A^\\prime\\\\ \\end{pmatrix},\\,\\, (b,b^\\prime).", "be953d821d0725186589aa056301b426": "p \\star 1 := p + 1 \\mod 2^n", "be958f7f88e767bbb948c8ace8bb2cfd": " \\dot{z}(t) = \\dot{u}(t) \\left\\{A - \\left[\nC_1(\\dot{u}(t),u(t),z(t),\\beta_1,\\beta_2,\\beta_3) \n+\nC_2(\\dot{u}(t),u(t),z(t),\\beta_4,\\beta_5,\\beta_6)\n\\right]|z(t)|^n \\right\\} ", "be95e17134c6ac025e45e4380131b0ab": "V_n(x_1, x_2,\\ldots x_n):=\\sum_{i0 ", "bea296a82c8755b1ec22caa511ec9e7b": " u \\wedge v = \\begin{vmatrix}u_1 & u_2 \\\\ v_1 & v_2 \\end{vmatrix} {e}_1 \\wedge {e}_2", "bea2e42f9484d02a3706a123b0a33665": "A,C>0", "bea2f943b1d5c07edbcf9231e4ddfecd": "d^{5}", "bea2fa4b1bcdeec688c830247d78424a": "[0,e^{-e} \\approx 0.066]", "bea36b93c65f1a56be8221c4d714653a": "r = f r_0 + g r'_0", "bea37992f693fa4f48c06ba273103bad": " \\boldsymbol{\\Omega}_\\text{T} = \\boldsymbol{\\omega} (\\gamma-1),", "bea3a1ca51c71418ecaebff78a7b2639": " D_\\mu := \\partial_\\mu - i g \\, A_\\mu^\\alpha \\, \\lambda_\\alpha ", "bea3a5e7a9e9e173788e2e73e2715c16": "C \\subseteq DB", "bea3cf0919418a0d6a290f545fd3330f": "\\neg A \\wedge (B \\vee C)", "bea3faf6119c7f595ffcd425a7e38b09": "s_1, \\dots, s_n", "bea44d68ebf658f3b63ac0572b9df9da": "y_{unknown}", "bea44f813339ba2c0c7291b8d69218f6": "K_{\\rm I}", "bea45fe8ed9cb31b9952e1d10482462d": " d\\Omega = - S dT - \\langle N_1 \\rangle d\\mu_1 \\ldots - \\langle N_s \\rangle d\\mu_s - \\langle p\\rangle dV .", "bea4aebaf115e15e14d9f11930d3d49f": "P=$608.02", "bea4b90215a6c3316ca2686ebdcb8280": "A_{[\\alpha\\beta\\gamma]\\delta\\cdots} = \\dfrac{1}{3!} \\left(A_{\\alpha\\beta\\gamma\\delta\\cdots}\n+ A_{\\gamma\\alpha\\beta\\delta\\cdots} \n+ A_{\\beta\\gamma\\alpha\\delta\\cdots} \n- A_{\\alpha\\gamma\\beta\\delta\\cdots}\n- A_{\\gamma\\beta\\alpha\\delta\\cdots}\n- A_{\\beta\\alpha\\gamma\\delta\\cdots}\n\\right)", "bea4f3c679deee8cd0d42c12bd7c011d": "g(s, t) = \\frac{f(s) - f(t)}{|f(s) - f(t)|}.", "bea5003a7f29bb13df2616614cc01115": "\\sigma_i\\in \\{-1,1\\}", "bea557bf9ccae94bcd49b4f72a073d7d": "\\frac{1}{\\rho}\\frac{\\,d\\,P}{\\,d\\,X}+\\tfrac{\\,f}{2\\,D}\\,W^2+\\left(\\frac {2-\\beta}{2}\\right)\\frac{\\,d\\,W^2}{\\,d\\,X} \\,=\\,0", "bea567cb3894c4c86387a46968eb63ce": "u^+(\\mathbf{x})", "bea56febd6bef26dd243448e7adaf58c": "\\phi(u) = u", "bea5b30770e5f42f0765356f5de02d26": "\\text{NegativePredictiveValue} = \\frac{TN}{TN+FN}", "bea5ce787cf8bb5c52de1da1f6993970": "X_{\\mathrm{i}}=d_{\\mathrm{ki}}E_{\\mathrm{k}} \\;", "bea667267591f0c5dce437f83ee28d94": "\\,(1.0583)^{2}=1.12", "bea69c94e81d5b3a94b0c6111e03f2cb": " \\varepsilon \\circ \\mu \\circ (\\mu \\otimes \\mathrm{id}_H) = (\\varepsilon \\otimes \\varepsilon ) \\circ (\\mu \\otimes \\mu) \\circ (\\mathrm{id}_H \\otimes \\Delta \\otimes \\mathrm{id}_H)= (\\varepsilon \\otimes \\varepsilon ) \\circ (\\mu \\otimes \\mu) \\circ (\\mathrm{id}_H \\otimes \\Delta^{op} \\otimes \\mathrm{id}_H) ", "bea6ab59bd1c2fe97d9c87ebd9c04c72": "m=0.2025", "bea6f3daf5548e960dfbf46d85ef1e83": "\\tau_1 \\curvearrowleft \\tau_2 = \\sum_{s \\in \\mathrm{Vertices}(\\tau_1)} \\tau_1 \\circ_s \\tau_2", "bea700ed533f781055d0bd6debdcc2c2": "(1 - c - b c + b c)Y_{p} = (C_{0} + I_{0})", "bea72021c5712fb8e11b91adec3f5c6b": "\n\\mathbf{g}_{-1} = \\left\\{\\left.\n\\begin{pmatrix}\n0&^tp&0\\\\\n0&0&J^{-1}p\\\\\n0&0&0\n\\end{pmatrix}\\right| p\\in\\mathbb{R}^n\\right\\},\\quad \n\\mathbf{g}_{-1} = \\left\\{\\left.\n\\begin{pmatrix}\n0&0&0\\\\\n^tq&0&0\\\\\n0&qJ^{-1}&0\n\\end{pmatrix}\\right| q\\in(\\mathbb{R}^n)^*\\right\\}\n", "bea77c2bf2c9f52977572fbcce2db044": "\n\\begin{align}\n C \\frac{dV}{dt} & ~=~ I - g_\\mathrm{L} (V-V_\\mathrm{L}) - g_\\mathrm{Ca} M_\\mathrm{ss} (V-V_\\mathrm{Ca}) - g_\\mathrm{K} N (V-V_\\mathrm{K}) \\\\\n \\frac{dN}{dt} & ~=~ \\frac{N-N_\\mathrm{ss}}{\\tau_{N}}\n\\end{align}\n", "bea793dd6aa823e5b3cba501fed3d364": " \nO^{-1} = O^T = \\begin{bmatrix}\n \\cos \\left( \\theta \\right) & \\sin \\left( \\theta \\right) \\\\ -\\sin \\left( \\theta \\right) & \\cos \\left( \\theta \\right)\n\\end{bmatrix}. \n ", "bea7a26a306313c3bf167b9723f8898e": "Itd \\sum_{n=1}^\\infty \\frac{(1-d)^{n-1}}{(1+i)^n}", "bea7ab38717a89a2f5385c1ddc7d6611": "O(p(\\tau+\\sigma m)) ) ", "bea7dd31998a5877f7733c5510d6e70e": "\\mathrm{FWTM} = 2 \\sqrt{-2 \\ln 0.1}\\ c", "bea7e1b515239d75fb329ddfe6acc5b8": "\\lambda_\\mathrm{tot}", "bea7e40bbccb3457d384ae3867e2c626": "Q_{hot}", "bea7e8b0ad8bfdbfe481a06843024ee6": "f([G,G]) \\leq [H,H]", "bea86168b357b2953937d9f273aa7633": "\\eta = B \\xi", "bea8881c0aec8660d1a8c4a10a782c32": "\\nu_L", "bea8a0f39699c5816541e69b1d6e7531": "R_{vs}=\\frac{1-h_1}{h_2-h_1}R_2-R_1", "bea8a9d979f4cbc8f9b883c4e63c617b": "M \\equiv_{m} N", "bea9239f0924e12dc28fd29f2f6be50e": "\\mathbf{r}=\\mathbf{r}_\\perp+\\mathbf{r}_\\|", "bea9318f5b97666f478f72dea8791cc4": "p,\\ a,\\ b,\\ c,\\ d", "bea957c642f913322ca0690eab04b4e8": " dS = \\frac{\\delta Q}{T} ", "bea979d2e1bb5fdc3d29fdb52f68566f": "\n{\\mathfrak{T}}^\\alpha_\\beta =\n\\left\\vert \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right\\vert^{W} \\, \\frac{\\partial {x}^{\\alpha}}{\\partial \\bar{x}^{\\delta}} \\, \\frac{\\partial \\bar{x}^{\\epsilon}}{\\partial {x}^{\\beta}} \\, \\bar{\\mathfrak{T}}^{\\delta}_{\\epsilon}\n\\,.", "beaa45a0878c6bd470b4c4d5b7994c7a": "\\mathbf{L}=\\mathbf{x}\\wedge\\mathbf{p}", "beaad0cae91b486ca1c250f721ccefbf": "\nm_p = m(\\pi/2).\\,\n", "beab2e5f5567f328151f717cbdfb86e0": "f(x) = \\left(x + \\frac{1}{2}\\right)^2 -\\frac{1}{4}", "beab3f673269b1e49260ee8c9943c3f9": "e_i, e_j", "beab4657fe203ec568fde84d85345850": "\\frac{h}{v} - \\frac{c}{v}", "beab5c7fdcb61012aff6c65d5aebbf9b": "{h_k^{\\nu}}={h_{\\nu -k}^{\\nu}}", "beab6fd14b73d4e3683eadc6c3727ccb": "s_it_{i+1}-t_is_{i+1}=s_it_{i-1}-t_is_{i-1},", "beab8d96b1befb80c2e6fef18e776aee": "\\sup_{t > 0} \\mathbf{E} \\big[ N_{t}^{-} \\big] < + \\infty.", "beaba1cce808a91b6c0ce162d7e88cea": "\\cos (2 \\sigma_m) ", "beabca3f3900b81abd03c928d4e79c2a": "~y~", "beac3011330493e8050c30bd5199a4bf": "\\Gamma=PSL(2,\\mathbb{Z})", "beacce35c60888103797034f92b77926": "u_1=v-p ", "beacd4e7061f5ea42fc62d196ecd8dbe": "\\beta = { v \\over c} \\,.", "bead001d089c51fa3f092b4e3f29afda": "\\rho(u)", "bead0e77c609bcec51a7fbc35c4db8c9": "\\frac{\\partial E(x)}{\\partial x} = 0 = 2 \\left(\\sum_i \\hat n_i \\hat n_i^\\top\\right) x - 2 \\left(\\sum_i \\hat n_i \\hat n_i^\\top p_i\\right) ", "bead15e67f2223a31346fcb9b2be4117": "f_t:X_r\\to X_r", "bead3332c80062261c63577c8cda9fd9": "G M_E \\,", "bead3bdb390852f44bf7358e580fef0a": "\\delta M ", "bead7a98293fb69523638ea8009b4554": "t \\equiv \\sqrt{1-\\frac{1}{\\xi^2}}", "bead949b802c9262d0afdb7799cd2a2a": "dim K_n = n-1", "bead9d5b3d499777c1627edda654b3ad": "\\Gamma(T)", "beadc36bb5605ecbde71d6b9476e3488": " Q(\\alpha_i, y_i) = 0 ", "beadd3cec72b251bb9018391f37af348": "( \\Delta t)", "beae7d02b51183be09dd9a1bfd583af7": "x^*(t)", "beaeff16ad40ffd628a6687766affc00": "TS=(\\alpha+1)+\\ln\\left(\\frac{\\tau^\\alpha}{\\zeta(\\alpha)}\\right)", "beaf347333ccb6af5bd8187892591c2c": "\\mathrm{amplitude\\ of\\ scattered\\ wave} = A \\mathrm{e}^{\\mathrm{i}\\mathbf{k} \\cdot \\mathbf{r}} S f(\\mathbf{r}) \\mathrm{d}V", "beaf552b14a0816bb008e81355f366c7": " n \\not\\subset \\{p, q, m\\} ", "beaf746f4f2599ba7b6b5402af255967": "C_h(f) = f\\circ h", "beaf8796670abd49c0b1c58d609af805": "y(i,j)", "beafb59abdb6c33d402084998c2aebf6": "v_{ix,iy,iz,jx,jy,jz} = \n\\sqrt{\\frac{2}{n_x+1}} \\sin\\left(\\frac{i_x j_x \\pi}{n_x+1}\\right)\n\\sqrt{\\frac{2}{n_y+1}} \\sin\\left(\\frac{i_y j_y \\pi}{n_y+1}\\right)\n\\sqrt{\\frac{2}{n_z+1}} \\sin\\left(\\frac{i_z j_z \\pi}{n_z+1}\\right)\n", "beafb6389d1b87ed5251f127fb3383c0": " N \\to \\infty ", "beafebc67755e09315de1e9ff6d13dee": "D \\left( x, y, \\sigma \\right)", "beaffcbb7b3613a8e814b42c727041f2": "P\\cap(-P) = \\{0\\}", "beb195239568309b3e3a0ce19bb5b987": "Q_j", "beb1963b1cdc6f2b5c3db76401b9f50f": "\\begin{Bmatrix} 3 \\\\ 5/2 \\end{Bmatrix}", "beb1d05793dc259c64224fd24b8264e6": "k^\\mu k_\\mu = \\left(\\frac{\\omega}{c}\\right)^2 - k_x^2 - k_y^2 - k_z^2 \\ = 0", "beb1df882531c1d42a016b0dc31b0ee6": "r \\leftarrow p,\\ q", "beb1e7eff4776a30b1d54bc0ea5ef8ac": "z_n = \\exp(i2^n\\theta)", "beb1fa12300ca4f76b81e0c27916d9e1": "\\mathbf{CP}^{k - 1} \\subseteq \\mathbf{CP}^k", "beb20271413b2a0d26111e3da1322279": " \\lim_{x \\to 0^{-}}\\frac 1 x = -\\infty", "beb202be0c66022a078efdd98aa308de": "K_2(A) \\rightarrow K_2(A/I) \\rightarrow K_1(A,I) \\rightarrow K_1(A) \\cdots \\ . ", "beb23e097a4915989cc7937aa15f5e5f": "R_2\\,", "beb25909564eaf2def7ed5dfcb0c279f": " H[(X_1,\\dots,X_d)] \\leq \\frac{1}{r}\\sum_{i=1}^n H[(X_j)_{j\\in S_i}]", "beb286d13de82999459b537653301e87": "\\boldsymbol{\\nabla}\\times\\left(\\boldsymbol{\\nabla}\\Phi\\right)=0", "beb2f668447ed68a7a661cb27552483d": " T_1 = {{m_1 g (2 m_2 + {{I} \\over {r^2}} + {{\\tau_{friction}} \\over {r g}})} \\over {m_1 + m_2 + {{I} \\over {r^2}}}}", "beb3792a704de8b39fe436d9411310dd": "x_0\\lim_{n\\to\\infty} y_n + \\sum_{i=1}^\\infty x_i y_i.", "beb39720c82f3b73094ef500511bd6ae": " d = \\frac{a}{ \\sqrt{h^2 + k^2 + l^2}} ", "beb3abccd2d26ba8156379f9dd474f42": "\\mbox{dist}_{\\pi_N}(P_N,Q_N) \\to 0 \\quad\\mbox{ as }\\quad N\\to\\infty", "beb3c684fb08cb3d94be5e24f6db646b": " \\cot(z - a_1)\\cot(z - a_2) = -1 + \\cot(a_1 - a_2)\\cot(z - a_1) + \\cot(a_2 - a_1)\\cot(z - a_2). ", "beb41a5c74a3e8d0a3056224df63f8cf": "x_1,\\ldots,x_k", "beb41ae466a5faf401abe990435acf16": "B ={1 \\over 4} \\left[ \n \\begin{array}{rrrr} 1 & 1 & 1 & 1 \\\\[6pt] 1 & -{1 \\over 2} & {1 \\over 2} & -1 \\\\[6pt]\n 1 & {1 \\over 2} & -{1 \\over 2} & -1 \\\\[6pt] 1 & -1 & -1 & 1\\\\[6pt] \\end{array}\n \\right].", "beb41f350356b66f9df8b92c702ada64": "S^{-1} M^* S = N^*. \\,", "beb4454a9b6d9ced4d3f10ee4b6c76d5": "\n\\mathbf{Q} = \\frac{\\partial G_{4}}{\\partial \\mathbf{P}}\n", "beb4ad6ff5d16fe29b1a447383af88fd": "\\kappa(\\kappa(v*)*) = v", "beb51225096ef8b06652cc78eb68c8aa": " 1 = \\int_0^t e^{-ra}\\ell(a)b(a) \\, da. ", "beb52063af348c8b6e831ee1ca91f548": "{v_{\\perp}}", "beb55bc55fb42afe899be0c86c02c867": "C_\\bullet(X)", "beb565acdd2de26d37fb9053a068676b": "\nI(t) = \\frac{1}{L}\\int_0^t E(t-\\tau) e^{-\\alpha\\tau} ( 1 - \\alpha \\tau ) \\, d\\tau\n\\text{ in the critically damped case }(\\omega_0 = \\alpha)", "beb57764fbcdcd1d7104b2e8846dd42d": "\\phi_{\\alpha_n,\\beta}", "beb58f78809f4909825862ac40547322": "\\pi \\approx {2^9 \\over 163} \\approx 3.1411", "beb59291fd264a82c99fad35db2e23fa": "\n\\begin{align}\np_n & = [x^n] P(x) = [x^n] \\frac{1-\\sqrt{1-4x}}{2} \\\\[6pt]\n& = [x^n] \\frac{1}{2} - [x^n] \\frac{1}{2} \\sqrt{1-4x} \\\\[6pt]\n& = -\\frac{1}{2} [x^n] \\sum^{\\infty}_{k=0} {\\frac{1}{2} \\choose k} (-4x)^k \\\\[6pt]\n& = -\\frac{1}{2} {\\frac{1}{2} \\choose n} (-4)^n \\\\[6pt]\n& = \\frac{1}{n} {2n-2 \\choose n-1}\n\\end{align}\n", "beb5dc79e80378d06f6dc9f68a4b15c9": "\nd_{12} = \\ln \\left( \\frac{\\nu_2}{\\nu_1} \\right)^{aa} + 1 + \\left( \\frac{\\nu_2}{\\nu_1} \\right)^{aa}\n", "beb614bf9d3fa44522952dd12aa78503": "\\overline{\\mathcal{X}}", "beb680053f1970f18d7376496315b19e": "(1/r)", "beb6c3df4b49906a6b94e3e9fed2ad9f": "\\ \\log k = \\alpha*\\log(K_a) + C", "beb6e00781e16a70ce8a6c793afc8cfe": "\\coprod_{c \\in \\mathrm{ob}C} X(c)", "beb75bc17fa6cd96a568cbd3ab53c76d": "\n H_{\\Phi}(\\mu) = \\Phi\\left(\\frac{\\sqrt{n}(\\mu-\\bar{X})}{\\sigma}\\right) ,\n \\quad\\text{and}\\quad\n H_{t}(\\mu) = F_{t_{n-1}}\\left(\\frac{\\sqrt{n}(\\mu-\\bar{X})}{s}\\right) ,\n", "beb77d97ee5eac39bcf21753ef3e7ed4": "L(\\hat{\\theta})", "beb780f45968382345c8753a5d82c9c1": "g : \\mathcal{A} \\to \\mathcal{B}", "beb79842ec7c4b8da6c0fc888a00209c": "\\,e_x\\!", "beb7a99ee746adf8a0da4eaf21c12a09": "P\\left(t\\right)=\\frac{K}{1+Ae^{-kt}}", "beb7b276f9514708500a8c025313697c": "1 \\le i \\le \\left|O(v)\\right|", "beb7f58d93fbbce0fe19e80badc03ad1": "BW(F)", "beb82263d16c3826cb8f736cc37f2671": "Profit=PF(K,L)-WL_RK", "beb8e52d83a5a4f6ce2fd3c562761dc6": "y_i(\\mathbf{w}\\cdot\\mathbf{x}_i - b) \\ge 1, \\quad \\text{ for all } 1 \\le i \\le n.\\qquad\\qquad(1)", "beb911c0ab9283c1f108ea4584616c9a": "p_x^0=", "beb9c2c8de3b3d58f00dc4aec4c7d437": "\\overline{A}.", "beba7db230ac45ef6510f77ba59946dd": "T_{\\mathrm{amortized}}(o) = T_{\\mathrm{actual}}(o) + C\\cdot(\\Phi(S_{\\mathrm{after}}) - \\Phi(S_{\\mathrm{before}})),", "beba7f9e18983d50d6c72e7280bd8ad1": "v_1 \\mathbf e_1 + v_2 \\mathbf e_2 = \n\\frac{\\mathbf{e}_{1}}{h_{2} h_{3}} \n\\left[\n\\frac{\\partial}{\\partial x_{2}} \\left( h_{3} \\psi_{3} \\right) - \n\\frac{\\partial}{\\partial x_{3}} \\left( h_{2} \\psi_{2} \\right)\n\\right] + \n\\frac{\\mathbf{e}_{2}}{h_{3} h_{1}} \n\\left[\n\\frac{\\partial}{\\partial x_{3}} \\left( h_{1} \\psi_{1} \\right) - \n\\frac{\\partial}{\\partial x_{1}} \\left( h_{3} \\psi_{3} \\right)\n\\right] + \n\\frac{\\mathbf{e}_{3}}{h_{1} h_{2}} \n\\left[\n\\frac{\\partial}{\\partial x_{1}} \\left( h_{2} \\psi_{2} \\right) - \n\\frac{\\partial}{\\partial x_{2}} \\left( h_{1} \\psi_{1} \\right)\n\\right]\n", "beba8405cbccbd1713d48b110fbc33c4": "\\beta : B \\longrightarrow FB", "bebaa0835f152922a49f74ebe1008f48": "\n\\begin{align}\n&\\chi_X(t\\mid\\nu,\\sigma) \\\\\n& \\quad = \\exp \\left( -\\frac{\\nu^2}{2\\sigma^2} \\right) \\left[\n\\Psi_2 \\left( 1; 1, \\frac{1}{2}; \\frac{\\nu^2}{2\\sigma^2}, -\\frac{1}{2} \\sigma^2 t^2 \\right) \\right. \\\\[8pt]\n& \\left. {} \\qquad + i \\sqrt{2} \\sigma t \n\\Psi_2 \\left( \\frac{3}{2}; 1, \\frac{3}{2}; \\frac{\\nu^2}{2\\sigma^2}, -\\frac{1}{2} \\sigma^2 t^2 \\right) \\right],\n\\end{align}\n", "bebb6c1c597ef1a310a84370c7067619": "|f'(x)| \\leq \\delta .", "bebb7124d2e8f05ab8618114aa1cd9a2": "\\beta^{a}", "bebbca6cfd447790d67fc551b5bcf9c6": "w_i(t+1)=vq_i(t)\\cdot y(t)+[1-vq_i(t)]\\cdot w_i(t)\\,\\!", "bebbfe02eb237bf9e41c4bf7d176db52": "\n\\leq B + C + VP(\\alpha^*(t), \\omega(t)) + \\sum_{i=1}^KQ_i(t)Y_i(\\alpha^*(t), \\omega(t)) \n", "bebc40e0cdc248136ace6e691f6922d9": "a_{-m} = - \\frac{1}{2 \\pi i} \\int_C (\\lambda - z) ^{m-1} (z - T)^{-1} d z", "bebc61baa2b37490e544ec0992565f76": " E_\\mathrm{k} = \\left ( \\gamma - 1 \\right ) m c^2 ,", "bebc71867d7ffc66539a169a176b6f68": "X[x,y]= x-y'\\frac{x'^2+y'^2}{x'y''-x''y'}", "bebcd560d6e524dcb4d4ad410a682dd3": "\\{\\lambda_i\\}_{i=1}^N", "bebce288cc2019b9a9e83e583332d2b2": "\nM_{ab} = \\{\\tilde{\\phi}_a,\\tilde{\\phi}_b\\}_{PB}.\n", "bebd116a52a6fb1a741b975ef99d065c": " \\mathbf{y} = (y_1\\,y_2\\,0)^\\top ", "bebd1de360cb3d1c26c22047de0bfa2d": "\\psi(x) = \\int h(z) G(x, z; \\lambda) dz.", "bebd77c39706cf8e0a21a8ade23355a1": "\\int_{r}e^{\\epsilon q(d,r)}\\times\\mu(r)\\,\\!", "bebd85e0afb4aecc624087615e78dd31": "\\mu_i^*", "bebda4a4ae609a426afdd980ac5196c1": "C(a)=\\int_0^a f(t)\\,dt-\\frac{1}{2}f(0)-\\sum_{k=1}^{\\infty}\\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0)", "bebe09a849e142217198138c96b8f2b2": "\\int_M \\mathrm{Tr}(F \\wedge *F)", "bebe22e46dc8f99736a8effef2ff6fc6": " \n\\{ e(t) : \\ {0 \\le t \\le 1} \\} \\ \\stackrel{d}{=} \\ \\left \\{ \\frac{|W((1-t) \\tau_{-} + t \\tau_{+} )|}{\\sqrt{\\tau_+ - \\tau_{-}}} : \\ 0 \\le t \\le 1 \\right \\} .\n", "bebe47f855713db02070be26c0090466": "\\geq\\kappa^{+}\\,", "bebe66b3968b4b32b07f441602aad03e": "t+n", "bebe8e4c37f68475f8624892ca0a53a8": " f(z) = \\sum_{i=1}^n { g(x_i) \\lambda_i x_i }", "bebef02e5c6a2bc75eea6b766ca7a019": "\\mathcal{A} = \\mathcal{S} \\times_{n=1}^N U^{(n)}", "bebef5dd532b520bd6e09c1991b841af": "\\frac{a(z)+ib(z)}{2}M_+(z) + \\frac{a(z)-ib(z)}{2}M_-(z) = c(z),", "bebf674f0091838c9ff2dc3d33784d0c": "\\begin{align}\n\\mathbf{F}_\\text{eff} &= -\\nabla U_\\text{eff}(\\mathbf{r}) \\\\\n &= \\frac{L^2}{mr^3}\\hat{\\mathbf{r}} - \\nabla U(\\mathbf{r})\n\\end{align}\n", "bebf79d6c09a86f73ec19790173dbe2d": "\\operatorname{ch}(E) = \\sum e^{\\eta_i}", "bebf7d72ee68be957e9b8d7f4c6eb123": "Ih \\ge 0", "bebf82c701eb774593128afeee2aeeb3": "\n\\begin{align}\nL&=\\|e_{m+1}\\|^2-\\lambda\\left(\\sum_i\\ c_i-1\\right),\\\\\n&=\\sum_{ij}c_jB_{ji}c_i-\\lambda\\left(\\sum_i\\ c_i-1\\right),\\ \\mathrm{where}\\ B_{ij}=\\langle\\mathbf e_j|\\mathbf e_i\\rangle.\n\\end{align}\n", "bebfd5281b5a93fcf183ce226567f181": "{dy \\over dx}=\\frac{1}{1+x^2}", "bebfe33a857b2ba995053c353d6fba25": "\\mathbf vdt = (v_xdt, v_ydt, v_zdt),", "bebfe340eaa51487f5853227fb6bb4cb": "{1\\over (n+1,q-1)}q^{n(n+1)/2}\\prod_{i=1}^n(q^{i+1}-1)", "bec04066195b6cf11dd711aa7704c342": "m_2 = [12.3, 7.6] - [1.697, 0] = [10.603, 7.6]", "bec07efe2661657da04a620ff91ee224": "M_L=\\sum_i {m_l}_i.", "bec11271d5620810262b28c33300c677": "\\left\\langle \\ldots \\right\\rangle_{\\varphi}", "bec114f72a42f0bbaae688b2803788ed": " \\mathrm{T} = \\mathrm{V{\\cdot}s{\\cdot}m^{-2}} = \\mathrm{N{\\cdot}A^{-1}{\\cdot}m^{-1}} = \\mathrm{Wb{\\cdot}m^{-2}} ", "bec149bd76dbee5638fa3444ecc2af8a": " \\textstyle P= \\frac{A+I}{1-q} \\,\\ ", "bec19236197e7eee0a9e0bce8de9846e": " x^b=a ", "bec1a86657a9714ee8f8aabc8a4dc1a1": "\\bar x(t) = \\frac{1}{N} \\sum_{k=1}^N x(t,k) = s(t) + \\frac{1}{N} \\sum_{k=1}^N n(t,k)", "bec1e2976c4ca2f0639b89fcfb72f6a5": "w_r=k_r \\prod_{i=1}^n a_i^{\\alpha_{ri}} \\, ,", "bec1ef6cd855f66d967013117a78714e": "e^{-\\pi R_{\\text{vertical}}/R_S}+e^{-\\pi R_{\\text{horizontal}}/R_S}=1", "bec24b4b6eb477ba19c7d0f6082aab4e": "\\displaystyle q = [a_0; a_1, a_2, \\ldots, a_k] = [a_0; a_1, a_2, \\ldots, a_k-1,1]", "bec2e24c8941361d0046c3dba9d32572": "d = \\frac{i}{1+i}\\approx i-i^2", "bec2e9f3be8b4c4a90797a72b19a16ec": "\nI_1 = \\begin{bmatrix}\n1 \\end{bmatrix}\n,\\ \nI_2 = \\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\end{bmatrix}\n,\\ \nI_3 = \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\end{bmatrix}\n,\\ \\cdots ,\\ \nI_n = \\begin{bmatrix}\n1 & 0 & \\cdots & 0 \\\\\n0 & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & 1 \\end{bmatrix}\n", "bec327c8bd7bdab41fbc61654874d85d": "f_\\rho=(-\\mu/T)", "bec375a6b1654602f1158057931c9e3e": " A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \\,", "bec396ce379ed4936fc2e8ae40bc3e71": " p=\\rho(\\gamma-1)e-\\gamma p^0 \\,", "bec3c8492fb2cee35d50009200fff4ed": "\\delta=\\frac{P}{P_{0}}", "bec3d0a94dde438ce70102c526351326": "\\tau=\\tau^\\mu\\partial_\\mu", "bec3fc66e391847ceb810a9e85be3def": "\\boldsymbol{\\psi} :", "bec40c9fadccca2edcd27d30013899ce": "y \\succeq x ", "bec4373014f275bad8e70f723c05252a": "p\\left(\nm\\right) ", "bec45c6db42b8561c5f5cca70454b323": "\\mathbf{PSPACE} = \\bigcup_{k\\in\\mathbb{N}} \\mathbf{SPACE}(n^k). ", "bec45f22f6eacfc45366b33283d1ba38": "f(a)=L_a", "bec4b93bbcfb22f00cc2fb1ddb42b553": " Y^2+h(X)\\cdot Y=X^3 ", "bec4be589f09a31248963ec656b92b2b": "u_{n | \\partial \\Omega}", "bec4de14cc2c98afb2cff5dc22a43319": "\n\\sum_{xy} A(x,y) |x,y\\rangle\n\\,", "bec4f7eb23d60fa2b4acfc932c04abb2": "[\\rho, z, \\phi]", "bec53d7b5c31c1ecc7f0c3294c82d82b": "\\ln(3) / \\sqrt{2} \\pi, ", "bec5508bc8bf3ed621c034c832cd1ed3": "3_0", "bec5949e422ff929d465e605e0756767": " A \\mathbf ( \\mathbf{r} ) = \\frac{-i}{2\\lambda} \\iint_\\mathrm{aperture} \\frac{e^{i \\mathbf{k} \\cdot \\left ( \\mathbf{r} + \\mathbf{r}_0 \\right ) }}{ \\left | \\mathbf{r} \\right |\\left | \\mathbf{r}_0 \\right |} \\left [ \\cos \\alpha_0 - \\cos \\alpha \\right ] \\mathrm{d}S \\,\\!", "bec5d64f66fc7acd40204c46b3b2cae2": " \\left ( {\\partial S\\over \\partial V} \\right )_{N,U} = { p\\over T } ", "bec653efe99371af43f63237bfd5f5fb": "F_N \\otimes F_N \\in H(N,N^2),", "bec66c39020dcd5cff8b4ffe4cc39531": "\\inf\\, \\{ (-1)^n + 1/n : n = 1, 2, 3, \\dots \\} = -1.", "bec6708a9ac8146cffffbe3be6325f58": "\\,i", "bec675e24a38adcdf133d35bb65e47da": "\\mathcal O(\\mathbb C^*)", "bec6817cf346a9877e4dda2353fb5033": "\\mu(f) = \\infty.", "bec68821e669d6daca52d67ab0a75097": "C_Y\\, ", "bec6f671aceec3fb1d851be5d55cb947": "\\begin{align}\n \\sum_n \\mathcal{L}(n)\n &= \\frac{\\binom{m - 1}{k - 1}}{1} \\sum_{n=m}^\\infty {1 \\over \\binom n k} \\\\\n &= \\frac{\\binom{m - 1}{k - 1}}{1} \\cdot \\frac{k}{k-1} \\cdot \\frac{1}{\\binom{m - 1}{k - 1}} \\\\\n &= \\frac k{k - 1}\n\\end{align}", "bec70c25e376a95439bec8c9c81c4759": " \\mathbf{P} \\left\\{\\left|\\;\\frac{1}{n}\\sum_{i=1}^n X_i\\;\\right| > \\varepsilon \\right\\} \\leq 2\\exp \\left\\{ - \\frac{n\\varepsilon^2}{ 2 (1 + \\varepsilon/3) } \\right\\}.", "bec727803a834cc42df685ecadd8c5ac": "\\, \\sigma = t_s - t_e. \n", "bec72daeb5a61e78e71fec675c3e6563": "\\hat{x} = \\frac{1}{N}\\sum_{i=1}^N y_i,", "bec76d429a13664055595b8941e76ae7": "k'=nk\\,", "bec7941f44e2fd0987e47488ac137328": "\\Pr\\Big(n \\text{ coin tosses yield heads at most } (p-\\epsilon) n \\text{ times}\\Big)\\leq\\exp\\big(-2\\epsilon^2 n\\big)\\,.", "bec83889c47c2bbe30203c14f20aa670": "MU_*\\to R", "bec8638b6b2a9cade6bd91def3a66575": "x_{0}^{'}", "bec935e2eb3922000b904e569cee5bf7": "gN=N \\Rightarrow g'N'=N'", "bec973e214c8c969d63bad65254435df": "\\mathrm{fold}: M^{*} \\rarr M = l \\mapsto \\begin{cases} \\varepsilon & \\mbox{if } l = \\mathrm{nil} \\\\ m * \\mathrm{fold} \\, l' & \\mbox{if } l = \\mathrm{cons} \\, m \\, l' \\end{cases}", "bec9b1615d8ba51662b7420bd5473cc1": "\\left(1+\\frac{x}{n}\\right)^n \\rightarrow e^x", "bec9c62dd05a992750ed4833d6862b6f": "GE({\\rm SI}) = (1+L_{\\rm B}) GE({\\rm TDB})\\,", "beca60e3272500478699d764f070d41a": " B \\succeq C", "beca7f43489f61b90e73a31a5829454c": " \\nabla V = \\pm \\nabla \\times \\boldsymbol{\\omega}, \\quad V = \\sum_{i=1}^2 \\frac{1}{|\\mathbf{x}-\\mathbf{x}_i| }.\n", "beca983d5e76ba57fadec188b2f10074": "f: BGL(R) \\to B(S^{-1}S)^0.", "beca9be1eb8cfe0703123f836cf04531": "\\hat{\\sigma}_+\\approx \\hat{b}^{\\dagger}", "beca9f3d2c2c5302f8d8e4fbc29cd3aa": " O(h^p) ", "becad3d770e2eaf639c8c72c3b8d529a": "\\mathbf{r}\\cdot\\mathbf{A}=0", "becb279ee40bbd8dfd499555df1a5a67": "\\mathrm{i^{th}}", "becb63f4ec2ce39231994aeb61b1e823": "\\begin{bmatrix}a+bi & c+di \\\\ -c+di & a-bi \\end{bmatrix}.", "becb85ddceb9a933594c9a13b6af99c1": "F(x) = x + n + a \\, ", "becbdce33a103518896ba1c2428c10ad": "p(\\mu|\\mathbf{X}) \\sim \\mathcal{N}\\left(\\frac{n\\tau \\bar{x} + \\tau_0\\mu_0}{n\\tau + \\tau_0}, n\\tau + \\tau_0\\right)", "becc1ec1b6a91e855c34d76f25069105": " r_{01}", "becc53b1fd3932eeaaf1c31306231036": "S/N = \\sqrt{A}\\times\\frac{N_{s}}{\\sqrt{N_{s}+N_{a}}}", "becca702f654420e37e8b5912264cea8": " \\Psi(\\mathbf{r}_1,\\mathbf{r}_2\\cdots \\mathbf{r}_N,t) = e^{-iEt/\\hbar}\\psi(\\mathbf{r}_1,\\mathbf{r}_2\\cdots \\mathbf{r}_N) ", "beccb06afa2bf1b9927eb92b9678e993": "\\,\\delta : Q \\times \\Sigma \\times \\Gamma \\rightarrow S", "beccb0aee30e9aa122ea7428ea03373c": "s,t : \\prod_{i\\in\\mathrm{Ob}(J)}F(i) \\rightrightarrows \\prod_{f\\in\\mathrm{Hom}(J)} F(\\mathrm{cod}(f))", "beccd83c83e4acf52007b81135ad6835": "T_A = [A] +\\sum [A_\\alpha B_\\beta \\ldots]= [A] +\\sum \\left(\\alpha K_{\\alpha \\beta}\\ldots[A]^\\alpha [B]^\\beta \\ldots\\right)", "becd126502565eb0626f20b2a9a6108c": "A_m(x,y)\\,", "becd2a5f133619d957c165713cdbb951": "f(m) \\leq f(p)", "bece03ed6f5b48fadcf5e75585abc574": "1 + 2\\cos(\\theta) \\,", "bece4b7eb2c79641f2d8bf6d4b66216d": " I_{\\frac{x}{1+x}(\\alpha,\\beta) }", "bece80a79e5db71744ffca4411a010fd": "\\frac{F_n}{\\dot{m}} = (V_{jfe} - V_a)", "bece8e5797aa06c6abc6c8dafa40ada5": "g_1,\\dots,g_k", "bece8f9445cf2c3705c0661379cc8a79": "\n\\begin{cases}\n\\boldsymbol{X}_1'=\\boldsymbol{G}_1'-\\boldsymbol{V}_1'\\boldsymbol{X}_2^{(t)}\\text{,}\\\\\n\\boldsymbol{X}_j'=\\boldsymbol{G}_j'-\\boldsymbol{V}_j'\\boldsymbol{X}_{j+1}^{(t)}-\\boldsymbol{W}_j'\\boldsymbol{X}_{j-1}^{(b)}\\text{,} & j=2,\\ldots,p-1\\text{,}\\\\\n\\boldsymbol{X}_p'=\\boldsymbol{G}_p'-\\boldsymbol{W}_p\\boldsymbol{X}_{p-1}^{(b)}\\text{.}\n\\end{cases}\n", "beceb1763ab289f11fd6d77764407b8f": "j|^2 \\delta[\\Epsilon _c (k') - \\Epsilon _v (k) - \\hbar \\omega]", "bee9f9463acb7f24874fb5767c7a2659": "R_W", "beea15e9ab96435bb70c4c431b40ae18": "\\left(0,\\ \\pm1,\\ \\pm2,\\ \\pm2,\\ \\pm2\\right)", "beea1ccef3be0d5024613b3637025368": "\\frac{D\\mathbf{V}}{Dt} = \\mathbf{S}(\\mathbf{V}),", "beea2ef62ea3e314fa739f67b346a39d": "\\mu_r = \\mu_r'+j\\mu_r''", "beea7ba86b0fe3f401bd2642da685aed": "q^{-1} + {O}(q)", "beeabd3e02c6c0044d17bb1d5c29210c": " \\alpha \\in )0,2\\pi(", "beeae23428d8a8900f058bf7add0bd36": "\\Gamma(N)=\\left\\{\n\\begin{pmatrix}\na & b\\\\\nc & d\\\\\n\\end{pmatrix} : \\ a, d \\equiv \\pm 1 \\mod N \\text{ and } b, c \\equiv0 \\mod N \\right\\}.", "beeb42348d1f72a4bba776ee43ff8727": "\\pi \\approx 0", "beeb5a391576cbaa90c06104b3e26f2a": "19683^3", "beeb932264b9d909b59cb6ce2f64eeda": "\\bar{\\Delta}_- \\cong \\sigma_-\\otimes \\Delta_-^*.", "beeba0d6d2f030a2abff9ae3e25fad0e": " S_5 = S_1 - 2NS_1 + \\frac {6(\\sum_{i} \\sum_{j} w_{ij})^2} {1} ", "beebbe509ac572742d309cd35aba90f4": "\\eta_{\\mu\\nu}=(-1,1,1,1)", "beebc16d8412da33da3b4cde828cc718": " c \\cdot (a + b) = c \\cdot a + c\\cdot b ", "beec007f9915942c15d1f29a80dfd980": "6^5 = 7,776", "beec2997df679c08e291434641f302df": " \\beta_c^{XY}\\ge 2\\beta_c^{\\rm Is}~;", "beec4d7ac8c659f9744baac2cb584077": "\\scriptstyle\\int_{V \\backslash W} d(f \\, dz) - \\int_{W \\backslash V} d(f \\, dz)", "beec74bda05b120f4c271fa23cf2b562": "\\mathcal{E} = \\oint_{\\partial \\Sigma}\\left( \\mathbf{E} +\\mathbf{ v \\times B}\\right) \\cdot d\\boldsymbol{\\ell} = -{d\\Phi_B \\over dt},", "beeca8b52f5b5f042985b5e80366a705": "F(x_0;\\mu,\\sigma)\n=\\int_{-\\infty}^{x_0} \\frac{\\mathrm{Re}(w(z))}{\\sigma\\sqrt{2\\pi}}\\,dx\n=\\mathrm{Re}\\left(\\frac{1}{\\sqrt{\\pi}}\\int_{z(-\\infty)}^{z(x_0)} w(z)\\,dz\\right)\n", "beecaebaa470aeea4839c08523d60269": "\\operatorname{sink}[(\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ \\operatorname{sink-test}[(\\lambda p.p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x)), X]] ", "beeccaef733b5360da45c6c694b6a92f": "t_{a}", "beecf499d82a32e407e72cd25de94baf": " {\\rm det} (I+A)\\neq 0", "beed584371120e11bf20723d0f22e52e": "N\\,", "beed80a7ac5e5535752bdcb57b63850b": " \\int e^{M_{ij}{\\bar\\psi}^i \\psi^j} D\\bar\\psi D\\psi= \\mathrm{Det}(M) ", "beedc65c1da86763c04451c4545962ff": "z/0 = \\infty", "beedcd756062a155b3b83d6b238c163d": "g_J", "beee182a0337892748184ae979f662c0": "\\widetilde{K}(X)\\cong[X,\\mathbb{Z}\\times BU]", "beee20529895448fca2754a1321f91ca": "F(y)=\\sum_{n=2}^\\infty \\left(\\Lambda(n)-1\\right) e^{-ny}", "beef277904a077726080f41662233e57": "\\langle \\hat{A} \\rangle = \\int A(x, p) W(x, p) \\, dp \\, dx.", "beef2e0023f928a13a8fe2eb4a781aae": " \\textbf{x}_{k-1\\mid k-1}^{a} = [ \\hat{\\textbf{x}}_{k-1\\mid k-1}^{T} \\quad E[\\textbf{w}_{k}^{T}] \\ ]^{T} ", "beef5e85ad2c98e7f78aca6f438bc868": " \\ln \\left( {P \\over P_0} \\right) = {-{\\alpha + 1 \\over \\alpha}} \\ln \\left( {V \\over V_0} \\right). ", "bef00d7becc36a592a25ca93bab31d22": "x_j.\\,", "bef04830ccb70c64a6364bc70761a6bf": "^{(0)}_0 = -4 ", "bef079fdb1da6b6d702d10a09ab1f6e1": "\\log(k) = s_Es_N(N + E)", "bef0de420172bb3db997a75a357fefc3": "\\begin{matrix}\\Delta L = 0, \\pm 1, \\pm 2, \\pm 3 \\\\ (L=0 \\not \\leftrightarrow 0, 1, 2;\\ 1 \\not \\leftrightarrow 1)\\end{matrix}", "bef0eb0ca3ce8f6e4767b794fcb76abf": " m(r_0), m(r_1), m(r_2), \\ldots, m(r_n). ", "bef11f81abc6dbcc978aebf08a58f87a": " \\langle\\phi(k) \\phi(k')\\rangle = \\delta(k-k') {1\\over k^2} ", "bef14b32fee2ea54e422366784aa0741": "E_k = m c^2 - m_0 c^2. \\,", "bef16b1ed7c92952b55ae096f6c64d38": "a > \\infty", "bef17c8ac1195758d214d4a1393183b4": "\n\\sqrt[5]{100} = \\cfrac{5}{2}+\\cfrac{3} {250+\\cfrac{12} {5+\\cfrac{18} {750+\\cfrac{27} {5+\\cfrac{33} {1250+\\cfrac{42} {5+\\ddots}}}}}} = \\cfrac{5}{2}+\\cfrac{5\\cdot 3} {1265-3-\\cfrac{12 \\cdot 18} {3795-\\cfrac{27 \\cdot 33} {6325-\\cfrac{42 \\cdot 48} {8855-\\ddots}}}}.\n", "bef1bb6f1db6ba2d45747610955dc9c4": "\\chi (-1) = 1", "bef1d5b756c509a52fbba7173ce25b88": " - vp_{air}", "bef274deac945853bc927096eda82f49": "f_o=\\frac {f_s} {\\gamma} \\,", "bef2ad4a1871b3c819b18f11bcd69e50": "\\partial\\alpha=\\sum_{|I|,|J|}\\sum_\\ell \\frac{\\partial f_{IJ}}{\\partial z^\\ell}\\,dz^\\ell\\wedge dz^I\\wedge d\\bar{z}^J", "bef2cc64d0b83698af1f06d39b569535": "\nL=4\\pi R^2 B\n", "bef3095072fd644654e0b60d34bac50e": " s+t \\leq 1\\,", "bef34c7640f4622a87216547d391c3df": "\\ f(u,v) \\le c(u,v)", "bef352cfc2646aaf6e902d528111329b": "dz/d\\zeta", "bef355059605f24c674f1c75555b262d": "\nH = \\hbar \\omega \\, \\sum_{i=1}^N \\left(a_i^\\dagger \\,a_i + \\frac{1}{2}\\right).\n", "bef3862a2853978c38dd8b9cb59e41f0": "{x^2 \\over a^2} + {y^2 \\over a^2} - {z^2 \\over b^2} = 0 \\,", "bef3c19b442df1b75cc62479bbb1a5de": "H=\\rho_T/\\rho_B", "bef3d6a1dfdd85834929a2e602ba5021": "M = \\left(1-\\left[\\sinh\\left(\\log(1+\\sqrt{2})\\frac{T_c}{T}\\right)\\right]^{-4}\\right)^{\\frac{1}{8}}", "bef3f267b5fb72ccc8e45daacb747e2e": "P(x_{1},x_{2},\\ldots ,x_{n})", "bef48c8f897edec3d5a19410d669103e": "K(\\mathfrak{g},[\\mathfrak{g},\\mathfrak{g}])=0", "bef4c19f8019e0e78033b09b4d11ad7b": "1\\leq i <4", "bef4d8a0d70cd1eb428d2540f554c748": "\\sqrt{ \\left| e^{i\\theta} \\right|^2 + \\left| e^{i\\phi} \\right|^2 } = \\sqrt{2}.", "bef4e1b621b30e3728302283a71b6333": "8m+1", "bef55c016f695902005c01f6da205299": "S \\Rightarrow aSa \\Rightarrow aaSaa \\Rightarrow aabSbaa \\Rightarrow aabaabaa", "bef56175b66bac48dadf82cfc5dba7bf": "d^j_{m'm}(\\phi) =\\left[ \\frac{(j+m)!(j-m)!}{(j+m')!(j-m')!}\\right]^{\\frac{1}{2}} \\left(\\sin\\frac{\\phi}{2}\\right)^{m-m'} \\left(\\cos\\frac{\\phi}{2}\\right)^{m+m'} P_{j-m}^{(m-m',m+m')}(\\cos \\phi).", "bef5e135a7cf07a1dd2b9761586a39a0": "\\mathbf{v}_\\textrm{op}", "bef5f35e2af2ab907ad9262e20412e7b": "F = (\\frac{v}{N})^n", "bef63d91629fd60537d62668fe40a03e": "\n\\dfrac{|b| - \\alpha}{n + \\theta}.\n", "bef648a4506e73919ce84e5743bb1a4c": " r_{no~disease \\rightarrow hypercalcemia} = 0.0014 ", "bef67f85a99e3c5c1794bddd919ecec8": "K_- \\subset K_{-0}", "bef6d1f81915f67e5db315eee622e689": " \\sqrt{1-x^2}{P_\\ell^m}'(x) = \\frac12 \\left[ (\\ell+m)(\\ell-m+1)P_\\ell^{m-1}(x) - P_\\ell^{m+1}(x) \\right] ", "bef6e0fa1a04263902b71de8a2c09011": "\\{ P \\} S \\{ Q \\}", "bef6e3041ec0d0697e440e4da9d24c11": "\\chi ^2 _{0.95}", "bef726a343e68c520b8471cbb79a814a": " B \\frac{dB}{dz} = \\mu_0 \\, \\rho \\, \\frac{g}{\\chi} ", "bef7974cf76160ac27e4a0277c51f253": "\\boldsymbol{\\psi} = (0,0,\\psi)", "bef83480ac79873391886db771376d61": "\\begin{matrix} \\ & light & \\ \\\\ Pc(Cu^+ ) + Fd_{ox} & \\Longrightarrow & Pc(Cu^{2+}) + Fd_{red} \\end{matrix}", "bef85f6f9912dbfe035bcbe67b8f65f9": "m_i \\,", "bef876c8feae9e54d232792597e93780": "(\\nabla I)(x; t)", "bef958aad084ce7858c85e1b0b02978b": "H^{s}_{p}", "bef9c98597d6c167a95bac888b31bc0a": "Q^{d}", "bef9ecb85d972acd69a5f9d70e863bfb": "G_r=G_r^+ + G_r^-", "befa3c3af46da9829934b28c43e635ed": " K = \\frac{\\det \\mathit{II}}{\\det I} = \\frac{ LN-M^2}{EG-F^2 }, ", "befa5f5909ebacf3931a2017e55cfd56": "a=96", "befaaaf398e96ca883756799544c1ce5": " \\hbar \\omega = E_r(J+1)-E_r(J)=2B(J+1) ", "befad54619a16a80557fed6e3795cb91": "T = \\Box x \\rightarrow x", "befaebb454d7c8ab40643f583cc26dc5": " c_{ij} ", "befafea8b731e006a8754e9e57ff911f": "\\lbrace q \\ :\\ q q^* = 0 \\rbrace = \\lbrace w + xi + yj + zk \\ :\\ w^2 + x^2 + y^2 + z^2 = 0 \\rbrace ", "befb4561a4a9126cbfb999e4823836cc": "\\frac{12.9k\\Omega}{M}", "befb4eb7e2f4e4437e46eb634847ba6b": "\\begin{align}\n\\dot{\\hat{\\mathbf{x}}}\n&= A \\hat{\\mathbf{x}} + B \\mathbf{u} + L M \\operatorname{sgn}(\\hat{x}_1 - x_1)\\\\\n&= A \\hat{\\mathbf{x}} + B \\mathbf{u} + \\begin{bmatrix} -1\\\\L_2 \\end{bmatrix} M \\operatorname{sgn}(\\hat{x}_1 - x_1)\\\\\n&= A \\hat{\\mathbf{x}} + B \\mathbf{u} + \\begin{bmatrix} -M\\\\L_2 M\\end{bmatrix} \\operatorname{sgn}(\\hat{x}_1 - x_1)\\\\\n&= A \\hat{\\mathbf{x}} + \\begin{bmatrix} B & \\begin{bmatrix} -M\\\\L_2 M\\end{bmatrix} \\end{bmatrix} \\begin{bmatrix} \\mathbf{u} \\\\ \\operatorname{sgn}(\\hat{x}_1 - x_1) \\end{bmatrix}\\\\\n&= A_{\\text{obs}} \\hat{\\mathbf{x}} + B_{\\text{obs}} \\mathbf{u}_{\\text{obs}}\n\\end{align}", "befb5810f550eed3c32a7ba6dd34a118": "uxv\\stackrel{*}{\\rightarrow}_R uyv", "befb8f33842e65437001c712f38c4ed5": "x_1 \\neq x_2", "befb93ee5986c279a8f528ca11b8173d": "\\omega_{0} = 2 \\pi f_{0} = 1 / \\sqrt{R_3 R_4 C_2 C_5} ", "befbd5cfa0cc771c3acebe6cd2977efc": "\\textrm{dim}(\\mathcal{H}_B) = 2", "befc2ba6fa61d026f97e54fa9ac48aa6": "p \\neq q", "befc3f8ae9b73f3897fd46cd4c23dcb6": "\\textstyle X^-", "befc68dad5846952fa2b31745b2d9192": "J_v", "befc97e10457b155afec14740004fe90": "f(p_1,\\ldots,p_n)", "befcb9d5a83ac30e8fdb7430b856da07": " t = \\mu - x ", "befcd32e1d9b9cdbc07d31f4c5a5999d": "\\scriptstyle\\log_2\\left(\\prod_{k=1}^n f_k\\right)", "befcd5d67c8f8a5aadfbd8574d5d718f": "w=\\frac{iz+1}{z+i}", "befd2491f0a58abd0ebcfaf4a8985cd3": "M_z(0)=-M_{z,\\mathrm{eq}}", "befd4ee507d221166c9012b44378ba31": "\\psi_i", "befd74481410b70fbb3f2fc2756d6bc6": " \\nabla \\times \\nabla \\times \\mathbf{H} = \\varepsilon_o \\frac{\\partial } {\\partial t} \\nabla \\times \\mathbf{E} = -\\mu_o \\varepsilon_o \\frac{\\partial^2 \\mathbf{H} } {\\partial t^2}\n", "befdd57eeea1e198e0d6591e84ab1ce5": "-\\boldsymbol{e}_k\\, \\frac{1}{k\\, h}\\, a\\, \\sin\\, \\theta\\,", "befde16d963ccbc0f839d46344733d19": "d=0,1,2,3,4", "befe19b8f0e9f8754c23e5cdb346199e": "\\partial_t \\eta + \\partial_x \\eta + \\tfrac32\\, \\eta\\, \\partial_x \\eta - \\tfrac16\\, \\partial_t\\, \\partial_x^2 \\eta = 0.", "befe76bda156a2df21dad906c5e525d4": "\\mathrm{Re}(f'(u)) = \\mathrm{Re}\\left( \\frac{f(b)-f(a)}{b-a} \\right),", "befe987427cacc816091a06609314710": "A = \\frac{dE(r)}{d\\sigma}", "beff64c08a9aa20207ae418ef2e2f46c": "\n \\left(\\cfrac{\\sigma_6}{\\sigma_0}\\right)^2 + \\left(\\cfrac{\\sigma_2}{\\sigma_{90}}\\right)^2 + \\left[ (p + q - c) - \\cfrac{p\\sigma_1+q\\sigma_2}{\\sigma_b}\\right]\\left(\\cfrac{\\sigma_1\\sigma_2}{\\sigma_0\\sigma_{90}}\\right) = 1 \n ", "beff67e4f2392ab6666bfad39544e874": "n\\not\\in \\mathfrak{p}.", "beffb72cc82a988ac9a35cb7d3ae04fe": "\\xi_t=(\\xi_{1t},\\dots,\\xi_{nt})", "bf00b440b867703cba471989a9bca411": "E_\\text{surface}", "bf00e8016f8c98eed81fd3225128c725": "A \\cap A^{c} =\\empty .", "bf00edda3641fb3df162b9b313ec1ddf": "M^rf(x) = \\int_K f(gk\\cdot y)\\,dk,", "bf01227820210dda7863727c9a4bb556": "R_{\\mathfrak{p}}/R", "bf01552cf759e254d2d1991eba9b5412": "\\frac{2}{\\pi}", "bf017a3d9a3453aab0da2cec4a78e097": " \\phi(\\vec r)", "bf01a847f5c0aea129b50c46a65145c6": "a \\land \\neg (b \\leftrightarrow c)", "bf01aac53faf9ef09776f408c0c8cd8d": "(\\operatorname{Spec}\\ \\mathbb{Q})", "bf01b3fcbcd5917fc9056abd1fa63f5c": "RSBI = \\frac{f}{V_T} ", "bf01b42e04ec7c9354a40bd4a51b5b9d": "\\lesssim10^{-5}", "bf022b767ea9096f489550ec9df3c21a": "\\frac{d}{dz}\\log v(z) = - \\frac {c-z(a+b+1)}{2z(1-z)} =-\\frac{c}{2z}-\\frac{1+a+b-c}{2(z-1)}", "bf0237b645e7780a2c6da21e06deacc1": "\\mathbf\\Sigma_i = \\alpha\\mathbf\\Sigma_e", "bf02555628dccee2e77f26c31743a5a6": "f(t)=\\frac{1}{8}\\sin(2t)-\\frac{t}{4}\\cos(2t)", "bf027365805b4e0fd4f0e1a1066692e1": "\\sigma>5", "bf028eca9d082ceb7a506994fe5f01ce": "\\left( \\int_A \\right)", "bf02eded54c8b0f5092238d7c17a2b55": "a^{(n)}=a(a+1)(a+2)\\cdots(a+n-1)\\,", "bf0323d970bc4c1f943573eb6a86b198": "\\{x,p\\}", "bf038ea0b8c5782463c93b04c39895aa": "S_N(f;t)", "bf03beec48276ac36b07711678e45492": " \\frac{\\epsilon_0}{2}\\mathbf{F}^{\\dagger}\\mathbf{F} = \\frac{\\epsilon_0}{2}\\left( \\mathbf{E}^2 + c^2\\mathbf{B}^2 \\right) + \\frac{1}{c} \\mathbf{S},", "bf03f10a34ee39b6c1b0f6f1bade33f0": " \\{ 0 \\}", "bf0464474e41b08c834f3d56d9c18b76": "p(\\hat{x_0})", "bf047e809a5dbb96f4ff443686f67ba4": "\\mathrm{low} = \\ln(2) \\cdot R_2 \\cdot C", "bf048a7c85bbd25521a0fa91ce22c14f": "Cxy\\rightarrow Cyx.", "bf04a8cb175c04da0992f45e1b3b5bfe": "\\exists_f", "bf04b22a7f56a05951ece0da7b123a70": "\\langle u,v\\rangle", "bf05741547e0e2ba1bcb04265cb07b01": "Z_{oc}", "bf05fcbb654cbe6e49b7f489aa4b5682": "b\\in f(a)", "bf060789eb25045d3f439a96ce9871d0": "N^{3}", "bf062324d43929102732182689ceed59": "f=f_1\\ldots f_k ", "bf0659fba4197ce81e7374e58b1333f6": "\\text{s.t.} =\n\\begin{cases}\n g_{1}\\left(x,y\\right) & = x^{2} + y^{2} \\leq 225 \\\\\n g_{2}\\left(x,y\\right) & = x - 3y + 10 \\leq 0 \\\\\n\\end{cases}\n", "bf067c5baf1c804e9e11c3861b81abea": "\\mathbb{S}^d", "bf06ad0024dad03a0ce70cd2a3ae7626": "\\leftarrow \\gets, \\nleftarrow, \\longleftarrow\\!", "bf06c4779dda2c547e655b11783a56c9": "= 2\\gamma^\\rho \\gamma^\\sigma \\gamma^\\nu - 4 \\gamma^\\nu \\eta^{\\rho \\sigma} \\,", "bf06cae5f70ce8798ef0e8faa37bfb79": "U(1) = Z(H)", "bf07083c42bd0c1be1927711e0942f95": "s_0/2", "bf071fdd2d4fb94a2243148341539ea7": "\n\\begin{align}\nn & \\to p+e^-+\\overline\\nu_e \\\\\n\\mu^- & \\to e^-+\\overline\\nu_e+\\nu_\\mu.\n\\end{align}\n", "bf076d801a92a08a07cc82cc3b4c4e38": "-\\frac{40}{39}", "bf07b34aaf67f3083dd5a95a4c156697": "\\xi_{inf}(\\alpha)=\\Phi^{-1}(\\alpha)", "bf07e7bac753262b17ac690a560545ef": "\\alpha_A=\\sum w_i \\alpha_i", "bf07f7fa7155e185d96525c8fab2c226": "\\sigma_{12}=\\sigma_{31}=\\sigma_{23}=0\\!", "bf07f86365aff20a2abe75247e98ce7f": " \\frac{x^3}{(1-x^2)(1-x^3)(1-x^4)} = x^3 + x^5 + x^6 + 2x^7 + x^8 + 3x^9 + \\cdots. ", "bf08025d0ee6bd4dc3794c2a1fa9fb68": "\\displaystyle k", "bf082d3069aa7c53b7fcc42cf79e41a0": "{\\ell_{AD}}^2 = x^2 + h^2", "bf0839b65c36253578140f8e45e2fb2c": "\\int x^2\\arctan(a\\,x)\\,dx=\n \\frac{x^3\\arctan(a\\,x)}{3}+\n \\frac{\\ln\\left(a^2\\,x^2+1\\right)}{6\\,a^3}-\\frac{x^2}{6\\,a}+C", "bf0844c1c4c628cdda71d0d39f521819": "2^{\\mathrm{intermediate\\ result}} = 2^{I+F} = 2^{I\\,+\\,(1-1)\\,+\\,F} = 2^{(I-1)\\,+\\,(1+F)} = 2^{I-1}\\,2^{1+F}", "bf08ba13a8f9dd3f204289dafd66b98e": "q(A) = \\min(q(B),q(C),q(D))+c(A) \\, ", "bf08ca138249983187bc3649bdef2aad": "\\Omega > T(\\Omega) > T^2(\\Omega) \\ldots", "bf090f2ab58289f110d2930961a4f3e1": "\\ 2 + \\cfrac{1}{3}", "bf094c6bcf073c5daef8f5c601362472": "\\,n^2", "bf09b21567ec20bb2a25293bf6cb15d9": "\\text{Quality} = \\frac{\\text{Results of work efforts}}{\\text{Total costs}}", "bf09cacc0c98d5756873c4ac8a9e1ab1": "1 + \\operatorname{dim} R \\ge \\operatorname{dim} R[x]", "bf09f7a0ab05e8f7d90b5a7f7e879fb9": "\\frac{64}{81} \\sqrt[3]{4}", "bf0a0350dfbf26976bcde928b1fe919e": "\\Delta{}f\\,", "bf0a5ea2088dab3c3ad26aac4b71fe31": "L^2(G) = \\underset{\\pi\\in\\Sigma}{\\widehat{\\bigoplus}} E_\\pi^{\\oplus\\dim E_\\pi}", "bf0af0131ac82fc893e844373815564d": " q_2=\\lambda_1c+\\lambda_2d \\le \\max(c,d) 0.", "bf18fad54ae863e519e6e576b363025e": "\\boldsymbol{\\psi} ", "bf1928e46d9656849a8edfde3160b89c": "\n f(\\theta x + (1 - \\theta) y) \\leq f(x)^{\\theta} f(y)^{1 - \\theta}\n ", "bf19318dc74079082024861c0cb83bd8": " \\alpha \\ge 2 ", "bf1934fb82581f97e6717f984842492e": "\\partial x^*_i/ \\partial q_j, i=1,...,n, j=1,...,m", "bf194a18f0f341a9927c2e07965037a4": "{\\bold M}", "bf1a268bfa0a9fa6dbb4a3b4e69108ca": " U(d,d_1) - P \\ge u^* \\,", "bf1a6ce7209354817f6e29061f50b01b": " \\mathbf{B} ( \\mathbf{r}, t ) = \\mathbf{B}_0 \\cos( \\omega t - \\mathbf{k} \\cdot \\mathbf{r} + \\phi_0 ) ", "bf1a91180a7b049d77f2870201268ba6": "z\\approx 0.49a", "bf1abd5308c2c60e071376a905b553f7": "10^{-pH_i} v_i", "bf1ac080f4d07dea8285e99e4d2183d7": "\\delta(x-a)\\,", "bf1add3886cfe36897fae8dbbee00471": "y= C + \\beta k + \\gamma \\ell + \\varepsilon. \\, ", "bf1b523c956a4ad228a45e26292b06ab": "(d-1)m.", "bf1b959765affb075c0a9d54187e4284": "\n\\pi=3+\\textstyle \\frac{1}{7+\\textstyle \\frac{1}{15+\\textstyle \\frac{1}{1+\\textstyle \\frac{1}{292+\\textstyle \\frac{1}{1+\\textstyle \\frac{1}{1+\\textstyle \\frac{1}{1+\\ddots}}}}}}}", "bf1ba06fd90412aa985287142589b283": "\nAx=b\\qquad (\\text{with given }A\\in\\R^{m\\times n}\\text{ and } b\\in\\R^m)\n", "bf1baedcb697ba468a7a8b8b34a22f30": "\\mathbf{w}_{(1)}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\{ \\Vert \\mathbf{Xw} \\Vert^2 \\}\n = \\underset{\\Vert \\mathbf{w} \\Vert = 1}{\\operatorname{\\arg\\,max}}\\, \\{ \\mathbf{w}^T \\mathbf{X}^T \\mathbf{X w} \\}", "bf1c020fc34daeb54ea7a64e0527de73": "\\frac{D}{dt}J=\\sum_k\\frac{dy^k}{dt}e_k(t),\\quad\\frac{D^2}{dt^2}J=\\sum_k\\frac{d^2y^k}{dt^2}e_k(t),", "bf1c2aaec419ce1159e555dec46b2b19": "\\left| x- \\frac{p}{q} \\right| < \\frac{\\phi(q)}{|q|}.", "bf1c4d6fd7b356ff9f844ff4275f6691": "h(x,y)", "bf1c698b9ed7e2f8c454b70e644844c0": "s(a, b)", "bf1cd63eaa4cb4c66aff3755d4f0e327": " v \\,\\!", "bf1ce0ff7d53b10675a4f750a2d42d9e": "\\{i\\gamma_0, i\\gamma_1, i\\gamma_2, i\\gamma_3\\}", "bf1cf062432719592756803294c87932": "\\lambda+\\mu\\ge\\kappa", "bf1cfd1c91f6689586f7d9ff7bca6d2c": " G_y ", "bf1de09d4323c8fd8aa8b2f69a0cdd2a": " V = f(I)", "bf1e8a402880fb2b05ceab019f79a2d5": "\\bold r = \\bold {r}_0 + s \\bold{v} + t \\bold{w},", "bf1eb9a139b356025556cfd264d71721": "r 0", "bf1f7bcc8b89c171492d63d522ab765d": " D_j = A_{jj} - \\sum_{k=1}^{j-1} L_{jk}^2 D_k ", "bf1f8380b732bd6110f6875c49f03f3f": "K-S_{T}<0", "bf1f86a268500e1dc1c0b31f7ec9876e": "r=\\frac{p}{1+\\varepsilon\\, \\cos\\theta},", "bf1f8fa56f9bf6cbbca931bfc06d3c74": "\\tau = \\frac{\\mbox{System capacity to hold a substance}}{\\mbox{Flow rate of the substance through the system}}", "bf1f9037998cd11b1891fee823ba95ff": "\\Sigma_{i+1}^{\\rm P} := \\mbox{NP}^{\\Sigma_i^{\\rm P}}", "bf1fa1eb55274c7017364c99cd133148": "\n T_m = \\cfrac{4\\pi^2 m \\nu^2 c^2 a^2}{k_B} .\n ", "bf1fc2b632f770766efd7ec3937500c5": "\n\\begin{align}\n & \\left(\\mathtt{init},\\ \\left[ n \\mapsto 4,\\ x1 \\mapsto 0,\\ x2 \\mapsto 0,\\ t \\mapsto 0\\right]\\right) \\\\\n\\rightarrow & \\left(\\mathtt{fib},\\ \\left[ n \\mapsto 4,\\ x1 \\mapsto 1,\\ x2 \\mapsto 1,\\ t \\mapsto 0\\right]\\right) \\\\\n\\rightarrow & \\left(\\mathtt{fib},\\ \\left[ n \\mapsto 3,\\ x1 \\mapsto 1,\\ x2 \\mapsto 2,\\ t \\mapsto 0\\right]\\right) \\\\\n\\rightarrow & \\left(\\left\\langle\\mathtt{halt},\\ 3\\right\\rangle,\\ \\left[ n \\mapsto 2,\\ x1 \\mapsto 2,\\ x2 \\mapsto 3,\\ t \\mapsto 0\\right]\\right)\n\\end{align}\n", "bf20193137085b07680e64a4ed4a7666": "K-1", "bf2064f78c639e2958aa9313b74a428e": "a \\otimes h, b \\otimes g \\in K", "bf20686f350ebf992f984b9b948c6a8a": "\\mbox{Inventory Turnover}=\\frac{\\mbox{Cost of Material − Change in inventories (of 1/2 and 1/1 goods)}}{\\mbox{Inventories}} ", "bf2079b5f4d4263c28f80bb91fb434e1": "V = \\frac{nRT}{p}", "bf20b029327d90f50ab6a630d93b235f": "\\nabla \\cdot \\mathbf{D} = 4\\pi\\rho_\\text{f}", "bf20b908fe416364e02fbdea12c761e4": "\\omega^{2} = \\frac{T}{\\mu}k^{2} + \\alpha k^{4}", "bf20c20788c561839a784ba7237308f7": "c_4=", "bf20fc5d7797c020a792c7c5d6411734": " FF=\\frac{a \\cdot b}{c \\cdot d} ", "bf211ab440127ef3240ad753d16d1121": "A_{\\mu} = \\left(-\\phi, A_x, A_y, A_z \\right) \\,", "bf21216d08cadad48047a40c3dbd53ed": "X \\prec Y", "bf215e4b59e0da27a08642f915c68638": "\\partial_t^m u = F(t,x,\\partial_x^\\alpha\\,\\partial_t^k u)\n= \\sum_{\\alpha\\in\\N_0^n,0\\le k\\le m-1, |\\alpha| + k\\le m}A_{\\alpha,k}(t,x) \\, \\partial_x^\\alpha \\, \\partial_t^k u,\\,", "bf21b651d4bf924146066c1034a5c1bb": " Y = UK +Ql + r ", "bf21db4598c2fabddb76f90ba1bda4c9": "g(X)=\\sum b_i^{p}X^i", "bf21e6aef048b05a6e8908adb7cfc824": " |g'_k(x)| ", "bf2200647146c202dbcc2073b7c1ff01": "\\langle Qx_1\\dots Qx_n\\rangle", "bf224db9f829c8b999b8d05d7d66dd0b": " \\frac{\\operatorname{d}}{\\operatorname{d}\\!\\theta}\\,\\sin\\theta = \\lim_{\\delta \\to 0} \\left( \\frac{\\sin\\theta\\cos\\delta + \\sin\\delta\\cos\\theta-\\sin\\theta}{\\delta} \\right) = \\lim_{\\delta \\to 0} \\left[ \\left(\\frac{\\sin\\delta}{\\delta} \\cos\\theta\\right) + \\left(\\frac{\\cos\\delta -1}{\\delta}\\sin\\theta\\right) \\right] . ", "bf227576fd80d68e97d614f27285319a": "\\ \\displaystyle R(q,u) \\ ", "bf2281a60b26af1b8a057b05cc091ae5": "w^{T}Bw=1", "bf22863091cfe180815d20be5d8a8e5a": "\\begin{align}\nx&=uv\\cos\\phi\\\\\ny&=uv\\sin\\phi\\\\\nz&=\\frac{1}{2}(u^2-v^2)\n\\end{align}", "bf2293ee31a24cc520864052a49b292a": "R = Y + Cr + (Cr>>2) + (Cr>>3) + (Cr>>5)", "bf2374fe3f6faec8627628f719c752de": "(~)", "bf23b8fafe821044cacb826f5ab087f7": "(j(y))(x) := d(x,y)\\,", "bf240c04dfae32c9ee7a2dcb84b916a0": " a_{01} = \\mathcal{L}(p_5) + p_3p_5+p_2p_6, ", "bf241ebcac6cb272892942af7a6b0d9f": " dX_t =\\left(rU_t\\sqrt{1-X_t} - \\delta X_t\\right)\\,dt+\\sigma(X_t)\\,dz_t, \\qquad X_0=x", "bf2463801fe49b72ee2aa1b4ba63f281": " \\partial_n \\colon C_n \\to C_{n-1},", "bf2473c5f11b8c79ed1f25f6ee8edfc5": "\\nabla ^2 \\Phi=0", "bf248fff42501ac4c94eea50a27a0e26": "U \\mapsto H^q(f^{-1}(U), F)", "bf24e89e02d88f47f80270c59df5f4c2": " x_n", "bf252573d902694fa17d7a1499c7dfdc": " \\lambda f.((\\lambda p.f\\ (p\\ p\\ f))\\ (\\lambda q.\\lambda x.x\\ (q\\ q\\ x)) ", "bf2599e9afc30b6c789f772fb273f7e2": "(2vx)^2+(2yz)^2 =\\, (v^2+x^2-y^2-z^2)^2 ", "bf2628ec400260bc7781a8b233f71d5f": " p(0|1) = 0.5\\, \\operatorname{erfc}\\left(\\frac{A-\\lambda}{\\sqrt{N_o/T}}\\right)", "bf2645f97dcc3a66722ac7740405e603": "\\begin{align}\n a_0 &= 1 ,\\, &&a_1 = \\frac12 ,\\, &&a_2 = \\frac14 ,\\, &&a_3 = \\frac18 ,\\, &&a_4 = \\frac1{16} ,\\, &&a_5 = \\frac1{32} ,\\, &&\\ldots ,\\, &&a_k = \\frac1{2^k} ,\\, &&\\ldots \\\\\n b_0 &= 1 ,\\, &&b_1 = 1 ,\\, &&b_2 = \\frac14 ,\\, &&b_3 = \\frac14 ,\\, &&b_4 = \\frac1{16} ,\\, &&b_5 = \\frac1{16} ,\\, &&\\ldots ,\\, &&b_k = \\frac1{4^{\\left\\lfloor \\frac{k}{2} \\right\\rfloor}} ,\\, &&\\ldots \\\\\n c_0 &= \\frac12 ,\\, &&c_1 = \\frac14 ,\\, &&c_2 = \\frac1{16} ,\\, &&c_3 = \\frac1{256} ,\\, &&c_4 = \\frac1{65\\,536} ,\\, &&&&\\ldots ,\\, &&c_k = \\frac1{2^{2^k}} ,\\, &&\\ldots \\\\\n d_0 &= 1 ,\\, &&d_1 = \\frac12 ,\\, &&d_2 = \\frac13 ,\\, &&d_3 = \\frac14 ,\\, &&d_4 = \\frac15 ,\\, &&d_5 = \\frac16 ,\\, &&\\ldots ,\\, &&d_k = \\frac1{k+1} ,\\, &&\\ldots\n\\end{align}", "bf266201dd8205956ac3ae7a592855c4": " \\mu_t =M\\mu_{t-1} +(1-M)(I_t\\rho +(1-\\rho)\\mu_{t-1}) ", "bf267274fad12fbf501a3d719a7c572c": "G=\\frac{4 \\pi}{3} r^3 G_v + 4 \\pi r^2 \\gamma", "bf267caba0b6ace448883dc0d113e694": "I_{n,m}= \\int \\frac{dx}{x^m(x^2+a^2)^n}\\,\\!", "bf278f29d9c2517a592127ca65fa3fb0": "\\phi(\\omega)", "bf27c6ad4156ac41afbb2002393203f7": "r_{xx}(\\tau) = \\operatorname{E}\\big[\\, x(t)x^*(t-\\tau) \\, \\big] \\ ", "bf27cabef87db23c6e739f3f1f2f4137": "\\delta B = \\frac{1}{\\gamma} \\sqrt{\\frac{2 R_{tot} Q}{F_z N} }", "bf280916ce0ee75e5c416f7678b50eef": "\\mathcal A \\xrightarrow{\\;\\; S\\;\\;} \\mathcal C\\xleftarrow{\\;\\; T\\;\\;} \\mathcal B", "bf28154cc525a95b2b07cb28def84c71": " D\\theta = \\partial \\theta - e A.\\,", "bf28378a185a8635eb153cacd6be1d1d": " \\frac{\\partial^2 u}{\\partial x^2}-\\alpha^2 \\frac{\\partial^2 u}{\\partial y^2}=-2\\alpha^2 \\mathbf{S}\\cdot\\left(\\frac{\\partial \\mathbf{S}}{\\partial x}\\wedge \\frac{\\partial \\mathbf{S}}{\\partial y}\\right).\\qquad (1b)", "bf283d0b7a76b9ade95a54ae3568af0b": "A^{*2}=A^2\\left(1+\\frac{0.75}{n}+\\frac{2.25}{n^2}\\right) .", "bf284ebf611840cc6a843788ca3a6af4": "P_{n-1}(b,a)\\neq 0", "bf284f0dd453d4f9eb53334d5bf15649": "x*y=z", "bf2859f4bf9cb146583de9ea07938766": "\\begin{align} \ndU &=& TdS-PdV \\\\\ndH &=& TdS+VdP \\\\\ndA &=& -SdT-PdV \\\\\ndG &=& -SdT+VdP \\\\\n\\end{align}\\,\\!", "bf286e5d6b06b1e1439eed9ba75f6a10": "\\ 10 \\times M_{heel_{max}} = M_{pitch_{max}} ", "bf2896b9b19dced441e6245106fb6357": "\\mathbf x[k] \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathbf x(kT)", "bf28a6ba4ef56506cd9c7a674bcdf77e": "kT\\left(\\frac{\\partial \\rho}{\\partial p}\\right)=1+\\rho \\int d r [g(r)-1] ", "bf28b12fdc32bca97e18fd6faef8d49c": " A_h", "bf28c799ad4b85026a88f6e2d1a1e518": "\\phi(\\vec{x}, t)", "bf293d39c91495603a5907e9cf116959": "\\frac {\\phi} {{\\phi}^{\\ast}}\\ ", "bf295d0d1554eb2d937e7419579586b0": "(Bu|v)=(u|Bv)\\, ", "bf2961e28336a6b69e2b0af495087f91": "dz = i \\exp(i t) \\,dt", "bf29d33f8029e1c1b19e2fb4a03f6bb5": "\\mathbf{\\mathit{Y}}", "bf2a4f5b08c4a161c23c10e6d3bfc3e0": "\\omega-\\omega_B", "bf2aa55a95351230948ab94f5703bb33": "\\frac{1}{2}\\left[\\boldsymbol{\\nabla} \\mathbf{u}' + (\\boldsymbol{\\nabla} \\mathbf{u}')^\\mathrm{T}\\right]", "bf2ab2d73f3beaedc609ff7ea52a48fd": "\n\\mbox{posterior numerator (male)} = \\mbox{their product} = 6.1984e-09\n", "bf2aea04b8917bf0521db0469ef3dee9": " g(z) = z + b_0 + b_1 z^{-1} + b_2 z^{-2} + \\cdots", "bf2b040e5b8b5bfc8ada01febd0cae21": " \\begin{matrix} \\frac {(r_O+R_C\\|R_L)r_E} {r_O+r_E +\\frac {R_C\\|R_L} { \\beta +1}} \\end{matrix}", "bf2b2c546ecea00c170b04d335c812b5": "t(t-1)(t-2) \\left (t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352 \\right)", "bf2b511dbe7d5f425aa4a9ba0aa54cc1": "r=s=0", "bf2bc2545a4a5f5683d9ef3ed0d977e0": "jj", "bf2bd89f18548674830a6fa278b94f94": "(\\phi \\to (\\chi \\to \\psi)) \\to ((\\phi \\to \\chi) \\to (\\phi \\to \\psi))", "bf2c16841d9dfa1305a7de0f3c38192d": "N_{p}", "bf2c26718cbe19b56a2fea395a186f18": "\n\\left| \\mathbf{c}_{1} - \\mathbf{c}_{2} \\right|^{2}\n= \\left( r_{1} + r_{2} \\right)^{2}.\n", "bf2c5966f43ec36ba2d4c8be3bb8e5f5": "(xy)^n = x^n y^n ", "bf2c84c891dcb375f1ab1c97db4ce2e9": "\\frac{x-t}{\\sqrt{1-2xt+t^2}} = (1-2xt+t^2) \\sum_{n=1}^\\infty n P_n(x) t^{n-1}.", "bf2cb8bc6b1a95a650d0f64b1b67ee89": " \\displaystyle \\frac{c_g}{c_p}", "bf2ce7c125327dc862f2e339398845af": "P = (P_x, P_y) = r K_B", "bf2d64f16964b912e2bc51c348c23b45": "\\tilde{Y}^n", "bf2d8c5624b0549294760991e57d0575": "S(G) = \\frac{s(G)}{s_\\mathrm{max}},", "bf2db3aa00c0e8c5c24a44a28e15db5c": "r(t,k) = \\sum_{v_i \\in t^{-1}(k)} s(i).", "bf2dedb77b09dae47b97eb6bfb2ff504": "\\mathfrak{P}^{53}", "bf2e1b3560df14f1ec0d710b89f69fac": "x_t^{(4)}", "bf2e263ae004a04a1cc468367d354c85": "x'=l_3\\|r_3", "bf2e3bdab35b0c06a77afe6d25ac5a0b": "\\neg (\\alpha_1 \\rightarrow \\overline{\\alpha_2})", "bf2e56455b02315b2b334002167fca55": "\\psi=0\\,\\!", "bf2e653aadd9fb0bb3dfeaa4560c9576": "A^*\\subseteq A", "bf2e6768ff6303e01637676f7bc3b53b": " \\vec \\tau = \\vec r \\times \\vec F ", "bf2e8835f2fe4b8489c14b267feeee9d": "f_n(x) = \\bar x_n = (x_1 + \\cdots + x_n)/n", "bf2ecba196e8954fb2cb2140ea251f82": "(\\mathbb{Z}\\rtimes\\mathrm{Diff}(S^1))\\times(\\mathbb{Z}\\rtimes\\mathrm{Diff}(S^1))\\, ", "bf2ed0418689d279192c4e38cc71c8ac": " \\chi = \\sum (-1)^n \\, \\mathrm{rank}(A_n) ", "bf2edba43239208768ba46e103b8bded": "n_3=\\pm1\\,\\!", "bf2f60ce8f7b059a0930111875a2caca": "\\Delta f = \\nabla^a \\nabla_a f", "bf2fa780ddecdb513556cd634ee804d1": " \\beta = \\frac{ \\gamma^2 + 1 }{ \\kappa } ", "bf2fcfeb19d083074009a31d434b50fd": "x^5- x^4 - x + 1 = 0", "bf309bcdbf2dcedefbfa9debc6b7d040": "\\scriptstyle \\mathcal{F}(X_0, X_1)", "bf30d66c7a00bbf7b67b064b41af550b": "(\\alpha_2^2 - \\alpha_2) x=\\lambda (\\lambda+1) x", "bf311135e91ebe096f26a3d8efcafd1b": "\\sigma (z_1+z_2)=\\sigma(z_1)+\\sigma(z_2)\\,", "bf312bf4bb293eadf2600161c4d98d86": "2^{9}3^{-28}5^{37}7^{-18}", "bf3165f7ab5189343f64324c19324c24": " v(0,x,y,z) = \\psi(x,y,z), \\quad v_t(0,x,y,z) = 0. \\,", "bf31e55da2d5f95a589264164298fcc2": "B(s) = \\frac{\\mu_0 I}{ 2\\pi}\\left(\\frac{1}{s}+\\frac{1}{d-s}\\right)", "bf3251edb7e1b9eca001071f56d149df": "\n b_s \\equiv \\operatorname{E}[(\\hat\\theta_\\mathrm{mle} - \\theta_0)_s]\n = \\frac1n \\cdot I^{si}I^{jk} \\big( \\tfrac12 K_{ijk} + J_{j,ik} \\big)\n ", "bf329d4d79d1e7f58f4bba4c359ab5f5": "p \\ll q", "bf32c8334204149d4ab347c4bd24d1e3": "\\sin \\theta = \\cos \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\csc \\theta}", "bf3300afa87f6a59ae8e8efa689f31b9": "\nW_\\lambda\\chi_E(x)=\\int_{\\mathbb{R}^n}g_\\lambda(x-y)\\chi_E(y)\\mathrm{d}y = \n(\\pi\\lambda)^{-\\frac{n}{2}}\\int_Ee^{-\\frac{(x-y)^2}{\\lambda}}\\mathrm{d}y\n", "bf331dfe731d2789eabeb840ce1c9f97": "\\langle v_j,v_k \\rangle=\\int_0^1 v_j v_k\\,dx", "bf33e7d72f9a991c3bd68b327cfa5c46": "\\sigma_\\theta = \\frac{pr}{t}", "bf34269974cc5840f252b2e7897b8109": "\\Sigma = \\|\\mathbf { A \\times B } \\| = \\sqrt{ \\|\\mathbf A\\|^2 \\|\\mathbf B\\|^2 -(\\mathbf{A \\cdot B} )^2} \\ . ", "bf3436db7c32b0070f074361b9961034": "f(\\theta) = P(D|\\theta,M)", "bf343a2e68ee9ec7234e7330bf87c935": "\\frac{1}{r^{N-1}} \\frac{\\partial}{\\partial r} \\Bigl(r^{N-1} \\frac{\\partial f}{\\partial r} \\Bigr).", "bf346e3b5e157eae6703e61f3f5c14e6": " \\frac{\\epsilon - \\epsilon_0}{\\epsilon + 2\\epsilon_0} \\cdot \\frac{M}{d} = \\frac{\\epsilon_\\mathrm{r} - 1}{\\epsilon_\\mathrm{r} + 2} \\cdot \\frac{M}{d} = \\frac{4\\pi N_A \\alpha'}{3} = \\frac{N_A \\alpha}{3\\epsilon_0}", "bf3481db934d9a5b86d48ef9e2ad8135": "B=N(0,1)", "bf349fa046b5e215076a4fe26d7a1e96": "\\rho_{-n}", "bf34b498976bdd74ab2e4f7247541cdf": "Q(x'\\mid x_t)\\,", "bf34dba3860fae12af23d9bf3ca275d6": "A_{i,j}", "bf358676359adb1dc860c469e9472ae2": "\\mu=\\frac{m v_{\\perp}^2}{2 B}", "bf358d91b5f8e5b084607c60adaad38e": " \\rho_b = -\\nabla\\cdot\\mathbf{P} ", "bf35a1a26866b951b3c68608d2ecc836": "b\\neq 0", "bf3616ea52074b7a162b75ead76dd824": "\\ \\kappa_0(\\mathcal B)", "bf3633c15d869377cb6218b61912dccb": " X = \\int_{\\lambda_0}^{\\lambda}\\,\\frac{d \\lambda}{gC}\\,+\\,k\\,\\varphi(x), \\,\\, Z = \\int_{\\lambda_0}^{\\lambda}\\,\\lambda\\,\\frac{d \\lambda}{gC}\\,+\\,k\\,\\varphi(x) ", "bf3657fdaf33bf27f18734d2db764888": "(\\bar\\omega\\gamma-1)", "bf369fca2b4b0b12204e3061a97e073b": "C\\ell_3", "bf36a473a71d8bfb2cbf5cabd6db00d3": " r(z) ", "bf37145a4548aaf52f796deb66ff4b33": "f^*:T(A)\\to T(B)", "bf373a1ee4bb2bc9063279361ea5849a": " u_i(p) \\approx u_i(p_0)+\\sum_j\\frac{\\partial u(p_0)}{\\partial p_j}\\Delta p_j ", "bf373f41055eb1d1c4947ff6fbe73768": "\\tfrac{G(E-2G)}{3G-E}", "bf375a6fedfbd7e4f8bf0b6da49bf131": "|0\\rangle \\otimes |0\\rangle = \\frac{1}{\\sqrt{2}} (|\\Phi^+\\rangle + |\\Phi^-\\rangle),", "bf3767118ab6ce263b118d01769f61cb": " E^2_{p,q} = H_p(B, H_q(F)) \\Rightarrow H_{p+q}(X).", "bf3779a1a0439e8f36178e2ae8b97603": "X \\sim \\Gamma(\\alpha, \\theta)", "bf37897a978936507952b54eb4f4a5d1": "\\tan\\frac{\\pi}{12}=\\tan 15^\\circ=2-\\sqrt3\\,", "bf3790273d90122007440630080269f4": "\\scriptstyle p(x,x+1)=p(x,x-1)=\\frac{1}{2} ", "bf37c0ec0245b74300f83007ae692f6e": "c_2=", "bf37c3dfe8b6488c72fa7c8bed6f582c": " r(x)=\\frac{1}{\\sqrt{1-x^2}} ", "bf38732eef27b93a40d80b01bc6ce4f8": "\\frac {80 \\cdot (mean\\ arterial\\ pressure - mean \\ right \\ atrial \\ pressure)} {cardiac\\ output}", "bf38a2e2013b589a2ac8f7b3f70f3a29": " u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1].", "bf38ac472fd9118cddc25519cacba8f6": "T^{i_1\\dots i_n}_{j_1\\dots j_m} \\equiv T(\\mathbf{\\varepsilon}^{i_1},\\ldots,\\mathbf{\\varepsilon}^{i_n},\\mathbf{e}_{j_1},\\ldots,\\mathbf{e}_{j_m}),", "bf38bf649d47e2aabed1fa9e0591ae6b": "x^3+62x^2+696x=38448,\\quad x=18; ", "bf395889598a2715b9a9641dc2668c92": "\\boldsymbol{\\hat\\beta}", "bf397408bf2a93041c52bcbaecbfd4f1": "\\frac{1}{4}(2 + 7z - (2 + 5z)\\cos(\\pi z)).", "bf399cee5bd3b9a7271e6cbfce26149b": "g \\Delta \\rho L^3", "bf39a4a8e3e823c107569713145bd3ea": " \\dot Q_{c} ", "bf39e818b28cf92abe333195b085b213": "\\dot{\\mathbf{x}}=F(\\mathbf{x})", "bf39f3d3cdfd6973e2ad4e8e8baa7b88": "\\{ \\alpha(n)_{k} | k = n, \\dots, N(n) \\}", "bf3a19971451723b7c50d7a0e38c89ba": "CEncode(l,s,u_1,u_2,v,e) \\stackrel{\\mathrm{def}}{=}I_{W^{\\ast}}^{B^{\\ast}}(s)||pad_l(I_{Z}^{B^{\\ast}}(u_1))||pad_l(I_{Z}^{B^{\\ast}}(u_2))||pad_l(I_{Z}^{B^{\\ast}}(v))||e \\in B^{\\ast}", "bf3a4a45dd5eec9c6457bb16ae2b56f6": "1) \\ x^2+2x-2=0", "bf3a7ae9e5fe86ee77cb7ad6210e2160": " \\mathrm{Gr}_L = \\frac{g \\beta (T_s - T_\\infty ) L^3}{\\nu ^2}", "bf3a94ec54844d9959a55c995ab0f1ac": "\\left(\\frac{\\mathit{W}_{1-2}}{{m}}\\right)=\\mathit{u}_1-\\mathit{u}_2", "bf3aadca5b5802084cd835e140e23d9d": " Z(s)\\sim(4\\pi s)^{-n/2}\\sum^\\infty_{m=0}a_ms^m. ", "bf3adc18f732b11a0ebfb42be486a0a2": "\\sum_{n=0}^\\infty[S^{[n]}]t^n=\\prod_{m=1}^\\infty Z(S,{\\Bbb L}^{m-1}t^m)", "bf3afeea5929d555ca24b1734547b5c7": "\\bar{y}=\\frac{1}{A}\\int_0^1 2y\\sqrt{1-y^2}\\;dy=\\frac{-2}{3A} (1-y^2)^{\\frac{3}{2}}|^1_0=\\frac{4}{3\\pi}", "bf3b29ba96262f7bf5791770934e6e0b": "L_{j+1}", "bf3b3442f06b9faeb3e9a8a8f357c863": "2\\mbox{N}_2 \\mbox{O}_5 (g) \\rightarrow \\; 4\\mbox{NO}_2 (g) + \\mbox{O}_2 (g)", "bf3bb885c032014d3fff55bdd814b0b2": " \\Psi(\\bold{r})", "bf3bd38021677a9543c5e3d825939eeb": "\\ \\displaystyle u\\in \\mathcal{U}(\\alpha,\\tilde{u}) \\ ", "bf3be67ae09c734c8bcf667745c19aa3": "{1 \\over 1}-{1 \\over 2}+{1 \\over 4}-{1 \\over 8}+{1 \\over 16}-{1 \\over 32}+\\cdots = {2\\over3}.", "bf3c08256d9af61a4941fbdd0c44a205": "|\\mathcal{Z}|={5\\choose 3}=10", "bf3c49159fc6b0a4fd6689d060d82bae": "\\oint_{\\gamma} f(\\zeta)\\,d\\zeta\\, = \\oint_\\tau f(\\zeta)\\,d\\zeta.\\,", "bf3c4bd8a0fc08be9a6cfba2249597b5": "p(n, t) = e^{-t} \\frac{t^n}{n!}", "bf3c6a876a3948c9917afffdc6e6c960": " \\mathbf{X}^{new} = \\mathbf{X}^{tr} \\odot \\mathbf{W} + \\mathbf{B} ", "bf3ca97e7b969e0df0cc527b292083a3": "P(Class(X_i) = Class(X_j)) = p_{ij}", "bf3cf16dd44d697d8db036b96ec6709a": " n_{x+1} = s_xn_x", "bf3cf21b8dceacbff4a19820aaa1dc1f": "\\scriptstyle t_r", "bf3d0a7f0497f3b76fbed970cdc7d801": "\n\\begin{pmatrix}1 & 1\\\\0 & 1\\end{pmatrix}\n\\begin{pmatrix}1 & 0\\\\1 & 1\\end{pmatrix}\n\\begin{pmatrix}1 & 1\\\\0 & 1\\end{pmatrix}\n=\n\\begin{pmatrix}0 & 1\\\\1 & 0\\end{pmatrix}\n", "bf3df19b2a3e427979bd3ebd04151e33": "v=F(I-C)^{-1}(Dp-b(v))=A(Dp-b(v))", "bf3e5a814e84e07affb8b361b9ae3b57": "S = \\{2,...,100\\}", "bf3e73049357684d9a06715b08c0e519": " R_{TP}= R_{T} + \\left( -s_{Oy}-s_{Ty}\\right) \\cdot \\left( k_{TOF} \\cdot R_T+t_{Delay}\\right) \\,\\!", "bf3e85d14816b5a1d2a135a5c67a18d3": " \\theta =\\frac{\\alpha \\cdot p}{1+\\alpha \\cdot p}", "bf3ebdd5018c0058fdc9d37f79b2859f": "X_w(a,b)=\\frac{1}{\\sqrt{a}} \\int_{-\\infty}^\\infty x(t)\\psi^\\ast \\left(\\frac{t-b}{a}\\right)\\, dt", "bf3ec249347b7ec0d585a4ed79caa2ae": "d\\mathbf{X}^2 = dX^\\mu \\,dX_\\mu = \\eta_{\\mu\\nu}\\,dX^\\mu \\,dX^\\nu = -(c dt)^2+(dx)^2+(dy)^2+(dz)^2\\,", "bf3f4d809eced2c75f85c0b8f4c548ed": "q:(x_1,\\ldots,x_n) \\to \\left( \\sum_{i=1}^n q_i(x_i)^p \\right)^\\frac{1}{p}.", "bf3f9b67d263fad15208c1e22554b2e0": "G=\\left[\\sum_{nm} G_{nm}^{-1}\\right]^{-1}", "bf3fc32c9b241602ea2abdb5c4ae9054": " \\max_{x,y} \\ \\{3x + 2y\\} \\ \\ \\mathrm { subject \\ to }\\ \\ x,y\\ge 0; cx + dy \\le 10, \\forall (c,d)\\in P ", "bf3fdfaa0ac99e0e6ecd483dce88beb2": "[\\mathbf{a}]_{\\times} \\stackrel{\\rm def}{=} \\begin{bmatrix}\\,\\,0&\\!-a_3&\\,\\,\\,a_2\\\\\\,\\,\\,a_3&0&\\!-a_1\\\\\\!-a_2&\\,\\,a_1&\\,\\,0\\end{bmatrix}.", "bf3fe121737c0104818a4fbe9b633718": "a - mx", "bf3fe1a2f69d1e14df12558868c4a613": "u(c)=\\frac {c^{1-\\theta}-1} {1-\\theta}", "bf4056ee1df6dd8d7612ca47801544c9": "\\omega^r = \\{(e,f) \\, :\\, fe = e\\} ", "bf405d246f6dc89624e4a61e047f8448": "\\operatorname{Li}_{-4}(z) = {z \\,(1+z) (1+10z+z^2) \\over (1-z)^5} \\,.", "bf407cd19b0940c5a04e2973c0817794": "\\nu_j", "bf4087dd93355747d87043f6e78875c6": "d = 2^{\\bar{k}}", "bf40c3d938d1fdd3e1011e9fceb8bd76": "\\|x + y\\| \\leq K(\\|x\\| + \\|y\\|)", "bf40c7d4514149598bd4f82fa98e75c4": "\\big( \\mathrm{d} f(x) - \\mathrm{d} f(y) \\big) (x - y) \\geq 0.", "bf40e91249d78ca2392562f39f413d73": "g_2^n", "bf410e17e7fc590eb433e9e872cd498f": " a_i = \\begin{cases}n & \\text{for } i = 0 \\\\ f(a_{i-1}) & \\text{for } i > 0 \\end{cases}", "bf41673d5305d2dbbb94e5d8e643d406": " \\Theta'' +n^2 \\Theta =0, \\,", "bf4172c2ecdef0293f189ae45292b331": "\\phi(x)e^{-i\\omega t}", "bf417dfa192456115a1574c9b06a25f5": "x(t)=\\begin{cases}\n\\cos (2 \\pi 10 t/\\mathrm{s}) & 0\\,\\mathrm{s} \\le t < 5 \\,\\mathrm{s} \\\\\n\\cos (2 \\pi 25 t/\\mathrm{s}) & 5\\,\\mathrm{s} \\le t < 10\\,\\mathrm{s} \\\\\n\\cos (2 \\pi 50 t/\\mathrm{s}) & 10\\,\\mathrm{s} \\le t < 15\\,\\mathrm{s} \\\\\n\\cos (2 \\pi 100 t/\\mathrm{s}) & 15\\,\\mathrm{s} \\le t < 20\\,\\mathrm{s} \\\\\n\\end{cases}", "bf41defd0cc4b3debf2ad6adaacf0244": "\\Delta Q = 0, \\quad \\Delta U = -\\Delta W\\,\\!", "bf422cabe168b2cfe7c64dba614e5998": "\\mathbf{X}\\text{ and }\\mathbf{D}", "bf424bde3b30275b1ba3bf4ee9efda93": "\\Delta S = R * interception.", "bf424c680f0e437aa9c68c4d06542e0e": "T = \\frac{2(Z/3)}{Z/3+Z} = 0.5", "bf42883b376cb84a6ace536962def502": "Q(z) = {\\sqrt z/2\\over \\sinh(\\sqrt{z}/2)}= 1 - z/24 + 7z^2/5760 -\\cdots.", "bf4291f1b7207b39efd563ff20ab5d2c": " \\mbox{erf} ", "bf42a58836f33f228e48035171d06c0a": " \\log( \\frac{k_X}{k_0}) = \\rho\\sigma^' ", "bf42dbfd58f9087fee65d9590435b359": "F(x;\\lambda) = \\mathrm (1-e^{-\\lambda x}) H(x)", "bf42eeca4c8cc86e442e9b5d1f0b896b": "a_r=\\ddot{r} - r\\dot{\\theta}^2", "bf42f7cf68332c3d17c1e195a2b957bf": "d^3 \\vec{p} = |\\vec{p}\\,|^2\\, d|\\vec{p}\\,|\\, d\\phi\\, d\\left(\\cos \\theta \\right). \\,", "bf42feea1cef5520cab00fbc060d9ffd": " \\mathcal{P}_B (A) = (A \\;\\big\\lrcorner\\; B^{-1}) \\;\\big\\lrcorner\\; B", "bf430e460979307dd5eef416c8f708ce": "[S_0] = [S] + [A_{ad}]\\,", "bf432f2733038be22d3e9342403dffc1": "E_{kin}^{2} - \\left(m_0c^2\\right)^2 = m^2v^2c^2", "bf434510e63c4ece0ac9e0308bac52e2": "RPM = {Speed \\over \\pi \\times Diameter} = { 1000 \\times 30 m/min \\over 3 \\times 10 mm} = {1000 revs/min}", "bf4470bc7f8e2db1859394d5911ad2ae": "\nP(n=0,t)=\\exp(-\\overline{\\nu} t),\n", "bf449ae8d9a1f95a35db2249703aacd3": "j \\in [q]", "bf4544bdf3e39d4d111151df5e73409d": "\\sum_{k}(n)", "bf45524d024862c244773967d5fd5c60": "\\mathbf{x}_i^{t+1}=\\mathbf{x}_i^t + \\beta \\exp[-\\gamma r_{ij}^2] (\\mathbf{x}_j^t - \\mathbf{x}_i^t) +\\alpha_t \\boldsymbol{\\epsilon}_t ", "bf4558e96c8b4458963036ed48552777": "\\scriptstyle n\\geq 1 ", "bf45634edcffaabcd83c66d0e72bf9c6": "\\limsup_{n\\to\\infty} \\int_S |f-f_n|\\,d\\mu \\le \\int_S \\limsup_{n\\to\\infty} |f-f_n|\\,d\\mu = 0,", "bf45d132f5f3da994f2a2dd57aed831b": "\\left( \\frac{\\partial f}{\\partial x} \\right)_{y,z}.", "bf45e1e5bfc87be88bf18c6e83592124": "L_x = (L_x/T)^a\\,(L_y/T)^b (L_y/T^2)^c\\,", "bf4655aa3032ab2f0e3536456e0f2068": "x+y=2, \\,\\,\\,\\,\\, 2x+2y=4, \\,\\,\\,\\,\\, 3x+3y=6", "bf46c6053f73af01eed975a4b558e864": "\\mathrm{O}(n+1,1)", "bf46d061dfd43d0bee1095d51323ed67": "\\scriptstyle I_1,\\, \\ldots,\\, I_n", "bf4726e094cb8e914b1139407af9cfff": "\\{f(u),f(v)\\}\\in E'", "bf4728a3911ceb3275b7e46397c04bd7": "\\mathbb{R}^5", "bf4757a2da739a7e77cedb39df9587a3": "X_{N-k} = X_k^*,", "bf47e834672a7b7ef16611a09dccb500": "\\left(\\frac{\\partial\\ln\\left(\\Omega\\right)}{\\partial x}\\right)_{E} = \\beta X +\\left(\\frac{\\partial X}{\\partial E}\\right)_{x}\\,", "bf4808752094b11ded1845d05ec854b6": "\\beta=\\pm\\pi/2", "bf48288b53caedce1bd5925b7a0693e0": "c(t)=\\textbf{m}(t)\\cdot\\textbf{d}(t)", "bf4842e134e3e08486ef21216c21ea33": "{t_{far}}", "bf485049c456bfa17b73831bbab046b5": "\\scriptstyle a \\,=\\, 0", "bf485b3aa62d39b9147fde1443be60a5": " \\Delta G_{micelle} = RT* ln (x_s) + \\frac{RT}{N} ln (\\frac {x_m}{N}) ", "bf48bd223e43bb0189c5b2456e385efd": "G(\\nu)", "bf48d957e8b7658430b6835a8287a7a1": "\n B(r) = \\frac{\\Phi_0}{2\\pi\\lambda^2}K_0\\left(\\frac{r}{\\lambda}\\right)\n \\approx \\sqrt{\\frac{\\lambda}{r}} \\exp\\left(-\\frac{r}{\\lambda}\\right),\n", "bf49584a4143802e58740155ae040a17": " \\frac{B}{B_w} = \\frac{B_{c}}{B_{wr}}", "bf4989b49a3e795515af767f5e6dc4ce": "u=w", "bf499bccf1be1a57972a1909f156f11c": "\\scriptstyle a x + b y = c", "bf4a11f8fc803074a6e7430ec9baae81": "\n\\varphi _j^{n + 1} - \\varphi _{j - 1}^{n + 1} = \\sum\\limits_m^{M} {\\gamma _m \\left( {\\varphi _{j + m}^n - \\varphi _{j + m - 1}^n } \\right)} \\ge 0 . \\quad \\quad ( 4)", "bf4a4218b704217b176dab214f6533cb": "K_{13} = K_{23} = K_{31} = K_{32} = 0", "bf4aa4318f40615e862b7ec70bc106b0": "\\begin{align}\n&a_0 (c(c-1) + \\gamma c) x^{c - 1}+ \\sum_{r = 1}^\\infty a_r(r + c)(r + c - 1) x^{r + c - 1} -\\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1)(r + c - 2) x^{r + c - 1} \\\\ \n&\\qquad + \\gamma \\sum_{r = 1}^\\infty a_r(r + c) x^{r + c - 1}-(1 + \\alpha + \\beta) \\sum_{r = 1}^\\infty a_{r - 1}(r + c - 1) x^{r + c - 1}-\\alpha \\beta \\sum_{r = 1}^\\infty a_{r - 1} x^{r + c - 1}= 0\n\\end{align}", "bf4af487859b9a5ae897ed5b1bb7455a": "\\int\\limits_{-\\infty }^{\\infty }{{{P}_{V}}f(u,\\xi )}.du={{\\left| \\hat{f}(\\xi ) \\right|}^{2}}", "bf4af9b14a5c4764c983cc61796969ae": "\n\\left.\n\\begin{matrix}\n(x+y)+z=x+(y+z)=x+y+z\\quad\n\\\\\n(x\\,y)z=x(y\\,z)=x\\,y\\,z\\qquad\\qquad\\qquad\\quad\\ \\ \\,\n\\end{matrix}\n\\right\\}\n\\mbox{for all }x,y,z\\in\\mathbb{R}.\n", "bf4b09e6da747abeba86ddf384be063a": " \\min_{x \\in C} f(x) ", "bf4b0e4585232efdde36fc963559973b": " \\frac{dB}{dt} \\le 0 \\mbox{ is equivalent to } \\frac {dS_{total}}{dt} \\ge 0 \\qquad \\mbox{(1)} ", "bf4b3a2b373a7f5459fff4786e4924d2": "\\langle X \\rangle = X - \\operatorname{E}X,\\,", "bf4b9b1cb563e0257b9132364c41805e": " \\scriptstyle \\tau = RC ", "bf4bb3bb3215d0bacc8bcbcffae0b7a0": "~k~", "bf4bd3f70583bd46d31971da7a8c7cc0": "d(n) < 2 \\sqrt{n}", "bf4c06d9e4b864578deab9fc9df83ee3": "Z(f) = \\{ (x,1-x) \\in \\mathbf{C}^2 \\}.", "bf4c24f9764b86db941423d725f71f34": "k=1,\\ldots,j-1", "bf4ce324c4c2ce3ea0e23384c64015bb": "I_i = \\frac{3}{2}^+ \\Rightarrow I_f = \\frac{3}{2}^+ \\Rightarrow \\Delta I = 0", "bf4d0652e39af596e4e7ce2a584c96de": "g^* (R^r f_* \\mathcal{F})_t = H^r(X_s, \\mathcal{F}) = H^r(X'_t, g'^* \\mathcal{F}) = R^r f'_* (g'^* \\mathcal{F})_t.", "bf4db7db985b9b8896671263ad9d7b03": "P = \\left\\langle -\\frac{\\partial \\varphi }{\\partial V}\\right\\rangle _{t}", "bf4dd8270b0a51703acf4e3247518ad1": "E[F|x'^{(t-1)}]", "bf4e03e8376be6e02c86df5b9c7a55d5": "\\Phi_M: \\mathbb{\\hat{C}}\\setminus M \\to \\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}", "bf4e55afd01c0ba8183e94cb83569802": "A=1\\text{m}^2", "bf4e68800b7902c27be5f6f2436d0e3a": "\\frac{c_{0}}{c_{a}}\\sim 10^{5}", "bf4f185f6aeeda0d377fbe667dc86f27": "path_{min}", "bf4f75f7880f0c31b738310297da7c1a": "a\\sin(z^2)/z+b\\sin(z^2)/z^2", "bf5012ca08cec87c29ccb7381413116c": "\n \\boldsymbol{\\sigma} \n = 2\\left[\\left(\\cfrac{\\partial W}{\\partial I_1} + \n I_1~\\cfrac{\\partial W}{\\partial I_2}\\right)~\\boldsymbol{B} - \n \\cfrac{\\partial W}{\\partial I_2}~\\boldsymbol{B}\\cdot\\boldsymbol{B}\\right] - p~\\boldsymbol{\\mathit{1}}~.\n ", "bf502e72bbeefb4df5b3df7e6772292e": "L^\\infty(X)", "bf505e102fc12c51d31661f3f8ca0d9e": "\\displaystyle{\\lambda_k=k\\sum_{n=1}^N \\alpha_n z_n^k.}", "bf50b95491e2dcab57cad8e3efc47373": "(p_{n+1}-p)^2\\approx(p_{n+2}-p)(p_n-p)", "bf50f5f0ef54ebf1a9b8f19876941c0a": "[1_K,\\chi\\downarrow^G_K] = [1\\uparrow_K^G,\\chi]", "bf51462285f6e553f06f990c53b38486": "[M_{\\mu\\nu},D]=0", "bf51a007c3172e8fd7c47196a26dff83": "\\frac{\\partial \\rho}{\\partial t} + \\nabla\\cdot\\left(\\rho\\mathbf{v}\\right)=\\frac{D\\rho}{D t} + \\rho\\nabla\\cdot\\mathbf{v}= 0,", "bf51e28d4963db88e847999f7b9b8456": "A_{n}^{x}+B_{n}^{y}=C_{n}^{z};", "bf5236f08da9981ece2885f54d99e6ed": "F_3(a, b) = a^b = a^{2^{\\log_2(b)}}", "bf523f645c9530308270c6a5f008ec52": "\\varepsilon_{i_1i_2\\cdots i_n}", "bf5264ba07935eb4d135e52bf6d25a98": "{I_{ion}} = {g_{ion}} ({V_m}-{E_{ion}})\\,", "bf533ef2d7e7f72c44ff56811949bab4": "\\Omega= H^2+1+2XY+2YX.", "bf53568afc7b78fcac4546b9da27f338": "X^\\vartriangle", "bf536c665023e95cdcfd3c43988bbb26": "d^I_{p,q} + d^{II}_{p,q} :\nH^{II}_q(C_{p,\\bull}) \\rightarrow\nH^{II}_q(C_{p+1,\\bull})", "bf53aa146f640566bde7dd8a474f0efe": "C'=C_1'\\ ||\\ \\dots\\ ||\\ C_{n-1}'\\ ||\\ C_n", "bf53e28c445a6f56045a9e37aea9b038": " S(f) = \\left( |M_0|^2 + \\sum_{n=0}^{\\infty} |M_{n+1} - M_n|^2 \\right)^{\\frac{1}{2}}. ", "bf5414080ebd7344e1f2d849732d437b": "|R_T| \\leq \\frac{(x_1-x_0)^2}{8} \\max_{x_0 \\leq x \\leq x_1} |f''(x)|. \\,\\!", "bf54b5139ef62661307c3b2020bfddc3": "=\\frac{1}{2}\\frac{m}{3}v^2", "bf54d4e22af17e1692ee7d53f7ada63a": "\\sum _x f(x)= \\int_0^x f(t) dt - \\frac12 f(x)+\\sum_{k=1}^{\\infty}\\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x) + C", "bf54d8f758acbf6c66bc8f68db0a2757": "\n\\begin{align}\n\\operatorname{E}_Y \\left( \\operatorname{E}_{X\\mid Y} (X \\mid Y) \\right) &{} = \\operatorname{E}_Y \\Bigg[ \\sum_x x \\cdot \\operatorname{P}(X=x \\mid Y) \\Bigg] \\\\[6pt]\n&{}=\\sum_y \\Bigg[ \\sum_x x \\cdot \\operatorname{P}(X=x \\mid Y=y) \\Bigg] \\cdot \\operatorname{P}(Y=y) \\\\[6pt]\n&{}=\\sum_y \\sum_x x \\cdot \\operatorname{P}(X=x \\mid Y=y) \\cdot \\operatorname{P}(Y=y) \\\\[6pt]\n&{}=\\sum_x x \\sum_y \\operatorname{P}(X=x \\mid Y=y) \\cdot \\operatorname{P}(Y=y) \\\\[6pt]\n&{}=\\sum_x x \\sum_y \\operatorname{P}(X=x, Y=y) \\\\[6pt]\n&{}=\\sum_x x \\cdot \\operatorname{P}(X=x) \\\\[6pt]\n&{}=\\operatorname{E}(X).\n\\end{align}\n", "bf5518542643b9cbd114cabe3d0a39fe": "q(0)", "bf552de30a86656d265fc95e7bf81d25": "\\left| u-\\bar{u} \\right| < 8\\times10^{-5}", "bf552f2a7ffdd26ccccf2129e6dabb64": " E(k') = E(k) \\pm \\hbar \\omega_q \\pm \\Delta E_C \\, ", "bf557adca2024e34b5f80aaf2fe98e44": "S_N:=\\sum_{n=1}^NX_n,\\qquad T_N:=\\sum_{n=1}^N\\operatorname{E}[X_n]", "bf55974300301ed0a9f81f3569af2ae8": "op_1", "bf55b8aae0833cab5ab5502d83140741": "P_{y,w_0}", "bf55c839c94e8bf99ad0197bfc8161d8": " V(t) =\\sup_{x\\in X\\left( t\\right) }f(x,t) ", "bf55d3e91b6c7b5184198ba49a6cd2d9": " c = g' ", "bf55db6b02a8a522fbbba80ef037a6cb": "\\sum_i \\gamma_i c_{V,i} = \\alpha V K_T", "bf55ef7768b9041ce4c98481d563605f": "c_1 = -x c_2\\,", "bf5620158b4311a3fbf5ef736ba35d58": "\\displaystyle{\\widehat{f}(t)= {1\\over (2\\pi)^{n/2}} \\int_{{\\mathbf R}^n} f(x) e^{-ix\\cdot t} \\, dx.}", "bf56466f4377e193a6eff9b41677959f": "g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3)", "bf565f5d0daaef0f26496e1859ec693f": "p_W(w|a) = \\frac{a^2}{\\pi^2(w^2-a^4)} \\ln \\left(\\frac{w^2}{a^4}\\right).", "bf569fbec08600fab1a7aa37a8d59d9f": "\\frac{1}{\\pi} \\arctan\\left(\\frac{x-x_0}{\\gamma}\\right)+\\frac{1}{2}\\!", "bf56db0f830b123887ff851e1f717f94": "xy - wz = 0", "bf57434164d31a257de80d0604e410dd": "M=\\{0,1\\}^n", "bf57962f52484c21d191b8298fb929bf": "\\mathcal{L} = \\mathcal{L}_X", "bf57aa7d0f3f3b8a79fa7b479678d1e2": "\\mid\\ \\subseteq2^S\\times2^S", "bf58963be38adb10cc21582c524b9595": "(\\mathbf{8},\\mathbf{1},0)\n\n", "bf58acbe1ed3bf57b70960af2f2ee6be": "\\begin{align}\n & \\log \\left( \\lambda \\right)=c^{\\lambda }+a_{i}-d_{j}-\\gamma \\cdot \\Delta _{i,j} \\\\ \n & \\log \\left( \\mu \\right)=c^{\\mu }+a_{j}-d_{i}+\\gamma \\cdot \\Delta _{i,j} \\\\ \n\\end{align}", "bf58adb848eb23885633467796324d40": "[\\quad]_p", "bf5937f9d67560fd960c15bb9e65e9d0": "\\scriptstyle P\\,\\sim \\,\\rm{W}(k,\\alpha)", "bf593b504e63b05f9e01a4a0630257c6": "\\mathbb C\\otimes_\\mathbb{R}\\mathbb O", "bf59b7228864195dfa173ecf988c0a55": "p = (x+yi)(x-yi)", "bf59c5852ecc3ef3ae655ba452ecff04": "j_2 = 1, m_2=0", "bf5a1ba01a3285539c856fb467563d0d": "z\\mapsto \\frac{1}{z-z_0}", "bf5acfb8540ccbf4c8bd1673cfb30a4d": "\\sum_{m=0}^n P(3m+1)=P(3n+3)-1", "bf5afb32e944fd4d85907808545b57b5": "\\cos\\left(2\\pi\\frac{f_1-f_2}{2}t\\right)", "bf5b096cae51753c4e13a60a604d9a28": "C_{n, m} = \\frac{(mn)!}{(m!)^n n!}", "bf5b2ebcebf8d97497718105bfae1798": "\\; \\log r", "bf5b4ee5cfdb1efd62ced7136df8c364": "\\ r_{c} \\le R(q,u) \\ ", "bf5b609de0929271d7e251f577782378": "((x_1,y_1),\\ldots)", "bf5b6edd244016a9ccb9204a5d2d8a6c": "\\left / \\frac{a}{b} \\right \\backslash", "bf5ba4d37b2e0ade18449a204616c222": "[\\mathbf{A},\\mathbf{B},\\mathbf{C}]\\mathbf{D}=\\left(\\mathbf{A}\\cdot\\mathbf{D}\\right)\\left(\\mathbf{B}\\times\\mathbf{C}\\right)+\\left(\\mathbf{B}\\cdot\\mathbf{D}\\right)\\left(\\mathbf{C}\\times\\mathbf{A}\\right)+\\left(\\mathbf{C}\\cdot\\mathbf{D}\\right)\\left(\\mathbf{A}\\times\\mathbf{B}\\right)", "bf5bd4d9d45749193bb21757e7091dde": "\\frac{2 \\nu^2 \\tau^4}{(\\nu-2)^2 (\\nu-4)}", "bf5be33667a5ad42931496e03c5ec5a4": " \\int_{-\\infty}^{\\infty} \\exp\\left( -{1 \\over 2} a x^2 + Jx\\right ) dx ", "bf5c08f762bfc1c2825015390f1371e8": "\\varphi_X(t) = \\operatorname{E}\\left[\\exp \\left ({i\\int_\\mathbf{R} t(s)X(s)ds} \\right ) \\right]. ", "bf5c184c6a75d67f33c3cb9fa3d09d6b": "\\sigma^2/\\mu", "bf5c25390f60bf614921e101183b3620": "D_r(z_0) = \\{z \\in \\mathbf{C} : r > 0, |z - z_0| < r\\}", "bf5c2b952870a71b05b27df0d01255e1": " S_n ", "bf5c70d58a557a8a69cedd35dd8c4800": "Rf(L) = \\int_L f(\\mathbf{x})\\,|d\\mathbf{x}|.", "bf5cd671a9660023a8e1b15d57f7e67e": "x_j=1", "bf5d561dfe214bae7cd86b2c2cedd95b": "1+\\frac 1x\\,", "bf5d7525ad400fc106d734e1badbcc7d": "\nR_{\\mu\\nu}=\\partial_\\alpha\\Gamma^\\alpha_{\\mu\\nu}-\\partial_\\nu\\Gamma^\\alpha_{\\mu\\alpha}+\\Gamma^\\alpha_{\\mu\\nu}\\Gamma^\\beta_{\\alpha\\beta}-\\Gamma^\\alpha_{\\mu\\beta}\\Gamma^\\beta_{\\alpha\\nu}.\n", "bf5d93e527c1f1f3c187d357e8ffe7c1": "|\\varphi'(z)| = \\lim_{k\\to\\infty} \\frac{|z'_k|}{|z_k|d^k},", "bf5da5521e13827fb4840a2e59ccfabd": "\\chi_{T}(G)\\geq \\chi(G)", "bf5dad0d3e350dbd19f690e8d30caed1": "P(r)", "bf5db64a7d3568c5ffacf3b9e544527b": "G^p = \\langle g^p | g\\in G\\rangle", "bf5e89bc33f337e218403103f3768564": "\n V_\\tau^B = \\sum_{i=t(\\tau-1)+1}^{t(\\tau)} V_i Z\\left(\\frac{S_i-S_{i-1}}{\\sigma_{\\Delta S}}\\right) \\; .\n", "bf5e8f136aa1f53b9b0ce134e14e657e": "A^c \\cap B~~~~=~~~~B \\smallsetminus A", "bf5ebe1fcca17c81058fd4628027d6df": "v = 0{.}8c", "bf5ed342974b8a1437af8cd24de76715": "\\ pV = nRT", "bf5f0e3468615497b2c77d03b209f7b2": "s_{\\beta} - s_{\\alpha} = \\frac{\\mathrm{d} P}{\\mathrm{d} T} (v_{\\beta} - v_{\\alpha}),", "bf5f3e6e933f8a6b6877b91fd579ba42": "\\boldsymbol X \\sim \\mathcal{N}(\\boldsymbol\\mu,\\,\\boldsymbol\\Sigma)", "bf5f6e97393f4509fee1e9800d0e4b27": "r=\\sqrt[n]{p}", "bf5f90eede3345a81b6d39884fa1c8e2": " f^n \\,", "bf5fb1861d51d2ed3b2e9641267ce588": "(D^\\mu F_{\\mu\\nu})^a=0.", "bf5fc2961f8f67b9fd6073a065b3dcea": "\\mathit{q}_i \\in \\{G, C\\}", "bf6004a06e991640ff4b396b10d94c67": "\n\\mbox{Grade Level} = \\left ( 0.141 \\times \\mbox{Average sentence length} \\right ) + \\left ( 0.086 \\times \\mbox{Number of unique unfamiliar words} \\right) + 0.839\n", "bf60496b32ae7eebe42c86683c78c067": "X|Y", "bf6052d4b4803a024816fbc9e85f61b3": "t_{{\\nu_0}'-p+1}\\left(\\tilde{\\mathbf{x}}|{\\boldsymbol\\mu_0}',\\frac{{\\kappa_0}'+1}{{\\kappa_0}'({\\nu_0}'-p+1)}\\boldsymbol\\Psi'\\right)", "bf605a071bc9a417adccbdec4882559d": "\\mathbb{C}\\cup\\{\\infty\\}", "bf60ba533b7705c3a5eb9eed1e9acfe8": "\n \\hat\\beta = (X'\\Omega^{-1}X)^{-1} X'\\Omega^{-1}Y.\n ", "bf60c05472f16a02c7add90efacd7d94": "\\int_{-\\infty}^{\\infty} |p\\rangle \\langle p| dp = 1 ", "bf60c83c1271b048d68a695dba7f218c": "W = dm \\times \\left ( \\frac{A}{M} \\right )\\,\\!", "bf6100dffe829311c69dddef1cf96576": "p = \\frac{h}{\\lambda} = \\hbar k\\;,", "bf61365415248b6c1abe28821a83d73d": "V_{CB}", "bf615185bbf20e569ff1d75090b7c0ed": "\\varnothing \\!\\,", "bf616600008bc1ed1a67ee172b872a63": "v \\ge 0", "bf61a5d42a756b1e67d731096fae9327": "\\forall a,b,c \\in Q, a\\neq b", "bf61ce70fda3b2bdc6e2b27d2b2a2912": " d = \\frac{|ax_0 + by_0 + c|}{\\sqrt{a^2 + b^2}}.", "bf61d87c254213b2341d1bbcb24a6d09": "S(x)=\\alpha^{4}+\\alpha^{-7}x+\\alpha^{1}x^2+\\alpha^{1}x^3+\\alpha^{0}x^4+\\alpha^{2}x^5,", "bf62360195d9eb0a7e59da695bde5d35": "u(x,t)=\\frac{1}{\\sqrt{4\\pi kt}} \\int_{0}^{\\infty} \\left[\\exp\\left(-\\frac{(x-y)^2}{4kt}\\right)-\\exp\\left(-\\frac{(x+y)^2}{4kt}\\right)\\right] g(y)\\,dy ", "bf62d907430ff81ef68efdcadcb29dbb": "{\\tilde{\\rm{SL}}}(2, \\mathbf{R})", "bf63165ebb22bcb21f0f2961a5b46eb6": "\\alpha_{n-2}^{ }", "bf6325167ede1a6c13814535e3f4bb17": "p_{{\\mathrm{CO}}_2}", "bf63558a7a5a7004be92dce6bece01ee": "f(n) = (1+o(1))g(n)\\quad(\\text{as }n\\to\\infty).", "bf6379d5778a28672b5a78578bdd3e4a": "d \\left( f(s), f(t) \\right) \\leq \\int_{s}^{t} m(\\tau) \\, \\mathrm{d} \\tau \\mbox{ for all } [s, t] \\subseteq I.", "bf63afcedb9bef87edac6e6b83f748ca": "A_R^2", "bf63ca8725a928ec730c7bb9bce65fdd": "F(F^{-1}(y)) \\geq y", "bf63e09df16af0b5277d1d3b6cf0a9d9": "\\scriptstyle\\ \\epsilon ", "bf64261dbba643a219d7fdcae620b367": "D^{p+1}\\times S^{q-1}\\;\\cup\\;D^{p+1}\\times S^{q-1}=S^{p+1}\\times S^{q-1}", "bf644f17bc515760764faf3869280200": "\\mathbf Q(\\sqrt{p^*}),", "bf646947fcac9586bd73dfe6aa5792fb": "\\sqrt{8}=2.828", "bf64791ccdbaafa2107fd464c61a20e1": "\\log w\\in BMO", "bf64ba6aa4c2d7bb776f5b41188a0c1b": "\\digamma", "bf64ff4af3ef76ad5bc514e340281b03": "\\sum_{i=1}^{n} \\left( \\frac{1}{r} \\right)^{\\ell_i} \\leq 1.", "bf654d5f3f254e7abdedaa29c7ffb13c": "q \\equiv \\bar{q} \\pmod l", "bf65781c3d3bbedcf8077303da3aa94a": "f(x, y)=d \\,", "bf667f47fdccec0996e3313d1e911a51": " \\dot x = f(x,u) ", "bf66a62ec438c095159ac2b7edff1787": "\n-d(n)(2\\gamma+\\log n)=\n\\frac{\\log^2 1}{1}c_1(n)+\n\\frac{\\log^2 2}{2}c_2(n)+\n\\frac{\\log^2 3}{3}c_3(n)+\n\\dots\n", "bf66df6358c551974c7bcf568d69f9b2": "\\langle 0|\\Phi(x)\\Phi^\\dagger(y)|0\\rangle = \\int\\frac{d^4p}{(2\\pi)^3}\\int_0^\\infty d\\mu^2e^{-ip\\cdot(x-y)}\\rho(\\mu^2)\\theta(p_0)\\delta(p^2-\\mu^2)", "bf67595bfb0c64663c9b5a2d8d24692e": "\\tbinom n r p^r q^{n-r} \\!", "bf6766bd5ad61fc7e797e68d7cd47afc": "T_{1A}=R_{1A}(x)", "bf6782749e6f26761aaa94146773f3de": "= \\frac{0\\cdot g(x) - 1\\cdot g'(x)}{(g(x))^2}", "bf679d0061d9f58523f30a6ad8fc617e": "X \\succ 0 \\Leftrightarrow A \\succ 0, S = C - B^T A^{-1} B \\succ 0", "bf67b5d7d7fb97f5b317c511cdd4caf0": "\\frac{V_2}{V_3}=\\sqrt{\\frac{Z_{I2}}{Z_{I3}}}e^{\\gamma_2}", "bf67f2d68fbd065baa04db4db6ed0d2c": "\\{\\mu_{k+1},\\ldots,\\mu_n\\}", "bf683671905d82ea2021a0b4a20f6dbe": "\\mu\\colon Q\\times M \\to Q", "bf68ef3d5c15893307c712029c12fdb0": "|\\psi_j\\rangle = a|0_j\\rangle + b|1_j\\rangle", "bf690bc9852eef559f50f8f4149dc7c9": "A_+", "bf693d4eaceda4ad75c61bc579f4b857": "\\exist a \\in A \\psi (a, b, p)", "bf69c8d9dc15ab4349271cd073439537": " \\vec x_1 ", "bf69e823eb81bd63077ef0fa02b8c01f": "(s^*,s)", "bf69ef2d18b5c11266f32b727f12f844": "\\scriptstyle \\mathcal{N}=\\{-1,0,1\\} ", "bf69f5520a2ecb1310b6db942c1b7209": "\\int_{-\\infty}^\\infty f(x)\\delta(x-x_0)\\,dx=f(x_0).", "bf6a6446c247ab07478176ae89ecc4d7": "\\omega_\\phi.", "bf6a82dbae12d1242637ce5bd119c67d": "\\ell(y)", "bf6ab63a86d0f2c5a3b3e51a9c4cee1f": " [a,b]=a * b-b * a.\\ ", "bf6b1f051acd635db785ee1aa4f7a7e7": "c(\\lambda)=\\begin{cases} l(\\lambda) & \\text{ if } \\omega(\\lambda)=0 \\\\ \\mu(\\lambda)-\\omega(\\lambda) &\\text{ if } \\omega(\\lambda)>0.\\end{cases}", "bf6b757b3606f11ef80a39d0bb2c1f3e": "(a,a) \\in D", "bf6c747ab35121aa9b898dce813eac9d": "L_{f, P} \\le R \\le U_{f, P}.\\,\\!", "bf6c8ad87771dbe1639780d1f9f275d8": " c_{i_1i_2..i_N}=\\sum\\limits_{\\alpha_1,..,\\alpha_{N-1}=0}^{\\chi}\\Gamma^{[1]i_1}_{\\alpha_1}\\lambda^{[1]}_{\\alpha_1}\\Gamma^{[2]i_2}_{\\alpha_1\\alpha_2}\\lambda^{[2]}_{\\alpha_2}\\Gamma^{[3]i_3}_{\\alpha_2\\alpha_3}\\lambda^{[3]}_{\\alpha_3}\\cdot..\\cdot\\Gamma^{[{N-1}]i_{N-1}}_{\\alpha_{N-2}\\alpha_{N-1}}\\lambda^{[N-1]}_{\\alpha_{N-1}}\\Gamma^{[N]i_N}_{\\alpha_{N-1}}", "bf6ca6c064573b1d3969e9602660f15a": "\n\\{\\phi_1,\\phi_2\\}_{PB} = c ~.\n", "bf6ce983df3314e54ec73cc6aa453aa1": "F_{x,0}", "bf6d0698aa19ad56aa92001a09244c48": "\\sum_{j=1}^{\\infin} jM_j = N", "bf6d1a288a597902dd77b941f69ae706": "y_i\\in Y", "bf6d37d3275ca6b3e7af74b7c8d396b9": "t\\geqslant0\\,\\!", "bf6d41ce18e88e073b8ade5e047a0c91": "V_{\\text{S}-}", "bf6d6c8e3cd4f648c5e957b2559832c8": "\\begin{align}\n p({\\mathbf{X}},z) = \\frac{1}{iz\\lambda} \\exp\\left(ik\\frac{{\\mathbf{X}}^2}{2z}\\right)\n \\end{align}", "bf6d7b2adc6b1bc1e105f53f17a740d7": "\\displaystyle{[{\\mathcal L}_m,J_n] =-nJ_{m+n} +{c\\over 6} (m^2+m)\\delta_{m+n,0}}", "bf6d7d7b96ca217809f7c279b7d2e6c8": " h(z)=a+c(z-a)^n+O((z-a)^{n+1}) ~,", "bf6df52990a0a58f68febbca9c557a19": "\n{dH\\over ds} = i[L,H] = 0\n\\, .", "bf6e231bc5411b625f9c8f7a1fd07612": "\n D_e(p \\parallel q) = \\int p(x)\\big( \\ln p(x) - \\ln q(x) \\big)^2 dx\n ", "bf6e4bcfee68fb98969d476cb4673a5c": "\\Phi_A^{-1}", "bf6e5fa156ec87c0d66491aee9787d8d": "Y(t+\\Delta t).", "bf6e6b071b3f1f192a17bb5dc859fa7c": "y(t)=\\int_{-\\infty}^\\infty \\sin (t+\\tau) x(\\tau)\\,d\\tau", "bf6eb047db907e80d1ad4534cc9927c0": " \\mu_B = \\mu_{B}^{\\ominus} + RT \\ln a_B \\,", "bf6edf74dfa50606ce7f854b38338fa8": "\n\\begin{align}\n\\mathrm{Slerp}(q_0, q_1; t) & = q_0 (q_0^{-1} q_1)^t \\\\\n& = q_1 (q_1^{-1} q_0)^{1-t} \\\\\n& = (q_0 q_1^{-1})^{1-t} q_1 \\\\\n& = (q_1 q_0^{-1})^t q_0\n\\end{align}\n", "bf6f0c6e85d5109254c30aef6263bb21": " \\Sigma_s^{(j+1)} = \\frac{\\sum_{t =1}^N h_s^{(j)}(t) [x^{(t)}-\\mu_s^{(j)}][x^{(t)}-\\mu_s^{(j)}]^{\\top}}{\\sum_{t =1}^N h_s^{(j)}(t)} ", "bf6f85b7de3b0339634989a8ef09bf0d": "\\mbox{STRUC}[\\tau]", "bf700b3ef5fc313355c3998a88d8abe7": "Q_B = 10^{R_B/400}", "bf702040349a0b86f303db196e40dd61": "Z_T = \\int_0^\\infty Z(\\lambda)I(\\lambda,T)\\,d\\lambda", "bf70782a018bfd3f4764da0d2ac14c61": "\\scriptstyle {\\it EUROP}^A_E", "bf708e5b66298575b5b38bde5b80190a": "a(1) = 1,a(2) = 2", "bf70e4678b707d8d50ba721540417555": "\\mathrm{E}[D]>0", "bf71040e1369962936d3854b1bb812b2": "\\frac{k_{z1}}{\\varepsilon_1} + \\frac{k_{z2}}{\\varepsilon_2} = 0 ", "bf7107281c896ea808c5ba34de7f7f85": "-\\log p(s|m)", "bf7137a23aa686ee65df5462ad943b15": "t = 1 \\ldots ", "bf7179a165c6114421ec855c9f8777ff": "\\mathrm{SO}(p+q)\\,", "bf717aba7026f7ce117e754e6ba8668f": "\\varphi _j (i)", "bf71f0196556b003df56c336bbcf2726": "\\Phi(x,y,z,t)", "bf720c202e6b827b7d18a06d2884ceb0": "A_1, A_2, A_3, A_4, B_1, B_2, C_1, C_2", "bf726d5a78a929e901e573a2ed019dea": "S_{r_1, r_2}'(r_1)", "bf728caa8c48a0ea3010e1c807214aaa": "\\sqrt[3]{8}\\,=\\,2\\quad\\text{and}\\quad\\sqrt[3]{-8}\\,= -2.", "bf72d8b994d90a8777d5be6e3ba0501a": " f(x)=\\frac{1}{x}", "bf72ea46ca212b3b8fe97c82c2f16d70": "d< \\frac{1}{3}N^{ \\frac{1}{4}},2d<3d,", "bf73051bf33e39b1646238e255480077": "P + \\tfrac12 \\rho v^2 = P_0,", "bf733a44a92778df16c53f1d830653fe": "p_1 \\approx -\\frac{1}{u},\\ p_2 \\approx \\frac{1}{u},\\ p_2 \\approx 1-\\frac{1}{u^2},", "bf73703aa858404c3ae2f373718c272d": "H_9(x)=512x^9-9216x^7+48384x^5-80640x^3+30240x\\,", "bf739cde54dd45a0e263f1983522941a": " H(z) = \\frac{1}{z-1}", "bf73af788b8929cb7d6bf5f459d49b94": "Q(x, p(x)) \\equiv 0", "bf73b32c3c7bbae4afbc204871516e66": "\\mathcal{F}(M) \\times \\mathcal{X}(M)", "bf74690b2029a7dbe29cdd35b5768de2": "x^i(X^\\alpha)", "bf74b6359eed9cbf28571cf4187881ed": "\\,L\\prec M\\,", "bf74fd3edef02c25f487c4f19403f41f": "f : U \\to Y", "bf758319655fb58f2efa4048232cb79f": "\\alpha=0.5, \\beta=1.0, \n\\delta=-0.1, \\epsilon=-0.6, \\zeta=-0.3, \\eta=-0.6", "bf75af6617d1d2f628408d6b61f6109e": "x = g^r \\pmod{p}", "bf75beb2c7c245b458ac7d46e6ca8abd": "\\frac{\\partial}{\\partial t} \\left (\\sqrt{-g} u_0 \\varepsilon^{\\frac{3}{4}} \\right ) = 0,\\ 4 \\varepsilon \\cdot \\frac{\\partial u_{\\alpha}}{\\partial t}+u_{\\alpha} \\cdot \\frac{\\partial \\varepsilon}{\\partial t} = 0,", "bf75f8e970e381758b6f3bfc2f612cae": "n\\omega_n^{1/n} L^n(\\bar{S})^{(n-1)/n} \\le M^{n-1}_*(\\partial S)", "bf7608930807ef8d52433efc451b05f0": "dz^j=dx^j+idy^j,\\quad d\\bar{z}^j=dx^j-idy^j,", "bf763bce370b4f608bec6ba4f8b36908": "i^*", "bf768b5ac37ba6853b08cdf33f6a4b8a": "I_{sp} = \\frac{C_0 \\cdot V_e}{g_n}", "bf76ab02e576a55397ef739e41fb4f67": "\n\\{x_i, x_j\\}_{DB} = 0,\n", "bf76e9c006c8b80b3d3bd5e4cc77d313": " \\psi_1(z + 1) = \\psi_1(z) - \\frac{1}{z^2}", "bf770a3ffd477b796e43a274ac990529": "\\mathbf{e}_x = \\dfrac{\\partial\\mathbf{r}}{\\partial x}; \\;\n\\mathbf{e}_y = \\dfrac{\\partial\\mathbf{r}}{\\partial y}; \\;\n\\mathbf{e}_z = \\dfrac{\\partial\\mathbf{r}}{\\partial z}.", "bf772cfedd33f113518304e527727018": "(\\phi\\leftrightarrow \\psi) \\leftrightarrow ((\\phi \\to \\psi)\\land(\\psi\\to\\phi))", "bf776cb20201221985a94ec8c584b264": "q_y=F^{-1}(y)=\\left(-\\log_e y \\right)^{-\\frac{1}{\\alpha}}", "bf77e587e21ee4e83c8d2b09d784f128": "\\int_{C_{0}} \\left\\langle \\nabla_{H} F (\\sigma), V (\\sigma) \\right\\rangle_{L_{0}^{2, 1}} \\, \\mathrm{d} \\gamma (\\sigma) = - \\int_{C_{0}} F (\\sigma) \\operatorname{div}(V) (\\sigma) \\, \\mathrm{d} \\gamma (\\sigma)", "bf78160d1e80f7e088d33905ec8810d7": "<,>", "bf7883cff7f1b3a3fa8e36c5a30ad1de": "\\vec S_o", "bf789544ba7822655e2ad410625cffec": "\n\\Psi^{(S)}_{n_1 \\cdots n_N} (\\cdots x_i \\cdots x_j\\cdots) =\n\\Psi^{(S)}_{n_1 \\cdots n_N} (\\cdots x_j \\cdots x_i \\cdots)\n", "bf78dbbebb19465c37848e85db3dcd86": "\\frac{x ( 180 - x )}{2 \\times 30 \\times 150} = \\frac{x(180-x)}{9000}.", "bf793aa530b294e7418d93724ec41ab5": "\\textstyle A ", "bf798f39334f8dd335d939d91b61ad73": "A\\xrightarrow{\\cong}A\\otimes I\\xrightarrow{\\eta}A\\otimes (C\\otimes C^*)\\xrightarrow{\\cong}(A\\otimes C)\\otimes C^*\n\\xrightarrow{f}(B\\otimes C)\\otimes C^*\\xrightarrow{\\cong}B\\otimes(C\\otimes C^*)\\xrightarrow{\\epsilon}B\\otimes I\\xrightarrow{\\cong}B.", "bf79f972f98fffa759eccf1ea8146d2b": "\nG = 10^\\frac{3}{10} \\times 1\\ = 1.99526... \\approx 2 \\,\n", "bf7a76cba3e898395e74a152e8a6bbd0": "P_{E_{7}}(x) = (1+x^3)(1+x^{11})(1+x^{15})(1+x^{19})(1+x^{23})(1+x^{27})(1+x^{35})", "bf7a860f4dc48443229afcd49d9412fe": " L_0=\n\\begin{pmatrix}\n\\sigma & \\tau \\\\\n-\\gamma & \\alpha \\\\\n\\end{pmatrix}\n", "bf7ab994fe124f6802d10fb308147479": " H_n(x) = n! \\sum_{\\ell = 0}^{(n-1)/2} \\frac{(-1)^{(n-1)/2 - \\ell}}{(2\\ell + 1)! ((n-1)/2 - \\ell)!} (2x)^{2\\ell + 1} ", "bf7aea27f7be7393bc304cb7356e91f5": "\\mathrm{tr}\\left(e^{-tD^2}\\right)<\\infty", "bf7b135c9c87d41ee8dc060ab7d0938e": "\\mathcal O_X(U)", "bf7b1b3fe72976c46943c2b75a59dcc7": " h = v_0 t_h \\sin(\\theta) - \\frac{1}{2} gt^2_h ", "bf7b210437bc9dbc878d19b9b1b5af1b": "m_{T}^2 = m^2 + p_{x}^2 + p_{y}^2 \\,", "bf7b28553305f695725669e5885c9b6f": "\\vec{\\xi}_1 = \\partial_t, \\; \\vec{\\xi}_2 = \\partial_z, \\; \\vec{\\xi}_3 = \\partial_\\phi", "bf7b2dca8993227e598b3fe60f5c835f": "\\star d \\star = d^*,", "bf7b4f261ee60300bed31232191fed62": "f_0 = { \\omega_0 \\over 2 \\pi } = {1 \\over {2 \\pi \\sqrt{LC}}} ", "bf7b500d0f47511b2beecfb31c0d16d5": " \\left.\\right. F(\\zeta) ", "bf7b8eae1da784af8e853c0653894109": "h\\rightarrow 0^-", "bf7b9906653de817e5dc98700f886470": " \\{\\psi: V \\rightarrow {\\mathbf R}^2\\}", "bf7c3a9d3027cfe4315ab65c9986dc4b": "\\begin{matrix} {n \\choose n-1}{4 \\choose 1}{3 \\choose 1} = 12n \\end{matrix}", "bf7c8911d26e769e340eaf38fb41c07d": "s = x/x_0", "bf7c8c9a571c7680574432192f7e50c3": "H(Y|X) = H(Y)\\text{ and }H(X|Y) = H(X) \\, ", "bf7d4e1fd4ea7aec8f638926313f2353": "L_t=AL-LA\\qquad (2)", "bf7d58d6e13781c6e92ae02d13ea3ef7": "C \\lor D", "bf7d761e3abb738f2999667683d15380": "\\displaystyle{\\alpha(t)=e^{-t}\\kappa(t),}", "bf7db49d4472c3a108306142a0cc67fc": "\\scriptstyle F=\\mathbf{Q}(\\sqrt{\\ell^\\ast})", "bf7dba00c17ab2030d646cdf5041b2fc": "\\Phi_{m_2K}:M\\longrightarrow W_{m_2}(M)", "bf7dba5a9aa3490ed808b2dfbce62b25": "h : A \\rightarrow C", "bf7e19a70267395f8e24476c4f230797": "R = \\frac {\\mbox {flux emitted from surface}} {\\mbox {flux incident upon surface}}", "bf7e3eb49b5eb0af56ddd92bc4e4a1ce": "[SU(4)\\times U(1)]/\\mathbb{Z}_4", "bf7eb4528ddfa8e99376fe176861ab9c": "\\frac{dx}{dy}\\,\\cdot\\, \\frac{dy}{dx} = 1. ", "bf7eba1ea942b8d0fe6acb5af6a6d460": "\\mathrm{We}^*=\\frac{E_\\mathrm{kin}}{E_\\mathrm{surf}}", "bf7f392af00d59880cb827b110fe7f63": "\\{r,r+1\\}", "bf7f47c691dc84f271dae8163908e38f": "(Y,X,\\langle , \\rangle')", "bf7fdffafe4225df3cf92e76ae9b814c": "\\operatorname{Aff}(n,K) = K^n \\rtimes \\operatorname{GL}(n,K)", "bf8101efc3e64709615847f56493742b": " A = \\frac{r_2 - r_1 }{2} \\quad (8)", "bf810b463c328a290f455c2854497a4f": "b^2-4d", "bf81389aec3b30457c8fd60f6db15384": "z = h+f", "bf813e5d812be3e16e626f35b479bf2f": " \\Delta : C \\to C \\otimes C", "bf81452dea3a1ee64cbeb9a7bbb535f0": "\\begin{align}\nz^2\\sum_{k=0}^\\infty &(k+r-1)(k+r)A_kz^{k+r-2} - zp(z) \\sum_{k=0}^\\infty (k+r)A_kz^{k+r-1} + q(z)\\sum_{k=0}^\\infty A_kz^{k+r} \\\\\n& = \\sum_{k=0}^\\infty (k+r-1) (k+r)A_kz^{k+r} + p(z) \\sum_{k=0}^\\infty (k+r)A_kz^{k+r} + q(z) \\sum_{k=0}^\\infty A_kz^{k+r} \\\\\n& = \\sum_{k=0}^\\infty (k+r-1)(k+r) A_kz^{k+r} + p(z) (k+r) A_kz^{k+r} + q(z) A_kz^{k+r} \\\\\n& = \\sum_{k=0}^\\infty \\left[(k+r-1)(k+r) + p(z)(k+r) + q(z)\\right] A_kz^{k+r} \\\\\n& = \\left[ r(r-1)+p(z)r+q(z) \\right] A_0z^r+\\sum_{k=1}^\\infty \\left[ (k+r-1)(k+r)+p(z)(k+r)+q(z) \\right] A_kz^{k+r}\n\\end{align}", "bf822d11108289714ba3a937eb7e57a8": "\\rho^*:D^*\\to[0,\\infty]", "bf823af1bcf7ca7a6c66968e73b326bf": "{2L} < \\rho < {R^2+L^2 \\over 2R}", "bf8266c05852824cc31cd337ff575d73": "E= L^\\infty(X)", "bf826824a3c36099ff55e34cd33b9206": "\\lim_{n\\to\\infty}\\frac{n!}{(n-k)!k!} \\left(\\frac{\\lambda}{n}\\right)^k \\left(1- \\frac{\\lambda}{n}\\right)^{n-k}\n= \\lim_{n\\to\\infty}\\frac{n(n-1)(n-2)\\dots(n-k+1)}{k!} \\frac{\\lambda^k}{n^k} \\left(1- \\frac{\\lambda}{n}\\right)^{n-k} ", "bf82a6f0c85a42ae84cba9e99b1ae2da": " |\\cdot|: P\\to\\{0,1,\\ldots,n\\}", "bf82c63b51292299c5a12d0a341bd896": "\\rho_{x}", "bf839f3597126555b9b70fc473b3e75e": "S =p(\\mathbb{N}) \\cap \\mathbb{N}", "bf83cbcf1ebc24cfcda4d6dc3b785006": "\\psi \\in B^{\\ast}", "bf83fea97a86fc9b5c9abcfe67b29337": "p_2 \\equiv 3 \\pmod{4}", "bf8419ed69c47a8725e0b9743fcf454a": "{{V_1^2y_1^2} \\over {gy_1y_2}} = {{V_1^2y_1} \\over {gy_2}} = {{y_2 + y_1} \\over 2}", "bf841b537f5e9369ce9c276c9708c56c": "i<<1", "bf84245bbcfdc61f8076d5a3e31d9bce": "N^a = \\left( \\nu, \\nu \\mathbf{n} \\right)", "bf842f4e7b33320cff71f9e3957763e4": " = r \\frac {d \\theta} {dt} \\left( -\\mathrm{sin}\\ \\theta \\ \\mathbf{i} + \\mathrm{cos}\\ \\theta \\ \\mathbf{j}\\right)\\, ", "bf84de3958a09d7ae3a5f8898cbd7ed3": "c_2 = \\frac{1}{16}", "bf85037c4adff6772f76e5628f908926": "\\forall\\,\\!", "bf8528d565e1da0996b9ebbd0a20a0c4": "\nx = x_1+2x_2+4x_3+\\ldots+2^{\\lfloor \\log_2U\\rfloor}x_{\\lfloor \\log_2U\\rfloor+1}.\n", "bf8544044b77011dc3af204480d15de9": "f = \\frac{\\sqrt{1+\\alpha^2}}{NT},", "bf859646cd41660c1a4062dac91b81c5": "1.25^{-1} + 3.9^{-1} = 1.056", "bf8646e1dab0b95d62e80c08336563e9": "\\scriptstyle\\hat\\Omega\\;=\\;\\hat\\Sigma\\,\\otimes\\,I_R", "bf864a1b4f4116f980d1fce3d2f0e4e8": "\n\\langle 0|\\varphi(x)|p\\rangle=\n\\sqrt Z \\langle 0|\\varphi_{\\mathrm{in}}(x)|p\\rangle\n", "bf8672798ff24f4d8a9e1c7bc41a04f0": "f''(x)\\,", "bf8681b9701ce73a91eb5434cef32e91": " 1 \\le p_u(n) \\le k^n \\ , ", "bf86aae94723c30939a83bba34d88220": "Cij", "bf86b6dd5b9e7dcea96950b1ea5dcad0": "V_U", "bf873b5e2f9f1930e191efc60fc8a5d2": " x_1^2 x_2 x_3 + x_1^2 x_2 x_4 + x_1^2 x_3 x_4 + x_2^2 x_3 x_4. \\, ", "bf87d3fd04c11da410da9622c9412a21": "\\mathbb F_p", "bf88d01486ba51b91247b05c8d0f7373": "(x^2-1){P_{\\ell}^{m}}'(x) = -(\\ell+m)(\\ell-m+1)\\sqrt{1-x^2}P_{\\ell}^{m-1}(x) - mxP_{\\ell}^{m}(x)", "bf88daa20309f72bafaf40b6a20fe1b8": "\\lim_{k\\rightarrow\\infty}Dq(x_{2k+1})=0", "bf892167a8b0c67e2d8cbb5a62b39c5c": "\\displaystyle{f(e^X)=\\sum \\lambda_i (\\mathrm{Ad}(e^X)e_i,e_i)_\\sigma \\ge (\\min \\lambda_i)\\cdot \\mathrm{Tr}\\,e^{\\mathrm{ad}\\,X},}", "bf892394f512c457261a1ee17b256671": " \\lang s'|s\\rang = \\sqrt{\\frac{N-1}{N}} ", "bf89d67bbb897883316552c969aed55b": "\\frac{d^2B_1}{dx^2}=1, \\frac{d^2B_2}{dx^2}=-2,\\frac{d^2B_3}{dx^2}=1,", "bf89fc55e1c16361ddc31385d40b9090": "\\mathfrak{M}(f(z)) = M(f(z), 1).\\,", "bf8a144d254b9c6eddd0ec3e97cb6053": "\\gcd(m,N) = 1", "bf8a55650b7c1eb916f41693e6f43703": "\\bot_i", "bf8a89c23191190f94a107a4da8af5e9": "x\\mapsto \\operatorname{ad}_x", "bf8ade72751bcba74d93ecc2f9b08638": "a^2I_n=-ax^n \\cos{ax} + nx^{n-1} \\sin{ax} - n(n-1) I_{n-2} \\,\\!", "bf8b1abb30a15f9eca31349002a91688": "I_0 = (P_0 A)/(\\lambda^2 f^2)", "bf8b44ca73dbc025ffd544910868617a": "A_o=\\mathrm{max}_{S_o}(\\mathrm{min}_x(U(S_o,x))-T_oS_o)", "bf8b65ae668a4ecf865098ae2c1ec4ed": "G=\\mathbb Z^d", "bf8b67642e521e437b692d3a4944cc94": "\\zeta=h-H_N", "bf8bd35c9c16bd9bab9f7ce852240f87": " ((12)(3)(4)) (1 \\lor 3) \\to 2 \\lor 3. ", "bf8c1afb18784db6857235bce89c8268": "A+A", "bf8c1ea0289dfb0fb783302937772057": "K\\left(2\\,\\sqrt{-4-3\\,\\sqrt2}\\right)=\\frac{\\left(2-\\sqrt2\\right)\\pi^{\\frac32}}{4\\,\\Gamma\\left(\\frac34\\right)^2}", "bf8c6d71312c8254c7330febdd3d3dce": "\\mathbf{M} = \\ -\\mathbf{e_y} EI {d^2w \\over dx^2} = \\ -\\mathbf{e_y} EI {d^2z \\over dx^2} ", "bf8c813038abf2ed95487c6a73e8a8ba": "\\begin{align}\nF_{0k} & = (\\eta_{0i} \\eta_{k0} - \\eta_{00} \\eta_{ki} ) F^{i0} + \\eta_{0i} \\eta_{kj} F^{ij} \\\\\n& = (0 - (-\\delta_{ki}) ) F^{i0} + 0 \\\\\n& = F^{k0} = - F^{0k} \\\\\n\\end{align}", "bf8d14bac9460b22f6d4e09581cc606f": "\\displaystyle{\\|h\\|_{(k)}^2=\\sum_{j=0}^k {k\\choose j} \\|\\partial_x^j\\partial_y^{k-j} h\\|^2,}", "bf8d342e7dbecb8f08a2b10ceaa29ac2": "\\textstyle L", "bf8e1a5ddeb8f04ed9ad1ec17fa645a7": "E_2 - E_1 = h \\, \\nu_0", "bf8ec14cb48b4b608a9ffec3b19f99b8": "Ce^{C(|S||\\Sigma|)^{1/d}}", "bf8f0b8556bfd95d6ac67e43e297d91c": " \\alpha = V/S ", "bf8fc6abf4a67707b809507720b78f8d": "s\\in(1,\\infty)", "bf900d20134d361d6ab9358e95124153": " |i| = \\alpha ", "bf9015a1234539c9268c3023df948f72": "\\bigcap_{i \\in I} A_i \\neq \\emptyset", "bf901acb9ddd08e8d36e26662ed676e8": "\n \\mathbf{v}\\cdot\\mathbf{b}_i = v_k~\\mathbf{b}^k\\cdot\\mathbf{b}_i = v_k~\\delta_i^k = v_i ", "bf9078b6fd6db54836fb0a37f3e71557": "\\sin^3\\theta \\cos^3\\theta = \\frac{3\\sin 2\\theta - \\sin 6\\theta}{32}\\!", "bf90cd26bd092f83e9b2ef283cccb936": "A=\\prod_{i=1}^nA_i", "bf9127337e317a53a6015e9631bf2ded": " a=r_0, b=r_1 ", "bf9165b8293ddff07f087a4f77e0190e": "(\\forall k\\in\\mathbb{Q}\\setminus\\{0,-1,-2,\\ldots\\}):\\frac{x J_k(x)}{J_{k-1}(x)}\\notin\\mathbb{Q}.", "bf9186851e46a25b0979cba35bf02a53": "B \\cap C \\in \\mathcal{B}", "bf91aaa246135b9046acde2272e94c89": "\\sum F_y=0=F_{BD}\\sin(60)+R_B=F_{BD}\\frac{\\sqrt{3}}{2}+5 \\Rightarrow F_{BD}=-\\frac{10}{\\sqrt{3}}", "bf91bf6701aebe094bd3c5b965cb5b98": "2(\\sqrt{2}-1)", "bf91c60da25cd9824e8d332c61e9bace": "\\Psi(x,0) = (\\phi(x), 0)", "bf91e08180a72f3203159471d326fff5": "p^4 = (p-1)^2 = p^2 -2p +1 = -p", "bf925d12c3486be1fac1579fbd14ffdc": "\\operatorname{MSD}(\\hat{\\theta}) = \\sum^{n}_{i=1} \\frac{\\hat{\\theta_{i}} - \\theta_{i}}{n} .", "bf9270230c67a798f2fea2416b959425": "\\frac{d^2y}{dx^2} \\,=\\, \\frac{d}{dx}\\left(\\frac{dy}{dx}\\right).", "bf929e69e408c8125c0de34dea96b6bb": "F=c_p P_{br}", "bf92d5085e0cbd168ce396afac39ac07": "a_r(n), a_{r-1}(n), \\ldots, a_0(n) \\in \\mathbb{K}[n]", "bf92e02cf23e918332ef360d3bbe61cd": "{t} = \\frac{T}{d}", "bf930cae2dd182e1286295527a5a9dfd": "\\sin \\left(x+y+z\\right)=\\sin x \\cos y \\cos z + \\sin y \\cos z \\cos x + \\sin z \\cos y \\cos x - \\sin x \\sin y \\sin z, \\,", "bf931bfcf085e47e747038ec59b6d9b4": "\\mathfrak{q} \\supsetneq \\mathfrak{p} R[x]", "bf9399f8e9acd786346d9953869ede90": "y_{k+1}+\\Delta t y_{k+1}^2=y_k", "bf93b95c863c83f9fa876deae11f3499": "\\lambda_0=0", "bf93d995b0c2ed5484bee98d11ce2e0f": "\\mathrm{(Chromophore)^* + {}^3O_2 \\ \\xrightarrow{} \\ Chromophore + {}^1O_2}", "bf93e9d4720601dcdaac585e9aa29d36": "B_k(x)", "bf9411fcb71d121e0dab8bac9c6ea6c9": "\\text{FA} = \\sqrt{\\frac{3}{2}} \\frac{\\sqrt{(\\lambda_1 - \\hat{\\lambda})^2 + (\\lambda_2 - \\hat{\\lambda})^2 + (\\lambda_3 - \\hat{\\lambda})^2}}{\\sqrt{\\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2}}", "bf9452285270e116facb48fa4b74ed95": "\\langle 1 \\rangle\\,\\!", "bf94ed6438dba62861a3b32e69e87718": " \\mathbb R ^n ", "bf94f68c60eb07895179149fad54c863": "\\left({\\mathit{u}_2-\\mathit{u}_3}\\right)=\\left(5 - 9\\right)=-4", "bf955f84dd7782f538cc8315d45b5d64": "g = h", "bf95c5fbc2743b3560ec4081ecf1b4ea": " E_\\text{k} = m \\gamma c^2 - m c^2 = m c^2\\left(\\frac{1}{\\sqrt{1 - (v/c)^2}} - 1\\right) ", "bf9698430056c8ada3dc4c92c6582c11": "1/\\sqrt n", "bf969e6995cdec80627a4de34d843020": "y^*_i = \\hat{y}_i + \\hat{\\epsilon}_i v_i", "bf96e6946b0991a9f648bc578b02f4ba": "\\scriptstyle x - x_0 \\ = \\ \\frac {1} {2} H \\cosh^{-1}\\left(\\frac {H}{h}\\right) - H \\sqrt{1 - \\frac{h^2} {H^2}}", "bf97105281e49e603c770f9ccf8664e6": "\n\\phi_{a} - \\phi_{b} + k\\pi = k\\pi = \n\\int_{\\zeta_{b}}^{\\zeta_{a}} d\\zeta \\ \\frac{d\\phi}{d\\zeta} = \n\\omega_{k} (\\zeta_{a} - \\zeta_{b}) \n", "bf9799446e536774873128c832ed03b5": "[A]=[A]_0 e^{-k_1 t}", "bf97c12b5f0d2c4910c02075a20b3e46": "p, q, r", "bf9809c0c46afdd2bb9bb68da8268360": "{\\frac {|AB|} {|BD|}} = {\\frac {\\sin \\angle BDA} {\\sin \\angle BAD}} ", "bf982976cffa261016e3476f1fcbd02a": "N \\cdot N^r \\cdot S \\cdot N^l \\cdot N \\leq S", "bf983763a4089b05121eab5a74f0c76e": "g^{x_1}, g^{x_2}", "bf987d23cccaf3cf295f6322f64e947e": "T_s = 1-R_s\\,\\!", "bf98d57517cee5b085f98d6a4fc4cdb8": "\\Delta J=0,\\pm 1 (\\text{except } J=0 \\rightarrow J=0)", "bf98dd50bc950c95df7527b938bc7070": "C_S = C_O k_O (1-f)^{k_o - 1}", "bf9920e44124beee5f27ed225db6932e": "\\frac{dy}{dx} = -\\frac{x_1-a}{y_1-b}.", "bf9950cf21bcbf4f925b5980a11a327a": "(A\\cap B)/N = A' \\cap B'", "bf996bf75cf005b887f4e523fa42c625": "(t_\\text{max}-t,y,x),", "bf996c8c37493d6330eb95a640255d72": "\\frac{u_{i}^{n + 1} - u_{i}^{n}}{\\Delta t} = \nF_{i}^{n}\\left(u, x, t, \\frac{\\partial^2 u}{\\partial x^2}\\right) ", "bf9984a0f5b8dfb880042d149bca0d14": "v \\neq v'", "bf99b12dbac297505e67e05844e358ac": "\\tan{\\beta_m}", "bf99bf496c08fe9b6dca12e1032a1360": "-\\boldsymbol{\\alpha}(tI+S)^{-1}\\boldsymbol{S}^{0}+\\alpha_{0}", "bf99c95f8a49ba843203da4126b71e34": "\\sum_i \\frac{n_{i+1} - n_i}{r_i n_{i+1} n_i},", "bf9a04da5cf30155ab4543336cb377b2": "E_2 = 4.00 + 2.00 = 6.00 \\text{ ft}", "bf9a20661ed7986debea5e28d2c4a3f2": "p=2^m+1", "bf9a2947fda1a3fc909c400589eb4c2a": "\\log \\nu", "bf9a4ad22bac064d9768e2ba20971178": " (\\forall x) A(x)\\ \\equiv \\ A(\\epsilon x\\ (\\neg A)) ", "bf9a7725b02e7f8d73345aa416442c2a": "D=P-\\infty_1", "bf9aa96570d4741cc7c2659307c623fd": "\\vert\\rho_{m,t}\\vert", "bf9aad155cae7e91c3aaa17d2a15bb8a": "\nR_{\\kappa \\nu} = \\Delta U \\Psi_{\\kappa \\nu} \\,\n", "bf9ae45b37a0f55df631a3a8de6037ad": "S = \\langle A, R\\rangle", "bf9b15b47a8adc516066c4901c9bf97a": "B(\\lambda,T)", "bf9b2b0a9d649762230c030f1eff0e77": "rank[{f_2(q, \\dot{q}, t)}] < dim[q]", "bf9baa674cd20114680753669d8cb1fd": "I = {\\rho \\cdot \\ell \\over A} \\,", "bf9bb91d614ea10a6e5ad63be2824994": "L(\\gamma) = \\sup \\left\\{ \\left. \\sum_{i = 1}^{r} d \\big( \\gamma(t_{i-1}), \\gamma(t_{i}) \\big) \\right| a = t_{0} < t_{1} < \\cdots < t_{r} = b, r\\in \\mathbb{N} \\right\\}", "bf9bc4a227a32315a5057430b7c7460a": "\\text{span} \\{ |\\psi_{1n}\\rangle ,|\\psi_{2n}\\rangle\\}", "bf9c5f6a2d549f2f137e12fd3e29950b": " \\phi (i) = \\sum_{E \\subseteq \\mathbf{X} \\backslash \\{i\\}} \\frac{(n-|E|-1)!|E|!}{n!} [g(E \\cup \\{i\\}) - g(E)]. ", "bf9c7637c7e0e314a1abc9c472b68642": "\\log(1) = 0 + 2 \\pi i k", "bf9cc056a2e2c1bcf1e6a676dc66b4d1": " +", "bf9cf5fca3d9a72b6c2322afa8b34290": "\\text{IDF}(q_i)", "bf9d0f53e2199313a6811c535ba5794c": "\na_{RWG}=a_{SIW}-1.08(2r)^2/p + 0.1 (2r)^2/a_{SIW}\n ", "bf9d2215d0652460ce51e6fc6acf7e41": " A + R \\rightarrow \\ C ", "bf9db06f701afde66bb2f5afb75b4c6a": "\\operatorname{prox}_f(x)", "bf9e4e5dd51d65cb1438124f0e52cf52": "\\mathcal{M}_i", "bf9e9c57d1d0d1cad1de8f874ae6a219": "\\mathfrak{H}_{SB}=\\mathfrak{H}_B\\otimes\\mathfrak{H}_l\\otimes\\mathfrak{H}_r\\otimes\\mathfrak{H}_U", "bf9eb72fc38b3cd33a13287f7dad43b5": " \\left(\\frac{\\tau}{p}\\right)=-1 ", "bf9efc2a28f957859c46bdeead288534": "H_B(s)= {k_B \\cdot s^3\\over(s+129.4)^2\\quad (s+995.9)\\quad (s+76655)^2}", "bf9f2d4f463c58aaaad19ba3c497cc7a": "\\tfrac{a}{0}", "bf9f42ca475bc91a845b311c20ca2fd4": "U_{k + 1} \\subseteq U_{k} \\mbox{ and } \\bigcap_{k = 1}^{\\infty} U_{k} = \\{ x \\},", "bf9f61463abba3234c5b2b3fc3244178": "T_0 = \\sum_p \\frac{1}{p(p-2)} \\approx 0.749 \\ . ", "bf9fba9db94b37783f58c5778f60c28c": "f \\circ C\\colon V \\to W\\,", "bf9fc2980e38810ca357305c38eb7418": "\\sum_{(i,j) \\in E}\\Omega_{i,j}=N-1", "bfa02e5cba3a70e2f6fed66c5af2da37": "w_{ji} \\,", "bfa0445ae59f645e745e6720d1315e2f": "1 + G(s)", "bfa05a69ecf497640a4f58e61b7c6766": "1-\\exp(-\\epsilon t)/\\mu(S_{t})\\,\\!", "bfa05e0c0571c8843671173d18da9db3": "J_{ij}=L_{ij}+S_{ij}", "bfa05e5c942f1dce020aea5404e6474c": "\\langle u,v\\rangle = \\int_\\Omega uv\\,dx", "bfa0800b45accba33e2f910cb87759d5": "B_{X^{**}}", "bfa0916bd8e51a5d21325c2eadb5fbdf": "\nf^{e}_{\\mathbf{k}} = \\langle \\hat{a}^\\dagger_{c, \\mathbf{k}} \\hat{a}_{c, \\mathbf{k}} \\rangle \\;.\n", "bfa0a427507fbcd3eb300855ce4e0033": "h(x)=(f\\star g)(x) = \\int_{-\\infty}^\\infty \\overline{f(y)}\\,g(x+y)\\,dy", "bfa0fad472478594ce674347274d6559": "Q=\\sum_{i=1}^k x_i^2", "bfa114eab5860aa284bbc83b604baf02": "z \\in V", "bfa1508922ad0bf53fbd3732287d4478": " im [L(\\hat{x}, \\hat{p}), \\hat{x}] = 0 = \\hat{p} ", "bfa18f9288836777cb202f92cecc7a20": "b = -1", "bfa199830ec61d0b0053ae9703475803": "dx^1\\wedge\\cdots\\wedge dx^n", "bfa19ef44291465249945f5026bd44de": " \\vec{p}_1 = \\partial_z, \\; \\; \\vec{p}_2 = \\partial_r ", "bfa20bd4dd13fd7ff79ecd77063be620": "\\frac{\\displaystyle \\sum_{k=1}^N k^2}{a}", "bfa22eca9b69d09f6c839d083f6bd7f8": "\\mathbf{F}_d = -6 \\pi \\eta r\\, \\mathbf{v}.", "bfa24bb60e7a6a95566ec1acb9e4c8cf": "P_{11}", "bfa2768964bb6424d8bf1499938a53f4": "\\rho\\, g\\, a\\; \\text{e}^{\\displaystyle k\\, z}\\, \\cos\\, \\theta\\,", "bfa2b87f67e45bc26c84b9f1aff17a2e": "a<\\frac{\\theta_1-\\theta_0}{\\theta_0 \\theta_1} \\sum_{i=1}^n x_i - n \\log \\frac{\\theta_1}{\\theta_0}1 ", "bfa974bffe8a8945fc8893987ba299f6": "\n\\frac {N^2} {t} = \\frac {E \\cdot S} {C} \\,,\n", "bfa9a8746af63b518e72336f5369e159": "p(\\theta|D) = \\frac{p(D|\\theta)p(\\theta)}{p(D)}", "bfa9b329780eb568175e8ec141733a64": "\\delta[\\phi] = -\\int_{-\\infty}^\\infty \\phi'(x)H(x)\\, dx.", "bfaa3f93ae8e539fdbe85ca9b4aa9c34": " \\mathcal{F}^\\mathbf{W}(t) \\triangleq \\sigma\\left(\\{\\mathbf{W}(s) ; \\; 0 \\leq s \\leq t\n\\}\\right), \\quad \\forall t \\in [0,T]. ", "bfaa677cb00f8ce1ec0f4706cdfce929": "(\\alpha,\\beta)=(\\text{out},\\text{in})", "bfaa7555b0cb4cbf615dbd30b0b6dcdc": " f/g", "bfaac4a912a734a0da72eae43ffc7058": "\\frac{3}{2}\\sin(2 \\theta) = 1", "bfab2b6dcf01e38dd80fd16f097ec321": " R_{S,t+1} = {{S_{t+1}-S_t+D_{S,t+1}}\\over S_t}", "bfab376e07a3c5376f43f33e885cd200": " \\displaystyle{T=(A+I)(A-I)^{-1}.}", "bfab46270f0f349075e10b93e6d2ee52": "\\phi *\\psi :A*B\\rightarrow S(W)", "bfab5aa1fbac65a6d43aa599ffc0b8dc": "(\\theta_i)^2 = 0\\,", "bfab71dc01bada8810fefccff9af2e4b": "P_2=1-cR/d", "bfab877b0479def981f033ab9ce59703": "F'(x) = f(x).", "bfaba0b9ed892032a182e9c00536670a": "P(W) = p", "bfabd2e38e3f89a9d07c1541f32e0199": "v_0 \\leftarrow 0 \\, ", "bfabe21f32a149bf3144331c215b2530": "M(u)_{,\\,u}<0", "bfabe8030941b22175f8ee9328a29c49": "\\sum_{k=0}^\\infty ar^k = a + ar + ar^2 + ar^3 +\\cdots", "bfac08b862d1141860e739fdd8dc36f9": "\\frac {r_N}{h-r_N} = \\tanh\\frac{Nb}{2}.", "bfac401e16bbfb332b17c3ce4e601a7e": " \\text{Input} + \\text{Generation} = \\text{Output} + \\text{Accumulation} \\ + \\text{Consumption} ", "bfac55ff0929aa5c99c720333c405cf2": "{d^2x \\over dt^2}+f(x){dx \\over dt}+g(x)=0", "bfac6df0a3740ecd393b22ad9bd7ef6b": "Y=\\exp\\left(\\frac{\\epsilon}{kT}\\right) - k_2", "bfac95f44cd054b2cbdd2ac188956d03": "M/R \\leq 1/RR,", "bfad332a540ca7b9aa9fd9844212f64b": "\\begin{align} X_K(z) &=\\sum_{n=-\\infty}^{\\infty} x_K(n)z^{-n} \\\\\n&= \\sum_{r=-\\infty}^{\\infty}x(r)z^{-rK}\\\\\n&= \\sum_{r=-\\infty}^{\\infty}x(r)(z^{K})^{-r}\\\\\n&= X(z^{K}) \\end{align}", "bfad3962e15a5c0cc49bb6447edfc2a3": "X\\setminus \\{p\\}", "bfad4cfec230ca3e41f7e84e29cf31ac": "\\mathfrak{P}^{83}", "bfad6b4c2e90bb3104546894d9fe4b10": "\n D \\approx 2E^ffh(f+h)\n", "bfad81de1f5ebfb0861fd35e741a2062": " (x_{1}+y_{1})^{n_{1}}\\dotsm(x_{d}+y_{d})^{n_{d}} = \\sum_{k_{1}=0}^{n_{1}}\\dotsm\\sum_{k_{d}=0}^{n_{d}} \\binom{n_{1}}{k_{1}}\\, x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\\;\\dotsc\\;\\binom{n_{d}}{k_{d}}\\, x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}. ", "bfad84cd5b0478abcf4a7a413bade588": "e_n(x) = x^n = \\sum_{k=0}^n \\delta_{n,k} x^k.", "bfada5af75154a7bf484b94894274915": "\n\\sum \\frac{(\\text{observed}-\\text{expected})^2}{\\text{expected}} = \\sum_{k=0}^8 \\frac{\\left((\\text{count in cell }k)- 20\\left(\\frac{\\lambda^ke^{-\\lambda}}{k!}\\right)\\right)^2}{20\\left(\\frac{\\lambda^ke^{-\\lambda}}{k!}\\right)} + \\frac{(0-a)^2}{a}\n", "bfadba4243ce94ca8334244dc0ebaf03": "\\int \\exp\\left[-{1\\over 2} \\theta^T M \\theta + \\eta^T \\theta \\right] \\,d\\theta = \\begin{cases} \\sqrt{ \\det M } \\exp \\left[ -{1\\over 2} \\eta^T M^{-1} \\eta \\right] , & n \\mbox{ even} \\\\ 0 , & n \\mbox{ odd} \\end{cases} ", "bfadd658ce18cc8260a7e5a37bf15650": "\\left(\\frac{b-a}{b+a}\\sqrt{ab}, \\frac{2ab}{a+b}, \\sqrt{ab}\\right) = \\left(\\frac{b-a}{b+a}\\sqrt{ab}, H(a,b), G(a,b)\\right),", "bfade04d88ebf1ab19f86d4aade04f87": "\\beta = \\ln\\left [ \\coth \\left ( \\frac{ R_{db} }{ 17.37 } \\right ) \\right ]", "bfae2d8c5ef6b8ecb74f653ed622f8f2": "f:(M,d)\\longrightarrow(X,\\delta)", "bfae42bcb36cf0318bbe1cdbe84834a3": "\n K_{\\rm I} = \\frac{F_y}{2\\sqrt{\\pi a}}\\sqrt{\\frac{a+x}{a-x}}\\,,\\,\\,\n K_{\\rm II} = -\\frac{F_x}{2\\sqrt{\\pi a}}\\left(\\frac{\\kappa -1}{\\kappa+1}\\right) \\,.\n ", "bfae482e6a8367282479da18a9d97738": " z \\mapsto \\int (T_zf)g \\, d\\mu_2", "bfae64f7120a870b028ae892abdbd315": "M(x) = \\alpha", "bfae84083964b9d3d9b4c57712bba34c": "a=n(m^{2}+k^{2}) \\, ", "bfaed1f5c53fe6829f12863431ac5871": "a \\otimes^n b\\,\\!", "bfaefd58fe21cbd1ef0b7bf049d86ef3": "m\\neq 0", "bfaefd733824c8e159324b3366950da7": "\\alpha, \\beta \\in \\mathcal{A}", "bfaeff7d0bcd9a64c670a676dca87da7": "R_\\beta", "bfaf93405b421f87404dd2dfd14c997f": "\\forall\\lambda\\in\\Lambda: \\phi_\\lambda", "bfafba00bf6b7c071a49570b68169850": "\\gamma_{t}", "bfafdce9985396b9e31094eca9bf10e9": "\\epsilon (r)= e^{ k_0 r}", "bfafea27c47d3b00838cc5823649f00c": "y[n] = \\sum_{q=0}^{M}x[n-q]\\beta_{q} - \\sum_{p=1}^{N}y[n-p]\\alpha_{p}.", "bfb0049daa898e6c31c1a47800a5723f": "\\alpha_1 \\geq \\alpha_2 \\geq \\cdots \\geq \\alpha_k. \\, ", "bfb0213e8f5454b7a265c33308a0f622": "\\hat{\\mathbf{r}} = \\mathbf{r}/r", "bfb024fa92253ca0eaeaee5dd0f0bf80": "M \\equiv Q^c Q", "bfb07a5e9a95eab857d559686fc2edae": "\\frac{25927}{(1+0.10)^5}", "bfb07f72dfac5851d7976af0c3800b86": "(1) \\,", "bfb0cf648a69ef68eff78a0ed27ed64a": "x_1,...x_N", "bfb0e70671464c4334722f7281884cda": " p_n(x) ", "bfb12f1bfaaec56d6475fd9f5b674c65": "p_{ij} \\approx \\sum_t^T p_t \\, p_{it} \\, p_{jt}.", "bfb1724c07b61d32c05ea8c9ed2af2d0": "(A.7)\\quad \\theta_{(l)}=-(\\rho+\\bar\\rho)=-2\\text{Re}(\\rho)\\,,\\quad \\theta_{(n)}=\\mu+\\bar\\mu=2\\text{Re}(\\mu)\\,,", "bfb17d5256a70f45086c4c58102a50f2": "a_r(x), a_{r-1}(x), \\ldots, a_0(x) \\in \\mathbb{K}[x]", "bfb22cf8e7eb9dd394e8f7a2b2ebc861": " P_i ", "bfb24d2f3dea57afc0025d91950bfc3f": " \\theta_i(x) = \\theta_i(y), \\forall~ i < k ", "bfb2920ccd89cfefff50c3c0983ad43d": " \\text{E}\\left(\\boldsymbol{\\varepsilon}\\right) = \\mathbf{0} \\; ", "bfb2a19a5651f278132dcbe20b9e7220": "\\alpha\\in \\mathcal{O}_k,\\;\\; ", "bfb2ac18eec369658a379261a3ca0a96": "\\mathrm{Lip}_\\alpha(\\mathbb{T})\\subset A(\\mathbb{T})\\subset C(\\mathbb{T})", "bfb2be02382c095d97526f743eea102e": "\\mu \\rightarrow 0, \\infty ", "bfb32d78584e2bc2f696c3494b8b5c1f": "V^{\\mathbb C}", "bfb361641e516aa0b4f3e78e351c24d4": "A^{-1}=\\frac{1}{\\det(A)}\\begin{pmatrix}\nA_{22}&-A_{12}\\\\\n-A_{21}&A_{11}\\\\\n\\end{pmatrix}", "bfb3659523d4bebbbdf6dc6500fc9f74": "\\bar{s}, \\bar{t}", "bfb37b6c2095b58bb8214154720c91e3": "\\text{subject to: } \\operatorname{Tr}_A(E_{AB}) = I_B", "bfb3bd615cb78135d67f3e2620751f20": "p!\\,", "bfb3f58609ca2927d0d2592bbf559564": "\nH_\\mu (s,t)=\\sum\\limits_{n=0}^\\infty e^{-sn}P_\\mu (n,t). \n", "bfb409145b254c2069448cfaab32134d": "X=\\{X_1,\\cdots,X_m\\}", "bfb455118d2283bb926a39d07df19cc0": "\n\\Delta := \\begin{vmatrix} A_{xx} & A_{xy} & B_{x} \\\\A_{xy} & A_{yy} & B_{y}\\\\B_{x} & B_{y} & C \\end{vmatrix} = 0\n", "bfb4953c237b0892c403a28d3b576acc": "{1}/{\\sqrt{r}}", "bfb4a189953c7b6be89fb3e165323982": "V= \\sqrt{H/d}", "bfb4a1e1841254fa56a444b75087b328": " E = \\hbar \\omega \\left[(n_1 + \\cdots + n_N) + {N\\over 2}\\right]", "bfb4acb1f7ce0fca00ec038b9b96d362": "\\mathrm{SML} : E(R_i) = R_f + \\beta_{i}[E(R_M) - R_f]\\,", "bfb4e0bde385b57842e1f8e0bbfd51f7": "1-2x+x^2-yz^2=(1-x)^2-yz^2=u^2-vw^2", "bfb4f3dd81a0b2c76a1c1fb9a17789c6": "k=\\frac{k_L+k_C}{1+k_Lk_C}.", "bfb54a99ce2c19d89be27e7ab7af537c": "\\scriptstyle i_\\mathrm i", "bfb56f0dc5642be09c2930d84279b450": "|\\phi'(z)|", "bfb578a94c7ad122d450b396834de903": " y_{p}(x) = c_{1}e^{r_{1}x} ", "bfb599a0be6c4d45e4ddc11fefad1f80": "Z = \\frac{1}{s}", "bfb5ad966942ef81dfbed27ff951c1f1": "V_{\\mathrm{int}}^{ij}", "bfb5ba9185b37b9967190548dbfd3183": "\\mathcal{R}\\mathbf{r}_{io}", "bfb5ed4851371e11de8227edddf3b4d0": "I(x_2;y) \\geq I(y_1; \\hat y_1 | y) ", "bfb5f5c2a451d7db87f9f57bb61b4a1f": "\n\\phi ^{2}(P) \\mp \\bar{t} \\phi(P) + \\bar{q}P = O\n", "bfb6488d6c250ac5aeed1bbf139baaa5": "y\\,\\!", "bfb649403e639909826b26883b18d039": "U^\\circ", "bfb6774f08ce3f4e1c8a3648da5dc0a3": "n! = n (n - 1)! = n (n - 1)\\cdots 1", "bfb726d5067d7efee07437b2f3a5aa48": "K f_0\\,", "bfb782b15f608bf25ac39b4dec861171": "T(x,a)", "bfb78680a3512cd27584aa208dba3013": " d=8 k+6 ", "bfb8158d29b977d384fd028bb66c6290": "A_*", "bfb81bf0b50e1d0dd0a736ba4c81f55c": "{\\rm kg/m s}", "bfb86c5f39bbb3bdf254f4f2ab9af485": "\\phi(N)", "bfb86d3fdf30af3d2892ecddb4ff0816": " \\tan^2 \\psi = \\frac { 2 g y_0 + v^2 } { v^2 } = C+1", "bfb88c1c312378254f99111f285d348d": "F_{p+1}=G_{p+1}+H\\wedge C_{p-2}", "bfb8d117e27c0c574db1ee2dae3de686": "\\textstyle -\\frac{\\hbar^2}{2m}", "bfb8d6f79a7a78522be9044ab3789605": "I_n = \\int \\cos^ {n-1} x \\cos x dx , \\,\\!", "bfb91ef6def5f4fec596e10b4ba5d600": "\\sigma(x) \\not\\in \\mathfrak{p}_1", "bfb970b2f679b966ebe13aab11619b34": "T = \\frac{dq_\\mathrm{rev}}{dS}.", "bfb99ad5edd59a8ef4d96a7ac1ea8d68": "T _* P = P\\big(T^{-1}(\\cdot)\\big).", "bfb9dc480f7971eb5419eb5bcecea2d1": "A_1, A_2, ... A_n", "bfba1ebe54521386299239d81379388d": "\\frac{5}{7}", "bfba297dc94384a4f20543d347c2e253": "\\mathit{g(x)}", "bfba2e5353ea328392e775480dea519d": "\\begin{array}{rlll}\n\\text{mean} & = e^{\\mu + \\sigma^2 / 2} & = e^{0 + 0.25^2 / 2} & \\approx 1.032 \\\\\n\\text{mode} & = e^{\\mu - \\sigma^2} & = e^{0 - 0.25^2} & \\approx 0.939 \\\\\n\\text{median} & = e^\\mu & = e^0 & = 1\n\\end{array}", "bfba6c35dbbcd8b89c6a29b1ffd6f517": "q = e^{i\\pi\\tau}", "bfbaa2e5bb0b303c55e5d74b44031b7a": "\\Psi(t)=\\frac{2}{\\sqrt{5}}\\pi^{-\\frac{1}{4}}(1-t^{2}+it)e^{-\\frac{1}{2}t^{2}}.", "bfbae90ea1ea14753c443412d79ec73f": " |\\delta| < |b_{k-1} - b_{k-2}| ", "bfbb0f7fc987f5116804feb4a13d2fa5": "t=\\frac{\\sin{\\theta}}{\\sin{\\theta_m}}", "bfbb1662d7a0312d1248ad9f0c166c7b": "r_1^2 = -1 = r_2^2", "bfbb99d4f3a753e896c031747883e46f": " 1 \\times n", "bfbba1466f1fe5c4c05c66eaa777961a": "\\displaystyle \\sin\\theta_c={[-K_2\\pm (K_2^2-3K1K_3)^{1/2}]/3K_3}^{1/2} ", "bfbbe0cd4148f4cfb498f13719a1d97d": "0<\\omega<\\omega_c\\,\\!", "bfbbfcaa9bedc1716e2072fd669465aa": "vknbxc'l;k\n", "bfbd955a9252bf42c52597ece4729765": "\\delta T=\\frac{cM}{C_p}[v_{Ar}\\cos(\\omega t-kx)-v_{Al}\\cos(\\omega t+kx)].", "bfbda4303fad9b2b23416a16484e90cd": "f_i:U\\rightarrow \\mathbb{R}", "bfbdb7b65971f54d6c8b59e671be7bb9": "I_1 \\times I_2 \\times I_3", "bfbdd7d089006253c9a32f7c78c15270": " n ", "bfbdfb4d4d06891429c094788384fd46": "v = \\frac{ds}{dt},", "bfbe1b16af66783030c13846c8f1d16c": "\\frac{\\partial \\mathcal{L}}{\\partial A_\\mu} = -e\\bar{\\psi} \\gamma^\\mu \\psi \\,", "bfbf032b13fe24ac1ed7b6d439e75272": " H_2^{+} ", "bfbf2f002071e28db32c4913c13400ac": "c_D", "bfbfe4a50d6b37f063aae0770dd199c5": "x_2 \\succ x_3", "bfc01eee66892400907134fc7272cfc8": "\\left\\{{3'\\atop3}\\right\\}", "bfc1027c5734560eb8cdc4edd6bf078a": "\\beta < \\varphi_\\gamma(\\delta) \\,", "bfc11263418c2169fcb8390a7d78a5f3": "\\geq \\sum_i p_i (\\log p_i - \\log (\\sum_j q_j P_{ij})", "bfc12f3be75d07ca852923700d81d21e": "EF\\phi \\equiv \\phi \\lor EX EF \\phi", "bfc14db688b3a62a3c777e35866c237b": "\\frac{1}{6} + \\frac{1}{6} = \\frac{1}{3}", "bfc1be91a58e32baa387528e8fa5779b": "M \\# S^m", "bfc1dd6e81d1d06ce9eb673d18fc4063": "\\left(\\frac{-1}{\\sqrt{10}},\\ \\sqrt{\\frac{3}{2}},\\ 0,\\ \\pm2\\right)", "bfc1e618ff4accee312a68e733bc4167": "\n{\\pi \\frac{dN}{dx} (x) = \\frac{1}{2i}\\frac{d}{dx}\\bigl(\\log(\\zeta(1/2 + ix)) - \\log(\\zeta(1/2 - ix))\\bigr)- \\frac{2}{1+4x^2} - \\sum_{n=0}^\\infty \\frac{2n + 1/2}{(2n + 1/2)^2 +x^2}}\n", "bfc1e812cb5eac96b6ea9108f9a652c3": "\nx^N - x + t=0 \\,\\!\n", "bfc248d5bdb349c6dea507b2e5569d5b": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi(\\mathbf{r},\\,t) = \\hat H \\Psi(\\mathbf{r},t) \\,\\!", "bfc3b7d3aac0250c7090e9163f7da55e": "\\hat{H} = -\\boldsymbol{\\mu}\\cdot\\bold{B} ", "bfc3c1918492c6d5d7b9e89760365fe3": "\\lambda_k=\\max\\limits_C\\min\\limits_{\\binom{\\| x\\| =1}{Cx=0}}\\langle Ax,x\\rangle,", "bfc3ddc0e94fd782c66d7f42d7f61ea0": " V = \\frac{1}{\\rho} \\left| \\frac{1}{f} \\frac {\\partial p}{\\partial n} \\right| ", "bfc3fcca9d58486cba75dee5e3c4a97e": "M=\\prod_{n}c_{n}", "bfc423fe0364dd961dca7b9634c34f95": " y_1'=-0.04y_1+10^4y_2\\cdot y_3", "bfc42706df31c2333b0b0a5fb84ca38f": "\\,\\gamma\\neq 0\\,", "bfc43f1f8b158971ae53595ef0b70c71": "\\Omega^8O\\simeq O ,\\,", "bfc4444ac33a1232119e8b13ed87a5b3": "M_C=\\begin{bmatrix}0&1&2\\\\0&1&2\\\\0&1&2\\end{bmatrix}\\quad\\Rightarrow\\quad M_{C^*}=\\begin{bmatrix}0&0&1\\\\0&1&0\\\\1&0&0\\\\0&0&1\\\\0&1&0\\\\1&0&0\\\\0&0&1\\\\0&1&0\\\\1&0&0\\end{bmatrix}", "bfc44ba7c64cf6c994a0e4769aa12391": "\\mathcal{M}^*/\\mathcal{O}^*", "bfc44d4dd44fbc6af687cdf684ff2927": "n^{2\\gamma}", "bfc46fb80c4905cce74f6f489984c711": "M \\le N", "bfc4d5b9df2fa83b3e1558c787aa030d": "\\mathrm{\\frac{1\\,statohm}{1\\,abohm}}=\n\\mathrm{\\frac{1\\,statvolt}{1\\,abvolt}}\\times\\mathrm{\\frac{1\\,abampere}{1\\,statampere}}=c^2", "bfc4df3835ff5d4d334195e935b168b8": "ABV = 131 \\left( \\mathrm{Starting~SG} - \\mathrm{Final~SG} \\right)", "bfc540c586562221a6ac916ade6db4df": "\\frac{d^2y}{dx^2} = F(y) \\,\\!", "bfc54177ea0f9a2629bba2a06edc18cf": "(V,\\triangleleft)", "bfc551c4106dcb7ee9590a22dd8b0bbf": "\\displaystyle{K^*(z,w)=K(w,z).}", "bfc57706290588e7d9d4ed0c44cd14cb": " \\hat{\\textbf{x}}_{k} \\leftarrow \\hat{\\textbf{x}}_{k} + ( \\alpha )\\ \\textbf{r}_{k} ", "bfc5aa1a7a54f1044d3da94f455c46a9": "\\begin{align}\\int_V \\dot{\\mathsf{L}} \\left(\\dot{\\nabla} dX;x \\right) &= \\int_V \\langle\\dot{F}(x)\\dot{\\nabla} dX I^{-1} \\rangle \\\\\n&= \\int_V \\langle\\dot{F}(x)\\dot{\\nabla} |dX| \\rangle \\\\\n&= \\int_V \\nabla \\cdot F(x) |dX| . \\end{align}", "bfc5ec2ab19c4beeb1bafa721e3da9dc": "f,g: X \\to Y", "bfc6552320b17337c1267c35f4734497": " v = [v]_E, \\,", "bfc6f4fe9730c75643858455878ca462": "\\sum_{d|n} f(d)\\;", "bfc7728eca7e89fd53b6353d4e26bb27": "\\int\\mathbf{Y}_{lm}\\cdot \\mathbf{Y}^*_{l'm'}\\,\\mathrm{d}\\Omega = \\delta_{ll'}\\delta_{mm'}", "bfc7ae7e6090ce8650da562e3a024c01": " G(A)=\\{(x,Ax):x\\in D(A)\\} \\subseteq H\\oplus H", "bfc7c571995802ab98bb178ed48f3ab5": "S \\to VP\\ NP_{subj} \\to TV\\ NP_{subj} NP_{obj} \\to saw\\ NP_{subj}\\ NP_{obj} \\to saw\\ John\\ NP_{obj} \\to saw\\ John\\ Mary", "bfc88c8a8b060f39ec716cf9f5058dd8": "(a_0,\\ldots,a_n)\\!\\!(x,y)^n", "bfc8c4e7e207ac3a812cffd99b398a5f": "\\exp_2^0(x)=x", "bfc911df96130844ad22416c72d5031a": "i_2=\\left[\\left(1+\\frac{i_1}{n_1}\\right)^\\frac{n_1}{n_2}-1\\right]{\\times}n_2", "bfc91bf6a91779c2126efbf2136bc8b0": "z^2 + c \\to z", "bfc944692f0b52f87271e3bb447c3264": "\\theta^i[\\mathbf{f}](X_j) = \\begin{cases} 1 & \\mathrm{if}\\ i=j\\\\ 0&\\mathrm{if}\\ i\\not=j.\\end{cases}", "bfc96016b8e0b93db451d17b38a9a9c0": " \\frac{4}{\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{6}}=2.", "bfc985b08d0a9df3babb6eb41bb99d51": "A \\rightarrow B: \\{A, K_{AB}\\}_{K_{BS}}, \\{N_B\\}_{K_{AB}}", "bfc9b9ee88cf8a70530e61f106957194": " \\gamma = \\frac{E_1 + m_2 c^2}{E_\\text{cm}} ", "bfc9c0967bf821639351e0b04e376331": "(\\hat{x},\\hat{y})", "bfca02e2b2e330bf81b61b04cb71eebf": "\\frac{\\partial}{\\partial \\mathbf{x}} \\|\\mathbf{x}\\|_2 = \\frac{\\mathbf{x}}{\\|\\mathbf{x}\\|_2}.", "bfca1c994d5c5f941f9a5ddef6151a11": "\\operatorname{Fit}(G) = \\bigcap \\{ C_G(H/K) : H/K \\text{ a chief factor of } G \\}.", "bfca501e8b13b4d096cc9c5d6cdb24af": "\naA + bB \\rightarrow yY + zZ\n", "bfcaa84b6466921757ba4e619e2034a2": "x_{11}' = (1-c)\\,x_{11} + c\\,p_1 q_1", "bfcb012ba3a1d0463121a0848e2b3e77": "\\varphi_i(n) = \\frac{1}{n}", "bfcb597f4e8910874191c8fd5b9dfcc5": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon a} \\sum_{k=0}^{\\infty}\n\\left( \\frac{r}{a} \\right)^{k} P_{k}(\\cos \\theta ) \\equiv \n\\frac{1}{4\\pi\\varepsilon} \\sum_{k=0}^{\\infty} I_{k}\nr^{k} P_{k}(\\cos \\theta )\n", "bfcb8f4ccf07774b9e09d0b4a42f8477": "o_i = s_i \\oplus r_1 \\oplus r_2 \\oplus ... \\oplus r_{t-1}", "bfcbaa3e63695caabfaadeac38904e2b": "AVG=\\frac{H}{AB}", "bfcbec695d709e41f3fbcc0c6fe587b7": "\\mathbb{C}^4\\otimes \\mathbb{C}^2", "bfcbfd26506b7b42e68f9b2e510527a1": "\\alpha=3/2", "bfcc1bb224115fb3503fed0ff25a4338": "\nE^{(t-1)} ~=~ \n\\Pr[x'_{s'}=0] E^{(t)}_0\n+\n\\Pr[x'_{s'}=1] E^{(t)}_1.\n", "bfcc6d9b035ec91069a2f7cd27793370": "\\ -\\ Bx^\\prime y^\\prime\\sin ^2\\theta\\ -\\ B{y^\\prime}^2\\sin \\theta\\cos \\theta\\ +\\ C{x^\\prime}^2\\sin ^2\\theta\\ +\\ 2Cx^\\prime y^\\prime\\sin \\theta\\cos \\theta\\ +\\ C{y^\\prime}^2\\cos ^2\\theta", "bfcca4dd12200fc02835c0f31f22bafd": "(A_b)", "bfccd9905bdbc7c2686927831518212e": " e + c = 180", "bfccedd50bf988ac0dfadb4db871df1f": " \\pi(x^*) = \\pi(x)^* \\,", "bfcd0aa502a23a6e5eb8fdf7513d2ae3": "I = [a,b]", "bfcd1685cae5b03db3edc44173642377": "\nP^{\\eta}[\\eta_t(x)\\neq\\eta_t(y)]=P[\\eta(X_t)\\neq\\eta(Y_t)]\\leq P[X_t\\neq Y_t]\\rightarrow 0\\quad\\text{as}\\quad t\\to 0 \n", "bfcd79c3ecd06a87c6789a0d5fe244ff": " F_{\\text{centrifugal}} = m \\Omega^2 \\rho ", "bfcd7c15f597eca946688be374cc263f": "x^7-x^6-x^5+x^2-1", "bfcdccebb60cc9dc2c0ccdf272a0e7bb": "q_{\\mathit{right}}", "bfce518f81ebdb0fa10b46320874c2d9": "\\omega_1^{\\mathrm{CK}}", "bfce5bd4c4905119688fc87feb32e160": "X_{k-m}", "bfceac6840930a2837c409166df06dfe": " \\xi \\approx \\sqrt{1-\\alpha^3} ", "bfceb8e576ddf07e43e3a2896017920f": "\\left|SCS(X,Y)\\right| = n + m - \\left|LCS(X,Y)\\right|.", "bfcec5037b487046bd1c188eb244aaef": "m^{th}", "bfcecab95c62fb89e6daacb3042e57c2": "\\sum_{k=0}^{n-1} e^{2 \\pi i k/n} = 0 .", "bfcee2a5d538971093753be22dd8354f": "\\delta_{ij} = [i=j ].\\,", "bfcf3e533b3b18403f5745b9d4fadd32": "\\{1, 2, \\ldots, 3^{h+1}-2\\}", "bfcfcb5bd9ab47972ab617b4d86c061a": "\\rho(t) = \\frac{|\\gamma'^3(t)|}{\\sqrt{|\\gamma'^2(t)| \\; |\\gamma''^2(t)| - (\\gamma'(t) \\cdot \\gamma''(t))^2}}", "bfd00b3bf6b67c211409c42860fe015a": "\\begin{cases}\ni_1 - i_2 - i_3 & = 0 \\\\\n-R_2 i_2 + \\epsilon_1 - R_1 i_1 & = 0 \\\\\n-R_3 i_3 - \\epsilon_2 - \\epsilon_1 + R_2 i_2 & = 0 \\\\\n\\end{cases}\n", "bfd011a3f7967874378e28a3df8576c9": " \\mathbb Q (\\zeta_n) ", "bfd0290ca781ff05226027d59831012f": "R\\left( \\mathbf{p,z}\\right) =\\sum_n p_n z_n (p_n).", "bfd064614286ed17908492533e9c4d5c": "R_a,R_b", "bfd0bbaf0424269293303dd4a1df93f8": " A = LU, \\, ", "bfd0c9f6f3f437a5ed32d7926c3c15fd": "n(\\mu)", "bfd0c9fe4826e98f6f5762742b041073": "I=\\emptyset", "bfd0ef1a6a47b4f9b30453a2e28ecb44": "D = \\frac{S}{mil} ", "bfd1056acd4b97bb6e92e71447adfdda": "{d^2\\theta\\over dt^2}+{g\\over l} \\sin\\theta=0. ", "bfd1285f989ddd286472b9ed42483265": "\\langle\\mathbf{S}\\rangle", "bfd166d5370cd77c6d3e9f5f2f5ee39c": "(0=0 \\vee 0=1) \\wedge (1=0 \\vee 1=1)", "bfd1b1d08463b38e15e27b6a2acf9e46": "\\R^n \\setminus \\{0\\}", "bfd1c7224517a5530c4f4e98015da0bf": "\\textstyle a^{*}(\\theta_{k})W_{k}=\\mu a(\\theta_{k})^{*}R_{x}^{-1}a(\\theta_{k})=1", "bfd1da71abf9e955d1caffd612314547": "P_{cc}(\\theta;\\zeta)= \\frac{1}{2\\pi } \\frac{1 - |\\zeta|^2}{|e^{i\\theta} - \\zeta|^2}", "bfd1de177693e497aeaca41c78d486a6": "G = \\lbrace g : g g^* = 1 \\rbrace \\!", "bfd1f8dccef2cd5cfad985dd24c80afa": "\\left( (1-w_i \\overline{w_j}) K(z_j,z_i)\\right)_{i,j=1}^N.\\,", "bfd227652841b2d2be092286cfad6264": "N_s = \\frac { n \\sqrt Q } { (gH)^{ 3/4 } } ", "bfd24868c0f5b4bf2608f19fd78cd60a": "n'_i", "bfd2590b7ee911d60455aa873f2650e8": "\\varepsilon_{\\color{Orange}{2}\\color{Violet}{3}\\color{Orange}{2}} = -\\varepsilon_{\\color{Orange}{2}\\color{Violet}{3}\\color{Orange}{2}} = 0", "bfd259b63ced58919935dc82c6e0a063": "a_n = e^{i\\frac{2\\pi}{q}x_n} \\, ", "bfd26071069dde5c092975e45bbd680b": " \\left|\\frac{(b-a)^5}{6480} f^{(4)}(\\xi)\\right|, ", "bfd260ba6b54a4515ff312b54eaadafe": "r_k[n] = r_k(nT) \\,", "bfd27cf7579bcd43d1871b3a49d933bc": "\\frac{1 + z}{1}", "bfd281d0aa74d5f13c33d58e100bb6e5": "\\{\\Phi_i{t}\\} ", "bfd2e3df6c5895542ae51ac3f36a0b16": "g:\\mathbb{R}^d\\rightarrow\\mathbb{R}", "bfd32cffcdfbd5f68f44115fb923ff82": " \\{A_1, A_2\\} \\subseteq P \\cup L ", "bfd334c9732ebec143c2a517dc261726": "|\\nu\\rangle", "bfd34614102a9b7bbbf7b359c58c2100": "w_E", "bfd34de2ebfa15e538d679b1d7304e69": "\\scriptstyle S=0", "bfd365fe5c45db60b96eceddf88c7e62": "\\sin\\theta_3=\\sin(\\theta_3+\\theta_2)\\cos\\theta_2-\\cos(\\theta_2+\\theta_3)\\sin\\theta_2 \\, ", "bfd3d4902b036e1eb5690b52e7381f51": "I=n/2\\sqrt{\\epsilon_0/\\mu_0}|E|^2", "bfd3f5079844c540154337a629286c67": "p,q \\ne 0", "bfd40807ee11ad071d7e62bf0a55752a": "\\operatorname{Var}\\left(\\overline{X}\\right) = \\operatorname{Var}\\left(\\frac {1} {n}\\sum_{i=1}^n X_i\\right) = \\frac {1} {n^2}\\sum_{i=1}^n \\operatorname{Var}\\left(X_i\\right) = \\frac {\\sigma^2} {n}.", "bfd40ba57efa2100e91b2d4526b476b2": " \\phi = \\omega T_D \\, ", "bfd474cd07ed64e5cc47a952fa1caa39": "\\Theta^{m}(\\mu,a)", "bfd4ccab15e5e092e8836a926df65d5a": "n = \\infty", "bfd4e84248abed27270626d54001ca63": "(\\psi,\\gamma)", "bfd506be1750c371fb5581d12ed70778": "\\mathbf{\\nabla}^2\\psi-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}\\psi = \\frac{m^2c^2}{\\hbar^2}\\psi", "bfd5a76cc8bf47a0f886dca27c8238b8": "\\prod _x f(Tx)=C f(Tx)^{x-1} \\,", "bfd61c857bdd99222f65b0a449afdf7a": "\\lim_{n\\to \\infty }\\, 2^n \\underbrace{\\sqrt{2-\\sqrt{2+\\sqrt{2+\\cdots +\\sqrt{2}}}}}_n", "bfd63aa018104a49f34baf94178e40ef": " \\phi = \\rm p\\cdot\\lambda - \\rm q\\cdot\\lambda_{\\rm N} - \\rm m\\cdot\\varpi - \\rm n\\cdot\\Omega - \\rm r\\cdot\\varpi_{\\rm N} -\\rm s\\cdot\\Omega_{\\rm N}", "bfd6846bc1ab88f3e5334dbf60d0dbb4": "\\ \\displaystyle \\hat{q} \\ ", "bfd6d9fa555ddcf62282c127a46b6fc5": "b > 0\\,\\!", "bfd6dde4dcc0ab74873b8236bc40e800": "\\int_s^\\infty P(t)dt = \\sum_p \\frac{1}{p^s\\log p}", "bfd71ed8d3455c6c988a26499d95c93a": " v= \\frac{E_i}{\\omega}(1+\\frac{2}{\\gamma^{2}}) ", "bfd78338d61455e66e6bad16abf94726": "S_{n}^{(1)} =\\left[ \\Omega _{n-1},A \\right] ,\\qquad S_{n}^{(n-1)}= \\mathrm{ad} _{\\Omega _{1}}^{n-1} (A) ~,", "bfd79ee29da4210cdf20ed6ee7195799": "-\\tfrac{1}{2}{\\rm Tr}\\rho KK^*", "bfd7b7b7c79a0971a3ff5f9c792824af": "\n k1 < f(e)< k2.\n", "bfd7f8dc9768ca9f847f5a5905dfd903": "\nR_0 = \\sqrt{x^2+y^2+z^2}\n", "bfd7ffbce69c86a241bf8427a4f66c83": "{a\\pi\\over 5}\\ {b\\pi\\over 5}\\ {c\\pi\\over 5}", "bfd859fb1874d3690c53ee32d24b6109": "\n\\cfrac{dN}{dt}=N\\bigg[r(1-cN)+\\cfrac{baM}{1+aT_H M}\\bigg]\n", "bfd894188c6ccdeea6ceadc74cc8753a": "2^{222}", "bfd8a1eebf199d5fff6e0f64a6ddcff1": " w = d + [2.6m - 0.2] + 5R(y,4) + 3R(y,7) + 5R(c,4) \\mod 7. ", "bfd8cef906597efac2be2dbbc8d3d46b": " \\bold{T}(r,\\theta) = ( ra \\cos\\theta , rb \\sin\\theta ). ", "bfd8cfea742fe1f1265dcc4de83cfabd": "0.9 \\approx 0.5 \\times \\left( 1 + e^{ - 0.2 } \\right)", "bfd8e947eb13c026f46e5a2ee6ce12e7": " kT / 2", "bfd903572a8e76e429abeaf7a9aa230e": "(xf,g)=(f,xg)", "bfd92c06592c3b5cf56c182673165891": "F^{+}", "bfd9526e0787afe07c7b98afdf225a7e": "-(s_{\\beta}-s_{\\alpha}) \\mathrm{d}T + (v_{\\beta}-v_{\\alpha}) \\mathrm{d}P = 0. \\,", "bfd9d0bcafd4286d0dd8a336a64ceda6": "\\gamma(g)", "bfdba22166c65a6c7d563ad426eb4d39": " \\mathfrak g ", "bfdc7d5728c2245c6c3d3f09f16d9e93": "U(r)\\equiv r \\psi(r)", "bfdd19199d950b31c97a36c644dab1eb": "\\psi_{f}", "bfdd409eb3b237da96c3ea075c6d3b9e": "\nM = \\oplus_{i \\geq 0} ( M_i \\ominus N_i ) \\quad \\oplus \\quad \\oplus_{j \\geq 0} ( N_j \\ominus M_{j+1}) \\quad \\oplus R\n", "bfdd5d2e5563af03bc06a67c6ab8de67": "R/M \\geq RR;", "bfdd896fe07dba0f909e07ad486c98d0": "R(fg) + R(f)R(g) = R(fR(g)) + R(R(f)g)\\,", "bfddc2d88ed54416ce5c629f642e5053": "\\scriptstyle (\\hat x \\hat y)^2 \\;=\\; \\hat x \\hat y \\hat x \\hat y", "bfde9f343a30bddb173c9be3aec68046": "\n\\frac{d}{dx}\\left((1-x^2)\\ \\frac{dP_{n}^{m}}{dx}\\right)\\ +\\ \\left(n(n+1) -\\frac{m^2}{1-x^2} \\right)\\ P_{n}^{m}\\ =\\ 0\n", "bfdf1bcca0f94e5304580072087d06d4": "a_nb_m", "bfdf3e033c25f7fdd44401cf4a13e735": "\\forall n\\delta>0.", "bfe3542f5bc5eb687de7c05a2691109b": "\\mathbf{e}_i\\cdot\\mathbf{e}_j = \\delta_{ij} ", "bfe36cce9f8d7a88aa2815011418e3a2": "{\\text{Bandwidth}}_{E}", "bfe3c4bb8220fe0b082cb1cfb7155966": " P( X \\ge k ) \\le \\frac{ 1 }{ 2 } - \\frac{ k }{ 2 \\sqrt{ 3 } } \\quad \\text{if} \\quad 0 \\le k \\le \\frac{ 2 }{ \\sqrt{ 3 } },", "bfe4141ec7090330ce8ece11ebf0ac25": "m_1>m_2", "bfe4148ce99b3fc3e590a7410bec1856": "L_{f}^{2}h(x) = L_{f}L_{f}h(x) = \\frac{\\operatorname{d}(L_{f}h(x))}{\\operatorname{d}x}f(x),", "bfe463798cef5da1d5dc7ef31887de9c": "PH=\\Sigma_2.", "bfe4894ec6a402750a4d93cbbf2f91f5": "\\Theta=2\\theta_{0}-\\pi=2\\arctan b\\kappa=2\\arctan\\frac{Z_{1}Z_{2}e^{2}}{4\\pi\\epsilon_{0}mv_{0}^{2}b}.", "bfe4900c61a63a27ec01232be7eed4a6": "\\frac{dX}{dt} = 0 ~.", "bfe4c20aae685db93f068196eb99649f": "(A \\or C)", "bfe567427a6977866f1fb0990de17ae0": " \\mathbf{x}_j = 1 ", "bfe5730c8335ac370db4e9419223c79d": "\\partial^n : k^* \\times \\cdots \\times k^* \\rightarrow H^n\\left({k,\\mu_\\ell^{\\otimes n}}\\right) \\ ", "bfe5a2dbd27381c133c4de3c87a9289d": "l_a=du", "bfe5e618c7d4587ee0971b2718ef58ad": "2x+4=0", "bfe61817fce3cba636465bfc941fb6a3": "~\\gamma~", "bfe65e9ebdc53269796fef83d87586e9": "\\limsup_{n\\to\\infty}\\frac{\\log d(n) \\log\\log n}{\\log n} = \\log 2\n", "bfe7c96b3782800722b6ebf537b59ed7": "D_{3h}", "bfe86d3af80f07b92c9875a9883c2e2b": "\\frac{\\partial G(x,s)}{\\partial n} = \\widehat{n} \\cdot \\nabla_s G (x,s) = \\sum_i n_i \\frac{\\partial G(x,s)}{\\partial s_i}", "bfe8bf74d41d441caf06bb2b9f07f76b": " [R|t] = \\begin{pmatrix}\n & & & \\Delta x \\\\\n & R & & \\Delta y \\\\\n & & & \\Delta z \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}", "bfe8c7fe427ed2a60966fe891648a45e": " C = C_{noskew} - Vega_v * Skew", "bfe8e26d5fd25f338a426e9fd05506d3": "\\emptyset '", "bfe9a7fa5035598e501d0d512a1575da": "D_{xx} + D_{yy}", "bfe9c03012cd6cc50076e08ba5b4bf40": "\n[\\hat{p}_x, \\hat{p}_y] = -i\\tfrac{\\hbar q B}{4c}~.\n", "bfe9ccd21ceb0570f8fb323958b6b59e": " \\lim_{x \\to 0}\\frac{1 - \\cos x}{x} = 0 ", "bfe9dffde29b78169330a5ac6d2a7a73": "\\eta_\\mu\\mathbf{v}", "bfe9e545ad99af201eb7c795128f727f": "\\mathbf{Z}_1 \\dots \\mathbf{Z}_M", "bfea9e99993555dc37247e6ba0c4eba6": "{\\hat x} = x", "bfeabe50993d4108eb56265c8305d979": "\\frac{4}{3}\\chi_\\mathrm{w}^{3/4} = 2^{3/4} \\mathfrak{M}^{1/2} d", "bfeac5de723ab051d815269d8ab425d7": "\\mathbf r,", "bfeaee320ece32d63a678dd580fd8fdd": "{\\Theta_o}", "bfeb0450f00c201cd8d0d4f61d71d9dd": "\\hbar \\omega_{q}", "bfeb48575baba93bad67ecf420cc9fa6": "c_1\\frac x{\\log x}1", "bffc1217155a9a072fa88adbe34f6757": "\\lambda_1 = \\lambda,~ \\lambda_2=\\lambda_3", "bffc5379aabad79cf9952faa4a49da7a": "\\!V = \\int_{-r}^{r} \\pi y^2 dx.", "bffc6b131d9e6d53a837a49a64329831": "\\tilde\\lambda_\\star", "bffc79f27c49657aaee11d699db9ed9b": "\\tan A = \\frac {\\textrm{opposite side}}{\\textrm{adjacent side}} = \\frac {a}{b} =\\frac {\\sin A}{\\cos A}\\,.", "bffcb7d0b68d3f49e89d6e5af9c73a03": "\n\\textstyle\\left(10^{189} + 10^9 + 1\\right)^{20}\n", "bffccb29c148f516464bc66ff403b3d5": "\\displaystyle y_{k+1} = x_1 y_k + y_1 x_k.", "bffd6af0ad5482836539ea6c0891c4a0": "\\lambda \\mathbf{u}", "bffd8fa5239e12c4bc23dd643b4b5654": "H_C(0) = \\frac{K}{M_S}", "bffd9b14a66e3b6ab5316f9bbc70648a": "\\Gamma(E)\\ \\stackrel{D}{\\to}\\ \\Gamma(E\\otimes\\Omega^1M)\\ \\stackrel{D}{\\to}\\ \\Gamma(E\\otimes\\Omega^2M)\\ \\stackrel{D}{\\to}\\ \\dots\\ \\stackrel{D}{\\to}\\ \\Gamma(E\\otimes\\Omega^n(M))", "bffdc87579d2a28d40a4c5c279ef82a8": "(5 - 2\\sqrt{2})\\pi/24", "bffdcde29e454929baf1e5656e3af7fe": " G_\\text{II} = b \\sqrt{ig} ", "bffddf44f09a8bc4c85810a1de4973c6": "T = \\int _{(r_{1},\\theta_{1})}^{(r_{2},\\theta_{2})}\\frac{n(r)}{c} \\sqrt{(r d \\theta)^{2} + d r^{2}} = \\frac{1}{c} \\int _{\\theta_{1}}^{\\theta_{2}}n(r) \\sqrt{r^{2} + \\left( \\frac{d r}{d \\theta}\\right)^{2}} d \\theta", "bffde71e5840cf494977774985f0882a": "Z_\\mathrm{R}\\,", "bffe1ee6e7aba19ff6c6258654b9bb6e": " \\mu=m ", "bffe464a4aad8a4e7f73c88983ada8ce": "\\beta=e^{2\\pi\\lambda}", "bffe4e36901d2df5578d000d9cffcf7d": "\\sqrt{!~}~", "bffe751e3ba4dbe5135f1324960b9a75": "(l,j_l)", "bfff260fbb9112a16fb83aff09861257": "\\scriptstyle R_{ws}[m]", "bfff27113b64bad828fe9b05ed2b88e1": "z=1.450795 + 0.7825835i", "bfff4cddfe7e0aa100f05a7e1804b520": "V = x^5 + ax^3 + bx^2 + cx \\, ", "bfff52c86c7a242cc4964b14abd57bbe": "\\scriptstyle \\frac{1}{2}cT", "bfffd4c4aadc55d6c82fa5acafa31491": "Q_s/Q_t", "bffff256a6f0a5a74cd798014e8f7eae": "\\frac {dy}{dx} = f(x)g(y)", "c000236bcd0ccf24561ed03297e6a4b7": " \\neg A", "c0007193ac11d1a4b7dbbc1e353e252a": "a_{0,i}", "c000a39b670b000a36f6603df4a6ab69": "\\lambda=\\frac{h}{p}", "c000b6ff1c879fb256b50cd07062c683": "N_\\textrm{1}", "c0011e3ea5981eff3e7bde9a1a299b51": "\\alpha_\\text{standardized} = {K\\bar r \\over (1 + (K-1)\\bar r)}", "c00179db3c9fc398ad3080699ea915c1": "g^\\prime", "c001d719bafe07d8cd5dfef9d1f8ce30": "n=(b+c)/2", "c0022e73a18b28c222761e6e07548629": "\\mathbf{T}^{(\\mathbf{e}_3)}= T_1^{(\\mathbf{e}_3)}\\mathbf{e}_1 + T_2^{(\\mathbf{e}_3)} \\mathbf{e}_2 + T_3^{(\\mathbf{e}_3)} \\mathbf{e}_3=\\sigma_{31} \\mathbf{e}_1 + \\sigma_{32} \\mathbf{e}_2 + \\sigma_{33} \\mathbf{e}_3,", "c0029330e6a0d4928c1c24f6e9d85523": "h(t) = \\mathrm{sinc}\\left(\\frac{t}{T}\\right)\\frac{\\cos\\left(\\frac{\\pi\\beta t}{T}\\right)}{1 - \\frac{4\\beta^2 t^2}{T^2}}", "c0029c9cb660474f8364caaea0eb8459": "\\tau(I)=\\Sigma^{-1}R_nC_n\\Sigma(I)=\\Sigma^{-1}C_nR_n\\Sigma(I)", "c002c71c8c2a3cf1047ccb30eae1fc4a": "\\log \\gamma_{\\pm}= \\frac{-A_\\gamma|z_+z_-|I^{1/2}}{1 + \\rho I^{1/2}}+\\frac{(0.06+0.6B|z_+z_-|)I}{\\left( 1+\\frac{1.5}{|z_+z_-|}I \\right)^2} +BI\n", "c002de2ad2373ffb187ad07997b34340": "\\scriptstyle (R_k)_{k \\in K} ", "c0032d4350ae0a781cfbe49f2a6bd473": "R_0(f)", "c0033ef384fe9b0aa6e278f13b1bd5e6": "\\alpha_{H\\beta}^\\text{eff}", "c003fb8014cb03db3d2da451758ad77d": " I_1, \\dots, I_n ", "c004818de00f2759750bee69d3fb61ec": "P(D|H)", "c004acfe7f298067c95de8be96f4216a": "\n\\vec S \\ =\n\\begin{pmatrix} S_0 \\\\ S_1 \\\\ S_2 \\\\ S_3\\end{pmatrix}\n=\n\\begin{pmatrix} I \\\\ Q \\\\ U \\\\ V\\end{pmatrix}\n", "c0052d82dda86d03cfb39c9aa40b7bc7": "\n\\left( X_{\\mathrm{sol}}| X_{\\mathrm{sol}} \\right) = \\left( X_{\\mathrm{sol}}| X_{1} \\right) = \\left( X_{\\mathrm{sol}}| X_{2} \\right) = \\left( X_{\\mathrm{sol}}| X_{3} \\right) = 0\n", "c0053dda5c53ebefca6eead2228ad1fd": "\\succ_P", "c0054fb90fe4baaca1cd8a2303572df8": "-\\frac{\\hbar^2}{2m}\\nabla^2 \\Psi(\\bold{r},t) = E \\Psi(\\bold{r},t) ", "c0056ae7cd6435d56b99be30d9dd668b": "\n\\begin{align}\nRR_{cost} &=\\left[ \\textrm{Call}(K_c,\\sigma(K_c))-\\textrm{Put}(K_p,\\sigma(K_p)) \\right] - \\left[ \\textrm{Call}(K_c,\\sigma_0)-\\textrm{Put}(K_p,\\sigma_0) \\right] \n\\\\\nBF_{cost} &= \\frac12 \\left[ \n\\textrm{Call}(K_c,\\sigma(K_c))+\\textrm{Put}(K_p,\\sigma(K_p)) \\right] - \\frac12 \\left[ \\textrm{Call}(K_c,\\sigma_0)+\\textrm{Put}(K_p,\\sigma_0) \\right]\n\\end{align}\n", "c00625466e7a45d90b8bf576465e6ae2": "1(1-\\varepsilon) + 2\\varepsilon = 1+\\varepsilon", "c0064d76a49ffe2508104e1769cd20c6": "A = \\ell w\\,", "c0069df7cbdfc7dae103c46ed9657b36": "j_{\\alpha} = q n_{\\alpha}(x) v_{\\alpha}(x)", "c006c8dd91b1e04a61eb631df2749bce": " h : B \\rightarrow A ", "c0071e854ed3f3aec98958cdd097cdaa": "\n\\begin{align}\n x_i \\ &\\sim\\ \\mathcal{N}(m_k,\\sigma_k^2 C_k) \n \\\\&\\sim\\ m_k + \\sigma_k\\times\\mathcal{N}(0,C_k) \n\\end{align}\n ", "c00725ac45d253d20e714a4d10e45819": "(7.a)\\quad \\nabla^2 \\psi =\\,(\\nabla\\psi)^2 +\\gamma_{,\\,\\rho\\rho}+\\gamma_{,\\,zz}", "c007ac8b6de34ca1a53244084013b896": "t\\in{\\mathbb N}", "c0080873beebe912dec1c8980870371a": "\\Delta{x}=x_M-x_U", "c00881f575a1cee9749e1de07e9afbc6": "D_h(f(x)) = \\frac{d_h(f(x))}{d_h(x)} = \\frac{f(x + h) - f(x)}{h}", "c008e71394e4a37ebf9ef748c8ed14e2": "{d\\over dt} Q = {d\\over dt} \\int_x J^0(x) =0 ~.\n", "c0091fb761d0f34329a047cc62d32a19": "I M_n = M_{n+1}", "c0094fcdae7f338ed347fe13dcc8d34b": "\na^2 = \\frac{q_{xx}+q_{yy} + \\sqrt{(q_{xx}-q_{yy})^2 + 4q_{xy}^2}}{2}\n", "c009da0e0a499c5b97d4c084e0d6f1e1": " \\tau \\subset \\sigma", "c00a09ce3371bf46a59776f8f1c47b6a": "V_{II_{1,1}}", "c00a0ec2c522933be8c74faa44597038": "\\frac{\\mbox{Long-term Debt + Value of Leases}}{\\mbox{Average Shareholders Equity}}", "c00a16d5109ed88b3648da62f50c919e": "0=f(x)=-1 + 10 x + 6 x^2 + x^3", "c00a1890df2e8f07dea257e68fc4e0a8": "Q_{t_0}", "c00a1c28d7a82b3b4810f5e2cc09148c": " q(e^{ix}) \\triangleq p(x). \\, ", "c00a1cf0fa2b110a73c2b80213df51e6": "\\varphi\\in\\mathcal{F}(\\pi)", "c00a2b3115c2823f89f42f4d1d03537b": "f:S\\to \\Gamma", "c00a38db687b63239901c1fd4c53f8c6": " \\int_{|x|>1}\\frac{1}{|x|}\\,d\\mu(x)<\\infty", "c00a6fb0081b70de923ab5b69b879da7": "D \\frac{\\mathrm{D} u_y}{\\mathrm{D} t} = -D\\frac{\\partial p}{\\partial y} + \\nabla^2 u_y", "c00a790aba6b2c8c0fa6fb39904f9931": " X = \\sum x_{ ij } ", "c00ac494a7abcec0eef45b80d58ec786": "\\Psi(x_1,x_2)=U(x_1)V(x_2)", "c00ad941377875601355d1dad3ea60c2": "d \\in B^{64}", "c00b2bc89595c2f795894dd7fd394f6b": "\\begin{align} \\delta^{\\mu_1 \\dots \\mu_p}_{\\nu_1 \\dots \\nu_p}\n & = \\sum_{k=1}^p (-1)^{p+k} \\delta^{\\mu_p}_{\\nu_k} \\delta^{\\mu_1 \\dots \\mu_{k} \\dots \\check\\mu_p}_{\\nu_1 \\dots \\check\\nu_k \\dots \\nu_{p}} \\\\\n & = \\delta^{\\mu_p}_{\\nu_p} \\delta^{\\mu_1 \\dots \\mu_{p-1}}_{\\nu_1 \\dots \\nu_{p-1}} - \\sum_{k=1}^{p-1} \\delta^{\\mu_p}_{\\nu_k} \\delta^{\\mu_1 \\dots \\mu_{k-1} \\; \\mu_k \\; \\mu_{k+1} \\dots \\mu_{p-1}}_{\\nu_1 \\dots \\;\\nu_{k-1} \\; \\nu_p \\; \\nu_{k+1}\\; \\dots \\nu_{p-1}},\\end{align}", "c00ba6535d82ec04f67b6690df2ce334": " \\frac{dp_2}{dt} = m_2 p_2 (1 - p_1 - p_2) - e p_2 - m p_1 p_2 ", "c00ba93b574a270d9bd82c82a8c9f833": "K_c = 2 \\gamma", "c00bac3c02fc9eddb8e6de4aed994fbb": " \\lim_{t\\rightarrow \\infty} P_t(S\\rightarrow S' | E) = P_t(S\\rightarrow S')", "c00bc41b385fad65449372e0c3f73509": " x_{n+1} = f(x_n) ", "c00c0a7003c36d1edebc1882067855ae": "q(n_2) = c", "c00c1b302f7925a25c890633f6da5e89": "\\scriptstyle\\widehat\\varepsilon", "c00c36b158aa6e457e08415b8333dc0d": " \\text{jiva}= s - \\Big[ s\\cdot \\frac{s^2}{(2^2+2)r^2} - \\Big[ s\\cdot \\frac{s^2}{(2^2+2)r^2}\\cdot \\frac{s^2}{(4^2+4)r^2} -\\Big[ s\\cdot \\frac{s^2}{(2^2+2)r^2}\\cdot \\frac{s^2}{(4^2+4)r^2}\\cdot \\frac{s^2}{(6^2+6)r^2}-\\cdots\\Big]\\Big]\\Big] ", "c00c89c82e3c73d8e2d26cf157b637bf": "\\Sigma_{g,k},", "c00c9ff3054ba8cb789ea46c29482550": "I_r \\, ", "c00d9be5b29d2dce99a57fedf38fec81": "J=2.46\\times10^{-22}", "c00db8cd6957ef56d469feade70c573e": "E^2-p^2=m^2", "c00dd66c3b88f73956a984d192d9ea11": " \\mathrm{Ca} = \\frac{v^2}{a^2} = \\mathrm{M}^2", "c00df26b710d9c7901a623fa867ce6a0": "D=2", "c00df3e1efbf1909657f239dbdc96cf7": "-L \\mathbf{u} = \\mathbf{b}", "c00df549d7944dd014101be55050f251": "\\Gamma_{e2} \\left (\\phi_3 - \\phi_2 \\right )", "c00e3fbf323652916b0d45a5de28dd16": " \\beta_1 ", "c00e6cf026a88fa6814785cc097bac03": "\\{x \\mid \\phi\\}", "c00ea6ae72dc358a811d28833a41602c": " \\operatorname{E} \\langle X_1,\\dots,X_k\\rangle = 0\\mbox{ for }k \\ge 1. \\,", "c00eeb3bb466f759caad8dc761b25b20": "\\begin{pmatrix} 6 & 24 & 1 \\\\ 13 & 16 & 10 \\\\ 20 & 17 & 15 \\end{pmatrix}^{-1} \\equiv \\begin{pmatrix} 8 & 5 & 10 \\\\ 21 & 8 & 21 \\\\ 21 & 12 & 8 \\end{pmatrix} \\pmod{26}", "c00f148d450bac1b00c8aa8c8385eb91": "\\begin{align}\n \\frac{\\partial I_1}{\\partial \\boldsymbol{A}} &= \\boldsymbol{\\mathit{1}} \\\\\n\\frac{\\partial I_2}{\\partial \\boldsymbol{A}} & = I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T\\\\\n\\frac{\\partial I_3}{\\partial \\boldsymbol{A}} & = I_2~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T~(I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{A}^T) = (\\boldsymbol{A}^2 -I_1~\\boldsymbol{A} + I_2~\\boldsymbol{\\mathit{1}})^T \n \\end{align}", "c00f38aced5454fe880b5b08ef3fb060": "R_{\\alpha}^{0}=\\frac{1}{2} \\left (\\varkappa_{\\alpha;\\beta}^{\\beta}- \\varkappa_{\\beta;\\alpha}^{\\beta}\\right )=0,", "c00f4294787a8ba71c00389d563e1316": "\\,\n\\begin{align}\n\\frac{((n+1)-1)^x((n+1)-1)!}{(x+(n+1)-1)(x+(n+1)-2)\\cdots(x+1)x}&\\leq \\Gamma(x)\\leq\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\left(\\frac{n+x}{n}\\right)\\\\\n\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}&\\leq \\Gamma(x)\\leq\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\left(\\frac{n+x}{n}\\right)\\\\\n\\end{align}\n\\,", "c00f7f63b4b7134455740a019f60077a": "\\operatorname{det}(A) = \\prod_i \\lambda_i", "c00f9b44e771bdcf378dca3f0ac3de62": "{|n\\rangle}", "c00fb4d2d3f6fb49c1c296aad5215b32": "dt_{c}", "c00fb79fd80309e30cfa0c0651650d2f": "\\frac{\\mathrm{d} \\Phi(\\mathrm{d}A,\\theta,\\mathrm{d}\\Omega,\\mathrm{d}\\nu)}{\\mathrm{d}{\\Omega}}= B_\\nu (T)\\,\\mathrm{d}A\\,\\mathrm{d}\\nu\\,\\cos\\theta.", "c00fbda39eeb5938445d7fc186e0b9c4": "{}^q\\!D_{\\gamma}={1 \\over{\\sqrt[q-1]{{\\sum_{i=1}^S p_i p_i^{q-1}}}}}", "c00fd54471f38ad4d05496c09abd1b39": "E(R_i) = R_f + \\beta_{im}[E(R_m) - R_f].\\,", "c00fe29ec162130e4a6d08d204f416ab": "\n\\frac{\\mu(n)n^s}{\\phi_s(n)\\zeta(s)}=\n\\sum_{\\nu=1}^\\infty \\frac{\\mu(n\\nu)}{\\nu^s}\n", "c0103e12efb99b385582d8c0ac3ba3e1": " a_n(t) = \\langle n | \\psi(t) \\rangle. ", "c0104b3609d6aacb3bfa62d785d554ad": " (-\\frac{\\hbar^2}{2m}\\nabla^2+V+2gn-\\mu-\\hbar\\omega)u-gnv = 0 ", "c01061b0934d2bae67cdd280b115ef62": "^k M_0 \\in MIP", "c010985b02a15f6f3f48ca4bae4dc459": "\\mathbf{e}_i={\\partial\\over\\partial x^i}", "c0110b8f398dfd2826b53f673d6e2f9e": "1.01/1.04 - 1 \\approx -2.88%,", "c0111c7355d207f3c8a94a41b6f442c6": "\\scriptstyle V ", "c01151a4ade606322cc462554af72b63": "i_s = i_1\\sin(\\Delta\\varphi^*).", "c011a2fa6d262198e0a38d12bc77a31e": "\\mathcal D", "c011bbd9d24194d046a22997feb850fc": "m\\mathbf{v} = \\mathbf{P} - e\\mathbf{A} ", "c01288007386797bd3a1941e0f590176": "W_t = W_t-W_0 \\sim N(0,t).", "c012e61f7bc76a5b37e5638e464be356": "\\alpha\\in\\Omega^1(TM)", "c014518534295aca46a59ab61e790d11": "f(x)\\partial_x", "c0148c8bd43474ba45bc36cc200a8efa": "3 x^2+210x -20325 ", "c014b1fba31e124131fd1fb52824ecdb": "n = 10000\\, ", "c014e716526d12b9e30559ab94308c57": "f(x;\\alpha)=\\frac{1}{B(\\alpha,\\alpha)}\\frac{\\exp(-\\alpha x)}{(1+\\exp(-x))^{2\\alpha}}, \\quad \\alpha > 0 .", "c0151b68182b80afa1bae003bf167cc0": "n>d+2", "c0152fec5b96960bf62544cd10e86487": "\n \\mathit{AIC}=24989.74.\n", "c0153e7c09b10c8893d1dfd8453b6849": " \\ \\Delta x /c = \\pi /4 ((A_1-A_2)/C_L)", "c0154fdf4ff7486eacae32606bf1b0d9": "\\mathrm{SQNR} = \\frac{P_{signal}}{P_{noise}} = \\frac{E[x^2]}{E[\\tilde{x}^2]}", "c0155bf09d6245eae676bf0bcfa1f6d0": "S_z = 2\\pi r_z l_z ", "c0159a963250e2ffb9414229773f6d90": "(n - 1)(n - 2)/2.\\ ", "c01611f454031fb0c29e972ccda0d181": "w \\in L \\Rightarrow \\Pr[V \\leftrightarrow P\\text{ accepts }w] \\ge \\tfrac{2}{3}", "c0162ba199a9512694844a1c01aff9ad": "f_R(x) = \\int_{E_R}\\hat{f}(\\xi) e^{2\\pi ix\\cdot\\xi}\\, d\\xi, \\quad x \\in \\mathbf{R}^n.", "c0164102969a2fb367453c4ce6ae049d": "X\\in TS", "c0164f2c5ab1bdd426dff9d5567c0358": "X(u,v) = (x(u,v), y(u,v), z(u,v)) ", "c01673bd6b4913892e3bd98ab0066ad6": "X^a \\sim \\operatorname{Log-\\mathcal{N}}(a\\mu,\\ a^2 \\sigma^2).", "c01689525972987d7eb04575d31927a8": "\\langle e,t\\rangle", "c0175e624d2a22022cba5206fd8c6144": "X_i= \\beta_{ik}x_k", "c017668e1e4bb4ced510a0b37f099807": " \\Delta \\omega = \\frac {1}{C R_\\mathrm L} ", "c01771f831e8b48435acd48aa9dc1e11": "m = \\frac{5\\beta_{2}-9}{2(3-\\beta_{2})}", "c017e607bf7da4c1753d2d532743d538": " \\mathcal{\\dot{H}}_1 = \\{\\mathcal{H}_1,\\mathcal{H\\}=}\\lambda _1 \\{\\mathcal{H}_1, \\mathcal{H}_1\\mathcal{\\}+}\\{\\mathcal{H}_1,\\lambda_1\\mathcal{\\}\n\\mathcal{H}}_2\\mathcal{+\\lambda}_2\\{\\mathcal{H}_2,\\mathcal{H}_1\n\\mathcal{\\}+}\\{\\mathcal{\\lambda }_2,\\mathcal{H}_1\\mathcal{\\}H}_2.", "c017fcea7a84d2102adfee5775f4c96b": "\\!\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = \\delta_{mn} k_{B} T,\n", "c0185e7e0e51f8b2ad885edcb16725a7": "f_Z(z) = \\int^{\\infty}_{-\\infty} f_X \\left( x \\right) f_Y \\left( z/x \\right) \\frac{1}{|x|}\\, dx, ", "c01862c4a98422ccc16876579dcd6c46": "10^k \\equiv 1^k \\equiv 1\\pmod{9},", "c01901c919a6171f0bee7803770fc913": "f(x,y)=xy", "c0192a4afe52ecf49725ea59c42cacff": "g:\\omega\\to\\omega", "c0194184be7b82a0ce8d8c01378eee22": "H_2(\\mathbb{C}^+)", "c019866dc59e220481fb6824288286dd": "x^2 - 9x - 27\\,", "c01991a339c232978d9db71fa62b29b7": "\\leq \\frac{d-i}{|S|}+\\frac{i}{|S|}=\\frac{d}{|S|}.", "c01a0525709c2284982fa933a5dab6e6": "\\nabla'_X\\xi=\\top (\\nabla'_X\\xi) + \\bot(\\nabla'_X\\xi) = -A_\\xi(X) + D_X(\\xi).", "c01a2a71d9a4cdae4fc59c32d2c58d8b": "x_{0}, x_{-1},\\dots, x_{-k}", "c01a9d8d22c30ed10f2f2ab6938e2ba0": "\\mathbf x = \\begin{bmatrix} x_1, x_2, \\dots, x_m \\end{bmatrix}.", "c01a9fd45420d2bf626286d47f3e8378": "\\scriptstyle D_F(1\\rightarrow 8)= 4(2)-1+1-3=5", "c01af6da212330e429091a53c14cc5ac": "r \\rightarrow 0", "c01b3008882b32b3a9799ddcb6ff6bbb": " \\frac{1}{b-a} \\int_{a}^{b} f(x) \\, dx \\approx \\alpha f(a) + \\beta f\\left(\\frac{a+b}{2}\\right) + \\gamma f(b).", "c01b4990efd42660a65bdd5e52dcf511": "k=0,1,2,\\ldots", "c01b5393e280a0867595be394ca2573c": "E_{\\text{g}}", "c01b69a0994698089d4eb2c41f4617e8": "x^x(1+\\ln x)\\,", "c01b8793bbdff11346b8b0921fece40b": "m=\\frac{\\langle Ty, y \\rangle}{\\langle y, y \\rangle}", "c01b8a77ba5efed18eca8892f0d6093f": "\\sigma = \\pi/\\sqrt{3}", "c01bc56267b1ea4f3740709e7bb0f14f": " a_0 = 1, a_1 = 0 ", "c01cc6a40dafad543487d0e7aff674e5": "f'(x)-(f(x)-f(a))/(x-a)=f'(x)-(f(a)-f(x))/(a-x)\\quad", "c01cddd18a15761df504a2d56d71e77b": "x_\\mathrm{min} = 1.461632144968362341262\\ldots\\,", "c01d329c19bf49ce88d62ffb8d405e31": "\\bar A=S\\setminus A", "c01d7572b450ceda82b7466bb754c2c3": "\n\\displaystyle\n\\text{Performance rating} = \\frac{1000 + 400 \\times (1)}{1} = 1400", "c01d8fb185e272eae20de8a0292dfbf4": "(\\exists x \\ \\neg \\phi(x)) \\to \\neg (\\forall x \\ \\phi(x))", "c01dab36d71fe2e34fe3009204a48b77": "\\displaystyle X^{\\alpha+1}=(X^\\alpha)'", "c01dafaa72b45562e4c06868cc12f67e": "-y(\\gamma - \\delta x) = 0\\,", "c01dd0e07e0c6e642db5dc6d4ff61ff6": "\\widehat{V}_a = {U(R)}^\\dagger \\widehat{V}_a U(R) = \\sum_b R_{ab} \\widehat{V}_b ", "c01e12ed2db12fd203cdaa79aad308ba": " \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\left(\\rho\\mathbf{v}\\right)=0.", "c01e762f5240da6f98fcff9e3cbd8b4d": "I_2 = K{I_1}", "c01e9555dee73f431fffc20c7ee0db33": "\\mathbf{x}(k+1) = A \\mathbf{x}(k) + B \\mathbf{u}(k)", "c01e95f97ae6aa51ccbcd26f9be18208": " V=(v_1,..., v_n) ", "c01ec027dc12e16ae52400dd69027f06": "\\varphi \\vdash \\psi\\,\\!", "c01ec812581580aaad72dcd9ac205fe4": "C(k)", "c01ed03343b2cd7cf6fe02816c8ed9e0": "m(t) = \\mathbb{E}[N(t)]", "c01ed1d571f6b784f1695e23f78207e9": " Z_N(K,L) = 2 e^{N(K+L)} \\sum_{ P \\subset \\Lambda_D} e^{-2Lr-2Ks} ", "c01ed516e111552e4954c8ea4ac7dcc9": "\n\\sqrt{\\bar {v_{n}^2}} = \\sqrt{4 \\cdot 1.38 \\cdot 10^{-23}~\\mathrm{J}/\\mathrm{K} \\cdot 300~\\mathrm{K} \\cdot 1~\\mathrm{k}\\Omega} = 4.07 ~\\mathrm{nV}/\\sqrt{\\mathrm{Hz}}.", "c01edb2563ed4164bc10ae5d016921b3": "U_{21}", "c01f07ce375e3e814fd9c2451b5d69ab": "\\Omega_{k,m}=\\begin{pmatrix}\nz/r & (x-iy)/r\\\\\n(x+iy)/r & -z/r\n\\end{pmatrix}\\Omega_{-k,m}", "c01fa6e38b40205e29316f2c3d3b14f5": "\n\\begin{bmatrix}\ndu\\\\dv\n\\end{bmatrix}\n=\\begin{bmatrix}\n \\frac{\\partial u}{\\partial u'} & \\frac{\\partial u}{\\partial v'}\\\\\n\\frac{\\partial v}{\\partial u'} & \\frac{\\partial v}{\\partial v'}\n\\end{bmatrix}\n\\begin{bmatrix}\ndu'\\\\dv'\n\\end{bmatrix}\n", "c020006fa70a6c947ff4f00eac4afa15": " c_n = \\langle \\phi_n | \\psi \\rangle ", "c020bc03dcfac336faddb5a99192a2f7": "h^n", "c021499aa305c3be954027ef0f410ad3": "I - Q", "c021a16726e0a3a4461789ce7b02384a": "\\textstyle\\cosh x = \\sum_{n=0}^\\infty\\frac{x^{2n}}{(2n)!}", "c021cb1927371f734681f5fe991ce535": "\\phi(xv) = x\\phi(v) \\forall x \\in \\mathfrak{g}, v \\in V", "c0222e5b68a1620c8214b6b9c99f559b": "\\tbinom n k = 0", "c0224bbe33c99c2d37086111134c5fa0": "m_r = E/c^2", "c0229ff1fefa85e60737d8c83d5cc464": "L_y/T", "c022a373bfc67d0c5c2c0abdda651f0d": " \\tilde{\\mathbf{n}} ", "c02369be40ddeb6f9c03645d014df1eb": "\\mathcal {Z}:=\\{g\\cup\\{\\infty\\} \\mid g \\text{ line of } {\\mathfrak A}(\\R)\\}", "c0236bc0681daccc80fe6ad7949dda68": "\\lim_{p\\to 1} S_p(x,y)", "c023b8260d821f8d6a53876180d16289": "Z_0 = R_0\\!\\,", "c023dab519ce48264a3827bd8365fb65": "n_a=-dv", "c0241ec0ec052e3a4a7a9b2b82e7eb6f": "M \\times \\{t\\}", "c02422ab1fd979e079cb07b3fda99923": "(\\forall x \\in V_{\\alpha}.\\psi(j^{N-i}(x)))", "c0247d12f2f09226dbc5066f0d319b29": "t^{-1/2}", "c02493517b69b32ba9cceaa816e6a1b4": "A_{22} \\in \\mathbb{R}^{(n-1) \\times (n-1)}", "c024ba91e691fa56eb77dfddda64e0bd": "R_1 = 1", "c025108d89194829f976647548f48b6b": "\\langle (T - \\lambda)x, \\varphi \\rangle = \\langle x, (T^* - {\\lambda}) \\varphi \\rangle = 0.", "c02551aa529bb7c275b6c082208f679d": "A\\in f(B)", "c0257fd454b63c33b50e71aaf924ba30": "dS_n", "c0258993d8cdd70a3c20aad3074b034e": "C(1,u) = C(u,1) = u ", "c025911260d7f2b8b0acae3810c1a96f": "g(x;\\alpha,\\beta) = \\frac{\\beta^{\\alpha} x^{\\alpha-1} e^{-\\beta x}}{\\Gamma(\\alpha)} \\quad \\text{ for } x \\geq 0 \\text{ and } \\alpha, \\beta > 0", "c0259942296c8d47412969d201ff75ce": "u_4 = \\tfrac{(x_1^2+x_2^2+x_3^2+ax_4^2+x_5^2+x_6^2+x_7^2+x_8^2)x_{12} - 2x_4(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +bx_4 x_{12} +x_5 x_{13} +x_6 x_{14} +x_7 x_{15} +x_8 x_{16})}{c}", "c025a9c7d970e5628ec4fb6de8f224b9": "T = t g", "c025eb43c439e0437b1d9289c0561c2e": "\\tilde d_i = d_i \\tilde b_{i - 1} - \\tilde d_{i - 1} a_i.\\,", "c026229dd05be1df1b1b60fa82817d86": "\\frac{1}{2d}> \\frac{1}{N^{ \\frac{1}{4}}}", "c0263ec05921f78e4b85c07324d3f1a5": "\\frac {x}{\\ln x - (1-\\varepsilon)} < \\pi(x) < \\frac {x}{\\ln x - (1+\\varepsilon)}.", "c02682ea9fc22f59524e9eaac9a334cf": "P_e = 2 \\left( 1 - \\frac{1}{L} \\right) P^+", "c0269c08f9fb263acb8e4eb1e43d4480": "\\epsilon'=\\delta x/a", "c0269c636196c2820b162f6185abe0f0": " \\nabla\\cdot(\\nabla\\times\\mathbf{A})=0 ", "c026a1df706414591996530f53b3f8dc": "\\sin 30 \\times \\cos 30", "c026aa7a5207c67d5cce349ed72494da": "\nA=s_1^3+s_2^3=2E_1^3-9E_1E_2+27E_3\\,.\n", "c026bf0dcca47d66c8c713a2a3d71d53": " T(u+v)=T(u)+T(v), \\quad T(av)=aT(v) ", "c026e7a1d8c11f7fa7473f21bdef8da4": " \\gamma_n = \\lim_{m \\rightarrow \\infty}\n{\\left(\\left(\\sum_{k = 1}^m \\frac{(\\ln k)^n}{k}\\right) - \\frac{(\\ln m)^{n+1}}{n+1}\\right)}. ", "c0278947a88cc0b215b1ecd861f83758": "Adv(A) = |\\Pr[A(F)=1] - \\Pr[A(G)=1]|", "c0279dc4fcf75283342a4b1368ab24c1": "y=z/(1-z)", "c027db671294184032695db6735fdff9": "\\forall x\\,(N(x)\\to P(x))", "c027f656cc16dfe0186d252688c6e84d": "\\vec u = (u,v,w)^t", "c0281655c4162996a181bdd628c7cd39": "T_{C}", "c02875524891ea79b9b722ceadfbf992": " \\mu\\to\\infty ", "c0289f99ab111089820102c50af17a91": "\\lim_{n\\to\\infty} \\Vert x_n - x\\Vert = 0 ", "c028f2caafdc9ca7c721c7af691bf97e": " \\Delta g_{ji} ", "c0294377b915e69cc004e5214f1af36b": " (x-x_1)^2\\cdot (x-x_2)=x^3-(2\\cdot x_1+x_2)\\cdot x^2+g(x)= x^3-(l^2+a_1\\cdot l)\\cdot x^2+g'(x) ", "c02988b49d1b9bbb0b6e6f2d77441d76": "(P,C)", "c0299f3f9d9f5198e4f98b0a1ec87a07": "q \\in C_{i} ", "c029a1fdac61f7bf764ee9f880e62fde": "107 = 53", "c029e1a92ef3839b20546f31492c1927": "L = \\frac{1 - \\alpha - \\beta}{\\alpha + \\beta + \\gamma}", "c029ebb313fa25cbfabcf849ac72e760": "\\beta_k := \\frac{\\mathbf{z}_{k+1}^\\mathrm{T} \\left(\\mathbf{r}_{k+1}-\\mathbf{r}_{k}\\right)}{\\mathbf{z}_k^\\mathrm{T} \\mathbf{r}_k}", "c02a3660dad83364336ede1261aaa909": " \\underset{\\delta} {\\operatorname{arg\\,min}} \\ \\mathbb{E}_{\\theta \\in \\Theta} [R(\\theta,\\delta)] = \\underset{\\delta} {\\operatorname{arg\\,min}} \\ \\int_{\\theta \\in \\Theta} R(\\theta,\\delta) \\, p(\\theta) \\,d\\theta. ", "c02ac8c71cfa058b7eb7f3d2135f920c": "\\operatorname{Log}{\\left({z_1}^{z_2}\\right)} = z_2 \\operatorname{Log}(z_1) \\pmod {2 \\pi i}", "c02ae81393c6b1d7735e4a8640bcb56b": "\\frac{1}{r} = \\frac{1}{h_a} + \\frac{1}{h_b} + \\frac{1}{h_c}", "c02aec3bf9b33df37304e1fc81f4d4ea": "(x\\le z \\and z\\le y) \\rightarrow x\\le y.", "c02b67c6a58668bf3bc6213fb4a8a187": "z^\\prime", "c02bd3688e7bcb9c75c35c37b6d37d7e": "(P \\lor Q) \\land (P \\lor R)", "c02c4a71b77353b6618d5fb75c880ad7": "o_i", "c02c4b4ac341785f8f87c5b889af0cb2": " c = f \\lambda.", "c02c9974fe559d843e9838081a7b52c5": "x_{1}^{m}", "c02c9b5864dd2a978967401f21e3f89a": "v = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}", "c02d0d05f34786f1284038e2c410bff7": "M_x(y) \\equiv xy \\pmod{12}", "c02df3028af56d74fb301fa739eb7d42": "\\dot{g}\\ =\\ \\frac{\\partial g }{\\partial x_1}\\ v_1\\ + \\ \\frac{\\partial g }{\\partial x_2}\\ v_2\\ + \\frac{\\partial g }{\\partial x_3}\\ v_3\\ + \\ \\frac{\\partial g }{\\partial v_1}\\ f_1\\ + \\ \\frac{\\partial g }{\\partial v_2}\\ f_2\\ + \\ \\frac{\\partial g }{\\partial v_3}\\ f_3", "c02e03457fa99fb03ec162beafea0126": "\\mathcal F = \\mathbf b + \\mathbf F", "c02e18e8b48fe003566c4a72816340cf": "\\Theta\\colon \\mathbf{\\mathcal{C}}_{\\bullet {\\text{ }\\Box_{S_\\ast(B)}}}S_\\ast(E)\\rightarrow S_\\ast(E_f,k)", "c02e22f04a9732f2bb03264dabf6033b": "C(a) \\geq C(b)", "c02e492aa6ae8e63adba5063c4c529c1": "\n{\\mbox{GEN}}[d : y \\leftarrow f(x_1,\\cdots,x_n)] = \\{x_1,...,x_n\\}\n", "c02e49e1cf01554ca5dae0f4136c9986": "s = 1/2 + i t", "c02efc56ebd59568925c2d09d344145d": "f_{xz^2}", "c02f226ecf3f20e24e102b8f4809dc1b": " | ", "c02f58ca464cd823c587cfb15ff55f1a": "\\operatorname{var}(\\mathbf{M}^{1/2}\\mathbf{X}) = \\mathbf{M}^{1/2} (\\operatorname{var}(\\mathbf{X})) \\mathbf{M}^{1/2} = \\mathbf{M}.\\,", "c02f92389fb3df0daa92f0ba2a0ebcb3": " r_{1}, \\ldots , r_{n} ", "c02fa737ba0cba15921fc9fd075c80eb": "\\lfloor n^2/4 \\rfloor.", "c02fe30e30a4ff85beb07e6a0007f1d2": "\\begin{align}\na + 0 &= a ,\\\\\na + S (b) &= S (a + b).\n\\end{align}", "c0301518ec1b57a51cdce9b8e0625d2e": "\\det\\begin{bmatrix}1&-1\\\\1&2\\end{bmatrix}=3\\neq0.", "c03018417e82ede97d8143efd6f28451": "0+ix,", "c0301eda1a6703ace7c55c4cb30009ec": "\\nabla_{\\sigma , \\mu}V_\\nu ", "c030c42563d1ff2d8b72216bc5fc1933": "\\begin{align}\n\\sqrt k&=\\frac mn\\\\[8pt] &=\\frac{m(\\sqrt k-q)}{n(\\sqrt k-q)}\\\\[8pt]\n&=\\frac{m\\sqrt k-mq}{n\\sqrt k-nq}\\\\[8pt] &=\\frac{nk-mq}{m-nq}\n\\end{align}", "c0310eb0a6858e1fb5803b4a33b5dd0a": " \\vec V = \\left[ v_1,\\ v_2,\\ v_3, \\cdots \\right] \\ , ", "c0311cfbe8670bc1f9da7e2169d496a3": "\nA=\\left[\\begin{array}{ c c } 1 - \\kappa - \\mathrm{Re}[\\gamma] & -\\mathrm{Im}[\\gamma] \\\\ -\\mathrm{Im}[\\gamma] & 1 -\\kappa + \\mathrm{Re}[\\gamma]\\end{array}\\right]\n=(1-\\kappa)\\left[\\begin{array}{ c c } 1-\\mathrm{Re}[g] & -\\mathrm{Im}[g] \\\\ -\\mathrm{Im}[g] & 1+ \\mathrm{Re}[g]\\end{array}\\right]\n", "c031692f6824a021f22043b603abb241": "\\begin{align}\n -R_l^l & =\\frac{(\\dot a b c)\\dot{ }}{abc}+\\frac{\\lambda^2 a^2}{2b^2 c^2}=0,\\\\\n -R_m^m & =\\frac{(a \\dot b c)\\dot{ }}{abc}-\\frac{\\lambda^2 a^2}{2b^2 c^2}=0,\\\\\n -R_n^n & =\\frac{(a b \\dot c)\\dot{ }}{abc}-\\frac{\\lambda^2 a^2}{2b^2 c^2}=0.\\\\\n\\end{align}", "c03219d5a02b1231e83d84bf1f9aba90": "BEDAC", "c0321bf5ae4726dcdc381ae63c69cdd9": " f^{\\mu} \\rightarrow f^{\\mu} + u^{\\mu} u_{\\nu} f^{\\nu} ", "c0325903a452150b7689f93c3143d1a9": " \\dot E = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\dot m \\cdot \\left(v_1^2 - v_2^2\\right) ", "c032628c1730a2ed701dc63b6b558103": "\\mathrm{id}_y f = f", "c0333266a9d8b40944adc2b807e21822": "f_w(\\theta)=\\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty \\phi_n e^{-in\\theta}", "c033367a1746561fbdd87f307fc3232f": "i=0,\\ldots,m-1", "c03379adc90b198459c9c9c129e9416f": "\\tan \\left(\\frac{\\pi}{4}-\\phi\\right) = \\frac{1- \\tan \\phi}{\\tan \\phi +1}", "c033c0e49251990c2ac7e0a34350d43c": "V_n(R) = \\frac{R^n}{n} \\frac{\\Gamma(\\frac{n-1}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{n}{2})} \\frac{\\Gamma(\\frac{n-2}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{n - 1}{2})} \\cdots \\frac{\\Gamma(\\frac{2}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{3}{2})} \\cdot 2 \\frac{\\Gamma(\\frac{1}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{2}{2})}.", "c033c77a2e1295b54190db53b4086892": " \\Delta \\varphi = \\frac {2 \\pi d} {\\lambda} = \\frac {2 \\pi x \\sin \\theta} {\\lambda}", "c033d19b324632e5892d69669c26ffe8": "A_2=S(a^{-1})M \\setminus B", "c0347b5fce5bcd754d69fb745a686ca4": "\\frac{1}{2}< t\\leq\\frac{131}{208}=0.6298\\ldots,", "c034a6c5b5abd491d4065de2aadac012": "P_1 P_2 CA", "c034c8485d8bbfd228fff63e71c48486": "[x(\\phi),y(\\phi)] = [r(\\phi)\\cos\\phi, \\, r(\\phi)\\sin\\phi], \\, ", "c034d771c648c9036725a412b7b005ac": "\\aleph_0^2", "c03523fbf18e99a424105145240ea163": "X_M\\,\\!", "c03545b3083eed412f5bbb766de2d861": "\\frac{\\ln\\, \\mathcal{L} (\\alpha, \\beta|X)}{N} = (\\alpha - 1)\\ln \\hat{G}_X + (\\beta- 1)\\ln \\hat{G}_{(1-X)}- \\ln \\Beta(\\alpha,\\beta)", "c0357ac6bd408f416f048db88578b617": "\\scriptstyle\\hat\\Pi", "c035a0a1952a797af68c953fafb95385": "A^j v, \\, j=0,1,\\cdots,n-2", "c035a5a14f37967374c703a8a7ff198c": " \\frac{1}{2} ", "c035b92047c50807390f49901cff84a6": "i=1, 2,..", "c035f4d6db735dd735c1d127e8048d97": "\\Gamma_1(r) := \\left\\{\\begin{bmatrix} a&b\\\\c&d \\end{bmatrix} \\in \\Gamma : a\\equiv d\\equiv 1,~c\\equiv 0\\mod r\\right\\}.", "c035f7d660f7cea9dd482bf57fdae0a7": "\\sum_{i=0}^{k} {\\binom{n}{i}} =O(n^k)", "c03641e72a99c2789fa6110f62d42e48": "\\theta \\colon \\mathcal{N}^{TOP} (S^n) \\to L_n (1)", "c0364244da761ba79bff122985cc2cbd": " \\pi_x\\mu_{xy} = \\pi_y\\mu_{yx} \\ ", "c03659a3af1143e06c1a40739ef5609f": "G_{i_0 + p i_1 + \\cdots + p^{n-1}i_{n-1} + 1} = 1 = G^{i_0 + \\cdots + i_{n-1} + 1}.", "c03696d4f6b0cb321ba70e42b3fb3ebd": "S_1 = 2\\pi. \\,", "c0379fa21fb4776d7dd86c8db8461252": "NPOT", "c037b2a25274e081efd192a21588cb8c": "\\frac{\\mathrm d^k}{\\mathrm d z^k} P_n^{(\\alpha,\\beta)} (z) = \\frac{\\Gamma (\\alpha+\\beta+n+1+k)}{2^k \\Gamma (\\alpha+\\beta+n+1)} P_{n-k}^{(\\alpha+k, \\beta+k)} (z).", "c03810fb10683d88b1f5ea8422636f40": "(\\mathbf{S})\\int_{t_0}^tg(t^\\prime)\\mathrm{d}B(t^\\prime)=(\\mathbf{I})\\int_{t_0}^tg(t^\\prime)\\mathrm{d}B(t^\\prime)+\\frac{1}{2}\\sqrt{\\gamma}N\\int_{t_0}^t\\mathrm{d}t^\\prime\\,[g(t^\\prime),c(t^\\prime)]\\,.", "c0381e6e3ebb1c11f6d2fb4127c58ae3": "X(t) = B", "c038754877b5b321c6924860e4daa3e1": "F{{=}}\\frac{1}{2}{\\operatorname{d}C\\over\\operatorname{d}z}V^2", "c03969db6b92d9b252f088ce51566347": "\\begin{matrix} {12 \\choose 1}{4 \\choose 3}{44 \\choose 1} \\end{matrix}", "c03979efb64041fdb3faf9fa4bda13e8": "\\Bigg(\\frac{\\alpha}{\\pi}\\Bigg)_3=\\Bigg(\\frac{\\alpha}{\\theta}\\Bigg)_3", "c039b795ae58933fc0bd6313a3188503": "\\mathcal{U}(\\hat{\\alpha}(q,r_{c})+\\varepsilon,\\tilde{u}) \\subseteq \\mathfrak{U}\\ ", "c03a00615242515c4e412feb4d518251": "\\begin{align}\\mathcal{L}_{Y} =\n&-\\lambda_u^{ij}\\frac{\\phi^0-i\\phi^3}{\\sqrt{2}}\\overline u_L^i u_R^j\n+\\lambda_u^{ij}\\frac{\\phi^1-i\\phi^2}{\\sqrt{2}}\\overline d_L^i u_R^j\\\\\n&-\\lambda_d^{ij}\\frac{\\phi^0+i\\phi^3}{\\sqrt{2}}\\overline d_L^i d_R^j\n-\\lambda_d^{ij}\\frac{\\phi^1+i\\phi^2}{\\sqrt{2}}\\overline u_L^i d_R^j\\\\\n&-\\lambda_e^{ij}\\frac{\\phi^0+i\\phi^3}{\\sqrt{2}}\\overline e_L^i e_R^j\n-\\lambda_e^{ij}\\frac{\\phi^1+i\\phi^2}{\\sqrt{2}}\\overline \\nu_L^i e_R^j\n+ \\textrm{h.c.},\\end{align}", "c03a0fbf5375fd3e75241d551b6c91ae": "K_\\alpha", "c03a1b499264664c08fe937166a8dba6": " \\text{SNR} = 20 \\log_{10} \\frac{\\text{signal}}{\\text{RMS noise}}\\,\\mbox{dB}", "c03a1ea8e210489285f161e454a4306a": "\\text{MUF} = \\frac{\\text{critical frequency}}{\\cos\\theta}", "c03a2a5f6cbec416761ff42bfc3bcfce": "\\tilde{s}=argmin_{s \\in QPSK}|\\hat{s}-s|^2,", "c03a6558cfa2253e38b932a6de8b855f": " \n(q, \\omega, (\\delta_{ext}(q,x), 0)) \\in \\Delta.\n", "c03a693248df43ffd202120e3ca4ce74": "D(f) \\leq R_0(f)^2", "c03a8fa0e4237243f90f7d8e4b95a1c1": "SSTr = n(\\bar{X}-\\bar{M})^2 + n(\\bar{Y}-\\bar{M})^2 + n(\\bar{Z}-\\bar{M})^2", "c03ae536a791d4ff82355025b5118c55": "M - \\tfrac{4G}{3}", "c03b2a1978f95b8ced10d6359852b627": "\\phi(x) \\leq p(x)\\qquad\\forall x \\in U", "c03b34b44648406113249417954c6c52": "r = \\rho^{-1}", "c03b5ece5bf4b10a384b1c9827cf4093": "\\omega > 30", "c03c29b8a816950bb7b64b2463f509b1": " Z_\\text{in} = {v \\over i} = {j \\over {\\omega C}} ", "c03c594ee327986a0c82ed9a189eb38f": "\nC =\n\\begin{pmatrix}\n0&1&0&\\cdots&0\\\\\n0&0&1&\\cdots&0\\\\\n0&0&\\cdots&1&0\\\\\n\\cdots&\\cdots&\\cdots&\\cdots&\\cdots\\\\\n1&0&0&\\cdots&0\n\\end{pmatrix}\n", "c03c5ecb22c5718e165dcf484710d586": "p_1^2 c^2 = E_1^2 - m_1^2 c^4 ", "c03cf6d51a39f41226b83a5fc0c1d1af": "\n(\\mathbf{M}_i)_j = \\frac{1}{l_i}\\sum_{k=1}^{l_i}k(\\mathbf{x}_j,\\mathbf{x}_k^i).\n", "c03cf7c4bd443e7f5e40786ea7ec4341": "Q_3 = \\text{CDF}^{-1}(0.75) ,", "c03d0b42a407a61953fbb48be7e3dbb3": "c = S_0\\exp(-r_f T)\\N(d_1) - K\\exp(-r_d T)\\N(d_2)", "c03d2bb1c97b460ff9b1efffe7d349fe": "K(K-1)/2", "c03d4a1713199ef1ce90a638d971d528": "p = mv \\gamma \\,.", "c03d8916ea8fe133886502ae8c590023": "\n\\begin{pmatrix}\n2 & 0 & 0 \\\\\n0 & 6 & 0 \\\\\n0 & 0 & 12\n\\end{pmatrix}\n", "c03d8a7572be895c72ddaf0657ee1148": "\\sigma = \\arccos((R^2-a^2)/Q)", "c03d9fa989defebf9a4202e5fdbb5445": "BP'", "c03da5e3290d122469eef40cc9b65ba5": "\\{ \\hat{Z_s} | s \\in \\mathbb{F}_d \\} ", "c03dec12a1e22ce7914cd3d2962e79f2": "\\int af(x)\\, dx+\\int bg(x)\\, dx.", "c03df42e04ee1d054a9cf6e86e09451a": "|z|<1", "c03e45e4154491a744a30f8e8d3eddd6": "(\\mathbf{1},\\mathbf{2},-1)", "c03e6324e86a4f4841a2d88131abe0f0": "S_e,in", "c03e6f7bbdda943808b56d22957c4083": "Eq.7", "c03e7bede0a41a2cb84973cf1549f4fa": "1+\\cfrac{q}{1+\\cfrac{q^2}{1+\\cfrac{q^3}{1+\\cdots}}} = \\frac{G(q)}{H(q)}=1+q -q^3 +q^5-\\cdots", "c03f31ec393f04e6b979ece798187067": "\ns = \\frac{2\\ k\\ T_e^2}{E_i\\; (T_e - T_g)} \\times \\frac {1}{1 + \\dfrac{3}{2}\\ \\dfrac{k\\; T_e}{E_i}}\n", "c03f6599ec0ad95c85bdd285e288b309": " \\bar{H'} ", "c0402dde74ffe1e2a5afb0df73e7f117": "\\mathbf{B}=\\int \\operatorname{div}\\sigma \\, dV = -\\int \\mathbf{f}\\, dV = -\\rho_f \\mathbf{g} \\int\\,dV=-\\rho_f \\mathbf{g} V", "c040b04e673afb550aed0e2a7b2190fe": "m=s", "c040c550d7d0de541548553aefa501ca": "\\begin{alignat}{3}\n\\beta_1 + 1\\beta_2 &&\\; = \\;&& 6 & \\\\\n\\beta_1 + 2\\beta_2 &&\\; = \\;&& 5 & \\\\\n\\beta_1 + 3\\beta_2 &&\\; = \\;&& 7 & \\\\\n\\beta_1 + 4\\beta_2 &&\\; = \\;&& 10 & \\\\\n\\end{alignat}", "c040e13e95bc62c3b0b238014cf85f5b": "\\frac{1}{T_{g}} = \\frac{1}{T_{g,\\infty}}+\\frac{K}{T_{g,\\infty}^2}\\frac{1}{M_{n}}", "c0413c46f42e18c97f5762d2d467d33c": "\\mathbf{r}=\\left( \\mathbf{X}^{T}\\mathbf{X} \\right)^{-1}\\mathbf{X}^{T}\\mathbf{y}", "c0419a4bba8499d56e9095266fe54dbc": "-\\frac{1}{2}\\!\\,", "c0419f6c92168e22a9d7fd88e5e1d704": " 1 - \\sqrt{2R} ", "c041a342f175b765ea6a8c53b6225f4c": "\\,X^*(s+jm\\omega_s) = X^*(s)", "c0423b9981625bc582b128e84ce6589c": " \\sigma(p, \\omega) \\propto \\frac{1}{R} |\\Delta p|^t \\Phi_{\\pm} \\left(\\frac{ i \\omega}{\\omega_0}|\\Delta p|^{-(s+t)}\\right) ", "c042e391a46fa72c69b8465b1ff6a0bf": "{V_{r1}^2} = {V_1^2-U^2-2UV_1\\cos\\alpha_1}", "c04339998ec23b623d9eaa307077708e": "U_{++}", "c0437c10cdc742091a94ff2faff95eaa": "\n\\mathbf{A} =\n\\left[ {\\begin{array}{ccc}\n \\hat{\\mathbf{u}}_x & \\hat{\\mathbf{v}}_x & \\hat{\\mathbf{w}}_x \\\\\n \\hat{\\mathbf{u}}_y & \\hat{\\mathbf{v}}_y & \\hat{\\mathbf{w}}_y \\\\\n \\hat{\\mathbf{u}}_z & \\hat{\\mathbf{v}}_z & \\hat{\\mathbf{w}}_z \\\\\n\\end{array}} \\right]\n", "c0444523fa7c3bec9c8322d6829a4fe2": "\\sum_{n=0}^\\infty \\phi_n(\\alpha,\\beta)t^n = \\exp \\left[ \\sum_{n=1}^\\infty \\frac{t^n}{n} \\left( u_n (\\alpha) + v_n(\\beta) -1 \\right) \\right]", "c0446a68c118187999c4aea001f59d68": "n_2 [n_2^{-1}]_{n_1} + n_1 [n_1^{-1}]_{n_2} = 1", "c0446d10a41af5a84a1b3d5f30c48ba1": "0\\rightarrow M^\\prime\\rightarrow M\\rightarrow M^{\\prime\\prime}\\rightarrow0", "c044c28b53f18388412de239b71f76ff": "X_1^n(1)", "c044c8120c4fd8760e1d283fec0e6859": "\\scriptstyle x_1 \\;=\\; 1", "c045053ebe6b163839385279db7d38ba": "\\textstyle \\sum_{i=1}^k x_{i} = N", "c0452b26b057e57fd35c5a3d549e979c": "\n \\begin{matrix}\n \\underbrace{2_{}^{2^{{}^{.\\,^{.\\,^{.\\,^2}}}}}} \\\\\n 65536\\mbox{ multiplied copies of }2 \\end{matrix}\n", "c045536432f8b8d214c0689f8886ea6c": "\\mathrm{4 \\ Fe^{\\,2+} + O_2 + 4 \\ H^+ \\longrightarrow 4 \\ Fe^{\\,3+} + 2 \\ H_2O}", "c045a28b8f220b1a61f48b45d887e178": " |f(x)| ", "c045a40bfb0b5242693f65fc43c2f7f0": "{u}_{n+1}=u_n + {\\Delta}t~\\dot{u}_{n}+\\begin{matrix} \\frac{1}{2} \\end{matrix}{\\Delta}t^{2}~\\ddot{u}_\\beta ", "c045b0cec3820a4985efb7fd166a090b": "J^1\\Sigma", "c045c3635deb0a40ca6aa1de2a1fc68a": "\\lambda \\sum_{t=1}^{\\infty}(1-\\lambda)^{t-1}g^i_t", "c045da2c7f2f19729e8fbceb8e2d7b15": " \\dot \\gamma \\gg 1/\\lambda ", "c045feaf4822dd8a9e9c03f07da83bbb": "\\lambda_{01}=2.40483", "c046572f7215d5d2904fe3ae6372d87d": "\\phi_A^{}", "c0467faea7cc05904a8050543cc6746a": "|H_1(e^{j\\Omega})| = |H_0(e^{j(\\pi - \\Omega)})|", "c046cfc5804e4a4e6ef1dbc25ea0fb30": "V_{2k+1}(R) = \\frac{2^{k+1}\\pi^k}{(2k+1)!!}R^{2k+1} = \\frac{2(k!)(4\\pi)^k}{(2k+1)!}R^{2k+1},", "c047a7231bcd11f4f5d87d154080dd05": "T_G(2,0)", "c047ecf64518f3606c4d5640a217cfc3": "\\scriptstyle [0,\\, 1]", "c047f1badb33778463ba62e788c24e7c": "\\sigma_0", "c047fc30ce9f40a7f8263e4b8826295b": "m_{0}=\\tfrac{4}{3}\\tfrac{E_{em}}{c^{2}}", "c048219dc186194f4cde637519346f74": "c\\left(x\\right).P", "c0483ba0d19758cd9709296f5fb248df": "g_{\\mu \\nu} = \\eta_{\\mu \\nu} + fk_{\\mu}k_{\\nu} \\!", "c0494e44b41f2c1ee9fb75442dd5e0ba": "x > a", "c04963ce3533ee8d6df4cb6672c5818e": "\\Bigl\\| \\sum_{n=1}^\\infty x_n \\Bigr\\| \\le \\sum_{n=1}^\\infty \\|x_n\\|.", "c0497521d35d66866e3af408094864c3": "G1", "c049d4c86ea336fda33f7ea8d2d3bdae": "uv^T", "c04a4475bd17b4e8984f94c4e5a19bc9": "\\overline v = \\operatorname{Re}\\{v\\} - i\\,\\operatorname{Im}\\{v\\}", "c04a6d483dc84aa0af429895371b1a43": "w \\models \\Box P", "c04ac206534c5d3b64591f79804ccb20": "\\# X (\\mathbf{F}_q) = \\tau(G) \\prod_x {1 \\over \\operatorname{vol}(G(\\mathcal{O}_x))}", "c04ac4050f8270f3839184343ac3e970": "(X, \\mathcal{A}, \\mu)=(Y,\\mathcal{B},\\nu)^\\mathbb{Z}", "c04b27069cabe7dde59408dca0e40956": " z=e^{\\beta\\mu}.\\,", "c04b52bb903f67b8eb9eeea0a71eeafc": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log (\\frac{\\varepsilon}{3.7D} + \\frac{5.74}{Re^{0.9}})\n", "c04b58b99b70b3f41190911dc7062a70": "\\mbox{sgn}(\\rho_n) = (-1)^{\\lfloor n/2 \\rfloor} = \\begin{cases}\n+1 & n \\equiv 0,1 \\pmod{4}\\\\\n-1 & n \\equiv 2,3 \\pmod{4}\n\\end{cases}", "c04bb10dc88aa6a70a585f0891bffb30": "\\binom{n+2}{2}-2n-1", "c04bdb38e170c5f0063e0e8b92566b4e": " G \\,\\!", "c04c0e4f05d80928731d714e2401127c": "\\forall a.a \\to a", "c04c41ac753be11a69b53a16dd50eea5": "( \\lambda, \\mu, \\nu )", "c04cc475de2161cb9d6c257b2ad5346b": "A(\\lambda) = B - \\lambda I", "c04ce3f9fc0fb1e6c6739aba9924c24b": "\\langle B_E u | v \\rangle_E = (u|v)_E", "c04ce5bbcf0b34afa4487b552a897d34": "B_0", "c04d5b3821f8c9ee801022e19324eee4": "\nI_2=\\frac{4 A_1^2 \\chi_0^2 L^2}{m^2 \\pi^2}.\n", "c04d72fec67930d4906e1e8f7e646b01": " 0\\le e\\le n", "c04ddb9af846d8d7a3481b0a1d5761e1": "f\\neq g", "c04de0b9652dbc07ab7f26725a667ce8": " mg\\sin\\theta = m\\frac{{\\rm d}^2 (\\ell\\theta)}{{\\rm d} t^2} + 0 \\Rightarrow g\\sin\\theta = \\ell\\frac{{\\rm d}^2 \\theta}{{\\rm d} t^2} \\,\\!", "c04e65f43ed335fe9253f9eca9960b30": "\\sqrt{\\frac{49}{120}}\\!\\,", "c04e8930ef6262e3e1625745719cd20b": "SDR=\\frac{d_o}{s}", "c04f61af521e339cdfbe06a372275590": "f=\\int_0^x \\! {\\Delta f_{cb}} = A(10^{ax}-K)", "c04fc212f3d4d58659bd57e1539118ac": "\\widehat{0}", "c04fc674c0456bc1cdc12537ad3fe3fe": "\\operatorname{cont}", "c05005f0343411e27ac541a51704d7f4": "\\left(\\frac{c-\\sqrt{c^2-4ab}}{2}, \\frac{-c+\\sqrt{c^2-4ab}}{2a}, \\frac{c-\\sqrt{c^2-4ab}}{2a}\\right)", "c0503e83819eebd2bdb63bcb7b1aff46": " dx:dy:du:dp:dq = F_p:F_q:(pF_p + qF_q):(-F_x-F_u p):(-F_y - F_u q). \\,", "c050bb196ff615ea814b96d5dbc1c291": "\\dfrac{dR}{dP} > 0 \\!\\ ", "c050fa7761638b226bf982d45d20a3fe": "\\ f(u,v) = - f(v,u)", "c051b96037dadbd54680f6dcd3b0439e": "C^n(u_1,\\dots,u_d) = \\frac{1}{n} \\sum_{i=1}^n \\mathbf{1}\\left(\\tilde{U}_1^i\\leq u_1,\\dots,\\tilde{U}_d^i\\leq u_d\\right).", "c0522b81522563d316266030a96edadd": "m'= c_2 \\cdot s'", "c05271d4557a4f76685dc1422346b3c8": " \\Psi(\\text{space coords},t) = \\psi(\\text{space coords}) e^{-i{E t/\\hbar}} \\,.", "c052c6cc583e58a4361bf32008fbf9d7": "\\nabla (\\nabla \\cdot \\textbf{A}) - \\nabla^{2} \\textbf{A} - k^{2}\\textbf{A} = \\textbf{J} - j \\omega \\epsilon \\nabla \\Phi \\,", "c052eba02c7068c5845d624cbf3f6f60": "w_i = \\frac {\\rho_i}{\\rho}=\\frac {c_i M_i}{\\rho}", "c05308e6eb6b86f31a8f1fe77e99686c": "{{\\sigma }_{scatt}}=\\frac{8\\pi }{3}{{k}^{4}}{{R}^{6}}{{\\left| \\frac{{{\\varepsilon }_{particle}}-{{\\varepsilon }_{medium}}}{{{\\varepsilon }_{particle}}+2{{\\varepsilon }_{medium}}} \\right|}^{2}}", "c05312ba3d7d030aa571ed334a0eb6f3": "S = R[t_1, \\dots, t_n]", "c05316f242fce091d0d7a888a564e40e": "\\textstyle{\\frac{\\log{3}}{\\log{2}}=1.5849...}", "c0532c4e2a377ed929cfffb6da924025": " P_1;P_2;\\dots ;P_k ", "c053379b77e7fea06439277fc76b5590": "\\langle f_1 \\circ \\pi_1, f_2 \\circ \\pi_2 \\rangle", "c0533ee86d92b5d24420ec726a865bbd": "0\\leq r_s < 1 ", "c053c7ba4e9c044b97b8c2ab7198aa97": "\\overline{\\Lambda}_n(T) = \\frac{2}{\\pi} \\log(n + 1) + \\frac{2}{\\pi}\\left(\\gamma + \\log\\frac{8}{\\pi}\\right) + \\alpha_{n + 1}", "c05454525bab8b42a15924b398921479": "\\mathrm{NapLog} (x) \\approx 23025850 (7 - \\log_{10} x).", "c0549a44896a1944919607348f0cd775": "2^{\\log^* n}", "c0551fcbcf61ad169df6fffd603c6dbe": "C= \\{ z \\in \\mathbb{C} ~:~\\Re(z)>-2 \\}", "c05639272bf89f768737aae3c5118ff0": " \\theta[\\vec{X}]_{\\hat{i} \\hat{j}} = \\frac{\\omega}{\\sqrt{2}} \\, \\frac{C^\\prime( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u)}{C( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2 \\omega^2}, \\omega u)} \\, \\operatorname{diag} (0,1,1)", "c056486654fe364b73833e7f06c1f855": "\\Omega_{M}", "c05649b4cb7b103af9e679004ebf0588": " \\{ \\Psi^n \\}_g ", "c0567e63f1b783ddbbdc72cdb199fa27": "\\exists i, y \\in A_i \\wedge (\\forall j, z \\in A_j \\Rightarrow i < j)", "c0568db9087bb98d352322f1da65aaa0": "\\ P_d ", "c0569b1c7d395a5d124ef19006debd73": "\\tau_{if}", "c056a8989374738e20a65a64f4699bed": " \\det \\begin{bmatrix} \n 0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & d(AE)^2 & 1 \\\\\n d(AB)^2 & 0 & d(BC)^2 & d(BD)^2 & d(BE)^2 & 1 \\\\\n d(AC)^2 & d(BC)^2 & 0 & d(CD)^2 & d(CE)^2 & 1 \\\\\n d(AD)^2 & d(BD)^2 & d(CD)^2 & 0 & d(DE)^2 & 1 \\\\\n d(AE)^2 & d(BE)^2 & d(CE)^2 & d(DE)^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 1 & 1 & 0\n\\end{bmatrix} = 0, ", "c056b0dff3ab85791d7e9c6266c0cd81": " j = 1, \\ldots, n ", "c0570599c4dcde0535e1b1124039907e": "B\\operatorname{GL} = \\varinjlim B\\operatorname{GL_n}", "c057548b5bd071d84db5206b8be0e4ce": "z\\mapsto e^{1/z}", "c057bbef5e4c252faef779726140276b": "S\\left(t\\right) = \\int E\\left(t^{'} \\right)E\\left(t^{'} -A\\left(t\\right)\\right)\\, dt^{'}", "c058837c4f3ab80a381dfd4f562595dd": "\\omega(x)=\\lim_{n\\rightarrow\\infty} x^{p^n}", "c05891c10b6470e3b635ea2c6245ed71": "W_{n} - W_{n + 1}= \\int_0^{\\frac{\\pi}{2}} \\sin^{n}(x)\\,dx - \\int_0^{\\frac{\\pi}{2}} \\sin^{n + 1}(x)\\,dx = \\int_0^{\\frac{\\pi}{2}} \\sin^{n}(x)\\, [1 - \\sin(x)]\\,dx \\geqslant 0", "c05917907bfc0ccd01bd025cff412ef9": "{\\rm SQNR}= 10\\log_{10}{\\frac {\\sigma_x^2}{\\sigma_q^2}} = 10\\log_{10}{\\frac {(M\\Delta)^2/12}{\\Delta^2/12}}= 10\\log_{10}M^2= 20\\log_{10}M", "c059c429ae9e0c15a9bfdaa3753d2e35": "S \\leq \\frac{2 \\pi k R E}{\\hbar c}", "c059ec887e2462e53bbbc562da44eeee": "X \\sim \\mathcal{N}^{\\textrm{R}}(\\mu,\\sigma^2) ", "c05a4ff450614e93e89ac29ae6f9b308": "b \\to 0", "c05a549b9da1cdedb2d0e78be7417cc2": "k_{\\eta}(t,t_{i}) = m\\left(\\frac{t}{t_i}\\right)^{\\beta(\\eta_{i})}", "c05a6cc484f685d303a879d3574f082b": "d_{448} = d_{558} = d_{668} = d_{778} = -\\frac{1}{2\\sqrt{3}} \\,", "c05a74f9c6830c6e70a93a355278c0b6": "\\langle A , \\Phi(\\rho) \\rangle = \\langle \\Phi^*(A) , \\rho \\rangle .", "c05a9d9bcf7a305d6459bc8a03afcffc": "(((P \\rightarrow Q) \\land (R \\rightarrow Q)) \\land (P \\vee R)) \\rightarrow Q \\, ", "c05aac896ddf6194b2de5fa28f0b80a4": "n \\not \\in Z", "c05aaf8deec1b1fcb0cdb17cc26ac624": "\\{b_n\\}_{n\\ge 0}", "c05ab160091255efd3e794ca18aee92e": "\\langle\\text{ }\\rangle", "c05b6184cd7d0189f9e000a6d080228d": "s_{\\sigma^2}=46.07", "c05bfdd7323b1f47e19c0ac95834fe25": " A, B, C \\in Ob(\\mathbf{CRing}) ", "c05c109e5844a990bc8b1f7c24572548": "A = 2 \\pi R h \\,", "c05c717ef0deb817dc9699c4ffa16973": "H(S) = 0", "c05c7eabad0a88e2ed9d698948980919": " c^2n^2 \\in O(n^2) ", "c05c7fad7e1554aa53c32b22d9851363": "\\delta \\rho(\\mathbf{r}t)= \\chi(\\mathbf{r}t,\\mathbf{r'}t')\n\\delta V^{ext}(\\mathbf{r'}t')", "c05c874f1ebd6e9da1460f9148098423": "|V|=n", "c05ca832014891039623658589a211df": "\\begin{align}\n 2\\alpha_{\\tau\\tau} & =\\left (\\mu b^2-\\nu c^2\\right )^2-\\lambda^2 a^4=0,\\\\\n 2\\beta_{\\tau\\tau} & =\\left (\\lambda a^2-\\nu c^2\\right )^2-\\mu^2 b^4=0,\\\\\n 2\\gamma_{\\tau\\tau} & =\\left (\\lambda a^2-\\mu b^2\\right )^2-\\nu^2 c^4=0,\\\\\n\\end{align}", "c05cd5e4e545a97aac37a8e58206a3d7": "\\frac{5}{3 \\pi}", "c05d180bb57c7aa441f01e926b486cab": "\\tan \\left (\\frac{4\\pi}{\\lambda} x\\right)=0", "c05d2fe9c4259e0a2ddd269ab96f2b4e": "x^p -1 = 0", "c05d4244e5608cb6004d7cfe8fe459f2": "q \\in C", "c05d44e6f35b185dcd3ac3c69ec34b2e": "\\ H(X) \\ge H(X|Y)", "c05d5f09970ebceec0a03594e0828732": "C_{4,3} = (25 + 1) / 2", "c05d99e88fb47615eebcefc65b8284f4": "\\int \\left(f + g\\right) \\,dx = \\int f \\,dx + \\int g \\,dx", "c05da6ca6ff64d9627f7942a70023298": "p(t)I_n=t^nI_n+t^{n-1}c_{n-1}I_n+\\cdots+tc_1I_n+c_0I_n,", "c05dc36f8e832396e526e66cfe10685a": "y*", "c05de82d71ea1a521c09a87898fc8f98": " [u,v,w] = -[v,u,w] ", "c05df9e22814fdb4851b1e3d8e6f7fd6": "R_{12}(u) \\ R_{13}(u+v) \\ R_{23}(v) = R_{23}(v) \\ R_{13}(u+v) \\ R_{12}(u),", "c05e2f0163fe714e852463cf0cba11c4": "S(x)=-t\\mbox{ln}(t)-(1-t)\\mbox{ln}(1-t)", "c05e4723282ac534f2f15b5851d6da8e": "d^{146} = d^{157} = -d^{247} = d^{256} = d^{344} = d^{355} = -d^{366} = -d^{377} = \\frac{1}{2}. \\,", "c05e5e2759f4e6d384b255b96df9ea80": "2\\pi E/\\omega", "c05eb16e3dfa8a837e5b725126f905b9": "3^{480}+3 \\equiv 0 \\pmod {3^{479}+1}", "c05eb34244b51c6cb12ce0a73d10076b": "f(P_i)=0, i=1, \\dots ,n", "c05eb42deb740f9782c5e6ee476ec60d": "Re = \\frac{4 \\dot{G} \\rho}{n \\pi D_h \\mu}", "c05ede4fef74b0b91d2d8417681ece5c": "1 - H(p) - \\frac{\\epsilon}{2}", "c05eedd9141c8c4a5e519b9cd07642ce": "\n\\begin{align}\n\\textstyle {n \\choose n, 0, 0} &, \\textstyle {n \\choose n - 1, 1, 0}, \\cdots\\cdots, {n \\choose 1, n - 1, 0}, {n \\choose 0, n, 0}\\\\\n\\textstyle {n \\choose n - 1, 0, 1} &, \\textstyle {n \\choose n - 2, 1, 1}, \\cdots\\cdots, {n \\choose 0, n - 1, 1}\\\\\n&\\vdots\\\\\n\\textstyle {n \\choose 1, 0, n - 1} &, \\textstyle {n \\choose 0, 1, n - 1}\\\\\n\\textstyle {n \\choose 0, 0, n}\n\\end{align}\n", "c05eee302c35df936537395b35ddb229": "y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6. \\, ", "c05f07695b3e7aaff7294dd19dbcb123": "\\mathbf{p}= m \\mathbf{v}", "c05f12d0f77c5826d313974f08bc7595": "a_0=0.20 \\,\\;", "c05f914276bb764ad5b57a7e1b91680a": " \\sum _{v \\neq v _{0}} (d(v) -1) q _{v} + d(v _{0})q _{v _{0}}", "c05fbf0f4d59cde74fe3eabbd22362dc": "\\alpha(|f_1(x)|^p,\\ldots,|f_n(x)|^p)=\\beta(|g_1(x)|^q,\\ldots,|g_n(x)|^q)", "c05ffa71d378496612b3450838e02cf8": " x_1, \\dots x_k ", "c0601456e9c7771883e320c8c951dd89": " \\ln \\Lambda ", "c0605654ae916afa9b35ca6e7cb60320": "\n\\frac{1}{R}\\frac{d}{dr}\\left(r^2\\frac{dR}{dr}\\right)\\ =\\ \\lambda\n", "c06059e983074468ef3364b4421cfed7": " f(x+iy)=\\varphi+i\\psi\\, ", "c060676ff13a0862d626087d9264f4fb": "\\tbinom{k}{2}-m", "c0611c0b31334864e1805d8ed81c956e": "1 - (1 - T2) * (1 - T1)", "c06131f1449e2b707bfbcc7d4bb3c9af": "\\mathit{N} - \\mathit{n}", "c061480471b01a94a9368867b3497efb": "H=\\frac{1}{2m}[\\mathbf{P}-q\\mathbf{A}(\\mathbf{R},t)]^2+V(R)-\\frac{q}{m}\\mathbf{S} \\cdot \\mathbf{B}(\\mathbf{R},t)", "c06175648be8a69c49e38071d80cbe3b": " P_{n} = N_{n} \\cdot S_{n} = 3 \\cdot s \\cdot {\\left(\\frac{4}{3}\\right)}^n\\, .", "c061ad909c0de61c4b4eff26338bc06b": "\nx_3 = \\frac{9}{4{y_1}^2{x_1}^4}+\\frac{9}{{y_1}^2a{x_1}^3}+(\\frac{9}{{y_1}^2a^2}+\\frac{9}{{y_1}^2a}){x_1}^2+(\\frac{18}{{y_1}^2a^2}-2)x_1+\\frac{9}{{y_1}^2a^2-3a}\n", "c061f73eeaf1f18e1534640c711f80ed": "[1 \\ldots n]", "c06252241845179c521e1cc7bbda5f0e": "T = \\frac{1}{f}.", "c062f919acd54fdedf4d937f6ba2e50d": "c \\equiv \\prod_{i=0}^{n-1} \\left( b^{2^i} \\right) ^ {a_i}\\ (\\mbox{mod}\\ m)", "c06341fab9dfc639b3f7b464a60570cb": "M\\vec{\\xi}_j= K_j\\vec{\\xi}_j.", "c063b237c7bf41faa40c048dfac44f3b": "12593338795500743100931141992187500.", "c063e14c0749dca06a39b9e8572bdc57": "\n2 H = c^{2} = \\frac{p_{t}^{2}}{c^{2} \\left( 1 - \\frac{r_{s}}{r} \\right)} - \\left( 1 - \\frac{r_{s}}{r} \\right) p_{r}^{2} - \\frac{p_{\\theta}^{2}}{r^{2}} - \\frac{p_{\\varphi}^{2}}{r^{2}\\sin^{2} \\theta} \n", "c064106f0bd63fab07bbd89a1a3fe030": "{n\\choose k_1,k_2,\\ldots,k_r} =\\frac{n!}{k_1!k_2!\\cdots k_r!}", "c0643f0b2a4e9fb3cbd6bf5a520f4cb8": "\\left(\\frac{\\omega}{c}\\right)^2 = \\mathbf{k}\\cdot\\mathbf{k} \\,,", "c06449320f6cec9db9831b16fbcd1ccf": " \\gamma_{1-2} > \\gamma_{1-l} + \\gamma_{l-2} ", "c0645b376125e94b25cba846f2c2bd61": "J_0(z)\\approx\\sqrt{\\frac{2}{\\pi z}}\\cos \\left(z-\\frac{\\pi}{4}\\right)", "c06462b44d3f737ef961d67e10cd2cd7": "\\kappa_5=\\mu'_5-5\\mu'_4\\mu'_1-10\\mu'_3\\mu'_2+20\\mu'_3{\\mu'_1}^2+30{\\mu'_2}^2\\mu'_1-60\\mu'_2{\\mu'_1}^3+24{\\mu'_1}^5\\,", "c0649d615f6d52db541739f79161e7be": " V_{LJ}(r) = \\frac{A}{r^{12}} - \\frac{B}{r^6}, ", "c064a3d830df684c5b6b6fd421dc76c3": " \\mathbf{F}=-\\oint_C p \\mathbf{n}\\, ds, ", "c064f59fb237cf99c07c25ebf6d575d4": " I^m_\\ell(\\mathbf{r}+\\mathbf{a}) = \\sum_{\\lambda=0}^\\infty\\binom{2\\ell+2\\lambda+1}{2\\lambda}^{1/2} \\sum_{\\mu=-\\lambda}^\\lambda R^\\mu_{\\lambda}(\\mathbf{r}) I^{m-\\mu}_{\\ell+\\lambda}(\\mathbf{a})\\;\n\\langle \\lambda, \\mu; \\ell+\\lambda, m-\\mu| \\ell m \\rangle\n", "c06518076fefba2de8fe44477b85cbb7": "\\vec{s} = s_1, \\ldots , s_m", "c0652577fbd473fd0c19c878fbda9bdd": "J(x \\oplus y) = -y \\oplus x", "c0652cec862e2ce96f1aec61892bb484": "q_{1} - q_{4}", "c0657ae89e647d556f727449cb30a1bc": "\\{a,\\,b\\}", "c065c53d1ca94cfff18e65aeaaa49669": "\\eta= \\left(\\frac{1\\, {\\rm atm}}{p_0}\\right)\\left(\\frac{273.15\\, {\\rm K}}{(273.15+20)\\, {\\rm K}}\\right) {\\rm amg}=0.932\\, {\\rm amg}", "c06608de79b14daad5ca2a75f02b5446": "\\lim_{n\\rightarrow\\infty} \\frac{t(n)}{b(n)} < \\infty.", "c066809e0b06f1bd6653442f542b3f93": "b \\times X + Y", "c066ab8dca686738cbc8eae2403e6aa3": "A_i^*", "c06765938141af53c0e2fa94b118f7ac": "t\\in Y", "c06787e407ab5430184ecabc9aa659bc": "\\scriptstyle R \\;=\\; R^{ab}g_{ab}", "c0678803a39a4b046a424044c88da0d8": "c_p\\, =\\, \\frac{\\omega}{k}\\, =\\, \\frac{\\lambda}{T}.", "c067ea3d4978f26d98c00cad5bd0d990": "6\\langle R(u,v)w,z \\rangle =^{}_{}", "c067f3bcd7c4839111d9c7e8fc1cb559": "g_B = g_{F} - \\delta g_B", "c068592204dcf00e0d21195494582808": "{\\rm multiply}", "c0687ca2db16cab366481d307ce5f9fd": "\\sigma(M) \\leq \\mathcal{E}(g_0)", "c068aba0178225aa849133cedb1c52df": "R_{\\rho\\sigma\\mu\\nu} = g_{\\rho \\zeta} R^\\zeta{}_{\\sigma\\mu\\nu} \\,.", "c068ce9b81318c7fffd679273828b0c5": " \\begin{pmatrix} 1 & 1 & -1 & -1 \\\\ \\alpha & -\\alpha & -\\beta & \\beta \\\\ e^{i(\\alpha-k)(a-b)} & e^{-i(\\alpha+k)(a-b)} & -e^{-i(\\beta-k)b} & -e^{i(\\beta+k)b} \\\\ (\\alpha-k)e^{i(\\alpha-k)(a-b)} & -(\\alpha+k)e^{-i(\\alpha+k)(a-b)} & -(\\beta-k)e^{-i(\\beta-k)b} & (\\beta+k)e^{i(\\beta+k)b} \\end{pmatrix} \\begin{pmatrix} A \\\\ A' \\\\ B \\\\ B' \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix}. \\,\\! ", "c068d185920bc4baf1b580a029e7a2ad": "\nU^\\dagger U = I\n\\,", "c068fdf141f0f95430aec97f4318da22": "V(x) : \\mathbb{R}^n \\rightarrow \\mathbb{R} ", "c0690e629d1c6daf654a25738b3e89ed": "-7x^5", "c0690f0267190d1fd05e161ae9110648": "76\\times 10^{-6}\\Omega \\cdot \\text{cm}", "c069e798d65640f72b3f24c639451309": "P_1\\cdot x_1 + P_2\\cdot x_2 + x_5 = P", "c06a650773fd021400d1f804c5e73201": " x \\in [0; {1 \\over {\\lambda(1-q)}}) \\text{ for } q<1 ", "c06a9c17cc950736e198c90fb8e32681": "\\ K_m = \\overline{\\xi'^2}\\left |\\frac{\\part\\overline{u}}{\\part z}\\right |", "c06a9d9c83d484f60c6c6e33865827d5": "\\mathrm{\\delta ^{13}C} = \\Biggl( \\mathrm{\\frac{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{sample}}{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{standard}}} -1 \\Biggr) \\times 1000\\ ^{o}\\!/\\!_{oo}", "c06aebcea1cbc9faa9cb4aed936f3180": "\\Im j", "c06aed421c0693c8d669af85d625e557": "C = \\left(\\sum x\\right)^2/n", "c06b42e12898afcd3a8e4ff412cf712d": "(0,0, W)", "c06b5fb180d5e491055684fbcebb79a1": "\\lambda \\phi(x)- \\int_a^b K(x,y) \\phi(y) \\,dy = 0", "c06bc4f69c7c51993c781f7465cbac71": "2\\theta\\,", "c06c0c569bf74f4856af712afb453c08": "c=s+2", "c06cfc5ea057e024dbad27e6cf78645d": " 0 < \\ln K, \\ln K^* < \\ln N ", "c06df0b2f6cb0f34a64fe41142c78271": "c=-1", "c06e0c1706be727a00d0ec3f26ca64a8": "[f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0", "c06e1dee46ba52122622f7756495d77b": "Q=\\{(s,t_e)| s \\in S, t_e \\in (\\mathbb{T} \\cap [0, ta(s)])\\}", "c06e42915defe7761a798b4211cb5a82": "[0,N]", "c06ebbfebe239b49c94a2b545dead494": "\\langle u_j ',c'\\rangle = -K_{jj}{\\partial (c)\\over\\partial x_j}", "c06ec8fc14eaef79aa49ff10cb021ba4": "\\frac{d\\ \\operatorname{Im} \\{V_c \\cdot e^{i\\omega t}\\}}{dt} \n= \\operatorname{Im} \\left\\{ \\frac{d\\left( V_c \\cdot e^{i\\omega t}\\right)}{dt} \\right\\}\n= \\operatorname{Im} \\left\\{ i\\omega V_c \\cdot e^{i\\omega t} \\right\\}", "c06ed303e54dfa8d35b785e22ab5b70f": "\\textstyle \\overline{ \\overline q }=\\left[[ q_{jk}]\\right] ", "c06ef87a215b2f2cc703a33bf88a8052": " d = \\operatorname{sign}(\\det(W V^T)) \\, ", "c06f7e76e033bab062fb236cf0090a9c": "h[n]=\\delta[n-3] + \\delta[n-2] + \\delta[n-1] + \\delta[n]", "c06fa351e4bf36796aad57cda2036971": "\\Bigg[\\frac{\\alpha}{\\pi}\\Bigg]=\\Bigg[\\frac{\\beta}{\\pi}\\Bigg]", "c06fdcbf8856f3a0b751902b6f6f87f0": "R^{(I)}", "c0702e40cf5b59b8b047ae7a6d6ab79a": "\\Omega = (\\lambda x.~x~x) (\\lambda x.~x~x)", "c07057454bd2c24407c148f42cb3472d": "\\phi_F \\colon C\\nrightarrow D", "c0705d7b284b9c90935748b5b212eacb": "\\sin \\alpha = \\frac {\\textrm{opposite}} {\\textrm{hypotenuse}} ", "c0706e11446e59882594ab21e095b61b": "x = \\lambda\\,", "c0708533b5d3ad12db4137c32b9fec4a": "(\\phi\\land\\psi) \\leftrightarrow (((\\phi\\lor\\psi)\\leftrightarrow\\psi)\\leftrightarrow\\phi)", "c0708679af6196b8ecf2fcf3e96e16c8": " \\ln \\Gamma(\\eta_1) + \\ln \\Gamma(\\eta_2) - \\ln \\Gamma(\\eta_1+\\eta_2)", "c070a9e26ab00945676ccd310821cbf0": "(1+\\sqrt{-19})=(4)q+r", "c07111485c9fd8368834e946afa8ad19": "\n{\\mbox{LIVE}}_{out}[final] = {\\emptyset} \n", "c07158d18d26015df7bdda3bb5574ad5": "V_2=V_1", "c071ae9af671c4477444a6b127957dd8": "\\mathbf{x=u_1G_1+c_{s1}}", "c071fa5c2fd5b3a48000af0960e2de9a": "(a_0-a_1) + (a_1-a_2) + \\cdots", "c07216de9a00b32e0ba84ac712340dad": "\\boldsymbol{y} \\leftarrow \\alpha A \\boldsymbol{x} + \\beta \\boldsymbol{y} \\!", "c0722fe32f6695f39966e06ade7b61b9": "\\sin(11\\tfrac14 ^\\circ) = \\frac12\\sqrt{2-\\sqrt{2+\\sqrt{2}}};", "c07336fbc502383d55e25da5664bcc10": "n \\times 1", "c0733caeba0d76464f2fc0c2062f66e2": "\\binom{a}{p} = (\\pi,a)_p", "c07378ea8e3d9a1cc9da3e77f5af68d8": "F_M(x) = P(M\\le x) = \\prod P(X_k\\le x)= \\left(F(x)\\right)^n", "c07392397b6b8b45fb855cb0a01b7bac": "\\sum_{i=0}^{m+n} v_i X^i = Q(X).", "c073e58daa16bb9f813d5e7be1aa8290": "\\mu^{s} (E) = \\lim_{\\delta \\to 0} \\mu_{\\delta}^{s} (E),", "c073e83a81092e90d6655fb4224db586": "\\color{Black}\\tfrac{\\bar 4}{m}", "c0742b853f1688396f6b918d212ae2a1": " \\acute{{R}^{\\mu}}_{\\alpha \\nu \\beta} \\acute{u}^{\\alpha} \\acute{x}^{\\nu} \\acute{u}^{\\beta} = - \\acute{f}^{\\mu} ", "c074affa239f6195a545e1f0b9e5af17": "V_\\nu^I", "c074f1bbd47661d5c953a218bb381369": "p_{U}", "c074f92f949d112d1a5149335de06a4b": "a_i\\,\\!", "c0751d35f7a8351e1ea0c168d3dea6e9": "e^{(1 + o(1))\\sqrt{\\ln n \\ln\\ln n}} =L_n\\left[1/2,1\\right]", "c0752579e671f57eaeee74b7c46c2158": "L=\\frac{\\hbar}{mc},", "c07558da7a78375c655428a5e22f17bf": "R_B - R_A\\,", "c0757b340f85083e73f05fd0813d29f9": "a^{\\varphi (n)} \\equiv 1 \\pmod{n}", "c07581d92c4fe46b80499f7c87e3e043": "0 \\le\\ \\gamma\\ < 1", "c075a66ad94071a1f0fea8db67073b3f": "\\Delta G = RT \\log \\frac{C_2}{C_1}", "c075b42dda9d21ea52fbe8127e9fcca1": " (\\pi[w] \\psi)(x) = \\psi(x+w y_0).\\quad ", "c075f6a467079c26c0ec7e90455b17c8": "nhf/e", "c07637b2584f8831304ac8506fa711c1": "t_8=-68 \\equiv 0\\pmod {17}\\, u_8 = 42 \\equiv 8\\pmod {17}.", "c07660a919157fd327ba98a4e96db850": "\n \\omega_c = {e B \\over m c}\n,", "c076bcbc13877158f115006f9f6b2f93": "\\sqrt[4] R", "c076e4fd1559721e171432e66b70a487": "Motivation", "c076e7f8120cfeb978eb6745f494a816": "{\\tbinom{2n}{n}}", "c076f92a62dc7c20bfd982fe685d1560": "z_\\mathrm t", "c07717f29a52139c97273a1897112f8e": "l'", "c0774a58dc3b12679be5196be74fc8bd": "\\left(a, p, u\\right)\\succsim \\left(b, p, u\\right)", "c077640af534f4851727e34dec29e5b4": "m_i \\in T(\\mathcal{M}, \\theta)", "c0777394684a93665289751d129f90e3": "\\delta[n-n_0]", "c07787bc4b0d8b704b888e70786d5545": " Q_{ab}=R_{ab}-\\frac{1}{4}g_{ab}R", "c077b86530087399a38fd58301d368a7": "g_x", "c078082976f1cd25e1c74b629ffca658": "B=\\rho gR^2/\\gamma", "c0784f0a18b04de26d47ccc60928a2c8": "F = \\frac{1}{4\\pi\\epsilon_0} \\frac{Q_1Q_2}{r^2}", "c07897f2283d406597d16522b7f7850c": "\\tfrac{33953}{3628800}", "c078a96f4ab5eb77b411ac04156b3490": "\\Delta G^\\ddagger", "c078bcc07fbcc21e6cb3f293157aba18": "\\eta=a\\mbox{ }\\exp(i\\theta)", "c078d2fa290989773f13b7603af1785b": "M_0=\\mu AD", "c07975bf0ef5df2932261fda42f31de1": "3x=10-x-6", "c079925a8b8b1683f6637bf171f21e46": "\\,^{238}_{92}\\mathrm{U} + \\,^{70}_{30}\\mathrm{Zn} \\to \\,^{308}_{122}\\mathrm{Ubb} ^{*} \\to \\ \\mbox{no atoms}.", "c079a66c7ed51ede7a34cc71d34cb2f9": "D(E_F)", "c079d3feaa4379a7384ce8dcae0fe6ea": "-5\\le x_{i} \\le 5", "c079d9819bc697581c6d7a58daa932fe": "K=\\tfrac{1}{2}\\sqrt{(m+n+p)(m+n-p)(m+n+q)(m+n-q)},", "c07a8f5db55ea46aa2371abc2d6ae7a7": "X(\\omega) = |\\omega|", "c07a9ec159d3fc28896cf47c52692442": "b = \\frac{R\\,T_c}{8p_c}.", "c07ad9649797663d0574775cfe9283b7": "g(x,y) = c", "c07b0a84b683afd3f099e847ff96c8bf": "S(a_{(1)})a_{(2)} = 1_{(1)} \\epsilon(a 1_{(2)})", "c07b0b4d7660314f711a68fc47c4ab38": "GF", "c07b24ed0602f2e398a82d5e2bc765ef": "\\Delta f=\\nabla^2 f=\\nabla\\cdot \\nabla f", "c07b73632445b2cc57626f718ddcf62e": "\\beta_{S}=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_{S}\\,", "c07bdd0952dc898cba7e10f95a5be30c": "\\underline{y} = (\\mathbf{C}-\\mathbf{D}\\mathbf{K})\\underline{x}.", "c07beae5a719c540e6ab10b8e78be773": "a^21 ", "c08bc7910a4a8832ea7e3a1dbafd5ae8": "\\sum_{t=0}^n g(t){n \\choose t}\\left(\\frac{p}{1-p}\\right)^t = 0. ", "c08c01a7aabdeca78cfb4b859a6ebbfc": "N(T)", "c08c0e45a937a414cd20dbdf046ca4b5": "O(|V|)", "c08c3c24c504ccca57fa3bbd8c04dcd3": "\\textstyle \\left\\vert \\alpha\\right\\vert =m", "c08c44718b6e8403a765a580f9a48d60": " A_1,A_2,\\ldots \\in \\mathcal{F}. \\,", "c08c45223539e89efe74bbc56e912ba0": "P(X \\le 74) = P\\left(Z \\le \\frac{74 - 80}{5}\\right) = P(Z \\le - 1.20) = 0.1151", "c08c6feee4c7b93281720b05121c0c20": "\\vec{p}_3", "c08c9c300058193ac0f814179e7043d9": "x_n > K", "c08c9def4e2aeda64dec92243b54bade": "\\sqrt{\\tfrac{32}{5}}", "c08ce887d099d848b454d2afa89746a1": "r=1,2,\\dots,K", "c08cff9ec70a201bbbcbd29459e14d88": "E_{i+1}=E_i\\times E_i=\\{2^{2^i}\\}\\times\\{2^{2^i}\\}=\\{(2^{2^i})^2\\}=\\{2^{2^i\\times2}\\}=\\{2^{2^{i+1}}\\}", "c08d04907b4dad690b1ee0dc14355f09": "\\frac{\\partial \\mathbf{u^*}}{\\partial t} = \\frac{1}{Re} \\nabla^2 \\mathbf{u^*}.", "c08d3d89e0afa6c3540220a0ca16b34e": "\\forall a\\in\\varnothing : x \\le a", "c08d454565d9ee3aa2052f55078d1b0f": " T|e_i \\rangle = \\omega^i |e_i \\rangle ", "c08d7461b5062bcb961a921eba2d9c75": "p\\in a", "c08d8b48c43143e3130274202ef42b48": "x \\triangleleft (y \\triangleleft z)", "c08d9e35e6b2ea706d08919fe6408892": " f(f(x))=x ", "c08dae94cd8810b1b156725531bafb57": "= p(C) \\ p(F_1\\vert C) \\ p(F_2\\vert C, F_1) \\ p(F_3,\\dots,F_n\\vert C, F_1, F_2)", "c08eb593f9d9792ec54bf16234aa053c": " V_{50} = (U^* )^{1/3} f\\left(\\frac{A_d}{A_p}\\right).", "c08ed005202eaed245cfb846065d2e45": "z=0,", "c08f9a059777d22f56a3269abaf3766f": " 0 < \\int_0^1 \\frac{x^4(1-x)^4}{1+x^2} \\, dx = \\frac{22}{7} - \\pi. ", "c08fc5aae09ed81e514a2363336fc8ce": "ATC=\\frac{C_0 + \\Delta C}{Q}", "c08fe44f6f895b84aa247cfdcbaaa744": "\ndV = a^{3} (\\zeta^2+\\xi^2)\\,d\\zeta\\,d\\xi\\,d\\phi\n", "c08ff9eb4a836c1bfa92de01def50b1e": "1/2 \\le \\exp (H(|f|^2)+H(|g|^2)) /(2e\\pi) \\le \\sqrt {V(|f|^2)V(|g|^2)}~.", "c0901b2dfa7ed9c5171637dc3e5b55c0": "P(T) \\approx 10^{-21.94} ~~~~~~~~~~~(10^{5.75} < T < 10^{6.3} K) ", "c0904f4f6a136fb102644d8d02f26d23": "\\psi^{(n)} (1-z)+(-1)^{n+1}\\psi^{(n)} (z) = (-1)^n \\pi \\frac{d^n}{d z^n} \\cot{(\\pi z)} \\,", "c09072584b866940277164a453d05052": "d \\det(A) = \\sum_i \\sum_j {\\partial F \\over \\partial A_{ij}} \\,dA_{ij}.", "c09081a65bca3d3aeb275bb8e8ce0fc7": "w\\Delta z = -a\\operatorname{var}\\left(z_i\\right)", "c090966874337bc20a28d00104755955": "\\beta_3 = \\alpha_1 \\alpha_4 + \\alpha_2 \\alpha_3.", "c09108d0acd92a96016c7215f292c30c": " a \\triangleright (a \\triangleright b) = b ", "c09116c9b0ac66f7421cf74b05308f51": "L_{yyy}", "c0912886405000ebd54e07f9724bbe0c": "\\delta = \\frac{\\alpha}{\\sqrt{1+\\alpha^2}}", "c091a21511a388f9754edbbb59733172": "e^{\\pi|x|^2}f\\in\\mathcal{S} '(\\R^d)", "c091ae473d380ac7b7a8258a31862753": "(\\sigma^2, \\nu, \\gamma)", "c091c75eae5b6064cd71af7e5292a871": " 0 \\to \\Omega^1_{\\mathbb P^n_A/A} \\to \\mathcal{O}_{\\mathbb{P}^n_A}(-1)^{\\oplus n+1} \\to \\mathcal{O}_{\\mathbb{P}^n_A} \\to 0.", "c091dc5f6b443884c44537b7cf976a86": " \\langle A \\rangle = \\frac{1}{M} \\sum_{\\mu=1}^M A_{\\mu}. ", "c091df61fed7b9364bb2f8d83a243170": "\\frac{1}{8g}\\sqrt{\\frac{\\pi}{ch}}(h-1)N^{3/2}+O(hN)", "c09203ed115e12aa101887974d9410df": "F(x) = A(x) + i B(x)", "c09207e3495afbe700a8a73bd5935ebc": "1065353216 \\cdot 2^{-23} - 127 = 0", "c0920b982b6b4888a2d172f3fdf0e56b": "F = \\frac{1}{2} \\sum_{i=1}^N ( k_x v_{i,x}^2 + k_y v_{i,y}^2 + k_z v_{i,z}^2 ).", "c092245792cd670b28048a44ed3904ac": " \\operatorname{ var }( \\tau_y ) = \\tau_y^2 \\operatorname{ var }( r ) ", "c0924a0eb3b61f2448347d64a44271a7": "\\phi_z", "c092507e16014f911e73087c808bfabd": "K = \\mathbb{R}^n_+ = \\{x \\in \\mathbb{R}: \\forall i = 1,\\ldots,n: x_i \\geq 0\\}", "c0929e9f25ff6dadfa7e06be46a94879": "y^+<5", "c092c746dfaa8ddb94099d0b868a407a": "N^J", "c092f6604afd7db524a222ac963cbf47": "\n\\sum_{\\alpha=x,y,z}\\sum_{\\beta=x,y,z} r_\\alpha r_\\beta v_{\\alpha\\beta}(\\mathbf{R})\n= \\frac{1}{3} \\sum_{\\alpha=x,y,z}\\sum_{\\beta=x,y,z} (3r_\\alpha r_\\beta - \\delta_{\\alpha\\beta} r^2) v_{\\alpha\\beta}(\\mathbf{R}) ,\n", "c093babc30c00367ec01bcb03cd704f1": "(T',J')\\to (T'',J'')", "c093e81ad62bf394782cea71f2a56839": "\\mathbf{f}(\\mathbf{v})", "c0944b6b7a2cae3c8bfd5ab80ef1974e": "\\dim \\mathrm{Hom}_K(\\pi, \\mathbf{C}) \\cdot \\dim \\mathrm{Hom}_K(\\tilde{\\pi}, \\mathbf{C}) \\leq 1", "c094a27a550937494620d42d22c4ff4a": "\\mathbf{a_i}", "c094fcd5c7d5558758c04e428dffec9a": "\\int_{-\\infty}^{+\\infty}\\psi(t)dt=0", "c09562a47ce967edfbcc327762f878bd": "I_{in} = \\frac{V_1 - V_2}{Z} = \\frac{(1 - K)}{Z}{V_1} = {(1 - K)}{I_{in0}}", "c095d737291d513672a4e5ba878ee91e": "\\mathfrak{b}", "c095d97de0d6d68ea1d053db9f646d78": " \\left|G\\right| \\cdot M\\left(t\\right) = \\sum_{g\\in G} \\frac{1}{\\det(I-tg)}", "c095ef777bd28f9f63412f9e32ccabe9": "\\mathrm{ {}^{238}_{92}U + {}^{1}_{0}n\\ \\xrightarrow\\ \\ {}^{239}_{92}U\\ \\xrightarrow[23.5\\ min]{\\beta^-}\\ {}^{239}_{93}Np\\ \\xrightarrow[2.3\\ days]{\\beta^-}\\ {}^{239}_{94}Pu\\ \\xrightarrow[2.4\\cdot 10^4\\ years]{\\alpha} }", "c096152dd95807cdd538aeda7aa068ef": "\\frac{1}{k_n(k_{n+1}+k_n)}< \\left|x-\\frac{h_n}{k_n}\\right|< \\frac{1}{k_nk_{n+1}}. ", "c0961da833a0940790ef9db62faf7025": "I(K_m)=\\theta(K_m)\\mathbf{Z}[G_m]\\cap\\mathbf{Z}[G_m].", "c0961dc7402269978e4c91fdfc0f3ead": "SL(2,\\mathbb{Z})", "c09634ee1ddffd58fda5f28e748d39cc": "\\,_nd_x = d_x + d_{x+1} + \\cdots + d_{x+n-1} = l_x - l_{x+n}", "c0967759fb662346a6b12c41c29f6663": "|b_{22}| \\ge |b_{21}| + |b_{23}|", "c096c9abcf220791ccc20e5493a69002": "\\scriptstyle f:X\\to X'", "c097214f72a6d114f920a37df5309775": "a=\\gamma c", "c0972bad0594f81e4a0881dd62b3cc6b": "X_f", "c09738058233ceca5c69754c23562726": "M \\cap F = \\varnothing", "c0976468aab482ad930ccfdf912ee412": "{{P}_{\\theta }}f(u,\\xi )=\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{{{P}_{V}}f({{u}^{'}},{{\\xi }^{'}})}}.\\theta (u,{{u}^{'}},\\xi ,{{\\xi }^{'}})d{{u}^{'}}.d\\xi '", "c09767ae029d722e174bd05b614d98fb": "\\lceil\\log_2(n!)\\rceil", "c097681791af87a5310c069cce8d1029": "A[i+1,n]", "c097a00ad11e7c067747175fa8352196": "\nA_2=A_1 \\chi_0 \\sum^{N-1}_{n=0} (-1)^n \\int^{\\Lambda (n+1)}_{\\Lambda n} e^{i \\Delta k z} \\partial z\n", "c097e64f6889e40b511ba5a94c07646d": "\\mathcal Z(\\beta, \\mu_1, \\mu_2, \\cdots) = \\operatorname{Tr}(e^{- \\beta (H + \\mu_1 N_1 + \\mu_2 N_2 + \\cdots)}) ", "c097ef77efd6c71281b58a619d713b67": "\\frac{1}{a}a = (-a)a = 1 = a(-a) = a\\frac{1}{a}.", "c097f6236769fbdfba0b70a58846a69f": " v = u + a t ", "c0986fb9e2746641125b1ecce91a3c45": " {\\phi} ", "c09877676d4111c49670b22667203065": "\n\\frac{\\partial \\bold m}{\\partial t}\n+ \\bold A_{x,0} \\frac{\\partial \\bold m}{\\partial x} = {\\bold 0}.\n", "c098a6e32537f8c38c33793e399ae8a2": " Q^{(2)} ", "c098a9280be91b3d3188efe0f673b6c6": " \\textstyle L_i ", "c098cd67d79243c824ebf630f07d0419": " \\neg (\\operatorname{true} \\and \\{n\\} \\subset F ", "c099332f11b312531f733c9f71e842c1": "G(\\overline{\\chi})=\\chi(-1)\\overline{G(\\chi)},", "c0994bc35f24baa02e718d2e0d60767e": "\\left(\\!\\!{n\\choose k}\\!\\!\\right) = \\left(\\!\\!{n\\choose k - 1}\\!\\!\\right) + \\left(\\!\\!{n-1\\choose k}\\!\\!\\right) \\quad \\mbox{for } n,k>0", "c09950027ceed8b778d4bf585d847d6c": "T(E)", "c099ba544b828eafcb1aa87d0a956a42": "\\ln(2\\pi)", "c099bb32bb7577293164ef41a9ed6d3a": "\\mathrm{Hom}_\\mathbf{Z} (A,T)", "c099ddadd8b7dc35af8a5ebbccf83e7e": "L=m \\dot\\theta r^2", "c099e17ee408abcea361952d4c3e4b5d": "O(n^{4k-3})", "c09a46a0cfcb92de9ce0c54f5a314e7b": "\n\\omega_{r}^{2} = \\left( \\frac{c^{2} r_{s}}{2 r_{\\mathrm{outer}}^{4}} \\right) \\left( r_{\\mathrm{outer}} - r_{\\mathrm{inner}}\\right) = \n\\omega_{\\varphi}^{2} \\sqrt{1 - \\frac{3r_{s}^{2}}{a^{2}}} \n", "c09a759976b802dde99e572ee8d05233": "f(\\pi_1,\\pi_2, ..., \\pi_m)=0\\,.", "c09a7dc29f8ee8086b437abcfce3fb64": "a_{m} = \\sum^{A}_{n\\neq m} \\frac{H_{mn}}{E-H_{mm}} a_{n} + \\sum^{B}_{\\alpha\\neq m} \\frac{H_{m\\alpha}}{E-H_{mm}} a_{\\alpha} ", "c09ad9333eb53b054ad9f264ab0eb753": "U(\\mathbb{Z}/n\\mathbb{Z}),", "c09b4e858fddf7c6efc19e70f7599e24": " P = \\begin{pmatrix}\n0 & 0 \\\\\n1 & 0\n\\end{pmatrix}.", "c09b5baca421718eda1cdb7e3ad5157f": "\\frac{dy}{dx}=f'(x),", "c09b7865cc7f3acbed103de719d323f3": "c'=c\\,(1-r_s/r)\\approx c\\,(1-\\ell^2_{P}/r^2)", "c09b791a0a50b2e2180b35cf6b4420a6": "\\operatorname{minus}\\ m\\ n = (n \\operatorname{pred})\\ m", "c09c7ece91104395556325b0550a5073": "\\geq\\,\\!", "c09c9639e4573a091201f3fafad0e3b0": " W = - \\alpha P_1 V_1 \\left( \\left( \\frac{V_2}{V_1} \\right)^{1-\\gamma} - 1 \\right) ", "c09c9a8b9737289e278783c82876d324": "r_{t+T} = cY", "c09cc97ec0476911cc84f4cc8777b2e9": "3 - 5 x + 2 x^5 - 7 x^9", "c09d50cd68501cf9734b445044a4efe2": " L^p(Y,\\nu) ", "c09d9aec20d979430a906d7f6919f642": "\\mathrm{Bo} = \\frac{\\rho a L^2}{\\gamma}", "c09d9da0d34d579ef44dfead29950971": "x\\in H_p(M)", "c09dcd673506c6d1755cf49ebf179b90": "I_{\\text{E}n}", "c09e0ef08a1030546ad46cbb2e13d6dc": "\\big \\langle E_1(l, m, t) E_2^*(l, m, t) \\big \\rangle = \\Bigg \\langle A (l, m, t) A^* \\left( l, m, t - \\frac{R_2 - R_1}{c} \\right) \\Bigg \\rangle \\times \\frac{ e^{i \\omega \\left( \\frac{R_1 - R_2}{c} \\right)}}{R_1 R_2}", "c09e19f4cf39644250913908249c0685": "2.9708", "c09e7804a55fe4191d96dba26fa193db": "B-B'", "c09e89846d1cf64b1d9cf64f2c5ee7e9": "O(\\sqrt n)", "c09e9916fd7c64659fe22cea0c67087d": "\\oint_{C} \\frac{f'(z)}{f(z)}\\, dz = \\oint_{f(C)} \\frac{1}{w}\\, dw", "c0a09472b1e844d6d71245362d289276": "\nA = \\sum_u \\sum_v w(u,v) \n\\begin{bmatrix}\nI_x^2 & I_x I_y \\\\\nI_x I_y & I_y^2 \n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\langle I_x^2 \\rangle & \\langle I_x I_y \\rangle\\\\\n\\langle I_x I_y \\rangle & \\langle I_y^2 \\rangle\n\\end{bmatrix}\n", "c0a0d59c25ca405209b650c99fecea4b": "\\cot 0\\text{ is undefined}\\,", "c0a11d4ccca5ccca2dc98e4e0eb37935": "\\left[\\begin{matrix}\n\\varepsilon_{xx} & \\varepsilon_{xy} & \\varepsilon_{xz} \\\\\n \\varepsilon_{yx} & \\varepsilon_{yy} & \\varepsilon_{yz} \\\\\n \\varepsilon_{zx} & \\varepsilon_{zy} & \\varepsilon_{zz} \\\\\n \\end{matrix}\\right]\n=\n\\left[\\begin{matrix}\n \\frac{\\partial u_x}{\\partial x} & \\frac{1}{2} \\left(\\frac{\\partial u_x}{\\partial y}+\\frac{\\partial u_y}{\\partial x}\\right) & \\frac{1}{2} \\left(\\frac{\\partial u_x}{\\partial z}+\\frac{\\partial u_z}{\\partial x}\\right) \\\\\n \\frac{1}{2} \\left(\\frac{\\partial u_y}{\\partial x}+\\frac{\\partial u_x}{\\partial y}\\right) & \\frac{\\partial u_y}{\\partial y} & \\frac{1}{2} \\left(\\frac{\\partial u_y}{\\partial z}+\\frac{\\partial u_z}{\\partial y}\\right) \\\\\n \\frac{1}{2} \\left(\\frac{\\partial u_z}{\\partial x}+\\frac{\\partial u_x}{\\partial z}\\right) & \\frac{1}{2} \\left(\\frac{\\partial u_z}{\\partial y}+\\frac{\\partial u_y}{\\partial z}\\right) & \\frac{\\partial u_z}{\\partial z} \\\\\n \\end{matrix}\\right] \\,\\!", "c0a13e072945e5c3279a026ed8bacd5d": "\\Phi(\\phi)= A * cos (m*\\phi + \\phi^0)", "c0a158b3368489d2f57bf461ea9a715f": "X\\mapsto C_n(X)", "c0a16c8166292a9c4cd977d8bcdf49ad": "\n\\mathbf{B}^\\mathrm{int} \\mathbf{M}^{-1} (\\mathbf{B}^\\mathrm{ext})^\\mathrm{T}\n= \\mathbf{0}.\n", "c0a17b05620e7a2ff163f7fbf04fd181": " A_m {(f)}^2= \\left[\\sum_{i=-f}^f \\operatorname{FFT}(-f,i)^2+ \\sum_{i=-f}^f \\operatorname{FFT}(f,i)^2+ \\sum_{i=-f+1}^{f-1} \\operatorname{FFT}(i,-f)^2+ \\sum_{i=-f+1}^{f-1} \\operatorname{FFT}(i,f)^2 \\right] ", "c0a1d82a363fbb0c565e19e2a67c5cc5": "\\lambda=1s", "c0a1ef3e017535b69700401982a255cc": "\\eta_\\mu=\\frac{p_\\mu}{1-P_0/\\eta}", "c0a2c63ed8a1d031e2896b587db66fa4": "a/q,q1", "c0a899a6db32eb9ab0c114464c932512": "d(X)", "c0a913669d5ab5141de11d5f8eeee808": "\\frac{1}{32} \\, S^{cd} \\, S^{ef} \\, C^{aghb} \\, C_{acdb} \\, C_{gefh}", "c0a952c2c872682a6e79fcd11df5103e": "\\begin{matrix} \\frac{12}{1326} \\approx 0.00905 \\end{matrix}", "c0a95afe62f41af1cea7e9674e4f9137": " (a:b:c) \\times (d:e:f) = (b f - c e : c d - a f : a e - b d) ", "c0a9a9d0e40dc4f826b0c51c01e4f3dc": "\\Phi_{J}", "c0a9e20e31a8ceabcbcf6b78f0505cc5": "\\scriptstyle A_1 \\times A_2 \\times \\dots \\times A_d", "c0a9e40fd706b54c80246e56561073ec": "A_m(4,1) = 1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260,\\ldots ", "c0aa99687cfc9e3d909e3e46c790956b": "\\textstyle\\sum_{i=1}^k p_i=1", "c0aacfd1cf40856c666a29501148436c": "\nx\\,y'' + (1 - x)\\,y' + n\\,y = 0\\,\n", "c0aad8e185b9ffd275aaeb961bc12853": " \\Rightarrow(y_2 + \\frac{10^2}{2(32.2)(y_2^2)} = 8.04)", "c0ab0bdbfd711d160cb4f12af4fb1fcc": "\\delta r_{yield}^{steepening} = \\sum\\limits_{i = 1}^m {KRD_i \\cdot \\delta y_i^{steepening} } ", "c0ab1e4bf6ad9218775689b57add25fc": "\\{Z\\}", "c0ab910e3eced14018bce4ee92e49fcf": "x_3-x_2", "c0aba74853b1829efcecc4962027b0f5": "Q = -\\frac{d}{d x}\\left(EI\\frac{d^2 w}{d x^2}\\right)", "c0abaf180f64c6d918a175cf8840cc6f": "\\operatorname{Cl}_{2}(2\\theta) = 2\\operatorname{Cl}_{2}(\\theta) - 2\\operatorname{Cl}_{2}(\\pi-\\theta) ", "c0abd32d6da914e96d4390ab16d25073": "\\begin{Bmatrix} q , p \\\\ r \\ \\ \\end{Bmatrix}", "c0abda4cb329c68cf677c243bc4e2d12": "a_1x_1 + a_2x_2 + \\cdots + a_nx_n < b\\ ", "c0abed8e72674063d53dbfa99c3ec853": " \\frac{d}{|A|}\\ ", "c0ac769b25ed18f5297e4f146720a4db": " \n\\frac{\\mbox{Luminance difference}}{\\mbox{Average luminance}}.\n", "c0ac8b12bab590e951e0afb93938cdcb": "P_3", "c0acd6b36abda26cb1326e062473b3c6": "\\sigma_{x_{i}}", "c0ace7ab30b8b79c3d178a63cd499705": " q \\geq 2 ", "c0ad214fdc23203165026d0cf9a1f44a": "\\kappa = (2\\pi f_n)^2 I\\,", "c0ad310ad750a2044f056146319a1bc7": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) \\ge 1.", "c0ad53f610dbd41b4806e6d35103db4b": "\\frac{\\partial\\eta}{\\partial t}=w'\\left(0\\right),\\,", "c0ad81afb7a5660194be023d9f55a60e": "\\{x_k:k 0.4", "c0b796b71a375747f1ae405d500ee93f": "E_\\text{inst,relax}", "c0b7d9c6e1fbc9050cd685c377412eb0": "\\left( a_{i}, b_i \\right)", "c0b7ed09d2ff597863b093d1134385dc": " c = \\frac {2 n ^ 2} { \\left( n + 1 \\right) \\left( 2 n + 1 \\right) } ", "c0b8343807a7b895a02e2dee8143e8b1": "\\,[-\\omega;\\omega]", "c0b888d958dce8c1104d1b4d185fa2f0": "3\\frac{\\ddot{a}}{a} =-4 \\pi G (\\rho+3p)=-4\\pi G(1+3w)\\rho, ", "c0b8b474619760e61a57bba6f06ce883": "\\int \\left| \\tan{ax} \\right|\\,dx = \\frac{-1}{a}\\sgn(\\tan{ax}) \\ln(\\left|\\cos{ax}\\right|) + C", "c0b94205aece5bdfd664ad47af2f79c2": "~\\gamma \\gg 1~", "c0b975cada43fca49699e3b8866a8bb0": "x_1 : \\sigma_1, \\; \\dots, \\; x_n : \\sigma_n", "c0b98143631661f07a869d89b8e52618": " Z = \\frac{ r_o - r_e } { SE } ", "c0b9b9f0d656764319d845abb0fbc1ba": "\\textstyle \\left(\\frac{a}{n}\\right) = 1", "c0ba1d86f99cc5b398d280a7b835d37e": " I \\otimes T (\\rho) ", "c0ba400ebbc3bd0d817acc6965411059": "k_{L} = -1", "c0ba4c3c3de4fbb1906855df33155f6e": " \\Box A^{a} = {{A^{a; }}^{b}}_{ b} ", "c0ba94d1f387bf677005a06823ac7f6a": "\\,K_iCK_j", "c0bab1034e40774420325e09aeb2e001": "\nS(A:B|\\Lambda) =\nS(\\varrho_{A, \\Lambda})\n+S(\\varrho_{B, \\Lambda})\n-S(\\varrho_\\Lambda)\n-S(\\varrho_{A, B, \\Lambda})\n\\,", "c0bae56212a5fb9d7e67be515675ea0f": "\n\\dot{Q_{ij}} = A_i F_{ij} (J_i - J_j) = \\frac{J_i - J_j}{R_{ij}} \\qquad \\text{where} \\quad R_{ij} = \\frac{1}{A_i F_{ij}}\n", "c0bb008f622c6509bb68a0b5389a815b": "\\overline{\\bold{u}}_p = \\frac{1}{\\overline{\\theta_p \\rho_p}}\\int\\!\\!\\!\\int \\!\\!\\! \\int \\phi\\Omega_p \\rho_p \\bold{u}_p \\; d \\Omega_p d \\rho_p d \\bold{u}_p", "c0bb77aa87298335dff53dc20141e23b": " 1 \\leq i \\leq t ", "c0bb8a66663e49ee7e8eab535be81782": "S/ D", "c0bb98d3a21aa837b46e8e7a6d61416a": "\\sim \\forall x \\sim F(x)", "c0bba5805a8e15fd8059fe2e275cea4a": "\\Gamma(V)", "c0bbabe6f07b7bfa1662c131d8accd7e": "\\sum_{i} \\frac{\\lambda_i^2}{1+\\lambda_i^2} = 1,2 or 3", "c0bbf3bfe4db55839ecf81bc27709fbc": "C^* = C_{out} \\circ C_{in}", "c0bbf95f078fcfe8abca0723ca782aa0": "\\lim_{n\\to\\infty} \\, \\max_{1 \\le k \\le n} \\; P( \\left| X_{nk} \\right| > \\varepsilon ) = 0 \\text{ for every }\\varepsilon > 0.", "c0bc1994fdda9476322fd87578258d62": "\\pi_1 x_1 + \\pi_2 x_2+\\cdots+\\pi_{k-1}x_{k-1}\\geq x_{k}\\,\\!", "c0bc787cf8ccb50efbbcdf41665fe633": "x =\\text{mode} - \\kappa = \\frac{\\alpha -1 -\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "c0bcb6216ab1c4056cd6c816b18ff255": "\\mathfrak{p}\\supseteq I", "c0bcdf0eb7818b93e386ffd5a0b7047a": "\\frac{27379}{8658}=3.162277662\\dots", "c0bcfe66aef592516d280918567d0d78": "\\theta(\\alpha_c(n))", "c0bd0253f196e54cefb3b7db83bab8bf": "SBP_\\gamma", "c0bd5592927dd80009cd6891b5530d79": " y-f(a) = k(x-a).\\,", "c0bdd8b6b5d3a78b3da146831aecd67c": "\\lim_{x \\to c} \\, f(x)^{1 \\over n} = L_1^{1 \\over n} \\qquad \\text{ if }n \\text{ is a positive integer, and if } n \\text{ is even, then } L_1 > 0", "c0be070b5f6d33c6d97c49f1d748fa69": "\\begin{bmatrix} Z_1 & Z_2 \\end{bmatrix}^{*} A \\begin{bmatrix}Z_1 & Z_2\\end{bmatrix} = \\begin{bmatrix} \\lambda \\, I_{\\lambda} & A_{12} \\\\ 0 & A_{22} \\end{bmatrix}: \n\\begin{matrix}\nV_{\\lambda} \\\\\n\\oplus \\\\\nV_{\\lambda}^{\\perp}\n\\end{matrix}\n\\rightarrow\n\\begin{matrix}\nV_{\\lambda} \\\\\n\\oplus \\\\\nV_{\\lambda}^{\\perp}\n\\end{matrix}\n", "c0be17d4cb166c5ce5cf3f806bbe379f": " V^\\mu U^\\nu {}_\\sigma = (V \\otimes U)^{\\mu \\nu} {}_\\sigma. ", "c0be226b95be5cf9f81138c78c40c87d": "\n (\\or R_2)\n ", "c0be7c8248709289edbda8e1082ad6e4": " \\langle \\phi, \\psi \\rangle = \\int_G \\phi(t) \\psi(t^{-1})\\, dt. ", "c0bec85897d7e4358c8391273156efc0": "F_{ab} = \\begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\\\ -E_x/c & 0 & -B_z & B_y \\\\ -E_y/c & B_z & 0 & -B_x \\\\ -E_z/c & -B_y & B_x & 0 \\end{bmatrix}", "c0bf2ce22ad11dd24c909f604d942303": "P(R_{i})=\\text{TRUE}", "c0bf54d6c4a746a6ad9639835d12e7c4": "S(abbabbabb) \\Rightarrow A(abb, abb, abb) \\Rightarrow A(bb, bb, bb) \\Rightarrow A(b, b, b) \\Rightarrow A(\\epsilon, \\epsilon, \\epsilon) \\Rightarrow \\epsilon", "c0bfa52dddbbc565cdb032935ce07513": "\\{\\,ae_\\lambda:\\lambda \\in \\Lambda\\,\\}", "c0c02285359af0780cc6be1f2d197d72": "\\begin{align}\n p(z) &= \\frac{1}{2}\\cdot \\|Z P(z)\\|_F^2 \\\\\n q(z) &= \\left|\\det [Z P(z)]\\right|^2 \\\\\n \\|P\\|_2 &= \\max\\left\\{\\sqrt{p(z) + \\sqrt{p(z)^2-q(z)}} : z\\in\\mathbb{C}\\ \\land\\ |z| = 1\\right\\} \\\\\n \\|P^{-1}\\|_2^{-1} &= \\min\\left\\{\\sqrt{p(z) - \\sqrt{p(z)^2-q(z)}} : z\\in\\mathbb{C}\\ \\land\\ |z| = 1\\right\\}\n\\end{align}", "c0c0232c66e49e59bb2fd635e25796c0": "X_{k+1} = \\tfrac12 (X_k + A X_k^{-1})\\,.", "c0c03c362f132653d4395e9ff32ac933": "\\operatorname{D}_2 (z)", "c0c044dbc23de9ba302719bdb1e5e75f": "T = \\frac{1}{2} \\sqrt{\\begin{vmatrix} x_A & x_B & x_C \\\\ y_A & y_B & y_C \\\\ 1 & 1 & 1 \\end{vmatrix}^2 +\n\\begin{vmatrix} y_A & y_B & y_C \\\\ z_A & z_B & z_C \\\\ 1 & 1 & 1 \\end{vmatrix}^2 +\n\\begin{vmatrix} z_A & z_B & z_C \\\\ x_A & x_B & x_C \\\\ 1 & 1 & 1 \\end{vmatrix}^2 }.", "c0c083deee03e40128f7fbf9bcbe610f": "\\int_{\\mathbb{R}^{n}} \\prod_{i = 1}^{m} f_{i} \\left( B_{i} x \\right)^{c_{i}} \\, \\mathrm{d} x \\leq D^{- 1/2} \\prod_{i = 1}^{m} \\left( \\int_{\\mathbb{R}^{n_{i}}} f_{i} (y) \\, \\mathrm{d} y \\right)^{c_{i}},", "c0c0996329604bf35f90f68569d37b32": " f_\\pm(k\\xi ,\\dots)", "c0c0b738a30d86443de42a923f732a9d": "y\\in \\mathrm{conn}(x)", "c0c1331fba7c20b749add663ec3d246d": "s=\\frac{hq}{f}+f\\ln\\left(\\frac{h+q}{f}\\right)", "c0c1a6b544c911db0dbb14e9666ac14d": "q^m", "c0c20cb9c098bf1506f17477e6ccf24e": "\\liminf_{n\\to\\infty}x_n := \\lim_{n\\to\\infty}\\Big(\\inf_{m\\geq n}x_m\\Big)", "c0c272b61135dc8277de2bf2f9e5922f": "\\rho_1\\ :\\ f(g(x), y) \\rightarrow y", "c0c2937a9a48553c5d1dd70e515a7e7d": "g(E)", "c0c29bbc1d93f505c1ca098b6b4b311d": " F_2(w) = \\sum_{n=-\\infty}^{+\\infty} F_1(w+n f_x) ", "c0c2c25e6514bc74d917517abc8c01ad": "\\Phi_{02}=\\delta\\tau-\\Delta\\sigma-(\\mu\\sigma+\\bar{\\lambda}\\rho)-(\\tau+\\beta-\\bar{\\alpha})\\tau+(3\\gamma-\\bar{\\gamma})\\sigma+\\kappa\\bar{\\nu}\\,,", "c0c2c4e2c97899a9e95361fac2bf8b7a": "r = \\frac{1}{Ns^2} \\sum_{n=1}^{N} (x_{n,1} - \\bar{x}) ( x_{n,2} - \\bar{x}) ", "c0c2d412a8a0881110b570b5bd6af007": "\n\\ell^\\prime(\\beta) = \\sum_{i:C_i=1} \\left(X_i - \\frac{\\sum_{j:Y_j\\ge Y_i}\\theta_jX_j}{\\sum_{j:Y_j\\ge Y_i}\\theta_j}\\right),\n", "c0c2fb15a163ece885810dde2572e938": "\\gamma v t' = \\gamma^2 v t + x \\left ( 1 - \\frac{1}{1-\\beta^2} \\right )", "c0c316197d9157bd988d5b345828812d": "\\mathfrak{P}^{9}", "c0c31c9e6aef46c1c4f619a6e613ebca": "L \\subset M", "c0c38e345fa8db2b541b6703f85ccabd": "\\int d^3\\mathbf{x'}\\mathbf{J}(\\mathbf{x'})=-\\int d^3\\mathbf{x'} \\mathbf{x'}(\\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x'}))", "c0c3a6f4093689fe61a55ced90e41a55": "E[Z]", "c0c405ad82f07afccb3929b71ccb42b8": "\\frac{\\partial \\mathcal{H}(q,p)}{\\partial q} = \\frac{q(t)}{C} = -\\dot{p}(t) \\ ", "c0c406a446fc053348c5746df709e48d": "C_\\beta ^{(i)} ", "c0c470b50a7762e2b37fd44a9c88e134": "f(x_1,x_2,\\ldots,x_{i-1},0,x_{i+1},\\ldots,x_n) \\ge f(x_1,x_2,\\ldots,x_{i-1},1,x_{i+1},\\ldots,x_n).\\,", "c0c4774ba31cae0b7d93c5e7e4c77a5b": "-\\tfrac{8}{5}", "c0c48eb8e595edca99d5418318fdddfd": "\\operatorname{env_1}(A) = \\inf_{p \\in C} p(A)\\,\\,\\,\\,;\\,\\,\\,\\,\n\\operatorname{env_2}(A) = \\sup_{p \\in C} p(A)", "c0c4deb6506d77baff166c7c8454e13d": "\\sum_j a_{ij}x_j=0,\\qquad \\sum_j a_{ij}y_j=0", "c0c4dedb1cd3182cf60262f9665373ea": "\\int_0^\\infty \\frac {e^{-ax}(1-\\cos x)}{x^2}\\ dx=\\cot^{-1} a-\\frac{a}{2}\\ln(a^2+1)", "c0c4e0485ffad68f1c0c2177fa8a14bf": "N \\cong M_\\alpha", "c0c51c6bceaf456b61b50bb287bbf588": "\n \\nu =\n{2\\pi N \\hbar c \\over eBA}\n", "c0c5346169e67307e14713491de94902": "\\{ v_1, \\ldots, v_m \\} \\subset H_2", "c0c5e002344e514320305de4c90ae66a": "|D\\rangle=\\cos\\theta|1\\rangle-\\sin\\theta|2\\rangle", "c0c5ef118c0d5b740668f8f74c0208f0": "\\textstyle \\coprod_{i=1}^N x_i", "c0c5f88d440ddf86042b6ab032c2a95e": "n-", "c0c61934d1ffeddcc50aa2bbd3ff3711": "r_\\text{min}", "c0c67f6e2edece3636b5610b1abe5d6d": "\n\\begin{align}\n\\int_0^\\infty \\frac{\\sin(x)}{x}\\frac{\\sin(x/3)}{x/3}\\cdots\\frac{\\sin(x/15)}{x/15} \\, dx\n &= \\frac{467807924713440738696537864469}{935615849440640907310521750000}\\pi \\\\\n &= \\frac{\\pi}{2} - \\frac{6879714958723010531}{935615849440640907310521750000}\\pi \\\\\n &\\simeq \\frac{\\pi}{2} - 2.31\\times 10^{-11}\n\\end{align}\n", "c0c6a169c2aea46c38b89e4cbc99aa0e": " f(W) ", "c0c76314c0bacea45891817aa8bf97d9": "\\mathcal{H}^1(\\partial E) = 5", "c0c783b3c9bde937abc0c4c9d89b79af": "\\begin{align}\n(\\mathcal{F}f')(\\xi) &= \\int_{-\\infty}^\\infty e^{-2\\pi iy\\xi} f'(y)\\,dy \\\\\n&=\\left[e^{-2\\pi iy\\xi} f(y)\\right]_{-\\infty}^\\infty - \\int_{-\\infty}^\\infty (-2\\pi i\\xi e^{-2\\pi iy\\xi}) f(y)\\,dy \\\\\n&=2\\pi i\\xi \\int_{-\\infty}^\\infty e^{-2\\pi iy\\xi} f(y)\\,dy \\\\\n&=2\\pi i\\xi \\mathcal{F}f(\\xi).\n\\end{align}", "c0c789e24e6f093cceb81fc0166c340b": "L = \\frac{r^2N^2}{8r + 11d}", "c0c7918e4473e66290506229ceaf4241": "\\mathbb{C}^N", "c0c7c76d30bd3dcaefc96f40275bdc0a": "50", "c0c7ccb4e8c909f2c4401b8cece938df": "\\frac{f(x_i)}{\\sum_{j=1}^d f(x_j)}.", "c0c8079fb7d40d204108f3e88442d8a5": "\\Phi(t_0,\\tau)B(\\tau)", "c0c8156de7a5455113e67f33c15182fb": "M'", "c0c8467e79e588868acb848b5a868da4": "-\\frac{R_1}{R_2}{V_s}", "c0c8771843358f24e256d5404d7fcc0c": "4(a^2+a b+b^2) = d^2", "c0c8957625fec695feb20f7ebef4840b": "\\Phi(z,s-1,a)=\\left(a+z\\frac{\\partial}{\\partial z}\\right) \\Phi(z,s,a)", "c0c8af3f136985a9ddd5f2ead5473129": "(n_\\mathrm{obs} - 1)^2", "c0c8e0a3b85287b8b12cf4cb24a738ef": "\\Delta \\vec{p} = \\vec{p}_\\mathrm{in} - \\vec{p}_\\mathrm{out} = q (1 - \\cos \\theta) \\hat{z} - q \\sin \\theta \\cos \\phi\\hat{x} - q \\sin \\theta \\cos \\phi\\hat{y} ", "c0c9114f0d5ad2e18c04f560c03f01dc": "F_A = * D_A \\phi", "c0c9139c702f1a842950ed91c63e5ef9": "z = n - c = (10^m \\cdot n_m + 10^{m-1} \\cdot n_{m-1} + .. + 10^0 \\cdot n_0) - (n_m + n_{m-1} + \\cdots + n_0).", "c0c92ded587c27bbe4bf1f97347ce6cf": "\n X_n \\ \\xrightarrow{p}\\ X,\\ \\ \n X_n \\ \\xrightarrow{P}\\ X,\\ \\ \n \\underset{n\\to\\infty}{\\operatorname{plim}}\\, X_n = X.\n ", "c0c939424f7d6698de908628370e7c29": "h = 90^\\circ - z \\,.", "c0c99bf784fb0c74816859edeee234f2": "f: \\mathfrak{X} \\to \\mathfrak{S}", "c0c9e49a25ce4d9afe93c529b8f19dc7": "\\Rightarrow \\ q_2^* = \\frac{a + \\frac{\\partial C_1 (q_1)}{\\partial q_1} - 2^*\\frac{\\partial C_2 (q_2)}{\\partial q_2}}{3}", "c0ca23a1eb48f32d85a115d896108965": " p_{0,2}(x) \\, ", "c0ca350e24ff95f257a0f91e153f631a": "a_\\max=\\sqrt{2\\Lambda|p_1(u)|}", "c0ca626e8fe69af7ce764edae22cf7dd": "{\\mathbf{}}\\hat{P}(t)\\hat{S}(t)", "c0ca8ffe61e174ada8cd3e2350b35de4": "\\begin{align}\n v_{n+1} &= v_n - \\omega^2\\,x_n\\,\\Delta t \\\\[0.2em]\n x_{n+1} &= x_n + v_{n+1} \\,\\Delta t.\n\\end{align}", "c0cb46a7a6acf294121c8ea508f4cfb6": "\\sum_{n=0}^{\\infty}x^n={1\\over1-x}.", "c0cb5f0fcf239ab3d9c1fcd31fff1efc": ",", "c0cb91ae3275f8c6160bc4257ef67ab6": "m_0^2 c^2 = p^a p^b \\eta_{ab}= \\frac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2", "c0cbb6485c3a3ca87fdbc91d9bdcb53f": "(1 - x) (1 + x + x^2 + \\cdots + x^{n-1}) = 1 - x^n = 1.\\ ", "c0ccb3f964a445df67bb70c65c7eac71": "\\omega_n^2 = n^2\\omega^2+4\\rho^2. ", "c0ccc4b6094999c4c048fc2909f72a07": "n_{r2}", "c0cceb67df280ae7d9cb0cfe59a255fd": "R=e^{-\\frac{t}{S}}", "c0cd1c0618bb8a049edea253ac922e5b": "\\textstyle \\{ [x] : x \\in \\Gamma_{\\mathrm e}(V) \\}", "c0cd846b8d88bcece857ba303f785dc3": "\n\\begin{align}\np_{n,t} &\\ge \\prod_{i=0}^{n-t-1} \\Bigl(1-\\frac{2}{n-i}\\Bigr) = \\binom{t}{2}\\bigg/\\binom{n}{2}\\,. \n\\end{align}", "c0cdd339073453c2cea506438410ca85": "\\mathbf{z}\\,", "c0cebce5fd4bab108858473c8c719299": "\\hat{b}^\\dagger", "c0cf1fe0c96edd76016a79d4fd8a05ec": " x < p - 2p^2 + \\frac{1}{4} ", "c0cf5dd2d2557df8f82467fb540b045c": "\\left|\\tau_{cap}(w)\\right|^2 + \\left|\\tau_{ind}(\\omega)\\right|^2 = 1", "c0cf9b5f4a9ce14834de4aee059e977b": "Z=-mU\\frac{d(\\theta-\\alpha)}{dt}", "c0cfa93e5e9158eb22cce206dacd21e6": "T_RS", "c0cfc3dfb07cdb904fd6726ddae7a372": "M(P) \\ge M(x^2 -x - 1) \\approx 1.618 . ", "c0d063ce00ce1f80bf094d4ca81767d6": "1.28 > \\beta \\ge 1", "c0d0693da6e6ae2cafa19ea02b4ba6ac": "\\lambda_{1}= -0.0000046 + 5.4280259i", "c0d0bafa786ddace7cb3a8a209484454": "X\\,\\!", "c0d159b83291dae1b3e268e3c0621663": "T\\left(E,eV,r\\right)", "c0d17accc2259d283708222f343b993d": " h(k, m_2, m_1 )= (L(m_1 ) +_{32} L(k) )\\cdot(U(m_1 ) +_{32} U(k) ) +_{64} m_2 \\, ", "c0d18d74cb383585627cb2edf9dd4ca2": "ARV_{total} = ARV_1 + ARV_2 + ... + ARV_n", "c0d1ce642b281c77d35a849831c9bb85": " \n \\begin{align}\n \\frac{\\partial \\eta}{\\partial t}\\, &+\\, u\\, \\frac{\\partial \\eta}{\\partial x}\\, -\\, w\\, =\\, 0 \n \\\\\n \\frac{\\partial \\varphi}{\\partial t}\\, &+\\, \\frac{1}{2}\\, \\left( u^2 + w^2 \\right)\\, +\\, g\\, \\eta\\, =\\, 0,\n \\end{align}\n", "c0d1e8a33f398c946bc90e311c8554af": "U_t\\cap U_{t'}.", "c0d1f47e0a0855a0203b919ada293146": "\\cdots \\triangleleft G^{(2)} \\triangleleft G^{(1)} \\triangleleft G^{(0)} = G", "c0d22a13fea10945f480d96ac0dd5be5": "n{\\rm C} + (n - x + 1){\\rm H}_2 \\rarr {\\rm C}_n{\\rm H}_{2n - 2x + 2}", "c0d28f082c5948736ccb9dbd16ebeead": "d_\\text{f}", "c0d312d3ca00c8114e3ebb353b2d708c": "E \\subseteq L_{x} \\implies \\mu (E) = 0.", "c0d465af910ad40b2a5ad2c681227a7b": "\\mathbb{P}_x", "c0d4e2848f9e63aad8b54936367d567e": " n_{i,j,k}", "c0d54788571a522adc1209baa66eaef3": " \\psi(\\sigma \\frown \\varphi) = (\\varphi \\smile \\psi)(\\sigma)", "c0d5a3fe5c44cecdd50da781def0cf3a": "\\lbrack\\mathbf b\\rbrack = \\lbrack\\mathbf a\\rbrack^{-1}", "c0d5c16b19a77a239186151628148eff": "\\frac{\\partial}{\\partial x_i}F_j - \\frac{\\partial}{\\partial x_j}F_i = 0", "c0d6106a1db91f83c4a634bb9e292e8c": "r\\underline{v}\\in\\overline{\\underset{=}{A}(kU)-\\underset{=}{A}(kU)}\\subseteq\\overline{\\underset{=}{A}(2kU)}", "c0d63708c3a2c5838654d8d5d4998257": "\\sigma_n(x)=\\frac{1}{n}\\sum_{k=0}^{n-1}s_k(x)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi f(x-t)F_n(t)dt,", "c0d69d060b89ba43d6eacd0eb6736ea3": "P = NkT/V", "c0d6fc7e356a1fa3ac61351d029ede50": " v = \\sqrt{ Zk_\\mathrm{e} e^2 \\over m_\\mathrm{e} r}. ", "c0d713989bc57e9f61f25432d1c55ea5": " h'(x) = -\\frac{f'(x)}{[f(x)]^2}.\\ ", "c0d73e3a27128654dd7ab96196d902f6": " j=j+1.\\,", "c0d76bbbcabca8d19fda97e3e3ce6333": "D_\\mathrm{max} = \\frac{1}{2} \\cdot c \\cdot t_0 = \\frac{1}{2} \\cdot 300\\;000\\;000\\;\\frac{\\mathrm{m}}{\\mathrm{s}} \\cdot 0.000\\;000\\;05\\;\\mathrm{s} =\\!\\ 7.5\\;\\mathrm{m}", "c0d8215890da64002dcb13c2f695045b": "\\left(\\sum_{i = 0}^{\\infty} a_i x^i\\right) \\left(\\sum_{i = 0}^{\\infty} b_i x^i\\right) = \n\\sum_{i = 0}^{\\infty} x^i \\sum_{j = 0}^i a_{i - j} b_j", "c0d8234f3e2ce67558b10a2f2e368701": "F[x,y] = \\int_a^t\\sqrt[3]{y''}\\, dt ", "c0d84fd54f7b69d901702b1bf1c10e7a": "\\frac{648,000\\ \\mbox{MW·h}}{(30\\ \\mbox{days}) \\times (24\\ \\mbox{hours/day}) \\times (1000\\ \\mbox{MW})}=0.9 \\approx{90%}", "c0d891f733a4714d1e84fb8de4892308": "\\forall a_0,\\dots,a_{n-1}\\;\\exists s\\; \\left(f\\left(a_0,\\dots,a_{n-1},s\\right)=0\\right)", "c0d8b4b800fcd5f789307e7b57bd6c2f": "d\\times d", "c0d8c562283f4ce55e49e755c5058a69": "\\deg(A)=a", "c0d9127cb9f56a95b46544efcc77f3dc": "\nT_{\\delta}^{X^{n}}\\equiv\\text{span}\\left\\{ \\left\\vert x^{n}\\right\\rangle\n:\\left\\vert \\overline{H}\\left( x^{n}\\right) -H\\left( X\\right) \\right\\vert\n\\leq\\delta\\right\\} ,\n", "c0d92ca4c11858f2ddef538b362bd4ec": "\\lambda=\\,", "c0d963c9f31a207c2898a585e46ae206": "sum(A[i,j]) = (a_i+\\ldots + a_j)", "c0d988aaeac88dbe6800ac5c47be9221": "\\mathbf{e}_7 \\times \\mathbf{e}_1 = \\mathbf{e}_3, \\quad \\mathbf{e}_1 \\times \\mathbf{e}_3 = \\mathbf{e}_7, \\quad \\mathbf{e}_3 \\times \\mathbf{e}_7 = \\mathbf{e}_1.", "c0d9a93bd96609bb25290e82054a265d": " \\begin{align}\nF(x,y) & = \\int^y M(x,\\lambda)\\,d\\lambda + \\int^x N(\\lambda,y)\\,d\\lambda \\\\\n & + Y(y) + X(x) = C \n\\end{align} \\,\\!", "c0d9aeb20aff9981a28e148ccbdd08bb": "{\\mathfrak b}({\\mathbb P})=\\min\\big\\{|Y|:Y\\subseteq{\\mathbb P}\\ \\wedge\\ (\\forall x\\in {\\mathbb P})(\\exists y\\in Y)(y\\not\\sqsubseteq x)\\big\\}", "c0d9da54d3b02476dc22026a3c2da486": "h_{11} = \\left. \\frac{V_{1}}{I_{1}} \\right|_{V_{2}=0} ", "c0da00c9fb54a833b4f346a7e192069f": "\\int_0^\\infty e^{-ax^{2}-b/x^{2}}\\ dx=\\frac{1}{2} \\sqrt \\frac{\\pi}{a}e^{-2 \\sqrt{ab}}", "c0da1c6eb01b23556de2d080f342c561": "K=-\\Delta \\log \\lambda.\\,", "c0da4eaee1d8ae16c73215345d645ae7": "M + e^- \\to M^{+\\bullet} + 2e^-,", "c0da62af72ae110cf0df2d615adf893e": "\\frac{\\pi^{d/2}}{d2^{d-1}\\Gamma(d/2)}\\rightarrow 0", "c0dad7417b8ec262e35434c385714c9b": "b_{i}^{*}:= b_{i}^{*} - \\mu_{i,j}b_{j}^{*}", "c0dbd1b5e2c74395d4fee05d5c3efe85": "x_n=q_+^{\\;n}", "c0dbe7fde95a39e411875c6655d34058": "(\\alpha, \\beta, f)\\mapsto \\alpha", "c0dbeb9700c6b7f91a4ff194824e3767": "f_{os} = 203 250 000 + \\frac{15626\\cdot 8}{12} \\approx 203260417", "c0dc3bc74927a0c8d95bc4c269d0c394": "u^\\alpha = \\gamma(c,\\bold{u}) \\,", "c0dc4fe9a6d9f8d7aed138e17a28e813": "g_{\\mu \\nu ; \\gamma} = 0 \\,.", "c0dcb97257b4b84b0e9f43c0718c5964": "C^{n-i} M", "c0dcd62af218e21d2da4becbe73cbe6a": "E\\left[e^{tX}\\right] = \\sum_{n=0}^\\infty \\frac{t^n\\lambda^n}{n!}\\Gamma\\left(1+\\frac{n}{k}\\right).", "c0dd57e94bfbe5670e2e4e5b760e5ac8": "FRS_{\\mathbb{F},\\gamma,m,k }", "c0dd6e870d310bb5ed223031d94c574c": " \\mbox{orb}(f) = \\{ \\psi \\circ f \\circ \\varphi^{-1} : \\varphi \\in \\mbox{diff}({\\bold R}^n), \\psi \\in \\mbox{diff}({\\bold R}) \\} \\ . ", "c0dd73fae1c008b4f8934ecc206f7be4": " D'(c)=0", "c0dd7af9ca00cc99c59437bb72069285": "\\scriptstyle \\sqrt{17/9}", "c0de075d30f56203aa13987ce6fe2548": "d(0),d(1),\\ldots,d(k)\\!", "c0de1d3761f3f90b1c05294005ddae6f": "\\int_E \\mathrm{D} F(x) (i(h)) \\, \\mathrm{d} \\gamma (x) = \\int_E F(x) \\langle h, x \\rangle^\\sim \\, \\mathrm{d} \\gamma (x).", "c0dea85ea4dd65b7e2065fd1d8506b37": "\\mathsf{i}", "c0dea91ca3cd718cc2cbfce7b01f3d48": "\\bot_{\\mathrm{nM}}(a, b) = \\begin{cases}\n \\max(a,b) & \\mbox{if }a+b < 1 \\\\\n 1 & \\mbox{otherwise.}\n\\end{cases}", "c0dedf54e3a3c83793e3acba202f95fe": "\\chi_{D_{0}} \\big(p \\big) = 1", "c0deeb5bc5e399db640da79f4be84cc3": "\\ h = \\frac{1}{\\sqrt{s^\\mathrm{H} R_v^{-1} s}} R_v^{-1} s. ", "c0df1f9fd5a071df2aa7cc3255506e7a": "\\textstyle J(r(i,j))", "c0df4322e7cf15ff821981a35a4b6a7e": "q_i(F_S)= 1-F_S", "c0df7869dff85909d9c5a77eb1c82177": "F^\\mu", "c0dfc4a422fed24a9b93a8e04d021a77": "\\neg(P \\or Q) \\vdash (\\neg P \\and \\neg Q)", "c0e03f1b88c179137c2bbe76a5d3f3d9": "[\\mu_1,\\mu_2]", "c0e0ccbb543a30d58aa7dded93c95fdc": " L^2_\\nu(\\widehat{G}) \\rightarrow L^2_\\mu(G) ", "c0e0d520fc076ad81f54ac1c5f907936": "\\alpha^{jk_i} \\in \\mathrm{GF}(p^{m_i})", "c0e0e10dabec970a68ea10c7ac75f6c6": "\\textbf{G}(s)=\\frac{\\textbf{P}(s)}{\\textbf{Q}(s)}", "c0e0f461941c0d8c56ee391030c4b0f2": "A^p(D)", "c0e12ea987f0f88770ec4ee8311ad999": "~A \\cup B", "c0e13df65ea4b0cc5c58aa3064fab44a": "\\mathrm{III}_T(t) = \\frac{1}{T}\\sum_{k=-\\infty}^{\\infty} e^{i 2 \\pi k t/T}.", "c0e153884fe358df762f07a05bdc5dd2": "TR=P \\cdot f(P)", "c0e16308c4ac78aa80b1c583b09dbedf": " \\sum_{i=1}^n a_{ij} x_i \\geq b_j \\text{ for }j=1,\\dots,m", "c0e16d79faa0c87cfb95d2d0dcf98a27": "h_0,\\!\\ h_1, h_2", "c0e17a3c7d8e0c28e051b441b5723266": "\\exp\\left(-S_{\\Lambda'}[\\phi]\\right)\\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{\\Lambda' \\leq p \\leq \\Lambda} \\mathcal{D}\\phi \\exp\\left[-S_\\Lambda[\\phi]\\right].", "c0e1867bcc8494f99a50758d4162ca58": " \\overline{a}\\cdot z = \\sum_{j=1}^n \\overline{a_j}z_j. ", "c0e1c5cd6fc1bc8aa8fcc07e2c99c8c6": "-x^4+763200x^2-40642560000=0", "c0e1db1d06f9f720488c97e5f0ca5375": "\\frac{\\alpha}{\\beta-1}", "c0e212f1eea8eae6e38136b3482809ec": "\\sigma_{ij}=\\sigma_{ji}\\,\\!", "c0e2ab6472bac4c60fd5970699f7700c": "U(x_0) = I", "c0e2db57c0a521eba020573741521a16": "\\scriptstyle A_k", "c0e2f8c3e6e485948f3cc5e23e1ad2a5": "2^{S}", "c0e3a786466ba222353b8118f397ee13": "\\textstyle \\prod_{} y \\hbox{, } (y \\hbox{ in } S)", "c0e4343d6ec85bfcc9258d72039ff08c": "BS'C = BAC - \\pi/3", "c0e46d4c908766f6783b14291952c9c5": "\\sqrt[4]{x}", "c0e4fcda898ff4847d212b4e65fcee36": " W v_z = m\\dot{V}V,", "c0e534f02a41e189e46de7b1685feb61": " A:\\mathbb R \\times M \\to M ", "c0e550631ecece9cb93098d526029a82": "\n\\sqrt{2} P(0) = \\sqrt{\\frac{h}{2 \\pi}}\\;\n\\begin{bmatrix}\n0 & i\\sqrt{1} & 0 & 0 & 0 & \\cdots \\\\\n-i\\sqrt{1} & 0 & i\\sqrt{2} & 0 & 0 & \\cdots \\\\\n0 & -i\\sqrt{2} & 0 & i\\sqrt{3} & 0 & \\cdots \\\\\n0 & 0 & -i\\sqrt{3} & 0 & i\\sqrt{4} & \\cdots\\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\end{bmatrix},\n", "c0e5d7fc210b76554f6172eae1e5d5fd": "u=w/t", "c0e5dc393e08abec53b06d8d6b660f8a": " \\operatorname{E}[X] = x_1p_1 + x_2p_2 + \\dotsb + x_kp_k \\;.", "c0e5dcd7d72306c06a725cce6729ea29": "X= e^{\\mu + \\sigma Z}\\,", "c0e5e8bcee9f6321c590b0984e005776": " \\exists{x}{\\in}\\mathbf{X}\\, P(x) \\to\\ ((P(c) \\to\\ Q) \\to\\ Q)", "c0e614fb91c9a82c4c1ebf0ee6dfbf23": " -i \\sigma_z \\frac{\\omega_r}{2}e^{-i \\sigma_z \\omega_r t/2}\\psi + e^{-i \\sigma_z \\omega_r t/2}\\frac{\\partial \\psi}{\\partial t}=i\\left(\\omega_1\\sigma_x \\cos{\\omega_r t} + \\omega_1\\sigma_y \\sin{\\omega_r t} + \\omega_0 \\sigma_z\\right)e^{-i \\sigma_z \\omega_r t/2}\\psi ", "c0e69e0a246e0f31625ae8cbe10ded52": "{\\mathfrak{T}_{\\mu}^{\\nu}}_{; \\nu} \\, + \\, f_{\\mu} \\, = \\, 0 \\,", "c0e72d20a644fc6498a087a0f1fc71b6": "P(\\{\\alpha_{\\mathbf{k},s}\\})=\\prod_{\\mathbf{k},s} \\frac{1}{\\pi \\langle\\hat{n}_{\\mathbf{k},s}\\rangle} e^{-|\\alpha|^2 / \\langle\\hat{n}_{\\mathbf{k},s}\\rangle}.", "c0e76b820bdb34ebbad920218b9791e8": "\\rho_{\\parallel,\\perp}", "c0e80a45d4ccaf43c7c8bcb4ce8cebfd": "h_1 = c \\sin A\\text{; } h_1 = a \\sin C \\,", "c0e81be6dae1f81f011010bbc7e02689": "\\phi=GM/r\\,", "c0e84e6513655105005e64d99b13ea4d": "R(t) = \\left| \\frac{\\Big( x_1'(t)^2+x_2'(t)^2 \\Big)^{\\frac{3}{2}}}{x_1'(t) \\cdot x_2''(t) - x_1''(t) \\cdot x_2'(t)} \\right|\\qquad\\qquad \\mathrm{and} \\qquad\\qquad Q(t)\\,=\\,\\gamma(t)\\,+ \\, \\frac{1}{k(t)\\cdot|| \\gamma'(t)||}\\cdot\\begin{pmatrix} -x_2'(t) \\\\ x_1'(t) \\end{pmatrix}\\,. ", "c0e8ba8d281fb87cde2930410db44d3c": "\\{ x \\in K(M) \\mid nx \\in M,\\ n\\in\\mathbb{N} \\},", "c0e8e18f51899d4e4a09da84b773f570": "*333", "c0e8ebf716c7077962504ff57b02469b": "S^{n+m+1}", "c0e926dfba8e1361e2b996500ae3a7fa": "\\epsilon=100 \\tfrac{\\theta - \\sin\\theta} {\\sin\\theta}", "c0e9349769d67c9216fcb70b55df13cb": "\\log_2(1/\\epsilon)", "c0e939752bf8a4120fa9e0d0f2ca9cda": "\\gamma(t) \\colon [0,1] \\to \\mathbf{R}^2,", "c0e947dc3c2170d0b48bd35fcbbb7726": "e=\\frac{a \\pm \\sqrt {a^2-b^2c}}{2},", "c0e96e69ad7c804eda2453fd5b9c81cf": "R = R_1 \\times R_2 \\times \\ldots R_n \\,,\\quad R \\subseteq \\mathbb{R}^n \\,,", "c0e9b54ae2652fcebd164ccd96f4bb6a": "F(x) - F(a) = \\int_a^x F'(t) \\,dt.", "c0ea15e10d1018620c968be8a55689a3": "f(z)=1, g(z)=z^k", "c0ea8233cbe37cf05bec93799d4b6a83": " \\int g_k \\, \\mathrm{d} \\mu =\\int 1_B \\, \\mathrm{d}\\mu = \\mu(B) = \\mu\\left(\\bigcup_n B_n\\right) .", "c0ea98163fe024022cffdd9c01ac22d1": "\n\\rho(r) = {4\\pi\\over 1+r^2/r_a^2} \\int_\\Phi^0 dQ \\sqrt{2(Q-\\Phi)}f(Q).\n", "c0eaaae73dab6af3860b23d85197cbe0": " A = A_{i+1}(h) + O(h^{k_{i+1}}) ", "c0eb031206ae195b84de56409ba34c31": "O(H_k(T) + {{loglog u}\\over{log^\\epsilon u}})", "c0eb719c968cb60e88c048e1a1158504": " H_{j, j}^{ } \\ne 0 ", "c0eba153543f0957aed49c049cec9f70": " q = x i + y j + z k = r_1 + h r_2 .", "c0eba7c91ed96813b8ab7a4a84bb81cf": "FAI = 100 \\times (\\frac{Total\\ Testosterone}{SHBG})", "c0ebcd45bfb8bcbc5022c60e551a31e7": "\\lim_{k \\to \\infty}\\left | \\frac{q_{k+1}}{q_k} \\right \\vert = 1.45607\\ldots", "c0ec6bfd3f0e64fc2c8e416322419578": "P^{-1}(Ax-b)=0,", "c0ec7ea1a282c06edf325ccb5a7e2bd7": "\n\\frac{\\partial\\operatorname{atan2}(y,x)}{\\partial x} =\\frac{\\partial\\arctan(y/x)}{\\partial x} = -\\frac{y}{x^2 + y^2}\n", "c0ec7f504059c571d8030dc0f9b34c08": "b^x = (e^{\\ln b})^x = e^{x \\cdot\\ln b}", "c0ec81ce4912595d9e8010d56250e2d0": "t > \\frac{D} {r}", "c0ec99377b4a6ccca8a46c40b8d47207": "\\begin{align}\n \\mu_{X \\cup Y} &= \\frac{ N_X \\mu_X + N_Y \\mu_Y }{N_X + N_Y} \\\\\n \\sigma_{X\\cup Y} &= \\sqrt{ \\frac{N_X \\sigma_X^2 + N_Y \\sigma_Y^2}{N_X + N_Y} + \\frac{N_X N_Y}{(N_X+N_Y)^2}(\\mu_X - \\mu_Y)^2 }\n \\end{align}", "c0ecd2a3cb19456dc6f18d4e9c0e5693": "\\Delta(ab)(v_{(1)}\\otimes v_{(2)})=(ab)v=a[b[v]]=\\Delta a[\\Delta b(v_{(1)}\\otimes v_{(2)})]=(\\Delta a )(\\Delta b)(v_{(1)}\\otimes v_{(2)}).", "c0ed16ce1e2a866af78f261c42bad4eb": "f(N_j)=g(N_j)+h(N_j)", "c0ed32c3ff243fe6e11c4cbcf6a6fba8": " \\pm\\frac{1}{2} ", "c0ed762d6ef4f0d125cb55834a910ded": " \\left(\\frac{\\partial T}{\\partial P}\\right)_S = +\\left(\\frac{\\partial V}{\\partial S}\\right)_P = \\frac{\\partial^2 H }{\\partial S \\partial P}\n", "c0ede6f4a83dbf88d6dedf3971da6de8": "\\mathbf{p}=u_1 \\nabla q_1+u_2 \\nabla q_2+u_3 \\nabla q_3", "c0edf226e82ab2531c5e323d5390e61d": "\\,\n\\begin{align}\n\\mathcal{M}(n-1,n)\\leq\\mathcal{M}(n+x,n)\\leq\\mathcal{M}(n+1,n)\n\\end{align}\n\\,", "c0ee441983a34c38ac31d17ce42023bd": "\\frac{\\partial \\epsilon_k}{\\partial k_i} = \\frac{\\hbar^2 k_i}{m} ", "c0ee8cf6318845b98f1abc61536e4fa6": "\n({\\mathbf\\Omega}\\cdot\\nabla){\\mathbf u}={\\mathbf 0}.\n", "c0eec1b20d9b109e87efa1627bbfb785": "\\mathbf{F}=\\psi \\nabla \\varphi ", "c0ef02f289264e31ef443bd72bff1eca": "C_{out} \\circ (C_{in}^1,..,C_{in}^N)(m) = (C_{in}^1(c_1),C_{in}^2(c_2),..,C_{in}^n(c_N))", "c0ef21b4587ea6f56f7dab8fd1c39ca1": "q_s* = 8\\left(\\tau*-\\tau*_c \\right)^{3/2}", "c0ef5b1630b2bf02c605f0061905c535": "\\varphi: \\vec x \\rightarrow \\vec x-\\frac{f(\\vec p,\\vec x)}{\\rho(\\vec p)}\\vec p", "c0efcc1f5b40f8148e5e24d2e1622ab3": " x_j \\in \\{0,1\\}", "c0efd79f433c260d979995bc5c87d9d0": "a = c_1\\frac{Du_1}{u_1}+\\dotsb+c_n\\frac{Du_n}{u_n} + Dv.", "c0f0123e273678425e65f59f9078dc55": "x_1,x_2,x_3, \\dots ", "c0f049f760c0ee228b523653e2e91ab3": "\\mathrm{wt}_0(c)=6", "c0f061f31b5a7815e8ac827518c06519": "\\theta_{obs}", "c0f0c5eb159c558e95a63471e900e389": " X:\\Delta^{op}\\rightarrow Sets", "c0f10de768b5ff8c39afff6a1df42152": "f,g : X \\to Y", "c0f166a648fdf746e178c4fd95b7df9b": "\\eta ^{\\prime }=-\\frac{\\pi Z_q^2}{\\rho _{\\mathrm{Liq}}\\,f}\\,\\frac{\\Delta\nf\\Delta \\left( w/2\\right) }{f_f^2}", "c0f16d112cf81f32eaac16084e7f959d": "H_{0,S}", "c0f1831dbd45a554b197eeee98e3e7f0": "i_{max}", "c0f19fface8599215e7b9332d7a77dd4": "t - t'", "c0f1b7be09b0c50d63ccfaa83b24b101": " y_0 = 1 ", "c0f2212d6913b93b4a1a22e9210b0212": "\\omega_\\mathrm{sig}+\\omega_\\mathrm{LO}", "c0f244818b071e3c3cb11faf6386a2b7": "\\frac{(\\tan x - 1)^2}{\\tan x}= 0.", "c0f2a6d095188b06ad58f3f422a1d4c9": "\\displaystyle S(3,2)", "c0f2c2ee50e8172d98a3d308226d8c23": "\n\\Rightarrow \\left(\\frac{\\partial T}{\\partial p}\\right)_H\n= -\\frac{1}{C_p}\n \\left(\\frac{\\partial H}{\\partial p}\\right)_T\n", "c0f2d63d8aab76e17bd74635cc49a900": "\\Delta G = \\Delta G^0 - (T_H-T^0)\\sum_{p}^{}{\\Delta S^0_i} ", "c0f2e44b887dcba59238b2bbc50f5d6d": "V^*\\propto (H m)^{-1}", "c0f3487042616646c9df152e17d1efd9": "k(-i) = -j\\,", "c0f38a1c68ea8edfda000241695bef00": " P^\\alpha(x):E \\to F", "c0f3a6a48a9412e60535b1dd1b542bf3": "\nx\\,y'' + (\\alpha +1 - x)\\,y' + n\\,y = 0~.", "c0f3a7957925749f16c311bbca664648": "\\operatorname{log}\\frac {\\bar{X}_m}{\\bar{X}_f} ,", "c0f3d54583a86fca9098da3aef23cdc6": "\n\\begin{align}\n C_{0} = & [0,1] \\\\\n C_{1} = & [0,1/3]\\cup[2/3,1] \\\\\n C_{2} = & [0,1/9]\\cup[2/9,1/3]\\cup[2/3,7/9]\\cup[8/9,1] \\\\\n C_{3} = & [0,1/27]\\cup[2/27,1/9]\\cup[2/9,7/27]\\cup[8/27,1/3]\\cup \\\\\n & [2/3,19/27]\\cup[20/27,7/9]\\cup[8/9,25/27]\\cup[26/27,1] \\\\\n C_{4} = & \\cdots .\n\\end{align}\n", "c0f3ecf1b7000c5386d7b460496db4dc": "\\ g\\ ", "c0f40ee5b385626b3f5d5adb09433a22": "a^n_f", "c0f4637b48742a17985f4b7d24bb43fd": " \\eta = X \\beta + v ", "c0f48c56772df07924f85b3559b6c565": "{\\tilde{D}}_9", "c0f4fdeb4e49f388229dd1b07b291c45": " \\tan(\\tfrac{1}{2}\\phi) = \\exp(-a) ", "c0f513dc7fe8ad2ff3f2802029110ed3": "1/\\rho \\epsilon", "c0f52b32a961d67a30288d474de5d158": " \\mathbf{f}=\\nabla p.\\,", "c0f561a907b8fde9d787e18426187eb9": "\nq = \\frac{\\omega^{2} R_{e}^{3}}{GM} \\approx 3.461391 \\times 10^{-3}\n", "c0f56f4133492f03a63d36146fddbd94": "f : \\mathcal{X} \\to \\mathcal{Y}", "c0f582773fdbd168bbab09a1e6159c46": "p\\,\\!", "c0f5f45d54cca69136e50847315eb801": "d^Q_i", "c0f62c6a8d67fd568db3d7c9cb12efe4": " \\operatorname{ord}(ab) = \\operatorname{ord}(ba)", "c0f62dfcea39252b144d87032dbd721d": " \\mathcal{H} \\otimes \\mathcal{H} ", "c0f63fbe52cfe34bb414151d68c24267": "\n(0.171010,\\ -0.030154,\\ 0.336824,\\ 0.925417)\n", "c0f64e43eb5549a24547cf574782b3e8": " C \\subseteq \\{1, \\dots, n \\} ", "c0f736a1a9b8722cc9e238bd25f7db24": " \\left | \\frac{d^k f}{dx^k}(x) \\right | \\leq C^{k+1} k!", "c0f791099e4945ceb2d5da2fc14adfb1": "T_{i_1i_2\\cdots i_q\\cdots i_n}n_{i_q}\\,dS .", "c0f7a6aacdc34b87ab38ecd05f11b241": " \\xi_i = \\frac{x_{max}}{2 \\pi} \\cos(\\theta_i) ", "c0f7ae8af64448089db21a3560ef6027": "x \\tau", "c0f7b5509b714df12147534ba3e619de": "a^ib^{p^2-i}=a^db^{p-d}", "c0f7b5ac89b9ec055fd2c6cd8b24be83": "s_1=\\alpha^{4},", "c0f80cf7c10f8bfe8736076af37f2f8e": "C_*({\\tilde X})", "c0f81672d969d30af4945f8477ccf291": "b(x) = x-a", "c0f822114342ebce16c3d7a1b33404c9": "\\sum_i T_{\\mathrm{actual}}(o_i)", "c0f8919d758a50b809fd0d657e9210b8": " u(x+a)=u(x) \\,\\! ", "c0f8aa00094bb1c9d4c4375bd9e5ba9d": "\\bold{j}(t)=\\frac{d^3\\bold{r}}{dt^3}", "c0f8cb85176b1cecedd4268691d2fb9c": " \\gamma^5 := i\\gamma^0\\gamma^1\\gamma^2\\gamma^3 = \\begin{pmatrix}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\end{pmatrix} ", "c0f8f3a79d8f9bb149a618443ab1204e": "\\mathfrak{a} = \\mathfrak{p}_1 \\mathfrak{p}_2 \\dots \\mathfrak{p}_n \n", "c0f90924b8a3d8668f2fe10253247bf4": "x_i \\ge x_j", "c0f90bb3b19bfda44f12a52aa7934998": "T(n) = 2 n^2 - n", "c0f938b64e188386bd9f4f01159fc7d0": "L = \\Theta(R)", "c0f97f5842f5b686a0ff8efac93dc72c": "G = (I_k | A)", "c0f9b521f31bcce78e16690f2affadd0": "\\langle W,R\\rangle", "c0f9c1a93cc40824e934e8bbe84aa157": "\\begin{matrix}\\text{If } y(t)=e^{i2\\pi f_0t}x(t)\\text{ then }\n\\\\ W_y(t,f)=W_x(t,f-f_0) \\end{matrix}", "c0fa03a3a23d7eb15ed6657890ae9d81": "A^c =\\{1,2,3,\\dots,n\\}\\backslash A", "c0fa23fd5dc4c4201a87a784dbfa9a48": "|F_{if}-F_{bfo}|=F_b\\,", "c0fab5f0858a0d61f5c9a7f07256240c": "\\deg(f)=\\deg(g).", "c0fac7a4f887bc6c849107e7f0df48ee": "\\rho(g):h\\mapsto hg^{-1},\\text{ for all }h\\in G.\\ ", "c0fb05ce87ca03adfc0478186c2ad99c": "I |\\psi \\rangle = |\\psi \\rangle = \\sum_{i=1}^{n} | e_i \\rangle \\langle f_i | \\psi \\rangle = \\sum_{i=1}^{n} \\ c_i | e_i \\rangle ", "c0fb15177c6ada359da54f176b459f88": "a_{\\sigma (i)} ", "c0fb393b0b82bebb572b07695d6c2825": " e ^ { - \\beta (\\Delta F - C)} = \\frac{\\left\\langle f\\left(\\beta (U_\\text{B} - U_\\text{A} - C)\\right) \\right\\rangle_\\text{A}}{\\left\\langle f\\left(\\beta (U_\\text{A} - U_\\text{B} + C)\\right) \\right\\rangle_\\text{B}} ", "c0fb72bb4da4f44c8251af6c5295d779": "J^{\\prime\\prime\\prime}(u_{0}) = \\frac{\\beta^2 (1 - \\beta^2) (1 + \\beta^2)}{u_{0}^2}", "c0fc09c0545b25261c7945606cbd6557": "Pr_{h \\in H} [h(a) = h(b)] = \\phi(a,b)", "c0fd18d0ad6da4c87f718c14e9e61f32": "A\\in\\mathbb{R}^{n \\times n},\\; b\\in\\mathbb{R}^n.", "c0fd7adff16d2eb7bd330ddf464bcb46": "X \\wedge A = 0", "c0fde37997db82b259a0b3050e077238": "\nz = a \\ \\frac{\\sinh \\tau}{\\cosh \\tau - \\cos \\sigma} \n", "c0fdf83f8888a1b1e304e5c71ed5108c": "(\\sqrt [12]{2})^7", "c0fe1578dd17663f9cd015849015a378": "q_{\\mathbf{R},ij} = \\langle \\phi_{\\mathbf{R},i} | \\phi_{\\mathbf{R},j} \\rangle - \\langle \\tilde{\\phi}_{\\mathbf{R},i} | \\tilde{\\phi}_{\\mathbf{R},j} \\rangle", "c0fea854124e2074fb9bad6dde36911a": "U \\subset T \\times M", "c0fedca0038dee4ca12ac94deffc9db4": "A=24.20", "c0ff067e718273bcebdb9e03177f8e2d": "\n[P,f(X)] = -i f'(X)\\, .", "c0ff15d07513b512da07c7c8634a4567": "i \\in \\{1,2,\\dots,n\\}", "c0ff238ce12c4960a42beccb246cbf9f": " \\boldsymbol{\\omega} = {1\\over 2}\\kappa \\mathbf{B} + {1\\over 2}(\\kappa \\mathbf{B} + \\tau \\mathbf{T}) + {1\\over 2}\\tau \\mathbf{T} = \\kappa \\mathbf{B} + \\tau \\mathbf{T}, ", "c0ff24802a4db7db5b20464e53682186": "n \\in \\mathbf{N}", "c0ffb0f0725ef6203065801aeaefe019": "R_{b}", "c0fffc8747dc974e1f19eac7a4e8e212": "y=\\ln x", "c1002c08931fd4f1d52b4f35c84a900e": " \\bar{x} = \\frac{1}{10} (x_1 + x_2 + \\ldots + x_{10})", "c10032b17f2286810c1a37bc4748e455": "y_1, y_2, ..., y_n", "c1009961380ba2118dadb25008aadfcf": "\\mathrm{Le} = \\frac{\\mathrm{Sc}}{\\mathrm{Pr}}", "c100c486172058547e49e1cc7919ef88": " \\epsilon^{\\mu \\nu \\rho \\sigma}", "c100d890c69395284a4586b89cc87a41": "x = y = 0,\\,", "c1010b9c38fb6a6334e97d3e3bb11606": "|q(T_w)(x)|>1", "c101142e20068f2928620115dff1f043": "\n\\text{If }p\\equiv q \\equiv 3 \\pmod4, \\text{ then}\n", "c10131847fb48cf2f5283843acfdd9b9": "\\mathrm{ CH_3OH_{(g)} + H_2O_{(g)} \\;\\longrightarrow\\; CO_2 + 3\\ H_2 \\qquad} \\Delta H_{R\\ 298}^0 = 49.2\\ \\mathrm{kJ/mol} ", "c1013584313fdb620453ed79dd81a14a": "\\left(-3\\sqrt{1/5},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "c1015f14836165504ccbb2a42b2c150b": "{x_2}", "c1016befaee537a177d9b6cf560ff7be": "\n2Q = v_r^2 + (1+r^2/r_a^2)v_t^2 + 2\\Phi(r)\n", "c101c1b33d3c63485d3f805f3dab8847": "f(x) = \\frac{p(x)}{q(x)} = P(x) + \\sum_{i=1}^m\\sum_{r=1}^{j_i} \\frac{A_{ir}}{(x-a_i)^r} + \\sum_{i=1}^n\\sum_{r=1}^{k_i} \\frac{B_{ir}x+C_{ir}}{(x^2+b_ix+c_i)^r}", "c101e49848f117f1df09b247afa47675": "m = 4", "c1022dd3f8abab96c9dd80b6c0c46d50": "E[J]=-\\ln Z[J] \\,", "c1026ab5aebf81489d8430e47ec4cb58": "\\frac{\\tau_\\tau}{\\tau_\\mu} = \\frac{B \\left ( \\tau^- \\rarr e^- + \\bar{\\nu_e} +\\nu_\\tau \\right )}{B \\left ( \\mu^- \\rarr e^- + \\bar{\\nu_e} +\\nu_\\mu \\right )}\\left (\\frac{m_\\mu}{m_\\tau}\\right )^5.", "c102bc2ac2f8d1eae3f128bbe5c243c6": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {l_1} {l_2}", "c102fcf64961bec60bebc77fa37ccccb": "F(r)", "c10357f20c25779c514aa5f0561f92ee": "Y_n\\ \\xrightarrow{L^r}\\ Y", "c103f4e4c96a9c3008e3df779ac351cf": "\\dot{n}_i=\\sum_j w_{ij}n_j-n_i\\,", "c10418211fee9178b76634ddaf2b5601": "\\Pr(\\cdot)", "c104626db4c1cf62e249164f0d80e41c": "\\int_{\\partial \\Omega}\\omega=\\int_{\\Omega}d\\omega", "c10471db9f197e4a2ee22cc56eaae41a": "I_{n,k}", "c1047e340dfd473102ffab572a08dc0b": "\n\\rho_{\\alpha\\beta} = 2\\pi \\delta(\\xi_\\alpha - \\omega)\\delta_{\\alpha\\beta},\n", "c104800ea84f7ff0d076181738b0e4d1": "I_\\text{P}=\\rho^E_\\text{P} c=\\frac{P_\\text{P}}{l_\\text{P}^2}=\\frac{c^8}{\\hbar G^2}", "c105339730e98451e2670b5ed3e089b5": "2\\min\\left(Pr(X \\leq x |H),Pr(X \\geq x |H)\\right)", "c1058ba1f28bcf4c69f1a716d6be4aaf": "0 \\leq r < |n|.", "c105983ec591ef902207bea782602a6a": "P(1\\mbox{ or }2) = P(1) + P(2) = \\tfrac{1}{6} + \\tfrac{1}{6} = \\tfrac{1}{3}.", "c1059a5977f1afe199143852de68f863": "\\boldsymbol{\\widehat{R_i}}", "c105f9fc3f051f2fdcd14875835c3bcd": "k = \\frac 1 {\\sqrt{1-\\varepsilon^2}}", "c106097e5a1d17f30e685fd5d4421928": "\\Delta a^* = a^*_1 - a^*_2", "c106b680b3cf355ce3d939332b53cd56": " \\mathbf{v}\\cdot\\mathbf{b}_i = v_k\\mathbf{b}^k\\cdot\\mathbf{b}_i = v_k\\delta_i^k = v_i ", "c106e65083d7874e9980ed28048d1d1a": "R(x,G)=MSE-MSE^o=Tr(GC_wG^*) + x^*(I-GH)^*(I-GH)x-\\frac{x^*x}{1+x^*H^*C_w^{-1}Hx}.", "c106ef8948e35c91eba183296a8417ad": " \\sigma_{A}^{2} = \\langle f|f\\rangle\\, .", "c106f4dd6a0f954af5df22fac0d6b4ce": "d^{118} = d^{228} = d^{338} = -d^{888} = \\frac{1}{\\sqrt{3}} \\,", "c1070828e6c399ec335c230b33004458": "\\left(\\frac{a-1}{ab-1}\\right)^{1/a}", "c107a79cbb26fbf78c1fd35b81af0f22": "H^0(M,\\mathbf{K}^*)\\xrightarrow{\\phi} H^0(M,\\mathbf{K}^*/\\mathbf{O}^*).", "c107a837dc6ee28b87314ab724912fcc": "\\operatorname{Spec}(A^G)", "c107b0abebee6a4687f9996a75fd3832": " R(1 + 5R((A-1),4) + 4R((A-1),100) + 6R((A-1),400),7)", "c107ba44da18e30a3e04bf77d2faa814": "\\mathcal{I}=\\mathcal{R}^T\\mathcal{R}", "c107c7dd4f0b4dcb8e160b3d4f936996": " \\operatorname{Arg}(x + iy) = \\operatorname{atan2}(y,x) = \n \\begin{cases}\n \\arctan \\left( y/x \\right) & \\qquad x > 0 \\\\\n \\pi/2 - \\arctan \\left( x/y \\right) & \\qquad y > 0 \\\\\n -\\pi/2 - \\arctan \\left( x/y \\right) & \\qquad y < 0 \\\\\n \\pi + \\arctan \\left( y/x \\right) & \\qquad x < 0, y \\ge 0 \\\\\n -\\pi + \\arctan \\left( y/x \\right) & \\qquad x < 0, y < 0 \\\\\n \\text{undefined} & \\qquad x = 0, y = 0\n \\end{cases} ", "c107f3769ec9a9c85ea203786a39956b": "\\hat{H} = \\frac{1}{2m}(\\boldsymbol{\\sigma}\\cdot(\\mathbf{p} - q \\mathbf{A}))^2 + q \\phi", "c1080a8ebda307f901732835337abb6b": "B_r", "c1082ac7f1c8a2c29f0cf8c0a1c1e857": " \\mbox{DL} = (y\\bmod 4\\times 2 + y\\bmod 7\\times4 + c \\bmod 7 + 2) \\bmod 7, ", "c108377b095f112457f587c4f5095c4f": "\\lim _{n\\to\\infty}P_n\\Bigl(\\cos{z\\over n}\\Bigr)=J_0(z)", "c1088ef681a2caef364e0bdb4b1e584a": "w(e_j)", "c109681b3db6a6efd9af268a6a5dcfca": " \\frac{dm}{dr} = 4\\pi r^2 \\rho ", "c10a16a6167fc86763f8064fce999b6e": "\\mathcal{F} \\left \\{ \\mathbf{x} \\right \\}[m] = \\sum\\limits_{k=1}^n x_k \\cdot e^{{-2\\pi i}\\tfrac{mk}{n}}, \\mathbf{x} = \\left \\{ \\gcd(k,n) \\right \\} \\quad\\text{for}\\, k \\in \\left \\{1 \\dots n \\right \\}\n", "c10a1998db7f52970bf067e53f1536a4": "E^i_c", "c10a1f3a7b50333b46d5010bcbb9c965": "\\left(\\sum_{n \\in N} a_n X^n\\right) \\cdot \\left(\\sum_{n \\in N} b_n X^n\\right) = \\sum_{n \\in N} \\left( \\sum_{i+j=n} a_ib_j\\right)X^n", "c10a22ccd3cea42a6b3cf10f50255588": " H(t) = \\begin{cases} 0, & \\text{if } t \\le 0; \\\\ 1, & \\text{if } t > 0. \\end{cases} ", "c10a50479f961c0ad49eb6332214da58": "\\Phi(m,b)=\\Phi(m,b-1)-\\Phi\\left(\\frac m{p_b},b-1\\right)", "c10a5fe09a765fc0d3c2a066aed6d54f": "\n\\left(\\frac{1}{\\mu_{0}\\rho}\\right)\\boldsymbol{(B\\cdot\\nabla)\\delta B} =\n\\left(\\frac{ikB\\boldsymbol {\\delta B}}{\\mu_0\\rho}\\right)= -{k^2B^2\\over\\mu_0\\rho}\\boldsymbol (\\xi)", "c10a7f6d89c749ca3d67aa63b46f11df": " \\sum_{\\mu\\uparrow\\lambda}\\frac{e_{\\mu}}{e_{\\lambda}}=1", "c10a8ed717be3be3561e790d3156a82b": "\n\\psi(k) | ...,n_k, \\ldots \\rangle = \\sqrt{n_k}\\, |...,n_k-1,\\ldots\\rangle\n", "c10aaea376d76a0e84bbb14a00d7b92f": "n \\in {\\mathbb N}_0", "c10abad78cf92d129f9f5a906d848ebf": " S_D: \\vec{r}=\\vec{r}(u,v), \\quad (u,v)\\in D ", "c10ac3e21ab016a9d6029de5922f8e2a": "\\vec{v}_1^T", "c10ae344d0971516456d4ea43426f312": "Q=4\\pi\\omega\\int\\limits_{0}^{\\infty}\\phi^2(r)r^2dr", "c10b0134460508bd6635807dbafa8c70": "\\beth_{\\omega} = \\bigcup \\{ \\beth_{0}, \\beth_{1}, \\beth_{2}, \\ldots \\} = \\bigcup_{n < \\omega} \\beth_{n} ", "c10b30af2ca82b997c02b71a76e1d330": "n=-1", "c10b485cefc0762947e2b8b4dcdd3cfb": "\\mathrm{id}(x) = x", "c10b485ff7b0d377ff9c44dc81049f65": "X>0", "c10b4880460e37e5de59253d55c60022": "\\omega_{i}", "c10b5116f2dba7bfc6bcb0b84b4ff483": "\\mathrm{P}(A \\cap B \\cap C) = \\mathrm{P}(A)\\mathrm{P}(B)\\mathrm{P}(C),", "c10b778b59ae64239e50a6b33f481cd2": " X^\\alpha ", "c10be7b9a24ed6ca5efcf221938e8cfb": "\n\\begin{array}{lll}\n\\text{Weak law:} & \\overline{X}_n \\, \\xrightarrow{P} \\, \\mu & \\text{for } n \\to \\infty \\\\\n\\text{Strong law:} & \\overline{X}_n \\, \\xrightarrow{\\mathrm{a.\\,s.}} \\, \\mu & \\text{for } n \\to \\infty .\n\\end{array}\n", "c10c44e79bd503f07690423512df2e60": "\\sigma_x = \\sigma_1\\,\\!", "c10c5f1bbb40ed5bdf16ea73ee517af7": "| \\cosh (i \\nu \\pi) | <1", "c10c6d4b7d0ade913f60bacb0f7d50f2": "\\hat{V}=n^{-1}\\sum_{i=1}^S n_s \\bar{z}_s \\bar{z}_s^\\top", "c10c7e941302059a371dbd358abbd6aa": "Y_{\\ell |m|}^{sgn(m)}", "c10d2379e271e6ad0471977eaee6e6ab": "P=x-\\frac{f(x)}{(x-Q)(x-R)(x-S)}.", "c10d2523806020d5a9d8dac0720fae5f": "\\gamma\\approx 0.577", "c10d270bd1b3da95094dc3868bf4491d": "e^{i \\pi} +1 = 0 \\, ", "c10d558a1f064ddb74129df58c00ab82": " 1/ \\zeta (s).", "c10d76a07c8b84bb53343fde64e5f0ca": "g(z)=e^{-z/2}", "c10d8d28dd3cf31415ffad8d3f4a8f1a": "\\textstyle p \\mid k-p", "c10dabfaee4f99c270d08d4eecd00611": " \\sum_{t=0}^{O(\\log\\log h)} O\\left(n \\log(2^{2^t})\\right) = O(n) \\sum_{t=0}^{O(\\log\\log h)} O(2^t) = O\\left(n \\cdot 2^{1+O(\\log\\log h)}\\right) = O(n \\log h).", "c10dc1c5fc763f00f0fa9506de78298a": "\\mathbf{Q}_{j}(\\gamma_{ij})", "c10e4abb11e00f28b1a74d73c3e22a42": "|ab| = |a||b|", "c10e8d046eb456d2eca6355cbddbe815": "e_2\\circ e_1", "c10ec78c5e7e4a3d82e40a14db14513c": " \\vec{e}_0 = \\frac{1}{\\sqrt{1-2m/r}} \\partial_t, \\; \\vec{e}_1 = \\sqrt{1-2m/r} \\partial_r, \\; \\vec{e}_2 = \\frac{1}{r} \\partial_\\theta, \\; \\vec{e}_3 = \\frac{1}{r \\sin(\\theta)} \\partial_\\phi", "c10f15a004d58411174f718fb0bff577": " \\left( \\ddot r - r\\dot\\varphi^2 \\right) \\hat{\\mathbf r} + \\left( r\\ddot\\varphi + 2\\dot r \\dot\\varphi \\right) \\hat{\\boldsymbol{\\varphi}} \\ = (\\ddot r - r\\dot\\varphi^2)\\hat{\\mathbf{r}} + \\frac{1}{r}\\; \\frac{d}{dt} \\left(r^2\\dot\\varphi\\right) \\hat{\\boldsymbol{\\varphi}}", "c10f7ca09f85ac9e71f1d622aefa32a6": "\\operatorname{Tr}_\\omega(T)= \\lim_\\omega a_N", "c1100fd8e8d371ae691b3b9060d29331": "z=\\mu^t c.", "c1103b3bd5fc099c0316cea0a35bd376": " {d \\Gamma \\over dt } = k c B(\\Gamma) ", "c1107284516b8bee5119fab5ddea67b6": " f_{yy}(x,y) \\approx \\frac{f(x,y+k) - 2 f(x,y) + f(x,y-k)}{k^2} \\ ", "c110872984d2ed6178483dd05d6996b2": "\\langle\\hat{T}\\rangle", "c110e6e6a3ca1eb5f0c3e13f1b08b936": "r=max(T)", "c1114581f55405cc77a588e24aff5200": "u_i:X\\longrightarrow \\mathbb{R}", "c1116c0dca808143e156d28ea9e81ce3": "\\left[ \\mbox{H}_2\\mbox{O} \\right]", "c1118ea53bd6e65011c515e673fa5005": " Q = r \\times [Q]", "c11193e81b7de83896a3e5772425b225": " \n\\frac{\\partial Q}{\\partial t}+\n\\frac{\\partial F}{\\partial x}+\n\\frac{\\partial G}{\\partial y}+\n\\frac{\\partial H}{\\partial z}=0\n", "c11265f09990597f57329e4843837bb4": "\n\\begin{array}{llr}\n\\min\\limits_{y\\in \\mathbb{R}^m} & q(\\xi)^T y & \\\\\n\\text{subject to} & T(\\xi)x+W(\\xi)y = h(\\xi) &\\\\\n\t\t & y \\geq 0 &\n\\end{array}\n", "c112888adecafdc7252632bb1e8c8716": "\\begin{align}\nx &= \\rho\\cos\\phi \\\\\ny &= \\rho\\sin\\phi \\\\\nz &= z \\end{align}", "c11291f34d999b813ed152a3e2e5d9ae": "\\sqrt[n]{x_1 x_2 \\cdots x_n} \\le \\alpha.", "c112d21322a928738fd0ce2122a13117": "u_0(x),\\, u_b(t),\\, u_c(t)", "c112d80553d4d902dd728862cb3199b1": "\\rho(x,p;t) = \\left| \\langle x,\\, p |\\Psi(t) \\rangle \\right|^2", "c1133c2d16ab73d32c450509472038c9": "a,b\\in \\Sigma", "c1135e0ef8412e281ad3dca02a283b8a": "nB_z\\in A", "c113667a0a86e05b7972ecf8c581adf1": "\\lambda = 375 \\ \\mathrm{mm}\\,", "c1136de17d31f65564e057dde9b6f85c": "\\kappa_1=\\mu'_1\\,", "c113b52c9bcc906fe2bacc3b50eb309c": "\\scriptstyle{[L]\\Phi}", "c113fdc727f68e15bfd5ad3c316b148b": "s_1=\\sum_{i=1}^m \\log x_i, s_{2}=\\min\\{x_i\\}", "c11412f68c739e5e454bb60f1a4d8fbd": "\\mathbf{g}(\\mathbf{r}) = \\frac{\\mathbf{F}(\\mathbf{r})}{m} = -\\frac{G M}{r^2}\\hat{\\mathbf{r}}.", "c1147499a4de3520025ee4648189e98c": "\\Delta G^* = \\frac{16 \\pi \\sigma ^3}{3(\\Delta G_v)^2}", "c114dfb14d9dd5bcad4b2151a86efdaa": "M''", "c114e5e0353cc502752ef23123a5297b": "g = -(1-2m/r) \\, dt \\otimes dt + \\frac{1}{1-2m/r} \\, dr \\otimes dr + r^2 \\, d\\theta \\otimes d\\theta + r^2 \\sin(\\theta)^2 \\, d\\phi \\otimes d\\phi", "c1154a03a4949d94877ac4b9024433f4": " B = (\\beta_1, \\beta_2, \\ldots, \\beta_r) ", "c1158b96f8c9f3510a6150be93306553": " 1 - SD[A,B] ", "c1161c1f271ed55bfeab5e3396486c94": "q(t) = \\int_V \\frac{1}{c} J^0(\\mathbf{r},t) \\, dV ", "c11624424627a735411b49a3a5c8f7a8": "\\langle f|g\\rangle ", "c1164e440e29bd9abb4e722cd0a34c49": "P_{i}(r_{k}^{A})", "c11679d1692d04229cdbd15b3b59346b": "Q \\left[ \\int_N \\mathcal{L} \\, \\mathrm{d}^n x \\right] \\approx \\int_{\\partial N} f^\\mu [\\phi(x),\\partial\\phi,\\partial\\partial\\phi,\\ldots] \\mathrm{d}s_{\\mu} ", "c1167afc050344eac0bbba1482567fda": "\\mathcal{A}^B", "c11695ddff4a01e46e55106d8625c661": " d\\,\\omega^{(k)}=\\sum_{j=1;\\, i_1<...>2) + (Cb>>4) + (Cb>>5)) - ((Cr>>1) + (Cr>>3) + (Cr>>4) + (Cr>>5))", "c12b00e246a0d7db046c3d27ee89de42": " [x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0 \\quad ", "c12c0f04c419cca9b0ba24bfd4200caa": " Dp(x) = P'(x) \\quad ; \\quad [D] = \n\\begin{bmatrix}\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 2 & 0 \\\\\n0 & 0 & 0 & 3 \\\\\n0 & 0 & 0 & 0 \\\\\n\\end{bmatrix} \n", "c12c1a9561f988332222747869d4ec99": "\\mathbf{c}, \\mathbf{d} \\in \\mathbb{R}^n", "c12c1c3671565f6bb4fc008056c9b5df": "32^{2}", "c12c1f5ac04df5a10b967b31e3564d28": " \\overline{H}", "c12c6ef007721fe57432f37581f2687f": "\\forall y ( R(x, y) \\rightarrow (\\forall z ( R(y, z) \\rightarrow P(z))))", "c12c7f8e70138bdc58697fec8e351e94": "\\psi_{nlm} = R_{nl}(r)\\, Y_{lm}(\\theta,\\phi)", "c12c9bd79419fd2f583e63842ac7c0cb": "\\mathbf{v} = (r, \\angle \\theta)", "c12cf2217764381c42e011f09244f307": "\\frac{\\partial \\phi}{\\partial x^{\\mu}}", "c12d1112233b8c9db6d7bbed2ba8ea51": "K(t)", "c12d2056f0d93080c6150e48aef077aa": " \n\\frac{I_\\mathrm{max}-I_\\mathrm{min}}{I_\\mathrm{max}+I_\\mathrm{min}},\n", "c12d39c3369d34d5a04f487f26385ddb": "\\displaystyle A(t)", "c12d3e9a5443e5ca9f16bd78883cfbf1": "M = m_1+m_2 \\ ", "c12d466da9e0768a04ca216a36b23dab": "f([y_1, y_2]) = \\left[\\min \\big \\{f(y_1), f(y_2) \\big\\}, \\max \\big\\{ f(y_1), f(y_2) \\big\\}\\right]", "c12d4f4d46bfbacbef2cff60e7dfa0c4": " G = 2\\pi / d ", "c12d643df39555ada4a7702f6dad0878": "RGV = \\cfrac{V*}{P} ", "c12d8286ea9796c7e54da1735ca5998d": " \\mathbf{y}'_{1}, \\mathbf{y}'_{1}, \\mathbf{C}_{1}, \\mathbf{C}_{2} ", "c12d8ae5eeccbd9bdb9f8e7f102b9b6f": "\\mathrm{net}", "c12de5800547eb89f33f1fdac73891cd": "T_2 = \\sum_F\\text{(number of links traversed where the buckets are different)}", "c12e0ac1109c57c3b8e589a0a60393b3": "{F}_{4}", "c12e68856cd03db08205d840f4e38dff": "\nB =\n\\begin{pmatrix}\n1&0&0&0\\\\\n0&i&0&0\\\\\n0&0&-1&0\\\\\n0&0&0&-i\n\\end{pmatrix}\n", "c12e68ff69a334dea4e7f3bc8a6db3c5": "\\log_{10}(2^{113}) \\approx 34.016", "c12e9bfa42f3390661b92c07c3b430b3": "v_r(\\mathbf{p})", "c12f4225e3e2c4d2041c921b29b84e83": "\\frac{ \\left(\\sqrt{s^2+ \\omega^2}-s\\right)^{n}}{\\omega^n \\sqrt{s^2 + \\omega^2}}", "c12fd9d577b558d0d0d2d755513e4ca8": "S = S \\cup \\{k\\}", "c12fdaaf80b16ef21fbe2f3ad870d3f3": "\n B \\or C \\vdash B , C\n ", "c130574de2ec4e23b7084d0d7191850d": " -2\\pi\\ \\frac {J_2}{\\mu\\ p^2}\\ 3 \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)\\,", "c13061167e38e6f59c4226defa5a74a1": " \\hat{g}_{-i}\\left(X'_{i}\\beta\\right) ", "c13089b363aac89fb4613fe3c83ec736": "u^{\\tau} a_{\\tau}=0", "c130f27da9ef1a3f7c6224168855d7ce": "\\frac{dp}{dt}=-\\frac{\\partial H}{\\partial q}, \\qquad \\frac{dq}{dt}=+\\frac{\\partial H}{\\partial p}, \\quad \\text{ with } \\quad H = \\tfrac12 p^2 + \\tfrac12 q^2 + \\tfrac14 \\varepsilon q^4,", "c131101cc03457eeba014d5842d591c0": "\n\\widehat{\\Omega}_{OLS} = \\operatorname{diag}(\\widehat{u}^2_1, \\widehat{u}^2_2, \\dots , \\widehat{u}^2_n).\n", "c1311903a69509d4d06c8b10d9188b3a": "T_{i,m+j}=\\sum_{r=1}^{m+n}\\sum_{s=1}^{m+n}{t_{r,s}\\cdot x'_{r,s}}", "c13123f3a2beec3e336117c421c87aa9": "\nV(\\mathbf{R}) \\equiv \\sum_{i=1}^N \\frac{q_i}{4\\pi \\varepsilon_0 |\\mathbf{r}_i - \\mathbf{R}|}\n=\\frac{1}{4\\pi \\varepsilon_0} \\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^{\\ell}\n(-1)^m I^{-m}_\\ell(\\mathbf{R}) \\sum_{i=1}^N q_i R^{m}_\\ell(\\mathbf{r}_i),\n", "c1312997c6aec4fa16c28ca21f49acb6": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 6.665868\\log_e(T+273.15) - \\frac {6530.97} {T+273.15} + 60.47398 + 3.522382 \\times 10^{-6} (T+273.15)^2", "c1313d92c048d926f8a6d927b202dae3": "z_{n+1}", "c1313e41fd17e18d5b9cfeaedc8e3717": "!\\,P", "c1316e4ba20f3d6793d97c844f7ac859": "\\text{Asymmetrical short-circuit current and kva:}", "c131b95992efa8f2397ace96804891ea": "\\chi(a)", "c131fd32e7e58e244229a9cbaa859072": "MRS_{SG} = \\frac{2\\text{ sheep}}\\text{goat}", "c1320d81a9f7804ac0789c866ab2e1db": "\\sum_{k=0}^n \\tbinom n k ^2 = \\tbinom {2n} n.", "c1323e2b9de098def80a8d1d9009ab73": "\nf(y) = \\frac{1}{\\left(1-I(f<0)-\\sgn(f)\\Phi(0,m,\\sqrt{s})\\right)\\sqrt{2 \\pi s^2}} \\exp\\left\\{-\\frac{1}{2s^2}\\left(\\frac{y^f}{f} - m\\right)^2\\right\\}\n", "c132ac0acf3be64ddc87e2a56dac8b6c": "(\\mathbf{Q},|\\cdot|_{\\ast})", "c132c23ff3ce9273f56c7644da160a15": "\\tau _{Auger1_{doped}}(t,x,n) = \\frac{2\\cdot \\tau _{Auger1(t,x)}}{1+(\\frac{n}{n_{i}(t,x)})^{2}}", "c132d2fbf1b35819ef0ea7f2bb73940e": "m \\ddot x \\ = -k x + F", "c132d8b91733dcf0a0959f08c442f75b": "2^4P[S_4=k]", "c1330c3e1ad46463ede710cbc15af052": "\\,M[f^{-1}] = M[f]^{-1}", "c1332f527b40eaa6fdc8130db99aeee9": " e^+e^- \\to 4 \\gamma, ~~ e^+e^- \\gamma \\gamma , ~~ e^+e^- e^+e^- ", "c1341bf90753693ad03592f5f9dcab2d": "E = pc\\,.", "c1343b7b1c7c8563f59594f82175f980": "V\\mapsto \\overline V", "c13480f5e449e0142d2ea62ac97a715a": "\\lim_{n\\to\\infty} a_n = \\lim_{n\\to\\infty}(s_n-s_{n-1}) = \\lim_{n\\to\\infty} s_n - \\lim_{n\\to\\infty} s_{n-1} = s-s = 0.", "c1349780774afdb4a286a9ccfcb368fc": " f_{-2}\\,", "c134acf0ad0416083f718cdd92157f60": "\\left\\lceil\\sqrt{2d\\ln2}\\right\\rceil+1", "c1350b9754047d4d61373fc3de9ad08e": " h_{\\alpha+1}(n) = h_\\alpha(n + 1),\\, ", "c1352bc45c1cbd62a35943ec118ed426": "2,\\;1,\\;3,\\;4,\\;7,\\;11,\\;18,\\;29,\\;47,\\;76,\\;123,\\; \\ldots\\;", "c1352be28fc8249103a79a896b75337d": "{U}_0", "c1357a8573e166de84d8afcee1065a13": "\\tilde{D} = D - \\frac{1}{N} d \\, \\mathbf{e}_{1\\times N}", "c1358d3af59ce8c697e85e4c6650b613": "\\gamma(1).", "c1358f40ee8ae013a4e71a22b9355454": "\\eta_{\\xi\\mu\\eta\\zeta}\\;", "c13593e169456344987aeaa866222024": "\\frac{\\epsilon_x}{\\beta_x} + \\frac{\\epsilon_y}{\\beta_y} + \\eta_s \\frac{\\epsilon_z}{\\beta_z } ", "c135d2d519b6af1ce6e680a9c748fb11": "I: W -> H", "c135de0de684e81745fb62115e9a450a": "R_e<1", "c1360a74d1d62cfa8bb26a68e402aba4": "D(p||q)+D(q||r)-D(p||r)=(\\theta^i(p)-\\theta^i(q))(\\eta_i(r)-\\eta_i(q))", "c1362f65633de475064f60794335661d": "\\cos \\theta = 1,\\,", "c13661f91fad78415e0f163af3170724": "2^{\\omega}", "c13665b6fa5a4191f095156b98449857": "\\dot Q_L", "c1368e8de8b12fa1f72874b2b2574d95": "V'\\equiv V(i+1)", "c136a2f9ef653f5ad4731a87c54d753c": "\\phi = 1\\,\\!", "c136fa2211b52f884ab69443214f247e": "\\hat{f}(0) = \\int_{-\\infty}^{\\infty} f(x)\\,dx", "c1372f70a479e61231a02b7777fe1510": " \\int_{-1}^1 f(x) (1-x)^\\alpha (1+x)^\\beta \\,\\mathrm{d}x \\approx \\lambda_1 f(x_1) + \\lambda_2 f(x_2) + \\cdots + \\lambda_n f(x_n), ", "c1375cda604a073f74d67e26d45cba0f": "\\langle 0,1,2 \\rangle", "c1376cbb0511272b4e7ddf822b79dc04": "\\frac{1-\\cos\\alpha}{1+\\cos\\alpha}", "c137b1c9c331812d9d1c538bdabb766a": "\nE(v,J) = \\sum_{k,l} Y_{k,l} (v+1/2)^k [J(J+1)]^l,\n", "c137d11b0a1ddae487bb9a7163a52949": "f(0, \\bar{u}) = g(\\bar{u})", "c137ef8ccd18e9f1886b10509bc7b74d": "f\\colon S^{n-1} \\to G", "c138071964af7b59a72fc74b30c46b8b": "g=\\|\\vec{g}\\|", "c13819a9fa04850c177f8d20e03a44d8": "b_1=b_2=b_3", "c1385bbf88ce1fbd15effd932c7889a9": "\\Delta^n_h[f](x) = \n\\sum_{i = 0}^{n} (-1)^i \\binom{n}{i} f(x + (n - i) h),\n", "c13879189dbe4c29af20a3bfdfec6bd8": "G_{ij}", "c138fab7e206fcdb3345b0623f4390f4": "\\sum_{i=1}^n x_{ij} = 1,", "c1394a65380f75f3d2c1afc321ca1fb8": " H_n(p_1,\\ldots,p_n) \\le H_n\\left(\\frac{1}{n}, \\ldots, \\frac{1}{n}\\right) = \\log_b (n).", "c1394e6bb026505fee74a484a2e1cb74": "C_2 = tuv + wtx + ywu + zyt + vzw + xvy + uxz ", "c1395df283ce0c02f7dbc9070af1e117": "A \\to w", "c1399ee4937f0b72d8f755008b272a06": "(D-\\tilde{\\lambda} I)^{-1}", "c139c002411e0cd724e28b5c7d777549": " y - l = US . \\frac{Y}{K} - (1 - Q)l + r ", "c139f6cf13b9a362b5a6a0554c5ee666": "ds^2 = d\\mathbf{Z} \\otimes d\\overline{\\mathbf{Z}} = dZ_0 \\otimes d\\overline{Z_0} + \\cdots + dZ_n \\otimes d\\overline{Z_n}", "c13a4734fe74d03ae105530927d8277d": "z ", "c13a548b93eb4a8b0952014ce043ed96": "s(t) = \\int_{t_0}^t \\vert \\mathbf{\\gamma}'(x) \\vert dx.", "c13aa8994d7681eb8f72bea069891ea8": "\\frac{d[\\mbox{A}]}{dt}=\\frac{d[\\mbox{B}]}{dt}=-k[\\mbox{A}][\\mbox{B}].", "c13ab86db0a8296eedf2afa85b60699e": "u_2 = t_1u_1 + t_1u_1 = 3+3 \\equiv 6\\pmod {17}.\\,", "c13abd3d5ac34bc39992fe438fb68c8f": "I (A; B) = H (A) + H (B) - H (A,B)", "c13ac8300da89ec8f0cd47a9e877acc7": " \\mathbf{S}_i \\cdot \\mathbf{S}_j", "c13afd55fe0d2fea76e045a1373532c3": "\\rm 1\\ A=1\\tfrac C s.", "c13b08f3dfbd52848aaee21f4a021d84": "1931 = [35, 41]_{54}", "c13b7dc9b2d70c91919729b2179a0cd2": "\\forall x \\in X \\setminus \\{0\\} \\quad \\exists y \\in Y : \\langle x,y \\rangle \\neq 0", "c13b9b73db7fe567b6dfcdeda6a398c6": "\\color{VioletRed}\\text{VioletRed}", "c13bcd433b868f590b8340eb74072fa0": "p^*(V)\\cong O_P^{\\oplus r}", "c13c3a47dd802b8a45a0aeb868c481e9": " x_{i+1} = x_{i} + (-2ff^{\\prime} ) / (2{f^{\\prime}}^2 - ff^{\\prime\\prime}) ", "c13c720dde0c0eb5e89680e2ba9d8edf": "\\mathcal{Y}\\in \\mathcal{R}^{L_1\\times L_2\\times \\ldots L_O}", "c13c807ebb6f0f084964a938c2198ba6": " P + P^* = -1 ", "c13c868335f72947c3c23412b548e6f1": "\\mathbf{\\omega'}_i", "c13cc2166b1ddd2aae5169e30192b5f6": "x(t) = \\cos(\\omega t+\\theta) = \\begin{matrix} \\frac{1}{2} \\end{matrix} (e^{j (\\omega t+\\theta)} + e^{-j (\\omega t+\\theta)})", "c13d0f6a50738582432036560d97da28": "\\scriptstyle Q\\,\\sim\\,\\rm{Exp}(\\beta)", "c13d2bb9d043dc0525e7ca38d625ee42": "X_i \\in \\mathbb{R}^m", "c13d6d88fde831506989069f83a01370": "\\overrightarrow{CA}", "c13e3db6754ec3fe50343ecc3df06cca": "\\Gamma_{12}(l, m, 0) = \\lambda^2 \\iint I(l, m) K(l, m, P_1, \\nu) K^*(l, m, P_2, \\nu) \\, dS", "c13e4f8a732d2bcecc1511d7c5374398": "\\mathbf a = \\sum_i a^i \\mathbf e_i \\quad \\mbox{and} \\quad \\mathbf b = \\sum_i b^i \\mathbf e_i", "c13e548560a626bd065e49f3ad692546": " BL_H = \\frac {RCM_H} {H_m}", "c13e72b2aaef33b4fedbc6ad95eaeb0c": "\\gcd(a,b)=\\gcd(a',b')=1,\\text{ then}", "c13e96b02ef7c0509e1f19d4b48bb48b": "\\left|\\Psi_{l,\\mu}^{k}\\right\\rangle = \\left|\\Phi_l^{-k} \\Psi_{\\mu}^{v+k}\\right\\rangle", "c13ec7182fcbe7c7a510b7287731d233": "\\left( e^{ \\frac{2}{n} \\pi i } \\right) ^k = e^{ \\frac{2}{n} \\pi i k }", "c13f63405b82eb6964c8e05a8773e002": " b^1 = b\\!\\, ", "c13f736ce92b7c5f2c6bb8f2f28e2869": "\\epsilon_{\\rm A}", "c13faacc0a569c122787d84884c81072": "R=\\left(P^{*}\\right)^{T}P,", "c14020201210d50764d05a7cbf28b4f4": "\\delta x^\\mu = \\epsilon_r X^\\mu_r \\,", "c1403921a635536fb1ae5e89e7001713": "\\ M_{heel} = D_{heel} \\times lift \\times (cos(\\beta) +{(L/D)_{\\alpha}} ^{-1} \\times sin(\\beta))", "c1404f4c5b2be117ddfa80d76d40b999": "V_{t,a}", "c140a213c0e2aa4a1f3fe0d663f8cc5a": "\\ f(\\mathbf A)=g(I_A,II_A,III_A). \\, ", "c14116333112e7506fdd58af06a9a8a5": "t_1 \\rightarrow t_2", "c14124d85ac2a3d3cec86c105b5f7bad": "=-(ED_t+BN_t)[CD_t+AN_t]^{-1}", "c1415efc2c4ab001fd5c0c679ffbf501": " \\sigma= \\frac{x^2}{m} ", "c141a33c0ed27f99e6fb2ccda4bee116": "R\\;", "c141e0c06ad8affc97ad2093fd503db4": "\n\\begin{pmatrix}\n \\cos\\theta & \\sin\\theta\\\\\n-\\sin\\theta & \\cos\\theta\n\\end{pmatrix}\n", "c14255c8c128f57b4e151fba3b2d544b": "F[x,y]= \\frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/3}}-\\frac{1}{2}\\left[\\frac{1}{(x'y''-x''y')^{2/3}}\\right]''", "c14298a0cfc3a4a5f63c63aac361a00c": "K_2 = -1 - \\alpha - \\sqrt{\\alpha(\\alpha+2)}", "c142ba95925dc60dbbd74c867583572b": "f(x) = h(g(x))", "c142d1182b9371f2213e3411cdfd3e4a": "^{\\;}c^{i}(\\xi,0) = \\xi^{i}", "c14335745a772decd0cd7eb574b1f816": "O(\\tfrac{d}{\\epsilon}\\log^2\\tfrac{1}{\\epsilon})", "c1439845aa52a0df9fbf4a5745c96c88": "f\\circ f^{-1}=f^{-1}\\circ f =id_{X}", "c144617a16feeb38c5308830e3a6b157": "\\, N_k=z(z-1)^{k-1}\\text{ for }k > 0. ", "c144701e2996bd9b6d8e3dbb6aa18d6a": "v\\Vdash A", "c1448b80ae4ba5bc4eede14405bc414e": "f\\hat{\\boldsymbol{z}}\\times \\boldsymbol{u}=-\\nabla\\phi+\\frac{\\partial \\boldsymbol{\\tau}}{\\partial z},", "c145344db4037b620c30b0218a84b1c1": "m \\ddot{x} = -kx + -c\\dot{x}.", "c14566de176ec675d1792c181a05366c": " (1 - \\frac{1}{n^2})", "c1457c7dea4a633912fab94a980a93f4": "\\left(\\tfrac an\\right)=\\pm1", "c14586d5c657a476a7e91e4142cdab16": " \\vec{j} \\times \\vec{B} ", "c14597b788ef75b30a12b9f5cec431bf": "\\frac{d\\mathbf{L}}{dt} = \\mathbf{r} \\times \\mathbf{F} + \\mathbf{v} \\times \\boldsymbol{p}. ", "c145a91c974f83e21766195dd2dca938": "\\mathfrak{gl}_V", "c145aced4e607c36b3b89091be5c3216": "M^* \\Omega M = \\Omega\\,.", "c145b85d177aa39709a0d8688ba3484b": "E_n^* = \\frac{n^2\\pi^2\\hbar^2}{2ma^2}", "c146016984ba04865662977c2ad1395a": "\\mathcal{O}(N^3)", "c1465219e79079ce8ffa45b52917c607": "b_{1}^{*}= b_{1}=\n\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix},B_{1}= \\langle b_{1}^{*}, b_{1}^{*} \\rangle =\n\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix} \\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}= 3", "c14670204b360e1e89d4ed0882a8c966": "\\frac{2x^4}{3x^2+1}, \\mbox{none}", "c146888019835e86d74cc1bc0c0ad354": "\\mathbf{R}\\equiv \\{\\mathbf{R}_A = (R_{Ax},\\,R_{Ay},\\,R_{Az})\\}", "c1469132e4f257ea77cd3c276bdfb503": "x_n = a + b i", "c146b457460e7c03f1a00eb6661b52dd": "\\frac {\\mathrm d x'_{\\eta}} {\\mathrm d t'_{\\sigma}}=V'_{\\sigma\\eta},", "c14774ad8cc5ed3d341273519c5c506a": "H_n(x)^2 - H_{n-1}(x)H_{n+1}(x)= (n-1)!\\cdot \\sum_{i=0}^{n-1}\\frac{2^{n-i}}{i!}H_i(x)^2>0", "c147f18882fb50af094c3450fb7415a4": "Q(y_1, y_2, y_3, y_4)", "c1480d00ca72f2653e862e97c0cb2c9f": "\\frac{\\mathrm{d}P}{\\mathrm{d}T} = \\frac{s_{\\beta}-s_{\\alpha}}{v_{\\beta}-v_{\\alpha}}.", "c148f3dcbee81e1edd5bac9e215522bf": "\\langle S,\\varphi\\rangle = 0", "c14989e4c2d3a55e219d7051fc2e5b30": "\\hat{\\xi}", "c1498cbaf19dee49f825aa287c4d8cf2": " \\tau^{\\pm} ", "c14993c653d31bf0e079685eb5565153": "= \\left( {\\partial A_z \\over {\\partial y} } - {\\partial A_y \\over {\\partial z} } \\right) \\mathbf{i} + \\left( {\\partial A_x \\over {\\partial z} } - {\\partial A_z \\over {\\partial x} } \\right) \\mathbf{j} + \\left( {\\partial A_y \\over {\\partial x} } - {\\partial A_x \\over {\\partial y} } \\right) \\mathbf{k}", "c1499aff1d0f0d030d2fdffd4ca3ec9c": "A=x^{(0,0)}", "c14a1266da7baa87b8b329f3d8aebe2e": "f(x)=72 \\Rightarrow f'(x)=0", "c14a1336cf0124bc51325105f45071e6": "\\scriptstyle f_i\\left(1\\right) \\;=\\; 1 \\;=\\; f_j\\left(1\\right)", "c14a3283a1cd4503a15caae3e587bb74": " \\int_0^{\\pi} \\frac {\\sin t}{t}\\, dt =\n\\sum \\limits_{n=1}^\\infty (-1)^{n-1} \\frac{\\pi^{2n-1}}{(2n-1)(2n-1)!} ", "c14acbd47ea6785da3ca9d00ee1a7cf3": "3/2\\ ", "c14b862038be9935eb7f33449ce2f451": " S_x(\\omega) = \\frac{2k_\\text{B} T}{\\omega} \\,\\mathrm{Im}[\\hat\\chi(\\omega)].", "c14b987c51b19ef684b8275ca741db8b": "\\textbf{A}_O = \\ddot{\\textbf{d}}", "c14bc01b47110a4de12c9e2f0cc2e1c9": "6-3=3", "c14bdd06baeb4be082feb7105b857bae": "V(\\rho,\\varphi,z)=\\frac{1}{\\sqrt{\\rho^2+z^2}}=\\int_0^\\infty J_0(k\\rho)e^{-k|z|}\\,dk.", "c14be571d434af01472a37b8382e95ff": "A^2\\xi_0=\\ln \\frac{t_0}{t_1}.", "c14bec4f1eb9242f4c1c47503e31004f": "\\mathfrak {sl}_{n+1}", "c14bfb61a2a8b4b61b4517fb7e1d93e1": "\\begin{align}\n \\int_k^{k + 1} f(x)\\,dx &= uv - \\int v\\,du &{}\\\\\n &= \\Big[f(x)P_1(x) \\Big]_k^{k + 1} - \\int_k^{k+1} f'(x)P_1(x)\\,dx \\\\[8pt]\n &= -B_1(f(k+1))-B_1(f(k)) - \\int_k^{k+1} f'(x)P_1(x)\\,dx\n\\end{align}", "c14c221f49560b9f60c85a42c745e033": " \\hat x + i\\hat p w_0^2", "c14c3e77c9812a4279400f928e036f01": "P(i)=\\frac{1}{M}", "c14c75023afc68a60ea0c68d74235eea": "ik_0(A_r-A_l)=ik_1(B_r-B_l)", "c14c7e96528175ba70406550cc7bb4f8": "\nk_p (\\tau _1 , .\\,.\\, ,\\tau _p ) = \\frac{E\\left\\{ {y(n)x(n-\\tau_1)\\cdots x(n-\\tau_p)} \\right\\}}{{p!A^p }}.\n", "c14cd60ccac5ce4a0bfdc081b51ffabd": "\\frac{dN}{dt}=\\frac{N\\ln\\left(\\frac{1}{2}\\right)}{T_{1/2} }", "c14cf1cdb13db70a15599868530e3a86": "\\overline{\\mathbf{CP}^2}", "c14cfbc342959b853a06e8d5dcc4475c": "X(t)=(X_1(t),X_2(t),...,X_N(t))^T", "c14d477045f8bd3937a9c8785f30b81a": "\\sigma\\in S_{n-1}", "c14d68e3306fdfabb2ab2de32e81d45c": "\\rho_{\\max}", "c14d7077adeb747a14b2cca103b4a7a1": " i=1,\\dots n", "c14db043b6753e1f0feded6ff89496fa": "X_{(1)}", "c14dde02e40b9b424b0abdbb62a16140": " 0 < | x - 5 | < \\varepsilon / 3 \\ \\Rightarrow \\ | x - 5 | < \\varepsilon / 3 ,", "c14deaad692043cf05ae927ab46ee27b": " \\gamma_{nl} |A|^2 A ", "c14e094b4d0798e4943057ead44b61db": "sen(\\sigma)(\\varphi)", "c14e1bbb6d4c047b5db77078c765d681": "\\underline{\\mathsf{f}}(X) = RXR^{\\dagger}.", "c14e4d1355248a9bcc734911dd853e03": " arg(c) \\, ", "c14ed80f6c452f249c0ed57cf4b3926c": "\\delta_{\\epsilon} P =\\bar{\\epsilon} \\gamma_{5} \\psi", "c14f2a16a5b5c9a47809f271b8e8ff8f": "K(\\sin \\varphi) E(\\sin \\theta ) + K(\\sin \\theta ) E(\\sin \\varphi) - K(\\sin \\varphi) K(\\sin \\theta) = {1 \\over 2}\\pi.\\!", "c14f592f2e10961cead983b726916ea6": "k[G]", "c14f606d42dd2c73732377e74358f56d": "V[\\rho] = \\int \\frac{\\rho(\\boldsymbol{r})}{|\\boldsymbol{r}|} \\ d\\boldsymbol{r}.", "c14fb1d0b4806d840557533a0a3774e3": "S_3(s)=s(3-4s)^2=t(s)", "c14fc372fe46f394f83be95e9d0f2592": "A\\colon V \\to W", "c14fe724485748a7f3c6c84afd9e4069": "B \\rightarrow A: \\{N_A, N_B, B\\}_{K_{PA}}", "c15033c476a4033b913a9e3149d86966": "\nG(\\boldsymbol{x} - \\boldsymbol{r}) = \\frac{ \\sin{( \\pi (\\boldsymbol{x-r}) / \\Delta )} }{ \\pi (\\boldsymbol{x-r}) }.\n", "c150ff54a6bd0df9d6513fec028d6788": " \\mathcal{L}_X \\mu = \\operatorname{div}(X) \\mu ", "c151188e6a0b2b4c18de716e84aaaf7a": "\\nu_{\\mu}", "c1516a75464998fef5dea337528ebaeb": "e^-\\to e^- + Z^0", "c151acb7f5e06a565930f612e2ed193d": "\\phi(\\mathbf{x}) = \\phi(\\|\\mathbf{x}\\|)", "c151b510eb6ce5e5d9defe6c1b457229": "[-128, 127]", "c151dbb3492a08cd04a8bedf2b8e7f01": "\\mathbf{B}(\\lambda\\mathbf{x},\\lambda t)", "c151e0158e0b27c1709e13d8286d3570": "H = - \\frac{\\partial\\mathcal{S}}{\\partial t}", "c151ee269ff7d170d4db5c9f5cb1baf7": " m_1, \\, \\dots \\, , \\, m_{n - 1} ", "c151f6420b6cd8573ac44774c6aeb0ff": "t_k = 0 ", "c1520a0083ab8ac933370fce928a8810": "F_4(s/\\sqrt{2})=\\cosh\\left(\\frac{1}{2}\\int_s^\\infty q(x)\\, dx\\right)\\, \\left(F_2(s)\\right)^{1/2}.", "c1522c8a941a5264153fb4ef3e7748a4": "\\R_+^2", "c152348fd05175700a78dbc77ebdd714": "r_L", "c1526e68e16efce45f9d696f39195f68": "B\\mapsto A\\otimes B", "c152c0ff3b3e1fa7b95c1757ef44d411": " |S| ", "c1537723a02b8fa1ed8ac711a00c9c60": "|S[\\phi]| \\le C_N \\sum_{k=0}^{M_N}\\sup_{x\\in [-N,N]}|\\phi^{(k)}(x)|.", "c153e3e73ec3239c31477d28e623e68a": "C_{n+1} = 2 \\sum_{i=0}^n d_i C_i C_{n-i} - \\sum_{i=0}^n \\frac{2i+1}{i+1} d_i C_i C_{n-i} = \\sum_{i=0}^n \\frac{d_i}{i+1} C_i C_{n-i}.", "c15424501cf9878a7e06fd7366294011": "\\textit{e}_{ex} = \\left[\\begin{array}{c}\\cos \\phi_{ex}\\sin \\theta_{ex}\\\\\\sin \\phi_{ex}\\sin \\theta_{ex} \\\\\\cos \\theta_{ex}\\end{array}\\right]", "c1543056b242e6218af37585ceaa1fa5": "1 \\; \\mathrm{C}\\sqrt{4 \\pi /\\epsilon_0} = 3.7673 \\times 10^{10} \\; \\mathrm{statC}", "c1549900bf96bad217d089104873596a": "P(N \\leq 40[200000]) = \\frac{39}{40}", "c1551b76f606485171a0229e23ae4b91": "w_i = \\frac {2^{n-1} n! \\sqrt{\\pi}} {n^2[H_{n-1}(x_i)]^2}.", "c1552058e8b93eec15c9d42092156b96": "X(z)=\\mathcal{Z}\\{x[n]\\}", "c1557d6871ae9f68e86d2135c0ff6e7d": " f(x) = \\begin{cases} 1-x^2 & \\text{if}\\ |x| < 1 \\\\ 0 & \\text{if}\\ |x| \\ge 1 \\end{cases}", "c1558e5de459da16db047a21f19610b1": "v_\\mathrm{p} = \\frac{\\omega}{k} = \\frac{E/\\hbar}{p/\\hbar} = \\frac{E}{p}. ", "c155a719e6fbd6ec9783e43c48146067": "\\displaystyle{C=B\\oplus Bj, \\,\\,\\, (a+bj)^*=a^* - bj, \\,\\,\\, (a+bj)(c+dj)=(ac -d^*b) +(bc^*+da)j.}", "c155c33b904651657e2c0a06e5dec5ae": "r(\\lambda)", "c156392eefa18872ec464d0826cc44c5": "Tr(g^{bk})\\in GF(p^2)", "c1565e74207793747b59774e418654f9": "H\\psi(x)=\\left[-\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}+V(x)\\right]\\psi(x)=E\\psi(x),", "c1565efbe80267264db678e1fc0b838f": "\\int x^n e^{cx}\\; \\mathrm{d}x = \\frac{1}{c} x^n e^{cx} - \\frac{n}{c}\\int x^{n-1} e^{cx} \\mathrm{d}x = \\left( \\frac{\\partial}{\\partial c} \\right)^n \\frac{e^{cx}}{c} ", "c15666034e721782962e268b27a342cb": "\n\\vec J_1\\left( \\vec x \\right) = a_1 v_1 {1\\over 2 \\pi r L_B} \\; \\delta^ 2 \\left( r - r_{B1} \\right)\\;\n\\left( {\\hat b \\times \\hat r }\\right)\n", "c156785d3c2433ba662270ae134ef8c5": " \\Phi(x,y)", "c156859c1340abb45b09404d383ca15c": "\\phi = \\arccos\\left(\\frac{\\Delta_1}{2\\sqrt{\\Delta_0^3}}\\right).", "c156ba102dc31af1203efb1db593cbb6": "\\Theta_{bg}", "c15795cbd16fbd4767fc45ba4aefc558": "\\,(q,\\gamma)\\in Q\\times\\Gamma^*", "c157a2d54deb75f19aaf3db49d1b17ff": "U_o(S_o)=\\mathrm{min}_x(U(S_o,x))\\,", "c157a8edf39b6b9119a6b0cb4268a105": "\\color{blue}\\mathcal{S} \\color{blue}\\rightarrow \\color{blue}\\mathcal{I} \\color{blue}\\rightarrow \\color{blue}\\mathcal{R} \\color{blue}\\rightarrow\\color{blue}\\mathcal{S}", "c157baad1f376c86bdf72ddac5326bed": "\\Tau", "c157e221a000491d4eb2b5362aa9c94f": "{C}_{2}^{(1)}", "c158420d2aa64d1aa97eb264e17e3278": "W' \\subseteq X^*", "c15871d00a6c10a5191b03156f0a0920": "\\gcd\\{f(n) : n \\geq 1\\} = \\gcd(a_0,a_1,\\dots,a_d).", "c15922c4d17f1c9b08faf4f04aa862e0": "\\sum_{n=1}^\\infty \\frac{1}{(n+2)^a}\\sum_{n=1}^\\infty \\frac{H_n^{(c)}(-1)^{(n+1)}}{(n+1)^b}=\\zeta(a,\\bar{b},c) ", "c15924b4dbe2f1854b825b9104242dd6": "\\Delta z^M=z-\\sum_i x_iz^*_i.", "c1596d865d3bf591528030c5ff151071": " \\text{development} = \\text{drive wheel circumference in metres}\\times\\frac\\text{number of teeth in front chainring}\\text{number of teeth in rear sprocket}", "c1597be8a1d8ca14f881e3294d71c833": "y = \\cos(t) \\left(e^{\\cos(t)} - 2\\cos(4t) - \\sin^5\\left({t \\over 12}\\right)\\right)", "c1598e3cc2ee95ee79b7509e19790553": "B_k=B_0\\bigg(1-\\frac{r_k}{r_n}\\bigg)=B_0\\bigg(1-\\frac{\\lambda_k}{\\lambda_n}\\bigg)", "c159baad0620c535b40bd8fad0c22880": "78 \\rightarrow 45_0 \\oplus 16_{-3} \\oplus \\bar{16}_3 + 1_0. ", "c15a8131037e01687149307a3041a2a0": "G[", "c15b315eb9dbfc696809882acc20a2a3": "f(x)=x^2-n", "c15b83c152327b8b451aa5a0bcddbcf7": "P_{i'}\\,", "c15be3a19862b5dc63526895ee917504": "|\\psi\\rangle = \\int c(\\phi){\\rm d}\\phi|\\phi_i\\rangle ", "c15bf271caef2cdef5a575a5a00b04a9": " \\mathrm{CG_{p}} = \\sum_{i=1}^{p} rel_{i} ", "c15bf66c4d89c18091ae703d0909c11c": "\\sum_{i = 0}^{n - 1} d(g(u_i), g(u_{i+1}))", "c15c50fa07fc74d3c961532a99ccfecb": "y^m = f(x),", "c15c9879a4fac8bff87bcd1ba68b94ef": "\\mathbf{F} = \\mathbf{1}", "c15c9cbacefdc458545601d2ce84aa7c": "\\lnot(x \\wedge y)=\\lnot x \\vee \\lnot y \\mbox{ for all regular } x, y \\in H,", "c15d36b0a761bb805b27bc8b84135bd2": "| z - x | > \\varepsilon.", "c15da82607af4855fff9593694735aca": " \\frac{ 9.80665 \\ \\mathrm{m} / \\mathrm{s}^2 }{ 8.870056 \\ \\mathrm{mm} } \\left( \\frac{6375416 \\ \\mathrm{m} }{299792458 \\ \\mathrm{m} / \\mathrm{s} } \\right)^2 = \\left( 1105.59 \\ \\mathrm{s}^{-2} \\right) \\left( 0.0212661 \\ \\mathrm{s} \\right)^2 = \\frac{1}{2}.", "c15db7650b735d16483ef5aa87a9543a": "F_1(x)=-x", "c15db9a8b210763895278593e1021967": " R_q = R/\\langle q \\rangle = \\mathbb{Z}_q[x]/\\langle f(x) \\rangle ", "c15e1619ca9a07941fb6c970f69ab9bc": "\\mathbf{E}(\\mathbf{r}) = \\frac{Q}{4\\pi \\mathcal{E}_0} \\frac{\\hat{\\mathbf{r}}}{r^2}", "c15e91c375acd4ebec2717558b0bc2b6": "L^r", "c15ed67aeac816c025614ba76da69ea0": "\\begin{bmatrix}1&0\\\\0&0\\end{bmatrix}:\\mathbf a", "c15eddd78acf97fdb0964f74bb9d48f6": " 2\\cos \\lambda=1+(l-l^{-1})/m", "c15ef02fb039d5e405d251399936aac8": "w(a)=w_0+w_a(1-a)", "c15ef0e0e022845a1a23cafc8ff56f13": "3^{(F_n-1)/2}\\equiv\\left(\\frac3{F_n}\\right)\\pmod{F_n}", "c15f1e0d5497d6c9b2b6f85682a4e160": "\n k_1 = +\\sqrt{z_+} ~,~~ k_2 = -\\sqrt{z_+} ~,~~ k_3 = +\\sqrt{z_-} ~,~~ k_4 = -\\sqrt{z_-}\n ", "c15fb44ca5df9be976d0cbdf496d94bb": "s^{-1}f(y/s,\\alpha,\\beta,c,0)", "c160531bfff435b2520b3160b1e142fb": "\\mathbf{u}\\cdot \\mathbf{n} = 0", "c160e8663fa8a6367fda47be9c829bc8": "\\langle\\mathbf{e}_i,\\mathbf{a}_i \\rangle = \\|\\mathbf{u}_i\\|", "c161312e572341cdfb5f365580d1eb21": "h(t)=\\frac{f(t)}{1-F(t)}=\\frac{f(t)}{R(t)}.", "c1619371c0dfe290eb0e3efce1798635": "V_i = \\frac{r_i}{\\sum_j r_j x_j}", "c161a094682b4359fb83b3e9e938b9a8": "X_1, X_2", "c161d535dd01e1258fdd3900c928be87": " c_{s_1s_2}(\\lambda) =c_{s_1}(s_2 \\lambda)c_{s_2}(\\lambda)", "c163351795a6f3aa425b7be9118d0344": "\\ |X_n(\\omega)| \\le |Y(\\omega)|,\\ Y(\\omega)\\ge 0,\\ E(Y)< \\infty,", "c16352ebe616126a68b5bf5e0ac08127": "\\scriptstyle f(x)^{g(x)}", "c1639c21407e65a21e2a51940e883047": "(x^2 - 5)^2 - 24.\\ ", "c163d31aad6246163fadf2ca47ef3ea7": "\n F_i~\\sigma_i + F_{ij}~\\sigma_i~\\sigma_j \\le 1\n ", "c164018a10b0a074478a8019feee6d4e": " \\left|1,V\\right\\rang = {1 \\over \\sqrt{2}} \\left|1,45\\right\\rang + {1 \\over \\sqrt{2}} \\left|1,135\\right\\rang ", "c164827d0cea7cbf7cc2bc264ec9d92c": " \\vec\\mathfrak r_i = \\vec r - \\vec r_i ,", "c1648799c5dcf40c08ac9ae508d1baf8": "f=k / r^p", "c164b82347a985f51a2889d9b2c56c48": " (1)\\qquad x_{n+1} = r x_n (1-x_n) ", "c164e630313c7e71508c5c046f83c6f5": "X'", "c1650eb3406cf447dba65b3f71863da4": "g^1(q;\\tau)= \\sum_{i=1}^n G_i(\\Gamma_i)\\exp(-\\Gamma_i\\tau) = \\int G(\\Gamma)\\exp(-\\Gamma\\tau)\\,d\\Gamma.", "c165560ff1838554839f63052bb511a7": " \\ \\Gamma ", "c165b3873206f1b6d1de26759fa9f31a": "M_n(x,\\beta,\\gamma) = \\sum_{k=0}^n (-1)^k{n \\choose k}{x\\choose k}k!(x-\\beta)_{n-k}\\gamma^{-k}", "c165f2c9dfdf5e0a7269631dacf82200": "SV > UCC", "c16618b7a1cea37ecfb14e9bb3b09d33": "x*(x\\Rightarrow y)\\le y.", "c166466a33e1f7204f147d487ed04bce": "\\gamma = \\arccos\\left(\\frac{\\cos c-\\cos a\\ \\cos b}{\\sin a\\ \\sin b}\\right),", "c16655f18ace42da0f3fb4ac012e9c6a": "L' = aL \\,", "c16747713b66aa8ee93af905a6d8b6eb": "\\operatorname{Tr}", "c167a4bf996ba9514c54fe492122592b": "f(x) = x^5 - 2x^4-7x^3 + 8x^2 + 12x = ", "c167b8a2a0b1dda13c06f6262cfd42c6": " fx_1 \\cdot \\cdot \\cdot x_n = A ", "c167cbe73628174eeac977f926f7a41d": "v = {{V_\\max [S]} \\over {K_m + [S]}}", "c167fccb3d211ce541da39435a1a2e72": "B_{2n} = (-1)^{n+1}\\frac {2(2n)!} {(2\\pi)^{2n}} \\left[1+\\frac{1}{2^{2n}}+\\frac{1}{3^{2n}}+\\frac{1}{4^{2n}}+\\cdots\\;\\right]. ", "c16806117c8069cd8ca9d39469ba8164": "\\vec{s}_i \\in \\mathbb{R}^3, |\\vec{s}_i|=1\\quad (1)", "c1683f36610ddb3e936fe57adf1d4657": "o = \\frac{1}{n}\\sum_{i=1}^n \\left (p_i - q_{\\pi(i)} \\right ),\n A = (Q^+ P)^t", "c16861f96fad2d66f7033007549f51ac": "\n\\overline{H}\\left( y^{n}|x^{n}\\right) \\equiv-\\frac{1}{n}\\log\\left(\np_{Y^{n}|X^{n}}\\left( y^{n}|x^{n}\\right) \\right) ,", "c168a75bc922248c865eb9c0c37a8255": "\\frac{n^2+n}{2}+\\frac{k^2+k}{2}", "c168b5a05c2d80d40cd485b288235a5b": "I(X; X)=H(X)", "c168c87a887fdf36f8685950b250beaf": "X = C + B", "c168ddfa671cdf68753340b5482a1b10": "\\scriptstyle \\sqrt[2]{10}.", "c1690f0cc648b2d8fb37e8586d5f5181": "A V = constant \\Rightarrow \\frac{dA}{A}=-\\frac{dV}{V} ", "c16969a4212b7bc5ae1a5794f3b1d655": "F_{T}(x)=F_{S}(\\rho^{\\lambda}x)", "c16a21f5c85d100d13cee08d6ab8b27e": "e^{-r^2}r^\\nu U\\left(a,b,r^2\\right)\\,", "c16a5320fa475530d9583c34fd356ef5": "31", "c16a7663832c61923b16f699b88f900e": "\n0.3<\\frac{M_1}{M_2}<20\n", "c16a8770582fda0b5e4ab1bd9964ae6a": "{x}_{k+1} = \\frac{1}{2}\\left(x_k + \\frac{ n }{x_k}\\right), \\quad k \\ge 0, \\quad x_0 > 0.", "c16aa53ffb0fd86e6d0fb11520f781ba": "g=r_{rs}/r_{rr}", "c16aafd2897bc21de8ff24a5dc95581f": "p=prob_A=\\frac{2+1+2+0+1+2+2+1+1+2}{2*10}=0.7", "c16ab2ca7fc87e11c28e903c5f2ebba5": "e=d_i-(b+wx_i)", "c16acd17a86ed3602304d51d5df696da": "\\gamma(t)=te^{i\\theta}", "c16b13102f0d74bc25bd257df96bd077": "\n\\sigma(s) = \\sqrt{\\epsilon \\cdot \\beta(s)}\n", "c16b4e874c7d8f15218bb2fce4b35840": "2 \\pi h\\over\\lambda ", "c16b5f270dc587cf509db2e5289c0cae": "C = 0", "c16b67ebc9f09d1a5d50fc29bc0759d7": "\\quad\n\\beta^{1} =\n\\begin{pmatrix}\n0&0&-1&0&0\\\\\n0&0&0&0&0\\\\\n1&0&0&0&0\\\\\n0&0&0&0&0\\\\\n0&0&0&0&0\n\\end{pmatrix}\n", "c16bc9ab51d44feb8113e6dabfdc1cbe": "\\int_{-\\pi}^\\pi \\cos(\\alpha x)\\cos^n(\\beta x) dx = \\begin{cases}\n\\frac{2 \\pi}{2^n} \\binom{n}{m} & |\\alpha|= |\\beta (2m-n)| \\\\\n0 & \\text{otherwise}\n\\end{cases} ", "c16bdce24f0afad8a51bee6716017e03": "\nU_X = \\nabla_{\\theta} \\log P(X|\\theta)\n", "c16caa4a61281b00b18fbb097120b66e": "\\sum_{k=0}^n S(n,k)(x)_k=x^n.", "c16cb3d3da6f96897b364bf3c2b364a0": "Q_\\text{surf}", "c16ce3cf79149393ea0dd1201ae8137b": " \\vec{J} ", "c16ce79b8afde914b1a42e4628ee0985": "\\mathrm{Er}=\\frac{\\mu v L}{K}", "c16cf693dc834bf13739b9c4b95b2579": "G_\\infty=\\bigoplus_{i\\in I}F_{Q_i}\\;", "c16d29c776b1b1854783af9fa8b2ef47": "y'=\\frac{y_d}{y_c}", "c16d4f94cd8106d0b320e4a47470dd8e": " \\frac{dR}{dt} = \\int_0^\\infty \\gamma(a) i(a,t) \\, da - \\sigma \\lambda R - m_R R,", "c16d65eef5066c6201289ad65c36728f": "P(\\lim_{n\\rightarrow\\infty} X_n=X)=1", "c16d6f04e79f99d6fc93bb5206f684ce": "\\begin{bmatrix}\\rho \\\\ \\theta \\\\ \\phi \\end{bmatrix} = \n\\begin{bmatrix}\n\\sqrt{x^2 + y^2 + z^2} \\\\ \\arccos(z / \\rho) \\\\ \\arctan(y / x)\n\\end{bmatrix},\\ \\ \\ 0 \\le \\theta \\le \\pi,\\ \\ \\ 0 \\le \\phi < 2\\pi,\n", "c16d9c1d72d9f99a247424e8f255e1af": "\\frac{1}{2^nn!}\\int_0^1(1-z^2)^n\\cos(xz)\\,dz=U_n(x),", "c16de1f33f1f5b42151049e8100beebe": "\\mathcal{L}[t] = - m c^2 \\sqrt {1 - \\frac{{\\dot{\\vec{x}}[t]}^2}{c^2}} - e \\phi [\\vec{x}[t],t] + e \\dot{\\vec{x}}[t] \\cdot \\vec{A} [\\vec{x}[t],t] \\,.", "c16e069f782151dd8057a46e80290465": " \\beta(v\\cdot\\phi,\\psi) = (-1)^m(-1)^{\\frac12 m(m+1)}\\beta(v\\cdot\\psi,\\phi) = (-1)^m \\beta(\\phi,v\\cdot\\psi)", "c16e571b9f71e740fda5018a37c10f17": "n0", "c16ea8b2a6c09b6890e0e9d72bfa95dc": "x^2 + K_a x - K_a F = 0", "c16eba0952cc5e4ed2922663ca869cdb": " {\\mathcal F}_c (f) ", "c16edfc45930f2611fe67422c9242fc0": "{A}_{13}^{(2)}", "c16efd7838b82f6f5527a931e38e25fa": " \\overline{\\rho} = {\\dot m \\over \\overline{U}A},", "c16f49a4d0a0ba354d59413d3e2fd498": "0\\to U\\to V\\to V/U\\to 0.\\,", "c16f4c7d91e5dac8b95b9720646009cf": "f(x)\\pm g(x) = \\sum_{n=0}^\\infty (a_n \\pm b_n) (x-c)^n.", "c16f4e85c563e4275677a44aebfea856": "I(X;Y) = \\sum_{y\\in Y} p(y)\\sum_{x\\in X} p(x|y) \\log \\frac{p(x|y)}{p(x)} = \\sum_{x,y} p(x,y) \\log \\frac{p(x,y)}{p(x)\\, p(y)}.", "c16fadcc62d6cab1a6416479c6fd819e": "R = r \\, \\cos \\phi", "c16fc7ccbbf6686c77eacb7722c0e673": " \\sin x = \\frac{a+bx+cx^2}{p+qx+rx^2}", "c16fd2404586459013af67efe64b8d83": " (x\\lor x)", "c17011318807b16baf5201af476d331c": "\n\\left(\\frac{\\partial S}{\\partial p}\\right)_T =\n-\\left(\\frac{\\partial V}{\\partial T}\\right)_p\\qquad=\n\\frac{\\partial^2 G }{\\partial T \\partial P}\n", "c17087e3d075ef112c2fef92e32679ae": "_{1 \\nleftarrow p=0}\\!", "c1709b4455f7b250e24ff93ca3a60bca": "\\hat\\phi_j", "c170cf33ef334b4f9f8a8f8a1c85c301": "\\hat{X}^{opt}_i(z^n)=\\hat{X}_{Bayes}\\left(\n\\Pi^{-\\top}\\mathbf{P}_{Z_i,z^{n\\backslash i}}\\odot\\pi_{z_i} \\right)", "c170f4e1059332d767f367abc981ef9d": "\\xi_i,\\quad i=1,2,\\ldots,m", "c170fa56c82f900bdae3429d34d1ce6d": " \\lim_{x \\to c} f(x) = 0,\\ \\lim_{x \\to c} g(x) = 0 \\! ", "c17104694c349ad86ceb792cc9e2c6b3": " \\nabla_{\\theta} J(\\theta) = \\nabla_{\\theta} \\int f(x) \\; \\pi(x \\,|\\, \\theta) \\; dx ", "c17109a10a718d3b79fe1b491ec39dce": "\n \\operatorname{E}^*h(X_n) \\to \\operatorname{E}\\,h(X)\n ", "c17133325d9515f0000ef84373482dde": "\\rho = 2RD", "c1714c5bbe569582ac5393800a4b168b": "0 \\leq \\mu^* \\leq 2.45", "c171f169915b9e4badfa1fe0dabb1bfe": "s(t)=\\beta e^{-\\alpha t^2} \\sin(2 \\pi f_{max} t)", "c1720ee3a7e9d929b31d69a2d1bc8271": " \\frac{\\partial F}{\\partial f} - \\frac{\\mathrm{d}}{\\mathrm{d}x} \\frac{\\partial F}{\\partial f'} = 0 \\ . ", "c172124fd0e88b90d8331e7faffcfd2d": "\\tau_{\\geq n}, \\tau_{\\leq n}", "c172541f77a147fcf545237fefa03643": "\\textstyle g", "c1726a67c2d3b2df100f7e58eb4666c4": "\\Delta A=A_L-A_R \\,", "c1729b10be28b60720efa01dbbd8fed0": "dH_{n-1}^2 = d\\xi^2 + \\sinh^2\\xi d\\Omega_{n-2}^2", "c172bbd3cd8d0a376b9783b8406b48b3": "\\frac{\\mu_1-\\mu_2}{\\sigma}.", "c172c9825591e8a593a782ab17879740": " \n\\sum_{n=0}^{\\infty}(-1)^{n}\\frac{\\ln(2n+1)}{2n+1} \\,=\\,\\frac{\\pi}{4}\\big(\\ln\\pi - \\gamma) -\\pi\\ln\\Gamma\\left(\\frac{3}{4}\\right)\n", "c172e309535f6ff639b845bddf5e5319": "w_2", "c173b2da9b4a358e60041a66b6c8bd83": "\\varphi(\\mathbf{x_i})", "c173c16bb14e3bf6e702bfb4c425b414": "B_k=B_0-r_k(B^*-B_0)\\;", "c1742b510e7adfff1f6a36efe654b528": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{1}{N} \\sum_{k=1}^{N} \\left( \\mathbf{r}_{k} - \\mathbf{r}_{\\mathrm{mean}} \\right)^{2},\n", "c1745c2d673ca9e892e7d57b00011ffd": "KC(\\mathcal{S} \\cup T(\\mathcal{M}, \\theta)) = C - 2 \\operatorname{cost}(\\mathcal{S}, \\mathcal{M}, \\theta)", "c17466c9b7d80c2dc58b5ed95f54d6ae": "{\\Delta P} = P_{ATM} - P_A", "c1747e63a9a6b7f10cb0b19a2bdb0a82": "a=[a_n]", "c174e35ff901325d7a5922651e804209": " v_i = \\left\\{\\begin{matrix} \n-1 & \\mbox {with prob. } 1/2 \\\\\n1 & \\mbox {with prob. } 1/2\n\\end{matrix}\\right.", "c174ed5dc6d78d47fcff6186da7b4e89": "\\frac{c}{d} - \\frac{a}{b} = \\frac{bc - ad}{bd},", "c1753c36ab4eb582f1420d5178cb4bc5": "p,q", "c175bab20b74214a753115cb10d0e0f8": "(\\hat{O} \\mathbf{E}_1, \\mathbf{E}_2)", "c175c1c3bff18902cf41f16ba9242e8f": "(S_s)_m = \\frac{1}{m_a}\\frac{dm_w}{dh}", "c1762498da2610a6c2b3ad3ddee50ec5": "x_{n-\\ell}\\,", "c1768932bcc86ee45ff942b69b3e22c4": "\\frac{d}{dq} q- q \\frac{d}{dq} = 1 ", "c1769db99949d0b63c67b57dc722063f": " d\\mu =c_\\Lambda\\exp(-\\text{tr} X^2\\Lambda/2)dX", "c1773d0ef5b4d7f7a984f663a36c7981": "\\Delta\\nu=\\frac{c}{2n_gl}", "c17747a43afc804036276635fd49a17e": "E_{\\vec{n}} = E_0 + \\frac{\\hbar^2 \\pi^2}{2m L^2} |\\vec{n}|^2 \\,", "c1778f2e4ed814796780e96f325c3f7f": "\\mathfrak{F}\\{f*g\\} = \\mathfrak{F}\\{f\\} \\cdot \\mathfrak{F}\\{g\\} ", "c1779c7281e44fe97e28347ba0021de6": "\\Gamma \\vdash_{\\mathcal {FS} } A", "c177cb2744aa19f073bcfde8e31cec1d": "\\operatorname{var}[S]=\\sigma_{B}^2", "c177f5525772c16e6e817a6ab771007a": " 1_B(x) =1", "c178139cbb8525b00b997c5c70748d92": "\\beta_i \\equiv ", "c1783d78de855590941caaee360cb33c": "f(x)=x^3-2x^2-11x+12", "c17879df3bfd1233e036bb6824f5389f": "\nH(t+T) = H(t) \\, .\n", "c179a228f9f83ebbb5bb43bf14837d5c": "Tv_{env}", "c17a07f076c4ebeac9d490a197bdc4bf": "^4C_1", "c17a23f798b2fa04227072c9eae4ca37": " y' = f(t,y), \\quad y(t_0) = y_0. ", "c17a8f87aa80962b6ec4cf0c3633ae28": "\\scriptstyle \\mathbf{H}_w", "c17ae98ae48d8066e08f0e98d0ec0fe7": "\\phi(r,\\theta)=U\\left(r+\\frac{R^2}{r}\\right)\\cos\\theta.", "c17b3fa5d95153ae339ea0269b188997": "O \\left (\\deg(f)^2(\\log(q)+\\deg(f)) \\right )", "c17b47981436b7e90118981cc56d61c8": " \\bold{H} ", "c17b52f10ceb500f5c560f798084aeb8": "{ G_{\\mu \\nu} = 8 \\pi T_{\\mu \\nu} } \\ ", "c17b65df29c6951451d78969866aa9fe": "f(\\theta,\\phi) = \\sum_{l=0}^\\infty\\, \\sum_{m=-l}^{l}\\, C^m_l\\, Y^m_l(\\theta,\\phi).", "c17b719feb7f8ffaf16320249fd011b1": " M(t) = G (e^t) = \\exp(a_1(e^t-1)+a_2(e^{2t}-1))", "c17b725d06c3cab3373530a217888e1b": "\n\\left(\\frac{1}{\\mu_{0}}\\right)\\boldsymbol{(\\nabla\\times B)\\times B} = -\\boldsymbol{\\nabla}\\left(\\frac{B^2}{2\\mu_{0}}\\right) + \\left(\\frac{1}{\\mu_{0}}\\right)\\boldsymbol{ (B\\cdot\\nabla) B}", "c17bab907950df67a34cb249f09cb6e3": "Z_{in} = \\frac{5}{3} \\cdot \\frac{1+\\frac{3}{5}s}{1+\\frac{5}{3}s}", "c17bd08cf6c23bc03520c101ac20105e": "R = 101111 + 110001 = 1100000", "c17d2323df200e20a7f68be8e26bd215": "V = \\bigoplus V_i", "c17d33ce64ca8ef39347a4910d0b939c": "\\phi(\\alpha x+\\beta y)=\\alpha\\phi(x)+\\beta \\phi(y)", "c17d44ddf603f5cd5811ca5aa23d9f87": "\n\\frac{dG}{dt} = \\frac{1}{2} \\frac{d^2 I}{dt^2} = 2 T + V_\\text{TOT}\n", "c17d59d4599b561a2d4e6ec225743fd6": "\n (PR)\n ", "c17dad0f29077aa1d652857619bd2341": "r_{ex} = \\frac{a}{(3+\\sqrt{5})\\sqrt 3} \\approx 0.1102641 a.", "c17dbfb95ffe5caf82723ad94350c276": "G_{ii}(\\tau) = -\\langle T c_i(\\tau)c_i^{\\dagger}(0)\\rangle ", "c17e049691f2eb2d93102b7d4b309572": "\\sqrt{a}", "c17e57af4016f750f08575d1a4eb4d99": "\n\\mathbf{F}(\\mathbf{r}, \\mathbf{m}_1, \\mathbf{m}_2) = \\frac{3 \\mu_0}{4 \\pi r^5}\\left[(\\mathbf{m}_1\\cdot\\mathbf{r})\\mathbf{m}_2 + (\\mathbf{m}_2\\cdot\\mathbf{r})\\mathbf{m}_1 + (\\mathbf{m}_1\\cdot\\mathbf{m}_2)\\mathbf{r} - \\frac{5(\\mathbf{m}_1\\cdot\\mathbf{r})(\\mathbf{m}_2\\cdot\\mathbf{r})}{r^2}\\mathbf{r}\\right]\n", "c17eaab89b239d28ab91fc6eb4007118": " X_1(X_1'X_1)^{-1}X_1' ", "c17f3766b67cf00d44fbfa581402a5f0": "z = n_1(10^1 - 1) + n_0(10^0 - 1) = n_1 \\cdot 9 + n_0 \\cdot 0 = 9n_1", "c17f4a97f61a20b314f59b9ac0052250": "f(s)=\\sum_{n=1}^{\\infty}\\frac{a_n}{n^s}=\\prod_{k=2}^{\\infty}\\frac{1}{1-k^{-s}}.", "c17f5c262eb5f765b5bd182c7ae39e10": " \\frac{\\partial }{\\partial t}|\\psi(x,t)|^2 + \\nabla \\cdot \\mathbf{J(x,t)} = 0 ", "c180200b2169b4da26ab073177d4de23": "[a\\;\\|\\;M]_m \\rightarrow [v\\;\\|\\;M]_m", "c1802158b7480810a780d0b959dfc536": "\\scriptstyle Z_{\\mathrm {i\\Pi}m}", "c18047fe20d84e6e9495e9a974e15237": "\\nabla\\,", "c180b15a898a4ffb80ef9df19ff5681f": "\\beta \\equiv \\alpha - 2", "c180cf9c798fc780fa9d00962d9c7ea1": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\sum_\\alpha m_\\alpha x^i_\\alpha", "c180e67a521f173b9d581a25903bdb93": "h(\\mathbf{x})=H\\mathbf{x+f}", "c18121ceffd624501e37f7c9c7960f35": "\\begin{align} \nF(n,k)&=\\frac{(-1)^k {n \\choose k} {2k \\choose k} 4^{n-k}}{{2n \\choose n}} \\\\\nG(n,k)&=R(n,k)F(n,k-1)\n\\end{align} ", "c1812d8b97fb36b665f80273f540cf13": "-\\frac{\\mu\\Delta x}{T}\\frac{\\partial^2 y}{\\partial t^2}=\\frac{T_2 \\sin(\\beta)}{T_2 \\cos(\\beta)}+\\frac{T_1 \\sin(\\alpha)}{T_1 \\cos(\\alpha)}=\\tan(\\beta)+\\tan(\\alpha)", "c18138b76a825fc47e45bb193ab0ee8b": "\\frac{c}{d}>0.33", "c18185a8bce3a7c70b8d91c16ab9cc5a": "\\frac{d}{d\\omega}\\mathopen{}\\left(k(\\omega) x - \\omega t\\right)\\mathclose{} = 0", "c181961b5a35a66622cf4de14f51553d": "m,n\\in\\mathbf{Z}", "c181c6b2f0c54672cbcebd2b7c066318": " = (3 \\lambda : 0 : 0 : 1) + (0:0:1:1) \\ ", "c181cea3d621b13325072803e65cd916": "a_1x_1^2 + \\cdots + a_nx_n^2.", "c1825b7bbbf171ade25b357c6b5f66f7": "Ca_{O_2}", "c1826b08151570173ab965de311cad01": "\\varphi_1,\\dots,\\varphi_n\\in\\Gamma", "c18288163dbf9fdcd8a48df98e507451": "\\delta \\psi(\\vec{x})=iq\\alpha(\\vec{x})\\psi(\\vec{x})", "c182a9fccf79135d10aa315487a280ed": "\\mathrm{adj}(\\mathbf{I}) = \\mathbf{I},", "c182ef1a0bdf38e0d530e3d4a830443b": "m(c) = c^{2753} \\; \\operatorname{mod}\\; 3233", "c182fdfb4fa0468ab02121923ccbed66": "D^{\\epsilon}(\\rho||\\sigma) \\geq D^{\\epsilon}(\\mathcal{E}(\\rho)||\\mathcal{E}(\\sigma))", "c183747564e5cc9d528dfc9e96ba2715": "\\frac {I}{G_{Eq}} = \\frac {I}{G_1} + \\frac {I}{G_2}", "c183ab260bf369569d8397fede736b8b": "C_L = M_L \\pm Corr \\times \\frac {M_L}{N_L}", "c184c62f9b3e13f1acda15cbf256c8ab": "\n-(\\lambda-1){(\\lambda +1)}^2\n", "c1853a3a3666f106d1bf0b05d23767c7": "\\Omega_b", "c18577c01659ec72ca683e027096aed0": "B\\geq0 ", "c1858e37571b9f5ce0578d650d4c28c7": "D_\\mathfrak{p}", "c185942760466b96722cbf9d568e298e": "{ c_{\\mathrm{gas: monatomic}} \\over c_{\\mathrm{gas: diatomic}} } = \\sqrt{{{{5 / 3} \\over {7 / 5}}}} = \\sqrt{25 \\over 21} ", "c185ff1947e036a0d1db9e1f591fb3eb": "\\begin{cases}\n \\mu + \\frac{b\\zeta(n-1)}{(n-1)\\zeta(n)} & \\text{if}\\ n>2 \\\\\n \\text{Indeterminate} & \\text{otherwise}\\ \\end{cases}", "c18616a4b870d40ac02b2bad18320fd9": "\\mathfrak{P}^{68}", "c1864bade9f0ec0acc2108a19b552ad2": "P_y(x)=\\frac {y}{x^2 + y^2}.", "c18668939144574569e1ff64a855d94f": "f(2,2) = x - 2y + 2 = (2) - 2(2) + 2 = 2 - 4 + 2 = 0", "c1867e364974b4baa785e7e7b0fcea5c": " Z(a;\\varepsilon_{1},\\varepsilon_{2},\\Lambda)=exp(-\\frac{1}{\\varepsilon_{1}\\varepsilon_{2}}(\\mathcal{F}(a;\\Lambda)+O(\\varepsilon_{1},\\varepsilon_{2}))\\,", "c186aedccb826009114923bc44f25bf1": "K(\\omega)", "c186af428a757af2971b3d482722ba33": "\\scriptstyle v^2\\,", "c186d9f1bd3e6961a0ef6aa2581bcbe2": "j_{\\mu}^{3}", "c187527e232b072d78eedb10d1115c23": "\np_{1}(\\Delta U) = \\frac{\\int ds^{N}exp(-\\beta U_{1})\\delta(U_{1}-U_{0}-\\Delta U)}{q_{1}} = \\frac{\\int ds^{N}exp[-\\beta(U_{0}+\\Delta U)]\\delta(U_{1}-U_{0}-\\Delta U)}{q_{1}}", "c187ba4974ceca9c65b28ff57424d5ad": "F = G \\frac{m_1 m_2}{r^2}", "c18827f2dfce31d9162b8b1300a324bc": " t\\mapsto [g(t),f(t)]. \\, ", "c18869048fcc9c5ae4092c39a86fad9d": "\\tfrac{25+3\\sqrt{69}}{2}", "c188d183a635104acb4002b3d8207d5d": "B_{\\epsilon}(x^\\ast) \\subset P", "c188eeb5bb111cfca4cdf3f06508fd57": "\\begin{matrix} {12 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "c189406e72400927cb2362165505fb0e": "T_d = \\frac{\\tau_d}{\\alpha}", "c189941e4539b8d2fd8e3c7dfd6049f3": "v = \\sqrt{xy}", "c189a4a0d8c3d77fec53de9b3a683881": "\n\\varphi = \\varphi_{0} + \\frac{L}{\\sqrt{2m}} \\int^{r} \\frac{dr}{r^{2} \\sqrt{E_{\\mathrm{tot}} - U(r) - \\frac{L^{2}}{2 m r^{2}}}}\n", "c189bd8d825845af7e825bebbfc86da8": " X(\\omega) ", "c189f758bc542da8e39632b07e4b4ad4": " \\operatorname{de-lambda}[p]\\ \\operatorname{de-lambda}[f], \\operatorname{de-lambda}[x]\\ \\operatorname{de-lambda}[x], \\operatorname{de-lambda}[f]\\ \\operatorname{de-lambda}[x]\\ \\operatorname{de-lambda}[x] ", "c18a4dd48a68fe4d04cb549ba43693c7": " dS_t = rS_t\\,dt + \\sigma S_t\\, d\\tilde{W}_t. ", "c18a97ff612a2fb9032ea1f1445160d8": "\\pi_{0}", "c18a9a045a8a6de9b98ae4f87ff09e77": " \\alpha, \\alpha', \\beta, \\gamma, \\gamma' ", "c18aa641bcc48799a0b317357ffe10ff": "\\log{\\frac {S_0}{S}} = K \\cdot C_a", "c18aef0bac9fa68e5c127b0519333588": "F_{\\alpha\\beta} = A_{\\alpha;\\beta} - A_{\\beta;\\alpha} = A_{\\alpha,\\beta} - A_{\\beta,\\alpha}\\!", "c18beb171fbdd75081392da0f0dc38dc": "\\scriptstyle {\\tfrac{S}{R} =\\ 2 \\sin{\\tfrac{\\pi}{7}} \\approx 1-(\\tfrac{4}{11})^2}", "c18c2b2fdf3de452e9d14f0c9946e49e": "\\mu_{2,1}", "c18c2fd98995409e2f850b5fba8b83ff": " i = \\; A_{\\mathrm{r}} a {\\phi^{-1}} {\\beta}^2 V^2 \\mathrm{exp}[- v(f) \\;b \\phi^{3/2} / \\beta V ], ..........(33) ", "c18cb0229166ceb40e98582ff50b9e90": "-\\frac{\\hbar^2}{2m} \\frac{d^2}{dx^2} \\Psi(x) + V(x) \\Psi(x) = E \\Psi(x)", "c18d0a7944ae4c9e2b29e3d7b5a24c43": "\n(-\\hbar ^2\\Delta )^{\\alpha /2}\\psi (\\mathbf{r},t)=\\frac 1{(2\\pi \\hbar\n)^3}\\int d^3pe^{i\\frac{\\mathbf{pr}}\\hbar }|\\mathbf{p}|^\\alpha \\varphi (\n\\mathbf{p},t),\n", "c18d213323b1dc557ff85bc7e2f05c15": "\\text{Diff} \\to \\text{PDiff} \\to \\text{PL}.", "c18db637047f91eab765f1d39344da57": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left( X \\right)", "c18ea24dbc4e7f6953ebda4cb640d393": "f(\\epsilon)", "c18ecc70941a8ddc392522e1f3ac7ce1": "\\neg a \\vee c,", "c18ecce5d8acace5fdf04008dbfe2955": "I^n", "c18eed0f7ad067223fd1a76a5a24c57a": "|\\psi(x)|^2 = \\frac{1}{\\ell \\sqrt{2\\pi}} \\exp{\\left( -\\frac{x^2}{2\\ell^2}\\right)}", "c18f2781cbdbdc43ac68f38f69c4845a": "i \\frac{\\partial \\psi}{\\partial z} - \\frac{\\beta_2}{2} \\frac{\\partial^2 \\psi}{\\partial t^2 } + \\gamma |\\psi|^2 \\psi = 0", "c18f5b3e9f93ab3c2238e9080236b432": "\\int_0^\\infty \\frac{\\cos x}{x^p}\\ dx= \\frac{\\pi}{2\\Gamma(p)\\cos (p\\pi/2)}, \\quad 0 G_0 \\Rightarrow K, < G_0 \\Rightarrow H.", "c197dbe3ea56aff65a9756878136ee4e": "-|z_A|", "c19915c5097ef26d4d2d878e196c34a1": "\\beta^{'}(\\alpha,\\beta,1,1) = \\beta^{'}(\\alpha,\\beta)\\,", "c1991f87590e9d76f6d46ae229294f0b": " {\\textstyle \\sum} a_kz^k = a(z) \\,(\\boldsymbol B) ", "c1996056f980edebf8f701328a04e2bb": " \\frac{1}{|A| |B|} \\sum_{a \\in A }\\sum_{ b \\in B} d(a,b). ", "c19a4bf52d6eee3657a5fbd4ba1296ed": "\\nu < 0", "c19a76c63d0265eb8951cdd2ec5ed87a": "L_n^{(\\alpha)}(x)=\n{x^{-\\alpha} e^x \\over n!}{d^n \\over dx^n} \\left(e^{-x} x^{n+\\alpha}\\right)", "c19a8efc4e44f63a3fdadf5a31d0eb23": "\n\\mu _i = \\mu _i^\\circ + RT\\ln x_i = 0 \\Rightarrow \\mu _i^\\circ = -\nRT\\ln x_i.\n", "c19aba4d1126866f940fb6fae2c99f77": " \\left( 1 + j \\omega { \\tau}_1) (1 +j \\omega {\\tau}_2 \\right) = 1 + j \\omega \\left(C_2 (R_1+R_2) +C_1 R_1 \\right) +(j \\omega )^2 C_1 C_2 R_1 R_2 ", "c19b6629b9ba31e0378a112ac229d0ad": "\\psi_i(x,p) = (x,\\varphi_{i,x}\\circ p).", "c19b98c9419c0e1d1931f0a41d2ba894": "A = \\sum_{n=1}^\\infty a_n", "c19b9ceae9693710953a8386b539ff7a": "Y_k X_k^{j+\\nu}", "c19c0392f411bc4df61a39c528362650": "= 2\\,y\\, \\left( \\frac{1}{1} + \\frac{1}{3} y^{2} + \\frac{1}{5} y^{4} + \\frac{1}{7} y^{6} + \\frac{1}{9} y^{8} + \\cdots \\right) ", "c19c3847a3181aa99c2e7990d39e4e34": "p=-\\mathrm{e}^t", "c19c4114dfdb7df0a89f2df969002f2e": "\n W(j_1j_2Jj_3;J_{12}J_{23}) \\equiv [(2J_{12}+1)(2J_{23}+1)]^{-\\frac{1}{2}}\n \\langle (j_1, (j_2j_3)J_{23}) J | ((j_1j_2)J_{12},j_3)J \\rangle.\n", "c19c4f50f329b0dc17d650bd2263ec03": "\\begin{align}\nV_{QPSK}(t) &{}= A(t) cos(\\omega_{RF}t + n\\frac{\\pi}{2}); n = 0,1,2,3 \\\\\nV^4_{QPSK}(t) &{}= A^4(t) cos^4(\\omega_{RF}t + n\\frac{\\pi}{2}) \\\\\nV^4_{QPSK}(t) &{}= \\frac{A^4(t)}{8}[3 + 4cos(2\\omega_{RF}t + n{\\pi}) + cos(4\\omega_{RF}t + n2\\pi)]\n\\end{align}", "c19c55f6e9a353afaede2329fb42a0b2": "\\scriptstyle\\circ", "c19c74ec36c7aff63645275b481dd327": "X(s\\otimes s)=X(n\\otimes n)=n", "c19cb9b731a8ab580d5d702eaf8c5280": "H_i = - \\sum f_{a,i} \\times \\log_2 f_{a,i} ", "c19cc6d4c585e922d69244afdfec63af": " H^{'}_{ij}=\\langle \\psi_{i}|H^{'}|\\psi_{j} \\rangle ", "c19cdea4df0718f0b74994660e1275b1": "\\Delta h = \\Delta f g(x_0) + f(x_0) \\Delta g + \\Delta f \\Delta g", "c19d0cde40edb13b33ab5c5972f14694": "\\gamma_1, \\gamma_2 \\colon S^1 \\rightarrow \\mathbb{R}^3", "c19d6236cde6b9b9cd1d727165b3524f": "W^{a}_{\\mu\\nu}", "c19d7d700a3626dcdc5f865f841b17d5": "{(-1)}^{1/2} = i", "c19dbc2405eda64e3aa5d35049bf5e53": "h=\\sqrt{r-r^2}", "c19de110e327d2965c0ec4a79127d089": "Y^2Z^4=X^6+3X^5Z-5X^4Z^2-15X^3Z^3+4X^2Z^4+12XZ^5", "c19de7e6d5869e3c6989ea3606ea7e66": "(\\alpha^2-\\beta^2)u_{j,ij}+\n\\beta^2u_{i,mm}=-F_i.\\,\\!", "c19e04cb5a08a5665ff3495797a937f2": "m \\in \\{ 13, 20, 57, 64 \\}", "c19e399664c999e385ff079a081e2aeb": " L_{{N}}=e^{-\\int_{\\textbf{R}^d}(1-e^{ f(x)})\\Lambda(dx)}, ", "c19eca9986e51deb14edbd94b359ce6a": "A \\xrightarrow[f]{} \\alpha", "c19f06b9f55c583b9ba4c665454ec02d": "\\frac{\\beta }{z}\\,\\,\\, = \\,\\,\\,0", "c19f335eff8e787d1a046a72121800ff": "\\mathrm{EV}_{400} = \\mathrm{EV}_{100} + \\log_2 \\frac {400} {100}\n= \\mathrm{EV}_{100} + 2 \\,.", "c19f6b6a7bae1fd5b14f578c6edc3454": "P(x)", "c19f8e26d5d7cbc3063f8c364e14b90c": "\nK^{i}_{j}= N^{i}_{j}-\\frac{1}{2}\\delta^{i}_{j}N = -8 \\pi \\kappa T^{i}_{j}~~~~~~~~~~~~~~(3)\n", "c19fa2db771d37f6c1723c6aeba4f5cb": " g_0=\\frac{g_{obs}}{1-\\beta_2 g_{obs} \\ln \\Lambda/m}\\,. \\qquad\\qquad\\qquad (2) ", "c19fb3acf511a449e7ec8a805c5bd21f": "\nI_0=\\frac{I \\cos(\\theta)\\, d\\Omega\\, dA}{d\\Omega_0\\, \\cos(\\theta)\\, dA_0}\n=\\frac{I\\, d\\Omega\\, dA}{d\\Omega_0\\, dA_0}\n", "c1a032ca4b1726a178c3f4e60631282e": "\n\\quad G=e^{\\frac{B}{127}}", "c1a03451d8be1e022b5832f7cb275d02": "r:R\\to rcl(R)", "c1a0e7b7609093ae7e478802d573e265": "\\psi(\\mathbf{r},t) = \\begin{pmatrix} \\psi_{+}(\\mathbf{r},t) \\\\ \\psi_{-}(\\mathbf{r},t) \\end{pmatrix}", "c1a15cf88dcb8608fcc6fd5f14e5ed39": "R^{(p)} = A[X_1, \\ldots, X_n] / (f_1, \\ldots, f_m) \\otimes_A A_F,", "c1a197c773391f59eff2eb391c794187": "[1, f(x,y)]", "c1a199b557b0f5123fe80eb525429024": "f(z)=\\sum_{n=0}^\\infty f^{(n)}(0)\\,\\frac{z^n}{n!} = \\sum_{n=0}^\\infty (D^n_q f)(0)\\,\\frac{z^n}{[n]_q!}", "c1a1a197c86849fba9fe006cf2dfc225": "\n{{\\partial \\hat g} \\over {\\partial T}}\\,\\, = \\,\\,\\,{{- 2k} \\over {T^3 }}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{{\\partial ^2 \\hat g} \\over {\\partial T^2 }}\\,\\,\\, = \\,\\,\\, - 2\\,k{{- 3} \\over {T^4 }}\\,\\,\\, = \\,\\,{{6\\,k} \\over {T^4 }}", "c1a1be4a5d960cfe9a7cd90bc7117a09": " \\hat{f_j} \\leftarrow \\hat{f_j} - 1/N \\sum_{i=1}^N \\hat{f_j}(x_{ij})", "c1a2203d04b556a8ce7936584afed08f": "\nW_{\\max} \\le \\inf \\{w : \\sup_{\\tau \\ge w} \\{ E(\\tau-w) - S(\\tau) \\} \\le 0 \\} = \\inf \\{w : (E \\oslash S)(-w) \\le 0 \\}.\n", "c1a229a243c9b251b7b66c37e1d5b962": "\\lambda\\ x.x:\\forall\\alpha.\\alpha\\rightarrow\\alpha", "c1a2500dd41a41810e4250455aa6d126": "D^* = [2+0.2 \\times (bulk\\ density)] \\times X_s", "c1a25b1441dc0711275d5edaf49ca517": "\\pi=\\frac{1}{2^6}\\sum_{n=0}^\\infty \\frac{(-1)^n}{2^{10n}} \\left (-\\frac{2^5}{4n+1}-\\frac{1}{4n+3}+\\frac{2^8}{10n+1}-\\frac{2^6}{10n+3}-\\frac{2^2}{10n+5}-\\frac{2^2}{10n+7}+\\frac{1}{10n+9}\\right )\\!", "c1a2668cff6938f4394c71787110d221": "\\mathbf{I}_{C}", "c1a288fd4a268ec780e16259d0c9a369": "\\,\\!E", "c1a2b6c89c17f9c26acf012f74063496": "2t - 1", "c1a2d0409230458182cd21c440eee0be": "d_1, e_1,e_2 \\in K[x]", "c1a307f363db8d4906f5829920765e60": "|\\langle{\\psi_{D}}|\\psi_{D}\\rangle|^2 = 1 - \\frac{\\epsilon_l}{1-\\epsilon_l} = \\frac{1-2\\epsilon_l}{1-\\epsilon_l}\n", "c1a340867d7e10bf64dff614243a2a37": "\\widehat{\\beta}_{RE}", "c1a36dd8b89c1f30ae515a50d929bf5c": " |T| \\leq d ", "c1a3b792bcb8fa46d612486ee7e0212e": "\\det\\mathbf{T} = \\det\\left(T^a_{\\ b}\\right)", "c1a3c1202dbee818313eae409e44cfb6": "\\sum_{n = 1}^{\\infty}{\\frac {F_n}{10^{(k + 1)(n + 1)}}} = \\frac {1}{10^{2k + 2} - 10^{k + 1} - 1}", "c1a4a2da0cecbce8718ba5886b95a856": " \\frac{f(x_0+h)-f(x_0)}{h}", "c1a4bcae6a3ed828c2e0d84f6e54e8db": "f(z)=(z-a)g(z)\\,", "c1a511e522e9f316291b8ff0e2d4b523": "\nP = {\\frac 1 2}\\,q\\,I\\,\\Delta f R.\n", "c1a5d1a706c878121d83e6e3eb0ac546": "L_iL_ju(x)-L_jL_iu(x)=\\sum_k c_{ij}^k(x)L_ku(x)", "c1a6581adc126de10c9b343532ad7d43": "\\vec Q_h", "c1a68ef2948a17acfbe681f823026064": "t_1=x_1/c", "c1a6d68502bd2d48bc828975d0edb734": "d\\mathbf{A}=\\mathbf{v} \\, dt \\times d\\boldsymbol{\\ell}", "c1a71406ec66583e6c3347b8fbb68cfa": "\\sum_{m=0}^n P(3m)=P(3n+2)", "c1a729e3cb1fb2c383f2e0df0362fdf6": "10^{720}", "c1a73f83d384762ea6ad169b15e6b57e": "\\rho_\\mathrm{in}\\,", "c1a757bac95b3e2b891e7319a5fc21e6": "\\cfrac{\\cfrac{stC \\qquad \\overline{s}tD}{tCD} \\, \\operatorname{var}(s) \\qquad s \\overline{t} E}{sCDE} \\, \\operatorname{var}(t) \\Rightarrow\n\\cfrac{stC \\qquad s \\overline{t} E}{sCE}\\, \\operatorname{var}(t)", "c1a77fdb913d5252b0d837cc0a2e0e92": "A(x)>x^{1/2-o(1)}", "c1a7cca33a030188e7059d40cfcb0781": "\\int d^dx\\, \\overline{\\psi}iD\\!\\!\\!\\!/ \\psi", "c1a84c4334a26b883c1a8ec979a5b43b": "\ni\\hbar \\frac{\\partial \\psi (\\mathbf{r},t)}{\\partial t}=\\widehat{H}_\\alpha\n\\psi (\\mathbf{r},t)", "c1a85862e0cf2152674cbf94bb8f07a1": " \\sum_{j=0}^n (-1)^j\\tbinom n j = 0", "c1a8642e502fcc8bd54e84a1667c4bc9": "f : V \\rightarrow [0, 1]", "c1a86ff1aa8da0e30673eb9d760193ac": "\\preceq\\ ", "c1a872e7ddc16df11b704fb9cfa297e8": "ds^2=(1-2GM_0/rc^2) c^2 dt^2 - (1-2GM_0/rc^2)^{-1} dr^2 - r^2(d\\theta^2 + \\sin^2 \\theta d\\phi^2) \\;", "c1a8b29afc39c6e225d2e2b3dbbd03ae": "Z_{I^n}", "c1a904254393e45111a7cff55b704af0": "\\mathbf{A} = \\mathbf{U} \\boldsymbol{\\Sigma} \\mathbf{V}^*", "c1a958b361e6be32c7b6c18167fe2935": " \\Phi(t - \\tau) ", "c1a9aeb0983210fa898451a9bb36130d": "M_0\\times10^{25}", "c1a9f73d94264b5c3c8742e68ef41cc7": "\\mathcal{O}_k:", "c1aa4fa74b49bb0109495bdacae8ffd5": "\\frac{d M/M}{d t}", "c1aac8df3ee2dbf07177f3557a4dc361": " (f \\, * \\, g)(n) = \\sum_{d|n} f(d) \\, g \\left( \\frac{n}{d} \\right)", "c1aaeb4bcf6735c01607594645cb2bc6": "Y = bY - btY + tY + I ", "c1ab1adf6fb9ad3cecd695973f42c6d2": " c = \\frac\n {d \\ln \\left(\\displaystyle \\frac{df}{dx_1}/ \\displaystyle\\frac{df}{dx_2} \\right)}\n {d \\ln (x_2/x_1) }\n = \\frac{\\displaystyle \\frac{d (\\frac{df}{dx_1}/\\frac{df}{dx_2})}{\\frac{df}{dx_1}/\\frac{df}{dx_2}}}\n {\\displaystyle \\frac{d (x_2/x_1) }{x_2/x_1}}. \n", "c1ab28ec8ddc943a7bed965d9707d503": " A \\subseteq A^* ", "c1abbe0973e843154178ad79250262e5": "\\sigma_i(A)=|\\lambda_i(A)|", "c1abdd34c0aab89b66191ee14a55d80e": "\\left(\\frac{\\partial S}{\\partial P}\\right)_{T}= -\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}= -V\\alpha\\,", "c1ac2ba6064030dccc5d28710cdf5477": " E \\left[ \\exp \\left( -\\int_0^1 w'(t)H(t)w(t) dt \\right) \\right] = \\exp \\left[ \\tfrac{1}{2} \\int_0^1 tr (G(t)) dt \\right], ", "c1ac6b60e54cff9325e9b14ecf329285": "X^*_{\\gamma(X^*, X)}", "c1acbda89618b6c20c01a1000ffd9e9a": "R_I", "c1ace90600c89c579a958bc210842a45": " \\mbox{Weekly } Kt/V = \\frac{7 K_D [l/\\mbox{day}]}{V [l]}. \\qquad(7b)", "c1acfebba9e0059f4aeec3d2a28b7f5c": " \\begin{cases} \\psi : E \\to TE \\\\ (x,u) \\mapsto \\psi(x,u) = V \\end{cases}", "c1ad398a7599b08131ba81d7faad4433": " \\frac{\\partial f_j}{\\partial x^i} - \\frac{\\partial f_i}{\\partial x^j}=0.", "c1ad8480587a14524c64bb8a71e4aca9": "f(x) - \\sum_{n=0}^{N-1} a_n \\varphi_{n}(x) = O(\\varphi_{N}(x)) \\ (x \\rightarrow L)", "c1ada1968b389bd65ca32bb2887ec32b": "R\\subseteq \\mathbb{C}", "c1ae37e98e0d9e4aeb2cae36bb4f75fe": "\\Phi_1(s) = \\frac{\\sqrt{2 \\alpha_1}} {(s+\\alpha_1)}", "c1ae6eedb00da93036f9f66c99ecfebf": " \\tilde{\\beta}(M) = (-1)^{r(M)+1}\\beta(M), ", "c1b092c2f7363a67e51a080ecf7ddd21": "\\Gamma=\\frac{V_pC_e}{A_pe^{\\frac{W_aA_c}{RT}+V_pC_e}}", "c1b09b91f0ce30bd40c95bdea9e4e2ac": " K_p = \\frac{1}{K_a}", "c1b0b8fa4a7debe8fd85f28d9190cabd": " \\eta_{B} = \\frac{\\dot{W}_{net}}{\\dot{m}_{fuel} \\Delta G^{0}_{T}} \\qquad \\mbox{(3)} ", "c1b0e19adf92b282c78b5e6172d18154": "i \\geq 0", "c1b0e34e3e231a9ab61bfd953d8cc526": "r\\sigma^2", "c1b0e947b9989a3d65f0cd7d0022dc4a": "7 * 2 = 14", "c1b126d36a952f54d3d2804da2aead0f": "T^{ab}", "c1b141afdc8f38d3660d9fcdf5cb193c": "e^{\\alpha}_{\\ I}", "c1b1ae28598ba510b15b187a3cdf2680": "\\{a_{k}\\}_{k=0}^{k=\\infty}\\,", "c1b1e3155ac140329f5d42ae6599f3ff": " \\dot{x_n} = \\frac{dx_n}{dt} = f_n(x_1, \\ldots, x_n) ", "c1b20aaa63924d3fa35133084d7f262a": "n! [z^n] g(z, -1, v) = \nn! [z^n] \\exp(-z + vz) (1 + z) = \n\\sum_{\\pi\\in S_n} \\sigma(\\pi) v^{\\nu(\\pi)}", "c1b24b2699340b7c3c445a88f25fd7e6": "\\hat{D}(\\alpha)\\hat{D}^\\dagger(\\alpha)=\\hat{D}^\\dagger(\\alpha)\\hat{D}(\\alpha)=I", "c1b2d54536afb02ad6748323f47bcced": "\\begin{bmatrix} -i & 2 + i \\\\ -(2 - i) & 0 \\end{bmatrix}", "c1b344d2c1d4b0ccea1531b40e3befae": "E_{y}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{k_{o}^{2}-k_{z}^{2}}[-\\frac{k_{z}}{\\omega \\varepsilon _{o}\\varepsilon_{r} }\\frac{m\\pi }{a}(A \\ e^{jk_{x\\varepsilon }w}+B \\ e^{-jk_{x\\varepsilon }w})-jk_{xo}(C \\ e^{jk_{x\\varepsilon }w}+D \\ e^{-jk_{x\\varepsilon }w})]e^{jk_{xo}(x+w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ (46) ", "c1b3632adca5b8f46567d591bc828315": "\\mathbf{N} = m \\left( \\mathbf{x} - t \\mathbf{u} \\right) = m \\mathbf{x} - t \\mathbf{p} ", "c1b3c180df01f215b3aa20f023e21f0a": "{}^3D_4(2): 3", "c1b4bb441d021d5ee8e8a3844adfb8d5": "\\sin \\theta = {O\\over H}\\approx{O\\over A} = \\tan \\theta = {O\\over A} \\approx {s\\over A} = {{A*\\theta}\\over A} = \\theta", "c1b55fb00b16b3b852d549949c360bb6": "P_1(c) = c \\,", "c1b590eb045c773646bf650f181d530b": "\\langle R^2 \\rangle", "c1b5d0a7b7f34c123c584a49c9831b94": "(N + 2)p - 1/2\\,", "c1b61b8667c10752a797abcffaaf45de": "2b^2 = 4k^2", "c1b6ed1fb6a7d741a5fc239605e80666": "h(a)=a", "c1b6f526eb01f5261dabfbb38118bd22": " \\frac{\\mathrm{d}}{\\mathrm{d}t}\\int\\ddot{r_i} {\\partial r_i \\over \\partial q_j} \\mathrm{d}t = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left({\\partial r_i \\over \\partial q_j}\\dot{r}_i\\right)-\\frac{\\mathrm{d}}{\\mathrm{d}t}\\int\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left({\\partial r_i \\over \\partial q_j}\\right)\\dot{r}_i\\mathrm{d}t= \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\dot{r}_i{\\partial r_i \\over \\partial q_j}\\right)-\\dot{r}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left({\\partial r_i \\over \\partial q_j}\\right)", "c1b6fe6b80950add071b5f322dbff2c5": "\\int_0^\\infty e^{-t} \\mathcal{B}A(tz) \\, dt. ", "c1b7b270894e8dcd96fd927baca17c6c": "\\frac{z-z_0}{\\overline{z_0}z-1}, \\qquad |z_0| < 1,", "c1b8474ba21ed444b4ce8e283dc5a274": "\n\\begin{align}\n& {} \\qquad (513 \\cdot 537)^2 \\mod 84923 = (275481)^2 \\mod 84923 \\\\\n& = (84923 \\cdot 3 + 20712)^2 \\mod 84923 \\\\\n& =(84923 \\cdot 3)^2 + 2\\cdot(84923\\cdot 3 \\cdot 20712) + 20712^2 \\mod 84923 \\\\\n& = 0 + 0 + 20712^2 \\mod 84923\n\\end{align}\n", "c1b84a0944f704d66d4c9c6420d7e05c": "\\omega = \\begin{bmatrix} \\phi \\\\ \\chi \\end{bmatrix} = \\begin{bmatrix} \\phi \\\\ \\frac{\\vec{\\sigma}\\vec{p}}{E + m} \\phi \\end{bmatrix} \\,", "c1b88d4662457058b7fd557001816f74": "\\Lambda_{Pillai} = \\sum _{i=1...p}(\\lambda_{i}/(1 + \\lambda_{i}))", "c1b8a108f17ad805f6109178314b58e3": "F = \\frac{m c^2}{\\frac{\\hbar}{m c}} = \\frac{m^2 c^3}{\\hbar}.", "c1b8b8564fc37e7a204e713c2c89dffa": "\\lim_{n \\to \\infty}a_n \\ne 0", "c1b8e87e70b0c21667b219f613c345d2": "\\sum_{j=1}^J \\mu_j = 1", "c1b90495bd492eaa8103d35e27e70a99": " E_Q = \\frac{\\partial Q}{\\partial \\textbf{x}} \\cdot \\frac{\\textbf{x}}{Q} ", "c1b925f34a8a551f2bb5fdf484c8b0bb": "(7=0\\times 2^3+1\\times 2^2+1\\times 2^1+1\\times 2^0 = 4+2+1)", "c1b9989daeed46e2288df5ff055d3e32": "y_{(1)}(t)", "c1b998b3a3791fa9cac147eddd90ab3d": "\\operatorname{perm} (A) = \\sum_{\\sigma \\in \\operatorname{P}(n,m)} a_{1 \\sigma(1)} a_{2 \\sigma(2)} \\ldots a_{m \\sigma(m)}", "c1b9d3ba1413f2f989d28793ac444ed8": "r_n = \\sqrt{(n+\\alpha) \\lambda f + \\frac{(n+\\alpha)^2\\lambda^2}{4}}", "c1ba1523c96bb1aaa86a0023b2f7ac2a": "u(t,x) \\approx a(t,x)e^{i(k\\cdot x - \\omega t)}", "c1ba2953290677cf069a8e63bfec3772": "\\eta, b>0,", "c1ba3933fcc81f6c7287f0fd677a936b": "E = h\\nu = E_i-E_f=\\frac{m_e q_e^2 q_Z^2}{8 h^2 \\epsilon_{0}^2} \\left( \\frac{1}{n_{f}^2} - \\frac{1}{n_{i}^2} \\right) \\,", "c1ba94f362205e06f7f136c0615d7adc": "\\sum_{k=0}^{N-1} \\cos_k(i)\\equiv. ", "c1bac467832ab861299a284ade8e42cf": "\\hat{P^T}B\\hat P=\\hat{D_B}", "c1bb35e315d64a88dcaf9768f1ac22cd": "c_k \\,", "c1bb6cfbea00fb05ba2d14e53a024589": "\\sum_k \\mid \\langle \\mathbf{e}_k|(0,1)\\rangle|^2 = 0 + 1 + \\frac{1}{2} + \\frac{1}{3} +\\cdots = \\infty", "c1bb7f7b0de81bc6a52da33b30227039": "\\color{YellowOrange}\\text{YellowOrange}", "c1bb9785ded9ad8adcdc9c803851f3da": "\\frac{1}{\\alpha }+\\frac{1}{\\beta }=1", "c1bb984e1f487db35dd296c5ad9ea02e": "\\begin{align}\n\\Gamma(t) &= \\lim_{n \\to \\infty} \\frac{n! \\; n^t}{t \\; (t+1)\\cdots(t+n)}\n= \\frac{1}{t} \\prod_{n=1}^\\infty \\frac{\\left(1+\\frac{1}{n}\\right)^t}{1+\\frac{t}{n}} \\\\\n\\Gamma(t) &= \\frac{e^{-\\gamma t}}{t} \\prod_{n=1}^\\infty \\left(1 + \\frac{t}{n}\\right)^{-1} e^{\\frac{t}{n}}\n\\end{align}", "c1bbe7d0fc5cbaef06a4be46be67e11c": "\\forall x\\in W\\,(x\\Vdash B)", "c1bc00097fbed8db6eaef106da324fb1": " \\Delta w := \\frac{\\partial^2 w}{\\partial x_\\alpha \\partial x_\\alpha} = \\frac{\\partial^2w}{\\partial x_1^2} + \\frac{\\partial^2w}{\\partial x_2^2} ", "c1bc515d687b582d0c16ffaa1a6f073e": "S(f),", "c1bc627ce99d3cee396ffc21ac093ae8": "OSIN3_i=max(INDEXB_i,0)", "c1bc65f26234230562fd8ec7b787c2c6": "\\omega=p/q", "c1bc9410041b6f493985a8271f6127e9": "\nrr_{xy, w} = \\frac{\\sum w_i x_i y_i}{\\sqrt{(\\sum w_i x_i^2)(\\sum w_i y_i^2)}}.\n", "c1bcbaee9e0b243aa401afddc21f676e": "M_1;M_2 \\in \\mathbb{Z}_N", "c1bdab5bef2930f16933366521aa4b17": "\\Gamma_p(s)=(-1)^s\\prod_{0b", "c1c38d24c1ab4656e8599fb11f07779c": "GPA = {(1.8\\times OBP)+SLG\\over 4}", "c1c3ab93d299540a13ed04a2434ed9b6": " \\ \\alpha_j", "c1c40a4556cebc64d4c6d63635c2a839": "\\mathrm{III}_T(t) = \\frac{1}{T}\\sum_{n=-\\infty}^{\\infty} e^{i 2 \\pi n t/T}.", "c1c446efe733d1dfded81b738fd97342": "\\sum_{p\\text{ prime }}\\frac1p = \\frac12 + \\frac13 + \\frac15 + \\frac17 + \\frac1{11} + \\frac1{13} + \\frac1{17} +\\cdots = \\infty.", "c1c468a3da2846dffbf9524a4911c4c9": "r_3 = s(\\alpha + \\beta)", "c1c471cd7d4e32ee6f44fa676ea1dbca": " F(k;n,p) \\leq \\exp\\left(-2 \\frac{(np-k)^2}{n}\\right), \\!", "c1c48728d7b8943d37ea347c93eba53e": "\\textstyle \\ge n - 1", "c1c4ded35c9fa62827fd2a3137f43415": "(5 + 5i)", "c1c50431ef96388bd1a6e6c33cb1723b": "\\mathbf{V} = E[(\\mathbf{X} - \\mathbf{M})^{T} (\\mathbf{X} - \\mathbf{M})] / c", "c1c50905ae994bf0eb2785799a06fb49": "\\mu = 2", "c1c50d43f71de02d14581d1416247f07": "x^n / n!", "c1c5608ff622c019c98e32d31ddb8aad": "Cl_2 \\xrightarrow{h\\nu} 2Cl\\cdot ", "c1c5c6c451770439d825e814d1f0eaa4": " n(n+1)~r^{-n-2}~\\sin(n\\theta) \\,", "c1c60cce7c59b2564dbb81991fb552d8": "(x_1,x_2,\\ldots,x_n)", "c1c64a04eb431e13df468380f98ff179": "\n{E_1}^2+{E_2}^2+{E_3}^2 > 0\n", "c1c66c9f93e75faca64eeafc4f1e59f6": "\\chi (\\omega ) = - \\sum\\limits_k {\\frac{{\\left| {d_{cv} } \\right|^{_2 } }}{{L^3 }}} (f_{v,k} - f_{c,k})\\left( {\\frac{1}{{\\hbar (\\varepsilon _{v,k} - \\varepsilon _{c,k}+ \\omega + i\\gamma )}} - \\frac{1}{{\\hbar (\\varepsilon _{c,k} - \\varepsilon _{v,k} + \\omega + i\\gamma )}}} \\right)", "c1c68b27396bbd67d8fa0d3841857803": "V(\\frac{N\\omega}{2\\pi})", "c1c6d7b5cb1d9fcd3a3373a873d08633": "\n\\begin{align}\ng(x) & {} = {\\rm lcm}(m_1(x),m_3(x),m_5(x),m_7(x)) \\\\\n& {} = (x^4+x+1)(x^4+x^3+x^2+x+1)(x^2+x+1)(x^4+x^3+1) \\\\\n& {} = x^{14}+x^{13}+x^{12}+\\cdots+x^2+x+1.\n\\end{align}\n", "c1c771fa584c43d459d693d02e499e33": "\\hat \\beta _0", "c1c7a924797f2b43d4219498f071d41c": "\\mathsf C=\\{c_1,c_2,c_3,c_4\\}", "c1c81853fd554000b5ca3bb761819186": "8\\pi-\\tfrac{4}{3}\\pi^2\\approx 11.973", "c1c895275358505015465cbcf0122c79": "\\text{Find }\\min_{\\alpha,\\,\\beta}Q(\\alpha,\\beta),\\text{ where } Q(\\alpha,\\beta) = \\sum_{i=1}^n\\hat{\\varepsilon}_i^{\\,2} = \\sum_{i=1}^n (y_i - \\alpha - \\beta x_i)^2\\ ", "c1c89756c660c178c50b7773be9e43e3": "\\partial_t L(x, y, t) \\geq 0", "c1c8c2addaf8ef4ccf8451cc8cf4b0fd": "Tx\\leq h", "c1c8ce99dd14d0b977752a9d4fb590c8": "v_1,v_2,v_3,v_4", "c1c8ef7c3fac1d3a6f8b6cbb5fc8d436": " \\mathfrak{su}(2) = \\operatorname{span} \\{ i \\sigma_1, i \\sigma_2 , i \\sigma_3 \\}.", "c1c981f37a6b4d0892c86a13e43aeef2": "\\bar{\\epsilon}=\\cfrac{1'''}{2}\\sqrt{2}[({\\epsilon}_{11}-{\\epsilon}_{22})^2+({\\epsilon}_{22}-{\\epsilon}_{33})^2+({\\epsilon}_{33}-{\\epsilon}_{11})^2]^{1/2}", "c1c9a246b654a021b03e2ece27371c70": "T^* Q", "c1c9e8577069c1c51fc987ade3602fca": "\n{r_{\\rm w}} > {r_{\\rm c}}\n", "c1ca33126431f6ca27265cd8532894a1": "v = V - A\\omega\\, ,", "c1ca4977c807e9fd3e6183d0e162cb07": "{}= \\frac{b-a}{2N}(f(x_1) + 2f(x_2) + 2f(x_3) + \\ldots + 2f(x_N) + f(x_{N+1})).", "c1cae700d54e5ca497e37e5ae7abdcfd": "= 12kn^2", "c1cb3b8c99b98ca0e40b0ca5d6713684": "h(\\mathbf{x},t)", "c1cb5a8cc411b9843ccfaebadae8fc3c": "x^2+6y^2=103", "c1cb80982c4321723a7a81d2dab86e81": "\nS=\\int{({\\mathcal L}_G+{\\mathcal L}_\\phi+{\\mathcal L}_S+{\\mathcal L}_M)}~d^4x,\n", "c1cbbdc2d6a49c40251f92179a1f1ed7": " \\frac{z^{-1} (1 + 4 z^{-1} + z^{-2} )}{(1-z^{-1})^4} ", "c1cc085ff94696a4d3f6cfaa1d2db667": "y=\\frac{1}{\\sqrt{x}}", "c1cc614b732911a2ad912a404bbd8ca9": "{\\dfrac {a-b}{c}}={\\dfrac {\\sin \\left( \\alpha/2-\\beta/2 \\right)}{\\cos \\left( \\gamma/2 \\right) }}", "c1cc7b247320886d871e82eb5ca1a20a": "S_p(n)=(-1)^{p+1}\\sum_{j=0}^p\\binom{p}{j}\\frac{B_{p-j}}{j+1}(-n)^{j+1}", "c1ccb3428b7620a2e7bd7c3bf7a56d0a": "2 d \\sin{\\theta_B} = m \\lambda", "c1cd1c2e5c56a85540a572600f9bc9c7": "H = \\int_{0}^1 - f(X;\\hat{\\alpha},\\hat{\\beta}) \\ln (f(X;\\alpha,\\beta)) \\, {\\rm d}X ", "c1cd3feffeb14eb5bcf3553dcea8f068": "\\lim_{n\\to\\infty}a_n = -\\infty", "c1cd43b5ab7f0dcbb897828b7091dda5": " ms^{-1} ", "c1cd6774ed5bee9ce8a62c061efbd068": "\\boldsymbol{U}_f", "c1cd992e819a97ca767d5ba1ba9faa3d": "L(s,a)=\\prod_p\\biggl(1+\\frac{a(p)}{p^s}+\\frac{a(p^2)}{p^{2s}}\\cdots\\biggr)", "c1cd9f16d9114b30f03c645b081a0f3c": "C_\\text{V}(x) = \\frac{F_\\text{B}(x) - F(x)}{F_\\text{B}(x)}", "c1cdaae512c9dfcbc2621f062ff68fda": " c = 2b", "c1ce007fb47f726333bd71c3bcf88cdb": " i \\xi ", "c1ce33d7a60e20fbfeb3e95e6540001c": " \\cos t = \\begin{matrix}\\frac12\\end{matrix} e^{i t} + \\begin{matrix}\\frac12\\end{matrix} e^{-i t} \\,", "c1ce56aea5b19ff315e7aa79c58835e8": " A = { I_0 \\over 2 } e^{j \\phi } ", "c1ce578aa8a54b244effd16d6687e122": "SB%=\\frac{SB}{SB + CS}", "c1ced53f6598bd024c757a5592c8847f": "\\rightarrow \\bigoplus_{j\\in I_i}K_{n-1}(RH_j)\\oplus\\bigoplus_{j\\in I_i}K_{n-2}(RH_j)\\rightarrow \\bigoplus_{j\\in I_i} K_{n-1}(RH_j)\\oplus KR^G_{n-1}(X^{i-1}) ", "c1cef9bc999a133a4568d1584c10d5b5": "\\scriptstyle \\lnot{(1 \\,\\oplus\\, 0)} \\;=\\; 0", "c1d0234fc84c7c051750a42cda67db8e": "S_k \\equiv 0 \\pmod{N}", "c1d05542ff117618c8fdf8eb161497a3": "SEncode(l,d,w,y,y^{\\prime},\\tilde{k}) \\stackrel{\\mathrm{def}}{=}d||pad_{21}(I_{Z}^{B^{\\ast}}(w))||pad_l(I_{Z}^{B^{\\ast}}(y))||pad_l(I_{Z}^{B^{\\ast}}(y^{\\prime}))||\\tilde{k} \\in B^{\\ast}", "c1d0c27d11107b7343e82e1a38c365ca": "\\int dz h(z) \\int dy \\langle x, (O-\\lambda I)y \\rangle G(y, z; \\lambda)=\\int dy \\langle x, (O-\\lambda I) y \\rangle \\int dz h(z)G(y, z; \\lambda) = h(x), ", "c1d10b2fe2a8083aba6130b89af0e2a1": "\n\\mathbf{a \\times b} =\n \\begin{vmatrix} \n \\mathbf{e_1} & \\mathbf{e_2} & \\mathbf{e_3} \\\\\n a^1 & a^2 & a^3 \\\\\n b^1 & b^2 & b^3 \\\\\n \\end{vmatrix}\n= \\sum_{i=1}^3 \\sum_{j=1}^3 \\sum_{k=1}^3 \\varepsilon_{ijk} \\mathbf{e}_i a^j b^k\n", "c1d12a495a0872db917796a005c0208b": "\\frac{1}{2} (A - A^*)", "c1d1350ec8a0e20aa97d1df92e7e6de6": "\\mathfrak{sl}_3(\\mathbf R)\\oplus\\mathfrak{sl}_3(\\mathbf R)", "c1d1672b879906e7fd5f8a16cc35f644": "(x-7) (x+1)^7 (x^2-7)^8", "c1d1accab7c4051ff75dd48b7ef3499d": "T_{l} = u^{l+1}\\left(a_{0}+\\frac{a_{0}}{2(l+1)}u\\right).", "c1d244df8347c38eecbc41bdfbb3ef92": "L=H\\cap K", "c1d2d2941eadb11ecfd25c626577de55": "h(x) = f(x) - f(a)", "c1d30bec2cd280ca5c3a3822316deda7": "r=\\frac{\\tbinom{n-1}{t-1}}{\\tbinom{k-1}{t-1}}", "c1d313e5f0fc3803cc1464562666feab": "V_Q", "c1d3437afb2a3bfba08d66fe77a54898": "1 = \\pi_{xy}\\left(x, y, z\\right) = \\left(x, y, 0\\right)", "c1d388a3d46d8bc38fafe901fac8db8f": "-2x\\,", "c1d395f58ff22a897294ad8107ca97ff": "G(n,a,b) = H_n(a,b)\\,\\!", "c1d3f3889e7b467eaa2e3714ff925e2a": "\\tbinom n q", "c1d41d54cc30cc662d9d445aa66e7956": "\n\\begin{align}\n\\tan^{-1}(x)&=\\int\\frac{dx}{1+x^2}=\\int\\frac{dx}{1-(-x^2)}=\\int\\left(1 + \\left(-x^2\\right) + \\left(-x^2\\right)^2 + \\left(-x^2\\right)^3 + \\cdots\\right)dx\\\\\n&=\\int\\left(1-x^2+x^4-x^6+\\cdots\\right)dx=x-\\frac{x^3}{3}+\\frac{x^5}{5}-\\frac{x^7}{7}+\\cdots=\\sum^{\\infin}_{n=0} \\frac{(-1)^n}{2n+1} x^{2n+1}\n\\end{align}", "c1d4592f6b4afae8237c4e4b3fd70ddc": "\\mu^{\\text{s}}", "c1d4838683af45350a49cdbaf64f6879": "\\pi_x(\\pi_y(R))=\\pi_x(R)\\,", "c1d4e20f7df686ce21df20b35b99e2bc": "\\begin{matrix} {10 \\choose 1}{4 \\choose 4}{44 \\choose 1} \\end{matrix}", "c1d4eab4f855718ee3cdb83bbc7bb7a5": "\\partial p^{i}_a / \\partial x^{i} = -\\partial H / \\partial y^{a}", "c1d513e63a5f4dac6c5682282d3a42fd": "z_\\mu", "c1d54648da0ad345e9dbd910b6c5fdbb": "E_{1j}", "c1d61caceef6ebc0b6d86792ab420af5": "\\textstyle \\cos(a x) = ", "c1d679998e89c739748925f4c5236647": "\\ z", "c1d6ca2773feb3fb38bba4e44eeb625f": "\nP_l = \\frac{R_s}{2}\\iint_S |\\vec{H}|^2 dS.\n", "c1d6db968dca38cc422c0795edcbeb66": "\\mathbf{V}_i=\\mathbf{V}+\\boldsymbol\\omega\\times\\mathbf{r}_i.", "c1d710376f9e913172f6022694456a09": "s(i) = \\begin{cases}\n 1-a(i)/b(i), & \\mbox{if } a(i) < b(i) \\\\\n 0, & \\mbox{if } a(i) = b(i) \\\\\n b(i)/a(i)-1, & \\mbox{if } a(i) > b(i) \\\\\n\\end{cases}\n", "c1d7a0e9a35be3848f0904230042d249": " x^3 + y^3 + z^3= Dxyz ", "c1d7bf578642153fc82efdae30372975": "\\binom{n}{l}", "c1d7c788e8a6c5d7dcee6167a403a99e": "(X_{2}=\\cos\\Omega t,Y_{2}=-\\sin\\Omega t,Z_{2}=0)^{T}", "c1d7d3bbe9d87296f27d5521a782a303": "\\frac{d^2\\alpha}{dt^2}-\\left(\\frac{Z_\\alpha}{mU}+\\frac{M_q}{B}+(1+\\frac{Z_q}{mU})\\frac{M_\\dot\\alpha}{B}\\right)\\frac{d\\alpha}{dt}+\\left(\\frac{Z_\\alpha}{mU}\\frac{M_q}{B}-\\frac{M_\\alpha}{B}(1+\\frac{Z_q}{mU})\\right)\\alpha=0 ", "c1d85bb4c3bbce1e25a0c56237de8543": "\\Diamond \\Diamond p \\rightarrow \\Diamond p", "c1d8a6e06903adcf6d6d9bad42315e72": "F(p)", "c1d8a91b86d3324ff44c17c63036bfdf": " a = mq+R^{-1} \\cdot r ", "c1d8c06e5e0b06ba51d98d087f5f8d54": "\\mathbf{\\delta}: [0,T] \\times \\mathbb{R}^N \\rightarrow \\mathbb{R} \\in L_2[0,T] ", "c1d8cc3f04ba295a9a9980fe7c15b8dd": "\\displaystyle{F^\\vee(x,y)={1\\over \\sqrt{2\\pi}} \\int_{-\\infty}^\\infty f(t,y) e^{-itx}\\, dt}", "c1d982b887e24335c22492ad86ad1d47": " A_w = \\{ \\langle max(t_{ij} | i = 1,2,...,m)| j \\in J_- \\rangle, \\langle min(t_{ij} | i = 1,2,...,m)| j \\in J_+ \\rangle \\rbrace \\equiv \\{ t_{wj} | j= 1,2,...,n \\rbrace, ", "c1d985e5e73f8d027146bb1be68425cc": "f(x_k+\\Delta x) \\approx f(x_k)+\\nabla f(x_k)^T \\Delta x+\\frac{1}{2} \\Delta x^T {B} \\, \\Delta x, ", "c1d98c96d1c2f14af338f5423a132408": "\\| . \\|_{p}", "c1d9ba522afb2678c308e73c2f2e87df": "\\mu_{eff}", "c1d9f50f86825a1a2302ec2449c17196": "H", "c1da1e2aae668aa11235c5d52a4d5d96": "y_i = a + bx_i + e_i\\ ", "c1da38db5cd2065be3b2ac04501f94ea": "r=rk(V)", "c1da92122f7e7329dda4d14db8849dc1": "\\lim_{\\varepsilon\\to 0^+}\\int_{-\\varepsilon}^\\infty.", "c1db0597a1bd2c3e3310bb91a810a534": "\\Pi_U(e^{iX}) = e^{i\\pi(X)}, \\quad X \\in \\mathfrak{so}(3;1).", "c1db3d458345c68bdc5ea692ee2daad9": "\\tan \\chi = \\frac{\\delta}{\\kappa E_0}", "c1db4bd66efb0ee46347f566758add4a": "{\\tilde{D}}_{8}", "c1db95dc525f999a3df791f3c7636c56": "\\mathbb{Z}_p", "c1dbc3972eb8c78aa10662ec26e1fe7b": "M = \\frac{\\pi abc\\rho}{6}\\,\\!", "c1dbe47f0918d709a917a2c1a3d633fe": "a^2 + b^2 + c^2 = g^2.\\,", "c1dc0c569cf7361951e255ce254986cf": "\\mathrm{E} [X] = \\{ \\mathrm{E} [V] | V \\mbox{ is a selection of } X \\mbox{ and } \\mathrm{E} \\| V \\| < + \\infty \\}.", "c1dc1d9924b6470b6833c9c83fb5cd5c": "\\partial n/\\partial \\mu", "c1dc44b43f442ea6c2b70b4cdfa55f89": "\\psi,\\tilde\\psi", "c1dc707a8274fc239663e0298ac6efb3": "i = 1,...,(n-1): d[C(i),C(i+1)] > d[C(i+1),C(i)]", "c1dc8ab336b02dcde2bfee3d51a054d2": "\\mathcal{N}.", "c1dc8e05697f6a2022cefaa57f9f7472": "T_{old} - T_{new} = D_{a} + D_{b} - 2c_{a,b}", "c1dc909eefa5f2188b5f1b37a2aaa3be": "\\mathbf{E}^{x} \\big[ f(X_{\\tau+h}) \\big| \\Sigma_{\\tau} \\big] = \\mathbf{E}^{X_{\\tau}} \\big[ f(X_{h}) \\big].", "c1dc9e963105c25fe4ad5cf92c2da743": "H_\\mathrm{e} = E_\\mathrm{e} \\cdot t", "c1dca98322c4e6fb7b05ba9a647278ae": "d_1(x,y)<\\delta \\Rightarrow d_2(f(x),f(y))< \\varepsilon.", "c1dce1e39944a91a4e6d8c75f3964a22": "7(2^4/7!!) \\pi^3 = (16/15)\\pi^3 ", "c1dcfe1b0430df6f1dcb0ad40ecf1690": "R_{\\text{INIC}} > R_s", "c1dd0c363b1f894b7c8e32acc74336db": "N_{\\rm A} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e})}{m_{\\rm e}} = \\frac{M_{\\rm u} A_{\\rm r}({\\rm e}) c_0 \\alpha^2}{2 R_{\\infty} h}", "c1dd2bdb6a57d03c761a6d057cdca26a": " d = p_1^{n_1} ", "c1dd3aba900aa78c433638806c86b147": "[(n:=n+1)^0] \\Phi(n) \\land [(n:=n+1)^1] \\Phi(n) \\land [(n:=n+1)^2] \\Phi(n) \\land \\ldots\\,\\!", "c1dd870a39d6b19492799e0cfb37b918": " \\tbinom nr ", "c1ddbd1aa61c27685e22c57bbedf4a41": "\\mbox{cash-on-cash return} = \\frac{\\mbox{annual before-tax cash flow}}{\\mbox{total cash invested}}", "c1ddc229230fda8866f29d2841a9f881": "\\displaystyle{\\mathfrak{p}= \\bigcup_{k\\in K} \\mathrm{Ad}\\, k \\cdot \\mathfrak{a}.}", "c1ddffc100c3b4afecff6b736e79ba25": " b(s) = 2e^{\\frac{1}{2}s i \\pi}\\int_{0}^{\\infty} \\frac{st^{s}}{1-e^{2\\pi t}} \\frac{dt}{t} ", "c1de0b4f05802dcc36a8933fd17bf2ac": "\\frac{d}{dx}\\left( \\ln x \\right) = \\frac{1}{x}.", "c1df246fd3650ce0ebddde44d616b77b": "\\sqrt[3]{2}\\approx 1.2599", "c1df4176056bb798f5a82a0af8540aaa": "\\mu(k)", "c1df752b24902ee100d97ada8c24c209": "\\scriptstyle 1.6 f_s", "c1e02f370124e29b8ddd941f714097f3": "\\phi \\to (\\neg \\phi \\to \\chi)", "c1e07c10c00215b53adbe9042721ef6b": "x_e=0", "c1e0831d622544b422b5cb8d8a3c8f1a": "\nX_i = w_{ ATM}\\, { ATM_i} + w_{ RR}\\, {RR_i} +\nw_{BF}\\, {BF_i} \\, \\, \\, \\, \\, i\\text{=vega, vanna, volga}\n", "c1e09c55b415b70a00b0120a4f948225": "\\begin{align}\n\\mathbf{v} & = \\int \\mathbf{a} {\\rm d}t = \\mathbf{a}t+\\mathbf{v}_0 \\quad [1] \\\\ \n\\mathbf{r} & = \\int (\\mathbf{a}t+\\mathbf{v}_0) {\\rm d}t = \\frac{\\mathbf{a}t^2}{2}+\\mathbf{v}_0t +\\mathbf{r}_0 \\quad [2] \\\\\n\\end{align}", "c1e13490186d45de68c392c45b041758": "\\langle\\phi| \\; \\bigg( c_1|\\psi_1\\rangle + c_2|\\psi_2\\rangle \\bigg) = c_1\\langle\\phi|\\psi_1\\rangle + c_2\\langle\\phi|\\psi_2\\rangle. ", "c1e13958102e2bef8cc060863f5e3432": " f(x) = \\frac{P(x)}{Q(x)} ", "c1e1a8d8efb3bdfc06685b678496e08a": "f(X)\\geq\\frac{1}{6}", "c1e1bb3b6baec3ab7ff9db3b1bb1c50c": "\\forall x_1,\\ldots,x_n . F", "c1e1ceeb210b29c0f4ca091414cdbc17": "G(r) = {C \\over r^{d-2} }", "c1e1ebaab50c051ac8b38cad2e18312e": "\n\\gamma_n^2B_2^2T_{1n}T_{2n} \\geq {1}\n", "c1e20e23faa03ecc17ef3f3af5fc02e3": " F \\ge 0,\\quad Q(t) \\ge 0,\\quad R(t) > 0, ", "c1e20f89a0d7cad02300e7fc5b56adab": "\n Q = -EI\\cfrac{\\mathrm{d}^3w}{\\mathrm{d}x^3}\n ", "c1e2409f865ba2d301cc5c4bc03cd742": "= H_a \\left( \\frac{2}{T} \\cdot \\frac{e^{j \\omega T/2} \\left(e^{j \\omega T/2} - e^{-j \\omega T/2}\\right)}{e^{j \\omega T/2} \\left(e^{j \\omega T/2} + e^{-j \\omega T/2 }\\right)}\\right) \\ ", "c1e2b6249a5599116f89480b8915b62d": "3+4\\cos(\\theta)+\\cos(2\\theta) = 2 (1+\\cos(\\theta))^2\\ge0.", "c1e2bf5c5486fea9ed4bca90cac3523d": "\\alpha - \\gamma", "c1e2c01dd53390e02407191f6876bbcd": "T = \\frac{\\ln(2)}{\\ln(1+r)}", "c1e2ccb3f3d5a48676736e1a579e3ae5": "X\\leq_{st} Y", "c1e2ef371bcc4f7f8652f6b78ed63092": "[T_A^2]", "c1e3b032107a70339932bc0e4f32e4f1": "Circ(F; Initiates, Terminates) \\wedge\nCirc(G; Happens) \\wedge H", "c1e49614a81305d73559d3cd1fa5800b": " \\hat h(P) \\ge c(K) \\log(\\operatorname{Norm}_{K/\\mathbb{Q}}\\operatorname{Disc}(E/K))\\quad", "c1e4d992adb2ab11793fd7cf2d84f8b8": "v \\ \\overset{\\sim}{\\mapsto} \\ \\phi_v \\in V^* \\quad \\text{such that} \\quad \\phi_v(u) = v^\\mathrm{T} u", "c1e4f2c6985bd996b3520b2a1cd3c7f1": "E(R_m)~", "c1e4ff911d68e90479e9535d2d1e607a": " \\vartheta_1(z,\\tau) = 0 \\quad \\Longleftrightarrow \\quad z = m + n \\tau ", "c1e56b1fd99d5fd8d49704a473c98d1e": "g_{k\\cdot A}(t) = \\log \\left(\\sum_i e^{t\\cdot (k\\cdot A_i)}\\right) = \\log \\left(\\sum_i e^{(t\\cdot k)\\cdot A_i}\\right) = g_A\\left(k\\cdot t\\right).", "c1e581b1471c55777068e81e0a535dc4": " X_{\\sigma(1)}, X_{\\sigma(2)}, X_{\\sigma(3)}, \\dots", "c1e5b2bf5d07db685d010aa8e6d2e34a": "p(t)=p_{0}\\cos(\\omega t)-m\\omega\\!x_{0}\\sin(\\omega t) ", "c1e5da6cc87ff893fbf5bcf0cd06acf0": "\\, y(t) = Cx(t) + Du(t)", "c1e5ea578859e13d91132d551905dec7": "\\mathcal Z(T,V,\\mu) = e^{-\\beta \\Phi} \\,\\; ", "c1e5ffe605f6333f6f9f04d18966469d": " \\mathbf T^{n-1} \n \\begin{bmatrix} x_1^{n-1} \\\\\n x_2^{n-1} \\\\\n \\dots \\\\\n x_{n-1}^{n-1} \\\\\n \\end{bmatrix} = \n \\begin{bmatrix} y_1 \\\\\n y_2 \\\\\n \\dots \\\\\n y_{n-1} \\\\\n \\end{bmatrix}.", "c1e6361a9a1f4fadd17f9c9179478729": "\\{ \\alpha_1 a_1 + \\ldots + \\alpha_n a_n \\mid \\alpha_1,\\ldots,\\alpha_n \\geq 0, \\alpha_1,\\ldots,\\alpha_n \\in\\mathbb{Z}\\},", "c1e652f8138521dffc682e411ecaa05f": "\\left(\\dfrac{\\delta-(\\dfrac{\\lambda}{d})}{1-(\\dfrac{\\lambda}{d})}\\right)^2", "c1e679a15425eb508221a92bd1473d21": "\\frac{1}{\\sqrt{\\det S}} = \\int_V e^{-\\pi\\langle x,Sx\\rangle}\\, dx", "c1e6ad941f7a5d3609c4eb363752f9dc": "r_i < \\text{CR}", "c1e6f907bc97e84b7ab9cde671cc7a3d": "\\textstyle H(X_r)", "c1e75ac603846f3d7d6fd75555bcdc62": " L_{p,\\infty} = \\{ A \\in K(H) : \\mu(n,A) = O(n^{-\\frac{1}{p}}) \\} ", "c1e768526acc6b287b28170f82fe8aa6": "(16 - 8\\sqrt{2}) r^3 \\, ", "c1e79f5f38ecbea952f7c8767ad35ef7": "1\\over{\\sqrt 2}", "c1e7b078aefe8f144a2412fc8c035790": "f = h \\circ g", "c1e7b4cca4ec3b5d1254b8151e4feb11": "\\theta(N)\\geqslant1\\,\\!", "c1e8198f6d31e0ab8aed9a1ec3f94420": "f(q\\mid e)", "c1e841abdf8db091fe4b30a9fa574809": "F_{eachAnchor}=F_{load}\\frac{Sin(\\alpha )}{Sin(\\alpha+\\alpha)} \\,", "c1e85cda0d0d53e38e02f759b45d9b11": " r_1 ", "c1e86af522964ba173202569108d7209": "X = (X_t)_{t\\ge 0}", "c1e870d46c1ddb814242a63812c16ceb": "j,k\\in \\mathbb{N}", "c1e93aba82501c8d623d7c87a0557f8f": "A = \\frac{R T}{p} \\frac{n^2 - 1}{3}.", "c1e97ae167dfe5e587bda4eaf6a33aed": "\\frac{\\partial f}{\\partial x} = y", "c1e9f5c34fb5e07072d736b5c4d7c108": "Lu \\leq 0", "c1ea1464bf0cbb0f85299d76a92955ca": "\\Delta H_\\text{L} = {2\\mu_\\text{B}\\over \\hbar m_\\text{e} e c^2}{1\\over r}{\\partial U(r) \\over \\partial r} \\boldsymbol{L}\\cdot\\boldsymbol{S}. ", "c1ea3d71fdfd7e0d5e549f08183c84da": "\\mathbf{B}(s) = \\frac{\\mu_0 I}{ 2\\pi s}", "c1ea70209e276a557cbbde5e16410ff2": "\n\\frac{1}{r}\\frac{\\partial^2}{\\partial r^2}r F(r) = \\frac{\\ell(\\ell+1)}{r^2} F(r)\n\\Longrightarrow F(r) = A r^\\ell + B r^{-\\ell-1}.\n", "c1eab11790261f15eda492944c12e596": "\\varphi:\\mathcal{E}\\rightarrow\\mathcal{T}", "c1eb116442dfca49c8e5d29bd5df2b18": "F_N = \\dot m\\, g_0\\, I_{sp-vac} - A_e\\, p \\;", "c1eb21f207d20a37db98807571a21fba": "f_C=f_F", "c1eb2d97aeb3d615f80c340c763fa4ce": " \\Pi(x) = \\pi(x) + \\frac{1}{2} \\pi(x^{1/2}) + \\frac{1}{3} \\pi(x^{1/3}) + \\cdots. ", "c1eb9716c4ef466950f0a3d28c05b22d": "\\left|\\frac{AF}{FB}\\right| \\cdot \\left|\\frac{BD}{DC}\\right| \\cdot \\left|\\frac{CE}{EA}\\right| = \\left| \\frac{a}{b} \\cdot \\frac{b}{c} \\cdot \\frac{c}{a} \\right| = 1.", "c1ebaf0e2f5dc568de39022ba05615d5": "c^2+d^2 = yx", "c1ebb0e3ea3820e64e03949a547db0a2": "\\partial_t g_{ij}=-2 R_{ij}.", "c1ebcf1414e2812837cdb9289150b52c": " u^{(e)}_{h}(x) = u^{e}_1 N_1^{(e)}(x)+u^{e}_2 N_2^{(e)}(x), \\ \\ v^{(e)}_{h}(x) = u^{e}_1 N_1^{(e)}(x)+u^{e}_2 N_2^{(e)}(x) ", "c1ebcf5d4200ffed0f469026e66f852d": "[0,1/A^{1/\\gamma}]", "c1ebf4cff565b1e1e56ea7fb5cbca814": "E(z)=\\psi(z)\\otimes \\psi^\\dagger(z)", "c1ec2535f0de9c3035bac06186344046": "v_p", "c1ec3e40815d7c66588f1caf65b273a9": "\\sqrt S = \\lim_{n \\to \\infty} x_n.", "c1ec787a8e89baecf0904d01513c5ce4": "|a+b\\omega|^2 = a^2 - ab + b^2. \\,\\!", "c1ec865371dd904fc95f0a457d6fa6ed": "\\frac{\\mathrm{e}^{itb}-\\mathrm{e}^{ita}}{it(b-a)}", "c1ecdc19167d04434ed1d1b7e046cbd4": "~~ g\\!=\\!|G|^2~~", "c1ed6b91b982b2b60933608d017d238d": "\nQ = v P^{-1} = \\Delta_v^{-1} P^T \\Delta_\\mu, \\quad(14)\n", "c1ed7dd12a1b987df86a369b8130fc93": "g(L, R):=f(LR^{-1})R ", "c1ed88947ff9cff01eb4adafd96a9b7e": "f(\\mathbf{x})\\,", "c1ee280ea8250f988f43327840b213d8": "{\\neg}", "c1ee5b3d163cfa68c29c79cd1c461a94": "\\mathrm{\\tfrac{u\\bar{u} - d\\bar{d}}{\\sqrt{2}}}", "c1ee5cf1987f7d3f1756780a444bc85c": "O(3/\\epsilon) = O(1)", "c1ef2751db25fe330c4b5b781398d0b4": "\\frac{\\mbox{d}}{\\mbox{d} x} ( \\alpha \\cdot f(x) + \\beta \\cdot g(x) ) ", "c1efd0c13e569a96d0caa65658d3f97c": "\\begin{align}\n|p(z)| &< |q(z)| + r^{k+1} \\left|\\frac{p(z)-q(z)}{r^{k+1}}\\right|\\\\[.2em]\n&\\le \\left|a +(-1)c_k r^k e^{i(\\arg(a)-\\arg(c_k))}\\right| + M r^{k+1} \\\\[.5em]\n&= |a|-|c_k|r^k + M r^{k+1}.\n\\end{align}", "c1eff258953b8a365dc7d0cf7d203864": "\\sigma_I^{(k)}", "c1f0d945e30bfea0ada488e9cb203ce4": "\\alpha=\\tfrac12", "c1f1232268b8ce6e1dca617c7eca18d8": "U_E = \\frac{1}{4\\pi\\varepsilon_0} \\frac{q Q_1}{ r_1 }", "c1f15b5460759c1ca43c389273ce391a": " \\mathrm{Ei}(a \\cdot b) = \\iint e^{a b} \\, da \\, db", "c1f1ba6a79c5575ce7762f5db61d9f32": "|W(S,T)|\\geq\\gamma |S||T|", "c1f24771e34fde0a3e10340ab43d594b": "p_3=\\frac {1}{q_3(1+m_1)}\\ ,", "c1f2571100bb042355892a78aaff0d2e": "\\mathbf{S}^2", "c1f29bba9df8e2d967092d5edc7ad43c": "\\frac{d^2u}{dz^2}+Q(z)u(z) = 0", "c1f31c6ad6308deedc83a882337ceb81": "f\\in L^2(\\mathbb{R})", "c1f337a23a66afc20cf6941cdb9d5f4e": "R \\to R \\times S", "c1f341dba9942eb6718e5cdafe9563d6": "\n2T = \\mu R^2\\big [\\dot{\\theta}^2+(\\dot\\varphi\\,\\sin\\theta)^2\\big]= \\mu R^2 \\big(\\dot{\\theta}\\;\\; \\dot{\\varphi} \\Big)\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & \\sin^2 \\theta \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n\\dot{\\theta}\\\\ \\dot{\\varphi} \n\\end{pmatrix}\n= \n\\mu \\Big(\\dot{\\theta}\\;\\; \\dot{\\varphi} \\Big)\n\\begin{pmatrix}\nh_\\theta^2 & 0 \\\\\n0 & h_\\varphi^2 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n\\dot{\\theta}\\\\ \\dot{\\varphi} \n\\end{pmatrix},\n", "c1f34a17982abc554fdd49c6800b7c00": "(x+1)^2.", "c1f3dd3672796f33bb1bdcb56cfdadcd": "p_1=\\textstyle \\frac{1}{6}\\ ,", "c1f413ce16212229d385baa87d001b67": "\\operatorname{Lie} (G) = \\operatorname{Lie} (K) \\otimes_{\\mathbb{R}} \\mathbb{C}", "c1f4357feefaf0b7ee7d2abd0054a926": "\\!\\!=\\!\\!", "c1f4bd104af01d822387698bc190d2f0": "|B(x,r)|", "c1f4c85399bb9a0d04c39dcada6f1b8e": "k={1,2}", "c1f4dfc327155a424297e0d5ed6a8c9c": "0 \\leq \\delta \\leq \\frac{1}{4}", "c1f5347ac6ed5d9ecc16b48994d7e91b": "\\dot{q}_p", "c1f56239daf5889e95c366903a797c2b": "\\ s_{X_1X_2} = \\sqrt{\\frac{1}{2}(s_{X_1}^2+s_{X_2}^2)}", "c1f5837956f056119ff514b939ce5795": " \\mathrm{Re}(V) = \\mathrm{Re}\\left [ V_0 e^{j \\omega t} \\right ] = V_0 \\cos \\omega t.", "c1f586a613b0ec018648ac493a6f4de0": "m = T / V_e", "c1f5b64ae98c9f4cbceec348ed0564a0": "n!=\\Gamma(n+1).\\,", "c1f5c948c0cd6cad225f3f29d9c1dab5": "\\rho = \\rho_0", "c1f5dd6df1a0095021b048d2940e95ab": "\n\\langle \\Psi \\rangle_I \\rightarrow\n \\frac{1}{2}\n \\begin{pmatrix}\n \\psi_{11}-\\psi_{11}^* & \\psi_{12}-\\psi_{21}^* \\\\ \n \\psi_{21}-\\psi_{12}^* & \\psi_{22}-\\psi_{22}^*\n \\end{pmatrix} \n", "c1f5eeb24599c5e7892504328231fdd9": "\\frac{\\mbox{total} \\; \\mbox{votes}}{2+\\mbox{total} \\; \\mbox{seats}}", "c1f60f380347a622036964b87e46f875": "[2^n,2^{n+1}-1]", "c1f678cc88f4a52655c1709a8ddfdec0": "\\epsilon((-1)^F)=1", "c1f67df3a849e3653258eab40f96da62": "\\operatorname{Hom}(-, U)", "c1f7a6f9f28d42d955e9c2ab69ac1786": "P= Clock \\backslash tick", "c1f7be1ba3f5da1d3d4f4974d66101c0": "\n\\begin{align}\np(Y) &= \\int_{-\\infty}^{\\infty} p_U(YZ)\\,p_V(Z)\\, |Z| \\, dZ \\\\\n&= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-Y^2Z^2/2} \\frac{1}{\\sqrt{2\\pi}} e^{-Z^2/2} |Z| \\, dZ \\\\\n&= \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-Y^2Z^2/2} \\frac{1}{\\sqrt{2\\pi}} e^{-Z^2/2} |Z| \\, dZ \\\\\n&= \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi} e^{-(Y^2+1)Z^2/2} |Z| \\, dZ \\\\\n&= 2\\int_{0}^{\\infty} \\frac{1}{2\\pi} e^{-(Y^2+1)Z^2/2} Z \\, dZ \\\\\n&= \\int_{0}^{\\infty} \\frac{1}{\\pi} e^{-(Y^2+1)u} \\, du \\quad\\quad \\text{(let }u=Z^2/2\\text{)}\\\\\n&= -\\frac{1}{\\pi(Y^2+1)} e^{-(Y^2+1)u}\\Bigg]_{u=0}^{\\infty} \\\\\n&= \\frac{1}{\\pi(Y^2+1)}\n\\end{align}\n", "c1f7c401a78d6f1ff5f44cb3d00d76f7": " v(R) = \\frac{1}{4 \\eta} R^2 \\frac{\\Delta P}{\\Delta x} + B = 0 ", "c1f7d3f42562748372d189802a247926": "\\{A_n:n\\in \\mathbb{N}\\}", "c1f7f003100114261576cc6a8c9f85f2": "V_x(\\mathbf{x}) > 0", "c1f8020d50dbf3c5be90cb06f1ac34ab": "i \\tau", "c1f82f8b93d32381366f3a2ed91391ad": "\\ S_i ", "c1f8456c7f7eb81504870934aba823b0": "\\mathcal{L}:\\mathcal{H} \\to \\mathcal{H}", "c1f88f38157cfa1c4ec89d6a9d7a4743": "\\varphi^n = F_n\\varphi + F_{n-1}.", "c1f8bf04182ace3a741f64b8acb1cb26": "\\psi_n({\\mathbf{x}})=e^{i{\\mathbf{k} {\\mathbf{\\cdot x}}}}u_{n{\\mathbf{k}}}({\\mathbf{x}})", "c1f8ca316b9270b08c53a71f7d5b7f3b": "K_L=\\frac{\\sin(\\alpha)}{\\sin(\\beta)}", "c1f91ebe8b5e5dbc5002c47c78f315cf": "p_6 = C\\rightarrow c", "c1f977aa95441290796b62081465dd76": "\\Delta G_{rac}<\\Delta G^\\ddagger_R<\\Delta G^\\ddagger_S", "c1f9aec75d23f917bb58c61ebefb871c": "\\Phi'(0)\\neq 0", "c1f9c7a2db86b86d21e79e60e2ca5b32": "\\theta_{n+1} = \\theta_{n} + p_{n} + {\\frac K {2\\pi}} \\sin(2\\pi\\theta_{n})", "c1f9ec4d22a1a806279af578009181df": "v>c", "c1f9eeb97030e35714544c339869e62b": "G_{-1} = G \\supset G_0 \\supset G_1 \\supset \\dots \\{*\\}.", "c1fa11a6435321139e4ff3e4ae9e5298": " \\frac{T''}{c^2T} = \\frac{v''}{v} = -\\lambda. \\,", "c1fa5dd7486962eac195aa4274493757": "t = \\frac{\\overline X - \\mu}{S/\\sqrt{n}} = \\frac{\\frac{1}{n}(X_1+\\cdots+X_n) - \\mu}{\\sqrt{\\frac{1}{n(n-1)}\\left[(X_1-\\overline X)^2+\\cdots+(X_n-\\overline X)^2\\right]}} \\ \\sim\\ t_{n-1}.", "c1fa8e2971301fddc83f5cbf84b952a4": "\\omega_0\\rightarrow 0", "c1fada8298be17b374562ab6f7f13df5": "\\mathbb{E} \\left[ \\int_{0}^{T} | X_{t} |^{2} \\, \\mathrm{d} t \\right] < + \\infty.", "c1fb4c486e51fb580fe74f110684cf0f": "\\sigma_{xy}\\sigma_{xz}\\sigma_{yz}<0", "c1fb7539f3d0ee5923a1b4ecbfa0d3d9": "0 < \\mu(\\mathbf{B}_{r}(x)) = \\mu(\\mathbf{B}_{r}(y)) < + \\infty", "c1fb8d8fbda440d38e7722b6bd46ae31": "\\{\\{a\\},\\{a,b\\}\\},", "c1fbbb19de36b42e5f1e9c9255317231": " (X,Y) ", "c1fbc697a6c34cba2a5c858f5949d346": "O(\\log^2 n)", "c1fbe26c9914fdbeb3c992b9b72c91e4": "2\\sqrt{n}", "c1fbe8b58520a9ca3930fea18b8426d4": " {\\mathbf{S}}({\\mathbf{p}}(t))\\in\\R^{L_1\\times L_2} ", "c1fc38513dbb5b992b2a50b3bf1d9915": "\\alpha^*_{g}", "c1fc4d3e09b6dd09797a400e61efcd59": "na$", "c1fc64c86ebabfdbe1b1541bd9ec50ac": " R \\approx \\frac{N^{5/3} \\hbar^2}{2m GM^{1/3}}.", "c1fca49379f5054257f266df3f76254d": "\\mathrm{e}^{- N \\gamma_{\\mathrm{rad},0} \\, t}", "c1fcfcd1377b34a8594b7be277b34058": "\\mu_{\\mathrm{N}} = e \\hbar / 2 m_p", "c1fcfd8a1adbd3a0203323c12e45b4bc": "z=i\\omega_m", "c1fd09366d8b8cb2d955c71e7793d63e": "\\langle E_\\mathrm{pot} \\rangle=-\\frac{Ze^2}{4\\pi\\epsilon_0}\\left\\langle\\frac{1}{r+\\delta r}\\right\\rangle.", "c1fd094c512b9756c41209d23d7f71a9": "\ny = \\sigma \\tau \\sin \\varphi\n", "c1fd0fa08af2a6d026c8f554248d905f": " X = E[p] ", "c1fd2d91aea62e3f01ad140afe4d1419": "R^2_\\text{L} = \\frac{D_\\text{null} - D_\\text{model}} {D_\\text{null}} .", "c1fdae5f7d66d1c431420ffe2fedbdbf": " L(u) = F(x, u, (\\partial^\\alpha u)_{|\\alpha| \\le 2k})\\,", "c1fdd053fbc8ddb069f35ccc25ecd26c": "{7}\\, ", "c1fdd28c0f2a374624db41c0a11b3fdc": " \\beta(w_1)\\neq 0. \\,", "c1fe81afd937d35226d4ae5c642a518f": " f = 10Hz ", "c1fea4300e5bd62023eaf5b7196749e5": "\\operatorname{El}", "c1fefc72296ab7612e29171518518989": "D > 0 ", "c1ff1c9f24b00b2e431a243fb20670fa": " -\\log P(x) = \\log |S|+O(\\log n)", "c1ff2a8cda8d42dfb2eda8548ce82b4d": "[\\frac{\\sum_{p=1}^{number of ordered products}(CLIP_{weekly,p})}{number of ordered products}]", "c1ff2e55f05472dc47184f4074b7730e": "a,b \\in Y", "c1fff7406f6576040bfa0904c4cd5b32": "\\|E\\|", "c2002068fa3c95b591e725a620bb34e9": "g(X) = \\sum_{n=0}^\\infty b_n X^n = b_0 + b_1 X + b_2 X^2 + \\cdots,", "c2005e8498003e4c80017134a36a61c1": "y(t) = \\frac{Y(t)}{A(t)L(t)} = k(t)^{\\alpha}", "c2005f4cda59875d160765fc2d515dbc": " \\frac{2\\pi \\varepsilon l}{\\Lambda }\\left\\{ 1+\\frac{1}{\\Lambda }\\left( 1-\\ln 2\\right) +\\frac{1}{\\Lambda ^{2}}\\left[ 1+\\left( 1-\\ln 2\\right) ^{2}-\\frac{\\pi ^{2}}{12}\\right] +O\\left(\\frac{1}{\\Lambda ^{3}}\\right) \\right\\} ", "c2006ecfde459efc17bb5db260d392f2": "(\\emptyset^c aa \\emptyset^c)^c", "c200c3f842ff829f7b8a81793915d6b4": " b^*(f)P_a(g_1\\otimes g_2\\otimes\\cdots\\otimes g_n)=P_a(f\\otimes g_1\\otimes g_2\\otimes\\cdots g_n) \\, ", "c200ecdcb41193bd38f29b8f2774122b": "\\vdash\\Sigma", "c2011331b4136260ca78d4ead207a5f0": "V_{Th}", "c2011781a6594af8e4286b8712405fed": "\\beta = \\frac{1}{2}", "c2011d597d004dbf6292ee27050439cb": " 1/x ", "c201eb1b7dd0113c04cbaa8fbb1f1da9": "\\sqrt{(12 + (-3)^4)\\over11}-22", "c201fee8767b6dff2ddcea60261e566e": "y-x", "c2020df1175f9e6f2de6e036a06e1dc4": "C_\\mathrm{MIS(max)}=\\varepsilon_0\\varepsilon_r \\cdot { {A} \\over {d} }", "c20238c84149ec7a05d74d06a05cb894": " \\lfloor \\log_2 n\\rfloor", "c2026cd2940d0f866f878a32ff65ad6f": "(S_1,\\dots,S_p)", "c20282b0f21faf5e8c95b49ac83c445d": "K_m = D_m + K_B^2(R_g^a + R_m^a)", "c202ce78df4d918e6dc2e2275a752c95": "g(x) = x-a\\,", "c202dcb8e4f68a59164df41112e1c6d7": "f(k) = k", "c202e2308c1388769fcb180f46340491": "D_\\mathrm{BL} < 3.57\\,(\\sqrt{h_\\mathrm{B}} + \\sqrt{h_\\mathrm{L}}) \\,,", "c20303a35b9d99a02352709c2e6a99fd": "D_x f(x)\\,", "c20309dcaff4e92e1f2e34f9c9f2a18f": "\\langle \\vec f \\rangle = - k_B T \\frac{3 \\vec R}{N l^2}~", "c20318e69874dff094a75bfe1b0c0357": "\\ell(t)", "c20371be015b59971a89e8a839668a6e": "R_r", "c2037cfb9a62e565d4a5c956a90c1100": "a_{12}+b_{12}+c_{12}=c_{1}", "c203ee76898f1d074270976241bfa9f9": "\n\\left(\\frac{\\partial T}{\\partial p}\\right)_S =\n+\\left(\\frac{\\partial V}{\\partial S}\\right)_p\\qquad=\n\\frac{\\partial^2 H }{\\partial S \\partial p}\n", "c203f0c237632628995295edb23823e0": " \\mathfrak{g}_0 = \\mathfrak{k}_0 \\oplus \\mathfrak{p}_0 ", "c2048d2c1ee7ca6c4f930041b49fcc48": "(v + s)n", "c204a6ceb740017cdb52b38629c82071": " W(x)= a \\frac{f(x)}{f(x/a)} \\,", "c204bb519e8625b20d5660801901329b": "\\gamma_s(0)=P", "c20510c7626f90fc70e7112442f246ba": "mn > k^2 \\ge m^2n/(2m+n) \\, ", "c2051575d5d96b50ec1c69486e03299a": "\\begin{align}\nx&=a\\cosh u \\cos v\\\\\ny&=a\\sinh u \\sin v\\\\\nz&=z\n\\end{align}", "c205244dc7f7091515e8e1d2327169b2": "{\\mathcal L}_{xy}^3: L=Lclm(k,l).", "c20561c4605965267e368e05b3cf2b22": "g(p)", "c20568d116448b45b8f8191d903b729b": "f(\\cdot |\\theta_0)", "c20581a0f30ea5a56767487136356f97": "\\alpha_x = f_1(x) \\, dx_1 + f_2(x) \\, dx_2+ \\cdots +f_n(x) \\, dx_n", "c205ac1a32b862cb08a562922639e43f": "\\mu_{eff} = \\sqrt{3 k \\over N \\mu_0 \\mu_B^2} \\sqrt{T \\chi} \\approx 797.727 \\sqrt{T \\chi}", "c205b7e3d8881defa26c0b04b0f7468c": " \\begin{align}\nVar( X(t+1) | X(t)=i) &= Var(X(t)) + Var(\\Delta(t+1)| X(t)=i)\\\\\n&= 0 + E[\\Delta(t+1)^2| X(t)=i] - E[\\Delta(t+1)| X(t)=i]^2\\\\\n&= (i-1-i)^2 \\cdot P_{i,i-1} + (i-i)^2 \\cdot P_{i,i} + (i+1-i)^2 \\cdot P_{i,i+1} - E[\\Delta(t+1)| X(t)=i]^2\\\\\n&= P_{i,i-1} + P_{i,i+1} - E[\\Delta(t+1)| X(t)=i]^2\\\\\n&= \\frac{(N-i)i}{(r i + N -i)N} + \\frac{(N-i)i(1+s)}{(r i + N -i)N} - E[\\Delta(t+1)| X(t)=i]^2\\\\\n&= i (N-i)\\frac{2+s }{(r i + N -i)N} - E[\\Delta(t+1)| X(t)=i]^2\\\\\n&= i (N-i)\\frac{2+s }{(r i + N -i)N} - (p s \\dfrac{1-p}{p s + 1})^2\\\\\n&= p(1-p)\\frac{2+s (p s + 1)}{(p s + 1)^2} - p(1-p) \\frac{p s^2(1-p)}{(p s + 1)^2}\\\\\n&= p(1-p)\\dfrac{2+2 p s + s + p^2 s^2 }{(p s +1)^2}\n\\end{align}", "c205e98fb4e2453a39236d3a11f2774c": "P_\\infty", "c20638030dcdf7b0ea74d21c281c39fc": "\\mathfrak{H}_r", "c2063ab92cc7b89871141338591a4bf7": "\\eta - 1", "c2064168fc0f7eecdeee72a62bb94850": "Y(t) = J_{N(t)+1} - t. \\,", "c20653cbe4441b7daf0ca3a36a1a3967": "2^{|k|-b}", "c206724b70a09091bdc068bf2fca9256": "r = r_0 \\cos(\\theta - \\phi) + \\sqrt{a^2 - r_0^2 \\sin^2(\\theta - \\phi)},", "c20694f4639c057de1509896982d690f": " M_x = \\left\\{ f \\in A(\\mathbb{T}) \\, \\mid \\, f(x) = 0 \\right\\}, \\quad x \\in \\mathbb{T}~,", "c206e422e4a327d8073b9c70923102ad": "syst_k(M)", "c206f9c5ace44103b47a654ff7b94be9": "\\Delta = -b_2^2b_8 + 9b_2b_4b_6 - 8b_4^3 - 27b_6^2", "c207796aabe7f9af174b69e0da779c37": "a \\otimes h", "c20799c9bd60ab76fbf27a81b73643ea": "\\beta_{ik}", "c207a5d8ff5ac07846e50a81abbdede0": "\\Delta_1=\\{amount(lactose,hi), amount(glucose,low)\\}", "c207a95d435bf3e8f643352731f2b488": "a^\\dagger_{ij} = \\langle\\psi_i | {a}^\\dagger | \\psi_j\\rangle", "c207f0a62f60d2ebeafc8194e65b66fc": "\\varphi_1 = \\bigwedge_{i\\ne j} \\neg (x_i = x_j)", "c208271e5d70ec327f55470e4e66f63e": "d V", "c2083c63959ce9f58f61a3ca977e62e0": "\\alpha \\beta^* = \\mu \\beta^* \\alpha,", "c20858cd19e3719941ec93fbe7d2b24f": "\\scriptstyle \\hbar k", "c2085fd1e2ede882fd7493d79cefdfb8": "\nj = \\rho + \\mathbf{j}\\,,\n", "c2086ec3f7c3442ddf6746bbca39be1e": "E = 0~,~~ t_B =", "c208a75a5a383169a3a876b43b38dd8a": "(f_1,\\dots,f_n)^t", "c208d446cfac2c087cfd54ecbf660838": "x^5 - x + a = 0\\,", "c208e8ed9ef795ae05c528e571aaad80": "\\left\\{5,{3\\atop3}\\right\\}", "c2090578e4d9933dc739955171be06ff": "\\sigma(T)=T^{-1}(\\mathcal{A}')", "c2090cc5fd6d83e3139badccff1448ac": "\\epsilon = -{\\mu \\over{2a}}\\,\\!", "c209bcd7ab3f2218d3700950c80b2b6d": "A_{eff} = \\frac {\\lambda^2}{4 \\pi} \\,", "c209c19af47c9df23f076294d365b49d": "(E_1,E_2,E_3,B_1,B_2,B_3)", "c20a01e71331f1db946026eb8ca02ba5": "|r_1| + |r_2|", "c20a0df3f801e6c19257f72204d1bfdb": "r\\ll r_a", "c20a174621accbddb929627b13b0c149": "\\Omega = \\{1,2,...,n\\}", "c20a1da3e9a4cf0298c6bf2b1dd457c9": "\nA =\n\\begin{bmatrix}\n ~D & -I & ~0 & ~0 & ~0 & \\ldots & ~0 \\\\\n -I & ~D & -I & ~0 & ~0 & \\ldots & ~0 \\\\\n ~0 & -I & ~D & -I & ~0 & \\ldots & ~0 \\\\\n \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n ~0 & \\ldots & ~0 & -I & ~D & -I & ~0 \\\\\n ~0 & \\ldots & \\ldots & ~0 & -I & ~D & -I \\\\\n ~0 & \\ldots & \\ldots & \\ldots & ~0 & -I & ~D\n\\end{bmatrix},\n", "c20a23e43bae2e32145a7b4acf931ce8": "(d\\mathbf{X})\\otimes\\mathbf{Y}+\\mathbf{X}\\otimes(d\\mathbf{Y})", "c20a2b90e96af5e39b23c9cb7d2db7e7": "\\mathit{dr}(n!)=9 \\Leftrightarrow n \\ge 6.", "c20a33e2363cb80f40672ca6a305f210": "\\Sigma_A", "c20a693ba20767b0e7bf363f93b99be9": "\\left| A_n^\\epsilon \\right| \\geq (1-\\epsilon)2^{n(H(X)-\\epsilon)}", "c20a84c8ffeffcd99aceed5ee7d0d627": " \\frac {4 \\beta A_0} { \\tau_1 \\tau_2} = \\left( \\frac {1} {\\tau_1} - \\frac {1} {\\tau_2} \\right)^2. ", "c20ad4d76fe97759aa27a0c99bff6710": "12", "c20b4b6a5284904ea5b10d4e5b818f5a": "\\tau_w", "c20be02de3f5e8d56305825cfe5094df": "f\\colon\\mathbb{R}^n\\rightarrow M", "c20c01e216eba6f71e145b5418e5e9c7": "Z e", "c20c4f7a36da2a315ad7ceb2baa3a5be": "\\frac {C_{13}^1 C_{4}^4 \\cdot C_{12}^1 C_{4}^1} {C_{52}^5} = \\frac {13 \\cdot 1 \\cdot 12 \\cdot 4} {2{,}598{,}960} \\approx 0.024\\%", "c20c75e268bd58a24ec0bf4506a971f5": "\\begin{align}\nor\\end{align}", "c20cd0676f31d8dd5697f311c4e410be": "\\beta = (1-\\rho)\\sqrt{s}", "c20d0b126ca07e0be4545c4302b16476": "\\frac{1}{Y_{TS}} = \\frac{1}{Y_1} + \\frac{1}{Y_2} + \\frac{1}{Y_3} + ... \\,", "c20d2632018467f4f0b97d2aac6be597": "f_s(x) = f(x/s) \\times 1/s = f(g(x)) \\times g'(x). \\!", "c20d2ccbbc1fe6495698f1b6e2e0a788": "\\vartheta_{00} (z; \\tau) = -i \n\\int_{i - \\infty}^{i + \\infty} {e^{i \\pi \\tau u^2} \n\\cos (2 u z + \\pi u) \\over \\sin (\\pi u)} du", "c20d44b71332a8075d0433a199a63d34": "\\cos (2 \\theta) = 1 - 2 \\sin^2 \\theta\\,", "c20d70c4479ade07762661a96bcbe0f7": "\\mathcal{F}_\\alpha", "c20d97e410509a47946cdf8775d7079d": "Pu = u", "c20e2740c7b29d71f1a1e41b6de4f2c2": " H^q(X,L\\otimes\\Omega^p_{X/k}) = 0", "c20e2df0e7478c6108ffb1e9405262e3": "\\check{\\mathbf{q}} = \\left[\\begin{array}{c} q_1\\\\q_2\\\\q_3\\end{array} \\right],\\ \\ \\ \\mathbf{\\mathcal{Q}} = \\left[\\begin{array}{ccc} 0 & -q_3 & q_2\\\\ q_3 & 0 & -q_1\\\\ -q_2 & q_1 & 0 \\end{array} \\right]", "c20e31a848e54863695d0acb0f9bcc68": "\\zeta(s,q)=\\frac{1}{\\Gamma(s)} \\int_0^\\infty\n\\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt", "c20e3482bac473f7cb64e9074058dc04": "h \\theta_c", "c20f9af681c7bf8e0cc2468fac54e977": "\\left\\langle a_0,\\dots,a_{n-1}, a_n\\right\\rangle \\longmapsto \\mu a . \\left[ a_0 = n \\land \\forall i < n \\; \\left(\\beta\\left(a,i+1\\right) = a_i\\right)\\right]", "c20f9ce286ed2c7ebfb9790a2f86605d": "(a_1,\\ldots,a_n)", "c20fcdfe7b24ca58f4ed6f4a42927d42": "A\\in I\\times J\\iff \\{x\\in X|\\{y|\\langle x,y\\rangle\\in A\\}\\notin J\\}\\in I", "c210021dfb9bca6ee7e4d2a29d63e56f": "s_0=\\underset{i \\in D}\\times q_{0i}", "c210320f96c5684f1b934ebfb0d4ad81": "\\frac{5\\cdot\\pi}{6}", "c21067d99cdb38dcc719980cb010b33c": "n\\pi p^2 \\sigma =2\\pi GM_{12} n a/\\sigma ", "c2116440408fa5d4681fdf1696a5ceab": "\\mathbf{W} \\in\\mathbb{R}^{D\\times D}", "c211813c09e0b6476311edcc3037bc88": "\\|\\alpha\\|_1", "c211b95509ca1bbede307824f5873150": " \\Gamma (\\mathbb P, \\mathcal O (m))", "c21221aa561e8a792a1a6b487083606a": "+ \\sum_{j=1}^3 x_j \\ \\frac{d^2 \\mathbf{u}_j}{dt^2}\\ . ", "c212880c33896106d8fae2a71d0018c8": "\\prod_{i=1}^n A_{i, \\sigma_i}\\ ", "c212b953091650856cb411b9a2297505": "z = 1/\\bar{z}", "c212eceafcf8cfd6222ee7bc25f6dc82": "(a^2 + b^2)(c^2 + d^2)", "c2130a9e5cca310acd24b8b36f52976e": " {d \\mathbf p_1\\over dt} = {q_1 q_2 \\over r^2}{\\hat{\\mathbf r}}\n\\; + \\; {q_1 q_2 \\over r^2}{1\\over 2m_1 m_2 c^2} \n\\left\\{ \\mathbf p_1 \\left( { {\\hat{\\mathbf r}}\\cdot \\mathbf p_2} \\right) \n+ \\mathbf p_2 \\left( { {\\hat{\\mathbf r}}\\cdot \\mathbf p_1}\\right)\n- {\\hat{\\mathbf r}} \\left[ \\mathbf p_1 \\cdot \\left( \\mathbf 1 +3 {\\hat{\\mathbf r}}{\\hat{\\mathbf r}}\\right)\\cdot \\mathbf p_2\\right]\n \\right\\} ", "c213183ae6be21f3ca7855176fadeae9": " x_0,x_1,...,x_{Ne} ", "c213247348bf0bb7240cc43c0ff14acf": "wfe/({\\delta}.\\sqrt{\\theta}) = wfe/[(P/101.325).(\\sqrt{T}/\\sqrt{288.15})]", "c2135a289e47f91d26df09e1113b1542": "\nF(x) = \\frac{3\\pi\\mu_0}{2} M^2 R^4 h^2\\frac{1}{x^4} = \\frac{3\\mu_0}{2\\pi} M^2 V^2\\frac{1}{x^4} = \\frac{3\\mu_0}{2\\pi} m_1 m_2\\frac{1}{x^4}\n", "c213692742fe3f6f8deea0790cc82d44": "f_\\text{Cf} = {1 \\over 2 \\pi R_\\text{f} C_\\text{f}}", "c21388550f76d08cf8217edb6523d36b": "\\int{\\pi(\\theta)d\\theta}=\\infty.", "c2138ef6c4396411a68ea8a1a39d9923": "D^{KL}(a||b)\\,", "c213c8a1f46dfb60ff25bf0d7839e792": "9 \\over 100", "c213f01fa538f69332e83980d88130b7": "S \\left(\\mathbf{R}^n\\right) = \\left \\{ f \\in C^\\infty(\\mathbf{R}^n) \\mid \\|f\\|_{\\alpha,\\beta} < \\infty\\quad \\forall \\alpha, \\beta \\right \\}, ", "c2144e7999f601efae62f1d5082124ec": "\nn(M) = \\sum_{\\vec{m}\\in M} c(\\vec{m})\n", "c21479e1f4a29d8e1016ffb6410ab871": "0(~)", "c214cb15d99cf75b4a7449c0f1811701": "\\operatorname{E}[S_N]=\\sum_{i=1}^\\infty\\operatorname{E}[S_i1_{\\{N=i\\}}]\n=\\sum_{i=1}^\\infty\\sum_{n=1}^i\\operatorname{E}[X_n1_{\\{N=i\\}}].", "c2158ff6f6d1f5cb900b411199482221": "(\\tan^2\\phi,2\\tan\\phi)", "c215b77cb5ea966681101c4645bb3bae": "\\tilde{G}=\\{g_c:g_c(a)=a + c,c\\in \\Bbb{R}^1\\} .", "c215c8cff27158bddc502e82e62e20da": "\\left|\\sum_{n=M+1}^{M+N}\\chi(n)\\right|=O\\left(\\sqrt q \\log \\log q\\right).", "c21618269924a93188acd66630b76184": "p = \\frac{h}{\\lambda} = \\hbar k", "c2162f8d4a4d76d17a442fb19f4cd3bf": "M_n = M", "c2165d3f06dfdc28e552f46787edad90": "(1 + i_\\$) = \\frac {E_t(S_{t + k})} {S_t} (1 + i_c)", "c216687f45f3d1e755bda8fd248644ef": "\\begin{array}{rcl}\nA_0 & = & \\emptyset \\\\\nA_{k+1} & = & \\left\\{\\ y \\in U\\ | \\ ((E \\Rightarrow wp(S, x \\in A_k)) \\wedge (\\neg E \\Rightarrow R))[x \\leftarrow y]\\ \\right\\} \\\\\n\\end{array}", "c2169c99809e8cef911b439fc6743e27": "a+b\\sqrt{ - 1}", "c216bf0df46c7e704d0108bdda73aea4": "a_{3}=(5/6)d", "c216e7cf52c198fec6674bd08289662f": "-1 G > R\\ ", "c217ec9db2ebac57b4794524a8275c93": "E[X_n]=2^n\\,p^n", "c218663e07a25f061581fc483be38791": " \\phi^2 = \\sum_{i=1}^r\\sum_{j=1}^c\\frac{(\\pi_{ij}-\\pi_{i+}\\pi_{+j})^2}{\\pi_{i+}\\pi_{+j}} ,", "c2187133b41de423f91395148d5f1f7b": " w_1 ", "c218ec92fd3aec73d611019e53752d77": "x'(t) = P(x(t)),\\quad x(0) = x_0\\in U", "c218ecb4eb6ffe976860473b4dba6d00": "r = a \\frac{\\sin ((\\alpha+\\theta)/2)}{\\sin ((\\alpha-\\theta)/2)}", "c2192651c784ab3c6aa3e14a7315a050": "I_\\perp", "c21988890f343680ba9b4d1d2441c22a": "\\Sigma \\Sigma^T", "c21989cfbcc2252cf837c19234d0e98a": "B_0 \\mathbf{\\hat z}", "c219976768182d8ee529d610dfbc2887": "\\langle F,R\\rangle", "c219af1f18168447921578b91cd73b90": "N_i \\bar{r_i}", "c219be3a518215efb9417ae1a77ecf99": "\\bar t = (C^T)^x \\bar c_{i}. \\, ", "c219be6abeff3b6c845a8e36caae1daf": " \\left| \\nu_{\\alpha} \\right\\rangle", "c21a5b5c11850cb5a0d404901e498e9a": "\\Delta E_{CB}=\\frac{\\mathrm{d} E_{CB} }{\\mathrm{d} V_{0}}\\Delta V_{0} = V_{0} \\frac{\\mathrm{d} E_{CB} }{\\mathrm{d} V_{0}} \\frac {\\Delta V_{0}}{V_{0}}=Z_{DP}\\cdot \\bigtriangledown u(r,t) \\; \\; (12)", "c21a66225f071011c3347c7d1f177c25": " \\pi_T([\\eta +i]^{-1}) = [T + i]^{-1} \\quad ", "c21a9d479ebf9c94be282fef6b9c3c7f": "m_1< m_2< ...< m_n", "c21b005cfdb531790661606e4536e8d0": " p = 1 - e^{ - m \\log( a m^{ b - 1 } )( a m^{ b - 1 } - 1 )^{ -1 } } ", "c21b34a55d751464f855b7b9abca0fd1": "L_2 \\,\\!", "c21b388a48eea039c6761e4bac9e5bb4": "\\oint \\!\\,", "c21b408090d52fc36ab220269bb06368": "0 \\,", "c21b4f0c76dee2657588b8107c9f6e8a": "e^{i\\varphi} = \\cos \\varphi + i\\sin \\varphi.\\,", "c21b776ea3ca4df6153329751aa8d38c": "(\\alpha_1, \\cdots, \\alpha_K)", "c21b8e06b1b546b0c5997e908f234dcb": "P(\\theta) = P_0 [ 1 - J_0^2(ka \\sin \\theta) - J_1^2(ka \\sin \\theta) ]", "c21baa8945c673f9a4ff8fa41048cebb": "a^{n},b^{n}\\in\nT_{\\delta}^{\\mathbf{p}^{n}}", "c21bf2d95b68c54843ccfc7fa4de7ffe": "S_n = S_0 \\times u ^{N_u - N_d}", "c21c5a9f9daae719a104026f2552c2b3": "\\deg(A_i) \\leq D\\text{ for }i = 1,2,\\ldots,s\\text{ and }\\deg(A_0) \\leq D + k -1", "c21ca86348bd3cbcfe65d7684d8e3f2e": " |x| < 1 ", "c21cd5448b83c6080057c31795b7b815": "dr/dt=0", "c21d09770d1a09083c5d1bc864fa6ce8": "2m |p-1", "c21d52e6028374d1db80f82593cac715": "\\widehat{\\mathrm{Fr}}=\\frac{v^2}{gd},", "c21d8396b9432f32a9486d8fdefe592b": "q\\leq 8", "c21d991ce9dfd1c6b0198959f1145846": "\\vec{c}_1", "c21dd90daf8c7974e3940fa9fd5f86ff": "t\\to 0", "c21ed98110ce43c6d7de2700b4c5e3ce": "f(x-yr^*) = u(x,-y) + r^* v(x,-y) = u(x,y) + r \\ v(x,y) = f(q).", "c21ee75af240efa507cf34a44d8b461c": "V_m(\\lambda) = (1-x) V'(\\lambda) + x V(\\lambda)", "c21f1d7926d390441cbb7a95870fa4a1": "\\sum_{w\\in N^{(t)}(u)\\cup\\{u\\}} \\frac{1}{d(w)+1} \\le (d(u)+1) \\frac{1}{d(u)+1} = 1 ", "c21f21bb974eab1be2e647c26b3ace91": "\\begin{matrix}\\Delta L = 0, \\pm 1, \\\\ \\pm 2, \\pm 3, \\pm 4 \\\\ (L = 0 \\not \\leftrightarrow 0, 1)\\end{matrix}", "c21f7c05b933cedf7423a06187b12d94": "t=\\dfrac{l^2 n^3 \\mu}{n_{e} k T}", "c21f8506bee7f57ba78ef63249ee5dbf": "\\delta_{s}f={{\\rm det}\\, s \\, f\\circ s + \\sum_{t0, \\, s^{-1}\\alpha<0} \\alpha}", "c21fda3018fc86e53b6cf4952cf8a1af": "\\gamma = \\frac{Z}{Z_0} = \\sqrt {ZY}", "c21fdba28ed2271fa7581c9f9766b412": " \\sup_z g(z) \\leq \\sup_z f(z, w) , \\forall w ", "c2206fdb6e02207e864535df17154789": "\\left(-\\sqrt{12/7},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "c2207ed94f0b703f1bfb353e4287c1b9": "\nA=\\begin{pmatrix}\n3 & 3 & 1 & 4 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 19 & 16 \\\\\n0 & 0 & 0 & 3\n\\end{pmatrix}\n\\qquad\nH=\\begin{pmatrix}\n3 & 0 & 1 & 1\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 &19 & 1\\\\\n0 & 0 & 0 & 3\n\\end{pmatrix}\n", "c220a8ea14796f8ccfc74842ac16270d": "\\int_0^1 x^3\\ln(1+x)\\ln(1-x)\\,dx=\\frac{13}{96}-\\frac{2}{3}\\ln 2.", "c220aaeff1f2d678de7bcd542492cd41": "e^{i(N-1)t}", "c220dd838bfea6e44c8c235cc669038f": " \\log m = c + d \\log p_0 ", "c220ed17b9ecb7a34948a430c8aeb31f": "B = N R.", "c2210c69074288b1b788d1ec3d2369f8": "\\dot{\\textbf{x}}=f(\\textbf{x},u,t) ", "c22177f6ba6562e604d0efb6cdb8b71b": " \\ \\textbf{f} \\cdot \\textbf{h} = \\textbf{g} \\pmod q ", "c221e1dc019905568141e4e017c52137": "\ns_\\mathrm{a}(t) = A\\cdot e^{i \\left(\\omega t -\\begin{matrix} \\frac{\\pi}{2}\\end{matrix}\\right)} \n\\,", "c221ed6ee476a72e4edcd85b9e36ceb7": " \\varphi(A) = {\\rm f}(\\lambda(A)) ", "c2222c8434958f64cd22f6d7fdd98b70": "\\mu=\\frac{c}{1-\\varphi}.", "c222485ddfcd4df36d6f926468bbdeef": "g \\ ", "c2225344dd27f9537752b617bbc665f8": "\\sigma_\\alpha(\\beta) = \\beta + n\\alpha", "c222cbe816017cc2562acbec700f3ba5": "Wg(\\sigma,d) = d^{-n-|\\sigma|}\\prod_i(-1)^{|C_i|-1}c_{|C_i|-1} +O(d^{-n-|\\sigma|-2})", "c223b419e7f880c67c7956dc1b9f5a22": "1-Q_1\\left(\\frac{\\nu}{\\sigma },\\frac{x}{\\sigma }\\right)", "c223cfee49425af2088eec91282a16be": "p_k=\\sum_i x_i^k", "c2240a496173d76e06a7208dbddc3591": "\\rho = 1 dx\\wedge dy\\wedge dz", "c224273ee9fe98c6ff3ecc44332c919f": "\\lim_{c\\to b^-} \\int_a^cf(x)\\, \\mathrm{d}x,\\quad\n\\lim_{c\\to a^+} \\int_c^bf(x)\\, \\mathrm{d}x,", "c224515e19260bd89486653c7960fd8a": " \\sdot \\frac {1} {1+j \\omega (C_M+C_i) (R_A//R_i)} \\,\\! ", "c224678287880278a59c77443691df66": "\\textstyle \\rho_c ", "c2246b49a27e54d88988c86729184472": "p''", "c2249cd50df0c11f3e20a6cfa94dd00d": "t_n=n\\Delta t", "c224b243c507d45917a5b114ddd8c5d8": "\nL_\\mathrm{I}=10\\, \\log_{10}\\left(\\frac{I_1}{I_0}\\right)\\ \\mathrm{dB} \\,\n", "c2250877414772bf2ddcc7d33cdd81cc": "(\\mathcal{F}_t)_{t \\ge 0}", "c225b4da5e5d74fa9b4a416818a24919": "\\varprojlim R_i", "c225e551abea59c292229350b545b090": "i_3(t)=K_1i_1(t)+K_2i_2(t)\\,", "c225fba29f1f7017506aeea9746ba97e": "\n=\\sqrt{2\\pi}\\sum_m i^m e^{i m\\theta_k} R^{m+2}\n\\sum_t f_{mt} \\int_0^1 x^m (1-x^2)^t J_m(kxR)x\\operatorname{d}\\!x\n", "c226225c431d4b5a36ec7b43a970736b": "\\operatorname{End}_R(U) \\simeq \\bigoplus_1^r \\operatorname{M}_{m_i} (\\operatorname{End}_R(U_i))", "c2263a2610b9ed8e8252be383ec18651": "\\mathfrak{P}^{60}", "c226567d811362db7109f0bcfe3e647f": "M' = \\mathbf{r}_1\\times \\mathbf{F}_1 + \\mathbf{r}_2\\times \\mathbf{F}_2 + \\cdots = M", "c226606ca88546eaf34813684b16e59c": "r \\in \\mathbb{R}_+.", "c226ae2917fe86ffc9165a513b041288": "c_2P_2(x)+c_1P_1(x)+c_0P_0(x)= {5 \\over 2} (6 \\cos{1} - 4\\sin{1})\\left({3x^2 - 1 \\over 2}\\right) + \\sin{1}(1)", "c226d93d5af8729ada12a8bff5f10d8b": "dp = L(p) dt + p h^T dz", "c227094af5ad867a977a84a65140bf3b": "F(\\varnothing)=\\frac{(T_1-T_0)}{T_0}", "c2273400c71cc9f73dad283644dfb1f6": "(S ; \\wedge, \\vee, /, 0)", "c2276885f773db0b5334ae36a9f98372": "f(x)=\\lfloor x \\rfloor", "c22841d09aa9010488d201e4bb49b33b": "0< p_0 < q_1 \\leq \\infty", "c2285515637e82328935936a32e162ff": "\n\\begin{align}\nU(x,z)\n&=\\hat {f} [\\delta {(x-W/2)}+ \\delta {(x+W/2)}]\\\\\n&= e^{- i \\pi Sx/\\lambda z}+e^{ i \\pi Sx/\\lambda z}\\\\\n&= 2 \\cos \\frac {\\pi S x}{\\lambda z}\n\\end{align}\n", "c22869133034bcac7c0a25fd93575801": "f(tx+(1-t)y)\\geq t f(x)+(1-t)f(y).", "c22879daca341942b91afffb69850680": "\\sum 2|a_n|", "c228a691d84c02b3a088317058da9ad7": " \\max_{x \\in J} |p(x)| \\leq e^{\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J} \\left( \\frac{C \\,\\, \\textrm{mes} J}{\\textrm{mes} E_\\lambda} \\right)^{n-1} \\sup_{x \\in E_\\lambda} |p(x)|~,\n\\leq e^{\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J} \\left( \\frac{C \\,\\, \\textrm{mes} J}{\\textrm{mes} E_\\lambda} \\right)^{n-1}\\lambda ", "c228b177feb96c71054d08f474e17fe7": "C_k^{-1}", "c228e358c494c674845f34f4f25416ea": "\n P = \\begin{bmatrix}\n 0.9 & 0.1 \\\\\n 0.5 & 0.5\n \\end{bmatrix}\n", "c22a01dae6c410496c0c064f6590152b": "k[x,y]/(x^2,xy)", "c22b2ec7e63a8279f6a480c6c1b89e72": "{}_{-\\frac{1}{3}}", "c22b99b27036ebb65f85f473fab6de3a": "1000 g C_{rr}", "c22bd096434b91d29f4018e0f389679d": "-\\textrm{p}K_\\textrm{a} = -\\textrm{pH} + \\log_{10} \\left ( \\frac{[\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "c22be475315c20a40f0d2a41653d71ba": "(y,z)", "c22be6819fa3b60f0d3f4201c12448a8": "\\mathfrak{B}(N_X)", "c22c219e94ae3204f309c896f8716d32": " \\lim_{\\delta \\rightarrow 0} \\varphi_\\delta(E) = \\varphi_0(E) \\in [0, \\infty]", "c22c43a94054ada7c339cfac4ffe9131": "X = C X_1^a X_2^b X_3^c \\cdots X_n^m \\,", "c22ce9b2e178733e11cac016f4cd4108": "Y_\\ell^{m*} (\\theta, \\varphi) = (-1)^m Y_\\ell^{-m} (\\theta, \\varphi),", "c22d012720fff759b1053e43b75e88b3": "\\alpha=\\omega^{\\xi}", "c22d0bbfdb9d8a4571aaa213241f1c9c": "\\left(\\begin{matrix}j \\\\ \\alpha\\end{matrix}\\right)", "c22d596e2d7aa07ba17394ce9195e4d5": "N(r)\\,", "c22da2752cbae98315143c98a10dd5ea": "\\textstyle C[0,\\infty) ", "c22e9f01471fc97a36f0542aabf9ac52": " T_{\\mathrm{min}} \\leq S \\leq T_{\\mathrm{max}}. ", "c22ede59fdc0d7bb590914a720769c92": "|E_n, l, m\\rangle", "c22f46851de935955430aaf4e5061316": "\\frac{\\mathrm{d} \\gamma^{n}}{\\mathrm{d} \\lambda^{n}} (x) = \\frac{1}{\\sqrt{2 \\pi}^{n}} \\exp \\left( - \\frac{1}{2} \\| x \\|_{\\mathbb{R}^{n}}^{2} \\right).", "c22f7ec42ffb3929023bf0cebee50ad4": "R2 = \\frac{V_{Z} - V_{BE}}{I_{R2}}", "c22fbfb8f4763cca948ab6cedcc04236": "M _{DC} ^f =\\frac{PL}{8} =\\frac{10 \\times 10}{8} = + 12.500 \\ kN\\cdot m", "c22fe106cb168ea82730c0b34dc2563b": "h\\in H\\}\\leq 4\\Pi_{H}(2m)e^{-\\frac{\\epsilon^{2}m}{8}}.\\,\\!", "c22fe960a47b6a3278e5b0e6e9cdb840": " MPGe = vehicle\\ efficiency \\times {GGE}", "c23084b19a63623c6be2376c654b0a22": "y^2 = f(x)", "c23085670dab9d483751b0c7af77dfa4": "EL(\\Gamma)=EL(\\Gamma^*)", "c231448919b1466d30cad0d750f56dc3": "\\Phi_{mK}", "c231449b086936e534eaeb224d289f8f": "a_0, a_1, a_2 \\cdots", "c23166d823042d61051b947ec1e1550a": "f_d= f_n\\sqrt{1-\\zeta^2}.\\,", "c231ae2e535778986ca476b7cd29705b": "d_Y(f_0(x),f_0(y))\\leq \\frac\\epsilon3+\\frac\\epsilon3+\\frac\\epsilon3=\\epsilon", "c2325fbb2199bc66d8d23edfcf74a128": "\\tilde{O}(V^{2.376}) ", "c2329b645242aa62221b0af69dbba4b3": "A - 3I = \\begin{bmatrix} 1 & 3 \\\\ -2 & -6 \\end{bmatrix}, \\qquad A + 2I = \\begin{bmatrix} 6 & 3 \\\\ -2 & -1 \\end{bmatrix}.", "c232bce44707c16aa58c54b79444da57": "d(D):=\\lim_{n\\to\\infty} d_n(D)", "c232bd590d1738038659518c520dca96": "\\frac{1-X}{2}", "c232e6e6212127a3aef64c126ab4baf7": "\\text{TM-score}=\\max\\left[ \\frac{1}{L_\\text{target}}\\sum_i^{L_\\text{aligned}}\\frac{1}{1+\\left(\\frac{d_i}{d_0(L_\\text{target})}\\right)^2} \\right]", "c232e94f2bceb740a2b23507941e6bcb": "\\alpha < \\beta \\Rightarrow \\gamma + \\alpha < \\gamma + \\beta", "c233800d65ce2d01623bff729e413ebc": "\\int_{\\mathbf{S}^2}K_g=4\\pi", "c2338d1ef7de9909bf1cd1b157883f6d": " \\frac{v^2}{2} + \\frac{p}{\\rho}= C", "c233e1466c89a04f75f264c6636f7fb9": "c = n/V", "c234319947d60eba226606607e61cd26": "\\Psi_c = \\Phi_{c}^{-1} \\,", "c234768336fcc49bfe3493d5e2708ff4": "F_B", "c234a1b98c1c24a3afb7c22e59a95d90": "I_{tsc} = {V_{svc}\\over{ X_{tsc} }}", "c234d3caa589a49544d45c4c4afa8d59": "x^n P(x) = \\pm P^{*}(x) \\, ", "c2350f6b646fc91ce8c40807d8632764": "\n\\mu_{\\operatorname{eff}}", "c2351172cbb38d9d9096d015003d1da8": "\\omega^g = \\alpha", "c2355d5324f9d6b74e9e49f133a7bb7d": " d(A)=\\lim_{n \\rightarrow \\infty} \\frac{a(n)}{n} ", "c235e45b7c12b1113f5865deb5c14d71": "\\phi(\\mu_0,\\mu_1, \\ldots ,\\mu_n)", "c235f7ddba280c0ce73072d2395366fb": "\\omega_1=(2,0)", "c2361ee4abad8d9e4ac91bbf85e11d5f": "R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}", "c23667a682f3a0ab84225cd4da3435d7": "\nC \\frac{dV}{dt} = I - \\epsilon \\left(\\frac{V^{3}}{3} - V \\right),\n", "c2367280651d6cc82a0e72e9185bf58e": "\\cot\\frac{\\pi}{3}=\\cot 60^\\circ=\\tfrac{1}{\\sqrt3}\\,", "c2370cbc88c464cac78dac89f4e7e566": "\na_1 = \\frac{P(x')}{P(x_t)} \\,\\!\n", "c2373b744b71859980660a186f7d0fee": "* [F , G]^{IJ} = [F , *G]^{IJ}.", "c237b0e139951df02b15b07259746186": "A v - \\lambda v = 0,", "c237b8a8b22435bddc52523486bd99d2": "Y_n(t) ", "c237cad926db3624df24ca85cf7b5074": "\\tilde{\\mu}(t) = \\underset{\\tilde{\\mu}}{\\operatorname{arg\\,min}} \\{F(\\tilde{{s}}(t),\\tilde{{\\mu }})\\} ", "c237f0fd1a4fcee013abbed5f760fdd6": "H^i(X, \\mathcal F), i \\geq 0.", "c237fbb704f6cfc4bbee7cf0fbdacd4c": " U: H \\rightarrow \\int_X^\\oplus H_x d\\mu(x) ", "c2384b7cc800e8ad9307e21c3a55e180": "K_r = \\frac{4}{\\pi} \\sum\\limits_{k=0}^{\\infty} \\left[ \\frac{(-1)^k}{2k+1} \\right]^{r+1}.", "c238981c1f8a9978cdec1166aa68ff8d": "\n\\begin{cases}\nR_1=\\{x\\in \\mathbb{R}^2\\,|\\,\\,d(x,p_1)\\leq d(x,R_j),\\,\\text{for all}\\, j\\neq 1\\}\\\\\n\\vdots\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\vdots\\\\\nR_n=\\{x\\in \\mathbb{R}^2\\,|\\,\\,d(x,p_n)\\leq d(x,R_j),\\,\\text{for all}\\, j\\neq n\\}\n\\end{cases}\n", "c2392ed9a789ecbb09f0cee5da936b85": "\n D_J(p \\parallel q) = \\int (p(x) - q(x))\\big( \\ln p(x) - \\ln q(x) \\big) dx\n ", "c2398f7e840e40695264b7b1349883db": "\\int_0^\\pi f(x)\\sin(x)\\,dx\\le\\pi\\frac{(\\pi a)^n}{n!}", "c239aeab2e0313f6422ea0989ccf2a22": "O_{h}", "c239cf2b6fbeda0fef7cdb853aaa8b32": " ( \\frac{dB_0}{dt} \\cdot \\frac{1}{B_1} \\ge \\gamma B_0 ) \\wedge ( \\frac{dB_0}{dt} \\cdot \\frac{1}{B_1} \\gg \\frac{1}{T_1} ) ", "c239d91f91630b68bd25ac98ca99bad6": "\\eta=", "c239e2ef7a3cc0919b22d294b1c52a49": "\\sigma^2=\\frac{a^2(3 \\pi - 8)}{\\pi}", "c239e34d4a74f2255c705cb37c8450b7": "\\ell > m", "c23a14a4eaadec3d8f9e26c051e65f81": "X_H=-\\partial/\\partial p^i; ", "c23a357859def73174e5dfc2a95397a7": " \\operatorname{value}\\ v\\ h = h\\ v ", "c23a3ae67be7785583eaecfb419a71a8": "{D \\over e^D - 1} = {\\log(\\Delta + 1) \\over \\Delta} = \\sum_{n=0}^\\infty {(-\\Delta)^n \\over n+1}.", "c23a40c671ba390c947d526a23829419": "\\sigma_j = \\langle\\sigma_j\\rangle f_j = \\frac{Q_j}{S_j}f_j.", "c23a5da55de6e4446bb1bd618b937e80": "|_\\text{corr}", "c23a61a27ac2e3107327cad9f22aebce": "\n f(I_1, J_2, J_3) = 0 ~.\n ", "c23ab000f960f7cfd1716e28d3e8ddf5": "q=p/(p-1)", "c23b1caf56b9a7089797a43ea210303b": "I=-G^{32}\\dot{X}_2", "c23b292261e86eefe76c561b469863f8": "\\bar{h_i}=h_i^*", "c23ba9872e7e22314166d388b68f2fa3": "\\kappa = \\frac{|\\gamma' \\times \\gamma''|}{|\\gamma'|^3}", "c23bb358d01e6eaa1c6f066bf39dd4fb": "\\hat n_1:= \\begin{bmatrix}0&-1\\\\1&0\\end{bmatrix} (x_2-x_1) / \\|x_2-x_1\\|", "c23bf8a873a9b2de0b30a6e6a834eb30": "\\varphi: 2^X \\rightarrow [0, \\infty], ", "c23c063dcf7c0830f151a1c5fa99148f": "G_{\\alpha_1\\ldots\\alpha_{2N}}q^{\\alpha_1}_\\tau\\cdots q^{\\alpha_{2N}}_\\tau=1.", "c23c08688eac39bd57c9a56d5b348b7a": "f_n(z)=z+\\frac{z^2}{2^n}\\Rightarrow F_n(z)\\to \\frac{1}{2}\\left( e^{2z}-1 \\right)", "c23c2c8cb7f02fa70683506e0f958674": "\\displaystyle L = \\prod_{d=1}^D \\prod_{w=1}^W P(w|d)^{n(w|d)}", "c23c2fff68bcf4ed05ae347bb952fd8a": "X \\sim \\chi^2_k", "c23cd5c63091d639e8a18f1a66c4ae89": "\\log\\,(1-\\alpha)", "c23cf7f113e25b791cd9a1e2e233bdb8": "\\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau_{D}} \\big| f(X_{t}) \\big| \\, \\mathrm{d} t \\right] < + \\infty.", "c23d1b8ed3754f97583598f59aa2542f": "q^{ab} = \\sum_{i=1}^{3} \\delta_{ij} E_i^a E_j^b = \\sum_{i=1}^{3} E_i^a E_i^b,", "c23d1cf65b1261727a607c5c2eeb1019": "f(\\vec{x})\\text{ is }O(g(\\vec{x}))\\text{ as }\\vec{x}\\to\\infty", "c23d225a20b2c83a405630cbbfe9ede8": "f_i(r_1, \\dots, r_{i-1}, z)", "c23dd1d9d66b818aa36090828787ad5b": "\\sigma={\\varepsilon_r\\varepsilon_0\\phi_0}{K},", "c23dde50ebd71e8075e7854e76d735bd": "u_{it} = \\mu_i + \\nu_{it}.", "c23df48e5b1c853d359038124e319898": " \\left\\{\\omega_j\\right\\} ", "c23dfb243cf8b1e5ca5c29f8ba0bc07d": "N_{s} \\geq \\mathbf{E} \\big[ N_{t} \\big| F_{s} \\big].", "c23e1c4ee11d4c2c5f59e9dc07fe5ac0": " \\int f \\, \\mathrm{d} \\mu \\geq \\lim_k \\int f_k \\, \\mathrm{d} \\mu,", "c23e537664dffd0366de37284cb3d96a": "\\,z", "c23e6e7d8c0ddfff4057c602fc00aa01": "V(\\mathbf{X},z)", "c23e7429d8d53cedf2f294927be8a346": " F_X(a-\\varepsilon) \\leq \\lim_{n\\to\\infty} \\operatorname{Pr}(X_n \\leq a) \\leq F_X(a+\\varepsilon),", "c23ea04f3bf09ecd4efd494b86a7b9ca": " \\bar{n} ", "c23ea1fa0089079e070789e58e1faca7": "\\lim_{x\\rightarrow \\infty}x\\partial_x((x+1)R_n(x))=0", "c23f0af3dee37e1286af6fdf64d7122c": "_{metric} \\delta_{c=k}^2 = 0", "c23f2738a3fc2820f6bebe980dc6b845": "\\int_{-1}^1 f(x)\\, dx = \\int_0^\\pi f(\\cos \\theta) \\sin(\\theta)\\, d\\theta . ", "c23fcde7591fc6728ed1cf5a98d31211": " y_{n+1} \\approx y(t_{n+1}) ", "c23ff410e5ee30c6d0515a0cf40ce721": " u_e = \\frac{dU}{dV} = \\frac{1}{2} \\varepsilon_0 \\left|{\\mathbf{E}}\\right|^2.", "c2405bb541ab788720c9417fd7f3c7e6": "s \\cdot \\varepsilon = s = \\varepsilon \\cdot s", "c240795c3fd57c9ffa6778e71d3e3086": " [x,[x, \\ldots, [x,[x,y]],\\ldots]] = 0", "c240c6467efb744649ff025da442b71d": "\\tan(1) = [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, \\dots]\\,\\!.", "c240f0ebefcf92ca6083d04da442df45": "x = a \\cos(t)\\,", "c240fd95b173ab7c1e533d1fc868bcc2": "f_i: X \\to Y_i", "c2410f70fe3dd49528116ec7eb7e8364": " V = \\oplus_{i=1}^N V_i ", "c24115c487c72cb87f07ed78f50b0e27": " R(3,k) ", "c2411bfd7f470ea4dde96106ad867ef1": "\\quad n=2\\,p+1", "c24123102d25f735eb7e88b2e426e1d8": "hx = x", "c2417e2ad7fb469ed736497e850bd415": "\\scriptstyle i~=~ \\sqrt{-1}", "c242269c737ab09f714b7145f568e117": "\\nabla f = \\frac{\\partial f}{\\partial \\mathbf{x}} = \n\\begin{bmatrix}\n\\frac{\\partial f}{\\partial x_1} &\n\\frac{\\partial f}{\\partial x_2} &\n\\frac{\\partial f}{\\partial x_3} \\\\\n\\end{bmatrix}.\n", "c242ba2e9ea2a27ab1a4fbb5a5ac4ab3": "f(\\xi,t')=-\\frac{1}{\\tau} \\int \\exp \\left[-i\\left(\\frac{\\xi}{\\omega}+\\omega t'\\right)\\right] \\frac{d\\omega}{\\omega^2}", "c242d074c86b7db90238495a68812316": "e^I_\\gamma e^J_\\delta", "c2430856cfe91cad90c94f7bd6cfe32b": "\\tfrac{1}{T}", "c243886a288804343eee2af0ad8dcebc": "\\hat x", "c243a190c1bf471360e97ae148bce68f": "X = A+B+AB+E \\,", "c244228296285acab30e23641c608803": "V_{s}", "c24473fa38a059a16513b086b5664b5a": "V(a_1, \\ldots, a_n) = \\sup\\limits_x f(x) ", "c24474a26cb5d33f2d9931f3e14bf9f6": "*\\colon[0,1]^2\\to[0,1]", "c24482ffd4c9c39aefc3550539a0abd3": "=\\frac{1}{3}\\cdot\\left(\\frac{1}{2^0}+\\frac{1}{2^{1}}+\\frac{1}{2^{2}}\\right)", "c244aa984c7689500b1ecf657937c126": "3: \\quad tmp \\quad = \\quad term \\quad / \\quad (2*n+1)", "c244de33a1e4a2dd452f6d84b2dca939": "f(x)>g(x)", "c244f04aac9cb7575929fe8f9185d9d9": "B(1,2)=\\langle a,b| b^{-1}ab=a^2\\rangle", "c2450bc479fb2579bf6d0c5f6cd795f2": "S_z", "c24522ca19a0400b051dff8f5544bd69": "\\displaystyle{\\mathbf{v}_s(t)=\\mathbf{v}(t)+s\\mathbf{n}(t),}", "c2454fd476946fbccf65d466685ebce9": "\\varphi=\\arccos\\left(-2I_{c1}/I_{c2}\\right)", "c24558c39f79b786e5e680c9b2e8a377": "\\displaystyle{EP=P,\\,\\,\\,PE=E.}", "c24599cad21db7686362a915142e84be": "\\left(\\frac{1}{2}\\right)^{(2 \\times 2)} = \\frac{1}{16}", "c245e3fd7f04684ccf5b329d15f77270": "S(q,E)", "c246618b383bc17a11d3b479d727565b": "h_i=\\sqrt{g_{ii}}.", "c24667bf7607b21edc57c8992947439c": "dQ = dU-dW.\\,", "c2467049fdf32c55620177d19503a427": " f(x)= \\sum_{I\\subseteq [n]}\\hat{f}(I)\\prod_{i\\in I}x_i. ", "c2469ac165cd0242852cb031362fc0a0": "\\dot{\\theta} = 1-\\cos\\theta + (1+\\cos\\theta) \\Delta I", "c246b78879ba61f5439a8a6cd0def711": "f:\\Z_p\\rightarrow \\Bbb Q_p", "c246be7404e0a0c0a2ed8e0c2987c4a2": "\\;p(1) = p(0) r - A = P r - A", "c246c09e3da06d927fb0eab638de9ed0": "b \\vee c", "c2472b046ffdabb6f9075636c3dbd038": "{\\color{Blue}~6.9}", "c247eea8c6c41e6d27fc6afd2cc7242f": " \\int_{a}^{b} f(x)\\,dx.", "c24828b17795bb3ba8d22889a7cb66e7": "I(k+r)A_k+\\sum_{j=0}^{k-1} {(j+r)p^{(k-j)}(0)+q^{(k-j)}(0) \\over (k-j)!} A_j=0", "c2484c378c4d39ecc6fabf94ec6eaeaf": "\nV(a,z)=\\frac{1}{2^\\xi\\sqrt{\\pi}\\Gamma[1/2-a]}\n\\left[\n\\sin(\\xi\\pi)\\Gamma(1/2-\\xi)\\,y_1(a,z)\n+\\sqrt{2}\\cos(\\xi\\pi)\\Gamma(1-\\xi)\\,y_2(a,z)\n\\right]\n", "c248531b57057703346590a6e9563390": "S \\Rightarrow_{f} AA \\Rightarrow_{g} aAaA \\Rightarrow_{g} aaAaaA \\Rightarrow_{h} aabAaabA \\Rightarrow_{k} aabaab", "c248588a81414263da5f8518d458c4ec": "\\scriptstyle (-1)^{1/2} \\;=\\; \\sqrt{-1}", "c2487cb52f6989e9cd0c05a15fcdef96": "t \\mapsto x", "c248b325650b057baff705ec13a7d6b6": "\\left | \\bar{D}_n \\right | =\\left | n- \\bar{\\eta}(E_n) \\right | ", "c248bb1ee5d28ede8ae971c8e5512062": "\n\\int \\int \\psi(x,y) \\phi(x)\\phi(y)\\,dx\\,dy\n\\,", "c248efe5f1fd79745f4a148044537331": "\\mathrm{Re}(e^{-i\\omega t})", "c2499cb41670b172b46e2a82b5013f1e": "E: X\\to\nX", "c249b4909d8b09e7983854455b1867ea": "a=-0.5, p=z^3-1", "c249cab8e5dfb128e6bcf2169cacadb2": "H_{1,I}^{\\text{RWA}}=-\\hbar\\Omega e^{-i\\Delta t}|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\Omega^*e^{i\\Delta t}|\\text{g}\\rangle\\langle\\text{e}|.", "c249f0ceb007ae07a9e9fc3f1e511ef9": "\\lim_{x \\to c} \\frac{f(x)}{g(x)} = \\lim_{x \\to c} \\frac{f'(x)}{g'(x)}", "c24a26a10cce3d7048f92252ce785e56": "\\Lambda\\}", "c24acfa927df0b139a5457a67f2924ee": "{\\displaystyle}P_{1}=(x_{1},y_{1})=(\\sin{\\alpha_{1}},\\cos{\\alpha_{1}})", "c24bea5a214d70b83b9df3048b3865dd": "\n \\int_0^{2\\pi} d\\alpha \\int_0^\\pi \\sin \\beta d\\beta \\int_0^{2\\pi} d\\gamma \\,\\,\n D^{j'}_{m'k'}(\\alpha,\\beta,\\gamma)^\\ast D^j_{mk}(\\alpha,\\beta,\\gamma) =\n \\frac{8\\pi^2}{2j+1} \\delta_{m'm}\\delta_{k'k}\\delta_{j'j}.\n", "c24bf3e38dee2e0c03799aeeabf79eb0": "n\\in{\\mathbb Z}", "c24bff7670f181429235a906bb75c15b": "\\prod_{p\\leq x} \\frac{N_p}{p} \\approx C\\log (x)^r \\mbox{ as } x \\rightarrow \\infty ", "c24c297b92e297fbe6a14aa1ea4d3199": "0=SdT-Vdp+\\sum_iN_id\\mu_i\\,", "c24c58451d0c10c1bcb8dc13eb9b06cb": "\\phi_{k}(a_i)=\\frac{1}{n-1}\\displaystyle\\sum_{a_j\n\\in\nA}\\{P_{k}(a_i,a_j)-P_{k}(a_j,a_i)\\}", "c24c92d9948de698711b18d41bf547cb": " ACH_{natural} = {ACH_{at 50 Pascal}\\over20}\\,\\!", "c24d5a79ace635b9dd2b025af78f9d0b": "dU = - p [\\text{Differential}] + T [\\text{Differential}]", "c24d6b9ab489cac038a3765a252bc207": "[\\cdot, \\cdot] \\circ (\\mathrm{id} + \\tau_{A,A}) = 0", "c24db4804a3eb66d62911105f4d543f3": " dS_t = \\mu S_t\\,dt + \\sigma S_t\\,dW_t ", "c24dcab915b5c2a8660c5eae84aa5f1e": " E_i E_{i\\pm1} E_i=E_i \\text{ and } G_i G_{i\\pm1} E_i = E_{i\\pm1} E_i,", "c24dd940cdaf8b2ca504895c10445ac6": "[n,k,d]_q", "c24e203908bc77b3475c91601325e792": "\\mathbf{q} ", "c24e2e995299981a844262db77572aaa": " g(x)=\\min(\\frac{x}{\\alpha},1)", "c24ebbc3d7507b1cbce287ef00ac4a57": "R_r^'\\gg{R_{TE}}", "c24eda4d43f5a97c5b366646427c794f": "t\\chi=\\sum_{s\\in W} t(\\lambda_s)\\,\\,a_s=\\sum_s M_{t,s}a_s.", "c24f17b8e3901b526db7dfb58b8631ee": " \\langle f,h_k\\rangle=a_k", "c24f1fa09876f35aecb8f81ae8648124": "\\operatorname{Der}(\\mathfrak g)", "c24f5c642cd06dd23ed38949d5069a90": "\nt^2=n(\\overline{\\mathbf x}-\\boldsymbol{\\mu})'{\\mathbf W}^{-1}(\\overline{\\mathbf x}-\\boldsymbol{\\mathbf\\mu})\n", "c24f6a0c5945ca9637ea37795bbad13d": "\\scriptstyle X_{\\text{eq}}", "c24f6e0c4a5f4994629cb857c1e7d7f5": "\\lim_{n \\rightarrow \\infty} a_n = 0", "c24fa8cf59385dcad0f36270b2b77d67": "W_D(x,s_p)", "c24fe467bf175f81bb9cd08eee73794e": "k_0=1.0004", "c24febe43f59d0117d7b8050055acaf6": "y_L+F_2\\cdot y_F+P_2\\cdot y_P\\geq S_2", "c2505526810ed96ef5479f74d9ad5a4f": "d_1(z) = a + (1-a)z.", "c2507b64709560b2a6eb2d0786bffd97": "\n\\pi = \\cfrac{4}{1 + \\cfrac{1^2}{3 + \\cfrac{2^2}{5 + \\cfrac{3^2}{7 + \\ddots}}}}\\!\n", "c250bcae2dc789f305ec35f3147f7a09": "\\tilde{f}_i u = \\sum_{n=0}^\\infty f_i^{(n+1)} u_n = \\sum_{n=1}^\\infty e_i^{(n-1)}. v_n", "c250cf1c43c19e88e9fceb6937d0735f": "tr(1/d) = 1 \\hbox { and }d \\not\\in GF(4)\n\\hbox{ if } h \\equiv 2 (\\bmod 4)", "c250e9f319825571f1e71b438d6cca27": "\\gamma(\\tau) = \\alpha_1 \\gamma_1(\\tau) + \\alpha_2 \\gamma_2(\\tau)", "c25105c9c8bfcd1a317c85e2dc540c41": " \n\\left\\{\\begin{array}{c}\n{dm/dt}=k_d C_m \\rho - \\lambda_r m_r(t) \\\\\n{dm_r/dt}=k_d C_m \\rho - \\lambda_r m_r(t) - \\lambda_c \\cdot m_r(t)\n\\end{array} \\right.\n", "c25122afce7f1caef5e35a7a4a7203c8": "\\left(\\frac{\\partial p}{\\partial T}\\right)_{V}= -\\frac{\\left(\\frac{\\partial V}{\\partial T}\\right)_{p}}{\\left(\\frac{\\partial V}{\\partial p}\\right)_{T}}= \\frac{\\alpha}{\\beta_{T}}\\,\\,\\text{ (3)}\\,", "c25131aaf658efb8c1d74863adf0dc47": "H_20", "c2513cedb6de0f9df1245d72a2720d18": "\\log{n \\choose k} \\approx (n+0.5) \\ln\\frac{n+0.5}{n-k+0.5} + k \\ln \\frac{n-k+0.5}{k} - 0.5 \\ln(2 \\pi k)", "c251839249fe6fe9e3363f6f8791851a": " \\mathrm{Sp}( v_1 ,\\ldots, v_n) := \\{ a_1 v_1 + \\cdots + a_n v_n : a_1 ,\\ldots, a_n \\subseteq K \\}. \\,", "c251d8d34edbf57d67b855b5cf082c8b": "m\\cdot Q_{P}(h)\\,\\!", "c251e664b77d598997774974acd69b1a": " y_g + \\frac{q_\\mathrm{trans}^2}{2gy_g^2}= E_{u/p} ", "c2522a298bbc1320870cd7efeacd8124": "f : P^\\ast \\rightarrow A", "c252420235ed70c96e0a404fdaa33391": "z=ax^2+bx+c", "c2524d8ab87f634a6a5702a20083f9d7": "x_{p\\times n}", "c2524f04c9df864c0cdc68317222f632": "\\kappa_n", "c25266148c373e0c407e9615fb36eb31": "\\zeta(s_n)\\,", "c2529eed21b8a0688b5332ad0478bc8d": "k \\leq r-2", "c2535683bcbddc403b99c2e5488e28f2": "\\kappa = \\kappa_0 + \\kappa_1 \\left(1+T_r^{0.5}\\right) \\left(0.7-T_r\\right)", "c253860a0d21c1d2aeec882bf0be4f1b": "\\Pi=C - wL \\,", "c253c68add720c520b83b7152fdfdc03": "V + \\Delta V = L^3+L^3\\alpha_V\\Delta T=L^3 + 3L^3 \\alpha_L \\Delta T + 3L^3\\alpha_L^2 \\Delta T^2 + L^3\\alpha_L^3 \\Delta T^3 \\approx L^3 + 3L^3 \\alpha_L \\Delta T", "c253ecc51fcc313e3f734ace3efad078": "0(1/0!)\\pi^0 ", "c254671f964f808632a665d0d75558e8": "Q'_0 = A_{in} \\epsilon _{s.s.} \\sigma \\left ( T_b'^4 - T_{surr}^4 \\right )", "c2546adf57f5e935e020ef024c12b2fb": "H = \\sqrt{\\frac{8\\pi}{3}G \\rho}", "c254a1faab2f0da01ba6f89e6cdab08c": "q=\\sigma^{2}_y/[\\sigma^{2}_x+\\sigma^{2}_y]", "c254e2e381fda66f0288baed7b75f83d": "\\varepsilon: A \\rightarrow \\mathbb{C}", "c254ee4d7c68dcb266ac0ddf0c2ced94": " D[g] = [x, S_8, A_8]::[o, S_6, A_7]::[y, S_6, A_6]::K_1 ", "c2551a9dff88dd9cfb8b22a2855b2b4b": "x\\le y\\Rightarrow z.", "c2552004a3b748b8555d4edbb6a0d57a": "P_l(x) = {1 \\over 2^l l!} {d^l \\over dx^l } (x^2 -1)^l, \\text{ for } \\left|x\\right| \\le 1 ", "c255296de5448b095354c558e7218f89": "u_{xy} = {\\part^2 u \\over \\partial y\\, \\partial x} = {\\partial \\over \\partial y } \\left({\\partial u \\over \\partial x}\\right). ", "c2554ef144b1488e5d5eaec8881604f3": "\nU_L = \\begin{bmatrix} \\rho_L \\\\ u_L \\end{bmatrix} = \\alpha_1 \\begin{bmatrix} \\rho_0 \\\\ -a\\end{bmatrix} + \\alpha_2 \\begin{bmatrix} \\rho_0 \\\\ a \\end{bmatrix}.\n", "c2554f1eef6ef227c895faa409852e49": "Nr", "c25590ae8d75e095285803560f995f1b": "\\textstyle d\\theta_j\\left(\\frac{\\partial}{\\partial \\theta_k}\\right) = \\delta_{jk}", "c255a2a2db582890a3721a5ac88f88b9": "\\textbf{x}(t)", "c255ae4161fd5791f99992aba2029b19": "{\\,\\!256^{256^{256^{256^{256^{257}}}}}}", "c255d6a4599547437c729c071983d4b3": "M_c(x, y; t, s) = \\operatorname{det}(\\mu(x, y; t, s)) - \\kappa \\, \\operatorname{trace}^2(\\mu(x, y; t, s))", "c2566e102351e3e43122e470e965083e": "G \\approx *m", "c256def1159fd814defa0967dd4a0116": "\\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left(r^2\\, \\frac{\\partial \\phi}{\\partial r}\\right) = 4\\pi G \\rho(r)", "c256e8ec831aea9c358fb9afd2d80b77": "C = (c_{ij}) = A \\star B", "c25705b89c389fd8c51d32b9c1edf521": "V_o^2 / Z_o = \\eta V_i^2 / Z_i ", "c2573cb6f7a69a885834f510514b1415": "\\|Mf\\|_p^p \\leq p \\int_0^\\infty t^{p-1} \\left ({2C \\over t} \\int_{|f| > \\frac{t}{2}} |f|dx \\right ) dt = 2C p \\int_0^\\infty \\int_{|f| > \\frac{t}{2}} t^{p-2} |f| dx dt = C_p \\|f\\|_p^p", "c2577cbfc8dc21393e76f55f292a64a9": "\\mathcal{O}_{X,x}^\\text{sh}", "c25783248c78d222bad5235f7699e89b": "\\left(\\frac{dS_{\\theta}}{d\\theta}\\right)^{2} + \\left(\\alpha E_{0}\\sin\\theta - \\frac{L}{\\sin\\theta}\\right)^{2} + \\alpha^{2}m^{2}\\cos^{2}\\theta = K", "c2579e9c77690e729027ae6191930007": "P \\to (Q \\leftrightarrow R) \\leftrightarrow ((P \\to Q) \\leftrightarrow (P \\to R))", "c257bfc24307ddd0f3150af9ef826688": "max_{deg(G)}", "c257ca87df7310553deca3f04918a179": " {\\rm p} K = -\\log_{10} K \\,\\!", "c257fa5066c9c420b5853fe3254973b8": " n-m ", "c2580d6c424c6527e14941a4245bb48e": " \\mathbf{\\hat T}(\\lambda) ", "c2583dc3223765e707f3b63615368c28": "\\{ \\phi , \\lnot \\phi \\} \\vdash \\psi", "c2586fbac27a07fcefd0092acf698452": "[z^q] G(z)", "c258974922126c399c50afac94142071": "\\sup_N \\Bigl| \\sum_{n=0}^N b_n \\Bigr| < \\infty, \\ \\ \\sum |\\lambda_{n+1} - \\lambda_n| < \\infty\\ \\text{and} \\ \\lambda_n B_n \\ \\text{converges,}", "c258c6b93fc940ecd1fc307879472cd4": "\\mathbf{u}(\\mathbf{x}(t))", "c258ec1b458a1d150dc135cdf9922781": "\\mathfrak{tri}(A)", "c258ffb0b043196dc8d7bc8edf294a9a": "{_uM_1}=\\frac{q^2}{gy_1}+ \\frac{y_1^2}{2} =\\frac{q^2}{gy_2}+ \\frac{y_2^2}{2}={_uM_2}", "c259050ef621feecbc7a475d7d1d873a": "P_3=2P_1", "c2591da4856c49a23214abd74f2a9066": "\\sum_{t=1}^{t=N} 11-r_t", "c259255c7aa4e4770cfcc202aad7d71b": "y_{L}\\left(t\\right)=\\int_{0}^{\\tau}u_{L}\\left(\\sigma\\right)w\\left(t-\\sigma\\right)d\\sigma", "c25968894095aad24c3f5ff796035f19": "\\phi^*L = L^{\\otimes d}", "c25a16e11e90aec836e70e1184191a78": "R^i f_* \\mathcal{F}", "c25a4866640065b9620f8db0ced93235": " \\mathbf{r'}(t), \\mathbf{r''}(t) ", "c25ada817958cc327cbe7d5f6514592d": "\\frac{(-y)\\bmod x}{y\\lceil y/x\\rceil} = \\frac2{y(y+2)/3}", "c25b0d434f9a1278b3b09b935d85fa04": " \\mathbf{X} = [x_1,x_2,\\ldots,x_n] \\in \\Omega \\subset \\mathbf {R^D}", "c25b0eb8d878c576fc84f18725a7cd53": "[X,Y] \\equiv XY-YX", "c25b55ad6182c363d47ec4500da6f0c8": "\n\\begin{array}{l}\n {\\rm Var}[z]\\,\\, \\equiv \\,{\\rm E}\\left[ {\\left( {z\\,\\, - \\,\\,{\\rm E}[z]} \\right)^2 } \\right]\\,\\,\\, \\approx \\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)^2 {\\rm E}\\left[ {\\left( {x_1 - \\bar x_1 } \\right)^2 } \\right]\\,\\, + \\\\\n \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right)^2 {\\rm E}\\left[ {\\left( {x_2 - \\bar x_2 } \\right)^2 } \\right]\\,\\,\\,\\,\\, + \\,\\,\\,\\,2\\left( {\\frac{{\\partial z}}{{\\partial x_1 }}} \\right)\\left( {\\frac{{\\partial z}}{{\\partial x_2 }}} \\right){\\rm E}\\left[ {\\left( {x_1 - \\bar x_1 } \\right)\\left( {x_2 - \\bar x_2 } \\right)} \\right] \\\\\n \\end{array}", "c25b6358fc698e3494ce64f9cc3f8904": "\\nleftarrow", "c25b8a617f7bb8bb0fc30b1d2911ab1e": "p, q \\in M", "c25bda227399c63f9de01a997fb31766": "2^{k} < n < 2^{k+1}", "c25c07f5671d6c2b3e75f79bcc5192f3": "\\scriptstyle \\log_{10} P_{mmHg} = 6.95464 - \\frac {1344.8} {T+219.482}", "c25c716ada128320453d1b4cb0458581": "x \\times x < 2\\,", "c25c82ee1ee8e960d59ab2fbc8575f42": "\\rho \\frac{\\partial \\bar{u_i}}{\\partial t} \n+ \\rho \\bar{u_j} \\frac{\\partial \\bar{u_i} }{\\partial x_j}\n= \\rho \\bar{f_i}\n+ \\frac{\\partial}{\\partial x_j} \n\\left[ - \\bar{p}\\delta_{ij} \n+ 2\\mu \\bar{S_{ij}}\n- \\rho \\overline{u_i^\\prime u_j^\\prime} \\right ]\n", "c25c842028e796a0bd3b8e8e3c644b80": "\n\\tau=\\frac{Q}{2\\pi f}.\n", "c25c900cfd06ec90e6f324391b787e9c": "\\phi:\\bar{\\mathcal{S}}\\to \\mathbb{R}", "c25c975c7ee9911df536559a0ccb04e8": "x = a \\ln(\\sec \\varphi + \\tan \\varphi) + \\alpha,\\,", "c25c9fdd244f72326d7a4ba4007cede6": " \\sum_{j\\in J'} \\lambda_d(5B_{j}) = 5^d \\sum_{j\\in J'} \\lambda_d(B_{j})", "c25ca147ede37f878032a7f2eb837230": "\\dot = ", "c25d35752111cff81c63a96504135bbe": "\\ddot{\\theta}_1 \\left(\\hat{J_0} + \\hat{J_2}\\sin^2(\\theta_2) \\right) \n+ \\ddot{\\theta}_2 m_2 L_1 l_2 \\cos(\\theta_2) \n- m_2 L_1 l_2 \\sin(\\theta_2) \\dot{\\theta}_2^2 \n+\\dot{\\theta}_1 \\dot{\\theta}_2 \\sin(2 \\theta_2)\\hat{J_2} \n+b_1 \\dot{\\theta}_1\n= \\tau_1", "c25d4740514e018dad91e329616d867c": "I_{L_{Max}}=\\frac{V_i D T}{L}", "c25d5314f40b5ae17be719e2deddcae5": "\\left( \\sum_{k=1}^n |x_k + y_k|^p \\right)^{1/p} \\le \\left( \\sum_{k=1}^n |x_k|^p \\right)^{1/p} + \\left( \\sum_{k=1}^n |y_k|^p \\right)^{1/p}", "c25d682f1a479c9628877b658301cb29": "\\hat{f}(\\xi)", "c25e1a8ee4bc1ccc4cd429e10e14d6ec": "\\|A x\\|_2 = \\|x\\|_2 \\,", "c25e4c52aaad2fa87b2414a9ccd528d2": "N = \\{1, \\ldots, n\\}", "c25eb7c42373919b4bd9bdd53e76a7cc": "B^n x", "c25ee32551c0ab4323d63b559dbb643e": "G: \\text{Mod}_{R[S^{-1}]} \\to C", "c25f8afcadf230beed847e9a2cebc7e9": "\\Phi(r,\\theta,\\phi)", "c25fd8f681c60ac98c10d56968a95e21": "\\sigma^2 = \\displaystyle\\frac {\\sum_{i=1}^N x_i^2 - (\\sum_{i=1}^N x_i)^2/N}{N}. \\!", "c26053df39f3f68bbf511e68f873426f": "e^{i\\beta _{j}}", "c260980edd49c2c6d161f32ad0d9968d": "\n M_{ijklmn} = C_{ijklmn} + C_{ijln}\\delta_{km} + C_{jnkl}\\delta_{im} + C_{jlmn}\\delta_{ik},\n ", "c260ba9d3a38fde3d588a8ba4f868eda": "\\{z \\mid \\pi_1(z) R \\pi_2(z)\\}", "c260bf65fcbba882c0c2a17a34590420": " \\mathbf{C} = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\frac{1}{f} & 0 \\end{pmatrix} \\sim \\begin{pmatrix} f & 0 & 0 & 0 \\\\ 0 & f & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} ", "c260edee20a3236245713a0f0a4df8dc": " p \\mapsto p(T) ", "c2616985b7df273eadcfcd8f917c3248": "n!=\\int_{0}^{1}(-\\ln s)^{n}\\,{\\rm d}s\\,,", "c2616bd79e793d8a3434d581708c21f1": " \\phi_i \\, ", "c2618f22140db6b432a2394e2491cb54": "N \\triangleleft G\\,\\,\\Leftrightarrow\\,\\forall\\,n\\in{N},\\forall\\,g\\in{G},\\, gng^{-1}\\in{N}.", "c2619d1e4421c775f8e54663ba227a2d": "\\,\\! 2^{2^{2^2}} \\ne \\left[{\\left(2^2\\right)}^2\\right]^2 = 2^{2 \\cdot 2 \\cdot2} = 256", "c261bfaf2cf09611416ff0c216882c9e": "\\frac{m+n+4}{2d}-\\frac{1}{2}", "c261e71721f2ab8f56a35f8cf87587af": "C= \\frac{Q}{V}", "c26261214dced765a7790e3f5d45c372": "(CH_2(COOH)_2)", "c2626c0c42138a9488bec2182698e3c2": "T:X \\to X", "c26276aa525c9e7bc424dbd4a927cc86": "\\left(X^{\\rm T} W X \\right)\\hat{\\boldsymbol{\\beta}} = X^{\\rm T} W \\mathbf y.", "c2628924d5f10546bdf33f2110fabc7d": "y(x,t) = A\\sin(kx + \\omega t + \\phi ) + D\\,", "c26306febca98f4281c0f221a167400d": "d\\mu = \\frac{1}{2\\pi i}\\phi \\, dz\\wedge d\\bar{z}", "c26354ed8c39dceb13360312d62a4730": "\\int e^x \\left( \\frac{1}{\\ln x}- \\frac{1}{x\\ln^2 x} \\right)\\;dx = \\frac{e^x}{\\ln x} ", "c26394881373ebf480a8268c8097e92b": "y_i^2 = 1", "c263b2a57813997ce5db8e34d8a06305": "\\frac{\\mathrm{d} u}{\\mathrm{d} x}(x) = 0,", "c2642519ece73cb381e8a3de1ef50b3f": "\\mathcal{PC}", "c264b3a7f0b8db970dac0d5dbb0a64cb": "\\frac{X}{Y} \\sim \\mathrm{Pareto}(1,n)", "c264bdb22a754961ef6021b6c1914161": "X_{1}", "c264cc8d5930dc6d7def11d7d1271cf0": "\\ (n_x,n_y,n_z)=(n\\cos \\theta \\cos \\phi,n\\cos \\theta \\sin \\phi,n\\sin \\theta )", "c264cca20bc028a2c795b4d55bd6f056": "U_{\\gamma}\\,", "c2656032b8820942488eb05076d39a6c": "\\alpha = \\frac{1}{4}\\,Z_0\\,G_0", "c265af1c352b51f9d74975f91e6702b3": "P : C^\\infty(X) \\to C^\\infty(X)", "c265bcc22bed79f2839c9529214c3674": "SU(M) \\times SU(N)", "c265bd5f1c591a1f9407ba74dbf78e6d": " \\lambda_0 ", "c266254d625d8e055008304f08706ca4": "R_{\\mu }\\left( t \\right)", "c2663a435372a1ef37ae0b3d2d53a0e1": "P_n(z) P_n(1/z) + Q_n(z) Q_n(1/z) = 2^{n+1} ; \\, ", "c26651a68e5f6769b03f62f14297109a": "(k_{-1}+k_2)/k_1", "c266802649458b3408f183d00feb5398": "\\alpha_G = \\frac{G m_e^2}{\\hbar c} = \\left( \\frac{m_e}{m_P} \\right)^2 \\approx 1.7518 \\times 10^{-45} ", "c266c05a68564eeacdb0a67ae8423dfc": "\\tilde d_1 = d_1\\,", "c266cedb000dbf72634b2b30dbfba6fc": "\\mathrm{inv}(f)f = \\mathrm{id}_x", "c26723c07070e9e69a622afc350ab939": "u^\\mu = (\\dot{t}, 0, 0, \\dot{\\phi})", "c267a124d2e4921a8a3684a19bbc9a9b": "x\\mapsto K(x)(B)", "c267bb1622dccc78c501f672b107ce86": "\\gamma_2=\\frac{-6\\Gamma_1^4+12\\Gamma_1^2\\Gamma_2-3\\Gamma_2^2\n-4\\Gamma_1\\Gamma_3+\\Gamma_4}{[\\Gamma_2-\\Gamma_1^2]^2}", "c267d17b856c1d069e4a04db35612a5a": "\\beta_{0} = 0", "c2681e99df3c60b78317eb9a0f55e3e9": " \\pi / 2 ", "c268d773da33bc852f055714a65eab5c": " (\\Omega + \\omega_r) ", "c26966795a22cabbb5616de2b49a8caa": "\\mathbb{E}_m[\\mathbb{E}_E[\\Pr_{e \\in BSC_p}[D(E(m) + e)] \\neq m]] \\leq 2^{-\\delta n}", "c26974591266fde55e9da6774a5c2ae7": "\\dot{m}_{air} V", "c269831971eee08a97469fb93611d015": "\\Delta L", "c269e8b5ebabf28a4c1c6703773d4481": "\\lambda = \\mu = 2", "c269ed847fd06aea522dc49a9bf31afd": "h(v_1,v_3)k(v_2,v_4)+h(v_2,v_4)k(v_1,v_3)\\,", "c26a88d939c78336d9538ea8475214a0": "c_n = O(\\beta^n)", "c26a9a59754e8ce50d16b9f804e23c22": "\\textstyle (1, 0)", "c26aaf2e4b3e37f267104fb35ec703e2": "K=-\\lambda^2,", "c26aca8d0152e4a11f296c5d5d37b737": "x_2 \\neq x_1", "c26ae403ece8b58c4d3f760a95f0971f": "\\mathcal{O}(d^2\\log^2{n}) ", "c26b26774eeccb92def1c5a1a08c2d30": "\\Omega = \\Omega_0 \\cup \\Omega_1 \\cup \\ldots \\cup \\Omega_j \\cup \\ldots \\cup \\Omega_m.", "c26b36bf1bbc911e394238cc37c43d3a": "P(N \\leq 20n) = \\frac{19}{20}", "c26ba2ab510869e4f3978af23d04307c": "\\scriptstyle \\mathfrak{a}_1, \\cdots, \\mathfrak{a}_n", "c26beca57bf6c66688eb864ae2d7e95a": " f_X(x) = \\frac{d\\mu_X}{d\\lambda}(x) = \\frac{1}{(2\\pi)^n} \\int_{\\mathbf{R}^n} e^{-i(t\\cdot x)}\\varphi_X(t)\\lambda(dt).", "c26bf08a720f2c13120573db42681013": "(\\mathbf B^T \\mathbf B)^{-1}", "c26bfd451925765624bc8eb8db038e0d": "1 = A_0\\leq A_1\\leq \\cdots \\leq A_\\omega \\leq A_{\\omega+1} = G", "c26c3b5c0575f5762d3d86f595b54a7c": "Z = R + jX\\,", "c26c3bfeddadeb92f8e926189907b072": "\n\\frac{\\partial}{\\partial t}(\\nabla \\times(\\nabla \\times \\vec \\psi))\n + \\nabla \\times\\left((\\nabla \\times \\vec \\psi) \\cdot \\nabla (\\nabla \\times \\vec \\psi)\\right)\n = \\nu \\nabla \\times (\\nabla^2 (\\nabla \\times \\vec \\psi))", "c26c96266b42eb6b02d144e9ce41ff2a": "V_{rsi}^{-1}", "c26ca89c8fee932aea37cd69624b1bee": "f(t).", "c26cb5319e6b09e12d6112127438a1db": "{q = ({{{1 \\over f_o}{\\nabla^2 \\Phi}}+{f}+{{\\partial \\over \\partial p}({{f_o \\over \\sigma}{\\partial \\Phi \\over \\partial p}})}})}", "c26cf7bf13c8bb617f84871fb0c7811a": "\\sum_{k=0}^n{n\\choose k}B_{n-k}(y) x^k = \\sum_{k=0}^n{n\\choose k}L\\left((2y)^{n-k}\\right) x^k = L\\left(\\sum_{k=0}^n{n\\choose k}(2y)^{n-k} x^k\\right) = L\\left((2y+x)^n\\right) = B_n(x+y).", "c26d3182101fcbb65a7989e133d340b6": "-R_0^0=\\frac{\\ddot a}{a}+\\frac{\\ddot b}{b}+\\frac{\\ddot c}{c}=0", "c26d3a9fbe55c1c7a299b6d3356d702c": "\\lim_{n\\to\\infty}\\frac{\\ln|a_n|}n=-\\infty.", "c26d7bc8aa7b9ce70e2026a5c1749212": "\\mathcal Q\\setminus {\\mathcal R}", "c26dc8a0a185abd81d1c919b85594a91": "\\mu(A \\times B) = \\int_A \\mu\\left(B|x_1\\right) \\, \\mu\\left( \\pi_1^{-1}(\\mathrm{d} x_{1})\\right).", "c26dc8af51097a82be3310111b3673df": " \\frac{K \\cdot t}{V} = -ln (1-URR)", "c26de16224bbc1af9190c0522e698fa9": " \\sum_i C^J_{v_i} = 1 ", "c26e071b57ddece0b9834fc1f4de65cc": "\\ T", "c26e24e14a42ff5a084b27ff52d51401": "\\overline{G}", "c26e329879f86ccfa2f9724f07792d4a": " d(uv)= dt'\\;\\;\\;du = dt+dt'\\,,", "c26e5e674eddc7dcf4993cda58f1e226": "\\{n\\}", "c26eabe776b974f0f8be3461a952f881": "\\operatorname {sn}\\; u = \\sin \\phi\\,", "c26f448f28d3b8877497d8d0295a6138": "p_i = a_i ^* a_i,", "c26f4847289542e0092af2911744f1e6": " \\limsup_{N \\to \\infty} S_N f(x_N) \\leq f(x_0^+) + a\\cdot (0.089490\\dots)", "c26fba5660054e0f75831f8cd0c7a1d3": "x \\mapsto T_x", "c26fff22c8476b8bab94bd055dd519ac": "\\begin{pmatrix}\n0 & 1\\\\\n-1 & 1\n\\end{pmatrix}", "c27058b1fbac8b058e3b7fe73ce1c2b9": "j = \\hat e_y \\cdot ( P3 - P1 ) ", "c270f4a1a37178c46a9a3f77aa46cfcf": "z = T(x)", "c2711e85414c057bef24afa26fc53547": " f(x) = \\int_{-\\infty}^{\\infty} \\frac{1}{2}|x - y|\\sin(y)dy", "c2715d42251a4dde981e0b6f8c5f3f5f": " Ric'(X, W) = Ric(X,W) + \\sum_{j=1}^k \\langle R'(X,e_j)e_j,W\\rangle+ \\left(\\sum_{i=1}^m\\alpha_j(X,E_i) \\alpha_j(E_i,W)\\right) - H_j \\alpha_j(X,W) ", "c2717f5e8d62d838edb9684c9c7c3dfb": "{\\mathfrak l}_m", "c2718413ae4daffd0027725a9647d7e0": "P_D = \\frac {\\frac{1}{n}\\cdot\\sum (p_{t})}{\\frac{1}{n}\\cdot\\sum (p_{0})}\n= \\frac {\\sum (p_{t})}{\\sum (p_{0})} ", "c271855dbf3f71ea807a2c7e4034b017": "f(z)=\\frac{e^{iz}}{1+z^2},\\qquad z\\in{\\mathbb C}\\setminus\\{i,-i\\},", "c271b729e24f147b025a92b0f2e35185": "\\mathcal{L}^{-1}(x)", "c271f312cf2876e8066cdba5864d3cee": "(P_3/P_2) \\,", "c2724812caa79d3542434de48b68ae2d": "\\frac{\\partial \\, {\\rm tr}(\\mathbf{g}(x\\mathbf{A}))}{\\partial x} =", "c2725258cb7b4e32d36861124a530b04": "\\ Z_{\\text{capacitor}} = \\frac{1}{\\omega C} e^{-j \\frac{\\pi}{2}}", "c2725d6922e783aad7f83ff0fa301a18": "I(X;B)", "c272c1f270bf9b08f74f2a5ec9976a79": "k = \\frac{\\det(\\gamma',\\gamma'')}{\\|\\gamma'\\|^3},\\ \\ \\ \\kappa = \\frac{|\\det(\\gamma',\\gamma'')|}{\\|\\gamma'\\|^3}.", "c2731ff8cd486db546bd9e6e368ddaea": "f \\le g \\quad\\iff\\quad \\forall x: f(x) \\le g(x).", "c273ef5c8dbff85d9e4f897d8962d0e1": "E (\\mathcal{A}f)(W) \\approx 0", "c27419957d91c713b4d95dbefb69666b": "(f\\times g)'=f'\\times g+g'\\times f. \\,", "c27482fb97d3dccc857088accf4579a1": "\\begin{align}\n\\|T(u) \\|_Y &= \\varepsilon^{-1} \\left \\|T \\left( x_0 + \\varepsilon u \\right) - T(x_0) \\right \\|_Y & [\\text{by linearity of } T ] \\\\\n&\\leq \\varepsilon^{-1} \\left ( \\left\\| T (x_0 + \\varepsilon u) \\right\\|_Y + \\left\\| T (x_0) \\right\\|_Y \\right ) \\\\\n&\\leq \\varepsilon^{-1} (m + m). & [ \\text{since} \\ x_0 + \\varepsilon u, \\ x_0 \\in X_m ] \\\\\n\\end{align}", "c274b6684a85d5893ae2659bce7d72d4": "x \\notin N", "c2753c1d83182820dc47807f8cbe71bb": "D=2t+1", "c2754e5f8691be9e0efd0c1d8df7eb49": "\ndS_t = \\mu S_t\\,dt + \\sqrt{\\nu_t} S_t\\,dW^S_t \\, .\n", "c275666ed6a881fe9e0aafb9273418f7": "\\varepsilon>0", "c275cc8065b0cd865bd18a3012862c6b": "\\quad i(L_n -L_{-n})", "c275f98567f3a34def045decd657071b": "|X_k|^2", "c2761ea6076a3d399ec60325cc77aa57": "(N,e)", "c276bc0fd8f38f4ffed01b2727d8c595": "\\operatorname{pd}_R k = \\operatorname{gl.dim} R", "c277057983eae27b2e63b119caebce8f": "\\log{z} = \\ln{|z|} + i\\left(\\mathrm{arg}\\ z\\right).", "c27723bd1400d3e7e41ab727e673d9b7": "\\therefore \\delta \\approx 0.72973 \\,\\mathrm{rad} \\approx 41.81^\\circ", "c2773236a915e843dcb576b5884312de": "\n\\begin{align}\n\\mathbf{f}_{n,m} & = \\frac{\\mu R_O^2}{R^{n+m+1}} \\left(\\frac{C_{n,m}\\mathcal{C}_m+S_{n,m}\\mathcal{S}_m}{R}(A_{n,m+1}\\mathbf{\\hat{e}}_3 - \\left(s_{\\lambda} A_{n,m+1}+(n+m+1)A_{n,m}\\right)\\mathbf{\\hat{r}}\\right) \\\\[10pt]\n& {}\\quad {}+ mA_{n,m} ((C_{n,m}\\mathcal{C}_{m-1} + S_{n,m}\\mathcal{S}_{m-1})\\mathbf{\\hat{e}}_1+(S_{n,m}\\mathcal{C}_{m-1}-C_{n,m}\\mathcal{S}_{m-1})\\mathbf{\\hat{e}}_2))\n\\end{align}\n", "c27751e33387a40eb89dfec0b34350f3": "{}^{14}_7 \\text{N}^*_{7}", "c27762917c059145075eb8ce696c22c6": "(x_1x_2 + Ny_1y_2 \\,,\\, x_1y_2 + x_2y_1 \\,,\\, k_1k_2)", "c277b1cc8b29bc9ca45f65e996833e11": "\n\\phi(t) = \\mathrm{arg}[ s_\\mathrm{a}(t)] \n.\\,", "c277ec43b40a5da1d5e9d6f4db02c729": "\\left(D_s\\right)", "c2788141002a2a825153c21a29755c12": "u_{22} = -1.5.", "c2788837a539f232733fc6bf6f4674da": "3.\\ \\mathrm{2H^+ + 2CrO_4^{2-} \\rightleftharpoons Cr_2O_7^{2-}+H_2O; \\beta_2=\\frac{[Cr_2O_7^{2-}]}{[H^+]^2[CrO_4^{2-}]^2}; \\beta_2=K_1^2K_D }", "c2789ba80c6704a54a7c6f61e3db2e3b": " \\hat{H}_\\mathrm{el} = \\hat{T}_e + \\hat{U}_{en}+ \\hat{U}_{ee}+ \\hat{U}_{nn}.\n", "c278ada17abe0e91058783017e789926": "\\frac{\\partial r_i}{\\partial\\beta_j}", "c279798992099c1d5d9e8491e9674764": "M_{T}", "c27982e7997fc63f9425bbc831846b34": "a^{n-1}\\neq 1 \\pmod n", "c279b42cede9d51c433602b9acc4da8d": "\\eta^l_A:I\\to A\\otimes A^l", "c27a33eddf327d3e36be5201f3e3ebf2": "P(v|h) = \\prod_{i=1}^m P(v_i|h)", "c27a4b0e4696c7cbb3410a7ad20cda8f": "\\,\\sum_{i=1}^{k} f_i(u,v) \\leq c(u,v)", "c27abf73414519c8ef91bb6129223748": "\\Gamma(1/2) = \\sqrt{\\pi}; \\Gamma(1) = 1; \\Gamma(x + 1) = x\\Gamma(x)", "c27ae01ad227ee5bd5a0afd961f44392": "\\alpha^3", "c27af2ee5c4ea9d133f518951b7cf908": "\\tau=\\lambda t", "c27b245ab0267a0bbf95c428877b5067": "a^{-\\frac{1}{x^2}}\\,", "c27b718a20d34e4c57748b93489e9425": "\\langle ax,y\\rangle= a \\langle x,y\\rangle.", "c27c331188a516e5798d4e79f0b788ff": "-{v dP\\over P dv} = (1- \\gamma)K + \\gamma", "c27c471792f7e5cd8e3629a504355723": "dU\\left(x_0,y_0\\right)=U_1\\left(x_0,y_0\\right)dx+U_2\\left(x_0,y_0\\right)dy ", "c27d5e3fb0b5a6f47f5bbffb1e9c69b0": " \\dot{M_{w}}=\\dot{M_{T}\\dot{m_{w}}} \\,", "c27de328426ff5f35aa55c68b3e2a9df": "^{\\;}f(\\xi)", "c27e07922c2f6aa6aae397f0ba299b4a": "A_W", "c27e8c9b4874da0608e8604837c70224": " \\begin{align}dY_t &= g(t,Y_t,Z_t) \\, dt - Z_t \\, dW_t\\\\ Y_T &= X\\end{align}", "c27ea6cf8ea3bc6768efd840ec5b90ee": "N0\\,\\!", "c297f4f56621aee502696c0a0cc5b69b": " 1 + \\left(\\sum_{j=0}^\\infty E_j\\frac{x^j}{j!} \\right)\\left(\\sum_{i=0}^\\infty E_i\\frac{x^i}{i!} \\right). ", "c2982c81bc171010376892162bb7f1b8": "P' = {P / {{\\rho}V^2}}\\,\\!", "c2983fb0b2fa02b0ec274634a674071c": "\\quad (1) \\qquad \\qquad \\frac{\\partial\\rho}{\\partial t}+\\frac{\\partial f}{\\partial x}=0,\\quad t\\ge0.", "c29859b120268327c050e31e62894a44": "[A^-]_i \\approx \\frac{v_i[OH^-]_0}{v_0+v_i}", "c2989ef033ddac798e59d8087c6ee01e": "r_j = x_j / \\overline{x}", "c298a7210642fc2a6302f35b599d44d8": " \\begin{matrix}\nZ_k & = & \\left[ X_k \\left( Y_k + Y_{N-k} \\right)\n + X_{N-k} \\left( Y_k - Y_{N-k} \\right) \\right] / 2\n\\\\\nZ_{N-k} & = & \\left[ X_{N-k} \\left( Y_k + Y_{N-k} \\right)\n - X_k \\left( Y_k - Y_{N-k} \\right) \\right] / 2\n\n\\end{matrix}\n", "c298bf5846c9921d830619106e023cc6": "m_0 - m_1", "c298cb249cdc3945ac35d269fcb58571": "a\\in Tr(\\langle g\\rangle)", "c298d23297ff4e1d8d904dc3b5551025": "0.733 \\pm 0.008", "c298dc3895f3e85b8b7977b8b2ed6deb": " \\phi_M = - {1\\over 4 \\pi} \\int {\\rho (x, y, z)\\, dv \\over r} \\; ", "c299625f278b73bd0363dd8b896ac187": "g(4,2) = 1", "c299be36e427e8fa87c9255d0fe27007": " D_{\\alpha} \\vec{e}_{\\beta} ", "c29a221ab3fba6c069bf96e7abebec48": "f^{-1}(X_2)=:Z_2", "c29a6646ea0c7d6538af4b979bab7fb8": " \\tfrac 1{1+\\tfrac 1{0+\\tfrac 1{8+\\tfrac 1{4+\\tfrac 1{1+\\tfrac 1{0+1{/\\cdots}}}}}}} ", "c29af7986010d6b53ed0e9fe1a678dfb": "\\left\\{d\\right\\}", "c29b24e466c0970f26112b080e39207f": "G(x) = \\prod_{n\\ge 1}\\frac{1}{(1-x^{5n-4})(1-x^{5n-1})}", "c29b25279f55711bb8a36badb896f56a": "(b\\bar{g}+g\\bar{b})/\\sqrt{2}", "c29b33b386b4ae165255a7e21cb03e2d": "x_j'=0", "c29b4e9400ff082a7d36e1cd7410981e": "|A| = |A \\cap B| + |A \\setminus B| \\le 2|B \\cap A| + 2|B \\setminus A| = 2|B|.", "c29b86296d563d8836a24b8dcfd7d446": "= C_{\\alpha IJ} + C_{\\alpha JI} .", "c29bb8f91b6b5fbc2a978c83273c5ff5": "H^0_c(X;k_X)^{\\vee}\\cong H^n(X;k_X).", "c29bbbeaf45d82563bd63d93b894c88d": "P(a,b) = P(a-1, b) + P(a, b-1)", "c29bd7f4c58fa023ec7147e1e5d58c33": "(x-1)(x^2+y^2)=ax^2 \\,", "c29be38e7370d5c90cd92a92d89cac08": "c=002001200", "c29c58eccd9c24102991847bd57f6e81": "x = ( \\lambda - \\lambda_0 ) S\\,", "c29c695c93f2cc683670f29fe811615c": "dw/dx = 0", "c29c73b350c8d549034939e4a60ac77d": "2^{(100/\\mathrm{log}_{10}2)}", "c29c947acebfded432dec1b74bc3ce9e": " G \\approx \\frac{\\Psi\\;\\bar\\Psi}{P^2-M^2},", "c29ce88688834f191f5a3fa638bd5708": " D[m] = \\_ ", "c29d09a6faf0720d384fd7208844d037": "=\n\\begin{bmatrix}\n0&\\kappa_g \\, \\mathrm{d}s&\\kappa_n \\, \\mathrm{d}s\\\\\n-\\kappa_g \\, \\mathrm{d}s&0&\\tau_r \\, \\mathrm{d}s\\\\\n-\\kappa_n \\, \\mathrm{d}s&-\\tau_r \\, \\mathrm{d}s&0\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}\n", "c29d2bfe02111807d96d8a278716f9e1": "U(1)=O(2)", "c29d38b931e1844368411700c872b1d2": "A + A\\,", "c29d4861109eb98d003cf96108b92ad2": "\\frac{dv_f}{dt}=\\frac{dU}{dt}\\sin(\\beta+\\psi)+U\\frac{d(\\beta+\\psi)}{dt}\\cos(\\beta+\\psi)", "c29d4a27cfbaa41bf18ff4fb2c08e1b9": "\\left(X, \\Sigma, \\mu\\right)", "c29d8ad55fbcbb286cf0c2c3d1e8876e": "-\\cos(t)", "c29dd199b1ee0db27b8bfc5a484d4fa6": "z = z_0 + r e^{i \\theta_0}", "c29e797169c8ac027e74f67a0313d5d4": " \\lim_{x \\to0^+} x^0 = 1 , \\! ~~ (7) ", "c29e9043d4be63aa240040728137b468": "9n^3", "c29e9e487edd30cb5c76cfb2235bdb77": "[G/\\mathbf{Z}_2]\\cdot[G/\\mathbf{Z}_3] = [G/1],", "c29ea6bc384fa83c4b7ef6eca478eba1": "\n\\begin{align}\np_2 & = 2p_1+1, \\\\\np_3 & = 4p_1+3, \\\\\np_4 & = 8p_1+7, \\\\\n& {}\\ \\vdots \\\\\np_i & = 2^{i-1}p_1 + (2^{i-1}-1).\n\\end{align}\n", "c29eae95f7f1a8734355d63c5857dd5b": "{\\mathbf E} \\propto \\frac{\\partial^2}{\\partial t^2}{\\mathbf P}", "c29ece46c38bb0b623d375576c32e719": " q_{max} = 45.4 \\text{ ft}^2/s", "c29edd0d63bd5810c80a535251f115fd": "\\displaystyle \\phi(\\alpha)", "c29f21e4ed4c967561b97c9ed32bd575": "R(t) = R(0) ( 1 + A*t + B*t^2).", "c29f4f089463108c639fb0c19499bc93": "x_0x + y_0y + z_0z = r^2", "c29f998e2e85f53766e879b935becee9": "x_1 = \\frac{1-2A_{12}/A_{21}} {3(1-A_{12}/A_{21})} ", "c29fb3362c47668d18b6f5494bc2aa51": "RF = \\alpha ln(C/C_0)", "c2a00f20b19127e2c94f2bd8601e1cfa": "\\left(\\frac {1}{2}\\right)^{t/t_{1/2}} = e^{-t/\\tau} = e^{-\\lambda t}", "c2a02675848d3d33bab8a67f0834b978": " c \\left \\| \\sum_{k=0}^N \\alpha_k b_k \\right\\|_V \\le \\left \\| \\sum_{k=0}^N \\alpha_k c_k \\right \\|_W \\le C \\left \\| \\sum_{k=0}^N \\alpha_k b_k \\right \\|_V.", "c2a03130995178113331d6e17a909567": "\\frac{dm}{dt} = \\rho A {v}", "c2a069267d62e17fe2d75a561d94ce3b": "\\left(\\wp'\\left(z\\right)\\right)^2=4\\left(\\wp\\left(z\\right)\\right)^3-g_2 \\wp\\left(z\\right)-g_3", "c2a078d18a3a03e03965333a102d7742": "V_{bc}", "c2a0d984728cbfe3c05799859b992d0e": "Q(y_1)", "c2a16a20468a9ca6f8a2804ac30ba418": "\\Diamond p ", "c2a214f280b620b6a3851d70428c48d2": "\\frac34+\\frac{3}{4^2}+\\frac{3}{4^3}+\\frac{3}{4^4}+\\cdots = 1.", "c2a27a04c7f14925004d7bf6787d333d": "\\tan A = \\left ({\\frac{\\left ({\\frac{-\\partial z}{\\partial y}}\\right )}{\\left ({\\frac{\\partial z}{\\partial x}}\\right )}}\\right )", "c2a3114f205fc9b75138a9251ce8d0e8": "\\mu_1, \\mu_2, \\dots, \\mu_n", "c2a32cfe2f66f14d3c6e0205944954e8": "c_\\mathrm d\\,", "c2a33b1009bc27ad291fe0b713bd77c3": " P(\\mathbf{x},t\\mid \\mathbf{x_0}) ", "c2a466c2863c5aa1b4a4ff293e8d9c08": "\\vert 1 - E_n(z) \\vert \\leq \\vert z \\vert^{n+1}.", "c2a471ea8f251bb88de9926e1e533aa6": "\nr_\\mathrm{out} \\approx {R_\\mathrm{source} \\over \\beta_0}\n", "c2a477d231ff1c46b298bfb68b9997cd": "{\\rm Ind}", "c2a48152ec79f617086e4940a0263a7a": "\\chi\\alpha^{-1} \\not= \\alpha\\chi", "c2a4e4a3c1807f2dc061d38e5b517c68": "\\langle a b \\rangle_i=0\\,", "c2a52ed60e610368c9fdb1a913b13918": "n(i) \\geq 0", "c2a542e2f43392362b6870f1cf3e1ea5": "= H(p,q) \\cdot G(p,q) |_{p = {x \\over \\lambda z} , q = { y \\over \\lambda z } }", "c2a545ecb7ce036ae56cc66544f795e1": "\\frac{\\hat{dx_i}}{dt}=\\hat{v_i}\\equiv i[\\hat{H}_0,x_i]=\\alpha_i", "c2a585a9761211cc4b211f7f619e302e": "\\oint_\\gamma d\\phi", "c2a58a581297a8c86163edfb56f7a13a": "\\prod _x \\sec x \\cos (x+1) = C \\cos x \\,", "c2a59faa0a3887cca7397e1e5f0aa615": "\n \\mathbf{M}_1 = \\int_{-t/2}^{t/2} \\mathbf{r} \\times (\\sigma_{11} \\mathbf{e}_1 + \\sigma_{12} \\mathbf{e}_2 + \\sigma_{13} \\mathbf{e}_3)\\, dx_3 \n \\quad \\text{and} \\quad \n \\mathbf{M}_2 = \\int_{-t/2}^{t/2} \\mathbf{r} \\times (\\sigma_{12} \\mathbf{e}_1 + \\sigma_{22} \\mathbf{e}_2 + \\sigma_{23} \\mathbf{e}_3)\\, dx_3 \n ", "c2a6466785024144dacba6e02080260f": "f : R \\to S ", "c2a6492dd4925145dce6793b6132717b": "\\beta(r,s)", "c2a7437221341b5cf33f5ab3f00d7679": "\\Box A \\to A", "c2a76129dae07c97231fa3e5999c2a94": "\\displaystyle\\mathbf N", "c2a76c551aabec6c60879e4a3853b6ca": "\\alpha = x/\\sqrt{2} - \\pi/8", "c2a7a87ecd0acae5b22c9bc26671804f": "G\\in \\mathcal{P}", "c2a8238f8b13a0c487875cb66330b9b0": "\\scriptstyle P_{\\mathrm{rot}} = \\frac{2\\pi r}{v_{\\mathrm{rot}}}", "c2a8cd60913b53c142fcf8f074f2305c": "u_{t} = \\Delta u", "c2a90a9a4418c0a63e22f99356629d66": "{d \\over dx}, D,\\, D_x,\\,", "c2a917253af13e5b051298e274e3b4c3": " M = \\begin{bmatrix} x_1 & \\cdots x_n \\end{bmatrix}", "c2a921c3558e2964e8d346d3235dc107": "g(x,z) = f(x) - z", "c2a9818946bb0f5ca73e23a0f61590d2": "B_{12}", "c2a9af10abdf80b5d89aaa8609221ecb": "+\\left[\n\\frac{\\alpha\\alpha' (a-b)(a-c)} {z-a}\n+\\frac{\\beta\\beta' (b-c)(b-a)} {z-b}\n+\\frac{\\gamma\\gamma' (c-a)(c-b)} {z-c}\n\\right]\n\\frac{w}{(z-a)(z-b)(z-c)}=0.", "c2a9e5be8d4d3c7467b216efcb2d2021": "\\cos\\varphi=0.71", "c2aa226ab4c0cbe5e7bc4fdf63929a04": "c_{3,2}(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta, \\zeta \\widehat{z} \\eta) = \\alpha x \\beta \\gamma y \\delta \\zeta \\widehat{z} \\eta", "c2aa7d263fca46bb539c9a03a0366155": "\\phi \\, (t)", "c2aa9b2ea22dff0ba8f067d6ee21de77": "\nE_{\\mathrm{bend}} = E ", "c2ab06a44d43b17dced63d9b200764ed": "\\displaystyle{P(f_{\\overline{z}}) = f(z) + z}", "c2ab23c62a8554742282899ef857b74d": "\\tilde u = \\$100,", "c2ab7a66402f24f24942c7d22e9dea4f": " \\frac{1}{C}\\|\\varphi(y)-\\varphi(x)\\| \\leq d(x,y) \\leq C\\|\\varphi(y)-\\varphi(x)\\|.", "c2ac05f80e76df84a2251d58d5b4063d": "P(x_1,x_2) = x_1^2 x_2 + x_1 x_2^2 + x_1x_2", "c2ad05810518ed320159d239642a9ad1": " \\langle\\psi |\\phi \\rangle=\\int {\\rm d}x\\,\\psi^*(x)\\cdot\\phi(x)\\,.", "c2ad06bc1df074ad611c5bfec8d6ca1a": "E^M{\\rm (abs)} = \\phi^M + \\Delta ^M_S \\psi", "c2ad142d2d32ac4a9ad2accd6f19ab85": "\\Omega = 12000", "c2ad59fa70a30e05d43523496e06c6f6": "S^z", "c2ad625a5245088fa127a02cb3ad4ee1": " n =1 ", "c2ad7cb71df6a83e8372508a65eee2e1": "q = f(\\varepsilon) q^*.", "c2ada636a85654b40a2a6fe575c02d93": "\\overline\\Phi:(X\\sqcup X^\\dagger)^+\\rightarrow S", "c2adac3173ab6275302fa59fec08d70c": "\\text{DOL} = \\frac{\\text{Total Contribution}}{\\text{Operating Income}} = \\frac{\\text{Total Contribution}}{\\text{Total Contribution} - \\text{Fixed Costs}} = \\frac{(\\text{P}-\\text{V})\\times \\text{X}}{(\\text{P}-\\text{V})\\times \\text{X} - \\text{FC}}", "c2adce49e8faa79e636d70b5c6016642": " \\lang\\omega|s\\rang =\\lang s|\\omega\\rang = \\frac{1}{\\sqrt{N}} ", "c2add18cc654577a5a2e336de7662a8f": " \n\\psi_{+} \\nabla^2 \\psi_{-} - \\psi_{-} \\nabla^2 \\psi_{+} = {} - 2 \\, \\Delta E \\, \\psi_{-} \\psi_{+} \\; .\n", "c2adf02915af7d6d48428d0ea70232e2": "\n\\frac{1}{k}\\sum_{i=1}^k x_i^2 \\leq P,\n", "c2ae2329632cc86715f8a4de7a056880": "m+1", "c2ae87f1358c69b59a2597003477efaa": "\\,_0F_1(;a;z) = 1 + \\frac{1}{a\\,1!}z + \\frac{1}{a(a+1)\\,2!}z^2 + \\frac{1}{a(a+1)(a+2)\\,3!}z^3 + \\cdots\\ ", "c2ae8ed4d91c0bf3849fa7febfa7ebce": "\\chi_\\nu(z) = \\frac{1}{2}\\left[\\operatorname{Li}_\\nu(z) - \\operatorname{Li}_\\nu(-z)\\right].", "c2aed9d1bf8860bed7289a9b31d2f8fd": "\\nu_1 \\le \\nu_2", "c2af20c02d987464a9e893ac1410bbee": "T_e = [ { i_A ( i_a -\\frac{i_b}{2} - \\frac{i_c}{2} ) + i_B ( i_b -\\frac{i_a}{2} -\\frac{i_c}{2} ) + i_C ( i_c -\\frac{i_b}{2} -\\frac{i_a}{2} ) } \\sin \\theta_r + \\frac{\\sqrt{3}}{2} {i_A ( i_b - i_c ) + i_B ( i_c - i_a ) + i_C ( i_a - i_b ) } \\cos \\theta_r ]", "c2afa32a1bad68c8d8981c17832f7998": " I(s) = \\mathcal{L} \\{ i(t) \\}, \\, ", "c2affefff8a4a4f7b093954c75869d79": " \\frac{d^2r}{d\\tau^2}+\\Gamma^{r}_{\\mu\\nu}u^{\\mu}u^{\\nu}=0. ", "c2b03be1ed2b220d43f59ec2bea7ab4e": "\\mathbf{Z}(s)=\\left(sI-A\\right)^{-1}\\mathbf{z}_0.", "c2b0a415ae65d2a381ff729a4452478f": "|\\gamma_n| < \\frac{4 (n - 1)!}{{\\pi}^n,}", "c2b0bb903ae2215e6ef0d1ec591ec9a6": "f^\\mu=x^\\mu\\mathcal{L}.\\,", "c2b0e67e1e8e8584bc5418f633df1808": "\\partial L/\\partial q_i ", "c2b124527c5106aa313f4ded87c17fdf": " A^{\\times}_A + X^{\\times}_X \\Leftrightarrow A^{\\bullet\\bullet}_i + \\frac{1}{2}X_2(g) + 2e^{'} ", "c2b1287b6eac3defe273e493d3b6ff53": "s=A*u", "c2b164e8bb4abd53ab73b00029e8bcdc": " \\sigma_D = \\eta \\frac {d\\varepsilon} {dt} ", "c2b18a7c64e289c6e19253f136b23e5f": "f:S \\rightarrow S", "c2b1973b93e39501ba6a3e38d35b863b": " {^*f}(x) \\cong {^*f}(\\operatorname{st}(x))\\,", "c2b233ae928ac3c41eb308cdbbf2b285": "\\partial /\\partial\\xi_{j}", "c2b23946e17d58633ba5c98ea25b1d16": "\\frac{T_s}{T(1)}", "c2b255cd0da51789396f108e36aa9742": "A^k(e_1)", "c2b298e73494f4b28cf889200c12dbaa": "\\ \\beta = 90 ", "c2b2ab2380fad114153e85341dbeed7a": " i = i_L = i_C \\,", "c2b2d84881fbfe2f4ae809ed56c86083": "\nc^2 {d \\tau}^{2} = \n\\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} dt^{2} - \\frac{dr^{2}}{1 - \\frac{r_{s}}{r}} - r^{2} d\\theta^{2} - r^{2} \\sin^{2} \\theta \\, d\\varphi^{2}\n", "c2b2dfaf667f6def52190a7e63c58bf0": "| \\psi_{S}(t) \\rangle ", "c2b31400bf6298655057a650d5d0ed22": "\\mathfrak{B} = \\{U_a(b)\\, |\\, a,b \\in \\mathbf{Z}^+, (a,b)=1\\}", "c2b38da9b19033b5b465b7cae2d08664": "\\eta^{\\mu \\nu}", "c2b3b4736efc644924ff9881eb01fd45": "\n\\dfrac{\n \\dfrac{\n \\dfrac{\n \\dfrac{\n \\dfrac{the}{NP/N}\n \\dfrac{dog}{N}\n \\qquad\n }{NP}>\n }{S/(S\\backslash NP)}T_>\n \\qquad\n \\dfrac{bit}{(S\\backslash NP)/NP}\n }{S/NP}B_>\n \\qquad\n \\dfrac{John}{NP}\n}{S}>\n", "c2b3d414cb1456a9e2b9883d634712ba": "y_0=\\sqrt{1-\\|x\\|_2^2}\\qquad\\mbox{ and }\\qquad y_n=x_{n-1}\\quad\\mbox{ for }\\quad n\\geq 1.", "c2b4029f7439f6a9cddf9679a438dc31": "P=(X:Z)", "c2b41a92b616c7b939549c6235696e03": "\\tbinom{n-k}{k}", "c2b4576e6e08542fac00a1787ef8a4f0": "\\frac{1}{\\tau_M}=\\frac{V_0 \\Gamma \\omega^4}{4\\pi v_g^3}", "c2b47120bbe8aac120afb22c850239af": " Z=\\lim_{\\nu\\rightarrow\\infty} T ", "c2b4c36bfd09197ddefd8f3c9a6b4daa": " \\begin{matrix} \\frac {r_{ \\pi }+ \\beta r_O} {r_{ \\pi} +( \\beta +1)r_O} \\end{matrix} \\begin{matrix} \\end{matrix} ", "c2b521373a83702b7d73768e7412f080": "\\scriptstyle{n = 0}", "c2b524160cb042169500df9f1da1fd26": "p = 0.41 \\cdot f^2 \\cdot a", "c2b53609bbb799e27f96a1ddf5c3c17c": "{\\rm Dehn}(n)=\\max\\{{\\rm Area}(w): w=1 \\text{ in } G, |w|\\le n, w \\text{ freely reduced}.\\} ", "c2b53c3f65ba979827c66f2c4db4f855": "(p,0011,Z) \\vdash (p,011,AZ) \\vdash (p,11,AAZ) \\vdash (q,11,AAZ)", "c2b5752c5dfe43e9580c6d72a2878d13": "p_1 = 2, p_2 = 3, \\ldots", "c2b5791a4b749f24a377dc698b525028": "~w=z \\theta_f = z \\lambda /d", "c2b5caaf52b78b84352f254558fae369": "\\displaystyle{M^{-1}\\|x\\| \\le \\|x\\|_0 \\le M\\|x\\|}", "c2b5fdb0c7dfeb8ea6d894f74636553e": "\n(-1)^{n+1} \\left( 1 - \\frac{1}{n+1} \\right) = (-1)^{n+1} \\frac{n}{n+1},", "c2b65bb52566ef4326259091197086fa": "\\mathbf{B} = \\mu_0 (\\mathbf{H}+\\mathbf{M})", "c2b666a9adaa59d35ffe95300999fc7d": "\\frac{d^{2}z}{dt^{2}} + \\omega_{0}^{2} z = 0", "c2b6b3dbd403a1a32efb02f1736f3cde": "\\ln 10", "c2b71fe3ab05a4de9293cb336fa25ff1": "b_1(\\theta)\\sin(\\theta)", "c2b753f7672516e39d6b15a999ec414f": "\\frac {\\mathrm{d}} {\\mathrm{d}k} \\left[ k (1-k^2) \\frac {\\mathrm{d}K(k)} {\\mathrm{d}k} \\right] = k K(k)", "c2b7ac76af1b1c7597d57aa3a4bd4102": "\\dot{x}_i=-\\lambda_{ik}x_k", "c2b7bfb1a7ec77fc526996ce3890f14b": "(\\mu, \\sigma)", "c2b7c0068fe0c6e9c3a6a992fc1ff85f": "\\varsigma", "c2b7cf47320c818166d8f43ff9fb919b": "S_{uvw}", "c2b85a079357067328e6ceff9ec81a81": "\nh_{ij} = -\\left ( \\frac{\\partial E_i}{\\partial S_j} \\right )^D\n = -\\left ( \\frac{\\partial T_j}{\\partial D_i} \\right )^S\n", "c2b8a5935e218550879dc8fd689d71e3": "\\widehat{\\mathcal{C}}_{Y|X} ", "c2b8bbdee12aa86af31ba63826a32122": "(\\textrm{CURVE}, G, n)", "c2b8e1e0be73186092598fab0440bb8a": "\\ S = S_i + S_e ", "c2b91e53766dbaeab2d4853f59b4910b": " a \\wedge b", "c2b94514c1ca7537dfde6f120ca7aa04": "\\nabla_{f{\\mathbf v}+g{\\mathbf w}} {\\mathbf u}=f\\nabla_{\\mathbf v} {\\mathbf u}+g\\nabla_{\\mathbf w} {\\mathbf u}", "c2b981cb7f62a76160ea1877eab9448e": "r(x_i) = p(x_i) - q(x_i) = y_i - y_i = 0", "c2b9857953204c477651185c5a858b2b": "N \\geq \\frac{p_f}{p_f - \\frac{1}{2}}\\left( C + 1\\right)", "c2b9a6b27020f7fa5e8e146dab86c3be": " \\hat{R}_I=\\varprojlim (R/I^n) ", "c2b9b8d96435867b137d7006dba81f67": "\\operatorname{pf}(A)=\\sum_{\\alpha\\in\\Pi} A_\\alpha.", "c2b9ca2b26e8b25174a9b8d6ab580ae1": "\\alpha_k(\\theta) = 2\\cos\\theta, \\quad \\beta_k = -1.", "c2b9d5ab8bc9ccd867337194c150b44d": "\\biggl(\\sum_{i=0}^m a_ix^i\\biggr) \\biggl(\\sum_{j=0}^n b_jx^j\\biggr)\n= \\sum_{r=0}^{m+n}\\biggl(\\sum_{k=0}^r a_k b_{r-k}\\biggr) x^r,", "c2ba53075877771f6a9afacf5bdfa7e7": "\\mathbf{q}_{M \\times 1} = \\mathbf{f}_{M \\times M} \\mathbf{Q}_{M \\times 1} + \\mathbf{q}^{o}_{M \\times 1} \\qquad \\qquad \\qquad \\mathrm{(2)}", "c2baa6c2166e6c14075f3bde794ebd08": "\\varphi_k(n) = \\sum_{i=0}^n i^k - \\frac{n^k}{2}", "c2baaad2487e4c9112a98e808ebb418e": "(H_2SO_4)", "c2baf015e9b897f0736cc6e4517574fd": "J:V\\to\\bar{\\mathbb{R}}", "c2bb431e145fd89b4d8d491d92fa2b1c": "=\\left|\\mathcal{F}^{-1}\\left\\{\\mbox{log}(\\left|\\mathcal{F}\\left\\{ f(t) \\right\\}\\right|^2)\\right\\}\\right|^2", "c2bbf985d1647a7380bf95ce21bd68ea": "0\\le \\psi \\le \\pi, \\;\\; 0 \\le\\phi\\le 2\\pi,\\;\\; 0 \\le \\theta \\le \\pi.", "c2bc008ad76e06179aaf288ffb01f72c": "\\mathrm{Reg}([0, T]; X) = \\overline{\\mathrm{BV} ([0, T]; X)} \\mbox{ w.r.t. } \\| \\cdot \\|_{\\infty}.", "c2bc7b5dc7b185cec35ab0f01a6e076a": "L^2(\\mathbb{R}, \\mu),", "c2bc906552091860d410750ce85e20a2": "x=r \\, \\sin\\theta \\, \\cos\\varphi", "c2bcaf6c6c15250c71f21e90a7a4b5c8": "\\sigma=\\frac {2 \\eta_0 \\dot \\epsilon} {1-2\\lambda \\dot \\epsilon} + \\frac {\\eta_0 \\dot \\epsilon} {1+ \\lambda \\dot \\epsilon}", "c2bcc2b5ccad6a3e5fd9aeaa51c39eb5": " m={A \\over \\ell} ", "c2bd1cdd81d9df22f8ecb15b55c7f007": "(x,y)\\in R_{k}", "c2bd469a7af547c2c844155560d888bf": "\\text{Typical Price} = \\frac{H + L + C}{3}", "c2bd6003b52ddd25403af0feae4f2633": "k \\times 1", "c2bd78a5a30d5824894594b0ce1721fe": "^{^{^{4}4}4}4", "c2bdd135c747307208ffb195b44c7d95": "B=A/c.\\,", "c2bdd5c07f1b0cc16508d0a7a669465d": "\\mathbb R^2\\,", "c2bddb547c7c90d83e55acab9604f320": " M = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} ", "c2be0f8ce4779ab10bf4dc258fc186d6": "I_{\\text{b}} = \\frac{ V_{\\text{cc}} - V_{\\text{be}} }{ R_{\\text{b}} + ( \\beta + 1 ) R_{\\text{c}} }", "c2be147c85a7fa01b5951be0322552fe": "a_{E}\\,= D_{e} - F_{e} ", "c2bea2221e4ceea06b4309a6bc1fb5b7": "D\\{\\emptyset\\}=\\{\\emptyset,\\Omega\\}", "c2bf1bfd4178f51735ff3968edd18063": "(\\partial T)(\\omega) := T(d\\omega)\\,", "c2bf3f2d8041fd3fac06d69033bd130e": "\\varphi(z) = \\lim_{k\\to\\infty} \\frac{\\log(1/|z_{kr} - z^*|)}{\\alpha^k}. ", "c2bf502b7332dc0181c5341cfb574843": "lnK_{eq}=\\frac{-\\Delta {G^o}}{RT}\\,", "c2bf637b4468c32b0036fd76598efca6": "u,v \\in [0,1], \\;\\;\\; (u+v) \\leq 1,", "c2bf6c4fb8aca3f1dc3a773759762933": "2^{\\kappa}", "c2bf75b56497dd6dbd56a29a7ff11387": "\\varepsilon:FU\\to \\operatorname {id}_\\mathbf{C}", "c2bfc273690fa98819bb7ae66d610927": "\\mathrm{sinc}(x) = \\frac{\\sin(x)}{x}.\\,\\!", "c2bfcc79ed6f69778f4df6af12e557ac": "(i_-, i_+),", "c2bfd79d2c8eab4bbf44b07f61414997": "O(\\log p)", "c2c036fefdb7d0634c647dca875e5d92": "\\mathbf U\\,\\!", "c2c0854cf6b7c6506a5e92ae3111ba02": " \\neg (\\neg P \\vee \\neg R) ", "c2c0ac2fa27a0146dc4bdb47100cd19e": " B_v = {h \\over{8\\pi^2cI_v}}; \\quad I_v=\\frac{m_A m_B}{m_A+m_B}d_v^2", "c2c0af910ad0b766e5f8823c8717d8f9": "t \\equiv n^Q \\pmod p", "c2c0cd56301cf0a76f168173bdd76d4b": "-\\pi < \\theta \\le \\pi", "c2c131049bba5abd1b2d5a75cfd75c88": "B_{\\emptyset} = A_I", "c2c1386fb80c9b364184c46e57937b8e": "\\textbf{v}_{k} \\sim N(0, \\textbf{R}_k) ", "c2c17500a921ed5956656aa5da60804a": "\\sigma_p = \\sigma_n", "c2c17c8eb831b5160a668f4e4c944bf1": " s^G ", "c2c181b5958b4b98b7edf69c40c0dfb3": "0\\leq 1", "c2c2b8eeb2650f63c1888e518146d949": "\n\\gamma^0 = \\left(\\begin{array}{cccc} I_2 & 0 \\\\ 0 & -I_2 \\end{array}\\right),\n\\gamma^1 = \\left(\\begin{array}{cccc} 0 & \\sigma_x \\\\ -\\sigma_x & 0 \\end{array}\\right),\n\\gamma^2 = \\left(\\begin{array}{cccc} 0 & \\sigma_y \\\\ -\\sigma_y & 0 \\end{array}\\right),\n\\gamma^3 = \\left(\\begin{array}{cccc} 0 & \\sigma_z \\\\ -\\sigma_z & 0 \\end{array}\\right).\n \\,", "c2c2d5640661b24f9a411f5a03627ecc": "\\hat H \\Psi = E \\Psi ", "c2c32092edef39a471f68c6106c6706e": "\\Delta T_s = \\lambda \\cdot RF", "c2c4139a87d07aba5dd4134a0b19da30": "\\textstyle R[[t]]", "c2c41411b3ad1472c5e9850c098f34ea": "\\sigma^{n} = \\frac{1}{H^{n}(\\mathbf{S}^{n})} H^{n}.", "c2c4558ec43f46c7c4e8c839160c1705": " \\preceq", "c2c475f7cc158247f74c8e5458d4e7ad": "\\frac{\\partial p_i}{\\partial x_k} \\approx \\frac{p_i(x_{0,k} + h) - p_i(\\hat{x_0})}{h}", "c2c48e95c69dd8c96ce7c1478795a300": "MO=\\frac{AC}{2 \\tan \\alpha}", "c2c4af19e9752d5f57400b7dd90131a2": " f = d0*z(t) + d1*z(t - \\tau ) - d2*\\sin(z(t - \\tau ))", "c2c561bb31e23dad8794181184ed2908": "\\hat{\\cdot}", "c2c567aad91cc4005392d0c0cfce4aab": "\\langle\\hat{A}\\rangle = \\langle\\psi|A|\\psi\\rangle", "c2c58988a9305c708868c2e25c9ebeb0": "\\mathbf{\\tfrac{n}{m}}", "c2c5ed9e6b7c58017a55180e7e1711ad": "\\operatorname{ad}(\\mathfrak{g})", "c2c5f4dc27498468416db64d4ce51583": "{\\mathbf{p}}_{\\text{crystal}} = \\hbar {\\mathbf{k}}.", "c2c63a2204fa0f4fac3c54346b31a072": "\\zeta\\left(\\tfrac{1}{2}+it\\right),", "c2c684988fbefe373c589caf73c0f8d3": "z=\\pm H", "c2c6916abfbbd4009aa8b780e81cf3a5": "\\mathbf{G}_1 = \\mathbf{F}\\begin{bmatrix}\n\\mathbf{0}_{NxN} & \\mathbf{0}_{NxN} \\\\\n\\mathbf{0}_{NxN} & \\mathbf{I}_{NxN} \\\\\n\\end{bmatrix}\\mathbf{F}^{-1}", "c2c6bbbf291177a1d926d538a2e29a09": "\\Psi \\left ( \\mathbf{r},\\mathbf{r'},\\mathbf{r}_3,\\mathbf{r}_4,\\cdots \\right ) = \\left ( -1 \\right )^{2s} \\Psi \\left ( \\mathbf{r'},\\mathbf{r},\\mathbf{r}_3,\\mathbf{r}_4,\\cdots \\right )", "c2c714e76424d55acf1264ac970ba4f4": "\\left(\\frac{\\partial ^2S}{\\partial X^2}\\right)_U < 0", "c2c7697e71d866de0c5edb65184d6088": "\\mathbf{m} = NIA\\mathbf{\\hat{n}}\\,\\!", "c2c7acf7ece75a34d49c43b47e04f383": "P\\rightarrow (P\\and Q)", "c2c82137bf1968462c183a78ed45b168": "\\int x\\,\\operatorname{arsinh}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{arsinh}(a\\,x)}{2}+\n \\frac{\\operatorname{arsinh}(a\\,x)}{4\\,a^2}-\n \\frac{x \\sqrt{a^2\\,x^2+1}}{4\\,a}+C", "c2c83ebd67e27c88f57a78672b9bd0ff": "H_\\mathrm{eff} (E) = QHQ + \\lim_{\\varepsilon \\to 0} QHP{1 \\over E + i \\varepsilon -PHP}PHQ = QHQ + \\Delta(E) - i \\Gamma(E)/2, \\, ", "c2c8449ca62ce1a63f573424e85ae516": "1.149782767465722 \\times 10^{-8}", "c2c876288bef68b1c5c1990a6abe80e4": "P(\\vec y|\\vec x)P(\\vec x)", "c2c87aa48bbc054c1293b75c5a292c7a": "\\int_0^\\infty \\frac{\\sin x}{x}\\,\\mathrm{d}x = \\infty - \\infty", "c2c886d0bf1571f243f88846964c72ab": " d_A", "c2c8c57038be7af54ef7b439f1616fd2": "\\frac{PV}{T}= C ", "c2c8e4f4be1ac024ba70405c3a365059": "\\tan \\theta = \\frac {v'} {S} \\,;", "c2c931e221df4d97e3456792ea24f1e5": "\\Vert v\\Vert >r", "c2c95cee7eec7545e64c2889beaca2f0": "\\frac { \\pi^2}{6 \\ln 2 \\ln 10} \\approx 1.03064083", "c2c9ab3d702716340580b9fde6b2fe8a": "\\overline{\\mathbf{Q}_p}", "c2c9b17c27b53e87888d7f6a3e9db76c": "\\hat{\\beta}_{FD}", "c2c9e555b4f9cddd402c82285328efc0": "\\scriptstyle \\mathbf x_{k\\alpha}(t)", "c2ca2019525b3d2bee36e6a37a146783": "x_\\nu", "c2ca722d6022cb9e5f87e034eb35d990": "V_j^2=2 C_{p0} (T_4-T_5)", "c2ca9a65bcd752366de49a64e7bd188e": " \\triangle ABD, \\triangle ABC, \\triangle BCD, \\triangle ACD ", "c2cad4c9aff6c2a2894bb36a910cb84f": " \\mathbf{P} = \\varepsilon_0 \\boldsymbol{\\chi} \\mathbf{E} .", "c2caee96205a3ed1ef59979f804cb6e1": " {\\mathbf A}_{11}^{-1}{\\mathbf A}_{12} ", "c2cafd1d398af2217d9618284ec4ea29": "p(\\vec{x},y)", "c2cb4d9acf41c89aa604d435b0bde7fc": " \\mathbf x \\neq \\underline 0", "c2cb787f6b079a22a34d6ef33a8163d5": "\\sqrt{\\frac{16}{35}}\\!\\,", "c2cb8a0d097f32cc5e3c590653c18672": "\\Psi(\\Omega)", "c2cc1c967d1d83139cb51893de78ae08": "bz-cy+3ay-2az+2xc-3xb", "c2cc37c5329eeb4e453345ee09e9fc07": "\\rho^{ABC}", "c2cc4c25b9a60988f05726671574db5a": "kWs = \\frac{W}{2g} v^2,\\quad\\mbox{or}\\quad v = \\sqrt{2ksg}.", "c2cc4f2e9d5a5a16a027e419e71efe6b": "\\frac{j_\\nu}{\\alpha_\\nu}=B_\\nu(T)", "c2cc888ab5712222299cb94f8645d70c": " G=X^T X ", "c2cc88fa48e2b3c2fd9f5267eaf2e168": "\\mathbf{H}(\\mathbf{r}, t) = \\frac{1}{\\mu_0} \\mathbf{B}(\\mathbf{r}, t) - \\mathbf{M}(\\mathbf{r}, t),", "c2ccca50569bc0dcf79afc3fbb5f9fb9": "\\tan{\\frac{B}{2}}=\\sqrt{\\frac{cd}{ab}}=\\cot{\\frac{D}{2}}.", "c2cd65eaab1e0054585e625313f1e23b": " i^i_{ion} = \\frac{V_i}{r_{Mi}}", "c2cd7b80cb552f1ad30445ff16879797": "\\begin{align}\n x &= 10 \\log_{10} \\frac{P}{ 1 \\mathrm{mW}} \\\\\n x &= 30+10 \\log_{10} \\frac{P}{ 1 \\mathrm{W}}\n\\end{align}", "c2cdf1b762a31730a615472df4c7a660": "\\frac{1}{x} \\,", "c2cdf3c01b79387e64d19db2fedf4798": "\\sim_{\\mathcal{B}}", "c2ce303c2756971aa9293dc9fbb53b80": " K(p)u(p)=Q(p) ", "c2ce61ed96dc9bb6000a27f958625934": "y = f^{-1}\\left(\\frac{1}{n}\\left[f(x_1) + f(x_2) + \\cdots + f(x_n)\\right]\\right)", "c2ce68b8d0368173f41cef96d488e7fc": "R_{ik} = R_{(a)(b)} \\xi^{(a)}_i \\xi^{(b)}_k", "c2cef0a206235b470930c95ec97b18f2": "CV = e(p_1, u_0) - e(p_0, u_0)", "c2cf02e4c21b100934079f09bbaeb659": " \\in \\mathbb{Z}_2", "c2cf72a2d8593c7330cbb330ff4ed4ee": "\\int_0^\\infty x^{\\alpha'-1} e^{-x} L_n^{(\\alpha)}(x)dx= {\\alpha-\\alpha'+n \\choose n} \\Gamma(\\alpha').", "c2cf8e300c180506653857585529e6ec": "B=\\operatorname{diag}(b_1,b_2,\\ldots,b_s),\\, M=BA+A^TB-bb^T\\, Q=BA^{-1}+A^{-T}B-A^{-T}bb^TA^{-1}.", "c2cfd6a9ae8ae2fae2031114b2e12134": "x^{*}_{i}", "c2d019600a48c55b7224990da60585ef": "\\scriptstyle d^{}_{ij}", "c2d0fe1a619d455592d356a1d43be73a": "A \\rightarrow U^* A U.", "c2d11bc52d09fdd8c35369dc1d816117": "G^{\\hat{a}\\hat{b}} = \\omega^2 \\, \\operatorname{diag} (-1,1,1,1) + 2 \\omega^2 \\, \\operatorname{diag} (1,0,0,0).", "c2d19ba141a62bd393859e80e843923f": "L(f)<1", "c2d202f5b2ed4bafb080ba926ea508a7": "c \\cdot \\sum a_\\alpha X^\\alpha = \\sum F(c) a_\\alpha X^\\alpha = \\sum c^p a_\\alpha X^\\alpha.", "c2d26993f9dfcfaf2dbe4c38b5284840": "S_{11} = {(Z_{11} - Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \\over \\Delta}", "c2d27d49b38ae48204f93d0f6c1a61c2": "\\mbox{EOT}=\\alpha_M-\\alpha", "c2d2a1c5860b7e92445223993d44e881": "0 < x < 1", "c2d379db6e131a9032d0e978c1c79e8d": "2q+3", "c2d3808ea17ce79b146a16fc2cdcee84": "(d+e\\,x)^m \\left(a+b\\,x+c\\,x^2\\right)^p", "c2d394903e9232b6b5f14745ca94fbdb": "{\\lambda}<10{\\mu}m", "c2d3b5b2d1189ced4e56f77c7824e761": "f(x) = \\frac{1}{(2\\pi)^{n/2}} \\int_{\\mathbf{R}^n} \\hat{f}(\\omega) e^{ i\\omega \\cdot x}\\,d\\omega. ", "c2d3f99aa29277920e422c3871107b96": "\n\\psi (\\mathbf{r},t)=e^{-(i/\\hbar )Et}\\phi (\\mathbf{r}), \n", "c2d4023c3e50e5663cea143a0cd52ebc": "\n\\begin{align}\n & \\varepsilon\\, \n \\left\\{\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right\\}\n \\\\\n & + \\varepsilon^2\\, \n \\left\\{\n \\frac{\\partial^2 \\Phi_2}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_2}{\\partial z}\n + \\eta_1\\, \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right) \n + \\frac{\\partial}{\\partial t} \\left( |\\mathbf{u}_1|^2 \\right)\n \\right\\}\n \\\\\n & + \\varepsilon^3\\, \n \\left\\{\n \\frac{\\partial^2 \\Phi_3}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_3}{\\partial z}\n + \\eta_1\\, \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi_2}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_2}{\\partial z}\n \\right) \n \\right.\n \\\\ & \\qquad \\quad \\left.\n + \\eta_2\\, \\frac{\\partial}{\\partial z} \n \\left(\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right) \n + 2\\, \\frac{\\partial}{\\partial t} \\left( \\mathbf{u}_1 \\cdot \\mathbf{u}_2 \\right)\n \\right.\n \\\\ & \\qquad \\quad \\left.\n + \\tfrac12\\, \\eta_1^2\\, \n \\frac{\\partial^2}{\\partial z^2} \n \\left(\n \\frac{\\partial^2 \\Phi_1}{\\partial t^2} \n + g\\, \\frac{\\partial \\Phi_1}{\\partial z}\n \\right)\n + \\eta_1\\, \\frac{\\partial^2}{\\partial t\\, \\partial z} \\left( |\\mathbf{u}_1|^2 \\right)\n + \\tfrac12\\, \\mathbf{u}_1 \\cdot \\boldsymbol{\\nabla} \\left( |\\mathbf{u}_1|^2 \\right)\n \\right\\}\n \\\\ &\n + \\mathcal{O}\\left( \\varepsilon^4 \\right)\n = 0, \n \\qquad \\text{at } z=0.\n\\end{align}\n", "c2d42749464f54b02c4a0bf1d2135139": "\\beth_{\\lambda} = \\bigcup \\{ \\beth_{\\alpha} : \\alpha < \\lambda\\}.", "c2d45b175ff4e73cd0e51aa3ebfdd848": " \n\\mathcal{L} [\\varphi (x)] \n= {1\\over 2} \\left [ \\left ( \\partial \\varphi \\right )^2 -m^2 \\varphi^2 \\right ] \n", "c2d45f6fcc020c9b5966f335fc6712aa": " \\frac{P(Presentation~is~caused~by~condition~in~individual)}{P(Presentation~has~occurred~in~individual)} \n= \\frac {P(Presentation~WHOIFPI~by~condition)}{P(Presentation~WHOIFPI)}", "c2d4685d87d646b1437ea060b472a392": "\\{Q,Q\\}=\\{Q^\\dagger,Q^\\dagger\\}=0", "c2d4a0762a31aa8c3dd409b555dd0563": " \\delta(x) ", "c2d5371c807a39c047e780cd7d318dd4": "A\\in \\mathcal{G}.", "c2d5551bf3ef31b0ebc1a33ae4ed3c18": "\\sum (-1)^i K_nG^i\\epsilon G^{m-i}:K_nG^{m+1}\\to K_nG^m", "c2d55d790d2768a3367deeb5545e1b89": "D_1 + D_2", "c2d563bac58f10f93d10506b94a2e327": "\\rho=mn", "c2d603217c9522256e2f2ddf349e0e18": " m_w ", "c2d61fd9db44d6a3f9d37ffaf5eb8340": "\\mu_{30} = M_{30} - 3 \\bar{x} M_{20} + 2 \\bar{x}^2 M_{10}, ", "c2d62813b1baeaa1c94e6c8a37dd5da2": "\nF_{1} = \\cfrac{1}{X_{\\mathrm{f}}} - \\cfrac{1}{X_{\\mathrm{f}}^\\prime}\\ ,\\ F_{2} = F_{3} = \\cfrac{1}{Y_{\\mathrm{f}}} - \\cfrac{1}{Y_{\\mathrm{f}}^\\prime}\n", "c2d64210ced91190b47efdd8f6301104": "C_G \\varphi", "c2d685ae7b9a28781f7704154d9806e7": "C_\\mathrm{L} = \\frac{L}{q\\,S}", "c2d699e2593e0640af07197a66640848": "X=[\\mathbf{x}_1,\\mathbf{x}_2,\\ldots,\\mathbf{x}_n]", "c2d6cf0c01b4c0592f4874497313075e": " X = \\omega L \\,", "c2d70ae5c638977f78c34d83ce2615f7": "\\Omega_{\\perp}=\\gamma|\\mathbf{B}_{\\perp}|", "c2d714f109fc12e5ff06bdbc80254a8d": " R = \\frac{r^4(\\sin(2\\rho))^2}{4\\rho^2\\rho '^2 + r^4(\\sin(2\\rho))^2} ", "c2d71f83584094fad18e133ee7c03ea0": "F_{15}", "c2d744eaff891dafdc80f1c9f94e630d": "\\sum \\frac{\\psi(n)}{n^s} = \\frac{\\zeta(s) \\zeta(s-1)}{\\zeta(2s)}.", "c2d777ed4b23b1274dc981437ff5d6df": "y(x)=c", "c2d833309912a7f648ffd26083f680cf": "a_0=1\\!", "c2d8d61daa5bc9f98023813ad659e697": "\\sum_{\\underline{m} \\supseteq T \\supsetneq \\varnothing}(-1)^{\\left|T\\right|-1} g(\\underline{m} \\backslash T) = \\sum_{\\varnothing \\subseteq S \\subsetneq \\underline{m}}(-1)^{m-\\left|S\\right|-1}g(S) = g(\\underline{m})", "c2d8e23ad707283db2b282b4f6ba9171": "\\tau(i):=\\begin{cases}i&\\text{for }i\\in\\{1,\\ldots,j\\},\\\\\n\\sigma(j)&\\text{for }i=k,\\\\\n\\sigma(i)&\\text{for }i\\in\\{j+1,\\ldots,n\\}\\setminus\\{k\\},\\end{cases}", "c2d8f5ed33fa2e28a23bf5d68679dc6b": "\\{ (x,y) | x,y \\in \\mathbb{R} \\}", "c2d8f7d9f3f249b1bc86374f75296999": " v_1/c =\\mbox{tanh}(s_1)={\\frac{e^{s_1}-e^{-s_1}} {e^{s_1}+e^{-s_1}}} ", "c2d90e0b3f979886ebf25c0174826771": " -k\\frac{\\partial T}{\\partial n}\\bigg|_A =-k_e \\frac{\\partial T_e}{\\partial n}\\bigg|_A \\, ", "c2d927189c2cf6aba9eb824c398e6abb": "x_{t+1}=A_tx_{t}, \\quad A_t\\in \\mathcal M \\, \\forall t", "c2d952c015c5b4aaad4a053d6415af14": "\\overline{H}\\left( y^{n}|x^{n}\\right) ", "c2d9532a4fa1e1a16c23201a48e2da81": "Pc + Pm = \\tfrac{12}{25} + \\tfrac{13}{25} = \\tfrac{25}{25} = 1", "c2d963b47e0f00f1891a3cadaeba9dd5": "\\binom{n}{k}k!2^{\\binom{n}{2}-\\binom{k}{2}},", "c2d9b4823cfe4b0227a8af50988bd25d": " O(2^{4n}) ", "c2da1920fe6ab54b0d76088119be933a": "r(x)=a(x)S(x)\\Gamma(x)+b(x)x^{d-1}.", "c2da4d3d46c35ee171ecc91998f436e5": "2^{n-1}(x_1 + x_2 + \\cdots + x_n) \\ge 2^{n-1} n \\sqrt[n]{x_1 x_2\\cdots x_n}\\,", "c2da535cbac2978f254b1b7bbe17edd4": "D_pf : T_p M \\to T_{f(p)}N\\,", "c2da6e2e5eb9e115951134d129e1d2e8": "D=\\frac{C_D}{C_L}W", "c2da8287dca2fd3937ff642e4d85b168": "\n \\underline{\\underline{\\mathsf{A}_\\varepsilon}} = \\begin{bmatrix} \n 1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & -1\n \\end{bmatrix}\n ", "c2dab197d07a6eb7aff40459a201da35": "\n\\rho I\\frac{\\partial^{2}\\varphi}{\\partial t^{2}} = \\frac{\\partial}{\\partial x}\\left(EI\\frac{\\partial \\varphi}{\\partial x}\\right)+\\kappa AG\\left(\\frac{\\partial w}{\\partial x}-\\varphi\\right)\n", "c2dac77c6009bbaa59d718547918706d": " {\\rho}\\left({{\\partial}u\\over{\\partial}t} + {u{\\partial}u\\over{\\partial}x}\\right) + {{\\partial}p\\over{\\partial}x} = 0", "c2dacc5c799d28f072aec8ab113cd2a0": "\n\\vec{u} \\cdot \\vec{k} = 0,\n", "c2db1c502fa8073e85f0021d39574b87": "wfe/({P}.\\sqrt{T})= [wfe/({\\delta}.\\sqrt{\\theta})] * \\sqrt{288.15}/{14.696})", "c2db3512719291186a50b993cc1f54a1": "\\mathrm{STA} = \\tfrac{1}{n_{sp}}\\sum_{i=1}^T y_i \\mathbf{x_i},", "c2db90535840c3e7fb0688ee38ebdc6d": "\\Omega = \\Big(\\int_{\\delta_i} \\omega_j\\Big)_{1 \\le i \\le r, 1 \\le j \\le s}.", "c2db96c92e69e9cc05a0e38da5eb670a": "f'' - (d(x'') - d(bs^{-1}(x'')))", "c2dc98bd41c18927a90172ad157317fc": " \\forall z ( (\\exists x (\\phi \\lor \\psi) ) \\rightarrow \\rho )", "c2dc98d7400ee196b0ee109c6449ff08": " \\tilde{u}(x) = \\{ u(x) :k_x \\frac{\\partial^2 u}{\\partial x^2}+ k_y\\frac{\\partial^2 u}{\\partial y^2} +q =0 \\text{ for } x \\in \\Omega , u(x)=u^*(x) \\text{ for } x \\in \\partial\\Omega, k_x\\in{\\mathbf k}_x, \\ k_y\\in{\\mathbf k}_y \\} ", "c2dcdb4d8a06815c6eff39188e1576c5": "y\\in\\sum_n", "c2dce1035c6c50d617009531cd82d573": " P = \\sum_ { i_n = S_n, .. i_2 = S_2, i_1 = S_1} K_{i_1 i_2 .. i_n} ", "c2dd12bcbbbe81a8a1d1831c755ff8a6": "\\lambda=-i\\alpha{c}", "c2dd174e1ef7ec397cc69f2e05f8dd1c": "(\\phi \\lor \\psi) \\,", "c2dd44de52110db6a58ea982ef62f0b1": " \\cos \\varphi \\cdot 0,32^{\\prime\\prime}", "c2ddbcbf000d35ab795d0fad89c69726": "S_{R} := \\partial B_{R}", "c2ddc3261467ccf6ca7a9364e0afd642": "\\mathcal{L}_\\mathrm{gf} = -\\frac{1}{2}\\operatorname{Tr}(F^2)=- \\frac{1}{4}F^{a\\mu \\nu} F_{\\mu \\nu}^a ", "c2ddde21e39f333048f9a7628458651f": "\\omega_{pe} = \\sqrt{\\frac{n_e e^{2}}{m^*\\varepsilon_0}}, \\left[rad/s\\right]", "c2ddecb55fa0be2f8e4f805df98bd38a": "2(s^2Y-s\\cdot 0-0)+3(s Y-0)+Y=\\frac{5}{s}. ", "c2de1fa6094d73142823310d55776435": "(-\\infty,+\\infty)=\\R", "c2de2424960286dbc9d11e28b85cc528": "1959 = [1, 8, 39]_{40}", "c2de6cc35e4be0c35b43eccf46beca81": "y_0 = S-1", "c2dea71c2030c0a1b616c2e0a6b1e752": "\\sigma^2_f\\approx\\left| \\frac{\\partial f}{\\partial a}\\right| ^2\\sigma^2_a+\\left| \\frac{\\partial f}{\\partial b}\\right|^2\\sigma^2_b+2\\frac{\\partial f}{\\partial a}\\frac{\\partial f}{\\partial b}\\text{cov}_{ab}.", "c2df0ae04af93ae996dc67350efe43fd": "C_p^\\infty({\\mathbb R}^n,{\\mathbb R}^m)", "c2df71baf80e270db8de823a5e22c761": "f+g", "c2df83dc495b1b68e2d2d40952000960": "m \\ge \\dim \\operatorname{supp}F", "c2dfb599df888dea44ccb8abff761dc7": "j_{xy}\\ ", "c2dfd202a4f035a5968c97973ad6d1e3": "V_d \\times F_a = V_t \\times (F_a - F_e)", "c2dfe9f608ce05d5844cad343aece6b6": "K = 4.4934x^2 + 4.3034y^2 - 4.276xy - 1.3744x - 2.5643y +1.8103", "c2e029cf57d756aea672ea02cf498249": "\\operatorname{gr}_I R", "c2e07e1d5708e0049758f419470aeae5": "[x,y]:=\\frac{\\sum_{j=1}^n x_j\\overline{y_j}|y_j|^{p-2}}{\\|y\\|_p^{p-2}},\\quad x,y\\in\\mathbb{C}^n\\setminus\\{0\\},\\ \\ 1 \\zeta_G ( \\alpha_2 )", "c2e22473f3fdd2e324712c0c73256688": "\\mathbf{I}_2", "c2e24a48ef1dd1cd7c0e6e90a3d86501": "W+1", "c2e24d7bb59d40c3ad1761e1c54871e9": " \\overline{u'_i u'_j} = 2/3 k \\delta_{ij} - \\nu_t \\left( \\frac{\\partial \\overline{u_i}}{\\partial x_j} + \\frac{\\partial \\overline{u_j}}{\\partial x_i} \\right), ", "c2e262a35d5eeb726a65f2beed1b6c1a": "(\\omega^2 m + \\omega c + k)x = 0.", "c2e273350ec12e188beb24b217452d76": "k(f)|_{f=f_p}=k_p.", "c2e281652cccfcd68096ba0250297786": "\\left(3 \\frac{\\log N}{\\log \\log N}\\right) ^{1/3}", "c2e28b881b4cf5d3cf14dc029707ead3": " \\mu_q", "c2e2a97b0037d91ffe70176c7864ef02": "t_3 = 1213121", "c2e2b258514d991b7c989e69de620e63": "S : G \\to F_{S}", "c2e2b8a71ab3324c9d3c884462d9e2db": "r''_2", "c2e305b40379d96d54aa275a810049b7": "\n\\sin \\theta = \\frac{\\rho}{r + \\rho}\n", "c2e35f4b854884913feb42a77859d6c7": "\\ \\det(\\mathbf A) = a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + ... + a_{nj}C_{nj} = \\sum_{i=1}^{n} a_{ij} C_{ij} ", "c2e3d0c91a1b27eca701e5086e7d1b45": " \\left(\\sin(x)\\right)' = \\cos(x)", "c2e407507b7f00b2b51748226bef6fe1": " \\frac{d\\varepsilon(t)}{dt} = \\left ( E_1 + E_2 \\right ) ^ {-1} \\cdot \\left [ \\frac{d\\sigma(t)}{dt} + \\frac{E_2}{\\eta}\\sigma(t) - \\frac{E_1E_2}{\\eta}\\varepsilon(t) \\right ] ", "c2e4112973e7bf20f05b896262dd8e39": "\\ell^\\infty(\\mathbf{N}) ", "c2e48a5e3769d677f433947862d6b85a": "s_1^2 = \\frac{1}{n_1-1} \\sum_{i=1}^{n_1} (x_{1,i} - \\bar{x}_1)^2", "c2e4d3016c63397ca072577a3310a203": "ln \\frac{P_t}{P_{t-1}} = \\frac{1}{2} \\sum_{i=1}^{n} \\left (\\frac {p_{i,t-1}q_{i,t-1}}{p_{t-1}q_{t-1}} + \\frac {p_{i,t}q_{i,t}}{p_tq_t} \\right) ln\\left (\\frac{p_{i,t}}{p_{i,t-1}} \\right)", "c2e551584611b60611357f1c2347053c": "K \\to V^* \\otimes V", "c2e594c3657f2ea7ad7db5ec2a0e20ec": "\\left(\\tfrac an\\right)=\\left(\\tfrac bn\\right)", "c2e5b2db4fa5371f244ca77846c1c6e6": " \\xi + x := \\sum_{i=1}^N \\delta_{X_i + x} ", "c2e5eb28f26359e6b140732926ec4874": "(\\cos\\phi,\\sin\\phi)", "c2e5f024c4bc0a187b796ff45f1e53e9": "[HA]_i \\approx \\frac{v_0 [HA]_0-v_i[OH^-]_0}{v_0+v_i}", "c2e638cee76ab92c625a7e3a12328cad": "p_a \\cdot a = p_b \\cdot b = p_c \\cdot c = \\frac{2}{3} T. \\,", "c2e686c1adf53a56f956b7eaf721401f": "p \\mid m_i \\land p \\mid m_j", "c2e6971fd54c50ddf22875b6419ee4e4": "\\mathrm{e}^{-\\mathrm{i} \\epsilon_v \\, t/\\hbar}", "c2e6a8de87c0851b5f8e223ae6964de3": "t_{n+1}", "c2e7991e5c18705f4777bb5955be21ef": "\\displaystyle{(\\widehat{f},\\widehat{g}) = (f,g)}", "c2e8058c236987872ef549ee04e21e9e": "\\operatorname{succ} \\equiv \\lambda n.\\lambda f.\\lambda x. f\\ (n\\ f\\ x)", "c2e80f58be47363234401467406b2e9c": "\n\\frac{n!}{\\prod_{k=1}^n (k!)^{j_k}} \n\\prod_{k=1}^n \\left( \\frac{k!}{k} \\right)^{j_k}\n\\prod_{k=1}^n \\frac{1}{j_k!}\n=\n\\frac{n!}{\\prod_{k=1}^n k^{j_k} j_k!}.", "c2e82adf2849ed7efb6c5891616977ee": "f:\\mathcal{X} \\rightarrow \\mathbb{R}", "c2e846d3833e3623b8e5548bb21d2d20": " |\\Delta_n(s)| \\leq C \\, 0.7^n~. ", "c2e8e40410faf4f2b0f0dc464d2c8012": "\\{\\Omega,\\mathcal{F},P\\}", "c2e910059d0e4d17c680532f55949083": "(-B, B)", "c2e946eb3ad8acab984a9cfeac9faf9c": " a_2 ", "c2e98696f927b0690473ae0c12bd083e": "\\frac{V}{T}\\;", "c2e9b0df03f846b3aec397c5c8812a63": " R_{5} ", "c2e9b485d2dfc63762aff0864cefdaf4": "(-b_2,a_2)", "c2e9c3eec44fa0ec85ab6f170764e39e": "\nx + i y = a i \\cot\\left( \\frac{\\sigma + i \\tau}{2}\\right)\n", "c2e9c4e08b9818c59c4d16d532cc1d4d": "\\mathbf{e}_i \\times \\left( \\mathbf{e}_i \\times \\mathbf{e}_{i+1}\\right) =-\\mathbf{e}_{i+1} = \\mathbf{e}_i \\times \\mathbf{e}_{i+3} \\ ,", "c2e9cd753f63fa7cea5799194bd34b47": "x_L = x_{0}^{2^{L}}~mod~N", "c2ea07ab1ba647b0b055327be225ce0a": "p, q, \\dots", "c2ea2953a399018dc6ea331551e5f752": "\\star \\mathrm{d}y=\\mathrm{d}t\\wedge \\mathrm{d}z \\wedge\\mathrm{d}x", "c2ea2aef8ce064909c46381ba1a5fadb": "\\sigma/\\sqrt{n}.", "c2ea3ff285c8537c2e3d9a7db7c554e5": "G_k", "c2ea7316972202d1fe6108e44a325c29": "f\\left(\\frac{az+b}{cz+d}\\right) = \\varepsilon(a,b,c,d) (cz+d)^k f(z).", "c2eac7c20ac50fb571b744b616dc4e9f": "\\omega(f)(x)=\\limsup_{h\\to 0} \\frac{\\int^{x+h}_{x-h}|f(t)-f(x)|\\, dt}{2h} \\le f^*(x) +|f(x)|.", "c2eaed442ee5009155a53309347801a7": "S(A,P)= \\sum_{a\\in A}\\sum_{d\\mid a;\\,d\\mid P} \\mu(d) =\\sum_{d\\mid P}\\mu(d)|A_d|,", "c2eb418a677536edfac660ca5f4065f3": "G(p,q) = \\mathcal{F} \\left\\{ g(x,y) \\right\\} \\equiv \\iint_{-\\infty}^{\\infty} g(x,y) e^{-i 2 \\pi (p x + q y)} dx dy ", "c2eb617b9300c06fa6d8fce753c4769b": "{{i}_{IN}}-{{i}_{OUT}}=\\frac{2{{i}_{IN}}}{\\beta \\left( \\beta +2 \\right)+2}\\approx \\frac{2{{i}_{IN}}}{{{\\beta }^{2}}}", "c2eb97f9ca311d9dd7a361c98b316311": "S^{\\nu, \\lambda} (\\omega)", "c2eba25f7aeef91729a4c200ece94168": "\\mathbf{\\delta^1_5}=\\aleph_{\\omega^{\\omega^{\\omega}}+1}", "c2ebc95c4560212c6bcf70a286a6f125": "\\gamma = \\sqrt{(R + j \\omega L)(G + j \\omega C)}", "c2ebebf9260e7003157786260111bbf0": "\\varphi\\colon S\\to T", "c2ec2cc75f3fe0610ca7f9a12074aa9e": "{4 \\choose 2}^2 = 36", "c2ec4e8ee9339601f23373ff41996b08": "E_H = E^0 - \\frac{0.0592}{n} \\log\\frac{[C]^c[D]^d}{[A]^a[B]^b} \\ \\{ V \\} ", "c2ec689f91f733912e7068471c287c74": "\\{(x,y)\\in A^2 : ax^2+y^2=1+dx^2y^2\\}", "c2ec74129f15727b685c74edcb10ead4": "\\vert \\partial^\\alpha \\mathbf{v_0}(x)\\vert\\le \\frac{C}{(1+\\vert x\\vert)^K}\\qquad", "c2ec773faa351e186a74f3dc7cf519a3": "\\sigma = \\left| c \\right|{{\\Delta t} \\over {\\Delta x}} \\in \\mathbb{ N} , \\quad \\quad (11)", "c2ecfb2abf7f728a4ce0f1a0a089bc4a": "X_{th}", "c2ed3ae6e2b90b5cd5d55c2507bc1379": "T \\left(n \\right) = \\Theta\\left(f(n)\\right) = \\Theta \\left(n^2 \\right).", "c2ed63f1a266f44fe45cb707a22b12f5": "\\mathfrak{m}_S \\cap R = \\mathfrak{m}_R", "c2ed6ea3159c0f96140992e3b739192c": "f(x)=\\frac{2x}{1-x^2} ~~~~~ \\forall x\\in D~", "c2ed86ee657da5914fec9455a5a4fcbe": "E_1=\\frac{v_1^2}{2g}+y_1 = \\frac{v_2^2}{2g}+y_2=E_2", "c2ee06e36048819d0ecbde4831656397": "\\sum_{i=1}^k \\frac {n_{i}c_{i}}{m_{i}}", "c2ee9f54c8ce831db4d4348e8d5225df": "\\Delta x\\,", "c2eeb3773d0d078f7076c9d908532b1e": "\\!\\mathcal A \\models_{X}^+ \\phi", "c2eed42fcc30e9b29142850593c1834c": " A_{24576}=3.14159261864 < \\pi <3.14159261864 +0.0000001021", "c2eee02ac8418fdcdde181ac9de6bb4d": "\\mathbb{E}\\{ \\left(\\mathbb{E}\\{X|\\mathfrak{G}\\} \\right) |\\mathfrak{G}\\}=\\mathbb{E}\\{ X |\\mathfrak{G}\\}.", "c2ef45e465246e15f2d3f3bdd1f2efb0": "\\sqrt{4\\pi\\varepsilon_0\\alpha}.", "c2efa8ee2dc5d5f992b1b0129bccb5a2": "\\vec{x} = x^i(t) = \\begin{bmatrix} x^1(t) \\\\ x^2(t) \\\\ x^3(t) \\end{bmatrix} ,", "c2efe0daa58365bf6f1b622630768d50": "\\frac{a}{-b} = \\frac{-a}{b}.", "c2f03051a635fa1e2be1e50d6e655b9b": " E^*(z,s) = \\pi^{-s}\\Gamma(s)\\zeta(2s)E(z,s)\\ ", "c2f03ab731d7b6991c6b0e29934b329d": "\\hat{E} \\Psi = i\\hbar\\dfrac{\\partial}{\\partial t}\\Psi = E\\Psi ", "c2f065c1e292480f08cf1ddfc87ac605": "\\begin{align}\na & = \\left\\lfloor\\frac{14 - \\text{month}}{12}\\right\\rfloor && \\mbox{(1 for January and February, 0 for other months)}\\\\\ny & = \\text{year} + 4800 - a \\\\\nm & = \\text{month} + 12a - 3 && \\mbox{(0 for March, 11 for February)}\n\\end{align}", "c2f06b87f41d4ed1e93cc5c22ce4214d": "e^{i\\omega t}\\ ,", "c2f06c9e87ae3d15ce9d02a85e2d2359": "x_a(t)=\\int_\\R WT_\\psi\\{x\\}(a,b)\\cdot\\psi_{a,b}(t)\\,db", "c2f07fa1b6f7e9639fe1c5c40eadc297": "\\alpha \\delta \\twoheadrightarrow \\beta \\gamma", "c2f0ce5237cae63af83961bab0478c3e": "t+1:\\ z^{t+1}", "c2f12e4056f32600e5804618e1d60198": "\\frac{\\partial}{\\partial t}\\left\\langle \\frac{\\partial f}{\\partial t},\\frac{\\partial f}{\\partial s}\\right\\rangle=0", "c2f1378c3b51e5443eca65d57260cf8c": " t \\leq z = t + y < t + (w + 1) = \\frac{(w + 1)^2 + (w + 1)}{2} ", "c2f13d44bee77928b94b52c5566de389": "\\pi_\\Sigma(s)\\div a = \\pi_\\Sigma(s \\div a )", "c2f165f951516a97fc6cd1264440403a": "\\mathbf{X}_B", "c2f209ff6eaa8536318ead1dc01a86af": "P(A)", "c2f20b6139a9a6a34979dc57f88c03f0": "\n\\min\\ \\frac{1}{2}\\,{\\mathbf{v_d}}^T\\,\\mathbf{I}\\,\\mathbf{v_d} + (\\mathbf{-v_w})\\cdot\\mathbf{v_d} \\qquad s. t.\\quad \\mathbf{S}\\cdot\\mathbf{v_d}=0\n", "c2f251ade41e0f64bd909c172e262b06": "p_0 = n", "c2f276e90adf7ddba5309553356973a2": "\\int_{a}^{b} f(x|a < X \\leq b)dx = \\frac{1}{F(b)-F(a)} \\int_{a}^{b} g(x) dx = 1 ", "c2f2db512d0ede5a8a3f75a774aa6370": "{{\\left\\| f \\right\\|}_{\\Beta ,p}}", "c2f33e6c183e9fdbdf22c860242152c4": "|\\Psi(t)|", "c2f3a62e7ef2fb8eb48d79ba4d64f8fc": "\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s} = \\frac{1}{1^s} + \\frac{1}{2^s} + \\frac{1}{3^s} + \\cdots.", "c2f3b668f7b91c73e72a0a02ec554282": " E_0 \\le \\frac{\\langle \\Psi | \\hat{H}| \\Psi \\rangle}{\\langle \\Psi | \\Psi \\rangle}. ", "c2f4327714e9e53f03296752d9d7ea9c": "\n e^\\frac{x}{y} = 1 + \\cfrac{2x}{2y - x + \\cfrac{x^2} {6y + \\cfrac{x^2} {10y + \\cfrac{x^2} {14y + \\ddots}}}}\n", "c2f43a5b78510289d4c850a27fce78ba": "P^0", "c2f4518277dc0202940ffd3815318c93": " {} + a\\cdot c\\cdot 100", "c2f454fa571d9606d6577e9012545c06": " U \\subset X ", "c2f4acac0ac03257699c93d54e0a0989": "2/c^2", "c2f4c18db68c436489c4ebf3dea1b0e1": "\\sin(\\phi)", "c2f4e69f66436f4142f3c365b3c4720c": "\\textstyle A_m A_0^{-1}", "c2f537bc0b26ca60b152f4425313511b": "\\alpha(v_i) \\mapsto a_j", "c2f59bf5ee07bc77c0a5b0319fb77933": "\\hat{A}= a|\\psi^T\\rang\\lang\\psi^T| + b|\\psi\\rang\\lang\\psi|, ", "c2f5c4320ca16b028e51d6594dac7b4a": "\\,\\! \\sum_{n \\epsilon CW}V_n = \\sum_{n \\epsilon CCW}V_n", "c2f5e1d0f65a1da4be2d6121ccca33f3": "\\nabla^2 R(r) = \\left[{n (n - 1) \\over r^2} - {2 n \\zeta \\over r} + \\zeta^2 \\right] R(r) ", "c2f5f14623fbdf737857bdda51b74af1": "E[ \\epsilon ] = 0", "c2f5fc2c72f6e8657e4314930646a1f9": "[\\mathbf{e}_a,\\mathbf{f}^b]=\\delta_a^b", "c2f600caa924d0898aaf89614e927dc2": "r\\varepsilon", "c2f62548f340b03dc6d368d6a6c21ac6": "(G, L)", "c2f687e2cad607b73888d0661963fe07": "\\operatorname{cosh}(z)", "c2f6dbafc9d53b75d655126c49c986e3": "S_{ij}=r_{ij}-s_{ij},", "c2f6ef23ae50853cabbfb7f818af6f1d": "{\\mathit{He}}_3(x)=x^3-3x\\,", "c2f6fa16e1654c069aa21af241fd5517": "H=2t+1 \\mbox{ and } P=2s ", "c2f70478beb25bd06ccf81df434c4bd1": "\\beta (\\omega)=\\frac{2 \\hbar \\omega}{I^{2}} W_T^{(2)}(\\omega)=\\frac{N}{E}\\sigma^{(2)}", "c2f73c0ec9cfc01d1fff8d9fbf6c1503": " ~r^2={(x_0-x')^2+(y_0-y')^2+z_0^2} ", "c2f7610c70863eab48c4d94c5a8fea1a": "\\alpha \\in M[X]", "c2f7747ca4bc5df626b2672cd8d9f405": "\\vec{M}", "c2f7a0cd29162ebf276f0c9a0353d34d": "R_{\\mu}=\\frac{2G}{c^3}mc\\,U_{\\mu}=r_s\\,U_{\\mu}", "c2f836f597de9581bf1c3df22740157c": "N_i=\\frac{dX_i}{dX}\\,\\!", "c2f8671812437413382502753710d649": "D_{j}\\, ", "c2f88c52be5cde553e853dc95e9f9749": " \\bold x^{(m+1)} =\n\\begin{align}\n& \\frac{1}{12} \\begin{pmatrix}\n-1.2 & 4.4 & 6.6 \\\\\n-0.33 & 0.01 & 8.415 \\\\\n-0.8646 & 2.9062 & 5.0073\n\\end{pmatrix}\n\\bold x^{(m)} +\n\\frac{1}{12} \\begin{pmatrix}\n11 \\\\\n-36.575 \\\\\n25.6135\n\\end{pmatrix},\n\\end{align}\n\\quad m = 0, 1, 2, \\ldots \\quad (16) ", "c2f89db101ed20311636ab6a89f8be9a": "V_n(a+b,ab)=a^n+b^n \\,", "c2f8b847d136466fc578e9cc089b172c": "\\alpha = 2 \\arctan {d \\over 2 f}", "c2f8b9b41b1df0f98cde1ae0c42c8e39": "\\scriptstyle d=2 ", "c2f92a9ce80ae67bd48181aff2510db5": " x \\in \\mathbb{R}. ", "c2f9581fe67ef3f2e10c4ebeb1984d4e": "i=i_0 \\frac {nF} {RT} (E-E_{eq})", "c2f96a746a433e91f7f39711917abb3c": "\\ mg=2\\pi r \\sigma", "c2f96aadfb69ad6c2d94a7f1c4bc777e": "\n\\alpha_{A^{2-}}={{K_1 K_2} \\over {[H^+]^2 + [H^+]K_1 + K_1 K_2}}= {{[A^{2-} ]} \\over {[H_2 A]+[HA^-]+[A^{2-} ]}}\n", "c2f98029f9ee3446b0d113a146a3a15c": "V^{n+1}_X = V^n_X + \\Delta V^{n+1}_X", "c2f9804b0e458d6591bd5981d8ba7032": "V(\\tilde\\beta)- V(\\widehat\\beta)", "c2f9886410b54a8b4b5932ce87dd619b": "\\frac{dS}{dt} = \\dot S_{i}", "c2f9a05db995224c8a797601f43e809c": " a_1 \\mathbf{v}_1 + a_2 \\mathbf{v}_2 + \\cdots + a_n \\mathbf{v}_n = \\mathbf{0}. ", "c2f9adf43a4b604e1cf6b683e6445511": "u = e^{\\lambda t}", "c2f9e1c91046c71a22c9805b306082ee": "t_{rip} - t_{0} \\approx \\frac{2}{3|1+w|H_0\\sqrt{1-\\Omega_m}}", "c2f9e986567c22282aeec85010f5189f": "(a_1,\\ldots, a_n) \\mapsto \\langle \\overline{x_1-a_1}, \\ldots, \\overline{x_n-a_n}\\rangle,", "c2fa18ee173fe332725d597f4d72075b": "\\sum_{n=1}^{20} $50,000 {\\left( \\frac{1}{1+0.02} \\right)}^{n-1} = \\frac{1-{\\left( \\frac{1}{1+0.02} \\right)}^{20}}{1-\\left( \\frac{1}{1+0.02} \\right)} = $833,923.10", "c2fab34197a5d2319d0ddf517bbcf8f2": "S^3 \\subset \\mathbb{C}^2", "c2fb0bdbfd1d36177bf0c61887a67d84": "\\sum_{i=1}^N p_i = 1. ", "c2fb42b6ce2f70ffab56c9776eac011d": "\\Sigma_{n,m}", "c2fbd899da56f2eb559c15b755e1ffe5": " A + B = 1 ", "c2fbf6033d5a0ce01d7a90cb488d1474": "\\left\\{\\frac{\\partial}{\\partial \\theta}\\,,\\Theta\\right\\}=\\left\\{\\frac{\\partial}{\\partial \\theta^*}\\,,\\Theta^*\\right\\}=1", "c2fc7214bd5f662e712e787ff7f01b74": "U \\to \\mathrm{Lin}(X; X')", "c2fc80e3dc9d7b498a9a16ebf8bf4b7e": "= \\sgn( \\cot( \\frac{\\theta}{2})) \\frac{1}{\\sqrt{1 + \\cot^2 \\theta}}", "c2fc8fe69b220c19967b2100a755de5b": "\\therefore c_1 \\mathbf{w}_1 + \\cdots + c_n \\mathbf{w}_n \\in \\operatorname{ker} \\; T", "c2fc93fce52391d915ee93a3d617b58d": "\\mu(x,G)=\\bigcup\\{B(x,a)\\colon a\\in A(x,G)\\}", "c2fcab0aa50d853fa011caeb548123b9": "t_{1/n} = \\frac{\\ln n}{\\lambda} = \\tau \\ln n. ", "c2fd24fddee17c12d841d4e7c9adcf89": "\\mathbf{x} = (\\mathbf{I - A})^{-1}\\mathbf{y}", "c2fd915cc283070fe958adf56bc428b5": " \\frac{b^{2n+1}}{n!} \\to 0\\text{ as }n \\to \\infty. ", "c2fd999e6b61966368f41109b3bda57b": "Q_2(f) \\leq Q_E(f) \\leq D(f) \\leq n", "c2fe5340a5466e371645539454c531cc": "I\\propto \\left( E_\\mathrm{sig}\\cos(\\omega_\\mathrm{sig}t+\\varphi) + E_\\mathrm{LO}\\cos(\\omega_\\mathrm{LO}t) \\right)^2 = E_\\mathrm{sig}^2+E_\\mathrm{LO}^2+2E_\\mathrm{LO}E_\\mathrm{sig}\\cos(\\omega_\\mathrm{sig}t+\\varphi)\\cos(\\omega_\\mathrm{LO}t) ", "c2fe6dfdabb480d3c1014f62c66a28c9": " \\langle \\sigma_i \\sigma_j \\rangle_\\beta \\geq c(\\beta) > 0.\\,", "c2fec2ab873e085be9e2a656f9c3176c": "\\begin{matrix} \\frac{1}{3} \\times 2 = \\frac{2}{3} < 1 \\end{matrix}", "c2fef6bc21200b82ad92f680b4556f62": "p = \\left(\\frac{1-\\|u\\|^2}{1+\\|u\\|^2}, \\frac{2\\mathbf{u}}{1+\\|u\\|^2}\\right) = \\frac{1+\\mathbf{u}}{1-\\mathbf{u}}", "c30019000eed84e7960f4e0610840877": "\\left\\{\\left(y, x\\right) : xRy\\right\\}", "c300237abbbe71f8a4618e8d9f608689": " \\|\\mathbf{U}\\|^2 = {\\gamma(\\mathbf{u})}^2 \\left( c^2 - \\mathbf{u}\\cdot\\mathbf{u} \\right) \\,,", "c30049f6e71b435a1a9ba573dfc25646": "\\Delta v = c \\cdot \\tanh \\left(\\frac {I_{sp}}{c} \\ln \\frac{m_0}{m_1} \\right)", "c30052af9364af47bfdcf3525a37d6e4": "| \\mathrm{Ai} ( x + iy) | \\, ", "c3005537d52b9126e17c46ac261179c2": "E_{\\mathbf{p}}", "c300a4940ebdea00fd7f54b63f368614": " F=1 ", "c300c510c54ce18a950929d2ee1bda41": "x^2-3x-3", "c3014a0e9998729985af3f0aac0ac420": " cf(\\kappa) = \\omega ", "c30201b8bfc6896d5ed4ef2652f07ed8": "R_{modern} = 0.95R_{HOxI,-19} = .7459R_{HOx2,-25}", "c30282602c124b0b62d427ea55d6d405": "\\langle \\cos(\\textstyle\\phi\\,\\!) \\rangle", "c302aa950500345f5da19752365a8748": "z=ax^2+a", "c302de48833b51303bec99d79aa9d910": "|A| + |B| = \\{C \\cup D \\mid C \\sim A \\wedge D \\sim B \\wedge C \\cap D = \\emptyset\\}", "c302e101039d58869fa0a534e3473008": "e\\in \\mathcal U", "c303081f7a16f603112b0375bdc84883": "n=2", "c3033fc016940111b256a5442ca0ddda": "T = \\mathbb{N}", "c3034478174a12155c002598739f90e6": "\\mathbf{v} = \\mathbf{T}\\cdot\\mathbf{u}", "c30409f9eb3e9578c2cd856abcc17f73": "MAP \\simeq DP + \\frac{1}{3}PP", "c30421622f2ab1815b789e05c266abc8": " b_n \\ ", "c3049bc050db679d14d5c9fcc7897c74": "V_Q/V_T", "c304a4926bc6d84894a9fa92a63b7d31": "\\boldsymbol{\\delta}=-(\\mathbf{v}_0\\cdot\\mathbf{v}_0)\\mathbf{r}_0+(\\mathbf{r}_0\\cdot\\mathbf{v}_0)\\mathbf{v}_0+\\frac{\\mu}{r_0}\\mathbf{r}_0-\\alpha_J\\mathbf{r}_0", "c304c26e964539a700bd3e43d7582a75": "z=\\frac{(\\hat{p}_1 - \\hat{p}_2)}{\\sqrt{\\hat{p}(1 - \\hat{p})(\\frac{1}{n_1} + \\frac{1}{n_2})}}", "c304fb7ae13fb50a5ae015a12cfa817a": "T^k M", "c304fdc2fd7cf9c1e390f33fb45fff19": "{dy \\over dx}\\sqrt{1-\\sin^2 y}=1", "c305e609cabc21aa562538e852f461ea": "A^{2-} + 2H^+ \\rightleftharpoons H_2A :\\beta_2=\\frac {[H_2A]} {[A^{2-}][H^+]^2}", "c30659db237b31f9cd0aa2cc9ee093d0": "\\left[\\frac{b+c-a}{b+c}, \\frac{c+a-b}{c+a}, \\frac{a+b-c}{a+b}\\right]", "c3066ce21fffc5bc8621c349dab4c309": "v_{p,i}", "c306bb7fba57c63dd93189d763722ff4": " \\int_{a}^{b} f\\, dx = \\int_a^b{\\lim_{n \\to \\infty}{f_n}\\, dx} = \\lim_{n \\to \\infty} \\int_{a}^{b} f_n\\, dx.", "c306ce40b59891c07956ecca58102fea": "f_n(x) < 0 \\text{ for } x = \\cos \\frac{(2k + 1)\\pi}{n} \\text{ where } 0 \\le 2k + 1 \\le n", "c306d28ba8bd1e3a0986276c4221bb42": "_{metric} \\delta_{ck}^2", "c30787c07468ff7428148bc7a038edb4": " > ", "c30798fa27d380cfc82bb4dc08441264": "\n\\langle H_{\\mathrm{pot}} \\rangle = \\frac{1}{2} k_{\\rm B} T - \n\\sum_{n=3}^{\\infty} \\left( \\frac{n - 2}{2} \\right) C_{n} \\langle q^{n} \\rangle\n", "c30799252bd74648962234ac71bdbe4e": "L'_w", "c307a3fc73c08ba2f04689f62ff6a42b": "\\beta_k=0", "c30801dc88b27ab9392c3b377bb08207": "\\displaystyle A_4=S(b^{-1})M", "c308939651804f73350ea97e93d66d47": "\n\\mathrm{SINADR} = \\frac{P_\\mathrm{signal}}{P_\\mathrm{quantizationError} + P_\\mathrm{randomNoise} + P_\\mathrm{distortion}}\n", "c308a1430a13368c52743667f92fa0cb": "E \\left(r^N \\right ) = E_\\mathrm{bonds} + E_\\mathrm{angles} + E_\\mathrm{dihedrals} + E_\\mathrm{nonbonded}", "c30901842bdc6ed8dc2662d70e1357fa": "=\\frac{2n-1}{2n} \\cdot \\frac{2n-3}{2n-2} \\cdot \\frac{2n-5}{2n-4} \\cdot \\cdots \\cdot \\frac{5}{6} \\cdot \\frac{3}{4} \\cdot \\frac{1}{2} I(0)=\\pi \\prod_{k=1}^n \\frac{2k-1}{2k}", "c30902496e2cf0b9eb7cf340332ae738": " \\qquad \\mathbf{K}_{\\text{Poiss}} = \\frac{1}{2 \\pi} \\begin{pmatrix} \\frac{3A}{\\sigma_x \\sigma_y} &0 &0 &\\frac{-1}{\\sigma_y} &\\frac{-1}{\\sigma_x} \\\\ 0\n &\\frac{\\sigma_x}{A \\sigma_y} &0 &0 &0 \\\\ 0 &0 &\\frac{\\sigma_y}{A \\sigma_x} &0 &0 \\\\ \\frac{-1}{\\sigma_y} &0 &0 &\\frac{2 \\sigma_x}{3A \\sigma_y} &\\frac{1}{3A} \\\\\n \\frac{-1}{\\sigma_x} &0 &0 &\\frac{1}{3A} &\\frac{2 \\sigma_y}{3A \\sigma_x} \\end{pmatrix} \\ .", "c30904d7cb184fecfb82041f94261fad": "|\\vec{A}|>1", "c3090d0e0de9f7f25295acd2adaa5dbf": "\\frac{\\Gamma \\vdash A, A, \\Sigma}{\\Gamma \\vdash A, \\Sigma}", "c3093b1b95d84ef3e3727f20872ab928": "\\{({I}^{2},{\\psi}_{\\mu},{L}_{\\mu})\\}_{\\mu\\in M}", "c30968fd87acccaa0f00de3d4b68a083": "\\scriptstyle{\\sum_i\\mathbf{l}_i = \\sum_i\\phi^l_i\\mathbf{L}}", "c3098d031cae8784f0fd2b7d937e647e": "\\textstyle R[t]/t^nR[t]", "c309f0daf5910cf7ac2038ce9520448a": "2d", "c30a1fb1a2e0c8df7f0f809ed3be89e4": "r = \\rho \\, (1 - m/\\rho)", "c30a4409f92a86906dae6d31e51c5408": "p > 1", "c30a4cdf3d35d13cd6a9e1591bab0da0": "\\kappa = \\theta-\\phi \\approx v/c ", "c30a54719b3d9b27eb9fe9271c6fc563": "\\dot T(t)\\ ", "c30a8220e7416834864ead40cdbcdbbb": "\\Phi = \\Phi_a + \\Phi_s.", "c30aad53a2c837971aa366fda05a9ad0": "\\mathbf{T}^{(\\mathbf{e}_i)}= T_j^{(\\mathbf{e}_i)} \\mathbf{e}_j = \\sigma_{ij} \\mathbf{e}_j.", "c30ada376f7755f55f8bf9ebcd0fdbac": "\\forall X ((0\\in X \\land \\forall n (n\\in X \\rightarrow Sn\\in X)) \\rightarrow \\forall n (n\\in X))", "c30ae17b2a201eefa2034187d490d475": "\n A_k(t) = \\sum_{j=1}^{M} X(t+j-1) E_k(j).\n", "c30b4b58f29a8760d2cece6fb175c82d": "\n \\sigma_{11} - \\sigma_{33} = \\lambda_1~\\cfrac{\\partial{W}}{\\partial \\lambda_1} - \\lambda_3~\\cfrac{\\partial{W}}{\\partial \\lambda_3}~;~~\n \\sigma_{22} - \\sigma_{33} = \\lambda_2~\\cfrac{\\partial{W}}{\\partial \\lambda_2} - \\lambda_3~\\cfrac{\\partial{W}}{\\partial \\lambda_3}\n ", "c30bb4698f5ea35fc31b7a5cb6333e7b": "\\text{NC}(S_1)", "c30bba6d30a95134bd219266da4b2d8a": "x = (2+\\cos 3t)\\cos 2t", "c30c56c2270072656bcc6de78fd2df6c": "R(\\tau) = \\int_{-\\infty}^\\infty S(f) e^{j 2 \\pi f \\tau} \\, {\\rm d}f", "c30c5f605246717965ae9a1ea9c0e837": "M_\\max", "c30cd871d3956bfdbe08642ca5fa38bc": "|z| = \\sqrt{x^2 + y^2}.", "c30d1f75a032d85af378a0de98281679": "a = 0.38", "c30d20f7aae31ca93929192906ee0916": "M\\sqcup M=\\partial (M\\times [0,1])", "c30d386fcea08ee417ff284d1e20eef9": "(e^{-2\\pi\\mathit{i}l})", "c30db097b511cad3923001d1345ced56": "m\\frac{dU}{dt}", "c30dcad07831afec65ee224c5db7c5a8": " [v]_C = [M]_{C}^{B} [v]_B,", "c30ddb106e6a111c43e80a59f51f39e1": "[N]", "c30e27f025bc390246739ba5e225bce1": "V_n(P, Q)", "c30e459880f4aa78fc0bda2037a9a719": "T_m(t)\\frac{}{}", "c30e59f96f8cc6c78d8ad7ee69812080": "E_\\text{pot}\\, =\\, \\overline{\\int_{-h}^{\\eta} \\rho\\,g\\,z\\;\\text{d}z}\\, -\\, \\int_{-h}^0 \\rho\\,g\\,z\\; \\text{d}z\\, \n =\\, \\overline{\\frac12\\,\\rho\\,g\\,\\eta^2}\\,\n =\\, \\frac14\\, \\rho\\,g\\,a^2,\n", "c30e894161d6baccf8ca777769588f99": "\\sigma_2 - \\sigma_1 < 0", "c30f07fde1b900e7ff4e9028d2595caa": "g={1\\over 2}(d-1)(d-2).", "c30f246120ce4128022ab08f1d3fb2e5": " \\frac{1}{r}\\frac{\\partial}{\\partial r}\\left(r \\frac{\\partial u_z}{\\partial r}\\right)= \\frac{1}{\\mu} \\frac{\\partial p}{\\partial z}", "c30f41d99f66b5d71645f7781dac9a72": "j(\\lambda _A) > \\alpha", "c30f57f7c96bd49b871cee1e5aecd6dd": "n_{air} = \\frac{A_{av} P}{RT} = \\frac{(6.02\\ast 10^{23} \\frac {molecules}{mol}) \\cdot (101325 Pa)}{8.314 \\frac{J}{mol \\cdot K}\\cdot 273 K}", "c30f8a74c898ed699cecbff229e4f85a": "d(x) =\\widehat{\\theta}(x) - \\operatorname{E}( \\widehat{\\theta}(X) ) =\\widehat{\\theta}(x) - \\operatorname{E}( \\widehat{\\theta} ),", "c30fe0582ba6819355a7ec6b8e183f99": "H: X(t) = N(t)", "c310a4c0f1d5b3121f620e6d3fd5cdf5": "\\Delta G_A^\\circ ", "c310d0c7fd50bf19f35f4a8015d82fb0": "T(M,E) = \\exp\\left(\\sum_i (-1)^ii \\zeta^\\prime_i(0)/2\\right) = \\prod_i\\Delta_i^{-(-1)^ii/2}.", "c31166919ac2da6b3d120d4d8b3125ff": "A(t|\\nu) = F_\\nu(t) - F_\\nu(-t) = 1 - I_{\\frac{\\nu}{\\nu +t^2}}\\left(\\frac{\\nu}{2},\\frac{1}{2}\\right),", "c31167412efec01449482276274dbd02": " H |\\psi(t)\\rang = i\\hbar \\frac{\\partial}{\\partial t} |\\psi(t)\\rang", "c311905eab60237306c75dfad5d0ca29": "G = (V, E)", "c3119dafb84573ab49ab958a714442af": " \\frac{1}{\\lambda} ", "c311c5346c0729868082750cf3e85781": "|\\mathrm{cat}_o\\rangle \\propto|\\alpha\\rangle-|{-}\\alpha\\rangle\n", "c311d4e096669d1cf5b368c5b69aa560": "R_T = R_0 \\left[ 1 + AT + BT^2 \\right] \\; (0\\;{}^{\\circ}\\mathrm{C} \\leq T < 850\\;{}^{\\circ}\\mathrm{C}).", "c311ff8c21af741a0127a61c3730b30d": "dF = \\left(\\frac{\\partial F}{\\partial x} \\frac{\\partial x}{\\partial u} +\\frac{\\partial F}{\\partial y} \\frac{\\partial y}{\\partial u} +\\frac{\\partial F}{\\partial u} \\right) du + \\left(\\frac{\\partial F}{\\partial x} \\frac{\\partial x}{\\partial v} +\\frac{\\partial F}{\\partial y} \\frac{\\partial y}{\\partial v} +\\frac{\\partial F}{\\partial v} \\right) dv = 0", "c3120140a73aac51df0bbb838838b1db": "|f(z) - f(w)| > |z - w|", "c312107b09983382e73a32ecf835ba5c": "\\frac{4096}{2187}", "c3121226330b879f4c272b213b050428": "\\cos \\theta = \\frac{x'(s)}{\\sqrt{x'(s)^2 + y'(s)^2}} = x'(s) \\ .", "c31240c1d274cecab7a06af3bee37750": " (P \\rightarrow Q) ", "c31286120727c9188f3e5082ad65bd64": "\\begin{matrix} {2 \\choose 2}{2 \\choose 1}{46 \\choose 1} \\end{matrix}", "c3129ae5240397838d6b9f3718157028": "1.4826\\ \\approx 1/\\left(\\Phi^{-1}(3/4)\\right)", "c312ce3d9a361f61c7980db798eafbfd": "\\rho\\! = \\frac{ m }{ V_s\\! }", "c312d6b819ad20b5a8af78b811597f43": "\\frac{1}{R_{\\text{INIC}}} > \\frac{1}{R_s} + \\frac{1}{R_L}, \\quad \\text{(i.e., when} \\, R_{\\text{INIC}} < R_s \\| R_L \\text{)}\\,", "c3131baeca46ce37f586fdf21b333887": "T = \\{ 2 \\le \\rho \\le 3, \\ 0 \\le \\phi \\le 2\\pi, \\ 0 \\le z \\le 5 \\}", "c31343dd1010a4600c38106a5567c984": " A^{-1} ", "c3134e438848f30127a408fc375a5f3f": "y \\in S", "c3135122df82844e8a4954196d1e61e8": "\\scriptstyle\\Omega", "c31354cc74d23bef4b1d85208f488898": "\\hbar = h / (2 \\pi)", "c31385e0c685d8d608f40bc473a25c54": " \\mathbf{x}^{(1)}=\\begin{bmatrix}\n 0.0075 \\\\\n 0.002 \\\\\n -0.20944 \\\\\n\\end{bmatrix}", "c314d689a7447f4e7c6710d9a1f4f7b9": "X_{2\\pi}(\\omega) = 2\\pi \\sum_{k=-\\infty}^{\\infty} \\delta (\\omega +a -2\\pi k)", "c31543a2e52de47472416cbdfda4eb5f": "g_3(\\tau)=g_3(1, \\omega_2/\\omega_1)", "c315afb11f4862274c9425393ca36f55": "N_c = 10 N - d\\left(10^m-1\\right), \\, ", "c315d4f8b9a88e71a92e85bc85b0834d": "\\textstyle \\Omega=\\{A,A^c\\}", "c31619177fb6ad982ec08433bc0ea532": "\\scriptstyle g_{\\mathrm{Alice}}", "c3162067e039f430bb1ab443a8c4b741": "\\mathrm {DOF} = \\frac {2 f ( 1/m + 1/P ) }\n{ ( f m ) / ( N c ) - ( N c ) / ( f m ) }\\,.\n", "c31631ef02b1a2452773e49eb25e52ef": "z^{-k}", "c3165b4eb5459fe980d44a2703077f38": "\\color{blue}\\mathcal{S} \\color{blue}\\rightarrow \\color{blue}\\mathcal{E} \\color{blue}\\rightarrow \\color{blue}\\mathcal{I} \\color{blue}\\rightarrow \\color{blue}\\mathcal{R}", "c3166f4c549fc471a062a5a5d82f574e": " c = M_{2,1} = -2\\,", "c316ab9d453dd89c01a6fdb29cfb28de": "1/\\sqrt{x}", "c316daa83dc25635ea62f0d883fa48bd": " H = h A I_z J_z + \\frac{hA}{2}(J_+ I_- + J_- I_+) + \\mu_B B(g_J J_z + g_I I_Z)", "c317267c171075e93c5a6a6e8191c53e": "t_{1}=t_{2}\\,", "c3172e6c050720fd78e63550734e3750": "\\lim_{a\\rightarrow\\infty}\\int_{-a}^a\\frac{2x\\,\\mathrm{d}x}{x^2+1}=0,", "c3176c92907c72be07d58ab28b3aa150": "\\approx 2.0", "c31786492e9eec9fb7f73aa965e7ec6c": "{{v}_{MIRROR\\_OUT}}={{v}_{BE2}}+{{v}_{CE3}}", "c31787649d083a639f19932fffd21240": "S = I/c", "c317fc278ddaa6a8256b2b8291924ab6": "\\begin{align}\ng^{(j)}(t)&=\\frac{d^j}{dt^j}f(u(t)) = \\frac{d^j}{dt^j} f(\\mathbf{a}+t(\\mathbf{x}-\\mathbf{a})) \\\\\n&= \\sum_{|\\alpha|=j} \\left(\\begin{matrix} j \\\\ \\alpha\\end{matrix} \\right) (D^\\alpha f) (\\mathbf{a}+t(\\mathbf{x}-\\mathbf{a})) (\\mathbf{x}-\\mathbf{a})^\\alpha\n\\end{align}", "c31800e52dcc9ddc8eb520d983197c7d": " \\lim_{b \\to 0} \\mathrm{Benktander}(a,b) \\sim \\mathrm{Pareto}(1,a+1) . ", "c3181a13344068d3c5bcd8ed2bbfa626": "\ns_\\mathrm{L}^{\\circ} = \\sqrt{\\alpha \\dot{\\omega} \\left( \\dfrac{T_\\mathrm{b} - T_\\mathrm{i}}{T_\\mathrm{i} - T_\\mathrm{u}} \\right)}\n", "c3185e8e50111a9de9c4cbff7d4675c2": " F_{\\alpha\\beta\\gamma\\mu}=3\\partial_{[\\gamma}T_{\\alpha\\beta]\\mu}.", "c318f0757c0304fe971228b3e25d3ccc": "(\\nu, \\mu, \\sigma^2)=(2\\alpha, \\mu, \\beta/(\\lambda\\alpha))", "c31933d8328097e5fa36476344f3cd09": "[TN] + [T_{M/N}] = 0", "c3195681898e98f30852a6b042fffca2": " \\hat T_{2,1} = - \\frac{1}{2}( \\hat a_{z} \\hat b_{+} + \\hat a_{+} \\hat b_{z} ) ", "c3198a6dbef629ca31403b4ccdff3fc7": "2\\pi", "c31a5ff9edcdb680e0f5cb62067f65a5": "\\mathbf{s} = \\operatorname{col}(S_1,\\ldots, S_{3N-6})\n\\quad\\mathrm{and}\\quad\n\\mathbf{Q} = \\operatorname{col}(Q_1,\\ldots, Q_{3N-6}),\n", "c31a73a2df724163b2bf51b66c54005f": "(a_1,a_2,\\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \\ldots (a_m;q)_n.", "c31a7879fe9fc81124d5363d07866989": "8Fe + S_8 \\longrightarrow 8FeS", "c31aa25677c6c10dd89e11ac0a4dfb97": " c_p ", "c31acd6bae64dedaf2906e051f5ddaa1": "\\mathbf{k}\\cdot\\mathbf{p} = k_x (-i\\hbar \\frac{\\partial}{\\partial x}) + k_y (-i\\hbar \\frac{\\partial}{\\partial y}) + k_z (-i\\hbar \\frac{\\partial}{\\partial z})", "c31b4ff6f3f52575192f1cd6bd95ba91": " \\Psi_s = F_s(\\mathbf R)\\Phi_s(\\mathbf R,\\mathbf r_1, \\mathbf r_2, ...., \\mathbf r_N)", "c31b500d6c5f2ca441e2093b19eee754": " M_{i,k}= M_{i,j} M_{j,k} ", "c31b9df62ed9894713d8b988f2fe103c": "P(Z_{m,n}|\\boldsymbol{Z_{-(m,n)}},\n\\boldsymbol{W};\\alpha,\\beta)", "c31c9ab8e05979e102d53d6e4d81ce28": "\\delta_2>\\delta_1", "c31cd5e98cb8adb97d4c9b0f286171bc": "|U_2|^2", "c31d2b7df15fa7d119c2f8d13f69e10b": "[c,d]", "c31d2f99140c07753c1be35233f9cb8c": "\n\\overline{n_\\mu ^k}=(-1)^k\\frac{\\partial ^kH_\\mu (s,t)}{\\partial s^k}|_{s=0}.\n", "c31d594ebe6e39bc2d99c28e7a620bb0": "\n\\int_{z=0}^\\infty\\int_{t=0}^1 e^{-z} (zt)^{x-1}(z(1-t))^{y-1}z\\,\\mathrm{d}z \\,\\mathrm{d}t\n=\\int_{z=0}^\\infty e^{-z}z^{x+y-1} \\,\\mathrm{d}z\\int_{t=0}^1t^{x-1}(1-t)^{y-1}\\,\\mathrm{d}t.\n\\!", "c31d60a75c2c2ec3ee3177d7b7cc43f7": " p_{S} = (p_{i_1},\\ldots, p_{i_r})", "c31d7a179e1aba693c298dccc33dc965": "\\tfrac{1}{s^{2}+1}", "c31d7bac3f54ea68c0b7a89606829d38": "\\scriptstyle \\leq-9\\times10^{-16}", "c31d9473ff7e7badec05a4e4230eb1e6": "M = \\left(\\frac{\\partial z}{\\partial x}\\right)_y, \\quad\n N = \\left(\\frac{\\partial z}{\\partial y}\\right)_x", "c31dfa3ba208e1bd395c0fb562e51573": "S \\rightarrow A: \\{B, N_A, K_{AB}, T_B\\}_{K_{AS}}, \\{A, K_{AB}, T_B\\}_{K_{BS}}, N_B", "c31e0ac4dcc74c761e5b68c071e6c2f7": "\\| u \\|_{W^{k, p}(\\Omega)} := \\begin{cases} \\left( \\sum_{| \\alpha | \\leq k} \\| D^{\\alpha}u \\|_{L^{p}(\\Omega)}^{p} \\right)^{\\frac{1}{p}}, & 1 \\leq p < + \\infty; \\\\ \\max_{| \\alpha | \\leq k} \\| D^{\\alpha}u \\|_{L^{\\infty}(\\Omega)}, & p = + \\infty; \\end{cases}", "c31e78d1378e173da026d03345b50b90": "\\rho(\\mathbf{r}, t)= qn(\\mathbf{r},t) ", "c31e7e0c3951d61d397f6da60a2149ee": "d(a_0\\otimes \\cdots\\otimes a_{n+1})=\\sum_{i=0}^n (-1)^i a_0\\otimes\\cdots\\otimes a_ia_{i+1}\\otimes\\cdots\\otimes a_{n+1}", "c31e8c6ed78566e6eaa1a94c824f7c46": "\n =\n\\exp\\left( {- {i \\over \\hbar } V \\left( q_j \\right) \\delta t} \\right)\n \\int { dp \\over 2\\pi } \\exp\\left( {- {i \\over \\hbar } { { p}^2 \\over 2m} \\delta t} \\right) \\langle q_{j+1} |p\\rangle \\langle p |q_j\\rangle\n", "c31e9b5c3d6ba53256bd5231b9f77d9d": "\\chi(\\mathbf X,t)\\,\\!", "c31f4729cc5e57c3689d0486af648609": "TYPE\\triangleleft\\text{symbol}", "c31fa0c51e3cf2769f4c4746749ea000": "10^4-10^7", "c32034ad1e14f69caa056c8c0e56b2d4": "\\mu = \\bar{x}", "c32038026e6d2e99239fb9f5b5d7abe6": "g_{uc}^{-1}(\\{ \\langle A \\rangle, \\langle bb \\rangle \\}) = \\{ \\langle a \\rangle\\} ", "c32058b034d7f478a80553f4ab2b80ac": "\\sin^{-1} x = \\arcsin x.", "c320649d4aba7669659c1047597873ed": "\\mathrm{ber}(x) = 1 + \\sum_{k \\geq 1} \\frac{(-1)^k (x/2)^{4k}}{[(2k)!]^2}", "c3206dc13c9ffebcd419cf6ff330e06c": "a x^2 + b x + c = 0", "c320713e450a4590e960eaffadc2e824": "q \\,", "c320d4c77f3778a6c88becead91af005": " \\theta (1 + p\\alpha) = p\\alpha", "c320e22383769e0d02c689d2ad7d4fef": "a_{A1} + \\cdots + a_{An} = a_{G1} + \\cdots + a_{Gn} = 1 \\,", "c320fa48df84c170460d1cabc8126440": "N=1019", "c3210bf7bd2144bd6fc731b84edee8a3": "{}^t u", "c3212f5ba39ef91db9997dc018da3caa": "\\{0\\} \\subset \\{0,1\\} \\subset \\{0,1,2\\}", "c32255a41290dfa37c138099c6355b05": " J_n = \\sqrt { \\frac { n } { 2 } } ( a m^{ b - 1 } - 1 ) ", "c322ffdc45198044f2ab6f95bb0b8200": " y=Y/Z ", "c323065214a77f879b31fdaa19a48a91": "v^0 \\in V", "c3231b93a7ef2eb5c13f3b8eb33003dd": "I \\left( x, y \\right)", "c323331da632254e00b022a7529533ff": "g(x,y)", "c32343b81c00f074c0407b13df7b2f70": "\\left\\{ z \\in \\mathbb{C}\\ |\\ \\mathrm{Im}(z) > 0 \\right\\} ", "c323672baae45e70d962e88bb95c7a3a": "x^\\prime", "c323c90bb5ebd1525c553473b90d5c1a": "F_n=\\bigg\\lfloor\\frac{\\varphi^n}{\\sqrt 5} + \\frac{1}{2}\\bigg\\rfloor,\\ n \\geq 0.", "c323fdc4816e07e26b853b431e838e30": "{{P}_{V}}X\\left( u,\\xi \\right)=\\int\\limits_{-\\infty }^{\\infty }{\\left\\{ X\\left( u+\\frac{\\tau }{2} \\right)X\\left( u-\\frac{\\tau }{2} \\right) \\right\\}}{{e}^{-i\\xi \\tau }}d\\tau ", "c3242877b1062d5c239a22b18898a49d": "0\\leq y_4< n_{i,j,k}", "c3242bf940537dc780955f674044525c": "\\frac{d^2\\Gamma}{dx\\,d\\cos\\theta} \\sim x^2[(3-2x) + P_{\\mu}\\cos\\theta(1-2x)].", "c3242ef94510e1a8b7646d67162f6cf6": "(x_2, y_2) = (bx_1, y_1)\\,", "c324301aaaed142f8df0637da132ace8": "E = {{CV^2} \\over 2}", "c32449dc600637143d1065c81a95e081": "Y(a,z)Y(b,w)c, Y(b,w)Y(a,z)c, Y(Y(a,z-w)b,w)c", "c32467fd361f88d8e500e518b1ad2dbf": " \\theta_{r} = \\lambda_{p}(\\theta_{r,0}-w)+(1-\\lambda_{p})\\theta_{r,0} ", "c324a4753d0aa5e5912811bd3f370df8": "\\{\\rho_F : F \\in \\Sigma,\\ \\mu (F) < \\infty\\},", "c324b6aeebc2fecd55a9dcdeecea2783": " \\Delta G^\\circ_{form} = G(T)compound - \\sum \\left \\{ G(T)elements \\right \\} ", "c324c46d2970ad5466c756f69a24f573": "b^{e-(p-1)}", "c324db024b7ae545b9f54b58bd33dff4": " \\gamma\\cos\\,{\\theta^*}=\\sum_{n=1}^N f_{i}({\\gamma_\\text{i,sv}}-{\\gamma_\\text{i,sl}})", "c32529fa1805d82181f73d291d31be7d": "a_n = \\left(\\prod_{k=0}^{n-1} f_k \\right) \\left(A_0 + \\sum_{m=0}^{n-1}\\frac{g_m}{\\prod_{k=0}^m f_k}\\right)", "c325461fd19932339615c5d5cb339ed1": "nw(Y)\\leq w(X)\\,", "c3254cce41d8f1a48bc9380ba5d476a3": "n^{th}\\,", "c325560236f7060423a3cf8d5cb6ccc6": "K_i(C; G) = \\pi_i(B^+ C; G)", "c3255c87eb33915cf79c30148bba7374": "T=\\sqrt{p^{2}c^{2}+m^{2}c^{4}}-mc^{2},", "c325840922e8ec6ed7f7c242ac375302": " v = v_2 - v_1 = (v_2 - v_0)-(v_1 - v_0)", "c3259222e96ae17412bb169b36c8f350": " \\mathbf{p}=[\\underline p,\\overline p] ", "c32595b064159a26cf14cf4874eba51a": "\\scriptstyle 3 \\times 3", "c325a8ec15b0654d44b453440502ba89": "f \\in {\\mathbb Z}[x]", "c325cca89456bd6755882f3a3aa6c67f": "\\frac{\\alpha:\\beta_1,\\ldots,\\beta_n}{\\gamma}", "c325cfa04c70953ccc99d576e5dd257e": "\\oint_W\\,\\{d_{\\,total\\,}U\\}\\, \\stackrel{!}{=}\\,0.", "c325d11c6faab98084632e46a1a17a17": "\\varepsilon \\cdot (b-a)", "c32611badbb4f84808c3e0fb1e663ce5": "M=\\Pi_{i=1}^l e_i^{m_i}", "c3265078d496c5cfee800fc2b6431cfd": "\\mu=2", "c3267fae2e64369f9a4d38a968937759": "\\varphi(x)=2\\ln\\left(\\frac{x}{1-x}\\right).", "c326ef80b0fd05f36f76c28f62f5590f": "\\mathbf u \\times \\mathbf v", "c3272809fdede4760996f3d98f6c2d13": "\\left\\langle M_B\\left(l_B\\right)\\right\\rangle", "c3277121bda976cd3eb0b2d61d737884": "f(a,b) = (-a; ab)_\\infty \\;(-b; ab)_\\infty \\;(ab;ab)_\\infty.", "c327f223a8dcdd0aa11a6b8a6bb06901": "A_0 \\to A_1 \\to A_2 \\cdots \\to A_{k-1} \\to A_k", "c327f631384dbccaa2e958ebbc2d9465": " B = \\{ b_1, \\ldots, b_n \\}", "c3280d901d56e93320d9163044c07b9c": "\\{\\,Y(t) : t \\geq 0 \\,\\}", "c328124ce350402ef209b1e547475c21": "\\operatorname{E}(\\mathbf{X})", "c32816fcb19eccf284f4910f25ae371e": "2^{O(\\sqrt n\\log n)}", "c3282427561c2857c813ce5916fb3d18": "P_{(k)} \\leq \\frac{k}{m} \\alpha.", "c32834c518755b3fcb2e3fa854d18e96": " L = \\mathbb{Q}(\\sqrt{-3}, \\sqrt{5}) ", "c328409e4128c6f588d1e09ee4b83597": "\\textstyle P_y ", "c32852a682b988ce07fd65c434c7a085": "x \\mapsto x^2\\sin(1/x)", "c328a8347363da7e98e67c5379ed4a85": "\\chi^{(3)}_{xxxx} ", "c328b556827685dd0e1396b5371e9171": "m(y|\\eta)", "c328e137f362145df7947d556441e7a9": "\\theta_{\\text{min.}} = 6M", "c32903096936fa648f1b4ac00f9fe1ea": "(i-1,j-1,k-1,*)", "c3292a4dc782133f41fc33582a1697c4": "L_z=i\\frac{\\partial}{\\partial\\phi}", "c3294e208564579802d80e2345f4d64e": "R \\sim \\mathrm{Rice}\\left(\\nu,\\sigma\\right)", "c32b292fc671fb9a8072547fba188201": "\\operatorname{Br}(k)", "c32b67c07d33f82d065adc543a6b0cd7": "\\left[A\\rightarrow aA,B\\rightarrow bB,C\\rightarrow cC\\right]", "c32bcb3f349ba6c5c72c985ca645198f": "H=-\\frac{1}{3}nvclA\\frac{dT}{dz}.", "c32c4c5229465e5ee2964c145b239fc7": "M_F = F L \\sin\\theta", "c32c4d4226d7f04e52b4273a33ceeddc": "\\omega^n", "c32cd68d7e285591cd258082de0b926e": "Z_e", "c32ceabd8a55daa2d9bcd0587cd16d12": " \\cfrac{\\Gamma \\vdash A, \\Delta}{\\Gamma \\vdash A \\or B, \\Delta} \\quad ({\\or}R_1)\n ", "c32d0060fed87819ed874750ef623d8b": "\\sqrt{n(N+\\epsilon)}", "c32d07fec030e78927a687ae1778d174": "\\star \\circ t = \\star", "c32d46f4dca48ef819024bc26559b356": "\\,E[X|Y=y]-E[X]", "c32d989c20b71798f461b365995603df": "\n \\begin{bmatrix}\n \\sigma_x & \\tau_{xy} & \\tau_{xz} \\\\\n \\tau_{xy} & \\sigma_y & \\tau_{yz} \\\\\n \\tau_{xz} & \\tau_{yz} & \\sigma_z\n \\end{bmatrix}\n", "c32d9d0aeea0e3d8eb5f8fc44186e6a7": "h_A:\\mathbb{R}^n\\to\\mathbb{R}", "c32e0169a5263c0733b3b4882853d4fd": "\\ell_i < x_i < u_i, \\; F_i(x) = 0", "c32e1db88f3dcb7a63dd51fe53519694": "K \\epsilon \\{ 0, 1, 2, ..., N-1 \\} ", "c32e90432d45f7e59591ca638e364ec4": " s=\\frac{3}{4} \\!", "c32eb64ca4df7f2235107b84c103841e": "\\exp(t\\Delta) f(x) = \\frac{1}{(4\\pi t)^{n/2}} \\int_{R^n} e^{-|x-y|^2/4t} f(y) dy", "c32ecc701095bb911e93934c84ee5ea1": "\\beta_k = \\frac{(-1)^kk!}{\\pi} \\text{Im} \\left(\\int_{-\\infty}^{\\infty} \\frac{e^x}{(1+e^x)(x-i\\pi)^k} dx \\right)", "c32ef9653ede9387a27ee27b4610f432": "q \\le 2/3", "c32f3235e78c92e75e6932693216f4ea": "\\sum_{\\Pi\\succeq\\Lambda}(\\Pi)", "c32f3523f75fcdc4e95a7f81fab7a55a": "a_{2}+b_{3}", "c32f6ce590157f30967b60d53bf5bc8e": " \\sum_{k=1}^{n} |A_{k}|^2 = |A_{0}|^2 ", "c32f6f56ad77b2612b87cb4f4a68fa80": " b \\in I ", "c32f9f991219a5da993f74634e379537": "L_{x^m y^n}(\\cdot, \\cdot; t) = \\partial_{x^m y^n} g(\\cdot, \\cdot;\\, t) * f (\\cdot, \\cdot).", "c32fc81e8103df6160c5b3991d5cb55b": "\\{X_{i}\\}", "c330a51e2aa5ff5f6789ea22284d28e8": "M = M_0 + nt", "c330d2801d622ac366fb73cb76574888": " n, m, q, d ", "c33145929f0c78788451467d9a4beb0a": "\nV_C = e_2,\n", "c3316d66cb9da21aa6c1b87b603c4f4a": "\n \\ {Y^*}_{\\ell}^m (\\theta_0 , \\ \\phi_0) Y_{\\ell}^m (\\theta, \\ \\phi) \n\n", "c3316e24ab200c0e9a5557e73ca2ce83": "\\frac{1}{s+\\beta} = P + { R (s+\\alpha) \\over s+\\beta }.", "c331939ca5bc43e6b4f40330727a3149": " \\eta_{Xe} + \\left|\\eta_{Qe}\\right| > 1 ", "c331a0557d1768fd9638adf8a5d48051": "x \\sigma", "c331efb97de9d11e2424e20f2823a5ce": "\\tilde C(\\alpha,\\rho) \\cap \\Omega = \\rho", "c3320271690616a9bc748105e2f4f645": "\n \\boldsymbol{F}\\cdot\\boldsymbol{F}^{-1} = \\boldsymbol{\\mathit{1}}\n", "c33206387a013371d821207c5075e8fd": "X^p-X = T^{-1}", "c33252391061395556b54f53bf87cc59": "\\mathcal{X}=\\left( a_1 , \\ldots ,\na_{|\\mathcal{X}|} \\right)", "c3329e5526b5f8373508ea6eb9ba6644": " g_{obs}\\to 0 ", "c3329fa212e21dfdba139a35395dff77": "\\tfrac12Bh", "c33312d20d8d95207a74b3e97504dbec": "L = L'.x", "c33319c768c1baa2e2e837e8173512bf": "\\alpha_n = -\\delta_n \\Omega_n \\,", "c3331cf315288932796e2b6cc4547cde": " 2^{|k_{f_1}|}+2^{|k_{b_{n+1}}|}+2^{|k|-|s_n|}...", "c33332cb55fc5586cd62f24c459caf92": "\nf(\\beta, 0)=-\\frac{1}{\\beta}\\ln \\int_{-\\pi}^{\\pi}d\\theta'_j\\;e^{\\beta J \\cos\\theta'_j}\n", "c33347c18db5a59e1bf4c893ad316320": "(\\nabla f)_i = \\mathrm{d} f^\\mathsf{T}_i", "c33356fd64fce25888610d3f6c9adf53": "\\{K_n\\}", "c33394419887d3881eafce2e5402b39f": "\\lambda _{p} = (-0.244 + 1.556\\cdot x + (4.31\\cdot 10^{-4})\\cdot t\\cdot (1-2\\cdot x) - 0.65\\cdot x^{2} + 0.671\\cdot x^{3})^{-1}", "c333c6287835b579ee906aa34d8aa060": "c_i=\\lang {k_i} | \\psi \\rang", "c334048bf03bcb3c289da00acec00aed": " \\Delta f /f ", "c334151bb015a47c1e0e6f3595a96c88": "\\Delta \\mathbf{S}=\\nu \\Delta S", "c334f66966ac67fa33a3c34f005fb304": "\\underset{a_{i},d_{i},h}{\\mathop{\\min }}\\,L(a_{i},d_{i},h,i=1,..C)", "c33501e34655ac95f626ab02e77132a6": "\\mbox{Ann}\\,(S) := \\{ n \\in N \\mid \\forall s \\in S, F(s,n) = 0\\}", "c33583c11b7f03e9266fcaedd1a144cd": " \\beta \\approx 0.9 ", "c335a9f586ef8cf4152c841762e2f7c0": "G=\\langle a, b| ab a^{-1}b^{-1}\\rangle.", "c335ab3760b27c40707f1e952b28fbca": "L_n, \\bar{L}_n, -1\\le n\\le 1", "c33611574675a468c30e1292b27b3ce8": "X=\\mathbb{A}^1-0", "c3365a925a0c730293ffc620f70483e7": "\\frac{\\mathrm{positive\\ zero\\ crossings}}{\\mathrm{second}} = \\frac{\\mathrm{negative \\ zero\\ crossings}}{\\mathrm{second}}", "c3366238b99297bdad1ff2f6ce05d959": " \\max(0, x - const) ", "c33685ee4813823d82737f7e8d1c95c3": " \\mathrm{A} = \\frac{\\rho_1 - \\rho_2} {\\rho_1 + \\rho_2} ", "c3369f3d08ccf8d98d5498bfac41c771": "T^{a}", "c336fb3606a9c5136b99db10b2e31f5d": "\\epsilon \\rightarrow 0", "c33721d042f3d31a32878dd57e5104f6": " T = \\sqrt{5}:2. ", "c337328c31e56ec70fc2f389cac71cd1": "|1/2 + z|", "c33758d3f07f9a7512678ca1d9d632f1": "\\frac{\\sin (nx)}{(\\sin x)^n} = \\left[ {n \\choose 1} \\cot^{n-1} x - {n \\choose 3} \\cot^{n-3} x \\pm \\cdots \\right].", "c337c369d725f10fe50b3eae3cead4ee": " U = \\frac{1}{2}k \\Delta x^2\\,", "c3394cf44de070f7648da763260e0729": "\\|x\\| > \\|y\\|", "c33985802169f581c45e7eafc77ee280": "(\\mu, \\nu, z)", "c3399cf20c0d98532db9694ee97d8356": "\\mathrm{NTIME}(2^{2^{\\cdots{2^{O(n)}}}}) = \\exists{}HO^i", "c339bc3aef7c654277330518af8e783b": "\\scriptstyle p(d_k \\;=\\; i),\\, i \\,\\in\\, \\{0,\\, 1\\}", "c339bf7d00005eb1ff7f2d5a1fcc0e1c": "G=\\frac{N_s-N_d}{N_s+N_d}\\ .", "c339fbf6823da53067d982d716b7de9d": "\n-\\frac{d}{dx}\\frac1{\\varepsilon(\\ln_k(x))^\\varepsilon}\n=\\frac1{(\\ln_k(x))^{1+\\varepsilon}}\\frac{d}{dx}\\ln_k(x)\n=\\cdots\n=\\frac{1}{x\\ln(x)\\cdots\\ln_{k-1}(x)(\\ln_k(x))^{1+\\varepsilon}},\n", "c33a0d03e01218dee4bfa5c0d82b70ad": "f(\\mathbf{x}^*)\\leq f(\\mathbf{x})", "c33a3f9a203bfe0cc594aa88555058e6": "f(x)|_{x=4}", "c33a6bf61eb0b841be9984ff658571ce": "\\mathbf{P} (X \\le a) = \\mathbf{P}\\left (e^{-tX} \\ge e^{-ta}\\right) ", "c33a818b3ac22ecafa96ed99d1c0b656": "P_d = 0.4 \\cdot \\;P_{2007} \\cdot \\;{\\frac{19}{20}} + 0.6 \\cdot \\left( \\;P_{Ap-2008} + \\;P_{Cl-2008} \\cdot \\;{\\frac{19}{18}} \\right) ,", "c33bba2e325a7775844fd4b1adcccb7c": "V(q)", "c33bcefea6d7fe1d3a401a9b34ab9ebf": " u \\to \\pm i \\infty ", "c33c025cc22a69dee60319600db585ff": "\\bar{x} \\pm 1.96 s", "c33c1ca67a8504427a04e387c67e6dab": "\\mathrm d U = \\delta Q - P \\, \\mathrm d V\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{(quasi-static process)} ,", "c33ca8cb70303efdfd5495314977e592": "y_1-\\frac{a}{4}=y_1-\\frac{a_3}{4a_4},\\; y_2-\\frac{a}{4}=y_2-\\frac{a_3}{4a_4},\\;y_3-\\frac{a}{4}=y_3-\\frac{a_3}{4a_4},\\;y_4-\\frac{a}{4}=y_4-\\frac{a_3}{4a_4}.", "c33ce2817cdd0c3782152f36181e103c": " A(Q, \\tau) =\\bigcup_{t = 1}^{\\tau} (A(Q,t) - A(Q, t-1)) \\cup A(Q,0) ", "c33ce9a2ffe05db5a47d7fea4f79d495": "u(x,t)=\\int_{0}^{t}\\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{4\\pi k(t-s)}} \\exp\\left(-\\frac{(x-y)^2}{4k(t-s)}\\right)f(y,s)\\,dy\\,ds ", "c33cf4ca1318d0d9c2190f1c1d691257": "\\text{M}^+~~\\xrightarrow{k_{t}}~~\\text{M}", "c33d22ae7ca1e4c271d761605344797a": " q = {\\rm max}_i(e_i . \\deg f_i)", "c33d2765dd61206ebfd3b2bf23c56c31": "\\operatorname{Br}(\\mathbb{R})", "c33d64db08a08f2e217646840180116b": " u \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac { \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert \\big ) } { \\sum_{j=1}^N \\rho \\big ( \\left \\Vert \\mathbf{x} - \\mathbf{c}_j \\right \\Vert \\big ) } ", "c33d8d41fb29e9427f3f24cc6989f00e": "\\tilde{\\gamma}(1)", "c33db8b3bc15b42db631ff2fb13263e2": "K_3 = \\frac{1}{cot(C)-cot(\\gamma)}", "c33e1d63c9b225d06d55ae1143588188": "(s,t) \\in R_G", "c33e70004cf9001948513394be0b730b": "k[y_{i,j}]/(g_{l,r})", "c33ead232870c7085d15fe582062ba24": " f^{(n)}(t) \\ ", "c33ebbe3ab04bc69e09fd82b683769dc": "\\scriptstyle\\mathfrak{F}", "c33ebe95f4e4259265632329faaf5b0f": " L_{ni} (\\beta_{n}) = \\frac{e^{\\beta_{n}X_{ni}}} {\\sum_{j} e^{\\beta_{n}X_{nj}}} ", "c33ed46d32a0ebd589e7d11b9e2706a3": "\\tan(\\phi) = \\frac { h\\sin(\\theta)}{hv/c + h \\cos (\\theta)}=\\frac { \\sin(\\theta)}{v/c + \\cos (\\theta)}", "c33ee13cbf0598f666a70379f385d20c": "[X_f,X_g] = X_{\\omega(X_g,X_f)} = -X_{\\omega(X_f,X_g)} = -X_{\\{f,g\\}}", "c33f223d96c4af32faeae47a84b6665a": " d = v_0 t_d \\cos(\\theta) ", "c33f355d5a40469e1620598ed862ac0b": "\\pmod{n_1n_2}", "c33f857feee16ff6c64c15448af897be": "\\lambda_J=\\sqrt{\\frac{15k_{B}T}{4\\pi G \\mu \\rho}},", "c33f8ecc6fa08aee90c499b39320396c": "(VA", "c33fba039a0f2f0d15d0115ce25316dc": "(A_B,r_B)", "c33fec8f17b1bb067a8cd8216dcb8d0b": "\\theta_1,\\ldots,\\theta_k", "c33ffd322c8c2eef2ebbdbf4af58e98c": "E(\\ln(x))=\\mu,E((\\ln(x) - \\mu)^2)=\\sigma^2\\,", "c33fffbe075dbb5d61f2d2497e44d8b5": "|z|\\le \\tanh {\\pi\\over 4} ", "c3406ed2154278a70ff8268e0979700f": "\n \\delta U = \\int_L \\int_A (\\sigma_{xx}\\delta\\varepsilon_{xx} + 2\\sigma_{xz}\\delta\\varepsilon_{xz})~\\mathrm{d}A~\\mathrm{d}L \n = \\int_L \\int_A \\left[-z~\\sigma_{xx}\\frac{\\partial (\\delta\\varphi)}{\\partial x} + \\sigma_{xz}~\\kappa\\left(-\\delta\\varphi + \\frac{\\partial (\\delta w)}{\\partial x}\\right)\\right]~\\mathrm{d}A~\\mathrm{d}L \n", "c340aae5ac14c28ed3c9b5a58f74e0d8": "f\\left(P\\right)", "c340c1b3fe7ea13da0cfe8139afe4690": "R \\xrightarrow[\\#]{} \\epsilon", "c340ca5152f3eddbaf2e9ae7fce5edc6": "[x,x] = \\{x\\}", "c340ee53546ca8cbc57157595433836d": "\\lim_{n\\to \\infty}a_n = \\lim_{n\\to \\infty}\\frac{g_{n + 1}^2}{g_{n}} = \\frac{g^2}{g} = g", "c340ff11d22649c4d40c3a91e7ad6d06": " {a} \\sin \\theta = \\lambda (n+1/2) \\,", "c341182f4c92ffb0fdc39acbdb5f5ea1": "\\Delta v \\ll v_e", "c3411e36a3a9fad89bfde99ee6ff7943": "(x + yz)/y = x/y + z + 0y\\ ", "c3414d0cc040c9e019305afb9f3a9460": "C^T", "c341552bdb1d4169c8716a3c0b00543a": " R_1 = \\frac {V_{CC} - V_A}{I_{R1}}\\ . ", "c341a720aa4af19915454f6be236ae1d": "R^1 = R\\,\\!", "c341eb974b70eb7211d7e2e4c5860879": "\\displaystyle s={1\\over 2} + i\\tau,", "c342019456234cbec149fb8087fc0510": "r\\circ i\\simeq 1_X", "c34201cd19e71ec91da714ab82b5b68e": "c\\!\\,", "c3420b68b04b8a45b20b2e0271d76895": "M(M+1)/2", "c3426acb7b5377956af7fab1a2d26828": " V_{fb}(t) = \\frac {C_s}{C_{fb}} \\cdot V_s(t-T) + V_{fb}(t-T)\\, ", "c3429463db5b25ae88f2772af9292580": " y(t_0 + h) = y(t_0) + h y'(t_0) + \\frac{1}{2}h^2 y''(t_0) + O(h^3). ", "c342df8cda00027e3d21552828a1c90f": "\\neg\\forall x \\overline{\\delta_1(x)}", "c342e5b7e4ba95281ff77029ca13cc25": "{\\rm Ci}(x) = \\gamma + \\ln x + \\int_0^x\\frac{\\cos t-1}{t}\\,dt", "c342e9f99a4265586642acc48934121f": " |T_j| \\geq \\frac{t}{2d}] \\geq 1 - n^{-d}", "c342f22c5b5a3a7c1d64b5c84b6b48db": "d : \\Omega^{k-1} \\to \\Omega^k", "c342f63497c2b3d7744280b453397137": "\n\\begin{align}\n S_{xx} &= \\left[ \\frac{k_x^2}{k^2} \\frac{c_g}{c_p} + \\left( \\frac{c_g}{c_p} - \\frac12 \\right) \\right] E,\n \\\\\n S_{xy} &= \\left( \\frac{k_x k_y}{k^2} \\frac{c_g}{c_p} \\right) E = S_{yx},\n \\quad \\text{and}\n \\\\\n S_{yy} &= \\left[ \\frac{k_y^2}{k^2} \\frac{c_g}{c_p} + \\left( \\frac{c_g}{c_p} - \\frac12 \\right) \\right] E,\n\\end{align}\n", "c342fc06e653de671940c6c6e195c0f7": "_k\\mathbf{V}^r", "c34308e28b08028a230bdb9f83e6d589": "u_{t}=ku_{xx}+f.", "c343c5c2a9537e8d14df0755ffc3131c": "V^{\\otimes 4}", "c343e5aa27d9c065941c7b880c64026d": "(i,j = 1,2,3)", "c343f7278362433fe1461d1d5524f7cb": "FF_j", "c343fb0118a2b9da37772493cf5a0868": "C=\\pi_1\\pi_2^a=\\left(\\frac{\\dot{m}}{\\eta r}\\right)\\left(\\frac{p_x\\rho r^5}{\\dot{m}^2}\\right)^a", "c3440077f43fede179d28e7c04ffb071": "aI_n-A=0", "c3441ac12340e92fc76d5ab267cb0773": "G=0\\,", "c3444082d5ce65ae31915d61f7f65143": "x \\wedge x = x", "c3444ef64dabbfaa8e1123300b0bf69c": "\\mu_{water}", "c3451d72b461efdeebc67ac95eabc663": "wk\\rightarrow0", "c345985c1a15d67a367f81a6b5ca3db4": "c_{3,1}(\\widehat{a}, w(c_{3,1}(\\widehat{a}, w(\\widehat{\\epsilon}, \\widehat{b}c), \\widehat{d}), \\widehat{b}c), \\widehat{d})", "c345992b947e89433a83ec02fa0e3de5": " h_1(X_1,\\ldots,X_n),\\ldots,h_n(X_1,\\ldots,X_n) ", "c345a25d1ff3a6299e15a2f27c74a5ea": " a_1,a_2, \\text{ . . .}", "c34608d1f052028e2ba456dc2a648fa0": "\\underline{X}(u_1\\ldots,u_m).", "c346762143128a4e49b8052a8d4afd63": "Slovak Republic: 19% \\cdot \\euro 1,000,000 \\cdot \\left[ \\frac{1}{3} \\cdot \\frac{\\euro 150,000,000}{\\euro 200,000,000} + \\frac{1}{3} \\cdot \\frac{\\euro 5,000,000}{\\euro 8,000,000} + \\frac{1}{3} \\cdot \\frac{\\euro 65,000,000}{\\euro 200,000,000} \\right] = \\euro 107,667.", "c346bce3554d3856d47c30d53cbfdd14": " \\mathrm{d} \\mathbf{u}_{\\rho} = \\mathbf{u}_{\\theta} \\mathrm{d}\\theta \\ , ", "c346ec7c6b39574a16b3be37713bf30d": "\\psi(x)-x < -K\\sqrt{x}", "c34728247194cb56292039d151c521e3": "x+0=x", "c3476050a143d880a6916420bb331088": "C_1 = D_1", "c347a40bdd8bd145fd2f87b07ed2fe42": "t_{1}=\\frac{AB}{c+v}+\\frac{BC+DA}{\\sqrt{c^{2}-v^{2}}}+\\frac{CD}{c-v+\\Delta v} ", "c347c26758830137d1622b48008bdb50": "Az \\neq 0.", "c347d2734be9efdc82bd5018bb051da0": " s = \\sqrt{\\frac{1}{n} \\sum_{i=1}^n (x_i - \\overline{x})^2 } ", "c347d72b31a8588d34efd72aeb842a31": "\\{\\{v_1, v_2, v_3\\},", "c34812632a263485f864f1cdb453cfd7": "c_i(E) = 0", "c348465ab605f8f635435a574fbbb19d": "r(s-a)\\,", "c348518f48c46365f9ee919bfa06c932": "\\scriptstyle\\sqrt{\\pi}", "c348675e1831996e257d8c32a84b9f5d": "\\Gamma(z)\\Gamma(1-z) = \\frac{\\pi}{\\sin{(\\pi z)}}\\!", "c348c188c55f03298f2b4d08c6c8f6cf": "\\displaystyle \\alpha \\psi + \\beta |\\psi|^2 \\psi + \\frac{1}{2m} \\left(-i\\hbar\\nabla - 2e\\mathbf{A} \\right)^2 \\psi = 0 ", "c348dafd7787c9985c625c722ea1b797": "\\tan \\theta \\approx \\sin \\theta= \\frac{\\frac{L}{2}}{l}=\\frac{L}{2l}\\Longrightarrow\\frac{ \\tan \\theta_1}{ \\tan \\theta_2}\\approx \\frac{\\frac{L_1}{2l}}{\\frac{L_2}{2l}}", "c348e562e08d23a9b520edade8824885": "\\frac{G}{N} = \\frac{G}{N}^\\circ + kT\\ln \\frac{V^\\circ}{{V }}", "c348f41048c923363d16bedf6cb7f6eb": "a\\varphi^n+b(-\\varphi)^{-n}", "c3492b587620ce25175fd1d25bbbc9d4": "\\boldsymbol\\beta_1,\\ldots,\\boldsymbol\\beta_m", "c3496f0b4770905dc75239b0070da672": "\nJ \\ \\stackrel{\\mathrm{def}}{=}\\ K \\frac{J_{\\alpha}^{\\prime\\prime}}{J_{\\beta}^{\\prime\\prime}}\n", "c3497a58bbdfb16b5a90046eeb79b9cb": "\\mathbf{SS(p,M)} = \\sum_{i=1}^n e^{-\\tau \\left \\Vert \\mathbf{p-m_i} \\right \\|} = \\sum_{i=1}^n e^{-\\tau \\sqrt{\\sum_{j=1}^L (p(j)-m_i(j))^2}}", "c349c55409ed9a6b7dfddba6bf5d0954": "\\phi (t) = P(t)e^{tB}\\text{ for all }t \\in \\mathbb{R}.\\ ", "c34a2a8f936f127c1d96f0b332f84247": " \\left\\vert \\left( \\frac{\\text{Doppler Frequency} \\times C}{2 \\times \\text{Transmit Frequency}} \\right) - \\text{Ground Speed} \\times \\cos \\left( \\Theta \\right) \\right\\vert > \\text{Velocity Threshold}", "c34a5ec319db07beff3ac45f51b91a63": "\n\\begin{align}\nL'_s & = \\frac{L_s}{\\sqrt{2R_c L_s}} \\\\\n & = \\sqrt{\\frac{L_s}{2R_c}}\n\\end{align}\n", "c34a6ff9381ea7bd696c7c848847e87d": "T_{{\\mu}v}= \\rho \\begin{pmatrix} c^2 & v_x c & v_y c & v_z c\\\\ v_x c & v^2_x & v_x v_y & v_x v_z\\\\ v_y c & v_y v_x & v^2_y & v_y v_z\\\\ v_z c & v_z v_x & v_z v_y & v^2_z \\end{pmatrix}~", "c34a7d7f73dbbb2e9668ca297602b092": "\\displaystyle x \\in (0, +\\infty)\\!", "c34a8954ba4f1fa4d5c3d029e8de17ca": "\\!\\mathcal A \\models_X^- \\lnot \\phi", "c34abcb8882171e50c700c49aca79aba": "X_n \\to 0", "c34ac3be484bbf166ceb214ee65b18e0": "P_0\\tau = \\epsilon_\\mathrm{pulse}", "c34af7a59ad8b0a72856de2aea2c02ba": "\\scriptstyle\\Box", "c34b017eeeaa50851943e94401830377": "\nf(z)= \\frac 1z + \\sum_{n=0}^\\infty a_n z^n,\\qquad z\\in \\mathbb D\\setminus\\{0\\},\n", "c34b4481b7218054eb7f975d59d3cc22": "E(W) = \\sum_{j=1}^n w_j^2 .", "c34b9ab4510b68bc5abf85519db9ccc3": "f \\circ f^r \\le \\mbox {id}\\qquad\\mbox{(right counit)}", "c34ba43a08fbdc79616818a998afd95a": "\n\\tilde{V}(\\mathbf{k}) \\ \\stackrel{\\mathrm{def}}{=}\\ \\tilde{\\rho}_{uc}(\\mathbf{k}) \\tilde{\\Phi}(\\mathbf{k})\n", "c34ba9aaf2845bd8d42cfd1aaea8f8a9": "\\frac{m^3}{Kmol}", "c34ba9d0e7e7bcda3f174c7859e984e2": "{Q_2} = {Q_1}\\sqrt {P_2/ P_1} ", "c34be4f211c7d9e4790d8e843ef516e4": "\\Delta \\phi =\\frac{2 \\pi \\Delta d}{\\lambda}", "c34c2fb8ffd44c919f0027dc55355345": "\\mathbf{B}\\cdot\\nabla\\alpha= 0 ", "c34c772d4aa7dc8fcb810862e6c0d583": "\\ C", "c34c7d1f991b0552aa0df96e12e15825": " m = m_1 m_2 ", "c34c99c0c3824b75328d8aa8e0145756": "\\scriptstyle \\left|V_o\\right|=-\\frac{D^2}{2\\left|I_o\\right|}", "c34ca21849097e5722a15d536e0d3831": "\\mathcal M(Q)\\to Q\\,", "c34cafef269145899ac6f985bf956452": "\n-e(t)=\\lambda e''(t)\n", "c34cd344569fed9bce5357bd227fb638": "\\Box (p \\land q) \\leftrightarrow (\\Box p \\land \\Box q)", "c34cdf22fb0dbc7cc7c9eabca14291b2": "v_P", "c34d0a89a36dc3904760a66f277db382": "A-\\lambda I", "c34d109acfb653a26c79c8de77fde30a": "N \\triangleleft M_\\lambda", "c34d5c409fd35eef03edffc3b22f9ad2": " u(x,t) = E \\left [e^{- \\int_t^T V(X_\\tau)\\, d\\tau} \\psi(X_T)) + \\int_t^T e^{-\\int_t^s V(\\tau)d\\tau} f(X_s,s)ds \\Bigg| X_t=x \\right ]", "c34d8586f8bcf8e7bb1d326fbbb4ef32": "gf = f(g) = g+ \\sum_{n=2}^\\infty a_n g^n", "c34db275e42528c97d83c95b1e67ef77": "\\Delta^0_2", "c34dd2ddd907c0eaee0112bfe776461d": " p(n) = 1-\\bar p(n) \\approx 1 - e^{- n(n-1)/(2 \\times 365)}.", "c34e1278c84c014e130eacfb49fdfd30": "\n\\begin{align}\n\\nu(A) &=\\int_A g\\,d\\mu+\\nu_0(A) \\geq \\int_A g\\,d\\mu+\\nu_0(A\\cap P)\\\\\n &\\geq \\int_A g\\,d\\mu +\\varepsilon\\mu(A\\cap P) =\\int_A(g+\\varepsilon1_P)\\,d\\mu.\\\\\n\\end{align}\n", "c34e17a437c86f9ec96588509ad3ee49": "S \\rarr \\mathrm{M}(A \\times S)", "c34e8df85c24e6ca336bd970a258060e": "(e_1,e_2,\\dots,e_n)", "c34ea4aab21a5fe0e7879f013ba84054": "\\langle\\hat{b}^\\dagger \\hat{b}\\rangle\n = \\mathrm{Tr} (e^{-\\beta \\omega \\hat{b}^\\dagger \\hat{b}} \\hat{b}^\\dagger \\hat{b} )\n = \\frac{1}{e^{\\beta \\omega}-1}\n", "c34ea793d3227b381da7d1170329ce44": "[A]_e", "c34ec8174f16ed412a9bbf8f2ce85cab": "x \\equiv \\pm 18^6 \\equiv \\pm 8\\pmod {23}", "c34f7e0c8019e227f26c8dfbe9b43f4e": " K_m \\times K_n \\rightarrow K_{m+n}", "c34fa39772d92e79b1dada4965e25db2": "\\hat{\\mathbf{x}} = (\\hat{x}_1,\\hat{x}_2,\\dots,\\hat{x}_n) \\in \\mathbb{R}^n", "c34fb69dfd5d55e614f554bce68e7dca": " t \\mapsto T(t)x. ", "c35001b680222ed72751c9067e641000": "y^2 =3/4", "c3502767cc6d078e105c388bcdff8613": "\\mathbb{Z}_{p} = \\displaystyle \\lim_{\\leftarrow} \\mathbb{Z}/p^n\\mathbb{Z}. ", "c350b3167a62376305baeebd2eeab86a": "A \\subset R^n", "c350fc4ce449a00b2421783e31830b66": "|\\omega_{ij}| \\ll 1", "c350fe57e53b186e01115f90a1e2be8c": "\\mathcal A \\models_X \\vec{t_1} \\subseteq \\vec{t_2}", "c35166b9c373e1aaa3bc23a5d7d42a16": "B_n(0) = B_n(1) = B_n\\quad(n\\text{th Bernoulli number})", "c3517309a39ecacf49e96ef156274509": "[H,C]=i D, [C,D]=-2i C, [H,D]=2i H,\\,\\!", "c351802720ffcd1ee26bfe5cb4e8f67c": "s_1, s_2, s_3", "c351bbd3f4eb86a87ab7ad40bb00bcfa": "\\|f*g\\| _r\\le\\|f\\|_p\\|g\\|_q.", "c351e8eb7f56f091f879e1980ba9837c": "x \\in L^c \\Rightarrow \\mathrm{Pr}[A^c\\,\\mathrm{accepts}\\,x] > 1/2", "c352066249a2e4342a684fe5093399f2": " y(x) = u(x)e^{r_{1}x} \\, ", "c352906cfc2e798b3dd16239c50af63a": "T_H = \\frac{\\kappa}{2 \\pi},", "c352ac5291f880ddf38e92b81cf6ecf4": "\n G \\equiv \\frac{\\epsilon_j^{n+1}}{\\epsilon_j^n}\n", "c352d29bed02461b2794cf521acb1e48": " L_2\\,\\,=\\,\\,\\frac{1}{2G} ", "c353130967344aeceaca57ac2a21da2c": "\\Delta t_n", "c353e5c1f9d1f9be7515ab980e47d660": "f(\\vec r)", "c353efa2e71e4ead027df197ca0b83b5": " \\boldsymbol \\beta^{s+1} = \\boldsymbol \\beta^s+\\alpha\\ \\Delta", "c3545246a2fab5f8fb9ba72c4178d69e": "\\alpha = \\left(1 + \\kappa \\left(1-T_r^{\\,0.5}\\right)\\right)^2", "c354b6af2eea49dce13423e4f4bdf038": "Z \\approx 276 \\log_{10}\\left(2\\frac{D}{d}\\right)", "c354bdd39692a0ba3f80f7c733f4e0eb": "f_1", "c354c0df0749a5219764437b606bc684": "\nD_{\\mathrm{KL}}(P\\|Q) = \\sum_j (- \\log \\frac{q_j}{p_j})(p_j) \\geq - \\log ( \\sum_j \\frac{q_j}{p_j} p_j ) = 0.\n", "c354d9c56187e268959771adb4ccf550": "g_{ij}=-D[\\partial_i||\\partial_j]", "c354dccef8c77e4e2f166748394082c6": "R_s=R(1-\\cot\\theta\\tan\\alpha)\\sec\\alpha \\;", "c354dd9405c54dbe4d390c0f8560b01c": "G(m,n,r) = \\langle a,b | a^n = b^m = 1, a^b = a^r \\rangle ", "c35529be4d013090be7e172a4cad580f": " \\chi_T(t,0) \\simeq \\begin{cases} \n\t(t)^{-\\gamma}, & \\textrm{for} \\ t \\downarrow 0 \\\\\n\t(-t)^{-\\gamma'}, & \\textrm{for} \\ t \\uparrow 0 \\end{cases}\n\t ", "c355399cc352c0a5d667830ae1de8350": "\\, T_{\\overline{r}o}", "c3554cd37a75222809a0341c9817887a": "\\theta = 2 \\frac{r_\\mathrm{s}}{r}", "c3561b7e3eeed55b3b93cf1cd563d411": "[(X+E) \\; (Y+F)] \\begin{bmatrix}V_{XY}\\\\ V_{YY}\\end{bmatrix} = 0", "c35631ce4d00893689119d1b283c2a4e": "\\begin{align}\n\\phi_n(\\alpha,\\beta) &= \\operatorname E(\\exp\\left[ i(\\alpha R_n + \\beta(R_n-S_n)\\right])\\\\\nu_n(\\alpha) &= \\operatorname E(\\exp \\left[i\\alpha S_n^+\\right]) \\\\\nv_n(\\beta) &= \\operatorname E(\\exp \\left[i \\beta S_n^-\\right])\n\\end{align}", "c3564bc0010fdf67346c8c2e541fea9b": "\\tfrac{1}{3}+\\tfrac{1}{4}+\\tfrac{1}{6}+\\tfrac{1}{68}", "c3566dc43a64b131899d48478efd9e55": "M_{\\alpha\\beta} = M_{\\beta\\alpha}", "c356a3479f068ae3baf242364dcccc0d": "C_{V,m}=\\frac{f}{2} R", "c35706dca215375c1745be765b025eae": "N\\cos \\theta =\\mu_s N\\sin \\theta +mg", "c3571822018e132bc76b02db31584e2c": "f = \\sum a_I t^I ", "c3571e66e379263416f9afe2b981452f": "{\\mathbf{q}}", "c35742e3a78ada5c118ed0e24fb12df4": "A[\\sigma]", "c3578ebc9fae23f7964bd2a0d094864c": "\\tilde{\\theta} \\,\\xrightarrow{P}\\,\\theta", "c357c59e7c3442a3f6dc6f7c18a97cfe": "\\sum_{j\\in\\mathbb{N}}T_j", "c3585683be6b538aaa522d1e86a8500b": "\\phi_{13}(a \\otimes b) = a \\otimes 1 \\otimes b,", "c3586ee32e03cf315010e4bc77160d30": " \\phi = kt + B ", "c35872de3e2498315f6a465aa832f7f3": "\\Omega _{n} =\\sum_{j=1}^{n-1}\\frac{B_{j}}{j!}\\int_{0}^{t}S_{n}^{(j)}(\\tau)d\\tau ,\\qquad\\qquad n\\geq 2. ", "c358a40f4b24dedbf19acebee12935c5": "\\theta_1 = q \\theta + \\theta_0", "c3594fdd7ad49d584331f140ded470ac": "E = \\sum_n E_n\\,\\bar{N}(E_n)\\,,", "c3595b2f1fabd9470464155e20d1d1cc": "\\ u^2x^2+v^2y^2+w^2z^2+2vwyz+2wuzx-2uvxy=0", "c359cb10cf2d2f04e4ff2b84218edfc8": "\\frac{d}{dt}\\langle A\\rangle = \\frac{1}{i\\hbar}\\langle [A,H] \\rangle+ \\left\\langle \\frac{\\partial A}{\\partial t}\\right\\rangle ~,", "c359d32b62122459f0cee50aed2e5097": "\\alpha < \\alpha^\\prime", "c359e6dd743081e957b4c706623403cc": "GF\\left( {q} \\right)", "c35a3ea8fc5c660e55437d1e0342d25e": "\\scriptstyle(+1.5\\pm4.2)\\times10^{-9}", "c35a9e5124f6b43fc3bcfacc8273f1be": "(t_1,\\cdots,t_n) \\mapsto \\sum_{i = 1}^n t_i v_i", "c35b1efca0a49af0350dc35c4c8f1df6": "x = 0\\,", "c35bfe55baff0d17f79b6d496429114d": "\\tfrac{\\sum_{i=1}^n a_i}{V}", "c35c35a593c42b0ef3ec48665849003f": " r~\\ln r~\\cos\\theta \\,", "c35cbe3216f9a0a3434046b844322c28": "g(x) = \\inf\\{\\lambda \\in \\mathbb{R} : x \\in \\lambda K \\} . ", "c35d0756633a286765613ec6ae8c0b65": "s:D \\rightarrow D", "c35d1ee328f2958173e5961c4017c7c7": "R = 0.01 \\times \\overline{F}_\\mathrm{max}", "c35d6c61db9f531ff20421df3dd23f88": "\\ {\\omega} = \\frac{{\\Delta}{\\theta}}{{\\Delta}{t}} \\,,", "c35dfcdb45c0ed5860fce4ed2bd8a3dd": "\\displaystyle{H=G\\cdot \\exp(C) =\\exp(C)\\cdot G.}", "c35e2280644051f440425f13c8633e99": "\\textstyle f(\\alpha x)\\ = \\alpha f(x).", "c35e865ba8c6a1ca896c21740c734694": "W^s(f,p) =\\{q\\in X: f^n(q)\\to p \\mbox{ as } n\\to \\infty \\}", "c35e8e193495923774422cb22349e35f": "MVA = V - K_0 = \\sum_{t=1}^{\\infty} { EVA_t \\over (1+c)^t }", "c35f1b02b4efc42eb5db4244783a7bf5": "\\nabla_u", "c35f2efc02bc2781afa1cb267f535620": "\\sigma_{j}^{-}", "c35f388a341d32ccb62278942934513e": " {\\mathit l \\over \\mathit l^*} = {1 \\over 2} ", "c35f4dd959326f379485e54e14a29391": "\\mathit{SU}(n) \\supset \\mathit{USp}(n)", "c35f5b4e2d967f9e3cc415361f0ea801": "2^{(j-1)}", "c35f75cab6814111c5aa0b2d7a8b14ed": "C \\setminus (B \\setminus A) = (A \\cap C)\\cup(C \\setminus B)\\,\\!", "c35fb3a0dd514a0650b79d20cecadad4": "a,b,c,d \\neq 0", "c35fdad74848c6754234b98befc633d9": "\\mathbf{a} = \\mathbf{a}_1 + \\mathbf{a}_2 + \\mathbf{a}_3 = a_1{\\mathbf e}_1 + a_2{\\mathbf e}_2 + a_3{\\mathbf e}_3,", "c35fe4d444953b36864f7ce56591bfdb": "\\varepsilon(t)=\\frac {\\sigma_0}{E} (1-e^{-\\lambda t}), ", "c35ff2c5e0b97ebdc6b8a149ac5762b9": "f(x,y,z,w)=xy+yz+w", "c35ff3edd18eb47caabeb6635843093d": "\\vdash A", "c3600f1e6a1238dc3cdc7b85c378832d": "\\int\\sin ax\\cos ax\\;\\mathrm{d}x = -\\frac{1}{2a}\\cos^2 ax +C\\,\\!", "c36011423ee22f8c7957754748e58025": " m = \\sum k_i + n + 2g-2 ", "c360536c7539d6f39e18f89d1b25ec32": "\\mathbb{E}[X] \\geq \\mathbb{E}[Y]", "c3606345cfe805cf1ef974223c76766e": " \\left|\\frac{x}{A}\\right|^r + \\left|\\frac{y}{B}\\right|^s + \\left|\\frac{z}{C}\\right|^t \\leq 1", "c36079d34f2d80d3c979c00eba3cebd9": "\\mathit{h(x)}", "c360867ff3e753fb81135fecad5fb1f9": "Z=\\frac{p}{\\rho R_{\\rm specific} T},", "c3609abce56568c727558ee1b9010612": " z = x I + n ,\\quad x = \\frac{a + d}{2}, \\quad n = z - x I .", "c360a5128a3e6d09d01d83591be75108": "(x^2+y^2)^2=cx^2+dy^2", "c3612d018f47fcf0759f50cacaa9c472": " \n\\lim_{t \\to \\infty ,F_e \\to 0}\\frac{1}{t}\\ln \\left( {\\frac{{p(\\overline J _t ) = A}}{{p(\\overline J _t ) = - A}}} \\right) = \\lim_{t \\to \\infty ,F_e \\to 0}\\frac{{2A\\left\\langle J \\right\\rangle _{F_e } }}{{t\\sigma _{\\overline J (t)}^2 }}. ", "c3614fac2a295b6a5e315343d0d3200d": "F = zSF_z \\, ", "c3619010b599b51a192e6d17acf09cf8": "p_{n,k} (x)", "c3622c164507952fa035d26bbc084b27": "\\chi({\\ell(b)})=\\ell", "c3624a05e78a47551bc3d7a56371cd38": " \\mathbf{K}_{o^{t+1}} = (k(\\widetilde{o}^1, o^{t+1}), \\dots, k(\\widetilde{o}^T, o^{t+1}))^T ", "c3628b9d9e16ba402f854e6e3ddeb50a": "i_{n-1}\\,\\!", "c3629b211304703c4e64ae429e6d866b": "T_k(e, i_1, \\ldots, i_k, x)", "c362a34d73aad0d7fba4dd1bf1e21b96": "P(Y)", "c362d1f037efc97eda29e2d61fc68013": "a^L", "c362dcb397828f26f2100789c1cbd242": "C/K", "c363280d194c6b0df1df1a0b78b34e0d": "\\left(2x^2+y^2\\right)^2+2 \\sqrt {2} ax\\left(2x^2-3y^2\\right)+2a^2\\left(y^2-x^2\\right)=0", "c36364d81fc72e9acadd43bf8423cbcb": "c^2=c_\\infty^2-2\\phi\\,", "c363cf19ee9e24be927ed9a0058a1beb": "{\\dot \\rho} = - 3 \\frac{\\dot a}{a}\\left(\\rho+\\frac{p}{c^{2}}\\right)", "c363e353beb88fe709ece6a8e34b1366": "\\frac{\\mbox{Net Income}}{\\mbox{Fixed Assets + Working Capital}}", "c3640536bbe89d27770150ab939eebe2": "\\alpha\\|u-u_h\\|^2 \\le a(u-u_h,u-u_h) = a(u-u_h,u-v) + a(u-u_h,v - u_h) = a(u-u_h,u-v)\n \\le \\gamma\\|u-u_h\\|\\|u-v\\|", "c36415c1a75b3855404e550ed8152ed2": " y^m(\\mathbf{x},\\boldsymbol{\\theta})\\sim\\mathcal{GP}\\big(\\mathbf{h}^m(\\cdot)^T\\boldsymbol{\\beta}^m,\\sigma_m^2R^m(\\cdot,\\cdot)\\big) ", "c364912da12032cae218294e789a503d": "\\begin{align}\n\\neg q\\\\\np \\rightarrow q\\\\\n\\therefore \\overline{\\neg p \\quad \\quad \\quad} \\\\\n\\end{align}", "c364b697e696b35b071969b864eb6ab9": "I_\\mathrm{D} ", "c365101483bcf6929d4d285457038cb5": "T_{max}\\approx{K*V_s^2}/(2X)", "c3656dfe0d87c1208309e20c4c162961": "a<1/2", "c365727259f131aac467ae0ff034ff3a": " \\psi_{j,k}(t)= \\frac{1}{\\sqrt{2^j}} \\psi \\left( \\frac{t - k 2^j}{2^j} \\right) ", "c365af03d39113376dad41e6e5700e8b": "t=1,2", "c365bf945692f63785dc20114b62d827": "C_S = \\frac{1}{2}\\sum_{k=1}^K \\frac{[g(k) - h(k)]^2}{g(k) + h(k)}", "c365c99245164505b20227669007825e": "1\\le d K\\sqrt{c} ", "c370a5b1f1de77ba1061828c130d1854": "Y^{\\prime}[] \\to Y[]", "c370cad78a8b8bf5d90f518a09948b66": "\\lambda \\in \\bigwedge^k V", "c37142d06956f72ff2fcaa64c67eae6c": " W_\\mathrm{adh} = 2(\\gamma_1 \\gamma_2)^{1/2} ", "c371658b8c9bc3ea0db863507a229151": "= a_0 +a_1x+a_2x^2+\\cdots", "c37176f9149b1d6bd7bf732dd1d63c57": "\\scriptstyle \\nu(g) > 0", "c371b24a22d52e04298fb57724beea2c": "f(x,y,z)", "c371e6f8b1efd97cd95bb5f989c081c6": " K^\\mu=\\delta^{\\mu}_{0} \\,", "c3722072716bfd025c5d009ca285f63b": "\n \\sigma_{11} = \\frac{3}{2h^2}\\,M_{11} = \\frac{6}{H^2}\\,M_{11} \\quad \\text{and} \\quad\n \\sigma_{22} = \\frac{3}{2h^2}\\,M_{22} = \\frac{6}{H^2}\\,M_{22} \\,.\n ", "c372254548f4b22fb9d4865a9abbdca7": "\\Psi^{n,K} (r,R) = {1 \\over \\sqrt {V}}exp(iK \\cdot R) \\phi_n(r)", "c3722c322a8b925fe6ce509ef354fa50": "\n\\sigma^2\\ = \\frac{ \n \\sum_{i=1}^N{\\left(x_i - \\mu\\right)^2} \n}{ \n N \n}\n", "c3724de2c7d50b65bd5eb3d21f8c2fcf": "r_3", "c3728392fdbd18977026a9826aebec9e": "\\psi(x_1,\\dots,x_n) = \\sum_{i_1\\dots i_\\ell}\\psi_{i_1\\dots i_\\ell}x_{i_1}\\cdots x_{i_\\ell}.", "c37299fe0affca9a54a0b36f12918866": " f^*(x) = \\int_0^\\infty \\mathbb{I}_{\\{y: f(y)>t\\}^*}(x) \\, dt.", "c3729ca3a52c775e51773fdbba76a4fa": "A P = Q R \\quad \\iff A = Q R P^T ", "c372a6b361ca316c5ed0aa912a0c95bf": "\\mathbf{u}\\times (\\mathbf{v}\\times \\mathbf{w}) = (\\mathbf{u}\\cdot\\mathbf{w})\\ \\mathbf{v} - (\\mathbf{u}\\cdot\\mathbf{v})\\ \\mathbf{w}", "c372f0984d75a91f38feb0597f3a44c7": "\n\\begin{align}\ny[n] = \\left(\\sum_{k} x_k[n-kL]\\right) * h[n] &= \\sum_{k} \\left(x_k[n-kL]* h[n]\\right)\\\\\n&= \\sum_{k} y_k[n-kL],\n\\end{align}\n", "c3739af56c2664f43a24ede47eae53d3": "\\mathrm{Sh} = 2 + 0.6\\, \\mathrm{Re}^{\\frac{1}{2}} \\, \\mathrm{Sc}^{\\frac{1}{3}}, ~ 0 \\le ~ \\mathrm{Re} < 200, ~ 0 \\le \\mathrm{Sc} < 250", "c3739dd09968ea81fe7df90d05c0d281": "\\frac{dTR}{dP} = Q\\left(f'(P) \\cdot \\frac{P}{Q} + 1\\right)", "c373f2afdbc4a2cbca5a7c155b165d5e": "x' = \\gamma \\left(a - v\\right) t", "c3743f2c93822afd46cd3c66c47e1f3e": " E = - \\sum_{i\\neq j} J_{ij} S_{i}^z S_{j}^z \\,", "c3746e3d1210efa61236827ae03f5ff5": "\\beta = \\alpha^{q-1}", "c374e1b303e5e1c4d234c43205dee3fb": "\\displaystyle{g(z)=W(z)M(z)}", "c37577b2afe2fd664124edb538f7d844": " \\zeta^g = \\zeta^{\\omega(g)} \\text{ for } \\zeta \\in \\mu_p . ", "c375a0415c00421de9ccf839495a03f9": "\\phi: B \\to B[t]", "c375cacb6c8e3fd828c867a2a2791062": "\n\\exp\\left( {i\\over \\hbar} S \\right)\n", "c375cc985eed45b78ef14d259371f2c7": "C \\cong B/f(A)", "c376c1d5c7a4cc9de67beb1dafecbb44": " \\vec{B} = \\mu_0(\\vec{M}+\\vec{H}).", "c376f3b0df5d7e1e3a47aed61710737b": "f(x) \\sim \\sum_{i=0}^\\infty c_i T_i(x)", "c377570454a2056bfd68313971fc533f": "g(S)=\\min(f(S),c)", "c37781e7e959b7cdce052db8f71f8238": "Z_{base} = \\frac{V_{base}}{I_{base} \\times \\sqrt{3}} = \\frac{V_{base}^{2}}{S_{base}} = 38.1 \\, \\Omega", "c37784413db3b51f575978425963c437": "\\rho(x,y,z)=\\frac{1}{V}\\sum_h\\sum_k\\sum_l|F_{hkl}|\\exp(2\\pi i(hx+ky+lz)+i\\Phi(hkl)).", "c3780dee66c8a2bd17845d2bfa43c4c8": "\\lambda=\\max\\{|\\lambda_2|, |\\lambda_{n}|\\}", "c3785744aba23138e70ae1551ad32756": "h(x,t) = \\sin(n\\pi x/L)\\sin(\\omega_n t).\\ ", "c378733bce00f303dce891433416d3dd": "Z_G = {1 \\over G } \\iff Y_R = { 1 \\over R } ", "c37893214a065dba091a9335ea230985": " f\\left(\\sum_kA_k^*X_kA_k\\right)\\leq\\sum_k A_k^*f(X_k)A_k, ", "c3792bed8fc9aa6c8f51b118f2c177c0": "\\begin{matrix} {9 \\choose 1}{4 \\choose 3}{32 \\choose 1} \\end{matrix}", "c37962c9ad5a6e8cb7bb6038330f4137": "\\sum_{k=1}^n V_k = 0", "c3799fcaae06daeb033d068866bd0831": "\\mathrm{DOF} = \\frac {2s} { ( dm ) / c - c / ( dm) } \\,,", "c379ce13d860b8fbc1dcefdd982372e0": "\\Lambda = \\arccos\\left(\\sqrt{1/L}\\right)", "c37a3f85195582a34a6e7da77edd1285": "\\eta_c = \\frac{14}{C_p}.", "c37a4917f7c34e226abdfafed733316a": "q=4\\pi \\sin (\\theta ) / \\lambda", "c37a49d7ecd1e18ba4ceb6e3b7729b58": " g\\left(\\cdot\\right) ", "c37a56de0629561b7fb379d22cffcd30": "\\int_{0}^{1}b_{\\nu, n}(x)dx = \\frac{1}{n+1} \\forall \\nu = 0,1 \\dots n", "c37a5aa248635e9234694a40a0e37037": " L=\\left (\\mathfrak{der}(A)\\oplus\\mathfrak{der}(J_3(B))\\right )\\oplus \\left (A_0\\otimes J_3(B)_0 \\right )", "c37a8b5be2d9f8afc9a1ce0f0f5f4993": "(t^2, t^3) A", "c37aac4334078c9aca087d1d614d855c": "t_{k+1}=t_k+\\Delta t", "c37ad5a79c4a52757ec0de4f7b3b8229": "A\\subseteq (C\\ominus B)", "c37b229bc8348326b0748c4e6b745656": " V_\\mathrm{anode} + \\int_A^C I(y) \\, dR_t + R_\\mathrm{load} I_C + V_\\mathrm{emit} + V_\\mathrm{cathode} = V_\\mathrm{emf} ", "c37b37396a741eec182d36c8ee21fc72": "x_0 = \\alpha^b\\,", "c37b3e9472964f48cc6e0983ace9e8ed": "\\rho (I - |D^{-1/2}AD^{-1/2}|) < 1 \\, ", "c37b48ce91bbce5d10009ec99348d67b": "O(k+n\\log n)", "c37b49af7c6b7a5590b6431cb451e1a1": "P_b = B(E,m)\\,", "c37bc4a4e39491e567fd0d31c639b9ba": "\\rho_{initial} = |\\psi\\rangle\\langle\\psi|", "c37bd994415374090c61adc2824bbef9": "\\exp{(-\\Delta/k_{B}T)}", "c37c26982d78194aa5b91dac30dcd1f2": "s, s_1, \\ldots, s_n,t, t_1, \\ldots, t_n", "c37c894ae1dbc2b5abf65274749ed9b1": "\\frac{x^{(n)}}{n!} = {x+n-1 \\choose n} \\quad\\mbox{and}\\quad \\frac{(x)_n}{n!} = {x \\choose n}.", "c37d5344a0f7e425aaba6b5283db96b2": "\\exp ( \\epsilon c )", "c37d5675e72c5b5f7925c5d60e3fab73": "X_{\\gamma(0)} = e_0.", "c37d68e7be9e4bb003497580c4be768d": "\\mathbf{O}(\\mathbf{m}\\mathbf{n})", "c37d6bd1affd3911063fac80c0ca6e6d": " A_\\mu \\mapsto A_\\mu + {1 \\over e} (\\partial_\\mu \\Lambda) ", "c37d76e767445d8e84376ef493ade0f5": "f: \\mathbb{R}^m \\to \\mathbb{R}^n", "c37d97453fd2708c594aa43cbde7c3cf": "k[A]_\\text{eq}[B]_\\text{eq}=k'[A']_\\text{eq}[B']_\\text{eq}\\!", "c37dbdf73a38c78eed616523f40a6cfa": " (2)\\,", "c37dd4a5b97a51ca26ce6f182b253f1b": "\\frac{d}{dt}\\hat{\\boldsymbol{\\theta}} = (-\\cos\\theta,\\ -\\sin\\theta)\\frac{d \\theta}{dt} =- \\frac{d \\theta}{dt}\\hat{\\boldsymbol r} . ", "c37ded103f9b042fd049fcb83a211bbb": "r=(6.5/12)/100", "c37e73b467df5a0e2af7c0ae26f33c25": "f_{*}h = f \\circ h", "c37e9ad6a422fa88764259abce484254": "\\! P_{\\text{pulse}} = P_{\\text{sys}} - P_{\\text{dias}}.", "c37ecd77e809cc2f9a8dcc87b9b9377f": "\\Lambda'(x)=\\lambda_0\\sum_{j=1}^v \\alpha^{i_j}\\prod_{\\ell\\in\\{1,\\dots,v\\}\\setminus\\{j\\}}(\\alpha^{i_\\ell}x-1),", "c37ede3a4397f0bb5bd90e1bd27a24f7": " p\\left( \\mathbf{\\hat{x}}\\right) \\propto\\exp\\left( -\\frac{1}{2}(\\mathbf{\\hat{x}}-\\mathbf{\\hat{\\mu}})^{\\mathrm{T}}\\hat{Q}^{-1}(\\mathbf{\\hat{x}}-\\mathbf{\\hat{\\mu}})\\right) , ", "c37f1d87c85b322a087a0e27553ace96": "\\Psi(x) = \\tfrac{1}{2} \\kappa |x - m|^2,", "c37f72d3430fafb634301b26b625b9c1": "\\left(\\tfrac{a}{n}\\right)", "c37f7a223a5b25b5bcdef36bee1a3658": "\\mathrm{Tr}: B \\to A", "c37f955a9c5374dbafee843574ab7dc5": " \\mathrm{d}x = C'(t)\\mathrm{d}t = \\cos(t^2) \\mathrm{d}t \\,", "c37fe1285d599602b3b1f6bb2198f8a7": " j \\in C ", "c37fe8b939a1579e87d20f1fbb9bc732": "U^\\mu = \\frac{dX^\\mu}{d\\tau} = \\gamma(v)( c , v_x , v_y, v_z ) . ", "c37ff12ca7f5cc2e426491811fff9827": "\n\\varphi={1\\over\\sqrt{2}}\n\\left(\n\\begin{array}{c}\n\\varphi^+ \\\\ \\varphi^0\n\\end{array}\n\\right)\\;,\n", "c38005bf0dbf7643ba6d2d53b9caf66e": "csp_{ij}(k)=IFFT\\left \\{ \\frac{FFT[s_{i}(n)]\\cdot FFT[s_{j}(n)]^*} {\\left |FFT[s_{i}(n)]\\right \\vert \\cdot \\left |FFT[s_{j}(n)]\\right \\vert \\quad} \\right \\} \\quad\n", "c3803d8ca533c9e388e347252124f66b": "a = \\Vert C_2 h \\Vert_2", "c38044e90536990caf4bf8ccb6a454c3": "\\scriptstyle \\boldsymbol I_0", "c3804b9a5803124b41be73242cfd9cd5": "\ny = \\pm \\frac{1}{4c}\\sqrt{16c^2r_1^2-(r_1^2-r_2^2+4c^2)^2}\n", "c380c790d4c4eae538ed7e74aa59a0a5": "\\,\\! -\\omega^2 A + 2 \\zeta \\mathrm{i} \\omega A + A = \\mathrm{e}^{-\\mathrm{i} \\phi} = \\cos\\phi - \\mathrm{i} \\sin\\phi . ", "c380d2e5f891e2f741369e940550a1f6": "\\textstyle Z ", "c380ea9beb5e0efa9a7fbc512ed39ea2": "[A,B]\\cdot X=A\\cdot (B\\cdot X)-(-1)^{AB}B\\cdot (A\\cdot X).\\,", "c380ed5f20e1fb95be614144aa8ee9ee": "J \\colon BO \\rightarrow BG", "c3812d2a186a413d24a4f6c115b2de42": "X\\to aY", "c38155c7535b15552be9eddb82c77297": "N(d_1)\\,", "c38185d79ab4e2daa23695c2fe8162b5": "{1 \\over \\sqrt{\\mathit{f}}}= -4.0 \\log_{10} \\left(\\frac{\\frac{\\epsilon}{d}}{3.7} + {\\frac{1.256}{Re \\sqrt{\\mathit{f} } } } \\right) , \\text{turbulent flow}", "c38191f84d883539c9f42c445f4b7402": "\\scriptstyle v_c[n]", "c381c915708517a58496c2b3074688ae": " S_c ", "c381f8e87661db354c23e8e25513196d": "{n+m \\choose m}_q", "c382452fee5f0222430a2599cde842be": "\\frac{T}{2}b = \\int_{t_1}^{t_1+T}f(t)\\sin(kt)dt", "c382577d2070b9de1d6a3098239d2666": "p_{d}", "c382a1be072fb79f6a284219bc5ce55b": "H(X,Y) = -\\sum_{x} \\sum_{y} P(x,y) \\log_2[P(x,y)] \\!", "c382e193dd29c8e96d3cf1da8977b292": "h_{\\mathbf{a},b} (\\boldsymbol{\\upsilon}) = \\left \\lfloor\n\\frac{\\mathbf{a}\\cdot \\boldsymbol{\\upsilon}+b}{r} \\right \\rfloor ", "c383982a7dc86d981ae2eef158b8e2b3": "\\langle v_ih_j\\rangle_\\text{model}", "c383abd228ff9600ac917b1af068c709": "\\nabla\\times\\nabla\\times \\mathbf{T}=\\mathbf{0}", "c383ae4d8bfcc85f55844248253795fa": " \\theta_{n+1} = \\theta_n - \\frac{2 \\theta_n + \\sin \\left( 2 \\theta_n \\right) - \\pi \\sin \\left( \\varphi \\right)}{2 + 2 \\cos \\left( 2 \\theta_n \\right)}.\\,", "c3843be584e186f3909ca46a782214fc": "RAB_t = (1+inflation_t) \\times RAB_{t-1} +Capex_t - Dep_t ", "c384983170427d75d1c388bd3bc7bec4": "P_B = \\frac{B^2}{8\\pi}", "c384d218dd8ba2cafddad9c473d615aa": "\\sigma = \\bold{p}\\cdot d \\bold{A} \\, ", "c385668a361a4f9d688084f42979a610": "\\mathrm{{\\phi}P_\\mathrm{n}}\\,\\!", "c38582bf73c2a92a8efe8b97fa02257b": "\\psi(\\mathbf{r}) = a(\\mathbf{r}) e^{j \\phi (\\mathbf{r}) } ", "c38586d83fd6fa953746a4beffda2037": "A_6", "c3860b550278b64d369217c552e64704": "\\theta = \\frac{d-1}{2\\mu (2-d)^2}", "c3864ee69eab643f056e7d85cbc1d2e9": "k \\mapsto \\begin{cases}\n2k-1 & k\\leq n\\\\\n2(k-n) & k> n\n\\end{cases}", "c386528be5cfeec3a72de1b632edcab7": "\\text{Minimize } (y - X \\beta)^T(y - X \\beta) \\, ", "c386613fba792459c065e925d1ac10d6": "\\frac{d^2 u}{d \\varphi ^2} = - \\left( 1 - k^2 \\frac{Z^2 e^4}{c^2 p_\\mathrm{\\varphi}^2} \\right) u + \\frac{m_\\mathrm{0} kZe^2}{p_\\mathrm{\\varphi}^2} \\left( 1+\\frac{W}{m_\\mathrm{0} c^2} \\right) = - \\omega_\\mathrm{0}^2 u + K ", "c38664c6f54426fae9d4b0e67e293c39": " \\prod_{s\\in S} I_s ", "c3869fbc5301504813a241ffd425fcd8": "\\frac{223}{71}<\\pi<\\frac{22}{7}.", "c386ac913a92fefe3e5cd5cc09d8c0ff": "\\partial_\\alpha A^\\alpha = 0", "c38734a1a1df8a8f3f85b49f9b7b1785": "0\\leq m\\leq n", "c3873d49c9c2c7099fa9c576be5713b8": "u_{w} < 0", "c38762d421ac861c87e3e9128cf391fb": " S_k = \\sum_{i=1}^k x_i, ", "c387a6e2bcdba02d5b408be50a8c2072": "\\operatorname{\\rho}(T) = \\rho_{0}[1 + \\alpha_{0}(T-T_{0})]", "c387e8ce303f0db0342fed92575e3515": "100\\uparrow\\uparrow\\uparrow\\uparrow 2=(10\\uparrow\\uparrow)^{98}(10\\uparrow)^{100} 2.3", "c3880bc63c2b0fd10cdc024cf76a1924": "\\Box", "c38812ce499de7d449020863b79114f2": "D_M \\delta \\theta", "c3881897b6b33e9dd5699b5809ebb420": "\\mathrm{\\tfrac{J}{K}}", "c38859a755781c6aa75ffb8d6c4ed24a": "S_{f\\!f}(\\ell) = \\frac{1}{2\\ell+1}\\sum_{m=-\\ell}^\\ell |f_{\\ell m}|^2 ", "c3885fca041d5b49c700afac64dcab20": "\\mathfrak{sl}(2,\\mathbb C), 0", "c38872d38c2bc037b21071c4f62b089b": "\\theta_m", "c388a91f17de90fc473297f6c24f0852": "G^{\\alpha\\gamma} = \\left( R^{\\alpha\\gamma} - \\frac{1}{2} \\mathrm{g}^{\\alpha\\gamma} \\, R \\right) = \\kappa \\, T^{\\alpha\\gamma}~", "c388c88f5c16d31571806a23df253f97": "P_{\\pi}", "c388cba856ea7fe6f2ffe752d3f7305c": "\\bot^{\\mathcal{I}} = \\emptyset", "c3891ff4eeb99ac30ed6b04db3359ad7": "m >n\\!", "c3893f194f52491b5713c57d819ce9af": "\\Pr\\left(A_5|A_1 \\cap A_2 \\cap A_3 \\cap A_4 \\right)=\\Pr\\left(A_5\\right)=\\frac{1}{2}", "c38940c43b15c7821b448bd3d9f79c61": "\\frac{\\sqrt{a}} {\\sqrt{b}} = \\sqrt{\\frac{a}{b}}", "c3894581a0cb395ebf7e76dcc9c57185": "x^2\\equiv n \\pmod{p}.", "c389898a8676ed996d5f82003cabbd3c": " X_L=\\omega L_{\\mathrm{ESL}}", "c3899e888b2fdb35a2b133b13d080cb4": "\\beta = \\sqrt{1 - \\frac{1}{\\gamma^2}} ", "c389abdeab28aadbe551b9cfc8ceec7b": " B=\n\\begin{bmatrix} c & A_{1} & A_{2} & \\cdots & A_{p} \\end{bmatrix} = \n\\begin{bmatrix}\nc_{1} & a_{1,1}^1&a_{1,2}^1 & \\cdots & a_{1,k}^1 &\\cdots & a_{1,1}^p&a_{1,2}^p & \\cdots & a_{1,k}^p\\\\\nc_{2} & a_{2,1}^1&a_{2,2}^1 & \\cdots & a_{2,k}^1 &\\cdots & a_{2,1}^p&a_{2,2}^p & \\cdots & a_{2,k}^p \\\\\n\\vdots & \\vdots& \\vdots& \\ddots& \\vdots & \\cdots & \\vdots& \\vdots& \\ddots& \\vdots\\\\\nc_{k} & a_{k,1}^1&a_{k,2}^1 & \\cdots & a_{k,k}^1 &\\cdots & a_{k,1}^p&a_{k,2}^p & \\cdots & a_{k,k}^p\n\\end{bmatrix}\n", "c389ecae53642cccfcd96f6f203831ae": "|G:H\\cap K| \\le |G : H|\\,|G : K|,", "c38a1c4fbcb7193fc93ce3f2efc2fc8f": "c\\eta=\\Psi\\,", "c38a592266859680a38c0f670db5dce6": "1/\\sqrt{3}", "c38a79cbe793ebe7f7ca89c493fb0a3a": "\\int_{-1}^{1} \\frac{dx}{\\sqrt[3]{x^2}} = 6", "c38abdc920fd8ca4ab2054bfc57112a8": "\\varphi+\\theta_0=\\theta_1=q \\theta + \\theta_0", "c38b077c60b14f600c3caf9c2351c2ab": " E_{21} = E_{12}", "c38b2df313797fb6145d8278fb28a9a7": " IMM_{i-1}(S_{x,{i-1}}, c) ", "c38b2ef093c25ed3b25e0ae191f6cfee": "\\rho = 1/ \\sigma ", "c38b313dd37bfc28bcfc0eaa25245e79": "\n \\hat{m}(\\theta) \\equiv \\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\theta)\n ", "c38b51808a2425c437670edeeb742af8": "y(t)=w(t),", "c38c7c8d1948b5c9e233ae0132fff480": "q_0<0.5", "c38c807cb393778d2111616e267c2c4f": "P^*=\\frac{1+\\phi}{3+\\phi}", "c38cd4b671e31ce816d85f66ac18698c": "h_{00}\\,", "c38cdb20b708718cdd5515843e4a5594": "r_{\\mathrm{ion}}", "c38ce74af6f0ee5c38fd232458ae5806": "X_{0i} = Y_i", "c38d1bde7c0b587bfd922c711a8a6e41": "\n\\begin{align}\n L &= -\\partial_x^2 + \\phi, \n \\\\\n A &= 4 \\partial_x^3 - 3 \\left[ 2\\phi\\, \\partial_x + (\\partial_x \\phi) \\right]\n\\end{align}\n", "c38d3ab62b3a29fa1806ee2149d73c94": "\nf_1(\\text{a}) = f_1(\\text{aa}) + \\frac{1}{2} f_1(\\text{Aa}) \n = q^2 + p q = q \\left(p + q\\right)\n = q\n = f_0(\\text{a})", "c38d3f048c217ba72bc4db5987e7abe2": " W = -\\int_{1}^{2} \\mathit{pA} \\cdot \\mathrm{d}\\mathbf{s}", "c38d45a5eb425aebeecace42a778b658": "r+2^b-M", "c38d4f38c520347dce02ffe9f8ac055b": "\\alpha \\ne \\epsilon ", "c38d7754708ddeea2d68e958d8f04503": "\\begin{align}f^{(n+1)}(x)\n&=\\biggl(\\frac{p'_n(x)}{x^{2n}}-2n\\frac{p_n(x)}{x^{2n+1}}+\\frac{p_n(x)}{x^{2n+2}}\\biggr)f(x)\\\\\n&=\\frac{x^2p'_n(x)-(2nx-1)p_n(x)}{x^{2n+2}}f(x)\\\\\n&=\\frac{p_{n+1}(x)}{x^{2(n+1)}}f(x),\\end{align}", "c38e955b3761b8674f512221bb69c3db": "A= 20", "c38ec426b444575ffb12fd79e0bd8ad6": "(12 + 6n)/(n^2 -3)", "c38edb0e6e8220b83d2d4fb8f404453f": "A'B'=A''B''=L_0", "c38eef2f8782c9935381ea89d3737d38": "\\mathrm{Oxygen:} \\ 2a + b + c = 8", "c38ef04ce2641c4d849a7f4245c1ae51": "C - P = S - D\\cdot K. \\,", "c38fc3a2162213a804f7cd71faab7057": "\nD = \\int_0^T \\sqrt{\\left({\\partial \\vec{r}(t) \\over \\partial t}\\right)^2} \\, dt\n", "c390772cb2af5a66398a9a925bbc38f8": "Q'_x(b,a)x+Q'_y(b,a)y + P_{n-1}(b,a)=0", "c390b40e4011cbd37fd75eb86a922dc9": " \\ \\textbf{f} = -qK \\cdot \\textbf{m}^{-1} \\pmod p ", "c390bec613c4e5d73c5015082a7c83f7": "\\Gamma = \\frac{1}{\\sqrt{1 - \\beta^2}}", "c390e0200acb157b3f022cff369ee058": "H(u)=\\phi(x,\\lambda,t)u+\\cdots", "c3911faa9abeaef030ab9b17b0a10772": "\\frac{\\partial \\rho}{\\partial t}=\\frac{1}{i \\hbar}[H,\\rho]", "c391a3d11fd7962cf8152a6386fd3739": "c^2 = a^2 + b^2. \\, ", "c39228a3b4d244ff0dface189f31ec07": "m_{i,j}=2", "c3922bed522795eec0f1ede60ccc2ddc": "p_{j,t}", "c39244e19560cd47147d8ea5d12ba85e": "I_k = \\int f(x,k) \\,dx, ", "c3927f846ceeee1cdf0411ce1b07ef97": "\\oint_{C} f(z)\\, dz = \\int_{C_1}f(z)\\,dz + \\int_{C_2} f(z)\\,dz\\,.", "c392970dd12b4ce2a041a0ae5edad8fd": "45 \\cdot a_n", "c392cd1d03e9c1ba82a5c155b7489f2d": "p= (-1)^l", "c392e019c93d5bb67f28b12ba18ee42a": "\\nu_d = R \\left(\\frac{Z_{3p}^2}{3^2} - \\frac{Z_{nd}^2}{n^2}\\right) n=4,5,6,...", "c392e8075c4bf0662221b6b5e2cc7dad": "2\\lambda(1 + 2 \\beta^2/\\gamma^2)/\\gamma^2", "c392ea5e2898193539a6436193bff446": " f_{\\mathrm{FD}} (\\epsilon) ", "c39338b00c2a2f0033fb25b052799d7c": "A \\otimes_{K}\\bar{K}\\cong\\bar{K}\\oplus ...\\oplus\\bar{K}, ", "c3934bbd034f2aee647efa4e0a3d4ba0": "\n\\mu_{\\operatorname{eff}}(\\dot \\gamma) = \\frac {\\mu_0}{1 + ({\\frac {\\mu_0 \\dot \\gamma} {\\tau ^ *}})^{1-n} }\n", "c3937f83ab818d6b1610489990f7dda1": "T_n (x) = x+n \\pmod{12}", "c39457109f14e17c373cdd5b405d61d9": "\\phi\\left(x\\right) = \\int K \\left( x, z;\\lambda\\right) f\\left(z\\right)dz.", "c394678b8107152a57e4a2c81ba9e48d": " I_1 = I_{12} - I_{31} = I_{12} - I_{12}\\angle 120^\\circ ", "c394d2bf1d13422534fea1622e3d8628": "\\scriptstyle \\left(c'-v\\right) \\rightarrow c' ", "c3951fb66bda93c0880793fa9b7a0e75": "(\\mathbb R^2, d_1)", "c3951fe045acd2ad04e0485000ab7f2a": "\\frac{\\partial \\log B_\\nu(T)}{\\partial \\log \\nu} \\simeq 2.", "c3952168db91045f057f1999dece7275": "\\frac{\\partial \\phi}{\\partial t} = \\nabla \\cdot (\\,D\\,\\nabla\\,\\phi\\,)\\,\\!", "c395d3325d89b64e5d623f6f5fa32209": "\\mathbf{B}(\\mathbf{r}) = \\frac{\\mu_0}{4\\pi} \\nabla\\times\\iiint_V d^3r' \\frac{\\mathbf{J}(\\mathbf{r}')}{|\\mathbf{r}-\\mathbf{r}'|}", "c395f826675420b24fc9811a66b9cae9": "R((a^1_i)/{\\sim},\\dots,(a^n_i)/{\\sim}) \\iff \\{i\\in I\\mid R^{S_i}(a^1_i,\\dots,a^n_i)\\}\\in U. \\, ", "c3962c2e1fa6a41a73d704d844ba04a5": "\\tilde V_y = V_y \\cdot \\exp(-i \\arg V_x) \\in \\mathbb R", "c39666f854726f13b598f3ee9fa6550a": " P_m(x,t) = \\sum_{\\ell=0}^{\\lfloor m/2 \\rfloor} \\frac{m!}{\\ell!(m - 2\\ell)!} x^{m - 2\\ell} t^\\ell. ", "c3967418f6692473e86d8182e44dfce2": "|X| \\leq dim(Hom(U(d),s,s-1))", "c396bd57c3ab646683cfa3c063a598ea": "OA:OB\\,", "c396dbb6377ff67f1217c1fe0046723e": "\\sin(p\\phi)", "c396dc8cdbfd958a6f9996621d269151": "e \\ne \\tilde{e}_i\\rbrace", "c396e970b82dfdc07259536cb5c3c310": "\\lessdot", "c396fe3ae89e1000980a0415dddbb22c": "L(x, t) = \\sum_{n=-\\infty}^{\\infty} f(x-n) \\, G(n, t)", "c3976c8a42eede057127b1d1c467d64f": "S_H = 1+K_2 C^*_1", "c39776507bd0526496878b420939b939": " SL(6,\\mathbb C)", "c3985d08636f09d8066513ec527a5028": "\\forall a \\, \\forall s \\, [(Ma \\and \\forall x \\, [x \\in s \\leftrightarrow \\exist y \\, (x \\in y \\and y \\in a)]) \\rightarrow Ms].", "c3998ca18bcd093ce1af3659359542f7": "|f(x)| \\leq \\alpha", "c3998cd1265e91e96176505976912296": "g_{copper}=\\begin{bmatrix}\n-0.579 & 0.707 & 0.406 \\\\\n-0.579 & -0.707 & 0.406 \\\\\n0.574 & 0 & 0.819 \\\\\n\\end{bmatrix}", "c399edae4b481561f8990d349b72e760": "(c_{n-1},\\dots,c_0)", "c39a01a7d09f1078e322ed1a4524cd58": "\n\\, \\cos \\phi_\\mathrm{s} = \\frac{\\sin \\delta - \\sin \\theta_\\mathrm{s}\\sin \\Phi}\n {\\cos \\theta_\\mathrm{s}\\cos \\Phi}\n", "c39a1426b213c3db9b692e385e80b199": "\\frac {e}{pq} ", "c39b085ea0145677c0ec93fe9e0712bd": "p^{ij} = \\left|U^{ij}\\right|^2", "c39b1721869a455466d3cc2abbce1cd6": "V_k(\\mathbb R^n) \\cong \\mbox{SO}(n)/\\mbox{SO}(n-k)\\qquad\\mbox{for } k < n.", "c39b2e9326ea893444274f07c35bbd2a": "\\scriptstyle \\gamma=1/\\sqrt{1-\\beta^{2}}", "c39b57d3aa1d804e8c0d10dc7ca9c90f": "\\frac{\\partial \\theta}{\\partial z} < 0", "c39b6146de9461f7cacbef94422779ab": "f = \\frac{v}{2L} = { 1 \\over 2L } \\sqrt{T \\over \\mu} ", "c39b862f07bf4dea7c98b1bb602ab8a8": " I(X;Y) = D_{\\mathrm{KL}}(p(x,y)\\|p(x)p(y)). ", "c39b94c9411c084aeb8d8bbcdd3f742b": "\\mathbf{w_{ii}}=\\sqrt{\\mathbf{W_{ii}}}", "c39badd8256713b112c5a479cef95b96": " \\sum_{n=0}^\\infty \\frac{p_n(x)}{n!} t^n = A(t) \\exp(x B(t)) \\, ", "c39bb282226a02386c3a83fcc439e756": "f(r)\\,", "c39c530294b3e9c4350cc579c9e8af34": "\\textstyle (X_{4}=-\\sin\\delta,Y_{4}=0,Z_{4}=\\cos\\delta)^{T}", "c39c696397c53669e6061db298635c80": "g_j^T(z-x^{(k)}) + f_j(x^{(k)})\\leq 0", "c39c85a44c60028a2f672a37d4596170": "\\mathcal{M}_X^*/\\mathcal{O}_X^*", "c39c9bac15617d5ef3ba35e28b552594": "P\\left( {y,t} \\right)", "c39cbfc25729eeb6e35632f89dc967b2": "\\max_i |V_i| \\le (1+\\varepsilon) \\left\\lceil\\frac{|V|}{k}\\right\\rceil.", "c39d06c583489fa887bd82845584b2b7": "\\pi_i = (I-\\overline{A}_1)^{-1} \\left[ \\overline{B}_{i+1} \\pi_0 + \\sum_{j=1}^{i-1} \\overline{A}_{i+1-j}\\pi_j \\right], i \\geq 1.", "c39d620de1ebf80276898c92f2c93340": "\n \\boldsymbol{\\sigma} = 2\\left(C_1 + I_1~C_2\\right)\\boldsymbol{B} -\n 2C_2~\\boldsymbol{B}\\cdot\\boldsymbol{B} -\\cfrac{2}{3}\\left(C_1\\,\\bar{I}_1 + 2C_2\\,\\bar{I}_2\\right)\\boldsymbol{\\mathit{1}}\\,.\n ", "c39d7cc4d94afc7dedfac7ddccdf3ddd": " z (y w - z^2) - w (x w - y z)=0", "c39e114f15cca92e713816e093f05572": "\n\\begin{array}{lcl}\n\\left(\\cfrac{\\left\\langle{t}_{n}\\right\\rangle}{{Y}_{n}}\\right)^2 + \\left(\\cfrac{{t}_{s}}{{Y}_{s}}\\right)^2 = 1\n\\end{array}", "c39e1767788455ab404bb9b9d380dd4a": "BC_{Physics} = \\frac{M}{C_d \\cdot A} = \\frac{\\rho \\cdot l}{C_d}", "c39e3267ece901000cb4fd99d5978b04": "\\left\\| P_m T x - T x \\right \\| \\leq \\left( \\frac{1}{m+1}\\right)^2 \\| x \\|.", "c39e3c6a215bad35b6cba66275970bf7": "d^2 \\leq \\frac{\\pi}{4} \\operatorname{area} (S^2).", "c39e7341850954b5cc33c5b22eb1f7fb": "\\{s_1,...,s_{n-1}\\}", "c39e78cd33cda9505c681f19065f58cb": "\\alpha_0 n_h J(r)=\\frac{3 S_*}{4 \\pi r^3}", "c39ec3ed99a933f016239359a8dfcafe": "C_{TDQ} = \\frac{\\epsilon_0 S_{TD}}{\\lambda_0} \\approx 9.34 \\times 10^{-12}\\;\\mathrm{F} = 9.32\\;\\mathrm{pF} \\ ", "c39f17ec680d41f12a7e3af3947d920b": "C_\\kappa(x_1,x_2,\\ldots,x_n)", "c39f571b4c48d16b6450525518a19925": " x-3 ", "c39f5c8a0b7961e0464094202a973b47": "S(d)", "c39f8404a41ca0c3d72dc39a80aa06c8": " w_s^{(j+1)} = \\frac{1}{N} \\sum_{t =1}^N h_s^{(j)}(t) ", "c39fc5856f1a96b727adcdaba33ebc19": " K_1 = \\frac{k_1}{k_{-1}} = \\frac{[\\textrm{H}^+]_{eq}[\\textrm{HCO}_3^-]_{eq}}{[\\textrm{CO}_2]_{eq}}, ", "c3a07263a9d198eed229d95c60e28cec": "\\frac{\\partial \\mathbf{u}^{\\rm T}}{\\partial x} =", "c3a0e1dab98598ad923895f27df64b59": "\\textstyle (a_i)\\in\\prod_{j\\in I}A_j", "c3a0e5d0e6dcf13dde919fb344f1e6b9": "\\langle a, b \\mid a^2, b^2, (ab)^3 \\rangle", "c3a10491a712336129374fd2a3ff07fd": "(p,q) \\in C^+", "c3a11c78de7e101d9f2029959599c7e5": " (1+v)^{N-j}\\,(1-v)^j=\\sum_i v^i K^{(N)}_{ij} ", "c3a11fbce2410a7d0d35cc18af2811a8": "C_D = C_{D_0} + \\frac{(C_L)^2}{\\pi e_0 AR}", "c3a123efedb57dbe3efe3e4ddece5dba": "u(t,x,y) = Q(t,x,y) + P_t(t,x,y)", "c3a130ac78bbcb8898cee85821c57f03": "1/n!", "c3a14acace7b46550cfbf789e43551f2": "alive(t)", "c3a18de9a7a1cfd187d791cc1f844eea": "m \\geq d \\geq 1", "c3a1915eb5dc5b1af7e3f0ba8a456926": "\\sigma = \\cosh \\mu", "c3a2945de92105230b383b21e17763fb": "\\mathcal{R}=\\frac{1}{2}\\left[ 1 \\otimes 1 + (-1)^F \\otimes 1 + 1 \\otimes (-1)^F - (-1)^F \\otimes (-1)^F\\right]", "c3a3436d2e3706ea8b593ad6720e8ff7": "H_{\\min}^{\\epsilon}(A|B)_{\\rho} = \\sup_{\\rho'} H_{\\min}(A|B)_{\\rho'}", "c3a3524439ff38350a6762d2958e036d": "-d(N-Z)^2/A^2", "c3a37b26d72efd21a4f50e872da54756": " \\frac{\\partial \\nu}{\\partial \\theta} = -\\frac{\\partial H}{\\partial x} = -\\left( \\frac{\\partial V}{\\partial x}(x,\\theta) - \\frac{1-P(\\theta)}{p(\\theta)} \\frac{\\partial^2 V}{\\partial \\theta \\partial x}(x,\\theta) - \\frac{\\partial c}{\\partial x}(x) \\right) p(\\theta) ", "c3a3818c86a7d4bddfb953015b62a028": "x=x(t),\\ y=y(t)", "c3a393e8839882191ee8f0db4ea19e56": "u(\\xi, \\eta) = F(\\xi) + G(\\eta)", "c3a3c4c6985d179acd7db53fccd2d45b": "\\varepsilon/2", "c3a3e90c839d7c7a91b4573905f84fad": "I + J = R", "c3a417660dc91a5dd6eba29a4ffcee4a": "1 - \\cos \\theta = \\frac{\\theta^2}{2!} - \\frac{\\theta^4}{4!} + \\frac{\\theta^6}{6!} + \\quad \\cdots ", "c3a41c33880cac1e5a27c996071c900a": " d \\alpha = (g_x-f_y) \\, dx\\wedge dy\\,", "c3a44c920c760649fc84fa4acbbe12ab": " x_3 ", "c3a463f49e222467ef050b0bcbc60315": " \\phi_1 ", "c3a489b7e47d8b726a5643c2266f9f78": "m[1]", "c3a4cff1605ede674919348e506ebce4": " R=(A+I)^n ", "c3a4d0b5ff63860385398e5fc3cc0945": "\\sin{A}+\\sin{B}+\\sin{C}+\\sin{D}=4\\sin{\\frac{A+B}{2}}\\sin{\\frac{A+C}{2}}\\sin{\\frac{A+D}{2}}", "c3a5401028bea0c98c852b9a6dd06e95": "C\\operatorname{-}\\min_{x \\in M} f(x)", "c3a546fcb9ec2930aaa3e61ebb60f09b": "h_{\\alpha_{ij}}", "c3a55df9188756fed86a45b7d53db23e": "f^{-1}(S)", "c3a586b99867ff0406238e05d7538b58": "V_\\mathrm{mot}", "c3a5916fdc59e1839ec0330b0f55f8bb": "\\chi\\colon N\\to M", "c3a5ac0aff048b569e4316689fc0d06e": "A_{\\alpha\\beta}^{\\ \\ \\ IJ}", "c3a5b53a5a512b8ee98de142e3b7c503": "\nc' = {dl/dt} = \\left(1 + {2 \\Phi \\over c^2} \\right) c.\n", "c3a5c4b0dd32924d66fe54a4c1e849f7": "\\mathbf{e}_z", "c3a5f5026878419995cae7ecfa1b51ee": "RPI = {Retic Index \\over Maturation Correction}", "c3a61c9fbe2f490e043fbd8128496f07": "f(x) \\sim \\sum_{n=0}^\\infty c_n J_\\alpha \\left( \\frac{u_{\\alpha,n}}b x \\right)", "c3a67b11c6c578fbad7b50722aab304c": "r=\\sqrt{x^2 + y^2}", "c3a6c467efedcc5b1fb773093a82f6f8": "1 \\le n \\le 3", "c3a6d09e2663d078dbd859f03153ea49": " \\mathrm{Gamma}(\\lambda; \\alpha, \\beta) = \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\lambda^{\\alpha-1} \\exp(-\\lambda\\beta). ", "c3a6f3d80fdd223f207b3425f9ef33c8": "M(p)>\\mu", "c3a6f8179063ff819b9c4f1823225a72": "\\qquad v_j\\,x_j \\ \\ge v_i", "c3a6fe74cbd57b32732d1e10e300b148": "w\\Vdash A[e(x\\to a)]", "c3a7138758cc73f5da821f026ed9d822": "\nf^n\\left(x\\right) = a + (x-a) f'(a)^{n} + \\frac{(x-a)^2}{2!}(f''(a)f'(a)^{n-1})\\left(1+f'(a)+\\cdots+f'(a)^{n-1} \\right)+\\cdots\n", "c3a773f3b253879bdee03c0591347f84": "s_0\\mapsto s_1\\mapsto s_2\\mapsto\\cdots\\mapsto s_{k-1}\\mapsto s_k=s_0", "c3a78cbc755120ec27e81717c61e5cc0": "E(z) = E_r e^{ikz} + E_l e^{-ikz}\\,", "c3a7e619de025f2fc6952d219364b13f": "\\textstyle I(P, \\zeta)", "c3a81abd3661fd3976884e190edf88b8": "-{m\\over 2} (\\bar{\\nu}^C\\nu + \\bar{\\nu}\\nu^C)", "c3a81e26e8a24fbb60d0c07fa5edb5d5": "\\text{Semiperimeter}=m(m+n)(m^2+n^2) \\, ", "c3a82e10c1c8e5a8a3360ee603c40c65": " L_a= \\frac{L_sQ_s+L_bQ_b}{Q_s+Q_b}", "c3a84062b9b3b2a2f30e4d1cc294eabb": "\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta} = \\frac{c}{\\sin\\gamma}", "c3a86636f71cd2f2dd7410fed4345bdc": "\\left\\{ y = 0 , x = 0 \\right\\}\\,", "c3a87a248f655c213c3b1d8f806eadcd": "R_k(x) = o(|x-a|^k), \\quad x\\to a.", "c3a8d30532431ab1120d6785f5582b30": "\\mathbf{v} \\mathbf{v}", "c3a927b1c65b88ec28d8a8b120fc63b7": " R(3,k) \\geq (\\frac{1}{4} - o(1))k^2/\\log k.", "c3a93af5fc798d8b505e36a3cfe90b50": " \\delta \\vec{v} = (\\nabla \\otimes \\vec{v}) \\sdot \\delta \\vec{r} ", "c3a9621dc66f2444188d0ae7391ac105": "\n\\bar y=\\begin{bmatrix}\n \\hat e_1 & \\hat e_2 & \\hat e_3\n\\end{bmatrix}\n\\begin{bmatrix}\n Y_1 \\\\\n Y_2 \\\\\n Y_3\n\\end{bmatrix}\n", "c3a97633f332b5048911353cede1ee85": "g_{1},\\ldots,g_{n-k}", "c3aa06f78edede1094ca3e6b068f40d7": "\\mathbf{P}= \\hbar\\mathbf{K}", "c3aa122d52f349d34d5fe236a9d886d2": "\\mathcal{P}\\,", "c3aa971cf96ab9a098d3f33c3e009b4c": "T^{3} \\to T", "c3aabb70b5836c10fe0df4e0d3a3ce37": "\\epsilon = h \\sin \\theta", "c3aac301e2401da60390bff09145e9c1": "T_i=\\sum_{n=1}^i\\operatorname{E}[X_n],\\quad i\\in{\\mathbb N}_0,", "c3aac7c088c2e9a3d24242ef9bcf06f3": "m_{vehicle} \\,\\!", "c3aad6fa1b32b160ca638a9ce3ae0252": " \\sigma(\\theta) = |f(\\theta)|^2 = \\frac{1}{k^2} \\left | \\sum_{\\ell=0}^\\infty (2\\ell+1) e^{i\\delta_\\ell} \\sin(\\delta_\\ell) P_\\ell(\\cos(\\theta)) \\right |^2 \\;,", "c3ab4797f3d8d4770fab26cfad29ecba": " (X\\circledast Y)\\circledast Z\\cong X\\circledast (Y\\circledast Z),", "c3abd68bffd9b7e9e53ac8552b7c225c": "{\\rm GL}_n", "c3abf4bf6e71932fe66ad8a9e76cad09": " x^{\\alpha-1} \\leq x^\\alpha + 1 ", "c3ac1362ee1dae33423bb1567b7f3a6e": "\\begin{align}\n\\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{1.414}}}}} &\\approx \\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{1.63}}}} \\\\\n&\\approx \\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{1.76}}} \\\\\n&\\approx \\sqrt{2}^{\\sqrt{2}^{1.84}} \\\\\n&\\approx \\sqrt{2}^{1.89} \\\\\n&\\approx 1.93\n\\end{align}", "c3acd4dda102515629414e37b0ac6b54": "\n p = \\cfrac{2}{\\lambda}~\\cfrac{\\partial W}{\\partial I_1} ~.\n ", "c3ad065d820fb17b9ea354c0c44518ef": "\npV^{\\gamma} = \\text{constant}\n\\,\\!", "c3ada5074d72157a3c9967fbeab0209c": "ds^2 = - \\left( \\frac{M}{\\lambda} - \\lambda^2 uv \\right)^{-1} du\\, dv", "c3adc200465f80115d9407a010c4006e": "\\operatorname{Im}(\\log\\ z) = \\arctan(b/a) + 2\\pi k", "c3adf4f221f71908e0c205a1a7762ac5": "(1+x)^n \\,", "c3adfed447b63160aa452464f09bb731": "\\triangle\\,\\!", "c3ae77d68ad29d616902ac8c782b60c7": " M_d", "c3ae88df9f82389ffd297cd4c2798deb": "AC=\\frac{TC}{Q}", "c3aef0b73b9663302e137d58e2f2715d": "y_0 = \\beta\\,", "c3af0e0ccd5fb89dae6c692a0c31e8ba": " \\mbox{c in knots} \\approx 1.341 \\times \\sqrt{\\mbox{length in ft}} \\approx \\frac {4}{3} \\times \\sqrt{\\mbox{length in ft}}", "c3af2fa3daa38cfe937457f4dbe6c6be": "U=U(X,Y)", "c3afac6e66da2f22e208a4c8b50b971b": "{(x,y)^2\\over 4} = Dt", "c3afe003d00aec0bb5b48ca44ab622b5": "F_{L} = q \\left ( \\mathbf{E} + \\frac{\\mathbf{v}}{c} \\times \\mathbf{B} \\right )", "c3b04e4cc91e34539a4361a01b608cfc": "\\phi_{sl,v}=\\frac{M_{s}}{M_{s}+M_{l}SG_{s}}", "c3b0e66252755f3652cb153fb4bf1c78": "X = \\eta e^{-bx}\\,", "c3b1465bc2614ff4f14150c7bad087b6": "u^0_1, u^0_2", "c3b179d1d454909f86cd7f91c8b35315": " \\log {f(z)-f(w)\\over z -w} -\\log{f(z)\\over z} -\\log{f(w)\\over w} =-\\sum_{m,n\\ge 1} c_{-m,-n} z^m w^n", "c3b1fe1f90bfbf08147e9e8fa2e5ccca": "m'", "c3b204edf607d85d9283f09c17832e94": " \\underline \\varepsilon = \\varepsilon(\\underline u) = \\frac{\\underline P}{\\overline EA} ", "c3b20da251879d94942f7af2330fc6d2": "x (\\theta) = (R - r)\\cos\\theta + d\\cos\\left({R - r \\over r}\\theta\\right)", "c3b212d6b9b78d2e0afb96aa7a172b2e": "\\Delta(x_1*d)", "c3b21f3c6af1c5b8a65453129d03155a": "x^2+2xc+c^2", "c3b296bf14ff19f9914be76d3d9e93c0": " \\Delta S_{int} + \\Delta S_{ext} \\ge 0 \\,", "c3b2bfe14cb2e6f1a9fc6933cde68de3": "\\ln \\Gamma_k = Q_k \\left[1 - \\ln \\displaystyle\\sum_m \\Theta_m \\Psi_{mk} - \\displaystyle\\sum_m \\frac{\\Theta_m \\Psi_{km}}{\\displaystyle\\sum_n \\Theta_n \\Psi_{nm}}\\right]", "c3b2c25a96a4e24d9b3b9ec3ca4c7883": "\\bold{S}=\\frac{1}{\\mu_0}\\bold{E}\\times\\bold{B},", "c3b2d599acc2d599a0cfdce180439190": " K=- 3 \\lim_{r\\rightarrow 0} \\Delta (\\log r),", "c3b2d680a9a35a2f401d58d814556a89": " S_{\\frac {\\pi} {3}} = S \\cot {\\frac {\\pi} {3}} = S \\frac {\\sqrt 3}{3} \\,", "c3b3405343537a57d174412f14e30760": "T(F_n)", "c3b3bbe9e4b87b6e65b8254a2e5f91d7": "z(a)\\quad", "c3b3cd081bad0752e6a5b67c24326324": "\n\\left( x^2 + y^2 \\right)^2 + a x^2 + b y^2 + cx + dy + e = 0.\n", "c3b3e42ccbf4ec00418b83085a95c0d9": "A A^+A = A\\,\\!", "c3b40b912cb6db9414bd046f621f3aa6": " \\begin{align}\nF(x,y) & = \\int^y \\mu(x,\\lambda)M(x,\\lambda)\\,d\\lambda + \\int^x \\mu(\\lambda,y)N(\\lambda,y)\\,d\\lambda \\\\\n& + Y(y) + X(x) = C \\\\\n\\end{align} \\, \\! ", "c3b4165aaa73e8974024c8f82fcd266e": "\\{p_3,p_4\\}", "c3b41cdc537d2c627a104e52acc83015": "\\sqrt{10} (4 \\rho^4 - 3\\rho^2) \\cos 2 \\theta", "c3b436b05dbc2fe9e5dce505ddceea8d": " n_h/N_h=k S_h ", "c3b51f981111ba9c24bf9ddbec99e8d9": "P(x) \\to (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\to Q(y))", "c3b521c71a4df0e532f57f3d7a99e912": "h : (T \\cup \\overline T)^* \\to \\Sigma^*", "c3b540aea638dc3dc4e6864647238e3a": "B : [0, + \\infty) \\times \\Omega \\to \\mathbb{R}", "c3b568a68cbaf1e21d2a86ee76258f95": "0 \\to \\mathbb{Z} \\xrightarrow{\\left(\\begin{smallmatrix} 1 \\\\ 2 \\end{smallmatrix}\\right)} \\mathbb{Z}^2 \\xrightarrow{(-2, 1)} \\mathbb{Z} \\to 0,", "c3b584bc006d6c25313304614c463fb6": "\\mathbb{H}(x) := (x_1 \\cdots x_n)^\\frac{1}{n}", "c3b595d448b4c5dd5c656aaa343731ee": "\\partial : \\Omega^{(p,q)} \\rightarrow \\Omega^{(p+1,q)}", "c3b64b0191386894017105c68da44c0d": " M' \\rightarrow M' \\oplus M''\\rightarrow M'';", "c3b64e9f327e23eccf34909256afcaf2": "[f|\\partial D^2]\\notin N. \\, ", "c3b6680b3453b91ed87c72a1e47acaf2": "E_{fw}", "c3b66e4db66de9b6abf10f53dce78f52": "\\int f \\,d\\mu = \\int f^+ \\,d\\mu - \\int f^- \\,d\\mu", "c3b69236a0ab5fcc30e1243b7018b321": "\\Phi_n(w)= (w-b_0) \\Phi_{n-1}(w) -nb_n -\\sum_{0\\le i \\le n-1} b_{n-i} \\Phi_{i}(w)", "c3b6f17cab2a3465c30733b94cce79ab": "c_{i,j}:=\\operatorname{Reduce}(c_{i,j},S)", "c3b706f837c69e165bbab0a4f5c22d66": "e^{x_1 y-\\beta_1}, e^{x_2 y-\\beta_2}, e^{(\\gamma x_1/x_2)-\\alpha},", "c3b7795f44c663efd41acdd988a8e6fc": "U ( \\rho ) = - \\pi R^4 \\rho_{solution} g \t\\left ( \\frac{\\Beta}{\\alpha} \\right )^2 \\mathit{A} K_0 (\\alpha \\rho) +\n\n \\begin{cases} 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\rho \\ge \\ 2 \\\\ \\pi R^4 \\rho_{solution} g \\left ( \\frac{(2-\\rho)^2}{\\alpha^2} \\right ) ~~~ \\rho \\le\\ 2 \\end{cases}\n\n", "c3b7ab20c56d20f722b24df2dfca7ad8": "F(x; 0, 1)=\\frac{1}{2} + \\frac{1}{\\pi} \\arctan(\\ln x), \\ \\ x>0", "c3b7e61c79fa414ad16017c61117395b": "x + nb", "c3b838288afbbb8f00de48b75883fb72": "M = |\\phi\\rangle\\langle\\phi|", "c3b84282cb31d2922cf690138b54e5bd": " e^{-\\lambda}\\frac{ \\lambda^k}{k!}\\, ", "c3b85d01addc4220a394e4838e990357": "\\mathfrak{e}_8", "c3b86dad4b414268cdc49c3c4209ac62": " u(w) = w^{\\alpha}", "c3b8e17e9c55194e8d4a8b3f758f2552": "a=6P_cT_cV_c^2", "c3b911f2ffa88d558fccb4f42c38b773": "\\frac{d}{dt}\\mathbf{L} = 0", "c3b9715dabe226670b5114c2f9246d04": "\\begin{pmatrix}1& 0& 1& 1\\\\0& -1& -1& 1\\\\ 1& -1& 0& -1\\\\ 1& 1& -1& 0 \\end{pmatrix}", "c3b992fd1d6d190b1de2fc541743075f": " \\rho, \\omega \\to \\pi \\pi \\gamma", "c3b9f0e3b7b468a26a6adbe1fe499bc2": " \\int_0^z \\log \\Gamma(x)\\,dx=\\frac{z(1-z)}{2}+\\frac{z}{2}\\log 2\\pi +z\\log\\Gamma(z) -\\log G(1+z)\\, . \\, \\Box", "c3ba9dc1b950d28ff1469a908f037085": "\\delta G_r = \\sum_k m_k \\mu_k \\, - \\sum_j n_j \\mu_j ", "c3bb2ce2ee2493e352a582b7347c3b5d": "y^2 = x^3 - mx", "c3bb4252ce69b4ca99a1379f9429dc9c": "G(x,s)=0", "c3bb7cd81b4bb345be8d25fcf416335a": "U_{ee} = \\dfrac{e^2}{4\\pi\\varepsilon_0}\\sum_{i < j}\\dfrac{1}{|\\mathbf{r}_i - \\mathbf{r}_j|}", "c3bb8660e1bed89b02135041a4bbc797": "K_3=-R_3", "c3bba383858a85c876e246cb3e444561": "\\frac{2}{5} = \\frac{1}{3} + \\frac{1}{15}", "c3bbbd6f403987a8090e5700fdab9296": "c_{n,k}=\\sum_{m=1}^{n}\\frac{1}{m^{3}}+\\sum_{m=1}^{k}\\frac{(-1)^{m-1}}{2m^{3}\\binom{n}{m}\\binom{n+m}{m}}.", "c3bbd3795a0b7377d437131ec3f9fcc8": " k_2 = hf(t_n+c_2h, y_n+a_{21}k_1), \\, ", "c3bc99ecc6069ed272aada06b256802c": "\\eta: K \\rightarrow A", "c3bca00d27cb2aac2bda6f43a0f45adb": "\\eta=2.\\frac{R_{s}}{B_{OFDM}}", "c3bcad1f822f61b10071ee89f4b2029e": "V^\\mathbb{Z}", "c3bd42baf93f3b8011d4d8e6e08ba3e8": " \\frac{1}{5} \\begin{bmatrix} 3/7 & 3/7 \\\\ 4/7 & 4/7 \\end{bmatrix} - \\frac{1}{2} \\begin{bmatrix} 4/7 & -3/7 \\\\ -4/7 & 3/7 \\end{bmatrix} = \\begin{bmatrix} -0.2 & 0.3 \\\\ 0.4 & -0.1 \\end{bmatrix}. ", "c3bd8118b2f50f2068edb15b8b09347c": "=\\sum_{k=1}^{d}\\left( \\dot q_k \\ + \\sum_{j=1}^{d}\\sum_{i=1}^{d} q_j \\ {\\Gamma^k}_{ij} \\dot q_i \\right) \\boldsymbol{e_k} \\ , ", "c3bda733dade24a75a47c10fc4fc6857": "\\mathrm{Metabolic\\ Rate} = 70 M^{0.75}", "c3bdb967fceadb7b8683a67ea69f7746": "(a-c) = kl", "c3bdc92c5573daac9c0f9c90db798852": "M(i,j) = \\lambda A(i,j)\\frac{m(j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)} = \\frac{\\lambda A(i,j)}{Nf(j)}", "c3bdd065e6455a68b38c53f8123bc36f": "H \\in O(n/p)", "c3bdd26a1505f4b34102f20bdc631825": "tch\\_req\\_success\\_city = tch\\_req\\_success\\_bts1 + tch\\_req\\_success\\_bts2 + tch\\_req\\_success\\_bts3", "c3be1d154847f4682227279a0403ee4a": " \\mbox{SMAPE} = \\frac{1}{n}\\sum_{t=1}^n \\frac{\\left|F_t-A_t\\right|}{(A_t+F_t)/2}", "c3be6a660dcaacf825c1e3ab0bbcb380": "\nQ^T \\Delta_v Q=v\\Delta_\\mu, \\quad(12)\n", "c3be6e525455c0aa982abd7c7d551072": "\\mathrm{Res}\\colon K(\\!(X)\\!)\\to K \\, ", "c3bf15bb3f6c4540018c3b2ef0da9fcb": " \\mathbf{S}(x,y) =\n\\left[\\operatorname{comb}\\left(\\frac{x}{c},\\frac{y}{d}\\right) *\n\\operatorname{rect}\\left(\\frac{x}{a}, \\frac{y}{b}\\right)\\right] \\cdot\n\\operatorname{rect}\\left(\\frac{x}{M \\cdot c}, \\frac{y}{N \\cdot d}\\right) ", "c3bf487e1f13b2b3106919d174eeb555": "\\begin{align}\n{\\phi}P_\\mathrm{n(max)} &= 0.85\\phi[0.85f'_\\mathrm{c}(A_\\mathrm{g} - A_\\mathrm{st}) + A_\\mathrm{st}f_\\mathrm{y}]\\\\\n\\end{align}", "c3bf898e1b6d7f6e0074998dc64f5b99": "\\mathcal{L}_{m} = -m_u^i\\overline u_L^i u_R^i -m_d^i\\overline d_L^i d_R^i -m_e^i\\overline e_L^i e_R^i+ \\textrm{h.c.},", "c3bf9b56cc57f5639388e71647335df8": "\\Pr(\\mathbf{x)_0} = N! \\prod_{i=1}^k \\frac{\\pi_{i}^{x_i}}{x_i!}.", "c3bfa4324c8c1bc0a8fe98a23ea4eb02": "L(1,\\chi)", "c3bfc4029ce324d34ecbc2e24322a597": "m^{e^d} \\equiv m \\pmod{pq}", "c3bfdb9d97802b2833eda3b7c0b5bc03": "= \\left(2^2 + 17^2\\right) \\cdot \\left(58^2 + 7^2\\right) \\, ", "c3bfeb13a09ad896fcb0a7da95621887": "( Q(x,y) \\rightarrow (\\forall u )(\\exists v)(P) \\psi) \\equiv (\\forall u)(\\exists v)(P) ( Q(x,y) \\rightarrow \\psi )", "c3c07dc725e805d668789b33b381d959": "\\Vert f^{\\prime} \\Vert^{2}\\leqq4\\Vert f\\Vert\\cdot\\Vert f''\\Vert", "c3c0941de414d0a0da7535128ea31a02": "\\text{LIX} = \\frac{A}{B} + \\frac{C \\cdot 100}{A}", "c3c118870196ba0d020a595165115f52": "(ap)^2 + \\cdots + (ds)^2 = ", "c3c1299234647a02263cd0bcff4ff4c0": "\\cos A", "c3c1317aa9f3513422fedbbec9878cd3": "\\langle x\\rangle_0", "c3c136a6d02b642b75285baf7fdf1dd5": "[F , G] = [P^+ F , P^+ G] + [P^- F , P^- G].", "c3c13a1571ffcfbd252d0b9205711a09": "\\scriptstyle \\mathbb{R}^4", "c3c1acc75c9d8254b6a890cf1de34fad": "W(S)=\\{a_1a_2\\ldots a_n\\,\\vert\\; a_k\\in S\\,; \\,n\\mbox{ finite } \\}", "c3c1cac661db2f1bf6b8eb93bab86d35": "|b_n | \\geq |a_n| + 1 ", "c3c1e4ec9373875bf9bfe7554f5e3a2f": "\n \\sigma_\\alpha = \\cfrac{\\sigma_0~\\sigma_{90}}{\\sigma_0~\\sin^n\\alpha + \\sigma_{90}~\\cos^n\\alpha}\n", "c3c293020e17ec19995f9209d72f428c": " \\sum C \\psi^\\dagger_i \\psi_i. \\,", "c3c329236b92f32e05be63cc811be19a": " A \\vee B ", "c3c3324014ce830b06bc329f766c4eff": "dR=\\frac{dQ}{2Q^{1/2}}.", "c3c338c65291180b39b7e13431c19be4": "\ny = \\pm \\frac{1}{4a}\\sqrt{16a^2r_1^2-(r_1^2-r_2^2+4a^2)^2}\n", "c3c359f05fd5231ca0a18f2fe200c35a": "yli[ab]", "c3c3e6ce3c678cffd371a72914c92046": "n=2^k", "c3c3fbb8d5a0177794caedf54c42c4e2": "\\tfrac{1}{\\sqrt{2}} < \\text{median} < 1", "c3c40c47c2711b016dfac57851252b09": "P(Z)", "c3c4449096791e640b1ed4210fcea589": "payout=\\frac{1}{n}(36-n)= \\frac{36}{n}-1", "c3c4b09e204eef565463c52c30d7d88e": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = \\sum_{i=1}^{2} \\left[-10 \\exp \\left(-0.2 \\sqrt{x_{i}^{2} + x_{i+1}^{2}} \\right) \\right] \\\\\n & \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = \\sum_{i=1}^{3} \\left[\\left|x_{i}\\right|^{0.8} + 5 \\sin \\left(x_{i}^{3} \\right) \\right] \\\\\n\\end{cases}\n", "c3c4c505b427db493d778f6be6934b49": " \\|R_i\\|_0 \\leq q ", "c3c5ddcf19b46d092ac39bc46895f9c2": "t = \\sqrt \\frac{\\ln(c)}{-Lc}", "c3c5e1beabc413888d68702c5b1f8274": "X_1,\\dots,X_n\\,", "c3c613354a1549b518d5aebd0b912a0d": "f(x)=\\lim_{r\\to\\infty}\\frac{1}{i\\sqrt{4\\pi}} \\int_{x_0-ir}^{x_0+ir} F(z)e^{\\frac{(x-z)^2}{4}}\\;dz", "c3c62e9ee87bffe648d64f77a90a639b": "{\\mathbf e}_i \\wedge {\\mathbf e}_j", "c3c63de2c1c196dc4e3e035f9b6aae6c": "\\mathrm{return}: A \\rarr \\mathrm{M} (W \\times A) = a \\mapsto \\mathrm{return} \\, (\\varepsilon, a)", "c3c667992bd70acca7d8576a463c2979": "c=\\frac{(1 - e^{-aT})\\sigma^2}{2a}", "c3c6ad6546b24192282a838d5f9c8c13": "=\\frac{1-b^2}{4a}+c=\\frac{1-(b^2-4ac)}{4a}=\\frac{1-D}{4a}", "c3c6d71d069f21a99220db9b9c8cbd70": "=\\frac{(a_2\\cos\\alpha-a_1\\cos\\alpha)\\cos\\theta-(a_2\\sin\\alpha+a_1\\sin\\alpha)\\sin\\theta}\n{\\cos^2\\alpha\\ \\cos^2\\theta-\\sin^2\\alpha\\ \\sin^2\\theta}", "c3c78f8a072ad4f5b65074afb2c0f04a": " N_n = \\frac{ 2 } { \\pi } \\ln n + C + O( n^{ -2 } ) ", "c3c7ebc160a2c949e610b283342680e1": "k > 1", "c3c7fb509b421fdc2e1938bf6547ac4a": "\\sigma\\to(\\tau\\to\\rho)", "c3c856fc2494a127a4d8f92b9c34e7f6": "E \\subset \\mathcal{X}", "c3c8687b87415a514c41f183ed7eb4c7": "r = \\sqrt{(x_1-x_2)^2 +(y_1-y_2)^2+ (z_1-z_2)^2}", "c3c87896bf7bc0eeebdeed2e66305fc1": "\\mathfrak{a}=i\\mathfrak{t}", "c3c87fe731dfb97cbd98cc291b907107": "|x| < a\\quad", "c3c8f6e94cf8db508c6d106821bd50e8": "m_j \\in \\{ -j, (-j+1), \\ldots, (j-1),j \\}", "c3c94a415a1b8ff38542b52fde222d48": "|z_1 - z_2|^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2.\\,", "c3c96616eb0ec31af1ea68b75afdcfa4": "(\\mathbf{a\\times b})_3 = a^1 b^2-a^2 b^1\\,.", "c3c9d2a51d42387a47cb6fc9b3e6ed74": "\\left(\\frac{x}{a}\\right)^2+ \\left(\\frac{y}{b}\\right)^2 = 1", "c3ca0b0239ea0daf1ce7cd04ce61950f": "\nr_1+\\cdots +r_k=n\n", "c3ca39489efc8180b0665c36ec04e17c": "g(x) = {\\rm lcm}(m_1(x),m_3(x)) = (x^4+x+1)(x^4+x^3+x^2+x+1) = x^8+x^7+x^6+x^4+1.\\,", "c3ca4ac34cb69504bcf6703df58f8543": "I_M(\\tau)", "c3ca6dc30481b4cb6b57d3f9f111cb39": " i \\in D", "c3caaffa8f70272e461c725e3ae50211": "\\phi =0", "c3cabdc9d6228ede9ed3f2bf5e9f36cb": "\\left(1+x+x^2\\right)^n", "c3cacf3c6d7a9c3491203a207d88c52d": " \\kappa \\neq 0 ", "c3cb1e39f27a12892c42d48fafd62808": " p_g \\le \\frac{1}{2} c_1(X)^2 + 2. ", "c3cb36594d8fada431362a09ddd571a8": "V_t-V_d", "c3cbc9e0b02589976fadd312970ce1f7": "F \\to \\cdots \\to T_{k+1}F \\to T_kF \\to \\cdots T_1F \\to T_0F,", "c3cbd490db81e6d4587992eacc7e638e": "\\displaystyle{\\mathrm{Tr}(T(X(X^2)) -T(X^2(X)))=\\mathrm{Tr}\\,T(aI)=\\mathrm{Tr}(T)a=0,}", "c3cbf93b1b99ff8623d769d58d22c9cb": "\\delta_1 \\in E_n(i)", "c3cc2d41d05a3cdb7525cb9778557256": " \\rho, \\omega, \\phi \\to \\eta e^+e^-, ~~ \\pi^0 e^+e^-", "c3ccaa447db424f1ef4f7fde831e8c97": " \\xi(s) ", "c3cce5daf34df18e166ddcf8941e0616": " \\frac{d^2x}{dt^2} + 2\\zeta\\omega_n\\frac{dx}{dt} + \\omega_n^2 x = 0. ", "c3cd60d9cc4317fcecd70e6889cd5cd9": "v_a", "c3cdb474acc615415239c2738a3528e0": "w_i=w_0\\,b_i M_i,\\ b_i=\\frac{w_i}{w_0 M_i},", "c3cde956c3fa40431bd25ca7888918ea": " p_a=(-1)^n(\\chi(\\mathcal{O}_M)-1).\\,", "c3ce1549b326fd23e2e86cd1030539e0": "I=\\langle \\eta-2\\rangle", "c3ce54dc2d081ba2ae6166c3b84878b2": " 0 = \\frac{-gx^2}{2v^2} \\tan^2 \\theta + x \\tan \\theta - \\frac{gx^2}{2v^2} - y", "c3ce8efd072ca63ff39c2c4faded7a59": "TK_R''^{}", "c3ce9bc2964f311e3245260a9ebe04e6": "a(1) = 1; a(n+1) = 1 + a(n + 1 - a(a(n)))", "c3ce9cc1df9a943a187977ebe1c60b61": "\nC_y = c_l\\sin\\phi - c_d\\cos\\phi\n", "c3cea96d14ea23bf0a9fe1840fd768da": "\\frac{\\tau_b}{(\\rho_s-\\rho)(g)(D)}=\\frac{\\tau_{c}}{(\\rho_s-\\rho)(g)(D)}", "c3cf083e08c135db529be28df1bdb3d1": "\\mathbb{C} - \\{1\\}", "c3cf25c781664abb694296275e829681": "\\frac{P,Q}{\\therefore P \\and Q}", "c3cf2df7b642ddd259c0bc8922444134": " h_f\\equiv \\frac{\\delta \\Phi}{\\Phi_0},", "c3cf496a5e84fb040c68888ddf20afcb": "\\cos{\\chi} = \\frac{ d_{BO}^2 + d_{BS}^2 - d_{OS}^2 } {2 d_{BO} d_{BS}}.\\!\\,", "c3cfa6e1fb74e4186761dc079c8673e3": "N_k(n)={1\\over n}\\sum_{d\\mid n}\\varphi(d)k^{n/d}", "c3d0090abd5c0e2be967c77e3a7bc9bb": "\\{z|-\\infty\\leq\\psi(z)\\leq c\\}", "c3d02fb6a7b14f47619619e1936b37ac": "r(d,s):= \\max_{d\\,'\\in D} f(d\\,',s) - f(d,s)", "c3d070a252d9f329dae49c9c8d25f17e": "\\mathbf{F}_1=(0,-m_1 g),\\quad \\mathbf{F}_2=(0,-m_2 g)", "c3d0d45e65d50a6bb9672645c2fb880f": "\\alpha=0.5, \\beta=1.0, \n\\delta=0.33, \\epsilon=-0.38, \\zeta=-0.18, \\eta=-0.42", "c3d1125478c0657d8adb2fd25ac24a97": "\\gamma:\\; {\\Bbb C}^n \\mapsto {\\Bbb C}^n", "c3d121a9108e191edba4ec89a91f0820": "\\mathrm{T}=", "c3d12bd090753c40633ad4354631c6e5": "\\displaystyle{ f_t(z)=e^t(z+b_2(t)z^2 + b_3(t) z^3 + \\cdots)} ", "c3d142faee09cfeffdb6574ce898072f": " \\alpha_k (t) = e^{i H t / \\hbar}\\alpha_k e^{-i H t / \\hbar}\\,\\!\\;", "c3d1ab99b3d270f6690bcc0af0b9fa38": "P(S=T|R)", "c3d1f168e4fcec634469f28189573dc5": "a \\,\\ ", "c3d24cbaec5f3d7686f8847c89c7148a": " \\nu = 1/2 ", "c3d260074ebf1f68343d43b16b9fa2f1": "f'(x) f''(x) \\neq 0", "c3d2848d368c006d956ebc5401786260": "\\displaystyle{W^2u=\\Delta u - V^2u - Au,\\,\\,\\, VWu =WVu + Bu,}", "c3d2f035cc0e485ac3331b924966a7f4": "A=\\mathbb{C}\\Gamma", "c3d31fe88616218c640dc5c5f092ab95": "\\int x^2\\,dx = \\frac{x^3}{3} + C", "c3d38e9a2e89158293ec0adc0f0f95fe": "n I_{n-1} = x^ne^{ax} - a I_n , \\!", "c3d3b4f2b92f7e9f0463096e607c67c9": "a\\int_{-\\tfrac{\\pi}{2n}}^{\\tfrac{\\pi}{2n}} \\cos^{-1+\\tfrac{1}{n}} n\\theta\\ d\\theta", "c3d3c16fd0b50302740c1059adc47f22": " Var(X) = V(\\mu) = \\sigma^2", "c3d3c44534fcc3cc739d03f6a3ece93f": " \\psi_1,\\dots,\\psi_m ", "c3d3dde84fd9f46ffe158a01a9ed512b": "G \\twoheadrightarrow K", "c3d42b2ceed9a30d70af4cf2d8d3a2a8": "h(x) \\leftarrow \\sum_{x \\rightarrow y} a(y)", "c3d45b48440d37dccba085fc400646b0": "\np \\,\\xrightarrow{a(x)}\\,p'\n", "c3d49974a7ff0b98409e181a6185a7bc": "\\{ X \\geq x \\}", "c3d4ab19171f8fabb385d4944d1d6266": " P(F^3) ", "c3d4e0bd95bcd0e296eb126ff4aec63e": " \\boldsymbol{\\nabla}\\times\\boldsymbol{S} = e_{ijk}~S_{mj,i}~\\mathbf{e}_k\\otimes\\mathbf{e}_m\n ", "c3d4e7a4c08ac6e7029c669985b1a143": " C_n = \\frac{(2n)!}{(n+1)(n)\\cdots(3)(2)(n)(n-1)\\cdots(2)(1)} = \\frac{(2n)!}{(n+1)!n!} = \\frac{1}{n+1}\\binom{2n}{n}", "c3d4f4bed91eb0a5dfb9908366e70854": " T_a ", "c3d4fe23606fec47f7e6dd08095169fb": "[L_m,L_n]=(m-n)L_{m+n}.", "c3d553324502b54b1a25ac4261bd08c6": "L/\\gamma", "c3d57d823c3cc2db783d89b72b4ad75d": "g_n \\sim \\frac {\\sigma^{2^n}}{n + 2 + O(\\frac{1}{n})}. ", "c3d58cdfb55381f460f07626661f2e47": "\n\\begin{align} \nE \\left[ \\sigma^2 - s_{biased}^2 \\right] &= E\\left[ \\frac{1}{n}\\sum_{i=1}^n(x_i - \\mu)^2 - \\frac{1}{n}\\sum_{i=1}^n (x_i - \\overline{x})^2 \\right] \\\\\n&= \\frac{1}{n} E\\left[ \\sum_{i=1}^n\\left((x_i^2 - 2 x_i \\mu + \\mu^2) - (x_i^2 - 2 x_i \\overline{x} + \\overline{x}^2)\\right) \\right] \\\\\n&= \\frac{1}{n} E\\left[ \\mu^2 - 2 \\overline{x} \\mu + \\overline{x}^2 \\right] \\\\\n&= \\frac{1}{n} E\\left[ (\\overline{x} - \\mu)^2 \\right] \\\\\n&= \\text{Var} (\\overline{x}) \\\\\n&= \\frac{\\sigma^2}{n}\n\\end{align}\n", "c3d593ffd7fa677f7b91363a14970651": "\\{x|\\rho_i(x)>0\\}", "c3d5d30a0a496f466f3852a1b43dbd4f": " S(\\mathcal{D}_{\\eta} (\\rho)) = H_2 (\\left(1 + \\sqrt{(1- 2\\,\\eta\\, p)^2 + 4\\,\\eta\\, |\\gamma|^2} \\right)/2) ", "c3d67cc0a640fde94419328a09820c7d": "\\sum_{n=0}^\\infty a_n = \\sum_{n \\in \\mathbf{N}} a_n.", "c3d69d18ae280f005a30b3d93f8cef9b": "{}_3Q_1 - _3 W_1 = U_1 - U_3 ", "c3d6b964c07ab810599895d95802ac2f": " \\omega =\\frac{dx\\wedge dy}{g(x,y)} ", "c3d6d7c8b7eea31eec7f9aaf534d17b3": "I t = C \\left(\\frac{C}{I H}\\right)^{k-1},", "c3d6ee58cca15abd0167a9ee386a8eff": "\\frac{MN}{EF}=\\frac{1}{2}\\left |\\frac{AC}{BD}-\\frac{BD}{AC}\\right|", "c3d7386e8bfe9750fc1c0334c24db89a": " P(z) = p(z) + \\log(2\\pi)/2 - z + \\left(z+\\frac{1}{2}\\right)\\log(z),", "c3d73f7edcee053fe520ef6089c6757b": "\\textstyle = x^i(a(x) + x^bb(x)) + x^{i+r}b(x)(x^{g(2l-1)}+1)", "c3d74730f91d97b0364ce2e774b7136c": "g(x) = \\arg \\max_y \\; f(x,y)", "c3d756dba715e62e815b151262cc0ccd": "\\mathbf{a}\\cdot \\mathbf{b} = \\sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \\cdots + a_nb_n", "c3d77f6f947a17fa59e5bbf1ca66aa50": "\\mathbb{R}^n_+", "c3d7bbdf5ef739b371cd6fec6f1fcdb3": "\\left[\\frac{\\alpha}{\\pi}\\right] = i^k \\equiv \\alpha^{\\frac{N\\pi - 1}{4}} \\pmod{\\pi}.", "c3d84058a911b6944b3646443761f252": "\\theta_1(0,\\tau) = \\sum_{n=-\\infty}^\\infty q^{\\left({n+\\frac12}\\right)^2} \\mathrm{ and } \\ \n\\theta_3(0,\\tau) = \\sum_{n=-\\infty}^\\infty q^{n^2} ", "c3d88614f9fa94973125358ec5a206ec": "N_n = \\frac1n \\sum_{d\\mid n} \\mu(n/d) q^d,", "c3d8fed7d5fe8d0a23183b0c44b5c4e5": "\n(A|B)=\n \\left[\\begin{array}{ccc|c}\n 1 & 3 & 2 & 4 \\\\\n 2 & 0 & 1 & 3 \\\\\n 5 & 2 & 2 & 1\n \\end{array}\\right].\n", "c3d907198f7519813194fa12ec948a9a": "f_i(W)", "c3d95344db9f67548fa4be315f924f13": "(1-\\varepsilon)", "c3d9ad349f810d39c4a96b6f0218681b": "x_1:=x_0-\\frac{f(x_0)}{(x_0-Q)(x_0-R)(x_0-S)},", "c3d9b8326534d57a6b78b69f88eda32e": "|\\chi(\\tau,f)|^2 \\le |\\chi(0,0)|^2", "c3da3d6344d134c07a45b20f8d3b12a7": "\n\\begin{align}\n\\left[S_x, \\, S_y\\right] &=\n -\\hbar^2 \\Big( \\mathbf{e}_{y} \\otimes \\mathbf{e}_{z} - \\mathbf{e}_{z} \\otimes \\mathbf{e}_{y}\\Big)\\;\n\\Big( \\mathbf{e}_{z} \\otimes \\mathbf{e}_{x} - \\mathbf{e}_{x} \\otimes \\mathbf{e}_{z}\\Big) \n+ \\hbar^2 \\Big( \\mathbf{e}_{z} \\otimes \\mathbf{e}_{x} - \\mathbf{e}_{x} \\otimes \\mathbf{e}_{z}\\Big)\\;\n\\Big( \\mathbf{e}_{y} \\otimes \\mathbf{e}_{z} - \\mathbf{e}_{z} \\otimes \\mathbf{e}_{y}\\Big) \\\\\n&=\ni\\hbar \\Big[ -i\\hbar \\big(\\mathbf{e}_{x} \\otimes \\mathbf{e}_{y} - \\mathbf{e}_{y} \\otimes \\mathbf{e}_{x}\\big)\\Big]\n=i\\hbar S_z. \\\\\n\\end{align}\n", "c3da56b301264dde7d17e2f940b29ff0": " X_n ", "c3da5863b1d1df34020664327c55de15": "n_f = 1", "c3dae7077fc34bb32b0ed21286713939": "W_E = \\left( \\frac{\\sigma^2}{2} \\right) (c_{11} + 2c_{12} ) \\left( 3 - \\left[ \\frac{c_{11} - 2c_{12}}{c_{1'1'}} \\right] \\right)", "c3daf30e2e7ac9351842a7422748f5b9": "x^{3} - 3x - 1 = 0 ", "c3db0d4483e3c3b184370d5e99056985": "\\{ 1, 2, \\ldots, n \\}", "c3db3ba73988fea79da0e6dcbf38d151": "F= 6\\pi\\,\\mu\\,a U\\left(1+{3\\over 8} N_R\\right),", "c3db8872883f58abd3bde6e25f3e98e0": "b = \\frac{1}{2}\\, k (V - Z_{p}^{*} I)\\,", "c3dc49cf4cb492def1ef19ae367a03ef": "\\left\\langle A_\\delta: \\delta \\in S\\right\\rangle", "c3dc4b160ec0c9d373e18f9decb46c65": " \\text{ } (3) \\text{ } V_{n+1} \\equiv 2 Q \\pmod {n} . ", "c3dcb2c82091c46125ab6f8e28a7bd15": "\\pm\\sqrt{1-x^2}", "c3dcd191f587e0d193f6d041c516d277": " \\zeta = \\frac{1}{\\sqrt{1-\\beta^2}};", "c3dcea7957ca62a3456e2988f0da9da6": "{F_{p}} = L \\times sin(\\beta) -{D_{t}} \\times cos(\\beta) > 0", "c3dcff99fa7ff0383cce7dcf76313ec5": "K = \\frac{0.823}{d}[1-0.0203(t-32)]", "c3dd0dfa3ae37d318497387306db50e8": "v_h=v_v", "c3dd127184c2351e56bf28b9c8d5d431": "E(\\lambda)=\\frac{\\Delta\\Phi}{\\Delta A \\Delta\\lambda}", "c3dd45b29f16463c285649485ac2812f": " 0<\\mu<\\frac{2}{\\lambda_{\\mathrm{max}}} ", "c3dd4c22642349b206c95c619b7ef909": "\n[\\delta_\\mathcal{D}, \\delta_\\mathcal{S}] = \\lambda \\delta_\\mathcal{D}\n", "c3dd65f383b1688ed7bd4c81a82d0a79": "\\displaystyle 2\\beta e^{1-\\beta}=1.", "c3ddb89fca4132a2c9073fb8e3a0ea9b": "E_n^{(1)}=V_{nn}", "c3ddd8d7e848d860d50e75299a1b6a10": "(0,0) \\in M_1", "c3de0a3dc726cbdd6834f69518fed19b": " \\Delta:H\\rightarrow H \\otimes H", "c3de2db53f33af3db2d716c5cd020fe4": "\\tfrac{1}{12}", "c3de305b5d92002bf3c69d526240ad02": "f:2^{\\Omega}\\rightarrow \\mathbb{R}", "c3df677cfe0d060e0d74053dadd0d114": "G = G + \\alpha \\, ", "c3dfc585a62d45aac3f42d98ab4f19eb": "\\mathrm{STUVWXYZ} \\!", "c3dfe70dc05858ef0121e96799ac7ed4": " \\operatorname{E}[(X-\\mu)^n] = s^n\\pi^n(2^n-2)\\cdot|B_n|.", "c3e07b9b5efe67e29562bd4451309ef1": "c(x)=1+xc(x)^2;\\,", "c3e0d963c972cf23cdbb2247c05fbefe": " f_n(t) \\ ", "c3e0de95f882c6b844f3c953f602daaf": "\\begin{array}{rcrcl}\nx&=&x_c+\\hat{x}&=&(R-r)\\cos t+\\rho\\cos \\hat{t},\\\\\ny&=&y_c+\\hat{y}&=&(R-r)\\sin t+\\rho\\sin \\hat{t},\\\\\n\\end{array}", "c3e176ab80395aa221d07a4c96f0c801": "\\lim_{|x| \\to \\infty} x \\sin \\frac{1}{x} = \\lim_{y \\to 0} \\frac{\\sin y}{y} = 1.", "c3e1af1d4a43ada5eeea824a1819e96b": "|\\Sigma|", "c3e1c3975dc05477d7fc959870867efd": "x = v e^u ,\\quad y = v e^{-u}", "c3e25a5714d46eb7d48135fa63f85755": "(V_A)^i_i ", "c3e25f67545bf2b271cc9d1e0bf12b14": "(\\begin{matrix} \\frac{m}{s^2} \\end{matrix})", "c3e2e106106423720d6bf535c2c1c809": "\\ \\Omega \\times G \\rightarrow G : (\\omega , g) \\mapsto g^{\\omega}", "c3e3287b429d407772b98ac114068c7a": "(1-\\epsilon)^{1/\\epsilon} \\leq 1/e < 1/2", "c3e33ceba2450681e426f7ab83833510": "\\rho=\\sum_i z_i q n_i = \\sum_i z_i q n^{0}_i e^{-\\frac{z_i q \\varphi}{k_B T}}", "c3e354c1cebe210391f7774dd74cbb79": "\\delta x_{Pe}", "c3e3b6f2617a7734f89d9339f8c1f770": "w_1 ,\\ldots , w_k \\in W", "c3e436738f69214351f21cd0bc3f6f95": "\nD_t(x_i,x_j)^2=||\\Psi_t(x_i)-\\Psi_t(x_j)||^2 \\,\n", "c3e473ec4b52f35b8a871341b13891ea": "(\\beta x)^2, (\\beta x)^3...,(\\beta x)^k", "c3e4ccd5fda158000159a8e5c46657a6": "S_0 = I \\,", "c3e4e211a925c2c6647603c5b6a91913": "\\{x+iy:a\\le x \\le b\\},", "c3e501f0a6cb071ba2b470e2add3bf26": "\\{\\psi\\in\\mathrm{End}(A)\\otimes\\mathbb{Q}:\\psi'=\\psi\\}", "c3e593b2ce21efdb066cb742e27ebc77": " x \\in \\mathcal{N'}_k(x) \\subseteq X ", "c3e593fcb1a8ac9defad6bc619e61d1a": "a_{11}+b_{11}", "c3e5ec206ed01f26e807abe16af92c9e": "\\vec{F} = - \\nabla U", "c3e604211549f36e99eeee2b5bf92f89": "d_{ij}(X)", "c3e644b4f84793606ba3d6a1554ca9a1": "x \\to x + \\frac{i}{\\omega} \\int^x \\sigma(x') dx'", "c3e653562f14dbaabfa1c6102471c02e": "P(G^\\prime) = \\frac{1}{N}\\sum_{i=1}^N \\delta(c(i)) ; c(i): F_R^i(G^\\prime) \\ge F_G(G^\\prime)", "c3e6786e191ce4380491a088c8616d7a": "\\begin{align} \\mathbf{F} &= 6\\pi\\mu a \\left( 1 + \\frac{a^2}{6}\\nabla^2 \\right) \\mathbf{v}^{\\infty}(\\mathbf{x})|_{x=0} - 6\\pi\\mu a \\mathbf{U} \\\\ \n\\mathbf{T} &= 8\\pi\\mu a^3(\\mathbf{\\Omega}^{\\infty}(\\mathbf{x}) - \\mathbf{\\omega})|_{x=0} \\end{align}", "c3e67eb65933276bcd2638befba051d0": "\\Sigma_{res}^{-1}", "c3e6b61808fc93660c0bc2cda1a7756e": " V \\approx \\hat{A}_0\\left[ f^2 \\right] - \\hat{A}_0\\left[ f \\right]^2 ", "c3e6e7cb83a2208b62b7c168655c4cec": "F'^{\\alpha \\beta} = \\Lambda^\\alpha{}_\\mu \\Lambda^\\beta{}_\\nu F^{\\mu \\nu}", "c3e715b5706df199388e98bd83691d1b": "\\operatorname{Li}_2(z)\n", "c3e72d0f5648e7c322ec0cc186073b46": "gd_{\\mathbb{Q}}A_3=1", "c3e739fc34986db9efaa713072554321": "I(X;Y;Z)", "c3e7786bf72156a6cc0f0b108aebd0a2": "\\Psi_{n}(t)=(2n)^{-\\frac{n}{2}}c_{n}H_{n}\\left(\\frac{t}{\\sqrt{n}}\\right)e^{-\\frac{1}{2n}t^{2}}", "c3e78360960a4a03c6b13daa64015dd8": "H = T^{\\varepsilon}", "c3e791818560518772b1548ac2f7bd9a": "\\mathbf{f}_e \\in \\mathbb{R}^3", "c3e826d52d66a2fd1958e45a639745be": "\\int M ds \\sqrt{v}", "c3e828f20b5244b043c2cf5a5709ec45": "EAS\\,", "c3e836f6799e7555746d35e52615ec76": "\\gamma=(1/2)F/L", "c3e97dd6e97fb5125688c97f36720cbe": "$", "c3e9faf2fbaa0483f06cb7adde965502": "\\displaystyle{\\pi(g)e_W= \\mu(\\det(I+\\overline{A}^{-1} \\overline{B}W)^{-1/2}) e_{gW},}", "c3ea002c56499f418214f061453ef0d6": "v_0 = \\frac{e}{e + \\delta}", "c3ea277ed9378cafd71ea4bf48ba5b20": "{\\displaystyle R_{{s\\ normal}}={\\sqrt{{\\frac{\\omega\\mu_{0}}{2\\sigma}}}}}", "c3ea5349cd76818a18d72c38a01ea8e1": "10x^2+y^2=1+6x^2y^2", "c3ea864f8c7c0d3e3202f80fb5122824": "\\big.\nQ=1+\\frac{N}{V}\\alpha_1 + \\frac{N\\;(N-1)}{2\\;V^2}\\alpha_2+\\cdots.\n", "c3ea8b655157a5b1a44164cac2072298": "\\textbf{c}", "c3eadb9893127eceb8a50ab0dfca586b": "\\frac{1}{11}", "c3eb338beee48b24766c894a28523772": "tT^2=0.74(10.75/g_* )^{1/2}", "c3eb983f7eb71e64818a81942c192e2a": "X= \\frac{1}{k}F\\,", "c3ebb9e2d4b56d09e4c9148363c9e88b": "\n\\Lambda_p^*:=\\{0,\\dots,p-1\\},\n", "c3ebea8d228fd8f3c6dbfd6e8719d5f9": " \\langle Q(\\text{start}) Q(\\text{end}) \\rangle ", "c3ec03d1a2ef71acedfe1022be5e289c": "c = \\sqrt{g\\,h}\\, \\left( 1 + \\frac12\\, \\frac{H}{h} \\right)", "c3ec102e08585b24334e780c1f081e5e": "\\xi=\\sigma+i\\tau", "c3ec43a012c628e26e924061eb48fad2": "n_\\mathrm{ch}/A \\propto 1/\\alpha_s(Q_s^2)", "c3ec8938395f75db8a0f52f9090cd6e5": "C_i = \\frac{|\\{e_{jk}: v_j,v_k \\in N_i, e_{jk} \\in E\\}|}{k_i(k_i-1)}.", "c3ecf434aef8ef5df7749f8cd1fa1ff8": "\\mathit{ACC} = (\\mathit{TP} + \\mathit{TN}) / (P + N)", "c3ecffd6c504c208f50569b688bc8c4a": "A \\Rightarrow^{ac}_{p_{i_1}} w_1 \\Rightarrow^{ac}_{p_{i_2}} w_2 \\Rightarrow^{ac}_{p_{i_3}} ... \\Rightarrow^{ac}_{p_{i_n}} w", "c3ed1518e5e82c34221a20fb618445f7": "\\tau_ 2", "c3ed17a26eb4bc331d9455a58de58d6c": "H(X,Y) \\geq \\max[H(X),H(Y)]", "c3ed2971ccd81000a070e36e36d91d43": "\\dot{\\theta} \\leq - \\frac{\\theta^2}{3}", "c3ed85cd68487b50ff8096469f96d750": "c_\\lambda", "c3edad3371bd15e5e48f04da50d49954": "u= +\\frac{\\partial\\psi}{\\partial y}", "c3edd57c7928e2c4b75552965465ba1e": "\\omega_\\text{P} = \\frac{1}{t_\\text{P}} = \\sqrt{\\frac{c^5}{\\hbar G}} ", "c3ee1dc5fa3ac0d0ff50b8993000a8bb": "\\mathbf{U}", "c3ee389660123356679bd8605a4eb430": "f^{1}(x) = b + m \\times x = 6 + 7 \\times x", "c3ee39b3f678fdb8f4ffd2e42387553b": "|\\psi _i \\rangle ", "c3ee9b0bf5bafa024ae28ee8df5027b6": "\\mathrm{maj}=\\mathrm{th}^3_2=(x\\land y)\\lor(x\\land z)\\lor(y\\land z).", "c3ee9f2ffcdccb06c6d50082536484d3": "H\\left(X|Y\\right)=-\\sum_{i,j} P(x_i,y_j)\\log P\\left(x_i|y_j\\right)", "c3eee9693f6d1c0dfaa75eb0a5d810ca": "R^G", "c3eef401839d9cbe3b6cf00d504007db": "(x_0 + 1, y_0)", "c3ef4d904e1200d0d0e08c0e82d57757": "\\sigma_a=\\limsup_{n\\to\\infty}\\frac{\\log(|a_{n+1}|+|a_{n+2}|+\\cdots)}{\\lambda_n}.", "c3ef61e5b7e46abace603a3dbd58e145": "\n2^{(1^2)} \\, \\ne \\, (2^1)^2\n", "c3ef954bb5b965c5a362de0b2bd21ab5": "\nP^{\\prime }(x_1,x_2,\\ldots ,x_n)=\\frac{\\frac{\\frac{\\prod_{\\mathcal{T}\n_{n-1}\\subseteq \\mathcal{V}}p(\\mathcal{T}_{n-1})}{\\prod_{\\mathcal{T}\n_{n-2}\\subseteq \\mathcal{V}}p(\\mathcal{T}_{n-2})}}{\\vdots }}{\\prod_{\\mathcal{\nT}_1\\subseteq \\mathcal{V}}p(\\mathcal{T}_1)} \n", "c3ef99b3bf0f1ea4e8b7e8bef817fafe": "\\mathbf{A} = A \\mathbf{\\hat{e}}_{\\bot} \\,\\!", "c3efc43ffffe21fd5ff74010a95c7a81": " \\begin{bmatrix} A \\end{bmatrix} ", "c3f0945f0b4ac4c23fa30979d77b4fab": " x \\; = \\; \\frac {\\text{E}}{k_0 \\nu_1}", "c3f096f2c53c904875608c4c1c9d8e53": "2\\pi \\text{ rad} = 360^\\circ", "c3f09fdccfff0cfdd8847a0ce1090fe5": "S \\to conc(\\langle a \\rangle, S, \\langle a \\rangle)", "c3f0a54efb31cd3d13c6849be2c3cc6e": "A^g", "c3f0eedd8187d3c046c60c2c50e83c2b": "L_a\\subseteq\\Delta^*", "c3f0ef71f0ef329a775610046c759b5c": " y_0, y_1, \\ldots, y_{s-1} ", "c3f11000507c7280e10ed83a9e292fb9": "V_Y,", "c3f1340ee054464707e9229f87f740c5": "f(x)=x^2-cx-a \\in \\mathbb F_q[x]", "c3f149998fe08472890895cc30916cba": "\\tau_B < \\tau_L", "c3f202794117b41e4a1098771fb1e653": "\\mathbf{\\hat \\mu} = \\frac {\\sum_i x_i} {n}", "c3f292383419faede8f52121defacd20": "\\left( \\begin{smallmatrix} 1 & -7 \\\\ -12 & -2 \\\\ \\end{smallmatrix} \\right)", "c3f2ecece5b3fc466a1da97daefd4b21": "\n\\frac{V_P}{V_S} = \\frac{I_S}{I_P} = \\frac{N_P}{N_S}=a", "c3f38e22bef2ee0179c7f65ff0716241": "\\Theta_\\ast \\colon \\operatorname{Cotor}^{S_\\ast(B)}(S_\\ast(X)S_\\ast(E))\\rightarrow H_\\ast(E_f),", "c3f3b79e87369600b297f2eb27667094": "= \\| x \\|_{\\infty} \\sum_{k=-\\infty}^{\\infty}{\\left|h[n-k]\\right|}", "c3f3be62f7859c78927bf850a3112caa": "XDH(g^x,\\ g^y)=g^{xy}", "c3f4169abcbd51d7280c6d9bc143a881": "\\alpha_1 \\ne \\alpha_2 \\in \\mathbb{F}_{q^k}-\\{ 0 \\}", "c3f434fcc0ade326bcf5adfd9c8a64b1": "[\\alpha] \\in H^q(X) \\, \\!", "c3f4ad577040472610d8889b75cd286a": "\\sin 2\\theta = -1\\,\\!", "c3f4e7354ef8ad509e265cbdffea70d3": "\\varphi^k", "c3f506a8260bfdb438c3a7c1e05c9084": "A_{i-1}", "c3f50745036bc33fba7183b850944087": " J_z \\Psi_s = M_j\\hbar \\Psi_s ", "c3f5d57ca7deaa240dc5c8c2da25f7cd": "\\mathbf{w_p} \\leftarrow \\frac{\\mathbf{w_p}}{\\|\\mathbf{w_p}\\|}", "c3f624ad60783e53dc28d87afb4b898c": " c(K) \\approx 1 - \\text{const} / \\log \\log K. ", "c3f632b2ef292d22c5a1c8eaf99efa26": "w_1, w_2, \\ldots", "c3f6477dedcfd62be2727e6133e83be1": " \\Phi,_{i} = 0", "c3f7087c9eb0c8653153a24a3b9eadf6": "F_{i} \\subseteq \\mathbb{R}^{n}", "c3f74cf997e22e996cb595040a4bac5d": " \\tfrac{1}{c}\\eta(N^a,V) ", "c3f7b93dd3ed19a478242a29c1c06ce6": "\\ ( i\\gamma^\\mu \\partial_\\mu - m) \\psi = 0", "c3f7ff3356caf218c075f88a432fe1f3": "\\mathrm{C + H_2O \\ \\rightleftharpoons \\ CO + H_2}", "c3f810b6be3a2981a76488c7d3ac03e2": "\\left\\lfloor\\frac{\\max(|s_1|,|s_2|)}{2}\\right\\rfloor-1", "c3f87ab2bf01b9f9976e973504292cb8": "Q_m", "c3f8b7298dce198c2f700f75df0e438b": "(2\\pi)^{-\\frac{k}{2}}", "c3f96978fbdde965b5b35214f1a5bec6": "\\scriptstyle\\leq10^{14}(\\text{eV})^{-1}", "c3f97a4420c67227501e8aa037c1c616": "V\\,", "c3f9f441cc593571b38fc3b06e0ba9d2": "\n f := \\sqrt{J_2} + \\lambda(J_2,J_3)~(\\tfrac{I_1}{3} - B) = 0\n ", "c3fa08d647beb65aa553b1af30ffef30": "T_b = \\frac{a}{Rb}", "c3fa178d143f6ce594b5b4ff13b2db96": "\\gamma = \\gamma_1+\\gamma_2\\,", "c3fa338f7e48cc638dd21e1e63be79b0": "1, 2, 3, \\ldots, p-1.\\,\\!", "c3fac6ff712661ab15acdf50ae2af50c": "\\frac{F-(1-F)p}{3F-1}", "c3facceb43addb1419954419ac4dc5a2": "K_c=\\frac{[{CH_3CO_2}^-][{H_3O}^+]} {[{CH_3CO_2H}][{H_2O}]}", "c3fae9dd87f79f32fa43d232c6038eb2": "Y \\subseteq \\mathbb{R}", "c3faf9749508f2c7d3c70fbf3c4361c5": "\n\\begin{align} \n& A=\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}-\\frac{dv}{dr}|_{R_{0}}\\right) \\\\\n& B=-\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}+\\frac{dv}{dr}|_{R_{0}}\\right) \\\\\n\\end{align}\n", "c3fb15422c82ea710cb65f8f2b3c47d1": "\\beta(\\alpha) := (\\pi_Y)_{*}(\\beta \\cdot (\\alpha \\times Y))", "c3fb5f05c17b8a6aa08f5549038f3743": "f:\\mathbb{R}^n\\rightarrow\\mathbb{R}^m", "c3fb8b26583a3809e40bf3f5e182d90b": "2^{n + 1}", "c3fb9971d7d6ba5e41f33dccfaa77c58": "\\scriptstyle i,\\, j,\\, k,\\, \\ell", "c3fbb7dd087a784b177d628e2c785cfd": "S_\\text{baker-folded}(x, y) = \n\\begin{cases}\n(2x, y/2) & \\text{for } 0 \\le x < \\frac{1}{2} \\\\ \n(2-2x, 1-y/2) & \\text{for } \\frac{1}{2} \\le x < 1. \n\\end{cases}\n", "c3fbe5e41beebab9f4388c23e4c29151": "y_1^2 + y_2^2 - y_3^2 = -1", "c3fbff259820058f3a94f1249cf53165": "(2j_1+1)", "c3fc45d8a19e552acfe699d3803eaf29": "C_{Cr} = \\frac { U_{Cr} \\times V }{ P_{Cr} }", "c3fc4c2c9f800d0f587c80bf5bbef6f0": "k = B/L", "c3fc683841298c5cb1fc174f7525c3f2": " h(G(\\beta)) = h([G(\\alpha_1)...G(\\alpha_n)]) = F(h(G(\\alpha_1))...h(G(\\alpha_n))). \\;", "c3fca2637b9421efbf4da170dd7d8531": "\\left(\\begin{matrix}\n1 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 1 \\\\\n1 & -1 & 1 & -1 \\\\\n0 & 0 & 0 & 1\n\\end{matrix}\\right)^{-1} =\n\\left(\\begin{matrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1/2 & -1/2 & -1 \\\\\n-1 & 1/2 & 1/2 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{matrix}\\right).", "c3fcaf2f248ff28947a34d3cb3aefdc3": "\\frac{\\partial u}{\\partial x} = \\frac{\\partial u}{\\partial r}\\frac{x}{\\sqrt{x^2+y^2}} - \\frac{\\partial u}{\\partial \\varphi}\\frac{y}{x^2+y^2} = \\cos \\varphi \\frac{\\partial u}{\\partial r} - \\frac{1}{r} \\sin \\varphi \\frac{\\partial u}{\\partial \\varphi},", "c3fcb7ea2a2bd44c11d87a4ce6b111c5": "\\sum_{s,t \\in G}\\langle F(s^{-1}t) h(t), h(s) \\rangle \\geq 0 ,", "c3fcdff48185e877afbb1e3c9859e119": "\\theta_{k} = du_{k} - u_{k+1}dx \\qquad k=0, \\ldots, r-1\\,", "c3fd1c94094901cbb9222e3b0b1eb2e3": "{\\mathcal{S}}", "c3fd5d57d398dded6e6ac9f73eead6ba": "\\mathbf{v}'_0 = \\mathbf{0}", "c3fd66bb835a9ba55c9bd12a5a82a843": "{e}^{\\pi}", "c3fdb7703c54d7508640da1dfca46ca9": "A_{12}=A_{21}=A", "c3fe1630e977adb1f59bf344c42e7011": "P = 11101010\\,", "c3fe43417fa48c921c42a31d7e490447": "(a=|\\mathbf{E}|, b=0", "c3ff84b770257c4f8a3c452cb7aa5b8f": "\\operatorname{Res}(f,c) = \n{1 \\over 2\\pi i} \\oint_\\gamma f(z)\\,dz", "c40026f798d2813f202d96f1c2cac398": "\\ln(k)", "c40071e07f09f3cba2ab96b979ce359f": " \\overrightarrow{m}(\\overrightarrow{B}) = \\frac{V \\chi \\overrightarrow{B}}{\\mu_0} ", "c400a4d3b9acc7c3a979d6403a46904b": "y_2=\\varepsilon(x,\\bar{\\bar{C}}).", "c400ba3999ea399dd38f8a497c0b696e": "\nAB =\n\\begin{bmatrix}\n 2 & 3 \\\\\n 3 & 4\n\\end{bmatrix}\n\\begin{bmatrix}\n 1 & 0 \\\\\n 1 & 2\n\\end{bmatrix}\n\\stackrel{?}{=}\n\\begin{bmatrix}\n 6 & 5 \\\\\n 8 & 7\n\\end{bmatrix}\n= C.\n", "c400c9d17b02fd9b3f5ebe71402f7db2": "f(z)=c+\\sum_{k=1}^{\\infty}\\left(\\frac{z}{k}\\right)^{n_k}", "c400f88ed504d957bcec5e3582ee74f7": "s - f = \\frac {f} {m}", "c40141b7c4a223f1143f15b922a2a77d": "Q^{(g+1)}", "c401a8bd1d2fae230f56e93ef3270a3d": " 1/a > 1 ", "c401af6524498f2a6abafe8b5712e6ad": " (a:-c:b) \\times (d:-f:e) = (-c e + b f : b d - a e : -a f + c d) ", "c401b3f3825edca5dc96a5a895894f2f": "G_U\\;", "c401dec42ab0f6b77000bf07d67990da": "S(T,x) = S_l(T)+x(\\frac{L_0}{T}+C_p)", "c401e32ed11f93fa38cf2de642183a2a": "O(mn log m)", "c402580a1cd9f080107669a2ca58e66b": "X^2-2", "c402d7d9a302ab1559d7f3a251be7345": "g(z) = \\frac{z - \\gamma_1}{z - \\gamma_2}", "c402df9f7e02732d3acab4eb4e840cf5": "E\\in\\Gamma = [E_\\min,E_\\max]", "c402fd52268cc7471d81f39600ef07b7": "f(\\xi,t')=-\\frac{i}{\\tau} \\sqrt{\\frac{t'}{\\xi}} \\int \\exp \\left[-2i \\sqrt{\\xi t'} \\cos k\\right] e^{-ik} dk ,", "c402fe01d285eb64c7231e5a7ac90663": " P(M_N >x) \\approx \\exp[-NI(x) ],", "c40300e7ddcd091c73c196d46c8de446": "f^*(x^*)=\\begin{cases}2x^*-4,\\quad&x^*<4\\\\ \\frac{{x^*}^2}{4},&4\\leqslant x^*\\leqslant 6,\\\\3x^*-9,&x^*>6\\end{cases}", "c4030f8e39d897a949514309bf903aca": " F + 000 = \\begin{bmatrix}\n 0 & 0 & 0 \\\\\n 1 & 1 & 0 \\\\\n 1 & 0 & 1\n\\end{bmatrix} = F^{id}=F^{(2,3)_R(2,3)_C}.\n", "c4031082d20a06ec4f105c0baae8419a": " x * y = y * x \\,", "c40382e06451ba2f9608433321570eaa": "0\\ 0^1\\ 1\\ 0^2\\ 00\\ 0^4\\ 01\\ 0^8\\ 10\\ 0^{16}\\ 11\\ 0^{32}\\ 000\\ 0^{64}\\ldots", "c40389fda1e24ee0b07c3b71b158fcc0": "\nG_{\\alpha\\beta}(t|t') = \\int_{-\\infty}^{\\infty}\\frac{\\mathrm{d}\\omega}{2\\pi}\\,\nG_{\\alpha\\beta}(\\omega)\\,\\mathrm{e}^{-\\mathrm{i}\\omega(t-t')}.\n", "c403deed4d9bfd107d98013faa875189": "\\binom n0 = \\binom nn = 1 \\quad \\text{for all integers } n\\ge0,", "c404babb4f9f325ea6542057557aab26": "-\\nabla\\,\\varphi=\\vec{E}+\\frac{\\partial\\vec{A}}{\\partial t}", "c404dcd997b787b50daed8ac5d740279": "x \\leq y \\land y \\leq y", "c404e703f867a64a095da954071c5b73": "\\Gamma = \\omega d_{\\text{eff}}E_0/nc", "c404efec5df46dc4187f845667d6c4eb": "\nI =\n\\begin{bmatrix}\n \\frac{1}{12} m l^2 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & \\frac{1}{12} m l^2\n\\end{bmatrix}\n", "c40504aa7916be9dc1331e4c377fbf67": "\\max( |A+A|, |A \\cdot A| ) \\geq c |A|^{1+\\varepsilon} ", "c40534b5d937262354a0abab4417bcdc": "\\tfrac{9K(K-\\lambda)}{3K-\\lambda}", "c40572bad068121a785a65b65fc97264": "\\gamma_1=\\left.\\Delta/(2\\mu)^{3/2}\\right.\\,", "c405b12bea83a18fb0a346a71d8e8026": "\\textstyle 10^{-4}", "c405b87ba36d5dc5334c48874b460493": "K_6(x,y)", "c406798ac994c91561ac86aac9546754": "~(E/M_Z)^2", "c4067fe099a3b8c4d855b58f02806819": " M =M' + M''=K + \\frac{4}{3}G", "c4069ff78b54ef1739ffdbb49ff1f582": "\n\\overline{\\mathbf{P}_{n}\\mathbf{U}} + \\overline{\\mathbf{P}_{n}\\mathbf{V}} = \n\\left( r_{U} + r_{n} \\right) + \\left( r_{V} - r_{n} \\right) = r_{U} + r_{V}\n", "c406a1d487195940fee930436f595d0a": "\\gamma^\\mu\\gamma_\\mu=4 I", "c406b013f294e479b280beaa0d36d656": "= 3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow (\\cdots (3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3) \\rightarrow 1) \\cdots ) \\rightarrow 1) \\rightarrow 1\\, ", "c406e96ed5d114ad864328877db60bb8": "{ {2\\ln\\left(2\\right)\\over5} + {3\\ln\\left(3\\right)\\over 10} + {\\pi\\sqrt{3}\\over10} }", "c4070d203bd57842dfb7baa7bfcfc7c1": " = \\int_{0}^{L} dz' \\int_{x,y} dx dy \\, \\rho(x, y, z') \\,r^2 = I_z", "c40710bf4ac0469ecef0efc64953dc59": "L^{2}dx", "c40772f01757c85cb4321b6eeb9ea9b0": "I_{L1}", "c407b790fbeacf6b256ad166147f2c53": " {\\mathfrak gl}(A) ", "c407f133bfc25db8d4d99d53123c20a6": "\\scriptstyle M_n= -\\sum_{j=0}^{n}\\ln{((n + 1)(X_{n,i+1} - X_{n,i}))}", "c407fdaff3d81042a4a432c481b429a1": "A \\otimes_{C} B = \\left\\{\\sum_{i \\in I} (a_{i},b_{i}) \\; \\big| \\; a_{i} \\in A, b_{i} \\in B \\right\\} \\Bigg/ \\bigg\\langle (f(c)a,b) - (a,g(c)b) \\; \\big| \\; a \\in A, b \\in B, c \\in C \\bigg\\rangle ", "c40813d62eb7d91c5f3495fc28c2bcab": "S_{f\\!f}(\\ell) = C \\, \\ell^{\\beta}.", "c4084011d71f3f4c966f2c5a5249cf42": " a_i\\, (i = 0, 1, 2, ..., n) ", "c40878a5ca14352e5bdfa87f30aefc94": "f(re^{i\\theta})=Re^{i\\Phi}", "c40878f3b75929ea4e5bbb161a09bb65": "\\rho_{X,Y}=\\mathrm{corr}(X,Y)={\\mathrm{cov}(X,Y) \\over \\sigma_X \\sigma_Y} ={E[(X-\\mu_X)(Y-\\mu_Y)] \\over \\sigma_X\\sigma_Y},", "c40888d02d5b1afbb250f57066258da2": "Y(t) = [K(t)]^{\\alpha} [A(t)L(t)]^{1-\\alpha} \\,", "c408cb2d7091f4fd7c4f0603dd3872b5": "(X, Y, \\langle , \\rangle)", "c408fc53dbee5e58c7af832d9a9c0446": "\n1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+\\dots \\, = \\, 1\n", "c4091caf48f8595f065c06ee1f3c26ac": " \\hbar = 1 \\ ", "c4094af322862946678f1fb807ca50b3": "C,k<\\infty", "c4099f42088aa07b2fc7dea1f9faf643": "(x)_n \\!", "c409c5e35d98c40fb2806e4eece8ce72": "\\varepsilon_{i_1,\\cdots,i_n}", "c40a154688555a3fd4183318e9a52c8c": "\\mbox{forward reaction rate} = k_+ [A]^\\alpha[B]^\\beta \\dots \\,\\!", "c40a3bf3a766772d5608861c77b41355": "\\{a_i,b_i\\}, i=1...g", "c40a57af4185ad6a1ce6cdf12b5616fb": "A \\sqrt{\\alpha(x)} \\exp\\left(-\\int\\alpha(x)\\,dx + i\\beta_{w}x\\right)", "c40a8b04697320e4c2a0fd92f4cc324d": " (n-1) a_n = \\sum_{k=1}^{n-1} b_{n-k}a_k.", "c40b27ce6fdeb8e9990fa864d6d97578": "\\sum_{n=0}^{N-1}|X_n|^2=1", "c40b2f415ee94618bb0bfe21ce770dcc": "\n \\mathbf{x} = x_1\\boldsymbol{e}_1+x_2\\boldsymbol{e}_2+x_3\\boldsymbol{e}_3 \\equiv x_i\\boldsymbol{e}_i\\,.\n", "c40bad0c069edc836c771705119ff9f9": " s_{\\pm} = -\\rho \\pm j \\mu, \\, ", "c40bb22321ab899245e1fbc7f98edf4e": "\\mathrm{Re}(\\gamma) \\approx \\frac{\\sqrt{LC}}{2} \\left( \\frac{R}{L} + \\frac{G}{C} \\right) \\,", "c40bf943ed31c809c5157a01bd68bc50": "\\Phi\\,", "c40c20af40117ef93e4e8c6eeeb723f0": "t \\pmod N", "c40c4555f247e650adc732dea09d51e4": " Y^{(n+1)}(t) = R_n^{(n+1)}(t) - \\frac{R_n(x)}{W(x)} \\ (n+1)! ", "c40c63b4f41601b0f35d3491da45aad1": "\\{1, 2, \\dots, k\\}", "c40c7b35f11997c102e76b136b82573f": " \\bar \\psi \\partial_\\mu \\psi \\mapsto \\bar \\psi \\partial_\\mu \\psi + i \\bar \\psi (\\partial_\\mu \\Lambda) \\psi ", "c40ccba9e980116cbd5c80c2932841ff": "z\\infty = \\frac{z}{\\left | z \\right |}\\infty.", "c40ce2eae0c54de97ef6dd93acafba24": "\\sum_{i=1}^n \\alpha_i y_i = 0", "c40d3d4e78d4ecc1314c9b1d67e602cf": "\\boldsymbol{\\xi} : X \\rightarrow \\Xi ", "c40d59a9d393f9dcd6bfcc8e2e442bd2": " v = \\frac{R_p}{R_i} = \\frac{R_p}{R_t}", "c40da8a99fd6276ac8bd92013c84ae4d": "(5.a)\\quad \\nabla^2 \\psi =0\\,,", "c40dddf444a789a70a9a411deb51e697": "|S_1|,|S_2| \\leq d", "c40dfef2d877d2f85962dc6fe2e330b2": "g^{-1}f", "c40e4eceeb21478c9f369c162410c43f": "\\Pr(X \\le x)=e^{-\\left(\\frac{x-m}{s}\\right)^{-\\alpha}} \\text{ if } x>m. ", "c40e7d63306b7d0e7e2dedb7cffcb4f6": "\n\\Delta G = \\Delta G_{w} - RT \\Delta n \\ln \\left(1 + k [D] \\right)\n", "c40eddf0e587b609aadb3faa6b2d9694": "\\sum_{k=1}^{n} \\overline{U_{j,k}} \\cdot U_{k,j'} = \\delta_{j,j'},", "c40f2df8b1230649b5cc5fcb4b4bcbeb": "\\pi_k(U)=\\pi_{k+2}(U) \\,\\!", "c40f8e665c72d0027cf1ea9de6c78dfb": "x_1^s \\, = \\, \\frac{Kx_1^l}{1+(K-1)x_1^l}", "c40fdde5dc50c975f763ff49cad84349": " \\mu_w + \\sigma_w x ", "c40fe9b20a654325b5b23d34e4f46688": "W^{1, p} (\\Omega) \\subset \\subset L^{q} (\\Omega) \\mbox{ for } 1 \\leq q < p^{*}.", "c4104c3f2e9492350f3502a8f0632d89": "(\\bar{4},1,2)_H", "c41052339ec239ca44335588897765cf": " P = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}. ", "c410592be2d4a555ab9bd68d315f3264": "\n\\begin{array}{lcl}\n\\boldsymbol\\alpha &\\sim& \\text{A Dirichlet hyperprior, either a constant or a random variable} \\\\\n\\boldsymbol\\beta &\\sim& \\text{A Dirichlet hyperprior, either a constant or a random variable} \\\\\n\\boldsymbol\\theta_{d=1 \\dots M} &\\sim& \\operatorname{Dirichlet}_K(\\boldsymbol\\alpha) \\\\\n\\boldsymbol\\phi_{k=1 \\dots K} &\\sim& \\operatorname{Dirichlet}_V(\\boldsymbol\\beta) \\\\\nz_{d=1 \\dots M} &\\sim& \\operatorname{Categorical}_K(\\boldsymbol\\theta_d) \\\\\nw_{d=1 \\dots M,n=1 \\dots N_d} &\\sim& \\operatorname{Categorical}_V(\\boldsymbol\\phi_{z_{d}}) \\\\\n\\end{array}\n", "c410e8564a5b9dd383fcbb10951a13eb": "\\psi=0.\\,", "c411462fe4530cf13c8e6a94342044d9": "t_{m+1}", "c4118ddcbea605836b7c98c214c0270a": "\n \\hat F_n(t)\\ \\xrightarrow{a.s.}\\ F(t),\n ", "c411ecd96cd41c1c3b881615a4528249": " {\\bold U} = \\left[\n\\begin{array}{c}\n \\rho \\\\\n \\rho u \\\\\n \\rho v \\\\\n \\rho w \\\\\n \\rho e_t \\\\\n\\end{array} \\right] \\qquad \\quad\n{\\bold F} = \\left[\n\\begin{array}{c}\n \\rho u\\\\\n \\rho u^2 + p \\\\\n \\rho uv \\\\\n \\rho uw \\\\\n (\\rho e_t + p)u \\\\\n\\end{array} \\right] \\qquad \\quad\n{\\bold G} = \\left[\n\\begin{array}{c}\n \\rho v\\\\\n \\rho vu \\\\\n \\rho v^2 + p \\\\\n \\rho vw \\\\\n (\\rho e_t + p)v \\\\\n\\end{array} \\right] \\qquad \\quad\n{\\bold H} = \\left[\n\\begin{array}{c}\n \\rho w\\\\\n \\rho wu \\\\\n \\rho wv \\\\\n \\rho w^2 + p \\\\\n (\\rho e_t + p)w \\\\\n\\end{array} \\right] \\qquad \\qquad\n", "c411f8fe927d8d1c76068d1728c5d2b7": "\\frac{\\phi(x)-\\phi_0}{\\phi_L-\\phi_0}\\,= \\frac{\\exp(P_e \\frac{x}{L})}{\\exp(P_e)-1}", "c412188b02efd163085e55c0d351fe41": "\\,m", "c4129f4d569686684493ca18084311a0": " \\mu(A+zI) = \\mu(A) + \\real\\,(z)", "c412a43c2f9e469c62a685b6c53dc629": "\\hat{\\sigma}^2:=\\,\\frac{\\sum \\left(X_i-\\overline{X}\\right)^2}{n}", "c412bfa49153053b9f6aa4c69e0b6a95": " C(G) ", "c41329e8760a2fa8c5963ed710d300cc": "= q_eV_0", "c41366a9ed039906005d80f6c2467516": "u(t) = \\tan(\\theta/2)\\;", "c414292cb36bbedc52f7db485898c075": "k_{\\mathrm{H,cp}} = \\frac{c_\\mathrm{aq}}{p}", "c41431e85a27fa55b89d6a0fc5cb3328": "\\mathrm{VF_5 + 2H_2O \\ \\xrightarrow{0^oC}\\ H[V(OH)_2F_4] + HF }", "c414357106cf586a526f022debc4378d": "N_{nl} = \\sqrt{\\frac{2 \\, \\gamma^{l+{3\\over 2}} \\, (\\frac{n-l}{2})! }{\\Gamma(\\frac{n+l}{2}+\\frac{3}{2})}} ", "c4151c349f09cc72d4045ede842f402c": "\\frac{d}{dt}\\boldsymbol{f} =\\left[ \\left(\\frac{d}{dt}\\right)_r + \\boldsymbol{\\Omega \\times} \\right] \\boldsymbol{f} \\ . ", "c416760dbf8f8b485a0bb5b8abeccff9": "\\mathbf{x'}(t) = \\mathbf{Ax}(t)", "c41678466c7878ae504c901e9c6f827a": "R=\\frac{(Z_1 - Z_0)^2}{(Z_1 + Z_0)^2}", "c416ad4b0f129816993434ee54643d6a": "\\Rightarrow \\ln x = W(\\ln z)\\,", "c4170143dc2d59a628b1abaf1248d319": "\\hat{p}_{00}", "c4172ec449e07be827785fc73c820aab": "\\scriptstyle m ", "c41736092fcd33d21d3125d8c9fedb55": "{\\dot{\\gamma}}", "c4173fbc8ec60f210299ab7fd60de754": "\\beta\\frac{NK}{2}", "c417c059d2eea9f47ebfc3efcf5aa5d2": "\n\\begin{align}\nf_Y(y) &= 2\\frac{1}{\\sqrt{2\\pi}}e^{-y/2} \\frac{1}{2\\sqrt{y}} \\\\\n&= \\frac{1}{\\sqrt{2\\pi y}}e^{-y/2}.\n\\end{align}\n", "c417c2b2f7ab197b3fa8c4cfb4e1687e": "\\frac{Gr}{Re^2} \\gg 1 ", "c417dde512b70e3f307dcef5d3c0e5e2": "S(T_f)=\\int_{T=0}^{T_f} \\frac{\\delta Q}{T}\n=\\int_0^{T_f} \\frac{\\delta Q}{dT}\\frac{dT}{T}\n=\\int_0^{T_f} C(T)\\,\\frac{dT}{T}.", "c417f13c55c2edfa2b798815971ddd86": "52\\pi", "c4180aa939961e017818f34db4bbcd9f": "^*", "c4181b39fa51ad9a30657febc31540b2": "\\mathbb{Z}_n", "c41863b89e5428de41d513801d7c51ec": "V_n(R) = \\bigg(\\int_0^R r^{n-1}\\,dr\\bigg)\\bigg(\\int_0^\\pi \\sin^{n-2}(\\phi_1)\\,d\\phi_1\\bigg)\\cdots\\bigg(\\int_0^{2\\pi} d\\phi_{n-1}\\bigg).", "c41886775003d02e199a40817b0dd9fb": "\\gamma = \\alpha +i \\beta \\,", "c418a660b0431b6ad3ce90097e3fc446": " q = 1 \\ kN/m", "c418ed841200dbd2212c96e087bf92a8": "\\omega_{(x,g)} = Ad_{g^{-1}}\\pi_1^*\\omega(\\mathbf e_U)+\\pi_2^*\\omega_{\\mathbf g}.", "c41923be17a55fcbfe9ec7ac1d26ab56": "\\mbox{d} \\epsilon(t) = J(t, t') \\mbox{d} \\sigma(t')", "c419313b8c00048c04781eef7e54bdd6": "\\left(n-1\\right)k > 0 \\,\\!", "c4193ea6efbfd670e32a221632905855": "\\mathbb{Z}/n\\mathbb{Z} \\cong \\mathbb{Z}/{p_1^{k_1}}\\mathbb{Z}\\; \\times \\;\\mathbb{Z}/{p_2^{k_2}}\\mathbb{Z} \\;\\times\\; \\mathbb{Z}/{p_3^{k_3}}\\mathbb{Z}\\dots\\;\\;", "c419b06b4c6579b50ff05adb3b8424f1": "bd", "c419b338901d505e25d7e1e47ab1677b": "\\sum_i w_i \\left( p_i - m \\right) = 0", "c419bffce24553bf7843f46fb01d15c4": "\\frac{a+c}{b+d},", "c419c0d3884a6f77e296f4876089110a": "x^{-1} y x = y^2.\\,", "c419c505aa7b89a2f7a993e7ce05daaa": " \\vec{e}_2 = \\frac{1}{r} \\, \\partial_\\theta ", "c419dbbb90785eb8441766bc0f265232": "E_{m} = \\frac{P_{K^+}} {P_{tot}} E_{K^+} + \\frac{P_{Na^+}} {P_{tot}} E_{Na^+} + \\frac{P_{Cl^-}} {P_{tot}} E_{Cl^-}", "c419ed0715a19e84a4873f9d3d65b3e2": "A_1 \\subset A_2 \\subset \\ldots", "c41a5b8eed32549ec10b5e6ff4462b34": "\\bar F(D) = p(A) + p(B) + p(C) + \\frac 12 p(D) = \\frac 13 + \\frac 14 + \\frac 16 \\frac 12 \\cdot \\frac 14 = 0.875", "c41a8962e4937e98a15b36ec28915e12": " \\vec t_1=v_{11} \\vec r_u + v_{12} \\vec r_v ", "c41b45ce76df3547d9982d81918d2393": "\\begin{matrix}\\frac{7}{10}\\end{matrix}", "c41b6ce8032aeaa8f7c7a7b7b744e711": " \\begin{align}\n& \\frac{d}{dt} \\begin{pmatrix}\nx \\\\\ny \\\\\n\\end{pmatrix} = \\begin{pmatrix}\nA & B \\\\\nC & D \\\\\n\\end{pmatrix}\\begin{pmatrix}\nx \\\\\ny \\\\\n\\end{pmatrix} \\\\\n\n& \\frac{d\\mathbf{x}}{dt} = \\mathbf{A}\\mathbf{x}. \n\n\\end{align}", "c41b81422e6c5c4eb34293d1cbad50d2": "m<(\\ln A)^{c\\ln\\ln\\ln A}", "c41bbc487bb7fd0b5517d92ef88fcc5f": "p_M(\\lambda) = (-1)^{r(M)} T_M(1-\\lambda,0). ", "c41bfb435989a0d68aa782e1bbe4d927": "\\frac{1}{r^2+z^2}\\,", "c41c273fa0f84eb90f88cb2e617bd596": " \\bold x^{(m+1)} = \\bold B^{-1} \\bold C \\bold x^{(m)} + \\bold B^{-1} \\bold k, \\quad m = 0, 1, 2, \\ldots \\quad (5) ", "c41c301ac33cd9017bc79b3c95d8e234": "Dip = (1 - \\frac{SBP_{Sleeping}}{SBP_{Waking}}) *100% ", "c41c510d327f67a2c48be622e7d8508d": "Q \\gets S - {p_1,p_2,...,p_m}", "c41caba61c9e2bbdded56a71763e0688": "\\dot sU \n ", "c41cf9dc4aff1dd3d9e183f2af7c3603": "\\scriptstyle \\ell^2", "c41d1535ce636ea1fa674d678524a100": "f_0(x) = x^2 \\,", "c41d854c78883f1125b29d78dca5388f": "E \\to \\epsilon ~~~~~~~~~~~~\\text{end with a probability of 1}", "c41d9a34ee5cfd51013ec204641d31ed": "\\alpha x \\leq \\alpha y ", "c41df07f27ec66f35bcfcee0a6f203d2": "\n \\left[\\mathbf{e}_1,\\mathbf{e}_2,\\mathbf{e}_3\\right] = 1 \n", "c41e2ba975122fb96db53d73310777b4": "(\\alpha f)(x) = \\alpha f(x) \\,", "c41e31ff3ab585342ee8295ac212ffb5": "\\mu^+(A) = \\liminf_{\\varepsilon \\to 0+} \\frac{\\mu(A_\\varepsilon) - \\mu(A)}{2\\varepsilon},", "c41e5b310837e709df33e1562e80c5ae": "\nX_n\\equiv R_n(\\xi,x)\\qquad \nL_n\\equiv R_n(\\xi,\\xi)\\qquad \nt_n\\equiv \\sqrt{1-1/L_n^2}.", "c41e91c6a2546bf7c42d2f274deb9c98": "{d \\over 1+d}", "c41ef128ef3d168b2af9dff177863478": "r_i=\\Delta y_i- \\sum_{s=1}^{n} J_{is}\\ \\Delta\\beta_s; \\ \\Delta y_i=y_i- f(x_i,\\boldsymbol \\beta^k).", "c41ef805783119307ec663a8c70db7de": "\\exists k_1>0 \\; \\exists k_2>0 \\; \\exists n_0 \\; \\forall n>n_0", "c41f00b30c3c2b6a9ea9ae12f038e300": "\\alpha\\in R, f\\in C(E)", "c41f48d45a01e845a6d8c5128ce19c05": "C_1+C_2/(1+r) = Y_1 + Y_2/(1+r)", "c41f790b379b76efd4fe33aca99d720d": "S_n(s) = s\\sum_{k=0}^{n-1} {n \\over n - k} {2n-1-k \\choose k} (-4s)^{n-1-k}.", "c4203d57f9e2783098ebe62b8d3ab13c": "v = \\frac{d}{t}\\,", "c4205f602a0352e17d0a1c086ee749ae": "n_2(x)", "c42090a4ff0c472ce5a653ea4df31efc": "\\begin{align}\nT_i^{(n)} &= \\lambda n_i \\\\\n\\sigma_{ij}n_j &=\\lambda n_i \\\\\n\\sigma_{ij}n_j -\\lambda n_i &=0 \\\\\n\\left(\\sigma_{ij}- \\lambda\\delta_{ij} \\right)n_j &=0 \\\\\n\\end{align}\\,\\!", "c4210e7b0f573c4dfeb892d7ad3572c5": " \\mathcal{QS} \\vdash A", "c4212969abe7aa3d6499508e5150956b": "\\sin 3 \\phi = 3 \\sin \\phi - 4 \\sin^3 \\phi\\,\\!", "c4215189102f6f3079cbcd4abe82c785": "x \\vee y = x + y - xy ", "c421995ba521fe01c1d06eb86356d33d": "\\left[ n,k\\right] ", "c421a26189002645895015903609cd1b": "y\\in H_w", "c421b6c4e06c3e7c73cef745c7304700": "\\sigma^2_{b1}(t)", "c4221ede9edf87d789993660c746a0f4": " \\sigma_{DC}(p_c) \\propto \\sigma_m \\left(\\frac{\\sigma_d}{\\sigma_m}\\right)^u ", "c422224b9c3f82c2d2eb9a59249baede": "\\rho^{\\prime} = \\rho/{\\rm sdet}(\\partial_{i}x^{\\prime j}) ", "c4224fbf61a13fd3dbdb7e14f128007e": "q=e^{-z}", "c4226b80c5f88625491c6063b37d0b74": "\\operatorname{SU}(2) \\cong \\operatorname{Spin}(3) \\cong \\operatorname{Sp}(1)", "c42279e3c9f99b8bab7818d7b24d7dff": "x=a \\left(-a^2 d_a d_a^{\\,'} + b^2 d_a d_b + c^2 d_a^{\\,'} d_c^{\\,'} \\right)", "c4228e2393314003aea4b49bc08529f7": "Ax = b.\\,", "c422c85cfa4c379855e0a8318f21c60c": " \\geq \\int_{x_0}^{\\max_x \\in X} f_{\\theta_1}(x_0) f_{\\theta_0}(x_1) \\, dx_1", "c422c86f58a7bba22fd259725e71e023": "\\mid g \\rangle", "c422d079ab2e7ac04b145dbc424e7c34": "O(n^2M(n^3)/\\log{n})=O(n^{5+o(1)})", "c423339155151f47343ae06572bdd9db": "\\psi_k(x)=\\sum_{K}\\tilde{u}_k(K)\\,e^{i(k+K)x}", "c4233cfea93c67d719488fdce766cb49": "g(y)=f(-y)/F(0)", "c42368a6e47234c997e580b0a2c40ca3": "\\Omega_c", "c4237c33528c335a1a8a5286f4d67bf5": " C^{1+\\alpha} ", "c42393eb6061ef77fc7f6246bb18c45a": "\\eta\\colon R \\to A", "c423baef7710446f0e6a2fdeddf050a4": "\\frac{ 1 }{ i }[M_{\\mu\\nu}, P_\\rho] = \\eta_{\\mu\\rho} P_\\nu - \\eta_{\\nu\\rho} P_\\mu\\,", "c423ddedee212f910e972fa0bf570727": "E(X)=-\\frac{\\operatorname{Li}_2(1-p)}{\\beta\\ln p},", "c423f1a9696c24283f3c6e0ecf291908": "\n\\ \\ \\ \\ \\ \\ \\ \\ + \\ \\left( \\frac{1}{2\\pi\\epsilon} \\right) \\sum_{k=1}^{\\infty}\nS_{1k} \\int d\\theta \\int d\\rho \n\\left[ \\frac{\\sin k\\theta}{\\rho^{k-1}} \\right]\n\\lambda(\\rho, \\theta)\n", "c4242a566134017f8a05550bb9cffba1": "\\frac{\\varphi}{\\sqrt{5}} = \\frac{1}{\\Phi \\cdot \\sqrt{5}} = \\frac{5 + \\sqrt{5}}{10} = 0.72360679774997896964\\dots = [0; 1, 2, 1, 1, 1, 1, 1, 1, \\dots].", "c42432756204e040a1cd28884517ccb0": " p'(t_0+h) = f(t_0+h, p(t_0+h)). \\, ", "c4243d92aa2b2ede88793ea4a9b96a82": "f = 0\\,", "c4244813f6d7c0af80a259a0ef11f961": " {\\beta} ", "c4247113a61f39d3298e57975a38d4a5": "[F]", "c424ffa5bd4ce61d36125896f37ed3f0": "S_{\\alpha \\beta }", "c4252cb882c2b575741518701fd22457": "\\frac{a_{n+1}}{\\prod_{k=0}^n f_k} - \\frac{a_n}{\\prod_{k=0}^{n-1} f_k} = \\frac{g_n}{\\prod_{k=0}^n f_k}", "c425433d06b9c48f88c347ab2bb55304": "{\\scriptstyle \\Omega}", "c425ec9f15d659529722bb1e56310886": "x, y\\in R", "c42676f48fdf67b6de5069702ae53578": "[2^{k-1},k,2^{k-2}]_2", "c426cc002a476d1f83d8a86813e40320": "\n p_{n,\\theta}(x_1,\\ldots,x_n;\\,\\theta) = \\prod_{i=1}^n f(x_i,\\theta).\n ", "c426d8081e3ac165c3e2ba051ccc6878": "|\\nabla u|", "c427335dcc66c22222b9799d413e25cc": "SSE_p = \\sum_{i=1}^N(Y_i-Y_{pi})^2", "c427ae474ef7531b647c6e0c3e6d1a0e": "\\pi_*(K)=\\mathbf{Z}[t,t^{-1}]", "c42824f6783458dea6129dfa01cefc91": " E{\\lVert x_{k}-x \\rVert^2} \\leq (1-\\kappa(A)^{-2})^{k} \\cdot {\\lVert x_{0}-x \\rVert^2}. ", "c4283183bd64d73c6ef0ad36e12e8e1b": " w(L)=w_{min}\\,\\!", "c428b1db0fcd4c44b725f7d86e11380c": "\\{d_\\text{f},\\, \\nu\\}\\,\\!", "c428b2a6466e27501bd1cb0cba118af9": "\\{\\varnothing\\}", "c428ba3cdddc479ed4fc1e9e8c0414b0": "\\, x \\, dx + y \\, dy + z \\, dz = 0.", "c428dca392a485f2fdbcd1d4a6859e6a": " \\hat{S}^2 = S^2 + 2 \\sum_{j=1}^{q} \\left( 1 - \\frac{j}{q + 1} \\right) C(j), ", "c4290ce3e0772ece8cc2f46e44668969": "\n\\sum^n_{i=1}F^P_id^Q_i = \\int_\\Omega \\sigma^P_{ij}\\epsilon^Q_{ij}\\,d\\Omega\n", "c4292c02b9c99efd9bf663a4fe350905": "\\alpha^*_k", "c4293d6cbc328bc958a8b977577e458b": "\\le \\left \\vert \\frac{3k \\sqrt{N}}{Nd} \\right \\vert = \\frac {3k \\sqrt{N}}{\\sqrt{N} \\sqrt{N}d} = \\frac {3k}{d \\sqrt{N}} ", "c42952440ffcf8129a1bedeac4773415": "R_i", "c4297b035893078d2659deb70c662594": "t_a = \\frac{\\lambda_a}{2}\\,,", "c4299a518d01d5e21d8962c3e593d5b3": "y(t) = \\frac{Y(t)}{A(t)L(t)} = \\frac{K(t)^{\\alpha}(A(t)L(t))^{1-\\alpha}}{A(t)L(t)} = \\frac{K(t)^{\\alpha}}{A(t)L(t)} = k(t)^{\\alpha}", "c429d7b173a4a6d6d4a5b96d44011721": "\n [\\mathbf{e}_i'] = [L][\\mathbf{e}_i]\n ", "c429fb6108cb7ce568787c7398475923": "S_\\theta = \\{\\alpha \\in At^L | \\alpha \\models^{SC} \\theta \\}", "c42a2622d98df26bae2eca7bfc4f1d0d": "\\Pr(X = k) = (1-p)^{k-1}\\,p\\,", "c42a27a244320b1eaffdf13c5743a8df": "V_{GNM} = \\frac{\\gamma}{2} [\\Delta X^T\\Gamma \\Delta X + \\Delta Y^T\\Gamma \\Delta Y + \\Delta Z^T\\Gamma \\Delta Z]", "c42a280003a64774acf91ad359d199bc": "\\displaystyle t>0", "c42a4b7d30b08116c233962d6d3300ca": "R=0.5", "c42ab05c9ee9d45fe7266ff4e5b2d007": "j^{i-N}", "c42ab8ad6474a230622088ad5fe7a3fe": "G_i = A_i \\cdot B_i", "c42ad687988b9bb444498a2bd3c75cdf": "\\infty mm\\, ", "c42b0730dcd63a29aa3c3bd7c77897a2": " p_j = \\text{constant}\\,.", "c42b25013bd1da551a454d4554f67802": "{\\log\\left(\\frac{p_{11}p_{00}}{p_{01}p_{10}}\\right) = \\log(p_{11}) + \\log(p_{00}\\big) - \\log(p_{10}) - \\log(p_{01})}.\\,", "c42b3a61cfd370a0f822c35cfdb5a1a8": "\\lim_{x\\to+\\infty} \\left(1+\\frac{k}{x}\\right)^x=e^k", "c42b45f6267aa8b4027f277772976616": "\\mathbf{B} = \\mu_0 \\mathbf{H} \\,\\!", "c42b5cb9ec297f9bae17ca253550e157": "GL_n(R)", "c42b844b53cb01fdad41e47ca9db7123": "u^2=u_0^2-v^2", "c42b97c48b94940d3784a62fd0fd1cc0": "g(z, u, v) =\n\\exp\\left( -z + vz + \\sum_{k\\ge 1} u^{k-1} \\frac{z^k}{k} \\right)", "c42ba1304ef9ad1cf54948137bf7798f": "\\partial Q = \\{ q \\in D \\setminus Q: \\exists p \\in Q : qAp \\}", "c42bd495a56f4828af3b5b2dfcbbffb8": "\\chi ( \\mathcal{F})= \\Sigma (-1)^i h^i(X,\\mathcal{F}),", "c42c0e9d2a60bb3e9441353aa7913eaf": "|n_\\mathbf{p}\\rangle", "c42c320df182736fe6b241c0f7ced093": "\\Lambda(A_1:A_2|B) = \\frac{P(B|A_1)}{P(B|A_2)}", "c42c3bb68ed6985acef2e5af05cba3ea": "E = h\\nu = \\hbar \\omega \\,\\!", "c42c5468b4113e46112df45142506c06": "\\rho \\frac{D \\mathbf{U}}{D t} = -\\nabla p + \\mu \\nabla^2 \\mathbf{U} + \\rho_e \\nabla \\left( \\psi + \\phi \\right)", "c42c951a5cdafa7d77951697a4ce49f0": "\\mathbf{T_{max}}", "c42cc436ddd26c78a422e0f5cb0efb8e": "U(g(x)')| \\psi\\rangle", "c42dbb13a27706f673b95826f54258f5": " t_{1/2} = \\ln (2) \\cdot \\tau ", "c42dc805e0df3f5796a1145f2feae5b8": " \\sum_{m\\ge 0} (-1)^m g_m(z) w^m =\n\\sum_{m\\ge 0} \\sum_{n\\ge m} \\frac{(-1)^n}{n!} \ns(n,m) w^m z^n =\n\\sum_{n\\ge 0}\\frac{(-1)^n}{n!} z^n \n\\sum_{m=0}^n s(n,m) w^m.\n", "c42e469c3ac047820d2a4763d30c456a": "b = \\log(c)", "c42e4cca48611a9c39a1b65a26492387": "x^{(q^m-1)/(q-1)} - 1", "c42e5f7cb85e65edeb7a565b5c8206d9": "J^2/\\rho", "c42f5f07acd4fae2d01a774b49378aae": "\\cosh x = \\frac {e^x + e^{-x}} {2} = \\frac {e^{2x} + 1} {2e^x} = \\frac {1 + e^{-2x}} {2e^{-x}}", "c42f69ef2fa10e0a466ad2d5e0b69b2c": " \\langle \\phi(k_1) \\phi(k_2) \\phi(k_3) \\phi(k_4) \\rangle = {\\delta(k_1 -k_2) \\over k_1^2}{\\delta(k_3-k_4)\\over k_3^2} + {\\delta(k_1-k_3) \\over k_3^2}{\\delta(k_2-k_4)\\over k_2^2} + {\\delta(k_1-k_4)\\over k_1^2}{\\delta(k_2 -k_3)\\over k_2^2}", "c42f8363a9c497bb4e262f875746e422": " \\int_U \\left[ \\psi \\nabla \\cdot \\left( \\epsilon \\nabla \\varphi \\right) - \\varphi \\nabla \\cdot \\left( \\epsilon \\nabla \\psi \\right) \\right]\\, dV = \\oint_{\\partial U} \\epsilon \\left( \\psi {\\partial \\varphi \\over \\partial n} - \\varphi {\\partial \\psi \\over \\partial n}\\right)\\, dS. ", "c42fb9abb18f99db5645e70caa6ca086": " Q^{c} = M^{1/2}\\left(AM^{-1/2}\\right)^+(b-AM^{-1}Q), ", "c43035d00bb6be8aa44c9026369b27a2": "\n \\hat\\beta = \\big(X'Z(Z'Z)^{-1}Z'X\\big)^{-1}X'Z(Z'Z)^{-1}Z'y.\n ", "c43049071a88dcdfd8424bf5718176c7": "\\begin{bmatrix}\n 0.5 & 0 \\\\\n 0 & 0.5 \\\\\n\\end{bmatrix}\n", "c43093ef9ad6adb41ed750ba6d67c32b": "\n\\Delta \\hat g\\,\\,\\, = \\,\\,\\,\\,\\hat g\\left( {L + \\Delta L,\\,\\,\\,T + \\Delta T,\\,\\,\\,\\theta + \\Delta \\theta } \\right)\\,\\,\\, - \\,\\,\\,\\hat g\\left( {L,\\,\\,T,\\,\\,\\theta } \\right){\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(3)}}", "c430c5dbbbb8bb4efc51ba9113614954": "\n\\begin{align}\n\\int_{LB}^{UB} G(p) \\, dp & = \\int_{LB}^{UB} F'(p) \\, dp=F(UB)-F(LB), \\\\[10pt]\n& =F[LB,UB]\\Delta B, \\\\[10pt]\n& =F'(LB < P < UB)\\Delta B, \\\\[10pt]\n& =\\ G(LB < P < UB)\\Delta B.\n\\end{align}\n", "c430fa66956b12c6527def2b600c05c0": "\\operatorname{dim} X = \\operatorname{Ind} X = \\operatorname{ind} X", "c431622c8a6f64742c793e96a9c5c4b2": "\\begin{align}\n& K(\\mathbf{Q},\\mathbf{P},t) = H(\\mathbf{q},\\mathbf{p},t) + \\frac{\\partial }{\\partial t}G_1 (\\mathbf{q},\\mathbf{Q},t)\\\\\n& K(\\mathbf{Q},\\mathbf{P},t) = H(\\mathbf{q},\\mathbf{p},t) + \\frac{\\partial }{\\partial t}G_2 (\\mathbf{q},\\mathbf{P},t)\\\\\n& K(\\mathbf{Q},\\mathbf{P},t) = H(\\mathbf{q},\\mathbf{p},t) + \\frac{\\partial }{\\partial t}G_3 (\\mathbf{p},\\mathbf{Q},t)\\\\\n& K(\\mathbf{Q},\\mathbf{P},t) = H(\\mathbf{q},\\mathbf{p},t) + \\frac{\\partial }{\\partial t}G_4 (\\mathbf{p},\\mathbf{P},t)\\\\\n\\end{align}", "c4317e7a89e36bc1b16822e61435ddfc": "S^*", "c43189339061d0804b0aa43933b198b1": "d(x,y) := \\begin{cases} 0 &\\text{if } x = y,\\\\ 1 &\\text{if } x \\neq y. \\end{cases}", "c431ad303d976c0e0500ced6ae0ac8d7": "x=a^{1/10}.", "c431b6cefab01ce8d920525f2aaebd0f": "RPQ = \\tfrac{\\pi}{2} - RQP = \\tfrac{\\pi}{2} - (\\tfrac{\\pi}{2} - RQO) = RQO = \\alpha", "c431bb6c6f86de7d0be65e363099e310": "\\widehat\\delta ( q, wa ) = \\delta_a(\\widehat\\delta ( q, w )).", "c4324ca82ddf37edc863397ed03ba46c": "\\eta=1/\\sigma\\mu", "c432799e12712d821d3d8719da167103": "\\sigma_z^2", "c4329dbcc342749589bcaaecd6620cbc": "p_x(a) = \\sum_{i=1}^k x_i a^{i-1} \\,.", "c432df1527872f915563718a6a63d3f7": "O\\left(N^{3\\over 2}\\log^2N\\right)", "c432ea8cd03b3685df2163536e99dc6d": "R^{2} = {1-{\\textit{VAR}_\\text{err} \\over \\textit{VAR}_\\text{tot}}}", "c43307ad5ac32c00f39b3ca3d642f945": " T_{\\delta\\sigma}=(T_\\delta)_\\sigma.\\, ", "c433c1e6764c6dcb3c1ac1922cfce382": "\\Gamma^{(\\lambda)} (R)_{nm}^*", "c433cbc514df1397f39af9ca162703d2": "P_{\\text{Electric dipole}}=\\frac{c^2 Z_0}{12 \\pi}k^4\\|\\mathbf{p}\\|_2^2", "c43403a72cac252deca347b8a80593e2": "\\Sigma^*", "c4341f178417d4c49e3686e9012297e5": "\\psi(\\beta)", "c434eb5acadc312936b607b3ae588468": "(G\\times GL(n),V\\otimes \\mathbb F^n)", "c435175acf141bb01341b6398f575660": " h_{a,b}(x) = ((ax + b)~\\bmod ~ p)~\\bmod ~ m", "c43532e23f9f96cc677972ab2c492f14": "\\hat t_i", "c4358049354db36473becdbd0dcdb3ac": "dG_p = - SdT = \\left(\\frac{\\partial G}{\\partial T}\\right)_p dT \\quad \\rightarrow \\quad -S = \\left(\\frac{\\partial G}{\\partial T}\\right)_p. \\,\\!", "c435f4f24e5f2224a6c17edc9249c298": "\\displaystyle (i\\omega)^n \\hat{f}(\\omega)\\,", "c436389e80bc4ab5c8513c9e736e86f9": "\\beta(\\alpha_s)=-\\frac{\\alpha_s^2}{4\\pi}\\frac{3N}{1-\\frac{N\\alpha_s}{2\\pi}}.", "c43650bcbab6a953609fa3fb99be57d1": "{\\mathbf\\mu}_2", "c43680964868e9c33039abce2f99749b": "\\hat{T}_{1} = Q_{1} D_{1} Q_{1}^{T}", "c436a6e38f57093b7b8c5450a560bbcf": "SRM=12.7\\times D \\times A_{430}", "c436acc11c60cd2aa785e7c2d23a2cd3": "\n\\begin{align}\n\\operatorname{E}\\left(\\sum_{i=1}^n \\left[x_i - \\mu - \\left(\\overline{x} - \\mu\\right)\\right]^2 \\right)\n&= \\operatorname{E}\\left(\\sum_{i=1}^n (x_i-\\mu)^2 - n (\\overline{x}-\\mu)^2 \\right) \\\\\n&= \\sum_{i=1}^n \\operatorname{E}\\left((x_i-\\mu)^2 \\right) - n \\operatorname{E}\\left((\\overline{x}-\\mu)^2\\right) \\\\\n&= \\sum_{i=1}^n \\operatorname{Var}\\left(x_i \\right) - n \\operatorname{Var}\\left(\\overline{x} \\right)\n\\end{align}\n", "c436c789f191a52504d80c69f4185095": "440 \\rm{ Hz}\\cdot (\\sqrt[12]{2})^{-14} \\approx ", "c436eb7ec8d6f38015d094e6744e83d1": "\nf_{x,f} = S_f g\n", "c436fe6e409f72f20e7600ada28a63e1": "(x_\\mathrm{c},\\ y_\\mathrm{c})", "c43733be76c0dd0693dcc73a33d6f0aa": "C=\\sum_i \\sum_{j_H}{_Z} ", "c43e9c2b77813d968b0d2048c5bb4ddc": "\n\\begin{align}\n\\Psi_0(\\textbf{r}) &=& \\psi_0(z)u_{\\textbf{k}_{||}}(\\textbf{r}_{||})e^{-i\\textbf{r}_{||}\\cdot\\textbf{k}_{||}}\n\\end{align}\n", "c43eafba351b4ac3881e973f7dfc9146": "\\Lambda\\neq\\{i\\}", "c43edaae1b76bffbe0c3270fbe2cc51d": " c_2 \\frac{dV_2}{dt} + \\frac{V_2}{r_{M2}} = g_{2,1}(V_1 - V_2) + \\frac {I_{electrode}^2}{A_2}", "c43edba39664b814266fce2682dc6d11": "1/p_L(ik)", "c43f098bb14ee9741d9369de7dc77665": "b = \\tan^{-1}\\left(\\frac{v}{u}\\right) - \\frac{1}{2} \\left(\\frac{v}{u+1}\\right).", "c43f31ab85a2e68bc064e9c9cadeac31": "\\sigma^2\\,", "c43f886401c701190f79f3f6ccec7335": "a_i=k_{i-1}(x_i-x_{i-1})-(y_i - y_{i-1})", "c43f9f542e3aa5d74073a3af77db2d69": "OBP = \\frac{H+BB+HBP}{AB+BB+HBP+SF}", "c43fb44c7856b9906b9a3ef9e113b7b4": "\\textstyle D\\left(z_k+w_k,\\frac14 |w_k|\\right)\\qquad k = 1,\\dots, n,", "c43fb5947a41919472e14a6de9cb79d5": "f_3(x) = 0", "c43fb822456158f71d63c1fbd4d633d0": "\\mathrm{id}_X = \\eta_X", "c43ffaf12402c8e86667b6d6081521c7": " -\\frac{dx}{dt} = {k_f x} - {k_b ([A]_0 - x)}\\,", "c440025a2f67b0576bbd150d9727ad9d": "\\exp\\left(i\\frac{px_o}{\\hbar}\\right)x\\exp\\left(-i\\frac{px_o}{\\hbar}\\right)=x+x_0", "c440337ff99dc89442ff929c58ea7765": "\\alpha A + \\beta B \\rightleftharpoons \\sigma S + \\tau T\\,", "c4409667e3a606651e474a5d681d9b5d": "F = \\int \\mathbf{H}\\cdot d\\boldsymbol{\\ell}", "c440cdbd23afca00859e78f5a85a3052": "\n\\frac{y}{c} = \\frac{k_1}{6} \\left[ \\left(\\frac{x}{c}-r \\right)^3 - \\frac{k_2}{k_1} (1-r)^3 \\frac{x}{c} - r^3 \\frac{x}{c} + r^3 \\right]\n", "c44111252a90894c07a32cc37e624467": " \\cos\\left(\\sum_{i=1}^\\infty \\theta_i\\right)\n=\\sum_{\\text{even}\\ k \\ge 0} ~ (-1)^{k/2} ~~\n\\sum_{\\begin{smallmatrix} A \\subseteq \\{\\,1,2,3,\\dots\\,\\} \\\\ \\left|A\\right| = k\\end{smallmatrix}}\n\\left(\\prod_{i \\in A} \\sin\\theta_i \\prod_{i \\not \\in A} \\cos\\theta_i\\right) ", "c4415ab23bf103e5febfe3cd9f1cc4db": "R_{ab} = R_{acb}^{\\ \\ \\ c}", "c4417b0381573e2bc1aac96de62662cf": "3\\times 1 + 2\\times 2 + 1\\times 3 = 10.", "c441971f80d326e755e53be4ec1d0d38": "= C_\\text{ss} \\cdot CL", "c441d8e7b1877654434806b1d370b987": "\\Delta I_{\\text{L}_{\\text{Off}}}=\\int_0^{\\left(1-D\\right) T}\\operatorname{d}I_{\\text{L}}=\\int_0^{\\left(1-D\\right) T}\\frac{V_o\\, \\operatorname{d}t}{L}=\\frac{V_o \\left(1-D\\right) T}{L}", "c4422556350942eaba98fb4b92dce833": "\\overline{\\delta}(-1,i) = 0\\,\\!", "c4425a7825cc3289efcd94fd7b61f6b2": "\\int_\\mathbb{R}(1\\wedge|x^2|) \\, \\nu(dx) < \\infty . ", "c44298da5c17bd6a73c51179b4cc28e3": "\\mathbf{f} (\\mathbf{r})=\\rho (\\mathbf{r})\\mathbf{a} (\\mathbf{r})", "c4429aed79cb51ea8de313f8ba3e34f5": "\\frac{d}{dt}\\int_{S_t}d\\boldsymbol{r} = \n\\int_{S_t}\\frac{d\\boldsymbol{r}}{dt}\\cdot d\\boldsymbol{S} = \n\\int_{S_t}\\boldsymbol{F}\\cdot d\\boldsymbol{S} = \n\\int_{S_t}\\nabla\\cdot\\boldsymbol{F} d\\boldsymbol{r} = 0 ", "c442a6aeb2b28b8419bfc426bae7244f": "\\mbox{Dividend payout ratio}=\\frac{\\mbox{Dividends}}{\\mbox{Net Income for the same period}}", "c442b129e5c9f276b33471d862a74857": " \\left(\\eta_2 + \\frac{p + 1}{2}\\right)\\ln|-\\boldsymbol\\eta_1|", "c442ba892b7770e68de2c5a45862e5fa": "\\vec{v}_\\mathrm{B}", "c442ca4c8dc8b749ede526c6712f9c5f": "\\phi(\\mathbf{x};\\mathbf{c}) = e^{-b||\\mathbf{x} - \\mathbf{c}||^2}", "c442eb852807b19dc243e13b0172dd17": "\\|\\mathbf{x}\\|_\\infty := \\max \\left(|x_1|, \\ldots ,|x_n| \\right).", "c442fbed56911dab891ec5034ae267dc": "\n\\begin{align}\nf_0 &= C_{00 \\dots 0} \\\\\nf_j &= \\sum_{m_j \\neq 0} C_{0 \\dots m_j \\dots 0} \\exp\\bigl[2\\pi i m_jX_j \\bigr] \\\\\nf_{jk} &= \\sum_{m_j \\neq 0} \\sum_{m_k \\neq 0} C_{0 \\dots m_j \\dots m_k \\dots 0} \\exp\\bigl[2\\pi i \\left( m_jX_j + m_kX_k \\right) \\bigr] \\\\\nf_{12 \\dots n} &= \\sum_{m_1 \\neq 0} \\sum_{m_2 \\neq 0} \\cdots \\sum_{m_n \\neq 0} C_{m_1 m_2 \\dots m_n} \\exp\\bigl[ 2\\pi i \\left( m_1X_1+m_2X_2+\\cdots+m_nX_n \\right) \\bigr].\n\\end{align}\n", "c442fef435d8d5addd27b724ee7be55f": "\nf_{WE}(\\theta;\\lambda)=\\sum_{k=0}^\\infty \\lambda e^{-\\lambda (\\theta+2 \\pi k})=\\frac{\\lambda e^{-\\lambda \\theta}}{1-e^{-2\\pi \\lambda}} ,\n", "c4431fb65976222b1f03eb8a2e0d0d34": "r = \\frac{1}{u} = \\frac{ h^2 / GM }{1 + e \\cos (\\theta - \\theta_0)}, ", "c443395dbebb1fde0b4588a3a4ce02e9": "\\scriptstyle\\{ D_n \\}_{n \\in \\mathbb{N}}", "c44352c8ec96d60c63ecfe7fd29e4f8e": "U^*U=UU^*=I \\!", "c443edb648b7c3d9318122b8ceaac339": "\\mathrm{^{241}_{\\ 95}Am\\ +\\ ^{4}_{2}He\\ \\longrightarrow \\ ^{243}_{\\ 97}Bk\\ +\\ 2\\ ^{1}_{0}n}", "c4442d4c874710b0dd190955816426b7": "f(z,\\tau) = q^{1/12}(p^{1/2}-p^{-1/2})\\prod_{n\\ge1}(1-q^np)(1-q^n/p).", "c44457a0b77500218b996787c76c40db": "\\left({3 \\over 25}\\right)", "c444712e21f9d9848b40772ae57f5841": "\\sigma_\\text{mean} = \\frac{\\sigma}{\\sqrt{N}}.", "c4447131cb2eed084aaaf78ad489a994": "V_\\mathrm r = V_\\mathrm o - V_\\mathrm i = 2V_\\mathrm i \\frac {Z_\\mathrm L}{Z_\\mathrm 0+Z_\\mathrm L} - V_\\mathrm i = V_\\mathrm i \\frac {Z_\\mathrm L - Z_\\mathrm 0}{Z_\\mathrm L+Z_\\mathrm 0}", "c444720daff23c14e36c1a8e526f819b": "\\eta = \\big( 1- \\mathit{R}_{min} \\big) \\times 1 \\big(\\frac{- 4 \\pi d}{\\lambda} \\big)", "c44493cb128ef84627f79814c61adc4e": "\nz\\,\\, = \\,\\,a\\,e^{b\\,x} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\, \\sim \\,\\,N\\left( {\\mu ,\\,\\,\\sigma ^2 } \\right)\\,\\,\\,\\,\\,\\,\\,a,b\\,\\,{\\rm constants}", "c4449ad4d11ab49695cb8daa41bd57bd": "\\pi(x;q,l)", "c444b0ff321dd5dbfd8cac62182d8ed5": "\\,\\frac{d w(t)}{d t} ~ = ~ w(t) Q - \\mathrm{diag} [w(t) Q w(t)^{\\mathrm{T}}] w(t)", "c444c5f7d69c81bf69a2ab8a9968ac2e": "I_S[f] = \\frac{1}{n}\\sum V(f,z_i)", "c444cc613b7da473503728255c17a233": "Dg(x)=\\lambda g(x) ", "c444f796839d9e5b486e054a0093b738": "\\sigma_2=\\sigma_3=0", "c445593638d71474d27216e2bcf971cf": "\nf(p_1,\\ldots,p_m \\mid n_1,\\ldots,n_m,I) = \n \\begin{cases} { \\displaystyle \n \\frac{\\Gamma\\left( \\sum_{i=1}^m (n_i+1) \\right)}{\\prod_{i=1}^m \\Gamma(n_i+1)}\np_1^{n_1}\\cdots p_m^{n_m}\n}, \\quad &\n \\sum_{i=1}^m p_i=1 \\\\ \\\\\n0 & \\text{otherwise.} \\end{cases}\n", "c4457cf59f3b5e875d9bca292a5852e7": "\\omega = \\begin{cases} 1 - i\\tan\\tfrac{\\pi\\alpha}{2} \\beta\\, \\operatorname{sign}(t) & \\text{if }\\alpha \\ne 1 \\\\\n 1 + i\\tfrac{2}{\\pi}\\beta\\log|t| \\, \\operatorname{sign}(t) & \\text{if }\\alpha = 1 \\end{cases}", "c445ac11b203910f85460b9215fd278d": "q_y = s_y", "c445c069cb6eca04d219ef49980d0118": "A_\\text{pfd} = ", "c446062c28c745a952892036f19dc812": "S\\in\\tbinom{[n]}m", "c4464f2ab6ca7d7d3a5c9978cf54f3fd": "C_1 = C_2", "c446d6264d5d83c8dbaf56fa95c71f95": "M\\ {=_L}\\ M^\\prime", "c447146ce39eccd4242a7a2fa4aad53a": "\\varphi:A\\rarr\\mathcal{P}(B)", "c447a4fc9d1d02524e002d2cc6cf6b9b": "\\Pi_{q,q'}(\\partial_j)=(\\partial_j)_{q'}-d\\xi^i\\Gamma_{ij}^k(\\partial_k)_{q'}", "c447e660094bd873819f6fbc525a4477": " E\\!\\left[X^2\\right] = \\int_{-\\infty}^\\infty x^2 f(x) \\, dx = \\sum_{i=1}^n p_i\\int_0^\\infty x^2\\lambda_i e^{-\\lambda_ix} \\, dx = \\sum_{i=1}^n \\frac{2}{\\lambda_i^2}p_i,", "c447ebcc634584ad4d5c2fe0dc938a18": "H_{m_0} = 4 \\sqrt{m_0} = 4 \\sigma_\\eta, \\, ", "c4484970e76862c6bd8de8ee724fc028": "\\delta[n-M]", "c448ff503edb181f556c5e34072ef900": "(a \\cos \\theta)^{\\tfrac{n}{n-1}}+(b \\sin \\theta)^{\\tfrac{n}{n-1}}=r^{\\tfrac{n}{n-1}}.", "c4496763bd9cd97bcebf7709e7de77e5": "Z(j\\omega)=R\\,\\!", "c4497d01f98e403473c4666ecc7d80f3": "\n2\\pi \\hbar (n+\\frac 12)=\\frac 4{D_\\alpha ^{1/\\alpha }q^{2/\\beta }}E^{\\frac\n1\\alpha +\\frac 1\\beta }\\frac 1\\beta \\Beta\\left(\\frac 1\\beta ,\\frac 1\\alpha +1\\right).\n", "c44980506a7c2b4cc7f002ff4ddae48e": " y_c(x) = (x-c)^2 . \\,\\!", "c449c40ecefb5bfd63b71ed65cb3816c": "\\mathbf{NP} \\neq \\mathbf{PSPACE}", "c449e4640397e6b31d2a4d85dc4ab6d7": "\\int \\frac{\\mathrm{d}u}{a^2+u^2} =\\frac{1}{a}\\tan ^{-1}\\left( \\frac{u}{a} \\right)+C", "c44a2ef14122acbee5ba0a42c7ad691a": "\\nabla\\left(\\mathbf{P}\\cdot\\mathbf{Q}\\right)=\\nabla\\sum\\limits _{m}p_{m}q_{m}=\\sum\\limits _{m}p_{m}\\nabla q_{m}+\\sum\\limits _{m}q_{m}\\nabla p_{m}.", "c44a369ee99dea182f1ce870c555d93d": "\\mathcal{G'}", "c44aba547d1f3a573be92c876eb0c354": "\\int_E \\left|s_N(x)-f(x)\\right|^2\\,dx \\to 0", "c44aef3138bc5f43963af0161d858d81": "\\operatorname{Var}(X)=G''(1^-) + G'(1^-) - \\left [G'(1^-)\\right ]^2.", "c44b1bfc1758cdbb3367a2191cd5f9e6": "U_0", "c44c20bf8c0c42f31bfac5b7cf17be34": "O(f)", "c44c210f51120e9b063496135e2face7": "\\scriptstyle f_\\mathrm{image}(1)", "c44c3b96a0b51c2af58d40a9a675c3e4": " \\operatorname{build-param-lists}[g\\ q, D, V, T_7] \\and \\operatorname{build-param-lists}[p, D, V, K_7] ", "c44cdf91fb52daf460a0f30e61308613": "\nL\\frac{dI}{dt} = E - V - RI\n", "c44cf7d7077b9507c0bb321977593370": "\\beta \\colon d_{2} \\to d_{1}", "c44d5a3eaf7e25d869cc54b40283f919": "\\ ln(1-(1-\\beta)F_t)< -(1-\\beta)F_t", "c44d9e625af1c501c18b840a2c10a2d4": "\\overline\\psi \\to \\overline{\\psi}^\\prime = \\overline{\\psi}(1+i\\alpha\\gamma_{d+1}) = \\sum\\limits_i \\psi_ib^{\\prime i}.", "c44e1b2a6b2b60ceededf556e90b9e4f": " 2\\pi \\approx 6.28318530717958648, \\, ", "c44e6c40278076bd5028035482a0319e": "f_\\varnothing", "c44e975bc14d1eaddbccc3b15471ff3e": "\\hat {\\boldsymbol{ u}}", "c44ee25d3bca08e7e15303698ed99ea9": "\\|L_x\\|_{(\\ell^p)^*} = \\|x\\|_q.", "c44f3e632dc455e3581c68ba11dd9ff1": "\n \\frac{\\partial^{2} u}{\\partial x^{2}} + \\frac{\\partial^{2} u}{\\partial y^{2}} - b \\frac{\\partial u^2}{\\partial x} = \\rho(x, y) \\qquad (1)\n", "c44f72eaf10e48159895d6c0b2dd0af4": "\\mu_{MMM}", "c44f98318a33090d91784716774076ed": "\\displaystyle (i-1)", "c44fa42a3f6e5443f4eb639bac1482da": "\\sum_{n=1}^\\infty {a_n \\over n^s}.", "c44fb1ff60482360023e74485295d37f": "\\displaystyle t", "c4503ad1d27e32f2bcd9bd287a2e9e5b": "\\mathbf{s1, s2}", "c450a4a01aef52013cb951fd7f9e119e": "\\scriptstyle I \\,\\cap\\, J \\;\\neq\\; IJ", "c450ec679bdfe6fb48d0f7ca345ed73d": "\\mathbf{v}_2", "c451a9abfea6fc46daba09363e7504e3": " Q^\\mu = P^\\mu + q A^\\mu. \\!", "c451b65f061805a69d3023572b179756": "\\frac{1}{2} \\,+\\, \\frac{1}{4} \\,+\\, \\frac{1}{8} \\,+\\, \\frac{1}{16} \\,+\\, \\cdots", "c45222d3694fc0bd2cb31f97c731c82f": "P \\equiv Q", "c4526e1e4e5554c88f030b78ea901d37": "L U_3", "c452b67c0fbef57c15eac6272b1a343c": "\na_n = \\frac {f^{\\left( n \\right)}\\left( c \\right)} {n!}\n", "c452e74e955b2f112ab551ad4c2764f9": "|n^{(0)}\\rang", "c4530e8670797d0cba2097f5bff70be8": "\\epsilon_{IJKL}", "c4533a5f878d2fe1864fb9d4d5046e05": "y=2x", "c45370031d393b6137778a404d089e7d": "\\sqrt{2}^ {\\sqrt{2}^ {\\sqrt{2}^ {\\ \\cdot^ {\\cdot^ \\cdot}}}} = 2", "c453d0e9e4b8d52931049e039dc3d54d": "\\operatorname{Li}_s(z) = z \\,\\Phi(z,s,1) \\,.", "c453e5b5c085fa009dd8b65c37eecb3d": "b^{(k)}{x_k}^{(k-1)}x_{k-1} \\le n", "c453e9a99a3e0955e44305abb793d814": "\\bar{\\sigma}", "c453fe2b8cd04f56993e8980ed8a52fd": "\nT_T=T_{\\alpha=1}=\\frac{1}{N}\\sum_{i=1}^N \\left( \\frac{x_i}{\\overline{x}} \\cdot \\ln{\\frac{x_i}{\\overline{x}}} \\right) \n", "c4548013a212bcf4fcfeb83dd4d5ec15": "\n \\begin{align} \n \\frac{\\partial \\zeta}{\\partial t}\n &+ \\frac{1}{a \\cos( \\varphi )} \\left[\n \\frac{\\partial}{\\partial \\lambda} (uD)\n + \\frac{\\partial}{\\partial \\varphi} \\left(vD \\cos( \\varphi )\\right) \n \\right]\n = 0,\n \\\\[2ex]\n \\frac{\\partial u}{\\partial t}\n &- v \\left( 2 \\Omega \\sin( \\varphi ) \\right)\n + \\frac{1}{a \\cos( \\varphi )} \\frac{\\partial}{\\partial \\lambda} \\left( g \\zeta + U \\right)\n =0 \n \\qquad \\text{and} \\\\[2ex]\n \\frac{\\partial v}{\\partial t}\n &+ u \\left( 2 \\Omega \\sin( \\varphi ) \\right)\n + \\frac{1}{a} \\frac{\\partial}{\\partial \\varphi} \\left( g \\zeta + U \\right)\n =0,\n \\end{align}\n", "c454a0d71757ca6db8eb9b7a1179f294": "P_0 = ( x_0, y_0 )", "c454da6d4a4f3a3148bbc465c7b6c318": " \n\\max_{ \\vec v}\\ v_b \\qquad \\textrm{s. t.} \\qquad \\bold{S} \\, \\vec v=0\n", "c454ec158107a1079afb908e75a21dc6": "\\therefore d = a + c. ", "c454fcbc3b0806d168306660c1ada08e": " d\\ln(r) = [\\theta_t-\\phi_t \\ln(r)] \\, dt + \\sigma_t\\, dW_t ", "c4552788403be840d713776d33193d0d": "\\mathbf{m_{ab}} = \\begin{bmatrix} a(1) \\\\a(2) \\\\ \\vdots \\\\a(k) \\\\b(1) \\\\b(2) \\\\ \\vdots \\\\b(k) \\end{bmatrix} = \\begin{bmatrix} \\mathbf{a} \\\\ \\mathbf{b} \\end{bmatrix} ", "c455433cade0aaea8d195916de310579": "V(s)", "c4558f12c64b3abcd479e6b9d5a037c8": "1,\\dots,n-1", "c4560f061ff57e52c31ba277ac59bc67": " \\bold c_t \\in \\mathbb{C}^n, t = 1, 2, \\ldots , n ", "c45658af653b2e7e5716e82f8d5d23ec": " U_\\rho = v_r - v'_r +\\frac{ra'_r}{c} ", "c456ac827da2b3e0acb28ed20a9c096f": "f\\in L_1", "c456af8553ac905938842d238de804e0": "(r,\\theta,\\varphi)", "c456de7be8597a5403a8706acd5909da": "\\;a=\\prod_p p^{a_p}\\;\\;", "c4570e5d00ccb1078a239882f7b1f252": "\\begin{align}\nP_{\\text{r}} =&\\ P_0\\left|R(\\omega)\\right|^2+P_0\\frac{\\beta^2}{4}\\Big\\{\\left|R(\\omega+\\omega_\\mathrm{m})\\right|^2+\\left|R(\\omega-\\omega_\\mathrm{m})\\right|^2\\Big\\} \\\\ \n&+ P_0\\beta\\Big\\{\\textrm{Re}[\\chi(\\omega)]\\cos{\\omega_\\mathrm{m} t} + \\textrm{Im}[\\chi(\\omega)]\\sin{\\omega_\\mathrm{m} t}\\Big\\} + (\\text{terms in } 2\\omega_\\mathrm{m}).\n\\end{align}\n", "c45735b547aad4c34ac8abcf6371f0bd": "J=\\int_0^\\infty \\frac{x^{3}}{\\exp\\left(x\\right)-1} \\, dx = \\Gamma(4)\\,\\mathrm{Li}_4(1) = 6\\,\\mathrm{Li}_4(1) = 6 \\zeta(4)", "c4574e70efa69fa718b26e8af3877a24": "\\scriptstyle(x_0, x_1)", "c457601fb2cb0df013212531a6ea7998": "V_n=\\int_{\\phi_{n-1}=0}^{2\\pi} \\int_{\\phi_{n-2}=0}^\\pi\n\\cdots \\int_{\\phi_1=0}^\\pi\\int_{r=0}^R d^nV. \\,", "c45775a317b5816f521570d2eb6ab787": "\\sigma=1", "c4577706f4e7f071189937ca11c8e1c6": " P = F_x C_u ", "c4579e91d6e960879796bc66df14a54a": "\\theta_S/\\theta_E", "c457b4f293b514bfe0a660d0c1595a24": "\\|\\ \\| _{1} ", "c457db0ddfe4cefbb65b8dd7f3568fd6": "MSF=\\sqrt{\\frac{1}{T}\\sum_{t_j=1}^{T}(x_i(t_j) - \\tilde{x}_i)^2}", "c457e1c9e856280677d4db297b2e89c1": "~018", "c45b3e9696515de7c1cb446e803da3fb": "\\mbox{Saturday (6)} \\mod 7 = \\mbox{Tuesday (2)} + 2009 + \\left\\lfloor\\frac{2009}{4}\\right\\rfloor - \\left\\lfloor\\frac{2009}{100}\\right\\rfloor + \\left\\lfloor\\frac{2009}{400}\\right\\rfloor", "c45b5f3ba671bfeb4a744871bd165932": "\\frac{\\sigma\\Gamma(\\alpha-\\gamma \\alpha)\\Gamma(1+\\gamma)}{\\Gamma(\\alpha)}", "c45b91ae46fdaa78f0de6641c80b86d1": "\\ddot x(t) = x''(t) = -32, \\,\\!", "c45ba3cc21dde7ca6eb1ccf390e39c7e": "\\frac{v_0 [H^+]_0-v_i[OH^-]_0}{v_0+v_i} \\begin{cases} \n\\approx [H^+]_i \\text{ or } 10^{-pH_i} & \\text{ when } v_{0^{ }} [H^+]_0 > v_i[OH^-]_0 \\text{ (acidic region)} \\\\\n= 0 & \\text{ when } v_{0^{ }} [H^+]_0 = v_i[OH^-]_0 \\text{ (equivalence point)} \\\\\n\\approx -[OH^-]_i \\text{ or } -K_w 10^{pH_i} & \\text{ when } v_{0^{ }} [H^+]_0 < v_i[OH^-]_0 \\text{ (alkaline region)} \n\\end{cases} ", "c45baac04997214407339eaa98ecb5d8": "4m + 2k - 4", "c45bd2557af5df2559ddd594d618e31f": "(B \\leftrightarrow C)", "c45c172cd7227af2dc9c0270ddb3ded3": "\\operatorname{excsc}(\\theta) = \\operatorname{exsec}(\\pi/2 - \\theta) = \\csc(\\theta) - 1. \\!", "c45c225ac6d2913c995e1e04bd84134d": "g(x) - h(x) \\in O(f(x))\\,.", "c45c275c4e43bc5b86243403e03a31ec": "\\Delta \\langle\\hat{X}^\\dagger \\hat{X}\n\\rangle", "c45c397dc2a3e4401473e19bdc78eaf0": "\\langle\\tau^n\\rangle \\equiv \\int_0^\\infty {\\rm d}t\\, t^{n-1}\\, e^{ - \\left( \\frac{t}{\\tau} \\right)^\\beta} = \\frac{\\tau^n}{\\beta}\\Gamma \\left({n \\over \\beta }\\right).", "c45c3ec8de204f079b4f6ec6357f8b43": "\\lim_{n\\to\\infty} \\frac{1-\\left(\\frac14\\right)^{n+1}}{1-\\frac14} = \\frac{1}{1-\\frac14} = \\frac43.", "c45c51aca7567123e91b3ab92da40e46": "\\Omega S^{2n+1}", "c45cb66647a7a1a571a4f5123e0d483a": "\n\\begin{align}\n\\Pr(Y_i=1\\mid\\mathbf{X}_i) &= \\Pr(Y_i^{1\\ast} > Y_i^{0\\ast}\\mid\\mathbf{X}_i) & \\\\\n&= \\Pr(Y_i^{1\\ast} - Y_i^{0\\ast} > 0\\mid\\mathbf{X}_i) & \\\\\n&= \\Pr(\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i + \\varepsilon_1 - (\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i + \\varepsilon_0) > 0) & \\\\\n&= \\Pr((\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i - \\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i) + (\\varepsilon_1 - \\varepsilon_0) > 0) & \\\\\n&= \\Pr((\\boldsymbol\\beta_1 - \\boldsymbol\\beta_0) \\cdot \\mathbf{X}_i + (\\varepsilon_1 - \\varepsilon_0) > 0) & \\\\\n&= \\Pr((\\boldsymbol\\beta_1 - \\boldsymbol\\beta_0) \\cdot \\mathbf{X}_i + \\varepsilon > 0) & \\text{(substitute }\\varepsilon\\text{ as above)} \\\\\n&= \\Pr(\\boldsymbol\\beta \\cdot \\mathbf{X}_i + \\varepsilon > 0) & \\text{(substitute }\\boldsymbol\\beta\\text{ as above)} \\\\\n&= \\Pr(\\varepsilon > -\\boldsymbol\\beta \\cdot \\mathbf{X}_i) & \\text{(now, same as above model)}\\\\\n&= \\Pr(\\varepsilon < \\boldsymbol\\beta \\cdot \\mathbf{X}_i) & \\\\\n&= \\operatorname{logit}^{-1}(\\boldsymbol\\beta \\cdot \\mathbf{X}_i) & \\\\\n&= p_i &\n\\end{align}\n", "c45cdb4b933159f2fbf5488fa06e495f": "\\mu_w=\\left(\\sum_{i=1}^\\mu w_i^2\\right)^{-1}", "c45cfc55ac393b0ee07b02a6e61feb67": "\nP(I,saw,the,red,house) \\approx P(I|) P(saw|I) P(the|saw) P(red|the) P(house|red) P(|house)\n", "c45d24fd9f9873961c277abdb8e07fd5": "f\\!", "c45d260faea3e346e813d23c86ec35be": "L_3 = 0.152 \\lambda\\,", "c45d33f61d44ac425608c7045a2a4b6c": "\n \\qquad \\qquad u_x^+ = \\frac{-u_{i+2} + 6u_{i+1} - 3u_i - 2u_{i-1}}{6\\Delta x}\n", "c45d4a6b1246cad735d11309d1079239": "\\alpha \\simeq 1/2", "c45d929648a4dd7003bbbc62935257df": "\n \\Psi_R \\rightarrow \n \\begin{pmatrix}\n \\psi_{11} & 0 \\\\ \\psi_{21} & 0\n\\end{pmatrix}\n", "c45dee3fa0ca834b741db1fd6daeced5": " | \\langle e_n, x \\rangle |^2 \\rightarrow 0", "c45e880f62b57c3128d35e613ddd258e": "s_{\\mathrm{Euclidean}}:x \\mapsto 1, \\quad v_{\\mathrm{Euclidean}}:x \\mapsto 1.", "c45e92bd90e7ded68deb3bdaad33c8ce": "(\\phi,T)", "c45e9b7854494f591cc5871a081bb026": "\\langle i,j \\mid jij = i, iji = j \\rangle\\,", "c45eb95fe8466bbbfec143055b01e0fe": "S = \\sqrt {P^2 + Q^2 + D^2}", "c45ee549a29f760f3627e41c2ff4e254": "Z_\\mathrm A=\\frac{Z_1Z_0}{Z_1+2Z_0}\\ ,\\!", "c45f24a5c64188c9e538708ba8b6236b": "a^3\\,", "c45f27438f87594405675803cb969bd6": "f \\circ g \\circ f = f", "c45f61421c6b542d0534f91dcf772157": "P_{end}", "c45fa51549bca1b59d3730524b1201ed": "G_I= \\pi S \\sin^2 \\alpha \\ ", "c45fac475eeeeb780660eafce7a0be49": "\\rho = \\frac{\\rm{normalization \\; constant}}{(\\omega - K r \\sin(\\theta - \\psi))}", "c45fcd263f6dcced9c3486c0c5dd16f7": "D=E\\left[\\mathcal{A},\\mu\\right]", "c45fd032609f5580934c5d0b79ce4e0b": "\n\\begin{align}\n \\mathopen[y_0] &= y_0 \\\\\n \\mathopen[y_0,y_1] &= \\frac{y_1-y_0}{x_1-x_0} \\\\\n \\mathopen[y_0,y_1,y_2]\n&= \\frac{\\mathopen[y_1,y_2]-\\mathopen[y_0,y_1]}{x_2-x_0}\n = \\frac{\\frac{y_2-y_1}{x_2-x_1}-\\frac{y_1-y_0}{x_1-x_0}}{x_2-x_0}\n = \\frac{y_2-y_1}{(x_2-x_1)(x_2-x_0)}-\\frac{y_1-y_0}{(x_1-x_0)(x_2-x_0)}\n\\\\\n \\mathopen[y_0,y_1,y_2,y_3] &= \\frac{\\mathopen[y_1,y_2,y_3]-\\mathopen[y_0,y_1,y_2]}{x_3-x_0}\n\\end{align}\n", "c45fea05b6c6b2be5e998177a09fd0bc": "ds^2=-2dx^+dx^- + \\delta_{ij}dx^i dx^j", "c45ffa672d6fd1fbd9b14df9d975a60e": "d(t)", "c460398ab9e9d3950a4926dee666eb62": "\\Pi_1^1 ", "c460754018a659cd3d5382894cad3cb1": "\\operatorname{ad}(x)y = [x, y]", "c460e2a0aab3556c4d27e10b3b4a9e7b": "\\,x(t)", "c4610b416b4a20eddf672d86dcb2ac20": "F(x) = D_+(x)", "c4610d8c7d28d9ad388c1afcc2099344": "P(t) = \\frac{M_a}{r}(1 - e^{-r(T-t)})=\\frac{P_0(1 - e^{-r(T-t)})}{1 - e^{-rT}}", "c4615eab81be6060fb8f75e5acc1b775": "x=(x_1, \\ldots, x_n)", "c461836b131a49740c6cec80bb3f40cc": "SSW = \\sum_{i=1}^m\\sum_{j=1}^n (Y_{ij} - \\overline{Y}_{i\\bullet})^2 \\, ", "c461a0741333c5df683a342951df0a4f": "Bv^2 = u^3 + Au^2 + u", "c46207246e97709dd8911c7047f6964b": "\\phi=-1", "c4621e594cabbca13bd91604a7f25cf7": "O_1, \\dots, O_n", "c4622277e83c1b0a8f0658920070246c": " A_{\\alpha\\gamma}{}^\\gamma B^\\alpha C_\\gamma{}^\\beta \\not\\equiv \\sum_\\alpha \\sum_\\gamma A_{\\alpha\\gamma}{}^\\gamma B^\\alpha C_\\gamma{}^\\beta\\,", "c4622aeecaf2ea8f8e01f35f7a154261": "x_0 = \\frac{1}{2} (y_0 + y_1) \\, ", "c4622e2b431df9ef1b5afd8d52925fd5": "\\ W_{final}=n*(1-(1-\\beta)F_1)*(1-(1-\\beta)F_2)...", "c46274ad8b40b1f1bc19a026341fee66": "\\mathbf{a} = \\begin{pmatrix} a_\\text{x} \\\\ a_\\text{y} \\\\ a_\\text{z} \\end{pmatrix} ", "c462796463c5e6e296f8c6cbe143e54f": "\\psi^1(z)", "c462fdb4d02f95b9ec9df761424709fd": "\\mu \\over{2R}\\,\\!", "c4630c4248101e2fa666796e15ec121a": "x_{k+1}-x_k", "c4634b66c388ec2cf52e35eee24d528a": "\\begin{align} 2^{\\frac 6 {12}} & = \\sqrt 2 \\\\ & \\approx 1.4142 \\end{align} ", "c46358ab534b66525b768e953eb008ad": "\\mathcal E^{p,q}(E)", "c463782aedef920d0953b7bc3e76d415": "f(q,q) = \\sum_{n=-\\infty}^\\infty q^{n^2} = \n{(-q;q^2)_\\infty^2 (q^2;q^2)_\\infty} ", "c463905de1d74446340ed61317b7609e": "\n\\varphi(1)+\\varphi(2)+\\cdots+\\varphi(n) = \\frac{3n^2}{\\pi^2}+\\mathcal{O}\\left(n(\\log n)^{2/3}(\\log\\log n)^{4/3}\\right)\\ \\ (n\\rightarrow\\infty),\n", "c46399457a7da8b7c23aff8174ca0a1f": "\nP = \\frac {TP } {TP + FP }\n", "c46400748e30bcb23c5259410894ace9": "\\displaystyle{W(gz)=\\lambda_g(z)\\pi(g) W(z)\\pi(g)^*.}", "c46423eb457873822f02b2fa395baf02": " \\tan\\alpha_r=\\frac{p_r}{p_b} \\cdot \\tan\\alpha_b+\\left(1-\\frac{p_r}{p_b}\\right)\\cdot \\tan\\alpha_m.", "c46431b391bc0011efc9e91508d9dcf7": "y,", "c4645294094b6a6a8fdc95e0d7be2dc8": "d_i = x_i - y_i", "c464f5dfd8ed674d2e600fcfada7bf16": "-R\\leq x \\leq R", "c4654494ee35956457030eebcc63efd6": " p = (\\rho c^2 + q \\frac{d}{dt}) \\frac{dW}{dz} \\quad (2.4.b) ", "c4654fa6139d57582fa7e513384ced49": "{\\rho}=\\rho_b \\cdot \\left[\\frac{T_b + L_b\\cdot(h-h_b)}{T_b}\\right]^{\\left(-\\frac{g_0 \\cdot M}{R^* \\cdot L_b}\\right)-1}", "c46553d6c9481ab6d0de1dbdbb32ae23": "(\\overline{x2}\\vee gate2\\vee \\overline{gate1})\\wedge (gate3\\vee x2)\\wedge (\\overline{gate3}\\vee \\overline{x2})\\wedge (\\overline{gate4}\\vee x1)\\wedge ", "c4658a79d7fd6f170d42882ac8749f10": "\\log_a(x)=\\frac{\\log(x)}{\\log(a)}", "c4658fdc8ff2b73c0373137edb453976": "\\begin{bmatrix}1&1\\\\0&0\\end{bmatrix}:\\mathbf b", "c465a64041721c03685cf7e9d5709495": "E \\xi (B) := E \\bigl( \\xi(B) \\bigr) \\quad \\text{for every } B \\in \\mathcal{B}.", "c465b9261639df29c518a750cc203691": "\\frac{v_1\\qquad}{i_1 \\qquad}\\overset{\\textstyle R}{\\!\\!\\and\\!\\!\\and\\!\\!\\and\\!}\\frac{\\qquad v_2}{\\qquad i_2=i_1}", "c465bee925137d64a0973884e4e611b3": "k!/k", "c466388532d276eb2ea1859c4a3a0046": "D_{ddm}", "c466952fd04e33e3579fc8ad18b72020": "u \\cdot y", "c4669d6f2d610905ae8b9b534587c7ca": "E \\, = \\, - \\, p_{\\beta} \\, u_{\\text{obs}}^{\\beta} ", "c4669e0f598c70bab9a03abe197261e2": "K_1=M\\frac{\\pi}{180}\\,\\!", "c466ab01d14808d43a5c2b05a2e41cef": "P(A \\cap B)", "c466d1aa7e3e9d710855d6f2b8986ea8": "\\lnot \\neg, \\not\\operatorname{R}, \\bot, \\top \\!", "c466d3e6e599911e5db43688e702d1e8": " {\\det}_q\\begin{pmatrix}\\alpha & \\beta\\\\ \\gamma &\\delta\\end{pmatrix}=\\alpha\\delta-q^2\\gamma\\beta", "c466f6513c24a47c41db7a5fb3ab8c4f": "\\ q = \\min(m+n-u,\\,l-u,\\,n-w) ", "c4671e91debeaf3ed0aa47486cf37bd4": "\\rho_1, \\rho_2, \\ldots", "c4671ef234c18eeb366b35bc3de7d23d": "\\Pr_r(A'(x,r) = \\mbox{wrong answer}) \\le 1/3", "c46801ca966e9549f1844662bc17c764": "u_y = 0, y = 0 \\ ", "c468250c2b18c71bfd34c53c1bb21eeb": "R(t)=1-F(t)", "c468811aa30dbf16180c5cf4312a3b9e": "\\delta_\\mathcal{D}\\lambda = \\delta_\\mathcal{S} \\lambda = 0", "c468a551a9bbb61957bd2f740f1e1c6d": "addOne(2)", "c468ad644e445660b4beb7ac2d965d9a": "F_{}^c + F^{gi} = 0", "c46938f088e1c8c968cc60e605cc587c": "v \\in V(G)", "c469556426a6617dd57b84721b1e098d": "J^k_p({\\mathbb R}^n,{\\mathbb R}^m)_q=\\left\\{J^kf\\in J^k_p({\\mathbb R}^n,{\\mathbb R}^m)|f(p)=q\\right\\}", "c469e9ab9efb42a55f860d809731dc77": "\\textstyle a", "c469fbc406e7af2593cb874de52efeae": "\\psi(h(\\psi(0)))", "c46a2a6a4b4c755e236f947bcb347535": "0.5 < A \\le 0.7853.", "c46a746cebc11129b12d4b02c0de0823": "\\scriptstyle{\\pi / 2}", "c46ab6c049473d8c20f3c72089397b8b": "\\frac{\\binom{32}{13}}{\\binom{52}{13}}", "c46ace27960c01ac8b3f8ad4d7b521f7": "m_{p}\\,", "c46af4693baec38442a74cad045c0881": "|X_i| \\leq 1", "c46b781ae3955a1babddb479f11bd370": "E = \\frac{q^2}{2gy^2}+y", "c46c01bedb129b7cd255273acf8b1883": "A,C,B,D", "c46c13b74ef5ba0b8b5baee564974c30": "p(Z)\\in\\mathbb{C}[Z]", "c46c48537d6996ab8646c88593a36eef": "T_A = \\{ z=x+iy\\in\\mathbb{C}^n\\mid x\\in A\\}.", "c46cd6f9b7eddb87fa2db79a68a5fd12": "x \\vee y = x", "c46cfee4426b771994a78dbb2cebc509": " \\nu ", "c46d2dd9de6e731e4476fdcbbe448fa0": "b_n = B_n - B_{n-1} \\,", "c46d459b03873c6e3d4eb8fd4df7d5aa": "\\left[H,Q\\right] \\neq 0 ", "c46d5ae2cb3fac0b1090b63669ad9788": "\n\\frac {q_H}{T_H} - \\frac{q_C}{T_C} = 0\n", "c46d9b8bdcd872676e1960c7ec03897c": "-\\nabla^2 u = f, \\, ", "c46dc103d66a95fb561d75e1383eef19": "C = A \\oplus B", "c46e2f27708b19d0998405c16fe13420": ") \\lor (", "c46e6e10680764f7a9b747012de683da": "\nS(x)\\Gamma(x)(\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2)-\n(\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3)x^6=\n\\alpha^{-4}+\\alpha^{4}x+\\alpha^{2}x^2+\\alpha^{-5}x^3.\n", "c46e9221cd570cb3c6bc7862f7433935": "3.04 = y_2 + \\frac{10^2}{2(32.2)y_2^2}", "c46eabfafa9c60e8f8c7acaa16b996b7": "2^{341} \\equiv 2\\pmod{341}\\,", "c46ede32d43ac15f9b8d86c355b97eca": "m_{T}=\\frac{3}{4}\\cdot m_{em}\\cdot\\frac{1}{\\beta^{2}}\\left[\\left(\\frac{1+\\beta^{2}}{2\\beta}\\right)\\ln\\left(\\frac{1+\\beta}{1-\\beta}\\right)-1\\right]", "c46f1308f8c316a45e3e19c3c64b781b": "n_Q", "c46f135aeec8bc8683faca10edfb202d": " (a, a) = (a, -1) ", "c4707a9df856505727ff53492837aea8": "X \\subseteq \\Lambda", "c47094d4095107283404601f9e26a8b7": "=\\frac{(1+\\varepsilon\\cdot\\cos \\theta)-(\\varepsilon+\\cos \\theta)}{(1+\\varepsilon\\cdot\\cos \\theta)+(\\varepsilon+\\cos \\theta)}\n", "c470debfb177c6ad95b7bf6a71006043": "\\mathrm{soc}(R_R)=\\mathrm{soc}(_R R)=R", "c4718740e8ddd72d881dae079009412b": "\\operatorname{exp} \\ m\\ n\\ f\\ x = (n\\ m)\\ f\\ x ", "c4718ba0df4085717586f66e73be7df8": "p=5", "c471a3a35fdff0d1569241799e766e98": "\\spadesuit", "c472068c80f866a5017ae203b5cd0357": "S(A)=\\frac{1}{2}(A-A^\\dagger)", "c47322085373db701f3f1f5fa0ec7911": "x > b", "c47358c402238e0dc6b5e817f54f06b3": " \\hat A ", "c4737226361681119a5042f36b0b5b4c": "R_e:", "c473c574b4833f5db51a20c713ec6341": "|\\frac{1}{m}\\left(\\sum_{i}w^{j}_{\\sigma(i)}-\\sum_{i}w^{j}_{\\sigma(m+i)}\\right)|\\geq\\frac{\\epsilon}{2}\\,\\!", "c4742f94c3ea32e2b1c92f52d2735bf4": "\\delta^n:T^n(M^{\\prime\\prime})\\rightarrow T^{n+1}(M^\\prime)", "c4745739235fe1f7ae7a2a71635218b2": " p = R \\cos(\\omega t - kx) + (1-R) \\cos(\\omega t+kx) ", "c4745d8b522a5d4216b874798c27f5f3": "\\int_0^1\\int_0^1 f(x,y)\\,dy\\,dx \\neq \\int_0^1\\int_0^1 f(x,y)\\,dx\\,dy.", "c474e1c64b1c66f0bd21814cf5d32b24": " \\delta_{int}(s,t_s)=(s',\\tau(s')) ", "c474eae5b68ac6f998d80e2e8d2fefb8": " P=\\{p_i\\}_{i=1}^k ", "c47551e3744e179b83d76282288daf14": "\\bar{r},\\dot {\\bar{r}}", "c475b1c1fd865d05b86cac28956b45ff": "\\scriptstyle \\tfrac{1} {3} = \\{ y \\in S_*: 3 y < 1 | y \\in S_*: 3 y > 1 \\}", "c475b3df5e1865c7d4c6f41b48282c62": "f(x)\\sim \\left|x-r\\right|^{p}", "c4760fb9e3bc1c7d7b924fec5d275937": "\\vec{f}_0 = \\vec{e}_0, \\; \\vec{f}_1 = \\vec{e}_1, \\; \\vec{f}_2 = \\cos(\\theta) \\, \\vec{e}_2 + \\sin(\\theta) \\, \\vec{e}_3, \\; \\vec{f}_3 = -\\sin(\\theta) \\, \\vec{e}_2 + \\cos(\\theta) \\, \\vec{e}_3", "c4762754db3d22b79d97c6de61bdc14e": "\\frac{\\partial S}{\\partial \\beta_j} = 2\\sum_{i=1}^{m} \\left( y_i-\\sum_{k=1}^{n} X_{ik}\\beta_k \\right) (-X_{ij})\\ (j=1,2,\\dots, n).", "c47628d0171dd6bb26a4e508abb32eb9": "A_{FB} = \\frac {A_{OL}} {1+ \\beta A_{OL}}, ", "c476cfa2e7a19055cb73d2ff058c76c4": "f_{\\alpha}(\\vec{r},\\vec{p},t)", "c476ea9e0e760fb9e4b7746a8999b40c": "\\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\right) = 0 \\,", "c476fbba75944d99360fc006906bac69": "\n\\mathbf{x}_{0} = \n\\sum_{k=1}^{N} \n\\left( \\mathbf{l}_{k} \\cdot \\mathbf{x}_{0} \\right)\n\\mathbf{r}_{k}\n", "c4770438d345333d0c1a3cb392e0870d": "\\overline{OO_i} = R - R_i,\\, \\angle O_iOO_j = \\angle K_iOK_j", "c4771630ecae696b22e65f9283373064": "quot = \\frac{V}{s+1}", "c47782f184ae5758de5f7c199ba3ccc9": "10^{-25}", "c47788213c5517033fdb0bcb6af98940": "n_{\\rm e} T \\tau_E \\ge 10^{21} \\mbox{keV s}/\\mbox{m}^3", "c4779d380a9e25cd8956fad9c48093b4": "X\\times U", "c4779d6429451c9127cb1ed8d4acd011": "y\\frac{d^2y}{dt^2}=\n\\tfrac12 \\left(\\frac{dy}{dt} \\right)^2\n+\\beta+2(t^2-\\alpha)y^2+4ty^3+\\tfrac32y^4", "c477bb1253599fa9a801f58dded0c0a0": "\\scriptstyle \\Delta\\lambda", "c477e084449219b9224ec1bea807d412": "P(X|Y)", "c4780203eb1220f4d60f83a1737ce56d": "\\alpha \\in Z^{*}(X)", "c478323eaf4c4f4c69fc5ce241be8c44": "a_{C}\\frac{Z^2}{A^{1/3}}", "c4783a26ba3c6858e584e1e1b7027e93": "\\mathbf{e}_{31} = k ", "c4788f2e66254c1076698a5941a12326": "1+ z = \\frac{1 + v \\cos (\\theta)/c}{\\sqrt{1-v^2/c^2}}", "c478c9ef11ee30ba790e792c64618046": "\\displaystyle \\frac{\\sqrt{2\\pi }}{T}\\sum_{k=-\\infty}^{\\infty} \\delta \\left( \\omega -\\frac{2\\pi k}{T}\\right)", "c478d9d54d9392ef1beb15d7d62c3fb0": "(\\alpha\\cdot X)^\\pi = \\hat\\alpha\\cdot X^\\pi", "c478ed91b32f7200e099b361d4f27cb0": " P(x; k, \\lambda ) \\approx \\Phi \\left\\{ \\frac{(\\frac{x}{k} ) ^{1/3} - (1 - \\frac{2}{9k}) } {\\sqrt{\\frac{2}{9k}} } \\right\\} , ", "c47913dee6208c408e72bd4038290c21": "\\rho=\\rho_c \\Omega_m /a^3.", "c4791a8e4e14fa2bdbc20cbcabd51f33": "1 - \\frac{c^2}{2} + O(c^4) = 1 - \\frac{a^2}{2} - \\frac{b^2}{2} + \\frac{a^2 b^2}{4} + O(a^4) + O(b^4) + \\cos(C)(ab + O(a^3 b) + O(ab ^ 3) + O(a^3 b^3))", "c4791d83b008c88ce1be42bb2d125df3": "Y_s = f(P-P_{expected})", "c4794e8d951d033d3a7edc88287e71a5": "f'(\\alpha)", "c479c601c1feb3277738b4c6453f4e39": "\\mathbf{\\sigma}=0", "c479d1e3e3a53f76b91f3d01e8eebe9e": " \\lambda > 0 ", "c47a08da0758c16341dd7ca7fc81904d": "\\varphi(\\mathrm{lcm}(m,n))\\cdot\\varphi(\\mathrm{gcd}(m,n)) = \\varphi(m)\\cdot\\varphi(n).", "c47a1352c914bc3a84d46a20d4394b59": "\\displaystyle\\gamma", "c47a1b8181a6165ed3dd00dec71de89f": "\\sin^n\\theta = \\frac{2}{2^n} \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{(\\frac{n-1}{2}-k)} \\binom{n}{k} \\sin{((n-2k)\\theta)}", "c47ac1a1048310fa73ba4a5bd388c893": " g(t) ", "c47acbc27e5edc72fa5a636f10aa2d8c": "\n S_n(a,b) = \\tfrac{1}{n+1}\\ln(x_{(1)}-a) + \\tfrac{1}{n+1}\\ln(b-x_{(n)}) - \\ln(b-a) + \\sum_{i=2}^n \\ln(x_{(i)}-x_{(i-1)})\n ", "c47acf6a5066827945740bd688d46121": " \\frac{\\partial\\psi}{\\partial t} +{\\bold u}\\cdot\\nabla\\psi=0. ", "c47b255bfca8e6156a17d8c726a19737": "\\mathcal{S} = -\\frac{1}{4\\pi\\alpha'} \\int \\mathrm{d}^2 \\Sigma ({\\dot{X}}^2 - {X' }^2).", "c47b69cb7120cfeb6d24bf3d40a8a315": "T_e=\\frac{2\\pi\\,T^2}{1-R^2}\\,\\sum_{\\ell=-\\infty}^\\infty L(\\delta-2\\pi\\ell;\\gamma)", "c47b74b3e596d3e03ba68761e01fa53c": "SEncode\\,", "c47b8250dee362a37fb4397afb533d05": "4+\\varepsilon", "c47b89c6419464c441ad808fe8497e65": "{{f_n}}/2", "c47bb8f52336868b8e048f381bcd5dbc": "V_b = \\frac{V_{as}}{\\alpha}", "c47bd279d215cdf1e314948137beece3": "{\\mathrm {pr}_1} \\colon {\\mathbb R^n}\\times{\\mathbb R^{p-n}} \\to {\\mathbb R^n}\\,", "c47c31b54a00c6defc5c7a8608d15dfb": "R = k[X_1, \\ldots, X_n]", "c47c47950c1465f9ba067d3dbc3db979": "H= \\frac{A}{L}[I_k(T_H)-I_k(T_L)].", "c47c4e7dc4fc299573a503b8cbe46540": "\n\\Pi_{\\rho,\\delta}^{n}\\equiv\\sum_{x^{n}\\in T_{\\delta}^{X^{n}}}\\left\\vert\nx^{n}\\right\\rangle \\left\\langle x^{n}\\right\\vert ,\n", "c47c5c172ea797059dd06003c143ed2a": "\\theta = \\operatorname{atan2}(L_y, L_x).", "c47cca09eb950a15cf9b2265b2fc7c7a": "(z_1 - T)^{-1} \\sum _{n \\geq 0} \\left ((z_1 - z_2) (z_1 - T)^{-1} \\right )^n", "c47ce2017e30e4373d0fcdf68d20f088": "R_{\\alpha \\beta \\gamma}^{\\;\\;\\;\\;\\;\\; \\delta} V_\\delta = (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_\\gamma ", "c47d102aeb62bac08f2850489f0c1fb9": "M = \\bigoplus_{i\\in \\mathbb N}M_i ,", "c47d287626db931f69c3c734fdcde8a5": "K\\subset \\mathbb{R}^n", "c47d9c38bdec418300f7f70c57c8ab50": "L_\\phi=e^{(3\\phi+\\chi)/2}(-\\textstyle\\frac{1}{2} e^{-\\phi}\\partial_\\alpha \\phi \\partial_\\alpha \\phi -e^{-\\phi}\\partial_\\alpha\\phi\\partial_\\chi\\phi + \\textstyle\\frac{3}{2} e^{-\\chi} \\partial_0 \\phi\\partial_0\\phi)\\,", "c47e099bca3cee3422eb8e210d14f1b8": "(5x+8)\\pmod{26}", "c47e11f11c9e6374c91441b77fe7693d": "M_1 := \\{ (x, x^2) : x \\in \\mathbb{R} \\} \\subset \\mathbb{R}^2", "c47e41982977a7ee29c1374db630ab43": "w(n) = \\sum_{k = 0}^{K} a_k\\; \\cos\\left( \\frac{2 \\pi k n}{N} \\right)", "c47e42bffd1a33ad46b2c3f4c4421d49": "\\widetilde{K}^{-n}(X):=\\widetilde{K}(S^nX)", "c47e62b5fe6ccedef7ff77f931317c6d": "\\left(-\\frac{\\hbar ^2 }{2 m }\\frac{\\partial ^2}{\\partial r^2}+V(r)\\right)\\Psi(v)=E(v)\\Psi(v),", "c47e8e9ca1cce47647573d564f6bbce1": " r_{1} ", "c47ead839fd866985f1c3cde5be13193": "\nI(v)=-\\log_2 p(v)\n", "c47f0da76a8b13f238eb2658faa963d4": "u = \\sin(x) \\Rightarrow du = \\cos(x)\\, dx", "c47f48c905cc1489c9546717e9268bc8": "\\frac{d\\bigl(f(x)\\bigr)}{dx}\\text{ or }\\frac{d}{dx}\\bigl(f(x)\\bigr)", "c47f99803db4b24efccb457a0226e9fd": "\\liminf_{n\\to\\infty} x_n = \\limsup_{n\\to\\infty} x_n,", "c48044cdb2891cfadc4b65b30157e881": "U_K^{(n)}", "c48067f244429d45877215405f9a4cd1": " S_{ss}=LWy_{ss,us}=2000.0 ft.\\cdot 1 ft.\\cdot 6.68 ft.= 13,353 ft^3 ", "c480f5b0eee1ac493f0fa61c647aadda": " \n\\begin{cases} \n \\displaystyle \\frac{d \\vec{x}_{P} }{dt} = \\vec{u}_{P} (\\vec{x}_{P},t) \\\\[1.2ex]\n \\vec{x}_{P}( t = \\tau_{P}) = \\vec{x}_{P0}\n\\end{cases} \n", "c480f8ca20c9ac61f4b3234e5ac80768": " y_{1ss} = 4.20 \\text{ ft}", "c4813071c61d2516f62c03acbb239950": "e_4=-\\Omega(\\alpha^{2})/\\Xi'(\\alpha^{2})=(\\alpha^{-4}+\\alpha^{6}+\\alpha^{6}+\\alpha^{1})/\\alpha^{6}=\\alpha^{6}/\\alpha^{6}=1.", "c481980d7b48802d9fd2dfb1677f5996": " 10^{-24} ", "c481fb4328163309bef48bb11d3e9717": "\\beta(\\omega)", "c482404dc01fc9f45d46cbe31f582e9a": "\\mathbf{\\ddot r_i} = G\\sum_{j\\ne i} {m_j}{r_{ij}^{-2}}\\hat{\\mathbf{r}}_{ij} ", "c4828a3ce0ec4a2d723314fc05e5ac9f": "\\frac{1}{2}\\chi'^2 = \\mathfrak{M}^2 \\left[ \\left( 1 + \\frac{2\\chi}{\\mathfrak{M}^2} \\right)^{1/2} - 1 \\right] + e^{-\\chi} - 1", "c4828e4d4c82591ca261583801bdc480": "X \\succ 0 \\Leftrightarrow C \\succ 0, A - B C^{-1} B^T \\succ 0", "c482ab99659fc23dbd22c08c2efd393b": "u_2 \\ne u_1^w rem P", "c482e21ce4c0e7a541c778ef39883294": "\\frac{I_{3}}{\\sigma_{yy} \\sigma_{zz} - \\sigma^2_{yz}}", "c482e52ad728f799a9039e4a3a6330af": "f,\\,g\\in\\mathcal H(G)", "c48324ee661263e1d1994c618dccd091": "C.R = \\frac{log_2(M_o) + log_2(M_e)}{2log_2(M_o)}", "c4832bbc70ee7c58635eb36150587a8d": "(x-\\tau)2\\tau=(y-\\tau^2) \\,", "c4832f0b693cbbcd8caed1c1ac0b2670": "P_{FAF}", "c4835556d9c9ee36ce3a201d909c84a2": "i\\ge 0", "c4835920913c029714f0f620d7f344e2": "\\mathbf{u,v}\\in\\left( \\mathbb{Z}_{2}\\right) ^{2n}", "c483b5a0545f83000ebfef153ec0b602": "q = - \\frac{\\partial F_4}{\\partial p} \\,\\!", "c483c5c1f9e3dcf1acbe5b38f51b0dd2": "\nh_{\\lambda} = \\frac{1}{2} \\sqrt{\\frac{\\left( \\lambda - \\mu \\right) \\left( \\lambda - \\nu\\right)}{S(\\lambda)}}\n", "c483f6ce851c9ecd9fb835ff7551737c": "pp", "c4842f70c3ea31910cca3ee318860c91": " \\Pr(\\bar x >z_{\\alpha}\\sigma/\\sqrt{n}|H_0 \\text{ true})=\\alpha ", "c4848b57a4ce4868fe3f215ad1e8a5cb": " \\frac{1}{r} ", "c484bf1f0cf7d9628400f3abb8f9f10a": "y(x)=\\begin{pmatrix}1\\\\x\\end{pmatrix},\\qquad x\\in I,", "c484fdb368ccc1380bf8420816d39be7": "\\mathcal{O}_n ", "c48534ffa003dddc12d4b2e11a9715ba": "b \\Rightarrow a;", "c485391cec084cc90ae89dc0ede8b678": "P_1-P_2=\\tfrac12\\rho[(\\text{velocity at station} 2)^2-(\\text{velocity at station} 1)^2] +\\gamma [(\\text{height at station} 2)-(\\text{height at station} 1)] +\\ \n(\\text{rate of change of kinetic energy of the blood between stations 2 and 1}) +\\ (\\text{integrated frictional loss between station 2 and 1})", "c48568924aa651dcfbd461038aaac274": "u_y' = u_y/\\gamma(1+u_x v/c^2)", "c485c0d5ea62045187ca33492f66b4b3": "-[Ag^+]_010^{-b_0}/K_{sp^{ }}", "c485ea1a46d46cbb6b5935667a018f3d": "\\alpha + \\sum_{i=1}^n x_i ,\\ \\beta + n\\!", "c485f424c2ebb0d0bd7e26c0a05ab8eb": "d((M,\\varphi),(N,\\psi))\\ge \\min\\{\\tilde x-\\bar x,\\bar y-\\tilde y\\}", "c4860607956e1bbeaf33dab69cbc7ce3": "-\\Delta S+\\int\\frac{\\delta Q}{T}=\\oint\\frac{\\delta Q}{T}< 0", "c48615ef0b22acb6ebae0b156bb79fec": "h_{2} = -h_{1}^{2} \\frac{J^{\\prime\\prime}(u_{0})}{12\\beta^{2}}", "c486208a9acd9432aee1e0407a51628e": "C_{OA}=\\left\\lceil \\frac{N_x}{N-M+1}\\right\\rceil\nN\\left(\\log_2 N+1\\right)\\,", "c486ade24961d8792bb9abd4c3009cf2": " c = -2\\,", "c486c29b4be5dae33c93fed75f25debb": "\\alpha\\rightarrow \\beta_1", "c48711e35927c9d7f64fc064859105bb": "\\langle \\mathbf{u}, \\mathbf{v} \\rangle", "c4871c8ac0ce9baf3ba169c8cf1886fd": "\nE_{L}\\beta_{L} = E_{U}\\beta_{U}\\rightarrow\\beta_{L}=\\frac{E_{U}}{E_{L}}\\beta_{U}\n", "c4873c595bb95ff63ced36d62b54fed2": "\\rho(X) \\leq \\rho(Y)", "c4874abf45e2741255170029c28784d7": "\\mbox{average scene reflectance} =\n\\frac {\\mbox {average scene luminance} } {\\mbox {effective scene illuminance }}\n", "c4878285c2b4a48eb9ae56a286fe1441": " \\scriptstyle z_{nj} = \\{ w^1_{nj}, w^2_{nj}, \\ldots, w^J_{nj} \\} ", "c4878f87b744b2f02d6ccb9ff584091c": "\\frac{D}{Dt} \\equiv \\frac{\\partial}{\\partial t} + \\mathbf{v}\\cdot\\boldsymbol{\\nabla}\\,.", "c487ed31a095dcb6db4c4015da3ddd0a": "E=-N(\\Delta\\phi_B/\\Delta t)\\quad(2)", "c48811296c3b9190aec81070dcfcc9e8": "eV = E_{F_A}-E_{F_B}", "c48817d3072d8dc293b7e48bdb9bb53a": "\\mathbf{\\hat p}", "c488abeedd5b31f880c5fb761f633751": "(y-1)(4y^2+2y-1)^2=0\\,", "c488f87573f61e134287cc391460da40": "\\Delta\\varphi = 0 \\qquad\\mbox{or}\\qquad \\nabla^2 \\varphi = 0", "c488fd7d87d9b798d09ff6abc026ecb1": "x^1,\\dots,x^n", "c4897878837c167e6c180dd5274e2a35": "O_{2j}", "c4897af2390e545ccccea700975450e0": "\\displaystyle x_0^{-1}+x_1^{-1}+x_2^{-1}+x_3^{-1}+x_4^{-1}+x_5^{-1} = 0.", "c48997e1eff3fbe7fd008f964aa1b663": "HS_{R/\\langle h_1,\\ldots, h_{d}\\rangle}(t) = (1-t)^d\\,HS_R(t)", "c489d2b408b6fddf936df6f38ca66b4a": "18 < \\frac{S}{L} < 20.", "c489dbb15f7d1ce4d75dffecc7102bb2": "h(x)=x", "c48a4b129d0a894884a70fd2f9c49e88": " \\Sigma=\\int_{\\textbf{R}^d} f(x)\\Lambda (dx), ", "c48a5ad4a6ef482680bdf7196f3040ce": "R(t) = e^{-\\Omega(t)} = (L_1(t) L_2(t) L_3(t))^{1/3}, \\quad \\sum_{k=1}^3 \\beta_k(t) = 0", "c48ab2ac24be1f47a6eafb4405679457": "m=-j, -j+1,...j-1,j", "c48ab778d535aaee0f5cd96219f7d0e3": "O(kn + 1.274^k)", "c48ab7ec0b5d138f108f2943c6cec5b4": "a^2/12", "c48b0fb33f1f7d2bc3967b65ff5a3c5c": "A + \\overline{B_{\\delta}} = \\left\\{ x \\in \\mathbb{R}^{n} \\mathrel|\\ \\mathopen| x - a \\mathclose| \\leq \\delta \\mbox{ for some } a \\in A \\right\\}.", "c48b5b8ab144636bb20a937ccd5dd7e0": "= H_a \\left(j \\frac{2}{T} \\cdot \\frac{ \\sin(\\omega T/2) }{ \\cos(\\omega T/2) }\\right) \\ ", "c48b6f7ff4d46420da1703d860ff6fc7": "\\widehat{c_v}_{ln} = \\sqrt{e^{{s_{ln}}^2}\\!\\!-1}", "c48b8d822386d2a9e37b3ea97adfdabb": "\\mathbb{P}(E\\mid A) = 10%\\ , \\ \\mathbb{P}(E\\mid B) = 0.10%", "c48bab9833a0b457fef3b2fb10ae8d6e": "Y(t) = \\ln I(t)", "c48bcfee2889efbe4ee6421dd50f1281": "\\mathrm{O}(kn)", "c48c1e669783c7c77180cba0483064e6": "\\left \\langle I,\\beta \\right |S\\left | I,\\alpha\\right\\rangle = S_{\\alpha\\beta} = \\left \\langle F,\\beta | I,\\alpha\\right\\rangle", "c48c31a76aa0bce342a38590cf579418": "\\mathbf{y}_{m+1}", "c48c45e79ea5f483d5d9e9f43ea5cdc6": "\\textbf{E}(X^{2}(t)) = \\sigma^{2}", "c48c7746f28ef62168adcf21223b2506": "-P=(0,a_3)", "c48c7fe6b1427adddf2e0e991410efae": "\\scriptstyle\\tau_{_0}", "c48c951ce73ae0ad12ad326adffaf541": "I = \\frac{a}{2R}\\frac{\\partial^2V}{\\partial x^2} ", "c48cb37db702e42bd859c28796bbc372": "\\begin{alignat}{7}\n2x &&\\; + && y \\;&& &&\\; \\;&& = \\;&& 7 & \\\\\n&& && \\frac{1}{2}y \\;&& &&\\; \\;&& = \\;&& 3/2 & \\\\\n&& && && &&\\; -z \\;&&\\; = \\;&& 1 &\n\\end{alignat}", "c48cf0417a3a1d93350ab69212ba4ffd": "A = 5\\sqrt{3}a^2 \\approx 8.66025404a^2,", "c48d05ffd63a87ad2183a96bb86f740d": "\\mathit{x = e^{2 \\pi i/p}}", "c48d13f5a4b3ee0fbff4979a68c1b8c5": "I(u+x,v+y)", "c48d61194fa56eac7fadc47e039fe050": " (E_{u/p}) = (E_c + Z) = (2.20 + 4) = 6.20ft \\,\\!", "c48d7d9054b27cffc48cc2b7d0cd8cab": "\\alpha = \\frac{\\gamma}{\\delta}", "c48d8078ff06ffc35400f93ace3575f6": "\\!\\mu_1(v_2)", "c48d96f2c187873c77dce070b388e08a": "\\tilde{g}\\, =\\, g\\, +\\, \\frac{\\gamma}{\\rho}\\, k^2.", "c48df875abd5fda83a334cb9983db56d": " f(6) = 0.03072 \\, ", "c48e0f6be850dbad3908400ae77998d4": "r^4=dr^2\\cos^2\\theta+er^2\\sin^2\\theta+f. \\, ", "c48e268e7a8f823d506caf04493a8bcb": "\\alpha, \\beta\\in G", "c48e2bc335c5b9697e22431adfe4ebe6": ",x)", "c48e9f6a8ac6bc0e32c1bd203c3b9297": "\\Delta\\nu", "c48ea259f850c380816e3a6efedfbd91": "\\begin{bmatrix}\n 1 & & \\ldots & & 0 \\\\\n 1 & x_1-x_0 & & & \\\\\n 1 & x_2-x_0 & (x_2-x_0)(x_2-x_1) & & \\vdots \\\\\n \\vdots & \\vdots & & \\ddots & \\\\\n 1 & x_k-x_0 & \\ldots & \\ldots & \\prod_{j=0}^{k-1}(x_k - x_j)\n\\end{bmatrix}\n\\begin{bmatrix} a_0 \\\\ \\\\ \\vdots \\\\ \\\\ a_{k} \\end{bmatrix} =\n\\begin{bmatrix} y_0 \\\\ \\\\ \\vdots \\\\ \\\\ y_{k} \\end{bmatrix}", "c48ebe1170d78bae472d1b4cbf5ab306": "\n\\phi _{\\mathrm{ground}}(x)\\equiv \\phi _1^{\\mathrm{even}}(x)=\\frac 1{\\sqrt{a}\n}\\cos \\left(\\frac{\\pi x}{2a}\\right),\n", "c48ee5175f225be31516c0259a82c703": "R = V/I \\,\\!", "c48ee9da0d35b78d161364615577e656": "h'(t)=\\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(a+t(x-a)) (x_i-a_i),", "c48fa3ec445468304f0cf462362e14fc": "nx[n]", "c48fb007a92c5e677c9930ed113e4e53": "y_m", "c49017e8db38ce7e2f9f9de57b0bc369": "\\lim_{\\varepsilon \\to 0} \\frac{1}{\\varepsilon^2} L = 3x \\ . ", "c490296d15ba6e51b0821881e07ca048": "\\ d = Gm", "c4908328a0559f9f68d8727601d40e3b": "O(\\mathrm{smoke} \\to \\underline{\\mathrm{ashtray}})", "c490c644fc02bf82168a5dc3b326e2dc": "x_1 \\ge 5", "c49103130f13b550cf10ce582d3005ae": "\\frac{\\partial \\overline{u_i}\\, \\overline{u_j}}{\\partial x_j} = \\overline{u_j} \\frac{\\partial \\overline{u_i}}{\\partial x_j} + \\overline{u_i} {\\frac{\\partial \\overline{u_j}}{\\partial x_j}},", "c4912a33e4e79de3d8e7736e25c6383c": "\\xi_y = \\frac{V_H}{W}", "c491b17648a67aee4ca5d53820739402": " Q^{n+1}_i = Q^n_i - \\nu \\left( Q^{n}_i - Q^n_{i-1} \\right), \\quad \\nu = a \\frac{\\Delta t } {\\Delta x }, ", "c4921788c00ec06da3174fb4a92000ef": "\\gamma_\\theta(r)=e^{i\\theta}r", "c49265c826f98aa81c21aeedaa7c0003": " \\psi_{n\\ell m} ", "c492975eba2a95552f5f07a069363c1f": "\\scriptstyle \\bar{\\mathrm{B}}", "c492b0c9ee73f4609ba1b9360be7c3fa": "\n[\\mathbf{c}] \\equiv\n\\left(\\begin{matrix} 0 & -c_z & c_y \\\\ c_z & 0 & -c_x \\\\ -c_y & c_x & 0 \\end{matrix}\\right)\n", "c492b558b9bfcf607ef242319011c378": "{\\mathbf M} = \\oplus_i {\\mathbf M} P_i .", "c4931a09b1cd71a6debd279d55d82c11": "K_D = (K_X + D)|_D", "c493595325525409245e829639ff253f": " \\frac {\\Delta P} {L} = \\frac {f \\rho V^2} {2D} ", "c493c0215ea14ad8caca2c4238e7a40b": " (X,Y) \\mapsto - h(X,\\nabla_YZ) ", "c493ec6ed48c67bacad810f4cca7ad1a": " \\operatorname{equate}[A, E] \\equiv A = E ", "c494087620382870b1933622af195798": " \\{ s_1, s_2, ..., s_n \\} ", "c4943a6806402309c21e8c6b110171ba": "S^* = S' \\,\\!", "c49441523b58cc056263236d4a5bb576": "\\Delta_n=\\left[\\begin{matrix}\nm_0 & m_1 & m_2 & \\cdots & m_{n} \\\\\nm_1 & m_2 & m_3 & \\cdots & m_{n+1} \\\\\nm_2 & m_3 & m_4 & \\cdots & m_{n+2} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nm_{n} & m_{n+1} & m_{n+2} & \\cdots & m_{2n}\n\\end{matrix}\\right].", "c49446a801a700f8648bc3342a71bc8e": "\\begin{align}\n\\text{Total Contribution} &= \\text{Total Fixed Costs}\\\\\n\\text{Unit Contribution}\\times \\text{Number of Units} &= \\text{Total Fixed Costs}\\\\\n\\text{Number of Units} &= \\frac{\\text{Total Fixed Costs}}{\\text{Unit Contribution}}\n\\end{align}", "c49465cd8e35714709c56e7172dd7808": " g(d:e:f) = (d:-f:e), ", "c494e42f628e3da26be0d6087a7357aa": "D_v \\leftarrow \\operatorname{diag}(v_1^k,\\ldots,v_m^k)", "c4959bd80cb975319c1c0eef79ef7ff7": "\\delta_n(t)=0", "c495e11ab8d902a26d61a22ede110588": "\\{f_n\\}", "c496276efdce93368a6190f7a969ce7e": "\\cos z = \\sum_{n=0}^\\infty \\frac{(-1)^{n}}{(2n)!}z^{2n} = \\frac{e^{i z} + e^{-i z}}{2}\\, = \\cosh \\left(i z\\right) ", "c496356487e4e4f21c97aa835342a127": "d_i > b-a+d", "c4966c2e603d514ae658e2b652c1d145": "m > 1", "c4967c69049efb745ad90eb23ff7d843": "\\mbox{bullet drop}= \\frac{-11.0\\cdot(282-200)+ -2.9\\cdot(300-282)}{300-200}= -9.5 \\mbox{ cm} ", "c4968afb690fa6812ffa8a7000286c2d": "(Y, \\mathcal{B}, \\nu, S)", "c496d09a59231a5029fc4a69f8482271": "2\\mathbb{Z}/4\\mathbb{Z}", "c496d7c98010b49c59d1e4fba5330c2b": " \\mathrm{w}", "c4979038b0ae5cbfd97ccdccaa23b844": "\n \\displaystyle \n w(n,g) \n =\n \\sum_{k_1=0}^{n}\n \\sum_{k_2=0}^{n-k_1}\n w(n - k_1 - k_2, g-2)\n =\n \\sum_{k_1=0}^{n}\n \\sum_{k_2=0}^{n-k_1}\n \\cdots\n \\sum_{k_g=0}^{n-\\sum_{j=1}^{g-1} k_j}\n w(n - \\sum_{i=1}^{g} k_i, 0).\n", "c4979087272aafd3ce5bb0e9a102d03a": " \\mathbf{F} = \\frac{q_1 q_2} {4\\pi \\epsilon_0 r^2} \\left[\\left[1+\\frac{3-k}{4} \\left(\\frac{v}{c}\\right)^2 - \\frac{3(1-k)}{4} \\left(\\frac{\\mathbf{v\\cdot r}}{c^2}\\right)^2 - \\frac{r}{2c^2} (\\mathbf{a\\cdot r}) \\right] \\frac{\\mathbf{r}}{r} - \\frac{k+1}{2c^2} (\\mathbf{v\\cdot r})\\mathbf{v} - \\frac{r}{c^2} (\\mathbf{a})\\right] ", "c497d7adf934e1e589512072ce37cd5c": "\\{\\ B_1\\}", "c49886a1c05fe0827759ec372c166b85": "\\frac{dy(t)}{dt}= 1/3*((c-a-z(t))*x(t)-(a+1)*y(t))+((2*(a-1))*x(t)*y(t)+(a+c-z(t))*(x(t)^2-y(t)^2))", "c49894a91e1029384cca34629e0175ad": "A_i:i\\in I", "c498a446df4b269cb51b36a53e7e4f7e": "\\rho(\\psi(P_A))^b) \\in \\mathbb{F}_q^m", "c49917c50804fa9836e3e89cd409264c": "T_G(x,y)=R_G(x-1,y-1).", "c4992ee4d26ac122048f40186e2b8bd1": "M_i = D_K(C_i) = (C_i-K_i) \\mod {26}", "c49937763ff42112fb10aa9a43be09d3": " (\\nabla^2 + k^2)~f = 0 , ", "c499742c69314b89ffaacfd82b66cdc6": "a\\to b=\\bigvee\\{c\\mid a\\land c\\le b\\}", "c499b59d362ea18ab698391a53819df1": "f(u,v)", "c499b5a4d83b4860929c380039f083fd": " F = \\int \\Pi(x) \\, dA,", "c499fffbea96c411d643fc8ef844ef9b": "I = \\frac{V}{R},", "c49a4151010c3f38976233738f138bde": "X_{H_\\xi}", "c49a5337ff22c822735f1ca68b71be53": "v \\otimes h = (w \\mapsto h(w)v)", "c49a761781ef5ddfee0d253d6df829d4": "\\alpha = 2\\beta.", "c49ac9682d9d4f72a417f20f00f83d31": "\\hat g(f) = e^{-\\frac{f^2}{2\\sigma_f^2}}", "c49b0b38975a748ed063c8557b3e3d8f": "\\frac {\\sum Exports} {\\sum GDP}", "c49b3bcf0fd2bfd8e95aca527d10dc8a": "(\\psi * \\mu) * \\alpha \\not\\models \\neg \\mu", "c49b59bea7ab9f793d551ad11a8f2556": "\\bar{y}_i = \\frac{1}{\\tau}\\int\\limits_0^\\tau y(iT + t_v) \\, dt_v = \\frac{x(iT+\\tau)-x(iT)}{\\tau}", "c49bbcacd0594f832fd0d92ad30477c4": " R(D) = \\left\\{ \\begin{matrix} \n \\frac{1}{2}\\log_2(\\sigma_x^2/D ), & \\mbox{if } D \\le \\sigma_x^2 \\\\ \\\\\n 0, & \\mbox{if } D > \\sigma_x^2. \n \\end{matrix} \\right.\n\n ", "c49c1b4e94d93345ccc401bb1f4b881f": "\\hat{a_1}=-(\\bar{x}+2\\log(f_0))", "c49c2ab637541018b06c2109bff93ed1": "TM\\vert_{i(N)}", "c49c57398f571e25ff76efe7f295ca1c": "\\mathbb{Z}^{(-n)}", "c49cbc00d9e4f4ab0c7e6fb9585ce6ce": " \\mathbf{e} ", "c49cf3840751c60dc7a6b11ab840ebd0": "R(\\omega)= \\frac{1}{K-\\omega I}.", "c49d1364dbddd5fc4b9082d6666bf72a": "\\theta\\in[0,\\pi)", "c49d821375527a789baee4e31844d5e9": "(2i)^2=-4", "c49dbedd7e0a334efc67552f498d84f5": "E^{n+1}(e) = E(E^{n}(e))", "c49e2769637446bb93fb233867cce42b": "\\mathbf{E}^{x} \\big[ f(X_{t+h}) \\big| \\Sigma_{t} \\big] (\\omega) = \\mathbf{E}^{X_{t} (\\omega)}[ f(X_{h})].", "c49ea4b92e56cec1c1708082b36b7ed1": "s=\\frac{k_R}{k_S}=\\frac{\\log[R]-\\log R_0}{\\log[S]-\\log S_0}=\\frac{\\log[(1-c)(1-ee)]+\\log\\frac{1}{2}-\\log R_0}{\\log[(1-c)(1-ee)]+\\log\\frac{1}{2}-\\log S_0}=\\frac{\\log[(1-c)(1-ee)]}{\\log[(1-c)(1+ee)]}", "c49f28839fe383037afd4ac421754aeb": " F = \\frac{\\tanh[(1-\\alpha) \\cdot ht]}{\\tanh(ht)} \\cdot \\frac{1}{1 + \\alpha \\cdot ht \\cdot \\tanh[(1-\\alpha) \\cdot ht]} ", "c49f36a909e892322792f65a8bd0bff4": "-1 = 1 = 1^2", "c49f50c94a9059460bb10c8c807543bd": "B^T=B", "c49f52621a087ffeb2e7f46bc0838d66": "\\left(1-\\frac{3}{4\\nu-1}\\right)^{-1}", "c49f9a5220129f0e2915e8cc4afb0fbe": "L=I-D^{-\\frac{1}{2}}WD^{-\\frac{1}{2}}", "c49fb0164f8e558158534a67af82202e": "X^2 + 1,\\ Y^2 - 1,\\ XY - YX", "c49fd9b39f05b7ebfd970edcd8e3a010": "\\scriptstyle \\epsilon(k)", "c4a0379822d55157149468d72756d42a": "\n \\begin{align}\n R_c & = 300\\mbox{m} \\\\\n L_s &= 100\\mbox{m}\n \\end{align}\n", "c4a04ab49ec1ce89828409192e05a7c0": "\\frac{max\\ packetsize}{max\\ packetsize\\ +\\ frame\\ size} = \\frac{1024}{1024+5} = 0.9951 = 99.5%", "c4a0dc975ea9d3fa0c730aca5dd7e461": " \\begin{bmatrix} \\Phi(S_{11}) & \\cdots & \\Phi(S_{1 n})\\\\ \\vdots & \\ddots & \\vdots \\\\ \\Phi(S_{n 1}) & \\cdots & \\Phi(S_{n n})\\end{bmatrix} ", "c4a1023c31d8f8d94962df45f1dc89fc": "\\hat \\mu_{1} = g_{1}(\\hat{\\theta}_{1}, \\hat{\\theta_{2}}, \\dots, \\hat{\\theta}_{k}) ,", "c4a17e3dca34510bfbc174e24dcf0f6f": "\\hat H^n(G,A) = H_{-(n+1)}(G,A)", "c4a1970b4d2898d62c370d7b74c11fbb": "(id \\otimes \\Delta) \\circ \\rho = (\\rho \\otimes id) \\circ \\rho", "c4a1a4623fc57a2cb409cf890a1d5dae": "\\neg open_1", "c4a1f87ed7a1bf2aa11c7bc3114ad256": "\\sum_{n=1}^k\\,\\frac{1}{n} \\;=\\; \\ln k + \\gamma + \\varepsilon_k < \\ln k + 1", "c4a239dd96fdc59a9353baedd2e4ca8e": "\n\\mathrm i^{2m+1} \\operatorname{erfc} (-z)\n= \\mathrm i^{2m+1} \\operatorname{erfc}\\, (z)\n+ \\sum_{q=0}^m \\frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}\\,.\n", "c4a2669e3b837a3f6cdeafbb2e1c1b4f": "\n\\frac{d}{dt} \\begin{bmatrix} A \\\\ B \\end{bmatrix} = \\frac{\\varepsilon}{k} f\\left( \\frac{a}{k^2} + A \\sin (\\phi), kA \\cos (\\phi)\\right) \\begin{bmatrix} \\cos(\\phi) \\\\ - \\frac{1}{A} \\sin(\\phi) \\end{bmatrix},\n", "c4a2689a8c359e5df51613b0c5345870": " \\int^{\\phi_1(x)}_{\\phi_2(x)} \\left( \\frac{\\partial V}{\\partial x}(x,\\theta) - \\frac{1-P(\\theta)}{p(\\theta)} \\frac{\\partial^2 V}{\\partial \\theta \\partial x}(x,\\theta) - \\frac{\\partial c}{\\partial x}(x) \\right) d\\theta = 0", "c4a2a3b2e0b8aedf477c34c104ebe6da": "\\Pr(x|t,\\theta) = \\Pr(x|t).\\,", "c4a2c0e22511d853c20338d0771d3d4e": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_{tot}", "c4a2c6e9a1d5a81b4ba3e731bced6fa7": " s(v) = v - 2 \\frac{B(e_s,v)}{B(e_s,e_s)}e_s", "c4a2e63aeffdbb86d39a5c6b85d594bf": "\\mathbf{A} = \\mathbf{I}_3\\cos\\theta + (1-\\cos\\theta)\\hat{\\mathbf{e}}\\hat{\\mathbf{e}}^\\mathrm{T} + [\\hat{\\mathbf{e}}]_{\\times} \\sin\\theta", "c4a2ed9e3ad5feae3d4473039bc8a30e": "u_1 \\ge u_2 \\ge u_3 > 0", "c4a313cf10421a4adf7f7db3a123b151": "N \\to M''", "c4a32503317c6dae33e4f155c5a8103e": "V_x = \\cos \\theta \\cdot V_r - \\sin \\theta \\cdot V_t = -\\sqrt{\\frac {\\mu}{p}} \\cdot \\sin \\theta", "c4a3283bd1e40885a74130ac9a7de987": "\\tau = s - t\\,", "c4a340579716dc91ea46bd566f33bd7a": " \\mathbf{Q}^{oe} = \\int_{V^e} - \\mathbf{B}^T \\big( \\mathbf{E}\\mathbf{\\epsilon}^o - \\mathbf{\\sigma}^o\\big ) \\, dV^e \\qquad \\mathrm{(12)}", "c4a36ab8928cacb4e046f27b3784651c": "G=\\frac{(P/S)_\\text{ant}}{(P/S)_\\text{iso}}.", "c4a3ab131dbf568a8f0a18263ba3c469": " \\mathrm{I}_2(n)", "c4a3dcaf8ba038baf6944cb063249534": "\n\\left[ \\frac{\\partial }{\\partial t} + \\frac{p}{m} \\frac{\\partial}{\\partial x} - U'(x) \\frac{\\partial}{\\partial p} \\right] \\rho(x,p;t) = 0.\n", "c4a3fac83cea9c3ce0b019c93673893d": " 6s^2 \\, ", "c4a446172e925abe1f4de02c3f01bd36": "\\varphi(z) = \\lim_{k\\to\\infty} \\frac{\\log|z_k|}{d^k},", "c4a45c72396e81def91d91a2879b2cd5": "c(S_i)", "c4a5190c684951970ca3ad52d140b625": "\\scriptstyle (k \\,-\\, 1)\\theta \\text{ for } k \\;>\\; 1", "c4a5248ab8dbaf3da1976e6fec748fbe": "\\mathrm{1\\, sb = 1\\,\\frac{cd}{cm^2} = 10^4\\,\\frac{cd}{m^2}}", "c4a54cd326448cb044262cda33883a36": " {d \\over dx} \\left( { L - u'\\frac{\\part L}{\\part u'} } \\right) = {\\part L \\over \\part x} \\, . ", "c4a57df39e3da74e4303df06e5e63bb5": "C = z_j\\overline{X_j} + (1 - z_j) \\overline{X}\\,", "c4a5a39c644eb55ff537927e9c81775b": " \\frac {n(n+1)(n+2)} {6}.", "c4a5daf25b998f94da32eab824d817a1": " L_{q}\\left[1/3,\\sqrt[3]{64/9}\\right]", "c4a65931e5144ff7751399f7687222a3": " A \\or B", "c4a682e8febb3dac4adef11e56934661": "\\left\\{\\begin{array}{l} a + b = 0\\\\ \\varphi a + \\psi b = 1\\end{array}\\right.", "c4a6af01daee3022693c3897d21a46dc": "\\Omega_{tot}", "c4a6d96afd730762ece9a9240a05e261": "y = \\sqrt{a^2+s^2} + \\beta\\,", "c4a6f3966381640a1f28552c6f5427b7": "\\int\\mathrm{coversin}(x) \\,\\mathrm{d}x = x + \\cos{x} + C", "c4a708f57c2f18c32619b963da27b8e4": "\\mathbf{b}\\cdot\\mathbf{\\Delta k}=2\\pi k", "c4a70f433f9a43b9d4c274ed7adb6f87": "g=\\frac{f'}{f}", "c4a71f42bf5cd453ce5a5faf3d3cd7c5": "\\frac{4R\\sin^3{\\frac{\\theta}{2}}}{3(\\theta-\\sin{\\theta})}", "c4a74c95b1cd9f204403705b0aa729cd": "\n\\begin{align}\n\\hat{A}_{m_j} &= \\frac{1}{2q+1}\\sum_{k=-q}^q f\\bigl(\\mathbf X(s_k)\\bigr)\\cos\\left( m_j \\omega_j s_k\\right), m_j \\text{ even}\\\\\n\\hat{B}_{m_j} &= \\frac{1}{2q+1}\\sum_{k=-q}^q f\\bigl(\\mathbf X(s_k)\\bigr)\\sin\\left( m_j \\omega_j s_k\\right), m_j \\text{ odd }\n\\end{align}\n", "c4a74d1a61485e59b77229d385445e44": "\\dot{\\boldsymbol{x}}-\\boldsymbol{v}(t,\\boldsymbol{x})=0 \\qquad\\Leftrightarrow\\qquad \\mathfrak{{G}}\\left(t,\\Phi(t,\\boldsymbol{{x}}_0)\\right)=0", "c4a7510a616b9c8088d5ac5414e8ef86": "\\sum_{x\\in \\Omega} f(x) = 1\\,.", "c4a7988bf116132f1635c0d1f0fc11fc": "Q_c(t) = C \\, v_{\\text{out}}(t)", "c4a7ae1333b8bbd1abdbb8d88a07d685": "\n \\sum_{a=1}^N\\sum_{b=1}^N\\sum_{c=1}^N\\mu_{ab}^{(c)}(t)[Q_a^{(c)}(t) - Q_b^{(c)}(t)] \n", "c4a83ade617f13552f6bf15ec7419206": " \\epsilon_0 \\ ", "c4a8a73c78835473de31a0faaa4dd2da": "f: \\mathbb{R} \\to \\mathbb{R}^n", "c4a8e9d0aba92415dea9ef234d857940": "k_s < \\delta_\\nu\\,", "c4aa40b95ce0de3b6ece2c8650f15cfa": " E = \\frac{ 1 }{ D } / K ", "c4aa4ea6641950930b29c0fe56f133e3": "\\left( \\theta_{*} (\\mu_{\\cdot}) \\right)_{S} = \\mu_{S \\circ \\theta}", "c4aa4ea9d859ba06cb5ae0cac5629a09": "dc^a", "c4aa96a78777885fc3cb5c1067ecbbf0": "\\pi_0 \\ge 2\\pi_1", "c4aaadf2ff2aef7032dd434100429e4b": " \\Gamma_a ~,~ \\Gamma_\\text{chir}~,~ \\Gamma_\\text{chir} \\Gamma_a ~,~ \\Gamma_{a_1 a_2}", "c4aad097e60857fd44d03faea7276d69": "S_n = A", "c4ab0976568c481b78f9d61e63c06b01": "m=m_kp^k+m_{k-1}p^{k-1}+\\cdots +m_1p+m_0,", "c4ab144b21a24c9fc65f03e9d3bf3952": "\\varphi^{-1} = \\varphi - 1", "c4ab21ef6200bf6611a0fb3857e99b95": "G_0(\\gamma)=\\sum_{j\\in Z} |\\hat{\\psi}(a^j\\gamma)|^2", "c4ab81a81395380a686d89a4fda75688": "\\frac{\\operatorname{d}y}{\\operatorname{d}x}=\\frac{\\alpha}{y-x}", "c4ab93229c6e5b76f856b97fb61630c5": "\n\\omega^4 - [2(k^2B^2/\\mu_0\\rho)+\\kappa^2]\\omega^2 +(k^2B^2/\\mu_0\\rho)\n[(k^2B^2/\\mu_0\\rho)+Rd\\Omega^2/dR] =0", "c4abd18389266b30d7597cd80ba79692": "h=(g,g') \\in H", "c4abd224077e341af87605882420fcfb": "\\scriptstyle N \\,=\\, 1000", "c4abe9c1f9a03d32114c8a0ce7c4a134": "x ^ 2\\,", "c4ac730d491a341ac52473edd4417d5a": "\\| \\mathbf {L} \\boldsymbol {\\beta} - \\mathbf {d} \\| \\le \\rho ", "c4ac82c4b5f24bfed1edb6708871b4cf": " c = {|S_2 - S_1| \\over S_2} {f^2 \\over N(S_1 - f)} \\,.", "c4aca4a6fff1c5754adc259417c491a3": "X_1, ..., X_n", "c4aca59a93d12d1da81562e4333b4e01": "\\Delta \\phi = \\phi,", "c4acb22d4b31864afbf077e1cb6fa8b1": "3^n-1", "c4accf77160ec08432ae32b4ff89bedf": "\\! \\mathcal{H}", "c4ace8c48b7d01e7afb5252566e85072": "(p,a,A,q,\\alpha) \\in \\delta", "c4ad2a4ce88e92345d1515fccfd55586": "(I = \\Sigma m r^2)", "c4ad88e6a8be7ed3382bb1a9f24026c7": " \\vec{s} = (s_1, s_2, \\ldots,s_m) ", "c4ad96dead34f7a0a5381b988ec7585e": "q:\\Sigma\\rightarrow \\mathbb{C}", "c4adbfba39b9b1be0c4b3f2a2af41bc0": "\\hat{R} = \\varprojlim R/I^n", "c4ade61e8103719d3a2411f47df00cad": "\\bar\\eta_{\\mu\\nu}^a", "c4adf5dd3a9688ceb1ee74ac13a74af8": "f_i:S \\to \\mathbb{R} ", "c4ae01791f33931fd7cdc02ae3f8ffeb": "\\varepsilon \\ll 1", "c4ae75464778bbef5ed497b9bbccb364": "F^{*n}", "c4ae88fb17d22fd0831cbda10a76fddb": "gd_{F}G", "c4ae97f52136d72a9acf2aa002d17503": "\\Pr(N\\mid n) = \\frac{{\\alpha}n^{\\alpha}}{N^{(1+\\alpha)}}.", "c4aeb28d61043eda883f2b8832b5b252": " s_2=3v", "c4aeb4f0447a37f0940de83d1e23bb94": "D=D_0 \\times 2^{-\\phi}\\,", "c4aebe2854a4a68712eeb5a9c7d26eab": "E(k) = E(\\tfrac{\\pi}{2},k) = E(1;k).", "c4af9ab758d6dd11c05699048bdba740": "\\Delta t_1,\\dots \\Delta t_N", "c4afafc449f93e862b854065125e9c4f": "f(T) = \\sum_{\\lambda_i \\in \\sigma(T)} \\sum_{k = 0}^{\\nu_i -1} \\frac{f^{(k)}}{k!} (T - \\lambda_i)^k e_i (T).", "c4aff224e18bfb95123eb4be66ded54a": "= 10^{\\left ( 1.18\\times 10^{98}\\times\\log_{10}\\left ( \\frac{1.18\\times 10^{98}}{2.718...} \\right ) \\right )} ", "c4b04c3018f91564b4bad00bc686ee14": " \\lim_{N \\to +\\infty}\\frac{K}{N}=p \\! ", "c4b04d849971926445ed8a1e057468eb": "\\left\\langle n_i\\right\\rangle", "c4b060e66d8b14a24095dd4db23c3ca6": "P {-\\!\\!\\ast}\\, Q", "c4b06e5c8f6141d2e43d4c2dbee01b9c": "\\displaystyle{\\kappa=\\ddot{\\mathbf{v}}\\cdot \\mathbf{n}=\\ddot{y}\\dot {x} - \\ddot{x}\\dot{y}.}", "c4b087f285b64332b680dc18957b16f7": "4x^2 + 2xy - 3y^2", "c4b093431b6c9718383da5f7b2352e4b": "(2j_1+1)(2j_2+1)(2j_4+1)(2j_5+1)", "c4b0c0a2adffcc3fecb86fd57c6518f3": "\\frac{J_\\nu(z)}{J_{\\nu-1}(z)}=\\cfrac{z}{2\\nu - \\cfrac{z^2}{2(\\nu+1) - \\cfrac{z^2}{2(\\nu+2) - \\cfrac{z^2}{2(\\nu+3) - {}\\ddots}}}}.", "c4b132f3d56246afb7fdd74dc95e0036": "\\nabla \\times \\mathbf{v} = \\mathbf{0}.", "c4b16fdc30e90cc4eeaa8ecc180e7889": " r(\\tau)= \\sqrt{2GM\\tau} ", "c4b18b08eab4b9312022a0314b34e638": "g_{S}", "c4b19c1dae42625bd3fb67180946cbe5": "R_{\\mu\\nu} - {1 \\over 2} g_{\\mu\\nu}R = \\kappa T_{\\mu\\nu}", "c4b1ee01f3e6bcac296db405c019ad3e": " \\mbox{Course Handicap} = \\frac{ ( \\mbox{Handicap index} \\times \\mbox{Slope Rating} )}{113} ", "c4b278dbd384c0f4765f159b206f326d": "(m_i,m_j)", "c4b2abd115f0a5315171e03fd5d1bbca": "a = (b + c+ f * g) * (d + 3)", "c4b35b15cd53b964d055618b8f15ddae": "y^2 = x^3 + ax^2 + bx + c \\ ", "c4b383d048a4e444ce92f55a84754578": "\\vartheta_L(z) = \\sum_{\\lambda\\in L}e^{\\pi i \\Vert\\lambda\\Vert^2 z} ", "c4b39a57aa05d5228fbefff04c30913a": "r=-\\frac {dC_A}{dt}=k_2 C_{AS}=k_2 \\theta C_S ", "c4b39d3478383176f86cf016a5a88894": "P_c(z)=z^2+c\\,", "c4b3a942424e1447fe4c22ddda02ee31": "P(N)_{t+1}", "c4b42635ecf6379ebcb92a26a7cb3b9c": "\\sqrt{1 - x^2 - y^2}", "c4b439cde15ea0dcba3083bd9c527132": "(2+i)(3+i)=5+5i. \\,", "c4b4871002ef1f8e27d1c5f9f1ab7935": "(x^6+3x^5-8x^4-21x^3+27x^2+38x-41)^2", "c4b4c19ffca962b73e831d2b371c1103": "\\pm 1.96/\\sqrt{n}", "c4b4e3ad679a0739d09e7f855e005ca6": "K_\\theta", "c4b553b356b01e1eba1387c4c9f4300b": "\n I_0(\\lambda) \\approx f(x^0)\\exp[{\\lambda S(x^0)}] \\int\\limits_{\\mathbb{R}^n} \\exp\\left( \\lambda \\sum_{j=1}^n \\frac{\\mu_j}2 w_j^2 \\right) dw = \n f(x^0)e^{\\lambda S(x^0)} \\prod_{j=1}^n \\int_{-\\infty}^{\\infty} e^{\\lambda \\mu_j y^2 /2} dy.\n", "c4b58b7e213d0c50b8c82d5e6cb461a8": "a_n=n2^n-1\\, .", "c4b59e58179e449be559483c99946aa7": "\\mbox{FileSize} \\approx 54 + \\mbox{PixelArraySize}", "c4b5e9c659a8237a242d96d31c7c8b02": "A,D,E,G", "c4b6316f5246eb7d7a1a663ba43382f8": "\\ddot{y}(t)", "c4b6ca538afce2d750606eb9e7257779": "O(n^{d/2} \\log n)", "c4b6d0044ad509cff6d3bc4be7a6bc48": "\\mathcal O_x", "c4b7a23c4c9e70a3fcdadc1bd8b40437": "\\mathrm{\\tfrac{u\\bar{u}-d\\bar{d}}{\\sqrt 2}}\\,", "c4b7b25d44d162a32723b1aec14a7784": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\nabla^2 + V_0 ", "c4b81946fdebbc10ee365e2c5362c8a6": "\\int_{\\mathbf{R}^d} f(x)d\\lambda(x) = \\int_{\\mathbf{R}^d}\\int_{\\mathbf{R}^d}f(x+y)\\,d\\mu(x)\\,d\\nu(y).", "c4b8238b7d80c18aa15b9d81838929e2": "{\\color{Blue}~5.9}", "c4b85277d845d7becfe0ef1c37ca0dec": "(k\\bmod n)+1", "c4b86f1d7cd3c372caad9287d8148ab4": "\\mathop{\\varphi\\mbox{-Var}}_{[0, T]} (f) := \\sup \\sum_{j = 0}^{k} \\varphi \\left( | f(t_{j + 1}) - f(t_{j}) |_{X} \\right),", "c4b8d24f39c36e1da2638ca5e63f5033": " t", "c4b8eab3f78e4a6fb1ccab65a4cff4fa": "\\mathbf{\\hat{n}} ", "c4b93b11969b0ae857ffbd2a43f2f28b": "S(\\rho_{AB} || \\rho_{A} \\otimes I_{B}/n_B) = \\mathrm{log}(n_B) + S(\\rho_{A}) - S(\\rho_{AB}) = \\mathrm{log}(n_B)- S(B|A), ", "c4b99157ee7facde1bee84fdda8117d8": " d\\sigma^3 = \\sin\\theta \\, dr \\wedge d\\phi + r \\, \\cos\\theta \\, d\\theta \\wedge d\\phi = -\\left( \\frac{\\sin\\theta\\, d\\phi}{g(r)} \\wedge \\sigma^1 + \\cos\\theta \\, d\\phi \\wedge \\sigma^2\\right)", "c4b9bcd1e1551817d874c42992939b96": " C := \\{b \\in B : \\exists\\, p(x) \\in A[x]\\,, \\hbox{ which is monic and such that } p(b) = 0\\}\\,.", "c4ba5b29a3dc5b0a4d347691a7220768": "Pr(c_{A} = c_{B} = 1) \\leq \\frac{1}{2} + \\epsilon", "c4bade1cab1f9eed1b15b88d6fcd75e4": "d\\Phi_3 = \\frac{1}{(2\\pi)^5} \\delta^4(P - p_1 - p_2 - p_3) \\frac{d^3 \\vec{p}_1}{2 E_1} \\frac{d^3 \\vec{p}_2}{2 E_2} \\frac{d^3 \\vec{p}_3}{2 E_3} \\,", "c4badfba8d71795ae9c92be7c7d22608": "\\omega_m / (2\\pi).\\,", "c4baff3903854d934dca5158b7ebcb6c": "\\Omega_{h}", "c4bb3e4ae287c15455409907ff767420": "C_i\\cap C_j\\ne\\emptyset", "c4bba5237fdd109140b50060fbf9093c": " E_{\\textrm r}=\\frac{Z_2}{Z_1+Z_2}\\,E_{\\textrm{out}}=\\beta\\,E_{\\textrm{out}}, ", "c4bc1550ad4a7443e80bd1b35af14273": "\n \\nabla\\varphi = \\sum_{i} {\\partial\\varphi \\over \\partial q^i}~ \\mathbf{b}^i = \\sum_{i} \\sum_j {\\partial\\varphi \\over \\partial q^i}~ g^{ij}~\\mathbf{b}_j = \\sum_i \\cfrac{1}{h_i^2}~{\\partial f \\over \\partial q^i}~\\mathbf{b}_i ~;~~\n \\nabla\\mathbf{v} = \\sum_i \\cfrac{1}{h_i^2}~{\\partial \\mathbf{v} \\over \\partial q^i}\\otimes\\mathbf{b}_i\n ", "c4bc3d3ae353cde4914217b59e9a405d": " \\ r ", "c4bc546ede921d7b896f47d2ab065248": "z=c\\;T\\;\\log(P_o/P),", "c4bc88c0804d2c1ac4cdc728e0d1f189": "n \\gg 1", "c4bc972a530dc6eb3d02e32f6c49e84c": "n=\\tfrac{1}{2}\\sqrt{a^2-b^2+c^2-d^2+p^2+q^2}.", "c4bcb8ee3f28d0fe745e258e8f0beec5": "\\{L,R,N\\}", "c4bcf4f721cb7ad0fe3d978634b18159": " f\\left(a\\right) = f\\left(b\\right) = 0 ", "c4bd93c6b62c13f4175a7a40b85f02ee": "\\displaystyle \\langle a , b \\rangle = \\sum_{\\mu\\nu} \\eta^{\\mu\\nu} a_{\\mu} b^\\dagger_\\nu", "c4bdcee2f3db78893406639de192c5a4": "50% \\times 10% = 5%,", "c4bde98eb5193337bea2fc242f292087": "\\Delta U = -\\Delta W ", "c4bdfddd64af296108ca5fa48f36c7bb": "\\frac{P(x)}{Q(x)} = \\frac{c_1}{x-\\alpha_1} + \\frac{c_2}{x-\\alpha_2} + \\cdots + \\frac{c_n}{x-\\alpha_n}", "c4be6d33884d5694565d528a3a03835b": " -\\frac{\\partial C_v}{\\partial K} = -\\frac{\\partial (S\\Phi(d_1) - Ke^{-rT}\\Phi(d_2))}{\\partial K} = e^{-rT}\\Phi(d_2) = C_{noskew}", "c4be83308b07e78a28ea86a8269b5033": "\\,\\mathcal{L}\\,", "c4beacb99b8d3c67feb06207f276499b": "simil(x,y) = cos(\\vec x,\\vec y) = \\frac{\\vec x \\cdot \\vec y}{||\\vec x||_2 \\times ||\\vec y||_2} = \\frac{\\sum\\limits_{i \\in I_{xy}}r_{x,i}r_{y,i}}{\\sqrt{\\sum\\limits_{i \\in I_{x}}r_{x,i}^2}\\sqrt{\\sum\\limits_{i \\in I_{y}}r_{y,i}^2}}", "c4beae65afce9ca2601b8a06c5089429": "\\bar{G} = H", "c4beb4deb46a0d05b0d399b8e60e57cd": "| \\cdot | = \\sqrt{Q(\\cdot)} ", "c4bf4bc2f86d84369af759df5904bbbf": "A(t) = e^{W \\cdot t}", "c4bf8725a6ecea99db1777e162ccdfa9": "a \\triangleright c = b", "c4bffb4b63779bb901332f8a41b2cf4c": "\\{0\\}\\times[0,1]\\cup[0,1]\\times\\{0\\}", "c4c1a22a38c4f9d9d4ae4ca788c722ee": "P=(ar^{\\frac{n}{2}})^{n+1}", "c4c216ba6b3d6cea4277b2343d93da12": " \\boldsymbol{s}\\in\\boldsymbol{S}_{\\boldsymbol{\\psi}} ", "c4c22d0d0b08ecf7e913c6be43350aaf": "S_x(f) = \\frac{1}{(2\\pi)^2f}h_1", "c4c246ffd5748687a415b45382a3e167": "(II) \\quad \\forall \\mathcal{R}' : s_{V_1}(\\mathcal{R}) > s_{V_1}(\\mathcal{R}') ", "c4c25726040ccd0e2be9285c2b0ce985": " U(R)^\\dagger \\widehat{T}_{pq} U(R) = R_{pi} R_{qj} \\widehat{T}_{ij} = R_{pi} \\widehat{a}_i R_{qj} \\widehat{b}_j ", "c4c3457cda871031dadd4cd08585c8f2": "B \\cdot \\cos(\\theta)=\\frac{k \\lambda}{D}+ \\eta \\cdot \\sin(\\theta),", "c4c345b5651d875db910d95738923b92": "a\\in \\Sigma", "c4c3e834a797fe6f7edd18df047beb2e": "\\vec v_0\\times\\vec B=-\\vec E_\\perp", "c4c417553b680cf203765de254be0350": "\\alpha = 0", "c4c45001ca94eb75b94301c99e6c1414": "y = (q - k)t + k.\\,", "c4c45815d29b9f292605ebeac81028de": "j = \\sqrt{-1} \\ ", "c4c4d1848cad364deb8ce5bfa2269f50": "t=\\tan(\\theta/3)", "c4c4dc574a072c147a83ce3c9810caf9": "\\phi_n(r)", "c4c4e2d8bd8dcc806a35c14f5b591ac0": "\\mathbf{1}_{A\\cup B}(x) = \\max\\{{\\mathbf{1}_A(x),\\mathbf{1}_B(x)}\\}. ", "c4c52167f0dadcb4575d761488822ed6": "{x}^i", "c4c52d6f69a65fe65459392f52c45085": "\\frac f g \\colon \\{ x \\in I| g(x) \\neq 0 \\} \\rightarrow \\mathbf R, x \\mapsto \\frac{f(x)}{g(x)}", "c4c57f19dd8538e41ddfd293b07c0747": "\\alpha_\\infty", "c4c5848d4d9460c15e6e76c18cfafbbe": "e \\cdot \\sin \\theta = \\frac{V_r} {V_0}", "c4c5b920f0958b2b529cbdaa6ec6fcb3": "X\\subseteq\\{0,1\\}^\\infty", "c4c5bf5fade84cc5223e38231a40d8b2": " t(d,n) \\leq \\mathcal{O}(d^2\\log^2{n}) ", "c4c5d0703527e66f88852e6c68490d31": "\\mathbf{J}_A = -D_{AB} \\nabla c_A", "c4c6e2f4f4aa5b26162307f2332bf5f4": "\\displaystyle{PU(g)P=\\Phi(g),}", "c4c71525c65a97bdea0fb4765323a897": "\\Omega_*^{\\text{SO}}", "c4c71ca42acf5500a41b61e45ee95493": "0.728^{+0.015}_{-0.016}", "c4c729c623bf0d11cdf09fc901877746": "Z({\\Bbb A}^n,t)=\\frac{1}{1-{\\Bbb L}^n t}", "c4c732d43454cb08d84f64fc45e94f87": "T = \\rho_{x_1/c_1,\\ldots,x_m/c_m}(S) = \\rho_{x_1/c_1}(\\rho_{x_2/c_2}(\\ldots\\rho_{x_m/c_m}(S)\\ldots))", "c4c7a70953d14dce5860babfcb02c18f": "R \\subseteq X \\times Y", "c4c7e1417961a279448b96563415dd70": "\\vec{r}_{i}", "c4c858375aa670bca75eb1622efe2069": "m \\frac{d^2(x)}{dt^2} = -kx", "c4c87e99197b1afb9434ad19fc9072db": "X=1/D", "c4c89c1788390d87f34b25563a508e23": "\n\\mathrm P(\\mathit{R}=T \\mid \\mathit{G}=T)\n\n=\\frac{\n \\mathrm P(\\mathit{G}=T,\\mathit{R}=T)\n}\n{\n \\mathrm P(\\mathit{G}=T)\n}\n\n=\\frac{\n \\sum_{\\mathit{S} \\in \\{T, F\\}}\\mathrm P(\\mathit{G}=T,\\mathit{S},\\mathit{R}=T)\n}\n{\n \\sum_{\\mathit{S}, \\mathit{R} \\in \\{T, F\\}} \\mathrm P(\\mathit{G}=T,\\mathit{S},\\mathit{R})\n}\n", "c4c8a773c7bb5f098911bff65fe5a8c5": "d_\\varrho\\,", "c4c8fd0a605287c1bbf3cf2e05f6328e": "\\Delta V_{BE}=\\frac{KT}{q}\\cdot\\ln\\left(N\\right) \\,", "c4c907a1a4bcfbbcb2dac834c7066c16": "\\bold p_1", "c4c92875ca95d6e4ff0c9b5f4184648c": "\\gamma_{\\mu}", "c4c938e15ad5cd32f109cf4dd2f63937": " \\mathit{k} ", "c4c954cedc4158c4b5d52e9d5563928a": " \\bold{p} = \\nabla S ", "c4c9a15065198cde2f3e4d233bb3ddd9": "Population_{t+1} = Population_t + Natural increase_t + Net migration_t", "c4c9a594f6ce7ebba51ac265f72ce9bc": "N_p:\\mathbb R^n \\to \\mathbb R", "c4ca1a16f598e041bdb3ca40c1b8c844": "\\delta_{ext}:Q \\times X \\rightarrow S \\times \\{0, 1\\} ", "c4ca4238a0b923820dcc509a6f75849b": "1", "c4ca85e9d75cb61719540a27fd44f20f": "\\left[X_a,X_b\\right] = i f_{abc}X_c", "c4ca8f91adda75548399bca88e223a66": "D_{\\mathrm F} \\approx \\frac {H s} {H - s} \\text{ for } s < H \\,.", "c4cae6f79e0384e2f92e7cf78719c8c8": "\\xi(x(t)) = \\xi(x(0)) - \\int_0^s \\nabla_x H(x(s),\\xi(x(s))).", "c4cb44892c5d7b381a5c68310915036f": "\\scriptstyle{}\\int\\sec^3 x\\,dx", "c4cb520a9aa6cfab61c6340bac8262ca": "Q = (\\text{Kinetic energy})_{\\text{after}} - (\\text{Kinetic energy})_{\\text{before}}\\,\\!", "c4cb81bd48ad24b5eb5b480305a218d2": "v_1,v_2,v_3,\\ldots ,v_M", "c4cbc6d6c213a7d7f7ddf744ed4b044b": " A(x) = \\frac {16V}{3L\\pi}[4x(1-x)]^{3/2} = \\pi R_{max}^2[4x(1-x)]^{3/2} ", "c4cc1a82b7d298618461f7d95fa5c618": "(s,a,c,t)", "c4cc3e1ead3b330210e6e403f57576cf": "\\mathbb{D}^q f(x) = \\lim_{h \\to 0}\\frac{\\Delta^q_h f(x)}{h^q}.", "c4cc8403b852b3cf2f5673d5becdba49": "= 4", "c4ccace9f27de08d62cea525d831207a": "\\int_0^{\\theta}\\log(\\cos x)\\,dx=\\tfrac{1}{2}\\text{Cl}_2(\\pi-2\\theta)-\\theta\\log 2", "c4cd1566a7d90cc40477f4b74ce9240a": "X_1,X_2,\\ldots,X_n", "c4cd196d139b46b7fe73e442c86f2099": "|\\mathbf{x} \\times \\mathbf{y}|^2 = \\begin{vmatrix} \\mathbf {x \\cdot x} & \\mathbf {x \\cdot y}\\\\\n \\mathbf {y \\cdot x} & \\mathbf {y \\cdot y}\\\\ \\end{vmatrix} = |\\mathbf{x}|^2 |\\mathbf{y}|^2 - (\\mathbf{x} \\cdot \\mathbf{y})^2 ,", "c4cdf245f17f37ae2b7c481a3413abde": "F_N \\otimes F_N\\otimes F_N \\in H(N,N^3).", "c4cdfd8bb24571cd540f46a86875965c": "\nF \\left( x - x_{0} \\right) - E \\left( y - y_{0} \\right) = 0\n", "c4ce0a9c58802cf2bf088d120395b903": "\\operatorname{E}(z,k)", "c4ce24f160c7220e9aad0b2a5f33518a": "O( n^2 / k^3 + n/k).", "c4ce320be8dfc94741c3c29a5b95b366": "{dx_S\\over u} = {dy_S\\over v} = {dz_S\\over w},", "c4ce585b969801e9007aa6d170775eff": "x \\div 4", "c4ce6bff989bdd961ab48bed8ff3c497": "(N - 1)/q", "c4ce8bd355417c63dcef245d4073264b": "\\alpha = |\\alpha|e^{i\\theta}~~~,", "c4ced77c7c44c422617391776eadbd1f": "V_{nom-load}", "c4cee5501df2ec9478a54c2c6bb58b31": "\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "c4cf1c6e45142caa5913dec5f4729582": "(p_{r+1},p_s)=(xp_r,p_s)-a_{r,s}(p_s,p_s)=(xp_r,p_s)-(xp_r,p_s)=0.", "c4cf4d5aaa256d589199aef1aaf916f0": "\\langle a, b \\mid b a^m b^{-1} = a^n \\rangle.", "c4cf754403099ee1aed9ae5edafaaff5": " D = \\frac{\\left | a x_1 + b y_1 + c z_1+d \\right |}{\\sqrt{a^2+b^2+c^2}}. ", "c4cf75a4a1654bf82a20d3967cd17075": " L = I_1 \\supseteq I_2 \\supseteq I_3 \\cdots \\supseteq I_n = \\{0\\}", "c4cf77f5167c6e3ec411d88187c9e27d": "h = \\frac{\\Delta{}f}{f_m} = \\frac{f_\\Delta |x_m(t)|}{f_m} \\ ", "c4cfc9381558708b5368cf727d4c5ce1": "F_d=\\frac{1}{2}\\ \\rho\\ C_d A\\ v^2.", "c4cfd89dad4555fcd5b85637f02557f0": "m = 3", "c4cfdf26a466927c21f72ffa98380579": "\\text{Gal}(L/\\mathbf{Q}_p)", "c4cffd9704dc130aad00176c59b13b46": "[U_i]_m=[V_{\\sigma(i)}]_m", "c4d0466475c858f4a3f096eed8d4c865": "c = m^2 + n^2, \\, ", "c4d06d6af01251700ad69f7199f28a45": "(23)\\quad ds^2=-\\frac{L^2}{(L+M)^2}dt^2+\\frac{(L+M)^2}{L^2}(d\\rho^2+dz^2+\\rho^2d\\phi^2)\\,,", "c4d0b647d3fbb5d45c9df2239aed34fe": "a^{15} = x \\times (x \\times [x \\times x^2]^2)^2 \\!", "c4d0ba3edc675ca8decb8ea55ca9ff3a": "\\|Tx_n - \\lambda x_n\\| = \\sqrt{\\frac{2}{n}} \\to 0.", "c4d13bbd41fae3e937dc55b4d6dd97e4": "{{}\\over{}} (E_{1,0}-E)^2 - d^2 \\epsilon^2 = 0 ", "c4d2260f3a40d28224b524a50bf94160": "f_{i-1}(\\Delta(P))=\\sum_{|S|=i}\\alpha_P(S), \\quad\nh_{i}(\\Delta(P))=\\sum_{|S|=i}\\beta_P(S). ", "c4d28f3cec44bd25dfea37495ed030d4": "\\lambda \\ll T/|\\nabla T|", "c4d29383564e50c53a2c55965325f5c6": "\\frac{1}{\\det(1-T^{deg(x)}F_x|E^k)} =\\exp\\left(\\sum_{n>0}\\frac{T^n}{n}\\text{Trace}(F_x^n|E)^k\\right) ", "c4d2a583182c7d09fa296718119fb4d0": "\\operatorname{perm}(A) = \\left[\\sum_\\delta \\left(\\prod_{k=1}^m \\delta_k\\right) \\prod_{j=1}^m \\sum_{i=1}^m \\delta_i a_{ij}\\right] / 2^{m-1},", "c4d2cda182c7a33182f7998489463d67": " \\Box p \\rightarrow \\Box \\Box p", "c4d30396142d2edc7273e0ffaf8d6832": "R=c\\sqrt{\\frac{ab+c^2}{4c^2-(a-b)^2}}.", "c4d3402a0af45053be19349cc1eb46e9": "\\beta = \\frac{p}{p_{mag}} = \\frac{n k_B T}{(B^2/2\\mu_0)}", "c4d368260b8b7881ba1a8348b50a5b22": " T(v) = \\lambda v ", "c4d390015123d2bc5df7596ff523cd75": "f(x) = \\begin{cases} 0, & \\mbox{if }x =0 \\\\ x^2 \\sin(1/x), & \\mbox{if } x \\neq 0 \\end{cases} ", "c4d4532bbe264b647fa700695e0442c8": "(R_1,G_1,B_1)", "c4d4544d4486683f4bbd731f3151892c": "\\frac{d W_i(t)}{d t}=\\frac{1}{\\tau([Ca^{2+}]_i)}\\left(\\Omega([Ca^{2+}]_i)-W_i\\right),", "c4d55fc973ba1923be79a22a540f944d": " S \\,", "c4d5b4883f763d1fc091b3393e5c56cd": "p_\\mathrm{r} = m \\dot{r}", "c4d60b68415337d40edfff637f2f2d50": "\\hat{t} = \\operatorname{argmaxminlocal}_{t} \\nabla^2_{norm}L(\\hat{x}, \\hat{y}; t)", "c4d67cc3fffc854ba23eaba1359e6834": "\\sqrt{-x} = i \\sqrt x.", "c4d6808103e5357d20f5b0a5642db7ba": "h\\, = r\\, v\\, \\cos \\phi", "c4d6d77ce04700d61fea8b706a4b11a9": "R_f \\parallel R_1", "c4d6f60c8377eb1f3ca069652defe677": " \\frac{\\partial \\psi}{\\partial t}=ie^{i \\sigma_z \\omega_r t/2}\\left(\\omega_1\\sigma_x \\cos{\\omega_r t} + \\omega_1\\sigma_y \\sin{\\omega_r t} + \\left(\\omega_0+\\frac{\\omega_r}{2}\\right) \\sigma_z\\right)e^{-i \\sigma_z \\omega_r t/2}\\psi", "c4d6f8070435f661dc2e8d7025092217": "n \\ne 0", "c4d728e5f19e0df8754a84b98e7a7821": "\\sum_{i=1}^n n_i=0", "c4d74a64d4551ab197dc06be8dab9a78": "W^*=25Y^{1/3}-17", "c4d7660e7956a37a9815e379ea85fba8": "\\underline{X} : U \\to \\R^n", "c4d77878531159aae83d2d41e2d6e2d2": "\\text{number of octaves} = \\log_2\\left(\\frac{13}{4}\\right) = 1.7", "c4d807fc9e61cf567bdd478ed95f788a": "\\delta[x]", "c4d86163cb807b753f4d4ea0b3aa69af": " v(r) ", "c4d87466cf0d63d88061db0e348d557b": "j \\in n_k", "c4d96cc13420e39f92e76d241ad9c90c": "\\begin{array}{cc}\n \\begin{array}{rr} \\\\ &3 \\\\ \\text{-}1& \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & \\text{-}12 & 0 & \\text{-}42 \\\\\n & & & \\\\\n & & & \\\\\n \\hline \n \\end{array}\n\\end{array}", "c4d98929851ad641e3e3d4ba2a63c6ce": "K(r) = \\langle (X_n-\\bar{X})(X_{n+r}-\\bar{X}) \\rangle = \\langle X_n X_{n+r} \\rangle -{\\bar{X}}^2,", "c4d9ce1e5e7fa6640143cef7d33b13b4": "\\scriptstyle{ [T,S] = TS - ST}", "c4d9e743a6ee0cd98ea9092dbd9d007c": "a^1 = \\frac{a_1 - \\sin(\\phi) a_3}{\\cos(\\phi)^2},\\,", "c4d9f807c8d221aaa7b15147c2456127": "\\varliminf_{n\\to\\infty}x_n:=\\liminf_{n\\to\\infty}x_n", "c4da1dc5578624d8c92a31dcd411bacd": "\\tau \\frac{\\mathrm{d}M}{\\mathrm{d}t} + M = \\tau \\chi_b \\frac{\\mathrm{d}H}{\\mathrm{d}t} + \\chi_{sp} H", "c4da4469511566fc518f2345dc04553d": "\\Delta t \\leq 2/\\omega", "c4da60f83554a7b1e4d065c3fe10bd39": " \\mbox{ad}_A(X) = AX-XA", "c4da86df442a02810a327b82428b8039": " \\hat H = \\sum_j \\vec{S}_j \\cdot \\vec{S}_{j+1} + \\frac{1}{3} (\\vec{S}_j \\cdot \\vec{S}_{j+1})^2 ", "c4da97ae4bdb7f7f0d42bb9257d528a3": "({\\mathbb {S}^1}\\times {\\mathbb {R}^1},{\\mathrm {pr}_1},{\\mathbb {S}^1})\\, ,", "c4da9a79ba148a2e9a7bcd92255680be": "\\theta \\in \\Theta", "c4daaf50f63c4a657a4cfa9003de7ec0": "\\frac{n_A+n_B+n_C+n_D}{\\frac{n_A}{f_A}+\\frac{n_B}{f_B}+\\frac{n_C}{f_C}+\\frac{n_D}{f_D}}", "c4daeeb1011077ca98173fcafb3c775f": "PG = P_0 \\cdot P_1 \\cdot P_2 \\cdot P_3", "c4daf08975dfc451796797c2c4ef6620": "T_{j',j} = t_{j'}\\dots t_j", "c4db26d1d228541f6a906b289057edcb": "\\textstyle s^\\alpha+t^\\alpha=1", "c4dbc5a11e36d5b201d3536e3251960b": "\\textstyle\\lim_{h\\to 0} Q(h)", "c4dbded35cb35bacc7f3f3d36d8f9e39": "(\\;2)\\quad \\quad\\frac{\\partial\\rho u}{\\partial t} \\, = -\\frac{\\partial}{\\partial x}\\left(\\rho u^{2}+p\\right)", "c4dc184f853e5ecc3271c655fd89d098": "\\hat{H} = -\\bold{\\hat{d}}\\cdot\\bold{E} = -q\\bold{E}\\cdot\\bold{\\hat{r}}", "c4dc27f5828e55d5c97dea78544e299f": "m = \\rho_\\mathrm{new} (V - A \\Delta x)\\,", "c4dc707c935e7df64e1978c9edec54d3": " f_\\text{square}(t) = \\frac{4}{\\pi} \\sum_{k=1}^\\infty {\\sin \\left ( (2k-1)t \\right ) \\over 2k-1}.", "c4dc8c72cb9adc5e93a7ea85983547c9": "\\alpha<\\beta<\\gamma\\implies M_\\alpha\\prec_K M_\\beta", "c4dc972926b2ef717c09c295e3e5148b": "\\mathrm{Res}_0 {e^z \\over z^5},\\ \\mathrm{or}\\ \\mathrm{Res}_{z=0} {e^z \\over z^5},\\ \\mathrm{or}\\ \\mathrm{Res}(f,0).", "c4dcc233497744d71f27dcb825c58e33": "dC = (0.439 \\cdot 0.5) + \\left(0.0631 \\cdot \\frac{0.5^2}{2} \\right) + (9.6 \\cdot -0.015) + (-0.022 \\cdot 1) = 0.0614", "c4dccbb79aeda777b6717e83c1de6b5e": "\n\\mu^-(E) = -\\inf_{B\\in\\Sigma, B\\subset E} \\mu(B) \n", "c4dd19bce4ae70ff87572bac1c473ca7": "E_{\\mathrm B}", "c4dd2bb69de81939f9f087c5c6cd7e5c": "\\sum_{\\sigma,\\tau\\in S_q}\\delta_{i_1i^\\prime_{\\sigma1}}\\cdots\\delta_{i_qi^\\prime_{\\sigma q}} \\delta_{j_1j^\\prime_{\\tau1}}\\cdots\\delta_{j_qj^\\prime_{\\tau q}}Wg(d,\\sigma\\tau^{-1})", "c4dd3c976de6440b86459fe9af7a6b88": "B \\cdot h", "c4dda46d95f15844e41340a1970eed2e": "1+2+4+\\ldots+2^{n-1}", "c4ddf85f7584fa508bea17c6eac3bfe7": "p \\approx 0.03", "c4de14c0ad30ad0cb113784ec8d24c3c": "\\frac{23}{16}=1+\\cfrac1{2+\\cfrac1{3+\\frac12}} = [1;2,3,2],", "c4de278d6eb6bf4ff6a35accd4ceac17": "G^{(n)} := [G^{(n-1)},G^{(n-1)}] \\quad n \\in \\mathbf{N}", "c4de758ab30f6fc78d02aeae611b7df9": "f' = f ( c-v ) = f \\left( 1-\\frac{v}{c} \\right)", "c4de7c2733bc399696d78d20b4bc9c8e": " \\left| \\lang \\omega | s \\rang \\right|^2 = 1/N", "c4debdd8bc4f5df532a507540c0f495c": "\\Box \\neg F", "c4debe8b914e6228869a134e7005c400": "c_\\mathrm{d} = \\dfrac{2 F_\\mathrm{d}}{\\rho v^2 A}\\, ,", "c4ded42dd82259245a768b30ef1b0062": " Z^{}_{}=XYX^{-1}Y^{-1},\\ XZ=ZX,\\ YZ=ZY ", "c4deeabd320cb823af0e550131888419": "\\boldsymbol\\beta", "c4def73028270daef113d41f9c59d7a3": "10^{48}", "c4df09747459bcbf3526f7550f86d07d": "end\\!\\,", "c4df64192abf572184192becbb24e3fe": " \\lim_{\\epsilon\\rightarrow 0}\\int_\\epsilon^{\\infty} \\frac{\\sin(x)}{x} dx = \\int_0^{\\infty} \\frac{\\sin(x)}{x} dx = \\frac{\\pi}{2} ", "c4dfad3a738ca9c361ce84d7f3625196": "F_2F_1", "c4e024e71ba191d41926d9fcfa7cde3a": "Input:", "c4e02ee9e37edbeba660579e9ce86d0f": "\\textstyle \\Omega_2 \\setminus A_2 ", "c4e04b8d144c266ca567f4f4ece18fc0": "G/G_x \\to M, \\, [g] \\mapsto g \\cdot x", "c4e08cb9c97eba4d0d051b1ed177692b": "f(I) =\n\\begin{cases} \n 0, & I \\le I_\\mathrm{th} \\\\\n {[} t_\\mathrm{ref}-R_\\mathrm{m} C_\\mathrm{m} \\log(1-\\tfrac{V_\\mathrm{th}}{I R_\\mathrm{m}}) {]}^{-1}, & I > I_\\mathrm{th} \n\\end{cases} ", "c4e0cac96b61c9a136fdbacf196aa3c2": "\n \\mathrm{J}_\\pm = \\mathrm{j}_\\pm \\otimes 1 + 1 \\otimes \\mathrm{j}_\\pm\n", "c4e1292b8110f0a0eb1d67b570fceac1": "D=\\left\\{f \\in C([0,1]) : f \\text{ is not differentiable at any point of } [0,1] \\right\\}", "c4e21b711ce2e0e61c17befab5887b3f": "\\mathsf{KK}", "c4e2320842dad852d527432e8c71825d": "\\operatorname{Ext}^n_R \\left (A,\\prod_\\beta B_\\beta \\right )\\cong\\prod_\\beta\\operatorname{Ext}^n_R(A,B_\\beta)", "c4e237a21e5b37095e5f8c4d6ecee7f0": "n=2\\alpha/\\pi", "c4e24db9940a56f6f574f451ebe07b00": "\\mathrm{d}G = \\sum_{\\alpha,\\beta,S}\\,\\left( \\mathrm{d}U +p\\mathrm{d}V\\,+ V\\mathrm{d}p\\,-T\\mathrm{d}S\\,-S\\mathrm{d}T\\,+\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}n_i\\,+ \\sum_{i=1}^k \\mathrm{n}_i \\,\\mathrm{d}\\mu_i\\,\\right) +A\\mathrm{d}\\gamma\\, + \\gamma\\mathrm{d}A\\,,", "c4e283f9e7086b49f692076d3bd2acbf": " B_{n}=\\sum_{k=0}^{n}(-1)^{k}\\frac{W_{n,k}}{k+1}\\ =\\ \\sum_{k=0}^{n}\\frac{1}{k+1}\\sum_{v=0}^{k}(-1)^v \\left(v+1\\right)^{n} {k \\choose v}\\ . ", "c4e2950110bbabc0286591d160fdf880": "\\partial a/\\partial b)_c = 1/(\\partial\nb/\\partial a)_c", "c4e32aec4c5c5a0621636c4c64f248a7": " \\frac{B_{k(p-1)+b}}{k(p-1)+b}\\ \\equiv \\ \\frac{B_{b}}{b} \\pmod{p}. ", "c4e37b29c87e6a9601443b4854a0e3e0": "n \\equiv i \\bmod s", "c4e387333bfe2684e093bdc91e6777f0": "\\Sigma'\\,\\!", "c4e3e4f2140463ab8abd3e82c2bb486d": " L = -g^{mn} \\, \\psi_{;m} \\, \\psi_{;n} ", "c4e3f0d1b0da554aff22f514dff41b6c": "\\sum_{n\\in\\Z} a_n a_{n+2m}=2\\delta_{m,0}", "c4e443f67a323dab6605ea7b813b0e66": "B(a):\\mathcal{U}", "c4e459af1282afd3fd1ba6fba752c903": "a = (c+d)/2", "c4e513109b6382692ec1330ba9a6a1b5": "\n \\bar{I}_1 = J^{-2/3}~I_1 = I_3^{-1/3}~I_1 ~;~~\n \\bar{I}_2 = J^{-4/3}~I_2 = I_3^{-2/3}~I_2 ~;~~ J = I_3^{1/2} ~.\n ", "c4e519b3b7ab4c890a70216974b758f9": "\\nabla \\cdot (u,v) = u_x + v_y", "c4e582a91565b2e614bec4c172415cb1": "h(k), h(k)+1, h(k)+3, h(k)+6, ...", "c4e60283f853d41c675e0d12cfc1c37e": "\\textstyle{\\frac {1} {1+\\varphi}}", "c4e6164322c9d1a556448b071007b623": "\\Gamma_i = \\frac{{n_i}^{\\text{TOTAL}} - {n_i}^{\\alpha}\\, - {n_i}^{\\beta}\\,}{A}\\,,", "c4e6d47004ef15cf7bd2c4d17b5a88b1": "\ns = \\left(\\frac{1}{1}\\frac{y \\cdot r}{x} + \\frac{1}{5}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}+\\quad \\cdots\\right) - \\left(\\frac{1}{3}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2} + \\frac{1}{7}\\frac{y \\cdot r}{x}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}\\cdot\\frac{y^2}{x^2}+\\quad \\cdots\\right)\n", "c4e71bd1bdff8b0712e50e365a6509b2": "\\operatorname{NWScore}(X,Y)", "c4e73dc4684d35f3e34a4dee083c5400": "\\rho_3", "c4e74bf72cfa624902c8532a49e1720c": "\\ \\|x\\|_\\infty=\\sup(|x_1|, |x_2|, \\dotsc, |x_n|,|x_{n+1}|, \\dotsc)", "c4e76048fd7430fb0493c48c7f026e3b": " \\psi(y) \\geq \\psi(-x) = - \\psi(x), ", "c4e7e43676b6db3067e1408c47fcbb8f": "U(z)", "c4e859a51d148a70f8ec57535c6dbe00": "\\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}+u=\\frac{k u}{m h^2} = C u.", "c4e873cf0262d4574c58ad1174418140": "\\ w_r = \\frac{l_e}{l_{us}} w_u ", "c4e887a2c9fec0601fec473e456f9574": "P_{sf}", "c4e8d9f1d39fd78ba94680c4a2cf70a5": "\n\\pi^* = \\underset{\\pi}{\\mbox{argmax}} J^\\pi(b_0)\n", "c4e94d5bed62b6ca4350ea595a694f51": " \\sum_{n\\in A} \\frac{1}{n} = \\infty ", "c4e9a40c24a9c75304c139f1f2aebaa7": " \\mathbf{P}(X \\ge a) \\leq \\min_{t>0} \\frac{\\prod_i E[e^{tX_i}]}{e^{ta}}.", "c4e9ca5c895faa198b7e85a0c8e31977": "a \\ne b", "c4ea03533fa68ae204d6c6c9c31b9033": "\\operatorname{H}^*(\\mathbb{R}P^n; \\mathbb{F}_2) = \\mathbb{F}_2[\\alpha]/(\\alpha^{n+1})", "c4ea4a25da91476fd539432947e82800": "\\mathrm{sinc}\\,\\alpha", "c4eac5f88979b333cd95aa877c909e88": "\n\\xi=\\frac{1}{2}a+\\frac{1}{4} .\n", "c4eb1ddfb0025bb8b93404dcf217c1b6": "[\\mathbf u]_{\\times}", "c4eb6d5149bc10c3e68e49a54e099a69": "P=O", "c4ebcf87964b0ead773032218e4b1ccd": " \\mathcal F^\\mathrm{an} ", "c4ebd39036c8a17532cbf9a3d6abda5d": "\\ln c, \\lg d = \\log e, \\log_{10} f \\!", "c4ec37eabd5584165837e4972cf6f6a1": "\n\\Delta t < \\frac{\\sqrt{2}}{\\omega} \\approx 0.225p\n", "c4ec3a40a029fcac09c3959f6be764ed": "\\delta( q, \\sigma, d)", "c4ec6e441f5ef60354c84d4faa656e90": "\\boldsymbol{u} = w\\hat{\\boldsymbol{z}} = w_0 (1-r^2/a^2) \\hat{\\boldsymbol{z}}", "c4ec9fa8bbc69cb60cac512c245defed": " u\\cdot \\psi = \\left\\{\\begin{matrix}\n\\psi&\\hbox{if } \\psi\\in \\wedge^{\\mathrm{even}} W\\\\\n-\\psi&\\hbox{if } \\psi\\in \\wedge^{\\mathrm{odd}} W\n\\end{matrix}\\right.", "c4eca62d92df711989416522b3f91320": "\\begin{align}\n & AI=\\sum\\limits_{i=160}^{8000}{SN{{R}_{i}}*W{{1}_{i}}} \\\\ \n & SN{{R}_{i}}=(V{{S}_{i}}-S{{S}_{i}})+(VR{{S}_{i}}-VM{{S}_{i}}) \\\\ \n & PI=100*(1-AI) \\\\ \n\\end{align}", "c4ecc36b3cc0ae2a0c50e878de5f3cbe": "\\| f - \\varphi_{\\delta} \\|_{\\infty} = \\sup_{t \\in [0, T]} \\| f(t) - \\varphi_{\\delta} (t) \\|_{X} < \\delta;", "c4ecc4079faa5752746c1877f9c09e0b": "\n H^{(\\lambda)}(X)\n =P_1(X)\n +O\\left(\n \\left|\\frac{\\alpha_1}{\\alpha_2}\\right|^M\n \\cdot\n \\left|\\frac{\\alpha_1-s}{\\alpha_2-s}\\right|^{\\lambda-M}\\right)\n", "c4ecd3306e96b0925486d6acbbb939a3": "\\operatorname{dist}(v)", "c4ed503bcb62447b6d4d2d40a264f698": "y = C \\left \\{ {{}_2 F_1}(\\alpha, \\beta; 1; 1 - x) \\right \\} + D \\left \\{\\sum_{r = 0}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r^2} \\left(\\ln(1 - x) + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{\\alpha + k} + \\frac{1}{\\beta + k} - \\frac{2}{1 + k}\\right)\\right) (1 - x)^r \\right \\}", "c4edcac3e4e24c6fb44204fc0915b637": "x \\;\\ ", "c4ee0bf44a925ee2ad36dd49936b6f54": "l+1", "c4ee3e9a4c6308a4a31da4c89a764216": "1/(n+1)", "c4ef2896dbf21e9b9734d2620b141d8a": "L_2=\\ln(-\\ln(-x))", "c4ef48baf0a9a9b179c5d96ffeea120c": " r = |z| = \\sqrt{x^2+y^2}", "c4ef942ef3d7350e7a009df4bddf66f2": "U(a,b,z)=z^{1-b} U\\left(1+a-b,2-b,z\\right)", "c4efbda4bccd9e97203abc6e3730ca66": "\\sum_{}", "c4efec9f711cc16e434d5a13ccb0dd44": "\\left[\\begin{matrix}0 & 1 \\\\ 2 & 3\\end{matrix}\\right]\\left[\\begin{matrix}1 & 2 \\\\ 3 & 4\\end{matrix}\\right] =\n\\left[\\begin{matrix}3 & 4 \\\\ 11 & 16\\end{matrix}\\right] =\n\\left[\\begin{matrix}3 & 4 \\\\ 1 & 1\\end{matrix}\\right]", "c4f0032fd95d358ec248bb8de52a6294": "E_{tgu} = 15.46J \\,", "c4f1316337ed0551a732c1a0c20f3229": "\\psi(x)=\\sum_{n=1}^\\infty \\vartheta \\left(x^{1/n}\\right).", "c4f140ef2b7c8a70b3c9c9f7c9645689": "L(n) = 2 \\frac{\\varphi^{n+1} - \\psi^{n+1}}{\\varphi - \\psi}- 1 = \\frac{2}{\\sqrt 5} \\left(\\varphi^{n+1} - \\psi^{n+1}\\right) - 1 = 2F(n+1) - 1", "c4f142bbac77eb95ab436eb50e8c7f58": "C_{D_0}\\;", "c4f1fa028aa970dc61252645039eafe0": "\\begin{align}\nP_n&=P_0(1+r\\Delta t)^n-\\sum_{k=1}^{n} M_N\\Delta t(1+r\\Delta t)^{n-k} \\\\\n&=P_0(1+r\\Delta t)^n-\\dfrac{M_N\\Delta t[(1+r\\Delta t)^n - 1]}{r\\Delta t}\n\\end{align}", "c4f20f737db5df125160d47f4e082788": "[g_{ij}] = \\begin{pmatrix}\n\\pm 1 & 0 & 0 & 0 \\\\\n0 & \\mp 1 & 0 & 0 \\\\\n0 & 0 & \\mp 1 & 0 \\\\\n0 & 0 & 0 & \\mp 1 \\\\\n\\end{pmatrix}", "c4f234680754c0847661742b37cf8c57": "\\sum_{i=1}^{2} \\log \\frac{N}{n(q_i)}.", "c4f284e6e25e1f4788523f9c11d067e6": "| 1 1 \\rangle", "c4f28bbd0f21181784fb0a026dcbbb0d": "h_{Preucil\\ circle} = 60^{\\circ} \\cdot \\left( 6 - \\frac{B - G}{R - G}\\right)", "c4f296a234e3f2419b7ccf4e0f04f0ce": " \\lambda=\\frac {1} {2} \\sigma^2 \\theta ", "c4f2984114e0126079ca7f07c5053e39": " \\nabla_i ", "c4f317566352879be83491a4407f1275": "k = \\dim E \\geq c(\\epsilon) \\, \\left(\\frac{\\int_{S^{n-1}} \\| \\xi \\| \\, d\\sigma(\\xi)}{\\max_{\\xi \\in S^{n-1}} \\| \\xi \\|}\\right)^2 \\, N. ", "c4f332ac3026e88ef871222cd8e2589b": " \\sum_1^\\infty \\frac{\\mu(n)}{n^s} = s \\int_1^\\infty \\frac{M(u)}{u^{1+s}} \\mathrm{d}u \\,.", "c4f34285b55672f7f6c79f61c2366781": "V_n(P,Q)=\\frac{(P^2-4Q)\\cdot U_{n-1}(P,Q)+P\\cdot V_{n-1}(P,Q)}{2}. \\,", "c4f348d7c126904e6e7fad715b745cc2": "m<3n", "c4f3ca3687d45f85d66a2cb84e3b8596": "Q_{t+1}(s_{t},a_{t}) = \\underbrace{Q_t(s_t,a_t)}_{\\rm old~value} + \\underbrace{\\alpha_t(s_t,a_t)}_{\\rm learning~rate} \\times \\left[ \\overbrace{\\underbrace{R_{t+1}}_{\\rm reward} + \\underbrace{\\gamma}_{\\rm discount~factor} \\underbrace{\\max_{a}Q_t(s_{t+1}, a)}_{\\rm estimate~of~optimal~future~value}}^{\\rm learned~value} - \\underbrace{Q_t(s_t,a_t)}_{\\rm old~value} \\right]", "c4f3cc008f92918d337c7ddd87b69229": "V_i\\!", "c4f3e95be92629a218856924509ecf70": "\\sum_{k=1}^{b} W_{k} F_{k} = 0.", "c4f3f7b012dce812242b8508f4ee083d": "T^{-1}(op_1,op_2)", "c4f41dda8fff074bd1720ebb368a53ae": "\\frac{C_V}{Nk} \\sim 3\\,.", "c4f41eb463e752a0dfee64b8fb95f7df": "q_2=q^c", "c4f4446220a9fede344e3a04236ebedd": "Cl_2 = \\{medium\\}", "c4f46fbde1a6a88917e2e6ef01a73ca5": "x = Xy\\,", "c4f4b9b0ab0a2eb771bf7decb3a53c8d": " v ", "c4f4c3516ae8626abc95626b5686a2b1": "r\\,\\!", "c4f4c46788069ce2e23474939d768a4f": "p_2=\\frac{20}{61}", "c4f4cf41333dd0ba145d7883d486e7a8": " \\left(-\\frac{b}{2a}, -\\frac{\\Delta}{4 a}\\right). ", "c4f50f90b2f186e6fbd55ffde636ca86": " \\lambda = - P/\\left (\\mathbf{A} \\cdot \\nabla T \\right ) \\,\\!", "c4f63532678af6100c5722d54051898a": "\\Delta_4^{\\prime\\prime}F(J) = \\bar \\nu [S(J-2)] - \\bar \\nu [O(J+2) ] = 4B^{\\prime\\prime}(2J+1)", "c4f71962318bc8ac92d3a858e26e5ade": "\\langle\\mu\\rangle=0", "c4f729d02a67ac278165d81c624944ca": "m\\times n", "c4f741251ed8932a24b2ec729f989a3b": "P_\\ell^{m}(x)", "c4f745422b1b59d0ac6455101b564c39": " {^*\\mathbb{N}} \\setminus \\mathbb{N} \\, ", "c4f751555ec5a7c864b3f50edee6f025": "A^T A = I_n, \\,", "c4f753b0d2d2278f3a883d68ba27ba8d": "p=7", "c4f7bbbf67a46730e803e9bdb7de4cf8": "\\mathbf{STFT}\\{x(t)\\}(\\tau,\\omega) \\equiv X(\\tau, \\omega) = \\int_{-\\infty}^{\\infty} x(t) w(t-\\tau) e^{-j \\omega t} \\, dt ", "c4f7c07d33befae09178de34b1502526": "\\frac{\\mathbf{J}_n}{-q} = - D_n \\nabla n - n \\mu_n \\mathbf{E} ", "c4f7f7fbe266e1a320fdc8995ac0fabb": "X_2=\\left\\{\\begin{pmatrix}x \\\\ y \\end{pmatrix}\\in \\mathbb R^2 : |x|<|y|\\right\\}.", "c4f83559711e1b6180d2e4a1fcc2653f": "LP(T)::\\sum_{i=1}^{m} \\sum_{j=1}^n c_{ij} x_{ij} \\le C ", "c4f85ff2aaafa16b3d607f223b51bb43": "\\ ( a\\ \\alpha_{i,j,k} + b\\ \\beta_{l,m,n} ) ", "c4f880635c8f0d21e0ebfec62433ce53": "\\ E\\{ |h^\\mathrm{H}v|^2 \\} = E\\{ (h^\\mathrm{H}v){(h^\\mathrm{H}v)}^\\mathrm{H} \\} = h^\\mathrm{H} E\\{vv^\\mathrm{H}\\} h = h^\\mathrm{H}R_vh.\\,", "c4f97f13c5100326aaa25d1371b1ce8c": "{DE}_{6}", "c4f9b574aa41ddea8d0ed68d18d1bf06": " h(r_{12})=g(r_{12})-1 \\, ", "c4f9b6bd8f3e1a00ade5efdec90de642": "\\lfloor(\\alpha + k)(n + 1) + \\rho\\rfloor - \\lfloor(\\alpha+k)n + \\rho\\rfloor - \\lfloor\\alpha + k\\rfloor = a_n.", "c4f9d30bcbcb079a3a1f12224f52596c": "\n\\ddot{s}_{\\overline{n}|i} = (1+i)^n \\times \\ddot{a}_{\\overline{n}|i}\n", "c4fa8a6a7ca2b7ba836c80635b15e1b1": "k^2(k-1)", "c4fa9ad6bc77c3d274dfea5780e8fbf7": "\\ dU= MU_xdx + MU_ydy ", "c4faa57e1def3539496d4759bc35d5a7": "L^{p,w}", "c4faa63f077c1a97ad5c709e947c8fef": "b_1 = a_1, b_2= b_1 \\cdot a_2 = a_1 \\cdot a _2, b _{d-1} = b _{d-2} \\cdot a_{d-1} = a_1 a_2 \\cdots a_{d-1}", "c4fab1659eecd0901dbc5ae64cf2d117": "\\ V", "c4fb5cd86b2cdf8da7d5f01970dcc095": "n=0,", "c4fb669795e91783c6c16e8f0eed587d": "f(x;\\alpha,\\beta,p,q) = \\frac{p{\\left({\\frac{x}{q}}\\right)}^{\\alpha p-1} \\left({1+{\\left({\\frac{x}{q}}\\right)}^p}\\right)^{-\\alpha -\\beta}}{qB(\\alpha,\\beta)}", "c4fb83b0ed070f90f1815f2958dd246b": "\\Phi _N \\left( x^\\mu \\right)", "c4fca95abbd833c434352c1cafd35780": " R(s,t) = C(s)+t\\mathbf{N},\\quad -1\\le t\\le 1.", "c4fd0dac17ae31a3a25cbddb32095f34": "I_\\nu(s)=I_\\nu(s_0)e^{-\\tau_\\nu(s_0,s)}+\\int_{s_0}^s j_\\nu(s')\ne^{-\\tau_\\nu(s',s)}\\,ds'", "c4fd13142b1a57fca03836e6b173d428": "{}_s\\langle X \\rangle_N", "c4fdda8946d75e32bfff0a42b53d9581": "F, \\text{CR}", "c4fe1433bacca88afc5a87e3853df562": " \\displaystyle P^{1} = I - 2v^{(1)}(v^{(1)})^\\text{T}", "c4fe1c41d35a7f01f36049209c521227": "\\varphi(\\cdot)", "c4fe2a592031460086a45a50bddcf133": "f^* : map (Y, W) \\to map (X, W)", "c4fe922591be922f85c13f1182507410": "GF\\left( {q^N } \\right)", "c4fe93da703fb9ac426946a784eb7746": " x_{n+1} = x_n + a_n \\bigg(\\frac{N(x_n + c_n) - N(x_n -c_n)}{c_n} \\bigg) ", "c4feac735606a4d8df26401f9d39db49": "\\int_{S^{n-1}} f\\bar{g}\\,d\\Omega = 0", "c4fef3c63e78fe3463c54f90c2f6ecf0": "\\sqrt{y} = \\begin{pmatrix}\n \\sqrt{y_{01}} & \\\\\n & \\sqrt{y_{02}} \\\\\n & & \\ddots \\\\\n & & & \\sqrt{y_{0N}}\n\\end{pmatrix}\n", "c4ff35b1877a8e0bd2e9aaad195a9579": "\nC = \\nabla^2I - c|\\nabla I|^2,\n", "c4ff4b4f1c4da8a39b34083410964e2d": "{13 \\choose 1} = 13", "c4ffa64c2d1986e06f29733cb54dbc76": "Re[\\lambda] < -\\overline{\\lambda}", "c4fff9ace3cac64aa6424a9fe3121b0a": "\n \\begin{align}\n I[f_1,\\ldots,f_m] & = \\int_{\\Omega} \\mathcal{L}(x_1, \\ldots, x_m; f_1,\\ldots,f_p; f_{1,1},\\ldots,\n f_{p,m}; f_{1,11},\\ldots, f_{p,mm};\\ldots f_{p,m\\ldots m})\\, \\mathrm{d}\\mathbf{x} \\\\\n & \\qquad \\quad\n f_{i,\\mu} := \\cfrac{\\partial f}{\\partial x_\\mu} \\; , \\quad\n f_{i,\\mu_1\\mu_2} := \\cfrac{\\partial^2 f}{\\partial x_{\\mu_1}\\partial x_{\\mu_1}} \\; , \\;\\; \\dots\n \\end{align}\n", "c5001b3752afa6116534497de9b6b083": "\\operatorname{sech}\\, x", "c500852e6b01a09ddd36bfc3ffdf0186": "r<(y+1)^n-y^n", "c501ca203fce88f8fb40a07dab1f4b87": "\\Phi = S - \\frac {U}{T}", "c502556bc0211b077be8a6bc06e5e06e": " k = k_0 \\exp \\left ( \\frac{-Q}{RT} \\right ) \\,\\! ", "c5028853c7552ba64e065d448b9f97d9": "\\ |q| = \\frac{4 \\pi n}{\\lambda_0 }\\sin\\left( \\frac{\\theta}{2}\\right) \\qquad(5)", "c502aade5f23f39f459b28846e1260a1": "[x(t_{1}), x(t_{2})]=\\frac{i\\hbar}{m\\omega}\\sin(\\omega t_{2}-\\omega t_{1}) ", "c502b990beef7c137450e0c5f61339a7": "k_i=q_i'(x_i)=q_{i+1}'(x_i) \\quad i=1,\\dotsc ,n-1", "c502dfd4f36d244b5b83fbe748712332": " l_- = {l\\sqrt{1-v^2/c^2}} ", "c502f6fc077b1d5c109ee335559eca61": " sA>\\, ", "c503047c548a97af0adc6c7f82fab7aa": " \\big| \\langle u,v \\rangle \\big|\n\\leq \\left\\|u\\right\\| \\left\\|v\\right\\|, \\, ", "c5031756cd3eff26cf7252bf1c4547ad": "(x-a)^2+(y-b)^2 = r^2", "c50335e6992affa63ea7f7bd3178fbd1": "S = \\{S_{j}\\}_{j=1}^M ", "c5036a005a34c209bfed088098f14da6": "\\,\\!x = b", "c503de66d24cab22c91513cbb741db43": "\\hat{H}'", "c503f3e7c7203d9e54d178f2071f7b10": "R(\\lambda)=\\frac{S(\\lambda)}{M(\\lambda)}", "c5046151cc4e283eba05005c94840046": "\\displaystyle u_i(a_i,a_{-i})", "c50477802c3425020da25962f65d3258": " \\text{Pulse Spacing} = \\frac{\\text{Propagation Speed}}{\\text{PRF}}", "c5047d1cc53629b667494a074e8b9e84": "-2 \\boldsymbol\\Omega \\times \\mathbf{v}_{\\mathrm{r}}", "c50493003c4242e49d415b9e36953d3a": "y_T = \\frac{Y_T}{Y_0}\\,", "c504d54cbff3ab3531de6940341293b2": "\\alpha \\gtrsim 1", "c504e52ef30483b2b63bf1d1fa0bdf2b": "P(X_t) = \\sum_{i=1}^K \\omega_{i,t} N \\left ( X_t \\mid \\mu_{i,t} , \\Sigma_{i,t} \\right )", "c5050a8bf185a0aed905c927483e5fdf": " \nP_{ni} = {exp(\\beta z_{ni}) \\over \\sum_{j=1}^J exp(\\beta z_{nj})}, \n", "c505841c04c32291def660caa9cd2e9b": "R^\\mathsf{T}R = I\\,", "c5058b0a908018660d170d22add6e2ce": "\\mathbf{Q}_d = (\\mathbf{A}_d^T)^T (\\mathbf{A}_d^{-1}\\mathbf{Q}_d). ", "c505bb7ff38fa92a158ceb039d01fbda": "{Ax} \\geq {b} \\,", "c505cf8d7fc80dab72737c846b961999": "\\varepsilon_e(n)", "c505eb59cba123f05dde5e59aac46f9e": "\\gamma(s,z)= \\sum_{k=0}^\\infty \\frac{(-1)^k}{k!} \\frac{z^{s+k}}{s+k}= \\frac{z^s}{s} M(s, s+1,-z),", "c505ff89542002e6ed1056300adfc9f0": "\\displaystyle \\kappa", "c5061d5271cb0d7f56d87d90a43cedf5": "\\mbox{tr}\\,\\mathfrak{GHG}^{-1} = \\mbox{tr}\\,\\mathfrak{H},", "c50643cb02907894ed6ddc505b881b97": "D(-,d) \\colon D^{\\mathrm{op}}\\to\\mathrm{Set}", "c50662a302ae06d6f07580c5f71d0bc2": "\\tilde{d}=\\frac{ncp}{\\sqrt{\\frac{n_1 n_2}{n_1+n_2}}}.", "c506aa059bb4da137848a89389d73183": " \\frac{\\partial \\boldsymbol{u}}{\\partial t} + \\left( \\boldsymbol{v} \\cdot \\boldsymbol{\\nabla} \\right) \\boldsymbol{u} = \\boldsymbol{0}. ", "c506c26a13a288e746d925e97de52d44": " x = \\frac{4a(1-u^2)}{(1+u^2)^2}, ", "c506e516572d71d42c5e94ba0ff3f0cc": "I_\\text{z} = \\frac{1}{2} (n_\\text{u} - n_\\text{d})", "c5070147502ae787a97d672a8f646aaa": "\\frac{P(I \\and H1+)}{P(H1+)}", "c50711eca545d49dac9ff0f62f77a36b": "\\sum_{i=0}^{\\infty} (1-\\operatorname{P}[E_1]-\\operatorname{P}[E_2])^i\\operatorname{P}[E_1]\n= \\operatorname{P}[E_1] \\sum_{i=0}^{\\infty} (1-\\operatorname{P}[E_1]-\\operatorname{P}[E_2])^i\n", "c50717c5fae5a47ef25ce1753ed4e77a": " [B]\\mathbf{y} =\\mathbf{b}\\times\\mathbf{y}.", "c5074b7172f0aef02aa3202b35b624a5": "f_1(n) = \\frac{1}{2n}, \\ f_2(n) = \\frac{n/2}{n^2+1}, \\ f_3(n) = \\frac{(n/2)^2+1}{(n^2+5)n/2}.", "c5078d5aa6406f2c2e2dc94fe21c7b5c": "NC+\\frac {Yi} {2N}", "c5079e2fd7323aee67d0fa30b4a40968": "\\alpha_{i}\\not \\perp X_{it},Z_{i}", "c5079fd66610f412fc5ab2383f43ed06": "\\mathrm{Re} = \\frac{vL\\rho}{\\mu}", "c507a8f0923c1ecb674fcaa8dec43b07": "\\forall\\varepsilon > 0\\;\\exists \\delta >0 \\;\\forall x \\in I \\;(0 < a - x < \\delta \\Rightarrow |f(x) - L|<\\varepsilon)", "c50814a2b6344071181951ae8f9eaa94": "\\alpha_F = \\frac{I_{\\text{C}}}{I_{\\text{E}}}", "c5081885bd2760ef8147db48032288fe": "\n\\omega_n=\\omega_{1}+ \\omega_{2}+ \\cdots + \\omega_{n-1},\n", "c50835e48c28f74b79c326acd1c07388": "\\sec a = \\sqrt{1 + \\tan^2 a }", "c508953675749419eed1dd93412edb2c": "\nw_{k^*j}^{L(new)} = w_{k^*j}^{L(old)} - \\eta (x_{ij}^p-z_{k^*j}^{old})\n", "c508afc1774f9c626d1c26ea38198145": "[L_x, L_y] = i \\hbar L_z", "c50943b6fc05f3e80b748dd19b14899a": "Y \\sim N(\\mu_Y, \\sigma_Y^2)", "c5096070fc159f49c54264a33bf4ac95": "x = (a + \\mathbf a) + \\ell(b + \\mathbf b)", "c509a96b21081483da93b16c648e9fe3": "{\\it object~evolution}", "c50a23529365638627daecb399713c7e": "\\tfrac{6}{\\pi^2}.", "c50a4b54a614b99974ffcfefc5c00c04": "J_{1}", "c50a7f89f0125749905c5a31099cf62d": "a^n = \\frac{V_n + U_n \\sqrt{D}}{2}", "c50ab724a0f83a81a26cc9e230a643f0": "D_{KL}(P\\|Q_\\theta) \\ge 0", "c50af913fdc7b7192dd5214ac204bf8d": "\\displaystyle{G_T=K_T A_T K_T.}", "c50b49d0f33eaf61e3afbbc1d002f682": " G( A(a,b), H(a,b) )=G\\left({{a+b}\\over 2}, {{2ab}\\over {a+b}}\\right) = \n\\sqrt {{{a+b}\\over 2}\\cdot {{2ab}\\over {a+b}}} = \\sqrt{ab} = G(a,b) ", "c50b6bef0f29b1a6cbdcf6c7641319aa": "\\psi_1(\\Omega_2+1) = \\varepsilon_{\\zeta_{\\Omega+1}+1}", "c50b966d2037ddc856ce1aec6024ede2": "\\left| \\frac{z^n}{n!}\\right| \\le \\frac{\\left| z\\right|^n}{n!} \\le \\frac{R^n}{n!}", "c50b9e82e318d4c163e4b1b060f7daf5": "\\epsilon ", "c50d218d36c7bfefd6819cb249056162": "y = zxz^{-1}.\\,\\!", "c50d2f3ac7b297c9c6b8bbedcff4e651": "\\operatorname{OE} \\mathrel{:} \\left(K \\to A\\right) \\to \\left(K \\to A\\right). ", "c50d8f219274bd041372e76f0ee25325": "\\mathcal{S}[\\phi]\\,", "c50deaab3526e30b99cb23e573d40ea2": "\\|Tx\\|", "c50e1523cbbb8028f114d23d8bb5d42b": "\\varphi=\\varphi \\quad", "c50e41b438d145094410c714ab03f00c": "1\\,\\!", "c50e4f9449b0b7ffdc7f089650a1dda0": "A = R K L S C P", "c50e6c63a219cc79c616014666120277": "x_1^{(t+\\Delta t)}=\\tilde{x}_1^{(t+\\Delta t)}+\\frac{1}{2}d_1d_3\\,", "c50e934baf08c0210d465a2e70015442": "J_i=\\sum_{k=1}^n\\beta_k I_i(\\alpha_k)=\\sum_{k=1}^n\\left(\\beta_k e^{\\alpha_k}\\sum_{j=0}^{np-1}f_i^{(j)}(0)\\right)-\\sum_{k=1}^n\\left(\\beta_k\\sum_{j=0}^{np-1}f_i^{(j)}(\\alpha_k)\\right)=\\left(\\sum_{j=0}^{np-1}f_i^{(j)}(0)\\right)\\left(\\sum_{k=1}^n \\beta_k e^{\\alpha_k}\\right)-\\sum_{k=1}^n\\left(\\beta_k\\sum_{j=0}^{np-1}f_i^{(j)}(\\alpha_k)\\right)", "c50f10ba241add04d26cff9d50cbf9e7": "\\overline{B_{\\delta}}", "c51037e0ed4d1ff6aee7de2885a78510": " e^x = \\sum_{n=0}^\\infty \\frac{x^n}{n!} = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\cdots,", "c51040d22d67b03dcb7b859dbd477fad": "=2(1/b^{2})(1/s^{2})\\beta^{s} {_3\\text{F}_2}(s,s,s;s+1,s+1;1-\\beta) ", "c5107da0174415a95090d9efc0b511f7": "p{\\rm d}V", "c510cb8aca5b03b5ca5a3693b65e5c42": "\\beta^-", "c5110a5e69175ca4719fb57ae568deb1": "\\dot{{\\mu }}={\\dot{{\\mu }}}'=0\\Rightarrow \\partial_{\\mu } F=0\\Rightarrow \\delta_{\\mu } S=0", "c5111c8458d8f0d6c9154c21d85863c9": "i^{\\ast}_{\\mathrm F_{SO}(M)}(T\\mathrm FM) \\to \\mathrm F_{SO}(M)", "c5115fdcf999a4e865f81bfa0eead578": "M =\\begin{bmatrix}a & b-a \\\\ 0 &b \\end{bmatrix}.", "c5116a89c992b82444a76be4f8629692": "\\phi,\\;\\lambda,\\;r,\\ ", "c51183d71b2238a04c8cd65d7fb60d2a": "\\sum_{\\{i\\}} \\vert \\lambda_i \\vert^p < \\infty", "c51187ce4cf6acd8cfeeb8328686f5b2": "\\operatorname{recc}(A)", "c511c8d4b5a56a604908fc9d6b5c23ed": "0 < F < 1", "c5123078fdace5aca84831925aea89fe": " \\operatorname{p.\\!v.} \\left( \\frac{1}{x} \\right) \\,:\\, {C_{c}^{\\infty}}(\\mathbb{R}) \\to \\mathbb{C} ", "c51244cd25d1c0eed7f0eebd45c78487": "\\|A^k\\|\\sim\\rho(A)^k", "c5128f579b83322a464b5b5065364dd8": " ABC ", "c512906a43f33e66f0c1365bed981e9c": "\\mathrm{d}(TS) = \\mathrm{d}U - \\mathrm{d}F = \\mathrm{d}H - \\mathrm{d}G ", "c512e90f27e77c819b6075aa3d1beb73": " \\ \\textbf{a} = \\textbf{f} \\cdot \\textbf{e} \\pmod q ", "c512fe22e1ee1e1cbe96680f81f8e765": "c^2=\\frac{g}{\\alpha}\\frac{\\rho_L-\\rho_G}{\\rho_L+\\rho_G},\\qquad \\sigma=0,\\,", "c513087e35f3d6630c617e0f4f128f1d": "L_{loc}^1(U)", "c5130eb32ef20ab0a16d581d1735a4c9": "\\scriptstyle {}^{(n+1)}i \\;=\\; a'+b'i", "c5131bb100ae0df005e79ad8a55dd8c2": "\\frac{A_2}{A_1} = \\left(\\frac{b_2}{b_1}\\right) ^ {1-1.12 \\cdot 10^{-3}\\sqrt{b_2/b_1}(b_1A_1)^{0.55}}", "c5137d5ad34132ded8b192da0570522c": " \\frac{u_{j}^{n+1} - u_{j}^{n}}{k} = \\frac{u_{j+1}^n - 2u_{j}^n + u_{j-1}^n}{h^2}. \\, ", "c513b3abd450052ff9f276149c50e3e7": "\\forall n\\in \\N,\\, u_{n + 1} = u_n", "c51433efd3f499b7b5a298336929acad": "N_\\text{sample}(S)", "c514d168ccf15d1a2dc430e094f8706f": "|p\\rangle", "c51571a2a4960a1752114d646d2d8c7a": "\\textit{animal}", "c515c8b8cd406a4b5395b4e3543eaf81": "\\frac{\\partial f}{\\partial\\bar{z}} = \\phi(z,\\bar{z}).", "c5160e74919f52eb6bd0c2726e8a1bd8": "L=5", "c5162c7366df77e228542e5ad748faec": "\\kappa=\\nabla^2\\eta=\\eta_{xx}.\\,", "c5163a2e301909c8b518da6027664025": " x_j \\geq 0, x_j", "c5165b223633f1958d65775d3b3379d1": "\\frac{1}{M_s}", "c5167a9492ec894f68db61bd9370a8c8": " i\\in\\mathcal{I} ", "c5168e2449592a82ded1cbd9332ab948": "\nP_{\\infty} + \\frac{1}{2}\\rho v_{\\infty}^2 = P_{D+} + \\frac{1}{2}\\rho (v_{\\infty}(1 - a))^2\n", "c5168feefff0b1bc2aa55b6091fc3c19": "n^2 \\approx 1", "c516949a25c538bada288877d5d618ad": "H=T^{0.5}", "c516cd7e8569871b6cf3d3c92ca0d88f": "\\beta_m = \\frac{2\\pi}{\\lambda_0} n_p \\cos \\theta_m", "c516ecc5f7a761a5673f5d0dbd9c2dec": "\np^2=r_p^2\\left(1+2GM_{12}/V^2r_p\\right) \\approx 2GM_{12}r_p/V^2;\n", "c517211205c9e82fb8532f1ee9445fd6": "[\\eta_j]", "c517666404297a1aabe5bd1369904d00": "\\begin{align}&\\nabla_X(\\sigma_1 + \\sigma_2) = \\nabla_X\\sigma_1 + \\nabla_X\\sigma_2\\\\\n&\\nabla_{X_1 + X_2}\\sigma = \\nabla_{X_1}\\sigma + \\nabla_{X_2}\\sigma\\\\\n&\\nabla_{X}(f\\sigma) = f\\nabla_X\\sigma + X(f)\\sigma\\\\\n&\\nabla_{fX}\\sigma = f\\nabla_X\\sigma.\\end{align}", "c5178bf27ca4695b749a285f1c9f1ea0": "(x,y),(y,z) \\in \\mathcal{M}", "c517e482b0010bc3a31e9eeb836d7fc8": "(S,T,W,M_0)\\!", "c517fa692b61b0884ba5362029822d12": "1^3+2^3+\\dots+n^3 = (1+2+\\dots+n)^2=\\left(\\frac{n(n+1)}{2}\\right)^2.", "c518148117272e06546e93a6b464d2ba": "f(z)=\\frac{z^2}{z-z_2}", "c51883dce8d5911a25971c2c661cccb5": "S_{\\mu}(z)=a\\left(z-c_1-\\frac{1}{S_{\\rho}(z)}\\right),", "c51884db35bb082eddec54932ffff9ee": "N_D = \\frac{1}{2}\\int_{-\\infty}^{+\\infty} D(E)f(E-U_{DF})\\,dE ", "c518b40968fe0ae1c05a29cd8fcba12e": " b_k ", "c518e72b6494bd98b4b07d22bc37c41e": "dRA", "c518f3184ca91e4c6c4c6f3a7c2f3658": "V_i = \\frac{e}{4 \\pi \\epsilon_0 } \\sum_{j \\neq i} \\frac{z_j}{r_{ij}}\\,\\!", "c5192f9d24d4454906b68a95aa96115c": "x=0.179487179487179487\\ldots=0.\\overline{179487} \\mbox{ has }4x=0.\\overline{717948}=\\frac{7.\\overline{179487}}{10}.", "c51960257606b4f60c5657cdd3efc6e9": "0<|u|", "c5197ddecbaa83573db682bcdccd84e1": "\\frac{}{\\Gamma \\vdash K: \\alpha \\rightarrow (\\beta \\rightarrow \\alpha)}", "c51a2b6f0541ca2e1499af71cd6c7354": " p_1=1, ", "c51a339c66989e9f05be11b50024b380": "\\Omega_n(E)", "c51a3a514be22a87e199a430dd37fce0": "|f_{1j}\\rangle", "c51a421b898fa90d2c6eb925f69da667": "1 \\over 2", "c51a6060e1329bf236df2b1e495479d4": "H_n=H_{n,1}=\\sum_{k=1}^n\\frac{1}{k}=(-1)^{n-1}n!\\sum_{k=1}^n\\frac{1}{k^2\\Pi_k(1,\\ldots,n)}=\\sum_{k=1}^n(-1)^{k-1}\\frac{1}{k}\\binom nk.", "c51a6319f51e6a615f4abb48c4bf8337": " = r \\frac{\\mathrm{d}\\theta}{\\mathrm{d}t} \\mathbf{u}_\\mathrm{\\theta} \\, ", "c51b45a59c135204358d29b4adde8c0f": "\nT_{\\delta}^{X^{n}}\\equiv\\left\\{ x^{n}:\\left\\vert \\overline{H}\\left(\nx^{n}\\right) -H\\left( X\\right) \\right\\vert \\leq\\delta\\right\\} .\n", "c51b57a703ba1c5869228690c93e1701": "XX", "c51b6eb813ae912d4f5dce02c4081ef3": "G_T\\;", "c51c180e16cc73d77974dff24d978a90": "4\\pi\\epsilon_0", "c51c481219abb373f468bb159e00d742": "X^{**} = (X^*_{b})^*", "c51c7121cc5b16da66e5d0d32f3e7b8c": "X-A", "c51c882e5c26dfd4d5fbf82de2f519a1": "\\sum_{i=0}^n i^p = \\frac{(n+1)^{p+1}}{p+1} + \\sum_{k=1}^p\\frac{B_k}{p-k+1}{p\\choose k}(n+1)^{p-k+1}", "c51c8c7a3661406307eb24164f6943db": "P(\\operatorname{deg}(v) = k) = {n-1\\choose k}p^k(1-p)^{n-1-k},", "c51ce410c124a10e0db5e4b97fc2af39": "13", "c51d37f176e2b5382d41a99aa107004e": "s(F)=1", "c51d3dc4b8345d5c44b9fd11f491eaba": "\\sum_{i=1}^{n}A_i=0.", "c51e2dbeba7128f222f564bd1ea5cc12": "m \\geq \\sqrt{\\frac{A}{16\\pi}}.", "c51ed41283b788d2848ecfd8cd1bc108": "x^{t-1},x^{t-2},\\dots,x^1,x^0", "c51efd5976599b69cd799eb64e5efdce": "\\mathbf{p}_{uc}", "c51fa5744cb885dce3eadc00f68d1e25": "d_I(x,y)\\,", "c51ffec3e12f30d03a63ae09fc47f0d1": "~w_f=z \\theta_f = z \\lambda /d", "c5208fe593701ca8fba2ddd3f290cc65": " \\operatorname{Var}(\\mathbf{Y}) = \\operatorname{V}( \\boldsymbol{\\mu} ) = \\operatorname{V}(g^{-1}(\\mathbf{X}\\boldsymbol{\\beta})). ", "c520d69606c1569e353d100c594167f6": "k(k-1)", "c520d84b4f11003486829ed56710d030": "RT\\ln \\frac{f}\n{P} = \\int_0^P {\\Phi dP}", "c5210080b3cd1309d9622c19fd7d1a21": "m_i \\equiv m \\mod p_i", "c52161b563b646236e63ab0176c80240": "f : E \\to E'.", "c5221282f0226b4ae31a5588906a6735": "(a * b)_0 = a \\smile b.", "c5223176ad3884126704963d59ca1a84": "\\ C_D", "c522352322fcabe52230c1589e954874": "\n\\sigma_5(n) = \\frac{1}{21}\\left\\{10(3n-1)\\sigma_3(n)+\\sigma_1(n) + 240\\sum_{00, b,s,\\beta>0", "c530651f9d2599dc79b93ed6a9356edc": "\\left(\\Omega-\\Omega_{0}\\right)=\\left(R-R_{0}\\right)\\frac{d\\Omega}{dr}|_{R_{0}}+...", "c530956aa7b4299435a3bb7d6850b71b": "s_n\\not\\in L_n", "c53100a921ace44bae579811f7492eac": "a_1 \\leq b_1", "c53108397ceaf1ba9cf914a664a4f0bf": "\\mathrm{GDP} =\nC + I + G + \\left ( \\mathrm{X} - M \\right )", "c531884a343e0db77c17c7a33e1b7675": "2/3", "c53191173d224aa0c56749fb3bce82f7": "\\frac{n}{2}", "c531b8ba274eac54104ba263259fe69e": "t=-1\\,", "c531f0444da1d938a133300fed3ad3e1": "\\hat{G}_j \\Psi (A) = - i D_a {\\delta \\Psi [A] \\over \\delta A_a^j} = 0.", "c5320c5ca2f2da019c43f93b471f6d8c": "A \\equiv B", "c53246ea955f6529a715ada4c9796c2a": "GVM(\\lambda_{0}) \\equiv \\left ( \\frac{1}{\\nu_{g}(\\lambda_0/2)} -\\frac{1}{\\nu_{g}(\\lambda_0)} \\right )", "c5327b2beda1a66630a6c1aa944b34eb": "100\\uparrow\\uparrow n=(10\\uparrow)^{n-2} (2 \\times 10^ {200})=(10\\uparrow)^{n-1} 200.3=(10\\uparrow)^{n}2.3<10\\uparrow\\uparrow (n+1)", "c532a45062a3b5c0238a383f9305d79c": " \\left[ \\operatorname{p.\\!v.} \\left( \\frac{1}{x} \\right) \\right](u) = \\lim_{\\varepsilon \\to 0^{+}} \\int_{\\mathbb{R} \\setminus [- \\varepsilon;\\varepsilon]} \\frac{u(x)}{x} \\, \\mathrm{d} x = \\int_{0}^{+ \\infty} \\frac{u(x) - u(- x)}{x} \\, \\mathrm{d} x \\quad \\text{for } u \\in {C_{c}^{\\infty}}(\\mathbb{R}) ", "c532ccfb504db2939f72711e739536ef": "P(X=115|M_1)={200 \\choose 115}\\left({1 \\over 2}\\right)^{200}=0.005956...,\\,", "c532f422c778628369b79d772fe16ff6": "\\vdash p\\!", "c533576850b5328b825f1c8dd96aeb39": "R_1, \\, W_1", "c5337b42fc28f8e3ac2ebddd48c9c4d6": "F(x; 0,1)=\\frac{1}{\\pi} \\arctan\\left(x\\right)+\\frac{1}{2}", "c53397fd165cf3595e9cef92df63aae9": "\nX_{i}\n", "c533d52887f2de31863f99c56b70b86b": " d = 1.26\\; R_m\\left( \\frac {M_M} {M_m} \\right)^{\\frac{1}{3}} ", "c53425527207c9e5bd2d72e3b1fa2d36": "\\begin{bmatrix}\nS_1 & S_2 & \\cdots & S_{\\nu} \\\\\nS_2 & S_3 & \\cdots & S_{\\nu+1} \\\\\n\\vdots & \\vdots && \\vdots \\\\\nS_{\\nu} & S_{\\nu+1} & \\cdots & S_{2\\nu-1}\n\\end{bmatrix}\n\\begin{bmatrix}\n\\Lambda_{\\nu} \\\\ \\Lambda_{\\nu-1} \\\\ \\vdots \\\\ \\Lambda_1\n\\end{bmatrix}\n= \n\\begin{bmatrix}\n- S_{\\nu+1} \\\\ - S_{\\nu+2} \\\\ \\vdots \\\\ - S_{\\nu+\\nu}\n\\end{bmatrix}\n", "c534b9b7467eb8967de4518ff143e380": "\\left(\n\\begin{smallmatrix}\n0 & 1 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 1 & 0 & 0 \n\\end{smallmatrix}\n\\right)", "c534d742e3dac31d1a8d0878005f823d": " \\mathbf{J} \\cdot \\mathbf{\\hat{n}} = \\frac{\\mathrm{d}I}{\\mathrm{d}A} \\,\\!", "c5350e6517c9f856493645d5f9b62498": " S(\\rho^1)+S(\\rho^3)-S(\\rho^{12})-S(\\rho^{23})\\leq 0.", "c535153de23ef058b739bc530e54e406": "2 \\frac{q_{\\mathrm e} q_{\\mathrm m}}{\\hbar c} \\in \\mathbb{Z}", "c5351e30f6d2a4c934fb2ddcae6c740b": "j_1+j_2 + j_3\\text{ is an integer} \\, \\text{(or an even integer if} \\,m_1=m_2=m_3=0)\\, ", "c53543008cbb677631f6793c25a65bd1": "h_0(t)", "c5354308f0469a4b3e134e8b2fb52f1b": "p_1^2 = p_2^2", "c5356191fdbf0dbd81167a38ab35584d": " f' ", "c5357dc8b1636b2fbfd96bb2eeb76c00": "c\\cap{\\boldsymbol S}(c)=\\emptyset", "c536a29ab602b88fa5c96118c16a9dd3": "f(x)=\\log(1-x)", "c536d131f6727b7426b61849572033dd": "\nf(z) = \\sum_{n=0}^\\infty a_nz^n\\,\n", "c5372f2c5ba048bcf07b6801db1b1ccc": "\\lambda_{\\mathrm{vac}}", "c53823969d536db5167e73fa62623c64": " QS = \\frac{2C}{A + B} = \\frac{2 |A \\cap B|}{|A| + |B|}", "c5386cad2aa3ba74135820c4f22947ad": "\\mathbf{A}^{-1} = \\frac{1}{\\det(A)}\\begin{bmatrix}\n A_{11} & A_{21} & \\cdots & A_{n1} \\\\\n A_{12} & A_{22} & \\cdots & A_{n2} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\ \n A_{1n} & A_{2n} & \\cdots & A_{nn}\n\\end{bmatrix} ", "c53878fc7bb1bc742024cc8f912b7745": "f^{(-1)}(y) = \\begin{cases}\n \\sup \\{ x\\in[a,b] \\mid f(x) < y \\} & \\text{for } f \\text{ non-decreasing} \\\\\n \\sup \\{ x\\in[a,b] \\mid f(x) > y \\} & \\text{for } f \\text{ non-increasing.}\n\\end{cases}", "c5389f575c8ca97b1fa585def5e0026e": " c- ab -a_x=0 \\quad \\text{and} \\quad\nc- ab -b_y =0,", "c538b9dbf2e5e8c33a0e9353d2f2626b": "n_3 > n_2", "c538cad38bd227ad802bc9f95064bbac": "2^{-18}", "c5390fe125eca5926ccae53e1dc40daf": "A(\\lambda)=c_1\\ \\varepsilon_1(\\lambda)+c_2\\ \\varepsilon_2(\\lambda).", "c539308caf93b9b9e36cc9c226241190": " \n\\theta_{TP} = \\theta_{T} + \\left( \\frac{s_{Tx}+s_{Ox}}{R_T} \\right) \\cdot \\left( k_{TOF} \\cdot R_T + t_{Delay}\\right)\n", "c53933d75246679cfb1fa24970355208": "(x_0, x_2, \\ldots, x_{N-2})", "c53955daa37ec7ea579342d4532a7803": "e(H')", "c53967251a2fba4af3c0bfe0a1585541": "r\\leq n-1", "c5398c22483e408cb2b001cbedb7440e": "G(a_{m,n};x,y)=\\sum_{m,n=0}^\\infty a_{m,n}x^my^n.", "c539e08b6c0f868f432d7db5521a59e4": "\\nabla\\cdot(\\mathbf{F}\\times\\mathbf{G})\n= (\\nabla\\times\\mathbf{F})\\cdot\\mathbf{G}\n- \\mathbf{F}\\cdot(\\nabla\\times\\mathbf{G}).", "c539e94b9beb691f4b14aeeff29d73e0": "f(z) = z", "c53a0313c63973d1becc8f637559a589": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "c53a43e4ac0e86409e9fcc95cbddb038": " P(D) := \\sum_\\alpha a_\\alpha \\, D^\\alpha ", "c53a58ff5dc30afa5adc330511e064ae": "l_2(\\theta) = (\\theta + \\alpha)/2", "c53a64ae356fbdee8f64ac1e3a05a81b": "b = \\frac {fm_\\mathrm s} {N} \\frac { x_\\mathrm d } { \\frac { m_\\mathrm s + 1} {m_\\mathrm s} f \\pm x_\\mathrm d }\n= \\frac {fm_\\mathrm s ^2} {N} \\frac { x_\\mathrm d } { \\left ( m_\\mathrm s + 1 \\right ) f \\pm m_\\mathrm s x_\\mathrm d } \\,.\n", "c53aa182e22a69ba23c89710523e3adf": "\n \\frac{F_D}{\\frac12\\, \\rho\\, A\\, u^2}\\, =\\, f_c\\left( \\frac{u\\, \\sqrt{A}}{\\nu} \\right)\n", "c53add1210eed07dd3f033353710f619": "= V + Z \\,", "c53ba961e7487e922e667d6f7c0eec96": " u_{tt} = \\partial_x \\sigma'(u_x) - f'(u) ", "c53bdffde641fa25dc0b16b340a58699": "\\psi(q) = \\sum_{n\\ge 0} {(-1)^nq^{(n+1)^2}(q;q^2)_n\\over (-q;q)_{2n+1}}", "c53c6e2fd0b453f6d00be5e7ae691033": "\\mathcal{O}_X(X_f)", "c53c7fa189b4363c57db63fd2746b9a2": "u, v, x, y", "c53d5a91f4e625f66fb6246857e6b9c9": "F=dE/dz\\,", "c53ddd03efc11504ba883d1be3c882be": "\n= \\frac{p(X,S,A|\\mbox{Object})p(\\mbox{Object})}{p(X,S,A|\\mbox{No object})p(\\mbox{No object})}\n", "c53e27b62e04775e75d5f9f8ea95b1c3": "2*Pi*f*L", "c53e6ca79eebe5712249aad8f55528fa": "x + x^2 + \\cdots + x^n = \\frac{x}{x-1} \\prod_{d\\,\\mid\\,n}^n \\Phi_d(x)", "c53e94bb6363294edf96ac0f37bb8fd0": "K = e^{-\\frac{\\Delta G^\\circ}{RT}}", "c53e98fb92477fa6ac0df494b1835ca7": "\n \\begin{align}\n \\hat{S}_{11} & = S_{11} =: S_{rr} ~;~~\\hat{S}_{12} = \\cfrac{S_{12}}{r} =: S_{r\\theta} ~;~~ \\hat{S}_{13} & = S_{13} =: S_{rz} \\\\\n \\hat{S}_{21} & = \\cfrac{S_{11}}{r} =: S_{\\theta r} ~;~~\\hat{S}_{22} = \\cfrac{S_{22}}{r^2} =: S_{\\theta\\theta} ~;~~ \\hat{S}_{23} & = \\cfrac{S_{23}}{r} =: S_{\\theta z} \\\\\n \\hat{S}_{31} & = S_{31} =: S_{zr} ~;~~\\hat{S}_{32} = \\cfrac{S_{32}}{r} =: S_{z\\theta} ~;~~ \\hat{S}_{33} & = S_{33} =: S_{zz}\n \\end{align}\n ", "c53ed07cc7c8f5dce1e89874a9d94095": " B_1,\\ldots,B_n ", "c53ee01134cd84cfc524e500c86f19dc": "r^2-r+1\\text{ and }r^2+1", "c53f2396d00752c26ed1b95d75e02659": "\\therefore \\!\\,", "c53f3e9ab3a6f306e497c80c4742c636": "\\tilde{U}", "c53f44f7dcaa2871e4490a54c61afa03": "\\tfrac{2^{t}-1}{p}\\,\\bmod\\,p", "c53f5ca8cc45c8f9d145537f63caa343": "p=mv", "c53f5f8dee351e1289d1ba44a107913f": "1\\in\\mathcal{S}", "c53f89287f98c635bb21c3e6714629a9": "\\zeta(3) = -\\frac{1}{2} \\, \\psi^{(2)}(1).", "c53f9e333d5edcdfbbb61179fa8925f8": " \\ln P(V) = \\sum_H Q(H) \\ln \\frac{P(H,V)}{P(H|V)} = \\sum_{H} Q(H) \\Bigg[ \\ln \\frac{P(H,V)}{Q(H)} - \\ln \\frac{P(H|V)}{Q(H)} \\Bigg] ", "c53fb06057a900065ad225d9c0d17867": "1-\\rho", "c53fbf189994fa733bb1209ac421139f": "\\textstyle \\lambda(a_\\diamond\\mid A^c)", "c53fdd562a7fe654f8cd43754edc1877": "\\widehat{\\theta} = \\arg\\min_{\\displaystyle\\theta}{ \\left( -\\sum_{i=1}^n \\log{( f(x_i, \\theta) ) }\\right) }.\\,\\!", "c53fe2f947722a5a23457cbdafcf8d05": "\\tbinom{n}{0}", "c53ff923d2684355ee6808e7d8ef6aa8": "x_3=1", "c5402be7e76934629afe6a4acf856f2c": "W^-_{\\mu\\nu}", "c540a8e52943b6cb0988fa659395b570": " 2.512^{-m_f} = 2.512^{-m_1} + 2.512^{-m_2} \\!\\ ", "c540bde527dcf7f90bfaabcbe52a4668": "S={Q_{permeate}\\over Q_{feed}} = 1-{Q_{concentrate}\\over Q_{feed}}", "c5414dd4e8aa4cb9672fa70592795e51": "2*Kp*NB + NB*NB", "c54193d8781b00d800b24fabf6921aea": "\\{\\Phi_{00}\\,,\\Phi_{01}\\,, \\Phi_{10}\\}", "c54240cd0cad28d31bb3f47ece2d393a": "\\lambda_{11}\\lambda_{22}\\neq\\lambda_{12}\\lambda_{21}.\\,", "c542f602fb78c5d9319ea4871b045401": "\\Pi_{\\omega,\\mathbf{k}}", "c5431bc1b49a0ba7f4389570ff6df3ec": "y=f(x_1, x_2, \\ldots, x_n) ", "c5434b7ad6bef5db25781792786aad4a": "\\textstyle t", "c5437b33458d64bfd853dc73b01b2d07": "C^\\phi_{MX}", "c5439062ee7d15b7cafabbb91a47f509": "N_e = N + \\begin{matrix} \\frac{1}{2} \\end{matrix}", "c543b5be792a93ba56cedd24ec3d9005": "\\frac{T^2}{C_v}\\left(\\frac{\\partial P}{\\partial T} \\right)_V", "c543bda6de01d38d4dfcbde0073e1eab": "\\zeta(s) = \\frac{1}{\\det(I-s K)}.", "c543e12160b001d6ef8e9b787b14d39c": "E[F']", "c543ed44e6f5db1abdf9c7792b76c3bb": "S = \\sum_{i=1}^m r_i^2.", "c5441617f0ce1f1f0ebb81ac02beead0": "\n\\begin{align}\nx_{n-1}^2\\Delta^{n-1}(p_1r_1)]_a^b & = \\int_a^b (x^\\prime_{n-1})^2 \\Delta^{n-1}(p_1) - \\int_a^b x_{n-1}^2 \\Delta^{n-1}(q_1) \n- \\sum_{k=0}^{n-1} C(n-1,k)(-1)^{n-k-1}\\int_a^b p_{k+1} W^2(x_{k+1},x_{n-1})/x_{k+1}^2,\n\\end{align}\n", "c544176010331adf317cf1a82fb2b82a": "dx^{123}+dx^{145}+dx^{167}+dx^{246}-dx^{257}-dx^{347}-dx^{356},", "c544501931b59feb9715047dd917a518": "\\phi^{\\epsilon}(x)", "c5446a8f7ae259a2132abf80b7c60717": "1(n) = 1", "c544e44ac2cf00905f1558539e0e7208": "\\textstyle\\epsilon=\\frac{R_d}{R_v}=\\frac{M_v}{M_d}", "c5452f3aa2e243b2d29d06284753e1e1": " \\omega_c = \\frac{1}{2}(\\omega_s + \\omega_a).", "c54544801874d86b48c6538926543396": "y\\in E,E'", "c545464ca8bb00784ad4118c73fc3265": "\\left(z-c\\right)^{-k-1}", "c5465a3a4ae25379ca15e667b44e9558": "\\Phi_\\mathrm{s}", "c546739a16384236b0058626d9ce3f0b": "f(z)=a_0^\\nu J_\\nu (z)+ 2 \\cdot \\sum_{k=1} a_k^\\nu J_{\\nu+k}(z)\\!", "c5467b7da7b77ecaa718457c22508411": " X_0 ", "c546afcb6ad32979dc0ba646cb1c55f5": "\\nabla \\cdot \\left(\\frac{\\mathbf{r}}{|\\mathbf{r}|^3}\\right) = 4\\pi \\delta(\\mathbf{r})", "c546ca498046e1310283f5943ab14493": "\nG_{p+2,\\,q}^{\\,m,\\,n+1} \\!\\left( \\left. \\begin{matrix} \\alpha, \\mathbf{a_p}, \\alpha' \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\n(-1)^{\\alpha'-\\alpha} \\; G_{p+2,\\,q}^{\\,m,\\,n+1} \\!\\left( \\left. \\begin{matrix} \\alpha', \\mathbf{a_p}, \\alpha \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n \\leq p, \\; \\alpha'-\\alpha \\in \\mathbb{Z},\n", "c546ebc6dbf1110f131b0da730ce5272": "\\scriptstyle \\, (r,\\, s)", "c547b1f50b70eb232a2cb4232e77b8b5": "\\Delta H_r\\,", "c547b34fe96723c9520cc42711bb7672": " A^*A = I_n\\, ", "c547f08873d85ec4bb4b8a8c754a42dd": "{\\mathrm{T}}", "c548132306867ef2a1f55c64728d812f": "\\tbinom{2n-2}{n-2}", "c5481f7aee39b21ad3a4297067a983d4": "\\nabla \\cdot (\\mu(|\\nabla\\Phi|/a_0) \\nabla\\Phi) = 4 \\pi G \\rho ", "c548865a082e761d2386d9d2a9f7deb6": "E \\subseteq F \\Rightarrow g(E)\\leq g(F)", "c54895a6b6e01192262322b07a90d4d3": "c_6 = 7.28898 \\times 10^{-3},\\,\\!", "c548fe26553ce8fb62b009588d33bd63": "2\\alpha = \\theta", "c549008f591312f5a3f9095874459069": " =\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\}\n\\right\\} +\\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\hat{\\Pi}_{\\rho,\\delta}\n^{n}\\mathbb{E}_{X^{n}}\\left\\{ \\rho_{X^{n}\\left( m\\right) }\\right\\}\n\\right\\} ", "c5491e931b2f893d12cb2dd2c13fedd3": "\\beta_{ }^{ }", "c5497cce405d909a75038621798b3095": "\\textstyle L_1 L_3 L_5 R_1 R_3 R_5 P_1 P_2 L_2 L_4 L_6 R_2 R_4 R_6", "c54983a422e4772351cb2aabedda2e81": "E_{M_{J}} = -M_{J}g_{J}\\mu _{B}H", "c54995f08875f447e0ec49f1e6bfe889": "\n\\overline{u_i u_j} = \\tau_{ij}^{r} + \\overline{u}_i \\overline{u}_j\n", "c549fc321dc941530928950e8d095249": "\\xi \\in T_xM ", "c54a511e3f19ee3ec7c7f1e023459af0": "0<\\lambda_1\\leq \\lambda_2\\leq\\cdots", "c54a51543dd792187eb3e4a691840209": "k_1^+k_2^+k_3^+=k_1^-k_2^-k_3^-", "c54a954436ec3dea296940bfa7d54dbf": "\nq_4=\\cos \\frac{\\theta}{2}\n", "c54abbafbe5ad346cac8fb83ec268c8e": "t=6", "c54ae881ebbf30cfb145ff494506284e": "P_+=\\frac{1}{2}\\left(I+\\frac{A}{\\alpha}\\right)", "c54b0dee7658a6f981fa2f2f8d517163": "\nn^2 = n_1^2 + n_2^2 + n_3^2\n", "c54b1e5c866f69997c0c4c15d6d24d67": "\\mathbf{x} \\in \\mathbb{R}^n", "c54b7991c7df72caafd9613184016612": "S_\\pm |s,m\\rangle = \\hbar\\sqrt{s(s+1)-m(m\\pm 1)} |s,m\\pm 1 \\rangle", "c54b9321a53ded9ace41ee97602ffdd7": "\\bigg. R_e = \\frac{R_mem}{A} \\bigg. ", "c54ba2975a4688818ff0dcb1c078f65b": " G = \\frac{1}{2\\pi}B( \\tfrac{1}{4}, \\tfrac{1}{2})", "c54ba6e6d18ea7ae077aab1436b70980": " E f( a_i b_i + \\cdots + a_n b_n ) \\le \\inf \\left[ \\frac{ E ( e^{ \\lambda Z } ) }{ e^{ \\lambda x } } \\right] = e^{ -x^2 / 2 } ", "c54bdb30cf908556dd5bb971316f61b5": "\\mathbb{JKLMNOPQR} \\!", "c54be9992c4554a3e83ed593ce9b6fdb": "\n\\sum \\mathbf{F} = 0\\; \\Rightarrow\\; \\frac{\\mathrm{d} \\mathbf{v} }{\\mathrm{d}t} = 0.\n", "c54bf62965765af10ef7eef4344863e0": "Y=AL^{\\beta}K^{\\alpha}", "c54c2474ce54e98ab393a2a2d7caf8f0": "P \\rightarrow Q ", "c54c403e06df5a2543f2195dac34f404": "\\! e^{ita}", "c54c8cdea6a7d1bab33af9d199517cd8": " K_F(z,w)={F^\\prime(z)F^\\prime(w)\\over (F(z)-F(w))^2}.", "c54cac17d0660caf71fa4fdf036dd010": "[\\hat p,\\hat q]=\\frac1{2\\pi\\sqrt{-1}}.", "c54cdf440f6558e661230aa66c25d8b7": " \\varphi _j^{n + 1} - \\varphi _{j-1}^{n+1} = \\sum\\limits_m^M {\\gamma _m } \\left( {\\varphi _{j + m}^{n} - \\varphi _{j + m - 1}^{n} } \\right) = \\left\\{ {\\begin{array}{*{20}c}\n {0,} & {\\left[ {j + m \\ne k} \\right]} \\\\\n {\\gamma _m ,} & {\\left[ {j + m = k} \\right]} \\\\\n\\end{array}} \\right . \\quad \\quad ( 6)", "c54d09ff3bfc2e5035da889edde8f4d1": " \\sum_{n=0}^{\\infty} \\mbox{PL}(n) \\, x^n = \\prod_{k=1}^{\\infty} \\frac{1}{(1-x^k)^{k}} = 1+x+3x^2+6x^3+13x^4+24x^5+\\cdots. ", "c54d37a631cd15f9f612f850f5210d10": "\\bar{X}=KXK\\,", "c54d6f75f20caed25a3531b0abfb5fe3": "f(n) = 2f(n/2)-1=2((2l_1)+1) - 1=2l+1", "c54db6c7d0304525706edeba02b393ad": "\\tan 0=0\\,", "c54de11a0530c548693eaad0cccbe738": "x'_{1}=\\gamma\\left(x_{1}-vt_{1}\\right)\\quad\\mathrm{and}\\quad x'_{2}=\\gamma\\left(x_{2}-vt_{2}\\right).", "c54de92c577db9f94906faf618390861": "Q = 2\\pi K \\frac{\\left(D_b - D_m\\right) \\left(D_w - D_m\\right) }{\\ln \\frac{R_i}{R_w} }", "c54dff2e0d2010369af2bfa36366c72a": "\\operatorname{Dom}(n_o) = \\left \\{ n_o \\right \\}", "c54e01b313aee7bf20b1d6cc9a16648b": "\n \\text{(b)}\\quad\n \\tan\\beta=\\frac{\\delta x}{\\delta y}\n =\\frac{a\\delta \\lambda}{\\delta y}.\n", "c54e1db0a11d677ac30ca708bf3d1c13": "P(x,t) = |\\psi(x,t)|^2", "c54e3ab098920b507c418a02a94fce53": " d_Y(f(x_1), f(x_2)) \\le K d_X(x_1, x_2).", "c54e53bcce3e571ce1603dc0b04007c3": " \\nabla u ", "c54e8efc2dae231b55b735eb5ddcd34a": "\\xi \\in H^2({\\Bbb C} P^n)", "c54f15190ed50d85adf4ab701a7d11f8": "\\varphi_{\\beta}", "c54f59934a8ccf5e023cebca62587155": "n> d", "c57376c57a00047816ad180f685c3641": "f'(g(a)) g'(a)h + [f'(g(a)) \\varepsilon(h) + \\eta(k_h) g'(a) + \\eta(k_h) \\varepsilon(h)] h.\\,", "c57385430c8834e10b7c84d03d50684a": "\\Phi_{\\rm B} \\approx \\frac{1}{2} E_{\\rm bandgap}", "c573aa025f3342553ef329545b26a22b": "\\color{Black}\\text{Black}", "c574b1b82c28a5090dadc811b3d3693b": "x \\text{ such that } f(x) = 0\\,.", "c574fda554f5a9a338ec901a3facd499": "VT^{\\hat{c}_v}", "c575350ee66128c100efd1b5e087ea53": "\\, p_1 \\,", "c575881d9cfb9d37ac55996f07fee093": "a + b = P\\, ,", "c576382cce50e11c4d3371852d5b0658": "\\mathbf{E}_{\\text{Magnetic dipole}}(\\mathbf{x},t)=\\frac{-k^2 Z_0}{4 \\pi}(\\mathbf{n}\\times\\mathbf{m})\\frac{e^{i k r - i \\omega t}}{r}", "c5764531c47552915894d3197b9b2dde": "C_{i,j} = \\frac1{F_{i+j-1}},", "c5765209ef4748d12945f6de37f48b40": " y(1+y'^{\\, 2}) = 1/C^2 ~~\\text {(constant)} \\, , ", "c57668c65ab4700cea6327a7148051a4": "h^*(n)", "c5767aadd453d7f1187556c4916d98f3": "\\Omega_* = \\Omega(*) = MSO(*)", "c576a9cb29e89ec9a8b67b51604aca3c": "\\varepsilon \\log_2 n", "c576d31ef655c7612fb65863a7b5d7a2": "-\\infty < a < b < \\infty \\, ", "c576f14287c17f100ce008e9a669d390": "\\frac{8 \\alpha_\\text{G}}{\\alpha} \\,", "c5779398b3939f2a02da3b4dd2459eea": "\\dot\\gamma(t)=d\\gamma(t)/dt", "c577b2c385bcf167eb71e9c03fbb35f9": "I_z = \\frac{m r^2}{2}\\,\\!", "c5782d02d81a846613625d4e9db56533": "A(x\\rightarrow x') = \\min\\left(1,e^{S - S'}\\frac{g(\\boldsymbol{x}'\\rightarrow \\boldsymbol{x})}{g(\\boldsymbol{x}\\rightarrow \\boldsymbol{x}')}\\right)", "c578501695825a6091c5438057d59709": "\n\\omega_{n}^{2} f(t) \\ \\stackrel{\\mathrm{def}}{=}\\ \\omega_{0}^{2} \\left\\{h(t) - \n\\frac{1}{2\\omega_{0}} \\left( \\frac{dg}{dt} \\right)\n- \\frac{b}{2} g(t) - \\frac{1}{4} g^{2}(t)\\right\\}.\n", "c57878853e267fbbc0781aeea912dea6": "X^{a} = (\\xi^\\mu,1)", "c5791edd332d44ea68ac3e8b456bba8c": "O(\\exp(n))", "c5794afe34fbc1d2d524a373f8264492": "\\begin{matrix} 432 \\times {4 \\choose 1}{3 \\choose 1} = 20,736 \\end{matrix}", "c5794fe66765d927a9b3d911b5bc538d": "\\mathrm{C + 2 \\ H_2 \\ \\rightleftharpoons \\ CH_4}", "c57993cb8bab2ffa87a65fdcdb9e8a34": "H(e^{j\\omega}) = \\sum_{k=-\\infty}^\\infty{H_c\\left(j\\frac{\\omega}{T} + j\\frac{2{\\pi}}{T}k\\right)}\\,", "c5799c010c876b44bfda32fb9e118de4": "\\begin{cases}z, z+1 & \\{z \\in \\mathbb{Z}\\}:\\; z = (\\alpha-1)\\beta+\\lambda\\\\ \\lfloor z \\rfloor & \\textrm{otherwise}\\end{cases}", "c579bf60e2cc078f4c5cc8385c21132f": "OPSBI = ((OBP + SLG) * 1000) + RBI \\,", "c57a2374a8f9d04783aa60a9d826bde5": "-2\\frac{(b\\!-\\!c)e^{iat}\\!-\\!(b\\!-\\!a)e^{ict}\\!+\\!(c\\!-\\!a)e^{ibt}}\n{(b-a)(c-a)(b-c)t^2}", "c57a29889fe538cbbb7c88c6c8efbf4f": "\\tfrac{1}{\\sqrt{\\xi}}", "c57a49959ebd9ae7f947f182f62bdab2": "\\mathcal{F}=\\phi \\mathcal{R}_m", "c57a4bc902063579fb424e5020d2a8da": "0!=1", "c57a72cbff4bba1f7b357d29ecca17e6": "\\delta =", "c57a852a5af4d30c7a5a61ae050a11de": "\\Alpha \\Beta \\Gamma \\Delta \\Epsilon \\Zeta \\Eta \\Theta \\!", "c57a86bac44254610d64cb8139f55284": "g(x)=\\int_a^b K(x,y) f(y)\\,dy.", "c57a894b2d11a3104387e45de9b170ef": "\\overrightarrow{r_2}", "c57a8cc6295a62b78d3f13d656f28e15": " -{1 \\over 2} \\operatorname{tr} \\left[ - \\Sigma^{-1} \\{ d \\Sigma \\} \\Sigma^{-1} \\sum_{i=1}^n (x_i-\\mu)(x_i-\\mu)^\\mathrm{T} - 2 \\Sigma^{-1} \\sum_{i=1}^n (x_i - \\mu) \\{ d \\mu \\}^\\mathrm{T} \\right]. ", "c57b4955bbd67a21cdb2a47846d2a8ef": "\\mathbf e_n = (0, 0, \\ldots, 1)", "c57b6180c25ea518de043dd7313622f5": " p = (\\kappa(iw) + \\frac{4}{3} \\nu(iw)) \\frac{dW}{dz} \\quad (2.4.a) ", "c57b64552e3b9262a4ff97ec7a089f23": "\\mathrm{RP}", "c57b8e020d91f8b9c6d81122e5da3b83": " X = \\begin{bmatrix}\n\n 1 & X_{11} & \\cdots & X_{k1} \\\\\n\n \\vdots & \\vdots & & \\vdots \\\\\n\n 1 & X_{1N} & \\cdots & X_{kN}\n\n\\end{bmatrix}. ", "c57bef40b634a6ae4771d8f1878549f8": " \\scriptstyle f ", "c57bfc64d25415099370ba89e931940d": "X=\\bigcup_n (S_n(X)\\oplus nB)", "c57c0c69fb77fe4be9c460020670bb62": "\\eta \\geq 1,\\,", "c57c37bf89b3ccf814faca228f8b80de": "\\int_X (g+\\varepsilon 1_P)\\,d\\mu>\\int_X g\\,d\\mu=\\sup_{f\\in F}\\int_X f\\,d\\mu.", "c57ccd80e408982e8b1124821e7cde62": "(\\{0\\} \\times [0,1] ) \\cup (K \\times [0,1]) \\cup ([0,1] \\times \\{0\\})", "c57cf71e040db7f276635ee491bb3172": "q=1/2", "c57dda2b94c9005b3aa0ee6966e058f0": "\\Gamma_2 ", "c57e1583f8cbee462ed3243c7cb11294": "S \\otimes_R S/ S\\otimes_R R ", "c57e2720c11da4d9ed125dbef3cb083e": "\\pi_{\\mathbf{f}}|_{SO(2n-1)}= \\bigoplus_{f_1\\ge g_1 \\ge f_2\\ge g_2\\ge \\cdots \\ge f_{n-1}\\ge g_{n-1}\\ge |f_n|} \\pi_{\\mathbf{g}}", "c57e5c64ecde9de27e5e00d7133281cc": "c_x", "c57e729b790ce3906b61ee8575929dd3": "K_{\\rm{eq}} = \\frac{[{\\rm{D_3O^+}}] [{\\rm{OD^-}}]}{[{\\rm{D_2O}}]^2}", "c57ed52931f8e2896696f2907f83b3df": "(A\\to B)\\to((B\\to C)\\to(A\\to C))", "c57f0e87fb29dff3f226690bb8fa85f4": "\\sum_i m_i \\mathbf{R}_i=0\\,", "c57f18249d1c73047ab3d4042d913339": " b = b_1 + i b_2 ", "c57f8798c2f282acc4555e549f49a75b": "\\mathbf{r}_1 = (x_{11},x_{21},\\dots,x_{N1})", "c57fc29e9d83e4e2468d34798d431582": "a \\in N^n", "c5800dee674f7ddca3febad26ec3b3e4": "\n \\begin{align}\n & \\frac{\\partial \\sigma_{rr}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{r\\theta}}{\\partial \\theta} + \\cfrac{1}{r\\sin\\theta}\\frac{\\partial \\sigma_{r\\phi}}{\\partial \\phi} + \\cfrac{1}{r}(2\\sigma_{rr}-\\sigma_{\\theta\\theta}-\\sigma_{\\phi\\phi}+\\sigma_{r\\theta}\\cot\\theta) + F_r = \\rho~\\frac{\\partial^2 u_r}{\\partial t^2} \\\\\n & \\frac{\\partial \\sigma_{r\\theta}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{\\theta\\theta}}{\\partial \\theta} + \\cfrac{1}{r\\sin\\theta}\\frac{\\partial \\sigma_{\\theta \\phi}}{\\partial \\phi} + \\cfrac{1}{r}[(\\sigma_{\\theta\\theta}-\\sigma_{\\phi\\phi})\\cot\\theta + 3\\sigma_{r\\theta}] + F_\\theta = \\rho~\\frac{\\partial^2 u_\\theta}{\\partial t^2} \\\\\n & \\frac{\\partial \\sigma_{r\\phi}}{\\partial r} + \\cfrac{1}{r}\\frac{\\partial \\sigma_{\\theta \\phi}}{\\partial \\theta} + \\cfrac{1}{r\\sin\\theta}\\frac{\\partial \\sigma_{\\phi\\phi}}{\\partial \\phi} + \\cfrac{1}{r}(2\\sigma_{\\theta\\phi}\\cot\\theta+3\\sigma_{r\\phi}) + F_\\phi = \\rho~\\frac{\\partial^2 u_\\phi}{\\partial t^2}\n \\end{align}\n", "c58034f9f23906a0dcea076c138d0353": "\\operatorname{Perf}_s(f,r)", "c580708479729d6f3d852d3d61a8af26": "\\log(1+x)=\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}x^n}{n},", "c58079cf75fb310387f8911f84d861e6": "{\\mathit{He}}_8(x)=x^8-28x^6+210x^4-420x^2+105\\,", "c580a81f873d57cae3cfa9c6b3faca6f": "{dt \\over t} = \\Bigl(2 - {B\\lambda \\over 2}\\Bigr) b ", "c580bacedcb7543340e4248ceeeefcd3": "\\left( \\begin{smallmatrix} 2 & 0 \\\\ 0 & 2 \\\\ \\end{smallmatrix} \\right)", "c580fb0e55372ec7a6de6968243282c8": "{j^n(\\kappa)}", "c58123fb6e2e568754d873788ded13fa": "{\\displaystyle}x_{3}=\\sin({\\alpha}_{1}+{\\alpha}_{2})=\\sin{\\alpha}_{1}\\cos{\\alpha}_{2}+\\sin{\\alpha}_{2}\\cos{\\alpha}_{1}=x_{1}y_{2}+x_{2}y_{1}", "c58154c386b51d78454db8fdd39a66c0": "\\textstyle r ", "c5819538e5b63f993312bf2e27e80935": "\\hbar c/(\\mu c^2)", "c581a02657ecc5ae6e848f9183425aaf": "F1={2 * P * V \\over P + V}", "c581c0afb34412c1a09b09c6e2c6ec71": "\\scriptstyle x(t) \\,=\\, s(t) \\,+\\, n(t)", "c581d6ad0ee56bb0d3612d58194e9b6e": "\\Sigma_n b_n", "c5823c1b633faa9b6c861ab837c616f9": " y^2+xy=x^3+a_4x+a_6", "c5824b700cdcb2867ea539affee87e36": "\\sum_{i = 1}^n \\alpha_i w_i", "c58297738784f8e72168a41e070056ca": "h\\approx\\left(\\frac{8.7}{3.57}\\right)^2", "c582d73695f0e113ce86f955d16d3202": "R_p", "c582dec943ff7b743aa0691df291cea6": "ar", "c58306d4a37a3c7e9362d8e1316b6940": " \\frac{z - z_0}{\\overline{z} - \\overline{z_0}} = u", "c583bde2d2a78fbbb40b1988555756cd": "\\hat{v_i}'\\equiv i[\\hat{H}'_0,x_i]=i \\beta [p_0,x_i] ", "c5840cf7864a11e0865200ea8eed4fb8": "\\sqrt{\\frac{1}{n} \\sum_{i=1}^{n} x_i^2} = \\sqrt{\\frac{1}{n}\\left(x_1^2 + x_2^2 + \\cdots + x_n^2\\right)}", "c5840d0124fde2fe7f674aa144ea6937": "r_1 = s(\\beta)", "c5842cc067daaf8e85ad6e3ac7466cf8": "\\begin{pmatrix} I & M \\\\ 0 & I \\end{pmatrix} ", "c58459ec7d17c6c9ffc6e933ff76a156": "(x-x_1)^2+(y-y_1)^2=r_1^2 ,\\ \\quad (x-x_2)^2+(y-y_2)^2=r_2^2", "c5846c60c6153591e64c03aeb22f1610": "\nd\\tau^2 = [(x^2 - y^2) \\cos (2u) + 2xy \\sin(2u)] du^2 - 2dudv - dx^2 - dy^2\n", "c584877f21e11029b2299a2eb463672a": "f = f(x)", "c584ba7b6597e3ef90e8cba65f8e5999": "\\xi \\sim |p-p_c|^{-\\nu}\\,\\!", "c585113534295df6f9aa4981ef0c7757": "W(3^{7(2 \\cdot 3^7+1)},2) \\leq 3^{7(2 \\cdot 3^7+1)}+1.", "c5851d3807e341f1e46bf05a1dbce2c9": "\\scriptstyle H_0 ", "c58588c9e817a9595b97487d0c9f3780": "(3,1)_{-\\frac{1}{3}}", "c585d0613487800a15088831057d867c": "\\displaystyle{L_0={1\\over 2}A^*A={1\\over 2} z{\\partial\\over\\partial z}}", "c58629138c1b0ebfb9e5bf0ee3015e21": "n\\mathbb{Z} = (n)", "c5866a62af2a0ad3578740b6f6417493": "\\phi_{A,B}\\colon U(A)\\times U(B)\\to U(A\\otimes B)", "c586807d8f827a1a31ebc6c8ecbb7953": "f^{\\mathrm{F}}_p(x) = \\begin{cases}\n -\\log x & \\text{if } p = 1 \\\\\n 1 - x & \\text{if } p = +\\infty \\\\\n \\log\\frac{p - 1}{p^x - 1} & \\text{otherwise.}\n\\end{cases}\n", "c5868928f8c116e6701af56ec2f5af6b": "F\\to \\operatorname{Gode}(F)", "c586c2b6770907ebab5905947516e793": "\\Bigg(\\frac{a}{n}\\Bigg) = \\left(\\frac{a}{p_1}\\right)^{a_1}\\left(\\frac{a}{p_2}\\right)^{a_2}\\cdots \\left(\\frac{a}{p_{\\omega(n)}}\\right)^{a_{\\omega(n)}}.", "c5875bcc7a44f1c85c6f29f2810ee49b": " E = \\dfrac{C_L}{C_{D0} + k C_L^2} \\Rightarrow \\begin{cases} E_{max} = \\dfrac{1}{2 \\sqrt{k C_{D0}}} \\\\ (C_L)_{Emax} = \\sqrt{ \\dfrac{C_{D0}}{k} } \\\\ (C_{Di})_{Emax} = C_{D0} \\end{cases} ", "c58780f4131ec07dd45e0bea8859f264": "\\frac{ \\{P\\}\\ C\\ \\{Q\\} }{ \\{P \\ast R\\}\\ C\\ \\{Q \\ast R\\} }~\\mathsf{mod}(C) \\cap \\mathsf{fv}(R) =\\emptyset", "c587b2e6d5d749434b7554e25c9f0560": " \n\\mathbf{G}_1, \\mathbf{G}_2, \\mathbf{G}_3\n", "c587cb2ae89da54a18c688a0b2d20299": "\\begin{array}{rcl}\n\\frac{1}{N}\\log W \n&=& \\frac{1}{N}\\log \\frac{N!}{n_1! \\, n_2! \\, \\dotsb \\, n_m!} \\\\ \\\\\n&=& \\frac{1}{N}\\log \\frac{N!}{(Np_1)! \\, (Np_2)! \\, \\dotsb \\, (Np_m)!} \\\\ \\\\\n&=& \\frac{1}{N}\\left( \\log N! - \\sum_{i=1}^m \\log ((Np_i)!) \\right).\n\\end{array}", "c587ceeba554ae493ec76ead10020b7f": "T_{0} = 0", "c5880b3f20525622c40d55752c2a205c": " \\partial S ", "c5885e56f8f3d9a50576ec79c22e0e2e": " H_0(A) := \\mathrm{log}_b \\vert A \\vert .", "c588f03ac0031a033038474557064de6": "\\max_{x \\in X}f(x)", "c5892ccc6b43d328c1ef2633857d3423": "N + S = J", "c589be9e81d550d5cbcd09bf30319270": "3:m\\ ", "c589f4d3e966bb320a02a5566f424110": "\\begin{align}\n V_1 &= Z_{11} I_1 + Z_{12} I_2 \\\\\n -Z_L I_2 &= Z_{21} I_1 + Z_{22} I_2\n\\end{align}", "c58a4f7bda9ac67cef84f4757d09fafc": "\\mathbf{J} = \\int_{t_1}^{t_2} \\mathbf{F}\\, dt = \\Delta\\mathbf{p} = m \\mathbf{v_2} - m \\mathbf{v_1}", "c58a644561dd593bb7b45afca0749342": "K>1\\,", "c58a9cc9ab8cba72fbd2523a27351fb8": "[X:Y:Z:W]", "c58ac8cafd80332c450f82de83f67048": " \\lim_{N \\to \\infty} S_N f\\left(x_0 + \\frac{L}{2N}\\right) = f(x_0^+) + a\\cdot (0.089490\\dots)", "c58add292eb7540b9294fc8fc6fc0e4c": "\n\\psi^{(4)}(z)\n", "c58b04667bcf6700fa38285f33640500": "f(x_0)", "c58b1a8d1c167575c891932e66354a19": "g: B \\to Q", "c58b1e5d351767c08639c5e924eefbf0": "\\mathbin{\\ast'}", "c58b32dcbde341b3a401ffc4c8944fb7": " \\operatorname{tail} \\equiv \\lambda z.\\operatorname{second}\\ (\\operatorname{second} z) ", "c58b3b4d9681421a8abf2d002624c7b8": "\n\\frac{98.4\\,^{\\circ}{\\rm F} - \\text{rectal temperature in Fahrenheit}}{1.5}\n", "c58b4021b2827f8a5224834abf7dbcc1": "a = b = 1", "c58b89d1f3d857fdbc0b13afe6982418": "\\Lambda(1-s,\\overline{\\chi})=\\frac{i^ak^{1/2}}{\\tau(\\chi)}\\Lambda(s,\\chi).", "c58c285d0ee8a8b3c39f67b0753f064d": "\\mathbb{PT}^-", "c58c2dc5cc7d75e012133eaff7c9bac7": "\\ \\displaystyle r_{c}\\ ", "c58c48a11eaa44255c1ee1a0446d6956": "\\hat{\\lambda}_{k-1} = \\textbf{F}_k^T\\tilde{\\lambda}_{k}", "c58c5f0e26014e89ebc1c3a62bc70629": "(\\forall \\epsilon >0) (\\exists \\eta >0) (\\forall x) \\;|x-a|<\\eta \\Rightarrow |L-f(a)|<\\epsilon,", "c58cc7b0b92db1013fb950effe2802d2": "X^3 + aX = b", "c58cdda7ec824c5b331b4658bad1d5aa": "\\int_{-\\infty}^{+\\infty} e^{-x^2}\\,dx", "c58ceacf49bdb95ddc93f0bcda76a1a0": "\\mathbf E_i\\,\\!", "c58cf4598f20629e63b47741089ee64d": "\\int_{\\mathbb{R}^n}\\!\\varphi(x)\\mathrm{d}x=1", "c58d20565a6e95079003a4da572fd4b3": " \\vec{f} = \\vec{j} \\times \\vec{B} ", "c58d3a4398be4166c968756e524f0da7": "x \\vee y \\vee x = x \\vee y", "c58d65e10ec71eab80533ad01d791181": "v_{1}'=-i\\,\\xi\\,v_{1}+q_{1}\\,v_{2}+q_{2}\\,v_{3}", "c58d808c6def2672f689cd7d2f1c195f": " F(s) \\cdot G(s) \\ ", "c58db10a5c7b36d00766448c928fd27f": "\n \\begin{align}\n EI\\dfrac{dw}{dx} &= \\dfrac{Pbx^2}{2L} - P\\cfrac{\\langle x-a \\rangle^2}{2} + D_1 & &\\quad\\mathrm{(v)}\\\\\n EI w &= \\dfrac{Pbx^3}{6L} - P\\cfrac{\\langle x-a \\rangle^3}{6} + D_1 x + D_2 & &\\quad\\mathrm{(vi)}\n \\end{align}\n ", "c58e5c01312bc3cde9923e3b969167d4": "w,x,y,z", "c58e806e96cd04a2eeeee8fa15c3cc24": " \\frac{\\partial f(x,t)}{\\partial t}=\\frac{\\partial g(f(x,t))}{\\partial x}\\,", "c58efe7195df1ef6f95456eb3b095c44": "\\displaystyle{f_0(h) \\ge f_\\varepsilon(h) \\ge f_0(h) - C(\\varepsilon) |h|,}", "c59034f99c138863ba68dfb37b6c8d50": " G(X)/G^0(X)=G", "c5903be58cc73fbebe4ebbede7f2daea": "E_{obs|ref1}=300 mV + 197 mV=497 mV", "c5907ece66cd06857a7798b07c13dbbb": "E\\left(\\epsilon(x_0)\\right)=0 \\Leftrightarrow \\sum^{N}_{i=1}w_i(x_0) \\times E(Z(x_i)) - E(Z(x_0))=0 \\Leftrightarrow", "c590a11dc6b48c96eb4db57f27b44e54": "\\sum_{n=0}^N \\nu_n(t_m)S_n(t_m)", "c590ac89f8804554dadf5e5525dc2260": "\nf(x) =\n\\begin{cases}\n1 & -1 \\le x < 0 \\\\\n\\frac{1}{2} & x = 0 \\\\\n1 - x^2 & \\text{otherwise}\n\\end{cases}\n", "c590c78475377771e85d7ee75a762932": "O(n^c),\\;c>1", "c5910dc9b981ad7f49ae9740c5048aae": "\n \\hat{p}^{\\mathrm{b.a.}}(t_j) = \\hat{p}_j^{\\mathrm{b.a.}} - \\langle\\hat{p}_j^{\\mathrm{b.a.}}\\rangle\n = \\frac{2}{c}\\bigl(\\hat{\\mathcal{W}}_j - \\mathcal{W}\\bigr) \\,,\n", "c591734520371a603fe877d38300e03e": "\\mathcal{L}:B\\to B", "c59189864aece8e9a9cedfe845364030": "\\tilde{\\mathbf{Gr}}(r, n).", "c591ad58794c37db612587603f86a2a7": "\\mathbf{g}=- {G M \\over r^2}\\mathbf{\\hat{r}}", "c592182c83fb0ff4f3e1589d7ca13f32": "s_{T,P}-s_{T,P}^{\\mathrm{ideal}}=R\\left[\\ln(Z-B)-2.078\\kappa\\left(\\frac{1+\\kappa}{\\sqrt{T_r}}-\\kappa\\right)\\ln\\left(\\frac{Z+2.414B}{Z-0.414B}\\right)\\right]", "c59236f47a747cb1dc7b8f0414ebded1": "c_i \\in \\mathbb{Z}, \\sigma^i \\in S", "c59254dfc7c8a03490721e0e94803781": "H(A)=\\bigcup_{i=1}^Nf_i(A).", "c59257ef299283f572afcfe9e061fa24": "\\text{EMA}_{\\text{today}} = { \\alpha \\times (p_1 + (1-\\alpha) p_2 + (1-\\alpha)^2 p_3 + (1-\\alpha)^3\np_4 + \\cdots ) }", "c59262292164839aa956ef679e42788d": "F\\colon C^\\mathrm{op}\\to\\mathbf{V}", "c59276110c2452e06684c5c2094b7552": "p(x^n)", "c592a4af9d5f61c03309fee17ecc94d2": "\\mathbf{C}_{i\\mu} \\ ", "c592b8369c1dfb4eeb95636760e70446": " \\omega = \\frac i2 \\partial \\bar\\partial \\rho ", "c5933572239739dc5232c0eda503d0ff": "T,", "c59372eb29742b3ae3bf70328e86dbac": " x^j (x, y, z, \\dots) = \\mathrm{constant}\\ , ", "c593ae692832e500a04c7f47900f689a": "\\mu_k", "c593ca2cfbe1cc2a2b0e8c02ac19263c": "\\hat{\\theta} = \\frac{1}{kN}\\sum_{i=1}^N x_i", "c594122d171635ac3e483fef71ef430b": " I = \\sum_{y \\in Y} \\operatorname{E}_y. ", "c5941e32e365a8be39419a36c7410141": " | \\psi^{(\\pm)} \\rangle = | \\phi \\rangle + \\frac{1}{E - H_0 \\pm i \\epsilon} V |\\psi^{(\\pm)} \\rangle. \\,", "c5941fae92a9ecac7f9caeffb080e6fc": " \\boldsymbol{\\sigma}^{dev}", "c5948312b6f5c5ed9fad30c514bfcf92": " x^{(q + m - 1)/m} + ax ", "c594d1159b6e56904cbc701dbd2df1c6": "\\mathbf{x}^1, \\ldots, \\mathbf{x}^{hms}", "c5957c4ff8605e13e364b96d45c4cf06": "Y^m_\\ell = Y^m_\\ell(\\theta,\\phi) ", "c59589ad88312e7a4b11a174669055ee": "\np(f(\\mathbf{x}')|S,\\mathbf{x}',\\boldsymbol{\\phi}) = \\mathcal{N}(m(\\mathbf{x}'),\\sigma^2(\\mathbf{x}')),\n", "c5958ecec71071086d0408599405f5f0": "Z_{ij} = \\left\\{\\begin{matrix} \n|Y_{ij} - \\bar{Y}_{i\\cdot}|, & \\bar{Y}_{i\\cdot} \\mbox{ is a mean of i-th group } \\\\ \n|Y_{ij} - \\tilde{Y}_{i\\cdot}|, & \\tilde{Y}_{i\\cdot} \\mbox{ is a median of i-th group } \\end{matrix}\\right.", "c5963de54a8bf47e3a85db40449525cc": " \\operatorname{E}(X) = \\int_\\Lambda X(\\lambda) \\rho(\\lambda) d \\lambda ", "c59678f193a7f05697f8912bafd6a8d6": "\\scriptstyle{\\arccos}\\tfrac{1}{\\sqrt{5}}", "c596e07b7120ccedecaf769c3f17a3d8": "(-1)^n a_n.", "c596e1b80f2b879819d4e7778201bb11": "\\scriptstyle{\\frac{\\log(4)}{\\log(2(1+\\cos(a)))}} \\in [1,2]", "c597544308e06fbc4a08265c6b3ff110": "V_a = \\frac{m_e}{2}{\\omega^2}(x^2+y^2+z^2),", "c59770951cfe44f72d7b3593ba25b8c8": "\\scriptstyle x^n + y^n = z^n", "c597a2b6cc333999919b0a0dc8447a71": "dI_\\nu=j_\\nu\\,\\rho\\,ds", "c597aba645cb0ca5e930207f2d8bf2e3": "\\operatorname{pos}(U)", "c597df01efa1f0e4b7ab185ca1ea7ebf": "\\frac {E_2} {E}", "c597ff3e353d2b53d1ba621fcfe192ba": "T_A = 0 : \\frac{a-b+c}{b} : \\frac{a+b-c}{c}", "c59801296d804f7bbcccbc2d0fb81f31": " \\left(\\frac{M L}{T^2}\\right)", "c5989bf6fcc3043a42e130f97dbba14f": "\n\\mathbf{P} = \\varepsilon_0\\chi\\mathbf{E},\n", "c598a5dbeb4bf47c16bad4b061343256": " \\eta \\in \\Gamma \\left( \\bigwedge^{2} TM \\right) ", "c598c57dd1d7b13a066bcc8b2f45e791": "\\Phi(x\\vert(-\\infty,t])=\\begin{cases} \\displaystyle\\frac{\\Phi(x)}{\\Phi(t)}, & \\text{if }\\Phi(x)\\le\\Phi(t)/2 \\\\ \\displaystyle\\frac{\\Phi(x)+\\Phi(t)-1}{\\Phi(t)}\\or 0.5, & \\text{if }\\Phi(t)/2\\le\\Phi(x)<\\Phi(t) \\\\ 1, & \\text{if }\\Phi(t)\\le\\Phi(x) \\end{cases}", "c598d7cc3e5eda729f78fa53d678b8e2": "\\ell_i \\in \\{0, 1\\}", "c5990569bd17876052580d9c09a48f85": "\\mathbf{F}_{\\mathrm{ext}} + \\mathbf{F}_{\\mathrm{thrust}} = m\\mathbf{a}_{\\mathrm{cm}}", "c5998064d4de8a16a584dd82a7a4920f": "S_{ij} = { }_{i+j}\\mathbf{C}_{i} = \\frac{(i+j)!}{(i)!(j)!}.", "c599a0608bdba0d0613fd1f500435396": "a = 0.1", "c599f21265f968c635c23353fcf286cb": "\\rho = \\frac{i\\hbar}{2mc^2}\\left(\\psi^{*}\\frac{\\partial \\psi}{\\partial t} - \\psi \\frac{\\partial \\psi^*}{\\partial t}\\right)\\, ,\\quad \\mathbf{j} = -\\frac{i\\hbar}{2m}\\left(\\psi^* \\nabla \\psi - \\psi \\nabla \\psi^*\\right) \\quad \\rightleftharpoons \\quad J^\\mu = \\frac{i\\hbar}{2m}(\\psi^*\\partial^\\mu\\psi - \\psi\\partial^\\mu\\psi^*) ", "c599fe675f4d9792cb15d45290a3e3ae": " Y_{21} = \\frac {1} { R_z } \\qquad Y_{11} = \\frac {1} {R_x} + \\frac {1} { R_z } \\qquad Y_{22} = \\frac {1} {R_y} + \\frac {1} { R_z } \\, ", "c59a0b7d5957b09106fd1adf9feed6e4": "C\\frac{de_2}{dt} + G(e_2 - e_1) = 0", "c59a3bf8e21696ae4f1dd63b6157f46a": "H_j = \\frac{ {\\displaystyle \\sum_{k=1}^T \\omega_j^k / T} }{ {\\displaystyle \\sum_{k=T+1}^N \\omega_j^k / (N - T)} }", "c59a3cd06a28c04ec5497f672795c8b7": "(n/w)^w", "c59ad306a21561e1f0971b024f4fc90f": "(\\sqrt{5}-1)/2", "c59ae5083f04beaa931ef9a06c21de6f": "\n \\begin{align}\n \\sigma_{11} & = -p + 2C_1\\lambda^2\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right] \\\\\n \\sigma_{22} & = -p + \\cfrac{2C_1}{\\lambda}\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right] = \\sigma_{33} ~.\n \\end{align}\n ", "c59b0c8998cfc6ea0a52d1cd9e77cd7b": "\\phi_0\\left(x\\right) = f\\left(x\\right).", "c59b15e365a482f5dc8e5a78eb033ac8": "\\sum_{i=-\\infty}^\\infty a_i \\delta_{ik}=a_k.", "c59b435bf3f929813dd491916b1d2a7d": " F = \\frac{mv^2}{r} ", "c59b632746ae1d38e2a6b3bb40ea5d2a": "\\mathbf{s} = \\tfrac{1}{2} \\left(\\mathbf{v} + \\mathbf{u}\\right) \\mathbf{t}", "c59b678886b3ea31388775e8cf95cc88": "\\Gamma=dx^\\lambda\\otimes(\\partial_\\lambda +\\Gamma_\\lambda{}^\\mu{}_\\nu\\dot x^\\nu\\dot\\partial_\\mu).", "c59b82e3906e2ec7bf6c084b59c4b4d9": " S, T \\subseteq N ", "c59c722fa4a5e621eecaf4839bae99a6": " \\left(\\boldsymbol{u} \\cdot \\boldsymbol{\\nabla}\\right) \\boldsymbol{u} = \\tfrac12 \\boldsymbol{\\nabla} \\left(\\boldsymbol{u} \\cdot \\boldsymbol{u}\\right) - \\boldsymbol{u} \\times \\boldsymbol{\\nabla} \\times \\boldsymbol{u} = \\tfrac12 \\boldsymbol{\\nabla} \\left(\\boldsymbol{u} \\cdot \\boldsymbol{u}\\right) \\qquad (1)", "c59c9123f6f83f44deee2f270bd1aecb": "\\operatorname{pd}_R M > 0", "c59daf2711b7af0e6d29395c6504f872": "{\\hat r}_t(i)", "c59db2191a5cd96107b2768e3a5ce17c": "A \\rightarrow CDC: B", "c59de1974807c92c1210c4490aed2f02": "\\sigma_\\pi^{2}=\\sigma_m^{2}-\\sigma_f^{2}=\\sigma_m^{2}-K .", "c59e010195943733dd94013cb242265e": "x=a\\cos\\left(u\\right)\\sin\\left(v\\right)", "c59e1b22738782577c2b7a7048ebe085": "F \\leq F_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\langle \\mathcal{H} \\rangle_{0} -T S_{0}", "c59e23b0567cc1084580827bb1630630": "\\delta_0 > \\| f(t_0) - f_e \\| ", "c59e2507e709c0079c9c9b18bbf7fc77": "X / \\approx_X \\; \\to \\; Y /\n\\approx_Y", "c59e2e35b6d7e03ae8d80460b9c33ae6": "L+2H \\rightleftharpoons H_2L:\\log \\beta_{012} =\\log \\left(\\frac{[H_2L]}{[L][H]^2} \\right)=pK_3+pK_2 ", "c59e859298457b15bfc473b0bf808fd8": " g(z)=\\frac{e^{iz}}{z +i\\epsilon} ", "c59ef6242eba55ed173312d62dd06e97": "\\theta\\in[0,\\pi).", "c59f2a89264a4faf7333be4776512663": "\\begin{bmatrix}0 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 0\\end{bmatrix}", "c59f553f69d3b07d36b28d2e33f61113": "\\underset{\\boldsymbol{w}}{\\min} \\quad \\|\\boldsymbol{w}\\|^2 + C \\sum_{n=1}^{\\ell}\n \\underset{y\\in\\mathcal{Y}}{\\max} \\left(\\Delta(y_n,y) + \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y) - \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y_n)\\right)", "c59f7489e2fe7c4f0d93c69051a401e3": "F(c,X)", "c59fff2e44881c7e28f708f6d1882548": "Y_0 = x_0;", "c5a011f111448ab16abc52a08b0925ab": " \\cup A_\\alpha", "c5a01f2ba7f74589e63a4426ca3e141a": "\n\\begin{align}\n\\Gamma(s,z) &= e^{-z} U(1-s,1-s,z) = \\frac{z^s e^{-z}}{\\Gamma(1-s)} \\int_0^\\infty \\frac{e^{-u}}{u^s (z+u)}{\\rm d}u =\n\\\\\n &= e^{-z} z^s U(1,1+s,z) = e^{-z} \\int_0^\\infty e^{-u} (z+u)^{s-1}{\\rm d} u = e^{-z} z^s \\int_0^\\infty e^{-z u} (1+u)^{s-1}{\\rm d} u.\n\\end{align}\n", "c5a032779338862da975051876dee988": "E^{\\mathrm {pot}}", "c5a08c8f28a900748bad6848d3c7eeb9": "\\vec{s}_a^{\\;2}", "c5a0dfc02b3c53bbda2fc0a5d9c9c405": " h_i := S_{(i)} ", "c5a0f66abb41232d4d6e6e79954e3ee2": "\\sigma \\tau \\upsilon \\phi \\chi \\psi \\omega \\!", "c5a1187475a00b56dd11331a586465fa": "d=L", "c5a16d76e44a5e5db609682f54869a60": " \\binom{n-1}{x-1},", "c5a1b48ad1296f7e0801b638ee3cfa02": " \\ \\textbf{f}_2 ", "c5a1c63b2607c5bbb7b7f9c535b369be": "q_1-1", "c5a1d87431ad0f25164190277b1545d5": "\\scriptstyle \\mathcal{R}_1,\\ \\mathcal{R}_2,\\ \\dots", "c5a1dc9ba561c517e79f2fad6f455c07": "\\sigma_t = \\sigma_s + \\sigma_\\gamma + \\sigma_f + ...", "c5a1ec3a949d5a6aefc0fb14dc04f933": " = h \\left[ \\beta_k f(t_{n+k},y_{n+k}) + \\beta_{k-1}\nf(t_{n+k-1},y_{n+k-1}) + \\cdots + \\beta_0 f(t_n,y_n) \\right]. ", "c5a20f158d72d9c50a70660a2c806062": " \\ \\cos (0) = 1 ", "c5a2d53b814fc47919541872c60244c3": "R_{bef}", "c5a2db26d419ab01048ea08c75706442": "\n \\cfrac{r^2}{R}~\\cfrac{d^2R}{dr^2} + \\cfrac{r}{R}~\\cfrac{dR}{dr} + \\cfrac{r^2\\omega^2\\rho_0}{\\kappa} = -\\cfrac{1}{Q}~\\cfrac{d^2Q}{d\\theta^2}\n ", "c5a30b9d4bb956cf06f6000917d61e0e": " \\operatorname{de-lambda}[E = F] ", "c5a34a0be0ff83a027568a9aeee1b4fc": "\\vec v_{T|E}", "c5a3bd5000a43a87187e4c4bb1ed6381": "\n\\begin{align}\ny_1 &= x + z_1 \\\\\ny_2 &= x + z_2.\n\\end{align}\n", "c5a3e44ddace795f36ad7be867f67e0d": "T(T(X)) \\rightarrow T(X) ", "c5a414b894e14f26dd1ed031a70e6429": "\\lim_{n \\mapsto \\infty} t_n=\\tfrac{1}{4}, \\quad \\lim_{n \\mapsto \\infty} s_n =\\tfrac{1}{2},", "c5a464518c5ed3dcca9b4cd260cd41cc": "M_G:=M/DM, \\, ", "c5a46b2e271bc6fb6e7762763ea383b2": "\\mathbb{Z}\\times\\mathbb{Z}", "c5a4c0b0b65d89dbf61fe8b98d14fdea": "\\rho_{\\mathrm{second}}", "c5a56825e420306c5ce39a4dd251a817": "|k-k'| \\leq \\epsilon k", "c5a639bf3605b125833629f34ec5fdf4": "\\Sigma||g_k||_u", "c5a6436afdbcc4c734a61bfeade7f71d": "g_c(E) = \\sum_k T_k \\sum_{n=1}^\\infty\n\\frac{1}{2\\sinh{(\\chi_{nk}/2)}}\\,\ne^{i(nS_k - \\alpha_{nk} \\pi/2)}.", "c5a6f48171e2c289c0b8bdf7e7046c13": "\\begin{align}\n A(z) & = \\sum_{k=0}^\\infty (\\omega_1^k + \\ldots + \\omega_m^k)z^k \\\\\n& = \\sum_{i=1}^m \\frac{1}{1-\\omega_iz},\n\\end{align} ", "c5a712d0a46d1cb24826b7f0ca82e312": "P_S\\left(n\\right)\\,\\!", "c5a8398165f02f1f2caea50946e47154": " s_{xy} = s_{yx} \\ ", "c5a862e0c3f5aed0005e7b26fd60655c": "\nQ_{r} = \n\\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\left( \\frac{\\partial \\mathbf{r}_{k}}{\\partial q_{r}} \\right) =\n\\sum_{k=1}^{N} m_{k} \\mathbf{a}_{k} \\cdot \\left( \\frac{\\partial \\mathbf{a}_{k}}{\\partial \\alpha_{r}} \\right)\n", "c5a86baaf35120fc0dfdf5f462c1f893": "\\begin{align}\nF(z, m) =& \\mathcal{Z} \\left\\{ \\cos \\left(\\omega \\left(k T + m \\right) \\right) \\right\\} \\\\\n =& \\mathcal{Z} \\left\\{ \\cos (\\omega k T) \\cos (\\omega m) - \\sin (\\omega k T) \\sin (\\omega m) \\right\\} \\\\\n =& \\cos(\\omega m) \\mathcal{Z} \\left\\{ \\cos (\\omega k T) \\right\\} - \\sin (\\omega m) \\mathcal{Z} \\left\\{ \\sin (\\omega k T) \\right\\} \\\\\n =& \\cos(\\omega m) \\frac{z \\left(z - \\cos (\\omega T) \\right)}{z^2 - 2z \\cos(\\omega T) + 1} - \\sin(\\omega m) \\frac{z \\sin(\\omega T)}{z^2 - 2z \\cos(\\omega T) + 1} \\\\\n =& \\frac{z^2 \\cos(\\omega m) - z \\cos(\\omega(T - m))}{z^2 - 2z \\cos(\\omega T) + 1}\n\\end{align}", "c5a8ee23d46b086c7c607dd28500f103": "pV = \\frac{2}{3} N \\bar{E_{\\rm k}}\\,", "c5a949557d2041d66d8ba6161f74a23b": "W(\\hat x)=\\bigl(w_1(\\hat x),..,w_n(\\hat x)\\bigr)", "c5a959b79524c43ae6220aa562901375": "\\sigma_3 = (1,2,3)", "c5a95c802a8183de24141eaaa111136f": "p(n)=\\frac{1}{2 \\sqrt{2}} \\sum_{k=1}^v \\sqrt{k}\\, A_k(n)\\,\n\\frac{d}{dn} \\exp \\left({ \\pi\\sqrt{\\frac23} \n \\frac{\\sqrt{n-\\frac{1}{24}}}{k} }\n \\right)\n", "c5a96e43944043127220edb49ca492e4": "T < T_c", "c5aa23d48996134e4fa541351a05d300": "A\\subseteq X \\,", "c5aa87634d09bf4460ce7ccf90592754": "a^2 x^2 + b^2 y^2 = (x^2+y^2)^2.\\,", "c5aab57e3b91faa2731f6c91e8e909de": "T(4k+1)", "c5aae2bc39c340693f68a52216124acb": "d=360", "c5aae9ee7e0f4e23cfe8b49c656d63c8": "(d_A, Q_A)", "c5aaf03b0f4141fd996e377f2790a16a": "p-q=\\frac{1}{\\pi}\\Delta\\arg f(iy)= \\left.\\begin{cases} +I_{-\\infty}^{+\\infty}\\frac{P_0(y)}{P_1(y)} & \\text{for odd degree} \\\\[10pt] -I_{-\\infty}^{+\\infty}\\frac{P_1(y)}{P_0(y)} & \\text{for even degree} \\end{cases}\\right\\} = w(+\\infty)-w(-\\infty).", "c5ab6773a8829bdd8fc8ab0b3fb1e8db": "|S_j|", "c5aba6d2da19e53e5834819e04c89ac5": "g(x, y, t) = h(x^2+y^2, t)", "c5abfbaab6b18d308289e1d2d68bb3ee": "P(c), P(f(c)), P(f(f(c))), \\ldots", "c5ac3bdb22767993721b45fee62876dc": " g (a\\cdot\\psi) = (ga)\\cdot (g\\psi)", "c5ac3f297c199889ec8b1419645cd39c": "\\begin{align} \ndV & = |(h_1 \\, dq_1\\hat{\\mathbf{e}}_1)\\cdot(h_2 \\, dq_2\\hat{\\mathbf{e}}_2)\\times(h_3 \\, dq_3\\hat{\\mathbf{e}}_3)| \\\\\n& = |\\hat{\\mathbf{e}}_1\\cdot\\hat{\\mathbf{e}}_2\\times\\hat{\\mathbf{e}}_3| h_1h_2h_3 \\, dq_1 \\, dq_2 \\, dq_3\\\\\n& = J \\, dq_1 \\, dq_2 \\, dq_3 \\\\\n& = h_1 h_2 h_3 \\, dq_1 \\, dq_2 \\, dq_3\n\\end{align}", "c5ac5c409bddc2fe78dc85a15777ba6d": "\\int_{0}^{\\infty} e^{-ax}\\cos bx \\, \\mathrm{d}x = \\frac{a}{a^2+b^2} \\quad (a>0)", "c5ac73d214418f75d7e4d98126fcb33c": "x\\mapsto x+1", "c5acb5b75ae5f3f51514f6f33d1183b0": "\\int\\frac{dx}{R} = \\frac{1}{\\sqrt{a}}\\ln\\left|2\\sqrt{a}R+2ax+b\\right| \\qquad \\mbox{(for }a>0\\mbox{)}", "c5acba8e3c95ddbd5895acb3bf8e3c04": "c>b>a", "c5accc69791b469f2ce6bde7e27a4506": "\\Gamma(z)", "c5acd211812a73cc071c46262c0d9bec": "1.177\\sqrt{p}", "c5ad456953807960c6b77e25ed0f2583": "Q(\\mathbf{x}) = \\lambda_1c_1^2+\\lambda_2c_2^2+\\dots+\\lambda_nc_n^2,", "c5ad5245d12cc4b5df43ef3e0c0202eb": "\\mathbb{E}\\left[g(X_1,\\dots,X_d)\\right]=\\int_{[0,1]^d}g(F_1^{-1}(u_1),\\dots,F_d^{-1}(u_d)) \\, dC(u_1,\\dots,u_d).", "c5ad6a043a6bb84e0e033919639753b8": "E[X]_{\\theta \\theta}, \\, E[X]_{\\phi \\phi}", "c5add31777a8e95dc92d278d9088b6ee": " = 5 \\times 5 + 2 \\times 2", "c5ae1fe1b26373fab4b374393329b022": " i=0, 1, ... n+1 ", "c5ae20cf770cc073b39a92d233a67171": "\\tau_1 = 5", "c5ae35ffabfc30194ff57ef6d7338b84": "h(\\lambda,\\,\\phi)", "c5ae6abe9e60231f9484b33657e18a8f": "\\tfrac{b}{\\|\\mathbf{w}\\|}", "c5af11dbeba47668c097f394d12c5698": "\\mu = \\lim_{n \\rightarrow \\infty} c_n^{1/n}", "c5af188d925e6b25ce2cf0bfbe051f4b": "\\Delta(\\tau) = 16\\pi^{12} \\left[\\vartheta(0; \\tau) \\vartheta_{01}(0; \\tau) \\vartheta_{10}(0; \\tau)\\right]^8", "c5af1daab33eceb226444500b0f3db74": "\nG(x) = \n\\begin{cases}\n\\frac {\\sin (x)}x & \\text{ if }x \\ne 0\\\\\n1 & \\text{ if }x = 0,\n\\end{cases}\n", "c5afb7dbb36d9abc2237f392cb69d461": "\\lambda,\\mu,\\nu", "c5afbbbd7fc062d7825e5a1ba6ee2735": "\\frac{y_n}{z_n}Z-Y", "c5afcaa7e14bd870191659576701e330": "y_2=v(t)y_1(t)\\,", "c5b02bda84f924c843650f187282276a": "(\\overline{C} \\vee (A \\vee B)) \\wedge (C \\vee (\\overline{A} \\wedge \\overline{B}))", "c5b061064fc91d831bfa2f26175b9b82": "x^2 + 1 = 0", "c5b06365a3730f04d91771a50dff31d3": "0 a^2 / \\lambda", "c5b472884f7f26910ebd19d232ca6e4d": "ds^{2} = -dt^{2} + e^{2t/\\alpha} dy^{2}", "c5b48ee3fdcbdbb77b957038bd63f85b": "n \\ge 1,", "c5b4ad76a4b83eba6b08f37756aab45f": "b = l + t \\ ", "c5b4b9e92d407e010024014554d11326": "\nR=|\\langle z \\rangle| = e^{-\\gamma}\n", "c5b4db051030c17b008ab46ffd14e242": "A \\equiv U-TS\\,", "c5b4e113502b7bd46dfce548ccb81324": " \\psi = 1- J(\\phi)\\, ", "c5b5496cc4080e320e9788fc896c6aa0": " \n\\frac{P(A \\rightarrow B)}\n{P( A \\leftarrow B)} = \\exp [ \\beta ( W_{A \\rightarrow B} - \\Delta F\n)]. \n", "c5b5654a3250b2c167c3ba3bac2f781d": "\\hat e = y - X \\hat \\beta.", "c5b5679ddc8f49dad884c3affc39a6df": "\\textstyle S_r", "c5b577b4e0cdda5c049e0d13fb891a27": "n! [z^n] \\exp\\left(\\log \\frac{1}{1-z} \n- \\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty\\frac{z^k}{k}\\right)\n\\sum_{k>\\lfloor\\frac{n}{2}\\rfloor}^\\infty \\frac{z^k}{k}", "c5b5c57bf6b1088a8b863b740a16c6d9": "\\textrm{GEV}(\\mu,\\,\\sigma,\\,\\xi)", "c5b6086c2733897f173fa0c9d15e6248": "\\textstyle \\lceil \\log_2(255) \\rceil = 8", "c5b67c7307707156d2be0909bdbc5ae8": "\\delta=\\frac{N-1}{N}\\frac{\\pi}{2}\\ ", "c5b6dc591a08f8571a2b7369b6e2c883": "M_{sys}", "c5b7457a64881b21c748e583298e5b53": "\\begin{align}\n&{} \\left( {\\sum_i{D_{**}(X_i, Y_i)}^p} \\right)^{\\frac1p} + \\left( {\\sum_i{D_{**}(Y_i, Z_i)}^p} \\right)^{\\frac1p}\\ \\ge \\\\\n&{} \\ge \\left( {\\sum_i{D_{**}(X_i, Y_i) + D_{**}(Y_i, Z_i)}^p} \\right)^{\\frac1p} \\ge \\\\\n&{} \\ge \\left( {\\sum_i{D_{**}(X_i, Z_i)}^p} \\right)^{\\frac1p}\n\\end{align}\n", "c5b75f45c940bb4ba9929755bb79104b": "d= \\frac{N m_n}{\\cos \\psi}", "c5b7c898a1405a25582df4d0b25d34b3": "X_c", "c5b82b88002ac195952c297c35824948": "r= \\frac{K}{K-1} \\left[ 1 - \\frac{\\sum_{i=1}^K p_i q_i}{\\sigma^2_X} \\right] ", "c5b82d18be4ce4311e3f9155dea6b610": "\\textstyle a a = 0 ", "c5b845aa2373916b6d15dbfe5ce5aae3": "R[x]", "c5b850351fd9065c6eb0e637774dc2eb": "\\epsilon_0 = \\text{initial bracket size} = b-a .", "c5b87f8c825fba9175add27242e354cc": "n_2\\times n_2", "c5b8c8036b6dc3735cc02b00a8f69fc4": "\\mathrm{Ai}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\cos\\left(\\tfrac13t^3 + xt\\right)\\, dt,", "c5b9025b241ea39030203607042ef2e3": "\\hbox{apotome} - \\hbox{limma} \\approx 113.69 - 90.23 \\approx 23.46 ~\\hbox{cents} \\!", "c5b945a8a8f7281c213d7c3efe24626a": "y^2=\\frac{-(am^2+bm+c)}{a}", "c5b9602e4439a0d0eeba26313c193039": "V = \\frac{k}{2}x^2 = \\frac{m\\omega^2}{2}x^2 ", "c5b998436608cdaf74f2e4b86594b514": "Z = R + j X \\,", "c5b99b975f7e65502981c5803affb333": "\\left(\\mathbf{A}\\right)_{m'n',mn} \\equiv \\left[\\left(A_x\\right)_{m'n',mn}, \\left(A_y\\right)_{m'n',mn}, \\left(A_z\\right)_{m'n',mn}\\right]", "c5ba329aab8001b56f4d45546bfcdb15": "g_1,\\ldots, g_{\\mu}", "c5ba650acc88d104722decd554aa2094": "\\dot{\\mathbf{z}}(t)=A\\mathbf{z}(t),", "c5ba7d49143a882d9757c2444060c345": " L_2 = T_2 - \\frac{1}{2} \\left( (V_R)^1_2 + (V_A)^1_2 \\right).", "c5bb01037ac4c721f7935abc6d535911": " X = \\sqrt{Y} \\,", "c5bb155662a796606910c4033e302977": " \\lambda_j = -\\frac{(j - 1)^2 \\pi^2}{L^2}", "c5bbb9b4bb3f1136fc9fcb6794a5c2c2": "f_W(w),", "c5bbca109f02fed4d4b07603974fb38b": "\n k_s^2\n= {6\\pi n e^2 \\over \\epsilon_F}\n", "c5bbd6c1ad84751952e8b8991f4387f6": " b_3 = f(0,1)-f(0,0) \\,", "c5bbed4e5ed70b89f78bd65b50a5533e": "\\frac{J_s\\left(\\sqrt{z^2-2uz}\\right)}{\\left(\\sqrt{z^2-2uz}\\right)^{\\pm s}}= \\sum_{k=0}\\frac{(\\pm u)^k}{k!}\\frac{J_{s\\pm k}(z)}{z^{\\pm s}}", "c5bc0c047a1393e2f13e97c4a0b32251": "\nx\\,y'' + (\\alpha +1 - x)\\,y' + n\\,y = 0", "c5bc20655bf294000455585527260326": "{{\\text{ }\\!\\!\\varepsilon\\!\\!\\text{ }}_{1}}", "c5bc282fd17c0fd5d7edfbca3ad68e5f": "EI = VWS \\times (DEF + CVG) ", "c5bc5b9b4870cbb07504df4f9a08e244": "\\displaystyle{b(s,t)= {(\\mathbf{v}(s)-\\mathbf{v}(t))\\cdot\\mathbf{n}(t)\\over |e^{is/L} -e^{it/L}|^2}}", "c5bc8596842774870de6fc899eb77296": "f(S)+f(T)\\geq f(S\\cup T)", "c5bcb164bac5aeafe7a6e93c614a5a1c": "\\vec Y", "c5bd31a44cce44b4cd4c6719d352c352": "\\| x(t) \\| < k_1 \\|x(0)\\| + k_2, \\, \\forall t \\ge 0 ", "c5bdc1805e56f267db0afe84691281a6": "L(x) := \\sum_{j=0}^{k} y_j \\ell_j(x)", "c5bdd418ca33e1bcc4489b4ed1bf90fa": "\\mathit{q}_i", "c5bdd833f44b167e9f3edaf6cf872596": " f(q,p)= 2 \\int_{-\\infty}^\\infty \\text{d}y~e^{-2ipy/\\hbar}~ \\langle q+y| \\Phi [f] |q-y \\rangle. ", "c5bdf36e825b8e170b72bd07f380a5f9": "\\lVert A \\rVert = \\sup \\left\\{\\,\\lVert Ax \\rVert : \\lVert x \\rVert \\leq 1\\,\\right\\}.", "c5be0db91fe5d121bccaa5b9c28b3245": "f \\left(x_1,\\cdots, x_{K-1}; \\alpha_1,\\cdots, \\alpha_K \\right) = \\frac{1}{\\mathrm{B}(\\alpha)} \\prod_{i=1}^K x_i^{\\alpha_i - 1},", "c5be86a62d6b9d12df6cc61915dcfb73": "\\mathbf{}V_{i}, W_{i}", "c5beb3723e4186c642ba3f71948408fa": "f_i=x_if_i^*", "c5beb7483d901678bc8becb18db1d593": " T_{\\perp}=T_y ", "c5bf1faef822b69337263b65c0fa1b63": "|\\Psi_\\epsilon\\rangle", "c5bf31f9b5a80c4af544f2e77c4200a5": "q=F^{-1}\\left( \\frac{p-c}{p}\\right)", "c5bf3ef1394405fa36ffd780ef8de121": " 6~r~\\sin\\theta \\,", "c5bfa04d151657a43e3b83e74c241cc7": "D^{\\epsilon}(\\rho||\\sigma)", "c5bfc93409cc15ab489c8c1695716207": "\\mu_z=g_L \\mu_\\mathrm{B} m_l", "c5bfcba758aa2f97c36953d3947bbe71": "C2 = 0", "c5bfec463d39cf851ac540b93fdd029b": "R_{\\mathfrak{p}}", "c5bffe5c42c2342120d9e7640e1a4338": " \\theta_{c} = \\sin^{-1} \\frac{n_\\text{clad}}{n_\\text{core}}", "c5c005cb55765ffbbed47f142a307667": "M \\times I", "c5c02031d12fc65953f281721a634704": "f_p(x)=\\frac{1}{1-f(p)x}.", "c5c02b4f8a4bfb4a57118e569e5babbe": "~I_{\\rm s}~", "c5c04249d2f32057f9d95d979a714445": "e \\in \\Lambda^{e_i}", "c5c071904c3c5fb61aec2bb3498384a6": "\\boldsymbol{B}=\\boldsymbol{F}\\cdot\\boldsymbol{F}^T", "c5c0c59c011172a84f935246d32f267b": "G_N\\;", "c5c11f9d0569537b98bbb99b251b5813": "c/d", "c5c121efadc836ec8a36cf689696519c": "\\mathbf{\\Phi}_{00}= \\mathbf{0}", "c5c125170d1556018186d135d8fe34b9": "B = e^\\frac{1300 - Z_A}{150} - 1 ", "c5c1382fdfe7db404616c1331778541d": "P\\lor(Q\\land R)=(P\\lor Q)\\land(P\\lor R)", "c5c14074336e0164accfa80270629c75": "a \\rightarrow \\left(\\left(a \\rightarrow STOP\\right) \\sqcap \\left(b \\rightarrow STOP\\right)\\right)", "c5c159a8c530f6d05a062215a2800884": "C=J^1P/G\\to M", "c5c15e828aecc42b571c8bb9b47bdae1": "E_2 = \\Im [E( \\omega )] = \\eta \\omega. ", "c5c17ef692eb2df088f093f83a2090c3": "R > a^2 / \\lambda ", "c5c1c41eb5c94e122823cfc4c20b3b2a": "u(z) \\ge \\frac{n}{2\\pi}\\log \\frac{H}{e}", "c5c1ccbedb80c7e2c7559d886b63785a": " Q = I_3 + \\frac{1}{2} (B+S).\\ ", "c5c1ce592d2cb431aad40822bc9f1d4f": "\\sum_{r=0}^n a_r\\left(\\overline{\\zeta}\\right)^r = \\sum_{r=0}^n a_r \\overline{\\zeta^r} = \\sum_{r=0}^n \\overline{a_r\\zeta^r} = \\overline {\\sum_{r=0}^n a_r\\zeta^r}.", "c5c1f8d4352a6137208d30230b634ee1": "\\langle C_i\\rangle_{i<\\alpha}", "c5c27fb190a64685ccd9e9dd551e76d4": "I<8%", "c5c2a9248deba296d0b6fd55e0c46139": "\\{a_1(\\mathrm{mod}\\ {n_1}),\\ \\ldots,\\ a_k(\\mathrm{mod}\\ {n_k})\\}", "c5c2bc6feb5d7aa3e5fc274c9b62dfb0": "\\mathbf{r}_1 = (a/2)(\\hat{x} + \\hat{y})", "c5c2c2c3f3f7504ec1b9004db66107a0": "r_b= \\cos i \\ \\sin u\\,", "c5c3412abbe5edc65e5c7e4666ecdb63": "S_x(f) = \\frac{1}{(2\\pi)^2f^3}h_{-1}", "c5c391275cacae365557166a9b2029c6": "\\mathsf{DTIME}(O(n)) \\neq \\mathsf{NTIME}(O(n))", "c5c3c54cf1ac0082792985af680bcc38": "\\tfrac{\\mathrm{d}\\mathbf{H_{eff}}}{\\mathrm{d}t}=\\tfrac{\\mathrm{d}\\mathbf{H_{eff}}}{\\mathrm{d}\\mathbf{m}}\\tfrac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}", "c5c3c7a44605e9349858fc8d49490848": "|P_1(A)|<|P(A)|", "c5c3c8e418b7e307e1e3eee382c432a1": "|\\psi(f(z))| = |\\psi(z)|^{2}", "c5c3f115463b23652e8d35daaa548639": "\n\\begin{align}\n\\cos x&=1-2\\sin^2\\frac{x}{2}\\\\[8 pt]\n&=1-2t^2\\cos^2\\frac{x}{2}\\\\[8 pt]\n&=1-\\frac{2t^2}{\\sec^2\\frac{x}{2}}\\\\[8 pt]\n&=1-\\frac{2t^2}{1+t^2}\\\\[8 pt]\n&=\\frac{1-t^2}{1+t^2}.\n\\end{align}\n", "c5c42d6ab64833e69ea4b5194b79725d": "\\textstyle \\alpha=\\left( \\alpha_1,\\alpha_2,\\ldots,\\alpha_n\\right) ", "c5c488832f1f26e3d0bb11578ec28cec": "td = \\sum_{i=1}^m \\, dx_i", "c5c4c730fa8ae0883840a49180d83d7c": "\\Sigma=(\\mathbb{R},\\mathbb{R}^q,\\mathcal{B})", "c5c570dd63463cc539be5ed96ac8d988": "q + 1 - \\sharp E(\\mathbb{F}_{q})", "c5c620ad91269190b385dcef909f5d24": "\\mathcal{E}=\\oint_{C} \\boldsymbol{E \\cdot } d \\boldsymbol{ \\ell } \\ ,", "c5c6a47b31e34ea80070524e76a67bf8": "R_{1}> R_{2}", "c5c6b36796a5cfb3b845cd7881e3fee4": "SX", "c5c6c0ab3e943b7223fc90b228f52a12": "\\det S''_{zz} (0) = \\mu_1 \\cdots \\mu_n", "c5c6d1a0053e5bb2f997696752b1a1a8": " \\begin{pmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{pmatrix}.", "c5c6eab38071acc9d029224e005ee003": "X\\to Y", "c5c73de6254c0a5b4b6421fcd1ba1ac5": "\nv = \\frac{I}{p} \n", "c5c76b188e3872ef455e744c3fa6016f": " \\hat{p} = - \\hat{z} \\,", "c5c7783399665c704515b23a57b500a0": "f(m) = O(m^n)\\,,", "c5c79488a037e4965840f9351713d894": "\n \\Phi(x, \\mu, \\sigma)", "c5c7aa8170062b21240065872a9bf230": "\\tilde{t}", "c5c7c90ec5f14f9e2ab574eed337b071": "\n Q\\ \\sim\\ \\chi^2(k)\\ \\ \\text{or}\\ \\ Q\\ \\sim\\ \\chi^2_k .\n ", "c5c7fa174744cf3159575339d496e5b1": "\n\\mathbf{g}_M = \\mathbf{g}_N \\sqrt{\\frac{a_0}{\\left\\| \\mathbf{g}_N \\right \\|}}\n", "c5c851d48aae2d4ca175dd1b5ac1d8ab": "f(\\Gamma (t),0) \\ne 0,\\;\\forall \\Gamma (0) ", "c5c8af10c878533c223f9ae3a6b207e5": " 1 - \\cos(t)\\over t", "c5c8d1cb8777fa95a829e2bcaafea425": "[x,[y,z]]", "c5c8d6c20974f23a88e4aff3199c4713": " z^{nad}_i= \\sup_{x\\in X\\text{ is Pareto optimal}} f_i(x) \\text{ for all } i=1,\\ldots,k ", "c5c8db5e19c3baab6b26357414d5d344": " \\begin{bmatrix}2 & -1 \\\\ -1 & 1\\end{bmatrix}\\begin{bmatrix}x_2 \\\\ x_3\\end{bmatrix} = \\begin{bmatrix}1\\\\2\\end{bmatrix}", "c5c8dc6dc485c29ed96971b6b50a612f": "-1.1", "c5c8eb6c1fe3c56f60c8dfe46cabd197": "\\phi(a_i)", "c5c9487a8e082ffce5b2f4742d400dc8": "{Q_r} = {\\eta_o}.. {I_c} . {A_c}", "c5c974f8d18d7c9a6491473bdaf3277b": "q' = { {1+q} \\over {3-q}}", "c5ca7fd85e2972511dab236b2d57563a": "z=c_0", "c5caaa3a5489175ffe97bf7a29ce975e": "\\vec{v}+a", "c5caccce4ea5a6e5c4a9fdfaf5213d42": "\\mathbf{y} \\ge 0", "c5cb02cfd0ec82524fd7a7ccf0e35e92": "\\frac{1}{(i\\omega-\\xi_1)^2-\\xi_2^2}", "c5cb0bcbc9ab9d27bb65742b9d042438": "\\Lambda(\\theta) = - \\int_0^\\theta \\log|2 \\sin(t)| \\,dt = \\operatorname{Cl}_2(2\\theta)/2", "c5cb3c8c194ce508e9c21aebb634be69": "\\; \\; N \\;", "c5cb7d65cf562089a7e0246b1c497836": "(2^\\nu,2^\\nu-\\nu+\\varepsilon),\\quad\\nu=1,\\ldots,k.", "c5cbc1026fca49fbc1c1d98e0bc9387c": "\\scriptstyle(1+r/K)^{NK}\\approx e^{Nr}", "c5cbe7403684c69438733920035810f6": "\\,\\begin{bmatrix} T_{r\\overline{o}} & T_{ro} & T_{\\overline{ro}} & T_{\\overline{r}o}\\end{bmatrix}", "c5cc1417bebda8d349325b03bcff0f57": "2^{\\tilde{O}(n^{1/3})}", "c5cc19a247d8f216939a2e7333a3ea8a": " c_{n-1} + c_{n-3} + \\cdots + c_r", "c5cc3d6af24ccd7a0d737dc2c45544bb": "\\forall p:\\exists q:\\mathcal{B}(q \\equiv ( \\mathcal{B}q \\to p)) ", "c5cc3dd430ad2f2d707bba04ee1234c6": "\\quad h\\;=\\;\\dfrac{\\delta y}{a\\delta\\phi\\,}=\\,1", "c5ccdc7362fa368da47ee5e8d603e938": "\\operatorname{tr}(\\gamma^\\nu) = 0", "c5cd706a95480724f3cc1253e95df268": "p(sober) = 0.999 ", "c5cdd09a033921edfcec49f359689c13": "M(z)=\\frac{z_1-z}{1-\\overline{z_1}z}, \\qquad \\varphi(z)=\\frac{f(z_1)-z}{1-\\overline{f(z_1)}z}.", "c5ce3748cad9e19b8c2028c187489927": "4\\pi \\epsilon_0 \\hbar^2 / (m_\\mathrm{e} e^2) = \\hbar / (m_\\mathrm{e} c \\alpha) ", "c5ceb902776829d9cb45bdf6279c2783": "\\mathrm{S_2O_3^{\\,2-} + 2 \\ O_2 + H_2O \\longrightarrow 2 \\ SO_4^{\\,2-} + 2 \\ H^+}", "c5ced882530ab1b8bda9587cbc0d48a9": "G_0\\cong H", "c5cf3461b0eb5d8a8af5cb8c9d1b3b06": " m=0,1... ", "c5cf6f146f2f45a2a502d712af9e1e2e": "f(x+\\Delta x)\\sim f(x) + df(x,\\Delta x) + \\frac{1}{2}d^2f(x,\\Delta x) + \\cdots + \\frac{1}{n!}d^nf(x,\\Delta x) + \\cdots", "c5d01f06ef2f26afa4b84ed67e69aa91": " \\mathbf V = \\begin{matrix} \\frac{1}{2} \\end{matrix} \\pi^2 R^4", "c5d02d06f661dc0836cd97363098281d": " P(x) = \\left| \\langle\\mathbf{x_0}, \\mathbf{x} \\rangle \\right|^2", "c5d05a93812bb225bba5ee49c095bb93": "F = C^2", "c5d0648b08ed676b6753495d4f489cfa": "\n\\hbox{If}\\quad k = \n\\begin{cases}\n j+m: &\\quad a=m'-m;\\quad \\lambda=m'-m\\\\\n j-m: &\\quad a=m-m';\\quad \\lambda= 0 \\\\\n j+m': &\\quad a=m-m';\\quad \\lambda= 0 \\\\\n j-m': &\\quad a=m'-m;\\quad \\lambda=m'-m \\\\\n\\end{cases}\n", "c5d091078fecc0e7ed89cd1288736bd0": "U\\cap N", "c5d15b76186bc51c800f0b15eaaed5ea": "B = k_1(1 - \\tfrac {3}{8}(k_1)^2)", "c5d1692a66cc240c87a9712feda6f29e": "d\\mathbf F\\,\\!", "c5d187e4b841202dd0d5ec577b02f568": "\\dot{J}(\\mathcal{T})", "c5d1b9ade787312f405adcdfd9455a65": "\\mathbf{x}_{(i)}", "c5d22d0639abefec71b0c585e866a04c": "\\frac{e^x}{x^x \\sqrt{2\\pi x}} \\Gamma(x+1) \\sim 1+\\frac{1}{12x}+\\frac{1}{288x^2}-\\frac{139}{51840x^3}-\\cdots\n \\ (x \\rightarrow \\infty)", "c5d23115d15b84da14919382f488f15d": "x\\cos\\theta + y\\sin\\theta = r\\ ", "c5d23e15d528e6b8e6cc5f6ad1f4f099": "z(x,y) = \\frac{1}{2xy}\\sqrt{x^2 + y^2}", "c5d25fa00d7b2153ec15c495b7a6519e": "p ,\\, e ,\\, \\theta,\\, \\hat{x} ,\\, \\hat{y}", "c5d28a6263ddb12594e868bcacc0a0a9": "\\operatorname{var}(p' \\mid p)= {p(1-p) \\over 2N}.", "c5d2f09921dc7ca9394a2898472bab7d": " \\sum_n i_n j_n, \\ i_n \\in I, \\ j_n \\in J ", "c5d2fe4869d0fb0dd738dfac9d7049e3": "F = 2 P A_t", "c5d304508e3c979c9d0be59491e82b44": "P_2 \\,", "c5d3280e5a4fd26a6ba66a4414b9cf98": " \\mathrm{Da} = \\frac{K}{d^2}", "c5d3450cf906151f8fa36ddc1ad53b97": "\n\\begin{align}\n&E(x,v)E(u,y)\\theta\\left(z+\\int_u^x\\omega\\right)\\theta\\left(z+\\int_v^y\\omega\\right)\\\\\n-\n&E(x,u)E(v,y)\\theta\\left(z+\\int_v^x\\omega\\right)\\theta\\left(z+\\int_u^y\\omega\\right)\\\\\n=\n&E(x,y)E(u,v)\\theta(z)\\theta\\left(z+\\int_{u+v}^{x+y}\\omega\\right)\n\\end{align}\n", "c5d3883072e8be78e9e3274e9e93a363": "a < r < c", "c5d3a42182ea9b243374ebbc265df89b": "sp(S, R)", "c5d3cd133b72cfaaa152c91ea90fd7ad": "\\frac{\\sqrt 3}{2} a", "c5d4067df2210cdb56a31504606ea70e": "\\mu(t)", "c5d4860b3a3fcd940feb663f67def4c7": "V_X-V_Y", "c5d515f053a84ca43e88c8b2632d8327": "{\\Bbb L}", "c5d52caedd731523e18ad55cd5be6d8e": "b = \\Omega_{FB} T_{FB}", "c5d5379cb0989ba515eb91ae823c3099": " \\gamma(s,x) =(s-1)\\gamma(s-1,x) - x^{s-1} e^{-x}", "c5d559bab3335ec6fb5728556f37e384": "\\mathcal{B}(x) \\subset \\mathcal{V}(x)", "c5d5f2d827e0a68f33ed49dd407366d2": "l+16\\left\\lceil l/1024 \\right\\rceil", "c5d6b60da82691d1b5642a095c0ade34": "V = \\pi \\int_a^b f^2(x) \\,dx \\qquad (1)", "c5d707ac75f8f4048c49e1c5771878d8": "\\mathcal F", "c5d71e0a4d45cfa6392181300d9aa031": "u\\!", "c5d722f869117dba240d7bc5eb21550a": " C_{i + 1}^{j+1} = C_{i - 1}^{j+1}. \\, ", "c5d72861ae94010b43c9b3231097bd68": "L(\\alpha)+\\alpha = K(x)", "c5d7393a571738594dd2ce7760000b4d": "82.0\\pm 0.5", "c5d77c1bee404075c3c7e2391a3655d0": " \\cos(\\varphi) = \\frac{r_2}{x} = \\frac{r_2}{\\left(\\dfrac{P r_2}{r_1+r_2}\\right)} = \\frac{r_1+r_2}{P} \\,\\!", "c5d7b7769ab76679e1e8e3bbcf15148b": "1 = 2.\\,", "c5d7cc391110f06187a2c910e28c8096": "\\frac{2\\pi}{\\Delta\\Omega}", "c5d81341d6f5a6f7108f6675f2b7b0c6": "E_{y}=\\left [- \\frac{k_{z}}{\\omega \\varepsilon _{o}\\varepsilon _{r}}\\frac{m\\pi }{a}(Ae^{-jk_{x\\varepsilon }x}+Be^{jk_{x\\varepsilon }x})-jk_{x\\varepsilon }(Ce^{-jk_{x\\varepsilon }x}-De^{jk_{x\\varepsilon }x}) \\right ]cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (34) ", "c5d85217e8619751661b2b408e29fc84": "\\frac{df}{dx} = \\frac{dg}{dh} \\frac{dh}{dx}", "c5d8933eaee55693c9f9af72ade72b5a": "\\mathrm{A} = \\begin{pmatrix}\\cos\\gamma & - \\sin\\gamma\\\\\\sin\\gamma & \\cos\\gamma \\end{pmatrix}", "c5d8efe72e613d9ecd7ac3f2d75f42fc": "x \\log_b \\left(x\\right) = a", "c5d928764e6956e3e8a3421c3559dd19": "\\alpha d=\\ln\\left(1+\\frac{1}{\\gamma}\\right)\\qquad\\qquad(5)", "c5d96f84aac61f2f9ea7372746a73224": "V_{M-1}", "c5d9a5f54f3f4074b5fb20bc0e1ad5f9": " \\exists X ", "c5d9aba070c3ae8070a237ef99e1a47c": "T_{recoil}=\\frac{\\hbar^2k^2}{2mk_b}", "c5d9bb2eccd1f0f54b0d9bff33b4d901": "\\mathfrak M=({\\mathcal P},{\\mathcal Z},\\in)", "c5d9d07debc847c4f3802b4b863de348": "K \\cos(a x) + M \\sin(a x) \\!", "c5d9e37155f7c0026b193259a7acb7fc": "\\operatorname{Alt}(3) \\cong [\\operatorname{GL}_2(\\mathbb{Z}/2\\mathbb{Z}),\\operatorname{GL}_2(\\mathbb{Z}/2\\mathbb{Z})] < \\operatorname{E}_2(\\mathbb{Z}/2\\mathbb{Z}) = \\operatorname{SL}_2(\\mathbb{Z}/2\\mathbb{Z}) = \\operatorname{GL}_2(\\mathbb{Z}/2\\mathbb{Z}) \\cong \\operatorname{Sym}(3),", "c5d9f7fbe8f7eb6c4efea6007199f830": "\\omega(v) = (L_{g^{-1}})_* v,\\quad v\\in T_gG.", "c5da0faed3f25903c451d52b27a47df2": "\n \\mathbf{x} = \\sum_{i=1}^3 x_i~\\mathbf{e}_i\n ", "c5da63d48215a2b0c296225efdd6237f": "K_{12},", "c5da986d56b5ad0d0c10bd10e644e0d7": " AD = C + I + G + (X-M) \\ ", "c5daa1e9c4002b72e9e94851304d6dcc": "V_+ = \\frac{V_{out}}{2}", "c5daac3e5c7be9239054d07b3d1079d5": "P \\left ( { {a, b}{|}{A, B} } \\right ) \\ge 0 \\quad \\forall {a,b,A,B}", "c5dac041cbb2ae58de3aff8011f0bfc7": "y'' + q(x)y = 0\\,", "c5daeb7cd564d50cad501389ec9358ba": "\\ \\theta", "c5db04afa7ce51daa27c54593811cafa": "\\forall x. S(x) \\to P(x)", "c5db06d2b1ec9806e5c54154df58b5d4": "\\sqrt{2+\\sqrt{2}\\ }", "c5db8eae1332ae3a6b79f034dd504e4c": "x=P_W x", "c5db98a8bf50023ffc921c52deda253d": "\\scriptstyle v_i ", "c5dbadb0153132cb9f85db8a0340328b": "\\operatorname{R}(x) = \\sum_{n=1}^{\\infty} \\frac{ \\mu (n)}{n} \\operatorname{li}(x^{1/n}) = 1 + \\sum_{k=1}^\\infty \\frac{(\\ln x)^k}{k! k \\zeta(k+1)}", "c5dbdd63b551cefe91afe42ee2f1b7fd": "\\left (a_0 + a_1x^1 + a_2 x^2 + \\cdots \\right ) \\left (1- c_1x^1 - c_2 x^2 - \\cdots - c_dx^d \\right) = \\left (b_0 + b_1x^1 + b_2 x^2 + \\cdots + b_{d-1} x^{d-1} \\right )", "c5dc63c0f9904acd31b147bad6efed43": "\\frac {\\tbinom{2n}n}{n+1}.", "c5dc8653d322618d545a6f76588fa55e": "\\psi / \\|\\psi\\|", "c5dc9e6e6b183d42b64d426fabbcbae2": "\\displaystyle{\\varphi_{s,t}\\circ \\varphi_{t,r}=\\varphi_{s,r}}", "c5dca07cc4ac88e0d01f732f97f3e16b": "T=\\frac{1}{4}\\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}", "c5dcc887a0d7f28deafeadb199ef431f": "\\langle\\hat a\\rangle", "c5dcd0cc06242f0de00cfeca12d6e74c": "t \\circ id = t", "c5dd11aaee650cd734af54d11180949a": " \\lim_{x \\to c} f(x) = \\infty,\\ \\lim_{x \\to c} g(x) = \\infty \\! ", "c5dd3dafba597ca70d29041140bb6146": "X_q", "c5dda52c3f234201f9f5a7baf7fcb893": "q_\\infty", "c5ddb5151170297f263e5ce0cf89f043": "t\\equiv \\sqrt{1-1/\\xi^2}", "c5dddc0df64d23d0b936bf1a148ae47a": "\\textstyle {4!\\over 4!\\times 0!\\times 0!} \\ {4!\\over 3!\\times 0!\\times 1!} \\ {4!\\over 2!\\times 0!\\times 2!} \\ {4!\\over 1!\\times 0!\\times 3!} \\ {4!\\over 0!\\times 0!\\times 4!}", "c5de140b40e1073b722948d8f3143855": " (u_t+uu_{x})_x+u_{yy}=0,\\qquad (1)", "c5de681615f09343d81edc9e0b46136b": "U_\\nu U^\\nu", "c5de9279c5b9af2dd19a7f9d56e8d515": "r\\in(0,\\infty)", "c5de947d5dbed7dc54460149458454dd": "C_f(z)", "c5dece33aaafa6dfa8e001da8d0cdac3": " = d_0(m + 2 {d_1} (m + 2 {d_2} (m + 2 {d_3} (m)))),", "c5dedf1b3ca11fcf6cd2f3aac1c691eb": " \\{0\\} \\subset \\ker L \\subset \\ker L^2 \\subset \\ldots \\subset \\ker L^{q-1} \\subset \\ker L^q = \\mathbb{R}^n", "c5df084b3f145e53a76ed4d7c45b37e9": " { c_B } = { \\left( { E I \\omega^2 \\over \\rho A } \\right)^{1 \\over 4} } ", "c5df28e4a2a28be92ef53ffe9f7de491": "\\langle x \\, p|", "c5df3c544d63ee58b2d836da47836574": "1 \\rightarrow 2", "c5df43a41c297ec0643c9f3dcd2a4a28": "{n \\choose r} = {n-1\\choose r} + {n-1\\choose r-1}", "c5df4dcd72ce95d492a94e13c997548a": "\\Pi (t,f) = \\delta_0 (t)\\,W_h(t,f) ", "c5df8b38452ca7ced6f966c3c17ae54d": "A=\\left(20+\\sqrt{\\frac{5}{2}\\left(10+\\sqrt{5}+\\sqrt{75+30\\sqrt{5}}\\right)}\\right)a^2\\approx27.7711...a^2", "c5dfbaa47ab24d03b28bbc58a984ae82": "V_n(R) = \\int_0^R \\int_0^\\pi \\cdots \\int_0^{2\\pi} r^{n-1}\\sin^{n-2}(\\phi_1) \\cdots \\sin(\\phi_{n-2})\\,\nd\\phi_{n-1} \\cdots d\\phi_1\\,dr.", "c5dfe1a29634f7d504ff093d5c96cd4b": "c_n \\approx \\mu^n n^{\\gamma-1}", "c5e03d65a0d3a6653c933fc5b4118ae7": "M \\simeq GL_{n_1}(\\mathbb{F}_q) \\times \\ldots \\times GL_{n_r}(\\mathbb{F}_q)", "c5e0870fa38b3d6b268c53997f1eb57a": "\\lambda\\!", "c5e090ab9b7d1ed443888432d6cb1341": "X(t)-c_1=r_1.\\,", "c5e093487c4ceaadd9d7391c14e0d8b0": "{{i}_{e3}}", "c5e1648ae749883be0514d959d073405": " e^{i\\theta} \\leftrightarrow \\begin{bmatrix}\n\\cos \\theta & -\\sin \\theta \\\\\n\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}.", "c5e1f91ea898c81fa0ef2d6e9bc234f5": "\\{ y_1, \\ldots , y_l \\} ", "c5e20130c9f4c739c7dd67cb7794348a": "dF(0,0; a, b)=\\begin{cases}\n\\frac{a^3}{a^2+b^2} & (a,b)\\not=(0,0)\\\\\n0 & (a,b)=(0,0).\n\\end{cases}\n", "c5e2c35e805443ef5efc4974b1bc0c32": "\\leq \\Pi_{H}(2m)\\cdot max\\left(Pr[|\\frac{1}{m}\\left(\\sum_{i}w^{j}_{\\sigma_{i}}-\\sum_{i}w^{j}_{\\sigma_{m+i}}\\right)|\\geq\\frac{\\epsilon}{2}]\\right)\\,\\!", "c5e2d64a9285e880309132b89a379dbd": "Index = {AGPA*50 \\over 4.5}+{LSAT-120 \\over 1.2}", "c5e31b70df32af7bbd99f5946716f7fc": "\\mathbf{u}\\oplus_M\\mathbf{v}=\\frac{1}{2}\\otimes\\left({2\\otimes\\mathbf{u}\\oplus_E 2\\otimes\\mathbf{v}}\\right)", "c5e31caf9e177d77f573698c7ed4be36": "R_{s1}", "c5e35e4da2d69ed6059c1cd338d57c4e": "\n\\begin{array}{ll}\nM_{\\text{lm}}(x,y)\n&=\n\\lim_{(\\xi,\\eta)\\to(x,y)} \\frac{\\eta - \\xi}{\\ln \\eta - \\ln \\xi},\n\\\\\n&=\n\\begin{cases}\n0 & \\text{if }x=0 \\text{ or } y=0 ,\\\\\nx & \\text{if }x=y ,\\\\\n\\frac{y - x}{\\ln y - \\ln x} & \\text{otherwise,}\n\\end{cases}\n\\end{array}\n", "c5e37da6d0e21f5ae11ea8ef882fafec": " \\frac{1}{2{\\pi}i}\\int_{c-i\\infty}^{c+i\\infty}g(s)t^{s}\\,ds = \\pi(t)", "c5e3a2351be3c4a7a964cd124638b104": "2^r", "c5e3f027152306dc681926d9501bf019": "H_{\\frac{1}{2},2}=4-\\tfrac{\\pi^2}{3}", "c5e3f913f938c7665f9c94b3f1571ebf": "\\displaystyle \\nabla^4w = a+b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy}", "c5e428485c9ae3bb4536426ba1164e92": " {Z}_{{{\\mathbf{k}}}}(S) \\ = \\ {Z}_{{{\\mathbf{k}}}}(S_1) \\ \\cup \\ \\cdots \\ {\\cup \\ Z}_{{{\\mathbf{k}}}}(S_e) ", "c5e489fb86d387664e19e83813412f72": "V_l", "c5e49a7989527b8243bd66ffe0665b45": "\\mathrm{\\tfrac{u\\bar{u} + d\\bar{d} + s\\bar{s}}{\\sqrt{3}}}\\,", "c5e4b861c304fe681c758f7d8042511e": "P_\\text{fusion}", "c5e4c0ba1c323fc8ee87a433f4246bce": "\n\\lbrack \\mathbf{S}_{1},\\mathbf{S}_{2}]\\psi =0,\n", "c5e4c378ba3af5a6b619dabb5f0a7fbf": "m=M_f(x_1,\\dots,x_k)", "c5e50593225fb7ba1d0f55bed678450a": "y\\otimes x\\rightarrow \\pm x\\otimes y", "c5e51a6bf6869c04efd5114df6f53ff5": "\n\\begin{align}\nA \\times (B \\vec{r}) - C\\vec{r} & =\n\\begin{bmatrix}\n 2 & 3 \\\\\n 3 & 4\n\\end{bmatrix}\n\\left(\n\\begin{bmatrix}\n 1 & 0 \\\\\n 1 & 2\n\\end{bmatrix}\n\\begin{bmatrix}1 \\\\ 1\\end{bmatrix}\n\\right)\n-\n\\begin{bmatrix}\n 6 & 5 \\\\\n 8 & 7\n\\end{bmatrix}\n\\begin{bmatrix}1 \\\\ 1\\end{bmatrix} \\\\\n& =\n\\begin{bmatrix}\n 2 & 3 \\\\\n 3 & 4\n\\end{bmatrix}\n\\begin{bmatrix}1 \\\\ 3\\end{bmatrix}\n-\n\\begin{bmatrix}11 \\\\ 15\\end{bmatrix} \\\\\n& =\n\\begin{bmatrix}11 \\\\ 15\\end{bmatrix}\n-\n\\begin{bmatrix}11 \\\\ 15\\end{bmatrix} \\\\\n& =\n\\begin{bmatrix}0 \\\\ 0\\end{bmatrix}.\n\\end{align}\n", "c5e5344e058735c85eb733c3c2aadb90": "g\\cdot z", "c5e572317273e04949a9ca57567aacde": " W = - \\alpha P_1 V_1^{\\gamma} \\left( V_2^{1-\\gamma} - V_1^{1-\\gamma} \\right) ", "c5e5de1b713345e6ab262f5b5dacbd62": "a_{ij} ", "c5e5e8cada3c963d2d978a700a732ea3": "{m_A}", "c5e681eadaa8e43ec3fc670a931fa759": " f = 1 - \\exp \\left (- \\frac{ \\pi\\ }{3} \\dot NG^3t^4 \\right ) \\,\\! ", "c5e6b05e8acc201c1c8b856743df8c8e": "[a \\cup b]p \\equiv [a]p \\land [b]p\\,\\!", "c5e6ca85bd73d624854fb4dddf0b13fd": "w_{j,t}", "c5e6d55f490793efdfacf9313c2078ef": "\\arccos\\left({1 \\over \\sqrt{3}}\\right) = \\arctan(\\sqrt{2})\\,", "c5e7102adc2f50b0016f098864493dcf": " \\psi_i^* = \\lang \\psi|i \\rang ", "c5e725569a579274b86b199108ed0334": "\\frac{ \\partial (\\varphi f) }{ \\partial x } + \\frac{ \\partial (\\varphi g) }{ \\partial y }", "c5e7861d7ff19fee10556536d14c56cf": "\\underline{t}", "c5e7c0f1ba6907a945766455039983b7": " a = \\frac{q}{m} \\nabla \\phi", "c5e821422f81175938379f2292a3246a": "\\bigg\\{ \\Pr(h) \\geqq 0 \\bigg\\}", "c5e855ae49263d62c87b8f840d4a99ec": "x ^{\\star} = x + u + \\frac{\\partial u}{\\partial x}\\Delta x + \\frac{\\partial u}{\\partial y}\\Delta y", "c5e8673c056036d798c829427fad891e": "\\mathbf{c} = \\mathcal{F}^{-1}(\\mathcal{F}(\\mathbf{a})\\mathcal{F}(\\mathbf{b})).", "c5e8fbf6ed42d448e6160bc056f15a96": "\n\\vec{S}_t \\equiv \\operatorname{row}(\\sqrt{M_1}\\;\\vec{s}^{\\,1}_{t}, \\ldots, \\sqrt{M_N} \\;\\vec{s}^{\\,N}_{t})\n\\quad\\mathrm{for}\\quad t=1,\\ldots, 6.\n", "c5e99b3d1d40c9762aa1004a9c27a9db": "EV = e(p_0, u_1) - e(p_0, u_0) ", "c5e9b15897ae136efeeacf5defbc31cf": " F : \\mathbf{x} \\mapsto n_\\text{o} + \\mathbf{x} + \\tfrac{1}{2} \\mathbf{x}^2 n_\\infty ", "c5e9ecdca3b8034a75e1d68996cf1283": " R(X_1, \\ldots, X_{n}):= \\tilde{Q}(\\sigma_{1,n}, \\ldots, \\sigma_{n-1,n}) \\ .", "c5e9ee8fc3e8308803b4b701b7f3837f": "C^2(\\Omega) \\cap C^1(\\overline{\\Omega})", "c5ea0ef16a817c716a5d88c7b0e4d925": "h_1 \\,\\bot\\, h_2", "c5ea3e40ef33cf9d22f55d69b8524387": "\\mathbb{E} (e^{- \\lambda X_\\alpha}) = \\frac{1}{1+\\lambda^\\alpha},", "c5ea63e49da3b39403c30e7bb3ab52e5": "\\scriptstyle \\left(\\mathbf{A}^T\\mathbf{B}\\right)^T\\left(\\mathbf{A}^T\\mathbf{B}\\right)", "c5ea72b4f7545953dfd615202d27c57b": "H(\\tilde{x}|\\alpha)", "c5eabe201d9d982d2f75d2ee0ed32071": " [\\psi(x),\\psi(y)] = [\\psi^\\dagger(x),\\psi^\\dagger(y)] = 0 ", "c5eae0f550809fe88026a6a0d1c65466": "\\Delta t = d_{min}\\sqrt{\\rho/E_0}", "c5eb7708474bd1d8dc422a541b9ded57": "x_j\\in A", "c5eb8f651f38a8f36ebc1f32de8d7548": "v_{0}\\,", "c5ec05db0a1ddc6323ea3c0d0cee1395": "\\Omega_0^{+}", "c5ec3f43f182764816d3dbbf64b687a5": "\\mathbf{C}(n,k) ", "c5ec839730d9ea3dbfe55f076b44cb89": "\n{\\dot p_{\\theta_2}} = \\frac{\\partial L}{\\partial \\theta_2}\n = -\\frac{1}{2} m \\ell^2 \\left [ -{\\dot \\theta_1} {\\dot \\theta_2} \\sin (\\theta_1-\\theta_2) + \\frac{g}{\\ell} \\sin \\theta_2 \\right ].\n", "c5ecb297465870bee795e2c00d20fa29": " M = \\begin{bmatrix} 0.5 & -0.1 & 0.7 \\\\ 0.1 & 0.5 & -0.5 \\\\ -0.7 & 0.5 & 0.5 \\\\ -0.5 & -0.7 & -0.1 \\end{bmatrix} ", "c5ecdca30c7260a072c370d7a73bea2f": "K_{\\mathfrak{p}}", "c5ecf87a2f7ef25630b9f0169f115c1e": "\\alpha \\approx (dw/dx) (dz/c)", "c5ed92fbab2a28417b74620cb6636317": "n-\\lfloor n/r\\rfloor", "c5eddfd627831665f92a1caef50925df": "\\hat{\\mathcal{H}}_0 = \\sum_{lk} \\left| \\Psi_{l}^{k}{}^\\prime \\right\\rangle E_{l}^{k} \\left\\langle \\Psi_{l}^{k}{}^\\prime \\right\\rangle + \\sum_{m} \\left| \\Psi_{m}^{(0)} \\right\\rangle E_{m}^{(0)} \\left\\langle \\Psi_{m}^{(0)} \\right|", "c5edf89fa7c550ea0753b6db00d1fa25": "B_n(y+x)=\\sum_{k=0}^n{n\\choose k}B_{n-k}(y) x^k.", "c5ee574dbec786c4cd361f4e35410bb3": " k_{b_1} ", "c5ee58a8fbdf38b51466e98398dd0490": " \\frac{{\\delta f}}{{f_{res} }} = 1.2\\left( {\\frac{d}{W}} \\right) ", "c5ee6851700f062efb2d708aa077620f": " \\delta_{m} < 0 ", "c5eeb75329fcef6e62cbac02bbf21cf0": "\\displaystyle f(x) = A_0x_1^n+\\binom{n}{1}A_1x_1^{n-1}x_2+\\cdots+A_nx_2^n", "c5eeeec0e3fe6c166dda0e40d06cf90c": " \\quad -f(D+h)u-Rv-g(D+h)\\frac{\\partial h}{\\partial y}=0 \\qquad \\qquad (2)", "c5eef63d316877689bba70ad4e05eefb": "\n\\frac{L(\\theta_0+h|x)}{L(\\theta_0|x)} \\geq K;\n", "c5ef31fa3cebeee869f854f438aa080d": " -y_1 ", "c5efc59fdd371ac0520690c927a39c4b": " \\mu_A (x) := \\left\\lfloor {x - x_0 \\over L} \\right\\rfloor, ", "c5efe631c1d8fdd2bbd7c960f3f7ddde": "h(x,y,t)", "c5efe7393ddc711b358ee4f34fc695c1": "P = \\frac{\\mu_0 \\ddot{m}^2}{6 \\pi c^3}", "c5eff918c8e2e766bf86c2396138e1e2": "G = \\frac{4Lmf_t^2} {bt}\\left( \\frac{B} {1+A} \\right)", "c5effc8f39797b47db4483d20d03f33b": "\\mathcal{B}(X^*_{b(X^*, X)}, Y_{b(X^*, X)}; Z)", "c5f09b4d637505c3d1de3e0ac13e916b": "\n \\mathbf{y} = \\begin{pmatrix} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{pmatrix}, \\quad\n \\mathbf{X} = \\begin{pmatrix} \\mathbf{x}'_1 \\\\ \\mathbf{x}'_2 \\\\ \\vdots \\\\ \\mathbf{x}'_n \\end{pmatrix}\n = \\begin{pmatrix} x_{11} & \\cdots & x_{1p} \\\\\n x_{21} & \\cdots & x_{2p} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n x_{n1} & \\cdots & x_{np}\n \\end{pmatrix}, \\quad\n \\boldsymbol\\beta = \\begin{pmatrix} \\beta_1 \\\\ \\vdots \\\\ \\beta_p \\end{pmatrix}, \\quad\n \\boldsymbol\\varepsilon = \\begin{pmatrix} \\varepsilon_1 \\\\ \\varepsilon_2 \\\\ \\vdots \\\\ \\varepsilon_n \\end{pmatrix}.\n ", "c5f1285215efa0b32cc52c03abaed653": "\\textstyle (e^{-1})^e \\le x \\le e^{e^{-1}}) ", "c5f13c7c2c8e6bcc1f51b24715ba0689": " E_{external} ", "c5f169f9880fd81d91bd61f33d0ef62c": "(n+O(n^{1/2}))(n + O(\\log n))^2 = n^3 + O(n^{5/2})\\ ", "c5f198f54c7d47c6f14ebbf199961727": "T = \\frac{\\sqrt{n}\\bar{X}}{\\hat{\\sigma}} = \\frac{\\sqrt{n}\\frac{\\bar{X}-\\theta}{\\sigma} + \\frac{\\sqrt{n}\\theta}{\\sigma}}{\\sqrt{ \\frac{(n-1)\\hat{\\sigma}^2}{\\sigma^2} \\frac{1}{n-1} }}", "c5f1d7e3b339a023f430af9bb1bcef9a": "x_1 = 1/3", "c5f211eab9af13ff806dcde283c9d4e1": "R_\\mathrm{srgb}", "c5f239fe12b70554500eb7596f0e196c": "\\sigma\\sqrt{r_t}\\, ", "c5f2534b210531f4ffc45e23b23d11bc": "C_1,C_2", "c5f257ab30ba7b0030bca94b788bba2a": "\\varepsilon_0 ! =1", "c5f287d00c27ddeb1075ac4368d6d8b0": "\\pi0,\\;\\; \\forall y, z", "c6006ebf1ace9a1b4911d253118219c8": " H\\subset G ", "c600b665f43cbd5805be1eb446a92b0c": "\n\\begin{align}\nm & = 2\\times 1000 + 4\\times 100 + 3\\times 10 + 0 \\text{ arcminutes}\\\\\n & = 2430 \\text{ arcminutes} \\\\\ns & = 5\\times 10 + 1 \\text{ arcseconds}\\\\\n & = 51 \\text{ arcseconds}\\\\\nt & = 1\\times 10 + 5 \\text{ sixtieths of an arcsecond}\\\\\n & = 15 \\text{ sixtieths of an arcsecond}\n\\end{align}\n", "c600e41f51129203cc6714c4ee26c1ba": "L_x/T", "c6011b0dd3006520ed9ac472e6767d21": "\\alpha\\in\\mathbb{Z}", "c6017e7944b6b13e4b9b9e546c43303b": "(Q,+,\\cdot)", "c601c807887f6d11b754119d779c0e9e": " W_{ADK}=|C_{n^{*}l^{*}}|^{2}\\sqrt{\\frac{6}{\\pi}}f_{lm}E_{i}(2(2E_i)^{\\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}e^{-(2(2E_i)^{\\frac{3}{2}}/3F)} ", "c6022a62ad0bcf95bf0ff2b9ac75c32c": " q = \\int \\rho_n \\mathrm{d} V_n ", "c6027e02b92c01f3ac0087cb2231cc83": "W\\,\\ ", "c6030ea79eff45699e379fb0631c32d1": "U_i ", "c603246890daea6e1ffea936edc6bba5": "\\kappa_5=\\mu_5-10\\mu_3\\mu_2\\,", "c60384d7c018049f8f98dbd0f84c3c53": "\\begin{pmatrix}\n-1 & 1\\\\\n0 & 1\n\\end{pmatrix}", "c603b1e9f3d856d8ef259be06a9aee2c": "(\\pm 8)^2 = 64 \\equiv 13\\pmod {17}", "c603ca4401fb82c82f0d7973c0735c5c": "P_{2}^{-2}(x)=\\begin{matrix}\\frac{1}{24}\\end{matrix}P_{2}^{2}(x)", "c60461a078c6e4c4a004a0b9ebed6f82": "u=\\frac{u_{\\star}}{\\kappa}\\ln\\frac{z}{z_0},", "c6046b2bc6cb9bfe9dd1341fa2db9f59": "2=\\tfrac{8}{4}", "c6046ce3734ae51e5c662eaeb19ebb96": "T_{Ti}", "c60487381d9a02c66e45a4952b93f236": "\\left\\{ \\frac{1}{\\sqrt{2\\pi}}, \\frac{\\sin(x)}{\\sqrt{\\pi}}, \\frac{\\sin(2x)}{\\sqrt{\\pi}}, \\ldots, \\frac{\\sin(nx)}{\\sqrt{\\pi}}, \\frac{\\cos(x)}{\\sqrt{\\pi}}, \\frac{\\cos(2x)}{\\sqrt{\\pi}}, \\ldots, \\frac{\\cos(nx)}{\\sqrt{\\pi}} \\right\\}, \\quad n \\in \\mathbb{N}", "c604a3fbdbd5977f5c9cc5ff80633e8e": "\\prod A", "c604b726b6167a469d9c7659150fd4ac": "A = \\begin{bmatrix}-2&2&-3\\\\\n-1& 1& 3\\\\\n2 &0 &-1\\end{bmatrix} ", "c604cf0071ea7e40924adccce4c2bf46": " kX \\sim LL(\\alpha, k \\beta)\\,", "c605379d8a4938470a306c4deed78a16": "{\\Bbb Z}^3", "c605bb6ec1dbac63ef4fa6395701cf2e": "\na_0 = \\frac{4\\sqrt{2}\\kappa c^2}{l}\n", "c606120bb6e46f6e8b02493ddae40df2": "1 < \\sigma_0(n) < n", "c6063979a84082494c0452095a6f7b05": " a^{(n-1)/2} \\equiv \\left(\\frac{a}{n}\\right) \\pmod n", "c6064ed1c693280294b51ad25f6d416d": "(F\\varphi)u_1 = u_2.", "c6064f76dd752fc41ee1244c53bb3fde": "\\sigma_n(A) = 1/2", "c606aece3ed75fb7a68c1ed1d73ffb7f": "t = \\frac{d}{\\sqrt{2U}} \\sqrt{\\frac{m}{q}}\\,", "c60744b2568a882aab1ef84bb4d04a7d": " L(\\theta,\\widehat{\\theta}) = a|\\theta-\\widehat{\\theta}| ", "c60760e2ef78b3d2ad09052c34d416e9": "(n+1)p-1 \\leq M < (n+1)p.", "c6076e3a2ac5511de1343e675e04d4f2": " \\begin{align} D \\left ( ka \\right ) = & \\frac{\\mu_A}{\\mu_B} \\left \\{ ka I_0 \\left ( ka \\right ) - I_1 \\left ( ka \\right ) \\right \\} \\Delta_1 - \\frac{\\mu_A}{\\mu_B} \\left \\{ \\left ( k^2 a^2 +1 \\right ) I_1 \\left ( ka \\right ) - kaI_0 \\left ( ka \\right ) \\right \\} \\Delta_2 \\\\\n& -\\left \\{ ka K_0 \\left ( ka \\right ) + K_1 \\left ( ka \\right ) \\right \\} \\Delta_3 - \\left \\{ \\left ( k^2 a^2 +1 \\right ) K_1 \\left ( ka \\right ) + ka K_0 \\left ( ka \\right ) \\right \\} \\Delta_4 \\end{align}", "c607799fc2dc684f9926a25bfea01381": "\n e^2 = 1 + \\cfrac{4}{0 + \\cfrac{2^2}{6 + \\cfrac{2^2}{10 + \\cfrac{2^2}{14 + \\ddots\\,}}}} = 7 + \\cfrac{2}{5 + \\cfrac{1}{7 + \\cfrac{1}{9 + \\cfrac{1}{11 + \\ddots\\,}}}}\n", "c6078578f732821ae3bd261a79376cb3": "\\overline t \\!", "c60798c7505991188e1050cb1ae9236b": " U \\neq 0, V = W = 0.\\,", "c607ac204a2583d94a2d274032f118f3": "\\sum_j x_{ij} - \\sum_j x_{ji} = \\begin{cases}1, &\\text{if }i=s;\\\\ -1, &\\text{if }i=t;\\\\ 0, &\\text{ otherwise.}\\end{cases}", "c607d36dbe5c9d68f49d69dbc3c69501": "c_{ij} \\in {0,1}, i = 1, 2, ..., P, j = 1, 2, ..., W", "c607e8b97df94e7a76d577b3c45e1893": "10 n", "c60887634617d902f638691962d9e2e5": " b > 1 ", "c608bc471824c0a06ee05e7295e59aa0": "P_{r} = \\frac{\\lambda^{2} G_{r} G_{t}}{(4 \\pi r)^{2}} P_{t}", "c608d2ddf1b7efdcefe03c288f175d36": "r = (1 + R)^{1/t} - 1", "c608d35cc2ee2cfb84ad85ba5fcebc14": "A = TSD^T", "c6097bfd58099d6b33497aa43148881d": " \\sum_{i,j} de_i de_j' a'_r = 0 ", "c609b3c59ffe0699ec9cd9c31f5859ec": " \\frac{R}{R_w} = \\frac{R_{c}}{R_{wr}}", "c609e8400ee3350bda430f317c8b6394": "(M_3,M_4)", "c60a4ed196d9c27cbc24988afffffc5b": "C_M(k) = w_1 \\cdot P(k) + w_2 \\cdot P(k + 1) + w_3 \\cdot P(k+4)", "c60ae0bf90bcf65b7492e598d545f183": "{\\Pr}_{h \\in H}[h(a)=h(a')] \\le \\epsilon ", "c60b106245bdb96e758fc090cad0264d": "\\ln |y| = \\left(-\\int f(t)\\,dt\\right) + C\\,", "c60b644e0bb599b2233a218ee7bce989": " I = \\hat{U}^{\\dagger} \\hat{U} \\approx \\left ( I + i\\hat{H}^{\\dagger}\\tau \\right ) \\left ( I - i\\hat{H}\\tau \\right ) \\approx I + i\\hat{H}^{\\dagger}\\tau - i\\hat{H}\\tau + \\hat{H}^\\dagger \\hat{H} \\tau^2 ", "c60b97cec31fd77a6859aa4845356bef": "N + \\tfrac{1}{N}", "c60b9b3c9afd729e0c01179910cd2b63": "x_{n+1} = \\frac{1}{2}(x_n + \\frac{2}{x_n})", "c60ba19b7fa40cab0c287b7515ca4dba": "dV=dx_1dx_2dp_1dp_2=Constant \\ ", "c60ba246c00bb667c0d30832dead29fd": " \\mathcal{E}\\cong\\mathcal{O}(a_1)\\oplus \\cdots \\oplus \\mathcal{O}(a_n).", "c60c55e520e5c1b4cbf1d9e9e9f73c62": " \\overline{\\theta} = \\mathrm{atan2}(-\\overline{\\zeta},-\\overline{\\xi}) + \\pi ", "c60cff708445ba89b6a7a6477bf818bb": "dE=|\\vec{\\nabla}E|\\,dk'_z", "c60d23608832a678d95f04968c95bee4": "(4) \\quad \\int_0^\\delta e^{-xt} \\phi(t)\\,dt = \\sum_{n=0}^{N} \\frac{g^{(n)}(0)}{n!} \\int_0^\\delta t^{\\lambda + n} e^{-xt}\\,dt + O\\left(x^{-\\lambda-N-2}\\right)", "c60d390954ba50e474428b8d08ef1d28": "P_1 = r_1P", "c60d5d2e37c4a242dfc02049e93e39c2": "\\displaystyle{(x_{ij})^*=(x_{ji}^*).}", "c60d9872ecddbe60c60204ed895e72f6": "p_{\\mathrm{out}}=P(\\mathrm{SINR}\\leq t)", "c60da36641d760351933ace6a9927b41": "Holds", "c60e0fbcd1c345eb5f71fe45839d3a8d": "\\overline T = \\frac{T}{T^{\\text{*}}}, \\text{ } ", "c60e4cbf504385ed11433873b3fd6d72": "b=\\frac{Z_{1}Z_{2}e^{2}}{4\\pi\\epsilon_{0}mv_{0}^{2}}\\cot\\frac{\\Theta}{2}.", "c60e91d2f134ffc58f84b55a6a995570": "\\text{pre } Op", "c60ed395039781b5376e3a9686376ccb": " \\frac{\\zeta^4(s)}{\\zeta(2s)}=\\sum_{n=1}^{\\infty}\\frac{d(n)^2}{n^s}.", "c60ed4b473846e890689a8ed7e8449d8": "x^2+y^2 = (x+iy)(x-iy),", "c60eee62e98d26a81fca7eddb37b36a4": "(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0.\\,", "c60ef9e780bf62475d722445d0538fcf": "[u]_{k,\\alpha;\\Omega} = \\sup_{\\stackrel{x,y\\in \\Omega}{|\\beta| = k}} \\frac{|D^\\beta u(x) - D^\\beta u(y)|}{|x-y|^\\alpha}", "c60f21da21fa19cd17a05a90350de8fe": "\\psi_{nlm}(\\bold{r}) = R_{nl}(r) Y_l^m(\\theta,\\varphi)", "c60faa48c78c96b6d72756da3fd83a4f": " \\displaystyle{-iHf= C_-f|_{\\partial\\Omega} - C_+f|_{\\partial\\Omega}.}", "c60faf3ee93b92c9bd0b34ed4fc91a89": "(p_r)=r", "c60fdae522b973e3016144af318f9bbc": "m^{e^d} \\equiv m^{e d} \\equiv m^{(e d - 1)}m \\equiv m^{k(p-1)(q-1)}m \\equiv 1^{k(p-1)}m\\equiv m \\pmod{q}", "c61027470021d4e6400c62d6cb895900": " \\int_\\Omega \\varphi_i\\cdot f \\, dx = -\\int_\\Omega \\varphi_i\\nabla^2u^h \\, dx = -\\sum_j\\left(\\int_\\Omega \\varphi_i\\nabla^2\\varphi_j\\,dx\\right)\\, u_j = \\sum_j\\left(\\int_\\Omega \\nabla\\varphi_i\\cdot\\nabla\\varphi_j\\, dx\\right)u_j.", "c61079bbe12903b59894fd6443ffddcc": "Q_n", "c6108856765f7c3f2929aaf81692ad2d": "z = -x - i\\epsilon", "c610b3168529ae88d6e728295d3510bb": "10^{10^{10}}", "c610f2efe00d126bd9bfa74a474693d1": " (\\csc x)' = -\\csc x \\cot x \\,", "c611242aa4b7305d634c5869d46c38b1": "\\imath ^k", "c611363cf5b86cfb587bd9e56e622ef4": "\\phi_{n}", "c6115cf86132f7172693a38863172e9e": "1=\\varepsilon\\circ\\eta", "c6119c438e5a58a31ecfe011f461c3e1": "\\sigma_{\\min}(A) ", "c611af4dc9226b41fd5e6ac751aa697c": "\n\\frac{1}{\\sin\\theta} \\frac{\\partial}{\\partial \\theta}\\left( \\sin\\theta \\frac{\\partial Y^{m}_{l}}{\\partial \\theta} \\right) + \n\\frac{1}{\\sin^{2} \\theta} \\frac{\\partial^{2} Y^{m}_{l}}{\\partial \\phi^{2}} = -l(l+1) Y^{m}_{l}\n", "c611d087a015f357c438c29c9dc0db7c": "F(x)=\\|A\\mathbf{x}-\\mathbf{b}\\|^2.", "c611db5614d40deb192fa8d1fd88b162": "153 = 1^3 + 5^3 + 3^3", "c611f18ecabbc25b35a86f6643fbd26d": "T_{b'} = T_c + \\frac{T_{surr} \\Delta S}{c_p}", "c6122021407b8858a36c718ab1a16610": "m = S_2/S_1", "c612285ad8748cbf3d31aa099274ba08": "M_0(x_1,\\dots,x_n) = \\lim_{p\\to0} M_p(x_1,\\dots,x_n) = \\sqrt[n]{x_1\\cdot\\dots\\cdot x_n}", "c612cb9976d37c06d84432e9dd52dad0": "\n\\forall z\\,\\neg is\\_ carrying(z,S_{0})\n", "c612ea4f4f8061aa4fe1fff9c6251fc3": "c(E)=c(E')\\smile c(E'')", "c613262b2c2b514f48e8d181e94143ee": "\\Delta D-D\\Delta=(\\gamma+\\bar{\\gamma})D+(\\varepsilon+\\bar{\\varepsilon})\\Delta-(\\bar{\\tau}+\\pi)\\delta-(\\tau+\\bar{\\pi})\\bar{\\delta}\\,,", "c6135a5fdff2babbe9594b90b17ae2aa": "q =\\,\\mathbf{S}q + \\mathbf{V}q", "c6139a12b4e330ee35fa9555952b0ddf": "P_{LOSS} = V_{IN} \\times I_{Q} ", "c613b1cad0801359f025ca061b4be34e": "\\theta_{DA}", "c613bac69f49b9f429cd14ee1770f372": "f(-q,-q^2) = \\sum_{n=-\\infty}^\\infty (-1)^n q^{n(3n-1)/2} = \n(q;q)_\\infty ", "c613ddba4a237165ff4204fed9e73d9d": "\\operatorname{Res}(f,a)", "c61407fbcf72fe9ee311bdea3a15f779": "R_2 =(57-35) \\times 0.0078", "c6141c5960298128f16f42047d55b695": " h_r ", "c6144eb50c5e4fb32dcf7fa71b5e2e69": "BH \\to BG", "c614c36501cca9a2ca0814e230ec9d03": "\\overline{a}\\langle x \\rangle", "c614d99ff3094e69b3c02332eaf8e124": "\\mathbf{j}_2", "c614f41e2923ddce66a1bacf9f4b25f5": "ChR = 100 (1 - \\frac{T_o}{T_s} ) = 100 \\frac{T_a}{T_s} ", "c614fa3204bd7f26ff40c3f0724bed52": "p \\colon T \\to A", "c6151bc89b883c630052a17ea752cdd6": "\n w(0,y) = \\cfrac{d w}{d x}\\Bigr|_{x=0} = 0 \\qquad \\implies \\qquad\n w_x(0) = \\cfrac{d w_x}{d x}\\Bigr|_{x=0} = \\theta_x(0) = \\cfrac{d \\theta_x}{d x}\\Bigr|_{x=0} = 0 \\,.\n", "c6157000058fb5461c6293ed76f03f0a": "0 \\le \\ell \\le n_0-1", "c615aff1334434f8e271d8ebe39036a8": "\\displaystyle{S^2=Z,\\,\\,\\, (SR)^3 =Z,\\,\\,\\, Z^2 =I.}", "c616185219f0d7462a99635a030bcf03": "P A_1 P^{-1} = A_2.", "c6162b3baf1eb2b71135b84b7f0e6627": "\\rho^\\pi", "c61684695c625fbd9c0149e2e5075d22": "R \\ \\Delta(x) = (T \\circ \\Delta)(x) \\ R", "c616b6d570a30898e84c3bee4085feff": "\\hat K", "c616ceb0f9f053e1b051177d68b2b25f": " d \\det (A) = \\mathrm{tr} (\\mathrm{adj}(A) \\, dA).", "c616e85c76daa2113d1bb299e43cc719": " -\\infty < u,v < \\infty, 0 < r < r_0, -\\pi < \\theta < \\pi", "c617beaf149cd9e05e7750f35be9754e": "LG(x,s)=\\delta(x-s).", "c617ed0c629661a31383e4d1c652fbca": "T = n\\Delta t", "c61819b5eda4ee943a275b647d9c118b": " \\sum_{k=1}^{\\infty} \\frac{(-1)^{k}}{s_k-1} = \\frac{1}{1} - \\frac{1}{2} + \\frac{1}{6} - \\frac{1}{42} + \\frac{1}{1806} {\\,\\pm \\cdots} ", "c6184d88d3ad8a0c693628229ea57a31": "\n \\begin{matrix}\n 0 & \\mbox{for }k}\\cdot\\operatorname{sock}\\right).", "c623b8e1b7b0433a4651d41ba0ba8aeb": "\\mathfrak{m}_R S \\subset \\mathfrak{m}_S", "c623de92d025eb7b611250a6b058bca7": "(C^5_{13} - 10)(4^5 - 4)=1{,}302{,}540", "c623e5834adb2601d93174624f435937": " \\mathbf{A}_\\text{pent.} = {1 \\over 2}|x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - x_2y_1 - x_3y_2 - x_4y_3 - x_5y_4 - x_1y_5|", "c623ea2574455504ca08d9201cd3d58d": "\\scriptstyle{D(ab)=D(a)b+\\epsilon^{|a||D|}aD(b)}", "c6240734596ffd496e7d662771f55a12": "\\star\\lambda", "c6243fa281417639afba7e3407973393": "v_B \\in [x_b,y_b]", "c6244dc56e687c3ddbdab807998745c7": "\nH_{\\text{pot}} = \\tfrac 12 a q^2,\\,\n", "c6247a032e560709a7aa3cd02df649fb": "\\varphi_n \\varphi_n^\\dagger, ", "c62489665d292b2fb3ed24a9287159e3": "h\\otimes \\mathbb{Q} : \\pi_i(X)\\otimes \\mathbb{Q} \\longrightarrow H_i(X;\\mathbb{Q})", "c624deed0de80592db84b6b266044e86": "4 \\pi G \\rho (\\vec{x},t) = \\nabla^2 \\phi(\\vec{x},t). ", "c624ff5f003a4259ded41c0e9a5957ec": "\\bar{f}(\\bar{x}^j) = f(x^i(\\bar{x}^j))", "c62500007039e4406c80b4e22fc29387": " 2^j ", "c6250ff83e0e57a9d038428a8e60fd76": "\n\\begin{align}\nG & =\n3 \\sum_{n=0}^\\infty \\frac{1}{2^{4n}}\n\\left(\n-\\frac{1}{2(8n+2)^2}\n+\\frac{1}{2^2(8n+3)^2}\n-\\frac{1}{2^3(8n+5)^2}\n+\\frac{1}{2^3(8n+6)^2}\n-\\frac{1}{2^4(8n+7)^2}\n+\\frac{1}{2(8n+1)^2}\n\\right) \\\\\n& {}\\quad -2 \\sum_{n=0}^\\infty \\frac{1}{2^{12n}}\n\\left(\n\\frac{1}{2^4(8n+2)^2}\n+\\frac{1}{2^6(8n+3)^2}\n-\\frac{1}{2^9(8n+5)^2}\n-\\frac{1}{2^{10} (8n+6)^2}\n-\\frac{1}{2^{12} (8n+7)^2}\n+\\frac{1}{2^3(8n+1)^2}\n\\right)\n\\end{align}\n", "c6254f9f19d7f15b58685314a52c4919": "\\scriptstyle{\\mathrm{Pr}(s|I) \\;\\propto\\; 1/s}", "c625a26362a20ea3c9366dc8cf6a92ec": "\\mathcal{S} = -\\frac{1}{2\\pi\\alpha'} \\int \\mathrm{d}^2 \\Sigma \\sqrt{{\\dot{X}} ^2 - {X'}^2},", "c625d19e22e113106e5b14204b57820b": "\\varphi_0(\\gamma+1) [n] = \\varphi_0(\\gamma) \\cdot n = \\omega^{\\gamma} \\cdot n \\,.", "c625e038d04fa5338c3723277dbf6988": "\\kappa^2 = K", "c625f384a03b783ea943d1eba4e32877": "x^{k}", "c626012b9b3bdacddc7f0602895e37c0": "H^1(X, \\mathcal O(E))", "c6265614e49b70d87ff30c4c23554959": "X_n\\ \\xrightarrow{L^r}\\ Y", "c626819b1b2dc44447e0cf7aed60ff27": "R_H=300 \\mbox{ meters} \\cos(20^\\circ)= 282 \\mbox{ meters}", "c626c0ad26b905798204cf58fa61509c": "\\bar M", "c626fc8ebb73212ff564360db909e8ff": "\n\nf(T_1,T_3) = \\frac{q_3}{q_1} = \\frac{q_2 q_3} {q_1 q_2} = f(T_1,T_2)f(T_2,T_3).\n", "c6272f6e57d6802517931933c6fc432f": "\\Pi^0_{n+1}", "c62760e0e8bb718ce686146d3e1b9b91": "\\frac{\\sum_{i=1}^n w_i }{ \\sum_{i=1}^n \\frac{w_i}{x_i}}. ", "c62773295ae87d5048c4a06e10452fba": " p_g - 1 \\le h^0(K|_D). ", "c6278f76f33aefbb41931bea52058e64": "((p \\rightarrow q) \\wedge (q \\rightarrow r)) \\rightarrow (p \\rightarrow r)", "c627c74e4d16b3f10fcc04fdafae5198": "e_ie_j = -e_je_i \\qquad i\\neq j. \\,", "c627d1243462dd7c246d9524e134fe0f": "p(I | \\theta_{bg}).", "c627f8786a23f88a4c94bf2b6524df22": "\\,i^{(2)}", "c6285c653e2bef927399155ed36d4727": "\\frac{A_1,\\dots,A_n}B\\qquad\\text{or}\\qquad A_1,\\dots,A_n/B,", "c62863ed266d5cf32f44de02c0249c43": "\\overline{b}(\\lambda)", "c6288237b0bdf4fe481989e5ab66a2b6": "W_{AB} - \\Delta F", "c628aa92643d3107d1958eaef5435ccb": "\\textstyle (x^{2l-1}+1)", "c628ba2b1047de93f66cb815d986e107": "x\\in [0,1]", "c628cb5bd2b440d0beef8434e4c0cadb": "g(p_i)", "c628dd96f7000e0d5149e75413235eb7": " \\lim_{n \\to \\infty} \\frac{1}{s_n^2}\\sum_{i = 1}^{n} \\operatorname{E}\\big[(X_i - \\mu_i)^2 \\cdot \\mathbf{1}_{\\{ | X_i - \\mu_i | > \\varepsilon s_n \\}} \\big] = 0", "c62905924caeded6e2c890d331ae3af4": "\\ \\Delta S^\\ddagger ", "c6291bc825c762df9f5eb6e722faef4b": "\\scriptstyle{d_i}", "c6292b31b268353741fb0b553675b09e": "\\mathbf{B}_{\\perp}\\cos(\\omega t)", "c62936f0b85ee6dfa1fe37c93349edc8": "\\hat{\\mathbf{C}},\\quad\\overline{\\mathbf{C}},\\quad\\text{or}\\quad\\mathbf{C}_\\infty.", "c629a19f200707411a8683f2188314b6": "\n \\boldsymbol{\\sigma} \n = \\cfrac{2}{J}~\\left[\\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{F}\\cdot\\boldsymbol{F}^T+\n \\cfrac{\\partial W}{\\partial I_2}~(I_1~\\boldsymbol{F}\\cdot\\boldsymbol{F}^T - \\boldsymbol{F}\\cdot\\boldsymbol{F}^T\\cdot\\boldsymbol{F}\\cdot\\boldsymbol{F}^T) + \n\\cfrac{\\partial W}{\\partial I_3}~I_3~\\boldsymbol{\\mathit{1}}\\right]\n ", "c629bd07489e871f469548b6facf0c6c": "G^{\\alpha \\beta}_{\\gamma \\delta} (t,\\vec{x};t',\\vec{x}') = \\frac{1}{4\\pi} \\delta_{\\gamma}^\\alpha\\, \\delta_{\\delta}^\\beta\\, \\frac{\\delta(t\\pm|\\vec{x}-\\vec{x}'|-t')} {|\\vec{x}-\\vec{x}'|}", "c629c2d4f5a9cfba1c4bd34ded588250": "t = (t_1+t_2)/2", "c629f085950cb573ed90431eb4241983": "{\\rm d}S = \\frac{{\\rm \\delta}q}{T}.", "c62a0c44ff3b3fda2ad825240655f554": "\\{x_n\\}_{n=1}^N", "c62a2ba5e481d5bbe490867c32b49468": " \\frac{\\partial F}{\\partial t}(t,{\\bold x}) = \\kappa (t) ({\\bold x}-\\gamma(t))\\cdot {\\bold N}(t) - 1 \\ , ", "c62a6412b276cf6f95f7f1c750cc8e7b": "v'(x) = \\text{False}", "c62aa5e4dfdf684c7a1b4de84cba399e": "\\prod_{i=0}^{n-1} \\left( b^{2^i} \\right) ^ {a_i}\\ (\\mbox{mod}\\ m)", "c62aad460683560119b7ab8c6ef57cc8": "\n\\vec{D_{\\alpha}}=|\\vec{C_1}.\\vec{X_{\\alpha}}-\\vec{X}|\n", "c62abd8ff661fd0e1ef16724965c714a": " P = \\frac{1}{T \\cdot R}\\int_0^T e(t)^2 dt\\,\\!.", "c62af7c264f9f8c2587c71e15bfe4ffe": " \\widehat{A}\\psi = \\widehat{\\Omega}^\\dagger\\widehat{A}\\widehat{\\Omega}\\psi \\quad \\Rightarrow \\quad \\widehat{\\Omega}\\widehat{A}\\psi = \\widehat{A}\\widehat{\\Omega}\\psi ", "c62b00b5c0d1dd6ee64f5651300b4929": "L = \\log_{(1-10^{-7})} \\!\\left( \\frac{N}{10^7} \\right) \\approx 10^7 \\log_{ \\frac{1}{e}} \\!\\left( \\frac{N}{10^7} \\right) = -10^7 \\log_e \\!\\left( \\frac{N}{10^7} \\right),", "c62b1d45bb438145bfb4f571cbd8508f": "R_j \\to R_i, j \\ge i", "c62b56309d8b912eb24eddbf857b50f5": "V_{r1}", "c62c0bbadb51db394a5b4b05b6362b6e": "\\langle u,u'\\rangle -\\langle v,v'\\rangle = \\langle u-v, u'-v' \\rangle ", "c62c52a66a5ebbe9e85f6df463a60bbe": "E_{\\rm d}({\\rm AB}) =\\sqrt{E_{\\rm d}({\\rm AA}) E_{\\rm d}({\\rm BB})}+1.3(\\chi_{\\rm A} - \\chi_{\\rm B})^2 eV", "c62c67fe1513bc82a7f596379e4f0543": "\\gamma_{F}^{+}", "c62c7e2e0d7db27ecb122edce823b239": " \\rm \\stackrel{GCE} {\\rightleftharpoons} ", "c62cc84f81d5eaadb4a1655c2bccbdb6": "\\oint_C \\bold{E} \\cdot {\\rm d} \\boldsymbol{\\ell} = \\frac{\\partial }{\\partial t} ", "c62d9ab2b9611aed074ac8e99f952d76": "3 \\omega_{p}", "c62d9e44b8a29abb903fc7a0fa1c25e0": "BAA = \\frac{H}{BF-BB-HBP-SH-SF-CINT}", "c62dba0fb75b5df0df82457fca711b8e": "(a_1, a_2, \\ldots, a_n) = (b_1, b_2, \\ldots, b_n)", "c62dec11d94adc7a293cd2847de1df6e": " R = e^{\\frac{-i\\theta}{2}} = e^{\\frac{-\\theta\\mathbf{e}_{12}}{2}}, ", "c62e07c87eabd63f21ab539f892fd9a5": "u'''(c)>0", "c62e0a5aa6fbedc9cb3e0a782e44f1c8": "{\\sigma}_i", "c62e4e6e0f67de6ed9da6fa3847f1fac": "\\frac{\\mathrm{d} (T_{h})_{*} (\\gamma^{n})}{\\mathrm{d} \\gamma^{n}} (x) = \\exp \\left( \\langle h, x \\rangle_{\\mathbb{R}^{n}} - \\frac{1}{2} \\| h \\|_{\\mathbb{R}^{n}}^{2} \\right),", "c62e84b9db5692ba1a5aabd32f869fb9": " X \\pm ( \\frac{ 1 }{ \\alpha } - 1 ) | X - \\theta | ", "c62e9c2376da9e2d11c556423d5da92c": " \\text{value} = (-1)^\\text{sign}\\left(1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i} \\right)\\times 2^{(e-127)}", "c62eb80f47fdeff1e75269b0e3551446": " (x+y) \\otimes z = x \\otimes z + y \\otimes z ", "c62edd86890f02a1ae13160a388380b6": "v_{\\rm rms}", "c62ef43e5b0117c5b582571adda2c833": "n=d+2", "c62f5e5da30ac83e9797f15d5397e3ea": "x^5-10x^3y^2+ 5xy^4\\, ", "c62f8b888f5fc83b9f56ee70b82a390f": "g^{}_i", "c6305d8f348b3a1488e753dd250d08a0": "P=X^2-X=X(X-1)", "c63060ac79ea5e0a7183c41f65600d25": "\\nu_i = \\partial N_i / \\partial \\xi \\,", "c630b8644676c58c51b321be020ab380": "f_{\\text{cu}}", "c630cac2a13b6cb3884526c6ad99af99": "-\\frac{\\hbar^2}{2m} u''(r)=E u(r),", "c630ee55658b5a55450746b709f1b1c6": "\\left[ [ \\mbox{min} ] [ \\mbox{ed} ] \\right]", "c6310eab20a977b6f66642f4ab44fb49": "e^{-x^2/2}", "c6314ed027f41dd3b4d09a42066651dc": "S_p = \\frac{T_1}{T_p}", "c631530cbdf55853e4d015afdf72e882": "\\begin{bmatrix} \\dfrac{1}{g_{11}} & \\dfrac{-g_{12}}{g_{11}} \\\\ \\dfrac{g_{21}}{g_{11}} & \\dfrac{\\Delta \\mathbf{[g]}}{g_{11}} \\end{bmatrix}", "c631559cf6c91b7c7a6fb7bd2cf836f1": "\\begin{matrix} {2 \\choose 1}^2{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "c6318404e254f6ceb0bb092c9bd1ba3e": "\\chi(\\omega) = {1 \\over i \\pi} \\mathcal{P} \\!\\!\\!\\int \\limits_{-\\infty}^\\infty {\\chi(\\omega') \\over \\omega'-\\omega}\\,d\\omega'. ", "c631a67f2c16c33e9c99ab40c77f8e13": " 0 = (x^2+y^2+z^2 + R^2w^2 - r^2w^2)^2 - 4R^2w^2(x^2+y^2) , \\,\\!", "c632056e88a3a4aa2dbe1b99ea087ae3": "\n\\begin{cases}\n \\tan \\alpha_m=\\frac{p_b+l \\cdot \\tan\\alpha_b}{l} \\\\\n \\tan \\alpha_r=\\frac{p_b-p_r+l \\cdot \\tan\\alpha_b}{l}\n\\end{cases}\n", "c6321397506355dd6de675a9afb59df1": "\\sqrt[3]{-\\frac q2+\\sqrt{\\frac{q^2}{4}+\\frac{p^3}{27}}}+\\sqrt[3]{-\\frac q2-\\sqrt{\\frac{q^2}{4}+\\frac{p^3}{27}}},", "c6322a6bc16b7f89adec51f72976b54c": "x = s/(1+s)", "c6323045d12d06c6faae510ba0081f9a": "{G_1}", "c63237922503c634c61a912e8dd5f303": "u \\,", "c6324bc8e3cd2ff69abcf7b372ce3b6e": "F(x) = \\sum_{n=-\\infty}^\\infty a_n\\cdot x^n", "c6325d0512837253f72f5952c5a8a236": "\\mathcal{B}(m,n) + o(\\mathcal{B}(m,n))", "c632e74decd75c82c854afe4f5a53773": "c\\leftrightarrow d", "c63449987481baf5c3aa09d72feb29fd": "\\tbinom{d-1}{n}", "c6344b6384d98ed29ff03df369a09d28": "X(e^{i \\omega}), 0 \\le \\omega \\le 2\\pi", "c634712ec0fe59f1f1f87097fa83789c": "R(x) := \\int_{-\\infty}^{x} H(\\xi)\\,\\mathrm{d}\\xi", "c634f0d08cf2058d35ab7647db2a3e70": "(x_1, ..., x_m,x_{m+1},..., x_n) ", "c6352bd15bc387259ce2e302f25da39d": "E\\cap\\overline{E}=\\Delta\\otimes\\mathbf{C}", "c6353c7b0b5895c7624dae73316c849c": "(1-c)^{-1} \\mathrm{OPT} + c\\mathrm{WORST}", "c635970d39870334893864d52e9a57ac": "\\left[ \\frac{1}{2} (n^2 + n) \\right] T_6 + \\left[ \\frac{1}{2} (n^2 + 3n) \\right] T_5 + (n + 1)T_4 + T_1 + T_2 + T_3 + T_7 \\le cn^2, n \\ge n_0", "c6361d549f9d89452f608dc586361564": "E=N\\varepsilon_0+Mk_BT. \\, ", "c636764521b5e3f237d8b32dd2a02552": "F(\\bullet, k)", "c63751a6688239139c4256ec4049d342": " \\vec{e}_2 = \\frac{1}{C\\left(\\frac{q^2}{\\omega^2}, \\, \\frac{2q^2}{\\omega^2}, \\, \\omega u \\right)} \\partial_x ", "c63783f16dfbc7c56ab33a0068af92b6": " K_{ij}=\\sum_{\\alpha = 1}^{N}(\\lambda_{\\alpha } V_{\\alpha i} V_{\\alpha j}) \\,\\!", "c637dd4c34dbd57eb1427453c9bf54a1": "(i_{1} \\times i_{2})_{*} (\\gamma^{H_{1} \\times H_{2}}) = (i_{1})_{*} \\left( \\gamma^{H_{1}} \\right) \\otimes (i_{2})_{*} \\left( \\gamma^{H_{2}} \\right),", "c637df2566988d9192182c85bbec3c22": "{\\pi^2\\over6}", "c637e777d23bd4a7d52a6d88c4581d46": "D_{n,n'}=\\sup_x |F_{1,n}(x)-F_{2,n'}(x)|,", "c638098d43c8fe7d5469b3928aaf63b0": "\\begin{align}\n & \\widehat{\\boldsymbol\\Sigma} = {1 \\over n} \\sum_{j=1}^n \\left(\\mathbf{x}_j - \\bar{\\mathbf{x}}\\right)\\left(\\mathbf{x}_j - \\bar{\\mathbf{x}}\\right)^T \\\\\n\n & A = {1 \\over 6n} \\sum_{i=1}^n \\sum_{j=1}^n \\left[ (\\mathbf{x}_i - \\bar{\\mathbf{x}})^T\\;\\widehat{\\boldsymbol\\Sigma}^{-1} (\\mathbf{x}_j - \\bar{\\mathbf{x}}) \\right]^3 \\\\\n\n & B = \\sqrt{\\frac{n}{8k(k+2)}}\\left\\{{1 \\over n} \\sum_{i=1}^n \\left[ (\\mathbf{x}_i - \\bar{\\mathbf{x}})^T\\;\\widehat{\\boldsymbol\\Sigma}^{-1} (\\mathbf{x}_i - \\bar{\\mathbf{x}}) \\right]^2 - k(k+2) \\right\\}\n \\end{align}", "c6383ec442085e15aad3d71a744e4800": "\\text{cl}:(\\mathcal{P}(X),\\subseteq)\\to(\\mathcal{P}(X),\\subseteq)\\,", "c63843ba5656f73e522a23dd9ea78e81": "T = \\frac{1}{4} \\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}", "c6385ca7f5610d53053a3bc3ab5c636b": "\\theta \\in \\bigwedge^{n-k} V", "c6386ed03133b14b845071fa5d6d1cab": "\nS(z) = \\frac{1}{z+\\lambda}\n", "c63892e2347b001a7bba066325879067": "w(\\pi(v),v)", "c638cf3e9f9368462452bbb315db7b87": "\n\\chi\\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_v + Y^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_v\\Big) = \\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_{p(v)}, (v^1,\\ldots,v^N,Y^1,\\ldots,Y^N) \\Big)\n", "c638f314bffd2867d04e96f9f7740cf0": "\\bar y \\pm \\sigma_y (n-1)^{1/2}.", "c639b4d3cbd52417c7a0b2838fedba8a": "\\dot{N}\\,\\!", "c639e85d889c39dabd6ee943fbc1a411": "b^{m+n} = b^m \\cdot b^n", "c63a2686d242b0cb6c184f10d84e4129": "f(t x,y) = t f(x,y),\\quad f(x,t y) = f(x,y) t^{\\alpha}", "c63a496da95d4aeeb2fcd8818185dfe1": "\\int_{g\\in G}\\chi(g^2)\\,d\\mu", "c63ae4e25a4e34460d328ce5add7f3ca": "a_{65:\\overline{10|}}", "c63b1da4bc99dbbc6991650f745993be": "c_{\\lambda,\\mu}^\\nu", "c63bdf91fca2ae0b71a1368b15c2c4a0": "\n\\operatorname{DE9IM}(a,b) = \\begin{bmatrix}\n\\dim(I(a) \\cap I(b)) & \\dim(I(a) \\cap B(b)) & \\dim(I(a) \\cap E(b)) \\\\\n\\dim(B(a) \\cap I(b)) & \\dim(B(a) \\cap B(b)) & \\dim(B(a) \\cap E(b))\\\\\n\\dim(E(a) \\cap I(b)) & \\dim(E(a) \\cap B(b)) & \\dim(E(a) \\cap E(b))\n\\end{bmatrix}\n", "c63c43add7c4bfd5eca210ff0a0d86b6": "\\textbf{F}_{k}", "c63c60d56d011695e2e6a2ad62657ed2": " \\mathrm{Res}(f, \\infty) = -\\lim_{|z| \\to \\infty} z \\cdot f(z)", "c63c7315963d17a9e83b90242033f7a5": "net.assets \\div net.premium.written ", "c63ca440dcfaffa7ca10427660a57d37": "\\boldsymbol{\\Phi} = \\left(\\phi(y_i),\\dots, (y_n)\\right), \\boldsymbol{\\Upsilon} = \\left(\\phi(x_i),\\dots, (x_n)\\right) ", "c63d4e0a661fd8878fd9665682c2805c": "\\alpha(\\langle v\\rangle)=\\langle \\beta(v)\\rangle", "c63d874c9116262d8a9d730b1c5edcdc": " \n\\rho \\rightarrow { \\rho \\over {4 \\pi } }\n", "c63d8b362af28b7302363c90fb42b6b1": "\\boldsymbol{k}\\cdot{\\boldsymbol{p}}", "c63df598ed113fcb9cb81f95a52c0c16": "i\\in D_2", "c63df9adddcfaa78d1c5980e49484bb3": "(a,b) = (b,a)", "c63e00919eb8231eb1cc54d2a8e8b169": "g(X,Y)", "c63e437482443c2911a84fbf88c7d413": "\\hat{\\theta}_{\\mathrm{ML}}(x) = \\underset{\\theta}{\\operatorname{arg\\,max}} \\ f(x | \\theta) \\!", "c63e5ee735969e4f2fbec1d9ef734f8c": "\nx_1 = \\frac{\\ell}{2} \\sin \\theta_1,\n", "c63e7738668629d6a4868f51056345d5": "[n,m]", "c63e8a97c9b00bff8478a2aedb74e3a2": "\\psi^{(0)}(n) = -\\gamma\\ + \\sum_{k=1}^{n-1}\\frac{1}{k}", "c63ea8ca0cf79eb31592c0b69d5d66d5": "0 < a < 1", "c63eb4ca10990eb239dfb7be1e5472e7": "\n\\begin{align}\n\\text{Area }&= r(s-a) + r(s-b) + r(s-c) = r(s-a + s-b + s-c) \\\\[8pt]\n&= r(3s - (a+b+c)) = r(3s - 2s) = rs \\\\[8pt]\n\\end{align}\n", "c63f98319b5696eb1d9da64058fb5e9e": "\\widetilde{\\Gamma}", "c63fdba90189b061dcb3ea05e7eef57b": "-cf+\\frac{d^2 f}{dX^2}+3f^2=A", "c640194918b8e6324e21fa9d7977a180": "p \\in A", "c64086f19ba25fb8de6e8873bc37ecce": "\\langle u,u'\\rangle /\\langle v,v'\\rangle = \\left\\langle \\frac{u}{v}, \\frac{u'v-uv'}{v^2} \\right\\rangle \\quad ( v\\ne 0) ", "c640885de84ab0eaa10e3cd246642161": " x_P,y_P,z_P", "c640b7d664bf74cb29fe47e3dffa5a12": "\\scriptstyle f: R \\setminus \\{0\\} \\rightarrow \\N", "c640f233980ab6fe7e6ce6726cd955de": "\\frac{d^2{T}}{d{t}^2} + \\omega^2T = \\left( { d^2 \\over dt^2 } + \\omega^2 \\right) T = 0,", "c6413ef3b0cd11e20acaad60af356c00": " dx \\wedge dy ", "c6414e9d3c1e19f7660ef74181f41970": "~\\setminus~", "c641835ab1d10c7b0df83bcb4ee582ca": " \n \\left|\\begin{array}{cc} \n a_{11} & a_{12} \\\\\n a_{21} & a_{22} \\end{array}\n \\right|_{11} = a_{11} - a_{12}{a_{22}}^{-1}a_{21}\n\\qquad\n \\left|\\begin{array}{cc} \n a_{11} & a_{12} \\\\ \n a_{21} & a_{22} \\end{array}\n \\right|_{12} = a_{12} - a_{11}{a_{21}}^{-1}a_{22}.\n", "c641b4ed30a01f513f7188e87a7e0843": "\\textstyle{{2 \\over 3} \\div {2 \\over 5} = {2 \\over 3} \\times {5 \\over 2} = {10 \\over 6} = {5 \\over 3}}", "c641edd9dfa2f75587ab1df33f1e959e": "m_\\mathrm{Ag} = \\left(\\frac{16.00 \\mbox{ g }\\mathrm{Cu}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{Cu}}{63.55 \\mbox{ g }\\mathrm{Cu}}\\right)\\left(\\frac{2 \\mbox{ mol }\\mathrm{Ag}}{1 \\mbox{ mol }\\mathrm{Cu}}\\right)\\left(\\frac{107.87 \\mbox{ g }\\mathrm{Ag}}{1 \\mbox{ mol Ag}}\\right) = 54.32 \\mbox{ g}", "c64258117e9830d936843123650debd8": "C(n,d)", "c64272d2266a200ef2aac2670b916c39": "s, s' \\in F_L(x)", "c64279a14cf8dd3c702f655f3244fad3": "S^+ = T^+", "c6436f2992c7eee9c75001f8bb4ea962": "\\displaystyle \\Psi(x,t) = \\sum\\limits_{n} A_n \\psi_{E_n}(x) e^{{-iE_n t}/\\hbar}. ", "c6439195e0abbcbaeddb16e1982c990e": " S(A,P,z) \\le \\frac{X}{U(z)} + \\frac{2}{U(z)} \\sum_{p \\mid P(z)} \\left\\vert R_p \\right\\vert +\n\\frac{1}{U(z)^2} \\sum_{p,q \\mid P(z)} \\left\\vert R_{p,q} \\right\\vert . ", "c643c2fa8f353cb098368ad803a01912": "M = \\chi_0 H + \\alpha_R \\mu_0 H^2.", "c64459cbf46db8b99415557ea2bdfc63": "100\\uparrow\\uparrow 4=(10\\uparrow)^2 (2 \\times 10^ {200}+0.3)=(10\\uparrow)^2 (2\\times 10^ {200})=(10\\uparrow)^3 200.3=(10\\uparrow)^4 2.3", "c6448f2311ad8e329547c1d3f16fc0b0": " \\begin{bmatrix} x \\\\[5pt] \\dfrac{1}{x} \\end{bmatrix} ", "c644961e61f0819d8b6761850de8812e": " gz(\\rho_2-\\rho_1)=2\\gamma\\left (\\frac{1}{R_A}-\\frac{1}{R_B}\\right)\\!", "c644b1f87fe09ebe604c61529a5fdfaf": "\nV^*(b) = \\max_{a\\in A}\\Bigl[ r(b,a) + \\gamma\\sum_{o\\in O} \\Omega(o\\mid b,a) V^*(\\tau(b,a,o)) \\Bigr]\n", "c644d4afcdcee76caa99a3483f269445": "N = q^k-1", "c64505e2ae4481688efb93a5d086201f": "\\phi \\lor \\lnot \\phi ", "c64540573160e23330f1f6bb4bea4bc8": "(g,h) \\mapsto g \\cdot h", "c645620a5fe173c3f379b4a1763757f2": "H= \\Delta S_x + \\sum_i\\left(\\frac{p_i^2}{2 m_i}+\\frac{1}{2}m_i \\omega_i^2 q_i^2\\right) + S_z \\sum_i{C_i q_i} ", "c64572919b0a395b2008e12ca51a319e": " I_{|X|\\geq K} = \\begin{cases} 1 &\\text{if } |X|\\geq K, \\\\ 0 &\\text{if } |X| < K. \\end{cases}", "c645b2e9795d0709001abf7d492ffb3d": "p_p = 1 - (1 - p_e)^N", "c64618b5fcadbfa47a648ecffadc7127": "X_1, X_2, X_3, \\dots \\!", "c646609d544e5c54ccc1cb4f544fff6e": "S(x) = \\phi_0(x)a_0 + \\phi_1(x)b_1(x) + \\beta_1(x)\\phi_0(x)b_2(x).", "c646d58a5c6d820087a278638a7052fe": "\nR_\\theta = \\frac{\\text{var}[\\theta]}{\\text{var}[\\hat{\\theta}]} = \\frac{\\text{var} [\\hat{\\theta}] - \\text{var}[\\epsilon]}{\\text{var}[\\hat{\\theta}]}\n", "c647227738617e22ef2624cf79854201": "|z_{kr} - z^*|\\alpha^{k}", "c6480cbec8a4cefd95938ae08dc5b469": " \\,c'=\\lambda_1 c+\\lambda_2 d ", "c648bc7192c7355d563b346a1e788b93": "O(n^2\\log\\log n/ \\log n)", "c648ce255f40f1f1d54a4a3976a5f770": "\\color{OliveGreen}\n\\mathbf{P}\\cdot\\nabla^{2}\\mathbf{Q}-\\mathbf{Q}\\cdot\\nabla^{2}\\mathbf{P}=\\nabla\\cdot\\left[\\left(\\mathbf{P}\\cdot\\nabla\\right)\\mathbf{Q}+\\mathbf{P}\\times\\nabla\\times\\mathbf{Q}-\\left(\\mathbf{Q}\\cdot\\nabla\\right)\\mathbf{P}-\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}\\right].", "c648fa8307a6c70f4704c170c06e1103": "a = \\frac{5(4\\nu+3)}{\\nu^2+1}", "c649130bacbf197da85b1825228e191e": "V_i^{}(t_n^{})", "c6495a705bf0b96b3935eb0dd1134aa3": "F_{\\mu\\nu} F^{\\mu\\nu} = \\ 2 \\left( B^2 - \\frac{E^2}{c^2} \\right) ", "c64989057f63ba993e4cd98c57a40d34": "rf:(x_1,\\ldots,x_n)\\mapsto rf(x_1,\\ldots,x_n)", "c649946cda9866c1c74e044f26af8013": "\\langle\\Psi(0)\\Psi(r)\\rangle\\propto r^{-\\gamma}", "c649b28018c61da37ea70bda100b798a": "S(a)", "c649e2a021b7812e0e0b8e6396c4eca3": "\\sigma_\\mathrm{elliptical\\ crack} = \\sigma_\\mathrm{applied}\\left(1 + 2 \\sqrt{ \\frac{a}{\\rho}}\\right) = 2 \\sigma_\\mathrm{applied} \\sqrt{\\frac{a}{\\rho}} ", "c64a12ad5fd0463e91c58ec3448f49b1": "\\begin{align}\n& J_\\parallel' = \\gamma ( J_\\parallel - v\\rho)\\\\\n& \\rho' = \\gamma (\\rho - v J_\\parallel /c^2)\\\\\n& J_\\bot' = J_\\bot\n\\end{align}", "c64a2706d4ddc435934e029030d0053f": "\\displaystyle y^{\\prime\\prime}=2y^3+ty+b-1/2", "c64a3a38fed25546cda2d9724ae54a49": " k_\\text{u} ", "c64a5e6fa0b874e06839f0360dd8b0f2": "\\frac{1}{2\\mu}\\cdot\\frac{2-\\rho}{1-\\rho}.", "c64a6619c24736e4eebb3595f9626f2d": "(d-1)", "c64aa9b3473402c3eeca5e8c246f5ca9": "N[\\frac{I(T_M)}{I(T_0)} - 1]", "c64b1834e2af0598b23a38175a383e72": "\nf_X(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} }.\n", "c64b2c2bc29fa9ce4494475d1b526c37": "\\textrm{Call}(K,\\sigma)", "c64b504371dc9166cb12a1f48b29c268": "\\operatorname{gr}R = \\operatorname{gr}\\widehat{R}", "c64b5ff2ed952b8511a6f11af02cb402": "\\boldsymbol{\\mathsf{A}} =\\frac{\\mathrm{d}\\boldsymbol{\\mathsf{U}} }{\\mathrm{d} \\tau} = \\gamma \\left( c\\frac{\\mathrm{d}\\gamma}{\\mathrm{d}t}, \\frac{\\mathrm{d}\\gamma}{\\mathrm{d}t} \\mathbf{u} + \\gamma \\mathbf{a} \\right)", "c64b85d4fb651fe2c690614f0080eef8": " c = \\frac{1}{P(A) \\cdot P(B|A) + P(\\neg A) \\cdot P(B|\\neg A) } .", "c64b8dcb4dc67ba8fcd1b3921fa6d1e4": " \\chi (T) = \\sum _{v \\in V} d(v)q _{v} - \\sum _{e \\in E} q _{e}", "c64b93746de1d638ca52fde6d80c9974": "\\{A_i \\mid i \\in \\underline{m} \\backslash A\\}", "c64b993cb648b7aec98ee45868f6754f": "T^3", "c64bda49919c0f55cba1322590077a3c": " \\Gamma = \\frac{q_e^2}{4\\pi\\epsilon_0 kT_e}\\sqrt[3]{\\frac{4\\pi n_e}{3}} ", "c64c88ccf28bd37345e5d64c3a5b1025": "K(x,t;x',t') = \\int \\exp \\left[\\frac{i}{\\hbar} \\int_t^{t'} L(\\dot{q},q,t) dt\\right] D[q(t)]", "c64cc759fdf8b1650b34406aa25f5455": "\\theta^8", "c64d411f995c3a4bdad737aa41209cd2": "h\\left(6\\right)=6\\mod 11=6", "c64d7e43353ce8694b7ada3fbf003ee8": "\\frac{l}{R}", "c64e46ac8e495f116b6c0636e34f7744": " u(x,t) = e^{-(x-ct)^2 +ik_0(x-ct)}.", "c64e75942a133c3f996a9bb7c444dd21": "\\tilde * = e^{(n-2p)\\varphi}*", "c64e7802ebd3fd60298c6728ac020b3f": "4) \\ \\mbox{New adopters}=\\mbox{Innovators}+\\mbox{Imitators} ", "c64e9f63779d3b1a950d786e50283abc": "R=\\left\\{\\mbox{roots of }P(x)\\in\\mathbb{Q}\\right\\}\\,\\!", "c64ec3748b1d5c76efe5120befb442eb": "\n{\\mathbf A}^2 = {\\mathbf B}^3 = ({\\mathbf A}{\\mathbf B})^7 = \n({\\mathbf A}{\\mathbf B}{\\mathbf A}{\\mathbf B}{\\mathbf B})^{12} = 1.\n", "c64ef405da72a6879e4db7d57ada8134": "d\\Omega_\\Sigma", "c64efff8aef494bd64e28b8a875333f9": "L=\\{(q,p)\\in T^*N|H(q,p)=E\\}", "c64f436a7320ef5d3672af5573f72ea8": "\\phi=\\sin^{-1}(\\tanh(m (\\lambda-\\lambda_0))),\\,", "c64f6b62ca96faed8ee28d4e5fd5d261": "(a,b)=(\\sigma_\\mathrm{avg}, 0)", "c64fa95e0fd5d78c6d96453ed898f7ff": "\\sum_{n\\in S}1/n = 1.", "c64fc8e2a48bf6ea6780ced7593ed91d": " \\psi(x) \\rightarrow \\psi_0 + \\phi(x) ", "c64fed2e2e3020cee6f555e67c3eddb9": "...0011010101q00110101010...", "c650723d0df7a6c004f470784326c892": "F_z = \\frac12 \\times \\rho \\times S \\times C_z \\times V^2", "c650d64ad49319670be36c638deae91f": " [T(\\phi, \\mathbf{d})] = \\begin{bmatrix} A(\\phi) & \\mathbf{d} \\\\ 0, 0 & 1\\end{bmatrix} \n= \\begin{bmatrix} \\cos\\phi & -\\sin\\phi & d_x \\\\ \\sin\\phi & \\cos\\phi & d_y \\\\ 0 & 0 & 1\\end{bmatrix}.", "c651412b94fc60c95e365c7f094718bd": "U_x = ((A_x^2 + A_y^2)(B_y - C_y) + (B_x^2 + B_y^2)(C_y - A_y) + (C_x^2 + C_y^2)(A_y - B_y)) / D,", "c6528e78773cb78bfc7042f86c5086e3": " \\alpha= 1 ", "c652947da16ab583850fbd2fc52a00e9": "\\textstyle w = (0,1,u,0,0,0)", "c652cf8bed316d16800160a7573a285f": "T=T_{eff}", "c652f3bcd3beefabac50c2fb5cb3f2d8": " \\nabla_l \\left(R^{lm} - {1 \\over 2} g^{lm} R\\right) = 0.\\,\\!", "c653796bb2939b5e3c3b5b1b7d728163": "z_B = {n_c - n_d \\over \\sqrt{ v } }", "c653c81ca7c343504859ed44394039dd": " i ^ {th}", "c653cea76001da3e85ada1841ebae6ba": "\\chi_\\mathrm{red}^2 > 1", "c653d9c721bc16ec9d2d59b90d9a4fe0": "e-\\operatorname{cr}(G) \\leq 3n", "c654302916c106744f63fcf5e500d606": "N=-{{1}\\over{2\\pi i}} \\oint_{G(\\Gamma_s))} {1 \\over {v+1/k}}\\, dv", "c654311f425ef909f5dfacecb3a9a9aa": "D = \\rho +\\nabla^{bas}+R^{bas}.", "c654731a358562e111f7f82288e1d232": "\\varphi_{\\beta+1}(\\gamma+1)", "c654bdadcbe8e4890fff0de4760cead6": "P_{m,n}=\\binom{n}{m}2^{-n},", "c654cf4c32840406337e3796f69c0253": "eV=h\\nu-A\\,", "c654dd73be5aef335e7dea2922774d4b": "\\boldsymbol{\\tau}=\\mu_0 \\mathbf{m} \\times \\mathbf{H}", "c6552cf8960f56f1fb15c1e477aa69cc": "\\gamma V^{2/3} = k\\left(T_C - T - 6\\right)", "c6552d38edf6d1e2fa04e8a2e5bfcfe5": "z_i \\in \\mathbb{C}", "c65554a1cf57d5e167dd1d306b95bc33": " A = \\frac{1}{2}(A - A^{\\mathsf{T}}) + \\frac{1}{2}(A + A^{\\mathsf{T}}) . ", "c6559fea8daa85c8b7b7911976e6f34e": " \\sum N_i=n", "c655c07b0d28e4bcecb8418e89a7b3f4": "||\\mathbb{P}_n - P||_{\\mathcal{F}} = \\sup_{f \\in \\mathcal{F}} \\dfrac{1}{n} \\left|\\sum_{i = 1}^n [f(X_i) - \\mathbb{E} f(Y_i)] \\right| \\leq \\mathbb{E}_{Y} \\sup_{f \\in \\mathcal{F}} \\dfrac{1}{n} \\left|\\sum_{i = 1}^n [f(X_i) - f(Y_i)] \\right|", "c655e8a2ec31e04d4d9799a5cbd24eb9": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log [\\frac{\\varepsilon}{3.7065D} - \\frac{5.0452}{Re} \\log(\\frac{1}{2.8257}(\\frac{\\varepsilon}{D})^{1.1098} + \\frac{5.8506}{Re^{0.8981}})]\n", "c6561e3ee83961fea9b423138f346f62": "A=2(5+\\sqrt{3})a^2\\approx13.4641...a^2", "c6567d00097727bda0e535a922aad50d": "0=x+f'\\left(\\frac{dy}{dx}\\right),", "c656b2789ff087e7839a6037be283b2d": "\\langle N\\rangle_{BE} = {1\\over e^{E/kT}-1}\\,.", "c657a70c1234fa440f880ce8c037e51e": "J_{m-m^\\prime}(\\ell\\beta)", "c657d8e41e463c8daca96028869d93d3": "\\pi =E\\left[p\\min (q,D)\\right]-cq", "c657f9a8db2d701d1468d2b3ff633be7": "\\frac{355}{113} = 3.14159\\ 29^+", "c65854268aa150f7165c4b49182326a1": "\\textstyle \\frac{235.215}{x}", "c6585ce65dd984141a10cb96e03d37bb": "\n\\mathbf{C}=\\mathbf{A}\\mathbf{B}\n", "c6586176e9b317ac32c72935e2a8455c": "\\hat{J_{z}}", "c6587dafe1bee4c6f58763329aa91763": "C_N= \\frac{1}{2} (C_{Ni}^{j+1} + C_{Ni}^{j})", "c658864d8c69eb27a97a7f00c38f9bf8": " |f_k' \\rangle = |f_k \\rangle ", "c6588a6fb805bb4f555ea910eab7199a": "\\tilde{\\mathbf{a}}_{i}=\\left(\\frac{1}{\\sqrt{1-Z^2}}\\right)_{ij}\\mathbf{a}_{j}+\n\\left(\\frac{1}{\\sqrt{1-Z^2}}Z\\right)_{ij}\\mathbf{a}^{\\dagger}_{j}", "c658cf548cf48548e36d965978b8d649": "\n\\exp(\\beta_x) = \\frac{P(Y=1|X=1, Z_1, \\ldots, Z_p)/P(Y=0|X=1, Z_1, \\ldots, Z_p)}{P(Y=1|X=0, Z_1, \\ldots, Z_p)/P(Y=0|X=0, Z_1, \\ldots, Z_p)},\n", "c659bff1a01cda1182da86159f78708e": " S\\mapsto\\alpha_P(S) ", "c659ca0b8498c24a031438fdf32fde8b": "P(X) - \\{ \\varnothing \\}.", "c65a6adb8c940e962cdf13d74753855d": "F(X,Y,", "c65a8452a9f986757ab7df4c1d380280": "\\delta_n(t)", "c65aa702d9cfd7b05a9abcc2f80ec78b": "sX(s)-x(0)=AX(s)+BU(s)+EW(s) ", "c65abc7be44fdd6bd3552ed28804c1fd": "(f * h) \\star g = h(-)*(f \\star g)", "c65ae6a040442fff7b3ec830e231e4bc": " \\operatorname{init} ", "c65b05b83c5f92e1c8cf9b2fd5ee0df2": "d_i \\ge 2", "c65b2c75e34d3cedf64158d9b0773e8c": "\\mathbf{j} = (0,1,0),", "c65b5702d90ce67e42ce709d0886cf9f": "\\bigcap \\mathbf{M} = \\{x : \\forall A \\in \\mathbf{M}, x \\in A\\}.", "c65b75092988e9f12b5f68d3ae39f823": " \\beta \\sim H ", "c65bb3c1891171dddce11ff4c990eb23": "F = \\kappa X", "c65be1fd864122a3e4a0ae205556a80f": "\\begin{align}\n\\left(a^2 + b^2\\right)\\left(c^2 + d^2\\right) & {}= \\left(ac-bd\\right)^2 + \\left(ad+bc\\right)^2 & & & (1) \\\\\n & {}= \\left(ac+bd\\right)^2 + \\left(ad-bc\\right)^2. & & & (2)\n\\end{align}", "c65c1dd354ff78b5ab5e899e95739cd1": "\\frac{\\mbox{Accounts Receivable}}{\\mbox{Total Annual Sales ÷ 365 Days}}", "c65c22e3321db1193d8274e8b7fdab67": "\\scriptstyle D_F(1\\rightarrow 5)= 4(1)-1+0-3=0", "c65c4e95407afb100ae63dceae9d9338": "f_1 \\otimes f_2 : (X_1, \\alpha_1) \\otimes (X_2, \\alpha_2) \\vdash (Y_1, \\beta_1) \\otimes (Y_2, \\beta_2), \\qquad f_1 \\otimes f_2 := \\pi^{*}_1 \\gamma_1 \\cdot \\pi^{*}_2 \\gamma_2", "c65ccebd5b163e2529a8d95125f43823": "(\\mathcal{P}(E),\\mathcal{P}(F),V)", "c65ce416b5710888e27ee0809aea38e1": "F'(x_1) = f(x_1).\\ ", "c65cf47b01b69c74e3a0ef3e5d6141e9": "N = 0, F = 0.", "c65dcac610cc3ceb150b98f61f72f48c": "e(\\mathbf{x},t)", "c65dd2ca840999a41500a919e0dd86b3": "\\phi(x_1 x_2 \\cdots x_n) =0 ", "c65e1d4aa977ed395a07a41bb866eb22": "\\begin{matrix} {52 \\choose 5} = 2,598,960 \\end{matrix}", "c65e4cc47b235138e6a8e911b7b481f4": " \\pi/\\sqrt{8}", "c65e4db810a967906517099653621a26": "1\\le j\\le n", "c65e594ad994adebd476ac372ff23da0": "H\\ge T^{1/2+\\varepsilon}", "c65e6f00d62889201c2b21de63aa198a": "WCI", "c65e75138c6f2cb4d6af96705d228832": "\\dot H_k = h_k\\dot m_k = H_m\\dot n_k", "c65e7ceecfb63fc42347ce6ccac3344e": " \\eta_{Xe} - \\eta_{Qe} - 1 > 0 \n\\Rightarrow \\eta_{Xe} - \\eta_{Qe} > 1 ", "c65ed2279ffa9ec9c8a0546d0bf632c4": "F^*: H^1(E, \\mathcal{O}_E) \\to H^1(E,\\mathcal{O}_E)", "c65ef4092b43cae1928cc8e80e2485de": "P^{\\alpha}_{I}(x^{i},u^{\\alpha})\\,", "c65f28c5ae7d1479591e03af91ea0f42": "\\textbf{R}_k \\equiv \\textbf{R}^{a}_k", "c65f2d24c15a22dc47afc12855fa5429": "\n\\begin{align}\n\\frac{d^3F(P)}{dP^3} & =\\frac{d^2F'(P)}{dP^2}=\\frac{dF''(P)}{dP}=\\frac{F''(P_1)-F''(P_0)}{dP}, \\\\[10pt]\n& =\\frac{d^2G(P)}{dP^2}\\ =\\frac{dG'(P)}{dP}\\ =\\frac{G'(P_1)-G'(P_0)}{dP}, \\\\[10pt]\n& {\\color{white}.}\\qquad\\qquad\\ \\ =\\frac{dH(P)}{dP}\\ =\\frac{H(P_1)-H(P_0)}{dP}, \\\\[10pt]\n& =\\frac{G(P_2)-2G(P_1)+G(P_0)}{dP^2}, \\\\[10pt]\n& =\\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{dP^3}, \\\\[10pt]\n& =F'''(P)=G''(P)=H'(P)=I(P);\n\\end{align}\n", "c65f47417ef36e3fdca61adbc2047984": "\\breve{\\boldsymbol\\theta}_i", "c65f7f477a1350adedb91b5a9880a6a9": "T_{hot}", "c65fac1f1d0bc9453c27232d631378c0": " P_{nj}", "c65fbedeffb02fc8a7dedc3e29169b9a": " Tr(X)Tr(Y)Tr(XY) + Tr(XYX^{-1}Y^{-1})+2= Tr(X)^2+Tr(Y)^2+Tr(XY)^2 ", "c6601bd96f33668ccaf48aa65b3388dd": "Ax=b.", "c66034957d481a832bd8f2610b328e7e": "\\dot x(t) = x'(t) = -32t + 16, \\,\\!", "c66047cda04e3a02c2a05627e985af00": " s_\\Lambda= \\sum_{j=1}^m x_j ", "c660c16a3107afb4bb7cbb7dbf0ed962": "P \\equiv_{b} P' \\mbox{and } P' \\rightarrow_{b} Q' \\mbox{and } Q' \\equiv_{b}Q \\mbox{implies } P \\rightarrow_{b} Q", "c6610be40412b47969f8af0103bca1a8": " c(u) \\alpha = \\left\\{\\begin{matrix}\n\\alpha&\\hbox{if } \\alpha\\in \\Lambda^{even} W\\\\\n-\\alpha&\\hbox{if } \\alpha\\in \\Lambda^{odd} W\n\\end{matrix}\\right.", "c6615abd6acdc3509804ae00d67ef6d2": "\\frac {\\part g_{\\mu \\nu}}{\\part t}=0", "c66199efd6837cb8e64596813621e314": "\\rho(x,y,z,t)", "c661d266d9ec25a7e888d3fb32dc3e06": "\n\\nabla\\times\\vec{B}=\\frac{4\\pi\\vec{j}}{c}+\\frac{1}{c}\\frac{\\partial\\vec{E}}{\\partial t},\\quad \\nabla\\times\\vec{E}=-\\frac{1}{c}\\frac{\\partial\\vec{B}}{\\partial t}\n", "c661dc1f9077355ffc33234e33157c04": " S_x = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix},\n S_z = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}", "c661f73e7e848f68b9771efd1b4113af": "A_5 \\times Z_{11}.", "c66214e40b2cca2f16c285c405b35f36": "q(x_1,\\ldots x_n) = x_1^2+ x_2^2+\\cdots+x_p^2-(x_{p+1}^2+\\cdots +x_n^2).", "c66215fb07bc90245fd67d4ac5a7ddfb": " \\left(\\frac{\\partial U}{\\partial V}\\right)_{T}", "c6622fb98208cae2a878e0f13d053ec2": "| b(x) - b(y) | + | \\sigma (x) - \\sigma (y) | \\leq C | x - y |", "c66234702b1823f30e52c17af6607a21": "w(e)", "c66266adfc94bb102f7916bcd6373f62": " M''= \\frac{2 \\rho \\alpha V^3}{\\omega}", "c6629c8fd464942d2e6792bb8db78a89": "\\scriptstyle {\\Gamma^{\\alpha}_{\\,\\beta\\mu}}", "c662b386d5094360ef944d186ff55901": "E^{b9}_6", "c662ba855a1706741f087548b8182369": "x = 5.13562... ", "c6632641f3fb9ce34ae6d14adce2f1fa": "\\Gamma_{\\beta} [n] = \\Gamma_{\\beta [n]} \\,.", "c6636e0a91af2ad6c58d16abeb943388": "\\begin{align}\nz\\frac{dM}{dz} = z\\frac{a}{b}M(a+,b+)\n&=a(M(a+)-M)\\\\\n&=(b-1)(M(b-)-M)\\\\\n&=(b-a)M(a-)+(a-b+z)M\\\\\n&=z(a-b)M(b+)/b +zM\\\\\n\\end{align}", "c663df987550323ffe178873c334b11a": "r(B_p,\\ B_q)=2q+3", "c663e76c721dce9b3d80c5944c044e10": "G_{xy}=(C_{11}-C_{12})/2=C_{66}", "c6640b8e2f497e9f25826228fb4ad1a7": "\\left(\\sqrt{1/28},\\ -\\sqrt{12/7},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "c6644748dcc8d81019482d0e152882ab": "\\int f(t) x^t \\; dt", "c664669f08d07479526bf159e98b1b1c": "h \\in (0,\\delta)", "c664d34cb95cfa96601a743688900ac1": "\\begin{array}{ll}\n\\dot x = Ax + b\\xi(t),& \\sigma = c^*x,\n\\end{array}\n", "c664e743eb3c0c95c2e4f6c20c04f644": " \\frac{\\partial}{\\partial u} g(z, u) \\Bigg|_{u=1} = \n\\frac{1}{1-z} \\sum_{k\\ge 1} k^2 \\frac{z^k}{k} =\n\\frac{1}{1-z} \\frac{z}{(1-z)^2} = \\frac{z}{(1-z)^3}.", "c6650181404df41ff4f6796022d8a244": "f = \\sum_{s \\in G} f(s) \\delta_s.", "c665050a7ad30ab97c60942622c9176e": "\nd\\nu^2_t = \\kappa^2(\\theta^2 - \\nu^2_t)\\,dt + \\xi^2 \\sqrt{\\nu^2_t}\\,dW^{\\nu^2}_t \\,\n", "c665476d35511f4aa3f1cc0b8c47887a": "3 Pmf - \\tfrac{4}{3}Pmf = 1", "c6667c64d42eb63f06e3337dd7ec1ba9": "g^n=h^{-1}\\circ f^n\\circ h,", "c666846fd18d243333e7f1583847695c": "(f_{\\text{high}} = c/2 \\pi r_1)", "c6670bdab8647b603831f4940e09e0c9": "\\gamma \\approx 0.9891\\dots", "c6670e3e50e1ed747a4bb99af69fc23f": "u^{(0)}_{k,j} \\equiv \\partial u^{(0)}_k / \\partial x_j", "c6670f9062aaa86f72d969660cacc11c": "\\scriptstyle| \\phi \\rangle_B", "c6673a9e812c1e802d3cb8d177fa5f13": "C^*={C \\over nR} = {C \\over {Nk}}", "c66770c6b498e966397591f8469dda1b": "\\forall x,y \\in A", "c66772a88ecb0820658ad328e6854909": "R_e^2=\\frac{N}{N-1}\\left(\\overline{R}^2-\\frac{1}{N}\\right)", "c6677b14e24e32db0ecc5a774fde7e66": " dI_z = - \\sigma N\\,I_z\\,dz .", "c668dcfd5370ef128cbdd3dd89052023": "M=[m_{i,j}] \\qquad (i=1,\\ldots,k \\text{ and } j=1,\\ldots,k) ", "c6691e4e02a25ac7f9748202f2de793e": "T_G(1,2)", "c669767bd1adb40964076d8d1869cbc9": "\\mathbf{K_1}\\plusmn\\mathbf{K_2}", "c669e0e8c55211ecfd67306cf73fbc8f": "k[x_1,\\ldots,x_n].", "c66a2ad725d1022596be76d62787a317": "\nW_n = \\frac{1}{2} \\Beta \\left( \\frac{n+1}{2},\\frac{1}{2} \\right)\n", "c66a4ed163cb3552ab24f1bd7ef13b20": "\\int \\frac{1}{x^{n} S} dx = -\\frac{1}{b (n - 1)} \\left( \\frac{S}{x^{n - 1}} + \\left( n - \\frac{3}{2}\\right) a \\int \\frac{dx}{x^{n - 1} S}\\right)", "c66a66ff336f178a238d8c8bab51e9de": "T \\,", "c66a7d5977ef06f1ab66e70b557713dc": " x^{0}=1", "c66a91bc21a7231b88c76b8c71b6ec18": "x=ka", "c66b640073807d2bc09923812a8174d2": "MA = \\frac{F_B}{F_A}", "c66b7ad8ae6c3b284d9c4fcab6c9dfc1": " \\mathbf{u} = \\mathbf{u}(x_1, x_2) ", "c66b9b1ada70c9c4f24bddda6a328bf5": " \\omega_ \\mathrm c = \\frac{1}{\\sqrt {LC}} ", "c66b9ef9969072438c7d32e57a7073f3": "\\mathbf{T}^{(\\mathbf{n})} \\, dA - \\mathbf{T}^{(\\mathbf{e}_1)} \\, dA_1 - \\mathbf{T}^{(\\mathbf{e}_2)} \\, dA_2 - \\mathbf{T}^{(\\mathbf{e}_3)} \\, dA_3 = \\rho \\left( \\frac{h}{3}dA \\right) \\mathbf{a},\\,\\!", "c66babbaa9657bd4cb6ebf2e91e5e2ac": "\\frac{\\delta R}{\\delta g^{\\mu\\nu}} = R_{\\mu\\nu}.", "c66baf8d74f4d2df2f541152d1319fce": "p = r(1-q^{-1})", "c66bd5470972a32d485f51982acf2b20": "\\neg \\phi \\,", "c66bfa5c1341f7a949f84fc8ecb1bbb3": "g_{ij} = \\sum_{kl}\\delta_{kl}{\\partial x^k \\over \\partial q^i} {\\partial x^l \\over \\partial q^j} = \\sum_k\\frac{\\partial x^k}{\\partial q^i}\\frac{\\partial x^k}{\\partial q^j}.", "c66c35eeb818835984f606b127acc116": "\n \\varphi\\Delta = \\tfrac{h}{\\surd\\pi}\\, e^{-hh\\Delta\\Delta} .\n ", "c66c4bb3232536b90107f6387b57fea5": "d = \\gcd(d_1,v_1 + v_2 + h)", "c66c6e1d97a7a3c4faf46314ecdb484d": "PS_k(X) = G_{k}(X)-G_{k+1}(X)", "c66ce77ffa967b227bb59f93603adf26": "\\psi(n) = \\frac{J_2(n)}{J_1(n)}", "c66d112b8c3336fdd6a9c3a93ea4c5d7": "\\lambda(t_0)", "c66d3e4775ed6974ffe729db2d4b9e75": " \\frac{1}{(\\beta + 1)r_O } ", "c66dada8057e8ae69a6b1cbe0c94a5fa": "D m^a=(\\bar{\\varepsilon}-\\varepsilon)m^a+\\pi l^a-\\bar{\\kappa} n^a\\,,", "c66dfa320a4eb7bab960176ceeb15099": "(x_n) \\sub X", "c66e33d334180c5185f9432ca7c2c796": "\\mathbf{j}\\cdot\\mathbf{\\hat{n}}= j\\cos\\theta ", "c66e411f25e2f62d3306bc8ad05931f6": "\\delta(t)\\,", "c66ea37beb2d8f7941a672ec7cd386c1": "\\begin{bmatrix}\n1 & \\lambda_1 & \\lambda_1^2 & \\cdots & \\lambda_1^{n-1} \\\\\n1 & \\lambda_2 & \\lambda_2^2 & \\cdots & \\lambda_2^{n-1} \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n1 & \\lambda_n & \\lambda_n^2 & \\cdots & \\lambda_n^{n-1}\n\\end{bmatrix}\\begin{bmatrix}\nb_{m,0} \\\\ b_{m,1} \\\\ \\vdots \\\\ b_{m,n-1}\n\\end{bmatrix}=\\begin{bmatrix}\n\\lambda_1^{n+m} \\\\ \\lambda_2^{n+m} \\\\ \\vdots \\\\ \\lambda_n^{n+m}\n\\end{bmatrix}", "c66eda2b2b913e90d9fbe9d80dc8c0f5": "v \\approx \\left(q_1 - q_0 \\right)/\\left(t_1 - t_0 \\right)", "c66eea5e43d903e1ccaa930cdf350ac8": "p\\in\\operatorname{cl}(A)", "c66f0f7b28833a743617cfeac5549c5e": "\\langle\\langle e,t\\rangle,t\\rangle", "c66f293f7fa2cb99695396d9d91e0bbe": "\\alpha = 2: \\quad \\operatorname{E} \\left [- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial a^2} \\right ]= {\\mathcal{I}}_{a, a}", "c66f48424b8176f446e6d3eb22e2adab": "\\Psi_L\\left(\\eta\\right)=\\Psi_G\\left(\\eta\\right).\\,", "c66f67693f36160e84fbd94b6619b0cf": "\\varepsilon\\cdot (M-m),", "c66f92ef7f31f07c40efe29283677120": "|f_k\\rangle", "c66fd7f55dba836188ce08c01b205353": "\n\\begin{align}\nTv_1 & = v_2\\\\\nTv_2 & = v_3\\\\\nTv_3 & = v_4\\\\\n\\vdots & \\\\\nTv_{n-1} & = v_n\\\\\nTv_n &= -c_0v_1 -c_1v_2 - \\cdots c_{n-1}v_n\\\\\n\\end{align}\n", "c66fe0b4e2bbeb21f58f085807a5f712": "\\left\\{\\frac{1+x_2}{x_1},\\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3},\n\\frac{1+x_2}{x_3} \\right\\},", "c670072c992b5e3757e0345b01d4d790": " \\psi(\\bold{r}+N_i \\bold{a}_i)=\\psi(\\bold{r}), \\, ", "c670174046b0334dc70ea26544514b8a": "F_{w,n} = gV( \\rho_w - \\rho_a) ", "c67023e0f5c2768ba0f7634e615e94ef": "f_i=f(x_i)", "c670257c85a6c2b487a19daf5cdbd167": " \\mathbf{ \\hat T} (a)|\\psi(0)> ", "c6707e87ef6a8088bb4c58004443e877": "\\{Q_i,Q^\\dagger_j\\} = \\mathcal{H}\\delta_{ij}.", "c670835d05e1d3836ad9bbcbb21f68cb": "H=\\frac{P_{max}} {A_{r}}.", "c671107a79e0d375ca8cbb927d48ee55": "A \\mapsto f(A)", "c671348fb62cba142124b7ade469d451": "r_1,\\ldots,r_i", "c671b3a3affce3269c666c74a8be8ebc": "U(a,b)\\,", "c672148ea42e1bc02b1ff910bd324f9b": "grlex", "c6722145578ca6e357db74144d35c96b": "Y={1\\over N}\\mathbf{1}", "c6722326693b776f4d0524c4be3e56f7": "(m,n) \\sim (r,s) \\leftrightarrow m+s = n+r", "c67231eeab7a9271ff69f2d943a0cb54": "\\mathbf{X}^{-1}", "c67242b03e69ebd6d54aa6694ad2b180": "\\vec{e}_0=\\partial_t, \\; \\vec{e}_1=\\partial_z, \\; \\vec{e}_2=\\partial_r, \\; \\vec{e}_3 = \\frac{ \\sqrt{2} \\omega }{ \\sinh( \\sqrt{2} \\omega r ) } \\, \\partial_\\phi - \\frac{\\sqrt{2}\\sinh(\\sqrt{2} \\omega r)}{1+\\cosh(\\sqrt{2} \\omega r)} \\, \\partial_t", "c6727d828a8c1f643691d84b3042e3db": "2^{n+c}\\times2^{n+c}", "c672a2a00a56fc8108db81dd73e2421a": "|f+g|^p \\le \\frac{1}{2}|2f|^p + \\frac{1}{2}|2g|^p=2^{p-1}|f|^p + 2^{p-1}|g|^p.", "c672d7bd113e96726ba79dfe457e834a": "\\mathcal{N}\\equiv 0", "c67344baeff82e29204e901033ad3f03": "Y_{8}^{2}(\\theta,\\varphi)={3\\over 128}\\sqrt{595\\over \\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(143\\cos^{6}\\theta-143\\cos^{4}\\theta+33\\cos^{2}\\theta-1)", "c6735326f059be5490b64ab8168f67ac": "K=\\frac{\\frac{k_f}{k_f+k_b}[A]_0 }{\\frac{k_b}{k_f+k_b}[A]_0}=\\frac{k_{f}}{k_{b}}", "c6735eccb87b48482ef39e93bd2431eb": "\\neg P(t_1,\\ldots,t_n)", "c67366e568463c35d807a639fcdc5bb4": "N(t) = N_0 e^{-\\lambda t}. \\,", "c67383b45fca8764acb0d4c7d92e1b0c": " \\zeta(x)=\\sum_{k=1}^{\\infty}\\frac{1}{k^x} ", "c673a425f1c40bda9238373cf91e8e2b": " Q_o = \\frac{G} {R_s} \\cdot ", "c673a7ee0e59d6e5f86035bd93f5bdc5": "-\\sum_{k=1}^n u_k \\phi (v_k,v_j) = \\sum_{k=1}^n f_k \\int v_k v_j dx", "c6741e761515ef9a6cc91c1202f518e0": " \\sqrt{\\sigma_S^D} ", "c6748b375d033fda7107a894bf540400": "f_i(a_1, \\dots, a_i) = \\begin{cases}\nf_i(a_1, \\dots, a_m) = 1 & Q_{i+1}x_{i+1}\\dots Q_mx_m[\\varphi(a_1, \\dots, a_i)] \\text{ is true}\\\\\n0 & \\text{otherwise}\n\\end{cases} ", "c674a6a661b9d6353117ba6a9c45211e": "T \\gg T_c\\,", "c67524c66711be1d102ddda2c49a948b": "y(x) = \\left|4\\left(\\left(\\frac{x}{2\\pi} - 0.25\\right)\\,\\bmod\\,1\\right) - 2\\right|-1", "c675902be2c07d2fcf70cef2b471f299": "K'(x)", "c675a6b423ca3e73acefe2283c7b800c": "V(x(t), t)", "c675c432a7897879b316f10af1fff385": "\\log_b (a+c) = \\log_b a + \\log_b (1+b^{\\log_b c - \\log_b a})", "c675e459019d3c34153d0b20d3da0548": "x\\in f^{-1}\\left(Z\\right)", "c675edd5cc49d85e653e87249227e68d": "\nds^{2} = -dt^{2} + e^{2\\sqrt{\\Lambda}t} dz^{2} + \\frac{1}{\\Lambda}(d\\theta^{2} + \\sin^{2}\\theta d\\phi^{2})\n", "c676308582b64af08621bcd621eb0281": "\\eta_C = \\pm 1", "c6767456a005676421197939e43e3f48": "|f(n)| \\leq g(n)\\cdot k", "c676f6478a944d0475484a1359899ade": "\\frac{p_0}{\\rho.g} = \\frac{V^2}{2.g} + \\frac{p}{\\rho.g}", "c67701e494b4dc785e9994b29bab3cb1": " \\forall t_{0} \\in \\mathbb{R}, ~ \\xi \\in \\mathcal{H}: \\quad \\lim_{t \\rightarrow t_{0}} U_{t} \\xi = U_{t_{0}} \\xi ", "c6771a2a031695859f04702fdc348dcd": " E = p, V = p, Y = (\\lambda f.\\lambda x.f\\ (x\\ x)), X = \\{\\} ", "c6773a5aa830b5d0cd764b954890be02": "f \\star g = f \\, \\exp{\\left( \\tfrac{i \\hbar}{2} (\\stackrel{\\leftarrow }{\\partial }_x\n\\stackrel{\\rightarrow }{\\partial }_{p}-\\stackrel{\\leftarrow }{\\partial }_{p}\\stackrel{\\rightarrow }{\\partial }_{x}) \\right)} \\, g", "c6775959ef171448011ace411478e408": "\\frac{}{}_S", "c677901d8a198d5d03e9df3769c73f8c": "\\tau_\\text{felt} = \\tau_\\text{applied} + n_\\text{dislocation} \\tau_\\text{dislocation} \\, ", "c677bc90428cd22f3de9d163f3e1a346": "\n\\ \\beta \\, = \\, \\tan \\theta\n", "c677d23dde13f01a28cd8137af111bd2": " (f \\cdot(\\ln g)^n)' = f^\\prime (\\ln {g})^n + n f \\frac{g^\\prime}{g} (\\ln{g})^{n-1} ", "c677fdcb1710c7f788cf87d683cb6a7e": " |\\phi \\rangle \\, \\langle \\psi | =\n\\begin{pmatrix} \\phi_1 \\\\ \\phi_2 \\\\ \\vdots \\\\ \\phi_N \\end{pmatrix}\n\\begin{pmatrix} \\psi_1^* & \\psi_2^* & \\cdots & \\psi_N^* \\end{pmatrix}\n= \\begin{pmatrix}\n\\phi_1 \\psi_1^* & \\phi_1 \\psi_2^* & \\cdots & \\phi_1 \\psi_N^* \\\\\n\\phi_2 \\psi_1^* & \\phi_2 \\psi_2^* & \\cdots & \\phi_2 \\psi_N^* \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\phi_N \\psi_1^* & \\phi_N \\psi_2^* & \\cdots & \\phi_N \\psi_N^* \\end{pmatrix}\n", "c6781195f0e9bf1a4f4a06e44972c2b6": "M_B = 2 \\frac{EI}{L} \\theta_A + 4 \\frac{EI}{L} \\theta_B = 2 \\frac{EI}{L} \\theta_A", "c67928c3d59f9a70c57a4742219d453b": "\\displaystyle{X_i= E_{\\psi_i} + E_{-\\psi_i}}", "c6793331c09c9c161ccb42a4e633f31c": "\\operatorname{d} V", "c67a3a2a7c8b88095be00e40bea60c33": " \\mathrm{Re} = {{\\rho {\\mathbf v_s} D} \\over {\\mu \\epsilon}}.", "c67a55526458cd29bb67bc5cf86e5294": "\nI(\\theta)\n=\n\\left(\\frac{\\partial\\boldsymbol{\\mu}(\\theta)}{\\partial\\theta}\\right)^T{\\boldsymbol C}^{-1}\\left(\\frac{\\partial\\boldsymbol{\\mu}(\\theta)}{\\partial\\theta}\\right)\n= \\sum^N_{i=1}\\frac{1}{\\sigma^2} = \\frac{N}{\\sigma^2},\n", "c67a58b6b54dc87db8ceb9c0377e3ee8": "i \\in [0, s)", "c67a9431e1a397ad80e74cb30f393c36": "r_s= \\left(\\frac{3M}{4\\pi \\rho N_A}\\right)^\\frac{1}{3}\\,,", "c67acc3db3548414c2d95f5c42dfea01": "\\frac{\\lambda_\\mathrm{now}}{\\lambda_\\mathrm{then}}=\\frac{a_\\mathrm{now}}{a_\\mathrm{then}}\\,.", "c67aea3dee4461b39a5258e075b94a47": "\\scriptstyle \\mathcal{N}(\\mu,\\, \\sigma^2/n)", "c67b481a81143a8d60ec9c0d216ab2d5": "\n(I|C^{-1}) = \n \\left[\\begin{array}{cc|cc}\n 1 & 0 & 0 & -\\frac{1}{5} \\\\\n 0 & 1 & \\frac{1}{3} & \\frac{1}{15}\n \\end{array}\\right]\n", "c67b750453740dbcf1f8c053cce3a5c1": "\\frac{PR}{PQ} = \\cos \\alpha\\,", "c67b9109b6d9277875fc654c4041caa9": " \\sigma^2_t = d ^ 2 \\rho + (1 - \\rho) \\sigma ^ 2_ {t-1} ", "c67bae384c19ebaf9ce07de893add5c8": "[r]", "c67baf1270331249726684a3cdd9aac6": "m = \\sum_{i=1}^n\\phi(i)", "c67c2b239fd13fe5b077a7c744ec9457": "P_\\mu(n)= \\Pr(X=n)= \\frac{e^{-\\mu n}(\\mu n)^{n-1}}{n!}", "c67c7a7f60e12f31c91cc9fa938a597f": "p_i = \\sum_j^n p_{ij} \\quad ", "c67c9e0502622b84da72fa9f7edf92ef": "\\mathcal M = \\int_S \\mathbf r \\times \\mathbf T\\,dS + \\int_V \\mathbf r \\times \\rho\\mathbf b\\,dV", "c67ca5e424859c9a702ab1d869620f9c": "\\rho(\\boldsymbol\\beta,\\sigma^{2}|\\mathbf{y},\\mathbf{X}) \\propto (\\sigma^{2})^{-k/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\boldsymbol\\beta - \\boldsymbol\\mu_n)^{\\rm T}(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\mathbf{\\Lambda}_0)(\\boldsymbol\\beta - \\boldsymbol\\mu_n)\\right) ", "c67caaf9ffae920207e4a455fda4e28b": "C_M= \\frac{1}{2} (C_{Mi}^{j+1} + C_{Mi}^{j}).", "c67cb7a87a72cffee1a92748dcb54a73": "P_B=\\frac{B^2}{2 \\mu_0} ", "c67cdbd46027a796d2379450755be652": "0\\le x_{4} \\le 6", "c67cef27170de96eb637eb5abacdab19": "\\Gamma=\\{c\\in C([0,1],X)\\mid c\\,(0)=0,\\,c\\,(1)=x'\\}", "c67d14488c1b4cc7f82e98eb360accf2": " W = \\int_{t_1}^{t_2} \\mathbf{F}\\cdot\\dot{\\mathbf{X}} dt = \\int_{\\mathbf{X}(t_1)}^{\\mathbf{X}(t_2)} \\mathbf{F}\\cdot d\\mathbf{X}. ", "c67d39637a2103aeef976182ca84d672": "\n\\frac{d}{dz} \\left[ z^{1-a_p} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) \\right] =\n- z^{-a_p} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_{p-1}, a_p - 1 \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right), \\quad n < p.\n", "c67d7126c600e84a3ab82af61d443d80": "\\left( 1 -\\sqrt{3} L + L^2 \\right)", "c67d7a986b1e1948cdf41a197c64d35a": "f_{n} (x) = \\begin{cases} - n^{2} x + n , & 0 \\leq x \\leq \\tfrac{1}{n}; \\\\ 0, & \\mbox{otherwise.} \\end{cases}", "c67da0a210fc4ed5805fad8a85f26b51": "\\lim_{n\\to\\infty} {n \\choose k}\\frac{1}{n^k} = \\frac{1}{k!}.", "c67dcebc2e5093759cb50ec1d51c533d": "D=D(x)", "c67df1c7643a3d86e066df8264ddf111": "|t|<\\frac{1}{2}", "c67df2595b9a0680e8261053a62ecbf5": "(x-u)P+(y-v)Q = 0", "c67e067ebcbc7a8dcbc677fa07dee8b7": "\\langle \\alpha^\\mu_\\tau(A)B\\rangle=\\langle B\\alpha^\\mu_{\\tau+i\\beta}(A)\\rangle", "c67ea7a5b5a1de9741171f82d38b3d83": "3\\sin((p-q)\\phi)", "c67ec3b3e60f28b2d707fea57d093fcb": " \\beta_k (\\tilde{T}) = \n\\begin{bmatrix}\n0 & \\; & & & & \\; & & z \\\\\n\\frac{1}{2} & \\ddots & & & & & & 0 \\\\\n\\; & \\ddots & \\ddots & & & & & \\vdots \\\\\n\\; & \\; & \\frac{1}{2} & 0 & & \\; & & \\\\\n & \\; & & 1 & 0 & & & \\\\\n & & & \\; &\\frac{1}{2} & \\ddots & & \\; \\\\\n\\; & & & &\\; & \\ddots & \\ddots & \\vdots \\\\\n\\; & \\; & & &\\; & \\; & \\frac{1}{2} & 0 \n\\end{bmatrix} \n.\n", "c67ee67afd569e814ec5b2abc9b6e9da": "R_H = a\\left ( \\frac{m}{3M} \\right )^{\\frac{1}{3}}", "c67f137dc5b277ed90a29785c9b34ab1": "dV = \\left(\\mu S \\frac{\\partial V}{\\partial S} + \\frac{\\partial V}{\\partial t} + \\frac{1}{2}\\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2}\\right)dt + \\sigma S \\frac{\\partial V}{\\partial S}\\,dW", "c67f7ba024e306e2e25891ea1826d67a": "Rm", "c6800275455cb00b0c7b5ce1d10298fe": "\\theta(x)\\big|_{x=j\\infty} = \\angle(x-r_1)\\big|_{x=j\\infty}+\\angle(x-r_2)\\big|_{x=j\\infty}+\\cdots+\\angle(x-r_n)\\big|_{x=j\\infty} = \\frac{\\pi}{2}N-\\frac{\\pi}{2}P \\quad (13)\\,", "c680405612eef2e93e72d9ad6ffd9306": "0\\leq\\frac{1}{2}\\int_{\\mathbb{R}^4}\\operatorname{Tr}[(*\\bold{F}+e^{-i\\theta}\\bold{F})\\wedge(\\bold{F}+e^{i\\theta}*\\bold{F})]\n=\\int_{\\mathbb{R}^4}\\operatorname{Tr}[*\\bold{F}\\wedge\\bold{F}+\\cos\\theta \\bold{F}\\wedge\\bold{F}]", "c680609c58755beb7da03730b1923bf1": " \\sigma_P(x,\\xi) = \\sum_{|\\alpha|= m} a_\\alpha(x) \\xi^\\alpha,", "c680b2c8a7d9dae6bcd51c8f2da945cf": "F(h) \\approx 2 \\pi \\left( \\frac{R_1R_2}{R_1+R_2} \\right) W(h)", "c680b83a9944a66d17dd09d28a226ec4": "c = 2, \\phi = 20^\\circ", "c680ec288b01f92c8f5c7ad760c3b442": " -\\log\\left(\\frac{n_0}{n}\\right) < \\bar{x} < -2\\log\\left(\\frac{n_0}{n}\\right) ", "c6811745cf9deede908e8407cb804563": "0,...., d - 2", "c68136daccdc243f33e0c96e4d15b3bb": "M \\leftarrow M||(\\Bigl[e\\Bigr]_i^{i+r}) \\oplus mask_e)", "c681727dcf41908b7548a0501b0da43f": "\\|x\\| := d(x,0)", "c6818e8902a19c60e23770ab5bcb2bd9": "\nCIQ_t = \\mathcal{A} e^{\\mathcal{B} \\int\\limits_{0}^{t} \\mathcal{C}_x \\, dx}\n", "c681efcb64058ef70f0068b79a541dda": "p_{v} =", "c682173185399e95d349226c8ee19dbf": "F^2", "c682331a04c348689ace482da1216829": "\\theta_n(x)=\\frac{(-2n)_n}{(-2)^n}\\,\\,_1F_1(-n;-2n;-2x)", "c6824dc07a7382dc0705e846fbe3b501": "a^2\\left(\\frac{\\Gamma((d+2)/p)}{\\Gamma(d/p)} - \\left(\\frac{\\Gamma((d+1)/p)}{\\Gamma(d/p)}\\right)^2\\right)", "c68271a63ddbc431c307beb7d2918275": "out", "c6828a2967af488bed44e719d66a8e61": "x + 1\\,", "c682e348dcef3af11ef6d780154d6612": "E_a(f)", "c6830fcc9b54992c4ebc60015b99b6af": "L\\;=\\;L_{FSL}\\;+\\;A_{MU}\\;-\\;H_{MG}\\;-\\;H_{BG}\\;-\\;\\sum{K_{correction}}\\;", "c6832ce30861dbd464e178083062bb99": "\\frac{1-x^2}{2}", "c683510d50697db59f79b7841f255eda": "\nT = \\frac{1}{2} \\left( \\frac{ds}{dt} \\right)^{2}\n", "c68375a690d19eca4d21090db1b813dd": "\\operatorname{Ric} = R_{ij}\\,dx^i\\otimes dx^j.", "c68398ae81024ceac8857a8c43d215c8": "\\text{150 km} = \\frac{30 \\times C}{2 \\times 30,000}", "c683d24c39eded943ef8c16b5365bb75": "\\top^{\\mathcal{I}} = \\Delta^{\\mathcal{I}}", "c683d36782b5695e64d337bf78947214": " \nE[\\Delta(t) + Vp(t)] \n", "c683fa40a66d418cc63eb47e8dd9eae1": "\\mathcal{F}_{\\mathsf{Pseu}}", "c6840a581f0588f5023702c21d63d143": "nat = \\mu \\alpha. 1 + \\alpha", "c6840d91ff94abd31e28f306c718ce28": "\\sum_{k=0}^{\\infty}\\frac{1}{(2k+(j/p))^{2m}}+ (-1)^q\\, \\sum_{k=0}^{\\infty}\\frac{1}{(2k+1+(j/p))^{2m}}=", "c6841d5d6877aa01bbc40e6ad2871834": "\\{H\\}^\\perp:=\\{D\\in Num(S)|D\\cdot H=0\\}.", "c6847901dd09a7efd9dd75477a521329": "Z_1", "c6847f0aaf9c404fd8c5ceaf1c49a769": "\\scriptstyle V_2 = \\frac{1}{1 - D}V_1", "c684e89979479c3614774339e18cd338": "A A_i = a_i A_i + b_i A_{i+1} + c_i A_{i-1} .", "c68526dc6aa170027eb6ceccd82640de": "H_{inv}(s) = \\frac{1}{H(s)}", "c68553dc87b5486a4de83627b536beee": "\\textstyle\\vec{F}_{i}", "c6856fd4d19a314a1811e73dac899506": "\\tan \\delta = \\mbox{ESR} \\cdot \\omega C", "c6858124ec6916af20dbad41afdd16f1": "n^{\\mathrm{th}}", "c685aae4e7c4c76c4a45a38f0b663739": "((N,h,x,e^{\\prime},k^{\\prime},s),(p,q,a))\\,", "c685b66b85cded1b9810d9c4d0e4e7b0": "| X(t,\\omega) |^2", "c685dceb5649b59d2b9a6bcfdf4e8ae1": "v-w", "c68618a3f4914d91e0ece7f7e3a21b23": "\\Lambda^{n}.", "c686328d678d96b07b5d254bf84d3d3d": "{m}", "c686ad7b67cafefaa0ec6007129da031": "J_N = \\frac{C\\bar u}{4}", "c686bbd8d99308cab829b4b8c0a329e7": "s^2 = \\frac{1}{N-1} \\sum_{i=1}^N (x_i - \\overline{x})^2.", "c686dc43d1d8dac51cc906315f03a4f3": " \\frac 1 {\\sqrt2} ", "c686f89cb35d5fccd781d977fcbc6a1d": " \\operatorname{sink}[(\\lambda p.\\operatorname{sink}[(\\lambda q.q\\ p)\\ (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f))])\\ (\\lambda f.\\lambda x.f\\ (x\\ x))] ", "c6873a9121895107ec2b158f130db165": "\\phi_m", "c6877bcb618b4f491b178f2f8e3b4809": " r=\\sum_{i=0}^\\infty \\frac{a_i}{10^i}", "c68785faffefbd7144e38cb9981b64cf": "r_1 x_1 + \\cdots + r_n x_n, \\quad r_i \\in R, \\quad x_i \\in I.", "c687e8c0e31a0c18433d66fea46f7032": "l = 0, 1, \\ldots, n-1.", "c6880697dd3bfc3fdb99e6b2268fcd8c": " \\pm \\sqrt{i} = \\pm \\left( \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{2}}{2}i \\right) = \\pm \\frac{\\sqrt{2}}2 (1 + i). ", "c6880a4557fb73236761ba7e23527c43": "\\mu=4\\pi^2\\frac{\\text{distance}^3}{\\text{time}^2}\\propto\\text{gravitational mass}", "c68830b9da1b9322297ea82790b8ebd0": "M=\\begin{bmatrix}\nf_1(\\alpha_1) & f_2(\\alpha_1) & \\dots & f_n(\\alpha_1)\\\\\nf_1(\\alpha_2) & f_2(\\alpha_2) & \\dots & f_n(\\alpha_2)\\\\\nf_1(\\alpha_3) & f_2(\\alpha_3) & \\dots & f_n(\\alpha_3)\\\\\n\\vdots & \\vdots & \\ddots &\\vdots \\\\\nf_1(\\alpha_m) & f_2(\\alpha_m) & \\dots & f_n(\\alpha_m)\\\\\n\\end{bmatrix}", "c688bc068779c2c2ba9126e609bdca09": "\\{x_2,x_3,\\dots,x_6\\}", "c68908becce230637b52e22ca92cbb36": "|a_n|", "c6891669746d2d229a8e5956eb41bc25": "y^{+}", "c68933941f2cbff60a75726f82c81cfe": "u_{xy}=u_{yx}=0", "c6894eab45093b30e8259681d6acfa3b": "f(x+1) = \\left( 1 + \\frac{i}{\\sqrt{x+1} }\\right) \\cdot f(x),", "c68965104f306a9bad8907f14f1258ab": "(\\pi \\oplus \\sigma)^{-1} = \\pi ^{-1} \\oplus \\sigma^{-1}", "c689797b9cce5c5802fca4067f907e28": "G:=M\\times M", "c689e179b47c46e7c45ae4b2cad609dd": "\\mathrm{Hom}(Q_M, B_M)", "c68a1d20eb9b6acb5bf251e6841c8e02": "\\sigma_y^2(\\tau) = 2\\ln(2)h_{-1}", "c68a35f3d2cfb4c365ca30684278f3c2": "\n H^2(p,\\, q) = 2 \\int \\Big( \\sqrt{p(x)} - \\sqrt{q(x)}\\, \\Big)^2 dx\n ", "c68a9f0030b568b4fcbc42113c1f8fea": "\\Delta S_{vap} = \\frac {\\Delta H_{vap}} {T_{vap}}", "c68aba84ec623ac250ce236ffca8a706": "L_{\\text{YES}}", "c68b32ac1a4cfe6ae89f61ebc9e47688": "\\Delta T_{\\rm b} = T_{\\rm b}(solution) - T_{\\rm b}(solvent) = i\\cdot K_b \\cdot m ", "c68b8368bb17bb618abc0a565d8278fb": " \\mathcal{L} \\supset m_{3/2}\\Psi_{\\mu}^{\\alpha}(\\sigma^{\\mu\\nu})_{\\alpha}^{\\beta}\\Psi_{\\beta} + m_{3/2}G^{\\alpha}G_{\\alpha}+h.c. ", "c68b9a79cf488024f5a7761e3ce76395": " \\omega^2\\,\\!", "c68bdac867558ac1f5066574ae5ce991": "a_{3}=(5/17)c_{1}", "c68beb6a65f6bcad8ccc778d802ee00f": "(x,y(x))", "c68c9a9a3397c50f36fc407bcfed7969": "g_0=1_{\\Omega'\\backslash U'}g", "c68ce116867f254483b7e2730d2d9c94": "Y_{8}^{1}(\\theta,\\varphi)={-3\\over 64}\\sqrt{17\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(715\\cos^{7}\\theta-1001\\cos^{5}\\theta+385\\cos^{3}\\theta-35\\cos\\theta)", "c68cfbe7d16003f6ea2745f8bc37fdb4": "C_{2} = T_{2} + \\frac{1}{2}(T_{1})^{2}", "c68d04be1d814a92068d730123e73580": "t\\ne0", "c68d5120856ab7edb08b0d0fd08e62b9": "= \\frac{1}{2}(x^2 + y^2)^{-1/2}\\frac{d}{dt}(x^2 + y^2)", "c68dd03ec5ad137136016406579315d4": " \\Sigma_1", "c68e2f05ccceedec527e3d0fbf84804e": "\\mathbf{J}_1=\\mathbf{J}_2", "c68e2fa112176fe6bcd7a487d713cc01": "\\alpha=3/2,", "c68ea8c0920a2094ebda3501adc89fff": "\\frac{\\mathrm{d}^2 x}{\\mathrm{d}t^2} + b \\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\omega^2 x = 0 \\,\\!", "c68ebab7a1e74618506a0a1fabe54186": "3x", "c68ed3697e0f41b282814a9ff91bcc54": "\\omega^2 + \\omega + 1 = 0. \\,\\!", "c68f14907c8345bc9dcb0396ba91f2e6": "\\langle f_n,g \\rangle \\to \\langle 0,g \\rangle = 0.", "c68f64451954d093575c6041b0b10014": "-\\hat \\beta", "c68fa7a33b272b2e915edb201dd8cd64": " Lift(\\Delta L) = \\frac {1}{2} C_L \\rho w^2 (ldr) ", "c68fcfeae86662a2daea89fc9cc1ff99": "f(x_1, x_2, \\ldots, x_m) \\in B", "c68fe3a0e9434f9c597a90dec6ca01c0": "L_a(a) = f(a)", "c690122b062d92a73d2cbd1b9ad480d0": "\nf^\\prime(x)=f(x).\n", "c690158ed86b8456b1e3df9d40cb56a3": "C_{t-\\Delta t,i} = e^{-r \\Delta t}(pC_{t,i+1} + (1-p)C_{t,i-1}) \\,", "c69035f898507defa1659bf8e32fc60f": "\\eta (x_1) = \\eta(x_2)", "c6904509a502bdcbff04ce846c980698": "\\mathbb{C}^* = \\mathbb{C}-\\{0\\}.", "c69056483ae9f772008ef617cf3fd0b8": "2\\times 10^{8}", "c69056c6217b1d4bdbaca9cdc471e653": "M_{k,t}", "c6909c6779b10e3a5d05b66bd9ee901d": "j = 1/2", "c690a5249bc5491cf418425b1c2929ec": "1=\\sum_{n=2}^\\infty \\left[\\zeta(n)-1\\right]", "c6912d915ee0e90bbb76d18ce2ca25e3": "\\mathbf{Z}\\left[ \\frac{1+\\sqrt{-163}}{2}\\right]", "c69135862ee8fc92c7a4f8da372a4d00": "\\frac{\\theta(t)}{\\pi}", "c6915c3f465c77dc79e06090b82eed57": " \\mathbf x = \\begin{bmatrix}\n\\mathbf x_1 \\\\\n\\mathbf x_2 \\\\\n\\end{bmatrix}\n", "c691906e2b63716e92a9ea7fe3441c66": "|\\psi\\rangle = \\sum_{i \\in \\mathbb{N}} \\langle e_i | \\psi \\rangle | e_i \\rangle,", "c691940c1b835735514b062df7d32437": "{x_0}^{n_0} \\cdot {x_1}^{n_1} = {\\left( x_0 \\cdot {x_1}^{q} \\right)}^{n_0} \\cdot {x_1}^{n_1 \\mod {n_0}}", "c691cff43bca7f7f03632da49c00dce3": "d^2r=\\rho\\,d\\rho\\,d\\varphi", "c691dc52cc1ad756972d4629934d37fd": "\\varepsilon ", "c691ee997fbf8801e9916843cfeb324b": "\\mathrm{abcdefghijklm} \\!", "c69233527bbcec34fd0da5567128035d": " {apparent\\ [E]_0} = \\frac{[E]_0}{1+\\frac{[I]}{K_I}}", "c6924f5b80cc3ae30191ca54ee968994": "\n\\left(\\frac{\\partial}{\\partial y_j}\n\\left(\\frac{\\partial \\Phi}{\\partial y_k}\n\\right)_{\\{y_{i\\ne k}\\}}\n\\right)_{\\{y_{i\\ne j}\\}}\n=\n\\left(\\frac{\\partial}{\\partial y_k}\n\\left(\\frac{\\partial \\Phi}{\\partial y_j}\n\\right)_{\\{y_{i\\ne j}\\}}\n\\right)_{\\{y_{i\\ne k}\\}}\n", "c692bdc5b260b6b917b23b26ab6686c4": "\\int e^{cx}\\ln x\\; \\mathrm{d}x = \\frac{1}{c}\\left(e^{cx}\\ln|x|-\\operatorname{Ei}\\,(cx)\\right)", "c692e6b4a30e36d13f65a64b32965766": "\\wp(z)", "c693035d5cb24ef38b51df488184292c": "Z_0\\,\\in\\Gamma\\,", "c69307c681eb0af385b3c1a19bd0ede8": " \\rho = \\rho_f + \\rho_b \\,", "c69360fb8a90bd528b53a7b60c87fa93": "\n\\begin{align}\n \\gamma_1 \n &= \\operatorname{E}\\bigg[\\Big(\\frac{X-\\mu}{\\sigma}\\Big)^{\\!3} \\,\\bigg] \\\\\n & = \\frac{\\operatorname{E}[X^3] - 3\\mu\\operatorname E[X^2] + 3\\mu^2\\operatorname E[X] - \\mu^3}{\\sigma^3}\\\\\n &= \\frac{\\operatorname{E}[X^3] - 3\\mu(\\operatorname E[X^2] -\\mu\\operatorname E[X]) - \\mu^3}{\\sigma^3}\\\\\n &= \\frac{\\operatorname{E}[X^3] - 3\\mu\\sigma^2 - \\mu^3}{\\sigma^3}\\ . \n\\end{align}\n ", "c6938ffdbdcbc8df7b380bcb750b8909": "m c^2\n", "c693ad8c7e7fdca3ac699d616addd88f": " \\alpha_{k} ", "c694a4c164bd252def1ebb558f935bc3": " x \\mapsto g x g^{-1} ", "c69520e43047eec8aee29302af41e48a": "\\frac{d}{dx} \\ln x = \\frac{1}{x}.", "c69531988a1cff2881a063ea7041e16e": " \\frac{1}{m!}\\sum_{k=0}^m (-1)^{k} \\left[{m+1\\atop k+1}\\right] B_k = \\frac{1}{m+1}, ", "c695350e40e31db87769f7f016be4a49": "a=b+c\\,\\!", "c6953bab1e5693c9dcd7711171139e70": "a \\uparrow \\uparrow 4 = a \\uparrow (a \\uparrow (a \\uparrow a)) = a^{a^{a^a}}", "c6953e8aed459bdcb12b9ca0e849bf21": "-2\\pi\\ \\frac{J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos i\\,", "c6954b925e21296e0c724e4756afac6e": "{\\color{Blue}~5.6}", "c6957e61aeb034d46ba4b11580a1036a": "ID = \\log_2 \\left(\\frac{D}{W}+1\\right).", "c6958f99caca48e82dc851c5eb0b8f2e": "C_1,\\dots,C_k", "c695f4cde08e571644cd68db0313cfa5": "K_b(A)>1", "c6960ada204cec3966abda29f9d1211c": "\nD = \n\\psi_a(1)\n\\begin{vmatrix}\n\\psi_b(2) & \\psi_b(3) \\\\\n\\psi_c(2) & \\psi_c(3)\n\\end{vmatrix}\n-\\psi_a(2)\n\\begin{vmatrix}\n\\psi_b(1) & \\psi_b(3) \\\\\n\\psi_c(1) & \\psi_c(3)\n\\end{vmatrix}\n+\\psi_a(3)\n\\begin{vmatrix}\n\\psi_b(1) & \\psi_b(2) \\\\\n\\psi_c(1) & \\psi_c(2)\n\\end{vmatrix},\n", "c6961cc4adbd6f330e4c9ecf7e1337db": "\\mathit{x K}", "c69635a4034c5857ccaf96091007f380": " \\mathbf{v}_j ", "c6963a475a62318108c8aadff7e8d7c5": " \\{\\ a^nb^n : n \\in \\mathbb{N} \\} ", "c6966f732856f606dc58673bb7c79ca4": "F^n \\leftrightarrow \\exist +F^n", "c6968be17806241d11e19a7216c04b41": "\\lim_{\\Delta x \\to 0} x_1 + \\Delta x = x_1.\\,", "c6969599beccde173f2913dcc7f5e7d8": "\\frac{d^2 T_n}{d x^2} \\Bigg|_{x = 1} \\!\\! = \\frac{n^4 - n^2}{3},", "c696b9648048137a22d6ef518ed027ed": "F \\left (x_1, \\ldots, x_n, u, \\frac{\\partial u}{\\partial x_1}, \\ldots, \\frac{\\partial u}{\\partial x_n}, \\frac{\\partial^2 u}{\\partial x_1 \\partial x_1}, \\ldots, \\frac{\\partial^2 u}{\\partial x_1 \\partial x_n}, \\ldots \\right) = 0.", "c696c430d214c1e1638e9da45ae2dd39": "N=3\\,\\!", "c697136311811943df7195db8a1b0039": "M = KDF(m)", "c6974d256b69688d174e70aa1052cb29": "(x_1, x_2, x_3, \\dots)\\text{ or }(x_0, x_1, x_2, \\dots)\\,", "c6978527fcbf2ab10da04746ff24e2b7": "DR_{p,c} = \\frac{\\sum_{p,c}(DR_{T}^{V})}{count_{p,c}(DR_{T}^{V}<>NULL)}", "c697895236a1cda9a2def1865d9af5a6": "A \\cup B \\in \\mathcal{R}", "c6978a4e7f06e950cd71dd47e6e7b214": " \\tau(mn)=\\tau(m)\\tau(n) \\quad \\text{ for } (m,n)=1. ", "c6979dde10013f889dafa3fa9c3df159": "F=b(\\dot{v}_2-\\dot{v}_1)", "c697afdf9742cbd9be2803110b600276": "\\varphi_{\\beta}(\\gamma+1) [n] = \\varphi_{\\beta [n]}(\\varphi_{\\beta}(\\gamma)+1) \\,.", "c69837dce534ca2e97a02e62125e19a8": "r_1 >> 1 >> r_2 \\,", "c69897c5dee057586d966358bb61f16b": "M + HL\\rightleftharpoons ML + H", "c698985e1fe94cad65be743b94c64b36": "\\langle 1, \\omega \\rangle", "c698fb6024869cf020a134f1cafeeff4": " \\hat{S}_d^{\\dagger} \\rightarrow -i { \\partial \\over \\partial \\theta}. ", "c69926d56d7a68470ff894305d633779": "F(\\mathbf{x}_0)\\ge F(\\mathbf{x}_1)\\ge F(\\mathbf{x}_2)\\ge \\cdots,", "c6994c5b6215af1f94abed1956391d69": "\\hat h^{ab}", "c699b134350b28b0106647bbbaffa00f": "\\alpha = (a+d)/c", "c69a17f14b1ab74d97d1d78097df1451": "b_X(y) =\n\\begin{cases}\n 1, & \\mbox{if }y \\in \\left [0,1/2 \\right ) \\\\\n 0\\text{ or }1, & \\mbox{if }y = 1/2 \\\\\n 0, & \\mbox{if } y \\in \\left (1/2,1 \\right ]\n\n\\end{cases}", "c69a58a3ac51db83f7456e159c282e64": "\\frac{q^k}{q^{2\\varepsilon k}}", "c69af44837a68f0ee6a34d0eb5b2cd73": "o(D)", "c69b35f385ec760e9c61fc62f31e08da": "a, y", "c69b4f2e6ca8f14db537a3e591424462": "\\Gamma(\\cdot)", "c69b6e8f647cedd39a185c14def7e72c": "a = c\\ \\frac{\\sin\\alpha}{\\sin\\gamma}; \\quad b = c\\ \\frac{\\sin\\beta}{\\sin\\gamma}.", "c69b93dd9c41f02b19e3ecd79db47bcb": "w=u_1r_1u_1^{-1}\\cdots u_m r_mu_{m}^{-1} \\text{ in } F(X),", "c69bc9a01c38da808e9e8d20e4f24f25": "\\theta = \\phi\\,", "c69bca76c1ba63ef632e962557ff21e1": "\\ \\begin{align} u^* &= y^*-x^* \\\\\n&=c(t)+Q(t)y-c(t)-Q(t)x \\\\\n&=Q(t)(y-x) \\\\\n& =Q(t)u. \\end{align} ", "c69be1ea8ac7df7fe999effc34710e84": "H = \\frac{1}{2q^2} + \\frac{p^2 q^4}{2},", "c69be2547f64aad7c4a955928f5c24d2": "k = A \\to \\epsilon", "c69bec8217234cfe1d3d1a51d2e446f9": "u_i(\\mathbf{s}) = \\sum_{k_1=1}^{m_1} \\ldots \\sum_{k_n=1}^{m_n} a_{i\\, ,\\, k_1\\ldots k_n} f_1(s_1)\\ldots f_n(s_n)", "c69c889212f4c54e8626bb32f0160a49": "\\oint_C f_n(z)\\,dz = 0", "c69cae10d7440f91ed24ada55c8573cb": "a \\mapsto \\mathrm{E}(\\left|X-a\\right|).\\,", "c69ce0e2c9c269f3840b88fa66e30b76": "\\mathbf{v} := 1 - \\frac{f_m}{N},", "c69cf9706c19f242e217f59151d50a42": "E_1 = \\frac{\\hbar^2\\pi^2}{2mL^2}.", "c69d0665ede3a5ca6e4594f6ae0e7476": "\\frac{d^2\\sigma(p+A\\rightarrow \\pi^+ + X)}{dpd\\Omega}(p,\\theta) = ", "c69d067b3f6a9a891810b25ff666e571": "\\cos^{-1}\\left(\\frac{b}{a}\\right)", "c69d2f111701566e309c592814e4fab1": "\\vec y_n = \\sum_1^n {c_i\\,\\lambda_i^n\\,\\vec e_i}", "c69d61c9f0f9b4377571d7ee6aedf814": "\\mathrm{tf}_{t,d}", "c69de9e6e85ea1d3556e9ee6243ea5b8": "F_n(x)=\\frac{\\alpha(x)^n-\\beta(x)^n}{\\alpha(x)-\\beta(x)},\\,L_n(x)=\\alpha(x)^n+\\beta(x)^n,", "c69df78163560bb6f8f2651f71b199b9": "\\int_0^\\infty \\cos ax^2\\cos 2bx\\ dx=\\frac{1}{2}\\sqrt{\\frac{\\pi}{2a}}(\\cos \\frac{b^2}{a}+\\sin\\frac{b^2}{a})", "c69e711a53878e99ddc0cf4abfa48a52": " a = b ", "c69e7e04f32dc8ff8bf4bc0e1d6f6759": "a^\\dagger(\\phi)\\,", "c69eaa3fed7f4b5ba0ce35a88488d99e": "U=U(S,M,I_D) \\,", "c69ee16effc2e3a0ccc0b14b3a1bc354": "\\mathbf{E}^{x} [f(X_{\\tau})] = f(x) + \\mathbf{E}^{x} \\left[ \\int_{0}^{\\tau} A f (X_{s}) \\, \\mathrm{d} s \\right].", "c69f1f71183e25c912cc6cafc75dfac7": "\\epsilon_{Si}", "c69fe9f7e77181478f5e5a520e08eef2": "\nW =\n\\begin{bmatrix}\nW_1 & & & \\\\\n & W_2 & & \\\\\n & & \\ddots & \\\\\n & & & W_k \\\\\n\\end{bmatrix}\n", "c6a015d92cce7f70a64cae8a47c0acb9": " K_{\\,0} ", "c6a0344fbec92d5a7469e19f82c3a6eb": "M_\\mathrm{int}", "c6a05e36b86c0f552f9aef0e946cd028": "g=m w_i", "c6a05fb5616c4bcc02fad0acf398a498": "\\,k=5", "c6a0d6b013bc4f89ed5a281996970d7d": "\\int \\sin^2 x \\cos 4x \\, dx.", "c6a10262cb89fdccba6ee95bb5240048": "\n \\sigma = \\int_0^{y_\\mathrm{atm}}\n \\frac {\\rho \\, \\left ( R_\\mathrm {E} + y \\right ) \\mathrm d y}\n {\\sqrt {R_\\mathrm {E}^2 \\cos^2 z + 2 R_\\mathrm {E} y + y^2}} \\,.\n", "c6a107300a285e5c9da6797a614c1882": "R_{ij}", "c6a1c5afa491d1dbfcf6cd044f924ac6": "Y = X", "c6a1cc0b85c682a0ba9647f7ea83cc0a": "i\\in \\{1,2,\\dots,n\\}", "c6a1ea602383224cb7bff61adfe94cc5": "\\sum_{k=\\alpha}^n {n \\choose k} (-1)^{n-k} f(k) = \n\\frac{-n!}{2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty} \\frac{f(z)}{z(z-1)(z-2)\\cdots(z-n)}\\, \\mathrm{d}z", "c6a2731ecd456d51643512c8f984316e": "\nq \\xrightarrow{a(x)} q'\n", "c6a276bb0303d5d4704329167cd70255": " \\langle \\sigma_A\\sigma_B\\rangle-\n\\langle \\sigma_A\\rangle \\langle \\sigma_B\\rangle=\n\\langle\\langle\\sigma_A(\\sigma_B-\\sigma'_B)\\rangle\\rangle~.\n", "c6a2caea2c6bcd1847d2e58df4a71d23": " e^+e^- \\to 3\\gamma , ~~ e^+e^- \\gamma", "c6a2d7c03b77af86985579a5f454f72c": "q=20/38", "c6a3044e1980adf10c3e2a9309a4c456": "\\frac 1x\\,", "c6a31c2caf424e94f4ab0a42cb90d13c": "\\det g = \\left|\n\\frac{\\partial \\varphi} {\\partial u_1} \\wedge\n\\frac{\\partial \\varphi} {\\partial u_2}\n\\right|^2 = \\det (\\lambda^T \\lambda)", "c6a35aedf3f09727f04323263794590c": "\\lim_{n \\to \\infty} \\Pr(X_n = 0).\\, ", "c6a41dd4377a930c55c9d29957e357b9": "A^T J = J A", "c6a42e82e315c93d65db3f7e6feb168b": " s_1 = { v_1 + u_1 \\over 1 + v_1 u_1 } ", "c6a4307aaf9dcf68594c9359a8d2867c": "E^p", "c6a44b7978203e5f4b942f83128503f3": "t=\\tan\\frac{x}{2}", "c6a4b33b871442cdb35035337ce5f310": "\\frac{1}{2k}", "c6a4d10544473a52a5d35f972b5c113c": "f(x)\\cdot\\left(1-\\frac{1}{1-g(\\vert x\\vert)}\\right)", "c6a4d531e69b212d395a44a0e1d3df8e": "\\pi(x)\\,\\!", "c6a55b401e6ff2205c60a931b6b924bc": "MSE = E\\left(||\\hat{x}-x||^2\\right) = Tr(GC_wG^*) + x^*(I-GH)^*(I-GH)x.", "c6a58b9f8864f58b5af3cd7ac9c389c0": "\\int_0^L a(x)v(x)\\dfrac{\\mathrm{d}u}{\\mathrm{d}x}\\mathrm{d}x-\\int_0^L b(x)\\dfrac{\\mathrm{d}v}{\\mathrm{d}x}\\dfrac{\\mathrm{d}u}{\\mathrm{d}x}\\mathrm{d}x + \\left[b(x)v\\dfrac{\\mathrm{d}u}{\\mathrm{d}x}\\right]_0^L = \\int_0^L v(x)f(x) \\, \\mathrm{d}x ", "c6a5eac6129588e852e23ec720d1a26b": "\\widehat{\\theta} = \\arg\\max_{\\displaystyle\\theta}{ \\left( \\prod_{i=1}^n f(x_i, \\theta) \\right) }\\,\\!", "c6a5f60a5b621ec3cd72db7af8894c2d": "c_{T}(k) \\, = \\, Ak^a", "c6a67105dc5ec08dab2502ea3987c276": "U = AP^{-1}.\\,", "c6a6da36a72c7fb0888f6b9cd717473d": "X \\sim \\mathrm{DP}\\left(\\alpha, H\\right)", "c6a6e69fd5e2b8cd694fdf6117fa4517": "m(f)\\cdot \\|u-v\\|^2 \\leq \\langle u-v, f(u)-f(v)\\rangle \\leq M(f)\\cdot \\|u-v\\|^2,", "c6a6eb61fd9c6c913da73b3642ca147d": "\\lambda", "c6a726115bd79db5b6ee47f79e948b98": "4H_{abc} \\approx h_{ac,b}-h_{bc,a}-\\frac{1}{6}(\\eta_{ac}{h^d}_{d,b}-\\eta_{bc}{h^d}_{d,a}) .", "c6a72a3ffb29c7b2e5f6d819601de890": "|x_1 x_2 \\cdots x_N; A\\rang = \\frac{1}{N!} \\sum_p \\mathrm{sgn}(p) |x_{p(1)}\\rang |x_{p(2)}\\rang \\cdots |x_{p(N)}\\rang ", "c6a7350a0a1379f89b45cf69401bce28": "\\lambda_{\\max}", "c6a7fd971958cefe11a86d055ceef436": "O\\left(e^{\\sqrt{2\\log n \\log\\log n}}\\right)=L_n\\left[1/2,\\sqrt{2}\\right]", "c6a8136327f1fd93674769796ad7219c": "4*y^2-8*x*y+3*x^2-8*y*z+6*x*z+3*z^2=0", "c6a8147e19473a60021942d482c6c21d": "f(x) = \\sum_{k=0}^{\\infty} M[f]_{1,k} x^k. ~", "c6a82123a3be3ef0d48daf95aa8432a1": "\\Delta S'_{system} = S'_{in} - S'_{out} + S'_{gen}", "c6a85cf0c32725e7fef3d03a75b23e44": "\\phi \\circ F_R = F_S \\circ \\phi.", "c6a88ca18ee720049cb81afbc79a5309": "\n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & -4 & 1 & 0 \\\\\n0 & -2 & 0 & 1 \\\\\n\\end{bmatrix}.\n", "c6a89a751ebf77505257501ef3da3cc2": " \\ell^{-1} = n \\sigma ", "c6a8a29fa2a30d387750e5fbbe4a96ef": "H=\\bigoplus_{\\lambda\\in\\sigma(T)}H_\\lambda.", "c6a8a38f99251161f9d3b13fdd9a1ef3": "(\\mathrm{D} u)(x) = u'(x) \\, ", "c6a8a4c4b5f4b7a6b92cc4ad6b99f15c": " \\qquad \\qquad \\mathrm{translational }\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ E_{f,t,n} = \\frac{\\pi^2\\hbar^2}{2m}(\\frac{n_x^2}{L^2}+\\frac{n_y^2}{L^2}+\\frac{n_z^2}{L^2}) \\ \\ \\ \\mathrm{and} \\ \\ \\ Z_{f,t}\\sum_{i = 0}^\\infty g_{f,t,i}\\mathrm{exp}(-\\frac{E_{f,t,i}}{k_\\mathrm{B}T}) = V (\\frac{mk_\\mathrm{B}T}{2\\pi\\hbar^2})^{3/2},", "c6a8ff138f1152579ea3171f738fe7b3": "f_{H} = \\frac{v}{2\\pi}\\sqrt{\\frac{A}{V_0L}}", "c6a92a011417fd5408c3ba35ec595b9b": "\\{ \\psi_{n,k}(t) \\; ; \\; n \\in \\mathbf{Z}, \\; k \\in \\mathbf{Z} \\}.", "c6a9453799ef7f6c21283e54094404ff": "\\pi r", "c6a94b9d183c69785455a7eed0a1383a": "X_i \\sim N(\\mu_i, \\sigma_i^2), \\qquad i=1, \\dots, n,", "c6a9779b7dd76d8ac14328c9a9b7f93a": " y^{(5)} + y^{(4)} - 4y^{(3)} - 16y'' -20y' - 12y = 0 \\, ", "c6a986ac69d4ba755729d7156998685c": "K_{n+1} = \\{ \\langle e, a_1, \\ldots, a_n\\rangle : \\exists x T(e, a_1, \\ldots, a_n, x)\\}", "c6a997fe1336b5d051bfdf6da8a34eaa": "D_2\\,\\!", "c6aa388e7f47653d04bc5023365d1e11": "\\langle s,t \\mid (st)^2 = s^3 = t^5 \\rangle.", "c6aa3e2549aaf3fc74256e468ae0e880": "\\eta(A,B)", "c6aabe780d275de45db116294620b58b": "\\rho_2(x_0, x_1, x_2) = \\frac{x_0 - x_2}{\\rho_1(x_0, x_1) - \\rho_1(x_1, x_2)} + f(x_1)", "c6ab10571fec44f8a492446e76b0d59e": " [\\mathbf{\\hat Q,\\hat T}(\\lambda)] ", "c6ab510059ec4cbdbb6f09b1a00a342e": "\\scriptstyle \\sqrt{\\pi}", "c6ab5c40084505d36301726871a1f54c": "\n T\\biggl(\\sum_{\\alpha\\in A} f_\\alpha \\mathbf{e}_\\alpha\\biggr) = \\sum_{\\alpha \\in A} f_\\alpha T(e_\\alpha) = \\sum_{\\alpha\\in A} f_\\alpha \\theta_\\alpha.\n ", "c6ab7255adb4fc2e32fbd9d4d048a6b2": "\\forall i\\in I\\quad A_i1,", "c6bdf038218be08e19bbdee35a62ed9b": "bx + py = 1. \\,\\!", "c6bdf8939bf8f3bc8cecf61bfecad824": "-j\\varepsilon _{r}k_{xo}cot(k_{x\\varepsilon }w )+k_{x\\varepsilon }=0 \\ \\ \\ \\ (4) ", "c6be0296cded01cff8e8d3f45b5bac46": "i^{-1}(U)", "c6be409b322503ff2aa26979bbabc2a5": "T_{Doppler} = h \\gamma /2k_{B}", "c6be4ecd7b0b752cf69c33bb054e8bc2": "A={\\Bbb C}[x]", "c6be8c8d426aa32c2ddcd0f19af5c104": "S(m,n) := Z_m \\wr S_n", "c6be962400f9174067ea3cf9e39aa9ae": " \\operatorname{build-param-lists}[n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n), D, V, R] \\and \\operatorname{build-list}[\\lambda x.\\lambda o.\\lambda y.o\\ x\\ y, D, V, D[g]] ", "c6beb3654d340055c3f6474c487bfe97": "s_{i}(n)", "c6bf5e4b055e0fc3d261bd6193c0e353": "\n\\frac{z^{2}}{a^{2} \\cos^{2} \\nu} - \\frac{x^{2} + y^{2}}{a^{2} \\sin^{2} \\nu} = \\cosh^{2} \\mu - \\sinh^{2} \\mu = 1\n", "c6bf7c4a70d37bf777b1115c1ddc2f88": "\n M_x(x) = \\int\\int z~\\sigma_{xx}~\\mathrm{d}z\\,\\mathrm{d}y = -\\left(\\int\\int z^2 E(z)~\\mathrm{d}z\\,\\mathrm{d}y\\right)~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} =: -D~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\n", "c6bf9a2fe5bf55a88f21db104a02d68e": "T^2/S_2,", "c6bfa7ecd6fb2a2de9e65a4f977a2722": "\\{a_i\\in P \\mid i\\in I\\}", "c6bfb9bc29377a0f0cb7133b8a24a41a": "dn_2/dn_1", "c6bffa0212b1cc714e7049e4748bd1a2": " \\frac{\\partial \\mathcal{L}}{\\partial f} - \\sum_{i=1}^{n} \\frac{\\partial}{\\partial x_i} \\frac{\\partial \\mathcal{L}}{\\partial f_{x_i}} = 0. \\,\\!", "c6c02709f2d41640f2b604860a50d64a": " \\exp ( \\lambda_t x ) \\bold c_t \\rightarrow 0 ", "c6c04622adc46d37ccbe2fdb176e291f": "k_{i\\perp}=\\frac{1}{\\hbar}\\sqrt{2m(E_f \\cos^2\\theta+V_0)}", "c6c0700c86d5325ff088a07de531d88c": "\\textstyle (x, y) ", "c6c0757c0b913848c571ddde1da22641": "\\mu^nf(p, q, r) + \\lambda\\mu^{n-1}\\Delta_Q f(p, q, r) + \\frac{1}{2}\\lambda^2\\mu^{n-2}\\Delta_Q^2 f(p, q, r)+\\dots", "c6c0c27d84f686eddd8b00f51c8fce62": "\nH= \\rho c_p \\int_{h2}^{h1} T(z) dz\n", "c6c0d55a44f1193e5aaad8a8dc20743b": "\\alpha_i > 1", "c6c0f39697d2962632828d990efc5e13": "F_2\\;=\\;\\frac{G_B}{4}", "c6c13f76fa211adc70e57771489a1546": "O(W*10^d)", "c6c18e1d1c2c8df3429b51f29f43155b": "m_1= \\frac{I_1(\\kappa)}{I_0(\\kappa)}e^{i\\mu}", "c6c1ca42815605bf0899fb1a1d51d076": "(\\tau_i)_{i\\in I}", "c6c1cfad7cc7f01fa91c6377315b1161": "L_{0}=Tv", "c6c1e67dc91eedb5a8147977e27efbd4": "X:=(x^2-y^3=0) \\subset W:= \\mathbf{R}^2.", "c6c1f8d3fb4f7e385042f8ebd0eea4c0": "T = \\sum_i T_i.\\,", "c6c2140eefe8d330efc70e037e8cc304": "n^{\\ominus}", "c6c2c8416f68a54161219a78bd956484": " \\left|-z\\right\\rangle \\otimes \\left|+z\\right\\rangle ", "c6c2e35f185e4e4b393b844083238c67": "N = \\max(m,n)", "c6c2f49a7595e7eb11704ce88aeb41e3": "S=\\sum_{i=1}^{n}{r_i(\\beta)}^2", "c6c2fa2e02b47d1ee84147259a3b468f": "h_{\\mu\\nu}\\;", "c6c3213d528b40f69c79ff46829cf8df": "\n\\text{Yield to maturity(YTM)} = \\sqrt[\\text{Time period}]{\\dfrac{\\text{Face value}}{\\text{Present value}}} - 1\n", "c6c32dec346ede82b6c11f5c4ebdbbc4": "\\lim_{a \\to \\infty} \\frac{\\pi a}{a^2-1} = 0.", "c6c35357e83f2076702d94da31e74f90": "\\mu_{z} (\\{ t \\}) = \\| z_{+}(t) - z_{-}(t) \\| \\mbox{ and } \\mathrm{rd}(z)(t) = \\frac{z_{+}(t) - z_{-}(t)}{\\| z_{+}(t) - z_{-}(t) \\|}.", "c6c3657fd5a0e3eeefefca6c444c797a": "1 \\cdot (a \\cdot b)", "c6c3ceb1828b3e5ae5ffa26ecef56416": " \\operatorname{mse}[\\lambda x.f\\ (x\\ x)]", "c6c3e0634597b53dab1788d003e8fbdd": "\\ D_\\mu = \\partial_\\mu + i g A_\\mu ", "c6c4304799a086460bad8504b0b3f021": "= \\left\\{ x \\in X \\left| \\begin{matrix} \\mbox{for all open neighbourhoods } U \\mbox{ of } x, \\\\ U \\cap A_{n} \\neq \\emptyset \\mbox{ for large enough } n \\end{matrix} \\right. \\right\\};", "c6c49ad9fdbc25adab1b025ef51fa44b": "\\textstyle p(x) = c\\;d(x)", "c6c4f7b5be0f2e64c10d4477aa30e4ba": "\\partial_n\\sigma_n(\\Delta^n)", "c6c4f83a303e3e9b60e1af547e243aad": "T_{t}=\\frac{2L}{c}", "c6c527dad7cf0c0644dff500432d308c": "\\Delta T_{sat}", "c6c531930eac081bb64c1ace82fb59b9": "\\textbf{P}_{k\\mid k} = \n(I - \\textbf{K}_k \\textbf{H}_{k}) \\textbf{P}_{k\\mid k-1} (I - \\textbf{K}_k \\textbf{H}_{k})^\\text{T} +\n\\textbf{K}_k \\textbf{R}_k \\textbf{K}_k^\\text{T}\n", "c6c54fee107c94391bea2d9efe6f2973": "E={E_1,E_2,\\dots,E_m}", "c6c55625bd50f59457cdd3e9e6ae0c2b": " S1 = \\sum_{j=1}^{k} n_j m_j s_j", "c6c5f1b07633356a9b6dfe9d9bb9c776": "W_{F}", "c6c614961e775c95c0f534cb4adab079": "\n\\sum_{d | n } J_k(d) = n^k. \\, \n", "c6c652452fed2e5d2f01d30e672a94ef": "H^++\\text{HCO}_3{}^- \\Longleftrightarrow H_2\\text{CO}_3\\Longleftrightarrow \\text{CO}_2+H_2O", "c6c6aa9b314f0b77c1e5fd9aeeafa6ed": "z=2", "c6c6da8b594ac497787b85f42db91bda": "\\delta_{\\bar k}<0", "c6c6efbf6816f15904b220ee8eb3e318": " P_0 ", "c6c710b2909fd0a9dc733bf84a2269a9": "\\sum_{i=1}^n y_i", "c6c75fec0ba9571195915c8d459f813a": "\n\\operatorname{Li}_{-n}(z) = {1 \\over (1-z)^{n+1}} \\sum_{k=0}^{n-1} \\left\\langle {n \\atop k} \\right\\rangle z^{n-k} \\qquad (n=1,2,3,\\ldots) \\,,\n", "c6c7ac976b1532cf1ab9d18d78e67d30": "X_R \\supseteq X", "c6c7d26651a76f18efb30dc9dcc14e23": "\\vec{v}_s=\\frac{1}{m}(\\frac{h}{2\\pi}\\vec{\\nabla}\\varphi-q\\vec{A}).", "c6c801b451c5120d54e7f6e66b38b71a": "G^2", "c6c80c93c627e6b30d63694f791e6067": "\n\\nabla \\times \\mathbf{F} = \\frac{1}{r \\sin \\theta} \\left( \\frac{\\partial F}{\\partial \\varphi} \\right) \\hat{\\boldsymbol\\theta} - \\frac{1}{r} \\left( \\frac{\\partial F}{\\partial \\theta} \\right) \\hat{\\boldsymbol\\varphi} = 0\n", "c6c826f3b09f4d08757cdfa146c4cb05": "{M_{air}}=\\frac{e}{p}M_v+\\frac{p_d}{p}M_d \\, ,", "c6c85266fd6e62469b1423c15d8e1d3f": "P(cancer|smoking)", "c6c87e23ba98f98a492dd4d5bd9be024": "t_2\\geq t_1 \\geq 0", "c6c90e9c9e2558bf4104e68075b3a718": "X^8-16", "c6c9790366dfce19d4caf284b404fb8d": "\\int_{-\\infty}^\\infty\\left|f(x)\\right|^2\\, w(x) \\, \\mathrm{d}x <\\infty,", "c6c9a618ac94047d07b858fe1d114ba8": "\\log_b N", "c6c9fb2553477da6bdc1bb8fdbca7c00": "\n\\begin{align}Q(x;\\;y_0,\\;y_1,\\ldots,\\;y_n)=P\\left(x+1;\\;xy_0,\\;xy_1+y_0, \\;xy_2+2y_1, \\;xy_3+3y_2,\\ldots,\\;xy_n+ny_{(n-1)}\\right)\n\\end{align}\n", "c6cabdecd34b7dddb0c35457cbc6777b": "I_{i j}", "c6cac2e4967d390d7e6ea7b1e39bb0c5": "\\exists \\kappa. (x\\wedge \\exists \\kappa. y) \\Leftrightarrow (\\exists\\kappa. x) \\wedge (\\exists\\kappa. y)", "c6cb06d6577a3a4c1d15b6fec52b5c0c": " \n\\begin{bmatrix} 1 & \\epsilon^n_b \\\\ \n0 & 0 \\\\\n\\vdots & \\vdots \\\\\n0 & 0 \\\\\n\\epsilon^n_f & 1 \n\\end{bmatrix} \\begin{bmatrix} \\alpha^n_f & \\alpha^n_b \\\\ \\beta^n_f & \\beta^n_b \\end{bmatrix} \n= \\begin{bmatrix} \n1 & 0 \\\\ \n0 & 0 \\\\\n\\vdots & \\vdots \\\\\n0 & 0 \\\\\n0 & 1 \n\\end{bmatrix}.", "c6cb16ff4d7b944aa0518ab13f2cbcf6": "f_3(\\omega) = \\frac {a_0}{b_0}f_2(\\omega) - f_1(\\omega) \\quad (29) \\,", "c6cb33be68967b80ea87564b8ffec84e": " a - 3b = 0 ", "c6cba76d96b32f5a93b9e97798357864": "s=-1", "c6cbb54a4ec4713ee88c92031a69e885": "y(n)=\\sum_{k=0}^{N-1}{h(k) x(n-k) }", "c6cbcb675002d647e9c67108a767ba24": "\\boldsymbol{p}_{k+1}", "c6cc315d43f3382f31efa40d3766efbc": "(d_{i}-1)", "c6cc32f23ec75f88686381c595b40549": " i\\hbar \\frac{\\partial}{\\partial t} \\Psi\\left(\\mathbf{r},t\\right) = \\hat{H} \\Psi\\left(\\mathbf{r},t\\right) \\,\\!", "c6cc706faac12bcaa54f602a0023dc54": "U = e^{-iH(t)/\\hbar}", "c6cca857c06e184f49d81cc3d8ca4867": "(a_i)_{i\\in I}", "c6ccb3feac54c97f04c6d893755c4a0a": "\\left(r,\\theta,\\zeta\\right)", "c6ccc1ebcbdf521e9a5db16477c563ca": "{\\mathcal O}(M)", "c6cced5621cb68cb865ab485fc273ef6": "\\lim_{\\theta \\to 0}{\\sin \\theta} = 0\\,", "c6ccfdf3488678eb864b9fe1f833b312": "[h,f]=-2f", "c6cdc916b0bf6e73add9fd920927b4cf": " a_P\\phi_P = a_W\\phi_W + a_E\\phi_E + S_u ", "c6cdea0e18f5d8aaca650af9e18878b1": " = \\operatorname{sgn}(\\sigma)\\cdot\\operatorname{sgn}(\\tau)", "c6cdf461165fe6e07f504c41b7ab4e89": "\\sqrt{2} = \\frac{m}{n}=\\frac{m(\\sqrt{2}-1)}{n(\\sqrt{2}-1)}=\\frac{2n-m}{m-n}", "c6ce08c529220d99340dfe406945d343": "l=\\frac{4V}{S}", "c6ce1389417f9ac36a3fc26cc75d0a33": " D_v(f) = (f\\circ\\gamma)'(0).", "c6ce5d2b010c8aa4edf2e0d58559dfb2": " b^*_s ", "c6ce69a06b2d09bcfae167e12dffb777": "(p \\leftrightarrow q)", "c6cea827f2986c6900b4b12b0ee335d3": "(\\vec{p}\n^ {2}-\\varepsilon ^{2}+m^{2}+2mS+S^{2}+2\\varepsilon A-A^{2})\\psi =0~", "c6ceb4f2bb6aabdbb1e0a06f954e7e9a": "d(x, y) = (x, z)_{y} + (y, z)_{x},", "c6ceb8267755b18f2b1dc13b61eccc92": "x(n-i)", "c6ced963d66fd3dc9b443403b1f921da": "\\left[\\hat{F}_j,\\hat{H}\\right] = 0 ", "c6cf163bad201f364db1b0db0b9829d7": "f(x)=(2\\pi)^{-p/2}\\, \\det(\\Sigma)^{-1/2} \\exp\\left(-{1 \\over 2} (x-\\mu)^\\mathrm{T} \\Sigma^{-1} (x-\\mu)\\right)", "c6cf3cb81024e08b6d88d4982661a966": "\\vec r_B=r_{Bi}+ \\vec v_B t", "c6cf996761690ec763b73e437125e329": "y = mx + b\\,", "c6cfbba0e7c13b24185caee2821023b8": " \\left(\\mathbf{A}+\\mathbf{B}\\right)\\cdot\\mathbf{C}=\\mathbf{A}\\cdot\\mathbf{C}+\\mathbf{B}\\cdot\\mathbf{C} ", "c6cfcdf0626b048b221adfc83b976609": "\n\\frac{1}{r} = A + B \\cos \\theta_1\n", "c6cfd9063407cb30fe193370bea590b6": "\\ \\|x\\|_p=\\left(|x_1|^p+|x_2|^p+\\cdots+|x_d|^p\\right)^{1/p}.", "c6cfdf1c159be58c6fe50466878b07c0": "\\mathrm{2\\,ZnS + 3\\,O_2 \\rarr 2\\,ZnO + 2\\,SO_2}", "c6cff1311457210bff3b82e7a8424c37": "af[n + 2] + bf[n + 1] + cf[n] = 0.", "c6d0083e14d8238f533536ee7930fa33": "\\alpha ^{\\pi/2}", "c6d058796e0799ba2da9c11149367a5e": "DAF = 1 + e^{-c\\pi}", "c6d05ec05ac56bae5118ac6457d620f7": " r = y - Ax", "c6d0a208507750b0b0e1d169ba533888": "K(x, y)", "c6d0a4a19aefae7a7cf80e203d974f03": "dH_\\xi", "c6d0ffe80770460935f8d1cd386c6546": " \\beta < 1 - \\frac{c}{(2+|q|) \\log q} \\ ", "c6d10ef1ff24bd1edf6770421917effe": "dy = \\mathbf{a}\\,d\\mathbf{x}", "c6d122b6c0ad710f0765248e166c71e8": "\\langle E \\rangle=\\langle |\\mathbf{p}| \\rangle c.", "c6d122d57b8f684745ca98e298a2e034": "\n\\begin{align}\n|\\frac{a}{2} \\sqrt{1^2+0^2+(-1)^2}|^2 >& |\\frac{a}{6} \\sqrt{2^2+(-1)^2+(-1)^2}|^2+|\\frac{a}{6} \\sqrt{1^2+1^2+(-2)^2}|^2\\\\\n\\frac{a^2}{2} >& \\frac{a^2}{6}+\\frac{a^2}{6}\n\\end{align}", "c6d14ed06f9e042db4d060ce8fe62cd2": "A_r+A_l=B_r+B_l", "c6d16b9a4f0b7544294cb8d6c14cfec4": " \\operatorname{Tr}(\\alpha^*(S) E) = \\operatorname{Tr}(S \\alpha(E)).", "c6d18713793058e63b2a1e9d127f4046": " (X,Y) \\mapsto -d_Y(h(X,Z)) = -(\\nabla_Yh)(X,Z) - h(\\nabla_YX,Z) - h(X,\\nabla_YZ) ", "c6d1b84d7567db475a080525cb6740f0": "P_\\mathrm{avg} = {(V_\\mathrm{RMS})^2\\over R}.", "c6d1d6c93131d0cacd35e1fc636f8efc": "m_p = 1", "c6d1d8a356dc466f945626f48a4e3027": "p = a_1 q_1 + 1", "c6d223eafd71470302a8b78e6d137648": "a+b=c", "c6d2352bfdce4a57724fd38ffad51b08": "\\mbox{creep}=\\frac{(\\mbox{actual displacement} - \\mbox{rolling displacement})}{(\\mbox{rolling displacement})} \\,", "c6d28d742b2b350b3b011d8b9642cca8": "A \\cdot \\overline{B} + \\overline{A} \\cdot B", "c6d297414cae2e4bb6fc710e974ba6e7": "OR=\\frac{D_{E}/H_{E}}{D_{NE}/H_{NE}}\\,,", "c6d2efb80edfd13dd11020c6d45da7cb": "{(\\eta_b)_{impulse}} = {\\cos^2\\alpha_1}", "c6d32b2d747e17dc2c186ee71fc487fc": "X^{\\tau}", "c6d33e11c1fbb65b368de53475b3a731": " \\int f(r) dx dy = \\int f(r) \\int d\\theta \\delta(y) |{dy \\over d\\theta}| dx dy ", "c6d3774f087739c406f9109c4ce2b346": "B_n = \\sum_{k=0}^n b_k", "c6d38e0f17ea0a25ab76b3fc445df7ad": "{B\\left[ \\vec{u} \\right]}_{\\hat{m}\\hat{n}} = 0.", "c6d3c518e34e5cef0d73eb8ce67aa78f": "\\theta = n \\times 137.5^{\\circ},\\ r = c \\sqrt{n}", "c6d3f2367b9eb275f94294dd2ef060e3": "k\\propto\\exp\\left(\\frac{\\Delta^\\ddagger S^\\ominus}{R}\\right)\\exp\\left(\\frac{-\\Delta^\\ddagger H^\\ominus}{RT}\\right)", "c6d416545071b90a158307f4a6529ca5": "2 \\zeta = \\frac{B}{\\sqrt{mk}}", "c6d4222c5e35232b037b30b9c87bd507": "B_r(\\underline{c}) \\subseteq (\\overline{\\underset{=}{A}(kU)})^\\circ ", "c6d4db13f4d824040f67cad99554203a": "X=m/\\sqrt{n}", "c6d4dd74aaff987cc2d7b99fdbec23f1": " \np_1 = 1. ", "c6d52352619a6994a1fcb835f6e6f96b": "I_0(\\cdot)", "c6d53c3d2a35151226b9e51e0b2ab522": "H(q, p, t)", "c6d5639b760424c577777933405218f3": "{2n-1 \\choose n-1} \\equiv 1 \\pmod{n^k},", "c6d591da24ebc3808378af1364292bb2": "Q(Mv)=Q(v)", "c6d5a7816344e04c1d8803e50083295f": "\\pi=D^{a_1}T^{a_2}V^{a_3}", "c6d5b38599d098df72bee52727ff2a24": " I_{\\textrm c}=\\frac{E_{\\textrm{out}}}{Z_1+Z_2} ", "c6d6a88e1420429b299859229df679ab": "\\sup_{x\\in K}\\left|\\frac{\\partial^{|\\alpha|}}{(\\partial x_1)^{\\alpha_1}\\cdots(\\partial x_n)^{\\alpha_n}}f_\\varepsilon(x)\\right| = O(\\varepsilon^{-N})\\qquad(\\varepsilon\\to 0).", "c6d6e8fd6090c977167911cacb679bbf": "E_{s,s} = V_{ss\\sigma}", "c6d7052b4010722335f5a2e52e4dc8a3": " f: \\mathcal F^\\mathrm{an} \\rightarrow \\mathcal G^\\mathrm{an} ", "c6d705a686641ffc5eb105fa27e240b9": "\\scriptstyle r(t) \\;=\\; \\mathbf{A}(t) r_0", "c6d7e5e6bed0c08e9a082342a94d115d": "\\omega_{\\mu}^{\\ IJ} = {1 \\over 2} e^{\\nu [I} (e_{\\mu , \\nu}^{J]} - e_{\\nu , \\mu}^{J]} + e^{J] \\sigma} e_\\mu^K e_{\\nu , \\sigma K}).", "c6d7eba8440d696c6f52f150d7ad403a": "e^{i\\theta_1}e^{i\\theta_2} = e^{i(\\theta_1+\\theta_2)}.\\,", "c6d8067b9b4f4cf679de095d8295ed8f": "B=\\frac{1}{b}\\sup_{|\\gamma|\\in[1,a]}(G_0(\\gamma)+G_1(\\gamma))<\\infty", "c6d865443794c660d03398578f9577fb": "\n\\frac{\\displaystyle 1}{\\displaystyle 10}\n\\begin{bmatrix}\n3 & 7 & 4 \\\\\n6 & 1 & 9 \\\\\n2 & 8 & 5 \\\\\n\\end{bmatrix}\n", "c6d86a26fed8518cc3813f5faa31369b": "B_1 \\in \\Sigma_1", "c6d8b85a2c0c7bdb035461dc1d8685fb": "\\textstyle \\mathbf{Q}", "c6d8e54acb7d337d55a453a6a20e1b3d": "p_3=\\frac{9}{61}", "c6d91fe21a57e28666ba2ad371f2660d": "(f_n)_{n\\in\\N}", "c6d95c672ac7df826e17e5e9a17992ff": "\\ k(T)/k(T_{Ref}) ", "c6d995389aa862d5c1f9448a9ad4c254": "\n\\tau = { L \\over R }\n", "c6d99fbf16f063707cd4cb788e453f16": "\\left(\\frac{\\ell^\\ast}{p}\\right)=\\left(\\frac{p}{\\ell}\\right).", "c6d9fc3c43b3e43974e3385a2a39c24a": "f(a) < 0", "c6da1afd28bed47dab293ea857770d12": "A = \\lim_{x \\to 1}{\\frac{1}{x^2 + x + 1}} = \\frac{1}{3}", "c6da9dd4885c09ba9bae2cd9cb1dc09e": "L(R,Y)", "c6dab44bd7208b1bf6df6a5c25e49c15": "{\\text{YG} = 2.5\n+ \\left(2.5 \\times \\text{adjusted fat thickness}\\right)\n+ \\left(0.2 \\times \\text{percent KPH}\\right)\n+ \\left(0.0038 \\times \\text{HCW}\\right)\n- \\left(0.32 \\times \\text{REA}\\right)}", "c6dab9ad6259828829af12b13b4e52ff": "n+\\tfrac{4n+3}{4n+4}", "c6db3650fcba8c1a71a43c12d4413edd": " Y = \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix}", "c6db791a446131597ba8af3072b1b9a8": "\nH_\\alpha(p_F(x;\\theta)) = \\frac{1}{1-\\alpha} \\left(F(\\alpha\\theta)-\\alpha F(\\theta)+\\log E_p[e^{(\\alpha-1)k(x)}]\\right) \n", "c6dc38a628b908e8cf122ddb513632e3": "\\int_{|\\xi| \\leq R} \\hat f(\\xi) e^{2\\pi ix\\xi} dx", "c6dc4bb3ca107ea55467abdde2c6ab63": "\\{\\varepsilon_1,\\dots,\\varepsilon_M\\}", "c6dc9847d0495630b7c0e193a935b705": "\\searrow", "c6dc9f0b86b7b46e7956440e345431f4": "\\frac{df}{da}=\\frac{d}{da}\\int_0^\\infty e^{-a\\omega} \\frac{\\sin \\omega}{\\omega} d\\omega = \\int_0^\\infty \\frac{\\partial}{\\partial a}e^{-a\\omega}\\frac{\\sin \\omega}{\\omega} d\\omega = -\\int_0^\\infty e^{-a\\omega} \\sin \\omega \\,d\\omega = -\\mathcal{L}\\{\\sin \\omega\\}(a).", "c6dcc26e0eba4472e1809876530bfea4": "R/P' \\simeq Q'/Q", "c6dce1911051355fe184304b53e194fe": "P \\cap I", "c6dcfd29862a91c74d1ff2f39bc1566c": "\\mathbf{H}_{-n-1/2}(z) = (-1)^nJ_{n+1/2}(z)", "c6dd9d481dfeafc04b7226eb9bae0450": "|\\Psi^+\\rangle_{AC} \\otimes (\\beta |0\\rangle_B + \\alpha|1\\rangle_B)", "c6de603f382af2e42ceaa9698289db80": "p(a/2) < p(a)", "c6de9c93f5a6a7d4614410f80bdc437d": "c_i = 0", "c6deb8e560dd19d617fa2a293f10ad6a": "\n|p|=\\left( \\frac{1}{D_{\\alpha }}(E-q^{2}|x|^{\\beta })\\right)\n^{1/\\alpha }\n", "c6dedb4665e353a2d81b30cc1201fd0d": "\nf(g) = \\frac{0.12}{0.12+0.88 \\exp(-4.2 \\times 10^4 g)}\n", "c6def931acf598667bd57f9a421ce5f9": "56.5 \\pm 1.6", "c6df105cefdcc9b08dd3ad4531b804db": "H(l)", "c6df6d4c3b281205ed3863d98355f0e3": "X+\\xi\\longrightarrow X+\\xi+i_XB", "c6dfb13bc4b5e4a90c1c5f209276f0ef": "\\int \\frac{dx}{x(\\ln x)^n} = -\\frac{1}{(n-1)(\\ln x)^{n-1}} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}", "c6e000e9128044b178c24a918898bdbc": "\\operatorname{var}[\\ln (1-X)] = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta)", "c6e003208ac4e1752a0a82db4759766b": " E_r = \\frac{\\hbar^2}{2\\mu {R_0}^2} J(J+1) = \\frac{\\hbar^2}{2I_0} J(J+1) = BJ(J+1) ", "c6e03dcd4817057332a1cb53e91310d5": "\\log \\bar{\\sigma}_N", "c6e08773e28f124cec860246489f9077": "\\sigma_\\varphi(R)", "c6e0bdc1f7d58f0d53f0925226c1a88e": "z = E/V_0", "c6e0c5955e22c7c2de36c3ad56aad861": "t_r\\cong\\frac{2.197}{2\\pi f_H}\\cong\\frac{0.349}{f_H}", "c6e0e09cd4738bb1127a6dca1d40a68c": "B = Attr_{1-i}(W'_{1-i})", "c6e0e9ae08edd0c3a0985ce3525c8b46": "\\tilde{S}_t", "c6e1187453200bd76dd783767d8f7306": "\\phi_2=-66.53^\\circ", "c6e151a71e20f9c90a556a2d9df2d0e8": "H=(H\\ or\\ E)\\ and\\ (H\\ or\\ \\overline{E})", "c6e15da219ff059c5e31e9e0084a1afe": "\\gcd (|x-y|,n)", "c6e1d14321ed2bec5767364eaa223aa3": "\\sigma^2 \\rightarrow \\sigma^2/N", "c6e207c73651dca271e91b1b3a01f072": "4\\pi R^2 \\,", "c6e2094be7dc30b83dc378321098ef0f": "X_{lc}(\\bold{r}) = Y_l^0", "c6e21cf05ecef32787e75070d853dc23": "\\forall a, b, c: a R b \\wedge b R c \\Rightarrow \\neg a R c", "c6e27628dfbdc2e97ff7a9e0c5007951": " \\mathrm{Vol}_n(K + t B) = \\sum_{j=0}^n \\binom{n}{j} W_j(K) t^j~,", "c6e2946021a0579861bf1a483c409c36": "{\\mathbf{u}}", "c6e2a2956a472722f3f224d965645130": "p\\in [1,\\infty)", "c6e2c8655ed7b9357bcd3cc0cff32272": " y=\\frac{a}{x-b}+c,a\\ne0 ", "c6e3252c874448b72b7c3c9e7d03df08": "q^{-1} + 4372q + \\cdots", "c6e392706cdf47bf0b4d4033d398bec2": "S_{NBR(n)} = 2\\cdot S_{RRB(n-1)} +1 + S_{RBN(n-1)} + 1 + S_{RBN(n-1)}", "c6e39452830767c836690e7a3c9d85de": "a = n \\times q + r\\,", "c6e3be24c8624c1f458fba83ed05ca4f": "e^{\\mu-\\sigma^2}", "c6e41a045bccdd98ecec2168581bf470": " c_1 = \\frac{ h_1 - h_2(\\bold {n}_1 \\cdot \\bold {n}_2) }{ 1 - (\\bold {n}_1 \\cdot \\bold {n}_2)^2 } ", "c6e44a3972332d8ed1d80f1e9febca26": " p_0", "c6e450b197053ca3b86d3d473fd07d7d": "R(r)", "c6e46d45965bfd0a95ad6d1e91196b18": "\\pi_1(X)=0", "c6e4a52689da50830b1a493708ce579a": "\\gamma=\\int_0^1\\int_0^1\\frac{1-x}{(1-xy)(-\\log(xy))}\\,dx\\,dy.", "c6e515762a188f57a391fecedd046421": "c=\\bigoplus_{i=1}^d K_{i(b_i+1)}", "c6e518b01ce56fda8096f1b2274ee22e": "z(x_1)", "c6e5531ccf40fbf8729c77090eea14cd": "H=\\Delta+P", "c6e559fc50231bcf201141247d8559cc": " \\mathbf J\\left( \\mathbf k \\right) \\equiv \\int d^3r \\exp\\left( -i\\mathbf k \\cdot \\mathbf r \\right) \\mathbf J\\left( \\mathbf r \\right) = q_2 \\mathbf v_2 \\exp\\left( -i\\mathbf k \\cdot \\mathbf r_2 \\right).", "c6e55b64fdd0b52b388a95416a602070": "\\Gamma(t) = \\int_A \\boldsymbol{\\nabla} \\times (\\boldsymbol{u} + \\boldsymbol{\\Omega} \\times \\boldsymbol{r}) \\cdot \\boldsymbol{n} \\, \\mathrm{d}S = \\int_A (\\boldsymbol{\\nabla} \\times \\boldsymbol{u} + 2 \\boldsymbol{\\Omega}) \\cdot \\boldsymbol{n} \\, \\mathrm{d}S", "c6e57dcf300a4817343c9a1444591c86": "\\rho_2 = \\gamma_2 / \\gamma_0 = \\frac{\\varphi_1^2 - \\varphi_2^2 + \\varphi_2}{1-\\varphi_2}", "c6e5b06f2d7d8b3a923a4e6aee640f98": "MD(p) = 0", "c6e5d763a77a2341570903e147f22a22": "p(X|\\mu,I)=f(X-\\mu)", "c6e5f17cca6d862c174cbe53a69ef210": "\\begin{align}\nY' &= & 0.299 \\cdot R' &+& 0.587 \\cdot G' &+& 0.114 \\cdot B'\\\\\nP_B &= -& 0.168736 \\cdot R' &-& 0.331264 \\cdot G' &+& 0.5 \\cdot B'\\\\\nP_R &= & 0.5 \\cdot R' &-& 0.418688 \\cdot G' &-& 0.081312 \\cdot B'\n\\end{align}", "c6e6060b3fe320bcf37e8f045696bb23": "\n\\gamma_n(t)=i\\int_0^t dt'\\,\\langle n(\\mathbf R(t'))|{d\\over dt'}|n(\\mathbf R(t'))\\rangle=i\\int_{\\mathbf R(0)}^{\\mathbf R(t)} d\\mathbf R\\,\\langle n(\\mathbf R)|\\nabla_{\\mathbf R}|n(\\mathbf R)\\rangle,\n", "c6e6276a3c8c4ee915a6fa6ac152ffc2": "\\mathbf{p} = \\hat{\\mathbf{e}}\\tan\\left(\\frac{\\theta}{4}\\right)", "c6e632cd74dc527d44b68af3b7811db4": "1,0,1", "c6e66cca65806cf949649844a0d71b9d": " \\vec a_3 ", "c6e66fc8ec230507f4062eb0f2fbf9fa": "\\mathcal{L} = \\, R \\sqrt{-g}", "c6e6b8d0de4f59f808a33ff49fbb93ed": "\\left|\\frac{x - a}{r_a}\\right|^n\\! + \\left|\\frac{y - b}{r_b}\\right|^n\\! = 1,\\,", "c6e6cbdd47f27aeffeb4b4a6d376e0df": "ax^2+bx+c=0,\\,", "c6e6f256af8ea4ad93b2e88f3c55b761": "dx\\,\\!", "c6e6fad9985863b38890b67f27c9b364": "=\\operatorname{st}\\left(\\frac{x^2 + 2x \\cdot dx + dx^2 -x^2}{dx}\\right)", "c6e713a06924729fe3fdef7e7f02770f": " 2R_c L_s = 60,000 \\,", "c6e7cb970ebd3829412e10a9b34b3aac": " c(\\psi) < \\infty", "c6e80b7b54ca49de6ac327115f5c1892": " {\\omega = {Dp \\over Dt}} ", "c6e83c9b479cee375d413c9d20393eaf": "\\frac{NK}{2}", "c6e84cf14595473c4f0eb028d9fc4011": "1 \\le g^{(2)}(0) \\le \\infty", "c6e8c472e34719fbcf2b40bdb221bc34": "F\\ \\hat{z}\\,", "c6e947857dfa93e4362fc05ba4b585bb": "\n\\omega(q) = \\sum_{n\\ge 0} {q^{2n(n+1)}\\over (q;q^2)^2_n} \n", "c6e949b1db45569b59feabfa6aa7b76f": "p_\\text{e}", "c6e9ba7b32d19f684ce58b0d879fc33a": "a,b\\in Y", "c6ea2d7e73d8f41fbed9480103fd0fac": "f\\star g = f*g.", "c6ea2eb2e4445927e8875f2fba96f903": "C_V\n= \\left ( {\\partial Q_{rev} \\over \\partial T} \\right )_V\n= \\left ( {\\partial U \\over \\partial T} \\right )_V\n= T \\left ( {\\partial S \\over \\partial T} \\right )_V ", "c6eaa79d1fa1c2f1979cca0ad5c48c48": "a_1,a_2,a_3,\\ldots", "c6eaadbfbcd09bd3c9312629bb4624ae": "P = (x ,\\, y)", "c6eb1aebd31a8995ee3f77011420f0b6": "\\rho\\, g\\, a\\, \\cos\\, \\theta\\,", "c6eb23ceea6c0c67e9a2d61b839661a2": "y\\not \\succ x", "c6ebbcea29ab1bcbb29adc4463633bd6": " C_k = C_ig_j ", "c6ec01f52b60b69c7573c7025e44d13e": " \\mathit{WER} = \\frac{S+D+I}{S+D+C} ", "c6ec5a7760342e5535de84181e00c64b": "\\log Z", "c6ec9add8402796db01b154afcf7c5c3": "\\textstyle\\{(\\vec{x}_1, y_1), \\dots, (\\vec{x}_n, y_n)\\}", "c6ecb234ec461a5ee47d8f71beca52e1": "n_in_i=1\\,\\!", "c6ecc6e58c39bd2ecdb4fc5ccc3e5ea9": "\\mathbb{C}^n\\times \\mathbb{H}_n.", "c6ecf3002264a5c2874332d0c95eac60": " f( s_1 \\sqcup t_1, \\ldots, s_n \\sqcup t_n ) ", "c6ecf64d35db53d2497d4b414fb34da0": "Y_{6}^{4}(\\theta,\\varphi)={3\\over 32}\\sqrt{91\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(11\\cos^{2}\\theta-1)", "c6ed7a848e46b0aa598480661273e633": "disc(\\mathcal{H}) \\leq \\sqrt{2n \\ln (2m)}.", "c6ed92e3b364f77645c5e86ce60086ea": "\n\\mathbf{F}_{\\mathrm{fict}} = \n- 2 m \\boldsymbol\\Omega \\times \\mathbf{v}_\\mathrm{B} - m \\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{x}_\\mathrm{B} ) - m \\frac{d \\boldsymbol\\Omega}{dt} \\times \\mathbf{x}_\\mathrm{B}.\n", "c6ee7f694733892d234b4b30506ec5a2": "e^{ix} = \\cos(x) + i\\sin(x) \\,", "c6eeaa52e3cee950bbf6b1a962f06f41": "|\\Psi \\rangle \\,=\\,\\sum_i a_i\\, | \\psi_i \\rangle. ", "c6eec52e28919594732c4a7aadc48a39": "D=E=0", "c6ef4edf929bdc31bafdf259a9952f67": "\n{\\rm E}[z]\\,\\,\\, = \\,\\,\\,\\int {z\\,\\,{\\rm PDF}_z } \\,\\,dz\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm Var}[z]\\,\\, = \\,\\,\\int {\\left( {z - {\\rm E}[z]} \\right)^2 \\,\\,{\\rm PDF}_z } \\,\\,dz", "c6ef6bccda45c047dda53007c9fd9c45": "\\mathcal{Z}(R_R)=J(R)\\,", "c6ef91942a57604e7024c68693db8361": "\\varphi=(x^1,\\dots,x^n)\\,", "c6efead2e4c75d9f6c066ee3d2fecf7b": " \\pi_n\\,\\!", "c6f0262955e6e5a96f12945f61d65ae8": "{L_k}^{(l+{1\\over 2})}(2\\nu r^2)", "c6f056de7bbaa064752b38ca00c37f21": "-1.4", "c6f057b86584942e415435ffb1fa93d4": "000", "c6f06d8295562d74b0be49fcf74d2b5b": "x=a \\sinh v \\cos\\theta ", "c6f07b6af1a47190cbbe62ef3dadc3d4": "\\langle \\Psi , \\Psi \\rangle = \\int\\limits_{\\mathrm{ all \\, space}} d^3\\mathbf{r} \\, \\left | \\Psi(\\mathbf{r},t)\\right |^2 = 1,", "c6f0a01b655b0681a2a03a8a01f723f3": "f^\\leftarrow:\\mathcal{P}(Y)\\rightarrow\\mathcal{P}(X)", "c6f0d02554b2a5fe7abd9a25c95752d5": " \\mathrm{d}U = T\\mathrm{d}S - p\\mathrm{d}V + (H_m-TS_m) \\mathrm{d}n.", "c6f118c5da82b18adb46c153cb7ec46c": "1 \\to V \\to V \\rtimes \\operatorname{GL}(V) \\to \\operatorname{GL}(V) \\to 1", "c6f13766ad76a708271b5a891abf8c0d": "\\nabla\\times\\left(\\psi\\mathbf{z}\\right) = \\psi\\nabla\\times\\mathbf{z}+\\nabla\\psi\\times\\mathbf{z} = \\nabla\\psi\\times\\mathbf{z} = \\mathbf{z}\\times\\nabla\\psi'.", "c6f17ac11ec129fff8f5b13d741441bb": " \\textrm{Spec} (\\overline{K}) ", "c6f1a0d3fe953857325e895525105939": "x_n=x(n\\Delta t)", "c6f1c1cc9e9d05b77cdd08cec9971dbc": "\\lambda_{st} ~:~ \\lambda_{22} - \\lambda_{21} \\neq \\lambda_{12} - \\lambda_{11}", "c6f1e7f52a584c5bf18db6a3d23ea9be": "A B C A^{-1} B^{-1} C^{-1} = 1", "c6f2612104fa04292e1fe423a94db13b": "\\overline {\\Delta v}_{t-16}\\,", "c6f279bad0ba2d324caad92466b39673": "F_\\Delta(x-1)=h_0x^{d+1}+h_1x^d+h_2x^{d-1}+...+h_dx+h_{d+1}", "c6f2f18bb797918a4987ab760e2ac6b6": "\\lambda(A) = 0", "c6f31ad5a88e64ab95f87ad97f61bfdf": "x_1 + x_2\\leq L", "c6f3447423bfe36f4b5231dda725a726": " Df=-f^{\\prime\\prime} + qf ", "c6f37e228091bf882c9884a3f0cd79ba": "\\frac{\\left|x\\right|}{-0} = -\\infty\\,\\!", "c6f3f6d15cd187c3008aec7e4d2cc386": "\\hat{v}", "c6f475a0dd862fd477cfaf9bb50f002d": "\n\\gamma = \\left|\\gamma\\right| e^{2i\\phi}\n", "c6f4cbdc7acc6572cc0a75a4e3638870": "z\\to {e^{i\\theta}}{\\frac{a+z}{1+a\\bar{z}}}", "c6f4dbcb325ce05dbf34526e1cb2c2d5": "\\theta_{n+1}", "c6f518e9e9693cf43f9343fce3872469": "{SS(\\mu_1,\\mu_2,\\dots,\\mu_K)}\\over{K \\times \\sigma^2},", "c6f532ebca03eeda2d8310c897a68d37": " \\sqrt{- g} \\approx 1 + \\tfrac12 h_{\\alpha \\beta} \\eta^{\\beta \\alpha} + \\tfrac18 h_{\\alpha \\beta} \\eta^{\\beta \\alpha} h_{\\gamma \\delta} \\eta^{\\delta \\gamma} - \\tfrac14 h_{\\alpha \\beta} \\eta^{\\beta \\gamma} h_{\\gamma \\delta} \\eta^{\\delta \\alpha} \\,.", "c6f5d649501d98060f7ee13a608b6cd3": " [( D_e - F_e ) + D_w + ( F_e - F_w )] \\phi_{P} = D_w \\phi_{W} + ( D_e - F_e ) \\phi_{E} ", "c6f5e082beb5131f77782b4795652e20": " \\wedge^2 F ", "c6f6782d5ba467ea6d62fd6b8af39ece": " \\mathbf{Q}^{te} = -\\int_{S^e} \\mathbf{N}^T \\mathbf{T}^e \\, dS^e \\qquad \\mathrm{(18a)}", "c6f67ccfd7cb2bbf07f0c547e75c84a1": "\\mu_n'= \\begin{cases} \\infty & \\alpha\\le n, \\\\ \\frac{\\alpha x_\\mathrm{m}^n}{\\alpha-n} & \\alpha>n. \\end{cases}", "c6f690a3f0e4c6c3dab289ab3094b245": " z = r \\sin \\theta \\sin \\phi \\,", "c6f6e7a8d7877df6c55626a1dfe7eab6": "\\Gamma_Y : \\mathbf{Ab}(Y) \\to \\mathbf {Ab}.", "c6f720523bb28f53af9358c85580ee50": "\\leq{|A\\cap(g^{-1}F_i\\,\\triangle\\,F_i)|\\over|F_i|}\\to0", "c6f740aeb9fc106c7ca581f4ddde21c2": "B(s)=8s^3+24s^1.\\,", "c6f7684237c3010b78aece2c89a20487": "\\mathbf{s}_k^{\\mathrm{T}} \\mathbf{y}_k", "c6f76c1b4fc022cd80a6cc9bda87b7ee": "r=\\sqrt[4]{efgh}.", "c6f7ba6262e67b5c744bf724cf4a5ab6": "((1+iz/n)^n+(1-iz/n)^n)/2", "c6f7c2ec18b8cfa2b2144eb9f783edc2": "J_{G \\circ H} (p) = J_G (H(p)) \\cdot J_H (p).", "c6f7c47f4e3f789139b6352fb5f0d3c6": "\\mathbf{v} \\left(\\frac{\\partial \\rho}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\rho + \\rho \\nabla \\cdot \\mathbf{v}\\right) + \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = \\mathbf{b}", "c6f7e4772f108b4527a2f227dd28f1b4": "s \\rightarrow_R t", "c6f8113ba7533d8284eefb2a349efbf2": " Z(C_n) = \\frac{1}{n} \\sum_{d|n} \\varphi(d) a_d^{n/d}.", "c6f86fe1f5fffbb0ad9d7e2901a7f293": "dr = dr_1 \\left(1-\\frac{(Gm)^2}{4c^4 r_1^2}\\right)", "c6f8d9a360a4c49638836c7ca62fcd95": "T_1 V_1^{\\gamma - 1} = T_2 V_2^{\\gamma - 1} \\,\\!", "c6f94f72fd0fac894e09d193dc54bafd": "u_2 - u_4 = 0", "c6f9c9df9561a2a3235ffc9acbee2c77": "(-1)^{\\ell(w)}", "c6fa1ea004907243445f63ae6804f3e5": "1/d_\\mathrm{o}", "c6fa343129962dd7949182eab6eea321": "F = A + BC\\overline{D}", "c6fa37143d4c61ca1e43ee26cb7a30a4": " \\pi = pq - (F_n + wq) \\qquad\\qquad (1) ", "c6fa560b151168e00bd5545a7e700ff3": "\\begin{align}f_X(x,1,\\lambda) &= \\frac{1}{2\\sqrt{x}}\\left( \\phi(\\sqrt{x}-\\sqrt{\\lambda}) + \\phi(\\sqrt{x}+\\sqrt{\\lambda}) \\right )\\\\ &= \\frac{1}{\\sqrt{2\\pi x}} e^{-(x+\\lambda)/2} \\cosh(\\sqrt{\\lambda x}),\\\\ \\end{align}", "c6fa66fdcc2bac52ebb12938659e0d58": "\\exp(-X) = (\\exp X)^{-1}.\\,", "c6fabe8c7a9ec60a9ff844d8d654e856": "\\left.n=|m|\\right)", "c6fae351ddafd75fa854febc9dbc880c": "\\scriptstyle \\text{Pad}_n^s=\\lbrace\\mathbf{\\xi}=(\\xi_1,\\xi_2)\\rbrace", "c6faf2ff3d3e2150a4255f8b0588d2e2": "\n\\mathrm{var}_{\\boldsymbol{\\theta}}\\left(T_m(X)\\right)\n=\n\\left[\\mathrm{cov}_{\\boldsymbol{\\theta}}\\left(\\boldsymbol{T}(X)\\right)\\right]_{mm}\n\\geq\n\\left[I\\left(\\boldsymbol{\\theta}\\right)^{-1}\\right]_{mm}\n\\geq\n\\left(\\left[I\\left(\\boldsymbol{\\theta}\\right)\\right]_{mm}\\right)^{-1}.\n", "c6fb2821cff7518b24fe5a1417fe286d": "a, b \\in \\mathbb{Q}", "c6fb371f2a72544b25212fc82db92c07": "\\mathbf{E}_{\\mathbf{P}} ( | Y_{t} | ) < + \\infty;", "c6fb4a93aa078dfad44143ad4fd5cdc0": "\n\\begin{align}\nO = & \\{ \\{x_1, x_8, x_{10}, x_{11}\\},\\{x_1, x_9, x_{10}, x_{11}, x_{14}\\},\\\\\n& \\{x_2, x_7, x_{18}, x_{19}\\},\\\\\n& \\{x_3, x_{12}, x_{17}\\},\\\\\n& \\{x_4, x_{13}, x_{20}\\},\\{x_4, x_{18}\\},\\\\\n& \\{x_5, x_6, x_{15}, x_{16}\\},\\{x_5, x_6, x_{15}, x_{20}\\},\\\\\n& \\{x_6, x_{13}, x_{20}\\}\\}.\n\\end{align}\n", "c6fb78e27fff24575310e4c2aba9210f": "\\varepsilon (a^\\mu) = 0", "c6fb7d38c108db917ceb9447a33cf50a": "\\kappa<\\lambda", "c6fb884bc3d7714712657de63a0bd1a5": "y_{C_{1}}(x), \\ldots , y_{C_{k}}(x) ", "c6fc636d5eea1770341ad0d40d267e79": "= p(C) \\ p(F_1\\vert C) \\ p(F_2\\vert C, F_1) \\ p(F_3\\vert C, F_1, F_2) \\ p(F_4,\\dots,F_n\\vert C, F_1, F_2, F_3)", "c6fcd268c280f6f8ab09da08250cb619": " DF(T) = \\frac{1}{(1+r)^T} ", "c6fd156889994bb5c8a464d015a7eb05": " \\nabla \\mathbf{v} = \\begin{pmatrix} 0 & 0 & 0 \\\\ {\\dot \\gamma} & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} ", "c6fd67a5300e2c6a0159b63a798e2133": "\\Delta M_i^{-1} = - \\alpha \\sum_{n=1}^N D_i \\left[ n \\right] \\left[ \\sum_{j \\in C \\left[ i \\right]}^{} F_{ji} \\left[ n -1 \\right] + Fext_i \\left[ n^{-1} \\right] \\right]", "c6fd87e98488f927f4445f7e9083794b": " \\frac{\\partial\\psi}{\\partial t} = i\\left(\\omega_1\\sigma_x \\cos{\\omega_r t} + \\omega_1\\sigma_y \\sin{\\omega_r t} + \\omega_0 \\sigma_z\\right)\\psi.", "c6fd8ec65ece9eee4853df2389b70bd4": " E(x;q) = \\max_{(a,q) = 1} \\left|\\pi(x;q,a) - \\frac{\\pi(x)}{\\varphi(q)}\\right|", "c6fd8efc81e1cefd9f9aa30e101bfaf6": "\\ y(t)", "c6fdcb44015d3d4147ad99e97d256fbe": "B=\\gamma. \\,", "c6fddc80998696edda7bb1f7794531c2": "f(x), g(y)", "c6fe0629a1e1f9b3bad7412137551c2a": "\\left(1 - 0, 1 - {9 \\over 10}, 1 - {99 \\over 100}, \\dots\\right)\n= \\left(1, {1 \\over 10}, {1 \\over 100}, \\dots \\right)", "c6fe5e51f6c036f7db73487b2487d842": "M_\\mathrm{L} = -2.53 + 2.85 \\log_{10} (F-P) + 0.0014 \\Delta^{\\circ} ", "c6fe848d754ec72aeb89e6759c1925ec": "\\phi (x) = \\frac{1}{x} \\,,", "c6ff8255c1421b163e763f1112451006": "x_{t+1} - x_t = - u_t\\!", "c6ffaabde1d68ccda98ba0572de922eb": "Pmf = \\cfrac{Pmo + \\tfrac{1}{3}Pmf + 2 Pmf}{3}", "c700ab51d81eca826e1c895c0da9f566": "S(X,Y)=\\tfrac12\\left(\\Delta(X,Y)+\\Delta(Y,X)\\right)", "c700b3a40333b98af1896f7917cc1453": "v/u", "c7012277df591ed267384b5961290d6f": " \\mathrm{Re} = 10^6", "c701542de1b1337766a5b8b337abee9a": "c(s + t) = c(s) c(t)\\ ", "c70161e7befd21e604e10cedad856236": " s_{\\lambda} ", "c70174313f10110d348ddaf9432f0840": "\n L_0 = -\\rho\\, \\left\\{\n \\zeta\\, \\frac{\\partial\\varphi}{\\partial t}\\, \n +\\, \\frac12\\, F\\, \\left[\n \\left( \\frac{\\partial\\varphi}{\\partial{x}} \\right)^2\\, \n +\\, \\left( \\frac{\\partial\\varphi}{\\partial{y}} \\right)^2\n \\right]\\, \n +\\, \\frac12\\, G\\, \\varphi^2\\,\n +\\, \\frac12\\, g\\, \\zeta^2\\, \n \\right\\},\n", "c7018bc8a7485286569997dac529fd21": "1/t > 0", "c70194b3562fefcfd71fc13049e4d08d": " f'(x)= \\operatorname{st} \\left(\\frac{{^*f}(x+h) - {^*f}(x)}{h}\\right) ", "c7020875bb74f8fd0cd405fb73cc2220": " \\mathbf{B} ", "c7022df708b01ce80521098936758d9f": "\\delta=\\pi/2", "c70264b5aa5176710e5fab00ad71349d": "\ng \\sin \\theta = - k^2 s \\,\n", "c702adda38a1f2c3b1aff466a50cdf6a": "\\Omega G", "c7031cb2c15da51787de90548f6e7582": " \\langle f \\rangle_\\beta = \\sum_\\sigma f(\\sigma) P_\\beta(\\sigma) \\,", "c7039a42b01b655fef279829894c5158": "f(t) = F'(t) = \\frac{d}{dt} F(t).", "c704086a723d3b356133162bc043598b": "u_1 = \\left[\\begin{array}{c}1/2\\\\ \\sqrt{3}/2\\end{array}\\right] ", "c70431c0f4dc9457f4ef6ad36ca17d0e": "b(\\mu)", "c7047d45c5dea6458d593948d0b51b23": "\\ I_4 \\,", "c704bfd078e65be0a659fbacbd4c5b77": "~C~", "c705597138ee82525de4e5fc0791d50d": "s_1\\subseteq s_2", "c7057b841d0e9e7b5d9554939eed5c44": "\\int_{T_\\text{load}}^{T_\\text{hold}} \\frac{1}{T} \\cdot{dT} = \\int_0^\\phi -\\mu \\cdot {d}\\varphi", "c7061db162720004a4edb8a68772b262": "\\phi = \\sum_{l=0}^\\infty \\sum_{m=-l}^l \\phi_{lm}(r) Y_{lm}(\\theta,\\phi)", "c7066ca9e2b4037b8633f33ff7142bac": "[M_1]_{a\\;\\|\\;u} \\;\\|\\;[N]_{\\overline{a}\\;\\|\\;b\\;\\|\\;v} \\rightarrow [[[M_1]_{u\\;\\|\\;x}]_b\\;\\|\\;N]_{v\\;\\|\\;y}", "c706877786dc8690c3eddb77aed7d07a": "Y | (X = k) \\sim \\mathrm{Binom}(k, p)", "c706d75cfffed5681a897f04405df71c": "r_s>R", "c706d9d5b3ec876a8349e3204ab59ee8": "f:[0,1]\\rightarrow X", "c70707015f3e92dd191a130dd8077c7c": "V_{Y0}t = h.", "c707381cd77b99473cd0e570c0da16d2": "T^4 = \\frac{3}{4}T_e^4\\left(\\int_0^z (\\alpha) dz + \\frac{2}{3}\\right)", "c707788d13419b056ae5daffd7abe386": "S\\left| k\\right\\rangle=\\left| k\\right\\rangle", "c7078d877ca6f63228dd20e44628e85c": "r_{\\text{out}} \\triangleq \\frac{v_{\\text{out}}}{i_{\\text{out}}}\\,", "c707e079c07c611b08916575ebd0e20a": "\\tilde{x}_0\\in \\tilde{M}", "c708055b6bd9cab3231221ea34963ceb": "z_1 = f_c(z_0) = c\\,", "c7082ec75c6bedc7b2734652c01c93db": "g \\colon A \\to B", "c7086cad2453babdfd2f5ffbaae54f96": "R=i\\Omega", "c7087b4108ee9b8694f51108a0269e71": "\n w(x) = w_b(x) + w_s(x)\n", "c708a652aa0b4040128a89bfc29c362d": " a_{t} \\in \\Gamma (x_t)", "c708ae846955ba5fa83329966804aac0": "V_d/V_t = \\frac{FaCO_2 - FeCO_2}{FaCO_2} \\times \\frac{Ptot}{Ptot} ", "c708afd197e7e2f19d771d8cba805e77": "\\sigma^2 =\\sum\\limits_{i=1}^n (1-p_i) p_i", "c708d2bf5d670e4d76267049223141ce": "\\begin{align}\n&&\\dot x\\,x+\\dot y\\,y&=0\\\\\n\\Rightarrow&& u\\,x+v\\,y&=0,\n\\end{align}", "c708d76be7118fd52a42b5539de74ba3": " E = \\langle E \\rangle_c", "c708f2d45b00fb9d56d14ee9bcb6f385": "\\sigma_\\mathrm{Total}", "c709197f62885a338d8fb804676e2c9c": "F(f)(x_1)=x_2", "c70919935dd2f8bc37dc10736ec0ce2d": "\\frac{dr}{dt} = \\mathbf{\\omega}(t)\\times r(t) = [\\mathbf{\\omega}]_\\times r(t)", "c7091c5f185d011e8ca07541c25d8e53": "C_n(x)=2T_n \\left(\\frac{x}{2}\\right)", "c7092db0c34e6f50b691ffef0bd77fcf": "\\mathbf{U}\\,", "c70934e9808cde463c661d1850111df9": "M_X^{gi} = \\overrightarrow{XG} \\times mg - \\overrightarrow{XG} \\times ma_G - \\dot {H}_G", "c7093fc0e5babfa0c678f5eab25fe14b": " k X \\sim \\mbox{Inv-Gamma}(\\alpha, k \\beta) \\,", "c7094ced6cba6c05dd88a6ada5f17b4f": "X_L = \\omega L = 2\\pi f L\\quad", "c7094fc159682629af25e1e6b60f10fd": "\\psi(\\Omega^{\\Omega^2 \\omega^2 7})", "c70a95656278b5aedc28c778b42e4476": "\\mathbf{e}_{12}", "c70ae0806541c82628ea241f638e6204": "\\mathbb{R}^{24}", "c70b10d374083456353529ee6eee0415": "S = \\frac{r^2}{2 \\times \\text{ROC}}", "c70b26ab544bf8a442e40cab7f577ee9": "\n\\mathbf{e} = \n\\frac{1}{mk} \\left(\\mathbf{p} \\times \\mathbf{L} \\right) - \\mathbf{\\hat{r}} = \n\\frac{m}{k} \\left(\\mathbf{v} \\times \\left( \\mathbf{r} \\times \\mathbf{v} \\right) \\right) - \\mathbf{\\hat{r}}\n", "c70b9ef55b5ec453c1d7163df823e094": "f(x) \\leq", "c70bac8e9f4c5497ea5fe646b90bfdc3": "5_H \\bar{5}_H", "c70bdb5b1e2ece3001c64214d21dd382": "\\zeta = \\langle 0|(1 + \\frac{i}{\\hbar}\\tau\\bar{H})(1 - {i \\over \\hbar}\\tau\\bar{H})|0\\rangle - \\langle 0|(1 + {i \\over \\hbar}\\tau\\bar{H})|0\\rangle\\langle 0|(1 - {i \\over \\hbar}\\tau\\bar{H})|0\\rangle", "c70c009a2105a41629e1fe543e3b5938": "\\exp(d_i \\tau D_V)", "c70c11157008df1305277c0e70fecec8": "\n \\Beta(x,y) =\n \\sum_{n=0}^\\infty \\dfrac{{n-y \\choose n}} {x+n},\n\\!", "c70cbb8e9c7b2192441e7a42ef8ab2a4": "r^2 = +1.", "c70d43c28a388bec410d11a24cf09fe6": "\\mathbf{W}^T \\mathbf{Q} \\mathbf{W} \\propto \\mathbf{W}^T \\mathbf{W} \\, \\mathbf{\\Lambda} \\, \\mathbf{W}^T \\mathbf{W}\n= \\mathbf{\\Lambda} ", "c70d755520a124a46f7d7f6a0e0b1bf2": "h_{max}", "c70ddb95e66fbfb20be8184f9b929026": "\\mathrm{End}(S)", "c70e085185da1e1981fb49771be1c1b2": "\nc_{0,0} = 1 >0, c_{1,0}=- \\mu = 0, c_{0,1}=-1 <0. \n", "c70e2db0f5966b2119f7f87b9b58f5f6": "z' = z' \\,,\\quad p_z' = p_z ", "c70e596017121b2edc412f0d61c9d209": "\\theta = \\int_{t_0}^{t} \\dot{\\theta}\\, dt", "c70ef57006a75cce7868a435d94c8e5d": "\\forall n [0=n \\lor \\exists m [Sm=n] ].", "c70f0b0271ad98639ed0093744f014be": "P = \\exp(+ \\beta Q) ", "c70f267ce1ce97551a045ec976a53cc9": "\\sum_{k=0}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}(x)_k=x^n.", "c70f508ebdd0c608c9509236a222f0b2": " \\left[\\begin{array}{c}A\\\\\\hline I\\end{array}\\right]=\n\\left[\\begin{array}{cccccc}\n1 & 0 & -3 & 0 & 2 & -8 \\\\\n0 & 1 & 5 & 0 & -1 & 4 \\\\\n0 & 0 & 0 & 1 & 7 & -9 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \n\\end{array}\\right]. ", "c70f73112929a3cef9d2c602bbd17c8e": "\\sum d^2", "c70f853164b6c84ad5d782226b151b7c": "M \\times M \\rightarrow M", "c70f899614afdb7477cdd7ee7cbc995b": "\\lambda >> a", "c70f8d478d3fbc3546e2341172c2a56f": "\\delta Q = T dS~", "c70f98c797017ee77d8eb42895b0452c": "\nH=\\int_{V}\\mathbf{u}\\cdot\\left(\\nabla\\times\\mathbf{u}\\right)\\,dV \\;.\n", "c70fa28cea75da56977c88a88842805a": "0x^3 + 0x^2 + 1x^1 + 1x^0", "c71007a9b4a3e58612f8d568cab682c3": "\\Pr(\\varepsilon_0=x) = \\Pr(\\varepsilon_1=x) = e^{-x} e^{-e^{-x}}", "c71015a317df40836440a5546cf202e2": "\\mathrm{MMH}^*_{32}", "c7105f19d6e51e880aae136c1050a1eb": "X_1,X_2,\\ldots,", "c71099ce3aa5d1b84b277a55b5a13468": "\\Pr(A)=\\sum_n \\Pr(A\\mid X=x_n)\\Pr(X=x_n) = \\operatorname{E}[\\Pr(A\\mid X)] ,", "c710d9140131b3865234e0b9207b6743": " \\frac{d}{dt} P = \\sqrt{ \\dot X \\cdot \\dot X} \\nabla n, \\,", "c710ee182b60b48dda4aae750af9acf3": " \\cos(k a) = \\cos(\\alpha a)-P{\\sin(\\alpha a) \\over \\alpha a} \\qquad \\left( P={m V_0 ba\\over \\hbar^2} \\right). \\,\\! ", "c71123db9111430764b3f4f45257f1b0": "\\lceil (n-1)/\\mathrm{diam}(G) \\rceil \\le \\varphi(G) \\le n - \\mathrm{diam}(G)", "c7116412b089cf15001dc324f095daac": "E_{POT}", "c711ce277ecf8cd98a3055ea292122ac": " \\frac{d \\mathbf{P}_\\text{mech}}{d t} = \\int_V \\left(\\rho\\mathbf{E} + \\mathbf{J}\\times\\mathbf{B}\\right) dV\\,.", "c71224710c57e0572f20fef1028aee1d": "\\csc^2(A/2)\\,:\\,\\csc^2(B/2)\\,:\\,\\csc^2(C/2)", "c7122af63089e117966447d390ae5711": "Fm_{uc} = \\frac {R'_s} {R_{modern}}", "c71247020531baefa8112dd1e6c3bac5": "V_{gt}", "c7125ef180245ef30881dc98a6f0f434": "C_a (x) = 0", "c7127540b4a7e0df70ffc7cd863bc46f": "\n\\sqrt{n}\\left(h(B)-h(\\beta)\\right)\\,\\xrightarrow{D}\\,N\\left(0, \\nabla h(\\beta)^T \\cdot \\Sigma \\cdot \\nabla h(\\beta) \\right)\n", "c71278dee346822a62c3374c30821c5f": "44^2", "c712af36c0813298d6b903858f57a703": "E(R_i) - R_f = \\beta_{i}(E(R_m) - R_f)\\,", "c713019e84b076a6e2bae791c9975a09": " g(x) = L_k^{(l+\\frac{1}{2})}(x).", "c7133a20f5d1d0bcdcc54de86314fd2b": "y'=f(y)", "c713590d425a716ba78d509a1fc81ba0": "({n+1})^{k_{n+1}}", "c713730471859783de43ca979b38ba08": "\\text{expi}(z) = e^{i z} = \\cos(z) + i \\sin(z), \\, ", "c713a5ed35847d33b159d8bc2c6e8042": "\\gamma : [0,1] \\rightarrow M", "c713c07c4f03073c7fc1adba139ae639": "\\eta_{th} = 1 - \\frac{T_{heat sink}}{T_{heat source}}", "c7141c2788f8d74686dfe5a211b3e359": "\\rho=\\lambda/\\mu", "c7143168fd4f86dfd1a7bb7843226b6f": " a = r \\alpha.", "c71454b834d6c6c23c65a9a9e2e0ccd4": "V[n_0] = \\int V(\\vec r) n_0(\\vec r){\\rm d}^3r. ", "c714a1d9cdcd7205a995f1250e98b60f": "\\Delta = \\frac{1}{256}(abc)^{2p}", "c714b467f91d14cf30bf304cb2479b38": "P=Q", "c714c520c50fee3d05f4ef20d2aca89d": "\nd\\varepsilon_x=\\frac{dx}{x}\\qquad d\\varepsilon_y=\\frac{dy}{y}\\qquad d\\varepsilon_z=\\frac{dz}{z}.\n", "c714e12796ea52b406449dadd399f818": "\\cot{ \\frac{A}{2 }} = \\frac{s-a}{\\zeta }", "c714e93e61d957d28c18fe670b5eba49": "n_B(\\xi)=\\frac{1}{2}\\left(\\mathrm{coth}\\frac{\\beta\\xi}{2}-1\\right)", "c714ea80c9f5dc689cbac1d7768cea63": "(1-x^2)^{1/2}\\,", "c714f5c529a426f04cc0be0ab8937c19": "\\frac{1}{24} + \\frac{1}{48} = \\frac{1}{16}", "c714fb5cd6ff762c25f34264cb9eb5db": " e_n (X_1, X_2, \\dots,X_n) = X_1 X_2 \\ldots X_n", "c715275da5da3c5f4f232e92df649566": "E = hc\\tilde{\\nu}", "c7158b7c556ed28bf8070510916e6065": "r_B \\colon B \\to A_B", "c71600edf15dc009e18c7abf436a2967": "w = W\\left( \\frac{I_SR}{nV_T} e^{(V_s+I_sR)/(nV_T)}\\right) ", "c7160143bf94d39cd55724e536819d26": "d_Y(f(x),f(x'))/d_X(x,x')", "c7160c6914385c519797b607401f2fd0": "\\pi_b,\\pi_a", "c7160d3c8f4a339ee057e588fa685182": "m-n", "c71696cdcb02c91763cf9e9b7039dc2e": "\\frac{\\zeta_K(s)}{\\zeta_{\\mathbf{Q}}(s)}", "c71696d979cdd21a1bb4dd8133f0fa20": "\\pi_{\\neg x}(o)", "c716c32ae39c201088cf5b3b5a853982": "\\{\\ell_i(\\cdot),\\,i=1,\\dots,n\\}", "c716c38ef1f00d924f930b5c38e70d1a": "\\operatorname{Dom}(T)", "c7174f0055546acc61bc2f98a1445a2f": "a+b=b", "c71756f74454994a2e77aac82846edea": "\\sum_{n=1}^\\infty \\frac{(-1)^{n}}{(n+2)^a}\\sum_{n=1}^\\infty \\frac{\\bar{H}_n^{(c)}(-1)^{(n+1)}}{(n+1)^b} =\\zeta(\\bar{a},\\bar{b},\\bar{c})", "c71777fcf7908e9bc7648eb0e421223a": "r_a=\\langle\\hat{f}| \\hat{b}\\rangle-\\langle\\hat{e}| \\hat{c}\\rangle", "c717bd734cb8d823a5f7c911eb556199": "G(A,B) = A \\cdot B", "c717c7fbd7b2b122fe65c0667743dc5d": "\\mathbf{N} ", "c717d4feb21d08d35f909ded6ef17da8": "\\text{barrels of crude oil per metric ton} = 1\\div\\bigg[\\frac{141.5}{\\text{API gravity}+131.5} \\times 0.159\\bigg]", "c717d7f5ac15d79f540e8257aae8315e": " \\Psi^{\\dagger}(\\bold{r})={ 1\\over \\sqrt{V}} \\sum_{\\bold{k}} e^{-i\\bold{k\\cdot r}}{a^{\\dagger}}_{\\bold{k}}", "c717e5d5e710812b231dd0f9a698622b": "e_i^k.v = f_i^k.v = 0", "c717f09a6d87b358e478c7a4a5a5eff7": "\\mathfrak{P}_m(\\mathfrak{C}(\\mathcal{Z}))", "c718180e705efa2f0db22e625da7efd0": "L_e", "c718873c35b9b97508bdae6fecad3b89": "\nf_{t1} = C_{t1} g_t \\exp\\left( -C_{t2} \\frac{\\omega_t^2}{\\Delta U^2} [ d^2 + g^2_t d^2_t] \\right)\n", "c718a1a8fe3ec8158af6664f3ffecab0": "\\mathrm{SL}(2n, \\mathbb C)", "c7192ccb0a3fd290f7a1fb5d4dd55994": "\\sharp := \\flat^{-1} : \\Gamma(T^*M) \\to \\Gamma(TM)", "c719d770487d7b22453f75bf6320f684": "\\Delta \\phi = \\phi - 2.\\ ", "c719f69721c879df4f5fda67555acf14": " \\sum_{I \\subseteq J} (-1)^{|J| - |I|} |A_J|.", "c71a3bcc07cf5b1c6ae64947b08b3e94": "\n f(k; r, p) = (1-p) \\cdot p^k \\!\n ", "c71ae0583b052ce34968e60bf883e0ac": "\\Gamma(\\omega)", "c71b3155ae35b78cb8616acf5023afb8": "{\\Delta}_{\\rho}", "c71b4da2a4608ae9c95fe8d2b73b794b": "\\phi : (M,x) \\to (M,x)", "c71b8118fca4ba7aafd6f6f9b0d23eef": "\\pi / 60", "c71bb3cd5093790c2f0d5cbccb7a2835": "ds^2=(\\hat{h}_{AB}G^A G^B-F)r^2 dv^2+2dvdr- \\hat{h}_{AB}G^B r dv dy^A -\\hat{h}_{AB}G^Ar dv dy^B+\\hat{h}_{AB} dy^A dy^B", "c71be704f957eb016ceca8aec43a70f1": "(4)~~ ~~ \\frac{\\mathrm{d}u}{\\mathrm{d}t}=\\frac{-u}{1+u} ", "c71c2b285a732cb86290de442905d05e": "\\mathbf{c}^T\\mathbf{x}", "c71c441c32153c2dfa983508efe596b7": "x_2 = x_1 - f(x_1)\\frac{x_1-x_0}{f(x_1)-f(x_0)}", "c71c8412274b7485826f5f6e5d58e1c9": "\\begin{array}{lrl}\n P & := & x_i := x_j + c \\\\\n & | & x_i := x_j - c \\\\\n & | & P;P \\\\\n & | & \\mathrm{LOOP} \\, x_i \\, \\mathrm{DO} \\, P \\, \\mathrm{END}\n\\end{array}\n", "c71d07756b7587cfb8de34820e98f3b1": "\n\\begin{bmatrix} D_1 \\\\ D_2 \\\\ D_3 \\end{bmatrix}\n=\n\\begin{bmatrix} 0 & 0 & 0 & 0 & d_{15} & 0 \\\\\n0 & 0 & 0 & d_{24} & 0 & 0 \\\\\nd_{31} & d_{32} & d_{33} & 0 & 0 & 0 \\end{bmatrix}\n\\begin{bmatrix} T_1 \\\\ T_2 \\\\ T_3 \\\\ T_4 \\\\ T_5 \\\\ T_6 \\end{bmatrix}\n+\n\\begin{bmatrix} {\\varepsilon}_{11} & 0 & 0 \\\\\n0 & {\\varepsilon}_{22} & 0 \\\\\n0 & 0 & {\\varepsilon}_{33} \\end{bmatrix}\n\\begin{bmatrix} E_1 \\\\ E_2 \\\\ E_3 \\end{bmatrix}\n", "c71d70af84123671d1ee84704fcb585a": " m + s(-\\log(X))^{-1/\\alpha} \\sim \\textrm{Frechet}(\\alpha,s,m)\\,", "c71dad244c42ca28d1c9d078b897c0dc": "{\\ddot{a}} = {\\mathrm{d} ^2 a \\over \\mathrm{d} t^2}", "c71dca7ae92c3cd93426de3a3464164f": "12\\mathbb Z\\,", "c71df0ff6de5c5d56217f6f903a58b1e": " e ^ { - \\beta \\Delta F} = \\frac{\\left\\langle M\\left(\\beta (U_\\text{B} - U_\\text{A})\\right) \\right\\rangle_\\text{A}}{\\left\\langle M\\left(\\beta (U_\\text{A} - U_\\text{B})\\right) \\right\\rangle_\\text{B}} ", "c71e0bcb68e782b1d4eff7bc97a9683f": "S(u)", "c71e28852c33608526fcd752301b6b23": "\\Theta _{cw} [K]", "c71e80dd81ee6b34258ad95f9800e1a0": "\\mathbf{X} = X^i~\\boldsymbol{E}_i", "c71ea806edd7500ab6c7f08d503a99ae": "\\{\\left|\\phi_{i}\\right\\rangle\\}", "c71ed9fbb51ee0d53fb44ca226331c2f": "\\displaystyle{(ST_gS^{-1} x, ST_gS^{-1} y)=(x,y).}", "c71f05dee47a2b830c4cb71ade423bfa": "\\Delta(u_{ij}) = \\sum_k u_{ik} \\otimes u_{kj}", "c71f451cb8a40e7982cc5fbc16c0e7ca": "\\tau_c*=f \\left(\\mathrm{Re}_p* \\right)", "c71f734df4b67d0735058ce6587bf7f5": "f=\\sum_{\\{x\\mid f(x)\\ne 0\\}} f(x) e_x", "c71fa3667100e91ec9986b796865aa26": "A_{x} = \\int_a^b 2 \\pi y \\, \\sqrt{ \\left( \\frac{dx}{dt} \\right)^2 + \\left( \\frac{dy}{dt} \\right)^2} \\, dt", "c71fe42f4cf75cd7d80a0be3fdd03b16": "\\sigma_1 \\sigma_2 +(\\sigma_h - \\sigma^0_h )=(e^2/(2h)^2 ", "c71feb4cac44d082850f4537a6ac715a": "H^i(j_x^* IC_p) ", "c720077756b0aed583bb3dc36449274d": "Q = 2\\pi \\times \\frac{\\text{Energy stored}}{\\text{Energy lost per cycle}}.", "c7206a35705f4d71e5be402b30f9672c": "L_z' = x' p_y' - y' p_x' = \\gamma(V) [ (xp_y - y p_x) + V(y E/c^2 - p_y t) ] = \\gamma(V) [L_z + V (m y - p_y t) ] ", "c72075a545c53ef26df07a9aaebc01c1": "\\mathrm{bind}: \\mathrm{M} \\left( A^{?} \\right) \\rarr \\left( A \\rarr \\mathrm{M} \\left( B^{?} \\right) \\right) \\rarr \\mathrm{M} \\left( B^{?} \\right) = m \\mapsto f \\mapsto \\mathrm{bind} \\, m \\, \\left(a \\mapsto \\begin{cases} \\mbox{return Nothing} & \\mbox{if } a = \\mathrm{Nothing}\\\\ f \\, a' & \\mbox{if } a = \\mathrm{Just} \\, a' \\end{cases} \\right)", "c720c8f313d9856af5e854fc3d207df8": "\\beta \\rightarrow \\gamma", "c72106e586987900f023142f9c876d71": "d_n = 0.005~\\mathrm{inch} \\times 92 ^ \\frac{36-n}{39} = 0.127~\\mathrm{mm} \\times 92 ^ \\frac{36-n}{39}", "c7212c5a3bdbf45b7b0e00dddbd941a3": "1 \\leq k \\leq n", "c72184445b922be5b5e5e34c61a7bda1": " \\tfrac{\\partial \\rho}{\\partial t} = 0 ", "c721bc9e9c830dcc591b5bc92e40e325": "d\\operatorname{Ad} = \\operatorname{ad}.", "c721eea73741eca8251002cc3879c5cf": "D\\subseteq \\{z: |z-c_0|\\leq 2 c_1\\}\\,,", "c722b7bd913b9452f9992e324c053f18": "\\vec{r}=\\vec{r_1}-\\vec{r_2}", "c722c3507a38abc07417259b5199ee9a": "\n\\begin{matrix}\nA \\Rightarrow B & \\equiv & \\forall x:A . B & (x \\notin B) \\\\\nA \\wedge B & \\equiv & \\forall C:P . (A \\Rightarrow B \\Rightarrow C) \\Rightarrow C & \\\\\nA \\vee B & \\equiv & \\forall C:P . (A \\Rightarrow C) \\Rightarrow (B \\Rightarrow C) \\Rightarrow C & \\\\\n\\neg A & \\equiv & \\forall C:P . (A \\Rightarrow C) & \\\\\n\\exists x:A.B & \\equiv & \\forall C:P . (\\forall x:A.(B \\Rightarrow C)) \\Rightarrow C &\n\\end{matrix}\n", "c722c74d7904a33c93a8decee60507c8": "\\alpha= P(R_{NP}, \\theta_0)=P(R_A, \\theta_0) \\,.", "c722ff0c3d6c2cf0b4cfa8605807dcd6": "T_{raise} = \\frac{F d_m}{2} \\left( \\frac{l + \\pi \\mu d_m}{\\pi d_m - \\mu l} \\right) = \\frac{F d_m}{2} \\tan{\\left(\\phi + \\lambda\\right)}", "c72307c3d8d67c9440af570f8cd8068d": "{\\mathcal C}_n(z) = z^{-n/2} I_n(2 \\sqrt{z});", "c7234e80b38da741acb45611a138119b": " r(N) = \\int_0^1 S(\\alpha)^3 e(-\\alpha N)\\;d\\alpha", "c723b1db50454ebcebd96f9a8fa93a22": "1/(4 \\pi \\epsilon_0)", "c723d0f002bd53a0dc018bfd2057f44e": "v_i \\neq 0", "c724110e5a76b5f4044a11db711405f1": "\\frac{X}{\\log X}", "c72525c8d86170b526ce661a153d0d8d": "T[\\mu\\alpha.T/\\alpha]", "c725b4819dcb96a07dd00c74cc764995": " f(x) = \\begin{cases}\n \\sin(1/x) & \\mbox{if } x \\neq 0,\\\\\n 1 & \\mbox{if } x = 0,\n \\end{cases}", "c725b5519eb2dde5ec474513bd44798b": "x^{\\prime}=\\gamma x", "c72669797c350ad6c5c4215625be8985": " (2')\\quad x_1\\cdot (x_2 \\cdot v) - x_2\\cdot (x_1 \\cdot v) = [x_1,x_2] \\cdot v ", "c726d6c2d8c1ba9285587039b4ae0434": " dU = -\\mathbf{p}_{uc} \\cdot \\mathbf{dE}", "c72726eacc79317a9b7dcd09c0a4d368": "S(X) = \\sup (\\lambda : ((1-\\lambda)X_0, \\lambda X_1) \\prec X)", "c727ed0f4eca1fa240b03ef1e50d0487": "aI + bJ + cK \\, ", "c7280161908810a8f6bd39ec4ace4620": " \\Gamma^l{}_{jk} = \\tfrac{1}{2}\\sum_r g^{lr} \\left \\{\\partial _k g_{rj} + \\partial _j g_{rk} - \\partial _r g_{jk} \\right \\} ", "c72916495c8a417e69d67a46a90c28ef": "\\mathcal{F}^{-1}", "c72925a7baafe02022be24b7ba7dfaf3": "f(k;\\rho) = \\rho\\,\\mathrm{B}(k, \\rho+1), \\,", "c72979169b1bc00f037491bf588a3e2a": "\\mathbf{\\hat X} = \\mathbf{D}\\mathbf{X},", "c729b35e06dd5235ac28f9de08e2954e": "\n L = -\\rho\\, \\left\\{\n \\int_{-h(x,y)}^{\\zeta(x,y,t)}\n \\left[ \n \\frac{\\partial\\Phi}{\\partial t} \n +\\, \\frac{1}{2} \\left( \n \\left( \\frac{\\partial\\Phi}{\\partial x} \\right)^2\n + \\left( \\frac{\\partial\\Phi}{\\partial y} \\right)^2 \n + \\left( \\frac{\\partial\\Phi}{\\partial z} \\right)^2\n \\right) \n \\right]\\; \\text{d}z\\; \n +\\, \\frac{1}{2}\\, g\\, (\\zeta^2\\, -\\, h^2)\n \\right\\},\n", "c729c381fc05f12cc21dc2b0fa06b262": " i{\\partial \\psi_j(t) \\over \\partial t} = H_{L} \\psi_{j+1}(t) - H_{R} \\psi_{j}(t) + H_{R} \\psi_{j-1}(t) - H_{L} \\psi_{j}(t) + H_{jj} \\psi_{j}(t) ", "c729c6ee993d8bbaa9a55c4adbdcf359": "f_n \\in \\mathcal{H}", "c729f63c1bc0638bebfa23e81cd1452a": "dx/\\sqrt{n+1}\\,", "c729f941ec700d7e0ea7a4fb81fa3112": "\\psi(\\alpha) = \\psi(\\beta)", "c72a4620b17d8aba77415e8c27a06b14": "\\displaystyle{TJT^t=J=T^t JT.}", "c72a4d2733ca21d63b8010c8dbfbae64": "R = D_\\mathfrak{p}", "c72a4e064d7dbe2f1d7019ddbc1033d5": "|Q_h(\\widehat{D})-Q_h(D)|\\leq \\alpha\\,\\!", "c72a8e1e228f79956001a366bb33f4d5": "e^{-\\frac{\\lambda}{2}}\\frac{\\Gamma\\left(\\alpha + 2\\right)}{\\Gamma\\left(\\alpha\\right)} \\frac{\\Gamma\\left(\\alpha+\\beta\\right)}{\\Gamma\\left(\\alpha + \\beta + 2\\right)} {}_2F_2\\left(\\alpha+\\beta,\\alpha+2;\\alpha,\\alpha+\\beta+2;\\frac{\\lambda}{2}\\right) - \\mu^2", "c72af341ae5774af6d5b3b90ac476bf1": "\\cos(c) = \\cos(a) \\cos(b) + \\sin(a) \\sin(b) \\cos(C). \\,", "c72b39577f91560fba6a3bc30d6b0680": "X= \\begin{bmatrix} X_0 \\\\ X_1 \\\\ \\vdots \\\\ X_n \\end{bmatrix}. ", "c72b633022bbb575e40f20181c84bab8": "P=\\frac{(1-\\kappa ^2)^2}{2\\kappa (1+\\kappa ^2)}\\, ", "c72b6d8d7215864fe859a92a97334abc": " f_1\\ v_1 \\ldots v_{A_1} ", "c72ba8b9ebd319b7f1b15d030f45a24f": "(c_i,c_j) \\in r ", "c72bb71f94775609ab30a42b323f7780": "\\operatorname{Tr}^*", "c72be56ff1653986c31a222b05cab91d": "x_\\mathrm d = \\left | D - s \\right | \\,.", "c72c92fa86265f3b016e51b98fc37fc7": " C^* := \\left \\{ v\\in V^*\\;:\\;\\forall w\\in C, v(w) \\ge 0 \\right \\}. ", "c72c9959f9763c68c036573ffbc66fd4": "f(a,b) = g^a x^{-b} \\pmod{p}.", "c72c9b5832344cab228be62b32b6e1e1": "x\\Vdash A", "c72cd98cf06179103da19714636fee5d": "O(|A|^3)", "c72d4ad949f0fd5aaefb987383f3aa25": "\nE_\\mathrm{spring} = \\int k_\\theta \\theta \\mathrm{d} \\theta = \\frac{1}{2} k_\\theta \\theta^2\n", "c72d834e560ebbfabd496d74f80b43de": "E(\\ln^n(X))", "c72dac8896e6157d7470daf923643612": "P\\left(n\\right) \\approx {p^n} ", "c72df3cc139b06ec304b216eb3e82f4a": "H(p,N)", "c72e948fb6dfe7de1d0015eadc6f5f33": " v_p=\\frac{\\omega}{k} = \\frac{E}{p} = \\frac{p}{2m} = \\frac{v}{2} ", "c72ed69d2db00c2bd3632261b13cdca9": "\\varepsilon\\approx3.18", "c72ed928db1e1b244c292bd50d281824": "x = \\left(\\lambda - \\lambda_0\\right)\\,", "c72f07e9db28109ca3e38d16cee122e3": "\\delta(x-y)=\\sum_n f^{(0)}_n(x) f^{(0)}_n(y) ", "c72f168bfc54b6b51d7d8ce8ab9da667": "\\scriptstyle \\bar x", "c72f213928a9fab8da78cba237bdec85": "\\mbox{MD} = \\frac{C_p CL}{F }", "c72f6b969452e863049a8cfca4d2df86": "\nA \\geq B \\Rightarrow f(A) \\geq f(B),\n", "c72f7b24f99c7579b5a5dadcada9c44d": "g - A^T \\lambda = 0~~~~~~(5)", "c72f8043ba8279be4746c7c1383e8026": "(x,\\lambda) \\rightarrow (x+ \\alpha p_x, \\lambda + \\alpha p_\\lambda)", "c72fae3541a2eecd78b553b2c9b8ea03": "x = -\\ln[(1 + \\cos\\theta)/2]", "c72fc10918c792791831d9932b971005": " V_{\\rm f} ", "c73009131ef6157962ca8afa9cf9ce45": "x = t^n + \\cdots", "c73053b715a05ce85f07662bad861c3b": "{A}_{11}^{(2)}", "c7312c51fd3f56c8759099cb89180062": "\\scriptscriptstyle O(E\\min(V^{2/3},\\sqrt{E})\\log(V^2/E)\\log{U})", "c7314cd98689039375f4da67f639666f": "|\\lambda(W_X)| = \\sqrt{\\lambda(W_C W_O)}. \\, ", "c73158253471a0fc8291be456f181f41": "T(f)=f'(0)\\,", "c73162e47938c238e6e7e0dda1892beb": "|\\langle\\phi\\rangle |", "c7317668bd7765bfd8025ecef1ffa441": "\\lambda_{MFP} \\approx 1{\\mu}", "c73190f7d177831c9ae67c6318f39a36": "\\rho_\\ell", "c731918d6224c232ff19d35a3dac6aa3": "{q}\\, ", "c731f18652e6b0955abdd665259b4e9d": "\\displaystyle \\int\\limits_{-\\infty}^{\\infty}|\\Psi(x)|^2\\,dx = \\int\\limits_{-\\infty}^{\\infty}\\Psi(x)\\Psi^{*}(x)\\,dx = 1. ", "c73263fc070eeccb49f115f73dc99546": "nR\\ln\\frac{V_2}{V_1}\\;", "c7327b750d3505af06d6b9ae5a1a6305": "h(x,y) = \\left (\\sin(x^2 + y^2), \\cos(x^2 + y^2) \\right ).", "c732dc4f3a686bb1a423359ab75b7ed7": "B=(D,R^B_1,\\ldots,R^B_n)", "c7330828423f8d8edd469d7ebc27f163": "\n\\frac{x^{2} + y^{2}}{a^{2} \\sigma^{2}} + \n\\frac{z^{2}}{a^{2} \\left(\\sigma^{2} -1\\right)} = 1\n", "c7330cb15e72a710f38c9139157865c1": "x_1 x_2~+~x_3 x_4", "c7333910b2984115a35a12c6dc6c1928": "\\partial \\Omega_D \\cap \\partial \\Omega_N =\\varnothing ", "c73339565cab30b541fbf829fec8c985": "n \\in \\mathbb{Z^+}", "c73369d26fb54ceeec6270c892afc8c1": " V(x) = \\begin{pmatrix}\n V_1(x), & V_2 (x), & \\ldots, & V_n(x)\n \\end{pmatrix}^T. ", "c7336af7dcc1204bebe1f80a9fb6bde7": "A-A := \\{x-y: x,y \\in A\\},", "c7339f7631d46196a88b3a5b8480e4ea": "\\scriptstyle 2^{\\#bits\\ per\\ pixel}-1", "c733b20b8dd1ebab3bdc62b8f868112d": "4 \\pi a^2", "c733d7218c9599b5414474acac2a60c6": " e^+e^- \\to \\omega \\pi^0 ", "c73423eca514959ae0d2eee00548e3fe": " U_{n \\times p} = [\\mathbf{u}_1,...,\\mathbf{u}_p] ", "c7342d04f257c7aab4f9da5e8bf566f7": "{\\bar x}", "c734e14bc62e5324daf4bb19665b9521": "\\alpha_0 > \\alpha_1", "c734f9f1e5d0dd7684020ec3cd1bc1bc": "|x_n - c| = 0 < \\epsilon", "c73524992bc7e4113ca4e4f609c201fa": "i_{i} = \\frac {R_S} {R_S+R_{in}} i_S \\ , ", "c7352b67423ef7509d8dd539b761a047": "\n\\begin{align}\nU(x,z)\n&=\\hat {f} [\\mathrm {rect} (x'/W)] \\hat {f} [ \\sum_{n=0}^N \\delta (x'-nS)]\\\\\n&=aW ~\\mathrm{sinc} \\frac {\\pi Wx}{\\lambda z}{1-e^{-i 2 \\pi S \\sin \\theta / \\lambda}}\n\\end{align}\n", "c73567ae89a2669eca34b3b4e790d1d5": " \\operatorname{build-param-lists}[x\\ (q\\ q\\ x)), D, V, K_1] ", "c735e9e20a4589ff04fdfa63df04f1ac": "\ne^z = \\cfrac{1}{1 - \\cfrac{z}{1 + \\cfrac{\\frac{1}{2}z}{1 - \\cfrac{\\frac{1}{6}z}{1 + \\cfrac{\\frac{1}{6}z}\n{1 - \\cfrac{\\frac{1}{10}z}{1 + \\cfrac{\\frac{1}{10}z}{1 - + \\ddots}}}}}}}.\n", "c73629ea41aa2fc5abec1e32a77c300b": "\\varphi'", "c7362c2fd93719c7edbab1d3fb38dc18": "I:\\{\\mathbb{X}\\subseteq\\mathbb{R}^n\\}\\rightarrow\\{\\text{Min},..,\\text{Max}\\}", "c7363bc559d7f65e1be287b01333a0c4": "\\scriptstyle (x_1,\\, x_2,\\, \\ldots,\\, x_n)", "c7363f1b4b1b8dd5ccda235fbeb0b3ca": "\\begin{align}\n\\left(\\frac{x_{j+1}+x_{j-1}-2x_j}{h^2}\\right)&=\\frac{u(x+h,t)+u(x-h,t)-2u(x,t)}{h^2}\\\\\n&=u_{xx}(x,t)+\\left(\\frac{h^2}{12}\\right)u_{xxxx}(x,t)+O(h^4).\\end{align}", "c73692b293e6d16b27e09e328061af64": "9\\circ a = 6.", "c7369aac2854da1c0f41c08166f19df1": "FOV_C", "c7369b540cf6a7075e5527f3441f2158": " A_{fb} = \\frac {i_L}{i_S} = \\frac {A_{loaded}} {1+ {\\beta}(R_L/R_S) A_{loaded}} \\ . ", "c736c07f8335e68cf7daa653d0f2bc20": "\n\\mathcal{G}_{\\alpha\\beta}(\\omega_n) = \\int_{-\\infty}^{\\infty} \\frac{\\mathrm{d}\\omega'}{2\\pi}\n\\frac{\\rho_{\\alpha\\beta}(\\omega')}{-\\mathrm{i}\\omega_n+\\omega'}\n", "c736d3b46f51165432c7ea0672d9fa31": "M^2 alpha \\equiv \\overline{D} \\times \\frac {\\sigma_B} {\\sigma_D}", "c736fe1c82da9399a8efd41a35526554": "I=\\cfrac {V}{Z} ", "c7372a8747951207ba872cae15e6295b": " (\\delta_{kl} \\partial_{tt}-A_{kl}[\\nabla])\\, u_l \n= \\frac{1}{\\rho} F_k\\,\\!", "c7373635fb887dd015c9d73eaf418b35": "\\psi(x):=\\sum_{k=-N}^N (-1)^k a_{1-k}\\phi(2x-k).", "c7373ef38e2e56f51e88d52e1cd5390a": "\\Delta\\sigma = \\sqrt{s^2} = \\sqrt{\\Delta r^2 - c^2\\Delta t^2}", "c73771daa6291d835f8fa396ebc6b0c0": " L^*", "c737b274a317b41fb81169b6455b7152": "\\mathcal{L(\\tau)}=-(m+S(x))\\sqrt{-\\dot{x}^{2}}+\\dot{x}\\cdot A(x) \\, ", "c73819be46eeb9e58423767a97985f9d": "f_\\sigma(x;\\mu) = \\frac{1}{\\sqrt{2 \\pi \\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}.", "c7383522c97e4ea2fe8f4eec1dac9eca": "J^{2}", "c73877cfb2abb16fb0b031ad13deccc9": "( -200 + 200.0000001 ) / 2 = 0.00000005.", "c7387ef787d6623cde795be4a25f9442": " \\sigma=\\frac{\\Delta\\rho\\omega^2}{4}R^3 ", "c738871b00b8b6ea948101d1c48035d9": "X=\\sum_{\\{i\\}} e^*_i e_i \\in B^*\\otimes B", "c7389543af3ad35a78a4d1f4331ccd5a": "u = \\pi \\vartheta^2 z", "c739203075328cb0bb847ba47bd2343a": "\\langle p|J^0(0)|p\\rangle =\\lim_{p'\\rightarrow p}\\langle p'|J^0(0)|p\\rangle", "c739259c003a730805bfbbefd9adc4e4": "\\langle \\Psi_{1} |x|\\Psi_{0}\\rangle", "c73932ff565625c13e80ceb884354503": " \\frac{\\mathrm{d}e_s}{\\mathrm{d}T} = \\frac{L_v(T) e_s}{R_v T^2} ", "c7396886dc445537c59af466fc149ac2": "f_i(x) <_{K_i} 0, i = 1,\\ldots,m", "c7397ad5f7ddabd41f2157acff3893cd": " \\mathbf{E} = \\frac{q}{4\\pi \\epsilon_0} \\frac{1-v^2/c^2}{(1-v^2\\sin^2\\theta/c^2)^{3/2}}\\frac{\\mathbf{\\hat r}}{r^2}", "c7398a6a1e35ac202957e0747b6785b4": "\\phi_P = \\sum_{j = 1}^{n}\\frac{1}{4\\pi\\epsilon_0}\\int_{S_j}\\frac{\\sigma_j da_j}{R_{j}}", "c739e7b1285679250be773ba8e183209": "\\mathcal{O}", "c73aab3cd8138be5cdd1d19a288e12c4": "\\exists{x}{\\in}\\mathbf{X}\\, \\lnot P(x)", "c73acb83deec9170566f5ce2113fd178": "\\Bbb Z_3", "c73ae8442ad6776c235b71af84434718": "EL(\\Gamma)\\le w/h", "c73b00076efe2c4d50f37dd704cfb491": " P_n(K)", "c73b2a3811b7487cd2e7877c3775a743": " P^{tr}(Y) \\neq P^{te}(Y)", "c73b7f3070410a06c427715785c34783": "\\epsilon_b", "c73b85fef6ce50cbcd1c41006a67e827": "q=1/\\sqrt{p}, \\quad m = q n", "c73ba90f61556dc643c11b4b01f19526": "\\bar{r_w}", "c73ba96a72c507c38cdb91e625f977d2": "\\mathbf{\\hat{\\boldsymbol{\\beta}}= {}(X^\\top X )^{-1}X^\\top Y}.\\,", "c73bf54f6435451da27146f35264dde3": " P_i", "c73bfa55a5e9413a59f2321a90dd23db": " \\arccos x = y \\, ", "c73c81714e18cb2311e01ce94fb037d1": "\\{(x_1, \\dots, x_n ) \\mid x_1 > x_2 > \\dots > x_n \\text{ and }\\sum_{i=1}^{n} x_i = 1\\}", "c73cdf79bf670fadbd552d1101109cce": "(-1)^n E_n(-x) = -E_n(x) + 2x^n\\,", "c73cf4fb1fad5405855a9aa6ec657ed6": "\\displaystyle (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220,", "c73cf7e1b420f260400db1ab3eb9bc59": "Nu_c", "c73d0bf54e4e5905c2cb2b71d2e2111b": "\n\\begin{array}{lcl}\n\\#\\mathbb{W}_v^{k,(-dn)} &=& \\text{number of words having value }v\\text{ among documents with label }k\\text{ excluding }w_{dn} \\\\\n\\end{array}\n", "c73d23c2d586678b895dbd5511f37695": " \\mu\\left(\\{x : Tf(x) > \\alpha\\}\\right) \\leq \\left( \\frac{C_{p,q} \\|f\\|_p}{\\alpha} \\right)^q ", "c73d6ad0b669d217d20fb06d3efe6754": "\\textstyle R(a_P\\mid[x]) = \\lambda_{PP}P(A\\mid[x]) + \\lambda_{PN}P(A^c\\mid[x]),", "c73d74f53bc05c9bdd5f5e26b2545971": "F=\\nabla\\Phi", "c73df4bb7b31a742ddc5faf9270831d3": "L_6(x)=x^6+6x^4+9x^2 + 2. \\,", "c73e30ae7e23c83ece2f04ea554f134f": "Sf(x) = \\frac{1}{f(x)}f'(x) = \\frac{d \\ln f(x)}{d x}", "c73e3acb140f7723f63cb3d2a1a49e2c": "M_{\\pi \\ominus \\sigma}", "c73e82ec6f0a1fa646eab6f3fa7419e3": "\\ddot{\\rho} = n^2 \\rho \\ ;", "c73e9cccaca0bdd171c7d5126fba51e7": "(7.b)\\quad \\nabla^2\\psi =\\,e^{-2\\psi} (\\nabla\\Phi)^2 ", "c73ead4760ca644cebd83ecfad19e7b0": "J_{ij} = \\delta_{i,j+1} ", "c73ec673c19cdddda330b926163ca96a": "\n\\begin{align}\n f&=\\frac{a-b}{a}.\n\\end{align}\n", "c73ef6beb9d03ba0f845412b712ddf03": " \\vdash A ", "c73efda32999f4243f0ee158987a7991": "(k+1)^p \\equiv k^p + 1^p \\pmod{p}.\\,", "c73f61a01a8f2cbe8a35eebf00ec20c7": "\\left(\\!\\!\\!\\binom{n}{k}\\!\\!\\!\\right)=\\left(\\!\\!\\!\\binom{k+1}{n-1}\\!\\!\\!\\right).", "c73f6a4876af9e7a6152c9f146e0b512": "v^2=u_0^2-u_0^4", "c73f7668f8cf5034f1bd315075c0d2b4": "dQ/dt = C (dT/dt)", "c73f8f7cc92adc75a94030b9b6a00ab9": "\nr_{\\mathrm{outer}} \\approx \\frac{2a^{2}}{r_{s}}\n", "c7401f9aa0441fe94c1d2542ba96b242": "\\frac{\\partial S}{\\partial t} = 2D\\ H(x,y) \\sqrt{1 + \\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y}\\right)^2}\n", "c7404fecae7d48695deca7e60b108d67": "p_{k,S}^C", "c7410a16bcae70a5020dd94ac30414c6": " \\langle \\Omega | T(\\psi(x)\\bar{\\psi}(0))| \\Omega \\rangle = \\int \\frac{d^4p}{(2\\pi)^4}\\frac{ie^{-i p\\cdot x}}{p\\!\\!\\!/-m - \\Sigma(p) +i\\epsilon} ", "c7413a822ff270b9a72c6723412287b3": "\n\\begin{align}\n y[n] & = \\frac{1}{a_{0}}(b_{0} x[n] + b_{1} x[n-1] + \\cdots + b_{P} x[n-P] \\\\\n & - a_{1} y[n-1] - a_{2} y[n-2] - \\cdots - a_{Q} y[n-Q])\n\\end{align}\n", "c7418fad8e6a2fd22986290bc65f93d4": " J_{1z}", "c741c0319cf3b870e43df5ad62963cfd": "\\frac{80\\sqrt{15}(5^4+53\\sqrt{89})^\\frac{3}{2}}{3308(5^4+53\\sqrt{89})-3\\sqrt{89}}", "c74210c718fce011692285705eea437a": "M[f \\circ g] = M[f]M[g] ~,", "c74244f16694c9363bc8f2612180568c": "\\{ x ~|~ x = x^2 \\} \\,\\!", "c7425f59147880293e9b035340a459f9": "NaN \\times a = NaN", "c7427e9225da7b6b10e52dea5daaac0a": "\\frac{\\partial f_t(z)}{\\partial t} = \\frac{ 2f_t^\\prime(z)}{\\zeta(t)-z}", "c7428277a6746bc9bb3d6dcae60bc02e": "\\mathbf{w_p} \\leftarrow \\frac{1}{M}\\mathbf{X} g(\\mathbf{w_p}^T \\mathbf{X}) - \\frac{1}{M}g'(\\mathbf{w_p}^T\\mathbf{X})\\mathbf{1} \\mathbf{w_p}", "c7429073858534592b1b3cf8f2690fa4": " \\left(\\sum_{j=1}^k p_je^{it_j}\\right)^n", "c742917e2db19433ddafd71fe10dfbc3": "c_n g = (-L)^{(n-1)/2}R(R^*g).\\,", "c742a1efa6c0c998d0733d186e1af380": "T_{-I}<\\cdots < T_{-1}", "c742b0990d1bbb832161c7008c40d875": "M:= \\max_{z\\in\\Gamma}|f(z)|", "c742da8225904d4e45e6e1e3cd530757": "Q(\\theta) = \\begin{bmatrix}\nxxC+c & xyC-zs & xzC+ys\\\\\nyxC+zs & yyC+c & yzC-xs\\\\\nzxC-ys & zyC+xs & zzC+c\n\\end{bmatrix}", "c742e9184d11fc7323ab9e26ec2d47ad": "C_s = \\frac{C_1C_2}{C_1 + C_2}", "c74306d5e53fe198575ea17b2ff79fa2": "F_{ij} = G \\frac{M_i M_j}{D_{ij}}", "c7430ce2b7f1ca346ee4d88d19f17dbe": "\\tilde{H}_n\\left(S^2\\vee S^2\\right)\\cong\\delta_{2n}\\,(\\mathbb{Z}\\oplus\\mathbb{Z})=\\left\\{\\begin{matrix} \n\\mathbb{Z}\\oplus\\mathbb{Z} & \\mbox{if } n=2 \\\\ \n0 & \\mbox{if } n \\ne 2 \\end{matrix}\\right.", "c743295282e2a0fcae8bdd33735ed0cb": "\\displaystyle | Y(s=i \\omega) | = \\frac{1}{\\sqrt{ R^2 + \\left ( \\omega L - \\frac{1}{\\omega C} \\right )^2 }}. ", "c743bf9c3a4a73ad58a7aaf2dcbf51a7": "\\clubsuit_{\\omega_1}", "c743c8856db5a759dd7294239ab87a8a": "I(z) = I_{in} + { \\gamma_0(\\nu) \\cdot z \\over \\bar{g}(\\nu) } I_S", "c74411bdd0593e0d3828a82cc836de24": "\\ T_{c} = \\frac{a}{4Rb}, \\ p_{c} = \\frac{a}{4b^{2}e^{2}}, \\ V_{c} = 2b.", "c744327bacade6b32d069fa6d1666a86": "\\delta_\\mathbf{k}", "c7448fe3e4c0528bf2b37ce913767d78": "\\scriptstyle c^2\\rho D\\Psi=g\\Psi\\rho+\\sigma k^2\\Psi.", "c744feeefa9f4be50a1963ebbd5eda66": "\\textstyle{\\frac {\\log(2)} {\\log(2/\\sqrt{2})} = 2}", "c745204dd7d201179dbf41f334758de3": "\\mathcal{S}_{1}\\psi =(\\cosh (\\Delta )\\mathbf{S}_{1}+\\sinh (\\Delta )\n\\mathbf{S}_{2})\\psi =0\\mathrm{,} ", "c74526f1eb6bf801455b30707485e512": " ds^2 = {dx^2 + dy^2\\over y^2}.", "c74533acc2c08b76d2de30fbe8234c05": "E(x,y,z) = \\frac{e^{i k z}}{i \\lambda z} e^{i \\frac{\\pi}{\\lambda z}(x^2 + y^2)} \\mathcal{F}\n\\left. \\left\\{ E(x',y',0) e^{i \\frac{\\pi}{\\lambda z} (x'^2 + y'^2)} \\right\\} \\right|_{p = \\frac{x}{\\lambda z}; q = \\frac{y}{\\lambda z}}\n", "c7455b9d6ce7204d0c360ddddb95f575": "\\begin{bmatrix} \\dfrac{\\mu}{\\sigma^2} \\\\[10pt] -\\dfrac{1}{2\\sigma^2} \\end{bmatrix} ", "c745850be627468652aa73c9ccb89049": "V=\\frac{5}{12}(11+5\\sqrt{5})a^3\\approx9.24181...a^3", "c74590e1a266f24409f53c85cbd2d931": "\\Pi (t,f)", "c745ce1fa0efaeb8b33d25db0a92a130": "E_f=p_f^2/2m_f", "c745d6cf79bc5a1573dd18c824378bc8": "\\boldsymbol{f}(s)", "c7460a06a0ab600d7885382b7c76aae3": " y \\in N ", "c7461c9b024005b15befbc50725751dc": "\\frac{\\pi^2}{6}\\,\\beta^2\\!", "c7467623756b05c76546ea2526a4c219": " (1.100011)_2 \\times 2^{3} ", "c746f5aa90453fa2e9282a9b914764c2": "e_1 = \\mu f_2 \\,", "c747694d8b0d5b96d43d48c5af62c307": "\\hat{\\theta}_{1}, \\hat{\\theta_{2}}, \\dots, \\hat{\\theta}_{k}", "c74786f47337229d6cfa3bc6049a6bc0": "b_1>0", "c748572b66a0a9cdf34895384546f771": "c_g E", "c749006df32fb473a78cb453caf88163": " \\mathrm{R}\\,\\mathrm{Na_{\\text{z}}} = \\mathrm{R}^{\\text{z-}} +\\mathrm{z}\\,\\mathrm{Na^{\\text{+}}}\\,.", "c74917b208c5b3ffb301123e6a7346df": "V_n(\\sqrt{R},Q) = a^n+b^n", "c74959b70650b9e50aef27e64a4d2b8b": "\\aleph_n", "c749b0dc8b5589792798c8965683fdb9": "\\psi^{\\langle2\\rangle}=\\psi", "c749c7ed6de3ed4d7a5ba5f351f87a02": "R \\to A, r \\mapsto r 1", "c749ec5230441e50da6c1623f28670c8": "\\bar{x}_j = x_k(\\boldsymbol{\\mathsf{L}}^{-1})_j{}^k", "c74a56546d602cff6ccf67e4a70aca7d": "\\displaystyle{\\partial^\\alpha F=\\sum_{m\\ge 0} \\partial^\\alpha F_m.}", "c74b35f4e9b8b193e94bb5621056b95d": "\\overline{x} \\pm 3\\frac {\\overline{MR}}{d_2}", "c74b3d936cc42e92a0e5d36776f88dd8": "\\frac{{6 \\choose 3}{42 \\choose 2}}{{49 \\choose 6}}\\approx\\frac{1}{812.07}", "c74b83c1eb05b5bcfd146948f1a22734": "F(c,\\ X)", "c74bb48f04e3de8b11a27e6b09ac6c14": "[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0.", "c74bcc97fa6ac934d225b60e23502875": "Y[x,y]=\\frac{(yx'-xy')x'}{x'^2 + y'^2}", "c74c148fad2f24ba52c8a34a0246c9b4": " \\delta_p ", "c74c3831d98c5c47e9f6960c4a6a56f9": "i\\geq 0", "c74c387e93b661d2d1fe686745d07ca1": "\\int\\limits_{\\partial B(x, r)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\\, u(y) \\, \\mathrm{d} S(y). ", "c74c86d17a0d38a0dd2533cc7b61f9a7": "A_n =A_m", "c74cf6591863d9e2c02a2161fc60f4e0": "\\mathit{FDR} = \\mathit{FP} / (\\mathit{FP} + \\mathit{TP}) = 1 - \\mathit{PPV} ", "c74cfeaa076270493576a1c0dda3627d": "|E(\\mathbb{Z}/N\\mathbb{Z})| = N+1+a", "c74d312a9ee1ae3dedc45a9414962602": " -\\beta F = \\frac{\\log(2)}{2} + \\frac{1}{2\\pi}\\int_{0}^{\\pi}\\log\\left[\\cosh\\left(2 K\\right)\\cosh\\left(2 L\\right)+\\frac{1}{k}\\sqrt{1+k^{2}-2k\\cos(2\\theta)}\\right]d\\theta", "c74d97b01eae257e44aa9d5bade97baf": "16", "c74dbb523532de4fea74b19c0b487b09": "y \\in B_0", "c74df6ebe834fba6709769d5bf86161e": "\\mathcal{H}_{0}", "c74e183913a7235d99af8309eb1520c9": " 6 = \\underbrace{3+3}_{\\text{2 parts}}. ", "c74e4b482f0db5810faabdb293c6106a": "\nP_{r}(\\omega)= \\sum_{n\\in \\mathbb Z} r^{|n|} e^{i n\\omega}\n", "c74e60234b480f002e9e4c097b56f782": " \\frac{d}{dz} B(z) = \n\\exp \\left(\\exp z - 1\\right) \\exp z = B(z) \\exp z,", "c74ebb0a5b6c2a61c9b14990711bbdcc": "CB = \\frac{d(-DB)}{dr} = (-D)(-DB) + \\left(-\\frac{dD}{dr}\\right)(B),", "c74eec3770ba65e27cdee0ead20da4aa": " \\frac{p(O_{fg})}{p(O_{bg})} ", "c74f504752b0a317430a7a9437c9f21f": "\\mathbf{p_0}", "c74f858133fb4bb7479aa9da9b54a42f": "\\bigwedge \\left( A \\cup \\emptyset \\right)\n= \\left( \\bigwedge A \\right) \\wedge \\left( \\bigwedge \\emptyset \\right)\n= \\left( \\bigwedge A \\right) \\wedge 1\n= \\bigwedge A", "c74fd8bf4e6e39e3a1aea1138bb509a0": "\\frac{dy}{d\\varphi} = \\frac{dy}{ds}\\frac{ds}{d\\varphi}=\\sin \\varphi \\cdot a \\sec^2 \\varphi= a \\tan \\varphi \\sec \\varphi.\\,", "c750326237e9b7bf35bfa34e3e84c6a3": " \\text{R} = \\operatorname{Tr} \\left( \\textbf{H} \\right)^2/\\operatorname{Det} \\left( \\textbf{H} \\right)", "c7504dc5b79db9321df9de03874a71ae": " \\begin{align}\\left [ \\hat{A}, \\hat{B} \\right ] \\psi & = \\hat{A} \\hat{B} \\psi - \\hat{B} \\hat{A} \\psi \\\\\n& = a(b \\psi) - b(a \\psi) \\\\\n& = 0 .\\\\\n\\end{align} ", "c75126dd13dc450e9a59625c6b17421d": "T_\\text{out}", "c751ba143a5c226d861a2905d14986e4": "10^{10^{50}}", "c7524cbb35f254dc3f92e4c8045009d1": "S^\\alpha", "c7526be9349d8d84f08b2b3db77282ff": "\n g_1\n = 3 \\uparrow \\uparrow \\uparrow \\uparrow 3\n = 3 \\uparrow \\uparrow \\uparrow (3 \\uparrow \\uparrow \\uparrow 3)\n = 3 \\uparrow\\uparrow (3 \\uparrow\\uparrow (3 \\uparrow\\uparrow \\ \\dots \\ (3 \\uparrow\\uparrow 3) \\dots ))\n", "c75279d549ce7f3db6f80fec866cc637": " TM ", "c7529d5cba372dbc14e0b9301b6c5ae5": "2 \\log_{4/3} n", "c752c0e17caf36a210e8193174bef0f0": "M = \\frac{2300}{50 + 100 + (10 \\times 8)} = \\frac{2300}{230} = 10", "c752c242855d3def64fc5c35bd1cb9c7": " \\frac{x}{\\sigma} ", "c752efd99f3060d133d84240196e1aca": "A(s)=2s^4+12s^2+16.\\,", "c75359541a10e3a3337ad68cdbb6e2af": "-$5.26+3\\cdot$9.99", "c753857def4e349e5976e609b6d041ab": " f(x) = \\frac{1}{x}\\, ,", "c7539bf0ff11c62efdd303594cd32b8e": "\\forall (a,b), S(a,b)\\geq 0", "c753e778a9a7de97b5336fd978135042": "g_1 \\mathbin{\\ast'} g_2 = g_2 * g_1", "c753f6dde93d6efdcbe04949dbecb0aa": "a_n=2^{2^n}+1\\, .", "c75430756e9a477c5aadf0cf31346225": " \\bold p = 4 \\pi \\varepsilon_0 \\left(\\frac {\\kappa-1}{\\kappa+2}{R^3} \\right) \\bold{E_{\\infty}} \\ ,", "c75435267f36d4b377f21c9fb25a9cdf": "\\big[ M + A(\\omega) \\big] \\ddot x + B(\\omega) \\dot x + C x = F(\\omega)", "c7543b13e26b2d536293b3b7a614295b": " \n\\Delta^1_{\\rm LONG}= \\frac{\\pi}{180}a \\cos \\phi \\,\\!\n", "c75466283b785df15e3e2ddba2622fe8": "\\displaystyle{B(z)=\\prod \\left[{|\\lambda_i|\\over \\lambda_i} {\\lambda_i -z \\over 1-\\overline{\\lambda}_i }\\right]^{m_i},}", "c7549f59bd4257cdf27c6a55f0b05727": "X_{n+1};", "c7556098eda48b8e584ee64676af3cb0": "R_A", "c7558796d2e312b7882cca618a15a631": "\\scriptstyle y\\,\\in\\, Y", "c7559617a8c67eefe63f2baca0d4fa55": "\\Delta_{Q1}", "c755dea22c9430fd20091c70b509ed47": "Val(\\mathrm{scalars}) = s^TG(\\mathrm{vectors})", "c755f38dd3f971e29326b8530cde4113": " \\lambda + 1/2 ", "c75632a6fd7a05f74d47d76297a73f66": "\\scriptstyle \\hat n", "c7568363ba6c66773db5dfeca81f7a94": "\n= {1 \\over 2} \\cdot 15 \\, \\mathrm{V} = 7.5 \\,\\mathrm{V}\n", "c7571e7550a79efb650a7c2540f2ef83": "y(n)=ax(n)+bx(n-1)+cx(n-2)", "c75787929dcf4f5b8f543679fed6e2e5": "{{(z_1-z_3)(z_2-z_4)}\\over{(z_1-z_4)(z_2-z_3)}}=1-{{(z_1-z_2)(z_3\n-z_4)}\\over{(z_1-z_4)(z_3-z_2)}}.\\,", "c75792357c1a0e8f5114eeeac41e432b": "\\mathrm{P}(A \\cap B) = \\mathrm{P}(A) \\mathrm{P}(B).", "c757a456071dfb953f6edf3635a3443b": "\\mathbf{T}^{(\\mathbf{e}_2)}= T_1^{(\\mathbf{e}_2)}\\mathbf{e}_1 + T_2^{(\\mathbf{e}_2)} \\mathbf{e}_2 + T_3^{(\\mathbf{e}_2)} \\mathbf{e}_3=\\sigma_{21} \\mathbf{e}_1 + \\sigma_{22} \\mathbf{e}_2 + \\sigma_{23} \\mathbf{e}_3,", "c758096620355ce54cb33599a8d2d609": "dx=a\\cosh{u}\\,du", "c758351b9c1803cf155cc1e4a003a72e": "L[H(s) e^{s t}] = e^{s t}", "c75843bdab926aa0ea83db19480f673d": "\\dot{z}_i", "c758adff7ff24d5581c92cb9e5cb3f32": "(g^b)^c = g^{bc}", "c758d1af28de3ad48d3d7cbf94601db8": "t_1 \\xrightarrow{*} t_2", "c75927243513d234cc2fa87f6872fd17": "\n\\frac{M}{10^8M_\\odot} \\approx 1.9\\left(\\frac{\\sigma}{200~{\\rm km}~{\\rm s}^{-1}}\\right)^{5.1}.\n", "c759272968a94512671b38d15f65c550": "\\displaystyle \\beta_k(t) = \\int_t^\\infty u^{-k} e^{-\\pi u} \\,du", "c75937434e8c6754e1d6ae4651c87377": "dy = {\\rm tr}(\\mathbf{A}\\,d\\mathbf{X})", "c7593f24b186ded5655ad8ce2e204a84": "\\scriptstyle \\log_{10} P_{mmHg} = 6.74808 - \\frac {882.80} {240.0+T}", "c75962fe6a8bac62ada320f204e4732a": "\\ MAD = \\frac{\\sum_{t=1}^{N} |E_t|}{N} ", "c7598fe814f2fe010acfa52f393b59e5": "?\\,", "c75a0aed2f70db5d4156f25fc523d7f9": " \\operatorname{let} x : (x = \\lambda f.f\\ (x\\ f))[x:=x\\ x] \\operatorname{in} x\\ x ", "c75a0bc85a28c845db6188b70d57c023": "{\\text{Φ}_{org}}", "c75a16eb4774333c8078ee61ec91e6c9": "\\int f(\\theta_i) \\, d\\theta = \\int f(\\theta_i(\\xi_j)) \\det(J_{ij})^{-1} \\, d\\xi.", "c75a2882329ee2fb29074340d9273271": "f(\\emptyset)\\geq 0", "c75a6f13016bf67eb90c52f08d8a94b3": "C_i \\in S", "c75a80dbb40620f3f6d38178e325ecf7": "\n(\\nabla \\times T)_{ij}= \\epsilon_{ilk} \\partial_l T_{jk}\n", "c75b0b4b40b209875f245a173d56d688": "f(C)", "c75b36349ee17fbf78eb92985b3cccd0": "d(A,B)=\\inf_{x\\in A, y\\in B} d(x,y).", "c75b80f7523218d14efb6273b163ed8e": " - \\,\\! ", "c75b8888989250e1fd59b18f00e28f89": "\\Phi(\\tau+32)\\,", "c75badde695e9bb3f63e66886ef7f16f": "K_{11} = K_{22}", "c75be23b3901f91405e58d68761f5f3e": "\\dot{e}_2 = h_3(\\hat{x}) - m_2(\\hat{x}) \\operatorname{sgn}( e_2 )", "c75c033596e30abfaa12c674906b7bd8": " G(\\beta) = [G(\\alpha_1)...G(\\alpha_n)] \\;", "c75c195553cb03a1bfe0a3fe9ae73d33": "(y_1,y_2,....,y_m)", "c75c22c0876976bad2957525698cc870": "x \\in [0,1]", "c75c340c41c8c61333fb71ca6277dbb5": "(\\log^{3}(n))", "c75c6715962a25af6eddd9df7529a426": "q' = \\Delta t \\times (q/2m)", "c75c8b4e448db91ffdbf1374e2fb6514": "K'_{AB}", "c75cccaf5125c0e8197c3bf757484792": "s_{\\mu,\\nu}(z) = \\frac{1}{2} \\pi \\left[ Y_\\nu (z) \\int_0^z z^\\mu J_\\nu (z)\\, dz - J_\\nu (z) \\int_0^z z^\\mu Y_\\nu (z)\\, dz\\right]", "c75cde1e780cea237a20b74fea1e3939": "r \\in (0,1)", "c75ceb73b8ac2c3e627b8040a7d8259d": "B^2 - AC = 0", "c75cefbad6b96acbd3bd4eea0b016dc0": "\\le W(u,v) - 1", "c75d1155bdc97104448762ffbbdc60ce": " \\operatorname{U}(n) /(\\operatorname{U}(n-1) \\times \\operatorname{U}(1)). ", "c75d59c46f1aa3976251af22a04ded26": "\\sdot \\left( 1-A_v \\frac {R_L} {R_L+R_O} \\right) \\,\\!", "c75da06be310dba0277de4e31df595c2": "CVR = (n_e - N/2)/(N/2)", "c75e26acd692c1dc0de74d88d17e2de0": "S_F(x-y) = (i \\partial\\!\\!\\!/ + m) G_F(x-y)", "c75e5b764b5fb5bb43410a5f9548b4d9": "HLB = 7 + m*Hh - n*Hl", "c75ec13211f25d449727557fc397aa34": "_{P}(f) >0", "c75ec9b4eb4082e218eaeee7afc49ad8": "x \\leq M \\land y \\leq M \\land (x = M \\lor y = M)", "c75f1474deaf307fa73734556b74101b": "W^{\\perp} = \\{v\\in V \\mid \\omega(v,w) = 0 \\mbox{ for all } w\\in W\\}.", "c75f2276b952cd9fe675f5833f6ce157": " n_adx^a=\\frac{1}{\\sqrt{2}}(\\sqrt{g_{tt}}dt-\\sqrt{g_{rr}}dr)\\,,", "c75f2b99a807e49ad121d976031fe5d3": " K = \\frac{48 G^2 M^2}{c^4 r^6} \\,.", "c75fe4d3657877c0d2d15a40e3944162": " I \\ ", "c7606412e47df5c59954ffd800f8d32d": "O(\\mathrm{smoke} \\rightarrow \\mathrm{ashtray})", "c760a952e1032901b4373ef145598ac8": " \\Delta u = u_{xx} + u_{yy} + u_{zz} = -\\delta(x-x',y-y',z-z'),", "c760c81b503b91791052f57c1a9fd13e": "F_{n+1}=F_n\\left(2^{1/2^n}\\right)", "c761441f6154214a2fd43d70a2a4b57c": "\n\\langle z^m \\rangle = \\phi(m).\n", "c7615ebc7b7e73bc2d1975d04f6e295e": "\\textstyle A_2 \\subset \\Omega_2 ", "c761f10f08812637932c6b5d2e082d9a": "\\|BA\\|_{op} \\le \\|B\\|_{op} \\|A\\|_{op} .", "c761f9beba1d59062de6acfee15e2e3e": "\\arctan \\frac{a_1}{b_1} + \\arctan \\frac{a_2}{b_2} = \\arctan\\frac{a_1 b_2 + a_2 b_1}{b_1 b_2 - a_1 a_2},", "c7620f406770b91da6382fc77983b95c": "F_i(\\mathbf{x}(t)) = \\frac{\\mathrm d}{\\mathrm{d}t} m \\dot{x}_i(t) = m \\ddot{x}_i(t),", "c7621feedef6a90abc89ff2564ca69b5": "w_r = \\mathbf{1}_{[0,N-1]}", "c76221d6a40234c9fa8fb2adc634015f": "x = y \\ ", "c7622eb33bac36a1100abbe385c3f6ab": " L_* \\mathbf{x}^{(k+1)} = \\mathbf{b} - U \\mathbf{x}^{(k)}, ", "c76266d4ac9ce9fad558f8516908c4c9": "\\forall A \\, \\forall B \\, ( \\forall X \\, (X \\in A \\iff X \\in B) \\Rightarrow A = B)", "c7627eca66c28769a72f651916352afe": "\n\\Psi(\\mathbf{x}_1,\\mathbf{x}_2) = -\\Psi(\\mathbf{x}_2,\\mathbf{x}_1)\n", "c762860666b4ba32a74f1c646cc26139": "J = {TMP \\over \\mu R_t}", "c762be2357865b269cb143bb50043ab4": "\\zeta(s)=\\sum_{n=1}^\\infty\\frac{1}{n^s}=\\frac{1}{1-2^{1-s}}\\sum_{n=1}^\\infty \\frac{(-1)^{n+1}}{n^s}\\,,", "c762d1d39cba035d5d844d5c2baf8435": "SX = (X \\times I)/\\{(x_1,0)\\sim(x_2,0)\\mbox{ and }(x_1,1)\\sim(x_2,1) \\mbox{ for all } x_1,x_2 \\in X\\}", "c762eb7db625a68d146d9df92b7df4d8": "P_0 = \\sum_{y=x_\\min}^{x_\\max} \\binom{m_1}{y} \\binom{m_2}{n-y} \\omega^y", "c762fc5bcf7d7067b886c29fd5cafd69": " \\begin{matrix}\\frac{2\\pi}{c}\\end{matrix} ", "c762fc69df1f2f21ddf81462715ebb93": "\\ \\sim r", "c76369d472f522023ab0476ab817f3f9": "\\eta_l", "c76377c446ad164ae89e81c4de6a936b": "\\phi(\\ln(x)) = y(x)", "c763812f6245384bf25948e1546731c3": "a = x_0 < x_1 < x_2 < \\cdots < x_{n-1} < x_n = b. \\, ", "c763d66093735731dba88682d9b60ca9": "F_1 \\,\\!", "c763d831c83a41b1be265f2a356fb2ad": "\\mathbf{T}q=\\sqrt{w^2+x^2+y^2+z^2}\\,", "c763db8fc905d8f56a63f684d520bd2b": "|\\det M|\\le 1,", "c7641384c9fb9383f38c091a0c8fe92e": "\\displaystyle{H^2=-I.}", "c7642efeb8a617ff44b87a2ce94bc9b3": "V(x) / I(x) = Z_0", "c76497501b4f245d27d79fa7cfc6c794": "k_{x}=\\frac{\\omega}{c} \\left(\\frac{\\varepsilon_1\\varepsilon_2}{ \\varepsilon_1+\\varepsilon_2}\\right)^{1/2}.", "c76497575ca05dda459462748189dcd1": " \\frac{\\sin \\alpha}{\\sin \\beta} < \\frac\\alpha\\beta < \\frac{\\tan\\alpha}{\\tan\\beta}.", "c76499d53a7e121631cbea391b6c4923": "\n \\boldsymbol{S} = J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T} ~.\n", "c7653ef0a4f1d2add6092081fc29ebeb": " \\mathrm{ CO_2 + 2\\ H_2S \\rightarrow CH_2O + H_2O + 2\\ S }", "c7654231686bf244472a6055700c4917": "\\operatorname{tr} (\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma) \\,", "c7654311e0f60b7fb736b5e1e225876d": "\\langle \\Psi_{1} |", "c7655279e138e783ceaa224a6a52815a": "R^{2}", "c76584057da31847aafcdaec7996ac52": "T_{bH} = e_H T", "c765bbf10cc05734830dff473c150673": " \\chi^2", "c765e30a6fbcef0c74b5940768035b85": " \\mathbf g = g(r)\\mathbf{ \\hat n }", "c766027576bfe40b30bf1b21600f1149": "\\ln 2 \\approx 0.69314718055...", "c766e0e768bbd597700fe92811aa38a9": "\\dot\\varepsilon ^+ - \\dot\\varepsilon ^- =\\frac{1}{2}\\left(\\textbf{g}\\otimes\\textbf{n}+\\textbf{n}\\otimes\\textbf{g}\\right), \\qquad {(3)}", "c767a3f749e737e9d9f64c6027fe66a8": "dw_{p}={\\sigma}_{i,j}d{\\epsilon}_{i,j}^p={\\sigma}_{i,j}s_{i,j}d{\\lambda}", "c767c780029735dcc5290afe1a9fe4b0": "U_i(\\{x_i, x_j\\}) = x_i - \\frac{\\alpha_i}{n-1} \\times \\sum{\\max(x_j - x_i,0)} - \\frac{\\beta_i}{n-1} \\times \\sum{\\max(x_i - x_j,0)},", "c767de4be1e2ed676d10777959683f68": "+f_0.", "c767e3bc976500685edbe4ff9f211ff7": "\\infty-\\infty", "c767ec394a1f4b49c27c45cd33fcdbf6": "\n\\begin{matrix}\n\\phi_1 = p_{11}Q_1 + p_{12}Q_2 \\\\\n\\phi_2 = p_{21}Q_1 + p_{22}Q_2\n\\end{matrix}.", "c76890796418a2bd3fff356a4e606cf1": " \\lang n^{(0)} | n^{(1)} \\rang=0.", "c768a67d766459381823cc18f7ecf293": "L=\\frac{d^{2}}{dx^{2}}+u,", "c768d0b8ec468308aa525498889d4b22": " \\boldsymbol\\Sigma = [\\operatorname{Cov}[X_i, X_j]], i=1,2,\\ldots,k; j=1,2,\\ldots,k ", "c768e3e4343ccc851f8ac3c245485b99": " MA = \\frac{F_B}{F_A} = \\frac{a}{b}.", "c769509d22dd40320e5633ac73aa3b22": "\\vartheta_a", "c769b2ba1b38c87deda27d6d34186e9b": " \\frac{\\partial^2}{\\partial x^2} f(x) = \\sin(x) ", "c769ce5232fdd2c79e4a52bad061d0af": "p[D,E] > p[E,D]", "c769d08e81720eda55bd89c6513326ba": "\\frac{\\partial Y}{\\partial r}", "c769fb4625919ef17585de62dd612746": "\\nabla^\\perp u(x,y)=\\nabla v(x,y)", "c76a3d515a7af1c0d0d853e31882b917": " f_X(x) = \\int_{-\\infty}^{+\\infty} f_{X,Y}(x,y) \\, \\mathrm{d}y = \\int_{-\\sqrt{1-x^2}}^{+\\sqrt{1-x^2}} \\frac{ \\mathrm{d}y }{ 2\\pi\\sqrt{1-x^2-y^2} } \\, ; ", "c76a49a6a9b02d3755180488231b38c4": "M_Z^{gi} \\times n = 0", "c76af145ceef3a01d3c91869638dcbaf": "\\alpha \\cdot d", "c76af2c3f04c8aed2817d94ad331a37d": "x * y = \\scriptstyle{DTFT}^{-1} \\displaystyle \\big[ \\scriptstyle{DTFT}\\displaystyle \\{x\\}\\cdot \\ \\scriptstyle{DTFT}\\displaystyle \\{y\\} \\big],", "c76b35ecbd641b608f4bf096f2e70c74": "\\xi\\to\\infty", "c76b4f3a3e432eb564b6c8d8b8ec4849": "\\psi(\\mathbf{x},t)", "c76b623b676df8ea0fa13661fbe20c4c": "-196884/e^{\\pi \\sqrt{163}} \\approx 196884/(640320^3+744)\n\\approx -0.00000000000075", "c76b8a2381f965c5537be6364aa621ee": "S'(0) =\\ 0", "c76c22636cd1a56f44a867eefa986e5a": "X^w", "c76c8e5b52dbcc6c380e164d7aec04ad": "\\frac{N + \\Xi}{2} = \\frac{3 \\Lambda + \\Sigma}{4} \\, ", "c76cb8b8bd89fc65cec59ee70db71c2f": "\n\\mathcal F_\\infty = \\sigma\\left\\{ X_s^{-1}(B) : s\\in[0,\\infty), B \\in \\mathcal E\\right\\}, \n", "c76cefa9ceabfc24ca7dd13c84099295": "i_n = i^*_n + rp + lp\\,\\!", "c76d517da406f731c3d397f8be08dabf": "q \\mid a^2 - b^2p^*.\\,\\!", "c76dc9ad1633f575a6b19836981803d9": "M_\\mathrm{L} = \\log_{10}A - 2.48+ 2.76\\log_{10}\\Delta", "c76dff26f5fd1beb9d8083e8a61f3d7f": " (L-W)^2 ", "c76e6606d8ae949a9e7c19ca037c0cbc": "H(p,q)=\\tfrac 12\\langle p,M^{-1}p\\rangle+V(q)", "c76e6afba272e3093d64120e2da30ec4": "C=I_1\\times I_2\\times \\cdots \\times I_n", "c76e81729e16f07b1d9f2794d2ff3cbc": "\\sqrt{\\frac{4}{7}}\\!\\,", "c76ec8e0351c1a76017430354ce56f53": "\\phi_L\\,= \\phi |_(x=L) ", "c76ede717a68b176a84de344611a95bd": "\\begin{align} (1 + x)^\\alpha &= \\sum_{k=0}^{\\infty} \\; {\\alpha \\choose k} \\; x^k \\qquad\\qquad\\qquad (1) \\\\ &= 1 + \\alpha x + \\frac{\\alpha(\\alpha-1)}{2!} x^2 + \\cdots, \\end{align}", "c76f07aa94fb11b905907ef8f10c2665": "\\phi=\\frac{\\pi}{2}", "c76f32bd62587f0a278538cadf5f445a": "N \\hookrightarrow W", "c76fecc3f1c8839ef274c01e2a7de198": "B_n+C_n=B_{n+1}.\\,", "c77032a413ad1ea812fd2300409743e6": "\\bar\\lambda_q", "c770c5539faa71fc2193f5aa0c774b9a": "\\begin{matrix}\\frac12\\end{matrix} (5x^3-3x) \\,", "c770cd1e0195ee81b37b35f1e5303040": "\\{t,r\\}", "c770d59c2a6f453a56edb8b38ca651df": "\n\\mathrm{E_1}(ix) = i\\left(-\\tfrac{1}{2}\\pi + \\mathrm{Si}(x)\\right) - \\mathrm{Ci}(x)\n\\qquad (x>0)\n", "c771041f28769993f3c8b4e88a3496a5": "x_1=x_2 \\ ", "c77108f83215d303a622094b3b23b602": " \\frac{n + a}{n}\\, , ", "c7717fbb7cee2104d65c4743a77ca91e": "N = \\frac{V_s \\times c \\times r_s}{n_r} ", "c771ab907ba8f17228996985dd673364": "x = (x_1, x_2, \\ldots, x_n) \\,", "c772000136f6d3ba6dec3c7c5a35458b": "Y_1", "c77241aea8f2a9e93a1fd368d167b8e8": "\\theta_{ex}", "c7724b9c131002354204744220528e62": "\\exp_{10}^2(3.33931)", "c77260a96efb7ae5721349bfc15a4bdf": "\\varphi_{X_1+X_2}(t)=\\varphi_{X_1}(t)\\cdot\\varphi_{X_2}(t)", "c77277b24f058b4fb8ac7f5aba9c8898": "\\Big|\\prod_{i} r_{ii}\\Big|=\\Big|\\prod_{i} \\lambda_{i}\\Big|,", "c7728490ead659031d83ddbf96cd8bb3": " {\\sqrt{1-(2v/c)^2}} ", "c772a7307efecf252a4f7440c174e2e7": "V : [0, T] \\times \\mathbb{R}^{n} \\to \\mathbb{R},", "c772efbdb154b713c8503e746dba7861": "\\Leftrightarrow H_0 = \\frac{1}{2} \\frac{\\theta_0}{\\pi r_0^2}.", "c7730d13a43810215dc8a911a7d44f1b": " \\Phi_{1}\\left(\\mathrm{R}_{i+1}\\right)", "c7733e28b6c018c3671597ee1bcb380a": "{dQ_p \\over dt} = F_p (C_{art} - {{Q_p} \\over {P_p V_p}})", "c7734c0a7ca77fb52ce43867bdc3268f": " x_i = M x_o \\, ", "c7737a8f93df75ebeb63042e62ecf873": "E(Y|X)", "c7745bbc120505ea6ddfe1425c438d60": "G\\underline{A}", "c77470a8571c4d45c3f8cfbfae27b01b": "\\ \\displaystyle d\\in D \\ ", "c774e4b5c5bd34b85b04729781278e61": "\n C_0^N=(1+r)^{-N}\\sum_{n=0}^{N}\\left(\\frac{q^n{(1-q)}^{N-n}}{\\sum_{k=0}^{N}q^k{(1-q)}^{N-k}}\\right){[S_0{(1+b)}^n{(1+a)}^{N-n}-K]}^+\n ", "c77515a289317d8cc7c29acfebe74422": "m=-\\ell,\\ldots,\\ell.", "c775db50740e1659c1fbaa3ca6bbeff5": " \\lambda \\rightarrow \\infty ", "c775e9b963eb085fc173eb16331884bc": "D(\\theta)\\,", "c7762d753df71f9f7e649f71583efe35": " \\Delta I \\approx \\Delta t [{\\alpha IS \\over N} - {\\beta I^2 \\over N} - {\\beta IR \\over N}]", "c7766d55920ce59ded321ce203026e61": "\\nabla V", "c776d22bb05c4af2182dad04205f8937": "\\Omega_X^* (E) = \\Omega^*_X \\otimes E", "c776dabc8c8cdc2ca53e9ce0a204c42d": "L_{ij}=\\phi (v_i,v_j)", "c777303d41678350900016cc7f6888b0": "w^k=\\frac{\\partial y^k}{\\partial x^i}(0) v^i", "c77744e8e27e9d959ea6320654804f20": "\\mbox{If } 1 \\to y = 1, \\mbox{ then } y = 1 ,", "c7778a1fe695e1086079956ed02965c7": " I_{C, \\text{ball}} = \\int_{-R}^{R} \\frac{\\pi \\rho}{2} r(z)^4 dz= \\int_{-R}^{R} \\frac{\\pi \\rho}{2} (R^2 - z^2)^2 dz \n=\\frac{\\pi \\rho}{2}(R^4z - 2R^2z^3/3+z^5/5)\\bigg|_{-R}^{R} \n", "c77791bedc6ca1bfa0be8771afe75933": "dB(Ratio_{i+1}) = 2 \\cdot dB(Ratio_i)", "c777f711146529f437b7876b79c24943": " \\mathbf{y} = \\mathbf{c} + \\mu \\mathbf{d}", "c778004a6df6566dc69b8088344c0bb1": "\n\\begin{align}\n\\chi(D) &= h^0(L) - h^1(L) + h^2(L)\\\\\n&= \\frac{1}{2} c_1(L)^2 + \\frac{1}{2} c_1(L) \\, c_1(X) + \\frac{1}{12} \\left(c_1(X)^2 + c_2(X)\\right)\n\\end{align}\n", "c778073b76325f17352ca6e4fe66e886": "\\frac{\\operatorname dC}{\\operatorname dQ}=\\frac{\\operatorname d (C_0 + \\Delta C)}{\\operatorname dQ}=\\frac{\\operatorname d \\Delta C}{\\operatorname dQ}", "c77808a998779d6a5114b356a78c3b68": " d-1 ", "c77837d0ddd74751f06aba6f95b0905b": " \\tau_k = \\begin{cases} \n\\sigma_k, &k\\neq j, \\\\\n- \\sigma_k, &k = j.\n\\end{cases}\n", "c778387c6025f1976a56ed5e4c1eabdd": "\\mu/\\mu_0", "c7784600f9dd99c784607ef439f90ffa": " I_y(I_xu+I_yv+I_t) - \\alpha^2 \\Delta v = 0", "c778a92c64c393f775951115891aa8d3": "R(\\lambda)= \\frac{1}{I-\\lambda K}.", "c77974564ad42c8f8c3b7ea3294c1985": "Y_{11} = {((1 - S_{11}) (1 + S_{22}) + S_{12} S_{21}) \\over \\Delta_S} Y_0 \\,", "c779c5f5c8de52140113044c0f5489b6": "\\Sigma Y", "c77a0e36320ebcae37481ed902bd3f8e": "t\\text{ and }t+\\Delta t", "c77a225f7404032f2ab185aedb5c361c": "M^{t}", "c77a95d18f925d7d18430aeeb77e1d81": " \\frac{1}{(m - 1)n}E(SSB) = \\frac{\\sigma^2}{n} + \\tau^2.", "c77acf353fa67142bb71d3aca81e7808": "\\Delta\\lambda/\\lambda \\approx 10^{-5}", "c77ba97576cc67844d1ca4094f4fb9e4": "z!! = z(z-2)\\cdots (3)\n= 2^{(z-1)/2}\\left(\\frac{z}{2}\\right)\\left(\\frac{z-2}{2}\\right)\\cdots \\left(\\frac{3}{2}\\right)", "c77bb70daf4133de831224410235824a": " \\Bigl[ \\prod_{i=1}^n u_i(x) \\Bigr]_a^b = \\sum_{j=1}^n \\int_a^b \\prod_{i\\neq j}^n u_i(x) \\, du_j(x), ", "c77c64d5496da967a87faf1ab516d039": " C = \\frac{1}{p_{11} + p_{22} - 2p_{12}}.", "c77c68413a897bdf15490e6e0a97e312": "1-(1-r)^{n}-nr(1-r)^{n-1}-{n\\choose 2}r^{2}(1-r)^{n-2}", "c77c921af4034c5b4798fd1db3860876": "-3x^2-1800x+37200", "c77cfcbc998febdaf1e18b282092e606": "P(\\omega_s\\mid\\xi)", "c77d1fb6f1151e628c510aa7323ec39c": "S(\\rho) = H[\\{\\lambda_i\\}] \\,", "c77d2263275390528fc55bd4dac4e255": " q_{0}A+h A {(T_{\\infty} - T_{0} )}+\\epsilon \\sigma A {(T_{sur}^4 - T_{0}^4 )}+k A\\frac{(T_{1} - T_{0} )}{\\Delta {x}} + \\frac {e_{0}}{2}A \\Delta {x} = 0 ", "c77d5981e3f4ee46f56f5f1b67828d15": "\\begin{align}\n\\phi(t)\n& = \\phi_0 + 2\\pi \\int_0^t f(\\tau)\\, d\\tau \\\\\n& = \\phi_0 + 2\\pi f_0 \\int_0^t k^{\\tau} d\\tau \\\\\n& = \\phi_0 + 2\\pi f_0 \\left( \\frac{k^t - 1}{\\ln(k)} \\right)\n\\end{align}", "c77d63cb99f9bb651281d7637430e55e": " J=\\{j\\}, \\alpha=1, x_j=\\lfloor \\frac{w_i}{w_j}\\rfloor", "c77d70d081adea677a2d319c6184505a": "\\hbox{NAPE} = -\\int_{\\rm SFC}^{\\rm LFS} B\\,dz", "c77da002673641dd763bc56baa0827bb": "S_B = 4\\pi a_B^2 \\ ", "c77e0bb539ba3c06ad411303187ce44b": "k^1=\\frac{n_c}{n_w}", "c77e6afee4f1f2e32b619bd66e77ac3f": "\\scriptstyle E=\\frac{1}{8}\\rho g H^2", "c77e984767a3514709a30c7a50925efe": "\\mathcal{A}_\\lambda", "c77ebe2a79e708a5797fa93d65611859": "b \\geq 1", "c77f8ef04e328a7fcdad89fa5f0235c0": " \\nabla S ", "c77fc9750faf6b29185fb12274e54a27": "U(t+\\bigtriangleup t,w)=U(t,w)\\exp \\bigg(|\\frac{w}{w_r}|^{-\\gamma} \\frac{|w|\\bigtriangleup t}{2Q(w)}\\bigg) \\exp \\bigg( i|\\frac{w}{w_r}|^{-\\gamma} w\\bigtriangleup t\\bigg) \\quad (1)", "c77fd1ef5f86a25daf5ca30bebfdb7c9": "\n \\gamma = {\\frac{ C_p}{C_v}} ", "c77fe3385f6e5279c129f1619dd7415a": "\\rho (z)", "c78024f23144db57e8e53c8210e027e5": "B(x)\\le {{B}_{x}}(x)", "c78054d5880f3972be3ae3f4a7c5c912": "y=a\\left(\\frac{1-b}{2a}\\right)^2+b\\left(\\frac{1-b}{2a}\\right)+c", "c78074723d8add1294745e443ac510c4": " \\alpha > \\alpha_c", "c780e038b0b533741e67464700e61abe": "X\\,\\sim\\,\\textrm{Levy}(0,c)", "c780f52ace87592235dfd7de17333fae": "-2 < \\beta \\le -1", "c781a3ba13d5bb086ff2e3ea162123f0": "\\begin{align} H^{'} = H + P= \\begin{pmatrix}E_{1}&0\\\\0&E_{2}\\end{pmatrix} + \\begin{pmatrix}0&W\\\\W^{*}&0\\end{pmatrix} = \\begin{pmatrix}E_{1}&W\\\\W^{*}&E_{2}\\end{pmatrix} \\end{align}\\,\\!", "c781bb4a227b070221af580c3d80b999": "f(\\Phi, Q)=0", "c781fff660407613cf1c7987c7f546d9": "\\scriptstyle v \\in V \\setminus \\{s, t\\}", "c782025b405987fca26a899eb512524c": " \\delta\\ \\mathbf{r}^T \\mathbf{R} \\qquad \\mathrm{(16)}", "c782551757fc6d84716da8aeb1f891f7": "\\mathbf{x_i}", "c7828de518690a6e4cfd54a8f4025f61": "\\frac{1}{\\tau}\\ll\\epsilon_p", "c782a8ea83562d2d13b5c3d9323ff54e": "N! = \\Gamma(N+1)=\\int_0^\\infty e^{-x} x^N \\, dx. ", "c782aa7498cd4418c73a05131cff5b04": "\\text{free}(\\ \\Gamma\\ )", "c782b3a5a5eb1b8404e45ebcdcffb20b": "\\mu (=k\\theta)", "c782d1c1e53fce09e85f86516324528f": "\\mathbf{R}^m", "c78350ff5db7f15719756895313d7a65": "w_20 \\; \\end{cases}", "c79857f5afb23e31c07ad4388810410c": "K^1(A)", "c7987f1bbfe2dffe0c3c9f73214a1f69": "m(\\varnothing) = 0. \\,\\!", "c7989b89e7b42a7aaf0119f37c2bd90e": "\\langle a, b \\mid aba = bab \\rangle", "c799681b2a0fb7c2094ebe82855ed99c": "M_0", "c79a2b6d5f6288f8bb48b82b9a5bbc58": "2d=21 cm", "c79a2d10ce59e2c7e6bb722de28333a2": "\\widehat{X} \\neq X", "c79a6cb75e223b467e667fbb6ea603c4": "\\{f(x_1,\\ldots,x_n) \\mid \\phi\\}", "c79a73bd5a0f0c62ad265fd48f96a140": " \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{a}| \\, |\\mathbf{b}|} = \\cos \\theta \\,", "c79af5d0f4c25e878a5c89150f5bb6be": "E_2=x_0x_1+x_1x_2+x_2x_0", "c79af8e8786b06d48ce27bc7ed1ac4c8": "f={\\rm slog}_{\\rm e}(z)", "c79b0d4b224b0c7d7fdb7e237018ede6": "(p_x, p_y, p_z)", "c79b119d9d9078ea5a96fffb76c27ce1": "|\\psi_{01}\\rangle = |+\\rangle = \\frac{1}{\\sqrt{2}}|0\\rangle + \\frac{1}{\\sqrt{2}}|1\\rangle", "c79b1cedc837420ec152077ebb934a46": "\\widehat{X}", "c79b54dc58357ba7eccc3178459bc9b4": "f_{\\rho}", "c79b55176e6352e5fcd8f01fae36c45f": "((x_n),(\\phi_n))", "c79baa0adc413bf8072a2be5b450c2da": "(K, \\langle \\cdot,\\,\\cdot \\rangle, J)", "c79bda0e66f07bee662a6840b907bf39": "BV", "c79c1fda5b0d3af05e966a300bab33dc": "P^{1-\\gamma}T^\\gamma \\sim M^{2 - \\gamma} R^{3\\gamma - 4}", "c79c287553fd761e73af1999b68753aa": " \\mathbb{R}\\smallsetminus\\mathbb{Q} = \\mathbb{J} ", "c79c2dfff10ae061ef3a921192093bdf": " m_t = m_g ", "c79c54b529fbe89d63c8cfccdff8245c": "\\sin(z), \\cos(z), \\exp(z), \\exp(-z), \\text{and }\\exp(-z^2).", "c79c6c0cdcbcd45ddc5a84aa73335c99": "\\begin{align}\n p\\left( \\tilde{s},\\tilde{x},\\tilde{u}\\vert m \\right) & = p\\left( \\tilde{s}\\vert \\tilde{x},\\tilde{u},m \\right)p\\left( {D\\tilde{x}\\vert x,\\tilde{u},m} \\right)p(x\\vert m)p(\\tilde{u}\\vert m) \\\\ \n p\\left( \\tilde{s}\\vert \\tilde{x},\\tilde{u},m \\right) & = \\mathcal{N}(\\tilde{g}(\\tilde{x},\\tilde{u}),\\tilde{\\Sigma}(\\tilde{x},\\tilde{u})_s) \\\\ \n p\\left( {D\\tilde{{x}}\\vert x,\\tilde{{u}},m} \\right) & = {\\mathcal{N}}(\\tilde{f}(\\tilde{x},\\tilde{u}),\\tilde{\\Sigma}(\\tilde{x},\\tilde{u})_x ) \\\\ \n \\end{align}", "c79cdee92179ed31e947358175062d92": "\\hat p = - i \\hbar {\\partial \\over \\partial x} \\, .", "c79d0fdf947ee342bfa31d150083c987": "\\nu(A) = \\int_A f \\, d\\mu", "c79d20f7633de2f004d97e1c5a5d1f9b": "MM_n(mendo,mexo)", "c79d2248d70d43887ab5a3fc571a5af4": "\\Theta=\\frac{v_s}{c}\\frac{x_s}{r_s}\\frac{df}{dr_s}", "c79d59e4f91ee13896b45c50b5d9f0ae": "d=-12,-16,-27,-28.\\ ", "c79d8b6aa2d29a518e1e279a7551bb46": " y_0 + hf(y_0) = y_1 = 1 + 1 \\cdot 1 = 2. \\qquad \\qquad", "c79dcc257d555d88cbcb29ed0f7d3cf2": "10! = [(1\\cdot10)]\\cdot[(2\\cdot6)(3\\cdot4)(5\\cdot9)(7\\cdot8)] \\equiv [-1]\\cdot[1\\cdot1\\cdot1\\cdot1] \\equiv -1 \\pmod{11}.\\,", "c79dfd04d4d9ccd7c6745ff959606414": "10^{k-1}", "c79e0bc084ca8eaa958f03589505831c": " \\operatorname{build-param-lists}[o\\ x\\ y, D, V, L] ", "c79e1fc215f22905d69086b43c99dafb": " m = \\sqrt{\\frac{h c}{2 G}} ", "c79e28709315c87470fd4cdd1939466c": "\\scriptstyle E \\;=\\; E_0 \\,+\\, \\rho^n E_1", "c79e4986c599e22194a278dcfe36fe9b": "s\\in S^{(t)}", "c79e6e2077bc0846b0fe802b7f9912e7": "\n\\begin{array}{lcl}\np(\\tilde{x}|\\mathbf{X},\\boldsymbol{\\chi},\\nu) &=& p_H\\left(\\tilde{x}|\\boldsymbol{\\chi} + \\mathbf{T}(\n\\mathbf{X}), \\nu+N\\right)\n\\end{array}\n", "c79e8c6c5f3750c4f4491f50a9258cf4": "\\mathbf{a_{\\mathrm{Cor}}} ", "c79ef23fe451b537f26a2696fc97c7e8": "f:\\mathcal{X} \\mapsto \\mathbb{R}", "c79efceab74cb3b4c0f57e7166e8b7fb": " \\dot{q} \\equiv \\frac{\\mathrm{d} q}{\\mathrm{d} t} \\,\\!", "c79f409b4ad57a6fc0b4408991f43731": "\\ Y = AF(K,L)", "c79f8a67d3a4f1fbd2e45321c2a55901": "x[T^*]y", "c79fd10b764182e8b9acf7d965c42f64": " g(x)=x+1.", "c7a08a4e7bbc468ce875b507478ece5d": "L=Ad = \\dfrac{nl\\sqrt{n_{e}}}{n_{e}} = \\dfrac{nl}{\\sqrt{n_{e}}}", "c7a092e2e227e33b9b307f77aca0a53f": "\\alpha^m \\cdot \\alpha^n = \\alpha^{m+n}", "c7a0cc9724e9b9d14f1c5b20a7dfd12f": "T2'", "c7a0ee377a0c10aa142e22c7f4581661": " X=X^i \\partial_i ", "c7a0f8b07cd629fd4935d6bc04fd8e8e": " M = \\sum_{i=1}^j\\ f_i - 6 ", "c7a1442cf85a297c7df19480f84fecb5": "\\frac{2(\\boldsymbol\\Sigma \\otimes \\boldsymbol\\Omega)}{\\beta(2\\alpha-n-1)}", "c7a1a1b02dc2466362a1bd40c64549c2": "X^{1-o(1)}", "c7a202c1c5299c8f55ead6c40684a37d": "\\bold{\\nabla} \\cdot \\bold{E} = \\frac{\\rho}{\\epsilon_0},\\quad \\bold{\\nabla} \\times \\bold{B} - \\frac{1}{c^2} \\frac{ \\partial \\bold{E}}{\\partial t} = \\mu_0 \\bold{J} ", "c7a26bab8543a7e7aecb41ec36c957f6": "\\begin{matrix}x_{1,i} & = & \\mu_1 & + & \\ell_{1,1}v_i & + & \\ell_{1,2}m_i & + & \\varepsilon_{1,i} \\\\\n\\vdots & & \\vdots & & \\vdots & & \\vdots & & \\vdots \\\\\nx_{10,i} & = & \\mu_{10} & + & \\ell_{10,1}v_i & + & \\ell_{10,2}m_i & + & \\varepsilon_{10,i}\n\\end{matrix}", "c7a2a780e8c62fc782e04b96453ca6db": "{\\boldsymbol S} (c)\\cap c'=\\emptyset", "c7a2f9f8d3f5b056b50ec03fff6bc2d8": " \\frac{v_{k+1} -2v_k + v_{k-1}}{h^2} = \\lambda v_{k}, \\ k=1,...,n, \\ v_0 = v_{n+1} = 0.", "c7a2fc8aa9134a039dcf23a49fbfde5c": "\\zeta=2(z-z_{c})/h_{z} ", "c7a30128350fff2a21582a1ddeadfeae": "a_{k,j_t} \\neq 0", "c7a31b6aa78bde2bce2b540d2759f09d": "\\psi'(x_i, t) = V(x_i) \\psi(x_i, t)", "c7a33d7c515a8690992871fc2469aa43": "v(0) = h(0) = 0", "c7a36834eba77b493e008675a0e0e43a": " \\sum_{S_i} x_i \\geq y_{ij} ", "c7a36e65d90f1db52033915989cf8a6b": "C(a) < C(b)", "c7a3b7c0dba9bfdf0d9f292482ba4e1d": "{\\rm Fm}/{\\equiv_{T}}", "c7a3c62ab5ada2a0a7bde3b2cf4b57bc": "\\int \\sin ax \\tan ax\\;\\mathrm{d}x = \\frac{1}{a}(\\ln|\\sec ax + \\tan ax| - \\sin ax)+C\\,\\!", "c7a41c86cfaa4b531cafc9024376851e": "x+S(y) = S(x+y)\\ ", "c7a41ebf4ce876ced88ba2a9e6f7a8d7": "\\textstyle S=\\sum_k \\sum_j r_k W_{kj} r_j\\,", "c7a4226e2ba22dbd24c819f4baa863b3": "G_B", "c7a43b6dd9724b572e6cf13392c4640e": "\\rho_{\\rm H_2O}", "c7a4509ac1eee53a67ed9ac27616efb7": "y^2 = x^3 - n^2x", "c7a459581eabea51ae3e99bbf0c5cf93": "p(x+5)", "c7a4a483a7d7606e1146c301c5592d6c": "\\begin{align}H_p &= - \\int |\\phi(p)|^2 \\ln (|\\phi(p)|^2 \\cdot \\hbar / \\ell ) \\,dp \\\\\n&= -\\sqrt{\\frac{2 \\ell^2}{\\pi \\hbar^2}} \\int_{-\\infty}^{\\infty} \\exp{\\left( -\\frac{2\\ell^2 p^2}{\\hbar^2}\\right)} \\ln \\left[\\sqrt{\\frac{2}{\\pi}} \\exp{\\left( -\\frac{2\\ell^2 p^2}{\\hbar^2}\\right)}\\right] \\, dp \\\\\n&= \\sqrt{\\frac{2}{\\pi}} \\int_{-\\infty}^{\\infty} \\exp{\\left( -2v^2\\right)} \\left[\\ln\\left(\\sqrt{\\frac{\\pi}{2}}\\right) + 2v^2 \\right] \\, dv \\\\\n&= \\ln\\left(\\sqrt{\\frac{\\pi}{2}}\\right) + \\frac{1}{2}.\\end{align}", "c7a4cd5bbfd4486e80dffbe8f8d3363d": " u\\in C", "c7a4fe1ba8b9f474f16b32a0199e6771": "\n 2) \\quad \\alpha \\subseteq \\beta: H(\\alpha) \\leq H(\\beta)\n", "c7a505f24b003a0a4793192303d7aa06": "\\alpha, \\beta \\in \\mathbb{N}^n_0", "c7a5490cbb6530c919ad2322dde48467": "u(t)\\leq G^{-1}\\left(G(\\alpha)+\\int_0^t\\,f(s) \\, ds\\right),\\qquad t\\in[0,T],", "c7a5621358e762e74a1ac3e6e0f1c160": "\\operatorname{E}[|T_N|]=\\sum_{i=1}^\\infty |T_i|\\operatorname{P}(N=i).", "c7a590a3ba14ad65d43d764087f84c92": " -\\int dx^\\mu dx^\\nu B_{\\mu\\nu} ", "c7a5c9b33423b94c9b0fd8e14f386aeb": "{}^{i-1}T_{i} = [Z_i][X_i] =\n \\operatorname{Trans}_{Z_{i}}(d_i)\n \\operatorname{Rot}_{Z_{i}}(\\theta_i)\n \\operatorname{Trans}_{X_i}(a_{i,i+1})\n \\operatorname{Rot}_{X_i}(\\alpha_{i,i+1}),", "c7a5cb41fd8eaaf6b65e12675e8cb20f": "\\textstyle x_{+j} = \\sum_i x_{ij}\\,", "c7a5de4ca9b63aeb65dba9162099c32e": "s_1 = \\frac{(x_2 - x_2)^2 + (y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n= \\frac{(y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2},", "c7a64c57518534c3b6ae87fefc5543f8": "\\tau_\\mathrm{n}=0\\,\\!", "c7a6758e66f4eb3141be7c47cc44b19a": "\\left[ \\hat{ E}_k (\\boldsymbol r ) , \\ \\hat{ B}_{k'} (\\boldsymbol r') \\right] \n= -\\epsilon_{kk'm}\\frac{i \\hbar}{\\varepsilon_0} \\ \\frac {\\partial}{\\partial x_m} \\delta (\\boldsymbol{r-r'}) \\ , ", "c7a6c85e719819ec785621ab117ea06d": "\\operatorname{E}[X_i] = \\frac{\\alpha_i}{\\sum_k \\alpha_k}", "c7a6cab7260889162d3f860dca656224": "\\scriptstyle O(n^{\\lfloor d/2\\rfloor})", "c7a6cbaea65172ffecb32271108b78c5": " \\operatorname{Vol}^{s_1}(V)\\otimes \\operatorname{Vol}^{s_2}(V) = \\operatorname{Vol}^{s_1+s_2}(V). ", "c7a7040f8fb65034a7028055b8387c5a": "w_{i}>0", "c7a7435b4bf84e53e9d0d51b3f00778a": "u_x = v_y, \\quad v_x = -u_y.", "c7a7711ecead4bfd5f91cc077500fca0": "(1/p)^n", "c7a7bc53d27bd47212f647c1aa9fecb6": "e^{i\\omega t}", "c7a80b532f19c4d9ee9fe492e3a776e0": "\\mathrm{P}(E_i | \\rho)", "c7a825bc613aad8e8e76df773516fd5f": "a^2 - n", "c7a83f309f70eeeab25c85747b86ddef": "F(E_*).", "c7a86abb179cf7b66d7e6e626ed5b1b8": "|U_{e3}|^2=0", "c7a88ab007fbedfefdf5614439c9ea8f": "F_z=B_IA_{mag}\\int_z^{z+h_{mag} } \\frac{dHz}{dz}dz", "c7a8921967636968ec0a735bc1f65a92": "\\frac{x_1 + x_2 + \\cdots + x_n}{n} \\ge \\sqrt[n]{x_1 \\cdot x_2 \\cdots x_n}\\,,", "c7a8994c38039eda99c694a5691d2e7c": "\\eta \\geqslant 0.5", "c7a8abf9bcdf86033f1ab2014ce16777": "\\rho^{'[DK]}=Tr_{JC}|{\\psi'}\\rangle\\langle{\\psi'}|=\\sum_{j,j',\\gamma,\\gamma'}\\rho^{jj'}_{\\gamma\\gamma'}|{j\\gamma}\\rangle\\langle{j'\\gamma'}|.", "c7a8f86bdf929dc170bf3ebc4606d47a": "10^{27}", "c7a9a5948e1212e92c66ce25593799f2": "s= \\tfrac{1}{2}(a+b+c)", "c7a9bb0b05fed3a720757f11861604c5": " \\nabla_{\\lambda X}v=\\lambda \\nabla_Xv", "c7aa2e2efaab042625fd66071bd5c3be": "\\frac{d}{dx}\\left(\\frac{1}{g(x)}\\right) = \\frac{- g'(x)}{(g(x))^2}", "c7aa77883eed5098c6d430b8258275ba": "f(x)=x^6+3x^5-5x^4-15x^3+4x^2+12x=x(x-1)(x-2)(x+1)(x+2)(x+3) \\,", "c7aa826086e9675ef772ecd4e192bf57": " V_v = 1 - V_f - V_m = \\frac {v_v}{v_c} \\!", "c7aaae73b6f94d63b2a211c8589273ca": "r \\approx R \\frac{7M_2}{12 M_1}", "c7aad0c0a95fe59827c856ea39f0c982": " t \\rightarrow \\infty", "c7ab033c0e650e6b446da1682b116a83": "a(t)\\propto t^{2/3}", "c7ab078521ca54ae8fe94c28057d684c": "\\alpha_1, \\alpha_2, \\alpha_3, ..., \\alpha_n ", "c7ab4b2e7089b9434c07bee8c11d4a1d": "r, \\phi_1,\\ldots,\\phi_{n-1}", "c7ab74504bd7b498fafd84b94dcb91a2": "Pf(z) = \\int_{\\partial\\Omega} f(\\zeta)\\overline{k_z(\\zeta)}\\,d\\sigma(\\zeta).", "c7ab77381814e37acfe00c91b4d0920b": "A(\\mathbb{R})", "c7abaea49554451a9d3f3d00ea737438": "\nr_8(n) = 16\\sigma_3^*(n).\\;\n", "c7ac0e2097897a74db55b26f9f672938": "e^{m\\sqrt{-1}}=\\cos m+\\sin m\\sqrt{-1}", "c7ac8ae6784f0cefab26f6a57e59a4e1": "Y=V_{sig}\\sin\\theta", "c7ad0b06d3d736ddc11596d504568747": "L=2S-\\frac{200(\\sqrt{h_1}+\\sqrt{h_2})^2}{A}", "c7ad3117f39a4c440eb389449bb00539": " \\bar{u} ", "c7ad9275f2a1c28f5025ec413b3c8d6a": "\\phi(x) = \\Phi e^{i\\theta(x)}", "c7ae5ccaead2d76dc5e2e1c1b170ac18": "b^2=4c,", "c7ae68c7186b80c9d40c09146e9988fa": " R \\otimes_S R \\rightarrow R ", "c7ae6e6f7e65becd158c470c0b717ce6": "\\left(1-\\frac{2Gm}{c^2 r}\\right) = \\left(1-\\frac{Gm}{2c^2 r_1}\\right)^{2}/\\left(1+\\frac{Gm}{2c^2 r_1}\\right)^{2}", "c7ae9cda818b00ec56ec1676e0a7d762": "\\scriptstyle h,\\, g", "c7aed7a6e7bf9ceda30a1d3c927afde0": "\\rho_{fw} \\,", "c7af6c5a821f6eb91605261d611aedfe": " |\\psi\\rangle = a_R \\exp \\left ( i \\alpha_x -i \\theta \\right ) |R\\rangle + a_L \\exp \\left ( i \\alpha_x + i \\theta \\right ) |L\\rangle ", "c7af744c1d2b7621e63b13e71944bf09": "i\\hbar \\frac{\\partial\\Psi}{\\partial t} = -\\frac{\\hbar^2}{2m} \\nabla ^2 \\Psi + V \\Psi + m \\Phi \\Psi ", "c7af85340f0d27afe0ae93575c67d2e1": "|x-y|", "c7af956a833f3ed03cb4b50cca8e126f": " \\psi(L) = 0 = C\\sin kL.\\!", "c7af95c5e983b853836d6cf7407ff023": "\\varphi\\colon \\mathcal{O}_X^n|_U \\to \\mathcal{F}|_U", "c7afb19a6cd88d9bfa5e0c4100d51a5e": "\\phi^1=\\phi^2=\\phi^3=0", "c7afcb4b1e84cb772e2896d391e5215c": "de_2=R \\cos{(\\alpha)} d\\alpha \\wedge d\\theta", "c7b0085bd03801ad03540ce7feb4fddb": "\\omega(x_i, x_j) = \\omega(y_i, y_j) = 0.\\,", "c7b01cf881d08824dd75fae1b1a8bfef": "c_s.", "c7b05fc27461935fcb2b5d4ef5e925e3": "y = \\pm x / \\sqrt2", "c7b08fd4170c51e5be073b14d1b77347": " \\begin{align}\n\\tan(\\theta_1 + \\theta_2) &\n= \\frac{ e_1 }{ e_0 - e_2 }\n= \\frac{ x_1 + x_2 }{ 1 \\ - \\ x_1 x_2 }\n= \\frac{ \\tan\\theta_1 + \\tan\\theta_2 }{ 1 \\ - \\ \\tan\\theta_1 \\tan\\theta_2 }\n,\n\\\\[8pt]\n\\tan(\\theta_1 + \\theta_2 + \\theta_3) &\n= \\frac{ e_1 - e_3 }{ e_0 - e_2 }\n= \\frac{ (x_1 + x_2 + x_3) \\ - \\ (x_1 x_2 x_3) }{ 1 \\ - \\ (x_1x_2 + x_1 x_3 + x_2 x_3) },\n\\\\[8pt]\n\\tan(\\theta_1 + \\theta_2 + \\theta_3 + \\theta_4) &\n= \\frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\\\[8pt] &\n= \\frac{ (x_1 + x_2 + x_3 + x_4) \\ - \\ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \\ - \\ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \\ + \\ (x_1 x_2 x_3 x_4) },\n\\end{align}", "c7b0bc0479e0643208c8ae5294434bb3": " f_0(x)", "c7b0be4ef172d8c6ecc49fa1eb8d585c": "\\pi pq", "c7b128e80e20799ceb942afb1dcf5f2e": "\\{a^i,b^i\\}", "c7b186aa5df0bb197b9bce82381b5dc0": "(\\Sigma, D)", "c7b1c6d93585b69846c38c59b710e092": "\\mathcal A = \\{A: \\{0, 1\\}^n \\to \\{0,1\\}^*\\}", "c7b2e2dd882eef15ea9b71ffbcab3d66": "V_n = \\frac{2 \\pi}{n} V_{n-2}", "c7b2e486b4cfdf3bd4e97e3a6aa61d63": "\\alpha = \\sum_{i,j}a_{ij}e_i\\wedge e_j", "c7b2f3a746ba8fb5a2d6129ad0d643f7": "p_1,p_2,\\dots,p_k", "c7b30f1dcba76d094fcc3f0549faa202": "I(\\mathbf{v}) = \\int dx P_{\\mathbf{v}}(x|spike)log2[P_{\\mathbf{v}}(x|spike)/P_{\\mathbf{v}}(x)]", "c7b460dac32b07e3e875fc44b2234c23": "\nP[X>x] \\sim x^{- \\alpha},\\ \\text{as} \\ x \\to \\infty, 0< \\alpha <2\n", "c7b46d6d77c9904b30ec87597007f050": "i \\leq \\lfloor n/2 \\rfloor", "c7b49aa190b2f18494f83fc75f1ccb3d": "\\scriptstyle x \\;\\equiv\\; y \\pmod{n_i}", "c7b4c55d5a1cda4d2e23a50868166aaf": "Y_t=\\sum_{j=0}^\\infty b_j \\varepsilon_{t-j}+\\eta_t,", "c7b4d82f7c5fa901f806155b433fe45c": "\\begin{matrix} {11 \\choose 1}{4 \\choose 3}{44 \\choose 1} \\end{matrix}", "c7b4d8341a25e3a43e74c44e2fe5feaf": "\\phi = \\hat{f} \\circ f.", "c7b5000497704a8c230c06a22145f670": "7/8+\\Omega(1/B)", "c7b50529dabda495f674571b80be8ef1": "V''\\subset Y", "c7b54441e8921604cac70b8f8ba26af4": " \\omega^2_p(f,t) = \\sup_{|h| \\le t} \\| \\Delta^2_h f\\|_p", "c7b561340880435f28c11b0fe1afd270": "\\mathcal{N}_\\partial (X \\times I)", "c7b5614a9ef1f0f8282b434c1239811d": "J_\\nu(x) = \\sum_{k = 0}^\\infty \\frac{(-1)^k \\; (x/2)^{\\nu + 2k}}{k! \\; \\Gamma(\\nu + k + 1)}, \\text{ for } x \\ge 0", "c7b5c2fcb64d75923de9d42353f4a17d": " \\text{Sl}_3(\\theta)= \\frac{\\pi^2\\theta}{6} -\\frac{\\pi\\theta^2}{4}+\\frac{\\theta^3}{12} ", "c7b5e714ad2d1ece8603d173d48a977a": "O((\\log n)^\\frac32)", "c7b64b3a5c8b710fbc039a39903d8eb0": "\\Omega_{e_{i}}", "c7b6ac4723e09880fc3907f4d4d6740a": "1-\\gamma=\\sum_{n=2}^\\infty \\frac{1}{n}\\left[\\zeta(n)-1\\right]", "c7b6b584e79fc84fabc5692594b604e5": " c_{1}, \\ldots , c_{n}", "c7b7100fc69a0b13e7c24782926a2743": "z \\ne 0", "c7b78f7c385ff6834474b527ecb3f327": " V^{-1}(x) = \\sqrt{4\\pi} \\frac{d^{1/2}N(x)}{dx^{1/2}} ", "c7b79ad6f357803dd388c56a641c9321": "V \\approx \\sum A(r) \\cdot \\delta r.", "c7b7d617589ff6b6ba3c88d9904e7086": "T^{\\mathrm{Y}}_p (x,y) = \\begin{cases}\n T_{\\mathrm{D}}(x,y) & \\text{if } p = 0 \\\\\n \\max\\left(0, 1 - ((1 - x)^p + (1 - y)^p)^{1/p}\\right) & \\text{if } 0 < p < +\\infty \\\\\n T_{\\mathrm{min}}(x,y) & \\text{if } p = +\\infty\n\\end{cases}\n", "c7b7ecbe6eba43f751655edcfa9e6431": "\\mathrm{Domain}(c(\\nu)) := \\{\\langle n, k \\rangle | n \\in \\mathrm{Domain}(\\nu)\\}", "c7b7fef03c5b0de65a229b9d2aec2b82": "p(N,M;n) - p(N,M-1;n)", "c7b85f242388ed6f84aff8b37c01fce7": "\\begin{align}\nc_1(t)&=c_1(0)\\left[\\frac{\\Omega_c ^2}{\\Omega^2}+\\frac{\\Omega_p ^2}{\\Omega^2}\\cos\\frac{\\Omega t}{2}\\right]+c_2(0)\\left[-\\frac{\\Omega_p \\Omega_c }{\\Omega^2}+\\frac{\\Omega_p \\Omega_c }{\\Omega^2}\\cos\\frac{\\Omega t}{2}\\right]\\\\\n&\\quad-ic_3(0)\\frac{\\Omega_p }{\\Omega}\\sin\\frac{\\Omega t}{2}\\\\\nc_2(t)&=c_1(0)\\left[-\\frac{\\Omega_p \\Omega_c }{\\Omega^2}+\\frac{\\Omega_p \\Omega_c }{\\Omega^2}\\cos\\frac{\\Omega t}{2}\\right]+c_2(0)\\left[\\frac{\\Omega_p ^2}{\\Omega^2}+\\frac{\\Omega_c ^2}{\\Omega^2}\\cos\\frac{\\Omega t}{2}\\right]\\\\\n&\\quad-ic_3(0)\\frac{\\Omega_c }{\\Omega}\\sin\\frac{\\Omega t}{2}\\\\\nc_3(t)&=-ic_1(0)\\frac{\\Omega_p }{\\Omega}\\sin\\frac{\\Omega t}{2}-ic_2(0)\\frac{\\Omega_c }{\\Omega}\\sin\\frac{\\Omega t}{2}+c_3(0)\\cos\\frac{\\Omega t}{2}\\end{align}", "c7b8b4653c2865e5b5afa1a38e6555c3": "\\, v_{rel} = v -(+w)", "c7b91cb71571ee199423a7d080404e87": "avail_i=\\sum_{i=0}^{|s_i|}\\sum_{j=i+1}^{|s_i|} conf_i.conf_j.diversity(s_i,s_j)", "c7b91e90aea7f29d60c03042dc413c5d": "\\Phi(\\mathbf{r}_3) = \\Phi_1(\\mathbf{r}_3) + \\Phi_2(\\mathbf{r}_3) = \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_1}{r_{31}} + \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_2}{r_{32}}", "c7b936797c13050891f6f7c9f2da52d1": " J_\\text{tot} = J_{\\text{c}} + J_{\\text{d}} = \\sigma E - i \\omega \\varepsilon' E = -i \\omega \\widehat{\\varepsilon} E ", "c7b960d2c8042b60395d29db98f2d48a": "Y\\to \\Sigma", "c7b9e59a746148ea488f4650fc4f074a": "{1 \\over p(1-p)}", "c7ba47ed522f302d9080caa976351ecf": "\\forall k, \\alpha", "c7bb26ca7f23038c05d47c3dd2f1a3eb": " \\sgn (X_2 - X_1)\\ = \\sgn (Y_2 - Y_1)\\ ", "c7bb4937d38109b542f040cc16c03b82": "p(x)=x^n+p_{n-1}x^{n-1}+\\cdots+p_0=(x-z_1)\\cdots(x-z_n)", "c7bb9ebbf41353bf03894731ca4dbd25": "m = a/\\text{d}(P,Q)", "c7bc120bc3352085c793ea4cbabc7b13": "\\Delta\\omega\\,", "c7bc14f212c6d75376c921132153fe81": "s(a, b) = 0", "c7bc540937fb45736d01d0c962555d63": "a = -(u^2 - v^2) (1 + u^2/3 + v^2/3),\\ ", "c7bc96ecceb2456476854882e76c01b4": "C_{0}, C_\\text{ss}", "c7bccebf2549584485e6727f8aa76881": "d(\\mathbf{x}) = \\mathbf{x}", "c7bcd6e9b67e81af103cb7663e5a41fc": "\\mathrm{Set}", "c7bd25d043141fe34e4251df48ebefe0": "3+2\\sqrt{2}=5.82842\\ldots", "c7bda662d237440fc7fa376b492f0beb": "\\hat{r}\\leq\\hat{p}, \\hat{r}\\leq\\hat{q}.", "c7be021053b5273b9d87b224776f25f2": "IJKL", "c7be037011d8060769c3420c723c1229": "e\\cdot ln(N)", "c7be147eabf1ce434236efc91d58b947": " \\ a_i = a ", "c7be1ed226d5413a469a31676f5a0692": "f(0)=1", "c7be2e59f0811178cd8de10f11ffc6f5": "\\delta(\\mathbf{x})", "c7be345be95b2c34a97d644b57a27b3e": "H : \\{0,1\\}^* \\rightarrow \\{0,1\\}^k", "c7be4e40e9ae021090a354a1d2259707": "\\frac{a}{2^b}\\times \\frac{c}{2^d} = \\frac{ a \\times c}{2^{b+d}}.", "c7be596faa1dfb10e673f73f0e8ee471": "B_1,\\ldots,B_n", "c7bec4d628683de3ef19a59532ca7c92": " \\varepsilon i = (\\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_4) \\mathbf{e}_2 \\mathbf{e}_3 = \\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_4 \\mathbf{e}_2 \\mathbf{e}_3 = \\mathbf{e}_2\\mathbf{e}_3 (\\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_4) = i\\varepsilon.\\!", "c7bee8990085a3747c315835dd818ae2": "\\frac{1}{2}+\\frac{2}{3}\\;=\\;\\frac{3}{6}+\\frac{4}{6}\\;=\\;\\frac{7}{6}", "c7bf0feeb6cc05505be14fd777528bfa": "i'^*\\mathcal{F}", "c7bf53f11481bae8dfaed0de7c4afd82": "H(f) = \\begin{bmatrix}\n\\frac{\\partial^2 f}{\\partial x^2} & \\frac{\\partial^2 f}{\\partial x\\,\\partial y} & \\frac{\\partial^2 f}{\\partial x\\,\\partial z} \\\\ \\\\\n\\frac{\\partial^2 f}{\\partial y\\,\\partial x} & \\frac{\\partial^2 f}{\\partial y^2} & \\frac{\\partial^2 f}{\\partial y\\,\\partial z} \\\\ \\\\\n\\frac{\\partial^2 f}{\\partial z\\,\\partial x} & \\frac{\\partial^2 f}{\\partial z\\,\\partial y} & \\frac{\\partial^2 f}{\\partial z^2}\n\\end{bmatrix},", "c7bf9874607053c0934bae118cf26d60": "\\tfrac{O_1O_2}{BE} = \\tfrac{BE - BO_2 - EO_1}{BE} = 1 - \\tfrac{BO_2}{BE} - \\tfrac{EO_1}{BE} = 1 - \\tfrac{1}{2} - \\tfrac{3}{13} = \\tfrac{7}{26}.", "c7bfff78e52206ba1507f269e19ba2c2": "\\begin{align}\n\\underline{\\int_{a}^{b}} cf(x) &= c\\underline{\\int_{a}^{b}} f(x)\\\\\n\\overline{\\int_{a}^{b}} cf(x) &= c\\overline{\\int_{a}^{b}} f(x)\n\\end{align}", "c7c0830f12cae5d3fa534104022739aa": "\n \\displaystyle \n \\frac{V}{N\\Lambda^3} \\le 1 \n \\ , {\\rm or} \\ \n \\left( \\frac{V}{N} \\right)^{1/3} \\le \\Lambda\n", "c7c115ce5add4b42e65f460afa271445": "g(x)=e^{-x^2}", "c7c136ab6469ec85800910363c5d642b": " \\forall xA \\land E!t \\rightarrow \\exists xA", "c7c1459e8b00df1d57e9777da2977b89": " x_1 := x_1 - \\alpha p_1.\\, ", "c7c1547aa55b7a1d2c96e173b99f1636": " A_q(n,d,w) \\leq \\lfloor \\frac{n q^*}{w} \\lfloor \\frac{(n-1)q^*}{w-1} \\lfloor \\cdots \\lfloor \\frac{(n-w+e)q^*}{e} \\rfloor \\cdots \\rfloor \\rfloor ", "c7c1c001dbb370a4e73547fadacc75a4": "X \\in H(E_n \\; | \\; n<\\omega)", "c7c1d99c7351b66ff04f8a57c6d35807": "\\dot{p}_i = - \\frac {\\partial H}{\\partial q^i}.", "c7c1f3b0b7695633f3d1fe622fe1c0a5": "X_1=\\left\\{\\begin{pmatrix}x \\\\ y \\end{pmatrix}\\in \\mathbb R^2 : |x|>|y|\\right\\}", "c7c24813f04408d8ce4afd6da3c7c636": "f(x,y) = x - y.", "c7c24a63210a2d7a68ca205e55f47391": "\\varphi \\,", "c7c25d2881b819c16a48648908cf1373": "\\Gamma_2", "c7c26500aafe792569524468e6982c02": "\\, \\varepsilon_r", "c7c2666ce59e68f43b0f856dee1c13a8": "(X_1, \\ldots, X_n)", "c7c29aeab02a583841bd25eb25ac97aa": "S/D", "c7c2b6d1a782e67dd04527b8d17b36d9": " \\alpha, \\beta, \\gamma, \\delta ", "c7c2ebda041439dfa6b81727235aadef": "(x)_{n}=x(x-1)(x-2)\\cdots(x-n+1).", "c7c2f6f15ebdb874027ab23d15e0109e": " 2 - g ", "c7c31b6446ccec965a2ae68274e7b3c9": "x(t)=\\phi\\,(t){\\phi\\,}^{-1}(0)x_0", "c7c35e52af9ea4ee7c1e90bde618c468": "F_2=\\frac{F_{load}}{\\left [\\frac{Sin(\\beta )}{Sin(\\alpha )}Cos(\\alpha )+Cos(\\beta )\\right ]} \\,", "c7c365a74d082846ca5d386310ff5452": "\\sum_n c_n \\lambda^{-n} \\, ", "c7c378b0b161550b62945d3f0b488ecd": "\\,4^3 + 0^3 +7^3 = 407", "c7c37e74eda141425b02773d73e02942": "\\pi=\\frac{\\alpha}{\\alpha+\\beta} \\!", "c7c38204359d87bf819f6c8b209154e6": "p_i=\\frac{1}{n}\\ {\\rm for\\ all}\\ i\\in\\{\\,1,\\dots,n\\,\\}.", "c7c3a04f066867acba82aba1cf751527": "HIAT \\to \\begin{pmatrix} H \\\\ I \\end{pmatrix}, \\begin{pmatrix} A \\\\ T \\end{pmatrix} \\to \\begin{pmatrix} 7 \\\\ 8 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 19 \\end{pmatrix}", "c7c3ac0a675ec4c720859dcc19a6d02d": "I_{D}", "c7c3bb854548cc83748d02d68f381d1b": "a_{\\mathrm{prim}}(Z)=2Z^d\\,\\left(\\frac{1+Z}2\\right)^A\\,q_{\\mathrm{prim}}(1-(Z+Z^{-1})/2)", "c7c3cb9a0354d4a66eeefca2dca17a26": "X = 1", "c7c40b9031303c7941ec04d47d84184d": "E \\times_X G \\rightarrow E \\times_X E ", "c7c418083c54fc093900b53a6fb6d992": "{\\rm Tr}\\,\\gamma=1,", "c7c42309c359c25dfe77f397d5261997": "x\\in[a,b]", "c7c453790167ed0497139340b4f7663b": "p_i\\ ", "c7c4586e1af81049151b1f51a920f8b0": "f(z)=\\frac{g(z)-g(0)}{g'(0)}.\\,", "c7c458751c53aefbf9cd0c0889e08328": "\\tfrac{EO_1}{BE} = \\tfrac{3}{13}", "c7c47161dd8e5b162e35a242deb47cce": " \\triangle MXX'' \\sim \\triangle MYY'',\\, ", "c7c4a1a7f8e2df1870ed91f0c34bdff4": "\\frac {v_\\mathrm N - v} {v_\\mathrm N} = \\frac {v- v_\\mathrm F} {v_\\mathrm F} = \\frac {N c} f\\,.", "c7c4ab19b8e7b1ff3c023c12c0a956b5": "Rings + \\pi Bonds = C - \\frac{H}{2} - \\frac{X}{2} + \\frac{N}{2}+1\\,", "c7c4ec14d3a10bc696f9bd15abc50ea2": "(m_{1} + m_{2})(m_{1}u_{1})^{2} = (m_{1} + m_{2})(m_{1}v_{1})^{2}\\,\\!", "c7c4ed9702ef0c2cbc24999b17af3898": " \\psi = \\frac{1}{2}[(I/I_0)^a + (I/I_0)^{-a}] ", "c7c52d7c59773cb02bb2470a95e885d2": "U\\,d\\beta = d\\left(\\beta U\\right) - \\beta\\, dU\\,", "c7c537a14c051201381d02b9a967765e": "\\mathcal{M}(P(x))", "c7c5508f70423c751d09d0f80e841330": " G=H\\ast_C K=H\\ast K/{\\rm ncl}\\{\\alpha(c)=\\omega(c), c\\in C\\}.", "c7c5ac92ee8911dcd9f80b974da4d8e7": "H^p(X_s, \\mathcal{F}_s) = 0", "c7c5b9285a6dd48509ed00924bf5b4fc": " \\hat{G} = \\hat{A}\\hat{C} = (\\hat{a}_0 + \\mathsf{A})(\\hat{c}_0 + \\mathsf{C}) = (\\hat{a}_0 \\hat{c}_0 - \\mathsf{A}\\cdot \\mathsf{C}) + (\\hat{c}_0 \\mathsf{A} + \\hat{a}_0 \\mathsf{C} + \\mathsf{A}\\times\\mathsf{C}).", "c7c6040ffdc2d7fb87e9e010922ecfe5": " Hg_1^{-1},\\ldots,Hg_n^{-1}", "c7c63510d6b7db04aab39ea2b5d09739": "W = \\sum_i f(\\tfrac{H_i - E}{\\omega}).", "c7c65d1074e44c238a2b44730cd97ee0": "s_n \\to s_{n+1} \\to \\cdots", "c7c66e25e1d85f0a559f7f9665adf085": "\\mathbf{s}_{u}", "c7c671bd6cbba0ebe0e6c8e7ec8ff9fc": " \\left\\langle {\\exp [ - \\overline \\Sigma_t \\; t ]} \\right\\rangle = 1,\\quad \\text{ for all } t .", "c7c69231e3b62d7f112a6f3f5663ca7c": "S = k[y_1, ..., y_d]", "c7c6ce2029df4caac99c43a66a83cd95": "\n\\mathbf{B}(t) =\n\\frac{\n\\sum_{i=0}^n {n \\choose i} t^i (1-t)^{n-i}\\mathbf{P}_{i}w_i\n}\n{\n\\sum_{i=0}^n {n \\choose i} t^i (1-t)^{n-i}w_i\n}.\n", "c7c6d577cf3e4ac1c360e8d55137a9f8": " \\mathbf{D} = \\epsilon\\mathbf{E} = \\epsilon_0 \\mathbf{E} + \\mathbf{P}\\,", "c7c6e8680f4af370a6293a0c15e075fa": "\\mathbf{E}^{(e)}", "c7c7300e30cf204a70f6151c91f49fe4": "\\left\\{\\begin{array}{l}p\\\\q\\\\q\\end{array}\\right\\}", "c7c73102ce2149b20be0dffc518278eb": " U(0) = U^0 ", "c7c7772a1a5b0816b23a9a2a57cffb67": "r_{\\rm IT}", "c7c77835fe95623d7aca6f3c590f3420": "t_i \\in \\{ 1,0 \\} ^{|r|} \\ ", "c7c7c7379084f8cb08445a5928e012be": "s \\in S, r \\in R", "c7c7d672df9da5207510562e3dbaa7a7": "\\lambda_1 \\geq \\lambda_2\\geq 0", "c7c8111d59b3915168fe4a9a263a816b": "x = \\sqrt{ \\beta A_0 } + \\sqrt { \\beta A_0 +1 }. \\, ", "c7c87de9739bc1a02b00e602e26b015d": "\\vec{r}_B", "c7c8ec9ee8cec5ee6b4f2fe097892eff": "s_{zj}=2", "c7c95b6b53362feeb1d601695c25f0c3": "B(\\alpha ,\\beta )=\\frac{\\Gamma (\\alpha )\\cdot \\Gamma (\\beta )}{\\Gamma (\\alpha +\\beta )}", "c7c96c4b581a5d5e6188e8fbafbed02f": "\ndV = a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right) d\\mu d\\nu dz\n", "c7c9796ff7887f631697397b1e7a547d": "Q(x,y) = 0,", "c7c981638b60e4280270b8cc5aeb3e10": "\\Phi(x,t):=(4t\\pi)^{-\\frac{n}{2}}\\exp\\left(-\\frac{|x|^2}{4t}\\right).", "c7c996ae378a99270960baeaac3a4e2f": "M_{a}=", "c7c9aca557ec3fc12ac36cd1f3cba81c": " \\leq2^{2n\\left[ H\\left( \\mathbf{p}\\right) +\\delta\\right] }2^{-n\\left[\nH\\left( \\mathbf{p}\\right) +\\delta\\right] }2^{-\\left( n-k\\right) }", "c7c9bdfed0a9b78dfebee1f71ba64549": "v:X \\rightarrow D", "c7c9e7ee7fa243e5f78fe8d1c46698c1": " \\bigcup_n{\\mathcal{E}^n} = RP ", "c7ca1fd42a1fb4f1aaba1eeb40744dc3": "V_{n+1}(\\mathbf{S}) =V_{n}(\\mathbf{S}) \\cup\n2^{V_{n}(\\mathbf{S})}", "c7cab8d3b8a1b7ad8911a151f92af2c0": "\\hat{\\rho} = \\int f(\\alpha,\\alpha^*) |{\\alpha}\\rangle \\langle {\\alpha}| \\, d^2\\alpha", "c7caf9b20ec34beb08f1ab7203ed5bc9": "\\sqrt{2}\\sqrt{3}", "c7cb613c8b1c14b117390eb34508fb8f": "X_1,\\dots,X_n \\, ", "c7cb6391bab290ee96bd7f72397ec9e1": "\\mbox{(real power)} = \\mbox {(apparent power)}\\cos(\\theta)", "c7cbe05ed6dd7fbf3067ea98dd595008": "\n\\begin{pmatrix}\n2 & 4 & 4 \\\\\n-6 & 6 & 12 \\\\\n10 & -4 & -16\n\\end{pmatrix}\n", "c7cc128c3fd57f9bea81feaaf799d028": " \\Leftrightarrow V^{'}_{C} + V^{\\bullet}_{A} \\Leftrightarrow V^{'}_{M} + V^{\\bullet}_{X}", "c7cc318e07d671f74c85f821dbbd6f7a": "\\text{MA} = \\frac{F_w}{F_i} = \\frac {\\text{Length}}{\\text{Rise}} \\,", "c7cc61856926935d3930c2a3ea282ebd": "\\rho_j=\\frac{\\lambda_j}{\\mu_j}", "c7cc6b78bc90565e826207ee46201594": " \\eta > 0", "c7ccc94d71030952bad7a56764e246fa": " (h_1(t), \\dots, h_n(t)) ", "c7cce792f23da7c5c21c6d61b1f297f3": "6\\, ", "c7cceaca849219b1acd74334197bb9f1": "\\tau=1+\\eta+\\eta^2", "c7ccee24cf8062756655b5a48d7459a1": "S(uS(u))=uS(u), \\varepsilon(uS(u))=1, \\Delta(uS(u)) = \n(\\mathcal{R}_{21}\\mathcal{R}_{12})^{-2}(uS(u) \\otimes uS(u))", "c7cd33ec68d383f80a59bb3c96b07043": "\\beta_2 >3", "c7cd3f184bd31387a61808eaa47bd38a": "{}_pF_q", "c7cd66cd4faa10fe75c9eeab7154abd4": "a_{n+1,k} =2 a_{n,k-1} - 2 n a_{n-1,k} \\ \\ k>0", "c7cd8aab1f3c42e93414777efc14de2a": "\\frac {a} {b} \\times \\frac {d} {d} = \\frac {c} {d} \\times \\frac {b} {b}", "c7cd8f6007637f4f0b0befce67170d57": "f_4(x)\\,", "c7cdbc94ee7163e7becad763d7bf472e": "(5+2)+1=(2+5)+1 \\,", "c7cdcba062630144b9bd95926fcd5e63": "s\\to\\infty\\,\\!", "c7ce4134696fb6d4f21034fa349148ab": "\\epsilon:G\\to Id", "c7ce43b09fcba0c85e99c07c11d742fd": "\\alpha + 2 y \\not=0.", "c7cef85f428fe5c3156a0523f5cbb2cb": "1 + \\tfrac13 + \\tfrac1{3\\cdot4} - \\tfrac1{3\\cdot4\\cdot34}", "c7cf1b9b9c957554447d062a3d1ab89c": "O(2^n)", "c7cf7cd0c68c8dea2e209c91c363c106": " b=0.02 ", "c7cfa2cea702ca848e52ef9a218a6df0": "\\tfrac34", "c7d0778a5d23c47e49a10b6041f2c130": "f: \\mathbb{N}^k \\rightarrow \\mathbb{N}", "c7d08dc6ec9c5bcafdbeca2dd67eecaf": "\n-\\mathbf{q} \\cdot \\dot{\\mathbf{p}} - H(\\mathbf{q}, \\mathbf{p}, t) = \n\\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t) + \\frac{\\partial G_{3}}{\\partial t} + \\frac{\\partial G_{3}}{\\partial \\mathbf{p}} \\cdot \\dot{\\mathbf{p}} + \\frac{\\partial G_{3}}{\\partial \\mathbf{Q}} \\cdot \\dot{\\mathbf{Q}} \n", "c7d0a81333b345f628605c3c0eb1c319": " \\lim_{x \\searrow a}\\,f(x)", "c7d0bcc66b9f1dc29def32a11057bc15": "I_{b-}=0", "c7d103949205c78d39aef0d58cb0bc1f": "u_{\\mathrm{rms}}", "c7d128ba0a5ac9d692f7f6ea388befb2": " \\sup \\, \\{ \\mathrm{rad}(B_j) : j \\in J \\} <\\infty ", "c7d13bfbee45474f3c08d5c0b4295f9f": "\\ pressure ", "c7d14dd9c67b64b1da81c825e4f4d36e": "\\displaystyle m_{i-1}", "c7d15f93b16e90dfb084c2d174a2efd3": "F_E = q E \\,", "c7d17d87582a0ce2c9d8434794ba9fcf": "\\tfrac{\\pi}{4}", "c7d1972a0f3625b010c63ca5a84ee755": " \\langle \\beta, \\alpha \\rangle \\langle \\alpha, \\beta \\rangle = 2 \\frac{(\\alpha,\\beta)}{(\\alpha,\\alpha)} \\cdot 2 \\frac{(\\alpha,\\beta)}{(\\beta,\\beta)} = 4 \\frac{(\\alpha,\\beta)^2}{\\vert \\alpha \\vert^2 \\vert \\beta \\vert^2} = 4 \\cos^2(\\theta) = (2\\cos(\\theta))^2 \\in \\mathbb{Z}.", "c7d1a41203957b843e39cd11fa3876d2": "\n \\quad (1) \\qquad \\frac {\\mathbf{u}^* - \\mathbf{u}^n} {\\Delta t} = -(\\mathbf{u}^n \\cdot\\nabla) \\mathbf{u}^n + \\nu \\nabla^2 \n \\mathbf{u}^n\n", "c7d25d4603b9875b126d2c4866b2e127": " \\frac{R_1 + \\cdots + R_n}{n} \\rightarrow \\frac{\\mu(X)}{\\mu(A)} \\quad\\mbox{(almost surely)}", "c7d271fc225e386242ca7a1cd8d98444": "x = D\\alpha", "c7d280df3e595c3a1e50b99ebd0cc720": "1 - \\frac{ 2 \\cdot \\cos^{-1}( \\text{similarity} ) }{ \\pi }", "c7d28a23defb49c9c88a1adff30c2a6e": "\n\\begin{align}\n\\int \\sec^3 x \\, dx &{}= \\int \\cosh^2 u\\,du \\\\\n&{}= \\frac{1}{2}\\int ( \\cosh 2u +1) \\,du \\\\\n&{}= \\frac{1}{2} \\left( \\frac{1}{2}\\sinh2u + u\\right) + C\\\\\n&{}= \\frac{1}{2} ( \\sinh u \\cosh u + u ) + C \\\\\n&{}= \\frac{1}{2} \\sec x \\tan x + \\frac{1}{2} \\ln|\\sec x + \\tan x| + C\n\\end{align}\n", "c7d28ea17b5536f0dbc3fe150d842bf4": " \\mathfrak{f}_4^{\\mathbb C}, \\mathfrak{so}(8,\\mathbb C)", "c7d29b84ba55e46957bda9d2f609fb52": " (x_0,\\lambda_0)=(0,0). ", "c7d2f1c5affc00ecc923acf5c01b0062": "g(x, y, t)", "c7d30e5848e0566409ebab638bcdbec8": "c^2 u_{x x}(x,t) - u_{t t}(x,t) = s(x,t) \\,", "c7d34192df871ca7559807f028214708": "-2.076^\\circ\\sqrt{\\text{elevation in metres}}/60^\\circ", "c7d34cd58b4dd8005a722561578d0e80": "\\mathbf{g}=f\\boldsymbol{\\eta}\\,", "c7d34ed23760291c386298d66670ec30": "\\bar{\\rho}", "c7d359de27923b756a4e5f4277aea568": "\\hat{x}=[1,0,0]", "c7d3c631c7d15b1de95c4c427444c4e3": " O\\left(h_n^{2m+2}\\right). \\, ", "c7d41786494b471f0afb4ee8ff0170db": " \\frac{\\pi}{4} = 44 \\arctan\\frac{1}{57} + 7 \\arctan\\frac{1}{239} - 12 \\arctan\\frac{1}{682} + 24 \\arctan\\frac{1}{12943}\\!", "c7d43c48422d60661ee138411556bd98": "\\omega_0 = 1", "c7d49aa9c96628c28450aa8831af2251": "srs = r^{-1} \\, ", "c7d4a119c6020d41d7c19000703cfe99": "-i \\epsilon mc^2/\\hbar", "c7d4bc7597deb7dd8422db5132504410": " \\tilde{P}_{ni} = \\frac {\\sum_{r} L_{n}(\\beta^r)} {R} ", "c7d4d0436a51b59e05b78c142578ede9": "k>n", "c7d4f66d64a31abc3e2972636cd69b34": "=E(c_1|\\psi_1\\rangle+c_2|\\psi_2\\rangle)", "c7d506d64fa41f43af1a1426dd4c2bef": "\\frac{\\partial}{\\partial\\alpha}\\,f(x,\\alpha)", "c7d530b7604246d4b78a4c7394f3094e": "\\sum_{n=0}^\\infty \\frac{m_n t^n}{n!},", "c7d594bab42e9616c4f69a0d8772673d": " W'(i,x)_{coenergy} = \\frac{1}{2} ~ L(x) ~ i^2 ", "c7d5d61ab900dde71cf0dd25cb1cab44": "\\scriptstyle \\sqrt{2/\\pi} = 0.79788456\\dots", "c7d5ef10071789ff99026f74b5036b34": "|\\mu_{i,j}| >\\frac{1}{2}", "c7d613ed8cf5a7630cf77a7740c7c976": "R_2 =0.3489", "c7d6314d0990f52727aac7d86f8793f6": "{\\tilde{C}}_{4}", "c7d6492edac601225975b92815414185": "U^{(n)} = [u^{(n)}_{i,j}]_{I_n \\times I_n}", "c7d652815da56f97255d0c64793b8981": "y = c \\ln \\sec \\frac{x}{c}.\\,", "c7d659bf355ce7aff778d1c1c7641914": "\\mathbf{J=QR}", "c7d6855627ecf767fecabce2f823bb04": " b / 2 ", "c7d7599dfb28a07d5166096fef94a214": "YW-Z^2", "c7d7915e2f71983ff922f5a1596a1329": "F(X_{j+m}) - F(X_{j})", "c7d7a3f91f1ab50cb544293049cb5099": "t+i, i>0 ", "c7d7d8dd3545d8fd3dad34f88a0fb3d6": "\\mathbf{A}' = \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_x) \\mathbf{A}", "c7d7f8a3578c42c9c66ec5cd5719bbe9": "\\varepsilon\\rho = \\varepsilon|\\psi\\rangle\\langle\\psi|", "c7d83739ac547430d75e4523e8b17056": "\\left ( \\begin{matrix} 1 & 2 & 3 & 4 & 5 \\\\ 2 & 3 & 4 & 5 & 1 \\end{matrix} \\right )", "c7d848113e29e2869971c54281ea44f4": "-b_1E_{i^{ }}", "c7d8b0c2d8353724cc63b687344d4106": "\\langle \\alpha',j'm'|[J_{\\pm}, T_q^{(k)}]|\\alpha,jm\\rangle=\\hbar \\sqrt{(k\\mp q)(k\\pm q+1)}\\langle \\alpha',j'm'|T_{q\\pm 1}^{(k)}|\\alpha,jm\\rangle ", "c7d8bc5d9977d2a0859f331e54bf61e0": "A = \\begin{bmatrix}\nI_x(q_1) & I_y(q_1) \\\\[10pt]\nI_x(q_2) & I_y(q_2) \\\\[10pt]\n\\vdots & \\vdots \\\\[10pt]\nI_x(q_n) & I_y(q_n) \n\\end{bmatrix},\n\\quad\\quad\nv = \n\\begin{bmatrix}\nV_x\\\\[10pt]\nV_y\n\\end{bmatrix},\n\\quad \\mbox{and}\\quad\nb = \n\\begin{bmatrix}\n-I_t(q_1) \\\\[10pt]\n-I_t(q_2) \\\\[10pt]\n\\vdots \\\\[10pt]\n-I_t(q_n)\n\\end{bmatrix} \n", "c7d8dc9d7b7d53e04c35515a05867ce2": "\\phi^+_i", "c7d8ebb64e3499b55ca4a93cbb0cbe8e": " F = F_n + \\alpha |\\psi|^2 + \\frac{\\beta}{2} |\\psi|^4 + \\frac{1}{2m} \\left| \\left(-i\\hbar\\nabla - 2e\\mathbf{A} \\right) \\psi \\right|^2 + \\frac{|\\mathbf{B}|^2}{2\\mu_0} ", "c7d8ee0bf0717617fe7b914b7924d1e2": "\\mathcal S_n(\\mathcal X_{1\\cdots n},\\mathcal Y_{1\\cdots n})", "c7d90b50a3dc2cc1729ebdf357fbe011": "C^\\infty(S^1)", "c7d99f9c6402c87f6bd9e3e1fde51fa0": "m + n", "c7d9b32015d3bd4db6672e9f5125a995": "(z-\\cos\\psi)^{n-\\frac12}=\\sqrt{\\frac{2}{\\pi}}\\frac{(z^2-1)^{\\frac{n}{2}}}{\\Gamma(\\frac12-n)}\n\\sum_{m=-\\infty}^{\\infty}\\frac{\\Gamma(m-n+\\frac12)}{\\Gamma(m+n+\\frac12)}Q_{m-\\frac12}^n(z)e^{im\\psi},", "c7d9def550406bef60a6d811f6dca954": "C = \\,2\\pi r", "c7da074a503fcacac7af69ed91a3bca5": "V(N_{i,j})", "c7da142f6344882ef758f89a20068b8d": "\\rho \\le v_l - v_r", "c7da4c51e87d6785e76627714ce28a17": "_{interval}\\alpha \\ge r_{ii} ", "c7da6db889b37b51f06f93077b174557": "\\text{Opposition strength multiplier} = \\frac{200-\\text{ranking position}}{100}", "c7daa13f8cc92b148d4dab44409bf730": "\\mathbf{f'}(x), \\dfrac{dy}{dx} , \\dfrac{d^2y}{dx^2} , \\dfrac{\\partial {y}}{\\partial{x}}", "c7dae9e6a68ad4291862e3ff449d8f44": "\\begin{align} S\n& = -k_B \\langle\\ln P\\rangle \\\\\n& = -k_B\\sum_{x_i} P(x_1,x_2,\\dots) \\ln P(x_1,x_2,\\dots) \\\\\n& = k_B(\\beta \\langle H\\rangle + \\log Z(\\beta))\n\\end{align}\n", "c7daff6b3b2c728947adc02b9e4d0ccc": "\\left( \\Phi \\cup \\{\\lnot \\phi\\}\\right)", "c7db255dba3c12bd4ed7de3437b6676d": "\\beta \\ge 0", "c7db4cc2fb7075518a7bc1c3fd3d25c0": "|\\psi(t)\\rangle = e^{-\\frac{i}{\\hbar}\\lambda V(t-t_0)}|\\psi_F(t)\\rangle", "c7db7e18daba76d1e7e5a2f60656b00f": " X\\subseteq Y_\\epsilon \\ \\mbox{and} \\ Y\\subseteq X_\\epsilon", "c7db841082b643d917b74e913b8b0b04": "0 = -\\nabla p + \\rho \\mathbf{g}", "c7db87662182d83eb49f3f49a6e20ba8": "\\scriptstyle \\frac{a^2 \\,+\\, b^2}{ab \\,+\\, 1} \\;=\\; q", "c7db91d7f6dc86cf6dbd53c545bbed02": "- \\otimes A", "c7dbc552d451be8eb81b5a03719cc554": "\\begin{pmatrix}T_2\\end{pmatrix}\\,", "c7dbefe5dc64d4adb00bd42abbc0e872": " \\Delta F\\uparrow = \\Delta\\epsilon \\left( \\sigma T_a^4 -\\sigma T_s^4 \\right) ", "c7dc19209eb71e2968d2e3b4fb4a17f2": "\\tilde{\\kappa}_{tr}", "c7dc239002e6d3ed0c4928f6d4d6bf8e": "(\\lambda 1 (\\lambda \\lambda 2) (\\lambda 1 (\\lambda \\lambda 1)) (\\lambda \\lambda \\lambda (\\lambda 1 (3 2) (\\lambda \\lambda 3 1 (6 1 1 (4 ((\\lambda 1 1) (\\lambda \\lambda \\lambda \\lambda \\lambda 1 2 (\\lambda 1 (\\lambda \\lambda 1) (\\lambda \\lambda \\lambda 8 2) (\\lambda \\lambda 1) (4 (6 6) (\\lambda 4 (\\lambda 1 7 2)))))) 5) ", "c7dca56a590ffec846f8c1d5345cb4a1": "\\color{red}\\forall y", "c7dcab94c5e42f8306a8eee61e5bb11a": "\\tilde{Z}[\\tilde{J}]=\\int \\mathcal{D}\\tilde\\phi \\prod_p \\left[e^{-(p^2+m^2)\\tilde\\phi^2/2} e^{-\\lambda/4!\\int d^4p_1d^4p_2d^4p_3\\delta(p-p_1-p_2-p_3)\\tilde\\phi(p)\\tilde\\phi(p_1)\\tilde\\phi(p_2)\\tilde\\phi(p_3)} e^{\\tilde{J}\\tilde\\phi}\\right].", "c7dd9b5f779772e3929c00bdd4a8e9ff": "2\\,H_2O\\;+\\;2\\,e^-\\;\\rightarrow\\;H_2\\;+\\;2\\,OH^-", "c7ddb564c6af179d9363d9b33fb5984d": "\\begin{align}\ns_{\\infty}(x) &= \\sum_{n=-\\infty}^\\infty \\hat{s}(n)\\cdot e^{i\\tfrac{2\\pi nx}{P}} \\\\\n&= \\sum_{n=-\\infty}^\\infty S[n]\\cdot e^{j\\tfrac{2\\pi nx}{P}} &&\\scriptstyle \\text{common engineering notation}\n\\end{align}", "c7de03b463a75f3cbf3ea2311cc1359e": "\\gamma(0).", "c7de8acf9b822d281d300bd17324e592": "\\eta=\\frac{\\gamma -1}{\\delta -1}", "c7deb8b5405b9f06b48d8c3b556013c7": "\nP_x \\left[ \\alpha_0\\left(T-T_0\\right)+\\alpha_{11}P_x^2+\\alpha_{111}P_x^4\\right]=0\n", "c7debe55357b7fbbe7522a95496c9faf": "f(n)/g(n) \\to 1", "c7dee51397fa48c5c0dc0b94cc675433": "Q(N, V, T, \\lambda) = \\sum_{s} \\exp [-U_s(\\lambda)/kT]", "c7defdd47c87db0ec3e72b219598f1af": "\\left( \\mathbb{Z}_{p} \\right)^\\times \\times \\left(\\mathbb{Z}_p\\right)^\\times", "c7df0f3642550241a0fa063ec48bcdf3": "\\delta^{n-1} tr \\rho(\\sigma)", "c7df3a6daa2b5505aae620e655257b70": " \\begin{align}\\mathcal Z & = \\sum_{N=0}^{\\infty} \\exp(N(\\mu - \\epsilon)/k_B T) = \\sum_{N=0}^{\\infty} [\\exp((\\mu - \\epsilon)/k_B T)]^N \\\\\n& = \\frac{1}{1 - \\exp((\\mu - \\epsilon)/k_B T)}\\end{align}", "c7df580ea1dd7dbe6c98f5cdde03b109": "\\phi_1\\,", "c7df67fbcfc3d79a2e765db481461dc3": "{\\tilde{u}}", "c7df9d85ece84943b8680f602d72440e": "f^{\\mathcal A}:|\\mathcal A|^2\\rightarrow|\\mathcal A|", "c7dfe22bbd4d00930caf84567222cf9e": "\\mathfrak{so}(3)\\cong \\mathfrak{sp}(1)", "c7dfe8ef681ccba9b3b242dc709a5a5c": "{1/(1 + D)}", "c7dfea36a324305656e5505d82434042": " \\operatorname{build-param-lists}[p\\ p\\ f, D, V, K_2] ", "c7e012b14f6cead3410b91351a39c63b": " \\nabla \\cdot \\mathbf{j} + { \\partial \\over \\partial t} |\\psi|^2 = 0.", "c7e05a7dd73e40b334b1ff4d5cfc4ab7": "W = \\sigma \\cdot A \\cdot T^4", "c7e05b0396804460fc80c6b1eb1d26d8": "g(x,p)\\equiv\\int_{-\\infty}^\\infty dy\\, \\langle x-y/2| \\hat{G} |x+y/2 \\rangle e^{ipy/\\hbar},", "c7e067362feac09b114d37a2eb9bae76": "\\rightleftharpoons", "c7e0b78c0864b55250cd1d8a070189ad": " {\\delta^*}= \\int_0^\\infty {\\left(1-{u(y)\\over u_0}\\right) \\,\\mathrm{d}y}", "c7e0b7d9ff70d680e479e02dfff1271c": "T = S/D", "c7e0c309ded07f0abfe583b22397fa7b": "K^*_p(s;\\theta,\\lambda) = \\lambda\\kappa_p(\\theta)[(1+s/\\theta)^\\alpha-1]", "c7e120602d98bda389fc471ad8a79b41": "\\cos(A + B) = \\cos(A)\\cos(B) + \\sin(A) \\sin(B)\\,", "c7e1702d632422e5eb2a40a85b7b0e3f": "\\hat{h} = \\arg \\min_{h \\in \\mathcal{H}} R_{\\mbox{emp}}(h).", "c7e194397fb4c18b2a03a610276bafe7": "~m_{max}", "c7e1b691664c8eeb2cc07fdf5d86f094": " f'(5) = \\lim_{h \\to 0}\\frac{f(5+h)-f(5)}{h} = \\lim_{h \\to 0}\\frac{log_e(5+h)-log_e(5)}{h} = \\lim_{h \\to 0}\\frac{1}{h} log_e\\left(\\frac{5+h}{5}\\right)", "c7e2272fbddd3ac9db572883ff653f31": "W_{n} = diag\\{w_{n}\\}", "c7e228b8ae253e098664cdee7227dcfd": " {\\textbf{L}}_k =\n\n\\begin{bmatrix}\n\\tfrac{\\scriptstyle\\sigma_2^{2}{\\textbf{P}}_k}{\\scriptstyle\\sigma_2^{2}{\\textbf{P}}_k + \\scriptstyle\\sigma_1^{2}{\\textbf{P}}_k + \\scriptstyle\\sigma_1^{2} \\scriptstyle\\sigma_2^{2}} & \\tfrac{\\scriptstyle\\sigma_1^{2}{\\textbf{P}}_k}{\\scriptstyle\\sigma_2^{2}{\\textbf{P}}_k + \\scriptstyle\\sigma_1^{2}{\\textbf{P}}_k + \\scriptstyle\\sigma_1^{2} \\scriptstyle\\sigma_2^{2}} \\end{bmatrix}.", "c7e2631a68f2e74e350fa734ce67edbf": "I_t = I_{t-1} \\times \\prod_{j = 1}^{N(t)} \\left( \\frac{e_{j,t} \\cdot \\frac{p_t}{p_{j,t}}}{e_{j,t-1}\\cdot \\frac{p_{t-1}}{p_{j,t-1}}} \\right)^{w_{j,t}}", "c7e41b2c92e266969102d2b917d784d0": "\\vec r=\\vec n_1\\times\\vec n_2", "c7e423f3cbf0adca8929d79548f02deb": " \\left(\\chi_\\text{e}, \\chi_\\text{m}\\right) ", "c7e424a119b9873f81276b424f0d12cc": "\\Delta\\tau_g = \\frac{g}{c^2} \\sum_{i=1}^{k} (h_i - h_0) \\Delta t_i", "c7e4955dce9de6fa623d0ce7187bbc60": "\\operatorname{and} \\operatorname{true} \\operatorname{false} = (\\lambda p.\\lambda q.p\\ q\\ p)\\ \\operatorname{true}\\ \\operatorname{false} = \\operatorname{true} \\operatorname{false} \\operatorname{true} = (\\lambda a.\\lambda b.a) \\operatorname{false} \\operatorname{true} = \\operatorname{false} ", "c7e4d90d72b85bd8d5f56679494e3680": "\n\\begin{align}\n\\mathbf{u}_1 & = \\mathbf{v}_1, & \\mathbf{e}_1 & = {\\mathbf{u}_1 \\over \\|\\mathbf{u}_1\\|} \\\\\n\\mathbf{u}_2 & = \\mathbf{v}_2-\\mathrm{proj}_{\\mathbf{u}_1}\\,(\\mathbf{v}_2),\n& \\mathbf{e}_2 & = {\\mathbf{u}_2 \\over \\|\\mathbf{u}_2\\|} \\\\\n\\mathbf{u}_3 & = \\mathbf{v}_3-\\mathrm{proj}_{\\mathbf{u}_1}\\,(\\mathbf{v}_3)-\\mathrm{proj}_{\\mathbf{u}_2}\\,(\\mathbf{v}_3), & \\mathbf{e}_3 & = {\\mathbf{u}_3 \\over \\|\\mathbf{u}_3\\|} \\\\\n\\mathbf{u}_4 & = \\mathbf{v}_4-\\mathrm{proj}_{\\mathbf{u}_1}\\,(\\mathbf{v}_4)-\\mathrm{proj}_{\\mathbf{u}_2}\\,(\\mathbf{v}_4)-\\mathrm{proj}_{\\mathbf{u}_3}\\,(\\mathbf{v}_4), & \\mathbf{e}_4 & = {\\mathbf{u}_4 \\over \\|\\mathbf{u}_4\\|} \\\\\n& {}\\ \\ \\vdots & & {}\\ \\ \\vdots \\\\\n\\mathbf{u}_k & = \\mathbf{v}_k-\\sum_{j=1}^{k-1}\\mathrm{proj}_{\\mathbf{u}_j}\\,(\\mathbf{v}_k), & \\mathbf{e}_k & = {\\mathbf{u}_k\\over \\|\\mathbf{u}_k \\|}.\n\\end{align}\n", "c7e508a8364473795e2548450b899bbc": "P_{x}", "c7e569a67ec764481f1ac43cae4e36de": "A = L_{x}L_{y}", "c7e57ad33725ded02314bcca59c72639": "\\mathbf u(\\mathbf X,t)=u_i\\mathbf e_i=u_i(\\alpha_{iJ}\\mathbf E_J)=U_J\\mathbf E_J=\\mathbf U(\\mathbf x,t)", "c7e57b50c51e35fa0aa1116964170d8b": "\nT_n=m_k=T(F_n) = \\frac 1n \\sum_{i=1}^n (x_i - \\overline x)^k.\n", "c7e580bc68ad13956252ca1bed696a74": "T(z) = T_0 e^{\\frac{\\alpha}{2}z}", "c7e5b0d89265a82ae3d27a21a243e336": "\\overline{\\textbf{Q}}", "c7e6047074f27e71d367d3b9969ea015": "m \\neq 0", "c7e68f7a515a9521684d0ecd46aaa6b4": "\\ a", "c7e71a5c4930771c39153232e8ae894e": "g(x)=[Gf](x)", "c7e732c3d92fba1c894e6c8443cb2a17": "\\widehat{y}_i = f(x_i)", "c7e7569233e99720890d3c78d86e4f25": "\n\\ln I(R) = \\ln I_{0} - k R^{1/4}.\n", "c7e75ffaa37db08b643fba20b7f5fd30": " \\boldsymbol{\\omega} = \\mathbf{\\hat{n}} \\left ( \\mathrm{d} \\theta /\\mathrm{d} t \\right ) \\,\\!", "c7e79e7529f5bed3e170675af4c4b7b3": "\\gamma:[a,b] \\rightarrow \\mathbb{R}^n", "c7e7a0659ba51b86fc1c9aa98d4f60a2": "=\\lambda^{-1}\\mathbf{P}(n-1)-\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)", "c7e7b495aaa2ea5cd6cc035046f89e1d": " f^{ge} \\, ", "c7e7eaa409e4a8968fc59e441ac6ebb9": "N \\times d", "c7e862a5019587ca139a778fb25d0aaa": "\\frac{d t}{d x} = C_1 e^{-\\int f(x) dx}", "c7e8762144c7acee87578e223d138d95": "pV = nRT = kTN\\,\\!", "c7e87bcb5a6e7e208dd59cb10607c000": "f_{jk} \\circ f_{ij} = f_{ik}", "c7e88978827e18ca10b77301973a930a": "(a\\rightarrow b \\rightarrow STOP)", "c7e8e8be30aa4e7abe97f1fc52b7fafb": "\\left( \\gamma^k \\right)^\\dagger = -\\gamma^k \\,", "c7e8fc0c832f16163ba1483e3577ff8b": "\n \\frac{\\partial u}{\\partial t} = \\alpha \\frac{\\partial^2 u}{\\partial x^2}\n", "c7e907a15d360849c997f6fd29028cf5": "\\text{PI}=\\frac{mass}{height^3}", "c7e9ccfee4e37d018f2b54e4db66e352": "cos(\\omega t)", "c7e9f738dc01608d434a2b11ce247df7": "\\alpha\\in E\\setminus F", "c7ea6eb88bfa7e0b33eae81139a518f6": "\\cong (\\mathfrak{m}_{f^{-1}P}/\\mathfrak{m}_{f^{-1}P}^2)/\\mathrm{Ker}(k).", "c7eabdb3cb2ed302d83744b3aea74174": "60 \\cdot f_1 = 2^2 \\times 3^1 \\times 5^1 \\cdot \\frac{3 \\times 7}{2^2 \\times 5^1} = 3^2 \\times 7^1", "c7ead73bc95dd724099899719a5cdd5b": "0 \\le x,y \\le 2\\pi ", "c7eb231c4bb96bc3a4ea2fe90c4dd3c4": " -\\delta\\ \\mathbf{V} = \\delta\\ \\mathbf{U} ", "c7eb3aa53ef6aea7a95eba0e6c87386f": "[A]_{\\text{seq}} \\subsetneq [[A]_{\\text{seq}}]_{\\text{seq}}", "c7ebba1ac4ce5d6b45ce79502dcc7357": "\\kappa(X,Y,Z)=E(XYZ).\\,", "c7ebe4ab09933fe77de60169781b1a2a": "\\scriptstyle\\frac{c\\,\\,b\\,\\,a}{f\\,\\,e\\,\\,d} = \\frac{a}{d} + \\frac{b}{de} + \\frac{c}{def}", "c7ebf6e0c56e25688d63f789290d3754": "\\mathbb{CP}^\\infty", "c7ec205540e00457bb81d1e2cb520734": "\\Delta q\\Delta p \\ge 1/2", "c7ec5b563e0569cad4a7efc1ddab2265": "\\eta_{\\mu\\nu}\\;", "c7ec7805a0e509dcf309e09dd19f8ca8": "\\gamma = g_{\\mu\\nu}u^\\mu v^\\nu = \\left(1-\\frac{2M}{r}\\right) \\sqrt{\\frac{r}{r-3M}} \\sqrt{\\frac{r}{r-2M}} = \\sqrt{\\frac{r-2M}{r-3M}}", "c7ec8d716c7e7f3d17c284753eb6b59f": "\n\\langle a \\phi \\rangle = a \\langle \\phi \\rangle, \\,\n", "c7ecd8acacc3202947b29dc186cd6d82": "10^{\\,\\!10^{10^{34}}}", "c7ecfb461d8731680dd9575f1032c448": " \\frac{d P(\\mathbf{X}, t)}{dt} = \\Omega \\sum_{j = 1}^R \\left( \\prod_{i = 1}^{N} \\mathbb{E}^{-S_{ij}} - 1 \\right) f_j (\\mathbf{x}, \\Omega) P (\\mathbf{X}, t). ", "c7ed2e4aa421585af4a5310a4a20e574": "\\left(-\\frac{1}{2}\\nabla^{2}+v_{s}(\\mathbf{r},t)\\right)\\phi_{i}(\\mathbf{r},t)=i\\frac{\\partial}{\\partial t}\\phi_{i}(\\mathbf{r},t)\\ \\ \\ \\phi_{i}(\\mathbf{r},0)=\\phi_{i}(\\mathbf{r}),", "c7ed3a38c5f3a7702b55b67b0df3a5dc": " Q = a_S + b_S P + cX \\, ", "c7ed50f22a3ab6b371646c424504ce74": " (E,\\preceq) ", "c7eda90ba49bf2b5850cef221f2d6392": "V_k(X) \\hookrightarrow V_k(Y),", "c7eda9875516bd2ba63d0b460954fb34": "P_i = P_j", "c7edd863b6c94c3e177673d19d9a8fe6": "\n\nA_{33} =\n\n\\begin{bmatrix}\n\n\tA\t&\tB/2\t\\\\\n\n\tB/2 \t&\tC\n\n\\end{bmatrix}.\n\n", "c7edf188c86b23a4e3181fb68a1dea6e": "\\text{Inradius}=k(mn-k^{2}) \\, ", "c7eebcca7f57d961c29ff9d4fb068a4c": "\\mathbf{a} \\cdot \\mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 - a_4 b_4.", "c7eed9b1b288bf0a3ed7f7e7d5c42752": "(\\pm C_1,\\pm C_2,\\pm C_3)", "c7ef034320d041b6f980a12f9313ab57": "Q_{in} \\,", "c7ef317744e280e2c8dc47a14259676c": "O(n k)", "c7ef5cd2bcbdf2599af40479e6b10b22": "\\alpha^m + \\alpha^n = \\alpha^m \\cdot (1 + \\alpha^{n-m}) = \\alpha^m \\cdot \\alpha^{Z(n-m)} = \\alpha^{m + Z(n-m)} ", "c7ef9298ecced56bb8a8e088ce013091": "\\liminf\\Phi(x_n)\\le\\Phi(\\overline x)\\le\\limsup\\Phi(x_n).", "c7ef95ab99dafe5d28a772777bb8f89b": "\\ z_1", "c7efd34fa8c67aa6144315e07a6c23e6": "\\{(x,y)\\mid \\phi(x,y)\\}", "c7efedcbbcc03fdbfee05cafe8b6bab2": "k=0,1,\\dots,2N-1", "c7f01d5bf182b9ad77b34534f22b7731": "L_-L_+ = \\mathbf{L}^2 - L_z^2 -L_z", "c7f09494a19781f9b7d26ed01b63d0d6": "\\sqrt{P(X)-(P(X))^2}", "c7f0be3bed6efa66b0bb21b1ed639b02": "\nH_{\\mathrm{pot}} = C q^{s},\\,\n", "c7f0c712620401a30b9446d80e965a97": "v(t)=v(t_0)+\\frac{1}{C}\\int_{t_0}^{t}i(\\tau)d\\tau", "c7f0d22aac4092e43dbbe0215e6d8737": "\nu(r) =\n\\begin{cases}\n\\infty &\\hbox{when}\\quad r < d, \\\\\n-\\epsilon \\left(\\frac{d}{r}\\right)^6 & \\hbox{when}\\quad r \\ge d,\n\\end{cases}\n", "c7f108546a4bfbc7ee5fd35318cb73e5": "35 \\le t \\le 45", "c7f109b15253023a0b33c3612e409c3e": "\\begin{bmatrix}\n a & b & c & d \\\\ \n -b & a & -d & c \\\\\n -c & d & a & -b \\\\\n -d & -c & b & a \n\\end{bmatrix}= a\n\\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\ \n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 \n\\end{bmatrix}\n+ b\n\\begin{bmatrix}\n 0 & 1 & 0 & 0 \\\\ \n -1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 \\\\\n 0 & 0 & 1 & 0 \n\\end{bmatrix}\n+ c\n\\begin{bmatrix}\n 0 & 0 & 1 & 0 \\\\ \n 0 & 0 & 0 & 1 \\\\\n -1 & 0 & 0 & 0 \\\\\n 0 & -1 & 0 & 0 \n\\end{bmatrix}\n+ d\n\\begin{bmatrix}\n 0 & 0 & 0 & 1 \\\\ \n 0 & 0 & -1 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n -1 & 0 & 0 & 0 \n\\end{bmatrix}.\n", "c7f1b3d097a58afa878dc365ac7151a0": "{2^{2^{2^{65536}}}} - 3", "c7f247a78ddbe3619894bc44ce62da2e": "d=\\sqrt{\\sum_{i=1}^{9}\\Big[\\Pr ( X \\text{ has FSD}=i ) - \\log_{10}(1+1/i) \\Big]^{2}},", "c7f24b08dd3a7ac56e7046dddb8abb5d": " \\ E _{local} ", "c7f2b12cc9cbe35512e9d9e66ba489f3": "\\frac{(x-a)^2}{a^2}\\pm\\frac{y^2}{b^2}=1", "c7f394408a74d1e9509704ef411510b2": "X_{i+1} = X_i^+", "c7f39a422d4d83c2c568fa8021b6bda7": " t_{ik} = \\sum_{j} c_{ij} c_{jk} ", "c7f3ce4579fff2f565162a01220fcbc1": "(\\alpha, \\beta, f)", "c7f3cfc2949117ed4d8c152b4b394c1b": "\\; t > 1\\,", "c7f403747e013d7c980be028fbc78bb9": "S \\in W", "c7f4104d74b38e6573bbc94ebaede5cb": "\\! H", "c7f4433dd7c11c83e6a7137c226a7719": "|\\alpha\\rangle =e^{-{|\\alpha|^2\\over2}}\\sum_{n=0}^{\\infty}{\\alpha^n\\over\\sqrt{n!}}|n\\rangle\n", "c7f4921f40d04efc8d6c669e15d124d0": " a_H\\,\\!", "c7f493b8dae3627bf4213f54ea85cfe7": "n! = \\prod_{k=1}^\\infty \\frac{\\left(1+\\frac{1}{k}\\right)^n}{1+\\frac{n}{k}}\\,,", "c7f4cd8fea7565a75100cf302d147c39": "s(s-a)-(s-b)(s-c)=\\frac{1}{4}[(b+c)^2-a^2] - \\frac{1}{4}[a^2-(b-c)^2] = \\frac{1}{2}(b^2+c^2-a^2).", "c7f526b385068f32335b65cc3b0704fc": "z_{d=1 \\dots M,w=1 \\dots N_d}", "c7f571bb63e18312096495b59a96fec7": "\\text{Color harmony} = f(\\text{Col} 1, 2, 3, \\dots, n) \\cdot (ID + CE + CX + P + T)", "c7f59db85fe923720b456c25dad4c3fb": "A'(x)u_1'(x)+B'(x)u_2'(x)=Lu_G=f.\\,", "c7f64f39ec913dcc7ec0a9f479177fdb": " \\,(a_0,\\;a_1,\\ldots,\\;a_n)\\, ", "c7f65b0f9460c97dfb59fc8cab054b83": "\\epsilon_0=\\omega^{\\omega^{\\omega^{\\cdot^{\\cdot^\\cdot}}}}", "c7f68be3486958fb851c4f2d6c7e7e3d": "\\scriptstyle t \\;=\\; t_0", "c7f6a6c5eb0f98bf916a5eb9523bc185": "f(b)>f(a)", "c7f7135317fb782f7c45606757c8481e": "\\gamma_\\mathrm{SL}=\\gamma_\\mathrm{S}+\\gamma_\\mathrm{L}-2\\sqrt{\\gamma_\\mathrm{S,d}\\gamma_\\mathrm{S,d}}-2\\sqrt{\\gamma_\\mathrm{S,p}\\gamma_\\mathrm{S,p}}", "c7f795af78ae4acd05d5f174f1de5647": "\\left( \\frac{3}{2} \\right) ^5 \\times \\left( \\frac{1}{2} \\right) ^2", "c7f7c12e3e86b724faf5b13f79d6fb2b": "\\left(\\frac{7}{\\sqrt{10}},\\ \\sqrt{\\frac{3}{2}},\\ 0,\\ \\pm2\\right)", "c7f7c938750a2ca187c43f411a0758b8": " i\\in I\\backslash S", "c7f7e6fa3963f4dc46922a46e20e121a": "(0, t_2)", "c7f7f65fc403064bac386426c49398b7": "\\vec{R}_1", "c7f80a5dd16c40db17c338655d687fbf": " c =M_{23,2}\\,", "c7f825ab46738bebb4c0dccd1713caa3": "\\bar{5}_2", "c7f841574144f7835f98624d0f358e25": " \\rho_1 = \\rho_c \\frac{c}{h_1+c} ", "c7f85708640706debe8c454133916ae9": "v \\in V_i ", "c7f89b69e16c9346d148507969ef420c": "DP_{T}^{S}", "c7f8b6f56b63884ab3ccf4259f9b432b": " a^2 - 2ab + b^2 = (a - b)^2.\\,\\!", "c7f8c801e3eb446233670affcbd80e0f": "P_n(1) = 2^{\\lfloor (n+1)/2 \\rfloor}; {~}{~} P_n(-1) = (1+(-1)^n)2^{\\lfloor n/2 \\rfloor - 1} . \\, ", "c7f8ca4c78a31c0180f8716926bcf571": "\\frac{\\partial H}{\\partial x_k} =-\\frac{\\partial }{\\partial x_k} \\sqrt{n(x_3)^2-p_1^2-p_2^2}=0", "c7f9758955c71d66d6f24ab21be76575": "\nK_0 \\left( mr \\right) \\rightarrow -\\ln \\left( {mr \\over 2}\\right) + 0.5772\n.", "c7f9ba8dfea6dce7f57339c6ecd71c53": "[S_x] = \\frac{ \\hbar}{2} \\cdot\n\\begin{bmatrix}\n0 & 1 \\\\\n1 & 0 \\end{bmatrix}\n", "c7f9e2076e9c63022b08ab58a54a9dd3": "2 b^2 = (2k)^2", "c7f9e9a90ac6ca48563ac71c08b93d1f": "\\sigma \\in B^{\\ast}", "c7fa3381b320199b26eb48492a35d85e": "l_0/D", "c7fa3a0d00baedc54a944029007ab9fd": "\n\\Gamma (E, \\Delta E) = \\int_{H \\in \\left[E, E+\\Delta E \\right]} d\\Gamma .\n", "c7faab1c68d9e01e163a6eeb53756aa6": "f(\\text{st}(v)) \\subseteq \\text{st}(f_\\triangle (v))", "c7fada16eb81d943339f2df60ebcb1b3": "\\Delta f(x)\\,", "c7faee92924aba9b3a15836b705f6540": "a^{[x_1, x_2]} = [a^{x_1},a^{x_2}]", "c7fb173e5708f708c8686904ec9d13d8": "J_P", "c7fb4740c1c4418bac1a6787fc859bed": "\\Gamma_{12}(u,v,0)", "c7fc0783621c46f32af768a89e4d8d11": "\\Delta B_h", "c7fc5929f88bfab6b7a312996336507b": " W_J = \\langle J \\rangle", "c7fc97d227c4d080a454c65cc42d14b9": "\\lceil \\ldots \\rceil", "c7fca6b884ea89ba250e4065268be1f0": " v(z)=\\partial_t f_t(z)|_{t=0},", "c7fd32d44d3a88d5b45c8bde1d6bfbe4": "c (x, y) = | x - y |^p/p", "c7fd39b4319c4df82c562256b52a48d8": "(h, \\, h')", "c7fd4a2cbb67a768e1f135a02ceae077": " [wz + h + j - q]^2 - ", "c7fdabc58e32970a06ec9568247d2690": "dS=\\mu S\\,dt+\\sqrt{\\nu }S\\,dZ_{1}+(e^{\\alpha +\\delta \\epsilon} -1)Sdq", "c7fe2bda9ad8b255a391691e7b0d13d9": "x=\\frac{1-x}{2+\\phi}", "c7fe349db1f2cd205e4c0796524b6305": "dr_t = a(b-r_t)\\, dt + \\sqrt{r_t}\\,\\sigma\\, dW_t", "c7fe56c5826bcd1f0452ef2d1ca7cc13": "\n\\theta(t) = \\theta_{\\mathrm{eq}} + \n\\left( \\theta_{0} - \\theta_{\\mathrm{eq}} \\right) e^{-2\\alpha t}\n", "c7fe7228101042fa1aa4559402e31594": "\\chi_0", "c7feabeae54c378a20ebf523ed1b0187": "r_c, r_t", "c7fec95d736fd23958c5a0afe1c0497d": "X_{1j}", "c7feeecf9f4eb0edce05469b251013ed": "2.7526", "c7ff15bf832ac8205889d0443ad44c04": " \\operatorname{let-combine}[\\operatorname{let} x : \\operatorname{de-lambda}[x = \\lambda x.f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x)] ", "c7ff1c2f9d5fecb6aff82fbfd655757a": "\\Gamma_{i+1}= S_{\\Gamma_i}", "c7ff3b7d7cea55f71ecf953b55392799": "V(f)(t) = \\int_0^t{f(s)\\, ds}.", "c8000268ba8853d6d6aa596d8560674e": "\n E =\n\\begin{cases}\n\n\\displaystyle \\sum_{n=1}^\\infty\n {\\frac{M^{\\frac{n}{3}}}{n!}} \\lim_{\\theta \\to 0^+} \\! \\Bigg(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}\\theta^{\\,n-1}} \\bigg( \\bigg(\n \\frac{\\theta}{ \\sqrt[3]{\\theta - \\sin(\\theta)} } \\bigg)^{\\!\\!\\!n} \\bigg)\n\\Bigg)\n, & \\varepsilon = 1 \\\\\n\n\\displaystyle \\sum_{n=1}^\\infty\n{ \\frac{ M^n }{ n! } }\n\\lim_{\\theta \\to 0^+} \\! \\Bigg(\n \\frac{\\mathrm{d}^{\\,n-1}}{\\mathrm{d}\\theta^{\\,n-1}} \\bigg( \\Big(\n \\frac{ \\theta }{ \\theta - \\varepsilon \\sin(\\theta)} \\Big)^{\\!n} \\bigg)\n\\Bigg)\n, & \\varepsilon \\ne 1\n\n\\end{cases}\n", "c80038ec9f312a778f60a848d896f399": " \\frac{\\sin(x)}{x} = 1 - \\frac{x^2}{3!} + \\frac{x^4}{5!} - \\frac{x^6}{7!} + \\cdots. ", "c80060d998c3d2c8f367eaf8f72dea54": "\\scriptstyle 1/p^2", "c8007c36426c46ca10c1425c53839d45": "e^{i(\\pi/2)} = \\cos(\\pi/2) + i\\sin(\\pi/2) = 0 + i1 = i\\,\\! .", "c800fb744d906244fb6a4f1d118d6401": "(n/m\n)^3", "c80113e4b98820c39a7f92deaa28e2ae": "u(1/V(z))=\\frac{1+\\sqrt{1+4z}}{2z}+\\frac{1+2z+\\sqrt{1+4z}}{2z^2}", "c80144c1cdbd6f57f3ccb8e4895f3aad": "A_{OL} = \\frac {A_0} {(1+j \\omega \\tau_1) (1 + j \\omega \\tau_2)}, ", "c80147ea5ee4956d7b5f2afb10652821": "\\ln\\left(\\Lambda/Q\\right)", "c8014927cbc9540ab84e7b1ada5706a2": "K_{\\rm II}", "c8015254330be6d098074349591b20b3": " r_\\mathrm{ corr } = r - \\frac{ s_{ [ y / x ] x } }{ m_x } ", "c8015d53c58b994c81149478bb14773e": "k_n\\,", "c8018002726cda8fc0e591b9a9526354": "k-FDR", "c80188c5487fda13c87c99a321676d47": " \\theta \\,\\!", "c801902049864c8075bbbf5e8edef77d": "\\cdots\\rightarrow A\\otimes A\\otimes A\\rightarrow A\\otimes A\\rightarrow A \\rightarrow 0", "c801aabccbb254c8d49c940cf7d19ed8": "(\\theta_t,\\phi_t)", "c801ca0409486002e02aa99d535aa83c": "\\underline{a} \\ ", "c801ce5b21359270312f95b01040b08f": "\\mathrm{aff}(\\mathrm{aff}(S)) = \\mathrm{aff}(S)", "c801f387d6ed43b92afa0251ae763e5f": "\\Sigma = A_{\\Gamma\\Gamma} - A_{\\Gamma 1}A_{11}^{-1}A_{1\\Gamma} - A_{\\Gamma 2}A_{22}^{-1}A_{2\\Gamma}.", "c8020d33c071b828c34f3c43cf40bc8b": "o_i, r_1, r_2 ... r_{t-1}", "c802553380b044578096c80d96266221": "\\lVert \\mathbf{v} \\rVert \\equiv \\sqrt{\\langle \\mathbf{v},\\mathbf{v}\\rangle} \\, .", "c8026dfc595373d44b9027f9ce1421af": "d\\colon \\bigwedge^\\bullet E\\rightarrow \\bigwedge^{\\bullet+1} E.", "c802b84fa731644fca643e7fed8522d5": " A^n = A \\times \\ldots \\times A", "c802f256263fcba444832e9295a54156": "((\\and_{\\mu < \\gamma}{(\\lor_{\\delta < \\gamma}{A_{\\mu , \\delta}})}) \\implies (\\lor_{\\epsilon < \\gamma^{\\gamma}}{(\\and_{\\mu < \\gamma}{A_{\\mu ,\\gamma_{\\epsilon}})}}))", "c8033a8001b60ca7c25df073520a00b2": "M_1 = \\{\\, a\\mapsto 0, b\\mapsto 0, c\\mapsto 1\\,\\}", "c8034f703058cfe2d932ce213b3984b0": "\\begin{align}\n\\sin \\theta &\\approx \\theta\\\\\n\\tan \\theta &\\approx \\theta\n\\end{align}", "c803bd7412c9973f917005c68a30fc59": "S[\\sigma] \\to T[\\sigma]", "c803f94be33ae8341edac94a18aa494f": "G=\\mathrm{SO}(2)", "c8047b1d940ab0544fbda5853d0fc96b": " \\ln(\\frac{N}{N_0})=\\Lambda_{CW} C^n t \\!", "c804c3a15ebe05b5baf40ad5ee12be1f": "2s", "c80510e1c26fa612bc6f064f3812f5f8": "X^{w_1}", "c80530d9f9d9d47d214b62027248102f": "\\scriptstyle \\{\\varphi_n\\}_{n=1}^\\infty", "c8055c988540eafe7345cca3f24b7f44": " \\det (e^A)= e^{\\operatorname{tr}(A)}~.", "c8058e82e2a5252befb77b2423212d50": "\\theta \\in [-\\pi, \\pi]", "c8059264382e98604e97b37d6098de99": " \\sum_{i=1}^\\ell i \\binom{2m}{i} , ", "c805941e452e03a518d82b5451d28b76": " J_{ij} \\sim |i-j|^{-\\alpha} ", "c805aba9c3b5cc09511af11b27338463": "x_\\pm \\nrightarrow \\pm\\infty", "c8061820e35439fdf4a2878850633164": "\\Sigma_{model} = \\Sigma_{residual}", "c806627e2c9685c9264af981105b8a67": "g_1=g_0+\\Delta g", "c80688ddda7c0ff5e28d1e3bfd655d8c": "f(a/2)/(f(a/2)+2f(a))", "c806a809c207b5b69936aa2b4f61d0d8": "\\; \\mathfrak I_{\\Phi} \\vDash \\phi", "c806b29d6b6d068da0c22884ca72aa1b": "x(t) = \\cos(\\omega_0 t)\\,", "c806ccd1cfc1bed89eab229749d6c1d6": "N\\in \\mathbb{R}^*", "c807b3b73b9f23c8f9a23e723acc94c5": "\\textbf{G}(s)", "c8082407feb32ffe53c33e58ecb228f1": "(1..n)", "c8084dffcf2aaff135d9374d2ac1242b": " t \\ ", "c808d8c3782326076a2fc344beb7081d": "H_{y}=\\left [jk_{x\\varepsilon }(Ae^{-jk_{x\\varepsilon }x}-Be^{jk_{x\\varepsilon }x})+\\frac{k_{z}}{\\omega \\mu}\\frac{m\\pi }{a}(Ce^{-jk_{x\\varepsilon }x}+De^{jk_{x\\varepsilon }x})\n \\right ]sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (37)", "c8091b2086e8cd446d0aa989ef035287": "\\sin\\Theta", "c80952271167d2ba5668536d49acc003": " n\\hbar ", "c8099f8c8536aa60c686ae7b0cb98707": "\\gamma=(J-1)m/(1-m)", "c809aea4f41f245dec99eea17aa6be08": "e^{-a x} x^{ax}\\,", "c809b2354f76aed4802ddca824887715": "a_0a_1\\cdots a_{2n-1} a^{-1}_0a^{-1}_1\\cdots a^{-1}_{2n-1}=1", "c809fc606831fb270548818de198d783": "c^{\\prime} = m{\\hat G}", "c80a73d3aa056f5f4a8f15e61f196e49": "S_{xyz}", "c80a7a589faf738a47f93c29601399b7": "0\\leq j\\leq n_i-1", "c80a8772799731ba1ba00853103783cd": "M=\\{M_k\\}_{k=0}^\\infty", "c80aa54f83535aef6f132a6a0c1bd6a1": "\\varphi(44)=20", "c80acdf88f33563199e9edaa187ca1c1": "\\bold{x}=A^{-1}\\bold{b} + (I - A^{-1}A)\\bold{w} = A^{-1}\\bold{b} + (I-I)\\bold{w} = A^{-1}\\bold{b}", "c80aec8102c83edff36549bec7cc826b": " k_{s}\\tau=\\int_{0}^{t}k(\\theta)dt ", "c80b26c91747874f3e84a7f28b14c8c8": "\\operatorname{Cov}(X_i Y_i, X_j Y_j) = \\langle X_i Y_i X_j Y_j \\rangle - \\langle X_i Y_i \\rangle \\langle X_j Y_j \\rangle", "c80c242bde4a04ac89831d41ed8e99ea": "m^a\\partial_a\\,=\\,\\frac{1}{\\sqrt{2}}\\, \\Big(0\\,,0\\,,\\frac{1}{r}\\,,\\frac{i}{r\\sin\\theta} \\Big) \\,,\\quad m_a dx^a\\,=\\,\\frac{1}{\\sqrt{2}}\\,\\Big(0\\,,0\\,,r\\,,i\\sin\\theta \\Big)\\,.", "c80c84d60fdc76d6ad6f6f404cdaf257": "(s_i, t_i)", "c80d9b08d9cd157a0963a09b1e491745": "1 + R = (1 + r)^t", "c80e1bb73a1aa42c03f485ec31105f71": " \\frac{(b-a)^{2n+1} (n!)^4}{(2n+1)[(2n)!]^3} f^{(2n)} (\\xi) , \\qquad a < \\xi < b . \\,\\!", "c80e1ef80b342d560d2d2fcf796e9130": "AP=(C,D)", "c80e4560a67049ef2aee228739a984c0": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "c80ed6689d907148ce76df51f1b588ac": "[x_\\alpha = y_\\alpha] \\supset \\centerdot \\, h_{o\\alpha}x_\\alpha = h_{o\\alpha}y_\\alpha", "c80eda94a3541090c1a5cdc0c569e9a6": "\\sum_{i=1}^n a_{s,i} d q_i + a_{s,t} d t = 0~~~~(s = 1, 2, ..., k)", "c80f155a9822eb5d3dca641fc9094c56": " \\mathbf{b} \\prec \\mathbf{a} ", "c80f4900c7f9cd9609006aebc16b1459": "A\\alpha=x", "c80ff268f6372acccb2e2c3a4488fad1": "3\\$=6\\uparrow\\uparrow6={^6}6=6^{6^{6^{6^{6^6}}}}.", "c8102a32821625bac329296428d1bd10": "\\delta\\nu-\\Delta\\mu=(\\mu^2+\\lambda\\bar{\\lambda})+(\\gamma+\\bar{\\gamma})\\mu-\\bar{\\nu}\\pi+(\\tau-3\\beta-\\bar{\\alpha})\\nu+\\Phi_{22}\\,,", "c8106b547e8e8b7f6be2e7cbe9c33945": "\\Delta(a) = \\textstyle \\sum a_i \\otimes b_i", "c81086e94cc11392bc08344156ee454b": "P_c", "c810abd929c5d40b79a79d4ef25c8bef": "f(q,t)", "c810db0c525ce1c3492fc583cbd73677": "\n\\begin{align}\n\\delta m(\\phi) &= M(\\phi) \\delta\\phi\n= a(1 - e^2) \\left (1 - e^2 \\sin^2 \\phi \\right )^{-3/2} \\delta\\phi\\,\n\\end{align}\n", "c810df86d24f6fe9c7f0960c21841161": "z = S(x, y)", "c810e3b04fd7a9757dfa712ddbb752d8": "\\scriptstyle V_\\mathrm {iL}", "c81100ad2f438656cece85edf64ed479": "M_B = \\tfrac{Pab}{L}", "c8110a2ffea3d9c8aaea91a9048514f7": " {\\frac{1}{G}} = \\int_0^{\\pi/2}\\sqrt{\\sin(x)}dx=\\int_0^{\\pi/2}\\sqrt{\\cos(x)}dx ", "c81143b5aa710848814ee32022097b2d": "m{\\partial^2\\over \\partial t^2} u(x+h,t) = k[u(x+2h,t)-u(x+h,t)-u(x+h,t)+u(x,t)]", "c8118685fa8fe0923f9fbcaada2ebdf0": " D\\left ( k \\right ) ", "c811bd9babf5143f31185a07f00d5091": "\n S_n(\\theta) = M_n(\\theta)= -\\sum_{j=1}^{n+1}\\ln{D_j(\\theta)},\n ", "c811c8c43e6fceec9ac5e6e4dd4e1365": "\\|\\vec{p}\\|\\rho_f", "c8260717024703775b56ecb06aede05f": " \\overrightarrow{QP} = (x_0 - x_1, y_0 - y_1),", "c826360fce8b7c0e752b6bfbab08d0c2": "\\scriptstyle G_a", "c826411b17fd61f1344cadf23b402d7d": "\\Pr(\\theta|X=x) = \\Pr(\\theta|T(X)=t(x)). \\,", "c826688d587adce350a76f8e1370c8c3": "U_r(g) = \\{h \\in \\mathcal G : g \\ge h, |g_0 - h_0| < r\\}.", "c8269413a58f3aa1361877b2feb92576": "\\Delta f(x)", "c8269d733af1e09ba7125e4cd0979d6d": "\\Delta U = 0\\,", "c826af310ad107cc54123174ca615a3c": "\\mathcal{E}_p = kT", "c82714f1762359936a560947087e503b": "\\overline{AB}.", "c8276e4de1dc7103e4d0e195084749f1": "h_p(\\eta, \\bar\\zeta) = \\overline{h_p(\\zeta, \\bar\\eta)}", "c827a37c8f87244e59e694e139e0dec9": "(a_1,a_2) \\parallel (b_1,b_2)", "c827c143ffbe1ceabbc9e092cc9076d3": " \\mu_{T,v} (t) = a_0 + a_1 t + \\ldots + a_{d-1} t^{d-1} + t^d. ", "c827d0852493c4660eb2b9d9496cc179": "P(r \\ge 1 ; 6, 1/6) \\ge P(r \\ge 2 ; 12, 1/6) \\ge P(r \\ge 3 ; 18, 1/6)", "c827fd710fd2909f2461ec31e72f73f2": "m \\equiv \\frac{h_\\mathrm{i}}{h_\\mathrm{o}} = - \\frac{d_\\mathrm{i}}{d_\\mathrm{o}}", "c82801c85285b78059057e598a3624fb": "\\scriptstyle (G^{\\mu\\nu})", "c828258d888fec47729d2221898235c2": "\\varphi = {b \\over a} = {{1+\\sqrt{5}}\\over 2}.", "c828a46151cc77ac76fcd692f748f5f7": " \\sigma \\le \\omega \\le 2 \\sigma ", "c828e3f8edf0133dba54c3a9f9f9e148": "U[R(\\theta, \\hat{\\mathbf{n}})] = 1 - \\frac{i\\theta}{\\hbar}\\hat{\\mathbf{n}}\\cdot\\widehat{\\mathbf{J}} ", "c8290229fa7c09bd82b7a4dfea14689d": "\\Psi_1=D\\beta-\\delta\\varepsilon-(\\alpha+\\pi)\\sigma-(\\bar{\\rho}-\\bar{\\varepsilon})\\beta+(\\mu+\\gamma)\\kappa+(\\bar{\\alpha}-\\bar{\\pi})\\varepsilon\\,,", "c829349b22226327ecaa77c94e926077": "y(n) = y_r(n) + i\\ y_i(n)", "c82996f4271033acdf6df9b911856fb6": "t' = \\gamma(V) \\left(t -\\frac{Vx}{c^2}\\right)\\,,\\quad E' = \\gamma(V) \\left(E - Vp_x \\right) ", "c829ec1220fbf8241c7390144b8a5051": "L_\\nu", "c829f80c84010315a869994563d52ecd": " \\lfloor A^{3^{n}} \\rfloor", "c82a1978b0eca1800032aac61970cf44": " R\\subset T ", "c82a6f5fd6b6c3cf45c43ea9602ec970": "|\\mathit \\Gamma| = 1\\,", "c82a9740c53899d14c60476ca49e6dc7": "\\psi(U)", "c82aa3eeaac173761029f57e65b0cedf": "\\tilde{f}(e)=1\\ ", "c82ab32e8f2fc3263c04440e726359f9": "\\mbox{pH} = -\\log_{10} \\left[ \\mbox{H}^+ \\right]", "c82ad406f4e31ea6bbb1a3e0974ddd94": " \\geq 2t+1", "c82ad683005141847f710a5ef76d4b10": "\\mathrm{E}(W_I\\cdot W_J)", "c82afcc3e7c60b8db5bbb4d7072d4b60": "K_{--} \\cup \\{0\\}", "c82b63cacd6e3b622cf23fab87c02db2": " \\sigma_{A}^{2} = \\langle(\\hat{A}-\\langle \\hat{A} \\rangle)\\Psi|(\\hat{A}-\\langle \\hat{A} \\rangle)\\Psi\\rangle. ", "c82b9a10a8d10e37ee9c862cf4026d38": "\\textstyle b_{-1} = p(t_x, a_{(-1,-1)}, a_{(0,-1)}, a_{(1,-1)}, a_{(2,-1)})", "c82bd75112f377a8d0d96817ddb7790d": "\\mathfrak{p}R", "c82bedd2f6fc924f048877b5012d580c": "\n\\prod_{s\\ni e} 1-x^*_s\n\\approx\n\\prod_{s\\ni e} \\exp(-x^*_s)\n=\n\\exp\\Big(-\\sum_{s\\ni e}x^*_s\\Big)\n\\approx \\exp(-1).\n", "c82bf5c9e19cf10c1cf662398a2274d5": " H_i(1/z) \\,", "c82c7f4d62462f6ead4a57d26d9a9b5b": " x{\\in}X ", "c82cc7fec8385f000604176ab0336cdc": "k_1 + k_2 < 1", "c82cdc60c5e1b50dc226d7ae4787619d": "\\ \\displaystyle u\\in \\mathcal{U}(\\alpha,\\tilde{u})\\ ", "c82d2519562c98e23d1ddb95b2958a43": "\\int\\frac{\\sin ax}{x} \\mathrm{d}x = \\sum_{n=0}^\\infty (-1)^n\\frac{(ax)^{2n+1}}{(2n+1)\\cdot (2n+1)!} +C\\,\\!", "c82d75587cfca383dc933501d91e94fd": "\\{F,G\\}=\\sum_{i=1}^n \\frac{\\partial F}{\\partial q_i}\\frac{\\partial G}{\\partial p_i}-\\frac{\\partial F}{\\partial p_i}\\frac{\\partial G}{\\partial q_i}.", "c82de34e1fe3b02814f8d36f9c9b1e04": "\\scriptstyle \\mathcal R", "c82e62ec13eab4e13e27c7e5a18d7ec0": "72^2", "c82f2a54a8226406ae835ded0fe5d8f5": "\nL_n(x)=\\frac{e^x}{n!}\\frac{d^n}{dx^n}\\left(e^{-x} x^n\\right) =\\frac{( \\frac{d}{dx} -1 ) ^n}{n!} x^n ,", "c82f8f1b92ca8b5744bc6907a6d66b0e": "g\\geq 2", "c82f9bb6d4348e2bce2cc9afb5cf9a40": "fH_{n-i} M", "c82fe3663bf027e3cf5ba49f9b2ae622": "e_{x}", "c82ff33339d5622d9416c745c861ff66": " H = \\lbrace g \\in G : g + (A+B) = (A+B) \\rbrace. ", "c82ffeabeef77946fa13eb839ca184bc": "(1,1)^T", "c8306407842ab0efdc5842490f586411": " (1 + A \\cdot t + B \\cdot t^2) \\cdot e^{c \\cdot t}", "c830930a81e3d04d0d0cdd4e2c1bc1b3": "|x|_{p} := \\begin{cases} 0, & \\text{if } x = 0 \\\\ p^{-n}, & \\text{if } x \\ne 0. \\end{cases} ", "c830989c9e92b9adbf200f2b6c9dbf5b": "p_e = p(0|1) p_1 + p(1|0) p_0", "c830b3f6e3107f8c9574d86f9d512fc5": "-\\left ( \\frac{\\partial w}{\\partial x} \\frac{\\partial \\theta}{\\partial z} \\right )", "c830cf7c532612a1f74182751b90d7c8": "N\\in\\mathbb{N}\\,\\!", "c830dcab5577cc5edb5878c6a8398029": "i\\omega V_c \\cdot e^{i\\omega t} + \\frac{1}{RC}V_c \\cdot e^{i\\omega t} = \\frac{1}{RC}V_s \\cdot e^{i\\omega t}", "c831042ce7781b7d7686e0b1120400df": "\\operatorname{tr} A(g) = \\operatorname{tr} B(g)", "c8314ae68762632633f423a15caf0b2e": "g_{n,k}(r)", "c8316e7046bf8c7f15e653c153dd8dfa": "(e^{-s}+s-1)/s^2", "c831b374ce892b169380ddc1268213d3": "\\,K = i\\rho_2 + K_1\\mathbf{i}+ K_2\\mathbf{j}+ K_3\\mathbf{k}\\qquad ", "c831becfabebfd539ec4bda4f21185b5": "\\hbar \\Omega _i", "c83221b837b6ac9da93a327510dbfc4f": "\\Delta s = 0, \\Delta m_s = 0, \\Delta l = \\pm 1, \\Delta m_l = 0, \\pm 1", "c83223b2dda43bcbdccd68e5d779472a": "y \\left( t \\right) = 1 + x \\left( t \\right) \\cos \\left( \\omega t \\right)", "c8328a6c78c4a3cc42481c7a3b41463d": " \\Sigma_u ", "c832ce7bcf24f1cc98664d7fa3be344a": "\\mathcal{L}_{aux}=\\frac{1}{2}(A,A)+(f(\\varphi),A)", "c83300384a95b9444dac3a7dc1ddc5e5": "d\\theta/dh", "c833431e1fe66c26149317322196c2a9": "P_s \\approx 2Q\\left(\\sqrt{2\\gamma_s}\\sin\\frac{\\pi}{M}\\right)", "c8335612757cfd5f80d855e58203697f": "\\{f,g\\} = \\omega(X_g, X_f)= dg(X_f) = \\mathcal{L}_{X_f} g", "c833658769520f096f15e61b57c11d1a": "{\\mathit{c}_{v}}=\\left(\\frac{\\delta{\\mathit{u}}}{\\delta{T}}\\right)_{v}", "c833a167039bc53b035d656de903b035": " X = j \\omega I", "c833dc0089806501f048d95bd221962a": "R^{2} = {SS_{\\rm reg} \\over SS_{\\rm tot} } = {SS_{\\rm reg}/n \\over SS_{\\rm tot}/n }.", "c833e934f43d9e08f5f7fb857e3d51ac": " x=y=w \\!", "c834161a8b459e4b88d5a957606c4aae": "R=v^2\\sin 2 \\theta / g = 2v^2\\sin\\theta\\cos\\theta / g", "c8343ac4f985a3487f5ba2f7c3b7f384": "\\pi_i(S^4)= \\pi_i(S^7)\\oplus \\pi_{i-1}(S^3) , \\,\\!", "c834b645f1b4817fa2028f93ccd3cd8e": "-\\ln Z[J_{ij}]", "c834c053ad74b67b6e8c6ae0f9c29821": " \nA = \\begin{bmatrix}\n 2&1\\\\ 1 & 1\n\\end{bmatrix}\n ", "c834c915575f01786de186dd1bf91da0": " \\textbf{G}(s) = \\frac{\\textbf{N}(s)}{\\textbf{D}(s)} = \\frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}", "c834f9e5bb7cda27397d4fc7584802e3": " \\{ x_i \\}_{i \\in I} ", "c8352902684b7be1d71c9e67778ac820": " SubCipher_1=DEC_{b_1}(k_{b_1},C)", "c8357ddd9644288f89ef5604d1abda5f": "N_F(T)=d_F\\frac{T\\log(T+C)}{2\\pi}+O(\\log T).", "c835a81aa132cd2c488441162edd6a23": "-\\frac{\\hbar^2}{2m}\\nabla^2 \\Psi(\\bold{r},t) = i\\hbar\\frac{\\partial}{\\partial t} \\Psi(\\bold{r},t) ", "c835d73f715a9419a9f6199e52795099": "Y_{8}^{-2}(\\theta,\\varphi)={3\\over 128}\\sqrt{595\\over \\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(143\\cos^{6}\\theta-143\\cos^{4}\\theta+33\\cos^{2}\\theta-1)", "c835f3b0c25bc87f918ebd5050f6e6a6": "\\{h_1, h_2, \\dots h_p\\} ", "c836305cf154233e159265ab1ddfff02": " \\mu^* = n(\\mu - b)/c \\,", "c8365ce97eec8e95db6cc8c502de1637": "(x, k)\\in L", "c83692d1da47211e167aa6ec9e4db0af": "\\Delta w (\\text{cm}^{-1}) = \\left( \\frac{1}{\\lambda_0 (\\text{nm})} - \\frac{1}{\\lambda_1 (\\text{nm})} \\right) \\times \\frac{(10^{7}\\text{nm})}{(\\text{cm})} . ", "c836a4cb7d451e4cc327148e52c2d9bc": "(G,\\,\\nu)", "c8371043f29368322f54dd7b5811b216": " f(e) ", "c8371411438bc471d2bd8a72b68db087": "\\nabla^2 E - \\mu_0\\, \\varepsilon_0\\, \\frac{\\partial^2E}{\\partial t^2} = 0.", "c8374d5cbddf99b257ab6b0da73f7f0c": "G\\times X\\rightarrow X,", "c83755e3a62100f7d7777144f5c684e8": " x^n(x^2-x-1) + (x^2-1).\\ ", "c837e7b7890007a591add7ca8ccd9fb6": " 0 \\subseteq R^{d\\times d}", "c838153960ba8c05f88c4b7d5c724369": "\nS^\\pm = S^x \\pm i S^y\n", "c83838326bc338663919b25b8cb17235": "\nf(x) = \\left\\{ \\begin{matrix}\n 1 & \\mbox{if } x>0 \\\\\n 0 & \\mbox{otherwise}\n \\end{matrix}\n \\right.\n", "c83848765d6e6f55df8350118de25240": "w^+_r(c^{\\rm eq},T)=w^-_r(c^{\\rm eq},T)=w^{\\rm eq}_r", "c8388434b3bf00227329305ff4c91d96": " \nO = \\begin{bmatrix}\n \\cos \\left( \\theta \\right) & -\\sin \\left( \\theta \\right) \\\\ \\sin \\left( \\theta \\right) & \\cos \\left( \\theta \\right)\n\\end{bmatrix} \n ", "c8388c08b37683541a990fa360a93690": "\n\\overline{\\phi(\\boldsymbol{x},t)} = \\displaystyle{\n\\int_{-\\infty}^{\\infty}} \\int_{-\\infty}^{\\infty} \\phi(\\boldsymbol{r},t^{\\prime}) G(\\boldsymbol{x}-\\boldsymbol{r},t - t^{\\prime}) dt^{\\prime} d \\boldsymbol{r}\n", "c8389ecb628fb2751e6766401895c931": "c_0 = \\sin\\varphi\\!", "c838c65b5a6ef8754ffa71f7c44d7bb1": "S^{\\mathrm WZ}(\\gamma) = - \\, \\frac{1}{48\\pi^2} \\int_{B^3} d^3y\\, \n\\epsilon^{ijk} \\mathcal{K} \\left( \n\\gamma^{-1} \\, \\frac {\\partial \\gamma} {\\partial y^i} \\, , \\, \n\\left[\n\\gamma^{-1} \\, \\frac {\\partial \\gamma} {\\partial y^j} \\, , \\,\n\\gamma^{-1} \\, \\frac {\\partial \\gamma} {\\partial y^k}\n\\right]\n\\right)", "c838e43a0518d238e032cb4d3d6efaa1": "Y_{10}^{-1}(\\theta,\\varphi)={1\\over 256}\\sqrt{1155\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(4199\\cos^{9}\\theta-7956\\cos^{7}\\theta+4914\\cos^{5}\\theta-1092\\cos^{3}\\theta+63\\cos\\theta)", "c838e53e0f363f33acbe5768422cb185": "k_L(f)=\\frac{L_m}{\\sqrt{L_1L_2}}\\frac{2}{\\sqrt{(1+f_1^{-2}f^2)(1+f_2^{-2}f^2)}},", "c838f4d4f95ca67e4c5a19fb74dbcd32": " = \\mathrm{rect}(Tf)\\cdot \\sum_{n=-\\infty}^{\\infty} T\\cdot x(nT)\\ e^{-i 2\\pi n T f}", "c8393631971b42d6ed062d82fdd9e4d5": "\\cos\\theta=(H\\cdot V)", "c83937ad66a46e0280bb5d3e825a5196": "\\beta=-\\frac{1}{V}\\frac{\\partial V}{\\partial p}", "c8394d432b833d5c47a77b2e5e47dec0": " q\\phi \\,\\!", "c839ddc5fa3fe91b67e2f88516c1a9ee": "a_n\\!", "c83a39181aab04a453820f435af8cf88": " |\\eta| = 1", "c83a5988ad4ac0dd557514bb84963021": "v \\ne u_1^{x+{\\alpha}y} rem P", "c83b074e1c32ea92b5f77ba93bb3c14a": "S(\\rho_{ABC}) \\, + \\, S(\\rho_{B}) \\, \\leq \\, S(\\rho_{AB}) \\,+\\, S(\\rho_{BC}).", "c83b27591e01fafd64e58987ff776abf": "N=N(\\varphi)=\\frac{a^2}{\\sqrt{(a\\cos\\varphi)^2+(b\\sin\\varphi)^2}}\\,\\!", "c83c6931644b91c0e64d6c29787a1e87": "F(x_1, \\ldots, x_n)", "c83cf7366d82a5f69071c28a9d679ee7": "\\operatorname{Cov}(X_i^2, X_j^2) = \\langle X_i^2 X_j^2 \\rangle - \\langle X_i^2 \\rangle \\langle X_j^2 \\rangle", "c83d16f5e81177d8b370e9dfcc7de0b2": "f(i) := \\begin{cases} i+1, & \\mbox{if } 0 \\le i < n-1 \\\\ k, & \\mbox{if } i = n-1. \\end{cases} ", "c83db59455c111953039ac7d658db5e3": "\\cap\\backslash\\!\\!\\cup\\!\\text{-}", "c83eab0cfba4ec2c3d0781808c01274b": "A\\colon X \\to X", "c83f1f5e63ec8bc1b599459b5d501623": "\\xi_{-1}(-\\hat{z}) = \\begin{pmatrix}\n-1\\\\\n0\n\\end{pmatrix} \\,", "c83f567b08625dd66a8948cfff06ca2f": "\\mbox{EF }\\mbox{EG } p \\Rightarrow \\mbox{AF } r ", "c83f6083b828694da61a066fcfd0e77f": "\\Delta H_m = N_1\\phi_2z\\Delta w\\,", "c83f72d6a0a56c0d653a72bc089a1dce": "x_s(t)\\,", "c83f9aa26dd8bfec3c05fbe8f87b8d2e": " 1 = \\frac{0.3048 \\ \\mbox{m}}{1 \\ \\mbox{ft}}.\\ ", "c840001e6a10d748279039fc1af00f9b": "p^a \\le N", "c8400b14cd99d6f25a08208ee79b8f87": "F= k X\\,", "c84012f38fe9103179b0ba6474fb1e29": "U = B_0\\,\\sin^3(\\theta)", "c84035db91fd28d26e6c17c98e0edbe7": "q^{-1} = q^*\\,/\\,(ad - bc).\\,", "c84039ff5e7848666731954038df2757": " K = K_0 + P\\ K_0'", "c8405afd2d6fa2a923ace7ce4795038b": "\n\\begin{matrix}\nT_{a^p}(x) = & \\underbrace{T_a(T_a( \\cdots T_a(x) \\cdots ))}, \\\\\n& p \\, \\textrm{ times} \\\\\n\\end{matrix}\n", "c8407a3cdb1bd69d3bc0eb48a181ed28": "x = \\frac{\\chi (1 + 1)}{1} = 2\\chi", "c840a4689c95e0faa77d3defa7823f94": "d = \\sqrt{x_1^2+x_2^2+x_3^2} ", "c8410cb43a9dfb9692ff24fbafb36f20": "y_t =c + A_1 y_{t-1} + A_2 y_{t-2} + \\cdots + A_p y_{t-p} + e_t, \\, ", "c841460084460caf8316640fb7b6b546": " x_{n+3} - 3x_{n+2} + 3x_{n+1} - x_n = 0 \\, ", "c84160aebeba27a5959d24f54c6f46fc": "(A^{\\rm T} B)_{jk} = \\sum_i A_{ij} B_{ik}.", "c8420ebf42bb6ae3cf9201ce849fe0ca": "K_d = \\frac{k_{off}}{k_{on}} = \\frac{[Ab][Ag]}{[AbAg]}", "c84255c21b098b0c2b7165ef25564d2b": "(\\exists y)", "c84267f628f906b4de11fc9ce06b283d": "\\hat{\\boldsymbol{\\beta}} \\approx \\mathbf { (J^TJ)^{-1}J^Ty}.", "c8427d2ddf5d151293b44fd25d95a398": "(1/p)(1/p) = 1/p^2 = \\infty", "c8427dd4e47e7c4a2a0212a98d7ae698": "x_{T-2}", "c842ef26c3fd74c6a36363d2693604fe": "\\theta (u,\\xi )=\\theta (u\\xi ,1)=\\beta (u\\xi )", "c843ae81129ae4151f562f90c4a28271": "\\Omega_m", "c843c76d90b11cf85646b7e50b566d5a": " x_i \\ge \\lceil x_i' \\rceil", "c843df0bc243224e007ac45a856fcbd8": "\\Sigma p_i = 1", "c84419e6372dbcad38243dd83d60c6b8": "a^{d\\cdot 2^r}\\equiv -1\\pmod n\\text{ for some }0\\leq r\\leq s-1. \\, ", "c844604e1f2d6309c7683128cc803964": "\\scriptstyle \\frac {[\\mathrm {CO_2}]} {[\\mathrm {CO}][\\mathrm {O_2}]^{\\frac {1} {2}}}", "c8448a24e11870b1e5d9b1c98507f016": "I\\geq\\Lambda\\geq0", "c844a0c5c8aa5a6584776b44f28f2ed6": "\\textstyle T_{s,t}", "c844c84dc59bc679920fb709b646fbfe": "\\{1,\\dots,N\\}", "c844d2f5c07efbffe327fdfc66d7fda7": " \\|f \\| _{W^{s, p}(\\Omega)} := \\|f\\|_{W^{\\lfloor s \\rfloor,p}(\\Omega)} + \\sup_{|\\alpha| = \\lfloor s \\rfloor} [D^\\alpha f]_{\\theta, p, \\Omega} ", "c84520d8870434a7c188a4723e878e9f": " \\delta (r^{(2)})=r^{(2)}{}_{(-1)}\\otimes r^{(2)}{}_{(0)} ", "c8457d75b58e0ff620e717328ac19b5f": "\\int_1^{\\infty} \\frac {x^2-y^2}{\\left(x^2+y^2\\right)^2}\\ dy = \\left[\\frac{y}{x^2+y^2}\\right]_1^{\\infty} = -\\frac{1}{1+x^2} \\ \\left[x \\ge 1 \\right]\\ .", "c8458a4e2f497ddf432b50ecfcb32ba4": "1/200000th", "c845974c1bf59cabb9d3ab53fa631c75": "s = k^{-1} * (h + x*r)", "c845a6891f2f0b10f8ab8fc1e6f7a8eb": "p_1, p_2, p_3,p_4", "c845ad58caf6836d7267a4f4ea50aa4a": "2 N", "c845d5261fc4f9b4388c5ce79e3d7320": "[e_i,[e_i,\\ldots,[e_i,e_j]]]= [f_i,[f_i,\\ldots,[f_i,f_j]]] = 0\\ ", "c845dcc08e9d5474c1a24f06a4770e6f": "K(\\omega) = \\frac{\\epsilon^*_p - \\epsilon^*_m}{\\epsilon^*_p + 2\\epsilon^*_m}", "c845f23e07ea737cc26986b9883e3d57": "R, \\bar R", "c845fc65d2b37d4751bf35f3a91cb241": "\\operatorname{cr}(y_1,y_2,y_3,y_4)-1={{(y_1-y_3)(y_2-y_4)}\\over{(y_1-y_4)(y_2-y_3)}} -1={{(y_1-y_2)(y_3-y_4)}\\over{(y_1-y_4)(y_2-y_3)}}>0", "c8461dab185df17b6ceaee3d0e8d67d6": "E_\\mathrm{k} = \\tfrac{1}{2}mv^2 \\, .", "c8463252c08ffb6d0b0b0ea706e90283": "\\mathfrak{p} \\mapsto \\mathfrak{p}[S^{-1}]", "c8463ac7a74fc3418500435a9ee797a3": "\\left \\{ L^x_{T_a} + W_x^2 \\colon x \\geq 0 \\right \\} \\stackrel{\\mathcal{D}}{=} \\left\\{ (W_x + \\sqrt a )^2 \\colon x \\geq 0 \\right \\}. \\,", "c84644736cd503be6748b7f2befaaedd": " s(g,x) = \\bigg[\\frac{d \\mu}{d g^{-1}\\mu}\\bigg](x) \\in [0, \\infty) ", "c8464bba70639d1096211ccc04f929c6": " F_i^{-1}(u_i)", "c846b55ae17235d4e35e9a35dcd3e836": "\\begin{align} E_\\mu \\| h(x) - \\mu\\|^2 & = E_\\mu \\|g(x) + x - \\mu\\|^2 \\\\\n & = E_\\mu \\|g(x)\\|^2 + E_\\mu \\|x - \\mu\\|^2 + 2 E_\\mu g(x)^T (x - \\mu) \\\\\n & = E_\\mu \\|g(x)\\|^2 + d \\sigma^2 + 2 E_\\mu g(x)^T(x - \\mu).\n\\end{align}\n", "c847019b2e530a5b42c5e99b303d1551": "\\begin{align}\nx' &= \\gamma \\left( x - v t \\right)\\\\\nt' &= \\gamma \\left( t - \\frac{vx}{c^2} \\right) \\\\ \ny' &= y \\\\ \nz' &= z\n\\end{align}", "c84748dd0da2455b3c7b1d3225d15265": "\\widehat{\\theta} = \\left(\\widehat{\\mu},\\widehat{\\sigma}^2\\right).", "c847587201860c9a27f5fccf86499316": "\\Omega \\subset \\Re^d", "c8475a61cd6787beb1a28688041a82e9": "(\\mathfrak{m}_{X, p}/\\mathfrak{m}_{X, p}^2)^*", "c847aad780fd71a5f8ceab5aef4a5687": "\\Sigma^p_{ii}", "c84818ab7bf0c07541a1af4b7ef4ea69": "\\beth_{\\beta}(\\beth_{\\alpha}(\\kappa)) = \\beth_{\\alpha+\\beta}(\\kappa).", "c848b4b7aad92a0ffd9fb7534bc658fd": "\\lambda : X : Y", "c848b9adff1fbbebe00d24e1976dfe9e": " 1 - e^{\\tfrac{a(1-x^b)}{b}} x^{-1+b} ", "c848dc5fae59d0531de5a597b0a1e5fb": "\\lim_\\omega(X_n,d_n)", "c848e2cf0f1902cf75f641afb3dc4efa": "\\left | f^\\prime (p) \\right | =0 ", "c8491d2fea79f6eaed877df175c89e9e": "\\left( \\frac{3}{4 \\pi n} \\right)^{1/3},", "c84946892fa0d119d2943232c3cbfab4": "\\frac{0}{0}=\\,?", "c849749c04bcc8cb3a175534820ac63c": "Y = Y_n + a (P-P_e) \\, ", "c84ab497814155a1b32cb4dfe35ce5ec": " 2h", "c84ab836bab39bd5ccd08acce335fc05": "\nz = z\n\\!", "c84abbc60f68fc5023f35a2ef805baac": "\\gamma_3 = \\frac{4-\\pi}{2} \\frac{\\left(\\delta\\sqrt{2/\\pi}\\right)^3}{ \\left(1-2\\delta^2/\\pi\\right)^{3/2}}", "c84ac361ad942ae94cd9a9cdae793577": "u=\\frac{1}{2} m \\omega^2 x^2 + \\frac{p^2}{2m}", "c84b293cb7f540cead1b6310c4a89f79": "\\Gamma^{\\mu}{}_{\\sigma \\nu} ", "c84b2a84397ec65ca4b9f05e58d05ea9": "\\hat{\\delta} ~=~ (\\overline{y}_{11} - \\overline{y}_{12}) - (\\overline{y}_{21} - \\overline{y}_{22})", "c84b3d72858ed8192e7e58121274a553": "n \\ge F_{d+2} \\ge \\varphi^d", "c84b8b899d153e4b65695ab3cae4759d": "\\sigma_\\theta", "c84b934aabe8dccb0edcde79902c78c5": " \\left(\\frac{\\partial \\ln N_i}{\\partial P} \\right)_T = -\\frac{V_{i,aq}-V_{i,cr}} {RT} ", "c84bf5b6d258ea52b31fa6e5e2643588": "\\beta_k \\leftarrow {r_{k+1}^* M^{-1} r_{k+1} \\over r_k^* M^{-1} r_k}\\,", "c84c13b6fb5122c6138b7a3de56c0c83": "\\operatorname d f_p", "c84c20a9f5e5748c776213d4bf7d4706": "\nn_a\\,=-dv \\,,\n", "c84c2dc1c413a9904277542ced1e6f4c": "f':j'\\to k", "c84d4ebd392383806275a482d75d10fa": "\\mathrm{Log}(z_1, z_2, \\dots, z_n)= (\\log |z_1|, \\log|z_2|, \\dots, \\log |z_n|).\\,", "c84e06e360b10834c88c10911a6a912b": "\n\\frac{1}{R_\\mathrm{eq}} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\cdots + \\frac{1}{R_n}.\n", "c84e8a9a3428d69e83484139baf32e74": "y \\mapsto y \\cup \\{y\\}", "c84eadfc025d67b8baa9e8e179054f00": "\\mathbf{J=U \\boldsymbol\\Sigma V^T} \\, ", "c84ecc0317c34a072e59c6cfb2e944dd": "\n\\iint_{S} f \\,dS\n= \\iint_{T} f(\\mathbf{x}(s, t)) \\left\\|{\\partial \\mathbf{x} \\over \\partial s}\\times {\\partial \\mathbf{x} \\over \\partial t}\\right\\| ds\\, dt\n", "c84f919301fe5d63c5edd1830d347d0b": "p - 2^x", "c84fcd85875b3948eb4e8eddb04428f8": "\\sqrt{L}", "c8501c9be08c8a377531dfbe12e26f27": "L = C - H - 0.16T", "c8503841902ee71dba32a975526d9ba8": "H(x,y) = (h(x), h(y)). \\, ", "c850e7b6675db00dfe8a086e4fdf30f7": "\\delta W = \\sum_{i=1}^n \\mathbf {F}_{i} \\cdot \\frac {\\partial \\mathbf {r}_i} {\\partial q_1} \\delta q_1 +\\ldots+ \\sum_{i=1}^n \\mathbf {F}_{i} \\cdot \\frac {\\partial \\mathbf {r}_i} {\\partial q_m} \\delta q_m.", "c8514d9aa0eee7863606d4de235f9555": "y(x) = \\exp\\left[\\frac{1}{\\delta}\\sum_{n=0}^\\infty \\delta^nS_n(x)\\right]", "c851768f3cff5e150a6dfe1f30827ae0": " \\frac{1}{\\lambda}_\\mathrm{net} = \\sum_j \\left ( \\frac{1}{\\lambda}_j \\right ) \\,\\!", "c8519d6cb25a1e8ec59cdd8bafc49bc0": "\\mathrm{d}\\mathbf{\\Omega}", "c851cd3186c05794a7ab79ee7e9095c3": "\\lambda_5 = \\tfrac{1}{2}(1+\\sqrt{5}),", "c85254c9a0cb3aee3621c990f461cd6a": "s'=c_1^{{q-1-x}}=g^{y(q-1-x)}=g^{-xy}", "c852583d0e10a87795f99f0ddc0e5db0": " \\stackrel{\\triangledown}{\\mathbf A} = \\frac{D}{Dt} \\mathbf{A}-\\dot \\gamma \\begin{pmatrix} 2 A_{12} & A_{22} & A_{23} \\\\ A_{22} & 0 & 0 \\\\ A_{23} & 0 & 0 \\end{pmatrix} ", "c852965ce2c6f395e0d1ee20c2e8b1c9": "\n\\overline{a} = a,\n", "c852b24c99ca7ce32848e0b38585bea9": " \\phi(k)^* = \\phi(-k)\\,.", "c852b8e6855699bd39f56b3ae52b3965": " \\int_1^n \\log x \\, dx \\leq \\sum_{x=1}^n \\log x \\leq \\int_0^n \\log (x+1) \\, dx", "c8532326ba403e7f8c614681dcbc1b7c": "\\mathrm{A}_3", "c8533416fd8e4415bd40489ccc836c7e": "\\{\\xi_i\\}", "c8538c58d6daf1c55a6251f6bafd3cc5": "\\epsilon_s~", "c853bb0354434b745074100723e59d9f": "f_{S_t}(s; \\mu, \\sigma, t) = \\frac{1}{\\sqrt{2 \\pi}}\\, \\frac{1}{s \\sigma \\sqrt{t}}\\, \\exp \\left( -\\frac{ \\left( \\ln s - \\ln S_0 - \\left( \\mu - \\frac{1}{2} \\sigma^2 \\right) t \\right)^2}{2\\sigma^2 t} \\right).", "c85404512fef48c23d95059455dd2cff": " xe_\\lambda \\rightarrow x", "c8540a12b234b3cfe0ef09bc420116aa": "HH^{\\dagger} = N \\; {\\mathbb I} ", "c8546b679af45cbe84549d8b9f141969": "\\Sigma|g_k|", "c8556299a534e5139ee86a6facbdf84a": "\\!V = \\int_{-r}^{r} \\pi (r^2 - x^2)dx.", "c855afd6defb7880deb3bf04fcffe82b": "h = \\operatorname{gcd}(a+c,d+b)", "c855e2491e4247551377d62f63b01e1a": "\\tfrac{a + b}{2} h", "c85603fb6d887bb566eea7d1bbb21e5e": "Z_\\mathrm S = Z_\\mathrm L^* \\, ", "c8560fb144800ba07bd3fa68d65470e9": "\\scriptstyle g^{x_i}", "c85648fcc6f65a7777f0dc7cc55f588d": " \n\\phi(x)=\\frac{1}{4\\pi|x|} \n", "c8564d6bb0b473b1734e92ef25159dd4": "B_{T}", "c856a09413ad4f4496c6f9b99167d726": "Z = S^{k-1}", "c856c64544afe146d7d0e0b70a5f2ad5": "\\lambda_-^0", "c856ccb10635ea2d08c4e2c295086e87": "X \\subseteq V", "c8578ab64456055e8a5fc7536f2e21d4": "a \\Rightarrow b.", "c857ab592107b384c462918ade90c2fe": "\\int \\mathcal{D}\\phi Q[F][\\phi]=0", "c857fe9dbfadebb3f2012b49ca9da37f": "\\omega = 7\\,292\\,115\\times10^{-11}\\, \\mathrm{s^{-1}}", "c8582fedbf9ac2f404bb4d7245a1ab2b": "x \\leftrightarrow y ", "c858496d8a274c419fbf683701f5eb8a": "m_x", "c85851771970611baf2eaa89c4b6e2fe": "H_\\kappa \\!", "c85863207c62d0eefb5cb4659d5f1727": "s(t+x)\\ \\stackrel{\\text{def}}{=}\\ g(x),\\,", "c8589bdf043df43e0f6ef00e236ef416": "( e^{-\\pi (t-\\tau)^2} )", "c858f75a0f39c57efdc6af4fda0f321f": "\\scriptstyle\\mathcal{K}", "c8590d525f1600e4783a401a3d21a19e": "\\dot {\\epsilon} \\epsilon_0", "c8595e454bf1d4da683a9b677533452e": "\\frac{d\\mathcal{L}(x)}{dx} = 0 ", "c859716438b5ec70f6e9e399c82c9573": "B_A = \\frac{[ABC]_P . D_{IV}}{[ABC]_{IV} . D_P}", "c859948f587df138774dfe6faac54ceb": "V_{\\text{out}} = -V_{\\text{T}} \\ln \\left( \\frac{V_{\\text{in}}}{I_{\\text{S}} \\, R} \\right)", "c859e7d17491066f783da8b36aa845b1": "V^a V_a\\leq 0", "c85a09227bff8d9c78c8157ad5d5486a": "a_{\\overline{n|}i}^{(m)} = \\frac{1-v^n}{i^{(m)}}", "c85a14696437c129478214ee5d39fd4b": " {S_i} \\rightarrow {-S_i} ", "c85a2d1e7f85c85276bca46198f44cad": "C \\to \\frac{R'}{R} \\,C", "c85a6f50aed21b913e3ba2e427a58cc2": "\\sigma_{33} = 0", "c85aa6a0ee107b4d80bb8e89c2e7fb16": "y^\\prime", "c85ae61d1937c894a5c646a5121743d6": "1\\leq j\\leq n", "c85b0679628d61781e92954a968b427f": "A/\\mathfrak{i}", "c85b48a70f69265e8929d0dcb451ce9d": "\\left(\\dfrac{(\\log{n})}{\\log(2-\\alpha)}\\right)", "c85b51991c075c3ef1b8c44322da13c3": "\\text{ROE} = \\frac{\\text{Net income}}{\\text{Sales}} \\times \\frac{\\text{Sales}}{\\text{Assets}} \\times \\frac{\\text{Assets}}{\\text{Equity}} ", "c85b75dca8947de9b51db4cb81f0f363": "C\\ell_{p,q}^0 = \\{ x\\in C\\ell_{p,q} |\\, \\alpha(x)=x\\}.", "c85bac6fc1a0d1bdec319b19c60aa108": "\\Phi_{3\\times 5\\times 7\\times 11\\times 13\\times 17}(x)", "c85bd9f5709dd45e6ca2c33ba26e60fa": "E[D]", "c85be078741ac85519757a8945ad32bf": " \\widehat\\varepsilon_{ij} = Y_{ij} - \\widehat Y_i \\,", "c85c4e727cfe5cde1aa5a516372eaa9b": " G\\approx 6.674 \\times 10^{-8} {\\rm \\ cm}^3 {\\rm g}^{-1} {\\rm s}^{-2}.", "c85c79a8b2956ed0ce700446dbf28ac0": " e(k+1) = (A - LC) e(k)", "c85cfcb31f7ba7e77df12810588a2b30": "A^n+c_{n-1}A^{n-1}+\\cdots+c_1A+c_0I_n=\\begin{pmatrix}0&\\cdots&0\\\\\\vdots&\\ddots&\\vdots\\\\0&\\cdots&0\\end{pmatrix}.", "c85d04252fbfb7777a85178fe94ee181": " G_{ab} = g_{\\mu \\nu} \\partial_a X^\\mu \\partial_b X^\\nu ", "c85d2e4e20436754b4926f981ff5c7d7": "\\frac{1 + 2\\%}{1 + 10\\%} - 1 = -7.27\\%", "c85d39a3b101fe6c1bee74e5cbcad968": "\\gamma_{SG}\\ =\\gamma_{SL}+\\gamma_{LG}\\cos{\\theta_{c}},", "c85db88574e03dbd18fb8593c2067793": "AB + BA = 0, \\;\\ldots", "c85e20a400d2bf98684b17e93cf3e52b": "\nP A P^{-1}= \\begin{pmatrix}\nA_1 & 0 & 0 & \\dots & 0 \\\\\n0 & A_2 & 0 & \\dots & 0 \\\\\n\\vdots & \\vdots & \\vdots & & \\vdots \\\\\n0 & 0 & 0 & \\dots & A_d \\\\\n\\end{pmatrix}\n", "c85f072a5c7d3b67bbc36f60f060050f": "\\mathcal{M}'", "c85f0a1548d1460f52d0d8532037dc02": " a+b+c = \\frac {S} {r} \\, ", "c85f1765e3cf2c2a3fe87ccd5793f04c": "\\eta = \\left |\\frac{W}{Q_H} \\right|\\,\\!", "c85f574b5de6db6248db8016a4e042a3": "A \\models \\alpha \\Leftrightarrow B \\models \\alpha", "c85fb29c28cf87ab07a3e4e9b104dac8": "\\Delta f = \n\\mathrm{d}\\delta f + \\delta\\,\\mathrm{d}f = \n\\delta\\, \\mathrm{d}f = \n\\delta \\, \\partial_i f \\, \\mathrm{d}x^i", "c85fda186ce1f941c7ac10f638896775": "I_{in} \\ll I_S \\, ", "c8600a0cc45fe853cb446a96bb8eae35": "Rx", "c8602672bf693050cc11723157df1fb9": "\\phi _n^{\\mathrm{even}}(x)", "c8604981b36752e54faf058235833e29": " \\operatorname{build-list}[\\lambda x.\\lambda o.\\lambda y.o\\ x\\ y, D, V, D[g]] ", "c86080cb0aa236c715d38933373a9e3f": "\\frac{\\rho - \\rho_0}{\\rho_0} = \\beta \\left( T_0 - T \\right) ", "c860f70a2018b886ee001d905eaf181b": "\\tbinom {12}5", "c8616cf382b39f9a6f25bc7ac5ed5e18": "\\tilde{(F \\circ G)}(x) = Z_F( \\tilde{G}(x), \\tilde{G}(x^2), \\tilde{G}(x^3), \\dots)", "c8619d668993761fc7d60419b9d10da0": "\\eta_Y:Y\\to G(F(Y))", "c862755990e2d5cedcf6b456b944b405": " \\scriptstyle{b=\\infty} \\ ", "c8627a1f7c4c7490d8bedadf3431d93b": "\\textrm{If}\\quad {\\mid}m{\\mid} > \\ell\\,\\quad\\mathrm{then}\\quad P_\\ell^{m} = 0.\\,", "c8627eb193bc09462e5d0aaa01316c16": "y_i\\!", "c862802ba105ccfac55854926943c670": "\\rho\\;", "c862a926bac6dc6455a0e4eb21e2dfe8": "|\\mathbf \n{A}|_+", "c862d1c11562c4cb5613ce88132b1fab": "v_{th}=\\sqrt{\\frac{k_BT}{m}}", "c8632317474add791aea9933e168e0b8": "\\left [\n\\begin{smallmatrix}\n 2 & -2 \\\\\n -2 & 2 \n\\end{smallmatrix}\\right ]", "c863288d68ffba10a138ab14398069cc": "T_{d,min}", "c8634757c0d4ee245a173881d12602d9": "P(A, \\ B).", "c863535c3d658c6fc528b423367d5bf4": "\\boldsymbol{\\mathcal{A}}^n(\\mathbf{r}, t) = [ \\underbrace{\\mathcal{A}^n_0(\\mathbf{r}, t)}_{\\text{timelike}} , \\underbrace{\\mathcal{A}^n_1(\\mathbf{r}, t), \\mathcal{A}^n_2(\\mathbf{r}, t), \\mathcal{A}^n_3(\\mathbf{r}, t)}_{\\text{spacelike}} ] = [\\phi^n (\\mathbf{r}, t), \\mathbf{A}^n (\\mathbf{r}, t)]", "c863a1c893efb6add76713299753909c": "\\mathbf{P}[ A | F_{k} ] \\to \\mathbf{1}_A ", "c863a98eca52d853f53b9ca637a9d6c9": "\\frac{f_\\theta(x)}{f_\\theta(y)}", "c863d9375393adc136bfb1029b1cdb15": " z \\mapsto \\bar{z}^2 + c\\, .", "c86413a20cdb33ee455882ccd6a15c5b": " Q(a,b)=L(a)L(b)+L(b)L(a) - L(ab),\\,\\,\\, R(a,b)= [L(a),L(b)] + L(ab). \\, ", "c864942be7c1043bb5cedc54f9f6753f": " S_u ", "c864c3dce8882e09620be2cb616568da": "PR(A)= 1 - d + d \\left( \\frac{PR(B)}{L(B)}+ \\frac{PR(C)}{L(C)}+ \\frac{PR(D)}{L(D)}+\\,\\cdots \\right).", "c864e91c93d838310d35b25e2fa3104a": "(x-y)[x^t(x-z)-y^t(y-z)]+z^t(x-z)(y-z) \\geq 0\\,", "c865368db64423b2dd35c8f40a856747": "\n\\, \\sin \\phi_\\mathrm{s} = \\frac{-\\sin h \\cos \\delta}{\\cos \\theta_\\mathrm{s}}\n", "c8655121d3cbf634a9d40d817aa9de14": " \\frac{dy}{dx}+p(x)y=q(x)", "c8656e25a9d7705c080ad7ce27bb8288": "\\mathbf{\\mathit{L}}_{D}[\\rho] = 0", "c865c86738d9a779aaac64266562c7d8": "\\sqrt{\\frac{1}{2}}", "c865e1497bc8f5d5d1962e6197708a11": "\\psi(\\bar{x})", "c865f9db5845e7d78270cf8cdd94ccb1": "\\theta_2=62.25^\\circ", "c8661796942eb89d93723bc781c29faa": " \\dot u\\,,\\ddot u\\, ", "c86626a6242ebc4624d5afe26ee3a986": "\\star \\lambda \\in \\bigwedge^{n-k} V", "c86674a0fd95c55679062567f12871f8": "\\frac{\\partial I}{\\partial x}V_x+\\frac{\\partial I}{\\partial y}V_y+\\frac{\\partial I}{\\partial t} = 0", "c8668d8e8c3d129a63baef6cd0f88328": " E\\left(u^{2}_{i}|x,z\\right) = \\sigma^{2}\\left(x,z\\right) ", "c866bef054f61d36c89dd7a1e081d517": "0.2\\overline{6}", "c866eb4f18bee02db6f1cd346a7c5d2c": " E _{ext} ", "c8674ea9863b2e88b5205b4866be810d": "{1 \\over n} \\sum x_i + {1 \\over n}", "c867700360832c395bbe8079b99bc535": "[\\alpha \\otimes X, \\beta \\otimes Y] := \\alpha \\wedge \\beta \\otimes [X, Y]_\\mathfrak{g}", "c8677522ce16d202d86d6abcdcce630a": "{\\mbox{Rate}_1 \\over \\mbox{Rate}_2}=\\sqrt{M_2 \\over M_1}=\\sqrt{352.041206 \\over 349.034348}=1.004298...", "c867a54723e1f670eb729383252c9a55": "\\frac{\\tan \\theta}{\\theta} > 1\\ \\ \\ \\mathrm{if}\\ \\ \\ 0 < \\theta < \\frac{\\pi}{2}\\,", "c867c766d0d0ebaa904e945ef7b319ec": "\n\\frac{d}{dt} = \\omega \\frac{d}{d\\varphi} = \\frac{h}{r^{2}} \\frac{d}{d\\varphi}\n", "c867df816fb0abae0e428582233bd79f": " f(v) = \\sqrt{\\left(\\frac{m}{2 \\pi kT}\\right)^3}\\, 4\\pi v^2 \\exp \\left(- \\frac{mv^2}{2kT}\\right), ", "c868096518189c7758c4f67763e36ba2": "\\frac { dU(r,w)}{ dr} - ikU(r,w)=0 \\quad (1.1)", "c8680fa208df369c16fc14c68f7d5242": "s(x)", "c86866f3c46c0415ef2c6b2c9164550d": "\\bold{q}", "c86868e8b7eb107f7ac2445cf911fb5f": " \\tilde{\\varepsilon}_{ij}", "c868896c55800e7d489b20e7ca40cf84": "\\triangle_{S}", "c868a333c28f5d7ef47fc2ba5e9e1119": "A\\,X=C,", "c868dadc95be9b4d4efeb048d68b879d": " V_{\\omega C} \\, ", "c868f99632efd5d5c4bfef8e497c86c4": "\\Delta^n := \\{x \\in \\mathbb{A}^{n+1} \\vert \\sum_{i=1}^{n+1} x_i - 1 = 0\\}", "c8691a58fe4234f7d231e8af566577a7": "R_\\mathrm{ap}", "c86971b4329c52880bcfedcfad3fc254": "x_i < y_i\\ ", "c869843748fd14153a7b715b9bb21753": "\\theta_B = \\frac{-6.937}{EI} ", "c8699744128848a307afa8895c1e23f4": "\\check{H}^1(X;\\mathbb{Z})=\\mathbb{Z},", "c8699a6c0c67b30729ff3abe41cf3cee": "{{\\tau }_{1,2}}=\\frac{\\frac{T}{2K}}{Arccoth\\left( \\pm \\frac{\\sqrt{1-2{{u}^{2}}-\\left( 4uv+2{{u}^{2}} \\right)\\cos \\left( n\\omega \\frac{T}{2K} \\right)}\\pm 1}{2u\\sin \\left( n\\omega \\frac{T}{2K} \\right)} \\right)}", "c869afe5373f3628959b96d957cc15fd": "F(t)=G(t)/t", "c869d085c2504a7544a88db37b87789f": " W^{s,p}(\\Omega) := \\left\\{f \\in W^{\\lfloor s \\rfloor, p}(\\Omega) : \\sup_{|\\alpha| = \\lfloor s \\rfloor} [D^\\alpha f]_{\\theta, p, \\Omega} < \\infty \\right\\} ", "c869e39efe470235b14e118fa1937a15": "X \\subseteq N", "c869e6faf88729f1d2051ae3f043901d": "c=\\text{percent conversion of each invite } ", "c869fbdf73ca82cc400105de11a52a98": "\\mathbf{v}^{(t)} = \\mathop{\\textrm{argmax}}_{\\mathbf{v}} \\frac{1}{N} \\sum_{i=1}^N H(\\mathbf{X}_i)\\frac{f(\\mathbf{X}_i;\\mathbf{u})}{f(\\mathbf{X}_i;\\mathbf{v}^{(t-1)})} \\log f(\\mathbf{X}_i;\\mathbf{v})", "c86a7d21beb6c21dae19b481ddd2065b": " V \\to 0 \\,", "c86a814452fc315a25e99e41833e3531": "L(\\Delta f) = \\frac{1}{\\pi}\\frac{f_\\Delta}{f_\\Delta^2 + \\Delta f^2}.", "c86adc068dc84d1215de784910af54df": "J=\\operatorname{diag}(J_{1}, J_{2}, J_{3})", "c86b456fbf225a8587e311e5731e3a89": "\\mathrm{CH_2\\!\\!=\\!\\!CHCOOCH_3+ArCH\\!\\!=\\!\\!N\\!\\!-\\!\\!CH(CH_3)\\!\\!-\\!\\!COOC(CH_3)_3\\ \\xrightarrow[CPME,\\ 0^oC]{TBAB,\\ CsCl,\\ K_2CO_3}\\ ArCH\\!\\!=\\!\\!N\\!\\!-\\!\\!C(C_2H_4COOCH_3)(CH_3)\\!\\!-\\!\\!COOC(CH_3)_3}", "c86b4ce3024c405e8d53ac787fb142f6": "\\max_{i \\in I} \\{i\\}", "c86b5878e56cf202d67ca147e1f365d6": " 5 \\le x_2 \\le 12. \\, ", "c86b90e1fb837b13dc9e69d671999dfd": " \n\\gamma :\\left[ 0,1\\right] \\rightarrow T", "c86b9715abcf1ce7e9f5a48272c0592e": "\\rho = {\\text{muscle mass} \\over \\text{muscle volume}}.", "c86c17a67849e170ab653ac844b4d3d1": "W_\\lambda", "c86c289e263396b9a4392b18f4312e33": "\\sigma(\\bar x)= d \\sqrt {V_2}", "c86c3075bc42128d6149fd966f799114": "\\textstyle\\frac{10}{5}", "c86c4d999c9dc08e8433019f65c7c4b8": "\\text{ES} = \\tan\\left(90-\\arctan\\left(g+a_\\text{f}\\right)\\right)", "c86c7472f45483590425b8a4602c1ec5": " 0.5 < (S_1/S_2) < 2.0 ", "c86c7ede8cb6d44946bbb6e15f8ef3dd": "\\ \\gamma\\,", "c86cd49d246eaaaccc16a317b835583b": "f: M\\rightarrow BG", "c86ced7f67c7b8b94687ec49a332b972": "b,b^{-1}", "c86cfcbd3a42a78604a6ad05c1b12c8e": "\n\\begin{align}\ne_p(\\sigma,Q) & = e_p \\left(\\sum_{1 \\leq i \\leq D} (u_i s_i P_i), Q \\right) \\\\\n& = \\prod_i e_p(u_i s_i P_i,Q) \\\\ \n& = \\prod_i e_p(u_i P_i, s_iQ)\n\\end{align}\n", "c86d078e1c54a8ea06fd98109f739dc9": "(5)\\,", "c86d1a865b1c1a716ed77bb64613c5b3": "\\forall i\\in\\{1,...,m\\}, \\mathbb{P}_S\\{|I[f_S]-\\frac{1}{m}\\sum_{i=1}^m V(f_{S^{|i}},z_i)|\\leq\\beta_{EL}^m\\}\\geq1-\\delta_{EL}^m", "c86d413f201a212047cf46af94b72fa5": "\\mu _{M_{J}} = M_{J}g_{J}\\mu _{B}", "c86d59a8fe6e1bb1f8221d5a575d9b6a": "\\left|\\frac{x}{\\sqrt{1-x^2}\\arccos(x)}\\right|", "c86d5bbca1fdd4e3ddb39fad0a8453c6": "E_\\gamma + E_e = E_{\\gamma'} + E_{e'}.\\!", "c86d7ae1fb9bfb342713f137cd931e3e": " 0 = (\\mathbf{\\bar y}')^{T} \\, ((\\mathbf{T}')^{T})^{-1} \\, \\mathbf{F} \\, \\mathbf{T}^{-1}\\, \\mathbf{\\bar y} = (\\mathbf{\\bar y}')^{T} \\, \\mathbf{\\bar F} \\, \\mathbf{\\bar y} ", "c86e3e079bd96fddb612a3bf0a6f6948": "\\scriptstyle\\sqrt{6}r2/3", "c86e8aeda0dcaed032cca0f86f43388f": "e = 3", "c86ebb11df4e223e03c83ab62ef86618": "\\mathrm{horizon}_\\mathrm{km} \\approx 3.57 \\cdot \\sqrt{\\mathrm{height}_\\mathrm{metres}}", "c86ed1e209f3ab62025d40b5d07f1eff": "(F_n)", "c86f46eb4db8123b1a9dbe4c47b932e6": " m_2= \\frac{\\sum_i Z_i^2 }{N}", "c86f7af1006d704b5ba81a753bb1d007": "u_3(\\mathbf{x},z_1,z_2,z_3)=-\\frac{\\partial V_2}{\\partial \\mathbf{x}_2 } g_2(\\mathbf{x}_2)-k_3(z_3-u_2(\\mathbf{x}_2)) + \\frac{\\partial u_2}{\\partial \\mathbf{x}_2}(f_2(\\mathbf{x}_2)+g_2(\\mathbf{x}_2)z_3)", "c86f86de9a006468283c073910080c43": "\nc(\\zeta, \\tau) = \n\\sum_{k=0}^{\\infty} c_{k} P_{k}(\\zeta) e^{-\\beta_{k}\\tau}\n", "c86fac3a5b329d8d2424833b9e037d43": "f: R/I \\rightarrow R/I_1 \\times \\cdots \\times R/I_k", "c86fbcbb77a052228d711d7bafcb48d9": "g_0 \\ ", "c86fe04e5dd64fc1c5f11f29f2ad2ffa": "e=\\lambda_B \\,", "c86fe1470609d4cd6d411ca209a5b47e": "T^\\mu{}_\\nu = T^{\\mu \\alpha} g_{\\alpha \\nu},", "c8705714cdcd3f09daed5227bf29cc0f": " \\epsilon_{n+1} = \\frac {- f^{\\prime\\prime} (\\xi_n)}{2 f^\\prime(x_n)} \\, {\\epsilon_n}^2 \\,.", "c870b05b93dd13bc1420e8894c7eff1e": "\\exp(-U/k_\\text{B} T)", "c871c03747496327262b062a1c240700": "\\vec E_1", "c87210a395f729d22181548d98a75e09": "X_i = X(i\\Delta t)", "c8728db9c57144761b5a599842ab47ca": "\\begin{pmatrix}b_1 \\\\ b_2 \\end{pmatrix} = \\begin{pmatrix} S_{11} & S_{12} \\\\ S_{21} & S_{22} \\end{pmatrix}\\begin{pmatrix} a_1 \\\\ a_2 \\end{pmatrix}\\,", "c872aed1cd8f31c68a8d09b65a462a85": "f^{ijk}", "c8731feb3fd21a56938917e08ddf4919": "\nL^{2} = \\Delta x^{2} + \\Delta y^{2} + \\Delta z^{2}\n", "c8739a226d0615168ad826ea3f590867": "ab<_yce(ab)", "c8739b63b5f477610bcce09132f6485a": " S'=\\{(s,t_s): s \\in S,t_s \\in \\mathbb{T}^\\infty\\} ", "c873b17cdb7168e749b36d4cac4c3b8f": "\\ f_*:H^{BM}_* (X)\\to H^{BM}_* (Y) ", "c873c15de6b92ef31a3f4e5508f6a71b": " Q_2>\\frac{T_2}{T_1}Q_1", "c873c1f454d2532793970988d66f564a": "b^y = x. \\, ", "c87401640aeb34fdc580abab66250485": "p[r]", "c8741fa4f02acd439a060bc6a6770922": " \\cos(k a) = \\cos(\\beta b) \\cos[\\alpha(a-b)]-{\\alpha^2+\\beta^2 \\over 2\\alpha \\beta} \\sin(\\beta b) \\sin[\\alpha(a-b)]. \\,\\! ", "c8746538cea4e22976d819be2d2e215a": "x_{r}", "c874678aabc99d9a3daece9cc79ea4f3": "\n\\begin{align}\nA\\cdot \\cos(\\omega t + \\theta) &= \\operatorname{Re} \\left\\{ A\\cdot e^{i(\\omega t + \\theta)}\\right\\} \\\\\n&= \\operatorname{Re} \\left\\{ A e^{i\\theta} \\cdot e^{i\\omega t}\\right\\}.\n\\end{align}\n", "c875781e8759a70bd97c223708bf6249": "\\sum q_i = Nq = \\frac{N(a-c)} {b(N+1)},", "c8757a24992d693cf76b9e267cb73cb5": "\n\\psi=\\begin{bmatrix}\na & e & \\mu \\\\\n\\overline{e} & b & \\tau \\\\\n\\overline{\\mu} & \\overline{\\tau} & c\n\\end{bmatrix}\n", "c87593b366498d32656da234fe6da679": "\\ln{}r = \\ln{}k + 2\\ln\\left[A\\right] ", "c87598ef66565ec62a5016b0884093c7": "J_{1z}|j_1m_1\\rangle=m_1\\hbar|j_1m_1\\rangle", "c87605116458d9ebe8b7ad1571da4233": "p(I | \\theta, O_{fg}) ", "c8763a2428bf8eb57ed1fd1106969860": "\\scriptstyle t'=(t-vx/c^{2})/\\sqrt{1-v^{2}/c^{2}}", "c876536336aafdb90d809524d7e4a232": "{{}^\\star C}_{acdb}", "c8765d4936d752300a0e3f1de8afe642": "\\bigvee", "c876657b63873a5b92d8fdff46333999": "\n\\frac{1}{\\rho^2} \\nabla \\rho \\times \\nabla p\n", "c8766b64dc691361595024164211b5fd": " G = A*B", "c8769a45314c54f19f88c46b91f663ee": "\\|\\phi_v\\|\\leq M\\|v\\|", "c8769bdf2bbdb2585714fb17a674400d": "\\ell=1,\\ldots, L", "c87701b8c75f9feed84045849461e8f1": "{\\vec b} = (b_0, \\dots, b_{L-1})", "c8773e8516537be43a497d92d113fcfc": "F_f = \\mu F_n \\,", "c87777bd17f57c0b876c91815bcd2a41": "C_{\\mathrm x} = \\frac{1}{6A}\\sum_{i=0}^{n-1}(x_i+x_{i+1})(x_i\\ y_{i+1} - x_{i+1}\\ y_i)", "c877c144bb5faa4929cec02fb2f43ef7": "(t_1, ..., t_n)", "c877d9f20abdf7757cd65926dea1758e": "i=0,1,2,...,", "c878c1c77f8d641b561020967acbea7d": " StDev_1 = 1 - \\sqrt{ \\frac{ \\sum_{ i = 1 }^K( f_i - \\frac{ N }{ K } )^2 }{ ( N - \\frac{ N }{ K } )^2 + ( K - 1 ) ( \\frac{ N }{ K } )^2 } }", "c878e12841a6ac4b296af92ffe6f34f7": "10^{19}", "c878efcec8faf08835bc3fef23f4e1d6": " \\|b\\|_a := \\sqrt{a^{ij}b_i b_j} < 1,", "c879017f171f2100f7b8344f41b72861": "\nCoDIAK_{t \\to t+1} = e^\\mathcal{B}\n", "c87917c7f9f2c3cc60ff111a8a0c87ab": "(a\\lor\\lnot c)", "c879758500409eac8a3f3017dc344233": "\n\\rho_n R_{n\\rightarrow m} = \\rho_m R_{m\\rightarrow n}\n\\,", "c879efc8df0aa9afbe44dc10960bb4c6": "\\phi\\colon L\\rightarrow L", "c879f503a66bd9c7487ebdca7bdd04db": "[1,0,0]',", "c87a07c50b2c6016f6f57662bd655c5a": " \\dot{x}(t)=F[x(t),u(t)] \\, ", "c87a2c344ed407b0d6ed8bcee6534867": "\\displaystyle f", "c87a4617c10d10730974b8a69a5a7ffb": "\\sideset{_1^2}{_3^4}\\prod_a^b", "c87aae8d9f3c82d3b158dce6c7f38b7f": "F_i=\\frac{H R_i}{1-(R/L_{max})^2}", "c87afdbb41c65f3c585ea81707929b9b": " -9.056 \\times 10^{-4}\\ \\text{eV}", "c87b3ed194dfe5ec3ca4f4103b275361": "cost(splay(v)) \\leq 3 \\left( \\log{s(root(v))} - \\log{s(v)} \\right) + 1", "c87b5241cd7426b760a5067cdafc9985": " \\varepsilon = \\varepsilon_0 (1 + \\chi) ", "c87bfb2fd83a26eca17323f7ac2f1fee": "x^2 + xy + g(y)", "c87c0e6faa5abc852ee7d5ac0e30fd96": "\\left|\n\\begin{array}{ccc}\n 1 & 1 & 1 \\\\\n x_1 & x_2 & x_3 \\\\\n x_1^2 & x_2^2 & x_3^2\n\\end{array}\n\\right|=\\left(x_3-x_2\\right)\\left(x_3-x_1\\right)\\left(x_2-x_1\\right).", "c87c17d4263b432d66edb28f47b92fe2": "\\begin{align}\n\\operatorname{lb}\\,z &= 2^m \\operatorname{lb}\\,y \\\\\n\\operatorname{lb}\\,y &= \\frac{ \\operatorname{lb} z }{ 2^m } \\\\\n&= \\frac{ 1 + \\operatorname{lb}(z/2) }{ 2^m } \\\\\n&= 2^{-m} + 2^{-m}\\operatorname{lb}(z/2)\n\\end{align}", "c87c39fca561306c8d96443a672764a2": "P_4=(2\\sqrt{2}:8:0)", "c87ca9b0ee03f77769b6af99e031a9a0": "H \\propto -d(\\mathbf{\\sigma}\\cdot\\mathbf{E}) -\\mu(\\mathbf{\\sigma}\\cdot\\mathbf{B}) -a(\\mathbf{\\sigma}\\cdot \\nabla\\times\\mathbf{B}),", "c87cd661b26c0d15ef7f018721012686": "P_2=\\frac{8}{8+5}", "c87ce25b7dc209238dba3246155cecb4": "2\\times \\epsilon", "c87cf807d80cb34677b92221366f2271": "\\hat{B} = D_n^{\\frac{1}{2}} Q_n^T E_m", "c87d06e98b88c5a0e8ed4bb43abb53e3": "1 \\leq p \\leq + \\infty", "c87d49768cbcf8b5f32d926925068206": "A = (r^n -1)/B", "c87d51e60571e5a0e95b7abd43880e54": "\\Sigma^\\N", "c87d52c7306d9bc4724c156c60573b84": "\\tanh ^{-1}x=\\frac{1}{2}\\ln \\left( \\frac{1+x}{1-x} \\right)", "c87d65cb4311987b0bb877ca1e19c0a3": "P(\\theta|D,M1)", "c87df03a1a3f991ec80231cf2fd59fe3": "\\mathfrak{a} \\mathfrak{b}:=\\{a_1b_1+ \\dots + a_nb_n \\mid a_i \\in \\mathfrak{a} \\mbox{ and } b_i \\in \\mathfrak{b}, i=1, 2, \\dots, n; \\mbox{ for } n=1, 2, \\dots\\},", "c87e2828a151369d51ddf199ebd5b2e8": "K_{X_i}", "c87e283224bb15003848d359c879556c": "\\langle a, b \\mid a^2 = b^2 = (ab)^3 = 1 \\rangle", "c87ebbff7b4b4e08fc93126ffdf8a65b": "(x_1,x_2,x_3,t)", "c87f3341fcedfde753f354fe652420b0": "y' = p(t)", "c87f45030acbab78ad150b89363d8b46": "\\theta < \\pi", "c87f6b50453ab4cdfd72ddd8577541a1": " a^x\\,", "c87fa8164c01df835e011d1d16f95ca9": "V_{2}-V_{1}=V_{0}\\left( \\frac{R_{2}}{R_{1}}-\\frac{R_{1}}{R_{2}}\\right)", "c87fb9f9609fccf67d570cb1ba396ff2": "\\Delta \\rho=\\frac{\\pi}{\\lambda n}g_{lk}l_{l}l_{k}=\\frac{\\pi}{\\lambda n}\\Delta G=\\frac{\\pi}{\\lambda n}(\\tilde\\gamma _{lkm}P_{m}^{s}+\\tilde\\beta _{lkmn}P_{m}^{s}P_{n}^{s})l_{l}l_{k}", "c880245cc43ede64b71ddd5b30f6238d": "a = ", "c8802cf8f566fb881173a3857623a7c2": "\\text{hom}(X,-):C \\to C", "c8806809f4b3137c75218912be8211cd": " Q(v)=q(v), \\quad v=[v_1,\\ldots,v_n]^\\mathrm{T}\\in K^n. ", "c88106d98fd370f8832a8b3b4538acad": "m=\\cot(\\beta)\\,\\!", "c8811f08b2ffd70c5ffb49094a1c7e25": "\\langle\\tau^n\\rangle \\equiv \\int_0^\\infty dt\\, t^{n-1}\\, e^{ - \\left( {t /\\tau_K } \\right)^\\beta } = {{\\tau_K }^n \\over \\beta }\\Gamma ({n \\over \\beta })", "c8817384b9423f95cf0f73479f0d595f": " \\mathbb{S} ", "c8823234bacd201d6c5674d1f79faf63": "\\mathbf{r} = \\rho \\mathbf{u}_{\\rho} \\ , ", "c8823c271a4cc82cd402f7d21eee0867": "\\theta_{i=1 \\dots K, j=1 \\dots V}:", "c882410bda64cd6808f1abd2d0409c6f": "V=Y\\,", "c88274aa9b2785c630ea719590e3ee87": "(\\sigma, \\tau)", "c882a7c4da63fbb539e56940f6bb1458": " p(x)=z(x)^T Q z(x) ,", "c882f8e7af9fb02290eda849205a04a3": " k_{xy}\\ ", "c88303f8699a0bcd7f765b904e261316": "O_2^{(\\alpha)}(t)=\\frac {2+\\alpha}{t}+ 4\\frac {(2+\\alpha)(1+\\alpha)}{t^3},", "c8839e6083502790df7029759f4e0ba8": "\n \\begin{align}\n _{(x)}\\Gamma^m_{ij} & = G^{mk}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~\\frac{\\partial X^\\gamma}{\\partial x^k} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} + G^{mk}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k}~g_{\\alpha\\beta} \\\\\n & = \\frac{\\partial x^m}{\\partial X^\\nu}~\\frac{\\partial x^k}{\\partial X^\\rho}~g^{\\nu\\rho}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~\\frac{\\partial X^\\gamma}{\\partial x^k} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} + \n \\frac{\\partial x^m}{\\partial X^\\nu}~\\frac{\\partial x^k}{\\partial X^\\rho}~g^{\\nu\\rho}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k}~g_{\\alpha\\beta} \\\\\n & = \\frac{\\partial x^m}{\\partial X^\\nu}~\\delta^\\gamma_\\rho~g^{\\nu\\rho}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} + \n \\frac{\\partial x^m}{\\partial X^\\nu}~\\delta^\\beta_\\rho~g^{\\nu\\rho}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~g_{\\alpha\\beta} \\\\\n & = \\frac{\\partial x^m}{\\partial X^\\nu}~g^{\\nu\\gamma}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} + \n \\frac{\\partial x^m}{\\partial X^\\nu}~g^{\\nu\\beta}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~g_{\\alpha\\beta} \\\\\n & = \\frac{\\partial x^m}{\\partial X^\\nu}~\\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j} \\,_{(X)}\\Gamma^\\nu_{\\alpha\\beta} + \n \\frac{\\partial x^m}{\\partial X^\\nu}~\\delta^{\\nu}_{\\alpha}~\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j} \n \\end{align}\n", "c883f691f334465bda47f11b2b6ae2bd": "T=\\frac{|A_3|^2}{|A_1|^2}=\\frac{4E(E-V_0)}{4E(E-V_0)+V_0^2 sin^2 [\\sqrt{2m(E-V_0)}\\frac{L}{\\hbar}]} ", "c883fb68a3a1f0754f59998d33409b3d": "e^{\\pi}", "c884675bff1da1c224acea73ed94e223": "Y=0", "c884777193325630ff8cae8466392430": "m=y_1-(\\frac{y_2-y_1}{x_2-x_1})x_1", "c884a9b72ec3ce847d48ebbf5c53bea1": "2x^2-2(b_{2}-2(b_{2}-c_{5}))b_{2}=0", "c884c984f4fd59f93be8c9c0aeb33655": "b_{14}+b_{15}", "c884d3af88a652d8d9803e107287f945": "S^{T}AS", "c884f0b11c9fc16736e963997bbbc9e5": "\\dim(\\operatorname{char}(M))\\geq n", "c8856a4e00837bdd91512decdd8bae15": " \\mathrm{Input} = \\mathrm{Output} ", "c8858ff915f73b8be1e847566f6715a3": "\n= 1000 \\times (1+.069/4)^{({5\\ yrs}\\ \\times\\ 4\\ {qtrs\\ in\\ a\\ year} )}\n= 1000 \\times (1+0.069/4)^{20}\n\\approx 1407.84\n", "c885fa024b6dc37bf35fd1140fa48305": " S_y", "c8860fe1656769d2c05fd542cce34cfe": "x,y\\in A_1", "c88665199ef7aba0e2eaf145bbe1d951": "\\Sigma(k,i\\omega_n)", "c8868fa274408559849c2b12ec252dd7": "- \\omega \\delta", "c88697c3025d3f6f4b25b5b515304638": "\\vec{\\beta}=\\vec{\\theta}-\\vec{\\alpha}(\\vec{\\theta}) = \\vec{\\theta} - \\frac{D_{ds}}{D_s} \\vec{\\hat{\\alpha}}(\\vec{D_d\\theta})", "c886a7af98e2614c1ca535c7df458c18": "e(\\mathbf{s}_i)", "c886ba325e376203f1b1915243888683": "i \\approx r + \\pi", "c887a0509f69fab9b842d0900e5ddc96": "D_{AB} = 1", "c887f29f810c55854bdfee309a1bc3eb": "A = -1.88 \\times 10^4", "c88853a53d4aff4840a82f6fe7ebcdc2": "\\lambda, \\mu, A, B, C", "c8886e09ccb494a41d83dd3e689a35c1": "\\theta_A = \\frac{40.219}{EI} ", "c888a77d0c9777617d7b4f86fcc0f8f4": "R(\\hat{n},\\phi) = \\exp(-i \\phi J_{\\hat{n}}/\\hbar)", "c888a8c5aa3b5c9aed83db141618d300": "a e = c. \\, ", "c88905b2ea75a6dae368aa5cabf8eb42": "{l_D \\over l_B}", "c8893c222d5d4d9c53dd57fea698bfab": "\\textstyle {4!\\over 3!\\times 1!\\times 0!} \\ {4!\\over 2!\\times 1!\\times 1!} \\ {4!\\over 1!\\times 1!\\times 2!} \\ {4!\\over 0!\\times 1!\\times 3!}", "c889dc7180b39a8a036332a779e4c2da": "\\int (af(x)+bg(x))\\, dx=a\\int f(x)\\, dx+b\\int g(x)\\, dx.", "c889fb06f98d9f57203a80d613fd8e8e": "\\nabla(\\vec u \\cdot \\vec v) = \\vec u \\times (\\nabla \\times \\vec v) + \\vec v \\times (\\nabla \\times \\vec u) + ( \\vec u \\cdot \\nabla) \\vec v + (\\vec v \\cdot \\nabla )\\vec u ", "c88a9be89a5dc68c94ee2ffb04799ff2": "j\\in \\{0,1\\}", "c88bad01ef4b9e5b4affcd909b461d43": "tangent = \\frac{c}{L}.\\,", "c88bb91988cd700f08863c6c0f06ef8d": "\n\\mathbf{D} \\ = \\ \\varepsilon_0\\mathbf{E} + \\mathbf{P} \\ = \\ \\varepsilon_0 (1+\\chi_e) \\mathbf{E} \\ = \\ \\varepsilon_r \\varepsilon_0 \\mathbf{E}.\n", "c88be807aac756448f1d316a72d02405": "E \\ := \\ \\frac{p_1(x_{n+1}) - f(x_{n+1})}{p_2(x_{n+1}) + (-1)^n}.", "c88c2619572dd462a9f881ecf92417ef": "\\operatorname{Log}(z) = \\ln(|z|) + i \\operatorname{Arg}(z)", "c88c28719f32608c8b87508ca863a549": "p(k) = {n\\choose k}p^k(1-p)^{n-k},", "c88c939f08e5f45acbbd5fae4e9319bb": "S_{\\rm C} = \\frac{k}{-e} \\Big[ \\frac{E_{\\rm C} - \\mu}{kT} + a_{\\rm C} + 1\\Big], \\quad \\sigma_{\\rm C} = A_{\\rm C} (kT)^{a_{\\rm C}} e^{-\\frac{E_{\\rm C} - \\mu}{kT}} \\Gamma(a_{\\rm C}+1).", "c88cd80f5ab7b7eeaf8a58f6632b80cf": "\\mathbf x = \\sum_i x^i \\mathbf e_i = \\sum_i x_i \\mathbf e^i", "c88d5a8d70302902c76a8e90b1ffb47e": "(In)\\quad n[\\;in \\ m.A\\mid A'\\;]\\mid m[\\;\\overline{in} \\ m.B\\mid B'\\;] \\Rightarrow_{amb} m[\\;n[\\;A\\mid A'\\;] \\mid B\\mid B'\\;]", "c88d8b68b88f7655d857b4cb4e6685c0": " \\frac{\\partial u}{\\partial P} = \\frac{L}{EA} > 0 ", "c88d979dac2bf77ef0838c1b778446e9": "\\sum_\\nu \\frac{\\partial j^\\nu}{\\partial x^\\nu} = 0", "c88e117fac050ac2570538a2877e4d09": "\\{\\ldots, -2,-1,0,1,2,\\ldots\\}", "c88e81a79ef2cfc1d261f8732b21a263": " y \\over {y-1} ", "c88eb0edbd9684b80d1769abca23d488": "\\Beta\\,", "c88f2c405d1dc597ca608e7dfcf7d806": "t = {\\overline{X}_1 - \\overline{X}_2 \\over s_{\\overline{X}_1 - \\overline{X}_2}}", "c88f6e6f21def81817988c2455623741": "\\frac {n(7n - 5)}{2}.", "c88fae4418b8eb8e45dc7da67593c78f": "\n R_p = \\cfrac{1}{2}\\left(R_0 - 2~R_{45} + R_{90}\\right) ~.\n ", "c88fbadd9e2a81d61ccd2bfd285cf69a": "\\text{for }k > 0,\\ a_k = a_i \\pm a_j\\text{ for some }0 \\leq i,j < k.", "c88fc94ef266cf017910833e466ceafd": "\\operatorname{Cov}(X, Y)_n = n ( p_B - p_X p_Y ).", "c8901bf5794b51d0479dc37831973379": "x_1,x_2,x_3\\,", "c89026611e695b47a96e1d115820b691": "_{0}", "c8907e5ec385212c2478c11b49f1d532": "\\frac{1}{2}\\int_{\\mathbb{R}^4}\\operatorname{Tr}[*\\bold{F}\\wedge\\bold{F}]\\geq\\frac{1}{2}\\left|\\int_{\\mathbb{R}^4}\\operatorname{Tr}[\\bold{F}\\wedge\\bold{F}]\\right|.", "c8909538c26175b053606a05d7f9d2ea": "b + b = b \\,", "c890b07594a7bd13de2d0e410a092189": "w(\\mathbf{e}) \\leq t", "c890b0ce849372ee6b6312ec0f6e8c5a": "\\frac{g^2}{4\\pi}", "c890d4fc0048e294e7b47d64f4885447": "R_{k,i}", "c8911463c55507389334a20fb72c756f": "\\langle \\mathbf{S}^2 \\rangle ", "c891ed7ec80459b8407b757ce5598e03": "q = m+n-2", "c892311d45206c7b60e96f998591cea7": "x^2 - y^2 + x - y", "c8927141280b8d2c1d53492563808882": "\n\\sigma _z^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)^2 \\sigma _1^2 \\,\\,\\, + \\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)^2 \\sigma _2^2 \\,\\,\\, + \\,\\,\\,2\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)\\,\\,\\sigma _{12}", "c892f940584fc4ab1e6ef794cc6b984b": " cm^2", "c8931522c10949a6616a6c844a902fa5": "\\scriptstyle 0.5/M", "c893802262244d89babffd76f6bc5a44": "\\{k_l, pk_l, p^2k_l, \\ldots, p^{m_l-1}k_l\\},", "c893808171f028d1c95be3623f77aae6": "\\bar{\\omega}_0", "c893bfe5b1bb8a92c8773ef4e94a642c": "\\mathbf{e}_1,\\mathbf{e}_2", "c893f4a3e29534d485b287d0eb0303a1": "x_a=(x^1_a,x^2_a,x^3_a) ", "c894c9cd813c150d9f622bb743e4c99e": "H_\\bullet(C)", "c894cbd769232cd67ec4c7837c40d0fc": "\\Delta(a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}", "c894edf081f0f8591451000ece754d2a": " a^2 = b^2 + c^2 - 2 b c \\cos \\alpha ", "c8953398f5332a5b7e2cd122a3656a3c": "\n\\begin{bmatrix} \\sigma_1 \\\\ \\sigma_2 \\\\ \\sigma_3 \\\\ \\sigma_4 \\\\ \\sigma_5 \\\\ \\sigma_6 \\end{bmatrix} = \n\\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\\\\nC_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\\\\nC_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & C_{44} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & C_{44} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\tfrac{1}{2}(C_{11}-C_{12}) \\end{bmatrix}\n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\end{bmatrix}\n", "c89571e17c45524bec43850e95bf9131": " V( \\mathbf 0) = \\dot{V} (\\mathbf 0) = 0 ", "c895af0be1903cd01f6c9c5acddfb891": "y\\to t+A", "c895c2e9bd802d08562f8944dbea22ad": "\\frac{g\\left[ u_1(x_1, \\dots, x_n), \\dots, u_n(x_1, \\dots, x_n); \\theta \\right]}{|J*|}=g_1\\left[u_1(x_1,\\dots,x_n); \\theta\\right] \\frac{h(u_2, \\dots, u_n | u_1)}{|J*|}", "c895ffedd1657f398dd6d1c894a330ad": "\\textbf{R}_{k}^a", "c89609f858f235909a47888802d9a8cd": "\\ \\Delta ASA", "c89693438ae53cd6cec10621789e6cc2": "D \\cdot F = \\langle DF \\rangle_{r-1}", "c896b0e075953d50a8c27b59d12573a1": "\\left\\{ P_{(i_{0})}>\\frac{\\alpha}{m_{o}}=\\frac{\\alpha}{m-i_{0}+1}\\right\\} \\subseteq A", "c896cc822ba9a55d5a5abe7c3b53a5e3": " PL_0 = 40 log(d_0)-10 log(G h_t ^2 h_r ^2) ", "c8973a126018f0eb7578f2efdb029a5b": "\\vec{J}_p = \\frac {\\Psi_0^2}{m}(\\frac{h}{2 \\pi} \\vec{\\nabla} \\varphi - q \\vec{A}).", "c89779cc6ee36ee90146becfb72398ca": " S= \\int_k \\bar\\psi( i\\gamma^\\mu k_\\mu - m ) \\psi. ", "c89796cda139cabaf0c18e12a8baa2b0": "r=-\\frac{r_1-e^{-i\\delta}}{1-r_1 e^{-i\\delta}} ", "c897e69c25af239399e0c37fab803cc1": "{\\color{Blue}~6.4}", "c897f3c752c2604bf946cdeaaae70553": "\\mathrm{\\tfrac{d\\bar{s} + s\\bar{d}}{\\sqrt{2}}}\\,", "c898146e69a5f9501566cb0f6915ed97": "\\bar P_e = \\sum_c \\frac{n_c^2}{(mN)^2}", "c89843a8ea26fde55d357b9566ad0b40": " | 1 0 \\rangle \\mapsto | 1 \\rangle U |0 \\rangle = | 1 \\rangle \\left(x_{00} |0 \\rangle + x_{10} |1 \\rangle\\right) ", "c8984ab8483b74a957e0e0477f61bb43": "\n \\frac{\\partial^4 w}{\\partial r^4} + \\frac{2}{r} \\frac{\\partial^3 w}{\\partial r^3} - \\frac{1}{r^2} \\frac{\\partial^2 w}{\\partial r^2} + \\frac{1}{r^3} \\frac{\\partial w}{\\partial r} = -\\frac{2\\rho h}{D}\\frac{\\partial^2 w}{\\partial t^2}\\,.\n", "c8984cef342d8cca14d340097bcd56a0": " f(r) = 1 - {2a\\over r} - b r^2 \\,", "c898785d8eefd2ee616f2155d09baef9": "\\textstyle(x\\mp1, y\\pm1, z\\mp1)", "c898a554d49991e0be4a4a3f30b8d503": "\\hat{\\bold{\\Psi}}_4 (\\bold{G}) = n^{-2} \\sum_{i=1}^n \n\\sum_{j=1}^n [(\\operatorname{vec} \\, \\operatorname{D}^2) (\\operatorname{vec}^T \\operatorname{D}^2)] K_\\bold{G} (\\bold{X}_i - \\bold{X}_j)", "c8991500725a1f1e94624f8eb0a388f0": "F_{\\chi} = \\frac{F}{l} \\delta x ", "c899e63aa044f20ad3c29a010dae592f": "P_\\ell^m(\\cos\\theta)", "c899feb8b06fd0e0ddf13cee726afa2f": "\n\\begin{matrix}\n &\\mathbf{1_0}&\\mathbf{1_x}&\\mathbf{1_y}&\\mathbf{1_z}\\\\\n\\mathbf{1_0}&1_0&1_x&1_y&1_z\\\\\n\\mathbf{1_x}&1_x&1_0&1_z&1_y\\\\\n\\mathbf{1_y}&1_y&1_z&1_0&1_x\\\\\n\\mathbf{1_z}&1_z&1_y&1_x&1_0\n\\end{matrix}\n", "c89a2fa86ae752ec145be4863c2afb1e": "2 \\Phi_{,i} \\approx g^{i j} (- g_{0 0 , j}) \\approx - g_{0 0 , i} \\,", "c89a484074318d390f8f3ad575a59482": "x_2\\left(\\sigma\\right)", "c89a5365e3bdaac10d01b162673e6c8b": " e_x\\,", "c89a9979ca45b526149fe3d011357ef6": "\\top_{\\mathrm{nM}}(a, b) = \\begin{cases}\n \\min(a,b) & \\mbox{if }a+b > 1 \\\\\n 0 & \\mbox{otherwise}\n\\end{cases}", "c89af3a1db6d3c40a71c567f7fd90704": "M \\leftarrow q + 1 + 2mk \\mp j", "c89b0c0902bc4882cecb3e1456d77116": "\\boldsymbol{p}_0, \\boldsymbol{p}_0 + \\frac{\\boldsymbol{m}_0}{3}, \\boldsymbol{p}_1 - \\frac{\\boldsymbol{m}_1}{3}, \\boldsymbol{p}_1", "c89b26273183028e8c027d71fdf795e2": "1 = \\sqrt{(N_x^2 + N_z^2)}", "c89b4723e4bbc55560399a35df9f1223": "\\epsilon^{64/N}", "c89b4a847b3884382b35e879271c2f8f": "[-\\alpha, \\alpha]", "c89b56c28e0fafddd88c42f1488fd42d": "v_{i+1} = \\pi(E_{i+1}, F_{i+1}, v_i)", "c89b68a69e67776e133361e5509ee7a0": "\n\\operatorname{coker}(\\pi_3(\\operatorname{BGM}(K)^+) \\rightarrow \\operatorname{K}_3(K)) = \\operatorname{B}_2(K)/2c \n", "c89b79cb9b63b6c03c3db6e995f464e1": "n (n-1) \\over 2", "c89b8edbcc78490b4184ad232d2d8fa1": "\n\\begin{align}\n p_g &= \\frac{\\partial w_g}{\\partial y} \\\\\n q_g &= \\frac{\\partial w_g}{\\partial x} \\\\\n r_g &= -\\frac{\\partial v_g}{\\partial x}\n\\end{align}\n", "c89c1e9ee143fc666edc773e846ff938": "\\sum_{i = 1}^N f_j(X_{ij}) = 0", "c89cd4771a9167e855e8da67c1d3e8ac": "\\zeta(s) = \\mathfrak{D}^{\\mathbb{N}}_{\\mathrm{id}}(s)\n = \\prod_{p\\,\\mathrm{prime}} \\mathfrak{D}^{\\{p^n : n \\in \\mathbb{N}\\}}_{\\mathrm{id}}(s)\n = \\prod_{p\\,\\mathrm{prime}} \\sum_{n \\in \\mathbb{N}} \\mathfrak{D}^{\\{p^n\\}}_{\\mathrm{id}}(s)\n = \\prod_{p\\,\\mathrm{prime}} \\sum_{n \\in \\mathbb{N}} \\frac{1}{(p^n)^s}\n = \\prod_{p\\,\\mathrm{prime}} \\sum_{n \\in \\mathbb{N}} \\left(\\frac{1}{p^s}\\right)^n\n = \\prod_{p\\,\\mathrm{prime}} \\frac{1}{1-p^{-s}},", "c89cd96786da9101d588048e569fa71d": "\\mathbb{S}^3_{\\varepsilon} \\subset \\R^4 \\hookrightarrow \\C^2", "c89cf7645674ae1b26d4bd41bb1802ae": "\\mathbf{select}_q(x)", "c89d10748f8e6026bf1d773b76430908": " |f_n(x)| \\le g(x)", "c89d46a36b15676bd6ce25fa8bd79a94": " \\frac{\\partial u}{\\partial t}=\\Delta u+F\\left( u\\right) , ", "c89de930e5227b4d239c25e53172ac36": " (x-r) ", "c89dfc94559c491c5abc41471331543b": "\\sin \\theta \\sin \\varphi = {{\\cos(\\theta - \\varphi) - \\cos(\\theta + \\varphi)} \\over 2}", "c89e1d4c058570c3c51344aab85543cd": "N_f + N_V", "c89e31b13b823520f4e97669d509e35a": "\\tan ( \\alpha + \\beta ) = \\frac{\\tan \\alpha + \\tan \\beta} {1 - \\tan \\alpha \\tan \\beta} \\,,", "c89e3b14d90a578d9a322401fda5ca85": "dE=T\\,dS+\\mu\\,dN", "c89f25ad5eeab9fe8bb0fb14e62a36d6": "{\\pi\\over 2}", "c89fa74c403339e5cb21170f12cfdb6a": "a^4 \\neq 1", "c8a0b90d8c567560730d83db618efa7e": "B_{xx}", "c8a0c703642820f2ea67b723c3832cde": "\\lambda x_1.\\ ...\\ \\lambda x_n.X ", "c8a0ea991a5ae08af13d50a6bdfb24c1": "(x - a)^n - (x^n - a) = nf + (x^r - 1)g \\qquad (3)", "c8a0ec17eca9e00b377ea93bcea9b115": "(a^{1/n})^{n}=a", "c8a1401d53d67da349a339a70e67d7c9": " \\csc \\theta = \\frac {\\mathrm{hypotenuse}}{\\mathrm{opposite}} = \\frac {h}{a}", "c8a14417becdc846e86355cb005631e9": "\\Delta v V_\\text{esc} + \\frac{1}{2} \\Delta v^2", "c8a1af2e25b3b3321f5337dc78b584e3": "x+\\,\\!", "c8a20444ddc2bb59ba0da65815559d50": "E_{tgu} = 0.5 \\cdot [\\tfrac {11848.05} { 1000 } ]^2 / 4.54 =", "c8a209e9d11a253bc4446cbbd9e1adbf": "u= u^A\\partial_A + u^a\\frac{\\partial}{\\partial c^a}", "c8a23426424cecf8df3da3bfa371077e": "T^{-1} = 4 \\sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.", "c8a27fa0a4ddcae621dfa57051aeedcc": "{1\\over p} + {1\\over q} =1", "c8a307feea48864d759f4fee83578d5e": "\\scriptstyle R \\gg \\omega L", "c8a347d62455980503ab811d418112be": "\\!\\mu", "c8a367dcc505c1a5a3609b9b95b3848b": "D_1 (D_1 \\neq \\mathbb{C})", "c8a3e1d0132f10028e1e64328a089e52": "a=-1,\\;\\;b=0,\\;\\;c=x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2\\,.", "c8a3e8efecf8425ea8ab56e5f3856d2b": "\\tau_{+} ", "c8a44d99087e9610963de6f98dfc0ec9": "y z = 0 = V\\text{ and }z x = 0 = W,\\,", "c8a451baeaf69996ec957e230ce196ce": "f_0(x)=x^3\\,", "c8a45cecb712987a85c9d6bcae917ad2": " =\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\right\\} +\\text{Tr}\\left\\{\n\\hat{\\Pi}_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\right\\} \\right\\}\n", "c8a50216a502d174cab8123a81a48281": "\\|v\\|_2=\\left(\\sum_{i=1}^n v_i^2\\right)^{1/2}", "c8a526790d1702864b1090e5e35a94d0": "\\,p(x)>0", "c8a5aeedbabadf4292d2f392c4faed3b": " R_H = a = \\frac{k_b T}{6 \\pi \\eta D} ", "c8a5be85abc08f973eeee13dd69e1122": "\\begin{array}{ccccccccc} 0&\\xrightarrow{}&A_1&\\xrightarrow{f_1}&B_1&\\xrightarrow{g_1}&C_1&\\xrightarrow{}&0\\\\ &&\\alpha\\downarrow\\quad&&\\beta\\downarrow\\quad&&\\gamma\\downarrow\\quad&&\\\\ 0&\\xrightarrow{}&A_2&\\xrightarrow{f_2}&B_2&\\xrightarrow{g_2}&C_2&\\xrightarrow{}&0 \\end{array}", "c8a5c475ac3118e6f6f9239d51c55eec": "SL_2(\\mathbb{Z})", "c8a61c7badee37e65d00c3524014aeeb": "\\textstyle E_{1}=E_{2} ", "c8a6b0ec08ff82e71b01c9781f8eaa3c": "[x_{1}:T_{1} \\dots x_{n}:T_{n}]", "c8a7098cf11c5e713d3f6fe7ecb009de": "{R}_{MR}\\,\\!", "c8a79795086c5881613bbda66a624ed8": "F(x) = \\begin{pmatrix} \\ & \\ & \\cdots \\\\ \\ & \\ & \\alpha_{22} \\cdots \\alpha_{2k} \\cdots \\\\ \\ & \\alpha_{11} & \\alpha_{12} \\cdots \\alpha_{1k} \\cdots \\\\ \\alpha_{00} & \\alpha_{01} & \\alpha_{02} \\cdots \\alpha_{0k} \\cdots \\end{pmatrix}", "c8a7c538cc3de5c1e91ea4bb1e1f574b": "y_i = y^{(i-1)}.\\!", "c8a80646c2a3dc3c3c7cdbee3952e656": "[[M]]", "c8a80c37af12c849a3ca4babd3e34b0a": " Z[h] = \\int e^{iS} e^{i\\int_x h(x)\\phi(x)} D\\phi = \\langle e^{i h \\phi }\\rangle", "c8a865d6e908a17db1d97c17123016ed": "n_i=1 ", "c8a8923ee732a91a188a24414d777a9a": "\\frac{D}{Dt} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\partial}{\\partial t} + \\mathbf{v}\\cdot\\nabla ", "c8a89ed79d8c6471308439624714fa3a": "\\varphi = \\frac{q}{\\hbar} \\int_P \\mathbf{A} \\cdot d\\mathbf{x},", "c8a924c2766be711f86c4043f7471c58": " \\{ f,g \\} = {\\mathrm{d} f}(X_{g}) ", "c8a94b0c0feceb5efab48d74d67fe40a": "(x\\alpha^{i_j})^{d-1}", "c8a9a366cf45a78d8721e0849948e208": "T_{00}", "c8a9c8a13af23865a151904c018e032e": "\\sigma _{d}=e\\left ( N_{d}-N_{d}^{+} \\right )\\beta \\mu \\tau ", "c8a9dd1948cd910250725a91928a794d": "a(x-y)^2 + b(x-z)^2 = (a + b)\\left(x - \\frac{ay+bz}{a+b}\\right)^2 + \\frac{ab}{a+b}(y-z)^2", "c8a9f7012ce4be13299cdaf4a32fa0d5": "X = \\begin{matrix}\\bigcup_{n\\in\\mathbb{N}} (n-1,n) \\subset \\mathbb{R} \\end{matrix}", "c8aa0f179ed09a68fa3845d9da321d3e": "\\Pi = \\left(\\pi^{ij}\\right)_{1 \\leq i,j \\leq d}", "c8aa60f479420e7955a765b17a3a4ef7": " x\\,\\triangle\\,y = (x \\lor y) \\land \\lnot(x \\land y) = (x \\land \\lnot y) \\lor (y \\land \\lnot x) = x \\oplus y.", "c8aa664f98f5f67083f45c59cb2ce0cc": "\n S(z) = \\sum_{i=1}^n z_i g_i (z).\n", "c8aa7f4a9c83dd7e2da4d99599fd0da1": "\\partial f\\over \\partial x", "c8aab0e284b00a0ab60a1b921e33d9cc": " \\|f\\|_1 := \\int_G |f(s)| d\\mu(s), ", "c8aae006185aec6bb9aebb46b109f597": "\\alpha_1 = \\alpha,\\ \\alpha_2 = -\\alpha", "c8ab069c7f01c73d30dc27870a8d5697": "y' = y / z", "c8ab1069842750c765638b00d56ee91d": "R - r", "c8ab1456089dfd8574a2fe2e4593a6e7": "(q,\\epsilon,Z,r,Z)", "c8ab3f956263dd914617868363550fb1": "\\Delta H ^{\\circ}_{\\mathrm{c}}[\\mathrm{CH_3OH_{(l)}}]", "c8ab58087aae50563dbd29768497a837": "\\frac{f_2}{f_1}", "c8ab60c864754e08fd277510ded27d57": "\\hat{\\boldsymbol{y}}", "c8abc58bafcd686f02e746379c5c5e16": "(g,L)", "c8ac445ee6a9a4cdbb8810d1a4f4750c": " v_{\\pi} = -i_f \\ ( r_{\\pi}// R_S ) \\ . ", "c8ac64ea5fbc35cf0e3573386be38b75": "D_{\\mathrm{I}^{+2}}", "c8ac93fbafd76bad5250c55e6f2e6cac": "\\mathfrak{g}\\,.", "c8accbca3c38b2a56f4d13019995ca6a": "\\chi:E^r\\rightarrow\\{-1,0,1\\}", "c8aceda21ea7576b556e755c24ec2e98": " F_q(T)\\{\\tau\\} ", "c8ad1e39388fa6da8fde27800aa7ed01": "\nQ = \\frac{X_L}{R} = \\frac{\\omega L}{R}\n", "c8ad77fae39cb3d6de4919db19d8d0f5": "\\frac{\\partial}{\\partial{c}} P_c^n(c)", "c8addccf19af6d62fdc9c0a73e559f2d": "W=C_W \\,\\ ", "c8ae0cc198afb868da29e4b99e6f0c96": " \\lambda^2-2u\\lambda+u^2-c^2=0 ", "c8ae250192aa21d8c0c69baf018a5757": "|\\beta_n|^2 \\le \n\\exp\\left(\\sum_{k=1}^n(k|\\alpha_k|^2 -1/k)\\right).", "c8ae364064cf637a2aae3fbcd82f3582": "|s_1| = 6", "c8aea8d499854562993190877584ceb1": "\n\\hat{\\mathbf{x}}_1=[\\cos\\gamma_1\\,,\\,\\sin\\gamma_1\\,,\\,0]\n", "c8aeed1d1e1cd09628e36b1df18efb73": "0\\in S", "c8aef3e07b97bd2df7c24100f273116f": "e = (x, y)", "c8af172f890b6d00426ce36569350181": "w(n)=e^{-\\frac{1}{2} \\left ( \\frac{n-(N-1)/2}{\\sigma (N-1)/2} \\right)^{2}}", "c8af3bad83bb0441f71839e9cc052cd1": "w_{k,i} = \\log \\phi(c_{k,i})", "c8af42ccfc0b04b9fd9ed6bc65af7d19": " S(\\rho^{12} | \\rho^1)= S(\\rho^{12} )-S(\\rho^1)", "c8af9a5973935e5300633625eb2b0fce": "\\rho=\\sum_i \\rho_i", "c8af9f5e073866459137e63aa7a11452": "M_{\\rm u} = {{M({}^{12}{\\rm C})}\\over{A_{\\rm r}({}^{12}{\\rm C})}} = {1\\ {\\rm g/mol}}", "c8afbffe41b31cd8f51999f295437c72": "M_\\text{b}", "c8afef8bacfbcba4bb893d75cfced93a": "\\theta(\\Omega^\\omega)", "c8afef9b0e3ba4aea2fdb78e10350637": "\\frac {d \\vec r(t)} {dt} = W \\cdot \\vec{r}", "c8b006bab5c8cf6acc2d533154c67500": " \\text{mag}_{AB} = -2.5 \\log_{10} {S_{\\nu}[Jy] \\times 10^6 \\over [\\mu Jy]} + 23.9 ", "c8b02523ba7ff0a461cff99242408e41": "T_{raise} = \\frac{F d_m}{2} \\left( \\frac{l + \\pi \\mu d_m \\sec{\\alpha}}{\\pi d_m - \\mu l \\sec{\\alpha}} \\right) = \\frac{F d_m}{2} \\left( \\frac{\\mu \\sec{\\alpha} + \\tan{\\lambda}}{1 - \\mu \\sec{\\alpha} \\tan{\\lambda}} \\right)", "c8b05ee208a0973cdd35aeb8d9c2f971": "|z| > 1\\,", "c8b06a3e4f07df57608006756d2bcc2b": "\n\\begin{align}\n & 0010 & 0101 &\\quad\\text{25} \\\\\n+\\; & \\underline{0100} & \\underline{1000} &\\quad\\text{48} \\\\\n & 0110 & 1101 &\\quad\\text{6D, intermediate result} \\\\\n+\\; & & \\underline{0110} \\\\\n & 0111 & 0011 &\\quad\\text{73, adjusted result}\n\\end{align}\n", "c8b0b92cf353e170a89b84c4bbb6cbd0": "k = 0,\\ldots,N/2-1", "c8b0def4dc2d2527d48c233992c2bef4": " -4y^3 + 3y - \\sin (\\theta) = 0.", "c8b12108dcee5b7b1a4832334665e4b9": "\\tilde{P} \\equiv \\det(P) P^{-1}", "c8b13e81fb06a127ea84d07b05c9a74f": "\\mathrm{E}(X) = \\frac{1}{p},\n \\qquad\\mathrm{var}(X) = \\frac{1-p}{p^2}.", "c8b14369ef4b6034108c829994514b7b": "\\tfrac{10}{3 \\cdot 5} = \\tfrac{10}{15}", "c8b14f2d147b9f3f85ec11b91766ba10": " \\tau_c \\propto \\xi^{z} \\propto |\\epsilon |^{-\\nu z}, ", "c8b1872c85b16fe8f149e2147469b099": "f\\colon M\\to X", "c8b1a2af66182d32ef143ac4a431c97f": "\\{\\pm 1\\}", "c8b1c60ceb4a653a02196077044aa7e3": "I_\\mathrm{total} = V\\left(\\frac{1}{R_1} + \\frac{1}{R_2} + \\cdots + \\frac{1}{R_n}\\right)", "c8b1e782b0184a2d5d1485079fd527c8": "\\pi_i^{-1} Y_i", "c8b253c71a2cf98274bb25ae651f27ca": "-K\\sum_{i=1}^np_i\\log (p_i)", "c8b3016686afb4a2160601cc207cd074": "|x\\rang \\overset{U_{\\omega}} \\longrightarrow (-1)^{f(x)}|x\\rang", "c8b3029353b00d339c31fa1ca2e8a7f0": "\\ddot{x}^a + {\\Gamma^a}_{bc} \\, \\dot{x}^b \\, \\dot{x}^c = 0", "c8b328b3977175dc60a65bef805a465b": "l \\circ f = \\operatorname{id}_{X}", "c8b36d77ce80c3883b65d6393bee8432": "r={{h^2}\\over{\\mu}}{{1}\\over{1+e\\cos\\theta}}", "c8b384f9b2c442cc3e4a7028e4a293eb": "y = \\sqrt{x}", "c8b3c7427b35c675ae360d9da42397bb": "r_f", "c8b3d8a0303c88ff5f233dacf4c70d9b": "s_0, s_1, ...", "c8b3f1e16475ea2653380770b01cad03": "\nL = {MR^2 \\over 2} \\dot\\theta^2 \n", "c8b432fdb58e19cb540ccfccdd2c59f7": " R_{S}(t) = \\frac{ 465 \\ {_2^1}S }{ 14 [63 \\ {_2^0}S + 31 \\ {_2^1}S] }", "c8b43cbe4840629ad1cd893864b74c96": "\\text{fmap}: (A \\rarr B) \\rarr (C \\times A) \\rarr (C \\times B) = (c, a) \\mapsto (c, f \\, a)", "c8b47f1ef6dc912ac290b196156ed352": "\\nu, \\sigma_0^2", "c8b4a621f6720becf12ae65f3fe03f2b": "T_M = T_{Ref} \\left( \\frac{ P_M - P_{Ref} }{a} + 1 \\right)^\\frac{1}{c}", "c8b4cc63d03e7e52ebf71f16483c149e": "\\mathbf{a}(s) = \\frac{\\mathrm{d}}{\\mathrm{d}t}\\mathbf{v}(s) ", "c8b4ef9aa455197efb775263e3ebb031": "\\left (\\sum a_n e^{in\\theta}, \\sum b_m e^{im\\theta} \\right )=\\sum_{n\\ne 0} |n| a_n\\overline{b_n}.", "c8b56efc946f79ff5be82cbe20f51fc4": "10^{(a\\cdot2^b)}", "c8b5e799c1bd6702a857a15f5bc41a8d": " CC = \\frac{ 2c } { s_1 + s_2 } ", "c8b6056408422e28692c1dfb00bd6c88": " T_K: L_2(X) \\rightarrow L_2(X) ", "c8b613e5d3094055475d3eea6e0c4919": "\\Delta F_0", "c8b638d684fb702aa22926e774d34b4f": "f \\in Hom(U(d),t,t)", "c8b64a2bda83e15eaa06a37245e9088e": "\\mathcal{F}\\left[(-\\Delta)^{(n-1)/2}\\phi\\right](\\xi) = |2\\pi\\xi|^{n-1}\\mathcal{F}\\phi(\\xi).", "c8b69dcb78df8314d12af47680752750": " \\Omega = \\left(\\begin{array}{c|c} 0 & I_n \\\\ \\hline -I_n & 0 \\end{array}\\right). ", "c8b69fd45c5b57c79fb5b773eb725359": "\\psi = Q_1 x_1 \\dots Q_mx_m[\\varphi]", "c8b6c389540a35d64aba6a2afa504114": "\\left\\{ X_{1},X_{2},\\cdots,X_{N}\\right\\}", "c8b7246a63d65066c96ed88c0c2beb1f": "\\frac{I}{A}", "c8b76d647e8cded7dcb41542406d5689": "\n\\alpha^5 = \\alpha^4 \\alpha = (\\alpha^2 + \\alpha + 1)\\alpha = \\alpha^3 + \\alpha^2 + \\alpha = \\alpha^2 + 1 + \\alpha^2 + \\alpha = \\alpha + 1\n", "c8b7892245211ca8003e11f63f474421": "1\\,+\\,\\frac{a}{1-r}\\;=\\;1\\,+\\,\\frac{\\frac{1}{3}}{1-\\frac{4}{9}}\\;=\\;\\frac{8}{5}.", "c8b7d70f2a4c2aa9b19f20cb9a816632": "\\begin{matrix}\n 245 & 236 & 245 & 236 \\\\\n 245 & 245 & 236 & 236 \\\\\n 245 & 245 & 245 & 245 \\\\\n 245 & 236 & 236 & 236 \n\\end{matrix}", "c8b85e877ea7d0843a5f56d1ba46f962": "\\scriptstyle O\\left(n^{\\left\\lceil \\frac{1}{2}d \\right\\rceil}\\right)", "c8b87d7a896a8a28b02e3f5f11b37efa": "\\psi_1, \\ldots, \\psi_n", "c8b884632a6b807d6f5648554e250084": "\n \\int x^m \\left(A+B\\,x^n\\right) \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^pdx=\n \\frac{x^{m-n+1} \\left(A\\,b-2 a\\,B-(b\\,B-2 A\\,c) x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{n(p+1) \\left(b^2-4 a\\,c\\right)}\\,+\\,\n \\frac{1}{n(p+1) \\left(b^2-4 a\\,c\\right)}\\,\\cdot\n", "c8b89d13b3d4bf440ec5143eb2ed23c8": "\\scriptstyle H(W),\\, H(H(W)),\\, \\dots,\\, H^{n-1}(W)\\,", "c8b8a416a1fd4647c433d837b0f2116b": "\\widehat{f}(a/p)=\\widehat{f_p}(a)", "c8b8a4fed5548e4fa2f302d6eb0579a9": "\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{n-2}} = \\sum_{k=0}^\\infty c_{n,k} \\frac{|\\mathbf{x}|^k}{|\\mathbf{y}|^{n+k-2}}Z_{\\mathbf{x}/|\\mathbf{x}|}^{(k)}(\\mathbf{y}/|\\mathbf{y}|)", "c8b8ab01c14744d05d2c4325fefcb77c": "A^c", "c8b8d33f05a3e21ecaa21e5d80e5585a": "C_X", "c8b8e595a540bb33c08cda441ca229a3": "\\ell^\\infty", "c8b92c6464cd2d6f35c2869b6b09c0a0": "=4!\\,\\! = 24 ", "c8b95a187bace6cb663166e911a138bc": " ax =[1,4][1,1]=[1,4] ", "c8b99fc0a585f5f8e0dcdaf4b851aec7": "F^g_{X/S} = 1_X \\times F_S^{-1}.", "c8b9a2eb13a5e7ba020d88a2fc047fb8": "u = \\int_0^\\phi \\sec t \\,\\mathrm{d}t = \\ln\\tan\\left(\\tfrac14\\pi+\\tfrac12\\phi\\right)", "c8b9a878f01ccfd4bd05aa6136cc2899": "m_k = 3\\times \\Delta_k", "c8b9ca6140f1885b09f235dd4cdb76c6": " S^3\\mathbb C^2", "c8b9f182eca12bf3b04ad57027606621": "\\big\\{|j_{k}\\rangle\\big\\}", "c8ba088e239bc9d3efea4379f05e0fad": " \\left\\langle {J(t)} \\right\\rangle _{F_e } \\ne 0 ", "c8ba0ae4d9350b58a5c478585e740c0c": "q' \\rightarrow 1", "c8ba26ffea793ebc77c0ad7e73dbb2e1": "\\varphi_{\\beta+1}(0) [n+1] = \\varphi_{\\beta}(\\varphi_{\\beta+1}(0) [n]) \\,,", "c8ba7811e2304c83dd25610c2cc47a4d": " \\int_a^b f(x) \\, dx < \\int_a^b g(x) \\, dx. ", "c8ba99a5ca4e73e4d780d9c946e482dd": "\\langle e_i|\\phi\\rangle", "c8bac54059e9ea2ed354eb0a66b36d5d": "\nf(x,y) = \\sum_{n=0}^{\\infty}\\sum_{l=-n}^{+n}A_{nl}V_{nl}(x,y).\n", "c8bae41759dbead35f8f43aaf48f1927": "\\mathbf{OPD} = m \\lambda_{m}", "c8bb118d41e2358c5d559d20df113540": "\\frac{d}{dx}e^x = e^x.", "c8bb3fb1eab5e9eab991e75815c3be05": "(\\pi z)^{-1}", "c8bba0b7d3de632b789f730b1b3072d7": "C_{m,n} =\\frac{1}{n e_n 4^{m-1} ((m-1)!)^2},", "c8bc1de1e9082a76f50a7d085451bd05": "(1,2,3)\\prec (0,3,3)\\prec (0,0,6)", "c8bc315c1bd7d5bf367771a1b550a7ea": "\\mathrm{CINT}_x(f_{-1}, f_0, f_1, f_2) = \\mathbf{b}(x) \\cdot \\left( f_{-1} f_0 f_1 f_2 \\right)", "c8bc763b2d0cc43c5bfcf0ba00169f1e": " c^2\\, =\\, g h\\, \\left( 1\\, -\\, \\frac{1}{3}\\, k^2 h^2 \\right). ", "c8bcb5d3387dcade9edd04ca0280cfac": "|t|<1/b\\,", "c8bcf669a9197ebbe950bccfde9f0884": "\\text{MTF}= \\mathcal{DFT}[\\text{LSF}] = Y_k = \\sum_{n=0}^{N-1} y_n \\left[\\cos\\left(k\\frac{2 \\pi}{N} n\\right) - i\\sin\\left(k \\frac{2 \\pi}{N} n\\right)\\right] \\qquad k\\in[0,N-1]", "c8bda255383e9d252afc004bc271e64b": "P_{e}=1-\\left[ 1-\\left( \\frac{1}{T} \\right) \\right]^{n}", "c8be2b3fd66395fa0bf1425a22538858": " \\sqrt{S} \\approx 2^8 = 1\\;0000\\;0000_2 = 256\\, .", "c8be4376752cf8bb0ddab9b27b5938da": "F_0 = f_0, F_1 = f_1", "c8be6ec5ca97437fc23b3e20273799d8": "I (\\rho) = S (\\rho_A) + S (\\rho_B) - S (\\rho)", "c8be89873a1eca4c720492a9db249695": "f:S\\to R", "c8bec9119778f3e3f5f7783b9c3ac89e": " b_{s-1} = \\cdots = b_0 = 0 ", "c8bf05e4b46247d741c8c8bdecf6a5fa": "\\frac{2 \\langle T_\\mathrm{TOT} \\rangle}{n \\langle V_\\mathrm{TOT} \\rangle} \\in \\left[1, 2\\right]\\,,", "c8bf1b12d60dbaf4e8ed0438703b87a9": "\\tau_{A} (\\omega) := \\inf \\{ t \\in T | X_{t} (\\omega) \\in A \\}.", "c8bf44a8d312825551c6444dd42236f3": "\\left.\\begin{matrix}\n& {}+\\kappa(\\kappa(X_1,X_2\\mid Y),\\kappa(X_3,X_4\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_1,X_3\\mid Y),\\kappa(X_2,X_4\\mid Y)) \\\\ \\\\\n& {}+\\kappa(\\kappa(X_1,X_4\\mid Y),\\kappa(X_2,X_3\\mid Y))\\end{matrix}\\right\\}(\\mathrm{partitions}\\ \\mathrm{of}\\ \\mathrm{the}\\ 2+2\\ \\mathrm{form})", "c8bf654f523fe7ff0800e74ed7cdb9cc": "E_k = \\frac{\\hbar^2 k^2}{2 m} + \\langle V \\rangle ", "c8bfbf5b90ef3d260e51c8317b2d3d63": "p = (A, X_1\\eta_1\\dots X_k\\eta_k)", "c8bfbfe0b131b195a7413fc07a3e1c98": "\\left[1 + \\left(\\frac{x-\\mu}{\\sigma}\\right)^{1/\\gamma}\\right]^{-1} ", "c8bfd0dbc5bea56c1f1e70b0184df0ff": "f:\\mathbb{N}_1\\rightarrow\\mathbb{R}", "c8bfd14a96147db181a68c22b5919741": "w_3 = w_1 w_2", "c8bfec35dd929a7f07cdcec19b3c5420": "\\widehat{T}(\\theta, t) = \\sum_{n=-\\infty}^{\\infty} T(n, t) \\, e^{-i \\theta n} = e^{t(\\cos \\theta - 1)}. ", "c8bfeeb4d554cbb9c3779a2d8235e4ea": " {\\rm MCG}(K)= \\mathbf{Z}_2 \\oplus \\mathbf{Z}_2.", "c8c02f12d7df2c3ad5007464dda86539": " s \\in S", "c8c0eb7dcaf58088fd9cb7d876366ddd": "\\hat{f}\\,", "c8c0f2975ef5e0ee3cfd15ecafed80ca": " w_i' = \\frac{w_i}{\\sum_{i=1}^{N}w_i} ", "c8c1211c0abf9e9017a0f7d4cf0a24c2": "1-r_i", "c8c146c3b3285da18e1d5f9d85030429": "v = \\frac{v_A}{\\sqrt{1 + \\frac{1}{c^2}v_A^2}}", "c8c1b29d21c2aefc695fc710c5dd9e06": " \\operatorname{E}_T(\\lambda) = \\mathbf{1}_{(-\\infty, \\lambda]} (T) ", "c8c2088a91068157bba50b9dae50aad0": " \n\\begin{align}\n\\mathcal B A(t)&= \\sum_{l = 0}^\\infty \n\\left( \\sum_{k=0}^\\infty \\frac{\\big((2l+2) t\\big)^k}{k!} \\right) \\frac{(-1)^l}{(2l+1)!} \\\\\n&= \\sum_{l=0}^\\infty e^{(2l+2)t}\\frac{(-1)^l}{(2l+1)!} \\\\\n&= e^t \\sum_{l=0}^\\infty \\big(e^t\\big)^{2l+1} \\frac{(-1)^l}{(2l+1)!} \\\\\n& = e^t \\sin\\left( e^t \\right).\n\\end{align}\n ", "c8c226d789fcab2d6e0bb9e065fd130f": " \\text{Var}(\\Sigma)=\\int_{\\textbf{R}^d} f(x)^2\\Lambda (dx), ", "c8c230bfcced5a768a88560a68c369b9": "t \\approx G \\sqrt{ \\frac{ N_s+N_d}{n(1-G^2)} }\\ ,", "c8c23e8ea7ae67e879809c6d011dfa26": "y_{scaled} = {x_{scaled} \\over x_{orig}} \\cdot y_{orig}", "c8c28f10f906bd9078537ac26242d8a4": "\n\\limsup_{n\\to\\infty}\\int_S f_n\\,d\\mu\\leq\\int_S\\limsup_{n\\to\\infty}f_n\\,d\\mu.\n", "c8c2c4249004d7a07f13c604a615bdd5": "\nm_t + m_x u + b m u_x = 0. \\,\n", "c8c2ce28f59c2a930cbccc43c1829935": "f_{n} : X \\to Y", "c8c2d470b5b490311862aaa2aa81e3ef": " \\prod_{i=2}^6 \\left(1 + {1\\over i}\\right) = \\left(1 + {1\\over 2}\\right) \\cdot \\left(1 + {1\\over 3}\\right) \\cdot \\left(1 + {1\\over 4}\\right) \\cdot \\left(1 + {1\\over 5}\\right) \\cdot \\left(1 + {1\\over 6}\\right) = {7\\over 2}. ", "c8c2da48aa6b7f97f78c8ea1d1cac36a": "q^i, i=1,2", "c8c327f851b5b0ef6e5c3ca5cd80ae98": "\\psi(\\bold r | \\bold r') = \\frac{e^{ik | \\bold r - \\bold r' |} }{4 \\pi | \\bold r - \\bold r' |}", "c8c3316073986f9ddac7b86db61e4a04": "(Tr(g^b),\\ E)", "c8c33cfa62083ed39ca1740a1caec329": "\nf_X(x;n)=\\frac{n}{2\\left(n-1\\right)!}\\sum_{k=0}^{n}\\left(-1\\right)^k{n \\choose k}\\left(nx-k\\right)^{n-1}\\sgn(nx-k)\n", "c8c3833ec43041282060a6ab0935fc84": "\\mathcal{H}\\Psi =\\{\\lambda _{1}[-\\varepsilon\n_{1}^{2}+m_{1}^{2}+p^{2}+\\Phi (x_{\\perp })]+\\lambda _{2}[-\\varepsilon\n_{2}^{2}+m_{2}^{2}+p^{2}+\\Phi (x_{\\perp })]\\}\\Psi ", "c8c3cc7ca0f31fa02c0f0bc8e01e224f": "X\\cup X'", "c8c3e5d5b83ff7808cce94e8a9cc9e51": "r-c >0", "c8c3eec3ffe8ef2ebd3662d6bd2f5076": "D: \\mathcal{L}^2 \\to [0,+\\infty]", "c8c3fdc1431ae42e96a2ea15c272a213": "dy/dx, \\operatorname{d}\\!y/\\operatorname{d}\\!x, {dy \\over dx}, {\\operatorname{d}\\!y\\over\\operatorname{d}\\!x}, {\\partial^2\\over\\partial x_1\\partial x_2}y \\!", "c8c41d0ef6fe896b97c9c3e57dd22687": "0.5 < A_{12}/A_{21} < 2 ", "c8c4627d03d8c8bfea0d480090145993": "\\vec y = \\left(\\frac 1 {b+c} , \\frac 1 {a+c} , \\frac 1 {a+b}\\right) ", "c8c466c6f83f4cf12a157888db1ff8ac": "\n\\begin{array}{lcl}\nKL_{i,1} & = & {\\rm ROL}(K_i,1) \\\\\nKL_{i,2} & = & K'_{i+2} \\\\\nKO_{i,1} & = & {\\rm ROL}(K_{i+1},5) \\\\\nKO_{i,2} & = & {\\rm ROL}(K_{i+5},8) \\\\\nKO_{i,3} & = & {\\rm ROL}(K_{i+6},13) \\\\\nKI_{i,1} & = & K'_{i+4} \\\\\nKI_{i,2} & = & K'_{i+3} \\\\\nKI_{i,3} & = & K'_{i+7}\n\\end{array}\n", "c8c4a91a4e6d41a01f47abb9ce1d5b0f": "f\\colon A \\to A", "c8c53932f145e7fa713dc90bce2e83be": "\\left(\\frac{1}{2\\pi iz},\\frac{1}{2\\pi iz}\\right)", "c8c552810abf2ca70350fb379f0e3283": "\\int_{-\\infty}^{\\infty} \\psi (t)\\, dt = 0", "c8c5845dd4eb086beb5a21b595e75e85": "(1)\\ \\ {\\mathit{He}}_n(x)=(-1)^n e^{x^2/2}\\frac{d^n}{dx^n}e^{-x^2/2}=\\bigg (x-\\frac{d}{dx} \\bigg )^n\\cdot 1 ,", "c8c59fb546d49e16262600793ca0a1ee": "\\kappa = +1", "c8c5a5278b3d4864cae22dbdcd78dd6b": "\n\\begin{align}\n \\nabla\\cdot\\partial_t \\mathbf{v} &= -\\nabla\\cdot(\\mathbf{v}\\cdot\\nabla)\\mathbf{v} + \\nabla\\cdot\\nabla^2\\mathbf{v} - \\nabla^2 p\\\\\n \\partial_t \\nabla\\cdot\\mathbf{v} &= -\\nabla\\cdot(\\mathbf{v}\\cdot\\nabla)\\mathbf{v} + \\nabla^2\\nabla\\cdot\\mathbf{v} - \\nabla^2 p\\\\\n 0 &= -\\nabla\\cdot(\\mathbf{v}\\cdot\\nabla)\\mathbf{v} - \\nabla^2 p\\\\\n \\nabla^2 p &= -\\nabla\\cdot(\\mathbf{v}\\cdot\\nabla)\\mathbf{v} & (\\ast)\n\\end{align}\n", "c8c5c4dfca236b9fd86b65f9a1ef7aca": "v << c", "c8c5d790bfc4aa4cd809ed386cdb1c2d": "\\frac{N_x}{N}=(1-p)p^{x-1} \\,", "c8c5da1370110f700ea8fced20be45b3": "\\scriptstyle{\\left(\\frac{dk}{dt} \\rightarrow \\infty\\right)}", "c8c5dcf90ad28e3edc5b70eab41824f2": "C(x) = \\left ( \\sum_{i=n}^{2n-1} x^i \\right ) \\bmod G(x)", "c8c5ffcee0839ec6aeed1b0f22c212b8": "Px = \\lim_{T\\to\\infty}\\frac{1}{T}\\int_0^TU_tx\\,dt.", "c8c6126ae7bc3a234c68539aa1343fba": "-0.2 \\leq \\rho_{AB} \\leq -0.1", "c8c669717f6a72323d2a25e191566b7c": " \\mathbf{f} = f^{\\alpha \\beta} \\, \\vec e_\\alpha \\otimes \\vec e_\\beta ", "c8c69d2f646e655fcaab182095227049": "\\begin{align}\n\\det(O) &= 1\\begin{vmatrix}x_{B}&y_{B}\\\\x_{C}&y_{C}\\end{vmatrix}\n-x_{A}\\begin{vmatrix}1&y_{B}\\\\1&y_{C}\\end{vmatrix}\n+y_{A}\\begin{vmatrix}1&x_{B}\\\\1&x_{C}\\end{vmatrix} \\\\\n&= x_{B}y_{C}-y_{B}x_{C}-x_{A}y_{C}+x_{A}y_{B}+y_{A}x_{C}-y_{A}x_{B} \\\\\n&= (x_{B}y_{C}+x_{A}y_{B}+y_{A}x_{C})-(y_{A}x_{B}+y_{B}x_{C}+x_{A}y_{C}).\n\\end{align}\n", "c8c6d8beb38e6c8cbbdd5d181d49ba67": "I_{\\mathrm{out}}", "c8c6ef2f953fbd1f7edc0349463e041d": "(S\\otimes T)(v_1,\\ldots, v_n, v_{n+1},\\ldots, v_{n+m}) = S(v_1,\\ldots, v_n)T( v_{n+1},\\ldots, v_{n+m}),", "c8c6f59bbc4518c1a946357e912c7131": "\\xi\\in\\mathbb{R}^n", "c8c6fdbbbb441013d01312dc2d7bb741": "s = 2 D_\\mathrm N \\,.", "c8c704a3ca256de1b575cc5fb0aa8359": "E = U/ \\langle g \\rangle", "c8c715f312a78961b79338a4349b5435": "h\\nu_0", "c8c7267c437f8546aac3528ef0212b30": "\\frac{\\rm d}{{\\rm d}t}x(t)=f(t,x(t),x(t-\\tau_1),\\dotsc,x(t-\\tau_m))", "c8c7f2b634cd7eb96d6996bee46b14a3": " \\succsim\\! ", "c8c814025786aa854dd6fdcc4a517c38": "\\sum_i y_{ij}", "c8c818834a450206ac02cc16a93949ea": "a, b \\,", "c8c8323231c7ca50dbbdfe408b855efc": "\\int_0^a x^m (a^n-x^n)^p\\,dx=\\frac{a^{m+1+np}\\Gamma [(m+1)/n]\\Gamma(p+1)}{n\\Gamma [((m+1)/n)+p+1]}", "c8c83e822f1b285cad58f934340310de": "((Y_1 + Z_1) Z_2 \\cdots Z_n) = (Y_1 Z_2 \\cdots Z_n) + (Z_1 Z_2 \\cdots Z_n)", "c8c881c66c50e79f429ffa99d27e9929": "y_1 = x_1x_2 + x_1x_4 + x_3x_4", "c8c8cb1dc40390cdc9d93dc349d6f24c": "f\\left(\\bigcup_{s\\in S}A_s\\right) = \\bigcup_{s\\in S} f(A_s)", "c8c8da04469f9b3c84f1f27dc042867f": "\\mathbf{\\overline{5}}", "c8c8e3931d614e0e7ef7ad62d40b4323": " a'_{hk} = a'_{kh} = a_{hk} - s (a_{h\\ell} + \\rho a_{hk}) \\,\\! ", "c8c8e3c27c5f7f354723a090b214ecc7": "A_{\\text{path}}(x,y,z,t) = e^{i S(x,y,z,t)}", "c8c916cb31e3dc90d67bf8cd573cd182": "f_1,", "c8c9469e00db948fb067dc8f970257a9": " E^2 = (pc)^2 + (mc^2)^2 ", "c8c9696ac1072a7c64866d38ff48ee20": "\n\\begin{cases}\nX_1=\\alpha\\cosh\\rho \\cos \\tau\\\\\nX_2=\\alpha\\cosh \\rho \\sin \\tau\\\\\nX_i=\\alpha \\sinh \\rho \\,\\hat{x}_i \\qquad \\sum_i \\hat{x}_i^2=1\n\\end{cases}\n", "c8c9760262d228a372672c4d8883ced2": "\\sum_{n=1}^\\infty \\frac{1}{10^n}.", "c8c9dbe23b5562941fab1e7fbb957cbe": "\\Sigma^0_n \\subsetneq \\Sigma^0_{n+1}", "c8cab6501401632d796257a0d9bae8d3": "x^2 + x + 1 = 0", "c8cbf8edfd6df1b4f5af3ba906a80d20": "\\mathfrak{m}_+", "c8cc071f7876e797062cc7075cbe1b8c": "H(s) = \\frac{\\overbrace{\\left(1+\\frac{R_\\mathrm{b}}{R_\\mathrm{a}}\\right)}^{G} \\frac{s}{R_1 C_1}}{s^2 +\n \\underbrace{\\left( \\frac{1}{R_1 C_1} + \\frac{1}{R_2 C_1} + \\frac{1}{R_2 C_2} - \\frac{R_\\mathrm{b}}{R_\\mathrm{a} R_\\mathrm{f} C_1} \\right)}_{2 \\zeta \\omega_0 = \\frac{\\omega_0}{Q}} s +\n \\underbrace{\\frac{R_1 + R_\\mathrm{f}}{R_1 R_\\mathrm{f} R_2 C_1 C_2}}_{{\\omega_0}^2 = (2\\pi f_0)^2}}", "c8cc1f2b606fa8195e8281a67791e818": "P_3 = (0,-\\sqrt{3})", "c8cc2cda3ad36ce2f38293742ec91db3": "di_i=eZ_i S_F vdn(v)", "c8cc9a3e76e897054fddf421cf12ea91": "\\log_2(2n)=k", "c8cd1adf3c1ab433e4f54649fc3d216e": "\\omega_0 = \\sqrt {\\frac{1}{LC} - \\left ( \\frac{R}{L} \\right )^2}", "c8cd3f01035d18696559ea3b2ee507d3": "p(\\textbf{z}_k\\mid\\textbf{x}_0,\\dots,\\textbf{x}_{k}) = p(\\textbf{z}_k\\mid \\textbf{x}_{k} )", "c8cd58c0bdeb14fdb1f5ce2a53c5557e": "M(f)<0", "c8cd88999753538d652e0069e3952e02": "s_1^3", "c8cd99bcfe4be90974ee2614bad4efa9": "\\hat{S}_{k}^{lm}(f)", "c8cde82a02b1fa751fa59642e8555d19": "\\left(\\frac{5}{4}\\right)^2x = \\left(\\frac{25}{16}\\right)x = 1.5625 x", "c8ce2a04a7269af1dbab3e51b5e35d5f": "{\\mathbb P}\\biggl(\\bigcup_{i=_1}^n A_i \\cap A_{n+1}\\biggr) \\ge 0,", "c8ce51aee86cf7bbc260b53c7c6cf2c1": "M = -2.5 \\log_{10}L(10) ", "c8ce95542ce6b590b8de0fa5f85a98e8": "hfv=(\\lambda-2)fv", "c8cee77cdc4fd2c74e42e28c075c7647": "1.3 {a \\over d} \\sqrt{n}", "c8cf56e1bbf2442a5f3fa9a311507fc0": "(z_1, z_2; z_3, z_4) = \\lambda\\,", "c8cf7a036a4879c0e66baa0921e10018": "x+1=y", "c8cfc2f0126fd7cf46de2e60cdda3d71": "C_{10} = \\left( \\frac{21}{2} \\alpha_1 \\alpha_3 + \\frac{70}{4} \\alpha_2 \\alpha_2 \\right) \\hbar \\omega", "c8cff0d049b7fa6bb1ed80897aa0a886": "\\widehat{\\operatorname{var}}(p'\\mid p)", "c8d011c4112b5b84dde8dadd2380db90": "\n\\boldsymbol\\Sigma\n=\n\\begin{bmatrix}\n \\boldsymbol\\Sigma_{11} & \\boldsymbol\\Sigma_{12} \\\\\n \\boldsymbol\\Sigma_{21} & \\boldsymbol\\Sigma_{22}\n\\end{bmatrix}\n\\text{ with sizes }\\begin{bmatrix} q \\times q & q \\times (N-q) \\\\ (N-q) \\times q & (N-q) \\times (N-q) \\end{bmatrix}", "c8d021bd7b07bc727d088e37bf6ad0e8": " D_t^2(x,y) = ||p_t(x,\\cdot) - p_t(y,\\cdot)||^2 ", "c8d12cc34aa81885a3c546a548101738": "\\frac{1}{\\tau }\\propto \\left \\langle v\\right \\rangle\\Sigma ", "c8d18ea8b4619d993e140243d76cdb29": "\\mathcal{O}_X(Y)", "c8d1987f5ad9b429ac95b8bc8c596158": "d = {\\pi \\over 4} \\times b^2 \\times s \\times n", "c8d224ed6accc5c322f2f73c06a99177": "0\\leq \\alpha \\leq n", "c8d22ce1616d5181884c2ff33ad3eed6": "\\scriptstyle L/K", "c8d237331c55e8520a0bfaccbb85b5d9": "|f_i(x)|\\leq M \\qquad \\forall i \\in I \\quad \\forall x \\in X.", "c8d24c184da7a0385a6291d062444af4": "\\ G_{u,v}", "c8d28a813a63c6c35babf1c11c44e788": " \\Pr \\left[ \\overline{A_1} \\wedge \\ldots \\wedge \\overline{A_n} \\right]", "c8d2a583bd0f29b2b66da919d6c94673": "\\epsilon_1 \\ge \\epsilon_2 \\ge \\ldots", "c8d2e878e9bb42f2376cb567d0c52d90": " \\phi(r) = r^2 ", "c8d30f77503965af7f920e3d0ddbf90e": "E(A-B) = E(A) + E(B)", "c8d35a30d26ac83506a9bb0fa6aa1881": "\\mathbf M^{-1}", "c8d3a51ebc6505abc79661dde0ab43bb": "((a+b)c)'=(a+b)'c+(a+b)c'=(a'+b')c+(a+b)c'=(a'c+b'c)+(ac'+bc')=", "c8d3c655af5f9a961f518faaacf8006d": "x^2+y^2\\leq r^2.", "c8d3fd1f8f81ed5931f73c285035f676": " 0 = \\sum_{i=0}^3 p^{ij} z_i , \\qquad j = 0,\\ldots,3 . \\,\\! ", "c8d40f55ac94b9e465485d7602d9890a": "\\mathbf{x}^{(k)}", "c8d421a10ce023d5c7bb1b5cf2297f0b": "\\left(\n\\begin{array}{llll}\n \\left\\{\\Gamma _{tt}^t,\\Gamma _{tr}^t,\\Gamma _{t\\theta }^t,\\Gamma _{t\\phi }^t\\right\\} & \\left\\{\\Gamma _{rt}^t,\\Gamma _{rr}^t,\\Gamma\n _{r\\theta }^t,\\Gamma _{r\\phi }^t\\right\\} & \\left\\{\\Gamma _{\\theta t}^t,\\Gamma _{\\theta r}^t,\\Gamma _{\\theta \\theta }^t,\\Gamma _{\\theta\n \\phi }^t\\right\\} & \\left\\{\\Gamma _{\\phi t}^t,\\Gamma _{\\phi r}^t,\\Gamma _{\\phi \\theta }^t,\\Gamma _{\\phi \\phi }^t\\right\\} \\\\\n \\left\\{\\Gamma _{tt}^r,\\Gamma _{tr}^r,\\Gamma _{t\\theta }^r,\\Gamma _{t\\phi }^r\\right\\} & \\left\\{\\Gamma _{rt}^r,\\Gamma _{rr}^r,\\Gamma\n _{r\\theta }^r,\\Gamma _{r\\phi }^r\\right\\} & \\left\\{\\Gamma _{\\theta t}^r,\\Gamma _{\\theta r}^r,\\Gamma _{\\theta \\theta }^r,\\Gamma _{\\theta\n \\phi }^r\\right\\} & \\left\\{\\Gamma _{\\phi t}^r,\\Gamma _{\\phi r}^r,\\Gamma _{\\phi \\theta }^r,\\Gamma _{\\phi \\phi }^r\\right\\} \\\\\n \\left\\{\\Gamma _{tt}^{\\theta },\\Gamma _{tr}^{\\theta },\\Gamma _{t\\theta }^{\\theta },\\Gamma _{t\\phi }^{\\theta }\\right\\} & \\left\\{\\Gamma\n _{rt}^{\\theta },\\Gamma _{rr}^{\\theta },\\Gamma _{r\\theta }^{\\theta },\\Gamma _{r\\phi }^{\\theta }\\right\\} & \\left\\{\\Gamma _{\\theta t}^{\\theta\n },\\Gamma _{\\theta r}^{\\theta },\\Gamma _{\\theta \\theta }^{\\theta },\\Gamma _{\\theta \\phi }^{\\theta }\\right\\} & \\left\\{\\Gamma _{\\phi\n t}^{\\theta },\\Gamma _{\\phi r}^{\\theta },\\Gamma _{\\phi \\theta }^{\\theta },\\Gamma _{\\phi \\phi }^{\\theta }\\right\\} \\\\\n \\left\\{\\Gamma _{tt}^{\\phi },\\Gamma _{tr}^{\\phi },\\Gamma _{t\\theta }^{\\phi },\\Gamma _{t\\phi }^{\\phi }\\right\\} & \\left\\{\\Gamma _{rt}^{\\phi\n },\\Gamma _{rr}^{\\phi },\\Gamma _{r\\theta }^{\\phi },\\Gamma _{r\\phi }^{\\phi }\\right\\} & \\left\\{\\Gamma _{\\theta t}^{\\phi },\\Gamma _{\\theta\n r}^{\\phi },\\Gamma _{\\theta \\theta }^{\\phi },\\Gamma _{\\theta \\phi }^{\\phi }\\right\\} & \\left\\{\\Gamma _{\\phi t}^{\\phi },\\Gamma _{\\phi\n r}^{\\phi },\\Gamma _{\\phi \\theta }^{\\phi },\\Gamma _{\\phi \\phi }^{\\phi }\\right\\}\n\\end{array}\n\\right)", "c8d4389afb044aa336076583ff467c19": "\\delta_{m,m_1+m_2}\n\\sqrt{\\frac{(2j+1)(j+j_1-j_2)!(j-j_1+j_2)!(j_1+j_2-j)!\n}{(j_1+j_2+j+1)!}}\n\\ \\times\n", "c8d4b64d6df6fa680adc695131b11fe6": "u_{ac} , u_{ab} , i_A", "c8d4c573a24d49bd0e3ca97d4dd7a97e": "\n\\eta_{2n}(s)\n = \\sum_{k=1}^{2n}\\frac{(-1)^{k-1}}{k^s}\n = 1-\\frac{1}{2^s}+\\frac{1}{3^s}-\\frac{1}{4^s}+\\ldots+\\frac{(-1)^{2n-1}}{{(2n)}^s}\n\n = 1+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+\\ldots+\\frac{1}{{(2n)}^s}\n - 2(\\frac{1}{2^s}+\\frac{1}{4^s}+\\ldots+\\frac{1}{{(2n)}^s})\n", "c8d514736c0aef5c5940d731c1eb1c36": "\\bigg \\langle \\! \\! \\bigg \\langle \n\\begin{matrix}\n n \\\\\n k\n\\end{matrix} \n\\bigg \\rangle \\! \\! \\bigg \\rangle", "c8d59793deb04098ecbffeb967a42b51": "\\ \\displaystyle \\mathop{Opt} \\neq \\mathop{opt}\\ ", "c8d59bd7575d8cfc462b05f1a8405e1b": "(\\text{skewness})^2+1< \\text{kurtosis}< \\frac{3}{2} (\\text{skewness})^2 + 3", "c8d5e57d2b82e5b3d21f8c59d51473cc": "n_1,\\dots,n_r", "c8d5f8eb5412a88ca674f57ad48700c0": "\\boldsymbol{l} = \\dot{\\boldsymbol{F}}\\cdot\\boldsymbol{F}^{-1}", "c8d62ff9795770d97adf02a271d5ae09": " \\ \\psi_e (\\phi) = \\sqrt{\\frac{2}{\\pi}} \\sin (m \\phi) \\quad (11) ", "c8d6946898206c7338f96edf392b7af6": "\\varphi = \\frac{1 + \\sqrt{5}}{2} \\approx 1.61803\\,39887\\cdots\\,", "c8d6d16e14afdd3f65356858f34cb465": "v_1,\\ldots , v_k", "c8d6eafb41af28ea55c050e5c4f87c95": "\\Psi_n = \\Psi_n^*", "c8d70847bd5ec2bcac8e0d9bae323045": "d\\mathbf x\\,\\!", "c8d74cd5dbcc145b64a118e08c730786": "I_{DSS} = (2a) \\frac{W}{L} q N_d \\mu_n V_{DS}", "c8d7d0103a2aa1739a7089e10e71fb01": "\\text{s.g.}=\\frac{144}{144 - \\text{degrees Baumé}}", "c8d838b82c53c16d1e9f089458cd1fb3": "j_1 = j_2 = m_1 = {-m_2}", "c8d8544fc0dfecd8797ec55cc0b79695": "\\mathbf{z}(0)=\\mathbf{z}_0 \\in\\mathbb{C}^n,", "c8d89ba9e639db1f2e290a34864134ce": "\n p = \\cfrac{2}{\\lambda^2}~\\cfrac{\\partial W}{\\partial I_1} ~.\n ", "c8d8b4a883cc88c9425dda1750f720bd": "a \\cdot {1 \\over b}", "c8d8bf0bec35d10aa14e244548b0f459": "P(conscientious|neurotic)=P(neurotic|conscientious)", "c8d8fb87ad20e8e0456f3907c97b8b39": "\\left(\\beta mc^2 + c(\\alpha_1 p_1 + \\alpha_2 p_2 + \\alpha_3 p_3)\\right) \\psi (x,t) = i \\hbar \\frac{\\partial\\psi(x,t) }{\\partial t} ", "c8d913f4b21e2855cd6376c10a893de2": "C_V\\left ( T_2-T_1 \\right )\\;", "c8d93cce3e2c0bb34b2109a9d95d702b": "f \\wedge g", "c8d96d42f060dec5947510d3a7bacc9e": "\\mathcal{\\tilde{H}}_{S} = span\\big[\\big\\{|\\phi_{k}\\rangle\\big\\}_{k=1}^{N}\\big]", "c8d98424cd7f6033f91ec7c70412a775": "\\langle f, v \\rangle = \\int_{\\Omega} f(x) v(x) \\, \\mathrm{d} x.", "c8d996be5e7bae644e3c07528d7a51df": "u(.)", "c8d9c38fa0c96d0e8f55aaa6df36c293": "(f\\circ g)\\circ h=f\\circ(g\\circ h)=f\\circ g\\circ h\\qquad\\mbox{for all }f,g,h\\in S.", "c8da0346faf04ede88b14e34bf763201": "\\epsilon(u_{ij}) = \\delta_{ij}", "c8da95373a550bbea349f231d38ad280": "\\scriptstyle \\frac{R}{Q}", "c8daba9b9d51fbb9ec978e17bc7babea": "\n \\left(\\nabla^2-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial{t}^2}\\right)\\psi(\\mathbf{r},t)=0,\n", "c8dae81c4ae561bce687ffa252e11588": "=r_O \\left( 1+ \\frac { \\beta_2\\ln \\left(\\frac {I_{C1}}{I_{C2}}\\right)} {\\left(\\beta_2 + 1 \\right) + \\ln \\left(\\frac {I_{C1}}{I_{C2}}\\right)} \\right) \\ , ", "c8db0a3df922cdf1ba10923db8d4749e": "g^{efghcdab} = g^{abcdefgh}", "c8db246aaf135d9582fa47aad52251b0": " \\mathrm{depth}(M) \\leq \\dim(M), ", "c8db5f5d4a6fb1006a5d3ddb492730fe": "|d(A,X)-d(B,X)|=i", "c8db7802a5db9c45b787bd462e6bdd0a": "L(z) = f'(x)z - f'(x)x + f(x)", "c8dbaf588cd7a650cd648176da0470e9": "f(n)=2l+1", "c8dbbe1dfb3882ac05a451aebfdba342": "K(t,x,y) = 0, \\quad x\\in\\partial\\Omega \\rm{\\ or\\ } y\\in\\partial\\Omega.", "c8dbc1697eec4fe0be110bea04a15160": " \\sum_{i,j} de_i de_j' a'_x = 0 ", "c8dc2e394c40ffe74a6f920434f92b29": "J_{-n}(z) = (-1)^n\\, J_{n}(z),", "c8dc7a6e535e6d182a54e38273cd740d": "\\theta = 90", "c8dcbdf19dbc6e142df8aa0428674d0e": "a(x) = f_{\\theta | t}(x)", "c8dcd0ffcc4887dee881d7c5f475b8e1": "n \\rightarrow \\infty ", "c8dd2675c3e411d20df33785d4d80d14": "dV=dx\\,dy\\,dz", "c8dd96455c92aaab30ab6605737212f6": "\\frac{\\zeta(2n+1)}{\\pi^{2n+1}},", "c8dddf6c66f107db8172b4bc57f6e0cb": "X \\rightarrow \\varnothing", "c8dde1b2f3b96c79fbab1bc03711e20d": "\n \\mathbf{w}(\\mathbf{X}_B) - \\mathbf{w}(\\mathbf{X}_A) = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} \\boldsymbol{\\nabla} \\mathbf{w}\\cdot d\\mathbf{X}\n = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} (\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon})\\cdot d\\mathbf{X}\n", "c8de3a68678c674eb24f277924d48df0": "P(x_1,\\dots, x_c)", "c8de3ec215127927f0d994de8bf52379": "\\hat {x}", "c8de9fc2683219833862f136709888fa": "\\omega = v/r", "c8dec084f13f7fee476963577c8cd753": "aw^2+bx^2+cy^2+dz^2", "c8deddc31d61d89529d5c4ba97063ff0": "\\frac{\\alpha}{s} \\; \\left(\\frac{x-m}{s}\\right)^{-1-\\alpha} \\; e^{-(\\frac{x-m}{s})^{-\\alpha}}", "c8df09c83186be46b9fcf6b15cbfbdb3": " \\flat : \\Gamma(TM) \\to \\Gamma(T^*M)", "c8df41ee4c601a555e53975ba1a49d6a": " \\text{ } (1) \\text{ } U_{\\delta(n)} \\equiv 0 \\pmod {n}. ", "c8df44072ade858668cecd3fd3464087": " |\\psi\\rangle = \\begin{pmatrix} \\cos\\theta \\\\ \\sin\\theta \\end{pmatrix} \\exp \\left ( i \\alpha \\right ) .", "c8df789478a3c31c4d782e7d40c014a9": "\n\\lambda_k = -\\frac{4}{h^2}(\\sin^2(\\frac{(k-1) \\pi}{n})), \\ k = 1,...,n.\n", "c8df91bb6e6727201fa15fd63fa0be9a": "dE = T_{\\rm v} dS_{\\rm v} - \\langle P\\rangle dV,", "c8dfb14f6feace74952bfe9852e75963": " \\mathrm{Pr}[\\exists (j,k) ", "c8e006203e5a7cadb3eae5f6199076f8": "0=x_0 (\\rho(A)-\\epsilon)^k", "c8e2d8bf7428cd352826768f2efd56d0": "\\Omega_\\text{LT}=-2.2 \\cdot 10^{-4} \\text{ arcseconds}/\\text{day}. ", "c8e2eabe185b4cb8ef054f39e45a5ac8": "v_i / 2", "c8e2f5faba6847dfb5e1790d7bb4350a": "o_{k+1:t}", "c8e367ff5395afe11e214b2660bb0e8a": " kX+l \\sim \\textrm{Cauchy}(x_0{k}+l,\\gamma |k|)\\,", "c8e37013d6b9f1cf4208cca4c07c0b18": "\\textstyle \\Upsilon_2", "c8e380003c7ec77f19c014fae21a4c8e": "W_L = 0.5LI_L^2- \\ ", "c8e38ad7b44f8a1d346f8f74ef8433e1": "\\{U_i\\}", "c8e3dd4df572d39e5ab97aa1ee7b2977": "4K^2 = (pq + rs)^2 (1 - \\cos^2 A) = (pq + rs)^2 - (pq + rs)^2 \\cos^2 A.\\,", "c8e3f1bae47c61323a8b28d96a1ab067": " \\; x=0 ", "c8e44867559c1f4284d82e6593b51079": "\\frac{\\mathrm{d}\\varphi}{\\mathrm{d}\\alpha} = \\int_a^b\\frac{\\partial}{\\partial \\alpha}\\,f(x,\\alpha)\\,\\mathrm{d}x+f(b,\\alpha)\\frac{\\partial b}{\\partial \\alpha}-f(a,\\alpha)\\frac{\\partial a}{\\partial \\alpha}. ", "c8e48a1bac90c414284df5e5bbf24f88": "\\Delta V=3000 psi \\times 300 bbl \\times 3.5 \\times 10^{-6} psi^{-1}", "c8e499fe22f077022154a388bee840ba": "\\left|\\frac{x}{a}\\right|^m + \\left|\\frac{y}{b}\\right|^n = 1; \\qquad m, n > 0.", "c8e4b7958ec8d796dbf0881102b2e61b": "\n \\boldsymbol{P} = J~\\boldsymbol{\\sigma}~\\boldsymbol{F}^{-T} ~", "c8e4d46a65a05cfbb3ce7bfd57ef4dbd": "\\frac{3}{2}k_{\\rm B}T=\\frac{1}{n} \\int_c \\frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\\, dc", "c8e4d9cdc3b01848bca148dabaf2bb80": "\\Bbb F_p^\\times", "c8e54e8a3a722cdd62ee808da320315e": "\\displaystyle{A=\\{T\\in\\mathfrak{t}:\\, \\alpha_1(T)\\ge 0, \\dots,\\alpha_n(T)\\ge 0, \\alpha_0(T)\\le 1\\}.}", "c8e54f8681018ef8e0d224e02fbf1fbc": "\n\\Psi^{(A)}_{n_1 \\cdots n_N} (\\cdots x_i \\cdots x_j\\cdots) = -\n\\Psi^{(A)}_{n_1 \\cdots n_N} (\\cdots x_j \\cdots x_i \\cdots)\n", "c8e5b09aa3369cb31f88225c4d39b1dc": "\\left|\\sum_{r\\neq s}\\dfrac{u_r \\overline{u_s}}{\\lambda_r-\\lambda_s}\\right|\\le \\dfrac{3}{2} \\pi \\sum_r |u_r|^2\\tau_r^{-1}.\n", "c8e5b1e3435967ab810bf6cd5de6ddf9": " \\beta_p^{i,j}", "c8e5d9be3765a47c12fece4d4d89c688": " \\langle Ax , y \\rangle = \\langle x , z \\rangle \\quad \\mbox{for all } x \\in H,", "c8e5f3255d7eb959c746f39e3e9f5b5d": "p^2 = q^2 = 0", "c8e601cf27e000135a7abe4c2383090b": "\n\\left\\langle x,p_{j}\\right\\rangle =\\left\\langle \\sum_i c_{i}p_{i},p_{j}\\right\\rangle =\\sum_i c_{i}\\left\\langle p_{i},p_{j}\\right\\rangle.", "c8e61fdf0926f7eca682fc6c08f81279": "-x_n\\le\\cdots\\le-x_1.", "c8e631b3da73bacde2d685e413e5ca3e": "z=\\frac{x-\\mu}{\\sigma}", "c8e63d4834cf8d672b1b2b9bbcba77f8": "\\Delta H ^{\\circ}_{\\mathrm{c}} [\\mathrm{CH_{4 (g)}}]", "c8e63e4c339d8c3fdcff21f5e4950525": "\\boldsymbol{g}=(\\delta,\\nu,\\boldsymbol{\\lambda}^T,\\boldsymbol{\\mu}^T)", "c8e6f3212bdd14e573b07a2e8379c2e1": "\\mathcal{P}(L)", "c8e7271920546417163cd815419f892d": "M =\\;1.", "c8e7a45a897c8f2e7c6c48760a56ad3b": "C_t = C(1 + i)^{-t}\\, = \\frac{C}{(1+i)^t} \\, ", "c8e7be28851a1fd8233c0060895db543": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 10.20778\\log_e(T+273.15) - \\frac {7217.173} {T+273.15} + 83.18124 + 8.369119 \\times 10^{-06} (T+273.15)^2", "c8e8299fdf15d84569859c36973c94db": "\\alpha_i, x_i, y_i", "c8e861b909adb72ce66623d871b27fdd": " (Torque)\\Delta Q = r \\Delta F_y", "c8e884b0d78edfd958e7a2bb5be5d78d": "L_{D}", "c8e96a854c27668bd5d96ec434723fca": "\n\\mathbf{n}(x,y,z,t) = (\\cos\\varphi(x,t), \\sin\\varphi(x,t) \\cos\\psi(x,t), \\sin\\varphi(x,t) \\sin\\psi(x,t)).\n", "c8e98301f120ee2c61a0a10fe76f5f03": " C^k", "c8e9a039672c70544cbcfa5e032297df": "(p \\leftrightarrow q) \\vdash (p \\to q)", "c8e9acdf1b72b818c03bab1e5163133f": " \\mathbf{x} \\times (\\mathbf{y} \\times \\mathbf{z}) + \\mathbf{y} \\times (\\mathbf{z} \\times \\mathbf{x}) + \\mathbf{z} \\times (\\mathbf{x} \\times \\mathbf{y}) = 0", "c8e9bad79e235807475d7b2f5f583fbb": "\\log\\sigma_{t}^{2}", "c8ea07ac73af10653468bf2d78e9a134": "\n\\begin{align}\n\\left(\\begin{matrix}r(0) \\\\ r(1) \\\\ r(-1) \\\\ r(-2) \\\\ r(\\infty)\\end{matrix}\\right) & {} =\n\\left(\\begin{matrix}\n0^0 & 0^1 & 0^2 & 0^3 & 0^4 \\\\\n1^0 & 1^1 & 1^2 & 1^3 & 1^4 \\\\\n(-1)^0 & (-1)^1 & (-1)^2 & (-1)^3 & (-1)^4 \\\\\n(-2)^0 & (-2)^1 & (-2)^2 & (-2)^3 & (-2)^4 \\\\\n0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}r_0 \\\\ r_1 \\\\ r_2 \\\\ r_3 \\\\ r_4\\end{matrix}\\right) \\\\\n & {} =\n\\left(\\begin{matrix}\n1 & 0 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 1 & 1 \\\\\n1 & -1 & 1 & -1 & 1 \\\\\n1 & -2 & 4 & -8 & 16 \\\\\n0 & 0 & 0 & 0 & 1\n\\end{matrix}\\right)\n\\left(\\begin{matrix}r_0 \\\\ r_1 \\\\ r_2 \\\\ r_3 \\\\ r_4\\end{matrix}\\right).\n\\end{align}\n", "c8ea59a2da6e4aa8c2ed5f4d86343969": "\\lim_{n\\to\\infty} \\frac{\\Psi_{n+1}}{\\Psi_n} = 0.", "c8ea686965a99ac9913589135e64bdce": " S = -k \\langle \\log P \\rangle = - \\frac{\\partial A} {\\partial T}, ", "c8eac172ffa1b77be81755364c80b6f3": "i_{ion}", "c8eb03e9aca615ba2930acb9a65f6d40": "q = 2^\\nu\\frac{a}{b}", "c8eb32cbf22886483ca96aa2d045add6": "C \\in \\mathbb{Z}^*_{N_1N_2N_3}", "c8eb6fec66fa1d1f11a39548ab62f1a5": "\\frac {C_{13}^1 C_4^3 \\cdot C_{12}^1 C_4^2} {C_{52}^5} = \\frac{13 \\cdot 4 \\times 12 \\cdot 6}{2{,}598{,}960} \\approx 0.1441 \\% ", "c8ebb902235acb7505179bef1b2dd171": " \\begin{align} r & = \\frac{{a}t^2}{2}+2r - 2r_0 - vt +r_0 \\\\\n0 & = \\frac{{a}t^2}{2}+r - r_0 - vt \\\\\nr & = r_0 + vt - \\frac{{a}t^2}{2} \\quad [5] \n\\end{align}\\,\\!", "c8ebc8ae3b57edd38031fc61cbb5de6e": " \\mu = \\dfrac{mc}{\\hbar}", "c8ec9644c96b934c06829970f50268ba": "\\frac{f(a)}{f(b)} = \\frac{\\log_2(a)}{\\log_2(b)}.", "c8eccb827fb02468f1917ad24243553a": " \\alpha = 1.3 ", "c8ecee4b3884543d89255c9ad1f08061": "s_3 \\setminus s_2 \\setminus s_1", "c8ecf85ea152c165a2440521bc09bd4f": "\\left(\\tfrac{a}{p}\\right)", "c8ed5368d516641cb8a6b383ec68e9ef": "\\scriptstyle q_e^4", "c8ed8519ab42aecc9dbac89b52f617d8": "k\\alpha^n\\quad{}", "c8ed8bd6d90e57f3e3c10e176917babf": "M\\subseteq N", "c8ed8f47cebf1fb47985b266c334abfe": "a^2-x^2\\,\\!", "c8edf1c6a9c81e5011f35be02403e89a": " (\\partial A)_G=-(\\partial G)_A=-S\\left[V+P\\left(\\frac{\\partial V}{\\partial P}\\right)_T\\right]-PV\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "c8ee00aab296782b3ab2d75245d1043d": "P_2\\,", "c8ee07525dc53d66321e929e97d56f11": "\\delta'(t)", "c8ee7be8ccfa22e8ff4e37d07463d112": "\\scriptstyle T_i", "c8eecb285fd372ab91ba42b790d12abe": "\\mathbf{L}\\neq 0", "c8ef12423c090aca609245813e67de20": "^{1}\\Sigma^{+}", "c8ef23d4a5220c118ce04b091202d3ca": "\\frac{1}{24*24*2}=\\frac{1}{1152}", "c8ef32f1f44fa5e5fbe52c181bc84d0a": "p_0=\\pi(1-\\theta_1)^2+(1-\\pi)(1-\\theta_2)^2", "c8ef34c20b948daa6c18b4bf5c116394": "\nP_E~dE = P_p\\frac{dp}{dE}~dE\n", "c8ef40df794fe96dfc51b6efb938f024": "\\displaystyle\\mu_r=\\mu_c=0", "c8ef7eedd3d08896ce147d57771d52ce": "(q_1^*, q_2^*)", "c8ef8e04b0b7b76441af6a80118d1d7d": "1^k + 2^k + 3^k + \\cdots + n^k", "c8efcda7ada40e01b3effcc70e68e96d": " 0 \\to X \\to E \\to Y \\to 0\\,", "c8efe7054b49eaf3f607ec43dbce2bb4": "i[\\hat{H}_0,\\hat{v}_i] \\ne 0", "c8f0112669191b2715c9b07d1a6296c8": "(p^{k+1}-1)/(p-1)=1+p+\\cdots+p^k", "c8f07cbb8bf8f676d11a8ab48f2ba9d2": "c = \\sqrt{2}, \\sqrt{3}, \\sqrt{5}, \\sqrt{7}, \\sqrt{11}, \\ldots \\, ", "c8f0edcacd45f5a21738bd4056b2e2fc": "\\{g_\\alpha\\}", "c8f113e761f7be25248f4669ceca769b": "R\\,\\mathcal{H}om(Rf_!\\mathcal{F},\\mathcal{G}) \\cong Rf_{\\ast}R\\,\\mathcal{H}om(\\mathcal{F},f^!\\mathcal{G})", "c8f1443218a3018d41527f17d33a03fc": " \\mathbf{A} = \\overline{A} + \\frac{1}{c}V\\mathbf{e}_4,", "c8f15d61fc96535e758487de1cf02390": "\\exists y[y \\lambda\\}, \\,\\, f_\\lambda =\\chi_{E(\\lambda)} f,}", "c8f9feb501b4a9c58099c7862fa9fa85": "A \\circ B = (A \\ominus B) \\oplus B ", "c8fa96823e17989796a0f3b3824685cb": "\n\\mathrm{Ri} = \\frac{\\mathrm{Gr}}{\\mathrm{Re}^2}.\n", "c8fae6f938e95d65f876e0722a4a5cb9": "\\forall x, y \\in N", "c8fae840e5b96d397d2976c9792dffc2": "\\Delta{G^o}=\\Delta{H^o}-T\\Delta{S^o}\\,.", "c8fb4ebf5a921b9f43585f5c8b53e5bb": "\\beta = 0.25", "c8fb6e5f920fa4176b02a336bf0f44ce": "\\dot{k} =f(k) - c - (n + g + \\delta)k", "c8fc0e3a38baa1a3e4db181878f01c1e": "G = (N, T, S, P, R)", "c8fc1c5ad9e2d55a16b90b5bd3bfaaa3": " \\Rightarrow_3 aa\\boldsymbol{aB}Bbccc \\Rightarrow_4 aaaB\\boldsymbol{bb}ccc \\Rightarrow_4 aaa\\boldsymbol{bb}bccc", "c8fc226a304e1db3987c68a46a139a77": " \\nexists \\; k ", "c8fc333304acaaa9e859ad51bd5e6725": "q=e^{-\\sigma^2}.", "c8fc9836659b52406b3ba62c9d62c6d7": "X' \\widehat \\otimes_\\varepsilon X \\simeq \\mathcal{K}(X).", "c8fca72c317adc25d5a731a09e3027d7": "= \\int d^4 x \\; e (e^{I [\\alpha} e^{\\beta]}_N C_{\\beta J}^{\\;\\;\\;\\; N} + e^{M [\\beta} e^{\\alpha]}_J C_{\\beta M}^{\\;\\;\\;\\; I}) \\delta C_{\\alpha I}^{\\;\\;\\;\\; J}\n", "c8fcc5e938c2428971e53e77f655bdd9": "\\sum_i \\left( n_i\\frac{m_i C^2_i(x,t)}{2} + \\int_c \\frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\\, dc \\right)", "c8fd2ad1f8c27dd614cedf2878493999": " B = \\frac{ \\Sigma n_{ ii }^2 }{ \\Sigma n_{ i. } n_{ .i } } ", "c8fd44360816ea8fbf49ff3798bf5921": "R \\bar f_* R\\bar g^! \\cong Rf^! Rg_*", "c8fd9e42d3b3d15b9caee16236466b36": "\nT_L=T_{\\alpha=0}=MLD=\\frac{1}{N}\\sum_{i=1}^N \\left( \\ln{\\frac{\\overline{x}}{x_i}} \\right)\n", "c8fdbc2923453abc74847e1bfd098a52": "\\mathbf{r} = M \\mathbf{x}", "c8fde1162a2cd06ba7061fb5d0e4a0c2": "\\eta = -1", "c8fe3c50b7a2c09c1a59be20ae06cf49": "\\partial u / \\partial x", "c8fe5862de2da1d7341a3882bfab52cb": "\\beta_{j}^{+}=\\max(\\beta_{j},0)", "c8fe6f4c17ff55ecf6a67a43c44e4a33": " \\bar{n}_i(\\epsilon_i) = \\frac{g_i}{e^{(\\epsilon_i-\\mu) / k T} + 1} ", "c8feafc53a5843a3175c2bf1d2a5ca9b": " (\\mathbf{P} - \\mathbf{P}_0) \\cdot \\mathbf{A} t = \\left( \\mathbf{V} - \\mathbf{V}_0 \\right) \\cdot \\frac{\\mathbf{V} + \\mathbf{V}_0}{2} t \\ , ", "c8feb8a18b669199ae91037985ae6ba6": "\\operatorname{OE}[a](t) = \\prod_0^t e^{a(t') \\, dt'} \\equiv\n \\lim_{N \\rightarrow \\infty} \\left(\n e^{a(t_N) \\Delta t} e^{a(t_{N-1}) \\Delta t} \\cdots\n e^{a(t_1) \\Delta t} e^{a(t_0) \\Delta t}\n \\right)\n ", "c8ff5ade992a2c7f569edeb3ab5d7c25": " \\bar Q = \\bar B = \\bar Y = 0", "c8ff6710b933df09020a8aade330ac35": "\\operatorname{pd}_R K = \\operatorname{pd}_R M - 1", "c8ffb7b9a0a58709341459b27e75147c": "h[f[x]]=(\\alpha -1) f[x]", "c8fffbe88f23e3369346a88f6103a80c": " \\rho= \\frac{e^{- \\beta (H + \\mu_1 N_1 + \\mu_2 N_2 + \\cdots)}}{\\operatorname{Tr}(e^{- \\beta (H + \\mu_1 N_1 + \\mu_2 N_2 + \\cdots)})}. ", "c9000e726c2da0aaefda53fa219d09fd": " s_n(w) = w \\; (w-1) \\; (w-2) \\; \\cdots \\; (w-(n-1)) = (w)_n,", "c9004d93f067ddb06bcfa019661f153a": " G = Q = k_f - k_i", "c90099d09e7c5c4b401b79e1dca1118b": "\\mathbb D", "c900ababfa26974d1d040a2eb23a0be0": "\\text{var}(\\hat{\\beta}_j)= \\sigma^2\\left( \\left[X^TX\\right]^{-1}\\right)_{jj} \\approx \\frac{S}{n-m}\\left( \\left[X^TX\\right]^{-1}\\right)_{jj},", "c900f8ca7b33e243fec0ab6196bbd9af": " \\frac{a_{1}''}{a_{2}''} = \\frac{a_{1}}{a_{2}} ", "c9011cae55846967834ac7462c055cde": "\\hat{A}^\\bullet\\, \\hat{B}^\\bullet \\equiv \\hat{A}\\,\\hat{B}\\, - \\mathopen{:} \\hat{A}\\,\\hat{B} \\mathclose{:}", "c9015c85e1b4cf6a8b92c1d6a21f39dc": "dW_{input} = dW_{stored} ~~~~~~~~ (dW_{mechanical} = 0) \\;", "c9015d0f5534c64c17a20c1c6a351bff": "G(N)=\\left(\\prod_{p\\mid N}\\left(1-{1\\over{\\left(p-1\\right)}^2}\\right)\\right)\\left(\\prod_{p\\nmid N}\\left(1+{1\\over{\\left(p-1\\right)}^3}\\right)\\right).", "c90182e6d250879fcf73951ea5559bb9": " {\\lVert x_{k-1}-x_k \\rVert^2} ", "c901896544b77e2b65d9faaa74bb0f55": "\n\\eta\\rightarrow\\eta_x \\quad\\text{at rate} \\sum_{y:\\eta(y)\\neq\\eta(x)}p(x,y).\n", "c90199ef28ca5addb43e16022f5223c0": "x=e^{W(a\\ln b)}.", "c902072ed04ac209fbd1b21f8cdd9f6b": "x \\preceq y \\preceq x", "c9022c4ca6df1598986324e731ffe699": "C_\\text{out}", "c902666133d639e75d36dbd1c0511733": "\n{\\beta} = \\frac{2m}{m + 1}\n", "c90270517b5c65f5339e9cb9d91f5db8": "\\mathcal{D}=(V_1\\sqcup V_2\\sqcup V_n, E)", "c903443b6b1a4d562ae4d2b449dbd8e6": "\nT_{program} = \\sum_{i=1}^{n}{(T_{BB_i}*F_{BB_i})}\n", "c90345b38e507d11fa279f0e641a1e0d": "W=\\prod_i \\frac{(N_i+g_i-1)!}{N_i!(g_i-1)!}", "c9037738fb9de29416b5ce2938c1a0ab": "\\zeta(s,x+y) = \n\\sum_{k=0}^\\infty {s+k-1 \\choose s-1} (-y)^k \\zeta (s+k,x)", "c9037c26c6816175c1e3b93b3f54daf2": "a\\mathrm{inf}_b = \\nabla_b Z_2 \\qquad Z_2 = \\sqrt{g_{tt}}", "c903a9ceda349146e6dfa25586028fda": "\\mathfrak{a}_i", "c903bdc44f69fc4f1e8cd2ff0bc6886d": "gate2", "c903fd74fe3e5823dffd2041ea84ff60": " G = 1 \\ ", "c904b4ca092023dcf4a985d81d0a1892": " f(c_1) ", "c90504d8d0f78ef988f6b88e42e9b0be": "F(x_1+x_2+\\cdots)=F(x_1)+F(x_2)+\\cdots \\,", "c9051315ea4eb1101037c76c6714621f": " (\\cot x)' = -\\csc^2 x = { -1 \\over \\sin^2 x} = -(1 + \\cot^2 x)\\,", "c9053cc09e9d71f6372963ad592e9f41": "d = -P \\nabla f(x)", "c9053f04c48836aee269b5857230b54b": "x = a + ib", "c905942c6ebbb199f12dd03db866a97e": "\\{0,1\\}^N ", "c905aebc5d38187d02182e4a7487e320": "k_\\mu k_\\nu", "c905c1f5b6a217219b9b8c677f2d6713": "P_E~dE = 2 \\sqrt{\\frac{\\beta^3 E}{\\pi}}~e^{-\\beta E}~dE", "c905e637f191f045b07938f7ef5e3131": "\\Gamma'' = v_1:\\bar\\Gamma(\\ \\tau_1\\ ),\\ \\dots,\\ v_n:\\bar\\Gamma(\\ \\tau_n\\ )", "c905ea0142a042749c96476c9d8b34c6": "t_r = t-\\frac{|\\mathbf r - \\mathbf r'|}{c}", "c90697534aaa35e228ca3bf356525cd5": "\\theta_i \\in \\left \\{ \\omega_0, \\omega_1, \\ldots, \\omega_k \\right \\}", "c906bda7f0b8c3556ddd333b57a87bc2": "\\left[{3\\atop 2}\\right] = 3", "c907033892d03e08e7ed861db64f3192": "\\begin{align}\n\\mathbf{D}' & =\\gamma \\left( \\mathbf{D}+\\frac{1}{c^2}\\mathbf{v}\\times \\mathbf{H} \\right)+(1-\\gamma )(\\mathbf{D}\\cdot \\mathbf{\\hat{v}})\\mathbf{\\hat{v}} \\\\\n\\mathbf{H}' & =\\gamma \\left( \\mathbf{H}-\\mathbf{v}\\times \\mathbf{D} \\right)+(1-\\gamma )(\\mathbf{H}\\cdot \\mathbf{\\hat{v}})\\mathbf{\\hat{v}} \\\\\n\\end{align}", "c907098b1ee9ea8bffaad02bdaa62e2d": " \\text{Minimize}\\, wL + rK \\, \\, \\text{with respect to}\\,\\, L \\,\\, \\text{and} \\,\\, K,", "c907606adb85ece72d56d7cd25270c1e": "p(x) = Q(x,f(x))", "c9078b41e063641e948f1e39db132640": "\\scriptstyle \\left\\Vert \\mu \\right\\Vert \\;=\\; \\frac{e \\tau}{\\left\\Vert m^* \\right\\Vert}", "c907a2b974d85f06eaf4e2ff80583ff9": "2^{O(\\log n)} = n^{O(1)}", "c907c7a66353888fd6d56a9e1c91e2d4": "\\displaystyle{B(x,y)=-(Jx,y)_{\\mathbf{R}}.}", "c907f5faff8bbbaa72fddee571d5d6f0": "\\begin{cases}\nf: \\mathbf{S}^2 \\to \\mathbf{R} \\\\\n m \\mapsto \\langle Y(m), m\\rangle.\n\\end{cases}", "c9081cdc42f19befe5098f57cd63d367": "\\tau_c*=f\\left(\\mathrm{Re}_p*\\right)", "c90821c12860a18ea6df43b0fbbda179": " K: [a,b] \\times [a,b] \\rightarrow \\mathbb{R}", "c9082356295ec82c5940e3b9baf0dd8e": "\\alpha=0,1", "c90825b7fa1da35334b74785017364a7": "A\\in{\\mathcal A}", "c9083896b42f463d3acaa66a7f7efb87": "{{\\omega }_{p}}", "c90849fdc8090159409f02a6a8c0799e": "V = 2\\pi \\int_a^b x \\vert f(x) \\vert \\,dx", "c9088cee6ac960d75fcbd17cf7205922": "S_{12} = 0\\,", "c908cad878763bd9f00e58424177bdbe": "{1 \\over r^2}{\\partial \\left( r^2 A_r \\right) \\over \\partial r}\n+ {1 \\over r\\sin\\theta}{\\partial \\over \\partial \\theta} \\left( A_\\theta\\sin\\theta \\right)\n+ {1 \\over r\\sin\\theta}{\\partial A_\\phi \\over \\partial \\phi}", "c908e32caaba649b641a64386e7013b8": "a_n(t+\\tau)= l_n \\frac {\\left [ v_{n-1}(t)-v_n(t)\\right] ^k }{\\left [ x_{n-1}(t)-x_n(t)\\right] ^m}", "c9091639c95a002a05a2884593d4afc9": "|\\psi|", "c90916974237c6bba1eee808e1108b72": "\\xi_1,\\xi_2,\\ldots,\\xi_m", "c909470da36a0bc681e5023dce523722": "3^{3^3}", "c9096e1489749bda175b300c2770537c": "T(n) = 2 T\\left(\\frac{n}{2}\\right) + O(n)", "c90992aa9fbfef9c0adfa4a54a7ea7a0": " A_p(g) = A(g,p) ", "c909a22977aa4aeb659a006e87e4b685": "\\neg Q \\or \\neg S", "c909a2ea4b1ed10b4748f33c5ddbd3be": "t=\\left \\lfloor \\frac{d-1}{2} \\right \\rfloor.", "c909d5ec030a5ef5b164c5512fd5438b": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & -1 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 0 & 2 \n\\end{smallmatrix}\\right ]", "c909d8d6516c56c73880e1c551c73836": " M_{1} )", "c90a2ab278f31cb6e5314ae993d2f767": "\\alpha = (d(f(f(\\dots f(z))))/dz)_{z=z^*}.", "c90aa6d2a4a96a90965898941f4c8e05": "O(|E|\\log^3|E|\\log\\log|E|)", "c90ab355c042f7fcee1da7a004231935": "\n\\begin{matrix}\nx^2\\\\\n\\qquad\\qquad\\quad x-3\\overline{) x^3 - 2x^2 + 0x - 4}\n\\end{matrix}\n", "c90b409edcad95755d48ff11576f0f9e": " D(t) = 1 + D_1 t + D_2 t^2 + \\cdots\\, ", "c90b458e66fa264f4a2d09e36d254ee6": "\\,d_x", "c90b51d2d5c3ea679cc5432d8acaa91f": "h^{p^4}", "c90bcc87d091f131810c8bcc0bb9720d": "8x-5", "c90bf0a1c7edf22f88942d783583f71e": "v = \\sqrt {T \\over \\rho}", "c90bf45c512990198ba7b6ab8f360c09": "f_{\\#}\\left(\\sigma_X\\right) = f\\sigma_X", "c90c0b4ebee0d5f4a20e5e5d979fbc03": "\\{x_n\\}_{n\\ge1}", "c90c18137bf009f6270c34271fbda11e": "= x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + Z \\,", "c90c2ff6a7f568aa44035f25393e7d32": "A=CR", "c90c454047b22666daed81ec909ac80f": "(-n)!! \\times n!! = (-1)^{(n-1)/2} \\times n.", "c90c7df35fa13fb004e4d006c0b8c81c": "Q_1 = 0", "c90cdd87d5e214dbbaaf1e94c6e9f642": "PA+PB+PC\\geq 2(PU+PV+PW),\\,", "c90cebcc9b8459cafb49e4e7d5147826": "\\begin{matrix}\\mathrm{Cabtaxi}(8)&=&137513849003496&=&22944^3 + 50058^3 \\\\&&&=&36547^3 + 44597^3 \\\\&&&=&36984^3 + 44298^3 \\\\&&&=&52164^3 - 16422^3 \\\\&&&=&53130^3 - 23184^3 \\\\&&&=&57316^3 - 37030^3 \\\\&&&=&97290^3 - 92184^3 \\\\&&&=&218316^3 - 217350^3\\end{matrix}", "c90d69d14dc937f77a913b14e7723245": "\\operatorname{E}(X)=\\operatorname{E}(Y)=0", "c90d6d1e47f9e1a08d1e7f788797561a": "~{\\left\\langle\\hat a\\right\\rangle_{\\rm initial}}~", "c90d7c58aabb3db27dfd5b8f181a1f5c": "\\vec{S}_1", "c90d84f416005da9af3de9b8bd97463b": "n^3 + (n + 1)^3 = (2n+1)(n^2+n+1).", "c90d890232add521d7d15b6ed5413b1f": "\\rm \\ C_6F_5XeF + Cd(2,4,6-C_6H_2F_3)_2 \\rightarrow 2,4,6-C_6H_2F_3XeC_6F_5 + CdF_2\\downarrow ", "c90d9e44c5a80832149f9490cc74a67c": "Y_1 \\cap Y_2 = \\varnothing, G_1 \\in S\\left(Y_1\\right)", "c90db7063c875ee1d595a452fb06e56f": " V_0 = V_\\mathrm{rms} \\sqrt{2} \\,\\!", "c90dbdfa9be1485610f795db12fcbf0b": " \\{ \\cdot,\\cdot \\}_{M} ", "c90dccab88b0c9bd831a2cda6f5ef267": "~n_2\\sigma_{\\rm e}(\\omega) v(\\omega)D(\\omega)~", "c90e1f4bed1640dad361a2b3565d3943": "\\begin{Bmatrix} 5 \\\\ 5/2 \\end{Bmatrix}", "c90f0512558f25e3b50dbf81010896ff": "T = \\left(\\frac{\\partial H}{\\partial S}\\right)_P, \\quad\n V = \\left(\\frac{\\partial H}{\\partial P}\\right)_S", "c90f963d21bd3907701e9087b7cc0b78": "-y \\not\\in \\operatorname{recc}(B)", "c90fb39cad5aa3161dfcff335dbe7c07": "\\mathcal{H}=\\mathcal{H}_{0}+\\Delta \\mathcal{H}", "c90fe3304f672d09e480938eeb38a8fe": "y: \\mathbb{R}^n \\to \\mathbb{R}", "c91026e2ca3d27df817f5b3ec6be833c": "\\mathcal{O}_K\\subset \\mathcal{O}_L", "c91039db6def410930c9f42140a01d3c": "0.83333", "c910b2432c87b1e73ca78eb11e888de8": " a_1 > a_2 < a_3 > a_4 < \\cdots \\, ", "c910fa67a6f93de3c8baf4a09c1203d6": "A(X) = k[x,y,z]/(xy - z) \\cong k[x,y]", "c91135e0c4446cee4030cebb48d3ad38": " \\mathrm{Var}\\left(\\sum_{i=1}^{n}x_{i}\\right)=E\\left( \\left\\vert \\sum_{i=1}^{n}x_{i}\\right\\vert ^{2}\\right) =\\sum_{i=1}^{n}\\sum_{j=1}^{n}E\\left( x_{i}\\overline{x}_{j}\\right) =\\sum_{i=1}^{n}E\\left( \\left\\vert x_{i}\\right\\vert ^{2}\\right) =\\sum_{i=1}^{n}\\mathrm{Var}\\left(x_{i}\\right). ", "c911daafe12fd8c0ab8cb6761103ebde": "\\frac{\\pi}{4} = \\arctan 1", "c911ef9ea0955b0fe939ea630170fe1a": "E_6\\,", "c913417524296bbb3c6a29673730b17d": "r(m,n)=(m,m)", "c913e31b273b4101ce29c5731f33d8ef": " -F'(s) \\ ", "c913fd51f885d7a13b105f545702e120": "t=-t_1=t_2", "c914139809fb1d05572384c87a7235c8": "\\ell \\in \\{ 0,1,2,\\ldots \\}", "c914b48c3c54f98a81db11810659acca": "|i-j|\\le d(x,y)", "c914f173f0212689e23a73d0cc3b4e98": "\\Omega /G", "c914f21fad3cf49297b59ee95e55e74b": "\\sin\\, \\pi x", "c91500e5e33613f2d5d795e89d2c6da2": " A \\in \\mathcal{A}", "c9152464683a6006de67cbfd1ca26dcc": " I^k = I\\, ", "c91537458c78a02d90894546377ada0a": "\\frac{1}{7} + \\frac{1}{14} + \\frac{1}{28} = \\frac{1}{4}", "c9153e143e714aa0b339ba71f55eecc7": " u(t,x) = \\tfrac{1}{2} \\left[f(x-ct) + f(x+ct)\\right] + \\frac{1}{2c}\\int_{x-ct}^{x+ct} g(y)\\, dy.", "c915be8717c03159e484554d8d59381b": "GL_n \\to \\pm 1", "c915c339edef0cc8edf715afd14a3bd5": "\\Psi(\\mathbf{r}_{1},\\mathbf{r}_{2})=\\chi(\\mathbf{R})\\Phi(\\mathbf{u})", "c9160671b9eaa45dc4fb34092e4468d7": " C \\subset \\mathbb{C}^n. \\quad ", "c9163eaa0e06f78a30d62d4860c614a3": "\\chi(a)=\\left(\\tfrac an\\right)", "c91701d57118e49a3e062fde55528c51": "h[n] = T \\sum_{k=1}^N{A_ke^{s_knT}u[n]}\\,", "c917210290a4f242f70b38b51b05705f": "\\mathcal{}H_*(LM)", "c9176c82b75e67bc81dc46bd2525ab2e": "x=\\mu", "c9176cea4a633bc5151e4e3d4c682ba0": "x^2 + 2 x^3 + 3 x^4 + 4 x^5 + 5 x^6 + 6 x^7 + 5 x^8 + 4 x^9 + 3 x^{10} + 2 x^{11} +x^{12}", "c9178d2cadfb3357e7e8f549bf33f2ef": " M(p) = \\alpha^{-p}\\int^\\infty_0 (1+y)^{-\\gamma}y^{p-1}dy ", "c9179a5ee6935d801c2e139be48d79cf": "4K^2", "c91834cb196dd6314f4386f3ba0e3fbf": " \\textstyle k_0 ", "c918b10a09654ec431f032445951c369": "C_{np} = \\partial^2 Q/\\partial n \\partial T \\,\\!", "c918b259edda2f5b0274ab57e7696400": " \\pi \\left(r\\frac{(h-x)}{h}\\right)^2 = \\pi r^2\\frac{(h-x)^2}{h^2}. ", "c9191e22381dbdf53deff2e82a0f6b26": "\n\\partial_{t} \\phi = -\\frac{1}{\\tau}\n\\left(\\frac{\\delta F}{\\delta \\phi} \\right) + {\\eta}({\\mathbf\nr},t)\n", "c91931f9bf7ec2b59247eb93e075b13c": "{1\\over |G|}\\sum_{g\\in G}\\chi(g^2).", "c91932492fc1fb70f4bcd3ab7965a5b0": "\\frac {k}{dg}", "c9193768651dcf797a151ec6e74a4386": "s_2 ", "c919464eaf35de771555984088378b9e": "\\alpha + \\frac{n}{2},\\, \\beta + \\frac{\\sum_{i=1}^n (\\ln x_i-\\mu)^2}{2}\\!", "c919d26a90103c300db0935ddd550c4f": "\\phi(x)\\,", "c91a0d6676a767263f9254d46110f0a3": "\\frac{\\partial nb}{\\partial t} = \\nabla \\cdot (k b \\nabla h) + N. ", "c91a124dde8ea7eb8bfeb95a359bfb91": " \\left[ \\begin{matrix} 1+\\alpha^2/2 & 0 & \\alpha & -\\alpha^2/2 \\\\\n 0 & 1 & 0 & 0 \\\\\n \\alpha & 0 & 1 & -\\alpha \\\\\n \\alpha^2/2 & 0 & \\alpha & 1-\\alpha^2/2 \\end{matrix} \\right] ", "c91acff9a00dd03b820ccd9c33154c93": "i = 1, 2, 3, \\ldots, n", "c91aff9638640924c43e7a0dcfa8cb11": " \\nabla = \\sum_{i=1}^n \\hat e^i {\\partial \\over \\partial x_i}", "c91b0f4f442a49bb37458b9e3627e247": " dU = TdS - PdV + \\sum \\mu _i dN _i", "c91b5789d857998521e7a344f24d712a": "\\int\\operatorname{arcoth}(a\\,x)\\,dx=\n x\\,\\operatorname{arcoth}(a\\,x)+\n \\frac{\\ln\\left(a^2\\,x^2-1\\right)}{2\\,a}+C", "c91ba35851c4f346d0b4ce32d66c2812": "B_\\lambda(T) = \\frac{2 c k T}{\\lambda^4},", "c91bca7f679e9b3d356755e1896f4c86": "\n\\begin{align}\nF^{(0)}(s) & := f(s), \\\\\nF^{(1)}(s) & := \\int^s_a F^{(0)}(u)du=\\int^s_a f(u)du, \\\\\nF^{(2)}(s) & := \\int^s_a F^{(1)}(u)du=\\int^s_a \\left( \\int^t_a f(u)du \\right ) \\, dt, \\\\\n& \\ \\ \\vdots \\\\\nF^{(n)}(s) & := \\int^s_a F^{(n-1)}(u) \\, du, \\\\\n& {}\\ \\ \\vdots\n\\end{align}\n", "c91bd3864b779e6db6283cef16fe81df": "RT\\ln X_2 = - ( \\Delta G^\\circ_{fus})\\,", "c91bdff30bfc831ff645b8442bed761a": "y^2", "c91c6728e8271e90b44a86f5252732c6": "a_i=0\\quad", "c91c8fca8d1bb3f10ff89693085c5ad1": "dU = TdS - PdV + \\mu dN", "c91ce630c53b6fac3786263b53abe179": "S_p^2 = \\frac{1}{N-k} \\sum_i (n_i-1)S_i^2", "c91ce9b17c7f5639003967afa9b9c09a": "\\begin{align}\n\\alpha(x) &= \\min_{S \\in \\mathcal F, x \\not\\in S}w(S) - \\min_{S\\in\\mathcal F, x\\in S}w(S\\setminus\\{x\\}) \\\\\n &= w(B) - (w(A)-w(x)) \\\\\n &= w(x),\n\\end{align}", "c91cfafa8d338a1924792ef606c77852": "\n \\Gamma_0 = -\\frac{1}{2d}\\frac{\\Pi'''(a)a^2 + (d-1)\\left[\\Pi''(a)a - \\Pi'(a)\\right]}{\\Pi''(a)a + (d-1)\\Pi'(a)},\n", "c91cfec0d5cda11db66531392e9731f8": "f''(x)=12x^2", "c91d04a9d4d61acdf6f257e74c9e6cf5": "\\mathbf{P} \\in \\mathbb{R}^{2k \\times d}", "c91d23db45bcb3b03671029cc010bd31": "\\lim_{n\\to\\infty} a_n \\leq \\lim_{n\\to\\infty} b_n ", "c91d422bbf4850436c826612eb5ebd83": "\\Pi_k^{\\rm P}", "c91d446515453388fe2a673c01e51bd1": "O(\\log(\\ell))", "c91d7d390024b8c96d5cc187b1b139d9": "K\\sin(x(n))", "c91db3524b1c6a3d65749264d33e6ca4": "E' \\subseteq V'\\times V'", "c91df22fa0bf81dfaf683da8c0eb973f": "\\langle v, w \\rangle = \\eta_{\\mu \\nu} v^\\mu w^\\nu = - v^0 w^0 + v^1 w^1 + v^2 w^2 + v^3 w^3 ", "c91e0bf9407d0ed6e082549800051e63": "0\\,\\!", "c91e2452951f754ea83b62a83bd2d0a3": " u(r,t) = \\sum^N_{n=0}U_n e^{in\\omega t} \\, .", "c91e4a0ea81fd74d6ba0751cfb74d55d": "Z_{\\mathrm {in}} = Z_0 \\frac {Z_\\mathrm L + Z_0\\tanh(\\gamma l)}{Z_0 + Z_\\mathrm L\\tanh(\\gamma l)}", "c91e73df739ad31eb7535dcb1bcfabcc": "\\langle \\psi |\\phi\\rangle = 0", "c91e7d4d02c718ccde3e966ea3b7341f": "x=u+v,\\quad y=u-v\\,,", "c91e8970269769d597bec85a46138a37": "y = tx = \\frac{2at^3}{1+t^2}", "c91e8bca38e064ff8e7d127880aced65": " \\int x^s \\phi (x)\\, dx,", "c91e96c03f8463ff372d53d1ae7d9876": "0 < \\tau < \\beta", "c91eae28af544700e1af5e8f33d488f6": "\\#\\operatorname{Aut}_X(X_i) = \\operatorname{deg}(X_i/X)", "c91edc952bddfd3d0c92ab8ae53593b9": "\\alpha \\wedge d\\alpha = 0", "c91ee857705e7da9781109a3565116ed": "\\lambda\\to+\\infty", "c91f6db11f4c3275710f889425b7fe53": "Q = 2 n! \\cdot \\mu^n + 1", "c91f9045f890c963689de8f81b84ae8c": "\\left[ \\hat{A}, \\hat{B} \\right] \\equiv \\hat{A} \\hat{B} - \\hat{B} \\hat{A}", "c91fd69e9692941e1f371f163f893b0c": "P(N\\mid n) = \\frac{n}{N^2}", "c920a2a4b91551a64501a7fcc0f38107": "U\\subseteq \\mathbb{R}^{n}", "c920ad8db4b1d439ecfdfd2db98aa9a9": "\\tfrac{4}{4}", "c920afd1a369ffc99536ff489d77925e": "\\ln{x \\over {x-1}} = \\sum_{n=1}^\\infty {1 \\over {n x^n}} = {1 \\over x}+ {1 \\over {2x^2}} + {1 \\over {3x^3}} + \\cdots \\,.", "c920fc4d5c6089bdd8de183afa5d9dbf": "i = 1, 2, \\dots, n", "c921a194445b7d2738ea9d5f5adc48c6": " \\, _2F_1(a,b;c;z)", "c9222c387f569bb061f32af2387c04e3": "{}_1F_1(-n;b;z)", "c922444a5a986d37b5129f7c8a04ce42": "\\pi\\approx", "c922520d0555b1799d4d1019b7918198": "d(x,x_{n_k})<\\frac{1}{k}", "c92267e48c4e24f42a232c0c17ed828b": " 0 \\rightarrow A \\rightarrow B \\rightarrow C \\rightarrow 0 ", "c92358ef19f79f1f2c60cf90d8d6ea3b": "n^\\sigma e^{(\\log n)it}.\\,", "c923ed016c1bca421847801c5d5ab4e0": "(x+y)^z + (x+z)^y + (y+z)^x > 2.\\,", "c9244d3bd4389e8900768bf739d141a1": "\\scriptstyle {\\bull}", "c924c478f04d304a4e2fd0b3dffe752a": "\\alpha \\in H", "c924f37a835af016e2c56d63bb810b02": " \\vdash \\ \\ A \\rightarrow \\left( B \\rightarrow A \\right) ", "c9255cf51eebf1eb7d8099008a3629af": "\\operatorname{E}(X \\mid Y=\\ \\cdot)\\circ Y", "c9256a28e0506a789dd0a26cfc9ad7b6": " e^{x}>1", "c92580073e78834542ba19b2b3526846": "\\mathfrak s", "c925ad89b9a092137c0fa4b4c82adda0": "\\mathbf{b} = D\\mathbf{a}", "c9266e0c336503da49d94bd26b0543b1": "(\\phi_n) \\in \\Delta", "c92727ac7f2c7f1f7202e0575b1f7f11": " |\\psi(x,t)|^2", "c927334f500b042f00cb201a4dd417c8": "s_{n/2}", "c927683884f07a5b354f7988c41e6818": "q(\\mu, \\tau)=p(\\mu,\\tau\\mid x_1, \\ldots, x_N)", "c9278723fff1d34b566b6e1d87e1e1bc": " \\gamma^0\\gamma^1\\gamma^2\\gamma^3 = \\gamma^{[0}\\gamma^1\\gamma^2\\gamma^{3]} = \\frac{1}{4!} \\delta^{0123}_{\\mu\\nu\\varrho\\sigma}\\gamma^\\mu\\gamma^\\nu\\gamma^\\varrho\\gamma^\\sigma", "c9281d8d8532a5ba15a81a34fb1d17d0": "g = -\\sigma^0 \\otimes \\sigma^0 + \\sigma^1 \\otimes \\sigma^1 + \\sigma^2 \\otimes \\sigma^2 + \\sigma^3 \\otimes \\sigma^3", "c9282029870e22b660ecc75f113a7cfd": "\\bar c = \\mbox{Redc}(\\bar a \\times \\bar b).", "c9284f22eeda190266a8ef946b06f0ad": "C = \\frac{n}{\\left| \\alpha \\right|} \\cdot \\frac{1}{n-1} \\cdot \\exp\\left(- \\frac {\\left| \\alpha \\right|^2}{2}\\right)", "c928744422608287fdebe3f0e66525f2": "g_{\\alpha \\beta} = e^I_\\alpha e^J_\\beta \\eta_{IJ}.", "c928cccf43dbca2bea16ef22a8842c4a": "f(b)=d", "c928d267f44952b7b631f8673e896c59": "RR_\\mathrm{total}=TAT-SAT \\,", "c92947a4af1cf803c06a76c515346015": "P_u/P_n", "c9297332420c4ffd0e42a2ee49dffff2": "D = \\sum_{i = 1}^{k}{[P_i]} - k [O]", "c9299952b8fa270230f148c953142fec": "\\operatorname{E}[\\,\\nabla_{\\!\\theta} \\ln f(x_t,\\theta) \\,]=0", "c929a4b8b5d75a381afbc51c9ae903c2": "Y= 0.299R' + 0.587G' + 0.114B'", "c929d90035f286b6e62f4111e7debf8a": "\\nabla^2 |\\mathbf{B}|^2 \\geq 0.", "c92a1589726396c323773c08120b8a10": "x^4 \\equiv \\alpha \\pmod{\\pi}", "c92a1d39e5be8ada88ba4824bef3f047": "\\left\\lfloor\\frac{(m+1)26}{10}\\right\\rfloor \\mod 7,", "c92a2e250d4b894c504172790f580089": "\\frac{\\nu}{\\nu-2} \\boldsymbol\\Sigma", "c92a2e438d4e79a4c1f04d71507c6cf0": " C^p ", "c92a340bb7c71e610df7fd05fa632525": "(j^{k+1}\\sigma)^{*}\\theta = 0.\\,", "c92a436286794cb470bdbab1f5a328ce": "0 \\rightarrow A \\stackrel{q}{\\longrightarrow} B \\stackrel{r}{\\longrightarrow} C \\rightarrow 0 \\,", "c92a4d7bdbfa6179864216aa0b1cb7f0": " J S J^T = J A^T A J^T = J A^T J^T J A J^T = B^T B ", "c92a9438a560118d7b44d92fd4b151d8": "\\Theta(n^3 \\cdot |G|)", "c92aaa06fa0c7af2b5b947b53075a9bb": "(n/l) \\lg n = n / \\lg n", "c92abbb985fa301cf6588966d233c49a": "f(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}}.", "c92acb15b4a42133e2cbe27f6060b2b6": "v=k/n", "c92ad44d8211383e24a49d340716bc50": "D_s = \\frac{1}{\\sum_i \\frac{f_i}{d_i}}", "c92b29f46c51e82950836cbaa31edd74": " \\omega_R = \\frac {\\gamma}{\\hbar} H_R ", "c92b423faafe684f2cf45a345e6d778a": "a/b=\\sqrt{2}", "c92b926a2138df87aaaa85b9a31dd7aa": "\\gamma = \\sqrt{\\alpha^2 - \\beta^2} > 0 ", "c92bea48d0219eed4c71716f73b54cf1": "p_{n+1}^2-2p_{n+1}p+p^2\\approx p_{n+2}p_n-(p_n+p_{n+2})p+p^2", "c92bf44bd04988bf46e52120e867b611": "\n\\mathcal{L}(r) = \\frac{\\sum_{i=1}^{r^2} S_i^2 Q(S_i,r)}{ \\left( \\sum_{i=1}^{r^2} S_i Q(S_i,r) \\right)^2 }.", "c92bf591f51abe961c594aed76add240": "\\begin{align}\nx=(Ex'-By'+BF-EC)/(AE-DB)\\\\\ny=(-Dx'+Ay'+DC-AF)/(AE-DB)\n\\end{align}", "c92bfcb302283f0d8dd4513b2d0696fc": "M \\to S^{-1}M", "c92c672ca9f086061a8b12ee93ae48aa": "\\textstyle p(x) = x^p - 1", "c92c8f31f83bc9dc466c3e11f6439212": " E \\cap F = \\emptyset ", "c92d034048232ec43b05c4363fca88bd": "G_i(R) = K_i(R),", "c92da943df73dc077dbee5514376346a": "{2n \\choose n}", "c92dd552cbd3b14515f9099faf8ffaca": "0 \\le \\theta \\le 2\\pi", "c92ddeff049c7f0bc8f4e413c591b0d6": " s = b A(\\sigma - \\Delta \\sigma) \\,", "c92e1e2556f73ffa8ca2fcbbc250614a": "\\Rightarrow_{h} SB \\Rightarrow_{h} SS \\Rightarrow_{f} AAS \\Rightarrow_{f} AAAA", "c92e7bc9ff8932d05b166f0981183df6": "x_i=x_{(t)},", "c92ed50ffe77cdf1926e867339c6daf9": "E_{k}(r_{k})", "c92f4425fc0e78d345679cf86b60ebcc": " P_\\mathrm{POVM}=1-|\\lang\\phi|\\psi\\rang|.", "c92f6ab19c5ec3568fcf29f755d5c935": "e[n] = x[n] - \\hat{x}[n]", "c92fdd401c12cbe7a7d238b654ed1820": "V = \\frac{h(A_1 + 4A_2 + A_3)}{6}", "c93017792d851596d105effa1ddc59da": "a_{ij} = 0", "c9304e5568a74dd7589f0f5ec10fdc56": " \\operatorname{build-param-lists}[q\\ q\\ x, D, V, K_5] ", "c9305e71a78238d7e95bb902f7effe98": "c_{ijk} = \\begin{cases} \n\\dbinom{k}{\\frac{i-j+k}{2}}\\dbinom{n-k}{\\frac{i+j-k}{2}}, & \\text{if } i+j-k \\text{ is even,} \\\\ \n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;0\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;,\\;\\; & \\text{if } i+j-k \\text{ is odd.} \\end{cases} ", "c930724674dfe03bc1d4d2791c4df6ce": "(\\mathcal A_n)_{n\\in\\N}", "c930b8ca49ca7befe884e7200b245e24": "\\partial_t-k\\partial^2_x,", "c930dc1570734938ea91aedefee95692": "\\frac{-1 \\pm \\sqrt v}{2} , ", "c930de2b2bc9abaaa9d0218ff3c2c2a3": "\\varepsilon_{i_1 i_2 \\cdots i_n} = (-1)^p \\varepsilon_{1 2 \\cdots n} ,", "c9310c47dcdb83fa93ba940df0f660fb": "\n\\cfrac{A \\wedge \\left ( B \\wedge C \\right ) \\ true}{\\cfrac{B \\wedge C \\ true}{B \\ true} \\wedge_{ E_1}} \\wedge_{ E_2}\n", "c9319088a2c124ea875ea89e7ad68539": "\\vec{e}_1, \\, \\vec{e}_2, \\, \\vec{e}_3", "c931bd59450dc8bdd7a045a9f0b4eeb5": "\\frac{d^2C}{d(x/L)^2} = \\left (\\frac{2k_1L^2}{rD_c} \\right) C", "c932344bbbc696540e85aaa2c89a7317": "\\alpha = \\frac{E_a}{RT} ", "c9326cd9a75da4fa4f792d654559b80e": "{k \\choose r_1, r_2, \\ldots r_n}", "c93272e6ed8ee35237774e820718b652": "r(t)=r(nT)=s(\\vec r, nT)", "c9328f8c15436ce2c9e413fe8fa8fc3b": "s+t=71, st=880.", "c932d2c0a52b376f45a69a46c006f3ad": "\n\\mathbf{v} = c \\left( \\mathbf{v}_\\textrm{D} +\nc_\\tau \\cdot \\mathbf{v}_\\tau \\right) = \nc \\left( \\left[ \\mathbf{D} \\right] + c_\\tau \\langle \\mathbf{D} \\rangle \\right)\n\\mathbf{v}_\\textrm{op}\n", "c932e3c460f35a8fba5f7c81137dc272": "\\frac{\\alpha\\,x_\\mathrm{m}^\\alpha}{x^{\\alpha+1}}\\text{ for }x\\ge x_m", "c933069f33e2078e23138bc58967b803": "\\aleph \\geq \\aleph_1", "c9333c0bdd67a0d4e58ce6660834a83d": "\\left.\\Theta\\right.", "c9334e5292986f3a604733dabe18bbc3": "x^{(0)}", "c933588a3674d2a08a8262edd7c99096": " \\mathbf{C}_{0} ", "c9337c55d290bd6a6acafcdbec745bcd": "H_{2}=H_{1}=H_{f}+x_{2}H_{fg} \\,", "c933aef7c807f40ce12c4aeeb4425d9c": "\\max_p", "c933db065d57333e2ae4bd8a39f3f8c4": "c_f(u,v) > 0", "c93438ddbbfcee221673b2eac4f3cd24": "\\pi(X)", "c934843cb97901d32065ae02152f9224": "\\overline{F} \\colon A/B \\to C", "c934a2300de7cd7855a06cef13228c2e": " (G,k) \\in \\Pi ", "c9354520a72c2ffe7ba6a792dcfbf23a": "Q = \\{P_4\\}", "c93547d84b499fcfc8b1f7a97bead3a4": "(\\mathbb R,+,\\cdot,<)", "c93561a0b3f7dc22893e57736c94b046": "(f \\, x \\, y) = ((f \\, x) \\, y)", "c935a0d539bee2d06cc46cbb0e47b93e": "F_i^{-1}", "c935b42b87f7d079fb5a0c7631a8000d": " x_{r,s}\\geq 0 ", "c936089205e4a923d9d5d040a3016ec7": "e^{t_k f_k}\\circ \\cdots\\circ e^{t_1 f_1}", "c936c4afc67f2cd36ae6c829f6d2941f": "X \\sim \\mbox{Gamma}(k, \\theta)\\,", "c936de60aba5fd5d9e24c9c396d7859f": "AUC_{0 - \\infty}", "c93756efd3e94b9d129058d3bdb2ba93": "\\lim_{n\\to\\infty}\\frac 1n\\sum_{k=1}^{n}\\log(a_k)=\\int_I f \\, d\\mu = \\sum_{r=1}^\\infty\\log(r)\\frac{\\log\\bigl(1+\\frac{1}{r(r+2)}\\bigr)}{\\log 2}", "c937a577a65a8e4a5f25ec07ca9678f0": "\nG \\equiv \\mathbf{q} \\cdot \\mathbf{p} - \\mathbf{Q} \\cdot \\mathbf{P} + G_{4}(\\mathbf{p}, \\mathbf{P}, t)\n", "c937b6a99ba23969677ee9709426580d": " s(f) = f - \\langle f, e_s \\rangle e_s^\\vee, \\, ", "c937dabc1b0fe5fc7d57c9b004e1e443": "|aq-bp|", "c937f1c6e192c982053d8d85731d88a0": "\\scriptstyle \\frac{1}{2} \\sqrt{K^2-3R^2}", "c9383b32f19760a2e3395a7eac4be16e": " 4\\pi r^2", "c938650967bee8960476e129f8e2a2c1": "x \\in L \\iff \\exists y_1,\\dots,y_m \\in \\{0,1\\}^m\\, \\forall z \\in \\{0,1\\}^m \\bigvee_{i=1}^mA(x,y_i \\oplus z)=1.", "c9386626e50b2902f41800743e5bbd77": "{(\\sigma_i\\sigma_{i+1}})^3=1\\ ", "c93881fc4f4a1f23c71c62e7b5a9928b": "A^\\prime=\\bigcup_{n\\in\\mathbb{N}}A_n", "c9388a98c8d53ef076cf7a67183282d4": " Z = \\sum \\exp(-E/k_BT), ", "c938b24064972763f852bd76b9917955": "h \\mathbf{a}_1 + k \\mathbf{a}_2 + \\ell \\mathbf{a}_3 .", "c938bed622a82e7c77b9546356d80081": "\nP \\mathbf{H}(\\mathbf{x},t) P^-1 = \\mathbf{H}(\\mathbf{-x},t), ", "c938d5c8b6a6c69f6586281e332cacd0": "r^{2} - (1+b )cr + b c = 0", "c93938a1d2407840a030b33d9e13e3a8": "\\left(\\sqrt{\\frac{2}{5}},\\ \\frac{-2}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0 \\right)", "c93982447bbba70439df4e04eeb326c2": "\\omega = v_sk", "c9399340681a56fb632bc4ce106f50da": "\\textstyle s", "c939b1cf33c54531924d7b80bfd5a347": " \\rho = 0.011010100010100010100010000\\ldots_2 ", "c939dbb7cebb45c80bfea7e5dfa943e1": "\\frac{h}{3}", "c939eb37b16fc2c36a61078c71213fa9": "OA = OD = 1\\,", "c93a1cb0c42c1b2dd0a8d0298fb52abb": "n=0, 1, 2, \\dots", "c93a7b4b5e9ebf0b33da218b0fe2aea0": "\\frac12 \\; \\operatorname{sech}\\!\\left(\\frac{\\pi}{2}\\,x\\right)\\!", "c93ae27543313baaf57e2f220de97d29": "L_n(\\xi)=R_n(\\xi,\\xi)", "c93ba317acc0d23ff50725b2341ee292": " \\hat{E} = i \\hbar \\frac{\\partial }{\\partial t} \\,\\!", "c93ba4dca0efd3ea624405724598a560": "w\\,R\\,u\\land w\\,R\\,v\\Rightarrow\\exists x\\,(u\\,R\\,x\\land v\\,R\\,x)", "c93bf0e377d1f1b68204dc3b2714c2ee": "\\mathbf{X}\\boldsymbol{\\beta}=\\ln{(\\mu)}\\,\\!", "c93c3624b22ab1ef1b39b0009261e7ca": "C_{XY} = C_X A^T .", "c93d052d2d3a714fe100221d7a1eb9ed": "Y^a = (0,1)", "c93d25b765325370dd07f68cab24881c": "\n\\mathbf{B}^\\mathrm{ext} \\mathbf{M}^{-1} (\\mathbf{B}^\\mathrm{ext})^\\mathrm{T}\n= \\operatorname{diag}(N_1,\\ldots, N_6) \\equiv\\mathbf{N},\n", "c93d367b8501e61d81f8663754197cfd": "\\frac{t:\\!\\!-~~\\alpha ~\\vdash~ \\beta\\qquad u:\\!\\!-~~\\alpha ~\\vdash~ \\gamma}{(t,u):\\!\\!-~~ \\alpha ~\\vdash~ \\beta \\times \\gamma}", "c93d4cd4f081ef3ccb013fbd94016c48": "\\mathfrak M(K,\\rho) ", "c93d77de11b654cd86d88bb85ab6d3f6": " \\biggl(\\sum_{j\\in \\N} a_j \\tau^j\\biggr)\\biggl(\\sum_{k=0}^N b_k t^k\\biggr) = \\sum_{k=0}^N a_k b_k. ", "c93d9ba9efb29b8a9db71fcf56747da4": " |x_0-c|< \\delta ", "c93d9e74685af51129d44ab36d6cacd6": "\ns^2_N = \\frac{1}{N} \\sum_{i=1}^N (x_i - \\overline{x})^2,\n", "c93dab9b830cc820256bf7bf886fc967": "{C}_{2+}", "c93deb4d19c680373ffd3e9bb88b5b35": "[a, \\, b]", "c93e3d753c2a52584b9b9efea56ca39a": "[T_A^2]=T_B^2", "c93e6c72c75e64da819ec1074a73d1a6": "\\sin(\\theta) \\ll 1", "c93e6ca167bd9e17e911fd4ea01c2829": "\\delta S= \\delta\\int_{\\mathbf{A}}^{\\mathbf{B}} n \\, ds = \\delta\\int_{\\sigma_A}^{\\sigma_B} n \\frac{ds}{d\\sigma}\\, d\\sigma ", "c93e96febcfa4376deeeb0b1582f77f4": "h,m \\in\n\\mathcal{N}, u, \\overline{u}\\in V^+, v,v',w,w' \\!\\! \\in V^*", "c93f1ed63fa9207497c12cc786e102b6": "\\frac{\\partial f}{\\partial x} + \\frac{\\partial f}{\\partial x} = e^f", "c93f9b50058c95a458d4bbdbf002a62f": "\\|T(t)\\|\\leq M{\\rm e}^{-\\omega t},", "c93fc9c8a0731f526548296191389a53": "\nq_{yy} = a^2 \\sin^2 \\theta + b^2 \\cos^2 \\theta\\,\n", "c9400e1ab2de340e8fe40cb11b60b2ce": "dl=\\frac{da}{\\dot{a}} = \\frac{dz}{(1+z)} \\frac{a}{\\dot{a}} = \\frac{dz}{H(z)(1+z)}", "c9401011ee7a7bb60a635cc5883f4e34": "s_1, \\cdots, s_k", "c9401d47bef7e2ca12994fb5bffdd249": "\\begin{align} P(H_1|E) &= \\frac{P(E|H_1)\\,P(H_1)}{P(E|H_1)\\,P(H_1)\\;+\\;P(E|H_2)\\,P(H_2)} \\\\ \\\\ \\ & = \\frac{0.75 \\times 0.5}{0.75 \\times 0.5 + 0.5 \\times 0.5} \\\\ \\\\ \\ & = 0.6 \\end{align}\n", "c9403083ad82e3a22769b4c1377318d9": "\nY = Y(K, L).\t\t\n", "c9403857e436f46b6f6d1e560ea1a266": "e^{727.95133}", "c9407e16454ffb7b4e57bd3dd08b0704": "\n{\\mathrm{g}}{\\overleftrightarrow\\partial}_0 f = \\mathrm{g}\\partial_0 f -f\\partial_0 \\mathrm{g}.\n", "c940a029628d96b9326de39c1a3c9338": "Q = xi + yj + zk\\,", "c940a9846d43bc0cbb6ea50f5e1b1612": "\\varphi(p^k) = p^k -p^{k-1} =p^{k-1}(p-1) = p^k \\left( 1 - \\frac{1}{p} \\right).", "c940b972e7e6667ec99ab1d3fda88536": "E=\\frac{-\\mu^2}{2}", "c94110b02a041090fba55bcd02ae95c4": "(Q,P)", "c94118e805440dcc65b1cf58b43dfb12": " \\frac{1}{(1-u^2)^2} = \\frac{1/4}{1-u} + \\frac{1/4}{(1-u)^2} + \\frac{1/4}{1+u} + \\frac{1/4}{(1+u)^2}. ", "c941275a6171ed4d67db71fcd007693d": "\\mathbf{\\nabla} \\times \\mathbf{B} = \\mu_0 \\mathbf{J} ", "c94129cb7ee740cc7bf8dd9118e89a31": "\\frac{\\partial\\rho}{\\partial t}+\\nabla\\cdot(\\rho\\mathbf{u})=0\\,\\!", "c94131c6d853f55530be6123ba9c3f09": "F_{\\rm smooth}", "c941480d2f79cdb762d7a6125377e2e3": "\\sum_{i=0}^k {n \\choose i} \\leq (n+1)^k", "c9414db3448434b6526002aeb38c3085": "\\left(\\mathbf{I}+\\mathbf{uv}^\\mathrm{T}\\right)^{-1} = \\mathbf{I} - \\frac{\\mathbf{uv}^\\mathrm{T}}{1+\\mathbf{v}^\\mathrm{T}\\mathbf{u}}.", "c9415540ecd26fec457f411770c77db7": "\\hbar\\omega_k/2", "c94189e815f8ea505a6f70d2da4b7148": "\\mbox{d}r", "c941d13210b6354721d3530f25ca5606": "(x^*,y^*)", "c941edf270e4d5b5c0734a200ceb1837": "\\mathbb{R}/\\mathbb{Z}=S^1", "c94217c68f269d58c988b3acea7038b4": "2t = {X \\over R}", "c942f7ca8a133ae9eb56de674f5f7c4d": "\\operatorname{dCor}(X, Y; \\alpha=2)", "c9431ffa0e2809f2d2bcd7e9a01330d8": "*_M", "c9433cef19a70600cbcabb2949460f39": " u_1(X_1)=E(X_1)", "c943b765b903abee95bf68c8eb79413f": "\n \\text{d}\\boldsymbol{\\sigma}:\\text{d}\\boldsymbol{\\varepsilon}_p \\ge 0 \n ", "c943c893a9cad1688b296aa24c401d4e": "\\chi(\\Sigma)", "c943ce0c568bacdf95453f9ac3607ffa": "r_b^{}", "c9441509e8a20e82bdd46c966144c403": "c: V\\rightarrow \\{1, \\dots, k\\}", "c9441f7bc36c7afec1b06c7454b53277": "Y \\subseteq L", "c94497b5f7422a601ebae5b66adeff04": " x' \\,", "c944a6adf61dd1e192ba9bc36eea63d8": "W_{p} (\\mu, \\nu)^{p} = \\inf \\mathbf{E} \\big[ d( X , Y )^{p} \\big],", "c944c3d98ab773e2edc6b479aa1a78a4": "\n \\begin{array}{llrll}\n \\textrm{mono} & \\tau &= & \\alpha & \\ \\textrm{variable} \\\\\n & &\\vert & D\\ \\tau\\dots\\tau & \\ \\textrm{application} \\\\\n \\textrm{poly} & \\sigma &= & \\tau \\\\\n & &\\vert& \\forall\\ \\alpha\\ .\\ \\sigma & \\ \\textrm{quantifier}\\\\\n \\\\\n \\end{array}\n ", "c944e06160a8e422a3ea19222c958af8": "\\delta = 2r\\Omega = 2\\frac{I}{a}", "c944edd7d01242c2d8f040859fb9841d": "\\,x = K ( \\alpha ),", "c9455f6dcdd967e5df5d2940b70c6a14": "f(xy)=f(x)+f(y).", "c9457ca765d4ad7c41f36a115781c7ef": "\\scriptstyle \\leq1.5\\times10^{-28}", "c945d65dcbe649b7055b39e5b0251927": "\\bar{\\boldsymbol{B}} = J^{-2/3}\\boldsymbol{B}", "c9461b38cda208ea242a009629f2e548": "r_{t_1,t_2} ", "c9462cb5704a33ca7faad364008655bc": "t(d,n) \\leq \\mathcal{O}(d^2\\log^2{n})", "c9467f9b4f461fd9250b977c53dc2db2": "g(X)=h(X)", "c94700fe07bb772bf40d537dc4de7c28": "N=N_{0}+k_{0}A\\left(\\xi'+\\sum_{j=1}^{3}\\alpha_j\\sin\\left(2j\\xi'\\right)\\cosh\\left(2j\\eta'\\right)\\right),", "c947659065220261d2699747c03a30f7": "J(t) = \\Re(\\sigma(\\omega) E_0 e^{i\\omega t}).", "c947e2f0bd2b4528bf5373fd5ba2a822": "p-1=2m\\lambda", "c9482a7d4fdd5e6b04fbad486dfdb77c": "\\left({ \\frac{N}{2e(M+N)} }\\right)^{N-1}", "c9486c06617ee451afad138c79e34fa2": "\\beta_i(T)=1", "c9487cf871c916fa5b0275e2bc3c057c": "\\gamma\\,l_{m-1}", "c94889412f6bd5bdb27d41cb676867c9": "L_{\\mathrm{fiss}}", "c948a7adeccd46fb4135d868c20447c2": " \n\\begin{align}\nX(f) &= \n\\begin{cases}\n\\ \\ \\frac{1}{2} X_\\mathrm{a}(f), & \\mbox{for } f > 0,\\\\\n\\ \\ X_\\mathrm{a}(f) & \\mbox{for} f = 0,\\\\\n\\ \\ \\frac{1}{2} X_\\mathrm{a}(-f)^*, & \\mbox{for } f < 0\n\\end{cases} \\\\\n&= \\frac{1}{2} \\left( X_\\mathrm{a}(f) + X_\\mathrm{a}(-f)^* \\right) \\ .\n\\end{align}\n", "c948b980f599168de79f2f4557820545": " r^{-1}~\\sin\\theta \\,", "c948f3e2bfee6028bc394e0cf18fbb16": "f(x) - P_{n-1}(x) = \\frac{f^{(n)}(\\xi)}{n!} \\prod_{i=1}^n (x-x_i) ", "c949bed84e8bdc976204fdb77d451bf7": "\n\\Delta z \\approx {{\\partial z} \\over {\\partial x_1 }}\\Delta x_1 \\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_2 }}\\Delta x_2 \\,\\,\\, + \\,\\,\\,{{\\partial z} \\over {\\partial x_3 }}\\Delta x_3 \\,\\,\\, + \\,\\,\\, \\cdots \\,\\,\\,\\,\\, = \\,\\,\\,\\sum\\limits_{i\\,\\, = \\,\\,1}^p {\\,{{\\partial z} \\over {\\partial x_i }}\\Delta x_i }{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(6)}}", "c949ec3c6b0b4a6c08744315bdafc450": "\\Theta(n^3)", "c94a08b36ba853bac3023c094dc27036": "DS=100\\%", "c94a1f85bf230411fcd9eedb3e1ccc0c": "L-R", "c94a751b800f9f5084cf6d6c83d749f2": "\\Box(A\\equiv\\Box A)\\to\\Box A", "c94ab59be83d90fe0c00a1f2fbbd2013": "\nI =\n\\begin{bmatrix}\n \\frac{1}{12} m (h^2 + d^2) & 0 & 0 \\\\\n 0 & \\frac{1}{12} m (w^2 + d^2) & 0 \\\\ \n 0 & 0 & \\frac{1}{12} m (w^2 + h^2)\n\\end{bmatrix}\n", "c94afd7e5723856add922ebd0dfb3e58": "\n\\begin{bmatrix}\n\\mathbf A^T \\mathbf A & \\mathbf A^T \\mathbf B \\\\\n\\mathbf B^T \\mathbf A & \\mathbf B^T \\mathbf B\n\\end{bmatrix}^{-1}\n=\n\\begin{bmatrix}\n (\\mathbf A^T \\mathbf P_B^{\\perp} \\mathbf A)^{-1} \n & \n -(\\mathbf A^T \\mathbf P_B^{\\perp} \\mathbf A)^{-1}\n \\mathbf A^T \\mathbf B(\\mathbf B^T \\mathbf B)^{-1}\n\\\\\n -(\\mathbf B^T \\mathbf P_A^{\\perp} \\mathbf B)^{-1}\n \\mathbf B^T \\mathbf A (\\mathbf A^T \\mathbf A)^{-1}\n & (\\mathbf B^T \\mathbf P_A^{\\perp} \\mathbf B)^{-1}\n\\end{bmatrix}.\n", "c94b02cbaffcb6c56a123389a31b2609": " d_i ", "c94b052b6c520ec7d7f0439cc897c75a": "K_c = \\cfrac{\\Lambda_c^2}{(\\Lambda_0 - \\Lambda_c)\\Lambda_0} \\cdot c ", "c94b69bfa89422f0a3be31b7550dfd8b": "\\rho^-", "c94b851ea6388b4f3b3698e745516f9c": " s(x) = \\{\\emptyset \\} \\cup \\{\\{t\\} | t \\in x\\} ", "c94bdfc694d8d550182891bdaf2e0935": "\\gamma_kdt", "c94bf56221157d871a2a8321fd256997": " \\deg(\\textbf{N}(s)) = 4 = \\deg(\\textbf{D}(s)) = 4 ", "c94c2866f893e2031ec91500f7e3bd31": "\\left( \\begin{matrix}\n A_{s} & B_{s} \\\\\n C_{s} & D_{s} \\\\\n\\end{matrix} \\right)", "c94d1fd3f6d45d80dc5c480c9064af15": "O(N \\log N)", "c94d4664fcebf7bf9910a987422e52c8": " \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^\\infty f(\\tau) \\delta(t-T-\\tau) \\, d\\tau", "c94d60e9f6083808c74663434e48a5ab": "q=1+\\log_3(4/3) \\simeq 1.26", "c94d6fd6e5a9b7e29844730bb23aeb7c": " s=2 \\pi r_{m}/ Z_{b}", "c94d8faaeebd2bed5ff09b49761ece01": "\\forall i, h_i", "c94da048625835ff91cbde87ffc04df2": " m_w = \\frac{-m_a \\cdot c_p \\cdot \\Delta T_a}{L_c(T)} ", "c94db47e7dd2b07fd77a867831c56a5e": " n = \\frac{c_0}{c} = \\sqrt{\\frac{\\epsilon \\mu}{\\epsilon_0 \\mu_0}} = \\sqrt{\\epsilon_r \\mu_r} \\,", "c94dedafeb6701814c6faebd04af0a02": "N(s,\\mathbf{x})", "c94e0d416b33581f716452e371727e1a": "\\Rightarrow\\dfrac{1}{3} = -\\dfrac{2}{3}+1 = \\dots 3132_5.", "c94e3ba41f1371235b830698f2fe6c1a": "\n\\begin{align}\n\\dot x &= V \\cos (\\theta) \\\\\n\\dot y &= V \\sin (\\theta) \\\\\n\\dot \\theta &= u\n\\end{align}\n", "c94e4ad9f8a2fac7ddd97b89ec552c45": "X=3/2", "c94ea6b41d8c0c94c7455651c9f508c7": "\nf_n(x)=\\begin{cases}n&\\text{for }x\\in (0,1/n),\\\\\n0&\\text{otherwise.}\n\\end{cases}", "c94ebd9e5b5022eef82c09229b5c4bad": "q_5:=\\frac {q_2+q_1-2m_2}{2q_2}", "c94f0941ac61b967fff12553cc82e0ca": "\\phi_{a} \\left(\\mathbf{r}\\right)", "c94f2cae93d85d0fbd9d2e409a560659": "\\partial_j\\sum{p\\partial_i\\ell}=\\sum(\\partial_j p\\partial_i\\ell + p\\partial_i\\partial_j\\ell)=\\partial_j\\partial_i\\sum p=0", "c94f54a6acbbb9a72de6ccdaa81fc144": "\\textbf{K}_k = \\textbf{P}_{k\\mid k-1}\\textbf{H}_k^\\text{T}\\textbf{S}_k^{-1}", "c94f96f87c5e3bac5bf59a529f479565": "\\mathrm{CmF_3\\ +\\ 3\\ Li\\ \\longrightarrow \\ Cm\\ +\\ 3\\ LiF}", "c94f9cb05b377034b2a2d10c2ffffa93": "x,y\\in V,(x,y)\\in E", "c94fa93f2fa486a89775f342108bdb20": "\\gamma^2 n", "c94fd9923b883d51783ee056d893496d": "u''+p(x)u'+q(x)u=f(x)\\,", "c94fddc9b69742a6c7d1a0de850fb3dd": "A(m,1)=a", "c94ff7f5bf522df328451e82042c725f": "s * (t-1) * \\frac {e}{100}", "c9501b5a0953d86bd52ab759f54d2ecc": "x_{k,N}[n] \\ \\stackrel{\\mathrm{def}}{=} \\ \\sum_{k=-\\infty}^{\\infty} x_k[n - kN],", "c95053bcacaf513a44afe7c5bba075d2": "(E_n)^{\\infty}_{n = 1}", "c9508b2321a8177c08e23a75af169539": "2^{\\sqrt{2}}=2.6651441426902251886502972498731\\ldots", "c9509bc3b268be8eeb0ea7216c13f61a": "\n Q^\\dagger(\\mathbf{p})|n_\\mathbf{p} \\rangle \\;\n= \\sqrt{ (n_\\mathbf{p} +1)\n\\left( 1- {n_\\mathbf{p} \\over \\Omega} \\right) }\n|n_\\mathbf{p} +1\\rangle, ", "c950ad40a5e61a77e5c620942c8a68bf": "\\vec{F}(\\vec{q}) = \\vec{T} = a+bz\\!", "c950d93277d86c4efba04bcc1ab2f0b5": "k = \\frac{n\\pi}{L}\\qquad\\qquad n=1,2,3,\\ldots.", "c950efa192e9cf95bbd116943e215a72": " \\left \\|\\mu- \\nu \\right \\|_{TV} = \\sup_f \\left \\{ \\int_X fd\\mu - \\int_X fd\\nu \\right \\}.", "c9512565ef6194ca664dc41ec0de7a53": "B1", "c9512aaeae7ac52719b6aa90ef71e4a1": "X^i\\partial_i\\log p", "c95146b9d90ebf8d8fa98fbef727dca4": "n - k", "c951526ac44cc8e9ee3a7c8ce7d119dd": "2^n\\,", "c951c1a985ea901be0dc78f4c1b529a7": "\\mathbf{e_y}", "c951cc779aa26032a4fff2ed58144ea0": "3 \\times 5", "c951e071f650183382110606a5169d98": "\n\\hat O D\\left ( x - y \\right ) = \\delta^4 \\left ( x - y \\right )\n", "c951eba006eaa9d69dab32323f9573bf": "Q_n = 1 + \\binom{n}{2} + \\frac{1}{1 \\times 2}\\binom{n}{2}\\binom{n-2}{2} + \\frac{1}{1 \\times 2 \\times 3}\\binom{n}{2}\\binom{n-2}{2}\\binom{n-4}{2} + \\cdots.", "c9527da7083f920baa77f9388b08173d": " \\operatorname{drop-params}[g\\ q\\ p, D, V, [F_4, S_4, A_4]::\\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "c952aef6b34088b237c54d5a4b3ac86f": "\\tan 2\\theta = \\frac{2 \\tan \\theta} {1 - \\tan^2 \\theta}\\!", "c952d56e62542f08515ccc787daa0780": "\n\\dot{Q_i} = \\frac{A_i \\epsilon_i}{1-\\epsilon_i}(E_{bi}-J_i) = \\frac{E_{bi} - J_i}{R_i} \\qquad \\text{where} \\quad R_i = \\frac{1-\\epsilon_i}{A_i \\epsilon_i}\n", "c952f9fa64228b40e816423b65f5860c": " 1/T_m ", "c95322dff254d4b8c0a0deafcfaf24dc": " R_{i+1} = \\begin{cases}\n X_{i+1}/2,147,483,563 & \\text{for } X_{i+1} > 0 \\\\\n 2,147,483,562/2,147,483,563 & \\text{for } X_{i+1}=0\n \\end{cases}\n", "c953583decf8b37d6d720b7469206e32": "Q^{-1}", "c953985c263332826fa57129a42e1d6b": "P_1,P_2\\in \\mathcal{C}", "c953a86ba776ec7649959b67c99800d7": " \\blacksquare ", "c953c652188fa667c0bbc457ee3ad827": "(X_1,X_2,\\ldots)", "c95446f2e34f15ac61384265de39d72a": "\n\\begin{bmatrix}\n91 & 77 & 71 & 6 & 70\\\\\n52 & 64 & 117 & 69 & 13 \\\\\n30 & 118 & 21 & 123 & 23 \\\\\n26 & 39 & 92 & 44 & 114 \\\\\n116 & 17 & 14 & 73 & 95\\\\\n\\end{bmatrix}\n", "c9545ed991a80dec91a446db1ab24b56": "\n\\mathcal M=i\\sqrt{\\frac{2\\omega_p}{Z}}\n\\int \\mathrm{d}^4 x\\left\\{f_p(x)\\part_0^2\\eta(x)-\\eta(x)\\part_0^2 f_p(x)\\right\\}\n", "c9546928b38cf36145623b512ea6d7fb": "T^t= e^{t \\frac{d}{dx}}~, ", "c954a63700c07fb837e129b8ff77cdd2": "g^\\prime=g", "c955329bec75f052d2fa54aa759cb6d4": "\\partial F/\\partial t", "c9554a9da32d067216e90653a3675b88": "y-1", "c9555cd51dd4099551254e123f6d5916": "ee_{product}=f\\,ee_{max}ee_{catalyst}", "c9559d56d9b88728ae5b4f8d02f8c6c0": "\\{x,y\\}\\subset P", "c955aaef07ad64bd5185413073e075a4": "|\\alpha\\rangle=D(\\alpha)|0\\rangle", "c95640b1e8eb996025fb29cfdaf6cfe3": "\\mathcal{L}=U(x_{1},x_{2})+\\lambda(Y-p_{1}x_{1}-p_{2}x_{2}) ", "c95648e37fc33da5767f03025c36771c": "C_{YY}(\\tau) = \\sum_{k,l=-\\infty}^\\infty a_k a^*_l C_{XX}(\\tau+k-l).\\,", "c9564c1595f4599aceabd1d8997dcd0f": "\\mathrm{Td}_n(\\mathbb{CP}^n) = 1", "c9565fb559bad26251200f6c4cef6048": " m_1 \\ddot{\\mathbf{r}}_1 = \\frac{-G m_1 m_2}{r^2} \\mathbf{\\hat{r}}", "c9567e782010c41c4e156c2fbb718a6d": " O(n)/O(n_1)\\times\\cdots\\times O(n_k)", "c956d1f17b5407942ac8fdacc98233e9": "e_i \\,", "c956f43344d82281cc61609bfcd35a32": "\\Delta_n\\to X", "c957199199fb6246131b44e53bd5576b": "f \\circ g", "c95774c61fb36a66ccf07f7e12a87ac6": " 0\\leq \\lambda\\leq 1", "c9577531300bdafa33771d3c29b01650": "C(x,0)=I_n {\\rm~for~all~} x\\in X", "c957a6ccca090261c7f70bb1c0be8e71": "\\, T", "c957aae58a6f0cfde5bc120f897801f2": " \\cos ^2 \\theta = \\frac{1+\\cos 2\\theta}{2}, \\qquad \\sin ^2 \\theta = \\frac{1-\\cos 2\\theta}{2} \\qquad \\text{and} \\qquad \\sin 2\\theta= 2\\sin\\theta\\cos\\theta", "c957d89494986a1c22ce522275564558": "RL'(\\mathrm{dB}) = 10 \\log_{10} {P_\\mathrm r \\over P_\\mathrm i}", "c957dff78a16b68771fa0f8701476e47": "m_1b\\end{array}\\right.", "c96f25c7730a2130b74b4159c6519912": "\nX_1+ \\cdots + X_n \\stackrel{\\mathrm{d}}{=} C_n X + D_n, \\, \n", "c96f49817996d43668474e4d35740801": "F^H", "c96f54a628099b2917d3d65f237f5565": "\\sum_i \\hat{a}_i^{(\\eta)} = \\sum_i \\hat{a}_i^{(1)}", "c96f5743b43930bb0d3ad9c2f903f221": "x_2^3", "c96fd00c2b4fff8c5dde8c66b113c6ab": "\\mathcal{E} = -{{d\\Phi_B} \\over dt}", "c9703385f47c86c4a7545b9ce2866232": "\\left\\langle\\mathbf{P},\\mathbf{P}\\right\\rangle = P^\\alpha\\eta_{\\alpha\\beta}P^\\beta\n= \\begin{pmatrix}\nE/c & p_x & p_y & p_z\n\\end{pmatrix}\n\\begin{pmatrix}\n-1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nE/c \\\\ p_x \\\\ p_y \\\\ p_z \n\\end{pmatrix}\n = -\\left(\\frac{E}{c}\\right)^2 + p^2\\,,", "c970504d328215fef7275b4714b4c739": "\\beta = \\omega \\sqrt{LC}", "c970a5bfe5c70eb7b982cb126a41bfdb": "\\beta \\rightarrow 0", "c970acd900d55c1d2f9b7589d41c605d": "\\frac{\\mbox{Current Assets -- (Inventories + Prepayments)}}{\\mbox{Current Liabilities}}", "c970d7559edc2d5e1c82328bdb775ef2": "(u',v')", "c970f70bd43f3ceb08d05aa88e9cbd0a": "\\frac{D}{r -g} = P", "c97107b99d8a838ea6464bec3f7c6223": "H = \\prod_{k=1}^n {52 - 2k \\choose 2} \\div n!,", "c9719c95d6f1136ff1ea71fdbbbeb1cb": " \\left [z(j\\omega) \\right] ", "c971a7265d195fc6c86d640604b41cfc": " K(\\tau) = \\tau H + 2/3 H = 1/3 J(\\tau) ", "c971d69f7bd1c2235d5e6d37663c11b2": " \\mathbf{F} = \\nabla \\times \\nabla \\times \\Phi \\mathbf{r} ", "c971dc6086f8b2f983e5433a2524291e": "a=\\sqrt{2}^{\\sqrt{2}}", "c9720a84c4cfd34aebd96b6162780b89": "u_{12}", "c972744457123bf16967105cc9f3034c": "\\scriptstyle{\\varphi}", "c9727ea8fd09a8e6efaebc078a84d51d": "\\mu_{XXX}", "c9729f5714f953aa0bf755d1443efcd9": "\\textstyle\\sum_{k=1}^i x", "c972e0cd37585b3c0789884cd9ecc3d4": "\\Pr(\\mathbb{Z}\\mid\\boldsymbol\\alpha) = \\prod_d \\operatorname{DirMult}(\\mathbb{Z}_d\\mid\\boldsymbol\\alpha)", "c973045270e43500ed1bf75e77dffdf9": "\\updownarrow \\;\\;\\;\\;\\;\\; \\updownarrow", "c973048e0152f47f58b2f795166b3803": " \\mathcal{L}= \\sum_{k=1}^n C_k(I_k) + \\pi \\left [ L(I_1,I_2,\\dots,I_{n-1})-\\sum_{k=1}^n I_k \\right ] + \\sum_{l=1}^m \\mu_l \\left [ F_{l}^{max}-F_l(I_1,I_2,\\dots,I_{n-1}) \\right ]", "c97315f25cc751e1ab649766878b07f5": "\\Omega_i=\\frac{\\rho_i}{\\rho_c}", "c973b497552bd169162a654a5b427199": "d - d_i - v_it = a\\frac{t^2}2", "c973c42609ffcf26361f1ca2cfbf1d39": "t_\\text{A} = \\frac{\\hbar^3 (4 \\pi \\epsilon_0)^2}{m_\\text{e} e^4} ", "c973eeff7e907c13137dfb277fae4a99": " [7,4,3]_2", "c9742018c5575f06f59ce542b29ad79d": "g \\in G", "c974329a80050c8637dd38dce17091bf": "\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma^\\sigma \\gamma_\\mu \\,", "c974b34c613c3f96ed5d469c14c32e0a": "\\hat{g}_n", "c974cdd50b664dcb2b938118c2492f13": "P(n)=P(n-1)+P(n-5)", "c974d6f208f9f06881c22565ef90d381": "RPM = {30 m/min \\over \\pi \\times 10 \\, mm \\left ( \\frac{1 m}{1000 \\, mm} \\right )} = {1000*30 \\over \\pi*10} = 955 revs/min", "c97578b3e70e8cfe90d4d3125312924d": "U_n(\\sqrt{R},Q) = \\frac{a^n-b^n}{a^2-b^2}", "c975b66dd9877453a1b9edd241a395bd": "\\mathcal{J}(\\theta_0) = - \\sum_{i=1}^n \\left. \\nabla \\nabla^{\\top} \\right|_{\\theta=\\theta_0} \\log f(Y_i ; \\theta)", "c975c0898249b5598f2e938f295e8919": "e = \\sum_{n = 0}^\\infty \\frac{1}{n!} = \\frac{1}{0!} + \\frac{1}{1!} + \\frac{1}{2!} + \\frac{1}{3!} + \\frac{1}{4!} + \\cdots", "c975c3596c1699e3ddc29cb2bf2dc878": "(fg)\\left((a+b\\sqrt{2})+(c+d\\sqrt{2})\\sqrt{3}\\right)=(a-b\\sqrt{2})-(c-d\\sqrt{2})\\sqrt{3}=a-b\\sqrt{2}-c\\sqrt{3}+d\\sqrt{6}.", "c975e03c8fe9f790d4ae200728d2365f": "M_X(H)=(X\\cdot H(X)) \\bmod P(X)\\,.", "c9760a4e181a64b965e1d7c20539d3ab": "i \\rightarrow i\\pm 1", "c9769bff4e1009609f1dbbbf4f8afb39": "\\forall v", "c976af541c2795d491c143b9bf868d19": "\\frac{n^{(k)}\\alpha^{(n)}\\beta^{(k)}}{k!(\\alpha+\\beta)^{(n)}(n+\\alpha+\\beta)^{(k)}}", "c976b41668848700376df49d100d2c58": "(1 \\; 2 \\; 3 4)\\,", "c976bb03acc6bd88a988717de83c92ef": "e= \\lim_{n \\to \\infty}(p_n \\#)^{1/p_n} ", "c976df774acb00f6126f3df984fb7e11": "\\mathcal{L}(\\theta |x) = f_{\\theta} (x), \\, ", "c9770da044bb3af661bc274f8a74248f": "\\chi_e(0)", "c977450b4592762dce017351f597a7b4": "(d_B, Q_B)", "c9774a8fe5a4762a38c228a1d457ce40": "\\mu = (3 \\sim 4) \\cdot 10^{-3} \\, Pa \\cdot s", "c9775e2e3c4e8f4c05681bd266d4628e": "+ \\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+p-2)\\frac{q\\pi}{p}\\right]}{(kp+p-2)^{2m}} + \\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+p-1)\\frac{q\\pi}{p}\\right]}{(kp+p-1)^{2m}} +\n\\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+p)\\frac{q\\pi}{p}\\right]}{(kp+p)^{2m}}", "c977803d9ab6f9daa77ed9ba58eafb6b": "E = \\frac{1}{2} m |\\mathbf{\\dot{r}}|^2 + V(\\mathbf{r}) = \\text{constant}", "c977c51a731c8a10982afb50f726ae77": "q_0 = 1", "c97811b12c3a42dc75d17140400c4f35": "\\ R", "c9781cdad1d4502d446635ecf232efa8": "m_1=0.5\\ ,", "c978220d0883c8ecfff145151e364724": "S^{*}", "c9784bed4dfbe763adb866dea1d6a858": "X=X(t)=a+\\frac{k^2(x(t)-a)}{(x(t)-a)^2 + (y(t)-b)^2},\\ Y=Y(t)=b+\\frac{k^2(y(t)-b)}{(x(t)-a)^2 + (y(t)-b)^2}.", "c978e2aa907cbe56ddec4e73b1b8e7f6": "I_{L2}", "c978ea00b9741ecff8426d0c502017c9": "U=\\sigma_z", "c979178bf920e6e44b6d9067b1f0e6b5": " \\epsilon, \\delta, K > 0, ", "c9793fd1d265c8a5bc9ebb7e9f3b820b": "V_\\text{t}", "c97999d5a23e83ccc2b99fdb4e42303a": "{\\rm T}_{n,p}(\\nu,\\mathbf{M},\\boldsymbol\\Sigma, \\boldsymbol\\Omega)", "c979aa55011ea000b25a0cc85320e456": "1/l_j", "c979aec183cdce40a041e09113453225": "p^2 = \\mathbf{p}\\cdot\\mathbf{p} = \\frac{m_0^2 \\mathbf{u}\\cdot\\mathbf{u}}{1- \\frac{\\mathbf{u}\\cdot\\mathbf{u}}{c^2}} = \\frac{m_0^2 u^2}{1-\\left(\\frac{u}{c}\\right)^2} ", "c979c5012fc1e6cac1d5999e2e818682": "V \\backslash X", "c979e3f7d8e90be2a8d90cdcb1443a1e": "\\mathfrak{g} = \\operatorname{Lie}(G)", "c97a2eda661f2ea6e6207cb0a7b90cd7": "\\ F^* = \\frac{\\partial x^*}{\\partial x} \\frac{\\partial x}{\\partial X} = QF. ", "c97a38d07bd95b6cbe1927501ba15ff6": "\\exist A \\, \\forall X \\, (X \\in A \\iff P(X) \\, )", "c97a42123acd434bc7cba3bd415e6711": "W_\\mathrm{SWU} = P \\cdot V\\left(x_{p}\\right)+T \\cdot V(x_{t})-F \\cdot V(x_{f})", "c97b0008ecbc5c2094c8bc516d15b4f8": "\n \\begin{align}\n \\varepsilon_{\\alpha\\beta} & =\n - x_3~\\varphi_{\\alpha,\\beta} \\\\\n \\varepsilon_{\\alpha 3} & = \\cfrac{1}{2}~\\kappa\\left(w^0_{,\\alpha}- \\varphi_\\alpha\\right) \\\\\n \\varepsilon_{33} & = 0\n \\end{align}\n", "c97b12ad3b814f8602816214a62bd69d": "\n1~\\mathrm{W} \\times 10 \\times 10 \\times 2 \\times 1.26 \\times 1.26 \\approx 317.5...~\\mathrm{W}", "c97b278d6543ecbb55796ece6c76512d": "e^{ - t^\\beta} = \\int_0^\\infty du\\,\\rho(u)\\, e^{-t/u}", "c97b4159fdc331124d88c542509c4da0": "Nh > L,\\,", "c97b64cbf798f44e0adc956a92e365e4": "\\frac{\\ln\\big((2u)^6+24\\big)}{\\sqrt{3502}}", "c97bbc7e6fe4ece25d4236568c9b7b05": "ECT_0", "c97be39378b0561895dec035edc3d631": "{i} = \\hat{ u} \\hat{ v} = \\hat{ u} \\wedge \\hat{ v}", "c97c01585a8ddc6dc1f4f6c900b0dcfc": "L_{pp, \\gamma-norm}", "c97c1315654a36f5a0073a44fbb6cfb5": "A_{11}^{-1}", "c97cf64dba4636d7d4c6b78d462ce108": " -n(n+1)~r^{-n-2}~\\sin(n\\theta) \\,", "c97d0f0f505bfd02ca828bc416686d11": " 2\\gamma_0 -1 = 1 ", "c97d1ce435a09c61f5dd5449ddab54d1": "\nT(y_0) = \\int_{y=y_0}^{y=0} \\, dt = \\frac{1}{\\sqrt{2g}} \\int_0^{y_0} \\frac{1}{\\sqrt{y_0-y}} \\frac{ds}{dy} \\, dy\n", "c97d7dcc17535c40761dc61e33b58ae3": "L=n\\frac{ds}{d\\sigma}=n\\left(x_1,x_2,x_3\\right) \\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}=L\\left(x_1,x_2,x_3,\\dot{x}_1,\\dot{x}_2,\\dot{x}_3\\right)", "c97d89980d5d0d43e3f37cfcccaf6f07": "U(S,V,\\{N_j\\})=TS-PV+\\sum_j\\mu_jN_j\\,", "c97dd89d3152410d3d0c86a6d6a97659": "z^{n \\backslash i}=\\left( z_1,\\ldots,z_{i-1},z_{i+1},\\ldots,z_n \\right)\\in\n\\mathcal{Z}^{n-1}", "c97ddba0d3e5f26f01ec2c0dbaab88a5": "\\sum_{ij;i \\ne j} \\psi^*_i \\psi_j \\varphi^*_j\\varphi_i", "c97eb3109986f893f0a81c4fa2422ba6": " \\delta_{int}(s)=s'=(\\ldots,(s_i', t_{si}', t_{ei}'), \\ldots) ", "c97ede004289bc9075cf5cd6c92dbd14": "\\textstyle D^{\\alpha}v=0", "c97ee7f079f6850336c7887efbc7d9c2": "a_c + b_c(Y - T)", "c97f11e0ad10a91833b6d59a37d208d9": "\\lambda_2 = -5 \\,\\!", "c97f62f913992189daa5a38b3044be7b": "2x^5-5x^4-x^3-7x^2+2x+3=0\\,", "c97f7e2fff3f6cc738f135f80233c849": "y_{n+1} = e^{-Ah}y_n + A^{-1}(1-e^{-Ah}) \\mathcal{N}( y( t_n ) )\\ . \\qquad\\qquad (8)", "c97fa3e5f6b442dd7c0f9bced0eacafb": "(A\\backslash B)\\backslash C", "c97fd785961a697e709c20a210b2d362": "f_c'", "c98025dd200b6a862ee6724e1d978bf7": "|\\omega_c - \\omega_a| \\ll \\omega_c+\\omega_a", "c9805eb91cbbcf7c4eacfa6186417298": "1/h_c", "c9809c9499a5ff3345c84059e6b39924": "a\\mathrm{T}+\\mathrm{U}", "c980ac474a9bfb4bb9b3d687771898b8": "r = -\\ln{\\left[\\frac{u'(c_{t+1})}{u'(c_{t})}\\right]} - \\ln{\\beta}", "c980cb1cd106121b15c70889937120c8": "Tr_{ex}", "c980d5cc619b56965acf8042244b9a36": "\nP\\left(r\\right)=\\frac{1}{Z}e^{-\\frac{F\\left(r\\right)}{kT}}\n", "c9811d49ad807af2f3ae8b95d1de12a2": "\\wedge^m \\subset \\wedge^{m+1}", "c98121887fb212c3aa63aa702e2bc778": "v(S) = \\begin{cases} |S|/2, & \\text{if }|S|\\text{ is even}; \\\\ (|S|-1)/2, & \\text{if }|S|\\text{ is odd}. \\end{cases}", "c981307a5468a3d927b424381eb40035": "\\Delta F = W", "c9818d83ff1bc139a1becf5476a24344": "T \\cos \\theta = mg \\,", "c98199743a68a086e95a28e869a5d746": "\\mathbb{D}_8 = \\mathbb{Z}_4\\rtimes\\mathbb{Z}_2", "c9820fc6709fca446d2a7946be422f92": "[F,G]=\\frac{1}{\\pi} \\int_{\\mathbb{R}} \\overline{F(\\lambda)} G(\\lambda) \\frac{d\\lambda}{|E(\\lambda)|^2}.", "c9822290a1e59e76925f1b347723455a": "g(x, y, t) = \\frac{1}{2\\pi t^2} e^{-\\frac{x^2 + y^2}{2 t^2}}", "c98290e2fe665e84db77e54cfe8db015": "e^{i(3\\pi/2)} = \\cos(3\\pi/2) + i\\sin(3\\pi/2) = 0 - i1 = -i\\,\\! .", "c9829a5bd5788bb51e4f0bb006c63e2a": "g(x) = f(x)", "c98310f1cef51d24ef19fc61f6aaf962": " |f(Tx(N))-f(x(N))|={1\\over N+1} |f(T^{N+1}x)-f(x)|\\le {2M\\over N+1}. ", "c9835b949e19bb3bda193f8d48969069": "\n\\begin{bmatrix}\n31 & 53 & 112 & 109 & 10\\\\\n12 & 82 & 34 & 87 & 100 \\\\\n103 & 3 & 105 & 8 & 96 \\\\\n113 & 57 & 9 & 62 & 74 \\\\\n56 & 120 & 55 & 49 & 35\\\\\n\\end{bmatrix}\n", "c983b18ee1cf2ff2c50770d8a685b09d": "\\Delta F_{\\textrm{T}}", "c983d79f3ce92bcb6f02f2ca251bc59d": "R = \\mathbb{Z}_p[\\alpha]/(\\alpha^n + 1)", "c983f00b006ca0a4f936bd0cc8c98e93": "z = {x \\over \\sqrt[4]{-\\frac{d_1}{5}}}\\,", "c9845d5b7a80acd16e11aa57dc56e1b8": "b_{6}=b_{7}", "c984d819e2293c3d616dc60075eed6b8": "E_{n_x,n_y,n_z} = E_0 + \\frac{\\hbar^2 \\pi^2}{2m L^2} \\left( n_x^2 + n_y^2 + n_z^2\\right) \\,", "c984f0869685a511822c9ab93bd305f3": "q_1= m+\\frac{s}{\\sqrt[\\alpha]{\\log(4)}} ", "c9853eab552f75cb9c75adcd9316db1b": "\\left [\\begin{smallmatrix}2&-\\sqrt{3}\\\\-\\sqrt{3}&2\\end{smallmatrix}\\right ]", "c98575ee42188b5cd006e5df72ba7673": "f_n = \\frac{nv}{2L}", "c9858f245a5e40676697b36ca5493be4": "\\int_{-\\infty}^{\\infty} ae^{-(x+b)^2/c^2}\\,dx=a |c| \\sqrt{\\pi}.", "c9864122422d8254a4b53f75b53767a6": "SA=\\pi r^2+\\pi r l", "c98677b2e933602fd77ceb8c98d64a2e": "\\scriptstyle f(\\frac{1}{z})", "c98697fc79bff5995c76490ae11ea266": "A_0 = \\alpha - \\frac {1}{\\pi} \\int_{0}^{\\pi} (dy/dx) \\; d\\theta", "c986bea85d7962cfa1a98b0660b47d71": "f(E;\\beta)=\\frac{e^{-\\beta E}}{\\mathcal{Z}(\\beta)}\\Omega(E)\\approx\\frac{\\exp\\{-(E-\\langle E\\rangle)^2/2\\langle(\\Delta E)^2\\rangle\\}}{\\sqrt{2\\pi\\langle(\\Delta E)^2\\rangle}}.", "c986e8e996a97cd0c0672477c3448e31": "WXYUVPQ", "c9873aee745b9664774c3a578a7a61fd": "\\,i=0.12", "c98745d66db6d4419969737bc3836c83": "\\tfrac{1}{4},\\tfrac{1}{12}", "c9882f7f315e76e70069adaab04582a5": "R=\\left (\\frac{2\\kappa }{1+\\kappa ^2}\\right )^2+\\left (\\frac{1-\\kappa ^2}{1+\\kappa ^2}\\right )^2\\, ", "c988b83bfeacecaf7ff293ac319dd4e3": "{\\Delta h_m}", "c9894fa3f372f5650fac960e7b788296": "\\widehat{f^{(k)}}(n)", "c989952df0f1d81d7f73ccc060a2130a": "X_{(n)}=\\max\\{\\,X_1,\\ldots,X_n\\,\\}.", "c989ab81d8a74bc1008e8d75d27d9451": "\\ln(e^{i\\theta})=\\arg(e^{i\\theta})=\\theta", "c989c57c9eee4c8a1d542ce0cec01922": "V(\\vec{x})", "c98a070395196895836c6f307ddbf909": " \\mathbf{\\pi} = \\mathbf{\\pi P} ", "c98a1e04dbbd95d149faa2ee31a10244": "= \\frac{6 (4/3)\\pi \nr^3}{[(3\\sqrt{3})/2](2r)^2(\\sqrt{\\frac{2}{3}})(4r)} = \\frac{6 (4/3)\\pi \nr^3}{[(3\\sqrt{3})/2](\\sqrt{\\frac{2}{3}})(16r^3)}", "c98a1f7cd81c8ff47c5f1b9e5b15c679": " \\log{z} = \\ln{|z|} + i\\left(\\mathrm{arg}\\ z \\right)\n= \\ln{|z|} + i\\left(\\mathrm{Arg}\\ z+2\\pi k\\right) ", "c98ad91cfeefc1aa5636f5320f0bf0ac": "\n\\begin{align}\n\\left| 2i\\sin{\\textstyle \\frac{\\theta}{2}}\\left(S_p - S_q\\right)\\right| & = \n\\left| \\sum_{n=q+1}^p a_n \\left(z^{n+\\frac{1}{2}} - z^{n-\\frac{1}{2}}\\right)\\right| \\\\\n& \\le \\left[\\sum_{n=q+2}^p \\left| \\left(a_{n-1} - a_n\\right) z^{n-\\frac{1}{2}}\\right|\\right] +\n\\left| a_{q+1}z^{q+\\frac{1}{2}}\\right| + \\left| a_pz^{p+\\frac{1}{2}}\\right| \\\\\n& = \\left[\\sum_{n=q+2}^p \\left(a_{n-1} - a_n\\right)\\right] +a_{q+1} + a_p \\\\\n& = a_{q+1} - a_p + a_{q+1} + a_p = 2a_{q+1}.\\,\n\\end{align}\n", "c98af34153355253186568b4b265a577": "f(y;e).", "c98b13cab18f14e9b72a70e3ba2177c0": "\\mathcal{X}\\times \\mathcal{U} \\rightarrow \\triangle \\mathcal{X}", "c98bbb3a644623bce20a692310ac1cf0": "{\\operatorname df}=\\frac{\\partial f}{\\partial t}\\operatorname dt + \\frac{\\partial f}{\\partial x} \\operatorname dx + \\frac{\\partial f}{\\partial y} \\operatorname dy.", "c98bec8723fb088b5125b721da24121f": "\\left(\\frac{g_0}{g_1}\\right)^{1/3} = \\frac{T_1}{T_0}", "c98c432de3214c1ccfe5639bc828c75e": "\\mathcal{S}=\\int \\mathrm{d}^{D-1}x \\mathrm{d}t \\mathcal{L} = \\int \\mathrm{d}^{D-1}x \\mathrm{d}t\n\\left[ \\frac{1}{2}\\eta^{\\mu\\nu}\\partial_\\mu\\phi\\partial_\\nu\\phi -\\frac{1}{2} m^2\\phi^2 \\right]", "c98cb239ea035e5aa725937e84ae2b92": "A0_{11} \\ ", "c98cc9e01e8c9d2d25c22b623a074c9c": "x \\pm \\delta", "c98ce64963d1d995c238dec72865a515": "\\scriptstyle \\vec B", "c98d1845a66f76fc127a3cd948273442": "\\mathbf{R}^2", "c98d2ee7fef766d7c73e1bd585ab6701": "z_j=x_j+iy_j", "c98d73fb74ac037aef0fef62b197d16d": "\\frac{1}{4+1} = 0.2", "c98d98dfa16d1fcbe85ec5c0fa518810": "D = \\begin{bmatrix}\nT1 & T2 & T3 \\\\\nR(X) & & \\\\\nW(X) & & \\\\\nCom. & & \\\\\n & R(Y) & \\\\\n & W(Y) & \\\\\n & Com. & \\\\\n && R(Z) \\\\\n && W(Z) \\\\\n && Com. \\end{bmatrix}", "c98daa49d7c40b0687749c7976e4582b": "s(t)=\\int_0^t \\|\\mathbf{r}'(\\sigma)\\|d\\sigma.", "c98e80cac404f4ed9dfa7711c0d337e5": "([1-x^2]\\,y')' + \\lambda\\,y = 0.\\,", "c98edbd17f5b3d9b7adab5fee27fb996": "\\mathrm{kei}_n(x) = -\\frac{1}{2} \\left(\\frac{x}{2}\\right)^{-n} \\sum_{k=0}^{n-1} \\sin\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{(n-k-1)!}{k!} \\left(\\frac{x^2}{4}\\right)^k - \\ln\\left(\\frac{x}{2}\\right) \\mathrm{bei}_n(x) - \\frac{\\pi}{4}\\mathrm{ber}_n(x) + \\frac{1}{2} \\left(\\frac{x}{2}\\right)^n \\sum_{k \\geq 0} \\sin\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right] \\frac{\\psi(k+1) + \\psi(n + k + 1)}{k! (n+k)!} \\left(\\frac{x^2}{4}\\right)^k", "c98ee2b1beb1fd76854e48049db2142e": "z(x_i+h)-z(x_i)", "c98f07efbe4ae3c15c61dd015e1b2795": " \\bar V_t ", "c98f850022cc565b7e770ad0d80d4e54": "\\tfrac12 \\rho u^2 + P = \\text{constant} ", "c98f91cb225a5622d6c2fdd9d4c03912": " \\rm{Ad}(G)e ", "c99031b849b5b3a90fc310bc24c1dbf3": "r^2 = 1 - r", "c99059ec7f14fb72ef19a314a03af6c8": "d_Q = d\\text{ (mod }q - 1\\text{)}", "c990769fcb4ac02e5eaf8c3c44922a1f": "T = \\frac{1}{2}D^{2}(\\sin \\alpha)(\\sin \\beta)(\\sin \\gamma)", "c99093799d566764624b64aa000f0eb3": "f(z)=e^{iaz} g(z)\\,,\\quad z\\in C_R,", "c990c9dbbbbf4b29b332547de38bb063": "\\gamma^2 + v \\delta \\gamma = 1", "c990ea26f06926b08d17215caa56e6fb": "\\sum_{j=1}^{20}n(j)M(j,j) = \\sum_{j=1}^{20}n(j) - \\lambda \\sum_{j=1}^{20}n(j)m(j) = N - N\\lambda \\sum_{j=1}^{20}f(j)m(j)", "c990f74878a195094c401e1573b6dca2": "\\sum_{i\\in I}A_i<\\prod_{i\\in I}B_i.", "c9914366c2d17f47cf5d7c20216a250f": "\\nabla_i \\equiv \\partial_i \\equiv \\frac{\\partial}{\\partial x_i} ", "c9915d61612a7dcb0c551b83598d0f27": "\\phi \\land \\chi \\to \\phi ", "c9916afee810bbb2e9602b301bac8199": "e^{\\zeta/2}", "c991f2178ebee4d2f507488e1eb25369": "WXYZRSTUV", "c992200b05fec19c46cde24825039d43": " A ", "c992e41ff545b6b9c8eba134d6093bcb": "\\frac{1}{M}\\nabla_1\\cdot\\nabla_2\\,\\!", "c99339a159369267f226c63e9ac68da1": "\\theta = n \\times 137.5^{\\circ},", "c993528b71eaf56cbbec01202319b2a7": "p(\\tilde{s}(t)\\vert m)", "c9936287a31b415d444da909d4f7374b": "\\!\\ \\log E(s) = 11.8 + 1.5 M.", "c9938189b04d758f8428ca2b159b71c7": "\\textstyle\\sum_c \\frac{n_c(n_c-1)}{n(n-1)}", "c993aabcbf8d1d6b54e2dd33166106f3": "\n v_{s|r} = v_{s,r} - \\Gamma^i_{sr}~v_i\n ", "c993b3200039a8db1bc36c9afe235b3d": "\\Delta S = \\frac{q_{rev}}{T} ", "c993b987142180d9faefe5cb6475460d": "2g-2 = d_1(d_2-2) + d_2(d_1-2)", "c994372aee2327776dae856aa6c682c6": "\\frac{v}{[E_{T}]} = \\frac{k_{cat}}{K_{M}} (\\frac{[S]}{1+[S]})", "c99451432e07c7eba32df97e78cb5bbb": "{\\rm det} \\left( \\beta I - B \\right) = \\beta^3 - 3 \\beta - {\\rm det}(B) = 0.", "c99485d511b9247d52345e733b767207": "\\scriptstyle \\chi \\,", "c9948a3770a8aa1ddba2774f4902c363": "\\mathbf{k(1-\\theta)} = \\frac{d\\theta}{dt}.", "c994b7eed35ea471a50ae4b7e3200122": " H \\to K ", "c99547e86abb53ee1722269a75cb5635": "t = -i\\tau", "c995d3c0b5b686aec25ab0c45f72b45a": "q(t) \\approx \\frac{q_1 - q_0}{t_1-t_0} \\left( t - t_0 \\right) + q_0", "c995ffae3d485bd55c6309a13d6cab55": "\\frac{p_{02}}{p_{2}}=(1+\\frac{\\gamma-1}{2}M_{2}^2)^{\\frac{\\gamma}{\\gamma-1}}", "c9962ddf2d92eccd2ffd86cbde3ca4fa": "p_1-1,c_1,sid_1", "c9965c38f1d5a88aaa9db2d339d866a5": " H_4 = \\begin{bmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & -1 & -1 \\\\ 1 & -1 & 0 & 0\\\\ 0 & 0 & 1 & -1 \\end{bmatrix},", "c9967e65bc620083897e9e207aca5478": "\\liminf_{n \\to \\infty} \\| x_{n} \\| \\geq 1.", "c996a78aba3e35b6b4e198f2135ec8be": "F_{ij} = w_r", "c996badb297d6c1e82eaecd285239458": "\\mu = \\int x \\, f(x) \\, dx\\, ", "c996e675da6389fc0e1e586828be7047": "\\ \\sigma(n) < e^\\gamma n \\log \\log n + \\frac{0.6483\\ n}{\\log \\log n}", "c996ec5859edbe0c32708bbde751c56d": "\\ln{\\frac{\\Gamma(k/2)}{\\sqrt{2}}} - \\frac{k-1}{2} \\psi\\left(\\frac{k}{2}\\right) + \\frac{k}{2}", "c99755932573740149e73cecdfe0460f": "\\begin{align}\n\\textbf{J} &= \\int_0^{\\frac{\\pi}{2}} \\frac{\\sec^2(x)}{a +b \\tan^2\\,x}\\;\\mathrm{d}x \\\\[6pt]\n&=\\frac{1}{b} \\int_0^{\\frac{\\pi}{2}} \\frac{1}{\\left(\\sqrt{\\frac{a}{b}}\\right)^2+\\tan^2 (x)}\\;\\mathrm{d}(\\tan x)\\,\\\\[6pt]\n&=\\frac{1}{\\sqrt{ab}} \\left(\\tan^{-1}\\left(\\sqrt{\\frac{b}{a}}\\tan (x)\\right)\\right) \\Bigg|_0^{\\frac{\\pi}{2}} =\\frac{\\pi}{2\\sqrt{ab}}.\n\\end{align}", "c9976b7362608a601607e130bed37b16": "\\frac{P(I)}{P(H1+)}", "c997e3ae646a7ac2beb58b9b07d6e1c0": "J_n(x)=x^{2n+1}I_n(x),\\,", "c99810c55d9aa6b604728d91827b8d7c": "\\{g_1, g_2,\\ldots, g_n\\}", "c99881fe6b2724b6d7c7bd369d36d1ea": "\\ln (K^{A,0}_{eq}\\,p_A) = \\frac{\\Delta H^0_{ad} \\, \\alpha_T \\, \\theta}{k\\,T} + \\ln \\left( \\frac{\\theta}{1-\\theta}\\right)", "c998abf5cde5d8879dfa0cbe4d0b0e1b": "-\\gamma'(1).", "c998dc31832e79fda7e7a83609a28c81": "v \\rightarrow v - \\frac{\\dot{r}}{2r} (x^2+y^2)", "c998ff5955e7fa6964a822c0dedb9e08": "1/{}^\\infty\\!D", "c999461c8e61d13c08d6b52cbc9de49f": "\n[a] =[a]\\cap \\frac{[b]}{[x_3]} \n", "c999c1d17cd9ed4fcf0401b158aabe07": "10^{-2}", "c999c8dc2b0391efc4a9953981c06688": "L\\cap (-m,m)\\subseteq \\bigcup\\limits_{q=2}^\\infty V_{n,q}\\cap(-m,m)\\subseteq \\bigcup\\limits_{q=2}^\\infty\\bigcup\\limits_{p=-mq}^{mq} \\left( \\frac{p}{q}-\\frac{1}{q^n},\\frac{p}{q}+\\frac{1}{q^n}\\right).", "c99a30c5a696b2746affce02b57aebbc": " Var(X) = V(\\mu) = \\nu_0 + \\nu_1 \\mu + \\nu_2 \\mu^2,", "c99a658436772796502c3d5ba68f3f80": " (\\nabla \\times \\mathbf{F})^i(\\mathbf{x}) = \\varepsilon^{ijk}\\frac{\\partial}{\\partial x^j} F^k(\\mathbf{x}),", "c99a714c015ee4b709d745365e3f64de": "P_\\mathrm{tot}", "c99ad630374210a4728edf7d99789219": " MA = \\frac{F_B}{F_A} = \\frac{7}{13} = 0.54.", "c99b125217612bf2860020c33a32f002": "(-\\Delta)^m G(x,y) = \\delta(x-y)", "c99bc9cba64dc3776705c44cb1f6357a": "Drug\\ usage(DDDs) = \\frac{24 \\times 500mg}{3g} = 4", "c99c503f873fdfa6228465f2233f25a9": "\\varphi_1 \\varepsilon_1", "c99ce0ef2325c6a786f35273df04d104": "\\qquad \\dot{y}=g(x,y)", "c99cf92c608d2ae5c35eda9aa0417556": " v = \\mu_{X2} - 2\\mu_{X1} \\mu_{Y1} + \\mu_{Y2}\\, ", "c99d09d86a3ea9ae972e69236919f4d0": "\\mbox{Therapeutic ratio} = \\frac{\\mathrm{LD}_{50}}{\\mathrm{ED}_{50}}", "c99d5f9749c1d7a1329577e75b729b4b": "\nD_{cNp} = 100 \\cdot \\ln\\frac{V_2}{V_1} \\approx 100 \\cdot \\frac{V_2 - V_1}{V_1} = \\text{Percentage change} \\text{ when }\\left | \\frac{V_2 - V_1}{V_1} \\right | << 1 \\,\n", "c99def35c7529be54fb49c7cbb4ab133": "{\\mathrm U}({\\mathbf W})\\,", "c99df7a4f7c719e39829c35c0c737170": " \\ w_j = 0 ", "c99e3dced8e1f76604fab5a47abf0de5": "\\scriptstyle{\\Pi(\\phi,-n,k)}", "c99e8633266a6290a8dba0539fae149e": "\\mathbf{X'}", "c99ed4fe24681e607b222cc44bc15833": " A=dx^\\lambda\\otimes[\\partial_\\lambda + (\\Gamma_\\lambda{}^\\mu{}_\\nu(x^\\alpha)\\dot x^\\nu+\\sigma_\\lambda^\\mu(x^\\alpha))\\dot\\partial_\\mu].\\qquad \\qquad (3)", "c99f1090addf56d570536c9d5ac00017": "\\lim_{n\\to\\infty}\\frac{f(n)}{\\varphi_{1}(n)}=a_1.", "c99f42bedee2ec833895964b9841270b": "\\bigtriangleup_{SO,dB} = OSOI_{dBm} - P_{out,f,dBm}", "c99f835edb6580a1660728c6a9b729f4": "\\widehat{QP1A}=\\frac{1}{2}\\widehat{QO1R}=", "c99fda262eaea108d9ff80cb350920e0": "y=(y_1,y_2,\\ldots,y_m)", "c99ff3317f336340b5ef2b8d13b41c55": "p_2,c_2,sid_2", "c9a0087b8f095c31282a7cf95930b5b8": "x_0, x_1, x_2, \\ldots, x_n", "c9a139eb4cb31c381d86e523f72fdb4e": "(l_1, l_2, l_3)= (\\vec{L}\\cdot \\hat {\\mathbf{e}}^1,\\vec{L}\\cdot \\hat {\\mathbf{e}}^2,\\vec{L}\\cdot \\hat {\\mathbf{e}}^3) ", "c9a13a1909260a233dab3c6420bdc77b": "M^2", "c9a141b5e1795b34d6919ca32dd3d4fe": "\\omega_0 = \\sqrt{\\frac{k}{m}}", "c9a14ae8cffb1ea219a6d2f70493b6cd": "\\mathit{h}(\\mathit{x}) \\in \\mathbb{Z}_p[\\mathit{x}]", "c9a158fac0f9d240471f068bcb8acc23": "(\\mathbb{O}\\otimes\\mathbb O)P^2", "c9a15f67bf6ef5d4b5653170a028ef6d": " T(z) ", "c9a172940276096a43c33f95faca3252": "n,k\\in\\mathbb{N}", "c9a176a7f6a0454fe19165b0ace5d967": "N_{jj}", "c9a194ecfc8961095ee1b2ce224ad11d": "\n\\mathbf{e}_{j}(t) = \\frac{\\overline{\\mathbf{e}_{j}}(t)}{\\|\\overline{\\mathbf{e}_{j}}(t) \\|} \n\\mbox{, } \n\\overline{\\mathbf{e}_{j}}(t) = \\mathbf{\\gamma}^{(j)}(t) - \\sum _{i=1}^{j-1} \\langle \\mathbf{\\gamma}^{(j)}(t), \\mathbf{e}_i(t) \\rangle \\, \\mathbf{e}_i(t)\n", "c9a1b6fe26db6d445cd6fd73ed17a925": "\\pi_i(M)", "c9a1e26e68053f289255f24c29c3fbb3": "C_j = A_jG_j", "c9a1fed9233e05c68f3a58ebe8165447": "\\in\\mathcal{X}", "c9a2316bef8d9fba0596fcf2f0f66025": "\\scriptstyle \\frac{1}{4}\\,", "c9a2373ae6435fb2a6398f0b80e6b696": "D^0\\bar D^{*0}", "c9a24055f0c67ea4a36d7bd917d84aa0": "\\,\\!d(x,y) = d(y,x)", "c9a25f7560d2d73bace6d64aef964e64": "[M] + [M] = [\\emptyset]", "c9a2aa0573f4605e6d2a8703e863de5b": "(di^2-dc^1-dp^2) / (di_2-dc_1-dp_2) \\times 2 =20.", "c9a2de281634cbdce3be5f23ce3b27cc": "p(t,A)", "c9a2f41b5f1699c96694835dfae6b8e3": "X_1,X_2,", "c9a2ffe2fc1f90a52186dc9191a597c5": "w^1, w^2,\\ldots, w^t", "c9a33321300648260ff8a75e7323f269": "\\scriptstyle d\\geq 3 ", "c9a36e23bc54e311e9babc28fd6d13aa": "G' = \\bigl({V',E'}\\bigr)", "c9a39b0fa3b15b2dc95f84efeb50e814": "t_r \\approx 1.4 \\tau \\approx \\frac{0.22}{f_c}", "c9a3ba01252fc6221bffd622a8eb6dc1": "y=\\cos\\theta", "c9a3eac38513fad356c5e65197e395c0": "\\biggl|S \\setminus \\bigcup_{i=1}^n A_i\\biggr| =\\sum_{k=0}^n (-1)^{k}\\binom nk \\alpha_k.", "c9a40c7a3c902f5873b7b795c05ce593": " \\textstyle J_v=(D_2-D_1)\\frac{\\partial N_2}{\\partial x}", "c9a41038a0845dc4667fd5b862873199": "\\omega(x,f) = \\bigcap_{n\\in \\mathbb{N}} \\overline{\\{f^k(x): k>n\\}},", "c9a416e3deb67d8561a60d52f52944d6": " \\frac{f/1}{(\\sqrt{2})^2} ", "c9a4739ada35c81caf37e6eab5fd5435": "\\sigma_k ", "c9a4868297fe4156203be39328557926": "h_{r} = 1", "c9a4a14d60bebc0a208cc7cfefda5e74": "a=376", "c9a4a4c55e5b0e4e2fd6f34e38ec4b88": "\\lim_{x \\to p}{f(x)} = L", "c9a4a5e11ef451aedf43ee28ca6e37c8": "\\hat{\\mathbf{r}}=(\\cos(\\varphi),\\sin(\\varphi))", "c9a4c415604cab0c730689148f45cbed": "v = c_1x_1 + c_2x_2 + \\cdots + c_r x_r", "c9a4cd116e240c7ba99ba784e43a2b23": "\\Psi(x_1, \\ldots x_n) = \\frac{1}{\\sqrt{n!}}\\left|\\begin{matrix}\n \\psi_1(x_1) & \\ldots & \\psi_n(x_1) \\\\ \n \\vdots & & \\vdots \\\\\n \\psi_1(x_n) & \\dots & \\psi_n(x_n) \\\\\n \\end{matrix} \\right|\n ", "c9a4f921765a9a856f782409c07d83b3": "pz\\overline{z} + gz + \\overline{gz} = q", "c9a517f26bb6b5cf893053d864078862": "\\mathrm{[M^+] + [H^+] = [A^-] + [OH^-]}", "c9a5254029daf6ce3be15cbea0b167e6": "=z_k\\cdot b_k(X)+\\sum_{j=1}^n w_j\\cdot b_j(X)", "c9a5706dc31b34156f6380e8da1fa2a8": "\n\\begin{align}\n(J^\\alpha) (J^\\beta f)(x) & = \\frac{1}{\\Gamma(\\alpha)} \\int_0^x (x-t)^{\\alpha-1} (J^\\beta f)(t) \\; dt \\\\\n& = \\frac{1}{\\Gamma(\\alpha) \\Gamma(\\beta)} \\int_0^x \\int_0^t (x-t)^{\\alpha-1} (t-s)^{\\beta-1} f(s) \\; ds \\; dt \\\\\n& = \\frac{1}{\\Gamma(\\alpha) \\Gamma(\\beta)} \\int_0^x f(s) \\left( \\int_s^x (x-t)^{\\alpha-1} (t-s)^{\\beta-1} \\; dt \\right) ds\n\\end{align}\n", "c9a583f4e1444c86ed75e23a15268b66": "x=(x_1,x_2)", "c9a599238f7926de09bf27c91541284a": "\\operatorname{det}(\\mathbf{M}) = AD - BC = 1 ", "c9a5c8e4b397ce02a493832d79956c75": "(L_{i+1}',R_{i+1}') = \\mathrm H(L_i' + T_i,R_i' + T_i)", "c9a5dce0dc0dd2023c4201988a46cd12": "\\textstyle{2-\\frac{1}{2}}", "c9a5ea43fc66f8c46204db5ae2938a34": "(G, e)", "c9a61fd8457e68b96fc2f82d34f65bc0": "K = R + i \\omega L \\,", "c9a62c1ab8ffd641dd27a66745854945": "s_{\\Lambda}", "c9a64c9c8f3b03b5bec8c4dfbdbd0cad": "(K-1)P dv = c_{\\,v} dT", "c9a66c2322a33a77fbc2e89dec29443e": " \\phi_n : \\frac{1}{n}\\mathbb{Z}/\\mathbb{Z} \\rightarrow \\mathbb{Q} ", "c9a66f5258b694612bdb40a11ff5446b": "qr(\\varphi) = 0", "c9a6904fcf97377eadcdcc7fefd91f46": "p(\\mathbf{x} | \\mathbf{\\theta}).\\,", "c9a6ddaaf1edf768ef6b2e7c3c429eab": "\\mathbf{v}(x,t)\\in\\left[C^\\infty(\\mathbb{T}^3\\times[0,\\infty))\\right]^3\\,,\\qquad p(x,t)\\in C^\\infty(\\mathbb{T}^3\\times[0,\\infty))", "c9a747cb3944fea25cd6fc28d14ab048": "n \\leftrightarrow p + e^- + \\bar{\\nu}_e", "c9a7be009af3987c872f7fa6aca376f5": " H=0, t<0", "c9a7c07e2b9d497330150759ae5b8292": "\\rho=|3a\\cos t|.", "c9a7d4b919fb21261a19b4c4a1119436": "\\alpha(m) := \\lambda^{m} (\\mathbf{B}_{1}^{m} (0)).", "c9a8273608ca25c4194b60e5159fe3c6": "\\scriptstyle{E_0}", "c9a82a2c6fcbfafde6d7a317026aaafe": "x_{31}=x_z\\,", "c9a85b3c98824768428074ec613af84f": "{m^+}_2 = f(10.8, 6.40) = [12.58, 0.53]", "c9a872ea03fcab62d15d5af4b6b8a0ae": "\ny_1\\; =\\; y_0\\, \\sin(kx - \\omega t)\\,\n", "c9a8748f7c1ab43a8f55d52481487c5c": "a_1,a_2,\\dots,a_n\\in|\\mathcal A|", "c9a909f61673d0e0efd7daece527706e": "24/25", "c9a96c918f511d80e4c98b4200515b1a": "{U} = {U}_0 + gh(r) - \\frac{1}{2}\\Omega^2 r^2\\,", "c9a9a01a1b56d4df08a1c5c4580c2a65": "\\mathcal{ELRO}", "c9a9a60deceabdbe85ba98b1e2a8f76d": "S=\\sum_i (\\log y_i-\\log \\alpha - \\beta x_i)^2.\\!", "c9a9a7d1fbfcf6bfb0d33dd85d93b227": "s>0", "c9a9ef5314af97083f727a481cee133d": "\\alpha(s) \\equiv \\sup \\left\\{\\,|P(A \\cap B) - P(A)P(B)| : -\\infty < t < \\infty, A\\in X_{-\\infty}^{t}, B\\in X_{t+s}^\\infty \\,\\right\\}. ", "c9aa7288949c5ff8903e2c924f420356": "q=[x_1\\, x_2]^\\mathrm{T}", "c9aa7fd3377c55903a5d5da13753102f": "\\scriptstyle \\frac{\\partial N}{\\partial y} = \\,0", "c9aab079d70e57da46fa80f4381e256b": "\\mu_e=\\mu_i", "c9aab0ab6009ba62854c077522fbdabe": "a=\\pi(b)", "c9aad74b6454ae8825c3f9f61a750af9": "f(x) = {2x-1 \\over x}", "c9ab3edb70a5542b137612a11965fdb6": "[X]_t=\\sum_{s\\le t}\\Delta X_s^2", "c9abbc1f20632661bde6f5d4c0da786a": "\\text{Gross Margin Percentage} = \\frac{\\text{Revenue - COGS}}{\\text{Revenue}}*100%", "c9abe9b909361ae53a62f3bd4cf65fa6": "\\frac{1}{9}", "c9abef10841c5663283917c5719d21c9": "x_2,\\ldots,x_p", "c9ac1d74fe9e46b6b3bbf7c8e5970f8d": "L(\\theta|x)", "c9ac3e5447c26141596fc316b37ba265": "O(|E|+|V|\\log|V|)", "c9ac42edceb3ae4d814c236db870018c": "\\frac{P(z_i)}{Q'(z_i)} = \\frac{z_i^2 - 5}{4z_i^3}", "c9ac6a3a19dfa29f632b901c81f5439a": "+ 10000^2*0.9", "c9acd42e461deb74ea60d6f5ac03d1eb": "H(A)=\\bigcup_{i=1}^N f_i(A), \\quad A \\subset \\mathbb{R}^2.", "c9ad9e056f68df7b3c50b6112a21f53c": "\\frac{f'(x+h)}{h} > 0", "c9ada9738e612a85cc9b26efefbcb659": "=\\int \\mathrm{d}^{D-1}x \\mathrm{d}t \\left[\\frac{1}{2}(\\partial_t\\phi)^2- \\frac{1}{2}\\delta^{ij}\\partial_i\\phi \\partial_j\\phi -\\frac{1}{2} m^2\\phi^2 \\right],", "c9adb8f08109e2f085ddf17afb76fc49": " \\forall \\theta \\in \\Omega", "c9ae146b26a0958a6c7c231389210d5d": "\\rho(A)", "c9ae2dc9a96139edf2942fb71fe7161e": "(13)\\quad \\theta_{(\\ell)}=-(\\rho+\\bar\\rho)=\\frac{2}{r}\\,,\\quad \\theta_{(n)}=\\mu+\\bar\\mu=\\frac{-r+2M(u)}{r^2}\\;.", "c9ae8bc58187924297c802d0c326e415": "\\|Tx-x\\|\\leq\\varepsilon", "c9ae952c65697424399cb52dcafd6d8d": "|\\tau(n)|", "c9aea9da6ee73e3a38c8023f581393bc": " d\\omega_G = -(g^{-1} dg\\, g^{-1})dg = -\\omega_G\\wedge \\omega_G.", "c9aeb0d6a4f4c7f44c52bcf3c009cf3b": "2^{4n+2}+1", "c9aef9ba7bf42f24d77a9eb685a3882b": "86.7 \\times 10^6", "c9aefa146ae9c868fdf9fdd3f95df75f": " y_{n+1} = e^{-A h } y_n + \\int_{0}^{h} e^{ -(h-\\tau) A } \\mathcal{N}\\left( y\\left( t_n+\\tau \\right) \\right)\\, d\\tau. ", "c9af1b1b612a4f7cd5bf67e5f9b9b4ff": "\\overline {\\mathbf{Q}_p}", "c9af4951720683cb750c567f7a64b564": "Q_2=\\frac{i}{2}\\left[(p-iW)b-(p+iW^\\dagger)b^\\dagger\\right]", "c9af75bfd2306bc214226421b4d34851": "f_{xy}(1,1) = p_{xy}(1,1) = \\textstyle \\sum_{i=1}^3 \\sum_{j=1}^3 a_{ij} i j ", "c9afd838e4b9a540f710baa34863888d": "a_{\\mathit{vf}}", "c9aff6ed392b24dc97d2866011c18cc1": "\n \\begin{bmatrix}\n x_{equatorial} \\\\\n y_{equatorial} \\\\\n z_{equatorial} \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos \\epsilon & -\\sin \\epsilon \\\\\n 0 & \\sin \\epsilon & \\cos \\epsilon \\\\\n \\end{bmatrix} \\! \\cdot \\! \n \\begin{bmatrix}\n x_{ecliptic} \\\\\n y_{ecliptic} \\\\\n z_{ecliptic} \\\\\n \\end{bmatrix}\n", "c9affddb053ab298879734e19b605420": " y_{n+1}^{(1)}-y_{n+1}^{(0)} = \\tau_{n+1}^{(1)} ", "c9b0349798633a8bbc4e7a7efc85c784": " v = \\sqrt\\frac{2mg}{\\rho A C_d}", "c9b04d37d6aa71608b0a40f967982266": "= {1 \\over \\sqrt{2}}{\\left|1,45\\right\\rang \\left|2,45\\right\\rang + \\left|1,135\\right\\rang \\left|2,135\\right\\rang} ", "c9b06f1f1af00f3440c579ffe541e406": "Initiates(break, isopen, t) \\leftarrow HoldsAt(hashammer, t)", "c9b079dd6d57b44ef2f69b33076ed573": "\\eta = -\\ln\\left[\\tan\\left(\\frac{\\theta}{2}\\right)\\right],", "c9b0a69e804debabee4c0190cdd1ce7c": "f(f(x))=x,\\,", "c9b0c2a2ee7ec840ac507b7fdb2d5091": "\\limsup \\frac{\\log Q(x)}{\\log\\log x} \\le 1.71 \\ .", "c9b156d8b2ef3e299acca75219990b56": "\\,x", "c9b15843670a064ebb321a505f378513": "\\ Y=f(K)", "c9b163e8ba4859250ddc44ce63428972": "G = (\\{S, S'\\}, \\{a\\}, \\{ f, g, h \\}, S)", "c9b197c24693f8d89ad5bbd451eb9c97": "\n\\begin{align}\n&\\text{Let }s = 1 + r + r^2 + r^3 + \\cdots. \\\\[4pt]\n&\\text{Then }rs = r + r^2 + r^3 + \\cdots. \\\\[4pt]\n&\\text{Then }s - rs = 1,\\text{ so }s(1 - r) = 1,\\text{ and thus }s = \\frac{1}{1-r}.\n\\end{align}\n", "c9b1d37fd1036606f048bf93a55a7e17": "\\frac{\\pi^2\\sin(s t)}{st(\\pi^2-s^2 t^2)}\\,e^{i\\mu t}", "c9b22235c38cbb465751911dfd4c9336": "g \\in k[x]", "c9b23099abbb869a381656027a014fba": "L^{(0)} = L , \\, L^{(1)} = [L,L], \\, L^{(n+1)} = [L^{(n)}, L^{(n)}]", "c9b306f83140d03e9b401b21eece5d7a": "(g_1,[p_1])(g_2,[p_2]) = (g_1g_2,[p_{12}]),\\quad g_1,g_2\\in SO(3;1)^+,\\quad [p_1]\\in\\pi_{g_1}, [p_2]\\in \\pi_{g_2}, [p_{12}]\\in \\pi_{g_{12}},\\quad p_{12}(t) = p_1(t)\\cdot p_2(t).", "c9b311791353c6e69d7b0f9579283039": "\\textbf{S}_{x}(j\\omega) = \\frac{2\\sigma^{2}\\beta}{\\omega^{2} + \\beta^{2}}.\\,", "c9b35b1ec858bc509df9622a2e8d2b5c": " 2^5 = 32,\\ 6^2 = 36,\\ 7^2 = 49,\\ 2^6 = 64,\\ 4^3 = 64,\\ 8^2 = 64, \\dots ", "c9b3976feb145a8d9d22994c3be9910a": "\n Q_x = \\frac{\\mathrm{d} M_x}{\\mathrm{d} x}~.\n", "c9b3bba7eea1733e3bd05cbae5c7e33e": " A_{2}(2d,d) \\leq 4d. ", "c9b3f2ee6eef17cc8a56d1df4d942b29": "Fe\\;+\\;2\\,H_{2}O\\;\\rightarrow\\;Fe(OH)_2\\;+\\;H_2\\;", "c9b42f8668f76c393bcbd0d2d00eaf91": "f\\in H^{p}(D^{2})", "c9b43086b74116628622b5b0c02cb34d": "x_1w = y_1", "c9b47164fba165d93676dd9790ffb938": "B(d \\setminus f)", "c9b4900835d01bd6192991849406c247": "\\, \\theta\\, ", "c9b4a5911e602d32b1844164ede8be06": "\\mu_{v_x} = \\mu_{v_y} = \\mu_{v_z} = 0", "c9b4e033bd2e2939aa10755191542a17": "i_a(t)", "c9b544206272fc10ae2163603baad934": "T_{\\rm h} > T_{\\rm c}", "c9b5c5f0733a2d24f0a722ebed2657f9": "x_{r}, x_{w}", "c9b5d48eea979a11bed039414c184c3e": "S = \\int d^4 x \\; e \\; e^\\alpha_I e^\\beta_J P^{IJ}_{\\;\\;\\;\\; MN} \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; MN}", "c9b5eba9defd75bf81f12561114a86a4": "g_i = \\log_2 \\frac{n}{1+ \\mathrm{df}_i}", "c9b623a66fb03ed2c93d80d14551043d": "\n\\begin{array}{c|c|c|c}\n\\textit{Model} & \\textit{Info-Gap\\ Format} & \\textit{MP\\ Format} & \\textit{Classical\\ Format} \\\\\n\\hline \n\\textit{Robustness} &\\displaystyle \\max\\{\\alpha: r_{c}\\le \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} R(q,u)\\} &\\displaystyle \\displaystyle \\max\\{\\alpha: \\alpha \\le \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})}\\varphi(q,\\alpha,u)\\} & \\displaystyle \\max_{\\alpha\\ge 0}\\ \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})}\\ \\varphi(q,\\alpha,u) \\\\\n\\textit{Opportuneness} &\\displaystyle \\min\\{\\alpha: r_{c}\\le \\max_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} R(q,u)\\} &\\displaystyle \\displaystyle \\min\\{\\alpha: \\alpha \\ge \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})}\\psi(q,\\alpha,u)\\} & \\displaystyle \\min_{\\alpha\\ge 0}\\ \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})}\\ \\psi(q,\\alpha,u)\n\\end{array}\n", "c9b632cdfe1489c467635a551fe70178": "\\pi_\\Sigma(s) = \\begin{cases} \n\\varepsilon & \\mbox{if } s=\\varepsilon \\mbox{ the empty string} \\\\\n\\pi_\\Sigma(t) & \\mbox{if } s=ta \\mbox{ and } a \\notin \\Sigma \\\\ \n\\pi_\\Sigma(t)a & \\mbox{if } s=ta \\mbox{ and } a \\in \\Sigma \n\\end{cases}", "c9b63529e047fd7a9734b3035a3bf449": " -f(x, r) = f(-x, r)\\,\\,", "c9b65b39a17e9f482fbc3ef3741b82ce": "1/(\\pi\\,\\mathrm{sr})", "c9b6a4f500a66b96d7302dcb9a6ca0fd": "{{I}_{M}}", "c9b6aa5af947cd7ad713cec4d7ca6770": "M^\\beta_{ij}", "c9b6edbdb2c1a067857244e25a29fe92": "S_{xy}(\\omega)", "c9b720fa403de14f90263957a173e8d5": "S < \\prod_{i=1}^k m_i", "c9b766abea94cf49b14779d37ab377e2": "\\{\\varphi(\\vec{x}),\\rho(\\vec{y})\\}=\\delta^d(\\vec{x}-\\vec{y})", "c9b76962d210f0e92da609cebff9cd68": "\\mathbf{v}^{b}\\cdot\\mathbf{n}=\\mathbf{v}\\cdot\\mathbf{n}.", "c9b7b3ed2484b8cc00e3849e5fb1c0b3": "E=\\left \\lbrack \\frac{LOC . B^{.333}}{P}\\right \\rbrack^3 \\left ( \\frac{1}{t^4} \\right )", "c9b7fbba2ee450ce1c66c6810e8aba96": "\\mathcal{L}_{X_g} f", "c9b8522dfb49ded1e919647f0be19fa8": "(\\cot x + i)^n = {n \\choose 0} \\cot^n x + {n \\choose 1} (\\cot^{n-1} x)i + \\cdots + {n \\choose {n-1}} (\\cot x)i^{n-1} + {n \\choose n} i^n", "c9b866924c0e35822eddeb77752cbfeb": " -\\dot{S}(t) = A'(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B'(t)S(t)+Q(t)+\\tau'_\\perp\\Psi_2(t)\\tau_\\perp(t),", "c9b9576be941a68a275f5ae87b10daea": " \\log_b(xy) = \\log_b (x) + \\log_b (y). \\,", "c9b97a2f71463c2880af0f00f94e58a9": "D_{\\text{a}}", "c9b9be87e569b173f302eef7921e5b1d": "T : X \\to X", "c9b9cf7f8f7554eec83b1ea5737ce531": "a_{j,p_t}", "c9ba0f994079ef95d986f43547dac9b8": "X = \\phi \\cdot M. \\,", "c9ba1709b74006fa6b858a050f37596b": " \\mu_C (x,y,z) := \\nu_C (\\phi_C^{-1}(x,y,z)) ", "c9ba8fd763a496f5864504a2db11130c": "\\Omega_c h^2", "c9bac0e9bfeec32f7366def4588c425a": "\\mathbf{e}^i=e^{ij}\\mathbf{e}_j", "c9bb38169100a0ad2461bb326e812c03": "p(\\theta) = \\frac{p_0 J_1(k_a \\sin \\theta)}{k_a \\sin \\theta}", "c9bb3e637f2c71587a57ab9e5f5f2f12": "a = 0.00115965218073 (28)", "c9bb87b9d9f48c9df7672f6c72996a7b": "d(X,Y) \\le H(X,Y)", "c9bb892fa988f7d9e6387c940a308bc0": "\\int_a^bf(x)\\,dx=\\int_a^cf(x)\\,dx+\\int_c^bf(x)\\,dx.", "c9bbe068544ec19c86394abd896bd10a": "c : V \\rightarrow [k] ", "c9bbef68e29716ff94ad4cb2bcb226bf": "f^\\#:\\mathcal{O}_X\\rightarrow f_\\ast\\mathcal{O}_Z", "c9bbf3c4acaff697c774851d5a03d4fd": "\\tan \\delta", "c9bbfea64516ffac7cc0b9af0634cac2": " = I - {1 \\over 7}\\begin{pmatrix}\n1 & -3 & 2 \\\\\n-3 & 9 & -6 \\\\\n2 & -6 & 4 \n\\end{pmatrix}", "c9bc53f027409d9d4f95794c3f5ca645": "\\mathbf{a}_3", "c9bc7d43a4f89c1324be879f9dab89c4": "\\displaystyle\\int_{-1}^1 {1\\over 2} (1-x^2) \\, dx ", "c9bcd4b4daf721584a7d58eda2adef97": " \\left(\\frac{m_1}{\\rho_1 a_1 D_1^2 Cos(\\alpha_1)}\\right) = ", "c9bd71ae414c6cf3782f65ea45dc81b4": "\\deg(\\mathrm{div}(f))=0", "c9bd7df59e534a9c48d39105267910e7": "\n\\alpha_n^2 = \\kappa H /h_n - 1/4\n", "c9bdc6de2a2cc7f67be9d7de97f44930": "-q", "c9bdfcfd85b9a94f1f8d21fb31af8fbd": "K' \\subset \\partial V", "c9be123ce03bde95a6284683a935b448": " \\mathit{WAcc} = 1 - \\mathit{WER} = \\frac{N-S-D-I}{N} = \\frac{H-I}{N} ", "c9be1fc623e614e25f3a196e4a92a6b7": "{AE}_{7}", "c9be31755bac96ce1e575f56810a9c43": "\\mathrm{(Fe,Mg)_2SiO_4 + nH_2O + CO_2 \\rarr Mg_3Si_2O_5(OH)_4 + Fe_3O_4 + CH_4}", "c9be666d22562663c9be327310f44ec0": "f(x) = x + y", "c9bea50b05d58efb40b7268511a78e71": "t_1 = TR\\,e^{3ik\\ell/\\cos\\theta-ik_0 \\ell_0}", "c9bed026d0d8e7db331106a439510e85": "\\mathfrak{m_i}", "c9bed56639dd4909f6fd5418cd936c2c": "K_n = \\{y \\in M | d(x,y) \\le n \\} ", "c9bf2d03f4a19f18ab65653adb2fd9f4": "\\deg(\\bar{f})=2g+1", "c9bf5e28cc77564e4cf3fa82d37caff7": " {\\lVert x_k-x \\rVert^2}={\\lVert x_{k-1}-x \\rVert^2}-{\\lVert x_{k-1}-x_k \\rVert^2}. ", "c9bf6c4c013a8ccf57bcad735a5e967a": "L\\left ( N\\frac{\\Delta \\omega_k}{\\omega_\\text{res}(\\theta)} \\right )\n=\\frac{\\sin^2\\left ( N\\pi\\Delta \\omega_k / \\omega_\\text{res}(\\theta) \\right )}{N^2 \\sin^2\\left ( \\pi\\Delta \\omega_k/\\omega_\\text{res}(\\theta) \\right )}", "c9bfbbfb7a5ed394abcc15bb7c7e2fd9": "\\begin{align}\nA & = 30 \\left [ 1 + \\sqrt{ 2 \\left ( 4 + \\sqrt{5} + \\sqrt{15+6\\sqrt{6}} \\right ) } \\right ] a^2 \\\\\n& \\approx 175.031045a^2 \\\\\nV & = ( 95 + 50\\sqrt{5} ) a^3 \\approx 206.803399a^3. \\\\\n\\end{align}", "c9c035c8f8c23c7565bf0ea59ee4f6db": " \\Gamma_{\\theta} ", "c9c05e325a3dc11dc32a715693bfa03c": "[-1,1] \\times [-1,1]", "c9c0e7c37936dac279a75138e91fb20f": "\\langle f, g \\rangle = \\int_{x_1}^{x_2} f(x) g(x) W(x) \\; dx.", "c9c1152c2c5c55fde21c92a4653d4dfc": " \\Sigma X \\cong X \\wedge S^1. \\, ", "c9c22c2cee43c7c15a03e23053d1c3d6": "E (\\cos\\theta\\, \\mathbf{\\hat f} - \\sin\\theta\\, \\mathbf{\\hat s})\\mathrm{e}^{i(kz-\\omega t)} = E [\\cos(-\\theta) \\mathbf{\\hat f} + \\sin(-\\theta) \\mathbf{\\hat s}]\\mathrm{e}^{i(kz-\\omega t)}.", "c9c23070111551a273b62d6b77aeff4d": "q_p(-a) \\equiv q_p(a) \\pmod{p}", "c9c23a3374b7e6dac9eb2f7e83b492cd": "\\bold{j}_{\\rm m} = \\rho \\mathbf{u}", "c9c2a210154c84cebb554b1dd6fde32b": "i\\omega\\mathbf{J}", "c9c31895b77d37d6986fde762486bfbe": "P\\left(S^{t+k}|O^{0}\\wedge\\cdots\\wedge O^{t}\\right)", "c9c3416f93183491e88a0cecbc844303": "\\cot(x)", "c9c382814b34bb401de74068732b0b47": "x_0, y_0", "c9c3cbd76f9a326ee1cdd03b30cde791": "\\tau(h_*) := [A] \\in {\\tilde K}_1(R)", "c9c3da8d98a22200e9aa3266240a3d38": "\\pm\\sigma_1, \\pm\\sigma_2, \\pm\\sigma_3\\,\\!", "c9c49015df666cd7677bac7441a82bc2": " Z_{P} = (b^2t_f)/2 + 0.25t_w^2(d-2t_f )", "c9c4cc62e23d04f1ec84ff1361a52a6e": " \\left(\\frac{r_1 + r_2}{2}, f\\left(\\frac{r_1 + r_2}{2}\\right)\\right).\\!", "c9c5059dd12c4d46e6c938fa70cedf80": "\\textstyle Z = e^{-A/(k T)}", "c9c553bf020cb8569469dd38a07b41a1": " \\hbar \\omega = h \\nu ", "c9c5d0b0832256ac2329e2c2e5ce8df6": "t_n()", "c9c5f6fd8aac9e8316f4bad37be251f7": "T_qE\\,", "c9c67bc4b027d13d72ca0fae10e7a2cc": " b_1 ", "c9c683545a40c95c0049c0098ae20c1e": "1 \\otimes h", "c9c6fa296757725a3507beb9516083a9": "\\int_{-1}^1 f(x)\\, dx \\approx a_0 + \\sum_{k=1}^{N/2-1} \\frac{2 a_{2k}}{1 - (2k)^2} + \\frac{a_{N}}{1 - N^2} = d^T c,", "c9c705f456371e574c855435c73525d2": "\n\\frac{\\Delta V}{V} = \\alpha_V\\Delta T\n", "c9c75be14a3d2ac7265e1d8e1e01e625": "(x_1, x_2, x_3, \\dots)", "c9c76eb2b4ae78d829ae3e06e94df9e3": "\\mathbf{Triv}=\\mathbf K\\oplus(A\\leftrightarrow\\Box A).", "c9c7710bfbd6a99e5f45f96745def59b": "44 = 6 + 9 + 9 + 20", "c9c7b5a39a77dd06174abb92f0ed79c2": "r+R", "c9c7cbdd096cd5aa34342e2568c83022": "\n S_{\\text{free open}} (\\Psi) = \\tfrac{1}{2} \\langle \\Psi | Q_B |\\Psi\\rangle \\ ,\n", "c9c7f4d67ab4a3b78ca042afc299a8b1": "x \\gamma_0 = x^0 + \\mathbf{x}", "c9c8464a8453cf9d98bc74f251bab3ed": "m(T) := \\sup \\bigl\\{ |\\langle T x, x \\rangle| : x \\in H, \\, \\|x\\| \\le 1 \\bigr\\},", "c9c89a98a4cfddbaba7e32991c4e1545": " \\Sigma_0^{\\rm P} := \\Pi_0^{\\rm P} := {\\rm P} ", "c9c8b63df090ad0f1d131ab62c6918a2": "\\{c_n\\}", "c9c8ef9fc95f9a44bb329d2f501f517c": "T=|t|^2\\,", "c9c8f5504dde8391c983c8ebda58515b": " M = \\begin{bmatrix} 0.936 & 0.352 \\\\ 0.352 & -0.936 \\end{bmatrix} ", "c9c916ea77aed44b1699032c30919e9b": "\\left(HE\\right)^2=\\left(r_1\\right)^2+\\left(r_2\\right)^2.", "c9c92572d2d02129ec35e488a20c45db": "_{s.12\\ s.13 \\,}\\!", "c9c97c76eba110f760598a2bd2d29539": "R_\\delta(x)", "c9ca01ec4e66f4d37e5472226c2d2cf6": "w(x) = \\frac{1}{(1+x) \\ln 2}.", "c9ca4ca911219f8b87f47dfa3e510870": "E_\\text{F}", "c9caa87e4100f932bb637319a1ff2cbc": "\\int \\arccsc{x} \\, dx = x \\arccsc{x} + \\ln \\vert x \\, ( 1 + \\sqrt{ 1 - x^{-2} } \\, ) \\vert + C , \\text{ for } \\vert x \\vert \\ge +1 ", "c9cafd5c1f05465925de7efe9e68a34d": " h_{12} = \\left. \\frac{V_{1}}{V_{2}} \\right|_{I_{1}=0} ", "c9cb65d86b5d030946d3cab8b4fb5de5": "x_i^{(k)}", "c9cba1bcb9a8ca6cae6b637d6b7134c4": "\\otimes F", "c9cbd789bfe47b3b55c8a3a6943d4192": " \\log{(1+\\sqrt{2})}", "c9cbdfd0dffe86e375822fb040a73977": "\\prod_v (a,b)_v = 1", "c9cc0b128eddd4f657b22dbee2759be3": "\\rho = \\sum_i \\psi_{1,i} \\psi_{1,i}^* \\otimes \\cdots \\otimes \\psi_{n,i} \\psi_{n,i}^*", "c9cc989da754bb538ca30624880c1ffb": "\\nabla^2 p = 0.", "c9ccd70c3f017a0a6ded38819647f2b1": "\\operatorname{CD}[n, (n-1)k]", "c9cceb0976ea852b39095003ce3c043e": "\\mathbf{x}=[A]\\mathbf{y},", "c9cd2bbe35de64a10cc2a2e158b02cdf": "F_{Load}", "c9cd36773f97a590c5db27784db7e18d": "a_{21} x_1 + a_{22} x_2 \\le b_2", "c9cda1f8a6ef7059f75d568e2a1d7bcb": "P \\land (\\exists x Q(x))", "c9cdb2b7d668ba9f3e330abd374de738": "(q)_n = \\begin{cases} 1 & n = 0 \\\\\n q(q+1) \\cdots (q+n-1) & n > 0.\n \\end{cases}", "c9ce2968aceed8dc83dfd21ae463a0a7": "q_i = 1/n ", "c9ce6b457a5b173c1b81f6aa63830796": " \\sum_{k=0}^\\infty n_k = N ", "c9ce98cb80a5109217d0ef7a96b78bc4": "C_m=\\frac{M}{qSc}", "c9cece2f7b1638db3e25ca36ba8f882e": "\\omega^{A}_{\\overline{x}}\\;\\;=\\lnot\\omega^{A}_{x}\\,\\!", "c9cee8cc710b05e252f39ba43c08da8f": "x \\mapsto f(x)", "c9cef0a7c27684287bfe025c6a1d2027": "\n{\\partial^4 \\varphi\\over \\partial x^4 } +\n{\\partial^4 \\varphi\\over \\partial y^4 } +\n{\\partial^4 \\varphi\\over \\partial z^4 }+ \n2{\\partial^4 \\varphi\\over \\partial x^2\\partial y^2}+\n2{\\partial^4 \\varphi\\over \\partial y^2\\partial z^2}+\n2{\\partial^4 \\varphi\\over \\partial x^2\\partial z^2} = 0.\n", "c9cf13e30acae713eae7f6d235bb84a3": "2p(E\\oplus F)=2p(E)\\smile p(F)", "c9cf149956adb86e00e9fcc648caba78": "{\\mathbf{x}}^n = \\left( {x_{n1} ,..,x_{nd} } \\right)^T \\in \\mathbb{R}^d", "c9cfaf4c08d98a47230c1ffaa4eaec46": "\\sigma_{xy} = \\pm {4\\cdot\\left(N + 1/2 \\right)e^2}/h ", "c9cfc6d7597106d2de08562d4cf22236": "(T) \\frac{A_1;\\ldots;A_n;\\Box B}{A_1;\\ldots;A_n; \\Box B; B}", "c9cfedc61d36b21244eb1feb28f2bca8": "\\Omega A + A^T \\Omega = 0", "c9d04320e73d2752f2293d35b8fed46d": "m(\\vartheta)= \\operatorname E\\left [ X_{ij} |\\Theta_i = \\vartheta\\right ]", "c9d09a959d7c50e27517e8b49c30a713": "\\delta=\\frac{\\Delta V}{V_0} = \\varepsilon_{11} + \\varepsilon_{22} + \\varepsilon_{33}\\,\\!", "c9d0c902119c12681244235d25fd1600": "k(x)=-x", "c9d0efe0351e5fb56802df9d12759990": "(N_{1},D_{1},\\lambda_{1})", "c9d1253f7bcd4d6632f55418f8e1341a": "v_i \\in F", "c9d1284b7d7675fa7d771e015853741c": "z^{\\bar n} = z(z+1) \\cdots (z+n-1);", "c9d197d6f144a6550b21cf0a93ca07ca": "\\left(\\frac{dn_1}{dt}\\right)_\\mathrm{stim}=B_{21} n_2 I_\\nu(T)", "c9d1d4e6136a79f11ccc589744dc488e": "g_m \\ r_O = \\begin{matrix} \\frac {2I_D} {V_{GS}-V_{th}} \\frac {1/\\lambda +V_{DS}} {I_D} \\end{matrix} = \\begin{matrix} \\frac {2(1/\\lambda +V_{DS})}{V_{GS}-V_{th}} \\end{matrix} ", "c9d20b2a86f0668e7366e6b99dfc2de9": "E[L(a-\\theta)] = \\int{L(a-\\theta) \\pi(\\theta|x) d\\theta} = \\frac{1}{p(x)} \\int L(a-\\theta) f(x-\\theta) d\\theta.", "c9d21412307f95fb42643dc86b861926": "x \\times x + y \\times y", "c9d216277ab57f39f292a96481ad2d0a": "0 = d_0 < d_1 < d_2 < \\cdots < d_k = n,", "c9d22706549fca3852eafcd293479d1b": "u_{\\Omega} := \\frac{1}{\\mathrm{meas} (\\Omega)} \\int_{\\Omega} u(x) \\, \\mathrm{d} x", "c9d2726c3c015d93ae84119a6ff2e660": "\\psi'=-\\psi", "c9d282b20ca374e8f40bbc1d64673dba": " \\mathbf{x}^{(0)}=\\begin{bmatrix}\n x_1 \\\\\n x_2 \\\\\n x_3 \\\\\n\\end{bmatrix}\n=\\begin{bmatrix}\n 0 \\\\\n 0 \\\\\n 0 \\\\ \\end{bmatrix}", "c9d297dc6d122166938d5897268a9e0a": "y(t_n).", "c9d29c7d06056aa1136421453ab7d11c": "\\sum_{j=1}^{\\infty}|b_{j}|^2", "c9d2b77f0b929bed3f8119a8f5af8372": " h= H/kT ", "c9d2e8a10891dedec739c88f4dc1d601": "y(n) = s(n) - e^{-2 \\pi i \\omega} s(n-1)", "c9d2ecfa2dd065a82c49b6d53df00f9a": "n_1 = 4 \\pi \\int_{r_0}^{r_1} r^2 g(r) \\rho \\, dr, ", "c9d309601e835deab3b6dbeea118f9da": "n_{r,s}", "c9d3138d0dcd486f1acb2bea1e166f5a": "\\frac{P \\to Q, R \\to S, P \\or R}{\\therefore Q \\or S}", "c9d31cbf37b14ec21709c5e08a2e3c26": "h_*(X)\\cong \\pi_*(X_+\\wedge E),", "c9d35ef47df0eda3c8fa1eb961354ede": "Q = V_\\mathrm{rms}I_\\mathrm{rms}\\sin \\left(\\phi \\right)\\,", "c9d391447607423c49abb72d9ca1dbc4": " Y = \\alpha_0 + \\alpha_1 X + \\varepsilon. \\, ", "c9d39fbf76c1fade999693d89224c2b5": "y'=x \\sin 2\\theta - y \\cos 2\\theta.\\,", "c9d3a57a408b4e35134e439beca1a329": "\\mathrm{return} \\colon A \\rarr \\left( A \\rarr \\mathrm{M} \\, R \\right) \\rarr \\mathrm{M} \\, R = a \\mapsto k \\mapsto k \\, a", "c9d3d24661854fe88885b38f5f4fbaa2": "\\mathrm{d}P = \\sum_i \\omega^ie_i,\\, ", "c9d4316711ad0aa45d89b984ae36c43f": "\\|x\\| \\,", "c9d474736f9092a47629852c69a8b85c": "\n\\begin{align}\n& 1 + \\left(\\frac{1}{2}\\right) + \\left(\\frac{1}{4}+\\frac{1}{4}\\right) + \\left(\\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8} + \\frac{1}{8}\\right) + \\left(\\frac{1}{16}+\\cdots+\\frac{1}{16}\\right) + \\cdots \\\\[12pt]\n=\\;\\; & 1 \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\frac{1}{2} \\;\\;+\\;\\; \\cdots \\;\\;=\\;\\; \\infty.\n\\end{align}\n", "c9d4e4b0ca05a4659edf4aab2c5b40e5": "I = 4O^{1/2}", "c9d52c2eebbd7222fdade755cb9c6941": "O_n = 3n^2-2n", "c9d54bff73d93e4acb0d33b235cbbc90": " \\text{Inv-Gamma}\\left(a_n,b_n \\right) ", "c9d54e8191898755d823728b108a560a": " (q^2,q)", "c9d585c020232236282ccd7d3acebfb1": "\\vec p = m \\vec v = \\gamma m_0 \\vec v. \\,", "c9d5b38d357dc8c8cc283cab67efd7ba": "\\bigcup_{f\\in\\mathcal{F}}f(U)", "c9d5daaa013038f5e652d01599894a4d": "M_\\text{Na}=-M_\\text{Cl} = {\\sum_{j,k,\\ell=-\\infty}^\\infty}^\\prime {{(-1)^{j+k+\\ell}} \\over { (j^2 + k^2 + \\ell^2)^{1/2}}}. ", "c9d5f7b4ec5731bb123d7ececa3d65ef": "\nPr\\begin{cases}\nDs\\begin{cases}\nSp(\\pi)\\begin{cases}\nVa:\\\\\nS^{0},\\cdots,S^{T},O^{0},\\cdots,O^{T}\\\\\nDc:\\\\\n\\begin{cases}\n & P\\left(S^{0}\\wedge\\cdots\\wedge O^{T}|\\pi\\right)\\\\\n= & \\left[\\begin{array}{c}\nP\\left(S^{0}\\wedge O^{0}|\\pi\\right)\\\\\n\\prod_{t=1}^{T}\\left[P\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)\\times P\\left(O^{t}|S^{t}\\wedge\\pi\\right)\\right]\\end{array}\\right]\\end{cases}\\\\\nFo:\\\\\n\\begin{cases}\nP\\left(S^{0}\\wedge O^{0}|\\pi\\right)\\equiv Matrix\\\\\nP\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)\\equiv Matrix\\\\\nP\\left(O^{t}|S^{t}\\wedge\\pi\\right)\\equiv Matrix\\end{cases}\\end{cases}\\\\\nId\\end{cases}\\\\\nQu:\\\\\nMax_{S^{1}\\wedge\\cdots\\wedge S^{T-1}}\\left[P\\left(S^{1}\\wedge\\cdots\\wedge S^{T-1}|S^{T}\\wedge O^{0}\\wedge\\cdots\\wedge O^{T}\\wedge\\pi\\right)\\right]\\end{cases}\n", "c9d604551c74893f9e7d0f728ac158ff": "\\,\\setminus\\, \\{u\\}\\cup\\{(\\gcd(g,u),u/\\gcd(g,u))\\}", "c9d61811c307e6ced4299f57dab81fb1": "f(t,x)", "c9d61ae63ce32406c623706b3701127b": "k_{\\rm B}", "c9d6509663d0fb9742ba46b8da871b89": "\\ S(f) = \\mathbb{E}|X(f)|^2", "c9d666ed97b70b831219c68ee48c74dd": "\\begin{bmatrix} -\\frac12\\boldsymbol\\eta_2^{-1}\\boldsymbol\\eta_1 \\\\[5pt] -\\frac12\\boldsymbol\\eta_2^{-1} \\end{bmatrix}", "c9d6715331b507d547d7287f8dfbd4b8": "\\mathbf{y}_{i} = C_{i} \\mathbf{x}_i + \\mathbf{w}_i", "c9d6756dad0a5f1e08c641644447dfae": "w(x^{q}, y^{q})=(x^{q^{2}}, y^{q^{2}})", "c9d691913ce84e581996ce3b22af9efd": "\\underset{\\mathbf{S}} {\\operatorname{arg\\,min}} \\sum_{i=1}^{k} \\sum_{\\mathbf x_j \\in S_i} \\left\\| \\mathbf x_j - \\boldsymbol\\mu_i \\right\\|^2 ", "c9d6ff7f9a282d858ef661dbbaeb21ee": "E_i \\le \\frac{\\Delta^3}{24}\\,f''(\\xi) ", "c9d70c0710e008f7ef10713ebdbe3771": "c=o(1)", "c9d71577d1724d2bc687f8d81c5d402a": " \\|f_n\\| \\to \\|f\\|,", "c9d73a0afd7356a06fcd2eda6d98cb82": "s \\cdot t", "c9d73bd1d7a0b69cbc9bd6d6bbb1b4e7": "r=a \\sin (k\\theta)\\,", "c9d75653c5e99c6711ac81690c60f2f0": "L'_0 = \\mathrm{EF} =30\\ \\mathrm{cm}", "c9d7603b508bd2bf5fa8f5f999f2e640": "g(\\alpha)=f(\\alpha)\\,", "c9d76340ae0e19ac049b8713c5ce8d69": "\\Psi_{id}^{\\otimes n} = \\Psi_{id} \\otimes \\cdots \\otimes \\Psi_{id}.", "c9d7a4e21cfb2b85ba91d18505ab8bf1": "\\ x^3 + a = b x", "c9d7ae7b3d472efc6cc7341cc8511fb8": " \\frac{\\mathrm{ft^3}}{\\mathrm{lbm}} ", "c9d7c17c81bf6f2a10918e22c6ff4384": "d_L(z)", "c9d7e5f4841732330a575f8d96cff2c2": "\\aleph_{\\beta} \\ge \\operatorname{cf} (\\aleph_{\\alpha})", "c9d7f1d63a4c4386b516807f6fb9fb3d": "C = -R^{-1}T = -R^T T", "c9d85885f13dfa766bc91181bbd5fc5d": "HInd + H_2O \\rightleftharpoons H_3O^+ + Ind^-", "c9d8791ed3ece6f348fa154453655621": "\\ln(1+x)= x \\,\\left( \\frac{1}{1} - x\\,\\left(\\frac{1}{2} - x \\,\\left(\\frac{1}{3} - x \\,\\left(\\frac{1}{4} - x \\,\\left(\\frac{1}{5}- \\cdots \\right)\\right)\\right)\\right)\\right) \\quad{\\rm for}\\quad \\left|x\\right|<1.\\,\\!", "c9d8859b4eeb120d209c937fca3a53af": "(1,3)", "c9d8e3c2c3da09156af81170239879a1": "\\frac{6(\\alpha^3+\\alpha^2-6\\alpha-2)}{\\alpha(\\alpha-3)(\\alpha-4)}\\text{ for }\\alpha>4", "c9d8febb3d14e4698000844de6e330cc": "{\\mathcal L}^2_0", "c9d95fc6927072f9f2d160b73b1f9738": " H=H_0+H_1\\ ", "c9d98d83fd8b8c0d1f8c0b37c4c8bc61": "\\scriptstyle \\, (T_p)^r{}_sM", "c9d9c12c57ff1153307791c1ba8416ab": "D = \\frac{S / 100}{mil} \\cdot 1000", "c9d9c6c65e6192d5a18911fd6f24a1ac": " \\mathcal{L}_X ", "c9d9fad0d9254cd9827b43cae88ad658": "\n EI~\\cfrac{\\mathrm{d}^4 \\hat{w}}{\\mathrm{d}x^4} - m\\omega^2\\hat{w} = 0\n ", "c9d9fbe7bfed94c13d571fdd050400ee": "\\frac{4}{3}\\pi\\left(\\frac{h}{2}\\right)^3 = \\frac{\\pi h^3}{6}.", "c9da16320b5154f50dd47f9010455c88": "x=a, y_i=b", "c9da953ad99c8b1de2959c85fc81ec00": "f(x)=1/x", "c9db0818268102735575605d2e53818c": "L_1 \\,\\!", "c9db1aafcb6843dee801da4e555641f7": "{\\mathbf{s}}_i", "c9dbd0cec39285478ad184e50353a229": "\\Delta: H_*(LM)\\to H_{*+1}(LM)", "c9dbe1a9975c953d0f4d1566a1d71189": "\\zeta(s)=\\prod_{p\\in\\mathbb{P}}\\left(1-\\frac{1}{p^s}\\right)^{-1}", "c9dc4d121ad764d54d41717f8b996629": " \\ln \\Gamma(\\eta+1)+(\\eta+1)\\ln 2", "c9dca9c6afd08aa6a4ba934521bab6aa": "n_y^e", "c9dce5584c55d43a51211b3b5a638866": "a_{n+1}/a_n = (1/2)^{n+1}/(1/2)^n", "c9dcee6a10434ea27798961ed46f5cc5": "\n\\chi (R/P,R/Q):=\\sum _{i=0}^{\\infty}(-1)^i\\ell_R (\\mathrm{Tor} ^R_i(R/P,R/Q)).\n", "c9dd0d197b698278a5eb45c9c28cbd09": "\\int\\mathrm{hacovercosin}(x) \\,\\mathrm{d}x = \\frac{x - \\cos{x}}{2} + C", "c9dd4d84a888612326973056466d2aad": "\\Phi_B = \\Phi \\cap (B \\times B)", "c9dd6be410a7627657e2875616c8022c": "(0=0 \\wedge 1=0) \\vee (1=0 \\wedge 1=1)", "c9dd77ff84a0d599ff6177c0cb551efe": "\\det(I- \\Delta S) = 0", "c9ddb1218cbfa94a7f48c7af92b7a652": " \\operatorname{E} \\operatorname{tr} e^{\\mathbf{H} + \\log(\\mathbf{Y})} \n\\leq \\operatorname{tr} e^{\\mathbf{H} + \\log(\\operatorname{E} \\mathbf{Y})}. ", "c9ddb5690486998785bd40fb2ef6a423": " \n\\rho \\rightarrow { \\rho \\over {4 \\pi \\varepsilon_0 } }\n", "c9ddd88cd14a5c3a625ecb4f0d1a4dbc": "A\\mid B\\Rightarrow A\\cap B=\\varnothing,", "c9de826ab3f0491b1ce250f9f1c1b185": "\\begin{smallmatrix}M_{\\odot} \\end{smallmatrix}", "c9de9995570f14f7987e9bcf1ae4dece": "\\sum_{n=a}^{a+q-1} c_q(n)=0.\n", "c9df2e96d98346b30c2a358bcdb532b7": "\\mathbf{a} = (a_\\text{x}, a_\\text{y}, a_\\text{z}).", "c9df540c5c67c2516cc9be59f3afbfa3": " \\tan \\varphi = \\left(\\frac{c_2}{c_1}\\right), ", "c9dfc7e0ed7aa869d50d10222bc8a213": " \\begin{align} \n E + S \n \\underset{k_{-1}}{\\overset{k_{1}}\n {\\begin{smallmatrix}\\displaystyle\\longrightarrow \\\\ \\displaystyle\\longleftarrow \\end{smallmatrix}}}\n ES\n \\overset{k_2}\n {\\longrightarrow}\n EI\n \\overset{k_3}\n {\\longrightarrow}\n E + P\n\\end{align}", "c9e002408e9fab50d51d7a64e3ff2da0": "\\alpha'(G)", "c9e0517f2e0422106d9aa78410cc040a": "|0\\rangle = \\begin{pmatrix} 1 \\\\ 0\\end{pmatrix}", "c9e07abe688bef76069a3bdb2f7a0b16": "d=4:", "c9e0987516389147a7b80abbec77955c": "\\inf\\emptyset=\\infty. \\,", "c9e0e892053abe7089b3529948f5f4e4": "\\int_\\Omega f(x) w(x)\\, dx", "c9e1224f11042a6ce23ddf2aa7e699d1": "k*i + 1 + c", "c9e1625f358d16d1447a5fc9477970fe": "\n0\n=\n\\frac{1}{\\sigma^2} \\left[ \\sum_{n=0}^{N-1}x[n] - N A \\right]\n=\n\\sum_{n=0}^{N-1}x[n] - N A\n", "c9e177716150e63acb5af4e6f900f0b8": "\\{s_0,\\dots ,s_{d-1}\\}", "c9e1ad19e7b30a61be742447a5450241": "\n\\sum_{c\\equiv 0\\, \\text{mod}\\ N} c^{-r} K(m,n,c) g\\left(\\frac{4\\pi \\sqrt{mn}}{c}\\right) = \\text{Integral transform}\\ +\\ \\text{Spectral terms}.\n", "c9e1d76f350462b0332732d39d9099f0": "\\scriptstyle Q_i", "c9e21eee70d8d62ef87177392c37304b": " \\mathbf{E} = - \\nabla \\Phi =\\frac {1} {4\\pi\\epsilon_0} \\left(\\frac{3(\\mathbf{p}\\cdot\\hat{\\mathbf{r}})\\hat{\\mathbf{r}}-\\mathbf{p}}{r^3}\\right) - \\frac{1}{3\\epsilon_0}\\mathbf{p}\\delta^3(\\mathbf{r})", "c9e2299bf0bccd17d8cc4c499242f7fd": "\\sigma \\approx 5.6704 \\times 10^{-5}\\ \\textrm{erg}\\,\\textrm{cm}^{-2}\\,\\textrm{s}^{-1}\\,\\textrm{K}^{-4}.", "c9e22c6fc6ef9e76c6b9b3cdca93dd8f": "\nD^\\ell_{-m s}(\\phi,\\theta,-\\psi) =(-1)^m \\sqrt\\frac{4\\pi}{2\\ell+1} {}_sY_{\\ell m}(\\theta,\\phi) e^{is\\psi}\n", "c9e23d0af15ad6712da53fa016ce3a3e": "c(v\\otimes w):=v_{(-1)}\\boldsymbol{.}w\\otimes v_{(0)},", "c9e283523e827d2297b85ecfbdab1b15": "T_D^{-3}\\propto c_{{\\rm eff}}^{-3}:=(1/3)c_{{\\rm long}}^{-3}+(2/3)c_{{\\rm trans}}^{-3}", "c9e2a4cd3832c3a9a4eafc009679317b": "e^{-\\mathbf At}\\mathbf x(t) - e^0\\mathbf x(0) = \\int_0^t e^{-\\mathbf A\\tau}\\mathbf B\\mathbf u(\\tau) d\\tau", "c9e2a57391780b8885b423bb0cf73468": "\\ \\beta ", "c9e2f676a686f522f6315e25d6305376": "q_1 \\cong \\langle a_1, ... , a_n \\rangle", "c9e2fb1e2b9e23c5a3a72c431ae8d0a9": "r \\frac{\\partial}{\\partial r}= x \\frac{\\partial}{\\partial x} + y \\frac{\\partial}{\\partial y} \\,", "c9e32f55856330e41e54708d09fddf53": "\n\\frac{\\mathrm{d}}{\\mathrm{d}z}\\, \\mathrm{sn}\\,(z) = \\mathrm{cn}\\,(z)\\, \\mathrm{dn}\\,(z),", "c9e33bda0a10cadeb91ff50b7f85e332": "2T_{n}", "c9e365e1f326f3300adbdafae0c46043": "\\pi=\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\ 1 & 4 & 2 & 5 & 3 \\end{pmatrix},", "c9e3a0918e3fa54369a7392545e75f81": "z^a = z^b \\iff a\\equiv b \\pmod{ n}.", "c9e400d571227955f40d36332131b835": "\\left( \\mathrm{vol} \\left( \\mathrm{E} [X] \\right) \\right)^{1/n} \\geq \\mathrm{E} \\left[ \\mathrm{vol} (X)^{1/n} \\right],", "c9e4256a6b1758bb5584f79a51691b68": " c_V = (d\\delta_{ij} -n_{ij})_{ij} ", "c9e4356a5d51c9f1b7679834cfe67c2c": " \\left .b_n\\tau+\\sqrt{b_n^2\\tau^2-b_n\\left[ 2\\left[x_{n-1}(t)-s_{n-1}-x_n(t)\\right] -v_n(t)\\tau-v_{n-1}(t)^2/b\\right]} \\right \\}", "c9e45574fc5ca7b0c39b5cedad7366de": "||f||_{Q,2} = \\left(\\int |f|^2 d Q\\right)^{1/2}", "c9e47d0323b4d3ed7e10b8f1850ab636": "\\tilde \\Gamma^k{}_{ij} = \\Gamma^k{}_{ij}+ \\delta^k_i\\partial_j\\varphi + \\delta^k_j\\partial_i\\varphi-g_{ij}\\nabla^k\\varphi ", "c9e483f43831258c3c1972ff101d13cd": "S(P,T)=S(P_0,T_0)+\\int_{T_0}^T \\frac {C_P(P_0,T^\\prime)}{T^\\prime}\\mathrm{d}T^\\prime-\\int_{P_0}^P \\alpha_V(P^\\prime ,T) V(P^\\prime ,T)\\mathrm{d}P^\\prime.", "c9e486d10bebd50c8ab44af877d4f7fa": "K \\subseteq X", "c9e4c42809b8ff637b7154c5fb43c750": "P(\\theta | k, x_1, \\dots, x_N) \\propto \\frac{1}{\\theta} \\prod_{i=1}^N f(x_i; k, \\theta)", "c9e4deb04357c5717c40214ee51fe0e4": "f_i : D \\rightarrow \\{0,1\\}", "c9e53cbdffe795c0913cd927f13ffb9b": "\\hat{y_i}", "c9e587f8c1feaf904b121da41257f516": "\\begin{align}\n f(\\text{Allan})&=2005 , \\quad & f(\\text{Brad})&=2007 , \\quad & f(\\text{Cary})&=2001 \\\\\n f^{-1}(2005)&=\\text{Allan} , \\quad & f^{-1}(2007)&=\\text{Brad} , \\quad & f^{-1}(2001)&=\\text{Cary}\n\\end{align}\n", "c9e5d5c88ee653a8b7b3e2e9df3edcbc": " \\frac{\\delta V}{\\delta \\rho(\\boldsymbol{r})} = \\frac{1}{|\\boldsymbol{r}|} \\ . ", "c9e5eecafec8afa5193711b343631e9b": "1 \\le p < \\infty", "c9e6add8cd90d3cbfa67587d96f2e6ec": "\\pm2", "c9e6d67b06a9ea164ebbfa3a2f82887f": " C = A {|S_2 - S_1| \\over S_2} \\,.", "c9e6eb46f03cb1cd903e21f14aeb0317": "p_\\mathrm{ref}", "c9e7a152c5de594cd76dc614f55aeef3": "h(t')", "c9e7c9acd7d98e85019b4a2ee49b8fc6": " W_{NP}=W_{T}+W_{C}+W_{CT}+W_{HT}+W_{A} \\,", "c9e823bc06ae693de5322ad5eedfbbe6": "I/I_S=e^{V_D/nV_T}-1", "c9e8674d8a251bd4d2b99b591fa43544": "\\exp(it\\Delta)", "c9e882fae96c74df4aa01fecfa18823e": "\nX|Y\\sim GD_k\\left(\n{\\alpha'}_1,\\ldots,{\\alpha'}_k;\n{\\beta'}_1,\\ldots,{\\beta'}_k\n\\right)\n", "c9e8baba64e459b56f787170bf18f5a1": "\\lang \\epsilon_i|\\epsilon_j\\rang = \\delta_{ij}", "c9e8bbc3daa750fbdc9197227612486a": "\\left\\vert f(z) \\right\\vert \\le 1", "c9e9ceb8590c81c16a6eae6c7d63093e": "\\beta_2^2\\omega_m^2 + 2 \\gamma P \\beta_2 < 0", "c9e9d2cbacab376ddf0e65b110dde296": "\\frac{ax + b}{cx + d}", "c9e9e8c9a995727797dff58d8c2e965c": "\\Phi_n(x) = \\Phi_p(x^{p^{m-1}}) =\\sum_{i=0}^{p-1}x^{ip^{m-1}}.", "c9ea18f8058ded3222a0755282987c8d": "(\\vec r -\\vec a)\\cdot \\vec n = 0\\,", "c9ea2a9a35cf78d4d62486498b58f385": "\\frac{x^2}{9}+\\frac{y^2}{25}=1", "c9ea93360efc0741a820cfd76f5fbe5c": "\\| f_{n} \\|_{L^{1}} = \\int_{0}^{1} | f_{n} (x) | \\, \\mathrm{d} x = \\frac1{2}.", "c9ead1ba0d0d402391bae39022fde508": "\\begin{bmatrix} 4 & 1\\\\6 & 3 \\end{bmatrix}\\begin{bmatrix}x\\\\y\\end{bmatrix} = 6 \\cdot \\begin{bmatrix}x\\\\y\\end{bmatrix}", "c9ead3d8d2dac7680155de247cb813a3": "I_{max} = \\frac{|A_m|^2}{2 \\eta_0 / n} = \\frac{|\\beta_2|}{T_0^2 n_2 k_0}", "c9eb3a8422dcb08e4dcbf6b4e73bbd73": " D = \\mu \\, k_B T ", "c9ec9d9189574aea4824ecb9d520d8d6": " R_g ", "c9ecc81f658e6b2c2e2dd076dbb4b879": "\\text{Pad}_n^s=\\lbrace\\mathbf{\\xi}=(\\xi_1,\\xi_2)\\rbrace=\\left\\lbrace\\gamma_s\\left(\\frac{k\\pi}{n(n+1)}\\right),k=0,\\ldots,n(n+1)\\right\\rbrace.", "c9ed1201b8240205b17a5f3b2c29eb9c": "\n \\left[\\dfrac{PbL^2}{6} + C_1 L \\right] - \\cfrac{P(L-a)^3}{6} = 0\n ", "c9ed1ec784571ddbef29beb26c8e3552": "\\scriptstyle \\gamma(t)", "c9ed2a2f8740b7ca218cf0cda383065c": "k(\\theta) \\int_{-\\infty}^\\infty h(x)e^{x\\theta}\\,dx\\,\\!", "c9ed5b5d3cc2357f11e58493a119a897": "\n||y||_B=\\sup_{x\\in B}|\\langle x, y\\rangle|,\\qquad y\\in Y,\\qquad B\\in{\\mathcal B}.\n", "c9ed6acda7286e65d4b4e64a29b899df": " \\mathbf{r} \\equiv r \\mathbf{\\hat{e}}_{\\parallel} \\equiv \\mathbf{d} - \\mathbf{r}_0 \\,\\!", "c9ede97a164750d7842a6e081b371d73": "\n\\lim_{t \\rightarrow \\infty}\\frac{\\Lambda^{(k,k+1)}(t)}{t} = \\lambda^{(k,k+1)} > 0\n", "c9ee421b36818ec7858cbe4104ba0aae": "\\overline{x}=(x_1+\\cdots+x_n)/n.", "c9eeafde51764ed382d3133c550bb9f1": "y = {1\\over2}\\left[\\frac{\\hbar\\omega}{k_B T_e}\\right]^2. ", "c9ef0509acff2300d536e626bc65b751": " a_{11}-\\mathcal{L}(a_{30} \\omega + a_{21}) = p_3(a_{30}\\omega+a_{21})- \\omega p_7+p_8,", "c9ef39a3cdd96671d211f631fd54e954": "U(x,\\omega)= e^{-i \\Phi x} A(x,\\omega)", "c9ef4cfdc45b0f715a1fb44bd25bdc42": "u^2-(609)(7766)v^2 = 1 \\,", "c9ef9cf8733ebf3f233aa6eafda45d88": "q(t)", "c9efe7575ce7ad08179d2480c0539d13": "\\hat{f}_i \\,\\hat{f}_j^\\dagger = \\delta_{ij} - \\hat{f}_j^\\dagger \\,\\hat{f}_i .", "c9f01f65aff169af5212b07942b6a071": "F = P \\cdot A", "c9f0c0b544fa2c741b97dc51945b9b41": " (s, t_s, t_e)", "c9f0f895fb98ab9159f51fd0297e236d": "8", "c9f1093c794fa00d02cd098edcf7d530": "\\text{let } U(\\theta) = \\max_{\\theta'} u\\left(x(\\theta'),t(\\theta'),\\theta \\right)", "c9f136c330dbdb5e9f2425db022afbf7": "\\gamma_K\\,", "c9f1a7ba49c473c8da5277ba79aa5167": "\\omega_w = \\sqrt{\\frac{1}{L_wC_w}}. \\ ", "c9f203d4be3583c74a03a1f777f5a878": "\\frac{\\partial V(x)}{\\partial x} = -(R + j \\omega L)I(x)", "c9f27196ee198d1be5e344d16af3a6c7": "\\int_a^b\\! e^{M f(x)}\\, dx\\approx e^{M f(x_0)}\\int_a^b e^{-M|f''(x_0)| (x-x_0)^2/2} \\, dx", "c9f284be5ab4620437ab8ee4ba467daa": "\\phi(t),\\phi(2t),\\phi(4t),\\dots,\\phi(2^n t),\\dots", "c9f2b49f42dbc410f73959629b867c19": "n>3", "c9f2e2fe32c2cfc41f2eb71ecf724584": " 0\\in F_p", "c9f3113e87c798e12ac06af20366a69a": "k(T)=\\frac{1}{S} \\sum{\\alpha} {\\int_{0}^{\\frac{\\pi}{a_z}} \\lambda_\\alpha (k_z) \\frac{\\hbar\\omega_\\alpha (k_z)}{2\\pi}\\frac{df_B}{dT}v_z(\\alpha,k_z)\\, dk_z}", "c9f31da8cb8c18cb56b7bc317a3c7d69": "\\beta =\\gcd\\left(a_{t,j_t}, a_{k,j_t}\\right)", "c9f3433b5167ffe1916d64e7f52cc6ca": " \\operatorname{lift-choice}[M] \\ne \\operatorname{none} \\to \\operatorname{lift-choice}[M\\ N] = \\operatorname{lift-choice}[M] ", "c9f36bf7ade8c3da5ed9e22bc4ea3454": "T_0 = 2\\pi\\sqrt{\\frac{\\ell}{g}}", "c9f382d6ab39f9514fc4481cf8faad1b": "J_1(x)", "c9f3a9a8f07dc1d7b92cd0555448c299": "\\mathcal{F}_s \\subseteq \\mathcal{F}_t \\subseteq \\mathcal{F}", "c9f3d350c0b75b92665684225defefe6": "(R-r_{in})^2=d^2+r_{in}^2,", "c9f3eee30983151d65a10cc3816d9a12": "v > \\tfrac{2a}{1 + a^2}", "c9f3fa44c3f1f5af4c3511ab986a4fd5": "\\sum_{n=0}^{\\infty}(a_n)", "c9f45135ce2dff9cb86a23da7ea4a328": "\\hat{J_2} = J_2 +m_2l_2^2", "c9f49883404bb50eb681b342ff2b9c96": "\\pi_n(X,x_0)", "c9f4cf2912849ba50a3d9ecc53ea2213": "K_{4,2}", "c9f4d6799882cf39480a2ce63e1f7973": "\\ v_{ref}", "c9f50dfae92dc7c9b54610290b149dd1": " G_i,g_j: R^{n_x} \\times R^{n_y} \\to R", "c9f53f8a852cdfbc74d837354c8146bc": "R(C)", "c9f553f3a6b70a7c0890f9a55a332596": "{\\tilde{B}}_{4}", "c9f57b1720bbd798dd911c586b411af5": "RD = \\frac{\\rho_\\mathrm{substance}}{\\rho_\\mathrm{reference}}", "c9f58b0e445b4087c7551e4f279e6a28": "L_\\text{water}(T) = (2500.8 - 2.36 T + 0.0016 T^2 - 0.00006 T^3)~\\text{J/g},", "c9f5c5847ea57416dd5be41be1c84797": "\n \\Gamma_{ijk}\n = \\tfrac{1}{2}[(\\mathbf{g}_i\\cdot\\mathbf{g}_k)_{,j} + (\\mathbf{g}_j\\cdot\\mathbf{g}_k)_{,i} - (\\mathbf{g}_i\\cdot\\mathbf{g}_j)_{,k}]\n", "c9f649ad515a9a3f593ae407e3354e04": "\\lim_{h \\to 0} \\frac{ \\| f(x + h) - f(x) - A_x(h) \\|_{W} }{ \\|h\\|_{V} } = 0.", "c9f67086e3c7d379feaa01ca4fba2c2b": "\nD_0 = N - \\gamma \\, \n", "c9f68641e6f1b3e4a665902692715437": " \\bar{X}_2-\\bar{X}_1 \\pm A \\sqrt{\\frac{S_1^2}{n_1} + \\frac{S_2^2}{n_2} } ", "c9f68b377570a68423515eebf2f21458": "\\dot \\gamma(t)", "c9f690febaa367db62d8fb4e9131f202": "\\scriptstyle\\mathfrak{P}_2", "c9f6c36c9a6d9edadbfe19d120049f40": " R^2 = (R-y)^2+r^2.", "c9f6d8557ce40f989fa727b5c0bb1ddf": "b_{i}", "c9f6f794d3557b7094db2abc511ff2cf": "1.8\\overline{3}", "c9f70af82bb2314b7488b365d69a7fa7": "\\frac{\\partial^2}{\\partial x^2_1} f(x_1, x_2, \\ldots, x_n)\\,,\\quad \\frac{\\partial^2}{\\partial x_1 x_2} f(x_1, x_2, \\ldots x_n)\\,,\\ldots, \\frac{\\partial^2}{\\partial x^2_n} f(x_1, x_2, \\ldots, x_n) ", "c9f73560a49ac606526c538f8ee59657": "\\frac{\\operatorname{d}^2 \\theta}{\\operatorname{d}t^2} = -U^2 \\left(\\frac{2k}{rd}\\right) \\theta", "c9f73beaf315c07675d9614f35178b60": "|\\psi_{11}\\rangle = |-\\rangle = \\frac{1}{\\sqrt{2}}|0\\rangle - \\frac{1}{\\sqrt{2}}|1\\rangle.", "c9f7fbbd0960114976dcfbf271384814": "\\Pi_i = \\left(\\frac{a - c} {N+1}\\right)^2 \\left(\\frac{1}{b}\\right)", "c9f805b074f96369d6b9ec1d2fef869d": "z!=\\sum_{n=0}^{\\infty} g_n z^n.", "c9f84296ccb926f1c09325020785ebce": "\\int_{E}\\phi\\, d\\mu<\\infty", "c9f84492226d497c9329ba6217eb08ae": " 100MCalories=10*pesticide/transporational.impact.vegetarian.city ", "c9f85335464bb6c63d57f1d5cbd73048": "g_k=\\inf_{n\\ge k}f_n.", "c9f8ca3b883aff3fdf6a544d23aaecf0": "\\left(A, m\\right)", "c9f8f5c239dabed80993fad5ebd98325": "\nG_{p,q}^{\\,m,0} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) = 0,\n", "c9f8ffc57a37f2644c393b7b53888188": "E_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\varepsilon _{o}\\varepsilon _{r}}(Ae^{-jk_{x\\varepsilon }x}-Be^{jk_{x\\varepsilon }x})sin(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ \\ (35) ", "c9f918cd05c15458197e9131d595a8e8": "\\dot{v} = (F\\cos\\alpha)/m - g\\cos\\theta\\,", "c9f9f054f1f4c52ddee33c939284576e": "\\ COP = \\frac{T_L}{T_H - T_L}", "c9fa094416ba158959ac46f26d5d1e81": " {n + 1 \\over n} = 1 + {1 \\over n}. ", "c9fa0e1df2f00351fe06ac44b28b04f9": "\\scriptstyle -(t_i \\,-\\, t_{\\text{rec}})", "c9fa73eec2f241f70b8c8aa5b034a124": "d([L]\\mathbf{v}, [L]\\mathbf{w})^2 = d(\\mathbf{v},\\mathbf{w})^2,", "c9faccbe6d781f628e1f4f788d0b681e": "(x_3-x_2)", "c9faecdc3525a38e99c35f10ece60f68": "(\\mathrm{id}_C \\otimes \\Delta) \\circ \\Delta = (\\Delta \\otimes \\mathrm{id}_C) \\circ \\Delta", "c9faf6ead2cd2c2187bd943488de1d0a": "\\mu", "c9fb03807e5948dd21aabc1d9a674a68": "\\partial M = K \\colon S^1 \\hookrightarrow S^3", "c9fb41e1191cd3bd188f5d985344a78c": "\\overline{\\left|c_{n}\\right|}=O\\left(n^{n\\varepsilon}\\right)", "c9fbaa73edf22b0ceb36c44fb2b23f91": " \n = 2A \\sum_i \\left ( p_i\\sum_k p_{ik}x_{ik}x_{ik}^T - \\sum_{j \\in C_i} p_{ij}x_{ij}x_{ij}^T \\right )\n", "c9fc071d5f63c695c365b3cbe80f3c8c": " P V = n R T\\,", "c9fc2107a55a1905c20016b28a5cecb1": "P(Z \\le -1.20) = P(Z \\ge 1.20) = 1 - 0.8849 = 0.1151", "c9fd0d7b20de1c2ce80c5a3fe4411f86": "y = \\frac{1}{1+e^{-f(X)}}", "c9fd1f8227515d5a22ac35c4b7d80596": " v(x(S),\\tau(t)) ", "c9fd34196a32312249c955c2ca7a8040": "\\{\\breve z_1^j,\\ldots,\\breve z_m^j\\}", "c9fd6a6f2b2363083f916d1852a1b176": "\\Lambda = \\langle \\alpha, z\\rangle", "c9fd828f81361dbb2cf33d1148df0634": "f'(z_i)", "c9fd97d5dd4d1ab2a6731d46f8828acb": "\\deg(f,\\bar\\Omega,p)\\neq 0", "c9fdebd74abb1c174c39899d64ccad2b": " \\frac{d}{dt} \\det A(t) = \\mathrm{tr} (\\mathrm{adj}(A(t)) \\, \\frac{dA(t)}{dt}).\\,", "c9fe1896372109abad5322e6576219e7": " \\overline{u_i u_i} \n= \\overline{\\left( \\bar{u_i} + u_i^\\prime \\right)\\left( \\bar{u_i} + u_i^\\prime \\right) }\n= \\overline{\\bar{u_i}\\bar{u_i} + \\bar{u_i}u_i^\\prime + u_i^\\prime\\bar{u_i} + u_i^\\prime u_i^\\prime}\n= \\bar{u_i}\\bar{u_i} + \\overline{u_i^\\prime u_i^\\prime} ", "c9fe37cfa65085aeed89790b2d2de1e7": "p - 1 \\mid n - 1", "c9feb4923241913422fab22239fe8bfa": "M_C", "c9fed88bd0dd62a938ccd54b35394a7b": "\\eta_0 = \\left(\\frac{4 \\pi^2 F_s^3 V_{as}}{c^3 Q_{es}}\\right)\\times100\\ %", "c9ff04fac81e9fddb41c31293cc9b590": "\\displaystyle l", "c9ff05db5e5a5e1338a3eb610a14f697": "M_y=\\int^z_0 \\rho v dz.\\,", "c9ff4c32e8cea5599488d336c218e86e": "\\scriptstyle v_{\\text{in}}", "c9ffbab4bc27002a534928d2fea8f7af": "a_e, b_e \\geq 0", "c9fffd55da4c45160eb3e5dde05c0ebc": " \\tau_1 \\approx \\hat { \\tau_1} =\\ \\tau_1 + \\tau_2 = C_2 (R_1+R_2) +C_1 R_1 \\ , ", "ca000054c1f306b929f3a12841c6a2a6": "1/b_{1^{ }}", "ca0042a81ce16dcc2a990071ce717aa1": "13^{n^2}(1-1/13)(1-1/13^2)\\cdots(1-1/13^n).", "ca0051fa6190ae0251f5493af07eafa3": "w_a", "ca0099e2a950ae79421446aadb968c11": "F\\,'(x) = G\\,'(x)", "ca00a34859aed5be8611c55580b20c36": "j\\ne k", "ca01292403c32d820adeadfbacaf6228": "(a,t) \\in A\\times I", "ca01dbdef4209aed5e440f3483dafc77": " B_{n+1} = \\sum_{k=0}^n {n \\choose k} B_k,", "ca01e423ab2e83c38d1b56b4dc822f3f": " h = r_pv_p = r_av_a = const", "ca01f08d893ecc06b702c7d4b5bac524": "\n \\operatorname{E}[\\,x_1x_2x_3x_4\\,] =\n \\operatorname{E}[x_1x_2]\\,\\operatorname{E}[x_3x_4] +\n \\operatorname{E}[x_1x_3]\\,\\operatorname{E}[x_2x_4] +\n \\operatorname{E}[x_1x_4]\\,\\operatorname{E}[x_2x_3].\n ", "ca0265dd33c8f45c2509270cbbf10ae7": "\\frac{d\\psi}{dt}", "ca026eca2e108819a28d6312fe761675": "\\rho=\\sum_{\\sigma}\\sum_i f_i\\,\\!", "ca0299d6414a4123734a930a634c4960": "V \\ge .2 LC_{50}", "ca02ac5ef60b36ee3db4562bcb89bf3f": "Q(x)\\,", "ca03118a2eeab0d2d5836e849ba495ea": "\\mu_m\\,\\!", "ca033eb38560089eb0d9c130d81da3f6": "\\alpha = \\sqrt{-2\\ln(y\\sqrt{2\\pi})}", "ca03c68643a916e925ae2cc6aa31d7f2": "F_z", "ca03fe0232491fc619f2fffe0fb3af6a": "{}+ b_1a_2i + b_1b_2i^2 + b_1c_2ij + b_1d_2ik", "ca0421055349524db50b6a35a66eff32": "\\nabla^2(\\nabla\\times\\boldsymbol{u}) - \\frac{1}{\\beta^2}\\frac{\\partial^2}{\\partial t^2}\\left(\\nabla\\times\\boldsymbol{u}\\right) = 0", "ca0470b59852210ea3b7057cc33e3e24": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} I_{lm} r^{l} \n\\sqrt{\\frac{4\\pi}{2l+1}} \nY_{lm}(\\theta, \\phi) \n", "ca048799666942a71b203dba6a70132d": "\\left.\\right. A^2_\\omega", "ca04c1939b412f3f69a1706be218227a": "AB^T", "ca05002fa3e3e77145d40a7723d9b38e": "s_i = s_{i-1} + 0.5+ r_i \\pmod 1. \\, ", "ca0519cdd9c3f477a52c079f5e9bf35a": " G = 2 \\cdot N \\cdot \\left[ H(row) + H(col) - H(row,col) \\right] , ", "ca05201ab173066ca4612c0efa8e611b": "v_\\mathrm N", "ca0521eb8f8b5acb145ba2336e4cc18b": " \\begin{bmatrix} c & -s \\\\ s & c \\end{bmatrix} \\begin{bmatrix} a \\\\ b \\end{bmatrix} = \\begin{bmatrix} r \\\\ 0 \\end{bmatrix} . ", "ca0534f2231ee4517c5f17b53e030003": "\\mu = i = ", "ca0553abf648c5ea5396fe0f82750912": "P_s \\leq{} 4Q\\left(\\sqrt{\\frac{3kE_b}{(M - 1)N_0}}\\;\\right) ", "ca055dbb4710fa3e17334e8523efa3c3": "\\mathcal{Q}\\rho ={{e}^{\\mathcal{Q}Lt}}Q\\rho (t=0)+\\int_{0}^{t}dt'{e}^{\\mathcal{Q}Lt'}\\mathcal{Q}L\\mathcal{P}\\rho (t-{t}').", "ca0564b0beb07e149e55786fd2a9691e": " -\\frac{a}{3} u", "ca2fdeeda6da5cd9a8d48be56a1705b5": "\\rho, z", "ca2ff5bb05401c71f6f033f17666b0ff": "\n\\mathbf{L} = \\mathbf{I} \\cdot \\boldsymbol\\omega \n", "ca3013a63e265a64aad06c6605157513": "QH^*(X, \\Lambda) \\otimes QH^*(X, \\Lambda) \\to QH^*(X, \\Lambda)", "ca303d45c19e3a41011d8631afe9560d": "\\begin{matrix}\n&H_B=TH_{B-l}T^\\dagger\n\n&S_{x_B}=TS_{x_{B-l}}T^\\dagger\n\n\\end{matrix}", "ca305dc1070ea7157d672ea58b457d84": "\n W = C_1\\left[\\tfrac{1}{2}(I_1-3) + \\tfrac{1}{20N}(I_1^2 -9) + \\tfrac{11}{1050N^2}(I_1^3-27) + \\tfrac{19}{7000N^3}(I_1^4-81) + \\tfrac{519}{673750N^4}(I_1^5-243)\\right]\n ", "ca30685dbe1e4e24a6351155e51fa5c7": " Standard~score~(z) = \\frac{ | Mean - (individual~measurement) | }{s.d.}. ", "ca30a606cb97d97e4692a891c8db726c": " SubCipher_2=ENC_{f_2}(k_{f_2},s) ", "ca30af5e4a36b251dc6f2663fc6b9691": "p(\\mathbf{Z}|\\mathbf{X},\\boldsymbol\\theta)", "ca30cc33180978cb531a37d1b49bb664": "\\begin{align}\n\\det\\begin{bmatrix}\\frac{d\\beta}{dt} & \\frac{d^2\\beta}{dt^2}\\end{bmatrix} &= \\det\\begin{bmatrix}\\frac{d\\beta}{ds}\\frac{ds}{dt} & \\left(\\frac{d^2\\beta}{ds^2}\\left(\\frac{ds}{dt}\\right)^2+\\frac{d\\beta}{ds}\\frac{d^2s}{dt^2}\\right)\\end{bmatrix}\\\\\n&=\\left(\\frac{ds}{dt}\\right)^3\\det\\begin{bmatrix}\\frac{d\\beta}{ds} & \\frac{d^2\\beta}{ds^2}\\end{bmatrix}.\n\\end{align}", "ca311c1c3b05ff98ce62f9319b73c513": "\\scriptstyle r \\ll d", "ca31284209947ebede61b34d2a37f1a3": " \\pi(x,t) = x p(x) - C(x) - t x \\quad", "ca313a2e6353cba078e68e1062eff8ca": "\nq_{13} = f(T_1,T_3) = f(T_1,T_2)f(T_2,T_3)\n", "ca3144dd08fa52ec75c9a047bf25dccf": "\\mathbf v = (\\mathbf v \\cdot \\hat{\\mathbf u})\\hat{\\mathbf u} + \\hat{\\mathbf u} \\times (\\mathbf v \\times \\hat{\\mathbf u})\n", "ca3168b7393f2a606b87d77174aaa883": " \\mathbf{F} = \\frac{q_1 q_2 \\mathbf{\\hat{r}}}{4 \\pi \\epsilon_0 r^2}(1-\\frac{\\dot{r}^2}{2 c^2}+\\frac{r\\ddot{r}}{c^2}) ", "ca31aa65cc6d5502bddd4da7949f751c": "A_{rz}", "ca31bf76e4203a8d384da5c629a2cb05": " \\int_{t_0}^{t_0+h} f(t,y(t)) \\,\\mathrm{d}t \\approx h f(t_0, y(t_0)). ", "ca31e4005548b3b7ec949dd2f4ed19d0": " \\frac{\\mathrm{d} p^\\alpha}{\\mathrm{d} \\tau} = q U_\\beta F^{\\alpha \\beta} ", "ca31ee0fd445b67fadc26641bfa7ca59": "H_{n,1}", "ca326671c80e1fc4098e28436a0870bb": "\\simeq 10^{\\left ( 10^{100} \\right )} ", "ca32aef26e29ba6edd09dc4aad5ff839": "u_{static}=\\frac{F_{1,eq}}{k_{1,eq}}", "ca3387950abfe81bb73462866f7b0850": "4fh=r^2", "ca33e33fc4ea490f8773a91ef2939aa2": "\\varphi(z) = \\log \\left| f(z) \\right|", "ca3404321755fb5487dd6aa1b19db03c": "A + CH_4^+ \\to CH_4 + A^+", "ca340abf4b48dc6d816137fbadf58b53": "\\mathbf{X}", "ca3435b52f132ad819ccc5f1e82d3f58": "u_i = z_i q B_i = -\\frac{z_i^2 q^2 \\kappa}{4 \\pi \\varepsilon_r \\varepsilon_0} \\frac {1}{1 + \\kappa a_i}.", "ca34eb0f0054332c53e95df341b11708": "\\begin{align}U(a,b,z)&= e^{(1-t)z} \\sum_{i=0} \\frac{(t-1)^i z^i}{i!} U(a,b+i,z t)=\\\\\n &= e^{(1-t)z} t^{b-1} \\sum_{i=0} \\frac{\\left(1-\\frac 1 t\\right)^i}{i!} U(a-i,b-i,z t).\\end{align}", "ca34f72d168ae23248aa9dc27a3c726a": "(N+1)^2", "ca3510049563178df4c612de670f49f8": "r_{(t+1)}", "ca3511c234a346ebe9ea777272c3ec8c": "a_n = r_n \\cos \\left( \\varphi_n \\right) = \\frac{2}{T} \\int_{t_0}^{t_0+T} x(t) \\cos(2 \\pi n f_0 t) \\, dt \\ ", "ca35b17e3ecc704eed18222ffb0bf173": "{\\hat{\\boldsymbol\\alpha}}", "ca35d37831ca15a214cf9df466ab1935": "\\sigma_{Age \\ge 30 \\ \\and \\ Weight \\le 60}(Person)", "ca36b26c6fab5d460a63193ef6932064": "\\scriptstyle 4\\pi r^3/3", "ca36c744ceee798a009d3dc94f4f8c0d": "\nH = -\\sum_{i_1 < i_2 < \\cdots < i_r} J_{i_1 \\dots i_r} S_{i_1}\\cdots S_{i_r}\n", "ca36dc2391c3dc5d0303f930f2514107": "\\sum_{n=-\\infty}^\\infty |a_n|^2 = \\frac{1}{2\\pi}\\int_{-\\pi}^\\pi x^2 \\, dx", "ca37047542363c42bfd5057493b1919a": "10^2+5^2+2^2", "ca37af3dcd95401f287f1f3aeaa212af": "X_i(t+\\Delta t)\\frac{}{}", "ca37de0562a095a9e4420eca9d454a75": "\nik A_{m (m+k)} \\approx \\left({T\\over 2\\pi} {dA\\over dt}\\right)_{m (m+k)} =\\left({dA\\over d\\theta}\\right)_{m (m+k)}\n\\, .", "ca37f925c3d7014fba2392c8d44403a4": "w_n(x)", "ca37fdd86e98c3ddedec0e81c71ba7f8": " = \\int f(a,x)\\,dx + \\int f((a+1),x)\\, dx + \\int f((a+2),x) \\,dx + \\dots ", "ca381123670d00f81f30c0ac57dd603b": "\\text{Gain}=20 \\log \\left( {\\frac{I_\\mathrm{out}}{I_\\mathrm{in}}} \\right)\\ \\mathrm{dB}", "ca385d14b5e4cc1536a5f1da6b18b87c": "Ty", "ca38a4e7b543da7c40233091659e88de": " F_x ", "ca38a79d8d8e1bc9b73c5201a36bbb56": "\\hat{H}_Q = -eT^2(Q)\\cdot T^2(q) = -e\\sum_m (-1)^mT^2_m(Q)T^2_{-m}(q)", "ca38b0a4ccfa77766d4cca06351a1416": "\\mathbf{d}_{i}", "ca38de6f0df6e552d41786a486bf81b1": "\\overbrace{ 1+2+\\cdots+100 }^{5050}", "ca390e3b12dd31fd98fead52be2450f6": "t_{TOF'}\\,", "ca394704e08fdb3a4ac04715ffdd5f25": "84^2", "ca3956b18a30ba6c0eed55127f2e02bb": "(P \\or (Q \\or R)) \\Leftrightarrow ((P \\or Q) \\or R)", "ca396eda7f14dfb905de55ead08316d8": "f_c = \\frac{R}{2\\pi L}", "ca39d92c9ab15125755d236c2f0ae204": "= -k_1 x - k_2 x \\,", "ca39ff9c5c441805cb9722c244715607": "\\{ x_i \\}", "ca3a2d954a4e622b835ab6a7693973aa": "\\mathbf{p}\\rightarrow p_i = m w_i", "ca3a494cf696d7726780f466dc08b4dc": "\nc_2 = -\\frac{1}{6} \\quad ; \\quad c_3 = -\\frac{1}{108} \\quad ; \\quad c_4 = \\frac{7}{3240} \\quad ; \\quad c_5 = -\\frac{19}{48600} \\ \\dots\n", "ca3a5328710f4a178a6e809be7b96199": "\nCl^{\\geq}_t = \\bigcup_{s \\geq t} Cl_s \\qquad Cl^{\\leq}_t= \\bigcup_{s \\leq t} Cl_s \\qquad t \\in T\n", "ca3b0f172a8a85f387992a723f660e51": " \\scriptstyle \\phi_1", "ca3b5141a01266863b1b8ee548618dfc": " P_{absorb} = \\frac{4.54}{R^2} cos \\alpha", "ca3b7fc4c18dd06696e77e07d4847751": "\\Delta_1.", "ca3c157d99041a150a7060c4daa93c84": "H(W), H(H(W)),\\dots, H^n(W)\\,", "ca3c6b5f5fb898decf49c3b03dc25bf3": "\\tau^2 > 0\\,", "ca3cb1a5061b25a6429bff8db2d70bb7": " Floor ", "ca3cf5b3067d4eb24309280a9f1f00e5": "\\underline{\\mathcal{O}}^n", "ca3d3bdec1f3a566de92ba5f6c3e078a": "E \\in \\operatorname{FV}[G] \\and E \\not \\in \\operatorname{FV}[H] \\to \\operatorname{sink}[(\\lambda E.G\\ H)\\ Y, X] = (\\operatorname{sink-test}[(\\lambda E.G)\\ Y, X])\\ H ", "ca3d6e61da204ed7ce12987ecf394ab1": "L_{\\gamma+1}", "ca3db94a315c1d83827f480cea60f8a6": " k_y = k \\sin \\theta \\sin \\varphi \\, ", "ca3dbeb237208317f8948c58a2c6348a": "\\mathbf{v}_{pi}=\\mathbf{v}_i+\\mathbf{\\omega}_i \\times \\mathbf{r}_i", "ca3de9dbb5e4fb4e3bff9da5fd4de969": "\\left[{n\\atop 3}\\right] = \\frac{1}{2} (n-1)! \\left[ (H_{n-1})^2 - H_{n-1}^{(2)} \\right]", "ca3e0b73b2e3f51c76a8a358ec5778dd": "P(f)\\sim f^{-\\beta}\\!\\ ", "ca3e1373d2796cb1ce5ebede3db8de8f": "\\lambda = \\frac{2\\pi}{k}", "ca3e1d11dceec442bc72a01e2dde280a": "\\phi_{j}", "ca3e2efbd2215db39faecb773c99e679": "T_{uv}", "ca3e3a1c6c820ffe144363186ac2d53c": "T(N) = 2N + T(N/2)", "ca3e7b9baf648987da5ccae92943c5c1": "\\frac{\\partial E}{\\partial t}\\rightarrow 0\\,\\quad \\nabla \\times \\mathbf{A} = \\mathbf{B} \\,, ", "ca3e7dd15b63542a4945febc00e27e46": "\\xi = e^{2\\pi i / p}", "ca3ef6af3d9068fc91244cf127ba89f9": " \\boldsymbol{\\sigma} e_i = \\lambda_i e_i", "ca3f37bd8d3de6c8723735bfe0599ca1": "\\tfrac{r^2}{2}", "ca3fac171904eb9e3b69eb758bd84bc1": " \\sigma_m = \\infty ", "ca3fc7e8442d5a1f193a75e82b5a0ede": "\\delta t=0.8\\pm0.7\\ (\\mathrm{stat.}) \\pm2.9\\ (\\mathrm{sys.})", "ca400ab3f30b1ab90106538b85afbd2e": " [a,b] ", "ca40265493ab34dc463665b171173c0a": "a/D", "ca402af223179b50dc6332d3e470aff5": "\\dots \\to H^{i - 1}(C(f)) \\to H^i(A) \\xrightarrow{f^*} H^i(B) \\to H^i(C(f)) \\to \\cdots", "ca4030a2411ac3388d30b205050da7dd": "p_M = e^{-\\Delta U/k_B T} \\, ", "ca403bc5ab8bfe9ed8cf4a4455c36e02": "|\\zeta(x)^3\\zeta(x+iy)^4\\zeta(x+2iy)|=\\exp\\sum_{n,p}\\frac{3+4\\cos(ny\\log p) +\\cos (2ny\\log p)}{np^{nx}}\\ge 1", "ca4056ecf002db3499f6648197fb2883": " \\text{number of sidereal days per orbital period}=1+ \\text{number of solar days per orbital period}", "ca405d6de9067fa4efc8770a9ba6d06e": "\\dfrac{d}{dx}(u\\cdot v)=u\\cdot \\dfrac{dv}{dx}+v\\cdot \\dfrac{du}{dx}", "ca40911e670f6a940a13adf025808cad": "\n =\\frac {\\mu_0r^2N^2\\pi}{l}\\left( 1 - \\frac{8w}{3\\pi} + \\frac{w^2}{2} - \\frac{w^4}{4} + \\frac{5w^6}{16} - \\frac{35w^8}{64} + ... \\right)\n", "ca40e36597e8172b37547992668a8aa7": "\\mathbf{V}^{*} \\mathbf{V} ", "ca410954cf8187f8618781597c5f94c1": "\\begin{align}\nq_+\n&=1+\\tfrac12(wh)^2+wh(1+\\tfrac18(wh)^2-\\tfrac{3}{128}(wh)^4+\\mathcal O(h^6))\\\\[.3em]\n&=1+(wh)+\\tfrac12(wh)^2+\\tfrac18(wh)^3-\\tfrac{3}{128}(wh)^5+ \\mathcal O(h^7).\n\\end{align}", "ca410eed5a0e3f705e1cbdaf85754105": " P = Se^{-qT}\\Phi(-d_1). \\,", "ca41224a67dbc2bccc8bee26d38abd57": "\\forall G \\subseteq Seq(S):", "ca4134f42692d4f9132a74259d7dc7ef": " (b \\triangleleft a) \\triangleleft a = b ", "ca4197a43611ccc422218d5abe7f90ae": "\\mbox{d}\\sigma(t')", "ca41d6379993cad01fc483a45795e532": "xFIP=\\frac{13(xHR) + 3BB - 2K}{IP}+C", "ca427d46c5332a9f086b2b064304cfd2": " \\gamma = \\frac{C_P}{C_V} = \\frac{c_P}{c_V}", "ca42866aed34982a586a06481f0b8fd5": " \nL_n^o=\\Gamma_n^o =\\overline{\\Gamma_n}^{*o}= \\langle \\mathrm{Shannon}_n\\rangle ^+\n", "ca4296c65f08ff79cc3f6ee207571c66": "p(t,T)", "ca429d9e8f3fb8d4a20f4ddd1c44637a": " v \\to v^q-v \\pmod f", "ca42b8cd5877d08f5e7043b188078e96": "\\Delta v_{total} \n= \\Delta v_1 + \\Delta v_2. ", "ca430deab67b5d394d7033ee20e7bebc": " \\begin{array}{llll}\n\\langle\\mathbf{A}\\rangle & = A_{11} \\mathbf{i}\\times\\mathbf{i} + A_{12} \\mathbf{i}\\times\\mathbf{j} + A_{31} \\mathbf{i}\\times\\mathbf{k} \\\\\n& + A_{21} \\mathbf{j}\\times\\mathbf{i} + A_{22} \\mathbf{j}\\times\\mathbf{j} + A_{23} \\mathbf{j}\\times\\mathbf{k}\\\\\n& + A_{31} \\mathbf{k}\\times\\mathbf{i} + A_{32} \\mathbf{k}\\times\\mathbf{j} + A_{33} \\mathbf{k}\\times\\mathbf{k} \\\\\n\\\\\n& = A_{12} \\mathbf{k} - A_{31} \\mathbf{j} - A_{21} \\mathbf{k} \\\\\n& + A_{23} \\mathbf{i} + A_{31} \\mathbf{j} - A_{32} \\mathbf{i} \\\\\n\\\\\n& = (A_{23}-A_{32})\\mathbf{i} + (A_{31}-A_{13})\\mathbf{j} + (A_{12}-A_{21})\\mathbf{k}\\\\\n\\end{array}", "ca4347f711bafa42b98cfa4d8e81c5fc": "\n\\mathbf{\\Phi}=\\begin{bmatrix} \\varphi_x \\\\ \\varphi_y \\\\ \\varphi_z \\end{bmatrix}=\\mathbf{v}\\phi\n", "ca43a8f2d66ea834977aea66a4e5fcec": "\\frac{7^7}{4^9} = 3.14156^+", "ca43cbf31ab9f9e65d146237df244dc8": "I_{\\mathrm{end}} = \\frac{m L^2}{3} \\,\\!", "ca4437074ea08654b181816476013a41": "K=\\{\\,z\\mid z\\geq 0\\,\\}", "ca444225a4567bb5d899606be1d91ba0": "4^{th}", "ca447c09149e79e3eb8c2e424cd715f9": "M_{ii}", "ca448b6565679673870fed1de7e1fff1": "m_T", "ca45668dd28bd94857b3fcb7682a7e8c": "s^i_t\\in S^i", "ca45d9e119b9261b3c5a488f6aa435b9": "x_n = 2", "ca45e07325c8f14f7cfa98c2eeda3548": "\n\\int \\hat\\mu''(x)^2 dx = \\hat{m}^T A \\hat{m}.\n", "ca461b05e87fb5611895c45d8398bdd3": "\\bigg\\{ \\Pr(h_2) \\bigg\\}", "ca462c60b6d14147bd3425106a346964": "\\mu(A)=0", "ca46459d0f64793bd95d0128e2bc2a57": "n\\mathbf{V}^{-1}", "ca4666169fc1a79bd1463eda11c2ac0e": "so(3,11) \\oplus 64", "ca46af7181e7d2a1eb5f406e46299aa7": "k\\sqrt{\\frac{1}{2}}", "ca46daf99b8e2e945e578ea173358ecc": " E^2 = m^2 c^4 + p^2 c^2 \\;", "ca46f9ad673047cf64ff67a48e8b36be": "\\ r\\ ", "ca470f35ea70b3fd60907a54251476ea": " X_L = \\omega_d L \\,\\!", "ca4723637dd6cde6b7e466e651721196": "M_{1} := \\{ \\delta_{n} | n \\in \\mathbb{N} \\}", "ca476bf226283d866190a67bd055c317": "\n\\begin{align} \nProb(choosing \\, 2) \n& = Prob(a < U_n < b) \\\\\n&= Prob(a- \\beta z_n < \\varepsilon < b - \\beta z_n) \\\\\n& = {1 \\over 1+exp(-(b - \\beta z_n))} - {1 \\over 1+exp(-(a - \\beta z_n))}\n\\end{align}\n", "ca476df819e099dbef16e03f651b6ba3": "r(x,u)", "ca47b441a73f2134ea7892d0464be09c": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{F}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "ca484ba17345d0e113e64c44e765ed7e": " \\max\\;u(\\mathbf{f}(\\mathbf{x}))\\text{ subject to }\\mathbf{x}\\in X,", "ca48611494a7c8a68040f424624f1daf": "n \\ge l", "ca488d2ff29206b83bc360235558b319": "\\operatorname{pd}_R k = \\operatorname{gl.dim} R < \\infty \\Rightarrow \\operatorname{pd}_{R_1} k = \\operatorname{gl.dim} R_1 < \\infty \\Rightarrow R_1 \\text{ regular}.", "ca48b256ecfd1401e6e6c3afaf0c4da6": " = *(f\\,\\mathrm{d}*\\mathrm{d}h + \n\\mathrm{d}f \\wedge *\\mathrm{d}h + \n\\mathrm{d}h \\wedge *\\mathrm{d}f + \nh\\,\\mathrm{d}*\\mathrm{d}f) \n", "ca492fcb3cb7411f324531287aa7c0e8": "Hw_1=Hw_2", "ca493052542df4c07307086dd4105ae1": "\\{x_{n}\\}_{n\\in \\mathbb{N}}", "ca4948e41a3d25b1945a40c933148330": "F_i, F_{ij}", "ca4973b38ed00c9f4cec2fb5b5f86f60": "e_j = U(\\lambda)/\\sin\\lambda", "ca497fc932694c6c3357da052cc24a92": "\\Theta_n / bP_{n+1} \\to \\pi_n^S / J , \\,\\!", "ca49942fc0f78b39734cd5833b169c81": " c_{1} , \\ldots , c_{5} ", "ca49afaee4017b91c4bff72b031d0dae": " (2\\theta -\\theta^{*}_{(1-\\alpha)};2\\theta -\\theta^{*}_{(\\alpha)})", "ca49b9cd334a802bcbe87468afc2866b": "F = \\frac12 \\times \\rho \\times S \\times C \\times {V_{apparent wind}}^2", "ca49dd5a10a5f741d23110908e6e956b": " \\gamma_{ws}", "ca4a6323b6ffa3e2e56b91518c4a4ef6": "\\alpha=0.5, \\beta=1.0, \n\\delta=-0.35, \\epsilon=0, \\zeta=-0.35, \\eta=0", "ca4a7a198c10851f4a9ae5dfada87df1": "\\sigma_L(\\cos\\theta + 1) ", "ca4ab7257b4d46f1ccee1f6b0f0cf0f5": "I_D V_B < P_{\\mathrm{MAX}}", "ca4b520013a0d2be8a714ef24e65d583": "\\pi := \\pi_1(X)", "ca4b535d841e93d95f0095e9184a846b": "\\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2R + c_8 T R^2 + c_9 T^2 R^2\\ \\, ", "ca4b543ad5dea50324126dfb755c2190": " \\operatorname{first}\\ (\\operatorname{pair}\\ a\\ b) ", "ca4b5d5447384df05e4d4d683332e627": " (\\mathbf{a \\times b})\\mathbf {\\cdot}(\\mathbf{c}\\times \\mathbf{d}) = (\\mathbf{a \\cdot c})(\\mathbf{b \\cdot d}) - (\\mathbf{a \\cdot d})(\\mathbf{b \\cdot c}) \\ . ", "ca4b6de5f9f1fcafa619c7a5d8b213f7": "a\\neq d", "ca4badab9f61aeb5f03d03e3f392cc3d": "\\forall i:\\textit{int}. \\; (i+i):\\textit{even}", "ca4beb438b76b4295d4640efa6d7ce66": " \\frac{1}{(\\beta + 1)(r_O + r_E)} ", "ca4c17b1591d82dedfa5bed369d27d7f": " \\log\\Gamma(1+x)=-\\gamma x+\\sum_{k=2}^\\infty \\frac{\\zeta(k)}{k}(-x)^k \\!", "ca4c2d9f7d366214c8d0934519f6a5b3": "D(\\mu) := \\operatorname{E}[a_\\mu(X)]", "ca4c8d434236c7210474ecb69d4fa342": "\\begin{align}\n\\frac{\\delta B_{\\alpha \\beta }}{\\delta t}& = \\nabla _\\alpha \\nabla_\\beta C - CB_{\\alpha \\gamma }B^\\gamma_\\beta \\\\[8pt]\n\\frac{\\delta B^\\alpha_\\beta}{\\delta t}& = \\nabla^\\alpha \\nabla_\\beta C + CB^\\alpha_\\gamma B^\\gamma_\\beta \\\\[8pt]\n\\frac{\\delta B^{\\alpha \\beta }}{\\delta t}& = \\nabla ^\\alpha \\nabla^\\beta C + 3CB^\\alpha_\\gamma B^{\\gamma \\beta}\n\\end{align}", "ca4cb23f65d08b75512a483f5a911273": " [\\mathbf{a}]_{\\times} = \\begin{bmatrix} \n 0 & c_2 d_1 - c_1 d_2 & c_3 d_1 - c_1 d_3 \\\\\nc_1 d_2 - c_2 d_1 & 0 & c_3 d_2 - c_2 d_3 \\\\\nc_1 d_3 - c_3 d_1 & c_2 d_3 - c_3 d_2 & 0 \\end{bmatrix}\n", "ca4d3c584044ba75b8db9fe197582875": "B_\\nu(T) = \\frac{2h\\nu^2/c^2}{e^\\frac{h\\nu}{kT} - 1} \\approx \\frac{2kT\\nu^2}{c^2}", "ca4d4f89eec34b8def3ad043828c067f": "\nA_2=-i A_1 \\chi_0 \\left( \\frac{e^{i \\Delta k \\Lambda}-1}{\\Delta k} \\right)\\left(\\frac{1-(-1)^N e^{i \\Delta k \\Lambda N}}{e^{i \\Delta k \\Lambda}+1}\\right).\n", "ca4d6099e9e66b438a9d516ab694b861": "e^{\\overline{\\Lambda}}", "ca4d8c5dffa4d37ad2cdffe0f0991aac": "\\scriptstyle \\binom n k", "ca4d925db0a235039946b9bb4208160a": "\n\\hat{P} \\Psi = \\Psi \\Longrightarrow \\mathcal{A} \\hat{P} \\Psi = \\mathcal{A}\\Psi \\Longrightarrow -\\mathcal{A} \\Psi = \\mathcal{A}\\Psi \\Longrightarrow \\mathcal{A} \\Psi = 0.\n", "ca4d948c0dac74222b96e5d24eeb8461": "X=\\{X_1,X_2,\\dots,X_n\\}", "ca4dc8699b2caeb765f35d2d29399520": "q_j", "ca4de10b776cc0e936bf1854545dfb7d": "\\frac{\\beta}{\\alpha+1}\\!", "ca4e00c75399ad8211abeacdf3beee6c": "\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}'},", "ca4e4be207b5d9773f8a46ead985c3c0": "\\ P_i = A_i + B_i", "ca4e6d4a5af272e8291e3ac20585fcce": "(f\\ominus b)(x)=\\inf_{y\\in B}[f(y)-b(y-x)]", "ca4e79f05ea7544b0f983fe3f6a331bb": " a \\ \\ = \\ \\frac{1}{\\sqrt{2}} \\left(\\ \\ \\ \\frac{d}{dq} + q\\right)", "ca4eb63ad8c09e4e71185920142be52e": "w^{k+1} = \\operatorname{prox}_{\\gamma R}\\left(w^k-\\gamma \\nabla F\\left(w^k\\right)\\right),", "ca4eb98d68527f3b09e6f79f714708ff": "\\scriptstyle\\text{curry}(f) \\; x \\; y", "ca4ebe33a82dce8bdff9d61fdece3b64": " T(f)(x)=g(x) ", "ca4ed39bcaf0c69bd6cb13ce51515754": " \\Lambda^3\\mathbb C^7", "ca4ed4cf6494cb597ecd8bd2a52b98d5": "J(C)", "ca4ef4fb66e456a1cf339e724ba07048": " t_0 ", "ca4f0172a524d17194756ab9ad2c19f8": "|\\alpha|^2 = \\hbar^2j(j+1) - \\hbar^2m^2 - \\hbar^2m = \\hbar^2(j-m)(j+m+1),", "ca4f5662ab50389455374eaf5dc34479": " \\operatorname{ch}(V) = e^{x_1} + \\dots + e^{x_n} :=\\sum_{m=0}^\\infty \\frac{1}{m!}(x_1^m + ... + x_n^m). ", "ca4f966bf05804449af414912cb54b5f": "s_{mx}", "ca500f2bfe2ef7552d49bb0304fdf808": "u_1 = (h * w)", "ca500fe49dfb66d996ba25b14ec56b20": " 1 = 1", "ca505f41c119e360a29d1318bd74b801": "\\begin{align}\n \\frac{\\partial y}{\\partial c} &= b_0 s^c \\ln(s) \\sum_{r = 0}^\\infty \\frac{(c - \\beta)(c)_r (c + 1 - \\gamma)_r}{(c + 1 - \\alpha)_r (c + 1 - \\beta)_r} s^r \\\\ \n & \\quad + b_0 s^c \\sum_{r = 0}^\\infty \\frac{(c - \\beta) (c)_r (c + 1 - \\gamma)_r}{(c + 1 - \\alpha)_r (c + 1 - \\beta)_r} \\left (\\frac{1}{c - \\beta} + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + k} + \\frac{1}{c + 1 - \\gamma + k} - \\frac{1}{c + 1 - \\alpha + k}- \\frac{1}{c + 1 - \\beta + k} \\right) \\right) s^r\n\\end{align} ", "ca5086268f1e6dbfd281f045a31bdaaa": "\\frac{\\partial^2 U}{\\partial g^2}<0", "ca50c3573868d2a5a976b65b9973a9c7": "P = \\left ( \\frac {13.7516 m} {1 ~ \\mbox {kg}} + \\frac {5.0033 h} {1 ~ \\mbox {cm}} - \\frac {6.7550 a} {1 ~ \\mbox {year}} + 66.4730 \\right ) \\frac {\\mbox {kcal}} {\\mbox {day}}", "ca5106aa2921dff466481f4573f11ed0": "P \\in (0,1)", "ca5107a6bba2d057c3f1b87a8cebdfca": "u_vv", "ca512dac73e727f13e88631a0f896fb2": "\n \\bar{F} = \\bar{a}^3 - \\cfrac{4}{3}~\\lambda \\bar{a}^2\\left[\\sqrt{m^2 -1} + m^2 \\sec^{-1} m\\right]\n ", "ca513a2c971fdfc74865ab264298800d": "P_n \\xrightarrow{D} Q", "ca51599b8ab0c5d4024a71676154ecab": "\\frac {d M_y(t)} {d t} = \\gamma ( \\bold {M} (t) \\times \\bold {B} (t) ) _y - \\frac {M_y(t)} {T_2}", "ca51672204b4260104972c632f8415c6": "\nH = \\frac{p^2}{2m} + mgz - \\frac{\\lambda}{2}(r^2-R^2) + u_1 p_\\lambda + u_2 (r^2-R^2) + u_3 \\vec{p}\\cdot\\vec{r},\n", "ca519607aac71360857f2227a99f90fe": "e^A", "ca51fdf2a0d3da59a1f57af5b707caab": "b \\;", "ca525653b61e31a0fe7cbf927991d633": "\\frac{f_2}{f_1} = 10^{s/1000}", "ca5265122c259734ca6cda4371b0e1fc": " \\sin(X) := \\sum_{n \\ge 0} \\frac{(-1)^n} {(2n+1)!} X^{2n+1} ", "ca52b5edd1e2c94535b31df2ee35e1d4": "b_{3} = b_{3}- b_{2}= \\begin{bmatrix}3\\\\5\\\\6\\end{bmatrix}- \\begin{bmatrix}-1\\\\0\\\\2\\end{bmatrix}=\\begin{bmatrix}4\\\\5\\\\4\\end{bmatrix} ", "ca52dc5ee5d369ceb0b5dcc5152402ed": "f_\\theta(x)=\\theta^{-1}\\exp\\left(-x/\\theta\\right), x,\\theta>0", "ca5300b91c2afcda98956e1a3949b9ae": "\\lim_{kh \\to \\infty} \\mathcal{S} = \\frac{1}{2}\\, ka.", "ca5312c4e747cd089e0e57efb3be61bc": " \\tan \\psi = \\frac{b}{a} \\tan \\phi\n", "ca5317e8d815f3eb02c46b55bc34687c": "F_1A_1=H\\,", "ca5335cea6ce65c66382ef398d91673c": "R_2(f)", "ca53dc5797146b7941540b4de112bbf9": "U = \\sum_i \\frac{GM_i}{r_i}", "ca54158b54c806a54650ee7de46cf51d": "\\hat{e}_i=\\frac{1}{\\sqrt{|e_i \\cdot e_i|}}e_i,", "ca547086bf2161b750921119ea5b840f": " Y \\sim \\textrm{Frechet}(\\alpha,n^{\\tfrac{1}{\\alpha}} s,m) \\,", "ca55173f7dc262224254ed269a296fe9": "d = \\| P2 - P1 \\|", "ca5564205807b35c1684bd9e244b0ae1": "R_\\text{OC}", "ca556931b8fbf4b25e60c3b86363b8e4": "D_C(A,B) = 1 - S_C(A,B)", "ca558d870121223b417d460c339ea9c6": "N_{2j}", "ca55bb8dc60b5e639d49d64666e65d71": "\\gamma_s \\equiv \\gamma_1/s", "ca5608a10b47747a0e19b081af68e324": "\\varphi\\circ j^k(\\sigma)", "ca562afe694df1ab85181ea75ff1eb23": "\\sum_{i=1}^n \\mathrm{Exponential}(\\theta) \\sim \\mathrm{Gamma}(n,\\theta) \\qquad \\theta>0 \\quad n=1,2,\\dots", "ca5666bdc7d1727bff570a7a305e1965": " \\prod_{r=1}^3 \\Gamma(\\tfrac{r}{4}) = \\sqrt{2\\pi^3} \\approx 7.8748049728612098721", "ca5699aaded29179a9bc8578a3d8d6e9": "y^2 = x^3+a_4x+a_6", "ca56baed90a90f5b5f674ef50e071f48": " X \\in (a,b), \\; -\\infty \\leq a < b \\leq \\infty ", "ca572fd5b0226a93753f5c913310348d": "A(x)=\\pi r^2", "ca573fbbee83053cded6e50a9594f099": "\\theta'=\\phi-2\\pi k", "ca574ebf24a720b8664e78bec4e721ab": "\\begin{align}\n\\frac{\\partial}{\\partial L} \\langle h(r) \\rangle & = \\displaystyle \\frac{\\partial}{\\partial L} \\left\\{ \\frac{1}{T} \\int_0^T h(r) \\, dt \\right\\} \\\\[1em]\n& = \\displaystyle\\frac{\\partial}{\\partial L} \\left\\{ \\frac{m}{L^{2}} \\int_0^{2\\pi} r^2 h(r) \\, d\\theta \\right\\} ~,\n\\end{align} ", "ca57698bcd7598b2f58850eeaccaf7bb": "m_\\odot=-26.73", "ca57ef612ec5a21b81fc8777680e340a": "i_{n-1}-i_{n-2}\\,\\!", "ca58144ef4fd77b26b79627d462eabb6": "I_o={E_c \\over R_v}", "ca59d303f44401eeea46cb5c88296f41": "{\\nabla ({\\partial \\Phi \\over \\partial p})}", "ca5a00dee41e2525ec05b9a916880a32": "R\\;\\overline{3}\\;m", "ca5a072afb58370d3ac54dbeb866d993": "K\\otimes_NL", "ca5a125a23b5e818c07f66150d9a694b": " a'_{h\\ell} = a'_{\\ell h} = a_{h\\ell} + s (a_{hk} - \\rho a_{h\\ell}) \\,\\! ", "ca5a1e65911645e5548819565c85e7ce": " \\begin{align}\nx' & = Ax + Bct \\\\\nct' & = Cx + Dct \n\\end{align}", "ca5a6dcf44f509ecdf36fb3e2b966c19": " X(t) \\approx X_m(t) = \\mu(t) + \\sum_{k=1}^m \\xi_k \\varphi_k(t),", "ca5a9b18dee212fed17a9d627e17afe1": "\\Delta_n.\\,", "ca5acc86a689d558caba0310ab1c246b": "\\Gamma(n+1) = n!\\;, \\quad n \\in \\mathbb{N}_0,", "ca5ace809b5e046cacb0e3e2f8c201d0": "u=g\\text{ on }\\partial\\Omega,\\,", "ca5b20ff668e070ea665028bf561a22e": "\\beta_\\mathbf{w}", "ca5b347fe0d701786235203137112828": "=e/\\sqrt{4\\pi\\alpha} ", "ca5ba62ef7dd6f13671899c7c725256d": "K_\\pm \\subset K_{\\pm\\pm} \\cup \\{0\\}", "ca5babd25a4d1cee429ca428a9f18257": "\n\\sum_{\\tau=0}^{t-1} y_i(\\tau) \\leq Q_i(t) - Q_i(0) = Q_i(t) \n", "ca5bbab847da8c6d1290fdd29440f105": " \\operatorname{build-param-lists}[E\\ P, D, V, R] \\equiv \\operatorname{build-param-lists}[E, D, V, T] \\and \\operatorname{build-param-lists}[P, D, V, K] ", "ca5bc2179a93c82d6c1b6b43a987d8f8": "\\ AB ", "ca5bd3801e15ac710d3ecd71ac30feee": "\\overline{S_n} = a \\mathbb{I}\\{S_n > a\\}", "ca5bfa6107f481248297e2122a478e90": " n = 37 ", "ca5c568a781511c36630e8aa3dedf9eb": " \\mathbf{A}_\\text{tri.} = {1 \\over 2}|x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3| ", "ca5cb2c515c42b7fb8402730dbb53887": "\nj_{\\text{t}}(t)=j_{\\text{t}}(\\eta(t))=j_0\\,\\left(\\exp(\\alpha_{\\text{o}}\\,f\\, \\eta(t))-\\exp(-\\alpha_{\\text{r}}\\,f\\,\\eta(t))\\right)\n", "ca5d039bfe11aa061287e5de5ea6fc3e": "\\lambda>0", "ca5d03e52573803ab5ec645614653ccc": "\\dot{u} = -\\delta v", "ca5d0fccc590f0ded4215ca76a160c64": "\\mathbb{N}\\times \\{2\\}", "ca5d7220f6bfb59af934e217673887e5": "(s_n)_{n>0}", "ca5d93cfd92d8150d6ec3857934b94a4": "\n \\mathrm{P}(A=0,B=1)=P\\{3,5\\}=\\frac{2}{6},\\; \\mathrm{P}(A=1,B=1)=P\\{2\\}=\\frac{1}{6}.\n", "ca5dd60c10e8388d3df8839551bc5ed4": "\\dot{k}(t) = sk(t)^{\\alpha} - (n + g + \\delta)k(t)", "ca5de8cc8e003dc4283feebc4a531b3d": "\\sum\\limits_{n\\in \\mathbb{N}_0} p(n) x^n = \\prod\\limits_{k\\in\\mathbb{N}} (1 - x^k)^{-1}", "ca5dfa028f0b9e09b129a1f7adfda2c4": "W = \\frac{1}{T} \\int_0^T hi\\,dt", "ca5dffb9d864e72342543fd96a83fd79": "\\boldsymbol \\phi \\rightarrow \\boldsymbol \\phi + \\delta \\boldsymbol \\phi ", "ca5e492b0abbbd1cf98a0160fe013279": "\\{\\gamma_1, \\gamma_2, \\gamma_3\\}", "ca5ea2a14de0e270fa028ed87c152246": " i = 1 + \\alpha (n - 1)", "ca5efb88bfe1c40d11da7bb1a2538508": "[{\\rm R}]", "ca5f071bdbace7962b681b3accecf917": "a=0:", "ca5f10fbebc53ad195a376485f4dcb31": "\\frac{\\part u}{\\part t} - c\\frac{\\part u}{\\part x} = 0 \\qquad \\mbox{and} \\qquad \\frac{\\part u}{\\part t} + c\\frac{\\part u}{\\part x} = 0", "ca5f2bc85dff34a9e9ad4c9b2e7f899a": "x(1\\text{ hr})= 1 \\cdot 2^6 =64.", "ca5f55984c33c5f2e2be2695241c712d": "7^3 = \\sum_{x=1}^b P(8 - x, 4)K(3,x) = P(7, 4)K(3,1) + P(6,4)K(3,2) + P(5,4)K(3,3)", "ca5f5a528fb1b3247999a5dbe4d1b17b": "K_a = \\frac{x^2}{C_a}", "ca5f63e98f4f7d5e6756745ecf9f927d": "$d = $t >> $s + \\left(\\sum_{n=1}^{$\\text{s}}2^{32-n}\\right)\\cdot \\left($t>>31\\right)", "ca5f973492be228b776d3bee624c8b31": "\\hat{D}(\\alpha)\\hat{D}(\\beta)= e^{(\\alpha\\beta^*-\\alpha^*\\beta)/2} \\hat{D}(\\alpha + \\beta)", "ca5f9b0e3ac836db4af2da954a716615": "\\displaystyle{\\dot{\\mathbf{v}}_s(t) =(1-s\\kappa)\\mathbf{t},\\,\\,\\,\\,\\partial_s |\\dot{\\mathbf{v}}_s|=-\\kappa.}", "ca5fd61bd86bb42898dd4e8aa276377e": "\n I_\\theta = \\mathrm{E}\\bigg[{- \\frac{\\partial^2 \\ln f(X_i,\\theta)}{\\partial\\theta\\,\\partial\\theta'}}\\bigg] = \\mathrm{E}\\bigg[\\bigg(\\frac{\\partial \\ln f(X_i,\\theta)}{\\partial\\theta}\\bigg)\\bigg(\\frac{\\partial \\ln f(X_i,\\theta)}{\\partial\\theta}\\bigg)'\\,\\bigg].\n ", "ca5fe5a402e8107f1e4fa7fd82d8c509": "\\frac{d}{dt}\\int_{\\Omega} L \\ dV = -\\int_{\\partial\\Omega} L\\mathbf{v\\cdot n} \\ dA - \\int_{\\Omega} Q \\ dV", "ca61070b9ec9e983eb5b1b62ae8f012d": " \\frac{2 \\sqrt{bcs(s-a)}}{b+c}.", "ca61072394cb73a2a2fd5518f790ee50": "\\int x^2\\sin^2 {ax}\\;\\mathrm{d}x = \\frac{x^3}{6} - \\left( \\frac {x^2}{4a} - \\frac{1}{8a^3} \\right) \\sin 2ax - \\frac{x}{4a^2} \\cos 2ax +C\\!", "ca6128dc9fd659e655ad5dfb6a424c2a": "\\mathrm{1.08\\overline{3}}", "ca6133eb8e2aa2dde7a8fca0fe431a6b": "\n\\begin{align}\n0 & = \\frac{\\partial}{\\partial \\sigma} \\log \\left( \\left( \\frac{1}{2\\pi\\sigma^2} \\right)^{n/2} \\exp\\left(-\\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{2\\sigma^2}\\right) \\right) \\\\[6pt]\n& = \\frac{\\partial}{\\partial \\sigma} \\left( \\frac{n}{2}\\log\\left( \\frac{1}{2\\pi\\sigma^2} \\right) - \\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{2\\sigma^2} \\right) \\\\[6pt]\n& = -\\frac{n}{\\sigma} + \\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{\\sigma^3}\n\\end{align}\n", "ca6155ab8706dab2f2a085a1437c64a9": "\\beta_D", "ca61589185ffb4d675af25c71eadfa57": "t_2(n)\\ge\\lceil 6n/13\\rceil", "ca619f672ad8c9b16803f4b8d95867bb": "x_i = \\cos\\left( \\frac{2(i+0.5)}{n+1}\\pi \\right)", "ca61c38208cfcc00971221beabe9acb4": "C \\lambda^{-n/2} \\, ", "ca61e726c82cbb59476897aba98b06c4": "x(s,t), -\\infty < s < +\\infty ", "ca620431690de0409eb116bda264a92d": "{T_I}^3\\frac{dT_I}{dr} = -\\frac {3\\kappa\\rho}{16\\sigma}\\frac{L}{4\\pi r^2}", "ca6268a688b181250d478abb1455e001": " d = ({{4 \\rho \\omega^2 \\over\\ 3 \\mu}t})^{-1/2} ", "ca62a9c6370f0602424190c8de603682": "\\int\\frac{x\\;\\mathrm{d}x}{1-\\sin ax} = \\frac{x}{a}\\cot\\left(\\frac{\\pi}{4} - \\frac{ax}{2}\\right)+\\frac{2}{a^2}\\ln\\left|\\sin\\left(\\frac{\\pi}{4}-\\frac{ax}{2}\\right)\\right|+C", "ca62c60650c196e25c3869f68de588c9": "A \\subseteq B", "ca63094f7601c4f632ff75bf226a774b": "A_{\\sigma|\\sigma'}=\\delta\\left(\\sigma_{1},\\sigma_{1}'\\right)\\sum_{interior\\atop spins}\\prod_{all\\atop faces}w\\left(\\sigma_{i},\\sigma_{j},\\sigma_{k},\\sigma_{l}\\right).", "ca6346121226169bae3191b28d4def27": "2^{n+k}-1", "ca63bd30ec596d6191fda36d60186716": "f(\\lambda x) = (\\lambda x)^n = \\lambda^n f(x).", "ca63be838055715bb6019cfb97b4eb61": "R = \\eta/2 \\left(\\frac{c}{\\lambda \\sigma}\\right)^{1/2}", "ca641d44faf37cf26c4e7441e85db1b3": "\\vec y_n.", "ca6428b5c69d539adfea27fbfc465f52": "x =\\text{mode} + \\kappa = \\frac{\\alpha -1 +\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "ca642d9ec4598efc8715ab81d2d11b87": "\\beta_1(t_1,a) = \\left(\\frac{1}{(1+t_1^2)(1+a^2)},t_1 - \\frac{2t_1}{(1+t_1^2)(1+a^2)},\\frac{t_1a}{(1+t_1^2)(1+a^2)}\\right).", "ca6447ad3c7816ea5490ec4c7165936e": "\\,\\,\\boldsymbol{\\sigma} = -p\\,\\boldsymbol{I} + 2\\mu\\,\\dot{\\boldsymbol{\\varepsilon}} + \\lambda\\,\\text{tr}(\\dot{\\boldsymbol{\\varepsilon}})\\,\\boldsymbol{I}\\,\\,", "ca645e22bc8059400804491f406a4970": "P_c^{ 0}(z) = z_0", "ca64c53266aaf76b91066e992b556327": "-56\\pm 20", "ca6521115a01f63b9dd58e98197974d5": "\\alpha_{(a,b)}=a+\\alpha_{(0,1)}(b-a)", "ca653e9c63b5a8e1065c4d99983099b7": " p_i(\\infty)= p_j(\\infty) ", "ca65434a0a455c7ba2238b28fcab62fd": "X = cos\\alpha sin(\\omega t + \\phi)", "ca656dc3ed9f9c45af4583704a479232": "Q \\ = \\ \\frac{\\sqrt{2^N}}{2^N - 1} \\ = \\ \\frac{1}{2 \\sinh\\left(\\frac{\\ln(2)}{2} N \\right)} ", "ca65709efae45ac03d01998460bc26d1": "g^{ab} = (g^b)^a", "ca6570f884ee93f5a59df3ebc2234a47": "\ne = \\frac{p}{2a} = \\frac{p}{r_{\\alpha} + r_{\\beta}}\n", "ca65b3373f02b4e8f548c037557a3eb3": "x^{1/2}\\log^{-A}x\\leq Q\\leq x^{1/2}.", "ca6632f64fe33d60da43ac7eaa203b58": "\\mathrm{conf}(X\\Rightarrow Y) = \\mathrm{supp}(X \\cup Y) / \\mathrm{supp}(X)", "ca66432afd4246fa3352e17ad2de3110": "a_0 = \\alpha / 4 \\pi R_\\infin \\,", "ca668bc1c077d9f3f899b8cfb58ff2f8": "%D-Slow", "ca669eec44b440571352e5a5e729c113": "P(0),P(1),P(2),\\ldots", "ca6749f8123e710cf6ec1b5aa55c55d4": "\\scriptstyle A'B'C'D'", "ca674dbcc19ab5d2f1401d8f33fb2777": "\\sigma(w)", "ca6780bd345bb574ff2245149feca8da": "\\begin{bmatrix} 3 & 1\\\\1 & 3 \\end{bmatrix} \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} 3 \\cdot 0 + 1 \\cdot 1 \\\\ 1 \\cdot 0 + 3 \\cdot 1 \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 3 \\end{bmatrix},", "ca680046f4d2b6e170d01c86364d78cd": " ~s^2={(x-x')^2+(y-y')^2+z^2} ", "ca6829009c5222299a0ae28aa462c11e": "x=X", "ca686968be433defb9a137078d41befc": "X^{p^n}-a", "ca68752ba1ad8144ed68cea3605188b3": " g_3=140\\sum_{(m,n) \\neq (0,0)} (m\\omega_1+n\\omega_2)^{-6}.", "ca68d42fb33b5e2f923a6e79a21b8082": "\\omega = \\omega_0 = { 1 \\over \\sqrt{LC}}", "ca68e4f935c9e7efe2dae818acd86efa": "\\textstyle x\\approx 0.4964", "ca692632e2ca83b98d343b042986dc98": "x^\\ast \\in \\mathcal{X}", "ca69278c501caf0643f9166ce791c6e5": "\\tfrac{\\alpha-1}{\\alpha + \\beta-2} ", "ca694452d86056b222bbe8ed752de1aa": "\\arccos(x)", "ca695675f9e0a76c77d5b776105a3b72": "v_{2,2} = 3", "ca695e1d93ecf1fbbbc954986c24568f": "\\frac{1}{\\Gamma(k) \\theta^k} \\int_0^\\nu x^{ k - 1 } e^{ - \\frac{ x }{ \\theta } } dx = \\tfrac{1}{2}.", "ca697fba78e79f3ba8dc4ffa6c3ccbd8": " v_g = \\frac {\\hbar k}{m} \\ , ", "ca69f25df34b95352316a2ede92a495b": "A_1 \\times A_2 \\times \\ldots \\times A_n = A_1 \\times (A_2 \\times \\ldots \\times A_n)", "ca6a0742b328df744ec45695e228afba": "\n\\delta\\phi(x) = h^{\\mu}(x)\\partial_{\\mu}\\phi(x) \n", "ca6aa31c75bc7c537c8c6ed73b8ec48f": " x^* \\mapsto x^*(x_1) \\, x_2.", "ca6ab83b8e776a261590a65d6b14ea0a": "~w^\\alpha", "ca6bb7687225c74863b1fe1ddf007c29": "\\phi : C_n \\to KT_n ", "ca6bf5afdca91bb863240f110233ba1b": "g(\\mu)=g_{obs}", "ca6bfd12bbcb3b3a226fd0eca5a9b4ea": "\n \\begin{align}\n \\frac{\\partial^2 Y_0}{\\partial t^2} + Y_0 &= 0,\n \\\\\n \\frac{\\partial^2 Y_1}{\\partial t^2} + Y_1 &= - Y_0^3 - 2\\, \\frac{\\partial^2 Y_0}{\\partial t\\, \\partial t_1}.\n \\end{align}\n", "ca6c35e079088f385ccbf587e6e56774": "\\sum_{n=1}^{\\infty}f(n)", "ca6c428ad94c65dac029446b7a03a304": "Me=\\beta_{50} +\\beta_{51}X +\\beta_{52}Mo +\\beta_{53}XMo + \\varepsilon_5", "ca6c43064f6af7dba375ff04e4a8a806": "\\Omega=\\Omega_1\\cup\\Omega_2", "ca6c746b987cd108a01666fece18ef17": "\\ell(\\ell+1) \\rightarrow \\left(\\ell+\\frac{1}{2}\\right)^2", "ca6c79f3b6c86aada75a5f71e6dfaefd": " J", "ca6cdd751d1f121b77876a417c41c16d": "\\frac{T_m}{T_t}=\\left(1+\\frac{\\gamma-1}{2}M_m^2\\right)^{-1}", "ca6d1ddf8069764225d869280766a2a9": "V_\\mathrm{anode}", "ca6d5b0806c95993a93b13436e03979d": " V_i = Z_c I_i \\, ", "ca6d7277b22a906a1c358ac96845bf26": "r^2 = R \\mod n", "ca6d9634e82f0ed91686e22b3f61e14c": "\\cos z=\\lim_{n \\to \\infty}\\prod_{m=1}^n \\left(1-\\frac{z^2}{((m-\\frac{1}{2})\\pi)^2}\\right)", "ca6db17e3a91271cdbb3a7e5c0a9735e": "i : (X-U, A-U) \\to (X, A)", "ca6db9512bf09c94c9bc165d5a1bd9c3": "\\gamma(t)=\\exp_p(tv)", "ca6ddd4edf96f295a2e4921ffa92578e": " log[\\rho(m)]= log[\\sigma(E)] ", "ca6de80bd6f2e7c5ef2f5bc328622838": "{\\rm Tr} [ \\bold{A} ].", "ca6e2d063bb6f17a819744200be4db67": "\n\\left.\n\\begin{array}{rcl}\n \\mathbf{u}_j &=& \\boldsymbol{\\nabla} \\Phi_j,\n \\\\[1ex]\n \\nabla^2 \\Phi_j &=& 0,\n \\\\[1ex]\n \\displaystyle \\frac{\\partial \\Phi_j}{\\partial n} &=& 0 \\quad \\text{ at } z=-h,\n\\end{array}\n\\right\\} \\qquad \\text{for all orders } j \\in \\mathbb{N}^+.\n", "ca6e404b3b7df69e664477ff08fa64fb": " H_1 \\otimes H_2", "ca6f0a84b02b12461f3b3fdc144fb8b1": "\\{\\tilde{\\mathcal{M}}f\\}(s):=\\tfrac{1}{\\sqrt{2\\pi}}\\{\\mathcal{M}f\\}(\\tfrac{1}{2}+is).", "ca6f2c2c6acb3f0888b3d99bc4d7658a": "R=\\frac{ l}{A}\\rho.", "ca6f4a4ef6ffeb8a65665d5a3cdc2187": "\\frac{1}{\\lambda}", "ca6f6e048ed306db98f30973f955f5a8": "m_i \\dot{\\mathbf{v}}_i", "ca6fa6d4c9e5a0f0f9d0b193aa2dda04": "d\\mu = fd\\lambda", "ca7028fe945da14f5005839243a10b03": "X_{\\bar{y}}:=X\\times_Y \\mathrm{Spec} (k(\\bar{y}))", "ca70a89e89344a42aab69f3b202af983": "\\mathrm{CH_2\\ +\\ HBF_4\\ \\rightleftharpoons \\ (CH_3)_3C^+\\ +\\ BF_4^-}", "ca70d3f394b63e661e022017a90840c2": " \\textbf{Q} \\setminus B ", "ca711d14f540fbef3811886ec9d54d81": "\\left(\\frac{d}{dt}J_{\\varphi_{01}}(\\varphi_1^{-1}(P(t)))\\right)", "ca71763703ead1f67a7105209e501ec1": "\\mu\\, =\\, \\frac{A_3}{A_2},", "ca71c47f2e99500d69bf4a03a6ecb996": "\\pi((g^t)^{-1})f_\\tau= (c\\tau+d)^{-1/2}f_{g\\tau}.", "ca7201c84297f4563de19e0384400117": "\\scriptstyle t : (1-t)", "ca721e8b2579d291bae8bde596602623": "K_{icl} = 0.016 \\sqrt{\\frac{E}{H}}\\frac{P}{(c_0)^{1.5}}", "ca723b74416f768adb426b24b4bf2020": "{\\ p = \\rho (\\gamma-1)e}", "ca728b023e7e48018393c09c863433cc": "K_D = \\frac{k_{\\text{d}}}{k_{\\text{a}}}", "ca729dd0a01cf97a91b9aed674af8dc8": "Y = Y_{c} + Y_{p}", "ca72c59e5daeaac51545bb705bbcdcb9": "\\frac{Q \\and P}{\\therefore P \\and Q}", "ca72ef2fc69a1596805551794b10fb5c": "Z(t)", "ca72f13a63f379e326347f762c79344c": "\\{\\cdot,\\cdot\\} ", "ca732cbf18944fdd2f8fbf53c93edba0": "\n \\Delta \\varepsilon_{ij} = \\dot \\varepsilon_{ij} \\Delta t = d_{ij} \\Delta t\n", "ca73362ac18f6aa3a9e45354bcc1a0d7": "\\frac{f/1}{(\\sqrt{2})^1} ", "ca735f095b94cb8669844ae74ed7f642": "\\textstyle\\sum_{k=0}^x \\{{ n\\atop k}\\}", "ca73d83520e53ad6406a542406c72392": "\\left(\\tfrac{\\partial H}{\\partial t} \\equiv 0\\right)", "ca73eee653d7a4b07ad58789589ac8ef": "\\mathfrak{e}_{8(8)}", "ca74540144a03cc3820d17e98739824f": "\\bigtriangledown", "ca74a1c3bd389e65f56e754231b160dc": "\\displaystyle C(x) > C(y)", "ca751f40f711f00dcb9fdc83e84c04e8": "\\bar{e}^{ch}=\\left[ \\bar{g}_{F}+\\left( a+\\frac{b}{4}-\\frac{c}{2} \n\\right)\\bar{g}_{O_{2}}-a\\bar{g}_{CO_{2}}-\\, \\frac{b}{2}\\bar{g}_{H_{2}O(g)} \n\\right]\\, \\left( T_{0,}p_{0} \\right)+\\bar{R}T_{0}\\, ln\\left[ \n\\frac{{{(y}_{O_{2}}^{e})}^{a+\\frac{b}{4}-\\, \\frac{c}{2}}}{\\left( \ny_{CO_{2}}^{e} \\right)^{a}\\left( y_{H_{2}O}^{e} \\right)^{\\frac{b}{2}}} \n\\right]", "ca752f455dc58d8257b545ab2f769fdf": "\n\\overline{\\mathbf{OP}} \\cdot \\overline{\\mathbf{OP^{\\prime}}} = R^{2}.\n", "ca7597d64239bed3ba564e1317e83daf": "\\scriptstyle{u_{ij}^{(\\pi)}}", "ca75a2d3279a6eebe9f813ab53756e3d": "\n\\frac{\\mathrm{d} p}{\\mathrm{d} x} = \\rho g \\beta \\Delta T,\n", "ca75a3023ef776f633622a8e5b043ce5": " 2~r^{-1}~\\sin\\theta \\,", "ca75a3a088c11a19e29bb586ffc01e23": "v = \\sqrt {{3\\, k\\, T}\\over{m}}", "ca765d65c3b3a769edb342d83be57e16": " \\begin{Vmatrix}\\mathcal{M}^1_1 & \\mathcal{M}^2_1 & \\mathcal{M}^3_1\\\\ \\mathcal{M}^1_2 & \\mathcal{M}^2_2 & \\mathcal{M}^3_2\\\\ \\mathcal{M}^1_3 & \\mathcal{M}^2_3 & \\mathcal{M}^3_3 \\end{Vmatrix} = \\begin{Vmatrix} 1 & 0 & 0\\\\ 0 & \\cos(\\theta ) & \\sin(\\theta )\\\\ 0 & -\\sin(\\theta ) & \\cos(\\theta )\\end{Vmatrix} ", "ca765f0b05782b03130aac35a6e4261a": "\\scriptstyle \\tau \\;=\\; RC ", "ca7669efd3da975569aae33501559584": "{\\tilde{G}}_{2}", "ca76a2f19d8747a464990c435c1f8c49": " |z| \\leq (a_n - a_0 + |a_0| )/ |a_n| ", "ca773eaa93d5c34f0fffcaf63b97394c": "\\frac{M_1 L_1}{E_1 I_1}+ 2M_2\\left(\\frac{L_1}{E_1 I_1} + \\frac{L_2}{E_2 I_2}\\right)+\\frac{M_3 L_2}{E_2 I_2} = 6 [\\frac{\\Delta A - \\Delta B}{L_1} + \\frac{\\Delta C - \\Delta B}{L_2}] - 6 [\\frac{A_1 X_1}{E_1 I_1 L_1} + \\frac{A_2 X_2}{E_2 I_2 L_2}]", "ca779884b073aa0737bcded30fc43b7b": "\\hat{\\boldsymbol{\\jmath}}(t) = (-\\sin \\Omega t,\\ \\cos \\Omega t ) \\ .", "ca779e988420c6642985da472d978b6f": "\\displaystyle c_b", "ca77ac448b2f6e5fe4eb25febb146238": "\\mathcal U", "ca77ae516454393f3947a605d03b0ae3": "y'=\\frac{y}{|cz+d|^2}.", "ca77baa174984994b648741752abfe84": "p^2", "ca77bfb8c7921b69943a3ee546fb5a88": "2 N \\log_2 N - N + 2", "ca77c8c1713973b114beec11c0adf840": "a_{\\sigma (i)} \\geq a_{\\sigma (i + 1)},\\ \\forall i = 1, \\cdots ,n - 1", "ca77f487286a25eb37a3444448ff7af1": "T_{\\textrm{n}}", "ca7817a5f37c820c1f2b6a78668b6d01": "-\\frac{\\text{polylog}(2,1-p)}{\\beta\\ln p}", "ca78687498b37c9a7b3809d7b1671b4f": "\\hat{a},\\hat{b}", "ca78824e181302e76f91c4fd43c847be": " \\mathrm{FillRad}(M\\subset E) = \\inf \\left\\{ \\epsilon > 0 \\left|\\;\\iota_\\epsilon([M])=0\\in H_n(U_\\epsilon M) \\right. \\right\\},", "ca788e6653806895e9ee1a532b5a62c3": "\\psi [x] = \\int dk \\psi (k) exp (ikx) ", "ca7890a680d48f8d8795b1e08b55ad80": "I_\\alpha f = f*K_\\alpha\\,", "ca78b6593436864698708ac6f279de95": "2n=2(2k+1)", "ca78dceacbd3816404956d20472d0345": "\ng \\bar{N}\\gamma^\\mu \\partial_\\mu \\pi N\n\\,", "ca78e1b536fbbeea057676aaea505e83": "b = \\frac{r}{\\delta r + 1/a}", "ca7911e6d46c77ea6df4187a5a0b6f70": "\\operatorname{cl}: \\mathcal{P}(S)\\rightarrow \\mathcal{P}(S)", "ca791ce524ac9be55ac8d8db38623ab1": "\\max_{j\\neq i} b_j < b_i ", "ca792a12999dff3302c8b2377c339a27": "r l_A(Q_B A_B + Q_D a_D) = \\sigma (l_A a_B + l_B) (Q_B l_B + Q_D l_D)", "ca79977615065279d19aff34b4f53ddb": " V(T_1, T_2, T_3, \\ldots, T_n) \\geq \\sqrt{V(T_1, T_1, T_3, \\ldots, T_n) V(T_2,T_2, T_3,\\ldots,T_n)}.", "ca79f68b0449a31b8cefcbba8e503d77": "(\\theta,\\phi)", "ca7a0fefb4869442b17aef1b7f773f5b": " \\zeta (s-m) ", "ca7a14861b7b00f32c653d6a38dcb7a4": "\nV_\\text{cap}(t) = \\left[\\left(V_\\text{capinit} - V_\\text{charging}\\right) \\times e^{-\\frac{t}{RC}}\\right] + V_\\text{charging}\n", "ca7a3ae2e48b36cdc60240f4651dd3b0": "d=0 \\;mod\\; \\lambda", "ca7a41687138499d8e103d09716af703": "f(a) = \\lim_{x \\rightarrow a} f(x). ", "ca7a625096ac687f911941856be5988b": "\\ V \\,", "ca7aa598d673e5b31f6f6823c6b93584": " P(E) \\leq \\exp \\left( -n \\left[ \\rho R - \\ln \\left( \\sum_{x_i} P(x_i)^{\\frac{1}{1+ \\rho}} \\right)(1+\\rho) \\right] \\right). ", "ca7ac312a5d2034b9f67742a78a081a3": "\\rho_{liquid}", "ca7acb575a6e920d87ab4b71aabbefbd": "9n", "ca7b9581a77ba8425142c006b8bb49ba": "T_\\text{load} = T_\\text{hold}\\ e^{ \\mu \\phi} \\,", "ca7ba9a522d4bb38b4e5844242e3588a": "\\textstyle=min_{a^{*}(\\theta_{k}w_{k}=1)}\\ W_{k}^{*}R_{k}W_{k}+ 2\\mu(a^{*}(\\theta_{k})w_{k}\\Leftrightarrow 1) ", "ca7be328d2a8ef16fccb407c22ebee3d": "F_\\text{seq}(x)", "ca7c1ca5a5f4f18dc66aa29e652596b3": "R_k \\to R_j \\to R_i", "ca7cf84a4be7ecd51086ad5276ab37f8": " \n J_n = \n \\begin{cases}\n 0 & \\mbox{if } n = 0; \\\\\n 1 & \\mbox{if } n = 1; \\\\\n J_{n-1} + 2J_{n-2} & \\mbox{if } n > 1. \\\\\n \\end{cases}\n", "ca7d00c345eabb5fed02ad5e20334204": "d\\geq 3\\,", "ca7d2bfb196576d8ca46aeffb2daeecd": "F_{\\alpha\\beta} = \\eta_{\\alpha\\lambda} \\eta_{\\beta\\mu} F^{\\lambda\\mu} ", "ca7d566a14bb684de1d27f5f0000e436": "[t_0-\\epsilon, t_0+\\epsilon]", "ca7db91f56156390846ecbf62b8a5a66": " \\mathrm{B}^{+} + \\mathrm{e}^{-} \\rightleftharpoons \\mathrm{B} ", "ca7e678560ac70ebaa1354ed62d1cb14": "\\, C \\;\\big\\llcorner\\; D := \\sum_{r,s}\\langle \\langle C\\rangle_r \\langle D \\rangle_{s} \\rangle_{r-s} ", "ca7ecf1f9172775ec071f6aaf5fb35de": "p(D|\\log \\mu ,\\log \\zeta ,\\mathbb{M})", "ca7ee1843a630dfaeebab5742a648cb9": "\\mathcal{Q} = \\frac{\\lambda_{min}(\\mu)}{\\lambda_{max}(\\mu)}", "ca7ee7738ecd697a52e8d09cd7e39ba7": "r' \\in N(r)", "ca7ef1abe7f13e5a3b5b2459816a492c": "\\hat{\\mathrm{ch}}", "ca7f036866c52f2310cd4b07f16deb7e": "L + R \\ \\rightleftharpoons \\ L\\! \\cdot \\!R ", "ca7f03cb15cc05db3b1ee11e405bfca5": "Pmf + Pmo = 1", "ca7f837c15cecec35dac72173776dc48": "\\Lambda:=\\frac{R}{24}", "ca803372f5c7e471d315283ee88b45b0": "\\lambda\\in\\R", "ca80b962e74df3957d4f1b52938cc216": "\\mu_i=m_i/m_p", "ca80ffb313d893472443505b85abe5ce": "\\sigma_{1t},\\sigma_{1c},\\sigma_{2t},\\sigma_{2c},\\sigma_{3t},\\sigma_{3c}", "ca811dc879a31272b111cf599b1a8c54": " \\textbf{a} = - \\omega^2 \\textbf{r}. ", "ca8125d69148f4fcd86f602c6df650a6": "\nS = \\int{ - m c^2 d\\tau} = - m c \\int{ c \\frac{d\\tau}{dq} dq} = - m c \\int{ \\sqrt{-g_{\\mu\\nu} \\frac{dx^{\\mu}}{dq} \\frac{dx^{\\nu}}{dq} } dq}\n", "ca8132b4fc9d28858f840dfad8a06cf9": "(X(1)|X(0) = i)", "ca8140d2be298ac4024c4cfb44aac7b5": "=\\langle \\hat{A}\\hat{B}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle-\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle+\\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle", "ca8145e2f1e8511a1bcacaab5246e37e": "[C_i,C_j]=0 \\,\\!", "ca818211ac97aea95e58285691df2f2b": "\\; I(\\rho^{AB}) = S(\\rho^A) + S(\\rho^B) - S(\\rho^{AB}).", "ca821c6a6ab4823812bea9255a865045": "\\sum_{k=1}^n (a_k , b_k ) (u , v)^k \\quad = \\quad \\left({\\sum_{k=1}^n a_i u^k} ,\\quad \\sum_{k=1}^n b_k v^k \\right).", "ca8232753813da60f262fb59a0efa8ab": "H f(\\mathbf{x})", "ca82662783aded41d6fbe192cd956742": "\n\\oint_{\\partial \\Omega} \\hat{n} (\\varepsilon \\nabla \\varphi)\n= - \\int_\\Omega \\rho\n", "ca8332bb9ab9b7cc8189417209a8055a": "A\\in \\mathbb{M}_{n}", "ca835817bcefd4d8089fca30795e31fe": " f(x) \\frac{\\partial^2 u}{\\partial x^2} + g(x) \\frac{\\partial u}{\\partial x}+h(x) u= \\frac{\\partial u}{\\partial t}+k(t) u", "ca835d00a0713c9cc9db289752a2c07f": "\\scriptstyle \\eta \\equiv 0", "ca83a5932e1ce71686cd7519ab1b1f2d": "\\mathrm{n}+{}_{\\ 90}^{232}\\mathrm{Th}\\rightarrow {}_{\\ 90}^{233} \\mathrm{Th} \\xrightarrow{\\beta^-} {}_{\\ 91}^{233}\\mathrm{Pa}+\\mathrm{n} \\rightarrow {}_{\\ 91}^{232}\\mathrm{Pa}+2\\mathrm{n} \\xrightarrow{\\beta^-} {}_{\\ 92}^{232}\\mathrm{U}", "ca83bf6732985b6dc39d2c6f52fcf909": " H_\\cdot ({\\mathfrak gl}(A),k) \\, ", "ca83cc7e0ca5b6cc1b16d97a42ab1653": "r^{\\prime\\prime} = \\sigma_2 \\sigma_1 \\, r \\, \\sigma_1 \\sigma_2 ", "ca83d14f6ecbae9ed3cf0e40345f6c3c": "r_a^{}", "ca83fc306b38de13e6221b4f5b665476": "x(yz) = (xyx)(x^{-1}z).", "ca8405bd8058e87b3b1bf37f92f78d36": "(e,g') \\in N'", "ca840be40fb86a7816e1c015dbfcbe02": "a\\in S, e\\in E(S), a\\,\\rho\\,e\\Longrightarrow a\\in E(S).", "ca843b0fa9b4adb0167e4200b4423cf5": "\\begin{align}\n & \\hat\\beta = \\frac{nS_{xy}-S_xS_y}{nS_{xx}-S_x^2} = 61.272 \\\\\n & \\hat\\alpha = \\tfrac{1}{n}S_y - \\hat\\beta \\tfrac{1}{n}S_x = -39.062 \\\\\n\n & s_\\varepsilon^2 = \\tfrac{1}{n(n-2)} \\big( nS_{yy}-S_y^2 - \\hat\\beta^2(nS_{xx}-S_x^2) \\big) = 0.5762 \\\\\n & s_\\beta^2 = \\frac{n s_\\varepsilon^2}{nS_{xx} - S_x^2} = 3.1539 \\\\\n & s_\\alpha^2 = s_\\beta^2 \\tfrac{1}{n} S_{xx} = 8.63185\n \\end{align}", "ca848ac206d7f0a91465ad4bd3da3912": "N_i-1", "ca84f61b1ffbcc60dbfb716caee6e55f": "\\gcd\\{f(n) : n \\geq 1\\}", "ca85922f6236f27ed8061b260f1f63c2": " \\vartheta(G) \\vartheta(\\bar{G}) \\geq n. ", "ca85ec1bedd91692b5f42edaac40aa18": "\\Gamma(\\tfrac12 n) = \\sqrt \\pi \\frac{(n-2)!!}{2^{(n-1)/2}}\\,,", "ca8608c053bf7f255351193cf521ef3c": " N = \\frac{V }{8\\pi \\hbar \\omega}\\mid \\mathbf{E} \\mid^2 ", "ca86092679d71a48879a63bf577e444d": " \\frac{1-\\xi^2}{(1-\\xi)^3} = \\frac{1 + \\xi}{(1-\\xi)^2}", "ca86141cad5bd4c0e7f5c6f9a864a384": " L(int P, t) = (-1)^n L(P, -t). ", "ca865b11375e69e6cc686edd1050fab8": "\\frac{6!\\times 2^5\\times 4!\\times 3^6}{4} = 100,776,960.", "ca86aa437d9a89950c1315aaf0f0261f": " \\mathbf{I} \\cdot \\boldsymbol{\\alpha} + \\boldsymbol{\\omega} \\times \\left ( \\mathbf{I} \\cdot \\boldsymbol{\\omega} \\right ) = \\boldsymbol{\\tau} \\,\\!", "ca86e1be10857ec8c9a1a9acd03db03f": "Q_2(OR_n) = \\Theta(\\sqrt{n})", "ca870d2b6a3a2d684b8647cb5bbb1ace": "Z=\\sum_i g_i e^{-\\varepsilon_i/kT}", "ca8746bf50dfac3f97cb0cb5fd9e8b14": "(\\mathbb{P}, <)", "ca876d4c7ad6964cfade546c1ce86bb7": "\\ln\\left(\\frac{103.02}{100}\\right) = 2.98%", "ca877b6f06299d97f5e4fb95905bd079": "\\scriptstyle f:(\\mathbb{R}^n,0)\\to(\\mathbb{R},0)", "ca87c4294afa703840531fedd5ebbb6a": "J =\\sigma E + D q \\nabla n,", "ca87e19c93c2b0e417906eee7886fd18": " 1 + \\cot^2 \\theta = \\csc^2 \\theta\\!", "ca87f1f1b9c7bc71548aec081beadaa4": "p_{x}", "ca8808c4e42c1b1f3c4fd01c21d1ac54": " I_4 ~,~ \\gamma_\\text{chir} ~,~ \\gamma_\\text{chir} \\gamma_a ", "ca8846787e2fb417f5e4c8a9b4e61b09": "E(S_n)=\\sum_{j=1}^n E(Z_j)=0.", "ca88609a2a5506a6df156d748c94780f": "\\{0,1\\}^{k}", "ca888004f2ba0bdf79bd8b203ebdd832": "W^{\\mathfrak{p}}\\simeq W_{\\mathfrak{p}}\\backslash W", "ca889032aa71a8eb09c0c717e45e8005": "\\{-1, 0\\} \\cap \\{0, 1\\} \\notin \\tau_1", "ca88a3d37ba96a4b8f3fef0e1f138ddb": "\\hat{t}=-\\sin u\\ \\hat{g}\\ +\\ \\cos u\\ \\hat{h}\\,", "ca88a6dd7b68c9566c8ee5e5b3a4e4ca": "v \\ \\overset{\\sim}{\\mapsto} \\ x_v \\in V^{**} \\quad \\text{such that} \\quad x_v(\\phi) = \\phi(v)", "ca88c846c923d33423cdbe519317453e": "k_{\\infty} = \\frac{\\nu \\Sigma_f}{\\Sigma_a}", "ca88d74898a509156ffed2f455183669": "\\sum_{n=0}^{N-1} |x_n|^2 = \\frac{1}{N} \\sum_{k=0}^{N-1} |X_k|^2.", "ca8a0db97bb7846e41c8a4521802d27a": "\\nabla \\times \\mathbf{H} = \\mathbf{J}_\\mathrm{f} + \\frac{\\partial \\mathbf{D}} {\\partial t},", "ca8a3b6185b42410c9ea89c6ed709b1a": "l_ia_{ij}=m_jX'", "ca8a619806fb24e7e9ada33d24bc4504": "p(y|x) = p(x|y)", "ca8bd45ff151b3662dcd0bf557296e47": "\\; = \\sum_i p_i (\\log p_i - \\log (\\sum_j q_j P_{ij}).", "ca8bfb3eab7f8bbfef33afcf4b260c24": "O(\\log(B))", "ca8c31e55d71d8c54cd7451b1f372bd6": "R^{ab}", "ca8c3e9a866b70dd2464ab82f627f777": " M_{\\theta\\theta}\\ddot{\\theta_r} +\n K_{\\theta\\theta}\\theta_r +\n M_{\\theta\\psi}\\ddot{\\psi} +\n C_{\\theta\\psi}\\dot{\\psi} +\n K_{\\theta\\psi}\\psi =\n M_{\\theta}\n", "ca8c6c36299cd43ecf57028bc6acc3e7": "\\;\\mathcal{O}_n ", "ca8ca558912369a6f0671060b1e4349c": "\\xi_t", "ca8d40e458a9cc13273e0cb801622c31": "\\!v_3", "ca8d4852c1240ad2a045683484a959ee": "k_e = 8.987\\,551\\,787\\,368\\,176\\,4\\times 10^9\\ \\mathrm{N\\cdot m^2\\cdot C}^{-2}", "ca8d71ac330cce5b71f9a83e3afab45f": "\\begin{align}\nF(x) &= \\int_{-\\infty}^x \\!\\!f(u)\\,\\mathrm{d}u = \\begin{cases}\n \\frac12 \\exp \\left( \\frac{x-\\mu}{b} \\right) & \\mbox{if }x < \\mu \\\\\n 1-\\frac12 \\exp \\left( -\\frac{x-\\mu}{b} \\right) & \\mbox{if }x \\geq \\mu\n \\end{cases} \\\\\n&=\\tfrac{1}{2} + \\tfrac{1}{2} \\sgn(x-\\mu) \\left(1-\\exp \\left(-\\frac{|x-\\mu|}{b} \\right ) \\right ).\n\\end{align}", "ca8d81c84731e12dbc1320f7511ce5b4": "p(c_j|f_i)\\ ", "ca8dd2599261a27adc17922789503efc": "F'(x)", "ca8e17479f395806df2a9c8b3da52d8e": " \\omega_0 = { 1 \\over \\sqrt{LC} } ", "ca8e235f1a9bf380a04f8eb443fc6221": "d \\sigma", "ca8e5bfed62a7b60ce818421fadcd6e7": "f(z)=\\sum_{h=0}^{\\infty}a_h (z-z_0)^h", "ca8e608169b20a94570ac837e8ba0833": "h(x)", "ca8ead4acb339c5170207c820ec70d23": "\nG(p) = \\frac{1}{4}\\left[\\gamma\\Phi^{-1}(p) + \\sqrt{4+\\left(\\gamma\\Phi^{-1}(p)\\right)^2}\\right]^2\n", "ca8ef1e88a080afa24b6b9736803e228": "\\Phi_{M} = \\frac{1}{4}\\lambda^{-1}(\\lambda^{2} + \\omega^{2} -1),", "ca8ef7c5e5e509fb20f209d4637d59ec": "\\min_{\\rho}\\; \\int_{\\Omega} \\frac{1}{2} \\mathbf{\\sigma}:\\mathbf{\\varepsilon} \\,\\mathrm{d}\\Omega", "ca8f184cfd4056fc24dc138543228733": "= n\\log n - n + \\frac {\\log(1 +1/(2n) +1/(8n^2))} {6} + \\frac {3\\log (2n)} 6 + \\frac {\\log(\\pi)} {2}.", "ca8f274f7437b9043f86ce465cccb25f": "| X |", "ca8f57df455547cbd6111956925f7e5a": "\\mu' = T_{*} (\\mu) \\approx \\mu.", "ca8f7b86cecd41ec510d0516a89574d1": "\nh_{\\mu} = h_{\\nu} = a\\sqrt{\\sinh^{2}\\mu + \\sin^{2}\\nu} = a\\sqrt{\\cosh^{2}\\mu - \\cos^{2}\\nu}.\n", "ca8fa967185206e3ee67d818f6dcdeb8": "C_{t} = C_{0} e^{-kt} \\,", "ca8fac2fc88597f7b2d2c6141cfc2b31": "{}_{\\ 90}^{228}\\mathrm{Th} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 88}^{224}\\mathrm{Ra}\\ \\mathrm{(1.9\\ a)}", "ca9027e8c3e49ed24d92539470156eff": "B_n(x)={D \\over e^D -1} x^n", "ca904667a27c231e162760d6ee8feff6": "\\ H = \\frac{1}{r}", "ca9096e97dd545a764891eaeeaba867c": "au+bv-cw+c=0", "ca91370d1f5400510aed6d14590a9bac": "*442", "ca913cd2fd7770942a91d14c4e825514": "g_i := \\textstyle\\frac{1}{2}(3i^2-i)", "ca91748e8c3d5bb2abe49c7530e5eff3": "CI(t)=1-e^{-IR(t) * D}\\, .", "ca9189405e3744f071701ff6bbbda373": "-(I_1-I_2)/3", "ca91b153f41b06fd88acd4cc6fd54abc": "\n \\sum_{R\\in G}^{|G|} \\; \\Gamma^{(\\lambda)} (R)_{nm}^*\\;\\Gamma^{(\\mu)} (R)_{n'm'} = \n\\delta_{\\lambda\\mu} \\delta_{nn'}\\delta_{mm'} \\frac{|G|}{l_\\lambda}.\n", "ca920928efc9b70c4747c05146d0e727": " \\varepsilon_z = \\frac {1}{E} \\left [ \\sigma_z - \\nu \\left ( \\sigma_x + \\sigma_y \\right ) \\right ] ", "ca9215f92c40b850c46d1f82a19f1aee": "\\{A_\\alpha\\}", "ca921b7d106508ce01350c975af254e5": "W = \\int_{t_1}^{t_2} \\mathbf{F}\\cdot \\mathbf{v}dt = m \\int_{t_1}^{t_2} \\mathbf{a} \\cdot \\mathbf{v}dt = \\frac{m}{2} \\int_{t_1}^{t_2} \\frac{d v^2}{dt}\\,dt = \\frac{m}{2} \\int_{v^2_1}^{v^2_2} d v^2 = \\frac{mv_2^2}{2} - \\frac{mv_1^2}{2} = \\Delta {E_k} ", "ca92534d78658b843a917fb67004e5aa": " \\partial_t \\rho + \\nabla \\cdot \\boldsymbol{m} = S = d_r\n(q \\Delta q^{\\ast} + q^{\\ast} \\Delta q) + 2 l_r \\rho + 2 c_r\n\\rho^2 + 2 q_r \\rho^3 \\quad\\text{with} \\quad\\boldsymbol{m} = 2\nd_i \\text{Im}(q^{\\ast}\\nabla q).", "ca927a2727c812ebb77718eefecbaec8": "\n \\varphi(\\theta) = \\begin{cases}\n1+\\frac{2\\theta}{\\pi}, & \\text{if }\\theta \\in [-\\pi,0],\\\\\n1-\\frac{2\\theta}{\\pi}, & \\text{if }\\theta \\in [0,\\pi].\\\\\n\\end{cases}\n", "ca92ed08cffb494dd45f4a545d160dbb": "r \\mathrm{OPT}", "ca9378adcdb2f3bb5fc1cb475661046c": "\\int_{-\\infty}^\\infty \\psi_0^* \\psi_0 \\,dq = 1", "ca937c8bd18ebb7f4bb31d8a4746567a": "x^n + c_1 w(x)x^{n-1} + \\cdots + c_n w(x)^n ", "ca9382308bb1896e0eca7d2150323a62": "\\Sigma_I", "ca9423fdb7e60cf8b87a508ecb4e6040": "R(i,j)", "ca9430cbf642177d424ffa854d91d7f7": "{\\mathcal{A}_{AG}}", "ca944eb34928832320ca282071d7b994": "a(n,x):=p \\left (x_0^{n-1}|x_{-\\infty}^{-1} \\right ).", "ca949079fc9cc506cf00105e9433cf1d": "\\eta = \\mathrm{diag}(\\underbrace{1,\\cdots,1}_{p},\\underbrace{-1,\\cdots,-1}_{q}).\\,", "ca949c538df51c35cbc6e6990b8e014c": "\\det (I + N) = 1,\\!\\,", "ca94d984dc2525e4d40e72e1b0edd492": "\\alpha_{i+1}", "ca950bedafdb844e097ef7639ce6e0c6": " \\hat{S}_-|\\psi_{n_{\\nu_1}}(\\bold{r}_1)\\rang|\\psi_{n_{\\nu_2}}(\\bold{r}_2)\\rang\\dots |\\psi_{n_{\\nu_N}}(\\bold{r}_N)\\rang= {c^{\\dagger}}_{n_{\\nu_1}}{c^{\\dagger}}_{n_{\\nu_2}}\\dots{c^{\\dagger}}_{n_{\\nu_N}}|0\\rang", "ca951dff9e605971245b3b559c92c575": "|0|_p=0", "ca9539818dbeef465fae9a8c54bb4111": "\\rho(\\operatorname{Frob}(p)),", "ca9544f81c57175436d1fe040863fcca": "\\operatorname{crd}(\\theta)", "ca957f9e9e28afa2a19068ce4f188ac0": "\\exp_{10}^3(1)", "ca958b6dbea44521d4b0446dac6f4116": "{Q}\\,", "ca9619646313373a0d0672fe772726f0": " \\tilde{\\mathbf{x}}' = \\mathbf{R} \\, (\\tilde{\\mathbf{x}} - \\mathbf{t}) ", "ca962f5b7abdd2073f5cddc7a1b8e7ed": " v = \\frac {2\\pi r}{t}", "ca96a98bf1d078e6d8007c27cd7e9a0e": "\\mathrm{E_1}(z) =-\\gamma-\\ln z-\\sum_{k=1}^{\\infty}\\frac{(-z)^k}{k\\; k!} \\qquad (|\\mathrm{Arg}(z)| < \\pi)", "ca96ec30cd5c6e9bd3f403384f9e0264": "d\\colon M\\times M \\rightarrow R^+", "ca978b90de2e4bacd6fffaf1d592adab": "\n p \\sim \\textrm{B}(\\alpha,\\beta),\n", "ca97aba0ba7b5b63758fc2168b51ff26": "A\\backslash B\\,\\!", "ca97fc8921c76acc98a3d516a3229dec": "\\lambda = \\int_0^\\infty \\frac{\\rho(t)}{(t+1)^2} dt ", "ca9891cf44bc169b04bcd9ecf2dddfaa": " \\rho = r \\left(1+\\frac{ra'_r}{c^2}\\right)^{1/2} ", "ca98bbc9bc2aa8cb0289e2a3a34caa50": "\\mathbf{\\begin{bmatrix} x \\\\ y \\\\ z \\\\ \\end{bmatrix} =\n\\begin{bmatrix}\n a & b\\cos(\\gamma) & c\\cos(\\beta) \\\\\n 0 & b\\sin(\\gamma) & c\\frac {\\cos(\\alpha)-\\cos(\\beta)\\cos(\\gamma)} {\\sin(\\gamma)} \\\\\n 0 & 0 & c\\frac {v} {\\sin(\\gamma)} \\\\\n\\end{bmatrix}}\n\\begin{bmatrix} \\hat{a} \\\\ \\hat{b} \\\\ \\hat{c} \\\\ \\end{bmatrix}\n", "ca98be17ec3026b0270907bbf6aa54b0": "\\left(a^2 + \\frac{3}{4}b^2\\right)m", "ca9940ae830bf96862afee25642a5fd7": " M(f)(x) = \\sup_{r>0} \\frac{1}{r^n} \\int_{B_r} |f|,", "ca996bba008da20574f97e5f022a4bcb": "g_{2,1} = \\frac{a_2a_1^2}{r_LL_1(a_1^2L_2+a_2^2L_1)} ", "ca996d11bcd7342cde9853de2999de2f": "FM/O(1,3) ", "ca99a7aa7a6ea568b58fa5509be59cff": "J = J^{*}", "ca99b4c1fb3ba1212385674d616de85c": "1^3 + 2^3 + 3^3 + \\cdots + n^3 = a^2;", "ca99b8ea1e05c4ca088ade2ae3105236": "\\ R + S = 1 ", "ca9a2a278b534bfdf3b5a574fee9ae33": "{\\omega}", "ca9a59542c8738682f06679c56bd64b2": " S_{ab}=\\delta(a-b)-2i\\pi\\delta(E_a-E_b)(\\phi_a,V\\phi_b)\\,", "ca9a74439937b9392f18f73cf3406abb": "\\vec A\\!", "ca9a77ffb2d9978e023129ee98efd755": "\\mathbb{E}[X_t^\\tau]=\\mathbb{E}[X_0],\\quad t\\in{\\mathbb N}_0,", "ca9a82102812cb3a624efcc0a9b11372": "\\displaystyle \\overline{z}\\,", "ca9a969d366c09eb125381691eeeba42": "\\Gamma\\left(\\tfrac{1}{2}, x\\right) = \\sqrt\\pi\\,{\\rm erfc}\\left(\\sqrt x\\right),", "ca9ab8a0b9d2dae0fa7667db4c502092": "\\sigma_2 > \\sigma_1", "ca9b0474465d147536b6833e16b7502d": "W(z) = \\frac{A(z/\\gamma_1)}{A(z/\\gamma_2)}", "ca9b556b41c2bffa5ac0c43adeae4636": "I = I_{0} \\, e^{-\\alpha \\, x},", "ca9b5fdc06546fe897f50d23a89fef07": "V_\\infty", "ca9b6b3920bbafc0c20ac04a4133af4e": "s\\neq 1 \\Longrightarrow t\\leq s^2", "ca9bdc2fa4cab6b9095637871533f76d": " \\langle E_\\mathrm{k} \\rangle = \\frac{1}{2}kT\\,\\!", "ca9bde99a25844b7fff74917c412da34": "\\epsilon > 0,", "ca9c123298d79396cf140f147e683d2b": "\\textstyle (Z,m) ", "ca9c3b429946ac38991889803c9f2eff": " \\sigma_i = \\sqrt{ \\frac{1}{(r-1)} \\sum_{j=1}^r \\left( d_i \\left( X^{(j)} \\right) - \\mu_i \\right)^2} ", "ca9c6bd85dc5683abe05b7d027873c86": "\\Delta = V(-\\infty) - V(+\\infty)\\,", "ca9cbdd16d4c84e5847b86167188f7c9": "DEP(T_i).\\mathrm{add}(WTS(O_j))", "ca9cd0e1d78520295253b00663edd8f7": "\\mathbf{R}^{n+1}_+", "ca9cf742481b8b692b3635e8d51fcada": " \\mathbf{P}_\\text{field} = \\frac{1}{\\mu_0c^2} \\int_V \\mathbf{E}\\times\\mathbf{B} dV\\,,", "ca9cfe4666e03198a08de6dc8f6aff9d": "\\begin{cases}\np(c|x)=Kp(c) \\exp \\Big( -\\beta\\,D^{KL} \\Big[ p(y|x) \\,|| \\, p(y| c)\\Big ] \\Big)\\\\\np(y| c)=\\textstyle \\sum_x p(y|x)p( c | x) p(x) \\big / p(c) \\\\\np(c) = \\textstyle \\sum_x p(c | x) p(x) \\\\\n\\end{cases}\n", "ca9d9ba73206a2148e1ce4bdec6e93e6": "j: d_j=0", "ca9da31c93c5ae37cd322fce192c86cd": "\\sum_i e^{-y_i f(x_i)}", "ca9de5b6af225ad62f6f30fce68c20a6": " \\{\\partial_i\\} ", "ca9e8236c9d8caadba3045c5fa3ec482": "a_nb_n \\to ab", "ca9ed2fabab1e2c2eca816d7402c2839": "\\epsilon\\;", "ca9f2f6af44753be2316db5e28c9e556": "ab \\le \\frac{a^2}{2} + \\frac{b^2}{2},", "ca9f51ae3c06f0a7b3b52dc21c603808": "B/A", "ca9f5943eb33ff25f19ff771107c264b": "\\langle v_k \\cup x, \\mu\\rangle = \\langle Sq^k(x), \\mu \\rangle", "ca9f5bf377b60ed8e880c54788906c3e": "(A_2,i)", "ca9f64c01188700c759377a811d5bf1c": "a_{1}+c_{3}", "ca9f824b71a232c740fa683c4bb25b50": "\\, F_{ab}", "ca9f8597a12748aa4eb4b93b09f88626": "\n\\frac{d}{dx}\\left[R(f_m\\dot{f}_n-f_n\\dot{f}_m)\\right]=\nR(f_m\\ddot{f}_n-f_n\\ddot{f}_m)\\,\\,+\\,\\,R\\frac{L}{Q}(f_m\\dot{f}_n-f_n\\dot{f}_m)\n", "ca9fb8db9b218e9355e05db7cf9adb94": "\\varphi\\in\\mathcal S(\\mathbb{R}^n)", "ca9fcd9d792e2b13ed449ac986769e20": " M_{i} ", "ca9ffb00980e509700f4f262bcbfe1a9": "\\det S''_{xx}(x^{(k)}) \\neq 0, \\quad", "caa03930b44aa696808c50c5f789bab5": "\\mathcal{E} = V_\\mathrm{ter} + V_\\mathrm{load} \\,\\!", "caa0663a7c5dcd585209c789f8e4cc6f": "f(t) = \\sum_{n=-\\infty}^\\infty \\hat f(n) e^{int}.", "caa06ff74229680a055682276e9ea3f0": " d_2(P,Q) = K \\cdot d_1(P,Q) ", "caa078ab88061832166354a247740d0b": "g(u):=e^{-u} u^{y-1} 1_{\\R_+}", "caa0878d576f4abfe6c9277850af05e0": "\\left[ \\begin{matrix} 1 & \\alpha \\\\ 0 & 1 \\end{matrix} \\right] ", "caa088e40e1257d46bb9d1c89aa08c80": " z=a+ib ", "caa0fd6dbd06ea8b9d049c4ce6b5adf3": "C(3, 1) = 3", "caa1445f48fde8f38653d017af815051": "(18)\\quad\\quad \\rho_1 u_1 \\left( e_1 + \\frac{1}{2} u_1^2 + p_1/\\rho_1 \\right) = \\rho_2 u_2 \\left(e_2 + \\frac{1}{2} u_2^2 + p_2/\\rho_2 \\right).", "caa15c3216296142ec7fe92d0623525c": "~u=v=0~", "caa1634b9a413ba71bff8b5ce436ce92": "\\mathrm{HI}\\,\\!", "caa1694d32570c03a9c97815a23f91b2": "\\gamma_{\\rm CMB}+p\\rightarrow\\Delta^+\\rightarrow p + \\pi^0,", "caa1766e722f3800fdf89258b5d254fd": "\\phi (\\mathbf{r})", "caa19e149ceb6595673a4a97400b866b": " \\varphi: S \\to V ", "caa21e225b446fd9a645a9eb9229e6c9": "\\|P_n-P\\|_{\\mathcal F}=\\sup_{f\\in {\\mathcal F}} |P_nf- P(f)|", "caa2231a7459de3b48ca3ee4d06942fb": "\\Chi^2_r", "caa223e3b7cd863da578264dde79fbc5": "V'(T) = \\{x \\in X \\mid f(x) = 0, \\forall f \\in T\\}", "caa24be8bfbf3917736dda2c809f0b4e": "\nH|_{(x,v)} := v^i\\tfrac{\\partial}{\\partial x^i}\\big|_{(x,v)} - \\ 2G^i(x,v)\\tfrac{\\partial}{\\partial v^i}\\big|_{(x,v)},\n", "caa284fe46698c18b12af535cce6838b": " \\mu = \\frac{ze}{f} ", "caa2c7013927fc2292134abc2f215920": "id_V = |0\\rangle", "caa2cf31f81c7d0d5ba466b534840182": "\\gamma_2=0", "caa2e66eb3d09879001199b075b6121a": "[{\\rm RL}]", "caa2f12713a8084d188eb701ade61783": "{\\bold v}_0(\\varphi_0^{-1}(P)) = J_{\\varphi_{01}}(\\varphi_1^{-1}(P))\\cdot {\\bold v}_1(\\varphi_1^{-1}(P)). \\qquad (3)", "caa310aba5975426b6bd50c174ff5b1e": "a(2 \\rho^2-1)", "caa338f0277fe87653f170e34987775d": "y\\setminus(x \\setminus L) =(xy) \\setminus L", "caa35b37e8f8df7cb38b825a327f2be4": "\\scriptstyle (h - y)^2", "caa38efdc40d19e7b7ad65960109fc55": "\\text{E}\\,[Y(\\mu;t)] = t\\mu\\,\\!", "caa396f82d3c84efc2849c13a0600a6c": "c>0.", "caa398c2ddbcc01ffbb03deca11ecbf5": "\\theta_{j},\\xi_{j}", "caa39d2d5201f2f98866f5f1f0b64052": " q_t + ( f( q ) )_x = 0. ", "caa41520dbc4fd74ad61c636fdc480e5": " \\textstyle \\alpha ", "caa41a3ab7292feba2f99e8b83839dad": "\\overline f \\circ \\overline g = \\overline{\\,f \\circ g\\,}", "caa4c71b4fefedda46cd1ea542af04e7": "\\mathbf{\\ddot{r}}_i = \\sum_{\\underset{j \\ne i}{j=1}}^n {Gm_j (\\mathbf{r}_j-\\mathbf{r}_i) \\over r_{ij}^3}", "caa4d3b3d6ced0c47cee651f33f9f40d": "\n\\delta = \\frac{3\\sigma\\left(1 - \\nu \\right)}{E} \\left(\\frac{L}{t}\\right)^2\n", "caa4da26abdc03f24602c9c2445b87b3": "[2.75] = 3", "caa4e1d4681e8e8c8f3b420919792f71": "O( n k \\log k )", "caa4fb9dc8a37b19be8cf591c38fa565": "\\ \\displaystyle \\alpha^{*}=\\hat{\\alpha}(q,r_{c}) + \\varepsilon \\ ", "caa5e36f342b6a123beee647fb9b9e26": "(x,z) \\in R_0", "caa5ed0f2f7a4c0b9420000f6041380c": "\\{\\ x\\} = \\{\\ X\\} + \\{\\Delta \\ X\\}", "caa618d6adc274387d718d5fc85f3f20": "\n\\begin{array}{lcl}\n\\#\\mathbb{W}_v^{k,(-dn)} &=& \\text{number of words having value }v\\text{ among topic }k\\text{ excluding }w_{dn} \\\\\n\\#\\mathbb{Z}_k^{d,(-dn)} &=& \\text{number of topics having value }k\\text{ among document }d\\text{ excluding }z_{dn} \\\\\n\\end{array}\n", "caa62b28acc4fb25294ea58b2fd63d1e": "C^k[a,b]", "caa6efd3d5f8d8a2e773ad39e39749a5": "\n\\psi (\\mathbf{r},t)=\\sqrt{\\frac{\\alpha }{2\\mathrm{v}}}\\exp \\left[\\frac{i}{\\hbar }(\n\\mathbf{p}\\cdot\\mathbf{r}-Et)\\right],\\qquad E=D_{\\alpha }|\\mathbf{p}|^{\\alpha\n},\\qquad 1<\\alpha \\leq 2, \n", "caa73b0b6375afc2b6fe987ed00d04ff": "t_{1/2} = \\frac{2^{n-1}-1}{(n-1)k{[A]_0}^{n-1}}", "caa7817359790fc63b7eca19ca9c8cb3": "j_\\ell(kr')", "caa7bce749a32e0849667843b69ecb6f": "t_{2}=\\frac{AB}{c-v}+\\frac{DE}{\\frac{c}{n}+v}", "caa7f0b28065750b5c9d68b03db58e96": "\\vec S_{avg} = \\frac{(\\vec S \\cdot \\vec J)}{J^2} \\vec J", "caa84c711e52a52143674ed89694d412": "B_y(T) =\\frac{\\hbar c^2}{4 \\pi^3 y^5} \\frac{1}{e^{\\hbar c/(y k_\\mathrm{B}T)}- 1}", "caa868b48adb7b21a17b27bdc1318c10": "\\int_0^{1/2} \\ln\\Gamma(x)dx=\\frac{3}{2} \\ln A+\\frac{5}{24} \\ln 2+\\frac{1}{4} \\ln \\pi", "caa87526e50f9bc10596af6951837679": "Q_1 = L_{12} + L_{34},", "caa8846d8c686d16f7351821d4f8ab35": " \\lim_{h\\to 0}\\frac{f(x+h) - f(x) - f'(x)h - \\frac{1}{2} f''(x) h^2}{h^2}=0.", "caa89a19800662744d5e25936fa4b061": "l^rzr^k", "caa8cd1715a7c9bd9ed166e76fa5f885": " \\vec{r}_1, \\vec{r}_2", "caa8e0aea35d0252671734a58c0f4029": "\\phi(x)=(x\\Omega,\\Omega)", "caa95925199540bfae962347cf64b529": "|a| = \\begin{cases} a, & \\mbox{if } a \\ge 0 \\\\ -a, & \\mbox{if } a \\le 0 \\end{cases} \\; ", "caa9ac498d6a46fe2788c7a14e06e6ed": "V_0\\to\\infty,\\quad a\\to 0", "caa9d2a27644ad2d8f3a7934eb2605c6": "\\beta h", "caa9fc802ff5792d1a372d9c149c5895": "f(t) = t \\log(t)", "caaa498d329fe7404cb8424131829dc4": "v = \\frac{d}{t},", "caaa754a9bba5a4fbf3f6c685973908c": " U_{nit} = \\bar{\\beta} X_{nit} + e_{nit} ", "caaac992285d2ea6c729feebce8562be": "S(\\rho) \\,=\\, - \\sum_j \\eta_j \\ln \\eta_j ~.", "caab21f7f05547c224bbfa88448cc4d6": "\n \\Omega^2(k)\\, =\\, |k|\\, \\left( \\frac{\\rho-\\rho'}{\\rho+\\rho'} g\\, +\\, \\frac{\\gamma}{\\rho+\\rho'}\\, k^2 \\right),\n", "caab350aa7e823c81d4e620b66efa1ed": "\\theta_\\lambda(x)=\\theta(\\lambda x)\\,", "caab55b9c0b91468ebcda72adcef065a": " r_{t+\\Delta t}-r_t =\\theta (\\mu-r_t)\\,\\Delta t + \\sigma\\, \\sqrt r_t \\epsilon_t ", "caab7dbcf0898fdd51aa58a627374f3d": "q=19", "caac08ede8a1acb02606bd50840ff9fe": "p_1 \\times p_2 \\times p_3 \\times p_4", "caac6d399c5959ee89cce04fdb726fbb": "(4C)!(C!)^{12}\\sum_{a, b, c} {\\left( \\frac{C!^2}{a! b! c!} \\cdot \\sum_{k_{12},k_{13},k_{14},\\atop k_{23},k_{24},k_{34}} {{a\\choose k_{12}}{b\\choose k_{13}}{c \\choose k_{14}}{c \\choose k_{23}}{b \\choose k_{24}}{a \\choose k_{34}} } \\right)^2 }", "caaca1593f7ce1f464bf1856ecde6181": "L_{Q0} 8\\pi \\beta \\cdot L_{QY} \\ ", "caacd87d762288dfdb58128db5ddad28": "Q = P(\\theta_0)", "caad20a18b42deabb68d94f383d26afb": "(g_{(a,k)}, U)", "caadd12dd1a8268132f5a8a30eaee8aa": "\\mathfrak{f}_\\chi", "caaddf1d295c60a48f898d83596d6904": "I_\\mathrm{B}", "caadf5fce355c86fc316f5ec1172fd5b": "\\sum_i^m X_{i1} \\hat r_i=\\sum_i^m \\hat r_i=0.", "caae86bbc9446f8c4404954f6f3ed04e": "u_i v_j", "caaf4ec27cf7cff01b0759c7391cf5ac": "\\vec{p}_{i1},\\vec{p}_{i2}", "caaf4ee7c2e84b641068c2c8d96694e9": " x \\ll 1", "caafd24073d66de503b4253f069e6f94": " \\forall x \\in {}^\\star\\mathbb{R} \\quad x < x+1. ", "caafdc25bfba75290380d6570e830ae3": "\\cosh^2\\phi-\\sinh^2\\phi=1", "cab047a07127041d34f759c022f30614": "\\chi^\\ast", "cab1a00aea71df1ca186510ad4cde8a8": " x^{x^{x^{\\cdot^{\\cdot^{\\cdot}}}}} ", "cab1bbe9a6f07a267ce0f9a000999f6d": "(\\mathbb C\\otimes\\mathbb O)P^2", "cab1cdb7df918d242e6f598a4d446cf3": "q(D,\\widehat{D})\\geq -\\alpha/2\\,\\!", "cab1fe6d68a7401934495ba00b771ef9": "\\ell_X: G\\to\\mathbf{Z}, \\quad g\\in G\\mapsto \\ell_X(g)", "cab20cf54b86b3e702c589b6f558e4ed": "\\mathcal{P} (X) := \\{ m \\in \\mathcal{M}_{+} (X) \\mid m (X) = 1 \\},", "cab236048d0e80f405ec3bfd757339e2": " \\Rightarrow kL = 2\\pi n \\rightarrow k = {2\\pi \\over L} n \\qquad \\left( n=0, \\pm 1, \\pm 2, ..., \\pm {N \\over 2} \\right). \\,\\! ", "cab26b33749350201e5167f61e1f0e3f": "\\Gamma= P\\,\\Lambda\\,P^{-1}", "cab2fe795eb8a0a178069d983bd181cc": "F(\\mathbf u_0,\\lambda_0)=0", "cab380994bc4459ead75c9f76722826e": "\\operatorname{sl}_2", "cab3dda8b07e40591811d9664d657919": "\\mathbb{C}(j)", "cab408c24eef75c9dfd8d0adac651f82": "x_3= \\frac{x_1y_1+y_1x_1}{1+dx_1x_1y_1y_1}=\\frac{2x_1y_1}{ax_1^2+y_1^2}", "cab42820bf6975a3ebb8cb75b9bbd11e": " u^{\\alpha} ", "cab4472d44d62320322cc3e176abde10": " \\Xi = \\sum_{n}{\\lambda^{n}Q_{n}} = e^{\\left(pV\\right)/\\left(k_{B}T\\right)}", "cab46e7506372b1b9ea86a40fe33a597": "\\Q(\\sqrt{2}):\\Q", "cab4eb7e6634e2a8b61b49ea1de4a20e": "\n \\frac{\\partial G_{ij}}{\\partial x^k} = \\left(\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^k}~\\frac{\\partial X^\\beta}{\\partial x^j} +\n \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial^2 X^\\beta}{\\partial x^j \\partial x^k}\\right)~g_{\\alpha\\beta} +\n \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~\\frac{\\partial g_{\\alpha\\beta}}{\\partial x^k}\n", "cab54f95b1cc8994c3d1efb3d66d1bd8": "\\mbox{Doomsday} = \\mbox{Tuesday} + y + \\left\\lfloor\\frac{y}{4}\\right\\rfloor - \\left\\lfloor\\frac{y}{100}\\right\\rfloor + \\left\\lfloor\\frac{y}{400}\\right\\rfloor = \\mbox{Tuesday} + 5\\times (y\\mod 4) + 4\\times (y\\mod 100) + 6\\times (y\\mod 400)", "cab55bb088ad486f98361a62ceab068b": "f: X' \\to X", "cab56b19b20d30c5aee9cd6918fa410b": "f(a/2) \\cdot (h/2) \\cdot (1/2)", "cab5effcdb16808a2e4a44e5add5d653": "\\hat{w} (\\omega) = \\frac{1}{2} \\hat{w}_r (\\omega) - \\frac{1}{4} \\hat{w}_r \\left(\\omega + \\frac{2\\pi}{N-1}\\right) - \\frac{1}{4} \\hat{w}_r \\left(\\omega - \\frac{2\\pi}{N-1}\\right) ", "cab601b1586ee69cd1295b78f2e1eda9": "\\sigma=\\frac{\\Delta P_{max} \\times R_{cap}}{2}", "cab681ed1d9c07cc076b05f43208cefb": "G \\to \\mathcal{T}(G)", "cab69cda9db305462703ff128c596d6c": "s=\\Omega(n/ (\\epsilon^2 \\log (1/\\epsilon))", "cab6dfe6e5b0b9d59e6c98f856975503": "dN = f (\\mathbf{r},\\mathbf{p},t)\\,d^3\\mathbf{r}\\,d^3\\mathbf{p}", "cab6f0c9aa0e42aef04d8a67b529be34": "\\omega^{\\ IJ}_{\\mu}", "cab73cbf71db856cdee523b06b2b4557": "C\\left( u,\\tau \\right)", "cab79d0f28facbbdead8ed5fee9286e5": "AA_{11} \\ ", "cab7bfd1b524811547aa019d7ce37ce4": "\\mu=6", "cab830ea21ade287f799b2852dbbe1b3": "\\frac{1-\\epsilon}{\\epsilon}||\\rho-\\sigma||_1 \\quad \\leq \\quad D^{\\epsilon}(\\rho||\\sigma) ~.", "cab89ded029645321bb6f90e3e5dc64f": " \\begin{align}\n\\boldsymbol{\\partial} & = \\left(\\frac{\\partial }{\\partial x^0}, \\, \\frac{\\partial }{\\partial x^1}, \\, \\frac{\\partial }{\\partial x^2}, \\, \\frac{\\partial }{\\partial x^3} \\right) \\\\\n& = (\\partial_0, \\, \\partial_1, \\, \\partial_2, \\, \\partial_3) \\\\\n& = \\mathbf{e}^0\\partial_0 + \\mathbf{e}^1\\partial_1 + \\mathbf{e}^2\\partial_2 + \\mathbf{e}^3\\partial_3 \\\\\n& = \\mathbf{e}^0\\partial_0 + \\mathbf{e}^i\\partial_i \\\\\n& = \\mathbf{e}^\\alpha \\partial_\\alpha \\\\\n& = \\left(\\frac{1}{c}\\frac{\\partial}{\\partial t} , \\, \\nabla \\right) \\\\\n& = \\mathbf{e}^0\\frac{1}{c}\\frac{\\partial}{\\partial t} + \\nabla \\\\\n\\end{align}", "cab900ba8f406538909cedcdbde9ab1f": "\\phi(\\boldsymbol{x},t),", "cab924c858baeaefb272a9a74c77a8eb": "\\lambda\\sqrt{p} > (p^{1/4} + 1)^2 \\, ", "cab93141ef4072cb883e43e5fe6d54f5": " M(t) = M_0 + \\left(M_\\text{eq}-M_0\\right)\\exp(-t/\\tau).", "cab959245a897169f45f7b59d80356a5": "\\sum_{i=1}^n \\left(y_i - \\bar{y}\\right)^2 = \\sum_{i=1}^n \\left(y_i - \\hat{y}_i\\right)^2 + \\sum_{i=1}^n \\left(\\hat{y}_i - \\bar{y}\\right)^2.", "cab9b5e4d1d2de085fecc41ba37d05c2": "\\begin{matrix}\\frac{1}{6}\\end{matrix}", "caba03ca5794e140054f526e01a805f3": "y_{n_k}", "caba066db2828a5a3d8a0d151deae676": "V = k\\times A \\times\\sqrt{A}", "caba09042800e37e64f3b4fa817a70aa": "S(\\mathbf{x}-\\mathbf{X}),", "cabab3d300ace61bf120b77c4575f236": "\n\\begin{Bmatrix}\n\\begin{Bmatrix}FS:\\\\GS:\\end{Bmatrix}\n\\begin{bmatrix}{\\rm general\\;register}\\end{bmatrix} +\n\\begin{bmatrix}{\\rm general\\;register}*\\begin{Bmatrix}1\\\\2\\\\4\\\\8\\end{Bmatrix}\\end{bmatrix}\\\\\\\\\nRIP\n\\end{Bmatrix} +\n\\rm [displacement]\n", "cabac7f8b32cffacc3c8fd7df403919b": "(5.a)\\quad \\big(F^{ab}\\big)_{;\\,b}=0\\,,\\quad F_{[ab\\,;\\,c]}=0\\,.", "cabaf6f31a02eab2d2ede3274e168b8f": "\\mathbb{E}|W_i| < \\infty.\\, ", "cabb5a0e500510cec02a0c7db4983829": "\n\\mathcal{G}^{(n)}(1 \\ldots n | 1' \\ldots n')\n= \\langle T\\psi(1)\\ldots\\psi(n)\\bar\\psi(n')\\ldots\\bar\\psi(1')\\rangle,\n", "cabbc847af1afa5ddcbbcae1ec087fa9": " \\ J2: ((A)(B))C = ((AC)(BC)).", "cabc295e14c8867480a6e0a2fb16baeb": "(f\\times g)(a, b) = (f(a), g(b)).", "cabc44780f60186862cb6f86326f0666": "\\mathbf v = v_1 \\mathbf e_1 + v_2 \\mathbf e_2", "cabccc8d8374979090da933fc08e85f1": "PLCM", "cabccf13037bee90fd451bcb3ff246bb": "f_{s}, s=1,2,\\dots,k,", "cabd101965d50753032b594cd6b45ac6": "(R_0)", "cabd2311b4c81453a93d1c5897979283": "\\displaystyle{\\Pi(t) H_n = e^{i(2n+1)^{1\\over 2} t} H_n.}", "cabd325bd59f6541a56c7a09731d5fa9": "\\ y_n", "cabd3e5505d2556e5d44c0ba24348012": "2^n(2^n-1){b_n \\over n}", "cabd82838c9473b6eeba05bdbec7d7f8": " \\mathcal{S} = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-\\eta} \\eta^{ab} g_{\\mu \\nu} (X) \\partial_a X^\\mu (\\sigma) \\partial_b X^\\nu(\\sigma) = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\left( \\dot{X}^2 - X'^2 \\right) ", "cabe6333dc0aa783167b3ec12cace686": "T_pM", "cabe83cd9689b6f84db63d577a3d8ba9": "(1+r_1)^{d_1}(1+r_{t_1,t_2})^{d_2-d_1} = (1+r_2)^{d_2}", "cabe9396c02887f6378830ce1cfa71ee": "p={ip 60\\over LARK}", "cabec781e313c9987aa06c7e0f84905f": "\\sigma_{-\\!1}(n)", "cabee02742a31e7f2988e113434dc011": "\\mathcal{F}_t", "cabef1e01b085392612f3db84285ea48": "\\text{Append}\\; \\begin{cases} \\text{Size} > \\left\\vert X_s - (X_{o} + V_{x} \\cdot \\Delta T) \\right\\vert \\\\ \\text{Size} > \\left\\vert Y_s - (Y_{o} + V_{y} \\cdot \\Delta T) \\right\\vert \\\\ \\text{Size} > \\left\\vert Z_s - (Z_{o} + V_{z} \\cdot \\Delta T) \\right\\vert \\end{cases} ", "cabef430c471d0e07bdd86a8ef1c950e": " \\hat{\\textbf{v}}_{k} \\leftarrow \\hat{\\textbf{v}}_{k} + ( \\beta / [ \\Delta \\textrm{T} ] )\\ \\textbf{r}_{k} ", "cabfafc4a99690ea4345b9140b8f4472": "x_u+x_v\\geq 1", "cabfcf1e4fe6339cb4375b49a0d26a75": "\\psi_1 = Ge^{ \\alpha x} \\,\\!", "cabfe27b036b98e870d6453de5b3178f": "\\or, \\neg", "cac00a12270facb442db85667590af16": "2\\|\\mathbf v\\|^2 +2 \\|\\mathbf w\\|^2 = \\|\\mathbf {v + w} \\|^2 +\\| \\mathbf{v-w}\\|^2 \\ , ", "cac0632e6904f59d3d057249f8a19115": "(F \\circ G)[A] = \\sum_{\\pi \\in P[A]} (F[\\pi] \\times \\Pi_{B \\in \\pi} G[B]).", "cac06f7d863882edec46b0e1f8692841": "\\frac{1}{2\\pi i}\\oint_{\\partial\\Delta_i} \\frac{p_y(x_0,y)}{p(x_0,y)}\\,dy = 1.", "cac08c34e78553d4e6417637fe9fd750": " a=-\\delta ", "cac0dc692216f9ca1b0fda452e9d0d5c": "\\eta_{tot}", "cac0e02c96a4e1f6a81e1735faf0b420": "H = 0", "cac0f24f01132905305eea3b66fd5d6b": "X_1 = Y_1 + Y_3,X_2 = Y_2 + Y_3", "cac0fbdd60baef93f7c74e4e1c405faf": "\\operatorname{xor} = \\lambda a.\\lambda b.a\\ (\\operatorname{not}\\ b)\\ b", "cac10acdb3dfc936889c43253d4d4dac": "J_{\\alpha\\beta}=\\frac{\\partial (x_a,\\theta_i)}{\\partial(y_b,\\xi_j)}.", "cac1189d612f0cd407fdfc9a4a60b8e6": "\\alpha = 0.23", "cac1438280aaac8bbf2949948abacd9f": "{\\mathbf{}}+A'_iS_{i+1}B_i(B'_iS_{i+1}B_i+R_i)^{-1}B'_iS_{i+1}A_i", "cac18ca73ff9f567cd6dc57b2f63cff1": "r = \\theta^{-1/2}", "cac1b7a22f7bfe68811c989366db7db5": "\\displaystyle{\\lambda(z)= \\left[{\\mu_0 \\over 1 -\\mu\\overline{\\mu_\\infty}} {g_z\\over\\overline{g_z}}\\right] \\circ g^{-1}(z).}", "cac1b9b107c6b4b9be9c82cbafb3144f": "f(\\beta)", "cac1bc619419bd9e5d6fd21dfce967a3": "\\frac{1}{2\\pi i } \\oint_C R |\\varphi \\rangle d \\lambda = -\\sum_{i=1}^n |e_i \\rangle \\langle f_i | \\varphi \\rangle = -|\\varphi \\rangle,", "cac1e1f352d340b101bf0d51532e5e86": "\n\\quad Q(\\mathbf{r_1},...,\\mathbf{r_n},\\,t) = -\\frac{\\hbar^2}{2} \\sum_{i=1}^{n} \\frac{\\nabla_i^2}{m_i}\n", "cac1f029db6a10d39a74bdf29dc88b49": "M_{\\mu\\nu} = \\int d^3x \\, {M^0}_{\\mu\\nu}", "cac20ae2d049000b53c7257dcca2bf0d": "\\lambda \\in F", "cac213d3eb93292c3eeb9fdc74ee95a5": "B_{p,1}(z)^r=B_{p,r}(z)", "cac22b6f21cd770b573b66b5d6b41a35": "\\rho _{1}", "cac237b03837f4ca4a91e54639c70fcc": "A \\to \\epsilon ", "cac28c14e92320cb140ba1d1809e2645": "m^{(2)}/\\bar{x}^2 > 1", "cac2b5fd5a4ce87e0379fc423bd0b178": "\\scriptstyle p_3=(x_3,y_3)", "cac2c277c640ca2606eb1c7210277791": " \\mathcal{L}", "cac2d18de21f462b88891b0cd32dee4e": " \\min_x f(x) + g(x). ", "cac30517afe169fb967f87df824c01e9": "q:V \\to F", "cac30793bc6c1165d03297e808e04cd5": "T_b=p_b \\cdot \\cos\\alpha_b,\\ T_r=p_r \\cdot\\cos\\alpha_r,\\ T_m=p_m \\cdot\\cos\\alpha_m", "cac3b1f93b68f744f00922e166ddb74a": "\\!h-r", "cac3c54a12dd9a36198dd242aa0f4c15": " \n{\\nu}", "cac3dd933b177a3c574bc0728d1db500": "\n \\tau_m = \\tfrac{1}{8}\\left[-A \\pm \\sqrt{A^2 + 4B^2}\\right]\n ", "cac3f2f27b4e78cff8fbd567f9eff2b5": " \\mathbf{p} =n \\left ( \\sin \\theta \\cos \\varphi, \\sin \\theta \\sin \\varphi, \\cos \\theta \\right) \\ ", "cac4135b6d8dc25f615c6a62a49c1f85": "\\begin{align}\n p_1 &= e_1,\\\\\n p_2 &= e_1p_1-2e_2,\\\\\n p_3 &= e_1p_2 - e_2p_1 + 3e_3 ,\\\\\n p_4 &= e_1p_3 - e_2p_2 + e_3p_1 - 4e_4, \\\\\n & {}\\ \\ \\vdots\n\\end{align}", "cac415903da3ee3ae299df901470ce58": "\n\\frac{T_\\mathrm{total}}{T_{s}}={1+\\frac{\\gamma -1}{2}eM_a^2}\n", "cac46e922982c086695b1faef75376ad": "A \\to \\alpha A_1 A_2 \\cdots A_n", "cac4a31febee649ad171c22f125877eb": "\\gamma(t)=(x(t),y(t),z(t))", "cac59716a32b72900780aef97e01577c": "x^{1/3} = \\exp ( \\tfrac13 \\ln{x} )", "cac5c1acbfa0a679c58717bbcf69e890": "\\scriptstyle x=\\tfrac{-b}{2a}", "cac5c8ddab5c2e5103239c15f555d9c1": "(a_1,\\ldots,a_n ) \\in R \\Leftrightarrow \\mathcal{M} \\vDash \\phi(a_1,\\ldots,a_n)", "cac65fa6f95f68b7c3d25e7b3ec17889": "SS_{E}", "cac67fff5b798b4b8b0ace17f6829425": "\\kappa_n=\\mu'_n-\\sum_{m=1}^{n-1}{n-1 \\choose m-1}\\kappa_m \\mu_{n-m}'.", "cac692cec0c0401771043bc36aa7cae2": " a(x,\\xi) \\in S^m_{1,0}(\\mathbb{R}_x^n \\times \\mathrm{R}^N_\\xi) ", "cac6d69bb6ec03bbc1e759cfdad9d6d1": "T_1=L/(c-v)", "cac6db7d08665fda95137175fbc1cbdf": "\\Delta m = \\mathbf{m}^T\\Delta\\mathbf{b} = \\mathbf{m}^T\\mathbf{R}^T\\Delta\\mathbf{x} = \\mathbf{m}^T\\mathbf{R}^T(\\mathbf{I - A})^{-1}\\Delta\\mathbf{y}", "cac71d4cd50de0e354ef94b9386f5a26": " dF_x = k I I' ds' \\int ds \\frac {\\cos(xds) \\cos(rds') - \\cos(rx)\\cos(dsds')} {r^2}. ", "cac795810e108ee88cbf830c46057546": "y_k=y_{k+1}", "cac84516fc0ab1ca34500ed5868ed325": "S \\, ", "cac853f40ee1f34eb8ff68cd5ca9969c": "\n M_z = \\mathbf{e}_z\\cdot\\mathbf{M} = -Fx \\,.\n ", "cac889c46cf54df8993139e182be5634": "\\xi_{+1}(\\vec{p}) \n= \\frac{1}{\\sqrt{2 |\\vec{p}|(|\\vec{p}| + p_z)}} \n\\begin{pmatrix}\n|\\vec{p}|+p_z\\\\\np_x+i p_y\n\\end{pmatrix} \n= \n\\begin{pmatrix}\n\\cos{\\frac{\\theta}{2}}\\\\\ne^{i\\phi}\\sin{\\frac{\\theta}{2}}\n\\end{pmatrix}\\,", "cac88c530c65b385278ccaf12820b794": "\\mathcal{R}(C) - \\mathcal{R}(C^\\text{Bayes}).", "cac8a53977e7cd9b07db7eae8e5f8ed5": "U_\\alpha |q_1\\rangle = \\sum_{q_2\\in Q} \\delta (q_1, \\alpha, q_2) |q_2\\rangle ", "cac8ecc7c6f3365e4c53a33edd8dfc09": "\\langle\\rm{T}(x),y\\rangle=\\langle x, \\rm{T}^*(y)\\rangle", "cac8fd6238be5522f8d8fa398bc80f27": "\\forall t . \\neg \\textit{occludeopen}(t) \\rightarrow (\\textit{open}(t-1) \\leftrightarrow \\textit{open}(t))", "cac9529c2698a6310c7f23bd9f42516d": "\\mathrm{P}(C|AB) = \\frac{\\frac{4}{40}}{\\frac{4}{40} + \\frac{6}{40}} = \\tfrac{2}{5} \\ne \\mathrm{P}(C)", "cac96b60a54dec0b01ba234d580df9c1": " F(a) = y(t_1; a) - y_1 \\,", "cac98306dfecb1348c3cd4374cda0387": "\\mathbf{\\Xi}", "cac9c506c4236a4daf2e5e0736963d29": "\\frac{\\delta^2 J[\\rho]}{\\delta \\rho(\\mathbf{r}')\\delta\\rho(\\mathbf{r})} = \\frac{\\partial}{\\partial \\rho(\\mathbf{r}')} \\left ( \\frac{\\rho(\\mathbf{r}')}{\\vert \\mathbf{r}-\\mathbf{r}' \\vert} \\right ) = \\frac{1}{\\vert \\mathbf{r}-\\mathbf{r}' \\vert}.\n", "cac9cc85892244c6e1025d6db92016c8": "\\partial W = M_0 \\cup M_1", "cac9d87f9c0f65593096a92d91b71a56": " A = \\begin{bmatrix} 1 & 1\\\\ 0 & 1 \\end{bmatrix}. ", "cac9ec82094b692fd3b541c783ca99d8": "\\begin{align}\nN(x) &= [y_0] + [y_0,y_1]sh + \\cdots + [y_0,\\ldots,y_k] s (s-1) \\cdots (s-k+1){h}^{k} \\\\\n&= \\sum_{i=0}^{k}s(s-1) \\cdots (s-i+1){h}^{i}[y_0,\\ldots,y_i] \\\\\n&= \\sum_{i=0}^{k}{s \\choose i}i!{h}^{i}[y_0,\\ldots,y_i]\n\\end{align}", "cac9fbabc4bdc85e9fdbe948c50071ed": "g_{a b}", "caca6741dae2956faa23baffd2f9e2a5": "w((v|u)+r)=w(((1,1,...,1)+v)|u)=d-w(v)+w(u)", "caca736e8a0f74cbeb432d991d240cb3": "\n \\begin{align}\n \\overline{u}_S\\, \n &=\\, \\overline{u_x(\\boldsymbol{\\xi},t)}\\, -\\, \\overline{u_x(\\boldsymbol{x},t)}\\,\n \n \\\\\n &=\\, \\overline{\\left[\n u_x(\\boldsymbol{x},t)\\, \n +\\, \\left( \\xi_x - x \\right)\\, \\frac{\\partial u_x(\\boldsymbol{x},t)}{\\partial x}\\, \n +\\, \\left( \\xi_z - z \\right)\\, \\frac{\\partial u_x(\\boldsymbol{x},t)}{\\partial z}\\,\n +\\, \\cdots\n \\right] } \n -\\, \\overline{u_x(\\boldsymbol{x},t)}\n \\\\\n &\\approx\\, \\overline{\\left( \\xi_x - x \\right)\\, \\frac{\\partial^2 \\xi_x}{\\partial x\\, \\partial t} }\\, \n +\\, \\overline{\\left( \\xi_z - z \\right)\\, \\frac{\\partial^2 \\xi_x}{\\partial z\\, \\partial t} }\n \\\\\n &=\\, \\overline{ \\bigg[ - a\\, \\text{e}^{k z}\\, \\sin\\, \\left( k x - \\omega t \\right) \\bigg]\\,\n \\bigg[ -\\omega\\, k\\, a\\, \\text{e}^{k z}\\, \\sin\\, \\left( k x - \\omega t \\right) \\bigg] }\\,\n \\\\\n &+\\, \\overline{ \\bigg[ a\\, \\text{e}^{k z}\\, \\cos\\, \\left( k x - \\omega t \\right) \\bigg]\\,\n \\bigg[ \\omega\\, k\\, a\\, \\text{e}^{k z}\\, \\cos\\, \\left( k x - \\omega t \\right) \\bigg] }\\,\n \\\\\n &=\\, \\overline{ \\omega\\, k\\, a^2\\, \\text{e}^{2 k z}\\, \n \\bigg[ \\sin^2\\, \\left( k x - \\omega t \\right) + \\cos^2\\, \\left( k x - \\omega t \\right) \\bigg] }\n \\\\\n &=\\, \\omega\\, k\\, a^2\\, \\text{e}^{2 k z}.\n \\end{align}\n", "cacaeb1254aaa1e7d7af504039fba696": "\\alpha_k = \\gamma/\\lVert g^{(k)} \\rVert_2", "cacb14baddc6371f435524ffe9434962": "\\frac{< \\textrm{\\ is\\ well-founded},\\;[I \\land C \\land V=z ]\\;S\\;[I \\land V < z]}\n {[I]\\;\\mathbf{while}\\;C\\; \\mathbf{do}\\; S \\;[I\\land\\lnot C]},", "cacb3bfd9faa9948b9baec929b89cd61": "\\begin{align}G_{\\alpha\\beta} & = g^{\\gamma\\mu}\\bigl[ g_{\\gamma[\\beta,\\mu]\\alpha} + g_{\\alpha[\\mu,\\beta]\\gamma} - \\frac{1}{2} g_{\\alpha\\beta} g^{\\epsilon\\sigma} (g_{\\epsilon[\\mu,\\sigma]\\gamma} + g_{\\gamma[\\sigma,\\mu]\\epsilon})\\bigr] \\\\ & = g^{\\gamma\\mu} (\\delta^\\epsilon_\\alpha \\delta^\\sigma_\\beta - \\frac{1}{2} g^{\\epsilon\\sigma}g_{\\alpha\\beta})(g_{\\epsilon[\\mu,\\sigma]\\gamma} + g_{\\gamma[\\sigma,\\mu]\\epsilon}),\\end{align}", "cacb53dfb49bbf43475cc35c5f528780": "\\delta_i(fg)=\\delta_i(f)g + (f\\circ s_i)\\delta_i(g).", "cacbb18b8cb1d8f06563a014ef85bc4f": "(K \\otimes_N L) / R", "cacbd2b2ab1071caab50767fa1bb0844": "|\\lambda-\\mu|\\leq\\kappa_p (V)\\|\\delta A\\|_p", "cacbf213e288d6e25c6edf6f423f819b": "\\rho:=\\rho_2=\\sum_k\\lambda_k\\rho_k", "cacc0e0129e535184063529da72289d7": "\\textstyle v(x)", "cacc65bc88450fc7364381072ff6e08c": " x=2\\pi t + 2\\pi [\\cos( 6\\pi nt)\\cos(2\\pi nt)- 1], ", "cacc8c39a8d719a5141194e4978e0bcf": "M' \\to M \\to N", "cacca103545791667da9ef9689bcddaa": "x^+ := \\begin{cases}\n x^{-1}, & \\mbox{if }x \\neq 0 \\\\\n 0, & \\mbox{if }x = 0\n\\end{cases}\n", "caccc165537415be7ce9120e88e9d76e": " A \\otimes B \\otimes C \\rightarrow C \\otimes B \\otimes A", "cacce535b167dae0c2a00729607612d1": "\n{ \\delta \\left ( t' + { { \\left | \\mathbf{r} - \\mathbf{r}' \\right | } \\over c } - t \\right ) }\n", "cacd3a6672739c24cc055d32716d80d8": "a_{j,k}=-\\frac{b_{j,k}}{j^2+k^2}", "cacd75edc2e849be91c78798d7369095": "\\sum_0^\\infty a_i t^i, \\quad a_i \\in R", "cacd7e95eafcf01f5e3c2562e7febb25": "\\lambda_0(u)", "cacd97e6854301d7a0c16e99ad802c4d": "x \\in \\mathfrak{h},", "cacdc31703aecead1245031f64e5ce1a": "\\frac{11}{10}", "cacdc60d45e977add09804b897fb3d5e": "\\log_b (a+c) = \\log_b a + \\log_b \\left(1+\\frac{c}{a}\\right)", "cace152e6e3c3ad216554000a3da6f53": "p(x):=\\{ n \\in x \\mid n\\in_{\\omega} \\omega \\} ", "cace20bd6c776ab0a66ffee77a853029": "\\varepsilon = 0.1", "cace461a9124a1fc59b19c09a45a06cb": "v(y) = u \\left(y / \\sqrt{\\gamma} \\right)", "cace5602c0af825134efb5cdf2b2338e": " \\phi - 0.5", "cace898c9184cb5208a929e123b43bac": "\\log l", "cacedde0c3c1967fc1aee38c750d46fc": "E\\subset TM", "cacf268b0215c495d43d22a361abad10": "\\textstyle{\\mathrm{Var}}(X_i) = n p_i (1-p_i)", "cacf30468ff01dfcaf3633a25e356024": "\n\\begin{bmatrix} U \\end{bmatrix} =\n\\begin{bmatrix} u_{22}, u_{32}, u_{42}, u_{23}, u_{33}, u_{43}, u_{24}, u_{34}, u_{44}\n\\end{bmatrix}^{T}\n", "cacf6062dd13619fce3c2437933cd9b3": "\\sigma_{X+Y}^2=\\sigma_X^2+\\sigma_Y^2", "cacf6317b35434ca77998f4135d9a408": "1 +\\sum_{k=1}^{\\infty} \\left( \\prod_{r=0}^{k-1} \\frac{\\alpha+r}{\\frac{1-F}{F}+r}\\right) \\frac{t^k}{k!}", "cacff48087daf0e2d048c9f77d7eb58b": "AC_XA^T", "cad07c3fa5ede82f216a3ced818df0bf": "L(s,\\tau \\times \\pi)", "cad0867417c818dedcde42cdb74ccafe": "j=n-2^k", "cad0acce720e59a71cb7cfd47e111283": "H(SDN*PASSWD)", "cad0dbe4970259dc6448c14be3c20915": "Z_{|x|,|y|}", "cad11811a6fe686344e658913c972c7a": "\\int_U u D^{\\alpha} \\varphi=(-1)^{|\\alpha|} \\int_U v\\varphi", "cad1221eed65ac6ab0ec8f7bf5acd33a": "\n\\,\\mathrm{d} s^2=\\frac{\\alpha^2}{z^2}(\\,\\mathrm{d} z^2+\\,\\mathrm{d} x_\\mu \\,\\mathrm{d} x^\\mu) \n", "cad14de446aa5c5d981a30dcc2bc8f68": "r_{los}(t)=Re\\{\\frac{\\lambda \\sqrt{G_{los}}}{4\\pi}\\times \\frac{s(t) e^{-j2\\pi l/\\lambda}}{l} \\} ", "cad17e4c9f40c6b65d579cfe2d8d4b0e": "(b_1,b_2,\\dots,b_n)\\in R", "cad1b306388030cf0503cccc9b444bd7": "\\sum_{i\\in\\mathbf{N}}\\dim_K(V_i)t^i.", "cad266b0099f3ad76ccd02bbbb34d843": "x^py^q=k,", "cad28afb00bb2764b7e7f4c3dc2f5cc2": "e \\in K", "cad29374d27b3f8c7579344e92e4c16b": "\\vee S", "cad294845f840518cb832c48476d436a": "a\\mathrm{inf} = \\sqrt{g_{tt}} a\\,", "cad2dea715f19db08ccf56f2f8f81308": "1 + 3 + 5 + 7 + 9 = 25\\, ", "cad2f228980a35ba6ad3ca864e522512": "\\hat{f}_{k}(t)", "cad31ea38d7da15c2b3a9c9d8a5a486c": "\\phi'_A, \\phi_A, \\phi'_X \\, ", "cad386a0a568c5d5e4771ac457a0e612": "(x_n)_{n=1}^\\infty", "cad3894b3216fd1cdb5607067e53bc3c": "\\delta(P,Q)", "cad3b2cec8600d6e34c2d0f6b8458831": "\\,k>5", "cad3b737e635aa3da95392f3efd80532": "\\mathbf T^{(\\mathbf n)}\\,\\!", "cad3da6f26e7acef7013cdb1dda49ee5": " \\mathcal{L} \\, = \\, - \\frac{1}{4 \\mu_0} \\, F_{\\alpha \\beta} \\, F^{\\alpha \\beta} \\, \\sqrt{- g} \\, + \\, A_{\\alpha} \\, J^{\\alpha} \\,", "cad4964fd4eb23a2e6d8ceacc35fd2f1": "\\mathbf{G}\\cdot (\\mathbf{a}+\\mathbf{b}+\\mathbf{c})=2\\pi (h+k+l)", "cad4ac7761d393a7877e569bc641a675": "\\begin{alignat}{2}\n\\mathbf h_k&=-F'(\\mathbf x_k)^{-1}F(\\mathbf x_k)\\\\[0.4em]\n\\alpha_k&=M\\,\\|F'(\\mathbf x_k)^{-1}\\|\\,\\|h_k\\|\\\\[0.4em]\n\\mathbf x_{k+1}&=\\mathbf x_k+\\mathbf h_k.\n\\end{alignat}", "cad4b7b4f5cc14405bd0a9907cdf7311": "d = \\frac{(b+d)-(b-d)}{2}\\, ,", "cad4c7f6369a7340624386348e1682c6": " \\cos(20^\\circ) \\cdot \\cos(40^\\circ) \\cdot \\cos(80^\\circ)=\\frac{1}{8}.", "cad583360c4c4dbcf3949b8f5450a8df": "\\hbar k_i=", "cad647a5ff760eb63e03bc48077fa66d": "\\begin{bmatrix}\\dot{v}_4 \\\\ \\dot{v}_3\\end{bmatrix} = \\begin{bmatrix}-({1 \\over {C_4 R_6}} + {1 \\over {C_4 R_2}}) & {1 \\over {C_4 R_2}} \\\\ {1 \\over {C_3 R_2}} & -{1 \\over {C_3 R_2}} \\end{bmatrix} \\begin{bmatrix}v_4\\\\v_3\\end{bmatrix} + \\begin{bmatrix}{ 1 \\over {C_4 R_6} } \\\\ 0 \\end{bmatrix} v_1", "cad64cf63b37aaa4f1fc0a6c9c2e342f": "\\omega_c / (2\\pi).\\,", "cad68776e1c9fa609e06db5ef5445236": " \\frac{f(k+1,n,p)}{f(k,n,p)}=\\frac{(n-k)p}{(k+1)(1-p)} ", "cad72ba836650c427a64f12b09f99780": " e^2 = f(2-f) ", "cad77a21926c2ab576fdb63754fe9075": "{\\lambda\\over{c}} = {(600\\times10^{-9})\\over{(3\\times10^8)}} = 2.0\\times10^{-15}", "cad79e58b518354e7942ff8693fe2da5": "D = \\frac{2|E|}{|V|\\,(|V|-1)}", "cad7d8d0313f3de3eaca0a03cf168cb6": " X_{ij}= \\frac{\\partial f(x_i,\\boldsymbol \\beta)}{\\partial \\beta_j}= \\phi_j(x_{i}) . \\, ", "cad7df365bb089fc0caa13fcec8ae956": "\n \\int_{\\Omega} [\\boldsymbol{\\sigma}\\cdot\\nabla{\\mathbf{w}} - \\rho\\,\\mathbf{b}\\cdot\\mathbf{w} + \\rho\\,\\dot{\\mathbf{v}}\\cdot\\mathbf{w}]\\,\\text{dV}\n = \\int_{\\partial\\Omega} \\mathbf{t}\\cdot\\mathbf{w}\\,\\text{dS}\n ", "cad87ffd1365e37a62345ae9311eb5ed": "x_i \\in X", "cad897788e18d04931a0610f4861ba96": "P_j\\,\\!", "cad8d008e9dd711ce46819fee2486ef0": "G = ({\\rm GF}(q), +)", "cad92e1c1b8dbf6cb2c4a53d75898b81": "E(x,y) = \\frac{1}{2}\\sum\\limits_n\\left(x_n - y_n\\right)^2.", "cad956b20b56e2b56cb25e802ca6e78c": "\\beta_0^{(0)} = \\beta_0", "cad983a3f5ca553ad8ed1c00e2258d88": " \\zeta(u)T(u)Q(u)=\\phi(u-\\eta)Q(u+2\\eta)+\\phi(u+\\eta)Q(u-2\\eta)", "cad998c9669ecfccef3ce29bbfe29ac3": "A^* = \\mbox{argmax}_A f(A)", "cad9a396185042dc64391250306cc3ae": " \\rho_x = r_x + \\frac{r^2 a'_x}{2c^2} ", "cada0ec30165c7c7faa5e6977b0df827": "(C)-(E)-(F)-(H)", "cada21b50aaa41f79302075755bf644a": "W_1,W_2,\\ldots", "cada5be7f1448693285b713537f731bf": "\n\\begin{bmatrix}y_1 \\\\ y_2 \\\\ y_3 \\\\ y_4 \\\\ y_5 \\\\ y_6 \\\\ y_7 \\end{bmatrix} = \n\\begin{bmatrix}1 & 0 & 0 \\\\1 &0 &0 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 1\\end{bmatrix}\n\\begin{bmatrix}\\mu_1 \\\\ \\mu_2 \\\\ \\mu_3 \\end{bmatrix}\n+ \n\\begin{bmatrix} \\epsilon_1 \\\\ \\epsilon_2 \\\\ \\epsilon_3 \\\\ \\epsilon_4 \\\\ \\epsilon_5 \\\\ \\epsilon_6 \\\\ \\epsilon_7 \\end{bmatrix}\n", "cadaba9a9f9efc624fdb77474fd72a5c": "\nQ_k = \\sum_{A=1}^N \\sum_{i=1}^3 D^k_{Ai}\\, d_{Ai} \\quad \\mathrm{for}\\quad k=1,\\ldots, 3N-6.\n", "cadaeae01bd2271c660cf4bba962c7e3": "T_{\\odot}", "cadaf2ff73cbc3cfc1ced17ea75815aa": " \\ C_h", "cadb25cd2177a358bce1bb57407c1eb4": "\\mid \\psi_{x+} \\rangle", "cadb76ba005aa1638189c53a69ef714d": "\\quad\\quad f(x) = \\tan(x)", "cadc0a3e541aaca22cfc7efac742a4d2": "vol(\\mathbf A)=\\sum_{v\\in V} \\frac{1}{|A_v|}.", "cadc8ded9426cf6747e2949999c0a180": "a > 0,\\ a \\ne 1", "cadd78e13c020036d4d88b9d973c2f5d": "B^n E_t [ X_{t+j} ] = E_t [ X_{t+j-n} ] \\, .", "cadd7da72bb72e5830a0468027cb03b9": "H \\simeq 1", "cadd8cf4ec867563fe300f899aad47ee": "P(x_0,y_0)", "caddd3fc3a1bcd8c8bedcef135e73c42": "r=5, \\ \\theta={\\pi \\over 9}", "cadde572cd25ed6bfcc816514546e5ca": "\\begin{align}\n\\alpha > 2: \\quad \\mathcal{I}_{a,a} &=\\operatorname{var} \\left [\\frac{1}{X} \\right] \\left (\\frac{\\alpha-1}{c-a} \\right )^2 =\\operatorname{var} \\left [\\frac{1-X}{X} \\right ] \\left (\\frac{\\alpha-1}{c-a} \\right)^2 = \\frac{\\beta(\\alpha+\\beta-1)}{(\\alpha-2)(c-a)^2} \\\\ \n\\beta > 2: \\quad \\mathcal{I}_{c, c} &= \\operatorname{var} \\left [\\frac{1}{1-X} \\right ] \\left (\\frac{\\beta-1}{c-a} \\right )^2 = \\operatorname{var} \\left [\\frac{X}{1-X} \\right ] \\left (\\frac{\\beta-1}{c-a} \\right )^2 =\\frac{\\alpha(\\alpha+\\beta-1)}{(\\beta-2)(c-a)^2} \\\\\n\\mathcal{I}_{a, c} &=\\operatorname{cov} \\left [\\frac{1}{X},\\frac{1}{1-X} \\right ]\\frac{(\\alpha-1)(\\beta-1)}{(c-a)^2} = \\operatorname{cov} \\left [\\frac{1-X}{X},\\frac{X}{1-X} \\right ] \\frac{(\\alpha-1)(\\beta-1)}{(c-a)^2} =\\frac{(\\alpha+\\beta-1)}{(c-a)^2} \n\\end{align}", "caddf420a2223e681df8a1ecd06dfe77": "\\Delta G_{i}= \\sum_{j}\\gamma_j O_j \\,\\!", "cade27b12502beebb8290ff3c3e11a6c": "\\cfrac{\\cfrac{stC \\qquad \\overline{s}D}{tDC} \\, \\operatorname{var}(s) \\qquad s \\overline{t} E}{sCDE} \\, \\operatorname{var}(t) \\Rightarrow\n\\cfrac{\\cfrac{stC \\qquad s \\overline{t} E}{sCE} \\, \\operatorname{var}(t) \\qquad \\overline{s} D}{CDE} \\, \\operatorname{var}(s)", "cade47736346421b021689cf71db1a8b": "\\mathcal{FL^-}", "cade7406f8c82b9bbb13f57f655642d5": "t = 25 \\quad \\mathrm{seconds}", "cadedea7605b6cd619ecae01e06cd2e8": "n_{j,k} a_k", "cadefde9712e9fe9095cd60bdeea8e44": "f(x+h)", "cadf067ccf9f21b91be3feb6c1474b1c": "k_\\mathrm{DM}", "cadf0c1eff3b927afe32eb9cc7a922ae": "\\theta (z,\\tau)=\\sum_{m\\in Z^n} \\exp\\left(2\\pi i \n\\left(\\frac{1}{2} m^T \\tau m +m^T z \\right)\\right). ", "cadf1cb029bfcaf21996772f714c0854": "\\begin{align}\nM_{4,X} = M_{4,A} + M_{4,B} & + \\delta^4\\frac{n_A n_B \\left(n_A^2 - n_A n_B + n_B^2\\right)}{n_X^3} \\\\\n & + 6\\delta^2\\frac{n_A^2 M_{2,B} + n_B^2 M_{2,A}}{n_X^2} + 4\\delta\\frac{n_AM_{3,B} - n_BM_{3,A}}{n_X} \\\\\n\\end{align}", "cadf5570e6160884b1bc900a533ba51d": "R_2 =45 \\times 0.0362 ", "cadf5c9d32165e6dd208c6db9230c143": "S = S(U,V,\\{N_i\\})", "cadf7916ab99da6242c0fa307c25c277": " d \\star d h = \\exp(-2 p) \\, \\left( h_{xx} + h_{yy} \\right) \\, \\sigma^1 \\wedge \\sigma^2. ", "cadfa0139b788490f215d5ab3d1a2342": "P\\left ( v \\right )=4\\pi\\left ( \\frac{m}{2\\pi k_B T} \\right )^{3/2} v^2 e^{-mv^2/2 k_B T} \\,\\!", "cadfb20d89178ff40c6c0d8ea9220142": "\\mathbf{x}^J", "cadfb816343bc7e4eadf93b7041695ba": " \\int_{\\overline{\\mathcal{M}}_{1,1}} \\psi_1 = \\int_{\\overline{\\mathcal{M}}_{1,1}} \\lambda_1= \\frac1{24}.", "cadfb89f4dc4fd92e7dcc41467d3061c": "\\cot \\frac{\\theta}{2} = \\pm\\, \\sqrt\\frac{1 + \\cos \\theta}{1 - \\cos \\theta} = \\frac{1 + \\cos \\theta}{\\sin \\theta} = \\frac{\\sin \\theta}{1 - \\cos \\theta} = \\csc \\theta + \\cot \\theta.\\,", "cae015b0140a44cade06252e9f326048": "O^p(G).", "cae01ff7b37cb1b213e17722138833ff": "r=\\frac{b}{n}", "cae0b661e3d492c8482eba7611d054db": "m = - \\frac{1}{8 \\pi} \\int_S \\epsilon_{abcd} \\nabla^c \\xi^d ", "cae123b7435531b3a6310ecb021fd516": "F(e^\\lambda g) = F(g) - d\\lambda", "cae18813675eaebccb282465dde115d8": "\\begin{pmatrix}0&0&\\cdots&0&-a_0\\\\1&0&\\cdots&0&-a_1\\\\0&1&\\cdots&0&-a_2\\\\\\vdots&\\vdots&\\ddots&\\vdots&\\vdots\\\\0&0&\\cdots&1&-a_{n-1}\\end{pmatrix}.", "cae193772bb9ac4ee88c4baec24ee5b6": "\\oint \\vec{v}_s\\cdot\\vec{\\mathrm{d}s}=\\frac{h}{2\\pi m_4} \\oint \\vec{\\nabla}\\varphi \\cdot \\vec{\\mathrm{d}s}.", "cae1b0121d6082464561cf4a1c6ed1d2": "\\min\\{n_1,n_2\\} ", "cae1e501b2ce1462f24011204240ba8a": "A_q(n,d) \\leq \\frac{q^n}{\n \\begin{matrix}\n \\sum_{k=0}^t \\binom{n}{k}(q-1)^k\n \\end{matrix}}.", "cae1fbb75be6e3925beb6e6f4f9f2ea5": "Z_F = \\sqrt{\\frac{\\mu}{\\varepsilon}}", "cae231fa2bb15fb2d9342c244b380536": " {\\mbox{d} l \\over \\mbox{d} r} = 4 \\pi r^2 \\rho ( \\epsilon - \\epsilon_\\nu )", "cae2c3ff019194ebcdae7f11174b64a1": "\\sqrt{x}_s", "cae2d37619f1ce4ee3d277e9c949c90e": "\\boldsymbol s=\\boldsymbol s_{\\boldsymbol\\Theta}", "cae32b596064d58d261111e1787675de": "y(x_0-h)", "cae3474d8fffebae2e0690ebab32579a": " \n\\Gamma(\\Omega) \\propto -\\frac{\\mathrm{Im} \\Sigma(\\Omega)}{\\left[\\Omega-\\omega_{\\mathrm{c}}-\\mathrm{Re} \\Sigma(\\Omega)\\right]^2 + \\left[\\mathrm{Im}\\Sigma(\\Omega)\\right]^2} . \n", "cae35dc78272b4802aeab3f6b2c2b49a": "\\displaystyle{\\{G_r^+,G_s^+\\}=0=\\{G_r^-,G_s^-\\}}", "cae36f20f933ddb675efd9254c01dc31": " \\begin{bmatrix} 1 & -w \\\\ 0 & 1 \\end{bmatrix}. ", "cae3f05342ab9fc5a9acb85a8f44050f": "f \\mapsto f(x)", "cae403a663c8dfc6d40d7c5963ab7cfb": "x \\longrightarrow a \\otimes x - b \\otimes x ", "cae52b12ee8771870ced1bcb9bb96be0": "x=N_1/N_2", "cae70dff06aec3499bccf6cf2b043621": " u^n_i ", "cae786949031e32820743835e5de3619": "\\mathfrak g = \\mathfrak{m}_{+}\\oplus\\mathfrak l\\oplus\\mathfrak{m}_-", "cae7d523de7cb7109e943b51f9d08427": "\\mathrm{Ek} = \\frac{\\nu}{2D^2\\Omega\\sin\\varphi} ", "cae802fc5eda43024840a5ae38f181a6": " \\vec a = N\\vec V_r \\times \\vec \\Omega", "cae8037fb39b3ac73ffc544ff0a701e5": "\\textit{lassie}: \\textit{dog}", "cae809bedff22c5ecea94c64e8eac1eb": "\n\\lim_{x \\to \\infty} \\Pr[X>x+t|X>x] =1, \\,\n", "cae8244d957652d715dfaa976c863694": "\\#:\\Omega^{p+1}M\\rightarrow T^*M\\otimes\\Omega^pM", "cae8476c66e5e279e8b1019030add46b": "(v_1,v_2)", "cae8a3ac6ee5374871968c36662dc2af": " \\|f\\|_2 = \\sqrt{\\int_G |f|^2 d\\mu}", "cae8f245ca79aee5d7ea603b331a0a2b": "C_\\text{L}=2\\pi \\sin\\alpha \\quad \\text{and} \\quad C_\\text{D} = 0.", "cae934894c7f1c8d5770136a1e3caa63": "[A(t_n),A(t_m)] = 0", "cae954061671bb9efc8b46e510e2db0b": "c^{\\left[\\sum_{n=s}^t f(n) \\right]} = \\prod_{n=s}^t c^{f(n)}", "cae96f87613c0fdba4d5dc973b250e1f": "\\textstyle R(a_\\diamond\\mid[x])", "cae9743b2aa30af47283cd8d49c0b452": "a>1", "cae979ed1a2c74f8799b75f05d03f00c": "d^2(\\mathbf{l}\\cdot\\mathbf{l})+2d(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))+(\\mathbf{o}-\\mathbf{c})\\cdot(\\mathbf{o}-\\mathbf{c})-r^2=0", "cae9dca9bfac9e93bd88aed6633005d9": " \\ X^N-1 ", "caea043d8821c292a5b312e5044d5014": "\\frac{u_i^{n+1} - \\frac{1}{2}(u_{i+1}^n + u_{i-1}^n)}{\\Delta t} + a\\frac{u_{i+1}^n - u_{i-1}^n}{2\\,\\Delta x} = 0", "caea0e243329b93ae5d4de5196453e5d": " i_1 - i_2 - i_3 = 0 \\, ", "caea3e0b7c6ddeefe6cdcf835302058e": " f = \\frac{[sm_{fu} - m_{ox}] - [sm_{fu} - m_{ox}]_0}{[sm_{fu} - m_{ox}]_1 - [sm_{fu} - m_{ox}]_0} ", "caea5b509673ffb3d1989d9b51e1cc99": "\\lim_{t\\to\\infty}\\|T(t)\\|=0", "caea7072f0153062b4df7a255745bf52": "\\mathbb{F}_q ", "caeac68832cdc8b02117323acafdd714": "\\rho = 1/\\delta \\sigma", "caeb43ed799fac95ab8c5c33fff79c82": "\\{\\pm 1,\\pm i,\\pm j,\\pm k,\\tfrac{1}{2}(\\pm 1 \\pm i \\pm j \\pm k)\\}", "caeb5d62232b0a0dcf87f4d2a16334fe": "\nP_x = \\frac{\\begin{vmatrix} \\begin{vmatrix} x_1 & y_1\\\\x_2 & y_2\\end{vmatrix} & \\begin{vmatrix} x_1 & 1\\\\x_2 & 1\\end{vmatrix} \\\\\\\\ \\begin{vmatrix} x_3 & y_3\\\\x_4 & y_4\\end{vmatrix} & \\begin{vmatrix} x_3 & 1\\\\x_4 & 1\\end{vmatrix} \\end{vmatrix} }\n{\\begin{vmatrix} \\begin{vmatrix} x_1 & 1\\\\x_2 & 1\\end{vmatrix} & \\begin{vmatrix} y_1 & 1\\\\y_2 & 1\\end{vmatrix} \\\\\\\\ \\begin{vmatrix} x_3 & 1\\\\x_4 & 1\\end{vmatrix} & \\begin{vmatrix} y_3 & 1\\\\y_4 & 1\\end{vmatrix} \\end{vmatrix}}\\,\\!\n\\qquad\nP_y = \\frac{\\begin{vmatrix} \\begin{vmatrix} x_1 & y_1\\\\x_2 & y_2\\end{vmatrix} & \\begin{vmatrix} y_1 & 1\\\\y_2 & 1\\end{vmatrix} \\\\\\\\ \\begin{vmatrix} x_3 & y_3\\\\x_4 & y_4\\end{vmatrix} & \\begin{vmatrix} y_3 & 1\\\\y_4 & 1\\end{vmatrix} \\end{vmatrix} }\n{\\begin{vmatrix} \\begin{vmatrix} x_1 & 1\\\\x_2 & 1\\end{vmatrix} & \\begin{vmatrix} y_1 & 1\\\\y_2 & 1\\end{vmatrix} \\\\\\\\ \\begin{vmatrix} x_3 & 1\\\\x_4 & 1\\end{vmatrix} & \\begin{vmatrix} y_3 & 1\\\\y_4 & 1\\end{vmatrix} \\end{vmatrix}}\\,\\!\n", "caeb943be3ef993c5c34e4231592336e": " \\mu_{e} ", "caebbcf790d2117a0ecfd158f70a2845": "\\boldsymbol\\Sigma_{XX} = \\mbox{var}(\\boldsymbol{X}), \\boldsymbol\\Sigma_{YY} = \\mbox{var}(\\boldsymbol{Y}),", "caebfdde27cb25e9bf20468c26ce7104": "n+c", "caec17489d601639b85ff661556b2a34": "d_{graphite}=0.335 nm", "caec76bfa6f229534f3438fa08652a14": "df = 0 ", "caec83871cbb26571bfe08c2275691fc": "\\mathop{\\mathrm{ker}}(\\pi_1(S) \\to \\pi_1(M)) - N \\neq \\emptyset", "caec9d3a69071320d39c1e8ae0c485f3": "f:\\mathbb{R}^n \\to \\mathbb{R} ", "caeca1a152c0d7b1de9f0a98a9d61e53": " \\arctan z = z - \\frac {z^3} {3} +\\frac {z^5} {5} -\\frac {z^7} {7} +\\cdots\\ \n= \\sum_{n=0}^\\infty \\frac {(-1)^n z^{2n+1}} {2n+1}; \\qquad | z | \\le 1 \\qquad z \\neq i,-i ", "caeca4cd68454d8ec396066b856a8ef3": "\\int_0^\\infty \\frac{2\\arctan (\\tfrac{t}{z})}{\\exp(2 \\pi t)-1} \\,{\\rm d}t = \\sum_{n=1}^\\infty \\frac{c_n}{(z+1)^{\\bar n}}", "caecaf633bc456b4ca6d135baeb1531d": "p \\ge 2^x", "caed00a256879e7dc861227434312579": "F(x,y) \\mapsto gF(g^{-1}x, g^{-1}y)", "caed4c067cc3d69af6ac95c0c72b0508": "L_n=V_n(1,-1)", "caed89fb72f30953255afff7da4c77a8": "\\alpha \\bar \\alpha=1/\\beta", "caed9b77522a0c03eb0f906094a0cff9": "q_{1}\\stackrel{\\epsilon , T}{\\rightarrow}r_{0}\\stackrel{x_{1} , T}{\\rightarrow}r_{1}\\stackrel{x_{2} , T}{\\rightarrow}r_{2}\\cdots r_{m-1}\\stackrel{x_{m} , T}{\\rightarrow}r_{m}, r_{m}\\in A_{1}\\cup A_{2}, r_{0}, r_{1},\\cdots r_{m}\\in Q", "caedc0f54e1443e5b91db1c644e145cf": "L'=\\{x1^{2^{|x|^c}} \\mid x \\in L\\},", "caedc513aec0c97deff0e7fc8f1fc00c": "\n\\mathcal{G}(\\mathbf{x}\\tau|\\mathbf{x}'\\tau') = \\int_\\mathbf{k} d\\mathbf{k} \\frac{1}{\\beta}\\sum_{\\omega_n} \\mathcal{G}(\\mathbf{k},\\omega_n) \\mathrm{e}^{\\mathrm{i} \\mathbf{k}\\cdot(\\mathbf{x}-\\mathbf{x}')-\\mathrm{i}\\omega_n (\\tau-\\tau')},\n", "caedf626699817268d888f5faf796889": "P=\\{p, 1-p\\}^\\mathbb{Z}", "caedf65d81a82830bb7de8562766faf5": " \\nu = (r-\\delta)/\\sigma - \\sigma / 2, \\quad c = \\tilde{\\gamma} K / (\\tilde{\\gamma} - 1). ", "caee05fa81588b35bdf79fc02827f2e2": "|\\psi(x_0,t)|^2 \\, dx", "caee32f33e4b57d58e0fdbc721c72049": "\\ell\\in\\mathbb{Z}^{k}", "caee434bc99a1b6f039f2d905a7c3e34": "\\beta_T=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_T\n\\quad = -\\frac{1}{V}\\,\\frac{\\partial^2 G}{\\partial P^2}", "caeeb14dc7f1149f4d91f35ec97dc9bc": "b_n =2\\int_0^{2\\pi} e^{-int} \\Re g(e^{i\\theta}) \\,d\\theta.", "caeec166b9a7df592c0882379ff99a75": "[D]", "caeef7d367fd5de94d4b8da5398e48df": "\n\\mathcal{A}=\\sum a_{\\alpha}\\partial^{\\alpha},\\qquad \\partial^{\\alpha}=\\partial_1^{\\alpha_1}\\cdots\\partial_n^{\\alpha_n}.\n", "caef0d0c4106bcef8255c3a3dcbe506b": "(s)_{k}", "caef5c49b596a56e9b2e3d2bc993a30d": "\\epsilon = \\frac{1}{2 q 2^q}", "caef723a1a171358c4a53d42edd932f4": "b=\\tfrac{k^2(s^4-r^4)}{2}, \\, ", "caef73d8a83c0dc2f9b8b51f546a41e1": "|i\\rangle, |n\\rangle, |f\\rangle", "caef82a3869ef5c4c017b95c47ef9c2f": "\\rho = \\Omega^\\Omega \\psi(\\Omega^{\\Omega+1})", "caef8d826be55918a819249332d6cdae": "F=m a", "caefd12f9f0d57d2e860e34ae6d217c6": " h(u) ", "caf0425ffe98ce5d606cfb1ae62aab2d": " m_s ", "caf08fa2820e37a4e3482ec96984fbff": "b_1,\\ldots,b_{d-1}=0,", "caf1202a3c05d728d7205165b5b19735": "R/\\mathfrak{m}.", "caf139bfe402aa5b5bbfa251d7a2298b": "N_{A(f)} = N_{A(\\Delta)}\\,", "caf174d390781fa6b9e10f268364ab43": "\\prod_{i=1}^{k-1}\\left(1-\\beta'_i\\right)", "caf204104a9ee64ff097b45f6db20f81": "\\mathbb{F}_{167}", "caf2169d36b2dee91e1fff5c1ea566c3": "VV^T=I", "caf2366aeb6df3b4fe125b1e492122c0": "b = 1.7R^{1/3} + 8G^{1/3} - 9.7B^{1/3}", "caf2932446388b2769e77bf49a91a1f1": "\n M(\\phi) = \\frac{a(1- e^2)}{\\left(1-e^2 \\sin^2 \\phi\\right)^{3/2}}\n", "caf2da3af34e26ad8ce8bc31bf64aa64": "\\boldsymbol\\beta_0 = \\mathbf{0} .", "caf2fa6adf5b69c2ff58e83eca08516d": "\\scriptstyle h[n]", "caf301fd2ae8f3e2518af579daf7329d": "\\frac {v_e} {2}", "caf35fb86c1f89fd8ca0501fe41cee60": "[ \\hat{x}, \\hat{p} ] = i\\hbar", "caf36b030b16621e316b91fc27c64af6": "(s,\\pi_v,r_{i,v},\\psi_v)", "caf3f99ce7afbdb340b5bb874bc64f6c": "\\lim_{h\\to 0^+}\\sum_{n=0}^\\infty \\frac{t^n}{n!}\\frac{\\Delta_h^nf(a)}{h^n} = f(a+t).", "caf4288d379ff2c4d4296c6b848b4e77": "\\textstyle K \\subseteq \\mathbb{R}", "caf458e324c3ccc5d5936bfb7ea9dc19": "\n\\frac{v^2}{c^2} = \\frac{c^2p^2}{E^2} \\quad\\Longrightarrow\\quad E^2= \\frac{m_0^2c^4}{1 - c^2p^2/E^2}\n\\quad\\Longrightarrow\\quad m_0^2 c^4 = E^2 - c^2p^2.\n", "caf47bc937f771a57f44621802a704a2": "K_m ", "caf4a4c0e60a1ceae4f79a84c96ac745": "\\frac{\\partial f}{\\partial\\bar{z}} = \\varphi(z,\\bar{z})", "caf4b4651b7b9e0e4d0f8dd25fd1b19a": " = \\sum_{\\theta_{-i}} \\ p(\\theta_{-i} | \\theta_i) \\ u_i\\left(s_i(\\theta), s_{-i}(\\theta_{-i}),\\theta_i \\right) ", "caf4b565a06ff76cfb049ffc9843dcdc": "T_G= T_{G-e}+T_{G/e},", "caf5c768e8ea26ac48fd295c03f17070": "{x^2 \\over a^2} + {y^2 \\over b^2} - {z^2 \\over c^2} = 0 \\,", "caf5e267868f985b7ce2af8cca119cb7": " \\sigma^2 = \\lim_n \\frac{E(S_n^2)}{n} ", "caf6561f248260426d31b09fc536c32a": "t(y)^2 g_{ij} \\, dy^i \\, dy^j.\\, ", "caf6800224dc15d5207e22690fcfdc71": "E[F | x^{(t)}] \\le E[F | x^{(t-1)}]", "caf69a7d6c40ca38d54428eb1a049d41": " \\Delta E \\Delta t \\gtrapprox \\frac{\\hbar}{2}. ", "caf69d7f105b5241fba62b15f53ea9fe": "\\widehat{M} = \\frac{\\hat{\\mu}(1-\\hat{\\mu}) - s^2}{s^2 - \\frac{\\hat{\\mu}(1 - \\hat{\\mu})}{N}\\sum_{i=1}^N 1/n_i},", "caf70d08ab07a53428cb3bb996292b02": "\\left( T \\rarr R \\right) \\rarr R", "caf74b7c6221ce89ac79f409f666c708": "F_{14}", "caf754021e7dd481feef6d888e29033d": "\n\\sigma _z^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)\\sigma _{11} \\,\\,\\, + \\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)\\sigma _{22} \\,\\,\\, + \\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)\\sigma _{12} \\,\\,\\, + \\,\\,\\,\\,\\left( {{{\\partial z} \\over {\\partial x_2 }}} \\right)\\left( {{{\\partial z} \\over {\\partial x_1 }}} \\right)\\sigma _{21}", "caf77fd1b34b5817eac9ef9cd44d7831": "\\operatorname{Bun}_G(X)", "caf7ac4bd6fdaf69ee00fcfcb7b4a9cf": "N \\geq 1", "caf7ae0df503a94078f2bdd5b7cffe65": "\n\\begin{array}{rcccl}\n \\operatorname{grad}(f) &=& \\nabla f &=& \\left( {\\mathbf d} f \\right)^\\sharp \\\\\n \\operatorname{div}(F) &=& \\nabla \\cdot F &=& \\star {\\mathbf d} \\left( \\star F^\\flat \\right) \\\\\n \\operatorname{curl}(F) &=& \\nabla \\times F &=& \\left[ \\star \\left( {\\mathbf d} F^\\flat \\right) \\right]^\\sharp, \\\\\n \\Delta f &=& \\nabla^2 f &=& \\star {\\mathbf d} \\left( \\star {\\mathbf d} f \\right) \\\\\n\\end{array}\n", "caf7fee692c1f24b5099ceb1b4e4d064": " O(x_o,y_o) = \\int\\!\\!\\int O(u,v) ~ \\delta(u-x_o,v-y_o) ~ du\\, dv", "caf8089ab77dc3cac457df1152912af6": "=\\frac{e^4}{(k-k')^4}\\operatorname{Tr}\\left( (k\\!\\!\\!/' - m) \\gamma^\\mu (k\\!\\!\\!/ - m) \\gamma^\\nu \\right) \\cdot \\operatorname{Tr}\\left( (p\\!\\!\\!/' + m) \\gamma_\\mu (p\\!\\!\\!/ + m) \\gamma_\\nu \\right) \\,", "caf808e2c6c54ef4bbf88d8b3520c3dc": "\\alpha_\\text{ww} = \\frac{3}{2} \\sqrt{\\frac{g}{h}},", "caf810bdc6ac3bc90dbf49553b59d4e8": "\\delta= \\left({1 \\over \\omega}\\right) \\left\\lbrace \\left( {{\\mu\\epsilon} \\over 2}\\right) \\left[ \\left(1+\\left({1 \\over {\\rho\\omega\\epsilon}}\\right)^2\\right)^{1/2} -1\\right]\\right\\rbrace^{-1/2}", "caf8528a5a3f537a0f7f4d9c75e6fb00": "\nx L_n^{(\\alpha) \\prime\\prime}(x) + (\\alpha+1-x)L_n^{(\\alpha)\\prime}(x) + n L_n^{(\\alpha)}(x)=0,\\,\n", "caf8674269ddbb2b52d7edcc50010e1d": "H^2(X;\\mathbb Q)", "caf87ef4e9b2a6bf20c29682d90e62ac": "\\log(1/\\epsilon)^\\gamma", "caf8d5668cb188404fdc192334d4a285": " c_{t+n} = (1-R^{-1}) \\left[A_{t+n} + \\sum_{j=n}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j-n} E_{t+n} y_{t+j} \\right]", "caf8fcbfc916ee6efa0ac1ff4cf1ee2c": "\\binom{mn}{k} = \\frac{(mn)!}{k! (mn-k)!}", "caf951b5fccb66eedf3e8abcc8f8b07a": "\\dot \\epsilon", "caf951d68cc8a041ab56b32bd0fe6fc4": "\nf^\\star\\left(x^{*} \\right)\n= \\begin{cases} b, & x^{*} = a\n \\\\ +\\infty, & x^{*} \\ne a.\n \\end{cases}\n", "caf975c1163f7f91f267323d8a17d6fd": "f:X \\rightarrow Y", "caf9cd4db0f661c227919469758c2822": "L = C A^h\\ ", "caf9e41e112443e713a9609449fe78ca": "R_N(x):= \\frac{(-1)^N}{\\sqrt{\\pi}}2^{-2N+1}\\frac{(2N)!}{N!}\\int_x^\\infty t^{-2N}e^{-t^2}\\,\\mathrm dt,\n", "cafa08800e1e40f6bbc30797365722f4": "F_{rel} = 100 \\cdot \\frac{AUC_A \\cdot D_B}{AUC_B \\cdot D_A}", "cafa25a0c81b18cbccd84d6fce7a26cc": "M\\ddot{X} = - \\nabla U(X) - \\gamma M \\dot{X} + \\sqrt{2 \\gamma k_B T M} R(t)\\,,", "cafa605fa7c612342dc57c67217c2991": " E = QV", "cafac51e5c8e9e62a4a6b703406eec5d": "\n \\hat\\theta_\\mathrm{mle}\\ \\xrightarrow{\\text{a.s.}}\\ \\theta_0.\n ", "cafae146446e0289132624e53306512b": "A = \\frac{k_1 + k_2}{2}", "cafaea83c7470e039cc76f1c280ab2fc": "\\mu : X \\to \\mathbb{R}", "cafb0bd0eb7aa45f942db8e47a857853": "\\mathbf{x} \\in\\mathbb{R}^p\\!", "cafb2df13c309784607d5a3f3f56bdc3": "O\\left({2^{N/2}}\\right).", "cafba0eccf6b63ec2c6fc0c8f801dd53": "M_\\epsilon = \\sum_{i=1}^{N_\\epsilon}m_{[i,\\epsilon]} = ", "cafbd823d3ef74b4908074dcf2e7d9f4": " \\langle\\mu\\nu|\\lambda\\sigma\\rangle = \\delta_{\\mu\\nu} \\delta_{\\lambda\\sigma} \\langle\\mu\\mu|\\lambda\\lambda\\rangle ", "cafc17eee2b4ae8a7b6e02ff0ed7ecd8": " f(0)=1 ", "cafc3c9a77a7fafd902a5a2dd59c1d2c": "a \\le x \\le b \\!", "cafc7234d19bc84911bfd6922500781d": "\\lambda=1/T", "cafc8ff279999eba221997ce4de0ec77": "J_q(\\delta) \\equiv_ \\text{def} \\left(1-{1\\over q}\\right)\\left(1-\\sqrt{1-{q \\delta \\over{q-1}}}\\right) ", "cafcd42b46847959b87b0c755678bbbf": "F = \\frac{E A_0 \\Delta L} {L_0}", "cafcdee38c825a7a8d0472f1185c9fb8": " \\frac{\\frac{|SA||SD|}{|SB|}}{|CD|}=\\frac{\\frac{|SB||SC|}{|SD|}}{|AB|}", "cafd0712ee8f59a00968eea755cb4d27": "\\gamma= \\frac{1-\\lambda}{\\lambda +1} ", "cafd18e7503df9f24aea93e9937b4b00": "\\frac{dH}{dt} = \\frac {\\partial H}{\\partial x^a} \\dot{x}^a +\n\\frac{\\partial H}{\\partial p_a} \\dot{p}_a = \n- \\dot{p}_a \\dot{x}^a + \\dot{x}^a \\dot{p}_a = 0.", "cafdbc37c79f00f7dd3964540c302539": "S(a,P,z) = X \\prod_{p \\le z,\\ p \\in P} \\left( 1 - \\frac{w(p)}{p} \\right) \\{1 + O(e^{-u/2})\\}.", "cafe15fc152426bf667b13f5e6b44e58": "{\\mu}_s", "cafe41bad6ff57150acd572c9d5eb843": " \\lim_{x \\to c} \\frac{f(x)}{g(x)} = \\lim_{x \\to c} \\frac{f'(x)}{g'(x)} , \\! ", "cafe561104013f327c33be93a9c5e71c": "A_n\\to A_{n+1}", "cafef8934b6dc1b5cd6ab848bbbb2e22": " x_i = v( S_{\\pi(i)} ) - v( S_{\\pi(i) - 1} ), \\forall~ i \\in N ", "caff22c955a1d07dfc7bfdd7701d8d74": "s_{12}^{2}=0.307\\,,\\; s_{23}^{2}=\\begin{cases}\n0.386 & (\\mathrm{NH})\\\\\n0.392 & (\\mathrm{IH})\n\\end{cases}\\,,\\ s_{13}^{2}=\\begin{cases}\n0.0241 & (\\mathrm{NH})\\\\\n0.0244 & (\\mathrm{IH})\n\\end{cases}\\,,\\ \\delta=\\begin{cases}\n1.08\\pi & (\\mathrm{NH})\\\\\n1.09\\pi & (\\mathrm{IH})\n\\end{cases}", "caff52c7ec37f038042d4be5a4a66b7b": "S(\\rho^A,\\rho^B) = S(\\rho^{AB}) = -\\operatorname{Tr}(\\rho^{AB}\\log(\\rho^{AB})).", "caff53095c2b1587b9664b74c860d885": "((x-a)^2+y^2)((x+a)^2+y^2)=b^4.\\,", "caffbb603e69f7c38f1ddb62da136a02": "I = \\int_{t_1}^{t_2} L [\\mathbf{q} [t], \\dot{\\mathbf{q}} [t], t] \\, dt ", "caffc53e6e7d5e12d3135009ea0192fe": "K_6(x,y,z)", "caffd61af6b69de5789a81b361984824": "M->C", "cb0015da9173f0cba795810257390ba2": "\n \\limsup_{n\\to\\infty} \\frac{\\sqrt{n}\\|\\hat{F}_n-F\\|_\\infty}{\\sqrt{2\\ln\\ln n}} \\leq \\frac12, \\quad \\text{a.s.}\n ", "cb004e3dd1e9550df1988af15ee85c86": " T_A = [A] + \\beta_1[A][H] + \\beta_2[A][H]^2 \\,", "cb0051ee3fea0d4bde79455950e98afb": " y_0 ", "cb0072665108e047bc0c03b5bd3a35ef": "F_N", "cb00874e8ef12d2bffebe080beba2ab8": "|n^{(2)}\\rangle =\\left(\\frac{V_{k_1 k_2}V_{k_2 n}}{E_{n k_1}E_{n k_2}}-\\frac{V_{n n}V_{k_1 n}}{E_{n k_1}^2}\\right)|k_1^{(0)}\\rangle-\\frac{1}{2}\\frac{V_{n k_1}V_{k_1 n}}{E_{k_1 n}^2}|n^{(0)}\\rangle", "cb0090bcfe0fed8e2cbbe89ad2661fea": "srt[ab]", "cb01191c69505ae017a5bbabdb294924": "\\qquad \\frac{19}{360 \\lambda^3} + O\\left(\\frac{1}{\\lambda^4}\\right)", "cb01795d04b204bc7681ec1cef98bff8": "m_4 =19", "cb01a2fbacdb4560d09f62e70514da86": "\\scriptstyle n \\;=\\; 0", "cb01b671c07322fc493f5f6d9c70645f": "m = \\frac{V_{\\mathrm{out}}[c_{\\max}] - V_{\\mathrm{out}}[0]}{c_{\\max}}", "cb01c683063ced52a1ea13b4d5cf5e05": "e^{i \\mathbf{k}_{out} \\cdot \\left( \\mathbf{r}_{\\mathrm{screen}} - \\mathbf{r} \\right)}", "cb020ce1d3b3bbb9ff44cfecef8765cb": "S = k\\log Z + \\frac{U}{T} + c\\,", "cb024da3a5d7d0344bd08acfe8b6dc55": " x_1=\\frac{x-y^3\\cdot x}{a\\cdot y\\cdot x^3-y} ", "cb024f27f03e2870db0b397f3e81496d": "\\phi(t)= E[e^{itX}] = \\sum_{j=0}^\\infty e^{ijt}P[X=j]", "cb0262d98b9fe29dcee242d50b9452bf": "f(x_1,x_2) = c_1 x_1 + c_2 x_2", "cb02fcd93d1178869d881d1e2ec9ee87": " \\Pr( d0\\right\\}=\\tfrac{\\lambda_i}{\\mu_i}.", "cb1291520dbaeb96809bc87ba306fe6d": "\\frac{e^{2}}{m_e c^{2}}", "cb12d65444a38f28d6a5ec2de8c53f65": "\\scriptstyle{\\mathbf{n}}", "cb130ca787015bcf4aeaeeda268cb386": "\\zeta \\ll 1", "cb13230d80c64bb4e77f6c8536e4411d": "\\{ 1, x, x^2, x^3, \\dots \\}", "cb133f43be8b7c892a4741f45f5fcea8": "\n\\tau \\frac{\\partial V}{\\partial t} = \\lambda^{2} \\frac{\\partial^{2} V}{\\partial x^{2}} - V\n", "cb135523be533462ed6a084c70a32ffb": "u_n \\to u_0", "cb137564ffa58e871bcc7a9421cfcbc6": "\\pi_2(x)", "cb139ffd45872a9a5f17ece5cdb1d314": "\\log", "cb13dfbfdf579a4240a0d98a012a6fa1": "3^\\frac{8}{13}", "cb14244c8ccb222b4e0071044ce21ae1": "a_{11} X_1 + \\cdots+ a_{1N} X_N = 0", "cb143c46a151999473ada6943c01bf96": " (\\mathrm{Id} - T)^{-1} = \\sum_{k=0}^\\infty T^k ", "cb144aac5c936b00f802b0b90039c2fa": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\sum_{k=0}^{n-1}e^{2 \\pi i m b^k x}=0 \\quad\\text{ for all integers } m\\geq 1.", "cb146c671cf66a615c06b8703acd959e": "\\{p\\in M:\\varphi(p)\\le a\\}", "cb1520535008f3b81319f857da7336fd": "\\mathbf{r}(t) = [r_1(t), r_2(t), \\ldots , r_n(t)] ", "cb1569c98705933fe2acd6d77da179b4": "\\frac{\\partial {\\rm tr}(\\mathbf{g(X)})}{\\partial \\mathbf{X}} =", "cb157374ba3d7de40e21bfd171e683c5": "\n\\frac{1}{t} Y_t \\simeq \\frac{\\mathbb{E}W}{\\mathbb{E}S}\n= \\frac{ 4(1200t + 200) }{ t^2 + 4t - 2t^2 }\n", "cb1588c25c679355f55fbcaf4a72c113": "\\lim_{n\\to \\infty} \\frac{x_{n + 1} - z_{n + 1}}{(x_n - z_n)^2} = \\frac{f''(\\alpha)}{2f'(\\alpha)}", "cb1595cad0365c1dc6da81aa298460fa": "(P \\to (Q \\to R)) \\to ((P \\to Q) \\to (P \\to R))", "cb15a38d12ded16647400a1ccd37f641": " y=\\sqrt{x^2 - |w|^2} > 0.", "cb15b7298ea6c59fdfe54d771a3a5a98": "x^{\\{m\\}}", "cb15cbc73b551587286888be2f844972": "\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p", "cb15eeefcc23e6d8859d122139e9f0e9": "\n\\mathcal{G}_{\\alpha\\beta}(\\tau|\\tau') = \\frac{1}{\\beta}\\sum_{\\omega_n}\n\\mathcal{G}_{\\alpha\\beta}(\\omega_n)\\,\\mathrm{e}^{-\\mathrm{i}\\omega_n (\\tau-\\tau')}\n", "cb165c50dda50a750e794e72efec0f4b": "\n \\boldsymbol{A}:\\boldsymbol{B} = \\sum_{i,j=1}^3 A_{ij}~B_{ij} = \\operatorname{trace}(\\boldsymbol{A}\\boldsymbol{B}^T) ~.\n ", "cb168239593ac5fd066a1280243489b1": "\\hat 7", "cb16c3994749d73df37072b6d1e33deb": "e^{-i\\omega t}u.", "cb16c486b9caec9d8f0b97b4aab7147d": "\n\\begin{align}\n\\operatorname{tri}(t) = \\and (t) \\quad \n&\\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\max(1 - |2t|, 0) \\\\\n&= \n\\begin{cases}\n1 - |2t|, & |2t| < 1 \\\\\n0, & \\mbox{otherwise} \n\\end{cases}\n\\end{align}\n", "cb16f319de7a2bba8ba8048018d55364": "sR_{AW} = {R_{AW}}\\times{FRC}", "cb173d0ace526078c19c816d98266c18": "\\mathrm{QSym}_n", "cb175df4acfd12573bf4c057150e5a4b": "\\limsup_{x\\to a} \\left|\\frac{f(x)}{g(x)}\\right| < \\infty.", "cb177477520d893717520319d4904b78": "\\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\\\ 0\\end{pmatrix}", "cb1778da8b1106a467a34cee28076354": " k_x = \\frac{l\\pi}{L}", "cb17c57de182d18a4859373fa84349ce": "\\{f_j\\}_{j=1}^r\\subseteq\\mathbb{F}[X_1,\\ldots,X_n]", "cb17e234d7a852a1be22776b6d284933": "= x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + (1 + u_{1}u_{1})\\frac{\\partial}{\\partial u_{1}} + \\phi(x,u,u_{1},u_{2})\\frac{\\partial}{\\partial u_{2}} \\,", "cb17e2e830e3a2be8e022ae4b6ffb113": "m - R", "cb180b46e61bf299637937f6b25804ae": "\\scriptstyle \\alpha^2+\\beta^2+\\gamma^2=1", "cb180d798abd99274f8556e7f75f6d65": " {t(n) \\sim C \\alpha^n n^{-5/2} \\quad\\text{as } n\\to\\infty,}", "cb180fab77b12fdf5f9f883a1975000a": "z \\mapsto w", "cb18106824c15738b32f27c0841ce29e": "|\\mathrm{O}(2n,q)|=2(q^n-1)\\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).", "cb184edda4656209edc5e6d10a842aee": "\\lim_{X\\rightarrow\\infty}\\frac{D^-(x)}{D(x)}=1-\\prod_{j\\geq1\\text{ odd}}\\left(1-2^{-j}\\right).", "cb186589611a5348bbe7fd746f039612": "\\widehat{\\mathbf{v}} = \\frac{i}{\\hbar}\\left[\\widehat{H},\\widehat{\\mathbf{r}}\\right]", "cb1889061d8ef9186025ebc3516117ff": "p(t_i)", "cb189ca9cbd419ed624c04d580c93c64": "H_{z}=\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\mu}(C \\ e^{-jk_{x\\varepsilon }w}+D \\ e^{jk_{x\\varepsilon }w})e^{-jk_{xo}(x-w)}cos(\\frac{m\\pi }{a}y)e^{-jk_{z}z} \\ \\ \\ \\ \\ (44) ", "cb189d8688df1d82ea482f096d2472be": "(M,\\cdot)", "cb18a7cb3a2e7684bc367fad39920495": "\nR_{\\alpha\\beta} = {R^\\rho}_{\\alpha\\rho\\beta} =\n\\partial_{\\rho}{\\Gamma^\\rho_{\\beta\\alpha}} - \\partial_{\\beta}\\Gamma^\\rho_{\\rho\\alpha}\n+ \\Gamma^\\rho_{\\rho\\lambda} \\Gamma^\\lambda_{\\beta\\alpha}\n- \\Gamma^\\rho_{\\beta\\lambda}\\Gamma^\\lambda_{\\rho\\alpha}\n=2 \\Gamma^{\\rho}_{{\\alpha[\\beta,\\rho]}} +\n2 \\Gamma^\\rho_{\\lambda [\\rho} \\Gamma^\\lambda_{\\beta]\\alpha}\n.", "cb18bfaef93d4f477aa6c03bb887c898": "\\langle S \\mid R\\rangle.", "cb197eec229272d61fd715c7075651fd": "\\left|H^\\dagger D_\\mu H\\right|^2/\\Lambda^2", "cb199067f477aebd1fd0b71ff6a1d45b": " E=\\vec r_u\\cdot\\vec r_u, \\quad\nF=\\vec r_u\\cdot\\vec r_v, \\quad\nG=\\vec r_v\\cdot \\vec r_v.", "cb1994e36723bb57944fd1021e02d451": "R_c", "cb19ac6a1839c2e8eb11df67a8640363": "\\{x_i,y_j\\} = x_i y_j + (-1)^{ij} y_j x_i \\ . ", "cb19dc01420f905b712dc4aafdf04582": " \\langle \\cdot,\\cdot \\rangle ", "cb19dfcfd953d69260b184872dd0adfc": " f(v) = \\int_\\Omega f v \\,dx. ", "cb1a06a374b0bfce8091cdc6b887c4f7": "\\;ord_P(G)=2r+s", "cb1ab266b3450a58bea27551e5ae96e1": "u=I/I_0", "cb1ab725c18869b8f2b6388545b4644b": "\\mathrm{d}(U-TS) = - S\\mathrm{d}T - p\\mathrm{d}V\\,", "cb1b1bc1c582116b7e5e7badd6ea403e": "C_L = I/L\\;", "cb1b1e98cd40f56be345a43a6ca3d64b": "\nf(x) \\mapsto A_f := \\int_X\\mu(dx) \\,\\mathcal{N}(x) \\, f(x)\\, |x\\rangle \\langle x |\\, .\n", "cb1b42cb2cc6318574142681fed4da7f": "\\binom{3}{2}\\cdot 16 - \\binom{3}{1}\\cdot 4 + \\binom{3}{0}\\cdot 2", "cb1b438931b28fd341f41fe849ebe717": "\\frac {f^2} {Nc} = \\frac {90^2} {2.8 \\times 0.03} = 96.4 \\text { m} \\,.", "cb1b65a0e053e0d54dc31e11613e3539": "Y(t)=b\\,\\sin t", "cb1ba03f7a8233f1110904e2f825bb42": " H_{\\infty}(U_{\\ell}) = \\ell", "cb1ba8cf231ed151316a3af6a71a3529": "\nP(\\omega) = {2\\over 3} {\\omega^4} |d_i|^2 \n\\, .", "cb1bc22c6ff0442523c66a0bbe8ee84d": "W=W'+\\mu^{-1}", "cb1cb7b5c998b1d1f029bbcfea7f6250": "\\psi_E(x) = C_1 e^{i\\sqrt{2mE/\\hbar^2}\\,x} + C_2 e^{-i\\sqrt{2mE/\\hbar^2}\\,x}\\,", "cb1cf9813212ff38bd906a45b5d8944f": "\\mathbf{ABC} = \\mathbf{A}(\\mathbf{BC}) = (\\mathbf{AB})\\mathbf{C} ", "cb1d672c1a6e70dc3bad7629d9171bd7": "{\\rm MCG}(X) = \\pi_0({\\rm Homeo}(X))", "cb1d900ee14f569457a3da25a5c6ac81": "\n\\begin{align}\nBull(t=0) & = 1 \\\\\nBear(t=0) & = 0 \\\\\nStagnant(t=0) & = 0\n\\end{align}\n", "cb1da659b9e0f916f6d5ac1beea3bbad": "D_{\\mathbf{v}}{f}(\\boldsymbol{x}) = \\sum_{j=1}^n v_j \\frac{\\partial f}{\\partial x_j}.", "cb1e287646261f64dcbdde86f05bf890": "\\langle\\,\\,\\rangle", "cb1e5ce778a6457b1b1425b465b39a75": "\\begin{matrix} {11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "cb1e5fa90777b8d224be91b5dc2f887e": "(\\lambda x.u)v \\; \\triangleright_c \\; u[v/x]", "cb1e6ef1d14dee0c523a62ccd0602b1f": "n_\\Sigma^2 d S \\cos\\theta_\\Sigma d \\Omega_\\Sigma=n_S^2 d S \\cos\\theta_S d \\Omega_S", "cb1e99b6363fc8530d459cb3fd946da2": "\\tfrac{m}{n}=\\sqrt{5}", "cb1ee4817737acf3fad8760c8537d8a0": "~f~", "cb1ef082fb1ca3c3ec60a3843c1f0d88": "I(\\alpha)=\\int_0^2\\left[2+\\sin\\left(10\\alpha\\right)\\right]x^\\alpha \\sin\\left(\\alpha/\\left(2-x\\right)\\right)\\,dx", "cb1f390a455797fa66ac65a3e905c1be": "\\vec{x} \\in \\mathbb{R}^n_{++}", "cb2003f7826441c08dd4b8c6bd37f25e": "\\hat{x}^o", "cb200adc00e228b4c4d458e9953a5f2e": "\\mu^T \\mathbf{x} \\mu = 1 ", "cb203f5f1ff2b360f5f3ed678b71bb52": "\\langle\\sigma, p\\rangle \\in \\eta", "cb20bd211ee869a9ba173111ee64e365": "\\approx 640320^3+744.", "cb20be532e290da550432aa712cf1cbf": " \\langle \\Omega | T(A^{\\mu}(x)A^{\\nu}(0))| \\Omega \\rangle = \\int \\frac{d^4q}{(2\\pi)^4}\\frac{-i\\eta^{\\mu\\nu}e^{-i p\\cdot x}}{q^2(1 - \\Pi(q^2)) +i\\epsilon} = \\int \\frac{d^4q}{(2\\pi)^4}\\frac{-iZ_3 \\eta^{\\mu\\nu}e^{-i p\\cdot x}}{q^2 +i\\epsilon} ", "cb20e74811ab8a3c239de7aeb676f247": "c_i =c_0 M_0\\,b_i,\\ b_i=\\frac{b_0 c_i}{c_0},", "cb20ed79a37332d9f2d6eb8f46b42002": " \\sum_{\\langle i,j\\rangle} ", "cb210983397fcec2fd941013d7a0a7a7": "E_{kin} \\propto v^2", "cb212a739d777ba44d6d587e968a7436": "\\begin{matrix}\n\\omega_{n}(t) \n\t&= \\frac{\\partial}{\\partial t} \\arg\\{ x_{n}(t) \\} \\\\\n\t&= \\frac{\\partial }{\\partial t} \\arg\\{ X(t,\\omega_{0}) \\}\n\\end{matrix}", "cb2135aa9a882bb66d7bc2778cc2b9a1": "\\mathbf{r}_k", "cb2165888cb8ebcada4167e6e39ee3b2": " W= 1.540 \\pm0.001 ", "cb217669510f20b175f82d84f2135319": "\\frac{A}{x}+\\frac{B}{x-1}+\\frac{C}{x-2},\\,", "cb2211f358d546809dd00b5b36503911": "I=4M^3(\\cos(\\lambda/2)/\\cos(\\lambda/4))^2", "cb223bf6b11aaacbe2aef552eb32d363": "\\dot{x}", "cb22e990d0168628b015847d7f2ba108": "RTS(O_j) = \\max(RTS(O_j), TS(T_i))", "cb232bb5a8381a85f864c4c1966f0dc0": "S^1 = \\{ e^{i\\theta} \\, | \\, 0 \\leq \\theta < 2\\pi \\}", "cb232cfbbbd7355183e2c142fad4fadb": "\\mu_s(l,x_s,p+\\Pi)=\\mu_s^0(l,p)", "cb233bb690363ca6da849fd25f7a24e2": "\\begin{align}\n1 - \\Pr(\\limsup_{n \\rightarrow \\infty} E_n) &= 1 - \\Pr\\left(\\{E_n\\text{ i.o.}\\}\\right) = \\Pr\\left(\\{E_n \\text{ i.o.}\\}^{c}\\right) \\\\\n& = \\Pr\\left(\\left(\\bigcap_{N=1}^{\\infty} \\bigcup_{n=N}^{\\infty}E_n\\right)^{c}\\right) = \\Pr\\left(\\bigcup_{N=1}^{\\infty} \\bigcap_{n=N}^{\\infty}E_n^{c}\\right)\\\\\n&= \\Pr\\left(\\liminf_{n \\rightarrow \\infty}E_n^{c}\\right)= \\lim_{N \\rightarrow \\infty}\\Pr\\left(\\bigcap_{n=N}^{\\infty}E_n^{c}\\right)\n\\end{align}\n", "cb237775398763dd5c7e27e06d4bc7dc": "b = \\frac{0.08662\\,R\\,T_c}{p_c}", "cb23b179b90c0dfc62a8ac5e37fe2254": "|E| > \\Delta (G) \\lfloor |V|/2 \\rfloor", "cb24042867bcce6adcd0a0470b5baa5c": "V_a = \\frac{m_e}{2}{\\omega^2}(x^2+y^2+z^2)-{e^2\\over 16\\pi\\varepsilon_0 Z^3}\\left(\\frac{x^2+y^2}{2}+z^2\\right)+\\ldots", "cb24777dba7559693b8bad44f14c7410": "\\Delta=\\frac{\\partial\\varrho}{\\varrho}", "cb24a19df8c059d149043016488a3d76": "\\sqrt{5}.", "cb24cbf8a1cc1be3cecd0a055ad072da": "\\sigma_L", "cb24cdc3de5128faf826f77d22cd5b8f": " \n \\frac{\\partial^2 \\eta}{\\partial t^2}\\, \n -\\, g h\\, \\frac{\\partial^2 \\eta}{\\partial x^2}\\, \n -\\, g h\\, \\frac{\\partial^2}{\\partial x^2} \n \\left( \n \\frac{3}{2}\\, \\frac{\\eta^2}{h}\\, \n +\\, \\frac{1}{3}\\, h^2\\, \\frac{\\partial^2 \\eta}{\\partial x^2} \n \\right)\\, =\\, 0.\n", "cb250402da446e727603fe3f8453b62f": "y \\le \\; ?(x) < y + d,", "cb25b241f6a0719daa7e2b4adb5797ac": "(x-a)", "cb268ff37a8080459174156a999397f2": "\\mathrm{ker}(x) e^{x/\\sqrt{2}}", "cb26937663deee023cbf8bd4ca7e596f": "\\oint_{\\Gamma} \\mathbf{F}\\, d\\Gamma =0", "cb26948b841883dc0ff0ab0ddfcc82c7": "\\scriptstyle n_0,n_1", "cb26b59d1caa49bc375215613f1ee108": "X(t) = \\frac{K}{e} = K \\cdot \\lim_{\\nu \\rightarrow +\\infty} \\left( \\frac{\\nu}{\\nu+1} \\right)^{\\nu} ", "cb26cdee9929d209caac2276df1cf6e5": "\n\\omega_{n}(t) = \\frac{d \\theta_{n}(t)}{d t},\n", "cb26f599f70a0aa4802c957ce338fc4a": "(x,\\theta,\\bar{\\theta})", "cb274be5b1ce238c5b2659f5353ed95c": "Q \\left[ \\psi(\\mathbf{x},t) \\right] = - (1/2R) \\int d^3 x \\, \\delta^2 R / \\delta \\psi^2 ", "cb275c5cd503205d0537a981d24dd475": "\\Pr'(S|W) = \\frac{s \\cdot \\Pr(S) + n \\cdot \\Pr(S|W)}{s + n }", "cb278c446f5b22f90c4474c1c7a55c55": "C_0 + C_1", "cb27a65a0edbde5e68bcd160da0d6e4b": "\n\\theta_\\mu=\\mathrm{Arg}\\langle z \\rangle = \\mu+\\sqrt{c}\n", "cb27af9977e3afc4165ecd8a7cb6c66f": "|\\psi\\rangle\\langle\\psi|.", "cb27b924691cb0b43e4e6ef28f56ea35": "\\psi _\\mu (\\tau)", "cb283ee5f7764f77508df632521e26ed": "\n\\sum_{k=0}^{\\infty}\n \\frac{(-1)^{k}}{\\sqrt{k+1}}=1-\\frac{1}{\\sqrt{2}}\n +\\frac{1}{\\sqrt{3}}\n -\\frac{1}{\\sqrt{4}}\n +\\frac{1}{\\sqrt{5}}\n \\cdots=-(\\sqrt{2}\n -1)\\zeta(\\frac{1}{2})\\approx0.6048986434....\n", "cb284946663565de19cfe74bbb2a5547": " G_t - T_t \\,", "cb288f51006529550589edd8b6f8d19b": "N(f,\\epsilon)=\\{\\hat{f}: \\hat{f}=\\sum_{i=1}^{k}\\beta_i\\phi_i, \\|f-\\hat{f}\\|<\\epsilon\\}", "cb2906d36ebe913fb8a772b590f9b156": "G^{\\mu \\nu} = \\begin{pmatrix} 0 & -B_x & -B_y & -B_z \\\\ B_x & 0 & E_z/c & - E_y/c \\\\ B_y & -E_z/c & 0 & E_x/c \\\\ B_z & E_y/c & -E_x/c & 0 \\end{pmatrix}", "cb293e00fec4ef96d21b3f414dcc3b35": "\\left ( \\begin{matrix}1 & 2 & 3 & 4& 5& 6\\\\2&1&3&5&6&4 \\end{matrix} \\right ) = (1 2)(3)(4 5 6).", "cb297a46933c06a8ef888ae82dafac23": "r_a^2+r_b^2+r_c^2+r^2=AH^2+BH^2+CH^2+(2R)^2.", "cb29dd05f71d98374a5e40fc57134a6a": "\\mathfrak b = \\min(\\{|F| : F\\subseteq\\omega^\\omega\\land\\forall f:\\omega\\to\\omega\\exists g\\in F(g\\nleq^*f)\\}).", "cb2a01194964c0e6b8e532e9289e56b3": "R(x)= \\frac{\\sum_{j=0}^{m}a_j x^j}{1+\\sum_{k=1}^{n}b_k x^k}=\\frac{a_0+a_1x+a_2x^2+\\cdots+a_mx^m}{1+b_1 x+b_2x^2+\\cdots+b_nx^n}", "cb2a3716e3757055203a0991b6637846": " D_2 = \\partial_\\eta - \\frac{2\\alpha_{,\\eta}\\lambda}{\\lambda+\\alpha} \\partial_\\lambda", "cb2a57be4eee4769043dfaa06a90c544": "\\frac{\\mbox{Percent Change in Net Operating Income}}{\\mbox{Percent Change in Sales}}", "cb2a924d8f65d2d34b42177fc509111e": "\\mathbf{\\delta p}", "cb2ae15c6773fd06f3cc3a133a2a09fa": "f(u)\\,", "cb2b247dfa8b61f362947f6d224d20af": "w(3)=1", "cb2b40f5a69314a32ac76210c7f83972": " y_{n+1} = y_n + hf(t_n,y_n).", "cb2b92b18c7a8fcda4b577bbee9723df": "\\ F \\subset X ", "cb2b96a2c0a457def4d173b474ef3f6c": "\\phi V \\to V", "cb2bb2cff4151d04f2ddb4521ee9ccc9": "|T_n|=2^n", "cb2be7eff2d112495815b2889ce0257e": " H^{'}|\\psi_{+}\\rangle=E_{+}|\\psi_{+}\\rangle ", "cb2c140433edd0ac4caa5869ad8730f4": "\\leq_A", "cb2c5079c2de7b6ce70c9583e07da8d3": "\\det (T^{-1})^\\ast", "cb2d2406ad2c686c4df3a71ee4e417b9": "\\,2 G_F(x-y) = -i \\Delta_1(x-y) + \\epsilon(x_0 - y_0) \\Delta(x-y) ", "cb2d2e1692ed31be49d53ddf5aa8fb41": "\\theta = 2 \\pi f \\cdot T", "cb2d5e775924e1560c2ea24d74e2b9f7": "T = \\frac{L}{a} + t' = \\left(\\frac{1}{a} + \\frac{1 - av}{a - v}\\right)L", "cb2e1413f3c8414ecf23c4ad8dfdedf5": "x_{n+1}=\\frac{x_n+\\lambda_n}{4}, y_{n+1}=\\frac{y_n+\\lambda_n}{4}, z_{n+1}=\\frac{z_n+\\lambda_n}{4}", "cb2e6da3e087b0cb1d9b579d0583438f": "\\int_{\\Omega} fv\\,ds = -\\int_{\\Omega} \\nabla u \\cdot \\nabla v \\, ds = -\\phi(u,v),", "cb2efc91a88f7dd055b40d5667d3c6bc": "\\dot{H} + H^2 = \\frac{\\ddot{a}}{a} = - \\frac{4\\pi G}{3}\\left(\\rho + \\frac{3p}{c^2}\\right).", "cb2f01c39faf57796b9b032f08739110": "z\\log(2\\sin \\pi z)-\\frac{1}{2\\pi}\\int_0^{2\\pi z}\\log\\left(2\\sin \\pi \\frac{y}{2} \\right)\\,dy", "cb2f0cb2f1e60728624340a4b686913e": "\\frac{1}{q}=\\frac{1}{p}-\\frac{k}{n}.\\ ", "cb2f1b08d90507c8c94609130ec1fca0": "A(t + T) = A(t)", "cb2f3a725fb8cb2d630933ad74ff2538": "\\mu=1.1727", "cb2f494c6b1669ccaef5978710de2a0b": "\\tau^{-1}\\sigma \\in \\Sigma", "cb2f59ee560d2a770723249e2ad1e775": " H(s) = \\mathcal{L}\\left \\{ h(t) \\right \\} \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty} h(t) e^{-st}\\, dt ", "cb2f708875201e5109424d7a4acbde74": "{dy \\over h(y)} = {g(x)dx},", "cb2fdb7ea7156fe6ef12cd861ea94525": " f \\in \\mathbb{Z}_p[x] ", "cb2fe4652d030b5510493e4071209568": "\\psi(\\Omega^{\\Omega^2+\\Omega 2+\\psi(\\Omega^{\\Omega^2+\\Omega 2+\\psi(0)})})", "cb2fea057e2390ed656e9208678ff583": "v_1,v_2,\\ldots v_n", "cb30b41cdb04e22f54b6edf0cdcb8735": " x_i x^i=1", "cb30f6f8df9c506d2520323d958fef4e": " m \\leftarrow \\tfrac{1}{2}(a+b) ", "cb3114328ca6e7d77b216e9525b3d05f": "A \\| \\mathbf{v} \\|^{2} \\leq \\sum_{k} |\\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle|^{2} \\leq B \\| \\mathbf{v} \\|^{2}\n\\text{ for all }\\mathbf{v} \\in V", "cb3118d7cb39360076adaf301d618ad8": "\\widehat{Y_k}", "cb315fa78c5ff7c6d96684b2f907d0c2": "\\delta_x \\,", "cb31c70319ddac7e19ed2764b6dbab96": "(ts)", "cb31c71a8581adbb6b95c1791cbc49f0": "\\mathbf{A}_q^*", "cb31f791d3fcc8ebfb82d3f27f5bb13e": "\\Delta S = \\beta \\times S {1\\over N} \\Delta t", "cb3213fdf7fcf4f897d1e35d5b8d8f7c": " \\psi(x) \\in [\\ln(x-1), \\ln x]\n", "cb32cd617db437eec9b077d98d3ed31f": " exp e_i ", "cb32cdc571a79cd10af506be63dcf841": "\\det\\left(\\mathbf{A}\\right) = \\prod\\limits_{i=1}^{N_{\\lambda}}{\\lambda_i^{n_i}} \\!\\ ", "cb32de605c9000aa592beb40bd8a08e3": "f_{3}", "cb33153f1429cb632877a2196be55e82": "\n \\varpi'_{f} (\\delta) := \\inf_{\\Pi} \\max_{1 \\leq i \\leq k} w_{f} ([t_{i - 1}, t_{i})),\n ", "cb3346caac33a183c4c35d62ebf94942": "r_i \\leq r \\leq r_{o1}: B(r) = \\frac{\\mu_d I}{2 \\pi r}", "cb338a79f709ceaa5f22ccad758e0542": "a_{r,s}=(xp_r,p_s)/(p_s,p_s)", "cb3426b7834a732362324bc119256e18": "a_{n+1} = f_n a_n + g_n, \\qquad f_n \\neq 0,", "cb34423238d32fd8cf47b97d6d7b2e12": " 3, 2, 3, 0, 1, 2 ", "cb3448647d66232052059e2bbc018818": "f_n(x) = 1_{[n,n+1]}(x),\\qquad n\\in\\mathbb{N},\\ x\\in\\mathbb{R},", "cb34853a020c2b1e467e362c777ee628": "V=(\\frac{5\\sqrt{2}}{3}+\\frac{3\\sqrt{3}}{2})a^3\\approx4.9551...a^3", "cb34f0c3641ee9e39acedfef4ce6aee6": " c = c(z) ", "cb3517040cff5c5bdd52836d4fe755af": "\\Delta v_\\alpha := v_\\alpha - v_{\\alpha-1}", "cb353afb971c73a25f57d60ec911987a": "\\nabla_{\\dot\\gamma(t)}\\dot\\gamma(t)", "cb354faacc62b0240e1a8d1c2f668c92": "\n\\begin{align}\nT &= r\\big((s-a) + (s-b) + (s-c)\\big) = r^2\\left(\\frac{s-a}{r} + \\frac{s-b}{r} + \\frac{s-c}{r}\\right) \\\\[8pt]\n&= r^2\\big(\\cot(A/2) + \\cot(B/2) + \\cot(C/2)\\big) \\\\[8pt]\n\\end{align}\n", "cb35a0beef22e51cd0df7181705a0273": "\\displaystyle 3.83", "cb35a65ea4a6973d09ecfbc446a20216": "\\left\\lfloor\\frac{q}{p}\\right\\rfloor +\\left\\lfloor\\frac{2q}{p}\\right\\rfloor +\\dots +\\left\\lfloor\\frac{mq}{p}\\right\\rfloor\n\n+\\left\\lfloor\\frac{p}{q}\\right\\rfloor +\\left\\lfloor\\frac{2p}{q}\\right\\rfloor +\\dots +\\left\\lfloor\\frac{np}{q}\\right\\rfloor\n\n= mn.\n", "cb35c15cbd33990d559a3c23c4970a1a": "\\asymp", "cb3680efb248ac33413d49d768523c3e": "\\bigg({\\Bbb H}\\backslash 0\\bigg)/{\\Bbb Z}", "cb368e0d293bd8499350458bd50f11a5": "L \\; = \\; 20 \\; \\log f \\; + \\; N \\; \\log d \\; + \\; P_f(n) \\; - \\; 28", "cb369f1afc1eb9add95479282616879a": "\\operatorname{tr}\\colon \\mathit{gl}_n \\to k", "cb369f43a1b9430c2842b0b895411b00": "\\frac{60}{360} = \\frac{L}{24}", "cb36a3803d5c0dc5a125ae83f21b93f5": "\\mathrm {rad}(M) = \\sum \\{ S \\mid S \\mbox{ is a superfluous submodule of M} \\} \\,", "cb3700eaa58fd2dae28052c9509095ce": "2^2\\cdot (2^{2^2}+1)", "cb375f6d22ceb99184d6338b9657bc28": " S_j \\geqslant S_k ", "cb37aef928d06f9a29c82075dd4d6d2d": "V = (V_{DC} - V_{CPD}) + V_{AC} \\cdot \\sin (\\omega t)", "cb37d27d789f8f8534180dc9bd293015": " Coppock = WMA[10] \\; of \\; (ROC[14] + ROC[11])", "cb3861ad9869c0eb80ba9334b8cc1f17": "f(x)=0\\,", "cb388e2dbdd101999c252fa0d6f94404": " \\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\cdots\\otimes\\mathbb{C}^n ", "cb3891fcaf2c1c5fcdf12ab0e59a4d11": "IG=\\frac{M_{int}}{M_{frac}}", "cb38ab08bacd465ead2802db2b9ee3cf": "{\\frac{\\partial{S}}{\\partial{L}}} = const", "cb38bc8104888e1d2c6b8a66e4a60ffd": "2^{nT}", "cb38ed033a3e760f00875d54cad25c58": "p_x^c=", "cb38fe028bb099625c90d7159994ae9f": "\\gamma = 0.5772\\ldots", "cb391281364e6341d8e0ea8f0f1c7458": "\\scriptstyle W_n", "cb394bbcaef69e1b2ecfb720d7fab15c": "H_{\\omega + 1}(1) - 1", "cb397f3e7f49a7b2a65f9fe8b55c27eb": " \\left(\\Psi_1 * \\Psi_2\\right) * \\Psi_3 = \\Psi_1 * (\\Psi_2 * \\Psi_3) ", "cb399dc8b0ef6eb3475b87b79f308ff6": " {} + \\frac{(y-f_{n-2})(y-f_{n-1})}{(f_n-f_{n-2})(f_n-f_{n-1})} x_n. ", "cb39c3897732445fc1f523bcc3c002d8": "y^q", "cb39ecf51f5154442b7f6a02cd5d53cc": "\n\\begin{pmatrix}\n (\\mathbf{v_1})_1 & \\cdots & (\\mathbf{v_n})_1 \\\\\n \\vdots & & \\vdots \\\\\n (\\mathbf{v_1})_n & \\cdots & (\\mathbf{v_n})_n \\\\ \n\\end{pmatrix}\n", "cb3a26f1dd6a81114575d271dba3f92b": "\\begin{align} E\\{MI(U,V)\\} = &\n\\sum_{i=1}^R \\sum_{j=1}^C \n\\sum_{n_{ij}=(a_i+b_j-N)^+}^{\\min(a_i, b_j)} \n\\frac{n_{ij}}{N} \n\\log \\left( \\frac{ N\\cdot n_{ij}}{a_i b_j}\\right) \\times \\\\\n& \\frac{a_i!b_j!(N-a_i)!(N-b_j)!}\n{N!n_{ij}!(a_i-n_{ij})!(b_j-n_{ij})!(N-a_i-b_j+n_{ij})!} \\\\\n\\end{align}", "cb3aa11591449cb56146d72f91ff0e6b": " K a^n ", "cb3ab767f2c252dde8a600210f519053": " dh = c_{p}dT ", "cb3af6e9f59692db2b847c81b815f0f2": " \\psi_0(x) = K_a(x) = K_a * \\delta(x) \\,", "cb3b3761d5cd88d8e0b888dab8943841": "\\{Y,X\\}", "cb3bc6b18d973b62706c23099a7755f2": "m=2 \\cdot i ", "cb3c04e37f1a18bf6633db2d99311887": "I^m_{\\ell}(\\mathbf{R}) \\equiv \\sqrt{\\frac{4\\pi}{2\\ell+1}} \\frac{Y^m_{\\ell}(\\hat{R})}{R^{\\ell+1}}\n", "cb3c3577834068699ff6b791651e3f9a": "(a \\otimes b)(x) = \\langle x, b \\rangle a. ", "cb3c5bc312825b9b9b54f1fd6400ac75": " 10^{20}= 10^5 \\times 10^5 \\times 10^5 \\times 10^5 ", "cb3c91eddc64bb26bbf56c5cf570ac0a": "\n\\boldsymbol{\\Tau}_{\\boldsymbol{n}}(z) = \\tau_0\\circ\\tau_1\\circ\\tau_2\\circ\\cdots\\circ\\tau_n(z) =\nb_0 + \\underset{i=1}{\\overset{n}{\\mathrm K}} \\frac{a_i}{b_i}\\,\n", "cb3ccbea37be7ff924860b2feaf72263": "\\{X_p, Y_p, Z_p\\}", "cb3ced2f1ccec22066b66dd9804c9dad": "\\scriptstyle \\epsilon_2 = p_2 / p_{2m} \\,", "cb3d0e0ff68db90cb708f8a30dd41a54": "\\Omega = \\frac{Q}{\\sqrt{(\\vec{R}-i\\vec{a})^2}} \\,", "cb3d531dae5d84753dce1d229cadc855": "f_{\\Phi,\\Lambda}", "cb3daf6ce2d1aeb17b66f0e48909509d": "K_1C_A \\gg 1, K_2C_B", "cb3dbd38974b37dfef54d12b9b57cb6c": "\\big\\{|j\\rangle\\big\\}_{j=1}^{N}", "cb3e1280f1ab1ec7a5d15de567d8538a": "G(U, V, E)", "cb3e209110da97025179bb39a35d67c4": " \\begin{align}\ns_0 &= \\tfrac12(x_0 + x_1 + x_2 + x_3), \\\\\ns_1 &= \\tfrac12(x_0 - x_1 + x_2 - x_3), \\\\\ns_2 &= \\tfrac12(x_0 + x_1 - x_2 - x_3), \\\\\ns_3 &= \\tfrac12(x_0 - x_1 - x_2 + x_3),\n\\end{align}", "cb3e304e61370d08f96dfbec8c3c705d": "ATC=AVC+AFC", "cb3e5576a4e6bd43bfc5e870d94a7ac6": " \\chi^2_{ 2n } = 2 n \\alpha ", "cb3e663b953cc3babd543a8cbbc3cd59": "\\phi\\rightarrow\\lambda^{-\\Delta}\\phi", "cb3e96e745e9ce1497e5c228c360882c": "\\overrightarrow{QP}", "cb3eae2a520f0f448eda55ec02e65298": "\\alpha_1(k),", "cb3ec17ebf22ef93b498e43330c54f52": " lift = {(L/D)_{\\alpha}} \\times drag ", "cb3f17350c9deb1f9323dc777c995b71": "\\,\\! x^+=I_x^+/I_u", "cb3f593eee6bf3a8ca2a807055521c5c": "Sc", "cb3f5e806823f678b9d85b3d380e8c54": "e^{i k z}\\approx\\frac{1}{2 i k r}\\sum_{l=0}^{\\infty}(2l+1)P_l(\\cos \\theta)\\left[ (-1)^{l+1}e^{-i k r} + e^{i k r}\\right] ", "cb3f5f36cc2bb83ed4cb4d5b00f8ba98": "\\lim_{p \\to \\infty} M_p(x_1,\\dots,x_n) = \\lim_{p \\to \\infty} \\left( \\sum_{i=1}^n w_i x_i^p \\right)^{1/p} = x_1 \\lim_{p \\to \\infty} \\left( \\sum_{i=1}^n w_i \\left( \\frac{x_i}{x_1} \\right)^p \\right)^{1/p} = x_1 = M_\\infty (x_1,\\dots,x_n).", "cb3f6e527af2199d7f3baa6ee3c457d1": "3 x^2 + 210x + 3675- 4 x 6000 ", "cb3f95ea2a4819dfa17f5d0a1fce884b": "U \\subset T", "cb3fa6cbdb01223046020a0ad1c4142a": "\\mathbb Z_3", "cb3faf3c6de6d49b6ea8600e73418ae7": "X ", "cb401404a72d5a9f28f7adc03af0ec7b": "x=\\dot{M}_I - \\hat{\\dot{M}}_I", "cb40923c4dddff624ab13b5d2d7e0cd2": " \\mathrm{Slerp}(p_0,p_1; t) = (1-t) p_0 + t p_1.\\,\\!", "cb40ccc7b0881c8c0fb0da94351b3c07": " = 41.6638\\,\\mathrm{m/s} = 149.99\\,\\mathrm{km/h} = 93.20\\,\\mathrm{miles/h}", "cb40ea85185c3392479c2894ac3392b4": "ln[A]_{f}-ln[A]_{i}=-kdt-k(0)", "cb40ef6a707ce80863dbe8d208263838": "P=hg\\rho", "cb40f4105b7a40a8bd72c401e2d19234": "\\begin{bmatrix}1&1\\\\0&0\\end{bmatrix}:\\mathbf a", "cb41055c15ddf9b48b4738f5631a1f47": "+4586''\\sin(2D-l)", "cb4114f8dc5337fd4a42775ad6dd1d05": "\\sin_k(i)", "cb417281ed1e044f2c2ed62630fc115d": "\\Gamma \\vdash e \\Rightarrow \\tau", "cb4186c375b864ab27456537c4969436": "R_{abcd} \\, R^{abcd} = C_{abcd} \\, C^{abcd} +\\frac{4}{d-2} R_{ab}\\, R^{ab} - \\frac{2}{(d-1)(d-2)}R^2", "cb41e42030cdbbc91ec21513efe355e2": "\\beta = \\frac{v}{c}", "cb41e766a767c1b67bdead8f39ae6018": "\n p(\\theta|y) \\simeq \\frac{p(y | \\theta) \\; p(\\theta | \\eta^{*})}{p(y | \\eta^{*})}\\,.\n", "cb420c953855ef9a0e4d3145c32c2222": " \\int_S f\\,d\\mu \\leq \\liminf_{n\\to \\infty} \\int_S f_n\\, d\\mu_n. ", "cb420fb45a36e9ed0b53f392878bbbde": "\\psi(\\Omega^{\\Omega^\\Omega})", "cb429eeb6fa0fcf0e641f37c00368c95": "a^\\dagger\\,,", "cb42e58827567aad3f591842dc6f9876": "\\lambda = -\\frac{1}{2}\\ln \\left( 1 - k r^2 \\right)", "cb42e74ef0581d0a59ddf59d37f497ec": "\\dot S_i=0", "cb43b566088855c3e84ef58dd43950c0": "R_{ji}", "cb440f5cb49c5c5fd5a1568220489766": " \\sigma/{\\sqrt{n}} ", "cb4412c7c8cd9ab16547e8c3e7a4511f": "\\operatorname{logit}(p)=\\log\\left( \\frac{p}{1-p} \\right) =\\log(p)-\\log(1-p)=-\\log\\left( \\frac{1}{p} - 1\\right). \\!\\,", "cb44631559f63d0c4dff14681d81712c": "g(n,2) = 1", "cb4472701351a1f6614bba5721ae8a57": "\\sin \\theta = \\frac{x}{z}", "cb44813fcd8146588ab19d7828978d35": " \\sin\\left(a + b\\right) ", "cb44ab61c0b848e257fc4fd1f112a7c1": "\n \\mathbf{M}_{xz} = -\\left[\\int_z\\int_{-h/2}^{h/2} y\\,\\sigma_{xx}\\,dy\\,dz\\right]\\mathbf{e}_z \\,.\n ", "cb44c7de97f64fc9467401201d49b544": "\\frac{1}{3}+\\frac{1}{7}+\\frac{1}{11}+\\frac{1}{19}+\\frac{1}{23}+\\frac{1}{31}+\\frac{1}{43}+\\frac{1}{47}+\\frac{1}{59}+\\frac{1}{67}+\\cdots", "cb4518471699437166414ec35013d64f": "l(r)", "cb4548564f05a9128be87370c56297c3": "p_y= h/\\lambda\\cdot \\sin(\\alpha)", "cb4549e89dca3a930c76e7a4128cdd91": "Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2.\\,", "cb456c6f869ba25354a1d5c8f1d9b227": "\\boldsymbol\\Sigma|\\boldsymbol\\Psi,\\nu \\sim \\mathcal{W}^{-1}(\\boldsymbol\\Sigma|\\boldsymbol\\Psi,\\nu)", "cb4585eb0562155be499d7168cf66f2d": "NR=\\frac{\\rho v L}{\\mu}", "cb4590eee82f4f562bba694c7c4a01cc": "F'\\,", "cb461fd39f5535d8eebc9c23577f83cb": "20\\times log_{10}(2)", "cb463dbdc0c49cd5a6b8c992a347ef2f": "1 ^ x\\,", "cb463eb1aa1842049ca11f0eab735593": "\n\\tilde{K}(p;T) = \\tilde{G}_\\epsilon(p)^{T/\\epsilon}\n\\,", "cb468666b7ea163b7ce8c8fd9f9d4b79": " \\textstyle P=C_P+I", "cb468ed44faa663942da18807a78bbd4": "f(L) = {K \\over L}", "cb46f5f5b85550a19ac4e957b0996648": "y^+ = \\frac{y\\, u_\\tau}{\\nu},", "cb4720664251bbf652c92d5d86de5722": "\\left[\\tfrac{1}{2}(\\sigma_1 + \\sigma_3), 0\\right]", "cb47b1c66fb7873c8b3df4b4387f40eb": "((x, y), z) = (x, y, z)", "cb47bacaca6b70d356fef4dc221463bb": "k=\\frac{K_{12}}{\\sqrt{x_1x_2}}", "cb47d0c425f6117a5d8aed5568d3d642": "5\\cdot 5=25 \\le 27", "cb47e6d4ea84de7877be6ea404721e76": " \\epsilon E^2V \\equiv B^2V/\\mu \\,\\!", "cb47f030706e97d4520d6cb2f24ec416": "(\\varphi)^2 = (\\sqrt\\varphi)^2 + (1)^2. ", "cb483ecdf8d65fdec151846ccb75628d": " epd \\le 1", "cb48577f06dce48b88c875176a77dc6c": "f(s)=s^2", "cb487958ffa1a4ac27821085ce000b69": "y\\in\\Omega\\,", "cb4890d23ef21cec25c0c13c63ef3cbe": "\\mathcal{B} \\to \\mathcal{G}", "cb48d7664e04ce46982107a88f7da5d8": " \\gamma = \\; \\beta W. ..........(48) ", "cb48e391a13078ce7136e5a5ecedbb04": "P_{n,j} = \\emptyset", "cb48fd98ec40fd1dcec2707e4749446b": "\\sum_{k=1}^n H_{k,m}=(n+1)H_{n,m}- H_{n,m-1}", "cb4931026e1ec79b4e95f77d26f3efa8": "\\textstyle{\\operatorname{tr}(BA) = \\sum_{ij} b_{ij}a_{ji}}", "cb4943089a61082893745973453e9ae8": "\\lim_{x \\to 0}x^2 \\sin(\\tfrac{1}{x})", "cb49901ff481c60caa577cd56e21e381": "S^2 |s,m\\rangle = \\hbar^2 s(s + 1) |s,m\\rangle", "cb4a2ec0ffe4a4f66334c3a75ec735d9": " C(G) \\otimes C(G) ", "cb4a3d445b521696eda88ec4c7f37338": "z = {a\\over 2} + b e^{i\\theta} + {a\\over 2} e^{2i\\theta}.", "cb4a8b5a5b429b2e1652e5bb96416f80": "\\mathbf{p}_{\\perp}", "cb4a8c24947d93f8e38bfd62675ec761": " S_{source} = J\\Phi", "cb4aa4589e3ee8a1379081011a204851": " mol Fe = 0.297 \\ \\mbox{mol}\\,Al \\times \\frac{2 \\ \\mbox{mol}\\,Fe}{2 \\ \\mbox{mol}\\,Al} = 0.297\\ \\mbox{mol}\\,Fe\\ ", "cb4ac189d4f99f0140868c6983506e9b": " v_2 ", "cb4ad94ad4ecce60b19be4350ce2d0e5": "q_u\\,", "cb4aed0e03ef1e464d06e910a6f202a8": "q''_i(x_i) = q''_{i+1}(x_i)", "cb4b1e0421080a17a724d2bfbc0b0b77": " \\mathfrak g \\to \\Gamma(TM), X \\mapsto X^\\# ", "cb4b214655b60be82c86f6625e1d1b76": "R_S \\approx R_H \\sec(\\alpha)\\,", "cb4b86b6e59f1c08b8d58f1e9d603262": "I_\\nu(T)", "cb4b8c7beda0df111040c206800729de": " \\begin{align}\nS &= (1_{\\!N} - \\sqrt{z}Y\\sqrt{z}) (1_{\\!N} + \\sqrt{z}Y\\sqrt{z})^{-1} \\\\\n &= (1_{\\!N} + \\sqrt{z}Y\\sqrt{z})^{-1} (1_{\\!N} - \\sqrt{z}Y\\sqrt{z}) \\\\\n\\end{align} ", "cb4c3b2b8e894af189c8f6a2fef27a60": " 7^2= 49 = 10 .", "cb4c4c099a52f3cb032d2b7220ce9f41": "L(\\theta,\\hat\\theta)=(\\theta-\\hat\\theta)^2,", "cb4c5db69ec8ddf7c18425335edbe8f3": "\n y(t) = h_{0}+\\sum_{n=1}^{N}{(H_{n}x)(t)},\n", "cb4c6cad93b5df9808f61111ac026e0e": "n_{avg}", "cb4c71ca9b68ce73087024ea225de7a7": "c(E)=1+e(E_{\\mathbf R})", "cb4c7b29721d537273724c459a29f841": "2^{S'}", "cb4c7c742e5300661daae70586f26af8": "\\sum_{j = 0}^{\\infty} e^{-\\lambda/2} \\frac{\\left(\\frac{\\lambda}{2}\\right)^j}{j!} I_x \\left(\\alpha + j,\\beta\\right)", "cb4c936be8a0a132af70da0abfcd586d": "\\alpha = 0.306 \\ ", "cb4c973804e4ab2b773322f0f89dcd3d": "\\operatorname{det} H L(x, y; t) = t^2 (L_{xx} L_{yy} - L_{xy}^2)", "cb4d2a253172aa3dd215a8dfbe2d53b4": "Q_m=N|x_m|^2", "cb4d6d54621cf82d6db2ac44fc42268c": "\\nu = n-1", "cb4d8047fa3fc7caed5d52cd0f54d0ee": "\\ T \\ ", "cb4dc7d0713da3a6e794976bbabaf904": "k=1,\\dots,B", "cb4e13beaac7aea4551bb2c115f477fe": "2^{k-1}", "cb4e588258531891310c52f12df128b8": "j\\ge 1,", "cb4e8b19a081469892d69ed4e74e552c": "\nV(P) = V_0 \\left[1+ P \\left(\\frac{K'_0}{K_0}\\right)\\right]^{-1/K'_0}\n", "cb4ed4e874d83405cd0f56298dc7ff2b": "\\tilde{X}^n", "cb4efae84f23aaf41fa73a2bf19e9068": "\\Sigma \\,", "cb4f19353af556b57b432ff62f7b8fdc": "\n\\begin{align}\n\\left|{\\partial \\mathbf{x} \\over \\partial t}\\right| & = \\left| \\sum_k\\left(\\sum_i h_{ki}~\\cfrac{\\partial q^i}{\\partial t}\\right)\\mathbf{e}_k\\right| \\\\[8pt]\n& = \\sqrt{\\sum_i\\sum_j\\sum_k h_{ki}~h_{kj}\\cfrac{\\partial q^i}{\\partial t}\\cfrac{\\partial q^j}{\\partial t}} = \\sqrt{\\sum_i\\sum_j g_{ij}~\\cfrac{\\partial q^i}{\\partial t}\\cfrac{\\partial q^j}{\\partial t}}\n\\end{align}\n", "cb4f39a5e8c8f2ef5948eeb6330b716c": " \\int_0^t f(\\tau)\\, d\\tau = (u * f)(t)", "cb4fd4245ba5b7a8697488ef13eb1145": "z=\\alpha", "cb4fdb452726220e60b990f15d4629fa": "F \\leftarrow G", "cb4ff04438e0a98a7d167727bccd559f": "|H_{AB}|", "cb502b382c85ecc4a227c946b3a05774": "df_x", "cb5038f24daed90e3da5e34a3a9da834": "\\mathit{far}", "cb503fdcdcdbe765cd4b8191d9282f1e": "BQ = \\frac{2 AP}{1 + AP}.", "cb505080714925320bdfe3b424f424d8": "\\textstyle x^p(1 + x + \\ldots + x^{p-k-1})", "cb506857c313ba4c477abf715c37202f": "P_2(n) = \\frac{n(n+1)}{2} = {n+1 \\choose 2}", "cb50920800f8273edac9f2d1dc1fbeb0": "\\alpha:f\\Rightarrow g", "cb5094176bac94e2e26ec5d3f41891fb": "L = \\oplus_{\\lambda} L_{\\lambda}", "cb50d815d231c3b96e4f209216f43323": "\\mathfrak{M}\\{\\mathcal{B}\\} = \\prod_{\\beta \\in \\mathcal{B}} \\mathfrak{G}\\{\\beta\\}.", "cb50e391c456965e99ab17492ce727cf": "s_{nk}=s(n,k).\\,", "cb50f87b063298b3633b5ced4559c4a7": "\\sigma_{A\\otimes A^*} \\circ\\varepsilon^\\dagger_A = \\eta_A", "cb50fc1fa6c0a52432894cc032e8f29d": " B_t = \\sum_{k=1}^\\infty Z_k \\frac{\\sqrt{2} \\sin(k \\pi t)}{k \\pi}", "cb5141821e6a1d4bf2135351355792aa": "F_{n}\\,", "cb51747eb072b6bb9edbde4afedfed60": "\\mathrm{d}\\mathbf{r}(s) = \\left[ \\mathrm{d}x(s),\\ \\mathrm{d}y(s) \\right] = \\left[ x'(s),\\ y'(s) \\right] \\mathrm{d}s \\ , ", "cb517555c7394b621dee96814ec20bd1": "e^{i\\theta}|\\psi\\rang", "cb5184d75e92a193a860b37f15e2df30": "\\nabla_{a}X_{b}=\\nabla_{[a}X_{b]}", "cb51d657558ea4add2dc5577a6a1e703": "\\mathcal{K}_k(x; n) = \\sum_{j=0}^{k}(-1)^j q^{k-j} \\binom {n-k+j}{j} \\binom{n-x}{k-j}. ", "cb524d2d057860bb675a4d495e95bc81": "\\begin{array}{cc} \\begin{array}{rrrr} j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrrrrrr} a & b & c & d & e & f & g & h \\\\ \\hline \\end{array} \\end{array}", "cb52a431cb29bbedac3c6c018309b9c6": "f^*\\omega = \\sum_{i_1 < \\cdots < i_k} (\\omega_{i_1\\cdots i_k}\\circ f)df_{i_1}\\wedge\\cdots\\wedge df_{i_n}.", "cb52a767195873ab859fb7e6d3a2085c": "\\pi: P\\to Q", "cb53a96e42cac1e46097159b71f65b9d": "[L_m, L_n]a = (m - n)L_{m + n}a + \\delta_{m + n, 0} \\frac{m^3-m}{12}ca", "cb5403400aea5544ff7b8b6d1f8708ee": "P(X)E(X)", "cb54035378e0cfc81c7add40af8967cb": "(U_i, \\mathcal{O}_X|{U_i}), \\quad U_i=\\{ (x_0:x_1:\\cdots:x_n) \\in X | x_i \\ne 0 \\}", "cb5423a723e0ce7a9feb5d452ed8db96": "f \\colon A \\to B", "cb542453ac951c87baa2586dfe5e1c8f": "X(\\omega) = \\int_{a}^{b} x(t)\\cdot e^{-j\\omega t} dt,", "cb54376b797fbb6a823c1b20c6f9a550": "\\sum_k A^*_kA_k=1", "cb5449507dd97c6c5591c9ce1185cf3d": " \\Omega: E \\mapsto \\mathbf{1}_E(T)", "cb549604a13d5c8c36a6fb475891c766": "l_1,\\ldots,l_p", "cb55772e39e3bbe3f356290c597db0d3": "C_V=\\left(\\frac{\\partial U}{\\partial T}\\right)_V=\\frac{3}{2}N\\,k_B =\\frac{3}{2}n\\,R", "cb55b2698e7cf093a7e314a064ec8386": "S_L(z) = \\sum_{n \\ge 0} s_L(n) z^n \\ . ", "cb55ca4d36534aae2507594f87d51c9f": "x_{ij}=\\begin{pmatrix}z_2&z_1\\\\-\\bar{z_1}&\\bar{z_2}\\end{pmatrix}.", "cb56fe0f08e34d6944cba501717326a7": "\\bar{M}", "cb5738525218a55832fd85e3ab2a7618": " N_{i}^{e}=\\frac{1}{2}\\left(1-\\xi-\\eta\\right)\\left(1\\mp\\zeta\\right),\\qquad i=3,6 ", "cb574d6d4ac9e563425b92078fa2b644": "\\mu_n=\\sum_{i=1}^n \\lambda_i \\delta_{x_i}.", "cb57541c375d2d505bbfcee432c03a3c": "\\boldsymbol{\\rho}_i \\equiv \\sqrt{M_i} (\\mathbf{R}_i-\\mathbf{R}_i^0)", "cb57a314698f8da07b1a18cf5258a7b2": "a^2+b^2+c^2=2s^2-2(4R+r)r.", "cb58199a4b51a2abd01dc8f41169a9c6": " A[x]=\\exp\\left(\\frac{1}{\\hbar}\\int X(t)\\,\\mathrm{d}t\\right) \\text{.}", "cb58dc205d45e2af452ddb9ccb19d48e": "\\mathcal{O}_k := \\left\\{ x \\in k \\mid \\nu(x) \\geq 0 \\right\\}", "cb595ad3271062b7d707668d37717432": "\nD_{3D} = \\frac{k_B T}{6 \\pi \\eta a}\n", "cb59883f3a87fcd48bd5ca6f02d653ab": "q=(q_x,q_y,q_w)", "cb598daa1c7ca622e6653aeb5d444251": "H(X) = \\lim_{n \\to \\infty} \\frac{1}{n} H(X_1, X_2, \\dots X_n)", "cb59c85b09fa02aa1390cbb4a9e3c307": "K = \\sum_{B \\cap C = \\varnothing} m_1(B) m_2(C). \\, ", "cb59caa82a333d32375343b000343427": "|xy| = |x| + |y|.", "cb5a0069e3b64e17c7993896c9a8ca95": "\\mid", "cb5a0668fa1bdce5a9b32f802e6889b4": "n\\to \\infty", "cb5a15d6f95707c0f130bf19d2fda206": "\\sum_{j \\in S\\setminus \\{i\\}} \\pi_i q_{ij} = \\sum_{j \\in S\\setminus \\{i\\}} \\pi_j q_{ji}.", "cb5a644951ab56cb576d52cc6a673289": "\\mu=X^6-4X^4-2X^3+4X^2+4X+1", "cb5a7e47f2b20db934d558fdc3e35326": "\\Re(a)>0\\wedge\\Re(s)<0\\wedge z<1", "cb5aae998e859e19588aea170a5c67cb": "\\theta_{TP}\\,", "cb5ab75b634bdca928cc60b7bc7b2b95": "((a,b),[c,d])\\in I \\Longleftrightarrow T(a,c,d)=b", "cb5abe88fd46e7c2992f4e8818afe12b": "-R\\Omega^2(R)\\ ,", "cb5ae17636e975f9bf71ddf5bc542075": "0.1", "cb5b0cfda97dae22c9cd0d1f1985f8f2": "\\sum\\limits_{n=1}^{\\infty }T_{n}\\left( x\\right) \\frac{t^{n}}{n}=\\ln \\frac{1}{\\sqrt{1-2tx+t^{2}}}.", "cb5b596a5cb0d782a4e8c66653f7b07f": " \\displaystyle{|b_n|^2 \\le {1\\over n} \\sum m^2 |a_m|^2 |b_{n-m}|^2.}", "cb5baad0e71af990373a2236a903e766": " H=H_0+W(t) ", "cb5bdbacf51f13a7335834a523b81eb3": "O(n^2 \\beta_{s+2}(n) \\log^2 n)", "cb5bef5ae7230c5c181efb9c94fb275f": " f(z) = {1\\over 2} z^2 = f(x + i y) = z_1 (x,y) + i z_2 (x,y) ", "cb5c4875d5bcfdac9d09258f35d57a61": " \\varphi_\\lambda(c)=1,\\, \\varphi_\\lambda^\\prime(c)=0, \\, \\theta_\\lambda(c)=0, \\, \\theta_\\lambda^\\prime(c)=1.", "cb5c74185fc6ae775adcc22e2b8288b6": "\\Delta(N,T,P) = e^{-\\beta G} \\;\\, ", "cb5cd28965da5be4118dec27a05750c7": "\\bold{A}\\wedge\\bold{A}", "cb5cd297bc98e96fb8f3798207e16ff6": "t\\equiv \\bar{t}\\pmod l", "cb5cf98b7884ce9f98d1568823c8eea1": "i-th", "cb5d7312f6ffbcda59e9bac53bd71ea3": "\\Re(a)>0. ", "cb5db7c3cd667d85cef98ee60f2379bd": "C_\\alpha^{\\;\\; IJ}", "cb5dfeb5ca2566de7a39abc0faf97af5": "r_{\\text{s}} (1-A_{\\text{V}}) C_{\\text{CB}}\\,", "cb5e018b451fff0dedeafaa72354b47d": "I_{m,n} = \\begin{cases}\n \\frac{1}{a(n-1)\\sin^{m-1}{ax}\\cos^{n-1}{ax}}+\\frac{m+n-2}{n-1}I_{m,n-2} \\\\\n -\\frac{1}{a(m-1)\\sin^{m-1}{ax}\\cos^{n-1}{ax}}+\\frac{m+n-2}{m-1}I_{m-2,n} \\\\\n\\end{cases}\\,\\!", "cb5e40a4edc6b105902c1a814af8a341": "\\Gamma_{i}", "cb5f8c2c36d6726c385c8644f5a21974": " y = r ~ \\sin \\theta ~ \\sin \\phi ", "cb5ff42543f4707ea0ae237d4657dc46": "d = r_1 + r_2", "cb60609a429e0efb9ea391c06030f6f7": "\\eta = \\frac {A}{q_H} = \\frac{q_H-q_C}{q_H} = 1 - \\frac{q_C}{q_H} \\qquad (1)", "cb606bea408c0fcd2deeadb42a709057": " H(x)u ", "cb608b3671b23a21e4e36068b96a49a5": "\\curvearrowleft", "cb60b324920e8d12a8a95ad83afddd25": "\\lambda: I \\rightarrow K ", "cb60c52fa2ef01afe5f59159abf67a2e": "{d \\over dx}\\sin y={d \\over dx}x", "cb61291523b68055232af4c19c4f64c3": "A_\\infty = \\lim_{\\omega \\to \\infty} A(\\omega)", "cb6147e97a27ab0ce062ff27a4c96bd2": "\n \\mathbf{E}[X;P] = \\mathbf{E}\\left[\\frac{X}{L};P^{(L)}\\right].\n", "cb6148ba106535939df3a967b5683ca9": "\nT = \\sum_j {\\sum_i {T_{ij} } } = \\sum_i {T_i } = \\sum_j {T_j } \n", "cb6158576124d46d6d478c5b1cf01b1a": "-[H^+]_0/K_{w^{ }}", "cb620b2ddff6de346cdde257b6868b1f": "\\epsilon \\sim N(\\mu, \\sigma^2),\\!", "cb6221c2573968b1e948b8af1d14093e": "I_{CM} = \\frac{m s^2}{6}\\,\\!", "cb625e429362d84e77cf1d0bf87dbf46": "{\\tilde{E}}_{8}", "cb627a6580bd432c76164f5b2863695d": "\\|f + g\\|_p^p = \\int |f + g|^p \\, \\mathrm{d}\\mu", "cb62afeda991da9306d81ac64e5e7a9c": "E_{kk}(r_{k}^{A}, r_{k}^{A})", "cb62e06bdcfd5c4f3f3900fe640fc349": "=\n-\\boldsymbol{\\nabla}\\left[\\frac{1}{4\\pi}\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\frac{1}{4\\pi}\\oint_{S}\\mathbf{\\hat{n}}'\\cdot\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'\\right]\n+\\boldsymbol{\\nabla}\\times\\left[\\frac{1}{4\\pi}\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\times\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\frac{1}{4\\pi}\\oint_{S}\\mathbf{\\hat{n}}'\\times\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'\\right].\n", "cb62f97365a18003196dddeae6f62d46": " \\mathrm{O}(2n) \\supset \\mathrm{U}(n) \\supset \\mathrm{SU}(n) ", "cb631aba99ba5987015c5ce2ff2a1f76": "\\mathrm{su}(2)_L+\\mathrm{su}(2)_R \\,", "cb63d1192512536196a4948cd9fe0015": " {\\mu}_0 \\times 10^6 = {2.6693}\\frac {(MT)^{1/2}} {\\sigma^{2}\\omega(T^*)},", "cb641e3428651b122cb5554359063696": "\\|\\mathbf{p}\\| = \\sqrt{p_1^2+p_2^2+\\cdots +p_n^2} = \\sqrt{\\mathbf{p}\\cdot\\mathbf{p}}", "cb644c2199e8f46eb0e746e622db4f04": " \\frac {A^2} {T} = \\frac {I S_x} {C} \\,,", "cb647d077f3ea7bb0ea089783ac96bca": "\\begin{alignat}{10}\nq_0 & + & q_1 & + & q_2 & - & 4e_0 & - & 4 & = & 0 \\\\\nq_0 & + & 2q_1 & + & 4q_2 & - & 3e_0 & - & 6 & = & 0 \\\\\nq_0 & + & 3q_1 & + & 9q_2 & - & 4e_0 & - & 12 & = & 0 \\\\\nq_0 & + & 4q_1 & + & 16q_2 & - & e_0 & - & 4 & = & 0\n\\end{alignat}", "cb650a12b7eb7598d907532060b44e02": "\\alpha=2 \\cdot \\tan^{-1}(1/\\mathbf{e}_z)", "cb6550f3dda2ffe74ebab3f907eba6dc": "g_k =P[S = hk] \\,", "cb655115974f7c0d5ef6c37a00aa6e0f": " (a \\circ b)\\bullet (c \\circ d) = (a \\bullet c) \\circ (b \\bullet d)", "cb655cfb320d7b57fa7a934ec65074bd": "r(A\\cup B)+r(A\\cap B)\\le r(A)+r(B)", "cb6568a3f3a0a29fb2024259d7a96c36": "1/r^2", "cb658d437221f7e3ff4c673707bb24e9": "\\mathit{AIC} = 2k - 2\\ln(L)", "cb6590cbd597eb41f99fb66f60a032b2": "V_{T+1}(k)", "cb661c01943c98aabd8aa80758b6926d": "\n C = t^{0.9} \\sqrt {\\textstyle{\\frac{1}{100}} J} (1.64 - 0.29^n)^{0.73}\n", "cb66464f4bf689b311b561b3c04b05f1": "=2\\pi \\varepsilon a\\left\\{ \\ln 2+\\gamma -\\frac{1}{2}\\ln \\left( \\frac{d}{a}-2\\right) +O\\left( \\frac{d}{a}-2\\right) \\right\\}", "cb66bcd936ee63e751e046ea6fef9c4c": "\\widehat P", "cb66beb27182c7ef9d315c66d7163abf": "\n\\eta = R_1 / R_2\n", "cb675b4b16b26acedf4218dbb6ccfd64": "h(x)=(f\\star f)(x)=\\int_{-\\infty}^\\infty \\overline{f(y)}f(x+y)\\,dy", "cb678534602b6bda11ff14552e036363": "\\Phi:", "cb67e38cb62d902383127ae51e6900e4": "X^2_{2k} \\sim -2\\sum_{i=1}^k \\ln(p_i),", "cb680ff2fdcea79456d44e163b56a7ae": "\\begin{align}\n\\frac{\\mathrm{d}E}{\\mathrm{d}r} &= \\frac{z^2 e^2 M}{4 \\pi \\epsilon_0 r^2} - \\frac{n B}{r^{n+1}} \\\\\n0 &= \\frac{z^2 e^2 M}{4 \\pi \\epsilon_0 r_0^2} - \\frac{n B}{r_0^{n+1}} \\\\\nr_0 &= \\left( \\frac{4 \\pi \\epsilon_0 n B}{z^2 e^2 M}\\right) ^\\frac{1}{n-1} \\\\\nB &= \\frac{z^2 e^2 M}{4 \\pi \\epsilon_0 n} r_0^{n-1}\n\\end{align}", "cb682cbf65e5bdef7e0d99eb514cb79c": "u(t) \\le \\alpha(t) + \\int_a^t \\alpha(s)\\beta(s)\\exp\\biggl(\\int_s^t\\beta(r)\\,\\mathrm{d}r\\biggr)\\,\\mathrm{d}s,\\qquad t\\in I.", "cb68514a3c5063851afb18b0d4bcdbf6": "v(x,y)", "cb686659a3ac2ffc2fa7eaa51955497f": "\\mathbf{Q} = \\mathbf{P}^{-1}\\mathbf{X}", "cb6868883e5fb981540486a5d96fe767": "(2^x) - 1", "cb688ba8024fa260c01ae72acfb95ce5": "\\qquad \\qquad \\mathrm{H}_p = \\sum_x \\frac{1}{2m} \\mathbf{p}^2(\\mathbf{x}) + \\frac{1}{2}\\sum_{\\mathbf{x},\\mathbf{x}^\\prime}\\mathbf{d}_i(\\mathbf{x})D_{ij}(\\mathbf{x}-\\mathbf{x}^\\prime)\\mathbf{d}_j(\\mathbf{x}^\\prime),", "cb68abf52417ce2483d011a6d6661ba9": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 121\\cdot 7.271)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 189.2\\cdot R_{\\bigodot}\n\\end{align}", "cb6901546fe0044fff3f6ad107d09758": "N_p(f) = \\int_S |f|^p\\, d\\mu < \\infty", "cb692b787a2e4730ac89f774d925b7b2": " (V,\\boldsymbol{.}) ", "cb6954257075735c2ac8f53f757d542f": "a_{n+1}", "cb69817c679c39c9e89f288aa56b2ed5": "\\left[{3\\atop 3}\\right] = 1", "cb69bcea96c0809a1f2cad116556f772": "\\alpha_0 = \\sqrt{\\frac{2}{K^2}\\left( K\\sum_{k=0}^{K-1}\\cos ^2\\left(\\omega t_k-\\theta_0\\right) -\\left[\\sum_{l=0}^{K-1}\\cos\\left(\\omega t_k-\\theta_0\\right)\\right]^2\\right)},", "cb69cd67d958e520cbd6030d0b145058": " 2{n \\choose 2} / {I(t)^2} ", "cb69fbef0c11cf4f3525013f20b73ac0": "P = Q =0", "cb6a921017dbd57334eb4387d344ec4d": "h(L,A) \\leq \\frac{\\varepsilon}{1-s}", "cb6b535655ec15e4d0a04546556948df": "\\sqrt (l/c)\\,", "cb6b9270a7a3372cd00c212b6d51530c": "(\\lnot \\forall x P(x) \\to \\exists x \\lnot P(x))", "cb6bd0396379973dfa1a32de2bece2a0": " 1\\over{N_{k,p}(u)w_{k} + \\sum_{i=0, i\\ne k}^{i=n} N_{i,p}(u)w_{i}}", "cb6bd7816652a644abbbf4e95f6370a5": "\\scriptstyle\\mu(A)=\\infty", "cb6bec6b9ffdaf5052caa9244b850532": "\\mathcal{D}= \\{d\\}", "cb6c2c07a8553eda64cb8ac4bac95eda": " \\widetilde D: J^1Y\\to T^*X\\otimes_Y V_\\Sigma Y, \\qquad \\widetilde D= dx^\\lambda\\otimes(y^i_\\lambda- A^i_\\lambda -A^i_m\\sigma^m_\\lambda)\\partial_i, ", "cb6c66574988ff6e61a442edb04f8bae": "z+i", "cb6c9b1af662881a72f4414e101703ee": "\\{ (n, \\alpha) \\vert n \\in \\omega \\land \\alpha \\text{ is an ordinal } \\} \\,.", "cb6cd1ef9d85ee928fa6ec1197ecba14": "E_{\\lambda_i}", "cb6cf21efbfb87f08b952f93e52f157f": "A\\circ B \\subseteq C\\circ B", "cb6d40e20840b62011c25aa3bd0c91f0": " \\vec{F} = q \\vec{v} \\times \\vec{B}", "cb6d48f7fbe44511e920d3bbb16e18da": "PAGR = \\left[ \\left(\\frac{P_2}{P_1}\\right)^\\frac{1}{t_2-t_1} - 1 \\right] \\times 100%", "cb6e5441cc366bd6e8c0c3ed722bcd4d": "\\neg(a \\vee b) \\equiv (\\neg a \\wedge \\neg b)", "cb6e5a88e36251c7b146f5c998444303": "H \\vert{\\Psi_{0}}\\rangle = H e^{T} \\vert{\\Phi_0}\\rangle = E e^{T} \\vert {\\Phi_0}\\rangle ", "cb6e67eed705de860c87a3fe271a3274": "\\{J_1^2, J_2^2,J^2,J_z\\}", "cb6e69d21daf8f0f8f82f9dda4278629": "r_{d_2}", "cb6e7a520583104889cc86cdf0e373f6": "d_F=2\\sum_{i=1}^k\\omega_i.", "cb6ed6d4864b9184bc17681c65a1b45a": "\\Pi_D=f(S,\\bar{P},\\bar{E}) - \\Pi_0 - T) \\,", "cb6f67444f7fc78324fade7685b8bc3f": "P_{i} = d_{ijk}\\sigma_{jk} \\,", "cb6f7afd51f38cc148bf8879f9712771": "= 2\\ \\gamma^\\rho \\gamma^\\nu - \\gamma^\\nu 2 \\eta^{\\mu\\rho} \\gamma_\\mu + 4\\ \\gamma^\\nu \\gamma^\\rho \\, ", "cb6f860243ef48af62e8c0ebb7731af7": " WA_k = TH_k * FF ", "cb6fd356cb3e477a229b8f6b7fd1493c": "2 \\pi\\!", "cb6ffd2c46009633cea5c8386fd05615": " \\alpha\\ ", "cb70192e501ec3c5659f5438f07c9227": "\\nabla^2p(\\vec{r},t)-\\frac{1}{v_s^2}\\frac{\\partial^2}{\\partial{t^2}}p(\\vec{r},t)=-\\frac{\\beta}{C_p}\\frac{\\partial}{\\partial t}H(\\vec{r},t) \\qquad \\qquad \\quad \\quad (1), ", "cb70340de8d1e21d01ba296ddfd22f94": "d(A, A)", "cb7034660f422a58938a76b8f8e6fbff": "(-640320)^3=-e^{\\pi \\sqrt{163}}+744+O\\left(e^{-\\pi \\sqrt{163}}\\right).", "cb70436bc0435e1bd7e60d2e7a746019": "F_r = \\frac{GMm}{2 \\pi} \\int \\frac{ sin^2{\\theta} \\cos\\phi} {s^2}d\\theta. ", "cb70e141b108cb42a9eb15b3fc7a7c97": "W^{1,p}(\\Omega)", "cb70ea3d98179780aef435c809b013ec": "f(L) =\n\\begin{cases}\nL, & \\mbox{if }|L| = 1\\\\\n\\{s\\} \\cup f(L_{s}), & \\mbox{otherwise}\n\\end{cases}", "cb714b567cadf315d6e4125412612689": "T_xM", "cb71d18877a22f30e0c59ec8565084ec": "\\textbf{P}_{k\\mid n}", "cb7228ef175abb775ef35ea6c5e4a578": "\\frac{F({\\rm state1})}{F({\\rm state2})} = e^{\\frac{E_2 - E_1}{kT}}.", "cb72375ca5c34db758d52b9b1edc898c": "\\textstyle \\frac15 \\times \\frac{4}{10} = \\frac{4}{50} ", "cb724b9a640ed77f165f80103a9458d9": " V = \\frac{\\mathrm{d}W}{\\mathrm{d} q} , \\quad I = \\frac{\\mathrm{d}q}{\\mathrm{d} t} ,\\,\\!", "cb72a1bf19883f309d20d028d2fc5e6b": "s'_A=11/6c", "cb72daa4e3a4c9088b6045d4b3fea73a": "\\alpha^{-1}(G_r^\\pm)=G_{r\\mp {1\\over 2}}^\\pm", "cb7358794e5c3827c9702974dc550e8d": "\\vec r_B-\\vec r_A", "cb737aaeeb3cb21b1fe6832c77c500ad": "L^{p_\\theta}(\\mu_1)", "cb7421e2a4c65908dbadeeaf79068998": "(a,b) \\in [0,4] \\times [0,4]", "cb74414d5ab9746cdadda54e0729def5": "N^{th}", "cb74a2805862db305cb71911fd3f7e60": "\\omega_n\\ ", "cb74b7bca45bb1298982d4ac387d97cd": "2\\leqslant i\\leqslant k-1", "cb74d98e89f1a6dcab4a84e1c4575d7b": "\\ln \\left ( \\frac{\\varepsilon_{n+1}}{\\varepsilon_n} \\right ) = 2 \\left [ 1 - p_3 ( u_n ) \\right ] \\Delta_{n+1}.", "cb750e62501fc34a6ff29d3a838fc6ec": "r_v = g^s y^e = g^{k - xe} g^{xe} = g^k = r", "cb752b6d728c8d0d7946000991d82a4c": "\\int_a^b \\sin(k \\cdot t) dt", "cb75460b4d820a2160229dfc3dd5e155": "\\lim_{n \\to \\infty} a_n \\neq 0", "cb75e653ebf46dbb41a49fd165b90195": "\\int_0^\\infty P_{n_1} (x) P_{n_2}(x)\\cdots P_{n_r}(x) e^{-x}\\, dx,", "cb7631ee5ed1340e2912d906886715cb": "F = \\frac{1}{2}\\frac{nt\\in_o\\in_rV^2}{d}", "cb76372c831d2b165083997b19b7c395": "\\Delta \\cup \\{A\\}", "cb764f5344c96d2a1de312e392ab5383": "W = \\frac{a}{1-v}=", "cb769a70bf3d30cf77f794d004a8825b": "N(f)", "cb769f2f2475c601b85506dd47c37193": "\\omega_{p}({\\mathbf e}_j,{\\mathbf e}_k)=\\omega_{p}({\\mathbf f}_j,{\\mathbf f}_k)=0\\,", "cb76f0e87671f95c7910c17b7a93fb7c": "\\mathbf{E}", "cb76f122b8ba06a756e64cc830d54ca5": " x_{L} ", "cb76ff6dcb8fb1627e68e049b8187a9f": "\\operatorname{Vec}({\\mathbb R}, n)", "cb77544ee3ae416403f6acd1afb89280": "\\frac{1}{p!}\\Delta_Q^p f(p, q, r)=\\frac{1}{(n-p)!}\\Delta_P^{n-p} f(a, b, c).", "cb775d638f7a5eb7d99bc51a34e5a379": "\\left|\\frac{4}{9}\\right| < 1", "cb77607e90c845883d4df3412ea10463": "D = \\frac{S}{mil} \\cdot \\frac{mag}{10} ", "cb77804a7db9fbdfe8ffc191ff421616": " \\frac{az^{-1} }{ (1-a z^{-1})^2 }", "cb778db93f53ef3df42349bae6323624": "\\hat{f}(\\xi) = 0", "cb77a7e9a542b58df67ce89813c6887b": "T^{0i} = T^{i0}.", "cb77dca7921ad29309abb67e72fd18a8": "\\mathfrak{o}(n)", "cb780bcce0fd55daa9bdd96469e171ed": "\\mu = \\frac{B_0}{H_0}\\cos \\delta - j \\frac{B_0}{H_0}\\sin\\delta = \\mu^\\prime - j \\mu ^{\\prime\\prime}.", "cb781f3578524a3db20a0db5c484e449": "( \\sigma_1 \\otimes I ) |\\Psi^+\\rangle = |\\Phi^+\\rangle .", "cb783e8d709e1858f5f08f4d45bb53a2": "< \\beta, \\alpha > = \\frac{(\\beta, \\alpha)}{(\\alpha, \\alpha)} \\, \\forall \\alpha, \\beta \\in E", "cb78763650752faf488b25b87fe8263f": "H_\\xi", "cb7887258d2aeafa829d08887c30ab47": "\\begin{pmatrix} p & q \\\\ nq & p \\end{pmatrix}", "cb788de4b07cd2fec24df0e924734388": "\n \\text{FPC} = \\sqrt{\\frac{N-n}{N-1}}\n ", "cb78b0b805c08022fbbf3deb31559db1": "a^{n}b^{n}c^{n}", "cb78d0a3cbcf59ad691b8e067e541334": " \\mbox{2-EXPTIME} = \\bigcup_{k \\in \\mathbb{N} } \\mbox{ DTIME } \\left( 2^{ 2^{n^k} } \\right) . ", "cb78d74751baa2c848b1096f30b21f80": "S = \\{(x_i, t_i): a_i > 0\\}", "cb791f4c03d07bcee214544010fa2788": " \\left(E_n^{(0)} - H_0 \\right) |n^{(1)}\\rang = \\sum_{k \\ne n} |k^{(0)}\\rang \\langle k^{(0)}|V|n^{(0)} \\rangle ", "cb7924addd6212318ab258dade7e9533": " \\{F_r L\\}_{r\\ge 1} ", "cb795c812e558548b079d1ef0de8f457": "x_+^\\alpha[\\varphi\\circ\\mu_t] = t^{\\alpha+1}x_+^\\alpha[\\varphi]", "cb79a919bcc14368d30471893e207460": " \\delta.", "cb7a2d49fc5d3d2c8e7e60310bf61f22": "\\{a^nbab^n | n \\ge 0 \\} = \\{ba, abab, aababb, aaababbb, \\dotsc \\}", "cb7a85b2b244057d43252ee21e7b1745": "\\mathbf E_{1s}^{non rel}", "cb7a97ac93c4200036ef1c63e1f21274": "\\operatorname{I}(Y;X) \\ge \\ln ([1 - \\iota_{Y\\mid X}]^{-1/2}).\\,", "cb7abd00596a905f20fea71dd648aef1": "v_{m+1},v_{m+2},\\ldots", "cb7acae74312c5533da969a345206c20": " x - ct = \\text{constant,} \\quad x + ct = \\text{constant},", "cb7ad94cc90ca8ff023d6f340265b355": " |z_1-z_3| + |z_2-z_3| \\le C |z_1-z_2|.", "cb7b1c34c16e9c2613361dee69a967a6": " \\langle \\mathbf{r}_0, m | \\Psi \\rangle = \\sum_{s_z}\\int\\limits_R d^3 \\mathbf{r} \\, \\langle \\mathbf{r}_0, m | \\mathbf{r}, s_z \\rangle \\Psi(\\mathbf{r}, s_z) = \\sum_{s_z}\\int\\limits_R d^3 \\mathbf{r} \\, \\delta_{m \\, s_z}\\delta( \\mathbf{r}_0 - \\mathbf{r} ) \\Psi(\\mathbf{r}, s_z) = \\Psi(\\mathbf{r}_0, m) \\,.", "cb7c83b1b2f22124222019de397e4758": "\\mathcal{I} = \\{ 1, ..., I \\} ", "cb7ce0498579ac98d6892af295422e82": "V'(x)=\\bigl(W'(x)+W(x)p(x)\\bigr)\\exp\\biggl(\\int_{x_0}^x p(\\xi) \\,\\textrm{d}\\xi\\biggr)=0,\\qquad x\\in I,", "cb7d1a0628570c5d72af6688d25b8492": "\\mathrm {Ind} (\\mathbb P^n) = n+1", "cb7d1da97770eeef888bbd5a4f4528f7": " V_{ECF a} = \\frac{n_{ECF a}}{Osm_a} = \\frac{V_{ECF b} \\times Osm_b - n_{lost Na^+}}{Osm_a} ", "cb7d268cba5d49928b3c70ebc23c0f3b": "\\varphi_i:\\mathcal{P}_{n-i} \\times \\mathcal{P}_{m-i} \\rightarrow \\mathcal{P}_{m+n-i}", "cb7d457cdafacd0027c8fb4285894839": "MP_1", "cb7d5440ce0e5d95bca0e9133828b91a": "n_1^2+n_2^2+n_3^2=\\frac{T_1^2}{{\\sigma_1}^2}+\\frac{T_2^2}{{\\sigma_2}^2}+\\frac{T_3^2}{{\\sigma_3}^2}=1\\,\\!", "cb7d68c3ea6495a31e6afeec6cf9484d": "x=\\sum_{g\\in G} a_g g.", "cb7daa82eea433958e61ea5a295554b0": "\\hbar = h / 2 \\pi", "cb7dd77d2cb1ace2fff7d126367e1c18": "F_{X,Y}(x,y) = F_X(x) F_Y(y),", "cb7e07ea4b41b0e22276b6213871bccb": "\\mathbf{v}={d\\mathbf{r}\\over dt} \\text{,}", "cb7e7ac909f037f2a07bd01eca1872c3": "\\partial \\pi (p)\\partial p_{i}=x_{i}^{\\ast }(p)", "cb7ed1b4cd73fd49881955f9b5d40010": "\\Delta_\\lambda(\\rho)=\\sum_{n=1}^{2d^2(d+1)}c_nU_n^*\\Phi_\\lambda^{(n)}(\\rho)Un", "cb7ee56f898803cb70670b2e64a9fd3d": " Cov(e_{nit}, e_{njt}) = \\sigma^2 (X_{nit} X_{njt}) ", "cb7f0de0bffa41de5f86b471f78ecd34": " \n\\begin{bmatrix} \\rho \\\\ u \\end{bmatrix} = \\begin{bmatrix} \\rho_L \\\\ u_L\\end{bmatrix} \\text{ for } x \\leq 0\n\\qquad \\text{and} \\qquad \\begin{bmatrix} \\rho \\\\ u \\end{bmatrix} = \\begin{bmatrix} \\rho_R \\\\ -u_R \\end{bmatrix} \\text{ for } x > 0\n", "cb7f440db476e1f09f3ececd187a3184": "N_{L/K}(I)=\\mathcal{O}_K \\cap\\prod_{\\sigma \\in G}^{} \\sigma (I)\\, ", "cb7f645b957f1c1e42793254eb2c681b": "VC(y)_z", "cb7fb431d1f8ff2e9909116089d7cebb": "s_k \\equiv 0 \\pmod{m} \\!", "cb7fbe54e2dc635c663783d49458671f": " \\omega_0 \\to 0 ", "cb80059d57e2d8e6f172ea245983c676": "b \\in K_a", "cb8023b18a5d0909e38fe58fc27314d3": "(x_0, y_0, z_0)", "cb802a7c2473a846d3116f443aa5c0c6": "\\ d_H\\,", "cb8093e6ef8756c5a184dbd0d1dffe21": "C_\\alpha", "cb80c04c42a9274a6bc2fbe30fafbb9d": " \\omega = \\sqrt{\\frac{k}{m}} , ", "cb80ca3f4a3ab9400f6caf004fea1afd": " \\vec f = - k_B T \\frac{3 \\langle \\vec R \\rangle}{N l^2}~", "cb80fd923325e17c51870d4a831b86b8": "W_0(x)=\\ln x-\\ln\\ln x+o(1).", "cb8105fc9f1eb12fa516063980b6e9df": "\\nabla \\rho(\\mathbf{x})", "cb8143e68c41a469d56fc76c1d3396fa": "H(f)|_{\\beta=1} = \\left \\{ \\begin{matrix}\n \\frac{T}{2}\\left[1 + \\cos\\left(\\pi fT\\right)\\right],\n & |f| \\leq \\frac{1}{T} \\\\\n 0,\n & \\mbox{otherwise}\n\\end{matrix} \\right.", "cb818cd02ece3f790931badf3eb0f13b": "\n\\begin{align}\ns_i &\\mapsto s_i \\quad \\mathrm{for}\\quad i=1,2,3 \\\\\ns_i &\\mapsto s_i + \\Delta \\phi \\vec{f}_i \\cdot \\mathbf{I}^0\\cdot \\vec{n} \\quad \\mathrm{for}\\quad i=4,5,6, \\\\\n\\end{align}\n", "cb81a57e32e39ff34c77319d7c934216": " x^{(6)} =\n \\begin{bmatrix}\n 0.000 & -0.1875 \\\\\n 0.000 & -0.1193\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 0.8121 \\\\\n -0.6650\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0.6875 \\\\\n -0.7443\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 0.8122 \\\\\n -0.6650\n \\end{bmatrix}. ", "cb824416b8057be37a2239d1f98dd7a3": " \\forall x\\in M^n: \\exists I_n(x)=x\\wedge A(x) \\mid I_n(x)\\or M_n=x ", "cb8249cd3122f920223fada58a3e44f0": "\\displaystyle{T(t)=PU(t)P}", "cb8256c631348e8538af50d4693a741a": "\\mathrm{dist} (x, A) := \\inf_{a \\in A} d(x, a),", "cb8277126b3b775f3ad757511636455a": "m(x)\\leq \\frac{f(y)-f(x)}{g(y)-g(x)}=\\frac{\\frac{f(y)}{g(y)}-\\frac{f(x)}{g(y)}}{1-\\frac{g(x)}{g(y)}}\\leq M(x)", "cb830240ca8210558d4e589ca07f30ec": " C = a + b(Y-T)", "cb8306bfb6bf3a925158e4c573353c10": "\\lim_n \\langle x_n, v \\rangle = \\langle x, v \\rangle", "cb83270f453a95b2ce8a3a6fcb7c91b7": " r_M(Y) - r_M(X) \\leq r_N(Y) - r_N(X) \\text{ for all } X\\subset Y \\subseteq E. \\, ", "cb838c54e73a0ecbd1d14bf78603fb57": "\\mathfrak g=\\mathfrak h +\\mathfrak f", "cb83a404c6c30602d807f847edd6d1f3": "(\\nabla f, -1) ", "cb83f1cd563408e37c4492b84fa3a44e": " (x_1...x_k) ", "cb843171a99db6558817b7b682baf055": "\\begin{align}\nD_{EU}=\\sqrt{\\sum \\limits_{u}(X_u-Y_u)^2}\n\\end{align}\n", "cb84bc351a95e46c6b304f68548ca752": "\\alpha_i \\in (T \\cup V)^\\star", "cb85079b898cf697405c2feb41a91b2d": " \\frac{p(n+1)}{1-p} ", "cb851c0aadd521eacb94bdb3a0a94fba": "E = \\frac {F \\cdot 360^\\circ}{S}", "cb85a896ce929e987f46fc9564574b38": "I_x^n(M)", "cb85b8a57d562de64f9ceb40953d394c": "A, B\\in\\mathbf{H}_n", "cb85b952815947044432870f70770c5c": "\\textstyle \\dot{x}_2(t) = \\ddot{u}(t)", "cb85be6077c498b5374f7e606be73199": "y(r,\\theta) = -r\\sin(\\theta)(r \\cos(\\theta) + 1)", "cb85d9ab287543203bb8fdec0cba05f1": "\\mathbb{Z}\\oplus\\mathbb{Z}", "cb865d9ac858322b304d00c9009fc53d": "g = \\det\\left(g_{\\rho\\sigma}\\right) = \\frac{1}{4!} \\varepsilon^{\\alpha\\beta\\gamma\\delta} \\varepsilon^{\\kappa\\lambda\\mu\\nu} g_{\\alpha\\kappa} g_{\\beta\\lambda} g_{\\gamma\\mu} g_{\\delta\\nu}\\,,", "cb8662a4eed7100e7c28322e9acb4644": "\\boldsymbol{\\hat r}", "cb866b6dad109c5b8ca9e595ca1f6777": "d=\\frac{v t}{2},", "cb8693e1036c7858006372e487b2f5e3": "(w_2 \\sqrt{T_3}/P_3) \\,", "cb86dd8dec57f4a0899b221f5e54c248": "\\int (ax + b)^n \\, dx= \\frac{(ax + b)^{n+1}}{a(n + 1)} + C \\qquad\\text{(for } n\\neq -1\\mbox{)}\\,\\!", "cb86e240f13a03627163204ea5ef0f7a": "\n\\frac{d^2u}{dt^2} + k^2 u = a + \\varepsilon f\\left(u,\\frac{du}{dt}\\right)\n", "cb876b4690fbbeae0726ffd64039abbf": " \\omega \\in \\Omega_{Z,[t_l,t_u]}", "cb8798f2e7b03b004599d166b5ec2de8": "c_g \\ \\equiv\\ \\frac{\\partial \\omega}{\\partial k}\\ = U - \\frac{\\beta (l^2-k^2)}{(k^2+l^2)^2},", "cb879e649c5b80cc585ce0292cfd7663": "P_1\\parallel_+ P_2", "cb87a1933a9af9ab7e3e8f4d680488a4": "\\scriptstyle \\forall p \\,\\in\\, A,\\; \\forall v \\,\\in\\, V", "cb87c6d716d92d13dfcfa24ca759c67d": "{\\mathrm{d} \\over \\mathrm{d}x}\\sin x = \\cos x", "cb87e35f2070cdb8a11acd2a8dd7b881": "J^+(\\gamma) \\cap J^-(\\gamma)", "cb8825906910bd0ba16dad44ea80255e": "\\sum_{j=0}^n|\\Delta p_j|^2\\!", "cb885c67cc3c5d9d541109a2b962a308": "t^{\\prime}=t-\\frac{vx}{c^{2}}", "cb88a1312b8fc29bc0e42e4dbffa1241": " SubCipher_{1} ", "cb8961d784a23491f9b4dd537aef07f4": "f(x^\\ast) \\le f(x)", "cb8a14b98cee78a7c76cfce0962bff4b": "\\displaystyle{A= \\sup_x \\int_X \\|T(x)^*T(y)\\|^{1\\over 2} \\, d\\mu(y),\\,\\,\\, B= \\sup_x \\int_X \\|T(y)T(x)^*\\|^{1\\over 2}\\, d\\mu(y),}", "cb8a3df1f10957008f0f4fb8ee4305b1": "\\deg_y Q(x,y) = p", "cb8a43cd2ea86152033b12f31b795c22": " A\\to B=SAS^\\mathrm{T}.", "cb8a5e0dc19ca9048ededfd8a634ae1a": "\\forall x\\in\\mathbb{R},\\,", "cb8a8964bb5766dade3422e1308a57da": "P=\\frac{u}{n}", "cb8ad2867597690b136e2694cbc6fbd3": "L' = a^m b^n c^m d^n ", "cb8aee1c9923f1f95e19f32283549f61": "I_m", "cb8b161bcd4a3011c06a2fa0164649ee": "\\mathfrak{g}", "cb8b39a4a47122c3a89eab1b16547a33": "b : 1\\,", "cb8b40bdc02bf55c1ad1353918a371f5": "y=4", "cb8b483969f2a83772e65940facf0b41": "\\begin{array}[t]{rcl} \nwp(\\textbf{if}\\ x < y\\ \\textbf{then}\\ x:=y\\ \\textbf{else}\\;\\;\\textbf{skip}\\;\\;\\textbf{end},\\ x \\geq y)\n& = & (x < y \\Rightarrow wp(x:=y,x\\geq y))\\ \\wedge\\ (\\neg (xn_1", "cbd4f511fd6be0f1e34dc4b3bbb22750": "K_p = \\frac{p_C^c\\, p_D^d} {p_A^a\\, p_B^b}", "cbd5152f8d67b906bbc6f9b7fba50962": "\n h_i = \\beta_1 t_i + \\beta_2 t_i^2 + \\varepsilon_i,\n ", "cbd554a7d47c0770c110c998f0f9ef06": "E={\\hbar\\omega / 2}", "cbd55bd2f272a28cd82bd9c1240319bb": "A = \\tfrac12\\, \\exp \\left(\\tfrac38\\, i \\, t_1 \\right).", "cbd561891ae01f6682744ff720f5435c": "\\bigcup S=\\alpha", "cbd5a8be21ee6be817dda209b8898402": "\n \\gamma_{\\mathrm{oct}} = \\tfrac{2}{3}\\sqrt{(\\varepsilon_1-\\varepsilon_2)^2 + (\\varepsilon_2-\\varepsilon_3)^2 + (\\varepsilon_3-\\varepsilon_1)^2}\n ", "cbd5a91b03ededc948bd25a4e0baae90": "T = 2\\left\\langle K\\right\\rangle _{t},", "cbd625b8267d170feb02b27721a213f3": "\\vec{v}(x, y, z) = v_x\\mathbf{\\hat{x}} + v_y\\mathbf{\\hat{y}} + v_z\\mathbf{\\hat{z}}", "cbd65d3551be99aef176b7b1f12b04cb": "P_s=\\sqrt{\\frac{1}{2\\alpha_{111}}\\left[-\\alpha_{11}+\\sqrt{\\alpha_{11}^2-4\\alpha_0\\alpha_{111}\\left(T-T_0\\right)}\\right]}", "cbd666ba84b6eb4bac526de1d07198b9": "(f \\circ g)'(t) = f'(g(t))\\cdot g'(t).", "cbd69e8c5a83b1f92ae750210dbf3a6e": "h = (1-(P/P_\\mathrm{ref})^{0.190284}) \\times 145366.45\\mathrm{ ft}", "cbd79ba918d0da7516cb2013ee05e36a": "\\mathbb{HP}^{n}", "cbd7ba5d390540c8132b46fa56c6d275": "\n\\left( x_{s} - x_{1} \\right)^{2} +\n\\left( y_{s} - y_{1} \\right)^{2} =\n\\left( r_{s} - s_{1} r_{1} \\right)^{2}\n", "cbd7c0628bf7917df3fee7a25673307e": "Z_h = \\frac{\\sqrt{\\rho \\, B_\\mathit{eff}}}{A}", "cbd7e20dc02ad1b5e038e258dad60927": " a,b>0 ", "cbd84dcd8c8c1dce22b3b4e5db9fe67a": " I_0 = I_1 +I_2 + I_3 + I_4 ", "cbd890e5b141370dc9c21cab6835d1a8": "g(i)", "cbd8a5eeeab08358cfb06c74f7b471b5": "a+1", "cbd8b4c5fddc821039959838f8abbdce": "\\displaystyle{f(z)=g(\\varphi(z))}", "cbd8e77f04b801ca1f356d18412371bb": "\\nu_f^{\\sigma_j}\\,\\!", "cbd94f942161a77e4cfc4677b31ad15f": "\\scriptstyle \\omega=2 \\pi f", "cbd9e072415b09a67665202d2701a1a9": "f_{pm}", "cbda01e3645eb60cbf27a31ab825da42": " \\prod_{j=1}^m \\prod_{k=1}^n \\left ( 4\\cos^2 \\frac{\\pi j}{m + 1} + 4\\cos^2 \\frac{\\pi k}{n + 1} \\right )^\\frac{1}{4},", "cbda069dedcafc533c2ed20f660e01ff": "\\nabla \\mathbf{v}=\\frac {\\partial v_j}{\\partial x_i} ", "cbda0781a28546354fc6c65d8d2aa8e9": " e^{-1+\\sum \\limits_{k=2}^\\infty \\sum \\limits_{n=1}^\\infty \\frac{1}{n k^{n+1}}} = e^{-1-\\sum \\limits_{k=2}^\\infty \\frac{1}{k} \\ln \\left( 1-\\frac{1}{k}\\right)} ", "cbda5968a03632c94ce186b10955422f": "\\; (A-4I)^2 p_4 = 0. ", "cbdaae7e151c3921fb7e1db0549f96d2": "\n\\sigma^2 = \\bigg\\Vert \\sum_k \\mathbf{A}^2_k \\bigg\\Vert.\n", "cbdabf72a0453495dcb3ed3a6aa5d1ee": "\\frac{1}{(2\\pi)^2} ", "cbdb16ab08ab2023ffc4ec14e016a436": "\\omega >> (4\\pi n_ee^2/m_e)^{1/2}", "cbdb2b68d256ad300875712b865d16d4": "\\begin{matrix} {4 \\choose 1} \\end{matrix}", "cbdb510ee1c373c7cd6128d84d90ae8f": "\\scriptstyle{R_a^b}", "cbdb7add2b14a437c047831db15855c7": "(\\delta[x] + \\varphi(x))/2", "cbdbe36ad50793b204d720d7964842b4": "\\exists x P(x)", "cbdc13bcd333c67f9a1faad28681d4af": "I = H_z \\,\\Delta z", "cbdc92acde7112771d2c91857f0b38ee": "\\Sigma r Q^{n}", "cbdcf0e203060323ebcd0079f280f4b8": "|E|", "cbdd019f456cc279f3df8fb265d9bc35": "P(z)=P^\\prime(z)=0", "cbdd1a564707f5029592f0b9595bf35f": "B_0 / \\text{T}", "cbdd75a4c76abf604e40627ec0915eb4": " (5)\\,", "cbdd83df4f379c98136b3deb68aa41e5": "\\sum_{n=1}^\\infty \\frac {a_n} {n^s} = \\zeta(s)\n\\sum_{n=1}^\\infty \\frac{b_n}{n^s}", "cbddf42f83f2bfe25d93824e94f76a74": "f\\omega", "cbde02fcdb2140358ea015c32b5d7c4c": "\\mathbf{D_{zz}} ", "cbdeab744808fb0dbaca4b9a76e77aeb": " e_k (X_1 , \\ldots , X_n )=\\sum_{1\\le j_1 < j_2 < \\ldots < j_k \\le n} X_{j_1} \\dotsm X_{j_k}.", "cbdeb2005a7355b6df42594c5aa97578": "\\alpha = \\lim_{k\\to\\infty} \\frac{|w - z^*|}{|w' - z^*|}.", "cbdf18717e61b5d26d8dfb2e8b378dae": "F(\\varphi , k) = \\int_0^\\varphi {1\\over\\sqrt{1-k^2\\sin^2{u}}}\\,du\\,.", "cbdf1946557b13df9d0321caf79e230b": "p(a)/(p(a/2)+p(a))", "cbdf342467ae7e76b1970edc687fcab3": "r=-a \\frac {\\sin \\tfrac{2}{3}\\theta}{\\sin -\\tfrac{1}{3}\\theta} = 2a\\cos \\tfrac{1}{3}\\theta", "cbdf3563ce8cd7200a719eb7abfd53c9": "\\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\mathbf{f}.", "cbdfbea9fad589da20e5fde3f7698cca": "\n\\delta \\int_{t_{1}}^{t_{2}} \n\\left[ \\mathbf{p} \\cdot \\dot{\\mathbf{q}} - H(\\mathbf{q}, \\mathbf{p}, t) \\right] dt = 0\n", "cbe0182ae501fd2eda2070a4fce3daa8": "g \\leq h", "cbe0838f64eb17da315c0e39644bbc91": "(\\phi^{\\Rightarrow x})^{\\Rightarrow y} = \\phi^{\\Rightarrow x \\wedge y}\\,", "cbe08703dbb4fb08e84a8fac90551a23": " \np(t) = P(x_1(t), ..., x_N(t)) \n", "cbe0cbc2135e652d1aadc3f27183121a": "\\sum f(n)\\Lambda(n)", "cbe14a780bfed17e92e434a13dee83fe": "S=({\\mathcal P},{\\mathcal B}, \\textbf{I}) ", "cbe1d993d83a7d3d4abb3947f2a25957": "B^{l_p} {}_{j_p}", "cbe203b034aaf604484d3844559741fd": " i \\in \\{1, \\ldots,N\\} ", "cbe20be8c333d485e954db49118f6746": "\\sigma=\\frac{\\pi^2 E}{(\\frac{Kl}{r})^2}", "cbe21a11302b0e7bdcb49fee0a0103f7": " \\mathbf{Z}' = \\boldsymbol{\\Lambda}(\\mathbf{v})\\mathbf{Z}.", "cbe259a6ad171dc8b2979ca4a5c45c7e": "\\Sigma, \\Omega", "cbe2d9f85419b2aa615ba1cfb61c990f": "\\{p_1\\}", "cbe2e21fe1036fdb41c7b675f9405a6c": "\\omega^{A}_{x\\overline{\\land} y}=\\omega^{A}_{x}/\\omega^{A}_{y}\\,\\!", "cbe2e98c9736640f8418d2031de74e42": "\\gamma_\\lambda \\rightarrow \\gamma", "cbe364c4841e436f76ed4a43aa6b3b74": "m \\colon A \\to \\mathbb{N}_{\\geq 1}", "cbe39a156795404bb4abdbdc8a00ca17": "\\operatorname{BetaBin}(\\tilde{x}|\\alpha',\\beta')", "cbe3a3d4041a0775c654f107844ff5a8": "\\{P(M_m)\\}", "cbe477caa9b26ecb7ea9fbc86e1ba545": " f(\\gamma^{mi+j})", "cbe550d965c1ce4e973f2063b07050c3": "\\ g_{\\phi}= \\left(9.8061999 - 0.0259296\\cos(2\\phi) + 0.0000567\\cos^2(2\\phi)\\right)\\,\\frac{\\mathrm{m}}{\\mathrm{s}^2}", "cbe5cfb7d3202a47410bc59199963552": "4x^3 + 4y\\frac{dy}{dx} = 0,", "cbe5f01effa1eaf0d0c281705c093ccd": "\\mathbf E =\\frac{1}{2}\\left(\\nabla_{\\mathbf X}\\mathbf u + (\\nabla_{\\mathbf X}\\mathbf u)^T + \\nabla_{\\mathbf X}\\mathbf u(\\nabla_{\\mathbf X}\\mathbf u)^T\\right)\\approx \\frac{1}{2}\\left(\\nabla_{\\mathbf X}\\mathbf u + (\\nabla_{\\mathbf X}\\mathbf u)^T\\right)\\,\\!", "cbe66ae10b1febcc351cfe76186a89b0": "|u_-\\rangle", "cbe6d1570d03584af351fbcd9b5e2e37": "\\left[{d^2 \\over dy^2} - {l(l+1) \\over y^2} - y^2 + 2n - 3 \\right] v(y) = 0 ", "cbe6fbba1300fa5eb5efb786c0ff5be1": "\n|u_-\\rangle=\n\t\\begin{pmatrix}\n\t\\sin{\\theta\\over 2}e^{-i\\phi}\\\\\n\t-\\cos{\\theta\\over 2}\n\t\\end{pmatrix},\n|u_+\\rangle=\n\t\\begin{pmatrix}\n\t\\cos{\\theta\\over 2}e^{-i\\phi}\\\\\n\t\\sin{\\theta\\over 2}\n\t\\end{pmatrix}.\n", "cbe75da685b8ccde01c47578feb83f43": "\nW = \\tfrac{1}{\\sqrt{2}} \n\\begin{bmatrix} \n1 & 1 \\\\ 1 & \\omega ^{2 -1}\n\\end{bmatrix}\n=\n\\tfrac{1}{\\sqrt{2}} \n\\begin{bmatrix} \n1 & 1 \\\\ 1 & \\omega ^{d -1}\n\\end{bmatrix}.\n", "cbe76d9e4b371e4d0274813f6c6a5147": "\\log|z_n|/P^{n} = \\log(N)/P^{\\nu(z)},\\,", "cbe772ef45e9a1916bee0d41d1414d16": "R_{nm}", "cbe7ab71a6233070cf133cc4ea32f70e": "x_1 \\ge 0 ", "cbe810b606cdced8d5ddc83c2ce55217": "g^{j\\phi(n)/p_i}", "cbe8162fbffbc49cef8307a57c221574": "M_W = \\frac{v|g|}2 \\qquad\\qquad M_Z=\\frac{v\\sqrt{g^2+{g'}^2}}2.", "cbe87565bc5dbec3f91ba299519ab1a2": "\\overline{\\Theta}=s\\theta-\\theta\\left(\\tau=0\\right)", "cbe88fff6109316b68e6c5bdf15a83c3": "m=\\langle id, t_{i}, t_{j}, s\\rangle", "cbe895edf26efdd4255c01c5ec90735e": "\\displaystyle{a\\circ b = \\{a,y,b\\},}", "cbe8dccec2b453e1836febf39b71509e": "\\beta = \\frac{2\\pi}{\\lambda}\\,", "cbe8f572f0764426d09bd572391e3370": "0.227\\pm0.014", "cbe948df24f74b0c0e929218a6e6d18d": "(a_1b_3 - a_2b_4 + a_3b_1 + a_4b_2 + a_5b_7 + a_6b_8 - a_7b_5 - a_8b_6)^2+\\,", "cbe994dab4e67e7368aeb25e32dce098": "s_\\alpha = x_{\\alpha-1} - x_\\alpha - l_{\\alpha-1}", "cbe9a219d4f602c72518e6b333c92767": "{V_1 \\choose V_2} = \\begin{pmatrix} Z_{11} & Z_{12} \\\\ Z_{21} & Z_{22} \\end{pmatrix}{I_1 \\choose I_2} ", "cbe9abd3d8a8d895732554366442abae": " \\theta \\in \\mathbb{R}", "cbe9b2363aa4b1533e50c32f2440a3e7": "\\mathrm{C}_G(S)=\\{g\\in G\\mid sg=gs \\text{ for all } s\\in S\\}", "cbe9c9dd27ff6ca4e3598946b0928704": "\\delta\\,", "cbe9df0f7846c24925cb56e6fe09a477": "d\\dot{Q}_{conv}=Ph\\left (T-T_\\infty\\right )dx.", "cbea2910be206f8f28a1922532859414": "X_{s} (\\omega) := \\chi_{P} (s, \\omega)", "cbea370f358cb3c03fa81558fa07cad0": "tau_p", "cbea388e818112a7d37e5082b0ffe8fa": " T(iu_n)=-\\lambda_n iu_n", "cbea4efdc6ecc81f5746cd719788e26f": " m = {f \\over S_1 - f} \\,,", "cbea71f16b368ab4d54e348ad44df8c8": "f_2(x)=0", "cbea90a4857be295d6b79e4eb5869610": "n + 2 = 2 + (n + 3) - 3", "cbeaf9e822472ea18baf2c81323a84e2": "\\operatorname{LS}(K)", "cbeb4307ef1002f9cc133c94311bfcab": "O(n^{\\frac{19}{12}}\\log^{\\frac{3}{2}}n)", "cbeb569ff2791b425217665a5ec52d55": "\\limsup X := \\sup \\{ x \\in Y : x \\text{ is a limit point of } X \\}\\,", "cbeb96c63da65bfa9c7421e99dcfd8fc": "b-\\varepsilon", "cbebbc7c271fe110b5e95371a306aeb7": "T^2(n)", "cbebc619b38266b84631030f88f307c2": "\\frac{1}{\\zeta(2)}", "cbebf743eccb08f30da500927c7fdc37": "u_n\\,", "cbec0a70da0a6e1fc8b43a589da5155c": "\nX_{rms} = \\sqrt {{1 \\over {T}} {\\int_{t_0}^{t_0+T} {[f(t)]}^2\\, dt}}\n", "cbec1b32729d98391927716b49ad4023": "\\rho (\\mathcal M)= \\lim_{k \\to\n\\infty}\\max{\\{ \\|A_{i_1}\\cdots A_{i_k}\\|^{1/k}:A_i\\in\\mathcal M\\}}. \\, ", "cbecedc4c06ce641160fae6c7922af1f": "l(x)", "cbecf4275afd44dad4b312042088da7e": "condition_2", "cbed28370aeff74dc048ffc5097666b9": "\\tan 2\\theta\\ =\\ \\frac{B}{A\\ -\\ C}", "cbedeb414e5acae28a9e6d0a638a470d": "\\eta_\\varepsilon(x) = \\frac{1}{\\pi} \\frac{\\varepsilon}{\\varepsilon^2 + x^2}=\\int_{-\\infty}^{\\infty}\\mathrm{e}^{2\\pi\\mathrm{i} \\xi x-|\\varepsilon \\xi|}\\;d\\xi", "cbee64ff4d571b4de5af0397a4555da3": "|\\Sigma_{i \\in I}A_i|", "cbee73cbc2c32da9642cf5db1e3adf5f": "S = A + B", "cbeeb769a80d7934c8844a8715db0473": "z^3+3 \\overline{\\beta} z^2 \\overline{z} + 3 \\beta z \\overline{z}^2 + \\overline{z}^3", "cbef351929ef197d8ccd695c89ab2340": "\\pi_1(L)\\to\\pi_1(M^3)", "cbef82660d4b1636a5e83b818abae4fe": "\n\\begin{align}\n \\| \\mathbf{var}\\ x \\|_\\Gamma &= \\Gamma(x) \\\\\n \\| \\mathbf{lam}\\ (x, s) \\|_\\Gamma &= \n \\mathbf{LAM}\\ (\\lambda S.\\ \\| s \\|_{\\Gamma, x \\mapsto S}) \\\\\n \\| \\mathbf{app}\\ (s, t) \\|_\\Gamma &=\n S\\ (\\|t\\|_\\Gamma) \\text{ where } \\|s\\|_\\Gamma = \\mathbf{LAM}\\ S \\\\\n \\| \\mathbf{pair}\\ (s, t) \\|_\\Gamma &=\n \\mathbf{PAIR}\\ (\\|s\\|_\\Gamma, \\|t\\|_\\Gamma) \\\\\n \\| \\mathbf{fst}\\ s \\|_\\Gamma &=\n S \\text{ where } \\|s\\|_\\Gamma = \\mathbf{PAIR}\\ (S, T) \\\\\n \\| \\mathbf{snd}\\ t \\|_\\Gamma &=\n T \\text{ where } \\|t\\|_\\Gamma = \\mathbf{PAIR}\\ (S, T)\n\\end{align}\n", "cbefb6d824fb8b08983affc1c63054bc": " \\mathbf{x}^{(k+1)} = L_*^{-1} (\\mathbf{b} - U \\mathbf{x}^{(k)}). ", "cbf019cdac5e32411a0650b2b64a8c7e": "n = 2500\\, ", "cbf0583b7507108dfa5103bba6998d6d": "g_n < 2\\sqrt{p_n} + 1.", "cbf0bcb4574a670bcf4bccca557fa893": " K=1", "cbf0c02610eea1fa22b9279fb9bb6253": "\\overline{s} \\to S", "cbf0d871a5d1ecafabfffa2f1bb8762b": "\\begin{align}p(\\sigma) & \\propto \\sqrt{I(\\sigma)}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{d}{d\\sigma} \\log f(x|\\sigma) \\right)^2\\right]}\n= \\sqrt{\\operatorname{E}\\!\\left[ \\left( \\frac{(x - \\mu)^2-\\sigma^2}{\\sigma^3} \\right)^2 \\right]} \\\\\n& = \\sqrt{\\int_{-\\infty}^{+\\infty} f(x|\\sigma)\\left(\\frac{(x-\\mu)^2-\\sigma^2}{\\sigma^3}\\right)^2 dx}\n= \\sqrt{\\frac{2}{\\sigma^2}}\n\\propto \\frac{1}{\\sigma}.\n\\end{align}", "cbf0e40d5b146c09fcf0f41d30dc302d": "\\Delta r = \\frac {I_{sp}}{c} \\ln \\frac{m_0}{m_1}", "cbf141c11dc11fc8dbcea56f0918999d": "\\frac{1}{AB} = \\int_0^1 \\frac{du}{\\left[uA + (1-u)B\\right]^2}.", "cbf1a0d3e421722a744fde101e264ab4": "N=N_0\\left(\\frac{1}{2}\\right)^{t \\over T_{1/2} }", "cbf1ed83e1a5c3cceeff97d9a26612f0": "mm(x) = \\text{max}(0, \\text{min}(x, 2.375))", "cbf2147782826a68fe42beb525be34ee": "\\boldsymbol\\theta = \\boldsymbol\\theta^{(t)}", "cbf266d31557a711b4acaeeefeaf75df": "\\tfrac{n}{n-1},", "cbf275387c216965c6667ee5b842c946": " \\left| \\sum_{n=M+1}^{M+N} \\left( \\frac{n}{q} \\right) \\right| < \\sqrt q \\log q.", "cbf2ea4f77cd77aa1e9a8d078ccbb0da": "|O-Y|", "cbf2fedaffa4e2abc77dbc01a3ea2716": " \\left[S\\right]\\left\\{ \\theta\\right\\} +\\left[R\\right]\\frac{d}{dt}\\left\\{ \\theta\\right\\} =\\left\\{ B\\right\\} ", "cbf312bb1338eaa618ba7225d8e344ce": "{\\mathbb{R}/\\mathbb{Z}}", "cbf331347324804cb425006d706afb96": "\n\\ {R_0} \\times {S} = {1}\n", "cbf3557abe7a66781147a31b73a926a4": "\\sum_{i=1}^n \\chi^2(r_i) \\sim \\chi^2\\left(\\sum_{i=1}^n r_i\\right) \\qquad r_i=1,2,\\dots", "cbf3b007a36b0e42453e877ec15d476c": "h_\\mathrm{z}", "cbf41dcff03cb7d3904797bcd05c7335": "A\\otimes I", "cbf4a053893bac443c14a035ff593ade": "Q: S \\times A \\to \\mathbb{R}", "cbf4b9b4c3d25879b7e047bd6a785390": "\nV_{\\text{BE}\\_\\text{Q1}} = \\left(\\left[\\left(V_{\\text{BE}\\_\\text{Q1}} - V_\\text{CC}\\right) - V_\\text{CC}\\right] \\times e^{-\\frac{t}{RC}}\\right) + V_\\text{CC}\n", "cbf52b3dfdcc12b18df575a588862cb6": "G=G_U\\oplus \\left(R_{p_1}\\oplus R_{p_2}\\oplus\\ldots\\right)\\;", "cbf55dc1da89777cfdb24b06184a27ba": " c_{X} (x,y,z)=c(f(x),f(y),f(z)).", "cbf6202297cf4e65e355107cad537837": "\\eta^{\\mu\\nu}", "cbf637fbd01c3159650940430ce8c5a3": "c_{t+1} - c_{t} = - Y", "cbf66ca41949e70ba46947ff71eac9ea": "(4-3.5 (1-URR)) \\cdot \\frac {0.55 \\cdot UF}{V}", "cbf6a169b57fb7ed02e8bd94fe833f06": "\n \\hat{\\mathbf{b}}^i = \\cfrac{\\mathbf{b}^i}{\\sqrt{g^{ii}}}\n ", "cbf73b3a86a600d826c340c87914a796": "\\{\\varepsilon\\}", "cbf7813c1ed53709229b2810992aeae0": "E_h", "cbf78cfb89395a523c007fbf5bd9fccc": "- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N\\partial \\alpha^2}= \\operatorname{var}[\\ln (X)]= \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta) ={\\mathcal{I}}_{\\alpha, \\alpha}= \\operatorname{E}\\left [- \\frac{\\part^2\\ln \\mathcal{L}(\\alpha,\\beta|X)}{N\\partial \\alpha^2} \\right ] = \\ln \\,\\operatorname{var_{GX}} ", "cbf79d2a32e917f29b976b7f80bd5d72": "\\Delta(b) = 0 ", "cbf827f0bda25dc86bfc5814511db684": "\\rho_t", "cbf844a6e71302ca8a690b7a0421a60b": "\\mathfrak{P}^{95}", "cbf8b29fdd115ea8969ff19a06e03556": "(\\operatorname{arcsch}\\,x)' = -{1 \\over |x|\\sqrt{1 + x^2}}", "cbf8b4ff9c646aa35be413b5e2b96867": " n \\ge l+1", "cbf8f345867defc8705b9631ac743489": "y'=x \\sin \\theta + y \\cos \\theta.\\,", "cbf939f22efa2eff31cfa8675f7c3181": "\\sigma: G \\times X \\to X", "cbf979ae57644bae1b4524f859177f56": " D = X_1^3 ", "cbf991363022c1c6838016c732520c4b": "{\\Vert x \\Vert}^2 =\n x \\cdot w =\n\\frac{1}{(A_{u,v})^2} \\sum_{i\\kappa^+", "cbff2246e1719004c7cb6c8becc9d12e": "x\\in k\\left[M\\right]", "cbff3ba11fbe5f12e97365f9ba7d793f": "\\max E( \\sum_0 ^{\\infty} \\beta^t u (c_t) ).", "cbff46ddc2ca84d41e2f7601df65a735": "{hc\\over\\lambda kT }{e^{h c/\\lambda kT}\\over e^{h c/\\lambda kT} -1} - 5 = 0. ", "cbff77b17ee9c608ecb4df64603f0225": "\n \\begin{align} A & = \\frac{15}{4}a^2 \\cot \\frac{\\pi}{15} \\\\\n & = \\frac{15a^2}{8} \\left( \\sqrt{3}+\\sqrt{15}+\n \\sqrt{2}\\sqrt{5+\\sqrt{5}} \n \\right) \\\\\n & \\simeq 17.6424\\,a^2.\n \\end{align}", "cbff91bb5a72a1951abb15e89ab6b1d4": "\n\\begin{bmatrix}\nA_{11} & \\cdots & A_{1k} \\\\\n\\vdots & \\ddots & \\vdots \\\\\nA_{k1} & \\cdots & A_{kk}\n\\end{bmatrix},\n", "cbff9f95dd5f2be3e664ee7ad258c759": "\nF(x)=1-\\boldsymbol{\\alpha}\\exp({S}x)\\mathbf{1},\n", "cbffc43d832c571c1e480156f55d6d40": "dF = 0", "cbffc546099016c803b34d9644341bcc": "\\mu=19", "cbffc892b1542f81cc46a4e61c4969f4": "\\widehat K(P)(B)=\\int_XK(x)(B)\\,\\mathrm dP(x)", "cbffceb442aaae70d59907c98a98e630": "\\partial p / \\partial s =0 ", "cbffd0e35dc370b8c78138beedfd08b6": "s = \\infty", "cc0071da12c0c0ad1bf75f4e071ca069": "(\\alpha=\\pi/2)", "cc00d68774aacc916fa1c5fe2580aa6b": "{dx \\over dt} = {dx \\over d \\tau} \\Big/ {dt \\over d \\tau} = {\\lambda p/m \\over \\lambda} = {p \\over m}", "cc00e28ba5471db931e088e290f46675": "P(M|E)=1", "cc0119b5c8ba742e155658cabc99035d": "h(x)=x \\bmod{2^{L'}}", "cc012127ac5f584f9cff03d74f4b10dc": "\\textstyle F_2(x,Q^2)", "cc0129c50924373c7f9dff4e43aea9f4": "\\psi(\\psi_1(0))", "cc017867a616dd01b2349c294abdb5dd": "(Y)", "cc01998c2809e90c81653587e37c2a9e": "78^2", "cc01ac7a1487c8ce443fb200cc0790e1": "Q_{i,j}^{(s)}", "cc01b7b3b4c483a533589018eaeef2e1": "I_x = I_y = \\frac{3}{5}m\\left(\\frac{r^2}{4}+h^2\\right) \\,\\!", "cc02178d82b528c0cd2d82ab056df52a": " z^{nad} ", "cc0286431f0833b3614f57e192573804": "\\cos(c)", "cc02b1eac663b32df6601ae356901854": "L\\supseteq F", "cc02b32c47f3cf1e493ca5f640bdabbc": "\\Psi_A", "cc02dacf9ac6469ecfe4c670433f885b": " \\mathbf{a}. ", "cc02f5eaa9ef857adbe4f7d18acf6437": "\\left(\\boldsymbol\\Sigma_0^{-1} + n\\boldsymbol\\Sigma^{-1}\\right)^{-1}", "cc02ffc1715134667bda94eac6eac168": " (\\varphi^*\\alpha)_x(X_1,\\ldots, X_k) = \\alpha_{\\varphi(x)}(\\mathrm d\\varphi_x(X_1),\\ldots, \\mathrm d\\varphi_x(X_k))", "cc0312f116b36a52172333835355d026": "\\bar{x}_i", "cc0321661ae97f1f0bba5583c72683e4": "m, nym", "cc03364055c4dc4c36705612d095f41e": " E\\left[ \\frac{d\\ln L}{db} \\right] = 0\\text{ at }b=b_0, \\,", "cc03be370b5cff3fdf84446ff1c70920": "\\lim_{q \\to 0} se_r(\\omega,q)= \\sin {r \\omega}", "cc0412e71271b038a1d4726925684841": "0 = \\sum_{j=1}^m \\frac {x_j - y} {\\left \\| x_j - y \\right \\|}.", "cc044f4317f590a312bce2dc6b002d71": "w_C = \\tfrac{PL^3}{3EI}", "cc047db69359c9cc879febae22fadca1": "\\begin{align}\n\\text{ Velocity function:} \\, v ({\\mathbf r} , t)\n& =\n\\frac{1}{\\sqrt{\\epsilon ({\\mathbf r} , t) \\mu ({\\mathbf r} , t)}} \\\\\n\\text{Resistance function:} \\, h ({\\mathbf r} , t)\n& = \n\\sqrt{\\frac{\\mu ({\\mathbf r} , t)}{\\epsilon ({\\mathbf r} , t)}}\\,.\n\\end{align}", "cc048bbfe4c1918e90ec5fb66936e9ba": "AA^* = I_m\\,\\!", "cc04b307dad63297a5b3228aa008eed3": "\\oint_S (\\mathbf{E}_1 \\times \\mathbf{H}_2) \\cdot \\mathbf{dA} = \\oint_S (\\mathbf{E}_2 \\times \\mathbf{H}_1) \\cdot \\mathbf{dA}. ", "cc04e4288916c6ace57e85c9db6059d7": "\\Gamma(s) = \\lim_{|z| \\rightarrow \\infty} \\gamma(s, z), \\quad |\\arg z| < \\pi/2 - \\epsilon", "cc04eec21153924afcd5c1a042e8519b": "+\\ 8.1328\\times10^{-3}(10^{-3.49149(T_\\mathrm{st}/T-1)}-1)\\ +\\ \\log\\ e^*_\\mathrm{st}", "cc05495b9dfc0b734e9b7efd1e6b6889": "I_{sp} = \\frac {v_e}{\\sqrt{1 - \\frac{v_e^2}{c^2}}} = \\gamma_e \\ v_e,", "cc05684da93cfadbd644f0f71d087c49": " \\frac{d \\vec S_m }{dt} = \\sum_{n=1}^N { f(m|n) \\frac{\\partial{\\ln l(n|m)} }{\\partial{\\vec M_m} } \\frac{\\partial{\\vec M_m}}{\\partial{\\vec S_m}} } ", "cc05b0e3116f2a3220803046ce1e99e2": "\\epsilon^*", "cc0601e5b877f804eb4c4958e0f6efb0": "\\frac{d}{dt} \\langle X\\rangle = \\frac{d}{dt} \\langle \\psi|X|\\psi \\rangle = \\frac{1}{m} \\langle \\psi|P|\\psi \\rangle = \\frac{1}{m} \\langle P \\rangle ", "cc0607349afb11bc02efa840b6e25be4": "0_{K_{m,n}} \\, ", "cc06bf9e271cb74cbbd556dab2096545": "q \\rightarrow 1", "cc074ca501aeebc984a6e421796d2383": "W\\left(D\\right) = W(D)_A + W(D)_R \\, ", "cc0755f13c23a0690906154fd616797c": "\n\\begin{align}\n& E\\{y\\} = 0, \\\\\n& C_Y = E\\{yy^T\\} = \\sigma_X^2 11^T + \\sigma_Z^2I, \\\\\n& C_{XY} = E\\{xy^T\\} = \\sigma_X^2 1^T.\n\\end{align}\n", "cc07803335364b21f42da40376dc1a52": " f_{IVIM} ", "cc078b1cba3de4a53a1bfed5af3dbbb3": "L_{uu}<0", "cc07998b8ee259010e6f589afbe9853c": "\\Delta(x_2*d)", "cc07fd2938ce19e509b70ddd99d88bb8": "S=n(s\\otimes s)^T + s[(s\\otimes n)^T+(n\\otimes s)^T+(n\\otimes n)^T]", "cc08181cc6188dac38062ec6b9ccd6dd": "f(x) = ax^k", "cc0897a4ef0a1ba9c6d53ab54190e97d": "w=w_1i_1+ \\cdots + w_ri_r", "cc08dbf24d8c2bb5aeeb067128c69920": "d^n\\colon A^n \\to A^{n + 1}", "cc08f0851b767eb531837cc860fbcfbe": " V _o =- A_v V _i = A_v \\frac { V _A} {1+j \\omega C_M R_A}, ", "cc09129e218c8a99bb3ad34db62b6147": "t\\rightarrow\\lambda t,", "cc092a29777a6be2329b74632da47847": "\\sqrt{1+\\left(dy/dx\\right)^2} = \\sqrt{1+\\left(\\frac{d}{dx}\\left(a \\cosh\\left(\\frac{x}{a}\\right)\\right)\\right)^2} = \\sqrt{1+\\sinh^2\\left(\\frac{x}{a}\\right)} = \\cosh\\left(\\frac{x}{a}\\right)", "cc0952d002442879ab8f7943efd50239": "g(y)", "cc0968c3fddb9491c47aa96921911442": "R_1R_2 + R_1R_3 + R_2R_3 = \\frac{R_aR_bR_c^2 + R_aR_b^2R_c + R_a^2R_bR_c}{R_T^2}", "cc097ce2511e106826d06f122a5171c4": "P_v(v)dv = \\sqrt{\\frac{m}{2\\pi kT}}\\,\\exp\\left(-\\frac{mv^2}{2kT}\\right)dv", "cc09b9121688b3613496e8ae8a85b62b": "\\forall x_0, (P \\Rightarrow wp(S,Q[x \\leftarrow x_0][y \\leftarrow x]))[x \\leftarrow x_0]", "cc09d021140643700dd7aaa358c6474c": "y^n-p(x)=0.", "cc0a2484ee49b0dd65d9d26099446352": "X_q=\\sum_i X^i(q) \\frac{\\partial}{\\partial q^i}", "cc0a2ccd856c4f10b277a53babc3b922": "R^\\rho{}_{\\sigma\\mu\\nu} = \\Gamma^\\rho{}_{\\nu\\sigma,\\mu}\n - \\Gamma^\\rho_{\\mu\\sigma,\\nu}\n + \\Gamma^\\rho{}_{\\mu\\lambda}\\Gamma^\\lambda{}_{\\nu\\sigma}\n - \\Gamma^\\rho{}_{\\nu\\lambda}\\Gamma^\\lambda{}_{\\mu\\sigma} \\,,", "cc0a3b22b2da7b7f24c8ee6c99bd330b": "\\dot{G}\\,\\!", "cc0a610ed9f8cdbd1c20b8e483d9faea": "x \\, = \\, \\frac {\\theta_A}{1- \\theta_A} \\, = \\, \\zeta_{L} \\frac{N^{3D}}{\\zeta^{3D}} \\, = \\,\n\\zeta_L \\left ( \\frac {h^2}{2 \\pi mk_BT} \\right)^{3/2} \\frac{P}{k_BT} \\, = \\, \\frac{P}{P_0}", "cc0a69469c9a75d2c83cb63be34aec3d": "M = Qreg/Q = R/(R-Rneg)", "cc0a7768b55ce098748398412c1e81ff": "\\mathbf{x}=x_i\\mathbf e_i\\,\\!", "cc0b4918745c5c696978e26b9955f817": " \\beta \\left(\\left[X,Y\\right],Z\\right)+\\beta \\left(\\left[Z,X\\right],Y\\right)+\\beta \\left(\\left[Y,Z\\right],X\\right)=0 ", "cc0b941261dcadbbfd464be762520345": "u_i^{\\overline{n+1}}", "cc0bc521aad090ac39b628da685aa890": "(x_i,y_i)\\mbox{ and } (x'_i,y'_i)", "cc0bc7c9bd51b4174f2fcdda30abaec8": "\\frac{(ax;q)_\\infty}{(x;q)_\\infty} = \\sum_{n=0}^\\infty \\frac{(a;q)_n}{(q;q)_n} x^n.", "cc0bcc40e22d1b2ab44a7bc3ebc79ab1": "\\lambda_{\\mathrm{max}} = \\frac{b}{T} ", "cc0bead6c191d4c8ea0ebb89314a7be6": "p \\in C_{i-1}", "cc0c0d6a8c5b4d0317e96fe938a8c7c9": "\n\\begin{align}\n\\Pr(\\mathrm{H} = 49 \\mid p=1/3) & = \\binom{80}{49}(1/3)^{49}(1-1/3)^{31} \\approx 0.000, \\\\[6pt]\n\\Pr(\\mathrm{H} = 49 \\mid p=1/2) & = \\binom{80}{49}(1/2)^{49}(1-1/2)^{31} \\approx 0.012, \\\\[6pt]\n\\Pr(\\mathrm{H} = 49 \\mid p=2/3) & = \\binom{80}{49}(2/3)^{49}(1-2/3)^{31} \\approx 0.054.\n\\end{align}\n", "cc0c29afaff93ac01523b55703f40f7b": "\\frac{m^2K}{W}", "cc0c4d3e268bba503eb44c257d331fef": "\\hat{P}^2\\left|x_1, x_2\\right\\rangle = \\hat{P}\\left|x_2, x_1\\right\\rangle = \\left|x_1, x_2\\right\\rangle", "cc0c50dcc66914d0dfdf2341944c26c2": "[B,H] = BH - HB = 0", "cc0cd770f57a52e59eabf732c44de3bf": "\\begin{align}\n{d \\over dx}x^{n+1} &{}= {d \\over dx}\\left( x^n\\cdot x\\right) \\\\[12pt]\n&{}= x{d \\over dx} x^n + x^n{d \\over dx}x \\qquad\\mbox{(the product rule is used here)} \\\\[12pt]\n&{}= x\\left(nx^{n-1}\\right) + x^n\\cdot 1\\qquad\\mbox{(the induction hypothesis is used here)} \\\\[12pt]\n&{}= (n + 1)x^n.\n\\end{align} ", "cc0ce634c21a5c8af2093e893290035e": "\\alpha^{F} : [0, T] \\times C_{0} \\to \\mathbb{R}", "cc0d156026c0003e2d91f507523cdb80": "R[e_1, \\ldots, e_k]", "cc0d88358b45ed6c438ecab40aba285d": "{\\color{Blue}~5.5}", "cc0db37d406f3af27b859ebfd7b06a61": "R_{i,j}(u,v) = \\frac {N_{i,n}(u) N_{j,m}(v) w_{i,j}} {\\sum_{p=1}^k \\sum_{q=1}^l N_{p,n}(u) N_{q,m}(v) w_{p,q}}", "cc0dc1c91e85a1aaa141a5514b519c5e": "N\\ge 0", "cc0e4e94e07534a431985b461812c7aa": "\\bold g = g_{\\mu\\nu}dx^\\mu dx^\\nu~~~~~~~~~~~\\text{where}~g_{\\mu\\nu} = \\bold g(\\partial_\\mu,\\partial_\\nu) ", "cc0e78539bed2fd69ac3d9034af31024": "a^2 - 2ab + b^2 = x\\,\\!", "cc0e7bfcd7819ebe7f48062f1985dd35": "\\det\\begin{pmatrix}A& B\\\\ B& A\\end{pmatrix} = \\det(A-B) \\det(A+B).", "cc0e7cd44971b17774cf4855aafd5bfb": "\\mu X \\bullet \\mathbf{F}(X) \\equiv \\sqcap \\left\\{ X \\mid \\mathbf{F}(X) \\sqsubseteq X \\right\\}", "cc0e8f90f7b7de9d7e90aeb433850a94": "(p+q)^2", "cc0e9df3fdc6c3f9c84a5df33c2dd4ed": "Q = \n\\begin{bmatrix}\n\\sigma_x^2 & \\sigma_{xy} & \\sigma_{xz} & \\sigma_{xt} \\\\\n\\sigma_{xy} & \\sigma_{y}^2 & \\sigma_{yz} & \\sigma_{yt} \\\\\n\\sigma_{xz} & \\sigma_{yz} & \\sigma_{z}^2 & \\sigma_{zt} \\\\\n\\sigma_{xt} & \\sigma_{yt} & \\sigma_{zt} & \\sigma_{t}^2\n\\end{bmatrix}\n", "cc0ecabc6aa5930080af2dc16b903c38": "(p^m)^d=p^{md}", "cc0ed02002430a417c9e43a58ed8f21a": "\\gamma_{mol} = \\gamma V^{2/3}\\,", "cc0f20c4c59bfdb6b12b33973045aecb": "\n\\begin{align}\nH(z) & = \\frac{Y(z)}{X(z)} \\\\\n & = \\frac{\\sum_{i=0}^P b_{i} z^{-i}}{\\sum_{j=0}^Q a_{j} z^{-j}}\n\\end{align}\n", "cc0f5c35423f7b9f53856f2619a0d255": " 2 p^2 \\ge \\Gamma(S )^2 / N ", "cc0fc8e64fbed33c2017069883e9c436": "\n\\left( F(x) \\right)^m \\le \\sum_{\\ell=m \\cdot \\min}^{m \\cdot \\max} r^\\ell \\, x^{-\\ell} \\; .\n", "cc100b6e8383b161452405bbc8b53516": "E = pc", "cc1037b90c429d00c8290457381ba315": "\nP = \\left(\n \\frac{2\\left(\\frac{m}{n}\\right)}{\\left(\\frac{m}{n}\\right)^2+1},\n \\frac{\\left(\\frac{m}{n}\\right)^2-1}{\\left(\\frac{m}{n}\\right)^2+1}\n\\right) =\n\\left(\n \\frac{2mn}{m^2+n^2},\n \\frac{m^2-n^2}{m^2+n^2}\n\\right).", "cc10ced5d347b2c1c250c7746a3eea1f": "m \\mapsto 1 \\otimes m", "cc10d9a91bbb85cd5c3ed5cf61a02f80": "x \\in \\{0, 1\\}^k", "cc10f17de214247622bd714d64a332da": "\\sum w_i \\, d^2(X,P_i) = C.", "cc10f3a564dd05f92d363825862bfac9": " W_i\\,,", "cc10f8b4c653c97a94961ce64659dbfb": "g(t|\\theta)=g(t+\\theta)-g(\\theta).", "cc1138b8de1fb9bfde78f9316e2f0ac1": "\\mathbf x^*", "cc115642a92416bf572eef3734578bad": "\\Gamma^a_{bc}", "cc1167aed0957e897d845bb0d9371f19": "v^\\gamma", "cc11ce0a3eb102229aba2797afac056e": "\\lambda_6 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix}", "cc123f2aeb40748704e67be4d379e107": "f = f(0)\\cdot p_0 + f'(0)\\cdot p_1 + \\frac{f''(0)}{2!}\\cdot p_2 + \\frac{f'''(0)}{3!}\\cdot p_3 + \\dots ", "cc126600db5a9e59465d21103eb72824": "\\textstyle \\frac{dr}{dx} = 0", "cc12a5c4cd7e4cca4fbb0a3cdf9ee8a4": "m = \\prod p_i^{\\alpha_i},", "cc12aa312c27e418ddf3cb4f24995a99": "\\! \\gamma_m", "cc12fb09601a30503b3b37363796b149": "(x^2+y^2)^2+4b(b-a)(x^2+y^2)=4b^2x^2", "cc13055a9429fec144cf2e6cf30149a4": "p_e=m_pV_0 + m_g \\alpha V_0\\,", "cc1325455cbc9c4c4bc4c792e3febe3f": "Q = \\{ p,q,r \\}", "cc1327b8c04e6f8b7f393a12cfd711d4": "\\scriptstyle e^x \\,>\\, 1 \\,+\\, x", "cc134c0cfb7e01819aae73ad43a264fa": "\\scriptstyle I(t)", "cc13c30184c5333f63dd9b99c2e3311d": "4x^2+20x+3yx+15y \\,", "cc13f73704bf72ecbd3989083bdde5f5": "(n-2) \\in (n-1)", "cc141282572fb2f6f9602bbd25804b96": "S^2 \\to \\mathbf{RP}^2", "cc1526c55a5f1b890bf52e2dd9a72a82": "U_{SF} = E_F-qV_{SC}", "cc15ebecad9da56413b4191396ac430f": "e^{-\\frac{1}{\\hbar}\\int_a^b\\sqrt{2m(V(x)-E)} \\, dx},", "cc16c4f2922957354d08bb546a63e745": "f(x,y) = f(1,1)xy + f(1,0)x(1-y) +f(0,1)(1-x)y + f(0,0)(1-x)(1-y)", "cc16f35d36d96782c416dd27af8b389a": "(1-O(\\frac{1}{\\sqrt{b}}))\\cdot|MDS(C)|", "cc16f6f3b6b9673ee98ade542dd09ab5": "\\neg(P \\and \\neg Q)", "cc16fd0bfb7c36a486eb58d41707865f": "|x_1 \\bar{y}_1 + \\cdots + x_n \\bar{y}_n|^2 \\leq (|x_1|^2 + \\cdots + |x_n|^2) (|y_1|^2 + \\cdots + |y_n|^2).", "cc170ca0999bd73e113cb0d329836067": "\nc = \\frac {\\sum_{i 0", "cc1ae0d2f00c0974c53dea4cfada8f10": " \n\\mathrm{d} \\left ( \\sum_i p_i {\\dot q_i} - \\mathcal{L} \\right ) = \\sum_i \\left( - \\frac{\\partial \\mathcal{L}}{\\partial q_i} \\mathrm{d} q_i + {\\dot q_i} \\mathrm{d}p_i \\right) - \\frac{\\partial \\mathcal{L}}{\\partial t}\\mathrm{d}t\n\\,.", "cc1b1e11262297906ddc5b36ee68136c": "\\pm z_{0}", "cc1b539b3bdaa32476bc507417d2056e": "\\theta_C = \\frac{5.785}{EI} ", "cc1b70bcacacb9bb261435cf2a747a8d": "k_1,k_2\\,", "cc1b7cf0ac98698d0c364646501aa0bc": "V(u)= \\pm |x|^{-n} * |u|^2", "cc1b8a5b5e00f283ffe9fb4a2d1b784e": "W = \\frac{1}{2}a_{ijkl}E_{ij}E_{kl}", "cc1bdf4d2d6814c1d191e72b894d84e6": "\\Delta = 1- \\left (\\frac{n_A}{n_B} \\right)^2 \\left (1- \\left (\\mathbf{i} \\cdot \\mathbf{n} \\right )^2\\right)", "cc1c6ea2e489940e2cafa36755b61be8": " H(s) = H_{\\infty}(s) \\frac{1 + \\frac{Z_n(s)}{Z(s)}}{1 + \\frac{Z_d(s)}{Z(s)}} ", "cc1c9cacbc58af6c806e192f69045cab": " \\mathbf{X}\\mathbf{u} = \\mathbf{0} \\Rightarrow \\begin{pmatrix}\n-\\mathbf{B} \\\\\n \\mathbf{I}_{n-r} \n\\end{pmatrix}\\mathbf{u} = \\mathbf{0} \\Rightarrow \\begin{pmatrix}\n-\\mathbf{B}\\mathbf{u} \\\\\n \\mathbf{u} \n\\end{pmatrix} = \\begin{pmatrix}\n\\mathbf{0} \\\\\n \\mathbf{0}\n\\end{pmatrix} \\Rightarrow \\mathbf{u} = \\mathbf{0}\\; .", "cc1cc7beff28f21285872a1b64cf38e7": "V(x)=\\frac{1}{2}m\\omega^2x^2", "cc1d080d09271f3d509779590be7847e": "\\mathrm{Proof}^R_T(x,y) \\equiv \\mathrm{Proof}_T(x,y) \\land \\lnot \\exists z \\leq x [ \\mathrm{Proof}_T(z,\\mathrm{neg}(y))],", "cc1d10cc87d06258e6ff70a0531f4d7a": " \\mu_B = \\mu_{B}^{\\ominus} + RT \\ln x_B \\,", "cc1d17c95136c30b4e711f59a1dd2c30": "Re > Re_c = 5772.22", "cc1d29847568ca57dcde6d456d105898": "M\\left[\\phi\\right]\\,\\mathcal{D}\\phi", "cc1d6b7bd113bbd72e08c5c79de7af46": " \\mathcal{L}(A_{\\alpha},\\partial_{\\beta}A_{\\alpha})\\,", "cc1da7d450825f3189700628394f2bd2": "e^{-\\frac{\\Omega}{k T}} = \\operatorname{Tr} \\exp\\big(\\tfrac{1}{kT}(\\mu_1 \\hat N_1 + \\ldots + \\mu_s \\hat N_s - \\hat H)\\big).", "cc1db3c6d2b166f5637d8e0572002bec": " K_z * K_{z'} = K_{z+z'} \\,", "cc1e3dd12ff8a401e94d0fe4f6209893": "i,j=1,\\dots,d", "cc1e3e9a19f03a4ca9df2cf65d6904f3": "B_n=B_n(0)", "cc1e7241919f3e0769385fd1eb95beb5": "\\min E[Y - (\\alpha + \\sum_{j=1}^p f_j(X_j))]^2", "cc1e78d2534be49af290e669f94ae0a9": "ax+b \\equiv 0 \\pmod N", "cc1e86d69def6d33a0f694c27e86cc25": "\n{d\\over dt} K(x;T) = {\\rm i} {\\nabla^2 \\over 2} K\n\\,", "cc1e976fcede0463f584600d18577656": "Y_{0}^{0}(\\theta,\\varphi)={1\\over 2}\\sqrt{1\\over \\pi}", "cc1ed6eca488605c5092859ac3963256": "R \\oplus S", "cc1f1febd1a2da9d941b4e44be076a51": "d x^3 \\, d x^4 \\, d x^1", "cc1f291a06ef4f5ee2d6b2bf26bfcff4": "\\mathbb{E}[\\tau]<\\infty", "cc1f3cb3096cde779ae9292065fe4339": "\\mathcal{H}_{2}\\Psi =0", "cc1f848a765330b102f6aceec210c92b": "m/z", "cc1fa329c1d31949ee47acafa833d4d4": "\\int [af(\\theta)+bg(\\theta)] \\, d\\theta = a\\int f(\\theta) \\, d\\theta + b\\int g(\\theta) \\, d\\theta", "cc1fb003c693e425886c1e23cd2903a8": "\\beta_k( f_{ij}(z) ) = f_{ij}(\\Zeta_2).\\;", "cc1fb907464d47761decf281939b4f1e": "\np(t) = T^t p(0)\n", "cc1ffdc3051badf8200f224f22d939d2": " P_3 = g(L_1 \\times L_2). ", "cc2031fce82ee2c482efb8c0a3a394f4": "\\delta p\\ ", "cc2032a472d4b1658912e4017b8d4d8e": "\\|x-a\\|^2=\\|z-x\\|^2+\\|a-z\\|^2+2\\langle z-x, a-z \\rangle=\\|z-x\\|^2+\\|a-z\\|^2", "cc206526ee4992139c7429ea4a7ff6e7": "k=\\sec45^{\\circ}=\\sqrt{2}=1.414", "cc20a45257c918b792ed3eee62ddc20d": " P \\mapsto \\{ (a, p) \\in \\Lambda \\times S : \\exists q . (a, p, q) \\in P \\}", "cc20ca967e0f8e4bd8a65be540fa8929": " \\and (S_1 \\implies (\\operatorname{equate}[A_1, g\\ q\\ p\\ n] \\and V[F_1] = g\\ q\\ p\\ n)) \\and D[F_1] = K_1) ", "cc20ee538fac9a251dea7c50f3784d8a": "Y\\left(\\begin{bmatrix} p \\\\ q \\end{bmatrix},z\\right) = \\begin{bmatrix} p-qz & -q \\\\ q & p-qz \\end{bmatrix} = \\begin{bmatrix} p & -q \\\\ q & p \\end{bmatrix} - \\begin{bmatrix} q & 0 \\\\ 0 & q \\end{bmatrix} z", "cc214ee7c3ace34a6bce4448f6b58d6a": "X\\to \\epsilon", "cc21bc9d2a9dd5fb1f08c284ce9718df": "\\mathcal{E}^{(k)}", "cc21e58475f5151e6b16af07904808f4": "A_{U}", "cc222de713880a365278bf6d63560deb": " DR_{S}^{T}", "cc232b23b57f037b2f22b7b68b57f7d7": "\\textstyle \\mu^{(n)}(x_1,\\dots,x_n)", "cc238b674b835e94602bbdceca8bb0b4": "{\\psi} (\\tfrac14) ", "cc23c9d8072a728dd48b928acdf8d7a1": "L^{1}(\\Omega)", "cc243ccd97217fc57e6534997f360fa8": "\\sqrt{5}, 1/2,", "cc24a1175a2cee36cd8357294c31d6e2": "au+bv=g,", "cc24f6ee574e531bc30a0085865fb631": "\n\\delta_\\mathcal{D} = \\dfrac{\\partial}{\\partial t} + ((a-bx)x)\\dfrac{\\partial}{\\partial x}\\cdot\n", "cc2528c6e978d0432683083b7a990a29": " H = - \\frac{ \\mu_0 }{ 4 \\pi } \\frac{ \\gamma_j \\gamma_k \\hbar^2}{ r_{jk}^3 } \\left( 3 (\\bold{I}_j \\cdot \\bold{e}_{jk}) (\\bold{I}_k \\cdot \\bold{e}_{jk}) - \\bold{I}_j \\cdot \\bold{I}_k \\right) ", "cc252f1404e5ca7bc198eb06983e68ee": "R_\\text{x} = V_\\text{t}/V_\\text{x}", "cc257c91353cba76984eca72940c0f0f": "f(k,i) = \\boldsymbol\\beta_k \\cdot \\mathbf{x}_i,", "cc25847a55ecf6bf86ce6788ca7841d1": "U{}^{n-1}_n", "cc25df19e6ebd7bdbc235ca54a25896d": "\\mathrm{ot}(X^\\alpha_{n+1}) = \\sum_\\gamma \\mathrm{ot}(X^{\\beta_{\\gamma+1}}_n\\setminus\\beta_\\gamma) \\leq \\sum_\\gamma \\kappa^n = \\kappa^n \\cdot \\mathrm{cf}(\\alpha) \\leq \\kappa^n\\cdot\\kappa = \\kappa^{n+1}", "cc25f034ae5ff29f23e32c51729e8739": "\\scriptstyle \\text{H}_0, \\text{H}_1, \\text{H}_2, \\cdots", "cc2643080c73e54c95233e073ef29419": "\nL u + N u = 0\n", "cc26d6a82702e2cb518e716a6a08677d": "\\vec{r} ", "cc2745ebbc4df7974f23e55800aabf5c": "H_{\\mathrm{dR}}^{k}(\\mathbf{R}^n \\setminus \\{0\\})", "cc2777bc60604a0a554d4b71e90eeb76": "\n\\Delta z\\,\\, \\approx \\,\\,\n\\begin{pmatrix}\n {\\partial z \\over \\partial x_1} & {\\partial z \\over \\partial x_2} & {\\partial z \\over \\partial x_3} & \\cdots & {\\partial z \\over \\partial x_p} \\end{pmatrix}\n\\begin{pmatrix}\n {\\Delta x_1 } \\\\\n {\\Delta x_2} \\\\\n {\\Delta x_3} \\\\\n {\\vdots} \\\\\n {\\Delta x_p}\n\\end{pmatrix}\n{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(8)}}", "cc27ded8637322c14b97b072cb389f64": "\nn = n_0 e^{e\\phi/T_e}. \\,\n", "cc289d954427d28516734f6472543f42": "\\lim_{n\\to\\infty}\\frac{d(1) + d(2)+ \\cdots +d(n)}{\\log(1) + \\log(2)+ \\cdots +\\log(n)} = 1.\n", "cc28aacd542a559f3d32f7c3a60be3e9": "\\lambda=1/\\omega", "cc28af583406850d2acd4a305889c875": "\\mathbf{w}_n(p_n(t))", "cc2986419f1154d18d3037d537e52824": "\nA = \\iint_S \\,dS\n= \\iint_T \\left\\|{\\partial \\mathbf{r} \\over \\partial x}\\times {\\partial \\mathbf{r} \\over \\partial y}\\right\\| dx\\, dy\n", "cc29e3358347b77bb3f65cda890bc227": "dR=\\lim_{\\delta x \\to 0}(R\\delta x)=Rdx", "cc29e84152f151181d4813b7ea8adb81": "m=4", "cc29ec30e7139a9ba0433badbaa2b991": "\n\\begin{align}\n& {} \\quad 1 + \\cfrac{e^{-2\\pi}}{1+ \\cfrac{e^{-4\\pi}}{1 + \\cfrac{e^{-6\\pi}}{1+\\dots}}} = \\frac{1}{2}\\left[1+\\sqrt{5}+\\sqrt{2(5+\\sqrt{5})}\\right]\\,e^{-2\\pi/5} \\\\ \\\\\n& = \\frac{e^{-2\\pi/5}}{\\sqrt{\\varphi\\sqrt{5}} - \\varphi } = 1.0018674\\dots\n\\end{align}\n", "cc2a4deb1c5aac26552824611de63fe1": "B^2 - 4AC < 0 ", "cc2a512a45b3fc3e09aed092afcb77c0": "\\sum_{\\nu = 0}^n b_{\\nu, n}(x) = \\sum_{\\nu = 0}^n {n \\choose \\nu} x^\\nu \\left( 1 - x \\right)^{n - \\nu} = \\left(x + \\left( 1 - x \\right) \\right)^n = 1.", "cc2a53e3257ada38f652651bb91cc4c2": "- \\frac{\\hbar^2}{2m_e} \\nabla^2 \\psi(\\mathbf{r}, t) + V(\\mathbf{r}) \\psi(\\mathbf{r}, t) = i \\hbar \\frac{\\partial \\psi}{\\partial t} (\\mathbf{r}, t)", "cc2a586e1a78abc9df539023abf96cfe": "C_2 = G_1 + P_1 \\cdot C_1", "cc2a73c879eb062727a7ee35a69b6bf1": "\\sum _x x^a = \\frac{(-1)^{a-1}\\psi^{(-a-1)}(x)}{\\Gamma(-a)}+ C,\\,a\\in\\mathbb{Z}^-", "cc2a8613d2fc6fe9c1918e15d284bb8d": "Q_{n}", "cc2a91bd5ffe34f49adeaf96ebd0f4f5": " [z^{2n}][u] g(z, u) =\n[z^{2n}][u] \\frac{1}{1-z}\n\\left( 1 + (u-1) \\left( \\frac{z^{n+1}}{n+1} + \\frac{z^{n+2}}{n+2} + \\cdots \\right) \\right),", "cc2b4003d97c8a42490209efc3eae818": "W(f)", "cc2b62fa569b8d601c8520c33da2cddb": "\\mathit{d_H}(\\mathcal(O)) \\geq \\mathit{d_{min}}", "cc2bfb5abbba547f90289181d698fac1": "\\mathcal{X} \\subset \\mathbb{R}^n", "cc2c204bb7fbf4fe57d4ff4c22d79ee3": "f({x})={c}^T{x}+\\frac{1}{2}{x}^T{Qx}\\,", "cc2c46aa7fa504bfbc2c2bbcbadaede3": "P(P_3, t)= t(t-1)^2, \\qquad P(P_3, 3)=12.", "cc2c780faadbb432bdf2c64c185d2fc4": "\\theta\\in \\mathbb{R}\\backslash\\{0\\} ", "cc2cb403add35ef79fa142fa7505969f": "|\\{Mf > t\\}| \\le 5^d \\sum_j |B_j| \\le {5^d \\over t} \\int |f|dy.", "cc2cbaf31a63fd43fb0c6f7f4555ec55": "\\left(\\tfrac a{nm}\\right)=\\left(\\tfrac an\\right)\\left(\\tfrac am\\right)", "cc2ccb19699767f2e043717d63aceb54": "g_j\\,\\neq\\,e\\,\\forall\\,j,", "cc2e0c67185a3bd25b40039953b2dd03": "y(t-T)", "cc2e0eda3ad753b0d75112729b8c9405": "\nd^* = {\\arg\\max_d} {\\int U(d,x) f(x) \\, dx}\n", "cc2e4807d1e1c7690ad9e7f1bf313c17": " (\\widehat{\\mathit{G}}) ", "cc2e59acd808473f09fbf68ca3f91457": " E = U - T_R S + p_R V - \\sum \\mu_{iR} N_i ", "cc2e94eeeb465a784af7e07eb39cb4bb": "\n G_{\\rm Ic} = \\cfrac{1}{E}~K_{\\rm Ic}^2 \n ", "cc2ea140fbc2c5cc0452926b1071e780": "\\scriptstyle \\bar{x}", "cc2eb383765687ea1ea8f2e3a2dff4d9": "\\omega_N^n = e^{-\\frac{2\\pi i}{N} n }", "cc2eff801dc677aa8a79b6b8707548d7": "\\Phi_{ij}=0", "cc2f22871ebd5e7a6168f16ee673bec9": " { 1 \\over s+\\alpha } ", "cc2f58b816d0ac1f3db0a03b8bcae71e": "\\hat\\Delta_u=S^2_u", "cc2f5eb4ac0e3380fb7f38b2cc96d201": "\\{M_i\\}_{i = 1}^S", "cc2f8640ec2fa9d5abcf1131e8d92aa3": "\\lim_{\\theta \\to 0}\\frac{1 - \\cos \\theta}{\\theta} = 0\\,", "cc2fae62aac44ba12dcd5274132f28a3": "\nlog(T)=-log(\\theta)+log(T\\theta):=-log(\\theta)+\\epsilon\n", "cc2faedbdd323a45c0ab50147de20467": "m \\times 1", "cc300e610afdb8e6ac026c529466df2b": "S_{base} = 1 pu", "cc30434e70f2318c86859bde04d89290": "\\{Y_1<\\cdots F(v) - \\varepsilon d(v, w).", "cc365c807056f915ea7097a93dee5e9e": "\\mathcal{A}^A", "cc36924db143408a55c8c364a5cba3de": "e^+e^-", "cc3737b6833dad3b4d358412bc8ec9b8": " f(\\mathbf{x})", "cc374377a25e36ffbf605f0091c1f98c": "\n\\bar{\\sigma} = \\frac{\\tanh(\\kappa h / 2)}{1-p+p \\tanh(\\kappa h / 2)} \\sigma\n", "cc37817e8a2db1f24c8c5c1419513933": "GapSVP_\\gamma", "cc37b9f46bf41b03b455741cb320de2d": "\\scriptstyle d\\leq2 ", "cc37bce6e65d21dc2f4bbe7b1fb396f2": "\\varphi (x)", "cc37ccc068cdf1f7a57c19451dd2c9e2": "T = \\sqrt{H^2+V^2}", "cc37ea6a2391016228a7e9c1a019d1bb": " \\gamma = \\frac{1}{\\sqrt{1 - \\beta^2}}", "cc3820a8cc515b52b20d191d349b3bcd": "h[n]\\ \\stackrel{\\mathrm{def}}{=}\\ \\scriptstyle{DTFT}^{-1} \\big \\{\\displaystyle \\sigma_H(\\omega)\\big \\} =\n\\begin{cases}\n 0, & \\mbox{for }n\\mbox{ even}\\\\\n \\frac2{\\pi n} & \\mbox{for }n\\mbox{ odd}\n\\end{cases}", "cc383bfd8a727fc5a2c831e7185f573d": "n^\\theta f(x)=f(nx)=f(x+\\dots +x)=nf(x)", "cc38823503f8b37847ef5c8fa7a1c6d8": "s_5 = 0001,", "cc38a33216ea0226dc262b78959db974": "j=0,\\dotsc,m", "cc38a609de384f825152b1ff9fa24981": "M_{v} =", "cc38b10c0965e289b4898489e00aab54": "\n \\begin{align}\n N^{\\mathrm{topface}}_{xx} & = -f\\left(h + \\tfrac{f}{2}\\right)~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} = - N^{\\mathrm{botface}}_{xx} \\\\\n M^{\\mathrm{topface}}_{xx} & = -\\cfrac{f^3~C_{11}^{\\mathrm{face}}}{12}\\left(\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2} + \\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2}\\right) = -\\cfrac{f^3~C_{11}^{\\mathrm{face}}}{12}~\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d} x^2} = M^{\\mathrm{botface}}_{xx} \n \\end{align}\n ", "cc391aa3a2c8312b389b522f12fb61f8": "f:N\\rightarrow M", "cc39260d7a40a528bf5840a357da4892": "X \\simeq Y \\Rightarrow \\pi_1(X,x_0) \\cong \\pi_1(Y,y_0).", "cc3995c5a927281a32fac4af89299011": "g\\cdot a,g\\cdot b", "cc39b00e1bf6a375d8dbd10e3d70220b": "P_\\ell^{m}(\\cos\\theta) = (-1)^m (\\sin \\theta)^m\\ \\frac{d^m}{d(\\cos\\theta)^m}\\left(P_\\ell(\\cos\\theta)\\right)\\,", "cc39c31082f278d26de9e0830af58441": "1 - \\frac{10^9}{10^{10}} = 1 - \\frac{1}{10} = 90\\%", "cc3a0906f6bd621206c140a95cf6480e": "P=\\bigcap_{i\\in\\mathbb{N}}\\left(\\mathcal{C}^i\\right)^{*}.", "cc3a32e1ef613abc645dbb2bdb9601f6": "\\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2 R + c_8 T R^2 + c_9 T^2 R^2 + c_{10} T^3 + c_{11} R^3 + c_{12} T^3 R + c_{13} T R^3 + c_{14} T^3 R^2 + c_{15} T^2 R^3 + c_{16} T^3 R^3\\ \\, ", "cc3a3ba440c66b05064d8f34367c4412": "R_\\mathrm{load}=R_\\mathrm{source}\\,\\!", "cc3a618924b128e8e2d5cd4f573d05ad": "\\textstyle\\frac{\\sin C}{c}", "cc3a723a2cea6a37373c71f1ff7439c3": "\\begin{matrix} {4 \\choose 2}{3 \\choose 2}^2{36 \\choose 1} \\end{matrix}", "cc3acadc52e60203a8ae969769386157": "S_n = a_{n + 1} B_{n} + \\sum_{k=0}^n B_k (a_k - a_{k+1})", "cc3b33fa39b6fb144b8aec89716ffe7c": "\\pi^2/(2\\,\\pi)=\\pi/2", "cc3b5171f38984f7a0772823758c2d84": "\n\\frac{100}{\\text{Clean price}} \\times \\left(\\frac{100 - \\text{Clean price}}{\\text{Maturity in years} } + \\text{Spread}\\right).\n", "cc3ba4bad3c6fa5ff637a8a6416ef733": "loaded(2)", "cc3bb9e406ac145fe2786bf9e88523cf": "[I]", "cc3bd7a801a8fe58246abafccc1d5166": "B_{iy_j}", "cc3c08e42ba7db53a524a2cc5c447f45": "su(2)\\oplus u(1)", "cc3c0c37a2dfd4705d7de658d8bfe488": "P=\\frac{1}{3}\\,\\frac{U}{V}", "cc3c223551421b6616ad7da4a4ab86b8": "\\alpha \\,\\!", "cc3c37a7e3747731f1ae26754d2ebdf8": "\\textstyle\\alpha", "cc3c9604827030a163162d727df8dd6a": " K-S+ max(0,S-K) = max(0,K-S) ", "cc3cf2cf9cc7eb024095f6ae441759d2": "(\\mathbf{A})", "cc3da7ef29eba76f3ee18b39e473b5ef": "\n\\begin{align}E(x, y; u) &= \\sum_{n=0}^\\infty \\frac{u^n}{2^n n! \\sqrt{\\pi}} \\, H_n(x) H_n(y) \\, \\mathrm{e}^{ - (x^2+y^2)/2} \\\\\n& =\\frac{\\mathrm{e}^{(x^2+y^2)/2}}{4\\pi\\sqrt{\\pi}}\\int \\!\\! \\int \\Bigl( \\sum_{n=0}^\\infty \\frac{1}{2^n n! } (-ust)^n \\Bigr) \\, \\mathrm{e}^{isx+ity - s^2/4 - t^2/4}\\, \\mathrm{d}s\\,\\mathrm{d}t \\\\\n& =\\frac{\\mathrm{e}^{(x^2+y^2)/2} }{4\\pi\\sqrt{\\pi}}\\int \\!\\! \\int \\mathrm{e}^{-ust/2} \\, \\mathrm{e}^{isx+ity - s^2/4 - t^2/4}\\, \\mathrm{d}s\\,\\mathrm{d}t,\\end{align}\n", "cc3dffd4ffd80c08c95bf2c4827bc988": "\\omega = \\sqrt{\\frac{k}{m}} \\,\\!", "cc3e25842f8bf6684690b45620666dba": "Q_r ~", "cc3e41cee823eaf15b6c4e6b5d32d624": "\\xi(x) = \\nabla u(x), x\\in H", "cc3ec5805b42963c3a569c39c778a6b1": " \\prod_{1 \\leq k \\leq n-1} \\left( 2^k - 1 \\right) \\equiv n \\mod \\left( 2^n - 1 \\right). ", "cc3ed1173799e737fa8bef1de8c807c8": "\n \\cfrac{\\sigma_1-\\sigma_3}{2} = \\cfrac{\\sigma_1+\\sigma_3}{2}~\\sin\\phi + c\\cos\\phi\n ", "cc3f3ab8fd6afc3a0c4b5e654368936f": "\\begin{align}\nv_o & = \\frac{2\\pi a}{T}\\left[1-\\frac{e^2}{4}-\\frac{3e^4}{64} - \\dots \\right] \\\\\n & = 18.31\\ \\mbox{km/s} \\left[ 1 - 0.0177 - 0.00008 - \\cdots \\right] \\\\\n & \\approx 17.98\\ \\mbox {km/s} \\\\\n\\end{align}\\!\\,", "cc3f3dbb4ccee1ac4e1f87c27f1f53ef": "2.5K_p/T_u", "cc3f51688b9194c98cd8a2724db90e87": " u_{xy} = \\sin u.\\,", "cc3f70607cf0f490caef186979d69f71": "\\theta:H^{n}(-,\\pi)\\to H^{q}(-,G)\\,", "cc3fc55264537889c78fae94d84c893a": "m_d", "cc3ff539d944567c2b6d274cb42af752": "-\\log f_{\\pm} = 0.5 z_1z_2\\left(\\frac{\\sqrt I}{1+\\sqrt{I}}-0.15 I \\right)", "cc3fff7ed3d8d49a49044ff6d40f02bc": "c^{\\mu }", "cc4020dd9674f365ff92d37951ac36ba": "\\begin{pmatrix}a & b\\\\ b&c\\end{pmatrix}", "cc4045a28d25b4a373c7bbe9029ceff1": "A \\cdot B\\neq C", "cc40c2ad0e19edc4e9294930b2da46e4": "-1/k", "cc414451003acf3e2fea93d737d35bd4": "(x_3,y_3)\\,", "cc414df27343e5458188bcfc45a8077e": " y''+\\frac{1}{2} \\left [\\frac{m_1}{x-a_1}+ \\cdots +\\frac{m_{n-1}}{x-a_{n-1}} \\right] y' +\\frac{1}{4} \\left [\\frac{A_0+A_1x+ \\cdots +A_\\ell x^\\ell}{(x-a_1)^m_1 (x-a_2)^m_2 \\cdots (x-a_{n-1})^m_{n-1}} \\right]y=0. ", "cc418da1e8247af676dc186598e3c283": " = \\omega r \\mathbf{u}_\\mathrm{\\theta} \\, ", "cc420bfe0221a8ec6baefd6323e02e0a": "\\mathbf{X}^{\\rm T} \\sim {\\rm T}_{p,n}(\\alpha,\\beta,\\mathbf{M}^{\\rm T},\\boldsymbol\\Omega, \\boldsymbol\\Sigma).", "cc42f3863c6f72711a7b61fe53cfa5b6": "\\mbox{SG oil} = \\frac{\\rho_\\text{oil}}{\\rho_{\\text{H}_2\\text{O}}}", "cc431b6337450446eabc5fe65b016ca9": "\\triangle ABC ", "cc4378e0c61094f466b676e70a27b248": "x^\\mathrm{SOR}_{n+1}=(1-\\omega)x^{\\mathrm{SOR}}_n+\\omega f(x^\\mathrm{SOR}_n).", "cc438e95a86980f70b7ad2d0aa776909": "\\displaystyle{B(f,g)=\\int_{S^1} f dg.}", "cc43d846db64cb25c4d499a44a55ad4b": "H_{Ref}\\,\\!", "cc43dd3d369c467b08c8e48b5da6938e": "f(q|e)", "cc43ee4e67812eb7c8d4ea9669896433": " FX = \\text {exchange rate between currency1 and currency2} ", "cc442e014d7016fa04eaf3d0dcfa9a1d": "s^{(J)}_n:=2^J \\langle f(t),\\phi(2^J t-n) \\rangle,", "cc446b8fda044741bccd7f532c1b0b71": " \\operatorname{E}[\\ln X_i] = \\psi(\\alpha_i)-\\psi(\\textstyle\\sum_k \\alpha_k)", "cc44c7271f30bf77c290443daa6731a9": " \\mathbf{T} = \\begin{pmatrix} \nT_\\text{xx} & T_\\text{xy} & T_\\text{xz} \\\\ \nT_\\text{yx} & T_\\text{yy} & T_\\text{yz} \\\\\nT_\\text{zx} & T_\\text{zy} & T_\\text{zz}\n\\end{pmatrix}", "cc44e2785b4f82015d19ed5e1ac12d32": " an^2+bn+q=0 ", "cc45144f87f783b4ab6a6011c7d37070": "G_{0}", "cc452d9749a56353c697a7378a28de5f": "\\beta=\\frac{B\\theta}{B+H_k}", "cc454314cc900e3961e6842017134176": "x_0 - x.\\,", "cc456921591fd061336c89f89e293ea9": "=2n\\sum_{i=1}^n r_is_i - \\frac12 n^2(n+1)^2 ", "cc457aacb92714addbf180fa86f70715": "f^n(x) = f(f(\\ldots(f(x))\\ldots))", "cc457d793dd77b7c09466bda6c60364b": "E = \\frac {Z}{ 2 \\, \\sqrt{ 10000 } } = \\frac {Z}{ 200 } ", "cc45aed53f146be8ffc5624671745ef2": "\nT = \\frac{dq_\\mathrm{rev}}{dS}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(9)", "cc45e058aabbb1582a02a40886475b91": "T_P\\ = \\frac{a_P}{a} + 2\\cdot\\sqrt{\\frac{a}{a_P} (1-e^2)} \\cos i", "cc45e4016cc89fcbc78e1a4ff2fd73d5": "\\mathcal{N} (\\mathbb{C} P^n) \\cong \\oplus_{i=1}^{\\lfloor n/2 \\rfloor} \\mathbb{Z} \\oplus \\oplus_{i=1}^{\\lfloor (n+1)/2 \\rfloor} \\mathbb{Z}_2", "cc4600ad17f46e3b48f5cc0a3207adee": "\\begin{align}\n x(u,v) &{}= A c\\left(v,\\frac{2}{r}\\right) c\\left(u,\\frac{2}{r}\\right) \\\\\n y(u,v) &{}= B c\\left(v,\\frac{2}{s}\\right) s\\left(u,\\frac{2}{s}\\right) \\\\\n z(u,v) &{}= C s\\left(v,\\frac{2}{t}\\right) \\\\\n & -\\frac{\\pi}{2} \\le v \\le \\frac{\\pi}{2}, \\quad -\\pi \\le u < \\pi ,\n\\end{align}", "cc46308a4d8d50baed608a17c91255f9": "\\lambda = 1 - \\hat{R}_m \\cdot \\hat{V}", "cc46c6ad26cc3cf76d3e6e5cf2e3a4ec": "v\\ ", "cc471524cf2173113c264cb65b0b04a2": "\\sum\\limits_{x: f(x) = y} D_n(x) \\leq p(n)D'_{m(n)}(y)", "cc47250251ac0b5f6a17372d69cad9ff": "1\\to A\\to \\tilde{G}_0\\to G_0\\to 1\\,", "cc477aef29cf1c062d2e7a095940493d": "q'(x,y)=a'x^2 + x'y'+b'y^2", "cc47a9141b8b651618b109e8ebdc81fc": "\\displaystyle{\\omega(z_1,z_2)=zJz^t.}", "cc47cc588e7d4b33811e07d8d0db2413": " \\, \\sim \\, ", "cc47e74ec54b0344f94f3e624c17e608": "V_n(1,-2)", "cc47eab6685201f2332f04e8dd90d696": "AF = \\frac EV", "cc48931e077b0952f370b043e6d33300": "\\langle \\partial_{t} u , v \\rangle = \\langle \\frac{1}{2} u^2 - \\rho \\partial_{x} u , \\partial_x v\\rangle+\\langle f, v \\rangle \\quad \\forall v\\in \\mathcal{V}, \\forall t>0.", "cc49ede3a04fa31e4815a306b1f68033": "Ambiguous \\ Velocity = - 0.5 \\left (\\frac{ Doppler \\ Frequency \\times C }{ Transmit \\ Frequency} \\right)", "cc4a189f2420d303025769fc43239f9e": "\\mathfrak{sl}_3(\\mathbf K)", "cc4a4e6f1b12d0a32a5f60e22af86508": " u \\frac{dC_i}{dx} = \\nu_i r ", "cc4a54ccedb34a25078e560339aef1e2": "\\mathcal{L}_X = \\mathcal{L}_Y", "cc4abd18d562cea8e519d55f2ebb0728": "CMF = \\frac{\\sum_{t=20}^tCLV_{t} \\times volume_{t}}{\\sum_{t=20}^t(vol_{t})}\\!\\,", "cc4ae816d6f4d31c72a6206ed01aacf2": "S_\\varphi = \\lim_{\\delta\\downarrow0}\\sum_{m=0}^\\infty(-1)^m\\varphi(\\delta m) = \\frac12.", "cc4af531bbf74b7759cceabbfca53ee9": "\\frac{v_\\perp}{v_\\parallel} > \\frac{1}{\\sqrt{r_\\text{mirror}}}", "cc4b6b92d7858d8c82f13e25f488277f": "\\sum_{x=0}^7 \\frac{ \\binom 7 x }{ \\binom{10}x } \\cdot \\frac1{2^{10}} \\binom{10}x = \\frac 1 8 , ", "cc4ba95f6019c239a20da87cf5e30946": "\\scriptstyle \\Phi", "cc4bdfe4470c358c67017198253f27eb": "\\displaystyle{\\int_G f(g)\\, dg = |W|^{-1} \\int_T f(t) |\\Delta(t)|^2\\, dt,}", "cc4c23999fda6f23a5b699120cd59b11": "dm = \\lambda_m dl", "cc4c366532c11326699efd6711469794": "t = n\\Delta_{T}\\,\\, f = m\\Delta_{F}\\,\\, \\alpha = p\\Delta_{F}", "cc4c4730b05063bd5c8f2cd1eb48faac": "\\frac {\\sqrt 1} 2", "cc4c620fe373e052b502f934bf87acdd": " \\cos(\\pi\\nu) - y(\\pi = 0) = 0, \\, ", "cc4c8a902a2edd99b11a91df488712a2": "(2h_{n+1} h_{n+2}, h_nh_{n+3}, 2h_{n+1}h_{n+2}+h_n ^2)", "cc4d241e39bd9c3d51a00b946a042ac7": " \\frac{\\log 4}{\\log 3} ", "cc4d3adecbf18d699d69f3fccef84397": "v-2r^*", "cc4d6068c6275d973593b133ccfb5d31": "\\begin{align}\n 0 & = \\dfrac{\\partial E}{\\partial \\mathbf{h}} \\\\\n & \\approx \\dfrac{\\partial}{\\partial \\mathbf{h}}\\sum_{\\mathbf{x}}\\left [F(\\mathbf{x})+\\mathbf{h}\\left(\\dfrac{\\partial F}{\\partial \\mathbf{x}}\\right)^{T}-G(\\mathbf{x})\\right ]^{2} \\\\\n & = \\sum_{\\mathbf{x}}2\\left [F(\\mathbf{x})+\\mathbf{h}\\left(\\dfrac{\\partial F}{\\partial \\mathbf{x}}\\right)^{T}-G(\\mathbf{x})\\right ] \\left(\\dfrac{\\partial F}{\\partial \\mathbf{x}}\\right)\n\\end{align}", "cc4d617bda13d3a323a76e1f6a985952": "\\textbf{Set}", "cc4d7c03c65f41fa6fe0f66b19b095b0": "v_\\textrm{biomass}", "cc4e5312caf9804eca8d798bb140f9df": "\\mathbb R^m", "cc4e594af56a32cb6e7a16904ee71192": "\\operatorname{aff} (S)=\\left\\{\\sum_{i=1}^k \\alpha_i x_i \\Bigg | k>0, \\, x_i\\in S, \\, \\alpha_i\\in \\mathbb{R}, \\, \\sum_{i=1}^k \\alpha_i=1 \\right\\}.", "cc4f11e511e749c798507bd3f4e28e4a": " \\rho(c) = \\prod_{ \\text{neighboring bonds }b',b'' \\in c } \\rho(b',b'' ). ", "cc4f312c83a1de5c8028b1f5c7a65668": "\\tfrac{1}{20}", "cc4f3af0e98a75246f3a6c1d99c03445": "p_{i+2}", "cc4f973107ceebd332e63e1ed8706507": "s_\\max\\,\\!", "cc4f9ae0683a3ba2df6464834d20ac47": "P'(r)", "cc4fd4f6cade81ec4682d3a7c587e476": "\nE_{abs} = \\pi r_E^2 \\times E_{a_0}\n:", "cc4ff2b8f76f282e14e6ea9dcc05794c": "\n \\sum_{J=|j_1-j_2|}^{j_1+j_2} \\sum_{M=-J}^{J}\n \\langle j_1 m_1 j_2 m_2|J M\\rangle\\langle J M|j_1 m_1' j_2 m_2'\\rangle=\n \\langle j_1 m_1 j_2 m_2 | j_1 m_1' j_2 m_2'\\rangle\n = \\delta_{m_1,m_1'}\\delta_{m_2,m_2'}\n", "cc50270ac8aad626d2d1492de1eb0275": "E\\to P/G\\to X", "cc508db19af5ff482d55f79de4e4b1eb": "m = \\text{length}(a)", "cc50e4f737e11d5217b23bf9adea6337": " \\bigvee_i A(x,y_i \\oplus z)=0", "cc50ec09adb5ce86600eb8910b094e28": " w \\sigma_i \\sigma_j) ", "cc517873c9b807dd8a837309c0650a1f": "\\mathcal{L}_K\\alpha=[d,i_K]\\alpha = di_K\\alpha-(-1)^{k-1}i_Kd\\alpha.", "cc5198b752db58b6b802b6f269709e09": "f\\mapsto \\left(\\int|f|^p \\; \\mathrm{d}x\\right)^{1/p}", "cc51a67351041def45c54943f459163f": "\n\\dot{\\vec{r}} = \\{\\vec{r}, \\, H\\}_{PB}, \\quad \\dot{\\vec{p}} = \\{ \\vec{p},\\, H\\}_{PB}, \\quad \\dot{\\lambda} = \\{ \\lambda,\\, H\\}_{PB},\n\\quad \\dot{p}_\\lambda = \\{ p_\\lambda, H\\}_{PB}.\n", "cc51b428e332414a62bf7ef40b2afe2f": "\\Gamma = \\frac{\\sum_{i,j = 1}^n a_{ij}b_{ij}}{\\sqrt{\\sum_{i,j = 1}^n a_{ij}^2 \\sum_{i,j = 1}^n b_{ij}^2}} ", "cc51dedb20bc3c95bd6d0d0eaf7678bd": "\\delta_k=(T^kp)(0)", "cc51ebe77cfc63c0918a85d810e3e2b3": "\\text {P}= 112 \\text {erg} \\times110\\text {/s}=1.23\\times10^4\\text {erg/s}=1.23\\times10^{-3}\\text {W}", "cc52020d35c4efdec290bca341212751": " l_n = {2 \\over 3} (1+\\sqrt 2) 2^n - {1 \\over 3} (2-\\sqrt 2) {1 \\over 2^n} ", "cc525d8263c38c5f563f59c1a9b8e6c3": " Z_L = Ls \\iff Y_c = Cs", "cc52820adbad71b5de4a33c68451c58b": "\\pi=\\frac{\\sigma_{kk}}{3}=\\frac{\\sigma_{11}+\\sigma_{22}+\\sigma_{33}}{3}=\\tfrac{1}{3}I_1.\\,", "cc528c0ae50499b555f4e4c37f653ea1": "u(x).", "cc5311245b70f20f847aea9f732aec6e": "\\mathcal K \\equiv {d\\Pi \\over dT} - S", "cc53397357515dbe8db3a3e3ae6200af": "q'_x(a,b)=q'_y(a,b)=0", "cc537c6b23ebcba048588cb0e35a852e": "\\sigma = \\frac{1}{\\sqrt{\\xi}}\\sum_{n=-\\infty}^\\infty \\left ( A_{1n} e^{in\\omega\\xi}+B_{1n} e^{-in\\omega\\xi} \\right ) e^{in\\omega z},", "cc53d30ef82cc15144eaa451d2d4c988": "SER = \\text{sph.} + \\frac{1}{2}*\\text{cyl.} ", "cc53d43736018b5fa0c8da93cf7e2d90": "C^\\infty({\\mathbb R}^n,{\\mathbb R}^m)", "cc53f7a1d5030e103156bf332c957cf5": "k=1,\\ldots,m", "cc53f8d4d4d4fa344c93255f28f0a950": " \\boldsymbol{\\tau} = \\int_V \\mathrm{d} \\mathbf{m} \\times \\mathbf{B} ", "cc54112e3298c0d31ac173ccc90a1c81": "\\scriptstyle \\sum_{n \\ge 1}^{\\Re} f(n)", "cc5426481be6feb78d47be05f0f0f75f": "\\Lambda^{(m,m+1)}", "cc546767b7ae3c9f6671555a25e5aaf0": "a\\sin x+b\\cos x=\\sqrt{a^2+b^2}\\cdot\\sin(x+\\varphi)\\,", "cc5489bf2a2804d7140a50c1be9c6b16": "T_{sys}", "cc550ac3d1b6125b727097642ce5fd87": "T_r = T/T_c\\,", "cc554633b31a3c7eb521b4bbd95c7b4c": "\\Big( (\\mathcal{M}, s) \\models EX\\phi \\Big) \\Leftrightarrow \\Big( \\exists \\langle s \\rightarrow s_1 \\rangle \\big( (\\mathcal{M}, s_1) \\models \\phi \\big) \\Big)", "cc557ef9f360bdc0944797213d0ae886": "\\tfrac{177147}{176776}", "cc55df90e8887dedeb2f2af69f0b3ba5": "9604*x^2+1546244*x-138297600=0", "cc564d79d265c1ec3622fbaf838d3149": "\n\\boldsymbol{S} = \\lambda~ \\text{tr}(\\boldsymbol{E})\\boldsymbol{\\mathit{1}} + 2\\mu\\boldsymbol{E}\n", "cc56a8ac588294ccf2ebfcc6aef75d66": "\\Lambda(x, y, \\lambda) = f(x,y) + \\lambda (g(x, y)-3) = x^2y + \\lambda (x^2 + y^2 - 3). \\, ", "cc56ce1f4ec606e4763c25d6613f5995": " a \\tfrac{b}{c} ", "cc56eda9e4eb0aa15902f638eb63b5ff": "\\frac{d}{dt}\\mathbf{P}(t) = \\mathbf{F}(t)\\mathbf{P}(t) + \\mathbf{P}(t)\\mathbf{F}^{T}(t) + \\mathbf{Q}(t) - \\mathbf{K}(t)\\mathbf{R}(t)\\mathbf{K}^{T}(t)", "cc57171381336260d7ba6a72bab9c487": "a^x = b", "cc5756bde0d60717068674c2337041cc": "(X,\\mathcal{T})", "cc5770711fb32203e19ad7423e87b9f1": "\n P(x,t)=\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\left(-\\frac{(x-x_0)^2}{4Dt}\\right).\n", "cc57880d889b15a55e6167e669b7d83b": "\\scriptstyle d", "cc581025a0e91dedcfb4146c793dea1e": "Y_{9}^{-9}(\\theta,\\varphi)={1\\over 512}\\sqrt{230945\\over \\pi}\\cdot e^{-9i\\varphi}\\cdot\\sin^{9}\\theta", "cc582317bb8add3782267e926bb4f790": "F_t - S_{t + 1}", "cc585b672bf56bfac2d89fd0132c2d6f": " \\overline{\\frac{\\partial u_i}{\\partial t}} + \\overline{\\frac{\\partial u_iu_j}{\\partial x_j}}\n= - \\overline{\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x_i}}\n+ \\overline{\\nu \\frac{\\partial^2 u_i}{\\partial x_j \\partial x_j}}.\n", "cc5899c9d31d4b04de85207dbb2fd331": "\\frac{E_1}{E_2} = \\frac{k_2}{k_1} = \\frac{c_1}{c_2} ", "cc58a432371a9c11e3d8c9ad5aa4f334": "(X', \\| \\cdot \\|')", "cc58ad590bcac61212159f6999bd724b": "\\Sigma=\\sum_{j=1}^3S_j\\sigma_j,\\qquad (4)", "cc58c2935b732dbca9a69fd06a4bbd41": "f \\circ g_1 = f \\circ g_2 \\Rightarrow g_1 = g_2.", "cc593d35c828fe67ca4e1ccb5864a253": "e^{{-i\\delta}G^{[l]}}", "cc594783292c76d2d0d0f9a32e5dfecb": " \\exp(-\\gamma \\; r) ", "cc59dc1223d1abd5b4a5d3401b2d8783": "\\Delta\\rho-\\bar{\\delta}\\tau=-(\\rho\\bar{\\mu}+\\sigma\\lambda)+(\\bar{\\beta}-\\alpha-\\bar{\\tau})\\tau+(\\gamma+\\bar{\\gamma})\\rho+\\nu\\kappa-\\Psi_2-2\\Lambda\\,,", "cc5a2a4627ebcc24553e9d7f91b2b67c": " x_{ij} \\geq 0 \\qquad i=1, \\ldots, m, \\quad j=1, \\ldots, n", "cc5a64b2735d6076414a7c9a27d43f54": "Re = \\frac{\\rho v d_{30}}{\\mu}", "cc5af7d4f039db0c60ae07239846a67c": "G: \\mathcal{C} \\rightarrow \\mathcal{D}", "cc5b21d58ba97c6369e0671f1d314451": "\\rho_0 c^2", "cc5b65c8fb8251ae7c9e050018a9ba5b": "\\Lambda_{Wilks} = \\prod _{i=1...p}(1/(1 + \\lambda_{i}))", "cc5bed712ae6c88a0502bfc7de8e288d": "KE= \\frac12 I\\omega_\\text {max}^2", "cc5c222483afd34f2bdd27a23521a15c": "p_xp_y", "cc5c22b72143d330252210dcf7e8e661": "Q=\\sum_{n=1}^\\infty c_n D^n", "cc5c2cf1eebcfcde0fa31e0204a6b35c": "x_3, x_4, x_5", "cc5c8842d6f8740ea24bf598b656aa92": "\\psi=Ar^n\\sin n\\theta.\\,", "cc5cce8a8bc697675eb1e25b291103b2": "\\Rightarrow\\!\\Leftarrow", "cc5cfaec6774bd937dea038ae3d47de8": "\\varphi_\\alpha", "cc5cfb59b85b4a057092c517b43e6c9e": "B_J(x) = \\frac{2J + 1}{2J} \\coth \\left ( \\frac{2J + 1}{2J} x \\right )\n - \\frac{1}{2J} \\coth \\left ( \\frac{1}{2J} x \\right )", "cc5df489ff41ae339764feae58f83f9b": "c=2 \\sqrt{r^2- a^2}", "cc5e6566e4e06a67c19437b1cfc5800d": "\n\\Delta^{i}_{jk}=\\{^{i}_{jk}\\}-\\Gamma^{i}_{jk}~~~~~~~~~~~~~~(1)\n", "cc5e6f8316449fd196186bde43b12465": "f(z_0,\\dots,z_n)=0", "cc5e9ea0c8676e8d50bbe42f35306022": " G = \n\\begin{bmatrix}\n\\frac{K}{r_{11}^2} & \\frac{K}{r_{12}^2} & \\frac{K}{r_{13}^2} & \\frac{K}{r_{14}^2} & \\frac{K}{r_{15}^2} \\\\\n\\frac{K}{r_{21}^2} & \\frac{K}{r_{22}^2} & \\frac{K}{r_{23}^2} & \\frac{K}{r_{24}^2} & \\frac{K}{r_{25}^2} \\\\\n\\frac{K}{r_{31}^2} & \\frac{K}{r_{32}^2} & \\frac{K}{r_{33}^2} & \\frac{K}{r_{34}^2} & \\frac{K}{r_{35}^2} \\\\\n\\frac{K}{r_{41}^2} & \\frac{K}{r_{42}^2} & \\frac{K}{r_{43}^2} & \\frac{K}{r_{44}^2} & \\frac{K}{r_{45}^2} \\\\\n\\frac{K}{r_{51}^2} & \\frac{K}{r_{52}^2} & \\frac{K}{r_{53}^2} & \\frac{K}{r_{54}^2} & \\frac{K}{r_{55}^2} \n\\end{bmatrix}\n", "cc5ea5822c9e461616dfcf28b9801d8b": "Y_{8}^{-7}(\\theta,\\varphi)={3\\over 64}\\sqrt{12155\\over 2\\pi}\\cdot e^{-7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot\\cos\\theta", "cc5eb0b7793178ba466dd11a6b3ad2ae": "r \\dot{A}B + 2 A^2 B - 2AB - r \\dot{B} A=0", "cc5eb56a70ef00b5a03bb9501cf40c55": "P'\\subset P", "cc5ecb159b0a505e1ad96d0f865c0d41": "\n D := \\cfrac{2h^3E}{3(1-\\nu^2)}\n ", "cc5ed2d4d3542ffdc95408e59d96843a": "\\,q_\\textrm{F} \\in Q_\\textrm{F}", "cc5f10313cee71c1ea5b32aff75bdc22": "\\frac{\\Phi}{c^2}=\\frac{GM_\\mathrm{sun}}{r_\\mathrm{orbit}c^2} \\sim 10^{-8},\n\n\\quad \\left(\\frac{v_\\mathrm{Earth}}{c}\\right)^2=\\left(\\frac{2\\pi r_\\mathrm{orbit}}{(1\\ \\mathrm{yr})c}\\right)^2 \\sim 10^{-8} ", "cc5f687ca2d188ebea93d3db06fa2ade": "{^{131}_{53}\\mathrm{I}} \\rightarrow \\beta + \\bar{\\nu_e} + {^{131}_{54}\\mathrm{Xe}^*} ", "cc5f68fb341cbc9d8c2dc57708649117": "\\psi(0) = \\omega^\\omega", "cc5f8d4527bbcaddfae591ccf1a60fb8": " \\mathbf{E}_7 \\supset \\mathrm{O}(12) ", "cc5f93caa7aec79874589bd0dc56eef0": "\\lambda_{\\beta_c} = 1+\\sqrt{1-4c}\\,", "cc5f97acddfc0a1713b7d66a425e1baf": "a^\\dagger", "cc6015038576a859b1e976c5c8708636": "p^{\\rm{sat}}_r = \\frac{p^{\\rm{sat}}}{p_c}", "cc601f23351a369b7789f63cde43f11b": "p\\mapsto \\alpha_p(X_p)", "cc61a939bdc34ee08903626cd74ad78b": "\nE^{(1)} = -\\mathbf{F}\\cdot \\langle \\psi^0_1 | \\boldsymbol{\\mu} | \\psi^0_1 \\rangle =\n-\\mathbf{F}\\cdot \\langle \\boldsymbol{\\mu} \\rangle.\n", "cc622be778d0a47248be46166565d09d": "\\mathbf{P} \\left (\\sum_{i=1}^n X_i > t \\right ) \\leq \\exp \\left ( -\\frac{\\tfrac{1}{2} t^2}{\\sum \\mathbf{E} \\left[X_j^2 \\right ]+\\tfrac{1}{3} Mt} \\right ).", "cc623b1569e8500b2859d59f684a87c7": "d^2\\alpha=r \\, dr d\\theta", "cc625609c133d75d99a0398198e5f2dc": "(-1)^\\text{signbit}\\times 10^{\\text{exponentbits}_2-101_{10}}\\times \\text{truesignificand}_{10}", "cc62666918a3b9be522feeccc70d90c5": "2 (\\mu,\\alpha_i)/(\\alpha_i,\\alpha_i)", "cc628ad79b2ad3ce69310bf911341361": "p=f(\\mathbf{AA})+\t\\frac{1}{2}f(\\mathbf{Aa})= \\mbox{frequency of A}", "cc62adb86eb75c6c1833538fb6d1ea32": "[f]([x_1,x_2]) =[e^{x_1}, e^{x_2}]", "cc62afacbce479c316e81fc8e1af32fa": "|\\mathcal{U}|", "cc62c47cafd926dd5b630c916611ebfe": "D_m ^{s}=D_m (x_0,\\dots , x_m-1)", "cc62ee43e7e44c365c37ae71b8dc5b1c": "f^{-1}(k) \\subset \\cup_{\\lambda \\in \\gamma_k} U_{\\lambda}", "cc62f6cb16400c1f341e5d7c494cbde9": " \\lVert \\hat{s} \\rVert_\\infty \\leq 10 \\phi p^{1/m}n \\log^2n ", "cc632793045d9d6ed4a2a0b8e7e31499": "x^{\\ast}", "cc63415a3634599903bfd7cc941197ed": "H_{\\mathbf{k}}|u_{n\\mathbf{k}}\\rangle=E_{n\\mathbf{k}}|u_{n\\mathbf{k}}\\rangle\\;.", "cc63688b11287aefd93277e605406738": "\\Delta =\\frac{D-2}{2}", "cc63ee57c414870c911e400300a4f1c8": " a < r < b \\, ", "cc63ef0da22a98bd2d07b74821c8f235": "P = \\sqrt{\\frac{T^3}{2 \\rho A}}", "cc648e51897d213d2d8cd355195ac6ee": "m_{\\vec{G}}(x)", "cc64aa51594863100f77edc700d45e99": "\\,pL+(1-p)N \\preceq pM+(1-p)N\\,", "cc64da3aadf3ec469afb16f55645a1cc": "\\alpha = \\frac{B \\wedge(q-t)}{B \\wedge v} ", "cc659cb79ba80e85dfbb3fafc8cc9179": "\\nabla^2 \\phi=0", "cc65fe3fea700742fe8e35869747bc79": "\\hat{J}^2", "cc66d434883acae70286367ba2174feb": "\\bar{\\theta}_{Jack} =\\frac{1}{n} \\sum_{i=1}^n (\\bar{\\theta}_i)", "cc675ba59b5571b1e3b333746af22e9d": "r^\\ell", "cc67b223702de5c4dc1a69b625ab43fe": " I = \\begin{matrix}\\frac{1}{2}\\end{matrix}\\sum_{{i}=1}^{n} b_{i}z_{i}^{2} ", "cc67b347d2f6d945eb89c02cf2840aa4": "|\\zeta(\\sigma+it)| = \\exp\\Re\\sum_{p^n}\\frac{p^{-n(\\sigma+it)}}{n}=\\exp\\sum_{p^n}\\frac{p^{-n\\sigma}\\cos(t\\log p^n)}{n},", "cc67e250dde2fbecedb7a62a42f1f0e5": " \\begin{matrix} x_2 & = & x_1 + d\\theta_1 \\\\\n\\theta_2 & = & \\theta_1 \\end{matrix} ", "cc67e34529132cc3fc3c76948da93dba": "\\epsilon_{33} = 0\\,\\!", "cc6810f941722d5fbc3e519612299154": " i, j \\in C ", "cc682462fad7d67c6e0397f6cf9108e2": " t = \\left(\\frac{dy}{dx}\\right)/2 ", "cc682e113fdaa16babdaafdacc090d29": "n_{12}", "cc687e6c53bb97bee375aa50c46eab92": "f_a, f_b, \\ldots, f_N", "cc6887abf18d5eb1bd08549c776321b0": "\\frac{e}{pq} = \\frac{k}{dg} (1- \\delta)", "cc68decbeaffa120e76975502b81a4fa": " \\tilde{\\mathcal{D}}_\\eta (\\rho) = S \\mathcal{D}_{(1-\\eta)/\\eta} \\left({\\mathcal{D}}_{\\eta} (\\rho)\\right) ", "cc68f37709dedd41c90c40e2600a95ce": "(W-X^T1)", "cc68f930c932ef424d78a4067dd5e24c": "\\gamma = 43/32", "cc699a96bff1fb4d20f3739b9cfe04e2": "(P \\to Q) \\to (\\neg P \\or Q)", "cc69fd198b08b394a91a3abf0c7f1aa7": "y=x_1w_1 + x_2w_2", "cc6a0cfb519f574f2b2c231bbdf4647d": " \\Delta f", "cc6a4490e2523dd1bf8c8caf71204977": " Cf(x) = \\sup_N\\left|\\int_{-N}^N \\hat f(y)e^{2\\pi i xy} \\, dy\\right|", "cc6a4939a074cad9b80b555b7043d302": "\\|\\cdot\\|_1", "cc6afbba5b5619ca87ffc32d40295d7a": "S=\\{s_1,s_2,\\dotsc,s_n\\}", "cc6b0360ef232125b71cc1f6ad6adc0e": " w\\mathcal{C} ", "cc6b7a1a465d9a8afa0ffe009b5b4a30": "\\tau=0.", "cc6b85c06079ddd483d4b55bf873b772": "y=|2a\\{ft\\}-a\\,|", "cc6b90cb71889e6a18142a13a0a09713": "H_4.", "cc6b9f4f21b5cc1bdb5ae043aa9697a3": "F(x;p,\\beta)=1-\\frac{\\ln(1-(1-p) e^{-\\beta x})}{\\ln p},", "cc6bd29fef11dae24a66bd7e106b0fd0": "\\mathbb{R} \\times\\!\\, S", "cc6c619249bf4dc010ce27c04759dfa5": "s\\begin{Bmatrix} p \\\\ q \\end{Bmatrix}", "cc6c6730b1a20d640d877ea33484e8b9": "V_t \\geq V_d", "cc6c6c5156d8ab2ce93114321f54c62f": "\\alpha_0\\,\\!", "cc6c7c32629287bfdd8081431a5c6517": "x = 3 - \\frac{3}{2}y.", "cc6ce75bad7c23a96459049bf4b7514b": "\\Gamma(\\theta_1:\\theta_0) = 2 [ F(\\theta_1)-F(\\theta_0) ]\\,", "cc6d020fe81b41bf7eaaf873ebdca85f": "\\langle\\psi|", "cc6d0579cf49af23e0bc0f7b59087a73": "\\sigma(t) = t + 1", "cc6d15c02abc9a5d2d6d106a4eeae87b": "\\displaystyle \\psi = a\\psi_1 + b \\psi_2 ", "cc6d34bccd97c33b4306e3174a214970": "\\varepsilon=t^{-2(1-p_3)},", "cc6da02094ab069849773882f7c7227e": "\\mathbb{R}\\cup\\{\\infty\\}", "cc6da03ca37ebcfc890225103430fa32": "x_i = \\tfrac{2i}{n}", "cc6da4f321f9ee6fa68fdb01fd453d9a": "X \\times \\mathbb{R}/R\\mathbb{Z}", "cc6dc4e0ea62f80941f5d03556456c9d": "T_j=\\sum_{i=1}^{g_j} (t_i^3-t_i),", "cc6dcf96aeb36fcbd29a170184d97880": "n_0 \\cdot p_0 = n_i^2\\ ", "cc6dd6828046dafdd4b260ad83dba73a": "\\Omega \\mapsto A^T \\Omega A", "cc6e222b7c7734e13cbc239d86265ec6": "x - x' \\equiv \\pm c \\cdot r^j \\pmod{m}", "cc6e2809a1bdfd18eb9d4a29f9be6957": "\\theta_W", "cc6eae945ee6cba00e06ca5a667d3252": "(\\mathbb{Z}/2\\mathbb{Z})[T]/(T^2+T+1)=\\mathbb{Z}[T]/\\langle 2,T^2+T+1\\rangle", "cc6ec4e4acb916392d890ed0973b04ba": "\\sum_{k=0}^\\infty (-1)^k k!", "cc6ecf22df4f885f6ecf1f4296648928": "CD^3C", "cc6ef904bc1ef8ffa817613af7aa0a7b": "+g_{\\alpha \\beta }g^{ \\sigma \\rho }(\\sqrt{-g}g^{\\mu \\alpha }),_{ \\sigma }(\\sqrt{-g}g^{\\nu \\beta }),_{\\rho }+\\,", "cc6f00bf1a41b672c4eb30f978b0aa41": "(0,c)", "cc6f3f5127dda28d880e39a15d170765": "\\begin{matrix} {1 \\choose 1}{11 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "cc6fad416e60e6fbaa789cc1f91599db": "b \\,\\!.", "cc700bda8f9ccf494d7cec844b54f387": "\\bigoplus_{i + j = k} H_i(X; F) \\otimes H_j(Y; F) \\cong H_k(X \\times Y; F).", "cc701c11e6c0f906616ed02aa460dd41": "k \\cdot r < j \\text{ and } j + 1 \\le (k + 1) \\cdot r \\text{ and } m \\cdot s < j \\text{ and } j + 1 \\le (m + 1) \\cdot s \\,.", "cc702cc32a3d46eb00ccf88222e3e944": "\\alpha_t(s,a) = 0.1", "cc704a42d5f5b33830680809d969fae3": "[\\omega]\\in H^2(M)", "cc706b5cdd99e6e52bbdc002f5a58f77": "\\mathrm{SL}_n(\\mathbb{C})", "cc7112390844170597d4e2e7a58d5e8d": "W_{i i} = w_i", "cc712c4972c258adbda6c41d476a96e8": "S=\\{(d,\\sigma)| d \\in S', \\sigma \\in \\mathbb{T}^\\infty \\} ", "cc719f2a3238fa1f7ba5d390e6e6f30d": "\\sum\\limits_{i = 1}^N {e_{c,i}^2 } = \\sum\\limits_{i = 1}^N {(y_i e_{c,i}^{} )^2 } = \\sum\\limits_{i = 1}^N {e_i^2 } = \\sum\\limits_{i = 1}^N {\\left( {y_i - (w^T \\phi (x_i ) + b)} \\right)} ^2,", "cc71ab3fb7d1a8aba67319563d6ac1e4": "\\omega_{p}({\\mathbf e}_j,{\\mathbf f}_k)=\\delta_{jk}\\,", "cc71dcf23e3dd03386cb2aaca571bd3d": " y \\in \\arg \\min \\limits_{z \\in Y} \\{ f(x,z) = ( f_1 (x,z),f_2 (x,z),\\ldots,f_q (x,z) ) : g_{j}(x,z) \\leq 0, j \\in \\{ 1,2,\\ldots,J \\} \\}", "cc721a7ecda72457f8720ca3e3266e5d": "\\epsilon_1=\\epsilon_2=\\ldots=\\epsilon_6=0", "cc721c24e0caad2f21055f34fe47ab55": "E=\\alpha +x\\beta", "cc7222566200e8f76c227ed7eb27f000": "\\chi\\psi", "cc722e1c9fd579e54ae59e2e6a091bf4": "\\overline\\pi:Y\\to X", "cc723626a161718a2b63b1985e2157bb": "\\xi^a \\ ", "cc724a334d60fd9bfe5924192cc9aba4": "\\textstyle \\Lambda_0(P)", "cc7257f5d4c5f050382b55df5e8a54da": "\\rho(\\nu,T) = \\frac{8\\pi h\\nu^3/c^3}{e^\\frac{h\\nu}{kT} - 1} \\approx \\frac{8 \\pi kT\\nu^2}{c^3}", "cc729ec4bc200844b9516cb14ca1f3ab": "\n\\Lambda = \\left\\{ \\sum_{i=1}^n a_i v_i \\; | \\; a_i \\in\\Bbb{Z} \\right\\}\n", "cc731d499192325e7ad0a537560c864a": "\\begin{smallmatrix}{{h}_{\\odot}}\\end{smallmatrix}", "cc731f6d8a0068f96352d9fd9e8186a6": "{\\color{Blue}~2.12}", "cc7360bd769faf6ba0c1c75237f558c0": " (-1.00, 0.00);", "cc737d5096e749a8c74c05a3388f9552": "{x^0 =1}", "cc739ee6abd190c56cfc743180780718": "P(t)dt=fe^{-ft}dt\\,", "cc73a15cf83a1691b2e2094ad368b228": " \\frac {i_L} {i_S} = A_{FB} \\frac {R_{C2}} {R_{C2} + R_L} \\ . ", "cc7427cd707cd67ce8153b9573c61891": "\\alpha = C/2I\\,", "cc7462e51c916d30a2ce9af2b15aae46": " R=1/a ", "cc746a1315bc8ac98b6a4132890c179a": "\\rho U^2/2,", "cc7513d9c8d0ddd11db117b4c43b39f4": "P_{-i}", "cc75586493bf274a2eb1727b13325eab": "(\\tfrac{b}{a}) = 1", "cc75861bea913cda3d8ec755991acc26": "\\deg(v)\\,", "cc75fc6ae0c72197989239211b3f1180": "= -3 + 7 + 10 - (3\\times 5) + 7 + -4 + 0 + -1 + 0 = 1", "cc761e32d657529bc1048d0bcddb6a27": "x \\sim N(\\theta 1_n,I)\\,\\!", "cc76301c7ac29747600f65cd0d74ee91": "F=GMm/r^2", "cc76652bca06eff129ea27e847fb8ed7": "P_{j'}\\,", "cc766a243ad3c62d5e8272342806ff30": " {V_{true wind}}^2 = {V_{apparent wind}}^2 + {V_{boat}}^2 -2 \\times V_{apparent wind} \\times V_{boat}\n \\times cos (\\beta -\\pi) ", "cc7691aeccd63d93c89d4ae19e997662": "\\frac{n!}{m!(n-m)!}", "cc76ba5d2a470c6496dff08668a332c3": "\\mathbb{R}^{3N}", "cc771f6d1fd8ea65f44ad62befa37bb0": "s_m", "cc77350eac092f0d2b3caea4465fb4e0": "x(t) = \\frac{\\cos(2 \\pi B t + \\theta )}{\\cos(\\theta )}\\ = \\ \\cos(2 \\pi B t) - \\sin(2 \\pi B t)\\tan(\\theta ), \\quad -\\pi/2 < \\theta < \\pi/2.", "cc773ddbb744f26e4e0d266985b91f04": "\\bigcup_{i\\in I} \\bigg(\\bigcap_{j\\in J} A_{i,j}\\bigg) \\subseteq \\bigcap_{j\\in J} \\bigg(\\bigcup_{i\\in I} A_{i,j}\\bigg)", "cc77eab363c650b405adcec80e6102ad": "e^{(a_1i_1+a_2i_2+\\dots+a_7i_7)\\pi} + 1 = 0. \\,", "cc786e8c399b07183e7c5661669b20d0": "\\begin{align}\n \\mathbf{b}_1 & = \\cfrac{\\partial x_1}{\\partial q^1} \\mathbf{e}_1 + \\cfrac{\\partial x_2}{\\partial q^1} \\mathbf{e}_2 + \\cfrac{\\partial x_3}{\\partial q^1} \\mathbf{e}_3 \\\\\n \\mathbf{b}_2 & = \\cfrac{\\partial x_1}{\\partial q^2} \\mathbf{e}_1 + \\cfrac{\\partial x_2}{\\partial q^2} \\mathbf{e}_2 + \\cfrac{\\partial x_3}{\\partial q^2} \\mathbf{e}_3 \\\\\n \\mathbf{b}_3 & = \\cfrac{\\partial x_1}{\\partial q^3} \\mathbf{e}_1 + \\cfrac{\\partial x_2}{\\partial q^3} \\mathbf{e}_2 + \\cfrac{\\partial x_3}{\\partial q^3} \\mathbf{e}_3\n\\end{align}", "cc78931f752b674599aa843ba2df2726": "(X, \\mathcal{F})", "cc78eac91f402d47342655542e10d090": "L(M) = \\{ w\\in\\Sigma^* | (q_{0},w,Z) \\vdash_M^* (f,\\varepsilon,\\gamma)", "cc78f43512de1e7ab933b78331c67513": "x:\\Delta^n \\rightarrow X", "cc795b067ea95a05fda1274098344f37": "\\frac{360^{\\circ}}{5} \\times \\frac{1}{2} = 36^{\\circ}", "cc795e07b0b3bff69dffce0d605740e5": "w_{,2211} = w_{,1212} = w_{,1122}", "cc7968781c5014e6d85ee89660adbcad": "\\scriptstyle \\pi(S)", "cc79da40eb928bbbe9ac2c773ce5a293": "Q = b (d + c)", "cc79e54574e3b786ffa8bdcd704ba29f": "\nh(a)=\n\\begin{cases}\nf(a) & \\mbox{if }a\\in C, \\\\\ng^{-1}(a) & \\mbox{if }a\\notin C.\n\\end{cases}\n", "cc7a9c9b61ef0a0b9f90332074496a41": "\\begin{align}\n f(j\\omega) & = j(-1)^{(n-1)/2}\\big[a_0\\omega^n+a_1\\omega^{n-2}+a_2\\omega^{n-4}+\\cdots \\big] & {} \\quad (23)\\\\\n & + (-1)^{(n-1)/2}\\big[b_0\\omega^{n-1}+b_1\\omega^{n-3}+b_2\\omega^{n-5}+\\cdots \\big] & {}\\\\\n\\end{align}", "cc7b57aec5df30abdb36253b3a925cc0": "(16)\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;\\; \\rho_1 u_1 = \\rho_2 u_2 ", "cc7b692c4c4d21d51de5866c2edf3376": "|a_n\\rangle", "cc7b770176a94d55c90926d6b76cc812": "(x+3)^5(x-1)^{10}(x-5)", "cc7b954732282ac49ba5a1da669791db": "[x_w\\, y_w\\, z_w\\,1]^T", "cc7baa8a97ee74b1f67cfcf3e48c6e8a": "F^V", "cc7bbdac94a5433288a6623d20d80b0b": "Q_{s3} = q_{1} - q_{2} + q_{3} - q_{4}\\!", "cc7bc1ae39d22efac1f56671e1878c6e": "A\\cap B=\\varnothing", "cc7c3e399e67bf6ed3124d4722ede1e2": "{G}", "cc7cac8a2157793a6da5eb5cf8263866": "F[J;T]", "cc7cb142536053204c23dcdcad94a071": "x = g(p),", "cc7cefc700623bf9bfd80e074c98db37": "\\textstyle{\\frac{\\log{(3^{0.63}+2^{0.63})}} {\\log{2}}}", "cc7d61ddc1f1a3be88c469665d4627e2": "P\\;", "cc7d6ed744c6c23ef1fe2864b78967d1": "f(t) - \\frac{1}{2\\pi} \\int_0^{2\\pi} f(t)\\, dt", "cc7d6f9c6d082186dd1a05fa363d9eb9": "g^{\\varepsilon\\delta}", "cc7d76f20bd9b7357c8455a4911f90b5": "w_i,v_i", "cc7e0f170eb21a88870f227ff215a5fb": "m \\omega^2 r", "cc7e343464e8e5d5b482dbfc20511e00": "\nR^2=r^2+z^2\n", "cc7e45b3e7cac1a915f2ee4493199003": "\\vec{v}_s=\\frac{1}{m_4}\\frac{h}{2\\pi}\\vec{\\nabla}\\varphi.", "cc7e530f889a315ac5a811c13752c376": "u(j^{1}_{p}\\sigma) ", "cc7e9249048517ed1c2dacf705a04597": "\\Big( (\\mathcal{M}, s) \\models E\\phi \\Big) \\Leftrightarrow \\Big(\\pi\\models\\phi", "cc7ea1ea37df4fcd28494fb687821833": "D_\\mu \\eta_{IJ} = \\partial_a \\eta_{IJ} + \\omega_{\\mu}^{ \\ IK} \\eta_{KJ} + \\omega_{\\mu}^{ \\ JK} \\eta_{IK} = 0. ", "cc7eb20fb2b0a1a8edf50019a5cb2267": "\n\\frac{\\partial n}{\\partial t} + \\nabla\\cdot n\\mathbf{ v} = 0.\n", "cc7eb980f9fb37efbfc8521523c48116": "\\neg{x_i} \\cup \\neg{y_j}", "cc7f5953233a52cfef60f3717fe8f043": "\\left. y_{j+1}=\\left( \\sum_{i=1}^n \\frac{w_ix_i}{\\| x_i - y_j \\|} \\right) \\right/ \\left( \\sum_{i=1}^n \\frac{w_i}{\\| x_i - y_j \\|} \\right).", "cc7f8935435b8dce69549bdde5584d7a": "Z = Z_{0} \\sqrt{1 - \\left( \\frac{f_{c}}{f}\\right)^{2}} \\qquad \\mbox{(TM modes)}", "cc7fa86c97481627fc54120e21256216": ",\\overline{a} \\in V, u,v,x \\in V^*, uvx \\in V^+", "cc7fb3285de874429cca398a1e8b96e0": "\\scriptstyle\\tau\\ =\\ it", "cc7fbf01ee72335def1acb1108ab1357": "%D", "cc7fe676157778bde59abac43931cd76": "E = \\tfrac{1}{2}mv^2", "cc802e09bb634c13ff1b8f566aff7b11": "\\|\\,|u|+|v|\\,\\|_p\\geq \\|u\\|_p+\\|v\\|_p", "cc805784aeb452259265b1f70d178d6a": "P(x_t | y_{1:t} )", "cc8088850a10b32e276f26c5b19267ce": "N = N_0 e^{-\\lambda t}, \\,", "cc80953b6a835d43dc555d77e0f1f539": "(p) \\subset \\mathbb Z_p", "cc811b63e156fbe4de93871111968a0b": "\\operatorname{Hol}(C_3)", "cc81864621754a3fefaec7415822c538": "\n |G| = \\int_{x_1^0}^{x_1^1} \\cdots \\int_{x_r^0}^{x_r^1} \\omega(\\mathbf{x}) dx_1\\cdots dx_r .\n", "cc81b764a514e451d3eb71c8e9a48aef": "\\Pr\\left(\\bigcap_{i=1}^n A_i\\right)=\\prod_{i=1}^n \\Pr(A_i)={1\\over2^n}", "cc81d46e59489c7f60ff394055c6cea5": "\n\\mathfrak{S}_{s_r} \\mathfrak{S}_w = \\sum_{{i \\leq r < j} \\atop {\\ell(wt_{ij}) = \\ell(w)+1}} \\mathfrak{S}_{wt_{ij}},\n", "cc81ff7ca0faabf23b4c96aa4757fc9c": "p(\\xi)", "cc823ae0f99728e3db7f68406a2781cd": " \\scriptstyle \\beta_i =0 ", "cc82e40313cd3ef860a973032742b4b9": "0 + a = a + 0 = a", "cc831310d0bd95dde1b3a500efb03826": "\\lambda_{CWL}(M)=\\left\\vert\\mathrm{torsion}(H_1(M))\\right\\vert\\mathrm{Link}_M (\\gamma,\\gamma^\\prime)", "cc833fc3729cae172e86c587fed6abe0": " \\varepsilon^v_{S_1} ", "cc83a50c9f5e4c9f5294f91b57f9df55": "\\ D_{heel} \\approx D_{pitch} ", "cc8411477a9ce7178b0f3ce6d10cc2cf": "\\left.{\\begin{alignat}{5}\nC_1 &&\\; + &&\\; C_2 &&\\; = &&\\; \\frac{R_0-Q_0}{P} & \\\\\nB.C_1 &&\\; + &&\\; A.C_2 &&\\; = &&\\; \\frac{1}{P}.[B.R_0-A.Q_0]\\end{alignat}} \\right|{\\begin{alignat}{5}\nC_1 &&\\; = &&\\; \\frac{R_0}{P} & \\\\\nC_2 &&\\; = &&\\; -\\frac{Q_0}{P}\\end{alignat}}", "cc8415d7e3533c233d83189ed6395f80": "\\frac{n^{2}-1}{n^{2}+2}(1/\\overline{\\rho })=a_{0}+a_{1}\\overline{\\rho}+a_{2}\\overline{T}+a_{3}{\\overline{\\lambda}}^{2}\\overline{T}+\\frac{a_{4}}{{\\overline{\\lambda}}^{2}}+\\frac{a_{5}}{{\\overline{\\lambda }}^{2}-{\\overline{\\lambda}}_{\\mathit{UV}}^{2}}+\\frac{a_{6}}{{\\overline{\\lambda}}^{2}-{\\overline{\\lambda }}_{\\mathit{IR}}^{2}}+a_{7}{\\overline{\\rho}}^{2}", "cc841c66f92981f7b264844494132e12": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} = \n \\left\\{\n \\int_{-h}^h \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix}\n", "cc8424d97488cc36e59431404dcfcad0": "a_t = dv/dt \\,", "cc85166814f0d661b9ae4c21285292fb": "x_{0}(z) = 1", "cc853aacae03476fccfaa91fa0c53684": "\\mathbf{T}_L = \\mathbf{X} \\mathbf{W}_L", "cc85442e120917f13e1973ffd1460df0": " h(x,t) = c x ", "cc85773987db604d20d1e2fcb837ac21": " \\mathrm d\\varphi_x(\\gamma'(0)) = (\\varphi\\circ\\gamma)'(0).", "cc85a34897dc882ba8432783a55a1730": "Y = \\Gamma / \\{(f(b), 0) - (0, f'(b))\\;|\\;b \\in B\\}", "cc85c3dd5a50eef4bcef8ba9d7d6c4bc": "\n\\operatorname{Var}(\\overline{X}_n) = \\operatorname{Var}(\\tfrac1n(X_1+\\cdots+X_n)) = \\frac{1}{n^2} \\operatorname{Var}(X_1+\\cdots+X_n) = \\frac{n\\sigma^2}{n^2} = \\frac{\\sigma^2}{n}.\n", "cc86020e37af1bbd60852de7544200cf": "x_0 = r\\cos\\psi\\ ", "cc861db657e5a0a9e4e4e18ac8bd6e3b": "\\exists x\\,( P(x) \\, \\wedge \\neg \\exists y\\,(P(y) \\wedge y \\ne x))", "cc86e23d13ef6831d827c856fcb59869": "\\theta\n= t-t'", "cc86f0ff71098e3650045e541e2d8cd3": " y_i = \\beta_0 + \\beta_1 x_{i1} + \\cdots + \\beta_p x_{ip} + \\varepsilon_i ", "cc8755609ad61864910f145119713de9": "/x", "cc88562d115c2650309a281d04111354": "x_{m+1}(z) = x_{m-1}(z) + a_m\\cdot((2\\cdot m+1)\\cdot z + (2\\cdot m-1)\\cdot z^{-1}) \\cdot z^{(-1)^m} \\cdot x_{m}(z)", "cc885c69369a1aa9f4a192d8e579fb66": "z = x + iy\\,", "cc88b6c98114322535fea92f67e01bbe": "\\hat{g} ,\\hat{h}\\,", "cc88c53f882b7ce058838d046d9a27fc": " x \\to \\hat x , \\, p \\to i \\hbar \\frac{\\partial}{\\partial x}", "cc88c6b107029ef8500d29ed30bbcf08": "\\ln = ", "cc88d3b74abe2e6bb27e91ff3065ebb1": "d(g,u)=d(f,g)-d(f,u) \\quad ", "cc88d88c0f1103bd24a916f33458633f": "w_{2}<0", "cc88fabf1bbef9d3d891a6abd52365ac": "m(0) = 0", "cc8907c54e95eeb9d3a39b66df34f68d": "q_k = \\frac{(k + 1) p_{k + 1}}{z}", "cc896200d119b37a55d729662a0c8d62": "\\hat{\\theta} = \\arg\\min_{\\displaystyle\\theta}\\left(\\sum_{i=1}^n\\rho(x_i, \\theta)\\right) \\,\\!", "cc897278bc5cd7de0f5c7ef826393eef": "\\langle x, x' \\rangle = x'(x)", "cc89af74350ab8588514ae6380ff5741": " x^{(0)} =\n \\begin{bmatrix}\n 1.0 \\\\\n 1.0\n \\end{bmatrix}.", "cc89b456de890259237e4b5a7656e93e": "\n\\delta = 2 \\ln \\left( \\sec\\theta + \\tan\\theta \\right).\n", "cc8a7316ba5643697bcd44c3d2dcec05": "2^5P[S_5=k]", "cc8a7ed65cad2a9551d485056e8c9776": "\\{(x_1,y_1),\\dots,(x_n, y_n)\\}", "cc8b0de3d0c35809bf6cc588ff71758f": "L(Y)", "cc8b3ef24569380a24bcba048905955a": "(\\alpha^2-\\beta^2)u_{j,iij}+\\beta^2u_{i,imm} = 0.\\,\\!", "cc8bca0d0c51cfc8e551a5e0f7084bca": "f_1(z)=\\frac{(1+i)z}{2}", "cc8bf27dbe85776693ed0b12774dad6a": "\\begin{bmatrix}\\Psi\\end{bmatrix}^{T}", "cc8c0683fc87aa5f5f5bb341d20d61cb": "[g]([x_1,x_2]) =[{-\\infty}, {\\infty}]", "cc8c163118790b9976db9be0376c280b": "S=\\sum_{n=0}^{\\infty}{i^n\\over n!}\\int\\prod_{j=1}^n d^4 x_j T\\prod_{j=1}^n L_v(x_j)\\equiv\\sum_{n=0}^{\\infty}S^{(n)}\\;,\n", "cc8c250f831aecf73f2bb3b4e00c11f6": "\\theta = 2\\pi - \\theta^\\prime ", "cc8c93da8e028286c78c2dd654805628": " [\\mathbf{\\hat T} (\\varepsilon),\\mathbf{\\hat H}] = 0 ", "cc8cef394d958acf1e0e9476f92c90fb": " \\kappa^{-1}(\\mathrm{nm}) = \\frac{0.304}{\\sqrt{I(\\mathrm{M})}}", "cc8d1da80dcae9ee600d5c6a0d08c656": "c_0, c_1, c_2 ", "cc8d928cc40d4459f31526710d6b089e": "(\\phi \\to \\psi) \\to \\neg (\\phi \\wedge \\neg \\psi)", "cc8da90dab6fec5c6e3b4d5e568c3c68": "{\\mathbf{}}J/T", "cc8dd020c719b49531ab6c8d088de623": "\\phi(v_i) = (1+\\exp(-v_i))^{-1}", "cc8e74e676e0fbcda371ec1014438ef6": "(\\square_x + m^2)G(x,y)=-\\delta(x-y)", "cc8e78843d3a70ad5aab212a5889c061": " \\Phi(\\Psi) ", "cc8e8882bd8f208484e88203f1d7e17d": "\n\\begin{cases}\n w^T \\phi (x_i ) + b \\ge 1, & \\text{if } \\quad y_i = + 1 , \\\\\n w^T \\phi (x_i ) + b \\le - 1, & \\text{if } \\quad y_i = - 1 .\n\\end{cases}", "cc8ea92b1445f0a6f2dc36960d05ff53": " u_{n+1} = \\left( \\frac{x}{1-x} u_n \\right)^\\prime, \\quad u_0=1, ", "cc8f0347df43b8a5536e4d5ac74ec2f2": "V_i, V_j", "cc8f3e113033cd2c55506c99bcfa0901": "v^i =\\frac{\\partial Z^i \\left( t ,S \\right)}{\\partial t }", "cc8f4a5649878dd15c687467972d7068": "\\langle \\Psi_1 , \\Psi_2 \\rangle = \\int\\limits_{-\\infty}^\\infty d x \\, \\Psi_1^*(x, t)\\Psi_2(x, t) \\,,", "cc8f669fffb7079f4b30b7cfa6c2aaeb": "a''", "cc8fd70efdf9a86f1c63b0020ef16f90": "\\tfrac12", "cc901b4cf9e43d80d36f1e37ae82d3c7": "N_{\\mathrm{Fe}}", "cc90624f4e3643d7e8505fdb13d31057": "f/D", "cc9063dd69feb1b8c5421472779b8cd7": "\\operatorname{d}\\omega", "cc910772719ce2e5871afc4b9986ea66": "\\mathit{b}_0\\mathit{b}_2...\\mathit{b}_{2n-2}", "cc910841ca5ace57e1078ddb6921d567": "\\widehat{Q}=", "cc910ae6a6fa560c1f761bc3655c4888": " \\text{sum} = I(C) + I(A) - I(B) - I(D). \\, ", "cc91826fdb2b47199cafc39678f5d29d": "\\sum_{i=0}^\\infty x_iy_i.", "cc91aa2a82b220ba5b47e8ec3f24e33a": "1/C\\,", "cc91d68a04862a6f67fb4bea6bfa6d0a": "\n Q = \\sum_{j=1}^n \\overline{z_j'} z_j = \\sum_{j=1}^n \\| z_j \\|^2\n ", "cc923ed9278ae397fea6b4fc5151831d": "\\ \\mathrm {ln} ", "cc926efe8bc101a09b16875c127a9260": "\\tfrac{1}{a^{2}}+\\tfrac{1}{b^{2}}=\\tfrac{1}{d^{2}}", "cc92fe482941ab923ae2fb8b972d775e": "\\,\\text{ not} (a \\text{ and } b) = \\text{ not } a \\text{ or}\\text{ not } b", "cc935bce3e65759ef29c2e23f421d1bd": "m_k = \\sup_x p_{y|x}(y_k|x) ", "cc9363063e4c4a0eb281cb960fa57de9": "F(s) = \\frac{F_s\\left( \\frac{1}{s} \\right)}{s}. \\,", "cc93699887fe78e4c7fab5031efe3529": "\\eta_{00}=1, \\quad \\eta_{i0}=\\eta_{0i}=0,\\quad \\eta_{ij}=-\\delta_{ij}\\,.", "cc9376e1d4fe987ecd3c5a2a7f6364ab": " \\underbrace{\\left|x\\right|+\\cdots+\\left|x\\right|}_{n \\text{ terms}} < 1 \\text{ for every finite cardinal number } n.\\,", "cc944d7ec7b5922cd8747f714c348481": "P_k(m,n)", "cc946525403b12a8e9388569568a81e5": " v_z ", "cc94e9ffa64ca5fe3ed3ab86a720e103": "\\Delta_{S^{n-1}}^{-1}", "cc94faf8e232888e038523bca172e3e1": "\\displaystyle{\\Delta(z)=\\det (I -zT^2).}", "cc9595024551a8b508fbaa301550d22a": "f(x_{N+2})", "cc95b5b3277ea08d6daf43aedf13eac8": "\n\\boldsymbol{\\alpha}=(1,0,\\dots,0)\n", "cc95bacb78e2ebc28907b73e3245f098": " \\rho(f) = \\frac{1}{V}\\int_\\Omega \\delta(F(\\boldsymbol{r}) - f) \\, d\\boldsymbol{r}", "cc95f1673771ca5901cfab85cd8f9e09": " \\Phi^{0} := \\Delta(1) ", "cc9601a40fd2989e901e2712b3253444": "0 \\to \\ker f \\to V \\to W \\to \\mathrm{coker}\\,f \\to 0.", "cc96040fdc9b987bdb3d2ca049174ac4": " w_2 > w_1 ", "cc9624d895ec64562f59319588888c57": "O(l^2)", "cc9633cfc6ac004fbb06ab78424606fd": "\n\\begin{bmatrix}\n \\cos\\theta & -\\sin\\theta & 0\\\\\n \\sin\\theta & \\cos\\theta & 0\\\\\n 0 & 0 & 1\n\\end{bmatrix}\n", "cc965ee70d4c0964417ac3870531b619": "\\phi(n_2)", "cc969877ed118130aec8bd9f11c8b292": "\\nabla \\mathbf{f}=\\frac{\\partial {{f}_{i}}}{\\partial {{x}_{j}}}{{\\mathbf{e}}_{i}}{{\\mathbf{e}}_{j}}", "cc9704c949df7b459f7cbd1ecd4872b9": " \\nabla y_{t}=(\\rho-1)y_{t-1}+u_{t}=\\delta y_{t-1}+u_{t}\\,", "cc972cf0fe94717a9ed43509c48916ed": "2^{-\\sum_{i=1}^N \\frac{1}{N} \\log_2 q(x_i)}", "cc9766f7d306f2667a4c4d144e7083ca": "\\mathrm{V_{CE2} = V_{CE1} + V_{BE2} > V_{BE2}} \\Rightarrow \\mathrm{V_{C2} > V_{B2}}", "cc97e80baffe594b59a3b83d26ec6cd4": "\\mathrm{bei}(x) = \\sum_{k \\geq 0} \\frac{(-1)^k (x/2)^{4k+2}}{[(2k+1)!]^2}", "cc97f5cd56a4f55c55f332df5dc80e9c": " \\theta_0 = 2 \\psi_0. ", "cc983d6f0d852cebc6ecacfec4963600": "\\ \\mathbf u(\\mathbf X,t) = \\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad \\text{or}\\qquad u_i = x_i - \\delta_{iJ}X_J = x_i - X_i ", "cc98c3332f71a340a4771a652eb5a07e": "\\Pi^* ", "cc995096409deb6cd9bbcaafe263b5fc": "f(\\alpha x, \\alpha y)= \\alpha f(x,y)", "cc998a9b96f17c6fe29af5c7c1c66b1d": "e' = E(s')", "cc9990400b21299814691e1ed508edcf": "\\theta=\\operatorname{arctan}(\\rho/z)=\\operatorname{arccos}(z/r)", "cc99a79f7c3cbceb700ff33efcf444f0": "\\text{Tp}(\\text{Prim})=\\text{Prim}\\,\\!", "cc99c2cf5235dfcbb7309ea224f1a068": "\\operatorname{E}(e^{-a(w-\\operatorname{E}w)})", "cc9a2e5d6a1ba09507f153cc34e29efd": " Q(t_2) \\ = \\ Q(t_1) + Q_{\\rm{IN}} - Q_{\\rm{OUT}}. ", "cc9a42601305ebadd5801ca5925a5db2": "\\lambda_i = \\gamma_i + \\sum_{j=1}^m p_{ji}\\lambda_j.", "cc9a67228c0818f778bd4146c2e6a9cf": "v^b(k)", "cc9a874c1f66900abd4b88d159f46827": "= \\frac{-p(x)y(x)y'(x)|_a^b + \\int_a^b\\left[p(x)y'(x)^2 + q(x)y(x)^2\\right] \\, dx}{\\int_a^b{w(x)y(x)^2} \\, dx}.", "cc9a91f16516abfa67785bda30683125": "\n0_{K_{m,n}} = \\begin{bmatrix}\n0_K & 0_K & \\cdots & 0_K \\\\\n0_K & 0_K & \\cdots & 0_K \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0_K & 0_K & \\cdots & 0_K \\end{bmatrix}_{m \\times n}\n", "cc9ab83e18754b5f1d52fd6c1c0b832b": "G_{\\mu\\nu}^a", "cc9ae53d2e220d55402ed13834d8c284": "\\frac{dp}{dt} = -\\frac{\\partial H}{\\partial q}", "cc9b1a286ee822fb51a4f0fc82752604": " Cr\\lbrace A \\in B \\rbrace = \\dfrac{1}{2}(Sup_{t \\in B} u(t) + 1 - Sup_{t \\in B^{c}} u(t)) ", "cc9b322dcfe560582c45cf283f33f65f": "r = r_0 + r'_0 t + r''_0 \\frac{t^2}{2!} + (etc) ", "cc9b7e13646f4320fb68a4be065effe4": "\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{23} \\\\ \\sigma_{31} \\\\ \\sigma_{12} \\end{bmatrix} =\n \\begin{bmatrix} C_{11} & C_{12} & 0 & 0 & 0 \\\\ C_{12} & C_{22} & 0 & 0 & 0 \\\\\n 0 & 0 & C_{44} & 0 & 0 \\\\\n 0 & 0 & 0 & C_{55} & 0 \\\\ 0 & 0 & 0 & 0 & C_{66}\\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{23} \\\\ \\varepsilon_{31} \\\\\\varepsilon_{12}\\end{bmatrix}\n", "cc9c329b38547566fccc2040e9cbc8f4": "\\varphi^{n+1} = \\varphi^{n} + \\Delta t F( \\varphi^{n+1} ),", "cc9c4f6d6f9479ea7cf827678901691f": " \\lim_{x\\rightarrow x_{0^-}}\\!\\!\\!u(x) = \\!\\!\\!\\lim_{x\\rightarrow x_{0^+}}\\!\\!\\!u(x) ", "cc9c6e952b61f19c343eb495ea2dd243": "3\\cdot A_6,", "cc9cec27b0564116db7b912e945698bc": "\\scriptstyle F_i:\\, k\\left[M\\right] \\,\\to\\, k", "cc9d12d4ec2029e7046567a36e1dd6c5": " S = \\frac{ 1 }{ ( e^{ \\frac{ \\sigma^2 }{ 2 } } + 1 ) ( e^{ \\mu + \\sigma^2 } ) } ", "cc9d2fd8d552c60302c7d6b29f06257d": " \\mathbf{G}^i=[A(\\theta_i)]\\mathbf{g},\\quad \\mathbf{W}^i=[A(\\psi_i)]\\mathbf{w} + \\mathbf{C},\\quad i=1, \\ldots, 5,", "cc9d7942db5d41d60f41f16bf8c359f7": "m(x,y,y) = m(y,y,x) = x.", "cc9daba77f953d3722794e8a7202db78": "r > r^*", "cc9e0ba8b4adc7e41fb9389552ddc653": "\n\\frac{100 \\times \\text{hip circumference in m}}{\\text{height in m} \\times \\sqrt{\\text{height}}} - 18\n", "cc9e9fd15a8118d61f280d9a871f61bb": "\ne^2=\\frac{a^2-b^2}{a^2}\n", "cc9eb11134baa1fd098009e22e67bfcb": "d^2G := n^2 dS \\cos{\\theta} d\\Omega \\ ", "cc9ed14b7aa90ae77b937af18873d01b": "\\textstyle Mn2^{l-1}", "cc9edce508aa75ad83444aee30a45085": "{\\alpha}= (1-D) \\bar \\alpha(\\theta_i) + D \\bar{ \\bar \\alpha}.", "cc9ee7586024e61069fa7f16dc10b210": "x = \\rho \\sin[n (\\lambda - \\lambda_0)]", "cc9ef172a65b319c7c41335509cca75e": "N\\in\\N", "cc9f3373d2ed6b9c70149e23b4ecf64b": "A_i = 0", "cc9f5f5673e953c234b9c6feb414cbd7": "\\mathit{c}_{p}=\\left(\\frac{\\mathit{KR}}{\\mathit{{K}-1}}\\right)", "cc9fdb05c3401d49be4f29866c581910": "h(m,m_b) =h(m',m'_b)", "cca0015f78a95d757dd224f60e195cb4": "\\|R_n\\| < \\mathrm{threshold} ", "cca0260cc5ca55a9c355dbe2b038d375": "\\sum \\lambda_n^2 < \\infty, \\ \\ K(x, y) \\sim \\sum \\lambda_n \\varphi_n(x) \\overline{\\varphi_n(y)},", "cca03eb2d296ba47ccceecd2a2346f42": "V_B = \\gamma \\left[ e^{-r/r_0} - \\left( \\frac{r_0}{r} \\right)^6 \\right].", "cca04c293f41db559008585a840345d1": "(\\exists x)(x=\\lnot y)", "cca05ab2f01f614f10cfd4a45cd19876": "1 + 1 + \\cdots + 1. \\, ", "cca11376df7a03c3af5f8fa43eb23b09": "\\begin{align}\nx &= \\xi (s) \\\\\ny &= \\eta (s)\n\\end{align}", "cca1184fa95c9fa047d3bfb838569624": " \\iint_D \\left[ \\nabla u \\cdot \\nabla v + f v \\right] \\, dx\\, dy + \\int_C \\left[ \\sigma u v + g v \\right] \\, ds =0. \\,", "cca125feee0eb6270b50e7ce4d592669": "L: X \\times Y^* \\to \\mathbb{R} \\cup \\{+\\infty\\}", "cca143be389914cb83cbb8b665c400e6": "area = \\frac{\\sin\\theta}2 r^2 ", "cca169bda5c2dd48a2770e18740f62c9": "p||q=p b^{l(q)}+q", "cca19d9c5ddf095740f0cb2c1dcbe638": " m( x ) = \\frac { \\sqrt{ A( x ) C( x ) - B( x )^2 }} {\\pi A( x )} ", "cca1b6090ecebd6b40317d1fdc7b4314": "m=1,3,5,...", "cca1ba21507170f591b01e97fa15decd": "D_{xx}", "cca26b9a66e851975226a13a140f3768": " Loss = 10 \\log { \\frac { P_{pad} + P_{load}} { P_{load} } } = -20 \\log { \\frac { 2 \\sqrt{Z_1/Z_2} } { 1 + (R_a+Z_1)(R_b+Z_2)/(R_bZ_2) } } \\, ", "cca2735ca4c18bc948d0b6bf63bbcaa0": "\\psi_{1s} \\approx \\psi_{2s}", "cca301156b2826c7e501377f4fd4cf54": "\\scriptstyle y_{op}", "cca33f0274fcfc0aff2141b1ae2d3b2d": "\\omega,\\omega^\\omega,\\omega^{\\omega^\\omega},\\dots", "cca375ee0deaf58b0e647566b55d6ed5": "X_{ij}=\\phi_j(x_i),", "cca38a8a99409ecda8fca12c0b19f8b3": "\\exists f \\forall x \\, \\phi(x,f(x))", "cca3ef45510837379a907b6b5a5aa8a1": "x_{j=0 \\cdots n-1}^{\\prime} = x_0^\\prime - j d ", "cca3f3abada32f996aa1d201fde525d0": "f_n = \\chi_{[\\frac{j}{2^k},\\frac{j+1}{2^k}]}", "cca403fb03b20e3ba1f042ea19abe168": "P_1, \\ldots , P_D \\in E[p]", "cca427fae5726472f93c618f6e5120ed": "I = \\frac{1}{a} \\int _0^{\\frac{\\pi}{2}} \\frac{1}{\\sqrt{1 - k^2 \\sin^2(\\theta)}} \\, d\\theta = \\frac{1}{a} F\\left( \\frac{\\pi}{2},k\\right) = \\frac{1}{a} K(k)", "cca46344a1cc5749c03add657dd4eeb7": "\\angle PML = \\angle PCL = 180^\\circ - \\angle ACP", "cca47621d2368bfe26987d8bb0a0b04a": "f^{-1}f_*\\mathcal{F} \\rightarrow \\mathcal{F}", "cca47635e94d1bd7db33ad0e7feb2da0": "\\frac{\\Gamma \\vdash P\\to Q ~~~ \\Gamma \\vdash\\neg Q}{\\Gamma \\vdash \\neg P}", "cca4c90f8b3c7b0e23cdc2992764d4e6": " f\\,", "cca517f4cba0fcb2e18709cb5bf286da": "S_4 \\times Z_7,", "cca52696c8d498a0ddf47652190790df": "\\kappa\\theta\\ = LF \\,", "cca57bf7b154888ffbb91412becbed21": " \\cdot\\left(\\text{largest monomial of }s_2\\right)^{i_{n-1}-i_{n-2}}", "cca5802ff11722b290f540916e0dc349": "\\mathbb{U}-{\\overline P}X", "cca59b492fd171cdfef9311327764667": "r_{s} = \\frac{2GM}{c^{2}}", "cca5a783e3c4780481c7edc0de56814e": "n(t,k)", "cca5e71d973d61a105d174f5c649974a": "\\int x^2\\,\\operatorname{artanh}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{artanh}(a\\,x)}{3}+\n \\frac{\\ln\\left(1-a^2\\,x^2\\right)}{6\\,a^3}+\\frac{x^2}{6\\,a}+C", "cca632a4a43bf0bf1fb2ff4ca91930f4": "\n(2n-3)!! = \\frac{(2n-3)!}{2^{n-2}(n-2)!} \\,,\\,\\text{for}\\,n \\ge 2\n", "cca6401967762e4ba10401e761138036": "\n\\begin{align}\n\\int_V \\frac{\\partial u_i}{\\partial x_j} \\sigma_{ij} - u_i f_i dV\n &= \\int_V \\frac12 \\left[ \\left( \\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i} \\right) \n + \\left( \\frac{\\partial u_i}{\\partial x_j} - \\frac{\\partial u_j}{\\partial x_i} \\right) \\right] \\sigma_{ij} - u_i f_i dV \\\\\n &= \\int_V \\left[ \\epsilon_{ij} \n + \\frac12 \\left( \\frac{\\partial u_i}{\\partial x_j} - \\frac{\\partial u_j}{\\partial x_i} \\right) \\right] \\sigma_{ij} - u_i f_i dV \\\\\n &= \\int_V \\epsilon_{ij} \\sigma_{ij} - u_i f_i dV\\\\\n &= \\int_V \\boldsymbol\\epsilon : \\boldsymbol\\sigma - \\mathbf u \\cdot \\mathbf f dV\n\\end{align}\n", "cca65bd549d017d6df09f7c20b2c20af": "(2^{16}+1)", "cca6b7880532baece5137f5de8a39ccd": "\\mathrm{^{249}_{\\ 97}Bk\\ \\xrightarrow {+\\alpha} \\ ^{249,\\ 250,\\ 251,\\ 252}_{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 99}Es}", "cca71611e08895882747d250a3401c93": "{p \\over 1-p} = e^{-2\\beta}.", "cca71657fab004e89757a90fb8cc4dc9": "\\omega(g(n))", "cca7432c852a6028279270e12abbb690": " | p \\rangle,p\\in \\mathbb{R} ", "cca79bb88963c81c7b7cc5edce50c4c5": "\\widehat X", "cca800603df17126dd2955586ff24429": "\n M_2 = -50 + R_a(x-10) - \\frac{x^2}{2} \\,.\n ", "cca8094af9a30e2dfcba1819224d839c": "\n\\tilde{S}(q,\\omega) \\ = \\ \n\\frac{1}{\\nu Dq^2} \n\\left[\\frac{2\\omega}{1-e^{-\\omega/T}}\\right].\n", "cca8231b7131d9d136d3479d01bc0104": "\n\\bar\\eta_{a\\mu\\nu} \\bar\\eta_{a\\rho\\sigma}\n= \\delta_{\\mu\\rho} \\delta_{\\nu\\sigma}\n- \\delta_{\\mu\\sigma} \\delta_{\\nu\\rho}\n- \\epsilon_{\\mu\\nu\\rho\\sigma} \\ .\n", "cca887f5132610fda379010e99b4124b": "\\displaystyle x_{n+1}=x_{n}+\\alpha_{n} s_{n}", "cca899f1c78dd271924037866292e8f5": "\\sum\\limits_{n\\in A}\\exp{\\biggl(2\\pi i\\,\\frac{an^{*}+bn}{m}\\biggr)},", "cca9931a8e80d5c99c45cc055bfe870e": " \\underline{\\mathsf{f}}(A \\wedge B) = \\underline{\\mathsf{f}}(A) \\wedge \\underline{\\mathsf{f}}(B)", "cca9e10d1e602c08b8b4d6beab7c486b": "{{P}_{V}}f(u,\\xi )=\\int_{-\\infty }^{\\infty }{f(u+\\frac{\\tau }{2}).}{{f}^{*}}(u-\\frac{\\tau }{2}).{{e}^{-i\\tau \\xi }}d\\tau ", "cca9efc9bd671ed543fa9ceeda49adea": "A \\supseteq L^p_+", "ccaa514b753925296c3804ecedfdab27": " (q^3,q^2)", "ccaa600483a2cb169b9a302f32d9d471": "\\varphi(z) = \\lim_{k\\to\\infty} \\frac{1}{(|z_{kr} - z^*|\\alpha^{k})}.", "ccaa67070733e31e9379fe5f01e29794": "\\displaystyle{\\varphi=u_- - u_+.}", "ccaaa9fbf22e8892bc4890a6422f0dce": "P_{\\mathrm{ref}} = 20\\, \\mathrm{[W]} + 4.32\\, \\mathrm{[W/dm^2]}\\cdot A.", "ccaac04a3e4e5d22956e9bfa727d7d65": "X_\\text{Ci}", "ccaac9efa070a9e4b9445d666e631ada": "(X, \\mu, T)", "ccab45dcf9c1c0a0dcc78b77a130122c": "\\Box p \\leftrightarrow \\lnot \\Diamond \\lnot p", "ccac0948aaa06325b76c33fe1ecee294": "2^{10} \\approx 10^3", "ccac2af58355a235c579da5a6d4dfb10": "V_a \\sim \\frac{m_e}{2}{\\omega_1^2}(x^2+y^2)+\\frac{m_e}{2}{\\omega_2^2}z^2", "ccac33b743b5ba218307f1e979dc668b": "x_2:={dx \\over dt} + F(x)", "ccacc300cf937249ba4b18ed26964d8b": "(e^2-1)/2", "ccacc40bfa434ff00427cfdf0c3874f8": "h(s)=\\sum_{i=0}^{n-1}s[i] \\cdot 31^{n-1-i}", "ccace7f824f9b7f1e92d3564d400f5a3": "P dv + v dP = Z R dT", "ccacfc8524493aeb9481b82204dcd2ff": "\\tfrac{4}{3} \\div \\tfrac{5}{4} = \\tfrac{16}{15}", "ccad261053dcd690160c645bf7a459de": "\\tfrac{1}{6000}", "ccad7b79f362118df4321c1e1ca54cae": " \\mathbf{R}_{AB}+\\mathbf{r}_{Bj}+\\mathbf{r}_{ji}-\\mathbf{r}_{iA} = 0\n\\quad\\Leftrightarrow\\quad\n\\mathbf{r}_{ij} = \\mathbf{R}_{AB}-\\mathbf{r}_{Ai}+\\mathbf{r}_{Bj} .\n", "ccad7d4037baa9ddcec17a16a9894666": "V \\otimes W := F(V \\times W) / R.", "ccae003d7b720557b0f2a8550c173775": " {(Pa \\cdot s)} ", "ccae37ceb3fc424f585f57ea37fa3e31": "\\scriptstyle \\phi(z)\\, = \\,\\frac{1}{z}", "ccae4bc3459c5efd05c13566299f70eb": "\\begin{align}\n ~R_1 &= \\frac{n_1 + n_2}{2n_1 + n_2}\\\\\n ~R_2 &= \\frac{n_1 + n_2}{n_1 + 2n_2}\n\\end{align}", "ccaeb5c92ed85073fff4c9e44c3a8023": "s = \\frac{2u}{1+u \\cdot u}.", "ccaf1017525795fb9d79a944e38a58e6": "y \\equiv x \\pmod{ r^n -1}, \\frac{m}{r+1}", "ccaf4f04ba16060d124445a660109560": "\\mathbf{\\mathit{g}}_{l}", "ccaf6a2c9553b7db89088388a47c7270": " V_n=\\frac{S_n+S_{n+1}+\\ldots+S_{2n-1}}{n} ", "ccafbeea9765ec3754018ad59dd7b0f6": " \n(Eq. 2) \\text{ } \\Delta(t) + Vp(t) \\leq B + Vp(t) + \\sum_{i=1}^K Q_i(t)y_i(t)\n", "ccaff244f3d9f904bfb5fe17abc8c839": "\\neq2", "ccaff89a4b2220fcffa3b82a13de55d8": "\\varprojlim:C^I\\rightarrow C", "ccb021135a6319fb912955d1c1ce3f89": "X = (X_1,\\dots,X_n)", "ccb04c78e2b5e02923bd445113c922c4": " 2a+1 ", "ccb05c57904d9eaf4f1621df829215be": " (\\mathcal{L}_X X)^{\\rho} = [X, X]^{\\rho} = 0 \\,.", "ccb06e1ef947951b02a10244716e4647": "\\sum_{n=1}^\\infty\\frac{e^n}{n}.", "ccb07c3549f46f9b4d09f8824b69eb27": "Z_0 = \\frac{120}{\\sqrt{\\epsilon_r}}\\ln\\left(\\frac{2l}{R}\\right),\n", "ccb0e9a5756c7c8fb36a0e07ebb9ed4a": "\n (\\and L_2)\n ", "ccb0f771c7b000f2679a3fe4c176f2c2": " \\varphi_e \\not \\simeq \\varphi_{F(e)}", "ccb143e297503e641d2a602f8558c95a": "r=\\frac{dr}{d\\theta}\\tan \\psi.", "ccb160b2497fb4a427b53c81731fe68b": "T_1^{(4)} = \\partial_x, \\quad T_2^{(4)} = \\partial_y", "ccb16d9d09abd28b1e64e6b08de973c8": "\\mathrm{Ku} = \\frac{U_h \\rho_g^{1/2}}{\\left({\\sigma g (\\rho_l - \\rho_g)}\\right)^{1/4}}", "ccb196a6ea16bcf9aa2e03cfa0561933": "\\mathbf{n}_1,\\mathbf{n}_2,\\mathbf{n}_3", "ccb19a49979223453d8fd99846b4d96e": "\\sigma_{i j}=\\frac{1}{4\\pi}\\left(E_{i}E_{j}+H_{i}H_{j}-\n\\frac{1}{2}(E^2+H^2)\\delta_{ij}\\right)", "ccb1b8dcb289377961e048a7de93b7b4": "\\mathbf b(\\mathbf x, t)", "ccb1bd05148bbdf25a9aa4d1d935df60": "E_{nm}\\equiv E_n^{(0)}-E_m^{(0)}", "ccb2448e837a6e3e725ba224c534159c": "\\frac{(\\lambda-\\mu)\\beta}{(\\mu(\\alpha+\\beta)-\\beta\\lambda)(\\alpha+\\beta)}(\\mu,\\lambda-\\mu).", "ccb25cd3ecbfe43bfab10f36055c6d9f": "q^i_t=\\Gamma (t,q^i).", "ccb3020d2d02920ef081c5f5e8bdc595": "a(x).P", "ccb3545aa7ed3f965bf18df2eb7b5985": "\\displaystyle z=\\frac{-xH(x)^5}{G(x)^5}", "ccb374da60a1baefe15becdfb3e020db": " \\mathsf{W} = \\sum_{i=1}^n \\mathsf{W}_i = \\sum_{i=1}^n (\\mathbf{F}_i, \\mathbf{R}_i\\times\\mathbf{F}_i). ", "ccb37e92fd72b686f049365873ceae4f": "\\gamma = 17.6 \\cdot g \\ ", "ccb3b649a59db67ca577d22185e0952a": "\\frac{1}{2}(x_2 + x_1)", "ccb3be77a4f951dee03089118e6096f6": "(x_1, ..., x_n, x_{n+1})", "ccb3c25fc4f0f922248df23294d8ee7d": "\n\\sigma _{i\\mu \\nu }=\\frac{1}{2i}[\\gamma _{i\\mu },\\gamma _{i\\nu }]; i=1,2.\n", "ccb3c907d1bc3752137cb842b07e7ac2": " \\Delta S = Q\\left(\\frac {1}{T_2} - \\frac {1}{T_1}\\right)", "ccb40aa5c9b87f68f25bdbbea779bea6": "\\text{Conversion Percent}\\ (%)", "ccb43fab5ac8817ef0f8fdfc878deb78": "A=\\mathbf{Z}_3=\\{0,1,2\\}", "ccb47932030f53fe2beb7f9efdd79522": "r', r'' > 0", "ccb4e2cdb8557a13be2859d2a2c77eb4": "x^2-4x+4", "ccb51a0353658d47d321a7994ab7843b": "e^Z = e^X e^Y,\\,\\!", "ccb51c819b516cd657a1d4ef51faae25": "\\exp(i\\theta[\\sigma])", "ccb5d76d76558d59a5fa6d320ad278f5": "\\mu_i\\ (i = 1,\\ldots,m)", "ccb6607c9f3e3521a8aeb9e734cf8d60": "[\\mathbb{N}]^k=C_1 \\cup C_2 \\cup \\cdots \\cup C_r", "ccb662f63b49174e7259c658fbb42e14": "z^2 = x^2 +i2xy - y^2\\ ", "ccb6d3739dc0bd01f2d1bfc08542bf2a": "d(0,1)=0", "ccb6e3bbb8759cd1db526926aa3ef25b": "P(Y \\mid X) ", "ccb6ee048caeaa64e51fe2ed715f944a": "n^{1+c/\\log\\log n},", "ccb751f4d960ef4f1b74db4ab9aa18b8": "\\mathbf{m}=p\\boldsymbol{\\ell}.", "ccb7531976ac279cc84c3047847adc3b": "\\gamma_{zx}=2\\epsilon_{zx}", "ccb791c2c99ea8c017466fff12b8bbf3": "r_a = \\frac{2K}{c-a+b} = \\sqrt{\\frac{s (s-b)(s-c)}{s-a}}.", "ccb7cb94e9125569da9474a995b8e40e": "0.7G_Q", "ccb7d595bd134d4f2cd97d6453301054": "\\partial_t \\left( \\tfrac12\\, \\eta^2 \\right) + \\partial_x \\left\\{ \\tfrac12\\, \\sqrt{g\\, h}\\, \\eta^2 + \\tfrac12\\, \\sqrt{\\frac{g}{h}}\\, \\eta^3 + \\tfrac1{12}\\, h^2 \\sqrt{g\\, h}\\, \\left[ \\partial_x^2\\left(\\eta^2\\right) - 3 \\left( \\partial_x \\eta \\right)^2 \\right] \\right\\} = 0,", "ccb7ed3c208bd659e1e911692a51e2a5": "180 deg", "ccb7ff65904eb611e707642e5f2175a0": "\\begin{align}\n j\\left(i\\right) &= j\\left( \\tfrac{1 + i}{2} \\right) = 1 \\\\\n j\\left(\\sqrt{2}i\\right) &= \\left( \\tfrac{5}{3} \\right)^3\\\\\n j\\left(2i\\right) &= \\left( \\tfrac{11}{2} \\right)^3\\\\\n j\\left(2\\sqrt{2}i\\right) &= \\left( \\tfrac{5}{6}\\left[19 + 13\\sqrt{2}\\right] \\right)^3\\\\\n j\\left(4i\\right) &= \\left( \\tfrac{1}{4}\\left[724 + 513\\sqrt{2}\\right] \\right)^3\\\\\n j\\left( \\tfrac{1 + 2i}{2} \\right) &= \\left(\\tfrac{1}{4}\\left[724 - 513\\sqrt{2}\\right]\\right)^3\\\\\n j\\left( \\tfrac{1 + 2\\sqrt{2}i}{3} \\right) &= \\left(\\tfrac{5}{6}\\left[19 - 13\\sqrt{2}\\right] \\right)^3\\\\\n j\\left(3i\\right) &= \\left(2 + \\sqrt{3}\\right)^2\\left(\\tfrac{1}{3}\\left[21 + 20\\sqrt{3}\\right] \\right)^3\\\\\n j\\left(2\\sqrt{3}i\\right) &= \\tfrac{125}{16}\\left(30 + 17\\sqrt{3}\\right)^3\\\\\n j\\left( \\tfrac{1 + 7\\sqrt{3}i}{2} \\right) &= -\\tfrac{1}{7} \\left(40\\left[651 + 142\\sqrt{21}\\right]\\right)^3\\\\\n j\\left( \\tfrac{1 + 3\\sqrt{11}i}{10} \\right) &= \\tfrac{64}{27} \\left(23 - 4\\sqrt{33}\\right)^2 \\left(-77 + 15\\sqrt{33}\\right)^3\\\\\n j\\left(\\sqrt{21}i\\right) &= \\left( \\tfrac{1}{2}\\left[5 + 3\\sqrt{3}\\right]\\left[3 + \\sqrt{7}\\right] \\right)^2 \\left(\\tfrac{1}{2}\\left[65 + 34\\sqrt{3} + 26\\sqrt{7} + 15\\sqrt{21}\\right] \\right)^3\\\\\n j\\left( \\tfrac{\\sqrt{30}i}{1} \\right) &= \\left( \\tfrac{1}{2} \\left[10 + 7\\sqrt{2} + 4\\sqrt{5} + 3\\sqrt{10} \\right] \\right)^4 \\left( 55 + 30\\sqrt{2} + 12\\sqrt{5} + 10\\sqrt{10} \\right)^3\\\\\n j\\left( \\tfrac{\\sqrt{30}i}{2} \\right) &= \\left( \\tfrac{1}{2} \\left[10 + 7\\sqrt{2} - 4\\sqrt{5} - 3\\sqrt{10} \\right] \\right)^4 \\left( 55 + 30\\sqrt{2} - 12\\sqrt{5} - 10\\sqrt{10} \\right)^3\\\\\n j\\left( \\tfrac{\\sqrt{30}i}{5} \\right) &= \\left( \\tfrac{1}{2} \\left[10 - 7\\sqrt{2} + 4\\sqrt{5} - 3\\sqrt{10} \\right] \\right)^4 \\left( 55 - 30\\sqrt{2} + 12\\sqrt{5} - 10\\sqrt{10} \\right)^3\\\\\n j\\left( \\tfrac{\\sqrt{30}i}{10} \\right) &= \\left( \\tfrac{1}{2} \\left[10 - 7\\sqrt{2} - 4\\sqrt{5} + 3\\sqrt{10} \\right] \\right)^4 \\left( 55 - 30\\sqrt{2} - 12\\sqrt{5} + 10\\sqrt{10} \\right)^3\\\\\n j(\\sqrt{70}i) &= \\left( 1 + \\tfrac{9}{4}\\left(303 + 220\\sqrt{2} + 139\\sqrt{5} + 96\\sqrt{10}\\right)^2 \\right)^3\\\\\nj(7i) &= \\left( 1 + \\tfrac{9}{4}\\sqrt{21+8\\sqrt{7}} \\left(30 + 11\\sqrt{7} + (6+\\sqrt{7})\\sqrt{21+8\\sqrt{7}}\\right)^2 \\right)^3\\\\\nj(8i) &= \\left( 1 + \\tfrac{9}{4}(1+\\sqrt{2})\\sqrt{\\sqrt{2}} \\left(123 + 104\\sqrt{\\sqrt{2}} + 88\\sqrt{2} + 73\\sqrt{2}\\sqrt{\\sqrt{2}}\\right)^2 \\right)^3\\\\\n j\\left( \\tfrac{1 + \\sqrt{1435}i}{2} \\right) &= \\left( 1 - 9\\left[ 9892538 + 4424079\\sqrt{5} + 1544955\\sqrt{41} + 690925\\sqrt{205} \\right]^2 \\right)^3\\\\\nj\\left(\\tfrac{1 + \\sqrt{1555}i}{2} \\right) &= \\left(1-9\\left[22297077+9971556\\sqrt{5}+(3571365+1597163\\sqrt{5})\\sqrt{\\tfrac{31+21\\sqrt{5}}{2}}\\right]^2\\right)^3\n\\end{align}", "ccb813e429ce5b9a723127df2ffddaec": "-\\mu/T", "ccb8359ef91d2d234b2a969a012f8c4e": "\\Upsilon_v = \\exp { \\sum_{t=1}^\\infty \\frac{z^t}{t!} \\int \\left [1 - \\prod_{i = 1}^{t} \\chi^v_i \\right ] {\\mathcal U}^{(t)}_{1...t} d\\boldsymbol{r}_1...d\\boldsymbol{r}_t }, ", "ccb85225f73ad3eec06e945a193ffea9": "\\dot{\\mathbf{x}}(t) = \\left(A + B K \\right) \\mathbf{x}(t)", "ccb88d0f3b55840c0f53a8b7ccf92ae1": "\\mathcal{H}_2 \\approx 0", "ccb9012c94ae46fb9b1d39e5a813d6e9": "{3 \\choose 2}_q = \\frac{(1-q^3)(1-q^2)}{(1-q)(1-q^2)}=1+q+q^2", "ccb94d0415f8314005a6658ec50aa371": "\\boldsymbol{F} +m r\\dot\\theta '^2\\hat{\\mathbf{r}} -m 2\\dot r \\dot\\theta '\\hat{\\boldsymbol\\theta} +m \\left( 2 r \\Omega \\dot\\theta ' + r \\Omega^2 \\right)\\hat{\\mathbf{r}} - m\\left( 2 \\dot r \\Omega \\right) \\hat{\\boldsymbol\\theta} ", "ccb99ca281478441174c1c1c9f06b472": "|a| = \\sqrt{a^2}", "ccb9be84d0e65930b0162c6749fe354c": "p+r \\leq 1", "ccba01f564fecad5be1e8a495fd45c97": "\\alpha' (t) = X (\\alpha (t)).\\,", "ccbb4ccbbd40677f2de36082a09b9479": "\\mathbf{1}_{A} (x) := \\begin{cases} 1, & x \\in A; \\\\ 0, & x \\not \\in A. \\end{cases}", "ccbb59c61c600845aee50f920a6ab140": "\\sigma_1-\\sigma_2", "ccbb8515396ac036bfcd9b0d3b68b50d": "E =- \\frac{N_AMz^+z^- e^2 }{4 \\pi \\epsilon_0 r_0}\\left(1-\\frac{1}{n}\\right)", "ccbc494455848c952d33db52721dd3d9": "M1=Mb*mm \\,", "ccbc6ec13df477b0f5023bdc6dc000a5": "Y=(Y_1,\\ldots,Y_n)^T", "ccbd1147fd7126b33bc0c20d8be548fb": "f_0(975)", "ccbd557f518f6f7f83b75677a9b00e87": " \\beta_j ", "ccbd71603ac99751e2d00ec64971d890": "E(\\theta) - E(0) = N\\rho_s\\theta_x^2", "ccbdeba2b52b814959ae05506fc16de7": "\n\\varphi(n) > \\frac {n} {e^\\gamma\\; \\log \\log n + \\frac {3} {\\log \\log n}} \n", "ccbe966fc60ffe0fd0a063c7486fda49": "\\phi = \\frac{V_V}{V_T}", "ccbeb0354c8cba70a4a7691e5a2e5203": "C^0", "ccbebda6b2d0360600aa1f28530e8d19": " {LR_N(\\beta_{ML,1},\\beta_{ML,2})} = L^1_N-L^2_N-\\frac{K_1-K_2} {2} \\log N", "ccbec4d41ae0b7aeec8a0d39e9ed4c74": "\\overline{r}", "ccbf4b842a9ec358731f59dc75399f1a": "\\aleph_1 = \\beth_1", "ccbf571995f8a74625e3ba820ba2d655": "\\phi _j", "ccbf5b3b8ad7523bf6d46a659b526b7e": "(X, \\|\\cdot\\|)", "ccbf973a25145261bb7c7e698f5ad112": " \\operatorname{Ker}(A^+) = \\operatorname{Ker}(A^*)\\,\\!", "ccc0527e14b66f6e263754cbb039cd50": "\\mathcal{H=}\\left( \\mathfrak{p-}A\\right) ^{2}+(m_{w}+S)^{2}\\approx 0\\,,", "ccc0541ab1d39d2ffb4ee00eb5b65950": "Q^c\\,(1,\\overline{N_f})_{-1/N_c,(N_f-N_c)/N_f}", "ccc06e814d06b17877d05e430f2a8146": "\\eta=(1-R_1)(1-e^{-\\alpha d} )[ \\frac {( 1 +R_2 e^{-\\alpha d} )}{(1 - \\sqrt{R_1 R_2} e^{- \\alpha_c d})^2}] ", "ccc0babd139621dd97fb4d1add5596e7": "t^{\\prime\n},t^{\\prime \\prime }\\in \\lbrack 0,1]", "ccc0c86aea353ff1c6c3a83e7fba98e7": "\\sum_{i=1}^{n} a_i^k = \\sum_{j=1}^{m} b_j^k", "ccc0dacef1a934c8dbf0651386018bd1": "\\mathbf{\\Psi}_{11}= -\\sqrt{\\frac{3}{8\\pi}}\\mathrm{e}^{\\mathrm{i}\\varphi}\\left(\\cos\\theta\\,\\hat{\\mathbf{\\theta}}+\\mathrm{i}\\,\\hat{\\mathbf{\\varphi}}\\right)", "ccc0dd3965935ac655a8f99c61d16771": "u = 0\\text{ on }\\partial \\Omega.\\,", "ccc0f39a3dd3f4848dbf281969a7a7e4": "dX(t) = d\\Gamma(t) + X(t)\\left[r(t)dt + dA(t)\\right]+ \\sum_{n=1}^N \\left[ \\pi_n(t) \\left( b_n(t) + \\delta_n(t) - r(t) \\right) \\right] + \\sum_{d=1}^D \\left[\\sum_{n=1}^N \\pi_n(t) \\sigma_{n,d}(t)\\right]dW_d(t)", "ccc104ad4ae3a211ee9396535e2b920a": "V_s = V_{s_{0}} + G \\pi \\rho_m \\cdot f (\\epsilon) \\cdot \\Delta d^2,", "ccc15fe62f2cf3380f8251d1bf8e42be": "L^{p_1}", "ccc20f2f450fe7b627e4abd8dd15af9c": "\\scriptstyle a_0 = 1 \\quad b_0 = \\frac{1}{\\sqrt 2} \\quad t_0 = \\frac{1}{4} \\quad p_0 = 1", "ccc29067abe06c8000613079a3fbf28c": "\\mathbf{N}_\\parallel' = \\mathbf{N}_\\parallel ", "ccc2b3ded92f88df5aaf39a01602fb9c": "s_5(x)=-\\frac{15}{2}x-\\frac{15}{8}x^3-\\frac{1}{32}x^5;", "ccc308841adec167ce69b726f54d9dcd": "\\left [\n\\begin{smallmatrix}\n 2 & -1 \\\\\n -1 & 2 \n\\end{smallmatrix}\\right ]", "ccc308b6d0e33d27302ab87d830b9cdf": "\ns_N = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\overline{x})^2},\n", "ccc335866724fae2a332405f0dcd202a": "\\operatorname{probit}(0.025) = -1.96 = -\\operatorname{probit}(0.975)", "ccc43808209a0d01746a0ac2f98e1537": "(x-3) (x-1)^{14} x^4 (x+1)^{14} (x+3) (x^2-5)^3 (x^2-3)^{11}(x^2-x-3) (x^2+x-3)", "ccc48e7282f20b443fd011edd66febc9": " \\alpha = 2 / (N + 1) ", "ccc4910a5ef94c5bffa42c4412353fa9": " \n\\Delta^1_{\\rm LONG}= \\frac{\\pi}{180}a \\cos \\psi \\,\\! ", "ccc4a90bef097263c47464d1f28b0d38": "K(X)\\cong[X_+,\\mathbb{Z}\\times BU]", "ccc5a753bb6e093efc65e23b46599ccb": "x_n(t)", "ccc5b131a672eaefdbcd5b319d770b3c": " \\mathbf{M}{x_1 \\choose \\theta_1} = {x_2 \\choose \\theta_2} = \\lambda {x_1 \\choose \\theta_1} ", "ccc61d832a64b46b752889ce429e7a7f": "\\Delta -\\Sigma^* = \\Sigma^* - \\Xi^* = \\Xi^* - \\Omega = a_1 - 2a_2 \\approx \\, 147 ~\\mathrm{MeV}/c^2", "ccc633b1395eb39cd68cfa3d2e61ef43": "l) = 1 ", "ccc663f4b075a7abc153b3a256634698": "A - S", "ccc6b27e10bc69086a28a8262cea7bed": "H_4", "ccc6f1d63b7a1b0da9a610b65f12bf2c": "V_{-}", "ccc7041e554c70174bb8cfb10b3a7dd0": "d = r \\Delta\\sigma.", "ccc72c25125a9f05474707666da18e3a": "c_i,\\,i=1,2,\\ldots,s", "ccc72e5d728bcafd7debaf2f72a8678b": "\\textstyle A \\subset (0,1) ", "ccc797f785000782ca09a9a56792b225": " \\mathbf{j}(\\mathbf{r},t) = \\frac{\\hbar}{2mi} \\left [ \\Psi^{*} \\left ( \\nabla \\Psi \\right ) - \\Psi \\left ( \\nabla \\Psi^{*} \\right ) \\right ]. ", "ccc7d53914c9be11e3046f981d7faef8": "|s-b_k| < \\begin{matrix} \\frac12 \\end{matrix} |b_k - b_{k-1}|", "ccc86015aa97b6a6ade16105ad3cb62e": "\\displaystyle{f_z -1,\\,\\,f_{\\overline{z}}\\in L^p(\\mathbf{C}).}", "ccc88e8f67719e1f2c69a5ba76fecfbc": "\n(x_1-x_2)(y_3-y_4)-(y_1-y_2)(x_3-x_4)=0\\text{ if the lines are parallel}\n", "ccc89ce16d970958c1ad99e315113d55": "\\left\\{\\land, \\lor, \\neg, \\supset, \\bot \\right\\}", "ccc89f358f3ca1e03420c234d87b9999": "\\sum T", "ccc8e939dbb30d88dc5fd1583b4d7886": "\n\\begin{align}\n\\theta_{ji}|G_j &\\sim G_j \\\\\nx_{ji}|\\theta_{ji} &\\sim F(\\theta_{ji})\n\\end{align}\n", "ccc912815b4a1b38fe8ece549c4f382c": "0\\!\\,", "ccc932d67625a237e395c076f6515887": "\\scriptstyle t \\,=\\, T", "ccc93e58ba8f11b12eee94f8fd2e18e6": "\\mathrm P(G|S,R)", "ccc9482ff31b40d210e98dead1d828e7": "\\begin{smallmatrix}r_{ap}\\ =\\ (1\\ +\\ e)\\cdot a\\ \\approx\\ 124\\end{smallmatrix}", "ccc959735b2070f281a672a4cd4e7ca9": "[\\cdot,\\cdot]", "ccc9b3e02d6e10eae3eba2c526df540b": " T'(E) \\rightarrow (\\partial S(E)/\\partial E) T(E),", "ccc9cbef9389c219b019e898202da75c": "G \\leq \\Gamma{}L(1,p^d);", "ccc9f6b39c9c5d4a64c6350b991d8dfb": "\\mathop{\\text{M-lim}}_{n \\to \\infty} F_{n} = F \\text{ or } F \\xrightarrow[n \\to \\infty]{\\mathrm{M}} F.", "ccca4701694c1874e1c16aa23a66262a": " -\\hat{\\mathbf{t}} ", "ccca4dd2043d9f7f433b171cadcb5843": "P \\wedge \\lnot Q \\Rightarrow P \\wedge \\lnot P \\equiv \\bot", "cccae01a924bbbb2309cd87a752251b6": "O(r^{\\delta})", "cccae0faab73ff1787d2e0d7aa3e8490": "\\mathbb{P}^2", "cccaf5f2680e57c429e50e22b32128d0": "F_\\mathit{Hooke} = F_{x+2h} - F_x = k \\left [ {u(x+2h,t) - u(x+h,t)} \\right ] - k[u(x+h,t) - u(x,t)]", "cccaf9cf753f5113de36a50ebd37e735": "= 0 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; Eq.9", "cccb0397f01b45ade1eb063fab7373fb": "t = \\tan\\left(\\frac{x}{2}\\right),", "cccb03987f87c93c8c2e6db8d2ddb418": "\\Phi_{q}\\left(\\mathrm{R}_{i}\\right)", "cccb1748f0baadce9089fe5189996c7c": "\\Gamma (z)", "cccb7d4a0278c9105fcabea07a792742": " [(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)) := f\\ ((x\\ f)\\ (x\\ f))] ", "cccc6fa0dbe214f3304a7729a9609e25": "Z = n \\times [Z] = n [Z]", "cccc8e5b67ac0a125d536e3efb998fc3": "{\\mathfrak h}", "ccccb996307d77f067cff4d0242ff36d": " \\varepsilon ", "ccccc98d7731d0dd97bad89c75a42f38": "n=f_{c}/f_{ref}", "cccccd71e817dc856a4009658f2cb868": "c_{8}-(c_{1}-b_{1})", "cccce7f792f0629e1c9f2392c424a61c": "\\{P_1,P_2,P_3\\}", "ccccf823b041685767ec325ec083e75d": "\\boldsymbol{\\phi}_{\\mathcal{B}}(x) = (\\phi_1(x),\\phi_2(x),\\ldots,\\phi_i(x),\\ldots,\\phi_\\ell(x)),", "cccd038ae851d99178089a90ebc25534": "\\begin{bmatrix} z_{11} & z_{12} \\\\ z_{21} & z_{22} \\end{bmatrix}", "cccd1eb617987b3ab1ca63b7cfb6dad7": "\\bar{z}", "cccd3c26364c95a6dfff66290750652e": "\\begin{align}\n d_p &= d\\; \\operatorname{mod}\\; (p-1) = 2753 \\; \\operatorname{mod}\\; (61-1) = 53 \\\\\n d_q &= d\\; \\operatorname{mod}\\;(q-1) = 2753 \\; \\operatorname{mod}\\; (53-1) = 49 \\\\\n q_\\text{inv} &= q^{-1} \\; \\operatorname{mod}\\; p = 53^{-1} \\; \\operatorname{mod}\\; 61 = 38 \\\\\n &\\Rightarrow (q_\\text{inv} \\times q) \\; \\operatorname{mod}\\; p = 38 \\times 53 \\; \\operatorname{mod}\\; 61= 1\n\\end{align}", "cccd6a47e1462654d78025f1d6a304ba": "\\phi_{A,B}, \\phi", "cccd92b8233703e124aac8a420e4f749": "R < r + a", "ccceb9bc23e5629d6741c2a109329657": " \\operatorname{Var}(I) = \\frac{NS_4-S_3S_5} {(N-1)(N-2)(N-3)(\\sum_{i} \\sum_{j} w_{ij})^2} ", "cccedd38ed2fa9670bb772e6671c7f40": "\\delta r = \\frac{{\\delta P}}{P}", "cccedf5dcc9ec2bceeadab0317761c61": "{{2p-1}\\choose{p-1}} \\equiv 1 \\pmod{p^4}.", "cccee59e418c22082035794672ac4734": "\\mathbf{e}_{\\pm}\\times\\mathbf{e}_{\\mp} = \\pm i \\mathbf{e}_0", "cccf1d373965e06cbd11024589ce531a": "\\inf \\{s : \\mathrm{some\\ c.e.}\\ s\\mathrm{-gale\\ succeeds\\ strongly\\ on\\ } X\\}", "cccf1de4eef967626037821f9c0f0885": "f(x) = x^2 - S = 0\\,\\!", "cccfcb59d31f1d2b84b88e227be0f795": " \\chi^2 = {(44 - 50)^2 \\over 50} + {(56 - 50)^2 \\over 50} = 1.44.", "ccd015b9406ec7c7a855ea871f413d25": "G = G_0 \\supseteq G_1 \\supseteq G_2 \\supseteq \\cdots \\supseteq G_n \\supseteq \\cdots", "ccd05572a6fe7953575f79d2e62f92da": "\n\\quad N(r,a,f) = N\\left(r,\\dfrac{1}{f-a}\\right),\n\\quad m(r,a,f) = m\\left(r,\\dfrac{1}{f-a}\\right).\\,", "ccd06684a05162d81a4f6c05b3396a0e": "\n\\bar{g}_{k_i}(s;L)= \\binom{k_{i-1}}{k_i} \\left(-\\frac{\\bar{h}(s,i+1;L)}{\\bar{h}(s,i;L)} \\right)^{k_i}.\n", "ccd07e30e8d348c1de8dec881b6a543f": "S/ L", "ccd0e89327807f3c22c19905e74e5a2f": "A_{e}", "ccd12978125cf7f43db32dbe2ef156a4": "f_j=e_j(\\alpha^{k_1}-\\alpha^j)", "ccd1e6b8b500b205e624d69ccea3f3ce": "\\mathrm{ad}_{[x,y]} = [\\mathrm{ad}_x,\\mathrm{ad}_y]", "ccd210175a931dd9fe57329c891be335": "A_n \\sim \\frac{c\\lambda^n}{n}", "ccd244ff84532f827768ec5c35dd760e": "\\!X = \\{(x:0), (x:1)\\}", "ccd26ba218d5a9540d8e2305618e155e": "t_\\text{P}", "ccd2987ef002f39666b03d100a0e701c": "z_n = e^{\\frac{2\\pi in}N} I\\;,", "ccd2ed6861ebd0985d68ef66e2e97259": "h_{a}(x)", "ccd332b3786211e98139727fde221eb8": "\\{\\tau,\\rho\\},", "ccd3f2a1c61cb749fa4ca248164f450d": "n^{k_2}", "ccd40c166138e5bd39d2ec0e12872dd8": "\\ \\|x(t)\\|_{\\infty} < \\infty", "ccd417c2156070e5477cee58698b5583": "\\Box \\bar{h}^{\\alpha \\beta} = 0 \\,", "ccd4902c7ff74c6566ee9e58961c2d6d": "\\begin{align}\n\\langle x, y\\rangle&={\\|x+y\\|^2-\\|x-y\\|^2\\over 4}\\\\\n&=\\frac{1}{4} \\left[ \\sum |x_i +y_i|^2 -\\sum|x_i-y_i|^2 \\right]\\\\\n&=\\frac{1}{4} \\left[ 4 \\sum x_i y_i \\right]\\\\\n&=(x\\cdot y),\n\\end{align}\n", "ccd4f7038c919980cd005c8d7c2c601e": "\\sum_{i=1}^k n_i \\leq (k-1)\\pi", "ccd508cc425902bbde3787a7bece01c9": "O\\left(N \\prod_{i=1}^{N} n_i\\right).", "ccd50b766cb9008eb0715720ef16ce45": " a_{n+k} = \\lambda_{k-1} a_{n+k-1} + \\lambda_{k-2} a_{n+k-2} + \\cdots + \\lambda_1 a_{n+1} + \\lambda_0 a_{n} + p(n) ", "ccd55617e4d71e087a7f9e201d509c19": " \\frac{n!}{k_1! k_2! \\cdots k_{m-1}!K!} \\frac{K!}{k_m! k_{m+1}!}=\\frac{n!}{k_1! k_2! \\cdots k_{m+1}!}.", "ccd592e4fb5e86f3be2827740cb84574": "\\chi_a=0", "ccd6136cdc902ca5040da7dd062a2250": "\\Phi = \\frac{U}{T}+\\frac{P V}{T} + \\sum_{i=1}^s (- \\frac{\\mu_i N}{T}) - \\frac {U}{T}", "ccd69d32b14fc1bbc15c0ab693de732e": "i_{0}", "ccd6c5945b5bf1c575fe3f153f862481": "~m", "ccd6d45c379ae1bf1e3a8c2f8193d7f1": "P = - \\frac{\\partial F_1}{\\partial Q} \\,\\!", "ccd6f07efae47b8d1f544925419e4b76": "I = \\left | U(P_1) \\right |^2", "ccd700d9e3f92a846c3bce134c681313": "f(x) = \\sqrt[3]{x}.", "ccd7978be93af9196d2f38f9d1d807ba": "\\mathbb{Z}'", "ccd7c228b11085f6a8f514d495af53a8": "\\psi :B\\rightarrow S(W)", "ccd7c3f10931b192310461b1aa291a6b": "m=0, 1, \\dots ", "ccd7ef8404924db44c923ddcd7600e52": "E(|(\\Delta_n)_i|^{-1})", "ccd7fd4071d5f665f8f42a9e502994e6": "r_{Ace}=\\frac{\\partial{V_{CE}}}{\\partial{I_C}}\\Bigg|_{I_B=const.}", "ccd8774fced4a770c707dd830d25047f": "\\begin{pmatrix}\n\\cos2\\theta & \\sin2\\theta \\\\ \\sin2\\theta & -\\cos2\\theta\n\\end{pmatrix}", "ccd87ffadc92f6f256b1f4cb7361017d": "f(x) = x^2+\\sin(x+4)\\ ", "ccd8a099d1b0b5711664cd4bb8b3d9d1": "\\hat{D}(\\alpha) \\hat{a} \\hat{D}^\\dagger(\\alpha)=\\hat{a}-\\alpha", "ccd8eb35214b46017235c8499347091f": "y_p = A x e^x.", "ccd901c5d0b6bd4babc7594b453b4adf": "\\sum a_n = \\sum \\lambda_n b_n", "ccd927b3c7b39c6199ecb917061756d1": "a^{m+n}", "ccd9b17843c25376a4063e9159ef4f73": " \\prod_{r=1}^2 \\Gamma(\\tfrac{r}{3}) = \\frac{2\\pi}{\\sqrt{3}} \\approx 3.6275987284684357012", "ccd9bb403dd08515f505e99f45d1af4e": "\\Delta \\left(\\frac{2Gm}{c^2}\\right) \\Delta\\overline{\\lambda}_{C} \\ge \\frac{G\\hbar}{c^3}", "ccd9cbe64dab8bb4c9edcaa48f7e4bfc": "N = 8 \\ln(2) \\cdot \\left(\\frac{t_R}{W_{1/2}}\\right)^2 \\,", "ccda6efbab419b62ef203d6c93b1a4bf": "d+1 = n_1+\\cdots+n_p", "ccda9db7a9c6ecdc9cd897478659d671": "\\displaystyle{{1\\over 2\\pi}\\int_0^{2\\pi} (f+iHf)^{2n} \\, d\\theta = 0.}", "ccdab239752d2132d816fe60741d8a82": "125 + 250 =", "ccdacecc9881f59d49f13c44202028ff": " -(\\lambda+\\alpha)C_{i-1}^{j+1} +(1+2\\lambda+\\beta)C_{i}^{j+1}-(\\lambda-\\alpha)C_{i+1}^{j+1}-\\beta C_{Mi}^{j+1} = +(\\lambda+\\alpha)C_{i-1}^{j} +(1-2\\lambda-\\beta)C_{i}^{j}+(\\lambda-\\alpha)C_{i+1}^{j}+\\beta C_{Mi}^{j}.", "ccdadb3f98610cbcbc5090248f7f62db": "a_1Z_1", "ccdb3480a93a14974300b702daac2472": "r = z_0 1_G + z_1 a + z_2 a^2\\,", "ccdb45911391de65029d264450c7a095": "z\\mapsto z^n+c", "ccdb6b7b977da7217853631f76610b90": "\\,i=1, \\ldots,3\\,", "ccdb8d7694e6e6f6e30db9572fe28f8e": "\\mathbb{P}^n.", "ccdbafcea6bdb3ce91a8332a1506ab02": "=\\frac{\\rho}{(2 \\pi RT)^{D/2}}e^{-\\frac{(\\vec{e})^2}{2RT}}(1+\\frac{\\vec{e}\\vec{u}}{RT}+\\frac{(\\vec{e}\\vec{u})^2}{2(RT)^2}-\\frac{\\vec{u}^2}{2RT}+...) ", "ccdbb63f52a07ca348ff4a44dc01393c": "\n\\begin{align}\nL &= T - V \\\\ \n&= \\frac{1}{2} M \\dot{x}^2 + \\frac{1}{2} m \\left[ \\left( \\dot x + \\ell \\dot\\theta \\cos \\theta \\right)^2 + \\left( \\ell \\dot\\theta \\sin \\theta \\right)^2 \\right] + m g \\ell \\cos \\theta \\\\\n&= \\frac{1}{2} \\left( M + m \\right) \\dot x^2 + m \\dot x \\ell \\dot \\theta \\cos \\theta + \\frac{1}{2} m \\ell^2 \\dot \\theta ^2 + m g \\ell \\cos \\theta \n\\end{align}\n", "ccdbc3be468d4eec6926dfa2db822c61": " \\forall |\\psi\\rangle\\langle \\psi |, |\\phi \\rangle\\langle \\phi | \\in P, \\quad \\exists U_g \\in G, \\quad U_g |\\phi \\rangle = | \\psi \\rangle ", "ccdbfdbc62fbff7e126f4d7b233aebb6": "\\Delta S \\geq 0 \\, ", "ccdca03db7a1368cd7b0659d64149714": "\\mathrm{csch}\\,\\theta = \\frac{1 - t^2}{2t},", "ccdcbe6e0f1cf236ff4ffe42304dc5f9": "\\delta\\Gamma^\\rho_{\\nu\\mu}", "ccdd15dcc247119891cbb928743f81a6": "h=l/(N-1)", "ccdd89ae429120974165e3acac7564ef": "\\operatorname{Alt}(v_1\\otimes\\dots\\otimes v_r) = \\frac{1}{r!}\\sum_{\\sigma\\in\\mathfrak{S}_r} \\operatorname{sgn}(\\sigma) v_{\\sigma(1)}\\otimes\\dots\\otimes v_{\\sigma(r)}", "ccdda93b4e0f8d7c19c646650be0298b": "p_{r+1}(x)=(x-a_{r,r})p_r(x)-a_{r,r-1}p_{r-1}(x)\\ldots-a_{r,0}p_0(x)", "ccddd87ed7853c4477d03a469fcb8a97": "\\mathrm{P}(A|BC) = \\frac{\\frac{1}{16}}{\\frac{1}{16} + \\frac{1}{16}} = \\tfrac{1}{2} = \\mathrm{P}(A)", "ccde1d6a7e3784ed7921e728d5cc2bb4": "p(x) = {n \\choose x}q^x (1-q)^{n-x}", "ccde48e0f6da31eb631722259ceeb419": "(Y - T - C) + (T - G) = I", "ccde5a042bf609663c76d1059857d159": "c>a=b", "ccde69a488f561b326e188683f7d955d": "TMT^T", "ccde72c7aa70ba328bd637902399367c": "l_2 = y + ar - e", "ccde82442fb5662413ed683924cd3eef": "\\sigma(x)", "ccde8493a73d4c00f733b513666589f7": " \\frac{\\partial(x,y)}{\\partial(s,t)} ", "ccdeb07b4a25172075377502f439421b": "\\alpha_{r + 1} = Q", "ccdeee8e64b2c477f521486c5280a034": "{}^2 E_{0q} = {}^2 E'_{0q}", "ccdf331d4f8f854b4189272022d94985": "c' 0.", "cceba52b4c33d986fb2631f18d851545": "p(h|n,b)\\,", "ccebd103f7ccd96884513c21aa9401c9": "S\\to(X\\to X)", "ccec05ef9c8e2607e95601edf2771f24": "\\mathbf{R}=\\mathbf{r}-\\mathbf{r}'", "ccec8b84d32202edf73a2c81b97b5da9": "\\gcd(a,b)=\\sum\\limits_{k=1}^a \\exp (2\\pi ikb/a) \\cdot \\sum\\limits_{d\\left| a\\right.} \\frac{c_d (k)}{d} ", "ccec9a59f5ad0935dad9cb295738fdcb": "B_H (t) = B_H (0) + \\frac{1}{\\Gamma(H+1/2)}\\left\\{\\int_{-\\infty}^0\\left[(t-s)^{H-1/2}-(-s)^{H-1/2}\\right]\\,dB(s) + \\int_0^t (t-s)^{H-1/2}\\,dB(s)\\right\\}", "cceceb04a5aab319219349024accdccb": "\\nabla_c \\, g_{ab} = 0.", "cceceeb64d236955a73878451dce0fb9": "L_v", "cced45e6c2204dc4f896824f2f9c31f9": "\\vec R = \\int_{0}^{l}\\hat t(s) ds", "cced5d500b8931cc44a3c67d047f2755": "H(x^\\mu)= H(0)+x^\\mu F_\\mu.", "ccedd61062d08c89e117bc183904392b": "(pb-aq)(pb+aq) = p^2b^2 - a^2q^2 = p^2(a^2+b^2) - a^2(p^2+q^2).", "ccede6f0cf39eeb6ff93d2d0bc6bc64d": "\\psi(\\Omega)", "ccedea187c904e8b7190ed1f10b98dfc": "\n\\begin{align}\n4\\Phi_{11}(z)\n&=4(z^{10}+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1)\\\\ \n&= (2z^5+z^4-2z^3+2z^2-z-2)^2+11z^2(z^3+1)^2\n\\end{align} \n", "ccedec18778095f8dd10b71abaf811b8": "Q(H)\\approx \\sqrt{\\frac{\\pi}{2}H}.", "ccee87d7967aad3a575850f68094e6d1": "\\left| {A_\\varepsilon}^{(n)} \\right| \\geqslant (1-\\varepsilon)2^{n(H(X)-\\varepsilon)}", "ccee8d326f9b30a50e06ed542d002128": "P(\\textrm{spike}) \\propto f(\\mathbf{k} \\cdot \\mathbf{x})", "ccef21ece2924eec59ef704a848b5f9e": "\\text{cl}:\\mathcal{P}(X)\\to\\mathcal{P}(X)\\,", "ccef904528c9a052a2165e1f600ada5a": "price = y", "ccf02f30ef40d9d89ab0a58c39c71aca": "h = c \\Lambda^{4-d},", "ccf07321ef9d02a8340c251a875a594a": " r_\\mathrm{ corr } = \\frac{ r ( 1 - \\theta c_x^2 ) }{ 1 - \\theta c_{ xy } } ", "ccf0c2f77b82e5476aa468fa98ef2385": "T = R_0 \\, \\tan (\\Phi)", "ccf0ec6d291aa9f7386950974600dfad": " \\mathbf{\\tau} = \\mathbf{d} \\times \\mathbf{F} .", "ccf1018d9f96536ea67509177bba568e": "\\nabla T \\,", "ccf11cdf33fbcad21e736f1bb65f863e": "\n\\begin{align}\n\\mathbf{x} & = \\textbf{P} \\textbf{X} \\\\\n\\mathbf{x'} & = \\textbf{P}' \\textbf{X}\n\\end{align}\n", "ccf1be30a4d6b690157aca0c2dc0a186": " \\left(\\cot(x)\\right)' = \\left(\\frac{\\cos(x)}{\\sin(x)}\\right)' = \\frac{-\\sin^2(x) - \\cos^2(x)}{\\sin^2(x)} = -(1+\\cot^2(x)) = -\\csc^2(x)", "ccf202da4d84023dfa9570ae7cd5be1e": "t=t(\\theta)", "ccf2879e1a0bdf3e10c6298fea0b95a2": "i_m = i_r + i_c", "ccf3441720b528bfbf141181336c8c6b": " \\frac{1}{\\sqrt{N_\\text{ZC}}} ", "ccf34b4099cd17d27604eef3948a80f7": "\\Omega_k", "ccf34eeee918edc0fe0eac0e6522a8dc": "\nd\\mathbf{E} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\left( \\frac{-1}{4\\pi\\epsilon} \\right) \\frac{dq \\ \\mathbf{r}}{r^3} = \n\\left( \\frac{-1}{4\\pi\\epsilon} \\right) \n\\frac{\\sigma\\, dS \\ \\mathbf{r} }{r^3}\n", "ccf36946afb67153945b8b0aad896ca2": "\\phi_{\\tau}\\left(\\omega\\right)", "ccf381b70031675da3efd3b202a03b91": " \\,", "ccf3b366e8ca8aa4fcc9087edc10d9eb": "\\mathbf{P} = \\begin{bmatrix}\n\\mathbf{P}_{11} & \\mathbf{P}_{12}\\\\\n\\mathbf{P}_{21} & \\mathbf{P}_{22}\\end{bmatrix}.", "ccf3ec6ba33f86c6850af26862d2f8b5": "{\\Psi}", "ccf424aa3852ab04b8d9bf4a225de7be": "\\varphi\\colon G\\to {\\rm Aff}({\\Bbb R}^n)", "ccf43ab547ae4495e2e0891592c864f0": " C_1'", "ccf48c7644de4b346d75d48d5ea4a221": "\\scriptstyle =(2.3\\pm5.4)\\times10^{-43}", "ccf4bff489b69a77058fd0db4bbaf4f1": "\n\\frac{d}{dt}\\left(\\frac{\\mathbf r}{\\Vert \\mathbf r \\Vert}\\right)\n= \\frac{1}{{\\Vert \\mathbf r \\Vert}^3}\\left(\\mathbf r \\times \\frac{d \\mathbf r}{dt}\\right) \\times \\mathbf r\n= \\left(\\hat{\\mathbf r} \\times \\frac{1}{{\\Vert \\mathbf r \\Vert}} \\frac{d \\mathbf r}{dt}\\right) \\times \\hat{\\mathbf r}\n", "ccf4eaa4c2dee1a6b8f942341b0e1a3e": "\\Pi(n) = n!\\text{ for }n \\in \\mathbf{N}\\, .", "ccf533819152ee0b98f7c96cec2a4d66": "B_d (y)\\bigcap B_d (z)", "ccf5a7851657f382ef387e834000e6ee": "f\\mapsto \\varphi\\times f -T_\\rho (f)", "ccf5e6652dafeef624f2d4f072f352d7": "y=\\mathrm{argmax}_{c_j \\in C} \\sum_{h_i \\in H}{P(c_j|h_i)P(T|h_i)P(h_i)}", "ccf5f455f5de788b5d5eeaf339059de1": "j\\neq i", "ccf66afc0b6d254ce9a3399dca7e6306": "\\alpha(n) \\in o(1/\\sqrt{n}\\log{n})", "ccf673a3ab825c12eea4280ce270fd9a": "\\mathbf{j}(\\mathbf{r},t)\\cdot d\\mathbf{S} + \\Sigma(t).", "ccf6aafb5e56d14d0005c299012cb63e": "c = A_xB_y - A_yB_x\\,", "ccf6cb7a07e53d6a5c3e8449ae73d371": "\\mathbf{P}", "ccf6ccc40e56cdf9c1326e2b891b9c8f": "\\boldsymbol{\\xi} = \\frac{G }{2 c^2} \\frac{\\mathbf{L} - 3(\\mathbf{L} \\cdot \\mathbf{\\hat{r}} ) \\mathbf{\\hat{r}}}{\\left | \\mathbf{r} \\right |^3}", "ccf72a7865745817c9debe5b7dd3db3a": "\\gamma_iT_{i0} \\ll m_iv_s^2", "ccf7720d18d13438f2a2bd59daec7121": "\\frac{1}{e^{\\frac{h \\nu}{kT}}-1} \\approx e^{-\\frac{h \\nu}{kT}}", "ccf77d65dd35b06e09848a33943f2529": " = \\| \\lambda \\cdot Ux \\|^2 + \\| U(\\lambda \\cdot x) \\|^2 - \\langle U(\\lambda\\cdot x), \\lambda\\cdot Ux \\rangle - \\langle \\lambda\\cdot Ux, U(\\lambda\\cdot x) \\rangle ", "ccf7c4182e11be157437f41192d95287": "n = 4, \\;\\; z = -1, \\;\\; z^2 = z^4 = 1, \\;\\; 2 \\not\\equiv 4 \\pmod{4}.", "ccf7cec0c5d7e4f3e2d1ffaffdb7ff77": "\\beta \\subseteq \\alpha", "ccf7ddcec73a3910f062da643ac58052": " \\mathbb{Z}^2 ", "ccf7e364b5906094510f75d5967eb4d6": "\n C = \\frac{\\rho C_0^2 V_0}{2(1-s\\chi)^2} \\,.\n ", "ccf854e36d724f56e9fe810261aa4a5a": "\\mathfrak{sl}(n)\\colon", "ccf8e5d503c3aee934ceec554da8b2a1": " \\bigoplus_{j\\in I_i}K_n(R[H_j])\\oplus \\bigoplus_{j\\in I_i} K_{n-1}(RH_j)\\rightarrow \\bigoplus_{j\\in I_i} K_n(RH_j)\\oplus KR_n^G(X^{i-1})\\rightarrow KR_n^G(X^i) ", "ccf91382753b5c1cf643dbfc6da024e8": "\\Psi(r)\\propto \\int\\!\\!\\!\\int_\\mathrm{aperture} E_{inc}(x',y')~ \\frac{e^{ik | \\bold r - \\bold r'|}}{4 \\pi | \\bold r - \\bold r' |} \\,dx'\\, dy',", "ccf915ca60074c040f63a351cd44e108": "\\emptyset", "ccf9b9e9180c9faebc4570850c17cf19": " = f \\mapsto k \\mapsto \\left( f \\left( t \\mapsto x \\mapsto k \\, t \\right) \\, k \\right)", "ccf9d8d380b1be6867d2d94aec04aac8": " \\mathbf n ", "ccf9f60c1ad0cf83bed3f2008b8f3a9e": "{x^3 - 2x^2 - 4} = (x-3)\\,\\underbrace{(x^2 + x + 3)}_{q(x)} +\\underbrace{5}_{r(x)}", "ccf9fd2c8ff1c7cdcade32f65258f330": "\\textstyle B(\\mathbf{c}_i)", "ccfa7bc28f1a78738735f9ac9ce80b44": "\\beta \\geqslant \\gamma", "ccfa8308393a47429a08d8b1d7c63a36": "\\ f+g: x \\mapsto f(x) + g(x)", "ccfafbfd79fbc9bf1e6127b0f1981aaf": "\\mathbb{P} (X \\cap K = \\emptyset)", "ccfb14c74d7270956ecfed120c170dbb": "m_{max}", "ccfb71d490035a4a66bcf68a74b39964": " p = \\frac{P_0\\cdot r\\cdot (1+r)^n}{(1+r)^n-1} ", "ccfbcbf5261b02ea680ec86b4deac822": "\\frac{\\partial \\mathbf{y}}{\\partial \\mathbf{x}},", "ccfbf328fc6360cf8e55e8e193d74d2b": " \\,q_k = \\,1- p_k ", "ccfcd347d0bf65dc77afe01a3306a96b": "[0,1]", "ccfd4616919b785406d421574ff06bc2": " - \\log_a p_i \\leq s_i < -\\log_a p_i + 1 ", "ccfdccbbabda6adb27e981d04a134576": " X_{(n)} \\le L \\le X_{(n)} / \\alpha^{ 1 / n } ", "ccfe15abdd655815e8013f69e84d69bd": "W(t)=-q/m \\mathbf{P}\\cdot \\mathbf{A}(\\mathbf{R},t)-q/m \\mathbf{S}\\cdot \\mathbf{B}(\\mathbf{R},t)+q^2/(2m) \\mathbf{A}^2(\\mathbf{R},t)", "ccfe2faf36945ddd374e6df89bb40658": "\nc(\\vec{m}) = e^{-\\left(\\frac{I(\\vec{m}) - I(\\vec{m}_0)}{t}\\right)^6}\n", "ccfe48d3b8c557709941f9e452db7de2": "\\frac{n\\alpha\\beta(\\alpha+\\beta+n)}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}\\!", "ccfe713e308ba8b6d7e19286f1e8f6bb": "ds^2 = \\phi^2 \\, \\eta_{ab} \\, dx^a \\, dx^b", "ccfe7891d80c2c7d1dcca876290380fc": " \\forall \\ x,y,z\\in X \\ \\ (xRy) \\Rightarrow (xRz) \\or (zRy) ", "ccfe9f9c0ab1044ee747811bce42f8ce": " \\begin{align}\n& && A \\mathbf{x} = \\mathbf{0} \\\\\n\\end{align} ", "ccff1cb43ddfd576f3553fdc80ef1aac": "(a \\cdot b) \\cdot c", "ccff27d04fdb4f47241396d2e4deb952": "\\scriptstyle h", "ccff359ec54226ccda4e73a4e7681a63": "\\frac{1}{1+1} = 0.5", "ccff793cfadd65b7a7479e0eb49c166e": "\\int_0^{t_r}F_r(t)\\,dt=m_f v_f=-m_p v_p", "ccff847475c5238ea28c3dda13a70324": "Area_{octane}/Area_{nonane}", "ccffee2b6dc7f869c01e0e6ff05486c6": " Q = q_0 + q_1 \\mathbf{e}_2 \\mathbf{e}_3 + q_2 \\mathbf{e}_3 \\mathbf{e}_1 + q_3 \\mathbf{e}_1 \\mathbf{e}_2. \\!", "ccfffdd558b28e5a58c8b6dda617627d": "n=5; \\quad s^5+15s^4+105s^3+420s^2+945s+945", "cd0002bfa6957796ea4ff7c24714ab74": "\\mathbf{p}=(p_1,p_3)", "cd000bf7e311d6e77369d9b5cf3c3979": "\\Delta S = \\int \\frac {dQ_{rev}}T ", "cd0020046d68bcf0d79a66d4d8249e91": "Fr = \\frac{q}{\\sqrt{gy_1^3}} = \\frac{10}{\\sqrt{32.2(6^3)}}= 0.12\\text{ (subcritical flow)}", "cd002cf687224b2860c1907cdd004fc1": "\\mathit{E_i}", "cd00432362578068c81afee8f403579b": " \n {\\rm Ind} (-S_{xx}''(x^0)) = \\frac{\\pi}{4} {\\rm sign} S_{xx}''(x_0),\n", "cd0088a07b0abf5ad9e7b0fc3094e267": "{{d^2} \\over {dR_\\mathrm{L}^2}} \\left( {R_\\mathrm{S}^2 / R_\\mathrm{L} + 2 R_\\mathrm{S} + R_\\mathrm{L}} \\right) = {2 R_\\mathrm{S}^2} / {R_\\mathrm{L}^3}. \\,\\!", "cd010762cb3e607c5601eed2945e0707": " {\\color{Red}\\boldsymbol{H}_{k}} = \\left . \\frac{\\partial h}{\\partial \\boldsymbol{x} } \\right \\vert _{\\hat{\\boldsymbol{x}}_{k|k-1}} ", "cd010ffc3abf0e20e9221b71f0b64c10": "r \\neq r'\\,", "cd014731964c742c274df08d7cc238fb": "\\phi \\,", "cd016810d4e494f201a6ed6b62d1cbd1": "\\varphi_{1}", "cd01c354e1313c76d84166d9e3ce1ec0": "\\scriptstyle U_n ", "cd01d193b11b0ac8d0319aa74e310cdb": " S_n=\\frac{n}{2}( a_1 + a_n).", "cd02121baacf9bb6f1152be2c1a1c0e5": "\\left(1=\\int_{-\\infty}^\\infty g(x)\\,dx\\right)", "cd023392d22cb7d4b4a31f4f45b26e42": "F_{ab}^k", "cd03d91a520ad178a1c5e7fae5a6f240": "T\\subsetneq A", "cd03e7dd60ad641863c5a5059bb5ae1c": "C(\\alpha,0) = C(\\alpha,\\sigma)", "cd0444d34ff31da6c97c355cd48c45c4": "X\\setminus A", "cd044a368eb0f21d5eb108b6189969e6": "C(j,k)", "cd046f5e6d8fedd18fb45f8b744fe178": "\\prod_{k=1}^n \\left({x_k \\over x_k + y_k}\\right)^{1/n} + \\prod_{k=1}^n \\left({y_k \\over x_k + y_k}\\right)^{1/n} \\le {1 \\over n} n = 1.", "cd047c0da9060a20b23d7cf379dd573a": "\\cfrac{\\partial q^i}{\\partial x_j}", "cd0484bc31ac9b92f80d2b44c13a8657": "\\beta_o", "cd0491b77ed2c68b524955d3c08c34ba": " \\cos \\theta = \\frac {\\mathrm{adjacent}}{\\mathrm{hypotenuse}} = \\frac {b}{h}", "cd0545ba5644a91193403b58bebf1b84": "D^2", "cd05f4225b4ede7aee16d23b82c5d482": "\\lbrace u e^{ar} : 0 \\le a < \\pi \\rbrace", "cd060223d4be7473856051c7c3e7dc8a": "\\frac{1}{\\sqrt{\\rho}}", "cd0632bb04c2be39cd2da2dc48d6caac": "X^{n}\\left( m\\right) ", "cd0672aa22ec01587f7d7d8cbd63f5e4": "\nv =\\sqrt{1-\\cos^2(\\alpha)-\\cos^2(\\beta)-\\cos^2(\\gamma)+2\\cos(\\alpha)\\cos(\\beta)\\cos(\\gamma)}\n\n", "cd073317f1f7257e4b302e426e0953e5": "V_Z-V_Y=0", "cd078775d2ba2625a2d995d3fc30e938": "\\mathbb{E}[Y_n\\,|\\,\\mathcal{F}_{n-1}]= Y_n", "cd07e250afb68204d5aee04f1a2d213a": "10^m - 1\\,", "cd07ffb7c6babdd5c5bb6b482c96419b": "\\, \\left(\\begin{bmatrix}A_{ro} & A_{24}\\\\ 0 & A_{\\overline{r}o}\\end{bmatrix},\\begin{bmatrix}B_{ro} \\\\ 0 \\end{bmatrix},\\begin{bmatrix}C_{ro} & C_{\\overline{r}o}\\end{bmatrix}, D\\right)", "cd082d3ca8668f1fb4003fcfdf8a6375": "\\sqrt{\\frac{c}{2\\pi}}~~\\frac{e^{-\\frac{c}{2(x-\\mu)}}}{(x-\\mu)^{3/2}}", "cd087de74104ef80f71b4ba0fac320ab": "\\tilde{\\nu} = 1/\\lambda", "cd08932ec9ca49532f4bf9873ca3eb12": "\\rho = -i\\partial\\overline{\\partial}\\log\\det(g_{\\alpha\\overline{\\beta}})", "cd0929263ebd4435af9a596b5674665a": "dV_w", "cd096b66c8a3a024023ae819f032a7e2": " \\bigcup_{j\\in J} B_{j}\\subseteq \\bigcup_{j\\in J'} 5\\,B_{j}. ", "cd0976ff63150c67f40caf8f6c2f3f6e": " g^2 < 1 ", "cd098024721b6a8ee7ff4729d0e48260": "s_\\ell", "cd099816b499bccd52cdbbb8797cf66b": "\\bigcap_{i=1}^{n} L_i.", "cd09b2319d112558eaa6728f377ad99b": "G_n(\\zeta )=f_n\\circ \\cdots \\circ f_1(\\zeta )", "cd09f3f75d57570ab9f4d91d89d2a703": "y^2+z^2=1-a, w=0, x=0", "cd0a429214bc6d0811bc83e6d9c07522": "\\omega_{SP}=\\omega_P/\\sqrt{2}.", "cd0a8d8de2d5481fd42e254b3b2e32b1": "\\frac{P}{MC}=\\frac{PED}{1+PED}.", "cd0ab73ac9c6d639c7bcef3dd6260267": "f'(z_n)", "cd0ac658a99aab7c09b427c56d34ecde": "y_{[1]}\\leq y_{[2]}\\leq ...\\leq\ny_{[S]}", "cd0afbb371aa82ce30c400b0386dfef1": "M=|M|e^{i\\theta}e^{i\\phi}", "cd0b75151ad18d171eff4674a9b7b3a3": "e_{max}", "cd0b96a5a96bc7282878b4fe5b3386bf": " m_{k+1} = \\arg\\max_{m} \\sum_{i=1}^\\mu w_i \\log p_\\mathcal{N}(x_{i:\\lambda} | m) ", "cd0ba1e316db46afc8b2d4a782d04a59": "F(n,k)=\\binom{\\tfrac{n+k-1}{2}}{k}", "cd0ba79f6050f1b569b28148df5a9de3": "\\mathbf{\\mu}=\\frac{I}{2}\\int\\mathbf{r}\\times{\\rm d}\\mathbf{r}.", "cd0c0e10ae428150e583a509d715397a": "\\begin{align}\n f(z) &= \\frac{1}{\\pi^k\\sqrt{\\det(\\Gamma)\\det(P)}}\\, \n \\exp\\!\\left\\{-\\frac12 \\begin{pmatrix}(\\overline{z}-\\overline\\mu)' & (z-\\mu)'\\end{pmatrix}\n \\begin{pmatrix}\\Gamma&C\\\\\\overline{C}'&\\overline\\Gamma\\end{pmatrix}^{\\!\\!-1}\\!\n \\begin{pmatrix}z-\\mu \\\\ \\overline{z}-\\overline{\\mu}\\end{pmatrix}\n \\right\\} \\\\[8pt]\n &= \\tfrac{\\sqrt{\\det\\left(\\overline{P^{-1}}-\\overline{R}'P^{-1}R\\right)\\det(P^{-1})}}{\\pi^k}\\,\n e^{ -(\\overline{z}-\\overline\\mu)'\\overline{P^{-1}}(z-\\mu) + \n \\operatorname{Re}\\left((z-\\mu)'R'\\overline{P^{-1}}(z-\\mu)\\right)},\n \\end{align}", "cd0c3c1a62c7db942df472732d680d6c": "B_\\ell", "cd0c4d49ca35bd04a636c62618e7d8c5": "\\,\\int_0^{\\frac{\\pi}{2}}\\,\\frac{\\cos^2\\,x\\;\\mathrm{d}x}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^2}\\;=\\;\\frac{\\pi}{4\\sqrt{a^3b}}.", "cd0ce88cafd5f510d660d63ddefe03ee": "r^T A = 0", "cd0d1c2bc60955fff92fc259a938e896": "\\alpha^c,\\ldots,\\alpha^{c+d-2},", "cd0d3826824565fbb61517d72182b40c": "\\int \\nabla _{t}f(x^{\\ast }(s),s)\\cdot ds=0. ", "cd0d3963a5f5f218d97318e6e0274c8a": "S_k = C_1S_{k-1}", "cd0d480fceeb1f13d502ee24017473aa": " \\frac{dT_y}{dx}\\ =\\ \\frac{dT_y}{dt}\\ \\frac{dt}{dx}\\ =\\ \\frac{V_t}{V_d}\\ \\sqrt{{y'}^2+1} ", "cd0d6965b35cae5128fc47d086731bf8": " \\rho = 1- {\\frac {6 \\sum d_i^2}{n(n^2 - 1)}}.", "cd0d707a2aacf92f21253363c19f27eb": "ds_{IX}^2 = dt^2 - \\left [ \\left ( a^2 \\sin^2 z + b^2 \\cos^2 z \\right ) \\sin^2 y + c^2 \\cos^2 y \\right ] dx^2 - \\left [ a^2 \\cos^2 z + b^2 \\sin^2 z \\right ] dy^2 - c^2 dz^2 + ", "cd0d70e2ee868699f3aacc314b3ff6e8": " \\delta q ", "cd0d920f61613692e4e3d1ae1c83bbaa": "\\{v_2,v_3\\},", "cd0da2074c6176e95ef6eaa2258d682e": "N \\ge 2Q+1", "cd0de2fb861c117a995bb43eec2809fa": " \\hat T_{2,-1} = + \\frac{1}{2}( \\hat a_{z} \\hat b_{-} + \\hat a_{-} \\hat b_{z} ) ", "cd0de4127c7634ebd41d83d5be2ba4d3": "O(n ^ k)", "cd0df83e143316392f68d90a182bc823": "*[F , G]^{IJ} = [* F , G]^{IJ} = [F , * G]^{IJ} \\;\\;\\;\\;\\; Eq.6", "cd0e0c3be238f353c6a97d52e46f6639": "s(x)=\\exists y \\; \\psi(x, y)", "cd0e5a4f6f18f2eaa1b2dee583689b0c": "\\forall a, b \\in X,\\ a R b \\Rightarrow \\; a = b.", "cd0e9b5915d00eb61d7123fdbfc36828": " {s} \\,", "cd0eab88967ccb31988555f78a0e6fd4": "\\begin{array}{rcl}\n\\langle \\hat{A}(t)\\rangle &=& \\langle \\hat{A}\\rangle_0-i\\int_{t_0}^t dt'{1\\over Z_0}\\sum_n e^{-\\beta E_n} \\langle n (t_0)| \\hat{A}(t)\\hat{V}(t')- \\hat{V}(t')\\hat{A}(t) |n(t_0) \\rangle\\\\\n&=& \\langle \\hat{A}\\rangle_0-i\\int_{t_0}^t dt'\\langle [\\hat{A}(t),\\hat{V}(t')]\\rangle_0\n\\end{array}\n", "cd0eb794b11fce00e474e1e90d72b488": "\n\\alpha\\approx {a \\,t_0 \\,\\Delta_{SO}\\over \\Delta_{BG}}\n", "cd0edbaa124de61d009ffcac7abe8415": "X_j = i \\left. \\frac{\\partial g}{\\partial \\xi_j} \\right|_{\\xi_j = 0} ", "cd0f1069db14b3485b705eb04d3e58a4": "\\alpha_i", "cd0f4e5f8e7fc2296c806ba2b8cceb28": "81^2", "cd0fa047a22f29ba144a9aadfb0e2f9f": "\\ y[n] = x[n] + A_1 e[n-1] + \\mathrm{dither},", "cd0fa39fbab97e45eaf24d00d5318f4e": "B(x) = \\sum_{n=0}^\\infty \\frac{B_n}{n!} x^n = e^{e^x-1}.", "cd10134d6a70164126eceb50b3aea498": "\\mathfrak c", "cd10276a33de4c1a1fd748d411ccf1ef": "\\left\\| a \\right\\| = \\sqrt{\\sum_i \\left\\| a_i \\right\\|^2}", "cd105e295eff3acb00f744d4fdb952f3": "\\log(1-\\alpha) = -\\alpha -\\alpha^2/2 - \\cdots", "cd106d9b6eefbafcea700aa5a018b740": "e \\in \\text{End}(X)", "cd11c467d7f4bd4b74d314b2f9a2de45": "f: \\mathbb{Z} \\rightarrow \\mathbb{Z}", "cd11cdf0f5cc4ca9a3834fc8831fdeda": "A_{\\mathfrak p}", "cd1262cbec707cb7295181b5fb3d849c": "\\mathbf{b}=e^{2t}\\begin{bmatrix}1 \\\\0\\\\1\\end{bmatrix}.", "cd12687f32b6c664ef50028e5e9f3ce1": "A^TJA = J", "cd12b75e7031628eaecb95b5d7052570": "O(V^2)", "cd12c8dec9f1f91ddf0340f7f59b9fcf": " x^* ", "cd131c9e7196eb7bc49be697e79a4e89": "\\sigma'\\in S_n", "cd135b27b9f37cdef6315c22a0e1ca85": "(x-1)x(x^2-3x+3)^{(n-1)}", "cd13724c2650cf8bbedf47d942d1d983": "\\lambda_i = S_{ii}/T_{ii}", "cd137710175e986af851d2e1f305b1a7": "(A.1.d)\\quad \\frac{1}{\\rho}\\,\\gamma_{,\\,z} =\\,2\\psi_{,\\,\\rho}\\psi_{,\\,z}- 2e^{-2\\psi}\\Phi_{,\\,\\rho}\\Phi_{,\\,z} ", "cd1492c5f81279f2f140ab5446ceaebd": "p_{X_i}(x_i) = \\sum_{\\mathbf{x}': x'_i=x_i} p(\\mathbf{x}').", "cd14cb4682021c8713c151ac61c3f499": "\\Phi(G) = G^p.", "cd14d4c55d8bf1fe00b6cf0dad77c395": "r_1, r_2, r_3, \\cdots, r_n", "cd14f203f7fa35bf9bcb2abd0bf82247": "g^2", "cd14ff60533284cc9052dbba0ab4ad24": "\n\\alpha=f_1(d_a)= \\left\\{ \\begin{array}{ll}\n\\alpha_{max} & \\mbox{if } d_a \\leq d_1 \\\\\n\\frac{\\kappa_1}{\\kappa_2+d_a} & \\mbox{otherwise.}\n\\end{array} \\right.\n", "cd152610fcd555ce2f62e00de9be95d4": "A\\!", "cd15855a8a09683c68f17689ba9b8c1b": "\\sum\\limits_{i=1}^{N}n_i=N_{\\mathrm{2D}}", "cd1596cf8803e0cf2b23eafd3c259260": "V \\in (0,1]", "cd15bc634a3ec18e1b1339b2a05243e0": " (dG)_{T,P} = \\sum_i \\mu_i dN_i\\,.", "cd15c068050d0f124c3867e5c1828f38": "f(r_{1},r_{2},r_{3},r_{4})", "cd1637634fa964a11a8f06b90e49ca35": "\\tfrac{1}{n^3}\\left(1+\\tfrac{3}{5n}+\\tfrac{31}{420n^2}+\\cdots\\right)", "cd1672ed688a87ffed8639f1702da26a": "T_p(x) = (-1)^{(p-1)/2}\\ p\\ G(x)\\,", "cd169d857e658836b5413183ae5a00d4": "K(s)=\\log[\\text{E}(e^{sY})]=\\lambda[\\kappa(\\theta+s/\\lambda)-\\kappa(\\theta)]", "cd16a251474525c0d1a3d570ae60a4c6": "R = \\log |\\mathbb M| ,\\,", "cd16dc862379694b5328f6a030aca44c": " L(4,1) ", "cd16e8c28be3bf605e53f7294f4eb4a8": "LaAIO_3/SrTiO_3", "cd16eaf81f6c6666a58a4d1e924287c1": " \\ v_{ \\bar{x} } ", "cd171a4d1ac43563735ddb4a6e851941": "\\operatorname{P}(X \\neq Y) = 0.", "cd177ecade84782909695e39bd1b653c": "z'(a)=z'(b)", "cd17bd92cdb4eed2ca7e6be07afaf275": "Y^{1/\\gamma}_{\\alpha,\\beta}", "cd17d0058fb8b0ea7fcf8a445ef94c12": " c =M_{2,1}\\,", "cd184b5db428f33b0f7cf6e14ebc4c0c": "n\\ge 2", "cd1852d5ada371233636664136d0a670": "(x^2+x-4)^2(x^2+x-3)^2(x^2+x-1).\\ ", "cd186249afc7863c4d97e5d00b5f30a3": "disc(\\mathcal{H}) = O(\\sqrt{t \\log n})", "cd18749dda34a586ccaece3c72547876": "C_b(\\widehat{G})", "cd18a92b272e4cb3008dff9560b1581d": "(x)\\mapsto r", "cd18ad9e3d7c1ab754eaf9c56a60a80f": "j_{d+1}^\\mu \\equiv \\overline{\\psi}\\gamma^\\mu\\gamma^5\\psi", "cd190b28b0579f748e1bf09c4491e6d7": " T_1", "cd1924b0b746b0c919a41b2779284fbd": "\n\\bar\\lambda = \\lambda K / L ", "cd1962386e8ae8a1ea1457b98c38c901": " \\langle 2x, 2y \\rangle = \\langle x, y \\rangle ", "cd19842e35da810649bf789b88bbe02f": " z \\ \\stackrel{\\mathrm{def}}{=}\\ e^{s T} \\ ", "cd19a61fc159764d2bdfc5db26140384": "\\frac{\\mbox{Net Income}}{\\mbox{Average Total Assets}}", "cd1a0e4a80f7f8e8e567646f88a16d39": "\\mathrm{Pois}(\\lambda)", "cd1a26b7868afc7da5f8d98eeb0190d0": "a' = a + b \\, \\pmod{2^n}\\,", "cd1a2cce5c805c58b46e5507f853d655": "R^m_n(\\rho)=0", "cd1a5cb9696a9177ce49f12a3b029c07": "F(\\Omega) = \\Gamma = [\\Gamma_\\min, \\Gamma_\\max]", "cd1a6a28ab2ee38521d952325cdcf88d": "C_{Hb}\\,", "cd1a9e6fbdd96533e319db59e751437b": "\\varphi = {1 \\over 2}\\csc(\\pi/10) = {1 \\over 2}\\csc 18^\\circ", "cd1acb4167a55fe96af1c05f12b20226": "\\begin{pmatrix} H_n \\\\ P_n \\end{pmatrix} = \\begin{pmatrix} 1 & 2 \\\\ 1 & 1 \\end{pmatrix} \\begin{pmatrix} H_{n-1} \\\\ P_{n-1} \\end{pmatrix} = \\begin{pmatrix} 1 & 2 \\\\ 1 & 1 \\end{pmatrix}^n \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}.", "cd1aefde3663295bc33525a14b61c315": "\\Psi(x) \\approx \\frac{ C_{+} e^{+\\int \\mathrm{d}x \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}} + C_{-} e^{-\\int \\mathrm{d}x \\sqrt{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}}}{\\sqrt[4]{\\frac{2m}{\\hbar^2} \\left( V(x) - E \\right)}}.", "cd1af491826d28dcf57676a03f41bb4f": "\\left|\\rho_{out}\\right| = 1\\,", "cd1b106ab9c008067a66438f9ea5a8e1": "S_{33}\\,", "cd1b46d2d8b6575cee4d88ad713ad5d5": "\\frac {E_t(S_{t + k})} {S_t} (1 + i_c)", "cd1b4f4de6b9d36277b6ce39b303520d": "\\begin{align} V(x(0),0)=&\\text{max}_u\\int_0^T r(x(t),u(t))dt+D[x(T)]\\\\\ns.t.\\quad & \\frac{dx(t)}{dt}=f[t,x(t),u(t)]\n\\end{align}\n", "cd1b66cde27f4d5a6fceda8c89aaa1d5": " \\begin{align}\nL(t) f(n) &= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n), \\\\\nP(t) f(n) &= a(n,t) f(n+1) - a(n-1,t) f(n-1).\n\\end{align}", "cd1c1651afc8675ccd8907a2a844ec3b": "[H,\\mathbf{J}]=0", "cd1c509903043db4fb5e6837c042a302": "P(x_1,x_2,\\dots)", "cd1c5a3931bc2455b8b45b091ff40b5e": "\n t_{\\hat{\\beta}} = \\frac{\\hat\\beta - \\beta_0}{\\mathrm{s.e.}(\\hat\\beta)},\n ", "cd1ca8543481b2e360d2b5a75222d5c9": " Fr ", "cd1d492d8383b91c66fb9b2637f6221f": "B^\\ast_k := (-1)^k B_k", "cd1dd8478dbd0a2ee4f2a2b775e5e370": "H_{\\aleph_1}", "cd1de15740589d8f7bb81b06c239c720": "\\alpha < 2", "cd1e25fe63a26e22d0ba6cd0b22d0719": "\\begin{cases} \nu_{t}=ku_{xx}+f(x,t) & (x, t) \\in \\mathbf{R} \\times (0, \\infty) \\\\ \nu(x,0)=0 & IC \n\\end{cases} ", "cd1ecb38384a4128fa4990ecf873b248": "(n\\times s)", "cd1ee50a23c82ecf7a57fabbdc83564e": " \\overline{Q}^{\\text{day}} = \\frac{S_o}{\\pi}\\frac{R_o^2}{R_E^2}\\left[ h_o \\sin(\\phi) \\sin(\\delta) + \\cos(\\phi) \\cos(\\delta) \\sin(h_o) \\right]", "cd1f3d8ed5f7f67ecebb1e7b1dc6b27d": "s=\\max S", "cd1f5ac2e97c112e56736cd65d7095b5": "\\begin{align}\n L'_a &= \\frac{400 {\\left(F_L L'/100\\right)}^{0.42}}{27.13 + {\\left(F_L L'/100\\right)}^{0.42}} + 0.1 \\\\\n M'_a &= \\frac{400 {\\left(F_L M'/100\\right)}^{0.42}}{27.13 + {\\left(F_L M'/100\\right)}^{0.42}} + 0.1 \\\\\n S'_a &= \\frac{400 {\\left(F_L S'/100\\right)}^{0.42}}{27.13 + {\\left(F_L S'/100\\right)}^{0.42}} + 0.1\n\\end{align}", "cd1f5e11e7498ba7b46fe33d4871b718": "\\zeta_{\\mathrm{Ai}}(1)=\\frac{-3^{-2/3}\\Gamma(\\frac23)}{\\Gamma(\\frac43)}.", "cd1f71f73612956472a4c32b616eb14a": "\\varepsilon_{ijk\\ell\\dots} =\n\\left\\{\n\\begin{matrix}\n+1 & \\mbox{if }(i,j,k,\\ell,\\dots) \\mbox{ is an even permutation of } (1,2,3,4,\\dots) \\\\\n-1 & \\mbox{if }(i,j,k,\\ell,\\dots) \\mbox{ is an odd permutation of } (1,2,3,4,\\dots) \\\\\n0 & \\mbox{if any two labels are the same}\n\\end{matrix}\n\\right.\n", "cd1fcfe57d57a215ea040b8f6d6ed55d": "S(3.5, 1.4)=1.1^2+(-1.3)^2+(-0.7)^2+0.9^2=4.2.", "cd20539176705a1d09f2b17dcf441676": "\n\\pi = 3 + \\cfrac{1^2} {6+\\cfrac{3^2} {6+\\cfrac{5^2} {6+\\ddots}}}\n= 3 - \\sum_{n=1}^\\infty \\frac{(-1)^n} {n (n+1) (2n+1)} \n= 3 + \\frac{1}{1\\cdot 2\\cdot 3} - \\frac{1}{2\\cdot 3\\cdot 5} + \\frac{1}{3\\cdot 4\\cdot 7} -+ \\cdots\n", "cd2078b4f866e080eab2b54177e3abf2": "\\begin{align}N(a+b\\,\\omega)\n&=|a+b\\,\\omega|^2\\\\\n&=(a+b\\,\\omega)(a+b\\,\\bar\\omega)\\\\\n&=a^2 + ab(\\omega+\\bar\\omega) + b^2\\\\\n&=a^2 - ab + b^2.\\end{align}", "cd207a50c59ee41ab23096e920aba25c": "b_2=3\\ \\left(\\frac {y_1 - y_0}{{(x_1-x_0)}^2}+\\frac {y_2 - y_1}{{(x_2-x_1)}^2}\\right)", "cd20846d4bd74091a17e6567db2c0b8c": "v = \\frac{\\omega}{\\beta}", "cd209131055ee6e70c952063366ea319": " z = a (1 + 2e^{it} + e^{2it}) = a(1 + e^{it})^2. \\,", "cd21a620587611e8419bd3a9b88fa478": "v(A - 0)", "cd21e39551fa5ad12e3f8f3dcda1dbea": "A \\in \\Gamma", "cd21e833fc85447db30646aa015b16ee": " \\mathbb{E} (Y|X) = \\frac3{10} X. ", "cd21ec4bc128214704fc4154802e3195": " u(t,x,y) = \\frac{1}{2\\pi c} \\iint_D \\frac{\\phi(x+\\xi, y +\\eta)}{\\sqrt{(ct)^2 - \\xi^2 - \\eta^2}} d\\xi\\,d\\eta. \\,", "cd2202495035ef92f2e15d4330055b62": "1+{1 \\over 2^2}+{1 \\over 3^2}+...+{1 \\over (p-1)^2} \\equiv 0 \\pmod p. ", "cd221e2ed6bc3fe757bdea0f4710ead0": "\\frac{d U_n}{d x} = \\frac{(n + 1)T_{n + 1} - x U_n}{x^2 - 1}\\,", "cd22374d0c0c56bb3ad01f163dee3fda": " R = \\left[ \\frac {P_t \\, G^2 \\, \\lambda^2 \\, A }{(4 \\, \\pi)^3 \\, \\sigma_{min}} \\right] ^{1 \\over 4} \\,\\!", "cd223ef7ffa7010715b841505c385985": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n \\frac{B\\,x^{m-n+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^{q+1}}{b\\,d (m+n (p+q+1)+1)}\\,-\\,\n \\frac{1}{b\\,d (m+n (p+q+1)+1)}\\,\\cdot\n", "cd2241a76b8a629c6c0810b7764666dc": "\\langle f_i, f_j \\rangle=\\int_a^b f_i(x) f_j(x) w(x)\\,dx=\\delta_{i,j}", "cd2269e9f3222e18785068d288fc5e3e": "\\frac{\\partial S}{\\partial \\beta_1}=0=8\\beta_1 + 20\\beta_2 -56", "cd2277e6981f377794a7afde39db0eff": " L \\equiv ", "cd228c57f02a1bf98e029897a7824ef6": " D_\\mu \\psi \\mapsto e^{i \\Lambda} D_\\mu \\psi ", "cd22a31c90ebc5626ba36d44717eb3f8": "P = i_P\\circ j^k", "cd22a72676b0316130941d15fd9322b6": "\\textstyle V(s)=\\sum_{n = 0}^\\infty a[n] s^n", "cd234519b7a49d96dbe713c063622d5f": "f(x)=\\sum_{n=1}^{\\infty}b_{n}\\sin\\frac{n\\pi x}{L},", "cd235943477c4aa1b4a07bd1d6ebf2d5": "P^{\\mu} A_\\mu = \\eta_{\\mu\\nu} P^{\\mu} A^\\nu = \\eta_{\\mu\\nu} P^\\mu \\frac{d}{d\\tau} \\frac{P^{\\nu}}{m} = \\frac{1}{2m} \\frac{d}{d\\tau} \\|\\mathbf{P}\\|^2 = \\frac{1}{2m} \\frac{d}{d\\tau} (-m^2c^2) = 0 .", "cd2376327d353a028343a9f5e69cb649": "\\mathbf {\\beta_1}", "cd239f1836b495ab4575d88153a89469": "\\delta V=I_0 \\ln\\left( {M_0\\over M_1} \\right) ", "cd23bb834e15a9e4c3d8069ccc6edd14": "a(t)\\propto t^{1/2}. \\, ", "cd23e47cf94179949af1bfd036fe3da2": "U\\leq G", "cd248e4a6aa620f9a36db418ebdb04f1": " q = e^{i \\pi \\tau}= 0.1 e^{0.1 i \\pi}", "cd266619c35538fa28fc940d2e19da49": "a_{6}=a_{7}", "cd2672b1e982fd6977dd378f8550945d": "\\sqrt{c^{2}-v^{2}}", "cd267eeaf090143dbee39fb4cf4a199d": "\n\\sqrt{z} = \\sqrt{x^2+y} = x+\\cfrac{y} {2x+\\cfrac{y} {2x+\\cfrac{3y} {6x+\\cfrac{3y} {2x+\\ddots}}}} \n= x+\\cfrac{2x \\cdot y} {2(2z - y)-y-\\cfrac{1\\cdot 3y^2} {6(2z - y)-\\cfrac{3\\cdot 5y^2} {10(2z - y)-\\ddots}}}\n", "cd271c19303119a2305ddd00f3ab3b9b": "\\vec p = \\gamma m \\vec v ", "cd27804b590e50c0ccab4f050d1da3d8": "\\Lambda(f)(x_1\\wedge \\dots \\wedge x_k) = f(x_1)\\wedge\\dots\\wedge f(x_k).", "cd279f4e87bbbbaaa88d836a24f5c6f4": "=\\hat{c}_V NkT\\,", "cd27cf325f695669abfb43c650586e79": "\\tilde{V} = \\text{constant}", "cd27d599717fe67a81bcf198e12356b7": "2d > n", "cd27e492dbaf98234bc35758128adf7e": "\\boldsymbol{\\tau} = \\frac{{\\rm d}\\mathbf{L}}{{\\rm d}t} \\,\\!", "cd2800e228254c61f70b3e215aa30bbf": "O(n^{\\sqrt{k}})", "cd281bcc9887e02f7a05a7d26f416d7c": "\\lambda \\phi^4", "cd282a49b844f145272bc2dec1fe141c": "r_k^* = b^* - x_k^*\\, A ", "cd283b6cba41ba46756b2ce005cfbf6b": "(10^8)^{(10^8)}=10^{8\\cdot 10^8},", "cd2866c35b99c32ab01ba7afb6e526ab": "\\|x\\|\\ge0", "cd288c899d7f92508e14806d2c6015ad": "r^2 = 2a^2 \\cos 2\\theta\\,", "cd28a6c55aa573e4fe1e05fdca935c84": "\\lim_{x \\to 0} \\frac{1}{\\cos x} = \\frac{1}{1} = 1", "cd28bf5e7134e0bb768bb6172a20c45c": "B(x,y) = 2xy", "cd28dbc32374311355954621d8b43e20": " \\underline{V} = \\underline{I} \\underline{Z}", "cd291e3cda2c7a7987972e32c9000e77": "1 \\times a = a,\\,", "cd29562e34c138b1f94d0afe1baa2aa0": "\\mathfrak{m} = (a_1, \\ldots, a_n)", "cd29699054256bb7bbfc4bf0cb48216d": "|x| < \\frac{1}{\\varphi},", "cd296cf5f58d9a1a727aeedd27ecfbd9": "\\lim_{h\\to 0} \\frac{\\|f(x+h)-f(x) -\\nabla f(x)\\cdot h\\|}{\\|h\\|} = 0", "cd2972126ec482fc0189b8ec21de0020": "\\mathrm{0.\\overline{857142}}", "cd29b79ce736b80821623fc35f0e88c4": "T_H = [H] + [HA] + 2[H_2A] - [OH] \\,", "cd29bf7acdb2c7b33286d3c3b19e191f": " X \\mapsto \\left.\\frac{d}{dt}\\tau_t^\\gamma\\right|_{t=0} = \\theta(X) \\in \\mathfrak{g}.", "cd29d22df6b245f051bfe74e9435b060": "\\Lambda^{p,p}(M)\\cap \\Lambda^{2p}(M,{\\Bbb R}).", "cd2a3627e13aad64973bd3ff582946f0": "p(H2|y) = \\frac{p(y|H2) \\cdot \\pi_2}{p(y)} ", "cd2a464cb544e48063d39b3276802d9c": "I_{\\text{D}} \\simeq I_{\\text{S}} e^{\\frac{V_{\\text{D}}}{V_{\\text{T}}}}. ", "cd2aa0d20e6a5a6a6e11f32aba5a8302": "\\langle\\phi|", "cd2ab19c861ac6c432082f89b64151b9": "f(\\mathbf{x})=\\mathcal{S}\\boxtimes_{n=1}^N \\mathbf{w}_n(x_n),", "cd2b526eeb4655664ff6c73dec50c52d": " \\mathbf{J}_{\\rho} = D'\\,\\nabla(-\\mu/T) \\! ", "cd2bd824c592efd3a439b765d692aed4": "a(\\omega)", "cd2be17474d1a3bfd86aae7373c67ff4": "R(i,b) = 10^{\\frac{i}{b}}", "cd2bf8dc485e0660209a7d24a0ab5401": " \\langle m \\left ( {\\rm X} \\right ) \\rangle ", "cd2c024558e68a6554965693c8ef667a": "-123.456.\\,", "cd2c26cc9bd7b2da82c84371df25fb89": "\\begin{align} (\\mathbb{Z} / 5\\mathbb{Z})^\\ast & = \\{ [1], [2], [3], [4] \\} \\\\ & \\cong C_4 \\\\ \\end{align}", "cd2cc9adee7d8b68045429f174e137c9": " (K_0(A), K_0(A)^+) = (\\mathbb{Z}^m, \\mathbb{Z}_+ ^m).", "cd2d31f4876de2ca5afbffde34d0dade": "a_{j}", "cd2d4285acd664c9539464dc18127935": "\\frac{\\text{c}_\\ell + 0.5 f_i }{N} \\times {100%}\\ ", "cd2d7824f7b50276415e94da6e9fd1df": "\\varepsilon(x) \\delta(x)+\\delta(x) \\varepsilon(x) = 0~;", "cd2d89e1e85cf967cefb78a6a3c5b7aa": "\n-i\\hbar\\Big( \\mathbf{e}_{x} \\otimes \\mathbf{e}_{y} - \\mathbf{e}_{y} \\otimes \\mathbf{e}_{x}\\Big)\\cdot \\mathbf{e^{(\\mu)}} = \\mu \\mathbf{e}^{(\\mu)}, \\quad \\mu=1,-1, \n", "cd2d96d772c2d61eb8f97919c578cb39": "\\begin{array}{ccl}\nn_0 & = & n(n-1)/2\\\\\nn_1 & = & \\sum_i t_i (t_i-1)/2 \\\\\nn_2 & = & \\sum_j u_j (u_j-1)/2 \\\\\nn_c & = & \\mbox{Number of concordant pairs} \\\\\nn_d & = & \\mbox{Number of discordant pairs} \\\\\nt_i & = & \\mbox{Number of tied values in the } i^{th} \\mbox{ group of ties for the first quantity} \\\\\nu_j & = & \\mbox{Number of tied values in the } j^{th} \\mbox{ group of ties for the second quantity}\n\\end{array}\n", "cd2d9f2ad0fb124acea1276c915496a5": "\\mathbb{T}_r \\, \\mathbb{H} = \\mathbb{H} \\, \\mathbb{T}_r \\,\\, \\forall \\, r", "cd2db539143557eb1a2b779d619ab968": "D = \\left \\{ x^2 + y^2 + z^2 \\le 9a^2 \\right \\}", "cd2db8ca9e0760c1d546fc52e7e03bf6": "\\frac{dy}{dx} = \\lim_{\\Delta x \\to 0}\\frac{\\Delta y}{\\Delta x}", "cd2dce7effb7c2dbda019aa31f98a16e": "\\Omega^G_n(X)=\\sum\\limits_{p+q=n}H_p(X;\\Omega^G_q(\\text{pt})).", "cd2e565a0dd687c27ddca1e924b1d087": "\\int_{-N}^{N} e^x\\, dx", "cd2f2432c2c213266a9a78335c6057fb": "i=1, \\dots, n", "cd2fac040c703d003fbbd854c156651e": "V_{12}", "cd301f9130a43eb8b2f3eea88d7e23a0": "1-\\cos\\left(E_{n}T\\right)<\\frac{\\delta^{2}}{2}", "cd304bc158d6947f17a6f6fc355ee247": " \\frac { \\tau_1} { \\tau_2} = 4 \\beta A_0. ", "cd307a0bb5773c7b87ba7d1487d64cb8": "\\frac{\\mathrm{d} \\mu (x)}{\\mathrm{d} x} = \\exp \\left( - 2 \\Phi (x) \\right)", "cd30c44855d9ef38083bd1dee791e4b9": "f(y)=A_j", "cd3129cb44c96f6a0a7fd308615ba3ad": "L^{0} \\ \\stackrel{\\mathrm{def}}{=}\\ L \\cap L^{[\\perp]}", "cd31a0da4158768e58d31122a9614ad1": "g \\le \\frac1g \\sum_{{i\\le g}}c_{i}", "cd31b9bf0f42b5c0e35c70ad1b932f8f": "w_r^+=w_r", "cd31bffb5572f688d0fde41c56463c07": " |f(x)-f(y)|\\leq H|f(x)-f(z)|\\;\\;\\;\\text{ whenever }\\;\\;\\; |x-y|\\leq |x-z|", "cd320698d61b563f20b5888e937d861a": " T_i ", "cd321242689f5ee89a2809705bacf15f": "\\mathbb{R} \\rightarrow \\mathbb{R}^2", "cd322a0269b30952befe1f9ae7972bcc": "T\\,\\!", "cd32374c001d545824e85d793bcdae7a": " f_{ij} = \\gamma \\delta_{ij}+\\partial \\gamma /\\partial e_{ij} ", "cd325912a2339458d9387f69dc2175cd": "ax^3 + bx^2 + cx +d=0", "cd326631112f02b2f32232551bf5bacd": "{d^2x \\over dt^2}-\\mu(1-x^2){dx \\over dt}+x-A \\sin(\\omega t)= 0,", "cd328db59a1860b4cf67d96024b65c71": "T \\left(n \\right)", "cd3296b5ddf9f0d40cdfe3521ee0f4c4": "x_1, \\ldots, x_l \\in \\mathbb{F}", "cd329ceabf6b18225694cf7537f773dd": "A \\cap B \\neq \\emptyset", "cd32a05aea1cc8956a76b15f8999c22a": " ds^{2} = \\lambda (dt - \\omega_{i}\\, dy^i)^{2} - \\lambda^{-1} h_{ij}\\, dy^i\\,dy^j,", "cd32b6fd74518605a47691cbc72eb070": "\\ell_{i,j}:=\n\\begin{cases}\n\\deg(v_i) & \\mbox{if}\\ i = j \\\\\n-1 & \\mbox{if}\\ i \\neq j\\ \\mbox{and}\\ v_i \\mbox{ is adjacent to } v_j \\\\\n0 & \\mbox{otherwise}\n\\end{cases}\n", "cd3312e8397ec82048be842e744afb3a": "{11 \\choose 1, 4, 4, 2} = \\frac{11!}{1!\\, 4!\\, 4!\\, 2!} = 34650.", "cd334e2bb3d3c76edb4d0b88ff33d61a": "\n\\beta(t) = \\omega_{0} \\left[b + g(t) \\right]\n", "cd33568fc51be75383c9ec7216771d65": "\\scriptstyle \\mathbf{X}_n", "cd338ae2c4e1bd8704572520a10f434b": "\\frac{q}{A} = U_c(T_i - T_o) - \\frac{U_c}{h_o} {[a \\cdot I - F_r \\cdot h_r \\cdot \\Delta T_{o-sky}]}", "cd33c6fb833905baf98ce3ba9eb9159e": "{BCD} = 2 \\times \\frac {1} {2 \\times 0.58778} = 1.701", "cd33f936c6ac2e004798b9021215b0ce": "\\textstyle \\{(3,4),(4,3)\\}", "cd346262267e8e4e8e65ff6abc25faf7": " \\frac{R}{2}\\sqrt{2} = \\frac{a}{2} \\!\\, ", "cd34c23db5332d0b276b400a10fb719d": "[M + nH]^{n+} + A^- \\to \\bigg[ [M + (n-1)H]^{(n-1)+} \\bigg]^* + A \\to fragments", "cd34c456ce16af0bba2ebce87e9fe3c9": "x_i p^i=0", "cd34fbb4c5a77528242c6a865e7411f5": " r\\ r ", "cd35884f9685ec91cc9a99c220917487": " \\hbar\\omega = \\epsilon_\\boldsymbol{q} = \\sqrt{\\frac{\\hbar^2\\boldsymbol{q}^2}{2m}\\left( \\frac{\\hbar^2\\boldsymbol{q}^2}{2m}+2gn \\right)} ", "cd358db4299f81f2e60429bed97c0fca": "|{\\psi_0}\\rangle", "cd361013b2edb016b6b678c95af89d7d": "(X)_p", "cd368692411ef5e80544222868719992": "{E}_{7}^{(1)}", "cd36f4b0aa3f60a9b9b3920690c97cb8": "T_{w,n}(x) = \\sum_{k=0}^n w^k s(2^k x)", "cd370276183fc46a9962c2d9e8d510fa": "a(x) = q_0(x)b(x) + r_0(x)", "cd374184587d2c1b1606d12fae5039df": "\\displaystyle{ST^*=TS,\\,\\,\\,SH={1\\over 4}I - T^2,\\,\\,\\, HS={1\\over 4}I - (T^*)^2, \\,\\,\\, HT=T^*H.}", "cd375884e4082fa849cb9d67c1480643": "D_n(x,0) = x^n \\, . ", "cd375b827b12329bed207357e649eb03": " 1 + \\xi(x-\\mu)/\\sigma \\geqslant 0. ", "cd379f2cc5c393b316508b0c996500d2": "\n\\sum_{c\\equiv 0 \\mod N} c^{-r} K(m,n,c) g\\left(\\frac{4\\pi \\sqrt{mn}}{c}\\right) = \\text{Integral transform}\\ +\\ \\text{Spectral terms}.\n", "cd37a2bcb6c37ef0869268a3d2904187": "\\beta< 4\\pi", "cd37e4bb9457d9be70314c263b07c781": "\\mu_0\\epsilon_0 = 1/c^2", "cd381193f4e6c1a76800d4a0bfc24a51": "\\mbox{kg}\\,\\mbox{m}^2\\,", "cd382d488f947119f1c2a1d2557c2815": " \\mbox{Exp}_{mutual fund} = 0.5\\times900 + 0.3*600 + 0.2\\times(-200) = 590", "cd3838a65610e4954824a103ea651cb1": "\\sigma=[p_0,p_1]", "cd38787cb2e73efd5feaa8d0e8a3440c": "\\lambda=10n", "cd38bcd6be36d9bb6413c305f2c3de20": "C_{ijklmn}=C_{IJK}", "cd38cbfd29ad933ad44827c76b846b66": "f(\\Gamma , 0)", "cd39480befb290b5269e81954553b16d": "\\mathbf{R}_x", "cd398863a23ef419572add09b3520bbe": "m^2 = + I, \\quad z = x I + m \\sqrt{p}", "cd39cce756803ab915cfa8f5977db6cd": "X_f=m\\frac{du_f}{dt}=m\\frac{dU}{dt}\\cos(\\theta-\\alpha)-mU\\frac{d(\\theta-\\alpha)}{dt}\\sin(\\theta-\\alpha)", "cd3a6286ed353ebcbfb0a0b3cd6e394c": "p \\rightarrow \\Diamond p", "cd3a758946ff2311c0d0dccc9ee8fcb4": " \\lang \\psi| (A |\\phi\\rang) = (\\lang \\psi|A)|\\phi\\rang \\, \\stackrel{\\text{def}}{=} \\, \\lang \\psi | A | \\phi \\rang ", "cd3ad07525a77b5b7375c7881ccb015b": "\\int_0^\\pi P_k^{m}(\\cos\\theta) P_\\ell^{m}(\\cos\\theta)\\,\\sin\\theta\\,d\\theta = \\frac{2 (\\ell+m)!}{(2\\ell+1)(\\ell-m)!}\\ \\delta _{k,\\ell}", "cd3ad2f9c5500323bb4c7ef328018774": "CII = (270.795 + 0.1038T)-0.254565D+23.708 \\log \\log(V+0.7)\\,", "cd3b3e8fa31c5b65f42a5c00fbd417a3": "\\langle A,B\\rangle / N = \\langle A',B' \\rangle,", "cd3b404019247d3019f99814c40a051d": "n\\cdot n! = n\\cdot n! + n! - n! = (n+1)! - n!\\;.", "cd3b8495cbae2e7a7ffbca14503ea788": "E (\\mathcal{A}f)(Y) = 0", "cd3b90fb57b355e13753c44a89b2401c": "\\displaystyle{W_{\\mathcal F}(z_1)W_{\\mathcal F}(z_2)= e^{-i\\Im z_1\\overline{z_2}} W_{\\mathcal F}(z_1+z_2),}", "cd3bcf1d1a356c9bff4bd9510f5596a0": "j=0,...,k", "cd3c200974bb49e013cc9b02d0102477": "\\mathcal S \\leftarrow \\mathcal S - \\{s'\\}", "cd3c59dd7f89bc6bd4b8953bdf9bc9fc": "u_i(y)=u'_i(y)", "cd3c6630bb7908fe7b48c8e6870d44ae": "\\langle f, g \\rangle = \\int_a^b f(x) \\overline{g(x)} dx", "cd3c9bb8acb671dbd1faba3deaa1e03e": "RD", "cd3c9ce5d80a57f0817d78280f5caaae": "f_{i-1} - f_i = k_i z f_{i+1}", "cd3d22a10e21444fd996a50e058add75": "E(v+1) - E(v) = h\\nu_0 - (v+1) (h\\nu_0)^2/2D_e\\,", "cd3db03c658ee1e01558458f721ff0dc": "\\scriptstyle\\{\\partial_\\mu\\}", "cd3de1b86b7941c8d339f096e8207e7c": " O(\\log^2 n \\log\\log n)", "cd3e54bab45d8cc299a6700bea91e3b0": " \na_{00} = \\mathcal{L} \\left\\{\n \\frac{\\omega a_{10}+a_{01} - \\mathcal{L}(2a_{20} \\omega+a_{11})}\n{2a_{20}\\omega+a_{11}}\\right\\}+ \\frac{\\omega a_{10}+a_{01} -\n\\mathcal{L}(2a_{20} \\omega+a_{11})}\n{2a_{20}\\omega+a_{11}}\\times\n\\frac{ a_{20}(a_{01}-\\mathcal{L}(a_{20}\\omega+a_{11}))+\n(a_{20}\\omega+a_{11})(a_{10}-\\mathcal{L}a_{20})}{2a_{20}\\omega+a_{11}}\n", "cd3e890e429c7299a8573ebd8b4f14d5": "\\frac{dU}{dx}= MU_x + MU_y\\frac{dy}{dx}", "cd3f6302b73156e1a6bc870a80c2539f": "\\textstyle H_{0} (x|q) =1, H_{-1} (x|q) = 0", "cd3f6bcd470e00783be33abd95d9b1bb": "\nE=\\sum_{m=1}^{\\infty}\\sum_{n=1}^{\\infty} \\frac{1}{2^{mn}}\n", "cd3f8b15d8973e9774338709bf49bccd": "\\int_b^a", "cd3f9f41a3e1fae42b48fbd552a362ed": "{v_1^2 \\over 2}+{p_1\\over\\rho} = {v_2^2 \\over 2}+{p_2\\over\\rho}", "cd3fb6178b956759b2201a36f25c6f71": "1+\\sqrt{8}", "cd3febae8e24d155357c5cd3921a5888": "\n\\frac{d\\boldsymbol{S}}{dt} \n= \\frac{2G}{c^2}\\sum_j \\frac{\\boldsymbol{L}_j\\times\\boldsymbol{S}}{a_j^3(1-e_j^2)^{3/2}}\n", "cd4062f07bcb9d3ba2c5a0bb357a3ef0": "x_{2n-2}^2+x_{2n-1}^2=1/2,\\quad x_i|(i\\neq 2{n-2}, i\\neq 2{n-1}) = 0", "cd40af8e341831b1a48ebf82eb98d370": " \\frac{\\sin(160^\\circ)}{\\sin(20^\\circ)} = \\frac{\\sin(180^\\circ-20^\\circ)}{\\sin(20^\\circ)} = 1", "cd40b2bdf15df371709fd83dcd0b6b74": "f(1,\\ldots) = f(0,\\ldots) \\oplus f(0,\\ldots) \\oplus f(1,\\ldots)", "cd40c020c062dc2203ac74afaf9525ae": "\\sum_{i\\neq j} |\\langle x_i , x_j \\rangle|^2 \\geq \\frac{m(m-n)}{n}", "cd40c9f4662c12b2439064f997632b33": "\\mathcal M= \\mathbf M_B + \\mathbf M_C", "cd40da750abc65c9a26c957ef290dfee": "cos\\theta |\\uparrow_{A}\\rangle \\otimes |\\downarrow_{B}\\rangle - sin\\theta |\\downarrow_{A}\\rangle \\otimes |\\uparrow_{B}\\rangle", "cd4143b71b0f8142ddfa4fb9fa7e31b9": " x \\diamondsuit x \\diamondsuit x = ( 0 \\ ,\\ 0 \\ , \\ 6 x_1^3 \\ , \\ 36 x_1^2 x_2 \\ , \\dots ) ", "cd41d09a64530499d7dd83058d94df6b": "\\omega_{S,i}", "cd41dd2e92c6073ef908b8b603d3e143": "P_n(1) = 1. \\,", "cd422d93a17143e6fccf3a46850eb403": "f(x) = \\sum_{y \\in \\Lambda} |\\Lambda| f(y) \\check \\chi_\\Omega(y - x)", "cd4236ae89918a3f2807452066b4d90b": "\\bigwedge( GFR \\longrightarrow GFC)", "cd4287804db45d3c6a3646fe17e1b760": "C_{t} = C_{0} e^{-k_{e}t} \\,", "cd42c1e6ce48f1cd5ad6cfc3bd367a7c": "\n\\begin{align}\n\\nu(x) & \\equiv \\int_0^\\infty \\frac{x^t \\, dt}{\\Gamma(t+1)} \\\\[10pt]\n\\nu(x,\\alpha) & \\equiv \\int_0^\\infty \\frac{x^{\\alpha+t} \\, dt}{\\Gamma(\\alpha+t+1)}\n\\end{align}\n", "cd42db2b3d29bae9e2f8ee9f9a8ca672": "\\left [\n\\begin{smallmatrix}\n 2 & -\\sqrt{2} & 0 \\\\\n -\\sqrt{2} & 2 & -1 \\\\\n 0 & -1 & 2 \n\\end{smallmatrix}\\right ]", "cd42dea02090f743e9ac2764500b16da": "\\frac{m_d}{m_s} \\approx 9 \\frac{m_e}{m_\\mu}", "cd42e8c2379195c514c8e5fc96e0c569": " \\langle \\psi | A | \\psi \\rangle", "cd4335404a5093bfc4bcb2332724ef16": "2^{n-1} n \\sqrt[n]{x_1 x_2 \\cdots x_n}", "cd43b573811e222787217c9214fcb026": "\\rho:D\\to[0,\\infty]", "cd43fdc41de5b1f12d0ccab1c9d11709": "C_n^{(\\alpha)}(x) = \\frac{(2\\alpha)_n}{(\\alpha+\\frac{1}{2})_{n}}P_n^{(\\alpha-1/2,\\alpha-1/2)}(x).", "cd440795e4e08ce9c52cf9925501db89": "\np(\\theta|y)\n= \\frac{p(y | \\theta) p(\\theta)}{p(y)}\n= \\frac {p(y | \\theta)}{p(y)} \\int p(\\theta | \\eta) p(\\eta) \\, d\\eta \\,.\n", "cd449c87ad85b87ac88bdb65b467c892": "Im(\\omega_j)", "cd44beba21469697e981cd893735e5ee": "L^{q_0} + L^{q_1}", "cd450dcd2dcaddcf752adc3df673a5d1": "E \\cap A", "cd4531bbec93b8fd898cb1be99305883": "= c \\mapsto f \\mapsto k \\mapsto c \\, \\left( t \\mapsto f \\, t \\, k \\right)", "cd45ed129741cbd32d6ea502ff5b4803": "\\operatorname{ch}(L) = \\exp(c_{1}(L)) := \\sum_{m=0}^\\infty \\frac{c_1(L)^m}{m!}.", "cd463c19bc1ed99234cbbdf83bb64750": "\\mathrm{Si-OH + OH-Si \\ \\xleftarrow[slow fracture] \\ Si-O-Si + H{_2}O}", "cd46a7c5b16a58c4f41ae218f54221c3": "L = 0.00508 N^2 h \\ln\\left({\\frac{d_2}{d_1}}\\right)", "cd46ab8c3080f259397742dfcf92fa97": "I_2(6) \\cong G_2", "cd46bd5dfbee122f75c6dcf9404f784b": " y \\ = \\ {\\log ({FV \\over PV}) \\over \\log (1+i)} \\ = \\ {\\log ({200 \\over 100}) \\over \\log (1.10)} \\ =\\ 7.27 {(years)} ", "cd46f6786cd7f9453545e3413168cba6": "R_{j}", "cd470e83ecc78b2097c923cd8709b2e1": "I_{\\mathcal{B}}", "cd4711ee56735c6d7740d15e301c9abe": "\\mathbf{Q}(\\zeta_n)", "cd4713e03a382e78b022f9b5176aa59e": "x_d=g_i(x_1,\\ x_2,\\ x_3,\\ \\dots,\\ x_{d-1},\\ x_{d+1},\\ \\dots,\\ x_N,\\ t), \\, ", "cd474d2a39091c8658a5f1a8f64d65a8": "\\cot\\frac{\\pi}{15}=\\cot 12^\\circ=\\tfrac{1}{2} \\left[\\sqrt3(\\sqrt5+1)+\\sqrt{2(5+\\sqrt5)}\\right]\\,", "cd47c7f78a78e422770cd38d219078d8": "\\mathrm{Nu}_D = 0.027\\,\\mathrm{Re}_D^{4/5}\\, \\mathrm{Pr}^{1/3}\\left(\\frac{\\mu}{\\mu_s}\\right)^{0.14}", "cd47ea39cc8e45136c592f6972bb64d7": "\\sigma(b_1)=b_2", "cd4855237da03b653138afed752f5da5": "\ne_{ii}= \\sum_{j} \\frac{A_{ij}}{2m} \\delta(c_i,c_j)\n", "cd485f6709e269e68177ab8855c5c537": "\\sum_{i=1}^n n_i\\arg(z-z_i)", "cd48682a8ac8d80a30c68ae1e8340863": "\\frac{p_{AC}}{m_1}=w_{AC}= \\gamma_{AC} v_{AC} =\\gamma_{AB} \\gamma_{BC} \\left( v_{AB}+v_{BC} \\right) = \\gamma_{AB} w_{BC}+w_{AB} \\gamma_{BC}\\, ", "cd48a1afdd1f2972a2dfef6dc3a911b4": "l_1 m_2 + l_2 m_1 = \\sin y,", "cd48a8fd3d267bc7b46ca0aff2183ade": "p_n \\sim n \\ln n.", "cd48f679bcfbab416f6eb48081bdb891": "\\begin{align}\n \\Beta_a &= \\frac{1}{2} A_i e^{ \\phi_i j}, \\\\\n \\Beta_b &= \\frac{1}{2} A_i e^{-\\phi_i j}, \\\\\n \\lambda_a &= \\sigma_i + j \\omega_i, \\\\\n \\lambda_b &= \\sigma_i - j \\omega_i,\n\\end{align}", "cd4912c693b13fd7c8da21c8ff8ffccb": "(e^\\mathbf{X})^{\\rm T}", "cd4921840de3ea98148d9f87343b2f62": "\\Sigma(W,S)", "cd49301bb48d66d08494754a0f8e7d48": "x = q + x^m", "cd499a4df3aea76d2236d091596241b5": "J_n (x) = \\frac{1}{2 \\pi} \\int_{-\\pi}^\\pi e^{i(n \\tau - x \\sin(\\tau))} \\,d\\tau.", "cd499bbac4726c3e8c09cca2228b3a13": "\nm\\frac{d^2 r}{dt^2} - mr \\omega^2 =\nm\\frac{d^2 r}{dt^2} - \\frac{L^2}{mr^3} = F(r)\n", "cd49d7ca4b92dc9bc32365b0246d92c6": "\n\\frac{\n\\int{ f(a) f(b) e^{i \\int{ f(x) \\Box f(x) dx^4}} }[Df] \n}{\n\\int{ e^{i \\int{ f(x) \\Box f(x) dx^4}} }[Df] \n}\n= K^{-1}(a,b) = \\frac{1}{|a-b|^2}\n", "cd49da3e7820946d9ad5498772905d75": "\nG^{\\mathrm{R}}(\\mathbf{x} t|\\mathbf{x} 't') = \\mathrm{i}\\langle[\\psi(\\mathbf{x} ,t),\\bar\\psi(\\mathbf{x} ',t')]\\rangle\\Theta(t-t')\n", "cd4a363464171701fae218ec9d745804": "g(x)=\\sum_{n=0}^\\infty s_n x^n ", "cd4a8c56197cac2e1cbf5cb5462a70c0": "\\left[ \\begin{array}{ccc}a&b&c\\\\d&e&f\\\\ \\end{array}\\right]", "cd4a983b9d5f214f20a87931d63e9116": "Y_{5}^{-2}(\\theta,\\varphi)={1\\over 8}\\sqrt{1155\\over 2\\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(3\\cos^{3}\\theta-\\cos\\theta)", "cd4a9cc7136cc0a64e1962c072923148": "\\{ (x, v)\\in \\mathbf{P}(V)(k)\\times \\mathbf{Gr}(r, \\mathcal E)(k) \\mid x\\in v\\}", "cd4ad6a9e48c9037c4645c6aad08d698": "\n\\begin{array}{c|cccc}\n0 & 0 & 0 & 0 & 0\\\\\n1/2 & 1/2 & 0 & 0 & 0\\\\\n1/2 & 0 & 1/2 & 0 & 0\\\\\n1 & 0 & 0 & 1 & 0\\\\\n\\hline\n & 1/6 & 1/3 & 1/3 & 1/6\\\\\n\\end{array}\n", "cd4af0a24262d767bfa5123246a1cab8": " \\sigma_u=\\sigma_v=\\sigma_w=0.1W_{20} ", "cd4b32be4912c1858950d7abd947c7a7": "t_{discharge} \\propto {\\sqrt{LC}}", "cd4b3523596be860c62e09f513a6a542": "\\mu^2 < 0", "cd4b92ece605ec5efc573c1c795d69a0": "\\begin{align}\n\\dot{\\mathbf{x}} &= A \\mathbf{x} + B \\mathbf{u}\\\\y &= \\begin{bmatrix}1 & 0 & 0 & \\cdots & \\end{bmatrix} \\mathbf{x} = x_1 \\end{align}", "cd4bbac3fe71d7d730433a73acd022de": "E_Y", "cd4bcf22aac79832ccb76f393031109e": "\nU =\n\\begin{pmatrix}\n1&1&1&\\cdots&1\\\\\n1&\\omega&\\omega^2&\\cdots&\\omega^{n-1}\\\\\n1&\\omega^2&(\\omega^2)^2&\\cdots&\\omega^{2(n-1)}\\\\\n\\cdots&\\cdots&\\cdots&\\cdots&\\cdots\\\\\n1&\\omega^{n-1}&\\omega^{2(n-1)}&\\cdots&\\omega^{(n-1)^2}\n\\end{pmatrix}\n", "cd4bdbc93662764d4501ce57c5151ce1": "\\frac{d}{dt}p_r =Q_r + mr{\\dot{ \\theta}}^2 \\ , ", "cd4bef16fe020baf7784ff2bc16a5b2f": "\\langle x \\rangle_1 \\wedge \\langle y \\rangle_1 = x \\wedge y", "cd4bf78b5cdfe6c654b3326286a7d2cd": "\\mathfrak{e}_{7(-5)}", "cd4c2ed9bad6562f08128e08c617e745": "\\begin{align}\ny_N - y(t_N) &{}=\n y_N - \\underbrace{\\prod_{j=0}^{N-1} \\Phi(h_j)\\ y(t_0) + \\prod_{j=0}^{N-1} \\Phi(h_j)\\ y(t_0)}_{=0} - y(t_N) \\\\\n&{}= y_N - \\prod_{j=0}^{N-1} \\Phi(h_j)\\ y(t_0) + \\underbrace{\\sum_{n=0}^{N-1}\\ \\prod_{j=n}^{N-1} \\Phi(h_j)\\ y(t_n) - \\sum_{n=1}^N\\ \\prod_{j=n}^{N-1} \\Phi(h_j)\\ y(t_n)}_{=\\prod_{n=0}^{N-1} \\Phi(h_n)\\ y(t_n)-\\sum_{n=N}^{N}\\left[\\prod_{j=n}^{N-1} \\Phi(h_j)\\right]\\ y(t_n) = \\prod_{j=0}^{N-1} \\Phi(h_j)\\ y(t_0) - y(t_N) } \\\\\n&{}= \\prod_{j=0}^{N-1}\\Phi(h_j)\\ y_0 - \\prod_{j=0}^{N-1}\\Phi(h_j)\\ y(t_0) + \\sum_{n=1}^N\\ \\prod_{j=n-1}^{N-1} \\Phi(h_j)\\ y(t_{n-1}) - \\sum_{n=1}^N\\ \\prod_{j=n}^{N-1} \\Phi(h_j)\\ y(t_n) \\\\\n&{}= \\prod_{j=0}^{N-1}\\Phi(h_j)\\ (y_0-y(t_0)) + \\sum_{n=1}^N\\ \\prod_{j=n}^{N-1} \\Phi(h_j) \\left[ \\Phi(h_{n-1}) - E(h_{n-1}) \\right] \\ y(t_{n-1}) \\\\\n&{}= \\prod_{j=0}^{N-1}\\Phi(h_j)\\ (y_0-y(t_0)) + \\sum_{n=1}^N\\ \\prod_{j=n}^{N-1} \\Phi(h_j)\\ d_n\n\\end{align}", "cd4c348cd64dfd3d0ee657a28b0bed12": "((y,z),S)", "cd4c62ef44a45d41f98408ff2da71e9e": "\\varepsilon=\\varepsilon(\\lambda)", "cd4c6fa7d09da88370837188761d9ab4": " \\operatorname{E}[\\operatorname{tr}(\\epsilon^T\\Lambda\\epsilon)] = \\operatorname{E}[\\operatorname{tr}(\\Lambda\\epsilon\\epsilon^T)] \n= \\operatorname{tr} (\\Lambda(\\Sigma + \\mu\\mu^T)) = \\operatorname{tr}(\\Lambda\\Sigma) + \\mu^T\\Lambda\\mu.", "cd4c968686b9174a589ed7489371c35f": "G'' = G \\setminus B", "cd4ca5ca3ac9a1fbe51417dc2031b2ac": " \\cosh x = \\sqrt{\\pi} \\; G_{0,2}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} - \\\\ 0,\\frac{1}{2} \\end{matrix} \\; \\right| \\, -\\frac{x^2}{4} \\right), \\qquad \\forall x ", "cd4cd28ccb175a9ff0004c1901892e8d": "\\displaystyle a^2+b^2=c^2", "cd4ce5730ecd1ef3db0cc326e35f5459": "H = \\frac{5}{2} N k_B T + \\frac{N^2}{V} (b k_B T - 2a)", "cd4d77102a42b2956709b3c78e1a3a14": "[x,y,x] = 0", "cd4dc5b529518d128079805ae4933097": " E = E_{ext} + \\sum_i E(p_i,r-R_i) ", "cd4e1c3d87f41e38378fda6de758e03c": "(rm)s = r(ms).\\ ", "cd4e51ebc7ee876ff7276c454682f316": "\\left(\\nabla_X Y\\right)^k = X^i (\\nabla_i Y)^k = X^i \\left(\\frac{\\partial Y^k}{\\partial x^i} + \\Gamma^k{}_{im} Y^m\\right).\\ ", "cd4edaaf6414c3c466349f97415f28f4": " \\forall z \\, ( z \\in X \\leftrightarrow z \\in Y ) \\rightarrow X = Y ", "cd4ee38d37100a2510dce36196f4e6e7": " g\\otimes x\\mapsto \\pm x", "cd4ef37cd8e18ff357a8968a5bb87f89": "\\left [x_i,x_{i+1} \\right ]", "cd4f1445920a90f5285cb0f2e31d94ee": "(\\xi, \\eta )", "cd4f3278dcb40aaf63882bdcc31f4cf2": "B(y, s) \\subseteq B(x, r) \\setminus E.", "cd4f451c3b1ae8be8de894ec8644c920": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 2}{4 \\choose 2}^2 = 123,552", "cd4f6eddda2ae173054faeade112522a": "s=s'\\left[{1-\\frac{2(xx'+yy')}{s'}+\\frac{x'^2+y'^2}{s'}}\\right]^{1/2}", "cd4f76d2ef4c45e2d7f3e39d511886c3": "\\ln K_{eq} = - \\frac{{\\Delta H^\\ominus}}{RT}+ \\frac{{\\Delta S^\\ominus }}{R}. ", "cd4f8e1039607760f1bda5fc93bdd056": "x_c(t)", "cd4fb56bed4ae0b27cf38a889a2d119b": " \\ \\psi (\\pi) = 0 = B \\sin (m \\pi) ", "cd4fdafac62738d91db7549ff93ca1c2": "\\frac{\\mathrm{Ai}(x)\\mathrm{Ai}'(y) - \\mathrm{Ai}'(x)\\mathrm{Ai}(y)}{x-y}.\\,", "cd500b16e7d5855794042e1037c0d496": "C_{GDj}=A_{GD}\\frac{\\epsilon_{Si}}{w_{GDj}}", "cd501e0b48802d8286888f7e02c755a9": "\\mathcal{L}_{qp}=\\mathbf{p} \\cdot \\dot{\\mathbf{q}} - H(\\mathbf{q}, \\mathbf{p}, t)", "cd502b164bda2e2f5ea4d824dc81f1d6": " O | \\psi \\rangle = \\sum_{i=1}^{n} c_i \\left( O | e_i \\rangle \\right) = \\sum_{i=1}^{n} | e_i \\rangle \\langle f_i | h \\rangle , ", "cd5059da46679b2e8c2b369aa3499ad8": " (\\mathbf{u}\\cdot\\mathbf{w})\\mathbf{v}_z-(\\mathbf{u}\\cdot\\mathbf{v})\\mathbf{w}_z", "cd50600524b3254133ae70a488d99932": "\\arctan z = \\sum_{n=0}^\\infty \\frac {(-1)^n z^{2n+1}} {2n+1}.", "cd507b993ef6d3a800ae293a52f4d796": " \\kappa \\ ", "cd50bead489ed744d267cb967a999c9d": "x_1 = r\\sin\\psi \\cos\\theta\\ ", "cd50c27b2483e397258ebcf975edc4cc": " \\langle E, E \\rangle = \\operatorname{span} \\{ \\langle x, y \\rangle | x, y \\in E \\},", "cd50f4616448e3bf2fb88d5995d96ac8": "\\Delta G^\\ominus =-RT \\ln K^\\ominus", "cd5118ca24d02fd9f9a7394bfd883911": "\\alpha_{b,c} = \\sum_{n=c^k>1} \\frac{1}{b^nn} = \\sum_{k=1}^\\infty \\frac{1}{b^{c^k}c^k}", "cd5121687db9fa7750a8f7145b04f90a": " a \\circ \\left( b \\circ a \\right) = \\left( a \\circ b \\right) \\circ a .", "cd51228667cb6c56a1a3c58e43946841": "F_{0,\\lambda} = \\frac{\\int_0^\\lambda E_{\\lambda,b} d\\lambda}{\\int_0^\\infty E_{\\lambda,b} d\\lambda} =\\frac{\\int_0^\\lambda E_{\\lambda,b} d\\lambda}{\\sigma T ^4} =\\int_0^{\\lambda T} \\frac{E_{\\lambda,b}}{\\sigma T ^5} d(\\lambda T) = f(\\lambda T) ", "cd5143a78a0e19ec8ad4f9142b3b04f3": "\\frac{\\hbar^2}{2m} (\\nabla + \\tau)^2 \\Psi + (\\mathbf{u} - E)\\Psi = 0 ", "cd516da018c97f5dacd9a69230d0e351": "\\psi_2", "cd519c0968fd44f5355cdf034ce3d4d8": "\\sigma>\\sigma_a", "cd51d76affe99487d95303d00e19e5ab": " f(x) = o(g(x))", "cd51e9d6678a84668b6e2b969d208100": " ds^2 = \\sum g_{ij}dq^idq^j = dx^2 +dy^2 +dz^2 ", "cd520b6444b715143bff836c4bbbdcc9": "\n \\Pr(X = k) = \\left(\\frac{r}{r+m}\\right)^r \\frac{\\Gamma(r+k)}{k! \\, \\Gamma(r)} \\left(\\frac{m}{r+m}\\right)^k \\quad\\text{for }k = 0, 1, 2, \\dots.\n ", "cd52318d4369a99c7c16cc3a89a595ca": "\\eta \\colon \\mathcal{S}(X) \\to \\mathcal{N} (X)", "cd5234d4849cc2f8d4b5ad0856add791": "\nJ=\\begin{bmatrix}\n\\mathbf{I}_k &0 & \\dots & 0\\end{bmatrix} ,\n", "cd524a117b3f64ece9aea37f1da43e3f": "\\frac{\\hat{L} X(x)}{X(x)} = \\frac{\\hat{M} T(t)}{T(t)}", "cd525c41e2331a3eff260b7a600605c3": "\\frac{d}{dx} \\left(\\frac{1}{\\cos(x) }\\right) = \\frac{\\sin(x)}{\\cos^2(x)} = \\frac{1}{\\cos(x)} \\frac{\\sin(x)}{\\cos(x)} = \\sec(x)\\tan(x).", "cd52b74cade4eed782dd38e4413cd7e6": "p_t/p_0", "cd52c97634d9eca70746b6036d4a4d0f": "W^{k,p}(\\mathbf{R}^n)\\subset C^{r,\\alpha}(\\mathbf{R}^n).", "cd52ddea435e410e04e1b061985b9ffd": "t=t_1", "cd52e13aef7fbd24d4b9438b2cc123c3": "\\langle E \\rangle = \\frac{1}{2} \\sum_n E_n", "cd52e5731c43b1d8110f34e2510c2dc6": "h=x_g-0.5 cV_t\\!", "cd53474fecd0f370049a78d991e25ae2": "\\pi_1(D,*) \\to \\pi_1(A,*)", "cd536b32018243679a685b55f4b73c49": "\\psi(\\Omega^\\Omega) = \\Gamma_0", "cd53737b0bca3c361ab3af9549a290a0": "\\sqrt{2},\\ -\\sqrt{5}/5,\\ -\\sqrt{5}/5", "cd541ac62d64b0a9805045b8669a1248": " \\mathcal{H} = \\frac{1}{2m_\\text{e}}\\sum_{i=1}^N \\left(\\mathbf{p}_i - \\frac{e}{c}\\mathbf{A}_i \\right)^2 + e\\phi(\\mathbf{q}),", "cd54316eee3330748b84feef0dfc1403": "T_qE=T_qQ\\oplus \\mathcal M_u\\,,", "cd544fa1a214d0ffc2f4297a3f934c3c": "T_1\\cap T_2", "cd5459a9eb42c47155e372d21c6f42b4": "\\begin{align}\nH_{1,I} &\\equiv U H_1 U^\\dagger \\\\\n&=-\\hbar\\left(\\Omega e^{-i\\omega_Lt}+\\tilde{\\Omega}e^{i\\omega_Lt}\\right)e^{i\\omega_0t}|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\left(\\tilde{\\Omega}^*e^{-i\\omega_Lt}+\\Omega^*e^{i\\omega_Lt}\\right)|\\text{g}\\rangle\\langle\\text{e}|e^{-i\\omega_0t} \\\\\n&=-\\hbar\\left(\\Omega e^{-i\\Delta t}+\\tilde{\\Omega}e^{i(\\omega_L+\\omega_0)t}\\right)|\\text{e}\\rangle\\langle\\text{g}|\n -\\hbar\\left(\\tilde{\\Omega}^*e^{-i(\\omega_L+\\omega_0)t}+\\Omega^*e^{i\\Delta t}\\right)|\\text{g}\\rangle\\langle\\text{e}|\\ .\n\\end{align}", "cd546734058dac1436200b788514d8fd": " \\sum_{j\\in \\N} a_j \\tau^j ", "cd54a12d1448b4f04b53505c84e54d97": "r'(t) = r(t) - v t.\\,", "cd550dd2087085cabc02a6ba09250c67": "\\kappa\\, h = \\frac{2\\,\\pi}{\\lambda}\\, h,", "cd554682ee0d7c52a75a83f871772cd0": "m'' = m \\| [(m_1' \\oplus t) \\| m_2' \\| \\dots \\| m_x']", "cd56535cf7de5420ea670ce19ff77409": "(u_n)", "cd566757c8798d55eed5fd497ea2a4e5": "\\mathbf{f} \\,\\colon \\,f_1\\ge f_2\\ge \\cdots \\ge f_N\\ge 0", "cd56c6667a803d81c77c8672ba654d3b": "J=0", "cd56da285247841b7c8335aed1dd76b9": "t(t-1)^{n-1}", "cd56e4587a653af6424f27421010e549": "\\lambda<1", "cd572b81278f144a23eae0295fb94707": "R_{\\theta JC}+R_{\\theta B}", "cd57cd5ced0ad3b24185afd3d2dc477a": "T_{2L}=A+\\frac{Bk_2}{k}e^{-kL}", "cd57e466d4809355c14a02178ce30683": "{\\color{Blue}~6.8}", "cd57e758dbad4b90ece5c134f0b728bd": " F_q ", "cd5801fdd6b77419438f8818179f8ae4": "\\ln(k) - \\psi(k) = \\ln\\left(\\frac{1}{N}\\sum_{i=1}^N x_i\\right) - \\frac{1}{N}\\sum_{i=1}^N\\ln(x_i)", "cd58315a47c9e409e9dcab593027b3a9": " {D^2 \\bar h^i \\over Ds^2} + \\bar R^i_j \\bar h^j = 0 ", "cd5865c3fb2b4153468f0eaf0cc000d9": "\\sum_{k=0}^\\infty h_k(X_1,\\ldots,X_n)t^k = \\prod_{i=1}^n\\frac1{1-X_it}.", "cd5892e262ec8b01edb200a713f79b76": "\n \\underline{\\underline{\\mathsf{C}}} = \\underline{\\underline{\\mathsf{A}_\\varepsilon}}^T~\\underline{\\underline{\\mathsf{C}}}~\\underline{\\underline{\\mathsf{A}_\\varepsilon}}\n ", "cd58ca418edc0c64206c92606d495876": "e_3=1,", "cd58de915f61f2c07fb5a264c0d7df42": "h_\\ell(m_1,\\ldots ,m_\\ell)", "cd5926fa566c240928dbdee77a64e2ac": "\\tau_D", "cd5929c79c477a5164992a39e3be0feb": " \\cup A_i", "cd594a990cf28f0eb73de59a43c4db6c": "\n\\ H(e^{j \\omega}) = 1 + \\alpha e^{-j \\omega K} \\,\n", "cd5954252353a1fd9a71fac4e594e4c5": "\nD^j_{m'm}(\\alpha,\\beta,\\gamma) = (-1)^{m'-m} D^j_{-m',-m}(\\alpha,\\beta,\\gamma)^*.\n", "cd596df08ab9bae52205dbe6c721a6c8": "\n\t\\frac{\\delta Q}{T}=\\frac{\\delta Q_0}{T_0}\t\n", "cd59aa0c488a3f128e4d8a5368572fd2": "\\langle \\mathbf{r}_{i} \\cdot \\mathbf{r}_{j} \\rangle = 3 b^2 \\delta_{ij}", "cd59db2d2c70c9a71e649dae06dc1fe1": "a=\\lim_{x\\to x_0^-}\\frac{f(x)-f(x_0)}{x-x_0}", "cd59edb56eade2101b6988df542d08df": "\nx_{n-1} - x_n = \\frac{A_{n-1}}{B_{n-1}} - \\frac{A_n}{B_n} = \n(-1)^n \\frac{a_1a_2\\cdots a_n}{B_nB_{n-1}} = \\frac{\\Pi_{i=1}^n (-a_i)}{B_nB_{n-1}}.\\,\n", "cd5a08e3f4509c9eb9748b14765bc59a": "w[T^*]y", "cd5a80e7290cc5462bf619e73e75ceb8": " 0 \\le \\theta_1 \\le \\theta_2 \\le \\ldots \\le \\theta_k \\le \\pi/2", "cd5b1f3c0be621d399bec4e5dae9f0dd": "V_{DD}", "cd5b6e77cc8342e09825ef7a05074d50": "\\int_B \\operatorname{P}(A|\\mathcal{B}) (\\omega) \\, \\operatorname{d} \\operatorname{P}(\\omega) = \\operatorname{P} (A \\cap B) \\qquad \\text{for all} \\quad A \\in \\mathcal{A}, B \\in \\mathcal{B}. ", "cd5b9d7a32271edc886e392ffe13feea": "\\operatorname{lcm}(1,2,\\dots, n)", "cd5bef28ccb15fab5fbe0e79d728885c": "A\\in Y", "cd5c0761adf918973d10d601ec7ad092": " R_{\\rm eff}^{-1} = R_1^{-1}+R_2^{-1}.", "cd5c0a66195cf0170cd0e5cc99fb336f": " \\begin{cases} x = \\frac{c\\pm\\sqrt{c^2-4ab}}{2} \\\\ y=-\\left(\\frac{c\\pm\\sqrt{c^2-4ab}}{2a}\\right) \\\\ z=\\frac{c\\pm\\sqrt{c^2-4ab}}{2a} \\end{cases} ", "cd5c26fd0af9574dcd467c0bf0475c38": "\\scriptstyle |x\\rangle + |y\\rangle", "cd5c2b538915f4335063c3749ecb834d": "f:Y \\to \\prod_{i \\in I} X_i \\mathrm{ , } \\quad f(y) := (f_i(y))_{i \\in I}", "cd5c900d868f41270288bfedce2108b8": "y = c t^k + \\cdots", "cd5c9b465b476657b8e716bb9c16e8a3": "\\textstyle (a_n)_{n\\geq0}", "cd5d075efcf4e2072714ca07a5e80949": "[Q^\\dagger,f\\}=\\frac{\\partial}{\\partial \\bar{\\theta}}f-i\\theta \\frac{\\partial}{\\partial t}f.", "cd5d5b68eac9e2f80900eda578232af6": "\\Delta(t) = 1,\\quad \\nabla(z) = 1,\\quad V(q) = 1.", "cd5d7c959b63d4384ed822f5a3845bcb": "\\langle A_i,f_{ij}\\rangle", "cd5d8ea67fcf2abad11d71b28160dc22": "|\\phi\\rang+|\\psi\\rang", "cd5d94360569c1f279a344cb4a952fb7": "a_N", "cd5d96c0ea3e75e95c51f8aba3e92062": " \\prod_{p} \\Big(1 - \\frac{1}{p(p+1)}\\Big) = 0.704442... ", "cd5e3e1edb938ab1c6f914e47b567d5d": "pq = (st - \\mathbf{v}\\cdot\\mathbf{w}) + s \\ \\mathbf{v} + t \\ \\mathbf{w} + \\mathbf{v}\\times\\mathbf{w}.", "cd5e7416b09d4f193de78d404352c2c2": " \\int_{a}^{b} L(t,x,x')\\, dt . \\,", "cd5e74ea975f3d9dd10380fb933349cf": " \\theta_1 + \\theta_2 = 90^\\circ,", "cd5ec923c26ea689fd35b4c83b953cea": "x_1x_2x_3x_4", "cd5f00faa0fdfeb382e25d2c696e432d": " \\scriptstyle \\sin (\\theta + 120^ \\circ)", "cd5f6ea4c9c0a1ef5c1442b5c8a81e82": "y' = f(t,y)", "cd5f83ac90fdc17c891398e7a48af588": "\n\\left\\vert \\bar{A}_{j_1} - \\bar{A}_{j_2}\\right\\vert > s \\,t_{1-\\alpha/2}\\sqrt{\\frac{N-1-T_1}{N-k}}\\sqrt{\\frac{1}{n_{j_1}}+\\frac{1}{n_{j_2}}}\n", "cd5fd201241c55a542c9b25065d9e36a": "J_n = \\int \\cos^n{ax} dx\\,\\!", "cd5ff37d7c9a8aac221b63ec31e168c5": "|a| \\le b \\iff -b \\le a \\le b ", "cd6007ff0bb1b9c06c9bfb287760fdcf": "(a/2, 0)", "cd600ea82c8ffac88c5186f9f70d0389": " (t/t_{1/2})\\ln \\left(\\frac {1}{2}\\right) = -t/\\tau = -\\lambda t", "cd601cb7744b5e9645f05751eacae9a8": "\\frac{\\partial \\rho_e}{\\partial t} = 0", "cd6022c8678ac6cef8c5e74545ead9cd": "W = \\frac{{4Td}}{{L}}", "cd6047451767491337f5041b6768cdb6": "R_n(\\xi,1)=1\\,", "cd6081f5ed35607847d4c4fbb0596d0d": "\nC^J_{E_3} = \\varepsilon^{1}_1 \\varepsilon^{2}_2 / D\n", "cd608ca1754a27ea809cb91206ef50f3": "y \\in [q]^n", "cd60bbc1158f2e196f5eade286a54613": "g'_*", "cd60bfb5e60be4ee9cb6395f938a62cc": "|k-1|<0.0004", "cd60cfacbc3d165bb5312c90daa158d6": "S = \\sum_{i=1}^{n} v_i(x_{i}-x_{i-1}),", "cd60d373f85d3aefc26b932ac206f922": "T(\\mathbf{X})", "cd60ec73f0609791d4e6987bc462608d": "V = \\frac{q_1q_2}{4\\pi\\epsilon_0 |\\bold{r}|}", "cd6110408ea276004c349c742c6d5910": " \\sum_{i}^{j_1} x_{ij} \\geq 1", "cd612a5e01e99bcdca3d3de9eaf258b6": "\\left(\\frac{\\part V_m}{\\part T}\\right)_{p}=-\\left(\\frac{\\part S_m}{\\part p}\\right)_T", "cd61bf1ba69e12b47bdfbea5948c8e36": "t_{LL}^{\\mu \\nu} = 0", "cd61c1cf0a2a776a55f76c427dd9079d": " \\Im ", "cd6240ab9e4a7a9acc08b781e686d46c": "P/H\\to X", "cd62533349de5bdb9e25c8003f705c51": "\\rho_m\\;", "cd62afc2a372119212a17fb92ddec47e": "P=K\\rho_c^{1+\\frac{1}{n}}\\theta^{n+1}", "cd62dfba8d4f181cd373a704ba0a2a89": "V \\not\\subset \\Omega", "cd62f10e304eb8f40df4cda83b7836cb": "\\mathbb{Q}_\\nu", "cd63073d19de2ef3c54fb1af892ea748": "\\frac{V_O}{I_S} =\\frac{(R_S \\mathit{\\parallel} r_D)}{1+j\\omega (C_D+C_J)(R_S \\mathit{\\parallel}r_D)} \\ , ", "cd6360a658bb44113748979a14e57569": "2S=(A+B+C)", "cd636ab023df551914910977e79de4a0": "a_{1,j}\\in S_j", "cd638b4a6dfd7af3edd32a93fdd75b42": "\\sqrt[n]{|a_n|} > 1", "cd638cc136cff970d033882d12e8c90c": "\\scriptstyle| \\psi \\rangle_A", "cd63e33745ce42605fa12afcf34102fe": "x_{n+1} = x_n + d\\; \\frac {\\left(1/f\\right)^{(d-1)}(x_n)} {\\left(1/f\\right)^{(d)}(x_n)} ", "cd6423c1cdd524a83568d0683be81483": "\\Omega \\cap \\{x+y:y\\in\\Omega\\} = \\phi ", "cd6427853c790285636f845da5c038af": "a^2 - b^2 = (a+b)(a-b)", "cd6428956cbfde59e350a7641ee80894": "\\delta\\gamma-\\Delta\\beta=(\\tau-\\bar{\\alpha}-\\beta)\\gamma+\\mu\\tau-\\sigma\\nu-\\varepsilon\\bar{\\nu}-(\\gamma-\\bar{\\gamma}-\\mu)\\beta+\\alpha\\bar{\\lambda}+\\Phi_{12}\\,,", "cd649fa4dfc905626418a7d005989cdb": "MI=\\sqrt{(Q_1 Q_2)/(Q_3 Q_4)} ", "cd64a1e88df0fea481c7a32725fac93a": "\nR_{\\text{eq}} = \\frac{V_{\\text{p}}}{I_{\\text{p}}} = \\left(\\frac{N_{\\text{p}}}{N_{\\text{S}}}\\right)^2 {R_{\\text{L}}}\n", "cd64cc996fb55515eaac362bc221dcf5": "Z = \\sqrt{\\sum_1^k \\left(\\frac{X_i}{\\sigma_i}\\right)^2}", "cd653b962316db22006bb9565c5ec01d": "f \\colon U \\to \\bigcup_{p \\in U} S_{(p)}", "cd657385b5b62de877448d0747f9ad2e": " \\sum_{k=1}^n a_k \\exp(i \\lambda_k x), \n\\quad a_k \\in \\mathbb{C}, \\, \\lambda_k \\geq 0 ", "cd65743ec562dc1b142180d8d9989271": "\\hat{\\beta},", "cd659c88b59700bf84387267a5f90d5d": "v''+\\left(\\frac{2y_1'(t)}{y_1(t)}+p(t)\\right)\\,v'=\\frac{r(t)}{y_1(t)}", "cd6601cdb5fc12d5aa58ed4308d52254": "\\rho_q = \\sqrt{\\frac{L}{C}} = \\begin{cases} \\rho_w = 2\\alpha R_H, & \\mbox{ - wave impedance } \\\\ \\rho_{DOS} = R_H, & \\mbox{ - DOS impedance } \\end{cases}", "cd674c43faafe81c8818c8d7c974e4f2": "u_{1,2}^{0} = \\delta_{1}^{0}(2) = 4", "cd67ad6c07657a8908e3e4be2fee266a": "\\mu\\geq0", "cd67e382ab3fbf3e8440a804492e4945": "\\left(2+\\sqrt{-6}\\right)^7 = \\left(9+2\\sqrt{-6}\\right)\\left(2+ \\sqrt{-6}\\right) = 6 .", "cd67ef1e1a18985b4fc9f8b9918af819": "\n\\nabla\\cdot\\left(A\\left(\\frac{\\vec x}{\\epsilon}\\right)\\nabla u_{\\epsilon}\\right) = f\n", "cd68a543eef26be3860c6f4be5df166c": "c = r - \\frac{1}{T} \\ln( \\frac{F}{S} ) = ", "cd68af350c86e40c092ea8ea9c959e81": "=\\frac{1}{|\\Gamma_{m}|}\\sum_{\\sigma\\in\\Gamma_{m}}\\int_{X^{2m}}1_{R}(\\sigma(x))dP^{2m}(x)\\,\\!", "cd68f3c639b2d14683f726f66abd01c9": "g_0 = g(0) = \\frac{G m_e}{{r_e}^2}", "cd69202e331943c3cbde996c111c0a91": "n_\\eta", "cd6929785e0286d4f9e4e758d4a517f2": "+\\Gamma^{a_i}{}_{dc}", "cd6966874cda090fb3c4e651fb8f9030": "\n C_1 = - \\cfrac{F_1}{\\pi} ~;~~ C_3 = - \\cfrac{F_2}{\\pi} ~.\n ", "cd6994b216c7da25073aee20fa0786bd": "c_1(\\tau)=: -a", "cd69ceee65a39bbbd0dc18b163f150ea": " (t^{1/2} - t^{-1/2})V(L_0) = t^{-1}V(L_{+}) - tV(L_{-}) \\,", "cd6a3a8b1168db4a92d5ae12e68412b5": "L+3H \\rightleftharpoons H_3L:\\log \\beta_{013} =\\log \\left(\\frac{[H_3L]}{[L][H]^3} \\right)=pK_3+pK_2+pK_1 ", "cd6a433546462592d2ffc939dbfa3f24": "0.\\overline{01}_2 = 0.010101\\dots_2", "cd6a60a3502cdc94d908f5dd94e56466": "{\\lambda_i}^{-}", "cd6a7f5d42bb87e124571621854ad2aa": "1^2+2^2+3^2+4^2+5^2+27^2=28^2", "cd6a9bd2a175104eed40f0d33a8b4020": "GT", "cd6a9fad8660103a432d4d62aa4b4aba": "(X, f, \\alpha)", "cd6aa508ba1da9177f2e161b9a4897a4": "|r| < 1 \\,\\!", "cd6adfce916b8cb75a99a5d85f93815d": "\n\\begin{matrix}\n4 \\overline{)950} \\ \n\\end{matrix}\n", "cd6b4dea2983207d65d7df57d672f817": "p_{iy}=\\sum_{u=1}^{i}\\sum_{v=1}^{j}b_{uv}", "cd6bfeea0d7ec132a469b8a84c3ce224": "b_2=\\frac{2 \\beta_2-3 \\beta_1 -6}{10 \\beta_2-12\\beta_1 -18} .", "cd6c5fa312327dca33de3b38e475deba": "E_{1}", "cd6c7890846251eabc08872b0e0c8dd2": " \\mathbf{H}_n = \\left\\{ \\begin{matrix} \\mathbb{C}^n & \\mbox{ if } n < \\omega \\\\ \\ell^2 & \\mbox{ if } n = \\omega \\end{matrix}\\right. ", "cd6c94a72eef9ecf92dfd075088fbad6": "\nA^2 = m^2 k^2 e^{2} = m^2 k^2 + 2 m E L^2\n", "cd6ca3adc8978bf9151fb1c520749408": "\\frac{2}{4}=0.5", "cd6cb2ee70eaa0d07251fde50f3f6a3b": " \\mathbf{\\hat{r}} ", "cd6d7897737c0dadac4da94c2ec95643": "(2^{|k_{b_1}|}+2^{|k_{f_2}|})", "cd6d96861f69f8a7da38794197f8faa5": "xy * z * wv", "cd6daa221ca375cf5fbf050a62cd6c1f": "J^\\alpha", "cd6dbc82bc61ee7481d4a09da05f18a3": " -i \\hbar \\frac{d}{d\\phi} ", "cd6dcdba33dcbc7467ee81ab4cacff7e": "H(\\rho)", "cd6df5147bc4ddf505e9367275498dd0": "u:X \\rightarrow I", "cd6dfe99e160c9c08e56c5f27789429b": "\\mathbf{X^TWX}", "cd6e036da338fdf83247b3bdd7fe2f20": "\\frac{d(k \\cdot f(x))}{dx} = k \\cdot \\frac{d(f(x))}{dx}.", "cd6e3a60d352e364fc24447ae452e8eb": "\\scriptstyle\\iota x", "cd6e46d7ee4073af385d38648b5ec2ac": "C_V = 3NC_V", "cd6eb1c6861077322dcd2114073810c6": "C(n) = \\langle \\sin(x(n))\\sin(x(0)) \\rangle ", "cd6ecd1a58ecf8e2062a0b6ff57b720f": " U_\\theta | 0 \\rangle = e^{i \\theta} | 0 \\rangle \\quad U_\\theta | 1 \\rangle = | 1 \\rangle. ", "cd6eefa5694628e4302619e0161d50d4": "\\mbox{eGFR} = \\mbox{166}\\ \\times \\ \\mbox{(SCr/0.7)}^{-0.329} \\ \\times \\ \\mbox{0.993}^{Age} \\ ", "cd6f162e2dac3ee832ac6b93d7bb4cf1": "\\sqrt{\\frac{1-x^2}{1+x^2}}", "cd6f3930af30b7d7a4ad06a71ba743f2": "\\sqrt[\\infty]{2}_s = 2^{1/2} = \\sqrt{2}", "cd6fdc7ee496372b7f983c1c1ddfb729": "f_\\mathrm{n} = \\frac{1}{2\\pi} \\sqrt{\\frac{k}{m}} \\,", "cd70546c8e41c0aaa3050a15035322ed": "INDEXB_i=1-\\frac{\\text{Securities in currency i (regardless of the nationality of the issuer)}}{\\text{Securities issued by country i}}", "cd7060d71ebb17b55047771001977325": "f(\\underline{m}) = 0", "cd70c78cead1086e90dd4bff7b8ce8e2": "\n\\mu _{1,\\dots,N}(\\mathbf{x})\\ \\stackrel{\\mathrm{def}}{=}\\ \\mu _{r_{1},\\dots,r_{N}}(\\mathbf{x})\\ \\stackrel{\\mathrm{def}}{=}\\ E\\left[\n\\prod\\limits_{j=1}^{N}x_j^{r_{j}}\\right]\n", "cd70ee1246a1ba2a999c3db72940d71b": "\\bold{A}:\\bold{B} = \\sum_i\\sum_j A_{ij}\\overline{B_{ij}} = \\mathrm{tr}(\\mathbf{A}^* \\mathbf{B}) = \\mathrm{tr}(\\mathbf{A} \\mathbf{B}^*).", "cd70fda901592768825213408921dd4e": "q''=h\\left (T-T_\\infty\\right ),", "cd7109c0844d30402c3bb00d849422f5": "\\begin{bmatrix}\n 1 & 2 & 3 & 4 & 5 \\\\\n 2 & 4 & 1 & 5 & 3 \\\\\n 3 & 5 & 4 & 2 & 1 \\\\\n 4 & 1 & 5 & 3 & 2 \\\\\n 5 & 3 & 2 & 1 & 4\n\\end{bmatrix}.", "cd715fa7b732812e345e5155165ab858": "f(\\mu)=0 \\, ", "cd7169d2bf2c71b419b5e119751a958b": "f_i \\, f_j", "cd7182a7467d7835123b70d64d22469f": "\\textstyle \\frac{n-1}{n}(x_n - \\bar x_{n-1})(y_n - \\bar y_{n-1})", "cd718f3cbb8ec9c58d6cf337ae95712d": "\\textstyle E: y^2 = x^3 + 1", "cd71a8297559e138dbfc834e53e8e908": "Z_0 = \\sqrt{\\frac{L_{\\frac{1}{2}}}{C_{\\frac{1}{2}}}}", "cd71b71dec584cbf086e0fd944accfec": "e(p,t)", "cd71d032c3b064e7adb3f77509e43262": "\\frac{64,800\\ \\mathrm{N}}{(165\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=40.05", "cd71e31780da897c949e11f662784684": "\\frac{g(E)}{g_{av}} = \\Pi_{2}\\prod\\frac{(p-1)}{(p-2)}\\,,\\quad\\quad\\quad\\quad\\quad\\quad(1)", "cd723b1a5cb790cc958d8d43aadd0690": "x_0 \\,", "cd7263059b3a097b533e5ae01c8afeb8": "y^2+h(x)y=f(x)", "cd72c2b590f0ad73f42b208f8a079674": "\nc \\,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \\,F_3(a_1,a_2,b_1,b_2+1,c; x,y) + a_2 y \\,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~.\n", "cd72dc09ef318e1569393190c8c1dffa": "\\frac{ \\partial }{\\partial a_i }f(\\mathbf{a}) =\n\\lim_{h \\rightarrow 0}{\nf(a_1, \\dots , a_{i-1}, a_i+h, a_{i+1}, \\dots ,a_n) -\nf(a_1, \\dots, a_i, \\dots ,a_n) \\over h }\n", "cd730c2e7633c91c4bff5910d01e9e24": "f(x)=\\sum_{n=0}^{\\infty} c_{n}\\frac{x^{n}}{n!}", "cd7365e95bb7dafc00f6785d9b1280f3": "V_0 = d[P]/dt = k_2 [ES]", "cd7368ed344853f96208011f04ece3e1": "\\{w_i\\}_{i \\in \\mathcal{I}}", "cd736d71ce2eff1c9db11b4c4ecfe84d": "\\Phi^{1-b}", "cd7381c865281e956bdc68e6f654de82": " (1 2 3 4),\\; (1 2 4 3),\\; (1 3 2 4),\\; (1 3 4 2),\\; (1 4 2 3),\\; (1 4 3 2)", "cd73b2336bbdcc1ea2738363b3d5e711": "\\sum_{n=2}^\\infty (-1)^n n^m \\left[\\zeta(n)-1\\right] =\n-1\\, +\\, \\frac {1-2^{m+1}}{m+1} B_{m+1} \n\\,- \\sum_{k=1}^m (-1)^k k!\\; S(m+1,k+1) \\zeta(k+1)", "cd73d8768d62de42fe8be3eeaa59b671": "\\omega_0 = 2\\pi f_0 = \\sqrt{\\frac{Ne^2}{\\epsilon_0 m}}", "cd744b05c0ece360aeb1f17504c41167": "i+k", "cd7484f1264ac80031813289d78f09e7": "\\begin{align}\n(J_3^{(j)})_{a'a} &= a\\delta_{a'a},\\\\\n(J_1^{(j)} \\pm iJ_2^{(j)})_{a'a} &= \\sqrt{(j \\mp a)(j \\pm a + 1)}\\delta_{a',a \\pm 1}.\n\\end{align}", "cd75030ac8b183aeb587a086b39b0734": "\\hat\n{\\textbf{q}}_i, \\hat {w}_i; i = 0...n", "cd7504f882c1d797635152efafbdbbe4": "3 \\cdot x \\equiv 2 \\ \\ (\\operatorname{mod}\\ 7),", "cd75186792585e02ea7b6314e17d5ec2": "\\alpha\\prime > \\alpha", "cd7523711349147c5ce55920fd89f180": "q=M_1/M_2", "cd753de7d6f58ded82720cc424d52381": " {} = {AX.DX \\over CY.BY}, ", "cd756a91f3a16fd634b86df7841ea928": " \\frac{\\psi '(t)}{\\alpha \\psi (t)} = \\frac{\\phi '' (x,y)}{\\phi (x,y)} ", "cd7578b108109274f07878199a6d19ef": "\\,\n\\begin{align}\n\\Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\\cdots(x+1)x\\Gamma(x)\n\\end{align}\n\\,", "cd75982f4eda2e06fcdef27fa444f2d1": "p_n, \\Phi_n,", "cd75ac45208c20a7c4980210ac7e88f9": "\n\\begin{array}{lcl}\n\\neg (A \\and B) & = & \\neg A \\or \\neg B \\\\\n\\neg (A \\or B) & = & \\neg A \\and \\neg B \\\\\n\\neg (\\neg A) & = & A \\\\\n\\neg (A \\Rightarrow B) & = & A \\and \\neg B \\\\\nA \\Rightarrow B & = & \\neg A \\or B \\\\\nA \\Leftrightarrow B & = & (A \\and B) \\or (\\neg A \\and \\neg B) \\\\\n\\neg (A \\Leftrightarrow B) & = & (A \\and \\neg B) \\or (\\neg A \\and B)\n\\end{array}\n", "cd75b4d365a0d10b73d36d9c523507cb": "S_0 + h\\nu \\to S_1 \\to T_1 \\to S_0 + h\\nu^\\prime\\ ", "cd75f1f9208ccba8ad0f9cbaf926cbc0": "G_0 = \\frac { \\beta } {\n ( \\beta +1) \\left( 1 + \\frac{R_f}{R_1} \\right ) +(r_{ \\pi 2} +R_C ) \\left[ \\frac {1} {R_1} + \\frac {1} {R_2} \\left( 1 + \\frac {R_f} {R_1} \\right ) \\right] \n} ", "cd767e3e5f5745a673cfb44794f073c6": "\\Delta x' = \\Delta x/\\gamma. \\,\\!", "cd76ad72de90e0b60ddd1751d07419eb": " h_k:(a-r,a+r)\\to \\R; \\qquad h_k(x) = (x-a)\\sum_{j=0}^\\infty c_{k+1+j}(x-a)^j ", "cd77520a78e94030203e0ea1860be584": "\\frac{L_{n + 1}}{L_n}", "cd778bc634a3541993f1596328c4e21f": "\\Omega_*^G(X)", "cd77efc8ffc5f5f949a6f000ccbc63fe": "\\ \\kappa_t(\\mathcal B)", "cd784d2192d98404544a4fb0d7fe0973": " J_i n_i \\mathrm{d} A = \\mathrm{d} I \\,\\!", "cd78a18627819bd0a89805435ca5e129": "Y_{5}^{2}(\\theta,\\varphi)={1\\over 8}\\sqrt{1155\\over 2\\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(3\\cos^{3}\\theta-\\cos\\theta)", "cd78f36ca159633a53bce7d59d60d36e": "_c^d\\text{P}", "cd78f5f15170ff7b7a9e5cdba81f3e45": "D\\phi(X_0,X_1,\\dots,X_k)=\\mathrm{d}\\phi(h(X_0),h(X_1),\\dots,h(X_k))", "cd7959e7a79487bb853129ad70962d23": " \\phi = \\tfrac32\\, \\eta, \\,", "cd79ae9b3d9d9b6e92619593f504a922": "Q=0", "cd79ba0a30571f2c7d88101fa1bf9ab6": "\\begin{array}{ll} \n p_x(x,y) = \\mu p_0 \\left( \\sqrt{1-r^2/a^2} - \\frac{c}{a}\\sqrt{1-r^2/c^2} \\right) &\n 0\\le r\\le c \\\\\n p_x(x,y) = \\mu p_n(x,y) & \n c \\le r \\le a \\\\\n p_y(x,y) = 0 &\n a \\le r\n \\end{array}\n", "cd7a00a37cbc4ba1dca1c9886a3d75a5": "\\tfrac 1 6 \\,", "cd7a16e8f3f5ca58813fbd4e8c34f1f0": "\\lambda^* (A) = \\lambda^* (A \\cap E) + \\lambda^* (A \\cap E^c)", "cd7a2cad9d2a3a8ccab369eec8c815fd": " \\mu_0 = I_0 ", "cd7a36929f1e8f6121b98e9e79e4166b": "u_{\\star}=\\sqrt{\\frac{\\tau}{\\rho}}", "cd7a9c7b91a0003cd0b1b1379eaecbcb": "\\frac{dT(s)}{ds}=-t", "cd7af577a35d1a622b98b900659bc9a9": "Y_{lm}(\\theta,\\varphi)", "cd7af7d9301149b74ef1cb25ad3c086e": "\\cos(c) = \\cos(a) \\cos(b) + \\sin(a) \\sin(b) \\cos(C) \\, ", "cd7b5ec622299327eb87289365f85fa7": " {\\mbox{d} T \\over \\mbox{d} r} = \\left(1 - {1 \\over \\gamma} \\right) {T \\over P } { \\mbox{d} P \\over \\mbox{d} r},", "cd7b8c48813da9b56ca35f1e3e2a031d": "\\gamma \\sim H_n - \\ln \\left( n \\right) - \\frac{1}{{2n}} + \\frac{1}{{12n^2 }} - \\frac{1}{{120n^4 }} + ...", "cd7b965c4702545aa749e5e41eed3896": "a = 2r_1, c = r_1, d = r_2\\,\\!", "cd7ba2f0b268300ddc3eac772c21fc1e": "\n\\Bigg[\\frac{\\lambda}{\\mu}\\Bigg]_2 = \\Bigg[\\frac{\\mu}{\\lambda}\\Bigg]_2, \\;\\;\\;\\;\n\\Bigg[\\frac{i}{\\lambda}\\Bigg]_2 =(-1)^\\frac{b}{2}, \\;\\; \\text{ and }\\;\\;\n\\Bigg[\\frac{1+i}{\\lambda}\\Bigg]_2 =\\Bigg(\\frac{2}{a+b}\\Bigg),\n", "cd7c017a3453cb178d40b580f10dc29b": "p = (E_{\\mathbf{p}}/c, \\mathbf{p})", "cd7c2e0d9267f2a6980183c82b2e62da": " \\boldsymbol{\\mathsf{U}} \\cdot \\boldsymbol{\\mathsf{U}} = c^2 \\,\\!", "cd7c2e66b6864bca81a873162740bdce": " z^{10}", "cd7c31108718ef8d95f9a80eec35a8b4": "\\angle DAB = 180^\\circ - \\angle DCB.", "cd7cd25a0d95fb56646246b49fe94b19": "f_n(X_1,\\ldots,X_N)", "cd7d59f0613121750861aeb1912c6e6e": "(\\xi^1,\\xi^2,\\xi^3)", "cd7d683ff84f8c91f34bf756300b3155": "\\Omega \\times \\mathbb{R}_+", "cd7d6b17ffaffae9de0294148d36f230": "\\sum_{n=1}^\\infty {a_n \\over n^s},", "cd7e0fcb5722b467eb07f1b5b7b2f56d": "\\{\\phi^i(x),\\phi^j(y)\\}=\\{\\chi^i(x),\\chi^j(y)\\}=[\\phi^i(x),\\chi^j(y)]=0", "cd7ec9467846ff483b1aa3c10671420d": "z = r e^{i \\theta}\\,", "cd7f4c7f38fce6768b6c904a4c69e848": "a_{8}*b_{6} ", "cd7f4e0e2fa9ff17ad283a9d114dc17d": "\\operatorname{or} = \\lambda p.\\lambda q.p\\ p\\ q", "cd7f832a6422ec2342ab95c7d5aba95a": "x_2(t) \\,", "cd7fd5dc5e5e772db66e849b7a7e3f66": "p(A)=\\det(A I_n - A) = \\det(A - A) = 0.", "cd7ff7d49daaaca8408ffd516ab1021b": "\\vec{\\tau}_\\mathrm{L}", "cd800d0081b5effc11d4fe13b710a68c": "v_{\\text{out}} = -R I_{\\text{D}}.\\,", "cd8010c72c27c834c2aab7d6acaeb4d7": "f \\star g = f*g", "cd807b24865807a63eabf5138261d59e": "\\mbox{ch} \\colon K_0(X) \\to A(X, {\\Bbb Q}),", "cd80b2256deedee49fa8cd61baee96a1": "-\\tfrac {11}{15} \\pi^2 - \\ln^2(-\\phi) \\,", "cd81169fcb5f8865cda34bc006f00071": "\\displaystyle \\frac{\\pi}{a} \\operatorname{sech} \\left( \\frac{\\pi^2}{ a} \\xi \\right)", "cd81426fc097ddb751fa9d87b6831da5": "\nC_{3}(\\kappa)=\\frac {\\kappa} {4\\pi\\sinh \\kappa}=\\frac {\\kappa} {2\\pi(e^{\\kappa}-e^{-\\kappa})}. \\,\n", "cd81531715291ab086f85b939675a7b8": "\\Omega=E(R^2)", "cd818ac4796c5af3db6f22275acb2d77": "\\mu(A) = \\lim \\limits_{h \\rightarrow 0^+} \\frac{\\| I + hA \\| - 1}{h}", "cd81db513417e3575023e87204487074": "P = P_1 + P_2", "cd82591bf513158aadd472f6a5d762c8": "x^{\\prime}\\in\\mathbb{Z}_{2}", "cd829a61d616a69b9d62e6ed9f746a2b": "\\hat{H} (x) = 0", "cd82c52e8b49c5dfbb58c52017455d70": " \\frac{d}{dx}\\coth x = 1 - \\coth^2 x = -\\operatorname{csch}^2 x = -1/\\sinh^2 x \\,", "cd82d9fe79ff8f856b8f610c73ca1e72": "- x", "cd82dfff343544995239b97d25905613": "\\cdots \\to X \\xrightarrow{u} Y \\xrightarrow{v} Z \\xrightarrow{w} X[1] \\xrightarrow{u[1]} \\cdots,\\ ", "cd83025d45a01bf6fd4146332e5f5bbb": "Vf", "cd834ab2bff4c58b283d7cf18b99af3f": "x_2 = K,\\; y_2 = 0", "cd83590986830064edddad821112f5f9": "\\left\\|\\mathbf{a}\\right\\|=\\sqrt{\\mathbf{a}\\cdot\\mathbf{a}}.", "cd837087eef0f6a0962860baf228a19c": "x \\frac{d^2 y}{d x^2} + 2 \\frac{d y}{d x} + \\lambda^2 x y = 0.\\,\\!", "cd8483049aff2a9726e88d9cc9d315c3": "P(s)=\\frac{M_a}{s(r-s)}=\\frac{M_a}{r} \\times \\frac{(-r)}{s(s-r)}.", "cd8486b36d411d1c4abdf84cda488250": "\\ln(1) = 0 \\,", "cd848781656bea6415bdabdc67f1465c": "\\operatorname{pq}(u)=\\frac{\\operatorname{pr}(u)}{\\operatorname{qr}(u)}", "cd849441dd345d5871aa8b8451878fe4": "M_i = \\prod_{j=1}^{i} m_j, m_j > 1, M_0 = 1 ", "cd849ce55926478961d8a692c6e311af": "\n\\operatorname{Li}_s(z) + (-1)^s \\,\\operatorname{Li}_s(1/z) = {(2\\pi i)^s \\over \\Gamma(s)} ~\\zeta \\!\\left(1 \\!-\\! s, ~\\frac{1}{2} - {\\ln(-1/z) \\over {2\\pi i}} \\right) .\n", "cd84fadcbda0cc75e5c56c4130aa4a0d": "\\left ( \\frac{12345}{331}\\right )", "cd8504fa9dc9f1ff16dd276639fa4b6e": "\\ e_1 = (1,0)", "cd8539a55aa03bc68b49025f6b45c79f": "f(i) = \\frac{i^2 + 2}{i^2 + 1} = \\frac{-1 + 2}{-1 + 1} = \\frac{1}{0}", "cd858072b07eab51fe59a43a098d9f19": "\\mathbf{g} = \\sum_i \\mathbf{g}_i = G \\sum_i \\frac{m_i}{\\left | \\mathbf{r}_i - \\mathbf{r} \\right |^2}\\mathbf{\\hat{r}}_i \\,\\!", "cd858cc9d16211fa53bf5a3fb2d11521": "m_{pq}=\\int_{-1}^1\\int_{-1}^1 x^py^qf(x,y)dxdy", "cd85a39959ec3b1c7dd0cd2cb15f464a": " =", "cd85f9b4b38c379e319c55d2644bb983": "\\theta = \\arctan(m)\\!", "cd862e000fa9cbd5ea05e3b60b6237d1": " m = \\max_i polydeg (H_i(1/z)) \\,", "cd866567bc2998f2e39f8838f737892b": "\\lambda(H_\\gamma)\\geq 0", "cd872054ff5f2a22d84022f70383caec": " C_p - C_v \\ = \\ -T \\frac{{\\left( {\\frac{\\part V}{\\part T}} \\right)_P^2 }} {\\left(\\frac{\\part V}{\\part P}\\right)_T} \\ = \\ -T \\frac{{ \\left( {\\frac{\\part P}{\\part T}} \\right) }_V^2} {\\left( \\frac{\\part P}{\\part V} \\right)_T} ", "cd872c5772f827f2334584828cd675df": "L[u] = A(u_{xx}u_{yy}-u_{xy}^2)+Bu_{xx}+2Cu_{xy}+Du_{yy}+E = 0\\,", "cd8736a8f4a721ac8b8cf90931b7e0e6": "\\neg [\\exists x.\\ [D(x) \\rightarrow \\forall y.\\ D(y)]]\\, ", "cd874e6c0e80002544a6e861a71cb933": "\n\\begin{align}\ndv &{}= \\sec^2 x\\,dx, \\\\\nv &{}= \\tan x, \\\\\nu &{}= \\sec x, \\\\\ndu &{}= \\sec x \\tan x\\,dx.\n\\end{align}\n", "cd876aa0d74e1b4136758fbf38739932": "s=\\pi r.", "cd87867f7a25bff8ae613863d78f1008": "H_{\\mathrm{sat}}", "cd87883990d48af58d575bc97765c90e": "\\tfrac{1}{10}", "cd87a4c4d36c5c75f26f7eecba3a386a": "\\psi \\circ \\phi", "cd87cb851c7044d671e24f2e39392302": "(R, \\Theta) = \\left(\\frac{\\sin \\varphi}{1 - \\cos \\varphi}, \\theta\\right) = \\left(\\cot\\frac{\\varphi}{2}, \\theta\\right),", "cd87cbac104e4f3b31f86b76379e1c62": "\\operatorname{sock}(\\omega ,\\theta) = \\left[\\frac{\\cos^2\\left(\\theta - \\theta_0\\right)}{\\alpha_0^2}+\\frac{\\sin^2\\left(\\theta - \\theta_0\\right)}{\\beta_0^2}\\right]", "cd87dff2b8078915f4c2303c009b4caf": "\\log(a \\cdot b) = \\log(a) + \\log(b).\\ ", "cd88382a98685e21ca00bc49d90e6a1e": "Dt = {TK_b\\over 3 \\pi \\eta d} = Dt", "cd8859aba5d39420b05f907ab69454ff": " \n{1 \\over 2}\n\\begin{pmatrix} \n1 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\ \n1 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 0\n\\end{pmatrix}\n\\quad\n", "cd898fa54d63d9d16fd87bca9ad8eb9c": "\\mathbf E_{(m)}=\\frac{1}{2m}(\\mathbf U^{2m}- \\mathbf I) = \\frac{1}{2m}\\left[\\mathbf{C}^{m} - \\mathbf{I}\\right]\\,\\!", "cd8aaae998c494d72f959caf9b237496": "\\phi \\to (\\chi \\to \\phi)", "cd8ab84745baf26fa8ba539af27bc7e3": "V(x_0)", "cd8addd5520e0d8137bdb5b04cd3181c": "2\\pi/5", "cd8b19c20ddbac754288bd0511efc6cc": "(\\mathbf{d}^2)t^2+(2\\mathbf{v}\\cdot\\mathbf{d})t+(\\mathbf{v}^2-r^2)=0.", "cd8b245e5705c757c5cc50043260ad77": " S(T\\rho||T\\sigma)= S(\\rho||\\sigma),", "cd8b5a2e1961ff26c9bd39ce4bb8a0c7": "P(R_n | W)", "cd8bae416c6dadcf077cea1a0a02a64f": "E = \\cup\\mathcal{F}", "cd8bf686fe14d8f6963d3642dac00d35": "reject \\leftarrow 1 \\,", "cd8c146910bf2a4dbcffb10e8f2fdfb3": "n = \\frac{35}{100} \\times 5 + \\frac{1}{2} = 2.25,", "cd8c20dc733128dc1b65e4daa4cb6578": "\\cos (2 \\theta) = \\cos^2 \\theta - \\sin^2 \\theta\\,", "cd8c5b3ff8d8b86aab13045d42093b52": "k^{a}=(1,0,0,0)", "cd8c89c0a97f4eeacf8468ef9c32d5d5": "\\tfrac dc\\ =\\ \\tfrac ba\\ =\\ \\tfrac{d\\,+\\,b}{c\\,+\\,a}.", "cd8d3cb522011f0f0060fb6dd430124b": "H_n(M;\\mathbf{Z}/2)=\\mathbf{Z}/2", "cd8d676b311d8b80ca7870807a986299": "c_t=\\frac{2 \\theta}{\\sigma^2(1-e^{-\\theta t})}", "cd8d85d1ac7dc0c2f271f6c98d12eabc": "E\\subseteq\\mathbb{R}^d", "cd8d9224aeceddea67909fbeecc0cb65": "(f \\star g)(t)\\ \\stackrel{\\mathrm{def}}{=} \\int_{-\\infty}^{\\infty} f^*(\\tau)\\ g(t+\\tau)\\,d\\tau,", "cd8e0e75f3b907757fc6c3528d0e0249": "I_{4,1} = -10 \\log{\\left( \\frac{P_4}{P_1} \\right)} \\quad \\rm{dB}", "cd8e50901d130e5c2bce4ae68af53761": " E_{2c} = \\frac{3}{2} \\times y_{2c} = \\frac{3}{2} \\times 4.27 = 6.40 \\text{ ft}", "cd8e705d5e6ea9c29fcb44c3f247ace0": "v = -K_m{v \\over {[S]}} + V_\\max", "cd8e8500537a7ae5da6ced607f560184": "\\frac{x(y'x^2 - 2xy)}{x^5} = 0", "cd8ea1d377ce6d16353b4fded5f75d04": "{\\rm F}", "cd8eb41a4134219be4ea578b707ada86": "\\displaystyle s^2=3r^2+12Rr.", "cd8eca8b011bfa1bd37384182ef20390": "A^e = A \\otimes_K A^{\\rm op}", "cd8f11a2b9c703052302cf2b90b14580": "a\\left(k,t\\right)=U^{-1}(t)a_i\\left(k\\right)U\\left( t \\right)", "cd8fc8609e76f82088621ec26718a4f8": "Q = - (1/2m) \\, \\Box R/R", "cd8fe60f419e6d90bfada6beca000b4d": "L = c \\int_P \\sqrt{-g_{\\mu\\nu} dx^\\mu dx^\\nu} ", "cd8ff24026c58cf2292fb3c819106b28": " G(a,b,2^n)=0 ", "cd903b6f8b4d9a3fb6c1763c01a72609": "Y_{10}^{10}(\\theta,\\varphi)={1\\over 1024}\\sqrt{969969\\over \\pi}\\cdot e^{10i\\varphi}\\cdot\\sin^{10}\\theta", "cd90569456d13ef3f4a5d2595552a2fe": " g \\in G_1", "cd9066ba2593e65b8bc703f18d1aeaee": "\\tan \\psi = \\frac {u' + v'} {v'} \\tan \\theta = \\left ( \\frac {u'} {v'} + 1 \\right ) \\tan \\theta \\,.", "cd9086949268863ed00395182b26adba": "\\ln \\phi = \\frac{1}\n{{RT}}\\int_0^P {\\Phi dP}", "cd90cbf6c5967184af9ef9d370e9f03b": "s = \\tfrac{1}{2}(1+i+j+k) \\qquad t = \\tfrac{1}{2}(1+i+j-k).", "cd90e820496e941be62f5a00fb5f5123": "w_s", "cd91089645dcd2aabfa345bfdc8ecd62": "(\\lambda_n^{(c)} + \\epsilon 1_n^{(c)}) \\in \\Lambda", "cd91985f6a499c936ec691a524d2547b": "(p_2(x) v')' + q_2(x) v = 0. \\,", "cd922bcb875abfb9eab6090bf29b4637": "q(\\nu , T_X, T_Y) = \\alpha _{\\nu , X, Y}(T_X, T_Y)I_{\\nu , Y}(T_Y) - I_{\\nu , X}(T_X).", "cd92475390fac5a531ada3145fcc5e82": " x+y+z =180. ", "cd924b04b36171996685a74d08ff4229": " A = \\frac{V_n}{L} ", "cd925a4178bc636a27f6de5ab936ca50": "\\log \\frac{K_X}{K_H} = \\sigma\\rho ", "cd92d64a6585626a0df7f1d0b938e746": "{{i}_{C1}}={{i}_{C2}}", "cd92d81102d7c681d481583c7568c8be": "P/H\\to M", "cd92f5edc25d036d3146e8d65453ca54": "\\sqrt{\\frac{3n}{5}} < s_n < \\sqrt{6n} \\text{ for } n \\ge 1 \\, .", "cd9340e3d45f67d791daacd86ff624c2": "\\frac{c}{n} + \\left( 1 - \\frac{1}{n^2} \\right) v.", "cd934972a154eac52d6ed7ee17684f8a": "\\Sigma \\left( M^{f} + M_{member} \\right) = \\Sigma M_{joint}", "cd934d4ff36a48e899d86cceba420658": "\n\\lim_{n\\to\\infty}\\frac{f(n)}{1/n}=0\n\\quad\\text{and}\\quad\n\\lim_{n\\to\\infty}\\frac{f(n)}{1/n^{1+\\varepsilon}}=\\infty\n", "cd9355b001bd819ac4576b108c231eec": "I = I_0e^{-\\mu x}\\,\\!", "cd936936d90d98c6d0357c1ce9df453e": "\\left(u_1, u_2, \\dots, u_N\\right)", "cd93d547ffb7b8d2ed3dbc09708bce39": "\\textstyle (nd+1)(t-1)", "cd9417af6d173d18a3bdba4a33d1dc4c": "ae^{-\\frac{(x-b)^2}{c^2}}", "cd941d55fffd1ed19b1ae45d46a173d1": " = (q^2;q^2)_\\infty\\,(-w^2q;q^2)_\\infty\\,(-q/w^2;q^2)_\\infty ", "cd94322972ae03a5aa5ade8b4d8e53fe": " X_{u}[k]= x_{u}^{*}(\\tilde{u}k) X_{u}[0]", "cd947befb04ab32127faa1fbfd443a69": "3 \\log_2 N", "cd948cad67e0ee623ad1005bbec4f67c": "\\tan \\frac{\\pi}{10} = \\tan 18^\\circ = \\frac{\\sqrt{5(5 - 2 \\sqrt 5)}}{5} ", "cd94919892e573d497e64f6ef77cf57c": "\\textrm{dim}_k", "cd94a3641bb6ba72c90dd0d8f4d2e199": "N \\times N", "cd94ba4fbc02edb15ed94f7907bd071c": "\n\\begin{align}\nI(\\theta, \\phi)\n& \\propto \\operatorname{sinc}^2\\left(\\frac{\\pi W \\sin\\theta}{\\lambda}\\right)\\operatorname{sinc}^2\\left(\\frac{ \\pi H \\sin\\phi}{\\lambda}\\right)\\\\\n& \\propto \\operatorname{sinc}^2\\left(\\frac{k W \\sin\\theta}{2}\\right)\\operatorname{sinc}^2\\left(\\frac{k H \\sin\\phi}{2}\\right)\n\\end{align}\n", "cd94d95d71199388e3933d52bca0551b": "E = i \\hbar \\alpha = \\hbar \\omega ", "cd95369b71577c839f7ec053f9f509b7": "MUR = \\left( c * 0.5 * TSP \\right) ", "cd954fac6350960be156379949445852": "\\overline{\\mathcal M}_{g, n}", "cd955056079ad2339bad5d450ed1c539": "(\\phi_{P,x})(y)", "cd956602e512636ec560624a75d8b007": "\\textit{list}(\\textit{int}) \\subseteq \\textit{list}(\\textit{float})", "cd9602d0ebecf7347f9fdf52fa3fb13c": "x^n\\equiv a\\pmod{p}", "cd965856e6687a0020ca26a8ac48a01e": " Y_t", "cd969b39f4773d7c431190ef4b4650ef": " S_{i,j}^t =0 ", "cd96b33539eeec56fce78d734c70e94c": "f(x,y) = \\begin{cases}y^3/(x^2+y^2) & \\text{if }(x,y) \\ne (0,0) \\\\ 0 & \\text{if }(x,y) = (0,0)\\end{cases}", "cd96bd391ea7cb1d0d3bbf28a512de58": " c_6 = -0.020716198, \\,\\!", "cd96d37c098c07f14f0e5b34c76e7ba6": "\\theta_n(x)=\\sqrt{\\frac{2}{\\pi}}\\,x^{n+1/2}e^{x}K_{n+ \\frac 1 2}(x)", "cd96e8d6f7a829c62a646ca690d9c703": " \nA=\\{x\\in E |\\phi(x)>a\\}\n", "cd9747b41832fac3dd085336efd90a52": "n \\neq 0", "cd97524b7d727d8cc8817257aa3ea701": "\\mathsf{NEXP} \\not \\subseteq \\mathsf{ACC}^0", "cd97721f1bd71b53722368c9c2771517": " \\varepsilon = \\hbar\\omega", "cd97972d65bbd2a658ae662390bae6a5": "\\hat{r}_0 \\leftarrow \\hat{b} - \\hat{x}_0A^T ", "cd97b41e8577f0f116856f0b934d6b05": "g_{n+1} = \\sqrt{g_n h_n}", "cd97bbd4f460d1ea0e7799f3fbe8075f": "\\dot \\partial_\\mu= \\partial_\\mu", "cd97d0d1f61daf22488e87cf768adc71": "\\{x_{n_k}\\}", "cd97f9229eb89d27309c7370a3924957": " \\log_{10}(x) = \\frac{\\ln(x)}{\\ln(10)} \\qquad \\text{ or } \\qquad \\log_{10}(x) = \\frac{\\log_2(x)}{\\log_2(10)}", "cd984f20067bf5e0e392fc18b97a2b2f": "\\ \\ t_k ", "cd99352ce2dbe717268c6027e07ed2be": "y\\ll\\rho", "cd9943b3e599a4ae9380f5977a61ff96": "v \\triangledown v' ~ = - \\triangledown p + \\nu \\triangledown ^2 v',", "cd99c2d633224aced3fe369b7f1c3602": "\\frac{1}{i\\omega} = \\frac{e^{-i\\pi/2}}{\\omega}.\\,", "cd99ee6985df4d669e3b40131c03c53b": "\\text{1 MET}\\ \\equiv\\ 1 \\dfrac\\text{kcal}{\\text{kg}*{h}}\\ \\equiv\\ 4.184 \\dfrac\\text{kJ}{\\text{kg}*{h}}", "cd9a184e84944f4f126d454757e2e49a": "\nV(x)=\\lambda ^2\\left(e^{-2\\left(x-x_e\\right)}-2e^{-\\left(x-x_e\\right)}\\right).\n", "cd9a32d0190e7d8e17aa11442ded2081": "(f^{*}\\theta)(X_x) = \\theta(f_{*}^{}X_x)", "cd9a55aa28f39b5443dfda15a1922e2f": "A(u,\\varphi) \\geq \\alpha \\int_\\Omega \\nabla u \\cdot \\nabla \\varphi.", "cd9a8e6997defdff289cbf4d4ef95cb9": "\\mathbf{\\hat{\\boldsymbol{\\jmath}}}", "cd9ad3d2b03781b6243a508df515c8ad": " \\sigma= \\begin{pmatrix}1 & 2 & \\ldots & n\\\\\\sigma_1 & \\sigma_2 & \\ldots & \\sigma_n\\end{pmatrix}", "cd9ae7b3188b1b12d541ff44ebea0b0a": "g^{\\mu \\nu}", "cd9b391e10f9185c9af2e94416336249": "\\Phi(\\vec{r},t)=\\int_{4\\pi}L(\\vec{r},\\hat{s},t)d\\Omega (\\frac{W}{m^2})", "cd9b548e68bc55c6e34065992a924fa1": "\\mu \\in \\mathcal{P}_p(X)", "cd9b7e5e5256721b7eb4f040914847bc": "\\theta _{c}=\\arcsin \\left( \\frac{1.00}{1.50} \\right)=41.8{}^\\circ ", "cd9bbcceeff67f6e83949cec584f388b": "\\bold U(x_0,y_0) = \\frac{1}{j\\lambda z}\\int\\!\\int \\bold U(x_1,y_1) \\frac{\\exp(jkz[1+(\\frac{x_0-x_1}{z})^2+(\\frac{y_0-y_1}{z})^2]^{1/2})}{1+(\\frac{x_0-x_1}{z})^2+(\\frac{y_0-y_1}{z})^2}dx_1 dy_1", "cd9bec6683ae9e5368eefab7af77bb21": "\\gamma_2", "cd9c0cea20afc343018bcfd001b8f75e": "x_{jt}", "cd9c31df4b3e21cf0ec9434d80bcb974": "\\lim_{x \\to 0} \\frac{1-\\cos x}{x^2} = \\frac{1}{2}", "cd9c8f1d422cd3290a8777be6920a270": "\\underline{\\psi \\rightarrow \\chi}\\,\\!", "cd9cf038596754e92f10337a3da842f6": "P_k=\\sum_{i=1}^{k}(\\sum_{j=1}^{k}m_{i,j})^2-n", "cd9d75d72832bb5f1de76cd10682b59f": "nk \\leq (n-1)k,", "cd9da546402d2c37095fa406c93c334a": "p \\ll \\Lambda", "cd9da5a3d9ecd90bcc5c27faf5e65e41": "H = \\hat a^\\dagger \\hat a + \\frac 12", "cd9dbb23dcffa69a8ce96ec4d0fd0884": "\\frac{\\partial}{\\partial \\varphi} = -y \\frac{\\partial}{\\partial x} + x \\frac{\\partial}{\\partial y} .", "cd9dbbfddd5ce61986127b7b5f3cedd0": "b=\\frac{1}{(\\delta+\\frac{1}{ar})}", "cd9e3255dae8b8beac7098435a6bb1d2": "T\\cdot S", "cd9e66895a2de1b2128c91205616ed38": "\\frac{\\Gamma_p(\\alpha+n/2)}{(2\\pi/\\beta)^\\frac{np}{2} \\Gamma_p(\\alpha)} |\\boldsymbol\\Omega|^{-\\frac{n}{2}} |\\boldsymbol\\Sigma|^{-\\frac{p}{2}}", "cd9e6c214c9228c574026ec645cd4ab5": "\\Delta x = w/m", "cd9e7608c35511b02fdf07f04de80337": "HP^2", "cd9f25d66fc6524bb9e0409c30ec4970": " x_1^{l_1}\\cdots x_k^{l_k} ", "cd9f45f59320173cce3f0223bdc4a536": "\\tfrac{1}{3}=0.\\dot{3}", "cd9f58f968dcbf072ee67417da5a160a": " m = \\int_V \\left( 2T_{00} - T g_{00} \\right) K dV", "cd9f66f90c946ce9e28c826962f3a2ae": "\\mu_i=\\frac{B}{\\mu_{0}*H}", "cd9fa68c65938b50f347c8d33873ea05": "S^1(n, k) = S(n, k)", "cd9fad21b779b28c0da99f282277a607": "n_{tot}", "cd9fd84feade32330e892399d86b4e7f": "h \\circ (g_1, \\ldots, g_m) \\stackrel{\\mathrm{def}}{=} f(x_1,\\ldots,x_k) = h(g_1(x_1,\\ldots,x_k),\\ldots,g_m(x_1,\\ldots,x_k))\\,", "cd9ff0d1f17a0e6cf6251a2bbd002a27": "d_{n,k} = [t^n]d(t)\\left(t h(t)\\right)^k", "cda042d4fe076a4c571b37641301ea74": "\\ \\Delta^s(\\phi_{1,1,1}) = \\phi_{1,2,1} - \\phi_{1,1,1}", "cda076d466c5b83523487e837445ddfa": "|zsp|=\\frac b a \\cdot|zsx|=\\frac b a \\cdot|zcy|=\\frac b a\\cdot\\frac{a^2 M}2 = \\frac {a b M}{2}, ", "cda0a935c10e554641845f34e8cb23e9": "\\mathcal{L}={1\\over 2}g_{ab}(\\Sigma) \\partial^\\mu \\Sigma^{a} \\partial_\\mu \\Sigma^{b} - V(\\Sigma)", "cda0dd7772878f7c3aefcb0ed4069d88": "\\langle -,-\\rangle", "cda0e90103b6f895df6d138db015f570": "\n R = \\cfrac{(2^{m-1}+2) L - N + H}{(2^{m-1} - 1) L + 2 N + F} ~.\n ", "cda101c5a4e4d7f0f5d6084d14619f3a": "P/\\sqrt{32}", "cda1124c33277558df59d80601c46598": "\\langle \\mathbf{a}_0,\\mathbf{y}_0 \\rangle, \\langle\\mathbf{a}_1,\\mathbf{y}_1 \\rangle,\\langle\\mathbf{a}_2,\\mathbf{y}_2\\rangle,...,\\langle\\mathbf{a}_{n-1},\\mathbf{y}_{n-1}\\rangle", "cda131d35b4f1525cf7b36beccadac2a": "PQR", "cda17d396a52600ab6d6cbdec9da46df": "L\\forall.", "cda19efdd4cfdb90917e7b423a418288": "\n \\begin{align}\n \\lim_{n\\to\\infty}\\mathbf{E}\\left[ L_{\\hat{X}^n_{DUDE}}\\left( X^n,Z^n \\right) \\right]= \n \\lim_{n\\to\\infty}\\min_{\\hat{X}^n\\in\\mathcal{D}_n}\\mathbf{E} \\left[L_{\\hat{X}^n}\\left( X^n,Z^n\n \\right)\\right]\\,,\n \\end{align}\n ", "cda201f2162be864f9a5d153945835ee": "(x^2)^{-\\Delta}", "cda250a49f54cca4d871f58dcb07f202": " u(\\mathbf{x}) ", "cda27f4acb9c8b7e2cb1653dfce553b6": "\ne^{[\\alpha}_M e^{\\beta]}_N \\delta^M_{[I} \\delta^K_{J]} C_{\\beta K}^{\\;\\;\\; N} = 0 .\n", "cda300935c8bf78e699d84b5ee72b5ec": "\\mathbf{\\Pi}^0_{\\lambda(i)}", "cda303b2c38f161a685c5d174460ddb4": "g(\\vec{x})", "cda30ca794b99ab8a3142f1aabfab9e6": "1\\le j\\le \\lfloor k/2\\rfloor", "cda348ac283e2ec3a0cf7a351a1f272a": "\\propto \\exp(-z/H)", "cda365e1e11e725608f5827c8a7fc160": "D_{k} = L_{k} - PHM_{k}\\;", "cda3c615b606fe214480a54dc97bf5cd": "\\mathbf{F} = q(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}) = q(\\mathbf{E} - \\mathbf{B} \\times \\mathbf{v})", "cda448e73450be4ac15b7ec7d4853e03": "\\partial_\\mu J^\\mu=0", "cda4535ab0bedcc37c5104f35aec6b17": " N", "cda45595fa6c985903e1315ec78c5af5": "\\alpha (x, v) = \\sup \\big\\{ | \\mathrm{D} f(x) v | \\big| f : B \\to \\Delta \\mbox{ is holomorphic} \\big\\}.", "cda4593566d93f4bd4ff0b7aadc000c2": "f'(t)=(f_1'(t),f_2'(t),\\ldots,f_n'(t))", "cda461c8ae91cca4e0b96916f138f3cc": " \n=\\left(\\frac{29}{37}\\right) \n=\\left(\\frac{37}{29}\\right) \n=\\left(\\frac{8}{29}\\right) \n=\\left(\\frac{2}{29}\\right)^3 \n=-1.\n", "cda463d4d4d0fc06cb2f59d7b2a7032b": "\\mathfrak{k}=\\mathfrak{so}_n(\\mathbb{R})", "cda4e7bb74217eb4c806e035e22759e4": "Z_{\\pi}", "cda522d4353b166cc2dee84673307b4e": "R1", "cda5539bc1fd0f1c4250dde78077d2bc": "F_n(T^{1/n}) = e^{2\\pi i/n} T^{1/n}.", "cda555088b554c56f7b72174d612f0e9": " \\sqrt{ \\sqrt[3]{28} - \\sqrt[3]{27}} = \\tfrac13\\left(\\sqrt[3]{98} - \\sqrt[3]{28} -1\\right), ", "cda56c99246c7a0370b19414e2bbb146": "\\operatorname{E}\\bigl[(X)_r\\bigr] = \\frac{(K)!}{(K-r)!} \\frac{n!}{(n-r)!} \\frac{(N-r)!}{N!} .", "cda57513da29510254fff4726831b1a8": "\\left|\\frac{A_0}{A_x}\\right|=e^{\\alpha x}", "cda5ee265801b110fe1ede838425de01": "\\scriptstyle \\log_e P_{mmHg} = ", "cda6211aaaf3e46313f06d3788fd2a4d": "E_{i,f}", "cda64b001f43bff1d4f1e172e1fa3ace": "k+ \\sum_{i=1}^n x_i,\\ \\frac {\\theta} {n \\theta + 1}\\!", "cda6635e9176d0280bbeccb6dbf43ba6": "\\mathbf{w}, ||\\mathbf{w}||=1", "cda6cac7ad5dd14e610b32c87421e62d": "\\mu_0,\\, \\tau_0\\!", "cda7347575b6796b667b8f05d85d0b5a": "r_1 0", "cdc476c4946e7f0d2356500348bd71bd": " \\mu_0 = 4 \\pi \\cdot 10^{-7}\\,\\mathrm{N/A^2}\\!", "cdc4ae5d7ecd076ae34763788947b7dc": "|\\alpha\\rangle = |\\psi(t_0)\\rangle", "cdc4b17bbc7f69a5e112706f4c53247f": "~x_1 \\leftrightarrow x_2 \\leftrightarrow x_3 \\leftrightarrow ... \\leftrightarrow x_n", "cdc541556473f1fb62d8a5119ddb521a": "p_1, p_2, ..., p_N", "cdc5aea6b1104423d6bf3039a9ecf383": " \\arcsin z = z + \\left( \\frac {1} {2} \\right) \\frac {z^3} {3} \n+ \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {z^5} {5} + \\left( \\frac{1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6 } \\right) \\frac{z^7} {7} + \\cdots\\ \n= \\sum_{n=0}^\\infty \\frac {\\binom{2n} n z^{2n+1}} {4^n (2n+1)}; \\qquad | z | \\le 1 ", "cdc5d70a79f073d49088d5f1ecc443e9": "\\begin{pmatrix}\\forall x_1 \\exists y_1\\\\ \\forall x_2 \\exists y_2\\end{pmatrix} [(V(x_1) \\wedge T(x_2)) \\rightarrow (R(x_1,y_1) \\wedge R(x_2,y_2) \\wedge H(y_1, y_2) \\wedge H(y_2, y_1))]", "cdc5e6a4dfdcd88be27507fd816d3da2": "\n(3.1)\\quad\nE(\\mathcal{A}f)(W)=E h(W) - Eh(Y),\n", "cdc5f397f2080630fa12f6d29fa3163f": "Q_{ij} = -(\\lVert s_j - \\mathbf{t} - \\mathbf{A} m_i \\rVert^2 - \\alpha) = -\\frac{\\partial \\operatorname{cost}}{\\partial \\mu_{ij}}", "cdc6b17f49b3f5816a472cf1945d03a4": "Q_i(w)", "cdc6d58a91c4b99859757fe8a0efcda8": " B_{x,i} = \\# \\left\\lbrace c \\in C \\mid d(c,x) = i \\right\\rbrace . ", "cdc6db78f1c69599057fbd79bf113b00": " \\frac{X}{\\tau^2 \\nu} \\sim \\mbox{inv-}\\chi^2(\\nu) \\, ", "cdc7208438687403e2d8e526e50d038f": "q = \\frac{d\\phi}{d\\theta}", "cdc7a6da3bfbadb8ed418d6b3ca28181": "S_n(s) = - \\frac{1}{4} \\left ( \\left ( \\sqrt {1 -s} +i\\sqrt {s} \\right )^{2n}-1 \\right )^{2}\\left ( \\sqrt {1 -s} -i\\sqrt {s} \\right )^{2n}.", "cdc7b641dbc271be190b4a86f102f3ca": "2\\ell+1 ", "cdc7c013abb208ac2a61eaf8224bc6cd": "\\textstyle{1+2+\\cdots+(n-1) =\\binom{n}{2}}.", "cdc7eae8aed8d57c9ee71f04acc32214": "\n\\int \\exp\\left( - \\frac 1 2 A_{ij} x^i x^j \\right) d^2x\n=\n\\sqrt{\\frac{(2\\pi)^2}{\\det A}}\n", "cdc80c1fbcdd6ef467b9cbf5fc8e2762": "\\phi^4", "cdc8ab5ca0e424bbcacdb3608bcc4247": "\\Psi_0:= C_{abcd} l^a m^b l^c m^d\\,,\\quad \\Psi_1:= C_{abcd} l^a n^b l^c m^d\\,,\\quad \\Psi_2:= C_{abcd} l^a m^b\\bar{m}^c n^d\\,,\\quad \\Psi_3:= C_{abcd} l^a n^b\\bar{m}^c n^d\\,,\\quad \\Psi_4:= C_{abcd} n^a \\bar{m}^b n^c \\bar{m}^d\\,.", "cdc8bcbe41e5fb3172fcf55a4ffac77f": "p_{j|i} = \\frac{\\exp(-\\lVert\\mathbf{x}_i, \\mathbf{x}_j\\rVert^2 / 2\\sigma_i^2)}{\\sum_{k \\neq i} \\exp(-\\lVert\\mathbf{x}_i, \\mathbf{x}_k\\rVert^2 / 2\\sigma_i^2)},", "cdc8bd9ab46bcbab3a6cb9e64262910d": "\\lambda 1", "cdc98e542629c97b3ec47e405f8ea3f7": "\\kappa^2 = 4 \\pi \\lambda_B n", "cdc9d7c4aa7a1346ecff1dc35706653b": "\\prod _x \\csc x \\sin (x+1) = C \\sin x \\,", "cdca3f66dbb700b968293e2d5bd35b14": " {N}(B)", "cdca4905901757b9d1c3e495273a5e52": " \\textbf{r} = r \\cos(\\theta) \\hat{x} + r \\sin(\\theta) \\hat{y}. ", "cdca7c0849cfd7f08b14a13f8bf449dd": "\\exp(\\eta'T(x))h(x)\\rightarrow 0", "cdcab301a15934c9fc189fe9d79153b3": "Fr=\\frac{U_0^2}{gh},\\,\\,\\ We=\\frac{\\rho u_0^2 h}{\\sigma}", "cdcac10a5df03af53afe19a110cb609b": "v_{th}", "cdcb363dba3a916eeb3b85918f9b173c": "P = - \\frac{dE}{dt} = - \\left( \\frac{d}{dt} \\right) M c^2 = -c^2 \\frac{dM}{dt} \\;", "cdcb717547405509fde40d07ba86f8fd": "\\mu(\\cup_{n=1}^\\infty T^{-n}E) = 1", "cdcb73b3fe18e3f781ace2a09dd28d74": " W(t_\\text{max}) = \\sup_{0 \\leq s \\leq 1}W(s) ", "cdcc039b7ba6abb7e4846bd984c1e5c1": "\\mu(T,P) = g(T,P)=g^\\mathrm{u}(T,p^u)+RT\\ln {\\frac{f_i}{p^u}}", "cdcc45663b04939fe1a76c34c48c2509": "= (E_1+E_2)^2-\\|\\textbf{p}_1 + \\textbf{p}_2\\|^2 \\,", "cdcc51aa94491147ac03a35d991c08e3": " \\theta(L)= 1 + \\sum_{i=1}^q \\theta_i L^i .\\,", "cdcc8b56cd8724994c558b0555c4b8bf": "A_icriterion)", "cde8bcb3200f86038f12b164419eae4a": "D_t", "cde8d114b33687c3fa3ce727d5d67ab9": "K_{min}", "cde8f3bdfa29a8372b7a0c43adff8fb3": "Au = f", "cde8f4796c0781b3157492dd263730e8": "\\left[J_\\pm, V_\\mp\\right] = \\sqrt{2} V_0 ", "cde8fcd971dac0ffbd877fac9b016437": "\n \\sum_{i=1}^{N} s_{ij} X_i \\xrightarrow{k_j} \\sum_{i=1}^{N}\\ r_{ij} X_{i}.\n", "cde9347215c9c7c082182a3f161dfee4": "\\frac{d[E_2]}{dt} = a_2[W'][E_2] - (d_2+k_2)[W'E_2]", "cde93cc6e61bf14ae40acd0add2563f4": "\\mathbf {a} \\times \\mathbf{b} = (a^2b^3 - a^3b^2) \\mathbf {e}_1 + (a^3b^1 - a^1b^3) \\mathbf {e}_2 + (a^1b^2 - a^2b^1) \\mathbf {e}_3 ,", "cde974a031fc54007219463986ef5c8f": "[i - 1; i]", "cde9924e83079ceb992a719821eeff7e": "1\\times16^3+15\\times16^2+3\\times16+10", "cde9cc46f3f612f79fc1b645210f4eb7": "\n \\begin{align}\n \\left[\\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^3 \\right.& \n \\left.+ \\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda^2 + \n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}}~\\lambda\\right]\\boldsymbol{\\mathit{1}} +\n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_1}{\\partial \\boldsymbol{A}}~\\lambda^2 + \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_2}{\\partial \\boldsymbol{A}}~\\lambda + \n \\boldsymbol{A}^T\\cdot\\frac{\\partial I_3}{\\partial \\boldsymbol{A}} \\\\\n & = \n \\left[\\lambda^3 + I_1~\\lambda^2 + I_2~\\lambda + I_3\\right]\n \\boldsymbol{\\mathit{1}} ~.\n \\end{align}\n", "cde9e35d8b2d7066d5f0174ec45aff92": "B=S^2-{s}", "cdea41f0aa8f1556e4be8c2cfe3f2831": "V = (-m) \\begin{bmatrix}\\frac {-G}{\\sqrt (x+d)^2 +y^2} \\end{bmatrix} ", "cdea86f56ce455a9a443cfb86cda4a6a": "\\displaystyle -\\sqrt{\\frac{\\pi}{a}}\\sin \\left( \\frac{\\nu^2}{4 a} - \\frac{\\pi}{4} \\right)", "cdeaa1fdc6c9983c54d3a7f9d2dea604": "\\mathbf{a} = a_i\\mathbf{e}_i \\equiv \\sum_i a_i\\mathbf{e}_i \\,.", "cdeab29c14ac1e4a7f23a482e1d33a91": " b\\sqrt{c} = 2\\sqrt{de},", "cdeaf3d3ac734be72a410f42f239d9a0": " E_i=U_i(\\lambda) ", "cdeaf8cca60761ca54eba5dac1cc0c90": " [0,2/\\pi]", "cdeb2a8f76f11bc12c2044f9236f858c": "\\lnot \\textit{fem}(e)", "cdeb6c184a89fa8999faa214d8ef08ea": "A[t]", "cdebb5bdced1a0b08fb3e6e0e4a03c04": "\\Gamma_{i}(d) = 0", "cdebd885ad7d58509c1028f5c1712cb9": "W(x_c)", "cdebde180bee58483d0cad25464ee57c": "\n\\begin{align}\nP &= P_{el} + P_{re} + P_{in} \\\\\nP &= EV + R\\dot{V} + I\\ddot{V} \n\\end{align}\n", "cdec04f466e84c9509a14892ccfd9bd7": "\\left\\lfloor\\frac{J}{4}\\right\\rfloor - 2J", "cdec2237a01667d9ae02e5bfcb304c3c": "

x)=\\frac{F_X(x+\\Delta\\;x)-F_X(x)}{(1-F_X(x))}", "cdfc81c1cdaf7ad75754aa105b21c2d3": "\\|x-y\\|_{A}^{2} ", "cdfcca6910b499f8f26b9ac6cf048165": "Q_{ab}{}^{c}", "cdfcd5f8d33746dd30603708ac617269": "H|-k\\rangle=\\frac{{\\hbar^2}{k^2}}{2m}|-k\\rangle", "cdfd38c2c1b6ef4e2ee08b7f641abe2a": "\\theta \\sim p(\\theta|\\alpha),", "cdfd71609a2cdd35047eddb2968b6b44": "P=k_{\\rm B}nT", "cdfdd271afb519746e944d82c37f6a6c": "|\\det(T)|\\, \\lambda\\,(A)", "cdfdfab1127774534a5d0178b32fa504": "u_{a1}(\\mathbf{x},z_1)=-\\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x})-k_1(z_1-u_x(\\mathbf{x})) + \\frac{\\partial u_x}{\\partial \\mathbf{x}}(f_x(\\mathbf{x})+g_x(\\mathbf{x})z_1)", "cdfe02269eba18df7c79ae9c960aa072": "\\int_0^\\infty x J_\\alpha(ux) J_\\alpha(vx) \\,dx = \\frac{1}{u} \\delta(u - v)\\!", "cdfe35c4ea1eb5aa06a627b8ac983c1c": "\\frac{4}{4k+1} = \\frac{1}{k} - \\frac{1}{k(4k+1)}", "cdfe35c5ed51aafdd619842d1c9a8635": "\\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0", "cdfe5ff9f2075fe78631baff104bad79": "X_n/Y_n \\ \\xrightarrow{d}\\ X/c,", "cdff3c21253aa313a795eb4d09d39af4": "\\mathbf{V}_i = \\mathbf{V}+W\\mathbf{r}_{i}", "cdff4efbc1723d9e13e5b2f7e759b2fb": "\nS = \\frac{2e^3}{\\pi\\hbar} \\vert V \\vert \\sum_n T_n \\ ,\n", "cdff67efcc45d928a9ae5e0cafb3a843": "S_N", "cdff8a03b913f09396ef5faf2912d6a5": "\\mathbf{R}^n \\setminus \\{0\\} \\to \\mathbf{R}^n \\setminus \\{0\\}: x \\mapsto x/\\|x\\|^2.", "cdffc18ee7a1978b754b8cbcc23fb7c3": "\n(1-r_{rb})\\frac{n_1n_0}{2}\n", "cdfff37ca14489aa98c89237cb979a0a": " x_{k} ", "ce00084adab75213d77862dde8b70d35": "\\mathrm{Pr} [M_{i,j} = M_{i,k} = 1] = \\frac{1}{d^2} ", "ce00181b809e6e2f23d3cf15bbb12f49": "\\frac{dx(t)}{dt}=a*(y(t)-x(t)), ", "ce001bd872d1108288f9a0fd23917bb8": "\n \\begin{align}\n & N_{\\alpha\\beta,\\alpha} = 0 \\\\\n & M_{\\alpha\\beta,\\beta}-Q_\\alpha = 0 \\\\\n & Q_{\\alpha,\\alpha}+q = 0 \\,.\n \\end{align}\n ", "ce00677b13d7afb9a0804aadc6ce25fb": "X=\\mathbb R", "ce007dede34a4d60a9d262258dd93bc6": "H^{2i}(V)(i) = W\\ ", "ce00a5f3bcb3662ece6005dc97829b6d": "\\scriptstyle{R_{\\alpha}^0 = T_{\\alpha}^0}", "ce00b6a5338381f68b3d5189be95cf82": "R_{TOT}=k_Dn_An_B+k_Bn_{AB}", "ce01016f995aab9b6dac2966879d8160": " r(\\sigma) ", "ce01288d64fe70b4d321e45f4155e2c9": "G(x)\\geq G(y)", "ce014eb370232f470e71ac35fe49023e": "\n\\mathrm i^n \\operatorname{erfc}\\, (z) \n=\n \\sum_{j=0}^\\infty \\frac{(-z)^j}{2^{n-j}j! \\Gamma \\left( 1 + \\frac{n-j}{2}\\right)}\\,,\n", "ce015a925cff9d1d478c241c10eb07c6": "\\Beta(x,y)=\\frac{\\Gamma(x) \\Gamma(y)}{\\Gamma(x+y)}", "ce01621caab12b54cc1b17960fe413f7": " \\pi_{1}(X,x) = \\pi_{1}(\\Pi_{2}(X,x)) .\\!", "ce0201010f94fd2f8977383fd5cab57c": "\\,x\\,", "ce02505b2ffa276d7b86fa9e26d534a3": "\\forall \\beta \\in \\alpha: A \\cap \\beta \\in L_\\alpha", "ce02739bb0d986d9d9e77343c4d38c1f": "r = \\frac{r_p(1 + e)}{(1+e\\cos\\nu)}\\,", "ce027a4e1e5aa9d09c055aff7d00406d": "0{.}45224\\text{ }74200\\text{ }41065\\text{ }49850 \\ldots ", "ce0285a116433d9b156527515607a2a8": " (Y\\times_X T^*X )\\otimes_Y VY\\to Y. \\qquad\\qquad (4)\n", "ce02c92117fb1c8ab4c342897d21f80d": "\\Pi=\\sqrt{2E_{ij}E^{ij}}", "ce02f6c78990388fbc83238a25377bc3": "\\frac{|\\psi|^n}{\\sqrt 5} < \\frac{1}{2}", "ce0349b1a8e06c648a3663a49e061c9c": "(d, w)", "ce03a9325ed190d57ce079eddf2b4d9c": "\\chi_\\text{e}", "ce03d53025904092996ffd534a8ca359": "|{-}\\alpha\\rangle =e^{-{|{-}\\alpha|^2\\over2}}\\sum_{n=0}^{\\infty}{({-}\\alpha)^n\\over\\sqrt{n!}}|n\\rangle\n", "ce042fbe1e0f7d3ee9f25c2e589c934a": "\\varepsilon = (-1)^k", "ce044b6f32b37932b646152bad4e7eba": "\\mathbf{C}(t=0.5)", "ce04503674737bcaaf3bb78c6267cf43": "n\\leq 50", "ce04b2b380f04a1bf1850084da9bcd74": "E_\\text{k} = \\sqrt{p^2 c^2 + m^2 c^4} - m c^2", "ce04be1226e56f48da55b6c130d45b94": "A,B,C", "ce04d670137bb98930859d0fb63065b5": "f(z) = \\sum_{n=-\\infty}^{\\infty} a_n (z - c)^n,", "ce04ebf3783ecf898bc8010bab876fa6": " 1/a ", "ce04f22a9876f44a2d7af097ae98195d": "\\scriptstyle \\Leftrightarrow", "ce04fbf15367f703534f3b3da5f44dfd": "\\ln(|\\mathcal U|)+O(\\log\\log|\\mathcal U|)", "ce05000085d286a208bfff5901e94cda": "\\;_3F_2 (a,b,c;1+a-b,1+a-c;1)=\n\\frac{\\Gamma(1+a/2)\\Gamma(1+a/2-b-c)\\Gamma(1+a-b)\\Gamma(1+a-c)}\n{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a/2-b)\\Gamma(1+a/2-c)}.", "ce052ae8bf5ff9c7d2bba24c2e5831c4": "\\ln (x) \\approx \\frac{\\pi}{2 M(1,2^{2-m}/x)} - m \\ln (2).", "ce0532ae723ea73d0ca60a97bacbbb5a": "\n T(F) = \\int (F(x) - F_0(x))^2 \\, w(x;F_0) \\, dF_0(x),\n", "ce0540d115ab341ad770f99f75f07b4a": " \\frac{t_e * 100}{h_b * 2} ", "ce055a3a0a12c8e768d575bb9699e943": " {\\theta_m} ", "ce0594a36bd4939717973b80aab9547d": "E_\\infty^{p,q} = \\mbox{gr}_p E_\\infty^{p+q} = F^pE_\\infty^{p+q}/F^{p+1}E_\\infty^{p+q}", "ce05a4f246cb4833526e240c6ed12e58": "n^{s+1}", "ce06238c1681259b600f1b2432f7e9e6": "\\delta (v)=1\\otimes v", "ce0680fed9032419017bb7bedd6dc7b1": "\\mathrm{K_C} = \\frac{V\\,T}{L}", "ce069051d702c4b506442213928dda16": "\\varepsilon u = e(u) \\, \\mathrm{d} x + \\big( u_{+}(x) - u_{-}(x) \\big) \\odot \\nu_{u} (x) H^{n - 1} | J_{u},", "ce0695c4f01f1d5f7828007501567f91": "\\overrightarrow{S} \\Delta V= S_u + S_p\\phi_p", "ce06d6cb02311c27ae3993ab36c783d4": "2|\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|\\mathbf{X}^{-1}", "ce070390d5064ad9acea3a427e97a718": "\n\\begin{matrix}\n\\; x^2 + x\\\\\n\\qquad\\quad x-3\\overline{) x^3 - 2x^2 + 0x - 4}\\\\\n\\;\\; \\underline{\\;\\;x^3 - \\;\\;3x^2}\\\\\n\\qquad\\qquad\\quad\\; +x^2 + 0x\\\\\n\\qquad\\qquad\\quad\\; \\underline{+x^2 - 3x}\\\\\n\\qquad\\qquad\\qquad\\qquad\\qquad +3x - 4\n\\end{matrix}\n", "ce076a1eb3429634690d821a8fe4ecc0": "2\\lfloor \\log_2 k \\rfloor+1", "ce07c377250dfba6acf255585945c69b": "\\scriptstyle{a=6.112\\ \\mathrm{millibar};\\quad\\;b= 17.62;\\quad\\;c= 243.12^\\circ \\mathrm{C}:\\quad -45^\\circ \\mathrm{C}\\le T\\le +60^\\circ \\mathrm{C}\\quad (<-0.35^\\circ \\mathrm{C})}", "ce07ca5b00a209eac5b5ed16d833e4ec": "V \\times W \\to W \\otimes V.", "ce07d4c6f97dfe0cb03d87cef2995641": "a\\in\\operatorname{cl}(Y\\cup \\{b\\}) \\setminus \\operatorname{cl}(Y)", "ce07d79c9977ccc7c8acdf4048f645d1": "C = \\frac{100Y}{1.1X}", "ce089636231f37a7dc6c561c7aa114d9": "p,q \\in F \\implies p \\nmid q", "ce090b2e581ab13130014ef40054ce38": " i(t) = A_1 e^{-\\omega_0 \\left ( \\zeta + \\sqrt {\\zeta^2 - 1} \\right ) t} + A_2 e^{-\\omega_0 \\left ( \\zeta - \\sqrt {\\zeta^2 - 1} \\right ) t} ", "ce091209dc146dcc9e6852da94922253": "TM= \\frac{Q_1 + 2Q_2 + Q_3}{4}", "ce094293317edb7503908c2f8199bd8e": "z = x + y \\epsilon, \\quad \\epsilon^2 = 0,", "ce0966a3d46d816d720d7fe13984f2dd": "\nL = \\frac{1}{\\sigma}\\frac{dN}{dt}.\n", "ce0982141c17b8ee32b5260b50033c8c": "\\upsilon_{c2}", "ce099cac9d188ba27ff23a8d6faf8abe": "\n\nnl = \\frac{C}{2K} \\!\n\n", "ce09aa09357b9f5455170472f72f4f6b": "q_* \\approx 1/2 +1/\\sqrt{\\pi M} ", "ce09b0201b93508f098c177701c8bb82": "\\begin{align}\ny &=a_{0}s^{c}\\sum_{r=0}^{\\infty } \\frac{(c)_{r}(c+1-\\gamma )_r}{\\left( (c+1-\\alpha )_r \\right)^{2}}s^r \\\\\n&=a_{0}s^{c}\\sum_{r=0}^{\\infty }{M_{r}s^{r}} \\\\\n&=a_0 s^c \\left (\\ln(s)\\sum_{r=0}^{\\infty} \\frac{(c)_{r}(c+1-\\gamma )_{r}}{\\left( (c+1-\\alpha )_r \\right)^{2}}s^r + \\sum_{r=0}^{\\infty } \\frac{(c)_{r}(c+1-\\gamma )_{r}}{\\left( (c+1-\\alpha)_r \\right)^2}\\left\\{\\sum_{k=0}^{r-1}{\\left( \\frac{1}{c+k}+\\frac{1}{c+1-\\gamma +k}-\\frac{2}{c+1-\\alpha +k} \\right)} \\right\\}s^r \\right )\n\\end{align}", "ce09f3d752296afe420af057a66f96fd": "x_i = \\frac {n_i}{n_\\mathrm{tot}}.", "ce09fc4e47ba79c1feea53eac80d793d": "\\phi=\\psi", "ce0a0f0ea7494a20d333573255d7f9a0": "L = I_1 \\supseteq I_2 \\supseteq I_3 \\cdots \\supseteq I_n = \\{ 0 \\} ", "ce0a4048553676021a141869b3736b52": " m_{vapor} ", "ce0ada3b9e5c2c5a6b45660dc1a85510": " y_{n}\\equiv m_{1}y_{n}^{(1)} + m_{2}y_{n}^{(2)} + \\dots + m_{r}y_{n}^{(r)} \\pmod m ", "ce0b200fee35000f09e4acf223e4ef66": "\\rho_{T}^{\\pm} \\rightarrow W_L^\\pm Z_L^0", "ce0b4278d05a57b0d28f2751c2be08ec": "x^\\prime=L^{-1}(x-a)", "ce0bc4386484d21ebbc63bede0515e5d": "\nME_\\mathrm{dB} = 20 \\log_{10} \\bigg(1-\\rho_1\\rho_2\\,e^{-j2\\theta}\\bigg) \\,\n", "ce0bcbf48bfc800a80c5ef74f61b110f": "g * h = 0", "ce0be187c6f33f7e960a37501113bdf0": "\\varphi: U \\times \\mathbf{R}^{k} \\to \\pi^{-1}(U) ", "ce0c1bdf95ae33766ce47be91dd74456": "\\begin{align}\n\\frac{d[E]}{dt} &= - k_f [E][S] + k_r [ES] + k_{cat} [ES] \\\\\n\\frac{d[S]}{dt} &= - k_f [E][S] + k_r [ES] \\\\\n\\frac{d[ES]}{dt} &= k_f [E][S] - k_r [ES] - k_{cat} [ES] \\\\\n\\frac{d[P]}{dt} &= k_{cat} [ES].\n\\end{align}", "ce0c6589c46152edfca5652b183b7aca": "\\,w_i (n+1) ~ = ~ \\frac{w_i + \\eta\\, y(\\mathbf{x}) x_i}{\\left(\\sum_{j=1}^m [w_j + \\eta\\, y(\\mathbf{x}) x_j]^p \\right)^{1/p}}", "ce0ccc89311c623d8235aa2bb2df3647": "A f (x) = \\lim_{t \\downarrow 0} \\frac{\\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}\\ ", "ce0cd1be7adb58c652012e011848e6f2": "\\int_{-\\pi}^{\\pi} \\sin(mx)\\, \\sin(nx)\\, dx = \\pi \\delta_{mn}, \\quad m, n \\ge 1", "ce0cf94409bf00425f78b7a73f9bb054": "\\phi=t", "ce0d089825ae4b03185da600080afb2d": "f'(x-) \\le f'(x+) \\le f'(y-),\\qquad x < y.", "ce0d17205344c5a671de139acbcd7b16": "F_+(L_2(X,\\mu))\\,", "ce0d41e0602c862f4e343b2bc8d9601b": " W(f)\\left(1,g,\\frac{g^{\\otimes 2}}{2!},\\frac{g^{\\otimes 3}}{3!},\\ldots\\right)= e^{-\\frac{1}{2}||f||^2-\\langle f,g\\rangle }\\left(1,f+g,\\frac{(f+g)^{\\otimes 2}}{2!},\\frac{(f+g)^{\\otimes 3}}{3!},\\ldots\\right), \\, ", "ce0d7c28e0abc996179f1c6506925cf4": "Insert formula here", "ce0da3b34f39b374e49b73c754ea0ed9": "\\left( \\frac{d\\boldsymbol{f}}{dt}\\right)_r", "ce0db7ba4f1ac4b50696f234211487f8": "\\begin{pmatrix}\nx_1 & y_1 & z_1 \\\\\nx_2 & y_2 & z_2 \\\\\n\\vdots & \\vdots & \\vdots \\\\\nx_N & y_N & z_N \\end{pmatrix}", "ce0dd4164d21993a5a9a59cd9920d02d": "\\operatorname{var}(\\mathbf{X} + \\mathbf{Y}) = \\operatorname{var}(\\mathbf{X}) + \\operatorname{cov}(\\mathbf{X},\\mathbf{Y}) + \\operatorname{cov}(\\mathbf{Y}, \\mathbf{X}) + \\operatorname{var}(\\mathbf{Y})", "ce0e01bc146e450b812939ba466449d5": "\\mathrm{wt}(c)", "ce0e5b544956fa278d68635a6d27f392": "t\\ge 0", "ce0ead9d9a9a2a6e980049f7124eeb9b": "E=\\frac{1}{2}\\,\\,\\frac{1}{4\\pi\\varepsilon_0}\\frac{e^2}{r_\\mathrm{e}}", "ce0ec4ab7f47a8d6310460493e7268e1": "f\\circ \\rho = f\\,", "ce0eef46a36f8ceb0a241665fa372ecf": "\\begin{pmatrix} 1 & 2 & 3 & 4 & \\cdots \\\\\n1 & n+1 & 2 & n+2 & \\cdots \\end{pmatrix}", "ce0ef9fc1cf93352b08b88b998c1edbc": "\\mu_\\mathrm{N}", "ce0f040132fb683085b2000483fdb865": "I^p", "ce0f3eff43b9670e9e5d1084748c71a1": "N_\\mathrm{w}", "ce0f8ca6e059f58bc231c14ef83f602c": "(h,y)(g,x) = (hg,x)", "ce0faf2fd0f022027c4e5f8e55194a74": "\\{x^k\\}_{k \\in K}", "ce0ff0f55b2466800f74155945f27daf": "\\frac{d\\psi}{dx}", "ce1010847cd65b293dc2213936082fee": "\n\\frac{dx(t)}{dt} = \\Pi_K(x(t),-F(x(t)))\n", "ce1051b742abece853f6bf0a057d9381": "\n \\mathbf{t}_0 = \\boldsymbol{N}^T\\cdot\\mathbf{n}_0 = \\boldsymbol{P}\\cdot\\mathbf{n}_0\n", "ce10739bafe64fedbaa6ae1f59c7b5bb": "MacD = \\frac{\\sum_{i=1}^{n}{t_i PV_i}} {V} = \\sum_{i=1}^{n}t_i \\frac{{PV_i}} {V} ", "ce108a509cfaf69547baaf2e70acb6ea": "\\mu = (L(g^{\\lambda}\\mod n^{2}))^{-1} \\mod n", "ce10f2e4ab1ec2745ce8e63018fa90ec": "\\mathbb{E}^g[X | \\mathcal{F}_t]", "ce10f3779dd5aaccf39634b50f5cb9bd": " \\frac{a(T,\\rho)}{RT} = \n\\frac{a^o(T,\\rho)+a^r(T,\\rho)}{RT}=\n\\alpha^o(\\tau,\\delta)+\n\\alpha^r(\\tau,\\delta)\n", "ce112153f3589e9041c7642dcd374937": "\\tilde{g}(p)=p \\cdot \\phi(p)", "ce113133888e2a223dbeecb4a6348f11": "dx^{0(2)} = \\frac{1}{g_{00}} \\left ( -g_{0\\alpha}\\, dx^\\alpha + \\sqrt{\\left ( g_{0\\alpha}g_{0\\beta} - g_{\\alpha \\beta}g_{00} \\right ) \\,dx^\\alpha \\,dx^\\beta} \\right ),", "ce116f6349754cf933a5328af36d7a52": "H = (\\oplus_{i \\geq 0} M_i) \\oplus (\\cap_{i \\geq 0} H_i) = K_1 \\oplus K_2.", "ce12817f1e31eb3310b233f0e02115fd": "\\log 2.", "ce128d5f5eb664ae5b1c2440a2ad085c": "\\psi_i : \\pi^{-1}(U_i)\\to U_i\\times \\mathrm{GL}(k, \\mathbf R)", "ce12cd08b6973b003b409bbd603f6725": "\\sum_{r'} \\left(\\; {dB\\over dJ}[r'](k-r')A[r',k] - {dA\\over dJ}[k-r'] r' B[0,r']\\right)", "ce130d7314937340b3071c1677e9fa07": "\n{EL}_{gas}=\\frac{{WI}\\times{LOE}}{{NRI}[({P_o}\\times{Y})+{P_g}]\\times(1-{T})}\n", "ce132a76d8b3d16203c0be850397b13a": "\\theta = \\frac{TL}{JG}", "ce13387b3985572d7a616733ac33e23e": "Z_{t,d,n} \\sim \\textrm{Mult}(\\pi(\\eta_{t,d}))", "ce136343b2cb894ee2513612fce8e6fe": "\\lnot(x \\wedge y)=\\lnot x \\vee \\lnot y \\mbox{ for all } x, y \\in H,", "ce1386cebd8be9ab615acf57534b1a55": " \\Phi(r) ", "ce139bbe511a843c9024caa70dbe332e": "e^{-y^2}", "ce13c7b560ec34b5ac590a6c8bbc45d7": "\\theta(x) \\in (0,\\infty)", "ce13c7d23ea43b005131ecc354c9ba81": "1\\le p\\le \\infty", "ce13e0422f28a11b680f089ad21d2195": "\\sum_{j=0}^p c_{j,k}z^j = a_0 + a_1 z + \\cdots + a_p z^p", "ce13f2e969dec13245895901ace77a62": "w\\le u", "ce140af267a49255c05524dc952501de": "\\frac{E(r) - r_f}{\\sigma} = \\frac{E(r_M) - r_f}{\\sigma_M}.", "ce140f4a7b00df532be72f5bec9b9984": "f(z)=a\\,(z-z_0)+b+o(z-z_0)", "ce1419d849868fa058f657f403057825": "\\Sigma = O(\\sqrt{N}\\log N).", "ce14edc931113d11b2ab4fcda24ea34e": "(\\overline{C} \\vee A \\vee B) \\wedge (C \\vee \\overline{A}) \\wedge (C \\vee \\overline{B})", "ce14f0fb4a45c33f559db3ddbd39e568": "\\varphi(m, n, p)\\,\\!", "ce150f2f73d31a49d0b56401b53125e2": "\\begin{align}\nA &= A^* = \\mbox{constant} \\\\\nT_0 &= T_0^* = \\mbox{constant} \\\\\n\\dot{m} &= \\dot{m}^* = \\mbox{constant} \n\\end{align} ", "ce152ac886d18ba4b6f51bb70e930f29": "h=x_n-x_{n-1}", "ce15571a85e193852696d5e1c9563fce": "x \\frac{\\mathrm{d}v}{\\mathrm{d}x}=-\\frac{J(1,v)}{I(1,v)}-v.", "ce15ced563fece429d3b3333d92d4b7e": "-\\infty < U < \\infty\\,", "ce15f5c4c64274ce8fefdee78fe4b22f": "\\begin{array}{rl}\n\\min\\limits_{x} & f(x) \\\\\n\\mbox{s.t.} & b(x) \\ge 0 \\\\\n & c(x) = 0.\n\\end{array}", "ce165400c6948d7b71bfce57fb765348": "H_k(\\partial M;\\mathbb{Z}_2) \\to H_k(M;\\mathbb{Z}_2)", "ce167f2608375ff7998d7dc7cbc1798f": " I(d) = I_0 \\ e^{-d \\ / \\lambda(E)} ", "ce168a047c20d945aead8b037b0c8ffc": "Y = \\sum_{i=1}^n (X_i - b)/c \\,", "ce16ffe82a3601fb0c2b6bd412f0487a": "\\hat{H} = \\frac{g_s e}{2m} \\bold{S} \\cdot\\bold{B} ", "ce17032b6f937373472c15739e88ced3": "\\frac{1}{\\sqrt{2 \\pi}}", "ce171d43e32d520cdb94e2982c53e5ed": "z \\ \\sim\\ \\ \\mathcal{N}(0,\\,\\sigma^2I)", "ce171dc704c219a5f5fdda7c184596ee": " \\rho_{mn} = \\sum_i p_i \\langle u_{m} | \\psi_i \\rangle \\langle \\psi_i | u_{n} \\rangle.", "ce178e9bf842b1d58e2bcfe3d788d812": "\n\\begin{align}\n\\psi(\\phi)\n&=\\ln\\left[\\tan\\left( \\frac{\\pi}{4}+\\frac{\\phi}{2}\\right) \\right]\n+\n\\frac{e}{2}\\ln\\left[ \\frac{1-e\\sin\\phi}{1+e\\sin\\phi} \\right]\\\\\n&=\\tanh^{-1}(\\sin\\phi) -e\\tanh^{-1}(e\\sin\\phi)\\\\\n&=\\mathrm{gd}^{-1}(\\phi)-e\\tanh^{-1}(e\\sin\\phi).\n\\end{align}", "ce17a56e5de45c3fabe699bbe621bc53": "s_2 s_3 = s_8", "ce17f04b0bd7f21196ef63d2e3d23290": "\\scriptstyle f_\\mathrm{image}(0)\\,", "ce18356f7b790399b9ce9c7f3a3e8b89": "\\Delta {\\mathcal A}", "ce185b063f1462a83853fe7a040a878b": "x,y,z\\in[0,1]", "ce186a2c563ab7b2ceca526ebde1f61d": "(\\mathcal{K}, \\mathcal{B} (\\mathcal{K}) )", "ce1886b40ce206d71b910c479c890e6e": "\\Delta p = 3 p^{\\star}_{\\rm A} x_{\\rm B}", "ce18b9b2399f866f36502d7260f34262": "f(y_1, ..., y_m) = 0", "ce192e65f8b5c62f7a19a86a0347acd8": "\\scriptstyle j\\,=\\,0", "ce19b5f8d6d3faa7f9af23d9a83afaa5": "\\frac{1}{d}\\left(N-\\left(\\frac{D}{N}\\right)\\right),", "ce19d0dae1429ce031585189d3b7f464": "U_s(\\lambda)", "ce1a309e4b1a27f892af275f008a7b95": "R_e=\\frac{vD}{\\nu}", "ce1a5bab6b28cc9689883a1ab8c5f17d": "(h(x)-h(y)) ~\\bmod~ m", "ce1a7ba9c2474ea8cbd1b668e88ddb4b": "1 \\leq i \\leq n-2", "ce1a8a5e342d5e9f3e1ee77fcc3ce0a4": "\\frac{\\operatorname{d}}{\\operatorname{d}t}\\gamma(t)=X_{\\gamma(t)}", "ce1aac5e2dab556123d276c7f5c08374": "d_e(x_e+1)", "ce1ab9a599812220af9db7ec7eebda3d": "V_\\mathrm{swap} = B_\\mathrm{fixed} - B_\\mathrm{floating} \\, ", "ce1ac472099fca4e4b8e8632adb42264": "\\scriptstyle \\log_{10} P_{mmHg} = 6.94459 - \\frac {1295.26} {218.0+T}", "ce1aecb63a993249d1a35497283c5903": "0 < x\\cos x < \\sin x < x\\hbox{ for }0 < x < 1.\\,", "ce1b92926ede222c9a76eb41f5cccb8a": "x_1^2 + y_1^2 = 1 = x_2^2 + y_2^2. \\, ", "ce1c0ae7dc3bfcedf393b7936f0ac0c6": "Z^{-1}_3", "ce1c2fe5f19dcf04afb194a2a0f2e216": "\\qquad\\vdots", "ce1cc70752f5fd436f271782298339b0": " \\Gamma^i_j = 0 ", "ce1cf85b1a323aacc1d378207f38ca51": "M^{-2}", "ce1d1da8aed25532153eb0edca0c857a": "\\begin{align}\n G &= \\Re(Y) = \\frac{R}{R^2 + X^2} \\\\\n B &= \\Im(Y) = -\\frac{X}{R^2 + X^2}\n\\end{align}", "ce1d42013a3883ce5fed754dd0dcf86e": " v \\in V ", "ce1d58b1af4580b533576ddf31cf4c8f": "\\mathcal{O}_D(2K)", "ce1d7c53638bcfa8cd6bd6f4d576e056": "-\\frac{\\textrm{d}^2}{\\textrm{d} x_e^2}", "ce1dc956b3ba954a1138a64f0424b2cc": "[1.75] = 2", "ce1dd8830e295ea18bcfe772efb3097a": "\n \\int x^{m-n}\\left(a\\,B\\,c (m-n+1)+(a\\,B\\,d (m+n\\,q+1)-b (-B\\,c (m+n\\,p+1)+A\\,d (m+n (p+q+1)+1))) x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx\n", "ce1dd9685242ec8eb84ae6c854fd3da2": "\\text{Holant}(H, \\text{OR}_2, \\text{EQUAL}_3).", "ce1e13c202660514e8ab18f62aaa1bb1": "\\begin{align}\n\\operatorname{E}[\\,(\\hat\\beta - \\beta)(\\hat\\beta - \\beta)^T] &= \\operatorname{E}\\Big[ ((X'X)^{-1}X'\\varepsilon)((X'X)^{-1}X'\\varepsilon)^T \\Big] \\\\\n&= \\sigma^2 (X'X)^{-1}, \\\\\n\n\n\\end{align}", "ce1e8d68519bf70a08ff41fabdd2f9f2": "N_e", "ce1e9005bc5b5b1ad1aab5f3e4f6caad": " P = D + R \\quad (2.6) ", "ce1e9b8515de784698c00c8e3c62579d": "(1 + i)(1 - i)", "ce1f472365dd3efbb26fa0fb96c350c9": "V_{i}", "ce1f67718e4757907261e833f02c5a5d": "x_{n+1}=g(x_n)", "ce1fadcfb563a5003e7b94f873eccea9": "\n\\hat{e}_{\\phi} = -\\sin \\phi \\boldsymbol{\\hat{i}} + \\cos \\phi \\boldsymbol{\\hat{j}}\n", "ce1fd27d18fe8d2979e11370389ff4cb": "\\sigma=\\sigma^i_\\mu dx^\\mu\\otimes\\partial_i \\qquad\\qquad (5) ", "ce1fe7788aa2b7ff67a2c61795148d27": "n_{l}", "ce2015b17f77359801258429bcfff9cf": "\\tfrac{1}{2} \\boldsymbol{\\nabla} (\\mathbf{A}\\cdot\\mathbf{A}) = \\mathbf{A} \\times (\\boldsymbol{\\nabla} \\times \\mathbf{A}) + (\\mathbf{A} \\cdot \\boldsymbol{\\nabla}) \\mathbf{A} ", "ce20170fc98715bdcb406f2c6d4d9448": "Q \\times \\Sigma_{\\epsilon} \\times \\Gamma^{*} \\longrightarrow P( Q \\times \\Gamma^{*} )", "ce207ac966578160a253930fb4300a64": "{A}_{eq}{x} = {b}_{eq} \\,", "ce20a2828e71658d4a082eff4d58e251": " C'_w= \\left (q^{-1/2} \\right )^{l(w)}\\sum_{y\\leq w}P_{y,w}T_y, ", "ce20b748b201de4f050270ee121d040a": "x\\in\\mathbb{F}_q^n", "ce20c507f72991f66b06ee6f244e7f54": "h(x) = x^2", "ce2125e729a2bfd06551bc2c268ea7c2": "u^i", "ce21a2fe1367d14338cffff79f1379aa": "\\frac{dx_1}{dt}=f_1(t,x_1,x_2,\\ldots,x_n)", "ce2262667581c283b79d1178976e5985": "\\lim_{x \\to 0} \\frac{\\sin x}{x} = 1", "ce2266176becedeb496a25f20e1b3182": "\\textstyle K_{in} = C_{ss} \\cdot CL", "ce230f0210898b1220d30ca7b6fdef81": "j_i' \\,,", "ce234584b5546f0d6e0ffafdea4a7f1e": "x(t-1)\\approx u_3(t)", "ce237577fcf8a65393b328bed42a3229": "\\epsilon \\to 0", "ce23b6c94c4ca72be595cdd790066750": "\n(\\mathbf{I}^{A}\\otimes\\left( \\mathbf{A\\otimes I}\\right) ^{B})\\left\\vert\n\\Phi_{n+c}^{+}\\right\\rangle\n", "ce23f7b8badbe94eea1c01c0d99340c9": "S(n_1,\\ldots, n_l) = \\forall m_1\\cdots \\forall m_k R(n_1,\\ldots,n_l,m_1,\\ldots,m_k)", "ce240a58f5651fbc89c836593f59d8a4": "\\begin{align}U & = a\\ \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n + b\\ \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}\n + c\\ \\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix}\n + d\\ \\begin{pmatrix} i & 0 \\\\ 0 & -i \\end{pmatrix} \\\\\n& = a\\,I+ic\\,\\sigma_x+ib\\,\\sigma_y+id\\,\\sigma_z,\\end{align}", "ce2448ba3232e8fc2912de29dba45fb6": "\\tfrac{1}{X} \\sim \\mbox{Inv-Gamma}(\\alpha, \\beta)\\,", "ce24ab4dd15eb4ce78978e4b107eeb19": "\\mathbf{G} := \\begin{pmatrix}\n 1 & 1 & 0 & 1 \\\\\n 1 & 0 & 1 & 1 \\\\\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 1 & 1 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}, \\qquad \\mathbf{H} := \\begin{pmatrix}\n 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\\\n 0 & 1 & 1 & 0 & 0 & 1 & 1 \\\\\n 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n\\end{pmatrix}.", "ce24bd7aff00b785749fd62026a2464f": " |R\\rangle \\equiv {1 \\over \\sqrt{2}} \\begin{pmatrix} 1 \\\\ i \\end{pmatrix} ", "ce2518c496fa0bd9991dcf39332a3145": "\\phi(y)=\\alpha", "ce2543682e195e2b4e234eb011d1518e": " A^*\\to I\\otimes A^*\\xrightarrow{\\eta}(A^*\\otimes A)\\otimes A^*\\to A^*\\otimes (A\\otimes A^*)\\xrightarrow{\\epsilon} A^*\\otimes I\\to A^*", "ce25f900d40c0ea3ecb0bfaba91c25f2": "pmi(x;y)", "ce269aa9a6e77c7110e3f6f36e0ae4d3": "\n\\bar x =\\alpha \\ \\hat e_1 + \\beta \\ \\hat e_2 \\quad -\\infty <\\alpha < \\infty \\quad -\\infty <\\beta < \\infty\n", "ce2710a2badbf520335746aaf17236a1": "\n\\mathbf u \\wedge \\mathbf v = \\sum_{i0}X^\\alpha_n = \\bigcup _n \\bigcup _\\gamma X^{\\beta_{\\gamma+1}}_n\\setminus \\beta_\\gamma = \\bigcup_\\gamma \\bigcup_n X^{\\beta_{\\gamma+1}}_n\\setminus \\beta_\\gamma = \\bigcup_\\gamma \\beta_{\\gamma+1}\\setminus \\beta_\\gamma = \\alpha \\setminus \\beta_0", "ce2b09b279e4dab7c405c5d6679deb16": "H=2h=0.4", "ce2b1e5e91651ec76fb439fc56ef6b87": "f: X \\to \\mathbb{C}", "ce2b400b0749c8d19c87fc2db83f3fa7": "\\textstyle\\frac{p_1}{q_1}", "ce2b6fe79f7c4eb51232d80155cadb17": "\\operatorname{dim}(\\Lambda^k(V)) = \\binom{n}{k}", "ce2b70cd40d9eb03cb1dae1361158ea7": "\\delta_{ij}~", "ce2ba6e818b0954c52286b041c0f6da5": "[3.25] = 3", "ce2bc4a41501959cf1232831f5515db2": "F'(R,Q:AL < P < AU)=\\sum_{T\\!A=1}^{U\\!A=\\infty}\n\\frac{F'(R,Q:P_{(ta)})}{U\\!A};\\,\\!", "ce2bd9ebd592766f6e4d3da3f9f5862e": "\\boldsymbol{\\eta} = \\boldsymbol{\\eta}(\\boldsymbol{\\theta})", "ce2bf7a7584f6169ac6ce2ac30952d55": "E_{n}^{(1)}=-\\frac{1}{8m^{3}c^{2}}\\langle\\psi^{0}\\vert (2m)^{2}(E_{n}-V)^{2}\\vert\\psi^{0}\\rangle ", "ce2c27afec17308b06189f18386a01f8": "L^2(\\mathbb{R}),", "ce2c3e1e5f6e8fec214a42df293304dc": "\\tfrac{6}{5}", "ce2c580d05372a8a2081ca745ed80e0d": "\\int\\frac{x^2\\;dx}{s}\n= \\frac{xs}{2}+\\frac{a^2}{2}\\ln\\left|\\frac{x+s}{a}\\right|", "ce2ca918be619b18bb2345f707b197dd": "dV'", "ce2cabe0ff387c9799057118aa53a9ff": "\\scriptstyle d=1 ", "ce2cc21feac483a07c90fafa5396694d": "\\displaystyle -i\\pi \\frac{(-i\\nu)^{n-1}}{(n-1)!}\\sgn(\\nu)", "ce2ccc0e0ac17ccc26345c70c3cbf45f": "z=0\\,", "ce2cdc0a3ee05409d7789d888512dac4": "\\stackrel{\\alpha}{\\omega}", "ce2d271ffbcbfd0fddb9acdefcda53e5": " \\iota_{Y\\mid X} = {\\operatorname{Var}(\\operatorname{E}(Y\\mid X)) \\over \\operatorname{Var}(Y)} = \\operatorname{Corr}(\\operatorname{E}(Y\\mid X),Y)^2,\\,", "ce2d7bcbdde483530e49d51ff9caaa9d": "\\int_0^\\infty e^{-t} \\mathcal{B}_\\alpha y(t^\\alpha z) \\, dt", "ce2db5a0d1a40e369dc1b33c639592b2": "(q-1)/2\\,", "ce2e6773fa5ed79b61e3ff2150e2ea8a": "SE\\text{ of }gf_i=\\sqrt{C}/(2N)=0.00628", "ce2e7f848a795f7ff8c4d4d6542e810f": "\\sigma_y = \\sigma_0 + {k_y \\over \\sqrt {d}}", "ce2e8697b482a1f9a76d6a3149706130": "V_I(f) \\le V_I (f') + \\frac12.", "ce2f24f9ef73018bea549bafa36eff55": "\\omega_e = \\sqrt{\\frac{1}{L_eC_e}} = \\frac{2\\pi c}{\\lambda_0} = \\frac{m_0c^2}{\\hbar}. \\ ", "ce2f735c6d3c3cff6d05f5798884c462": "\\begin{align}P \n&= \\left(\\frac{C}{1+i}+\\frac{C}{(1+i)^2}+ ... +\\frac{C}{(1+i)^N}\\right) + \\frac{M}{(1+i)^N}\\\\\n&= \\left(\\sum_{n=1}^N\\frac{C}{(1+i)^n}\\right) + \\frac{M}{(1+i)^N}\\\\\n&= C\\left(\\frac{1-(1+i)^{-N}}{i}\\right)+M(1+i)^{-N}\n\\end{align}\n", "ce2f99461479211bac6a19a819959ad4": " D[g] = [x, S_5, A_5]::[o, S_4, A_4]::[y, S_3, A_3]::K_2 ", "ce2f9bd832201fd3dbf9b121196d7a75": "-\\ln(\\ln(2)) ", "ce2fa2186f6c85dc9c657c88cc44d9be": "\\theta_\\mathrm{(t)}=\\frac{1-e^{-R'(1+k_\\mathrm{E})t}}{1+k_\\mathrm{E}e^{-R'(1+k_\\mathrm{E})t}}.", "ce2fae2bbdffe7a96acb46d7a907b6e6": "D_n = (1-c)^n\\; D_0", "ce2fd54748e01630af1c4ff3c9fe1116": "\\lambda = \\frac{\\operatorname{E}(V)}{\\operatorname{stdev}(V)}\n=\\frac{\\sum_{i=1}^t c_i \\mu_i}{\\sqrt{\\text{Var}(\\sum_{i=1}^t c_i G_i)}}\n=\\frac{\\sum_{i=1}^t c_i \\mu_i}{\\sqrt{\\sum_{i=1}^t c_i^2 \\sigma_i^2 + 2\\sum_{i=1}^t \\sum_{j=i} c_i c_j \\sigma_{ij} }} ", "ce2fe1d9548ffe433ab1897a5ff9691c": "x^4+\\left(16\\frac{1}{2}\\right)^2x^2=\\left(164\\frac{14}{15}\\right)^2", "ce30092920fb7dbd9a12cc5c9e43bbd1": " B_1 > B_2 > \\cdots > B_N. ", "ce30330c087d8db9a89a466417608ac4": "\\frac{1}{\\alpha} \\ ", "ce30718c33f5ba96e44579bd47a0326e": "\na \\cdot \\nabla F(x)= \\lim_{\\tau \\rightarrow 0} \\frac{F(x + a\\tau) - F(x)}{\\tau}\n", "ce309c7ee3906f01dd00d07662fb1a79": " |c\\rangle = \\psi_R |R\\rangle + \\psi_L |L\\rangle ", "ce30f2e6c2c03d965f96ef86dedbc862": " \\lambda_{max} = 9.02a ", "ce31007407f7def702fa7b041130bea5": "g_{\\mu\\nu}\\,\\mathrm{d}x^{\\mu}\\,\\mathrm{d}x^{\\nu}=(-\\,N^2+\\beta_k\\beta^k)\\,\\mathrm{d}t^2+2\\beta_k\\,\\mathrm{d}x^k\\,\\mathrm{d}t+\\gamma_{ij}\\,\\mathrm{d}x^i\\,\\mathrm{d}x^j.", "ce31113847f4a2c7de441f02fb0140d9": " {\\mathbf u} = \\lozenge \\{u:K(p)u = Q(p),p\\in {\\mathbf p}\\} ", "ce31a61768779f9fa92662c4e07e6307": "\nB(E)=\\int_0^\\infty d\\beta f(\\beta)\\exp(-\\beta E).\n", "ce31e571a3aecf37b584d2c588698175": "\\mathcal{J}_i", "ce3221b7fbd616c5d708f18dd091289a": "(),", "ce32284e930aefac69917961da0b71f5": " \\frac {V_S-V_1}{R_1} = j \\omega C_1 V_1 + \\frac {V_1-V_O} {R_2} \\ . ", "ce326d14bc4900c3bcc3db5f136ebeb1": "V_{BB}", "ce3283cc0c53538ba91fa9f1d16b72d5": "\\binom{p+q}{p}.", "ce32871328cbf233ebcbc0241eaf8a70": " \\frac{1}{2}(f(x+) + f(x-)) = f(x) ", "ce329b8566b45497a70edda9feb3ea2c": "u(y) = \\frac{y - y^2}{2}.", "ce32de9f4e13cb3fffdfcd35f180a900": "gate8", "ce32f1c72fdc2daa87a8a39f1741e2c3": "\\|f\\|_B = \\sup\\nolimits_{x\\in X}|f(x)|", "ce33299439d22934ac15efa0ca460f7f": "U(x) = e^{\\int_{x_0}^x A(x)\\,dx}", "ce3336fe8484a1cdebe3db26bc78791b": "c_{full}", "ce335ed7cda4017234864a41e8bbf8f3": "t(n)", "ce3363952504b5b73ddef75f6bbe5adb": "c : S^1 \\to X", "ce3394f52da5b27b04fa57bed4495b80": "Z_d=\\frac{d_{80,1}}{d_{80,2}}\\,", "ce33ca126888dcc0ebef1d9ad0afb099": "\\lim\\sup F_j = \\bigcap_{n=1}^\\infty \\bigcup_{k=n}^\\infty F_k = \\mathbb{R}^n", "ce34b2ad478d0c9ae4ddd97999047924": "\\|x\\|_{\\infty}=\\|y\\|_{\\infty}=1", "ce34b524da61624b045b4b149cd92971": "\\overline{m_n}=\\frac{1}{N}\\sum_{i=1}^N z_i^n =\\overline{C_n} +i \\overline{S_n} = \\overline{R_n} e^{i \\overline{\\theta_n}}", "ce34e3940bc9a3e4705a57d1778f0e10": " \n\\text{(Eq. 3)} \\qquad (\\mu_{ab}(t)) \\in \\Gamma_{S(t)} \n", "ce34e42f9cbf99557d3a61eea1a0d80f": "g^2_{ETC} \\gtrsim 1", "ce353cafa2338c2978d3ac5422694b01": " \\lim_{h\\to 0} \\frac{\\delta^h_{n+k}}{h} = 0. ", "ce3547b13f5fa7371f47200952c32ae3": "R(t_r)", "ce354e1bbf61ac70119fb2b8237cf4ee": "S^{-1} M", "ce357a8a3c8a8cfdabe08fb592e1f99f": "\\displaystyle{X =Q-iP={d\\over dx} + x,\\,\\,\\,\\, Y=Q+iP =-{d\\over dx} +x}", "ce35969d6e1713293c54ba8974695e5e": "deg(\\chi_n^{\\alpha\\beta})", "ce35a9b73906bc7a26c08acfed4d469e": "G^{op}", "ce35b33bc17293239c796848567e7a36": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "ce35bbb3aa12601bf7bcc82733d2c41c": "E(\\ln(x^2+\\gamma^2))=\\ln(4\\gamma^2)\\,", "ce35e00122358b4a77888db0b9c3f8cf": "\\textstyle R=\\frac{2W_s}{\\rho C_L\\sin\\theta}.", "ce35f0752df818d1170f4cbb5add06cf": "\\cot \\theta = \\frac{\\cos \\theta}{\\sin \\theta} = \\tan \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\tan \\theta} ", "ce36191071a7c0fb4102d977119ab1f9": " \\nabla_{{\\mathbf e}_i} {\\mathbf e}_j = \\sum_k \\Gamma^k_{ij} {\\mathbf e}_k.", "ce36205cda97703d449b31b3b19c1a7f": "\np_1 + 2q_1 + r_1 = (p + q)^2 + 2(p + q)(q + r) + (q + r)^2 = 1\n", "ce36571a250a537e532170afb80d702c": " dg = g g^{\\mu \\nu} dg_{\\mu \\nu} = -g g_{\\mu \\nu} dg^{\\mu \\nu} ", "ce36727f69a797c8a4958212894ae021": "\\displaystyle\\mathrm{Nat}(H^n(-,\\pi),H^q(-,G)) = \\mathrm{Nat}([-,K(\\pi,n)],[-,K(G,q)]) = [K(\\pi,n),K(G,q)] = H^q(K(\\pi,n);G)", "ce3725669cc28ae0623f4b8fa22f45f8": "A_i\\rightarrow A_j", "ce3735d4900a6a9caa6680fa21d4bd34": "\\frac{\\Delta (C \\cdot V)}{\\Delta t} = -K \\cdot C + \\dot{m} \\qquad(6)", "ce37cba642b16d8bf313ba768c4fa3a9": "Y_m=Y_{m+1}=Y_{m + 2}=\\cdots", "ce37e14c882d34f158a6e05745584a52": "a^3+b^3=c^3", "ce3801df3641c3c5edb109d6db8ffcc8": " u_2(X_1,X_2)=E(X_1X_2)-E(X_1)E(X_2)", "ce388062f073eca77c95f81a289b0e7b": "y_{1}^{\\star} = F_{1}^{-1}(1-\\frac{R_{2}}{R_{1}})", "ce38a7ebb925f2238d1f8abc1aef298b": "U = 4\\sigma T^3 \\frac{1}{1/\\epsilon_1 + 1/\\epsilon_2 - 1},", "ce38adef3c2e25a17207741f500c2129": "\\scriptstyle\\pi(x)", "ce38c103862513a4d98290986e32d883": "e_\\mu = R \\gamma_\\mu \\tilde{R}", "ce38d8bbcac0481707774a9d5fea616b": "M_1, M_2", "ce395d0d5ade71e820bf3cfc009de745": "E[Y] \\leq E[X]/2", "ce39674b316fb86a9ac7c7ed0de6311d": "\\begin{align}\n |\\tau| &\\ge 1 \\\\\n -\\frac{1}{2} &< \\mathfrak{R}(\\tau) \\le \\frac{1}{2} \\\\\n -\\frac{1}{2} &< \\mathfrak{R}(\\tau) < 0 \\Rightarrow |\\tau| > 1\n\\end{align}", "ce39707beb6064dab9ae0a27f45a6f15": " \\prod_{k=1}^{\\infty}\\left\\{1-[1-\\prod_{j=1}^n \\underset{p_k: \\text{ prime}}{(1-p_k^{-j})]^2}\\right\\}", "ce397b9024e2ec6cb018c3147dc95dde": "\\begin{pmatrix}\n4 & 0 & 0 & 0 & 0 & 0\\\\\n0 & 3 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 2 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 3 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 3 & 0\\\\\n0 & 0 & 0 & 0 & 0 & 1\\\\\n\\end{pmatrix}", "ce3a032f5e4e9ed21a0a136ebdd91507": " T\\, v= \\tfrac12\\, \\dot m\\, {w^2}.", "ce3a123dc5f58c1fb5967aabf0453b86": "\\tan\\frac \\theta2=\\sqrt\\frac{1+\\varepsilon}{1-\\varepsilon}\\cdot\\tan\\frac E2.", "ce3a23a35359ebe16f077cfb651485e6": " L[A] = L^{+-}[A] + L^{-++}[A] + L^{--++}[A]. ", "ce3a2dc37907776cb1efa7d8667d8b98": " - \\bold{f} = - (\\rho \\bold{E} + \\bold{J} \\times \\bold{B})\\,", "ce3a4f92ff81b7daeca3a25b63eb796e": "q \\in Q \\}", "ce3aa7cdca29f6037a02be6aa8838b1d": "\\, \\sigma_0", "ce3ab22ceb4cbaa606127c6311dfc3be": "i^\\text{th} (i=1, 2, \\ldots, t)", "ce3acb93fee747450f92c6159603cc69": "\\vec{B} = B_{\\theta}(r)\\hat{\\theta} + B_z (r) \\hat{z}", "ce3b12e8b4bd1dd31c48937d9ae1d630": "d \\Xi = - U d \\frac {1} {T} + \\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i - \\frac{P}{T} d V - V d \\frac{P}{T}", "ce3b1a58d36792e37a34876b84e77473": "\\, |n(\\mathbf R(t))\\rangle ", "ce3b395cb4bb8fbcaa497bf94887c2fd": "m = m_r + \\delta m", "ce3b64d039489409edc5481696c65644": "\\forall x \\in a", "ce3b7d0c14a807a438b3cfea811434ac": " K\\cdot N ", "ce3b94ba43333fcf1e89eeca8726201a": "S=\\{1,2,\\ldots,M\\}", "ce3c0b9c6890be0cdc7f3de701269b9a": "\\mathbf{g} = \\begin{pmatrix} \ng_\\text{xx} & g_\\text{xy} & g_\\text{xz} \\\\\ng_\\text{yx} & g_\\text{yy} & g_\\text{zz} \\\\\ng_\\text{zx} & g_\\text{zy} & g_\\text{zz} \\\\\n\\end{pmatrix} = \\begin{pmatrix} \n\\mathbf{e}_\\text{x}\\cdot\\mathbf{e}_\\text{x} & \\mathbf{e}_\\text{x}\\cdot\\mathbf{e}_\\text{y} & \\mathbf{e}_\\text{x}\\cdot\\mathbf{e}_\\text{z} \\\\\n\\mathbf{e}_\\text{y}\\cdot\\mathbf{e}_\\text{x} & \\mathbf{e}_\\text{y}\\cdot\\mathbf{e}_\\text{y} & \\mathbf{e}_\\text{y}\\cdot\\mathbf{e}_\\text{z} \\\\\n\\mathbf{e}_\\text{z}\\cdot\\mathbf{e}_\\text{x} & \\mathbf{e}_\\text{z}\\cdot\\mathbf{e}_\\text{y} & \\mathbf{e}_\\text{z}\\cdot\\mathbf{e}_\\text{z} \\\\\n\\end{pmatrix} = \\begin{pmatrix} \n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{pmatrix}", "ce3c6e1c082505df2e444e2a18d196cc": "Tr\\left\\{ \\sigma\\right\\} \\leq1", "ce3c7c93fa56f39f6f9bc51106cc43a5": "h(p,u)", "ce3ca817b7c1435036d1f94e4e076412": "r'_1", "ce3cc7da80148137795e76691ed15408": "\\scriptstyle \\frac{1}{c+1}", "ce3ce6ea1f2eb01930265a678f6b5bb0": " e \\ ", "ce3cfcf584d7d7af53084facd02e4c5e": "\\textrm{pH} = \\textrm{p}K_\\textrm{a} + \\log_{10} \\left ( \\frac{[\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "ce3d04ef1e9fb8fbefef338f1a2e6e50": "\\int (x+y) \\, dx = \\frac{x^2}{2} + yx", "ce3d3597bf4dc27616c34c37986b46e8": "\\begin{align}\n\\int_{S_1}&\\int_{S_2}|f(x,y)\\,g(x,y)|\\,\\mu_2(\\mathrm{d}y)\\,\\mu_1(\\mathrm{d}x)\\\\\n&\\le\\biggl(\\int_{S_1}\\int_{S_2}|f(x,y)|^p\\,\\mu_2(\\mathrm{d}y)\\,\\mu_1(\\mathrm{d}x)\\biggr)^{\\!1/p\\;}\\biggl(\\int_{S_1}\\int_{S_2}|g(x,y)|^q\\,\\mu_2(\\mathrm{d}y)\\,\\mu_1(\\mathrm{d}x)\\biggr)^{\\!1/q}.\\end{align}", "ce3d76912d607ba866cc2899bd55dae9": "D_\\max = \\begin{cases}\n\\min(p_1q_1,\\,p_2q_2) & \\text{when } D < 0\\\\\n\\min(p_1q_2,\\,p_2q_1) & \\text{when } D > 0\n\\end{cases} ", "ce3d774f399d429f5e179da9504d0f19": "z_i \\,", "ce3d934b74cd9ec43641c6a82618be69": "\n -k_0^2 +\\vec k^2 + k_D^2 -{m \\over T_e} k_0^2\n=0\n,", "ce3db4c3fc58486637d0956595c87e48": "\n I_1 = \\mathrm{tr}(\\boldsymbol{B}) = 3 + \\gamma^2\n ", "ce3dc645fa4b8fce232a11667495fb32": " \\mathbf{F}_A = \\int_\\Sigma (-\\Delta p \\mathbf{n} + \\mathbf{f}) \\,d\\sigma ", "ce3dcb981c4553361dae0d64522a7f92": "\\frac{V_{Th}}{R_{Th}} = I_{No}\\!", "ce3dddff847b4f400ea4041e529e380c": "\\mathbf{Z}/p_1^{n_1}\\mathbf{Z} \\ \\times \\ \\mathbf{Z}/p_2^{n_2}\\mathbf{Z} \\ \\times \\ \\cdots \\ \\times \\ \\mathbf{Z}/p_k^{n_k}\\mathbf{Z}", "ce3dee85c718769cab3e57bb4985bee7": " W(C;1,1) = \\sum_{w=0}^{n}A_{w}=|C| ", "ce3dfaa14027c879a575bb1bd26d7243": "(h * h_{inv}) (n) = \\sum_{k=-\\infty}^{\\infty} h(k) \\, h_{inv} (n-k) = \\delta (n)", "ce3e1b14fce516c575a90fcf8dd28a41": "\\left| \\Psi_{l}^{k}{}^\\prime \\right\\rangle", "ce3ebf2f993b323055a25448105aa2d3": "v_p = \\sqrt { \\frac{2kT}{m} } = \\sqrt { \\frac{2RT}{M} }", "ce3ed4bf9f3fc07b3d7b1f0753adebf6": " z^{-1}\\delta\\left(\\frac{y-x}{z}\\right) (a \\otimes_x(b \\otimes_y c)) - z^{-1}\\delta\\left(\\frac{-y+x}{z}\\right)(b \\otimes_y(a \\otimes_x c)) = y^{-1}\\delta\\left(\\frac{x+z}{y}\\right)((a\\otimes_z b)\\otimes_y c)", "ce3efa189764b990ce563c27c0c875ef": "\\text{mplus} \\colon A^{?} \\to A^{?} \\to A^{?} = a_1 \\mapsto a_2 \\mapsto \\begin{cases} \\text{Nothing} & \\text{if} \\ a_1 = \\text{Nothing} \\and a_2 = \\text{Nothing}\\\\ \\text{Just} \\, a'_2 & \\text{if} \\ a_1 = \\text{Nothing} \\and a_2 = \\text{Just} \\, a'_2 \\\\ \\text{Just} \\, a'_1 & \\text{if} \\ a_1 = \\text{Just} \\, a'_1 \\end{cases}", "ce3f4714b337442601f58dc1f3769015": "z=\\frac{\\overline{x}-\\mu_0}{\\sigma}\\sqrt n", "ce3f5d52767e68571ca890c59374e4ca": "t_{down}", "ce3fc7fd9b6030eb5856fe68df9baa43": "\\begin{align}\n\\zeta_f(t) & = \\exp \\left ( \\sum_{n=1}^\\infty \\frac{2t^{2n+1}}{2n+1} \\right ) \\\\\n&=\\exp \\left ( \\left \\{2\\sum_{n=1}^\\infty \\frac{t^n}{n}\\right \\} -\\left \\{2 \\sum_{n=1}^\\infty\\frac{t^{2n}}{2n}\\right \\} \\right ) \\\\\n&=\\exp \\left(-2\\log(1-t)+\\log(1-t^2)\\right)\\\\\n&=\\frac{1-t^2}{(1-t)^2} \\\\\n&=\\frac{1+t}{1-t}\n\\end{align}", "ce3fd6652de6a8c75071fa54501e2958": "|{\\psi}\\rangle = \\sqrt{1-\\epsilon_n}|{\\psi_{D}}\\rangle + \\sqrt{\\epsilon_n}|{\\psi^{\\bot}_{D}}\\rangle", "ce3ff68579729e165ee17fe483be0398": "\\mathrm{ker}(A^\\mathrm{T}) = (\\mathrm{im}(A))^\\perp", "ce40173cf4f27817f72acb9129e4b420": " \\rho_{dc}^\\alpha\\,\\rho_{s0}^\\alpha/8 \\simeq 4.4\\,T_c ", "ce4031855e58064e847d96b212035d9c": " \\overline{E}_\\text{k} = \\begin{matrix} \\frac 1 2 \\end{matrix} kT,", "ce403b3cdf92c6f919359c279364347f": "< 0", "ce4040881d26723b84af8a9fb2f13015": "^{b_j}M_{S_j}", "ce405a269067444261aed7788a1c1f4e": " \\frac{\\part u}{\\part n} + \\sigma u + g =0, \\,", "ce406affae639b05ccc2c26e8d296bc8": "dw/dx", "ce4073af58ed65dad80415d8ce504515": "E^{0'}", "ce40866b9032b80278190a2a62a2ecf5": "h_{t_1}(t) - h_{t_2}(t)", "ce40937fdfbd06b8a15244e102a09356": " f ", "ce40da2938fba90078978dacbfbcdbeb": "pq=2r\\left(r+\\sqrt{4R^2+r^2}\\right).", "ce4107a8bcbe758a4c01847f56bc254b": " \\mu = GM_\\oplus = ( 398 600.4418 \\pm 0.0008 ) \\ \\mbox{km}^{3} \\ \\mbox{s}^{-2}.", "ce410b290e84d5f11d0e906aad59cb4d": "\\sum_{n=1}^\\infty\\frac{1}{|z_n|^{p+1}}", "ce4114af7c5ce1382b04e9fc1b676217": "(2\\epsilon\\Delta q)\\,\\!", "ce411ed6e46218c2224379818f29567d": "k \\geq \\lceil (1 - H(p + \\epsilon)n) \\rceil", "ce4140fa3aef4b6f5b5484fcdc3cd77c": "V^2_R = \\frac{\\big(\\frac{V_n}{V_0}\\big)^\\frac{P}{n} - 1}{\\sqrt{\\frac{\\sum_{i=0}^{n}{\\big(\\frac{V_i}{V_i^p}-1\\big)^2}}{n}}+1}", "ce4176dbd9644823d81a0811b6a92d65": "\\Delta_\\alpha(u) = \\frac{u-s_\\alpha u}{1-e^{-\\alpha}}", "ce4178f59bc5ab91e3abd04e5daf4041": " M < N_{\\rm min} ", "ce418137949b4019a2496ed8248e208b": " \\vec v \\cdot \\vec \\nabla \\vec v = \\vec \\nabla (\\tfrac{1}{2} \\vec v \\cdot \\vec v) - \\vec v \\times \\vec \\omega ", "ce41855be50e71bcd69d60e47a2e9565": " : \\hat{b} \\hat{b}^\\dagger : \\;=\\; : 1 + \\hat{b}^\\dagger \\hat{b} : \\;=\\; : 1 : + : \\hat{b}^\\dagger \\hat{b} : \\;=\\; \n1 + \\hat{b}^\\dagger \\hat{b} \\ne \\hat{b}^\\dagger \\hat{b}.", "ce418ce5121fcd7bbe9dd86c7bf0308b": "p(\\lambda)=\\det(\\lambda I_2-A)=\\det\\begin{pmatrix}\\lambda-1&-2\\\\\n-3&\\lambda-4\\end{pmatrix}=(\\lambda-1)(\\lambda-4)-(-2)(-3)=\\lambda^2-5\\lambda-2.", "ce41d01046c92e363de9d53f2396cbae": "H(M) = \\log |M|.", "ce41e1471c9307b7bb38b8fb6bfb52e0": " a_1 - a_2 = S_2 \\leq S_{2m} < S_{2m+1} \\leq S_1 = a_1. ", "ce4260cc7de783cada704c01fced5ae3": "\\mathrm{Pr}[ |T_j \\cap T_k| > \\frac{2t}{d^2}] \\leq e^{\\frac{-t}{3d^2}} = e^{-2\\log n} \\leq n^{-4} [", "ce429c65b62799156994a0669b7dc54c": "\\ddot\\theta \\to 0", "ce42aa8e89a8c7cd4e1e22e56fa1403f": "\\star \\mathrm{d}t=\\mathrm{d}x\\wedge \\mathrm{d}y \\wedge\\mathrm{d}z", "ce42aedafd36ce4ec031838ee7a920d6": "\\, \\begin{bmatrix} T_{r\\overline{o}} & T_{ro}\\end{bmatrix}", "ce4368e139868f243bb3ec1a53f6bc34": "\\sum_{n=1}^\\infty2^{-n}\\xi_n,", "ce43b8aa1ed5467b4cb3dd4a1bdc8429": "\n\\begin{bmatrix}\n\\boldsymbol{A}_1 & \\boldsymbol{B}_1\\\\\n\\boldsymbol{C}_2 & \\boldsymbol{A}_2 & \\boldsymbol{B}_2\\\\\n& \\ddots & \\ddots & \\ddots\\\\\n& & \\boldsymbol{C}_{p-1} & \\boldsymbol{A}_{p-1} & \\boldsymbol{B}_{p-1}\\\\\n& & & \\boldsymbol{C}_p & \\boldsymbol{A}_p\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1\\\\\n\\boldsymbol{X}_2\\\\\n\\vdots\\\\\n\\boldsymbol{X}_{p-1}\\\\\n\\boldsymbol{X}_p\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{F}_1\\\\\n\\boldsymbol{F}_2\\\\\n\\vdots\\\\\n\\boldsymbol{F}_{p-1}\\\\\n\\boldsymbol{F}_p\n\\end{bmatrix}.\n", "ce43c08517faec73e58b74665c13e793": "\\C^{n\\times n}", "ce43f9d9fc5b8454fe637735e28b6685": "\\vert A_i \\rangle_\\mathcal{S}", "ce4423ba4c342a9862bb86120f52b447": "\\phi:X \\to Y", "ce443d8149791bfc93ad6203cbcea9df": "\\mathrm{D}_{H} F (\\sigma) := \\mathrm{D} F (\\sigma) \\circ i : H \\to \\mathbb{R}, ", "ce444e5c89369e903653160b391dd321": "d z = \\left [ {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial y} \\right )}_z + {\\left ( \\frac{\\partial z}{\\partial y} \\right )}_x \\right ] d y + {\\left ( \\frac{\\partial z}{\\partial x} \\right )}_y {\\left ( \\frac{\\partial x}{\\partial z} \\right )}_y dz,", "ce447b6d8773f12793a3d6d0869566be": "2c_{44} - c_{11} + c_{12} ", "ce4482bfb79a0232aa92c93dd218ded6": "c_2 = 0.185212, \\,\\!", "ce452b46c90d0a8c78d30551ffa087ba": "A=r^2\\cdot\\frac{2}{1}\\cdot\\frac{2}{\\sqrt{2}}\\cdot\\frac{2}{\\sqrt{2+\\sqrt{2}}}=4r^2\\sqrt{2-\\sqrt{2}}.", "ce454ac5d4cd2e029e0ce14182d5ed6d": "E=\\frac{(20\\frac{ft^2}{s})^2}{2(32.2\\frac{ft}{s^2})(4.4ft)^2}+4.4ft=4.7 ft", "ce4555ff72424049691b30d9f4c3cc2e": "\\{y\\in\\mathbb{R} \\mid y=f(x)\\}", "ce458af4d89b1bbf99add3c144a3aa8e": "t\\partial/\\partial t.", "ce458c42249ea1e1360349b19e8e920a": "\\, I^+(x)", "ce458f356808feaf248a5441045d8f5d": " \\alpha n R (T_2 - T_1) = \\alpha n R T_1 \\left( \\left( \\frac{P_2}{P_1} \\right)^{\\frac{\\gamma-1}{\\gamma}} - 1 \\right) ", "ce45dd5bf4bb15e5c7d61d61a11fc2cc": "R_0 = \\frac{a}{\\mu+a}\\frac{\\beta}{\\mu+\\nu}.", "ce4671191e8eba7ba783a9bf7723a5b5": " \\Delta\\tau = \\cfrac{r_{\\rm{particle}}}{l_{\\rm{interparticle}}} \\gamma_{\\rm{particle-matrix}} ", "ce46b3903461eeda92e4930a72ba3cac": "C_{ij}^b", "ce46c67192720316e6f07fb26353ba3a": "a_1 x_1 + a_2 x_2 + \\cdots + a_n x_n = b,", "ce46fb56963d17c56072818d6828d03b": "| \\psi _{\\sigma} \\rangle", "ce47328e118bc0619f4b4bb118515149": "\nf(r,\\theta)= \\left\\| (x,y) \\right\\| \\sin\\left( 2 \\tan^{-1}\\left( \\frac{y}{x} \\right)\\right)\n", "ce474b56c3f37be9a7d0886fced2c866": "\\mathrm{\\frac{3}{2} O_2 + 3\\ H_2O + 6\\ e^- \\to 6\\ OH^-}", "ce4754048c9538f9d95237ef88424e21": "=1/2+\\epsilon_1+\\epsilon_2+2\\epsilon_1\\epsilon_2-1/2-\\epsilon_1-1/2-\\epsilon_2+1\\ ", "ce4755427fdfa9b83487269d75e295d0": " t \\approx \\frac{69.3}{r} \\times \\frac{200}{200-r}", "ce478b17642c7330a6f7e5504757f2b9": "e^{-i 2\\pi \\frac{kn}{N}}\\,", "ce4795891ed1b2100e8814d7f8df80e4": "\\exp(-2\\alpha l)", "ce47bd8243beb2377534783f1a815674": "\\frac{d\\mathbf a}{dt} = \\sum_{r=1}^{n}\\frac{\\partial \\mathbf a}{\\partial q_r} \\frac{dq_r}{dt} + \\frac{\\partial \\mathbf a}{\\partial t}.", "ce47c906d7b97d60d3362a417a71dc17": "\\operatorname{prob}(\\psi \\Rightarrow \\varphi) = |\\lang \\psi |\\varphi \\rang|^2 = |\\sum_i\\psi^*_i \\varphi_i |^2\n", "ce47ff4d0f1c73d90b500dc6b692b7a0": " H(3,q^2) :\\ s=q^2,t=q", "ce4806278c95b074fc2620cae9449b7f": "\\begin{align}\nk[X^k] g^n & =k\\mathrm{Res}\\left(g^n X^{-k-1} \\right) =k\\mathrm{Res}\\left(X^n f^{-k-1}f\\,'\\right) =-\\mathrm{Res}\\left(X^n (f^{-k})'\\right) \\\\[6pt]\n& =\\mathrm{Res}\\left(\\left(X^n\\right)' f^{-k}\\right) =n\\mathrm{Res}\\left(X^{n-1}f^{-k}\\right) =n[X^{-n}]f^{-k}.\n\\end{align}", "ce483da9b2c674c32bc474c81e030896": "A(T,V,\\{N_j\\})=U-TS\\,", "ce4862653173f670ae6e68ce101eb9a3": "\\liminf_{n\\to\\infty} J(x_n) \\geq J(y)", "ce489dcaf38db9a041c3808557fd9ff1": " \\frac{P_f}{P_f^{a}P_m^{1-a}}=\\theta", "ce48bc4ce048552823637bd288da76a6": "x = -0.270845 \\,", "ce48bdeae19c8230db522e03be9b54d5": "\\tfrac{vSPI\\ *\\ hSPI\\ *\\ area\\ *\\ color~depth~encoding}{8}", "ce491860f2ff820ccb99d3acafefe996": "\\scriptstyle f_s,", "ce499dea30cfce118f4fe85da0227e83": "T1", "ce49a691f3f5b48d071ba61fbbc38fd1": "T_ij", "ce49bd38cf02ba33ae8575e77cba367b": "|z| = a", "ce49fbcbb896499dd7fa0c9375e33b85": " \\Gamma=dx^\\lambda\\otimes (\\partial_\\lambda + \\Gamma_\\lambda^m\\partial_m). ", "ce4a29855a8c75c33a3add7c0847f393": "\\;_k\\psi_k \\left[\\begin{matrix} \na_1 & a_2 & \\ldots & a_k \\\\ \nb_1 & b_2 & \\ldots & b_k \\end{matrix} \n; q,z \\right] = \\sum_{n=-\\infty}^\\infty \n\\frac {(a_1, a_2, \\ldots, a_k;q)_n} {(b_1, b_2, \\ldots, b_k;q)_n} z^n.", "ce4a356864d88485e3fe6b021431df53": "\\mathbf{F}_\\mathrm{rad} = -\\frac{\\mu_0 q^2 r^2}{24 \\pi c^3} \\frac{d^3 \\mathbf{a}}{dt^3}", "ce4a71b28f01b12c0188329548f701dc": "N = N^{T} > 0", "ce4a9d6069d4d11cb85bbdea4e7cdde3": "\\Delta x = \\frac{b-a}{n}.", "ce4aa4879bdec0297eb928d86ad1f12d": "\\sin x^2\\ ", "ce4accc4932f08bd894f2b5f9efe61e0": " f_X(x|\\theta) = h(x) g(\\theta) \\exp \\left ( \\eta(\\theta) \\cdot T(x) \\right )", "ce4af912ee8d06183e4d954e010ca795": "\\nabla (\\vec u \\cdot \\vec v) = (\\vec u \\cdot \\nabla) \\vec v + (\\vec v \\cdot \\nabla) \\vec u + \\vec u \\times (\\nabla \\times \\vec v) + \\vec v \\times (\\nabla \\times \\vec u)", "ce4b3ed27f4e8ce82a4b7d8de9dc1edb": " v^2 = v_x^2+v_y^2+v_z^2\\,.", "ce4b8cf1c09ed13630a22bdbc6ccad7e": "A^* = \\{\\lambda\\} + A + AA + AAA + \\cdots,", "ce4bb795ce2831abd51270f2b77845f6": "[M^{\\mu\\nu},M^{\\rho\\sigma}] = i(\\eta^{\\sigma\\mu}M^{\\rho\\nu} + \\eta^{\\nu\\sigma}M^{\\mu\\rho} - \\eta^{\\rho\\mu}M^{\\sigma\\nu} -\\eta^{\\nu\\rho}M^{\\mu\\sigma})", "ce4bc3e4d6bb3d84211758628964b316": "L^{B \\otimes_A C/C} \\cong B \\otimes_A L^{C/A},", "ce4d1a8f1cdaaf9222958fd74cfef7d9": "\\alpha_n", "ce4d996cbbc0f863f66ebbe1d4df69ce": "\n{d\\over dt} Q_A = {d\\over dt} \\int_x e^{-x^2\\over 2A^2} J^0(x) = -\\int_x e^{-x^2\\over 2A^2} \\nabla \\cdot J = \\int_x \\nabla(e^{-x^2\\over 2A^2}) \\cdot J ~ ,", "ce4daba2647ebc12a166d79a9b9eabfe": "1 - \\sin(\\theta)", "ce4db739797ab78000f464bdbf84a6f1": "y^2 = x^2(x+1)", "ce4e31285551f7a92cd1dd2b521b88bb": "\\ \\nu=1+e", "ce4e8b547adde875606a71a8640626e8": "{\\rm QJSD}(\\rho_1,\\ldots,\\rho_n)= S\\left(\\sum_{i=1}^n \\pi_i \\rho_i\\right)-\\sum_{i=1}^n \\pi_i S(\\rho_i)", "ce4eb514b611905a5a8ae704994be1fc": "\\rho_i = \\rho_i^* \\frac{V_i}{V}\\,", "ce4ed71163786cfcf090fd64accc5b14": "(x\\Rightarrow y)=\\sup\\{z\\mid z*x\\le y\\}.", "ce4ee668d8a633349add3bfa82912f97": "a \\in \\mathbb{R} \\cup \\{ + \\infty \\}", "ce4efade2a36a3e948c380f74f20ab4a": "p(s,x,t,y) \\, dy := P(W^+_t \\in dy \\mid W^+_s = x)", "ce4f0252a4232856a088707d9ad4f92f": "c_d = a_d, c_{d-1} = a_{d-1} c_d = a _{d-1} \\cdot a_d, \\ldots , c_2 = a _2 \\cdot c_3 = a _2 a_3 \\cdots a_d", "ce4f52ebf4d043fe6f1ba190aba8e0e7": "D_x(t,f)=\\min\\left\\{|G_x(t,f)|^2,|W_x(t,f)|\\right\\}", "ce4f7669990da84199cafeea11426f71": "C(s) \\,", "ce4fd39e7f2299438775cbac24b43fde": " W_{k} ", "ce4fd8e8a8d209f859c8160606742f0a": "\\lim_{n \\to \\infty} P(|X_n| \\geq \\varepsilon) = 0,", "ce4fe7c08a83a989e9e94b1038f14043": "d_p \\neq \\infty", "ce501dd42cf404e318a4df2a83f2ffb8": "c_{u+v}=c_u c_v - c_v^p c_{u-v} + c_{u-2v} \\text{ for } u, v \\in \\mathbb{Z}", "ce502222b7ec3a13ab2df50855420616": "c=-1/\\omega", "ce502ef1d7bb3f3202862ec163213f48": "\\pi_l", "ce503d662ea05baa28c9aad983c2c407": "\\forall N \\in V: \\exists w \\in \\Sigma^*: N \\stackrel{*}{\\Rightarrow} w", "ce5074edf9331dc2f3eff7a9eb0b274a": "{\\color{white}.}\\qquad\n\\cos\\alpha\\,ds = \\rho\\,d\\phi = - dR/\\sin\\phi, \\quad\n\\sin\\alpha\\,ds = R\\,d\\lambda,", "ce507fb2d0888bddc1150ebde869152b": "s_X^2, s_Y^2", "ce50800152ffa8cc6097db36805039af": " \\iint_F f(x,\\ y) dA = \\int_c^d \\ \\int_{r(y)}^{s(y)} f(x,\\ y)\\, dx\\ dy \\ . ", "ce50918f759cb4b1b4d3392c3bdc67d0": "Y=Y_0 + \\frac {\\partial Y}{\\partial \\beta} \\beta +\\frac {\\partial Y}{\\partial r}r", "ce50d629db04655ee6b5b7aa6a2feec4": "e = r_a/a - 1\\,", "ce510a1b04d2c4de6622d806779e2365": "(f \\circ g \\circ h)'(a) = f'((g \\circ h)(a))\\cdot (g \\circ h)'(a) = f'((g \\circ h)(a))\\cdot g'(h(a))\\cdot h'(a).", "ce510aaac24a872a27413ee036961e75": "z\\,\\operatorname{erf}(z) + \\frac{e^{-z^2}}{\\sqrt{\\pi}}.", "ce511b9b767bc3bf2af3e3bb294d0572": "c_q(n)=\n\\mu\\left(\\frac{q}{(q, n)}\\right)\n\\frac{\\phi(q)}{\\phi\\left(\\frac{q}{(q, n)}\\right)}\n.\n", "ce514e73fb98d525406b083886502c65": "V\\otimes_A V,", "ce5171fc49f112e4936759bb814a46ae": " M_\\bigodot", "ce51d8b5dcb28f5f08d1b66cdff76190": "\\frac{d^2x^\\lambda }{dt^2} + \\Gamma^{\\lambda}_{\\mu \\nu }\\frac{dx^\\mu }{dt}\\frac{dx^\\nu }{dt} = 0,", "ce51f9a5d56d77f28ec55041b90f288a": "\\mathbf{r} = \\hat{\\mathbf{e}}\\theta", "ce5232b6644d12f9fb477da3ecbc5c8f": "f(-x)=a(-x)^3+b(-x)^2+c(-x)+d = -ax^3+bx^2-cx+d \\equiv g(x). ", "ce524f4eca32c4efc17dd344c2a4f81f": "C_1\\,", "ce5256f355d978826603a5098d5cbb37": " \\forall s \\forall t \\exist r[tN", "ce605238b6be2f2897be7fd59accb21c": "\\epsilon = 0", "ce60838ecd990b335453032579a67961": "\\mathrm{wt}_2(c)=2", "ce60add3e49725824afc27d12c30bcad": " {^*\\mathbb{N}} \\setminus \\mathbb{N}\\,", "ce613d8d89730c0a44fd6cc7643bc0b6": "S \\times \\{0\\}", "ce61ccd4b5ac798327be1657f39e51ab": "T, P", "ce61e4bc5dd2ca8b7a3a03e0a65e768b": " Z(\\lambda_1,\\ldots, \\lambda_m) = \\sum_{i=1}^n \\exp\\left[\\lambda_1 f_1(x_i) + \\cdots + \\lambda_m f_m(x_i)\\right],", "ce6238f99e41192e99a63caacaff039f": "\\textstyle H(x)", "ce6263be9d2040d238f3e64e870ceb7b": "Q = \\chi x_c, \\ t = \\tau t_c, \\ x_c = C V_0, \\ t_c = RC, \\ F = V.", "ce6268825d705d888d282a63f65a0465": "\n l_x = \\int_0^1 \\left| \\cfrac{d \\mathbf{X}}{d s}\\cdot\\boldsymbol{C} \\cdot\\cfrac{d \\mathbf{X}}{d s} \\right|~ds\n", "ce627bd00e98a731731daff60defacaa": "b\\, \\mathrm{XOR}\\, k = c", "ce62b869f587a485b382676bf75225bd": " 1 ", "ce62d50d8f62b0fcfd6598a4b8609f8f": "P(R)", "ce631639104959b042c5b66a2b9912da": "\\begin{align}\nX_q(s) & = \\int_{0^-}^\\infty x_q(t) e^{-s t} \\,dt \\\\\n& = \\int_{0^-}^\\infty \\sum_{n=0}^\\infty x[n] \\delta(t - n T) e^{-s t} \\, dt \\\\\n& = \\sum_{n=0}^\\infty x[n] \\int_{0^-}^\\infty \\delta(t - n T) e^{-s t} \\, dt \\\\\n& = \\sum_{n=0}^\\infty x[n] e^{-n s T}.\n\\end{align}", "ce63956b59c42ae125b325837922d0ef": " 3.046 mm/s \\le Rate \\le 5.331 mm/s~~~~~~~~~~~~(2) ", "ce643be915be8af2a9ecbf2c1d9846f0": "V_{ACDA(s)}", "ce657b7b70cfa14546a7443031a72b0f": "\\sum_i q_{ij}=1\\,", "ce657f9bf3677862625619e5b39ed1db": "\\ f(C,H) = (231.3*n_{C} + 52.1*n_{H})", "ce6583bfa6706c45b61167f3077395fa": "\\frac{23+12}{2}-10=7.5", "ce659420caa033416546f43ac4b7e168": "\n\\exp \\left (-a (x^2+p^2)\\right ) ~ \\star ~ \\exp \\left (-b (x^2+p^2)\\right ) = {1\\over 1+\\hbar^2 ab} \n\\exp \\left (-{a+b\\over 1+\\hbar^2 ab} (x^2+p^2)\\right ) .\n", "ce6595870b1ee200627434779955769e": " 1- 1/n", "ce65b600c2140eddda3a6004a2af2699": "\\frac{\\partial \\rho uv }{\\partial x} + \\frac{\\partial \\rho vv }{\\partial y} = \\frac{\\partial \\frac{ \\nu \\partial v} { \\partial x}}{\\partial x} + \\frac{\\partial \\frac{ \\nu \\partial v} { \\partial y}}{\\partial y} - \\frac{\\partial p}{\\partial y} + S_v ", "ce65edf7f54bc6aff53489a11b092391": "y_{i}^{\\prime} = D_\\text{in}(y_i), \\quad i \\in (0, N)", "ce660a2bca3acd6d54c886382b64e907": "y=x \\cdot w", "ce6618fc02a7c007e6d361dc05c2db3b": "TC(F(\\R^m,n))=\\begin{cases} 2n-1 & \\mathrm{for\\,\\, {\\it m}\\,\\, odd} \\\\ 2n-2 & \\mathrm{for\\,\\, {\\it m}\\,\\, even.} \\end{cases}", "ce662f0e87606fda25f1f8693abfdce0": "\n\\begin{align}\nL_0^{(\\alpha)} (x) & = 1 \\\\\nL_1^{(\\alpha)}(x) & = -x + \\alpha +1 \\\\\nL_2^{(\\alpha)}(x) & = \\frac{x^2}{2} - (\\alpha + 2)x + \\frac{(\\alpha+2)(\\alpha+1)}{2} \\\\\nL_3^{(\\alpha)}(x) & = \\frac{-x^3}{6} + \\frac{(\\alpha+3)x^2}{2} - \\frac{(\\alpha+2)(\\alpha+3)x}{2}\n+ \\frac{(\\alpha+1)(\\alpha+2)(\\alpha+3)}{6}\n\\end{align}\n", "ce665053fb1141e7da7771f963a10fd5": " U_\\text{A} = U(\\lambda = 0), U_\\text{B} = U(\\lambda = 1), ", "ce66b051fd3a6e6f21f56b793b04311a": "\\mathcal G^+(1,3) \\cong \\mathcal G(3,0)", "ce67085ed4236364f2489edb87f80f89": "c_S", "ce671b03c95dea2cef8b788fb84a9259": "c = \\frac{r P}{1-(1+r)^{-N}} = \\frac {Pr(1+r)^N}{(1+r)^N-1}.", "ce671e634d3efedb6a6e11f554b8ba1a": "\\lambda =\\frac{\\lambda _{o}}{\\sqrt{\\varepsilon _{r}}} ", "ce67a1bcff03db9438ef9f43a5b67599": "b = 2mn+n^2, \\, ", "ce67e4f88fcbcbcbe1925c3b14d9cca2": "L_b", "ce67f3828a5f97dcf5af6d7da3982df2": "\n\\begin{array}{rcl}\n\\Pr(z_{dn}=k\\mid\\mathbb{Z}^{(-dn)},w_{dn}=v,\\mathbb{W}^{(-dn)},\\boldsymbol\\alpha)\\ &\\propto\\ &\\bigl(\\#\\mathbb{Z}_k^{d,(-dn)} + \\alpha_k\\bigr) \\dfrac{\\#\\mathbb{W}_v^{k,(-dn)} + \\beta_v}{\\sum_{v'=1}^{V} (\\#\\mathbb{W}_{v'}^{k,(-dn)} + \\beta_{v'})} \\\\\n&& \\\\\n&=& \\bigl(\\#\\mathbb{Z}_k^{d,(-dn)} + \\alpha_k\\bigr) \\dfrac{\\#\\mathbb{W}_v^{k,(-dn)} + \\beta_v}{\\#\\mathbb{W}^{k} + B - 1}\n\\end{array}\n", "ce6843b0f109392f14bae136642a114c": "P (\\varepsilon _i) = \\frac{1}{Z} g_i e^{- \\varepsilon_i / kT}", "ce685409d8d054fde9f688c320722c2e": "\\overset{\\cdot}{x}_{n-1}(t) =x_{n}(t)", "ce687af8164c899ada713041613bc651": "\\sin\\gamma = \\frac{c}{b} \\sin\\beta.", "ce68a58b1e95d2872f61a63a41dd0fbd": "LE\\to LB", "ce68d13c8806c2db8b22c66dd7a9c667": "f_{x,y} =\n \\frac{1}{4}\n \\sum_{u=0}^7\n \\sum_{v=0}^7\n \\alpha(u) \\alpha(v) F_{u,v}\n \\cos \\left[\\frac{(2x+1)u\\pi}{16} \\right]\n \\cos \\left[\\frac{(2y+1)v\\pi}{16} \\right]\n", "ce68da2043a7df05c7421e80809bdaa7": "\\sigma(\\omega)", "ce68f78072fa0c22d47ed8ae7ba82a00": "|g\\cap {\\mathcal Q}|=2", "ce692237f49e1e3644e9fd611c11d03e": "2^{|k|-b}+2^{|k|-2b}+2^{|k|-3b}+2^{|k|-4b}", "ce69a3607766516b30447d5492886a1f": " Y_i ", "ce69db5f9442d29745436efb0f785eca": "x (v_1, w_1) = (x v_1, w_1)", "ce6a6da8a5b3903970a5957a6904bd49": "\n\\frac{\\beta }{z}\\,\\,\\, = \\,\\,\\,\\frac{1}{{2n}}\\left[ {\\alpha \\left( {\\alpha - 1} \\right)\\left( {\\frac{{\\sigma _1 }}{{\\mu _1 }}} \\right)^2 + \\,\\,\\,\\beta \\left( {\\beta - 1} \\right)\\left( {\\frac{{\\sigma _2 }}{{\\mu _2 }}} \\right)^2 \\,\\, + \\,\\,\\,\\,2\\,\\alpha \\,\\beta \\left( {\\frac{{\\sigma _{1,2} }}{{\\mu _1 \\,\\mu _2 }}} \\right)\\,} \\right]", "ce6a849d248a32aa3122d75ab4102952": " e_a := e(f_a) := e^\\mu_a \\partial_\\mu.", "ce6ab06cefc02e83a996f712883916ea": " \\dot{x} = \\phi(x) = A x + b,\\,", "ce6ade12eda1d147cda0ad5dc24b3c12": "a=b=.1", "ce6af588bc762286bb27fa19eb2f3adf": "y(t) = A_c\\sin\\left(\\omega_\\mathrm{c}t + m(t) + \\phi_\\mathrm{c}\\right).", "ce6b029de7da53981eb753476ae2a274": "(h_{\\mathrm{o}})_k = h_{2k+1}", "ce6b6d066b33112ab8557069dbee2837": "a_y=0", "ce6bc9339f80a24e1ee2858f941befbb": "L(G) = \\{ w \\in (N \\cup T)^{*} : S \\Rightarrow_{p_1} w_1 \\Rightarrow_{p_2} ... \\Rightarrow_{p_n} w \\}", "ce6bd9a3a0eedfaa3176e9f59542f362": "\\mathbf{J}_\\mathrm{M}=\\nabla\\times\\mathbf{M} ", "ce6c07cbcc40d465f64768b018603a89": "[x,y^{-1},z]=1=[y,z^{-1},x]", "ce6c501773ac30cac18fbb2a5a9cedc4": "\\Phi(A) = \\operatorname{Tr}A - A.", "ce6d2ca556d15e7f7f566dc5e9d93f6e": "u_y\\,\\!", "ce6d2f868791f7d6ee5424c7865c70fe": "M(n) > (4 - 15^{1/2})^{1/2} n", "ce6e0d53935dbd40acef6f92feea1802": "\\min_{d\\in D}\\max_{s\\in S} dist(d,s)", "ce6ecead801b35e25fa483787f924405": "\\phi = \\left[k(\\omega_0) x - \\omega_0 t\\right] + \\frac{1}{2} x k''(\\omega_0) (\\omega - \\omega_0)^2 + \\cdots", "ce6efce6537259c38f56a49319fb4a1c": " H(u,v) = {a^2 + b^2 + {4 u^2 \\over a^2} + {4 v^2 \\over b^2} \\over a^2 b^2 \\left(1 + {4 u^2 \\over a^4} + {4 v^2 \\over b^4}\\right)^{3/2}} ", "ce6f17fed3af9afd501a9c0e328642aa": "B_{k+1}^{-1} = B_k^{-1} + \\frac{(\\mathbf{s}_k^{\\mathrm{T}}\\mathbf{y}_k+\\mathbf{y}_k^{\\mathrm{T}} B_k^{-1} \\mathbf{y}_k)(\\mathbf{s}_k \\mathbf{s}_k^{\\mathrm{T}})}{(\\mathbf{s}_k^{\\mathrm{T}} \\mathbf{y}_k)^2} - \\frac{B_k^{-1} \\mathbf{y}_k \\mathbf{s}_k^{\\mathrm{T}} + \\mathbf{s}_k \\mathbf{y}_k^{\\mathrm{T}}B_k^{-1}}{\\mathbf{s}_k^{\\mathrm{T}} \\mathbf{y}_k}.", "ce6f1f18e6fd3a7746a4dd0dee71ea3d": "m\\ or\\ \\mu >= 0.5", "ce6f23bafe7f5a383de0a4d10415ad08": "\\frac{(2m + 1)\\#}{(m + 1)\\#} = \\prod_{p > m + 1}^{p \\leq 2m + 1} p \\leq \\binom{2m+1}{m} < 4^m.", "ce6f54086d46a933ab5910258775fc9b": "y_k[n] = \\textrm{IFFT}\\left(\\textrm{FFT}\\left(x_k[n]\\right)\\cdot\\textrm{FFT}\\left(h[n]\\right)\\right)", "ce6f5642f50d7dd1a053491a13defa3b": "\\Phi(\\lambda x) = \\lambda \\Phi(x)", "ce704e13589b5d13f44669aaad3f9588": "r\\left\\{\\begin{array}{l}p\\\\q\\\\q\\end{array}\\right\\}", "ce70c9c8929acabc78a0e3212d3dd4f8": "S \\rarr \\left(A \\times S \\right)^{?}", "ce70d2fc63574a1a361e7cc770025e51": "\\alpha, \\beta, \\gamma", "ce714f0f021c94656b59ff171db674c1": " \\boldsymbol\\mu = \\operatorname{E}(\\mathbf{Y}) = \\nabla A(\\boldsymbol\\theta). \\,\\!", "ce717a34e37a9ce59e2ba1a182ae4ba7": "\\ln (-1) = i\\pi.\\,\\!", "ce718bde333e860b6a6e17d0467d9b1f": "\\Gamma^i{}_{ki}=\\frac{1}{2} g^{im}\\frac{\\partial g_{im}}{\\partial x^k}=\\frac{1}{2g} \\frac{\\partial g}{\\partial x^k} = \\frac{\\partial \\log \\sqrt{|g|}}{\\partial x^k} \\ ", "ce71fa62bc15b4e756ebc98787dfbe4c": "C^{k,\\alpha}(\\overline{\\Omega})", "ce7208d6dedef9fd0675d8b1b8ca3bba": "v=\\frac{y}{B}", "ce72a16f7eb4b5b260d41499c6f3ebc9": "W^{1,p}(\\mathbf{R}^n) \\subseteq L^{p^*}(\\mathbf{R}^n)", "ce7310afb54fb0c9f01f8a4a2af544ed": "(10)\\;\\;\\quad L=r-M\\;,\\quad z=(r-M)\\cos\\theta\\;,\\quad \\rho=(r-M)\\sin\\theta\\;,", "ce733389d3581200c55adf06f282914a": "\\mbox{tr}^2\\mathfrak{H}", "ce73d34797b8406ede2ae480695d853c": "F^{\\#}(z) = \\overline{F(\\bar z)}", "ce742d4148cda1e92dd0246be4e08914": "{\\it phagocytosis}", "ce7447f54d7f7c94ee619f40ce53b4fd": "\\cos \\gamma =\\cos \\left( {{z}_{\\text{max}}}-90{}^\\circ \\right)=\\sin {{z}_{\\text{max}}} \\,.", "ce746d2e09070b466e85ce51a2d3a951": "q = UA \\Delta T_{LM}", "ce74ffd304bab98d457198067f3bc745": "Q^{c}", "ce7503963a0d5830db1e23e17c4e8ef0": " \\hat\\beta = \\Bigg( \\sum_{t=1}^T \\hat\\sigma_t^{-2}x_{(t)}x'_{(t)} \\Bigg)^{-1} \\sum_{t=1}^T \\hat\\sigma_t^{-2}x_{(t)}\\Phi^{-1}(\\hat{p}_t) ", "ce750e44fb866646459d04f16baf1cef": " \\xi = \\sum_n \\exp\\left[ - \\frac{\\varepsilon(n)}{kT} \\right].", "ce753e06e0cd1804f1431c2aa2ec0fac": "\\operatorname{ad}_{[x,y]}=[\\operatorname{ad}_x,\\operatorname{ad}_y].", "ce75558ea969dc0e0f4848f576eb7b8d": "{C}", "ce757d6ce6d604e0927df960e13e0800": " \\gamma =\\frac {\\tau_T} {\\frac{1}{2}\\tau_L}", "ce75baeb1ab0a62aa36d01911e213c1c": "\n\\begin{matrix}\n\\\\\n\\lambda_1 \\begin{bmatrix}1\\\\4\\\\2\\\\-3\\end{bmatrix}+\n\\lambda_2 \\begin{bmatrix}7\\\\10\\\\-4\\\\-1\\end{bmatrix}+\n\\lambda_3 \\begin{bmatrix}-2\\\\1\\\\5\\\\-4\\end{bmatrix}=\n \\begin{bmatrix}0\\\\0\\\\0\\\\0\\end{bmatrix}.\n\\end{matrix}\n", "ce762f209258eec616ca07cfffcf8942": "p_n(z)=\\frac{z}{n} {{z-\\beta n -1} \\choose {n-1}}", "ce765f37c1f06125389bc8ab6ef73415": "\\sqrt{(16/9)\\times (4/3)} \\approx 1.5396 \\approx 13.8:9 ", "ce7674106ab493dd7c9ee393549c5127": " (\\;6) \\quad\\quad \\frac{\\partial w}{\\partial t}+\\frac{\\partial}{\\partial x}f\\left(w\\right)=0.", "ce76a3db6b3e63c0b78737449c523252": "K_d = \\frac{[L][R]}{[L\\! \\cdot \\!R]}", "ce770ce5c494d1a9b7df4bb3849f7325": "\n\\mathcal{G}(\\tau,\\tau') = \\mathcal{G}(\\tau - \\tau').\n", "ce77136c1d91d945a1df60a184a9a357": "Q_{R}(r)", "ce774d9cab3ae0bdf522cd0839bed364": "SI", "ce77680be0dd3fce47ac8fd13bfac397": "M^2 _\\beta \\equiv \\overline{D} \\times \\frac {\\beta_B} {\\beta_D} + \\overline{R_F}", "ce77a6a6729dc6d1d635287040c43b82": "(m_i \\circ n_j) (m_i \\circ n_j)^T", "ce77b0cc05090fd03a6092d0a3c82e6e": "dl^2= \\left (a^2l_{\\alpha}l_{\\beta}+b^2m_{\\alpha}m_{\\beta}+c^2n_{\\alpha}n_{\\beta} \\right )dx^{\\alpha}dx^{\\beta}", "ce77ee3e75f0d0cff1aff3b9d80ee820": "e^\\gamma\\cdot\\log(y)/\\log(2).", "ce77f362d7dee31a92fc235789c2b328": "(k,X) \\mapsto k\\cdot \\mathrm{exp}(X)", "ce78043d4333a3002b089eb75ab3bb8c": "\\textstyle\\frac{\\ln 4}{\\ln 2}=2", "ce7814100e8bda9a0934b0ef5c0f84d3": "A_{i+j}=A_i \\sum_{k=0}^{2^j-1} (I-A A_i)^k.", "ce781a89bb745633231a815a741b67c3": "h(d)=\n\\begin{cases}\n\\dfrac{w \\sqrt{|d|}}{2 \\pi} L(1,\\chi), & d < 0; \\\\\n\\dfrac{\\sqrt{d}}{\\ln \\epsilon} L(1,\\chi), & d > 0.\n\\end{cases}", "ce78277f1e979c164828638b6156b4e5": "N = mp", "ce785edf5c393e0b45f63f3801e8a85d": "R_{\\mathrm{g}}^{2}", "ce786ece197502576aab49952b939bfe": "(x^\\lambda,y^i)", "ce78c885210e132ec98c3ef46aa0e58d": " | \\langle \\phi | U_g | \\phi \\rangle |^2 = \\frac{1}{d+1} \\ \\forall g \\neq id ", "ce78ee081997f1ba2b06a77b8af8bc52": "\n\\begin{matrix}\n\n\\begin{array}{|c||c|c|c|}\n\\hline\n x & f=\\Delta^0 & \\Delta^1 & \\Delta^2 \\\\\n\\hline\n1&\\underline{2}& & \\\\\n & &\\underline{0}& \\\\\n2&2& &\\underline{2} \\\\\n & &2& \\\\\n3&4& & \\\\\n\\hline\n\\end{array}\n\n& \n\n\\quad \\begin{matrix}\nf(x)=\\Delta^0 \\cdot 1 +\\Delta^1 \\cdot \\dfrac{(x-x_0)_1}{1!} + \\Delta^2 \\cdot \\dfrac{(x-x_0)_2}{2!} \\quad (x_0=1)\\\\\n \\\\\n=2 \\cdot 1 + 0 \\cdot \\dfrac{x-1}{1} + 2 \\cdot \\dfrac{(x-1)(x-2)}{2} \\\\\n \\\\\n=2 + (x-1)(x-2) \\\\\n\\end{matrix}\n\\end{matrix}\n", "ce7928aa8782ca0039269c6f77edced2": "\\begin{align}\n T''_n(1) & = \\frac{n}{2} \\lim_{x \\to 1} \\frac{\\frac{d}{dx}(n T_n - x U_{n - 1})}{\\frac{d}{dx}(x - 1)} \\\\\n & = \\frac{n}{2} \\lim_{x \\to 1} \\frac{d}{dx}(n T_n - x U_{n - 1}) \\\\\n & = \\frac{n}{2} \\lim_{x \\to 1} \\left(n^2 U_{n - 1} - U_{n - 1} - x \\frac{d}{dx}(U_{n - 1})\\right) \\\\\n & = \\frac{n}{2} \\left(n^2 U_{n - 1}(1) - U_{n - 1}(1) - \\lim_{x \\to 1} x \\frac{d}{dx}(U_{n - 1})\\right) \\\\\n & = \\frac{n^4}{2} - \\frac{n^2}{2} - \\frac{1}{2} \\lim_{x \\to 1} \\frac{d}{dx}(n U_{n - 1}) \\\\\n & = \\frac{n^4}{2} - \\frac{n^2}{2} - \\frac{T''_n(1)}{2} \\\\\n T''_n(1) & = \\frac{n^4 - n^2}{3}. \\\\\n\\end{align}", "ce79622a02f00f29e52601ef8404844e": "\nc_{\\mathrm{air}} = 331.3 \\ \\mathrm{\\frac{m}{s}} \\sqrt{1+\\frac{\\vartheta^{\\circ}\\mathrm{C}}{273.15\\;^{\\circ}\\mathrm{C}}}\\,\n", "ce7965da2d8297ce74f2036721771ddb": " T(p_n) = \\Box p_n ", "ce797a7f648ea3c5abbfd8b7469f1265": "g^{\\alpha \\beta}g_{\\beta \\gamma} + g^{\\alpha 0}g_{0 \\gamma} = \\delta_\\gamma^\\alpha,", "ce7a253e924cb2530ac7108b9fc1a965": "\\tfrac{d}{\\epsilon}\\ln\\tfrac{1}{\\epsilon}+\\tfrac{2d}{\\epsilon}\\ln\\ln\\tfrac{1}{\\epsilon}+\\tfrac{6d}{\\epsilon}", "ce7a28a5af80c2f14b2f246ae5d15ffc": "\\tau_\\mathrm{w} \\equiv \\tau(y=0)= \\mu \\left.\\frac{\\partial u}{\\partial y}\\right|_{y = 0}~~.", "ce7a45c876b761e0428e5cba076b2b72": "G={\\rm GF}(q^{n+2})^*/{\\rm GF}(q)^*", "ce7a70bbe5d12f00bed8638f4e59ebeb": "\\hat n", "ce7b130384d6d7ba54b6c13528292f15": "\\gamma a_i \\equiv \\zeta_n^{b(i)}a_{\\pi(i)} \\pmod{\\mathfrak{p}}, \n", "ce7b3f0cc198a7099b98f96bcfef52c7": "P+\\textstyle \\sum_{j={m+1}}^k P_j", "ce7b46f5df17747d24837fc4c37d899c": " \\hat{f}_{-}(k,y)=\\int_{-\\infty}^{0} f(x,y)e^{-ikx}\\textrm{d}x. ", "ce7b6d0462cf5f7b9632b3b3649f3c63": " \\textbf{x}_{k\\mid k-1}^{a} = [ \\hat{\\textbf{x}}_{k\\mid k-1}^{T} \\quad E[\\textbf{v}_{k}^{T}] \\ ]^{T} ", "ce7bfad281cb8ce08aaeaf7226550c1f": "SL_2(p)", "ce7ca6f75bc4b4d5524f2e64b3c81de3": "\\gamma_{0}", "ce7ced2bf7a521468674b2f5ea0cf278": "z(n,t) < (t-1)^{1/t} n^{2-1/t} + \\frac{1}{2}(t-1)n.", "ce7cfcab8e3a6e4ff574e13623666a4d": "\\frac{1}{g(x)}", "ce7d2c577f24afa797307d4eafcb3145": "-\\cos x \\, ", "ce7d368cbd13c686c70d311068b3b134": "S_a(Tr(g))=\\left(Tr(g^{a-1}),Tr(g^a),Tr(g^{a+1})\\right)\\in GF(p^2)^3", "ce7d62159152ab6144afb5a03606402c": "17^{10}\\ \\equiv\\ 1\\ \\equiv\\ 1 \\pmod {71}.", "ce7db640ab61fe1f3f422644d13a0da7": "F(\\rho,\\sigma)", "ce7db70ddbc5b14c447e81a8d2aa8838": "\\langle p_i|\\tilde{\\phi}_j\\rangle=\\delta_{ij}", "ce7dfdc79d25ded20ffe214fc562ee17": "P = K G (W - W_e)", "ce7e445c21cc34398f274bc242d58d1d": "B_\\text{max}", "ce7f2e54b16d5611000092c588657c7e": "\n \\begin{align} f(k|n,\\alpha,\\beta) & = \\int_0^1 L(k|p)\\pi(p|\\alpha, \\beta) \\, dp \\\\\n\n & = {n\\choose k}\\frac{1}\n {\\mathrm{B}(\\alpha,\\beta)}\n \n \\int_0^1 p^{k+\\alpha-1}(1-p)^{n-k+\\beta-1} \\, dp \\\\\n & = {n\\choose k}\\frac{\\mathrm{B}(k+\\alpha,n-k+\\beta)} {\\mathrm{B}(\\alpha,\\beta)}. \n \\end{align}\n", "ce7f3c2379b9f39fbd1b6ec30b7e7a88": "\\epsilon_{f}", "ce7f3de44962a0c72f0bbf5ffa2ef8f0": "\\begin{pmatrix}\n0 & i\\\\\ni & 0\n\\end{pmatrix}", "ce7f885b7ccae52d2b265fab06dc97ae": "a_2=\\frac{2}{x}-\\frac{2x-2}{x^2-2x+\\frac{3}{2}}", "ce7f90e468726be01462b6700226e588": "z\\colon I \\to \\mathbb{R}", "ce7f9a8ec4c2b62f9c35a0d4ab5698d6": "\\xi(t)", "ce800b39790fb32f9dac5a0a346cc6d0": "(j+1)P", "ce801ce34a046b7d19eb372bf048b282": "\\frac{\\Gamma(5/\\beta)\\Gamma(1/\\beta)}{\\Gamma(3/\\beta)^2}-3", "ce8081df7a54e804d2e185cf327e1869": "\nV = \\frac{\\Phi_0}{2\\pi}\\dot{\\phi} \n= \\frac{\\Phi_0}{2\\pi}(\\underbrace{\\dot{\\phi_0}}_{=0} + \\dot{\\delta\\phi})\n= \\frac{\\Phi_0}{2\\pi} \\frac{\\dot{\\delta I}}{I_c \\cos(\\phi_0)}.\n", "ce8095258669ef13cd8e27a46dedf6b9": "O(N e^\\sqrt{8\\ln(5/2)\\ln N})", "ce8113c7f7ed1b24ebad8f0e89ae5589": "\\frac{1}{k!} \n\\begin{vmatrix} \\operatorname{tr}A & k-1 &0&\\cdots\\\\\n\\operatorname{tr}A^2 &\\operatorname{tr}A& k-2 &\\cdots\\\\\n \\cdots & \\cdots & \\cdots & \\cdots \\\\\n\\operatorname{tr}A^{k-1} &\\operatorname{tr}A^{k-2}& \\cdots& 1 \\\\ \n\\operatorname{tr}A^k &\\operatorname{tr}A^{k-1}& \\cdots& \\operatorname{tr}A \\\\ \\end{vmatrix} ~.", "ce8157339e877aaff07e70a0e38fa813": "e^{{\\beta}}", "ce815c16cfda28efb5516880890cada4": "N\\trianglelefteq G", "ce816acd130f3902f1fc59aa07ae1e81": "\n\\psi(\\vec{\\theta}) = \\frac{2 D_{ds}}{D_d D_s c^2} \\int \\Phi(D_d\\vec{\\theta},z) dz.\n", "ce8170dccbbcfda705fba6cbb9a07fb4": "\\delta_{ext} ", "ce817b2081ed43c738379edf1547ae1b": "z = z_2:", "ce81bc9159d3d31cc906cf4ec12c376a": "A/A(\\Phi)", "ce81df90c3d2b2962b8385ceb99cfdfb": "\\vdots\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots", "ce81efe4fc650b432c8bbd254fd3c08d": "\\displaystyle{\\mu=\\psi\\mu + (1-\\psi) \\mu=\\mu_0 + \\mu_\\infty.}", "ce81ff139f7e563ad630ec8c94d07b79": "{\\mathcal E}^\\star(M)", "ce821cb315ea3db341cf0297c979f434": "K_{max}", "ce824ca14c6eead4e709de34a7176bd3": "\\,\nS = \\sqrt{ V^2 + U^2 - V^2 U^2 }\n", "ce82f8992fdb919492f87103f9b00656": "Y_l^m(\\mathbf{r})", "ce834c8ad42b5cebeb2926b5e85d88e8": "\\textstyle p \\mid k", "ce836a2ff5947260c8ca793ec167ec90": "{1 \\over 2} m\\overline{v^2}", "ce8401d571a0867d72e4b2ddc926eaf2": "\\mathrm{vol}\\,B(x,r)\\geq Cr^d", "ce841d8f34a9a01bde9541b31578295a": "H(f,g) = \\begin{bmatrix}\n0 & \\dfrac{\\partial g}{\\partial x_1} & \\dfrac{\\partial g}{\\partial x_2} & \\cdots & \\dfrac{\\partial g}{\\partial x_n} \\\\[2.2ex]\n\\dfrac{\\partial g}{\\partial x_1} & \\dfrac{\\partial^2 f}{\\partial x_1^2} & \\dfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_n} \\\\[2.2ex]\n\\dfrac{\\partial g}{\\partial x_2} & \\dfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_1} & \\dfrac{\\partial^2 f}{\\partial x_2^2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_n} \\\\[2.2ex]\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\[2.2ex]\n\\dfrac{\\partial g}{\\partial x_n} & \\dfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_1} & \\dfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_n^2}\n\\end{bmatrix}", "ce84221d03d6dbf9cd316a95d55ed5f7": "S(\\mathbf{x}) = 0", "ce844c1ac68a71526ad7ec51b73f9baa": "r_i = a\\, \\frac{\\varphi^2}{2 \\sqrt{3-\\varphi}}", "ce84702c9342a6c6b5baa0810554bc85": "\\left(\\!\\tbinom{n}{k}\\!\\right)", "ce84caf6edc1ccca02796ca2d74b105e": "Z_{n+1} = \\sum_{i=1}^{Z_n} X_{n,i}", "ce84f6c8ca1f395fce402ba343b7b73c": "B^\\lambda", "ce84fcc906f5b1c584eaf142f8686f8e": "{A,T,G}", "ce856d6eb1e3404ac4d0a37c788b486b": "\n\\begin{array}{rcl}\nf(x) &=& \\dfrac{1}{x} \\cdot \\left(\\frac{\n\\begin{array}{l}\n1 + 7.44437068161936700618 \\cdot 10^2 \\cdot x^{-2} + 1.96396372895146869801 \\cdot 10^5 \\cdot x^{-4} + 2.37750310125431834034 \\cdot 10^7 \\cdot x^{-6} \\\\\n~~~ + 1.43073403821274636888 \\cdot 10^9 \\cdot x^{-8} + 4.33736238870432522765 \\cdot 10^{10} \\cdot x^{-10} + 6.40533830574022022911 \\cdot 10^{11} \\cdot x^{-12} \\\\\n~~~ + 4.20968180571076940208 \\cdot 10^{12} \\cdot x^{-14} + 1.00795182980368574617 \\cdot 10^{13} \\cdot x^{-16} + 4.94816688199951963482 \\cdot 10^{12} \\cdot x^{-18} \\\\\n~~~ - 4.94701168645415959931 \\cdot 10^{11} \\cdot x^{-20}\n\\end{array}\n}{\n\\begin{array}{l}\n1 + 7.46437068161927678031 \\cdot 10^2 \\cdot x^{-2} + 1.97865247031583951450 \\cdot 10^5 \\cdot x^{-4} + 2.41535670165126845144 \\cdot 10^7 \\cdot x^{-6} \\\\\n~~~ + 1.47478952192985464958 \\cdot 10^9 \\cdot x^{-8} + 4.58595115847765779830 \\cdot 10^{10} \\cdot x^{-10} + 7.08501308149515401563 \\cdot 10^{11} \\cdot x^{-12} \\\\\n~~~ + 5.06084464593475076774 \\cdot 10^{12} \\cdot x^{-14} + 1.43468549171581016479 \\cdot 10^{13} \\cdot x^{-16} + 1.11535493509914254097 \\cdot 10^{13} \\cdot x^{-18}\n\\end{array}\n}\n\\right) \\\\\n& &\\\\\ng(x) &=& \\dfrac{1}{x^2} \\cdot \\left(\\frac{\n\\begin{array}{l}\n1 + 8.1359520115168615 \\cdot 10^2 \\cdot x^{-2} + 2.35239181626478200 \\cdot 10^5 \\cdot x^{-4} +3.12557570795778731 \\cdot 10^7 \\cdot x^{-6} \\\\\n~~~ + 2.06297595146763354 \\cdot 10^9 \\cdot x^{-8} + 6.83052205423625007 \\cdot 10^{10} \\cdot x^{-10} + 1.09049528450362786 \\cdot 10^{12} \\cdot x^{-12} \\\\\n~~~ + 7.57664583257834349 \\cdot 10^{12} \\cdot x^{-14} + 1.81004487464664575 \\cdot 10^{13} \\cdot x^{-16} + 6.43291613143049485 \\cdot 10^{12} \\cdot x^{-18} \\\\\n~~~ - 1.36517137670871689 \\cdot 10^{12} \\cdot x^{-20}\n\\end{array}\n}{\n\\begin{array}{l}\n1 + 8.19595201151451564 \\cdot 10^2 \\cdot x^{-2} + 2.40036752835578777 \\cdot 10^5 \\cdot x^{-4} + 3.26026661647090822 \\cdot 10^7 \\cdot x^{-6} \\\\\n~~~ + 2.23355543278099360 \\cdot 10^9 \\cdot x^{-8} + 7.87465017341829930 \\cdot 10^{10} \\cdot x^{-10} + 1.39866710696414565 \\cdot 10^{12} \\cdot x^{-12} \\\\\n~~~ + 1.17164723371736605 \\cdot 10^{13} \\cdot x^{-14} + 4.01839087307656620 \\cdot 10^{13} \\cdot x^{-16} + 3.99653257887490811 \\cdot 10^{13} \\cdot x^{-18}\n\\end{array}\n}\n\\right) \\\\\n\\end{array}\n", "ce85f1210fe03efd72aaf36d18d71024": "p(m_i)", "ce85fa42937401b558e2f0b33cf91d5c": "Z^A", "ce8625475f063e569fcc38072c488fb5": "\\ \\varepsilon ", "ce86333a843a4f27432d6697398cdd8e": "\\alpha(a, \\, b)", "ce86b64297b3a895360f3749381bd783": "K(k) \\approx \\frac {\\pi} {2} + \\frac {\\pi} {8} \\frac {k^2} {1-k^2} - \\frac {\\pi} {16} \\frac {k^4} {1-k^2}", "ce86e3ba8adb71ccf0dd7c230a032436": "\n\\delta\\theta\\approx \\sqrt{20m/M_{12}}.\n", "ce870a145d96c3075cece25b53735a97": " \\ C_c", "ce876c99cedf6f2595ab0f33fe47b26e": " \\left\\vert A_d \\right\\vert = \\frac{1}{f(d)} X + R_d . ", "ce87a3c3096955b04636f6ff71330d4e": "c_0 + c_1X_1 + c_2X_2", "ce87eb5067d402e1c34cce889ff02e6a": "e[{\\mathcal{M}}]=\\frac{1}{2} \\int_{\\mathcal{M}} H^2\\, \\mathrm{d}A", "ce87f350ff037ee24650d6fc9a31edd3": "F_1 = F_2 \\,", "ce87f3d46b38c2f9188411b056623a06": "\\vec{\\Omega }", "ce885a2288c3e663f053b3767239ba01": "{ p_1/(1-p_1) \\over p_2/(1-p_2)}={ p_1/q_1 \\over p_2/q_2}=\\frac{\\;p_1q_2\\;}{\\;p_2q_1\\;},", "ce8869d5c618378392b05e23758c8749": "Q_n^j", "ce886ff446eba75e99483157ab90d823": "(\\rho \\mathbf {u} \\phi)_r - (\\rho \\mathbf {u} \\phi)_l \\,= \\left( \\frac{\\Gamma}{\\delta x} \\delta \\phi\\right)_r - \\left( \\frac{\\Gamma}{\\delta x} \\delta \\phi\\right)_l.", "ce887ead0ae051d7ba3fb89833aa23ec": "M_\\perp=H\\chi_\\perp\\sin{\\theta}", "ce88c9baf72b6c51398d249cac8ff669": "s_{pm}^-", "ce894d06de8b12e354c3a9e2cfe2be35": " \\frac {dy}{d\\mathrm{net}} = y(1-y) ", "ce899c9f1c27bf0a7afb4cab7e4f62f6": "\\frac {1{,}302{,}540} {2{,}598{,}960} \\approx 50.11\\%", "ce89bcf711a5239d600e078028eda022": "m_1 - m_2 = \\Delta m", "ce89cdddfdec02f6ea5a0b60c56a543b": "Tx \\geq 0", "ce89cea239dac237aa0b913e0e4b67a9": "| f' (x^*) | < 1. \\, ", "ce89dcc669c1f4af753c23ef4d731e94": "{m_\\text{P}}", "ce89fb153ad9b543533da3241d728d92": "C=\\frac{\\kappa\\varepsilon_{0}A}{t} ", "ce8a454a1f7cf604725cc327b3c1a893": "V = ( xBe ^{..r^2} + 1) [ - G(m/r)]", "ce8a4e21bccec774ed29894d67f8112d": "F_j=\\sum_{i=0}^{N-1}f_i\\alpha^{ij}, 0\\le j\\le N-1, ", "ce8beff4b067de2e393c719654020ddb": " x_N \\rightarrow x", "ce8c0e5c29d863aa9f118b35597560dd": "N_{i,n}", "ce8c118f41733c48815dbdf351fdb081": "\\vec V = v^j \\frac{\\partial \\vec\\Psi}{\\partial x^j}\\quad", "ce8c3ddd5b2134b148aed5ed5a9acd3d": "\\psi^{\\langle1\\rangle}=\\psi_{SS}", "ce8c43cc52f804e706a30c16388f04df": "e_3", "ce8cef15290693ec9a69551e82246f7b": "R = 2\\cdot X\\cdot Z", "ce8d0ead0c3a9733c9525349d485e2c6": "c_0 = ", "ce8d204c5fa6ed352a6a3f9f017b4007": "t_{prt}", "ce8e98272306e098bc5c5e7d24d7eddb": "R \\subseteq S \\times S ", "ce8ed5111d6e3c52ee5667c302cc7283": "\\begin{align}\n \\text{Side }A & =2m_1 n_1 & & = 2(4x+5)\\text{ }(x+2) & & = 8x^2+26x+20 \\\\\n \\text{Side }B & =m_1^2-n_1^2 & & = (4x+5)^2-(x+2)^2 & & = 15x^2+36x+21 \\\\\n \\text{Side }C & =m_1^2+n_1^2 & & = (4x+5)^2+(x+2)^2 & & = 17x^2+44x+29\n\\end{align}", "ce8f103b35cdf010d75682b87f318a09": "COP = \\Iota \\cdot COP_{c}", "ce8f7f0f0b0200252f108430d3d586e2": "\\int_{{0}}^{{2 \\pi}}\\sin^{2m+1}{x}\\ cos^{2n+1}{x}\\;\\mathrm{d}x = 0 \\! \\qquad \\{n,m\\} \\in \\mathbb{Z}", "ce8fe7e520459c88b22866d32fb21a8c": "(A_0, A_1)_{\\theta, q} = (X_0, X_1)_{\\eta, q}, \\ \\ \\text{where} \\ \\ \\eta = (1 - \\theta) \\theta_0 + \\theta \\, \\theta_1.", "ce905b702cb7c5bbe657b3e7e11e703e": "\n\\frac{\\tan(\\beta d / 2)} {\\tan(\\alpha d / 2)} = - \\frac\n{4 \\alpha \\beta k^2}\n{(k^2 - \\beta^2)^2}\\ \\quad \\quad \\quad \\quad (3)\n", "ce90adce975ba31e74a11e207f9da8af": "\\ \\delta \\varepsilon=\\frac{\\delta \\ell}{\\ell}", "ce90e4edd9a53fa847d3d3e57ae61544": "s \\cdot (s' \\otimes m) = ss' \\otimes m", "ce910333422b44d081c995a32027d27c": "\\int\\frac{1}{x^2-a^2} \\, dx = \\begin{cases} \\displaystyle -\\frac{1}{a}\\,\\operatorname{arctanh}\\frac{x}{a} = \\frac{1}{2a}\\ln\\frac{a-x}{a+x} + C & \\text{(for }|x| < |a|\\mbox{)} \\\\[12pt] \\displaystyle -\\frac{1}{a}\\,\\operatorname{arccoth}\\frac{x}{a} = \\frac{1}{2a}\\ln\\frac{x-a}{x+a} + C & \\text{(for }|x| > |a| \\mbox{)} \\end{cases}", "ce910711ee548d7aa3c99e3a7dc8c544": "\\hat F[\\{\\phi_j\\}](1) = \\hat H^{\\text{core}}(1)+\\sum_{j=1}^{N/2}[2\\hat J_j(1)-\\hat K_j(1)]", "ce9135d05a24f5c61017f9164af25382": " \\tan\\delta = \\frac{\\text{ESR}}{\\left|X_c\\right|} = \\text{DF} ", "ce91415f9a389c11f7b55e166ecf3d50": "x \\in \\mathbb{R}^N ", "ce914177a994af3437e0be408be2900c": "\\mathbf{a}\\times\\boldsymbol{\\nabla}\\psi=\\psi\\left(\\boldsymbol{\\nabla}\\times\\mathbf{a}\\right)-\\boldsymbol{\\nabla}\\times\\left(\\psi\\mathbf{a}\\right),", "ce918e1539afe19f7c2e6b2e7c1d0232": "A = \\sum_{m=0}^\\infty a_m\\,", "ce9192029c34dccf384f413758d1f99f": " \\sqrt{x\\cdot y} < M_{\\text{lm}}(x,y) < \\frac{x+y}{2} \\qquad \\text{ for all } x>0 \\text{ and } y>0.", "ce919ee5f6a31d4f6c40e8bc4d1d6305": "X\\setminus C_n=U_n\\subset U_{n+1}=X\\setminus C_{n+1},\\forall n", "ce91a254a0c9d7a2b6ff4d95de60e667": "\\Delta f_{\\mathrm{pred}}/\\Delta f_{\\mathrm{actual}}", "ce91cdfca68db9f1c693dcc48c686516": "\\,x_1 0.\\ ", "cec7bda9ec77b24388affba7d52ea5ec": "y_1,\\ldots,y_n", "cec7f336aa0432e86dfbab911ab0937c": "f = \\begin{bmatrix} f_1 \\\\ \\vdots \\\\ f_n \\end{bmatrix} \\in C(X)", "cec803c61b84351f4a05ab699f9eaf0b": " \\xi = \\begin{Bmatrix} 0\\\\v \\end{Bmatrix}.", "cec8154ca4a62bf956a9969ae0eed656": "\\left[\\begin{matrix}\n\\varepsilon_{xx} & \\varepsilon_{xy} & \\varepsilon_{xz} \\\\\n \\varepsilon_{yx} & \\varepsilon_{yy} & \\varepsilon_{yz} \\\\\n \\varepsilon_{zx} & \\varepsilon_{zy} & \\varepsilon_{zz} \\\\\n \\end{matrix}\\right] = \\left[\\begin{matrix}\n\\varepsilon_{xx} & \\gamma_{xy}/2 & \\gamma_{xz}/2 \\\\\n \\gamma_{yx}/2 & \\varepsilon_{yy} & \\gamma_{yz}/2 \\\\\n \\gamma_{zx}/2 & \\gamma_{zy}/2 & \\varepsilon_{zz} \\\\\n \\end{matrix}\\right]\\,\\!", "cec82cb0c5ea63c044519c9b3f6464f3": "(\\pm P_i,\\pm P_{i+1})", "cec88d45def44aa23acb6be598b63042": "i : G \\rightarrow G", "cec8f1621c705bbf3b501494cbf97006": "V_U = V_S {AX \\over AB} ", "cec96f553d40cfb7b12f10a03fe9169c": "\\neg A \\or B\n", "cec98ba3d3b4da631b82b03ebae161d1": "ABCD=1110", "cec999ea132aa076da43c7529aece26c": "{HV}_{[i,n]}=\\sum_{k=0}^{n-i}\\frac{div(i+k)}{{(1+r)}^{n-i-k}}", "cec99fd11d3d7c7f528f530d2a2d6f70": " \nL(\\mathcal{G},\\infty)= \\{\\omega \\in \\underset{t \\rightarrow \\infty} \\lim\n\\Omega_{Z,[0,t]}: \\exists \\{q: (q_0, \\omega, q) \\in\n\\Delta, q_0 \\in Q_0 \\} \\subseteq Q_A \\}. \n", "cec9b0ff6e4e1e85124789dd0c83c31b": " dG = \\sum_{i=1}^n \\mu_i dN_i = \\mu_1dN_1 + \\mu_2dN_2 + ...\\,", "cec9b8e518d6e8d85787e1d29919c93d": "\\sin^2\\theta = \\frac{1 - \\cos 2\\theta}{2}\\!", "cec9c3abbf4933d2220f2bf1d5ab153e": "A^{2-} + 2H^+ \\rightleftharpoons H_2A; \\beta_2=\\frac{[H_2A]}{[H^+]^2[A^{2-}]}", "cec9d6cb59fa7bc5a85e03fa87c51f53": "n \\in \\mathbb{N}_0", "cec9d7d8e036b588a7d1213345b90b9c": "\\scriptstyle\\Phi\\,", "cec9e24b2d2dae8f624607e85e45817d": "\\bar{X}/(\\hat{\\sigma}/\\sqrt{n})", "ceca0808d172f7a4b8597a1e4e2c5974": "\n{\\mathcal L}_g=-\\frac{1}{16\\pi G}R\\sqrt{-g},\n", "ceca10c7ba9f0c285f5ecfe2c93af714": "S' / D' = \\tan \\theta\\,", "ceca119f226a27521479cfc8a02336e2": "\\lambda \\wedge \\theta \\in \\bigwedge^n V", "ceca40d63e2309a0cb820cabd8a7f591": "f(x,y,p)", "ceca62e6de10235f6d84f7ca92f7ea06": "\\Pr(P_i|X,{\\mathbf r}) \\propto \\Pr({\\mathbf r}|P_i,X)\\Pr(P_i|X)", "cecaa1a21c6506f52d235d649cc0167d": "\\lambda=", "cecac065569bd1b84e4c70a9f59cb866": " \\hat{f}_2^{(i)} = S_2 \\sum_{\\alpha = 0}^{i-1}(S_1 S_2)^\\alpha(I-S_1)Y + S_2(S_1 S_2)^{i -1} S_1\\hat{f}_2^{(0)} ", "cecaf36ddba17ddc102edca8745961d1": "I=\\epsilon v < \\mathbf{E}^2 >_T)", "cecb530d3eb343fad822a5a0afa150f3": "\\tau=i", "cecb616f42266be49f30faf5b3cff15c": "V(x) = \\max_{U_t \\geq 0} \\;E\\left[ \\int_0^\\infty e^{-\\rho t}(\\pi X_t-U_t^2)\\,dt\\right],", "cecb75bf23a7dc687d324568101ebc73": "(\\pm 1,0),(0,\\pm 1)", "cecb7c6521dfd57cc9a8642dfa619765": "\\, e_+ ", "cecb8ef7dc18e02ff14f06c02521ceaf": " F_\\text{P} = \\frac{m_\\text{P} l_\\text{P}}{t_\\text{P}^2} = G \\frac{m_\\text{P}^2}{l_\\text{P}^2} ", "cecbd931bdac3974daf7b7711c4f8cb4": "b = \\frac {p}{\\sqrt{e^2-1}} = a \\cdot \\sqrt{e^2-1}", "cecca0f535f8869fc8ecd66fc33e1625": "w = \\emptyset", "cecd1f33736d521882496fea00801007": " h(X,Z) = -d_X\\Delta", "cecd4d5f7606cf79329f129a7a4f0974": " \\hat{\\mathbf{n}} ", "cecdac8242cfd1849255ed7b2addef9c": "\n\\frac{d}{dt} \\mathbf{x}(t) = \\mathbf{A} \\mathbf{x}(t).\n", "cecdd3fa9c26a95a71a56d2094c865ce": "\\int\\ln ax\\;dx = x\\ln ax - x", "cecde297e9defa5a105630ed2df22c11": "{T}_0", "cece081f0dc6f908655fb9da3fe71bdd": "N_{m+1}", "cece1736cfe56049e9a41854549f8871": "\\frac{z+z^2+z^3+z^6}{1-z}.", "cece846b66701bded7a549a18c494fc1": "\\scriptstyle N_r \\times 1", "cece9867e362ca5ca5bd6301e001b2b1": "(8.b)\\quad \\nabla^2 \\psi =0 ", "cecee6f3665abe04830e25e81ec4e14b": "\\textstyle \\text{Slope}_{\\text{right}} = f(x_i + h, y_i + h f(x_i, y_i))", "ceceebc8fcb157cbafbaf57480164eb9": "T(op_x, op \\circ \\overline{op})=op_x", "cecf06cf219e9b14368b4e11dcf47a3f": "e_1, \\ldots, e_m ", "cecf133d8445c948dbeb1c096835ba8b": " \\phi(x) = B(x) + \\eta (x)", "cecf235fd6bfc23f8678ff30761f5101": " \\begin{align}\n\\mathbf{q} & = q_r + q_i i + q_j j + q_k k \\\\\n\\theta &= 2 \\cos^{-1} q_r = 2 \\sin^{-1} \\sqrt{q_i^2 + q_j^2 + q_k^2} \\\\\n(a_x, a_y, a_z) &= \\frac{1}{\\sin \\tfrac{1}{2} \\theta} (q_i, q_j, q_k)\n\\end{align}\n", "cecf2403e25ad541f2e7468a4947de34": "\\underset{x\\in[-5,5], \\; y\\in\\mathbb R}{\\operatorname{arg\\,max}} \\; x\\cos(y),", "cecf28ff44c6c4346a19ce307388ce7a": "e=\\frac{c}{\\mu}", "cecf396d998e1fbe77f588b68942d1ff": "\\mathrm{d}G = -S \\mathrm{d}T + V \\mathrm{d}P.", "cecf3c0e12f1ba59d4d9254a3689e752": "\\Gamma^\\alpha_{\\beta\\gamma}", "cecf49703b6039cf51c2998b0e11b428": "h = \\mathcal{L}_X g = 2c g", "cecf5d6f41c00a870baaa2e3696295e9": "\\tfrac{T}{100} = \\tfrac{4000}{100} = 40", "cecf71677167a0e56ad08107360288ec": "D\\subseteq X \\,\\!", "cecf9c083a173e0feb800d371c41680a": "f(t) = a_0 + a_1 (t - t_i) + a_2(t - t_i)^2 + \\cdots + a_h(t - t_i)^{h_i} \\, ", "cecfcb55926159b57b53c5d15e8ef37e": "\\tfrac34+\\tfrac{5}{12}=\\tfrac{9}{12}+\\tfrac{5}{12}=\\tfrac{14}{12}=\\tfrac76=1\\tfrac16", "ced00c44601f25097a0ab5bf89258c51": " \\exists x \\, P(x) \\equiv \\neg \\forall x \\, \\neg P(x). ", "ced0ec40d3bd59a58b3a67e95c8b64e8": "\\bar s_i = \\sqrt \\frac{\\sum_{j=1}^n \\left ( x_{ij} - \\bar x_i \\right )^2 }{n - 1}", "ced1462ebda1fea8d368fb1102c0d84f": "\\Omega=d\\omega +\\tfrac12 [\\omega\\wedge\\omega].", "ced1509ece1122df5e3a5666d9d5b58d": "h=(h_1,h_2,\\cdots, h_n)", "ced1c549a50befcd03d2eaa140781eaa": "R\\,y'' + \\frac{R\\,L}{Q}\\,y' + \\frac{R\\,\\lambda}{Q}\\,y = 0\\,", "ced1d9fab34d44d563d50d5d7b444c5c": "a^{15} = a \\times (a \\times [a \\times a^2]^2)^2 \\!", "ced279ad10e5b6b4d6ac63f9e833a1b4": "F_{i + \\frac{1}{2}}", "ced2828448a99eaa8bd379e2279cad12": "L(\\theta)", "ced295749e42c4d11cbd31f4376bf57a": "\\Delta N_s+\\Delta N_o=0\\, ,", "ced2b59268db439bb0ebacda32f7587c": "\n\\delta^i{}_j = g^{ik}g_{kj}\n", "ced2c1df5f41ac694b2c21f8696fd826": "\\epsilon_{ijk}", "ced31f8a36281779719bb68c9b3ed26b": "T_{zz}(z) \\sim \\sum_a j^a_z (z) j^a_z(z)", "ced325d92ed0ac5a04e5e4ef7cec5ae0": "\\wp(z; \\tau) = \\pi^2 \\vartheta^2(0;\\tau) \\vartheta_{10}^2(0;\\tau){\\vartheta_{01}^2(z;\\tau) \\over \\vartheta_{11}^2(z;\\tau)}-{\\pi^2 \\over {3}}\\left[\\vartheta^4(0;\\tau) + \\vartheta_{10}^4(0;\\tau)\\right] ", "ced3d343826c9ff7cc5cfcfb97fbb62c": "\\begin{array}{lcl}\n U\\cdot + AB & \\xrightarrow{k_R} & UA + B\\cdot \\\\\n \\bigg\\downarrow{k_r} \\\\\n R\\cdot + AB & \\xrightarrow{} & RA + B\\cdot\n\\end{array}", "ced3eb2c801f8964f4a27021b492422e": "\\sigma_{\\mathbf{f}}|_{\\mathrm{Sp}(N-1)}= \\bigoplus_{f_i \\ge g_i\\ge f_{i+2}} m(\\mathbf{f},\\mathbf{g}) \\sigma_{\\mathbf{g}}", "ced457ec8881f98a18eb4e72fd45bc0d": "\\tau_g^\\mathrm{iono} = -\\tau_p^\\mathrm{iono}", "ced56d2b77d3f86f1051613b9571ffee": "W(A) = \\left\\{\\frac{\\mathbf{x}^*A\\mathbf{x}}{\\mathbf{x}^*\\mathbf{x}} \\mid \\mathbf{x}\\in\\mathbb{C}^n,\\ x\\not=0\\right\\} ", "ced5760e0b7a6d879e9dd85d2f38ab88": "{DE}_{9}", "ced5ba60e05b3c974858c40e1487501c": "\\displaystyle{\\mathrm{tr}\\,(P_NK^*P_N)^2 =\\sum_{m=1}^N \\lambda_m^2 \\cdot \\mathrm{dim}\\, H_m,\\,\\,\\, \\mathrm{det}\\, [P_N - z(P_NK^*P_N)^2] = \\prod_{m=1}^N (1-z\\lambda_m^2)^{\\mathrm{dim}\\, H_m}.}", "ced5c58625765a68e2f8f011811a9079": "\\displaystyle S(4,3)", "ced65f65f3afb335f519fcad5d666a2d": "F=PA", "ced691224d1fcae5d48e5d790a3c523a": "{z\\choose n} = \\frac{\\Gamma(z+1)}{\\Gamma(n+1)\\Gamma(z-n+1)},", "ced69bea6d47294cdbf76f158db679ee": "B^n \\le \\alpha\\,", "ced6c49f4e7c7d1e635c33a73f89ccfe": "R_i, E_i, \\nu_i,~~i=1,2", "ced6d347b4d811d8be08ca5a1216ce18": "[C'_i,P'_j]=i M\\delta_{ij} \\,\\!", "ced70cab00b3da2e6ebe25fec9b073c0": "[-1,1]^2", "ced729d8c7d6ee329cadbbec62d5b4d7": "R^{2^n-1}", "ced747098c0c7adc2144e49ed9bbaf81": "f:\\mathbb R \\rightarrow \\mathbb C", "ced791d2e2f353af3df2ad20c00641bc": "e^{j\\omega C}\\,\\!", "ced799da1b6206727f7765a274f67c94": "a_0b_0", "ced7bb5b2f71d6b71e423ae9d05abdf6": "f(z) = W(e^z) - 1\\,", "ced7de59a587951feb6f333a5c5aa5ab": "HW=\\cos(A) \\cdot WS", "ced85936b4f62ba5b4eaa1c1fdb85c3f": "> k", "ced8aaa591f15b835974e436e115750d": "\\mathbb{\\bar{F}}_q", "ced8b480c9e9a6cb47ece6f350fcdee4": "\\delta_j=1", "ced8e2e6e514c505bc48634001510664": "\\left(\\omega^{2^{p-2}}\\right)^2 = kM_p\\omega^{2^{p-2}} - (\\omega\\bar{\\omega})^{2^{p-2}}", "ced903f07b05eb2d92f8991edaee370c": "p_i \\propto e^{- \\beta E_i } ", "ced912da24d50d277eb39731817b120f": "location(s)", "ced9141ba27ce40628ba25bcb359fbd8": "K(n)", "ced9298adbf00889b6bdbda5ad24cafd": "\\log_2(n)+ \\log_2(\\nu(n))-2.13\\leq l(n) \\leq \\log_2(n) + \\log_2(n)(1+o(1))/\\log_2(\\log_2(n))", "ced96b89f861ba1de98f32ea136a83cb": "O(\\lg^2 n)", "ced97d123a746bf7c0466e5c91b33571": "R_{tt}=8\\pi T_{tt}", "ced97d48331d4c3bad5c1e48f818f5f4": "\\lambda=q^n,\\ n\\le0", "ced9b8a2445df07ae27bd6221c6d51ff": "[x] = [-2,2]", "ced9bc08e42236df8458e3b5d2e7efcf": "4i \\,", "ced9f3d753dc2e5b73422f958211c19c": "\\sum_{n=0}^{\\infty}T_n(x) \\frac{t^n}{n!} = \\tfrac{1}{2}\\left( e^{(x-\\sqrt{x^2 -1})t}+e^{(x+\\sqrt{x^2 -1})t}\\right)\n= e^{tx} \\cosh(t \\sqrt{x^2-1}). \\,\\!", "ceda6b35789073608ceade3b2f4ed7db": "b_{PT} = (b_P + b_T) - 2(c - 2) - 2.\\,", "ceda71e88fd0079321b09e3ea4cfd9a6": "x\\subseteq d(R)\\,", "cedadd6f2d3f3f944d084f29e86ba2e8": "(-a;q)_\\infty = \\prod_{k=0}^{\\infty} (1+aq^k)", "cedaeb7b98eee255b7ea641698011e79": "a = 0.2 ", "cedb1947b900d95844e1223226173912": "\n\\begin{align}\n\\text{Maximize} &\\sum_{i\\in S}\\sum_{a\\in A(i)}R(i,a)y(i,a)\\\\\n\\text{s.t.} &\\sum_{i\\in S}\\sum_{a\\in A(i)} q(j|i,a)y(i,a)=0 \\quad\n\\forall j\\in S,\\\\\n& \\sum_{i\\in S}\\sum_{a\\in A(i)}y(i,a)=1,\\\\\n& y(i,a)\\geq 0 \\qquad \\forall a\\in A(i)\\,\\,and\\,\\, \\forall i\\in S\n\\end{align}\n", "cedb7241772a4209f3599e3aa71e6091": "C(g^{}_i,\\mu)", "cedb8076684e2359c49950b925544137": "P_N = P_{N-1}(1+r) - c", "cedb856dae2d968ac071b7a8cadf1834": "P_{1r} = \\mathrm{A_{1}}(\\theta,\\Phi) \\frac{G_{2}}{4 \\pi r^{2}} P_{2t}", "cedb9aceda53c8beeb46628a6d92b5ef": "\\mathbf{M} \\cdot[\\mathbf{A}]", "cedba588422575f1d90e61e4ad0db94e": "\n\\begin{align} \nE \\{ (\\hat{x}-x)(y - \\bar{y})^T\\} &= E \\{ (W(y-\\bar{y}) - (x-\\bar{x})) (y - \\bar{y})^T \\} \\\\ \n &= W E \\{(y-\\bar{y})(y-\\bar{y})^T \\} - E \\{ (x-\\bar{x})(y-\\bar{y})^T \\} \\\\ \n &= WC_{Y} - C_{XY}. \n\\end{align}\n", "cedc0c8a28276085b332f46af5865083": "d^2P(r)/dr^2 + [2(E-v(r)) - \\ell(\\ell+1)/r^2]P(r)=0.", "cedc3dc692231662c7a76c1211c95232": "\\psi(x) = \\begin{cases}\n\\psi_{\\text{L}}(x) = A_{\\text{l}}e^{\\kappa x}, & \\text{ if } x<0; \\\\\n\\psi_{\\text{R}}(x) = B_{\\text{r}}e^{-\\kappa x}, & \\text{ if } x>0.\n\\end{cases}\n", "cedc4c7636c88585cf3f48710715751d": " \\scriptstyle\\boldsymbol{\\phi} ", "cedc6f71e177445c87eea6cc15effa3a": "a=t_0\\le t_1\\le\\cdots\\le t_n=b", "cedc93fa02ebec406b2071c50ca46c5d": "k=0,1,\\ldots", "cedcfa1e761977b967b076cfa833adf6": "\\ X(z) = \\frac{1}{1 - 0.5z^{-1}}\\ ", "cedd478a99cdaffc0356869d9182fb09": "\\sqrt{ {p \\over \\rho}}\\,", "cedd668fcb9bce284852285eb05a113a": "\\bigcup_{k\\in\\mathbb{N}} \\mbox{DSPACE}(n^k)", "cedd9d76ac4b0f3dfba105d753e62aef": "\n\\mathbf{A}\\mathbf{P}=\n\\mathbf{A}\n\\begin{pmatrix}\n | & & | \\\\\n \\mathbf{v_1} & \\cdots & \\mathbf{v_n} \\\\\n | & & | \\\\ \n\\end{pmatrix}\n=\n\\begin{pmatrix}\n | & & | \\\\\n A\\mathbf{v_1} & \\cdots & A\\mathbf{v_n} \\\\\n | & & | \\\\ \n\\end{pmatrix}\n=\n", "ceddbcaf2c2ab2d3541b2caa362098b6": "m_1,\\ldots,m_n", "ceddf33069b123c9040a4a86d49dcdf1": "S_{n-1} = \\frac{2\\pi^\\frac{n}{2}}{\\Gamma(\\frac{n}{2} )}", "cede2a0175ae5cc34ca9405705dcb2a6": "\\vert\\mathcal{F}f(\\xi)\\vert \\leq \\frac{I(f)}{1+\\vert 2\\pi\\xi\\vert^k}", "cede364fa783b3e70e7e878f56d8cc00": "f:\\mathbf{B}^n \\rightarrow \\mathbb{R}", "cede5c5a2a18681a2426481964b714ec": "\\mathbf{E} = \\frac{1}{\\varepsilon} \\mathbf{D}", "cede5d4e2d519cc81a8c841444fd055e": "\\eta(\\boldsymbol{x},t)", "cedec65908ffa25cd407c3d12f5a98ea": "\\frac4n=\\frac1x+\\frac1y+\\frac1z", "cedee59487086114b86be3eef17d0ecc": " \\tau^2", "cedf8da05466bb54708268b3c694a78f": ">", "cee065dbc5b3dc23a884c7d942b0f1b3": "68\\to 6^2+8^2=100\\to 1^2+0^2+0^2=1.", "cee16bf0bcb4688de7b73f7fbd452ce3": " h\\longrightarrow 0", "cee1c901a7179ada34cb20f72316fa36": "Q = \\zeta\\, u_b(\\xi) - \\tfrac16\\, \\zeta^3\\, u_b''{\\xi} + \\cdots.", "cee21267b0ac28799e5a794de66fa8d0": " \\sigma = \\frac{V'_{w2}}{V_{w2}} ", "cee29819ddfb22e2d6ccd783cdd82aa5": " \\nabla \\times \\mathbf{E} = -\\frac{1}{c} \\frac{\\partial \\mathbf{B}} {\\partial t} \\,", "cee29a2a7c088515341ae45d190deb6c": " \\Delta Y' ", "cee2a83d9fd1fbc8988edffc518e9a9e": "\\frac{M}{P}=L(i, Y)", "cee34fddadd1b7273749c794c5ef7ce8": "\\mathbf{B} =\n\\begin{bmatrix}\n\\begin{pmatrix} \\\\ \\boldsymbol\\beta_1 \\\\ \\\\ \\end{pmatrix}\n\\cdots\n\\begin{pmatrix} \\\\ \\boldsymbol\\beta_m \\\\ \\\\ \\end{pmatrix}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\begin{pmatrix}\n\\beta_{1,1} \\\\ \\vdots \\\\ \\beta_{1,k} \\\\\n\\end{pmatrix}\n\\cdots\n\\begin{pmatrix}\n\\beta_{m,1} \\\\ \\vdots \\\\ \\beta_{m,k} \\\\\n\\end{pmatrix}\n\\end{bmatrix}\n.\n", "cee36f57ace8003debe46f9dc51cbef6": "n_h=(1-x)n", "cee3b0f3a1c646c1799b30e4d90b9d1c": "\\epsilon = \\left(1+\\frac{1}{q_{sig}}\\right)^{q_{sig}+1} \\cdot q_{sig} \\cdot \\epsilon'", "cee40024bd9776d64f0adb51a9706568": "\n\\begin{align}\nf(x,y,p)=xyp - \\kappa\\frac{e^{\\epsilon \\phi(1-z)^{\\mu}yx}}{1+e^{\\epsilon \\phi(1-z)^{\\mu}yx}}\n\\end{align}\n", "cee42c3e0efdec65358626a2294aa44d": " (n,\\theta) \\in \\{ n \\in \\Z : n \\ge 0 \\} \\times \\{ \\theta = 1/k : k \\in \\Z, \\ k \\ge 1 \\} \\, . ", "cee44a4736519848cd908612350c85fe": "(n-1)", "cee495a276035118f798dbb5ac380229": "\\mathbb{Z}[\\zeta]", "cee54a8f8d33207fa4733b306f4d7d1f": " \\mathrm{IDCG_{6}} = 8.69 ", "cee59d257ed07a58ccee9c512697b1e6": "F_\\text{n}", "cee5cbe59a58427e6bf1b83fc5d01e52": "g_K", "cee605448899b3f6215019f74f32cad7": "\\operatorname{Spec}(A)", "cee66444f51f05ada1b1241fb1863d86": "\n E_\\text{pot} = \\tfrac12\\, \\rho\\, g\\, H^2\\, \\left[ \n - \\frac{1}{3\\, m} \n + \\frac{2}{3\\,m}\\, \\left( 1 + \\frac{1}{m} \\right) \\left( 1 - \\frac{E(m)}{K(m)} \\right)\n - \\frac{1}{m^2}\\, \\left( 1 - \\frac{E(m)}{K(m)} \\right)^2\n \\right].\n", "cee69c708b4fa8268adbe49c3f2c936d": "\\alpha_{nk}", "cee746c323a418f3f924e5f8a3367e13": "q \\Rightarrow r", "cee748bc67c3b30807feb5b3de15b4f4": "\\lim_{n\\to\\infty} (a_n b_n) = (\\lim_{n\\to\\infty} a_n)( \\lim_{n\\to\\infty} b_n)", "cee782755bc81e71522773fea4d99244": "\\ \\beta ", "cee803f95b1c8b65b9257d77e7b19c23": "O(k^n)", "cee8064ec4ba0d98a9d1c851f92a335f": "a_n=n\\#-1", "cee876c8c27112f092834d8abf2e3050": "\\hat u_\\theta", "cee8a3cda1df266a59c3c0009cb94f68": "r=\\frac{\\cos\\theta}{\\sin^2\\theta}", "cee8b6ce451de93a295d8616cecc9d52": " KL = k(K \\cup L) ", "cee8beb7d7026c8a76ab4c14cb2bef04": "\\theta\\in \\Theta", "cee8d22071da92c53406a5358a766ad8": "\\ \\mathbb{Z}_{p} ", "cee91a411cc0af2fd99e95e55f6ae74d": " g(r) = \\exp \\left [ -\\frac{w^{(2)}(r)}{kT} \\right ] ", "cee95b6f248fc0ee58be682d32712cdf": "C(x_2)", "ceea4745c4381ff81225676441f4ff89": "[\\phi, f\\psi] = \\rho(\\phi)f\\psi +f[\\phi, \\psi]", "ceea501c33852cd6b26bacde91069d28": "Tx_k = x_{k+1}.", "ceea73efc717dd722bb79a7213ec3862": "\\displaystyle P_{Y_r}(y) = \\sum_{x\\in X} \\sum_{s\\in S} P(x)P_{S_r}(s)W(y|x,s)", "ceea8b9579ce142658d57ebac21ed06f": "u \\models P", "ceea8efe68e20fb832c9c2cd4c15a9cb": "\\varphi_i = N_i(\\chi_{iA} \\pm \\chi_{iB}), \\, ", "ceeafaf2718ecfa3a7a49b763a5d0b49": "\\left|\n\\begin{array}{ccccc}\n x_1 & x_2 & x_3 & \\cdots & x_n \\\\\n x_n & x_1 & x_2 & \\cdots & x_{n-1} \\\\\n x_{n-1} & x_n & x_1 & \\cdots & x_{n-2} \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n x_2 & x_3 & x_4 & \\cdots & x_1\n\\end{array}\n\\right|=\\prod _{j=1}^n \\left(x_1+x_2\\omega _j+x_3\\omega _j^2+\\ldots +x_n\\omega _j^{n-1}\\right),", "ceeb1b1047c14634d8a00eb3ff994222": "\\sum_{k=1}^{n-1}\\frac1{p_k}=\\frac1r.", "ceeb5308b818eb02104aa18115e064c7": " x f = 1 \\quad \\Rightarrow \\quad f = \\operatorname{p.\\!v.} \\left( \\frac{1}{x} \\right) + K \\delta, ", "ceeb551c8377f154232c84e647d7bfaf": "|X-O||O-Y|=c. \\, ", "ceeb6d52ec644dffa9948c1c032dbacc": "dU = \\left(C_{p}-\\alpha p V\\right)dT +\\left(\\beta_{T}p-\\alpha T\\right)Vdp\\,", "ceeb7117d712ddc7b36e53c845b517de": "\\frac{dy}{dx} = \\left.\\frac{dy}{du}\\right|_{u=g(h(a))}\\cdot\\left.\\frac{du}{dv}\\right|_{v=h(a)}\\cdot\\left.\\frac{dv}{dx}\\right|_{x=a},", "ceeb7a58213555a6cea6ca9cee373b9e": "\\begin{align}\n\\nu_0' &= \\nu_0 + n \\\\\n{\\sigma_0^2}' &= \\frac{\\nu_0 \\sigma_0^2 + \\sum_{i=1}^n (x_i-\\mu)^2}{\\nu_0+n}\n\\end{align}", "ceebd10ea04337bed12aa927853b1c13": " A[3]=S[4,7]=ana$", "ceebdb41f67407a43a5778656948a8a8": " \\neg \\neg P \\And \\neg \\neg R ", "ceec06cd5af4c0106602a2271c23a968": " \\{1, \\zeta, \\zeta^2, \\ldots ,\\zeta^{m-1}\\} ", "ceec0cebd51807cb495822e1d0e4a96c": "\\operatorname{tr}(\\gamma^5)= 0", "ceec661283a92ace82512e8480fba1e6": "\\frac{1}{4}\\left[\\begin{array}{cc} \\# & 2 \\\\ 1 & 1 \\end{array}\\right]", "ceecc100d0547dbdf76ce1f8148f7a22": "\\,a + i b", "ceecd5b0a02db52bb78706dfdc48800f": "P(e) \\triangleq \\frac{1}{M} \\sum_{x} \\mathbb{P} (X \\neq \\widehat{X}|X)", "ceed6b6bdc6fe2a14055ed95f7f307d0": " P = \\log R ", "ceedb66e8ac17e3def61095ef5645a48": "\\tilde{z}_{\\cdot j}", "ceedc11a03955558c9421f820e6f8773": "P_{i,2}", "ceedcee2b72596d05e31965841cceba2": "\\displaystyle{|Pf(w_1) - Pf(w_2)| \\le K_p \\|f\\|_p|w_1-w_2|^{1-2/p},}", "ceedd8bf3a2188937eaedb473f346fdc": "\\sigma \\leftarrow SEncode(L_B(N),d,w,y,y^{\\prime},\\tilde{k})", "ceee00c2f66f08565fff527aced3d935": " p_{1j} = p_{2j}", "ceee1ae0880c2b13feb03f207879a806": "\\mathbf{e}_3(t) = -\\mathbf{e}_1(t) \\times \\mathbf{e}_2(t)", "ceee4c68c7770a8772a9343457d73129": "\\mathbf{f}' = \\mathbf{f}J^{-1},\\quad J=\\left(\\frac{\\partial y^i}{\\partial x^j}\\right)_{i,j=1}^n.", "ceeebf0a34677026648101a14ed8b6c1": " W(u_1,\\ldots,u_d) = \\max\\left\\{1-d+\\sum\\limits_{i=1}^d {u_i} , 0 \\right\\}.", "ceef270f9cc7ddfb2342d265828d6798": "\\tan(z) = -\\sum_{k=0}^{\\infty} \\left(\\frac{1}{z - (k + \\frac{1}{2})\\pi} + \\frac{1}{z + (k + \\frac{1}{2})\\pi}\\right)", "ceef78b61bf01306cc7e80344c92c19d": "k=1", "ceef7ddbf133a293ba5838ef4b6fb345": "\\left( \\frac{x}{\\lambda z}, \\frac{y}{\\lambda z} \\right)", "ceef8606689e7ad0f0659572ca3b71a0": "\\mathbf{y} = [y_1, y_2]^T", "ceef9a2933f68334b5fdfb708482e29d": " N_a = \\frac{1}{ [ \\sum_{ i = 1 }^K p_i^a ]^{ a - 1 } }", "ceefce0015443a29e432a4cbadf8a236": "(t,\\rho,z,\\phi)=(t,r\\sin\\theta,r\\cos\\theta,\\phi)", "ceeff5f58a3fb6a173462a2f5cba3dfa": "\\iota_{X, Y} : \\begin{array}{rcl}\n\\mathrm{Hom}(\\mathbf{1}, X^{*} \\otimes Y) & \\longrightarrow & \\mathrm{Hom}(X, Y) \\\\\nf & \\longmapsto & (\\epsilon_X \\otimes id_Y) \\circ (id_X \\otimes f) \\\\\n(id_{X^{*}} \\otimes g) \\circ \\eta_X & \\longmapsto & g\n\\end{array}\n", "cef0a2c4efc4723772701d0b1d64de2b": "\\lim_{x \\rarr 0^+}{1 \\over 1 + 2^{-1/x}} = 1,", "cef0a92cddb895d25e5f1e7d828a6dfc": "F(\\bold{x})=\\sum_{S\\subseteq \\Omega} f(S) \\prod_{i\\in S} x_i \\prod_{i\\notin S} (1-x_i)", "cef13cf38f79d667dbd43908dff28a49": " I_{S} ", "cef162fab317bcb008749666e26ce4c0": "\\|\\mathcal Ff\\|_q \\le \\left(p^{1/p}/q^{1/q}\\right)^{1/2} \\|f\\|_p.", "cef1746c5fcd3e8ef42a6ba5faf34655": "w(2^j t - k) \\!", "cef1c791eaf655c7db38170527c0d454": "\\left(a, b\\right)", "cef1dba9cde7ceec93d8d66936af0516": "\\Delta V=k\\sin(\\omega t)", "cef203488bc0f93afad73428b2a8e106": "T:\\mathrm{ob}(C)\\to \\mathrm{ob}(C)", "cef297351f69cfaba9facf2753f1ff49": "s(t) = \\frac{1}{2}\\cos\\left(2\\pi 800 t\\right) - \\frac{1}{2}\\cos\\left( 2\\pi 1200 t\\right)", "cef2cafa076d7be7b590ad8bb9f242ce": "3\\left(\\frac14+\\frac{1}{4^2}+\\frac{1}{4^3}+\\frac{1}{4^4}+\\cdots\\right) = 1.", "cef2f6135663fa4bf6fae84c6f2bd3b2": " \\|A\\|_p = \\left( \\sum_{i=1}^{\\min\\{m,\\,n\\}} \\sigma_i^p \\right)^{1/p}. \\, ", "cef39472ee6d08fa81178ac33f60f42c": "\\lambda^2-\\tau\\lambda+\\Delta=0", "cef3e2a4a63aaab553a6fdd5a6d67b60": " E_{n_1} = - \\frac{1}{2} \\frac{Z^2}{n_i^2} \\text{ in a.u.} ", "cef46ea1137d2e24cec33d66c7b3fbfa": "1/z^2+2/z", "cef493419436f8204119a5b90391e8c6": "\\big(f(p),g(l)\\big) \\in I'", "cef4c5c8b47551c16f5cf1829f455852": "[0,1,0,0,0]", "cef4d032f98909e5737740da9b3ecb25": "\\begin{cases}\ny = mx+b_2 \\\\\ny = -x/m\n\\end{cases}", "cef4e17d616ea7a0437403358289d3e1": "Y_1 \\subset [-2,1)\\ ", "cef54b14583a977e6fc2ccef307c8677": " [N, K]_{Q} ", "cef57227187085ade40ca4bad8b43fbc": "\\delta_{i,j}\\,", "cef575cd2e44ae9fe48e3f79642b5228": "Y_0 = Y_1", "cef584bc12bdf51fdfec4eb1120c2eaa": "\nV_{\\infty}(t) = A\\frac{e^{j \\omega t}}{j\\omega +1/\\tau}.", "cef5bd70c799fce02b1ff492220ba805": "\\textrm{Im}\\left( \\operatorname{Li}_s(z+i\\epsilon) \\right) = {{\\pi \\mu^{s-1}}\\over{\\Gamma(s)}} \\,.", "cef5d103049bf2309e5d1c2d146008e0": "\\,n\\rightarrow\\infty\\,", "cef6095bd0ca89cea01f58c08eae3e55": "s\\not\\in S^{(t)}", "cef632d7ee0e7c89508d61815beb9f3c": "\\kappa<\\kappa^{cf(\\kappa)}. \\!", "cef6565d78da1927d1ac228a5aa66c31": "f \\; \\langle x, y \\rangle", "cef6b76eacb13b0e8541445d3fd5bc5b": " A^{(n)} := L_n A^{(n-1)}.", "cef6e94d4015931401f9e974b08137ad": "H_R", "cef76c6c22e8a2c768c791eb706a8b75": " \\mathbf{a} \\wedge \\mathbf{b} \\ne 0,", "cef79c354977ec320a700c37bc465619": "0 = \\nabla_\\nu T^{\\mu \\nu} = {T^{\\mu \\nu}}_{,\\nu} + \\Gamma^{\\mu}_{\\sigma \\nu} T^{\\sigma \\nu} + \\Gamma^{\\nu}_{\\sigma \\nu} T^{\\mu \\sigma} \\,", "cef7e8f9dbe11db741661415c1e43c56": " \\mathbf{F} = \\frac{d \\mathbf{p}}{dt}\\,.", "cef81993a162e4f3bc71ff4405c598a3": "\\alpha(z) M_+(z) + \\beta(z) M_-(z) = c(z)\\,", "cef81cdaa6525d1086f0c6dacf9f9f99": " \\sigma^{2}_{P} = \\mathbb{E}\\left[\\sum^{n}_{i=1}x_i(R_i - \\mathbb{E}[R_i])\\right]^2 ", "cef88fc600a8894691aea8cc8eeab36c": "R_{ab} \\, = {R_{acb}}^c", "cef8aad13c8c03721c19bccb2eb69079": "\\{X_1,\\dots,X_n\\}.", "cef8b67ce695ca25a038581ac7f061f2": "\\frac{\\partial T}{\\partial t} + u \\frac{\\partial T}{\\partial x} + v \\frac{\\partial T}{\\partial y} + \\omega \\left( \\frac{\\partial T}{\\partial p} - \\frac{R T}{p c_p} \\right) = \\frac{J}{c_p}", "cef917a9c5fb2a4c1e5b81a05c42c8db": "\\,Cov(X|Y)", "cef97e5ee2a7572bf210535cb398c5c2": "\\sigma^\\alpha", "cef99f0ef9519f3acde4291fa8b23b4f": "\n M_4 = 312.5 + R_a (x-10) + R_b (x-25) - 25 x = -625 + R_a(30 - 0.6x) + M_c(0.04x -1) + 12.5x\\,.\n ", "cef9b14bff781bb6b1d95bc47616ba4e": "\\pi=\\left(\\frac{1}{2},\\frac{1}{2}\\right)", "cef9c27463d12e54c0b649c23a67d255": "\\begin{align}\n\\Delta y &{}\\stackrel{\\mathrm{def}}{=} f(x_1+\\Delta x_1, \\dots, x_n+\\Delta x_n) - f(x_1,\\dots,x_n)\\\\\n&{}= \\frac{\\partial y}{\\partial x_1} \\Delta x_1 + \\cdots + \\frac{\\partial y}{\\partial x_n} \\Delta x_n + \\varepsilon_1\\Delta x_1 +\\cdots+\\varepsilon_n\\Delta x_n\n\\end{align}", "cef9d86b1fbe3e10c1517f88dd0c831f": "{\\partial f \\over \\partial \\rho}\\hat{\\boldsymbol \\rho}\n+ {1 \\over \\rho}{\\partial f \\over \\partial \\phi}\\hat{\\boldsymbol \\phi}\n+ {\\partial f \\over \\partial z}\\hat{\\mathbf z}", "cef9dab61517fb65be93b8bd8c3386cb": "\\mathcal{O}(n^3)", "cefab93af02de4ff18810d7245dc73b9": "p^2 \\ll k^2", "cefb09173eebfdac456f9b22882d122f": " \\pm \\lambda/4 ", "cefb1500e542b58fe3e37f87f4e5d1af": " \\epsilon ", "cefb1bbbd7b2cad39498c8d27c148a80": "\\begin{align}\n\\tan\\alpha_1&=\\frac{\\sin\\lambda_{12}}{ \\cos\\phi_1\\tan\\phi_2-\\sin\\phi_1\\cos\\lambda_{12}},\\\\\n\\tan\\alpha_2&=\\frac{\\sin\\lambda_{12}}{-\\cos\\phi_2\\tan\\phi_1+\\sin\\phi_2\\cos\\lambda_{12}},\\\\\n\\end{align}", "cefb5434d4ba8731f797836e10a82bda": "\\log\\min\\left(n_x,n_y\\right)", "cefb7bac0b1ebedc38e58842d85508cc": "\\text{Equivalent mass, } M_{eq} = \\int{M\\bar{u}^2} du ", "cefb969db448aa7350d9dca61af6852b": "\\Delta\\phi\\,\\!", "cefb96cd62f2b33924bf9273020cd00f": "\\sum_\\rho\\Re\\left[1-\\left(1-\\frac{1}{\\rho}\\right)^{-n}\\right]\n\\ge 0", "cefc2229588d8c4f5987926e129aa423": "\\sqrt{\\mathrm{1TT}}=\\mathrm{1T.1T0101010TTT1TT11010TTT01T1...}", "cefc877aa1f61ea49391e3a392cf6710": "\nx^{[i]} = x^{q^{i \\mod N}}. \\,\n", "cefcb76ea6aa1679e0aa53d8cb27f5dd": "\\lang \\mathbf x | \\mathbf y \\rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4.", "cefcba8ef8f54387da335b17e080717e": " \\| \\tilde{H}_n y_n - \\beta e_1 \\|. \\, ", "cefd3e995ae92d6c841cacd5f2030d22": "\\tfrac{5}{18}>\\tfrac{4}{17}", "cefd4237b32f9bc07c57b6e44684e715": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{T}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{F}&\\mathrm{F}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "cefd5a17dfcc1a08eaf5f08d9e59e143": "\\begin{bmatrix}\nt_1 & 0\\\\\n0 & t_2\\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nt_1 & 0\\\\\n0 & 1/t_1\\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\exp(\\theta) & 0 \\\\\n0 & \\exp(-\\theta) \\\\\n\\end{bmatrix} ", "cefd91912c9d0ef157dabdf2f5fdebb7": " \\operatorname{def}[y] \\and \\operatorname{ask}[\\_] \\and FV[p] \\subset \\{p, q, m\\} ", "cefdec7df1ffa2587b11df5241a76f30": "\n\\begin{array}{l}\n\\mathrm{dDTF}^2_{j\\rightarrow i}(f)= F_{ij}^2(f)C_{ij}^2(f) \\\\\nF^2_{ij}(f)=\\displaystyle\\frac{ \\left| H_{ij}(f) \\right|^2}\n{\\sum_{f}\\sum_{m=1}^{k}\\left| H_{im}(f) \\right|^2}\n\\end{array}\n", "cefdf0a9e027c2d114d492a37f542b97": "\\Delta\\lambda = \\lambda^{\\mathrm{state 1}}_{\\mathrm{observed}} - \\lambda^{\\mathrm{state 2}}_{\\mathrm{observed}}", "cefe10860597f68422152e87660488d2": " 0 \\leq \\theta \\leq \\pi", "cefe9678c80f874d75f738be5906e88a": " \\begin{pmatrix} \\cos \\alpha & -\\sin \\alpha & 0 & 0 \\\\ \\sin \\alpha & \\cos \\alpha & 0 & 0 \\\\ 0 & 0 & \\cos \\beta & -\\sin \\beta \\\\ 0 & 0 & \\sin \\beta & \\cos \\beta \\end{pmatrix}.", "ceff06a4162e6ac8f1a55969593f9ebc": " D_v f(x) = \\frac{d}{dt}f(x+tv)\\big|_{t=0}=\\sum_{i=1}^{n}v^i\\frac{\\partial f}{\\partial x^i}(x).", "ceff512fdc1ba8fe2d3dfe6e774668c7": "g(x) = \\ell(x) \\sum_{j=0}^k \\frac{w_j}{x-x_j}.", "cefff0d3988077ad39120f48d044f4fd": "\\textstyle 2x^{i+r}b(x)", "cf01229987cb62a3720931dace40f64d": "G(s)= \\exp \\left [\\lambda \\frac {\\alpha-1}{\\alpha} \\left( \\frac{\\theta} {\\alpha-1} \\right)^\\alpha \\left\\{ \\left(1- \\frac{1} {\\theta}+ \\frac {s} {\\theta}\\right)^\\alpha-1 \\right\\}\\right]", "cf01357708e0222c7aba96b7f6fcbd0d": "\\begin{pmatrix}0 & -1 \\\\ 1 & a\\\\\\end{pmatrix}", "cf0187b219094ecc31ee62e0bd0f1ca6": "R_{TE}, X_{TE}", "cf01b030268b1fd12c4bf8ae00332d6f": " S = 3/2,\\ L = 0 ", "cf01ef5075bba3806911e0636231a295": " \\frac{\\mathrm{d}\\mathbf{A}^{-1}}{\\mathrm{d}t} = - \\mathbf{A}^{-1} \\frac{\\mathrm{d}\\mathbf{A}}{\\mathrm{d}t} \\mathbf{A}^{-1}. ", "cf02c22fc164faf4976cae168d7d73bd": "M_i", "cf030115a8e54dacf5b5ff5275534593": " { {d u^{\\mu}} \\over {d\\tau}} + {R^{\\mu}}_{\\alpha \\nu \\beta } u^{\\alpha} x^{\\nu} u^{\\beta} = 0 ", "cf03720613c2533106db23f4af1b0284": "\\arg(f)=0,\\pm 1,\\pm 2,\\ldots", "cf038e1604cebc621216ce901a5d9059": "\\frac{1}{\\tau} = \\sum_{k'} S_{k'k}^{Ac}=\\sum_{k} S_{k\\pm q ,k}^{Ac}", "cf0391615791477cd1d3d9c350941df9": "\\tbinom{i}{j}", "cf03c6f7bba76111a3ef8bf0b85df3ed": "SO(3, 1)", "cf03f87a994418f32c81a3a7f6837219": " \\displaystyle{SK=K^*S.}", "cf0409eee06d3d81a935f52c1aab070a": "\\quad W_{2\\,p+1}=\\frac{2\\,p}{2\\,p+1}\\,\\frac{2\\,p-2}{2\\,p-1}\\cdots\\frac{2}{3}\\,W_1=\\frac{2^{2\\,p}\\, (p!)^2}{(2\\,p +1)!}~", "cf0486ab50f7cb191520712861575a84": "\\frac{_uM}{y_c^2} = \\frac{gy_c^3}{gyy_c^2} + \\frac{y^2}{2y_c^2}", "cf048f74f71721abd7b8df49453d1310": "\\mathbb{R}^n", "cf04ae8b3b10970fbb522c11e33c8fc2": " \\nu \\approx {(\\gamma_1 + \\gamma_2)^2 \\over \\gamma_1^2/(n_1-1) + \\gamma_2^2/(n_2-1)} \\quad \\text{ where }\\gamma_i = \\sigma_i^2/n_i. \\, ", "cf04c1125d18a5ea17a34fff265628cd": "\\mathbf{BB} = \\begin{bmatrix}\nBB1 & -AA3 & 0\\\\\n-AA3 & BB2 & -AA3\\\\\n0 & -AA3 & BB1\\end{bmatrix}", "cf04da9959110f02b8c56544c4f80f9c": "K \\cdot t", "cf054399b5732591bcf0eca7922aa71e": " \\theta_m = \\rm{arccos}\\frac{1}{\\sqrt{3}} = \\rm{arctan}\\sqrt{2} \\approx 0.95532 rad \\approx 54.7^\\circ", "cf0562dee1453fcf7ac1eea1dff5517b": "\\exist F^{n+1};", "cf05c80b662adfecbc9a217aada59e3e": "\n\\left(H_{\\epsilon, t=0}-E_0 \\pm i \\hbar \\epsilon g \\partial_g\\right) U_{\\epsilon I}(0,\\pm\\infty) |\\Psi_0\\rangle = 0.\n", "cf065a8706ea5d9a398e9e0467ff6dc2": "(P \\and Q) \\leftrightarrow (Q \\and P)", "cf06715c824a69d25884c206a089c407": "z=k", "cf06b58bf17909fc93f59442c72707be": "d\\in K\\setminus\\{0,1\\}", "cf06cd3da4326052f070caf3f77b7772": "\\frac{T}{W}=\\left(\\frac{\\eta_p}{V}\\right)\\left(\\frac{P}{W}\\right)", "cf0714c856747d96f37e9e67813902b6": "D(d) \\wedge \\neg D(f(d))", "cf07223eac05304aaa67e5b21c6685f3": "\\chi(v)", "cf076fbc762afff890aed927c786148a": "(\\tfrac{b}{q}) = -(\\tfrac{b}{p}) = 1", "cf07941e58ba52efdcd850a118335412": "X{_i^?} = 1", "cf07b6aaaef074aa07d8309ac6bbc199": "\\chi = {n_{\\rm s}\\varepsilon_{\\rm s} + n_{\\rm p}\\varepsilon_{\\rm p} \\over n_{\\rm s} + n_{\\rm p}}", "cf088af88547db8255377c404b74bdd6": "E_{n}", "cf096a7e782110fcc3da52d265017fd6": "\\chi = b_0 - b_1 + b_2 - b_3 + \\cdots.\\ ", "cf09743b3227ef00a4184db6628495d6": " z \\rightarrow \\pm \\infty ", "cf098559318f11cb793909b82f926fa2": "2,4,8,\\ldots,2^k", "cf09a3ae9ac6874c39174268c42e9f57": "\\mathbf{x}=(x_1,x_2,\\ldots,x_M)^T", "cf09db0bbc6e77f841f35b5c6b3d4455": "\nC_{4,1/2}=\n\\begin{bmatrix}\nc_1 & c_2 & c_3&c_4\\\\\n-c_2 &c_1&-c_4&c_3\\\\\n-c_3&c_4&c_1&-c_2\\\\\n-c_4&-c_3&c_2&c_1\\\\\nc_1^* & c_2^*&c_3^*&c_4^*\\\\\n-c_2^* &c_1^*&-c_4^*&c_3^*\\\\\n-c_3^*&c_4^*&c_1^*&-c_2^*\\\\\n-c_4^*&-c_3^*&c_2^*&c_1^*\n\\end{bmatrix}\n\\quad\\text{and}\\quad{}\nC_{4,3/4}=\n\\begin{bmatrix}\nc_1&c_2&\\frac{c_3}{\\sqrt 2}&\\frac{c_3}{\\sqrt 2}\\\\\n-c_2^*&c_1^*&\\frac{c_3}{\\sqrt 2}&-\\frac{c_3}{\\sqrt 2}\\\\\n\\frac{c_3^*}{\\sqrt 2}&\\frac{c_3^*}{\\sqrt 2}&\\frac{\\left(-c_1-c_1^*+c_2-c_2^*\\right)}{2}&\\frac{\\left(-c_2-c_2^*+c_1-c_1^*\\right)}{2}\\\\\n\\frac{c_3^*}{\\sqrt 2}&-\\frac{c_3^*}{\\sqrt 2}&\\frac{\\left(c_2+c_2^*+c_1-c_1^*\\right)}{2}&-\\frac{\\left(c_1+c_1^*+c_2-c_2^*\\right)}{2}\n\\end{bmatrix}.\n", "cf0a7a95770f693babfe1f5427e1c42f": "\\omega^2 r^3=\\mu", "cf0ab0468acef7c72a77acf65f64cb3f": "\\tilde{D}=(N_1D_2+N_2D_1)\\frac{\\partial ln a_1}{\\partial ln N_1}", "cf0af6d79126c799b11fb045005952ed": "x+7=10", "cf0b0a042e68726074d349f2dd425485": "\na = -\\frac{GM}{r^2},\n", "cf0b26124853c50a235379b74e296c7f": "x_1 = \\tfrac13(s_0 + \\zeta^2 s_1 + \\zeta s_2),\\,", "cf0b51cb2d2257e56f68d925dbbfecee": " b = \\iota x (\\pi (a,x) \\land x=b ) ", "cf0b830d3357039d7a337f818dbcc05b": "\\underline{\\mathsf{f}}", "cf0bc8215226f73d68ba5144d2425f56": " \\mathfrak{so}(5,\\mathbb C) \\cong \\mathfrak{sp}(4,\\mathbb C)", "cf0bf28f7ca8422b763c0338665409e6": " \\hat\\mu = 27.40, \\hat\\sigma = 3.81, \\hat\\nu = 2.13.", "cf0c0a187885cba279b1f2916fac9ee0": "{\\bar x}<{\\bar y}", "cf0c0bdfdf42bb68948152aea595e57e": "D_t^j", "cf0c11e191edcec0f418f7fdc661b732": "\\sum(r_j-r_i)(s_j-s_i) = 2n\\sum r_i^2 - \\frac12n^2(n+1)^2 - nS ", "cf0c4d95cf7ace6df53c2001be6c0808": "\\frac{b-a}{n}\\sum_{i=0}^{n-1} f\\left(a+i\\frac{b-a}n\\right) \\approx \\int_a^b f(x)\\ dx,", "cf0c77692690a08ff6810ead36f2a994": "3x^3+4y^3+5z^3=0.", "cf0c926a7176b2137fc776e9c63ecc8f": "\\Psi = \\sum_{j=0}^{N-1} C \\int_{-\\frac{a}{2}}^{\\frac{a}{2}} e^\\frac{ikx\\left(x^\\prime - jd\\right)}{z} e^\\frac{-ik\\left(x^\\prime - jd\\right)^2}{2z} \\,dx^\\prime", "cf0cab7a99de0d070918b774647d3bc2": " (I-Q)f(x_1+x_2(x_1,\\lambda),\\lambda)=0 \\, ", "cf0cb7115fb26f60fcfc980b15b978b3": "\\begin{matrix}\n\\mathrm{person} & \\mathrm{year} & \\mathrm{income} & \\mathrm{age} & \\mathrm{sex}\\\\\n1 & 2001 & 1300 & 27 & 1 \\\\\n1 & 2002 & 1600 & 28 & 1 \\\\\n1 & 2003 & 2000 & 29 & 1 \\\\\n2 & 2001 & 2000 & 38 & 2 \\\\\n2 & 2002 & 2300 & 39 & 2 \\\\\n2 & 2003 & 2400 & 40 & 2\n\\end{matrix}", "cf0cc7280a670edf6dfa2fc6bf597d95": "t':=t-\\frac{|\\vec r -\\vec r^{\\,'}|}{c}\\,,", "cf0cd8e11dba2ddf7fa64187331067a6": "\\ G(R) = \\sigma^+ - \\sigma ", "cf0cda72a6cdd2f1c5f03109ca966dc6": "\\operatorname{pmi}(x;yz) = \\operatorname{pmi}(x;y) + \\operatorname{pmi}(x;z|y)", "cf0d1f36be26afa212e2c05838ef134e": "\\delta\\theta=\\theta-\\alpha", "cf0d39a5a54d7f777e53d650ccc41058": "\n\\begin{cases}\n \\frac{dx_1}{dt} = x_2 \\\\\n \\frac{dx_2}{dt} = 1-2*x_2-x_1 \\\\\n x_0 = [p_1 \\ p_2] \\\\\n t_i = [1 \\ 2 \\ 3 \\ 5] \\\\\n x_1^m(t_i) = [0.264 \\ 0.594 \\ 0.801 \\ 0.959] \\\\\n |p_{1:2}| <= 1.5 \\\\\n\\end{cases}\n", "cf0d73328975fae0e3b398b84527e89f": "\\rho_{ef}", "cf0de54527fb9fe9038126d0dc576d5a": "x_1, x_2", "cf0de71fd738272d4c51131f0ffe5453": "c' \\equiv \\sum_{i = 1}^n \\alpha_i w_i\\pmod{q}.", "cf0df846a79b79a1f6a2684b72a1e182": "7 (1+0+0) + 3 (1+0+2) + 9 (1+0) = 25 \\mod 10 = 5.\\,", "cf0e09dc85c890948e45bfda27d055ca": "x^* = 0 \\,", "cf0e3f80b8ceddf483396071893e566d": "f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2", "cf0e6dfccce3ea39d307fed131cde47f": "\\prod_i p_i^{e_i}", "cf0e6e9b494dfd64135c9940f1580e84": " a_{02} = \\mathcal{L}(p_6)+p_3p_6+p_2p_8,", "cf0e74380dc3bf717481bcba4309621e": "J_{qi}=\\tfrac{1}{2}\\rho\\langle (w_i-V_i)(w_k-V_k)(w_k-V_k)\\rangle", "cf0eb1ec04c10ad58d239bba2dc8168b": "(5.395 \\times \\ln INR) + (1.485 \\times \\ln creatinine) + (3.13 \\times \\ln bilirubin) - (81.565 \\times \\ln Na) + 435", "cf0ee45948c77a2f62dc7e520ad778c7": "-dx;", "cf0f1bef73c1e6d3e81adcef690e5bb8": " {\\hat x}= i\\hbar {\\partial \\over \\partial p} ", "cf0f2d7819e948139e40272f6d78f8bb": " \\langle f,h\\rangle (x):=g(f(x),h(x)).", "cf0f31cba36a99bd3bdb1577a9f3d4dd": "\\int_{0}^\\infty R_m(x)\\,R_n(x)\\,\\omega(x)\\,dx=\\frac{\\pi c_n}{2}\\delta_{nm}", "cf0f4ea093700fcf83eab5944feb7a05": "C_i = E_K(P_i \\oplus P_{i-1} \\oplus C_{i-1}), P_0 \\oplus C_0 = IV", "cf0f6f05771bc68a7e1c48c76501aa4b": "M=L_w x_g-(l_t-x_g)L_t\\!", "cf0f81eeda1e53bed5acce04d991f961": "{\\mathfrak b}", "cf10135b0b9edadf1e5cc4d07bb82680": "\\scriptstyle \\frac{1}{z} = \\frac{\\bar{z}}{|z|^2}", "cf10aaa29ef7dde108850b2cd6b7c50c": "y_k=y(t_k)", "cf10b55979df34a475b461d2a85752f2": "I=[a,b),\\ b\\leq+\\infty\\ ", "cf10f1c7bee75d34fd0e988aa05cf635": "(P,\\le_P)", "cf112d1858db4b1d4f6941d08caf2d35": "\\Delta{K}=K_f-K_i", "cf113a24ab3ce113d3f89577056154ef": "\\rho\\left(u \\frac{\\partial u}{\\partial s} + v \\frac{\\partial u}{\\partial y}\\right) = \\frac{\\partial}{\\partial y}\\left(\\mu \\frac{\\partial u}{\\partial y}\\right) - \\frac{d p}{d s} + \\rho g", "cf113af99a5e0e8a5f708ffb2c342321": "|e\\rangle|0\\rangle", "cf1177bf742b9ad0d3d5eea841fd0033": " x_i\\in A", "cf11a70df8f6ad623d36bfe07e5fd59e": " x \\triangleq \\mathbb{E}_0\\left[\\frac{B}{S_0(T)} \\right] < \\infty ", "cf11ae776a39134146bf76b2ff5c2111": "\\mathbb{C}[\\partial]", "cf11bd30387d72e00112fe377c48bd6a": " \\Phi(g h, x) = \\Phi(g, h \\cdot x) \\Phi(h, x) ", "cf12389bd94b00493b927be279f2b32e": "\\ell_1 + \\ell_2", "cf12913d43f9ecda54e954ac3b965ac0": "\n\\arccos z = -i \\log \\left( z + \\sqrt{z^2 - 1}\\right), \\,\n", "cf1325a6172e37928741541d19eb7712": "q=\\exp(2\\pi i z)", "cf135259fb4351ec6f5b226b641f7685": "\\mathcal{L}_{Y} T", "cf147fdd167e0064067de2bac2d8882f": "\\bold{r} = \\pm \\pi / (2 \\bold{k})", "cf14ce35f791b97a72d673acbbc6084b": " (\\mathcal P,\\mathcal Z, \\in) ", "cf1530ea8cf47ec85d537be9793d704b": "\\mathcal{C}\\in\\mathbb{Q}[\\mathfrak{A}].", "cf1538b47e93f6f25cdf0c0bb35ce0ab": "S_2 = 2m \\left[\\frac{m^{(2)}-\\bar{x}^2}{2\\bar{x}}\\right]^2 = \\frac{m(\\tilde{d}-1)^2}{2}", "cf159ad12d3ee54c36bddab7aa91daf0": "w_i=(u_i p+v_i)", "cf15acb7cb23207c554ee0ec0fb0e551": "i_L", "cf169d013219e15fe527978cb4d8fccd": "\n\\dfrac{\\theta + |B| \\alpha}{n + \\theta},\n", "cf16b5659ff721d1ed8206a7080129ce": "f_{\\alpha \\beta} = g_{\\alpha \\beta} = \\forall x_\\beta \\centerdot f_{\\alpha \\beta}x_{\\beta} = g_{\\alpha \\beta}x_\\beta", "cf16e1b1ef62393ad84a1ec135640569": "\\tfrac{n+1}{n}", "cf179208aed1c7fc83186e255d53ce2c": "I(q)", "cf17c3c9fc590569d7b982a3c7a8b157": "[c, c+\\Delta c)", "cf1811ee19ec04510ff7bb985dc28d17": "t=T-1", "cf1843ea6dae67233326c2f62d0be736": "P(x) \\gets (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\gets Q(y))", "cf191b60a0f97cf01029f96a2e7b8a14": "\\neg(P \\and Q) \\vdash (\\neg P \\or \\neg Q)", "cf19d7568e4028dd0d5441661b4a4c00": "\\bar \\partial_{j, J} f := \\frac{1}{2}(df + J \\circ df \\circ j) = \\nu.", "cf19fbcd5c61b7777c115d278e7db5d5": "Z=\\sum_{n=0}^{\\infty } \\frac{(-1)^n(4n)!(21460n+1123)}{(n!)^44^{4n}882^{2n}}\\!", "cf1a0dab12a21731cd60057db6d70be8": "\\mathbf{e}_i\\otimes\\mathbf{e}_i = \\boldsymbol{\\mathit{1}}", "cf1a0f4e18dda280358858f2a5324db0": "p_i = \\left[ \\max_{a \\in A} \\sum_{j \\neq i} b_j(a) \\right] - \\sum_{j \\neq i} b_j(a^*).", "cf1a15eb3cd7272e3b6f1cd6f902b9c2": "\\mathrm{im}(A^\\mathrm{T})", "cf1a784e5f9f9c08d1e5acf36bc07581": "\\vartheta(x;a,k) = \\sum_{p0 && \\text{ for }\\varphi\\ne 0,\n\\end{align}", "cf27611b7c0e65827a7bbee559da3e06": "(S,\\,I,\\,F)", "cf27655a2131532367f88b51a11b4900": "U\\left(x,y\\right)", "cf279f311c8a54fdd0a7a0b2c950df8a": "a_{14}+a_{15}", "cf27b1366a4a6e36df50a8fed277d018": " E_6^{\\mathbb C}", "cf281c98aee7151c463a2dd2857755bf": "x_0=(-3,-4)", "cf283a15ff56b1c622d8987a40ccaad0": " T=\\frac {1}{2}(\\frac{1}{\\rho c_0}\\frac {d \\rho c_0}{dz} + \\frac{1}{\\rho c_0}\\frac {d Q}{dz}) \\quad(2.8.b) ", "cf283d55010da4f2267a2663b21c1ad1": "V_h \\subset V", "cf28a85cfacf4a443eccf33cfc3f8bc7": "\\int_0^\\infty \\frac{x^{m}dx}{1+2x\\cos\\beta +x^{2}}=\\frac{\\pi }{\\sin (m\\pi) }\\frac {\\sin (m\\beta)}{\\sin \\beta}", "cf28c45458b9b471310371929c936ffc": " A=\\begin{pmatrix}0&1\\\\4&0\\end{pmatrix} ", "cf292b8f4f769611953f3841c1f9509b": "\\mathrm{F_{\\rm thrust}}=v_{\\rm e} \\cdot \\dot m \\,", "cf297a5790db2df2c8f5444d7af32b31": "S\\cong \\operatorname{End}(P_R)", "cf29a053a19041a447f59dcff6bdea01": "\\begin{align}\n&\\text{Abraham} & \\phi(\\beta) &=\\frac{3}{4\\beta^{2}}\\left(\\frac{1+\\beta^{2}}{2\\beta}\\lg\\frac{1+\\beta}{1-\\beta}-1\\right)\\\\\n&\\text{Lorentz--Einstein} & \\phi(\\beta) &=(1-\\beta^{2})^{-1/2}\\\\\n&\\text{Bucherer--Langevin} & \\phi(\\beta) &=(1-\\beta^{2})^{-1/3}\n\\end{align}", "cf29bee2122127f95a10e85380022130": " \\cos^2 \\theta \\!", "cf29d9fee9356e6ca6a2a99dac9336d1": " [X+\\xi, Y+\\eta] = [X,Y]+(\\mathcal{L}_X\\,\\eta -i(Y) d\\xi +i(X)i(Y)H)", "cf2a088db9fd12872d4b4e4d070b33ef": "\\begin{align}\\gamma &= \\sum_{m=2}^{\\infty} (-1)^m\\frac{\\zeta(m)}{m} \\\\\n &= \\ln \\left ( \\frac{4}{\\pi} \\right ) + \\sum_{m=2}^{\\infty} (-1)^m\\frac{\\zeta(m)}{2^{m-1}m}.\\end{align} ", "cf2a38998d9d33d524599258973bae30": "k_Br_B=20", "cf2aad360d00e44f90bc6123808cc383": "2^{70}", "cf2ab6dfa2fd1ccf95742ad3e9224ab9": "f(P, V, T) = 0 ", "cf2b47ea618160524017aa72b878ac23": "80/81", "cf2b9353a42d342006feeeb572d80723": "\\begin{cases} x_1 + x_2 + \\dots + x_{n-1} + x_n = -\\tfrac{a_{n-1}}{a_n} \\\\ \n(x_1 x_2 + x_1 x_3+\\cdots + x_1x_n) + (x_2x_3+x_2x_4+\\cdots + x_2x_n)+\\cdots + x_{n-1}x_n = \\frac{a_{n-2}}{a_n} \\\\\n{} \\quad \\vdots \\\\ x_1 x_2 \\dots x_n = (-1)^n \\tfrac{a_0}{a_n}. \\end{cases}", "cf2bb5e6c36bcc38e0f5ae06ccf8b39c": " G = G(T,P,\\{N_i\\})\\,.", "cf2bbd6bcb4728b190cb8fab7cbe29d4": "(N/2) \\cdot S_{\\mathrm{min}}", "cf2bd700b07cebefd8a81e2ddd375875": "y'(x)+3y(x)=2", "cf2c6adb448bb97cbb73d4ae0cd129d6": "\n\\Pr(B_n = B) =\\dfrac{\\Gamma(\\theta)}{\\Gamma(\\theta+n)}\\dfrac{\\alpha^{|B|}\\,\\Gamma(\\theta/\\alpha + |B|) }{\\Gamma(\\theta/\\alpha)}\\prod_{b\\in B}\\dfrac{\\Gamma(|b|-\\alpha)}{\\Gamma(1-\\alpha)}.\n", "cf2c7b9b9ea5384f49aae97c6a84fdce": " \\psi^{(0)}(z) = \\ln(z) - \\sum_{k=1}^\\infty \\frac{B_k}{k z^k}\n", "cf2c808e27c63a7acee1d53b3ed75c8e": "5x^2+8xy+5y^2=\n\\begin{bmatrix}\nx&y\n\\end{bmatrix}\n\\begin{bmatrix}\n5&4\\\\4&5\n\\end{bmatrix}\n\\begin{bmatrix}\nx\\\\y\n\\end{bmatrix}\n=\\mathbf{x}^TA\\mathbf{x} \n", "cf2cbb5dd133615288040891faedc23f": "\\textstyle c = \\left(rP, m \\oplus H_2\\left(g_{ID}^r\\right)\\right)", "cf2cd727f44e8a3a05e9a525ad227729": "L_*(\\mathbf{Z}[\\pi])", "cf2d0ebfe6ad5deba65ac33a42e631bc": "\\ \\quad0\\le t\\le t_1\\quad", "cf2d983d4b7379c4730a55c2942f845e": "*1", "cf2d98cadfcaffea2b64e1fcdbfec2fe": "P = \\Delta^2 + \\delta \\left\\{(n-2)J - 4V\\cdot\\right\\}d + (n-4)Q", "cf2dc52be2a6c1619d28a2b302971a10": "\\uparrow \\uparrow", "cf2e6e32287b223625f11cba25692dd5": "= \\int_{-\\infty}^\\infty f(t-\\tau)\\, g(\\tau)\\, d\\tau.", "cf2ed7b30c6af503c8387c75cf933260": "M^T \\Omega M = \\Omega", "cf2f3aa7274b450d98f10cc615b9299e": "c \\leq r", "cf2fea8a1c04343b58634f9c424312df": "\\, R_{ab}", "cf305d58e170f6e892c0ff06525add46": "\\scriptstyle\\lambda_0\\,", "cf30c489ecfe0f067056cc592170b30c": "B \\to A", "cf30e5a91f2a03c2f774601066418113": "m \\frac{d^2 x}{d t^2} + B \\frac{d x}{d t} + kx = F_0 \\cos(\\omega t)", "cf315689e0642b3dd47c7b45139575c1": "\\delta_i;i=1,\\dots,n", "cf31a7b04ceeaf2ce92b63ed2c2a3198": "\\begin{align}&|DE|\\leq|\\widehat{DC}|\\leq|\\widehat{DG}|\\\\ \\Leftrightarrow &\\sin(x) \\leq x \\leq\\tfrac{\\pi}{2}\\sin(x)\\\\ \\Rightarrow &\\tfrac{2}{\\pi}x \\leq \\sin(x)\\leq x \\end{align}", "cf3203b8a6bf048294d7449ed4a4c8ba": "\\lambda_M(a,b)=\\langle a\\smile b,[M]\\rangle \\in \\mathbb{Z}", "cf323f415956f8cbc028119bf42d899b": "A \\wedge B \\equiv A \\otimes (A \\rightarrow B)", "cf32457b0b0d3e87c4fb4f14d383a595": "y R y", "cf328df089e2cedf48fa06573451a769": "f(.) \\,", "cf32e16bfb10426a660a5d8af2b015cb": "W_{\\beta,B} = b\\left(W_{TOT}-W_\\alpha\\right)", "cf32fab024741001bfb36cb68aaf67a1": "\n[a] =[x_1]+[x_2]\n", "cf332163ba36a8f223b46c9a264b22f1": "\\mathcal{M}(P(x))=1", "cf333bd4760ecd5b596a3e0b0ecadbb6": " \\sigma_0(p_n\\#) = 2^n \\, ", "cf3347eac5a5cab70e6e893685bd3359": "\\Omega_{\\Lambda} \\simeq 0.7", "cf336c1f2f9df65a47c32173a23d6d64": "a'", "cf349bf58592148dba23771b2541c7e4": "\\rho_{\\uparrow,\\downarrow}=\\frac{2\\rho_F}{1\\pm\\beta}", "cf34a69acca9ef34165268dcabd5a013": "\\chi = \\frac{c_1^2+c_2}{12} = \\frac{(K.K)+e}{12}", "cf34ffb3012aaa601ce28a53a8507b43": "\nm \\ddot{x} + A \\frac{d^r x}{dt^r} \\imath + k x = 0\n", "cf351d1365f3adb552b36baa99dc9638": "N_e^{(v)}", "cf353e86fb5c0d03cce0b6961f9fdb14": "C_0=", "cf3558398a6bb93cce9b327164939305": "\\sum_{k=0}^n \\left[{n\\atop k}\\right] = n!", "cf356d4bb15fdf57e23203c466adcfd1": " m_a = m ( 1 - \\frac { var( \\log( m_i ) ) } { 2 } ) ", "cf35ad3e01c5fe2b1bf6f0a31f0e2b2b": "E\\neq 0", "cf35fbec7c051c9c823677fec2a60067": "M_{f,C} x = C x \\cdot f^{-1}\\left( \\frac{f\\left(\\frac{x_1}{C x}\\right) + \\cdots + f\\left(\\frac{x_n}{C x}\\right)}{n} \\right)", "cf362e129b583b0e6304af487665e3b9": "\\max_{x\\in X,v\\in \\mathbb{R}} \\ \\{v: v\\le f(x,u), g(x,u)\\le b, \\forall u\\in U(x)\\}", "cf367571d02ce2a9930d1570f0869b6a": " q(z) = \\sum_{k=-K}^K c_k z^{k}, \\, ", "cf36c19a0a6e3d1808447ab9a003dd3b": "C \\subset \\mathbb{F}_2^n", "cf36c2b3aeb0ce3db768f3ef9489909e": "\\ c_f(u,v) = c(u,v) - f(u,v)", "cf36dec29bae9f427489e83a0aeea1d7": " \\lim_{t \\rightarrow \\infty} (J^t \\cdot \\mathrm{Transpose}(J^t) )^{1/2t} ", "cf36e8f399d464b157c0f519b2e26148": "p=(3.0, 0.0)", "cf37148ac29fef08738c0c2ebd5f3768": "R=|r|^2=\\cfrac{1}{1+\\cfrac{\\hbar^4k^2}{m^2\\lambda^2}}= \\cfrac{1}{1+\\cfrac{2\\hbar^2 E}{m\\lambda^2}}.\\,\\!", "cf3760b045ef8bfc87c71c094c5af4f8": "CDecode\\,", "cf376a16bfe28fa39be14c9c62fa9fa9": "\\underline{\\varphi \\quad \\quad \\ \\ }\\,\\!", "cf378506005aa63d36d9bd9251512f98": "\\delta m^a=(\\beta-\\bar{\\alpha})m^a+\\mu l^a-\\bar{\\rho} n^a\\,,", "cf3785ee6429783a518fe1baa5334778": "f(x)-f(x_0)\\ge v^*(x-x_0).", "cf37ad69f168212ef99ec4081440601f": "S/ R", "cf37c1895fe52b7344a7b7dc0bf9b46c": "\\mathbf e_i = \\alpha_{iJ}\\mathbf E_J\\,\\!", "cf37dbff40106411bf79d83c4d22ce92": "\\sin(\\theta) \\approx 1", "cf384492f34674daec5e8020d12818d8": " f\\circ \\alpha = \\beta \\circ F(f)", "cf38708e7a1b6167b3cda9b6da888718": "\\Delta\\left(\\frac{dS_{r}}{dr}\\right)^{2} - \\frac{1}{\\Delta}\\left[\\left(r^{2} + \\alpha^{2}\\right)E_{0} - \\alpha L\\right]^{2} + m^{2}r^{2} = -K", "cf38862d3dfbfc7284f1c3644dd3192e": "\\textstyle (X, Y),", "cf3892b0c98684f83d80367aaba65355": "(n - 2)^2", "cf38dd92cbee6c83b2b660bff1e779ba": "R(X)", "cf3904572b42c571060f5cea160eb990": "x |q-1|", "cf428998b812b231d29f5f8d2ba1f19f": " T: t \\mapsto -t.", "cf428a460545edc9546af75a2194696b": "\\begin{cases}\n\\dot{\\sigma} < 0 &\\text{if } \\sigma > 0\\\\\n\\dot{\\sigma} > 0 &\\text{if } \\sigma < 0\n\\end{cases}", "cf42e285d0e71b7d9d4320646f1ba34c": "\\rho_\\mathrm{out} = S_{22}\\,", "cf431afe3e3106e9db03a11fbb765c25": "\n\\varepsilon^v_{S} = \\frac{K_m}{K_m + S} \n", "cf43572b551826535f37520a652b0bf5": "\n\\begin{align} \n& A= \\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}+\\frac{v}{2r}|_{R_{0}}\\right)= \\frac{3V_{0}}{4R_{0}} \\\\\n& B=-\\frac{1}{2}\\left(\\frac{V_{0}}{R_{0}}-\\frac{v}{2r}|_{R_{0}}\\right)=-\\frac{1V_{0}}{4R_{0}} \\\\\n\\end{align}\n", "cf43707fd5846fe135a6851c6c676c14": "\\textstyle F_{t}=\\phi ", "cf43975f71a47e1c4c41fba75287db2f": " [x] \\geq [y] ", "cf439f2510af503135d378277cac260f": "\\mu = \\frac{(1-\\epsilon)\\operatorname{Tr}(Q\\rho)}{1 - \\operatorname{Tr}(Q\\rho)} ~.", "cf43f5faf4802229a0375e676a3deb83": "x\\in\\R", "cf447dc134d5ec8e50a83915ab901301": " (f \\cdot g)(a) = f(a)g(a)\\mbox{ and } (f \\cdot g)(b) = f(b)g(b). ", "cf44a21df14fa699995d02a6ad4b1e16": "p_{\\text{non-wetting phase}}", "cf44bcc973e210e4671f2fbf50279517": "\\operatorname{SP}(n) = \\mu(n),", "cf44e62d6f6ac6d240a49e8a33c6c06d": "\ne^{2i\\pi\\frac{k}{n}}\n", "cf450259e475304bc86c3cd555c22f19": "{D}/{E}", "cf45626c898b5854e001d1525cbccc46": "\\displaystyle{\\mu^\\prime(f(z))=-{f_z\\over \\overline{f_z}}\\cdot \\mu(z)}", "cf45a1278403ef0f2674d8ab37727a54": "p(U)^\\perp", "cf45d044dda7081f1c056613ca7f90b4": "T = 1+6.585\\left( \\frac{t} {L} \\right)^2", "cf45f5008a44bca8f581f1cb4fe1dc3a": "\\mathrm{AB} = R \\arccos\\left\\{\\sin \\lambda_\\mathrm{A} \\,\\sin \\lambda_\\mathrm{B} + \\cos \\lambda_\\mathrm{A} \\,\\cos \\lambda_\\mathrm{B} \\,\\cos \\left(L_\\mathrm{A}-L_\\mathrm{B}\\right)\\right\\},", "cf4616b6ff018509351b201f2b5ba07d": "\\scriptstyle X_j", "cf463506570c43200b8a00f2138cf5dc": " \\alpha \\approx {\\frac{2 \\lambda}{W}} ", "cf466e671576166a29bc4b092688460c": "\\bar s = \\frac {\\sum_{i=1}^m s_i}{m}", "cf4675048488ff0dd96405f0c28a6c6d": " N(\\mu, \\sigma^2) ", "cf46f503cb0fe4b7baac5d04d32ed3cf": "\n\\begin{align}\n\\dot{x}(t) & = A x(t) + B u(t) \\\\\ny(t) & = C x(t)\n\\end{align}\n", "cf46fbb908b6100bb49adeccd397ac59": "a+bx,", "cf47087b7e84c8b54b3a67e5216feecd": "R = \\frac {\\pi L} {E} = \\frac {\\pi K} {C}", "cf474def7767d492e1e674642a75a7ba": "H(x) = \\textrm{sign}\\left( \\sum_i \\alpha_{i} h_{i}(x) \\right)", "cf4759ee94e7c333a1dba849404a9b7d": "A \\# B", "cf475b94bef7afe5576540e908012aee": "q=\\frac{Aq+B}{Cq+D}", "cf478b3cbd60efc4be139f2a49b8d8cd": "nw(X)\\leq w(X)\\,", "cf479aee7c20e40fa4be85123bc1a703": "\\alpha/v", "cf47f2b4e5cbb912a2a98c943caf0d3a": "\n\\left(\\!\\!{-5.5\\choose 7}\\!\\!\\right)=\\frac{-5.5\\cdot-4.5\\cdot-3.5\\cdot-2.5\\cdot-1.5\\cdot-.5\\cdot.5}{1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6\\cdot7}={.5\\choose 7}=\\frac{33}{2048}\\,\\!", "cf482c5807b62034beeabdb795c5a689": "IV", "cf48a683298427cffeaaefcaf09c495c": "\\frac{2^{-\\nu/2}}{\\Gamma(\\nu/2)}\\,x^{-\\nu/2-1} e^{-1/(2 x)}\\!", "cf48d565d46cc498b1f72eefd81db6e3": "O(2^{N/2})", "cf48f26a7d2e607dbf10dcab6916b6b2": "\\angle PC_2C_1\\,\\!", "cf48fde528f3da1b1df95899d1912e3c": "v(z)=\\sum v_n z^{-n-1}.", "cf4921293399681174050a3450a176f0": "(g_{jk})\\,", "cf49c9576c263c80acdac4d223670b28": " H = -\\frac{d^2}{dx^2} \\ln |p(x)|= \\frac{1}{(x - x_1)^2} + \\frac{1}{(x - x_2)^2} + \\cdots + \\frac{1}{(x - x_n)^2}. ", "cf4a0b8916b34d49a9228310be6b2cf3": "RR= \\frac {p_\\text{event when exposed}}{p_\\text{event when non-exposed}} ", "cf4a37174ea44c687469cff3422c5f7b": "z_{xx}>z_{xy}>z_{yy}>z_x>z_y>z", "cf4a48a9128d06eacfdc320a43be5f63": "g= \\operatorname{E}(X \\mid Y)", "cf4ae7601e3b41430abc7c1dfbd14256": " \\sigma_1,\\ldots,\\sigma_{n} ", "cf4b204c6cce9a5e3375c068640f64e2": "k = 0, \\dots, N-1 ", "cf4b5fac46ebe5e573552e88fe0d5457": "\\frac{dP(r)}{dr} = -\\frac{ \\left( \\rho(r) c^2 + P(r) \\right) \\left((e^{\\lambda(r)} - 1)c^4 + 8 \\pi G r^2 e^{\\lambda(r)} P(r) \\right)}{2 c^4 r} \\;", "cf4b8bb69f459e5bd23f74eea48b2801": "+ x^4", "cf4b9b283e367f2ddac5223221385abd": "\\lambda = \\frac{\\hbar}{\\sqrt{2mE}}", "cf4bf34bd9344414740725e299108789": " v_2^2+v_3^2=2E_A \\qquad v_1^2 = 2E_B", "cf4c8a1a773e6dbaae04e494ba62ca79": "D \\ge 5 ", "cf4cbb2b6876b335814ceb099d78f97b": "\n\\begin{align}\n\\frac{D^2F(P_0)}{DP^2} & =\\frac{DF'(P_0)}{DP}=\\frac{F'(P_1 < P < P_2)-F'(P_0 < P < P_1)}{P_1-P_0}, \\\\[10pt]\n& {\\color{white}.} \\qquad \\ne\\frac{F'(P_1)-F'(P_0)}{P_1-P_0}, \\\\[10pt]\n& =F[P_0,P_1,P_2]=\\frac{F(P_2)-2F(P_1)+F(P_0)}{(P_1-P_0)^2}, \\\\[10pt]\n& =F''(P_0 < P < P_2)=\\sum_{TN=1}^\\infty \\frac{F''(P_{(tn)})}{UT}, \\\\[10pt]\n& =G'(P_0 < P < P_2)=H(P_0 < P < P_2).\n\\end{align}\n", "cf4cf26fa28ce4825716ffeb535243b6": "\\begin{bmatrix}\n 1&1&0&1\\\\\n 0&1&0&1\\\\\n 0&0&1&1\\\\\n 0&1&1&0\n\\end{bmatrix}", "cf4cfcfe829c278344306a1c233df46a": "X \\times_S H_X^P", "cf4cfe49d6bc29e37409d2cdbf70e2bd": "\\delta W = p\\,\\mathrm{d}V\\,", "cf4d23e487caeeddb197e458bf062ae0": " \\sigma(\\mathbf{x}) = \\mathbf{0} ", "cf4d549c7a5648cafff3bda0b53a3ade": "f_{0}/2", "cf4d8188c0462b0bb3736cf1f0cde710": "\\frac{a}{2^b}", "cf4e0dca7deaca8291ec10024f5afe8c": "m_n(\\cdot)", "cf4e6ef7eab73926bc069e86138f56ae": "\n\\Gamma(z) \\; \\Gamma\\left(z + \\frac{1}{k}\\right) \\; \\Gamma\\left(z + \\frac{2}{k}\\right) \\cdots\n\\Gamma\\left(z + \\frac{k-1}{k}\\right) =\n(2 \\pi)^{ \\frac{k-1}{2}} \\; k^{1/2 - kz} \\; \\Gamma(kz) \\,\\!\n", "cf4e856738aee46c8cd9347023dd0c28": "x_{22}=-x_{21}\\,", "cf4e8b68cb8bb92aa4e533cad080687c": "\\operatorname{util}(x,i)", "cf4f75f37390aace59d6bbffc31a2dc6": "F_b:=\\pi^{-1}\\left(\\left\\{b\\right\\}\\right), b\\in B", "cf4f7eb5c35f2e655ab334ebf57a7abb": "\\nu \\,", "cf4faa28df63ee7eea69d28be5a0e7d7": "R = k[x,y]/(xy) ", "cf4fc9735b4f09d1ac2f47f66f23b423": "k[x_{ij}] = \\bigoplus_{r\\ge 0} A_k(n, r).", "cf4fd7ed3d5da7034d6553675f56707b": "f^{-1}(a) = \\sum_{k=0}^\\infty \\binom{5k}{k} \\frac{(-1)^k a^{4k+1}}{4k+1} = a - a^5 + 5 a^9 - 35 a^{13} + ...", "cf50f68d8d3e3915e37fc56a10e30f4f": "\\sqrt{2\\cdot 8}=4", "cf511b9bff007cd3197bd1ecc701dee1": "\\sum_{k=1}^{+\\infty} T^{k+\\frac{1}{k}}", "cf512417b513dae4630aa446f7136c98": "\nE_{+,-}=E_{kin,+/-}+m_e c^2=\\sqrt{m_e^2 c^4+\\mathbf{p}_{+,-}^2 c^2}.\n", "cf513decf6e4ace0e25cb1c932aaa049": "|x|", "cf515279cc937167b4fe532d12a98dc6": "F=\\langle W,R\\rangle", "cf517e9029fba22311ec0e5fa8442921": "\\tan \\frac{\\theta}{2} = \\frac{\\sin \\theta}{1+\\cos \\theta} ", "cf5194f5f9a46fe115463fa6f29bcf65": "\\log\\ e^*\\ = ", "cf519aa1af7c612ed79625113dbf2df5": "Y \\mathbf{\\operatorname{di}} X", "cf51d4ccac4633495d83c06ce64bfb56": "|V|-1", "cf5235872cfab8b8b9ea047a72b8cdf8": "P_{i,i-1}=\\beta_i", "cf529d799a50ebcd1c020c80c78a789e": " E(m_{k+1}\\,|\\, m_k) = m_k ", "cf52a93ce80a3af53f7c794ad728143c": " x = 1.22\\ \\frac{\\lambda f}{d}", "cf53999e557951025f41c68a3a42e786": "\n=-E_{12} E_{22} + E_{11} E_{12} -\n(E_{11}- E_{22}) E_{12} - 0 +E_{12}\n ", "cf54028efb3fce64500876f8d4fa2d32": "E(\\vec{r}) = \\sum_{i=1}^N\\sum_{j\\in viz_i} (1 - J_{ij}\\sigma_i \\sigma_j)", "cf544376189fb393cfc331a5f4026900": "x(0) = y(0) = 1 \\,\\!", "cf5461e164791ee51f9def8620183f28": " \\Phi [f] = \\frac{1}{(2\\pi)^2}\\iint\\!\\!\\! \\iint f(q,p) \\left(e^{i(a(Q-q)\n+b(P-p))}\\right) \\text{d}q\\, \\text{d}p\\, \\text{d}a\\, \\text{d}b.", "cf5471728a8b554744b46aa162691140": " \\mathbb{P} \\left(\\sup_{0 \\leq s \\leq t} W(s) \\geq a \\right) = 2\\mathbb{P}(W(t) \\geq a) ", "cf54f1c596a82e45196b830074cb5f5b": "Q_s", "cf550a12c214b67b36ed6c47dd872ca8": "A^f = \\sqrt{I_0}", "cf551ec55e87e0ef731a66be99e88727": " \\frac{1}{\\Gamma} = \\frac{1}{\\Gamma_{max}} + \\frac{1}{\\Gamma_{max}Kc}", "cf55336855bbc0dd66c2631696e81776": " \\phi^4 ", "cf553f8fdd7c017fbddc39310e207ffe": "\\bigcup_{A \\in C}{^*} A", "cf555f954a4c831bf8ca403a185be998": "\\log_b \\bigg(\\frac{x}{y}\\bigg) = \\log_b(x y^{-1}) = \\log_b(x) + \\log_b(y^{-1}) = \\log_b(x) - \\log_b(y)", "cf5561d8833d5b5a5a6a05e2d864d9c9": "e^{-k_2t} \\ll e^{-k_1t}", "cf558b59dd78fe0fe79ac2cfdcf56793": "\\displaystyle{j T_a j T_b j = j T_c T_\\lambda j T_{\\lambda^{-1}} T_d \\circ j = \\lambda^2 j T_c j T_{-\\lambda} j T_d \\ j = \\lambda^2 T_{-c^{-1}} j Q(c^{-1})T_{-c^{-1} -\\lambda - d^{-1}} j Q(d^{-1})jT_{-d^{-1}},}", "cf55c0c3f131b0ec19da3b9e9cce581b": "SH_k X", "cf55ca381648c0e951cc8dae2fe2e1cb": "\\frac{dE}{dt} = E_0b \\frac{bt^{a-1}e^{-bt}}{\\Gamma(a)}", "cf55cc289659d30fc235f80b939479de": "\\phi^3", "cf55d9365b3ce436084d6659309cee09": "p,q\\in{\\mathbb Q}(x)", "cf561357095da10464eed7b7d9a66fe2": " \\overline \\Gamma=dx^\\lambda\\otimes(\\partial_\\lambda +\\Gamma_\\lambda{}^i{}_j(x^\\nu) \\overline y^j\\overline\\partial_i).", "cf5627a49acb9518b23257db6b926bac": "D_{\\mathrm{KL}}(\\lambda\\|\\lambda_0) = \\lambda_0 - \\lambda + \\lambda \\log \\frac{\\lambda}{\\lambda_0}.", "cf564ecc0a6add582af9053d97eaa2da": "x = 1468", "cf5651271d94acea94cf22a9fe97a8d4": "\\sum_{S\\colon e \\in S} x_S \\leqslant 1 ", "cf565515ddc98530292529dcd7634dca": "\\left(\\begin{smallmatrix} 0 & 1\\\\ 1 & 0\\end{smallmatrix}\\right)", "cf567120490e694b9c7d90b6224c1572": "c\\cdot s(n)", "cf568ff7ae3ebce07691c4a98a2f1e1a": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = x_{1} \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = g\\left(\\boldsymbol{x}\\right) h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) \\\\\n g\\left(\\boldsymbol{x}\\right) & = 1 + \\frac{9}{29} \\sum_{i=2}^{30} x_{i} \\\\\n h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) & = 1 - \\sqrt{\\frac{f_{1}\\left(\\boldsymbol{x}\\right)}{g\\left(\\boldsymbol{x} \\right)}} - \\left(\\frac{f_{1}\\left(\\boldsymbol{x}\\right)}{g\\left(\\boldsymbol{x}\\right)} \\right) \\sin \\left(10 \\pi f_{1} \\left(\\boldsymbol{x} \\right) \\right)\n\\end{cases}\n", "cf56906ef60023553d943fa47c19a501": " Z = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}", "cf569ff60f8f5310adb505ae48960119": "n \\times k", "cf56b1b100f41c0db9ee8c36ced06cbe": "H = \\sqrt{3\\varepsilon}", "cf56bdbacea2e7db099c93c0fd13c1e2": "d\\mathbf{l} = d\\rho \\, \\hat{\\boldsymbol \\rho} + \\rho \\, d\\phi \\, \\hat{\\boldsymbol \\phi} + dz \\, \\hat{\\mathbf z}", "cf56dd80adb895dd2577b5b60f6a6c59": "\\cos C= -\\cos A\\cos B+\\sin A\\sin B \\cosh c,", "cf5707aa0c3cfdcc5e64f38dad048424": "F_k \\cup \\Phi(F_k)", "cf573bc136eb5c7342748f844105a271": "\n\\dot{l}_a = \\{ H, l_a\\}, \\dot{n}_a = \\{H, n_a\\}\n", "cf574a521ca3dfb8e687f0be7ad4dcd1": "F \\equiv F_{\\mu\\nu}\\,dx^\\mu\\wedge dx^\\nu\\,.", "cf578940248a17113480e9c93798a1c6": "E=\\{a_1a_2\\ldots a_n\\,\\vert\\; e=a_1a_2\\ldots a_n\\,;\\, a_k\\in S\\,;\\,n\\mbox{ finite }\\}", "cf57d295129104f39283d4474e8c5105": "\\widetilde{E}^{\\pm ac}(\\widetilde{t}\\mp\\tau) = \\mathfrak{F}\\{ \\sqrt{I(\\omega)I(\\omega-\\Omega)}e^{\\pm i[\\phi(\\omega)-\\phi(\\omega-\\Omega)]}e^{\\pm i\\omega\\tau} \\}", "cf57f9b77d63ce3eee10144401c06922": "0 < F(a) = p < 1 ", "cf58148941141be06eb7923a89f29f2d": "(\\mathbb{C}\\otimes\\mathbb O)P^2", "cf581bb224ec0001c9f58243d2556179": "a_1,a_2,\\ldots,a_k", "cf5837be446d4ad5aa6a8bcbd6d325b7": " t_{j-1,j} = t_{j,j-1} ", "cf586b5b5fb8f7a8328d2ba363c6cbd8": " \\mathrm{idf}(t, D) = \\log \\frac{N}{|\\{d \\in D: t \\in d\\}|}", "cf58b05e03632aae43b26da2a4a08973": "K(S)+\\log |S| = K(x)+O(1)", "cf58e7d77e36bb5c3196b049d9819e50": " \\nabla \\cdot \\mathbf{H} = \\mathbf{0} \\,\\!", "cf592e397ce8c0c669814333651806ac": "\\exists i0", "cf59ea882c130b8f24ea19ff97a6cd2f": "\\boldsymbol{\\mathcal{B}}", "cf5a43bf3c3f147510b26c1bd902bb79": " \\lim_{s\\to 1}(s-1)\\zeta(s)=1.", "cf5a4c926b6cd40cda6af592fac9e8c5": "\\psi_3(1/4)", "cf5a62bd19d03a18e82d705163c124a5": "\\{u, \\; d\\}", "cf5a6a33760360a1848575cdaf1d8509": " e p (D+1) \\leq 1 ", "cf5ab955640e728ec7b92ace61763933": "\\bar{ \\bar \\alpha}", "cf5b11e72b771069ae5f47011484c7ae": "\\text{s.g.}=\\frac{145}{145 - \\text{degrees Baumé}}", "cf5b3d26bb623fcce3a673c3f6566324": "\\mathbf{P}_j = \\prod_{k=1}^{j}\\left(\\mathbf{A}-\\lambda_k \\mathbf{I}\\right)= \\mathbf{P}_{j-1} \\left(\\mathbf{A}-\\lambda_j \\mathbf{I}\\right), j=1,2,\\dots,n-1", "cf5b4d08d227b8eca4de6aea3dc5152a": "\\mathbf{E} \\approx \\mathbf{B} \\times \\mathbf{\\hat{r}}.", "cf5b500b3ee7d00fc232fce19d2eba21": "\\lambda \\left( f_{n(k)}(t) \\right) \\to \\lambda(f(t)) \\mbox{ in } \\mathbb{R} \\mbox{ as } k \\to \\infty.", "cf5bd860c062a69bc4c4557be25f23e9": "p^2 = 0, p \\ne 0", "cf5c284ecb56958c2374d00f976158b6": "N+H=M\\,", "cf5c4aad81e5777eadecae2a9a0a0164": "Q^{-1}A Q = \\Lambda", "cf5c8fecd12670f9a7d4e8cba2eae70f": "g_{i,n}(u) = {{k_{i+n} - u} \\over {k_{i+n} - k_{i}}}", "cf5c93039afc1f4066114d099ec96bfa": "D(\\vec x, t)", "cf5cdd3ba87d541f5753ce01ea6a685e": " P(M_N > x) \\approx \\exp(-NI(x)) .", "cf5d6e9d9459d445d1828a72ac7d481d": " R = 1.00014 - 0.01671 \\cos g - 0.00014 \\cos 2g ", "cf5d8ded8edcf86c73e0f3b04a5af211": "\\underset{x}{\\operatorname{arg\\,max}} \\, f(x) := \\{x\\ |\\ \\forall y : f(y) \\le f(x)\\}", "cf5d98f6c63fbddfc2747c57e9365cc1": "\\text{If }a, b, a' \\text{ and } b' \\text{ are positive and odd and} ", "cf5d9bceaa54a869165774b66e5011d6": "\\eta = 1-\\dfrac{n \\beta}{4}", "cf5e7428e60cf09cd9d8a9ab8715d5a2": "\\beta = \\frac{a + d - \\sqrt{(a - d)^2 + 4bc}}{2}", "cf5e92928bb0a7842eca295f18ac22c7": " x_1, x_2", "cf5ee524a7a9b164c1deb0671223a0ef": "\\overline{M}_i = \\sum_{j} \\frac{z_j}{r_{ij}/w}.", "cf5f02cdec1d10812a5fd882531c2b75": "m^{\\prime} = m(1 + a)\\,", "cf5ffc12029f6241db4bea0ce66ef261": "\\Delta E_{F=I\\pm1/2}", "cf60a2f6877c26cc69019e38270112ad": "k_{\\alpha}(t)", "cf614c2faa0969d3ed4c9bc4b699f539": "\\mathit{p_t}", "cf61bc86469e29c45c53ee05a6922461": " \\alpha_{m,l}(\\boldsymbol{R_n}) = \\int \\varphi_m^*(\\boldsymbol{r}) \\varphi_l(\\boldsymbol{r - R_n}) \\, d^3 r \\ ", "cf621d18d3589e930f3ab0d37f3ca5ff": "\\scriptstyle P(x) \\,\\in\\, \\C[x]", "cf62b9a8e3fe0cd78ec76585b2700dfc": "AC \\cdot BD = AB \\cdot CD + BC \\cdot AD.", "cf632c12b0a2b029e1bacaafab58be04": " G(\\theta) = (\\cos(3\\theta),\\sin(3\\theta)) , \\, ", "cf63753ff2d706ecf4546450743427fe": " M_s = -3.2 + 1.45 M_{L} ", "cf63b1359c07cc9c56f57d9eb64fade9": "u(x,0)=u_t(x,0)=0.\\,", "cf63bd9d79ce9a77486e7a4d482d1e63": "d = 0, 1, 2, \\dots, 2^w - 1", "cf63e050a09afea594c788720b1f4503": " z=\\zeta+\\frac{1}{\\zeta}", "cf640c1b0012ff752f23ef5ee159d550": "(\\hat{\\bold{e}}_1, \\hat{\\bold{e}}_2, \\hat{\\bold{e}}_3)", "cf6441787c13551c9c22e98bd0b96dc9": " \\ln 2b", "cf6477f52a95fa11e1f772320043d62d": "\\langle \\psi|\\hat{G}|\\psi\\rangle=Tr(\\hat{\\rho}\\hat{G})=\\int_{-\\infty}^\\infty dx\\, \\int_{-\\infty}^\\infty dp P(x,p)g(x,p).\n", "cf64ace12e2e7e68d1fc8489a489ace4": "\\begin{array}{ll}\n {\\rm (BL1)}\\colon & (A \\rightarrow B) \\rightarrow ((B \\rightarrow C) \\rightarrow (A \\rightarrow C)) \\\\\n {\\rm (BL2)}\\colon & A \\otimes B \\rightarrow A\\\\\n {\\rm (BL3)}\\colon & A \\otimes B \\rightarrow B \\otimes A\\\\\n {\\rm (BL4)}\\colon & A \\otimes (A \\rightarrow B) \\rightarrow B \\otimes (B \\rightarrow A)\\\\\n {\\rm (BL5a)}\\colon & (A \\rightarrow (B \\rightarrow C)) \\rightarrow (A \\otimes B \\rightarrow C)\\\\\n {\\rm (BL5b)}\\colon & (A \\otimes B \\rightarrow C) \\rightarrow (A \\rightarrow (B \\rightarrow C))\\\\\n {\\rm (BL6)}\\colon & ((A \\rightarrow B) \\rightarrow C) \\rightarrow (((B \\rightarrow A) \\rightarrow C) \\rightarrow C)\\\\\n {\\rm (BL7)}\\colon & \\bot \\rightarrow A\n\\end{array}", "cf64b09d3befc8e781a6228af7288331": "\\mathbf{A_0} = \\begin{bmatrix}\n0 & -\\mu & 0 \\\\\n-\\lambda & 0 & -\\mu \\\\\n0 & \\lambda & 0\n\\end{bmatrix}.", "cf64d4a13a7d049ef4b0ca5547215dc5": "X_{th} = - \\frac{1}{T} \\nabla T", "cf64e96f009c5884e7e18d79d1f981bb": "|P_1(V)| < |P(V)|", "cf650b2ad5adee6b2605130552c04cd0": "\\phi_n(x) = \\frac{1}{\\sqrt{L}} e^{-\\imath k_n x}", "cf65753fdd1c7856fdec0bbc2539c7e7": "\n\\begin{align}\n7.48181818\\ldots & = 7.3 + 0.18181818\\ldots \\\\[8pt]\n& = \\frac{73}{10}+\\frac{18}{99} = \\frac{73}{10} + \\frac{9\\times2}{9\\times 11}\n= \\frac{73}{10} + \\frac{2}{11} \\\\[12pt]\n& = \\frac{11\\times73 + 10\\times2}{10\\times 11} = \\frac{823}{110}\n\\end{align}\n", "cf657d510353c4bb93c07b8a0f2dc8fe": "(2\\pi n)^{-1}|\\sum_{j=1}^n X_je^{iwj}|^2", "cf657d61523d8ea864eeed30ad446059": "n_{\\rm coating}T,\\,\\forall T\\in[0,E_0]\\}", "cf6afe04cd0f09d197b5d7a0b5adf64d": "W^{-1}=e^{-D^2}. \\, ", "cf6b1165bd9b23604a7ec68900ae5f0a": "\\int_{0}^{\\infty} \\frac{\\cos px - \\cos qx}{x}\\ dx= \\ln \\frac {q}{p}", "cf6b6cfd7d040fc9c2f3a5b7100902de": " \\frac{1} {m}_{\\ell} = {{1} \\over {\\hbar^2}} \\sum_{ m} \\cdot {{\\partial^{\\ 2} E_{c} (\\boldsymbol{k})} \\over {\\partial k_{\\ell} \\partial k_ m}} \\approx \\frac{1}{m}+\\frac{2}{E_gm^2}\\sum_{m,\\ n} {\\langle u_{c,0}|p_{\\ell}| u_{n,0} \\rangle }{\\langle u_{n,0}|p_{m}| u_{c,0} \\rangle } ", "cf6b7241fcafcdecb824c9fdfe666203": "(s,t)\\in[-3,3]^2", "cf6b8916fb60db81579ef941b33cad31": "\\frac{1}{6}_{10} = \\frac{1}{2*3}_{10}= 0,16666..._{10} = 0.0(01)..._2\\,", "cf6b9e74bf49b41af3935efc148b58e9": "d_- < m < d_+,", "cf6bcde6b220d2e46992cab0f13fe1df": "\\int_{-\\infty}^\\infty | x(t) |^2 \\, dt = \\int_{-\\infty}^\\infty | X(f) |^2 \\, df ", "cf6bedb273c76cd1090a28faf789b2e5": "N/k", "cf6c6a8c05f388d6bac6c00654be38a7": " (A \\land B) \\lor C \\to (A \\lor C) \\land (B \\lor C)", "cf6c8d99301569e5613ff856de41d641": "\nI_{i,j} = \\frac{p_ip_j}{d_{i,j}^\\beta}\n", "cf6cdc9fbf21d0b79c118a4119e5dd30": "Q(m-\\sqrt{U(|\\Phi|)/|\\Phi|^2}\\Big |_\\min) ", "cf6d1afb4b3aee0b687b11795219f75b": "\\sigma=0", "cf6e1604a09a7070d3af1b5db0a4d0b1": " \\langle p | \\hat{x} | p' \\rangle = i \\hbar {d \\over dp} \\delta (p - p') ", "cf6e2d6c794f74786652be59aee1a2c3": "ax_0+ b = \\varphi(x_0).", "cf6e99172e2631f2417f9a2093993ff0": "\\ \\forall c\\in C : \\exists p_c\\in \\text{Sym}(n) : p_c(C+c)=C.", "cf6f375d36162c2d0f276ed64162335e": " \\mathrm{MA} = P_\\mathrm{E} \\times P_\\mathrm{O}", "cf6f53cbdedcd14abbff9bc397d7822a": "n(n+2)\\,", "cf7030b095e4f38bece0cfe674c097a4": "\\scriptstyle \\vec J", "cf70528cdb660c3c5e7a60594de56647": "\n\\left[ L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^{\\mu} \\right) - \nL \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) \\right] = \n\\frac{\\partial L}{\\partial \\phi^A} \\bar{\\delta} \\phi^A + \n\\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\bar{\\delta} {\\phi^A}_{,\\sigma}\n\\,.", "cf708f4c0a4677de271b21c11cde182a": "f: I \\rightarrow \\mathbb{R}", "cf709f7db2ca7bfb6f826addadd18402": "1/\\sqrt S = \\lim_{n \\to \\infty} 2h_n.", "cf71110bb2b5519f05898eaaf09e1603": "\\operatorname{var}[\\ln X] = \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta)\\!", "cf7130c86806d6b0e705fbd5c07a89aa": " \\frac{[drug][receptor]}{[complex]}", "cf716e96c46b684889a61d9311a2586d": "\\operatorname{Supp}(M) = \\bigcup_\\lambda \\operatorname{Supp}(M_\\lambda).", "cf71843cf24828988ccc1971bd5241de": "J(y):= \\int_a^\\infty dx \\ f(x,\\ y)", "cf718a429995340153fe24669c9dce76": "\\frac{k_{o}^{2}-k_{z}^{2}}{j\\omega \\mu} \\ T_{o}^{TE} =\\frac{k_{o}^{2}\\varepsilon _{r}-k_{z}^{2}}{j\\omega \\mu }^{TE}", "cf7190873b0e399715acd05d42bc0392": "\\frac{\\Delta t}{(\\Delta x)^4} < C u", "cf720cb8b57fc53f232a9c4f22b793f1": "\n \\sigma_{ij}(r, \\theta) = \\frac {K} {\\sqrt{2 \\pi r}}\\,f_{ij} ( \\theta) + \\,\\,\\rm{higher\\, order\\, terms} \n", "cf721396195f03c18d80e1a9dd906649": "(x,X)", "cf72fe0fe9fa54e4ff1eda89f877ff0c": "V_n(R) = R\\sqrt{\\pi}\\frac{\\Gamma(\\frac{n+1}{2})}{\\Gamma(\\frac{n}{2} + 1)} V_{n-1}(R).", "cf7316cd1117accb8359a63d0816e577": "\\neg F", "cf7330a0db8b0291e8ea7a11be1a4c3e": "P(k)\\propto k^n", "cf73a3858956469658b0d18e8c55db00": "D(p||q)=D^{(-1)}(p||q)=D^{(1)}(q||p)=\\sum{p\\log\\frac{p}{q}}", "cf73d966e6c438d3916aa735ab739a78": "\nb_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{a_i}{b_i} = \nb_0 + \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{d_i a_i}{1}\\,\n", "cf7419b96205cc27051829e89954e6c1": "\\left(1 - \\frac{1}{e}\\right)", "cf7420bde1791f4bf31d2bd2c1c3fd74": "A^j", "cf74528093717ca4bd9bc9e1c1758854": "f'_+ (t,d) \\triangleq \\limsup_{h \\to {0+}} \\frac{f(t + hd) - f(t)}{h}.", "cf746b0c9ea3812627a55fa4c006a027": "\\mathcal{U}(\\hat{\\alpha}(\\hat{q},r_{c}),\\tilde{u})=(-2,2)", "cf74bb8cb1c75a2db4bf028cbc30c9ac": "y \\cdot z = yz", "cf74c1214a81a931b689fe03b29c5991": "T\\to\\infty", "cf74c45e028d1314f023d5abb51204c8": "\\begin{pmatrix} 1 & \\{x\\} & \\{z-x \\lfloor y \\rfloor \\} \\\\ & 1 & \\{y\\} \\\\ & & 1\\end{pmatrix}", "cf74d17ca1e5052a9accc94923f50e0c": " \\bigg| \\mathbb{P} \\Big( a \\le \\frac{ X_1+\\dots+X_n }{ \\sqrt n } \\le b \\Big) - \\frac1{\\sqrt{2\\pi}} \\int_a^b \\mathrm{e}^{-t^2/2} \\, \\mathrm{d} t \\bigg| \\le \\frac C n ", "cf74dc46531048162efff82f854107a2": "n(d) = \\left\\lceil\\sqrt{2d\\ln2}\\right\\rceil", "cf7516dac59edf7d7e6a7ee235b1244f": "I_{r,A,i} = \\int_{E_{th}}^{E_0} dE' \\frac{\\Sigma_p^{mod}}{\\Sigma_t(E')} \\frac{\\sigma_a^i(E')}{E'}", "cf75c7dd79d7aaca49c1b2c0a69a21e7": "h l = Q", "cf75e54791dd1f49f918345fdfe2430b": "DC", "cf76277624605df77fa17e076af1c7ba": "B_{n+m} = \\sum_{k=0}^n \\sum_{j=0}^m \\left\\{{m\\atop j}\\right\\} {n \\choose k} j^{n-k} B_k.", "cf7635e2f06e13bf855aa4dede9ca757": "qR_g < 1", "cf769cb09da827b30592a99127452cc3": "f(\\xi) = \\frac{1}{(1-\\xi)^3} = 1 + 3 \\xi + 6 \\xi^2 + 10 \\xi^3 + \\dots", "cf76d41e416477922f70075feca7a02b": " v=\\pm |v| e^{i\\phi}.", "cf7741a378c9539dcc7a1c2b7a27da2a": "\\operatorname{Ind}_H^G \\pi = K[G] \\otimes_{K[H]} V.", "cf7826431dd32ff87d9b488714e36f80": "e^{(k-1)t/\\Lambda}", "cf7851fd7107a800af9b3a53217f25f5": "\n\\begin{array}{c|c|c}\n\\textit{Model} & \\textit{Classical\\ Format} & \\textit{MP\\ Format} \\\\\n\\hline \n\\textit{Maximin:} & \\displaystyle \\max_{d\\in D}\\ \\min_{s\\in S(d)}\\ g(d,s) &\n\\displaystyle \\max_{d\\in D,\\alpha\\in \\mathbb{R}}\\{\\alpha: \\alpha \\le \\min_{s\\in S(d)} g(d,s)\\} \\\\\n\\textit{Minimin:} & \\displaystyle \\min_{d\\in D}\\ \\min_{s\\in S(d)}\\ g(d,s) &\n\\displaystyle \\min_{d\\in D,\\alpha\\in \\mathbb{R}}\\{\\alpha: \\alpha \\ge \\min_{s\\in S(d)} g(d,s)\\}\n\\end{array}\n", "cf793f9c4cb50e274459061cc580a2cb": "B_i=E_{X_{i+1},X_{i+2},...,X_{n}}[f(\\vec{X})|X_{1},X_{2},...X_{i}]", "cf7942481084069fcd6ff5ef2c05d92b": "f^{\\alpha} = - {T^{\\alpha\\beta}}_{,\\beta} \\equiv - \\frac{\\partial T^{\\alpha\\beta}}{\\partial x^\\beta}.", "cf79587e6381008ca5478fd187c9356b": " \\epsilon^2 \\frac{d^2 y}{dx^2} = Q(x) y, ", "cf7976d9793a8b7a406cfad7bffb4a2b": " X^{283} + X^{12} + X^7 + X^5 + 1 ", "cf7986e052c9a3f7956584408d6489b4": " p = \\frac{u}{3}", "cf798f3f3e0143f39c84d2d6e8db08b1": "\\frac{\\partial \\varphi}{\\partial t}(x,t) = F(\\varphi).~", "cf79b3daf6bdf5731655d78e3932d22e": "i^{\\prime \\prime}(x) = enS_z \\frac{v_p}{2\\pi ^{3/2} (\\mathcal {E}_p/e)^2} \\frac {1}{\\sqrt{x}}\\int \\limits_0^\\pi (\\sqrt{x}- \\cos \\varphi) \\exp\\left ( -\\alpha ^2 (\\sqrt{x} - \\cos \\varphi)\\right ) d\\varphi", "cf79ce78f7924ed952653e5749d5230e": "\\frac{E(R_i) - R_f}{\\beta_i} =E(R_M) - R_f.", "cf79e2a63b28b7bb690a7e1da22303d2": " x=(\\,\\cos\\alpha,\\sin\\alpha \n\\,)", "cf7a273c518963f247cdff3987240ad8": "Pr(S)", "cf7b02fa6f0326141d2035a9943ed811": "p_1(x_i) = f(x_i), p_2(x_i) = (-1)^i, i = 0, ..., n.", "cf7b311fea10e49be9dfdf60248f75ef": "\\langle (\\delta \\vec{r} \\cdot \\nabla )^2 \\rangle _{vac} = \\frac{1}{3} \\langle (\\delta \\vec{r})^2\\rangle _{vac} \\nabla ^2", "cf7b3418a0c760084bf0596b61c8da2a": "x'_s \\in\\{0,1\\}", "cf7c65234480c04ef6e11fd1b2e837b1": "\\scriptstyle R = 0,\\ G = 0", "cf7c78144f2a069a1f5a334c6ffbde5c": "I(Y;X) = H(Y) - H(Y|X) \\, ", "cf7c7ac0bf5c8ed69c670aeeb0d1425e": " \\displaystyle dz=dx+i\\,dy ", "cf7ca9ebdce0344059bb4e055b1dada4": "b \\cdot \\overline{\\mathsf{f}}(a) = a \\cdot \\underline{\\mathsf{f}}(b).", "cf7cc4e31dd4599df38607d05a2ff886": "a \\ge vb", "cf7d6ec0a508a7d2d4b1493bc3266e13": "x_0\\,", "cf7ddd56325dcbe54f0bb8adc7c38d2f": "\\delta \\!\\,", "cf7e19a3a9ab56b0e96b5bd63091d2a8": "X^\\lambda=\\eta^{\\lambda\\mu}X_\\mu = \\eta^{\\lambda 0}X_0 + \\eta^{\\lambda i}X_i ", "cf7edd0df6bfb427bac2189027e44e9e": "\\gamma ( ct^\\prime + \\beta x^\\prime )", "cf7ee950cf61a6003c0ec4af7971d8a8": "x_t", "cf7f424635d25c91afb2834c4c5640cd": "\\mathfrak U, \\mathfrak X", "cf7faf822b24b2dce7f21fd2c3c17aaf": "\\tbinom {2n}n", "cf8049faf839cc8b8621b5bfff72ce27": "T(A)", "cf809027664b02a9a4896d2c40b50b86": " g \\in \\mathcal{H}: \\ \\mathbb{E}_{Y \\mid X} [g(Y)] \\in \\mathcal{H} ", "cf809cfc8019e316d62855f34470e112": "\\overline{F(\\bar{z})}", "cf80d0b6a168891cc3e5bf1f9c7b7e7d": "\\deg P_i\\leq d-i", "cf81442039a710221f7291949070bb26": "(7,4)", "cf815e97b11461786ab2d912957b76e7": "\\mathbf{e}_x, \\mathbf{e}_y, \\mathbf{e}_z", "cf819e736c829852c3b489f79a26bde0": "\\mathbf{B} = B_{12}\\mathbf{e}_{12} + B_{13}\\mathbf{e}_{13} + B_{14}\\mathbf{e}_{14} + B_{23}\\mathbf{e}_{23} + B_{24}\\mathbf{e}_{24} + B_{34}\\mathbf{e}_{34}", "cf81bebd16259153cc44540084d1f791": "\\chi(E) = \\sum_{v \\in E} \\chi(v)", "cf81c516a12fd3120270ca9d826963fb": "\\begin{align}\n\\mathbf{T}_\\mathrm{oct}^{(\\mathbf{n})}&= \\sigma_{ij}n_i\\mathbf{e}_j \\\\\n&=\\sigma_1n_1\\mathbf{e}_1+\\sigma_2n_2\\mathbf{e}_2+\\sigma_3n_3\\mathbf{e}_3\\\\\n&=\\tfrac{1}{\\sqrt{3}}(\\sigma_1\\mathbf{e}_1+\\sigma_2\\mathbf{e}_2+\\sigma_3\\mathbf{e}_3)\n\\end{align}\n\\,\\!", "cf81d68025a9e681676a87dd96267a36": " S_{ab}=\\delta(a-b)-2i\\pi\\delta(E_a-E_b)(\\phi_a,V\\psi^+_b).", "cf81f8c9b2cbaca9d3011ce2c0c1867f": "X = x + b (\\alpha - a) \\ ", "cf82019191ca82313f859baf3486e1ed": " = x + b f(\\alpha) - b f(a) \\ ", "cf822d26a8c6c7d0e66909cbbac44731": "I(x) = I_0 e^{-\\kappa_\\nu \\rho x}", "cf825ce21877245ea754dad1f0408061": " Bi= \\frac{hL}{k} ", "cf8298b0e273301afdd921e7e4cf6c2b": "a<0", "cf82aa6a6d283b132ecc013791bd330b": "x_{f}", "cf82cb1e52a69b972bfd44bd8c4c097f": "x(t)=\\begin{cases} 1 & |t|<1/2 \\\\ 0 & \\text{otherwise} \\end{cases} \\qquad", "cf83029c1a0561db2dc18ae15b589fbc": "H^{II}_q(H^I_p(P_\\bull \\otimes Q_\\bull)) = H^{II}_q(H^I_p(P_\\bull) \\otimes Q_\\bull)", "cf830dc3b7c8a0d5fd852e087ff691cf": "e^{-1/t}\\; \\operatorname{Ei}\\left(\\frac{1}{t}\\right) = \\sum _{n=0}^\\infty n! \\; t^{n+1} ", "cf835a4120046a41dcdcacb6abdd21b8": " \\, P(z;\\;f,\\;f',\\;f'')= x^2 f''+x f'+(x^2-\\nu^2)f=0\\, ", "cf836ab90d3368640c7a8f1613c35f55": "E^C", "cf83d359c86f4cfbedcd0ae20c2547c0": "H_n(\\rho) = 1 - \\Phi\\left(\\sqrt{n-3} \\left({1 \\over 2}\\ln{1+r \\over 1-r} -{1 \\over 2}\\ln{{1+\\rho}\\over{1-\\rho}} \\right)\\right)", "cf840e53a286fbfbe001a3dca4af4333": "P_{n} \\Rightarrow P", "cf842b2ec0af1dac9aac50b63d750edd": " e^{\\pi \\sqrt{58}} = 396^4 - 104.000000177\\dots. ", "cf844526b273dc3d0bdf850ee28791b1": "\nn\n= p_1^{\\alpha_1}p_2^{\\alpha_2} \\cdots p_k^{\\alpha_k}\n= \\prod_{i=1}^{k}p_i^{\\alpha_i}\n", "cf845268d008f00fdc5dff4274a1af23": "\\frac{1}{\\sigma}(1 + \\xi z )^{-(1/\\xi +1)} ", "cf8468795e83b95dfbd065b7ff869a09": "(x^3 + x^2 + x) = (x^2 + 1)(x + 1) - 1 \\equiv (x^2 + 1)(x + 1) + 1 \\pmod 2", "cf846fa59b60eb0aca7dc958b80cb9aa": " y(t) = Cx(t) + Du(t)\\,\\!", "cf848b9bbbfacb8ad703c8e0d8beac93": "f^{\\left(-1\\right)}\\!\\left(B\\right) \\in \\Sigma_X", "cf84e43d4d6d632044af5e2bbf18f42d": "\\,K_1K_2K_3K_4", "cf85103689a553ee54fe09b9585df2d7": "\n\\begin{align}\n\\frac{a}{2} (1) =& \\frac{a}{6}(2)+\\frac{a}{6}(1)\\\\\n\\frac{a}{2} (0) =& \\frac{a}{6}(-1)+\\frac{a}{6}(1)\\\\\n\\frac{a}{2} (-1) =& \\frac{a}{6}(-1)+\\frac{a}{6}(-2)\n\n\\end{align}", "cf85566de09f84690b96d1ceed5307fc": "G=\\mathbb{Z} \\!", "cf855f4c807a8f78247caf3bb72147b0": "\nW \\approx \\frac{GM}{R}\n", "cf85895869ff53357abe9ba82f1126b3": " \\langle P \\rangle_\\psi = i\\hbar \\int_{-\\infty}^{\\infty} \\psi^\\ast(x) \\, \\frac{d\\psi(x)}{dx} \\, \\mathrm{d}x", "cf85a5ac3298dcfe6878c4f3d1021f51": " \\sin \\alpha", "cf860070c682482fe0b1799a7c02607c": "\\left(\\frac{\\partial H}{\\partial S}\\right)_P \\equiv T(S,P)\\,.", "cf861973f3198b4fab63b2b9feaed12b": "\\nabla\\times\\mathbf{B}-\\frac{1}{c^2}\\frac{\\partial \\mathbf{E}}{\\partial t}=\\mu_0\\mathbf{J}", "cf8639db8d9a3ba0847bc433425321a0": "g:=\\sum_\\beta\\tau_\\beta\\cdot\\tilde{g}_\\beta,\\qquad\\text{with}\\qquad\\tilde{g}_\\beta:=\\tilde{\\phi}_\\beta^*g^{\\mathrm{can}}.", "cf8671a8bdd7be64567fca041e6f3b7b": "\\mathcal{E} = -{{d\\Phi_B} \\over dt} \\ ", "cf86e8ccac448d6eb0b26224db2a26e8": "T(|A|) = |P_1(A)|", "cf86ed8790b62407f1b69018a1c9c76d": "3 \\times \\sqrt{2}", "cf8722f76043a0dbfac8e78e0d4ea958": "(-\\Delta)", "cf87cadedbacb39f65d7ddf6323b27e7": " U_d \\equiv 0 \\pmod {n} \\text{ and } V_d \\equiv \\pm 2 \\pmod {n} ", "cf87fbda4b92538b2faf48e1350e87a8": "3+4+2=9\\neq b=10", "cf88183453d0949031f9613ce6099c42": "z \\mapsto {{z+1}\\over{z-1}}", "cf8866aaef147fab0c384fdafdcf419c": " \\theta = \\frac{1}{N}\\left(\\frac{s}{q}\\right)\\frac{dq}{ds} ", "cf8907d6434462fa4fce91e0b2382fa5": "\n\\begin{align}\n\\mu_N &= \\frac{\\lambda_0 \\mu_0 + N \\bar{x}}{\\lambda_0 + N} \\\\\n\\lambda_N &= (\\lambda_0 + N) \\frac{a_N}{b_N} \\\\\n\\bar{x} &= \\frac{1}{N}\\sum_{n=1}^N x_n \\\\\na_N &= a_0 + \\frac{N+1}{2} \\\\\nb_N &= b_0 + \\frac{1}{2} \\left[ (\\lambda_0+N)(\\lambda_N^{-1} + \\mu_N^2) -2(\\lambda_0\\mu_0 + \\textstyle\\sum_{n=1}^N x_n)\\mu_N + (\\textstyle\\sum_{n=1}^N x_n^2) + \\lambda_0\\mu_0^2 \\right]\n\\end{align}\n", "cf89349d0e3aede72a2ee94f75fae26a": "J ", "cf8a13c1e09c4cafc57d20ae88e32697": "\\forall x \\exists y \\forall z. P(x,y,z)", "cf8a1bea02671a09bf9eb7df0ce62eba": "\n\\Phi\\{\\zeta(s+h_1),\\zeta'(s+h_1),\\dots,\\zeta^{(n_1)}(s+h_1), \\zeta(s+h_2),\\zeta'(s+h_2),\\dots,\\zeta^{(n_2)}(s+h_2),\\dots \\}\n=0\n", "cf8a2c30822f85e687c5ee1256191229": " \\frac{dp}{dt} = m p (1-p) - e p ", "cf8a4a70fcad4fbfb35d3ad33760a8b0": "t_l=-2w", "cf8a4f2dd8e340272d99edadce6407f8": "G_P", "cf8a658b783268aca0cbae68c3e64601": " [H_i^{0+}] = \\left[\n\\begin{array}{rrrr}\n0 & 0 & 0 & q_{i,1}^0 \\\\\n0 & 0 & 0 & q_{i,2}^0 \\\\\n0 & 0 & 0 & q_{i,3}^0 \\\\\n0 & 0 & 0 & q_{i,4}^0 \\\\\n\\end{array} \\right],\n", "cf8a6bcb484d76eb9a571963c61fd428": "p(\\mathbf{Z},\\boldsymbol\\Theta\\mid\\mathbf{X})", "cf8a9297efc9da065a7ea0ce840c508a": "|E|\\geq nk/2", "cf8b10e80cf0223273f73daca415cff8": "\\pi_{0} \\ge \\frac{2}{3}(\\pi_0+\\pi_1)", "cf8b2a39493ce8a6dc3151501940f760": "\nf(\\mathbf{r})=\\frac{1}{(2\\pi)^2}\\iint F(\\mathbf{k})\ne^{-i\\mathbf{k}\\cdot\\mathbf{r}}\\operatorname{d}\\!\\mathbf{k} =\n\\frac{1}{2\\pi}\\int_0^\\infty F(k) J_0(kr) k\\operatorname{d}\\!k\n", "cf8b86c1a1b0edfbfd889d7f6bedd441": " V\\otimes V ", "cf8ba1c8ffa351beae361b9401e84769": "K_E(a,b;\\tau)=\\langle x=a|e^{-\\frac{\\mathbb{H}\\tau}{\\hbar}}|x=b\\rangle =\\int d[x(\\tau)]e^{-\\frac{S_E[x(\\tau)]}{\\hbar}},", "cf8c10b3296f0d3a3deb2bc8458680dc": "\\ N^2-1", "cf8c15280948342e9e7411182d6330bf": "\n\\nabla \\times \\mathbf{H}(x) = \\partial \\mathbf{E}(x) / \\partial t.\n", "cf8c22c28d838cc9e1c0d6b1fecc7878": "E_\\text{K} = \\frac{1}{2}\\boldsymbol\\omega\\cdot(-\\sum_{i=1}^n m_i [\\Delta r_i ]^2) \\boldsymbol\\omega + \\frac{1}{2}(\\sum_{i=1}^n m_i) \\mathbf{V}_C\\cdot\\mathbf{V}_C.", "cf8c3f999c98ba9d4b032cae967ae955": "\\hat{V} \\equiv \\hat{H} - \\left(\\hat{F} + \\langle\\Phi_0 | (\\hat{H} - \\hat{F}) | \\Phi_0\\rangle\\right),", "cf8cdb7bf5af8ecdfd34b989519f6911": "p(x) = 2x^2 - 3x + 2", "cf8d0982159fdaed0cce586813867c60": "\\hat{x}_\\mathrm{m}", "cf8d629dc48badee3bff7240ab28bda4": "[a,b] =\\,\\! :\\, a^{-1} b^{-1} ab", "cf8dc8421d677b2c9469c06bcd17a062": "\\mathrm{Re}(\\gamma) = (a^2 + b^2)^{1/4} \\cos(\\mathrm{atan2}(b,a)/2) \\,", "cf8dcd59663d8df53e5a02e432be024b": "f_{\\infty}", "cf8def930ae7e682882e216e06b5b88e": "\\displaystyle f_x=2fgc(x-t)=g_t", "cf8e208bea82e4598c7b046dddb87c36": " \\delta \\simeq \\sqrt {\\kappa \\tau_b} ", "cf8e3d42b5b25201bf12805c0be752d3": "\\hat{N}=\\frac{k+1}{k} m - 1 = m + \\frac{m}{k} - 1", "cf8ea0320ab5290d7c44368b71b61fad": "\\psi = \\pi - \\theta - \\phi\\,", "cf8f0e0e3e329a934a2bbc19e2ffbc24": "\\bar x_n = \\bar x_{n-1} + \\frac{x_n - \\bar x_{n-1}}{n} \\!", "cf8f341249d51ecfaf6bc66a150fe62b": " z = \\int X(x) e^{ax} \\,dx\\text{ and }z = \\int X(x) x^a \\,dx.", "cf8f5a215d26cdfa965096ca6a0327ec": "y(x)=(Ax+B)^2-n \\qquad A,B\\in\\mathbb{Z}", "cf8fbbfcfdc125b3dfeecba48cfe51a6": "\\scriptstyle\\gamma \\in (0,1]", "cf8ffcc97ec474eca4e75da98e7131e1": "\\epsilon \\sim exp(1)\\cdot q_{sig} \\cdot \\epsilon'", "cf901c24a63a55e2e67854c7f43ce804": "\\mathrm{Unadjusted MPG (combined)} = \\frac{\\mathrm{1}}{\\mathrm{ \\frac{\\mathrm{.495}}{\\mathrm{City MPG}} + \\frac{\\mathrm{.351}}{\\mathrm{Highway MPG}} }} + .15", "cf902b2dd688e098969bf8dbee6c4b09": " f : X \\to Y ", "cf90674dca48c52d0024c76b4423b8c3": "\\hat e_x = \\frac{ P2 - P1 }{ \\| P2 - P1 \\| }", "cf907b5f9e4f16c5fbcec97d150025d0": "\\vec{x}\\cdot\\vec{y}", "cf9085d58bec4a0d8807f00e32b2e4a3": " Z_{dp}= V_1 / I_1 = Z_{11} - {Z_{21}^2 \\over Z_{22}} ", "cf9179062aca8d092a4d5145c6345114": "\\{y,z\\}", "cf918f452fe157788adf6c30c801ce7e": "p'\\!", "cf91adb28ed61bc9eb9a48abdda02ae2": "\\frac{| \\text{mean} - \\text{mode} |}{\\text{standard deviation}} \\leq \\sqrt{3}", "cf91b1b0001ab812562873cc676db2bc": "\n = \\sum_{a \\in A_i} \\left( C -1 \\right) \\sigma^*_i(a)^2 > 0\n", "cf91b95f641b23550df33d7d1d49323a": "a^2 b^0 c^1 ", "cf9251de97a710e9f7d81a9858a1e6f1": "\\mathbb E[f(X_{t+s}-X_t)g(J_{t+s})|\\mathcal F_t] = \\mathbb E_{J_t,0}[f(X_s)g(J_s)]", "cf92b8ac3f0a869e325ea79cf19ea6c5": "f_Y(y) = \\frac{1}{\\sqrt{2\\pi}\\sigma_Y} e^{-{(y-\\mu_Y)^2 \\over 2\\sigma_Y^2}}", "cf92e6691bf4bf8c1c064e9420786146": "I[u]\\geq a", "cf92f05b568eca6f042154e3a9a6fe4b": "F(s) =\\sum_{n=1}^\\infty \\frac{f(n)}{n^s}", "cf930850797173abb7bfc895da77871c": "\\mathrm{Im}(\\tilde{\\epsilon}/\\epsilon_0) = \\frac{c^2}{(\\omega^2)(\\mu/\\mu_0)}(k\\alpha_{abs})", "cf931cd9749add9629520a99649a8eaa": "F\\left(k\\right) = {{\\varphi^k-(1-\\varphi)^k} \\over {\\sqrt 5}}={{\\varphi^{k}-(-1/\\varphi)^{k}} \\over {\\sqrt 5}},\\,", "cf933a5e8775baa6f3abb0564ef88321": "\\{1,\\dots,k\\}", "cf933d7def7cce60f5ec1ec6088f4709": "A + B = \\{ z \\in \\mathbb{R}^{n} \\,|\\, A \\cap (\\{z\\} - B) \\neq \\emptyset \\}.", "cf934c9125f2918e3fd0d8953ceb64b6": "S_b(Tr(g^a))=\\left(Tr(g^{a(b-1)}),Tr(g^{ab}),Tr(g^{a(b+1)})\\right)\\in GF(p^2)^3", "cf9382c3ee3869f311d15c8558320aa7": "\\bigcup_{n\\geq m} (S_n(X)\\oplus nB)=X\\circ mB", "cf93b09daf9698dc0be04b3b4b7eb5cf": "I = I_S \\left( e^{V_D/(n \\cdot V_T)}-1 \\right) \\Leftrightarrow", "cf93b622e4dc3cc11a3fe5813255d42d": "\\pm", "cf93b9889de91f594d5855f595fc4760": "\\textstyle Y = \\coprod_{x \\in X} \\{x\\}", "cf93d046d9742bcf43d080e23ac44e30": "8x\\,\\!", "cf942c8a67af20ca399b5c996bfb86f0": " \\Phi(\\varepsilon) = J[f+\\varepsilon\\eta] \\, . ", "cf9432d45db91a42900fdf8a82720442": "\\mathbf{C}^n = \\mathbf{C} \\otimes_{\\mathbf{R}} \\mathbf{R}^n.", "cf94426fad627c92f5ed1980fa2909dd": "P(3) = 6.13%", "cf94954f5f8b384e4f62d8b9f992a426": "r \\in [1, n-1]", "cf94ec930bc7e7d1ba9069a42e6f5d96": "\\Gamma\\vdash t\\!:\\sigma \\to \\tau", "cf95903fd3f9ddc8c9ec4df0d26cd704": "\\left(\\frac{3V}{4\\pi}\\right)^{1/3}", "cf95977719d7483f8f39c54954306ec9": " v_i + (1 + r) c_i", "cf95987f8765bb5042ed75756ffc82fb": " p_1\\ne 0, ", "cf95b0a869cff64172c6aaec1d7d2273": "\\begin{align}\n p' &= p\\prod\\nolimits_i x_i,\\\\\n q' &= q + \\sum\\nolimits_i x_i,\\\\\n r' &= r + n,\\\\\n s' &= s + n,\n\\end{align}", "cf96535175f7487616d8bde9a0b95492": "1 : \\tfrac{1 + \\sqrt{5}}{2}", "cf965c945fb987b2e038b3d4454c405e": "G : \\text{Hom}_S(_SM,N) \\to \\text{Hom}_R(M,N)", "cf9738ee29da89a912cac4bc22528d06": " t_g = \\frac{2 v \\sin \\theta} {g} ", "cf976398d71076f291e8843001ec158a": "S = N k \\ln \\left(\\frac{V}{N}\\right) + N f(T)", "cf977f03d56f5b33ef41aa3e441e9f51": "\n\\eta_{a\\mu\\nu} \\eta_{a\\rho\\sigma}\n= \\delta_{\\mu\\rho} \\delta_{\\nu\\sigma}\n- \\delta_{\\mu\\sigma} \\delta_{\\nu\\rho}\n+ \\epsilon_{\\mu\\nu\\rho\\sigma} \\ ,\n", "cf97ae49c7a82d79dcbc727ce557f28c": "z^{-k}X(z)", "cf9883fefdd77edbbd2c89cd7e13b60b": "B_\\nu:= \\mathbb{Q}_\\nu \\otimes_{\\mathbb{Q}} B", "cf98c9bab1b2feaf7cf3fa897d45a1a0": " \\left| S^{'} \\right| \\leq k", "cf990abed4d65c5a99676888b21c81e7": "x^2 = \\frac{1 + \\sqrt{1 - 4 U^2}}{2}", "cf992f4a6dee838c46af6f4146316318": "J_{2k}", "cf9930c246cd62da0cf3aae7cb95cf00": "u(x)=\\int G(x,s) f(s) \\,ds.", "cf99496d9f0d8c7c6f9140118fc145c2": "W^*U(t)W = e^{itQ} \\quad \\mbox{and} \\quad W^*V(s)W = e^{isP},", "cf9ab231ece1984bf25a525a48e611b0": "\\! \\chi = \\sinh^{-1} r", "cf9ad245864996b9a16dc35ee62a1780": " N_D = \\frac{N_1(0)}{\\lambda_D} \\sum_{i=1}^D \\lambda_i c_i e^{-\\lambda_i t} ", "cf9b5073b3046c7f8bfa791b14d68ba5": "-a\\,(s_0 + v_\\alpha\\,T)^2\\,/\\,s_\\alpha^2", "cf9b705e07a168791046d24f129683c9": "\\{ r, p, s \\}", "cf9b980380159782a726051989749600": "\\mathbf{z} = \\mathbf{Hr}\n= \n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 1 \\\\\n1 & 0 & 0 & 1 & 1 & 0 \\\\\n\\end{pmatrix}\n\n\\begin{pmatrix}\n1 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 1 \\\\ 1 \\\\\n\\end{pmatrix}\n\n=\n\\begin{pmatrix}\n0 \\\\ 0 \\\\ 0 \\\\\n\\end{pmatrix}.\n", "cf9b9e44eb03d38171db917f023e6567": "\\,A_x = 1-d \\ddot{a}_x", "cf9bf046a11266c4bf92e8b7dd1e05f5": "v_{e^{ }}=v_{i^{ }}", "cf9c2ea2b30620ea2e827ae1e8af2ec7": "\\rho=\\max\\{\\rho_1,\\rho_2\\}", "cf9c581c3d49277ef7ce4daddd25c66c": "E \\le \\frac{n\\Delta^3}{24}f''(\\xi) ", "cf9c826cad1281a038bff72ba2f02c9f": "\\approx \\frac{Nab}{(N-1)(m_1 b + m_2 a)}\\,", "cf9cac44868960fc73816f1921250556": "X_i,", "cf9cbc44b158fc7840040bc5975ef531": "H_1 P_n", "cf9ce41f2e684b72cb3fff0f7fdab527": " \\Lambda", "cf9d655193e9a3025c0faadf4f6c9c55": "h(x,y) := f(x,y) + f(y,x)^J \\varepsilon \\in R", "cf9d6c98f13e3e2fdcad95622d857e3f": "\\scriptstyle K_2 T^{\\frac {1}{3}}_c P^{\\frac {2}{3}}_c", "cf9da0de7fd2be708d04d7cb0d204269": "\\Delta F_x = \\Delta L(\\cos\\phi - \\frac{\\Delta D}{\\Delta L} \\sin \\phi) = \\Delta L(\\cos\\phi - \\tan\\phi \\sin \\phi) = \\frac{1}{2} C_L \\rho w^2 ldr \\frac{\\sin(\\beta - \\phi)}{\\cos\\phi}", "cf9db37f3d5b34780993c43603ce64f5": "e^{-i \\hat{H} t}", "cf9db579dfd502492db5a20e24ddf0a7": " \\lim_{\\rho,z\\to\\infty}e^{2\\psi}=1", "cf9de8d18da45056424fbbbabf7ee0e0": "B(q) = \\sum_{n\\ge 0} \\frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)^2_{n+1}} = \\sum_{n\\ge 0} \\frac{q^{n}(-q;q^2)_n}{(q;q^2)_{n+1}}", "cf9e2fe89a0fa4ac261378dcf1bd7299": "Q_\\text{sanger} = -10 \\, \\log_{10} p", "cf9e3ad728ef9895145682ed9efa7c3e": " \\frac{dy}{dx} = \\frac{f(x,y)+\\tan\\alpha}{1-f(x,y)\\tan\\alpha}.", "cf9f0fbe0a969290b6e8374d356f7666": "V = \\pi R^2 t", "cf9f8ca84a6b353acede293fc32bddef": "\\sin \\omega_{p}t", "cf9f93b6c8e3c3e3d7c425450098f7cd": "S^{2n-a(n)}", "cf9ff09f26c4b50cc28ef8f03f19cbcb": "I_{\\text{E}} = I_{\\text{ES}} \\left(e^{\\frac{V_{\\text{BE}}}{V_{\\text{T}}}} - 1\\right)", "cfa00dceabcd3e03010717b073e15747": "\\left(\\mathit{u}_1-\\mathit{u}_2\\right)=\\left(1 - 5\\right)=-4", "cfa04827954db5010089fd134de4098b": " \\nabla \\times \\mathbf{E} = -\\frac{\\partial \\mathbf{B} } {\\partial t} \\ ", "cfa065bd0583d754c71e32c8402c1684": "\\pi: Z \\to X", "cfa0dfdd681dd0547c7a5b641f40c998": "x>1\\,", "cfa0e030dbc11bba66845bf7caa3ae4b": "\\mathfrak{a}=i\\mathfrak{t}^*", "cfa11008580cad50f272d8631c41dd1c": "\\psi(\\vec{\\theta}) = - \\frac{2 G D_{ds}}{D_d D_s c^2} \\int dz \\int \\frac{d^3\\xi^{\\prime} \\rho(\\vec{\\xi}^{\\prime})}{|\\vec{\\xi}-\\vec{\\xi}^{\\prime}|}\n = - \\sum_i \\frac{2 G M_i D_{is} }{D_s D_i c^2} \\left[ \\sinh^{-1} { |z -D_i| \\over D_i |\\vec{\\theta}-\\vec{\\theta}_i | } \\right ] |_{D_i}^{D_s} + |_{D_i}^{0} .\n", "cfa157005bdaf6324895f377cb5df070": "k=a^2", "cfa165627802f650930bb147560a1631": "S=\\{(1,1)\\}", "cfa20f3836e32a703ab9a4ab3399f806": "\\lim_{\\epsilon\\to 0}S_\\epsilon\\cdot T=\\lim_{\\epsilon\\to 0}S\\cdot T_\\epsilon\\overset{\\mathrm{def}}{=}S\\cdot T", "cfa2174eaed1808ebdc254ecd751d4b1": "d_{\\mathrm H}(X,Y) = \\max\\{d(X,Y),d(Y,X) \\} \\, .", "cfa24110130070eb100fc1cd7b6b9e70": "{m_0}_\\text{tot} = \\frac {\\sqrt{E_\\text{tot}^2 - (p_\\text{tot}c)^2}} {c^2}", "cfa248979303010c6d37a7df07828ca6": "\\hat{f}^\\dagger\\, \\hat{f}^\\dagger = 0 ", "cfa24a16e42e373ec1c8d79a0828dd65": "\\Rightarrow c^2+c=(c^2+c)^2+c", "cfa26836d12c653ad0a11c536d7569e4": "{{z}_{\\text{max}}}=180{}^\\circ -{{\\sin }^{-1}}\\frac{{{R}_{\\text{E}}}+{{y}_{\\text{obs}}}-h}{{{R}_{\\text{E}}}+{{y}_{\\text{obs}}}} \\,.", "cfa2d4e7aa4e038c0c3365d98b655014": "{}_tq_x", "cfa30724c0a6292403400b18c04cc79c": "(E, \\mathcal A, \\mu)", "cfa32cfac3b0f2875f94c90cee520677": "s,t", "cfa357956fd15884aea0001b5a6ecef9": "A_iA_j \\sube A_{i+j}", "cfa3975634c4adb4695af0fa45934033": "\\mathcal{B_{\\mathit{w}}} = \\{u \\in \\mathcal{B} : \\mathit{w_H}(u) = \\mathit{w} \\}", "cfa450c89c6b1ddf4ff8fd1830eff006": "c > a", "cfa451b5fffc349440b18cbedf4635cb": "M^T", "cfa4a4abf53469f36c604239ead40cf0": "W_{\\tau_{n}}", "cfa4e0ce6e86f7ad175e3e3a82e20973": "\nx\\,y'' + (\\alpha+1 - x)\\,y' + n\\,y = 0~.", "cfa50d3493e2e4fb1614d0aebcaa2b1c": "\nT_K(x) = \\overline{\\bigcup_{h>0} \\frac{1}{h} (K-x)}.\n", "cfa510537c850afaf1f2c54f30e5b854": "P'(G, 1)", "cfa529c8221c49176bc721f59058894e": "f(\\gamma \\tau) = f(\\tau)", "cfa54a687c4a2f72dc60d8bee74c7e3c": "q(z) u_i(z)", "cfa56c920560f024c947cecbc4e17f11": "\\mu = 2C_1", "cfa5889f3c13c5b040610f4846a0fcd0": " p = p^0 + \\mathbf{p} ", "cfa5b127792b0dab0059ab4a145e1600": " x^2 + y^2 = (x+yi)(x-yi), ", "cfa5fe55aa0f8e58c37ef2d800a5020c": "\\scriptstyle\\varphi\\ \\in\\ \\mathcal{C}", "cfa64049b68e89d73628783737af049b": "V(x,k) = \\sup_{X_T \\in \\mathcal{A}(x)} \\mathbb{E}\\left[u\\left(X_T + k C_T\\right)\\right]", "cfa659d92745c07bd72b16dac14f9fe7": "\\mathbf{B}(\\hat{\\mathbf{R}}) =-\\beta(p,n)\\mathbf{R}", "cfa69bd8935ce11664a1b436a12e088d": "k[[t_1, ..., t_n]]", "cfa6a5bd3f3e8e769ad18dac15378364": "P_C", "cfa7116e8d377f6c498f11816e80d047": " \n\\frac{\\partial \\bold m}{\\partial t}+\n\\frac{\\partial \\bold f_x}{\\partial x}+\n\\frac{\\partial \\bold f_y}{\\partial y}+\n\\frac{\\partial \\bold f_z}{\\partial z}={\\bold 0},\n", "cfa75f4886ceda536ca1453c0cd25aca": " \\Delta G^{\\ddagger} = \\frac{\\Delta G^0}{2}+\\sqrt{\\Delta G^{\\ddagger}(0)^2 + \\left(\\frac{\\Delta G^0}{2}\\right)^2} ", "cfa77d4ce900a41fa0aae0c8f5bf9489": "V(x) = (1 - 2x) \\ln \\left(\\frac{1 - x}{x}\\right)", "cfa78af78e031363a17c843b516915c5": "\n\\alpha=\\mathrm{sgn}(\\mathbf{\\hat{n}} \\cdot \\mathbf{\\hat{j}})\n\t\\cos^{-1}\\left ( \\frac{\\mathbf{\\hat{j}} \\cdot \n\t\\mathbf{\\hat{n}} \\times \\mathbf{\\hat{v}}}\n\t{| \\mathbf{\\hat{n}} \\times \\mathbf{\\hat{v}} |} \\right ),\n", "cfa7a01f499232a502498edcced3cc60": "\\displaystyle{g(x)={f(x)-f(a)H_0(x)/H_0(a)\\over x-a}}", "cfa7e5cbdb6546a2840bd81d0358a12a": "u(x) \\to u(x) + a(x)", "cfa7ee9019485a4d17f86d4d69bc6201": "\\frac{1}{n/2+1}", "cfa805e22385ae117e0fd05968d92afd": "w'_4 = \\cos(w_1)w'_1 = \\cos(x_1) 1", "cfa80968818ff9d63b6847b00d97e949": "\n\\overline{N}(f) = \\frac{8\\pi}{3}lwh\\left(\\frac{f}{c}\\right)^3 - (l+w+h)\\frac{f}{c} +\\frac{1}{2}.\n", "cfa835c6ae5ea0f14ee92cdb8f9640b8": "\\rho\\frac{D\\mathbf{v}}{D t} = \\nabla \\cdot \\boldsymbol{\\sigma} + \\mathbf{f}", "cfa84eecf8c62370559ab562ab55f499": "{MR}_i = \\big| x_i - x_{i - 1} \\big|", "cfa8a20558aa30d06bde0db04975327d": "\\hbar/2", "cfa8d0240f7536d09ec9a8eface1e002": "\\scriptstyle 7 \\sqrt{h/g}", "cfa90adb6ba686edbf883f744dadfaee": "C\\psi(q)=-C\\psi(-q)", "cfa90e9cb0e9eb99b1ed5df211ac1307": "\nR(\\tau) = R(-\\tau).\\,\n", "cfa914003bc2af2c5e8b362ba5afa44e": "{\\bar{P}}_6", "cfa9217a55586a49866d694916651b8b": "{\\mathcal{P}} := \\{ z \\in G \\mid f(z) = - \\infty \\}", "cfa941e206445eef81c8f968fa8b0bc2": "c_{12}", "cfa9a43eda43132a3c0bd18b21627876": "\\int W(x)\\,{\\rm d}x = x \\left( W(x) - 1 + \\frac{1}{W(x)} \\right) + C.", "cfa9bb830bd4ac44289c07364db6cab7": "\\begin{align}\n\\sigma_{13}' = &a_{11}a_{31}\\sigma_{11}+a_{12}a_{32}\\sigma_{22}+a_{13}a_{33}\\sigma_{33}\\\\\n&+(a_{11}a_{32}+a_{12}a_{31})\\sigma_{12}+(a_{12}a_{33}+a_{13}a_{32})\\sigma_{23}+(a_{11}a_{33}+a_{13}a_{31})\\sigma_{13}.\\end{align}", "cfa9c6222959ce3a4af21015425019e7": "\\Phi(q,d)", "cfaa38ffc37b364e4700a1fe04e8535d": "a_i\\in A_{v_i}", "cfaa75e9d64c694995b867ef19b6f840": " \\rho = \\rho_1 \\otimes \\rho_2 , ", "cfaa85a7bb4d0685f88b5b95e2fd5f7c": "3^\\frac{9}{13}", "cfaadcf6a970a306320e532f0d9f952b": "\\operatorname{ev}_z : f\\mapsto f(z)", "cfab1ba8c67c7c838db98d666f02a132": "--", "cfab1fecd763bdf4ab37c9b763032a3d": "\\ln(N)=\\ln(N_0)+\\left(\\frac{t}{T_{1/2} }\\right)\\ln\\left(\\frac{1}{2}\\right)", "cfab3ba6ada89c3fd89a4fe1b056923e": "A = QR", "cfab7873a240c4b01fa5509f7fdd72a9": " \\nabla^2 U_\\text{out} = 0.", "cfabe57eada73264188f173b426db0e2": "f(A, B, C, D) = \\overline{A}BC\\overline{D} + A\\overline{B}\\,\\overline{C}\\,\\overline{D} + A\\overline{B}\\,\\overline{C}D + A\\overline{B}C\\overline{D} + A\\overline{B}CD + AB\\overline{C}\\,\\overline{D} + AB\\overline{C}D + ABC\\overline{D}", "cfabfaa27341f272c873e3d16b10d64b": "\\Delta{E} = \\gamma \\hbar B_0 \\ ,", "cfac06cbd52c0ed505d95a2003b9e848": "\\epsilon'_n", "cfac2a7bcbd1a01b182ff6a40f29d1ff": "\\operatorname{F}", "cfac6b6a690e6695a53093e3f60bb819": "\\prod_{x\\le k}(n+x)", "cfac76814b0fb15d1211378d05dccf10": " a_{n} ", "cfac9d3d2751058fb41730b48c285a5c": "\\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} = \\partial_\\alpha \\omega_\\beta^{\\;\\; IJ} - \\partial_\\beta \\omega_\\alpha^{\\;\\; IJ} + \\omega_\\alpha^{IK} \\omega_{\\beta K}^{\\;\\;\\;\\; J} - \\omega_\\beta^{IK} \\omega_{\\alpha K}^{\\;\\;\\;\\; J} \\;\\;\\;\\;\\; Eq(1).", "cfac9de132f75f99cd532e89874071cb": "K = \\frac {C_U \\cdot Q}{C_B} \\qquad (11)", "cface5f22f9831773fbccada62466740": "{1+K\\textbf{G}\\textbf{H}}=0", "cfad05497df246aa8712645dbb694308": "\\mathbb{R}^3.", "cfad0bd1a3784313b315d7711f986da6": " {S_2 \\over S_1} = {{135\\over128} \\div {25\\over24}} ", "cfad124bda4ba030cb6b6453288f9e91": "G_z", "cfad68e88c7d99ab26ecdd8bbb96b3fb": "\\Lambda_{ii}=\\lambda_i", "cfadb2806a481f49dcba336fad284e1a": "\\mathbf{H} = \\mathrm{Re}(\\mathbf{H}_0 e^{-i\\omega t})", "cfade240eae38ed4f164fd30ff037717": "(T, S, \\Phi)", "cfae44a64235dcb5160b54a47f166297": "\\pi_\\Sigma(s)\\,", "cfae562ad5baf9adf6f1fdee7af30001": "\\boldsymbol{\\theta}", "cfae9d0c4062dc45f89806e444b3530f": "\\begin{alignat}{2}\ndp & \n= \\frac{\\partial p}{\\partial t}dt \n+ \\frac{\\partial p}{\\partial x}v_xdt \n+ \\frac{\\partial p}{\\partial y}v_ydt \n+ \\frac{\\partial p}{\\partial z}v_zdt \\\\ &\n= \\left(\n\\frac{\\partial p}{\\partial t}\n+ \\frac{\\partial p}{\\partial x}v_x \n+ \\frac{\\partial p}{\\partial y}v_y \n+ \\frac{\\partial p}{\\partial z}v_z\n\\right)dt \\\\ &\n= \\left(\n\\frac{\\partial p}{\\partial t} \n+ \\mathbf v\\cdot\\nabla p\n\\right)dt. \\\\\n\\end{alignat}", "cfaeb1cea407b9c03b53ad5a829ef1dc": "T \\ll T_B", "cfaedc995849ad82f1067d335f58d8e5": "SO_2 \\,", "cfaf09d1854041a279d686a6b2b8a32d": "\\dot x", "cfaf56a4123e702d662d514fe25c4290": " e^{\\lambda_{\\max} \\theta \\mathbf{Y}} = \\lambda_{\\max} e^{\\theta \\mathbf{Y}} \\leq \\operatorname{tr} e^{\\theta \\mathbf{Y}}", "cfaf6a75141f24d52ebff92d673bd851": " 1 \\to A \\to H \\to G \\to 1 \\! ", "cfaf6ae82a7443a4b92ee4b2960d669f": "q = \\frac{\\gamma}{2} p\\, \\mathrm{M}^2", "cfaf8943cbc54d3d5a9fc67b6eba47f0": "(x, x') \\mapsto \\langle x', x \\rangle = x'(x)", "cfafb02f9f1d78c04fe6358eedbfa3a4": " \\nabla (F+R)(x) = 0", "cfafe665e09c2e205b705f1fde24839f": "\\sigma_y^2(\\tau) = \\int_0^\\infty S_y(f)\\frac{2\\sin^4\\pi\\tau f}{(\\pi \\tau f)^2} \\, df", "cfb04e42eb729ffc1f317e7125c79276": "p^4", "cfb05bc13d3ba235a3f032e3288f0c84": " \\rho\\,\\!", "cfb06a6407589200777dba961c8911cf": " \\mathbf{P} \\mathbf{B} *\\left ( \\frac {MR-PBR} {MR} \\right )", "cfb0dd6f590edf403fbc7e3035c5a223": "\\begin{align}\n\\bold{j} & = \\frac{\\hbar}{2mi}\\left(\\Psi^* \\bold{\\nabla} \\Psi - \\Psi \\bold{\\nabla}\\Psi^*\\right) \\\\\n& = \\frac{\\hbar}{2mi}\\left(R e^{-i S / \\hbar } \\bold{\\nabla}R e^{i S / \\hbar } - R e^{i S / \\hbar } \\bold{\\nabla}R e^{-i S / \\hbar }\\right) \\\\\n& = \\frac{\\hbar}{2mi}\\left[R e^{-i S / \\hbar } (e^{i S / \\hbar } \\bold{\\nabla}R + \\frac {i}{\\hbar}R e^{i S / \\hbar } \\bold{\\nabla}S ) - R e^{i S / \\hbar } (e^{-i S / \\hbar } \\bold{\\nabla}R - \\frac {i}{\\hbar} R e^{-i S / \\hbar } \\bold{\\nabla} S )\\right] \\\\\n\\end{align}", "cfb0f51b4182fb69f66c78421c2daac5": "K(k) = F\\left(k, \\textstyle \\frac{\\pi}{2}\\right) = \\int_0^{\\frac{\\pi}{2}} \\frac{d\\theta}{\\sqrt{1 - k^2 \\sin^2 \\theta}}", "cfb13a16a507882d8ea88e92f458df02": "S = S_{r}(r) + S_{\\theta}(\\theta) + S_{\\phi}(\\phi) - Et", "cfb1790f4092ed0cfac94a36e28d133a": " df = 1 \\, dx ", "cfb196b66179f873e58fe4bdf08413c5": "\\displaystyle(a;q)_n = \\prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\\cdots(1-aq^{n-1}).", "cfb1f98a05a7c6e5de2f86b9e939d6ec": "\\bar{R}_{pq}=\\mathsf{L}_i{}_p\\mathsf{L}_j{}_q R_{ij}", "cfb28e85b148844f525a0e8f87d1adbd": " x \\in R^n ", "cfb2a0e519ff55a6c339e2160b8750e9": "\\frac{ \\sqrt{2 - \\sqrt{2}} } {2}", "cfb2bbb4baf039f8fb80faae5558df2d": "\\qquad \\qquad k_i = \\frac{1}{3}n_ic_{v,i}u_i\\lambda_i,", "cfb2e0b99815b185911832a9e90e72c5": "E_a=E_0 + \\alpha\\Delta H\\,,", "cfb2f08edf3f1f5dc9332ee773b4a073": "\\phi(\\mathbf{x},t)=\\frac{1}{4 \\pi \\epsilon_0}\\int d^3\\mathbf{x'}\\int dt'\\frac{\\rho(\\mathbf{x'},t')}{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}\\delta\\left(t'-(t-\\frac{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}{c})\\right)", "cfb3309881a0b80d6b537ae0c27a8486": "8.\\mu_{2,1}(p_{2}) = \\Sigma_{p_{3}} \\alpha_{2}(p_{3},p_{2}).\\mu_{4,2}(p_{3}).\\mu_{5,2}(p_{3}) ", "cfb36ef472fd6ec80dc37898bd14cfa9": "d\\chi = e^{-k(\\log x)^2}x^{-1}dx", "cfb37e361f85d904e4602c4dec80a72f": "\\mathbf{F} = q\\left(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}\\right)", "cfb3a6b7399550cd5c4dd98fc614bbfa": "\\scriptstyle\\operatorname{erf}", "cfb3e14be210166b4cce040481114f03": "\n p_0(V) = -\\frac{de_0}{dV} = -\\frac{de_0}{d\\chi}\\,\\frac{d\\chi}{dV} = \\frac{1}{V_0}\\,(B + 2C\\chi + 3D\\chi^2 + \\dots) \\,.\n ", "cfb471ce1c11aa902f863d077f5836a8": "{m+n \\choose r}=\\sum_{k=0}^r{m \\choose k}{n \\choose r-k},\\qquad m,n,r\\in\\mathbb{N}_0,", "cfb51261db94eaf980791dac2232e4e5": "P(E_\\gamma,\\theta)\\rightarrow 1", "cfb5dfdd1e4c74123168cd789f13a81d": " \\zeta = \\exp \\left(\\frac{2\\pi i}{17}\\right). ", "cfb61daaa730bd396acee9be39f054c1": "(f\\times g)'=f'\\times g-g'\\times f. \\,", "cfb61fb30cc4e2c4478f4c11d113cce8": " K_T = -{ 1\\over V } \\left ( {\\partial V\\over \\partial p} \\right )_{T,N} ", "cfb64e68872280f819b695a5e9bc50d0": "\\alpha_{ij} = s_i (\\Delta) \\cdot s_{ij} (\\Delta) \\cdot s_j (\\Delta)^{-1}", "cfb666f98225db6338ddd436b4761ceb": " \\frac{1}{90} \\left(\\frac{b-a}{2}\\right)^5 \\left|f^{(4)}(\\xi)\\right|, ", "cfb66af9d86e60a7072682725128e873": "(10)\\quad \\hat\\theta = \\frac{1}{2}\\, k^a{}_{;\\,a} \\;.", "cfb69d784c726b04ed951629affa4e30": "k=1,2,3,\\ldots", "cfb6ab35242270202b1f7b50f951998f": "\\mathbf{J} = \\sigma (\\mathbf{E} + \\mathbf{v}\\times\\mathbf{B})", "cfb783a2f7f92bb6e362fbdf62e4cc68": "x \\,2^m > 2^{p/2}.\\, ", "cfb7d07b337880f04c656d28c48ebfec": "\nQ_a = \\frac{16\\pi^2 V f^3}{c^3 N_{a}},\n", "cfb887a49c001b13349326ac3b926bbd": "H = -T^2\\left(\\frac{\\partial \\left(G/T\\right)}{\\partial T}\\right)_p", "cfb9036b477afaeba52c3d913af9321a": "\\ln\\lambda", "cfb95eb40b252aa4ec1b924a78c80091": " S \\And \\neg S ", "cfb96ff7faea1b01680497d9f519a35b": "\\langle A | A \\rangle = |A_x|^2 + |A_y|^2 + |A_z|^2 ", "cfb975439348e288d2e60f0c59b802e5": "X_\\theta = Y_{t(\\theta)}(x(\\theta))", "cfb97c0fae281e53c807fb59b1790dc9": "\\sum_R\\frac{1}{\\dim(R)^s}", "cfb989b5e1133ef1e29ad53500454f65": " \\varepsilon : K[G] \\to k ~\\text{by}~ \\varepsilon(g) = 1 ~\\text{for all}~ g \\in G_1 ", "cfba1bbf0de4a674cc658ae2576a5519": "X_{k+N/2} = U_k - \\left( \\omega_N^k Z_k + \\omega_N^{3k} Z'_k \\right),", "cfba2c5b7de10aecf9ab8fa95f2188d4": "\\mathcal{N} = \\mathsf{id}", "cfba5c03dbba19a0368a88ae94fe1c4b": " \\Leftrightarrow V_{Ti}'''' + 2V_O^{\\bullet \\bullet}", "cfbae4a41404fd4eff15731ef1324357": "s_x", "cfbb48ce88f77bee446bde14b960429e": "G_{ab} \\, =R_{ab}-\\frac{R}{2}g_{ab}", "cfbba83b1a8cbf6de349f958a65825ae": "x_{\\sigma(1)},\\dots,x_{\\sigma(n)}\\,", "cfbc02dd86a1dac45540a69bd98e0424": "N_{\\max}", "cfbd3a70ca44b6673c61472c37ec2908": "\\begin{matrix} \\frac{n}{2} \\end{matrix}(n - 1)", "cfbd836173753322b4331a1bc1be8db6": "\\textstyle{\\frac {\\log(\\frac{1}{3})} {\\log(\\sqrt{5})}}", "cfbdb8c3e62a941ea7614954d7b34a0f": "\\mathbf{F}_1 + \\mathbf{F}_2 + \\cdots = 0.", "cfbe0d4577f0ec5b4fff56a5d93150c7": "r_i = a\\frac{1}{2} \\sqrt{\\frac{5}{2} +\\frac{11}{10}\\sqrt{5}} \\approx 1.113516364 \\cdot a", "cfbe4dd136d931bcccf1ae292b266804": "0 \\leq \\left( 1 - \\left[ \\frac{m}{r} \\right]_1 \\right)r < 1", "cfbeadca4d1021044c8bbc3a6fd9f2ed": " S_1 = V_1 U_0 + V_0 U_1 ", "cfbeb91f31b24c6afecdbea8a8828c3f": "f(t_1,\\dots,t_k; \\delta,\\nu,\\boldsymbol{\\lambda},\\boldsymbol{\\mu}))= \\delta^\\nu \\sum_{n=0}^\\infty \\frac{(1-\\delta)^n\n\\prod_{i=1}^k \\mu_i \\lambda_i^{-\\nu-n}}{[\\Gamma(\\nu+n)]^{k-1}\\Gamma(\\nu)n!}\\exp\\bigg\\{-(\\nu +n)\\sum_{i=1}^k \\mu_i t_i - \\sum_{i=1}^k \\frac{1}{\\lambda_i} \\exp\\{-\\mu_i t_i\\}\\bigg\\},\\quad t_i\\in \\mathbb{R}.", "cfbf5dd510de2085f1444ca49a12e858": "\\pi^{-1}(F_i)", "cfbfa83c36e202553584ec1943f8c575": "1/(\\lambda \\tau)", "cfbfb5052daa89871f9f33e309a09010": " ( X \\in X ) \\to ( ( X \\in X ) \\to Y ) ", "cfbfd6bc18163eec7789ad55ddf7ea91": "\\theta_e = \\theta_{L} \\exp \\left[ \\left( \\frac{3036}{T_L} - 1.78 \\right) r \\left(1 + 0.448 r\\right)\\right]", "cfc00685f5f0f057ba827780a2de447a": "m_1>>m_2", "cfc008d67971ac83970255b670b6378a": "p_\\alpha", "cfc01935b7454e03bf918aa42cdc0131": "k \\times (n-k)", "cfc02ddf3273252f03f7318760259008": "i = 1,2,\\dots,p-1", "cfc03239de065a1291c4c48717c60b93": "(0.5)^0 = 1", "cfc0a6d22bde76432f3a96afc59d3797": " \\left\\{X(t)\\right\\} ", "cfc0aeeddcf13e7807a985ba91fe5494": "\\,b^2m + c^2n = a(d^2 + mn).", "cfc0d4d5a5b81811554f5b6760834d87": "\nX^{\\{5\\}}=[1,9],\n", "cfc100da3e06556aa1e5c9841374239b": " \nA = \\{(x_1, x_2, ..., x_N) \\text{ } | \\text{ } x_{min,i} \\leq x_i \\leq x_{max,i} \\text{ } \\forall i \\in \\{1, ..., N\\}\\} \n", "cfc1be5bbb4c8840725d0a80cf279a38": "H_A \\otimes H_B", "cfc1d06c206813217ea19c1cd3d9a433": " e^{n\\Theta} ", "cfc1d1ead00dc4b7a3a4201f9d4ac9a6": "d=\\sqrt{(X_2-X_1)^2+(Y_2-Y_1)^2}", "cfc22f979a26b0a9592ac30788c4e8e9": "c_V = \\frac{R}{\\gamma - 1}", "cfc274acec48d1e891fd2e4bf61e6d6f": "EL(\\Gamma_0)\\ge EL(\\Gamma_0^*)=EL(\\Gamma^*).", "cfc2955d4c201091f4523ed7d5909491": "R_0 = \\tfrac{1}{2}\\, v^2 + g\\,h", "cfc2eb4499fc2889533eb9196084f466": "\\oint_S \\vec{B} \\cdot \\mathrm{d}\\vec{S} = 0", "cfc3108b9e785b1f34cf5a289d6d4bd0": "\nH(z) = \\frac{1 + 2z^{-1} +z^{-2}} {1 +\\frac{1}{4} z^{-1} - \\frac{3}{8} z^{-2}} = \\frac{Y(z)}{X(z)}\n", "cfc332b1ebf2fece2cbec28882c9c7f3": "f^{*}(x)=e^{xf'(x)\\over f(x)}.", "cfc3cecd90677e0f90f3076d97385154": "\\psi(\\Omega^{\\Omega^2})", "cfc3dec9ac2f1efa99ec253230a7449e": "\\left(\\frac{d\\mathbf{L}}{dt}\\right)_\\mathrm{relative}", "cfc3f6c703bad29ade0364eac0ba19cb": "\\mu \\colon MF \\to X", "cfc43e737634d81b4ebeb28715f4afbc": "(3^k - 1) / 2", "cfc471a1b51de55915642dc39065b296": "K(k) = \\int_0^{\\pi/2} \\frac{d\\theta}{\\sqrt{1-k^2 \\sin^2\\theta}}.\\!", "cfc4e7fa050cf1c9f3f185d68c76da67": "\\beta w_i \\in H^{i+1}(X;\\mathbf{Z})", "cfc50268a45fbdda08d0413c7cbe4d29": "\\int_{0}^{\\infty} e^{-ax}\\sin bx \\, \\mathrm{d}x = \\frac{b}{a^2+b^2} \\quad (a>0)", "cfc539388ddb083ea130935b4f9799b8": "\\boldsymbol{\\hat \\theta} = \\cos \\theta \\cos \\varphi\\mathbf{\\hat{x}} + \\cos \\theta \\sin \\varphi\\mathbf{\\hat{y}} - \\sin \\theta\\mathbf{\\hat{z}}", "cfc5438ad8431ca774b072ab26d4a33e": "(-\\infty,+\\infty),", "cfc5454c85ade4d4ee34f5e31ff3a4d1": "\\operatorname{mwnchypg}(\\mathbf{x};n,\\mathbf{m}, \\boldsymbol{\\omega}) = \\operatorname{mwnchypg}(\\mathbf{x};n,\\mathbf{m}, r\\boldsymbol{\\omega})\\,\\,", "cfc556fa096c119794fdd8a7197c49d5": " \\nabla^2 A = \\frac{1}{v_{\\parallel}^2} \\frac{\\partial ^2 A}{\\partial t^2}\\,\\!", "cfc57bff5d9bd5b4f129146664f070f2": "\\int\\frac{dx}{\\sinh ax} = \\frac{1}{a} \\ln\\left|\\frac{\\sinh ax}{\\cosh ax + 1}\\right|+C\\,", "cfc5e25f05762538b49a049233fb0e47": "\\kappa = 1/\\alpha", "cfc5ee9cb6a49318657ca9d1caa52733": "f\\colon x_0 \\mapsto y_0", "cfc60688e6b0f27e2bf0326c61f544c0": "\n\t\\frac{\\partial^2 S (\\boldsymbol{\\phi}(y))}{\\partial y_i \\partial y_j} = \n\t\t\t\t\\sum_{l,k=1}^n \\left. \\frac{\\partial^2 S (z)}{\\partial z_k \\partial z_l} \\right|_{z=\\boldsymbol{\\phi}(y)} \n\t\t\t\t\\frac{\\partial \\phi_k}{\\partial y_i} \\frac{\\partial \\phi_l}{\\partial y_j} \n\t\t\t+ \t\\sum_{k=1}^n \\left. \\frac{\\partial S (z)}{\\partial z_k } \\right|_{z=\\boldsymbol{\\phi}(y)} \n \\frac{\\partial^2 \\phi_k}{\\partial y_i \\partial y_j}\n", "cfc613fc02cf540a59477a569859adb8": "\\hat{f}_i", "cfc648bb620e853b1a5acf3cfa2acb6c": "\\sum_{k=1}^{n-1} \\sin\\frac{\\pi k}{n}=\\cot\\frac{\\pi}{2n}\\,\\!", "cfc6be6f5ff620514b42a4808d4bd0b2": "\\sigma_{atmf}", "cfc70dfa11bb425d09c86c47a65999d2": "x\\frac{d^2\\theta_n(x)}{dx^2}-2(x\\!+\\!n)\\frac{d\\theta_n(x)}{dx}+2n\\,\\theta_n(x)=0", "cfc715960a1e40744ea32b5748145983": "\\#R=\\{(x,y)\\mid y\\leq c_R(x)\\}", "cfc7d677b35f7ea8fd1a134243f0e389": "u \\in S", "cfc7e70e4e770d166b6a1ceec768dc2d": "d_k(a_i,a_j)=f_k(a_i)-f_k(a_j)", "cfc7f0b68cff4ca22a9144d1e4ac8862": "T=T_{0}\\cdot\\gamma", "cfc834c0fdc56bf644c5a7ace235a8fc": " \\text{d}_x{}^2 f(x) = f(x+\\Delta x) + f(x-\\Delta x) -2 f(x)", "cfc85b3b0724f41c4845be4fb064016e": " {\\rm cor}_{G/F}(c_G) = \\prod_{q\\in \\Sigma(G/F)}P(\\mathrm{Fr}_q^{-1}|{\\rm Hom}_O(T,O(1));\\mathrm{Fr}_q^{-1})c_F", "cfc871403601395aefce19b0037ff40d": "P(X \\le k) \\approx P(Y \\le k)", "cfc8811542837b7c622b853c347d5a20": "\\frac{A}{B}\\times \\frac{C}{D} = \\frac{(A\\times C)}{(B\\times D)}", "cfc8a3db0f967a9ddb78afdb9ad9897a": "\\scriptstyle \\max_x \\Pr[X=x]", "cfc90b5c77f4731b24dd70d31e59497e": "\\mathbf{q}~=~-k~\\square\\theta~=~-k~\\nabla\\theta~+~\\frac{i~k}{C}~\\frac{\\partial\\theta}{\\partial t}~\\mathbf{o} .", "cfc9e13e5703b81420179c5c18977850": "(h_1,\\cdots,h_{n-k})", "cfc9fce7a1452a28d72f1009ea74880c": "\\alpha^{n+1}>x_1x_2 \\cdots x_{n-1} x_nx_{n+1}\\,,", "cfca677f874494fb3ab475cc10aa0354": "g={1\\over 2}(d-1)(d-2)-\\delta-\\kappa.", "cfca7a3c1c46441d490b9564ab04934a": "\\scriptstyle I(n,\\, \\Lambda) \\,=\\, \\int_{0}^{\\Lambda}dxx^{n}", "cfcac2954a05cce10f838089a341692b": "S_1(c)", "cfcb12fc312a5410dc1a3d7ec91ace13": "\\neg \\Phi", "cfcb9663a2b147ef9b191fa7eb106762": " (ru)_{tt} = c^2 \\left[(ru)_{rr} \\right],", "cfcb9e02d114d0887854fd88f40198ab": "(m-1)!,", "cfcc3a96182169ed65a52f553ec23a49": "(\\tfrac{a}{m}) = 1,", "cfcc6b88a943b0e6a21f5940d4918254": "\\neg \\hat{P} := \\mathbb{I} - \\hat{P}", "cfcc79dc65842f8f4ab94f138e46c530": " D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2 ", "cfcc89561c0f92cce11e735f320b39d3": "\\Pr(\\text{next outcome is success}) = \\frac{s+1}{n+2}", "cfcc94ca2c3f2eb79eb3c8ab5f274af4": "\\mathbf{\\tilde{v}}", "cfccdb1f1997f57b51b88784147c6047": "z=e^{i(\\theta-\\mu)}\\,", "cfcceff143a3247267a21a1cfd540725": "\\sqrt{\\epsilon_r}", "cfcd208495d565ef66e7dff9f98764da": "0", "cfcda49382849c67509217013248fc54": "\\|\\mathbf x_{k+p}-\\mathbf x_k\\|\\le t_{k+p}-t_k.", "cfcdc63f4108474a79d2a7fb860a6482": "\\ln(ab) = \\ln(a) + \\ln(b), \\qquad\n\\ln\\left(\\frac{a}{b}\\right) = \\ln(a) - \\ln(b), \\qquad \n\\ln(a^n) = n\\ln(a)", "cfcdc918736530d22d7a5c0ab476c56b": "u=(p_1-p_4)^2=p_1^2 + p_4^2 - 2p_1 \\cdot p_4 \\,", "cfcddc7227a5e5d197d3937ee5288edb": "2^{bh(v')}-1", "cfce024d0bc76afe4f4db237d716035c": "-L \\mathbf{u} = M \\mathbf{f}.", "cfce95ec855242d957d214597ef1959b": "\n\\Delta \\tau = \\left(1-\\frac{2m}{r} - r^2 \\beta\\omega^2 \\right)^{1/2} \\, dt = \\left(1-\\frac{3m}{r}\\right)^{1/2} \\, dt.\n", "cfcf100255470977542c350b35cd8ffd": "\\int_c^\\infty dy \\ \\left(\\int_a^{\\infty}dx\\ f(x,\\ y) \\right )", "cfcf1a76050ffb1f9b3da1e20b95f132": "\\begin{align}\n\\left[\\nabla\\times(\\mathbf{A}\\times\\mathbf{B})\\right]_i & = \\varepsilon_{ijk} \\nabla_j (\\varepsilon_{k\\ell m} A_\\ell B_m) \\\\\n& = (\\varepsilon_{ijk} \\varepsilon_{\\ell m k} ) \\nabla_j (A_\\ell B_m) \\\\\n& = (\\delta_{i\\ell}\\delta_{jm} - \\delta_{im}\\delta_{j\\ell}) (B_m \\nabla_j A_\\ell + A_\\ell \\nabla_j B_m ) \\\\\n& = (B_j \\nabla_j A_i + A_i \\nabla_j B_j ) - (B_i \\nabla_j A_j + A_j \\nabla_j B_i ) \\\\\n& = (B_j \\nabla_j)A_i + A_i(\\nabla_j B_j ) - B_i (\\nabla_j A_j ) - (A_j \\nabla_j) B_i \\\\\n& = \\left[(\\mathbf{B} \\cdot \\nabla)\\mathbf{A} + \\mathbf{A}(\\nabla\\cdot \\mathbf{B}) - \\mathbf{B}(\\nabla\\cdot \\mathbf{A} ) - (\\mathbf{A}\\cdot \\nabla) \\mathbf{B} \\right]_i \\\\\n\\end{align}", "cfcf4d18e74a8f5a799f5f9dd31d38d1": " pFx_1...x_n \\leftrightarrow Inv inv Fx_1x_3...x_nx_2.", "cfcf594338a13dcdd61fce94aad932e4": "\\textstyle \\exp(x) = \\sum a_n", "cfcf970a2821c0974039a57a9c36cfa7": "\\displaystyle{\\widetilde{\\nabla} f=(\\partial_yf, -\\partial_xf).}", "cfcffc639e90f38c5a1eea5573bdd1ef": "i_{c_{0}} = asin \\left( {V_{0} \\over V_{1}} \\right) ", "cfd010635b4c6a4a499f3a0739546226": "\\tfrac{OD}{OE} = \\tfrac{25}{15} = \\tfrac{5}{3}", "cfd07535363876101cb481b7daa29a1d": "\\overline{c}", "cfd0a2a70e32c0c5e649e733d5adf226": " T(x,t) - T_i = \\frac{T_i \\Delta X}{2\\sqrt{\\pi \\alpha t}} \\operatorname{exp} \\left ( -\\frac{x^2}{4 \\alpha t} \\right ) ", "cfd0bc5f26339a397665996436663a3f": " {81\\over64} \\cdot {80\\over81} = {{1\\cdot5}\\over{4\\cdot1}} = {5\\over4}", "cfd0cba381b498026f62e3491d87d624": "N(\\lambda)\\sim\\frac{\\omega_n Vol(M)\\lambda^{n/2}}{(2\\pi)^n},", "cfd143d79bda4ad5e8718613c91a23a4": "2\\pi a^2", "cfd23cf051f08a7ceb78a5a9874d4632": "N/H", "cfd256dbebd7d7c9c86345599d202b61": "c_r = 0.264002", "cfd25d958437598112414f22d27539ae": "\\pi_A(\\{ \\langle A=a, B=b \\rangle \\} \\cap \\{ \\langle A=a, B=b' \\rangle \\}) = \\emptyset", "cfd26d7690a0131cad9bffc40771d5bb": "X_n\\,\\xrightarrow{P}\\,\\theta", "cfd27cc369e086308fa8006f0d797a50": "\\gamma/r^2", "cfd282dccd4258494061439ce42f42d0": "f(x)=\\sum_{n=0}^\\infty a_n x^n", "cfd2ff2a77cbf94b8d32f84bb5202e99": "U_{eN} = \\int n(\\vec{r}) \\ V_N(\\vec{r}) \\ d^3r \\, ", "cfd380b814d6928558b15c40daab9b24": "v_\\perp \\in N_p N := (T_p N)^\\perp", "cfd3ac00eed6256e9d8590dfe22eb4c2": "\\rightarrow \\!\\,", "cfd43970332e11dc6fa5165ccc19d9df": "\\wedge^m_n = \\vartriangle^{m-1}_n \\subset\\ \\vartriangle^m_n = \\wedge^{m+1}_n", "cfd44032e8460328d8bfa783bbc0a226": "P_\\star", "cfd44eba0a8ac7ae049fdbac5a69ae50": "B(xy) \\to a/x\\ b\\ B(y) c", "cfd48f443785d60257c56f947191753f": "\\partial_i\\phi dx^i", "cfd4a7fe07655c28294990490cef8f37": " A x = b.\\, ", "cfd4baf32a3fefdb87c66f0a86071b5d": "\nML_\\mathrm{dB} = - 10 \\log_{10} \\bigg(1-\\bigg(\\frac{VSWR-1}{VSWR+1}\\bigg)^2\\bigg) \\,\n", "cfd4bb53d866c3bb4ac32d69cae9fe3d": " q^{1}, \\ldots, q^{n} ", "cfd5308b0431f7fe4ad1456fd5bc95b5": "Q_k=\\sum_{j=1}^{k}(\\sum_{i=1}^{k}m_{i,j})^2-n", "cfd5acf7e6027201fefbd54de90b15ee": "\\Rightarrow \\triangle ABC \\text{ is isosceles}", "cfd611c313b4cffb971ef9dfd5726a0b": "(\\mathbf P^n)^{\\star}", "cfd637eeb2209830ed514b36e4a88ce8": " \\boldsymbol \\beta", "cfd64d189d23ca63a21bdc3fcd24aeb8": "CA(NX) = (private)S - I + (T-G)", "cfd6eee17aafde1a1a6113fd7dd33e9f": "(((P \\to Q) \\and (R \\to S)) \\and (P \\or R)) \\to (Q \\or S)", "cfd6ff183f550705f316a95fcf21a5d4": "\nr_{rb} = 2\\frac{M_1 - M_0}{n_1+n_0},\n", "cfd7407c61cc62031dfcbd76ff181874": "\ne = \\frac{u_{2} - u_{1}}{u_{2} + u_{1}}\n", "cfd754297ba313c0d1b330cfbc9ec7e4": "s_p^2=\\frac{\\sum_{i=1}^k (n_i - 1)s_i^2}{\\sum_{i=1}^k n_i }", "cfd75ba675722f310640153c8ac4e667": "\\varphi - 1 < r <\\varphi\\,", "cfd7801890ccd166ff7d222e2cc6f9a3": "A \\wedge (\\neg B) \\wedge (\\neg C)", "cfd7d73bae5d2a1c7e39e73326c7cda6": "\\frac{dy}{dt} = 3,", "cfd7e098a4eade95c8691f03f1a21f71": "D=\\{S^1,S^2, \\dots, S^d\\}", "cfd84753d5d69a2b4ca8b93f413b8ccc": " S(D)=\\frac{A(f^{-1}(D))}{A(D)},\\quad S(\\gamma)=\\frac{\\ell(f^{-1}(\\gamma))}{\\ell(\\gamma)}. ", "cfd90d9af3d27119306e6b982d7297f7": "q^{-1} = q^* \\lbrace q q^* \\rbrace^{-1}\\!", "cfd9333377681be77c73dc573ae8e3d9": "\\mathrm{rd}(z)(t) = \\frac{\\mathrm{d} \\hat{z}}{\\mathrm{d} \\tau} \\left( \\frac{\\hat{\\tau}_{-} (t) + \\hat{\\tau}_{+}(t)}{2} \\right).", "cfd97002b4cb4b139382f3eecc189b33": "G_n(\\omega,\\omega_0) = \\frac{1}{\\sqrt{1+ \\frac{1} {\\varepsilon^2 T_n ^2 \\left ( \\omega_0 / \\omega \\right )}}}.", "cfd9704186e51e94e2d19690e5974af4": "\\aleph_{0}\\,", "cfd9b74af97277f60ffc3982d30bf431": "\n\\det(I - TA) \\, = \\, \\det \\begin{pmatrix}\n1 & -t_1 & t_1 \\\\\nt_2 & 1 & -t_2 \\\\\n-t_3 & t_3 & 1\n\\end{pmatrix} \\, = \\, 1 + \\bigl(t_1 t_2 + t_1 t_3 +t_2t_3\\bigr).\n", "cfd9bde6f45e366b6d55df7cf5ed962b": "f''+2f'-3f=4x-1\\,", "cfd9bfb06c5e1823206bf0352ca66ff3": "Z \\cap T", "cfda02c15c62bed7daebbe71cfd7295e": "[L_{ij},C_k]=i\\hbar[\\delta_{ik}C_j-\\delta_{jk}C_i]", "cfdac4ce5ad328273014f671fc8ef918": "m=p_n-p_{n-1}+...+(-1)^{n-k}p_k", "cfdadbe62b38ab33c9d9ae3da495e697": "Cone_\\omega(\\mathbb R^m, d)", "cfdae0e07226fa715e2ce998e1565a56": "\\frac{dy}{dx}=\\frac{S_0 - S_f}{1-F_n^2}", "cfdafb9045009c91367d6686efc08b31": "\\mbox{Annual Depreciation Expense} = {\\mbox{Cost of Fixed Asset} - \\mbox{Residual Value} \\over \\mbox{Useful Life of Asset} (years)}", "cfdb64c3b87ac533d812eba2b0dcd6d2": "\\varepsilon \\,", "cfdb87208c2c41836c9aa043e958a4ce": "z \\not \\in L \\implies \\forall \\pi Pr[V^{\\pi} (x) = 1] \\le \\frac{1}{2} + \\epsilon", "cfdbab3f5939631e101feead729db4ae": "1/\\limsup_{n \\rightarrow \\infty}{\\sqrt[n]{|c_n|}},", "cfdbac10f97f56fcfa85939eb8777280": "V=(\\frac{1}{6}(5\\sqrt{2}+9\\sqrt{3}))a^3\\approx3.77659...a^3", "cfdbd284182dbddcbd407f4412aff8b3": "\\lambda/\\text{mm}", "cfdbdc40c8a8b7b77ced52e8e6e80401": "{\\mathbb N}\\to{\\mathbb R}", "cfdbe15771eb6b1b1035eb57b394ef2d": "\\,\\hbar\\omega_k", "cfdc23cbc4c252a0ef37513b6d0a0ffd": "\\hat P_{PHD}(e^{j \\omega}) = \\frac{1}{|\\mathbf{e}^{H} \\mathbf{v}_{min}|^2}", "cfdc2cdc641a857926bbb04f8e5dc922": "x\\approx 0.5036", "cfdc6ebd1b398d0f74ba84f78c1270a0": " K(x,y) \\ \\stackrel{\\mathrm{def}}{=}\\ \\overline{K_x(y)}. ", "cfdc85f9963252092835bfff04715081": "\\phi(x)=\\sum_{k=0}^{N-1} a_k\\phi(2x-k)", "cfdc8cb293666d6d5591b91ceb597618": " {\\mbox{d} T \\over \\mbox{d} r} = - {1 \\over k} { l \\over 4 \\pi r^2 },", "cfdc93a351d3d61c74f6945edb1cf6a9": "{\\rho = \\frac{P}{RT\\sum_{\\text{for all j}}{\\frac{m_j}{M_j}}}}", "cfdca267eacf49cf8d24b12e331e5b85": "\\bigcup_{n=1}^\\infty U_n = X", "cfdcb15f6e724217514de97d11bcd358": " M_n = M_o \\frac {[\\mbox{M}]_o} {[\\mbox{I}]} ", "cfdcb89bb5b9acabd84ab6cc93a7f253": "P_s\\;", "cfdcd48feb41958cc2c27e43345685cf": "\\gamma_n=1", "cfdd0beaf5e08a085573bf9b73bece91": "\\frac{C_{vL}-C_{v1}}{R_{vs}+R_1}=\\frac{C_{v1}-C_{v2}}{R_2}", "cfdd3a84084fa6f4691e3c2a154d579c": "\nH_n = \\sum_{h=1}^n {1 \\over h}, \\qquad H_0 = 0 \\,.\n", "cfddd0e8c2e3f1d0a7f29489fc8a5b05": "\n{\\mathfrak{T}}^\\alpha_\\beta =\n\\sgn \\left( \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right)\n\\left\\vert \\det{\\left[\\frac{\\partial \\bar{x}^{\\iota}}{\\partial {x}^{\\gamma}}\\right]} \\right\\vert^{W} \\, \\frac{\\partial {x}^{\\alpha}}{\\partial \\bar{x}^{\\delta}} \\, \\frac{\\partial \\bar{x}^{\\epsilon}}{\\partial {x}^{\\beta}} \\, \\bar{\\mathfrak{T}}^{\\delta}_{\\epsilon}\n\\,.", "cfde36ea67dc9e1fb02e08ee09ec7491": " \\mathbf{a}\\times \\mathbf{I}", "cfde51627c8a0ea0d867b1185999e23d": "(f\\circ g)^{\\mathbb C} = f^{\\mathbb C}\\circ g^{\\mathbb C}", "cfde8e7247f853b5fed1f8cff8a52735": "f_1(X_1,\\ldots, X_r, Y_1,\\ldots, Y_s), \\ldots, f_n(X_1,\\ldots, X_r, Y_1,\\ldots, Y_s) \\, ", "cfde9ba8c93afb87e0bda89a78ff288c": "\\Delta \\Phi_B = 2\\pi\\hbar/2e=h/2e.", "cfde9c24118031c9af6e275681082cdd": "\n\\mathcal{L} = \n\\frac{1}{2}\\left(\\sum_I^{\\mathrm{nuclei}}\\ M_I\\dot{\\mathbf{R}}_I^2 + \\mu\\sum_i^{\\mathrm{orbitals}}\\int d\\mathbf r\\ |\\dot{\\psi}_i(\\mathbf r,t)|^2 \\right)\n- E\\left[\\{\\psi_i\\},\\{\\mathbf R_I\\}\\right],\n", "cfdeaeee3c88b2811f88cb80e2e779d6": "|f_n(x) - f_n(y)| \\le K |x-y|,", "cfdeaff8e0d8d437118c5225c66ccb41": "e^{\\psi(x)}\\,", "cfdf177ff0e2e5bdf4a64ea979a06097": "C_3<0\\,", "cfdf7892e38fe3ffea0015442753b703": "\\forall x \\exist y Lxy", "cfdfcf933fbe6d4d07967de9935d9067": "\\{ \\omega : X(\\omega) > 2 \\}\\,\\! ", "cfe03219b722b8ef229ffeac59a83d75": "\\dot{Q_i} = \\frac{A_i \\epsilon_i}{1-\\epsilon_i}(\\sigma T^4_i - J_i)", "cfe0426fae2997f5a53e7d9fe79081ec": "\\scriptstyle{}\\leq\\sqrt n", "cfe0616b4bbaf6601e747cfe7b12eae0": "c_1y_1 + c_2y_2", "cfe07f81bdd3fd2d5bf095fc987294ca": "\\varphi_v : G \\longrightarrow V", "cfe0a4eb8be8c637404ecb2c032261c0": " Q(\\lambda)x = 0\\text{ and }y^\\ast Q(\\lambda) = 0,\\, ", "cfe156b5a547863298911654cbb55793": "f_{os} = f_{ch} + \\frac{f_L\\cdot p}{12}", "cfe17a529f7c36d54d04ed041e54fde1": "n = 1\\;", "cfe237807a385fff2e7d2d576c5b9873": "|B| = \\frac{B_0}{R^3} \\sqrt{1 + 3\\sin^2\\lambda}", "cfe24f2882b616c155f78330af4c4de3": "X_{b}(t)", "cfe26d594175acd4027da6526360768a": " x_1< x_2< \\dots < x_n ", "cfe2a0bb1df5f6637d9077958a091b97": "\\int{d^4x\\, F(x)}", "cfe2ba31e9d5157d9b416dae9718b407": "\\delta K ", "cfe2ba906716fd525ca637b3b3a3abc5": "A(y)=\\pi r^2", "cfe2c81ed9d96154df478f804192c858": " \n\\begin{bmatrix}\n \\mathbf{A} & \\mathbf{V}^T \\\\\n \\mathbf{V} & \\mathbf{0} \\end{bmatrix}\n\\;\n\\begin{bmatrix}\n \\mathbf{w} \\\\\n \\mathbf{v}\n\\end{bmatrix} \\; = \\; \n\\begin{bmatrix}\n \\mathbf{y} \\\\\n \\mathbf{0}\n\\end{bmatrix}\\;\\;\\;\\;\n", "cfe2d52204d7f6861dff43cd0a788162": "v(\\alpha) = 0", "cfe32da75f1cfe0150b803bd684658c6": "E-E0", "cfe33b68f3517596e454d7cd7357c067": "\\scriptstyle \\vec S", "cfe34b295b41e640de582c6ae7c5b1d9": "\\langle x,y\\rangle = \\tfrac{1}{4} \\left(\\|x+y\\|^2 - \\|x-y\\|^2 + i(\\|x+iy\\|^2 - \\|x-iy\\|^2)\\right).", "cfe37041191166b7465600d6198049fd": "\\frac{dT}{dz}=- \\frac{mg}{R} \\frac{\\gamma-1}{\\gamma}=-9.8^{\\circ}\\mathrm{C}/\\mathrm{km}", "cfe37211ab976c66682ba9bd6114822b": "y = -0.923039 \\,", "cfe3746306f5a9f93878cda50780066b": "\\pm \\det \\mathbf{A}", "cfe3cefe79b2905e2ea5b26569585b4f": "R = R_z(\\gamma) \\, R_x(\\beta) \\, R_y(\\alpha)\\,\\!", "cfe3d996237897ce3cf44f794643f86d": "\\sum_{j}h_j \\delta O_j = 0 \\,\\!", "cfe3eb5f63a203fe7dee9bc312d24b82": " a(p) \\ne 0 \\bmod p", "cfe40521f029e32f769091eeb63f9260": "[m/n]_f(x). \\,", "cfe4462866192df6920d0ac46b88ac01": "\\mathfrak{A}_i", "cfe4bc11e4c872a1aa36abbe1b450d55": "\\Box \\forall x Fx \\rightarrow \\forall x \\Box Fx", "cfe4e72026109b74ec89874c3e46c66c": "\\delta W=\\mathrm{d}(P_{out}V_{out})-\\mathrm{d}(P_{in}V_{in})+\\delta W_{shaft}\\,", "cfe4f1d53990ba1f674c578379035658": "\\lambda(L)=0", "cfe533b0f57474568d00b88642ad3fe6": "\\lim_{r\\to +\\infty}z(r)=\\frac{1}{\\sqrt{1-\\left(\\frac{2GM}{R^*c^2}\\right)}}-1", "cfe59c696c875b1dd812ec65b62b186a": "(k,n)", "cfe59cfbcd4e9579f5dbae477c62a2e9": "T' = \\frac{T_h-T_0}{T_h-T_c}", "cfe5d07c4791d53c14f04f18059255f6": "\\vec{M}(0)", "cfe5e2523e559bd41a0d01c827e08870": "\n \\begin{matrix}\n a\\ \\underbrace{\\uparrow_{}\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n}\\ b=\n \\underbrace{a\\ \\underbrace{\\uparrow\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\n \\ (a\\ \\underbrace{\\uparrow_{}\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\n \\ (\\dots\n \\ \\underbrace{\\uparrow_{}\\!\\!\\dots\\!\\!\\uparrow}_{n-1}\n \\ a))}_{b\\text{ multiplied copies of }a}\n \\end{matrix}\n ", "cfe5e784a8f06de68bcf797bab39dba7": "d(A,B)=\\frac{d^2(c_a,c_b)}{1/n_A + 1/n_B},", "cfe5ea6239538a4685ad32eaa183c326": "(A\\bullet B)\\bullet B=A\\bullet B", "cfe6055d2e0503be378bb63449ec7ba6": "LE", "cfe61f719af1d2983fd3581b5387135c": "\\frac{ \\partial f}{\\partial n}", "cfe6301ce5a361f976b77bc057eae68e": "TeX", "cfe70f9215cb4c4f4308e15b3a65e61d": "d_1>0,\\ d_2>0", "cfe731a4c82c2cf8e6843a4fcb25ab33": "(X_1 \\times X_2, \\Sigma_1 \\otimes \\Sigma_2)", "cfe787ae08231ae6d012506bf0a61132": "r \\approx +1.000", "cfe7c5735a0fb98b0e3cd339f84c4ec8": "\\textstyle = x^ia(x) + x^{i + g(2l-1) + r} + 2x^{i+r}b(x)", "cfe7f6bf0383695945ce5df068aa5035": "N_{M_2} V \\setminus V \\to N_{M_2} V \\setminus V", "cfe8164740d0da0f433becdc6e99eb69": "(G,k)\\in \\Pi", "cfe852d49ec43cf9e8914bbc87f219f2": "\\sum_i\\frac{z_i \\, (K_i - 1)}{1 + \\beta \\, (K_i - 1)}=0", "cfe8d552f55194a0aa7f9a8d879aa452": "W=\\int_C \\mathbf F\\cdot d\\mathbf s.", "cfe8e4ae33b97f68c9f039a9c9792f0c": "\\hat{\\mathcal{H}}^D \\left|\\Phi_l^{-k} \\Psi_{\\mu}^{v+k}\\right\\rangle = E_{l,\\mu}^{k} \\left|\\Phi_l^{-k}\n\\Psi_{\\mu}^{v+k}\\right\\rangle", "cfe98d7eff4f318466ebe5ae1a61ebb3": "a = 2r, c = r, d = -r\\,\\!", "cfe9fd170139347bfdb956835ae701c9": " \\forall w\\,\\forall u\\, \\neg ( w\\, R\\,u)", "cfea15a8db96e0f4fbf52a172a6ad737": "a\\in G", "cfea1c291239ed2d82cb8ebb2835a44b": "(Q_0\\nu^2)^{-1}", "cfea60869801f72597de9d0a7d9980e0": "\nV_n=\\langle \\Delta n^2\\rangle=\\langle n^2\\rangle-\\langle n\\rangle^2= \\left\\langle \\left(a^{\\dagger}a\\right)^2\\right\\rangle-\\langle a^{\\dagger}a\\rangle ^2.\n", "cfea8f1ac9b3138e8953306480387272": "-1 < \\beta \\le -0.5", "cfeaca3950bef7449c785bab4dc13f45": " \\frac{dP_t}{P_t} = \\mu dt + \\sigma(M_t)\\,dW_t,", "cfeaed546570733dead5eaa663d389bc": " \\textbf{a} = - \\omega^2 (r \\cos(\\omega t) \\hat{x} + r \\sin(\\omega t) \\hat{y}) ", "cfeaf02a8eb2a25d1e1b72de9e402897": "L(x_i) = \\sum_{j=0}^k y_j \\ell_j(x_i) = \\sum_{j=0}^{k} y_j \\delta_{ji} = y_i.", "cfeb1ae1c574614674bb4b2c567822c8": " E(C_{k+1} \\,|\\, C_k) = C_k ", "cfeb6574a354de0a6a3dc643d1460882": "f = \\left(1-\\frac{1}{n^2}\\right) \\ . ", "cfeb812a5871d2206efaa77019a0f18f": "f(x_1, \\ldots, x_n) \\in K[x_1,\\ldots,x_n]", "cfeb84cf563f1405d4b5e5149742e004": "(\\sqrt{\\pi}/2)\\langle v\\rangle = v_p", "cfeb88cac4d2b57c1f3af879cff2a174": " C_{ij} ", "cfebb02fb41e863951c1d31cc5c72977": "\\sqrt{2} + \\sqrt{3} = 3.146^+", "cfebb1a5cd4d075bcbc731315d367141": "\\overline{P} = (a, -b-h)", "cfebe9fae1983a654b83cc8c4650147c": "N/3-1", "cfec58d20e9bef6182e2a979e3e65d63": "\\frac{V_{cc}^{2}}{R_1}", "cfec7b89ba2c8b17a185488812bf880f": " f_X(x|\\boldsymbol \\eta) = h(x) \\exp\\Big(\\boldsymbol\\eta \\cdot \\mathbf{T}(x) - A({\\boldsymbol \\eta})\\Big)", "cfecc560a6a6870a61d6711c86b21c8f": "\\gamma \\sim H_n - \\frac{{\\ln \\left( n \\right) + \\ln \\left( {n + 1} \\right)}}{2} - \\frac{1}{{6n\\left( {n + 1} \\right)}} + \\frac{1}{{30n^2 \\left( {n + 1} \\right)^2 }} - ...", "cfecca107f24d2084e8769654fda1b6f": "\\left(\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12,\\pm\\tfrac12\\right) \\,", "cfeccb5113f376b1c344604fb56348ab": "M \\to M''.", "cfed59936a5e7fc8c56ba55fa15538e8": "\\lim_{n \\to \\infty}n \\min \\left (X_1, \\ldots, X_n \\right ) \\sim \\textrm{Exp}(1)", "cfed83ad87d7684160f7d84db69e7ec1": "\\text{John} : N \\qquad \\text{Mary} : N \\qquad \\text{the} : N \\cdot N_0^l \\qquad \\text{dog} : N_0 \\qquad cat : N_0", "cfed92f4e6e17a6aa8ecb6a0f1b797fc": "\\nabla\\cdot\\mathbf{w}^{\\perp}=0", "cfed945482b398e56c5e15c59d37156a": "R_\\text{t}", "cfeda44978be1459c9f1d73995783dba": "D_\\mathrm{KL}(\\mathbb P \\| \\mathbb Q)\n = \\int_{\\mathrm{supp}\\mathbb P}\n \\frac{\\mathrm d\\mathbb P}{\\mathrm d\\mathbb Q}\n \\log \\frac{\\mathrm d\\mathbb P}{\\mathrm d\\mathbb Q}\n \\,d \\mathbb Q\n = \\int_{\\mathrm{supp}\\mathbb P}\n \\log \\frac{\\mathrm d\\mathbb P}{\\mathrm d\\mathbb Q}\n \\,d \\mathbb P,\n", "cfedc5d56139faa4688e6eff42c06d34": " \\forall x\\in\\mathbb{R} \\ \\exists y\\in\\mathbb{R} \\ x+y=0.\\,", "cfedd77f061d8e2eca6bcf0b9d45f69d": "y_1,\\dots,y_N", "cfedebbcdb2e65944f259718fc0c5aa6": "\n\\frac{{\\sigma _z }}{z}\\,\\,\\,\\,\\, \\approx \\,\\,\\,\\,\\,\\frac{1}{{\\sqrt n \\,\\,\\,\\ln \\,(b\\mu )}}\\,\\,\\left( {\\frac{\\sigma }{\\mu }} \\right)", "cfee0be674e2296bea614962d907fa4c": "\\{\\tilde\\psi, \\tilde\\gamma \\}", "cfee223f818b2ebcdbf0fa1e2939b872": " ( (X_0, X_1)_\\theta )' = (X'_0, X'_1)_\\theta, \\quad 0 < \\theta < 1.", "cfee9b83f7a9fa235237e8a894676e57": "R_n(\\xi,x)=r_0\\,\\frac{\\prod_{i=1}^n (x-x_i)}{\\prod_{i=1}^n (x-x_{pi})}", "cfee9f41b91f8156acfc697a74b6422a": "\\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\gamma_\\mu = 4 \\eta^{\\nu\\rho} I_4. \\,", "cfeea111bc8683ff897d83defdad354c": "y(1) = e^{1 - 1/\\epsilon }", "cfeea26463225a055650d723274ab493": "T(n) = 8 T\\left(\\frac{n}{2}\\right) + 1000n^2", "cfeeef78ba57c5542fdd143199ce60d8": "f(x,0) = \\delta(x)", "cfeef1bf4b106c44d39f7c652b0529f0": "\nx^{2} + y^{2} + z^{2} = r^{2}\n", "cfef07570330134cff15311f4f97ab11": "T(4k+3)", "cfef0eabc960a8d19c44a5f9607bdf0c": " C_2 = 0.0231 ", "cfefc4cfe5a3114520680b6827a409ad": "A \\in K^{m \\times n}, x \\in K^n", "cff08840507b445fd88f7f3a23b04167": " \\varepsilon_{ij}=\\varepsilon_{ji}\\,\\!", "cff09be71c0db8f45e49b3a9dcb5f864": "\\bot \\; \\le \\; f(\\bot) \\; \\le \\; f\\left(f(\\bot)\\right) \\; \\le \\; \\dots \\; \\le \\; f^n(\\bot) \\; \\le \\; \\dots", "cff0aee85d6fc4aecef352033b7a9416": " p = N(\\alpha) = N(a+bi) = a^2 + b^2", "cff0e9187bfb60ee21f2e831f4defe06": "\\nabla^{2}\\left(\\mathbf{P}\\cdot\\mathbf{Q}\\right)=\\mathbf{P}\\cdot\\nabla^{2}\\mathbf{Q}-\\mathbf{Q}\\cdot\\nabla^{2}\\mathbf{P}+2\\nabla\\cdot\\left[\\left(\\mathbf{Q}\\cdot\\nabla\\right)\\mathbf{P}+\\mathbf{Q}\\times\\nabla\\times\\mathbf{P}\\right].", "cff0ffffb006d8fa42d6cfed764600e9": " U_{Web} = \\frac{q_1 q_2}{4 \\pi \\epsilon_0 r}(1-\\frac{\\dot{r}^2}{2 c^2}) ", "cff123cbe1ead7f131ea80aa041741c5": "\\sqrt{3}=1.732\\ldots=[1;1,2,1,2,1,2,\\ldots]", "cff19eeeeb9692cbf8a6b0864c461f5e": "p=1", "cff1ceae1d0fc1c0c32c6246fbb403b7": " \\mathrm{ ApEn}=\\log(\\Phi^2 (3))-\\log(\\Phi^3 (3))\\approx0.000033", "cff1d364db6554dbfe3a04c1b29eb19a": "y\\in\\pi^{-1}(x([0,1])) ", "cff1dddf6a9b8f362e2db04615498fcf": "\\begin{matrix} {11 \\choose 1}{4 \\choose 4} \\end{matrix}", "cff221a91849829afff4e821ad0b1d1d": "\\Delta_3(e_i) = k_i^{-\\frac{1}{2}} \\otimes e_i + e_i \\otimes k_i^{\\frac{1}{2}}", "cff28c9e871024acd3f327375d339ede": "\\sum_i \\; u(x_i) \\; P(x_i).", "cff2a87a43566dfbdde4347cdd179539": "A = \\mathbb{Z}[\\sqrt{5}].", "cff2d5d4dc3f56218957bc082158164d": "S, V, \\{N_i\\}", "cff2ebd617140078066d391477272558": "m_L", "cff31514c01cf80741839a76b9943934": "\nO = \\begin{bmatrix}\n\\cos \\phi & \\sin \\phi \\\\\n- \\sin \\phi & \\cos \\phi \\end{bmatrix},\n", "cff326dc7e4ce5b89e42d8a78c34f411": "\\sum_{i=1}^r (-1)^i\\dim(V_i) = 0.", "cff33ef40df69cfce80a417adbf7f4e8": " | \\psi(t_0) \\rangle = U(t_0,t_0) | \\psi(t_0) \\rangle.", "cff33f9af1b359ccbd31f93ab4ba4de1": "\\mathcal{K} = \\langle V, v^0, T, p\\rangle", "cff3433b5449cf8c62957b7e13a560a3": "x_2, x_3, x_4", "cff358544c7d051bca860239d9825c40": "\\left(x_1,y_1,z_1\\right)", "cff38a330387d98ce36c49951922c30a": "F := \\forall y \\exists x [R(y,x) \\wedge \\neg\\exists z S(x,z)]", "cff39318b69a85581f53e10c7e8ee34d": "(\\mathbb{N}, \\mathbb{N}, F)", "cff39470a3db1af60df4438811839c61": "\\frac{2 \\times 10^{-9}}{(25812.807) (483597.9)} \\ ", "cff3c62d4bc5a804eac88d3ac9fbd856": "\\beta\\in\\mathcal{O}_k", "cff3d3645abf5b48f4017ae19325659c": "Q=(1-\\mu)^L", "cff3f2b0d9bda13b3d5e5eda8c9103f6": "g = \\left(\\frac{\\alpha-1}{\\alpha-2}\\right)^{\\frac{\\alpha-1}{\\alpha}} \\frac{1}{T^{\\alpha}}", "cff447a9b012ab096a1709ec21a65aca": "x = \\text{mode} - \\kappa = 1 - \\frac{2}{\\alpha}", "cff4610608dd50e5a5231f486b239aa3": "\\lambda \\cup \\left\\{c\\right\\}", "cff49f359f080f71548fcee824af6ad3": "a/b/c", "cff4ce2e0957cb4f2e8a7f65fbdfcbb5": "f\\left( {C_{ij} } \\right)", "cff5003b50599ad604c88181bfdb34ab": "\\Sigma M_C = M_{CB} + M_{CB}^f + M_{CD} + M_{CD}^f = 0.4EI \\theta_B + 1.2EI \\theta_C - 4.167 = 0", "cff50a84be1c50afb7c1340c9d82d853": " \\left (\\frac{\\rho_2}{\\rho_1} \\right )^{(\\gamma - 1)}", "cff56fe61307e2f45241cb5d9e519934": " \\operatorname{lambda-anon}[X] \\to \\operatorname{lift-choice}[X] = X ", "cff57537ccbb4368cf5d29897ba30008": "\n\\mu_c = \\mu_c^0 + kT \\ln \\frac{[\\mathrm{Ox}]}{[\\mathrm{Red}]}.\n", "cff5e22c125ef6a5a3373c2c653457ad": "C_Y = AC_XA^T + C_Z,", "cff5fefba3b5ea8eaccbda8a75d1eef2": "\\Rightarrow \\delta = n^{\\gamma -\\frac{1}{2}}", "cff613616933bf14be7891f3aec17d76": "\\ A_{peak}", "cff620fcb3b3196268f911ef7b2dd21b": "\\alpha=-E[U^{(i)}]", "cff6816a34ab4b6b3dbdaf53fde3e1e7": "p = - i \\frac{d}{dq}", "cff6a0454d14f999df05e954187d3683": "n = 4", "cff72e199a7441bc4d55e61afc89891f": "25 = 5^2", "cff75c70fe05c5e24d5abe4ece801636": " \\boldsymbol{v} =\\sum_{k=1}^{d} v_k \\ \\boldsymbol{e_k} \\, ", "cff767de7c500343e9ae558d1c2f3802": "L_k(X)=\\frac{b_k(X)}{b_k(z_k)}", "cff76de28815c9b569a1936cc37ff5ee": " E = y + \\frac{v^2}{2g} \\,\\!", "cff7736b71a0b6e21b51c14a9b4373f6": "\\,MM(p) = 2^{2^p - 1} - 1", "cff7bf3b37b0bd9f36c7adf34b9165f7": "H_1(w, s) = H_1(\\tilde{w},\\tilde{s}) ", "cff7e7c7f03beb841ed986671929fc71": "\\Omega_n^{\\text{O}}", "cff838cadbebd32be4515ef512db1950": "\\text{sample mean(X)}=\\bar{x} = \\frac{1}{N}\\sum_{i=1}^N X_i", "cff86d897be7b91f305ce0fd1e780db4": "D = \\frac{1}{2}\\lambda \\,v", "cff8824649215accc0591c75a6c202e6": "(r-GM)^{2} = G^{2}M^{2}-J^{2}\\cos^{2}\\theta", "cff90885402f054fe1374b21ebd9d46b": "X = \\mathbb{R}", "cff9184d0e2a9613f1fb5b582904f413": "a_k=\\left \\lceil \\frac{1}{u_k} \\right \\rceil,", "cff92c2c9f225995c5a899749ef00edb": "r=\\frac{N_{AA}}{N_{BB}}", "cff94d35f7c461ff0c8a34b552aec828": "T(n) = 1001 n^3 - 1000 n^2", "cff994c43aa388cebd3f843e7f0f1668": " \\frac{\\partial L}{\\partial v} - \\frac{\\partial}{\\partial x}\\frac{\\partial L}{\\partial v_x} - \\frac{\\partial}{\\partial y}\\frac{\\partial L}{\\partial v_y} = 0", "cff9d9f2f9d76f65af997a3a3f10515b": " E_N = x_2x_4\\cdots x_{2N} ", "cffa2ba41765b04bb579457becc08816": "\\Delta U_0 \\,=\\,Q\\, -\\, W\\, -\\, \\sum_{i=1}^m \\Delta U_i \\, \\,\\,\\,\\, \\text {(suitably defined surrounding subsystems, general process, quasi-static or irreversible),}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, (1)", "cffa6c6324cc9a74ad4afeaa4b7ec4c3": "2x^2 + 3ab - x^2 + ab", "cffa8d76e44da31456f3cf0a069e00a7": "\\vert(2\\pi i\\xi)^k \\mathcal{F}f(\\xi)\\vert \\leq \\int_{-\\infty}^\\infty \\vert f^{(k)}(y) \\vert \\,dy.", "cffaadb93d57208cabd512bfeae95e4d": "\\mathbf{U}_i", "cffab3737aa3bb85e799065ae3fa2e48": "0.05q", "cffabf8625f973522366cc1f4b88c572": "W(\\mathbf{y})=\\sum_{s\\in \\Omega }h_{[s]}(\\mathbf{\\pi })u(y_{[s]}) ", "cffae47b8d5c1078dbb0d03ded353002": "F(p,q) = \\sum _i \\sqrt{p_i q_i}", "cffb201dac947b4408564e6d177357ad": "\\eta_{\\mu\\nu}{\\Lambda^\\mu}_\\alpha{\\Lambda^\\nu}_\\beta = \\eta_{\\alpha\\beta}", "cffb52bf749b1e07842fbc8bce4a2c39": "\\tfrac{1}{4}(a+c)(b+d)", "cffbba3db19b9a4fa7a70fa10eecf607": "\\displaystyle{V_i^2=-I,\\,\\,\\, V_iV_j=-V_jV_i \\,\\, (i\\ne j).}", "cffcfb8a5f6c3ccd3e6312a1dc50ac27": "\\begin{align}\n\\frac{1}{\\sqrt{2\\pi npq}} \\exp &\\left \\{-\\left(np+x\\sqrt{npq}\\right)\\left(x\\sqrt{\\frac{q}{np}}-\\frac{x^2q}{2np}+\\cdots \\right) -\\left(nq-x\\sqrt{npq}\\right)\\left(-x\\sqrt{\\frac{p}{nq}}-\\frac{x^2p}{2nq}-\\cdots \\right)\\right \\} = \\\\\n& =\\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\left(x\\sqrt{npq}-\\tfrac{1}{2}x^2q+x^2q+\\cdots \\right)-\\left(-x\\sqrt{npq}-\\tfrac{1}{2}x^2p+x^2p+\\cdots \\right) \\right \\}\\\\\n& =\\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\left(x\\sqrt{npq}+\\tfrac{1}{2}x^2q+\\cdots \\right)-\\left(-x\\sqrt{npq}+\\tfrac{1}{2}x^2p+\\cdots \\right) \\right \\}\\\\\n& =\\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -x\\sqrt{npq}-\\tfrac{1}{2}x^2q+x\\sqrt{npq}-\\tfrac{1}{2}x^2p-\\cdots \\right \\} \\\\\n& =\\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\tfrac{1}{2}x^2\\left(q+p\\right)-\\cdots \\right \\} \\\\\n& =\\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\tfrac{1}{2}x^2-\\cdots \\right \\}\\\\\n&\\simeq \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\tfrac{1}{2}x^2 \\right \\} && \\text{as } n \\to \\infty \\text{ we get } x \\to 0\\\\\n&= \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\frac{1}{2} \\left(\\frac{k-np}{\\sqrt{npq}}\\right)^2 \\right \\} \\\\\n&= \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ - \\frac{(k-np)^2}{2npq} \\right \\} \n\\end{align}", "cffd6bf3e2cfe13c083543af878754cb": "\\alpha \\ll 1", "cffd8a8209392489d7c39d750e2b62ac": " \\widehat{\\gamma}_k = \\left(W_{k}^{T}W_{k}\\right)^{-1}W_{k}^{T}\\mathbf{Y} \\in \\mathbb{R}^{k} ", "cffdc5fe1ee042661106ec4590809287": "f(x,t)=-\\frac{1}{\\tau} \\int e^{-i (k(\\omega)x-\\omega t)} \\frac{d\\omega}{\\omega^2 - (2 \\pi / \\tau)^2} .", "cffdf1860343a271c371ee4b86dcee47": "\\forall h\\in\\mathfrak{h}, [h,x]=\\lambda(h)x", "cffe80773e642e0cbf1b8ee89ba04622": "\nr_h = c_h/c_0 \\,\n", "cffe97fe2f3f0bd59b30d31853785856": " r \\, dt = n \\, V = n \\, \\sigma \\, v \\, dt ", "cffeabd7f82c482f66f3c09bda51f137": "E_0\\,", "cffec9f6f39049c661f2e448d9b8b06f": "q_j^{*}(\\mathbf{Z}_j\\mid \\mathbf{X}) = \\frac{e^{\\operatorname{E}_{i \\neq j} [\\ln p(\\mathbf{Z}, \\mathbf{X})]}}{\\int e^{\\operatorname{E}_{i \\neq j} [\\ln p(\\mathbf{Z}, \\mathbf{X})]}\\, d\\mathbf{Z}_j}", "cffee4651057c40c0a391fb073152d56": "L \\xrightarrow[-f]{} La ~|~ a", "cfff6aa9cef2b0192c8f4612b311b9b7": "2^{12/12} = {2}", "cfff9e9fdf3c4995c59ed15589f0cc3a": "Q = C\\; A\\; \\sqrt {2\\;g\\;h\\;\\frac{T_i - T_o}{T_i}}", "d0003edd75768341b6690c59050ba87b": "1- \\exp\\left( - F \\frac{\\pi r^2}{S}\\alpha L \\right) ,", "d0005ac2dec0075ae008800991b94cd1": "{n\\choose k}_2=\\sum_{\\textstyle{0\\leq\\mu,\\nu\\leq n\\atop\\mu+2\\nu=n+k}}\\frac{n!}{\\mu!\\,\\nu!\\,(n-\\mu-\\nu)!}", "d000a254e3306ce604c01ab4dfc86ae1": "t [\\mathrm{ns}] = \\cfrac{36}{1+12.8*p[\\mathrm{bar}]}.", "d000b0261bf8acb5c920f14499cf1a01": " \\mathcal{C}_\\varepsilon ", "d000b7697f17d7bb15bb974354e83a75": "\\boldsymbol \\omega", "d000e7a5e71089760d3476cd5b0006a5": "A^*=\\frac{q^2 x_{vc}^2 (2m_{\\text{r}})^{3/2}}{\\lambda_0 \\epsilon_0 \\hbar^3 n}", "d001131a8d607f253bb952fbed8d10b9": "\\mathcal{N}(P(\\rho_{i})) \\le \\mathcal{N}(\\rho_{i})", "d0016fe797fa6d47c142781a3f8f9fca": "\\frac{\\partial \\mathbf{u}}{\\partial x} + \\frac{\\partial \\mathbf{v}}{\\partial x}", "d00185974d0dad6802e9989cb2ac97d7": "S_1,\\ldots,S_k", "d001b369385ecda53d2841dde574f2ef": "\\operatorname{GL}(A)", "d001be74cb873725372eed4e1ec14d65": "\\scriptstyle S_{x,s}(s)e^{\\alpha s}", "d001cbd67ab5d76f420e3a580487302d": " \\bar \\Omega _t = - \\beta \\overline J _t VF_e.\\, ", "d00200eba05716d10a742534dfae8be8": "\\text{WAL} = \\frac{An-P}{Pr}", "d0033cdb8418a27cc02915458cfaa7ee": "\\Delta^E_k=\\mathrm E^{\\Sigma^P_{k-1}}", "d00404f5161802d3599fa428db962b97": " \\sqrt{\\langle v^2 \\rangle} = \\left(\\int_0^{\\infty} v^2 \\, f(v) \\, dv \\right)^{1/2}= \\sqrt { \\frac{3kT}{m}}= \\sqrt { \\frac{3RT}{M} } = \\sqrt{ \\frac{3}{2} } v_p ", "d0047de2e2f2f270d9e2a74529491f92": "dx'=dx-vdt", "d004b00d4d1e51321c03d26ca3271f70": "E = \\{ (x,y) : 0 \\leq x, y \\leq 1 \\} \\cup \\{ (x, 0) : -1 \\leq x \\leq 1 \\} \\subset \\mathbb{R}^2 ", "d004becbb1889b65e5757eb3d527aa8b": "\\{\\mathrm X_{i}\\}", "d004e9a9fa805fbcadefb377f1370f8f": "\\left \\lfloor \\frac{A}{121} \\right \\rfloor", "d0053b7c3591e9f03651d084d5a8b1ee": "\n A \\vdash A\n ", "d00549e1b9c6a70cc943af29c33a646c": "C\\left(\\partial X\\right)", "d005c96edfa5756f6d70b56c63dc4f96": "T = 300 \\mbox{ K}", "d005e3ea3731ab3a8480f8229b3d02ce": "i=N-1", "d006402cbe2d14fdf19b5d4a93973e77": "\\operatorname{max}\\{0, 1 - \\chi(S)\\}", "d006e3c41bab87e89a673a058166c0bb": "A^\\lambda := \\frac{DU^\\lambda }{d\\tau} = \\frac{dU^\\lambda }{d\\tau } + \\Gamma^\\lambda {}_{\\mu \\nu}U^\\mu U^\\nu ", "d00720c4b0decee37991200275106c95": "\\|xy\\| = \\|x\\|\\|y\\|", "d00723f27d56f6e60de5b1e5b864127b": "(e_1,e_2)\\mapsto (\\cos \\theta \\, e_1 - \\sin \\theta \\,e_2, \\sin \\theta\\, e_1 + \\cos \\theta \\,e_2).", "d00760c66c7c3f1be5787641613e5097": "\\sum_{k=2}^\\infty \\zeta(k) x^{k-1}=-\\psi_0(1-x)-\\gamma", "d007797a312462f296f6d0c76773b764": "\\operatorname{eval}", "d007b0916c9ab33e2d7752a333764c50": "\\mathcal{E}\\subseteq\\mathcal{D}\\left[\\mathcal{A},\\mu\\right]", "d0081415f63e9f1e7594ff13ea88006d": "\\prod {A_i}'", "d0085ffb9a50426fe0039242f6a5d583": "\\langle P,A,\\mathit{IC} \\rangle", "d008732dde9e1a0d2fc16a31aefb88ac": "\\frac{\\left|f'(z)\\right|}{\\text{Im}(f(z))} \\le \\frac{1}{\\text{Im}(z)}. ", "d008dc713c8272b901f8bd5eb542617b": "\\mathcal{A} ", "d009589008f26f4b22019e951cd57802": " E=-\\frac{d\\phi}{dx} ", "d00977e36928c2992fe238140509747d": "\n (\\exists L)\n ", "d009aeb6d5eeba7fc6f96ecad14fc989": " \\mathbb{Z}^n ", "d009b1333b475da55b732aea6e1f8d76": "\n s^2 = \\frac{1}{N} \\sum_{i=1}^N \\operatorname{var}\\left(\\frac{k_{i}}{n_{i}} \\right) \n = \\frac{1}{N} \\sum_{i=1}^N \\frac{\\hat{\\mu}(1-\\hat{\\mu})}{n_i}\n \\left[1+\\frac{n_i-1}{\\widehat{M}+1}\\right]\n", "d009b974e87eafd6c9d0ffcad7aa9c3c": "\\hat{B}^{-1}", "d009c26afd8886a5b872bccf30b9b7c1": "\n\\left(\\frac{a}{p}\\right) = \\begin{cases}\\;\\;\\,0\\mbox{ if }p \\mbox { divides } a\\\\+1\\mbox{ if }a \\mbox{ R } p \\mbox{ and }p \\mbox { does not divide } a\\\\-1\\mbox{ if }a \\mbox{ N } p .\\end{cases}", "d009f2996a1e236e69d5362c96cf886e": "\nJ_{\\beta}^{\\prime} = \n\\int_{0}^{\\infty} \\frac{dx}{\\left( x + b^{2} \\right)^{2} \\sqrt{\\left( x + a^{2} \\right)^{3}}}\n", "d00be4215a2e43d1deb920da6190d78b": "460MeV g^{-1} cm^{2}", "d00bfc068a9a720f9a7a33cd588a55ce": "\\nabla^2 \\psi_j(r) = -\\frac{1}{\\epsilon _0 \\epsilon _r}\\rho _j(r)", "d00bfd2afb7242ed465f4d1b3c00e8b3": "Q(-1)=0", "d00bfef144b5b98f484988f6fd0115fc": "n_1 + n_3\\,\\!", "d00c8989e35f1ef9d02de6a957ec8f0d": "c=2\\pi r", "d00cdad31c84ac32baac5f47b29262a7": "(p(x) - \\beta)^{v}", "d00d201c1687f9bdbcb61eeaec77f12c": " D^2 u", "d00dbf00237d656d1f17c18b94553495": "\n\\frac{\\partial^2 f}{\\partial x^2}\\ +\\ \\frac{\\partial^2 f}{\\partial y^2}\\ +\\ \\frac{\\partial^2 f}{\\partial z^2}\\ =\\ {1 \\over r^2}{\\partial \\over \\partial r}\\left(r^2 {\\partial f \\over \\partial r}\\right) \n + {1 \\over r^2\\cos\\theta}{\\partial \\over \\partial \\theta}\\left(\\cos\\theta {\\partial f \\over \\partial \\theta}\\right) \n + {1 \\over r^2\\cos^2\\theta}{\\partial^2 f \\over \\partial \\varphi^2} ", "d00e941c52c6fb77edb165fe89f9c3ab": "var[G|H] = E[G^2|H] - (E[G|H])^2 = \\rho", "d00ee7b398b443321a6d2ac6aae48626": "a^2/b", "d00f5250fe7a7989faf99a98f8fd1cd0": "d(a,b) = k", "d00fc68714d429656a47b797e9750a3c": "J = 1/2", "d01003e5fa13394b7e1e0d43bc938baa": "N-i", "d01010ab38a919c235f6659263f1b97f": "2d>n", "d010436da1ecabb0219f5ffb6fc4717a": "N(\\mathbf{0},{\\mathbf \\Sigma})", "d01045fe45801483a37a78fbac3a8ddb": "f\\in\\mathbb K(\\mathfrak g^*)^{Ad(G)}", "d0107bfb5959b2e7c03d09cafa4bdaf9": "\\delta(\\varphi) = \\left\\langle \\delta, \\varphi \\right\\rangle = \\varphi(0).", "d01093c04369f1615f0b8cb77bebef27": "string\\rightarrow Set\\ int", "d01098af30eb51fd88c2cb1c5ea528f8": "v\\cdot\\nabla v\\sim U^2 / L", "d010d6793a7bf5e8fbd2df617926076b": " A(r,\\theta) = R(r)\\Theta(\\theta), \\,", "d0111eff46b69d8271c64fc2228969d8": "a,b,c,d,e \\in A", "d011243c9cc44861f6f16445f117cfe2": "1/4 = 0.25", "d01142e446c936b23d58ccb7d158139b": "J(n,k)", "d0115314b757ca3f2f31730a0e27ce9e": "f_A", "d0115a93b2e2346c9add7251a1ed175d": "\\gcd(x_1-x_2,n)=1", "d0115ebcae2dee13fce5428f3b472a70": " K_{m2} ", "d01178e86876ae3978e30407d0ec20f7": " E_\\mathrm{h} / {a_0}^3 ", "d011bb8209c9228bba10d809673c50df": " 2 \\, m", "d011f3d271e07425cc6fce59d8afb1c5": "P^{m}(V)\\leq 2P^{2m}(R)\\,\\!", "d012906bd2f4017a89ac93350a61535c": "(B\\to C)\\to((A\\to B)\\to(A\\to C))", "d012ad8841e9b5587f9e5a4404f3848b": "t \\gg 1", "d0130907004ac1e744dcc0ba342b02bc": "\\sigma^2_1\\sigma^2_2\\sigma^{-1}_1\\sigma^{-2}_2.\\,", "d0131f4d970d8d3c5968d5d9710908d8": "\\scriptstyle v({\\mathbf P}_i) = ", "d013265a595f30b8a44ef40dc2d8e56c": "\\int_4^{10}\\left[ \\int_3^7 \\ 5 \\ dx\\right] dy.", "d013c269feae23bec595535be822092b": "p=\\frac{b^{2}}{c}", "d013cf5954699075db380c030d65ea31": "A_{\\nu}\\,", "d013d7dc32e3367985e2129d3cca310e": "P^I(t_j)", "d013db8a52f3129b33d9ca6492fbb204": "G=\\left[ \\begin{array}{ccc} \\langle x_1, x_1 \\rangle & \\cdots & \\langle x_1, x_m \\rangle \\\\ \\vdots & \\ddots & \\vdots \\\\ \\langle x_m, x_1 \\rangle & \\cdots & \\langle x_m, x_m \\rangle \\end{array}\\right]", "d0140108e2b87c9b75c92a6ab5feeb38": "D_{\\rm Chess} = \\max \\left ( \\left | x_2 - x_1 \\right | , \\left | y_2 - y_1 \\right | \\right ) .", "d0140f8d7d26817ba57048a33ece96c1": "(\\forall n\\in\\mathbb{Z}_+):U_{n+1}(x)=-\\frac{U_n'(x)}x.", "d014143e699fd3273654ab0f2db32080": "\\sum_{k=1}^\\infty \\frac{\\sin(k\\theta)}{k}=\\frac{\\pi-\\theta}{2}, 0<\\theta<2\\pi\\,\\!", "d0147b1363104f81b2ddf72f0dc9a17b": "24^{\\frac{M_p-1}{2}} = (2^{\\frac{M_p-1}{2}})^3(3^{\\frac{M_p-1}{2}}) = (1)^3(-1) = -1.", "d014bac23e966ec2337f60cb4d4fac5e": " \\begin{array}{ll}\n\\Delta x' = \\gamma \\ (\\Delta x - v \\,\\Delta t) \\ , & \\Delta x = \\gamma \\ (\\Delta x' + v \\,\\Delta t') \\ , \\\\\n\\Delta t' = \\gamma \\ \\left(\\Delta t - \\dfrac{v \\,\\Delta x}{c^{2}} \\right) \\ , & \\Delta t = \\gamma \\ \\left(\\Delta t' + \\dfrac{v \\,\\Delta x'}{c^{2}} \\right) \\ . \\\\\n\\end{array}", "d014f270da77c0cd2c31bab896e42136": "\\tau = \\frac{t}{t_c} \\Rightarrow t = \\tau t_c ", "d0151a95c90b3f0cfb7390458d689212": "H_\\alpha^{(1)}(z)\\sim\\sqrt{\\frac{2}{\\pi z}}\\exp\\left(i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)\\text{ for }-\\pi<\\arg z<2\\pi", "d01542e249728c5cfd23f44aff96a690": "\\left(I-H \\right)^\\top \\Sigma\\left(I-H \\right) ", "d015ca8454549921f82f60272a060824": " \\approx 2.512 ", "d015f51b48001f82fff63c1250784557": " y_2 = \\frac{2(4.4)}{-1 + \\sqrt{1 + \\frac{8(32.2)(4.4)^3}{(20)^2}}} = 1.4 ft", "d0161c23f96a4389a2c863be296e55c4": "P\\cdot p\\Psi =0,", "d0163493289d44209b3bd6668c6eff7c": "k\\propto \\frac{c}{\\sigma}v.", "d0168e14533e0ba552464dad3eaa8d85": "y_{t+1}= \\alpha - \\frac{\\beta}{y_t}", "d016b2d20f2f580857159d5aa8d779e5": "F[y]=t\\,y'^{[-1]} - y\\circ y'^{[-1]} ", "d0173224503cb8ff7efae70c45dfaf84": "\\left({\\sqrt {gy}}\\right)", "d01751acd6c7bbba8cb89064c356d25d": " V = [v_{ij}] ", "d01754818ac47d52f18505d488d3e0eb": "V_{i,\\alpha}\\subset U_\\alpha", "d017aa36a152c2ac4636957c4a01b41b": "P = Q + \\sum_j \\Delta\\theta^j Q_j", "d017c7d9d1376ef07ce27ceed7cd98b6": "\\gcd(a,b)", "d0181900d4d2e2c31bc521d18e846bc6": "[A,B]\\,\\!", "d018268506e2868537a478629b59e7c1": "par", "d018c8b05c39b39b0eccbb83ff8779c5": "k={ \\log (0.001) \\over \\log (1-\\alpha)}", "d0190cc4bf3d7a419fc42fa8590f4125": "\\lambda_c = \\sqrt{ \\frac{8 \\pi^2 \\kappa}{f''} }.", "d0195e50d80a56e3d1676c74f8987e90": "\n\\rho^A_\\mathrm{tot}(\\mathbf{r}) = \\sum_{\\alpha} Z_\\alpha \\delta(\\mathbf{r}-\\mathbf{R}_\\alpha) - \\rho^A_\\mathrm{el}(\\mathbf{r})\n", "d01973b70ca57ecb7eb0c7018e0f669c": "f\\left(A x\\right) = f(x), \\; \\forall x, \\; \\forall A \\in G ", "d01994725853b48d8f8978b9e8b6d6ea": "\\operatorname{var}[\\ln X]= \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta)", "d019bf1eda2cc0078eacbd77625048d7": "(\\tau_h S)[\\varphi] = S[\\tau_{-h}\\varphi].", "d01a54b184274b1dc21591365fac27b1": "(x + 1)^{3}p(\\tfrac{1}{x+1}) = 8x^3+104x^2+376x+344", "d01a56e8bd4ff5eff42b6d166062a907": " X_t = \\left(1 + \\sum_{i=1}^q \\theta_i L^i\\right) \\varepsilon_t = \\theta (L) \\varepsilon_t , \\,", "d01aae380a210d464d1e5adde36d1d41": "\\Delta = \\left( AC - \\frac{B^2}{4} \\right) F + \\frac{BED}{4} - \\frac{CD^2}{4} - \\frac{AE^2}{4}", "d01ad2b39915a61d1f2cda290a720f59": "\\mathrm{diag}(-1,1,1,1)", "d01aec9a070b31b29fc2934f6397e324": "\\int_{\\mathbb{R}^{d}} \\prod_{j = 1}^{d} f_{j} ( \\pi_{j} (x) )^{1 / (d - 1)} \\, \\mathrm{d} x \\leq \\prod_{j = 1}^{d} \\left( \\int_{\\mathbb{R}^{d - 1}} f_{j} (\\hat{x}_{j}) \\, \\mathrm{d} \\hat{x}_{j} \\right)^{1 / (d - 1)}.", "d01af714aa0f30d73b75b04c8c633809": "\\frac16n(n+1)(2n+1)", "d01b23a955275feee671a0b81b2280a6": " \\omega_{2/3}", "d01b8482406a204c54074a5be1acda5d": "\\omega(k) \\approx \\omega_0 + (k-k_0)\\omega'_0", "d01b866a422a2cc58e5b56cb0331183a": "p(\\theta|\\mathbf{X},\\alpha) = p_G(\\theta|\\alpha'),", "d01c31a50532bea70240989d6078ccf6": " \\lim_{t \\to \\infty} \\frac{1}{t} X_t = \\frac{1}{\\mathbb{E}S_1} ", "d01c44e8bccb6ac5b5e4d90034e0d8bb": " \\nabla^{(i)} \\cdot \\frac{\\partial f}{\\partial\\left(\\nabla^{(i)}\\rho\\right)} = \\sum_{\\alpha_1, \\alpha_2, \\cdots, \\alpha_i = 1}^n \\ \\frac {\\partial^{\\, i} } {\\partial r_{\\alpha_1} \\, \\partial r_{\\alpha_2} \\cdots \\partial r_{\\alpha_i} } \\ \\frac {\\partial f} {\\partial \\rho_{\\alpha_1 \\alpha_2 \\cdots \\alpha_i} } \\ . ", "d01c4e58ae749d4de8159498ee7c967b": "O(N^4)", "d01ca7e8d11b1b6ab26ac526dd9a980d": "\nX \\left( y {dz \\over dt} - z {dy \\over dt} \\right)\n + Y\\left(z {dx \\over dt} - x {dz \\over dt} \\right)\n + Z\\left(x {dy \\over dt} - y {dx \\over dt} \\right) \n = 0\n", "d01dabfe6cae9fff26b7f59ced961528": "\\rho = \\sigma", "d01dccc70eb3b2e14d62d245c3509bef": "v = \\frac{d [P]}{d t} = \\frac{V_\\max {[S]}}{K_m + [S]}", "d01e3ea2bea74f11eb491aabc0b4ec97": "y = \\sqrt{\\rho^2 - (L - x)^2}+R - \\rho", "d01e57edecf194260ef33570b1058547": "P(M|E) = \\frac{P(E|M)}{\\sum_m {P(E|M_m) P(M_m)}} \\cdot P(M)", "d01e7fd09053e33c53161b2cdc399c57": "y(t) = \\sin\\left( \\theta(t,x) \\right) = \\sin(\\omega t + kx) \\,", "d01ea6e6216a83e2f0203be317f6ecfa": "h_u, h_d", "d01f362a3e2dd8a90979621063638f7e": "F_n X = \\{(x_1,\\cdots,x_n) \\in X^n : x_i \\neq x_j \\forall \\ i \\neq j \\}", "d01f90c1b13b77a8b5932402a3bf00fe": "\\textstyle x = (x_1, \\ldots, x_n)", "d01fda763ca701d01128892bf255429b": " \\frac{D_{\\odot}^2}{2\\mu_0 r_c^6}=n m v^2", "d020d35548add6414bd047eb626e8f60": " \\mathbf{L} = \\sum_{i=1}^n m_i\\Delta\\mathbf{r}_i\\times \\mathbf{v}_i = \\sum_{i=1}^n m_i \\Delta\\mathbf{r}_i\\times(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i),", "d021123ffc79a849978393ab1bc3c3b7": "\\frac{1}{\\theta}\\,(t^{-\\theta}-1)\\,", "d021131fab9827dc480d6b50ab9f3c2e": " Z_1 + Z_2\\,\\!", "d0211b747e194739b8f6767a1290dff2": " Q = (I + A)^{-1}(I - A)", "d021735a97bcd53b4ffadc8f08df5a84": "\\mathrm{lcm}(k_1, \\dots, k_m) \\mid \\lambda(n)", "d021aa9bc46c0421100c196a2fb93bdc": " v \\ll c ", "d021f6b7304d0922a1fcaa6b3b8b9570": "P = e^{\\frac{A - E}{k T}},", "d0227c46f10345c241a8cdda2d3ac6fb": "(A:B):=\\{r\\in R \\mid Br\\subseteq A\\}", "d0229ec09db96a6940160b707a7093dd": "\\lambda_k = |\\{ j \\in \\mathbb{Z}_{R^n} : \\delta_j = k \\}| \\, , \\text { for } k \\in \\mathbb{Z}_R", "d022d0842177c4dd8de6936c5ec03117": " x_n = 0 ", "d022dba58b6b1f95a46bfe226f8ef432": " \\tilde B /cm^{-1} = {h \\over{8\\pi^2cI_B}}= {h \\over{8\\pi^2cI_C}}", "d022fcb621061a51f4fef1741ac23ff0": "f^t", "d0230e6d0d38512977344040b63ad9e3": "\\nabla FG = (\\nabla F)G.", "d0234351754d63c9f4c12f11da683413": "\\hat{f}(\\xi) := \\int_{-\\infty}^{\\infty} f(x)\\ e^{- 2\\pi i x \\xi}\\,dx, ", "d023540da8c3cb1ec2871d4eb5f7000b": "\\tau = \\frac{1}{\\Delta \\nu} \\approx \\frac{\\lambda^2}{c\\, \\Delta \\lambda}", "d02392bce62dc7cf155602129148950a": "n_x = 2, n_y = 1", "d023a2f9de3d527541b1a71f2288f6b0": "y_{(2)}''(t) = p(t)y_{(2)}'(t)+q(t)y_{(2)}(t),\\quad y_{(2)}(t_0) = 0, \\quad y_{(2)}'(t_0) = 1. ", "d023a771ee6f8a95655585389f7f8d7b": "\n\\begin{align}\nT &=\nR_2^2 \\,\\Gamma + \\int \\biggl(\\frac1K - R_2^2\\biggr)\\cos\\phi\\,d\\phi\\,d\\lambda\\\\\n&=R_2^2 \\,\\Gamma + \\int \\biggl(\n\\frac{b^2}{(1 - e^2\\sin^2\\phi)^2} - R_2^2\n\\biggr)\\cos\\phi\\,d\\phi\\,d\\lambda,\n\\end{align}\n", "d023b24da341f7029af406ebd26f7d1c": "\\land, \\lor, \\lnot, \\to", "d02409f18a8049eac8296ea5b01cd535": "\\mathrm{Nat}(\\Delta^{\\mathrm{op}}(\\mathbf{n},-), X) \\cong X(\\mathbf{n})", "d02431579804d5ad835e8530038e346c": "FDR \\le \\frac{{{m_0}}}{m}q\\left( {1 + \\frac{1}{2} + \\frac{1}{3} + ... + \\frac{1}{m}} \\right) \\approx \\frac{{{m_0}}}{m}q\\log \\left( m \\right)", "d024afdf8ec9fc9eb11925db5b4b51cd": " \\int_B s \\, d\\mu = \\int 1_B \\, s \\, d\\mu = \\sum_k a_k \\, \\mu(S_k \\cap B). ", "d024c7ada48e58b072ac727d03757834": "{1/a^2}", "d0251eae2a80b263a3a6508cbb701c6b": "\\frac{1}{2\\pi i} \\oint_C \\frac{dz}{z - a}.", "d0259a3064f7eaa279b46c8aa0c1cb3a": "\\begin{array}{rcl}\n \\dot x &= &Ax + b x_m(t), \\\\\n x_f(t) &= &c^{*}x,\n\\end{array}\n\\quad\nx(0) = x_0,\n", "d02624fd90c0c349c3b3b0efa051d533": "x_t = VD^tV^{-1}c", "d0263a4ccdce229590876c7d08a7b142": "\\dfrac{\\partial J^\\alpha}{\\partial x^\\alpha} = \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\mathbf{j} = 0", "d026633963eb4ce5bbda1e36b64bd4ef": "\\mathbf{m}_x", "d0272ec65d5d3af9829a636a0ace9cab": "S_{\\rm oblate} = 2\\pi a^2\\left(1+\\frac{1-e^2}{e}\\tanh^{-1}e\\right)\n\\quad\\mbox{where}\\quad e^2=1-\\frac{c^2}{a^2}. ", "d0275a1c5f2d97b9b9352791b6591a0d": "\\frac{4b^3}{27}", "d0277740e23d0872ee18ab0107967265": "\\nabla\\varphi", "d027b45a132935a26f41ccfee5442067": "3-2\\eta\\vartriangleleft O_K", "d0282023661a751a8127f95ddab70fe4": "\n \\delta U = \\int_{\\Omega^0} \\left[-N_{\\alpha\\beta,\\alpha}~\\delta u^0_{\\beta} \n + M_{\\alpha\\beta,\\beta}~\\delta w^0_{,\\alpha}\\right]~d\\Omega \n + \\int_{\\Gamma^0} \\left[n_\\alpha~N_{\\alpha\\beta}~\\delta u^0_{\\beta} \n- n_\\beta~M_{\\alpha\\beta}~\\delta w^0_{,\\alpha}\\right]~d\\Gamma \n", "d0291170b6425bc3114715773c436b7f": "p(x_i) - f(x_i) \\ = \\ -(-1)^i E,\\ \\ i = 0, ... , n\\!+\\!1. ", "d0291c6dbe0c6e8e18e91df94ae81242": "\\mathcal{}(W;M_0,M_1)", "d0298f99e144ef5d30a5b68b1ee1d39e": "-K_2/8", "d0298fc9b7f0547f8772c187929dde5a": "+ {13 \\choose 2}{4 \\choose 2}^2{11 \\choose 2}{4 \\choose 1} - 6,912", "d029b45e8567dcbe21ec15fb8f9b88cb": "\\mathrm{subject~to~}x_i\\ge x_j~ \\text{ for all } (i,j)\\in E.", "d029eaa731e09ea47e28bdd1a248a14b": "D = C^{-1} A {C^*}^{-1}", "d02a3677e38c1566ade01086f9b9be31": "(5)\\; \\frac{E_2}{E_1}=\\frac{(8 F r_1^2 + 1)^{3/2}-4 F r_1^2+1}{8 F r_1^2(2+F_1^2)}=\\frac{(8*4.5^2+1)^{3/2}-4*4.5^2+1}{8*4.5^2(2+4.5^2)}=0.55", "d02a7612fcae1f5650b072b5d4d32369": "x\\Vert A\\iff x\\in\\operatorname{cl}(A)", "d02ab7950231603be1b493a616eaa72a": "23\\cdot 47 = 100 (2\\cdot 4) + 10 (3\\cdot 4) + 10 (2\\cdot 7)+ 3\\cdot 7", "d02abf177e6d9b57226d1f48ac21e7d5": "P_i=(x_i,y_i)", "d02ad63af4429585d1b69a5aa836fe73": "\\vec{v}_{\\nabla B} = -\\frac{\\epsilon_\\perp}{q} \\frac{\\vec{B}\\times \\vec{R}_c}{R_c^2 B^2}", "d02adbc3e156ff43050ef46133945864": " \\vec{E} = -\\vec{\\nabla} \\phi = - {1 \\over 4 \\pi} \\vec{\\nabla} \\iiint_{\\vec{r}'} {\\vec{\\nabla}_{\\vec{r}'} \\bullet \\vec{E}(\\vec{r}') \\over \\| \\vec{r} - \\vec{r}' \\|} \\, d\\tau' ", "d02afb3a22140b600e714bdd13686e11": "\\scriptstyle \\phi:\\; U\\, \\rightarrow \\,\\mathbb{C}", "d02b0d18026a0d23549d28453dcbe8cf": "\\frac{x}{e^x-1}", "d02b5f968234861c432f74038bc99174": "\\beta := \\tfrac{1}{N} = \\tfrac{1}{\\lambda_m^2}", "d02b9fdbc9113fe447e7797e39541776": "\\frac{du}{dy}", "d02bb49b00cbbed6bc1fc39cd6106f50": "\\,\\!n_0", "d02c670d0bc0e93536888f0776831cdd": "\\Phi_i", "d02c8a32bd551278a48778f7bd4868ca": "\\displaystyle{f_n=g_{z_n}\\circ f \\circ g_{-w_n},}", "d02cb0f19f352cdd40d9351100e8eea1": "q=kiA", "d02cc47167bc0fd41941ebd47a126036": "\\text{duplicate}: ((S \\rarr A) \\times S) \\rarr (S \\rarr ((S \\rarr A) \\times S)) \\times S = (f, s) \\mapsto (s' \\mapsto (f, s'), s)", "d02d49578bb1c838f25179fbb107d562": "|\\lambda|^{1-2H}", "d02d5459279916032e1cb0a83b26c891": "(d)\\text{ }P(R_{i})=TRUE\\text{ for }i=1,2,...,n.", "d02d9182bdc4120388be0f995b9c6d0a": "B_0(a)", "d02d9aa7be30d6f4589fa3710bfc03bf": "\\ln (\\pi R) - \\frac12 \\,", "d02dab6886f29d53a480c86210e3f9a3": " \\dot{I} = 0 ", "d02ddb89105f15f105fcc60b6d0b1fff": "n\\ge2", "d02de294732dc553e4c637a84a6620d2": " \\vartheta_2(z,\\tau) = 0 \\quad \\Longleftrightarrow \\quad z = m + n \\tau + \\frac{1}{2} ", "d02e7af1e6f982ff81db24dd36224c70": "\n1\\leq p\\leq n_c\n", "d02eb44a4929e60b7cc0061755053f19": "m n n!", "d02ed0e88d721de7203f70d261572372": "\\vec{a}_j = \\sum_{i \\neq j}^n G \\frac{M_i}{|\\vec{r}_i - \\vec{r}_j|^3} (\\vec{r}_i - \\vec{r}_j)", "d02f2ceaad6c99d363b16904239811e3": "x^2y+xy^2=880", "d02fa3b52ced52aa798b674ea5710116": "\\aleph \\beth \\gimel \\daleth \\!", "d02fc4fa70f368b62da03cc51d7e446f": "A\\in\\mathcal{F}", "d02fd32b2e20477054204a650ce7b491": "M_A l + 2 M_B (l+l') +M_C l' = \\frac{1}{4} w l^3 + \\frac{1}{4} w' (l')^3.", "d02fe9a0b0c5fc5b334ea19c03da98db": "k:=k_n", "d02ffbdb62170a01f6c39c8d991c074c": "\\mathrm{SO}(10)\\times\\mathrm{SO}(2)", "d02fffa624b33c14596cc079d829ceb8": " 3^{63} \\cdot 16! \\cdot \\left(\\frac{4!}{2}\\right)^{15} \\cdot 4 \\cdot \\frac{96!}{(4!)^{24}} \\cdot 2^{95} \\cdot \\frac{96!}{(4!)^{24}} \\cdot 2^{95}", "d03034927faf59e5fb0ff7939e0cd471": "[P] + [Q] + [R] + [S] -4[O]", "d0303b49a3b97215a9470218675fb367": "\\mbox{d} \\sigma (t') =\n[\\mbox{d} \\sigma (t') / \\mbox{d} t'] \\mbox{d} t' ", "d030e0d3ffdf0c9ddc65ffaa6f21adba": "\\psi(x)<1.03883x", "d0316e6933541181a928466f31d962cd": "j+1", "d031e547b17503166dc33cc91067bd33": "\\mathbf{W}=\\begin{bmatrix}\n W_{11} & W_{12} \\\\\n W_{21} & W_{22} \n \\end{bmatrix}. ", "d032375228b0f6630f587e934163a159": "\\begin{align}\n g_1 &= -{3 \\over 2} \\mathrm{Im} \\left[ {z (1 - z^4) \\over z^6 + \\sqrt{5} z^3 - 1} \\right]\\\\\n g_2 &= -{3 \\over 2} \\mathrm{Re} \\left[ {z (1 + z^4) \\over z^6 + \\sqrt{5} z^3 - 1} \\right]\\\\\n g_3 &= \\mathrm{Im} \\left[ {1 + z^6 \\over z^6 + \\sqrt{5} z^3 - 1} \\right] - {1 \\over 2}\\\\\n\\end{align}", "d03246936dace46be47f2fcf63982661": "d=-(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c})) \\pm \\sqrt{(\\mathbf{l}\\cdot(\\mathbf{o}-\\mathbf{c}))^2-(\\mathbf{o}-\\mathbf{c})^2+r^2}", "d0329cec1eff5267346f6716cdeb90f6": "g_\\mathrm{e}", "d032c2589ea9785a804c4f81fa06eb59": "p \\rightarrow p - \\frac{\\Lambda c^{4}}{8 \\pi G}.", "d032f615beedf338030b750068f49202": "\\varepsilon < 1", "d03365844386aae4cf0e16239b36c4c0": "\\begin{align}g(t)\\mu(A_t)&=\\int_X g(t)1_{A_t}\\,d\\mu\\\\ &\\leq\\int_{A_t} g\\circ f\\,d\\mu\\\\ &\\leq\\int_X g\\circ f\\,d\\mu.\\end{align}", "d0337a6d529d3739eb293bb70d0dab3f": "{\\partial^2 V_{ij}\\over\\partial x_i\\partial y_j} = {-\\gamma\\over {s_{ij}}^2} {(x_j - x_i)}{(y_j-y_i)} ", "d033a800ad4f55f4817b423623492e86": "A= W_1 + i\\, W_2)\\,,", "d033f1d0f70d2fbe21086820ba561cb3": " (1+x)^{a/b} \\leq \\left(1+\\frac{a}{b}x\\right)", "d034c7792e8c76e93da7ff74a2db46a5": "q>3", "d0351b1760805ab26b1a67c982d33c57": "\\mathfrak{so}_{12}", "d035509ea175be97a05a4253b912197f": " \n{(F^{\\mu\\nu})^2 \\over 4}\n+{\\mu^2\\over 2} (A^\\mu)^2 \n", "d0358492d963a81327c6e527c105dea9": "Y(t) = \\sum_{i=1}^{N(t)} D_i", "d035bbb3241692871f8337204cf73d0d": "\n i_V \\circ T^* = T' \\circ i_V. \\,\n ", "d035e472df160f6c32705a2e13073184": "a=\\begin{bmatrix}2\\\\ 0 \\\\ -1 \\\\ 1\\end{bmatrix}.", "d035fc8bd41fc17fbdfa517b9d36eb0c": "\\R^{d_m}\\subset \\R^{d_{m'}}", "d036896e8305d096941d03432c764f99": "\\textstyle \\mathbb{R} ", "d036c070e4e9c3bfae4a8e69842feb9e": "E=\\gamma mc^2 \\,", "d036e5d92bf1d4fa2a2f772822b16595": " \\mathbf{A}_\\text{quad.} = {1 \\over 2}|x_1y_2 + x_2y_3 +x_3y_4 + x_4y_1 - x_2y_1 - x_3y_2 - x_4y_3 - x_1y_4| ", "d0371f5589cb17d538a92c151e5590ce": "\\Delta G^{TS-D}_{M}", "d0375b4c878f3ab82ce97f510250471f": "M = c \\mathbf{I} + \\sum_i a_i \\sigma^i", "d037c7cfbdf75752be6b5c0d26279719": "\\,\\mathrm{slog}_b(z) = \\mathrm{slog}_b(\\log_b(z)) + 1", "d037f74958d0a302d68212d1908b9e16": "(c_1,c_2)\\,", "d037fcd5fa24c8673763af9c9253cdb1": "n_x", "d0380e32ef6436523ef7c43f1b2d3dc3": "x=-a", "d0383718ba1daf302fbb379c597119bf": "\\partial_r", "d0386bf19ceeaa6adc1c586583fd5673": "\\frac{1}{\\textit{eff}} \\approx 1 + S \\frac{x}{100}.", "d038e45f8ad3885bed6d02ca26af7d39": "\\tan \\psi = \\frac {v'} {v' \\cos \\theta - f} \\sin \\theta.", "d038ecac3013f9f3cff8f74bebe885be": "X=T_{a}P_{a}^{'}+T_{b}P_{b}^{'}+T_{ab}P_{ab}^{'}+T_{e}P_{e}^{'}+E_{a}+E_{b}+E_{ab}+E_{e}+E \\,", "d039175ace385aea7b6b6643a407b8a9": "\\chi_\\perp", "d03962a995146b8bb03acb8191540c31": " \\delta \\mathrm{E}(e^2) = -2 \\int\\limits_{-\\infty}^{\\infty}{\\delta g(\\tau)\\left(R_{xs}(\\tau + \\alpha)- \\int\\limits_{0}^{\\infty} {g(\\theta) R_{x}(\\tau - \\theta)d\\theta}\\right)} d\\tau.", "d03a58f69ad3e10796357b984df275d7": "\\omega_1=\\tfrac{1}{2}(-1+\\sqrt3i)\\omega_2.", "d03aa7730398b925eefd1ba195f8211c": "Q_{ij}=\\int\\, \\rho(3r_i r_j-r^2\\delta_{ij})\\, d^3\\bold{r}", "d03ae22e1b4faa1492f95741a936e7bc": "\\Box_i", "d03b9a09e7e5ec0d9376b87be7758e55": " \\boldsymbol{D} = \\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P}\\ .", "d03b9b8a4ac9d360d5057502539c5061": "16:9", "d03baaea2d3a41df7b03f2346ab4747f": "\n1 - P_f = \\prod_{k=1}^N [1 - P_1(\\sigma_k)]\n", "d03bc2947ca6367124aaa558f5c46094": "p_1< p_0", "d03c0ba5cdeab941a0e762d5062c1cc9": "\\frac{dS}{dt} = - \\beta S I ", "d03c10579e5040977312ec42ae2f3b6d": "\nt \\quad = \\quad {\\; \\overline{X}_1 - \\overline{X}_2 \\; \\over \\sqrt{ \\; {s_1^2 \\over N_1} \\; + \\; {s_2^2 \\over N_2} \\quad }}\\,", "d03c47a3b14f8b046091e293f9b8eedf": "\\scriptstyle x_i", "d03c4a7bcb225a6d33e464c168cec700": "\\mathfrak{P}^{112}", "d03c6b1d2b6ffe5490c837b0434004a9": "27ay^2 = (a-x)(8a+x)^2", "d03ce254a595ae33aaef1f799e0cbd53": "\\nu(-\\gamma,z)=\\nu(\\gamma,z)", "d03d057b12f140cd1942e20686eb1784": " S^2 \\cdot X \\approx 1.1963 \\approx 310.3 \\ \\hbox{cents} ", "d03d95dbcdb499b8b9bd64697699e0d7": "1 + p + p^2 + p^3 + p^4", "d03e323b1452b4d6208c09737a3ba6dc": "\n \\hat{g}(x) = \\frac{\\hat{\\operatorname{E}}[\\,y_tK_h(x^*_t - x)\\,]}{\\hat{\\operatorname{E}}[\\,K_h(x^*_t - x)\\,]}, \n ", "d03e9088179c2db9aa337f1c8f769dc3": "\\ \\sgn(x) = -[x < 0] + [x > 0] \\,.", "d03f310621dc0f94a9e734adaff01af3": " {\\Delta}p ", "d03f35d101e711a8a0ad4adabf9c95c2": "p(\\hat{x}|x)", "d03f78dc143622bf0746bdf13edcf59d": "\\mathbf{F}_{k}", "d03f8589195ad3727f020c99e393a583": "{w_1,w_2,\\ldots,w_n}=", "d03fc966d20f6420b25d1f29b1a8a5a9": "10^{-12} \\mathrm{s}", "d03ffb5842de505a6e813ebf3f5d99b3": "{{z}_{in}}=\\frac{2}{{{g}_{m3}}}", "d04022f638fdecec659d790096c3ca49": "\\ell = (k-1)\\sum_{i=1}^N\\ln{(x_i)} - Nk - Nk\\ln{\\left(\\frac{\\sum x_i}{kN}\\right)} - N\\ln(\\Gamma(k))", "d04046e5eda9c17b74b85f088caca8bd": "\\mathcal{R}^{2}=R^{2}+R_{\\alpha \\beta \\mu \\nu }R^{\\alpha \\beta \\mu \\nu\n}-4R_{\\mu \\nu }R^{\\mu \\nu }", "d0405c93fff8d83c36dbf5db31362568": "T_{f+g} x = T_f x + T_g x", "d04061936dcfb751757df59ed56b671c": "\n\\langle \\rho^2 \\rangle_{R_s}=\\frac{7}{8}\\rho_0^2\n", "d0408843d1bd2fdcd6dcc07355581936": "R \\times D_R = E \\times D_E", "d04095debfca69165cfdb51e9c9ee4e6": "\\operatorname{E}_{p(\\sigma^2\\mid S^2)}\\left[\\sigma^4 \\left(c n \\tfrac{S^2}{\\sigma^2} -1 \\right)^2\\right] \\neq \\sigma^4 \\operatorname{E}_{p(\\sigma^2\\mid S^2)}\\left[\\left(c n \\tfrac{S^2}{\\sigma^2} -1 \\right)^2\\right]", "d04097af81c1015333b3db46d0215172": "-\\left ( \\frac{\\partial w}{\\partial y} \\frac{\\partial \\theta}{\\partial z} \\right )", "d040b69277e5f4f1190b9066eed875b6": "\\mathrm{SURE}(h) = d\\sigma^2 + \\|g(x)\\|^2 + 2 \\sigma^2 \\sum_{i=1}^d \\frac{\\partial}{\\partial x_i} g_i(x), ", "d040cae4d7f873b3507fb0b8db1219d8": "\\tan\\frac{\\theta}{2}", "d040d8eb33e52a1c5c2d821fa4aa02c0": "\\begin{array}{c|c}92&8\\\\87&13\\end{array}", "d0411cccd40108c915be0bfe21d42d06": "N_{B/A}: \\operatorname{Id}(B) \\to \\operatorname{Id}(A)", "d0417ef7f23531fda6f938681b23897a": " P(H_{TS}|D_pX)/P(H_{TL}|D_pX) = [P(H_{FS}|X)/P(H_{FL}|X)] \\cdot [P(D_p|H_{TS}X)/P(D_p|H_{TL}X)] ", "d0418946ad931b118cf3bf1d988eba61": " M(G \\setminus e)", "d0419c9af46e043e3d1ad6802f3c81cc": "\\frac{243}{50} = 4.86", "d041d21146fb9dd14d4d27b3292ad280": " (X_{j})_{j\\in S_{i}}", "d041e5bf91cdf254735ef8f9e0bdc639": "M,N,... ::=\\ v\\ |\\ [v]\\;M\\ |\\ (M)\\;N", "d042069eaefa686854407f40830e249c": "(1/\\tau_c)", "d0420e71f39f8dabcb0d32e37ba2cb87": "\\tau=T-t", "d0421881881685835fc5ee222cf2c6a2": "| \\psi \\rangle = \\alpha|\\uparrow\\rangle + \\beta|\\downarrow\\rangle", "d042dab14d774a0e0f63842afe150c12": " \\mathbf{R} \\times \\mathbf{F} = \\sum_{i=1}^n \\mathbf{R}_i \\times \\mathbf{F}_i, ", "d0437ebf0c527baed227dad868ce4599": "x=d_n\\dots d_2d_1d_0.d_{-1}d_{-2}\\dots d_{-m}", "d0438646c1f482faffdd1bac9a841799": "s\\,", "d043c9a8ddd4cfc257393f6e1cf3f65c": " 2r\\sqrt{3} = R\\sqrt{3} \\!\\, ", "d0440e66a15b34a6350004d3d02b431a": "\\hat{a_2}", "d04460b1de03dc289a1906a4e3982bab": "P(\\boldsymbol{D}|\\alpha) = \\prod_d P(d|\\alpha)", "d0447036e1954fc3ed7c106d5a2720f7": " 32 = 2^5 : 200 = (2^3)\\cdot(5^2),", "d0447683f4da0c888eceee608f479197": "p_0 = \\sqrt{2\\hbar \\rho_0} = \\phi_0\\sqrt{\\frac{2\\alpha}{\\pi}} \\ ", "d0447bf192bb1c32e6e78704b02ed59c": "\\textbf{let}\\ x = e_1\\ \\textbf{in}\\ e_2\\ ::= (\\lambda\\ x.e_2)\\ e_1", "d0451e1224cadec35ee4bba6030502b2": "T_2 \\vdash \\neg\\varphi", "d045212f04c1c4e833f82bab230d4247": "N_\\text{ZC} ", "d045409ee61d589ff7335de2c45f2a8e": "\na_{ij}\\sim P_{j}A_{i}\\qquad i=1,\\ldots n,~~j=1,\\ldots m\n", "d0455585f425cacfd0783f08cd4ced00": "\\mathcal{P} = \\{\\bullet\\} \\times \\mbox{Seq}(\\mathcal{P})", "d04563fff99a3daaff219d573db05a44": "(a-b)t+b", "d045cb5b1f4eef5d0c171d26d4e869e3": "\\beta(1)", "d04661d8f8b4626dcfa29a7882d5bcf9": "U \\subset \\mathbf{R}^2", "d0470f50a9d9f194797bfb5c8b409383": "E' \\simeq E^\\beta", "d0471df1144f5f522ac48569ee15293f": "\\int_a^b \\mathbf{y}(x) \\cdot d\\mathbf{r} = \\int_a^b \\mathbf{y}(x) \\cdot \\frac{d\\mathbf{r}(x)}{dx} dx ", "d04736c6f7052abf783fefbf15f4d36a": "e^{-\\frac{\\Omega}{k T}} = \\sum_i e^{\\frac{\\mu_1 N_{1,i} + \\ldots + \\mu_s N_{s,i} - E_i}{k T}}.", "d047492e56830608bd89908a08f441b3": " n> 0", "d047b993ed67fe927443acdd6a091866": "C_{1}, C_{2} > 0", "d04861bc6d8692981d6245f768da8a37": " \\mathrm{Ta} = \\frac{4\\Omega^2 R^4}{\\nu^2}", "d048675f1e886ad6ac984f2bb548dd94": "x,y \\in \\real^n \\ x \\preceq y \\iff x_i \\leq y_i \\ \\forall i=1,\\ldots,n ", "d048860409f45741f172c1cf91657229": "(\\varphi,\\lambda)", "d04913a17f8b3dad5759681cb2bbbfb2": "\n\\dfrac{\n \\dfrac{\n \\dfrac{the}{NP/N}\n \\qquad\n \\dfrac{dog}{N}\n }{NP}>\n \\qquad\n \\dfrac{\n \\dfrac{bit}{(S\\backslash NP)/NP}\n \\qquad\n \\dfrac{John}{NP}\n }{S\\backslash NP}>\n}{S}<\n", "d04941d68e3a1d9edd226d76692da152": "\\,\\delta : Q \\times \\Gamma \\rightarrow Q \\times \\Gamma^*", "d0494be5069220a91e32f913d188d750": "y_1(t) \\,\\!\\ne y_2(t)", "d049648e9311e10c6a8a3ded11f50986": "X_T = \\int_0^\\infty X(\\lambda)I(\\lambda,T)\\,d\\lambda", "d0497115ac4bc8e4d6b3e858e88b8f2e": "\\text {elasticity} = \\frac {\\partial f(x)}{\\partial x} \\frac {x}{f(x)} = \\frac {\\partial \\text{ln} f(x)}{\\partial \\text{ln} x},", "d049f7c48217fe76e65d89352cd3027b": "\\,\\ \\sin x", "d04a12fa315d77c1e0d5f424eee813ee": "V = 0", "d04a34315fc4da8fc15454c09969bb45": "H=\\hbar \\omega_c a^\\dagger a + \\frac{1}{2} \\hbar \\omega_m (p_m^2+x_m^2) - \\hbar g_0 a^\\dagger a x_m + i \\hbar E (a^\\dagger e^{-i \\omega_l t} - a e^{i \\omega_l t})", "d04ab430890ecf363bda0ef729f29bc9": "f_i(x) \\leq 1, \\quad i = 1,\\dots,m", "d04ac5cc2976e5e2f4e90257f863f849": " (T", "d04b6435cfe36000611d5d1bdc12eac2": "[P] + [Q] + [R] - 3[O]", "d04b6614ee93a582cef9e0db44de48f4": "g \\bar w = h", "d04b77e33bb2582d1bd91f0bb656a6c3": "A = \\langle E\\rangle -TS= - k_B T \\ln Z.", "d04bb5bcdda0979f87e70ba130f20f19": "X^{\\prime}=F\\left(X\\right)X, F^{\\prime}(X) \\le 0 ", "d04bbbbe9a21dd40329b4a621c232528": "c(t,s)", "d04cf4ed2c1dfdd7f7f2076eb20ac3e1": "\\top(x,y) = f^{-1}(\\top_{\\mathrm{Luk}}(f(x), f(y))).", "d04cf680b9619b8f293b79d05257e2ba": "\n d\\mathbf{f} = \\mathbf{t}_0~d\\Gamma_0 = \\boldsymbol{N}^T\\cdot\\mathbf{n}_0~d\\Gamma_0 = \\boldsymbol{P}\\cdot\\mathbf{n}_0~d\\Gamma_0\n", "d04d2dfbbdb69602b20c64d61a52a2c3": "1 F^{-1} = \\frac{1 V}{1 C} ", "d04e05ae30f4add67162c695ffef638b": "\\Phi_{N}(\\rho,z)", "d04e35e01ce7d91695b953bde37c3d58": "S_1 \\equiv_1 S_2 \\Leftrightarrow Ext_\\sigma(S_1) = Ext_\\sigma(S_2)", "d04f58693570b936442efae34184ef61": "B^{\\gamma}{}_{\\beta\\cdots} = g^{\\gamma\\alpha}A_{\\alpha\\beta\\cdots}", "d04f9bd958b1d22fcf1b089063669a7a": "10^{10^{56}}", "d04feb7358bfe75ae0d5ef2b40489902": "(X_0,X_1,...)\\times(Y_0,Y_1,...)=(X_0 Y_0,X_0^p Y_1+X_1 Y_0^p+p X_1 Y_1,...)", "d0503b89bbbf12f0d82187efe30c4e3a": "\\oint _{\\partial D}\\,g(\\beta)\\;d\\beta=-\\int_{\\mathbf{R}^d}\\,\\nabla_x\\mathbf{1}_{x\\in D}\\,\\cdot\\,n_x\\,g(x)\\;dx. ", "d050b20ef45c1a1a21cbbd0871f873cb": "B^2", "d051037ff5cae742435dfcd0c1b3ee13": "(x+3)(x+3)", "d0510794fa00f46f32a6d3679ef1871b": "x = \\operatorname{prox}_{\\varphi}(x)+\\operatorname{prox}_{\\varphi^*}(x)", "d0519bbbb81d1bec78e1f0c8f1967219": "\\sigma = \\sqrt{\\frac{B - N D^2}{N}} \\,", "d051a4695c5aa20a4aa8d880e19891c7": "R(x,y,z,w) = R(z,w,x,y),\\,", "d051bf4ddb4dfe749c1718f2a23b92c3": " \\det S \\propto \\left( \\int_V \\mathcal D \\phi \\; e^{- \\langle \\phi, S\\phi\\rangle} \\right)^{-2} \\,. ", "d05221ff6b14c85f2ae5c388efde9956": "\\left\\lceil \\log_2 \\frac {1}{p(x)} \\right\\rceil + 1", "d05261b46d34cecbe68b8407175b47f4": "G=F/{\\sim}", "d0531331cc135b9029a5ad84cdb0af0e": "a_{i_0}", "d05343092dd6c88188be03a29dd66dba": "\n\\begin{bmatrix} u \\\\ v \\end{bmatrix} = \\begin{bmatrix} m1 & m2 \\\\ m3 & m4 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} + \\begin{bmatrix} tx \\\\ ty \\end{bmatrix}\n", "d05355baa4a0390474ef715e7c873dd6": "M_X(t)\n=\\sum_{k=0}^\\infty t^k\\underbrace{\\operatorname{P}(X=k)}_{=\\,\\lambda^ke^{-\\lambda}/k!}\n=e^{-\\lambda}\\sum_{k=0}^\\infty \\frac{(t\\lambda)^k}{k!} = e^{\\lambda(t-1)},\\qquad t\\in\\mathbb{C},\n", "d053652f104971b634841e0838b08b33": "n(n+1)^2", "d053840e053d14c2a6fc5b847c009861": "K(m)=\\int_0^{\\pi/2}{\\frac{1}{\\sqrt{1-m \\sin^2 \\theta }}} d\\theta,", "d0547ecc7e8b0ef623f28ef4880fc4d7": "s \\models_K \\neg f", "d054a84c8466b64b76eefee13ca8103c": "\\displaystyle X", "d054e95fe398bcf2e061ebeb6bb776c0": "x'(\\mu - r_f \\cdot k) - \\frac{a}{2} \\cdot x'Vx.", "d0550d23f4f779d8022faaad14ea40a5": "00.25\\}", "d058834b91a44159927f78b8b6be53c6": "\\Pi_r=\\Pi - M \\,", "d05898df6f20841ba0a35421252098e7": "\n(1)\\cfrac{\n (2)\\cfrac{\n (1)\\cfrac{C_1 (1,3)\\qquad C_2 (-1,2,5)}{C_3 (2,3,5)}\n \\qquad\n C_4 (1,-2)\n }\n {C_7 (1,3,5)}\n \\qquad\n (4)\\cfrac{C_5 (-1,4) \\qquad C_6 (-1,-4)}{\\color{red}C_8 (-1)}\n}\n{\nC_9 (3,5)\n}\n", "d058f88bbee6f748d23e8e9b7818bf5c": " V=\\left\\{e,a,b,ab\\right\\}", "d0590bd495adb42bd157e2fd6539ecdc": "\\textstyle p(c_j) \\sum_{f_i \\in F} \\sum_{k=1}^m p(f_{ik})^2", "d0594e1138fb908ad50195b06517776c": "\\frac{\\operatorname{d} u}{\\operatorname{d} x} = -u^2\\,", "d05961a904b3ee2a2b003c1a1ad7611d": "\n\\frac{1}{42}, \\qquad \\frac{2}{42}, \\qquad \\frac{3}{42}, \\qquad\n\\dots\\dots, \\qquad \\frac{39}{42}, \\qquad \\frac{40}{42}, \\qquad\n\\frac{41}{42}.\n", "d0596e210450f68fc217caf7b251c25e": " E_{max}=I_p+3.17U_p ", "d05996a9c1e8530684dab8069d0e2f37": "\\log^2n", "d05a518a4e17faa5d251550f1804eecd": "\\begin{align}\n n_i &= \\sum_j n_{ij} \\\\\n z_i &= \\frac{1}{n_i}\\sum_j z_{ij}n_{ij} \\\\\n w_i &= \\frac{1}{n_i}\\sum_j w_{ij}n_{ij} = k + (b - a)z_i \\\\\n n_i'&= \\sum_j n_{ij}' = n_i[k + (b - a)z_i] \\\\\n z_i'&= \\frac{1}{n_i'}\\sum_j z_{ij}n_{ij}'\n\\end{align}", "d05a97990ea836fca14f8f3549236ee3": "\\ \\Delta S={a} \\sin \\theta", "d05ac2ded87e33e491d2608f10f75b7b": " \\left ( \\frac{d}{dx} - r_{1} \\right ) ue^{r_{1}x} = \\frac{d}{dx}(ue^{r_{1}x}) - r_{1}ue^{r_{1}x} = \\frac{d}{dx}(u)e^{r_{1}x} + r_{1}ue^{r_{1}x}- r_{1}ue^{r_{1}x} = \\frac{d}{dx}(u)e^{r_{1}x} ", "d05b0458069daecb2ce5bf522f60200f": "\\epsilon_r\\;", "d05b1d8f262dc5a7aa0cd1244e3cef40": "\nE_x=\\alpha_0\\left(T-T_0\\right)P_x+\\alpha_{11}P_x^3+\\alpha_{111}P_x^5\n", "d05b22b49ed815cb35e6ae099b7d04bf": "R_{\\mu \\nu} = \\Lambda g_{\\mu \\nu} \\,.", "d05b23fcee19225826bbb1bdd1a4e42d": "\\|Eg\\|_{(k)}^2=\\sum_{m,n}|\\widehat{Eg}(m,n)|^2 (1+m^2+n^2)\\le c_k^{-2}\\sum_{m,n} |\\widehat{g}(n)|^2 (1+n^2)^{2k-1} (1+m^2+n^2)^{-k} \\le c_k^{-2}C_k \\|g\\|_{k-1/2}^2.", "d05b315100d40860c1209e563ff86dc3": "y_{min}", "d05b5a064ac8d97339030a8fd2542577": "\\begin{cases} \\pi_{r, k}: J^{r}(\\pi) \\to J^{k}(\\pi)\\\\ j^{r}_{p}\\sigma \\mapsto j^{k}_{p}\\sigma \\end{cases}", "d05b93773e61b96443984728b1ecd1df": "\\xi_x", "d05b951d7d7c841e0f7af2375cfd4f1b": "d^* \\geq d", "d05bb45d43f25f3da0bc77d91ad832c0": "\\overline{H^{p,q}}=H^{q,p}.", "d05bc16258809f928b7f94fb6a6e19b1": "\\Delta \\bar S_{vap} = 10.5R ", "d05bfd1905df4f9df0f10890d5407aa1": "B(e_{i},e_{j}) = 0\\ \\forall i \\neq j.", "d05c263edb0faf08238d1357ad1d283a": "\\alpha(x)\\beta(y) = \\gamma(xy)\\,", "d05c5dffcb3dbf8919890e69716af516": "\\tau = i \\frac{\\,_2F_1\\big(\\tfrac{1}{6},\\tfrac{5}{6},1,1-\\alpha\\big)}{\\,_2F_1\\big(\\tfrac{1}{6},\\tfrac{5}{6},1,\\alpha\\big)}", "d05c81200da47f184a162f9ae2608ec6": "\\Delta W_e", "d05ce3fc7a697fab5b9c42bfdbf7569e": "E \\frac{f(x)}{g(x)} = E f(x) - E g(x)", "d05cff1bc58f63d5705e7bc04329c33a": "H(X_1, X_2) \\le H(X_1) + H(X_2).", "d05d11aa314607d5cf9b552e5f42ae63": "\\boldsymbol{Y}_v", "d05d255f9bdb56a327c7e35f199427fb": " \\Delta t", "d05d57ecbc38c2297633ece51e48dfe9": "f^{*}\\colon B\\to A", "d05db13d37493dc14b8682827b612e30": "\\eta_{xy}", "d05e3bd2ecba2fde940335a5210ed050": "\n\\frac{M(a+1,b+1,z)}{M(a,b,z)} = \\cfrac{1}{1 - \\cfrac{{\\displaystyle\\frac{b-a}{b(b+1)}z}}\n{1 + \\cfrac{{\\displaystyle\\frac{a+1}{(b+1)(b+2)}z}}\n{1 - \\cfrac{{\\displaystyle\\frac{b-a+1}{(b+2)(b+3)}z}}\n{1 + \\cfrac{{\\displaystyle\\frac{a+2}{(b+3)(b+4)}z}}{1 - \\ddots}}}}}\n", "d05e3eae5dac86585e73fdee2a905afb": "i_\\alpha:\\Lambda^k V\\rightarrow\\Lambda^{k-1}V.", "d05e65b0368b50015d6d80d124000416": "\\beta=s^{1/3}", "d05e82f87afbd07ae6affb408f5453ae": "a_i \\geq 0", "d05ea03d364bdc715406b7204bb8e21d": "\\sum_{j} p^{ij} = 1. \\ ", "d05ea7c86c1257f05d67781ae411b583": "\\varepsilon :TV\\to k", "d05ef823cf7f881688cefbc07624bb90": "x_{11}=0\\,", "d05f14b93db77c466ceddeb7a9cfe618": "\\xi, v, a, I, E, P_{ac}", "d05f353094f2715725c0e174c0959ab8": "\\frac{a}{\\sin A} = \\frac{c}{\\sin C}", "d05f5d8a590956eb287a243bacc2fa66": "\\beta^\\Phi", "d05f9614b8c3f2499621193d90002355": "\\cos\\omega t", "d06008e1b4af6d0f7747d608e4fc9c51": "D(fg) = D(f)\\cdot g(x) + f(x)\\cdot D(g)", "d060682b176f959d0fc27c66a3401dcc": "\\log(n^c)=c \\log n", "d0609e70f376366cf377e54e142c430f": "\\big\\}", "d060b17b29e0dae91a1cac23ea62281a": "[-1,1]", "d060cb615b9e5fc5e5739dcf49e45f38": "\\mathbf u' = F(\\mathbf u,\\lambda),\\, \\mathbf u(0) = \\mathbf u(T)", "d060cd2506d1c7cbd36adf9d1f27209d": " + \\,", "d060f228ea69a2ddc97306b327d66019": "P_2/P_1", "d061254ecee1a174f473c3403e68e161": "\n \\begin{align}\n \\sigma_{xx} & =-\\frac{2z}{\\pi}\\int_a^b\\frac{p(x')(x-x')^2\\, dx'}{[(x-x')^2+z^2]^2} ~;~~\n \\sigma_{zz} =-\\frac{2z^3}{\\pi}\\int_a^b\\frac{p(x')\\, dx'}{[(x-x')^2+z^2]^2} \\\\\n \\sigma_{xz} & =-\\frac{2z^2}{\\pi}\\int_a^b\\frac{p(x')(x-x')\\, dx'}{[(x-x')^2+z^2]^2}\n \\end{align}\n ", "d06136410f570132b3298fd6e414095a": "\\begin{align} P(hypercalcemia~WHOIFPI) = \\\\\n P(hypercalcemia~WHOIFPI~by~PH) + P(hypercalcemia~WHOIFPI~by~cancer) + \\\\\n P(hypercalcemia~WHOIFPI~by~other~conditions) + P(hypercalcemia~WHOIFPI~by~no~disease) = \\\\\n 0.00125 + 0.0002 + 0.0005 + 0.0014 = 0.00335 \\end{align} ", "d06194458ef43aa931d9753f34b04ffb": "\n \\frac{S_n}{n} \\ \\xrightarrow{p}\\ 0, \\qquad\n \\frac{S_n}{n} \\ \\xrightarrow{a.s.} 0, \\qquad \\text{as}\\ \\ n\\to\\infty.\n ", "d061a7b34b13cdc20ff17ec0c4af61a9": "300-3x", "d062beb66ddcd802512bb82935b5a1e6": "g^\\lambda_{\\mu,\\nu}(q)\\in\\mathbb{Z}[q]. \\, ", "d062c25a51ff7cab83f662ff09c5888b": "\np(t) = p_{t}(j) \\times p_{t}(j+1) \\times p_{t}(j+2) \\times p_{t}(j+3)\n", "d062dd367bcf48e7295e99310f35c299": "k_R\\,", "d0635806718dc025ded630e2c38d59da": "\n0 = \\frac{d}{dq} \\left[ \\ln \\frac{d\\phi}{dq} + \\ln r^{2} \\right]\n", "d0635ee569ff09ea551359fd1ed9602f": " U = \\frac {c}{n} ", "d063df115a1a1ea0ed5a9d635118a413": "\\pi_1(V,w)", "d063e86f955dac10116d8a19a9ff7abe": "\\begin{align}\n h_1(X_1,X_2)&= X_1 + X_2\\\\\n h_2(X_1,X_2)&= X_1^2 + X_1X_2 + X_2^2.\n\\end{align}", "d0641a62d71dcf07ef38d4b3699cdf52": "\\prod_{p_i\\in P} p_i^{a_i} \\equiv \\prod_{p_i\\in P} p_i^{b_i} \\; \\operatorname{(mod} \\; n \\operatorname{)}", "d06424a242869d74c5903f03246036a8": "v_n : \\Omega \\to V", "d06425f94cf71f15abd4dc85667b8c69": "\\, {\\hat{A}} = {T^{-1}}AT", "d06454d5a555a235eeaa407f7c97278c": "Y_3=\\frac{1}{T_3}", "d0648db76489ff4b6322654bc12ec803": " \\lambda_j ", "d0649c2a9fbced5b72735c2cc649a6ca": "L_n[\\alpha,c]=O\\left(e^{(c+o(1))(\\ln n)^\\alpha(\\ln\\ln n)^{1-\\alpha}}\\right),", "d064a3bc0865e9382105ab2ccce27c1f": "\\sum_i b_i(a^*)", "d064c133d3bfe2ba87dfc2fd5ba8deb8": "S \\to c_{3,1}(\\widehat{a}, T, \\widehat{d})", "d064fe0ee9a1548f1a0c1fccb509b110": "\\mathrm{APF} = \\frac{N_\\mathrm{atoms} \nV_\\mathrm{atom}}{V_\\mathrm{crystal}} = \\frac{2 (4/3)\\pi \nr^3}{(4r/\\sqrt{3})^3}", "d0655f0e75b2aa6978afd8e7b7d81ece": " \\mathbf{v} = \\mathbf{\\hat{e}}_{\\bot} \\left ( \\partial A/\\partial t \\right ) \\,\\!", "d0658e8cb85baf9a03de50c336d0d103": "\\Pi\\left(\\frac{z}{m}\\right) \\, \\Pi\\left(\\frac{z-1}{m}\\right) \\cdots \\Pi\\left(\\frac{z-m+1}{m}\\right) = (2 \\pi)^{\\frac{m-1}{2}} m^{-z-\\frac{1}{2}} \\Pi(z).", "d065954c086bc48efa65d44fc60b7047": "\\lambda^*(S) = \\lambda^*(S \\cap A) + \\lambda^*(S - A) \\, .", "d0660a56aac84ba28f88a1fb0359a94b": "\\displaystyle{\\pi_\\pm(g)F_\\pm(z)= (-\\overline{\\beta} z + \\alpha )^{-1\\pm 1/2} F_\\pm\\left({\\overline{\\alpha} z -\\beta\\over -\\overline{\\beta} z + \\alpha}\\right).}", "d06696abfdb5bc4b700bdee1de08298b": "\\ \\|x\\|_\\infty=\\max \\left\\{|x_1|, |x_2|, \\ldots, |x_n|\\right\\}", "d066accacf4db983fd34b08732a661c5": "\\mathbf{F}^2 = \\mathbf{P}.", "d06712abe4cf8b5068a51f41959c5656": " \\Delta \\leq xx^T ", "d067423909b6484d0dc4838344391ee9": "\\frac{dx_i}{dt} = r_i x_i \\left(1- \\frac{\\sum_{j=1}^N \\alpha_{ij}x_j}{K_i} \\right) ", "d06750c0ac32321242af9d718e051263": "e_2 ", "d0677079573f42b6155d75c775e64789": "Em_U= Tr_{theta}.Tr_{ex}.Ex_U", "d0677fbd65be63f921a1798980022a8b": "\\Rightarrow \\ - q_1 + a - (q_1+q_2) - \\frac{\\partial C_1 (q_1)}{\\partial q_1}=0", "d0680e64d5f687d25a2404b05819d743": "\\mu{\\frac{dy}{dx}} + y{\\frac{d{\\mu}}{dx}} = \\mu{q(x)}", "d06811af76102021b9ace49a52204117": "\\begin{align}H_x &= - \\int |\\psi(x)|^2 \\ln (|\\psi(x)|^2 \\cdot \\ell ) \\,dx \\\\\n&= -\\frac{1}{\\ell \\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} \\exp{\\left( -\\frac{x^2}{2\\ell^2}\\right)} \\ln \\left[\\frac{1}{\\sqrt{2\\pi}} \\exp{\\left( -\\frac{x^2}{2\\ell^2}\\right)}\\right] \\, dx \\\\\n&= \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} \\exp{\\left( -\\frac{u^2}{2}\\right)} \\left[\\ln(\\sqrt{2\\pi}) + \\frac{u^2}{2}\\right] \\, du\\\\\n&= \\ln(\\sqrt{2\\pi}) + \\frac{1}{2}.\\end{align}", "d0684b3278cf522adb05211ebbf102b3": "\\| K \\| = \\max \\left\\{ \\left. \\| v \\|_{\\mathbb{R}^{n}} \\right| v \\in K \\right\\}", "d068679e50a3524c7e27df2f5eda972d": " K(y,x) = \\sum_{i \\in \\mathbb{N}} \\lambda_i e_i(y) e_i(x) ", "d068b68e14852378ff1877e52e6c57d7": "u_{n\\mathbf{k}}", "d068f3c3395b16e4d14ab748cfe6c18f": "\n\\mathbf{G}=\\begin{pmatrix}\n\\mathbf{G}_1 \\\\ \\mathbf{G}_2 \\\\ \\mathbf{G}_3\n\\end{pmatrix}\n", "d0691be9603b4d08166ea776c5c6fd75": "(\\operatorname{div} T)(Y_1,...,Y_{q-1}) = \\operatorname{trace}(X\\mapsto \\#( \\nabla T)(X,\\cdot,Y_1,...,Y_{q-1} ))", "d0692faf6cb718e93d94fa0ede05c560": "X({\\pi/2},\\phi)=(2\\cos\\phi,2\\sin\\phi,1)\\,", "d0693510b566536d7f3b24b2f477d71d": "5F_{14}^2=710645\\equiv 0 \\pmod {29} \\;\\;\\text{ and }\\;\\;5F_{15}^2=1860500\\equiv 5 \\pmod {29}", "d0699aebef37bf62ae7d875781072eff": "\n\\forall (x_n),\\, x_n\\in U, \\, d(x_{n+1},f(x_n))<\\varepsilon \\quad \\exists (y_n), \\, \\, y_{n+1}=f(y_n),\\quad \\text{such that} \\,\\, \\forall n \\,\\, x_n\\in U_{\\delta}(y_n).\n", "d069a6f6c29e8d198ba24988396ccc19": "\\frac{E_P}{E_{ann}} \\cdot r \\cdot C_T \\cdot t = C_T \\cdot t", "d069dc53ea9108d7c52553a7bd2b69f9": "Z_t= \\sum_{i=1}^{t} Y_{i} \\quad \\text{ for } t=1,2, \\dots ,n \\,. ", "d069f745a8a4cd5e091db633dbef345e": "R(x, y) = \\begin{cases}\n 1 & \\text{if } x \\le y \\\\\n a_i + (b_i - a_i) \\cdot R_i\\left(\\frac{x - a_i}{b_i - a_i}, \\frac{y - a_i}{b_i - a_i}\\right)\n & \\text{if } a_i < y < x \\le b_i \\\\\n y & \\text{otherwise.}\n\\end{cases}", "d06a2341d76f72997e2ed312e2cae76c": "\\hat{\\alpha}", "d06a27c011df686b10d778912af09fb5": "\\omega=\\omega_0.", "d06a4dcf62d60a6a737a48640643daa8": "[\\mathrm{Q}]", "d06ab1b550780a7b2a06b8979fe0c1e0": " \\ x^2 - Ny^2 =k ", "d06abe5ef52118191a59a53dc844130b": "\\Delta_d=dd^*+d^*d", "d06ac28e481c43258bfddd3cdddc0385": " \\operatorname{ var }( r ) = \\frac{ 1 }{ n } ( 1 - \\frac{ n - 1 }{ N - 1 } ) \\frac{ r^2 s_x^2 + s_y^2 - 2 r \\rho s_x s_y }{ m_x^2 } ) ", "d06afac4059bab55dcf276e44072f073": "w_{2}=-.142,", "d06b092363e510ab39951272507a3c9d": "k=C.C", "d06b09a16c55e1c4ea6749b9cea9be44": "\\binom{a_{12}}{a_{22}}", "d06b0dc9269fd81626fa8353dd638265": " \\mathbf{r}_A - \\mathbf{r}_P = a\\mathbf{e}_A, \\quad \\mathbf{r}_B - \\mathbf{r}_P = b\\mathbf{e}_B.", "d06b8ec402f80c165ebc28d6b72e6458": "t_0 0. \\\\\n \\end{cases}\n", "d06f349cb3462fbbaffac06e97cdc21c": "E_\\infty", "d06f6eee4b32ce2f5b901987d0091bb3": "A = 4\\pi R^2 \\rightarrow dA \\approx 8\\pi R dR", "d06fc27ad2ff24762475fac9fb5355cb": "P_0=(R_0, L_0 \\bigoplus F(R_0,K))", "d06fd41f7ea5f56fec7597a36b02ba6a": "-k^2+\\frac{\\omega^2}{c^2}=\\frac{m^2c^2}{\\hbar^2}.", "d070206dac6f3609db0291d441c42bd2": "X = \\{1, \\cdots, n \\}", "d0709102ac10647cb2d527cc995f0f70": "\\frac{n}{u}\\cdot|MDS(C)|", "d0709592caa65ce8456c5eefaa767ee0": "\\mathrm{Sym} (\\mathfrak{g})", "d070de842531481b361c02663c6e9013": "\n\\operatorname{Li}_s(z) = \\sum_{k=1}^\\infty {z^k \\over k^s} = z + {z^2 \\over 2^s} + {z^3 \\over 3^s} + \\cdots \\,.\n", "d0712fab0ec3d679cbe3b1e23b62843c": "\\sum_{g\\in G}f(g)hgk^{-1}=\\sum_{g\\in G}f(h^{-1}gk)g", "d07140a1be02922fd2715ba4d259f672": "E=pc", "d071a63c472c9bd7821820028b23f50d": "\\sigma_{\\ell j}", "d071b7eb98367a0cc4fe939f89bb9c91": "W_2^T W_2 =I", "d071dbb93b456e40a8c9fb12b7e362d9": "\n\\begin{align}\n&b^2m + c^2n \\\\\n&= nm^2 + n^2m + (m+n)d^2 \\\\\n&= (m+n)(mn + d^2) \\\\\n&= a(mn + d^2), \\\\\n\\end{align}\n", "d071f72c24d1ba844366b094c1aeed8e": "T_B^{\\mu\\nu} = T^{\\mu\\nu} +\\frac 12 \\partial_\\lambda(S^{\\mu\\nu\\lambda}+S^{\\nu\\mu\\lambda}-S^{\\lambda\\nu\\mu})", "d072301f9e5b52438188f6cf5caae4a5": "Q_{s2} = q_{1} + q_{2} - q_{3} - q_{4}\\!", "d0726241020676b14aa6298ce6a18b21": "a|b", "d072a2027c903e9b6e57e48ee90d7e7c": "h = (B \\to S, S^{*}B^{+})", "d072eddfc3116f145a8a2aca3c66ed13": "Var(X^{(m)})\\to am^{-\\beta}", "d072f96f0afa36aa01cb7ab08be43034": "w = \\left (z\\frac{d}{dz}+a \\right )\\frac{dw}{dz}", "d073263ac057a6ee267aee500bb38d4e": "\\displaystyle{ \\|Rf\\|_{(s)}\\le C_s^\\prime \\|f\\|_{(s+1)}.}", "d07356725e2f50acf63a973bd6302c63": " f(x_1, x_2) = (a+1)a(\\theta_1 \\theta_2)^{a+1}(\\theta_2x_1 + \\theta_1x_2 - \\theta_1 \\theta_2)^{-(a+2)},\n\\qquad x_i \\geq \\theta_i>0, i=1,2; a>0.", "d07399f877ed1264573077b14b602e86": "\\sqrt{\\frac{5}{42}}\\!\\,", "d073d962d7cabb72a50ef520712053dd": "\\{ v_i, w_i \\}_{i \\in \\mathcal{I}}", "d074404a6e52abc60ac6e3e612f76fb0": "A=LUP", "d07457df603179e18a319105aa78f000": "\\psi_{p}", "d07476489ec60d54765e9231b83a76a9": "\\overline{V 'L}", "d074cf456d359435e61eed0131688921": "\\mathbf{q_0}", "d074d5de0396c43452f288994c1bd4a3": "Lu=f", "d074d5e649dd80d43ad8c14e319b4e31": "\\left(2\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ \\pm2\\right)", "d074e42acbd202de593fcdfe56cf9683": "b = 13.0", "d074fa475b8b06d28150afdd0089155b": "\\kappa(X,K_X)=-1", "d0754cd1659f5d00f072c33581c9cf39": "(w^i,w^i_j)", "d075d1d6a2d81a05a31aacf74f495eb7": "\\mathcal{D}_n = \\{(X_i, Y_i)\\}_{i=1}^n ", "d075f01b6ef5335bd11f9bcdf34ceb08": "W = \\frac{1}{2}c\\left( e^{b_{ijkl}E_{ij}E_{kl}} -1 \\right)", "d0763d3e76063a3aba33557e56f77623": "A \\in \\mathbb{R}^{m \\times n}", "d0763e7680bcf6b580b52faad5804245": " \\langle \\lambda^a_\\alpha \\lambda^b_\\beta\\rangle \\sim \\delta^{ab}\\epsilon_{\\alpha\\beta}\\Lambda^3 ", "d0766b056b32e01ce7707cfb9d8f111e": "\\theta_k^\\dagger", "d0767c2116da759232049c1626762406": " \\delta \\approx 0.382x/ {\\mathrm{Re}_x}^{1/5} ", "d076853fe33cbc1efa82568b75d2ec86": "i:G\\to G", "d0769ea527fdc9e9347a96e207537cc9": " ~\\epsilon_{t-1}^{-} = 0 ", "d076c47b8e09235ab6e9897130044ef4": "C \\left(A,W\\right) = \\left\\{c_{1},c_{2},\\cdots,c_{n}\\right\\}", "d076c7f9ad5f6b7e431dfa29b357aa04": "q_3:=\\frac {(1+m_1)(1+m_2)}{(m_1-m_2)^2}", "d07714ce0ae5af87919c4a12db63b5f0": "pV_m=RT~\\frac{\\textrm{Li}_{\\alpha+1}(z)}{\\zeta(\\alpha)}\n\\left(\\frac{T}{T_c}\\right)^\\alpha", "d0772680d653be191cdb6f0f01147199": "J^k_0 f\\circ J^k_0 g=J^k_0 (f\\circ g).", "d077290e84ce279af76ea38341184f12": "\n \\begin{align}\n & EI \\frac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} = F \\delta(x - L) \\\\\n & w|_{x = 0} = 0 \\quad ; \\quad \\frac{\\mathrm{d} w}{\\mathrm{d} x}\\bigg|_{x = 0} = 0 \\quad; \\quad\n \\frac{\\mathrm{d}^2 w}{\\mathrm{d} x^2}\\bigg|_{x = L} = 0\\,\n \\end{align}\n ", "d077b2b11d521de41543f31bb48e0c4a": "x^k \\equiv 1 \\pmod{n}", "d077d1efeb0945db39d7ba90e3be0353": "S=\\sum_{i=1}^m W_{ii}r_i^2.", "d0782365a6e6e093b31176a79bad5770": " \\Sigma \\equiv ", "d0786a4a414dec21af19fa8a33503b0e": "f_3 = \\frac{f_1 + f_2}{2}", "d07931c5822c1740fe1e65d6aa0db776": "I - 1", "d0796b0cb794411dfd25efcd6906714c": "\\sum_l \\mathrm{e}^{-\\mathrm{i} \\Delta \\omega_{l} t}", "d0797f76bdc8bd19843d06e6786c014f": " \\overrightarrow{OM} \\mapsto \\lambda \\overrightarrow{OM}. ", "d079bdccc6199adcf1ff5b61f5878a85": "\\boldsymbol{\\alpha}=\\mathbf{r}_0", "d079c599b6adedfe16b36f0021859cca": "\\exp(t\\Delta)", "d079d718ef2bd92a29cbd485f30b368f": "x \\in \\mathfrak{g}, v \\in V", "d07a3450de28f57945e867e3f38c4bc5": "\\xi=\\frac{a^2\\omega_0^2}{2c}x", "d07a36baa06f8dbccc881e6740ef20e4": "(1,1,3)\\rightarrow (1,1)_1\\oplus (1,1)_0\\oplus (1,1)_{-1}", "d07a4ad08fef8b2d0531b82cf9bedb91": "f(z)=a_0z^{n}+a_1z^{n-1}+a_2z^{n-2}+\\cdots+a_{n-1}z + a_n", "d07a6ebc7ef3ac87b0213a1ff85db701": "AC'=AC\\cdot\\cos\\theta=\\frac{2d}{\\tan\\theta}\\cos\\theta=\\left(\\frac{2d}{\\sin\\theta}\\cos\\theta\\right)\\cos\\theta=\\frac{2d}{\\sin\\theta}\\cos^2\\theta. \\,", "d07a9e3915f5ffd32fa4ff3730deb3eb": "f_{(U)} : X \\times_Y U \\to U", "d07b2fa3682f955cd65a276b2e053ee0": "Z=(z_1,\\dots,z_n)=(t,x_1,\\dots,x_k,\\theta_1,\\dots,\\theta_l)", "d07b363e86f5e53aa72ed69b49c7bc11": " \\left\\{ Y(t) \\right\\}", "d07b41180776bf9beeeb2d1659e83151": "(-2)^{3 + 4i} = \\left( 2^3 e^{-4\\pi} \\right) e^{i[4\\log(2) + 3\\pi]} \\approx (2.602 - 1.006 i) \\cdot 10^{-5}", "d07b47f8c0d01063c454fccf64669f39": "g(z_0)", "d07b7fe025d799eb950be82dd0e04089": "\\textstyle(x\\pm1, y\\mp1, z)", "d07b861beb6de8323625bd50c53024de": "\\Lambda(d_k) = \\log\\frac{p(d_k = 1)}{p(d_k = 0)}", "d07ba7556484f05d585b38ba6c899fba": "{A}_{7}^{(2)}", "d07bb125d494a803ef1ff4fc0e285ab9": "\\underline{\\underline{\\boldsymbol{\\varepsilon}}} = \\left[\\begin{matrix}\n\\varepsilon_{xx} & \\varepsilon_{xy} & \\varepsilon_{xz} \\\\\n \\varepsilon_{yx} & \\varepsilon_{yy} & \\varepsilon_{yz} \\\\\n \\varepsilon_{zx} & \\varepsilon_{zy} & \\varepsilon_{zz} \\\\\n \\end{matrix}\\right] = \\left[\\begin{matrix}\n\\varepsilon_{xx} & \\gamma_{xy}/2 & \\gamma_{xz}/2 \\\\\n \\gamma_{yx}/2 & \\varepsilon_{yy} & \\gamma_{yz}/2 \\\\\n \\gamma_{zx}/2 & \\gamma_{zy}/2 & \\varepsilon_{zz} \\\\\n \\end{matrix}\\right]\\,\\!", "d07bd33bad5f14f442b328d8d09a947e": "E_m^{(2)}\\left( S_{rsij}^{0} \\right) = - \\frac{N_{rsij}^{0}}{\\epsilon_r +\\epsilon_s - \\epsilon_i - \\epsilon_{j}}", "d07c327e726b27df2eb1930bccc888ee": "\\gamma\\in\\mathbb{R}", "d07c84d1bdbe2659aab9a536a398a771": "n^{\\nwarrow}", "d07c8fb61482a8cb0392fdf35e839664": "\\frac {1} {UA} = \\sum \\frac{1} {hA} + \\sum R ", "d07d279eb93bd454926dc7edd51be217": "z_k", "d07d2fe179a79aa9c79730ef051b8ff2": "\nP\\left(L_{k}|L_{k-1}\\wedge\\cdots\\wedge L_{1}\\wedge\\delta\\wedge\\pi\\right)=P\\left(L_{k}|R_{k}\\wedge\\delta\\wedge\\pi\\right)\n", "d07db0da01a7636a4ae756ce9d8de89c": " V_{2} = 10 ", "d07df93aadae3fe43a792455cafcce97": "P(D){u(x)} = \\delta(x),\\,", "d07e37d84e7abeb2e0f0978d5d8b4acb": "\\tau_b*=\\tau_c*", "d07e41cb6c4828705aae2442a0561d34": "E\\log\\frac{\\|m_k - x^*\\|}{\\|m_{k-1} - x^*\\|}\n \\;\\to\\; -c < 0 \\quad\\text{for}\\; k\\to\\infty\\;. \n ", "d07e546b89af0db89cc6160c036a4e73": "P_{\\text{ps}}", "d07eb1bd19dbb2b019c24ccbf38be0ee": "= - \\left ( t_j-y_j \\right ) \\frac{ \\partial y_j }{ \\partial w_{ji} } \\,", "d07eb5103b24d73c0408398ef21fd594": "M=(M_{ij})", "d07efb7525570f460134632f8aeb3575": "\\varphi = {1+\\sqrt{5} \\over 2}", "d07f1bf2a0dd27d12063efc3631505e6": "\n \\left[\\sqrt{3}~\\sin\\tfrac{2\\pi}{3} - \\sin\\phi\\cos\\tfrac{2\\pi}{3}\\right]\\rho - \\sqrt{2}\\sin(\\phi)\\xi = \\sqrt{6} c \\cos\\phi\n ", "d07f1c3b573453e6b9044ad75b24f651": "(b \\to [k]b) \\to (b \\to [k*]b)\\,\\!", "d07f65792ca260bb7b10b61eb7b1516a": "x - y = x + (-y)", "d07fef07a9bfda19d64429aa74196b24": "\n \\Gamma^{(\\lambda)}(R^{-1}) =\\Gamma^{(\\lambda)}(R)^{-1}=\\Gamma^{(\\lambda)}(R)^\\dagger\\quad \\hbox{with}\\quad \\Gamma^{(\\lambda)}(R)^\\dagger_{mn} \\equiv \\Gamma^{(\\lambda)}(R)^*_{nm}.\n", "d07fefa01f4730ce9f8ab7b2e5f99f02": "p_{i_{j}} 1\\leq j\\leq k(i)", "d080121691d763e394645e52b3a382ba": " \\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial}{\\partial \\theta}[\\rho \\omega + \\rho K r \\sin(\\psi-\\theta)] = 0. ", "d0803c26b6137b4cbd8203104480fc0f": "4^2\\ll\\frac{\\gamma P_0\\tau_0^2}{|\\beta_2|}=N^2 ", "d080625367a44065c7bb241cf11479ad": "\\mathrm{6 \\ CO_2 + 6H_2O + energy \\longrightarrow C_6H_{12}O_6 + 6 \\ O_2}", "d0806d6d40f6af826a5b057610b39064": "\\mathrm{Borel} (E) = \\sigma \\left( \\mathrm{Cyl} (E) \\right).", "d0806d991858cb138e9f04f854787c6a": "d \\approx \\sqrt{2 \\cdot k \\cdot R \\cdot h} ", "d0808e057fe53fc7ad49535a013604a8": "\\frac{E_{\\text{x}}}{P_{\\text{x}}}=R_{\\text{x}}\\ [1-T]", "d080d10e88c88059c13867ca058ee874": " = 90 ^\\circ - \\arctan \\left( \\frac {1} {\\alpha } \\right) = \\arctan \\left( \\alpha \\right). ", "d080ef28d9ce1db41046d1e24375ac17": "u\\rfloor \\phi=u^A\\phi_A + (-1)^{[\\phi_a]}u^a\\phi_a", "d0810a81ef2686d6fb2918fdfc6b5c77": "R_{ab} - \\frac{1}{2}g_{ab}R + g_{ab}\\Lambda = 8\\pi T_{ab}, ", "d0817a754152bb4fb8e0f187f182cf87": "\\sqrt{2\\log{\\frac{p}{q}}}", "d081941e282b107364d3aad2286202d9": "\\text{Gain}_i(\\sigma^*, a) > 0", "d0824772e1e480bb85ba82273bec1eb1": " {x^2\\over a} + {y^2\\over b} +{z^2\\over c}=1.", "d0824821649e6a183d4e456d62ef6a95": "\\displaystyle{X=\\sum \\psi_i\\cdot X_i.}", "d082b646abd246e89938d1ab8009573d": "z / 0 = \\infty\\quad\\text{and}\\quad z / \\infty = 0", "d082b6ee6686dca1ac88c5fa783cf4bb": "\\tan \\delta =\\frac{\\sigma}{\\omega\\epsilon_0\\epsilon_r}", "d082cd9768dfd38dc8313c3428498011": "\\mathcal{ALC}", "d08315263be39a77a4774fa60c49dad3": "z = A e^{j\\phi} = A(\\cos{\\phi}+j\\sin{\\phi})\\,", "d083209f7dcbe824795b7198e7d6c767": "(n_i, n_{k'})", "d0835c5697d6f0cafb6b5788ae774f44": "-\\eta n_\\eta(\\xi)-1/2", "d08379286e79a991740ef76879182443": "I(f+g) = \\int_a^b(f(x)+g(x))\\, dx = \\int_a^b f(x)\\, dx + \\int_a^b g(x)\\, dx = I(f)+I(g)", "d083a609125d4516ae74b9880194e0e0": "F_2 = \\forall x_1 \\dots \\forall x_n R(x_1,\\dots,x_n,f(x_1,\\dots,x_n))", "d083b179980d7bfa0e38da001f530d63": "\\ \\Delta S(T)=\\frac{\\Delta H(T_d)}{T_d}+ \\Delta C_pln \\frac{T}{T_d} ", "d084022502e1d19a0a3b6eb7dacf164c": " |\\mathcal{E}| = \\left|{{d\\Phi_B} \\over dt}\\right|", "d0853824aba8b81a3a32ed9060eb3f3a": "\\mathbf{x}_{n+1} = \\mathbf{x}_n - [H f(\\mathbf{x}_n)]^{-1} \\nabla f(\\mathbf{x}_n), \\ n \\ge 0.", "d0855ec2471fac6a79351d8ef32978e5": "\n\\mathbb{E}[\\log X!] = \\overline{\\log X!}\n", "d085753c479447a6cf2dfbbe730b80a5": " \\omega = E_n / \\hbar ", "d08595a869cefeabe6807ae386a2dc0b": " R^{\\rm T} R \\hat{\\boldsymbol{\\beta}} = X^{\\rm T} \\mathbf y. ", "d085a763de5d633e44721886ee4a84ea": "\\textstyle \\cos \\frac {\\pi}{8} = \\sin \\frac {3\\pi}{8} = \\frac{1}{2} \\sqrt{2 + \\sqrt{2}}=\\sqrt{ \\frac{\\delta_s}{2}} ", "d085bacb4743efd2f26a6a17b39c341f": " y(x) = \\frac{\\int g(x)e^{\\int f(x)\\,dx} \\,dx+c}{e^{\\int f(x)\\,dx}}.", "d085bb21b7e22d4c31891bba17b0710d": "1 \\dots K", "d085d9dd24ec9e06d0d4dc3433bbae70": "\\mathbf{U}^*(\\mathbf{x})", "d08646b0d68b48c537040b5e275fc969": "\\min(3,2,1) = 1", "d0866d24d7a25e0cf53de5f5505df05d": "\\varepsilon_n(\\mathbf R)", "d086976274c142ccfd6f28ca7f24d5ea": "=\\det(\\Sigma)^{-n/2} \\exp\\left(-{1 \\over 2} \\operatorname{tr} \\left( \\sum_{i=1}^n (x_i-\\overline{x}) (x_i-\\overline{x})^\\mathrm{T} \\Sigma^{-1} \\right) \\right)", "d086a3938500326c59057b5ebfbb3364": "U_i(\\theta_i,z_i)=-U_0\\left (S+\\alpha\\sigma\\cos\\left (\\frac{2\\pi z_i}{d}\\right )\\right )\\left (\\frac{3}{2}\\cos^2\\theta_i-\\frac{1}{2}\\right )", "d086b7fdfd21a7b5542ab2d3248f22c6": "R_{c,\\theta} = T_c \\circ R_{0,\\theta} \\circ T_{-c},", "d086dc7316db25a23a96dc14e937c403": "q_1 = 1+\\frac{k+1}{6N}. ", "d0876b897bf6dc5a7ad477a7efe3f42b": "c = 1.", "d087a13a16be3e0c31481e09de846f09": " (fg)(X) = f(X) g(X). \\, ", "d087a1c4806afcdf9cb4789ae5fedfc5": "\nm\\frac{d^{2}r}{dt^{2}} - mr \\omega^{2} = \nm\\frac{d^{2}r}{dt^{2}} - \\frac{L^{2}}{mr^{3}} = -\\frac{dV}{dr}\n", "d087c83863596b54d0860e82e448e592": "Q = 2s-1\\,", "d0886c8140df10e28fd05f58eaff56b4": "dt_{c} = \\frac{T_{c} - T_{p}}{S}", "d088e38e091f0b1c482c7445e2d80743": "Z_{eff}(F^-) = 9 - 2 = 7+", "d0891762c408be1fc3ad5fc6fcdf5243": " C_g \\gg C_v , ", "d08921531a08c39e6d958618174b6d94": "C = R - P \\wedge\\!\\!\\!\\!\\!\\!\\bigcirc g.", "d0892c81f69dd397ccd1039c2a4b1aa6": "\t\\begin{array}{rr|rr} \n 1 & \\text{-}13 & 16 & \\text{-}81 \n\\end{array}", "d089448fd22eeaa974ed31ba39977bca": "\\inf_{x \\in M} f(x)", "d089459825c1167f05edf5863191c25d": "S \\neq 0.", "d089744874f49da4cb2aec944e8576e4": "(a_1,b_1,\\dots,b_r)", "d0897c98c41eb2c34c6402d84cac2f56": " L = q \\frac{\\lambda}{2} ", "d08a119c373564eca3b56859aabb534c": "\\left(\\frac{a}{b}\\right)^{-n} = \\frac{b^n}{a^n}.", "d08a7e2aa40159a705bf4ea68eee77ea": "y=r \\, \\sin\\theta \\, \\sin\\varphi", "d08aacc3cc4b57998e00b26b70cc26c5": "r_t^k +\\lambda_kr_x^k=0, ", "d08b3fcbc3011b62d65b5035202a5c89": "\\omega_f(\\delta)=\\omega(f, \\delta)=\\sup\\limits_{x; |h|<\\delta;}\\left|\\Delta_h(f,x)\\right|.", "d08b5a8179af7985b598ca2d29463a68": "G_c = G ", "d08b62e799e1ff8f24464dc26a2daebe": "\\lambda x", "d08ba6576f35297944d2a4af0acf467f": "\\chi \\to \\phi \\lor \\chi", "d08bab2bdcd4b9735043433b6e6b4670": "\\xi(1-s) = \\xi(s).", "d08c37c53c6bb7d3738ceb9712cb2632": "R_2 = \\frac{R_aR_c}{R_T}", "d08c4a1bce49329ba0117d70b6a038f9": " V_r \\, ", "d08cae081bea66da29927fe7a058c0a3": "\\mathbf{J} = \\frac{1}{i\\omega} \\left[ \\left( \\nabla \\times \\frac{1}{\\mu} \\nabla \\times \\right) - \\; \\omega^2 \\varepsilon \\right] \\mathbf{E} \\equiv \\hat{O}\\mathbf{E}", "d08d28d17cfe2b1ef06456c2fde1fb89": "a_i-a_j=a_k-a_l", "d08d3bc0cab39c9ae3bdcf390babdceb": "n_\\text{piv} \\notin n_\\text{clause}^\\text{left}", "d08dc2986596fa8f862c51bbde397a93": "\\bar{7}", "d08ddc27fb74a5f7d3d1c1b732800779": "\\frac{dx_n}{dt}=f_n(t,x_1,x_2,\\ldots,x_n)", "d08de6ad3074976925af9d8223e2d629": "n = 255", "d08def83ba11924f01541e357c4e023d": "n = \\frac{\\lambda_0}{\\lambda} = \\frac{ck}{\\omega}", "d08df93ad793c0c77f232496cd7757a1": "V = \\varnothing", "d08e5672cd55a0f14d95fb2ab78b91eb": " \\frac{E_{\\textrm r}}{E_{\\textrm i}}=\\frac{\\beta\\,A}{1+\\beta\\,A} ", "d08e7b4049b923be95662d238a683e91": "\\psi=-\\frac{\\Psi}{r\\sin\\theta}\\boldsymbol{\\hat \\phi},", "d08efa50c50c44c2fd0325fe1693ffdb": "B_{jk}", "d08f0cbd2595c53b719bc1ef2397658b": "E_2^{pq}=H^p_\\text{etale}(\\text{Spec }A[\\ell^{-1}], Z_\\ell(-q/2)),", "d08f1bf2a1785bee41c494a87c4d5351": "\\frac{P_{t+1}}{P_t} = 1-\\frac{1}{2N}. ", "d08f42df75b4adf0ec233dbd198b3434": "Y_0(kr)", "d08fca849e095b0ebfe8cac2c56788ec": "\\int \\frac{dx}{\\ln x} = \\ln|\\ln x| + \\ln x + \\sum^\\infty_{k=2}\\frac{(\\ln x)^k}{k\\cdot k!}", "d08fd44caa48c263def57eba4198083b": "\\dot{\\tilde{\\rho}}=- \\sum\\int^t_0 dt' \\left(\\alpha_i(t)\\alpha_j(t')\\rho-\\alpha_j(t')\\rho(t)\\alpha_i(t)\\right)\\langle\\Gamma_i(t)\\Gamma_j(t')\\rangle + \\left(\\rho(t)\\alpha_j(t')\\alpha_i(t)-\\alpha_i(t)\\rho(t)\\alpha_j(t')\\right)\\langle\\Gamma_j(t')\\Gamma_i(t)\\rangle ", "d08fefcd5c30b3ff683cdfbaea0bb8b3": "H_{k+\\dim(N)-\\dim(M)}(N)", "d08ff98f0aaf6f0467e32ebc859b395b": "v^{*}:= \\max_{d\\in D}\\min_{s\\in S(d)}f(d,s)", "d0902e86c4f2ce290f4e0a57cc20ed37": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left[ (M + m) \\dot x + m \\ell \\dot\\theta \\cos\\theta \\right] = 0, ", "d0904fd99a2cfb13f897223b9213c6f1": "D.", "d09067625faf99d4e05661e373dc6406": "\\implies N(mx+y)^2+k(m^2-N) = (my+Nx)^2", "d09135713c4b32fa53f9a74a369029c5": "\nU_i(a_1,\\ldots,a_i,\\ldots,a_N) = U_{\\pi(i)}(a_{\\pi(1)},\\ldots,a_{\\pi(i)},\\ldots,a_{\\pi(N)}).", "d091606c60f54bd897c57cd3bc06aae0": "A_{nm}", "d0917f3b3fd24ae461217f343d5b1694": "l_P = \\sqrt{\\frac{\\hbar G}{c^3}}", "d09185d93a27e148e4346f148fd6fbce": "YA, YB, YC", "d091c4dabf8f3706143fa23227611517": "\\mathrm{\\frac{3}{2} O_2 + 6\\ H^+ + 6\\ e^- \\to 3\\ H_2O}", "d091ceb632b769e66b7361739fc6534a": "\\phi(t,q^{\\mathrm I},Q^{\\mathrm{II}}(t))=0", "d09208e2aad5d3e8a2337f59fd88436f": "\\mathbf{Z} = \\mathbf{P}^{-1}\\mathbf{Y}", "d09229206a86c4a9acf88d37769a3b29": "\\mu (E) = \\int_{X} \\mu_{x} \\left( E \\right) \\, \\mathrm{d} \\nu (x).", "d09237602924016867ee745ee504bf41": "n\\tau + \\tau_0", "d0925eb52552335d2b228c029540cef3": "S_4\\,", "d092983d433f2a13c559eeb3938f5658": "\\left| z_2 \\right| > \\left| z_1 \\right| \\;", "d092b1cb66db6b770c91f8f87d86b68e": "b=11", "d092b836d9b7102cfb29de4a7fd586cd": "M_{i,j}\\neq 0", "d09315477a37d1b341f191eed9930cea": "\\Delta u_n", "d0933e4b7e82815131f6159df63c6230": "Y^* ", "d093d85aa123579e6d376f6e359414d2": "\\hat Y_j \\,", "d094acd34b74166571eee12baa0cdc36": "q^*(\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k) = \\mathcal{N}(\\mathbf{\\mu}_k\\mid \\mathbf{m}_k,(\\beta_k \\mathbf{\\Lambda}_k)^{-1}) \\mathcal{W}(\\mathbf{\\Lambda}_k\\mid \\mathbf{W}_k,\\nu_k)", "d094de49b970094b82390cb8c4abfee4": "a_\\text{c}\\in \\Sigma_\\text{c}", "d095077683602f46671af4d468498c6a": " b_n = \\sum_{k=0}^n {n\\choose k} a_k, \\qquad (n \\geq 0),", "d095107b604d91286e31bbf5fc1d8823": "v_1 \\otimes v_2 = \\epsilon \\eta", "d09517fffa1c4598957be3f272dad344": " \\rho\\left(x,t\\right) ", "d09524d87836ec8474115d39dea608d9": "\\lim_{n \\rightarrow \\infty} n(\\sqrt[n]{2} - 1) = \\ln 2 \\approx 0.693147\\ldots", "d0954546afd5191926bbf1a6dfbcc824": "\\widetilde D", "d09598754fc42b31c0f6ad533581de34": "{\\nabla}^2 \\varphi = \\sum_i \\frac{z_i^2 q^2 n^{0}_i \\varphi}{\\varepsilon_r \\varepsilon_0 k_B T}", "d095bb01fa6dda3841ed27e7b62764ea": "\\bar\\psi .", "d095c11c053445d2e6f02ae27492bf36": "f_1(x_1,\\ldots,x_n)=\\cdots=f_k(x_1,\\ldots,x_n)=0,\\quad\\forall(x_1,\\ldots,x_n)\\in M.", "d0962b1f18a8aee15953512be476e54b": "\\operatorname{E}(3) = 3.5", "d0967770d30c2e0cedd15b23c6026ed5": "\\leq\\kappa", "d09688752c3f8a47e3a3b67b6907e5bc": " \\frac{dS}{dt} = \\mu N (1-P) - \\mu S - \\rho S - \\beta \\frac{I}{N} S ", "d09693e2254cfffaf39594cbcf1c4813": "f^\\text{pmi}(X_i^{w_1},w_2)>0", "d09694b77d8a3a03f6879fa37f09d0b0": "[0,1]\\,", "d096cba695873d85520ac7cab874234d": "I_n\\,", "d096eeb1538f9e3d5d408bdfaa256f30": "\\lambda^+_i = \\sum_j \\rho_j \\mu_j p^+_{ji} + \\Lambda_i \\,", "d096f61370e13566ca2424b1eb5362ec": "TEE_f", "d096fc15d57854ec89d746709b02e52e": "\\varnothing ", "d0975088d5bd5612276fcbb1f48c13c6": "R(\\theta,\\delta)=R(\\bar{g}(\\theta),\\delta)", "d097b3d22763003ae53436f82d01ce75": "1.05^{12} = 1.7959", "d097c74f217bf8493f6f738b639d74e1": "\nD_{\\mu \\nu}\\left ( k \\right )\\mid_{k_0=0}\\; = \\;\n \\eta_{\\mu \\nu}{1 \\over - k^2 + m^2}\n.", "d097d229f8cd107b281eaea0ff0043e9": "G_{V_1, E_1} \\times H_{V_2, E_2} \\rightarrow J_{(V_1 V_2), (2 E_1 E_2)}", "d0981af90be2b1010458d8e0828ca590": " Z_{01} = \\frac \\text {short circuit voltage per phase} \\text {short circuit current} = \\frac {V_{S}} {I_{S}} ", "d0981bea582ead43d40fdb8d609662bf": "\np(x_t|z_{1:t}) \\approx \\sum_{i=1}^M \\omega^{(i)}_t \\delta \\left( x_t - x^{(i)}_t \\right).\n", "d0982a24823122bfcc27369d9745c460": "\\mathcal{C}(R)^\\mathcal{K}", "d0985288768c460c0a70af0553cc0f7a": "\\frac{q}{A} = U_c(T_i - T_\\mathrm{sol-air})", "d09899c5c1f25b6f22f0c8818a03bf1f": "D^2+E^2<4(A+C)F", "d098affbfab376e6db7b03a9065f05fe": "\\|f\\|_p = \\int_0^1 |f(x)|^p \\, dx \\, ;", "d098c9f7af7aa8da73f8822daa468ff5": "\\{a ~|~ (\\exists p, q \\in \\mathbf{Z}) \\land (q \\not = 0 \\land aq=p) \\}", "d098dc50c9fd938c8173dea65d0a69f3": "S(n)=\\frac{1}{n}\\left((6n-9)S(n-1)-(n-3)S(n-2)\\right).", "d098ddaf4e591771583a0b37cfb5741c": " E^\\ominus\\left( \\mathrm{B}^{+} \\vert \\mathrm{B} \\right) ", "d0996a978b5fd51e28544582724584bf": "\\dot{v} = \\delta u + \\kappa E w", "d099a909010b2fa40168ab23bcb22fae": "\\frac{d\\boldsymbol{r}}{dt} = S_N \\cdot \\nabla_{\\boldsymbol{r}} H(\\boldsymbol{r})", "d099aa4794932e8b4cdcf314a634d8cf": "A[1,i-1]", "d099edaef99d324803a4b555d3054ffc": "\\eta=0.5,\\,\\!", "d09a06c28b2758dcd0db4ddeedb07e3a": "\\int_{-1}^1 f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i).", "d09a49328aa474bfe6d909306e31539f": "\\frac{\\Delta f^{*}}{f_f}=\\frac i{\\pi Z_q}\\,\\frac \\sigma {\\dot{u}}=\\frac\ni{\\pi Z_q}Z_{\\mathrm{ac}}=\\frac i{\\pi Z_q}\\sqrt{\\rho i\\omega \\eta }", "d09a4cb7d457130ae1247ca3e8fa32f6": "\\{ ad_g | g \\in A \\}", "d09a5d516ee0550c3a50f180bd8cd6f3": " \\operatorname{get-lambda}[F, F = E] = \\operatorname{de-let}[E] ", "d09aba86dd79b47658fdf4d47ceb6271": "\\begin{align}\nx' & =\\gamma\\left(x-vt\\right),\\\\\nt' & =\\gamma\\left(t-vx/c^{2}\\right).\n\\end{align}", "d09ac2499372377c75d466e597556a2a": "s_\\lambda(x_1,\\ldots,x_n)", "d09ae9dcd2ffd71d516864874130487d": "\\{c_{jk}\\}", "d09afac7c7c4ca08c4f51561c40f904f": " \\scriptstyle{\\phi (x)} ", "d09b088beddd628ed63fd48c7a0d7f9d": "\\mathbf{M}\\mathbf{M}^*", "d09b39903c20f840cac2bda7e23385ba": "P(x) \\nleftarrow (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\nleftarrow Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "d09bb7ef7d868ab98bfe44b771e5f38d": "K_\\mathit{rw}=\\frac{{K_\\mathit{rw}^o}{S_\\mathit{wn}}^{L_\\mathit{w}}}{{S_\\mathit{wn}}^{L_\\mathit{w}}+{E_\\mathit{w}}{(1-S_\\mathit{wn})}^{T_\\mathit{w}}}", "d09bd6c78282228ff3011be57bc0d8c4": "p_3=(1+m_1)^2\\ .", "d09be8f0c44eb4c95cfda6e86bee93ed": "\\mathrm{C^{\\alpha}}", "d09bf41544a3365a46c9077ebb5e35c3": "75", "d09bf8e35593a8268049eceb0f29cbd9": "(r-2)^2 + z^2 = 1", "d09c0b1b64bf4358c6f2fdd389e60834": " \\text{plane}_D = \\text{PointOnPlane}\\cdot(\\textbf{normal})", "d09c40bdad73fd07148801843a5f0d7d": "TL_j", "d09c86150548cbeba6c8b43f177bfd38": "r_+,~r_- ", "d09ce992642bf92832ddfe34f63fab1a": "\n\\left(\\frac{\\partial H}{\\partial q} + \\dot{p}\\right)\\delta q + \\left(\\frac{\\partial H}{\\partial p} - \\dot{q}\\right)\\delta p = 0 ~,\n", "d09d0d6ff368b3943189240da5a525bb": " \\sdot \\ [ 1+j \\omega C_i (R_A//R_i)] \\,\\!", "d09d14ddaf8f64863a4af57e5d726ee3": "O(c^2)", "d09d3816efb17f2f81131c7f6ff04b70": "\\sigma = \\sqrt{\\int_\\mathbf{X} (x-\\mu)^2 \\, p(x) \\, dx}, {\\rm \\ \\ where\\ \\ } \\mu = \\int_\\mathbf{X} x \\, p(x) \\, dx,", "d09d6475aba30494d200aebb1f29788e": "T_a(\\cos x)={}_2F_1\\left(a,-a;\\tfrac{1}{2};\\tfrac{1}{2}(1-\\cos x)\\right)=\\cos(a x)", "d09d80ce0239b4e35c31709c0fb26daf": " T_{\\varepsilon }^{TE}=cos(\\frac{m\\pi }{a}y)\\cdot(Ce^{-jk_{x\\varepsilon}x }+De^{jk_{x\\varepsilon }x}) \\ \\ \\ \\ \\ \\ m = 1, 2, 3, ... \\ \\ \\ \\ \t(20) ", "d09e05a16586a01262ac4818c1bc0e9b": "\\bold j = \\frac{\\hbar}{2mi}\\left(\\Psi^* \\bold \\nabla \\Psi - (\\bold \\nabla \\Psi^*)\\Psi \\right), ", "d09e4ca530c2b9780f1c65cb572945bf": " \\; \\pi_{ij} = {O_{ij} \\over N} \\;", "d09e7f4fff4b7b646e2c327a00fa4c63": "\\boldsymbol{H}_i=\\boldsymbol{V}_i^\\mathrm{T}\\boldsymbol{AV}_i", "d09ec0107db6ce08a8a0c3a8d6522f0b": "M_\\mathrm{L} = \\log_{10}A + 1.73\\log_{10}\\Delta - 0.83 ", "d09ec0394b3ed057c0e653d5284c669e": "d_e : \\mathbb{N} \\longrightarrow \\mathbb{R}", "d0a0682b3cdf2260007ea46cbb9341ed": "\nG =\\exp \\mathcal{G}, ", "d0a06c357d9b3f4aba61f03837d31cc5": "\\!\\gamma_{j-1}", "d0a09468834d1feba3e83520b1828c64": "V\\mathfrak{E}", "d0a0a4ae7bd6e15a7f1f62306940e27e": "x[n-p]", "d0a0aac8fad5ac051b397c720fcec954": "f'(t)", "d0a0ea54b6b9dd1eda13b311003649c9": " \\operatorname{build-param-list}[(\\lambda p.f\\ (p\\ p\\ f))\\ (\\lambda q.\\lambda x.x\\ (q\\ q\\ x)), D, V, \\_] ", "d0a0f2894642608e89f91464fdcd4740": "{s \\choose k}", "d0a0fdb1022141cd1ef6f628d1514164": "\\begin{matrix}\n\\times & cx & d \\\\\nax & acx^2 & adx \\\\\nb & bcx & bd\n\\end{matrix}", "d0a10554523c158fb796f718e151dfa1": "\\tilde\\theta", "d0a11067062e0b8c69c289b46b35ff48": "e^{i(\\mathbf {k}_{i}\\cdot\\mathbf {r} - \\omega_{i}t)}", "d0a11a926dd48e1e24ac258d99067e81": "A_i=\\mathbb{C}\\Gamma_i\\subset A", "d0a11e5eaee18ae20987b7e78903df56": "X_t \\to \\infty", "d0a13c4c8fbf68c1478935db05bc08d7": "P_E", "d0a1496386b9c5c7609831009becfa5b": "(\\nabla f(x))\\cdot \\mathbf{v} = D_{\\mathbf v}f(x).", "d0a14f99f197e2281075ee94fc182fde": "7^2 - 7 - 5", "d0a16b70f46fbb99b84801943c6d453c": "f(x) - f^\\star(p) = x\\,p~.", "d0a1863f2c57ee860aaec972c37dfa0f": "\\mathbf{J}_{tot}", "d0a19d01f1a8e957749c4a77331929df": "\\hat\\alpha=0.859", "d0a1a57df89ad8b677fc54fe706a4ad0": "{E_D \\over E_B}", "d0a1c0fe9e89a6da32863b8bee9491f1": "\\bar{R}", "d0a1c284a3dd2c19b4a9c66a3347bffd": "M_c'", "d0a2256dffa709a8be6bb8fe2acb31be": " Z_P=A_Cy_C + A_Ty_T", "d0a23ce69cd47b7935be077e7d84585f": "\\lbrace e_1, e_2, \\dotsc, e_n \\rbrace ", "d0a27071ac98577629767282c2d93c89": "\\mathbf{J} = \\sigma \\mathbf{E},", "d0a2a5c51f33b1aa54e83e0e952aa70d": "\\dot{V}_1 = \\overbrace{\\frac{\\partial V_x}{\\partial \\mathbf{x}}(f_x(\\mathbf{x}) + g_x(\\mathbf{x}) u_x(\\mathbf{x}))}^{{} \\leq -W(\\mathbf{x})} + \\frac{\\partial V_x}{\\partial \\mathbf{x}}g_x(\\mathbf{x}) e_1 + e_1 v_1 \\leq -W(\\mathbf{x})+ \\frac{\\partial V_x}{\\partial \\mathbf{x}} g_x(\\mathbf{x}) e_1 + e_1 v_1", "d0a2a958bd87f90ae00532aec534f2d9": "n!/n^n", "d0a2f6a5d05a6359a829e866944d05fc": "\\gamma_1", "d0a325261d140eaefe2ac518f133a1ce": "Y_N = N^{-1} X X^T \\, ", "d0a387ce545c99907630926d29be52b8": "X_{\\gamma(a)} = \\xi.", "d0a3be9c2d8e8dc2ddeff8faedb5e62a": "0.\\overline{90}", "d0a439a0447ea2113d8a7a88a1e90c9c": "\\,\\cos(z) = \\frac{e^{-i z} + e^{i z}}{2}", "d0a4674b73effd57d0b9f7b769f42851": "M(t_0,t_1) = M(t_0,t) + \\phi(t,t_0)^T M(t,t_1)\\phi(t,t_0)", "d0a4a9d7fcbec9bb22742e47ba7b366a": "N=2k(a\\phi(front)-b\\phi(rear))=2k(a-b)(\\theta-\\psi)-2k\\frac{(a^2+b^2)}{V}\\frac{d\\theta}{dt}+2ka\\eta ", "d0a4e1190dd3c6879b962b84204274ba": " \\begin{matrix}\\frac{1}{2}\\end{matrix} (m_\\textrm{b}+m_\\textrm{p})\\cdot v^2 = (m_\\textrm{b}+m_\\textrm{p})\\cdot g\\cdot h ", "d0a4fdc1333343294edb989b757bdca0": "V \\times W ", "d0a571e6f0a91c254e9e183416e292e8": "\\scriptstyle e = (u, v)", "d0a5d37d75f30be62251a39d2826e517": "\\hat F(i)", "d0a63c4114ff38bec2a4c11bcc0fb742": "A(w)=1", "d0a65a919b234902022103daa0626793": "\\textstyle f(x)=sgn\\left ( \\sum_{i=1}^{l}y_{i}\\alpha _{i}\\cdot k(x,x_{i})+b \\right )=\\left\\{\\begin{matrix}\n1, positive\\; inputs\\\\ \n-1, negative\\; inputs\n\\end{matrix}\\right.", "d0a66a8d68cc8ecf94ff5bbf458df58f": "size_i", "d0a72aab0c7244e4bcab7c6ed137039e": "y ^{\\star} = y + v + \\frac{\\partial v}{\\partial x}\\Delta x + \\frac{\\partial v}{\\partial y}\\Delta y", "d0a76aa7e3b24ffc56eccc4a0b9fea90": "C_\\text{aer}", "d0a7b77c473fd636908c5e5c64c5ee7f": "e(n) = x(n) - \\widehat{x}(n) = x(n) - \\sum_{i=1}^p a_i x(n-i) = - \\sum_{i=0}^p a_i x(n-i)", "d0a7c1671c48998f2f776ca785d685a5": " x ( u \\wedge v)", "d0a86ebffc01c3977f1ca87abda94bd6": " \\mathbf{z}'(t)\n = \\begin{pmatrix} z_1'(t)\\\\ \\vdots\\\\ z_{N-1}'(t)\\\\ z_N'(t) \\end{pmatrix}\n = \\begin{pmatrix} y'(t)\\\\ \\vdots\\\\ y^{(N-1)}(t)\\\\ y^{(N)}(t) \\end{pmatrix}\n = \\begin{pmatrix} z_2(t)\\\\ \\vdots\\\\ z_N(t)\\\\ f(t,z_1(t),\\ldots,z_N(t)) \\end{pmatrix} ", "d0a8ae838ff024939bdd0edd135a1658": " (D + i k) (D - i k) y = 0,", "d0a8b25a44d79fc2a66c4b9aa07a53b1": "\\rho_q", "d0a90f5d21f1cef4b96a021a6c52a4ce": "\n \\boldsymbol{P} = J~\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T}\n ~\\text{where}~ J = \\det(\\boldsymbol{F})\n ", "d0a93984159953e3e2f36fc86f8d0ed2": "e^{\\pm i\\pi/3}", "d0a93eec8ed7c15a09b2046b3a37802b": "\\kappa \\rightarrow \\kappa^{\\prime} = \\lambda \\kappa+(1-\\lambda)", "d0a970adfc7b3b2b2e73a2e681519e43": "\\text{ in the interval }[A, B]\\text{ is Poisson-distributed with}", "d0a97b4ec3bebe4a036af68f24f8c863": "y_2(x)=-j_{-3}(x)=\\left(-\\,\\frac{3}{x^2}+1 \\right)\\frac{\\cos(x)}{x}- \\frac{3\\sin(x)} {x^2}", "d0a9924cc58b33d58ea80afa6fa67549": "0 = \\operatorname{cov}\\left(\\frac{w_i}{w}, z_i\\right) = \\frac{f(2 - 2a - af)}{(1 + f)(2a + 2f + af)}", "d0a9c8d0ed8fc10a6fa682e78038f01b": " \\ \\textbf{e} = c\\textbf{h} + c ", "d0aa1b19c14396b751a1f8edf7e847e0": "\\int\\, 1\\, d\\theta = 0", "d0aa4ff58beb461774f9b44c6e0dc7d0": "\\Gamma^{(2)}_{k}", "d0aa5feaa5e848b8db88d89f277b81ba": "| 0 0 \\rangle", "d0aa871f9fb9a8101404d7980983c82f": "F_{ji}", "d0aaa18f2568db1b3db3073bb7e543e0": "\\Diamond A\\to\\Box A", "d0aacfda94fb28732bb51b6ca99acd2b": "H,G", "d0ab322794d8cfa0d533333640aaebb1": "\\alpha_2=1", "d0ab7004fc1c860b5fbbd153d3b63e84": "\\quad\nS_x=\\frac{1}{2}\\det\\begin{bmatrix}\n|\\mathbf{A}|^2 & A_y & 1 \\\\\n|\\mathbf{B}|^2 & B_y & 1 \\\\\n|\\mathbf{C}|^2 & C_y & 1\n\\end{bmatrix},\\quad\nS_y=\\frac{1}{2}\\det\\begin{bmatrix}\nA_x & |\\mathbf{A}|^2 & 1 \\\\\nB_x & |\\mathbf{B}|^2 & 1 \\\\\nC_x & |\\mathbf{C}|^2 & 1\n\\end{bmatrix},", "d0ab88c643999bd2bb263f3e8f1a7c4d": "t_0 = C/hA", "d0aba36d61d572f26148e8c48b5c17ef": "{\\rm ?}(x) = a_0 + 2 \\sum_{n=1}^m \\frac{(-1)^{n+1}}{2^{a_1 + \\cdots + a_n}}", "d0aba632c41945c0eacb749c56206fbf": "V \\otimes W. \\, ", "d0abba055e1d3eb7637fa1633978f336": "E = h \\nu", "d0ac80e62c777c7cf46a85de970c1f86": "x\\cos A+ y\\sin A", "d0ac92b1d921c4ee4048e990062a6a24": "\\frac{1}{V_m^2}", "d0aca068b1ded34aad9ea3cfe7975dff": "\\epsilon^T\\Lambda\\epsilon", "d0accaf51c344e44d6b2812902c9f3b3": " K = \\beta J ", "d0ad03607aa135716ea384fbb86570ea": "K\\subseteq_e N", "d0ad73db5e556f5c048c156845f37513": "a\\ne b", "d0ad79d5c367c226418a00697063769a": " 3 \\sqrt{25 + 10\\sqrt{5}}\\, s^2", "d0ad8ca87acff8206a389ae6432548d9": "V_{ab} = V_{bc} = V_{ca} = 240 V", "d0ada0431eb45c3d2cc48430239bde8c": "\\textbf{0} = -\\textbf{S}\\textbf{A}-\\textbf{A}^{\\text{T}}\\textbf{S}+\\textbf{S}\\textbf{B}\\textbf{R}^{-1}\\textbf{B}^{\\text{T}}\\textbf{S}-\\textbf{Q}", "d0adcb80346477aa7bdd7b941e96874c": "IL = -20\\log_{10}\\left|S_{21}\\right|\\,", "d0ade663daaac14cc40d75949f281ddc": " e_1 = 1 - \\Pr\\left\\{ \\bigcap_{i \\in \\overline{S}_p} E_i^c \\right\\}", "d0ae3003391f4bda62f54e0aabbcbb7a": "x_{l+1},\\dots,x_{l+u} \\in X", "d0ae3b58b94d489f4cabfc9a126a68ac": "\\mathbf{F}=\\begin{bmatrix} \\alpha & 0 & 0 \\\\\n0 & \\alpha^{-0.5} & 0 \\\\ \n0 & 0 & \\alpha^{-0.5} \\end{bmatrix}\\,\\!", "d0ae4d43f87948c5332be314f1c796bd": "y - x\\in S", "d0aeab0e7abcb160509b9c026c7c0e86": "\\text{AVC} + \\text{AFC} = \\text{ATC}.", "d0aec81eb1e69d8c8cd1523b6132f90d": "\\frac{d}{d x}\\left(\\frac{1}{2}\\left(\\frac{d t}{d x}\\right)^{-2}\\right) = f(x)", "d0aee2951d6c8c27ed49b699634a26da": "Lz=0", "d0af465108c807709ad897ad4bcd23b0": "P|n_1, n_2; S\\rang = + |n_1, n_2; S\\rang", "d0af64b2839ecaf91a28a60c4a305e17": " T[t] = {1 \\over 2} m \\dot{\\vec{x}}[t] \\cdot \\dot{\\vec{x}}[t] ", "d0af7a7c795e67606319d8cba050295d": "\\mathrm{error}\\bigl(x(t_0 + 5\\Delta t)\\bigl) = 15\\,O(\\Delta t^4)", "d0af87c3e9a818b4d68c0d92edd4f836": " R(z;A)= (A-zI)^{-1}. ", "d0af87c4ecd0cdf20380147dcf3cf6ef": "B_k=\\gamma^{2}+\\sin^{2}\\left ( \\frac{ k \\pi }{ n } \\right ),\\qquad k = 1,2,3,\\dots,n ", "d0afbae583806589762327841b93adac": "r=(\\sqrt{5}-1)/2", "d0b00ed1b85fffe448ec309f08938712": "t < 2N", "d0b04d51450a33ff210efedffde7f7e5": "V_o = DV_i", "d0b052a824ef24dc36673cad44136d6d": "-\\Delta x_2", "d0b08046927ecc546393bec6d88fb465": "\\,\\! q_s (\\tau) = A(\\zeta,\\omega) \\cos(\\omega \\tau + \\phi(\\zeta,\\omega)) = A\\cos(\\omega \\tau + \\phi).", "d0b08b73085f2254d0e98c38f6fbbb76": "\\int\\frac{\\cosh^n ax}{\\sinh^m ax} dx = -\\frac{\\cosh^{n-1} ax}{a(m-1)\\sinh^{m-1} ax} + \\frac{n-1}{m-1}\\int\\frac{\\cosh^{n-2} ax}{\\sinh^{m-2} ax} dx \\qquad\\mbox{(for }m\\neq 1\\mbox{)}\\,", "d0b096d01471237185d06e5277b548ad": "(2ad)^2+(2bc)^2 = (a^2-b^2-c^2+d^2)^2", "d0b0d29d8173a87b7c4d91c36cdf2e40": "\\frac{a}{\\pi}", "d0b0d65f30ffb949284fefb7cd7d8fc9": " \\mathbf{\\hat x} ", "d0b1fb7cbdf0cc127be2a7fc57726dc7": " 0.008883 \\times W^{0.444} \\times H^{0.663} ", "d0b295cb1995e56180b6bf4818b4b54b": "\\frac{W}{Q} \\le \\frac{T_H - T^0}{T_H} ", "d0b2a9d9733c8e70d30afb8065bd43f5": "\nh(r) = \\frac{kL^{2}}{m^{2}c^{2}} \\left( \\frac{1}{r^{3}} \\right) ~.\n", "d0b2b01b0e636787bfe3ea247816db51": "\n \\boldsymbol{U}^{-1} = \\left[\\begin{array}{ccc}\n1/\\lambda_{X}\\\\\n & 1/\\lambda_{Y}\\\\\n & & 1/\\lambda_{Z}\\end{array}\\right]\n", "d0b361eead34c96af32b85950a455a2f": "S = -\\sum_i \\lambda_i \\ln \\,\\lambda_i = -\\operatorname{tr}(\\rho \\ln \\rho)\\quad. ", "d0b37ae0247330f030c7427071132b4e": "A = \\frac{ \\sqrt{3}}{2} d^2 \\simeq 0.866025404d^2.", "d0b39e9f5c5b4212f63fdc3ada0e1b4e": "\nr_{xx} (\\tau) = \\int_{-\\infty}^\\infty e^{2\\pi i\\tau f} dF(f)\n", "d0b3b5839b35af3cc89b4ef540067424": " u_j, v_i ", "d0b456f7ffad8a7becf4bcabf78c61ab": "\t \n(V_{+}-V_{fl}) = (k_BT_e/e)\\ln 2 \n", "d0b465b0ab907e820aa9df2f535094b9": "T_M(n) = \\max\\{ t_M(w) : w\\in\\Sigma^{*}, \\left|w\\right| = n \\}", "d0b4aad68d18e8ac386f98fb84230bef": "XX_1 = X_1^2 = 4 \\, ", "d0b539f0d635e33ab83c3993d834fb3b": " \\sum_{k=0}^{\\infty} \\frac{\\sin(kx)}{r^k} = \\frac{r \\sin(x)}{1 + r^2 - 2 r \\cos(x)} ", "d0b53e5fdd305cc8634d667dec33a158": " \\cos(a)\\cos(b) - \\sin(a)\\sin(b) = \\cos(a + b)", "d0b5f92a7dd8a23a5d827c9163c8993c": "F[y] = \\int_E (t-\\int_E yt\\,dt)^2y\\,dt", "d0b671849611c4b8355573cb5c3409c9": "[I - A^+ A]", "d0b706e4bfe9a277e0803c906cbb174d": "\\sqrt[3]{e^{\\pi\\sqrt{d}}-744}", "d0b73839bae8a3ac7ef6ca385fc4606e": "\\mu_1 v \\le \\varphi(v,t) \\le \\mu_2 v,\\ \\forall v,t", "d0b738de753237c57ae83bb89d5bfe32": "\\sigma_{j,n}", "d0b73b94c73b8119ea0d2fbed23b9215": "{d_{mx}}", "d0b78c9750075eb572b24a3323db5e02": "\n{\\rm Var}[z]\\,\\,\\, \\equiv \\,\\,{\\rm E}\\left[ {\\left( {\\,z\\,\\, - \\,\\,{\\rm E}[z]\\,} \\right)^2 } \\right]\n", "d0b790d8f9aca14134123670ac5f1851": "L\\frac{\\mathrm{d}^2q}{\\mathrm{d}t^2} + q/C = \\mathcal{E} \\sin\\left(\\omega_0 t + \\phi \\right) \\,\\!", "d0b7acc8eaa320db179e0c0216c0600b": "\\tan(e)=\\left(\\frac{g}{2h}\\right)t", "d0b7b12912c834cfaec04aa0a81316f2": "\\mathrm{Var}[\\nu]", "d0b7fb7d4c0aec6ce746ae166fa10dfe": "\\begin{align}\nA_\\text{ellipse} &= \\int_{-a}^a 2b\\sqrt{1-x^2/a^2}\\,dx\\\\\n &= \\frac ba \\int_{-a}^a 2\\sqrt{a^2-x^2}\\,dx.\n\\end{align}\n", "d0b819a94c90df896502c03ccf1a65c8": "\\tan B \\tan C = 3.", "d0b85786f55a982f5ed91a2a9e8e2f39": "\\Sigma_{n \\mathbin{:} {\\mathbb N}} {\\mathbb R}", "d0b8aca60783652eee6d9ad67a0ce14d": "2p_i", "d0b8f27b45d1e9db01c48d60b0c9da40": " \\tau = \\frac{C_{A in}- C_{A out}}{(-r_{AF})(1-\\delta_{A}f_{A})}\\ ", "d0b93b86db5f8f9414c86790fa6b3381": " \\frac{x}{y+z}\\ ", "d0b956e0fe0d73a8a88fa118e2098417": "V = \\frac{4}{3}\\pi a^3 ", "d0b9c0fb671dcfdc1d78f2c4492901fa": "\\begin{align}\nx_1&=\\frac{p_1}{p_0}\\\\\n\\vdots\\\\\nx_n&=\\frac{p_n}{p_0},\n\\end{align}\n", "d0b9d8ed973b9a557cafab720f5d6706": "\\left(\\frac{dy}{dt}\\right)_k = \\frac{y_{k+1}-y_k}{\\Delta t} = - y_k^2", "d0b9eaa8fd821dcbecd5490c725c8640": "N_{\\ell,m}^\\text{SN3D} = \\sqrt{{2-\\delta_m \\over 4\\pi}{(\\ell-|m|)! \\over (\\ell+|m|)!}}, \\delta_m \\begin{cases}\n1 & \\mbox{if }m=0 \\\\\n0 & \\mbox{if }m\\neq0\n\\end{cases}", "d0b9f57154690d550a706fd10ba28421": "\n\\displaystyle\\mbox{Radiation Pattern (in units of dB)} \\propto 20\\log_{10}\\left(\\left|\\frac{\\sin(X)}{X}\\right|\\right)\n", "d0baa1d4a17a2ca884f58afbcd9aeaec": "\\sum_{m=1}^\\infty\\;\\big\\{\\exp\\;[-\\,(z - H - 2mL)^2/\\,(2\\;\\sigma_z^2\\;)\\;]", "d0baccaa8d457e4a75818500de9a800f": " E = \\{ (x,y) : x \\in C, \\; y \\in D \\}", "d0bb4dbf85efff2aaf65d76e322f7d6d": "\\mathbb{N} \\cup \\{ 0 \\}", "d0bb8d1d8cb193bbe887effdb4098f70": "8y^{3} - 6y - 1 = 0", "d0bba329a5f630ddca6bdbf67818b7b1": "\\omega_{nlm}", "d0bbb43e67b2d4bc98e2e4e96d7ad807": " b:\\mathcal{S} \\mapsto \\mathcal{B} ", "d0bbbd1c9989c5b5ce4e1de925edc559": "\\chi(M)", "d0bbc3c3b05e0cfc71343d0001d72e28": "C = \\frac{V_c^2}{9}", "d0bc33509c009569aa24c01e09265eba": "\\oplus \\!\\,", "d0bc345da40b8fd5a565b828b882cdc0": "FSR=\\frac {\\lambda^2} {2 n_{eff} (L + L_{eff,1} + L_{eff,2})}", "d0bc4c00e187ab5011918208dc74d696": "M=\\mathbb R^3", "d0bc568c916f742a564a0d8f3b969fea": "E(v)", "d0bc818be544e0b2188dda7fd6f58ed9": "\\Omega(\\alpha) = \\inf_\\theta\\left({ \\limsup_{n \\rightarrow \\infty} \\left\\lbrace{\\theta\\alpha^n}\\right\\rbrace - \\liminf_{n \\rightarrow \\infty} \\left\\lbrace{\\theta\\alpha^n}\\right\\rbrace }\\right). ", "d0bc82eb86a78b216674e9d1c6e3a23c": "\\langle a \\mid a^n \\rangle\\,\\!", "d0bd07b6ff37d4bcb6c03014ca024b0a": "\n (\\underline{\\underline{\\boldsymbol{\\varepsilon}}} - \\varepsilon_i~\\underline{\\underline{\\mathbf{I}}})~\\mathbf{n}_i = \\underline{\\underline{\\mathbf{0}}}\n ", "d0bd2043bf559ae74c9f30dc266a46c4": "\\rm \\ (CH_3)_2C=CH_2 + ClONO_2 \\rightarrow O_2NOC(CH_3)_2CH_2Cl", "d0bd3005f7e62fbd50e02dad4017eb8c": "w(v)", "d0bd4a5849ae1c66a9d0d36d6a9fe818": "\\frac{1}{2}(1+\\pi_1-\\pi_0)", "d0bd548bd14ce4e370445bec6f5ab88b": "L = \\lim_{n\\to\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right|\n= \\lim_{n\\to\\infty} \\left| \\frac{\\frac{n+1}{e^{n+1}}}{\\frac{n}{e^n}}\\right|\n= \\frac{1}{e} < 1.", "d0bd8465274bbd64a4c285bfb79cb0f4": "y(t)=2e^{-3t}+2t+1. \\,", "d0be9ff2042726765e4833837ed143f1": "\\phi_{p,i}", "d0bea5dd29c26b76062ddeaec3b611de": "K = \\overline{K} \\,", "d0bf0bd8b804d49e91178d696ea090b7": "G(x,s)", "d0bf24e8b4e561a2f290c6cf70d955fd": "\\cdot_{x_i}", "d0bfb370b781076ade7e7e5307b28ab9": "\\mathrm{MD} = \\frac{1}{n^2} \\Sigma_{i=1}^n \\Sigma_{j=1}^n | y_i - y_j | .", "d0bfd801b244ca82c20ad13d741d2b96": "\\Gamma_S", "d0bff11618bd67a93888af99af7e6b3c": "M(n,1,3) = n^n", "d0c18198b4d69d2c16e66195253eccdc": " =H_a \\left( \\frac{2}{T} \\frac{z-1}{z+1}\\right) \\ ", "d0c262ed378d02b0fd4bffae17ec17e1": "\\bold{P} = ", "d0c2da3fc7c288e1e10fe9921cae39db": "\\int \\frac{dx}{(\\ln x)^n} = -\\frac{x}{(n-1)(\\ln x)^{n-1}} + \\frac{1}{n-1}\\int\\frac{dx}{(\\ln x)^{n-1}} \\qquad\\mbox{(for }n\\neq 1\\mbox{)}", "d0c2fba01f12047c05faae7b427d16b7": "C_*(X) \\otimes C_*(Y)", "d0c30bd20ced4c16ed6fa6bd134e7b74": "\\overline{\\mathbf{P}_{2}\\mathbf{P}_{3}}", "d0c3145f9ed5e88b1492fe3aff683e7a": "\\varphi(a_1,\\ldots,a_m)", "d0c3280cf792393dfc57b826864bf2ac": "\\mathbf{k_{n_x,n_y}} = k_{n_x}\\mathbf{\\hat{x}} + k_{n_y}\\mathbf{\\hat{y}} = \\frac{n_x \\pi }{L_x} \\mathbf{\\hat{x}} + \\frac{n_y \\pi }{L_y} \\mathbf{\\hat{y}}", "d0c37777bc4a802dea32cd38aff62f6e": " y=\\sqrt{\\frac{r}{2M}}", "d0c3a8c3dda3d53029112f80335fcd68": "P [C \\ge x] = P [B \\ge x]", "d0c3ae8ea2c4051c1e7106f24ccd1cd4": "\\langle Ux, Uy \\rangle = \\overline{\\langle x, y \\rangle}=\\langle y, x \\rangle", "d0c4091074d320097789f0e343fe715f": "f \\approx f", "d0c431f5700d9b50393128aa0dc95fda": "\\frac{\\partial p}{\\partial t} = 0", "d0c462dfd0b19f0f9e4b5363522b9242": " z^+ ", "d0c471d2ac1345f54eb4b594c7ea8224": "G\\simeq H", "d0c4819c5f96948c0a73780a5a40848e": "\\mathfrak{su} (2) = \\left \\{ \\begin{pmatrix} ix & -\\overline{\\beta}\\\\ \\beta & -ix \\end{pmatrix}: \\ x \\in \\mathbf{R}, \\beta \\in \\mathbf{C} \\right \\}", "d0c4cb6b9f8690deae275e6bed9aa4b1": "z_{n+1}=z_n- a \\frac{p(z_n)}{p'(z_n)} ", "d0c5a2d392c92835dce843aa76e4d59e": "L = {\\mu \\over \\pi} \\, \\operatorname{arccosh}\\left({D \\over d}\\right)", "d0c5f53c28ba144d2320b027ccd11174": "\\begin{pmatrix}i\\\\u\\end{pmatrix}", "d0c62ec4aef5e448db7cc4655453f8b7": " \\mathbb{E}_X [f(X)] = \\langle f, \\mu_X \\rangle_\\mathcal{H} ", "d0c6449d317f927c0572f89e06a90fe4": "-\\sqrt{\\frac{32}{63}}\\!\\,", "d0c66013d60e8f74687793cd3af5228b": "x_3 = \\frac{B(y_2-y_1)^2}{(x_2-x_1)^2}-A-x_1-x_2=\\frac{B(x_2y_1-x_1y_2)^2}{x_1x_2(x_2-x_1)^2}", "d0c67185d01e96f6f5005aca21d7626e": "E_\\text{pair} = -\\frac{z^2 e^2 }{4 \\pi \\epsilon_0 r}", "d0c6a29fe9662a489636b1e4ac9bd52f": "H_{1,I}", "d0c6d172037740925121c7f32af10052": "\n \\| f \\| := \\sup_{t \\in E} | f(t) |\n ", "d0c70c5cfae61b90c0b3bd4fc6e33012": "\\begin{align}\n \\quad &(2) \\qquad& W &= \\frac{\\pi}{6} d^3 \\rho_s g \\\\\n \\quad &(3) \\qquad& F_b &= \\frac{\\pi}{6} d^3 \\rho g \\\\\n \\quad &(4) \\qquad& D &= C_d \\frac{1}{2} \\rho V^2 A\n\\end{align}", "d0c7379c7c3d748cc3a4994a5e708391": "c'(U' \\cap V') = (c \\circ \\phi)(\\phi^{-1}(U \\cap V)) = c(U \\cap V)\\;", "d0c73e1ef685e793b83ad05c7bdf9148": "E[\\vec{X}]_{ab} = R_{ambn} \\, X^m \\, X^n", "d0c751dadea5c6d6b09923b5823a2aee": "(p \\land q) \\vdash p", "d0c795d83f2b4af703ed44b31a41e3a2": "Mod(\\Sigma)", "d0c7e10df4b4d327b43d834ef1fa4d29": "\\beta(X^*, X)", "d0c80e0aefa571ed7962b0b3ed862acf": "c\\in GF(l)", "d0c819f86241984b9cf7299778240d3d": "BAa", "d0c84a8858006b969a31c02fac5c954c": "w = ms", "d0c85326140344391b29875f4a275778": "a_{ }^{ }", "d0c8536fd121a91bdfc53f4c53878486": "R^f \\oplus(\\bigoplus_i R/(q_i))", "d0c86728266ae1cf08fcce9690c37c04": "\\sum_i\\frac{b_i}{b}= 1", "d0c8bce4e32f6656ee18500fc1138e7d": "bP_{n+1}\\ ", "d0c8d7487d8d94290a1e22c22b75264e": " \\operatorname{build-list}[\\lambda P.B, D, V, [X, \\_, \\_]::L] \\equiv \\operatorname{build-list}[B, D, V, L] ", "d0c8d760f455adb31ad140a0ad5b1666": "\\alpha = D_{AB} = \\nu", "d0c8d9f80a834f45bad22b45417ba016": "\n \\mathbf{G}_i := \\frac{\\partial \\mathbf{X}}{\\partial \\xi^i} ~;~~ \\mathbf{G}_i\\cdot\\mathbf{G}^j = \\delta_i^j ~;~~ \\mathbf{g}_i := \\frac{\\partial \\mathbf{x}}{\\partial \\xi^i} ~;~~ \\mathbf{g}_i\\cdot\\mathbf{g}^j = \\delta_i^j\n", "d0c8ebf212a8fbe8ce64f39c52d2cb53": " {n-2 \\choose d_1-1, d_2-1, \\ldots, d_n-1}.", "d0c8efd6a4559a82434d4c2bb8ea0125": "r, s", "d0c8fe0fa370e4fc140993355dcc8ec1": "\\Delta^2", "d0c9466fd4b212fb0b00e205dcca136b": "\\Delta V\\ = v_e \\ln \\frac {m_0} {m_1}", "d0c972a1300deed2690b75610a25cc18": "Q(z) = A(z) - z^{-(p+1)}A(z^{-1})", "d0c992bf3945068116a7bba9c8bc4dcb": " f' \\circ g = g' \\circ f ", "d0c9fd3f7e6c2fddce4dcade42fb5f45": "\\mathbf{g}(\\mathbf{r}) = g(r)\\mathbf{e_r}", "d0ca8a63ab757736fbbd517b049ceaeb": "g_{\\rm indirect}(r)=\\exp^{-\\beta[w(r)-u(r)]}", "d0cab134061e76d6aad219000306df1e": "\\xi_{[T-1]}=(\\xi_{1},\\dots,\\xi_{T-1})", "d0cadc41ceaefed6dfb1195331572306": "\\Pr(X_{n} = i, X_{n+1} = j) = \\Pr(X_{n+1} = i, X_{n} = j)\\,.", "d0cb096b5f8944a0785430926ff31380": "=\\int_{S_1}^{S_2} \\left(\\frac{\\partial H}{\\partial S}\\right)_P \\mathrm dS\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\text{at constant pressure.}", "d0cb0a179ba227d6386ff237ba9bc51b": "\n\\theta_E=\\arccos\\left(e^{-\\frac{T_R}{T_1}}\\right).\n", "d0cb244468afb9a6d759245307920a62": "\nT(F) = \\sum_{i=1}^k \\frac{(\\int_{A_i} \\, dF - p_i)^2}{p_i},\n", "d0cb286554d31128d21a23b03e3a3c7b": " \\rho \\left(\\frac{\\partial v}{\\partial t} + u \\frac{\\partial v}{\\partial x} + v \\frac{\\partial v}{\\partial y}+ w \\frac{\\partial v}{\\partial z}\\right) = -\\frac{\\partial p}{\\partial y} + \n\\frac{\\partial}{\\partial x}\\left(\\mu\\left(\\frac{\\partial v}{\\partial x} + \\frac{\\partial u}{\\partial y}\\right)\\right) + \n\\frac{\\partial}{\\partial y}\\left(2 \\mu \\frac{\\partial v}{\\partial y} - \\frac{2\\mu}{3} \\nabla \\cdot \\mathbf{v}\\right) + \n\\frac{\\partial}{\\partial z}\\left(\\mu\\left(\\frac{\\partial v}{\\partial z} + \\frac{\\partial w}{\\partial y}\\right)\\right) + \n\\rho g_y", "d0cb8664dbfebc91ab2592e7f285d85c": "\\gamma q", "d0cc61f1bb5ffd9fa2e29a3ed340c187": "\\langle J'(u), u\\rangle = 0. \\, ", "d0cc684fc54aaa68974adb484ff93a56": "\\mathbb Z_p := \\varprojlim_n \\mathbb Z / p^n", "d0ccc96517ff4d0da803d72bcae19a78": "\\mathbf F\\times\\mathbf{G}\\cdot d\\mathbf{S}.", "d0ccc9a090d212e3ea8b966346502525": "\nE_{n}=\\left( \\frac{\\pi \\hbar \\beta D_{\\alpha }^{1/\\alpha }q^{2/\\beta }}{2\\Beta(\n\\frac{1}{\\beta },\\frac{1}{\\alpha }+1)}\\right) ^{\\frac{\\alpha \\beta }{\\alpha\n+\\beta }}\\left(n+\\frac{1}{2}\\right)^{\\frac{\\alpha \\beta }{\\alpha +\\beta }}.\n", "d0cd5345f8b878429b195aa031963dd0": "(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0", "d0cd8f3fa3134ea5e68f24974f744856": "A, A', B, B'", "d0ce14020925f0235e6ff38e3e68d1c5": "\\forall x, x \\le y \\implies F(x) \\le G(y)", "d0ce1ef5de7a65c2ab07b05d6f898a14": "a_n b_m", "d0ce6598b8c99846912450de57fce776": "A_{ij}=\\langle T(v_j), v_i\\rangle", "d0ce8885111bc33e4d4931dec357db71": "\\ M_{heel} ", "d0cea51d3393c79eb716cc0efdc8b6a5": "0 < a < b", "d0ceafa0eb986eb2a789b6ca3a7d4fd9": "L=\\frac{m}{2}\\mathbf{\\dot{r}}\\cdot\\mathbf{\\dot{r}}+q\\mathbf{A}\\cdot\\mathbf{\\dot{r}}-q\\phi", "d0ceb83e8678d99071dc1340f5f85647": "h[0]", "d0ceeca63658832fd1ff10420929dba0": "A \\cap (B\\,\\triangle\\,C) = (A \\cap B)\\,\\triangle\\,(A \\cap C),", "d0cef50653958d5181eff6497c06c446": "\n\\sum_{n=1}^\\infin \\frac{1}{n^2} =\n\\lim_{n \\to +\\infty}\\left(\\frac{1}{1^2} + \\frac{1}{2^2} + \\cdots + \\frac{1}{n^2}\\right).\n", "d0cf0faa755d04cf02b10b45a04f5a2c": "P(W_n|[Spam=true])=\\frac{1+a^n_t}{2+a_t}", "d0cf4d485627e76a29f63f80ab31c831": "n(p;d)\\approx \\sqrt{2d \\cdot \\ln\\left({1 \\over 1-p}\\right)}.", "d0cf53199a954ee4e2354e864da07567": "\\ K_o ", "d0cf5792bdb2764831bd2d90e3b2782a": " \\operatorname{build-param-lists}[g\\ m\\ p\\ n, D, V, K_2] ", "d0cffa103c5d67e1ce1e7ab10652aa52": "\\textstyle n-l", "d0d03e3ec12047cddd590ad068d45bc8": "\\Pi \\subset T_{\\gamma(t)} M", "d0d04960a95d247ba110088469c39f61": "s = \\mathrm{d}\\rho / \\mathrm{d}t \\,\\!", "d0d0dcdd0a6c42df3b3d7adf1dafcd98": " U(d,d_1) = u - r|d-d_1| \\,", "d0d16255fe57bd74940ea05bbc417264": "\\displaystyle{T\\psi=\\lambda\\psi, \\,\\,\\, T^*\\varphi =\\lambda\\varphi, \\,\\,\\, S\\varphi=\\psi, \\,\\,\\, \\partial_n D(\\psi)|_{\\partial\\Omega}=(\\lambda^2 -{1\\over 4})\\varphi,}", "d0d16cdd354e9872dc508763e5a4d491": "\nV_C(s) = V\\frac{1}{1 + sRC}\\frac{1}{s}\n", "d0d1ce6e2f001bb88f2e12f29b3748b2": "[\\mathbf{\\hat{p}}, \\mathbf{\\hat{A}}]=0 ", "d0d2198422dab0844f1b421c71803131": "K \\subset \\cup_{i =1}^s V_{k_i}", "d0d24062f2ac3cbe0686063cb60c8fb5": "\n\\hat \\sigma_\\mathtt{RK}^2 (\\mathbf{s}_0)\n= (C_0 + C_1 ) - \\mathbf{c}_\\mathbf{0}^\\mathbf{T} \\cdot \\mathbf{C}^\\mathbf{1}\n\\cdot \\mathbf{c}_\\mathbf{0} + \\left( \\mathbf{q}_\\mathbf{0}\n- \\mathbf{q}^\\mathbf{T} \\cdot \\mathbf{C}^{ - \\mathbf{1}} \\cdot\n\\mathbf{c}_\\mathbf{0} \\right)^\\mathbf{T} \\cdot \\left( \\mathbf{q}^\\mathbf{T}\n\\cdot \\mathbf{C}^{ - \\mathbf{1}} \\cdot \\mathbf{q} \\right)^\\mathbf{ - 1} \\cdot \\left(\\mathbf{q}_\\mathbf{0} - \\mathbf{q}^\\mathbf{T} \\cdot\n\\mathbf{C}^{ - \\mathbf{1}} \\cdot \\mathbf{c}_\\mathbf{0} \\right)\n", "d0d2705aae16cea428a1188e759ab955": "\\mathbf{e}_{\\pi(i)}", "d0d281e218c91df3d6a64ce5a210f4a7": "C_t[a_0, \\cdots, a_m]= C_t[a_0] \\,\\cap\\, C_{t+1}[a_1] \\,\\cap \\cdots \\cap\\, C_{t+m}[a_m] = \n\\{x \\in S^\\mathbb{Z} : x_t = a_0, \\ldots ,x_{t+m} = a_m \\}.", "d0d2bdb818a985e2e4610687cc55c6f4": " \\mu>\\mu_0 ", "d0d2c5b88972b327072b0608dc65c275": "\\frac{1}{2} \\begin{pmatrix}\n1 & i \\\\ -i & 1\n\\end{pmatrix}\n", "d0d3783c23dfc2d7c718ea785acb8d62": "10! = 6! \\cdot 7! = 1! \\cdot 3! \\cdot 5! \\cdot 7!", "d0d3840694545a365a35d77ea592693f": "\\scriptstyle \\log_{10} P_{mmHg}=6.82973 - \\frac {813.20} {248.00+T}", "d0d3c6dee3556c58cc90c7ec384c2131": " u \\le \\varphi\\text{ in }|\\Omega,\\quad \\nabla u = \\nabla\\varphi\\text{ on }\\Gamma. \\, ", "d0d400f4f43ef1981fcfe64e48bfbf19": "a_{s,i}", "d0d42073fcf32d03aca98b2acd4b668b": "K_0(BG)", "d0d432ac18302227eab279e909124dc9": "F^*", "d0d438f16038f85808fd0eef544acb7d": "n\\Delta{s}\\,", "d0d4c696382403153c77b6c0eb75a6d1": "2.065 < R < 3.97", "d0d4e75424c967beb2b5beb2f0a58b24": "B(x,y) = (Q(x+y)-Q(x)-Q(y))/2=x_0y_0 - x_1y_1 - \\cdots - x_ny_n.", "d0d5002055ccfebfcb3de8433bae09cf": "\\min (d, 1)", "d0d52d17da1a86274c49c3ba9befef12": "\\frac{1}{\\rho}\\frac{\\,d\\left(\\,P-P_e\\right)}{\\,d\\,X}+\\tfrac12\\,\\left[\\frac{\\,f}{\\,D}-\\left(\\,1-\\frac{W_0}{W}\\right)\\frac{f_e}{D_e}\\left(\\frac{F}{F_e}\\right)^2\\right]\\,W^2\n+\\left[\\left(\\,2-\\beta\\right)\\,-\\left(\\,2-\\beta_e\\right)\\left(\\,1-\\frac{W_0}{W}\\right)\\left(\\frac{\\,F}{F_e}\\right)^2\\right]\\,W\\tfrac{\\,dW}{\\,dX}\\,=\\frac{f_e}{\\,2D_e}\\,W_0^2\\left(\\frac{F}{F_e}\\right)^2", "d0d535f44fa78b899783076466627117": "\n\\frac{d{X}}{d{t}} = V.\n", "d0d5cbd2b9413896717463413801a6ac": "H^{p,q}_{\\bar{\\partial}}(X)", "d0d610354e0b236e332b28b329a5c180": "21.5~", "d0d618351ce0f3e04c4319888e0e35b3": "[k]_q=\\frac{1-q^k}{1-q}=\\sum_{0\\leq i1", "d0dac7be2f314ecee1fda8c202ce6e98": "T_{i_1} \\times \\dots \\times T_{i_k}", "d0db59c99ba5f13d5b24aa850096b434": "S= Nk\\ln\\left(\\frac{VT^{\\hat{c}_v}}{f(N)}\\right)\n", "d0dbbf77bd9193a2f84b4b9662e4b04c": "\nI = I_{ion}^{sat} \\tanh\\left( \\frac{1}{2}\\,\\frac{eV_{bias}}{k_BT_e} \\right)\n", "d0dbca76e160a5ef4b06eb75fcf8bae2": "\n\\frac{A\\hbox{ true}}{A \\vee B\\hbox{ true}}\\ \\vee_{I1}\n\\qquad\n\\frac{B\\hbox{ true}}{A \\vee B\\hbox{ true}}\\ \\vee_{I2}\n", "d0dc1c7893d6597f4395f045d2970e9c": " \\sigma_{\\max}(A)", "d0dc32480c0d0f1ac1e0534f16feed4f": "(\\mathbf{q}(t), \\mathbf{p}(t))", "d0dc67e08bb0f7cd2e22abb0ab743fbd": " \\max \\limits _{ \\| x \\| = 1} \\| AB x \\| = \\| AB\\| . ", "d0dcc0d754fdc5d0e01f7476e262b7a8": "J_{N(t)} \\leq t < J_{N(t)+1}. \\, ", "d0dcf9e7d20a840a43da4afd2c9be3d7": "u=(z-B)/(A-B)", "d0dd52a374d3d1642d993b32268703b8": "\n\\sin \\gamma = \\cos (\\lambda + \\chi) \\cos \\eta \n", "d0ddeac00654b883251df845a395168c": "H^d_\\delta(S)=\\inf\\Bigl\\{\\sum_{i=1}^\\infty (\\operatorname{diam}\\;U_i)^d: \\bigcup_{i=1}^\\infty U_i\\supseteq S,\\,\\operatorname{diam}\\;U_i<\\delta\\Bigr\\}.", "d0de2fc3d96fd2d64113e00df3d5ea16": "\\mathbf{x} ", "d0de49199e161d09d803e251f961dee7": "x(t)=x_{0}\\cos(\\omega t)+\\frac{p_{0}}{\\omega m}\\sin(\\omega t) ", "d0deda20e9bd1b3513e95e37446deb31": "(b_1 + a_1 i)*(b_2 + a_2 i)", "d0defa23cb377054ea20e9eb94d532ad": "\\liminf_{n\\to\\infty} g_n < 7\\cdot 10^7", "d0df02eb5d291085509ba8ba1cb9f4d0": "U_J\\,\\!", "d0df4d1868365b3666eb915839de5bf0": "\\hat{J}_a", "d0dfa60a748a33658fc82b28225f0449": "\\cos A - \\cos B \\cos C : \\cos B - \\cos C \\cos A : \\cos C - \\cos A \\cos B", "d0dfb6e138e2e13e9900819523d651fa": "x > 1", "d0dfbafaea441f6262b04edbf8e22499": " U_i = \\alpha P_i/Y + \\beta D_i +\\varepsilon_i\\, ", "d0dfdaecb07c4a170821a069fb55ff41": "0\\rightarrow1\\quad\\mbox{at rate }\\lambda\\sum_{y:y\\sim x}\\eta(y),", "d0dff207e831dfb9c8d91659e552f606": "g = A \\to aA", "d0e023223dec93fcae9ad6d4d7e6c063": "n = 3, 9, 21, 231", "d0e05623a5a50b0bc8168e7ce730925f": " \\|\\mathbf{P}\\|^2 = \\frac{E^2}{c^2} - \\mathbf{p}\\cdot\\mathbf{p} ", "d0e08138cd3c9ca5ed6022c32ff5ed34": "\\frac{128}{81}", "d0e0eddc95f259cdc51ac6b9aabdf313": " a n^2+bn", "d0e114a4e0adfa90f1f2364cd657c673": "1,208 \\times 600 = 724,800\\,", "d0e11a37917317d1326139e826c6d5c5": "\\left(\\frac{211.080-202.416}{202.416}\\right)\\times100%=4.28%", "d0e1547c99ada19bc348e2e2579ed7ea": "b \\ ", "d0e16ffff015fca4e63616e21282cb20": "\\mathbf{\\hat{r}}", "d0e1b8571128845c03a4cfac00d43b66": "K\\,", "d0e206befc6d77d85e38ba1a5b0b9c9f": "L_C=R/\\sqrt{B}", "d0e212f2abb927b315e7c6ef56247b5b": "U - T S -", "d0e2b9ec65982855e193bbb590edfd73": "(0,0), (0, -1), (1,-1)", "d0e2db845c1a59d5adf06b48fb5917c9": "d_\\min = \\tfrac{n}{2}", "d0e2f885b72601c5adc204eca9132a8d": "{\\mathbf P} (t) = \\sum_{c,v} ( {\\mathbf d}_{cv} p_{cv,0} \\, \n\\mathrm{e}^{-\\mathrm{i} (\\epsilon_c - \\epsilon_v) t/\\hbar} + \\mathrm{c.c.} )", "d0e34114d8ad3a75777f6d773cffcefa": "\\mathbf{\\hat n}_\\mathrm{out}", "d0e37ea7d0ebb88ce6fa686fa5d5a3e0": " \\frac{\\mathit Div} {\\mathit P} + \\frac{\\mathit XCF} {\\mathit P} ", "d0e3e08987ae1782ef7c454ff910853c": "v=1320", "d0e3e658c14887a9404e6d5506050f74": "\n\\Delta x \\sim \\frac{\\lambda}{\\sin \\theta}\n", "d0e4281d0d7e0f8b4b1b95c7c9016a91": "f = \\mu/s", "d0e45ddc61f9268a6fc4ad232e82ed7f": "H_{1}^{(i)}(x) = xG_{1}^{(i)}[H_{1}^{(1)}(x),...,H_{1}^{(n)}(x)]", "d0e466674bd6c9285617ef267543d246": " \\!\\ S_m^7 = S_{(m^7 + 7m^5 + 14m^3 + 7m)} ", "d0e4831af4db4d17e8c7ab57bd2711d1": "r(x) = r_1 + \\frac{x}{l}(r_2 - r_1)", "d0e4c432812c5790e7caa1544e11e65a": "D_\\mathrm{KL}(p(x|a)p(a)||m(x,a)) = D_\\mathrm{KL}(p(a)||m(a)) + \\mathbb{E}_{p(a)}\\{D_\\mathrm{KL}(p(x|a)||m(x|a))\\},", "d0e5254734ce11a2215c1709095bbb80": " {V(y) = \\gamma y(1- {y \\over b})} ", "d0e57580e02fee98e25dfe081b29d0af": "C_{qs}^-", "d0e57f3d2c3c3027aec29033bd6f464d": "l = 1", "d0e60ebd596d4f672d950235e4fef03d": "\\mathbf{X}, \\mathbf{X}_1", "d0e63ce6362efeac2fb91dde51efdbce": "T^n/S_n", "d0e6a6a859856744c9026787d23c3e2f": "b \\rightarrow \\infty", "d0e6ff5828d94c83ea9e97158ed9a337": "(-1)^{\\lfloor n/2 \\rfloor}=(-1)^{n(n-1)/2}", "d0e766dd2c438782fbe3ec3a243f9631": "\\mathbb{F}_{q^k}-\\{ 0 \\}", "d0e799c4ef9412257d855597ed0fd2bd": "F(x) = Z_F(x, 0, 0, \\dots)\\,", "d0e7e066324c0ddfd12f3c90bbdb281f": "empty(j) = \\{j\\}", "d0e7e2d7c5c1943ba201e4b681951dcd": "G \\subseteq \\mathbb{P}", "d0e7e76d56bb6eededa0bd8b09db0cf4": "D=3,4", "d0e835593238c3fb8af8442534f72c60": "H_0(K)\\simeq H_0(L)", "d0e848aa7eadbe605dbef83a5ec04374": "R \\subseteq S \\times S", "d0e84dd2fc8ca44c6261fc89686743c0": "[2,200]", "d0e8adfaab180f8975be535dcba3b570": "(X_t)_{t \\in M}", "d0e8e02acc626fe01014c6aca4f6cf01": "O\\left(w_\\text{kernel} h_\\text{kernel} w_\\text{image} h_\\text{image}\\right)", "d0e8f9299ea0a5cf78b76d5e1542fc38": "F(s;q) = \\sum_{m=1}^\\infty \\frac {e^{2\\pi imq}}{m^s}\n=\\operatorname{Li}_s\\left(e^{2\\pi i q} \\right) ", "d0e9222ee6653ebdd5e234531178255d": "\\scriptstyle(+1.2\\pm2.2)\\times10^{-9}", "d0e9289b6358be30b70554083bf78639": "\\tilde \\delta_i", "d0e95d27ad92e1aabc4e375c332d1b45": "\\mathrm{Ei}(x) \\,=\\, \\gamma+\\ln x - \\mathrm{Ein}(-x)\n\\qquad x>0\n", "d0e975cf09c4072c266059520b5c8bdc": "\\R^n.", "d0e97bdfc983bbbd42b6bbc671c21263": "\n\\delta'_{2s}(n)=\n\\frac{(\\frac12\\pi)^s}{(s-1)!}\\left(n+\\frac{s}4\\right)^{s-1}\n\\left(\n\\frac{c_1(n+\\frac{s}4)}{1^s}+\n\\frac{c_3(n+\\frac{s}4)}{3^s}+\n\\frac{c_5(n+\\frac{s}4)}{5^s}+\n\\dots\n\\right).\n", "d0e98cf7bd4ae132cc0105e1b58e7bde": "\\beta \\Leftrightarrow \\neg \\alpha", "d0ea57aa8f60bb079eab48fdf2d4ba22": "\\scriptstyle 0\\le x< 1", "d0ea8031c0c51ce997b657ad45f36d76": "S=1234\\,\\!", "d0eae19a74031dc82b3ae8dbb2150ae1": "\\sum_{n=0}^\\infty a_n = A", "d0eb0814ea5d99dc7c2ede08b98ec96b": "B_{n+1}(x) = B_{n+1} + \\sum_{k=0}^n\n\\frac{n+1}{k+1}\n\\left\\{ \\begin{matrix} n \\\\ k \\end{matrix} \\right\\}\n(x)_{k+1} ", "d0eb38684b9d14065b75ec4f4286581d": "Energy = area ~ OABO = W_{stored} = \\int_{0}^{\\lambda} i(\\lambda) ~ d\\lambda \\;", "d0eb781bc349eee3706677ca68af6a87": "\\omega = N \\cos\\Theta", "d0ec0b78d566a0d7a860780858fbce1c": "\\hat{N}_a \\equiv \\sum_{n=1}^{N} \\sigma_{n}^{+}\\sigma_{n}^{-}", "d0ec32d16aeb7aee3de08ab0115b692e": "\\int f(\\alpha,\\alpha^*) \\, d^2\\alpha = \\mathrm{tr}(\\hat{\\rho}) = 1 ", "d0ec5b00be4cfbabc6e74c82fb010987": "x\\,e^{-x}\\,", "d0ec6e7e501b75bf7f39ea30403d67bb": "\n\\begin{pmatrix}(1+\\alpha^{2})+(\\alpha^{0}+\\alpha^{-6})x+\\alpha^{7}x^2&\\alpha^{-1}+\\alpha^{6}x\\\\\n\\alpha^{3}+\\alpha^{1}x&1\\end{pmatrix}\n\\begin{pmatrix}\\alpha^{4}+\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3+\\alpha^{1}x^4+\\alpha^{-1}x^5\\\\\n\\alpha^{7}+\\alpha^{0}x\n\\end{pmatrix}.\n", "d0ec8bc77e73b23226373b517837494a": "\\lim_{r \\to 0} \\inf \\left\\{ \\left. \\frac1{\\gamma \\big( B_{s} (x) \\big)} \\int_{B_{s} (x)} f(y) \\, \\mathrm{d} \\gamma(y) \\right| x \\in H, 0 < s < r \\right\\} = + \\infty.", "d0ecd63c96332dd26d055fe1c0c2a477": "A=\\{A_\\gamma | \\gamma < \\delta <\\alpha \\}", "d0ecdf57b7c188f63d0f422336a2ec3e": "\\phi *\\psi (w)(1)=(g_1,g_2,\\dots , g_n). ", "d0ecf1ca5395595104c711da900349c0": "(e^z)^w = e^{(z + 2\\pi i n) w}\\,", "d0ecfdb73908dde47a6f451d50702a14": " {\\partial \\mathcal{L}\\over\\partial \\pi} = 0 ", "d0ed163e2bcba163d908e2084f5f329f": "d_H(\\mathbf{u^1G, u^2G}) \\geq 2t+1", "d0ed6f9007055522d82c9be9edaefd2b": "\\sum_{i=1}^{p-1} i^{p-1} \\equiv -1 \\pmod p.", "d0ed7a4c72d3e18d426df25748386de7": "\\beta > -3", "d0ed8ea810fde49fe6f5f7bdf44858b2": " \\Delta P= \\gamma (C_1 +C_2)", "d0ed960d19222805bcf7c7fbd398efc3": "u(x,\\,y)", "d0ede0e8c053e0f841902845033b4be1": "\\mathfrak{sp}(2l, F) = \\{ x \\in \\mathfrak{gl}(2l,F) | s x = - x^t s, s = \\begin{pmatrix} 0 & I_l \\\\ -I_l & 0 \\end{pmatrix}\\}", "d0ee193cb579fff1c6f37468609aa47b": "\n\\hat m = (I + \\lambda A)^{-1} Y.\n", "d0ee5d5728167804709c4a3aaf525b30": " d(f(x), f(y)) < d(x, y) \\quad \\mbox{for all} \\quad x \\ne y \\in M_1", "d0ee660dd65844c4cc7e0f28d5489036": "\\mathcal{G}(\\Omega)=0.", "d0ef66b2c300bfdd6c1e9b11d103e63a": "\\sum_{n=-\\infty}^\\infty \\delta(x-nP) \\equiv \\sum_{k=-\\infty}^\\infty \\frac{1}{P}\\cdot e^{-i 2\\pi \\frac{k}{P} x} \\quad\\stackrel{\\mathcal{F}}{\\Longleftrightarrow}\\quad \\frac{1}{P}\\cdot \\sum_{k=-\\infty}^{\\infty} \\delta (\\nu+k/P),", "d0ef834e0ca1bc1964c0b2db4b1c0fca": "w^0 = w^6 = 1", "d0efa87997c25303d612954d933a8430": "x = t \\dot u_0", "d0efdf6427e36b0ae76f88ab6b7c611f": "\\frac{\\sigma^2}{(1-\\xi)^2(1-2\\xi)}\\, \\; (\\xi < 1/2) ", "d0efe24ee0a07e15d3e1d7336e0b4c4b": " I_0 ", "d0f0005459f1048e91e5ba5635095265": "V^{1.3}", "d0f05107be8dfeac386e553ac5b14a34": "\np_{\\theta_2} = \\frac{\\partial L}{\\partial {\\dot \\theta_2}} = \\frac{1}{6} m \\ell^2 \\left [ 2 {\\dot \\theta_2} + 3 {\\dot \\theta_1} \\cos (\\theta_1-\\theta_2) \\right ].\n", "d0f05782d9ddeb13cb2e2227d2dcc864": "\\hat v^*", "d0f06dac84767f305b76ff6f702b5b53": "|\\gamma_n(s)| \\le \\frac{3}{(3+\\sqrt{8})^n} (1+2|\\Im(s)|)\\exp(\\frac{\\pi}{2}|\\Im(s)|).", "d0f0a429ff55ccccca29833acc3c479f": "J^1Q\\to\\mathbb R", "d0f0c0097e31d46a992fa9c6edfd8137": " E\\left[ \\Lambda(n+1) \\right] = E\\left[ \\left| \\hat{\\mathbf{h}}(n) + \\frac{\\mu\\, \\left( v^*(n)+y^*(n)-\\hat{y}^*(n) \\right) \\mathbf{x}(n)}{\\mathbf{x}^H(n)\\mathbf{x}(n)} - \\mathbf{h}(n) \\right|^2 \\right]", "d0f14356ef87addea12bcfd28dfe9b4e": "\n \\begin{align}\n \\sigma_{11} & = 2C_1\\left[\\cfrac{\\lambda^2}{J^{5/3}} - \\cfrac{1}{3J}\\left(2\\lambda^2+\\cfrac{J^2}{\\lambda^4}\\right)\\right] + 2D_1(J-1) \\\\ \n & = \\sigma_{22} \\\\\n \\sigma_{33} & = 2C_1\\left[\\cfrac{J^{1/3}}{\\lambda^4} - \\cfrac{1}{3J}\\left(2\\lambda^2+\\cfrac{J^2}{\\lambda^4}\\right)\\right] + 2D_1(J-1)\n \\end{align}\n ", "d0f171aeb1e573278fa41230bb991c4f": " \\dot{\\tilde{\\rho}}= - \\int^t_0 dt' \\operatorname{tr}_R\\{[\\tilde{H}_{BS}(t),[\\tilde{H}_{BS}(t'),\\tilde{\\rho}(t')R_0]]\\} ", "d0f19d84032e1980d97ce7a1c5a5e27c": " \\{x : \\dot{V}( \\mathbf x) = 0 \\}", "d0f1b884d1237b0af4bd2413cc1768a0": "f(A)=\\sum_{h=0}^{\\infty}a_h A^h", "d0f22623b3435a51f46fb52cc79b3981": "\\rho(A)=\\lim_{k \\to \\infty}\\|A^k\\|^{1/k}.", "d0f2a719fdb790449519bd35ded4d6fd": "G\\!", "d0f35fde97e74ab778136c6a547b70ee": "X^0", "d0f377f77aaec3cce8c0fd026ed4c16d": "V_\\text{oc} = m\\ \\frac{kT}{q}\\ \\ln \\left( \\frac{I_\\text{L}}{I_0}+1 \\right) \\ , ", "d0f39efad900a0b7ff223ad2a65ed345": "~\\hat{\\varphi}~", "d0f3eb81be27d57a9d3b32006c3bc10c": "\\operatorname{red}_1", "d0f40b2a06259298e90cdf65c1abb698": "h=0.25.", "d0f44b4cd0a846b9c7bebc9d487c5851": "\\mathcal{B} (M)", "d0f461911ae462651c510095f59ecc00": "P_{2}^{-1}(x)=-\\begin{matrix}\\frac{1}{6}\\end{matrix}P_{2}^{1}(x)", "d0f467efeca0c6e1e22e03b52a5ddc7b": "\\sum_{n=0}^{\\infty}\\frac{z^n}{n!}.", "d0f49236088ec52415186edf03b4eb89": " \\ = i \\lang 0|T(\\Phi(x) \\Phi(y))|0 \\rang ", "d0f4b347f869da43c610b9fa725e7554": " P_n e^{\\alpha t} \\cos{\\beta t} \\!", "d0f518520c7282be7caa3845577321cd": "\\ln\\frac{P_v}{P_{sat}}=-\\frac{2H\\gamma V_l}{RT}\\ ", "d0f5da9ceba70c99a8278a2cff7a0536": " \nCentroid = \\frac{\n \\sum_{n=0}^{N-1}\n f \\left ( n \\right )\n x \\left ( n \\right )\n} {\n \\sum_{n=0}^{N-1}\n x \\left ( n \\right )\n}\n", "d0f5e68e6a94f717d5161e5f83dd1fcf": " \\gamma = 4 for d,d_0 > d_c ", "d0f642b9d1b69602e6128d09d95bf113": "u \\in \\mathbb{R}", "d0f681ac6ee3b654a16dc8f391253665": "T(\\varphi)(x) = x^2 \\varphi (x) \\quad ", "d0f6b0e96b31f93b1d2f866c104e735c": "h=0.5", "d0f6de3b42f0baa0dfb04dc5e1c57f42": "\\hbox{ch}(V\\oplus W)=\\hbox{ch}(V)+\\hbox{ch}(W)", "d0f6e2b853f62282b5b3a17001b90c63": "\\rho_\\beta\\!", "d0f723adb43929e5b3c959bc1b31958d": "1.1,", "d0f782d9b82eba6143e2195c22a341cd": "y^2 = x^4 + 2ax^2 + 1", "d0f8529eb8ac0ce7fb4e03c5486a7913": " F = U_1 \\cdot U_2 \\cdot U_4 = \\sum_{i=1,2,4} U_i ", "d0f89a1297a0dbcba9095e3fcb998099": "Y_{9}^{-1}(\\theta,\\varphi)={3\\over 256}\\sqrt{95\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(2431\\cos^{8}\\theta-4004\\cos^{6}\\theta+2002\\cos^{4}\\theta-308\\cos^{2}\\theta+7)", "d0f8eb6b1dfb60c9e5594a597fc1a015": "u(t) \\le u(a) \\exp\\biggl(\\int_a^t \\beta(s)\\, \\mathrm{d} s\\biggr)", "d0f98092dd623043237b5775d6827aaf": " \\sigma = |d|\\sqrt{\\frac{(n-1)(n+1)}{12}}", "d0f98af4e3457d645c65725c9f12f38c": "\\begin{matrix} \\frac{3}{1} \\end{matrix}", "d0f9c9f10a8940b8df19f2c2b3aaca05": "\\begin{align}\n f(x) & {} = a_0x^n+a_1x^{n-1}+\\cdots+a_n & {} \\quad (1) \\\\\n & {} = (x-r_1)(x-r_2)\\cdots(x-r_n) & {} \\quad (2) \\\\\n\\end{align}", "d0fa3a3671ee7a577e3650194f87e108": "x_n\\ ", "d0fac14a11d7a8f3574216d7527a8ecf": " \\hat f(x) = \\lim_{n\\rightarrow\\infty}\\; \\frac{1}{n} \\sum_{k=0}^{n-1} f\\left(T^k x\\right).", "d0facd8e44e78949d54f543eae2fc13b": "\\lambda/\\mu", "d0fb0b18a6a90cbcfb9fe217a651fa43": " =\\text{Tr}\\left\\{ \\Pi_{\\rho,\\delta}\\ \\rho^{\\otimes n}\\right\\} ", "d0fb2df38d71c116d4763fcee392f505": "\\mathcal{H} =\\lambda_1[p_1^2+m_1^2+\\Phi_1(x_1,x_2,p_1,p_2)] + \\lambda_2[p_2^2 + m_2^2+\\Phi_2(x_1,x_2,p_1,p_2)] ", "d0fb94f6e46fe5e536ea4a59dc6ea734": "f_j^o[n]", "d0fc12f2671c81c61fda6080a54acef2": "x^{2p} + y^{2p} = z^{2p}", "d0fc227aac9e269ce9512dc495915877": "R_{ij}^{k\\ell} = \\exp(-\\beta \\varepsilon_{ij}^{k\\ell})", "d0fc23470a565c404d213bc64dc6d0aa": "1 \\le i \\le n-1", "d0fc9381a8fddf62a8fed0ab99ddd696": " \\nabla^2 \\bold A + k^2 \\bold A ~ = ~ -\\bold J ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (1.2) ", "d0fc9ae7bceeb1f7b8c5273ac6b73b14": "\\begin{bmatrix} \\dfrac{e^{\\eta_1}}{\\sum_{i=1}^{k}e^{\\eta_i}} \\\\[10pt] \\vdots \\\\[5pt] \\dfrac{e^{\\eta_k}}{\\sum_{i=1}^{k}e^{\\eta_i}} \\end{bmatrix}", "d0fcbee5c6f629576a1911b3ff8e5997": "p=N_v\\text{ exp}\\left[-\\frac{(E_F-E_v)}{kT}\\right]", "d0fce3c3ecbeda6e1f898081671095e2": "\nH= \\sum_{\\mathbf{k},\\mu} \\hbar \\omega N^{(\\mu)}(\\mathbf{k})\n", "d0fd3e2563a9ad2f7dfc7f8d295ca094": " \\int_X g\\,d\\mu = \\int_X g\\frac{d\\mu}{d\\lambda}\\,d\\lambda.", "d0fdaef420c5b8ad020e9118fe78cb28": "\n \\begin{align}\n \\sigma_{11}-\\sigma_{33} & = 2C_1\\left(\\lambda^2-\\cfrac{1}{\\lambda}\\right) -2C_2\\left(\\cfrac{1}{\\lambda^2} - \\lambda\\right)\\\\\n \\sigma_{22}-\\sigma_{33} & = 0\n \\end{align}\n ", "d0fdba37f16f8cf7d4936efaf1e57f42": "(\\pi_{2,1})^{*}\\theta_{0}\\,", "d0fe9077a79a145f34f4f4abb8364a27": "t=0\\!\\,", "d0fe9a891ee465f25460771bf9470223": "f(x)\\in I \\Leftrightarrow f_k(x)\\in I, ~ \\forall k\\in \\mathbb{N}.", "d0ff02e50d69e582b916158c6c90f56e": "\\mathbb{Z}[1/p] / \\mathbb{Z}", "d0ff2b3b4e60a139bc5bf557dbc47938": "\\Pi(dx)", "d0ff8611e533b353aaece8e74dc63a03": "\\mathrm{tri} = \\mathrm{rect} * \\mathrm{rect}.\\,", "d0ffb308fe8ccc1fd1ccbd2ba9fd0f0c": "f(n)=b_1 b_2 b_3 \\dots b_m 1", "d0fff85588f3d148b0036cf0cee6d136": "\\eta = \\frac {F_{out}/F_{in}}{d_{in}/d_{out}} = \\frac {F_{out}}{F_{in}} \\frac {l} {2 \\pi r} < 0.50 \\,", "d1000aa0c1d1b8e143005ef94cd9b848": "\\frac{(a+b) + (a-d) - 8}{2} = (a-4) + 0.5(b-d)", "d10018383441a70c077e4433efb2fc8f": "\\overline K", "d100311d422d0d1ccdf44d1d1b2c062a": "\\phi:F\\rightarrow G_1\\ast G_2", "d100ebd4090caa9e0e64331d3af66bb0": "\\sqrt{-2\\epsilon}=\\sqrt{\\mu/a}", "d1012faadbf3484337c4064af249da56": "C_{abs}=\\frac{2\\pi}{\\lambda}Im[\\alpha]", "d10171e6cb3841329e122408d55bf618": "\\begin{bmatrix} \\mbox{category} & noun\\ phrase\\\\ \\mbox{agreement} & \\begin{bmatrix} \\mbox{number} & singular \\\\ \\mbox{person} & third \\end{bmatrix} \\end{bmatrix}", "d10183f0514d21272b6128bf9b4d2df8": "2+\\sqrt{2}\\,", "d101f6d16974b8b3cebb3fac658e5ace": "X_n\\ \\xrightarrow{L^r}\\ X", "d102298ac6fb16e5e0bce71c85cd8623": "\n\\mathbf{L} = \\mathbf{r} \\times \\mathbf{p} = \\mathbf{r} \\times \\mu \\frac{d\\mathbf{r}}{dt}\n", "d1028da68fc88899f0d83c047d2bcc3f": "(0.85 \\cdot 1 + 0.14154 \\cdot 0.4 \\cdot 1) \\cdot (1+0.1030) = 1", "d10295245fe1123899c1f8f15361b705": "M_M", "d102fffa3f9c04f857e3caf29443a4b5": "\\varphi^{M}=\\psi^{M}", "d103291f5ac3789c91d7328128023ff8": "=-\\left ( \\frac{1}{3}\\right ) \\left ( \\frac{1}{5}\\right ) \\left ( \\frac{2}{7}\\right ) \\left ( \\frac{3^2}{23}\\right )", "d103bdf6d923bf6b1b584cf0178a4de5": "\\scriptstyle\\sqrt{\\tan ORH}", "d103f14fb8e56490811d73d1171d817c": "[2.50] = 2", "d103f92fc55ee4462683189e037e4e96": "\\mathcal{Z} \\left\\{ u(t - n T)f(t - n T) \\right\\} = z^{-n} F(z, m).", "d10448e61dd1f4b60a15a3790a4f319e": "\\frac{\\mbox{Net Operating Income}}{\\mbox{Total Debt Service}}", "d104531ea0abdfb812796c91038b018d": " (r_1,\\ \\vec{v}_1) + (r_2,\\ \\vec{v}_2) = (r_1 + r_2,\\ \\vec{v}_1+\\vec{v}_2)", "d10516830d6d773b62fc656bc5ba87e0": " H(x_0,\\phi(x_0),D\\phi(x_0),D^2 \\phi(x_0)) \\leq 0 ", "d10534281470066404dd8ed5de23e187": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(x,y\\right) & = 2 + \\left(x-2\\right)^{2} + \\left(y-1\\right)^{2} \\\\\n f_{2}\\left(x,y\\right) & = 9x + \\left(y - 1\\right)^{2} \\\\\n\\end{cases}\n", "d105716566a0354ca694cb90552b5603": "\\ N=51 ", "d105937beb07d0e1877a14d80172cb5c": "\\left|\\alpha q_n - p_n \\right| \\neq 0", "d106011bbf68766269873e6d4bce701d": "\\Omega(M)", "d10653246b8510daf15d33d41141919f": "f:X\\rightarrow Y", "d1066b3668e06bbebaa1f331ac122bc7": "\\,^{z_{14} = x_{14} y_1 + x_{13} y_2 - x_{16} y_3 + x_{15} y_4 - x_{10} y_5 + x_9 y_6 - x_{12} y_7 + x_{11} y_8 + x_6 y_9 + x_5 y_{10} - x_8 y_{11} + x_7 y_{12} - x_2 y_{13} + x_1 y_{14} - x_4 y_{15} + x_3 y_{16}}", "d106cbc3e529df9fac26034319ff6465": "r_2 = s(\\alpha)", "d106f039aaf186478a7a839057022678": "T=\\frac{1}{1+ma^2V_0/ 2\\hbar^2}", "d10701d172987023bdf05c9bf2bbfbee": "R=\\prod_{i=1}^nR_i", "d1071c81db0c780fbeb97f840a72a7ee": "L_\\delta (a) = \\delta^2(\\sqrt{1+(a/\\delta)^2}-1).", "d10722d990e4f4419b3e65d575b72f6e": "\n\\mathbf{G} =\n\\left(\\left.\\begin{array}{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 0\\\\\n0 & 0 & 0 & 1\\end{array}\\right|\\begin{array}{cccc}\n0 & 1 & 1 & 1\\\\\n1 & 0 & 1 & 1\\\\\n1 & 1 & 0 & 1\\\\\n1 & 1 & 1 & 0\\end{array}\\right)_{4,8}\n.", "d1075bacbace3567de07e2b9060f633b": "\n\\mathbf{b_{5:5}} = \\begin{pmatrix} 1.0 \\\\ 1.0\\end{pmatrix}\n", "d10780b20a718eb454b72d2910fced87": "z\\log \\Gamma(z)-\\log G(1+z)=-z \\log\\left(\\frac{1}{\\Gamma (z)}\\right)-\\log G(1+z)=", "d107ad11bf1aebc460ee2c540228054a": " L = | k_1 \\rangle \\langle b_1 |, ", "d107de586ad69d7cf3b43df7315f254c": "\n \\theta = \\cfrac{1}{3}\\arccos\\left[\\left(\\cfrac{r}{q}\\right)^3\\right] ~.\n ", "d10855113066e5245ec1373ee45816c6": " a = b. \\, ", "d1087c896de2748548411c8b51e227a5": "a_i\\to a", "d108930fc08e215760da0c09e4ee2adf": "+ \\!\\,", "d108967258a70984c82cd0f9b4b5fab0": "\\begin{align}{\\color{white}\\dot{{\\color{black}\n\\sin\\mathrm{gd}\\,x}}}&=\\tanh x ;\\quad\n\\csc\\mathrm{gd}\\,x=\\coth x ;\\\\\n\\cos\\mathrm{gd}\\,x&=\\mathrm{sech}\\, x ;\\quad\\,\n\\sec\\mathrm{gd}\\,x=\\cosh x ;\\\\\n\\tan\\mathrm{gd}\\,x&=\\sinh x ;\\quad\\,\n\\cot\\mathrm{gd}\\,x=\\mathrm{csch}\\, x ;\\\\\n{}_{\\color{white}.}\\tan\\tfrac{1}{2}\\mathrm{gd}\\,x&=\\tanh\\tfrac{1}{2}x.\n\\end{align}\\,\\!", "d108d30d779da74a909a7528c9060146": "\\tilde{y} = y - \\hat{y} = y - A\\hat{x}_1", "d1093b08e76cfb6b47ba3ab5b6e888d2": "\\mathrm{Sc}", "d1094332b246889b0697d3721ebf4ffe": "M_x\\ ", "d1094e6f1cd520abee3bc24149304302": " \\operatorname{E}(g(T)) = \\sum_{t=0}^n {g(t){n \\choose t}p^{t}(1-p)^{n-t}} = (1-p)^n \\sum_{t=0}^n {g(t){n \\choose t}\\left(\\frac{p}{1-p}\\right)^t} .", "d10956cffa0a0b5fa7de7f56f41c6f8f": "c>\\sigma_a", "d1095deae7f4cc234e6c37fc150e3281": " x_t=\\mu+{\\sigma\\over\\sqrt{2\\theta}}e^{-\\theta t}W_{e^{2\\theta t}} ", "d1096f8fafdd8d804931349fb17847ae": " \\left(\\arccos(x)\\right)' = \\frac{-1}{\\sqrt{1-x^2}}", "d109885ac2f617914c7c784c03627c08": " {{e^*}_w} = (1.0007 + 3.46 \\times 10^{-6} P) \\times (6.1121) e^{\\left(\\frac {17.502 T} {240.97 + T}\\right)}", "d109afa8e1ffa5b147e55cc1db964db1": "\n\\mu = \\frac{F E_{m}}{RT}\n", "d109e7f7ea3444eaf8ad5f2f5c7190b7": "A_m(4,2) = 1, 2, 9, 52, 340, 2394, 17710, 135720, 1068012, 8579560,\\ldots ", "d10a32a8150db3e965210a5fcddc9d5a": "\\int \\frac{1+\\cos^2 x}{\\cos x + \\cos 3x} \\, dx.", "d10a94866a9f8ec92e33c3aafcb9fd58": "p<0", "d10aabb794635f6a64c125193467bb78": " \\ell_{x+t} = (1-t) \\ell_x + t \\ell_{x+1}, \\qquad 0 \\zeta_0", "d116e7ad16756584f119a7968c0c63ba": "\nf(x)=\\begin{cases}\n\\sqrt{1-x^2}&\\text{for }x\\in[0,1],\\\\\n-\\sqrt{1-x^2}&\\text{for }x\\in[-1,0).\n\\end{cases}\n", "d1173eb6a717de123b019dbdb1e57ae8": "X \\times_S V_i", "d11777361d42dd51442f8cbcff0a251d": "I,J\\subset \\mathbb{R} ", "d1177afb580d4d8b2969940a2693acee": "G=c=1", "d117ae311f45f97ee1d34ecbc52ba06d": "p(x|\\alpha, \\beta) = \\frac{\\beta^\\alpha\\, x^{-\\alpha-1} \\exp(-\\beta/x)}{\\Gamma_1(\\alpha)}.", "d117ca43fffd7e42c14e05ea4e23b48b": "z\\mapsto \\frac{az + b}{cz + d}", "d1180e41204d29572bb55ff84cfaf057": "\\overline{F}(x) = 1-F(x)", "d1185c89fac1177e9e9089661bff2667": " = a_1 (1,0,0) + a_2 (0,1,0) + a_3 (0,0,1) \\,", "d1187616daee05a45d0324820c11eee2": "R_1(f)", "d118a89b436a5229633004a6c7959746": "\\{x_i\\}_{i=1}^\\infty", "d118b394e54ad9e21fb24489a4ccb244": " \\langle f, g \\rangle = \\int_0^\\infty f(s) g(s) \\, ds.", "d118d8b751bf0f14dc6a2a901e143e7f": " \\frac{2}{U_{2n}} = \\frac{1}{u_n} + \\frac{1}{U_n} . ", "d119680a84fe0028e73803de2d6dc44b": "N 1 \\}", "d11cf3398686c09afc4ccec18e9011fc": "\n{{\\partial ^2 \\hat g} \\over {\\partial \\theta ^2 }}\\,\\,\\, = \\,\\,\\,{k \\over {32}}\\left[ {9\\cos \\left( {\\mu _\\theta } \\right)\\,\\,\\, - \\,\\,\\,\\cos \\left( {2\\mu _\\theta } \\right)} \\right]", "d11d2e899b12881bb7f3e1f0434ae5f1": " \\det(M) = \\det(D) \\det(A - BD^{-1} C)", "d11d56b49525e679b4db5ab2e275b55b": "\\mathbb{P}^n", "d11d5754ed5c467fa0493f13da27e5a3": " \\lim_{\\lambda \\downarrow 0} \\int_{\\mathbb{R}^n} f(\\lambda^{-1}(z-x)) |D\\chi_E|(z) = \\int_{T_x} f(y) \\, d\\mathcal{H}^{n-1}(y)", "d11d9065b2861c90c3c9600fb4802d98": " I=A_{eff}\\left( \\frac{q^2 m_o}{8\\pi h m_{eff}} \\right )\\left ( \\frac{1}{t\\left ( E^2 \\right )} \\right )\\left ( \\frac{\\beta^2 V^2 }{\\phi d^2} \\right )e^\\left (\\left ( \\frac{\\left ( 8\\pi \\right )\\left ( 2m_{eff}q \\right )^\\frac{1}{2}}{\\left ( 3h \\right )} \\right )\\left ( \\nu \\left ( E \\right ) \\right )\\left ( \\frac{d}{\\beta V } \\right )\\left ( \\phi^\\frac{1}{3} \\right ) \\right ) ", "d11db1fe0d3eb76020813dc4bcabd976": "each truck type must go back and forth(miles are already X2); ", "d11ddc10d44b9290f679ce0421a0bbd6": " \\overrightarrow{uv} \\in A ", "d11df2f1f0d268179464a477568489a1": " W = \\mu + \\sigma(Y^{-1}-1)^\\gamma, \\qquad \\sigma>0, \\gamma>0,", "d11e397eb1c21063374ee2851f7b7b99": "\\phi(g(x) + \\langle f(x) \\rangle) = (g_1(x) + \\langle p_1(x) \\rangle, \\ldots, g_s(x) + \\langle p_s(x) \\rangle)", "d11e437a87d4f324bc5cc827ad9520a8": "\\operatorname{Spec}(R),", "d11e727f829da89e6967389d3075de3f": "SBV(\\Omega)", "d11eadf6fa3a3fb3262b2c1af271da8e": " G(z) = \\sum_{n\\ge 1} \\left(\\frac{1}{|C_{2n}|}\\right) g(z)^{2n} = \n\\frac{1}{2} \\log \\frac{1}{1-g(z)^2}.", "d11f390cf9e1c0e83b9611ed659422c5": "\\rho = 1/(2\\pi)", "d11f4a864a207d7e979af7f40c96d838": "\n{\\partial E\\over\\partial t}+\n\\sum_{i=1}^3 {\\partial((E+p) u_i)\\over\\partial x_i}\n=0,\n", "d11f56c79eeeedb6d2feaab67042e23a": "E_\\text{surface} ", "d11f69525eb03399576fc01a6b6c97a6": "p \\mid i-j \\lor p \\mid m", "d11f81c713c7c7d4a82fd5b3747aa6b4": "\\theta \\in [0,2\\pi],", "d11fee2755bec48af93a0545a3fa6c67": "\\Rightarrow {x} = \\frac{P r_2}{r_1+r_2} \\,\\!", "d1204764192d4059d690e17100055b75": "Y\\subset X", "d12056a2c34807f5faaacae2140d44be": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = \n \\int_{-h}^h x_3~\\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ \\varepsilon_{12} \\end{bmatrix}\n dx_3 = -\\left\\{\n \\int_{-h}^h x_3^2~\\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}~dx_3 \\right\\}\n \\begin{bmatrix} w^0_{,11} \\\\ w^0_{,22} \\\\ w^0_{,12} \\end{bmatrix}\n", "d12104061868e2d05b0fcc2d2e77f64e": "\nx_{s} = M + N r_{s}\n", "d1213fd34dd8851be637f1290984ee66": "x^5 -2 = 0\\,", "d12182785b6a784c2190a522cbb4162f": "\\le \\|x\\|^2 + 2|\\langle x, y \\rangle| + \\|y\\|^2", "d12182b3ab9f1eaf4cc6b856972a6824": "{}_{\\ 92}^{232}\\mathrm{U} \\xrightarrow{\\ \\alpha\\ } {}_{\\ 90}^{228}\\mathrm{Th}\\ \\mathrm{(68.9\\ a)}", "d1219420437238f2a877215f1daac6d5": "F_m(x) = F_{m-1}(x) + \\underset{f \\in \\mathcal{H}}{\\operatorname{arg\\,min}} \\sum_{i=1}^n L(y_i, F_{m-1}(x_i) + f(x_i)),", "d121b8a1cb097c260792f856ca7c5d83": "\\forall r, \\exists N, \\forall n > N, x_n \\in H_r", "d1225d5e9abf818f483baf3479c16e32": "\\ ND_e", "d122a602480a767ecfd20bb464d7804f": "\\scriptstyle\\omega^\\dagger \\omega \\;=\\; 2 E \\,", "d122bd6e0a79247e2f8fe1f5b82dd86c": "\\Delta_{S^{n-1}}f(x) = \\Delta f(x/|x|)", "d12312753a3790ecf7cef9687857d3c7": "\\,K_i=(s_i,t_i,d_i)", "d12373fdcc3c1afb681c5273cf09e621": "F(p) = - \\left. \\frac{\\delta \\langle E(s) \\rangle} {\\delta s} \\right\\vert_p.\\,", "d12396876a7747669f390c7db7b97ab6": "\\frac{L}{L_{\\odot}} \\approx 1.5\\left(\\frac{M}{M_{\\odot}}\\right)^{3.5} \\qquad (2M_{\\odot} < M < 20M_{\\odot})", "d123d9d801ba27893c0278cfcebba496": "p > 2", "d123e527f6960fe7c26efc0acfed6727": " \\sum_{jN+1}^{(j+1)N} Y_i=\\alpha+ \\beta \\sum_{jN+1}^{(j+1)N} X_i+ \\sum_{jN+1}^{(j+1)N} u_i ,", "d1240968cb73778d30d62e100e527098": "\\delta>1", "d12418b65511ebd9017cddc125d5b3e1": "W(t_0,t_1)\\eta = x_1 - \\phi(t_0,t_1)x_0", "d12429ddfa52119d9ffcb418e0910bf7": "Eq.1", "d124e4914315e43cd91aa0bca9606a57": "\nG_n(\\omega) = {1 \\over \\sqrt{1 + \\epsilon^2 R_n^2(\\xi,\\omega/\\omega_0)}}\n", "d12507464d0c069ddc355168d2c76f14": "P(h) \\ge \\varepsilon \\quad \\Longrightarrow \\quad S\\cap h \\neq \\varnothing.", "d1254e15ab50bd96f8df4f1ef78d0af7": " F\\;=\\;C\\;-\\;P\\;+\\;2 ", "d12567c871980c4eeb8880e3e72ce7f9": "\\nu=0 \\Leftrightarrow \\lambda=0", "d125898ecca64e2eae80ef38d613b0fb": "\\left(\\mathbf{w}_i, \\frac{\\partial\\mathbf{v}_j}{\\partial t}\\right) = -(\\mathbf{w}_i, \\mathbf{v}\\cdot\\nabla\\mathbf{v}_j) - \\nu(\\nabla\\mathbf{w}_i: \\nabla\\mathbf{v}_j) + \\left(\\mathbf{w}_i, \\mathbf{f}^S\\right).", "d125922725d948d9c5bff7a5a2d8e9c2": "\\scriptstyle B(x, r)", "d125b6845ddd66efe4ed79f3f46e29cc": " \\xi \\subset \\mathbb{R}^d", "d125fe1cd90151983bb83054d8eae2e2": "\\langle \\text{D}, \\text{R}\\rangle", "d12609da7dc3540a91d43fa2693666c4": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon r} \\sum_{l=0}^{\\infty}\n\\left( \\frac{r^{\\prime}}{r} \\right)^{l}\n\\left( \\frac{4\\pi}{2l+1} \\right)\n\\sum_{m=-l}^{l} \nY_{lm}(\\theta, \\phi) Y_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "d126b1ae5aa35880a27db1fa66b4889a": "\\mathrm{QUANTUM \\;DYNAMICS}", "d126ec59ef8495a9b009a84956e28c70": " I_P = \\frac{mgr}{\\omega_n^2} = \\frac{mgrt^2}{4\\pi^2},", "d126fd017965332a6d367f0309078170": " \\scriptstyle {\\color{white}..\\color{black} Do_1\\;\\; Do\\#\\;\\, Re\\;\\, Re\\#\\;\\, Mi\\;\\; Fa\\;\\; Fa\\#\\; Sol\\;\\, Sol\\#\\, La\\;\\; La\\#\\;\\; Si\\;\\, Do_2} ", "d127a54d5db1bad9b22acae932d0e79b": " R_0(a, b) =\n \\begin{cases}\n 1 \\mbox{ } , \\mbox{ } \\mbox{if } a = b \\mbox{ } , \\\\\n 0 \\mbox{ } , \\mbox{ } \\mbox{if } a \\neq b \\mbox{ } .\n \\end{cases}", "d1282c9fce3441f02b5f0db6260abb10": "\\limsup_n d(X|n)=\\infty", "d128543ef23d35f61cfb6cb82949ff4d": "x_n \\rightarrow x", "d1288ed37c0240dae6ef04eb933e1336": "\\chi_G", "d12912f48f77ec994c34e0e8defd6a6b": "\\displaystyle{Q(B(a,b)c)=B(a,b)Q(c)B(b,a).}", "d1297c40044e71e596ba9d1b08c9fa87": "\\nabla_{\\mathbf x}\\mathbf U\\,\\!", "d12a0aadcfba3c64c44deb4d6bcc166a": " \\beta ", "d12a0d0bebd0b8b2e837a382eda4cfbb": "x_n,y_n", "d12a97db1b1e9eb01b2d6e331df24bff": "M_{xy} = M_x + iM_y \\text{ and } B_{xy} = B_x + iB_y\\, ", "d12aa7d756aaa692f4a7551052505e62": "\n\\begin{align}\nA &= \\langle a-b^2c^{-1}\\rangle + \\langle c^{-1}\\rangle^{-1} \\langle bc^{-1}\\rangle^2 \\\\\nB &= \\langle c^{-1}\\rangle^{-1} \\langle bc^{-1}\\rangle \\\\\nC &= \\langle c^{-1}\\rangle^{-1} \\\\\nD &= \\langle d^{-1}\\rangle^{-1} \\\\\nE &= \\langle e\\rangle \\\\\n\\end{align}\n", "d12b78ee70283bae010e17b70b8335b2": "\\nu:\\mathbb{P}^1\\to\\mathbb{P}^n", "d12b9d19d3523a08ad8f32dae0139b4e": "|\\psi\\rangle=\\alpha_0|0\\rangle+\\alpha_1|1\\rangle", "d12bcc2e0ed4357bed182cbbf842c31c": "\\displaystyle S = \\sum\\nolimits_{\\alpha} D^{\\alpha} g_{\\alpha}", "d12bcf813c077364b62717f5b3361e14": " \\left(\\arcsin(x)\\right)' = \\frac{1}{\\sqrt{1-x^2}}", "d12beced200f68b56be83dd3235f7ae8": "(A,e)", "d12beffb56f6fd1c4d0a0ddb42524e4d": "\\kappa\\ln(R/a)", "d12bf4ea2e1fc845acddcd13fc0fe031": "a_n \\leq b_n", "d12c631775f438189eb84d5b15d13f35": "\\Delta E=\\Delta E^{o}- {\\mbox{0.05916 V} \\over \\mbox{n}} \\mbox{log Q}\\,", "d12c73fbb029d5b4f010721c4f9cb311": "\\sqrt{2}=1.4142135623731...", "d12c798def566a9f78f06f1b1a001eb7": "U(R)", "d12cee2b9d5be580a2ed49e26a7a78e5": "\\scriptstyle (u,v)\\,", "d12d126b89b71ef744dfc6428a05064e": "D_1\\left(E\\right) = \\frac{1}{c_k} ", "d12d2839a77895fdb0f4979af4266808": "{\\mbox{Div}}^0(E) \\rightarrow E\\,", "d12d4a64ea601e50c8d7d5ec5da819cc": "\\neg(p \\and \\neg q)", "d12d531c4e8f684c2059cce87b1308d6": " C_{(s)} \\Gamma_\\text{chir} C_{(s)}^{-1} = \\beta_{d} \\Gamma_\\text{chir}^T = s \\Gamma_\\text{chir}^T ", "d12d920782032e0c4e1a83a7554d219d": "X,Y\\subseteq O", "d12db5a4126359f24728d1ab23617c60": "(7)\\quad \n\\tilde\\gamma\\,=\\,\\gamma^{\\langle1\\rangle}+\\gamma^{\\langle2\\rangle}+2\\int\\rho\\,\\Big\\{\\,\\Big( \\psi^{\\langle1\\rangle}_{,\\,\\rho}\\psi^{\\langle2\\rangle}_{,\\,\\rho}-\\psi^{\\langle1\\rangle}_{,\\,z}\\psi^{\\langle2\\rangle}_{,\\,z} \\Big)\\,d\\rho +\\Big( \\psi^{\\langle1\\rangle}_{,\\,\\rho}\\psi^{\\langle2\\rangle}_{,\\,z}+\\psi^{\\langle1\\rangle}_{,\\,z}\\psi^{\\langle2\\rangle}_{,\\,\\rho} \\Big)\\,dz \\, \\Big\\}\\,.\n", "d12e5939fc0ede82177143ea53a94796": "\\theta = 0.003\\lambda", "d12e7c9853dd70d90963cfe75ebe554d": " I_t = 1000\\times\\frac{\\sum_{i=1}^N Q_{i,t}\\,F_{i,t}\\,f_{i,t}\\,C_{i,t}\\,}{K_t\\,\\sum_{i=1}^N Q_{i,0}\\,C_{i,0}\\,} ", "d12e9d73bdd4ea55e34b6ef63668a0d3": "\\,F_t+F_x X+X(-t,F)=0,\\qquad F(0,x)=x.", "d12eb3c668f4d41e0cc00020e78b49fc": "B(X, Y; Z)", "d12ef5c2fafc472ceee4620242b2cbd8": "f(t_0, y_0)", "d12f1b651f64bfadc56b80d5e929a8a3": "((p \\wedge (p \\rightarrow q)) \\rightarrow q", "d12f2c848d40cb6c7fdbc7789f8ef8a7": "ds^2 = g_{\\alpha\\beta}\\, dx^\\alpha\\, dx^\\beta + 2g_{0\\alpha}\\, dx^0\\, dx^\\alpha + g_{00} \\left ( dx^0 \\right )^2,", "d12f3dea01e8dbdf2e162193a66cd85a": "\\ x[n]", "d12f4d6ee8ca24f8419c287a44956bc1": "\\left(\\sqrt{\\frac{2}{5}},\\ \\pm\\sqrt{6},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "d12f6c63ad9deb9548e0fe672bf383b1": "G = E_{antenna} \\cdot D", "d12f6ed914ada8be17174d6de1c62090": "{d\\over dt} (g^T g)= \\dot{g}^Tg +g^T\\dot{g}= g^T(A^T +A)g=0.", "d12f74b9917ea5733c1f7e56f6d5d8ce": "M = \\frac{v}{a},", "d12f76675461a183eb80bed4f677c89c": "L(n,k)=\\left\\lfloor\\begin{matrix} n \\\\ k \\end{matrix}\\right\\rfloor.", "d12f791999da53a213b8e7d75343a8c3": "\\forall P[[P(0) \\land \\forall k \\in \\mathbb{N} (P(k) \\Rightarrow P(k+1))] \\Rightarrow \\forall n \\in \\mathbb{N} [ P(n) ]] ", "d12fbc4a9c480c58156079b5e61f80e4": "\\frac pq = \\frac rs \\leftrightarrow ps=qr", "d12fd54c8849732ddeb15675f65ab663": "\\ln (z) = 2\\sum_{n=0}^\\infty\\frac{1}{2n+1}\\left(\\frac{z-1}{z+1}\\right)^{2n+1}.", "d1300b3587affd01437003fd3af9d5d0": "S_2(s)=S_2(\\sin^2\\theta)=\\sin^2(2\\theta)=r(s)", "d1305099956d75ffa9a1b8fe64b448ee": "j_{12}^\\mu=\\left(\\frac{\\partial}{\\partial (\\partial_\\mu\\phi)}\\mathcal{L}\\right)(Q_1[Q_2[\\phi]]-Q_2[Q_1[\\phi]])-f_{12}^\\mu.", "d130ea41fe3e3459a33f8cb779822c21": "\\lfloor(Q-x_0-1)/r\\rfloor", "d13115f47faca5c5514078d0d2bff6ef": "E_f = \\frac{V + 8\\pi P_s a}{d + \\epsilon_f\\left(2a\\right)}", "d131460869096b13a773942008aa6f91": "\\tfrac{34}{35}", "d13153afb576d23c2f5e6f873cf9f4c9": "\\sigma_{CC}=E(\\cos^2\\theta)-E(\\cos\\theta)^2\\,", "d1315b56f094a0a8a6f65e017378c4f2": "\\operatorname{Aut}(G)=\\langle \\sigma\\rangle=\\{1,\\sigma\\}", "d131d82a541ea3502b19f0f49bf9fa22": "s_1 = (1,2), s_2 = (2,3), \\dots, s_{n-1} = (n-1,n)", "d1321e0722cca3b2bed565cc247ee82a": "\\rho_{a / b}(R \\setminus P) = \\rho_{a / b}(R) \\setminus \\rho_{a / b}(P)", "d132222c84c1963dc8dd2a7ffb60c268": "(q \\to r)", "d13235c51710598798f34f0d5146599e": "\\left\\{ \\frac{x - 0.5}{0.5-0}\\cdot\\frac{x-1}{0-1} ; \\frac{x-0}{0.5-0}\\cdot\\frac{x-1}{0.5-1}; \\frac{x-0}{1-0}\\cdot\\frac{x-0.5}{1-0.5} \\right\\}", "d132492130d75b28f6aaa8a739708b30": " M = \\frac{ N - B - C }{ N } ", "d13382f5878491b221ed1ee24861b1c6": "\\lim_{n\\to\\infty} \\frac {1}{n} \\sum_{k=0}^n \n |\\mu (A \\cap T^{-k}B) - \\mu(A)\\mu(B)| = 0.", "d133b7303d97acc214c2ef3b1fdf48e8": "T=\\frac{4 Z_0 Z_1}{(Z_1 + Z_0)^2}", "d13404c477b35c98961785d6aa96ba05": "\\hat{X}_{1}", "d13415d8e40ea27d55afd2944368455f": "dX = \\mu(X,t)\\,dt + \\sigma(X,t)\\,dW^Q,", "d1343571cb5ae5201a9d8913af6c72c6": "\n E_{ij} := \\frac{1}{2}\\left[\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i} \n + \\frac{\\partial u_k}{\\partial x_i}\\,\\frac{\\partial u_k}{\\partial x_j}\\right] \\,.\n ", "d1343980b14f9f41056ef24d4de4da41": "\\mathbf{x_i^T}", "d134472a77041287eb81d9c4e0111e16": "R = \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2}", "d13489778e71320d896f669851cc4781": "\\vec{p}_{1} =\\frac{m_{1}}{M}\\vec{P}+\\vec{p}", "d1349b46abecd095968b382e7ef566e3": " (\\ln f)_{,\\eta} = \\frac{(\\ln \\alpha)_{,\\eta\\eta}}{(\\ln \\alpha)_{,\\eta}} + \\frac{\\alpha}{4 \\alpha_{,\\eta}} \\operatorname{tr} (g_{,\\eta} g^{-1} g_{,\\eta} g^{-1}) ", "d134ff353de964493679e65946734e1c": "V_L = I_L R_\\mathit{DCR}", "d13508b31bdeba309fe45ebcf9bfeab4": "\\mu \\left( (1 - \\lambda) A + \\lambda B \\right) \\geq \\mu (A)^{1 - \\lambda} \\mu (B)^{\\lambda},", "d1351b58e89db370dff86d5e123fb18e": "M+1", "d13551b81b83db9d104407d1e1841776": "R=\\frac{\\eta_j}{c_p}\\frac{C_L}{C_D}\\int_{W_2}^{W_1}\\frac{dW}{W}", "d1359645b53f942100626fe82bf170f0": " \\operatorname{get-lambda}[F, G\\ V = E] ", "d1359d0bf3b25d90fa8675eebe183f70": "|\\psi\\rangle = \\sum_j |j\\rangle\\,", "d135e4127b9643be8e0209ff2bb8c23a": " B \\sub X", "d13612df3f53ba180c081de590ad36ee": "v_{QP}\\in[0,v_f]", "d13625eeb26f4f65dd6cd018cc7b6ff3": "\\mathbf{v}(\\mathbf{x},t)", "d1375550c3bd0d304379389fb4dee87b": "\\int_0^\\infty e^{n(\\ln y-y)} dy \\sim \\sqrt{\\frac{2\\pi}{n}} e^{-n}\n\\left(1+\\frac{1}{12 n}\\right)", "d1377dc15bf1a7981421dd3cbdcb3389": "\\hbar G/c^3", "d137a27c46984207f6a059f2d596e28b": "\\frac{\\partial \\overline{u_i}}{\\partial x_i} = 0,", "d137aba004822e3783f694305e05a6ab": "\\parallel", "d137b1d792e107759f9ef03ba28ce668": " (W^\\tau(t): t \\geq 0)", "d1380a6948f97a2a36c926481a628416": "\\mathbf{d},\\mathbf{f}", "d138201a6e43b5446fcbad8a8f0b9ad9": "x^3+px^2+qx=N, \\, ", "d1382e066206339e9d7f173fcab35358": "\\sum_L (2d)^L (\\varepsilon)^L ", "d138366881e3c7b9c593b6e457b07ebd": "C(x_i,x_j)=C(x_i+h,x_j+h)\\,", "d1388d5096f23ca0212b7e3d1d060ce9": "p_{B(G)}(\\lambda) := \\sum_{k=0}^n (-1)^k f_k (\\lambda-1)^{n-k},", "d138dc5ab6367ae2a57afbee20502534": "\\forall \\varepsilon>0\\;\\exists n_0\\;\\forall n>n_0\\;\\left|{f(n) \\over g(n)}-1\\right|<\\varepsilon", "d138f12b80f00f54cd25a6d9dcd453ed": "\\frac{1}{\\rho}\\frac{\\partial \\tau_x}{\\partial z} = -fv,\\,", "d139093091d03b52f1241ecccea61989": "\\mathbf{u}(x,t)", "d139744e62efee0f0517eb28752ebf62": "(\\mathbf{v}, \\mathbf{v} + d\\mathbf{v})", "d13996613d44cde0a0c6c977f6522a99": " D[q] = \\_ ", "d13999682cf85d63001b741e42f07d02": "P_C(k)", "d139ac4a0978ce2880b84e2016f9d32a": "a={\\ell\\over 1-e^2}.\\,", "d139b8de3423254366efb0a6576f6a85": "r_\\mathrm{O} = \\begin{matrix}\\frac {1/\\lambda+V_\\mathrm{DS}}{I_\\mathrm{D}}\\end{matrix} \\approx \\begin{matrix} \\frac {V_E L}{I_\\mathrm{D}}\\end{matrix} ", "d139dc80250b83f24516117ba83af840": "H=\\frac{t^2}{U}\\sum_{\\langle i,j\\rangle}\\left[4\\left(\\overrightarrow{S_i}\\cdot\\overrightarrow{S_j}\\right)(\\tau_i^{\\alpha}-\\frac{1}{2})(\\tau_j^{\\alpha}-\\frac{1}{2})+(\\tau_i^{\\alpha}+\\frac{1}{2})(\\tau_j^{\\alpha}+\\frac{1}{2})-1\\right]", "d13a54a9a8bb5cab5d9180d9e00fc454": "(c_0, \\dots, c_{L-1}), x_L", "d13aaff38a8936b76bb896815f2907cf": "F(x) = \\int_{0^-}^\\infty K(x-y)F(\\text{d}y) \\quad x \\geq 0", "d13b11ee36df3fec4612d6442d0d93b3": "P(D)=\\sum_i P(D|H_i)P(H_i) \\;", "d13b50bca98ad832f53e242ec336c1aa": " T^{\\alpha \\beta} {}_\\gamma = g_{\\gamma \\lambda} \\, T^{\\alpha \\beta \\lambda},", "d13b6def36c092d8ed8b204b6ab113ca": "1/M*cm", "d13b716aa7ce91860e5aaf4f348f4b1d": "\\langle\\Phi(x),\\Phi(y)\\rangle=\\langle x,y\\rangle", "d13b793d1802da471083e9b652f7f53e": "\n[\\wp(z)]^3|_{z=0}\\sim \\frac{1}{z^6}+\\frac{9}{z^2}\\sum \\frac{1}{(m\\omega_1+n\\omega_2)^4}+15\\sum \\frac{1}{(m\\omega_1+n\\omega_2)^6}.", "d13b9d6efa751886fe2fa0e2fd68e3e3": "I(f) = a_0 f(x_0) + a_1 f(x_1) + \\dots + a_n f(x_n)", "d13bb6e06e2bff69fabb7b5cc91f9e1d": "N(d_1), N(d_2)", "d13c00599883e0e93a8d5928955f50f8": " K_{ij} = 1 ", "d13c19f9f776c97cdaf295c8b57db5ca": " \\Delta : K[G] \\to K[G] \\otimes K[G] ~\\text{by}~ \\Delta(g) = g \\otimes g ~ \\text{for all} ~g \\in G_1 ", "d13c3326655f7d2dd6388a6cfb0fc576": "\\operatorname {volume} (f(S)) = \\sqrt{\\det(A^\\mathrm{T} A)} \\times \\operatorname{volume}(S).", "d13c4b5ba7dbc7348294fa2c01863fef": "f^{n_k}(x)\\rightarrow y", "d13cfd4782ab73037e1bce36a4cbed79": " \\Lambda_{\\mu'}{}^{\\nu}", "d13d58e113dfed6aadc13a20ee3f77c9": "j^a = \\left[c\\rho, j_x, j_y, j_z\\right]", "d13d731b2b0f10f505a8c2ead7a5f533": "(\\mathbf{a\\times b})_2 = a^3 b^1-a^1 b^3\\,,", "d13d80051cca041266513536512b6389": "k!/k^k > \\tfrac{1}{e^k}", "d13dd9164f84e41b09941f9e2e24790e": "\\left|e^-,\\, +\\frac12\\right\\rangle = \n\\begin{bmatrix}1\\\\0\\\\1\\\\0\\end{bmatrix}", "d13ddb9d87fc59038a3aff9450bc21f7": "D_0,...,D_{n-1} \\in GF(m)", "d13de5ed73dbb4362f6cc38d19442e9d": "P(\\textrm{spike}) \\propto f(K\\mathbf{x}),", "d13e1927719868257eceab1933e35b8f": "\\varrho : \\mathcal{L}", "d13e196b00164280686a172e8a74aa92": "S_{t,\\text{ alternate}} = \\alpha \\cdot Y_t + (1-\\alpha) \\cdot S_{t-1}", "d13e2fd0d98f8a22650094672ffdf2e4": "\\# X", "d13e5b523b739e864c8aad3ea83b3ea6": "\\begin{align}\n ((P \\and Q) \\or (R \\and S)) \\leftrightarrow (((P \\or R) \\and (P \\or S)) \\and ((Q \\or R) \\and (Q \\or S))) \\\\\n ((P \\or Q) \\and (R \\or S)) \\leftrightarrow (((P \\and R) \\or (P \\and S)) \\or ((Q \\and R) \\or (Q \\and S)))\n\\end{align}", "d13ec66477cfc80586cc3f5a185dd5fc": "\\ n = \\lambda^2", "d13ee06731d53377983756d466c4b801": "\\operatorname{hypg}(x;n,m,N)", "d13eefc1f9afff0f89cd9df78c9a1ea9": "\\sum_{k=0}^{n-1} z^k = \\frac{z^n - 1}{z - 1} = 0.", "d13f453d5a25ba83fd2f3ecc74880538": "y=g(x)", "d13f48b8f7f773581b4d1b43b3fb0f7d": "\\theta \\in \\Theta \\,\\!", "d13fb2d2d105197510715cbb92a470e5": " Precursor~molecule + ATP \\rightleftharpoons {} product~AMP + PP_i", "d13fe82a7a81510b6f715e8926cf8f04": "\n\\begin{align}\n N &= \\left\\lfloor\\frac{V}{M}\\right\\rfloor\\\\\n A &= V - MN\n\\end{align}\n", "d1404058cabf89682d6a6f5a3b795d47": "\\Delta u^*(x^*)=\\frac{R^{4}}{|x^*|^{n+2}}(\\Delta u)\\left(\\frac{R^2}{|x^*|^2} x^*\\right).", "d1404832fc8633a6aae9d4f17b5a7951": "BFD=F- {e \\over n}.", "d1407e89814671e03e84d9907fafdf9c": "Tr_A\\rho_{AB}=\\rho_1", "d140bd020b5d769befeea4d93b2a17cd": "\\omega/\\omega_c", "d140bfdd3905a3e28963a94b7ef7bdd6": "{p(\\mathbf{\\Sigma}|\\mathbf{X})}", "d140c592474fe9e3e07b9e5ceb33d8d6": "\\operatorname{E}(3) = \\frac{1}{7+1}\\sum_{i=0}^{7} i = \\frac{1}{8}(0 + 1 + 2 + 3 + 4 + 5 + 6 + 7) = \\frac{28}{8}", "d140da883ba03981a9142d29b33029e7": "\\int_0^\\infty \\frac{\\log(x)}{(1+x^2)^2} \\, dx = - \\frac{\\pi}{4}.", "d140f5f73bee3ee153fc755f4cd9febc": "\\mu(\\bigcup_nU_n ) \\le \\oplus_n\\lambda(U_n)", "d14101162e8caf2fd68cd41c9ad79a0d": "|\\mu|(E)=\\overline{\\mathrm{W}}(\\mu,E)+\\left|\\underline{\\mathrm{W}}(\\mu,E)\\right|\\qquad\\forall E\\in\\Sigma", "d1410868461a1a40a6882aa144c60497": "F_{n-1}F_{n+1} - F_n^2 = (-1)^n.\\,", "d14108e9cb83ff0fcc30586619e05a8c": "\\frac{\\partial a\\mathbf{u}}{\\partial \\mathbf{x}} =", "d1411a9ce765ac6506f6d240dc33f9f5": "\\sqrt{40n +9} +3\\over10", "d1411de1848a83391a13fb5a301c8b53": " \\dot{\\hat{\\mathbf{x}}}_r(t) = A_r(t)\\hat{\\mathbf{x}}_r(t) + B_r(t){\\mathbf{u}}(t)+K_r(t) \\left( {\\mathbf{y}}(t)-C_r(t)\\hat{\\mathbf{x}}_r(t) \\right),\\hat{\\mathbf{x}}_r(0)={\\mathbf{x}}_r(0),", "d14150f098b2567f0158f4b02b58fcee": "f \\in \\mathbb{F}_q[X]", "d141a8eabc38c6ca9d5fcebed9324cde": "\\int_C = \\int_\\varepsilon^R + \\int_\\Gamma + \\int_R^\\varepsilon + \\int_\\gamma.", "d141c525e271b0f29b09f1bafeb4fd4d": "Q(x, p(x))", "d141db98fbbb8453b68f7ef533526fcb": " \\Psi(x,t) = \\sum_k x^k P_k(t),\\quad ", "d1420a3b2965a4eed9d7b1676873c6dd": "|\\psi(t)\\rangle = U(t, t_0) |\\psi(t_0)\\rangle.", "d142a5b265bf691895ac13774ff48ed5": " b \\in\\mathbb{R}^m \\ ", "d142e9f5b63dd9ef4f62d4abeaae6bc0": "\\quad q^{*} =\\begin{pmatrix}d & -b \\\\ -c & a \\end{pmatrix}. \\,", "d14315cd2654dfc1b514760c7285bbb4": "U_{\\beta\\gamma}", "d1442441d6296396cb7809bb448623d0": "5 \\cdot 107367629.", "d1442ea5e3e9225be4c998ff00c5344d": "L^p(\\Omega,d\\mu)", "d1448ce7b7b4fe836fcfaaf68e275841": "z_1,z_2 \\in \\mathbb{R}^{m \\times d}", "d144bd4b28829ee77ad4856a343319f7": " (\\log uv)' = (\\log u + \\log v)' = (\\log u)' + (\\log v)' .\\! ", "d144c0c93dde4fec938d9746061f81cc": "\na(\\xi,\\zeta) = \\frac{4[\\cosh (3 \\xi) + 3 e^{4 i \\zeta} \\cosh (\\xi)] e^{i \\zeta / 2}}{\\cosh (4 \\xi) + 4 \\cosh (2 \\xi) + 3 \\cos (4 \\zeta)}\n", "d144c5235b3661b12372968f3cf8e4ae": "\\dot{M}_I - \\dot{M}_O", "d14568ff84cb373d17e5a30e964814a2": "V = {I_\\max - I_\\min \\over I_\\max + I_\\min} ,\\,", "d1457ed7257771d3ca988b0e303ea7cd": "t _2", "d146b998c95278d5c97beddd6b866e41": " \\varepsilon_{ijk} =\n\\begin{cases}\n+1 & \\text{if } (i,j,k) \\text{ is } (1,2,3), (3,1,2) \\text{ or } (2,3,1), \\\\\n-1 & \\text{if } (i,j,k) \\text{ is } (1,3,2), (3,2,1) \\text{ or } (2,1,3), \\\\\n\\;\\;\\,0 & \\text{if }i=j \\text{ or } j=k \\text{ or } k=i\n\\end{cases} ", "d146c0d19db43782334097e376f0d9ea": "u(t) \\le \\alpha(t) + \\int_{[a,t)} u(s)\\,\\mu(\\mathrm{d}s),\\qquad t\\in I.", "d146e82de567c6b8279ea60cf3151113": " \\scriptstyle T^r_s(V) ", "d146f4eacfb6cf8819ed28b99a7c786e": "\\ m=2 ", "d1470915f346a28f68230e864ac544e5": "\n0 \\leq c\\bar F(c) \\leq \\mathbb E(X) - \\int_0^c x f(x)dx \\to 0 \\text{ as } c \\to \\infty\n", "d1470ff52027f6cbba226656dd4cc2fd": "\\begin{align}J_n\\left(\\frac\\pi2\\right)&=\\int_{-\\pi/2}^{\\pi/2}\\left(\\frac{\\pi^2}4-y^2\\right)^n\\cos(y)\\,dy\\\\\n&=\\int_0^\\pi\\left(\\frac{\\pi^2}4-\\left(y-\\frac\\pi2\\right)^2\\right)^n\\cos\\left(y-\\frac\\pi2\\right)\\,dy\\\\\n&=\\int_0^\\pi y^n(\\pi-y)^n\\sin(y)\\,dy\\\\&=\\frac{n!}{b^n}\\int_0^\\pi f(x)\\sin(x)\\,dx.\\end{align}", "d147830806e80b7245610736917d35a9": "P_1 + \\frac{1}{2} \\rho _1 V_1^2 = P_2 + \\frac{1}{2} \\rho_2 V_2^2", "d1478a6b82386039e99fc89855d2badd": "A=\\begin{pmatrix} 0&1 \\\\ -1&0 \\end{pmatrix}", "d1479f4d25ebd5d3adbc38cf999d5530": "\n\\begin{align}\n1 & = p^2+t^2, \\\\\n1 & = q^2+u^2, \\\\\n0 & = pq+tu.\n\\end{align}\n", "d147c8f98d1dfbfa5417534034c68ad1": "f(x_i, \\boldsymbol \\beta)\\,", "d147f71f328e503f49e225c134e74c5d": " \\lambda_1 \\gg \\lambda_2 \\simeq \\lambda_3 ", "d1480884c6eb30d13c862428717d9cef": "\\Phi_{2^hp^k}(x) = \\sum_{i=0}^{p-1}(-1)^ix^{i2^{h-1}p^{k-1}}", "d1481694db242a622e0ebe9c5d48e138": " A - LC ", "d14837cebe802f9c9c4506b2c46b77ca": "x \\in C = \\{x_1 > 0\\}", "d14870e3cc1f34cd343dfa9d52eb1e32": "\\langle f,g\\rangle =\\frac{1}{\\pi^2} \\int_{[-1,1]^2} f(x_1,x_2)g(x_1,x_2)\\frac{dx_1}{\\sqrt{1-x_1^2}}\\frac{dx_2}{\\sqrt{1-x_2^2}}\n", "d1489815d8916b92a1efd8c0fd0d7512": "\\left(\\frac {dG}{d\\xi}\\right)_{T,p} = \\Delta_rG_{T,p} = 0 ", "d148b05c18ff5939f7615de3c85e83ef": "\nf(x;\\lambda,k) =\n\\begin{cases}\n\\frac{k}{\\lambda}\\left(\\frac{x}{\\lambda}\\right)^{k-1}e^{-(x/\\lambda)^{k}} & x\\geq0 ,\\\\\n0 & x<0,\n\\end{cases}", "d148c06bbe7ac3c8a40e80018ab33d64": "\\exists\\kappa. (\\kappa=\\lambda \\wedge x) \\wedge \\exists\\kappa. (\\kappa=\\lambda\\wedge \\neg x) \\Leftrightarrow \\mathit{false}", "d1492eff0b7ed00185081d9141d23397": "\\partial_y", "d149305bcf916ba345c7742c6280a023": "270^\\circ T\\,", "d1508b8e92c1ecc32bb6d2a4ec4dbd70": "R'(\\phi)>0", "d150b1d5ea345dbe0cf695e86fa5c5e1": "\\; \\varrho_{A_1\\ldots A_m}", "d150d5bdeaa2cd734b4bd3e192fc7435": "\\Re(s)>1", "d150f4d0aba764127242fa13ba1ed61d": "\\sum_{i=1}^k \\mu_i^\\ominus \\nu_i = \\Delta_rG^{\\ominus}", "d151138d9f65aee4791ee25d7aa2c569": "180^\\circ 0,\\, \\Delta_2 > 0, \\ldots, \\Delta_n > 0, ", "d155972351fc70cc6d02527db44a0d16": "t>m+ld", "d155e52931d507ea9076e5f5eb4a1f78": "\\left[\\tfrac{1}{2}(\\sigma_2 + \\sigma_3), 0\\right]", "d15613155ef10e30dab22f9f3bf64857": "\\lambda^2 - 2c\\lambda + 1", "d156f9beba463be9dbbde313d2c8fd7d": "H=H_0+H_{\\mathbf{k}}', \\;\\; H_0 = \\frac{p^2}{2m}+V, \\;\\; H_{\\mathbf{k}}' = \\frac{\\hbar^2 k^2}{2m} + \\frac{\\hbar \\mathbf{k}\\cdot\\mathbf{p}}{m}", "d1573e8d44a750b5b172d9e9d409bcf5": "2/\\widehat{\\theta}_0", "d15749504a79448767e70f9b6dce1860": "\\Delta x_{l,s}=\\frac{x_{l,o} x'_{s,o}}{k_1}", "d15792838a0949c9192beefa8e15f763": "n_i/V", "d15803e50836765434f93abd8503f0a0": "\\frac{\\partial}{\\partial x_k} \\|\\mathbf{x}\\|_2 = \\frac{x_k}{\\|\\mathbf{x}\\|_2}", "d1584c0702ec9e82e73e7e049c972aea": "\\Theta(N)", "d15855468308836e78f29796a8765c01": " g = \\cos^2( x / r) + \\sin^2(x/r) = 1. ", "d1585a7ab52f166623df311b916f75c5": "x=c \\ ", "d15888bfdf570dd0afba2ad026bd1c23": " \\Delta (M_{ij})= \\sum_l M_{il} \\otimes M_{lj} ", "d15897f8c7952e312b716b7e0f838ac3": "\\int {F(x, y, y^\\prime)} dx", "d158cd13ed4407fd01ea00c156c156e9": "i \\neq n", "d158e74aaf86431f54ad44d1d8dfa3c6": " \\int f \\, d \\mu = \\int f^+ \\, d \\mu - \\int f^- \\, d \\mu. ", "d158ef5b99b3c3f781c5f524217ce2eb": "\\Omega = \\{ z \\in \\mathbb{C} | x_1 \\leq \\mathrm{Re} (z) \\leq x_2 \\text{ and } \\mathrm{Im} (z) \\geq y_0 \\} \\subsetneq \\mathbb{C}. \\, ", "d1596eb59b7f57cd19f210ba9b3ad00e": "\n W_{mn}(x_1,x_2) = \\sin\\frac{m\\pi x_1}{a}\\sin\\frac{n\\pi x_2}{b} \\,.\n", "d15981491d79738d22f795a2513afac8": "|\\psi\\rangle_A |\\psi\\rangle_B |A\\rangle_C \\rightarrow |\\psi\\rangle_A |0\\rangle_B |A'\\rangle_C", "d159dac693e66a39079760dccb833858": "\nV = - \\frac{e^{2}}{4 \\pi \\varepsilon_0 } \\left( \\frac{1}{r_a} + \\frac{1}{r_b} \\right)\n", "d159ebb6d1be1c8818fa8537cfe6467a": "U \\times P", "d15a18823593f0cb264e86c0f3fba728": "E = Mc^2 \\;", "d15a25d590b5d7f5f54a25e2c74480a9": " f(x_0) = \\max_{[a,b]} f(x) ", "d15a507d085dbb1d1d2dfe8fa667d420": "\\psi({\\mathbf{r}})=\\frac{{\\mathbf{m}}\\cdot{\\mathbf{r}}}{4\\pi r^{3}},", "d15a6ffd9255e70c6b213fd82127876a": "\\rho_0 \\;", "d15ad6a6cf561d8da9c96acf32557386": "|i!| = \\sqrt{\\pi \\over \\sinh \\pi} ", "d15b3d0254126388ed5d852fa43e0426": "{}^bp_i", "d15b644f1a78ed0ff4d8742db6ad14fb": "e^k\\cdot O(V^\\omega)", "d15b7a5bac20115062aa3f33fc7ef477": "E = \\{(1,2),(2,1),(1,3),(3,1),(4,2),(2,4),(5,2),(2,5),(6,3),(3,6),(7,3),(3,7),(8,4),(4,8),(9,4),(4,9)\\}", "d15be78196733ba55a9b566e392f0391": "\nP^{-m}_\\ell(x) = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!} P^{m}_\\ell(x).\n", "d15bfe836d9115b2a917466a0b963c39": "\\cos \\frac{\\pi}{10} = \\cos 18^\\circ = \\frac{\\sqrt{2(5 + \\sqrt 5)}}{4} ", "d15c777253c41b27ddc9715b4446f2ac": " C = 1000000 \\times \\frac{2}{1-e^{-2}} \\approx 2.313\\times10^6 ", "d15cb76977a626bba97ad8c261f83478": " E_s ", "d15cd8b6d018cdc73741d42489bc2387": "|\\beta_1\\log\\alpha_1 +\\beta_2\\log\\alpha_2|\\,", "d15cea7b8ef72960d421a3df852e844c": "h_{a,b}(x) = ", "d15d1ea7fd20eb9db1fff3d4e38fd66c": "\n\\begin{align}\n\\mathrm{d}\\omega_1^2 & =\\omega_1^3\\wedge\\omega_3^2\\\\\n\\mathrm{d}\\omega_1^3 & =\\omega_1^2\\wedge\\omega_2^3\\\\\n\\mathrm{d}\\omega_2^3 & =\\omega_2^1\\wedge\\omega_1^3\n\\end{align}\n", "d15d1fd652dfc63a04cfcec05e313fc9": "p = \\frac {l} {t} ", "d15d21de0592fe2c0ae18b4d70f04100": "\nQ = \\begin{bmatrix}\n0 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & -1\\\\\n1 & 0 & 1 & 1 & -1 & -1 & -1 & -1 & 1\\\\\n1 & 1 & 0 & -1 & 1 & -1 & 1 & -1 & -1\\\\\n-1 & 1 & -1 & 0 & 1 & 1 & -1 & -1 & 1\\\\\n-1 & -1 & 1 & 1 & 0 & 1 & 1 & -1 & -1\\\\\n1 & -1 & -1 & 1 & 1 & 0 & -1 & 1 & -1\\\\\n-1 & -1 & 1 & -1 & 1 & -1 & 0 & 1 & 1\\\\\n1 & -1 & -1 & -1 & -1 & 1 & 1 & 0 & 1\\\\\n-1 & 1 & -1 & 1 & -1 & -1 & 1 & 1 & 0\\end{bmatrix}.\n", "d15e067bea0f46a86d61d3bbaf6bfae7": "\\chi_{\\text{e}}\\ = \\varepsilon_{\\text{r}} - 1", "d15e0ac11848f4e002fb54f7b689f72e": " p(x,t) = (x^2-t)^2, ", "d15e4b4244ba38a4fb56c151e557def9": "x_{n+1}=(ax_n+c)\\ \\bmod\\,2^{32},", "d15e6b97e2b2cd95f8c6667198b17339": "i\\}", "d15e703570540e3acb2b50906704b743": "(vt)^2 - (ct)^2 = 0 \\,", "d15eea42f35033684d16fc7f072865c2": "\\ w_i", "d15f4a8dff6acd178b5a804a7e4ca82a": " I_V \\otimes S ", "d15f53f7eff85a098baf07f70a9ad2b4": "n_{k}", "d15f6c9cc01c64ee121da35a21c31ef0": "\\prod_{\\text{prime } p} \\left(1-\\frac{1}{p^2}\\right) = \\left( \\prod_{\\text{prime } p} \\frac{1}{1-p^{-2}} \\right)^{-1} = \\frac{1}{\\zeta(2)} = \\frac{6}{\\pi^2} \\approx 0.607927102 \\approx 61\\%.", "d15fb2412b8076b273a80d053c17b5d1": "E = 2G(1+\\nu) = 3K(1-2\\nu).\\,", "d15fbc47f20ff6002992f8e0bfe213a7": "P=\n\\begin{bmatrix}\n 0&0&\\ldots&0&1\\\\\n 1&0&\\ldots&0&0\\\\\n 0&\\ddots&\\ddots&\\vdots&\\vdots\\\\\n \\vdots&\\ddots&\\ddots&0&0\\\\\n 0&\\ldots&0&1&0\n\\end{bmatrix}.", "d15fe2de85602cb0b100bc3c40c6ce8d": "\\mathbf{B})", "d15ffe18296753d8c23d66a7c1bac6ba": "\\sigma_{0}(n)", "d1606b5ac5730176d2fb6510b43887a1": "b_{11}", "d160a996d2bb045f14321239884bf326": " \\mathbf{q} = \\mathbf{A}\n\\begin{bmatrix}\n1 \\\\\ne^{\\lambda_2t} \\\\\n \\vdots \\\\\ne^{\\lambda_nt} \\\\\n\\end{bmatrix}\n", "d160c6f26246a2b0192a2a2662b8a49d": "1f, 2f, 3f, \\dots \\ ", "d160e90193c5c5ed955920a5e7607588": "n=N_{c}\\frac{e^{-(E_{c}-E_{F})}}{k_{B}T}", "d161804459cca5e884fdceb1f48fb9b8": "\\mathrm{SO}(n)\\,", "d161c804762c71540ae0945bc26930ad": "\\operatorname{pf}\\begin{bmatrix} 0 & M \\\\ -M^\\text{T} & 0 \\end{bmatrix} = \n(-1)^{n(n-1)/2}\\det M.", "d161cf2bd8c7c6ff4f0be77aa52b716f": "\\operatorname{\\Gamma L}(V),", "d1626d5eea79016c4ca83e22485b80eb": "(W,S)", "d16329f8a3d5d88879b718a91b67d068": "f_{\\text{r}}(\\omega_{\\text{i}},\\,\\omega_{\\text{r}},\\,\\mathbf{x})", "d163727e25acd00c49436cce9fbceeb1": "R_n^m(\\rho)=(-1)^n R_n^m(-\\rho)=(-1)^m R_n^m(-\\rho)", "d1639d82f14a139e5a35e2451f52c857": " {-{\\pi}(\\tau-t)^2} ", "d163e4dc80f522e1121855fc8c8dd3f1": "x = \\sum_{i = 1}^{m} c_{i} (x \\cdot u_{i}) u_{i}", "d163f9cbf5eec57165955b5590872ec6": " \\mathbf{\\bar F} ", "d163fb21cb5a5b674ce25f320518016c": "D(\\alpha) = D_F\\{x, \\alpha(x) = \\alpha\\}", "d16475dadb19a9e4dca983c6e3e003f2": "E = m_0c^2\\sqrt{1 + \\left(\\frac{p}{m_0 c}\\right)^2}", "d1647b9d9b6466ebb2cc88b83c7a23cb": "x_i = e_k", "d164aeefe1b0cf9ded998c5f6d55bb98": " H_{z}=\\frac{k_{t}^{2}}{j\\omega \\mu } L \\ T^{TE}=\\frac{k^{2}-k_{z}^{2}}{j\\omega \\mu } L \\ T^{TE} \\ \\ \\ \\ (32) ", "d164ee5aa6dc7d1d340cadbe188cf8e1": "v_x = v_{xo} e^{-\\frac{k}{m}t}", "d164f84f9a3b5f72c188c980dd3175cf": "S(q) = 0", "d164fc9ebe6cf9f583cd7137ad529802": " W(f)W(g)=e^{-i(f,g)}W(f+g), \\,", "d1650e55224af99bb4b50fd9e4e42cda": "y_2 = \\left.\\frac{\\partial y_b}{\\partial c}\\right|_{c = 1 - \\gamma}.", "d16557b38ad7f5b739ac6a0e86c571f7": "F(t,T) = S(t)e^{r(T-t)} \\,", "d1656ebea4d8a733a3b9b76a75f02ed0": " \\operatorname{H}(S) = -\\operatorname{Tr}(S \\log_2 S) ", "d1658f0db9dc4e84f601978b3dbe2d02": " B_n < \\left( \\frac{e^{-0.6 + \\varepsilon} n}{\\ln(n+1)}\\right)^n ", "d16595e660edb206abf3cb1a137a3265": "\\frac{1}{m}+\\frac{1}{n}+\\frac{1}{k}<1.", "d165b30e01352e5fae671ee813321bbc": "\n\\begin{align}\nd^4\\sigma &=\n\\frac{Z^2\\alpha_{fine}^3\\hbar^2}{(2\\pi)^2}\\frac{|\\mathbf{p}_f|}{|\\mathbf{p}_i|}\n\\frac{d\\omega}{\\omega}\\frac{d\\Omega_i d\\Omega_f d\\Phi}{|\\mathbf{q}|^4}\\times\n \\\\\n&\\times \\left[\n\\frac{\\mathbf{p}_f^2\\sin^2\\Theta_f}{(E_f-c|\\mathbf{p}_f|\\cos\\Theta_f)^2}\\left\n(4E_i^2-c^2\\mathbf{q}^2\\right)\\right. \\\\\n&+ \\frac{\\mathbf{p}_i^2\\sin^2\\Theta_i}{(E_i-c|\\mathbf{p}_i|\\cos\\Theta_i)^2}\\left\n(4E_f^2-c^2\\mathbf{q}^2\\right) \\\\\n&+ 2\\hbar^2\\omega^2\\frac{\\mathbf{p}_i^2\\sin^2\\Theta_i+\\mathbf{p}_f^2\\sin^2\\Theta_f}{(E_f-c|\\mathbf{p}_f|\\cos\\Theta_f)(E_i-c|\\mathbf{p}_i|\\cos\\Theta_i)}\n\\\\\n&- 2\\left.\\frac{|\\mathbf{p}_i||\\mathbf{p}_f|\\sin\\Theta_i\\sin\\Theta_f\\cos\\Phi}{(E_f-c|\\mathbf{p}_f|\\cos\\Theta_f)(E_i-c|\\mathbf{p}_i|\\cos\\Theta_i)}\\left(2E_i^2+2E_f^2-c^2\\mathbf{q}^2\\right)\\right].\n\\end{align}\n", "d165e60dc8a8b1a56b6ea81f6e59ef12": "H_\\lambda,X_\\lambda,Y_\\lambda\\text{ for }\\lambda\\in\\Delta", "d165e912c645852f59e8e17fdfdf8c0f": "P\\left(\\omega < \\frac{X}{a}\\right) = \\alpha .", "d16671d7211de9b1eb5f66e4c8cae8dd": "x '= x,, y' = y, ", "d166ab834f2c020829f12a8ee8fbfb88": "\\sqrt[n]{|a_n|}.", "d166b8ba44689c633f189894763ae38a": "C_o = \\alpha_a C_a + (1 - \\alpha_a) C_b", "d166cb45e654b0734d3c1d37a76442e3": "\\frac{du_1}{dt}=J_1 \\frac{dT_1}{dx}", "d1675dc95c879aa693385c1273d1884e": "C_D = c_{d_0} + c_{d_2} (C_L)^2 + \\frac{(C_L)^2}{\\pi e AR}", "d16774aec4bd9a74f8a41afd417f2458": "\\frac{x_1 x_2 \\cdots x_n}{\\alpha^n} \\le e^0 = 1,", "d16888f4ee0cfc2eae886cb61452c282": "\\mathcal{F}\\left[\\left(-\\frac{\\partial^2}{\\partial x^2} \\right)^{\\frac{1}{2}}f\\right](\\xi) = |2\\pi\\xi|\\mathcal{F}f(\\xi).", "d168925a6500c70294f0ed3b3cfe542d": "U_{\\mathrm{internal}}= \\frac{1}{P} \\sum_{t=1}^P V(\\mathbf{r}_t)", "d168a787495f5bcceaf2a99c63383b65": "z=(z_1, z_2, \\dots, z_n)", "d168d662e57f9fb28823afbc9721fb53": "\\{p(1+x),M(1+x)\\}", "d168e35ad09aa1843a0d07f2a67fc916": "\n\\mathbb{D}(A,B) := \\sup_{x\\in A} \\{ \\inf_{x' \\in B} \\|x-x'\\| \\}\n", "d1692b087b32e52958f4931e3c70a12c": "\\begin{align}\nd\\theta\n&= \\partial_x\\left(\\operatorname{atan2}(y,x)\\right) dx + \\partial_y\\left(\\operatorname{atan2}(y,x)\\right) dy \\\\\n&= -\\frac{y}{x^2 + y^2} dx + \\frac{x}{x^2 + y^2} dy\n\\end{align}", "d169527cf699a330bcd966ab77ea1106": "x^i \\to x^i + \\alpha^i x^-", "d169700b837c829b0cfccb235d11f84c": " \\delta_p", "d169c3780ecc5e87f1e37e06f4bd37e9": "a^{n-1} = 26^{220} \\equiv 169 \\not\\equiv 1 \\pmod{221}.", "d169c6fa69787e6d382cf3cc8f652947": "\\,V (\\in \\R^n)", "d16a21b9afab9405f7487912a8ddd493": "R_\\mathrm{load}=0\\,\\!", "d16a22ad825e560d5c9d5283e2176238": "\n\\begin{bmatrix}\n 1 & x_1 & x_1^2 \\\\ \n 1 & x_2 & x_2^2 \\\\\n \\vdots & \\vdots & \\vdots \\\\\n 1 & x_n & x_n^2\n\\end{bmatrix}\n\n\\begin{bmatrix}\n a_2 \\\\ \n a_1 \\\\\n a_0 \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n f_1 \\\\ \n f_2 \\\\\n \\vdots \\\\\n f_n \n\\end{bmatrix}\n", "d16a827566558ae2ec6b8bdd003a82c5": "I(c)=10", "d16a9eab2cef65203e9e123865bd46c5": "\\forall a \\in S :\\exists a' \\in S :\\forall b \\in S :\\exists b' \\in S :\\forall c \\in S :\\exists c' \\in S ... : (a,a',b,b',c,c'...) \\in G ", "d16ab71862829eab41c3b737e9f0a44e": "p=p(\\rho)", "d16add0c9ed0a5fd8d97c44ce14d17d7": "107\\frac{1}{5}", "d16ae04f3aec24069d1c27a9120cd2bd": "\\tfrac{x^3+x^2+1}{x^2-5x+6}", "d16b42ca77c86ec93a47afbdd0299a1e": "{\\rm Chi}(x) = \\gamma+\\ln x + \\int_0^x\\frac{\\cosh t-1}{t}\\,dt = {\\rm chi}(x)", "d16b638fb6961d1f3eb65118e19ce237": "\\,p_x = l_{x+1} / l_x", "d16bc2d9f286c45559bf29cbebc67ca7": "\\Sigma^1_{k+1}", "d16bf794c5938aa084bb186cb5dc1319": "\\scriptstyle \\mathcal {H} \\subseteq \\mathcal {F} ", "d16c25f738494e3b04b2ac660ebb76dc": "(M,x,n)", "d16c31f1c41c4d7a2aa6f5fcf84cfc1a": "\\dot\\rho=-{i\\over\\hbar}[H,\\rho]+\\sum_{n,m = 1}^{N^2-1} h_{n,m}\\big(L_n\\rho L_m^\\dagger-\\frac{1}{2}\\left(\\rho L_m^\\dagger L_n + L_m^\\dagger L_n\\rho\\right)\\big)", "d16c4387fe2c57af7781ca9da4e367b8": "{}^2E(f) = 1 \\quad", "d16c5b9309ad8bdd5d4ef8cadb786219": "MPL=w/P", "d16c6d55c0a44c42d61d8f0ab60e8526": "\n\\omega^2=\\frac{\\sigma}{\\rho+\\rho'}\\, |k|^3,", "d16d45b1f5cfaa2d4d8a13d8e04228b6": " \\delta_{ext}", "d16d8c60f9384e7515e334d8f437bab8": "\\pi_{M} : (x, v) \\mapsto x.", "d16d97741465b78ceeca2148cbac03aa": " d\\Gamma = 4 s V_\\infty \\sum_{n=1}^{\\infty}n A_n \\cos(n\\theta) \\qquad (5)", "d16e189b24c1868ceebf0f27771c6076": "v_{\\alpha-1}", "d16e1d26f470781d4e3db7c3e8698eaa": "\\, \\frac{e^{at} - e^{(b+1)t}}{(b-a+1)(1-e^{t})}", "d16e3d93f853860f8167c63893f255f4": "\\; \\Psi_i (A) = M_i A M_i", "d16e60e2123ba3e90d6a1d2cc97d39df": "\\left\\{1/\\sqrt{2},i/\\sqrt{2}\\right\\}", "d16e75503bd0d80390e8b721d084944a": "\\cos(\\theta(u,v))", "d16e86f04e1740daf9b626711563b0db": "\\binom{n/2}{n/4}^2 = \\Theta(\\frac 1 n 2^n)", "d16e8bf005f2b525e4bb41a2b017fd45": "\\langle [e_{n+1}], [e_m] \\rangle = A_{m,n+1} = m_{m+n+1} = \\langle [e_n], [e_{m+1}]\\rangle.", "d16f14caa736d7d3d31db89a8827e070": "\\begin{align} P^{-1} & =\\left( R+\\frac{1}{N-1}HA\\left( HA\\right) ^{T}\\right) ^{-1}\\ = \\\\\n& =R^{-1}\\left[ I-\\frac{1}{N-1}\\left( HA\\right) \\left( I+\\left( HA\\right) ^{T}R^{-1}\\frac{1}{N-1}\\left( HA\\right) \\right) ^{-1}\\left( HA\\right) ^{T}R^{-1}\\right] , \\end{align}", "d16f3a18b9ade9b4938102a2aa16fc40": "\\left (\\sigma_T\\right )", "d16f69ffbb7dfb7cbc7fc6fbb0b7034d": "\\Re,", "d16fb36f0911f878998c136191af705e": "xyz", "d16fb9f6783b5312881a7db204e48843": "P = V^2/R", "d16fbb54f40b5ce1f6f78468110c35c7": "\\ g_b", "d16fd35bf302002c6f193cf3083ba9fa": " k \\times 1 ", "d1700c489a7d3060dd285637a5322adb": "\\displaystyle{Q(a)=T_a \\circ j \\circ T_{a^{-1}}\\circ j \\circ T_a \\circ j.}", "d17042739cc04bd8efec6a8184f6abb3": "i:A\\to X", "d1704c098c5a1ed15904347374941009": "\\bullet \\overrightarrow{\\to} \\bullet", "d17091642dccd643ec0e680372ffd687": "a_{n+2} - a_{n+1} = a_{n+1} - a_n", "d170e9afb2b726c2161841dd208c961c": "v \\mapsto e^{\\log c t}v", "d1711b04cbff6197fd4ed330952c8a8e": "\\bar{\\mathbf{e}}_p\\otimes\\bar{\\mathbf{e}}_q = (\\boldsymbol{\\mathsf{L}}^{-1})_{pi}\\mathbf{e}_i\\otimes(\\boldsymbol{\\mathsf{L}}^{-1})_{qj}\\bar{\\mathbf{e}}_j = (\\boldsymbol{\\mathsf{L}}^{-1})_{pi}(\\boldsymbol{\\mathsf{L}}^{-1})_{qj}\\mathbf{e}_i\\otimes\\bar{\\mathbf{e}}_j = \\mathsf{L}_{ip} \\mathsf{L}_{jq} \\mathbf{e}_i\\otimes\\bar{\\mathbf{e}}_j", "d1712bd87a33f5409b4a81b72f101188": "\n N_P =\\min_L\\{N_L+\\sum_L(C_{AB})\\}\\qquad (4)\n", "d1717dbeac641dcb920b98ac3557dc6b": "\\overline {\\Delta q} \\,.", "d171e7603564620584341fae1250a054": "z_1(x,y)=2(x-y)F(y)+(x-y)^2F'(y)", "d17220ecbc75539e7611e431838dd120": "(a + ib)", "d17234057cdb208bf3a282a15d4d88dd": "\\ v_{tr} = \\frac{R_p}{R_t + R_{tr}}", "d17241f1cdd07a6546152286e8aeab94": "2\\pi\\sqrt{m/k}", "d17243bea2a43de9c6e7d54e09d3a364": "\\gamma_{\\pi}^*", "d172b78575ccdb04d8a9ea3636f813e6": "\n\\frac{y}{c} = \\frac{k_1}{6} \\left[ \\frac{k_2}{k_1} \\left(\\frac{x}{c}-r \\right)^3 - \\frac{k_2}{k_1} (1-r)^3 \\frac{x}{c} - r^3 \\frac{x}{c} + r^3 \\right]\n", "d17308ee2ffb87765c6ddbaaf27d823e": "\\scriptstyle 1-x", "d173145a7f4f8fc1251cf12f9e8b3711": "\n\\begin{array}{lcl}\n\\operatorname{Ack}(0) & = & \\operatorname{Succ} \\\\\n\\operatorname{Ack}(m+1) & = & \\operatorname{Iter}(\\operatorname{Ack}(m))\n\\end{array}\n", "d1734b1c4440235dd208bb50459d758f": "\\deg(g)", "d1739b44dd80e5b4e7b5356153610b16": "x \\cdot y", "d1744b3f418141feefdb1b620465c666": " M_1 = \\left\\{ \\frac{| 0 \\rangle+| 1 \\rangle}{\\sqrt{2}},\\frac{| 0 \\rangle-| 1 \\rangle}{\\sqrt{2}} \\right\\} ", "d174509eb210a83f81eeec9c0a2df47d": " h(1,2,3,4,5)=(2,3,3,4,5) \\, ", "d1754f5c94553f36ee0994a4dd85cdef": "h^* = \\arg \\min_{h \\in \\mathcal{H}} R(h).", "d175859936fd5ee352bde145d084ff16": "2,1,0", "d175ffaed1494417f9288f86e7e5277e": "\\int_{0}^{\\infty }\\frac{\\sin ax}{\\sinh bx}\\ dx=\\frac {\\pi}{2b}\\tanh \\frac{a \\pi}{2b}", "d17623e34b53ca2d076b71d4e9fd215d": "\\rho_2\\ :\\ g(x) \\rightarrow f(x, x)", "d1762cc821f592fffe14bd53a3ca3f22": "h = \\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix}", "d1762f74adff119775916483241a292b": "a_{(1)}S(a_{(2)}) = \\epsilon(1_{(1)}a)1_{(2)}", "d1767f342b979ac51fd63380e6835306": "\\displaystyle{W(a,T,b)=(Wa,WTW^{-1},(W^t)^{-1}b),}", "d176b30d5c972fc9c9a702957a55a24b": "\\textstyle \\operatorname{sign}", "d17715de4c0d8a177dc61edea1516664": "K>0,", "d17779a8156d77b8fee3e641109fa642": "|z\\rangle", "d177b3c0b70b147d5add2f6ae0854ea3": "p=\\frac 1 2 \\frac {{(\\sum\\lambda i)}^2} {\\sum\\lambda_i^2}.", "d178271d4a25d6e4d8472ee090dece16": "(\\mathbf{a} \\times \\mathbf{b}) \\cdot (\\mathbf{c} \\times \\mathbf{d}) = (\\mathbf{a} \\cdot \\mathbf{c})(\\mathbf{b} \\cdot \\mathbf{d}) - (\\mathbf{a} \\cdot \\mathbf{d})(\\mathbf{b} \\cdot \\mathbf{c}).", "d178a03ef9eb826e1929973102236bd2": "\\eta_\\mathrm{cp}=\\sqrt{2}\\pi/6 \\approx 0.74048", "d178ef05bb8c210db6a0e694e8ad3839": "\\sigma_{ij}=\n\\begin{bmatrix}\n\\sigma_1 & 0 & 0\\\\\n0 & \\sigma_2 & 0\\\\\n0 & 0 & \\sigma_3\n\\end{bmatrix}\n\\,\\!", "d1791f2a965c8f3c7925b47d07460aa9": "E_F = \\frac{\\hbar^2}{2m_e} \\left( 3 \\pi^2 \\ 10^{28 \\ \\div \\ 29} \\ \\mathrm{m}^{-3} \\right)^{2/3} \\approx 2 \\ \\div \\ 10 \\ \\mathrm{eV} ", "d1794b79b7d24ae80d73788ca3783a21": "\\bar{X}_2", "d17950c20b61c8f2f17b25e348ace50a": " \\theta \\equiv ", "d1795245742797187cd60e9f31368ae5": "b_{\\nu,n}(x)", "d17964b696d912901ce26fa4ba53cbee": "\\mathit{cornerness}", "d179ece07b4b48f9b43d44549e1a470f": "k=\\mathbb{C}\\;", "d17a4fa19c4f113cdfdb1c43833efe59": "\\bar V_{\\text{L}}=L\\frac{\\bar{dI_{\\text{L}}}}{dt}+R_{\\text{L}}\\bar I_{\\text{L}}=R_{\\text{L}}\\bar I_{\\text{L}}", "d17a6cc344b34e3a91fceacbe24f6ede": "\\vec l\\!", "d17ad62805959aeedb6b91ea1ba93473": "\\mathcal O(N)", "d17af43db57fdedfbe3440467c0835c9": " = \\alpha ( u_{xx} + u_{yy} + u_{zz} ) \\quad ", "d17b20d9f2e1f1af0137f6db9db909c3": "X\\setminus A_c", "d17badf0fa64a1decc1158e4bd4c28da": "112 \\times 10^3", "d17beabda33ab3b0d8fd065c333a9602": "\\underline{R}", "d17c81ed3e7e74285bd3aec7ff5bf473": "\\scriptstyle [0,1]^{Z^d} ", "d17ca6f1c4b1c0c7002225388e189778": "\\frac{1}{(4\\pi t)^{n/2}} e^{-|x|^2/4t}", "d17cae013f535b2e3f1e118f05059048": "y'' - xy = 0 , \\, ", "d17cb99aa496a2288dc104240b085945": "I=\\frac{m}{6}\\frac{\\sum\\limits_{n=1}^{N}\\|\\mathbf{P}_{n+1}\\times\\mathbf{P}_{n}\\|((\\mathbf{P}_{n+1}\\cdot\\mathbf{P}_{n+1})+(\\mathbf{P}_{n+1}\\cdot\\mathbf{P}_{n})+(\\mathbf{P}_{n}\\cdot\\mathbf{P}_{n}))}{\\sum\\limits_{n=1}^{N}\\|\\mathbf{P}_{n+1}\\times\\mathbf{P}_{n}\\|}", "d17ce1f4b592ad4f310ec9b0a5e767a7": "\\ \\mbox{FSPL(dB)} = 20\\log_{10}(d) + 20\\log_{10}(f) + 92.45", "d17ce6d5629fc4cd62df50e20e054410": " a_n", "d17dac12f906b39b28d5427751a1f0f4": "\\mathcal{P}_i", "d17df88539a4338b53bd753428a3e98f": "S = \\frac{\\mu_0LV_A}{\\eta} ,", "d17e007ae3661fc393e2b51b4a9b9b28": "\\beta_T = \\frac{v_T}{c} = \\frac{\\beta\\sin\\theta}{1-\\beta\\cos\\theta}", "d17e180bd71f4a21d9cf498b8c12ec1d": "b_n=(-1)^{a_n}", "d17e2154a21e0bce2a62842885f342da": "P[i,j,k]", "d17e2efd8f434e6be459b2b6f2555536": "\\mathbf J = \\sigma (-\\boldsymbol \\nabla V + \\mathbf E_{\\rm emf})", "d17e668efc770fd36ee02688db560c3c": "R_f.", "d17e9f9e771e6953a2773eeef4b251fa": "\\Omega^{\\psi(\\Omega)}", "d17ea1ebcc73552cf25d1abb00d95e6f": "\\mathbb S", "d17ef9fa5649bbc0baea279da0e7b0dd": "0 = -r \\sin \\phi \\, d\\phi + \\cos \\phi \\, dr ", "d17f586e8e1363a49c84c6c725e43257": " O(d^2)", "d17f740b42afcc953c20a6d9f68254ab": "d(0)\\ge d", "d17fc980ef7cfec2ef50430d74e22179": "a \\frac{x_c}{t_c} \\frac{d \\chi}{d \\tau} + b x_c \\chi = A f(\\tau t_c) \\ \\stackrel{\\mathrm{def}}{=}\\ A F(\\tau).", "d17fedb5eae1eb1a4ba4b5efd0ce5f19": "a := E \\equiv a' = E \\land u' = u", "d17ff673d92903f9e74157742369d7ce": "G_1,\\ldots ,G_k", "d18002b9efbc344188117e64224fb37e": "\\Delta Weight_{front} = a \\frac{h}{w}m", "d1805e389d25941e0c5f6fa1c272aed9": "V_R(s) = \\frac{R}{R + Ls}V_{in}(s)", "d180791a5d7d1fcc6a0006f6c6596568": "\\phi(-\\theta,\\tau)\\phi(\\theta,\\tau)", "d180ac7f7ecdd571633bc7a4390f3f0e": " \\mathbf{A} \\circ \\mathbf{B} = (\\mathbf{A}_{ij}\\circ \\mathbf{B})_{ij} = ((\\mathbf{A}_{ij} \\otimes \\mathbf{B}_{kl})_{kl})_{ij}", "d180c0b5486289d35c8d663d163eee0b": " \\{v_B \\mid B\\text{ is a flag in }\\mathcal{F}\\}. ", "d18105d4c0b729ddfc1190eed295ad9f": "S>0", "d1811017b041813bed0b4346d912f5ae": "\\mathbf{x} \\cdot \\mathbf{y} = x_0 y_0 - x_1 y_1 - \\cdots - x_n y_n \\, ", "d181165cae141626a58c637e86726672": "\n\\phi' =\\frac{e\\phi}{T_e}.\n", "d18153d9a43664882c96dfdf18216f5b": "\\mathbf{D} = \\varepsilon\\mathbf{E}, \\;\\;\\; \\mathbf{H} = \\mathbf{B}/\\mu,", "d18176f841416c5abba2470e66c57727": "N_{\\mathrm{A}} h \\,", "d181c853796b0e5988bf8435e3b09ba1": "\\varepsilon^y", "d181f672ba12333e26a5801f58f5bb36": "\nY_{V} = \\begin{pmatrix} \\begin{pmatrix} \nq_{1,1,1} d_{1,1} v_{1,1}^{\\alpha_{1,1,1}-1} x_{1}^{\\beta_{1,1,1}} \\\\ \\vdots \\\\ q_{1,1,i} d_{1,1} v_{1,1}^{\\alpha_{1,1,i}-1} x_{i}^{\\beta_{1,1,i}} \n\\end{pmatrix} & \\cdots & \\begin{pmatrix}\nq_{1,n,1} d_{1,n} v_{1,n}^{\\alpha_{1,n,1}-1} x_{1}^{\\beta_{1,n,1}} \\\\ \\vdots \\\\ q_{1,n,i} d_{1,n} v_{1,n}^{\\alpha_{1,n,i}-1} x_{i}^{\\beta_{1,n,i}} \n\\end{pmatrix} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\begin{pmatrix} \nq_{j,1,1} d_{j,1} v_{j,1}^{\\alpha_{j,1,1}-1} x_{1}^{\\beta_{j,1,1}} \\\\ \\vdots \\\\ q_{j,1,i} d_{j,1} v_{j,1}^{\\alpha_{j,1,i}-1} x_{i}^{\\beta_{j,1,i}} \n\\end{pmatrix} & \\cdots & \\begin{pmatrix}\nq_{j,n,1} d_{j,n} v_{j,n}^{\\alpha_{j,n,1}-1} x_{1}^{\\beta_{j,n,1}} \\\\ \\vdots \\\\ q_{j,n,i} d_{j,n} v_{j,n}^{\\alpha_{j,n,i}-1} x_{i}^{\\beta_{j,n,i}} \n\\end{pmatrix} \\end{pmatrix}\n", "d1821b4423ef0b37b209939b1d95144c": "x_1, \\, x_2", "d1823e7adf03fc2e90441cb045a5d2b7": "1 + \\sum_{i=0}^d F_i = F_{d+2}", "d18271e4941032b4293d55eec26098a6": "\\,i_t\\,", "d182910e5086c09291e5d2d23db8216c": "a_i\\neq b_j", "d18298f1b0cb6ad3c50257943cd64061": "x^{16} + x^{14} + x^1 + 1", "d182b44b73079b5dc1397b3eff3e5649": " \\sqrt{a^2+r} \\approx a + \\frac{r}{2a} - \\frac{(r/2a)^2}{2(a+\\frac{r}{2a})}, ", "d1834393b229fae2bdadec4543797fcb": "\\gamma \\frac{n}{2}", "d18366993d490c42a955ee9eb15c8244": "6) \\ x=-1\\pm\\sqrt{3}", "d18389482ac9927debe458e26a444119": " S_{ij}=k\\int_{v}\\nabla N_{j}.\\nabla N_{i}dV ", "d183ac9cf0a6f5a247098623486a223c": "\\beta = \\frac{b}{2 l_c}", "d183b6593f4b2e38b950026ef2e7dc65": " \\varphi(x) = 2\\rho(x)\\text{ln}\\left(\\frac{x}{1-x}\\right) - 2 \\int_0^1\\frac{\\rho(t)-\\rho(x)}{t-x} \\, dt", "d183b9bcac1606e4258127aec16520b7": "\\sqrt{10}^\\frac{1}{10}\\,", "d18422a24ad86a3a174f4f697175b919": "\n\\left\\{b+1,\\sum_{i=a}^b g(i)\\right\\} \\equiv \\left( \\{i,x\\} \\rightarrow \\{ i+1 ,x+g(i) \\}\\right)^{b-a+1} \\{a,0\\}\n", "d18550562ecfccd66ae4655fe0bddf68": "\\mathbf x\\cdot\\mathbf V \\cdot \\mathbf x\\ge 0 \\,\\!", "d18566ad9c42cb16a73a5fb39db1a4aa": "V(t)\\,", "d185807de1b45bc2da6230d99f49bef6": "\\sum_{i=0}^2 p_i = 1 \\quad \\mbox{and}\\quad 0\\leq p_i \\leq 1,", "d185854d536af4c661a217ed540b5f85": " \\frac{1}{x!} ", "d1860c52ffc104b6f6726ed8753bdec4": "E_{t-1}U_{t+\\tau}", "d186367e29d858fbe891ed1e2f25d592": "\\gamma = \\frac{1}{\\sqrt{1 - {(v/c_0)}^2}}", "d186807573fd96070ad33b0b3f006fa7": "\\hat{\\mathbf e}_i", "d186a46d8dafd09b03f7e9915c5881ba": "L(\\theta;X)", "d186a8aa73498273e1a7c83eb85bd4e3": "H (x)", "d186c6de74de5843f55a085264c5b138": "\\scriptstyle\\Phi:T\\times M\\longrightarrow M", "d18713cea4c67710275d84fa60a38686": "\\left|\\int_{\\mathrm{arc}}{e^{itz} \\over z^2+1}\\,dz\\right| \\leq \\int_{\\mathrm{arc}}\\left|{e^{itz} \\over z^2+1}\\right| dz \\le \\int_{\\mathrm{arc}}{1 \\over |z^2+1|}dz\\leq \\int_{\\mathrm{arc}}{1 \\over a^2-1}dz = \\frac{\\pi a}{a^2-1}.", "d187566245ca42f24051c15dafac2151": "P(\\operatorname{deg}(v) = k) = {n-1\\choose k}p^k(1-p)^{n-1-k}", "d1878d88716eb3474e6aa92d57f4d847": " S\\in sB(S_1,\\ldots,S_j) \\Longleftrightarrow \\bigcap_{k=1}^j S_k \\subset S\\subset \\bigcup_{k=1}^j S_k. \\, ", "d187c5af7a090cfdd89bf559a9c22958": "\\tilde\\epsilon=\\epsilon_1+i\\epsilon_2= (n+i\\kappa)^2.", "d187f9896ff171048bb400bed97d23e2": "\nC(d) = \\exp(-d^2/V)\n", "d18855ea2d0dfe5892a1cf1185ce1878": "h\\left( L, \\bigcup_{n=1}^N w_n (L) \\right) \\leq \\varepsilon,", "d188cc82c7b827f1628307a88ddb450d": "\\sigma_I", "d188dae1118a91c4cf09ce4e3d5530e4": "X_1, X_2,\\ldots", "d188fb596851c812625d6564c59583f7": "f(z)\\rightarrow\\infty", "d189029f37e21cc32ab12010e7831567": " \\mathcal{L}\\left\\{ R\\left( x \\right)\\right\\} (s) = \\int_{0}^{\\infty} e^{-sx}R(x)dx = \\frac{1}{s^2}. ", "d18976a5901f394107ca77e339976660": "{\\eta_b}", "d1897f59fe9c081d74f80d2e4ea574e2": "T_{02}", "d1898db0f29f1bee790b0d455212d0b5": "\\lambda^{-1}(q) ", "d189a0300644ae1769492f6b6120b3a0": "\\beth_0 = \\aleph_0", "d189b85b5653dae9c7f35e5608390353": "R = \\frac{T + T^*}{2}, \\quad J = \\frac{T - T^*}{2i}.", "d189baa43ea3d92c82e776ad4e3bb154": " \\Delta F\\uparrow ", "d18a04ba5bcd3220a50c5af54bbe1db3": "p(\\tilde{x}|\\mathbf{X},\\alpha) = \\int_\\theta p(\\tilde{x},\\theta|\\mathbf{X},\\alpha) \\, d\\theta = \\int_\\theta p(\\tilde{x}|\\theta) p(\\theta|\\mathbf{X},\\alpha) \\, d\\theta .", "d18a04c6fdaa723742a8880f44317945": "A / \\mathfrak{m} = k", "d18b091dbcd0c4d9e51390395826ed6e": "L(y) = \\frac{1}{w(x)}\\left(-\\frac{d}{dx}\\left[p(x)\\frac{dy}{dx}\\right] + q(x)y\\right)", "d18b114d2fcb793976c3fdb69c0e4e69": "f^{\\prime\\prime}(x) \\cdot f^{\\star\\prime\\prime} (x^\\star(x))=1,", "d18b37d25e0fd23756e22090eaa7dcd0": "\nF = G \\frac{M m}{r^2}\n", "d18b3b009a985e4878f098f5fee75c6b": "ds^2=- G dv^2+2dvdr+r^2 d\\theta^2+r^2\\sin^2\\!\\theta\\,d\\phi^2\\,,\\;\\;\\text{with } G\\,:=\\,\\Big(1-\\frac{M}{r} \\Big)^2\\,,", "d18b6390925145322506b77216cbc027": "P(c) \\to\\ Q", "d18c3658e8ce99d2249a38d734b73676": "\\tau(\\eta,\\sigma) \\equiv (\\sigma - \\eta)/(1-\\bar{\\eta})", "d18c6aa7010910151437895f8aa72d21": "p = {mv\\over \\sqrt{1-\\displaystyle{v^2\\over c^2}}}.", "d18c8edb0f2b586ec458e7bb9502540c": "\\mathrm{range} (A^\\mathrm{T})", "d18ca3f8f247f9266a656dca6f6cc5bc": "r^1", "d18ca9604e28c4529c604f6b37c1555a": " \\hat{p} = -i\\hbar\\nabla ", "d18caeedc38faee2cfc374f134b18f6c": "\\overline{X^i}", "d18cda84c64b25041ba2e3246019ab86": "f(S)=\\min(B,\\sum_{i\\in S}w_i)", "d18d0aba8327c12beb841185ec5c0a39": "DP_{D}^{S}", "d18d48f34ce1de3f20af5d0460c38ec0": " \\pi^{3^2}/e^{2^3}\\approx 9.9998\\approx 10", "d18debdf6f5332f8d9292d857eab0902": " T_{d} = (t_{2} - t_{1}) * \\frac{\\log(2)}{\\log(\\frac{q_{2}}{q_{1}})}.", "d18e0bbc474708b532428009ddda1750": "NF = 10 \\ \\log_{10} (1 + T_{eq}/290) ", "d18e215c129158c08235b0cf303a3cdd": " i = k \\, \\bmod \\, m + 1 ", "d18e313ca91c7fddca1a11f8b0ff3c7b": "\\Box (K \\rightarrow (K \\and \\lnot Q))", "d18e494b14e4ebab382c3075ec675034": "\\det(I + \\epsilon X) = 1 + \\operatorname{tr}(X) \\epsilon + O(\\epsilon^2).", "d18eb5830355575371c04c47a8bb8a0d": "x^{(k+1)} = x^{(k)} - \\alpha_k g^{(k)} \\ ", "d18ed8d1ee1010146f54f108aa4ec45e": "K_R", "d18edfb60f374504cdfc6a1e82c9f1df": "f^*\\gamma^n \\in Vect_n(X)", "d18f7d145ddc6b0941bb0f783ded170d": " \\begin{array}{lcl} \\textbf{g}(\\textbf{z}) & = & \\textbf{0} \\\\ \\textbf{h}(\\textbf{z}) & \\leq & \\textbf{0} \\end{array} ", "d190137b051ece15ca50779fc1ce6690": "\\ln r", "d19037777baca51bee2bf7c72b9acc69": " i \\,", "d1904e99a8acbf7d872c48f21910ba83": "b_1,\\ldots,b_m \\in \\mathbb D\\backslash\\{0\\}", "d1909345b9e73710020d5a28d67ae718": "u(r \\cdot m) = u(f(r) \\cdot m) = f(r) \\cdot u(m) = r\\cdot u(m)\\,", "d1912968bc5cacadd4bc097a0c826997": "\\operatorname{H}^k(B) \\simeq \\operatorname{H}^{k+m}(K)", "d1915f3f03f179ffdfcc067398f7df2f": "2n_{\\rm coating}d\\cos(\\theta_2)=\\left(m-\\frac{1}{2}\\right)\\lambda", "d191703d71772ba63d02e42a2ed1facb": " H=\\frac{1}{2} L_1 \\frac{dQ_1}{dt} ^2 + \\frac{1}{2} L_2 \\frac{dQ_2}{dt} ^2+ m \\frac{dQ_1}{dt} \\frac{dQ_2}{dt} + \\frac{Q_1^2}{2C_1} + \\frac{Q_2^2}{2C_2} ", "d19172929364901bed8a4bd9aece8ef5": " A = \\mathrm{d} N /\\mathrm{d} t \\,\\!", "d191a7ddf179bbc9d1a10c08151a78d9": "\\Delta Y", "d191cd2c93f29388435c66c8f540b914": "E \\in \\mathcal{F}", "d191ea7dcabb77680edf94fa5a19d5a3": "\\textstyle I(d)", "d1921b2a325fb848e92f27011879bb4c": "z_\\mathrm{n}", "d1927859c5b499a6f5e9f57ec345f4b1": "k' = k + K,", "d192c0491e2ffa30bc085b25ba09dd78": "\\sin \\theta = \\theta - \\frac{\\theta^3}{6} + \\frac{\\theta^5}{120} - \\frac{\\theta^7}{5040} + \\cdots ", "d192e0c4ad64a9c35fe32972477e4cd8": "0,1", "d192ea9d1a40a09ff1bd95ca07e7ec9e": "\\prod_{k=0}^{\\infty} (1+q^kt)=\\sum_{k=0}^\\infty \\frac{q^{k(k-1)/2}t^k}{[k]_q!\\,(1-q)^k} ", "d19349d457025e191bc4f9e3e1532e7b": "G'=G^\\frac{1}{2.19921875},", "d19354be93c2d2528123e71593584dde": "\\frac{dx+dX}{dX} \\approx 2\\,\\!", "d1937b97e8eff321a806b8562162ff48": "\\mathbf{v} = \\frac{d\\mathbf{x}}{dt} = g(t),", "d193c8a4642f1031167b938c61e5d633": "P_a = P_r", "d194796ff9a418d1d09c16c7a1befb09": "K^*(X)\\to H^{2*}(X,\\mathbf{Q})", "d1949dde7ef7a1e2092cae3cbd4a5f66": "\\textstyle \\tilde{D} ", "d194de58aa8df3bc8c808d7b76010af8": "\n \\qquad \\qquad \\frac{\\partial [(D+h)u]}{\\partial x}+\\frac{\\partial [(D+h)v]}{\\partial y}=0 \\qquad \\qquad \\qquad (3)\n", "d194f99a74da426b534d56f63f564db9": "\\Sigma_0", "d195027d77cb619ac78019ef77af5012": "Ch^p", "d1954c125c4bc3158ee18c2dddd4114b": "r_{k}, r_{j}", "d195582ee98a873eb58ec1aac559d5d6": "x^* \\in \\mathcal{E}^{(k)} \\cap \\{z: g^{(k+1)T} (z - x^{(k)} ) \\leq 0 \\}.", "d195823422acc170f151a8c1d3ff33cd": "\\Gamma\\models A,", "d196417b02814f6502528c3650d5464b": "\\sigma_\\text{a} = \\sigma_\\text{fat}\\times\\left(1-\\frac{\\sigma_\\text{m}}{\\sigma_\\text{ts}}\\right).", "d196bb801886d927d9ea469e034394ad": "(N \\cup T)^{*}", "d1975296fd648f2c22c3b66ea42c27be": " a = \\frac{n}{G \\plusmn \\sqrt{(n-1)(nH - G^2)}} ", "d19796b639153eedcc370922337e239e": "\\gamma(\\mathbf{r})", "d197c6844531bfb401aea19a666b29a5": "X\\subseteq Y \\Rightarrow \\operatorname{cl}(X) \\subseteq \\operatorname{cl}(Y)", "d1980eaca78c23ad3a6338832d934aeb": "\\Delta\\omega", "d19812f05fa2b8be415fc46459930daf": "1/RR.", "d1981aa3501314fff4e541965aba69fd": "\\textbf{P}_{k|k} = \\textrm{cov}((I - \\textbf{K}_k \\textbf{H}_{k})(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k-1})) + \\textrm{cov}(\\textbf{K}_k \\textbf{v}_k )", "d1988d0ce8f0db2ea74baa3da7de9a21": "u'_j=(-1)^{n+j}\\frac{W(y_1,\\ldots,y_{j-1},y_{j+1}\\ldots,y_n)_{0 \\choose f}}{W(y_1,y_2,\\ldots,y_n)}.", "d198d52e656e0be5e9cdd663057c0f44": " \\sum_{i=1}^n \\sum_{j=1}^n \\left( x_i y_j - x_j y_i \\right)^2\n\n= \\sum_{i=1}^n x_i^2 \\sum_{j=1}^n y_j^2 + \\sum_{j=1}^n x_j^2 \\sum_{i=1}^n y_i^2 \n- 2 \\sum_{i=1}^n x_i y_i \\sum_{j=1}^n x_j y_j ", "d198fbbbb0ca00b3304577b031252f79": "= \\frac{2L_{T} \\gamma (v)}{c}", "d199167ea8bbe5a4f963906c559611bf": "a_{21}", "d1992397991d6e692162285ba1da8ad0": " r = \\frac{\\bold{r}\\cdot(\\dot{\\bold{r}}\\times\\bold{H})}{\\mu + c\\cos(\\theta)} = \\frac{(\\bold{r}\\times\\dot{\\bold{r}})\\cdot\\bold{H}}{\\mu + c\\cos(\\theta)} = \\frac{|\\bold{H}|^2}{\\mu + c\\cos(\\theta)}", "d19964e09caed59fb61cc0f7ce744afe": "\\mathcal L_{\\Omega_\\alpha}\\omega = 0", "d199801e749af52b16f9f3804b4ce821": "\\mathfrak{H}^n", "d199e60cefb887374fde625309555a54": "S_N = \\sum_{n=0}^N a_n b_n", "d19a190bc54de4ef42c0b772a634ea3c": " (1-\\gamma_e \\lambda_{De}^2\\nabla^2)n_{e1} = \\sum_iZ_in_{i1} ", "d19a1e14cfd2510eaf1f3b1e150c4a13": "y=tx=\\frac{2at^3}{t^2+1}", "d19a62c315ef760aee1236ce7260db8e": "Z = X+Y \\sim N(0, 2).", "d19aa991b9d53339b96f6480875ae9a2": "(Q \\and P) \\vdash (P \\and Q)", "d19ac8d7069f4c1144f62bc9f8f87213": "\nU^* = \\begin{bmatrix} \\rho^* \\\\ u^* \\end{bmatrix} = \\beta_1 \\begin{bmatrix} \\rho_0 \\\\ -a\\end{bmatrix} + \\alpha_2 \\begin{bmatrix} \\rho_0 \\\\ a \\end{bmatrix}\n", "d19ae293f1cb56d5302382d59a179fda": "\\tfrac{m^2}{s^3}\\;", "d19b108faf5f203e1526bce1f6e28098": "\n\\mathbf{L}(\\mathbf{X} | \\tau, \\mu) = \\prod_{i=1}^n \\mathbf{L}(x_i | \\tau, \\mu) .\n", "d19b6e33c1c96c1a421df7a0cb2470fd": "\\frac{1}{2}\\log(2 \\pi e \\lambda) - \\frac{1}{12 \\lambda} - \\frac{1}{24 \\lambda^2} -", "d19b7ba53cb843c27407a5807e506e4c": "b^{p^q} = b^{(p^q)} \\ne (b^p)^q = b^{(p \\cdot q)} = b^{p \\cdot q} .", "d19b86a4cc218b73ede9bc4e0d862c9b": "p_{\\theta_{r}}\\left(\\theta_r\\right) = \\frac{1}{2\\pi}e^{-2\\gamma_{s}\\sin^{2}\\theta_{r}}\\int_{0}^{\\infty}Ve^{-\\left(V-\\sqrt{4\\gamma_{s}}\\cos\\theta_{r}\\right)^{2}/2}dV", "d19baea22b8397e06b6e8951cccb2599": "\\hat f(\\xi) = A(\\xi) e^{i\\varphi(\\xi)}", "d19c813e4e38006f57a247104d734a30": " u, c, t ", "d19cc651d829cc72ab10d3a0873a49a8": "0\\rightarrow H^2(E/F)\\cap H^2(E/E\\cap G_\\infty) \\rightarrow H^2(E/E\\cap G_\\infty)\\rightarrow H^2(F/F\\cap G_\\infty)", "d19cd362ffba637255b316f96b4bf721": "\\lambda = {\\varepsilon \\over 3B} ", "d19d34f6f9b956bd10368f3f0a833977": "e_a = i_a r_a + { {d \\varphi_a} \\over {dt}}", "d19d879ca15b882cfbca27dc5ea641e1": "789 \\times 345 = 272205", "d19daf9d7e88cbe5731515566c1fcd8b": " L_{1,\\infty} = \\{ A \\in K(H) : \\mu(n,A) = O(n^{-1}) \\}. ", "d19db775ba048337885402bfcceb6ff2": "p_{I}= r B", "d19dba4e3b1256329b9bafac72ff602e": "{{i}_{c1}}\\approx {{i}_{e3}}", "d19df3e42f2e36e3bc75214465b05645": "m=-\\frac{v}{u}=-\\frac{v-f}{f}", "d19e51e05edadc409623dc04f100e1a7": "\nV(A)=\\sum\\limits_{j} P(O_j | A) D(O_j),\n", "d19e83a19240ce010b51f94883330260": "\\alpha \\in k", "d19eabd35872a18332874ff83f70cb8b": " (\\neg \\neg A \\or A) \\rightarrow A ", "d19ec6958a4e179956e670d006abc995": "\n \\gamma_{n,c}= \\sum_{q=1}^{Q}\\gamma_{n,q} \\quad \\quad \\textrm{for} \\quad n=0,1,2,3,4\n", "d19f6630f26a3a663aa1bdc432eb1701": "\\mathbf{x} = \\mathbf{y} + \\mathbf{Ax}", "d19f9ee04886b9fb40c1a0abd847ada5": "(4 \\pi r^2 N/V) dr", "d1a01be44e4d6a622b72778af1be0430": "(\\pi_i^\\text{pr} (F, s))(f) = \\pi_i (F(Y), f^*(s))", "d1a042d1a934c631686bfc28ba2d9961": "\\varphi = \\cot \\varphi_1 + \\varphi_1 - \\rho\\,", "d1a124c6b9c62f857b8ce86c6277a8ec": " \\mu \\circ (S \\otimes \\mathrm{id}_H) \\circ \\Delta = (\\mathrm{id}_H \\otimes \\varepsilon) \\circ (\\mathrm{id}_H \\otimes \\mu) \\circ (\\sigma_{H, H} \\otimes \\mathrm{id}_H) \\circ (\\mathrm{id}_H \\otimes \\Delta) \\circ (\\mathrm{id}_H \\otimes \\eta) ", "d1a1951a8508626af8402f7b51f12608": "\\mathbf{C_1, C_2}", "d1a1c517eb148c2143b03a1ff0bc737a": "|V(H)|-1", "d1a1cc9e013edf29c8046c92040efc70": "E, X, D", "d1a22371d200f13f38920a8fdcab705b": "\\frac{S\\text{ in }A\\text{ planum }+\\text{ Rbis in }A\\text{ planum}}{R} \\text{ aequabitur }B\\text{ plano}", "d1a2296773e2a0c20e68200441e86958": "\\mu \\in \\Re ", "d1a2694636c7ea1c9abd6b44d298719d": "F_t(x,y,t)=-8tx+4y-1", "d1a26babf31782a519d2cacacd46086a": "\\frac{\\mbox{Net Income}}{\\mbox{Average Shareholders Equity}}", "d1a29aabdeae0178b5533d0cfe98fa87": "\\triangledown^2 v' = O\\left({a^3\\over r^3}\\right).", "d1a2df0a8a2c325dc19b8669bba2398d": "f^{(0)}_{n,k}", "d1a3074f5ffb2649cb4812a32e7f3cc3": "x_i \\in \\mathbb{R}^n", "d1a32d5717463d68726fd375909c6b9d": "S=k_B \\ln \\Omega_E ", "d1a372ae9d2a052cba05ca9f93870635": "y=\\lceil n/2k \\rceil", "d1a37814e17381b0919cc8d0500d1066": "a \\triangleleft S_0", "d1a388717d484545786076f16035448c": "S=\\{-1,1\\}", "d1a3935653b88ffb873bbf69ad55415a": "\n\\begin{bmatrix}\n1 & 2 & 1 \\\\\n2 & 4 & 2 \\\\\n1 & 2 & 1\n\\end{bmatrix}\n", "d1a3c55a807992761de0e120636175ba": "\nH(jf) = \\frac{1}{1+j2\\pi f R C}.\n", "d1a3cced5a01829da8ce6ea69b2f141a": "\\mathbf{Q}_{\\mathbf{XY}}", "d1a3f35be4c3065860e902b966a798ee": "|\\alpha_1-s_\\kappa|<\\min{}_{m=2,3,\\dots,n}|\\alpha_m-s_\\kappa|", "d1a3fa1aba369e7b2a03ad68c55a05dc": "s = 0", "d1a40ab591704560d43501f9deb0febe": "V_e = I_{sp} * g_n", "d1a52bf78818e5e77fa8cb1e0b7beb31": "m'=0.4134\\,", "d1a5e0e6d608e370f411fa502bd3accb": " \\operatorname{Arg}: \\mathbb{C} \\smallsetminus \\{0\\} \\to (-\\pi,\\pi] ", "d1a64a19b63ea5141125681311c0dca2": "v_h", "d1a6b26be266a63a71c604638cf8a3b1": "\\frac{d\\mu}{dn}", "d1a6f4e5f3d3b241a1ed40a97a55c48b": "\n \\varepsilon_{xz} = \\frac{1}{2}~\\kappa~\\left(-\\varphi + \\frac{\\partial w}{\\partial x}\\right)\n", "d1a7645517e064fb52ae90401e521c84": "(s,e)", "d1a786ef6f05385ec454f9281205dfec": "\\scriptstyle 1/\\sqrt{2}", "d1a797792abbe276aef200d33157a915": "\\scriptstyle y[n]", "d1a7c81b133cc72fa61f5339cf8e4781": "(X,\\mathcal{A},\\mu, T)", "d1a7ec191e41c943feb23fb1f01d1905": "\nD_{N}^{*}(x_1,\\ldots,x_N)\\leq\n\\left(\\frac{3}{2}\\right)^s\n\\left(\n\\frac{2}{H+1}+\n\\sum_{0<\\|h\\|_{\\infty}\\leq H}\\frac{1}{r(h)}\n\\left|\n\\frac{1}{N}\n\\sum_{n=1}^{N} e^{2\\pi i\\langle h,x_n\\rangle}\n\\right|\n\\right)\n", "d1a7ec9a1ad6ad3fbc0e84a77b94cecd": "\\begin{alignat} {2}\n\n \\bar{n}_i & = \\sum_{n_1,n_2,\\dots} n_i \\ P_{n_1,n_2,\\dots} \\\\\n \\\\\n & = \\frac{\\displaystyle \\sum_{n_1,n_2,\\dots} n_i \\ e^{-\\beta (n_1\\epsilon_1 + n_2\\epsilon_2 + \\cdots + n_i\\epsilon_i + \\cdots)} }\n {\\displaystyle \\sum_{n_1,n_2,\\dots} e^{-\\beta (n_1\\epsilon_1 + n_2\\epsilon_2 + \\cdots + n_i\\epsilon_i + \\cdots)} } \\\\\n\n \\end{alignat} ", "d1a803c613ad0464d55cc8e2e24347e2": " r < |z-a| < R.\\,", "d1a819ff19b54426dbc04c2c2ec0b94f": " uq\\equiv 1 \\pmod{p - 1}", "d1a8acaec8fda528796b94f91c44e677": "_{q \\nleftarrow p=p \\nleftarrow q\\,}\\!", "d1a8b28facf0e62de89fc6db8f458145": "\n\\left(\\frac{N}{c}\\right) \n= \\pm 1\n\\equiv N^{(c-1)/2} \\pmod c.\n", "d1a8b6f957ab6df9c13194332a7b2238": "\\phi_e \\notin F", "d1a8bd9e0fa6529a28cd8761f56e3eb7": "H_\\mu (s,t)", "d1a8cabf9a979e0bbf19fb882c157862": " \\frac{1}{2}(\\mathbf{ab} + \\mathbf{ba}) = \\frac{1}{2} ((\\mathbf{a} + \\mathbf{b})^2 - \\mathbf{a}^2 - \\mathbf{b}^2)", "d1a922f75925b72e89600feeffc1ee3b": "\\tfrac{G(4G-E)}{3G-E}", "d1a937a0b493e8d9cfd4685c6fc7cf9a": "\\begin{matrix}\n m l^2\\ddot \\varphi_0 = -\\frac{\\partial V_{\\mathrm{eff}}}{\\partial \\varphi_0} \\;,\n\\end{matrix}", "d1a9741f64a180bd4969f00c7dad07c0": "\\forall x \\isin V : \\forall y \\isin V : \\exists \\text{path}(x,y)", "d1a99aecbd33f489e139ad633fd1e10c": " \\text{angle in radians} = \\text{angle in degrees} \\cdot \\frac {\\pi} {180^\\circ}", "d1a9b937e4c1dc46ea6f216f045364c0": " V(t=0) = L \\frac{di}{dt}(t=0) = -\\omega_0 L I_0 \\sin( \\phi ) .\\,", "d1a9d3d2c18f84a7b022749db519e4d9": "\\left[-\\ -\\right] : C^{op} \\times C \\to C", "d1a9e279841b895ca70f57b6137dd3e4": " i-1", "d1aa50172de5f5fb6936468c92339e0c": "\\sum_{p_{S_{i}\\setminus S_{j}}\\in A_{S_{i} \\setminus S_{j}}} \\alpha _{i} (p_{S_{i}}) \\prod_{{v_k \\operatorname{adj} v_i},{k \\neq j}} \\mu_{k,j}(p_{S_k\\cap S_i})(1)\n ", "d1aa60c62bee8c8789195b28889a31f5": "\\lambda C", "d1aa736697f40251ef9734f79daca131": "w_2(M)", "d1aa9189fa97acf57c1de584328dc0aa": "\\alpha = \\int^{\\infty}_{G_0} N(0,\\rho)dG = 1 - \\Phi(\\frac{G_0}{\\sqrt{\\rho}}).", "d1aaf716ce75c866152e3cb9d597dc9c": "\\varphi_k", "d1aafde4a5c14ef2996c89b77f0f7337": "v_4", "d1ab6da24e7e1a261eb4447de004f7e1": "f': X \\to S'.", "d1acb51e55432393e308efa19f5e04e7": "\\tilde{p}=\\frac{r}{r+k},", "d1acd39da4e1140f49e5e2c4216bd1ac": " \\int_{\\overline{\\mathcal{M}}_{g,n}} (-1)^j \\lambda_j \\psi_1^{d_1} \\cdots \\psi_n^{d_n}, ", "d1acea886fdc7a1c8670e9445938e9a3": "\\boldsymbol{\\mathcal{Q}} = (\\mathcal{Q}_1,\\mathcal{Q}_2,\\cdots \\mathcal{Q}_N)", "d1ad6aadedc7f81663f2138216a7bdec": "\\displaystyle \\frac{2\\pi J_1\\left(\\sqrt{\\nu_x^2+\\nu_y^2}\\right)}{\\sqrt{\\nu_x^2+\\nu_y^2}}", "d1ad6ceb84045eb0d2e71b7c5bbffd85": " \\bar{x} = \\left(\\prod_{i=1}^n x_i^{w_i}\\right)^{1 / \\sum_{i=1}^n w_i} = \\quad \\exp \\left( \\frac{\\sum_{i=1}^n w_i \\ln x_i}{\\sum_{i=1}^n w_i \\quad} \\right) ", "d1ad990f264ed0798a510e747708bf26": "g^2(q;\\tau)= 1+\\beta\\left[g^1(q;\\tau)\\right]^2", "d1adbe487c526b7d85aa64ebfd9c4dad": "24^{2}\\tbinom{1,312,000}{2}", "d1adf5833cea752c13ed69885d4c1d52": "\\begin{bmatrix}\n1&0&0&0&1&1&1&1 \\\\\n1&1&0&0&0&1&1&1 \\\\\n1&1&1&0&0&0&1&1 \\\\\n1&1&1&1&0&0&0&1 \\\\\n1&1&1&1&1&0&0&0 \\\\\n0&1&1&1&1&1&0&0 \\\\\n0&0&1&1&1&1&1&0 \\\\\n0&0&0&1&1&1&1&1\\end{bmatrix}\n\\begin{bmatrix}0\\\\0\\\\1\\\\0\\\\1\\\\1\\\\0\\\\1\\end{bmatrix}\n+\n\\begin{bmatrix}1\\\\1\\\\0\\\\0\\\\0\\\\1\\\\1\\\\0\\end{bmatrix}", "d1adf961be9aafba78c1d6ca032b41b2": "\\theta(t) = \\arg \\left(\n\\Gamma\\left(\\frac{2it+1}{4}\\right)\n\\right) \n- \\frac{\\log \\pi}{2} t", "d1adfa648c4c4871726418d47f7fb247": "\\langle A, B\\rangle = \\operatorname{tr}(B^* A)", "d1ae0a01471443cf86b902fcb11f7cba": "u(a)=u(b)=0.", "d1ae2ec40b41a00b1f1fd5f3e51728f6": "( \\lfloor nr \\rfloor)", "d1ae4c8560db039173618cf04daa84cb": "\\cos_k^2(i)\\equiv (2^{-1}\\bmod{p})\\cdot(1+\\cos_k(2i)),", "d1aeaad784e91dabb2a9bcae3f1fdacc": "\\square\\cdot\\left(\\frac{\\mathbf{q}}{\\theta}\\right)~+~\\rho~\\frac{\\partial s}{\\partial t}~=~\\sigma ,", "d1aec060bbc95980b695b94219a7bfae": "\n\\lambda (x) =\\text{card} \\left\\{ i\\ |\\ x\\in X_{i}\\right\\}. \n", "d1aedbebd075423e649c549757e048e5": " (\\varphi^*S)_x(X_1,\\ldots, X_s) = S_{\\varphi(x)}(\\mathrm d\\varphi_x(X_1),\\ldots \\mathrm d\\varphi_x(X_s))", "d1aedeff406b9d115d5b4bb5e7959909": " \\varepsilon ^2 = (\\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_4)^2 = \\mathbf{e}_1 \\mathbf{e}_2\\mathbf{e}_3 \\mathbf{e}_4 \\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_4 = -\\mathbf{e}_1 \\mathbf{e}_2\\mathbf{e}_3 (\\mathbf{e}_4 \\mathbf{e}_4 ) \\mathbf{e}_1 \\mathbf{e}_2\\mathbf{e}_3 = 0,\\!", "d1af0e6da67b5036a05725ed7bc22f98": "T \\frac{\\mathrm{d}S}{\\mathrm{d}t} = \\dot Q + \\dot n TS_m.", "d1afb72c03c5d6e06399c090998dc85f": "\\gamma_r(\\theta):=f(r\\,e^{-i\\theta}),\\qquad \\theta\\in[0,2\\pi].\n", "d1b04d3b43d59b311e2dc102e6e909cb": "X' \\subseteq X", "d1b0a4f2ed718a91c81c90f0fc12010e": " f': B \\rightarrow A \\otimes_{C} B ", "d1b1620ed1d4972cb54f857667a9864a": "\\mu(T)\\,=\\,\\mu_0 \\exp(-bT) ", "d1b16b1f510e1f1f0cbc277d5a9f45d8": "\\mathbf{F}=0", "d1b17f829c6315e337452dedfd0badef": "V = (HH \\cdot TR \\cdot TU) + (HH\\cdot TR\\cdot MR \\cdot RR \\cdot RU)", "d1b1af1e382c647052ad4da2e0ce5b9e": "1 + \\frac 1 2 + \\frac 1 3 + \\cdots + \\frac 1 n = \\sum_{k=1}^n \\frac{1}{k},", "d1b1e1b3f5717f6503ba7a256161951d": "y = x \\cdot \\dot{f}^{\\star-1}(x) - f^\\star\\left(\\dot{f}^{\\star-1}(x)\\right).", "d1b1f81e1a4012534b16c818f2f45d76": "\\mathcal{H}_0", "d1b21919bec6881b8d5b05a409d4fda8": "\\operatorname{Aut}(A_n)", "d1b2196508f5da0f6602bc74b9b263f9": "\\Leftrightarrow", "d1b23246733f30670255de605530e15e": "\\lesssim10^{-8}", "d1b256eab5fcd69b72c39f557ad241ed": "\\dot{v}_a \\ll \\omega v_a", "d1b26b0f847792599a8f160b422fe60f": "\\chi(G)\\le\\left\\lceil\\frac{\\Delta+\\omega+1}{2}\\right\\rceil.", "d1b2e52808d71aa89a427c7b04b4ed9b": "\\sqrt{3}.", "d1b2e61bb1d21d48f6f2447d60595876": "f_a(o,s)", "d1b3033285f44d4f785f647002061913": "\\tau_\\mathrm{n}^2\\,\\!", "d1b31d4f1eb20937a67a53071f09753f": "\\scriptstyle9.4(8.1)\\times10^{-11}", "d1b31e7302417c3a79581abd10a07189": "i\\colon X \\hookrightarrow \\mathbf{R}^m,", "d1b3731492cbe1470dc502467202f16f": " K^{\\times}/N_{L/K}(L^{\\times})", "d1b3e4a5871e4556ff184075e5e81c5d": "L_{4k+2}", "d1b3faffb37d7d5707827e05d2f0b113": "\\mathbb{F}_{2^{1971}}", "d1b4033fca376da3e2acb99041bc801b": "F_t = A \\int f(c) + \\eta Y (c-c_0)^2 + K\\left(\\frac{dc}{dx}\\right)^2~dx", "d1b40d2078c2e642e3ace994468df994": "\\widehat X_i(t)", "d1b421da90c6e9c68a603355034801e2": "\n\\begin{matrix}\nC_{t3} &=& 1.2 \\\\\nC_{t4} &=& 0.5\n\\end{matrix}\n", "d1b42d16b60666ad1a2d48e9bc29ea6c": "H_k = \\sum_{n=0}^{N-1} x_n \\left[ \\cos \\left( \\frac{2 \\pi}{N} n k \\right) + \\sin \\left( \\frac{2 \\pi}{N} n k \\right) \\right]\n\\quad \\quad\n k = 0, \\dots, N-1 ", "d1b45a7da7780827bff89a331cbade20": " V(x_1,x_2,\\cdots x_N) = \\sum_{n=1}^N V(x_n) \\, .", "d1b4b61e1abdfa01845b297e1ea88b2b": "\\begin{align}\n E_\\textrm{confinement} &= \\frac{\\hbar^2\\pi^2}{2 a^2}\\left(\\frac{1}{m_e} + \\frac{1}{m_h}\\right) = \\frac{\\hbar^2\\pi^2}{2\\mu a^2}\\\\\n E_\\textrm{exciton} &= -\\frac{1}{\\epsilon_r^2}\\frac{\\mu}{m_e}R_y = -R_y^*\\\\\n E &= E_\\textrm{band gap} + E_\\textrm{confinement} + E_\\textrm{exciton}\\\\\n &= E_\\textrm{band gap} + \\frac{\\hbar^2\\pi^2}{2\\mu a^2} - R^*_y\n\\end{align}", "d1b4ff6f81bc02f7deec33c50c7deb23": "GF = 1 + 2\\nu ", "d1b504f35dd56f33f959a0d3f1c529bf": "P(X_1=X_2)=P(X_1=X_2=0) + P(X_1=X_2=1)=P(X_1=0, X_2=0) + P(X_1=1, X_2=1)", "d1b526f8cf6f5491622ec5cea747ecdb": "F^{-1}(Y_\\alpha) = X_\\alpha", "d1b5626bae42fe05b6892ace2fe71a05": "= AA^{-1} + uv^T A^{-1} - {AA^{-1}uv^T A^{-1} + uv^T A^{-1}uv^T A^{-1} \\over 1 + v^TA^{-1}u}", "d1b5af341fd9556966747d0ec07d8aff": "A(\\beta,t) = A(\\beta,0) \\exp[ R(\\beta) t] ", "d1b5fa0a5afebd4897574db17dd185f2": "G=\\frac{2}{\\pi}\\int_0^1\\frac{dt}{\\sqrt{1-t^4}}= 0.8346\\ldots.", "d1b605d0e1f6e20545f91f6b27848e48": "\\gamma_{ij}\\gamma^{jk}=\\delta_i{}^k", "d1b649d82ee0b107e6da361685706573": " \\sum_{n \\in \\mathbf{N}} \\|a_n\\| < +\\infty.", "d1b6a20450baf2b957a5430b64c10468": "\\left( \\frac{\\partial \\gamma}{\\partial T} \\right)_{A,P}=-S^{A}", "d1b6c7ea2f724fa44921b00abda70c8a": "y_k = \\mathbf{h}_k^H \\sum_{i=1}^K \\mathbf{w}_i s_i + \\mathbf{h}_k^H \\sum_{i=1}^K \\mathbf{e}_i s_i+ n_k, \\quad k=1,2, \\ldots, K", "d1b6f74432f909d28a47815b82212eaf": "z_1\\times z_2", "d1b735c33708daf89d886fb8397684a2": " y z = {y \\over z} ", "d1b750dd696279f94d2692b0d9134cb0": " (\\lambda x.\\operatorname{de-let}[f\\ (x\\ x)])\\ \\operatorname{get-lambda}[x, x\\ q = f\\ (q\\ q)] ", "d1b7631882045faa9439ffbf0f3d7f54": "\\mathbf{e}_{23} = j ", "d1b77cbaf9fb8dbd4da2ea40cb08a05f": "\\tfrac{\\lambda(1-2\\nu)}{2\\nu}", "d1b79306d2f6a9b64693bccb91ac1e75": "(q+1-2\\sqrt{q},q+1+2\\sqrt{q})", "d1b7e225b6d58ff25337506889cd433e": "\\cosh \\frac{x}{2} = \\sqrt{ \\frac{1}{2}(\\cosh x + 1)} \\,", "d1b805c1265757c14f4aee656ea671bf": "b=2mn", "d1b80c8cd3ef28ee468995a11a9da284": "X_t = \\nu t + \\sigma W_t\\quad\\quad\\quad\\quad", "d1b862a4a0356ab0546adf075db6c12d": "\\nabla P", "d1b8df3f5ab2a4335c0931c94bdcdf3b": "Q_n=(1+\\sqrt 2)^n+(1-\\sqrt 2)^n.", "d1b8ea6d6573d14455c12627132198ca": "\\exists^*\\forall^*", "d1b90bf2fc2722961395407d407b64e1": "\\dot{x} = -x", "d1b90d92aa2b01568cd6a193ea36704e": "x_i=X_i(\\omega)", "d1b90fcf46fceed3ae81e13ebe5d7ec3": "\\left(\\begin{array}{rrrrrr}\n 2 & -1 & 0 & 0 & -1 & 0\\\\\n-1 & 3 & -1 & 0 & -1 & 0\\\\\n 0 & -1 & 2 & -1 & 0 & 0\\\\\n 0 & 0 & -1 & 3 & -1 & -1\\\\\n-1 & -1 & 0 & -1 & 3 & 0\\\\\n 0 & 0 & 0 & -1 & 0 & 1\\\\\n\\end{array}\\right)", "d1b92d98abbeede0ef2f19df5247a71d": "\\nabla = \\left ( \\frac{\\partial}{\\partial x_1} , \\dots , \\frac{\\partial}{\\partial x_n} \\right ).", "d1b939d7faa2fc8d49d976ee05e3d1a5": "S_{slotted} = Ge^{-G}", "d1b93fcb5c23327a073bb5ab29af3c33": "\\tau_{c,D_i}", "d1b9611b2e104fdd2150bcf8eed29ca1": "\\sqrt 2\\approx\\frac{577}{408}", "d1b975ae4fdce37082c9ebb0a71cd74a": "\n\\begin{align}\n k(\\lambda,\\phi)&=\\frac{k_0}{(1-\\sin^2\\lambda\\cos^2\\phi)^{1/2}},\\\\\nk(x,y)&=k_0\\cosh\\bigg(\\frac{x}{k_0a}\\bigg).\n\\end{align}\n", "d1b98829f4c22d20044a3d3d0934376f": "\\ E\\{ |y_v|^2 \\} = 1.\\,", "d1b9bcb651f394b77375766e55da4958": "(d-1)/2", "d1b9c429147dbda1f522938db129e9f8": "\\mathbf F", "d1b9d33f919d3903086c8cffe621aed4": "2s^2\\,", "d1ba13180a0aabe2cb09c0253e6ab851": " P(N(D)=k)=\\frac{(\\lambda|D|)^k e^{-\\lambda|D|}}{k!} .", "d1ba1826b75c9582c603f8670ec785e4": "j\\,", "d1ba52144ed9f07b844d06b7a9f15c8a": "\\omega = 2\\kappa \\frac{k}{1+k^2},", "d1babe0c558262572630e33fb4a3afbb": "\nF = -\\frac{dV}{dr} = -\\frac{mc^{2}}{2r^{4}} \\left[ r_{s} r^{2} - 2a^{2} r + 3r_{s} a^{2} \\right] = 0\n", "d1bac18d1cbcf06f4f2411bbbaf16aec": " H_n^{(c)}=+1+\\frac{1}{2^c}+\\frac{1}{3^c}+\\cdots", "d1bb3a3effeb89338b345742efc95673": "r_i = \\frac{a}{6} \\sqrt{6} \\approx 0.4082482\\cdot a", "d1bb5e592842260edef58818350cfd8b": "\\gcd(p, q)=\\gcd(p, qr)", "d1bba36500fccb041957fa8faf351ccd": "Q^\\tau(i)=\\left\\langle \\sum_{t=0}^{\\tau-1} \\prod_{j=0}^t \\beta[Z(j)] \\right\\rangle_{Z(0) = i}", "d1bba52f8a4e7b2752aa7f1fd5749a0f": "{{(Y-Y^{*})}\\over{Y^{*}}} = -\\beta{}(u-\\bar{u})", "d1bbbea8fb8f7ee897b6223c65794a06": "\\alpha_\\mathrm{\\{per\\ comparison\\}}", "d1bbf4a9a80edb117bb048414354d6bc": "C_H = \\frac{[H] + [HA] -K_w}{[H]}", "d1bcfa075cf64b5dbdfac1fd8a00adfc": " \\and ((T_3 = [F_3, S_3, A_3]::K_2 ", "d1bd098f8291a660efd4474fd5dfb1ef": " \\int |f| \\, d \\mu < \\infty, ", "d1bd0ae3fa5898ea0d482c3e40eda48e": "{\\lambda}_n\\,", "d1bd10258aea843491f9fd7e2880d208": " \\displaystyle{m(g,z)=cz+d}", "d1bd83a33f1a841ab7fda32449746cc4": "2.0", "d1bd8ac211df24b10adf4159a33efffd": "\\Gamma^{(\\lambda)}\\,", "d1bdb3e708ae000d41d52720b5ea5ef5": "\\lVert x x^* \\rVert = \\lVert x \\rVert ^2", "d1bdf1e038f9bea5c4e37a0444873c8e": "q=-\\exp(-\\pi \\sqrt{163})", "d1be158bbb7a379f810ca13b83eb5bc8": "\\mathbf{v}_i\\in \\mathbb{R}^3", "d1be92fe69ad32fdbb329c57d67c2a48": "\\mathrm {EV} = \\log_2 {\\frac {N^2} {t} } \\,,", "d1be9b968294148678c1f9f050449e9a": "x\\mapsto\\sum_{uv=x}f(u)g(v)=\\sum_{u\\in G}f(u)g(u^{-1}x).", "d1c04407a051fabe2428691c7d87d986": "\\mathrm{CO_2 + H_2O \\xrightarrow{Carbonic\\ anhydrase}\nH_2CO_3}", "d1c0756dead59a48cb10cdbef9f6e37c": "z= u-\\frac {\\langle u, v \\rangle} {\\langle v, v \\rangle} v.", "d1c0c672981bf4e8e31b4ea176c7c28d": "f = \\frac{d X_*P}{d \\mu} .", "d1c0f9e2747d4acdf8e13ae61c5077be": "p_2 = x_1^2 + x_2^2 + x_3^2\\,,", "d1c112a5f50b51b61072f068020efe66": " \\int f(x)\\,dx . ", "d1c186d1389b829fa2caa84144629366": "\\tau = 0\\,", "d1c1b31f5667688a288bb3e6f3b2ce0c": "g\\in L^2(R)", "d1c1cd4ebb2714f365b515aaa5071b16": "E\\left({S \\over N}|n,s=0,N\\right)={1 \\over N}\\sum_{S=1}^{N-n}S P(S|N,n=1,s=0)={1 \\over N}{\\sum_{S=1}^{N-n}\\prod_{j=1}^{n-1}(N-S-j) \\over \\sum_{R=1}^{N-n}{\\prod_{j=1}^{n-1}(N-R-j) \\over R}}\n", "d1c23bee6d4fdef8012363c89457ddbd": "{V_{8}}", "d1c25f8367f423856e9a583f7d1a6e03": "(X,Y,Z,Z^2,Z^3)", "d1c27d545360ac3be80d14757deca4b1": "R_{k+1}(*, *)", "d1c2a152d941280b95fc9a6c454e79da": "(x, y, z) = \\left(\\frac{2 X}{1 + X^2 + Y^2}, \\frac{2 Y}{1 + X^2 + Y^2}, \\frac{-1 + X^2 + Y^2}{1 + X^2 + Y^2}\\right).", "d1c2ff0d117db81999a48b7079762990": "A=[a_1,a_2,..,a_n]", "d1c301f6f3c03669dca093befe423031": "M_m", "d1c31e9e218011598f3dd825246ed747": "\\cot 2\\theta\\ =\\ \\frac{A\\ -\\ C}{B}", "d1c3671aeab941bd1a28f416e4d02510": " ds^2 = (1 + 2 \\, \\psi) \\, \\eta_{ab} \\, dx^a \\, dx^b ", "d1c37b824c44956559befc60cd70d6c3": "| SA | : | AB | =| SC | : | CD | ", "d1c3aa202c65d720abebe4c8eb2cf002": "\n\\max E[U(W_T)].\n", "d1c3e65d4b2511d2b64169690ee969ec": "AB + C \\leftrightharpoons AC + B; K_{B,C}=\\frac{[AC][B]}{[AB][C]}\n=\\frac{K_{AC}[A][B][C]}{K_{AB}[A][B][C]}\n=\\frac{K_{AC}}{K_{AB}}", "d1c41223b1c1c74f81e6ff1aef2f5d44": "\\omega\\bar{\\omega} = (2 + \\sqrt{3})(2 - \\sqrt{3}) = 1", "d1c4154433e3faf4b46d0944606b4b32": "G^{ab} = 8 \\pi \\, T^{ab}", "d1c4dd4aeed3133e52009c40038d3987": "\\color{Bittersweet}\\text{Bittersweet}", "d1c4e1101515902cc3f571acb8c7516b": "\\cos{\\frac{\\phi}{2}}=\\sqrt{\\frac{(s-b)(s-d)(b+d)^2}{(ab+cd)(ad+bc)}}", "d1c4f657ca063e3abe33f53a8ad20bca": "\\scriptstyle\\mathbf{S}", "d1c50407e652fbdcb355696a71e8dd11": " f(z) ", "d1c515dcd324006995b3fab1b1200f19": "\\star \\mathrm{d}x=\\mathrm{d}y\\wedge \\mathrm{d}z", "d1c5303b28dfd020115344164aa67a2c": "0=\\boldsymbol{\\phi}(0)", "d1c548505fd699429319deb84b62483f": "\\min\\,\\{|z_i|\\}", "d1c551ec37b91c06c22cb8fd7f7af86d": "V(I) = \\{ p\\in \\operatorname{Proj} S\\mid p\\supseteq I\\}", "d1c55dc936ce33fe270788a7665d0397": "\n\\rho(\\mathbf{k},\\omega) = 2\\mathrm{Im}\\, G^{\\mathrm{R}}(\\mathbf{k},\\omega).\n", "d1c57e14f8cb6fd2e1a09bc1aaacd57b": " r^{n+2}~\\sin(n\\theta) \\,", "d1c58c65f7626fff207f545dfc1e838d": " N_1(r,f) = 2N(r,f) - N(r,f') + N\\left(r,\\dfrac{1}{f'}\\right) = N(r,f) + \\overline{N}(r,f) + N\\left(r,\\dfrac{1}{f'}\\right).\\,", "d1c59c20c7f1326ef5e149bc5f404924": "u(1)_A", "d1c603f505ed6f0597d1510d5b6e30a8": "\\scriptstyle \\int_P", "d1c61709d731db1daeadbdd648cae334": "\\left( a, b \\right) := \n\\left\\{\\left\\{ \\left\\{a\\right\\},\\, \\emptyset \\right\\},\\, \\left\\{\\left\\{b\\right\\}\\right\\}\\right\\}.", "d1c6655b7fe9c28a6ce57fb81d80f324": "\\frac{\\rho(12-\\rho)}{8(1-\\rho)}", "d1c67afebf00d890010db979d037c38e": "1.\\overline{27}", "d1c6a470694de358fb60710ae836d4eb": "xi,...,xj", "d1c6b780d44520a568020b505623e6eb": "|\\widehat{f}|_2=|f|_2", "d1c6d7dd3310319fa6a625dfee043bb0": "\\Delta = \\,b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.", "d1c6daf69b1626e2469b65ac37567608": "C_H \\; = \\begin{cases}\\;8.29 \\; (\\; \\log_{10} ({1.54 h_M}))^2 \\; - \\; 1.1 \\; \\mbox{ , if } 150 \\le f \\le 200 \\\\ \\; 3.2 \\; (\\log_{10} ({11.75 h_M}))^2 \\; - \\; 4.97 \\; \\mbox{ , if }200 < f \\le 1500 \\end{cases}", "d1c6ffd5d8f2688fa5be715c9f390640": " D(n, m) = {n \\choose m} D(n-m, 0) \\; \\; \\mbox{ and } \\; \\;\n\\frac{D(n, m)}{n!} \\approx \\frac{e^{-1}}{m!}", "d1c71cbf748d8d62038e32ce93eeaa63": "\\scriptstyle 2/\\sqrt{2\\pi n}", "d1c72cb449665f9bae2b00992431caa4": " u:\\mathbb{R}^N\\times[0,T]\\to\\mathbb{R}", "d1c76d4ff2f154d3edef67f759062475": "I t", "d1c7a641eb8c15b883905ffc179f50a0": " | 0 \\rangle ", "d1c7be82aee71d07df9e276d723e0fb8": "Q_B |\\Psi_i\\rangle = 0", "d1c7ea67e199c6cca60530bb5b43e227": "\nF(x) = \\sgn(x) \\begin{cases} {A |x| \\over 1 + \\ln(A)}, & |x| < {1 \\over A} \\\\\n\\frac{1+ \\ln(A |x|)}{1 + \\ln(A)}, & {1 \\over A} \\leq |x| \\leq 1, \\end{cases}\n", "d1c8136a7e36fc24050a0d6a89ec136e": " \\frac{1}{8.4797}\\leq F(\\textrm{square})\\leq\\frac{1}{4}-\\frac{1}{384},", "d1c8bf0173f7de71da21077656cf00eb": "h={v_i^2\\sin^2\\theta_i\\over 2g}", "d1c8ce10bd485152f30589d2263712c6": "V_\\mathrm{T} = \\begin{matrix}\\frac {kT}{ q}\\end{matrix}", "d1c9271241a80c2dc224683321280616": "g=\\sum_i (dx_i)^2+(dy_i)^2+\\theta^2.", "d1c92ffce2ad296bc31a029ba80cc430": " F = \\frac{{\\rm d} \\phi}{{\\rm d} x}.\\,\\!", "d1c949f16e972104b5f02cc13ac84594": "\\frac{dV_r}{dt}", "d1c970761164b1b320a8c0bde631a8a9": "(c^p \\smile d^q)(\\sigma) = c^p(\\sigma \\circ \\iota_{0,1, ... p}) \\cdot d^q(\\sigma \\circ \\iota_{p, p+1 ,..., p + q})", "d1c993ef0c4be46b532f139a8c0c84a3": " q(z) = \\frac{1}{2} \\times g(z, -1) + \\frac{1}{2} \\times g(z, 1)\n = \\frac{1}{2} \\exp(-z) \\frac{1}{1-z} +\\frac{1}{2} \\exp(z) (1-z).", "d1c99710d45ad22b8719fdba024071e4": " \\text{Phase3}: \\text{MP} > \\text{CIW} \\,", "d1c9ae2ccdb71aaf639ee3e672b3b949": " (a_1 + b_1x)(a_2 + b_2x) = a_1a_2 + (a_1b_2 + b_1a_2)x + (b_1b_2)x^2 \\equiv (a_1a_2 - b_1b_2) + (a_1b_2 + b_1a_2)x \\, . ", "d1c9e860982e6f80a53a80915d32908f": " |\\lang x | \\psi \\rang|^2 \\; d^3 x ", "d1ca4668348dc6adbdd55fa8c7c93d0a": "\\mathbb{E} \\big[ | Z |^{2} \\big] < + \\infty.", "d1ca6a3cc66657f1cdd265535d321f5e": "(\\sin c,\\,0,\\,\\cos c)", "d1cadf16eeab190b9ebea884a0052a24": "\\bar{x}{}_i=\\bar{x}{}_i\\left(x_1,x_2,\\cdots\\right) \\quad \\rightleftharpoons \\quad x{}_i = x{}_i\\left(\\bar{x}_1,\\bar{x}_2,\\cdots\\right)", "d1cb0fb9c35e701ad27694490cac25e2": "\\Gamma T^*M=\\Delta^*(\\mathcal{I}/\\mathcal{I}^2).", "d1cb3ee44d55b9b86f17e671eb110836": "{T_3}", "d1cb8b420f3e741ee1d88b5679379604": "f=p\\Phi", "d1cbebb8641ecc6db5906f862822e8ea": "(\\rho)", "d1cbf20d32f694315cd6d1251f608b89": "y(t + h) \\approx y(t) + hf\\left(t + \\frac{h}{2}, y(t) + \\frac{h}{2}f(t, y(t))\\right)", "d1cbf7cfa4497f70b0022e74ec7fdbc9": "\n|\\operatorname{SL}(m,\\mathbf{Z}_n)|=n^{\\frac{m(m-1)}{2}}\\prod_{k=2}^m J_k(n).\n", "d1cc9e0c8ca8d852771f144a940617e2": "U_{(i)}", "d1ccb5fd26a1baa36423dc3c9b53ea11": "\\|x\\|_{bv} = |x_1| + TV(x) = |x_1| + \\sum_{i=1}^\\infty |x_{i+1}-x_i|.", "d1ccba4052b51268f8bc2bb790b5b31e": "h_{+} = -\\frac{1}{R}\\, \\frac{G^2}{c^4}\\, \\frac{2 m_1 m_2}{r} (1+\\cos^2\\theta) \\cos\\left[2\\omega(t - R)\\right],", "d1ccd6900c97ce1c2f345e7247afe15d": "\\delta M=\\int_0^{r_B} 4\\pi \\rho(r) r^2((1-2GM(r)/rc^2)^{-1/2}-1) \\; dr \\;", "d1cd2903cd00179971c154c2234bab82": " E=\\{(a,e_1,e_2)\\colon (e_1 \\times e_2)\\cdot a=1, \\, \\|a\\|=1, \\, \\|e_i\\|=1, \\, a\\cdot e_i = 0, \\, e_1\\cdot e_2=0 \\}.", "d1cd3c46d870350613eea54deac3928d": "\\mathbf{A}_{\\text{Magnetic dipole / Electric quadrupole}}(\\mathbf{x},t) = \\frac{\\mu_0}{4 \\pi} \\frac{e^{i k r - i \\omega t}}{r}(-i k)\\int d^3\\mathbf{x'}(\\mathbf{n}\\cdot\\mathbf{x'})\\mathbf{J}(\\mathbf{x'})", "d1cd662f696b1701769368c7382a2024": "\\mathbf{r}_c", "d1cdb7a3980e6198e6910fce4efa7829": "A \\cup (A \\cap B) = A\\,\\!", "d1cdf19bfe946115d788146f8679e3fe": "Y=\\{x \\mid \\phi(x)\\}", "d1cea164146fddd6162d4ac47ab9d06b": "\\hat{B}|\\psi\\rang", "d1cea5379bd27c3064cea193fda1cbd4": "\\mathrm{H} = 0.40", "d1cee12826a44b0a6101ef31bfc7cbeb": "\\displaystyle{\\mathfrak{h}=\\mathfrak{z} \\oplus \\mathfrak{h}_1\\oplus\\cdots \\oplus \\mathfrak{h}_m.}", "d1ceffee9d45d3ad22d28c29bbfb1a08": "X,Y: \\Omega \\rightarrow R^n", "d1cf1b891fc1bb1a8a762cc0055850fb": " \\rho = \\frac{M P}{R T} ", "d1cf1c6b3bb7fd6e984fc38d4d9664ed": "\\cdots \\to H^q_{\\mathrm c}(U \\cap V) \\to H^q_{\\mathrm c}(U)\\oplus H^q_{\\mathrm c}(V) \\to H^q_{\\mathrm c}(X) \\overset{\\delta}{\\longrightarrow} H^{q+1}_{\\mathrm c}(U\\cap V) \\to \\cdots ", "d1cf1fa06bcf85a01a555c095d72f184": "\\displaystyle \\frac{-2}{\\sqrt{2\\pi}}\\frac{\\sin(\\pi\\alpha/2)\\Gamma(\\alpha+1)}{|\\omega|^{\\alpha+1}} ", "d1cf38bb2a6972bded97831061e6baa2": " S = E^T \\mbox{Diag} (e) E ", "d1cf3aa3719ae5234e0ebccec5ac4c84": "\\bar\\omega\\in\\Omega", "d1cfcb54f476b0495bdad6a1bff96513": "g_{ij}=\\delta_{ij}c_1\\,", "d1cfe074fd22164af29d4f132ce30105": "s^2=w^2-r^2", "d1cfed7495839c483739c793c8c7490d": "G = \\frac{2(OA-IB)}{S}", "d1d063fd1e05a8aa00ebf83eaf95272f": "w'_1 = 1", "d1d1a0eb206dbf4e605be357198d2fb0": "2^{m+1} - 2,", "d1d1eeb88f40f8004210fcca638fa6cb": "\\frac{d}{dx}({f} * g) = \\frac{df}{dx} * g.", "d1d1f4de10a0c8c76a6945277d84c5ec": "R = \\frac{|V_\\parallel|^2}{P}", "d1d2055a8f1d0798b37e2c032d5869ad": "L_2(5) \\cong A_5", "d1d20d7ef66eca8b52c63647cd62b6a4": "f(x_i)=x_{i+1}", "d1d25564ded462fa1b8a0e0f142ed3bf": "M=M(\\varphi)=\\frac{(ab)^2}{((a\\cos\\varphi)^2+(b\\sin\\varphi)^2)^{3/2}}\\,\\!", "d1d277d312c201af48e85b522c1f00a6": "\\bigcup_n \\left \\{ |\\lambda| > \\tfrac{1}{n} \\right \\} = \\bigcup_n S_n .", "d1d299b0d5635e2dfb2f05aea2b36d41": " p(n) = 1 - \\bar p(n). \\, ", "d1d30c677eb80b11f9644f57a553c698": "1/\\sqrt S = \\lim_{n \\to \\infty} y_n.", "d1d333ab91af326fe98a0345a9cf0a3b": "A=L L^T", "d1d354f9526f47ec075edcae88f42347": "-1.0559", "d1d3728fb9c2a12da9a03cbaf0d485c3": "M(\\theta)", "d1d433f6450fd4cd19bad77ebb4c9577": "\\forall x \\,,\\exists y \\,Rxy \\vdash \\exists y \\,\\forall x \\,Rxy", "d1d4b4e3019486aaf710d097297d9aa7": "(H, \\Delta, \\varepsilon)", "d1d4e0e671492dea7615e69b0d3f5d5e": "h_x = 0.332{k \\over x} Re^{1/2}_x Pr^{1/3}", "d1d55e1d4637f2cea8d9e478402d7fd3": "n_x^e", "d1d567ee989bd2cd4b4dc33fd3ba1a72": "(\\mathcal{X},\\Sigma)=(\\mathbb{R},\\mathcal{B})", "d1d616d9738e65bc531691c56c3f809f": "( \\neg p \\lor q )", "d1d64979867ce9df119326d882dd0eb8": "\\sum_{i=1}^{n} i{n \\choose i} = n2^{n-1}", "d1d6f38ed21129e08447ee0a9c80d411": "\n\\frac{1-E_4}{{E_1}^2+{E_2}^2+{E_3}^2}\n\\begin{bmatrix}\nE_1 E_1 & E_1 E_2 & E_1 E_3 \\\\\nE_2 E_1 & E_2 E_2 & E_2 E_3 \\\\\nE_3 E_1 & E_3 E_2 & E_3 E_3 \n\\end{bmatrix}\n+\n\\begin{bmatrix}\nE_4 & -E_3 & E_2 \\\\\n E_3 & E_4 & -E_1 \\\\\n-E_2 & E_1 & E_4 \n\\end{bmatrix}\n", "d1d759c572a790dd74c039f66f19f20e": " S(V,T)=S_0+nR\\ln\\left(\\frac{V}{V_0}\\right)+nC_V\\ln\\left(\\frac{T}{T_0}\\right)", "d1d79497f2d96b38587e91f6ff06e965": "\n\\begin{align}\n \\sigma(\\mathbf{x},\\mathbf{y}) \n & = \\operatorname{E}\n \\left[(\\mathbf{x} - \\operatorname{E}[\\mathbf{x}])\n (\\mathbf{y} - \\operatorname{E}[\\mathbf{y}])^\\mathrm{T}\\right]\\\\\n & = \\operatorname{E}\\left[\\mathbf{x} \\mathbf{y}^\\mathrm{T}\\right] - \\operatorname{E}[\\mathbf{x}]\\operatorname{E}[\\mathbf{y}]^\\mathrm{T},\n\\end{align}\n", "d1d7e1ed5c1243be3b7c0ba430ceb38f": "\n\\begin{align}\n\\mathrm{EV} & = \\sum_\\mathrm{Start}^\\mathrm{Current} \\mathrm{PV(Completed)}\\\\\n\\end{align}\n", "d1d7fe06a029e41fed79859cb0751f75": "Z = \\sum_k e^{- E_k / k_B T}", "d1d81231ccad85965212521837df7313": "G_{\\nu}(\\omega)=e^{j(\\nu-2)[ \\frac {\\omega - \\pi} {2}]}. \\frac {ce_{\\nu} ( \\frac {\\omega-\\pi} {2},q)} {{ce_{\\nu}(0,q)}}.", "d1d888ecc548c16a6f81308d757d6db2": "(1+i)", "d1d8ee24cae68dcd941bd79a5ae42553": "\n\\to abb T[fgg] T[fgg] \\to \\dots\n\\to abb abb T[fgg] \\to \\dots\n\\to abb abb abb", "d1d952a2ec225abf48baa1b7275ff068": " R^n {}_{n;l} - R^n {}_{l;n} - R^{nm} {}_{nl;m} = 0.\\,\\!", "d1d95364d8679aaacb6b918242bde1bf": "\\delta \\mathcal{L} = B\\,\\partial_{\\mu} A^{\\mu} + \\frac{\\xi}{2} B^2", "d1d9893d5660364a63e6e76c8a0bb007": "\\Delta z = \\frac{1}{w}\\operatorname{cov}\\left(w_i, z_i\\right) = \\frac{1}{3}", "d1d99ef83248ac73aa51eabbdd588fc3": " j^{\\star}_{W} = j^{\\star} / \\zeta(4) \\approx 0.924 \\, \\sigma T^{4} \\!\\, ", "d1d9d5bf1e393cfbe8542e4b309054f9": "X^\\mathrm{opt} = \\frac{W}{\\Delta}[(r^TV^{-1}r)V^{-1}1 - (1^TV^{-1}r)V^{-1}r] + \\frac{\\mu}{\\Delta}[(1^TV^{-1}1)V^{-1}r - (r^TV^{-1}1)V^{-1}1]", "d1d9f28ef7408be6ebf5f8986f20aa99": "\\langle\\;,\\;\\rangle_q:T_q\\times T_q\\to\\mathbb{R}", "d1da44a9b13c67bce4829b15d5fd4a67": "\\to aa\\ a/a\\ b\\ B(a)\\ c \\to aab\\ B(a)\\ c \\to aab\\ a/a\\ b\\ B()\\ cc \\to aabb\\ B()\\ cc\\ \\to aabbcc", "d1da54c1169e5cfeffc0a95fd0c96986": "a_n = 0", "d1da593959860bd93267aa53058ab189": "y = f(x) = \\pm\\sqrt{\\frac{8 - x^4}{2}},", "d1da5afa46f0b10da1caffee5077890b": "\n \\boldsymbol{\\sigma} = \n \\cfrac{1}{\\lambda_1\\lambda_2\\lambda_3}~\n \\left[\\lambda_1~\\cfrac{\\partial W}{\\partial \\lambda_1}~\\mathbf{n}_1\\otimes\\mathbf{n}_1 +\n \\lambda_2~\\cfrac{\\partial W}{\\partial \\lambda_2}~\\mathbf{n}_2\\otimes\\mathbf{n}_2 +\n \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3}~\\mathbf{n}_3\\otimes\\mathbf{n}_3\n \\right]\n ", "d1daeb0d776bca72099b383474cd26d9": "\\rho_I", "d1db0d9c696a8c056e7117dbbb4ef6db": "2^n", "d1db2bf51f69328885294d639e53dd56": "d = L + C_f", "d1db3a2bff912395317bc06642bc5cad": "(a+bi)i = ai+bi^2 = -b+ai ", "d1db4a7351957d68469b2e216c77e8f9": " = \\int_{-\\infty}^{\\infty} X(\\tau, \\omega) \\, d\\tau. ", "d1db67713ff711771832273cca8723cd": "b(1)e^{\\gamma(1)}+\\cdots+ b(n)e^{\\gamma(n)}= 0,", "d1db714a2a8245907f87452a1a42b344": "\\arg(z)", "d1dbe92ad56663350e67b443092e189b": " \\frac {M(T)}{M(0)} = \\left (1-(T/T_c\\right)^{\\alpha})^{\\beta},", "d1dd31b1abf22290da978d54ff9ea184": "\\mathfrak{J}^2", "d1dd65b3b72cbc4d27bff440abcd182b": "\n |j_1 m_1\\rangle|j_2 m_2\\rangle \\equiv |j_1 m_1\\rangle \\otimes |j_2 m_2\\rangle, \\quad m_1=-j_1,\\ldots j_1, \\quad m_2=-j_2,\\ldots j_2.\n", "d1ddf9d11949734ec92ca25d1d754f05": "\\lim_{c\\rightarrow \\infty} f(c)=0 ", "d1de2ab05a7d401426d06ae5286551aa": "\\Im \\tau", "d1de4eaa3e6f1ca8269cc5013aae7195": "\\sigma_f", "d1dea305dcac34343a7fcda347365ca0": " E_2", "d1dea9af20bd5c87b7321f8d2cee3459": "\n \\tilde{\\mathcal{A}}^{AB} \\Phi^A_n \\Phi^B_m\n", "d1deac834cd66135e30fe5c4528c19b1": "\\xi, \\rho, \\theta\\,", "d1deee4af438a79f038a8268a01a754c": "\\begin{vmatrix} 6 & 24 & 1 \\\\ 13 & 16 & 10 \\\\ 20 & 17 & 15 \\end{vmatrix} \\equiv 6(16\\cdot15-10\\cdot17)-24(13\\cdot15-10\\cdot20)+1(13\\cdot17-16\\cdot20) \\equiv 441 \\equiv 25 \\pmod{26}", "d1df29d5a134fc2bb41fc57b670467d2": " c^2 = a^2 +b^2 -2ab \\,\\cos \\gamma.", "d1df2b53171b3b30612ecc97682ee7cb": "\\mathbb{N}_{max}", "d1df4d75ceb6f5fd618576a53fca44b1": "\\scriptstyle{E_2}", "d1df70fb21e162c64a393e3dfa8631f0": "\\vec{r}(u,v) = ( x(u,v), y(u,v), z(u,v) )", "d1dfa1919bbd119ac8d8341637f93402": "\\mathbf{j}=0", "d1dfa3327f1ccae992c89485cfdc9e7d": " h(i) = i ", "d1e0be16ffd7481c3d6a66384fcce998": "\n\\frac{\\partial\\psi}{\\partial t} = {\\rm i}\\cdot \\left[ \\frac{1}{2}\\nabla^2 - V(x)\\right]\\psi\n\\,", "d1e0c4e439ee6243b4077b4714e62bb3": "|\\psi\\rangle \\rightarrow |\\phi_i\\rangle.", "d1e11b2661d0204b53e6b38cc6db8328": "\\Pr[X =x] \\leq 2^{-k}", "d1e126a5fd0f8d173827e71948911870": "x \\in \\mathfrak{i}", "d1e135fc547651eba237517123aacc3a": "D(a_{1},\\ldots,c a_{i} + a_{i}',\\ldots,a_{n}) = c D(a_{1},\\ldots,a_{i},\\ldots,a_{n}) + D(a_{1},\\ldots,a_{i}',\\ldots,a_{n}) \\,", "d1e1b1d492011d57b864828c717066c6": "\\mathbf{B}(\\mathbf{x})=\\frac{\\mu_0}{4\\pi}\\left[\\frac{3\\mathbf{n}(\\mathbf{n}\\cdot \\mathbf{m})-\\mathbf{m}}{|\\mathbf{x}|^3} + \\frac{8\\pi}{3}\\mathbf{m}\\delta(\\mathbf{x})\\right].", "d1e1d5d74573996d74b7a0d32626394d": "\\nabla\\cdot \\mathbf{g} = -4\\pi G\\rho, ", "d1e1e6832eb410cd93a23362c3f061d7": "(b^n-1)-y", "d1e210d879cd63f89631d2848aaf3065": "AP^2 = AR \\cdot AS \\, ", "d1e212e5f0a8ec9d72f1329396b898b0": "\\frac{[A]_{f}}{[A]_{i}}=37/100=37\\%", "d1e214decbdedf9fe502448de147c067": "\\sum_{i=1}^m \\sum_{j=1}^m |\\langle x_i , x_j \\rangle|^2\\geq \\frac{(\\mathrm{Tr}\\;G)^2}{n}", "d1e2297af4433e85c939f03465c3026f": "e_1\\circ e_2", "d1e23ba1bf2d8271ed7a5bbd6bbf6f1e": "\\mathbb C\\backslash \\{0\\},", "d1e2810ffbc375d118564b4dfc4cf4aa": "\\gamma=\\alpha+i\\beta=\\cosh^{-1}\\frac{\\omega}{\\omega_c}+i\\frac{\\pi}{2}", "d1e28d50e30e757899159cc4d5844794": "1 + 3 = 4\\, ", "d1e2a1af4eed9e0e0b6f1fb8987d7db3": " \n\\begin{align}\n\\beta_0^{(2)} & = \\beta_0^{(1)} (1-t_0) + \\beta_1^{(1)} t_0 \\\\\n\\ & = \\beta_0(1-t_0) (1-t_0) + \\beta_1 t_0 (1-t_0) + \\beta_1(1-t_0)t_0 + \\beta_2 t_0 t_0 \\\\\n\\ & = \\beta_0 (1-t_0)^2 + \\beta_1 2t_0(1-t_0) + \\beta_2 t_0^2\n\\end{align}\n", "d1e2c0dd9354f4cc1cb9b964fce82ee7": "a=\\det\\begin{bmatrix}\nA_x & A_y & 1 \\\\\nB_x & B_y & 1 \\\\\nC_x & C_y & 1\n\\end{bmatrix},\\quad\nb=\\det\\begin{bmatrix}\nA_x & A_y & |\\mathbf{A}|^2 \\\\\nB_x & B_y & |\\mathbf{B}|^2 \\\\\nC_x & C_y & |\\mathbf{C}|^2\n\\end{bmatrix}", "d1e2ec867dcced3237a7111ae4f6a6f8": "\\frac{1}{1-in/\\lambda}", "d1e2f8d2bbfc9c187ef1dfe4b5146826": "\\alpha_1 = \\beta_2 ", "d1e320921a91025cdfae107c86a79a7e": "f:\\tau_1{\\to}\\tau_2", "d1e36ccd616b78ccd3d790e6f533bede": "\\{a_1^n \\dotso a_{2^k}^n|n\\geq0\\}", "d1e45e065006074818ff4f8673ceaf3a": "\\scriptstyle\\mu_r", "d1e4807fcc449e97020a3e2fc43bbcb5": "~z~", "d1e49ef41eba2140001c87e96798740c": "\\approx_X, \\approx_Y", "d1e4d67832a31d56476a9504cbb8b57c": "2 ^{\\mathfrak c} > \\mathfrak c ", "d1e54d138291733e2e38472d391681b7": "\\frac {{N} {D}} {\\sqrt {RT_{01}}}\\ ", "d1e564bd4e0b29bef1389bd546de6760": " T_{c0} ", "d1e568e89aa658dd0fddbe56a30c79e1": "[1..k]:[k+1..N]", "d1e575075259745ccb044d639ab34c76": "E_{abcd}", "d1e57faa74279f65f918396e55f6c1e6": "y^{calc}", "d1e5b81a40e4ca3b2d879fcf0fef4ddc": "H_{*}\\left(X\\right)", "d1e5cd0ebdd66997e4bb81a1fa09eec8": "f(P_i)=0,\ni=1, \\dots ,n", "d1e5df0fdeb209572c264df87ea9b6a9": "V_\\mathrm{out} = -V_\\mathrm T \\ln \\frac{V_\\mathrm{in}}{I_\\mathrm{SO} R_1}", "d1e5f5cdc5620073e9f69f2dae528e11": "(A\\cap C)D/(B\\cap C)D.", "d1e635875455e9e64bb1688e3dfe7fa8": " T_a = { T_s \\over 2^{1/4} } \n = { T_s \\over 1.189 } \n", "d1e646069964a592cc0cd5e1feaf9398": "d=(1/2)b_{2}", "d1e6a7a870d400183158667e8f7d8dc1": " \\frac{dX}{ds} = P. \\,", "d1e71109b650d417ff358c8c99d36613": " n! \\sim e^{n \\ln n} n \\sqrt{\\frac{2\\pi}{n}} e^{-n}\\left(1+\\frac{1}{12 n}\\right)\n\\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n \\left(1+ \\frac{1}{12 n}\\right).\n", "d1e722b0c20b9717a4ee40968535058d": "\\overline{a}_{\\overline{n|}i}", "d1e7325a85fa097eac7b0a33f4416a9a": "f(\\tau)", "d1e769c75c9fb1eb744ebfb6d588785a": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_1+\\theta_2)\\sin(\\theta_3+\\theta_4)\\;", "d1e76efd25c1f85eb918ed85ddcc9a3f": "f(x_i)\\approx y_i.", "d1e77f8322a86054fa8690d737f7830c": " \\phi \\,", "d1e7a3305386b5bd08e7303f4d81aa30": "C(W,p)", "d1e7c85df1960175a567d28728b887bd": "\\vec L \\cdot \\vec J = \\frac{1}{2}(J^2 - S^2 + L^2) = \\frac{\\hbar^2}{2}[j(j+1) + l(l+1) - s(s+1)].", "d1e812ee8b9cb14e637964d4e3a8de7d": "l_i(u,v)=0", "d1e87de309111341e0a9c7c6a8a8e990": "s_1^3=x_0^3+x_1^3+x_2^3+3\\zeta (x_0^2x_1+x_1^2x_2+x_2^2x_0) +3\\zeta^2 (x_0x_1^2+x_1x_2^2+x_2x_0^2) +6x_0x_1x_2\\,. ", "d1e8dd7b33ade295550e6f098717a505": " \\vec{B} = \\nabla \\times \\vec{A}, ", "d1e8e545439db85c8bd5aed4be373b96": "f(z) = z^2 + c", "d1e8fce0b1caf6e359c0f8be6eeb3503": "\n\\begin{bmatrix}\n\\boldsymbol{I}_m & \\boldsymbol{0} & \\boldsymbol{V}_1^{(t)}\\\\\n\\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_1^{(b)} & \\boldsymbol{0}\\\\\n\\boldsymbol{0} & \\boldsymbol{W}_2^{(t)} & \\boldsymbol{I}_m & \\boldsymbol{0} & \\boldsymbol{V}_2^{(t)}\\\\\n& \\boldsymbol{W}_2^{(b)} & \\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_2^{(b)} & \\boldsymbol{0} \\\\\n& & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots\\\\\n& & & \\boldsymbol{0} & \\boldsymbol{W}_{p-1}^{(t)} & \\boldsymbol{I}_m & \\boldsymbol{0} & \\boldsymbol{V}_{p-1}^{(t)}\\\\\n& & & & \\boldsymbol{W}_{p-1}^{(b)} & \\boldsymbol{0} & \\boldsymbol{I}_m & \\boldsymbol{V}_{p-1}^{(b)} & \\boldsymbol{0}\\\\\n& & & & & \\boldsymbol{0} & \\boldsymbol{W}_p^{(t)} & \\boldsymbol{I}_m & \\boldsymbol{0}\\\\\n& & & & & & \\boldsymbol{W}_p^{(b)} & \\boldsymbol{0} & \\boldsymbol{I}_m\n\\end{bmatrix}\n\\begin{bmatrix}\n\\boldsymbol{X}_1^{(t)}\\\\\n\\boldsymbol{X}_1^{(b)}\\\\\n\\boldsymbol{X}_2^{(t)}\\\\\n\\boldsymbol{X}_2^{(b)}\\\\\n\\vdots\\\\\n\\boldsymbol{X}_{p-1}^{(t)}\\\\\n\\boldsymbol{X}_{p-1}^{(b)}\\\\\n\\boldsymbol{X}_p^{(t)}\\\\\n\\boldsymbol{X}_p^{(b)}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\boldsymbol{G}_1^{(t)}\\\\\n\\boldsymbol{G}_1^{(b)}\\\\\n\\boldsymbol{G}_2^{(t)}\\\\\n\\boldsymbol{G}_2^{(b)}\\\\\n\\vdots\\\\\n\\boldsymbol{G}_{p-1}^{(t)}\\\\\n\\boldsymbol{G}_{p-1}^{(b)}\\\\\n\\boldsymbol{G}_p^{(t)}\\\\\n\\boldsymbol{G}_p^{(b)}\n\\end{bmatrix}\\text{,}\n", "d1e8fed1d0a95c545eff44574457e1a4": "(A \\rightarrow (B \\rightarrow A)) \\rightarrow (A \\rightarrow A)", "d1e923f10e1dc01970093c2c5155955d": " \\mu _1 (\\Sigma _{11} )^{ - 1} ", "d1e94202c7437f6fc2a3cae9e812ff37": " Y = i \\omega C + \\frac {1}{i \\omega L}", "d1e94afd085b6442e2b2eb7b651e360f": " \\int{L(\\theta,a)\\pi(\\theta|x)d\\theta}", "d1eaa6571d26e068f1cf3a105180f6bf": "R_{AW}", "d1eac9e7a18bf330f4a0e4b11f6e81b5": "(x^2+y^2+z^2+b^2-a^2)^2 = 4b^2(x^2+y^2). \\, ", "d1ead05a78267172a636736a0ba108f5": "\n\\operatorname{Li}_s(z) = \\tfrac{1}{2}z + {\\Gamma(1 \\!-\\! s, -\\ln z) \\over (-\\ln z)^{1-s}} + 2z \\int_0^\\infty \\frac{\\sin(s\\arctan t \\,- \\,t\\ln z)} {(1+t^2)^{s/2} \\,(e^{2\\pi t}-1)} \\,\\mathrm{d}t\n", "d1ead128e68f360349e481b2c578a885": "-1-S", "d1eb519aa1b17974f5ba2d428b7fc0b0": "\n \\omega^2 = \\left( g k + \\frac{\\sigma}{\\rho} k^3 \\right) \\tanh (kh),\n", "d1eba58edef9a981c2aba73bd682c15a": "\\psi(s+1)=-\\gamma-\\sum_{k=1}^\\infty \\frac{(-1)^k}{k} {s \\choose k}", "d1ebbae32bb9e1d11fb8041b8fde6654": "C^k(\\mathbb{T})", "d1ebd42f0b2ef0ba0ce08925e2ef070e": "\\tilde{G}^{vap}", "d1ebec798069e58f5ecb7ed64e8e91c5": "a \\qquad b", "d1ec1291c140a0bb6859b75186ba9e2c": "\\rho > {R^2 + L^2 \\over 2R}", "d1ec6d079cfd51f445fac112c29db7ab": "x\\in\\mathcal{X}", "d1ec79321b62aafbf0e063063f913b43": "\\scriptstyle \\mathcal{V}", "d1ec7ea4296fb2320eb5a90a9222be07": "w_0 = w(0)", "d1ec8c369a468be5e9579a10c11638ed": "v_0(t)", "d1ec8e2cd6ab11ec56bb7671e8996fda": " SR(t) = \\frac {\\frac{ \\partial Y}{ \\partial t}}{Y} - \\left( \\alpha \\frac{ \\frac{ \\partial K}{ \\partial t} }{K(t)} + (1 - {\\alpha})\\frac{ \\frac{ \\partial L}{ \\partial t} } {L(t)} \\right) ", "d1ecbd4bfe985806e0d34364bbc04458": "k := 0 \\, ", "d1eccfc964ad9203904dca47aac3a9bd": "f'_-,\\,", "d1ed340bcf985dfdf752eba1051ae4ef": "\n\\overline{T}(x) = \\frac{t}{0.2}\\left(0.2969\\sqrt{\\overline{x}}-0.1260\\overline{x}-0.3516\\overline{x}^2+0.2843\\overline{x}^3-0.1015\\overline{x}^4\\right)\n", "d1ed46c13e1429e77b279efbaa0d5b02": "\\begin{align}\n\\det(O) &= (x_B-x_A)(y_C-y_A)-(x_C-x_A)(y_B-y_A)\n\\end{align}\n", "d1ed644172e1d32b216e1b903a429050": "w' = \\frac{-y'}{y^2}.", "d1edab5835de19ef505b1be9898eb74e": "(M, H)", "d1ede0fb95bd810e954f8141d2825e11": "\n\\ddot \\theta - {g \\over \\ell} \\sin \\theta = -{A \\over \\ell} \\omega^2 \\sin \\omega t \\sin \\theta.\n", "d1eebd20a28bed57a4008fbe35f7cd7c": "U(t,t_0)=\\sum_{n=0}^\\infty U_n(t,t_0)=\\mathcal Te^{-i\\int_{t_0}^t{d\\tau V(\\tau)}}.", "d1eee20eeece1c0864c19647cdf60329": "\\scriptstyle \\exp(x)", "d1ef0eba41ff6a2c496d05458657f0f1": "d-p+1", "d1ef3b920f06ff6e53f7cb46aa628654": "\\sigma^t ", "d1ef3c3e901f20172e721688c074c40b": "\\log \\frac{k_X}{k_H} = \\sigma_X\\rho ", "d1ef3d64f422ad8d0f61f263f22d92d6": "C\\times t = k", "d1ef660f2ba5cc2e4b326f11b8e137e9": "\n \\sigma_{11} - \\sigma_{33} = \\lambda_1~\\cfrac{\\partial W}{\\partial \\lambda_1} - \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3} ~;~~\n \\sigma_{22} - \\sigma_{33} = \\lambda_2~\\cfrac{\\partial W}{\\partial \\lambda_2} - \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3}\n ", "d1ef7df82378becefbd5466b785630c0": "\n\\mathrm{Pr}\\left(m \\vert n\\right) = \\mathrm{Tr}\\lbrace\\hat{\\rho}_{retr}^{[n]}\\hat{\\Theta}_{m}\\rbrace,\n", "d1ef8a635a2c360ca82f01316ce90a7f": "f\\colon (a, b) \\rightarrow \\mathbf R", "d1efdd66f53cec2fb50403594daed649": "\n \\begin{align}\n \\lambda_1\\cfrac{\\partial{W}}{\\partial \\lambda_1} & = 2C_1\\lambda_1^2 + 2C_2\\left(\\cfrac{1}{\\lambda_3^2}+\\cfrac{1}{\\lambda_2^2}\\right) ~;~~\n \\lambda_2\\cfrac{\\partial{W}}{\\partial \\lambda_2} = 2C_1\\lambda_2^2 + 2C_2\\left(\\cfrac{1}{\\lambda_3^2}+\\cfrac{1}{\\lambda_1^2}\\right) \\\\\n \\lambda_3\\cfrac{\\partial{W}}{\\partial \\lambda_3} & = 2C_1\\lambda_3^2 + 2C_2\\left(\\cfrac{1}{\\lambda_2^2}+\\cfrac{1}{\\lambda_1^2}\\right)\n \\end{align}\n ", "d1efdf811e2972eb6c349c5664a931f7": "2^{m}-1", "d1f05ad0f46701d1aeda76e0654ab087": "\\Delta u = -e^{2u} + K(x).", "d1f067c188e486647d4efaa5cc423545": "\\int\\frac{\\mathrm{d}x}{(\\cos ax\\pm\\sin ax)^2} = \\frac{1}{2a}\\tan\\left(ax\\mp\\frac{\\pi}{4}\\right)+C", "d1f0690eab92d3d43472c2f259733581": "Hom(H_p(M; \\mathbb{R}), \\mathbb{R}) \\simeq H^p(M; \\mathbb{R})", "d1f07d3c0251652ade193c171423c46e": "\n\\Gamma_{TL} = {Z_L - Z_c \\over Z_L + Z_c} = \\Gamma_L \\,\n", "d1f07e65c8737a6ed962e4b522800e30": "\n\\begin{align}\n&-\\int\\limits_{0}^{2\\pi}\\ 2\\ r_g \\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\ du\\ =\\ \n-6\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}\\ \\left(1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u\\right)^2\\ \\cos^2 u\\ \\sin u\\ du\\ =\\ \\\\\n&-12\\ \\sin^2 i\\ e_h \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^2 u\\ du\\ =\\ -2\\pi \\frac{3}{2}\\ \\sin^2 i\\ e_h\n\\end{align}\n", "d1f0adb3f1ce216a73bb319b23865e6e": "\n\\begin{pmatrix}\n {{\\mathbf{{\\dot{x}}}}}(t) \\\\\n {\\mathbf{y}}(t)\n\\end{pmatrix}=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}_n(p_n(t))\\begin{pmatrix}\n {\\mathbf{x}}(t) \\\\\n {\\mathbf{u}}(t)\n\\end{pmatrix},\n", "d1f0b02c836a641083443bc3cc1bab43": "Z=(Z_1,Z_2,\\ldots,Z_{n+1}) \\in \\mathbb{C}^{n+1},\n\\qquad (Z_1,Z_2,\\ldots,Z_{n+1})\\neq (0,0,\\ldots,0)", "d1f0b45356ef0dea65269e3672e37098": " [\\mathrm{Fe}/\\mathrm{H}] = \\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{star}} - \\log_{10}{\\left(\\frac{N_{\\mathrm{Fe}}}{N_{\\mathrm{H}}}\\right)_\\mathrm{sun}} ", "d1f138f846fb28c92b71e54adec8e4cb": "y(x) = e^{rx} \\, ", "d1f13f52c58fa2aed9fa5386faacfb5f": "g(\\xi)=\\operatorname{rect}(a \\xi) \\quad \\Rightarrow \\quad (\\mathcal{F}^{-1}g)(x)=\\frac{1}{|a|} \\operatorname{sinc}\\left(-\\frac{x}{a}\\right)\\!.", "d1f17971d81acf53d765ac4e6e84f2b5": "\\mu^*(X^0) = 1", "d1f1c5131c6c424b6950966b362f5c51": "F_{t,s}", "d1f1edd6a120b1f71f1bbe3061aec5f4": "x^2 - x + 2", "d1f203978a32496eaaba664da2d65f05": "\\sigma(x)\\,\\!", "d1f20bf4ffd2b213f433945dc1c720b5": "20+\\lambda", "d1f27a33e60c14fe0c75ed0aea5b385a": "\nSS_T \\equiv \\sum_{ij} (Y_{i j} - \\bar{Y}_{\\cdot\\cdot})^2\n", "d1f299f4829992f13311bfc9e18ce30b": "\\sqrt{N} \\,", "d1f2ba75433434534a32c3b9c5af19bd": "\n J~\\boldsymbol{\\sigma} = \\boldsymbol{F}\\cdot\\boldsymbol{N} = \\boldsymbol{P}\\cdot\\boldsymbol{F}^T~.\n", "d1f2edf9891e05417e8552640baa931d": "\\zeta(s)=s\\int_1^\\infty \\frac{\\lfloor x\\rfloor}{x^{s+1}}\\,dx", "d1f309ee5ecf941a49c51b0f3b5b0adb": "\\scriptstyle \\, J^a", "d1f34ea4f2d2d69f52db3192b2d441d9": "m=\\tfrac{1}{2}\\sqrt{2(b^2+d^2)-4x^2}", "d1f354785cd6a42a43142b55e3d3e615": "m^\\mu = \\frac{1}{\\sqrt{2}}\\left( \\hat{\\theta} + i \\hat{\\phi} \\right)^\\mu\\ .", "d1f38895c95b3e9845ac6f394f7f7016": " \\overline{r_{cf}} ", "d1f38d1dbbf0f8f5ea3059aaa5013a08": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}}(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{A}):\\boldsymbol{T} = \\boldsymbol{\\mathit{0}}\n", "d1f391364bfdea7fbf85ee60c7d59b51": "O(m^{2/3}n^{d/3}+n^{d-1}).", "d1f3cb3996035dd9cdb561b0d90bce5a": "\\lfloor ns \\rfloor", "d1f401595d8c437adc1215a8d86cc4ca": "\n\\zeta_{G, p}(s) = \\frac{1}{(1-p^{-1})^3}\\int_\\mathcal{M} |a_{11}|_p^{s-1} |a_{22}|_p^{s-2} |a_{33}|_p^{s-3}\\;d\\mu,\n", "d1f4124a86565838b0a6adc45e1b38d4": "e^{Ht}", "d1f42309e9a523c0b63d26190121341d": "\\mathrm H^i(M, \\mathbf R) = \\mathrm H^i( \\mathcal C^*(M)).", "d1f462cbd675d7b034b8e7e87bb0b383": " A = - \\frac{\\mathrm{d}N}{\\mathrm{d}t} = \\lambda N ", "d1f4ad4a89fdc2fd6a7e2ea76cd4c475": "F(x|\\boldsymbol{\\theta})", "d1f4c3daf38ba63c18488a831a67942c": "\\mathbf{B}_i=\\mathbf{P}_{ins,i} \\left \\langle\\exp \\left ( -\\frac{ \\psi_i }{k_B T} \\right)\\right \\rangle ", "d1f4cefdd6725057f88924aaa1c58280": "\\frac{1}{2}\\,", "d1f4eee5519eece58f4e7699c7b93112": " a = \\frac{g-2}{2} = F_2(0) ", "d1f530c906eba9e1ce847af134a0f368": " \\int_U \\left[ G(\\mathbf{y},\\mathbf{\\eta}) \\nabla^2 \\psi(\\mathbf{y})\\right]\\, dV_\\mathbf{y} - \\psi(\\mathbf{\\eta})= \\oint_{\\partial U} \\left[ G(\\mathbf{y},\\mathbf{\\eta}) {\\partial \\psi \\over \\partial n} (\\mathbf{y}) - \\psi(\\mathbf{y}) {\\partial G(\\mathbf{y},\\mathbf{\\eta}) \\over \\partial n} \\right]\\, dS_\\mathbf{y}.", "d1f5818a782557a4938feea41053ea40": "\\mathcal{W}^{-1}(\\mathbf{\\Psi},\\nu)", "d1f5a12b8c72ca1550579667bc304ccb": " |\\xi| ", "d1f5f13f8db548f86b2b46d540f839d5": "\\epsilon_{graphene}= \\frac {-lg( \\frac{I} {I_0})} {d_{graphite}} = \\frac {-lg( \\frac{97.7%} {100%})} {3.35 \\times 10^{-8}cm}=301655cm^{-1}", "d1f5fc57764decf961ef789732473520": "\\displaystyle{\\sum |b_n|^2 \\le \\sum c_n = \\exp \\sum_{m\\ge 1} m|a_m|^2.}", "d1f60cd98f6f33090e150452f24835cf": "r, m", "d1f62f32d2e2255f022ed2ce2f0dc4b5": "\\alpha = \\sqrt{\\cfrac{K_d }{c_0 }} ", "d1f671fd632cdf6c43e4a64e3ffb9d6c": "KLMNOP", "d1f7593f423721fc336e8bf559e1405c": " \\log_2 p +1-R", "d1f7e17ad5eeb7c8cd7e78e3caccde6b": "e = e_0\\,\\!", "d1f82abdab280341c910f46b9378e681": "X_t ", "d1f836d82d066c1bfb6611834856f806": "\\operatorname{pf}(BAB^\\text{T})=\\det(B)\\operatorname{pf}(A)", "d1f83794abd33b033602b35d3200ad55": "\\big. \\frac{1}{U} = \\frac{1}{U_1} + \\frac{1}{U_2} + \\frac{1}{U_3}+ \\cdots", "d1f839df76560c9d314ec18aea20b2f9": " \\partial M \\subset \\partial D^4 = S^3", "d1f8f537cc34a9ecc2d50c2675ac2005": "w = \\Phi_c(z) = \\lim_{n\\rightarrow \\infty} (f_c^n(z))^{2^{-n}}", "d1f9164368482ca31e764f45561b9c63": "F = (A \\cdot \\overline{S_0} \\cdot \\overline{S_1}) + (B \\cdot S_0 \\cdot \\overline {S_1}) + (C \\cdot \\overline{S_0} \\cdot S_1 ) + (D \\cdot S_0 \\cdot S_1)", "d1f93a15a0abdd5bd7dbb84efd57b5a7": " \\operatorname{U}(n-1) \\times \\operatorname{U}(1). ", "d1f96310924b68402b4516104592c18b": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i \\mathbf{v}_i\\cdot\\mathbf{v}_i = \\frac{1}{2}\\sum_{i=1}^n m_i (\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i + \\mathbf{V}_C)\\cdot(\\boldsymbol\\omega\\times\\Delta\\mathbf{r}_i + \\mathbf{V}_C),", "d1f96b906f2885b68eedfe3943d8e0cc": "\\Delta E=\\sqrt{(0.5 \\Delta V_J)^2+(\\Delta W_X)^2 + (0.4 \\Delta W_Z)^2}", "d1fa24864cf4000bb7d992793c244ff3": "x(k+1)=Ax(k)+Bu(k)\\,", "d1fab4f5860f88d3ea0986916d57a0b3": " X_{ki} = \\lambda_0 + \\lambda_1 X_{1i} + \\lambda_2 X_{2i} + \\dots + \\lambda_{k-1} X_{(k-1),i} + \\varepsilon_i ", "d1fadb9cb3e9771a0352655f3af63424": "\\ln \\varepsilon{}_n = \\frac{\\pi }{2} ( i_0 - i_n )", "d1faee9af0ea4e623f51913eb8b05cb9": "U=\\frac{d\\log m}{d\\log r}=\\frac{\\xi^3\\theta^n}{\\phi}", "d1fb0e247ae7a2085e40a6564a7de868": " w = s_1 \\ldots s_L ", "d1fb2324b632557cca3c8ac5e3a14535": "\\Phi+a(t)\\;", "d1fb320d3be62d73863a147f1be7a6f0": "2^4 = 16", "d1fb5e302c1dae22f4718bffef39c200": "a_n= a_{n-1}^2-a_{n-1}+1", "d1fbf0fa20f0db082cf5feddcb081c03": "O(n^2/m)", "d1fbff9a799510d8253fd8a6bf4ec850": "\\Psi (x) = \\langle x | \\psi\\rangle", "d1fc0cc826446a4cfe4803d3dd6dbdd5": "\\ \\alpha_{Ji}", "d1fc6e9edce18a70500de1e9e864a756": " X_{p} ", "d1fcb01fd4bad3ca8748aebe43c96a21": "\n\\psi _\\mu (\\tau )=\\nu \\tau ^{\\mu -1}E_{\\mu ,\\mu }(-\\nu \\tau ^\\mu ),\\qquad\nt\\geq 0,\\qquad 0<\\mu \\leq 1, \n", "d1fcedfc51a38ac382e700d44e026e8f": "(A,R)", "d1fd32d341e94f7e4c74446a43480ca3": "\\theta_E - \\theta_W \\,\\!", "d1fd740e95483b578ac15b47b1a3eaa5": "q = \\left\\lfloor{\\frac{v_{1}}{w_{1}}}\\right\\rfloor", "d1fd78a2b4b5c885bd672d5eaf255636": "P_n^m(z)", "d1fdba7516eb3d57a497d267ef1ab3f3": "\\dot{\\epsilon}", "d1fdd1b42d6e83425620b6d8b253e44d": "\\bold{e}_x", "d1fdfbc2ec7e310098f240ca96a48f87": "K_1 = I_1 + 2 \\, R_{ab} \\, R^{ab} - \\frac{1}{3} \\, R^2 ", "d1fe4fbe3bf3b858c57573918f3ca345": "\\scriptstyle 2k", "d1fe622286ab99443302ab05d4a4aca9": "\\max\\nolimits_{m_{j+1}} \\Pr\\left[V\\text{ accepts }w\\text{ starting at }M_{j+1} \\right] \\geq \\Pr\\left[V\\text{ accepts }w\\text{ starting at }M_j\\right]", "d1fe78882f7bfeda8e8738ffbedbff04": "\\mathrm{^{238}_{\\ 92}U \\ + \\ ^{1}_{0}n \\ \\longrightarrow \\ ^{239}_{\\ 92}U \\ \\xrightarrow [23.5\\ min]{\\beta^-} \\ ^{239}_{\\ 93}Np \\ \\xrightarrow [2.3565\\ d]{\\beta^-} \\ ^{239}_{\\ 94}Pu}", "d1fedcf346bd4d628580a5b488d1d928": "\\lambda\\in [0,1]", "d1fef3d0c17477249610e7bbf640e0a5": "\\frac{\\mbox{Accounts Payable}}{\\mbox{Annual Credit Purchases ÷ 365 Days}}", "d1fefe843dd501a34108b7b4e1fddf15": " C_D = \\sigma_D^2 I ", "d1ff20b2b9bc4f6b31d3725dee1a7ca0": "\nS(z)=\\sqrt{\\ln(1/R^2)}=\\sqrt{-2\\ln(R)}\\,\n", "d1ff4f39bcf20d3d1de0ec69d11d3052": "\\scriptstyle b \\;=\\; (b_1,\\, \\dots,\\, b_p)", "d1ff53212d612c251ca7f2a663532ac1": "F^* \\otimes F^* / \\langle a \\otimes 1-a \\rangle", "d1ff75416a3754285069de89041ce94b": "\\textstyle(x, y, z)", "d1ffa74723799db8c5ed3c15c6ef5ee2": "s - D_{\\mathrm N} = \\frac {Ncs(s - f)} {f^2 + Nc(s - f)}\\,,", "d1ffaa22b1bcce59d071826c8fb8a9eb": "\\overline{f}: R/I \\to S", "d1ffbfe96140641c3f95aa944f2d2c6a": "\\lambda\\;", "d1ffd571177b010216f439ee3cf6854b": "A = f^{-1}[B]", "d2000260081a06f8549fb33fef4bbb2f": "T_{aa}\\circ \\operatorname{id}_a=\\operatorname{id}_{T(a)},", "d2001ec306ec8077f510d6981b10e9b0": "p < \\frac{q (1 + q)}{1 - q + q^2}", "d2007b646f011455bfbad857af263704": "\\mathbf{T}=\\mathbf{v}\\frac{dm}{dt}", "d200846f0084ac27cad72da234b95ae5": "\nI = \\int { dp \\over 2\\pi } |p\\rangle \\langle p |\n", "d2009c5dbc25dba85893484e9d9b7acd": "\\mathfrak{p} \\subset \\mathcal{O}_k ", "d200a2da7a690943d810116b06e2311d": "Q_{P}(h)=P\\{(y\\in X:h(y)=1\\}\\,\\!", "d200bfae362c7c8cc13c0135dacb906b": "\\lim_{x\\to+\\infty} \\left(1+\\frac{k}{x}\\right)^{mx}=e^{mk}", "d201258b19fdda2cadb9b6c730921247": " n f_0(t) \\ ", "d20154c3e900b30ede1a475f642a9ce3": "x_n = 1/n^2", "d201dd81016c4e79d770a7fa418a7bf0": " j(X) \\chi_\\lambda (\\exp X) = \\int_{\\mathcal{O}_{\\lambda + \\rho}} e^{i\\beta (X)}d\\mu_{\\lambda + \\rho} (\\beta), \\; \\forall \\; X \\in \\mathfrak{g} ", "d201ff382e94b4a365debf470b8d0cbe": "n-\\ell", "d202084069632f5eb3d596ca374a0146": " = \\frac{(1-x)(1+x)(1-x^2)(1+x^2)...}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)...} = (1+x)(1+x^2)(1+x^3)... ", "d202304393a9e084dec5348d77d39f34": "\\sum_{k=1}^\\infty \\frac{1}{F_k^2} = \\frac{5}{24} \\left(\\vartheta_2^4\\left(0, \\frac{3-\\sqrt 5}{2}\\right) - \\vartheta_4^4\\left(0, \\frac{3-\\sqrt 5}{2}\\right) + 1 \\right).", "d2023a04bcd938c01847904b42e53938": "\\{x_n\\}\\subset X", "d202493751528274cf2dadd2adc18a50": " B_{2n}=\\sum_{j=0}^{2n}{\\frac{1}{j+1}}\\sum_{m=0}^{j}{(-1)^{m}{j\\choose m}m^{2n}} \\!", "d2027aa1f66f0c264f171e26b892adae": "\\neg q", "d202a927f525e81adb82e685c2a05002": " U^2 ", "d202c0c66e306c636aba8b0b7daec233": "\\bold{F}\\ \\stackrel{\\mathrm{def}}{=}\\ d\\bold{A}+\\bold{A}\\wedge\\bold{A}", "d202c2702a1cf8ea0a5a3dfb2bd37566": "X=(x_1,\\dots,x_k)", "d2030a5fb8233163e2950ac4e3813cb9": "(x+y)\\times(x-y)", "d203ce274f6b486f0edbd077ca000862": " \\kappa \\,", "d2040eb3af9ae8061bd53368e9d10f4b": "A_{n-1}(1) 2^{n/2 - 1} \\int_0^\\infty e^{-t} t^{n/2 - 1}\\,dt.", "d2047f3c516e4d1ee24c24f9076e4948": "p(x_1, \\ldots, x_n) = \\Pr[X_1 = x_1, \\ldots, X_n = x_n]", "d2050ed17f07037608a5b3018545609d": "\\forall x \\forall x' (x + x' = x' + x)", "d20526d7189304b6ac7080e807dbf594": "\n\\widehat \\beta_{WLS} ~\\sim N(\\beta , (X'\\Omega^{-1}X)^{-1})\n", "d205502571624ce3ff876a8aac059f7c": "\n\\int_{\\omega_1}^{\\omega_2}\\,S_{xx}(\\omega)+S_{xx}(-\\omega) \\,d \\omega = F(\\omega_2) - F(-\\omega_2)\n", "d205557871cffe4fe24fb779a5b354f1": "0.2079", "d205af4306070f594d08ed06eda39a8a": "f(x,\\cdot )", "d206a36b6269e21af57796da88965ce3": "\\|f\\|_\\infty = \\sup \\Bigl\\{ \\Bigl| \\int_S f g \\,\\mathrm{d}\\mu \\Bigr| : g\\in L^1(\\mu),\\,\\|g\\|_1 \\le 1 \\Bigr\\}.", "d206a41d15d136552af1e27f3f5e5826": "\n\\frac{1}{t} X_t \\to \\frac{1}{\\mathbb{E}S_1}\n", "d206bafae203eb687d764cd7009dc0e0": "\\vec{B}\\cdot\\vec{ds}=0", "d206e7382c2929538e992a0a659cfb18": "(a,b)_p = (-1)^{\\alpha\\beta\\epsilon(p)} \\left(\\frac{u}{p}\\right)^\\beta \\left(\\frac{v}{p}\\right)^\\alpha", "d207032b51aad83015ddfe5ce7b00c87": "{R^2}_1 = (C)", "d2070e667176f27f08fd40f75b374ecd": "F(EG, X)^G", "d2071af5cbe65d8b02752fbb3cc38c75": "\\mathbf{q}' = \\mathbf{q}_2 \\mathbf{q}_1 ", "d2073c561ab6aad5efa9c615efc7b018": "\n{1\\over 2} m v^2 \n= \\hbar \\omega_c\n.", "d2075cbc9fd7088d20ebf3d1ae03be86": "\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p\\ \\ K ", "d2076bf000c81f23be13ecf9fb8999ba": "x=a_{+}", "d207921d0054c56fcf93a6d1cffbc51c": "f \\mathbf{v}_T = \\mathbf{k} \\times \\nabla ( \\phi_1 - \\phi_0 )", "d207931012a604d85d0961d6cdb8af0f": "\\vec{S}(1) = \\sum_{j=1}^8 b_j K_j \\vec{\\xi}_j,", "d207a43c6f3c908e1c56ea439de7bc9f": "A_\\alpha^{\\;\\;\\; IJ} = (P^+ \\omega \\big)^{IJ}.", "d207ed3cc36e4da34f88a0fbdaaeb2fc": " - \\,", "d207fa40cf5c0286d962233c79766134": " \\cos(\\alpha) \\cos(2 \\alpha) \\cos(4 \\alpha) \\cdots \\cos(2^{n-1} \\alpha)=\n\\frac{\\sin(2 \\alpha)}{2 \\sin(\\alpha)} \\cdot \\frac{\\sin(4 \\alpha)}{2 \\sin(2 \\alpha)} \\cdot \\frac{\\sin(8 \\alpha)}{2 \\sin(4 \\alpha)} \\cdots \\frac{\\sin(2^{n} \\alpha)}{2 \\sin(2^{n-1} \\alpha)}. ", "d208293e95201e256d003a73499afc1a": "x \\in V \\land \\exists y ( y \\in x) \\rightarrow \\exists y ( y \\in x \\land \\lnot \\exists z (z \\in y \\land z \\in x)).", "d208443ef4d13ba2cb94dd743681e463": "\\scriptstyle \\mathbf{R}^4", "d2085ea45204f8e28570839d9a31eee0": "H+1", "d208834935f23d959c881967c10cde80": "P_1 V_1=P_2 V_2\\,", "d208a34bc91e76b36aa2c884f081b6a2": "x\\mapsto d(x,S)", "d208d75b35768b91ddb1fc0ec3da904d": "\\alpha_B", "d208f3c58402a6206bd6aedab862b866": "\\pi^2\\approx 227/23,", "d209b96c51a9b50f36a1532d567c8597": " \\frac{1}{\\gamma } \\approx 1-\\frac{3874^2}{2 \\left(2.998\\times 10^8\\right)^2} \\approx 1-8.349\\times 10^{-11} ", "d20a2e5700f58d1bc5cd9a7de560b280": "X_1 - X_2 \\,", "d20a52698867ee28878e6a2a9b36b2f2": "x<0;\\ 1 ", "d20aa8e5cf718f17aa8f92838a6056c6": "\n |SO(3)| = \\int_{0}^{2\\pi} d\\alpha \\int_{0}^{\\pi} \\sin\\!\\beta\\, d\\beta \\int_{0}^{2\\pi} d\\gamma = 8\\pi^2,\n", "d20aca18ddc4a98b76d4d224063a6e02": "C_{\\alpha I}^{\\;\\;\\; J} V_J = (D_\\alpha - \\nabla_\\alpha) V_I.", "d20b937c6ed3dc728449bed1643172e0": "dv(dr^{-1})dx\\cdot dt", "d20b96350020c9c199eaa5108a8193e1": "F^{-1}( 0.5 )", "d20ba9d73a17e6cab8c07c651787f6d5": "K_a", "d20bb7ff41f335b5d47a806ede074cb0": "\\hat{t}\\ ,\\ \\hat{z},", "d20bd987991677491ab0fa652a13ce99": "0 \\rightarrow R \\subset \\mathcal C^0(M) \\stackrel d \\rightarrow \\mathcal C^1(M) \\stackrel d \\rightarrow \\dots \\mathcal C^{dim M}(M) \\rightarrow 0.", "d20c067f91d6a56d978fc4191acc213e": "\\tan \\frac{\\theta}{2} = \\sqrt{\\frac{1+e}{1-e}} \\cdot \\tan \\frac{E}{2}", "d20c33c146dbd9cd05e989280c46cb3f": "E(\\varphi) = \\int_M \\|d\\varphi\\|^2\\, d\\operatorname{Vol}.", "d20c42ff4e5cb364b2fed7b83c0ec859": "\\frac{\\partial u}{\\partial t} = a(u) \\frac{\\partial^2 u}{\\partial x^2}", "d20caec3b48a1eef164cb4ca81ba2587": "L", "d20cc3eeaa4379228a390f81538a754a": "\\Delta y = \\Delta T * \\frac{-b_C * y}{(1 - b_C)(1 - b_T) + b_M}", "d20d62f5decaa1928d56c7dcf7a29ef0": "L_z = x_ip_{iy} - y_ip_{ix}", "d20db86803e69512e26338a009094029": "u_{h+i}=v_{k+i}", "d20de1fa124517c91ebb375b63ec56ee": "n \\in \\mathbb{N}", "d20dfef7a898493d1e7256a7c5f3ee46": "G\\cdot\\lambda.", "d20e1d96b00f0e2c0865334948ef5d08": " y_{n+1} = y_n + hf(t_n, y_n). \\, ", "d20e488cc73758439687b0872d83e956": "\n c(x,\\eta)= \\left\\{\n \\begin{array}{l}\n 1 \\quad\\text{if exists one}\\quad y \\quad\\text{with}\\quad |x-y|\\leq N \\quad\\text{and}\\quad \\eta(x)\\neq\\eta(y) \\\\\n 0 \\quad \\text{otherwise}\\\\\n \\end{array} \\right.\n", "d20e5c8f323448023c6f4f4b683c181f": "(B ^\\prime+B^{\\prime\\prime})m", "d20e750cf40341fe7097d3abf2469b88": "\n\\begin{align}\n\\frac{d(f^{*})}{dp}\n&{} = g(p) + \\left(p - f'(g(p))\\right)\\cdot \\frac{dg(p)}{dp}\\\\\n&{} = g(p),\n\\end{align}\n", "d20e898adc0c4b7f5319bac3eb73b3ce": "qc(x)", "d20ea0dfa3353b22d17fd3bba13d9cc5": "\\mathbf{\\Phi}(t, \\tau)", "d20eb4fb06aaaf4e1fc2186b0838ffbe": "V_j^2=2(h_4-h_5)", "d20eb8c38baff930d7e2d6d882b10b8f": "f(t,y)", "d20ed0244dc5ab667de0cf8981eb2d99": "\\frac{\\text{d}x}{\\text{d}t} = v", "d20ed1066e67ce68ebbfbb98d0cefdd0": "2f_1 - f_2, 3f_1 - 2f_2, \\ldots, f_1 - k(f_2 - f_1)", "d20ef94978c61475437b8a8192294295": "\np = (\\vec x - \\vec{b_j}) \\cdot \\nabla_{\\vec x} R |_{\\vec x=\\vec{b_j}} \\,\n", "d20f004ab8897908486fe9c4f84f6f4e": "\\eta = \\frac{E}{2K_uV} = \\frac{1}{4} -\\frac{1}{4}\\cos\\left(2\\left(\\phi-\\theta\\right)\\right) - h\\cos\\phi, \\,", "d20f1b06b823bde2e0dc88e55ec460cb": "(\\mathbf{k}_0)^2=(\\mathbf{k}_i)^2", "d20f501b38be58304f972baf7d5c7863": " (12.375)_{10} =(1.100011)_2 \\times 2^{3} ", "d20f524471d667976182a1750cad3278": " (d_\\lambda f)(c)= d_\\lambda f(c,\\cdot)=\\mu^{(0)}, \\quad (d_\\lambda f)^\\prime(c)= d_\\lambda f_x(c,\\cdot)=\\mu^{(1)}.", "d20f61782d5153e100537f53b1a33001": "\\bar{K}\\times ...\\times\\bar{K}", "d210222ea711114f445decc37c933cc4": "\nf(z) = \\sum_{n=0}^{L+M} c_{n} z^{n} = \\frac{a_{0} + a_{1}z + \\cdots + a_{L}z^{L}}{b_{0} + b_{1} z + \\cdots + b_{M}z^{M}}\n", "d2105d07a0636e07432935fe8fcf5742": "H(D^2 u, Du, u) = 0", "d210cb36c55444c561abc45a79d46a13": "\\frac{\\partial\\bar{e}}{\\partial V_t} = \\frac {1}{V_0} \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)", "d2110b007c5f4d76e2eb1dbb7df17161": "k_1=\\sqrt{2m E/\\hbar^2}", "d21135be604b5fb0a84d66392fb654af": "f(x+kp)=f(x)+2xkp+(kp)^2\\equiv f(x)\\pmod{p}", "d21140fe8098ed8784895d8494db7358": "y_1,\\ldots,y_m \\in \\mathbb{R}", "d211b2da184501cad03e1eda2b62ef3f": "\\forall i, 1\\leq i\\phi_i ", "d21848cdd835abcb491be1f151e9b6c6": "\\sqrt{2}", "d2185fe4f6b02e3bb56029dffdda2430": "N_k=p", "d21871bf52cd4b5fd64ff1552611ded5": "\\eta\\ = f_5(\\phi,\\beta),\\,", "d218bd5dc6e999aea33a981b2e2de916": "\\scriptstyle\\phi", "d21905bf55bd4897e275e021e1d4d403": "\\begin{pmatrix} \\alpha & \\beta\\\\ \\gamma & \\delta\\end{pmatrix}", "d2194ed2085f89bdc39711d69c010c39": "J_2(\\mathbb R), J_2(\\mathbb C), J_2(\\mathbb H),J_2(\\mathbb O)", "d219b3d76c6294472564a1941aaf8fa9": " \\|Af\\|^2 \\le 4\\|f\\|\\|A^2f\\|. \\, ", "d21b43d24f5d6b5aa9e130831e7176ac": "\\widehat{T}(\\Delta\\mathbf{r}) = I + \\frac{i}{\\hbar}\\Delta\\mathbf{r} \\cdot \\widehat{\\mathbf{p}}", "d21b75c74fb89d03ba0dc749e279f2af": "F_4 + F_2", "d21b7cbf4de69151b0349e180f80e76d": "\\frac{dN}{dt}=rN\\left(1 - \\frac{N}{K}\\right) \\qquad \\!", "d21b7cf14003d2762628538e6fa76247": "S_\\ast(X)", "d21b7eb6d1a805d9c2344003ec645e24": " \\mathcal D \\phi = \\prod_i \\frac{dc_i}{2\\pi}. ", "d21baf802fc154a1d3b86533e0937868": "\n \\begin{align}\n K_{\\rm I} & = \\sigma\\sqrt{\\pi a}\\left(\\cos^2\\beta + \\alpha \\sin^2\\beta\\right) \\\\\n K_{\\rm II} & = \\sigma\\sqrt{\\pi a}\\left(1- \\alpha\\right)\\sin\\beta\\cos\\beta \n \\end{align}\n", "d21c0d21d71a22d3c399d81b5592e7ef": "V = (1+L^2/r^2)/(1+m/r)^2", "d21c2c42b7d932b6e3240743f1cc2a56": "\n\\text{ Then } \n\\left(\\frac{p}{q}\\right) = \\left(\\frac{q^*}{p}\\right).", "d21c8c8504a16c63394bfbf4682c4db7": "t=t_2", "d21ca503c0a884b402e2c7030153a545": "\\left( A^T A \\right)^{-1} = Q", "d21ca84ad56f6e35c23406e62ef020cd": "\\det(t+E+(n-i)\\delta_{ij}) = \nt^{[n]}+\\sum_{k=n-1,\\dots,0} t^{[k]} C_k, ~~~~~ ", "d21cceafe3a867370111e930e8afef20": "\\sum_{n\\ge 0}\\sum_{\\textrm{Hom}(\\Delta_n,X)}\\Delta_n", "d21d130736347f5145c13d2e3f27030c": "P_{\\mathbf{r}\\in R,s_z=m} (t) = \\int\\limits_{R} \\, d ^3\\mathbf{r} |\\Psi(\\mathbf{r},t,m)|^2", "d21d179a39697151bce62423cddb3dd7": "H = \\sum_{i=1}^N \\Bigg[-\\frac{\\hbar^2}{2m} \\nabla_i^2 + V(\\vec{r_i})\\Bigg] + \\int \\frac{e^2\\rho(\\vec{r'})}{|\\vec{r} - \\vec{r'}|} d^3r'", "d21d5b21c8b3eb413d59df9a2477f3cd": "\\nabla ^2 \\psi = -\\omega", "d21db0dac2290ddf1f3acf2ada8460f1": "{SAR}_{n+1} = {SAR}_n + \\alpha ( EP - {SAR}_n )", "d21e11c86ba845407626aa8b382f6507": "cy(i)", "d21e1d0b08c6bc5814843e6aa54b7d12": "Y(n) \\approx \\frac{12}{\\pi^{2}} \\ln 2 \\ln n + 0.06.", "d21e73e0042f55904ee97d2ed4e4116f": " \\tau_m", "d21ec94756e2c3f6228aa03614aec870": "m,m_1,m_2 \\in M", "d21ecd2ddea7d3e127b7dcfda65c6487": "\\frac{f'(c)}{g'(c)}=\\frac{f(b)-f(a)}{g(b)-g(a)}\\cdot", "d21ef7c696cb55901f40adcf3d4e7cb6": "I(X;Y;Z) = - 1", "d21f1f88facaa4229a64af8da76e7f9c": "\\Omega(\\frac{\\log n}{\\log (S w/n)})", "d21fdb34690e95f182b265d36205139a": "A^{-1} = e^{-\\ln(A)}. \\, ", "d21fe586541835a5378ca3d220701aa3": "\\sum_{n=0}^\\infty a_n e^{-ny} \\sim \\frac{1}{y}", "d2202574295d5cb604ea2cdfdd92a9d4": "S(A,C) + S(G,G) + S(A,A) + (3\\times d) + S(G,G) + S(T,A) + S(T,C) + S(A,G) + S(C,T)", "d22029909cb5a8f9eae0b5aa1569386e": "D_{u}up_{k-1}(u)=(-n-2k+2)p_{k-1}.", "d22088958259eb1f9130bf18c47de4b9": " O(h^2) ", "d220bd5d84373c18876e31a5a1fc9937": "-g^{\\alpha \\beta} \\gamma_{\\beta \\gamma} = \\delta_\\gamma^\\alpha,", "d220ee3f1555a8822715edcfdd90f564": "p_n \\geq \\bigl(1- \\frac{2}{n}\\bigr) p_{n-1}", "d22116c7cc906c20f044e6ce5aa914ea": "\\scriptstyle \\frac{1}{\\omega C}", "d22129831409e9b8d4d3b333626716ca": "P_0=\\eta", "d2212aacce1c3420aa343235a9103b3e": "X \\sim \\textrm{Inv-Gamma}\\left(\\frac{\\nu}{2}, \\frac{\\nu\\tau^2}{2}\\right)", "d221353e286923de6bc3fdeca738abec": "\\delta ,", "d2214841bd5b6a7856025b921bb9ee8b": "score(X,Y) := \\begin{cases} d(X, Y) &, d(X, Y) > d(Y, X) \\\\ 0 &, else\\end{cases}", "d2214c509ac62db300aa9f9fa41c28a1": "\\|g\\|_{q,w}", "d22185aeb69ff28837860a5c70c4accf": "\\scriptstyle{R_a^3}", "d22190a91a5c674ee8e11938dfca66c7": "\n\\sum_{i,j,k,\\dots=1}^n \\varepsilon_{ijk\\dots}\\varepsilon_{ijk\\dots} = n!\n", "d2219e5573aaa7e95235ccd7c0fe6986": "\\tau(t)=I_n, \\tau_\\perp(t)=0", "d221cfb8d9678a993772f0d8933efc2d": "b = \\frac{0.077796\\,R\\,T_c}{p_c}", "d221f44d2fb628ad7d42c8a93b598309": "d(f(x),f(y))\\leq k\\,d(x,y).", "d2220f4416e3e52e56938e3ad5fcf5b5": "\n \\frac{\\nabla h}{\\left| \\nabla h\\right|} = \\left(\n\\begin{array}{c} \\cos\\phi \\cos\\lambda \\\\ \\cos\\phi \\sin\\lambda \\\\ \\sin\\phi\n\\end{array}\\right).\n", "d222232d6742a2fbf58f8000f35573de": "\\ \n \\frac{d\\epsilon(f)}{dG(f)} = G^*(f)N(f) - H(f)\\Big[1 - G(f)H(f)\\Big]^* S(f) = 0\n", "d222b121603ab2964b8326c375b3c1a0": "a = 0~ mod~ p", "d222dea569c2d8abdd42adb8ba197a2e": "\\Phi \\left(\\eta,\\tau \\right) = \\frac{\\sin \\left(\\pi \\eta \\tau \\right)}{ \\pi \\eta \\tau }\\exp \\left(-2\\pi \\alpha \\tau^2 \\right), ", "d222f1df028816fba24782bd284c51e5": "P_\\ell(x) = \\frac{1}{2^\\ell\\,\\ell!} \\ \\frac{d^\\ell}{dx^\\ell}\\left[(x^2-1)^\\ell\\right],", "d22349bd2381305a79f1047bd91dfb1f": "\\dot m = {\\rho AC}", "d22375c9f8d835c07197382599df153b": "f(x): \\mathbb{R}^n \\to \\mathbb{R}", "d223923781566e63b72f1bd66a359721": "F(\\mathbf k)=\\int_0^\\infty r\\operatorname{d}\\!r\\,\\int_0^{2\\pi}\\operatorname{d}\\!\\theta\\,\nf(r,\\theta)e^{i kr\\cos(\\theta-\\theta_k)}\n", "d223dcafd459cff0702196a187be3218": "\n\\Omega^2 = {GM\\over R^3}", "d223f533fcb3e68162cbe51d1951258a": "\\mathbf{L} = \\mathbf{x} \\times \\mathbf{v}", "d224216d301a9e08c44d973039a157fd": "\\begin{align}\n&1_{\\mathcal E} \\xrightarrow{\\eta} G F \\xrightarrow{G \\eta' F} G G' F' F \\\\\n&F' F G G' \\xrightarrow{F' \\varepsilon G'} F' G' \\xrightarrow{\\varepsilon'} 1_{\\mathcal C}.\n\\end{align}", "d22428461e4fe23021894e337ee1e888": "\\tilde{a}", "d2244b5c8af162ff2474b305af1c4cd2": "\\text{Var}\\left(Y\\right)\\ge\\frac{\\text{Cov}\\left(Y,X\\right)\\text{Cov}\\left(Y,X\\right)}{\\text{Var}\\left(X\\right)}.", "d224ac0efc8eb3e6e51498bbe7b39cfc": "\\liminf_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{P} \\big[ X^{\\varepsilon} \\in G \\big] \\geq - \\inf_{\\omega \\in G} I(\\omega).", "d224d04592887155afcabff48317e523": "k=1,\\cdots,N", "d225679b8996d54a5d0c666296fb325f": "(x + 1)^2 = 0. \\,", "d2257e63e85d86e670f83d7cccb9bb6b": " \\phi = \\frac{\\mbox{fuel-to-oxidizer ratio}}{(\\mbox{fuel-to-oxidizer ratio})_{st}} = \\frac{m_{fuel}/m_{ox}}{(m_{fuel}/m_{ox})_{st}} = \\frac{n_{fuel}/n_{ox}}{(n_{fuel}/n_{ox})_{st}}", "d2259325f75ef5e5630abc88d2878bc7": "P(L.M.) = T[\\log t_r + C]", "d225d13c261046d3bd91d80c11ca184f": "f(\\bold{x}) = f(x_1, x_2, \\ldots, x_n)", "d2265917b0ac71648a45214b2c819ad8": "\n\\mathbf{B}\n=\n\\begin{bmatrix}\n 1 & 0 & 0 & 0 & 0 & \\ldots & 0 \\\\\n 0 & 1 & 0 & 0 & 0 & \\ldots & 0 \\\\\n 0 & 0 & 0 & 1 & 0 & \\ldots & 0\n\\end{bmatrix}\n", "d2265a19eb703a993824cfba5736037e": "\\dot{e} = (A - L C) e \\, ", "d22673e2577ec106c335434e4987681d": "E = \\int_x \\left[ k \\left(\\frac{dy(x)}{dx}\\right)^2 + V(y(x)) \\right] dx,", "d226cea13b78642400bea4b3b2b60d6d": "\\displaystyle{(Q(a)^{-1} a)a=a Q(a)^{-1} a=L(a)Q(a)^{-1}a=Q(a)^{-1}a^2 =1.}", "d2273efe6bee9b477eecb3a675305c79": "~\\Leftrightarrow~", "d2274b26fd253d6b1dd9e53429e986ad": "\\text{Rank }(A) + \\text{Rank }(B)\\leq\\text{Rank }(F_1) + \\text{Rank }(F_2) = \\text{Rank }(F) = \\text{Rank } (A\\ast B).", "d22778f8437dfcfcb17ecc84ec1d76a3": "K = \\frac{a+b}{|b-a|}\\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},", "d22794ef4402c4f1511d9b5d4d167f4f": "4/\\pi^2", "d227bbbb20b25c58ef4690889b54098d": "\\int\\frac{e^{cx}}{x}\\; \\mathrm{d}x = \\ln|x| +\\sum_{n=1}^\\infty\\frac{(cx)^n}{n\\cdot n!}", "d227c507a05c98d3d5b6fdcf664d8112": "\n(d^2E_\\lambda/d\\lambda^2)\\vert_{\\lambda=1}=2 I_1+12I_2=-2I_1\\,<0.\n", "d2283339a79be4951c1f9d0c83171c50": "\\sqrt{\\frac{1}{20}}\\!\\,", "d228371a6c495874123d40eb8a989055": "\n\\begin{align}\nF_1(s) & = p_1 +\\frac{1}{2}(d_1-p_1)+s(d_1-p_1)^\\perp \\\\[8pt]\nF_2(s) & = p_1 +\\frac{1}{2}(d_2-p_1)+s(d_2-p_1)^\\perp.\n\\end{align}\n", "d2286850acc6a829b03d26d143308f56": " v_a = \\sum_i {\\sum_j {\\sum_r {\\alpha _{ij}^{ar} x_{ij}^r } } } ", "d228ab22cec8dafff7b110a2d6382c69": "a_1 < b_0 \\leqslant b_1", "d228b1cec5a46cdae9b2da4e80ec2356": "\\left( z_{i-k}, \\ldots, z_{i-1},z_{i+1}, \\ldots,z_{i+k} \\right) ", "d228cb0ccdad972ebca94e4fd8587a8b": "|a_i, b_j, c_k,...\\rangle", "d228ce5aeed2a1504305eaa015e0e939": "\\langle \\phi_i | \\hat{A} | \\phi_j \\rangle = \\langle \\phi_j | \\hat{A} | \\phi_i \\rangle^*.", "d2290d690ed2b5fe702a8e5ea60a8a16": " \\operatorname{build-param-lists}[p\\ p, D, V, T_3] \\and \\operatorname{build-param-lists}[f, D, V, K_3] ", "d229188e655d5c6f84bb2ff50bab2b9e": "\nV(x, y) = \\frac{-\\mu_1}{\\sqrt{\\left( x - a \\right)^2 + y^2}} - \\frac{\\mu_2}{\\sqrt{\\left( x + a \\right)^2 + y^2}} .\n", "d2292c71375d892415fabc1af3fb3c2e": "1+a", "d2292e6c5dfba4a0b26932073e5270d7": " d^*: \\Omega^{p+1}(M)\\rightarrow \\Omega^p(M) ", "d2299dd0963b4a24e11c0141cf3d34e0": "\\displaystyle{m(\\Omega)\\le \\alpha^{-1} \\|f\\|_1.}", "d229a42d0d4dbcd6585b47c36315b785": "GL_1", "d229e6af0883bd2ca6abbe33e6001500": "A \\in \\mathcal{B}", "d22a42cd4d61ea9506a53d53ba034f51": " K_{\\lambda 1^{(n)}} ", "d22a689d1a2f3594a2b3e2a5fb75b1c9": "\\varphi_{i}:W_{i}\\rightarrow U_{i},\\,", "d22aac8fab54de01380c2a472e6381ad": "\n\\sum_x \\rho(x) |x\\rangle\n", "d22ab6e2122bd752676d15fb32dfb79b": "\\mathfrak{so}(4)\\cong \\mathfrak{sp}(1)\\oplus\\mathfrak{sp}(1)", "d22ac5aa56ef34c4ea92ba2b0fbe770b": "q_n", "d22ae085d7792e3a349574055fa23aa0": "n\\times N", "d22b070b2be87bad133b2bd94b50a914": "\\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} [t] = \\frac{\\partial L}{\\partial \\mathbf{q}} [t].", "d22b0995844726acf185155eaeb38502": "\\gamma''(t)", "d22b2f3d2e079883c35ef34f612757de": " H\\psi(\\vec{r}_1,\\, \\vec{r}_2) = \\Bigg[\\sum_{i=1,2}\\Bigg(-\\frac{\\hbar^2}{2\\mu} \\nabla^2_{r_i} -\\frac{Ze^2}{4\\pi\\epsilon_0 r_i}\\Bigg) - \\frac{\\hbar^2}{M} \\nabla_{r_1} \\cdot \\nabla_{r_2} + \\frac{e^2}{4\\pi\\epsilon_0 r_{12}} \\Bigg]\\psi(\\vec{r}_1,\\, \\vec{r}_2)", "d22b3079886b5a1f64019e91700d488a": " \\hat{\\omega}^{ab}_{\\mu} = \\omega^{ab}_{\\mu} + K^{a b}_{\\mu}", "d22b3548839e10718529e39a501b0acb": "k \\equiv 3 \\pmod{7}", "d22b4f467b676791dd34ad376d8942ed": " F_r =\\, - k \\, x", "d22b6f11044c72fd0ac1d2d47f22db67": "L_{SU}\\; = \\; L_U \\; - \\; 2 \\big( \\log_{10} {\\frac{f}{28}}\\big)^2 \\; - \\;5.4 ", "d22bcdd1f81e04a2c4bf79c69ebd1327": "Z(\\theta) = K(1+\\cos\\theta)\\;", "d22c51f92588e15b7ffc35189e718ac4": "-(\\mathbf{X}^{-1})^{\\rm T}\\mathbf{A}(\\mathbf{X}^{-1})^{\\rm T}", "d22c817772a062e31bc537dd294497a3": "\\begin{align}\ni^0 &{}= 1, \\quad &\ni^1 &{}= i, \\quad &\ni^2 &{}= -1, \\quad &\ni^3 &{}= -i, \\\\\ni^4 &={} 1, \\quad &\ni^5 &={} i, \\quad &\ni^6 &{}= -1, \\quad &\ni^7 &{}= -i,\n\\end{align}", "d22c83d92defa5331e6d419a8b8bad34": " {K}_{kl} ", "d22cea2548ab4ace34a6394e64eb8f56": "i,j\\in E", "d22d1dc731b296ad1a895c9b8d5e0afc": "F_Q = - \\nabla Q", "d22d296f13d8d52d085667ba287fadf7": "\\mathbf{P}^5.", "d22d3e8536ae5d0ba919046bb55fc664": "(\\aleph_0)", "d22d7ec2d2dddf3233132c06680b589f": "\\{u,x,z\\}", "d22dca041c05378d1b48c9d205c538fb": "\nm\\ddot{r} = m r \\dot{\\varphi}^{2} - \\frac{dU}{dr} = \\frac{m h^{2}}{r^{3}} \\dot{\\varphi} + F(r)\n", "d22debb43a068b2181ef3b5d13428e56": "\\begin{align}\nP(A|B) &= \\sum_{\\omega \\in A \\cap B} {P(\\omega | B)} + \\cancelto{0}{\\sum_{\\omega \\in A \\cap B^c} P(\\omega|B)} \\\\\n&= \\sum_{\\omega \\in A \\cap B} {\\frac{P(\\omega)}{P(B)}} \\\\\n&= \\frac{P(A \\cap B)}{P(B)}\n\\end{align}", "d22e137a24f31a6eb07cbe58daf08fe2": "T = \\sqrt{ \\frac{\\phi^2}{\\sqrt{(r-1)(c-1)}} }", "d22e3099a1775ece62cb6d1cb1a4dea2": "A = \\begin{bmatrix} 3 & 2 & 6 \\\\ 2 & 2 & 5 \\\\ -2 & -1 & -4 \\end{bmatrix}.", "d22e31d955e5fe27a44bdb4cf9c9116f": "x_P-x_0=R_{11} (X-X_0)+ R_{21}(Y-Y_0) + R_{31} (Z-Z_0)", "d22e41400e3ed86bc5fe10a51d099aae": "\\frac{1}{a \\, \\cos \\varphi} \\, \\left( \\frac{\\partial u'}{\\partial \\lambda} \\, + \\,\n \\frac{\\partial}{\\partial \\varphi} (v' \\, \\cos \\varphi) \\right) \\, + \\,\n \\frac{1}{\\varrho_o} \\, \\frac{\\partial}{\\partial z} (\\varrho_o w') = 0", "d22e86d6c656cc1803ee34de537d0660": "C=[a_1, b_1)\\times [a_2, b_2) \\times \\cdots \\times [a_n, b_n)", "d22ec1c7e5a63aa62803963f13962f9d": "(b,a) \\in D", "d22ec7c60af4b2b3caa212c4b8e7a3f1": "\\operatorname{nil}(A)\\,", "d22ec7d1ad181184c5fab34295affd4d": "w=s_1 s_2 \\ldots s_n", "d22edb00bf46148f85b5c3f4b96aed98": "B \\subset X", "d22ef11cc68c9809a06675fd38904e90": "P \\subseteq N \\times (NF^\\ast \\cup T)^\\ast", "d22f7ae09fdf10da2c185c7f2538083e": "\\scriptstyle{k = \\frac{1}{3}(-1-\\frac{2}{\\sqrt[3]{17+3 \\sqrt{33}}}+\\sqrt[3]{17+3 \\sqrt{33}}) = 0.54368901269207636}", "d22fb464d5b6a6d861729560d7c4c979": " \\vec \\sigma (\\beta,\\lambda) = (a \\cos \\beta \\cos \\lambda, a \\cos \\beta \\sin \\lambda, c \\sin \\beta);\\,\\!", "d22fc13b0292131136edaac3c2606eec": " V = \\bigl\\{v_0, \\dots , v_n\\bigr\\} ", "d22fc48d6bfcf1116dbaf6430a255224": "\\sigma_{i j} = \\frac{1}{\\mu_0} B_i B_j - \\frac{1}{2 \\mu_0} B^2 \\delta_{i j} \\,.", "d22fe671a6e42b42914858b3e88d06a8": "\\delta(A,Z)", "d2300c333c5cd565c379a9eaa143221d": "( x - x_1 )( y_2 - y_1 ) = ( y - y_1 )( x_2 - x_1 ).", "d2306a3dd47a20e45f8a044c792df009": "C \\neq 0,", "d230841e9109833ce7d15d56a606dae0": " A_{OL}(f) = \\frac {A_0} { 1+ j f / f_C } \\ , ", "d230a62e4562cda5749531d8dc9f1082": "f(I) = 0", "d230ad6e24203282ab9a16623089a68b": "P=4\\ell ", "d230b06162f9437e6252104dbb51540d": "Q^TQ=I", "d230d8dd6e241631f23716f04e020f3b": "\\partial_{t}|\\psi\\rangle=-\\alpha\\sum(2a_{i}^{\\dagger}a_{i}-a_{i-1}^{\\dagger}a_{i}-a_{i+1}^{\\dagger}a_{i})|\\psi\\rangle=-\\alpha\\sum(a_{i}^{\\dagger}-a_{i-1}^{\\dagger})(a_{i}-a_{i-1})|\\psi\\rangle ", "d2315c5198a3e16887b85bca263bef40": "i\\ne 1", "d231e514817d440b7e0c28af6d6c21c1": "ax+by+c=0", "d231e72a193370537ac2f61488a42cb6": "H \\equiv \\frac{\\dot{a}}{a}", "d231f717ae993083805133b406c22b0c": "\\exp_{10}^2(4.55997)", "d232015628b92233a9f3168c187703ac": "\\frac{\\sin t\\,\\sqrt{\\hat{\\varphi} + g^2}}{\\sqrt{\\hat{\\varphi} + g^2}}\\; \\hat{a} = \\hat{a}\\; \\frac{\\sin t\\,\\sqrt{\\hat{\\varphi}}}{\\sqrt{\\hat{\\varphi}}} ,", "d2320f14fbecb80bd3fd4a629e8b17d5": "\\tan A=\\frac{\\textrm{opposite}}{\\textrm{adjacent}}=\\frac{a}{\\,b\\,}=\\frac{\\sin A}{\\cos A}\\,.", "d2322d3d46e2c3f519a15ac536912ffa": "\\frac{M}{C} \\left [ \\left ( 4 \\frac {N}{C} \\right ) + 1 \\right ] < \\frac{1}{2}", "d232d17de8be9ae97431822a62cbb382": "\n\\begin{align}\n\\frac{1}{4} \\frac{\\partial^2 J_0}{\\partial t^2} & = W_{\\perp \\text{kin}} + \\Delta W_{E_z} + \\Delta W_{B_z} + \\Delta W_k - \\frac{{\\mu_0}} {8 \\pi} I^2 (a) \\\\[8pt]\n& {} - \\frac{1}{2}G\\overline{m}^2 N^2 (a) + \\frac{1}{2}\\pi a^2 \\epsilon_0 \\left(E_r^2 (a) - E_\\phi^2 (a) \\right)\n\\end{align}\n", "d233265d1fba670cc6ffc24a7b425706": "f(x,y,z)=0\\ ", "d23343fee76be53b938fbaeff831791e": "\\Pr(3\\text{ heads}) = f(3) = \\Pr(X = 3) = {6\\choose 3}0.3^3 (1-0.3)^{6-3} \\approx 0.1852", "d2335742b706b10c69ea27862cc62354": "S_2 \\wr S_5", "d233a15d8bbfa82a31042fbc75054fb1": "x(n), n=1, 2, ...N", "d233b2bb72e5ee27e16ac3db795f5f12": " \\tfrac{\\lambda_\\max(U)}{\\lambda_\\min(U)} > t_\\text{diverge} ", "d2341e0dbeea21783c7a9d721dfdd20a": " \\delta \\approx 4.91 x/ \\sqrt{\\mathrm{Re}_x} ", "d23470a94be3e54d5b82007c5981f880": "\\overline{\\mathcal B}", "d23485012be9e684992266f876b50a26": "S = S(x,y)", "d2349fc336f22038f5d1d9df8adf1c16": "L \\varpropto M^3", "d234a9233d843ee2cdf7e9a8ecced9d3": "K_2 = -9K_1", "d234dd5e8928a5feacf773d673f22723": "V(x_1,x_2)", "d234e0a5904b96fc6b2b395f9be594b3": " {\\rm pOH} = -\\log_{10} [ {\\rm OH^{-}}] \\,\\!", "d234febd88f4458719b13ad7fb6895cf": "d_{min}", "d2353006d8e26246e9c9beabd4f48e54": "d_{A\\cap B}= min(d_A, d_B)", "d2353ab3e555c83ff3aa59d50f444667": "S(A|B)_{\\rho} = \\lim_{\\epsilon\\rightarrow 0}\\lim_{n\\rightarrow\\infty}\\frac{1}{n}H_{\\min}^{\\epsilon}(A^n|B^n)_{\\rho^{\\otimes n}}~.", "d235791cc9e1b2db784b7421622292c7": "\\pi i \\left({i\\over \\sqrt{2}}-i\\right)=\\int_0^\\infty {\\sqrt{x} \\over x^2+6x+8}\\,dx = \\pi\\left(1-{1\\over\\sqrt{2}}\\right).\\quad\\square", "d235d76647568d570c7bfd791e386d52": "\\mu\\left(\\limsup_{n\\to\\infty} A_n\\right) = 0.\\,", "d235e98c3c06b9476031c3bf9df9c708": "\n\\Bigl\\langle q_{j} \\frac{\\partial H}{\\partial p_{k}} \\Bigr\\rangle = \\Bigl\\langle p_{j} \\frac{\\partial H}{\\partial q_{k}} \\Bigr\\rangle = 0 \n\\quad \\mbox{ for all } \\, j,k\n", "d2365e39c46472527d25de5c2687e8c8": "\\operatorname{LCP}(i,j)=H[\\operatorname{RMQ}_H(A^{-1}[i]+1,A^{-1}[j])]", "d236d9a2bb7577b7714c4920874b56b1": "\\mathbf{x}^* = -\\mathbf{A}^{-1}\\mathbf{b}", "d236fb37b67d0f49a21062db7f567a3c": "\\operatorname{PGL}(2,\\mathbf{Z}) \\twoheadrightarrow \\operatorname{PGL}(2,\\mathbf{Z}/2).", "d23715b8562e4c4897e8bf2f53c66ae1": "\\langle\\!\\langle a,b\\rangle\\!\\rangle\\cong \\langle 1, a, b, ab \\rangle \\cong x^2 + ay^2 +bz^2 +abw^2.", "d237294477e4faa66d03944f142ebdee": "A_{\\mu\\nu}\\;", "d237546b6f765dce4759885888a07a60": "B_{j,k} = \\mathrm{round} \\left( \\frac{G_{j,k}}{Q_{j,k}} \\right) \\mbox{ for } j=0,1,2,\\ldots,7; k=0,1,2,\\ldots,7", "d23801639f2e921c2e05ad8d6090ac98": "\\cdot : Q\\times Q\\to Q", "d23815236ef8c92eb844a52f3857e425": "y = \\sinh \\ t", "d2381b49df4d3cb73ca1387d61adc110": "\\mathfrak{M}/\\mathfrak{P}", "d23825772a5ab54c133d601a768bfd18": "\n\\begin{align}\nx & \\leftarrow x + \\mu_x s \\\\\ny & \\leftarrow y + \\mu_y s \\\\\nz & \\leftarrow z + \\mu_z s\n\\end{align}\n", "d2384c1aa5a72f6716f09b9fffa586f2": "\n\\frac{d^{2}}{dt^{2}} \\langle r^{2} \\rangle + \\frac{1}{\\tau} \\frac{d}{dt} \\langle r^{2} \\rangle = \n2 \\langle v^{2} \\rangle = \\frac{6}{m} k_{\\rm B} T,\n", "d238e5452be94224851e5c7feec934ec": " \\mathbf{M} ", "d23956507fa1cd5a8cbbefe2992dfd67": "= {1 \\over \\sqrt{2}} \\left|1,45\\right\\rang \\left|2,H\\right\\rang + {1 \\over \\sqrt{2}} \\left|1,135\\right\\rang \\left|2,H\\right\\rang ", "d23963b0d0c93ecb4c73a57d97b09d97": "\\vec{B} = B_{z}(r)\\hat{z}", "d2398add5dc655703410a2390a8340dc": "{\\mathcal J}: \\mathbf{T}\\oplus\\mathbf{T}^*\\rightarrow \\mathbf{T}\\oplus\\mathbf{T}^*", "d239b8468d42a8673c9af36024721c89": "v = \\omega r \\approx 3.0746~\\mathrm{km}/\\mathrm{s} \\approx 11\\,068~\\mathrm{km}/\\mathrm{h} \\approx 6877.8~\\mathrm{mph}\\text{.}", "d239dc4c1b8fd4a7cdc9c046bb777993": "\\left( -\\sqrt{2 \\over 5},\\ -\\sqrt{6},\\ 0,\\ 0 \\right)", "d239f2c9b09d714017d571216c9e275f": "\\Gamma=\\oint_{C}\\mathbf{V}\\cdot d\\mathbf{l}", "d23a4ce8bca0f4891e037439a79b45a6": "X,Y", "d23a6f6f39e1e274bdbdf7d98e1dcdd2": "\\ s_x s_y = 1", "d23aa99a4045b4b1e8d7bfacd91e067d": " E_0 = m_0 c^2. \\ ", "d23ac5f696a3066a93913cc3bead0c6c": "\\int_M \\langle \\alpha,\\delta\\beta\\rangle := \\int_M\\langle d\\alpha,\\beta\\rangle", "d23ae56f9ac83fad3524b4848836493d": "G = \\mathbb{R}/2\\pi \\mathbb{Z}", "d23af5ff0855135676cc7e02049cbb19": "h=\\sqrt{\\frac{3}{4}-(\\!{\\color{Blue}R}-\\frac{1}{2}\\cot\\!\\left(\\frac{\\pi}{n}\\right)\\!)^{2}} + \\sqrt{{1}-(\\!{\\color{Blue}R}-\\frac{1}{2}\\csc\\!\\left(\\frac{\\pi}{n}\\right)\\!)^{2}}", "d23b013feeb0cc909ae954a6576825c9": "\\lang y_j|\\hat\\rho|y_j\\rang", "d23b11fef89a3f8556ac93486ecaadab": "\\mathrm{ker}(A^\\mathrm{T})", "d23b3770ea88b7e401a27b82ce9cbf1a": "\\mbox{backward reaction rate} = k_{-} {S}^\\sigma{T}^\\tau \\,\\!", "d23b4316e60373d5e776db7472514b34": "\\scriptstyle M(\\theta)= -\\sum_{j=1}^{n+1}\\log{D_i(\\theta)}", "d23b719025d2e95646e078b9b638ee2f": "0 < \\lambda < 1", "d23ba829947a9c2beefe84531d099d79": " 0 1", "d25fc54c2a299b3ea894f7d33adb986f": " \\lambda_1(M)\\geq \\frac{h^2(M)}{4}. ", "d260182fad7f7d05e0eb8619e0b14f3a": "|n\\rangle = |n^{(0)}\\rangle + \\lambda\\sum_{k \\ne n} |k^{(0)}\\rangle\\frac{\\langle k^{(0)}|V|n^{(0)}\\rangle}{E_n^{(0)}-E_k^{(0)}} + \\lambda^2\\sum_{k\\neq n}\\sum_{\\ell \\neq n} |k^{(0)}\\rangle\\frac{\\langle k^{(0)}|V|\\ell^{(0)}\\rangle\\langle \\ell^{(0)}|V|n^{(0)}\\rangle}{(E_n^{(0)}-E_k^{(0)})(E_n^{(0)}-E_\\ell^{(0)})} ", "d2603a3f255b2c81e179e4762927ec0c": "\\mathcal{G}(n,n)", "d26043a3bb2fde2351fc6726210256a8": "e^{zt}=\\sum_{n=0}^\\infty p_n(z) \n\\left[\\left(e^t-1\\right)e^{\\beta t}\\right]^n.", "d260664d27955ef2daebf3292cb5f1a4": "n\\le m", "d2606be4e0cd2c9a6179c8f2e3547a85": "\\rho", "d260ccacd5868b652b77c17cc4b15479": "w_{MP}", "d260dea92a35ce1dfb37dc39468dfd21": "A=\\{x\\in X|(\\exists y\\in Y){\\langle}x,y{\\rangle}\\in C\\}", "d260e0d01bf869a6c85158ba8ce17009": "E_\\pi \\{ L(\\theta, \\widehat{\\theta}) \\}", "d260e17e8d32517d0d7654aa139ee22e": "A=\\{x,y,z\\}", "d260ed012124d25e2f96feb4ce402a33": " a \\geq b ", "d260ef21f9bd503e5c51c51fcf9ba981": "\\,L \\preceq M\\preceq N\\,", "d260f7a77a81b6234ece2829ff7f8141": "\\tbinom {4+13-1}4=1820 ", "d260ff4906939fa6d50e70cfe4738dfa": "p(\\mathbf A)", "d2615170182b99c71332bc3b8ba4fd07": "\nQ [ i w ] = \\frac{\\int D \\mathbf{R} \\exp \\left[ - \\beta U_0 [\\mathbf{R}]\n- i N \\int_0^1 d s \\; w (\\mathbf{R} (s)) \\right]}{\\int D \\mathbf{R}\n\\exp \\left[ - \\beta U_0 [\\mathbf{R}] \\right]}. \\qquad (8)\n", "d2615fb65040b38ad8377de000d43291": "K = 2^{142}", "d26161b69c546a82eed7a065446c34b5": " p(k) = \\operatorname{Pr}(X = k) = \\frac{G^{(k)}(0)}{k!}.", "d261a0ef000c2d49cadf945405c75219": "(1.67 \\cdot 10^{-27} \\ \\mathrm{kg}) (1.22 \\cdot 10^{44} \\ \\mathrm{m}^{-3}) = 2.04 \\cdot 10^{17} \\ \\mathrm{kg} \\cdot \\mathrm{m}^{-3}", "d261e3114e9f3025858f7e56b5d23a77": "\\operatorname{spec} (A[f^{-1}])", "d2624645a52eac66bac600e5586d2c9b": "d\\rightarrow\\infty", "d262782a1a0a26b5506d947fe622f96b": "l=\\frac{2d}{3\\Phi Q_s}", "d262c32ba535120fbe5d1cf0507d6885": "R^\\prime", "d262cb1c38acb64376205295d3dd139c": "\\,L_t\\,", "d262cc6d38528b62bd019a526a987f6c": "\\frac{1-b}{2a}+\\frac{b}{2a}", "d262ccb16564fed33d36de7c9c7a0865": "\\int e^{cx}\\cos^n x\\; \\mathrm{d}x = \\frac{e^{cx}\\cos^{n-1} x}{c^2+n^2}(c\\cos x+n\\sin x)+\\frac{n(n-1)}{c^2+n^2}\\int e^{cx}\\cos^{n-2} x\\;\\mathrm{d}x", "d262ee199ad4239894d2ef43b961ac44": "c, \\sigma \\in \\mathbb{R}^+", "d263124514f0ad20a8775db1aacaf71f": "\n\\sum_{n=N}^Mf(n)\\le f(N)+\\sum_{n=N+1}^M\\underbrace{\\int_{n-1}^n f(x)\\,dx}_{\\ge\\,f(n)}=f(N)+\\int_N^M f(x)\\,dx.\n", "d263427ade4c2e4a70383e5749b94adb": "\n\\begin{array}{c}\\textit{Robustness\\ Model}\n\\end{array}", "d26354b46ae68dbd70a818891a286881": " P_i = AB ", "d26391caec8bedd2596552843be136fd": "i = \\;1.", "d2639959915bcce76bf5faa90a778a3c": " {n \\choose n - k}, ", "d263b4fd069ca6b414a45f41a37ed772": "(\\forall n (P(n) \\vee \\neg P(n)) \\wedge (\\neg \\forall n \\neg P(n))) \\rightarrow (\\exists n\\;P(n)).", "d263cf2387f69dfccaf5db2a6bd3703b": "\\sum_{\\vec{k}} \\rightarrow 2 \\frac{\\Omega}{(2\\pi)^3} \\int d^3 k ", "d2641e0cfb64a6c43404a1f0fd70e3f3": "\\neg\\neg P", "d2641e467dd1d82c36537ec3dec67cc9": "\\partial_\\lambda F_{\\mu \\nu} + \\partial _\\mu F_{\\nu \\lambda} + \\partial_\\nu F_{\\lambda \\mu} \\,", "d2648aa044fb631b232e693bc42e5a08": "\\frac{1 - 1/(1.05)^8}{0.05}", "d2648dd1e2092577cf97479ff6555c9e": "\\Omega = S^{\\mathbb{L}}", "d2650656664c2413e6ee4079b04481dc": "\\iint (Tf) g = -{1\\over \\pi}\\lim \\int_{|z-w|\\ge \\varepsilon} \\frac{f(w)g(z)}{(w-z)^2} =\\iint f (Tg).", "d26545ed9ef27ede2dd3792a3c00f67c": " k = k^{\\Dagger}K^{\\Dagger}", "d2655dd95c52679112d9cf71dfece05c": "E_{h-1}", "d265929b49c84683f913596e34b02856": "\\int\\frac{dx}{R} = \\frac{1}{\\sqrt{a}}\\ln|2ax+b| \\quad \\mbox{(for }a>0\\mbox{, }4ac-b^2=0\\mbox{)}", "d26621ab2c50c7e5e90fb523b1876566": "1+x \\le e^x", "d266228aeeef410977c83776665701b8": "N_A^-", "d2671040fdd5503b0376e018e68a4622": "\\top'\\colon (X,\\top)\\times(X,\\top)\\to(X,\\top)", "d2674a60b8f528475dc01063ee5b59ee": "1 / ( \\bar{d}-1 )", "d2676387482fd5004b9efe3976817501": "B_{ij}", "d267689fb74a58b3fcfbc38134382b80": "H^i_c(X;k_X)^{\\vee}\\cong H^{n-i}(X;k_X).", "d267b71528c99f571ab1a96477d9da80": "T^m(V)", "d267cf6d282d4ea4feb5e3e79a4efbbf": "\\scriptstyle\\mathbf{T}", "d267f009fb0f543be3c4ef7355adbe58": "\\mathbf{A}_{\\perp}", "d26875e601390533af775f29dbe89412": " D_H = \\frac {4 L W} {(L + 2W)}", "d2687c09f24f08040305929c9d257a1d": "(a, b) (c, d)\n = (a c - d^* b, d a + b c^*).\\,", "d268f3b571fb2b7c54eea0d7ad9aa564": " u:X\\rightarrow Y", "d2694de0a2d2dd539c07ab9ace570d3a": " d:S \\times S \\rightarrow \\mathbb{R}", "d269623644762f8effdb5e3ee7d639c5": "\\mathrm {DOF} = 3 N c", "d269a668c432319cd9bf335c6ae9ff8a": " (x, y)=(r \\cos \\theta, r \\sin \\theta) \\,\\!", "d269a9a9cfb20e5b237cf12497082b48": "\\frac{d}{dx}(u + v)=\\frac{du}{dx}+\\frac{dv}{dx}", "d269b70e0d610323808af7389459512f": " \\lambda = [2,1]", "d26a3947229263910f2031ad52389303": "\\mathbf{N} - 1 = (N_1 - 1, N_2 - 1, \\dots, N_d - 1)", "d26a98a09f894241b3469f4e21c9108d": "e = 10.1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1\\ 1..._!", "d26aa199dc34d88a7cd94faf1243834f": "\\chi \\sim \\frac{1}{(T - T_{c})^\\gamma}", "d26ac30f654d77c201faa327a171ad95": " d \\approx 2.423 R\\left( \\frac {\\rho_M} {\\rho_m} \\right)^{1/3} \\left( \\frac{(1+\\frac{m}{3M})+\\frac{c}{3R}(1+\\frac{m}{M})}{1-c/R} \\right)^{1/3} ", "d26acaafc5ce3f49adca30023527526f": "(R,\\oplus)", "d26ace8f1a73ec4128dfdad460d17d2f": " \\Delta v = v_e \\ln \\left(\\frac{M+P}{P}\\right). ", "d26acf03f22814649257a05b9cffe96b": "\n\\rho\\left(\n\\frac{\\partial}{\\partial t}+{\\bold u}\\cdot\\nabla\n\\right){\\bold u}+\\nabla p=\\bold{0}\n", "d26b5288600780375e8ab1a31136d0da": "1 - C_{xy}", "d26b64df7312287551b6da101d2eb7c8": "E(\\omega) = E_0 e^{-i \\omega t},\\quad P(\\omega) = P_0 e^{-i \\omega t}", "d26ba5f695b9d1941d93c0afbf7078fc": "\\mbox{QMA}\\left(\\frac{2}{3},\\frac{1}{3}\\right) =\\mbox{QMA}\\left(\\frac{1}{2}+\\frac{1}{q(n)},\\frac{1}{2}-\\frac{1}{q(n)}\\right)=\\mbox{QMA}(1-2^{-r(n)},2^{-r(n)})", "d26bc7ace9e2c486c625c9eb88a9f3ca": "T\\colon D(T) \\to Y", "d26bcc586ff72374b47069d20bedd664": "\n\\langle f_{thm}(s)F^T_{thm}(t) \\rangle = -2k_B{T}\\Lambda\\Upsilon\\delta(t - s).\n", "d26c06e784d6cd31c5084de0d19c3fca": "\\sum_{(i,j) \\in E} \\frac{1-v_{i} v_{j}}{2},", "d26c085dfd830515b58ecd24951d7351": "n^{k/2}", "d26c4c791defeb89ade9bb65026ce93e": "r = l + 1", "d26c68bfa7418a2e0f95d82621293e3d": "\\sigma_1 = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}", "d26c8a52f9b048eab259f37e7e84e37f": "\\chi = \\frac{m_{vapour}}{m_{total}}", "d26ce484783bc2b80e4a71283b3ac67d": "(\\cos x + i \\sin x)^n = \\cos (nx) + i \\sin (nx).\\,", "d26d1e9a6bb648aec70bb6d3a4e73aed": " a = -\\infty ", "d26d40de94ec8328f2c73eb6e6ef3f3e": "F(\\bar{z})=\\overline{F(z)}.", "d26d74e76927a2126ff48cb70c6780e3": "\\sigma_{zz}-\\frac{\\sigma^2_{xz}}{\\sigma_{xx}}+|\\sigma_{yz}-\\frac{\\sigma_{xz}\\sigma_{xy}}{\\sigma_{xx}}|", "d26da5737145f68a30772c6f54b7b4a0": "{E} =\\frac{1}{2} h \\nu. ", "d26dd3dc4fa7e1720845c8d9f4f931b8": "\n \\displaystyle \n w(n,g) \n =\n \\left\\langle \n \\begin{matrix} \n\t g \n\t \\\\ \n\t n \n \\end{matrix}\n \\right\\rangle \n = {g + n - 1 \\choose g-1}\n = {g + n - 1 \\choose n}\n = \n \\frac{(g + n - 1)!}\n {n! (g-1)!}\n", "d26e56592fed5f2e2941c2a6369db223": "2.9508", "d26e6ef33225befda24b2a4e48391ad6": "\\sum_{n=0}^\\infty \\frac{P(n)}{\\alpha^n} = \\frac{\\alpha^2(\\alpha+1)}{\\alpha^3-\\alpha-1}.", "d26ee3bd75c14365bf72a704c4a917a2": "\\mu\\left(\\bigcup_{i=1}^\\infty A_i\\right) =\\sum_{i=1}^{\\infty}\\mu(A_i)", "d26f7dab826251e8c8ac5a90400ccb18": "T\\approx T_c", "d26fe445cd7f3a1bd71229349c940570": " \\int_{-\\infty}^\\infty f' \\,d \\lambda = \\int_{-\\infty}^\\infty (gh)' \\,d \\lambda = \\int_{-\\infty}^\\infty g h'\\, d \\lambda +\n\\int_{-\\infty}^\\infty g' h\\, d \\lambda.", "d26ff343bd21eb45c02524f1230db8a1": " f_0(n) = n + 1,\\, ", "d270318efede45272f7ff58b745fcb40": "\nN^{i}_{j}= \\gamma^{\\alpha \\beta}\\left\\{(g^{hi}g_{hj, \\alpha}),\\beta \n-
(g^{hi}g_{mj}\\Gamma^{m}_{h\\alpha}),\\beta\\right\\} -\\gamma^{\\alpha \\beta}(\\Gamma^{i}_{j\\alpha}) ,\n", "d2704f200adb66c4ddff08a5505c5b23": "\\mathbf{\\nabla} \\cdot \\mathbf{E} = \\frac{\\rho_\\text{free}}{\\mathcal{E}}", "d2707a37a9e8064c05b57d756ec15f13": "\\mathbb{T}^n", "d27085f521cbecb807a127a151dc40bc": " \\psi^0_1, \\ldots, \\psi^0_g ", "d2710482b4b8da997ba532f4755600de": "\\sqrt{2m+1}", "d2712e175cb6f6c9e2a61fa6be1371ad": "F^{-1}", "d2713c638624e39db94dfe13aa18cdfd": " t_{\\frac{1}{2}}=\\frac{\\pi}{\\omega} ", "d271871df18660c71c87086f4c7efef6": "\\Delta_7", "d271a78554a42d8a5bb977f2bad37177": "A=VDV^T", "d271cedde6675e55152d3c7a4236f775": "f(x)=x^2", "d27208ee625d6156ff0151fc2bc99fea": "f_{dp}", "d2723a8dc7b765106ebd67698b01a46f": "H =\n\\begin{pmatrix}\n & - & A & C & A & C & A & C & T & A \\\\\n- & \\color{blue}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nA & 0 & \\color{blue}2 & 1 & 2 & 1 & 2 & 1 & 0 & 2 \\\\\nG & 0 & \\color{blue}1 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\\\\nC & 0 & 0 & \\color{blue}3 & 2 & 3 & 2 & 3 & 2 & 1 \\\\\nA & 0 & 2 & 2 & \\color{blue}5 & 4 & 5 & 4 & 3 & 4 \\\\\nC & 0 & 1 & 4 & 4 & \\color{blue}7 & 6 & 7 & 6 & 5 \\\\\nA & 0 & 2 & 3 & 6 & 6 & \\color{blue}9 & 8 & 7 & 8 \\\\\nC & 0 & 1 & 4 & 5 & 8 & 8 & \\color{blue}11 & \\color{blue}10 & 9 \\\\\nA & 0 & 2 & 3 & 6 & 7 & 10 & 10 & 10& \\color{blue}12\n\\end{pmatrix}\n", "d2725b580c04a3e5c1f27ea7c60406de": "L = \\rho + \\frac{\\rho^2 + \\lambda^2 \\operatorname{Var}(S)}{2(1-\\rho)}", "d2727d68ec260903823fef527a015aee": "\\delta([X,Y]) = \\left(\n\\operatorname{ad}_X \\otimes 1 + 1 \\otimes \\operatorname{ad}_X\n\\right) \\delta(Y) - \\left(\n\\operatorname{ad}_Y \\otimes 1 + 1 \\otimes \\operatorname{ad}_Y\n\\right) \\delta(X)\n", "d272838d9c483cd86b2491289f077e50": "\\frac{W}{2}.", "d2728dd6f13c2a2c78bf139fb11b0a5d": "P_\\mathrm{avg}", "d27298dae370d141f099ed91739dadba": "W=C^{-1}AC", "d272dbbced459a8508ad07ec67cdeeec": "h=\\sqrt{p^2-\\left(\\frac{a+b}{2}\\right)^2}=\\tfrac{1}{2}\\sqrt{4c^2-(a-b)^2}.", "d273079a4d263c332eba6b7b726dcc4a": "\\mathcal{D}_\\mu\\psi \\rightarrow D(\\Lambda) \\mathcal{D}_\\mu \\psi ", "d273890bb7079b7f0d603df138e8cf7b": "\n\\begin{align}\nG_0 &\\sim \\operatorname{DP}(\\alpha_0,H)\n\\end{align}\n", "d2738e9a6d1f253772882e7df3a50ffc": "M =", "d273975390bb5844a4f1ee3375a08354": " H_i(1, a) = H_j(1, a) = 1 ", "d273b255eab2c242011849da15e99273": "\\lim_{n \\to \\infty}n\\Beta(1,n) = {\\rm \\textrm{Exponential}}(1)", "d273e4953c8b995956b5ea48964969bc": "f(x)=(2+\\sin(1/x))x^2", "d273fa6078a4c8816f93dd0a86f65a28": "\\partial_{tt}", "d27417e15861ce231584252288b6ed73": "(i \\pm 1) \\mod N ", "d2741df547ca01f2cc5a48bac696e47b": "h(x) = \\inf_{y \\in C} f(x,y)", "d2744f31d98910979334d99e252644dc": "\\nabla \\times \\mathbf{B} = \\frac{1}{c} \\left(4\\pi\\mathbf{J} + \\frac{\\partial \\mathbf{E}}{\\partial t} \\right)", "d27474f04ceb48510618976bb44f6015": " p_i = 2^{i} a + 1 ", "d2747d6ceedcc5002befc2c26ea25930": "\\mathfrak{M}^{\\mathrm{CC}} \\propto J_{\\mu}^{\\mathrm{(CC)}}(\\mathrm{e^{-}}\\to\\nu_{\\mathrm{e}}) \\; J^{\\mathrm{(CC)}\\mu}(\\nu_{\\mathrm{e}}\\to\\mathrm{e^{-}})", "d274e1ffa8eb9a25ef78148aacc33605": "\\int\\frac{dx}{\\sinh ax} = \\frac{1}{a} \\ln\\left|\\tanh\\frac{ax}{2}\\right|+C\\,", "d2752860a543c50460daf55b4d353b5f": " \\tau_f = c' + \\sigma_f ' \\tan \\phi '\\,", "d27588566b8109a03a76c7b18cd92b57": "\\triangle\\delta\\;=\\;(\\cos \\delta)^2 \\cdot (\\tan \\delta' - \\tan \\delta) = (\\cos \\delta)^2 \\cdot \\tan \\delta \\left(1-\\frac{\\beta}{\\cos \\delta}\\;-1\\right) = - \\cos \\delta \\cdot \\tan \\delta \\cdot \\beta = -\\beta \\cdot \\sin \\delta ", "d27598f6d830d4f7046e5f48ecf9440b": "R_j f(x) =c_n\\lim_{\\varepsilon \\to 0} \\int_{|y|\\ge \\varepsilon} f(x-y){y_j\\over |y|^{n+1}}dy= \\frac{c_n}{n-1}\\int \\partial_j f(x-y){1\\over |y|^{n-1}} dy,", "d275c9b0eaefdbdbabd0f75879f4b513": " S\\subseteq\\mathbb{R}^3\n", "d2762a0f535810d39edc2932ee53efdc": "\\neg (X \\lor \\neg X)", "d2765301b382f7f60378be7beeb4f38a": "\\mathbf{P}(X < (1-\\delta)\\mu) < \\left[\\frac{\\exp(-\\delta)}{(1-\\delta)^{(1-\\delta)}}\\right]^\\mu.", "d2767cca904c605a5e648335fc669a8d": " {div\\, (k\\, grad\\, T )} = 0 \\, ", "d276c8092ce0f0329ce99ea76b64bcb0": " 2 \\int\\limits_{-\\infty}^\\infty f(t)\\sin\\,{2\\pi \\nu t} \\,dt.", "d276e6f8740882666dd356a6d42cfe07": "\\begin{smallmatrix}\\sqrt{87^2\\ +\\ 64^2}\\ =\\ 106\\end{smallmatrix}", "d276e88cec0dbd6ea4540fbbf1eb9ca1": "[1,2,3] + 10 == [11,12,13]", "d276fcd4dc00fb5a7573321157a71b48": " (r \\, \\cos 2 u, r \\, \\sin 2 u, r \\, \\cos u) ", "d27806cb87f320b6eb74bb6fbf1b385c": "C^{SL}=C_0", "d2787207a1c428f9a7ce1497d6a5b6fd": "DSH=\\frac{dv}{dx} + \\frac{du}{dy} ", "d278e431670222dc3a1f209b2e734640": "\\mathbf{r}_1 - \\mathbf{r}_0", "d278e4de491ce547c09863a180936c39": "\\eta = 1", "d2792f3088bea4ff64a1a173245e20a6": "\\rho(xy) = \\sigma(x) \\sigma(y) \\otimes \\mbox{Id}_W + \\mbox{Id}_V \\otimes \\tau(x) \\tau(y)", "d279cc96db2c5c3bbd0e8a77116f5b1a": " l_{\\rm turb} ", "d279ed2badfd7109357c8ecbe95c85f9": "\\displaystyle \\theta(x_1,...x_m;p)=\\theta(x_1;p)...\\theta(x_m;p)", "d27a0ca539ba5d2f864b4e2d1526793c": "\\operatorname{E}\\big[ (\\overline{X}-\\mu)^2 \\big] = \\frac{1}{n}\\sigma^2 .", "d27a2ec9984ae39c723d1e057bc8d066": "f\\left( x,t\\right) =t\\cdot x", "d27a3dafd0596db5932610d07ae42af3": "\\mathbf x_i", "d27aa394b640bcc7353f5708b40053cd": "|x'(s),\\ y'(s)| = 1", "d27b05a49300962c85391cef251ec307": "X \\subseteq \\mathbb{R}^n", "d27b47f29e94df679097e4ab315cb03e": " \\mathrm {rate} = - \\left( \\frac{d[A]}{dt} \\right) _T = k[A]", "d27bbdd6798d0281cfd0f5dce51d47a8": "\\mid n \\rangle", "d27c5f30551d9d59cf1fe392fa59c251": "\\displaystyle{|H_\\varepsilon f - T_{1-\\varepsilon}Hf|\\le 4f^*.}", "d27c7bb83828abf01906a77414eae835": "Z_\\alpha", "d27ca76dc08cc4226bbac019be31fc2d": "c^2 = b^2 + h^2.\\,", "d27cafc5621c969e8a26fe58ef92ef14": "\\mathcal{FL}_o", "d27ccdbe40559b5b0190d44906f68206": "= H_a \\left( \\frac{2}{T} \\frac{e^{ j \\omega T} - 1}{e^{ j \\omega T} + 1}\\right) \\ ", "d27ce0cc18ae00deaf8b06be48fa6325": "E[x^2] = \\mu^2 + \\sigma^2", "d27cec0d5a5b7c16e61622b278e16ee9": "\\mathrm{A}(M) = \\frac{TN + TP}{TN + FP + FN + TP}", "d27cfbb463c98d5f338656b304e65a86": "\\Lambda,", "d27d21de3fe18ccfd036c0c68207a887": "J_0(p)=\\frac {1}{2 \\pi} \\int_0^{2 \\pi} e^{ip \\cos \\alpha} d \\alpha ", "d27d3d50b4aa2560be73dbf1122a666e": "\\langle \\psi^* | \\psi \\rangle = 1", "d27d5383a2f51014eef8588e748c7a5d": "\\; \\frac{z_2}{z_1} > 1", "d27d763e436fc9ebba22a184d75789d4": "(B^\\prime-B^{\\prime\\prime})m^2", "d27db823b19f77ad016b8e35fe42943a": "\\Vert f\\Vert_{\\mathcal{F}^2(\\mathbb{C}^n)}:=\\int_{\\mathbb{C}^n}\\vert f(\\mathbf{z})\\vert^2 e^{-\\pi\\vert \\mathbf{z}\\vert^2}\\,d\\mathbf{z}", "d27e02af355c5fb16aca7a18ff731804": " \\operatorname{Var}(\\xi(B)) \\geq \\operatorname{Var}(\\xi_{\\alpha}(B)) ,", "d27e636f2d033fea97e90bbe45cc5dca": "\nZ_{2n} = (4X_nZ_n)((X_n-Z_n)^2+((A+2)/4)(4X_nZ_n))\n", "d27e7ba5a6c17400a08fe39109f49a8d": " E \\psi = -\\frac{\\hbar^2}{2\\mu}\\nabla^2\\psi - \\frac{e^2}{4\\pi\\epsilon_0 r}\\psi ", "d27eac4cb19f30103b302baddd9ca6da": "f(x,y) = 0.5 + \\frac{\\cos\\left(\\sin \\left( \\left|x^{2} - y^{2}\\right|\\right)\\right) - 0.5}{\\left(1 + 0.001\\left(x^{2} + y^{2}\\right) \\right)^{2}}.\\quad", "d27ed381476288adb38046425a4d5684": " L = v \\, dt \\, n \\, dV ", "d27ed3d04c94d9be6638bb213be7a64d": "S_L \\, = \\, 1 - \\gamma \\, .", "d27f13dce272854b0f1194c08c078f8a": "4 \\ \\mathrm{m}/\\mathrm{s} \\times (\\frac{7}{4} \\ \\mathrm{N} \\times \\frac {\\mathrm{s}}{\\mathrm{m}})= 7 \\ \\mathrm{N}", "d27f25ee3a3cf446109e349f701eb4f2": "X=(0,1]", "d27f2dfac894da07491d205ee2bc27b6": "\\scriptstyle \\frac{1}{NT},", "d27f3074e0c15300298ad857f6ab06ae": "\n\\hat{e}_{\\mu} = \\frac{1}{\\sqrt{\\sinh^2 \\mu + \\sin^2 \\nu}}\n\\left( \\sinh \\mu \\cos \\nu \\cos \\phi \\boldsymbol{\\hat{i}} + \\sinh \\mu \\cos \\nu \\sin \\phi \\boldsymbol{\\hat{j}} + \\cosh \\mu \\sin \\nu \\boldsymbol{\\hat{k}}\\right)\n", "d27fb82396f5c6d4d701e1d36778bb56": "\\gcd(q, r)=1", "d27fec34834e18ee9aadf996215df643": "R_1 = \\sqrt{R^2 + x_1^2 + y_1^2} \\, ", "d27ff6b1ae573a16faf4fad067aa28fc": "\\scriptstyle \\text{Pad}_n^s", "d27ffb1e6e28197c4129738ab4eed80c": " (\\phi,\\psi) \\cdot f = \\psi^{-1} \\circ f \\circ \\phi.", "d280747f6e66d69e1c3d4b223d8ba859": "f(a\\mathbf{i} + b\\mathbf{j} + c\\mathbf{k}) = u (a\\mathbf{i} + b\\mathbf{j} + c\\mathbf{k}) u^{-1}", "d2809c5c051e858faa23ca3d867956ba": "d_0 |t|^{\\beta_0}\\exp\\big(-|t|^\\beta/\\gamma\\big) \\leq \\varphi_X(t) \\leq d_1 |t|^{\\beta_1}\\exp\\big(-|t|^\\beta/\\gamma\\big) \\quad \\text{as } t\\to\\infty", "d280a4fe268786e2313cfe2cabf7d215": "\\exp(-z^2)=\\lim_{n \\to \\infty}(1-z^2/n)^n.", "d280bab717280e63b1452019df7b1c0a": "R = A \\times (e - \\frac{t}{C} - e - \\frac{t}{I})", "d2810b11141e1665c70005868a5a4dc5": "r_i = \\frac{n_i}{n_\\mathrm{tot}-n_i}.", "d2810b811054ccfc55e6956666c2b6ad": "\\mathbf{x}_{0}=\\underset{\\mathbf{x}\\in \\mathbb{R}^{2\\times 2}}{\\operatorname{argmin}}\\int_{\\mathbf{x'}\\in N}T_{\\mathbf{x'}}(\\mathbf{x})^{2}d\\mathbf{x'}", "d2812eded3d49830bdd3dd94ba41ed3c": "{j \\over r} < k < {j + 1 \\over r} \\text{ and } {j \\over s} < m < {j + 1 \\over s}. \\,", "d2813b031cc7b7623082ead3f0d398e2": "(I+W_1\\cdot dt)(I+W_2 \\cdot dt)=(I+W_2 \\cdot dt)(I+W_1\\cdot dt)", "d2818aa046d7002e904d72d0167ea1ea": "e^x + C \\,", "d281a28e19f0e3797c0723a2a57165d4": "\\mathbf{b}(\\boldsymbol\\theta')", "d281a64bf081900ef635e1a02d16aeff": "\\sqrt{2})", "d281d42a60d06d29bf1e7137ae93b733": "y(\\hat\\theta)", "d28265817919160501c7fd4f6318390b": "\\, T_{r\\overline{o}}", "d282fd3350bb3ec2d1e940574e8fd42d": "x^{*}", "d283a31be49374ef53c4cd7bdcafea1c": " f(\\gamma) = \\frac {\\gamma^2 \\beta }{\\theta K_2(1/\\theta)} \n\\exp\n\\left(\n- \\frac {\\gamma}{\\theta} \n\\right)\n", "d283c9b12a9008f6ce2ad66d4589732b": "2\\mu=2\\text{Re}(\\mu)", "d283e92b4cae68513d243b6fb42c5144": "\\hat{\\dot{M}}_I", "d2843ed9a50a7a44f44b46559dd75e2f": "q\\rightarrow 1", "d28444da6f922c7c12707bf201636d89": "\\|f-\\hat{f}\\|<\\epsilon", "d2844c568a37664eb7e7317818ff11f6": "\\lim_{n \\rightarrow \\infty } \\left( \\prod_{i=1}^n a_i \\right) ^{1/n} = \nK_0", "d2846ddbfecf442fa8bf27f0ec6df214": " \\Delta W_i = \\Delta h_i g_i\\ ", "d2848076378d94a7963a76424a17d827": "[Q_1,Q_2][\\mathcal{L}]=Q_1[Q_2[\\mathcal{L}]]-Q_2[Q_1[\\mathcal{L}]]\\approx\\partial_\\mu f_{12}^\\mu", "d2848557dbed5eaed98c69ae33449e3d": "\nZ (n,V,\\beta) = \\frac{1}{n! (\\lambda_T^3)^{n N}} \\prod_{j=1}^n \n\\int D \\mathbf{r}_j \\exp \\left( - \\beta \\Phi_0 \\left[ \\mathbf{r} \\right] \n- \\beta \\bar{\\Phi} \\left[ \\mathbf{r} \\right] \\right), \\qquad (1) \n", "d2849cf63cff0478bdccbd5bb1800fcf": "\\mathrm{CH_3OH + 6\\ OH^- \\to 5\\ H_2O + 6\\ e^- + CO_2}", "d284d07c7c913cd9cd957cead91a67fd": " B(z) = \\exp \\left(\\exp z - 1\\right).", "d2853311a376129a6850e3f022b42138": "\\int_S \\varphi(\\mathbf{x}) dS = \\iint_T \\varphi(\\mathbf{x}(\\lambda_1, \\lambda_2)) \\left|{\\partial \\mathbf{x} \\over \\partial \\lambda_1}\\times {\\partial \\mathbf{x} \\over \\partial \\lambda_2}\\right| d\\lambda_1 d\\lambda_2", "d2854d535bdcb7c3142752868c97b33d": "S = |\\langle e^{i\\phi} \\rangle|^2\n = |\\langle e^{i2\\pi\\delta/\\lambda} \\rangle|^2", "d28571dffe51b6bafb1f0779c76a159d": "\\gamma(a,q) = \\lim_{x\\to \\infty}\\left ( \\sum_{00 ", "d28d91214ebea7a738ebf268d4f0f91f": "2k-c\\log k", "d28d956d72c0be4a331176276fd48c0c": " \\{ (x_i^{tr}, y_i^{tr}) \\}_{i=1}^n ", "d28d960f6a0d26236556fa30b0690d10": "I <_K J", "d28d9b8ad7964ec89574d529a52c7463": "y_{it}=a+bx_{it}+\\epsilon_{it}", "d28e286c1690f095d666fcb0b587c1e6": "-\\frac{1}{\\frac{dy}{dx}}", "d28eb35ae921ad45f4051eb61fe65837": "\\tau \\in [0,1]", "d28ebc4522fe0f11681b51660de7ab8f": "\\theta R=\\alpha r", "d28ec726511563e78c3a8f4dd129e8a6": "\\begin{align}\n \\hat\\mu(u + 1, v; \\tau) &= a^{-1}bq^{-\\frac{1}{2}}\\hat\\mu(u + \\tau, v; \\tau) \\\\\n & {} = -\\hat\\mu(u, v; \\tau) \\\\\n e^{\\frac{2}{8}\\pi i}\\hat\\mu(u, v; \\tau + 1) &= \\hat\\mu(u,v;\\tau) \\\\\n & {} = -\\left(\\frac{\\tau}{i}\\right)^{-\\frac{1}{2}}e^{\\frac{\\pi i}{\\tau} (u - v)^2}\\hat\\mu\\left(\\frac{u}{\\tau},\\frac{v}{\\tau};-\\frac{1}{\\tau}\\right)\n\\end{align}", "d28ee45336f0e1376aa30c18e7cf016b": "n_c-n_d = n_0 - n_1 - n_2 + n_3 - 2 S(y)", "d28f5ad4234ba705f3970e5be1db1c00": "{}^{\\mathsf{T}} \\!\\,", "d28fcc9556e214ef3431729c0899a6b6": "\\Phi(a) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{a} e^{-\\frac{x^2}{2}}\\, dx", "d29021d5be273882babfb9a299c2cb6f": "\n\\sqrt[n]{z^m} = x^m+\\cfrac{2x^m \\cdot my} {n(2z - y)-my-\\cfrac{(1^2n^2-m^2)y^2} {3n(2z - y)-\\cfrac{(2^2n^2-m^2)y^2} {5n(2z - y)-\\cfrac{(3^2n^2-m^2)y^2} {7n(2z - y)-\\cfrac{(4^2n^2-m^2)y^2} {9n(2z - y)-\\ddots}}}}}.\n", "d290471fd2a8d2952f1222fb73a70e07": "h: \\mathcal A\\rightarrow\\mathcal B", "d2908437be577a7efdc8ae347e56cb7a": "\\psi^{(m)}(z)= (-1)^{m+1} m! \\zeta (m+1,z) \\ .", "d29089dca2c7cf2b59adf80f6441c55d": "(\\log n)^d", "d2909d2936a581babc696b9ed87cdcdc": "\\textstyle \\lambda(a_\\diamond\\mid A)", "d29190494a8c1cb7f7066f497fb18d51": "\\mathfrak{P}^{58}", "d291fa2c9d9282e6549589e533ebeb78": "T(f) = f(0).\\,", "d291fc489907fe84d34db2f1dc2539eb": " I = k_1 I_S + k_2 I_S + 2 \\sqrt { \\left ( k_1 I_S \\right ) \\cdot \\left ( k_2 I_S \\right )} \\cdot Re \\left [\\gamma \\left ( \\tau \\right ) \\right] \\qquad (1) ", "d29209a73c2958310cd67853f5647da0": "{u}_{1} (\\mathbf{q})", "d2921a8ae131e2b36ee01fd183d45e9b": "{}^{n}x", "d29236b7a414ffcd89f3a2e8e910ed4e": "Y \\sim \\chi^2(\\beta)\\,", "d292a3f67cc438f06dceb4ce4c3a0702": "\\frac{{\\tan\\theta_1}}{{\\tan\\theta_2}} = \\frac{{\\varepsilon_{r2}}}{{\\varepsilon_{r1}}}", "d2930afad6e0420d0e909463088f5663": " - \\tilde J = M \\frac{\\partial^2 f}{\\partial c^2} \\nabla c - 2 M K \\nabla^3 c ", "d2931f7f50e4a1596b3916ecb5aa7cd3": "{\\theta}=\\arctan \\left( \\frac{\\sqrt{x^2 + y^2}}{z} \\right)=\\arccos \\left( {\\frac{z}{\\sqrt{x^2 + y^2 + z^2}}} \\right)", "d29385ac9e6a10f15a429e52f8213268": "r_\\text{G} = \\frac{r_\\text{s}}{2} = \\frac{G m}{c^2}.", "d293c6a3f2331b420511985fc1d74ed6": "\\tfrac{4 \\pi \\kappa}{\\lambda_0}", "d294960ffcab8ff41a87cfa555946521": "y_0=1", "d294a90cf9698f8d17a4a4ff404c0500": " A_0(m) := \\frac12\\left(\\sum_{n < m} a(n) +\\sum_{n \\le m} a(n)\\right) = A(m) - \\frac12 a(m) .", "d294eb1122d80d9c167ffc26206506a1": "t' = E_K(P_n' \\oplus C_{n-1}')", "d295489105f962e7cbaaa8471f4bf9ed": "\nZ(\\omega) = \\frac{1-i}{cb}\\sqrt{\\frac{\\omega}{2\\pi \\sigma}}\n", "d295a1c2900b756616b676b32c2e2f1b": "s = \\frac{a}{b} 2 \\pi r", "d295c5a0d47dc4578721201e8b726007": "\\frac{A_0}{A_x}=e^{\\gamma x}", "d2961507a5d856ca10435a2785de59eb": "\\nu\\sim M+2e\\sin M+(5/4)e^2\\sin 2M", "d29644236b80e8b6a4fa214cd1c50f75": " \\langle \\psi \\mid \\operatorname{E}_A \\psi \\rangle. ", "d2966c63f9b0611241721ff8354a133d": "{\\mathbf{y}}(t)", "d296dbd04bf57b98d483123c6195fc45": "\n\\bar\\lambda=\\frac{\\sqrt{(\\rho+\\delta)^2+r^2\n\\pi}-(\\rho+\\delta)}{r^2/2}.\n", "d29709c27b74ed2ce9df6671abba890d": "\\kappa_{\\rm es} = 0.20(1+X) {\\rm\\, cm}^2{\\rm \\,g}^{-1}", "d2973868129d0d8fca04b2901dd4b31a": "p_{0,0,2k}", "d29749748316643ff8cc648bf42e903e": "h \\in L^2(\\mathbb{R})", "d297732b0751169bc81c698ce1e16063": "(a;q)_{-n} = \\frac{1}{(aq^{-n};q)_n}=\\prod_{k=1}^n \\frac{1}{(1-a/q^k)}", "d2978bbe4a752850416693e23a76cdd3": "p_2 = p_1 + (2i, 2j)", "d297a0a274257ae5b6abcdbaf8857997": "\n\\det \\vec{a} \\cdot \\vec{\\sigma} = - \\vec{a} \\cdot \\vec{a}= -|\\vec{a}|^2.\n", "d297a72e2803ef41f8e5ba412f69f51b": "(g^{jk})\\,", "d297a9db71cf24d067fad92c6be12d71": " \\sigma \\leftarrow \\sigma + a_{ij} \\phi_j ", "d297d074c2cad73105acba8126435f52": "\\overline{\\frac{\\partial u_iu_j}{\\partial x_j}}", "d2980c9cd27e432d2587527c6c1a8d47": "\\frac{Y(z)}{S(z)} = 1 - e^{-2 \\pi i \\omega}z^{-1}", "d29847589e949709ebafeb088435f26c": "\\sigma^{n}(A_{r}) \\geq 1 - \\sqrt{\\frac{\\pi}{8}} \\, \\exp \\left( - \\frac{(n - 1) r^{2}}{2} \\right).", "d298fc1a5efcef21ff5a3d51b549007c": " {\\rm det} (I+A):=\\prod_{n\\ge 1}[1+\\lambda_n(A)]", "d299615767381f3f5422a70bb7540d95": "\\tau_C={\\epsilon_r\\epsilon_0 \\over \\kappa}", "d29a423988c83d065d1c48a8003e1172": "P \\Rightarrow Q", "d29a57171bfdd4a07625c077f832d6e8": "f(X)=X^n+\\sum_{i=1}^n a_i X^{n-i} = \\prod_{i=1}^n (X-x_i),", "d29aa25511d8b6d15da4d7850861511d": "q =\\frac{\\alpha}{\\beta}", "d29ade2e40ce83e1806a98410fae2f88": "\\bar\\psi \\rightarrow -i(\\gamma^0 \\gamma^2 \\psi)^T", "d29b238e95ae6e28537ba4574c6e285d": " \\iint_F f(x,y) dA = \\int_a^b\\ \\int_{p(x)}^{q(x)} f(x,\\ y)\\,dy\\ dx \\ .", "d29ba668904c5cb214ef11deb53fddd7": "L_{\\omega_1 , \\omega}", "d29bb44c8f02d47a8eb8547136531849": "\\varphi = \\arg(z) =\n\\begin{cases}\n\\arctan(\\frac{y}{x}) & \\mbox{if } x > 0 \\\\\n\\arctan(\\frac{y}{x}) + \\pi & \\mbox{if } x < 0 \\mbox{ and } y \\ge 0\\\\\n\\arctan(\\frac{y}{x}) - \\pi & \\mbox{if } x < 0 \\mbox{ and } y < 0\\\\\n\\frac{\\pi}{2} & \\mbox{if } x = 0 \\mbox{ and } y > 0\\\\\n-\\frac{\\pi}{2} & \\mbox{if } x = 0 \\mbox{ and } y < 0\\\\\n\\mbox{indeterminate } & \\mbox{if } x = 0 \\mbox{ and } y = 0.\n\\end{cases}", "d29c35959aba4c4681c3c941f8cfaf01": "(G * H)/N.\\,", "d29c3bcbaa0f4fd4ee225e97881e54f1": "P \\to R", "d29c52a4054f2260ad2a1f9d25706111": "\\overline{\\rho} \\left( \\widetilde{\\phi \\psi} - \\tilde{\\phi} \\tilde{\\psi} \\right)", "d29c7230050408dd7a8d5f14d81ec777": "h = \\vec J\\cdot\\hat p = \\vec L\\cdot\\hat p + \\vec S\\cdot \\hat p = \\vec S\\cdot \\hat p,\\qquad \\hat p = \\frac{\\vec p}{\\left|\\vec p\\right|}", "d29c7a15dccb9d30263db1012ddd68ea": "\\ B=\\{", "d29ca030088135b03941c2e74506c925": "v_o \\approx \\sqrt{\\frac{GM}{r}}", "d29ce18ca2f19870f28624c4d2f3d0c0": " np(1-p) \\,", "d29d12f73282f9f3ba511c5724ab7095": " A(\\theta) = \\exp(\\theta)\\ .", "d29d185a35b0476d8ae7bbe85f759b25": " S= S(M, N^a) ", "d29d60bcd709105b97d73bf1c33ee09b": " \\hat{S}_+|\\psi_{n_{\\nu_1}}(\\bold{r}_1)\\rang|\\psi_{n_{\\nu_2}}(\\bold{r}_2)\\rang\\dots |\\psi_{n_{\\nu_N}}(\\bold{r}_N)\\rang= {b^{\\dagger}}_{n_{\\nu_1}}{b^{\\dagger}}_{n_{\\nu_2}}\\dots{b^{\\dagger}}_{n_{\\nu_N}}|0\\rang", "d29d792715a991c3f531d409578e994c": " \\alpha \\mu_A + \\beta \\mu_B = \\sigma \\mu_S + \\tau \\mu_T \\,", "d29d8a75ef39bdee629c6337c0fc96e6": "J_{ij} = -\\delta_{i,j-1} ", "d29debc07d7e8e4d7f354054bf0309d2": "\\mathbb{Z}\\left[\\sqrt{-5}\\right]", "d29e1885d8f5293f43a377a242558959": "\\overline Q", "d29e2d82db5ebee18a3effeb811e32b5": "\\displaystyle r_c=s", "d29e503c6b1e87efc1f1670a887a0f1b": "P,~L,~R", "d29e9f846b87d96ed902aa658e338ffb": "g:X\\rightarrow\\mathbb{C}", "d29ecca7337afa1d3ced84241af640ce": " \\beta \\in \\Big[\\ \\hat\\beta - s_{\\hat\\beta} t^*_{n-2},\\ \\hat\\beta + s_{\\hat\\beta} t^*_{n-2}\\ \\Big] ", "d29eecca420d791a3be745be188c7c58": " \\frac{I_0}{s} ", "d29f0b7d7e1eff0e26fea5a9ff36fa2f": " \\mathrm{SNR_{ADC}} = \\left (1.76 + 6.02 \\cdot Q \\right )\\ \\mathrm{dB} \\,\\!", "d29f3de3f9006d62f51279fb38823714": "(r\\xi^m)(s\\xi^n) = \n\\sum_{k=0}^m r (\\partial^k s) {m \\choose k} \\xi^{m+n-k}.", "d29f487e41c348126ccee45a462179df": "\\frac{\\operatorname{d}\\boldsymbol{r}}{\\operatorname{d}t} = \\left[\\frac{\\operatorname{d}\\boldsymbol{r}}{\\operatorname{d}t}\\right] + \\boldsymbol{\\omega} \\times \\boldsymbol{r}\\ ,", "d29fa1a85a18f9f7cb207ca4ee48224b": "x^2 + y^2 = r_1^2, z^2 + w^2 \\leq r_2^2", "d29fcd39811a379df5abf74de925d97e": "\\lim_{x \\to 5} (3x - 3) = 12.", "d2a02f5b7f8397d20130d2ea627818fd": "\\displaystyle P(w,x|z)", "d2a06012418f1f860583c66222f54f36": "s = \\sqrt{\\frac {\\sum_{i=1}^n {\\left ( x_i - \\bar x \\right )}^2}{n - 1}}", "d2a0ab87e3e46fcaeffcaeb36c268833": "\\sin(U-V)", "d2a0b0dbef7c0bd0d582c44af72e356b": "B(S'_i)", "d2a0fece5c2f45830e976333d24756e2": "\n\\begin{align}\n\\frac{d^\\acute{n}F(P_0)}{dP^\\acute{n}} & =\\frac{d^{\\acute{n}-1}F'(P_0)}{dP^{\\acute{n}-1}}\n=\\frac{d^{\\acute{n}-2}F''(P_0)}{dP^{\\acute{n}-2}}\n=\\frac{d^{\\acute{n}-3}F'''(P_0)}{dP^{\\acute{n}-3}}=\\cdots=\\frac{d^{\\acute{n}-r}F^{(r)}(P_0)}{dP^{\\acute{n}-r}},\n \\\\[10pt]\n& =\\frac{d^{\\acute{n}-1}G(P_0)}{dP^{\\acute{n}-1}} \\\\[10pt]\n& =\\frac{d^{\\acute{n}-2}G'(P_0)}{dP^{\\acute{n}-2}}=\\ \\frac{d^{\\acute{n}-3}G''(P_0)}{dP^{\\acute{n}-3}}=\\cdots=\\frac{d^{\\acute{n}-r}G^{(r-1)}(P_0)}{dP^{\\acute{n}-r}}, \\\\[10pt]\n& {\\color{white}.}\\qquad\\qquad\\qquad=\\frac{d^{\\acute{n}-2}H(P_0)}{dP^{\\acute{n}-2}}\n=\\ \\frac{d^{\\acute{n}-3}H'(P_0)}{dP^{\\acute{n}-3}}=\\cdots=\\frac{d^{\\acute{n}-r}H^{(r-2)}(P_0)}{dP^{\\acute{n}-r}}, \\\\\n& {\\color{white}.}\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\ =\\ \\frac{d^{\\acute{n}-3}I(P_0)}{dP^{\\acute{n}-3}}\n=\\cdots=\\frac{d^{\\acute{n}-r}I^{(r-3)}(P_0)}{dP^{\\acute{n}-r}}, \\\\[10pt]\n& =F^{(\\acute{n})}(P)=G^{(\\acute{n}-1)}(P)=H^{(\\acute{n}-2)}(P)=I^{(\\acute{n}-3)}(P)=\\cdots\n\\end{align}\n", "d2a12eb3115b30199c1fd52a4e16c270": " c = G = 1 \\ ", "d2a1955bef52cfb336c27d79ac2ef579": "d_1 = \\gcd(u_1,u_2)", "d2a1bfc9944f54866f67e0547034f929": "\n\\frac{x^{2}}{a^{2} + \\nu} + \\frac{y^{2}}{b^{2} + \\nu} + \\frac{z^{2}}{c^{2} + \\nu} = 1\n", "d2a1ea5476ddee2bb3b4475fec0ff1fb": "F_n=U_n(1,-1)", "d2a1eb73734fa25a51202b4b74be135b": "LSA = \\pi r l", "d2a1ff6c87e7445eaef577b6ebf4de47": "\\vec{p}_3 = \\sqrt{1-\\omega^2 \\, r^2} \\, \\frac{1}{r} \\, \\partial_\\phi + \\frac{\\omega \\, r}{ \\sqrt{1-\\omega^2 \\, r^2}} \\, \\partial_t", "d2a20e5599720cde780957a6882f5a09": "V_u", "d2a25ed9c288402873afa7989293068e": " V_r = \\Gamma_{TL} V_i \\, ", "d2a38316d17e3c61d8f9f4f36186e0e2": "\\phi_{ij} = 1", "d2a3cd8ffc6c504260a3c546b27c206a": "\\mathcal F = \\mathcal O(1) \\in \\mathrm{Pic}(\\mathbb P^N)", "d2a4176dad6bdab9c4ce07fff8af38a0": "\\ c(u, v) = 0", "d2a4514eb16a4ecad77f0422a8fb46c7": "x^* = x_0 + (A^T PA + Q)^{-1} (A^T P(b-Ax_0)).\\,", "d2a4637b2ac660bbd89cbc8e20dbcf20": "(\\Pi(R)f)(x) = \\sum_{l = 1}^\\infty\\sum_{m = -l}^{m = l}\\sum_{m' = -l}^{m' = l}D^{(l)}_{mm'}(R)f_{lm'}Y^l_m(R^{-1}x), \\qquad R \\in SO(3).", "d2a4d7afeed6f7bacbe79f096e9c7b02": "T_{\\rm b}", "d2a5005c0a6c61b3e676b23a8c6a9c58": " \\lim_{x \\to 0}\\frac 0 x = 0 ", "d2a528f973d3cc4444c1fab52d999287": " \\omega^2 = k^2 + m_0^2 \\,.", "d2a542f69f3d097d3ceba096bd7f9a72": " c \\in \\mathbb{C} \\,", "d2a54a17dde907ce77d5d0cc95178eb8": " \\theta'(y') ", "d2a5506677fe6aeeaf1eae015fb2b0e8": "ab.next", "d2a5890c31120b5b6fd25c6a215afd0d": "1 \\leq b 0\\mbox{)}}", "d2afb48ec46e4045fb617d181164f030": " X \\sim \\textrm{IG}(\\mu,\\lambda)\\,", "d2afc3e588209c1af55009ab427112b9": "\\sim2\\delta", "d2afe1191ee15649af7bb8fff58cfbfe": "q=b_m x^m +\\cdots+b_0", "d2aff18d304ecad894988b71db624fed": "\\frac{|2 - 5| + |2 - 5| + |3 - 5| + |4 - 5| + |14 - 5|}{5} = 3.6", "d2b03176d21d636adea48f50d468b3e7": "\\sigma_{12}=\\frac{1}{2}(\\sigma_1+\\sigma_2)", "d2b096915e9faa980d4eeda146c40f74": "\\scriptstyle \\frac {du}{d\\tau} = \\frac {dw}{d\\tau}, ", "d2b1155a1b75d5503cad2bdfa7336c70": " \\mathfrak{H}, \\mathfrak{H}'", "d2b15ce85045eae5a4111630802c8312": "S\\subset[\\omega_1]^\\omega", "d2b1718a989c4b753ca1bf5c3621f7db": "v_{8}", "d2b1781bc87d200308dfcfaf2df38288": "T(n-k,r-1)", "d2b19f9981da9fca87b15ca3134f6984": "\\beta = 8\\pi nkT/B^2 = 4.03\\times10^{-11}\\,nTB^{-2}", "d2b203da0f5758d225bda210509f7ca5": "\n[L_z,X+iY] = (X+iY)\n\\,", "d2b2459ff02207587c5fc29f5598bdcf": "\\begin{align}\n 0 &{}= \\det (\\lambda I - Q) \\\\\n &{}= \\lambda^3 - \\tfrac{39}{25} \\lambda^2 + \\tfrac{39}{25} \\lambda - 1 \\\\\n &{}= (\\lambda-1) (\\lambda^2 - \\tfrac{14}{25} \\lambda + 1).\n\\end{align}", "d2b24d12f95887217566c225f6e61ab5": "\\begin{cases}\nx' = ct' \\\\\nx = ct.\n\\end{cases}", "d2b2997af2b30caa1505d4b4ffeb0284": "w=\\varepsilon", "d2b2d9fec288403faf6e85ebf2c58972": "2n+1", "d2b2eb6cf8525376607fb4978b7a029e": "S^2\\times S^1 ", "d2b32196a546e62fed2b894a44e3f293": "\\mathbf{PTF(\\xi,\\eta)} = e^{-i 2\\cdot\\pi\\cdot\\lambda (\\xi,\\eta)} ", "d2b32958f460609dee1d071b0ff84766": "\\displaystyle{(a,b)=Q(a)Q(b)Q(a+b)^{-1}}", "d2b38b88845e879616844e9c0c8dff24": "\\mathbf{S}^{-1}", "d2b3a356374e6315341d24778b15cd14": "a_{\\overline{n|}i} < a_{\\overline{n|}i}^{(m)} < \\overline{a}_{\\overline{n|}i} < \\ddot{a}_{\\overline{n|}i}^{(m)}< \\ddot{a}_{\\overline{n|}i}", "d2b3b61aa5624007c627775f5627a13d": "\\pi:= \\bar{m}^aDn_a=\\bar{m}^al^b\\nabla_b n_a\\,, \\quad \\nu:= \\bar{m}^a\\Delta n_a=\\bar{m}^a n^b\\nabla_b n_a\\,, ", "d2b403238f792126b7cab7e041bbfe58": "\\begin{bmatrix}\n1 & 1 &\\cdots & 1\\\\\n\\end{bmatrix}\\mathbf{A}=\\begin{bmatrix}\n a & 0 & \\cdots & 0 \\\\\n \\end{bmatrix}", "d2b42b3e5e9d2f840337f69922491429": "c_w=\\left(\\frac{\\lambda_1-\\lambda_2}{\\lambda_1+\\lambda_2}\\right)^2", "d2b43e709bfe697a6210e87d84073ca0": "\\beta =\\alpha_y-\\alpha_x", "d2b45c68870d8e4b91bba9adda17684d": "\\scriptstyle \\sqrt{L^{2}+\\left(vT_{3}\\right)^{2}}", "d2b537e6413e48afbb2510793f7742c1": "n_i = m", "d2b547b0918a8fcc9106e1fd8d71c187": "T^{\\alpha' \\beta' \\cdots \\zeta'}_{\\theta' \\iota' \\cdots \\kappa'} =\n\\Lambda^{\\alpha'}{}_{\\mu} \\Lambda^{\\beta'}{}_{\\nu} \\cdots \\Lambda^{\\zeta'}{}_{\\rho}\n\\Lambda_{\\theta'}{}^{\\sigma} \\Lambda_{\\iota'}{}^{\\upsilon} \\cdots \\Lambda_{\\kappa'}{}^{\\phi}\nT^{\\mu \\nu \\cdots \\rho}_{\\sigma \\upsilon \\cdots \\phi}", "d2b57f86c4b3a0f728db0dab61b0fc6c": " y = a e^{b x}U \\,\\!", "d2b5cf0345f8728337047b9cd0062dde": "\nT={\\frac{T_{t}}{1+0.2M^2}}\n", "d2b6059a3c3bed343c8f49ee52a779fb": "u_1=x,", "d2b6b7c9a8ff2e109e9bdbc6aacbd519": "\\Phi (m) (n) := e(m,n) ", "d2b773c9c113cb8ff25aa3d3dfe420ed": "\\operatorname{Var}(Y) = A''(\\theta) d(\\tau). \\,\\!", "d2b7bf4765493b71de36ea2c3b4c8f11": " K_\\text{RQ}(x,x') = (1+|d|^2)^{-\\alpha}, \\quad \\alpha \\geq 0", "d2b7e762c04b3194d7adbb22cbfa4cf4": " \\dot{f}(x, y)=2x\\dot{x} + 2y\\dot{y} = 0.", "d2b7ff22ef5584b71366e86a53b580ec": "T_n(C_{\\bull,\\bull})^{II}_p = \\bigoplus_{i+j=n \\atop j > p-1} C_{i,j}", "d2b838f52bf3de4579046da417467868": "\\nabla \\cdot \\bold{P} = -\\rho_b \\ , ", "d2b84494357856dde4397401b2297fe3": "f^{\\#}(\\delta_p^v) = \\delta_{f(p)}^{df_p(v)}", "d2b84b95ad8f2f0ec2e8d17a04654337": "A_1.", "d2b8db348db25d4994a429de246238b9": "231 = (123)", "d2b8f6b413c6c1a562ef930f35af9b61": "T_\\text{std}= ", "d2b94fc0d999be9eb75eef675b181842": "Z[J_{ij}] \\sim e^{-\\beta J_{ij}}", "d2b95c728077c7867c8cc01b43fcb932": "t=-1", "d2b96c9915fe7d1368b87ab64b692766": "\\Delta W_{n} = W_{\\tau_{n + 1}} - W_{\\tau_{n}}", "d2b9ae3d3c3996f99a1c326b254c32af": "C^{oo} = \\operatorname{cl}(\\operatorname{co} \\{\\lambda c: \\lambda \\geq 0, c \\in C\\})", "d2b9ef407229b73ae4825a28a3997569": "\\Phi = \\omega \\, R_0 ", "d2ba3b784ba4e5cef1105b953a5e557e": "(z,y)\\in R_{j}", "d2baa25bda3b9e857e0aa114b1cb82e5": "\\Gamma \\vdash \\phi \\to \\psi", "d2bb2d17957067b49b69f485f80e9d6b": "\\,a^2 + b^2 = c^2 + 2ab\\cos\\gamma\\,.", "d2bba9466c52bc7403e82ee9cc4e35ef": "\\mathbf{L}= I \\boldsymbol{\\omega} \\,\\!", "d2bbb43bd0d7e850fad3c11b16cb04d2": "A \\subseteq B \\subseteq C", "d2bc74b64f89968db9266ab73c0f22b0": " \\operatorname{M}(a,b,c) = \\begin{bmatrix} 1 & a & c \\\\ 0 & 1 & b \\\\ 0 & 0 & 1 \\end{bmatrix} ", "d2bc9731331511a1182c3ddb2f3c41e1": " \\begin{align} \\Lambda_{00} & = \\gamma, \\\\\n\\Lambda_{0i} & = \\Lambda_{i0} = - \\gamma \\beta_{i}, \\\\\n\\Lambda_{ij} & = \\Lambda_{ji} = ( \\gamma - 1 )\\dfrac{\\beta_{i}\\beta_{j}}{\\beta^{2}} + \\delta_{ij}= ( \\gamma - 1 )\\dfrac{v_i v_j}{v^2} + \\delta_{ij}, \\\\\n\\end{align}\n\\,\\!", "d2bcf612b6f98f9104c598c9b2796c3f": "df_1(p) \\wedge \\ldots \\wedge df_r(p) \\neq 0,", "d2bd544c35b63b6c4c0976002a9e41fc": "\\alpha \\rightarrow 0", "d2bd556356046c7fcc0ba085439c66f4": "\\left(\\frac{\\sigma_f}{f}\\right)^2 \\approx \\left(\\frac{\\sigma_A}{A}\\right)^2 + \\left(\\frac{\\sigma_B}{B}\\right)^2 + 2\\frac{\\sigma_A\\sigma_B}{AB}\\rho_{AB}", "d2bd7e62df716f816cd61a1a5c244d40": "U_{ji}=\\varphi_{j}\\left(W_{ij}\\right)", "d2bd9081dfb3010649c353ed72eeedc7": " w_t= w_t(L_t, L_{t-1})\\,\\!", "d2bda62f4910906b7023a4df330626f1": "ce_r(\\omega,q)= \\sum_m A_{r,m} \\cos {m \\omega}\\text{ for }a=a_r(q)", "d2bdc3960da892b5da608cdfb7239a6a": " C = \\frac{(N-1) \\sum_{i} \\sum_{j} w_{ij} (X_i-X_j)^2}{2W \\sum_{i}(X_i-\\bar X)^2} ", "d2bdf1464abaa02d1527477420a69ad2": "\\{u,u'\\}", "d2bdf5eea9630de10d11a9c803c89b8a": "V^2 = {\\Vert (\\mathbf u \\times \\mathbf v) \\cdot \\mathbf w \\Vert}^2\n= {\\begin{vmatrix}\nu_1 & u_2 & u_3 \\\\ \nv_1 & v_2 & v_3 \\\\ \nw_1 & w_2 & w_3 \\\\ \n\\end{vmatrix}}^2\n", "d2be1e73d3f4e23b6058b20295a00484": " = -\\dfrac{5(\\sqrt{3} - 4)}{13}.\\,\\!", "d2be35feb55e7ce91e01f90bfbce1543": "O (n / \\log n)", "d2bea8e11bb8111d423d580d76894c45": "w = \\lceil e/\\epsilon \\rceil", "d2bed1ee3ef75724f8caf0d5f42d70e4": " p_\\text{up} = \\mid c_1 \\mid^2 ", "d2beee195639419f5a800fce5b9418f9": "F = \\mathbf{P}^{(1)}", "d2bef8ad08418ca94dcb749ff2422e27": "P(x)=a_0 (x-\\alpha_1)(x-\\alpha_2)\\cdots(x-\\alpha_D),", "d2bf2ef839dac39d96ddc048157a005d": "\\{c_r,c_i\\}", "d2bf47282336d90584f0d769f8ede077": "\\mathbf{B}_\\text{g} = \\frac{G }{2 c^2} \\frac{\\mathbf{L}}{r^3},", "d2bf7b6f33b1082b9f57f3a089cf711f": "\\frac{a}{bc}+\\frac{c}{b^2 d}\\;=\\;\\frac{abd}{b^2 cd}+\\frac{c^2}{b^2 cd}\\;=\\;\\frac{abd+c^2}{b^2 cd}", "d2bf9c7b67f8218387d143bd69d0d55c": "\n\\begin{align}\n \\partial_t \\rho &= \\nabla\\cdot(\\rho \\mathbf{v})\\\\\n \\partial_t \\rho &= \\frac{1}{c^2}\\partial_t p\n\\end{align}\n", "d2bffbfb2430b87110126bba406a27f5": "-D_e", "d2c03b2ba1d768b7fde7f73d7df0a75a": "f : G \\rightarrow H", "d2c103af71625de62dc744f94f356398": "-\\frac{6(n^2+1)}{5(n^2-1)}\\,", "d2c171bdaf9e42cf7e6b09c10fe1a552": "\\mathbb{Z}_{3}\\oplus\\mathbb{Z}_{3}", "d2c1a30928f8276d7417cf5efa91e58e": "\\mathcal{CC}", "d2c1f0e44f3f445fc5c1dd2b36009165": " \\textbf{Z}_{t} = \\left \\{ \\textbf{z}_{1},\\dots,\\textbf{z}_{t} \\right \\} ", "d2c1f37f5e4fef0d151917deaa9f17dc": " \\downarrow \\qquad \\qquad \\qquad \\quad \\downarrow ", "d2c2a212f99a2d3f15d873b09c541b8d": "\\sigma_{ij,kk}+\\frac{1}{1+\\nu}\\sigma_{kk,ij}+F_{i,j}+F_{j,i}+\\frac{\\nu}{1-\\nu}\\delta_{i,j}F_{k,k}=0.\\,\\!", "d2c37efbdacccd12a242f770bc5cfbd7": " \\bold g = g_{ab}e^{(a)}e^{(b)} = g_{ab}e^{(a)}_\\mu e^{(b)}_\\nu dx^\\mu dx^\\nu = g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}", "d2c38b6ed33c33e100c70270c31d23f5": "(f_0,\\ldots,f_{n-1})", "d2c3a7cd682d01aa0d92b1e81906156c": " \\nabla B = - B \\frac{\\vec{R}_c}{R_c^2} ", "d2c3b4f37803ba37ea93af8f43bb21c9": " \\cos(\\omega_0^{ } t ) ", "d2c3b5d13dedd706418ba536f0a5ca6e": " Y_i = \\begin{cases}\n1 & \\text{if }Y_i^{1\\ast} > Y_i^{2\\ast},\\ldots,Y_i^{m\\ast} \\\\\n2 & \\text{if }Y_i^{2\\ast} > Y_i^{1\\ast},Y_i^{3\\ast},\\ldots,Y_i^{m\\ast} \\\\\n\\ldots & \\ldots \\\\\nm &\\text{otherwise.} \\end{cases} ", "d2c3bd440fa32835b648759a04fd547d": "(t,\\overline q^i)", "d2c3c4fad194705e7f44c7609e7de4e7": "\\scriptstyle\\ \\hat{p} > \\alpha ", "d2c406ddbeff1066d91642bffbeb41cf": " a_n = \\frac{1}{n \\log^2(n)} , \\quad n \\geq 2 . ", "d2c4745bacbe01ed677f503676d5808d": "q=\\|q\\|e^{\\hat{n}\\theta} = \\|q\\| \\left(\\cos(\\theta) + \\hat{n} \\sin(\\theta)\\right),", "d2c47d21c44b3e17ca76b9915b8d1a13": "\\mathbf{y} \\triangleq \\begin{bmatrix} \\mathbf{x} \\\\ z_1 \\end{bmatrix}\\,", "d2c485b0018ee6b306b5207e9679921c": "i = -\\ln(p)", "d2c4a16e9b793c3c409b4b1af6b0f61a": "\\lambda_m=\\frac{m}{\\sum_{k=1}^K W_k(m) v_k}.", "d2c4c61a28950f01487422722689ee7d": "-3\\zeta(-1)=\\eta(-1)=\\lim_{x\\nearrow 1}\\left(1-2x+3x^2-4x^3+\\cdots\\right)=\\lim_{x\\nearrow 1}\\frac{1}{(1+x)^2}=\\frac14", "d2c4f65c67c0995dd22a01998b28d332": "\\rho(\\phi)\\langle \\psi,\\psi\\rangle= 2\\langle [\\phi,\\psi],\\psi\\rangle ", "d2c529dd6e0178f8e8e7cc297ddbfaf5": "\nS(x) := \\left\\{\n\\begin{matrix}\n S_0(x) := & \\sum_{\\nu=0}^{n} \\beta_{\\nu,0} b_{\\nu,n}(x) & x \\in [x_0, x_1] \\\\\n S_1(x) := & \\sum_{\\nu=0}^{n} \\beta_{\\nu,1} b_{\\nu,n}(x - x_1) & x \\in [x_1, x_2] \\\\\n \\vdots & \\vdots \\\\\nS_{k-2}(x) := & \\sum_{\\nu=0}^{n} \\beta_{\\nu,k-2} b_{\\nu,n}(x - x_{k -2}) & x \\in [x_{k-2}, x_{k-1}] \\\\\n\\end{matrix}\\right.\n", "d2c544fd787b4b5ea6acb37037f14d40": "\\lambda^*(E)", "d2c5456d6d46814c191c642ecb37f867": "\\tilde{A}=P \\hat{A} P", "d2c55ff6d1bbf4fc7ab556a6fcd9adba": "VF = { \\frac{1}{\\sqrt{\\kappa}} } \\ ", "d2c5972d2e5dee5568c71beed19098f5": "\\lceil{(\\ell +1 -2)/{2}}\\rceil \\log n +1 ", "d2c5b2e28bc10d8212d702dbb52640ed": "\\Sigma a_n", "d2c5b885b40673fea0cf782547d87b09": "\nq_{xy} = \\frac{\\sum (x-\\bar{x})(y-\\bar{y}) I(x,y)}{\\sum I(x,y)}\n", "d2c5e8a1dfd32cefea31f18098d8aff8": "C_{m,n}", "d2c5f26a93f0bb3f7e91703a46585a1d": "[(T_1 \\land T_2 \\land \\dots \\land T_n ) + ( C_1 \\land C_2 \\land \\dots \\land C_m )]", "d2c66112a95f107997b6721c36f1c2dc": "\\Sigma_{\\rm cr} \\kappa_{\\rm smooth}, ", "d2c6b7c43c9dbab87010169ca4f8c94e": "\\partial\\circ\\partial \\equiv 0", "d2c6b965b57454f5eaefce1437892948": "\\sum_{i = 1}^N \\frac{\\partial \\phi_i}{\\partial t} \\frac{\\partial \\Pi(\\mathbf{\\xi}, t)}{\\partial \\xi_i} = \\sum_{i = 1}^N \\sum_{j = 1}^R S_{ij} f'_j (\\mathbf{\\phi}) \\frac{\\partial \\Pi(\\mathbf{\\xi}, t)}{\\partial \\xi_j}. ", "d2c7049c0f33fe7447464fc567e67f5c": "\\mathrm{End}_R(P)", "d2c7ffd3a25269a2ca07535003d2b007": "e={{r_a-r_p}\\over{r_a+r_p}}", "d2c81be3b0acfa1063082f0b141426e2": "\\mathfrak h", "d2c844145958c108854d122ae77c42c7": "r\\cot\\psi=\\frac{dr}{d\\theta}.", "d2c8880f66318dca335c0b235a7d4ffe": "\\Delta _{j,0}=y_j,\\quad \\quad \\Delta _{j,k}=\\frac{\\Delta _{j+1,k-1}-\\Delta _{j,k-1}}{x_{j+k}-x_j}\\quad \\ni \\quad \\left\\{ k>0,\\ \\ j\\le \\max \\left( j \\right)-k \\right\\},\\quad \\quad \\Delta 0_k=\\Delta _{0,k}", "d2c88ca7a24733eb8045a134417f8d51": "\\frac{\\left ( \\dfrac{dr}{dt_r}\\right )_\\text{raindrop}} {\\left ( \\dfrac{dr}{dt_r} \\right)_\\text{light}}=\n\\frac{\\sqrt{\\dfrac{2M}{r}}} {1+\\sqrt{\\dfrac{2M}{r}}} < 1\\,\\!", "d2c8a49b5b075ea6ca2105818138da8e": " \n\\begin{align}\n & {} 2 x^6 -4 x^5 +5 x^4 -3 x^3 + x^2 +3 x \\\\\n & = (A + D) x^6 + (-A - 3D) x^5 + (2B + 4D + 1) x^4 + (-2B - 4D + 1) x^3 + (-A + 2B + 3D - 1) x^2 + (A - 2B - D + 3) x \n\n\\end{align}\n", "d2c8a4bda7345c329e43212f488d16a5": "h(p, u) = \\nabla_p e(p, u).", "d2c8aac8cbc0b55705ec126614f3f58e": "\\mathbf{x}_n", "d2c8de199c96a63a5b800c82b21de477": "X=\\sum_{y=1}^Y W_y.\\,", "d2c94681f056fd402f9f3f429f2d74e6": "\\sigma_a^F", "d2c993d51ea8d9a1fa23f2b78ec3b697": "K = (g^{y} g^{b}) ^ {x + a}= g^{(x + a) (y + b)}", "d2c9a6d16778f3ca3f23c3cb62c50834": " \n\n {1\\over 4 \\pi m^2r^3 } \n\\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right]\n .", "d2c9f9293800969b1feb42db6a44af94": "\\alpha = 1/N \\sum_{i = 1}^N y_i", "d2caa9307067a6d53cfcd7e196534842": " \\int_a^b \\frac{\\partial f(x)}{\\partial x}\\,dx=\\underset{x \\nearrow b}\\lim f(x)-\\underset{x \\searrow a}\\lim f(x),", "d2cb11ff2d24402e3f1edb9518e2cdae": "\\mathbf J_{\\mathrm{i}} = \\mathbf{J}_{\\mathrm{f}} + \\boldsymbol{\\lambda}", "d2cb7ad790344c3a93ce7389dd517b9b": "\\rho u_o {du_o \\over ds} = -{dp \\over ds}", "d2cbc45424fbe12a32bfee048eaae123": "\n V\\left( r_{12}\\right)\n=\n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_{B}^2 }\n\\; M \\left ( \\mathit l + 1, 1, -{k^2 \\over 4} \\right) \\;M \\left ( \\mathit l^{\\prime} + 1, 1, -{k^2 \\over 4} \\right) \\;\\mathcal J_0 \\left ( k{r_{12}\\over r_{B}} \\right)\n", "d2cc65b287eda82f5fe710ea5729b482": "\\nu_d: \\mathbb{P}V \\to \\mathbb{P}(\\rm{Sym}^d V)", "d2cc818b868f49fe69371a7ac538cbdb": "y_{i} = \\mathbf{x}_{i}^{\\rm T} \\boldsymbol\\beta + \\epsilon_{i},", "d2cc9eaa9983e519af159cdf15cdb304": "(x_1, x_2, \\dots , x_k)", "d2ccb4609d03c1b9cf9bd3e9f8b56818": "\\kappa\\setminus S", "d2ccca9e486c1838f929683ca9e60c0d": "\n\\begin{array}{llll}\n1: & x:\\alpha \\vdash x : \\alpha & [\\mathtt{Var}] & (x:\\alpha \\in \\left\\{x:\\alpha\\right\\})\\\\\n2: & \\vdash \\lambda x.x : \\alpha\\rightarrow\\alpha & [\\mathtt{Abs}] & (1)\\\\\n3: & \\vdash \\lambda x.x : \\forall \\alpha.\\alpha\\rightarrow\\alpha & [\\mathtt{Gen}] & (2),\\ (\\alpha \\not\\in free(\\epsilon))\\\\\n4: & id:\\forall \\alpha.\\alpha\\rightarrow\\alpha \\vdash id : \\forall \\alpha.\\alpha\\rightarrow\\alpha & [\\mathtt{Var}] & (id:\\forall \\alpha.\\alpha\\rightarrow\\alpha \\in \\left\\{id : \\forall \\alpha.\\alpha\\rightarrow\\alpha\\right\\})\\\\\n5: & \\vdash \\textbf{let}\\, id = \\lambda x . x\\ \\textbf{in}\\ id\\, :\\,\\forall\\alpha.\\alpha\\rightarrow\\alpha & [\\mathtt{Let}] & (3),\\ (4)\\\\\n\\end{array}\n", "d2cd3bbbadfe05c0e12818a59afa96d6": "\\mu_e = 0.5e\\nu_eS_e = 0.5e\\frac{m_0c^2}{h}\\frac{\\lambda_0^2}{2\\pi} = \\frac{e\\hbar}{2m_0}, \\ ", "d2ce2526f1a41110308903502052d2d1": " F: C_\\bullet\\to D_\\bullet", "d2ce2970f395516f0ec19a9371e972a4": "h=y-x", "d2cec7f234dc4ddea06d5ecb119e38a1": " b^{\\log_b(x)} = x\\text{ because }\\operatorname{antilog}_b(\\log_b(x)) = x \\, ", "d2cf0b397bd1634ad0b04511467cfe80": "\n\\mathbf{X} \\equiv \\frac{1}{M_\\mathrm{tot}} \\sum_{i=1}^N M_i \\mathbf{R}_i \\quad\\mathrm{with}\\quad\nM_\\mathrm{tot} \\equiv \\sum_{i=1}^N M_i.\n", "d2cf0ba47496f1cb80d9c662a230a5cb": " N_{+} \\rightarrow N_{-} \\,\\!", "d2cf2bd14ec7b6ce79688b1fd9cb0f7d": "2tx-y=t^2 \\,", "d2cf3d662d85c26043cb14b214774b89": " \\lambda_1+\\lambda_2 \\le 1. \\, ", "d2cf9632b2db9d68677957a087b8f1ac": "|\\mathrm{GHZ}\\rangle = \\frac{|0\\rangle^{\\otimes M} + |1\\rangle^{\\otimes M}}{\\sqrt{2}}.", "d2cfab7f163bd9700b971e1d3188aaa0": " {\\pi}_n\\,\\!", "d2cff2399aa5f8a9a5b685d33f530f2f": "g_{SP}(t,\\omega) = C_h(t,-\\omega)", "d2d018acab663e3a3d8108f8602ec7d6": "K_3/8", "d2d07ee96e5370f71e08a9ed4d7f87a8": "p \\equiv q", "d2d094acc7b325c11c75128a8de96409": "\\sum_{n=2}^{\\infty}\\varphi^n = 1", "d2d0dc62120993259bc9966f4a39b661": " [P_\\mathrm{c} - P_\\mathrm{i}] - \\sigma[\\pi_\\mathrm{c} - \\pi_\\mathrm{i}]", "d2d14813275b0a65a1900e4c161e042b": " f(t_0,y_0) = f(0,1) = 1. \\qquad \\qquad", "d2d14bc14fbc6f53e3cd1c689bebe0ec": "4\\%-2\\%=2\\%", "d2d162fd8c7fbff2853ba4c174dba4f1": " g_{\\alpha \\beta , \\gamma} \\, g^{\\beta \\gamma} = \\tfrac12 g_{\\beta \\gamma , \\alpha} \\, g^{\\beta \\gamma} \\,.", "d2d18be2ab2c3d7b6291aaf770c8ca4a": "\\scriptstyle\\sigma", "d2d1afed9b43510de1b50b39ad179af6": "\\mu _{G} (x) = \\mathop \\vee \\limits_{\\alpha :x \\in \\Phi _\\alpha \\left( {A_\\alpha ^1 , \\cdots\n,A_\\alpha ^n } \\right)_\\alpha } \\alpha ", "d2d1bfecd058281360a78bac79ef720f": "\\hat \\sigma:V\\to\\bar V\\,", "d2d1c999685e9f49b481f48790bc8d37": "\n \\Pr(tD_1", "d2e54a361959f402f39a252d615c69e7": "I=I_\\mathrm{S} e^{V_\\mathrm{D}/(n V_\\mathrm{T})}", "d2e55399cb6adfeabc31a87296f78958": "a_k=(a_{1,k},\\ldots,a_{m,k})^T", "d2e5e8221399070bcf2bae1031941906": "\\begin{cases}\\mathcal L^\\infty(X,\\Sigma,\\mu)\\to L^\\infty(X,\\Sigma,\\mu) \\\\\n f\\mapsto [f]\\end{cases}", "d2e5fa7b4626feecb8624566e835050d": "BD-C^2-AE > 0,", "d2e601fb5ea93d840da31204705fe7c4": "(Tx_n)_{n\\in\\mathbb N}", "d2e63917537816191e3b101f7c611687": "de_{ij}", "d2e64269212d852ba506c92916e4e975": "E = E_0 + \\frac{(\\hbar k)^2}{2m} \\ ,", "d2e64831d6a9336c72d44922a65408aa": "GL_n \\times GL_m ", "d2e6651ff1b4e4e344a1d33bb4a6cf64": "x_{n}=\\sin^{2}(2 \\pi y_{n})", "d2e671e652f40709cdb655c5fbccdfe8": "n=2,4,8", "d2e67cb8d7a1711846e46ec934e6ecd1": "\\hat{a} ", "d2e6acc6fc41038fb84cec7b55d51a25": "ax \\bmod 2^w", "d2e70d7a2de7144926bc8e4f8e211966": "R\\ddot{R} + \\frac{3}{2}\\dot{R}^{2} = \\frac{1}{\\rho}\\left(p_g - P_0 - P(t) - 4\\mu\\frac{\\dot{R}}{R} - \\frac{2\\gamma}{R}\\right)", "d2e77b980456bd92969e33214619f38f": "ax^2+bx+c = a \\left( x + \\frac{b}{2a} \\right)^2.", "d2e7800d519a997a606a44aa3e5b15ff": "a_{\\ell m}^{(E)}=\\frac{-ik^2}{\\sqrt{\\ell(\\ell+1)}} \\int d^3\\mathbf{x'} j_\\ell(kr') Y_{\\ell m}^*(\\theta', \\phi') \\left[-ik\\mathbf{\\nabla}\\cdot(\\mathbf{x'}\\times\\mathbf{M}(\\mathbf{x'}))+ik\\mathbf{x'}\\cdot\\mathbf{J}(\\mathbf{x'})\\right] + c Y_{\\ell m}^*(\\theta', \\phi')\\rho(\\mathbf{x'})\\frac{\\partial}{\\partial r'}(r' j_\\ell(kr'))", "d2e78d675321c767e8087462a85491e2": "1/\\cos(x)", "d2e79e19d5fb3abb962237e7f93e3801": "z = 5.2", "d2e7a035c8248c3a41b297684a9c2cf0": "\\displaystyle \\phi\\,", "d2e7aecc55e557cee9a8e041708fec0d": "\\frac{5}{12}+\\frac{6}{12}=\\frac{11}{12}", "d2e819df1dfadb367ccf6266f377f2aa": "\n\\Lambda = \\left.\\left\\{ \\sum_{i=1}^n a_i v_i \\; \\right\\vert \\; a_i \\in\\Bbb{Z} \\right\\}\n", "d2e82d5d6cd1103ee2dcb7ee39f51ba4": "\\mu_{3,1}= \\frac{\\langle b_{3}, b_{1}^{*} \\rangle}{B_{1}}=\n\\frac{\\begin{bmatrix}3\\\\5\\\\6\\end{bmatrix} \\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}}{3}=\\frac{14}{3}(> \\frac{1}{2})", "d2e8621dbf508c7a4f7fe9cce5188af8": " \\frac{[(x-x')^2+(y-y')^2]^2}{ \\lambda^4} \\ll 8 {z^3 \\over \\lambda^3}", "d2e8f321ab111e87965f0d85a20d7950": " t \\approx \\frac{72}{r} \\times \\frac{192}{200-r}", "d2e92112f702df2b9c00596580122032": "\n\\begin{align}\n \\langle Q\\rangle & = \\int_{\\Gamma_\\min}^{\\Gamma_\\max} \\int_\\Omega Q(F_\\boldsymbol{r}) p(F_\\boldsymbol{r}) \\delta(f - F_\\boldsymbol{r}) \\,d\\boldsymbol{r} \\,df \\\\\n & = \\int_{\\Gamma_\\min}^{\\Gamma_\\max} Q(f) p(f) \\int_\\Omega \\delta(f - F_\\boldsymbol{r}) \\,d\\boldsymbol{r} \\, df \\\\\n & = \\int_{\\Gamma_\\min}^{\\Gamma_\\max} Q(f) p(f) \\rho(f) \\, df \\\\\n\\end{align}\n", "d2e9345efddd9ac62cd2dcce8bdfa844": "\\frac{1}{n-c}", "d2e94b5d1176d988bdadfab90b7a252d": "F(x) = \\int f(x)\\,dx.", "d2e97a618ac5894d746eaa52d2841f05": "\\Pi^I", "d2e987a9b429b84118c3e0d3c0bb90b7": "T_yf(x) =c_n\\int_{\\mathbf{R}^n} \\frac{y f(x)}{\\left (|x-t|^2 + y^2 \\right )^{\\frac{n+1}{2}}} dt.", "d2e9a81fcaf01b6f4cbf982413aece88": "F_{X_{\\gamma}} = -Z_{\\gamma}\\left(\\int\\mathrm{d}\\mathbf{r}\\ \\rho(\\mathbf{r})\\frac{x-X_{\\gamma}}{|\\mathbf{r}-\\mathbf{R}_{\\gamma}|^{3}} - \\sum_{\\alpha\\neq\\gamma}^{M}Z_{\\alpha}\\frac{X_{\\alpha}-X_{\\gamma}}{|\\mathbf{R}_{\\alpha}-\\mathbf{R}_{\\gamma}|^{3}}\\right).", "d2e9d33a22ed311d9f6203564e300ad1": "\\mathcal{A}(\\mathcal{O})", "d2ea369a889d116a73580a887657e29e": " P = D^{-1}K. \\, ", "d2ea8d338ac1b5cccad1052a0c0af1af": "a*(1+e)", "d2eaa1f1575f9959cf15f60a26a0468b": "g_{11} = - e^{\\lambda(r)} \\;", "d2eabb3492d4d0a0529b83d2342558e4": " L_\\text{normalized} = I - D^{-1/2}AD^{-1/2} ", "d2ead69046b2346f44901321539ee6c4": "2^{\\aleph_0+n}\\,", "d2eb3b0eddb5caac3fea60ab154a1037": "p = 3n + 1= \\tfrac14 \\left(L^2+ 27M^2\\right),", "d2eb5d8fe34bba179be079f52f4462af": "\n Y_{ij} = \\mu + U_i + W_{ij},\\,\n ", "d2eb646d65ad645c4f223d9f9ae73be3": "n^{2^k-1}M^{2^k}", "d2ebd2589acea0081d8e47365135bb40": "K = h \\cdot S_b (X + \\bar{X}A)", "d2ebdb1622dccb7a0f00fa11374872d0": "\\mu'_s \\,", "d2ec26711d4daf8969247aa50ae912e2": " \\begin{cases} \\displaystyle\n\\int_{-\\infty}^\\infty \\left| G_x(t,f) \\right|^2df < e^{-2\\pi(t-t_0)^2}\\int_{-\\infty}^\\infty \\left| G_x(t_0,f) \\right|^2\\,df; & \\text{if } x(t) =0 \\text{ for }t>t_0 \\\\[12pt]\n\\displaystyle\n\\int_{-\\infty}^\\infty \\left| G_x(t,f) \\right|^2\\,dt < e^{-2\\pi(f-f_0)^2}\\int_{-\\infty}^\\infty \\left| G_x(t,f_0) \\right|^2\\,dt; & \\text{if } X(f) =FT[x(t)] = 0 \\text{ for }f>f_0\n\\end{cases}", "d2ec2775f27f9675195c5e37f937bdbb": "f_n(r_1, \\dots, r_n)", "d2ec3d3d87bb7b9ab0719240b886d6df": " \\epsilon_j", "d2ec80c5a8ec196ee7e2b50273875646": " = \\int_{0}^{L} adz' \\int_{x,y} dx dy \\, \\frac{\\rho(x, y, z/a)}{a} \\,r^2 ", "d2ecf0312f3cf21a2a21aa25c95e9f2e": "\\displaystyle m_a^2+m_b^2+m_c^2=6R^2.", "d2ed13c8dbd67f22fa47a3f69323073e": "[a, a+1, a+2, ... , a+p] \\bigcap S \\neq \\emptyset", "d2ed8442aaf6a5b6ea76c8c4af313100": " const=\n7.7750627...\\times 10^{-5}", "d2ed893e864ceba6768b2997cce0f956": "\\mathcal{P}=\\{P: R\\subset P", "d2eda47b863445ab739a1d48782aab2e": "\n \\underline{\\underline{\\boldsymbol{A}}} = \\begin{bmatrix} A_{11} & A_{12} & A_{13} \\\\ A_{21} & A_{22} & A_{23} \\\\\n A_{31} & A_{32} & A_{33} \\end{bmatrix}~.\n ", "d2edba3c52c76fc55fcdb0c1d773bc30": "\\mathrm{Tr}^U_{X,Y}((Y\\otimes g)f)=\\mathrm{Tr}^{U'}_{X,Y}(f(X\\otimes g))", "d2ede2fde7fe8f69d78985424e0e3d8b": "(m,f(x),a,b,G,n,h)", "d2ee92cba004ed9baba46962ae48fd35": "T \\hat{\\mathbf{L}} T^\\dagger = - \\hat{\\mathbf{L}} ", "d2eeb9e559cd4aec85190a3061321276": "\\ o(t) = s(t) - m(t)", "d2eee10bb81613c249a6c5d8da445825": "{}_1F_1(a;b;z) = 1 + \\frac{a}{b\\,1!}z + \\frac{a(a+1)}{b(b+1)\\,2!}z^2 + \\frac{a(a+1)(a+2)}{b(b+1)(b+2)\\,3!}z^3 + \\dots", "d2eeec3feef08eb6ec928c68d7a50d0e": " \\scriptstyle \\Psi^{*} \\partial \\Psi/\\partial t \\,\\!", "d2ef0585a4b938253e63ab08b8048921": " \\mathfrak{I}\\,", "d2ef05ee8d8f56a81c6529919aeef5be": "R_E \\frac{r_E + R_E}{r_\\pi + r_E + 2R_E}", "d2efa8585fd4454fa2202562eb5f5006": "f^o_k", "d2efe9b679fc3aaa730ae8930cbc0079": "\\mathbb{E}[2X_i^e + X_i^?] = 2 - {2\\omega_i \\over d} \\le {2e_i \\over d}.", "d2f00a4684b63e82144c8db9ca699c7a": "\\eta^{\\mu \\nu} \\,", "d2f072d4b1b97f6827c3166b8d10f610": "x = x_0 + x_1\\,i + x_2\\,j + x_3\\,k + x_4\\,\\ell + x_5\\,\\ell i + x_6\\,\\ell j + x_7\\,\\ell k,", "d2f0a6c137c7f9c902638cccf62a241f": "T(1) e", "d2f0cb2b901afa923577789302239316": "Action = [-1,0,1]", "d2f107c2640280cfb6e4426342e38c4a": " \n\\alpha(t) = \\text{drift-plus-penalty action for slot t} \n", "d2f133f2062a7de0788273d96cc7b590": "\\{ \\hat\\theta_{(\\ell)}\\,,\\hat\\sigma_{(\\ell)}\\,, \\hat\\omega_{(\\ell)} \\}", "d2f140d1f9a3508cd9b2e1f093aaad14": "\\Gamma(s,z)", "d2f16d9e7253c241d9ee797b2a4483f4": " \\eta_{batt}=\\frac{P_{out}} {P_{in}} ", "d2f1864d672c73f8a374581b017a95b7": "\\alpha(x) < 5", "d2f1ed6c77ad32bb8d2104472028c18b": "C_l \\ (l \\ge 2)", "d2f23f9b3456cb26bc2fef0dd30ecd49": "\n\\begin{array}{rl}\n{\\displaystyle\\min_{X \\in \\mathbb{S}^n}} & \\langle C, X \\rangle_{\\mathbb{S}^n} \\\\\n\\text{subject to} & \\langle A_k, X \\rangle_{\\mathbb{S}^n} \\leq b_k, \\quad k = 1,\\ldots,m \\\\\n& X \\succeq 0\n\\end{array}\n", "d2f29ba700c9c4d5650d25ad7cb200f8": "y_k=\\nabla f(x_{k+1})-\\nabla f(x_k),", "d2f2adb7127a4bcc433168610b987a68": "\\frac{d^2\\gamma^\\lambda }{dt^2} + \\Gamma^{\\lambda}_{\\mu \\nu }\\frac{d\\gamma^\\mu }{dt}\\frac{d\\gamma^\\nu }{dt} = 0\\ ,", "d2f32499cf24f16f7eb8435bf97739e8": " X_1^n ", "d2f324bc19b4f172d0e2572244f99619": "(\\mathbb{Z}/2\\mathbb{Z}) \\oplus \\mathbb{Z}", "d2f36f57f93f85c2f9bc7e66d4da4243": "F=\\{f_1 ,..,f_q\\}", "d2f3b5c4ada3e2dce5f24bc904dab0e2": " m_1 > m_2 > \\cdots > m_k, ", "d2f3cfc2e974ec2b8c981811f983088d": "\\mathrm{Range}(\\mathcal{A})\\,\\!", "d2f4003e7b069d0471398ac784841611": "\\dot H_k= \\dot n_k H_{mk} = \\dot m_k h_k ", "d2f401db3e704ce6cb16df8db10b56e8": "(I = \\int r^2 dm)", "d2f404f5ab39a3b91ec90b19f4de4c2b": " \\overline{O_L P}, \\overline{O_R p}", "d2f4133cdf90965c325a46f958fe7438": "\\mathbb{C} P^n", "d2f42d49b535eff653dbe0b4b9f467e8": "\\left\\{X\\mid \\; C\\hat{X}D \\; = \\theta\\right\\}.", "d2f4abf793a91f7253e1b639647f93bc": "\\theta - \\phi = v/c", "d2f4d73e158ca4a911f2ab575d7e996d": "\\rightarrow_1 \\cup \\rightarrow_2 = \\rightarrow", "d2f4ec20e5e7c3b8a1fe1fad70c803fe": "\\phi(x,y,t)", "d2f535cf1cc98a41745d14455734f318": "f(\\lambda x + (1 - \\lambda)y)<\\max\\big(f(x),f(y)\\big)", "d2f555fe41b9d2bc7bc6e3571fa3a903": "\\pi_n(X,A) \\to \\pi_n(X \\cup CA) \\,\\!. ", "d2f58736fb032b5d2809e1974717f296": "\n\\kappa(\\vec{\\theta}) = \\frac{1}{2} \\nabla_{\\vec{\\theta}}^2 \\psi(\\vec{\\theta}) = \\frac{4\\pi G D_{ds}D_d} {c^2 D_s} \\int dz \\rho( D_d \\vec{\\theta},z) \n== {\\Sigma \\over \\Sigma_{cr} } == \\sum_i { 4\\pi G M_i D_{is} \\over c^2 D_i D_s} \\delta(\\vec{\\theta}-\\vec{\\theta}_i)\n", "d2f59ed3191523b0c4f897b086c91cf3": "\\begin{align}\nm(\\mathbf{x}') & = \\mathbf{k}^\\top (\\mathbf{K} + \\sigma^2 \\mathbf{I})^{-1} \\mathbf{Y}, \\\\\n\\sigma^2(\\mathbf{x}') & = k(\\mathbf{x}',\\mathbf{x}') - \\mathbf{k}^\\top (\\mathbf{K} + \\sigma^2 \\mathbf{I})^{-1} \\mathbf{k}.\n\\end{align}", "d2f61462db23ea6f73e52e5786b03487": " = ac + bdi^2 + (bc+ad)i \\ ", "d2f6d33f8c9070a78813e590a8629288": "|a_n|\\le M", "d2f7195f4a92eea0685e5de7dae1c011": "g+i", "d2f74621396cdb68952f80370e2b8b23": "\\left[{n\\atop n-2}\\right] = \\frac{1}{4} (3n-1) {n \\choose 3}\\quad\\mbox{ and }\\quad\\left[{n\\atop n-3}\\right] = {n \\choose 2} {n \\choose 4}.", "d2f76d89b19c5756cc0d66102871d808": "G = M / N", "d2f774ef585cdee0b7177817ff1c27ca": "\\displaystyle{g=\\begin{pmatrix} \\alpha & \\beta \\\\ \\overline{\\beta} & \\overline{\\alpha} \\end{pmatrix}}", "d2f79e7e6f1606f298b9feffa4743871": "S^2_n", "d2f7a87ecb19b70a3ee6348b21b7dafa": "N(k,d)", "d2f81d83f2a4fc3a6c5f5e2cb3e57e58": "x_n\\uparrow x", "d2f89dd7c9cc5d8f493e5d43416d5106": "\\vec \\sigma=(\\sigma_x,\\sigma_y,\\sigma_z)", "d2f8af9cc20ce1361470a6eaf7ba58d7": "\\frac{b}{h^2} \\int_0^h (h-y)^2 \\, dy = \\frac{-b}{3h^2} (h-y)^3 \\bigg|_0^h = \\tfrac{1}{3}bh.", "d2f8bc28217c18befdd7130ed3ae05b4": " \\frac{C_V}{Nk} \\sim {12\\pi^4\\over5} \\left({T\\over T_D}\\right)^3", "d2f902421d1b7667e57bba984f2d45da": "t, (t x_{i_2} < x_{i_3} > \\cdots x_{i_k}\\qquad \\text{and} \\qquad 1\\leq i_1 < i_2 < \\cdots < i_k \\leq n.", "d30d421cc4a82b9fee6c3d98682014c7": "\\Lambda^{1/2}", "d30d81f7e608b4e3302a1d262c69ed36": " 27 \\times 33 = (30 - 3)(30 + 3)", "d30dd19a84660a946380bd5a1d9e34f2": "\n A = \\cfrac{6~c~\\cos\\phi}{\\sqrt{3}(3-\\sin\\phi)} ~;~~\n B = \\cfrac{2~\\sin\\phi}{\\sqrt{3}(3-\\sin\\phi)}\n ", "d30e374e44f83d7d91e0062ceae0e0b7": "t\\to y(t) ", "d30e778c021d29e0efc4147ca04386e5": "\\frac{1}{3}Bh", "d30ee030f22d47b622a14e7d8c80771e": "\\cos(\\theta_T)=\\sqrt{1-\\sin^2(\\theta_T)}=i\\sqrt{\\sin^2(\\theta_T)-1}", "d30f1d04fc9c9e05a8fed1c1126f4958": "\\sum_n \\mathbf{P}_n = \\sum_n (E_n /c , \\mathbf{p}_n )= \\left(\\sum_n E_n /c , \\sum_n \\mathbf{p}_n \\right)\\,,", "d30f23f23f8a5c1a66011221e8315f29": "\\scriptstyle \\sqrt{\\tau}", "d30f5e93f5adf066a234be73a31395d8": "D^{p+1}\\times D^q", "d30fa5df511320765763b972909c4a92": "Pc", "d30fb2efff624b6ba9046318c6d7b6a1": "f(x;a,b)=1/(b-a) I_{[a,b]}(x)", "d30fdaeaefcfaafa49453c704ce79b8d": "\\bigcap_{i\\in F}N_i=\\{0\\}\\,", "d3101bd1827aba22f44adcefadc2f6f7": "\\scriptstyle{{\\omega = 2\\pi f}}", "d310233ffb53119e4b717df42a2f0a94": "\\Sigma_k \\hat{\\textbf{t}}_p^T ", "d310ad2c97e6707c7c33398a3f6a4465": "\nw = g(z) = \\left(z^2 - 1\\right)^{1/2},\\,\n", "d310cb367d993fb6fb584b198a2fd72c": "0.5", "d310f074a0d4976716905bb581dedafa": "\\tfrac{360}{n}", "d31102fe4b9d9245b38983313d928ee2": "x_{0} < 0", "d311bc7a02a360e3e41f6febcd806d1c": "\\int \\frac{1}{x}\\,dx = \\ln |x| + C, \\qquad x \\neq 0.", "d311c37ef0df5c69b507dad474e4a7d0": "\\begin{array}{rcl}\n\\operatorname{mse}[x] & = & \\lambda a, b, c.a\\ x \\\\\n\\ \\operatorname{mse}[M\\ N] & = & \\lambda a, b, c.b\\ \\operatorname{mse}[M]\\ \\operatorname{mse}[N] \\\\\n\\ \\operatorname{mse}[\\lambda x . M] & = & \\lambda a, b, c.c\\ (\\lambda x.\\operatorname{mse}[M]) \\\\\n\\end{array}", "d311cd240793dfbdf076bd49e53cc5e9": "\\underset{p+1}{\\underbrace{\\alpha\\wedge\\cdots\\wedge\\alpha}} = 0.", "d311d05a2b8c7d34793fca5ec25e6b2e": "\\ f_i", "d311d5b005e6d768315a6ec97e32955f": "\\partial^\\mu A_\\mu=0", "d311dbe7d2cfeb153c81e287652b18a9": " z \\approx (t_0-t_e)H(t_0) \\approx \\frac {D}{c} H(t_0) \\ , ", "d311dd18402a7385a87ee60488a1d58f": "\\omega^2\\cdot\\omega=\\omega^3", "d311e760c3bbe788b8530b86d8f36c19": "m^* = \\xi m_0 - \\ ", "d3121b44ab658ca6c0a1cf8b0bc3747a": " x^{(1)} = (x_1,x_3), x^{(2)} = (x_2,x_5) ", "d3125825bccefc72ac79ed205a73ed03": "f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr}) \\,", "d3128210a5d6e292fe618c4aa1888458": "{2}^{15}", "d312b0e35c5453d634212bcb9dd7687d": "(\\sin(x+y)^2+ \\log(z^2-5))^3", "d3130177fefb270a275848eb471bf632": "f_n = \\frac{\\omega_n}{2\\pi} = \\frac{1}{2\\pi}\\sqrt{\\frac{\\kappa}{I}}\\,", "d3131dac460970639ca2c4daefc4093a": "u\\left(a_i\\right) > u\\left(a_j\\right)", "d3134ccc3d733386d72a9c45ef474182": " r'(z) ", "d3137d9d4608eeac172025b8c9443f2c": " - 4b) + b = 180", "d313c2a4f42f41d399c053b1985e9b21": "v(x_n) \\rightarrow 1", "d313c32e033a4673d725f0ad9c15e2c0": "\\left\\vert \\frac{\\partial m}{\\partial x} \\right\\vert", "d31415f5b6bbc47e392878b1a17dd423": "\\bold{Pr} = \\vert \\langle P\\vert \\psi\\rangle \\vert^2", "d31425d0c08651babe3493dbbcaaabe5": "1 + \\frac{x^1}{1!} + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\frac{x^5}{5!}+ \\cdots = 1 + x + \\frac{x^2}{2} + \\frac{x^3}{6} + \\frac{x^4}{24} + \\frac{x^5}{120} + \\cdots\\! = \\sum_{n=0}^\\infty \\frac{x^n}{n!}.", "d314458827195d8dbced83dc2e4687d8": "\\tilde C^i{}_{jkl} = C^i{}_{jkl}", "d3146713447063c11a12f6cef9a6ba9b": "N_y' = m' y' - p_y' t' = \\gamma(V)\\left(m-\\frac{V p_x}{c^2}\\right)y - p_y \\gamma(V)\\left(t-\\frac{Vx}{c^2}\\right) = \\gamma(V)\\left(N_y - \\frac{V L_z}{c^2}\\right) ", "d3149cf5a9402da9e957275687831e6c": "x^s \\not\\equiv 1 \\bmod\\,\\big(n,x^2-bx-c)", "d3149f86e05dfd7036b605192af59da5": "\\vec{ds}=(dX_2,-dX_1)", "d314ac37a236cb7f5c03d56c7d52c3ea": "\\mathbb{Z}_{n}", "d314d1e551a0f21214d60b814969c5db": "A = \\frac{1}{2}.(\\bar{c}-\\sqrt{\\bar{c}^2-4.\\bar{k}}), \\; B=\\frac{1}{2}.(\\bar{c}+\\sqrt{\\bar{c}^2-4.\\bar{k}}), \\; P=\\sqrt{\\bar{c}^2-4.\\bar{k}}, \\; P=B-A", "d314d6c0f1c56af58e08478c88fed503": "\n\\begin{align}\n \\nabla \\cdot \\boldsymbol{u} &= \n \\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho}\\Bigl( \\rho\\, u_\\rho \\Bigr) \n + \\frac{\\partial u_z}{\\partial z} \n \\\\\n &=\n \\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho} \\left( - \\frac{\\partial \\Psi}{\\partial z} \\right)\n + \\frac{\\partial}{\\partial z} \\left( \\frac{1}{\\rho} \\frac{\\partial \\Psi}{\\partial \\rho} \\right)\n = 0,\n\\end{align}\n", "d3156b25438c7f0e74be7831f418c1f2": "\\Bbb Q,", "d3156f57aa6b8f5f2cb0158bef8ffe57": "\\beta_c \\,", "d31580f558e60f0284c0247577884340": "(h,g')", "d31590250b89ca9c1ff9fa0623cb1631": "\ne_I^{[\\alpha} e^{\\beta]}_K C_{\\beta J}^{\\;\\;\\;\\; K} + e^{K [\\beta} e^{\\alpha]}_J C_{\\beta KI} = 0\n", "d315b2f5396515818aa5c1cfa9e12ac2": "\n \\left. \\left[ -2k_{c}\\frac{\\partial H}{\\partial\\mathbf{e}_2}+\\gamma k_n+\\bar{k}\\frac{d\\tau_g}{ds}\\right]\\right\\vert _{C}=0\n", "d315f827fc5c92692e20f100be59f833": "0=\\frac{h}{2\\pi}\\vec{\\nabla} \\times \\vec{\\nabla}\\varphi + 2e \\vec{\\nabla}\\times\\vec{A}.", "d316095c0fef30df628ffb1ea8a3bbbd": "\\Psi(w,cv)=c\\Psi(w,v)", "d316371def238ef76474e4c0a1bfde42": "H_{(1)} \\ldots H_{(k-1)}", "d316592b99dc5d4ffa68ed08d091a097": "x = r_1\\, \\sin(\\theta)\\, \\cos(\\phi) \\dots,", "d3168476910f8c4d1e857528bfb7815a": "\\rho_0 = 1 g/cm^3", "d316a37b7d58d80fed94964259270955": "f: \\mathbb{N} \\rarr \\{n: \\mathbb{N} | n > 5\\}", "d316a40cbb95912d215eaf8e28f450ee": "\n\\left[ \\mathbf{N}\\left( \\mathbf{u}\\right) \\right] =\\left[ \\mathbf{Z}\n\\left( \\mathbf{z}\\right) \\mathbf{X}\\left( \\mathbf{x}\\right) \\right] .\n", "d316bdfe9dcec1ef4f64b4291a260761": "\\mu(t)=e^{\\int(\\frac{2y_1'(t)}{y_1(t)}+p(t))dt}=y_1^2(t)e^{\\int p(t) dt}", "d316d0ca2ce934f2c6f0353333924899": "\\hat{\\Z}", "d317553207293725295330c8eaa18814": "\\Sigma(i\\omega_m)=-\\frac{1}{\\beta }\\sum _{i \\omega_n } \\frac{1}{i \\omega_m +i \\omega_n -\\epsilon }\\frac{1}{i \\omega_n -\\Omega }=\\frac{n_F(\\epsilon )+n_B(\\Omega )}{i \\omega_m -\\epsilon +\\Omega }", "d31823bc43507546ab909e3d4b00c406": "\\lambda(n) = n / (\\ln n)^{\\ln\\ln\\ln n + A + o(1)}\\,", "d3182a38c47e1ebb97e4fb0c5d0b7e6b": "\\begin{align}\n\\sigma_{ij} &= \\lambda \\delta_{ij} \\varepsilon_{kk}+2\\mu\\varepsilon_{ij} \\\\\n&= \\lambda\\delta_{ij}u_{k,k}+\\mu\\left(u_{i,j}+u_{j,i}\\right). \\\\\n \\end{align}\n\\,\\!", "d3182ed169db00ca9d8827acaf48afec": " \\boldsymbol{\\omega}_\\mathbf{B} = {1 \\over 2} \\ \\mathbf{B}(t) \\times \\mathbf{B'}(t). ", "d3186205a7f7a3fa4237e32933739750": "\nx - x_0= -c\\ \\frac{R_{11} (X-X_0)+ R_{21}(Y-Y_0) + R_{31} (Z-Z_0)}\n{R_{13}(X-X_0) + R_{23} (Y-Y_0) + R_{33} (Z-Z_0)}\n", "d31882afd3a189052e987889ed0ad297": "c_1=1,", "d3189cf599a5a24fed6fc4252c5b7381": "\n\\mathrm{Sim}(w_1,w_2)=\\frac{f(w_1,w_2,\\beta_1)}{\\beta_1}+\\frac{f(w_2,w_1,\\beta_2)}{\\beta_2}.\n", "d319253925d56307b0a79ad1cfd1a2cc": "\n{\\rm Dec}(y, z) = b(f^{-1}(y)) \\oplus z\n", "d31960eeb3ddfccb86b81b42df7b56b5": "S=\n\\begin{pmatrix}\n1 & 0 & 0 & \\lambda & 0 \\\\\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \n\\end{pmatrix}.\n", "d31961a934a18b2c7f8aeea64e663d76": " f(z + \\omega) = C f(z) ", "d31975a3f0d73c365668086137694fae": "|z_i|<1", "d319c9514da42c7a0e3f2a107acc8980": "\\rho_{AC} \\ ", "d319e6261216883f2222bda967c16a8c": "\\log n = \\sum_{d\\,\\mid\\,n} \\Lambda(d),\\,", "d31a21dd251040568b8f7ef5bbcd081a": "\\left | \\frac{\\mathrm{W}(-\\ln{z})}{-\\ln{z}} \\right |", "d31a59e019a1da60fecbbf1df0223059": "m = E/c^2", "d31a760f604533827f17368a52f1d5ae": "\\delta < \\omega^Y_1", "d31a86c237ab86368ea29eec28bc0488": "\\Gamma_q(x+1) = \\frac{1-q^{x}}{1-q}\\Gamma_q(x)=[x]_q\\Gamma_q(x)\n", "d31a9a36011dcc8279f903cd85ee11f9": "(j=1,\\dots,p),", "d31ab9ede46e5932c7282182f133cbc2": "\\tilde{m}_0", "d31ac38182d64f32bfc3b2a5a3d395e2": "\\frac{1}{T} = \\frac{dS}{dE},", "d31ad4aca154094052f4f10d4d780f61": "R_\\lambda = (T-\\lambda)^{-1}", "d31ada318698b149eeedc39cc366ad35": " A.0=0", "d31ae5e77ec73da68e3138961e55ba5e": "I(X;Y|Z) - I(X;Y)", "d31b0aa8c05c28d09a2e8b85ffd5c60c": "\n \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\left(\\frac{\\partial \\mathbf{f}_1}{\\partial \\mathbf{v}}\\cdot\\mathbf{u}\\right)\\times\\mathbf{f}_2(\\mathbf{v}) + \\mathbf{f}_1(\\mathbf{v})\\times\\left(\\frac{\\partial \\mathbf{f}_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} \\right)\n ", "d31b83822a0b5b9019fdd4b8daff6267": "d^nF(u;h) = \\left.\\frac{d^n}{d\\tau^n}F(u+\\tau h)\\right|_{\\tau=0}.", "d31ba27f0bec9eba9b480be0445ca72b": "\\pi a^2/2", "d31bbfb42bd17fd51afe0732fbe3cb7b": " C_{M0}^{j}", "d31c2abe2cf0b1119092f0f1d045fd9a": " {N}^{n}(B_1\\times,\\dots,\\times B_n)= \\sum_{(x_1,\\dots, x_n)\\in {N} } \\prod_{i=1}^n \\mathbf{1}_{B_i}(x_i) ", "d31c46082159bc44ede2500225923297": "m_n = \\int_{-\\infty}^\\infty M_n(x) \\,d\\mu(x)\\,\\!", "d31c51a90c35a95fc150b8378f461e02": "P=ND", "d31c7fe1853be53003a1b8d94133c388": "\\frac{1}{2} \\sqrt{(\\mathbf{AB} \\cdot \\mathbf{AB})(\\mathbf{AC} \\cdot \\mathbf{AC}) -(\\mathbf{AB} \\cdot \\mathbf{AC})^2} =\\frac{1}{2} \\sqrt{ |\\mathbf{AB}|^2 |\\mathbf{AC}|^2 -(\\mathbf{AB} \\cdot \\mathbf{AC})^2}.\\,", "d31c8c0b3cd97711b64f77036c067b3a": "a/(a+b)", "d31c92a960c720b00a0187983c4b02e1": "(-1)^n", "d31cf705210ac0b862c393c927aab2f2": "S \\subseteq Z_n", "d31d05d5fc05e18cb65affe3adf88a0b": "\\scriptstyle V \\to \\mathbf{R}", "d31d2e1abed6c621d92f0e10b7428630": "\\mu_{HA}", "d31d36ae8c87d3b314dc0e05386e83bf": "\\Delta p = \\rho g h - \\gamma \\left( \\frac{1}{R_1} + \\frac{1}{R_2}\\right)", "d31dcca0d3ec3a44b7c6b410ac4ea40e": "C = \\pi (a + b) \\left[1 + \\sum_{n=1}^\\infty \\left(\\frac{(2n - 1)!!}{2^n n!}\\right)^2 \\frac{h^n}{(2n - 1)^2}\\right].", "d31df9975fa5d433905a2a9b998e53db": "\n\\text{if }ax^2 + by^2 + cz^2 \\equiv 0 \\pmod{4abc/p} \\text{ has a nontrivial solution }\n", "d31df9fc7cd8296c92c69b45554205d2": "\\overline{y}=\\frac{\\sum_x n_x \\overline{y}_x}{\\sum_x n_x},", "d31ead18171e7708fe647bc27bc3ce77": "z\\neq 0", "d31ee6cbcc951ece785f33c583d28b38": "E_{3} = \\Delta x \\Delta y \\Delta z", "d31f43ca8d543377910ceb59b5948d26": "CO={1 \\over {L\\cdot N}}\\sum^{N}\\Delta S_{i,j}", "d31fb0d54f11c4d329464093fa57ebf2": "f(x)\\nabla^2f(x) - \\nabla f(x)\\nabla f(x)^T", "d31ff7d7ab3939e6b06f00e4186c8bc7": " +_\\mathcal{O} ", "d32055a32f4e412ae12f6094cefaecdc": "\\exists z \\exists w ( z \\not = w) ", "d320830f9ced030fd75565305449e906": "K_{BB}", "d320a4a58681943cf305ade271291a18": " \\text{d}\\sigma_i = \\frac{1}{2}\\epsilon_{ijk} \\sigma_j \\wedge \\sigma_k", "d320a4df84a4998ad22446f4f3891c15": "\n(\\%neutrophils + \\%bands)\\times (WBC)\\over (100)\n", "d320abd9955bcd3582842918982d4eb0": "np=5", "d320b0635c9e3ac6b0df66cf005b7558": " F_x v = m\\dot{v}v.", "d32158966065a556543c1e406a377148": "H= kA\\frac{T_H-T_L}{L}.", "d3220b95266613b46e273a6e5549f2bc": "R_2=\\tfrac{1}{2}(\\sigma_1 - \\sigma_3)", "d32225cddf23d9684f72d71488e7732f": "M=M\\#S^n.", "d32231596a12d31942cccaad1a1c462b": "r = a \\frac {\\sin (\\theta + \\theta_0)}{\\sin \\theta_0}", "d3223eb2492e228efa9768d66611478c": "(a, b, c) = (520, 576, \\sqrt{618849}).", "d3237462468a8f9d2e50e3ac9ee7f3dc": "\\langle E(t) \\rangle = \\frac{1}{2} \\sum_n \\hbar |\\omega_n| \n\\exp (-t^2|\\omega_n|^2)", "d323bd0111202591cc6bf49a0252b0a6": " \\deg(\\textbf{N}(s)) = 3 < \\deg(\\textbf{D}(s)) = 4 ", "d3242aeb465e6a502f44ce97582a5c29": "\\Sigma a_i {x_i}^m\\ ", "d324418413a08d029684618066e456a4": "H_n(X)=H_n(Y)\\,", "d3249a0e22b8c0d7f8b20189ad3a1e7a": " x \\to \\infty, \\bar F(x) \\to 0 \\ ", "d3249af616db7e84b598dd14759ff548": "[-(a+1),+(a+1)]", "d324bced968eda52e62e58cb90c82c2d": "\\mathbf{g}", "d324cd4db5a7aa48ede65edfefe72899": "\\,\\!(R,\\varphi,z)", "d324d1595a26b317804d91e34f4880c0": "L_{K}", "d325760ebb69cb5063358e6b47e05929": "\\theta _i", "d325ae50f5a9914eca72ea7f2fb0c28b": "\\Xi(\\lambda_0)", "d325b24ea1f03f9324da90b96c7e41d8": " CMC ", "d325b7380536d83194a90d4d9e43344c": "\\scriptstyle \\Pi \\;=\\; T S", "d325bc5142af81bef019e0f1308f1092": "\\langle a \\rangle p \\equiv \\neg[a] \\neg p\\,\\!", "d325cbef6b5169a5cd27237c15c9c5bc": "\\kappa\\!", "d32635980c55a79a8c9544826f5cd2af": "I_p(x,y) = 1", "d32766b5bc661c6a0609add1d32a7494": "\nm\\frac{d}{dt} \\langle x \\rangle = \\langle p \\rangle, \\qquad \\frac{d}{dt} \\langle p \\rangle =\\langle -U'(x) \\rangle.\n", "d327850178b581d3395a16bd4d921aa1": " \\left| \\sum_{n=M+1}^{M+N} \\left( \\frac{n}{q} \\right) \\right| < \\frac{4}{\\pi^2} \\sqrt q \\log q+0.41\\sqrt q +0.61.", "d327b7e494a99f23b99c589099aac731": "H(\\boldsymbol{q},\\boldsymbol{p},t) = H(\\boldsymbol{q},\\boldsymbol{p})", "d327f350b0bd025ef53e636168f38d56": "\\scriptstyle i ", "d3280fb9e71f78602b806c44aec3f9d5": "U(a,b,z)=\\frac{\\Gamma(1-b)}{\\Gamma(a-b+1)}M(a,b,z)+\\frac{\\Gamma(b-1)}{\\Gamma(a)}z^{1-b}M(a-b+1,2-b,z).", "d32893051e4add28fec0f40ed7f0713f": "D_{i}", "d3289a96da4c1cf6ce57b2b76b80b965": "x=-1", "d328a7d3c3a1b428cf9272ad28668f50": "m\\frac{d^2x}{dt^2} +kx=0,", "d3291559f95fe9781b8a9393ad8ea524": "\\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} - L \\right) T - \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\epsilon}", "d32935ecc2f6fa1d2219d18f9ec7d5c6": "M(j,j) = 1 - \\sum_{i=1, i\\neq j}^{20} \\frac{\\lambda m(j)A(i,j)}{\\sum_{i=1, i\\neq j}^{20}A(i,j)}", "d32954e87b21d08631904d3c52df7216": "D(f) \\leq Q_2(f)^6", "d329574997fa99cdf366db3cc8f75c4e": "\\cos\\alpha = \\frac{\\hat n_1\\cdot \\hat n_2}{|\\hat n_1||\\hat n_2|} = \\frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\\sqrt{a_1^2+b_1^2+c_1^2}\\sqrt{a_2^2+b_2^2+c_2^2}}. ", "d3299d95e5e6ad5471742c77b782d78a": " \\pi = C \\frac{d\\phi}{dt}", "d329d163ee5e9a1e79ed221983aa800e": " {k}_{ij}^e ", "d32a10c53e66c6bdece756c344dbfdc1": " |2^i|= \\alpha + 1 ", "d32a434254ce813942083f81bafd9906": "K/k.", "d32a59965f90f9411dbb7039221696e2": "x'_{v_{i1}j_1} < x_{ij_1}", "d32a5dd935da1591c34d96edb86a2058": "\\mathbf{RP}^{2k-1} = \\mathbf{P}(\\mathbf{R}^{2k})", "d32b0ece7e70848082161976c495977a": "(c-a)(c+a)=b^2", "d32b12d7c7276191c9a4a7db1cb0c711": "\\mathrm{3Mg_2SiO_4 + SiO_2 + 4H_2O}", "d32b2081165971d213ff2112f5eb7aaa": "\\frac{1}{10} + \\frac{1}{40} = \\frac{1}{8}", "d32b363744e2d8f8c5f66770788aaf39": "N > |s|", "d32bfae940827a036ecc3073dfdf6f21": "\\mu \\in \\mathcal X", "d32c32bb0e866478a102ef6b5ac5e388": " \\Delta \\circ \\mu = (\\mu \\otimes \\mu) \\circ (\\mathrm{id}_H \\otimes \\sigma_{H, H} \\otimes \\mathrm{id}_H) \\circ (\\Delta \\otimes \\Delta) ", "d32cacc8c10ffdca3e60b27b2599db9a": "S_0'=1\\,", "d32cd7fa2a1426115918587d59c172a0": "A_W={\\frac{R T}{An^2F^2\\sqrt2}}{\\left(\\frac{1}{D_O^{1/2}C_O^b}+{\\frac{1}{D_R^{1/2}C_R^b}}\\right)}=\\frac{R T}{An^2F^2\\Theta C\\sqrt{2D}}", "d32ce25b5459b664037c44c99e9987be": "26^2 = 676", "d32d4e4bf6de62ab48c252bf6e42f9df": "E[F_k]", "d32d629d0c2721e1171df5316bf28e6e": "v(x):=\\mathbb{E} [(\\tau_{x}-T(x))^{2}]", "d32de594e64c3d594860604d3e3b5ae0": "K_{--} \\ \\stackrel{\\mathrm{def}}{=}\\ \\{ x \\in K : \\langle x,\\,x \\rangle < 0 \\}", "d32deb9e541de2eca8dd0af2c6f6b735": "{\\mathbb{R}/\\mathbb{Z}}\n = \\{x+\\mathbb{Z} : x\\in\\mathbb{R}\\}\n = \\{\\{y : y\\in\\mathbb{R}\\land y-x\\in\\mathbb{Z}\\} : x\\in\\mathbb{R}\\}", "d32df979a41206bfdc3a26b381f990f4": "f(\\mathbb{C}): X(\\mathbb{C}) \\to Y(\\mathbb{C})", "d32e0204eda75873977e3c688c3bce02": " S = {Q \\over T} ", "d32e280a70a0d6ad366971e53c3f6024": " op \\mapsto m_1 \\mapsto m_2 \\mapsto \\text{bind} \\; m_1 \\; (a_1 \\mapsto \\text{bind} \\; m_2 \\; (a_2 \\mapsto \\text{return} \\; (op \\, a_1 \\, a_2)))", "d32e330c91c92582cba0c515b3a37d2f": "x^{\\overline{n}}", "d32e3bdb03fb6304f0b0b483fbcae8aa": "\\frac{d [Y]}{dt}= -k_I [A] [Y] - k_{II} [X] [Y] + \\frac{1}{2}f k_V [B] [Z]", "d32e6d72c902366e344c4ae3564da9f2": "k>3", "d32eaf8d5eb60c0a379dee9347eec72f": "\\Pr[C_i=C]\\geq \\frac{2}{n(n-1)}", "d32ef95ecfe12198fd900337088fbe0a": "\\Xi(x)=\\Gamma(x)\\Lambda(x)=\\alpha^3+\\alpha^4x^2+\\alpha^2x^3+\\alpha^{-5}x^4,", "d32f0c98489820dbeed6c9d39d74037f": "P=\\frac{nRT}{V}", "d32f17a78e4423cb30086b2c2e19dad8": "\\,\\psi_{\\theta}(t)", "d32f44bd0ed76ae45f1d0efceb978030": "\\left[1,f(x),f(x)^2,...\\right]^\\tau", "d32f6f12198266cd4c4d39ade6a7c109": "\nR = \\frac{p(\\mbox{Object}|X,S,A)}{p(\\mbox{No object}|X,S,A)}\n", "d32fc155c7971816867d38f65243810a": "\\int \\tan{x} \\, dx = -\\ln{\\left| \\cos {x} \\right|} + C = \\ln{\\left| \\sec{x} \\right|} + C", "d33016ea2ff1f6b6dbff14a6d0fd5371": "16K^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2.", "d33052ec5f1c426d5eadbdac61eedb46": "2 + \\sqrt{2} + \\sqrt{6}", "d33086d89a1836f63e86fdafaed65628": "\\chi_G(\\lambda) = 0", "d330956259dd5a9a9b22f0345798d048": "(4/3)\\pi r^3", "d330a828ae391677c5e4124ef4d9b604": " \\phi(z) = \\int (Tf_z)g_z \\, d\\mu_2", "d3310e847cf3b3ecb4c494cb74b0988d": "\\mathrm{CO_2 + H_2O\\rightleftharpoons HCO_3^{-}+ H^{+}}", "d33136a5742eee25c329345bbb19d55c": "x_i \\geq _i x_i^*", "d3313755efb58d483319d926ffb9593d": "\\frac{1}{\\sqrt{m}}\\|A\\|_1\\le\\|A\\|_2\\le\\sqrt{n}\\|A\\|_1.", "d33144d8266a877453bb6c09b0ab6d3c": "|\\alpha|=\\alpha_1+\\cdots+\\alpha_n", "d33208d2ddf262fb694dc214e7b717f2": "", "d3322ea8c76ec8ee7750a85b5cd2fb8d": " {}_RW_P = \\int\\limits_R^P {pdV} = 0", "d33232d2050aaf6de01a5d7f64cc10f8": "N_a", "d332565feeadb47b84fc0f5abbe17986": "\\vec{r} = \\begin{bmatrix}1 \\\\ 1\\end{bmatrix}", "d33274fef907013f7ef1423640e66fdb": "2+2\\lfloor\\log_2 n\\rfloor", "d332796598a8a044b825ffb2234bab1e": " \\hat T(V) \\times T(V^*) \\to \\mathbb{F},", "d332abca463506569cbdc7115ecf93a0": "p(x) = \\begin{cases} |x|, & x \\leq 0; \\\\ 2 |x|, & x \\geq 0; \\end{cases}", "d332dd31bb8c5f4ca14aac0ccd1b6b07": "\\psi(\\varepsilon_{\\Omega+1}+1)", "d33325957847c5c6e01bc18ccbd7428f": "\n\\begin{align}\n\\int_{x=0}^{x=2} x \\cos(x^2+1) \\,dx & {} = \\frac{1}{2} \\int_{u=1}^{u=5}\\cos(u)\\,du \\\\\n& {} = \\frac{1}{2}(\\sin(5)-\\sin(1)).\n\\end{align}\n", "d33369ea3dc781c738e97c3219da153a": "\\dim_ED_{B_\\ast}(V)=\\dim_{\\mathbf{Q}_p}V", "d333854f7ded7455adc9b9ac0e25a32e": "Q=3", "d333cdab0d723710eba62eed741b5d26": "\\rho_q = \\sqrt{L/C} \\ ", "d334434f6fb65bfe5b88adc0c5f91389": "\\left| s_{ij}- \\lambda\\delta_{ij} \\right| = \\lambda^3-J_1\\lambda^2-J_2\\lambda-J_3=0,\\,", "d3346b8ceaedafe7e2079e888d178604": "-vMv, \\quad v^2=1 ,", "d3346f63e16ce81ec28e39ed4444292f": "\\zeta^\\prime", "d33497c35ef8a76229582775a5b5b25d": "\n|\\theta| > \\sqrt{2\\Bigg( 1 - \\frac{k_\\theta}{F L} \\Bigg)}\n", "d334dec32faf0bd0b124338fec6af840": "\\phi:A\\to\\mathbb{C}", "d3354a6de04e854002b3dc15d3b19c6f": "Q(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{x}^{\\infty}e^{-t^{2}/2}dt = \\frac{1}{2}\\,\\operatorname{erfc}\\left(\\frac{x}{\\sqrt{2}}\\right),\\ x\\geq{}0", "d336275e8b55d9afaad7c7d8ee440455": "Q_i v_i", "d3362cbe72d857d7c18edd05c3c119b7": "= t_2 - t_1 + \\left( \\gamma \\frac{v}{f^\\prime} \\right) \\left( \\frac{u^\\prime - v}{1 - v u^\\prime / c^2} \\right)^{-1} ", "d3365097159e9e624e0ad678f7cda518": "D^{+}(\\mathcal{S})\\cup \\mathcal{S}\\cup D^{-}(\\mathcal{S}) \\not= \\mathcal{M}", "d336792052fe6b69dbf8acbd7e42778b": "P=\\frac{M\\cdot V}{Q}", "d336c8898b5f6bdb8b94477e089392f6": " \\mathbf{q} ", "d336db1b28cfcf37ded4999235761dee": "\\scriptstyle N_2=1 ", "d336e68aac640ba2b70cf35ef5a9fdb4": "F[r] = u ", "d3373c83a9ca4db6434f21c3f90022b9": "\\begin{align}\n P &= 0 \\\\\n Q &= |S| = V_\\mathrm{RMS} I_\\mathrm{RMS} = I_\\mathrm{RMS}^2 |X| = \\frac{V_\\mathrm{RMS}^2}{|X|}\n\\end{align}", "d3379a5c0cd0b9f6151bb222b097f5ac": "n = 4l", "d337a21d624bd85339730c47f42e19d3": "I_{0}= \\ \\mbox{IV}", "d337e5c96fea2b4eca6e438789ebfbed": "\\rho_j \\rightarrow 1", "d338a22eed6dc7c454b62659dc8c7879": "2 = 1 \\,", "d338b0c91d592b574ab93531c6bab27d": " \\frac {\\eta} {E_2} = \\tau ", "d33947c02c714ac0f21f87740a35009d": "\\overline{AB} = \\overline{B} \\,\\, \\overline{A}", "d33998fb411e955c40c6ddb5a9c46b4f": "A = \\frac z{\\exp(y)}, \\,", "d339ceb6cd07b2c46524fd5c0d40a71a": "{}_3W_1 = \\int\\limits_3^1 {pdV} = p_1 \\left( {V_1 - V_3 } \\right)", "d339d8da3d4ccdfcdd998596a80281f9": "\\{x[m-k];\\ m\\}", "d33a1bd21aeb3b8d9af00b075480347b": "\\kappa=k/h", "d33a6aafa6cf8b39f13ed26dd3d38fc6": " V(t)-V(t_{0})=\\int_{\\gamma }\\nabla _{t}f(x^{\\ast }(s),s)\\cdot ds. ", "d33abd8a26ee42647b7f27926be2b97d": "\\scriptstyle{G^{\\prime\\prime} + \\ddot G = 0}", "d33ae370a4caa76f2bdb53d9fa68bb47": "x\\cdot v:=\\chi_\\lambda(x)v", "d33b03d89dc432cf17e7dbd52e45bd77": "(W_t)", "d33b3f57aa545ca35cda4196edc8a444": "\\frac{1}{\\alpha+\\frac{1-\\alpha}{P}}", "d33b4783fb2da9c3911d878a439bab3e": "\\mu \\left (X\\backslash \\bigcup\\nolimits_{Q\\in \\Delta_{k}}Q \\right )=0.", "d33b4e95d7716973c34154e316748c36": "\\mu(A)\\leq 0", "d33b8d337010d2dc7bf5b737bc84c964": " S(\\Psi) ", "d33be4b28a135a11ff28d678633c83bc": " \\prod_{1 \\leq k \\leq n-1} \\left( X^k - 1 \\right) \\equiv n- \\left( X^n - 1 \\right)/\\left( X - 1 \\right) \\mod \\left( X^n - 1 \\right) ", "d33c0c99e1cbdedbb13c8f31510f0c43": "1.7177", "d33c4ca6d959548112fde13c80cef28e": "g=\\sum_{i=1}^d (E(r_i ) + \\sum_{j=i+1}^d E(r_i,r_j) )", "d33ca1a1e2542fd07b59f4546b5c8df9": " 1<\\omega, \\quad 1+1<\\omega, \\quad 1+1+1<\\omega, \\quad 1+1+1+1<\\omega, \\ldots. ", "d33ca42830e279f86e520783fe052919": "\\alpha t", "d33cada6836cdc7c128cde39250c4eab": " {\\rm {}}_{}^{2}\\Pi_{\\rm u}", "d33cc3643d249fc62c0837dfe449ae98": " F_{12} = -F_{21} ", "d33cdfea42a3bc451f1c010b8fb3c08a": "\\mathcal{E}=\\frac{\\hbar^2}{2m}\\vert\\nabla\\Psi(\\mathbf{r})\\vert^2 + V(\\mathbf{r})\\vert\\Psi(\\mathbf{r})\\vert^2 + \\frac{1}{2}g\\vert\\Psi(\\mathbf{r})\\vert^4, ", "d33d5f68432c73327d1aa2325867cfe1": "a\\ge x_i", "d33d6bd3e7dbb2657a7473651b946f2a": "I=I(V)", "d33daa80c65092a56b937c1671912f01": "\\{0,1\\}^h", "d33dcdbfc48943fb42bc7cf18a19eb7e": "V_n(k;\\rho,\\varphi,z)=P_n(k,\\rho)\\Phi_n(\\varphi)Z(k,z)\\,", "d33dcf150fa3d7bc65607f943d7ae115": "\ny = \\frac{a_{n}x^{n} + a_{n-1}x^{n-1} + \\ldots + a_{2}x^{2} + a_{1}x + a_{0}} {b_{m}x^{m} + b_{m-1}x^{m-1} + \\ldots + b_{2}x^{2} + b_{1}x + b_{0}} \n", "d33ddb3e2349354116f5b9e62e5afd7d": "V(x) = V^+ e^{-\\gamma x} + V^- e^{\\gamma x} \\,", "d33def0eb4933f91b88eb4e784adaf05": "u_1", "d33e16593a8007202f456b3ae7bdbbd3": "(X, Y)= \\left(-y'\\frac{x'^2+y'^2}{x'y''-x''y'}, x'\\frac{x'^2+y'^2}{x'y''-x''y'}\\right).", "d33e2a7be5de524c962a15da6bd7f687": "r_t^*", "d33e4fbc090f8c403b50cff35dc1cb2c": " [\\dot U] ", "d33ebfa99987ca2bf431e04405646256": "L^{\\lambda, p}(\\Omega)", "d33eeb4376d084bd9e59b87e0bbbb31b": "\\frac{\\Gamma(1+\\beta n/2)}{\\Gamma(1+\\beta/2)^n}", "d33f5afc4b0387f2cfdb82b66d9cd9cc": "k(\\mathbf{x_i},\\mathbf{x_j})=(\\mathbf{x_i} \\cdot \\mathbf{x_j})^d", "d33f6c2bdd50071ce2d111a011106c92": "f_{a}(5) = 4^{1^{1} 2^{0} 1^{1}} = 4^{1} = 4 ", "d33f8c8c5d88a9c6aa0c8f947417ae5e": "T =", "d33f8d2db8888df5a3d862ff45cbc691": "\\exist I", "d33f8d4fc4e0d33937ff3e1677748ecc": " G = c = \\hbar = k_\\text{B} = 1", "d33f9f5df0fea74d80fb09ddc7f8bc2c": "R_t^{} ", "d33fdb03399670f19b60a72090a51438": "\\hat{\\mathbf{S}^T}\\mathbf{P}^T\\hat{\\mathbf{M}}", "d3400d219ec1acd7758cf53e812ad0f7": "(F)", "d340279584dc250050b8f2bd924cc0a6": "\\mbox{power}=\\mbox{torque} \\times 2 \\pi \\times \\mbox{rotational speed}. \\,", "d34041dff33064a9e5fe88df0ac30bc9": "\\binom {n+k}{n}", "d3405414228de613629a1ad99d7074dd": "M=\\frac{P+Q}{2}", "d340642272b0f8973659bdc35e597c96": "{0\\choose k},\\;{1\\choose k},\\;{2\\choose k},\\;\\ldots", "d340743c6af0eaf2f5ce4b5106591b9d": "\\lim_{k \\to \\infty}A^k=0", "d3408d0a55ccb5acc1741e7e6d751192": "p_{\\rm CNML}(x_{n+1} \\mid x_1, \\ldots, x_n) = \\frac{ n^{n+1} \\left( \\overline{x} \\right)^n }{ \\left( n \\overline{x} + x_{n+1} \\right)^{n+1} },", "d34095f50aaf0558942fb2e41ac036ab": "a=\\mathbf{l}\\cdot\\mathbf{l}", "d3413f05324413c3742c5ddb43c5ba24": "d_T", "d34190e17c1c18a8f4c1b5a2cf5770e6": "M\\cdot V_T = P\\cdot (\\mathbf{p}_{real}^\\mathrm{T}\\cdot\\mathbf{q}) = P\\cdot T", "d3419a36f40e1cd392bc3ab6ffec89a1": "G_j = 0", "d341c70504a9397721cc5c25c092ff38": "\\pi: E \\to X", "d3424a410851724c04fbc1e5961bce8f": "z=Ax^2 + Bxy + Cy^2", "d3429a59d24fc9785b55ccdf5752805e": "x=u+v ", "d342bbfa652516d3f23ee21b36df3149": "m=2,3,\\ldots", "d342ce8eb00e7d83da19248f04b9c302": "g(\\theta). ", "d3431760e71a3bc754d8a71505d84903": "a_{\\pi}=a_y=0.5", "d3435805c589b164af772ffdbb301ae2": "\\begin{array}{lcl} \\sigma_1 & \\le & T \\\\ \\sigma_2 & \\le & T \\\\ \\sigma_3 & \\le & T \\end{array}", "d34361d8f98718210ed638ae11c3e292": "\\ \\mathfrak{f}(B/A)=\\operatorname{Hom}_A(B, A)", "d343974735deb05d4e71751dccbd20bb": "v \\sim w", "d3439ae46498c7b0ec80b37decf28f8d": "\\beta_{n}^{FR} = \\frac{\\Delta x_n^\\top \\Delta x_n}\n{\\Delta x_{n-1}^\\top \\Delta x_{n-1}}\n", "d3439e7a731b7bc456e86aab94c208de": "(1+r)(1+i)=(1+R) \\,", "d343ae5238f6bfe358467b343f45afd9": " f \\left( \\mathbf 0 \\right) = \\mathbf 0 ", "d34414e5de207f23fcb6ee81ee13435d": "\\nabla\\cdot\\mathbf{E}(\\mathbf{r}) = \\frac{1}{\\mathcal{E}_0} \\int \\rho(\\mathbf{s})\\ \\delta(\\mathbf{r}-\\mathbf{s})\\, d^3 \\mathbf{s}", "d344389080b7ad304f649a3e12c60011": "\n \\boldsymbol{N} = \\boldsymbol{S}\\cdot\\boldsymbol{F}^T \\qquad \\text{and} \\qquad\n \\boldsymbol{P} = \\boldsymbol{F}\\cdot\\boldsymbol{S}\n", "d34483f3928d44dc6408cf7be5280785": "-j \\frac {1}{\\sqrt 2}", "d344cc5af463d4a4bd3df6e52903c2c1": "L_{oc}^{pri}", "d3451cfb078e74f3c6d7fb5a944a510d": " a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2 ", "d3451d826d2354cb489c885780be00e8": "E^2 - (pc)^2 = 0 \\,\\!", "d3456094258862b5c2145e5da32f2f44": "\\Delta t = R_N^{-1} \\ln(1/u^\\prime)", "d345858a054bca47c0b9e355f1396e7a": "RN_i[j] == LN[j] + 1", "d34590284a15107c365777b486bf7cad": "\\frac {y_2 - y_1}{x_2 - x_1}", "d345badee3cad0aaa4c782b2aa6927c5": "P(H_0|D) \\propto P(D|H_0)P(H_0).", "d345bcbc6e5d80258f745b7f3b2f4a43": "h_2\\,", "d345cbd2ecdc9d489f92f120405225aa": "W^* \\to W", "d345d1af91d30d9577ab503018498532": "S_{k} = X_{1} + \\cdots + X_{k}.\\,", "d345d8799905564e058e1de88e9b6711": " \\mu(g S) = \\mu(S). \\quad ", "d345eba46386b1d0f190360b529932b8": "m_e v_e^2 > k_B T_i.", "d345ec5c717cc78764acd978339565b8": "\\le {e^{-n(1-w)}}", "d345f4321245249bc09b7e2318185131": "\\sum_{k=0}^\\infty \\frac{E_{2k}z^{2k}}{(2k)!}=\\operatorname{sech} z, |z|<\\frac{\\pi}{2}\\,\\!", "d3463abfd76fb1064a265ca593290fa9": "A=\\text{Angle of the wind from the direction of travel} ", "d34659cdda1775542ddd3283613db788": "r(x) = s(x) + e(x) = 3 x^6 + 2 x^5 + 123 x^4 + 456 x^3 + 191 x^2 + 487 x + 474", "d3466a72d041f9cd051a1e8252199820": "K_q(n,R)", "d346d898a962c5a790ecc83ac58ee15c": "\\kappa=\\sigma=\\tau=0\\,,\\quad \\nu=\\lambda=\\pi=0\\,,\\quad \\varepsilon=0 ", "d34707b6bd88e8f335917c485437bef6": "\\Delta \\phi \\approx \\frac{4 \\pi h_t h_r }{\\lambda d}", "d34723e00f5bef7c364a752935a462ae": "\\phi(r) = e^{-(\\varepsilon r)^2}\\, ", "d3475d44e312508083e569a8de0e2e4d": "\\star \\mathrm{d}x \\wedge\\mathrm{d}z = - \\mathrm{d}t\\wedge \\mathrm{d}y", "d3477828b6e4fefbd341c70ffeca39f3": "(Df)(a,b) = \\left [\\begin{matrix}\n -1 & \\cdots & 0 \\\\\n \\vdots & \\ddots & \\vdots \\\\\n 0 & \\cdots & -1 \n\\end{matrix}\\left|\n\\begin{matrix} \n\\frac{\\partial h_1}{\\partial x_1}(b) & \\cdots & \\frac{\\partial h_1}{\\partial x_m}(b)\\\\\n\\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial h_m}{\\partial x_1}(b) & \\cdots & \\frac{\\partial h_m}{\\partial x_m}(b)\\\\\n\\end{matrix} \\right.\\right] = [-1_m |J ].", "d347857686b3885a39031e63e2fdf2f7": "\nM(t) = \\frac16\\left(\\frac{1}{(1-t)^3} + \\frac3{(1-t)(1-t^2)} + \\frac{2}{1-t^3}\\right)\n= \\frac{1}{(1-t)(1-t^2)(1-t^3)}\n", "d34790d17c8f6fc3413e940d654650dd": "\\eta = \\left ( \\theta_0, \\theta_1, \\ldots, \\theta_k \\right )", "d3481bf707857cd11b06674bebef9c18": "x_0 = S y_0", "d34854f9267796fc5a9e9676a0797d56": "y = \\arctan\\left(\\frac{\\tan(\\phi)}{\\cos(\\lambda)}\\right)", "d34855b5c62f030b2d977bf30e4453d6": "\\lambda=\\frac{1}{\\left(\\phi_0 / \\phi\\right)^{\\frac{1}{3}} - 1}", "d3487ca7461c5ec86f1b5c214ed48e66": "(\\Lambda,i,M,C)", "d348b8d3133c26e25028ca69809d7329": "y_{i,j}=\\mu_j+\\varepsilon_{i,j}", "d348bdbb0ad3dfecdebb22aac6c1bd22": "\nAF_{\\mathrm{dBm}^{-1}} =\nE_\\mathrm{\\mathrm{dBV/m}} - V_{\\mathrm{dBV}} =\nE_\\mathrm{\\mathrm{dB}\\mu\\mathrm{V/m}} - V_{\\mathrm{dB}\\mu\\mathrm{V}}\n", "d348ff9918ef5504e79b1a70a947f8a9": " z \\in R ", "d3492c88f7ec87739a24190b3405c6c9": "\\lambda(G) = \\max_{|\\lambda_i| < d} |\\lambda_i|.\\,", "d3494592527bc76e65639c5f9de5df5b": " \\Phi = \\frac{dV}{dt} = v \\pi R^{2} = \\frac{\\pi R^{4} \\left( P_{i}-P_{o} \\right)}{8 \\eta L} \\times \\frac{ P_{i}+P_{o}}{2 P_{o}} = \\frac{\\pi R^{4}}{16 \\eta L} \\left( \\frac{ P_{i}^{2}-P_{o}^{2}}{P_{o}} \\right) ", "d349b121dc2aba5d3d7990449ccc60c8": "\\hbar = {{h}\\over{2\\pi}} = 1.054\\ 571\\ 726(47)\\times 10^{-34}\\ \\mathrm{J \\cdot s} = 6.582\\ 119\\ 28(15)\\times 10^{-16}\\ \\mathrm{eV \\cdot s}.", "d34a9bb682d846e171b28a190ffddc32": "f'(x) = {\\rm st}\\left( \\frac{f(x+\\Delta x)-f(x)}{\\Delta x} \\right)", "d34b2e105bac7ff19dd32602f27b71d3": "h = \\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\ 4 & 2 & 1 & 3 & 5\\end{pmatrix}", "d34b3d391f948ca4bbac76bd37a0340c": "\\ell_j(x_i)\n = \\delta_{ji} = \\begin{cases} \n1, & \\text{if } j=i \\\\ \n0, & \\text{if } j \\ne i \\end{cases}\n", "d34b6fc293fd23cf5bae063f8206b1a9": "B_1[p] = X", "d34b6fe9d58c327ffb20fef8bf680e98": "\\phi(b_n) = b_n t^n", "d34b7c5e2516ac7dbebfe0e0fdd960a6": "\\mathrm{Inequality} = \\Sigma_j \\, p_j \\, f(r_j)\\, , ", "d34ba8e4b67c169e7c8627934dd4d6a4": "(E,\\pi,B)\\,", "d34bcdc899ecab500f80e3faecc63c5c": "080", "d34c5830e1c7c3af8cc39b4f778ae710": "\\left(\\lim_{k \\to \\infty}A^k = 0 \\Rightarrow \\rho(A) < 1\\right)", "d34c68ddc92820143746555a15307006": "\\sum_{i=1}^{e_x} L_e(i) = L_e(1) + L_e(2) + \\cdots + L_e(e_x)", "d34cb677e15f74cb7b8c6e55d56f6494": "F_\\wedge(x,y) = \\min\\{x, y \\}", "d34d1b8c13c640b550a6ebf931b90e97": "\\vartheta_{L_8\\times L_8}(z) = \\vartheta_{L_{16}}(z),", "d34db8489ceefb929c8ba27b45312d2d": "2\\log_2(|b_{k-1}-b_{k-2}|/\\delta)", "d34dd2b383e7ec585a80ee485535aa86": "\\frac{\\mathrm{d}f}{\\mathrm{d}x} = 2 x", "d34e3650b0cabc340564443f40070ba6": "S(x){\\ne}1", "d34e759bc36bc676310a6c8317bd1568": "222 \\times {4 \\choose 1} = 888", "d34e94b47c9f14e7fa23ab916bbd4f00": " \\mathbf{x}_{k+1} = \\mathbf{x}_k + \\alpha_k \\mathbf{p}_k ", "d34ea21aacf9d421d6727d381d2a8b1b": "(\\iota x.\\phi)", "d34ef1689fd6b7bd3cfe9e6f65ebb455": "R_{out}=10 \\ln \\left[\\exp(R_{in}/10)+1\\right]", "d34f21ce7b361f04a38e85b2c8aeeade": " \\begin{matrix}\nI&=&|E_a|^2+|E_b|^2, \\\\\nQ&=&-2\\mbox{Re}(E_a^{*}E_b), \\\\\nU&=&|E_a|^{2}-|E_b|^{2}, \\\\\nV&=&2\\mbox{Im}(E_a^{*}E_b). \\\\\n\\end{matrix}\n", "d34f3ba856aded1214f3d3a40d37c70f": " x^n(x^2-x-1) + 1\\,", "d34f55c174b9d6343180d782a805adc2": "|S_{1}|+|S_{2}|+\\cdots+|S_{n}| = |S_{1} \\cup S_{2} \\cup \\cdots \\cup S_{n}| ", "d34f771d557395d2e255d18d3d6bd8be": "[B]_0", "d34f8f9db32205e2ff5bf9bf9c658ee6": "{\\frac{P_oy}{P_ox}=\\left(\\frac{\\frac{\\gamma+1}{2}M^2_x}{1+\\frac{\\gamma-1}{2}M^2_x}\\right)^\\frac{\\gamma}{\\left(\\gamma-1\\right)}\\left(\\frac{2\\gamma}{\\gamma+1}M^2_x-\\frac{\\gamma-1}{\\gamma+1}\\right)}", "d34fa2ce76ff94532d35146de1ccbba6": "\n\\frac{1}{\\Theta}\\ \\cos\\theta\\ \\frac{d}{d\\theta}\\left(\\cos\\theta \\frac{d\\Theta}{d\\theta}\\right)\\ + \\lambda\\ \\cos^2\\theta=\\ m^2\n", "d34fce1ed79d000a45549e65de73a74f": "\\textstyle x^2 \\equiv y^2 \\pmod{n}", "d3508cdcfcc2ff210c476a6967b7e016": "x \\equiv 0 \\pmod 2", "d3509cd982b15cda5f5adb2152bd054b": " S(M) \\in (1 + \\sqrt{M})^{\\log \\log M} + \\log \\log M \\cdot O( \\sqrt{M} )", "d3509d464a41c003f6242599d58be734": "\\varpi \\equiv \\omega + \\Omega\\,", "d350ae9aff8a25554ccc64a11ab061be": " -(n+2)(n-1)~r^{-n}~\\sin(n\\theta) \\,", "d35111ac51f5376180e3f8c8cd156705": "E_{ij} x_k = \\delta_{jk} x_i. ~~~~~~~~~~~~~~", "d3512ff5f497352e058c3749887a9aee": "x_r=\\frac{r\\pi}{2m+1}", "d3516a240bad4a58f240987621ea3475": "r_f=k_f[A][B]\\,", "d3519bd553f721090ccdb1967fa43fc2": "{N-1 \\choose n}\\,", "d351aa7819c372c431516d738c23301c": "\\operatorname{E} ((T^2 - aT - 1)^2)\\,", "d351bce3145a178ee19a4ce9608b2911": "\\begin{align}\np \\rightarrow q\\\\\nq \\rightarrow r\\\\\n\\therefore \\overline{p \\rightarrow r} \\\\\n\\end{align}", "d351c94b118ce47c738faaae26c128f3": "\\mathbf{W'}=\n\\begin{bmatrix}\n1-2k&0&0&0\\\\\n0&1-2k&0&0\\\\\n0&0&1&0\\\\\n0&0&0&1+k\n\\end{bmatrix}\n", "d3520827d6af7acb8b65af5cd1ff2495": "(E_f \\mathbf{\\hat f} + E_s \\mathbf{\\hat s})\\mathrm{e}^{i(kz-\\omega t)},", "d35212cbb75976940789724b8f0ee00e": "\\mathbb{P}_n^0", "d352bad17f9d2e4590500dce0502f5cb": "\\sigma^2 + 1/\\lambda^2", "d352d6bfb711879d057437761b7b757b": "Y_{lm}(\\theta,\\phi)\\,", "d3531d7d798d0633e9555f5017a97f33": "O(n (\\log n) (\\log \\log n))", "d3532dead928de54fe682b00dbd313d2": "\\textstyle \\mathrm i\\sqrt 2", "d3534929b63de87f8fca9a675f773cb9": "s \\mapsto \\chi(\\mathcal{F}_s)", "d353557b177aa5db33461464517cbe9b": " \\sum a_n = s.", "d3538fe9bc6cb14ecbe0c3a7d39cb5fd": "\na_0 = a_2^1 a_3^2 \\cdots a_k^{k-1}, \n", "d353d4746ab71f7a977a09fffaa26cde": "\nP_s = 1 - \\int_{-\\frac{\\pi}{M}}^{\\frac{\\pi}{M}}p_{\\theta_{r}}\\left(\\theta_{r}\\right)d\\theta_{r}\n", "d354046b054415bd87e9c199fe392789": "C_{d0}\\;", "d354184f1c05f089058c0ce4d780cc99": "f(r | H=h, T=t) = \\frac{(N+1)!}{h!\\,\\,t!} \\; r^h\\,(1-r)^t. \\!", "d35438b6d0a8de0d8d6837b778e106ca": "c_\\text{fw} = \\frac{\\nu^2}{\\nu^2+k^2},", "d35443af084302c97285698705b08f13": "B_r^{p,q}", "d3547876ef373cff4bba55fae6d73223": "R^{\\mathcal A}=I(R)\\subseteq A^{\\operatorname{ar(R)}}", "d354a1e924393d682e89663f9087b6b1": "\n[D_{0}]_{i,i}<0\\; \n", "d354ddcebecdce24315e83919aa65dbf": "n = |G|", "d354e22b156a580484640682fc3cc29b": "\\begin{Bmatrix} p \\\\ 2 \\end{Bmatrix}", "d354f4a453f79a0e991ca711c160003d": "\\# V \\geq 3, \\# N\\geq 2", "d35525636216d36d1a321326a1d424f1": "p_2=\\frac {q_5(q_1-q_2)}{2m_2}\\ ,", "d3552c803f7160152ed0f57b85de8b55": "\n\\begin{align}\n W_3 &= \\beta-t^2 \\\\\n W_5 &= 4\\beta^3(1-6t^2)+\\beta^2(1+8t^2) -2\\beta t^2 +t^4 \\\\\n W_7 &= 61-479t^2+179t^4-t^6+O(e^2) \\\\\n W_4 &= 4\\beta^2+\\beta-t^2 \\\\\n W_6 &= 8\\beta^4(11{-}24t^2)-28\\beta^3(1{-}6t^2) +\\beta^2(1{-}32t^2)\n -2\\beta t^2 +t^4 \\\\\n W_8 &= 1385-3111t^2+543t^4-t^6 +O(e^2)\\\\\n V_3&= \\beta_1+2t_1^2 \\\\\n V_5&= 4\\beta_1^3(1-6t_1^2)-\\beta_1^2(9-68t_1^2)\n -72\\beta_1 t_1^2 -24t_1^4\\\\\n V_7&= 61+662t_1^2+1320t_1^4+720t_1^6 \\\\\n U_4&= 4\\beta_1^2-9\\beta_1(1-t_1^2)-12t_1^2\\\\\n U_6&= 8\\beta_1^4(11-24t_1^2)-12\\beta_1^3(21-71t_1^2)\n +15\\beta_1^2(15-98t_1^2+15t_1^4) \\\\\n &\\qquad\\qquad +180\\beta_1(5t_1^2-3t_1^4)+360t_1^4\\\\\n U_8&=-1385-3633t_1^2-4095t_1^4-1575t_1^6\\\\\nH_2&= \\beta\\\\\nH_4&= 4\\beta^3(1-6t^2)+\\beta^2(1+24t^2)-4\\beta t^2\\\\\nH_6&=61-148t^2+16t^4\\\\\nH_3&=2\\beta^2-\\beta\\\\\nH_5&=\\beta^4(11-24t^2)-\\beta^3(11-36t^2)\n +\\beta^2(2-14t^2)+\\beta t^2\\\\\nH_7&=17-26t^2+2t^4\\\\\nK_2&=\\beta_1 \\\\\nK_4&=4\\beta_1^3(1-6t_1^2)-3\\beta_1^2(1-16t_1^2)\n -24\\beta_1 t_1^2\\\\\nK_6&=1\\\\\nK_3&=2\\beta_1^2-3\\beta_1 -t_1^2\\\\\nK_5&=\\beta_1^4(11-24t_1^2)-3\\beta_1^3(8-23t_1^2)\n +5\\beta_1^2(3-14t_1^2)+30\\beta_1 t_1^2+3t_1^4\\\\\nK_7&=-17-77t_1^2-105t_1^4-45t_1^6\n\\end{align}\n", "d35597cb878582b3259841c563a7d543": "w = \\frac{1}{f\\rho_o}\\left [-\\left (\\frac{\\partial \\tau^x}{\\partial x} + \\frac{\\partial \\tau^y}{\\partial y} \\right )e^{z/d}\\sin(z/d) + \\left (\\frac{\\partial \\tau^y}{\\partial x} - \\frac{\\partial \\tau^x}{\\partial y} \\right )(1-e^{z/d}\\cos(z/d))\\right ].", "d355ec8f6a72cc69606e57d6c2b77c1b": "\\tau\\to\\Gamma\\tau", "d355f8591d655c5cd294d3ab775130e8": "D^2 = D^2_{ij}:=(D_{ij})^2", "d35603d4a999a483cce699b0c33b2152": "\\max\\{|I_{yes}|, |I_{no}|, f(k)\\}", "d35613cbd3e681dab9afa98a6513ea98": "k = \\frac{\\log n}{\\log q} \\leq \\log n", "d3566bf55af01823debb1aad8b76ccd3": " (\\operatorname{E}(R_m) - R_f) ", "d357143b014a38a7ebedc08919f23ca6": "\\bigwedge \\left( A \\cup B \\right)= \\left(\\bigwedge A \\right) \\wedge \\left( \\bigwedge B \\right)", "d35729fa0604612ffb28eb2add7d189c": " e_s(T)= 6.1094 \\exp \\left( \\frac{17.625T}{T+243.04} \\right)", "d3578732f6c7ec32f195767ba6197d57": "\\frac2\\pi=\n\\frac{\\sqrt2}2\\cdot\n\\frac{\\sqrt{2+\\sqrt2}}2\\cdot\n\\frac{\\sqrt{2+\\sqrt{2+\\sqrt2}}}2\\cdots.", "d357eb5bb8ca78b37994eb21a0fc0113": "|\\psi\\rangle=c_0|\\alpha_0\\rangle+c_1|\\alpha_1\\rangle", "d35829ba9e95be47540de8708d2da6f1": "\\scriptstyle \\alpha_k", "d358618227e99a5720f7d5d6c13e4792": "(\\log n)", "d358814eb93d4f9f66e196a9dea36467": "\\mathbf {\\varepsilon}_{i}", "d35918835d644b3a098a3186ef8ed7c4": "23 = 17", "d3592ca9e00c661e6c36a5f9e406e59b": " \\tilde{X}=\\frac{\\partial{}}{\\partial{t}}+X", "d3593c0ebb72401cab4bdb30b6ac1362": "(p_1,...,p_d)", "d3597dcc2ec009a47aa79ce6eeeeb29c": "3\\delta-6\\delta^2<\\delta", "d35995d8173d7180877b10e0dee3f450": "\nX(x,y) = \\left( \\frac{y^{q^{2}} - y_{\\bar{q}}}{x^{q^{2}} - x_{\\bar{q}}} \\right) ^{2} - x^{q^{2}} - x_{\\bar{q}}.\n", "d35996ef986ee602f10b7646511cc17f": "Y_N", "d359c5adfe035723a0dcbffe3fc61c5c": " a= 0~ mod~ p_i^{k_i}", "d35a33c1071bcd2d41661db04a12ca0b": "\\frac{64}{Re}", "d35aa5e8ebe6369289d365c4c5353ada": "\\text{supp}(\\rho) \\subset U", "d35aada9cea591f4acb7e54a60350b15": "(\\mbox{background})+(\\mbox{constant})*e^{-(\\mbox{time})/{\\tau_{\\rm pb}}}", "d35abf52a6b8cd7b5cc6834764c34f01": "A = (-1, 0)", "d35ac3d8c3d0436d30bc88c9ab6965fb": "\\varphi\\in\\Gamma(V)", "d35acb7e57fe0bc84eb0dbda386eda15": "dz = Mdx + Ndy \\,", "d35ad0fd3b9f0bd76f6235259f9f22ac": "\\tau_0 = \\left( {\\partial v_x \\over \\partial y} \\right) _{y=0}=0.332 {v_\\infty \\over x} Re^{1/2}", "d35b373a4448fb3857e98f5b013609f8": "\\{e^{\\pm i\\theta} \\} = \\{\\cos(\\theta)+i\\sin(\\theta),\\ \\cos(\\theta)-i\\sin(\\theta)\\}. ", "d35b72af741c05f42b3f598c10707e6d": "\n\\quad \\quad =\\int_\\Omega\\int_0^1 f(u+s\\tau\\psi)\\psi \\,ds\\,dx.\n", "d35bca9d1fd61e30bde7fc9b7696a805": "\\chi_i\\times\\psi_j(a,b) = \\chi_i(a)\\psi_j(b), (a,b)\\in A\\times B", "d35bda221e618a48209bcb3eb2889133": "\\{ 1, 2, 3 \\}", "d35c25e6c48aad6121b3d34b8a6ba470": "\\,\\! f(t) = f(\\tau t_c) = f(t(\\tau)) = F(\\tau).", "d35ce749ef4ce06f4ac515ceb60a6460": "\\ f''(x) > 0", "d35cfc5b6a5406fc3d01cbadb77eaea8": "f^{-1}(U) = \\operatorname{Spec}(R/I)", "d35d1ffcb0896b642c827a18a383fa66": "\\boldsymbol{\\nabla}\\times\\mathbf{v} = e_{ijk}~v_{j,i}~\\mathbf{e}_k", "d35d379679eaf812bc846d83ee560da1": "\n\\delta \\int_{t_{1}}^{t_{2}} \n\\left[ \\mathbf{P} \\cdot \\dot{\\mathbf{Q}} - K(\\mathbf{Q}, \\mathbf{P}, t) \\right] dt = 0\n", "d35d95c544841b74784dbf734af21d86": " L = e^\\gamma \\,", "d35da8b20146a746d1d64c2a55aff056": "\nK(p) = \\int_0^{\\infty} e^{-\\Tau p^2 - \\Tau \\alpha} d\\Tau = {1\\over p^2 + \\alpha }\n\\,", "d35dbfc9179898e80141890f6271b476": "F_{23.6\\%} = \\left({\\frac{1 + \\sqrt{5}}{2}}\\right)^{-3} \\approx 0.236068 \\,", "d35dc9ae707cfc282b8db5789b3b2ef4": "\\mathbf{st}(f^*(x_{i_0}))\\geq \\mathbf{st}(f^*(x_i))", "d35e26ac9baf9a63db10e6a196a1a4b6": "\\arccos\\left({1 \\over 3}\\right) = \\arctan(2\\sqrt{2})\\,", "d35e2cb72823f04e194d36bc7091274a": "f_{a}(k) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}}", "d35e628d4924b45b5200ab2b56b1efb8": "q\\,", "d35ee9b5769915c08b5bffb8321566ce": "c_k = p(x_k) + h(u_k)", "d35f2c84a54fe5669bf552d6c82472eb": " S = ", "d35f70aef2306b7e8e6d1a04c0776feb": " Q_n ", "d35fae2285dd8b0ef929f02d74a3cca4": "(0,1,2,3)=(r,\\theta,\\phi,t)", "d35fb4a15a118098d16eaa6f1f5e6ccc": "W\\in\\mathcal{O}*\\,", "d36075a13ed4201323fb202ffc989b48": "\nx\\in X, r \\le f(x) \\ \\ \\ \\longleftrightarrow \\ \\ \\ x=\\arg\\, \\max_{x\\in X} I(x)\n", "d3607973728c7ba7f06036c736ea1726": " \\Delta_\\Omega= -{1\\over 12\\pi}\\left[\\|\\partial_z \\log f^\\prime\\|^2_D +\\|\\partial_z \\log g^\\prime\\|^2_{D^c} - 2 \\|\\partial_z\\log f(z)/z \\|^2_D - 2 \\|\\partial_z\\log g(z)/z \\|^2_{D^c}\\right].", "d360a386d1237237b93a9ba94856512d": "a_G", "d360bd20af628c7e95bec9ed6ca49e5a": "|\\mathbf{X^{\\rm T}}\\mathbf{A}\\mathbf{X}|((\\mathbf{X^{\\rm T}AX})^{-1}\\mathbf{X^{\\rm T}A}", "d36150ca6ccfbc53d999e600f5f67624": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ -\\sqrt{\\frac{3}{2}},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "d361622fb3b6ef1673a5a1b15f15d25f": "\\scriptstyle G_\\sigma\\cap G_\\tau= \\empty.", "d361e3ba54fbeeadf90d7c9edb30a706": "\\varphi\\left(\\frac{\\sum a_i x_i}{\\sum a_j}\\right) \\le \\frac{\\sum a_i \\varphi (x_i)}{\\sum a_j} \\qquad\\qquad (1)", "d361fa620c190e2cc1b41106fa68daa8": "a \\uparrow\\uparrow \\dots \\uparrow b", "d3626f51831d5445ba0a6c1c708ce298": " P(X,Y) ", "d362a11a049be9b2ada0774848b6d582": "dq = 2 \\gamma r \\, dr \\,\\;", "d362a4887f8124d193ad224e4f75918b": "Q_r = \\frac{\\prod (a_j)^{\\nu_j}}{\\prod(a_i)^{\\nu_i}}~", "d363d7b841af88cd4f97bbcbe167f630": "\\frac{1}{2} (A + A^*)", "d36444b07a87351e674ae0ea23eea041": "3x^2 (2x^3 + 1)", "d3644dc7cb1aec6c9875064b963ddd82": "M_z", "d36478becd78a54857a7b16c3c39d0bd": " 2 \\le j \\le n - 1 ", "d3648a439dd9b3f8305057f9f8266215": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{T}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{F}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "d3649048229eef75e5642e7a3b86cc48": "X \\sim \\mathrm{Maxwell}(1)\\,", "d364b8147e3b8143fd95b40cdf00cd88": "\\pi_{\\mathbf S}\\circ\\psi: M\\to M\\,", "d3650f770f920a3c39fd602b030b43fa": "f: \\mathbb{R} \\rightarrow \\mathbb{C}, f(x) = \\sqrt{x}", "d3656af698680a8f6360cc0171fc0acc": " D\\left ( x-y \\right ) ", "d365eca5be14284eb8b39aa837831977": "A < B \\to 2^A < 2^B \\!", "d36633d04e5617b68c66cb541f6b57ac": " w = Rec(w',s) ", "d36646f90426d935866f62ae11fbb61f": "T^1(V)=V", "d3664a330dec7752e1c41176ea76f174": "\\textstyle 4.\\ Search\\ for\\ all\\ possible\\ \\theta\\ such\\ that: \\left | P_{\\bold A}^{\\perp}a(\\theta) \\right |^{2} = 0\\ or\\ M(\\theta)=\\frac{1}{P_{A}a(\\theta)} =\\infty", "d366863c660fdac95327c6c96e99311d": "f(x)=A_i", "d3669e0f493918c74b9b1000e68254f8": " \\psi", "d366e18eeb1f2a219a8fe66064c55a2e": "\\boldsymbol{x}_i \\in \\mathcal{X}", "d36719653fb22de3a2acf52ce1335fdc": " \\int_a^b G(t)\\varphi(t)\\,dt = G(a+0) \\int_a^x \\varphi(t)\\,dt + G(b-0) \\int_x^b \\varphi(t)\\,dt. ", "d3674cc8ecb7e0ce6bf55ae639875a07": " E_0 = \\left(\\frac{Z_A}{1000}\\right)^4 + J", "d367928e2b2851fbb8d063774bce82d6": "E_p=E_0 \\log p \\, ", "d367ad579533b36806c20c513f5f1c4f": "CCT_{est.} = -449 n^3 + 3525 n^2 - 6823.3 n + 5520.33", "d367bfd7ab16325e5d14ac1b422c5bfe": "a \\ge i", "d367f1b2adcca877d8bb411121064f0b": "Tail^+(S) \\gtrdot $ ", "d36821bda69a8a650052d8c5b1fe696d": "A \\in \\mathbf{C}", "d36823b2a83217210052a6a279bb8c16": "X = E(M)", "d3687e2978f67254c69d310b87cbfe2e": "\\forall n > N , |c - a_n| = c - a_n \\leq c - a_N < \\varepsilon ", "d36961dade592292e8b6e411f5b9da76": "\\operatorname{gcd}(l,m) = 1", "d369b60ffb4447a1e4a0a2fb16b0381e": "\\left( \\begin{matrix}\n A & B \\\\\n C & D \\\\\n\\end{matrix} \\right)\\cdot \\left( \\begin{matrix}\n r \\\\\n r^{'} \\\\\n\\end{matrix} \\right)_{1}", "d369d559e0b7e07e2acb0357ed130b33": "{52 \\choose 5}{5 \\choose 3}{47 \\choose 2} = 28,094,757,600", "d369ff0b7f0606be0dde3b0ba4b599ca": "\\rho\\, v_j\\, \\frac{\\partial v_i}{\\partial x_j} = -a^2\\, \\frac{\\partial \\rho}{\\partial x_i}.", "d36a3565f686a2fd5699a053c287c6b9": " H_\\infty \\le H_2 ", "d36adb434f774a569b488ddc135f7ca3": "\\frac{\\delta }{\\delta t}\\left( S^i_\\alpha T^\\beta_j\n \\right)=\\frac{\\delta S^i_\\alpha}{\\delta t}T^\\beta_j + S^i_\\alpha \\frac{\\delta T^\\beta_j }{\\delta t}", "d36b1fdbbdb9e139b70b4fa6ff1f98a0": "\\omega^\\alpha\\!{}_\\beta = {\\omega_i}^\\alpha\\!{}_\\beta\\,\\mathrm dx^i.", "d36b502ec29f64ae8f5a249dbcfe12a5": "g_{mi} \\Psi_{jk}^{m} + g_{mk} \\Psi_{ij}^{m} + g_{mj} \\Psi_{ki}^{m} = 0 \\mbox{ for } 1 \\leq i, j \\leq n. \\quad \\mbox{(A)}", "d36b6724c6174681a98a8fe698c747dd": "(A + C) \\det A_Q < 0", "d36bb06bc7bf5ac3dddcbcf1fe157191": "\nK_{11}x_{1}-K_{12}K_{22}^{-1}K_{21}x_{1}=F_{1}\n", "d36bcd1cfd4a3c429e1bbae4d466765f": "\\mu_o= \\frac{\\epsilon \\zeta}{\\eta}", "d36bd7bd4ce406b4e4d3c3c044b02cfd": "OBP = \\frac{H+BB+HBP} {AB+BB+SF+HBP}", "d36bf78a3adca7bb421f2cb69e471791": "a'_i = \\sum_j (u_{ij} a_j + v_{ij} a^\\dagger_j)", "d36c45fa6d0926a792dd3a7864465830": "\\Omega^2BU\\simeq \\mathbf Z\\times BU\\,", "d36c5f0faae37906e176382c6fadd35c": "\\sigma = \\lambda/\\sqrt{2}", "d36cbe6aed0bcd8da0c39d9b1804fa88": " y_i \\cdot M^t x_j", "d36ce4d5cc3489fd00b8ac8879871d24": "\\Delta = g_2^3 - 27g_3^2 = \\lambda^2(\\lambda - 1)^2", "d36d3a2a4dc71c127dd00c5e089fbe45": "x(t)=at^2/2+t+D", "d36d4414c9dce588fbfeebf962057a55": "F\\;=\\;C\\;-\\;P\\;+\\;2", "d36d64c3d97391d79a6724f4250141b5": "\\sum_{t=1}^{\\infty} (\\frac {a_t}{c_t})^2 < \\infty ", "d36d65bf965358bcddcef45f04ea5fa8": "\\det\\begin{pmatrix}\\lambda-1&-2\\\\-3&\\lambda-4\\end{pmatrix}.", "d36d6b0005722e132e7a18e3fe8b6d61": "R(n,0) = \\frac{1}{2} R(n-1,0) + h_n \\sum_{k=1}^{2^{n-1}} f(a + (2k-1)h_n)", "d36e0a3e9415168db1366009a127d60b": "A_n \\hookrightarrow SO(n-1)", "d36e12b3ca7f0b83b3c75177f939ed96": "\\frac{\\partial C}{\\partial t} = D_x \\frac{\\partial^2 C}{\\partial x^2} - U_x \\frac{\\partial C}{\\partial x}- k (C-C_N)-k(C-C_M)", "d36e89f705a790b7f470ff5000dec826": "\\int\\frac{dP}{P\\left(1-\\frac{P}{K}\\right)}=\\int k\\,dt", "d36f2f4a101150689afbeed505083dce": " \\ \\textbf{f}_1 \\cdot \\textbf{h} = \\textbf{g} -\\textbf{f}_2 \\cdot \\textbf{h} \\pmod q ", "d36f7ea654595f9415e6cb5ab5186856": "\\sigma_c\\geq0", "d36f895cfd03a651053363269715f6a0": "\\Psi(u_1,u_2,\\dots,u_m, J^k_f)=0\\!\\,", "d36fc7972ff85abd44e33bc83734faab": "N = 2^c - 1", "d37016e1a32885f87f978a899bc031bb": "dU/dt=G(U)", "d370c6397de45c023aba88cfe0fdbd9e": "w := w - \\alpha \\nabla Q_i(w).", "d370d32faec4d48d3041cdf3613a6c90": "s \\ge \\operatorname{dim} R_\\mathfrak{p} = \\operatorname{ht} \\mathfrak{p}", "d3711e3d6a4968c49c7ffc69231f67c1": "V_D = \\frac{ n_D - 1 }{ n_F - n_C },", "d3716ecc52483a87b5efc7c81000edae": "M_{ba}", "d37187b9abcff66bbe037dc69eccd77c": "\\scriptscriptstyle(x_0\\lor x_2)\\land(x_0\\lor\\lnot x_3)\\land(x_1\\lor\\lnot x_3)\\land(x_1\\lor\\lnot x_4)\\land(x_2\\lor\\lnot x_4)\\land{}\\atop\\quad\\scriptscriptstyle(x_0\\lor\\lnot x_5)\\land (x_1\\lor\\lnot x_5)\\land (x_2\\lor\\lnot x_5)\\land (x_3\\lor x_6)\\land (x_4\\lor x_6)\\land (x_5\\lor x_6).", "d37193fb71aec01a1c0538254efae3de": " x^2 \\equiv p^* \\text{ (mod }q), \\,\\!", "d3719a71ee04dc23244b10e1ab3586cd": "g_{ik}\\,", "d371cef975dde707bf392fd9c865db1c": "Q*^2={\\frac{2DK}{h}}", "d371df46f0cf45aea4f052a2925d1c65": "\\,a_{\\overline{n|}i}^{(m)}", "d372328d43fa8e30e6963d79aa839be5": "((A\\to B)\\to C)\\to(B\\to C)", "d3724e1e7737337211d1937c4b24308c": "f = \\beta y", "d37253afbb5414b18704b69e9ce03f8c": "(n+d+1)", "d372773921e5a5cb017752366679d93e": "\\frac{H_s}{\\Delta D_{n50}}= \\frac{(K_D cot\\theta)^{1/3}}{1.27}", "d3728f6c5839c2fdd1cdf231b387b3d7": "X_0 = 0\\quad", "d3730d553c46f1feb6987d9a743c54cc": "F(x;\\mu,s)=\\frac{1}{2}\\left[1\\!+\\!\\frac{x\\!-\\!\\mu}{s}\n\\!+\\!\\frac{1}{\\pi}\\sin\\left(\\frac{x\\!-\\!\\mu}{s}\\,\\pi\\right)\\right]", "d3733d798b2b05d3b39c4ca25cfe2202": "\\frac{\\partial \\mathbf{x}_i}{\\partial M_{(k\\ell)}} = \n -\\mathbf{x}_{0i}\\frac{x_{0i(k)}x_{0i(\\ell)}}{2}(2-\\delta_k^\\ell) - \n \\sum_{j=1\\atop j\\neq i}^N \n \\frac{\\lambda_{0i}x_{0j(k)} x_{0i(\\ell)}}{\\lambda_{0i}-\\lambda_{0j}}\\mathbf{x}_{0j}(2-\\delta_k^\\ell).", "d373e8d7ff0cd5830c709ea7c2a69af2": " P\\left( \\left| \\sum_{ i = 1 }^n a_i X_i \\right| \\ge k \\right) \\le 2 \\left( 1 - \\Phi\\left[ k - \\frac{ 1.5 }{ k } \\right] \\right) = 2 B_{ Ed }( k ) , ", "d373ff09c672b6ebb678221b8dc741f0": "F(a,b;2b;z) = (1-z)^{-\\frac{a}{2}} F \\left (\\tfrac{1}{2}a, b-\\tfrac{1}{2}a; b+\\tfrac{1}{2}; \\frac{z^2}{4z-4} \\right)", "d373ffb5f6554dba443b85dde48accf4": "\\mathrm{Y}", "d37469b8ef22fc0fee78a6fec5976cb6": " \\overbrace{\\smile \\smile-\\smile}^{\\mathrm{Foot 1}} | \\overbrace{--\\smile}^{\\mathrm{Foot 2}} \\star \\overbrace{\\smile\\smile-\\smile}^{\\mathrm{Foot 3}} | \\overbrace{-\\smile-}^{\\mathrm{Foot 4}}", "d3748a8650000db86a24b5afe0fb316f": " a= 6^2-2^2 =32 \\,,\\ b = 24 \\,,\\ c = (6^2 + 2^2)=40", "d374e8528dc56336f7de63d460434d3a": "\\beta_{12}=K_1 K_2\\,", "d374fcccb24f38a1f01cdf7bd8636cdb": "\\frac{d}{dx} \\ln(f(x)) = \\frac{f'(x)}{f(x)}.", "d375092ba1f1bb2cfa888c7d8db59df5": "\\tbinom{n}{k}\\mapsto\\left(\\frac{n}{|n|}\\right)^k\\tbinom{|n|}{k}", "d3756dfd3e4456911a49cd7f311fc131": "\n\\begin{align}\n 3x^2 + 12x + 27 &= 3(x^2+4x+9)\\\\\n &{}= 3\\left((x+2)^2 + 5\\right)\\\\\n &{}= 3(x+2)^2 + 15\n\\end{align}", "d375820ad9cae3257631333b64ac014d": "= \\arctan\\left(\\frac{\\sin{(\\text{spring angle})}}{\\tan{(\\text{wall angle/2})}}\\right)", "d375f6fb1081b285280f57252fac70f5": " \\varepsilon / \\sqrt{h} ", "d3761355f7783a28aa349c92fd0a23b7": "f_{\\nu}", "d37618a3f9e6710ae506cd94e6f07980": "P_n( \\cos {\\theta})", "d3762f971817eac00d3203e425e6223a": "m_{rocket}(t) \\frac{\\mathrm{d}\\mathbf{V}}{\\mathrm{d}t} = -\\mathbf{v_{rel}}(t) \\frac{\\mathrm{d}m_{gas}}{\\mathrm{d}t}.", "d376f9fb291021d2256882bf60eaf293": " \\text{SNR} = \\frac{\\text{signal}}{\\text{RMS noise}}", "d377278725de15cbc671904bf2e5bcf1": " \\Delta_{\\psi} L_z = \\sqrt{\\langle {L_z}^2\\rangle_\\psi - \\langle {L_z}\\rangle_\\psi ^2} \\quad (9)", "d3776725d4dec293f3d498a54c0a8dff": "\\psi(0) = 0 = C\\sin 0 + D\\cos 0 = D\\!", "d3776b80b43b14a8687bc89ac9cba89a": "(I \\otimes \\Phi)(\\rho) \\geq 0.", "d377c31793cd1ca3e9505fadedd02fcc": "R = \\mbox{ either }\\sum_{y=0}^\\infty \\ell_y m_y\\mbox{ or } \\int_0^\\infty \\ell_x m_x\\,dx,", "d377f69a95e6ac105d5c63f4038724e2": "\\Delta(C_{out}(m^1), C_{out}(m^2)) \\ge D.", "d377f760b1da407382fe1d566ba2f49a": "\\sin x = \\operatorname{Im}(e^{i x}) \\,", "d377fc46c617af800dc72e674a17c1f4": "T=d\\gamma/ds", "d3782757d7e295d196b89c1305112921": "[A_i, A_j]", "d378b09f76fce0693aa34042941cc393": " W^+_1 ", "d378b4ea21f6c3eab31986eda5cd0036": "Ae^{i\\omega t + i\\beta\\sin(\\Omega t)}.", "d3790754efcc416881692def2c5a6a18": "A = (a_1, a_2, \\dots , a_n)", "d379470954432f13d00ce2652f25323b": "\n\\mathcal{Z} = \\mathbf{v}_{0} \\cdot \\left\\{ \\prod_{k=1}^{N} \\mathbf{W}_{k} \\right\\} \\cdot \\mathbf{v}_{N+1}\n", "d379b025acf37239e29a0a99a218dfba": "z_0 \\le z \\le z_0 + h_0", "d37a3496d73340d5fce1ef1dd6808644": "\\phi^{X_t}(z)", "d37a6d7acc87a997b8d09007fe804a02": "\\tan{2 \\omega \\tau} = \\frac{\\sum_j \\sin 2 \\omega t_j}{\\sum_j \\cos 2 \\omega t_j}.\n", "d37a7d5535d32b5f1eb0fd204acea35e": "\\nabla \\cdot \\mathbf{J}_\\mathrm{tot} = 0", "d37a9bf6e76b0a1e39d2dab50d194f69": "\\eta = \\frac{b}{4\\, V_m}", "d37acd05810d86cff27a90f69b32d121": "x \\mapsto \\lfloor x \\rfloor", "d37af2ed3dab80385741124082d66426": "p \\mapsto \\log \\|f\\|_p", "d37afbf748da8d66a0dba83e91958a68": " \\frac{p}{q} = \\frac1r + \\frac{i}{qr}", "d37b389a02a203de0e954b9d86e33f72": "K = \\{k \\in G : g^{-1}kg \\in H\\;\\;\\forall g \\in G\\}.", "d37b40ea6dae1d50dd24431efdcace74": " a \\cdot b = a*b", "d37ba3e6bf6e2f0590c8505d6d2f2984": "pf = {P_a + P_b + P_c \\over |S_a| + |S_b| + |S_c|}", "d37bd69c46393f96fd18799d131464a4": "\\tan(\\phi) = \\frac { \\sin(\\theta)}{v/(c/n) + \\cos (\\theta)}", "d37be4271034b9b678fd8f11e3cfedc5": "x e^{-n}", "d37c02f1142ce56c37f57600723cda81": "= \\pi^{-1}(Y \\oplus R) \\oplus N' ", "d37c0d918e9117937016dc3dd13c2faa": "k>1", "d37c7fdbcfafbe1ac8d9adcadd41e5a9": "\\vec p_{1,2} = P1 + x \\ \\hat e_x + y \\ \\hat e_y \\ \\pm \\ z \\ \\hat e_z ", "d37cc74a74ce1febc5605c043a301c3e": "x_{n+1}=f(x_n,\\lambda)\\,.", "d37ccdce7b57b2abb3114cc0d25af789": "a_{\\sigma(i)} ", "d37d0829009e8cee7793e4b2c7f3696a": "xy+x+y=71", "d37d2517cb32f38f81c9b454af80b820": "u = T(v)", "d37d6fb8b3ca0f764bbf59612f5906a7": "{v_g}", "d37d801f49d6cce455bb1c07bf3ccf33": "\\operatorname{E}(X^k)\\,", "d37de2afd529578c16fc66a8dc2143a2": "q \\equiv e^{i\\pi\\tau} .", "d37e1a30690229d4702cde5034c89204": "c_x:\\ M \\to M", "d37e212b6504b44cd095024e1b9caafb": "\\int \\cos ax\\, \\cosh bx\\, dx = \\frac{1}{a^2+b^2}\\left( a\\sin ax\\, \\cosh bx+ b\\cos ax\\, \\sinh bx \\right) + C", "d37e4220506a010238a8037e2b6f9272": "\\delta\\lambda=k\\left(\\sqrt{\\frac{I_0}{I_{th}}}-1\\right)", "d37e595cba652e37ff88ce436bf65dcf": "\\displaystyle x \\in X. ", "d37e744a73559e2516a648025040dd9e": "\\forall h \\in H, \\; T h = \\sum _{i = 1} \\alpha_i \\langle h, v_i\\rangle u_i \\quad \\mbox{where} \\quad \\alpha_i \\geq 0 \\quad \\mbox{and} \\quad \\alpha_i \\rightarrow 0 ", "d37ec07e06aaa8e6dc2e13749c178a92": "p(\\phi) \\qquad \\qquad (0\\le\\phi<2\\pi),\\,", "d37ec942588af7aee8e5e44b0215dd67": "X_1+X_2", "d37edeb9ae73e4a21947f2f737eec165": "\\chi_{ss}", "d37ef89f394cf970821341bdca7a5516": "\\pi_\\mathrm{e} ", "d37f3a55dd2ee5f827d3d1163905150c": "r = \\cos(q\\phi)+2", "d37f5c3e651edbc263d5ca0079033196": "v,w\\in V", "d37f64427e4af62156f6e3355e98fd91": " f^*(g)=\\overline{f(g^{-1})}", "d37f8253897958549db2bd673ad2b092": "\\sin(x + 2\\pi) = \\sin(x)", "d37f97994f4146a4c0a19ca41350f691": " \\hat{\\boldsymbol\\theta}=-\\sin(\\theta)\\hat{\\bold{x}} + \\cos(\\theta)\\hat{\\bold{y}} ", "d37f99004188a3b481c07665e06d20d5": "A = 1 + 2\\left(\\frac{M}{C}\\right)", "d37ff111e429c0bc52e4ac9617288f83": "\\gamma_{\\tau\\tau}\\left(\\alpha_{\\tau\\tau}+\\beta_{\\tau\\tau}\\right)=-\\alpha_\\tau\\beta_\\tau+\\frac{1}{4}\\left(e^{2\\alpha}-e^{2\\beta}\\right)^2.", "d37ffc54b67ce8de1f01efb1f2e33689": "x = 1", "d3806d601ab8d0948aca3545c9142d89": "E_8.", "d3809cbef516d8933d8bbb11b161a423": "\\scriptstyle\n\\sum_{n=-\\infty}^{+\\infty} T\\ s(nT)\\ \\delta(t-nT)\\ =\\ \\sum_{n=-\\infty}^{+\\infty} T\\ s(t)\\ \\delta(t-nT)\\ =\\ s(t)\\cdot T \\sum_{n=-\\infty}^{+\\infty} \\delta(t-nT).", "d380afae492608c723034ab4bafa6e05": "E(\\phi) = S(p_{A}) = S(p_{B})", "d380ca5291d973eb1abb89a0228de9bf": "\\Delta_k = 0", "d38104a9945a64145ab4da30e908977c": "\\Rarr\\!\\Larr", "d38271500550181459718ff5de56cd35": "\\lambda/k", "d3828039bc162deb9f3900fcb47a8bcf": "X_t=\\frac{c}{1-\\varphi}+\\sum_{k=0}^\\infty\\varphi^k\\varepsilon_{t-k}.", "d382a1bb45359aff3d4be6b47d08e019": "c(x,y)=i(x,y) \\otimes h^{*}(-x,-y)", "d382aaf544c03cbe084295fd06dc28b5": "\\ell(w)", "d382ac1c86eafb61b61a5338010d76f8": " \\frac {1}{2} < \\theta < 1 ", "d382d0ef06ab25924ba5eaa70f309a22": "\\sum_{i=1}^n x_i R = U", "d382d7d4feb2f95148407eff21d72f2d": "1+1=10", "d382ef09acd0618dafb7340523f217d3": " \\mathrm{exp} [-v(f) \\; b {\\phi}^{3/2} / F] \\; = \\;\\mathrm{exp}[-v(f) \\; \\eta /f] \\; \\approx \\; {\\mathrm{e}}^{\\eta} f^{-\\eta/6} \\mathrm{exp}[- \\eta /f] \\; = \\; {\\mathrm{e}}^{\\eta} f^{-\\eta/6} \\mathrm{exp}[-b {\\phi}^{3/2} /F ]. ..........(45) ", "d38319eb00f2acd32b5e3b3e74c570ea": "\n \\int_{\\Omega(t)} \\mathbf{f}(\\mathbf{x},t)~\\text{dV} = \n \\int_{\\Omega_0} \\mathbf{f}[\\boldsymbol{\\varphi}(\\mathbf{X},t),t]~J(\\mathbf{X},t)~\\text{dV}_0 =\n \\int_{\\Omega_0} \\hat{\\mathbf{f}}(\\mathbf{X},t)~J(\\mathbf{X},t)~\\text{dV}_0 ~.\n", "d383216d39323669989037ab4ad3f0d2": "\\Omega \\subset \\mathbb{R}^n", "d383578d2213dc0b55867e7ef9b9b873": "x_n=s_n+e_n", "d383d745de8abc86012c06fc6631bf0b": "\\gamma=a", "d384059592cc116f673310d5757bf715": "\n\\sqrt[n]{z} = \\sqrt[n]{x^n+y} = x+\\cfrac{y} {nx^{n-1}+\\cfrac{(n-1)y} {2x+\\cfrac{(n+1)y} {3nx^{n-1}+\\cfrac{(2n-1)y} {2x+\\cfrac{(2n+1)y} {5nx^{n-1}+\\cfrac{(3n-1)y} {2x+\\ddots}}}}}};\n", "d3840f08962f5ad2af0085dd09a0e566": "|v+w| \\ge |v|+|w|,", "d3842647fb678380ed468b464640489a": "w{}_{ji}", "d384a3c25f047ef431d7166cf45e63ed": "|S|\\geq\\frac{|MDS(C)|}{M}", "d3850c82325b6f6e7ef475f09049302b": " \\mathbf{G}, \\mathbf{K} ", "d385244b82cbe103652bfb9c78b7a61e": "\n \\delta W = \\int_L q~\\delta w~\\mathrm{d}L\n", "d38556785b35a91bfc08e214da7b3f29": "Y = \\log y,", "d38581663004353615c39e9ee6a246d7": "\n \\hat{w} = A_1~e^{k_1 x} + A_2~e^{-k_1 x} + A_3~e^{k_3 x} + A_4~e^{-k_3 x}\n ", "d385aa3a6d3b4b078223836794050619": "V_{\\rm w} = {4\\over 3}\\pi r_{\\rm w}^3 + \\pi r_{\\rm w}^2d", "d385e430ea94c992c4a2120c9711612c": "p_3(x)=9x-6x^2+x^3;", "d385e928978f137b58e17a936457bdf3": "\\bigcap_{X\\in H} X=\\varnothing", "d38633a6d891da7f821fde725b1e617d": "(u,v,x,y) \\longrightarrow (u, \\; v+ x \\, \\lambda + \\frac{u}{2} \\, \\lambda^2, \\; x + u \\, \\lambda, \\; y)", "d38634e2713d1a822124825bdbfc4450": "SU(N_f)_L^2 U(1)_R", "d386da1dfc1e6f598d88180b59217dfd": "\\varrho_{A,B} = \\sum_\\lambda \\varrho_A^\\lambda \\varrho_B^\\lambda w_\\lambda \\,", "d386e549016432fa44f0c9bdc6b8412e": "|\\sum_{k=0}^\\infty(-1)^k\\,a_k\\,-\\,\\sum_{k=0}^m\\,(-1)^k\\,a_k|\\le |a_{m+1}|.", "d387b4c915394abf5d72accddaaf7fd6": "c_{N-m} \\leftarrow c_{N-m} \\oplus b_0, c_{N-m+1} \\leftarrow c_{N-m+1} \\oplus b_1, \\dots ", "d387d5fff2a13ec8b8d71c1fd56302f9": "(\\Sigma^{-1})_{uv} =0 \\quad \\text{if} \\quad \\{u,v\\} \\notin E .", "d38806e1e07ce8df1c953ae05c378697": "f(\\theta) = \\tfrac12 a_0 + \\sum_{n=1}^\\infty \\left(a_n\\cos n\\theta + b_n\\sin n\\theta\\right)", "d38829e0f4ad5134795f435bf1fba562": "e^{idt\\hat N}\\psi(x, t)", "d38840f6229b504d3336e383a79a38f0": "dt\\leftarrow T", "d388901493f65678825eb00a746434fe": "2.9709", "d388a04dc5c72b31f9bb12b718bdfd02": "(a,b) - (c,d) = (ad-bc,bd) \\,", "d388a6fbbea59171c584e5ca1e238ba5": "\\rho(\\sigma)", "d389429c049480b57b53610600ea7b84": "w_{s}=-2e \\left(\\frac{dc_{1}}{dz}\\right)", "d38962eb20fb8e5d6247810f2dd7352b": " K_{\\phi(x)} = H_x \\quad \\mbox{almost everywhere} ", "d3898f2cdcbf132b9437e9b7fc126260": "b^2-4ac.", "d389b3441711a2d02596f57505abc985": " [\\mathfrak{g}, e] ", "d38a11f29a7c185bded49a217d4fa265": "i_1,\\dots,i_l", "d38a7155b12ffeb055167ed98d158608": " Z_3 = Y_1\\cdot (F-G) ", "d38ac61af578078f0851ce9478be7dbf": " f_W = \\frac {e'_w} {e^*_w} ", "d38acbd4e5579afb84a2acb43695a886": "\\frac{d Q}{d P} |_{\\mathcal{F}_t} = Z_t = \\mathcal{E} (X )_t", "d38b228dbe9278e2c1623c5f619b191d": "\\mathcal{L}_\\text{approx}(\\theta \\mid x \\text{ in interval } j \\text{ containing discrete mass } k)=p_k(\\theta) + f(x_{*}\\mid\\theta) \\Delta_j, \\!", "d38b2a45cb6e2f8482b6679e0d9632cb": "y = r \\sin \\varphi \\,", "d38b3573e6c6b90f20ec3ee2797b8b73": "\\frac{m_{\\text{e}} c^2\\alpha^2}{2} = \\frac{0.51\\,\\text{MeV}}{2 \\cdot 137^2} = 13.6 \\,\\text{eV} ", "d38b4aa268dfd1ac4ea6ec229c89eeb1": "F(x, \\ t) = \\sin \\left[ 2 \\pi \\left( \\frac {x}{\\lambda - \\Delta \\lambda } - ( f + \\Delta f )t \\right) \\right] ", "d38bd141c1545671d21ce5abaea93aea": " \\sum_{i=1}^J\\sum_{j\\ne i}\\frac{\\lambda_i}{\\lambda_j}p_{ij}\\mu_j(x_j) = \\sum_{j=1}^J [\\sum_{i \\ne j}\\frac{\\lambda_i}{\\lambda_j}p_{ij}]\\mu_j(x_j)=\\sum_{j=1}^J[1-p_{jj}-\\frac{\\alpha p_{0j}}{\\lambda_j}]\\mu_j(x_j). ", "d38be5a9e886a212345673fc855e145c": "\\tau_{x'y'}=\\tau_\\mathrm{n}", "d38bea3eed9eb2c02f97ab6a609cc6c5": "t_1>t_2", "d38bf62897b46ed28c43bdea91934db9": "f(t) = \\left\\{\n\\begin{array}{rl}\n0 & t<0\\\\\n\\sin \\frac{2\\pi t}{\\tau} & t \\geq 0\n\\end{array} \\right. ,\n", "d38c17684b7b7980ea610d42c0201940": "A_1,A_2,A_3,A_4", "d38c2692f6aeb095fd8ac20a2aefb0ad": "\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots", "d38c488b59eafb0b3a173cfe38743612": "(2n)", "d38c6df6785410ac99c32df9a92c5f75": "\\Pr(a) = \\frac{1 + 5 + 9}{45} = 0.333 \\!", "d38c949b9ae7ac336f46ae1398b35183": "\\nabla \\cdot (E + iB) = 0", "d38cb7ccb0d50c7c50c7b94456de83a0": "\\displaystyle y_1", "d38cf456378664857b0e22b9c647944d": "\\sigma_{A,A} \\circ \\delta = \\delta", "d38d4f479490b5294f3d318fcb45efac": "P_\\ell^{m}(x) = \\frac{(-1)^m}{2^\\ell \\ell!} (1-x^2)^{m/2}\\ \\frac{d^{\\ell+m}}{dx^{\\ell+m}}(x^2-1)^\\ell.", "d38d57469b8834fef6f773b205dbbd76": " R(X,Y) = (\\nabla_X \\nabla_Y - \\nabla_Y\\nabla_X - \\nabla_{[X,Y]}) ", "d38db4a71575d24adfed3a2f939a033c": "\n\\langle m_0, m_1 \\,\\vert\\, w = m_0 m_1 m^{-1}_0 m^{-1}_1, \\,w m_1 = m_1 w \\rangle.\n", "d38e151c1747fa419f35ae0353f153ce": " L_{j,j} = \\sqrt{ A_{j,j} - \\sum_{k=1}^{j-1} L_{j,k}^2 }. ", "d38e342381319d592900a4ffe96db212": "\\psi_{n,\\mathbf{k}}(\\mathbf{x}) = e^{i\\mathbf{k}\\cdot\\mathbf{x}} u_{n,\\mathbf{k}}(\\mathbf{x})", "d38e6e49703052a598aa90e897cf23a7": "\\displaystyle g(x) = \\frac{1}{n}\\sum_{i=1}^n g(x_i)", "d38e9861ed850d4ed2790c49106713b5": "c, d ", "d38ea4c923686e1702311e067fc97c10": "\\frac{d^2}{dy^2} = \\frac{d}{dy} \\left( 2 y \\frac{d}{dx} \\right) = 4 x \\frac{d^2}{dx^2} + 2 \\frac{d}{dx}. ", "d38ec31896acc6d2bc2042a0bf45f18b": "0.3>{\\alpha}>-0.3", "d38f07edd7703a698b4f2ce4af338d23": "c\\equiv m^2\\pmod{pq}", "d38f8ab95c2adaeb7163ff81391217ea": " \\sum_j p_j R_{ij}. \\, ", "d38f9164afba735a22f5a737e17bce35": "F = G \\circ \\varphi", "d38f935d9c92127c4570b912d870e67d": "\\mathbb{Z}^+", "d38fb19de8b74d4a2bea2990d42b6238": "\nB^{\\prime\\prime}-Y^\\prime=-0.299\\cdot R^\\prime-0.587\\cdot G^\\prime+0.886\\cdot B^\\prime\n", "d38fe28f46268b85c4469c27eb3a9226": "f(t_i,y_i)", "d38fe4674984459df15cf4d1d9defc71": "x[n] = -(0.5)^n u[-n-1]\\ ", "d39001a574dfa8fa420fc4115427e51a": "\\varepsilon\\!", "d3900ff8b0a5ecec90c5b78130a69c92": "{A}_{15}^{(2)}", "d39036aea033a02a34c701a5787ce18e": " Y \\approx BZ", "d3903fd826e8086d0a9287314e6ad443": "\\frac{dN}{dt}=\\frac{-0.693\\,N}{T_{1/2} }", "d39054424326eb5aa4c059ecece168ae": "1 \\over t^2 + 1", "d3911ab7ae0b8ca318a600471f004c7e": "B^\\prime=-\\left(n_\\text{b}-n_\\bar{\\text{b}}\\right); ", "d391707b3a59ac3a5e64d2023de2f872": "x \\sim \\mathcal{N}(\\mu, \\sigma^2)", "d391b48b991acdb527f698f15285c37b": "P_i = wl_i + (h + r) P_A a_i", "d391b686f35d53df7b2f25f3bb9b84b5": "\\mathit{c}_i = 0", "d39294e151196ee40d0d9bce67fcd3a3": "\\bar{y}_{i\\bullet}-\\bar{y}_{j\\bullet} \\pm \\frac{q_{\\alpha;r;N-r}}{\\sqrt{2}}\\widehat{\\sigma}_\\varepsilon \\sqrt{\\frac{2}{n}} \\qquad i,j=1,\\ldots,r\\quad i\\neq j.", "d392b7a519028072d2c911d0499eb8d2": "\\frac{30\\,\\alpha-66}{(\\alpha-3)(\\alpha-4)}\\!", "d392f7f63b01e2fd0d4b49cbaf0bf517": "\\operatorname{Supp}(M \\otimes_A N) = \\operatorname{Supp}(M) \\cap \\operatorname{Supp}(N).", "d392fbd7dbba17c4932bf50cf799d13c": "C(R) = 1/2", "d3930c6c3e0b7121c5e1f656ad6a02af": "\\hat{\\mu} = \\frac{\\sum_{i=1}^N k_i }{\\sum_{i=1}^N n_i }.", "d39316c42b70bc2820ee4a9d2a33a7d7": "n_e=10^3", "d3934563fde81a0cc38771c1c36fd383": "EP_f:=\\{x\\in \\mathbb{R}^S|\\sum_{e\\in U}x(e)\\leq f(U), \\forall U\\subseteq S\\}", "d393ba319387b0c29b54c3488101e21b": "\\varepsilon \\digamma \\varkappa \\varpi \\!", "d393cdf42ae9ca79f135dcd04e11d66a": "max_{x\\in X^{2m}}(Pr[\\sigma(x)\\in R])\\leq 4\\Pi_{H}(2m)e^{-\\frac{\\epsilon^{2}m}{8}}\\,\\!", "d3944ed95bb8e265cc909db282c83a4b": "I>10^{-1}", "d394b4097fd55747a3a9a862ee849824": "\\gamma_\\mu \\,", "d394b62fb84800aae50cdf5f15dbfa57": "\\mathbf{t_{i+1}} = \\begin{bmatrix} t_i(1)+\\epsilon(1) \\\\t_i(2)+\\epsilon(2) \\\\ \\vdots \\\\ t_i(r)+\\epsilon(r) \\end{bmatrix}", "d39553232cff16d1f06090f4c202effd": "dV = \\rho(u_1,u_2,u_3)\\,du_1\\,du_2\\,du_3", "d39560f13596b93d78ed3a3ffadc3166": "(x^2_1 - Ny^2_1)(x^2_2 - Ny^2_2) = (x_1 x_2 + Ny_1 y_2)^2 - N(x_1 y_2 + x_2 y_1)^2", "d395aa1b74fae75fb3351239b16bca33": "y_1= -1/4", "d3963176663ee1a990fdb08b52b81910": "O_n^{(\\alpha)}(t)= \\frac{\\alpha+n}{2\\alpha} \\sum_{k=0}^{\\lfloor n/2\\rfloor} (-1)^{n-k}\\frac {(n-k)!} {k!} {-\\alpha \\choose n-k}\\left(\\frac 2 t \\right)^{n+1-2k},", "d39644e465fb4d322f08532049f6c67f": "\\vert a \\vert > a_0", "d39651ad78e66a97134b93e5740992aa": "k \\equiv \\frac{R}{2}", "d3967d2f0549441c2b62c56873034e28": "v=C\\sqrt{R\\,i},\\,", "d3969eba27de034201d67f443a618886": "M \\geq \\sqrt{Q^2 + P^2},", "d39707ba7b185ad25ec58168e17e5165": " \\langle \\mathbf{u} \\rangle ", "d39747c6fc2b07e65780a195996ca40e": "d_i-c_i", "d397678a9e49edc07d5924cbe24a3c8d": "[L(z),L(w)] =\\left(\\frac{\\partial}{\\partial w}L(w)\\right)w^{-1}\\delta \\left(\\frac{z}{w}\\right)-2L(w)w^{-1}\\frac{\\partial}{\\partial z}\\delta \\left(\\frac{z}{w}\\right)-\\frac{1}{12}cw^{-1}\\left(\\frac{\\partial}{\\partial z}\\right)^3\\delta \\left(\\frac{z}{w}\\right)", "d397c1dc4c7d21dd267419bc9c2a0988": "\\displaystyle =\\frac{(1/2)_j(\\alpha/2+1)_{n-j}(\\alpha/2+3/2)_{n-2j}(\\alpha+1)_{n-2j}}\n{j!((\\alpha/2+3/2)_{n-j}(\\alpha/2+1/2)_{n-2j}(n-2j)!} ", "d397e16ee2b2ce3291c2333c4271b226": "[-\\pi/2,\\pi/2]", "d3983739dc029c967aa6e13994351c3e": "0 < \\lambda_1 < \\lambda_2\\le \\lambda_3\\le\\cdots,\\quad \\lambda_n\\to\\infty.", "d39862f4ab106d6caa45b522b4160ce9": "\\mu_{max}", "d398bc83ba2fa89d59e6f77b4a6abca5": "K[G]", "d39993866ef38db9f4055369995840f7": "\\rho = u", "d399b308a6fda136e148fa2c3f4d4e47": "\\beta v=f \\, \\partial{w}/\\partial{z} \\ ", "d39a5d295935faf252a0510271d8412c": "R=240", "d39a96951a8f8d3258533e2a1b45bdd8": " \\pm 1 ", "d39ae04e85b6315aff1d80eb4cb99849": "s^2 = \\frac {1}{n-1} \\sum_{i=1}^n \\left(x_i - \\overline{x} \\right)^ 2 = \\frac{\\sum_{i=1}^n \\left(x_i^2\\right)}{n-1} - \\frac{\\left(\\sum_{i=1}^n x_i\\right)^2}{(n-1)n} = \\left(\\frac{n}{n-1}\\right)\\,s_n^2.", "d39ae704c047d04e67220d6cc4e884f5": "\n \\begin{align}\n F_1\\sigma_1 + & F_2\\sigma_2 + F_3\\sigma_3 + F_4\\sigma_4 + F_5\\sigma_5 + F_6\\sigma_6\\\\\n & + F_{11}\\sigma_1^2 + F_{22}\\sigma_2^2 + F_{33}\\sigma_3^2 + F_{44}\\sigma_4^2 + F_{55}\\sigma_{5}^2 + F_{66}\\sigma_6^2 \\\\\n & \\qquad + 2F_{12}\\sigma_1\\sigma_2 + 2F_{13}\\sigma_1\\sigma_3 + 2F_{23}\\sigma_2\\sigma_3 \\le 1\n \\end{align}\n ", "d39b25f49c59c6a708a9090ebd363b04": "\\chi(\\hat{\\mathbf{C}})=2.", "d39b30c497704ecbb215f5a96a5b4d01": "\\Omega_\\text{LT} = -\\frac{2}{5}\\frac{G m \\omega}{c^2 R}\\cos\\theta. ", "d39b53cc34583e29974673cd9bc05fba": "N = \\frac {f}{D} = \\frac {1200}{254} \\approx 4.7", "d39b9fc1236a53fb0794dae3a574f213": "\\begin{matrix} (\\frac{1}{3}) \\end{matrix}", "d39bb66672749723e6fceb24ba8f3186": "D(f*g) = (Df)*g = f*(Dg).\\,", "d39be3295e918385a0f5f4990f3f8375": " 0 A, X=x) \\cdot P(B > A | X=x) \\\\\n&+ E(B | B < A, X=x) \\cdot P(B < A | X=x).\n\\end{align}", "d39f473d304eb2c538497aeea97cd150": "p\\in\\mathbb{C}", "d39f51014f49b9b5964611088b860ec0": "\\boldsymbol{\\mathcal{Q}} = \\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial T}{\\partial \\mathbf{\\dot{q}}} \\right ) - \\frac {\\partial T}{\\partial \\mathbf{q}}\\,,", "d39f718024d789810fdd98f8fbc3bf4d": " I\\subset \\Omega^*(M) ", "d39f9d274eae7d6483d124f3f28960a9": "\\hat{\\mathbf{n}}", "d39facf3ab83c30b1a78fddc1c232448": "O_F", "d39fbe7b07ac94c775eeb90534f1105c": "\\operatorname{diag}(A)", "d39feed6b8135da59da98cdcda1c273a": "\\textstyle=-\\frac {d} {dt} \\iint_{\\Sigma (t)} d \\boldsymbol {A} \\cdot \\mathbf {B}(\\mathbf{r},\\ t),", "d3a049138fdcb9c96b93ad1bbe6c4d15": " S_m = \\sum_{i=1}^k \\sum_{j=1}^m e_{ij}. ", "d3a050af978692cde94583934eddbd91": "\\begin{pmatrix}X\\\\Y\\\\ Z\\end{pmatrix}=A\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}.", "d3a0d7f9f6b25a248b544ad7783fa0b9": "(y_{11}-y_{21})-(y_{12}-y_{22})", "d3a0fbff9efd3d6013fc894fe4f6de7c": "\\phi(a) = 0", "d3a13581793c2714b62e510d04ff2d41": "O(m^{n})", "d3a14d71b5499bfd29b386b430a26a29": "h^*P", "d3a151fef4f0fd388ca19450227519c7": "\\frac{(7+2+1)^2}{3} + \\frac{(7+6)^2}{2} + \\frac{(11+6)^2}{2} + \\frac{(10+7+3)^2}{3} + \\frac{(5+3+4)^2}{3} + \\frac{(11+4)^2}{2}", "d3a155ee7dcb697b0aa2f0a1834c65fc": "p(drunk|D) = 1/50.95 \\approx 0.019627", "d3a1568528969bc08c4de1fbfd80d426": " \\frac{2\\mu}{\\hbar} ", "d3a1662d84ff804918af5f9b96d7ef70": " \\sin\\left(\\sum_{i=1}^\\infty \\theta_i\\right)\n=\\sum_{\\text{odd}\\ k \\ge 1} (-1)^{(k-1)/2}\n\\sum_{\\begin{smallmatrix} A \\subseteq \\{\\,1,2,3,\\dots\\,\\} \\\\ \\left|A\\right| = k\\end{smallmatrix}}\n\\left(\\prod_{i \\in A} \\sin\\theta_i \\prod_{i \\not \\in A} \\cos\\theta_i\\right) ", "d3a1ede22e032da167759b0ecffc5ffc": "A\\subset \\R^n", "d3a1f852cb44bfd705fa2ead3aa31b44": "\\mathbf{x} \\cdot \\mathbf{d}", "d3a21ddce135234907f65978c8e473e8": "\\frac{|MDS(C)|}{\\log{n}}", "d3a21ead2d769aa0fd56166f7ff75e0b": "A(n,n)", "d3a258f21492fb560247cacb23d8af3b": "\\frac {\\zeta^\\prime(s)}{\\zeta(s)} = -\\sum_{n=1}^\\infty \\frac{\\Lambda(n)}{n^s}.", "d3a25c7081a0fe493e9ff1c3c73fe062": "\\frac{1}{N} \\ln(\\mathcal{L} (\\alpha, \\beta|X)) = (\\alpha - 1)\\frac{1}{N}\\sum_{i=1}^N \\ln X_i + (\\beta- 1)\\frac{1}{N}\\sum_{i=1}^N \\ln (1-X_i)-\\, \\ln \\Beta(\\alpha,\\beta)", "d3a282178f217615abbef0887c23400e": "\\delta \\to d", "d3a28be184901964ecd090e7e681a97d": "\\sum_j q_{ij} = 0", "d3a2c154f4725b824a9a19c776f4f572": "d \\gets 1", "d3a2c524936f0ba1de1d8cad868baae0": "\\,\\hat{V}=n^{-1}\\sum_{i=1}^S n_s \\bar{z}_s \\bar{z}_s^\\top", "d3a2c70f91e6f13f35e8b259531ff2d3": "a + b + 90^{\\circ} = 180^{\\circ} \\Rightarrow a + b = 90^{\\circ} \\Rightarrow a = 90^{\\circ} - b", "d3a30e047e01fee83bf748791a39e1b9": "\ny_n = \\frac{32}{31} y_{n-1} - \\frac{1}{31} y_{n-2} + h \\left(\n\\frac{5}{31} f( y^* ) + \\frac{64}{93} f( y^*_{n-1/2} ) + \\frac{4}{31} f( y_{n-1} ) - \\frac{1}{93} f( y_{n-2} )\n\\right).\n", "d3a3191169df4a700ca51ae291e359da": " \nA\\subseteq \\bigcup_n A_n \\Rightarrow \\mu(\\bigcup_n A_n)=\\infty.\n", "d3a3a35e7193edfc7920e81453277166": "\\lnot A \\vee \\lnot B", "d3a3e0d1946c98ec91ec9c2a12075850": "{{I}_{R1}}=\\frac{{{V}_{CC}}-{{V}_{BE2}}-{{V}_{BE3}}}{R1}", "d3a3e408cf81613760cd75e54778845c": "(a_{n})", "d3a42455cfd51ab153240785fce7c5f3": "\\ \\sigma(n) < e^\\gamma n \\log \\log n ", "d3a42a67d3b2b9be899faea86fe5fd9c": "\\Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0\\,", "d3a42f74193054630556b175c4b14755": "\\eta, \\varphi", "d3a4377fe1c3797a737d207ce698443e": "n \\ge l\\,", "d3a45bd812aeb1e977fea263415d33ac": "\\frac {d M_x(t)} {d t} = \\gamma \\left ( M_y (t) B_z (t) - M_z (t) B_y (t) \\right ) - \\frac {M_x(t)} {T_2}", "d3a478b91c5f979c02543708936d4873": "\n \\begin{bmatrix} M_{xx} \\\\ M_{xy} \\\\M_{xz} \\end{bmatrix}\n := \\int_A \\begin{bmatrix} y\\sigma_{xz} - z\\sigma_{xy} \\\\ z\\sigma_{xx} \\\\ -y\\sigma_{xx} \\end{bmatrix}\\,dA \\,.\n ", "d3a4d36778c7196e31f96d351aea955a": " (\\lambda p. \\lambda f.(p\\ f)\\ (p\\ f)) ", "d3a52a2aecf5e84a184b9c3dc67fa7c7": "m_{inf}(R,T_i)", "d3a5a9d85dd62c0160a617f6b61af62e": " L_*=\n \\begin{bmatrix}\n 2 & 0 \\\\\n 5 & 7 \\\\\n \\end{bmatrix}\n", "d3a5e50fec1940fe504479a12f52dae5": "\nJ_{2}=\\begin{pmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{pmatrix};\\quad J_{3}=\\begin{pmatrix}\n0 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n1 & 0 & 0\n\\end{pmatrix};\n\\quad J_{n}=\\begin{pmatrix}\n 0 & 0 & \\cdots & 0 & 0 & 1 \\\\\n 0 & 0 & \\cdots & 0 & 1 & 0 \\\\\n 0 & 0 & \\cdots & 1 & 0 & 0 \\\\\n \\vdots & \\vdots & & \\vdots & \\vdots & \\vdots \\\\ \n 0 & 1 & \\cdots & 0 & 0 & 0 \\\\\n 1 & 0 & \\cdots & 0 & 0 & 0 \n\\end{pmatrix}.\n", "d3a60a2a66afc6ce33b884362b622b5a": "X_n\\ \\xrightarrow{a.s.}\\ Y", "d3a61ee29baf6aa896eabdb6c222773a": "\\nu \\rightarrow 0^+", "d3a62928f856a3b449ab07f0f15f886f": "A = \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix}.", "d3a663d3bbe7469ef48a7eaf5f520b46": "\\sin\\langle u,u'\\rangle = \\langle \\sin(u) , u' \\cos(u) \\rangle ", "d3a6ac4f366c5d61ff4e4d4e1038a041": "\\mu_3", "d3a75f8d2f19a15e6729cd45a53dcfad": "a_{i}^{j}", "d3a77dad262a1aae1e359a8d5d256ed7": " \\Sigma : \\Im \\rightarrow \\Tau ", "d3a7a4727b778f31fc936b4f8476d047": "|k| \\neq 1", "d3a7c1b064b0d39f8dc0e0459dd11c5f": "f(t)=\n\\begin{cases}\n(\\mu-\\lambda)e^{-(\\mu-\\lambda)t} & t>0\\\\\n0 & \\text{otherwise.}\n\\end{cases}", "d3a808b819ed47f3593e6c540fdd5580": "\n\\begin{align}\nt_n & =\\sum_{k=0}^n{n \\choose k}\\frac{x^k}{n^k}=1+x+\\sum_{k=2}^n\\frac{n(n-1)(n-2)\\cdots(n-(k-1))x^k}{k!\\,n^k} \\\\[8pt]\n& = 1+x+\\frac{x^2}{2!}\\left(1-\\frac{1}{n}\\right)+\\frac{x^3}{3!}\\left(1-\\frac{1}{n}\\right)\\left(1-\\frac{2}{n}\\right)+\\cdots \\\\[8pt]\n& {}\\qquad \\cdots +\\frac{x^n}{n!}\\left(1-\\frac{1}{n}\\right)\\cdots\\left(1-\\frac{n-1}{n}\\right)\\le s_n\n\\end{align}\n", "d3a84464e710115622227fe4fdbf38d1": "P_{I, M} = P_{I, M'} + P_{I, M''} - F", "d3a85fdfec7599243ca3725ecd2c04cb": "\\textbf{let}\\ x = x\\ \\textbf{in}\\ x", "d3a8d16938c189b68d67cfcde2c60736": "\\operatorname{InverseGamma}(\\frac{1}{b_2}-1,\\frac{a-C_1}{b_2})\\!", "d3a8e410385b18fc074826b875d71ca1": "a/b = \\sqrt{2}", "d3a8f18292f2cd91bd49b60f914ae8ed": "S_1 = \\sum_{i=1}^n x_i", "d3a97a77db28d5cf403d8e9aea983102": "\\mathbf k \\cdot \\mathbf J_t\\left( \\mathbf k \\right) = 0,", "d3a986eb9d3be94e3dcfa725bcaeae9f": "\\frac{\\partial q} {\\partial t} = 0", "d3a99a826b486b5f896d217e4b5975cb": "f(x) = \\frac12 \\; \\operatorname{sech}\\!\\left(\\frac{\\pi}{2}\\,x\\right)\\! ,", "d3a9ba561039dcedfffa5a16b7ad3c7a": "{2\\omega_i \\over d}", "d3a9fdb36650cc6365a0f460ad177d9d": "\\scriptstyle I_2", "d3aa099a02d598b6abfdaed852c8731b": "-35 \\sqrt {2} /256", "d3aa105d4a95051e81d3686baf8dd5b3": "\\Delta U = \\Delta Q - \\Delta W", "d3aa2cfe20bca5e19cde6f5aec1df9aa": "RD_{\\mathrm{new/ref}} = \\frac{1}{1 - \\frac{A \\Delta x}{m} \\rho_\\mathrm{ref}}", "d3aa5de64cb9110ab212dc5edd201b6a": "\\frac{d}{dx}\\log_a x = \\lim_{h\\to 0}\\frac{\\log_a(x+h)-\\log_a(x)}{h}=\\frac{1}{x}\\left(\\lim_{u\\to 0}\\frac{1}{u}\\log_a(1+u)\\right),", "d3aaa17bcd65e1d7b84f12eba27b4287": "S_+|-\\rangle=\\hbar|+\\rangle", "d3aadee4bd0068f95a6219d02d1a5d70": " \\hat{A}^{-1} ", "d3aaf88472ef591f847e9dba9637217a": " \\sum_{m = 1}^{M + 1} P_m x^{m - 1} = \\prod_{m=1}^{M} \\left(x - e^{\\lambda_m}\\right).", "d3ab21c1f20f3f3bb6feb2f3281b6450": "|n\\rangle ", "d3ab5a18c84078f7f625e328934a29a0": "{\\delta U}=T\\mathrm{d}S-P\\mathrm{d}V.", "d3abda4500595b47cea9720f4b598640": "c/4", "d3abfd36abce6575f88b227f88ef8449": "{\\mathbf{}}L_i^r=\\left( B'_iS_{i+1}B_i+R_i \\right)^{-1} B'_i S_{i+1}G'_i,", "d3ac1793bd7c67c96ac5bb4f47b85303": "Y = C(X)", "d3ac3e20e97577ff2d98309085b99385": "\\mathcal{L}_X(\\alpha\\wedge\\beta)=(\\mathcal{L}_X\\alpha)\\wedge\\beta+\\alpha\\wedge(\\mathcal{L}_X\\beta)", "d3ac48964424c6341d9bf308689d84f9": "\n\\varepsilon^v_{S} = \\frac{n}{1 + (S/K_s)^n}\n", "d3ac7422dc8a8fbd13ec0249e72c1ef2": " \\frac{}{} \\Delta I = \\left (I_0 \\cos \\delta_0\\right) \\Delta \\delta. ", "d3ac7bed29e9a40e36f76d9d702dcd15": "\\Theta(x)=x-1-\\ln x\\ge 0", "d3aca444e2f7a2f3451c418866b2257a": "= \\sqrt{\\pi}\\,", "d3acbc37c35dc01d1b77ef6155aad7df": " h\\otimes y\\mapsto -y", "d3ace6c669eb0943f2001fc33e68e73c": "\n\\begin{align}\nN! & = \\int_0^\\infty e^{-N z} \\left(N z \\right)^N N \\, dz \\\\\n& = N^{N+1}\\int_0^\\infty e^{-N z} z^N \\, dz \\\\\n& = N^{N+1}\\int_0^\\infty e^{-N z} e^{N\\ln z} \\, dz \\\\\n& = N^{N+1}\\int_0^\\infty e^{N(\\ln z-z)} \\, dz.\n\\end{align}\n", "d3ace8867a31e68cf1861743662b3da0": "(a_{i,j})", "d3ad064aa69c69b883a78ab41e6e658c": " r = R", "d3ad0f38ff1e7d2813ec2704e409640a": "x+y=0", "d3ad194a4ac030a6c7ea1f963fdcc7b8": "E^{d-4}", "d3ad71161d87b46f5419a285ac9cfed2": "C(t,\\omega) = \\dfrac{1}{4\\pi^2}\\iint M(\\theta,\\tau)e^{-j\\theta t-j\\tau\\omega}\\, d\\theta\\,d\\tau", "d3ad73a490bec7adab1cb5f72eaa3018": "_c", "d3ada459cb3433536b8e1e031d539b40": "B = \\{a, b\\}", "d3adb02c50d6c31e0bea195b2a34c071": "|\\uparrow_x \\rangle, \\; |\\downarrow_x \\rangle", "d3adbf7be019c37ab7ff9a68a4391d76": "\\mathbf{a} = \\frac{\\mathrm{d}v}{\\mathrm{d}t}\\mathbf{u}_\\mathrm{t}(s) + v\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mathbf{u}_\\mathrm{t}(s) = \\frac{\\mathrm{d}v}{\\mathrm{d}t}\\mathbf{u}_\\mathrm{t}(s)-v\\frac{1}{\\alpha}\\mathbf{u}_\\mathrm{n}(s)\\frac{\\mathrm{d}s}{\\mathrm{d}t} ", "d3ade12fdb606b2a3159088af7839155": "f\\;'", "d3ade8a400e4411fa67100fdf3f388de": "\\R^n-f(\\partial\\Omega)", "d3ae49564ec54fbb8c1547f06f1b7fdd": "1 < r < N", "d3ae4fa631f0dcb7f1331f6478808a0c": "A^\\text{T}", "d3ae697c9b643b4f000ac8d0b53d8db8": "q = \\begin{bmatrix} u \\\\ d \\end{bmatrix} ,", "d3ae800bb9552512caafb1f814ae3cd1": "A_\\lambda := \\bigcap_{\\alpha < \\lambda} A_\\alpha", "d3ae8e5aba09a2d830b8fc3b83811bbe": "\\mathbf{k} = k_x \\mathbf{x} + k_y \\mathbf{y} + k_z\\mathbf{z} ", "d3aeb03e0a9cb45154f8bf95a234012e": "S_{400}", "d3aeef0055c7ec3a8acd700daa525be9": "(x-1)(x^{10} + x^9 -x^7 -x^6 -x^5 -x^4 -x^3 +x +1) = 0 ", "d3af2550dd0a8acbce41fa967f075c81": "SU(N_f-N_c)", "d3af4e0341788858dfb6dc584d14b18f": "f(x) = \\sin(x)", "d3af8ba1cb9bb1a77f96b1beeced1697": " d\\tau ", "d3af8d90c272672fb5ef4a525f77b97a": "f(r,\\theta)=r \\sin(2 \\theta)", "d3afa8f7cbae373854e2fe2a48a59287": "c=c_\\mathrm{g}", "d3b0252215be317a932f99649d38cdd9": "\\mathbf{T} = a_i b_j \\mathbf{e}_{ij} \\equiv \\sum_{ij} a_i b_j \\mathbf{e}_i \\otimes \\mathbf{e}_j \\,,", "d3b0b6fb3d0d4898912349444b5d6e2a": "DGS_{\\sqrt{2n}\\eta_\\epsilon(L)/\\alpha}", "d3b0f2a509eae7c65094609f36c192e4": "\\left(\\frac{\\omega}{c}\\right)^2 - \\mathbf{k}\\cdot\\mathbf{k} = \\left(\\frac{mc}{\\hbar}\\right)^2 \\,.", "d3b1300e7fe43a49e98d2c288d6a650d": "\\Box = \\nabla_\\alpha\\nabla^\\alpha", "d3b13f312190e397e61344b02ca3660b": "\\mathrm{^{252}_{\\ 98}Cf\\ \\xrightarrow {(n,\\gamma)} \\ ^{253}_{\\ 98}Cf\\ \\xrightarrow [17.81 \\ d]{\\beta^-} \\ ^{253}_{\\ 99}Es}", "d3b2045b6c83415e8b72e9a15d2ce0f6": "(u,v,w)", "d3b2123f862db4f4c0c94a53b2b53c51": "L[y]=\\frac{y\\circ t-y\\circ -t}{2}", "d3b232b66d5ed5e9c123280726f5d9d9": "\\theta=0.282", "d3b23eb257fc6def40b080d4292790aa": "\\nabla \\cdot (\\vec u \\times \\vec v) = \\vec v \\cdot (\\nabla \\times \\vec u) - \\vec u \\cdot (\\nabla \\times \\vec v ", "d3b251172c0e5a56e3f729320b1170ef": "s_0 = x_0 + x_1 + x_2,\\,", "d3b27d69ece6c376c8b792fd284b091f": "\\scriptstyle \\mathbf{S}=\\textrm{diag}(s_1,\\ldots,s_{\\min(N_t, N_r)},0,\\ldots,0)", "d3b2b83b2ad37ce49770b1a2dbab4a20": "\\langle x, y \\rangle = 0", "d3b33de8589bcf778756400f9755fddc": "\\textstyle\\mathcal{E} = - \\frac{d\\Phi_m}{dt} ", "d3b33f401ffc0ef5c3384e32deb8b9be": "\n\\begin{align}\nV(\\tilde\\beta) &= V(Cy) = CV(y)C' = \\sigma^2 CC' \\\\\n&= \\sigma^2((X'X)^{-1}X' + D)(X(X'X)^{-1} + D') \\\\\n&= \\sigma^2((X'X)^{-1}X'X(X'X)^{-1} + (X'X)^{-1}X'D' + DX(X'X)^{-1} + DD') \\\\\n&= \\sigma^2(X'X)^{-1} + \\sigma^2(X'X)^{-1} (\\underbrace{DX}_{0})' + \\sigma^2 \\underbrace{DX}_{0} (X'X)^{-1} + \\sigma^2DD' \\\\\n&= \\underbrace{\\sigma^2(X'X)^{-1}}_{V(\\hat\\beta)} + \\sigma^2DD'.\n\\end{align}\n", "d3b4347562a8e513e5f895972c2809b3": "\\sum_{m=2}^{\\infty} \\sum_{k=2}^{\\infty}\\frac{1}{m^k}\n=\\sum_{m=2}^{\\infty} \\frac {1}{m^2} \\sum_{k=0}^{\\infty}\\frac{1}{m^k}\n=\\sum_{m=2}^{\\infty} \\frac {1}{m^2} \\left( \\frac{m}{m-1} \\right)\n=\\sum_{m=2}^{\\infty} \\frac {1}{m(m-1)}\n=\\sum_{m=2}^{\\infty} \\left( \\frac {1}{m-1} - \\frac {1}{m} \\right) = 1 \\, .", "d3b497b557aadb1006e1fdfbb5f403b9": "(\\mathbf v \\cdot \\nabla) f = \\left (v_x \\frac{\\part}{\\part x}+v_y \\frac{\\part}{\\part y}+v_z \\frac{\\part}{\\part z} \\right )f = v_x \\frac{\\part f}{\\part x}+v_y \\frac{\\part f}{\\part y}+v_z \\frac{\\part f}{\\part z} ", "d3b4d3e72b6fc70ef232f2869ebb81d3": "\\tfrac{\\lambda}{2(\\lambda + G)}", "d3b5072aad292997a4f675c827504940": "R_{13} = \\phi_{13}(R)", "d3b56a77a11b9017bc7b4b4897686edb": "e^{-\\frac{E}{k\\,t}}", "d3b56ccd3f78e9afd61ce043d957e16c": "RTC = \\frac{\\Delta Br}{Br \\Delta T} \\times 100", "d3b586b2bb4407c6508d13fe737e2c01": "(\\varepsilon_1, \\mu_1)", "d3b58e1ec5992329c2859c583e3d76d8": "F_j(x)", "d3b642d7e16e35f12cbf3abd00282ea0": "GF\\left(p^t\\right)", "d3b6448b1dd415597ed008ee4b4e1f44": "S \\choose C+1", "d3b644b0db506a5c2cb4ccbc9a823b94": "\\scriptstyle v({\\mathbf P}_1), v({\\mathbf P}_2), v({\\mathbf P}_3), v({\\mathbf P}_4)", "d3b678eb7f389bcdf17cb1338baca047": "d\\mu=\\frac{4 dx\\,dy}{(1-(x^2+y^2))^2}=\\frac{4 dx\\,dy}{(1-|z|^2)^2}.", "d3b6a2c0cfedddf006d1cc9571a6d2cc": "\\textstyle 2l", "d3b6d1b8c317552105de1f7e31cd5553": " \n\\mathcal{A}_2 = a_{20}\\partial_x^2 + a_{11}\\partial_x\\partial_y + a_{02}\\partial_y^2+a_{10}\\partial_x+a_{01}\\partial_y+a_{00}.\n", "d3b72edda6fd08a658eb82765bfb0823": "\nR(f) = h(\\lVert f \\rVert).\n", "d3b748127a8c5556a9e58b960e8cfa13": "\\alpha\\left(\\frac{\\beta-1}{\\beta+1}\\right)^{1/\\beta}", "d3b75063ca0fcd3c4e6c459f9a68f23e": "\\mathbf{F} = q\\left(\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}\\right) \\,\\!", "d3b760826a331c0af1842aa02a3c7f9f": "\\! v_{pec}", "d3b7aaadb09d41529f9f4e2617673873": "a \\triangleright b = {}^a b", "d3b82a2a547aaa5297396fae7a340ace": " (\\lambda q.(\\lambda p. q\\ p)\\ (\\lambda f.\\lambda x.f\\ (x\\ x)))\\ (\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f)) ", "d3b8743fc8eaa3a2a5036a20c933c2ac": "(21)\\quad\\quad \\frac{\\rho_2}{\\rho_1} = \\frac{\\frac{p_2}{p_1} (\\gamma+1) + (\\gamma-1)}{(\\gamma+1) + \\frac{p_2}{p_1}(\\gamma-1)} = \\frac{u_1}{u_2}", "d3b880e746f6d55e9f8c8c1308a5df6b": "\\scriptstyle a = (-r)c + d = (-r)c + d - r + r = (-r)(c + 1) + (d + r)", "d3b89f13441118fbe8212a9ae43cf91c": " \\mathbf{X} = \\mathbf{A} \\, \\mathbf{Y} ", "d3b93db59603b29c0f4b4050dc0f92cc": " \\textbf{b}\\,[\\,\\textbf{x}(t),\\textbf{u}(t),t\\,] \\leq \\textbf{0},", "d3b9630cbd207031c2c13600ca6e34a2": "\\operatorname{DP}_2", "d3b9729cfa16125e023f5e3bcbeb73b7": " [T \\psi](x) = f(x) \\psi(x) \\quad ", "d3b97840810945a8e0fdeb765b15a658": "a = \\log(t_2/t_1) / \\log(n_2/n_1)", "d3b9a3a8280c482a6fa48a44dae4c4e2": "\\omega = df + k\\cdot d\\theta,", "d3b9da149d00dc1abdcbdcfe0d3a1c60": "\\ln K = {{-\\Delta_\\mathrm{r} G^\\ominus} \\over {RT}}", "d3ba24d31d5ea464c9e17954ae1d1481": "= u_m(t) \\cdot e^{i(\\omega t + \\phi)}\\,", "d3ba5dd0a5ff32fbcdd74447b2039f25": "x = {hc_sn\\over 2LkT}", "d3ba71cb10af0a14428d92dae3686b56": "X(t) = \\sum_{n=0}^N \\pi_n(t).", "d3ba804966f075a885e0201ffd5153b3": "-R_0^0=\\frac{\\ddot a}{a}+\\frac{\\ddot b}{b}+\\frac{\\ddot c}{c}=0.", "d3baa1d207109ccc49fcb553fbb1b6e5": "\\begin{align}\nG_n(z) &=z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\\cdots + g_n(G_{n-1}(z)) \\\\ \n&= z+\\frac{1}{\\rho \\cdot 1^2}\\sqrt{z}+\\frac{1}{\\rho \\cdot 2^2}\\sqrt{G_1(z)}+\\frac{1}{\\rho \\cdot 3^2}\\sqrt{G_2(z)}+\\cdots +\\frac{1}{\\rho \\cdot n^2} \\sqrt{G_{n-1}(z)}\n\\end{align}", "d3baaa3204e2a03ef9528a7d631a4806": "x[n]", "d3babb2cadad0fd6c325e018654d34d4": " \\mathsf{R} = \\sum_{i=1}^n \\mathsf{W}_i = \\sum_{i=1}^n (\\mathbf{F}_i, \\mathbf{P}_i\\times\\mathbf{F}_i). ", "d3babfe112b0d3894939f8bf6746cbe0": "b_{2}^{*}= b_{2}- \\mu_{2,1}b_{1}^{*}= \\begin{bmatrix}-1\\\\0\\\\2\\end{bmatrix}- \\frac{1}{3}\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix}=\\begin{bmatrix}\\frac{-4}{3}\\\\\\frac{-1}{3}\\\\\\frac{5}{3}\\end{bmatrix}.", "d3bad988460b8435675a365f9c32527e": "\\omega\\wedge\\eta=\\frac{(k+m)!}{k!\\,m!}\\operatorname{Alt}(\\omega\\otimes\\eta)", "d3baf623f6b43262f03873fbcd2f2a34": " \\frac{\\mathrm{d}\\mathbf{r}}{\\mathrm{d}t} = \\frac{\\partial}{\\partial \\mathbf{p}}\\left ( \\sqrt{(\\mathbf{P}-e\\mathbf{A})^2 +(mc^2)^2} + e\\phi \\right ) \\,\\!", "d3bb09d922a719799eab08fcae136ba9": "c=g^x h^r", "d3bb39117e214eb4e7e868a793126429": "[c]", "d3bb40226a7bca2f559591b6c0847a3d": "R(3,t)", "d3bb7b6b463302049b13ec0bad087280": "\n\\begin{array}{lcl}\n\\text{Maximum stress failure criterion:}-X^\\prime_{\\mathrm{f}} < \\sigma_1 < X_{\\mathrm{f}}\\\\\n\\text{Quadratic failure criterion: }\\displaystyle\\sum_{j=1}^6\\displaystyle\\sum_{i=1}^6 F_{ij}\\sigma_i\\sigma_j + \\displaystyle\\sum_{i=1}^6 F_i\\sigma_i = 1\n\\end{array}", "d3bbac461694b35db1ab93c674ab105b": "\\alpha_{(X, d, \\mu)}(r) := \\sup \\{ 1 - \\mu(A_{r}) \\mid A \\subseteq X \\text{ and } \\mu(A) \\geq \\tfrac{1}{2} \\},", "d3bbeaf01138746785ec823bddfd483d": "f = bc", "d3bd0109e38499c9785b0a68b8c0a44f": " \\beta(g) \\propto g ", "d3bd45674554c00e92fefa7419ff734c": "\\frac{\\lambda_X}{\\lambda_Y}", "d3bde20ce3f7de639d1eca10e478beb2": " E(x;k) = \\int_0^x \\frac{\\sqrt{1-k^2 t^2} }{\\sqrt{1-t^2}}\\, dt.", "d3be09e249c542151034064dabc52b77": " t \\approx \\frac{70}{r} \\times \\frac{198}{200-r}", "d3be1869cbb8b275a4078a1aaefc19cf": "(e^{\\sigma^2}\\!\\!+2) \\sqrt{e^{\\sigma^2}\\!\\!-1}", "d3be781fab052ac834ec86a3546287ba": " p= \\tfrac{1}{2}", "d3bec921ef0f1d4c1266a76dc1a29884": "\\mathcal{C}=\\mathcal{C}^{\\dagger},", "d3bef0229f6ef828491722e33b474b43": "B_x^{(-n)} + B_y^{(-n)} = B_z^{(-n)}", "d3bef4c9ed89271ab829e0e0d6e4b7e0": "\\Lambda\\in\\mathbb{R}", "d3befc7c8a4bb8634f32820dbeb8fb91": "c_0 = \\hat X [0],", "d3bf220ccaef5a0575a4554a78788d28": "\\frac{d \\bar \\rho_{eg}}{dt} = - \\left( \\frac{\\gamma}{2} - i\\delta \\right) \\bar \\rho_{eg} + \\frac{i}{2}\\Omega^*(\\rho_{gg} - \\rho_{ee})", "d3c0878b530a48d2633bc2b9e11ecf80": "\n\\tau = \\int \\frac{dt}{y^{2}}\n", "d3c0da737f8894cbce86f8659efaf4cb": " a_n < (c + \\varepsilon)b_n ", "d3c0ef1b6afcbef9f152a4cf5e759411": "\n\\tfrac{ 1}{20},\\,\\tfrac{ 2}{20},\\,\\tfrac{ 3}{20},\\,\\tfrac{ 4}{20},\\,\n\\tfrac{ 5}{20},\\,\\tfrac{ 6}{20},\\,\\tfrac{ 7}{20},\\,\\tfrac{ 8}{20},\\,\n\\tfrac{ 9}{20},\\,\\tfrac{10}{20},\\,\\tfrac{11}{20},\\,\\tfrac{12}{20},\\,\n\\tfrac{13}{20},\\,\\tfrac{14}{20},\\,\\tfrac{15}{20},\\,\\tfrac{16}{20},\\,\n\\tfrac{17}{20},\\,\\tfrac{18}{20},\\,\\tfrac{19}{20},\\,\\tfrac{20}{20}\n", "d3c0fc86007d704f43428b3be9eef296": "\\textstyle R_j = a (j - (N-1)/2)", "d3c1723c3c29661c6e7862faa12463f1": "c=\\left(4.0-u-10.0v \\right)/v", "d3c17b7b5bb7f30e97b36b902daf1314": "\\widehat{\\mathbf{v}}", "d3c1a77847d69b944c2aaa34a95709fc": "\n\\begin{bmatrix}\n\\mathbf A, & \\mathbf A\n\\end{bmatrix}\n^{+} \n= \\frac{1}{2}\n\\begin{bmatrix}\n\\mathbf A^{+} \\\\ \\mathbf A^{+} \n\\end{bmatrix}\n\\neq\n\\begin{bmatrix}\n(\\mathbf P_A^{\\perp}\\mathbf A)^{+}\n\\\\ \n(\\mathbf P_A^{\\perp}\\mathbf A)^{+} \n\\end{bmatrix}\n= 0\n", "d3c294d8f2fa531ff2d5fa92d54cb56d": "\\mathbf{Q} (\\sqrt{21})", "d3c2ac450bd3d7a1b187218bac81cd0b": "\\mu_i = \\frac{100Z(Z-1)h(Z,w_i)}{g(Z, w_i)}\\,", "d3c2b2bfc5ff8fd6567f13ffc7a7a4d4": "P^2=(a_1 \\cdot a_{n+1})^{n+1}", "d3c2d6d8ee617df3481fb565f25c1e30": "\\Gamma(t):= \\varphi(\\gamma(t)) \n=(\\Gamma_{1} \\oplus \\Gamma_{2} \\oplus \\Gamma_{3} \\oplus \\Gamma_{4})(t)\n", "d3c2e3187bf1c7761776a50542e7330a": " V_{ss \\ wake} = V_{wake} \\left( \\frac{1} {1 - e^{-\\tau} e^{j\\delta}} - \\frac{1}{2} \\right) ", "d3c46d22d028262aed810328269efe53": "\\Delta(u_{ij}) = \\sum_k u_{ik} \\otimes u_{kj}.", "d3c479d83d3896cf0dd9f9d4c8c90e2c": " (a**b)**(c*b)=(a**c)**(b**c) ", "d3c4a492cb4da08440d80953d9ed3f15": "\n w_{f} (F) := \\sup_{s, t \\in F} | f(s) - f(t) |\n ", "d3c4c8db40e78629b2574619b1c6ef2c": " S_{(2,2,0)} = e_2^2 - e_1 \\, e_3", "d3c4f81f0e8bb6cd2aa178a04daa0bee": "G_{dBd} = 10 \\cdot \\log_{10}\\left(\\frac{G}{1.64}\\right)", "d3c4fce46e8a8c141910899beabee6f0": "\\lim_{x\\rightarrow\\infty}f(x)", "d3c52335595604183e97bf331368fa9c": "\\{\\theta,\\phi\\}", "d3c5c9ecf7a3609016eaefcd2f89e3dd": "1+z=\\frac{1}{\\sqrt{1-\\frac{2GM}{rc^2}}},", "d3c5ce2c1191ca89f1b6eff6a916f088": "\\scriptstyle n ", "d3c5e5679d08aa3ae1eb7167811a48c6": "L\\left ( \\mathrm{d} I/\\mathrm{d} t \\right )=-NV\\,\\!", "d3c65aa9bcf1bb0ec1bbbabbe9bd7f01": "g^{(2)}=1", "d3c66cb17ff00fa149bd424487fa4cd1": "p(x) = \\sum_{i=1}^n p_i \\delta_{x x_i}.", "d3c692e74ebfcfeb97255c3dcbffe780": " K'_n ", "d3c6e847db70f2308f64d3f378ac82e9": "\n \\lim_{n\\to\\infty}\\Pr\\big(|X_n-X| \\geq \\varepsilon\\big) = 0.\n ", "d3c718725461f606723acba5c343f075": "E = \\frac{1}{\\sum\\limits_{i\n", "d3cdec7e62069db846d3c4c8ddb2c92e": "(\\forall t)", "d3ce6fe62d5aa1b2a3d0082ac1705145": "AB \\ = \\ v\\delta t", "d3cea25cd1653fc226550eb2277a51c4": "\\alpha(T_r) = \\left[ 1+c_1 \\left(1-\\sqrt T_r \\right) +c_2 \\left(1-\\sqrt T_r \\right)^2 +c_3 \\left(1-\\sqrt T_r \\right)^3 \\right]^2", "d3cea6655a3466d66d69779316cb0b6a": "c^2\\left(\\rho_G D\\Psi_G-\\rho_L D\\Psi_L\\right)=g\\Psi\\left(\\rho_G-\\rho_L\\right)-\\sigma\\alpha^2\\Psi,\\,", "d3ceddc1affa0bcb5eaf3ee8ebec7bdb": "1 \\le s,t \\le n-1.", "d3cee16be366591d65bba2a8ead52c18": " \\varphi : \\mathbb{R}^n \\to \\mathbb{R} ", "d3cee66416b07976ad36444a4d975512": "\\Pi_D=\\Pi - \\Pi_0 - T \\,", "d3cf14f851323eb2bc99cf65805f1159": "\nV(\\mathbf{r}) = V(\\mathbf{0}) - \\sum_{i=1}^3 r_i F_i ", "d3cf349ebcef899ee5d449178e08abb4": "\\int_X p(x,\\theta) \\,dx = 1", "d3cf5bb5d2a5a0da7a967a6601239ebd": " n_2 m_2 \\equiv 1\\bmod m_1 ", "d3cfa1356bcb9644c4617a224762fc06": "\\pi_{i+1}", "d3d00bb64f52ab1483c6c8bfaef7b2d0": "\\Theta (u, v, s)", "d3d02e12ae20766cfd5e398b4a3e94ff": "\\begin{align}\n 1 &{}\\mapsto 2,\\\\\n 2 &{}\\mapsto 4,\\\\\n 3 &{}\\mapsto 6.\n\\end{align}", "d3d07bad058ba9633697b8f9f90b3202": "\\begin{align}\n c(\\omega,m) &{}= \\sgn(\\cos \\omega) |\\cos \\omega|^m \\\\\n s(\\omega,m) &{}= \\sgn(\\sin \\omega) |\\sin \\omega|^m\n\\end{align}", "d3d084b6781b7b8dcfe1b291524e06e5": "\\phi_1(\\phi_{\\omega+1}(\\phi_2(0)) + \\phi_\\omega(0)^{\\phi_3(0)}42)^{\\phi_1(1729)\\,\\omega}", "d3d090691ef33dcf9c0f62cb04d4d377": "f: \\oplus_1^n U \\to \\oplus_1^n U", "d3d09d1e513a93b840afbe68fb7f49b3": "\n 3~\\tilde{J}_2 + (\\eta^2 - 1)~\\tilde{I}_1^2 = \\eta^2\n ", "d3d0aca834dd016a0758e74cac216c67": "\nS_0(p)=\n\\begin{bmatrix}\n(I_x(p))^2 & I_x(p)I_y(p) \\\\[10pt]\nI_x(p)I_y(p) & (I_y(p))^2\n\\end{bmatrix}\n", "d3d0aea9e1d14b7c1fa4ce7977683083": "E[\\pi_G]=p B + (1-p) D", "d3d0b1d65f93a47f1a1f1ad943c52d1f": "\\Gamma^i=\\partial_t q^i(t,\\overline q^j)", "d3d0e7795a11cecf0c776afc375ca7ec": "\\sum_{k=1}^{135}\\ 2\\left (100 - \\left(\\frac{504-(k+1)}{504} \\cdot 100 \\right )\\right )(k+5) + ((k+1)\\cdot 15) \\approx 490921.43", "d3d168c9e9d152e428a264cd2622bc7f": "\\mathbf{x} = \\begin{bmatrix} x_1, x_2, \\dots, x_m \\end{bmatrix}^{\\rm T}", "d3d1ffdb5509bd7867e3dad117f371b6": "\\textstyle{\\in(0,2)}", "d3d2034dca50e9e002977fc2e731b83c": "I_i=\\frac{2e}{h}\\sum_{i}E_i\\frac{1}{1+e^{\\frac{E-E_f}{k_BT}}} ", "d3d2090a81f7c0320a0207bc81fdec09": "R_i=\\frac{1}{h_iA}", "d3d2114c7274cfefb8ec0da3c4365071": "\\mathit{\\Iota\\Kappa\\Lambda\\Mu\\Nu\\Xi\\Pi\\Rho} \\!", "d3d215b3f2166ab9fd4cbebd6ae60a2f": "S=\\sum_{i=1}^{n}{r_i}^2", "d3d2477fa0ab6d6c6c9ac07ce426d40f": " \\overline{vm_{pq\\mu\\gamma}} = \\frac {vm_{pq\\mu\\gamma}}{A * I} ", "d3d27a0e51919b701aee23e5c617ca9a": "\\tilde{A}(\\boldsymbol{U}_i,\\boldsymbol{U}_{i+1})", "d3d27f9199943953110158980b5e28ee": "\\frac{T_{\\text{f}}}{T_{\\text{i}}}=\\left(\\frac{p_{\\text{f}}}{p_{\\text{i}}}\\right)^{\\frac{\\gamma -1}{\\gamma}}.", "d3d28e7148101e1cc969fbdc3bd27277": "\\frac{v_\\text{s}}{c} \\ll 1", "d3d3115127cc0141f42a3559bdce6283": "\\bar{c}", "d3d3380f3c9b113ddd6f13ad8100c398": "\\mathit{L}\\,", "d3d357752f64708dd7347e0875e8669c": " H(C,f) = \\lim_{n\\to\\infty}\n\\frac{1}{n} H(C\\vee f^{-1}C\\vee \\ldots\\vee f^{-n+1}C). ", "d3d381c65c7f06ebf1e52310675f4925": "V_{i,j} = \\alpha_i^{j-1} \\, ", "d3d3821df7c02c5d2b71622e6eba98e7": "[R]=\\frac{(1-ee)(1-c)}{2}", "d3d3a389b9e8f0433065f4dc8d63dff4": "V_{P}^{(1)}", "d3d3b6a9d3f961ba6665a410b26674c2": "\\mathbf{x}'_i\\boldsymbol\\beta", "d3d3ddda55ca1ebe8118d07ab91ac61f": "\\Delta t > \\Delta \\tau\\,", "d3d4799807f0440ad57a1c764340b5a3": "\\lim_{\\underset{h\\in\\mathbf{R}}{h\\to 0}} \\frac{f(z_0+h)-f(z_0)}{h} = \\frac{\\partial f}{\\partial x}(z_0).", "d3d48d51a71675ba9a24f73fa9f23095": "\n\\overline{\\phi} = G \\star \\phi .\n", "d3d4a217c0fafaf0c0b0f37e47661dd0": "m_7(x) = x^4+x^3+1.\\,", "d3d50f8dcbc65c9dca345a2601f4d1da": " M7 = \\frac{ \\sum_{ i = 1 }^K \\sum_{ j = 1 }^L | R_i - R | }{ 2 \\sum R_i }", "d3d5270e342f820dcbeed3a5d938a2c3": "\\tilde\\psi", "d3d52bb3dd11ad14837fb6779b64b0e6": "\\displaystyle{\\partial_n u(z + a\\mathbf{n}_z)={d\\over dt} a(z+t\\mathbf{n}_z)|_{t=a}.}", "d3d5916af94ad10c511c9e7d3bfa4088": "X = B \\mathbb{G}_m", "d3d5bd507e0066cb6729f26bf3c92735": "\n \\delta K = \n -\\int_0^T \\left\\{ \\int_{\\Omega^0} \\left[\n J_1\\left(\\ddot{u}^0_{\\alpha}~\\delta u^0_\\alpha \n + \\ddot{w}^0~\\delta w^0\\right) \n - J_3~\\ddot{w}^0_{,\\alpha\\alpha}~\\delta w^0\\right] ~\\mathrm{d}A\n + \\int_{\\Gamma^0} J_3~n_\\alpha~\\ddot{w}^0_{,\\alpha}~\\delta w^0~\\mathrm{d}s\n \\right\\}~\\mathrm{d}t \n", "d3d5f66537b7c0f31212bde7e37b6482": "w_{ij}= - 2 \\cdot J \\cdot S_i \\cdot S_j ", "d3d6c4f4565e2ba8c76e66b606a8dab0": "p_k,\\gamma_k,\\xi_k", "d3d6ffef489a74ab812143b2a3123e51": "\n\\phi_Y(u) =\\exp \\left( i\\gamma u- \\frac{1}{2} \\sigma^2 u^2 +\n\\int_{-\\infty}^\\infty\n(e^{iux}-1-iux1_{|x|\\le 1} ) \\, \\nu(dx) \\right),\n\\sigma\\ge0,~~\\gamma\\in\\mathbb{R}\n", "d3d748b086a5e964dd9be67cb514e272": "u(x,t)=\\frac{1}{\\sqrt{4\\pi kt}} \\int_{0}^{\\infty} \\left[\\exp\\left(-\\frac{(x-y)^2}{4kt}\\right)+\\exp\\left(-\\frac{(x+y)^2}{4kt}\\right)\\right]g(y)\\,dy ", "d3d796a123cc547434c241005678e9e7": " \\boldsymbol{x} \\in \\mathbb{R}^n ", "d3d83200e2b90a6621ced9646bc39a2b": "I_k / I_{k+1} \\subseteq Z( L/ I_{k+1} )", "d3d836774384648dc36cc3537312b51b": "g_{0j}", "d3d85921b8a6c7a2e0881507a6f6db14": " \\Delta (I)\\subset I\\otimes TV+TV\\otimes I.", "d3d867cec4c59dc46ded9786784251af": " \n r^2 = \\mathbf r \\cdot \\mathbf r \n", "d3d86cf45383a43f333570908f4295dd": "\\overline{n_\\text{piv}} \\notin n_\\text{clause}^\\text{right}", "d3d878edb66a07fbad2e9ae84560b5ae": " \\mathrm{MSD}\\approx t^{\\frac{1}{2}} ", "d3d88f7ab46639816cd58d77cdafb112": " f(0) = (2\\pi)^{-n/2} \\int_{{\\mathbf R}^n} {\\mathcal F} f(t) \\, dt.", "d3d8d461d545722f43e28350f33f0c22": "\\mathbf{Z}/2\\mathbf{Z} * \\mathbf{Z}/2\\mathbf{Z}", "d3d8ecba5329706e0eaf7ab78bbb6779": "\\left( \n\\begin{smallmatrix}\n0 & 1 & 1 \\\\\n1 & 0 & 1 \\\\\n1 & 1 & 0 \n\\end{smallmatrix}\n\\right)", "d3d8f00acfe885b42e7409bcf155ee19": "\\beta_1,\\ \\beta_2,", "d3d8f6da5c781ff35e137b3e3bdc2c28": "\\frac{\\beta}{2\\alpha\\Gamma(1/\\beta)} \\; e^{-(|x-\\mu|/\\alpha)^\\beta}", "d3d8fb8cb005662218ef3b4c903f14b6": "\\lceil N/2\\rceil", "d3d8fd35e409aef44b93110e9adab162": "\\operatorname{d} U = \\left ( \\frac{\\partial U}{\\partial S} \\right )_V \\operatorname{d}S + \\left ( \\frac{\\partial U}{\\partial V} \\right )_S \\operatorname{d}V", "d3d90ddab8f30a965332c8d8bf0ffbba": "R_{ab} = {R^m}_{a m b}", "d3d911d024543627aabff4066a96c275": "\\begin{align}\n x(u,v) &= -\\frac{2}{15} \\cos u (3 \\cos{v}-30 \\sin{u}+90 \\cos^4{u} \\sin{u} \\\\\n &\\quad -60 \\cos^6{u} \\sin{u}+5 \\cos{u} \\cos{v} \\sin{u}) \\\\\n y(u,v) &= -\\frac{1}{15} \\sin u (3 \\cos{v}-3 \\cos^2{u} \\cos{v}-48 \\cos^4{u} \\cos{v}+ 48 \\cos^6{u} \\\\\n &\\quad \\cos{v}-60 \\sin{u}+5 \\cos{u} \\cos{v} \\sin{u}-5 \\cos^3{u} \\cos{v} \\sin{u}-80 \\\\\n &\\quad \\cos^5{u} \\cos{v} \\sin{u}+80 \\cos^7{u} \\cos{v} \\sin{u}) \\\\\n z(u,v) &= \\frac{2}{15} (3+5 \\cos{u} \\sin{u}) \\sin{v}\n\\end{align}", "d3d915f1c26e56a46c42d7ca6675abe2": "\\phi_g(h) = ghg^{-1} \\,", "d3d93a3c9d790555eb6ded5c8dc6010d": "\\sum_{n=1}^\\infty \\mu(n)\\,\\frac{q^n}{1-q^n} = q.", "d3d94384f607998c37d150d481c0e67b": "c'(N,P)=c(N,P)+h(P)-h(N)", "d3d9446802a44259755d38e6d163e820": "10", "d3d94d0274da434a451114471539e0b2": "0\\le f\\le 1", "d3d995f8f61d9d886112eaed0b5e2710": "\\operatorname{sink}[G\\ \\operatorname{sink}[(\\lambda E.H)\\ Y, X]] ", "d3d9c0e15ee537644ceedaf473644883": "k = A e^{-E_a/(k_B T)}", "d3d9eb434708fe6e931656de61cb7206": "\\scriptstyle{FX/\\mathrm{SO}(1,3)}\\to \\scriptstyle{X}\\,", "d3da64586ce142dd4286501e82a001c8": "a'=-\\frac{1}{r}\\mathbf{r}\\cdot\\mathbf{Q}sc_1-\\alpha_j'\\big[as^2c_2+2bs^3\\bar{c}_3+\\frac{1}{2}\\gamma s^4c^2_2\\big]", "d3da77dcc611d412dbe7b75a1da273db": " a_1, \\dots , a_m \\in R ", "d3daa8c140a59a1b139191174e54d65d": "75\\cdot 23", "d3db00a1b6d4f10fa66f64cf49c35a86": "S_{[a_1 \\dots a_p]} = \\frac{1}{p!} \\delta_{a_1 \\dots a_p}^{b_1 \\dots b_p} S_{b_1 \\dots b_p} .", "d3db0fb3bb7460b35d25ad28714ab818": "-\\sqrt{3}", "d3db349ad9390d36b4d563455f2dba30": " HRR = \\frac{P_1}{1-P_1} * \\frac{1-P_2}{P_2}", "d3db35add119698ade0d9a0376fd3153": "q^* = - \\frac 1 2 (q + iqi + jqj + kqk).", "d3db3f2891c830969eac43a0fa95b538": "K_0 = 0", "d3db40970f54965f9b882ed183c8f472": "\\tilde{C}(-1) \\quad \\mbox{and} \\quad \\tilde{C}(1)", "d3dba201408c7905869c092177c95836": "\\partial/\\partial\\theta_{i}", "d3dbc54567a8eb2d6fb6d30afb5dad20": "\\Delta G_{\\mathrm{m,mix}} = RT \\sum_i x_i \\ln x_i ", "d3dc346daadadaaa9c33594a828fd376": "f_{*}", "d3dc40ac13521ef2b014775d54f8c707": " 0 \\le x_1^k + x_2^k + \\cdots + x_N^k \\le n", "d3dc96e0ddceb7129d913d53e89ac812": "Z =\\{-1\\}\\cup\\{x\\in R: 0 < x < 1\\} = \\{-1\\}\\cup (0,1) ", "d3dcad92cd8dc319e3be7934f83fe08e": "\\begin{cases} \n-u''=f \\mbox { in } [a, b] \\\\\nu(a)=u(b)=0 \n\\end{cases}\n", "d3dcf429c679f9af82eb9a3b31c4df44": "BE", "d3dd157040bd575aecb813527fda5e9c": "f: A \\to B", "d3dd1d64d79b457764f040ac785f6845": "X(t)= V_0 * t + A*t^2\\,", "d3dd63ff460bd638caf19400de70d137": "\\hat{f}(\\xi) = \\int_{-\\infty}^\\infty f(x)\\ e^{- 2\\pi i x \\xi}\\,dx", "d3dd7032763de7500cf84647465ef8a9": "\\left\\{ \\begin{matrix}\n \\frac{1}{W} & \\mbox{if } 2n < W \\mbox{ or } 2(N-n) < W \\\\\n 0 & \\mbox{otherwise}\n \\end{matrix} \\right. ", "d3ddb23b6c7cc9f3d54adeea91aa0933": "\\gamma(i, j, v)", "d3ddf2d0ee30f22ae3f74dbd11a480a4": "n_1 = 70\\ ", "d3de135af9babd126be71c7246ce3c5b": "t_{l}=-\\bar{t}", "d3de2549c6dfe25f58193e827a430400": "\\sum_{1 \\leq i \\leq r} (a_iP_i) = \\sum_{1 \\leq j \\leq r} (b_jP_j).", "d3de48c5d8146c6f12deda734d7d7096": "\\ \\alpha(u)", "d3de61315c551763e2d2354ea400a809": "(\\Phi(n) \\land [(n:=n+1)*](\\Phi(n) \\to [n:=n+1] \\Phi(n))) \\to [(n:=n+1)*] \\Phi(n)\\,\\!", "d3de6179d292f6009b2e44e7400ebebe": "\n \\begin{align}\n & \\frac{\\partial^2 }{\\partial x^2}\\left(EI\\frac{\\partial \\varphi}{\\partial x}\\right) = q \\\\\n & \\frac{\\partial w}{\\partial x} = \\varphi - \\cfrac{1}{\\kappa AG}~\\frac{\\partial }{\\partial x}\\left(EI\\frac{\\partial \\varphi}{\\partial x}\\right) \n \\end{align}\n", "d3deb6e9543ab2331522673f8c7d67f8": "0\\leq x_1\\leq x_2 \\leq \\dots, \\leq x_N \\leq L", "d3dece59094df94b785cf8b179bb8aff": "\n\\underline{P}(Cl_t^{\\leq}) = \\{x \\in U \\colon D_P^-(x) \\subseteq Cl_t^{\\leq} \\}\n", "d3def7814e3b03b28a391a84567c3a3a": "\n\\begin{align}\n\\mathbb{P}(x \\mbox{ received} \\mid y \\mbox{ sent}) & {} = \\frac{ \\mathbb{P}(x \\mbox{ received} , y \\mbox{ sent}) }{\\mathbb{P}(y \\mbox{ sent} )} \\\\\n& {} = \\mathbb{P}(y \\mbox{ sent} \\mid x \\mbox{ received}) \\cdot \\frac{\\mathbb{P}(x \\mbox{ received})}{\\mathbb{P}(y \\mbox{ sent})}.\n\\end{align}\n", "d3df34939d30a32db3831e5e90276642": "{\\tilde{A}}_4", "d3df73d8c7fad42ae10a47182c0fc15c": "\\csc^2(x) - \\cot^2(x) = 1\\ ", "d3df9655ea4041f3b8afb760377ed355": "s_j = r_j \\, s_0", "d3dfcdde95864e94014d507711f4f8f0": "a_1+\\cdots +a_n=b_1+\\cdots+b_n.", "d3dfda81c7162758c665c9f463f092ea": " \\hat{A}_j \\psi = a_j \\psi .", "d3dfdaaa762496718235468dea5383ef": "\n\\boldsymbol{y}|\\boldsymbol{x} \\sim\\ \\mathcal{N}(\\boldsymbol\\mu_{Y|X}, \\boldsymbol\\Sigma_{Y|X})\n", "d3e0d4dc645b8d0a2799e247b0f57e9f": "\\xi = \\frac{3}{5} \\gamma \\int_{\\reals^3} \\rho^{5/3}(\\vec{r}) d^3\\vec{r}\\, + \\int_{\\reals^3} V(\\vec{r}) \\rho(\\vec{r}) d^3\\vec{r}\\, + \\frac{e^2}{2} \\int_{\\reals^3} \\frac{\\rho(\\vec{r})\\rho(\\vec{r'})}{|\\vec{r} - \\vec{r'}|}\\, d^3\\vec{r}d^3\\vec{r'} ", "d3e153a7bb9ab6d46591bcf369f12d9a": "s\\colon A\\rightarrow\\ V", "d3e1abc03c5ab6e5fa84fce429516a6b": " \\Delta f ", "d3e1ae71a53ebd89a869e6ecd31180d3": "\\scriptstyle \\hat{\\boldsymbol\\theta}", "d3e1dece425cff0612de283f8f1db71a": " \\alpha(z) = \\theta - \\beta ", "d3e24e6d55ec45bff18f5ca808a448e1": "m/2", "d3e28c93c73e6940258da1bc04777e05": "\\text{E}[-e^{-aW}] = - \\text{E}[e^{-a [x'r + (W_0 - x'k) \\cdot r_f]}] = - e^{-a[(W_0 - x'k)r_f]}Ee^{-a \\cdot x'r} = - e^{-a[(W_0 - x'k)r_f]}e^{-a \\cdot x'\\mu + \\frac{a^2}{2}\\sigma^2}", "d3e29cca6c482a3ee4b14491953d27c8": "\\ \\sum_{j=0}^Q a_{j} y[n-j] = \\sum_{i=0}^P b_{i}x[n-i]", "d3e2cc28935fe4881ebadb2a30241f03": "\\epsilon_Y = \\frac{\\%\\ \\mbox{change in demand}}{\\%\\ \\mbox{change in income}}", "d3e2dfc98e801bcf442a273358bbf959": "R_S=R_H \\left(1+\\frac{\\cos(\\theta)-\\cos(\\alpha)^2\\cos(\\theta)-\\cos(\\alpha)\\sin(\\theta)\\sin(\\alpha)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "d3e306f62157fa376aaa5deb3d20c064": "1s \\sigma_{\\rm g}^{}", "d3e3f8b48bbc3ac0cb13220886a1bc06": " 2 d_\\text{f} - d = 2 - \\eta\\,\\!", "d3e42288af19a17f277818347916e547": "\\sigma^{2}_{s}", "d3e45f5109e373c5521b52cabdad8d29": "\\mathsf{ACA}_0", "d3e4fd28d192d9b45e04ed7efdf820d4": "Q(h) = \\frac{f(a + h) - f(a)}{h}.", "d3e4fd8742baa28941afdde63ac3e2f8": "1-e^{-bsx}, \\beta=1", "d3e54984f0f1313ad2091859d367ebf2": "\\max_{S_k} \\min_{x \\in S_k, \\|x\\| = 1}(Ax,x) = \\lambda_k.", "d3e55a1d73b6e9ab08c379a9a0771844": "A_{\\alpha}\\in\\mathcal{P}(\\kappa)\\setminus\\bigcup_{\\xi<\\alpha}f(A_{\\xi})", "d3e5d38cf46e2a53c061de819b322e88": "T_f", "d3e60e180d5b548d7cfd276358f7ce04": "L=\\frac {AS^2} {120 + 3.5S}", "d3e6b267695ceba92813bd2ba9891644": "\\chi_{\\rho}(g) = \\mathrm{Tr}(\\rho(g))\\,", "d3e6b35820f6de0a3a57b810f8601e5f": "\nc_h = \\frac{1}{N}\\sum_{t=1}^{N-h} \\left(Y_t - \\bar{Y}\\right)\\left(Y_{t+h} - \\bar{Y}\\right)\n", "d3e73c41e3dd96e9f71197cf2ed7ef16": "\\Delta_Y = Y_{11} Y_{22} - Y_{12} Y_{21} \\,", "d3e7a1296567819731d1b7fb869f19bf": "\\delta \\mathbf{r} = \\mathbf{r} - \\boldsymbol{\\rho},", "d3e7af2412c1e3ce09d63aa0f6411ade": " BSFC = \\frac{r}{P} ", "d3e7f0f262e8aaf542c7919ff145c374": "\\left(\\frac{\\ln(\\frac{k_H}{k_T})}{\\ln(\\frac{k_H}{k_D})}\\right)_s=\\frac{1-\\sqrt{m_H/m_T}}{1-\\sqrt{m_H/m_D}}=\\frac{1-\\sqrt{1/3}}{1-\\sqrt{1/2}}\\cong1.44", "d3e7f49a5841c08284a375b9187078c0": "\\frac{1}{T_2}=\\frac{1}{T_{2b}}+\\rho\\frac{S}{V}", "d3e855e7ea8cbdf1e8d8b170dd27e4d5": "\n\\mathcal{L}_n^s f(\\mathbf{x})=\\sum_{\\mathbf{\\xi}\\in\\text{Pad}_n^s}f(\\mathbf{\\xi})L^s_{\\mathbf\\xi}(\\mathbf{x})\n", "d3e8ae5c721c72bc5e6aa5e67664b79c": "(g,\\Gamma)", "d3e8d82895a1a80433e16af047ee2157": "|n\\rangle = \\sum_{k \\in D} \\alpha_{nk} |k^{(0)}\\rangle + \\lambda|n^{(1)}\\rangle ", "d3e9165682bd39f79b1e1ece43180409": "\\delta_{ij} = q_{ab} E_j^b E_i^a", "d3e9ae5f517266a81ed863d6456d01ff": "u : X^*_{\\sigma} \\times Y^*_{\\sigma} \\to Z^*_{\\sigma}", "d3ea0399d029fecd93b22bf8e3e66701": "[n]_q", "d3ea08b174583e1bd3f762a2f541ccf9": "\n\\langle p \\rangle_{RS} = \\langle p \\rangle_{SR} \\equiv \\langle \\langle p \\rangle_R \\rangle_S\n", "d3ea8a928fac1e7d6b88b2d12d53bace": "\\frac{{{V}_{T}}}{\\beta {{I}_{IN}}}=\\frac{kT}{q\\beta {{I}_{IN}}}", "d3eac1365ce539eabdf869c92c430956": "x \\not = u", "d3ead5ae181602085f5c1f2ec3ce0dac": "\\cup \\!\\,", "d3ead834574924c63a17b6d7dba2a5a7": "\\left ( \\frac{\\Gamma_e}{\\delta x_{PE}} A_e + \\frac{\\Gamma_w}{\\delta x_{WP}} A_w - S_p \\right )\\phi_P = \\left ( \\frac{\\Gamma_w}{\\delta x_{WP}} A_w \\right ) \\phi_W + \\left ( \\frac{\\Gamma_e}{\\delta x_{WP}} A_e \\right ) \\phi_E + S_u", "d3eaf369522ce5e0797be9f3790a69e3": "[sc] \\vdash C", "d3eb3dfabdb6cd3aa31a7b6c8a8e4002": " p = \\sqrt{ \\left(x - \\frac{1}{4}\\right)^2 + y^2} ", "d3eb4c7f49c1df6a614397cc077e1339": "\n\\begin{array}{ccl}\n\\max\\{\\alpha: R(q,u) \\in C, \\forall u \\in \\mathcal{U}(\\alpha,\\tilde{u})\\} &=& \\max\\{\\alpha: \\alpha \\le I(q,\\alpha,u), \\forall u \\in \\mathcal{U}(\\alpha,\\tilde{u})\\} \\\\\n&=& \\max\\{\\alpha: \\alpha \\le\\displaystyle \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})} I(q,\\alpha,u)\\}\n\\end{array}\n", "d3ebf18527168f3ed2567a23a13ee64a": "\\mu^'_3=(k+\\lambda)^3 + 6(k+\\lambda)(k+2\\lambda)+8(k+3\\lambda)", "d3ec035d1fe435ad73534930c5c8ad9b": "\\overrightarrow{b_1}", "d3ec24c0d7d3acfc9fe7ec4b856522c9": "\n\\xi \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\sqrt{\\left| p^{2} - 1 \\right|}}{p}\n", "d3ecc135d433e9ba3115c91079e7b467": "dX_t = \\mu(X_t,t)dt + \\sqrt{2 D(X_t,t)}dW_t", "d3ecf54125bf0be3eee8e2513ccd9e47": "|g\\cap {\\mathcal Q}|=0, \\ |g\\cap {\\mathcal Q}|=1", "d3ed05146b7758dd9b084267b0b6161b": "\\sqrt{1+x}=\\sum_{k\\geqslant0}{\\tbinom{1/2}{k}}x^k.", "d3ed31bb615346856cf44a127ebb25d7": "\n\\max Z = \\sum_{k = 1}^u {\\sum_{i = 1}^n {\\sum_{h = 1}^m {x_{ih}^k \\left( {b_{ih} - c_{ih}^k } \\right)} } } \\quad x_{ih}^k \\geq 0\n", "d3ed45fb89e5c9e5af5c9d9945e6f0bb": "\\cot\\frac{\\pi}{6}=\\cot 30^\\circ=\\sqrt3\\,", "d3ed81bd9aa92eef443a73b92d7d59a8": " \\xi^n, ", "d3edeb02ec32b6c8e971c0c168e88090": " A \\vee B \\equiv (\\exists n) [ (n=0 \\to A) \\wedge (n \\neq 0 \\to B)]", "d3ee23afe5bd5a346788d4bc71c5b647": "H(A|B)_\\rho", "d3ee945a0d6a9753c9c3d9638d65ea1f": "M_{i,j}", "d3ef02222e409b5cd5f69411892af90b": "B \\to bB", "d3ef2e4145f0aff4828f3e06450b1d63": "{\\sigma}= \\frac{M y}{I_x}", "d3ef338ab823a3df2f57116fc966b847": "\\pi(\\sqrt{n}) \\approx {2\\sqrt{n} \\over \\ln n} ", "d3ef6eaeb80d6e19000514101b726c07": "\\frac {\\alpha_a} {\\nu} = \\left( \\frac {RT} {nF} \\right) \\left( \\frac {\\partial ln(|I_{ox }|)} {\\partial E}\\right)_{T,p,c_{i,interface}}", "d3ef990b300bcfcac08851bebcf3e022": "(25)\\quad \\sigma_{ab}=-\\sigma \\bar m_a \\bar m_b-\\bar\\sigma m_a m_b\\,,", "d3efdd5688ed8522537b50d9134d5db5": "\\langle \\Phi_1 , \\Phi_2 \\rangle = \\int\\limits_{-\\infty}^\\infty d p \\, \\Phi_1^*(p, t)\\Phi_2(p, t) \\,,", "d3efdd70d4693fbd4b44f42a99456ad9": " J_y = J_2 = i\\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & -1 & 0 & 0 \\\\\n\\end{pmatrix}\\,,", "d3efe5420e9c0e64f042686a0485c169": "\n p_0=\\frac{1}{\\pi}E^*\\frac{d}{a}\n", "d3efef7894cd82256d18e32943e8c41f": "b>0\\,", "d3eff2657882fda991ea11dcd87b76c7": "f_3(z) = \\,_1F_1(a+2;b+3;z)", "d3f009931773dee686425614403a8ee7": " P_{ni} = \\int L_{ni} (\\beta) f(\\beta | \\theta) d\\beta ", "d3f0bc1527cf1cd8ed75038690be93e0": "S/\\mathfrak{m}_S", "d3f0c1cc83a6a352895180e1a7ed331b": "\\nu_t \\,", "d3f0dc7234b3085e08e87381ac19b7fb": "y(b) = B", "d3f150cb55b77fc08896e26031c31f89": "\\partial V_x/\\partial \\mathbf{x}", "d3f1ad34c46a6db8bd11803fc6e5b58d": "dH = dU + pdV + Vdp\\,\\!", "d3f20771180b5b6b77f30b441bdcf08c": "e^{-a^2r^2/2}\\,", "d3f20c127d5fb47de84e7455f9ca1f15": "\\psi(t)=\\sqrt2 \\sum_{n\\in\\Z} g_n\\phi(2t-n).", "d3f2404f94ec2a4dece7c90e2c4e3b66": "k=1,...,N.", "d3f24b5e803dc4da283f093b63b3f776": " e(k) = \\hat{x}_L(k) - x(k) ", "d3f29b0e2fe09189b2510f6d77f2b0a6": " \n\\int_{-\\infty}^\\infty \\frac{- \\ln f(x)}{1+x^2} dx\n = \\int_{-\\infty}^\\infty \\frac{\\ln^2 x + \\ln \\sqrt{\\pi}}{1 + x^2} \\, dx < \\infty, ", "d3f2a603c65fdc57c0633baca428f2e1": "f_1,f_2,f_3,f_4", "d3f2ce56902b999106cf0995f8f0a2ed": "z=F_\\ell(\\mathbf{P},\\mathbf{K})\\,w", "d3f2e3fc4417324e8682855d63f7e324": "\\operatorname{var}(z_i) \\;\\stackrel{\\mathrm{def}}{=}\\; \\operatorname{E}(z_i^2) - \\operatorname{E}(z_i)^2", "d3f3680615121ec0905054fbf40419da": " {_1^0}\\text{S} + {_1^1}\\text{S} + \\text{E} \\overset{\\xrightarrow{\\text{k}_{1(2)}}}{\n\\xleftarrow[\\text{k}_{2(2)}]{} } \\text{C}_2 \\xrightarrow{\\text{k}_{3(2)}} u_\\beta {_2^1}\\text{P}^\\beta + u_\\gamma{_2^1}\\text{P}^\\gamma + \\text{E}, ", "d3f369d93176cfb5647d378b67396079": "\nP' = Mv - 2\\Delta P = \\left(M - {E\\over c^2}\\right)v.\n\\,", "d3f3de7e1f2d3462915d5a139e6e790a": "x=i", "d3f43bb5ecacec2029283484356bab70": "\\left( a_{i},d_{i} \\right)", "d3f44e55aeb131164195a1b61972e251": "3^2+4^2=5^2", "d3f460c6bbe0e1221534aefe763601e1": "dx^2 + dy^2 =\\, dl^2", "d3f4dd1fb8f1f9cda5bdde48c256397f": "K_{\\alpha+\\beta} (u,u') = \\int K_\\alpha (u, u'') K_\\beta (u'', u')\\,\\mathrm{d}u''.", "d3f51e684fdd6601e9a44b966ff175ff": " z\\mapsto z^d+c\\,", "d3f5c27e7625306e810a0d7e93ad3466": "x \\in (-\\infty,\\infty) \\text{ if } \\kappa=0", "d3f5d9e65ceec972a6a3ed072cbdf8fe": "\\text{minimize}(\\alpha)", "d3f602d346d0c1bbfcb0e6dc53a3cb9a": "\\Delta = x x^T ", "d3f641c33662cc4bcef7cda7e6065df5": "\\begin{align} \n &\\Pr(N\\le x\\mid M=m,K=k) \\\\\n = {} &1 - \\Pr(N>x\\mid M=m,K=k) \\\\\n = {} &[x \\ge m]\\left(1 - \\frac{\\binom{m - 1}{k - 1}}{\\binom{x}{k - 1}}\\right)\n\\end{align}", "d3f64a4bced405eacc9a8e281176ffe8": " \\tilde{f}(\\lambda)=\\int_S f(s) \\alpha^\\prime(s)\\, ds,", "d3f66ee397955675dc92a23910b8e1d6": "|\\rho|>1", "d3f6a5963bc98e95b1d4280c4491a6c0": "\\overline \\Delta_{21}(\\mathbf{p},\\mathbf{p})", "d3f7019d79d8177264a2733b450dc1c2": "\\Delta H ^{\\circ} _{\\mathrm{comb}}", "d3f773eec633519ef38814dd3ef4c1dd": "\\beta : \\mathbf A \\rightarrow R", "d3f7764a2a70661d307dd393fab9a6b9": " \\mathcal{C}_{Y \\mid X} = \\bigg( P(Y=s \\mid X=t) \\bigg)_{s,t \\in \\{1,\\dots,K\\}} ", "d3f78d23c74954d17e8e38f2d989de6d": " K_1 = \\frac{d}{d\\theta} A(t) \\, .", "d3f84a16d39b818bd343add0ffa98441": "\n\\begin{align}\n\\frac{D^\\acute{n}F(P_0)}{DP^\\acute{n}} & =F[P_0,P_1,P_2,P_3,\\ldots,P_{\\acute{n}-3},P_{\\acute{n}-2},P_{\\acute{n}-1},P_\\acute{n}], \\\\[10pt]\n& =F^{(\\acute{n})}(P_0 < P < P_\\acute{n})=\\sum_{TN=1}^{UT=\\infty}\\frac{F^{(\\acute{n})}(P_{(tn)})}{UT}\n \\\\[10pt]\n& =F^{(\\acute{n})}(LB < P < UB)=G^{(\\acute{n}-1)}(LB < P < UB)= \\cdots \n\\end{align}\n", "d3f85ab2af768543372eca5d92dbccce": " E = \\rho \\left ( U + \\frac{1}{2} \\mathbf{v}^2 \\right ) \\,\\!", "d3f87c2ce6bac6f4dd6fe91a6ef54bce": "\\pi \\approx A_{192}+ \\frac{1}{3} D_{192}\\approxeq 3.1410319509 +0.0016817478/3", "d3f90cece5f504e82dedb06013ebb9df": "\\frac{\\partial \\phi_i}{\\partial t} =\\sum_j {\\rm div}\\left(D_{ij} \\frac{\\phi_i}{\\phi_j} {\\rm grad} \\, \\phi_j\\right) \\, .", "d3f94d583b4904e8463a53e491aeb093": "\\theta \\mapsto f(\\theta | x) = \\frac{f(x | \\theta) \\, g(\\theta)}{\\displaystyle\\int_{\\vartheta \\in \\Theta} f(x | \\vartheta) \\, g(\\vartheta) \\, d\\vartheta} \\!", "d3f982e7db2fdd47c0932b3b9f430b28": "TT^*", "d3f9d1e5deec03464428289ee13008e8": "\\operatorname{var}(k) > 2.", "d3f9e0104bb863205cb2c12de278d43d": "\\rho = (m P)/(k T)", "d3f9e60800ffc1feb52b6d1768c63997": "v \\Delta v + \\frac{1}{2}(\\Delta v)^2 ", "d3fa231987b1ae0515cad6ef1347d4d9": "\\sqrt{g_{tt}}", "d3fa790a894dbbc547bca79492382fa7": "A = \\frac{1 + 2 \\frac{M}{C}}{1 + 2 \\frac{N}{C}}", "d3fa82f5b234b5c0ac62d5983a77b26e": " \\alpha \\centerdot q \\geq \\omega (\\sqrt{log n}) ", "d3fa83b6b79d18d3a875ab493705e02d": "c < d", "d3fa84db4b75e4d0aed929165c07d33d": "\\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\left(\\frac{\\partial \\mathbf{f}_1}{\\partial \\mathbf{v}} + \\frac{\\partial \\mathbf{f}_2}{\\partial \\mathbf{v}}\\right)\\cdot\\mathbf{u} ", "d3facde4643f63ed7d3ee719782f10ec": "\\bar3", "d3fad27759452b82128e84221ab79c9a": "\\mathcal{}BP_*BP", "d3fb3169db32cc7a357d5332890643bc": "e^+e^- \\to e^+e^-e^+e^-", "d3fb397fea373c58eaf8ddcde93ae549": "|x\\rangle", "d3fb41f42a7765f5856b7f358eab7bf4": "a_A", "d3fb61d6898a0e440127a9884173a324": "t_{i-1}", "d3fbe158984d2e0f29c3bc7acb78602c": "U^*=13W^*(u-u_0), \\quad V^*=13W^*(v-v_0), \\quad W^*=25Y^{1/3}-17", "d3fc296181fb7af6daee8d7d8a306774": "\\alpha S := \\{\\alpha x \\mid x \\in S\\} .", "d3fc51a8b86616febfd579536d1f72ec": "G_1', G_2' \\in \\mathcal{G'}", "d3fc51fa5b01c21d98517e5b816751e1": "C_4 = \\{U^2, SU^2, VU^2, SVU^2\\},", "d3fc53cbe2c1b33946afe28ebab33f3a": "E_{m} = \\frac{4U}{L}", "d3fc6597d50dc2cb5dba45cdb8c27a88": "\\lambda (\\lambda + 5) - 1 (\\lambda + 5) = 0 \\,\\!", "d3fd2fba30113520b58b7803cbc57c90": "{n}^{th}", "d3fd360f6a38e4a91e167e94d8f5acb7": "P(a,x)", "d3fe93a2e529f77ba69bd893cd8896db": "\\operatorname{tr}(AB) = \\sum_{i=1}^m \\left(AB\\right)_{ii} = \\sum_{i=1}^m \\sum_{j=1}^n A_{ij} B_{ji} = \\sum_{j=1}^n \\sum_{i=1}^m B_{ji} A_{ij} = \\sum_{j=1}^n \\left(BA\\right)_{jj} = \\operatorname{tr}(BA)", "d3fecb851099ec0247780bbd5c1f5b20": "{d \\over dt} p(t) = {i \\over \\hbar } [ H , p(t) ]= -m \\omega^{2} x", "d3fedad00a709a09e12c5d6b4d62cf4d": "47 = 9 + 9 + 9 + 20", "d3ff03330cf7c147dcb225a3917388f9": "[D,S]=\\frac{1}{2}S", "d3ff45a3f450bed8f8c67b2ffba617f3": "x_{1,2}=\\frac{-b\\pm\\sqrt{\\color{Red}b^2-4ac}}{2a}", "d3ff6ae73ce07c8dc3e8ac6d6a69d3da": "\\sum_{i} ( \\mathbf {F}_{i} - m_i \\mathbf{a}_i - \\dot{m}_i \\mathbf{v}_i)\\cdot \\delta \\mathbf r_i = 0.", "d3ffb2191949d5a99a181f7915bcb58f": "\n\\int d\\mathbf{q} d\\mathbf{p} = \\int d\\mathbf{Q} d\\mathbf{P}\n", "d3fffbecfeab276cf149a13a2a195b59": "{\\mathcal P}:=\\R^2\\cup \\{\\infty\\}, \\infty \\notin \\R", "d40017253f065ae065454004518ae605": "j_l", "d400955cb90ccda515d8a2fb2fab2734": "X_b \\hookrightarrow f^{-1}(U) \\cong X_0 \\times U \\twoheadrightarrow X_0", "d4009bb81eede2a4a08356a29eb20ced": "E_{j}", "d400a18a8f430281e30a350cd667e017": "\\{x+'y-'(x+y)\\mid x,y\\in M\\}", "d400c482e03c912005f6c72c8ca61db5": "h(\\psi(0))", "d400dcbcc123c500765f0b78e89e9b66": "u(x, y)", "d400e970d7dee1fc0fda611983389173": " E(k) = \\frac{\\hbar}{2} \\left(\\frac{k^2_l}{m^*_l} + \\frac{2k^2_t}{m^*_t}\\right) ", "d400f0888280ed081455112f93b308aa": " \\Delta(y,y)=0 \\;\\; \\forall y,z \\in \\mathcal{Y}", "d400ff1913507c9942a5708d35298255": "\\nabla u = (u_x,u_y)", "d40127c46dabb194cf16f16a22c8e004": "I_n = -\\frac{\\sin{ax}}{(n-1)x^{n-1}}-\\frac{a}{n-1}\\left [\\frac{\\cos{ax}}{(n-1)x^{n-1}}+\\frac{a}{n-1}I_{n-2}\\right ] \\,\\!", "d401679d9fe24d29ecd0bf4e4d88ebf4": "\\textstyle 2^{13{.}8}\\cdot n^{6{.}9}>(64\\cdot n^3)^2", "d40188d553044bddbfeff39b3a7c2b4c": " E( \\theta ) = \\frac{ N } { N - 1} ", "d4023aff653f0523e5f89e9dec330aa1": " \\Gamma(t) = \\int_0^\\infty x^{t-1} e^{-x}\\,{\\rm d}x.", "d40276d65b505bef48c3ff958251c975": " -z \\partial_y + y \\partial_z \\,\\!", "d402cf9259957a489c3ef912e3940d86": "z\\log(2\\sin \\pi z)-\\int_0^z\\log(2\\sin \\pi x)\\,dx", "d402dbea9c75e0a7d13983d383880314": "\\partial H/\\partial u", "d402dca3c12afdce7b42031068918a58": "F_{2,1} ", "d402fcca2dd14eb61408d4fb6230f528": " \\Delta \\mathbf{[x]} ", "d4031d8d36289785195b0c5a540ad134": "d_\\text{b}", "d40335c7d0e3acbf3685dace78e43f5d": "t\\in\\mathbb{R}", "d40383b733d4756091859e480aa492ed": "\n\\tilde{\\mathcal{A}} g= e^{-\\varphi}\\mathcal{A} (e^{\\varphi}g).\n", "d4038567049cce5eed12b0eca8491989": "px^2+(3p-2a)xy+(b-3a)y^2 \\, ", "d40385fe39aab3342e2f023493cccd6f": " L_{\\psi} = \\{ A \\in K(H) : \\frac{1}{\\psi(1+n)} \\sum_{j=0}^n \\mu(n,A) < \\infty \\}. ", "d404547bf9683a3e75b698f17ee34391": "\\mathcal{H}\\left(r, \\theta, \\dot{r}, \\dot{\\theta} \\right) =\n \\underbrace{ \\frac{1}{2} M_t \\left( R \\dot{\\theta} - \\dot{r} \\right) ^2\n + \\frac{1}{2} m r^2 \\dot{\\theta}^2 }_{T}\n + \\underbrace{ gr \\left(M - m \\cos{\\theta} \\right)\n + gR \\left( m \\sin{\\theta} - M \\theta \\right)}_{U},\n", "d40469e6bbf9d398c592ab2f05928755": "H_{ba}", "d4047e85ca4e9dced39d04a5520c6170": "\\,(1+i) = \\left(1+\\frac{i^{(m)}}{m}\\right)^{m} = e^{\\delta} = \\left(1-\\frac{d^{(m)}}{m}\\right)^{-m} = (1-d)^{-1}", "d404c98b046ed1ee726c17465892c298": " -\\omega \\propto -2 \\vec{\\nabla} \\cdot \\vec{Q} ", "d404f6e00690f693d7428037d45371ff": "\\, t_\\text{r} = \\tilde{t}_\\text{r} + b", "d4055155c7ecb12e1996701a775813f1": " (u,\\phi) \\mapsto E_{u,\\phi} \\quad u\\in U(G), \\phi\\in\\widehat{A(u)}, E_{u,\\phi}\\in\\widehat{W} ", "d4058fe209b6ef5c8204918f50fb5d9e": "\n \\begin{bmatrix}\n a_{11} & a_{12} & a_{13} \\\\\n 0 & a_{22} & a_{23} \\\\\n 0 & 0 & a_{33} \\\\\n \\end{bmatrix}\n ", "d4059b171870c0fa1d51ed15faba7516": "\\int_{\\Gamma} g \\in L(X), ", "d405b97bec9c6131030c45cbb4e28532": "\\begin{bmatrix} ^\\diagdown m_{r\\diagdown} \\end{bmatrix}", "d406199f121d0e155772324958c0f8c1": "\\pi(b_i)\\to x", "d40623bfd16070f08622524a22bbdbbe": " h(x;k,\\lambda) = {k \\over \\lambda} \\left({x \\over \\lambda}\\right)^{k-1}.", "d406441571de52043abb655596c55684": "M(256,1,3)\\approx 3.23\\times 10^{616}", "d40647a9371e77cfeb2cdaf679858701": "\\begin{align}\n& Z_{\\text{equil}}=\\sum\\limits_{J=0}^{\\infty }{(2-(-1)^{J})(2J+1)e^{{-J(J+1)\\hbar ^{2}}/{2Ik_{B}T}\\;}} \\\\ \n\\end{align}", "d406796f64fce753499b8b1b6b6c3bac": "\ni{d \\psi_n \\over dt} = c \\psi_{n+1} - 2c\\psi_n + c\\psi_{n-1}\n", "d4067da12b2cea260912a25064694125": "PA = LU", "d40720fea7142d06dd0544a42865e28c": "\\frac{\\sum_1^n x_n}{n} \\,\\forall x_n", "d40721a74470a58a745eb95281a159d4": "=2p_1p_2-p_1-p_2+1\\ ", "d407480cca28ea38d7b1808cc3d57ced": " \\boldsymbol{\\tau} = \\left ( \\mathbf{r} - \\mathbf{r}_0 \\right ) \\times \\mathbf{F} = \\mathrm{d} \\mathbf{L}/\\mathrm{d} t \\,\\!", "d4074c86abeeb4cc4117791d0eb7db2a": " J(f,x)\\in A_{p}", "d4079c7625a54b123687fc1627b3eea8": " f_j ", "d407c8230f6a75da0edd45c661b0cfc5": "Q_{q}", "d407f2acf21272732d4d75402490b12f": "\\int d^Dx \\sqrt{-g}\\, G", "d4081dc666e8640fb41a0fa1ef401463": "\\phi(u_1,u_2) = (r\\cos u_1\\sin u_2,r\\sin u_1\\sin u_2,r\\cos u_2).", "d408acfc5d34ad34ab31ba1d05d2962b": "R = \\frac{\\log(p_n)}{\\sqrt{p_{n+1}-p_n}}.", "d408e7eb126d98ebfbd573dc68e57c94": " Z(s)=\\sum^\\infty_{i=1}e^{-\\lambda_i s}. ", "d4092b93ceccecb231780dbf6fe16204": " f(x_i,\\boldsymbol\\beta)\\approx f^0+\\sum_j J_{ij}\\beta_j ", "d4092de2f028ea5e72dce288b92cdbee": "\\tilde{\\phi_i}, \\boldsymbol{\\tilde{\\mu_i}}", "d4094c4792b8980cd0b35698b4d85c8d": " \\phi^*(g) = \\omega(g) \\phi(g^{-1}) . ", "d4095364da34a864feeab849fb63b257": "a_1 + a_2 \\equiv b_1 + b_2 \\pmod n\\,", "d4097838ff144638cd09e350dfed8871": "t_{TF}[n] = \\frac{p^2}{2m_e} \\propto \\frac{(n^{1/3})^2}{2m_e} \\propto n^{2/3}(\\vec{r})\\ ", "d409c6d6bda4a3f7de0c6e6866982859": "\\vec{r}_{uu}, \\vec{r}_{uv}, \\vec{r}_{vv}.", "d409d5f5d2f9b74a49aeca9f146ebfcd": "wlp(a,p)\\,\\!", "d40a08a9356d88b7e751837420e91f5c": "\\frac{\\frac{ K^- + \\bar{K}^0 }{2} + \\frac{ K^+ + K^0}{2}}{2} = \\frac{3\\eta + \\pi}{4}", "d40a8e135fd38fc6c80c759620716e20": "\\pi_1(t) + \\cdots + \\pi_n (t) = 1", "d40b12144e26be919332faae8018adf2": "p\\ll T_\\mathrm{smax}", "d40bc6d9c1b6e280ab6f9830a4cd2d86": "\\textstyle\\ \n (1/f)''(x)=-\\frac{f''(x)}{f(x)^2}+2\\frac{f'(x)^2}{f(x)^3}", "d40bef687af39e59c6089c4b6c82ce5a": "\\nabla^{2}", "d40c17ea6ee1666dbc3b87c3c313e1fc": "N = \\{S, A, B, C\\}", "d40c31e87e6264a6325d03a6a4e71a7f": "A,A',B,B',\\dots", "d40c34c50c0094549852811253493f7d": "(x-a)^2 + y^2 + z^2 = R^2.", "d40c66a6c5101294c9c0e806ba52fff1": "r=\\epsilon a \\sin( \\omega t)", "d40ca739db64d5f5874cb9ca0a8e847f": "\\rho\\colon \\mathfrak g \\to \\mathfrak{gl}(V)", "d40cc044891ed7e7384a77f1e3242cfd": "x^{\\iota} = \\{(\\{a\\},\\{b\\})\\mid (a,b) \\in x\\}", "d40d690bab55cd13e2a9e38dd5ba4450": "\\triangleleft : V \\otimes V \\to V", "d40d71796949200ed6ce1e43d9569d80": " D_C = D_A (1 + z)", "d40dd249cfb8d5b242532bc931184504": "\\alpha(i)\\neq k \\nu(i), \\forall k", "d40e2dae6e5cb0afe628f478417526a9": "\\Delta q = \\int_{T_1}^{T_2} C_\\mathrm{v}\\mathrm{d}T", "d40e37d34ad860e1ae423c7727939768": "-\\boldsymbol{\\mu}_S = -\\mu_B g_S \\mathbf{S}", "d40e3ca548ad3037346a772ed28c2ec2": "x_h(t) = C_1.e^{ -\\frac{1}{2}.(\\bar{c}+\\sqrt{\\bar{c}^2-4.\\bar{k}}).t}+C_2.e^{ \\frac{1}{2}.(-\\bar{c}+\\sqrt{\\bar{c}^2-4.\\bar{k}}).t}", "d40e431c30bc5a53b3b67a38eb2361db": "(YW-Z^2)^2", "d40ea010599f3819c599c0c319b40857": "V_{GS}-V_{th}", "d40ea5eaf347e8ce5dfb3bb1ef1795d9": "A_f", "d40f160692ada07a7766d392d307c59e": " \\frac{p}{2-p} ", "d40f398a47ca0202461e8e5c727d6512": "O(n\\log^2(n))", "d40f4347f9e85bc660e1c48cdf03e8d3": " \\sum_{k=0}^\\infty r^{k} = \\frac{1}{1-r}", "d40feeae3a2e958032d54e810bb22307": "a+b\\sqrt{p^*} \\in \\beta\\setminus(q),", "d40ffe5a12cd386479a4061432219aa2": "L_c(s,t)=(1-t) c_0(s)+ t c_1(s) \\, ", "d410a628a5cb053e5907f8d67e6b774a": "W_i = \\rho K N_i", "d410ed664a74bf05081871549ba43277": "\\operatorname{depth} \\operatorname{M} = \\sup \\{ n | \\operatorname{Ext}_R^i(k, M) = 0, i < n. \\}", "d410f291016731027099e08a348ac4a4": " -P=(y_1,x_1)", "d4113971d49ea1c9a1a33a408238808e": "\\mathbb{E}^{-S_{23}} f(x_1, x_2, x_3) = f(x_1, x_2 - S_{23}, x_3)", "d41140a9b2d9aecf77caffdd60b73903": " \\dot{\\textbf{r}} = \\textbf{v} = - r \\omega \\sin(\\omega t) \\hat{x} + r \\omega \\cos(\\omega t) \\hat{y} ", "d41187c546cb62407c2b7e403b1febd3": "P=\\frac{RT}{v^2}\\left(1-\\frac{c}{vT^3}\\right)(v+B)-\\frac{A}{v^2}", "d411ea355c5a20585060139eb3e1d273": "\\mu((T^{-n}E)\\cap H)>0", "d411f8a703b195b76850c11b699a4417": "y^{-}", "d412611af66d7227cdc73935d683c5be": "\\lambda_B = \\frac{e^2}{\\varepsilon_r k_B T}.", "d4126702422117487716948d35a4963e": "\\theta = 0.25", "d4126dbf5198f46c1ab26638fac7fba6": "\\sum_{i=1}^n \\widehat{\\varepsilon}_i=0", "d412bee2731f75e8a60a863681f8f937": " \\sigma = \\sqrt{\\sigma_S^2+b^2\\sigma_M^2-2b\\rho_{SM}\\sigma_S\\sigma_M}", "d412cc8126faacf8c38bd34b7ea2c142": "= \\sqrt{ \\frac {p \\, (1-p) } {n} } = \\sqrt{ \\frac {0.5 \\times 0.5 } {n} } = \\frac {1}{2 \\, \\sqrt{n}}", "d412cfd17aab0d850b384fe2774e0c2d": "Q=\\begin{pmatrix}\n-\\lambda & \\lambda \\\\\n\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&2\\mu & -(2\\mu+\\lambda) & \\lambda \\\\\n&&3\\mu & -(3\\mu+\\lambda) & \\lambda \\\\\n&&&&\\ddots\\\\\n&&&&c\\mu & -(c\\mu+\\lambda) & \\lambda \\\\\n&&&&&c\\mu & -(c\\mu+\\lambda) & \\lambda \\\\\n&&&&&&c\\mu & -(c\\mu+\\lambda) & \\lambda \\\\\n&&&&&&&\\ddots\\\\\n\\end{pmatrix}", "d412fda2c2143dd93da308cbaed0f2d3": "\\hat{\\mu}_j=\\frac{x_{j,.}}{I}", "d4133ecd77fc9bf70856370f449f3e04": "\\tan(1/n) = [0; n-1, 1, 3n-2, 1, 5n-2, 1, 7n-2, 1, 9n-2, 1, \\dots]\\,\\!,", "d41357a403c29dc65f6c24da2f78357f": "\\upsilon_{t}", "d4139de232486acd86b56db0e889f836": "= 2 \\eta^{\\mu \\sigma} \\operatorname{tr} \\left( \\gamma^\\nu \\gamma^\\rho \\right) - \\operatorname{tr} \\left( \\gamma^\\sigma \\gamma^\\mu \\gamma^\\nu \\gamma^\\rho \\right)\\quad \\quad (3) \\,", "d413e898d6152d125e08cd1e27f3848c": "1 \\leq i \\leq m", "d4145b0f7b1c3f2c6c30dec22edb5f69": "N = \\frac{f}{(\\hbar\\omega\\beta_c)^3}~\\zeta(3)", "d4145d9d1cadc88e18d0ae5984f01ec3": "\n9.000 \\mbox{ metres} = \\frac{L + B + 1/3G +3d + 1/3\\sqrt{S} - F}{2}\n", "d414c49f229d314c410b6c0211f3baff": "P\\to P/G", "d414e280f292b5287bae2ae254139c09": " \\ell ", "d414fa5231bb3380379e2a650b18ae35": "u_i=u(x_i)", "d4151bef4534f8a2886504b5ac33ee46": "\\mathbf{Set} \\to \\mathbf{Top}", "d41522e5592a595b66757129e71ec7d8": "F(0) = a", "d415379bf3af4803cc5542a3f434dea9": "\\mathcal{S}_{0}", "d41540e2fb6cfe340ba8bf6624dd43ce": "[ T_{m_1,m_2} ~ , T_{n_1,n_2} ] = \n2i \\sin \\left (\\tfrac{\\hbar}{2}(n_1 m_2 - n_2 m_1 )\\right ) ~ T_{m_1+n_1,m_2+ n_2}, ~\n", "d415c50b01991eb0b023d2fa998589c7": "L_\\eta", "d4161070e3d2dc4b32a0ed6f4480b9cd": "\\int\\frac{x}{(ax^2+bx+c)^n} \\, dx= -\\frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-\\frac{b(2n-3)}{(n-1)(4ac-b^2)}\\int\\frac{1}{(ax^2+bx+c)^{n-1}} \\, dx + C", "d4161c2f4f034991dcca8a22fca1c97b": "Z_\\mu ", "d41652bc461d441287db6750d478286c": "\np + 2q + r = 1\n", "d4165743c8dd545c243c1a3d0ee68a46": "dY/dt=-3Y\\quad\\text{and}\\quad dZ/dt=Z.", "d416e904d018ac98228829ad904ef47a": "t\\in \n\\left[ 0,1\\right] ", "d4171f3e2da7d7780910ef88d5898bcb": "\\omega\\cdot\\gamma", "d4172aee16b67e1a2a1a1b9e4c5f5b59": "\\rho = \\frac {M_{\\rm air}p}{RT} ", "d417a3c91693e4cb9ea1848e7ab85f24": "G(s) = \\mathcal{M}\\left\\{g(\\theta)\\right\\} = \\int_0^\\infty \\theta^s g(\\theta) \\frac{d\\theta}{\\theta}", "d417bd91404c12468776651120fcb381": "\\operatorname{pt} \\stackrel{0}\\to R", "d41867292c94fe8e157fb652d6566bcd": "\\{F(x_n)\\}_{n\\ge1}", "d41878ef2cd1e9934616e520d877350f": "v=-\\frac{k}{\\mu}\\nabla p", "d418a2969fadfc311c65525baa8f1f1e": " -\\frac{691}{2730} ", "d418dec0d158fb3dd6153ea00d5a6964": "x_U", "d41903b267e2da2f656b9dd002178d1e": " e^{at} f(t) \\ ", "d41910f0c11ba38cbfa89acc055721ed": "(x,w) \\in \\mathbf{R}^2_*", "d4191a5b1a19bab8e8eddc5742d653c8": "\\langle(\\nabla_X Y)_x,x\\rangle = 0\\qquad (1).", "d419b0e1f7432ba0309bcb6f4871b3cf": "t = \\,", "d419ce7df239132fb7d6c1486374aedd": "\\eta=\\frac{W}{Q_H}=1-\\frac{T_C}{T_H}\n\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad(3)", "d41aa894bb0c329834e9fa2adf508f73": "z_6 = x_6 y_1 + x_5 y_2 + x_8 y_3 - x_7 y_4 + x_2 y_5 + x_1 y_6 + x_4 y_7 - x_3 y_8", "d41ad0cfa40fb4d9f422cfbd5816c882": " x_1 := A^{-1} (b_1 - B x_2) ", "d41ae01a9551370a544a72e1edaf6eae": "s = a \\tan \\varphi = a \\sinh \\tfrac{x}{a}.\\,", "d41b0aae143a8fd1016c932e5d8b9d6e": " \\mathbf{I} ", "d41b0acf374f5a3410f393c5b3cd2084": "\n\\frac{z}{c} = \\frac{y^2}{b^2} - \\frac{x^2}{a^2}.\n", "d41b591496180e024370baacc950d825": "\\gamma < \\varphi_{\\beta} (\\gamma) \\,,", "d41bad6732084673d4d75b7c3fb1b2d3": "q:X \\mapsto \\mathbb{R}", "d41c2ed6cbb61861afbf5c8740fa437d": "x_1\\left(x_3\\right)", "d41ca30cddeaea297cf29876c1e5a658": "\\left\\langle a_0,\\dots,a_{n-1}\\right\\rangle", "d41cd7eea28b123957260a76fcbace8f": "P(\\vec R ) = \\left ( \\frac{3}{2 \\pi N l^2} \\right )^{3/2}e^{-\\frac{3 \\vec R^2}{2 N l^2}}", "d41d3af276ab379a008e55e333bbb74d": "\\rho_{m_{0}}", "d41d536babf8af05ebc3ff4e152d1bc9": "0 = \\sum_i f_i^{(k)} \\vec{e}_i", "d41d8cd98f00b204e9800998ecf8427e": "", "d41d9d1503bd2873928dfdbf5236c2a8": "\\mathcal{L}=\\,i\\,\\bar{\\psi}\\partial\\!\\!\\!/\\psi+\\frac{\\lambda}{4} \\,\\left [\\left(\\bar{\\psi}\\psi\\right)\\left(\\bar{\\psi}\\psi\\right)-\\left(\\bar{\\psi}\\gamma^5\\psi\\right)\\left(\\bar{\\psi}\\gamma^5 \\psi\\right)\\right]=\\, i\\,\\bar{\\psi}_L\\partial\\!\\!\\!/\\psi_L+\\,i\\,\\bar{\\psi}_R\\partial\\!\\!\\!/\\psi_R+\\lambda \\,\\left(\\bar{\\psi}_L \\psi_R\\right)\\left(\\bar{\\psi}_R\\psi_L \\right).", "d41dd39c1bf118a82c14012e7e4a903d": "\\pi_1 T", "d41df9761579f86a701024fa234cfe24": "v_m = \\frac{1}{A+I}\\qquad (3)", "d41e230aafa65dd59bfa1a923cddc01b": "(N-1)N(N+1)/12", "d41e738e383f0cab1c4d983268d6b080": " (x^2+1) = (x+1)^2 ", "d41e77cb89a12485b858332d49e13d7c": "\nb_x = s_x a_x + c_x\n", "d41e8d1e96489a7695277bf1b8400237": "c_n=H_{n+1}P_n+P_{n+1}H_n.", "d41eac225b3a092f4de1e147c66b94a2": "\\sigma=C du/dz", "d41eb8306d80a7b49d8dcafd567c5c55": "\n\\hat{\\mathbf{x}}_{0\\mid 0}=E\\bigl[\\mathbf{x}(t_0)\\bigr], \\mathbf{P}_{0\\mid 0}=Var\\bigl[\\mathbf{x}(t_0)\\bigr]\n", "d41f35098cc4ede4ea46b8365ca602af": " x^5-20 x^3 +170 x + 208", "d41f9f4faae86e91ebb9abc376cd81c9": "A_s = A_{std} \\left ( \\frac{M_s - M_b}{M_{std} - M_b} \\right ) ", "d41fb86d24ebbc0e24fbd514ad633b27": "H' \\subseteq H", "d41fe0e4382b8249b988165c0fea6d1a": "\nT^7\\times E^7, T^8\\times E^8, T^4\\times F_4, T^2\\times G_2\n", "d4204d6d9651aba7868da7f4a5cd4cc7": "{\\it{O}}(M^3{\\cdot}{\\chi}^3)", "d42092f1d060671de78dbdae788b8b0c": "\\pi_x\\ ", "d420e1a755c63d45ea69f575f18fdfa1": "x\\le (y\\Rightarrow z).", "d420f0714c88a246134b97e64b1869ca": "\\tfrac{10}{20}", "d4210f6398ef7a6491e2519e17047efa": "L_f \\le U_f . \\,\\!", "d421a0d1fbd7c0d3d0ebcb0c691cb02a": "-248\\pm 2.8%", "d421b3b1a79b6737ab2e402bcdf04daa": "0 .. m-1", "d421c9513f1ede65b8d026ad6fbf1b22": "0 = P_{bottom} - P_{top} - \\rho \\cdot g \\cdot h.", "d421dc9b41872cb60030d5218363fd0c": "f_x\\in L^\\infty(X)", "d4225df29ee5efd68caf20614f27c22f": "i>1", "d422bb2e231977cb426426485456cf66": "a_j = \\int_a^b \\phi _j^* (x)f (x) \\, dx. ", "d422cacdf2faf9ac018ed806a19f21fb": "w \\models Q", "d42316e2ca7db96bcb21a37fe8fb97a4": "\\displaystyle{\\partial_{n-}u(z) = -{1\\over 2}\\varphi(z) + T_K^*\\varphi(z),\\,\\,\\,\\,\\, \\partial_{n+}u(z) = {1\\over 2}\\varphi(z) + T_K^*\\varphi(z).}", "d4231c404f1740264abc1f82556a3150": "\n\\sum^p_{i=1} ~\\beta_{i} +\\sum_{i=1}^q~\\alpha_{i} = 1\n", "d4235f4bb0bd6ca8c0434b37dc6b2830": "\\mathit{F}", "d42368dd6805d03fb03ad7ab9186c065": "S=S_1\\cup S_2 \\cup\\cdots\\cup S_k", "d4236d74c85de2951e9ce349cbb7429b": "RC = \\Delta_T \\left( \\frac{1 - \\alpha}{\\alpha} \\right)", "d42377c39529f9fc78a39ac3c9f653b6": "S=(x+e)^2", "d4238bcc5560a1c262f5368e95df015d": "f^*J \\to \\mathcal{O}_X", "d4238d598409aff4bb4f772afd2e0360": " \\rho = \\sum_i \\rho_i \\,", "d423963ced7297b646e3f96682d58993": " d = \\frac{|a(x_0 - x_1) + b(y_0 - y_1)|}{\\sqrt{a^2 + b^2}}.", "d4239b8b1c1b69146103ede643bcfc46": "\\left|x\\right\\rangle", "d4241f2dc3805e05a3c71adb79adcfeb": "\\beta = \\pi", "d4243ff8262cd00fbd7a0963a8b1bed5": "i=1,\\dots, 50", "d4245939b596a06f5c9535e32c9f1e94": " R\\otimes H ", "d424d05b8ca638bc2bc634d8b4dfb236": "x^2=5 \\Leftrightarrow x=\\pm \\sqrt{5}", "d424e1481e0378236558ee766c20d169": "{}^{j(\\kappa)}M \\subset M.\\!", "d424f63b3c4ae63ef0d82a94b2e2ea49": "P[1,7,R_S]", "d4254fab39e06ff5ed28c6a8267919a3": "\\rho = \\tfrac{\\pi}{2} - \\phi\\,", "d425830e50030ad71c36d70e6fbedcf1": "\\alpha^\\circ", "d425c55d57cd55fcab081d411d30c5a4": "k = 3", "d425d6f978575340331c52e5498f4463": "\\max\\left(x\\right)", "d42612ba2005a885b58ce0edb07ea3fb": " i = \\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 ", "d4267202da99e41c2d6e428f2119b2cc": "{\\mathbf P^{''}}=[{\\mathbf B} \\; | \\; {\\mathbf b}_4 ]", "d4269ff65b1c1108a4b4ab334b208302": "\n G_{ij} = \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~g_{\\alpha\\beta} \n", "d426c0ecf4f7908a1780c280f9cf9eab": "m + i \\leq n", "d4270c15d9df551ffe9179a47d7825ce": "c=(\\text{random},0)", "d42766fa2665b3832b18bcd618bd7414": "\\mu(t)=0", "d427d32c31278a84216870ff3294e6d6": "-NaN = NaN", "d428c3757e344613e5e48d4e384e3644": "\n\\begin{align}\n&\\frac{\\partial\\rho}{\\partial t}+\\nabla\\cdot(\\rho\\mathbf{u})=0\\\\\n&\\frac{\\partial\\rho\\mathbf{u}}{\\partial t}+\\nabla\\cdot(\\mathbf{u}\\otimes(\\rho \\mathbf{u}))+\\nabla p=0\\\\\n&\\frac{\\partial E}{\\partial t}+\\nabla\\cdot(\\mathbf{u}(E+p))=0,\n\\end{align}\n", "d428d533eaa8c20cd6e63c524fde7c6e": "\\frac{d T(A, R)}{dR}|_{R = T(A, R)}", "d429129dc5d422ce3c5124c817df36eb": "\\delta _\\nu^2=\\frac{2\\eta }{\\omega \\rho} ", "d42945708abcd557d27c41ec4f4651f2": "\\displaystyle{f=T_K\\varphi + T_K^*\\varphi}", "d429485b220ed0bf0e8531b0e12163f1": "\\Delta x\\equiv x_c-x_{0}", "d42961c7e605400edc9bca07b29da577": "\\sum_{k=1}^{n-1} k = \\frac{n(n-1)}{2} \\,\\!", "d429a5014d809158f4cad35a842b811b": "\\nabla J", "d42a21b343ac177e801ba1ab80d5eea3": "\\,F_{ab}", "d42a606546be4b2c6f0a0439be84eb30": "y = a \\sin(t)\\,", "d42ab533a56e15df3880434e531d7a09": "T_{MC}(s)/T_H(s)", "d42ac842316855cad0699d74109c0603": "\\,^{nat}_{68}\\mathrm{Er} + \\,^{136}_{54}\\mathrm{Xe} \\to \\,^{298,300,302,303,304,306}\\mathrm{Ubb} ^{*} \\to \\ \\mbox{no atoms}.", "d42ba06d6ca102d5306faff11d0f8f52": " \\mathrm{Operating\\ margin} = \\left ( \\frac {6,318}{20,088} \\right ) = \\underline{\\underline{31.45 %}} ", "d42c0bff92eca259c62f5df3c9945f18": "f_{ii}\\,", "d42c1d803e488fc932d60f9ddd703ab9": "v_{2,1} = 5", "d42c29037a33094b72d56b1e34d3b696": "Math \\leq bad", "d42c36bf8433c84797aceb8d07f0d7b8": "\npV = m R_s T\n\\,\\!", "d42c54dedf6b51bf127755df9351ca03": "f(z) = \\sum^\\infty_{k = 0} a_k z^k,", "d42c5d5d8536b715273e99a82758eac2": " {\\Pr}_{x_1 \\in \\!{F_p}}(x_1=0)= \\frac {1}{p} ", "d42c79508f860f12d3d71d6d979526af": "F(a_1, a_2,\\ldots, a_n) = 1", "d42ca04f5a6a50296f8b94786530610d": " f_x(y) = f(x, y)", "d42cd3fbd7c2364635403dcb766de29c": "P=(p_1 \\ldots p_k)", "d42d076517cd03a45e045c77e867c7c4": " \\Delta_T(t) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{n=0}^{\\infty} \\delta(t - n T) ", "d42d0b2f27ea456493dcdeb3151a97b6": "\\displaystyle \\sum_n(-1)^nTr(f|H_n(B))", "d42d49049a89f753a560a3c2f5c84437": "ds^2\\to h", "d42d600c34b685bfb606cb583919b069": "\\begin{align}\n\\sum_{n=1}^{\\infty} a_n & = a_1+\\underbrace{a_2+a_3}_{\\leq a_2+a_2}+\\underbrace{a_4+a_5+a_6+a_7}_{\\leq a_4+a_4+a_4+a_4}+\\cdots +\\underbrace{a_{2^n}+a_{2^n+1}+\\cdots +a_{2^{n+1}-1}}_{\\leq a_{2^n}+a_{2^n}+\\cdots +a_{2^n}}+\\cdots \\\\\n & \\leq a_1 + 2 a_2 + 4 a_4 + \\cdots + 2^n a_{2^n} + \\cdots = \\sum_{n=0}^{\\infty} 2^n a_{2^n}.\n\\end{align}", "d42d8e027e821f65ceefbe92f63439a1": "\\left (\\frac{\\pi}{\\sqrt{2}} \\right)", "d42dc12c453070081873c27de55c13d0": "\\int_{x_1}^{x_1 + \\Delta x} f(t) \\,dt = f\\left(c(\\Delta x)\\right) \\Delta x.", "d42e4beb13567115bf87097025eaaf82": "G_{ij}=\\langle v_j, v_i \\rangle", "d42e57e12840705717c6cd5ebabb1ce6": "\\cot\\frac{\\pi}{30}=\\cot 6^\\circ=\\tfrac{1}{2} \\left[\\sqrt3(3+\\sqrt5)+\\sqrt{2(25+11\\sqrt5)}\\right]\\,", "d42e782e916668aaa2c0d2f016ce8217": " \\phi(s) ", "d42e78d6652851f3b54fa8515f322833": " f(z) = f_i(z)f_o(z)", "d42e8c47157c5a8d9d016bbca401292e": "||x||_B \\leq C ||y||_V", "d42eaa44d77b8dd61745eddb2c8514b0": "\n\\delta S = k \\left[ 2N \\ln(2V) - N\\ln V - N \\ln V \\right] = 2 k N \\ln 2 > 0\n", "d42ee9d01476effe75cf8014237f5a58": " \\mathbf{A} \\, \\mathbf{x} ", "d42efb6c6c032c4d8b54a2e5283fe8e5": "e^x \\equiv \\lim_{p \\to \\infin} (1+1/p)^{px}", "d42f475abf7574cf2cee83538f584ec0": "W[f](x)=\\frac{1}{\\sqrt{4\\pi}} e^{-x^2/4} L[g]\\left(-\\frac{x}{2}\\right).", "d42f4804677a29bc9b918487fa742d8e": "M\\, ", "d43032a1b732547647e1fb138d8a4b51": " {\\boldsymbol{r}}_{\\text{SO}}=\\frac{\\hbar^2g}{4m_0} \\left(\\frac{1}{E_G}+\\frac{1}{E_{G}+\\Delta_0}\\right)(\\boldsymbol{\\sigma}\\times{\\boldsymbol{k}})", "d430f2158ca70ff4442bab16c657dc5e": "{}^{n}0 = \\lim_{x\\rightarrow0} {}^{n}x", "d4313e59930e0a0c9aed512842b46962": " E\\left[ \\Lambda(n+1) \\right] = \\Lambda(n) + \\frac{\\mu^2 E\\left[|e(n)|^2\\right]}{\\mathbf{x}^H(n)\\mathbf{x}(n)} - \\frac{2 \\mu E\\left[|r(n)|^2\\right]}{\\mathbf{x}^H(n)\\mathbf{x}(n)}", "d43141057d50d3214cea3226c6e27371": "2x^2 + y^2 + 32z^2 = n", "d431749bcb89cd02c0202cdc2d22d038": " \n\\begin{pmatrix} \nd & -b \\\\\n-c & a\n\\end{pmatrix}\n\\begin{pmatrix} \na & b \\\\\nc & d\n\\end{pmatrix}\n=\n\\begin{pmatrix} \nda-bc & db-bd \\\\\n-ca+ac & -cb+ad\n\\end{pmatrix}\n = \n\\text{if and only if }M\\text{ is a Manin matrix} \n=\n\\begin{pmatrix} \nad-cb & 0 \\\\\n0 & ad-cb\n\\end{pmatrix}.\n", "d43188ad089a78efafa5a6be37a4bc7a": "Q_1 x_1 Q_2 x_2 \\cdots Q_n x_n \\phi(x_1, x_2, \\dots, x_n).", "d431b4cfa61e81fcdaae6305cde36c76": "\\lbrace x_1, x_2, x_3, x_4\\rbrace", "d431ddac655634df1f4355f23af02a5b": "\\int \\langle x, (O - \\lambda I) y \\rangle \\langle y, G(\\lambda) z \\rangle dy = \\int \\langle x, (O-\\lambda I) y \\rangle \\left \\langle y, (O-\\lambda I)^{-1} z \\right \\rangle dy = \\langle x , z \\rangle = \\delta (x-z).", "d43299c130e59b2ac1e50a611323bfec": " \\overrightarrow{m} ", "d4329f83b088fc04ec047f70436c063b": "L(1-s,\\psi) = -{k_1^{s-1}\\over s} \\sum_{r=1}^{k_1}\\psi(r)B_s(r/k_1)", "d432f626f5ce9b13f25a956d01031efb": "\\mathcal{Y} = \\{0,1\\}", "d4332941ce597f285df1b749a1bd9ad9": "\n\\frac{\\partial}{\\partial t}\\left\\langle \\frac{\\partial f}{\\partial t},\\frac{\\partial f}{\\partial s}\\right\\rangle=\\left\\langle\\underbrace{\\frac{D}{\\partial t}\\frac{\\partial f}{\\partial t}}_{=0}, \\frac{\\partial f}{\\partial s}\\right\\rangle + \\left\\langle\\frac{\\partial f}{\\partial t},\\frac{D}{\\partial t}\\frac{\\partial f}{\\partial s}\\right\\rangle = \\left\\langle\\frac{\\partial f}{\\partial t},\\frac{D}{\\partial s}\\frac{\\partial f}{\\partial t}\\right\\rangle=\\frac12\\frac{\\partial }{\\partial s}\\left\\langle \\frac{\\partial f}{\\partial t}, \\frac{\\partial f}{\\partial t}\\right\\rangle.\n", "d43334c51d523984d9fb45b5bb2d75e2": "\\mathbb{C} \\to \\mathbb{C}^*, z \\mapsto e^z", "d433467e5f241c4f67a8f91efd6af48e": "p = {df \\over dx}(x),", "d433588ee0a728d4036af8105ec89b2e": "\\mathbf{P} := \\frac{1}{N} \\mathbf{1}", "d433a867ccb08b36ec83b86323dd0a38": "n \\mu (B) \\, ", "d4467b6dfec7b0cb280babf4bf6ffcbe": " L = \\frac{j^{\\star}}\\pi = \\frac\\sigma\\pi T^{4}.", "d4473341252589a7122499ff5badcb6f": "K_C = \\frac{V\\,T}{L},", "d44798bb3508b82db849a3d0e6c7f9d8": "x \\rightarrow x/r", "d44798d600397bd5efe53e7a4400d934": "\\begin{align}\nD_{a}=-\\ln{\\frac{J_{XY}}{\\sqrt{J_XJ_Y}}}\n\\end{align}", "d4479ba3dd2142580453bdb74b070cef": "P_1(X)", "d447a4e99dfda8155073b4a5cf20844b": "\\frac{\\partial \\nu}{\\partial \\tau} = \\frac{\\partial^2 V}{\\partial \\sigma \\, \\partial \\tau}", "d447cf32c129607302828c2db10b8db8": "a= \\tfrac{1}{2}", "d447f3a2ae604a2ffe6e446df235b9ad": "I=\\{i_1, i_2,\\ldots,i_n\\}", "d44813a9963f0f603d8235d98b02a9d3": " \\frac{\\sum_{a \\in A} f(a) w(a)}{\\sum_{a \\in A} w(a)}.", "d44816ae481348e484897dedd1d286fb": "1+(-1+1)+(-1+1) +\\cdots = 1.", "d44880fc0d38f07f67e734f75b7d3342": " \\displaystyle \\mathfrak{f}_2(\\tau) = \\sqrt2q^{-1/24}\\prod_{n>0}(1+q^{n})", "d44888a1cee1c3a092dafb7a0cd59bf7": "\\displaystyle{\\partial_n g = \\kappa - K e^g}", "d448ebf8c628b074f60ab9edb315c053": "\\mathit{IC} = -2\\mathbf{E}^\\theta[ \\log(p(y|\\theta))]+2p_D.", "d4493edc9f7149b47cd9853280f3170a": "K_\\text{c}=\\frac{{[R]} ^\\rho {[S]}^\\sigma ... } {{[A]}^\\alpha {[B]}^\\beta ...}", "d449430e869ee53bcc4ab9c9bdb8fa23": "p = (u_0, \\cdots, u_{L(p)-1}, u_0)", "d4496e85b122aa84d867b9f9aab109e8": "\\Delta z = \\operatorname{cov}\\left(\\frac{w_i}{w}, z_i\\right) + \\operatorname{E}\\left(w_i\\,\\Delta \\frac{z_i}{w}\\right)", "d449922bd5d933a3c3fe9525cb921fd8": "c=M", "d44a0e3cfb68700d54c1b63f959d751f": " |\\psi_n(x)| \\le K \\pi^{-1/4}", "d44a36b52b306d8a44f5cd623040fb2d": "\\partial_zX^\\mu-i\\overline{\\theta_L}\\Gamma^\\mu\\partial_z\\theta_L", "d44a48d32093e0d80c40600f88e3e53d": "Q_1A = \\begin{pmatrix}\n14 & 21 & -14 \\\\\n0 & -49 & -14 \\\\\n0 & 168 & -77 \\end{pmatrix},", "d44a51160ce9f4e304b2948d07adbf94": "Y\\ g", "d44a59bf9b6b0fc7327a200faf859c28": "\\frac{e^{-m\\tau}}{2m}+\\frac{\\eta}{m}\\cosh{m\\tau}\\;n_\\eta(m)", "d44a83333184ab0af351111253c0ef86": "agh+dfg=beh+cef", "d44a882df14b21667c9837f88f28d578": "10^{10^{120}}", "d44ade06c1fb6531cfc44617437a2a9d": "u(1)_C", "d44b4afda166734e6dcb22f4a2147a65": "A = CR", "d44b5ae5b4c70b879429b0b3f99f718d": "v(x) = u(x;\\varphi(x)),\\quad x\\in\\Omega,", "d44b7908910c3c58090fbdfe465a2d0b": " \\mathbf{x}^{(k+1)} = L_*^{-1} (\\mathbf{b} - U \\mathbf{x}^{(k)}) ", "d44bc090a7c08be56dcaafefc539f5c5": "\\Omega^*_X", "d44bc8516ca5fde6a0a2852978855ac3": " u \\to u^h ", "d44c09b3227c0fb4aa7b8cd54780782e": " \\mathbf{S}({\\mathbf{p}}(t)) ", "d44c17276124978406352284c6ae7fca": "\\displaystyle \\frac{1}{|a|}\\cdot \\operatorname{rect}\\left(\\frac{\\xi}{a} \\right)\\,", "d44c641ecc2cc02dcde473eae71e170b": "S(t) = p + [(1 -p) \\times S^*(t)]", "d44c75169a040931e6551976a0076c34": "\\scriptstyle \\nu(E)", "d44cce61e7a595adea26c48f70926113": "\n\\begin{bmatrix}\n25 & 16 & 80 & 104 & 90\\\\\n115 & 98 & 4 & 1 & 97 \\\\\n42 & 111 & 85 & 2 & 75 \\\\\n66 & 72 & 27 & 102 & 48 \\\\\n67 & 18 & 119 & 106 & 5\\\\\n\\end{bmatrix}\n", "d44d18a2ea41525ca5e5d8d0a812729f": "u(c)=1-e^{-\\alpha c}", "d44d591ed554968c9de7919e1e66e7fd": " \n \\begin{align}\n \\operatorname{E}(\\theta|\\mu,M) & = \\mu \\\\ \n \\operatorname{Var}(\\theta|\\mu,M) & = \\frac{\\mu(1-\\mu)}{M+1}.\n \\end{align}\n", "d44d8f050f8ee3b72415e6354ec9b638": " \\vec{\\theta} ", "d44da871fcd0ec38097b66d1f5a5d4ca": "\\mathbf{I}_i\\in \\mathbb{R}^{3 \\times 3}", "d44dc1c36e5119e32566a50507ec4608": "X = \\begin{pmatrix}\n 0 & -1 \\\\\n 1 & \\;\\;0 \n \\end{pmatrix} ", "d44dd9bfa8f7d7647c5274a5b9870f0f": "x_j \\in \\frac{[b_i]- \\sum\\limits_{k \\not= j} [a_{ik}] \\cdot [x_k]}{[a_{ii}]}", "d44def7c2199385ad1a1c01e19e2575e": "Mp", "d44e30cc54b363ec5220725249bac849": " H\\psi = i\\hbar \\partial_t \\psi", "d44e33e7e32c8f5d3bf8afa7254bd266": " \\beta_T ", "d44e3a29a098f71735779aada52758ac": "\\langle R, \\mathcal{E}(\\rho)\\rangle = \\langle \\mathcal{E}^{\\dagger}(R), \\rho \\rangle \\geq \\epsilon", "d44ec5cd77b72abdc5884f7ecb1ef5a7": "i_X", "d44f5b47ab377f0ebfe167c65eb27e24": "\\!X[A/x] = \\{ s[m/x] : s \\in A\\}", "d44fc9415ab4599d89671e8db1188200": "g_{SP} = g_{12}", "d44fdcce7df0bfb4565770ba81c53bbb": "\\begin{pmatrix} 1 \\\\ 0 \\\\ -1 \\\\ 0\\end{pmatrix}", "d45020632c692eeb084106134d443ef9": " e^{i \\varphi}=\\cos \\varphi +i \\sin \\varphi", "d45039a968d341dec7902e3b60385dd5": "a \\lor a = a", "d45039e4185b79014d892e5371660167": "\\ C_L = C_o ", "d4505f67e9254b32ab592ff1d8950022": "the\\ dog\\ barked\\ at\\ the\\ cat : (N \\cdot N_0^l) \\cdot N_0 \\cdot (N^r \\cdot S) \\cdot (S^r \\cdot N^{rr} \\cdot N^r \\cdot S \\cdot N^l) \\cdot (N \\cdot N_0^l) \\cdot N_0", "d45076e0a2dbd4a45f59f58bb11af3d0": "\\hat{a}_i", "d450bf87ed285dd589cce98e8e2f8c7f": " \\frac{a_{ph-e-p,e}(\\hbar\\omega-\\Delta E_{e,g}-\\hbar\\omega_{p})^2}{1-\\mathrm{exp}(-\\hbar\\omega_p/k_\\mathrm{B}T)} ", "d450c7ed637438f237f8d52609a2b573": "\\chi_{\\rm A} - \\chi_{\\rm B} = ({\\rm eV})^{-1/2} \\sqrt{E_{\\rm d}({\\rm AB}) - [E_{\\rm d}({\\rm AA}) + E_{\\rm d}({\\rm BB})]/2}", "d450db49ec933b46e5c6ae0f741a68c7": "O(n^2),", "d450e2daa61f974bae84e027d979d3a4": "\\theta^\\prime_p = Ad(h^{-1}_p)\\theta_p + h^*_p\\omega_H,", "d450f199cf2bd071dace845081a68591": "W = W_\\text{pressure} + W_\\text{gravity}. \\,", "d4513d99c186aa7b7a669b5a5822b4d1": "|z|^2+|w|^2=1", "d4516511ba45ca96d9f4edad6e987d77": " {e} ", "d451e22aefd457ad78ced8ec827e299e": "H(X_1)+H(X_2)", "d4520c8d861e339db1c11192e0c5d179": "(10\\uparrow)", "d4520d72c1b481115e94eabb0fcb1bd0": "vw=dt", "d45279a825830fd8aeac5c7d51cf0a22": "\\nu_0(X)-\\varepsilon\\mu(X)=(\\nu_0-\\varepsilon\\mu)(N)\\leq 0,", "d452d8b17f579737886cb63567eab964": " \\frac{\\zeta(s)}{\\zeta(2s) } = \\sum_{n=1}^{\\infty}\\frac{ |\\mu(n)|}{n^{s}} ", "d452e97ccc2da16906f1ade670a5e3aa": "C_1 = \\begin{bmatrix}\n0 \\end{bmatrix}\n", "d4530078b7a6edd0f9f735c7879427a7": "\n\\begin{align}\nQ_t & = \\int_0^{2\\pi}\\int_0^\\infty \\sigma\\left(\\rho\\right)\\, \\rho\\,d \\rho\\,d\\theta \\\\[6pt]\n& = \\frac{-qa}{2\\pi} \\int_0^{2\\pi}d\\theta \\int_0^\\infty \\frac{\\rho\\,d \\rho}{\\left(\\rho^2 + a^2\\right)^{3/2}} \\\\[6pt]\n& = -q\n\\end{align}\n", "d4531562aaeb1049c3c587ee287d8ad5": "\\lim_{a\\rightarrow 0+}\\left(\\int_{-1}^{-a}\\frac{\\mathrm{d}x}{x}+\\int_{2a}^1\\frac{\\mathrm{d}x}{x}\\right)=-\\ln 2.", "d45318a950a50be6e273bf3807f41aed": "\\mu_i \\sim \\text{i.i.d.} N(0, \\sigma^2_{\\mu})", "d45325b097f024a9f89109e14fdf79aa": " (X, Y)", "d45349ca5dff51ff2bcca39cc2ba0675": "\\frac{10}{9}", "d4534a6588d402caa6879d948f36541c": "\\frac{\\text{number of statements executed by the test data}}{\\text{total number of executable statements}}", "d45352e774bd00a52bee96c0b8a6eb16": "\\sigma_\\alpha(n) = (\\textrm{Id}_\\alpha*1)(n) = \\sum_{d\\mid n} d^\\alpha \\,", "d453727f5d0dbc81c976d6b068675221": "{\\partial A_{ik} \\over \\partial A_{ij}} = \\delta_{jk},", "d4538b2978b71a750531147a08511c6c": "{\\left \\langle \\frac{\\delta \\mathcal{S}}{\\delta \\phi(x)}\\left[\\phi \\right]+J(x)\\right\\rangle}_J=0", "d4538b9f487be27e96454cd9e2b21668": "\nV(\\omega_k) = H(\\omega_k)X(\\omega_k)\n", "d453d26c1aef7a214af73118ff385a0e": "{s}\\,", "d453f154c62462c5ad6f044f419ab6de": "\\vec{n} = \\vec{x}_1 / |\\vec{x}_1|", "d4544311db01b1deb1abf033b7ab9399": "\\psi'", "d4546bc9d2cd3163532b40c3bd109d01": "J_e", "d45482d6daa2e276ca0275b698cb32a1": "\n\\hbar\\omega=E_{+}+E_{-}.\n", "d454a4bc825c773c117aaeb5ba97e712": "\nF= \\dot{m} ( V_{w2} - V_{w1} )= \\dot{m} (V_{f2}\\tan\\alpha_2 - V_{f1}\\tan\\alpha_1)\\,\n", "d4553cfb31df81bc88c52e757b682d49": "a_2 \\leq\nb_2", "d4554fb122cf5752369972e767eb054c": "\\ F(t_i) = \\frac{1-e^{-(p+q)t_i}}{1+\\frac{q}{p} e^{-(p+q)t_i}} ", "d4556207793a3c060bc633da8cecc1bd": "a^\\dagger_j a_j = \\sigma_j^z+\\frac{1}{2}.", "d4556fbece9c82c91c4413eef60bbf43": "w(x) = \\begin{cases}\n \\tfrac{Pbx(L^2-b^2-x^2)}{6LEI}, & 0 \\le x \\le a \\\\\n \\tfrac{Pbx(L^2-b^2-x^2)}{6LEI}+\\tfrac{P(x-a)^3}{6EI}, & a < x \\le L\n \\end{cases}", "d455e8fbb4f825fa9711186560cdfff6": "x_1=a(x,t)", "d455eb62f2723d3e51d49accb9403c29": "\\Delta H_{vap}", "d456461a43571da9254a1207b6764efb": "i_b(t)", "d456471e49fe3f5f0dc0817759d211d9": "\\{x:= 4\\}[]x\\ \\dot{=}\\ 4", "d45656be1824cd8f874d8ba2aa004846": "\\lim_{n\\to\\infty} x_n =0.", "d456ecf93401f10e51d2ed7792bafa4a": "\\pi_1 (X, t)", "d45729004c99407a5ed2b1c976292f41": "\\frac{p_n}{p_0+p_1 + \\cdots + p_n} \\rightarrow 0.", "d4575fde92258314e22d6ea8750680c3": "\\subset A", "d4577c0ad354803e1c3e56a2405c4128": "\\{x\\in\\mathbb R^n : \\|x\\|_1 \\le 1\\}.", "d4579b2688d675235f402f6b4b43bcbf": "do", "d4580744f029728229be0b1b71fe0e72": "\\langle A,\\land,\\lor,-,0,1,\\Box\\rangle", "d4582dc42512b85cba991dbc34303bb9": "\\lim_{n \\to \\infty}n\\Beta(k,n) = \\textrm{Gamma}(k,1)", "d458e0c52d13b6e6adaa50ed14412e67": "\\phi : \\mathbb{F}_q^n\\to \\mathbb{F}_q^{n-k}", "d4590381623bd1ba2812380609ec5e25": "\\mathrm{FillRad}(C\\subset \\mathbb{R}^2) = R.", "d45939fc4f7498901fc0d06b86191e66": "\n X^{VG}(t; \\sigma, \\nu, \\theta) \\;=\\; \\Gamma(t; \\mu_p, \\mu_p^2\\,\\nu) - \\Gamma(t; \\mu_q, \\mu_q^2\\,\\nu)", "d4594965071293fb02ee40607b03c39d": " (c_1, c_2, c_3, \\dots) \\, ", "d459699ab031562a6c4fbb577db204bc": "a^2 + b^2 = c^2", "d459e01108c81fd999889d6b1f21b139": "\\rho\\ (x,y)", "d459ebeb40051c904f0364349bfda48a": "a+b = \\{q+r \\mid q\\in a, r\\in b\\}.", "d459fbcb662705b4951a9ec543ae645d": "\n\\begin{align}\nP(\\text{User}|\\text{+}) &= \\frac{P(\\text{+}|\\text{User}) P(\\text{User})}{P(\\text{+}|\\text{User}) P(\\text{User}) + P(\\text{+}|\\text{Non-user}) P(\\text{Non-user})} \\\\[8pt]\n&= \\frac{0.99 \\times 0.005}{0.99 \\times 0.005 + 0.01 \\times 0.995} \\\\[8pt]\n&\\approx 33.2\\%\n\\end{align}", "d45a1212cf9a6e61b4e28e462eb2be21": " E = ", "d45a402dd73113fb1e2e7e0bd29b0656": " TV ", "d45a59ebb15b47dbd03af8a4a45dccb0": "\nI_2=A_2 A_2^*=A_1^2 \\chi_0^2 \\Lambda^2 \\mbox{sinc}^2(\\Delta k \\Lambda/2) \\left(\\frac{1-(-1)^N \\cos(\\Delta k \\Lambda N)}{1+\\cos(\\Delta k \\Lambda)} \\right).\n", "d45a76a3a60b4c7aaf14f7de54ce06f8": "\\textstyle\\mu_{\\mathrm{B}}=\\frac{e\\hbar}{2m_{\\mathrm{e}}}.", "d45b59d774f3c21653b86cfcb8fca7dd": "SS_e = \\frac{1}{n(k-1)} \\sum_{i=1}^n \\sum_{j=1}^k (r_{ij} - \\bar{r})^2", "d45be87d56134c6b754f41364a28bc50": "k_BT/2", "d45cb36b861ff9362112ea00f2753174": "\\left[{6\\choose 2}+{6\\choose 3}+{6\\choose 4}\\right]2^{-6} = {25\\over 32} \\approx 78\\%.", "d45cf1885999eeaaa1763a5511683e19": "f^+:=f\\vee 0", "d45d3463b727aab1db4a07d058fa6518": "\\tau_p ", "d45daeb13abcd11b30092ef2ea7ddb14": "\\scriptstyle w", "d45dc227688c1d9de9565b2e257ce392": "B_I R = \\oplus_0^\\infty I^n", "d45dc65d8a4559579bee9902b05b3713": "y_1= g_1^{x_1} \\land y_2=g_2^{x_2}", "d45dcf11e4124ffa44fcab513ebd61f5": " {d^m x\\over ds^m} = \\sum_{|t|=m} \\alpha(t) \\delta_t,", "d45df25b25c9b7ed6545e3c43b5b7ab8": "\\scriptstyle 6n", "d45e0e8228f19f80392a6a8bc7a1db13": "(e^z)^w \\ne e^{z w}", "d45e222cda8dd7a75a1645e805c7bb3c": "\\scriptstyle \\lbrace x \\isin V : q(x)\\ =\\ \\text{nonzero constant} \\rbrace ", "d45e453bf26656f90bc48f469eefbf41": " \\xi{\\left(\\theta\\right)} = 2 + \\theta^2 - \\frac{\\pi}{8} \\exp{(-\\theta^2/2)}\\left[ (2+\\theta^2) I_0 (\\theta^2/4) + \\theta^2 I_1(\\theta^{2}/4)\\right]^2,", "d45e697b4c8eade6c17a65dc95ecc097": "\\begin{align}\ne^{i a(\\hat{n} \\cdot \\vec{\\sigma})} & = \\sum_{n=0}^\\infty{\\frac{i^n \\left[a (\\hat{n} \\cdot \\vec{\\sigma})\\right]^n}{n!}} \\\\\n& = \\sum_{n=0}^\\infty{\\frac{(-1)^n (a\\hat{n}\\cdot \\vec{\\sigma})^{2n}}{(2n)!}} + i\\sum_{n=0}^\\infty{\\frac{(-1)^n (a\\hat{n}\\cdot \\vec{\\sigma})^{2n+1}}{(2n+1)!}} \\\\\n& = I\\sum_{n=0}^\\infty{\\frac{(-1)^n a^{2n}}{(2n)!}} + i (\\hat{n}\\cdot \\vec{\\sigma}) \\sum_{n=0}^\\infty{\\frac{(-1)^n a^{2n+1}}{(2n+1)!}}\\\\\n\\end{align}", "d45ebf2ceb276bf2bc50f27dad2c50fc": " v^{\\alpha} = \\left(1, \\frac{d \\mathbf{x}_\\text{p}}{dt}(t) \\right) \\,,", "d45ec220eee8fb43bda12a1e18e0568a": "\\dim R + 1 \\le \\dim R[x]", "d45ec58fbe2c8ac99599efc121ffee97": "\n\\begin{align}\n\t|a|_p \\geq 0 & \\quad \\text{Non-negativity}\\\\\n\t|a|_p = 0 \\iff a = 0 & \\quad \\text{Positive-definiteness}\\\\\n\t|ab|_p = |a|_p|b|_p & \\quad \\text{Multiplicativeness}\\\\\n\t|a+b|_p \\leq |a|_p + |b|_p & \\quad \\text{Subadditivity}\\\\\n\t|a+b|_p \\leq \\max\\left(|a|_p, |b|_p\\right) & \\quad \\text{it is non-archimedean}\\\\\n\t|-a|_p = |a|_p & \\quad \\text{Symmetry} \n\\end{align}\n", "d45ef154bba3314c2a7cbb3f7d9e21d6": "\nH = H_{\\text{kin}} + H_{\\text{pot}} = \\frac{p^2}{2m} + \\frac{1}{2} a q^2.\n", "d45f2f265dbbd88e18e1bc440ce402b2": " \\operatorname{de-let}[\\lambda V.E] ", "d45f54a4894552d4c4b07492de71a488": "\\mathbf{X}_k", "d460767970cac76253ad4697ad121960": "\\boldsymbol{g}=-\\mu\\frac{\\boldsymbol{\\hat{R}}}{|\\boldsymbol{R}|^2}", "d460d5b23a7fa89ef7346d72c28cf762": " \\Gamma(z) = \\int_0^\\infty t^{z-1} \\mathrm{e}^{-t}\\,\\mathrm{d}t ", "d46142cded99ed6a275bec302f94d59a": "\n\\mathbf{F}=q\\left(\\frac{Q}{4\\pi\\varepsilon_0}\\frac{\\mathbf{\\hat{r}}}{|\\mathbf{r}|^2}\\right)=q\\mathbf{E}\n", "d46153a12372216ce70f6a13a11895c1": "R_0/W_0", "d461692ac172a65d150bbf45af4a5d81": "\\mathbb{G_A}", "d4619bfa4ae4781a79570a5d26d5735a": "x^*(q)", "d461a65056fce1bb8d35b1f05a56695b": "\n \\Omega^\\mathrm{spont}_{\\mathbf{k},\\omega} = \\mathrm{i} \\mathcal{F}_{\\omega} \\Bigl(f^e_{\\mathbf{k}} f^h_{\\mathbf{k}} + \\sum_{\\mathbf{k'}} c_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}} \\Bigr)\\,.\n", "d461c3e8dd824cc0d6211194d5c0052f": "i\\pi", "d4620697bcf885f17f5991194af62eb5": "\\tilde{z}_{\\cdot\\cdot}", "d462615fe1c4dd374cee1a0d66090c9d": "d + d^*", "d4627b427f2e8b75b7881e1a065249a6": "\\mathrm{cons}_\\alpha\\ x\\ l = \\mathrm{roll}\\ (\\mathrm{inr}\\ \\langle x, l\\rangle)", "d463301f93b269ba849664e416269f7a": "z \\in \\{0,1\\}^n", "d4634372f2dbf6ff46505b795344d189": " {\\stackrel{\\triangledown}{A}}_{i,j} = \\frac {\\partial A_{i,j}} {\\partial t} + v_k \\frac {\\partial A_{i,j}} {\\partial x_k} - \\frac {\\partial v_i} {\\partial x_k} A_{k,j} - \\frac {\\partial v_j} {\\partial x_k} A_{i,k} ", "d46371495eceae7ae47311f903c0bfe6": "k_C(f)=\\frac{-C_m}{\\sqrt{(C_1+C_m)(C_2+C_m)}}\\frac{2}{\\sqrt{(1+f_1^2f^{-2})(1+f_2^2f^{-2})}}", "d463853c5472d125d50eebcdd5c57cf7": "p(x) = \\sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots + a_n x^n,", "d46387956e5d8de6e8f4c8e94b36113e": "\\begin{align}\n&\\underset{x}{\\operatorname{minimize}}& & f(x) \\\\\n&\\operatorname{subject\\;to}\n& &g_i(x) \\leq 0, \\quad i = 1,\\dots,m \\\\\n&&&h_i(x) = 0, \\quad i = 1, \\dots,p \n\\end{align}", "d4639e4d5619f75048027e2c297a0f3c": "(N,NP, \\text{ and } S)\\,\\!", "d4640b0b04a4049b72c05988ccd6e551": "\n\\tilde{Q_s} = \\frac{3}{2} \\frac{V}{S\\delta_s} \\frac{1}{1+\\frac{3c}{16f}\\left(1/l + 1/w + 1/h \\right)}\n", "d4643873a78c7d6b973fc53d2330bd83": " R=I-P' ", "d46438abbd2d260a259e0ba3368d968e": "Q_1(f)", "d464853312d95aa4cad766219e3eba4b": " ^\\ominus ", "d464cc97dc0cc2e3f0e01ab7c07b47bf": "|H( \\pi)|=0", "d464eff1a5ece00a4358ad52eb430f67": "\\mathbf{I} =\\sum_{k=1}^Nm_k((\\mathbf{r}_k\\cdot\\mathbf{r}_k)\\mathbf{E}-\\mathbf{r}_k\\otimes\\mathbf{r}_k),", "d46528bcc81b5a3661f541f159628144": "\\alpha = \\beta = \\theta_B", "d46587027d97ab30a50e17c581de567c": "P_{A\\alpha}\\,", "d46588b362758646a7d4641b207d20ca": "\n \\underline{\\underline{\\mathsf{S}}} = \n \\begin{bmatrix}\n \\tfrac{1}{E_{\\rm 1}} & - \\tfrac{\\nu_{\\rm 21}}{E_{\\rm 2}} & - \\tfrac{\\nu_{\\rm 31}}{E_{\\rm 3}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm 12}}{E_{\\rm 1}} & \\tfrac{1}{E_{\\rm 2}} & - \\tfrac{\\nu_{\\rm 32}}{E_{\\rm 3}} & 0 & 0 & 0 \\\\\n -\\tfrac{\\nu_{\\rm 13}}{E_{\\rm 1}} & - \\tfrac{\\nu_{\\rm 23}}{E_{\\rm 2}} & \\tfrac{1}{E_{\\rm 3}} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\tfrac{1}{G_{\\rm 23}} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm 31}} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & \\tfrac{1}{G_{\\rm 12}} \\\\\n \\end{bmatrix}\n ", "d465910d50473ac568362d23ae2e6841": " V_{lost ECF} = V_{ECF b} - V_{ECF a} ", "d465a39a52274ac65092487bdc27b548": " = [\\textrm{CO}_2]_{eq} \\left(1 + \\frac{K_1}{[\\textrm{H}^+]_{eq}} + \\frac{K_1K_2}{[\\textrm{H}^+]_{eq}^2}\\right) ", "d46669c1aa19e8ec697d3bba4b99fb9e": "\\operatorname{BG}_p(n^2;x)=\\sum_{n=0}^\\infty (p^{n})^2x^n=\\frac{1}{1-p^2x}", "d46699b0b54841de409d837eac055f80": "\\frac{E}{2}", "d4669a3e3c9e46fea12b8224c2a75125": "B^0 \\leftrightarrow \\bar{B}^0", "d467846707c4dea16bdf3353f919ce54": "\\prod_{n = 1}^\\infty \\mathbb R = \\mathbb R \\times \\mathbb R \\times \\cdots", "d46805137e20d378ea0630ccd073be97": "{2^{2^{2}}}-3", "d468c45888826b0b82e62e2e6977b9be": "\\psi(x_1, x_2,\\dots) = -\\psi(x_2,x_1, \\dots) \\qquad \\text{for fermions} ", "d468ef76661f312c0be57560f53c997e": "\\forall k>0 \\; \\exists n_0 \\; \\forall n>n_0 \\; |f(n)| \\le k\\cdot |g(n)|", "d4690afabf213a58549dc053b97086c5": "\\exists M \\exists \\mu ~.~ ( M,\\mu \\models \\Phi)", "d46910ce8eefb0dce42655c9837e49a6": "f * g = g * f \\,", "d4691e52d4e251e02932f51de74961e3": "\\int_0^\\infty e^{-ax}\\sin bx \\, dx=\\frac{b}{a^{2}+b^{2}}", "d469570a5235ef6a100d60d99f1bd56a": "\\sigma_{\\pm} : \\operatorname{O}(p,q)\\to \\{-1,+1\\}.", "d469691dbfd50a88598dbe2448e454e9": "PdV\\ = \\gamma dA", "d469b5d5c0872540aa52f34bf837e763": "v_0 [S^0]_0 /[S^1]_{0^{ }}", "d469bfca2d775d8adb7196142aceba7c": "C_\\mathfrak{st}^\\lambda", "d469c90eb80149916620a287217263cc": "s^2 = 0", "d469e5adefc3ac0a513e1bebd4474fa7": "C=\\{ \\lambda_1 a_1 + \\ldots + \\lambda_n a_n \\mid \\lambda_1,\\ldots,\\lambda_n \\geq 0, \\lambda_1,\\ldots,\\lambda_n \\in\\mathbb{R}\\}", "d46a668c7bf4fd5868ef714e809355c5": "\\mu(tI)=t\\mu(I)", "d46a6cd7dc718f76e29cc2366b8b70d3": "\\displaystyle e^{- i a \\nu} \\hat{f}(\\nu)\\,", "d46a969c844277a459cf028f209e346f": "\\alpha=\\pm 0.9 ", "d46a9c915c9ebcd71311e247e8712111": "R_{aft}", "d46ac2c701bb0625b421e1ca382d7174": " a(T) = b \\cdot \\left( \\sum_i {x_i} \\frac{a_{ii}(T)}{b_{ii}} + \\frac{g^E_{res}} {-0.53087} \\right) ", "d46ac3c140435968658de591f238815b": "\\sigma_{ij} = C_{ijkl} \\, \\epsilon_{kl} \\, \\rightleftharpoons \\, \\epsilon_{ij} = S_{ijkl} \\, \\sigma_{kl} \\,", "d46aefc436d86b4be08ec8c8f4b590a2": "\n\\begin{align}\ny+ix&=f(\\zeta)=f(\\psi+i\\lambda)\\\\\n &= f(\\psi+i.0) + A_1\\lambda+ A_2\\lambda^2+ A_3\\lambda^3+ \\ldots,\n\\end{align}\n", "d46b032afdf6d4a01d3f520fba4211da": "\nx = \\frac{1}{2} \\left( \\xi - \\eta \\right)\n", "d46b141eefdfb73fe8e41800d192a75a": " M_t = \\sup \\{ W_s \\, \\colon \\, s \\in [0,t] \\}, ", "d46b8d26131da8c51a297f54bb617a17": " 0.0347 \\times \\sqrt{h} ", "d46be37163fdfc204e95d257d0515696": "[0,1],", "d46c04c65fb4341cb44a3300e50d71a9": "\\sum_{j=0}^\\infty C_j / \\lambda^{j(n-2)}", "d46c28b0a2d325b74bc57062ebd2b274": "-jY_{\\varepsilon } cot(k_{x\\varepsilon }w)+Y_{o}=0", "d46c46b9c7f41fd58c0c40b190da9825": "\\langle a \\mid a^n\\rangle,\\,\\!", "d46c7f55737736f992b9f75016a6bce7": "x^3 + y^3 - 3 a x y = 0 \\,", "d46cccf5fcb64f9928c0439ff500c599": "(M_L = 4, M_S = 1)", "d46d2b170cc2560d5a1077efce6315ae": " {\\mathcal L}^2_1: L=l^{(1)}_2l^{(1)}_1; ", "d46d34b8507cd27bafdda4cff9470a48": " x^{\\nu}", "d46d4c5dcccb305f5709725f57a3a11e": " J = \\frac{ H } {log_e( S ) } ", "d46da7b3fa1481daf3c60dde17e297a8": "X_6", "d46db4c421c2f05d3458e85ca0a5ca04": "0 \\le \\rho \\le 1", "d46dd57ec7c32cd3237629758261b12c": "\\{f_i\\}", "d46ddda2b7222137f628f8106bc78411": "Q(x) = 1-x^2\\,", "d46ded747a176a68ec9176c5702997d9": "\\mathbf{d}_0, \\mathbf{d}_1, \\dots, \\mathbf{d}_{p-1}", "d46ed831dde7e79f643d09c3a51bef0b": "(S \\subseteq X) \\; \\mapsto \\; \\forall_f S\n= \\{ \\; y \\in Y \\; \\mid \\; \\forall x \\in f^{-1} \\lbrack \\{y\\} \\rbrack, x \\in S \\; \\}", "d46f122c4f35c6995ae5037b27acce79": "\\frac{3\\sqrt{2}}{4} \\approx 1.0606601.", "d46f1969245f332ba8459dc474381835": "(0,\\tfrac{1}{2})", "d46f47f2e5dda9f8528c7c7edc11ae56": "Z \\subset {\\mathbb R}^m", "d46f4a22f75dea08376d8bba16b6e413": "\n \\sigma_{xx} = \\frac{12z}{h^3}\\,M_{xx} \\quad \\text{and} \\quad \n \\sigma_{zx} = \\frac{1}{\\kappa h}\\,Q_{zx}\\left(1 - \\frac{4z^2}{h^2}\\right)\\,. \n", "d46f6ee8cd1b81e00fbe9d36e4676475": "S\\left(Y\\right)", "d46f830021d8d137501bf133470ce5a2": "\\phi_x(t) = \\exp(a_1(e^{it}-1)+a_2(e^{2it}-1))", "d46fb132e3f37ea9b84837aaae698238": "P-\\frac{f(P)}{(P-q)(P-r)(P-s)} = P - 0 = P.", "d46ff51cdcb847964e9153234e47b61a": "\n\\begin{align}\nW & = \\sum_{m_1=1}^n \\left| A_{m_1} \\right| \\\\\n & {}- \\sum_{1 \\le m_1 < m_2 \\le n} \\left|A_{m_1} \\cap A_{m_2} \\right| \\\\\n & {}+ \\sum_{1 \\le m_1 < m_2 < m_3 \\le n} \\left|A_{m_1} \\cap A_{m_2} \\cap A_{m_3} \\right| \\\\\n & {}- \\cdots \\\\\n & {}+ (-1)^{p-1} \\sum_{1 \\le m_1 < \\cdots < m_p \\le n} \\left|A_{m_1} \\cap \\cdots \\cap A_{m_p} \\right| \\\\\n & \\cdots. \\\\\n\\end{align}\n", "d4700a23f59effd43772db6df1193bce": "\\Phi (A) = \\sum_{i=1} ^{nm} \\lambda_i V_i A V_i ^*", "d4701b6b55ea368ee58b758903c9716f": "1 +\\sum_{k=1}^{\\infty} \\left( \\prod_{r=0}^{k-1} \\frac{2r+1}{2r+2} \\right) \\frac{t^k}{k!}", "d47023972f661026a9f1d4059c10ddf1": "P+Q", "d4706805c5dcf34814f275b7883017f3": "\n\\begin{align}\n\\operatorname{dCov}^2(X,Y) & := \\operatorname{E}[\\|X-X'\\|\\,\\|Y-Y'\\|] + \\operatorname{E}[\\|X-X'\\|]\\,\\operatorname{E}[\\|Y-Y'\\|] \\\\\n&\\qquad - \\operatorname{E}[\\|X-X'\\|\\,\\|Y-Y''\\|] - \\operatorname{E}[\\|X-X''\\|\\,\\|Y-Y'\\|] \n\\\\\n& = \\operatorname{E}[\\|X-X'\\|\\,\\|Y-Y'\\|] + \\operatorname{E}[\\|X-X'\\|]\\,\\operatorname{E}[\\|Y-Y'\\|] \\\\\n&\\qquad - 2\\operatorname{E}[\\|X-X'\\|\\,\\|Y-Y''\\|],\n\\end{align}\n", "d4706c255af838ab7db2ecd27e66e040": "\n\\prod_{k=0}^{n-1}\\frac{(3k+1)!}{(n+k)!} = \\frac{1! 4! 7! \\cdots (3n-2)!}{n! (n+1)! \\cdots (2n-1)!}.\n", "d4706ce7163973997ec41bb83e1744b2": "\\epsilon=1\\;", "d470d22f3fdd1dc41423107bfd4ed16a": "\\overline{X}_{\\mu \\leq \\epsilon}", "d4712c5b378e2b32b6e118ed37fddf9b": " q_z = \\frac{4\\pi}{\\lambda}\\sin ( \\theta )", "d47136bef6a2fef79cdf79f184e68ff5": "{{\\left\\| f \\right\\|}_{\\Beta ,p}}={{\\left( \\sum\\limits_{m=0}^{\\infty }{{{\\left| \\left\\langle f,{{g}_{m}} \\right\\rangle \\right|}^{p}}} \\right)}^{1/p}}", "d4714a06b5c942a24e6d4a8ff3f855c4": "{t}_{n}", "d47166ed91ed394ae8b0114e94d0934f": "\n\\begin{array}{rcl}\nV_{1,k} &=& \\mathrm{P}\\big( y_1 \\ | \\ k \\big) \\cdot \\pi_k \\\\\nV_{t,k} &=& \\mathrm{P}\\big( y_t \\ | \\ k \\big) \\cdot \\max_{x \\in S} \\left( a_{x,k} \\cdot V_{t-1,x}\\right)\n\\end{array}\n", "d471835b5737901128599198c3ed3560": " \\mathbf{A}^{-1} = \\frac{1}{\\det (\\mathbf{A})}\\sum_{s=0}^{n-1}\\mathbf{A}^{s}\\sum_{k_1,k_2,\\ldots ,k_{n-1}}\\prod_{l=1}^{n-1} \\frac{(-1)^{k_l+1}}{l^{k_l}k_{l}!}\\mathrm{tr}(\\mathbf{A}^l)^{k_l},", "d4723e9b77b5426c7dd64575b23c5286": "\\phi(h)= \\rho_{X_0X_h\\cdot \\{X_1,\\dots,X_{h-1} \\}}. ", "d472ac6a93fa536d338f0e96e1a54bcf": " |w|\\ge {1\\over 4}.", "d472d7192baef6ca8e526e2d52b778a3": "\n\\begin{align}\nH(X) &= - \\sum_{i=1}^n p_i \\log_2 p_i \\\\\n&\\leq - \\sum_{i=1}^n p_i \\log_2 q_i \\\\\n&= - \\sum_{i=1}^n p_i \\log_2 a^{-s_i} + \\sum_{i=1}^n p_i \\log_2 C \\\\\n&= - \\sum_{i=1}^n p_i \\log_2 a^{-s_i} + \\log_2 C \\\\\n&\\leq - \\sum_{i=1}^n - s_i p_i \\log_2 a \\\\\n&\\leq \\mathbb{E}S \\log_2 a \\\\\n\\end{align}\n", "d4733eac329ef036dcb16e2db46ffdfd": "(D,V,s,R) \\nvDash P", "d4738184fa6738875d36045068338858": "pH = -log \\sqrt {10^{-5} \\left( 5 \\times 10^{-4} \\right) } = 4.15", "d473dcbe0354fe58be62a5fae85d197f": "\\begin{align}\n\\begin{cases} \\gamma_{1}:[0, 1]\\to {I}^{2} \\\\ \\gamma_{1}(t) := {\\delta}_{[1,2,0]}(t)=(t,0) \\end{cases}, \\qquad &\\begin{cases}\\gamma_{2}:[0,1] \\to { I }^{2} \\\\ \\gamma_{2}(t) :={\\delta}_{[1,1,1]}(t)=(1, t) \\end{cases} \\\\\n\\begin{cases} \\gamma_{3}:[0, 1] \\to {I}^{2} \\\\ \\gamma_{3}(t) := \\ominus{\\delta}_{[1,2,1]}(t)= (-t+0+1, 1)\\end{cases}, \\qquad &\\begin{cases}\\gamma_{4}:[0,1] \\to {I}^{2} \\\\ \\gamma_{4}(t) := \\ominus{\\delta}_{[1,1,0]}(t) =(0, 1-t)\\end{cases}\n\\end{align}", "d4744ab8d2731e02928346503f06090e": "\\scriptstyle\\vec{\\sigma}", "d47454a51132d222958a5c4a7bb04272": "(1 - \\cos m\\pi) = (1 - \\cos n\\pi) = 0", "d474b03a694044db501ee76626e9e671": "K(k(6))", "d475155473c520f7e6e39e3f7053dabf": "\\delta_\\varepsilon \\bold{F} = \\varepsilon \\bold{F}", "d475320ada5836d91610c2e20f1e306a": "\\eta_{1234}=\\sqrt{-g}\\;", "d475a8b14041a2ce97471c58c2a6472c": "f(\\mathbf{a}) \\leq f(\\mathbf{b}).", "d475e62f6c831e954aa562a56d00d04d": "4 \\sqrt{\\tfrac{2}{5}} = \\frac{4 \\sqrt{2}}{\\sqrt{5}}", "d4766064add511a2891f96ad74dc6115": " \n\\int_o^{\\infty} {k\\; dk \\over k^2 +m^2} J_1^2 \\left( kr \\right)\n\\rightarrow\n{1\\over 2 }\\left[ 1 - {1\\over 8} \\left(mr\\right)^2 \\right]\n . ", "d47681289e82d37052b3da7785add15d": "\n\\begin{bmatrix}\n\\cos{\\theta} & -\\sin{\\theta} & 0 & 0 \\\\\n\\sin{\\theta} & \\cos{\\theta} & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \n\\end{bmatrix}\n", "d476943ef0e7aca61f61b63319a73259": "D=E(f_{Y}(X\\beta)XX^{\\prime})", "d4769f02cc09069fb2f4df97de455ca8": "\\mathbf {D}", "d476c88490ce00d6c178454b16e222e1": "q_m\\left(\\sum_i\\xi^iX_i\\right) = (\\xi^1)^2+(\\xi^2)^2+\\cdots+(\\xi^p)^2 - (\\xi^{p+1})^2-\\cdots-(\\xi^n)^2", "d4774fbf4147ffe45ab143afe0892a78": "3 \\div x", "d477873c1afd700b57f56f21359d61f0": "Z_1=Z_2=1", "d477d1bd527801659a06c250e3531a39": "W_{S_1} \\ge W_{S_2}", "d477f71b83e613b9b26b8fd6e0f47d10": "(h_0, h_1, h_2, ..., h_{d+1}).", "d4780c03df8af17c57f3c239c813c1ae": "(c + d)/2 - \\sqrt N = (\\sqrt d - \\sqrt c)^2 / 2 = (\\sqrt N - c)^2 / 2c", "d4781673358542ea804753316282fc87": "X(t)=\\sum_{j=1}^n c_j+r_n.\\,", "d4781ff89a7232aa62ed606124983608": "\\gamma>0", "d4782968a8fa596872b873ed1ba6adf3": "Q = u b h", "d478656e70e86adcd238d83a03875dcc": "\\textstyle{f(x)=\\sum_{k=1}^\\infty \\frac {\\sin(b^k x)} {a^k}}", "d478afd9d972149cdf51d4adb5bae7db": "\\frac{t(t+1)}{2}=s^2\\, ", "d478f5db3f38248e8286b7b0c81ca4e4": "a(t,s)=0", "d479063d1799178d1ce3d6e0848096e5": "G= \\tfrac{\\pi}{8}\\ln(2)+\\sum_{n=0}^{\\infty}(-1)^n\\frac{H_{2n+1}}{2n+1}.", "d47983b0dbaaedcbcbf16d249eff558f": "\\phi^i", "d479b2e27875e76480223ee234ee786f": "\\lceil \\mu \\rceil", "d479e1e72d920d28449cd1e53e63acd4": "\\int_0^\\infty \\sin ax^2\\ dx=\\int_0^\\infty \\cos ax^2= \\frac{1}{2}\\sqrt \\frac{\\pi}{2a}", "d479e699d2e049f3bed2fb09d6e1ec3a": "\\mathit{q}^{RC} = \\mathit{\\bar{q}_n}\\mathit{\\bar{q}_{n-1}} ... \\mathit{\\bar{q}_1}", "d47a01023aadf5ff4b043e69f178455b": "\\left(\\tfrac{D}{n}\\right)=0", "d47a2d4d4c7b665be9ecb36f707c0320": " \\mathbb{D} ", "d47a3d1ffa25782c2c37ae4b9bde4a60": "E_{x,y} = \\frac{\\% \\mbox{ change in } x}{\\% \\mbox{ change in } y}", "d47a58e337ca287fa4bbdd3f86d52cf3": " \\mathrm{Id}_X - ST \\quad\\text{and}\\quad \\mathrm{Id}_Y - TS ", "d47a5d0ad2c6af951636b4ae2f4a4533": "\\sigma_1, \\sigma_2 \\in \\mathcal{K}", "d47a64fa7140fdd671fc0700e5df307e": "\\check{x}", "d47abac50a04f4054c752c60b1a4206a": "\n\\begin{align}\n\\int \\arcsin x\\,dx &{}= x\\,\\arcsin x + \\sqrt{1-x^2} + C\\\\\n\\int \\arccos x\\,dx &{}= x\\,\\arccos x - \\sqrt{1-x^2} + C\\\\\n\\int \\arctan x\\,dx &{}= x\\,\\arctan x - \\frac{1}{2}\\ln\\left(1+x^2\\right) + C\\\\\n\\int \\arccot x\\,dx &{}= x\\,\\arccot x + \\frac{1}{2}\\ln\\left(1+x^2\\right) + C\\\\\n\\int \\arcsec x\\,dx &{}= x\\,\\arcsec x - \\ln\\left[x\\left(1+\\sqrt{{x^2-1}\\over x^2}\\right)\\right] + C\\\\\n\\int \\arccsc x\\,dx &{}= x\\,\\arccsc x + \\ln\\left[x\\left(1+\\sqrt{{x^2-1}\\over x^2}\\right)\\right] + C\n\\end{align}", "d47ad7d97a9dedca055bcc0826ed4426": "\n\\Phi = \\frac{(\\Delta G^{TS-D}_{W} - \\Delta G^{TS-D}_{M})}{(\\Delta G^{N-D}_{W} - \\Delta G^{N-D}_{M})} = \\frac{\\Delta\\Delta G^{TS-D}}{\\Delta\\Delta G^{N-D}}\n", "d47ae93ea4855e0812b92cf5e825b28e": "\\begin{align}\n B_{n} &= (-1)^{\\left\\lfloor \\frac{n}{2}\\right\\rfloor }\\left[ n\\text{ even}\\right] \\frac{n }{2^n-4^n}\\, T_{n}\\ , \\quad (n = 2, 3, \\ldots) \\\\\n E_{n} &= (-1)^{\\left\\lfloor \\frac{n}{2}\\right\\rfloor }\\left[ n\\text{ even}\\right] T_{n+1} \\quad\\quad\\qquad(n = 0, 1, \\ldots)\n\\end{align}", "d47af9af795c4e6736bef1b7dc4c0321": "i_{th} (0)", "d47b17ed20e3f71646bab45d21c4b062": "\\mathrm{STA}_w = \\left(\\tfrac{1}{T}\\sum_{i=1}^T\\mathbf{x_i}\\mathbf{x_i}^T\\right)^{-1} \\left(\\tfrac{1}{n_{sp}} \\sum_{i=1}^T y_i \\mathbf{x_i}\\right),", "d47b29cfd4a6c8cb4a945951493e1d0f": "D = C_D {\\tfrac {1}{2} \\rho V^2 A}", "d47b2b5c8da55c166b3a9261074c70fd": "E = \\tfrac{1}{2}I \\omega^2", "d47b6beaabc7b436851117878e0f6852": " \\mathrm{e}^2 = 7.3891\\ldots ", "d47b77414912ae0ef25a062faffa6593": "\\nabla y_t =a_0 + u_t \\,", "d47b7c84a5b29c9b7298956308a22025": "4 \\,", "d47b8f5239528e276bb3d4c5b0fe527b": " J = \\sigma E \\,\\, \\rightleftharpoons \\,\\, E = \\rho J \\,\\!", "d47be8d1ea271ced699a15a932964580": "{\\tilde{F}}_{4}", "d47c0c12ef43d3621ec69772e3352e6c": "M = -\\int S dx", "d47c402b2cd8ac3fec4a400e81367d59": "(4)\\quad \\theta_{ab}=B_{(ab)}\\;,\\quad \\omega_{ab}=B_{[ab]}\\;.", "d47c587fe800aa151c7c1d9ae991166a": "\\nabla^2 \\phi = 4 \\pi G \\rho \\,.", "d47c69c46f5aa74dcb9e1962273f2732": "z_2=0", "d47c81b64f234406905c530d75a9cf5d": "\\alpha,\\ldots,\\alpha^{d-1}.", "d47c9e20223a8f5bb1c732267d8230be": "CFM = \\frac{Q}{Cp \\times r \\times DT}", "d47ce45f78599c0e5d8b44ba005c0087": "\\tau=2\\left({UG \\over V}\\right)^{1 \\over 2},", "d47d14afa57d32be46f114ad5b1ab940": "\\scriptstyle\\Omega\\in\\mathbb{R}^n", "d47d1f2cebec94a6f3e2ebc3f1556d9a": "\\Phi(\\mathbf{r}_2)", "d47d289b14b533e25fc857e956fecc1e": "\\xi=\\frac{x-\\mu}{\\sigma},\\ \\alpha=\\frac{a-\\mu}{\\sigma},\\ \\beta=\\frac{b-\\mu}{\\sigma}", "d47d582371fb282a391a667b1ee860b6": "\\begin{pmatrix} X_1\\\\ \\vdots \\\\ X_n \\end{pmatrix}\n = \\bar X \\begin{pmatrix} 1 \\\\ \\vdots \\\\ 1 \\end{pmatrix}\n + \\begin{pmatrix} X_1-\\bar{X} \\\\ \\vdots \\\\ X_n-\\bar{X} \\end{pmatrix}.", "d47d69f378e65f218eb57139c6c7ef27": "N{{u}_{b}}=\\frac{\\left( \\frac{q}{A} \\right){{D}_{b}}}{\\left( {{T}_{s}}-{{T}_{sat}} \\right){{k}_{L}}}", "d47d6b65816a23c367d63a3473a11602": "R_{\\text{t}i}", "d47d74d698976d3382ca7ab50e6b7b32": "\\rho_C^2", "d47d9dffbbc9526edbea1893d22804fd": "F_{1} P F_{2}", "d47e22db4625b450343bb8fdda66cff2": " f_*(\\mu) \\,", "d47e3576ea91ce1ed51690c0127198d4": "\\hat S(t) = \\prod\\limits_{t_i= 0.02 ", "d486512657361a6c607c882fd9abf9f3": "\\ell(x)-66", "d4865cba1a5c581e3b8e3527f3674d2c": "X\\times A+1", "d4869183aef58333dc102c2699c851a2": "fH_i M \\to \\mathrm{Hom}_{\\Bbb Z}(fH_{n-i} M,\\Bbb Z)", "d486a4deef563c08eac36f3eac9e3124": "\\hat{X}_{Bayes}(\\mathbf{v}) = \\text{argmin}_{\\hat{x}\\in\\mathcal{X}}\\mathbf{v}^\\top\n\\lambda_{\\hat{x}}", "d486a4e7cb1a0d0806c3b2b769d56382": "a \\in S^* \\implies a \\in (S \\setminus \\{a\\})^*", "d486bb67e4d764fb4d375cec40a31eda": "DPV = \\sum_{t=0}^{N} \\frac{FV_t}{(1+i)^{t}}", "d486c415d3429aed3e484c0101fc3df6": "||\\bigstar |\\bigstar \\bigstar", "d486f4c83cf1910e2c9bd32b4d349bf4": "\\beta(2[S_3/\\mathbf{Z}_2] + [S_3/\\mathbf{Z}_3]) = \\beta([S_3] + 2[S_3/S_3]).", "d486fd23a36eb93a57ed78c2c59fbf20": " o(M)", "d4870a294dc77aad67c62ac87bb83fed": "f:C\\rightarrow M", "d487c989e4301a0d565a66356cc8bba2": "H_{2} (x|q) =4x^2 - (1-q^n)", "d48854978c3f830924d862352cf74278": "(n-i)\\delta_{ij}", "d488eb52dcb44bc0cf266d621b300925": " \\mathbf{x} = \\sum_i \\alpha_i \\mathbf{v}_i ", "d489069155d3cccf24b42df4c8d6b6e8": "\\forall\\alpha.\\alpha\\rightarrow\\alpha \\sqsubseteq \\forall\\beta.\\beta\\rightarrow\\beta \\sqsubseteq\\forall\\alpha.\\alpha\\rightarrow\\alpha", "d48935b17c4f237ef1afda7ff237f045": " \\gamma \\colon l^2(G) \\to l^2(G)", "d4895c13d0b49b1c6339f1466452c8bc": "\\sqrt\\frac{\\hbar{}c}{8\\pi G}", "d489afdc276411fc7090b74702b49c0d": "\\operatorname{dim}_{\\operatorname{H}}(X):=\\inf\\{d\\ge 0: C_H^d(X)=0\\}.", "d489e2a74c742b92ca37eb6a759a407b": "e^{-t}\\leqslant (1+t)^{-1}", "d489e3db5109d67f30852ea2e447e30f": "\\mathbf{b_i}", "d48b03ddda1b04d084f9184e7aeb63ff": "R \\bowtie S = \\left\\{ t \\cup s \\ \\vert \\ t \\in R \\ \\land \\ s \\in S \\ \\land \\ \\mathit{Fun}(t \\cup s) \\right\\}", "d48b17bab18c994f4f82f44b7380ad0f": "{*F} = \\pm F\\,", "d48ba64159427f290cb8b79720370054": "\\ x^2+7x+8 = 0", "d48bada8350b099a4f53577237a5e427": "n = p", "d48bbcad7579611d5296c4565602d7b4": "\\nabla_c g_{ab} = 0", "d48bd8b33030af07185c11f564412baf": "+j1.14\\,", "d48bf23856d88c4a6491641b02ccf57f": "n^{polylog(n)}", "d48bf7a3fae375b95be897a11c75d229": "B_n^2", "d48c9cfb864bc17dccfc1e68ea1bd0aa": "(C \\rightarrow \\overline{(A \\vee B)}) \\wedge (\\overline{C} \\rightarrow (A \\vee B))", "d48cb230fe9185a9a9ccdfc547317d57": "\\Lambda_e= \\sqrt{\\frac{h^2}{2\\pi m_ekT_e}}= 6.919\\times 10^{-8}\\,T_e^{-1/2}\\,\\mbox{cm}", "d48cca54fd6c3c19081460e06dd63970": "\ng_{ab}(r) = \\frac{1}{N_{a} N_b}\\sum\\limits_{i=1}^{N_a} \\sum\\limits_{j=1}^{N_b} \\langle \\delta( \\vert \\mathbf{r}_{ij} \\vert -r)\\rangle\n", "d48cd1f25d335a74fc8f4056ed626d02": "M_{mn}\\,", "d48cdadbe821616c38105e046dacde50": "\\mu = \\mu_\\mathrm{s}\\,", "d48cf9fd6d3dfae4624ffa85f7ced2fd": " 3 \\lor 4 \\iff | 1 \\rangle ", "d48d25da28f653d696ecc50f7702db16": " \\langle s_{\\mu},s_{\\nu}\\rangle = \\delta_{\\mu\\nu} ", "d48dc5c8ef76259197d732878366392a": "K_\\alpha (u, u') = K_\\alpha (u', u)", "d48e331705aac0b853235792e50bf217": "l_c =\\frac{k T_s}{\\pi d^2 p_s}", "d48ea045606a03f57fa76749f197fa42": "(S-K)^{+} \\ ", "d48f81e9c9dd20358280964428c99754": "\\mathit{2} 7 \\mathit{52}\\, ", "d48f9753a32af776bb0c6ab799d5f78a": "S \\leq A \\leq \\operatorname{Aut}(S).", "d48fa2c6052429711c23dbc718f8c130": "\\pi_1(S^3\\backslash K)", "d48fb9acb98c4d5bdfde5a3d587d9683": "Y=\\sum_{i=1}^n Y^i \\partial_i", "d490678b232fbc3c2d203bfd80b8c4f0": "n={2m}", "d490bd52d3aff98b9649c6ed43385b28": "\n\\sigma _{\\hat g}^2 \\,\\,\\, \\approx \\,\\,\\,\\left( {{{\\partial \\hat g} \\over {\\partial T}}} \\right)^2 \\sigma _T^2 \\,\\,\\,\\, = \\,\\,\\,\\left( {{{ - 8L\\,\\pi ^2 } \\over {T^3 }}\\alpha (\\theta )} \\right)^2 \\sigma _T^2\n", "d490e70127f013487caa3e8089d0b02b": "F_{P}= \\frac{\\pi r_d^2 }{c} I \\mathbf{\\hat{e_i}}", "d490ff89298dee5c4d67417da0797099": "K\\in \\Bbb{R}", "d49126eab74acee982ea62e58e0aecf8": "w_{ij}=w_{ji}\\qquad \\forall i,j", "d491f6727f9f00cee843579e5bc2f27c": " \\mathbf E = \\mathbf E_{o}e^{j \\omega t}", "d491fa25c3096b318b2de54ca0eb0366": " \\int_{0}^{\\pi} \\left| \\psi ( \\phi ) \\right|^2 \\, d\\phi = 1\\ \\quad (5) ", "d49238fd46cd13e873f4b83ca7bf3547": "c+v\\,\\!", "d4929137d5816af192f44a667ff65a98": "E'= \\frac{E}{y_c}", "d49344a6840d107262948c866d141cc9": "V a = y \\, ", "d493473e084ed65dbb087a5149f204fc": "\\frac{R}{S}(n)=\\frac{1}{S(n)}\n[\\max_{0\\leq t\\leq n}(Y_t -\\frac{t}{n}Y_n)-\\min_{0\\leq t\\leq n}(Y_t -\\frac{t}{n}Y_n)]", "d4935919ab840d7fe21c9e6a45d0cc79": "(af)' = af' \\,", "d493771be30db7d0ed9bb27fdddf78f5": "|\\psi_\\text{NOON} \\rangle = \\frac{|N \\rangle_a |0\\rangle_b + e^{iN \\theta} |{0}\\rangle_a |{N}\\rangle_b}{\\sqrt{2}}, \\, ", "d4948d71e7c429ea4d8c320d9cce4677": "\\mathbf{L} = \\mathbf{L}_\\parallel + \\mathbf{L}_\\perp \\,,\\quad \\mathbf{L}' = \\mathbf{L}_\\parallel' + \\mathbf{L}_\\perp'\\,.", "d494da0ac600d053af23f5fe5a90f3a9": "{\\rm gcd}(\\varphi(p-1), \\varphi(q-1))=2", "d494fff4bb742eda9807f33385293eba": "r_0", "d49506c9dc5996df0cd841f0a6e1117f": "\\mathfrak{P}^{86}", "d4957a753e7a355a674498e44385f5d4": "\\mathcal{SN}(\\mu,\\,\\sigma_1,\\sigma_2)", "d49628330f42582fb091abf0fe1b0fa4": "\\langle\\psi_\\Lambda(t)|E_1^{(-)}(t)E_2^{(+)}(t)|\\psi_\\Lambda(t)\\rangle", "d4966afe1405c7ec840d1f18f6e776c6": " \\alpha = \\alpha' = 2 -\\frac{d}{p}=2-\\nu d ", "d4969dce0dcf9c0d66e621cadea3cda5": "M_{t}=\\sum_{j=1}^{n}x_{jt},", "d4978e5831f1f6224178cdfbfd1896a7": "\\Omega\\left(\\omega\\right) = \\left(\\omega / \\omega_o \\right) - \\left( \\omega_o / \\omega \\right) = \\left( \\lambda_o / \\lambda \\right) - \\left( \\lambda / \\lambda_o \\right)", "d497b5ce06a381877638b9006be9b184": "(B(y+1))^n \\le B^n x + \\alpha\\,", "d497c0feca9d103e91de9b8b79fcf7fe": "E_1 = \\bigoplus_{p,q\\in\\bold{Z}} E_1^{p,q} = \\bigoplus_{p,q\\in\\bold{Z}} \\frac{\\bar{Z}_1^{p,q}}{\\bar{B}_1^{p,q}}", "d49826ca963aa05c9c5caceead67180f": "q_{or}=k_1 \\lor^p k_2 \\lor^p .... \\lor^p k_t ", "d4983322fb3fd629c418d226a944688e": " \\gamma_1 = 2 ( m - a ) ,", "d4984f549d0d6cd9bd4de10419eb40b1": "L(P, -t) = (-1)^d (t - 1)^d = (-1)^d L(\\text{int}(P), t), ", "d49868528b9f01b7ddff48a3268b241c": "\\uparrow\\{1\\}", "d498aa142097a39d8583c5cd2c853fb0": "\n\\begin{align}\n{}_a\\mathbb{D}^q_tf(t) & = \\frac{d^qf(t)}{d(t-a)^q} \\\\\n& =\\frac{1}{\\Gamma(n-q)} \\frac{d^n}{dt^n} \\int_{a}^t (t-\\tau)^{n-q-1}f(\\tau)d\\tau\n\\end{align}\n", "d498b2d5803e0cc80fb8ece215fe5d0f": "k(f)", "d498e91589492fe5b31de8f54c761570": "e_{\\phi}=1", "d4990a0a2bba01c43cbe4c4afa7f352d": "k_{\\rm A} = 1", "d499625e2b0c55b1480225198477e53a": "\\omega \\equiv \\frac{d\\theta}{dt}", "d4996d7c8690bc7d9aa378d015496ddf": "\\scriptstyle P_0,...,P_n", "d499aa12be1b9c00783824e181cb9fb2": "a(n,k,x):=p \\left(X_0^{k-1} \\right)\\prod_{i=k}^{n-1}p \\left (X_i|X_{i-k}^{i-1} \\right )=j(k,x)\\prod_{i=k}^{n-1} c(i,k,x)", "d49a09c1e4b9abd74178f281e3360235": " m\\equiv 3\\bmod 4", "d49a16994f57b969db742b4f2ba8111e": "\\log \\left\\{{n \\atop K_n}\\right\\} = n\\log n - n \\log\\log n - n + O(n \\log\\log n / \\log n).", "d49ab6209d1b57f5bd79f34eec016cba": "\n(4.2)\\quad\nf(x) = -\\int_0^\\infty [E^x h(Z_t)-E h(Y)] dt.\n", "d49ac51da1a900fc54d206a9963ef82b": "EVIU = 313.7 - 151 = 162.7\\text{ minutes} \\,", "d49b55f6161481e4e5d92e14da349892": "\\overline{x}\\langle y_1,\\cdots,y_n\\rangle.P", "d49ba5d8e50bbc54fbab215f200be10a": "\\mu_e = {v \\over E}.", "d49befe7caeefd4623ecd8c508d6d373": "{\\mathcal L}=-\\frac{1}{2}(\\partial^\\mu \\phi^*)\\partial_\\mu \\phi +m^2 \\phi^* \\phi = -\\frac{1}{2}(-iv e^{-i\\theta} \\partial^\\mu \\theta)(iv e^{i\\theta} \\partial_\\mu \\theta) + m^2 v^2 ,", "d49c627ab6168d35c55c5f6305f85f25": "F' = x^3 Ma", "d49c8467694474ab321dfa83f966ef80": "B(x,y)", "d49ca5490c2cf43b5a081aa26e10a53d": "\\! K=1,0", "d49ce894e469526ceac59557d12bba13": "\\tau(H)", "d49cffe048e5ae3fca5d370e1d271416": "\\cos \\theta = \\frac{x} {r} =\\frac{e-\\cosh E}{e \\cdot \\cosh E-1}", "d49d21afc07cbfb8253102fe239c4112": "6 \\cdot V =\\begin{vmatrix}\n\\mathbf{a} \\\\ \\mathbf{b} \\\\ \\mathbf{c}\n\\end{vmatrix}", "d49d3bdba07b18c69aa6152c5fa791b9": "f(n) = 4 n^2 + b n + c", "d49d430f2e7929be06ee271252612384": "\\Delta_2^{\\prime}F(J) = \\bar \\nu [R(J) ] - \\bar \\nu [P(J) ] = (2B^{\\prime}-3D^{\\prime}) \\left(2J+1\\right)-D^{\\prime}\\left(2J+1\\right)^3", "d49dba37dd7ec81737c93f30821f4892": "p_A(a, \\lambda)", "d49e6bba5913d0505b11c89a7adc3db5": " \\rho \\to \\pi^+ \\pi^- \\pi^0 ", "d49e6d5e1aa1a35ec118a06207dc3595": "\\tilde\\psi(\\alpha) = \\psi(\\alpha)", "d49e72084e9127fe282f438aa2c706c4": " L[u] = \\frac{d^nu}{dt^n} + a_1\\frac{d^{n-1}u}{dt^{n-1}} + \\dotsb + a_{n-1}\\frac{du}{dt} + a_nu = r(t) ", "d49e8dc3187e74d0903289b8ab6f474a": "\\frac{df}{du}", "d49e916458db631e2c44d37b8a1d6adc": "\\sum_{i} \\sum_{j} \\dots \\sum_{k} \\mathrm{P}(X_1=x_{1i},X_2=x_{2j}, \\dots, X_n=x_{nk}) = 1.\\;", "d49eb2f6d2a5c5cac22cdb805e12ba04": " r_{n}, n \\geq 0. ", "d49ec952ec6529f202907799fd561a4b": "*m", "d49ed88e5c823283e17299662a72dbb4": "\\Rightarrow \\frac{P}{x} = \\frac{r_1+r_2}{r_2} \\,\\!", "d49eea4a52b8db85c16a5995b8c866b9": " P=-i\\hbar \\frac{d}{dq} ", "d49f06fdb2bd5672abdb09b68c8eeaed": " {f(x)\\over g(x)} = {\\sum_{n=0}^\\infty a_n (x-c)^n\\over\\sum_{n=0}^\\infty b_n (x-c)^n} = \\sum_{n=0}^\\infty d_n (x-c)^n", "d49f47d6409083b66c4d47e3013276dc": "\\delta^2 E(\\gamma)(\\varphi,\\psi) = \\left.\\frac{\\partial^2}{\\partial s\\partial t}\\right|_{s=t=0}E(\\gamma + t\\varphi + s\\psi).", "d49f9cf69182a440710480aca79dd3e6": " \\sigma _c > 0", "d49fec5d667ac48f6040b6b131b1ef5d": "X^* = \\frac{W}{1^TV^{-1}(r-1r_f)}V^{-1}(r-1r_f).", "d4a03daf9e022e82a7ce7666741b08c2": "\\{ A , \\sqrt{V_\\epsilon} \\}", "d4a0548bfecf285eea77fb9003513274": "\\Theta_0", "d4a08435b9fe6a38c7b07f86199ef875": "\\mathbf{P}(V) = \\operatorname{Proj} k[V]", "d4a13fc7fe918c99a6d69c643318b0dd": "\\scriptstyle{Rc>0.9}", "d4a1a3f78c18c5f2361f4ad22c6c7521": "\nE(Y_i^2) = P_i+N\n\\,\\!", "d4a1b9597fbe4185d62448eb761a3d11": " H(z)=\\frac{1+a*z^{-1}+a^{2}*z^{-2}}{1-a^{3}*z^{-3}}", "d4a1c77ddaddbea6d4780b8d63de2429": "n=H", "d4a1cdfa920f353f0c7f8173f1291095": "\\phi (t)", "d4a1e9418221ccfe33164839888e1ece": " \n\\Delta(t) + Vp(t) \n", "d4a2104ed06acad024f58077538fc4f8": "{2\\pi^{3N/2}(2mE)^{{3N-1}\\over 2}}V^N\\over {\\Gamma(3N/2)}", "d4a29d6945960468b482af72ff4f7b61": "(X, \\mathcal{B},T,\\mu)", "d4a305b3ddaf69306121473a22f7b880": " P(PH~WHOIFPI) \\approx \\frac {1}{8000} * 10 = \\frac {1}{800} = 0.00125\n= ", "d4a333e1b34f5da12c403ffd0e8fa5ef": "(S,\\mu^S,\\eta^S)", "d4a384649034b8b4280b8371b33db0c3": "\n = \\frac{A}{2} \\int_{0}^{\\infty} \\delta(u - \\omega) Q(u) e^{i u t} du =\n \\frac{A}{2} Q(\\omega) e^{i \\omega t}\n", "d4a3a65b8e32f7df4ba13565107696b1": "\\frac{X}{X_o} = e^{-k}", "d4a3dceec92a95986672fe18a2c4d69a": "f_\\mathrm{V}\\approx f_\\mathrm{G}\\left(1-c_0c_1+\\sqrt{\\phi^2+2c_1\\phi+c_0^2c_1^2}\\right)", "d4a44d3c910bfc240d48ae1650629544": "Q_i c_i", "d4a4525dc2866bc74b12b1f1b34e7053": "\\displaystyle K=\\sqrt{(e+f+g+h)(efg+fgh+ghe+hef)}.", "d4a4576fa67aa2d73e714faec73d4e96": " \\|f\\|_r = \\max_{|z| \\le r} |f(z)| = \\max_{|z| = r} |f(z)|", "d4a47992ab3df95b409046e14814684c": "a(x, y)=a(y, x)", "d4a52a2b95a9ae95239940c2f31aa84d": " \\tan^2 a + \\tan^2 b - 2\\tan a \\tan b \\cos C \\;\\;\\;=\\;\\;\\; \\sec^2 a + \\sec^2 b - 2 \\sec a \\sec b \\cos c", "d4a535ac942043373b0b5a46028edb29": "\\sum_{i \\in S_k} a_i \\leq V", "d4a552d70428272b7a00fdbcf88ef55b": " \\mathbf {P_{\\nu\\nu}} ", "d4a6067f5c7bc04cf844e354a9b6760b": "\n\\begin{align} \n& V_{\\text{obs, r}}=Ad\\sin\\left(2l\\right) \\\\\n& V_{\\text{obs, t}}=Ad\\cos\\left(2l\\right)+Bd \\\\\n\\end{align}\n", "d4a6461168a21766b09c14c9a66f9f79": "\\omega^8=\\left(e^{\\frac{2\\pi i}{8}}\\right)^8=1", "d4a6b916cc466c07c1643135f2eec8f1": "{d\\alpha_j \\over dt} = -{c(\\alpha_j)\\over p^\\prime(\\alpha_j)}. ", "d4a6e65d70b084ec51edb52c51f569be": "I/2 + I_s", "d4a6fe68ef6e8b174fbebfee8868ad3e": "\\mathcal C \\, | \\pi^+ \\, \\pi^- \\rangle = (-1)^L \\, | \\pi^+ \\, \\pi^- \\rangle", "d4a750fed13e8f018a9091d3c1e3ee29": " J^t(x_0) = \\left. \\frac{ d f^t(x) }{dx} \\right|_{x_0}. ", "d4a767721998ef258994ff3f99c11469": "d_1 + d_2 = 2a\\,", "d4a86f098b2214c6a9612760c76bb0b4": "\\mathrm{2\\ CmF_3\\ +\\ F_2\\ \\longrightarrow\\ 2\\ CmF_4}", "d4a8eb4f06108f374d67ce759d8dfa48": "I({\\vec x})", "d4a9218342f29901937a583e286a05fe": "c=91", "d4a9252d72f6a61f8a5ef374bae38103": "W_{out} = W_{in} \\,", "d4a94d23dabb04ad19f4a1918e22a28e": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{r}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\cos u\\ du\\ = \n\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\cos^2 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin u\\ \\cos u \\ du\\ = \\\\\n&\\hat{g}\\ \\left(\\int\\limits_{0}^{2\\pi} \\cos^2 u \\ du\\ +\\ \n3\\ {e_g}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^4 u \\ du\\ \\ +\\ \n3\\ {e_h}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^2 u\\ \\cos^2 u \\ du\\ \\right) \\\\\n&+\\hat{h}\\ 6\\ e_g\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^2 u \\ du = \\\\\n&\\hat{g}\\ \\left(2\\pi \\left(\\frac{1}{2}\\ +\\ \\frac{9}{8}\\ {e_g}^2\\ +\\ \\frac{3}{8}\\ {e_h}^2\\right)\\right)\n+\\hat{h}\\ \\left(2\\pi \\left(\\frac{3}{4}\\ e_g\\ e_h\\right)\\right)\n\\end{align}\n", "d4a95300b068eb4f6e5d0f4fe8e768cd": "(H, K)", "d4a956a91bdddfd16ff34860e05c3cb1": "\n\\frac{}{\\mathbf{s(-3)} \\,\\,\\mathsf{nat}}\n", "d4a9752fd7b7bc78f9979a8f4cc98cf0": "RN_i[n]", "d4a97dbd61a04adb7d505826ddd97c00": " \\sum_{d\\mid n} \\frac{\\mu (d)}{d}. ", "d4a9f446af705ee81b3585ec982ddf7e": "\\{E_{mb}T_\n{na}g\\}_{m,n\\in Z}", "d4aa439d1aa3fdbcfad9129d0a17f280": "\\delta(u, \\infty)=\\delta(\\infty, u) = \\frac{2}{\\sqrt{1+\\|u\\|^2}}.", "d4aa8a73a7ce6117f3e4bca3067a4ed0": "p^N", "d4aabe4cc8dd422d54e64fb59bcd11b5": " \\log G(1-z) = \\log G(1+z)-z\\log 2\\pi+ \\int_0^z \\pi x \\cot \\pi x \\, dx.", "d4aac9bcc27539c0475e1475cd92ecb5": " e^4 \\left| \\frac{(\\bar{v}_{k} \\gamma^\\mu v_{k'} )( \\bar{u}_{p'} \\gamma_\\mu u_p)}{(k-k')^2} \\right|^2 \\,", "d4aaf7ff520d58f5eaf6db15ac3696f8": "\\begin{cases}\n\\displaystyle v\\frac{\\partial v}{\\partial s} = -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial s},\\\\\n\\displaystyle {v^2 \\over R} = -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial n} &({\\partial / \\partial n}\\equiv\\boldsymbol{e}_n\\cdot\\nabla),\\\\\n\\displaystyle 0 = -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial b} &({\\partial / \\partial b}\\equiv\\boldsymbol{e}_b\\cdot\\nabla).\n\\end{cases} \n", "d4ab4bae0d183ae3095ee3f8be26e1b9": "\\xi =-J(t\\partial/\\partial t).", "d4ab64d93b52189c6df96d5d5af9f749": "\\hat{b}_j^{(\\eta)} = \\frac{x_{+j}}{\\sum_i \\delta_{ij}\\hat{a}_i^{(\\eta)}}", "d4ab7e72f828b2d27e4bc141ec9ad07b": "P_F = \\sqrt{P_P\\cdot P_L}", "d4aba8755318c93affd779663798e0e6": "dy= F(x) \\, dx\\,\\!", "d4abf39ac340aad4d29c15187679b990": "\\mathbb{F}_1, \\mathbb{F}_2 \\subseteq \\mathbb{F}", "d4ac1352309fa5f986c5b045c54851b3": "f(\\alpha^j)", "d4ac31195ac232976e7cada43f0a6797": "\\Phi [A] = \\hat{O} \\Psi [A] \\qquad Eq \\; 1", "d4ac7516e611ab946b7b560cf4a0e4ab": "|\\bigstar \\bigstar | \\bigstar|", "d4ac9039ba45cc297661e5c629822d84": "M \\times R \\to M", "d4acaa3275c181248ee799364be057cf": "\\xi=[\\xi^i]\\in\\mathbb{R}^n", "d4ace0d1eb036e25085055a346e254e5": "\\mu[f 1_E]", "d4ad708c676745ab7f080a589734893e": "(g^2-g+1,-g-(g-1)^2,1-g+g^2) = (2,-2,2)\\sim (1,-1,1)", "d4ad83915c19927b19762179570c518f": "\\ u_n \\sim n\\, W_n^2", "d4adfc49607d7e8e12217b9f23cefade": "X \\sim \\Gamma(\\alpha,1)\\,", "d4ae60d596f09d66c052bb5f6296d0b5": "(g,gs)", "d4ae749040f6e154d31917a6fae07f31": " T = k * (SLOC)^{(1+x)}", "d4aeb1a04b663cf20ff23f795537e3c1": "\\mathbb{Z}^{(B)}", "d4af016b53e76c2539fa0857a85c8c87": "\\ \\omega_{xy,G}^2", "d4af45ea35b4839bf0a40ebb0c7a4407": "\\mu\\,\\!", "d4af4905aa1227eb1348cf0fe97b11c8": "n/p^j \\bmod 1< 1/2", "d4af82c6d33b5770a0f2fe1d8c300478": "Y_{p}=\\frac{\\dot{n}_{p,\\text{out}}-\\dot{n}_{p,\\text{in}}}{\\dot{n}_{k,\\text{in}}}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "d4af9bd4d5a5f8e7bea93c2b5f342f2e": "\\int\\!\\!\\!\\!\\int\\!\\!\\!\\!\\int_V\\nabla f\\, dV =", "d4b0283960fda480e81963c98c42b0d0": "[V_{mn}(r,\\theta)]^*", "d4b02dd27bc6ca16f0ac29c6c8626b04": "\\prod _x a = C a^x \\,", "d4b047ff8e10036c81387c852542ce6b": "\\big(\\mathbf{\\mathit{Tr}}[\\rho]=1\\big)", "d4b0a634becf354bedc5c1bbc8336908": " x \\succeq y", "d4b0b905380350c7005d449703234c72": "{\\rm ATIME}(t(n))", "d4b0bcf82aa8dcfd4534a36c225725c6": " \\frac{d U_\\text{eff}}{dr} = 0 ", "d4b0c1cc42eda52bad4ef68a3f77e8bf": "W(r)", "d4b0dd26ca8c1b9afed918dcfd73eb0c": "g_{ij}=\\delta_{ij}", "d4b108194ecdf4d24ac8271225f16ee1": " C = \\arcsin {c \\sin A \\over a} \\text{ or } C' = \\pi - \\arcsin {c \\sin A \\over a}", "d4b10e447621429d836bcab76ec00f75": " \\beta_{j+1} \\leftarrow \\left\\| w_j \\right\\| \\, ", "d4b129d02234ffde3c079639a2e50241": "\nz^{2} +\n\\left( \\sqrt{x^{2} + y^{2}} - a \\coth \\tau \\right)^{2} = \\frac{a^{2}}{\\sinh^{2} \\tau}\n", "d4b17fc8c64e219ae557092df49f6359": "x_1^2+x_2^2+x_3^2=r^2", "d4b180d2df31fa7967884e4a6928b637": "\n\\begin{align}\n\\Psi^{(S)}_{n_1 n_2 \\cdots n_N} (x_1, x_2, \\cdots x_N) & \\equiv \\lang x_1 x_2 \\cdots x_N; S | n_1 n_2 \\cdots n_N; S \\rang \\\\[10pt]\n& = \\sqrt{\\frac{\\prod_j n_j!}{N!}} \\sum_p \\psi_{p(1)}(x_1) \\psi_{p(2)}(x_2) \\cdots \\psi_{p(N)}(x_N)\n\\end{align}\n", "d4b19d1f1091b10f6f0eb349969ceddf": "\\frac{1}{|X|}\\sum_{U \\in X} U^{\\otimes t}\\otimes (U^{*})^{\\otimes t} = \\int_{U(d)} U^{\\otimes t}\\otimes (U^{*})^{\\otimes t}dU", "d4b1acce91173e9a33cf3b2bfc74acf2": " y = x^{3/2} \\,", "d4b1b5bcfdea0c306356a09791dc239e": "\\rho = \\rho(0) \\in H", "d4b204085d31240b9cd4ab1ffc31b66a": "\\underset{i}{\\overset{3}{x_j}}(t)", "d4b207ac1952c12cb1fc90ec93242369": "I^2 = \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty e^{-(x^2+y^2)/2}\\,dx\\,dy\n =\\int_0^{2\\pi}\\int_0^\\infty re^{-r^2/2} \\, dr \\, d\\theta.", "d4b20aae350eedbc39a3e6aab9d2f1db": "\nP-49s_2^2-14s_1s_2 = x_1^2 + 2x_1x_2 + x_2^2\n", "d4b22ec0a5437776eac11304d1f1b7fc": " \\alpha \\ge 4 ", "d4b23f63a9835eeb7a4de8f1b4e16cca": "V(t;T)", "d4b247779f3e8f4d191ed45e4532970a": "\\delta P = \\frac{{\\partial P}}{{\\partial t}}\\delta t + \\frac{{\\partial P}}{{\\partial y}}\\delta y + \\frac{1}{2}\\frac{{\\partial ^2 P}}{{\\partial y^2 }}\\delta y^2 + O\\left( {\\delta t^2 ,\\delta y^3 } \\right)", "d4b2ba3d94563d834f3e018250bdd2f1": " \\mathbf{F} = \\frac{1}{c}\\overline{E}\\mathbf{e}_4 + \\overline{B}\\mathbf{e}_{123},", "d4b2e99a9dd84c86df31769829e7c505": "P(x_{t+1}|x_t,u_t)", "d4b309afa12f547f9b48d30a1e31e6fc": "T_n:(\\mathcal{X}^n,\\Sigma^n)\\rightarrow(\\Gamma,S)", "d4b34b69248768bd3582a2a53f802339": "\\overline{T}_2=A+\\frac{Bk_2}{k^2L}(1-e^{-kL}).", "d4b37d7e80a0cbe71dc3fd1c0d1434e5": "\\displaystyle S(x,\\alpha) = \\int_x^\\infty t^{\\alpha-1}\\sin(t) \\, dt", "d4b3a6025d95ddc3655562763723c973": " \\frac{\\mathrm d J_\\varepsilon}{\\mathrm d\\varepsilon}\\bigg|_{\\varepsilon=0} = \\int_a^b \\left[ \\eta(x) \\frac{\\partial F}{\\partial f} + \\eta'(x) \\frac{\\partial F}{\\partial f'} \\,\\right]\\,\\mathrm{d}x = 0 \\ .", "d4b3bbe2109135919ba5769c41e77777": "\\frac{\\partial \\mathbf{\\hat{r}}} {\\partial \\varphi} = -\\sin \\theta \\sin \\varphi\\mathbf{\\hat{x}} + \\sin \\theta \\cos \\varphi\\mathbf{\\hat{y}} = \\sin \\theta\\boldsymbol{\\hat \\varphi}", "d4b3ce174223909f21d86a9b166e19aa": "\\sigma_{red}", "d4b40e833afc196f1dc79feae85d00ed": "\n\\begin{align}\n& {} \\quad\n\\left(\\begin{array}{rr}\n \\cos\\alpha & -\\sin\\alpha \\\\\n \\sin\\alpha & \\cos\\alpha\n\\end{array}\\right)\n\\left(\\begin{array}{rr}\n \\cos\\beta & -\\sin\\beta \\\\\n \\sin\\beta & \\cos\\beta\n\\end{array}\\right) \\\\[12pt]\n& = \\left(\\begin{array}{rr}\n \\cos\\alpha\\cos\\beta - \\sin\\alpha\\sin\\beta & -\\cos\\alpha\\sin\\beta - \\sin\\alpha\\cos\\beta \\\\\n \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta & -\\sin\\alpha\\sin\\beta + \\cos\\alpha\\cos\\beta \n\\end{array}\\right) \\\\[12pt]\n& = \\left(\\begin{array}{rr}\n \\cos(\\alpha+\\beta) & -\\sin(\\alpha+\\beta) \\\\\n \\sin(\\alpha+\\beta) & \\cos(\\alpha+\\beta)\n\\end{array}\\right).\n\\end{align}\n", "d4b41899c1092cb790c2b1aba3d52174": "f\\ll p", "d4b437cbda222a93b9a4bf60e7722ee5": "A = -\\frac{3}{\\kappa +2} E_{\\infty} \\ ;\\ C=\\frac {\\kappa-1}{\\kappa+2} E_{\\infty} R^3 \\ , ", "d4b49c686b05a7ab701a2266a26eb8bd": " \\int {R(x)\\over P(x)} \\, dx = {T(x)\\over S(x)} + \\int {X(x)\\over Y(x)} \\, dx, ", "d4b52a5286b496f5fe5984bbd10df236": "k[\\epsilon]/(\\epsilon)^2", "d4b53a3ae163f20f38dc452ca5f8a32e": "(J+K)", "d4b56de3f60030216c1acb839af3f5ce": "F = \\frac{\\mu^2 N^2 I^2 A}{2\\mu_0 L^2} \\qquad \\qquad \\qquad \\qquad \\qquad (4) \\,", "d4b57f22faa7a61d152e7ead5df9787f": " a=b_1=\\sqrt{\\mu_2} \\sqrt{\\beta_1}\\frac{\\beta_2+3}{10 \\beta_2-12\\beta_1 -18},", "d4b59d2a3c809f6e74f140be8b4d4203": "\\begin{bmatrix}\na & b \\\\ -b & a \\\\ \n\\end{bmatrix}", "d4b5c72ac16c28c7f24ee15285b66c66": "var(Q) var(P) \\geq cov^2(Q,P) + (\\frac{\\hbar}{2})^2", "d4b62b05283d10e5253a3a24420ae662": " \\log\\sqrt{c/a} = (1.0576927 - 0.6192290) / 2 = 0.2192318", "d4b668f3ba239470e7c47905e4ea2ccd": "\\xi = \\sqrt[3]{\\frac{\\tau}{2} + \\frac{1}{2}\\sqrt{\\tau - \\frac{5}{27}}} + \\sqrt[3]{\\frac{\\tau}{2} - \\frac{1}{2}\\sqrt{\\tau - \\frac{5}{27}}}", "d4b6d3ed942c2ed9a1429ef5d4ef933e": "\n\t\\begin{bmatrix}\n\t\tx\\\\\n\t\ty\\\\\n\t\tz\n\t\\end{bmatrix} = c\\begin{bmatrix}\n\t\t -1\\\\\n\t\t-26\\\\\n\t\t 16\n\t\\end{bmatrix}.\n", "d4b6fa7cc0ea73fdc79ae43e2f5dd4c0": "\\left(x^\\prime ,\\ y^\\prime\\right)", "d4b724a248576af63b34c9a33fd6fd12": "1-1/n", "d4b72dbb44d2d281205758258a3e1505": "\\mathrm{6Cl^- + Cr_2O_7^{2-} + 14H^+ \\rightarrow 3Cl_2 + 2Cr^{3+} + 7H_2O}", "d4b76b7b6a5fbd74c775ed0ac5f1690c": "R_1 = (2a+b)/3", "d4b7749422c8949d027a7741095da870": " a^2 + b^2 =c^2 ", "d4b77ac54fdb77234f648d6addaa03e6": " \\mathbf{y} = \\begin{bmatrix}X & K\\end{bmatrix} \\begin{pmatrix} \\hat{\\boldsymbol{\\beta}} \\\\ \\hat{\\boldsymbol{\\gamma}} \\end{pmatrix} ,", "d4b7a49600eee63a4797fc7859bf90af": "S^2 = P^2 + Q^2", "d4b80363aec63e8f90c4785a1c796c10": "\n\\begin{bmatrix}\na_{11} & a_{12} & \\cdots & a_{1n} \\\\\na_{21} & a_{22} &\\cdots & a_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{m1} & a_{m2} & \\cdots & a_{mn} \\end{bmatrix}\n", "d4b8249610c669581cbf024bf1ee4e3a": "\nF_{2^3} = \\frac{1}{\\sqrt{2^3}} \\begin{bmatrix} 1&1&1&1&1&1&1&1 \\\\\n1&\\omega&\\omega^2&\\omega^3&\\omega^4&\\omega^5&\\omega^6&\\omega^7 \\\\\n1&\\omega^2&\\omega^4&\\omega^6&\\omega^8&\\omega^{10}&\\omega^{12}&\\omega^{14} \\\\\n1&\\omega^3&\\omega^6&\\omega^9&\\omega^{12}&\\omega^{15}&\\omega^{18}&\\omega^{21} \\\\\n1&\\omega^4&\\omega^8&\\omega^{12}&\\omega^{16}&\\omega^{20}&\\omega^{24}&\\omega^{28} \\\\\n1&\\omega^5&\\omega^{10}&\\omega^{15}&\\omega^{20}&\\omega^{25}&\\omega^{30}&\\omega^{35} \\\\\n1&\\omega^6&\\omega^{12}&\\omega^{18}&\\omega^{24}&\\omega^{30}&\\omega^{36}&\\omega^{42} \\\\\n1&\\omega^7&\\omega^{14}&\\omega^{21}&\\omega^{28}&\\omega^{35}&\\omega^{42}&\\omega^{49} \\\\\n\\end{bmatrix} = \\frac{1}{\\sqrt{2^3}} \\begin{bmatrix} 1&1&1&1&1&1&1&1 \\\\\n1&\\omega&\\omega^2&\\omega^3&\\omega^4&\\omega^5&\\omega^6&\\omega^7 \\\\\n1&\\omega^2&\\omega^4&\\omega^6&1&\\omega^2&\\omega^4&\\omega^6 \\\\\n1&\\omega^3&\\omega^6&\\omega&\\omega^4&\\omega^7&\\omega^2&\\omega^5 \\\\\n1&\\omega^4&1&\\omega^4&1&\\omega^4&1&\\omega^4 \\\\\n1&\\omega^5&\\omega^2&\\omega^7&\\omega^4&\\omega&\\omega^6&\\omega^3 \\\\\n1&\\omega^6&\\omega^4&\\omega^2&1&\\omega^6&\\omega^4&\\omega^2 \\\\\n1&\\omega^7&\\omega^6&\\omega^5&\\omega^4&\\omega^3&\\omega^2&\\omega \\\\\n\\end{bmatrix}.\n", "d4b8a1d40b8af3edc80fb41f3b23ec88": "\\theta=\\pi/2=90^\\circ", "d4b939e02c10fe8e015a8d6be9c7be68": "N(3,\\delta) \\leq C^{\\delta^{-1}(\\log(1/\\delta))^5}", "d4b97e4504d43040bcd537d5722d29c3": "\\textstyle \\Delta f = 0", "d4b9d3aebf6dd5c06d2dc68691627d60": "\\langle a* \\rangle p \\to p \\lor \\langle a* \\rangle (\\neg p \\land \\langle a \\rangle p)\\,\\!", "d4b9dd56d60dc64aa66eeb9d3a636263": "\\boldsymbol\\theta \\in \\mathbb{R}^p .", "d4ba1c0407eac210e3552e63fc96f871": "\\sqrt{k/M}", "d4ba3cf1ae1da3172eea87c83c34b5a7": "M_\\star(r Z^2 E_h", "d4bc2c43e02ddd534203b369218a3e5a": "K_C = \\frac{|Q_L|}{|Q_H|-|Q_L|} = \\frac{T_L}{T_H-T_L}\\,\\!", "d4bc59ca9a101b62bbfb9ea0ebfcdd02": "dA = r^2 \\sin\\theta\\, d\\theta\\, d\\phi", "d4bcb61ca50cf37e0bce2da323052e5f": "\\left|A\\cap B\\right| < \\kappa.", "d4bcc08599b9eca914a50c77361ae90b": "Y_{10}^{1}(\\theta,\\varphi)={-1\\over 256}\\sqrt{1155\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(4199\\cos^{9}\\theta-7956\\cos^{7}\\theta+4914\\cos^{5}\\theta-1092\\cos^{3}\\theta+63\\cos\\theta)", "d4bcddd00d4f01904b3787dfc56e9c90": "\\mathcal R=\\mathcal S", "d4bce7adf260361cfbefb7ea67457156": "n_{pas} = {3}\\times \\frac{3600}{2}", "d4bd4cf82c3d032cb125e03a3d0fa83e": "\\mathcal{O}(m)", "d4bd6a9e3fc2f872b0c64d463f07ff20": "2y^2-xy^2+2xy+x^3=0", "d4bd6baac5e9236537b657ec3bd440f1": "U |\\psi\\rangle_A |e\\rangle_B = |\\psi\\rangle_A |\\psi\\rangle_B", "d4bd7c1252c00ea766fa9b28ae4857bf": " P(z)= \\sum_{j=0}^{n} a_jz^j ", "d4be7b837eabd0510c92e3068ef0d1a2": "f(x) = x^{1/4}", "d4be9e395f98a21b28f1c899020edef4": "H^1(M)", "d4bec8a16d79a96e880d78ecf5265c4b": "\\Pi_{i \\in I}|A_i|", "d4bee1e3d8a4af4ca7409291d0471d22": "F(x) = \\begin{cases}\n 1, & \\mbox{if } x \\in S \\\\\n 0, & \\mbox{if } x \\not \\in S\n\\end{cases}\n", "d4bf4a6db3b0e679488149f7d98ea3de": "\\nabla\\cdot \\mathbf{E} = \\rho/\\mathcal{E}_0", "d4bf53c8be7169aa0d681e2c8f63b5ee": "\\mathrm I_p^{2 \\omega}(\\gamma)=C |s_5\\chi_{zxx}+\\cos^2{ \\gamma}\\ {(s_2\\chi_{xxz} +s_3\\chi_{zxx}+s_4\\chi_{zzz}-s_5\\chi{zxx})}|^2(I^{\\omega})^2", "d4bfd9e3bb583a72d2d71f0f88ff9957": "J_{\\lambda,n}", "d4c05c73de1b0498f2314f05af5606b7": "g^{(2)}(\\tau)", "d4c099d8c48cf424994cb577edadd930": " i \\in S ", "d4c0f4581d0cfeaf16a9f83a4cc33421": "\\theta = \\frac{v}{v_\\mathrm{mon}}", "d4c0fdd12842034ee49cf964c5572da6": " \\langle E\\rangle =-\\frac{\\partial\\ln\\mathcal{Z}}{\\partial\\beta}, \\qquad \\ \\langle(E-\\langle E\\rangle)^2\\rangle=\\langle(\\Delta E)^2\\rangle=\\frac{\\partial^2\\ln\\mathcal{Z}}{\\partial\\beta^2},", "d4c1d8183ad4f836cd606e6e21552337": "D_{RR}(X, Y) = \\begin{cases} \\frac{24\\sqrt{3}\\sigma^3-\\mu_{xy}^3+6\\sqrt{3}\\sigma\\mu_{xy}^2}{36\\sigma^2}, & \\mu_{xy}<2\\sqrt{3}\\sigma, \\\\ \\mu_{xy}, & \\mu_{xy} \\ge 2\\sqrt{3}\\sigma. \\end{cases}", "d4c28188c048c00926d115f29ac128be": "M_X(t) = M_Y(t),\\, ", "d4c287a9dd4acf055440d13a8defa999": " g(x) = \\sum i \\cdot T_i \\cdot g_i (x) ", "d4c2efeae4a3424da6bb388e41e87557": "\\sqrt{-1}", "d4c31316fb96cba69b84ef3994c0581d": " W_k = \\mathbf{X}V_{k} ", "d4c314439653ccf57b66e5fbb04b50d1": " z = (u^2 - v^2)/3.\\ ", "d4c315e29a532b424786a50398a069c0": " \\frac{\\Gamma}{c} = K\\Gamma_{max} - K\\Gamma", "d4c355c9a4989aa9a16eff7cefd43fb2": "p_4 \\in \\operatorname{Ker}(A-4 I)^2.", "d4c35703933a454ee8b7c737442e6dc6": "L_n(y) \\equiv \\frac{d^n y}{dt^n} + A_1(t)\\frac{d^{n-1}y}{dt^{n-1}} + \\cdots + A_{n-1}(t)\\frac{dy}{dt} + A_n(t)y ", "d4c37d89cff2213733969b4891773432": "\\begin{align}\n \\boldsymbol{\\dot{\\hat \\rho}} &= \\dot\\theta \\boldsymbol{\\hat\\theta} + \\dot\\phi\\sin\\theta \\boldsymbol{\\hat\\phi} \\\\\n \\boldsymbol{\\dot{\\hat\\theta}} &= - \\dot\\theta \\boldsymbol{\\hat \\rho} + \\dot\\phi\\cos\\theta \\boldsymbol{\\hat\\phi} \\\\\n \\boldsymbol{\\dot{\\hat\\phi}} &= - \\dot\\phi\\sin\\theta \\boldsymbol{\\hat\\rho} - \\dot\\phi\\cos\\theta \\boldsymbol{\\hat\\theta} \\end{align}", "d4c3cdb51857071b22a2b809c715ffd2": "D^- = \\mathrm{max} \\left[z_i-(i-1)/n \\right],", "d4c3d493af93fd5942662d3bc134e274": "E_{x}[\\rho_{\\alpha},\\rho_{\\beta}] = \\frac{1}{2}\\bigg( E_{x}[2\\rho_{\\alpha}] + E_{x}[2\\rho_{\\beta}] \\bigg)\\ .", "d4c3e4065bf70346ad535b2befd9371c": "\\frac{\\partial \\hat{H}_{l}}{\\partial l} = \\frac{\\hbar^{2}}{2\\mu r^{2}}(2l+1).", "d4c402cefa1200b20c5a209401ecdb5c": "A_n = A + \\sum_{p=1}^k \\alpha_p q_p^n.", "d4c48eee014acd5908e2878d3dd92ee8": "P_x(f)= \\int_{-\\infty}^{\\infty}R_x(\\tau)e^{j2\\pi f\\tau}\\, d\\tau,", "d4c4a49424559b6356ef3727929d2d73": "{\\mathbf \\Psi_{ij}} ", "d4c4fb75beaa338972c390806d8d155a": " T_{\\alpha \\beta \\gamma}, \\ T_{\\alpha \\beta} {}^\\gamma, \\ T_\\alpha {}^\\beta {}_\\gamma, \\ \nT_\\alpha {}^{\\beta \\gamma}, \\ T^\\alpha {}_{\\beta \\gamma}, \\ T^\\alpha {}_\\beta {}^\\gamma, \\ \nT^{\\alpha \\beta} {}_\\gamma, \\ T^{\\alpha \\beta \\gamma} ", "d4c568da4c78a74780c9047f3b052fea": "\\, r,\\varphi", "d4c586a5e9c0bb1542b68ad32082220a": "\\mathcal{E} = - \\frac{d\\Phi_\\mathrm{m}}{dt},", "d4c5f21c870e9ac2c16d322e5e6243ed": "\\mathbb{RFM}_I(R)", "d4c6643d1176b9d08e7528de86c1a1d6": "p(\\sigma[1] \\dots \\sigma[L] \\gamma[1] \\ldots \\gamma[L])", "d4c74d506d39a01e28a8fb5c555202f9": " \\sigma^2(z) ", "d4c77daaa8eaa4f81c490572dd205f4b": " \\operatorname{lambda-process}[S, L] = \\operatorname{lambda-apply}[\\operatorname{lambda-lift}[S, L]]", "d4c7ad3c96eb09db16d99de8c14e2365": " \\partial_x^3 + 2u\\partial_x + u_x, ", "d4c8163ce2adcebb032e2f75e4945da6": "S(A : B | \\Lambda)\\geq 0", "d4c8c28eb41825950889a7409d703151": "\\lceil6\\sigma\\rceil", "d4c8c822aa3c9e4d909c712e89ac7cd5": "L_k(z)=\\prod_{i\\ne k}(1-z_iz^{-1}),\\quad k=0, 1, ..., N-1", "d4c8db486d028225044d17f42b253eb2": "y=1/x", "d4c8e50f0e0d097f72b5efc19eb6d821": "\\left(-\\frac{\\hbar^2}{2m}\\nabla^2+v_{\\rm eff}(\\mathbf r)\\right)\\phi_{i}(\\mathbf r)=\\varepsilon_{i}\\phi_{i}(\\mathbf r)", "d4c8e8ec47e1ebcfe15698b7b73e473c": "C: \\{0,1\\}^k \\rightarrow \\{0,1\\}^n", "d4c90e3ecc5139aeaec1c205742d7b1e": "x:A", "d4c98d75e25f5d28461f1da221eb7a95": "\\pi_0", "d4c9af7a066cd6bde2d131c5e708014e": "k_{1T}", "d4c9d57bfae8f0b1f46d5032021764dc": " \\alpha_k, \\beta_k = k \\delta, \\quad \\delta = \\frac{1}{q}, q \\in Z^+", "d4c9e572955eb7e1a6f545793a35c1db": " Z_N(K,L) ", "d4ca0681ed52ff73fb56379dd34fcbad": "(g\\cdot f)(x,y)=f((x,y)g)=f(\\alpha x+\\gamma y,\\beta x+\\delta y)", "d4ca56199efae56c18dffc53d26b2853": "\\beta_*^{ }", "d4cab204287151be11232560f8a67bd4": "E=-\\frac{k^{2}m^{3}}{2L^{2}}", "d4cae9f4d5f7ae76f0786b38c8038224": " \\mathbf{a} = - \\omega^2 \\mathbf{r} = - { \\mathbf{v} \\cdot \\mathbf{v} } { {\\mathbf{r} } \\over r^2 } ", "d4cb26c7cf6b5e9eb85ef2a26c4f91e0": "b = (b_1, b_2, \\dots , b_n) ", "d4cb3f0680bac4aff8a0643a58904ae7": " \\frac {d \\vec {J_R }}{d t} = \\frac {d \\vec {J}}{d t} - \\vec {J} \\times \\omega ", "d4cb5ac8a48bd2c57608692e029cf41c": "| \\nu_i \\rang", "d4cc41436f0c3a00fda11a7e738d8793": "\\bold N = [n(t_{1}), ......, n(t_{M})]", "d4cc9cda71b5b4219496176720d36cd6": "s_{-i}\\in S_{-i}", "d4ccd277cbb832629184ab2e8e4bb447": " R_{ new } = R_{ old } + \\frac{ K }{ 2 }( W - L + \\frac{ 1 }{ 2 } \\frac{ \\sum_i D_i }{ C } ) ", "d4cd0dabcf4caa22ad92fab40844c786": "NA", "d4cd193dde1619d30224508094735d84": "\\{ f_{k_j} \\}_{j=1}^\\infty \\subset \\{ f_k \\}_{k=1}^\\infty", "d4cd69d392a8b264b721ee850566b906": "\n\n{\\eta} = y \\sqrt{\\frac{U_{0}(m+1)}{2{\\nu}L}}\\left(\\frac{x}{L}\\right)^{\\frac{m-1}{2}}\n\n", "d4cdae1211950fc717928736b21f63e2": " v(S) ", "d4cdd6e9954694abb297ca7eb95c73c5": "{\\color{Red}\\tfrac{7}{m}}", "d4cde9ea1ab92b4df7c0bf2784816fbd": "\\mathbf{A} = \\mathbf{a}i \\,,\\quad \\mathbf{a} = - \\mathbf{A} i. ", "d4cdf0b49438dbddfc515c5365072c51": "\\operatorname{MKL} (\\bold{H}) = \\int f(\\bold{x}) \\, \\operatorname{log} [f(\\bold{x})] \\, d\\bold{x} - \\operatorname{E} \\int f(\\bold{x}) \\, \\operatorname{log} [\\hat{f}(\\bold{x};\\bold{H})] \\, d\\bold{x}", "d4ce130a2effd0688010b4919f1e7d54": "|\\mathrm{in}\\rangle", "d4ce268936e91cb33294bc5d44f594e9": "c_1 \\mathbf{w}_1 + \\cdots + c_n \\mathbf{w}_n = d_1 \\mathbf{u}_1 + \\cdots + d_m \\mathbf{u}_m", "d4ce403f30cae735a63053ea48c0d517": "\\exists h(z)\\in\\mathbb{R}[z,z^{-1}]\\ h(z)\\cdot s(z) = s(z^2)", "d4ce484e4184bbcebde1690e70525a82": "\\omega_{D}", "d4ce9e43103b4282d1e7e4d4b746ccbe": "\\hat a^\\dagger ", "d4cea4a0bee54433aaa34f7010eda4e3": "A:V\\to V", "d4cee764794cebb1da359e0be231b3ff": " 1-p\\, ", "d4cf3ace86d3c02d6905e7e38941cc12": "v_{\\mathrm{Wind}}(z)", "d4cf78c2c5b1a984cd234014d665ed96": "y = b \\cdot \\sinh E", "d4cf86747cbd2f1b4ba1034fea5b8ab1": "\\displaystyle{\\omega(z_1,z_2) = e^{iB(z_1,z_2)},}", "d4cfe61e6e65e589e530878ec7843787": "\nd_{ij} = \\left ( \\frac{\\partial D_i}{\\partial T_j} \\right )^E\n = \\left ( \\frac{\\partial S_j}{\\partial E_i} \\right )^T\n", "d4d0177e16c7dd0d27de4daa22e5b2fc": "{\\Bbb Z}_2", "d4d0324341fa6301df4602577e9e8ca3": "\\alpha'\\,", "d4d07d35ce49199dd6eb95111346d714": "H^*(Y)", "d4d0c946fb8e182d7db4658f3883de20": "q_t (\\tau) = \\begin{cases} \\mathrm{e}^{-\\zeta\\tau} \\left( c_1 \\mathrm{e}^{\\tau \\sqrt{\\zeta^2 - 1}} + c_2 \\mathrm{e}^{- \\tau \\sqrt{\\zeta^2 - 1}} \\right) & \\zeta > 1 \\text{ (overdamping)} \\\\ \\mathrm{e}^{-\\zeta\\tau} (c_1+c_2 \\tau) = \\mathrm{e}^{-\\tau}(c_1+c_2 \\tau) & \\zeta = 1 \\text{ (critical damping)} \\\\ \\mathrm{e}^{-\\zeta \\tau} \\left[ c_1 \\cos \\left(\\sqrt{1-\\zeta^2} \\tau\\right) +c_2 \\sin\\left(\\sqrt{1-\\zeta^2} \\tau\\right) \\right] & \\zeta < 1 \\text{(underdamping)} \\end{cases}", "d4d0de8768da41973bca54c15a596b73": "P^{-1} = \\begin{bmatrix}\\cos(\\theta)&\\sin(\\theta)&1\\\\\n\\cos(\\theta - \\frac{2\\pi}{3})&\\sin(\\theta - \\frac{2\\pi}{3})&1\\\\\n\\cos(\\theta + \\frac{2\\pi}{3})&\\sin(\\theta + \\frac{2\\pi}{3})&1\\end{bmatrix}", "d4d101ffbfb988715f07a64399639eda": " [\\Delta,L_{a}] ", "d4d1ad331c3c14f2a61e792ab3218442": "\\dfrac{0.04m}{\\text{total mass}}", "d4d1e8270524e2881db3c8ca618c3260": "N_f-2 \\ge N_c > \\frac{2}{3}N_f", "d4d21667e0f7957eabfb3066de8e0acf": " \\int_C \\varphi(\\mathbf{x}) ds = \\int_a^b \\varphi(\\mathbf{x}(\\lambda))\\left|{\\partial \\mathbf{x} \\over \\partial \\lambda}\\right| d\\lambda", "d4d26f09c44a0dd802e3283c8fd9bea8": "G_i(R[t, t^{-1}]) = G_i(R) \\oplus G_{i-1}(R), \\, i \\ge 0, \\, G_{-1}(R) = 0", "d4d376a8afa8e2fe674ffd1b640d3400": "\\mu-s,\\mu+s", "d4d3be9d1bc25639e4e18ce21f5f3745": "\\displaystyle{\\|w\\|_{(2)} \\le C(\\|V w\\|_{(1)} + \\|W w\\|_{(1)}) \\le C^\\prime \\|\\Delta w\\|_{(0)} + C^\\prime \\|w\\|_{(1)}.}", "d4d3c2c1395d30c4b34006fa023053b3": " |A'\\rangle = \\alpha |A_0\\rangle_C + \\beta |A_1\\rangle_C .", "d4d3d045e26bb11167b2288c617b18b6": "x_1=0.9511", "d4d3ff833177e5949d0a29fe2f530a09": "\nU(x;q) = u_0(x) +\\sum_{m=1}^{\\infty} u_m(x) \\, q^m. \n", "d4d4764baf30970cc226068293ccbacc": "H = T + V", "d4d49bead125261b226eaa867bd016ce": "\\forall ", "d4d4b35cc174419e0c1086230a398489": "C^{\\Delta[2]} \\to C^{\\Lambda^1[2]}", "d4d4c85afd3908d3255f1d774fbbda6d": "\\tfrac{1}{25}(33+\\varepsilon)", "d4d4eefa606326942041260c584fe7b3": "{{pp-template}}{{documentation}}", "d4d523c0152dc78493afc5be32ddb527": "A=\\begin{bmatrix}\n1 & 0 & 0 & 0 & 0 \\\\\n3 & 1 & 0 & 0 & 0 \\\\\n6 & 3 & 2 & 0 & 0 \\\\\n10 & 6 & 3 & 2 & 0 \\\\\n15 & 10 & 6 & 3 & 2\n\\end{bmatrix}", "d4d5ade41287b05bb293fa2c4736f730": "\n U = \\left[ \\sum_{j=1}^n \\frac{X_{1j}^2}{\\sigma_1^2} \\right]^{1/2}, \\qquad\n V = \\left[ \\sum_{j=1}^n \\frac{X_{2j}^2}{\\sigma_2^2} \\right]^{1/2}.\n", "d4d5cfaadb4c72f19b25d1dbbb166f28": "\\tau_1 \\circ_s \\tau_2", "d4d5e11c733f80fe40d78044948bfb5e": "f(\\mathbf{x})=\\mathcal{A}\\boxtimes_{n=1}^N \\mathbf{w}_n(x_n),", "d4d5f75415ca2b9df0eb0d133a7bd1eb": "\\kappa\\rightarrow(\\lambda, \\mu)^n", "d4d63fc01e3781e74ddb4d18eb5b4b7d": "Q = v A \\cos\\left(\\frac{\\pi}{2}\\right) = 0", "d4d687a6af3c8d4490c5a743d3ab149e": " \\langle x(t) \\rangle ", "d4d6b82223ae91a094608f4d2349f1f8": " \\partial_\\phi, \\; \\; \\sin\\phi \\, \\partial_\\theta + \\cot\\theta \\, \\cos\\phi\\, \\partial_\\phi, \\; \\; \\cos\\phi \\, \\partial_\\theta - \\cot\\theta \\, \\sin\\phi\\, \\partial_\\phi", "d4d6c07d42508aaa27ed1397501c8b2f": "\\frac{\\partial \\psi}{\\partial t}(t,x)=P \\psi(t,x)", "d4d70dc625745a1fd0543458356a7657": "\\displaystyle [a]", "d4d74db6c50a101b5a8b10e1d886dc9a": "S_b(Tr(g))=\\left(Tr(g^{b-1}),Tr(g^b),Tr(g^{b+1})\\right)\\in GF(p^2)^3", "d4d7fdea46bafbb35e4531ffad57ee88": "q_0 \\in [0,1]", "d4d88633a26264df3c4fadc2a86d5fbe": "V(x) = \\max_{a \\in \\Gamma (x) } \\{ F(x,a) + \\beta V(T(x,a)) \\}.", "d4d8c81f21b8c5c784c38f0f1b9bc7f0": "2^m,2^m-1,2", "d4d96377ae2f43f22f3c76b5ae8b7791": "\n \\hat{w}_n = A_1 \\Bigl[\\cosh\\beta_n x - \\cos\\beta_n x +\n \\frac{(\\cos\\beta_n L + \\cosh\\beta_n L)(\\sin\\beta_n x - \\sinh\\beta_n x)}{\\sin\\beta_n L + \\sinh\\beta_n L}\\Bigr]\n ", "d4d98225e3da3a6cc268713c3a9263b1": "Z_2.", "d4d9dfef08e94b0b8ecaf57c52a462c1": " |V| = \\sqrt{ I^2 + Q^2 } ", "d4daa3c0edbeae553c7f7a80741b1e16": " k! = \\prod_{p\\text{ prime}} p^{f(p,k)} \\, ", "d4daad9ba7c70db5c709efb4aeedc4dc": "\\operatorname{cov}\\left [\\frac{1}{X},\\frac{1}{1-X} \\right ] = \\operatorname{cov}\\left[\\frac{1-X}{X},\\frac{X}{1-X} \\right] =\\operatorname{cov}\\left[\\frac{1}{X},\\frac{X}{1-X}\\right ] = \\operatorname{cov}\\left[\\frac{1-X}{X},\\frac{1}{1-X} \\right] =\\frac{\\alpha+\\beta-1}{(\\alpha-1)(\\beta-1) } \\text{ if } \\alpha, \\beta > 1", "d4db1ee6607a7f5b02bf5b8491e6c5e6": "\\mathbb{F}[G]", "d4db27fb6b2a0f5339826fbe553c8367": "(a_k)_{k=1}^\\infty, \\qquad a_k = k^2.", "d4db44f047f08c911d4521e2dc2030ec": "y'=\\underbrace{\\begin{pmatrix}1&-1/x\\\\1+x&-1\\end{pmatrix}}_{=\\,A(x)}y", "d4db97529f12135daa6d9a15ada4938c": "k\\ge 1", "d4db9b9ea65c687012c115f08060c797": "\\frac{m}{\\sqrt{1 - (v/c)^2}}", "d4dbd5b06d4be18fba0abffdcdb8bf3b": "Pr\\ne 1", "d4dbe753c1d6e2ddfa20a4a6b9e816ba": "\\Phi(r, \\theta)", "d4dc076ac4a00e7e39acce7825d64395": "C^{\\infty}(\\Omega)\\;", "d4dc4c9e63a63b086877b7d10fe94ad8": "\\frac{15+13+17+100}{4}=36.25", "d4dd8296ee5d9d2bf22bb3633ceb8259": "(a^2 -b^2)(a^2+b^2) = (c^2 -d^2)(c^2+d^2)", "d4ddbb019de66e793988630b24a06f1f": "\\left\\{\\sqrt{d^{(\\pi)}}u^{(\\pi)}_{ij}\\mid\\, \\pi\\in\\Sigma,\\,\\, 1\\le i,j\\le d^{(\\pi)}\\right\\}", "d4ddd8dfefcfaa3c912c990f91bd9790": "\\frac{1}{R}", "d4de039327be4f093b0c39b68d233f4c": "\\lambda^\\prime(kx) = \\Delta_{AN}(x)^{1/2} \\lambda(x)", "d4de3136c194a2045a8c532597e3a87d": "e_k=-{\\alpha^{i_k}\\Omega(\\alpha^{-i_k})\\over \\alpha^{c\\cdot i_k}\\Lambda'(\\alpha^{-i_k})}.", "d4de48813dfb3df2b8f8ca50da2b81a1": "=\\frac{{\\color{Periwinkle}\\mathfrak{\n4\\cdot5\\cdot6\\cdot7\\cdot8\\cdot9\\cdot10\\cdot11\\cdot12\\cdot13\\cdot14\\cdot15\\cdot16\\cdot17\\cdot18}}\n\\mathbf{\\cdot19\\cdot20\\cdot21}}{\\mathbf{1\\cdot2\\cdot3}{\\color{Periwinkle}\\mathfrak{\\cdot\n4\\cdot5\\cdot6\\cdot7\\cdot8\\cdot9\\cdot10\\cdot11\\cdot12\\cdot13\\cdot14\\cdot15\\cdot16\\cdot17\\cdot18}}},", "d4ded198d5c83d4abcbe68efe4631fa1": "m_n = \\int_{-\\infty}^\\infty x^n \\,d\\mu(x)\\,.\\,\\!", "d4dedf14c456d4836b67ca7851509519": "2{n} \\cdot \\pi", "d4df2216fe16d38f7eb936acebfce166": "\\mathcal{R}_\\mathcal{R}(C^{bnn}_{n,n'}) - \\mathcal{R}_{\\mathcal{R}}(C^{Bayes}) = \\left(B_1 \\frac{n'}{n} + B_2 \\frac 1 {(n')^{4/d}}\\right) \\{1+o(1)\\},", "d4df751019ccf1b50fc3051bb640c6f4": "I = \\lim_{\\Delta t \\rightarrow 0} \\sum_{i=1}^n f(\\mathbf{r}(t_i))|\\mathbf{r}'(t_i)|\\Delta t", "d4df8503f1ae4402d5ba50def87ab64a": "f(\\alpha) \\neq \\alpha", "d4df9e96ccf06440f128987e2eb03707": "\\exp_{\\mathbf{R}}(\\hat{\\mathbf{R}}) =\\mathbf{R}^{\\frac{1}{2}}\\exp\\left(\\mathbf{R}^{-\\frac{1}{2}}\\hat{\\mathbf{R}}\\mathbf{R}^{-\\frac{1}{2}}\\right)\\mathbf{R}^{\\frac{1}{2}}", "d4dfcade2fe0ad44eecec09b8ac38e64": " \\int_{\\partial \\gamma} \\phi = \\int_{\\gamma} d\\phi", "d4dfcb293ff7d3aa5c3b0c6eca68d816": "F(x + 1) = F(x) + \\frac{1}{x}", "d4e02d5f333afeb167ed0a641e135797": "\n \\left( A \\rightarrow \\left( B \\or C \\right) \\right) , \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\vdash \\lnot A\n ", "d4e0329595d9508d453a7ddaba862340": "{\\partial^2 \\phi \\over \\partial x^j \\partial x^j} = 4 \\pi G \\rho \\,.", "d4e039a9a365e50aa9a4f6914646db96": "R(f)", "d4e07accf04d57a9cc396f9858ab2262": "y_2''=v''(t)y_1(t)+2v'(t)y_1'(t)+v(t)y_1''(t).\\,", "d4e082aca0fcc633d8fbcd5fdd0ed724": "\\forall^\\mathrm{st}x\\,\\exists^\\mathrm{st}y\\,\\forall^\\mathrm{st}t\\,(t\\in y\\leftrightarrow(t\\in x\\land\\phi(t,u_1,\\dots,u_n)))", "d4e0b02a3a0ed25cfe558c82e4c1dc61": "\\angle BAE \\cong \\angle DCE", "d4e0e38e7ccf2ff20165077318edcb3a": "I_{(-\\infty,\\theta]}(\\breve\\theta_{j,i})", "d4e115bde2f38aff4f916802a0a40f84": "dY/dt= (\\text{negative})Y", "d4e1298e7e3981be04b09a644a921989": "\\sum_{k=0}^\\infty k^2 \\frac{z^k}{k!} = (z + z^2) e^z\\,\\!", "d4e1323331c263398fe20a40b541c3d1": "\\tfrac{E}{3(1-2\\nu)}", "d4e14d739e16d3e870e9d7e18814ded9": "j = \\sqrt{-1}", "d4e160e9e1c0068253781dfb18ec65fd": "x = exp(2\\pi j/p)", "d4e1c13eb54b3837e415b3bdbe510678": "(AB-BA)|\\Psi\\rangle=\\sum_{n}c_n(AB-BA)|\\psi_n\\rangle=0", "d4e2151a1a8f1697a580c6794b51e640": "\\scriptstyle { | \\mathbf{r},t \\rangle = | x,t \\rangle + | y,t \\rangle + | z,t \\rangle, | \\mathbf{p},t \\rangle = | p_x,t \\rangle + | p_y,t \\rangle + | p_z,t \\rangle, | E \\rangle, | s_z \\rangle, | L_z \\rangle, | J_z \\rangle, \\cdots } ", "d4e2593912f9767223e7cb58875e90e3": "{m_{vehicle}}", "d4e29f833ebd276773d61268e9d9b104": "\n\\mathbf{a}\\cdot(\\mathbf{b}\\times \\mathbf{c})=\n\\mathbf{b}\\cdot(\\mathbf{c}\\times \\mathbf{a})=\n\\mathbf{c}\\cdot(\\mathbf{a}\\times \\mathbf{b}).\n", "d4e2ec117fbeb0b5b8590ec40a1fec53": "\\boldsymbol{F}(\\boldsymbol{S}) = \\boldsymbol{F}_1(\\boldsymbol{S})\\cdot\\boldsymbol{F}_2(\\boldsymbol{S})", "d4e3c0c448b858427904103ed40d2bdc": "A(\\omega) \\approx i \\omega C R_0", "d4e4222559a8905185d70de1c44ca72b": "\\partial_\\mu\\partial_\\nu E_n=\\langle \\partial_\\mu n|\\partial_\\nu H|n\\rangle +\\langle n|\\partial_\\mu\\partial_\\nu H|n\\rangle + \\langle n|\\partial_\\nu H|\\partial_\\mu n\\rangle.", "d4e468bf23165b691c5c0bd4371c7909": " \\{ x_1, \\dots, x_n, p_1, \\dots, p_n \\} ", "d4e49a7d563f51ed3d9a8e05e3895594": "X \\,\\sim Pois (\\mu) ", "d4e4f108426e1362524b096608c068f0": "{\\mbox{Pin}}_{p,q}=\\{v_1v_2\\dots v_r |\\,\\, \\forall i\\, \\|v_i\\|=\\pm 1\\}.", "d4e5d025439f6b8eafa9d34bf650e36e": "\\dot S_i=\\dot Q\\left(\\frac{1}{T_2}-\\frac{1}{T_1}\\right).", "d4e5d1733480bb3bfbd28f8151d0b8c6": "\\frac{\\partial k_{i}}{\\partial t}=\\left[\\frac{\\delta}{2}-\\frac{1+\\delta}{1-\\delta}\\right]\\frac{k_{i}}{t}", "d4e5f7095d2571873f8dd38dc66cfa61": "\n\\eta_{rel}= \\frac{\\eta}{\\eta_{0}}\n", "d4e5f73dcb12b61b852b2d2dcc1529e3": "\\xi\\delta", "d4e5fb8449afdc8c6771c6b4e99e1d23": " (x(s),y(s))=(x_1+s(x_2-x_1),y_1+s(y_2-y_1)),", "d4e64feaf320326e18047d2c0f0c84be": "\\mu_i(0)", "d4e67a6062d43ba9897f54ca5daa4f9b": "\\psi(x) = \\frac{d\\rho(x)}{dx}", "d4e6ce80a10a3d5820809c276930f10d": "w=\\sqrt{1-\\frac{U}{E}}.", "d4e735dfb735a2421181d9243a8bbd88": "a \\uparrow \\uparrow \\uparrow 4 = a \\uparrow \\uparrow (a \\uparrow \\uparrow (a \\uparrow \\uparrow a)) = \n \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{a} }}", "d4e76017069f363064f16b1d42bcd0cd": "\nC_{2i} = \\frac{{12mi^2 - m\\left( {m^2 - 1} \\right)}}\n{{m^2\\left({m^2 - 1} \\right) \\left( {m^2 - 4} \\right)/15}}\n", "d4e76b29fbdebaf6416896fc3528d5ce": "-\\frac{\\partial^2}{\\partial\\theta_i \\partial\\theta_j} \\mathcal{H}\n= \\int \\left[ \\frac{\\partial^2 f(X; \\theta)}{\\partial\\theta_i \\, \\partial\\theta_j} \\left(1 + \\log f(X; \\theta) \\right) + \\frac{1}{f(X; \\theta)} \\frac{\\partial f(X; \\theta)}{\\partial\\theta_i} \\frac{\\partial f(X; \\theta)}{\\partial\\theta_j} \\right] dX\\,.", "d4e76bbef7f4a99fd8b776fee4184201": "\\lim_{z \\to a}\\frac{1}{f(z)}", "d4e78696a05ae64f5c0a2fa7260ef1ce": " \\underline x_i=min\\{x_i: Ax=b, A\\in\\mathbf{A},b\\in\\mathbf{b}\\}, \\ \\ \\overline x_i = max\\{x_i: Ax=b, A\\in\\mathbf{A},b\\in\\mathbf{b}\\} ", "d4e78d6689fa68cd1fa3ea4feda2d82e": "D = \\frac{w^{2}}{4t_{D}}", "d4e793ffc2815c2808df1a845ab48b99": "49 = 9 + 20 + 20", "d4e7f743c0b94301a25c3974d61c50dc": "\n\\begin{cases}\n\\mathrm{ERBS}(0) = 0\\\\\n\\frac{df}{d\\mathrm{ERBS}(f)} = \\mathrm{ERB}(f)\\\\\n\\end{cases}\n", "d4e7fdfbe00d073b4bbceb5594201ece": "\\prod p_i^{t_i-1},", "d4e871d7572bc05ef2382ba977d584be": "\n\\textrm{Span}(\\mathbf{D}) \\equiv \\left[ \\mathbf{D} \\right] = \\mathbf{D}(\\mathbf{D}^T\\mathbf{D})^{\\dagger}\\mathbf{D}^T\n", "d4e891dae7145cdf720c402b225a1765": "\\omega_{xy}", "d4e8bee50f2093fbced5daaf3c41980d": "EL", "d4e8ce101a3f4187fb22214dc3485cf3": "\nR = \\frac{XU}{YZ}\n", "d4e902143ea0d64bc37dcd52353ef551": "\n \\frac{1}{E^*}=\\frac{1-\\nu^2_1}{E_1}+\\frac{1-\\nu^2_2}{E_2}\n", "d4e9059dccbafc31d205e5636d0fbb74": "\\kappa(\\omega)=\\sqrt{\\frac{\\gamma}{2\\pi}}\\,.", "d4e917b13b591a763c18a560e4d6b129": "Q \\cap H_r", "d4e922db0b9b0e1de98dfe636da186ea": "\\frac{\\partial \\rho}{\\partial t} + \\bold \\nabla \\cdot \\bold j = 0", "d4e95dd2d19c66bf09fd9ffe36d78ce5": "{\\alpha}_{k} = -\\textbf{S}_k^{-1/2} \\textbf{H}_{k}\\hat{\\textbf{x}}_{k\\mid k-1} +\n\\textbf{S}_k^{-1/2} \\textbf{z}_{k} ", "d4ea1a9e6b23eb12aea4f1b1fec13844": "U_{a'a}", "d4ea28678ef80ff5ee945c65506870c0": "\\displaystyle{G_{\\mathbf{C}} = G\\cdot A \\cdot N}", "d4eaafb2473cdd35b69852390cd06121": "\n\\varphi \\mathbf{(r)} = \\frac{1}{4 \\pi \\varepsilon_0 } \n\\sum_{i=1}^{n} \\frac{q_i \\left( \\mathbf{r} - \\mathbf{r}_i \\right)} {\\left| \\mathbf{r} - \\mathbf{r}_i \\right|}\n", "d4eac1ef29ddbd9e1b1c4ff3fb9be9e3": "f'(0)=\\lim_{x\\searrow0}\\frac{f(x)-f(0)}{x-0}=\\lim_{x\\searrow0}\\frac{e^{-1/x}}{x}=0.", "d4eaff9e65e4dd1bf32358bc9f30b6b6": " (X^{(1)},X^{(2)},X^{(3)},...)", "d4eb039465e1a8037484d3c83235d4b3": " \\mathbf{A} = \n \\begin{bmatrix}\n a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m1} & a_{m2} & \\cdots & a_{mn}\n \\end{bmatrix}.\n", "d4eb2fca0cf429b2953cb1834d6f0907": "\\mathbf{B}(\\mathbf{m}, \\mathbf{r}) = \\frac {\\mu_0} {4\\pi} \\left(\\frac{3(\\mathbf{m}\\cdot\\hat{\\mathbf{r}})\\hat{\\mathbf{r}}-\\mathbf{m}}{r^3}\\right) + \\frac{2\\mu_0}{3}\\mathbf{m}\\delta^3(\\mathbf{r})", "d4eb9594ba5f6aa29810ae139d771846": "x \\mapsto (x_0, x_1, x_2, \\ldots)", "d4ec8a9007076bb35ef588496b3b3cba": "j_2(x)=\\left(\\frac{3} {x^2} - 1 \\right)\\frac{\\sin(x)}{x} - \\frac{3\\cos(x)} {x^2}", "d4ecae498125ad2325af38ee84dd339d": "\n y_i = Y_{-i}\\gamma_i + X_i\\beta_i + u_i, \\quad i=1,\\ldots,m,\n ", "d4ecbc6da6e60ec043c88e1f4bfccea6": "\\mathbf{h}=\\mathbf{H_{{2}/{1}}}/m_2\\,\\!", "d4ecdc6346a185aeb795c45c46cbbc52": "|\\psi \\rangle = c_{\\psi} |\\uparrow_x \\rangle + d_{\\psi} |\\downarrow_x \\rangle", "d4ecedaddc2c82c89420ac752be86c02": "R_3(\\xi,x)=x\\,\\frac{(1-x_p^2)(x^2-x_z^2)}{(1-x_z^2)(x^2-x_p^2)}", "d4ed2059bf100ee21ff606ce79ae8bf9": "\\rm \\ 2LiNH_2 \\rightarrow Li_2NH + NH_3", "d4ed34c8cb75d903aa4c66dae6f209ec": "1_{\\{X \\leq \\mathbb{E}[X]\\}} = \\begin{cases}1 & \\text{if } X \\leq \\mathbb{E}[X]\\\\ 0 & \\text{else}\\end{cases}", "d4ee0763034a8a46e96f3b781634c440": "m = 21600 = 2^5 \\times 3^3 \\times 5^2 \\, ,", "d4ee33f180b52e44282bb4cdc872cc64": " \\mathbf{E}(z,t) = \\mathrm{Re} (\\mathbf{E}_0 e^{i(k z - \\omega t)})", "d4ee82b3c24fc2e7e61bb1e52bf88b01": "\\Delta B_{max} = 2\\Delta B_{1s}", "d4ee9789bc2d79fce0974d29567ae4cc": "dU = T dS - p dV.\\,", "d4eeeeddb7b13d65f541779535e9ef0d": "V = V_{\\text{exterior}} + V_{\\text{interior}} = n + { n \\choose 4 }.", "d4ef580d534ebbd31a18e711a8b811e5": "{\\bold \\ L}", "d4ef866c757db088fbd6e4543f51ceb6": "-j0.80\\,", "d4ef97516a88a821e0237e40b3f8db67": "C(G) = \\frac {1}{N_2} \\sum_{i \\in V, d_i \\ge 2} c(i)", "d4eff2a8991acbeb4015312c4732887e": " V^\\alpha {}_{;\\alpha} = V^0 {}_{;0} + \\cdots + V^n {}_{;n} ", "d4eff4d0678b2d08cf5b2447685c0544": "\\tilde{O}(\\log^5 q)", "d4f009bb4871894cf745d9c9bbce26fa": "f(m) = f(m')", "d4f028f051a507d782bfb0218e032313": "\\int_S \\omega =\\int_D \\sum_{i_1 < \\cdots < i_k} a_{i_1,\\dots,i_k}(S({\\mathbf u})) \\frac{\\partial(x^{i_1},\\dots,x^{i_k})}{\\partial(u^{1},\\dots,u^{k})}\\,du^1\\ldots du^k", "d4f02fcdf87e8c5f349c9b7b32a270f6": "A\\,\\triangle\\,B = \\{x : (x \\in A) \\oplus (x \\in B)\\}.", "d4f04e958b79f4c5482d8f22bbd45bde": " \\exist r \\exist u [u p(x) \\ge 2^{-L(x)}", "d4fd4fd85c7469d5fae00c3ea9615331": "{\\mathcal M}B(z^2)=B(z^3) {\\;} {\\mathrm {or}} {\\;} {\\mathcal M}{\\;}{\\mathrm {green}}\\subseteq{\\mathrm {yellow}},", "d4fd942df44fd59f16585d149befc58d": " \\psi_1(z) ", "d4fda8db95fc18fd2be811d9194b5c4b": "N(f) \\le \\mathit{MF}[f],", "d4fdac8a6130fe21e11becd229915b7e": "\\bigg[\\frac{\\alpha}{\\beta}\\bigg]_2 = \\bigg[\\frac{\\alpha}{\\beta \\mathcal{O}_k}\\bigg]_2. ", "d4fdd275ac743eebc2ed21a8214d3301": "\\,k_{2,m}=cos\\phi_{2,m}/cos\\psi_{2,m}", "d4fdf5318dfa42cbfeb2298336a8d0c1": "\\theta \\circ ((\\theta,1) \\circ ((\\theta,1),1))", "d4fe166739c723e6d9543cfa2882c727": "\\mathbf{T} = T_{j_1 j_2 \\cdots j_p} \\mathbf{e}_{j_1}\\otimes\\mathbf{e}_{j_2}\\otimes\\cdots\\mathbf{e}_{j_p}", "d4fe83283f60bea565ad98c7d8e7218e": "[2^k,k,2^k/2]_2", "d4fea9bac6feb6de6a7009e99c623b49": "S\\subseteq V", "d4fed75713934a92b8f92e47c737378d": "Z_1\\,", "d4ff30078a6e41979b698c53582a0f4f": " x^* \\mapsto x^*(x_1) \\, x_2", "d4ff933b24b090fa0a1257ac9de9d2a1": "4\\frac{\\partial}{\\partial z} \n\\frac{\\partial}{\\partial \\overline{z}} \\Phi(z,\\overline{z})=\\lambda^2(z,\\overline{z}).", "d4ff9c52ce42c319debbfef327bfa58b": "\\delta_{x}(y)", "d4ffdd5b5c0a03457373a12dcb78e4f5": "\\mathrm{M}(W \\times A)", "d4fff161a43278cc251e19bb7a44da9a": "\nV(x_1,\\dots,x_N) = V_1(x_1)+ V_2(x_2) + \\cdots + V_N(x_N)\n\\,", "d4fffac8e7415e72eee2fb76b779dc34": "\\{z:|z|j. ", "d50630884f8ad91aeab5d68114000e2a": "U_1 \\cap U_2", "d50671351c441c2055c6dd7dab56f192": "A_1\\, v_1\\, = A_2\\, v_2", "d50688b99fdd129cf3fd8686cffb7a83": "E_{m,n} \\!", "d5069e3549439d04cbdabbc7df580fd9": "5\\rightarrow (1,2)_{1\\over 2}\\oplus (3,1)_{-{1\\over 3}}", "d506a17fa4734553171150bd17aaa092": "\\Phi_{PW} (r)=-\\frac{GM}{r-r_g}", "d506be0b514763f0e30c66a8a06f09f5": "\\pi:X \\times \\{T,F\\}\\to \\{T,F\\}", "d506c9cafd2d8df80ee25c2ffb08fea2": "\nE = A_{xx} x_{0} + A_{xy} y_{0} + B_{x}\n", "d50710d3daa2bcf128a706f7f359afb7": "\\ddot{\\delta \\mathbf{r}} = \\mathbf{\\ddot{r}} - \\boldsymbol{\\ddot{\\rho}}", "d5077bafaeb276cda5c6a3cdb8a753a4": "I(0)=0", "d5078df8ad1a4811074252c16f76be76": "\\Omega_0 = \\{\\bot, \\top \\} \\,", "d507b0d125dc07c9a304d5b122943afe": "\\mu\\left(\\bigcup_{n=1}^N A_n\\right)=\\sum_{n=1}^N \\mu(A_n)", "d507d92b8109cd79283c5165815ba5b4": "\\frac{d [A]}{d t} = -k_1[A][B]", "d5088d4db9e83907e48e5255ee3bcdb1": "F_{in}v_{in} = F_{out}v_{out}.\\,", "d508ec1130159e3b5652bda4210af31f": "\\pi_p(F_n(\\mathbf{C}^k)) = \\pi_p(F_{n-1}(\\mathbf{C}^{k-1})) = \\cdots = \\pi_p(F_1(\\mathbf{C}^{k+1-n})) = \\pi_p(\\mathbf{S}^{k-n}).", "d508f5bebdc065b56f7eec56f79da17d": "(2, 5),", "d509188bff3e54eeff6dad9badabe572": "=\\frac{1-q}{1-q} \\frac{1-q^2}{1-q} \\cdots \\frac{1-q^{n-1}}{1-q} \\frac{1-q^n}{1-q}", "d509432111298f0f9ee378bbe4fb29da": "\n \\frac{\\partial I_1}{\\partial \\boldsymbol{C}} = \\boldsymbol{\\mathit{1}} ~;~~\n \\frac{\\partial I_2}{\\partial \\boldsymbol{C}} = I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{C} ~;~~\n \\frac{\\partial I_3}{\\partial \\boldsymbol{C}} = \\det(\\boldsymbol{C})~\\boldsymbol{C}^{-1} \n", "d509a6d981b26ea90f98ecff8c1fd2be": "S, T, U \\subset G", "d50a2db0ad03e2e2b4995e7a2f177222": "Z = ( A \\cdot \\overline{S}) + (B \\cdot S)", "d50a52aba5847bd159b5f32e0a27b98f": "{\\rho^2_{XT}}", "d50a8b1ccbaedbc5259ab771cb1b4ceb": "f(x) = F'(x).", "d50ac3c417b88d568f50168ca9c1572f": "J^{\\prime\\prime}", "d50afcf92bb88aea1cb440eb47d3eb7b": "\\ge \\frac{2 \\pi}{P}", "d50b0a38b325bf1aab31b9c0d8104f29": "\\rho_n(e^{i\\theta}) = \\begin{bmatrix}\n\\cos n\\theta & -\\sin n\\theta \\\\\n\\sin n\\theta & \\cos n\\theta\n\\end{bmatrix},\\quad n\\in\\mathbb Z^{+},", "d50b32f05134aa34ac1c106d551c3582": "\\displaystyle{\\|\\pi_z(g)\\|\\le {1+|z|\\over 1-|z|}.}", "d50b53ad30a74072b6b3a28cc7c462d0": "\\frac{d}{dt}e^{\\mathbf At} = \\mathbf A e^{\\mathbf At} = e^{\\mathbf At} \\mathbf A", "d50baa151fdca08412e0a1641fdaca8a": "2+2+2=6 \\equiv 0 \\pmod {6}", "d50bf166a474853ab5721c4dbc26bb00": " \\left\\|\n\\begin{matrix}\n(1 + b^T x +c)/2\\\\\nAx\n\\end{matrix} \\right\\|_2\n\n\\leq (1 - b^T x -c)/2.", "d50c0c7cb763a16a48ecde7345d7750c": "f : R^n \\rightarrow R", "d50c0cbfe68ca13c180ff7185d2c6b67": "{\\textbf{P}}_k", "d50c1c73c7f854902130e89b98cd3c77": "\nV_\\text{S} = E_\\text{S} = N_\\text{S} \\frac{\\mathrm{d}\\Phi}{\\mathrm{d}t}.\n", "d50c60d5180c41267b0cc576f1631762": "\\binom{n}{\\lfloor n/2 \\rfloor}^{-1} H(P) \\le M(P) \\le H(P) \\sqrt{n+1} ; ", "d50ccab15f5d1d1e4d39fcf8f4eecfac": " | \\psi_i'\\rangle \\sqrt {p_i'} = \\sum_{j} u_{ij} | \\psi_j\\rangle \\sqrt {p_j}.", "d50cd5a39fc46bfa8b3acb23de9b3bf0": "K_1(A) = \\operatorname{GL}(A)^{\\mbox{ab}} = \\operatorname{GL}(A) / [\\operatorname{GL}(A),\\operatorname{GL}(A)]", "d50d933e96f226f0ec7868486fae1b2b": "\n \\left|\\psi\\right\\rang = -\\frac{1}{\\sqrt{2}} \\bigg(\n \\left|+x\\right\\rang \\otimes \\left|-x\\right\\rang -\n \\left|-x\\right\\rang \\otimes \\left|+x\\right\\rang\n \\bigg)\n", "d50def5688bee50a11439ff187491dfb": "0=-\\nabla p+\\mathbf{j}\\times\\mathbf{B}. ", "d50df52b395738d5857e0a198c14248b": "x \\leftarrow x + 1", "d50e0a7608fac044f29f4b5d3103b31f": "\\scriptstyle\\sqrt{gh}", "d50e219bac6e8189ecec7091f7f197e3": "\\frac{dy}{dx}=\\tan \\varphi = \\frac{\\lambda gs}{T_0}.\\,", "d50e8aa80e24bbd1eb36aae9a3fc3895": "\\Delta{m}_{15}", "d50eb89d6e99b0317c04fc359eb52bd9": "R_\\mathrm{xy} = R_xR_y\\sum_{i=1}^N \\frac{1}{R_i}", "d50edbe99b4f4ba5fc7f541282f87090": "\\int_0^\\infty e^{-x} x^{n-1} \\, dx", "d50ef092613cc40ad5f6e2b75c179812": " \\mu = \\frac{mc}{\\hbar}", "d50ef143c548522f64c58e69ae06d8c8": "h_{31}=h_{32}=0, \\; h_{33}=1.", "d50f0a076977d5f20842e8a15435992b": "\\Psi_{k,l}(x)\n= -\\frac{\\partial^{l-1}}{\\partial x^{l-1}}\n\\left(\\prod_{j=0,j\\neq k}^a \\left(\\lambda_j+x\\right)^{-r_j} \\right) .\n", "d50f3cc0250b4ad2bd14ad1c30bfd1c2": " \\frac{a+b}{a} = \\frac{a}{b} = \\varphi.", "d50f66da826d27e8efd85a6a4b67cb23": "w_k(\\mathbf{x}) = \\frac{1}{(D_{**}(\\mathbf{x}, \\mathbf{x}_k) )^\\frac{1}{2}},", "d50fc3cc0a9529b66f125525859e849d": "BGL(R) = \\varinjlim BGL_n(R) \\to B(S^{-1}S)", "d50ff7d83829dec2124b3498ee010873": "\\rho=\\sqrt{y^2+z^2}", "d5100cbf3bc6a9606cd0b6df885fa441": "{\\mathbb L}_{y^n}(L)\\equiv{\\langle\\langle} L,{\\mathfrak k}_n{\\rangle\\rangle}", "d5100f37239b65d8d39c77afebf7e797": " \\arctan x = \\frac{1}{2} \\; G_{2,2}^{\\,1,2} \\!\\left( \\left. \\begin{matrix} \\frac{1}{2},1 \\\\ \\frac{1}{2},0 \\end{matrix} \\; \\right| \\, x^2 \\right), \\qquad \\frac{-\\pi}{2} < \\arg x \\leq \\frac{\\pi}{2} ", "d5103591e7e24ce576d68ef0cf9af472": "x^q = x^{3 n + 2} \\equiv x \\pmod{q}\\;\\mbox{ and }\\;x^{q - 1} = x^{3 n + 1}\\equiv 1 \\pmod{ q},\\mbox{ so }\n", "d5103ae732b3a1c5b8b23bf0e6e4497d": "\\{0^n 1^n : n \\in \\mathbb N\\}", "d5108fdc118426f44c093b64127e756a": " G=(V,E) ", "d510b3a24b05f701f2892b05c356bfa4": " \\cos \\alpha \\,f(a) - \\sin \\alpha \\,f^\\prime(a)=0, \\qquad \\cos\\beta \\,f(b) - \\sin \\beta\\, f^\\prime(b)=0,", "d5111eb7facc1a8e85427a80791bbb46": " N^{(\\mu)}(\\mathbf{k}) \\equiv {a^\\dagger}^{(\\mu)}(\\mathbf{k})\\, a^{(\\mu)}(\\mathbf{k})\n", "d51133fe7ca3d7797633445c65e1010c": "\nM=D^{-1}L. \\,\n", "d5113d9cbe4b02f7a282cfbc587f44bb": " \\omega_0 = 2 \\pi f_0 = \\frac{1}{ \\sqrt{R_1R_2C_1C_2} } ", "d511cde663e75a8bde6cb58f32268e9f": " (a^b)^c.", "d511fa50741afd81f55fdc7d73bde824": "KE=\\frac{p^{2}}{2m^{*}}=\\frac{h^{2}n^{2}}{8m^{*}\\pi ^{2}r^{2}}=-\\frac{U}{2}=\\frac{q^{2}}{8\\pi \\varepsilon \\varepsilon _{r}r} \\; \\; (3)", "d512453861b9feeadcc11a2861875042": "(id,id)", "d5127b7fd595310e4ec08b8b43421c8d": "\\sum_{j=1}^m k_j = n \\,", "d512872509572ce81ed604695be38508": " \\left[-\\pi/2, \\pi/2\\right] ", "d512c0d5024f9822e455e05f25c2d09d": "{dQ_{pn} \\over dt} = F_{pn} (C_{art} - {{Q_{pn}} \\over {P_{pn} V_{pn}}})", "d512c7bccacc51e0e253570052319712": "\\textstyle M_t = f(t,W_t), ", "d512e23e45cd94eadb53daca5c6cc0dd": "\\alpha_{\\mathrm p}", "d512fbd1bbadf67ea1281387d8a3d700": "d_{2p+1}\\ ", "d513243483a8265c2de49db03f79a67a": "\\int_{X} f(y) \\delta_{x} (y) \\, \\mathrm{d} y = f(x),", "d51332b0f72fb63503d137269cf0fd86": "\\Rightarrow rP_0-M_a+M_ae^{-rT}=0", "d5138aa4934c9942a9dd99939ae50b29": "G_\\infty\\;", "d5138f0423fa5ccb86ded55474d20ce7": "\\delta/2", "d51390a6eb77a49680deb12aa99d4eb9": " T(0.833) ", "d51399c810c981185d40626d9caded6f": "d_0 |t|^{-\\beta} \\leq \\varphi_X(t) \\leq d_1 |t|^{-\\beta} \\quad \\text{as } t\\to\\infty", "d513b8cccaa72dc3f9718484b2778bf3": "(XY)^2 - 1 \\in B", "d513c93d5bec6dbd90eeb82aa42069bb": "\\beta\\in [0,\\infty]", "d514422d79c8b79bb2c8efa25a34fdab": " \\mathbf{A} \\mathbf{v} = \\lambda \\mathbf{v} ", "d5147f3b1a0cb400e908088e4010bc00": "K = H^{}", "d515329bd54a36867cf84de0d7c37df3": " \\rho_s", "d5154e721367991d8fa70e4f245fc233": "E +\\delta E", "d5157c00b093f4ec8e4cd5834e686835": "{\\mathcal D}'(M)", "d5158d14a124f3122eb56b743901ceeb": "(n_1, h_1)*(n_2, h_2) = (n_1\\varphi_{h_1}(n_2), h_1h_2)", "d5167c8c3a8b6dea9d49341535f4ae2c": "\\nabla_\\lambda (\\delta \\Gamma^\\rho_{\\nu\\mu} ) = \\partial_\\lambda (\\delta \\Gamma^\\rho_{\\nu\\mu} ) + \\Gamma^\\rho_{\\sigma\\lambda} \\delta\\Gamma^\\sigma_{\\nu\\mu} - \\Gamma^\\sigma_{\\nu\\lambda} \\delta \\Gamma^\\rho_{\\sigma\\mu} - \\Gamma^\\sigma_{\\mu\\lambda} \\delta \\Gamma^\\rho_{\\nu\\sigma}. ", "d51688e0c326e2fca5cc2077913a68c8": "G \\to G \\times_X G", "d51692348c7e95726dc69611365692cf": "\\rho_V", "d516a1225810504963101e8c500a6723": "|01\\rangle _{VH}\\equiv |0\\rangle _V |1\\rangle _H", "d516b54ab3881d60af495af11920a1c8": " \\mu : K[G] \\otimes K[G] \\to K[G] ~\\text{by}~ \\mu(g \\otimes h)= \\left\\{ \\begin{array}{cl}\ng \\circ h & \\text{if target(h) = source(g)} \\\\\n0 & \\text{otherwise} \\end{array} \\right. ", "d516bb961870f287a9099bd520dc5640": "H(z) = \\frac{P(z)}{Q(z)} = { G \\cdot \\displaystyle\\sum_{m=0}^M {b_m z^{-m}} \\over 1 + \\displaystyle\\sum_{n=1}^{N} {a_n z^{-n} } } ", "d516ea760e3abd30c9cbaeb300987346": "\\det (\\tilde{E}) = |\\det (E)|^2", "d5179d43083e8a9a5b990bd4bd96fd7c": "J=\\sqrt{\\frac{kT}{2hcB}}", "d517b5c206897d4dae1d50e548e623cb": "(A,b)", "d517b90052b27cfb104bd4d37eeaa2da": "\\exp\\left(\\left(1+o(1)\\right)\\left(\\tfrac{32}{9}\\log n\\right)^{1/3}\\left(\\log\\log n\\right)^{2/3}\\right)=L_n\\left[1/3,(32/9)^{1/3}\\right]", "d517c3d5aa566014a9f021e4b3866f77": "I = \\lim_{\\Delta t \\rightarrow 0} \\sum_{i=1}^n \\mathbf{F}(\\mathbf{r}(t_i)) \\cdot \\Delta\\mathbf{r}_i", "d517d2221ebf5207ce6afac2759567cc": "\\mathcal{L} = -mc^2\\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2} + e \\mathbf{A}\\cdot\\dot{\\mathbf{r}} - e \\phi \\,\\!", "d517e3a509a09010ea03e8b689dae5d0": "2^{nd}", "d517ee4f93616aa1889cfc232c0c24d7": "F F'' + 2 F'^2 + (\\eta - 1) F' + F' = 0 \\quad ; \\quad F(1) = 0 \\ , \\ F'(1) = -\\frac{1}{2}", "d517f138e70eb2a73da722e008b7b558": "r^\\prime", "d51824062119270816bc1d5949447059": "\\textstyle \\mathcal{C}", "d5186fe7f61ef57eb8eaf1f49611143e": "\\Rightarrow A + B = \\frac{a^2}{c^2}C + \\frac{b^2}{c^2}C\\, .", "d51895fdccbb13274a32dc40ebe9df2f": "\\langle v,v\\rangle \\geq 0", "d5189de027922f81005951e6efe0efd5": "location", "d518b44f926e28c7ae65d95ece61ec85": "\\alpha_1, \\cdots, \\alpha_K", "d518f394abb9ef2f384eb5e827586b36": "\\mathbf{\\hat r} = \\mathbf y- X \\hat{\\boldsymbol{\\beta}}= \\mathbf y- H \\mathbf y = (I - H) \\mathbf y ", "d519639b33d685281662f1c0965e95b5": "Z_{\\text{in}} = \\infty", "d5198fe5fdbaf87e580e3ac86406da0f": "m \\in \\{0,1\\}^{k}", "d519a679bc4c155aca4ad898f9554315": " H' = -\\sum_{i=1}^R p_i \\ln p_i = -\\sum_{i=1}^R \\ln p_i^{p_i}", "d519ce29d4f5a3737cb11c5b4177897f": "2n(n-1)+4n=2n(n+1)", "d51a1c9fdeca504dabc4c6bcdb11b580": "\\bar{B}\\otimes B", "d51a2ee3cd002108f5bf910fc042e0a6": "Z_{ave}", "d51a37f019cd957fe688a76bdd7f60ef": "s \\ne \\tfrac{\\pi}{2k}", "d51a58811f13f60a8c4ff1e5af5b84b5": "\\vec L_{avg} = \\frac{(\\vec L \\cdot \\vec J)}{J^2} \\vec J.", "d51a6f1dacd95a3e2daeaa2010179862": " f_k( x ) = \\frac{ 1 }{ 3k } \\quad \\text{if} \\quad | x | < \\frac{ 3k }{ 2 }, ", "d51aa1c108ceccf64e4830ee9891fa76": "K_+", "d51ad558b6cfb75d069012519425df2a": "\\int\\frac{dx}{xR}=-\\frac{1}{\\sqrt{c}}\\ln \\left|\\frac{2\\sqrt{c}R+bx+2c}{x}\\right|, ~ c > 0", "d51af29b501d8c412a87ae1fa105b055": "g_{\\rm D}(\\omega)=\\frac{9\\omega^2}{\\omega_{\\rm D}^3}", "d51b1641a7e0981412d8b4822aab8593": "T_g L_g", "d51b2e0efd14ea7099435defb8d5cf42": "x(n)", "d51b3d476bb05e54eef457c590c9ae27": "\\mbox{and}\\quad\\lim_{x\\to a}h_k(x)=0.", "d51b6edf1354a4db069d9d114d5173b6": "\\beta:", "d51c20dd3065c51ee57bd06e6827b4e2": "\\max(A_1(x_1, \\dots, x_{r-1}), \\dots, A_{n_A}(x_1, \\dots, x_{r-1})) \\leq x_r \\leq \\min(B_1(x_1, \\dots, x_{r-1}), \\dots, B_{n_B}(x_1, \\dots, x_{r-1})) \\wedge \\phi", "d51c3caf97f95c2041e13587faeb24e5": "\\frac{12\\tfrac{3}{4}}{26} = 12\\tfrac{3}{4} \\cdot \\tfrac{1}{26} = \\tfrac{12 \\cdot 4 + 3}{4} \\cdot \\tfrac{1}{26} = \\tfrac{51}{4} \\cdot \\tfrac{1}{26} = \\tfrac{51}{104}", "d51c4a3d3b25ca8abb84d7b5202b7c63": "f'''B\\triangleq\\{f''U:U\\in B\\}\\,", "d51c50834e705a9c83f1e3395950bec2": "[T+U]_\\beta^\\gamma=[T]_\\beta^\\gamma+[U]_\\beta^\\gamma", "d51c6c01153365ce0f3a811a59056aa2": "\\gamma = \\frac{F}{l \\cdot \\cos \\theta}", "d51c6fcacbb0e472719b1953b567969f": "t^2", "d51c7aa539af95480c6ef7ea11294db1": "\\hat{f}_{+}", "d51cb227382dfcdabea59984145271c6": "A \\cap B = \\varnothing", "d51d0201dfb892794c277b115752852d": "F_+(x) - F_-(x) = f(x)", "d51dac49d40a6f99a4c69926953b74d9": "w / h \\le 3.3", "d51daf3183c78d42a062ff69cf28440e": "R_\\mathrm{spatial}(\\hat{n},360^\\circ) = +1", "d51dc46364fe5b201328822642173df1": "\\hat{p}_{k+1} \\leftarrow \\hat{r}_{k+1} + \\beta_k \\cdot \\hat{p}_k\\,", "d51e152e5057f4912e42335c6ad4a29d": "\\boldsymbol{R}", "d51f06e0e4617194ce444172968069f1": "\n\\hat{\\boldsymbol\\varphi} = (-\\sin \\varphi ,\\ \\cos \\varphi)\n", "d51f0e6edcc0c4a67cbd02dfbc387202": "U_{B}(\\delta_{B})(U_{A}(\\delta_{A})-U_{A}(\\delta_{B}))\\leq\nU_{A}(\\delta_{A})(U_{B}(\\delta_{B})-U_{B}(\\delta_{A}))", "d51f628f3f33e137c7f3fa3ae5bc40a9": " T_f + T_p + T_p' + T_p'' ", "d51f6f282727bdddb25f918601e3257f": "\\scriptstyle x\\in\\mathbb{R}^n", "d51f8feca20ff8a0beeaf64ff0139ce1": "b_1 = 1a_0 + 2a_1 + 3a_2 + 1a_3", "d51fbb484e38555f47b6179aabb71505": "\\displaystyle T", "d51ffc72f5f9d6b2650cce9460916d88": "\\displaystyle{a^b = a +Q(a)b^a.}", "d5200723b55a30cc03094b53a5aeab6c": "\\begin{pmatrix}x\\\\ y\\\\ z\\end{pmatrix} = \\frac{1}{g_1^2 + g_2^2 + g_3^2} \\begin{pmatrix}g_1\\\\ g_2\\\\ g_3\\end{pmatrix}", "d52030d35527b0ac2e241ef9aa5a8ea7": "H(u)(t) = \\frac{d}{dt}\\left(\\frac{1}{\\pi} (u*\\log|\\cdot|)(t)\\right)", "d520a14aff7bc18320d8d07ad3b68a2d": " i_p ", "d521009ec0f97b048ba6f8fb362d741a": " 3, 3, 2, 2, 1, 0 ", "d5211f5478a49ce954871ced42d4f91e": "C_s = (1 + 2 a) {C_u} ", "d521368cdae02b61f81b3e032ce3a86f": "\\scriptstyle\\langle\\cdot,\\cdot\\rangle: \\boldsymbol{E}^*\\times\\boldsymbol{E}\\rightarrow\\mathbb{R}", "d521abd580343bf716d137aaae311244": "\\left(\\frac{x}{a}\\right)^{2}+\\left(\\frac{y}{b}\\right)^{2}+\\left(\\frac{z}{c}\\right)^{2}=1;\\,\\!", "d521bd30ca0dc9c6c4f838e2d0651b78": " T_n= (n-1)!. \\,", "d521c3cd2d3e818c6da3f57908d50140": "\\begin{vmatrix}\n e^{-2x} & xe^{-2x} \\\\\n-2e^{-2x} & -e^{-2x}(2x-1)\\\\\n\\end{vmatrix} = -e^{-2x}e^{-2x}(2x-1)+2xe^{-2x}e^{-2x} ", "d521e183fe91496f13defa5dd8e2dee3": " y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0. \\, ", "d521e2db03153d55f2bf08c04bdcc7e4": "p_x=m\\dot{x}", "d52273f41dade940b909324fc6a0d5ac": "1.2{K_p}/P_u", "d522c32c0cab08cc6819e83c9ed66cc6": "\\Gamma \\vdash_D\\ e:\\sigma \\Leftarrow \\Gamma \\vdash_S\\ e:\\sigma", "d522cf6f607376842ee6b3fa182a6afc": "\\int_x^{x+t} A + Bc^{y} dy = At + B(c^{x+t}-c^{x})/ln[c].", "d5231b8fd9d4efaa40c92439c9851131": " \\{ |e_i\\rangle \\}_i", "d523896a4ec59bff14f5c11c689e4312": "\\begin{align} \\mathbf{a} \\cdot \\mathbf{b} &= a_1b_1 + a_2b_2 + a_3b_3 \\\\ \\mathbf{a} \\wedge \\mathbf{b} &= (a_2 b_3 - a_3 b_2)\\mathbf{e}_{23} + (a_3 b_1 - a_1 b_3)\\mathbf{e}_{31} + (a_1 b_2 - a_2 b_1)\\mathbf{e}_{12}. \\end{align}", "d523d00b6f6716322f01394d084e2c28": "\\mathbb{E} f(n) \\rightarrow f(n+1)", "d523e628d4e80833a589224f37518b03": "\\ RI = \\frac {gland}{wall}", "d524582455b24d55dd4fe982b6fba983": "\\frac{2\\sqrt{2}}{9801} \\sum^\\infty_{k=0} \\frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\\frac{1}{\\pi}\\!", "d524784a8451478b9bcfa0902155e746": "\\forall x \\phi(x)", "d524cd4aa986ef390e8eee2a864d17b4": "e^{-\\epsilon\\Delta q}\\,\\!", "d524d4f99fa02b3931d412123f5ba5c1": "p(z) = z,", "d524d6690bb511afa661f8491cb9ba2f": "(x^\\lambda,\\overline y^i)", "d524f30f39ec8ca80e72668617fffb08": "\\displaystyle E/V = K_1 \\sin^2\\theta + K_2 \\sin^4\\theta + K_3\\cos\\theta\\sin^3\\theta \\cos 3\\phi ", "d52522665c1edee9283c1c5af97b8dd7": "(mask_m,genState) \\leftarrow GenWords(4,genState)", "d52558b155faf73f2ed588accb0539bd": "\\scriptstyle f", "d5257f4aacab02ea273f3d4332326471": "O(n\\,\\log(\\log(n)))", "d52586972e94435bcd0f8dc853b64bb9": "\\mathbf{u_1G}", "d5258d00ba46963bac62ebd0e3e6a2d7": " L = p r , \\,\\!", "d525a9e197ea145a63f79f90e5a71105": "\\|x\\| \\le \\|y\\|", "d525adf24bd64fbffdc6b0af52b132a1": "q(\\psi | \\mu)", "d525b1f3377caa02cae528d7cf16f6c6": "\\setminus, \\smallsetminus, \\times \\!", "d5261ceeb7cd224a9e698d7dc28dc668": "\\frac{\\sin(\\pi x)}{\\pi x}", "d52632dbc420d610b298df1538cd1917": " \\Gamma_{d-1}= i \\Gamma_\\text{chir} ", "d52691249c900eb9e8c3b7d64fa66766": "4.\\overline{3}", "d5269a6dd186048f9b785a8899775669": " = 2\\sum_{n=0}^{\\infty} \\cfrac {n!}{(2n + 1)!!}\n= \\sum_{n=0}^{\\infty} \\cfrac {2^{n+1} n!^2} {(2n + 1)!}\n= \\sum_{n=0}^{\\infty} \\cfrac {2^{n+1}} {\\binom {2n} n (2n + 1)} \\!", "d5269a78230a1ac01038a689f5e7e765": "l_1 + l_2 < m ", "d526f1c8ef6c1e4e980e2b8471352d23": "Bb", "d526fcdf9b10529b54820f729dbb25f4": "u_0", "d527121fbf1329eb3ff34b8b11a015dc": " Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0", "d52768c1844328c23465a82a9ae3f2ac": "v_\\mathrm{radial} = \\frac{2 \\pi}{\\lambda} \\frac{n \\sin \\alpha}{M_\\mathrm{tot}} r", "d52778efaca579277267cf937e0d79b6": " \\theta\\!_1 ", "d52783ddf364b765861ff73aade06ea5": "\\psi(\\mathbf{r}) = A(\\mathbf{k}) e^{-j \\mathbf{k} \\cdot \\mathbf{r}}", "d5279df2b9a3c9d40b5c8489962bcee3": "d\\colon E \\to E \\otimes E", "d527b915f393c41c312db3f548f1ff09": " Q(x) \\, f'' + L(x)\\,f' + \\lambda f = 0 \\,", "d527dac6f1fdfdd3da5be90b11e56dac": "W(\\mathbf{q})=\\mathcal{S}(\\mathbf{q},t) + Et ", "d527dcf4d0837c8a31e1275c4ee50c77": "\\displaystyle{K(w,z)=C\\cdot \\exp\\,{1\\over 2}(p z^2 + 2qwz + r w^2)}", "d52819d431d38e787bf2c03debc12275": "\\{|{b_j}\\rangle\\}_{j=1}^m", "d52868a1f845dab22a9ef125409288e6": "\\phi(t)=\\pi-\\arg\\left(\\sqrt {2}-1-\\frac {2}{\\left(1+\\sqrt {2}\\right)e^{it} -1}\\right)", "d528a2302c5d54af8b82316ed64e5c27": "B_0(x)=1\\,", "d528ac84ce573fb827d8b2094eb46563": "N_{\\mathrm{A}}, L \\,", "d528baf28e2e224617db25e3de06f105": "\\rho = \\lim_{n\\to\\infty} \\left(\\zeta_n \\frac{a_n}{a_{n+1}} - \\zeta_{n+1}\\right).", "d5292be22c189b089f14ce205798367c": " [A^\\ast] = \\sum_i (-1)^i [A^i] = \\sum_i (-1)^i [H^i (A^\\ast)] \\in G_0(R). ", "d52938657a7d0f1d8a34bc21eec0c710": "Q(i) = 0", "d52969394e84ed16c7ef790bda45480e": "\\eta(x,y)\\, =\\, a(x,y)\\, \\text{e}^{i\\, \\theta(x,y)},", "d52994d686b8599e77dd3e6392a0509d": "\\varepsilon \\left[ M \\right]=\\left\\{ {{\\left\\| f-{{f}_{M}} \\right\\|}^{2}} \\right\\}=\\sum\\limits_{k=M+1}^{\\infty }{\\left\\{ {{\\left| \\left\\langle f,{{g}_{{{m}_{k}}}} \\right\\rangle \\right|}^{2}} \\right\\}}", "d529c1e5c42915172b1dbac77f651d0b": "P_n (k) = \\frac{{n \\choose k} {80-n \\choose 20-k}}{{80 \\choose 20}}", "d529c2c3ec695309c3fa53fc006421ca": "P_1 \\uparrow S(X,J)", "d529d8cbcc694ea1289f12df6cfa9111": "P_3(c) = (c^2 + c)^2 + c \\,", "d529d8e55a64de39ce0f5056ca684725": "\nz^{2} = a^{2} \\left( \\sigma^{2} - 1 \\right) \\left(1 - \\tau^{2} \\right)\n", "d529e630769b893bf52c367eb302b8ea": "Y_{5}^{0}(\\theta,\\varphi)={1\\over 16}\\sqrt{11\\over \\pi}\\cdot(63\\cos^{5}\\theta-70\\cos^{3}\\theta+15\\cos\\theta)", "d52a1c8d206f3abe74ffdb0b860729be": " \\left | \\alpha - \\frac{p}{q} \\right | < \\frac{f(q)}{q} ", "d52ad477870b2930f9dfebc19583fad6": "[\\alpha f,g]=\\alpha[f,g]\\quad \\forall \\alpha\\in\\mathbb{C},\\ \\forall f,g\\in V,", "d52aecbfb6ec4e93cf61a9beadb977b1": "\\Pi^S\\,\\mathbf{F}(\\mathbf{r}) = \\frac{1}{4\\pi}\\nabla\\times\\int \\frac{\\nabla^\\prime\\times\\mathbf{F}(\\mathbf{r}^\\prime)}{|\\mathbf{r}-\\mathbf{r}^\\prime|} d V^\\prime, \\quad \\Pi^I = 1-\\Pi^S", "d52b0c0f2106b7cb270621c8c5f60c93": "\\scriptstyle f_s \\,\\gg\\, f_0", "d52b2663fa9c96c4da89196739c070b5": " \\text{OMA} = 2 P_{\\text{av}} \\frac{r_{e}-1}{r_{e}+1}", "d52b2d8f24bfe7be41a03ffe0bff22b5": "\\sum_{n=0}^m 1_{\\{N=n\\}}E^n\n=\\sum_{n=0}^m\\sum_{\\scriptstyle J\\subset\\{1,\\ldots,m\\}\\atop\\scriptstyle|J|=n}\n1_{\\cap_{j\\in J}A_j}\\Delta^n,\n", "d52b70a519f6914e095dd9157ebd5f17": "\\Delta \\omega", "d52b767b45956fdf8543f6b0dbf28af5": "\\Delta: Spaces \\to Spaces^I", "d52b919fd94f92bd90807f5f6275e01b": "\\pi_k(a_i,a_j)=P_k[d_k(a_i,a_j)]", "d52b9d5d02fe23c5d0321c35e332dc22": "P_{fission}", "d52be9b7884a099919f36fe6bd7f2238": "|\\nabla_i f(x + h e_i) - \\nabla_i f(x)| \\leq L_i |h|, ", "d52bf9e6bad399320ce51a3d018d40f0": "A_{ji}", "d52c06397c30b9c5094dd4e83d7025ec": "\\Sigma=\\{a,b,c,d,e\\}", "d52c5d5778bbc10507e6a69dc6a6c9f8": "R(r) = J_m(\\lambda_{mn}r),\\,", "d52cd5bb86b9ff0b311ca0d083e353cb": "\\int\\frac{dx}{s^5}=\\frac{1}{a^4}\\left[\\frac{x}{s}-\\frac{1}{3}\\frac{x^3}{s^3}\\right]", "d52d089631eeb1874355db17934ceba7": "(t,x) \\in \\Omega", "d52d65429bbcdcad41487bfc4e39f834": "\\bar{D}", "d52d7144035b6d00f069899f7a9f729b": "[1]_\\sim", "d52d92ef40989aed8f272de2a340e6ac": "\\rho\\tilde{\\rho}", "d52dbf71b4fb7859f087bdf28971d483": "R_\\mathrm{p}", "d52df469433b2d5302574225625d099f": "\nR_{0 0} = \\Gamma^\\rho_{0 0 , \\rho} - \\Gamma^\\rho_{\\rho 0 , 0}\n+ \\Gamma^\\rho_{\\rho \\lambda} \\Gamma^\\lambda_{0 0}\n- \\Gamma^\\rho_{0 \\lambda} \\Gamma^\\lambda_{\\rho 0}\n.", "d52e2e0a2805ff9645f90ce7f35f2f6d": " \\Pr(\\bar x >z_{\\alpha}\\sigma/\\sqrt{n}|H_a \\text{ true})\\geq 1-\\beta ", "d52e8eeaa2e0de7136b651c90fba85c0": "i u_t + u_{xx} + 2k |u|^2 u =0.\\,", "d52ea3bfb8434f11ad827a38a2a2f4e5": "\\left[\\frac{Fe}{H}\\right]\\ =\\ -1.07 \\pm 0.01", "d52ec2ddc5e93c7080f27c0b693c208a": "\\eta = (E - E_{eq})", "d52eeddb0885728743b974b81c9616c2": "\\Delta E\\Delta t\\ge\\hbar,", "d52eef76470e15704f9e62f29eec7cab": "\\{\\neg, \\land, \\lor\\}", "d52f2ff84e5cdbd09898ae2bf2c813db": "[P] + [\\overline{P}] - 2[O]", "d52fdef32d8975ecaa6ffa54f39355c2": "\\scriptstyle x^n \\,-\\, y^n", "d52fff85c7412f3b0d7da6bccf9d9319": "D_{\\mathrm{KL}}", "d52fffdeeed9c373a37d9a22850f04ee": "P \\,\\ ", "d5301ca9d188bf3d4cb0fda8b3d4887f": "n=\\sin( \\rm{latitude} \\times \\pi/180)", "d5303684c33abb3f5351ed658871c018": "M \\ ", "d5307bc4f02da3d6e20175b85206ad07": "\\int_{-a}^a =\\pi e^{-t}-\\int_{\\mbox{arc}}.", "d5310138aaf4ed9bbe24bc8be85ae18d": "{z}={z} \\,", "d531425a683bb96d4c71f39a5c4ea3c3": "\\alpha _{0}", "d5316181def920a4a4f2813c91559ea6": "((\\phi \\to \\chi ) \\to \\phi ) \\to \\phi ", "d531ea4b73481114f4cd79e7730c9c03": "a_0 < b_0 < a_1 < a_2 < ... < a_n < b_1", "d531f5e66842c07c126017ac5c1dd9f8": "V(\\lambda)", "d5323582f456384c0a44650912bb271d": " dx^\\mu = d\\varphi^\\mu", "d5323d13a91f763f86fc48a0a9bb7a44": " \\|x - \\sum_{y\\in B} \\langle x,y\\rangle y \\| < \\varepsilon", "d5327fa78cf2594312903f361488018c": "e^{-\\pi \\frac{p^2}{m^2}}", "d532880e5df1ee710fb2890c05557397": "\\mathbb{C}^+ = \\{z \\in \\mathbb{C} | {\\rm Im(z)} > 0\\}", "d532d7989d7caecaf8c2c8a859de5243": "A_1, A_2, \\ldots, A_n \\vdash B", "d532de73002192cc69919bba13e0ba95": "{ \\epsilon \\, }^R = \\epsilon \\,", "d532ee62566092c547a3ca9d46685eb5": "F^i_{~\\alpha}", "d533bd95f0f8cfec1d9ea4339fae91de": "\\displaystyle f(x)", "d5340b8168cfc259c08e9377c6111e03": "\\left(\\prod_{n=0}^{M-1} (M-n)\\right)^{N-1} = (M!)^{N-1}", "d53462b4edf01f707d3109d101ef2643": "\\delta W\\,", "d534a97bab2a345dd04c0bb130654d97": "\\Pi^{1,A}_n", "d534b85d1032cfdb59d07d66f560b476": "T(q) = g^* q g^{\\star}.", "d535068dda4d60b1d6403c738a2c3db5": " f(x-0) ", "d53507a68d0c75a65fe5f1fb3ff19cff": "\nD_T^2 = 0. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (6)\n", "d5361da92b4badd4972aaec34196e3d2": "\\Gamma^k_{\\ i j}", "d53625799b563815e675b6b74a2f3e1b": " R_{AB}\\,", "d536547942f720ca72f6cc98bc80c5a1": "L_\\rho(\\Gamma)=w", "d53685028aed302c2c367890992dc549": "1/d_\\mathrm{i}", "d536850520fd3bddca4700d1b1ff4048": " \\mathcal{P}_c = {N \\hbar \\omega \\over cV} = {N \\hbar k \\over V} ", "d536aba392c179a2e94d97ee654d6c75": "T' = T \\sqrt{\\frac{c-v}{c+v}}.", "d537091c0e2c2a9ba5a8ae0ff486f0e0": "\\cos(3\\pi/7)", "d5371f9a61cf684a1771f511826b4909": "\\sqrt{\\frac{5}{7}}\\!\\,", "d537497a05e5568be566290f9b8d1447": " \\mbox{cond} (A) = \\| A \\|_2 \\| A^{-1}\\|_2 ", "d537bd2dd4b3aa9c012fa403468d3096": "\\zeta(s,N) =\n\\sum_{k=0}^\\infty \\left[ N+\\frac {s-1}{k+1}\\right]\n{s+k-1 \\choose s-1} (-1)^k \\zeta (s+k,N) ", "d538331492b0b00dba83e07093f3ecea": "\\displaystyle{a_{\\pm 1}={1\\over 2\\pi} \\int_0^{2\\pi} f(e^{i\\theta})e^{\\mp i\\theta} \\, d\\theta,\\,\\,\\, b={1\\over 2\\pi} \\int_0^{2\\pi} f(e^{i\\theta})^2 \\, d\\theta.}", "d5383590699034c217db7eb5a6938397": "k\\geq \\max\\{n-m+1,1\\}", "d5384cfada4314ca972a1978b67477ba": "\\mathcal{I}=\\mathcal{R}\\mathcal{R}^t", "d53886b0f69d1c49e5380d6a514bfe38": "\\int \\left| f(x)\\right|\\,dx = \\sgn(f(x))g(x)+C,", "d538a99096c7fd7e3d750ef486348199": "\\scriptstyle T_pM", "d538b573a71e8d0915de9e1669b76588": "x_i \\in S_2", "d538bf2898dacbfcbb402e3806068e1a": " \\lambda_n(t) = {n \\choose 2} \\frac{1}{N(t)} ", "d538e5c77cbac2be4fb8645154a4b610": "\\gamma_{SL}", "d539011a86d2799ecee7a5a5bbc9a606": "A_2 = \\neg A", "d5393f0398524481134a6e6bcf9d4ee0": "P \\ll e \\approx \\rho c^2", "d539600df3a05451869b736f4e21e51e": "h^d", "d5398339ad2253aa0b6eb95d9710c092": "|.|_{r,p,\\Omega}", "d539b811205ac44b95f1dddb65ede7d7": "(S,T,W)", "d539f8a75128181415a71fae2500a7b1": "A.C_2+B.C_1=\\frac{1}{P}.[B.R_0-A.Q_0]", "d539fd6bb3a56c6d30250f5544107404": "\\mu (A) = \\sup \\{ \\mu (K) \\mid \\text{compact } K \\subseteq A \\}.", "d53a07e230253f4686bb6818314b3fec": "d_b", "d53a339501c37a8f566dca944eab3341": "E = \\sum_{i=1}^N E_i(X_i),", "d53a7a796476c519809e4f1264a6fd74": "\\omega_{max}", "d53a9ce77f40534cd0a1ee00cb6b8f3f": "w-x", "d53aa1ab6105bdc48bf6f0ccdf8236dc": "PPOT", "d53ac3b458d4348e1a32495c5fa65bc7": " \\mathrm{CVA} = (1-R) \\int_0^T \\mathrm{EE}^*(t)~d\\mathrm{PD}(0,t) ", "d53ae4495ccb4fa1be2232ff99f02882": "\\frac{100{\\left[ \\sqrt{\\frac{Score_a}{Score_b}} + 2{\\left( \\frac{Floor_a}{Floor_b} \\right)} + \\frac{Combo_a}{Combo_b}\\right]}}{4}", "d53aeb78abc83a52ab8982f5c82a3d5b": "SS", "d53b0174817d0a6c1d6dd1310b88f343": "\\mathbf{G_0} = \\frac {\\mathbf{W}} {\\mathbf{V_1}^2} ", "d53b1735e422caff3485cf3ee455bb72": "T_{rx}", "d53b4c326f9d53982acc68b9e8339840": "\\left\\{\\begin{array}{ll}1 & m = 1 \\vee n = 1\\\\ 2 & \\text{otherwise}\\end{array}\\right.", "d53b91e4bc12051c49f33a791c4dde7f": "M \\cap M' = \\emptyset", "d53bc21237d24c48c8d314a843ea6130": "E = D \\times V ", "d53be80347637dab355b01bee123a560": "(k,a) \\mapsto \\eta(k) a.", "d53c3c269412c6f6c94f3f97793b0aab": " \\mathbf{J}=- D\\nabla \\phi ", "d53cd8bf25f77b55fbf642490fb23159": "\n\\frac{\\sigma _z }{z}\\,\\,\\, \\approx \\,\\,\\,\\sqrt {\\,\\,\\frac{\\alpha ^2 }{n}\\left( \\frac{\\sigma _1 }{\\mu _1 } \\right)^2 + \\,\\,\\,\\,\\frac{\\beta ^2 }{n}\\left( \\frac{\\sigma _2 }{\\mu _2 } \\right)^2 \\,\\, + \\,\\,\\,\\frac{2\\,\\alpha \\,\\beta }{n}\\left( \\frac{\\sigma _{1,2} }{\\mu _1 \\,\\mu _2 } \\right)}", "d53cfe456c77bf31adab12edb76cc000": "\\frac{256}{243} \\sqrt[3]{4}", "d53d182d62a151382a9fa27e7083ebf0": "\\rho^n=1", "d53d2370e8882c715516bd424de9007c": "\\|Lv\\| = \\left \\Vert {\\|v\\| \\over \\delta} L \\left( \\delta {v \\over \\|v\\|} \\right) \\right \\Vert = {\\|v\\| \\over \\delta} \\left \\Vert L \\left( \\delta {v \\over \\|v\\|} \\right) \\right \\Vert \\le {\\|v\\| \\over \\delta} \\cdot 1 = {1 \\over \\delta}\\|v\\|. ", "d53d970dee9a9ff1ae11601101d74e97": "\\scriptstyle \\boldsymbol L", "d53d9fa7a395f9468da84ee4dbe27903": "\\frac{_1}{\\aleph_1}", "d53da985ca7c63a8f80e915b2e8db6c9": "M_S", "d53dd44314d17d30698ddcf5dfdb2880": "g_n \\leqslant a_n", "d53e6e888a4a7686931e1bcb4b494cab": "z/L = 0", "d53e8657f8621300c26e69bb92c01770": "\\Delta E=\\sqrt{(0.23 \\Delta V_Y)^2+(\\Delta W_X)^2 + (0.4 \\Delta W_Z)^2}", "d53e9df62ae00a278315fbb499a0dc97": "R_8(\\xi,x)", "d53e9e3fe7f9ec9a1c050ec9e82af815": "\\,\\Lambda", "d53eb99f26080eb918a3a364c6e72d60": "\\varepsilon = \\| W \\cdot (F_{I} - \\mathcal{F} \\{ f \\}) \\|^{2}", "d53edf7a743ff703baa7e5312bfe1576": "s^\\prime_i", "d53ee7d0fcd785192a026a3252fb783d": "\\overline{Y}_{\\bullet\\bullet} = \\frac{1}{mn}\\sum_{i=1}^m\\sum_{j=1}^n Y_{ij}", "d53f8b6a9c3f0e01d78bc97a2f489d59": "\\textstyle{1+2+\\cdots+n=\\binom{n+1}{2}}.", "d53fb617a3045efbdf6aeed05e18429f": "\\scriptstyle F(\\omega)", "d5403b78c4752caad8d0f00c4cd54e59": "[L_x,L_y]=i\\hbar L_z, \\;\\; [L_y,L_z]=i\\hbar L_x, \\;\\; [L_z,L_x]=i\\hbar L_y", "d5406be841c7bc4c3e1b2ef98422de1f": "\\mathfrak{c} = \\aleph_1 = \\beth_1", "d5408d51074106f3aff4fbd5ab04a6ca": "\\frac{v^2}{c^2}\\ll1.\\,", "d540ac8ce72b7ee64ef86242cb730198": " \\mathbf{e}_i\\otimes\\mathbf{e}_j,\\quad i,j=1,2,3,", "d540afa85ec2abc317569163bf5f9c46": " d=\\frac{\\lambda}{2n \\sin\\alpha} \\approx \\frac{\\lambda}{2\\,\\textrm{NA}}", "d54185b71f614c30a396ac4bc44d3269": "sc", "d5424767da8351484e39414d954b4a23": "(\\phi\\land\\psi) \\leftrightarrow ((\\phi\\to\\psi)\\leftrightarrow\\phi)", "d54251db40b576d6a6252bd3c5d5f4a4": "q_g", "d5426a496c8c8bdcd450592f7e00a111": "\\int_0^1 \\cdots \\, dx", "d54270db0cd1002ce33f4ba35e893935": "j\\ge 1", "d542d1fdd12a46d1a2b0b76ab000ff4c": "\\frac{ \\mathbf{A \\cdot B}}{\\|\\mathbf A \\| \\|\\mathbf B \\|} \\le 1 \\ ", "d542d3e76b15e81bf873ae227692de98": "h(k,i) = ( h(k) + c_1 i + c_2 i^2 ) \\pmod{m}", "d542eea572ff848b9f96bee592c9d66b": " \\hat \\epsilon^2 ", "d54390f4410d3697f4fd615b70792293": " k = \\frac{\\omega_o R} {2 Q_o} ", "d543db3cd9ec7f327a7f912b3337fedb": "\ne^{-\\zeta/2} P_{0}(\\zeta) = B e^{-\\zeta} = B e^{-m_{b}gz/k_{B}T}\n", "d543e1a1fb301f6ff035d3bf3ec0734f": " \\frac{d[C]}{dt}=k_2[A][R]", "d543f380531d91fe79fbabee445e7c94": " \\ W = [w_1, \\ldots, w_n] ", "d544099abfe05bc31a7945909986e3a2": "\\tau_1,\\tau_2", "d5452b8ec048c70fd796182e2533cf1d": "= (1 - \\rho(x,u,u_{1}) )dx + u_{1}du \\,", "d545345ec75760cfd2693cad45e78705": "\\Theta\\left(n^{2/3}\\right)", "d5457762b9d19ca00c164d9f0c254158": "C_{abcd}\\, k^bk^d=\\alpha k_ak_c", "d5459e4ce62bcdb5506deda6f01a060e": "r_1 e^{-j\\phi_1}", "d545a0394c60ea668f37629f23b5b2b5": "v + R\\mathbb{Z}", "d545ae5c9535cf076ead2a49a5aa96c1": "\\textstyle \\phi(\\mathbf{q}) = \\int_{V} \\phi(\\mathbf{r}) \\exp (-i \\mathbf{q} \\mathbf{r}) \\, \\mathrm{d} \\mathbf{r}", "d54622d4bde74bb72c021e631ceb4573": "\\scriptstyle (\\Omega, \\mathcal{F}, Pr )", "d5463b7b67de98124605e71083ea7e40": "x_1=1.4<\\sqrt{2}", "d5466f902d795eccb091e06c6afce764": "\\epsilon_{it}", "d5467589fa65a61de2fd8dc12f9cd232": "\\frac{1}{|\\mathbf{x}-\\mathbf{y}|^{n-2}} = \\sum_{k=0}^\\infty \\frac{|\\mathbf{x}|^k}{|\\mathbf{y}|^{k+n-2}}C_{n,k}^{(\\alpha)}(\\mathbf{x}\\cdot \\mathbf{y}).", "d546771a330c8ab73fa3af48c694bd42": "\\sum_{i=1}^m x_i", "d5467fe36197c3b0c1b550ba823cdc64": "L \\geq 1", "d546896ef1f2ae733b1eaeaec5c76ea4": " (x,y,z) = xe_1 + ye_2 + ze_3, \\, ", "d54692c7711e1b0864fee0742712fbb0": "\\dot{V}", "d546eff6d252cdcc9d45110eb5a22a96": " \\sum_i \\alpha_i = 1.\\,", "d5471ba468b8e8c1b441bcb4c423365a": "\\mathrm{d}^3\\mathbf{x}=\\mathrm{d}x^1\\wedge\\mathrm{d}x^2\\wedge\\mathrm{d}x^3 ", "d5473645cb79c52ebfdb0a956cce196b": "\\mu_{2} = (n-1)[(n^2+n)/12]^2", "d54756e5d16b60de4852baeb82a661fc": "\\scriptstyle +1", "d5479eae1f42c7bce59de2431206a266": "L[u]=h\\,", "d547d01cb7b6ed4b379f930f848078dd": "V_{L1-N}=\\sin \\left(\\theta\\right) * V_P\\,\\!", "d547f3dcecafacefe51ef636e92cd536": "(\\hat{E} + mc )^{2j} \\psi_{3,4}^{[2j]} = (\\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}})^{[2j]} \\psi_{1,2}^{[2j]} ", "d54861aeb37b758a5b0d7e666e2918e3": "\\; \\Phi(a) = V \\pi (a) V^*,", "d54882fc7cb9a4c0ad3cece36f53dff2": "\\exists k>0 \\; \\exists n_0 \\; \\forall n>n_0 \\; |f(n)| \\leq |g(n)\\cdot k| ", "d548f74dbf0a64a852c6104f7e378603": "\\psi(t) \\approx e^{-\\omega^{\\prime\\prime}t}\\cos\\omega^{\\prime}t", "d54971b37fab4df71466c7648748cc2d": "\\lim_{y\\to 0} F(x+iy)", "d549be2620f891d915501cfaa61661c9": "x \\vee y = (x \\rightarrow y) \\rightarrow y.", "d549e3052df1d79fc657ac361651e8b8": "\\textstyle \\lfloor\\frac{h}{\\lambda}\\rfloor", "d549f34b8256b685ae8db5eb6fbb40a8": " \\frac{\\partial^2 r} {\\partial s \\partial s'} = \\frac{-\\cos\\epsilon + \\cos\\phi \\cos\\phi'} {r} ", "d54a58763077d98d5f73ba44d9cad228": "\\scriptstyle A_1=a_1", "d54a971b3aaa106e3a5caf2472d072ae": "\\text{Base amperes }=\\frac{\\text{base kva * 1000}}{\\text{base volts}}", "d54ad6e4dba981f49c5f641601af42cb": "s\\left(x\\right)=f\\left(x\\right)\\delta \\left(x-b\\right)", "d54b38cd048f13c0dbb613f92d5cf63a": "c_P = \\frac{7 R}{2}", "d54b44bbecdb78d1fb81e4e801ccd3cb": "{\\vec p_{i}(t)}", "d54b4f110b021b56d92ac47764d4dc97": "E_L\\rightarrow e^{i\\beta}E_L\\text{ and }(e_R)^c\\rightarrow e^{i\\beta}(e_R)^c", "d54b62f64b2fcf27be776511d4a22a9c": "\\operatorname E(X\\cdot\\operatorname E(Y\\mid N)) = \\operatorname E\\left(\\operatorname E(X\\mid N)\\cdot \\operatorname E(Y\\mid N)\\right) = \\operatorname E(\\operatorname E(X\\mid N)\\cdot Y)", "d54b73eb82cbe2532a4944f4c2c1dfd8": " (a+N\\Z) + (b+N\\Z) = a+b + N\\Z", "d54be98ec037dc07b3748ccdc3a44239": "\\displaystyle{T(f_{\\overline{z}})=f_z,}", "d54bf14722355f21b7fc59e84b36d44a": "1/2^{|l|}", "d54c49450c2d9f6d4492566e1fe5f901": " f_\\pm", "d54c505e59272b13632b513839c704e0": "q = k - \\pi/a ", "d54c5b2cd18a0171040b651b2ac37cac": " J_k = h n_k. \\, ", "d54ca1aa1c6b351fd895fb37afd4e131": " \\mu_0 > 0 ", "d54ca40743a03f94e76cbb1e1c3e4c9f": "\\pi_{[0,1]} \\circ f_t", "d54cdf0ce818e5221c30fccf16f6c144": " g^{bn} (R_{bn;l} - R_{bl;n} - R_b {}^m {}_{nl;m}) = 0,\\,\\!", "d54ce289b65b124944c510a68908f8da": "\\Delta E_\\mathrm{Lamb}=\\alpha^5 m_e c^2 \\frac{1}{4n^3}\\left[k(n,\\ell)\\pm \\frac{1}{\\pi(j+\\frac{1}{2})(\\ell+\\frac{1}{2})}\\right]\\ \\mathrm{for}\\ \\ell\\ne 0\\ \\mathrm{and}\\ j=\\ell\\pm\\frac{1}{2},", "d54dedf35c2417365a7183c7ef20f1dc": "\nF_{2}(r) = F_1(r) + \\frac{L_1^2}{mr^3} \\left( 1 - k^2 \\right)\n", "d54e329372e67900920926545c62fb8d": "dx^2 + dy^2 + dz^2", "d54e84465d7119667be7dfb3e9dfbc6f": "\\bar{x}_i(x_1,\\ldots,x_k)", "d54ecda43c5f2d439ece67bc635e886e": "V^* \\otimes V ", "d54f6ce67b75f58dac81e59a9f85fc22": "\\frac 1 2", "d54f7485c52c72e4173601838af23b38": "S_{\\phi} ", "d54faa335a44937aaa8ac369c796177b": "q_1,...,q_m", "d54fae2f0e793d88e77a01711f366cfa": "\\mathbb{E}[x(t)] = m_x(t) = m_x(t + \\tau) \\,\\, \\text{ for all } \\, \\tau \\in \\mathbb{R}", "d54fe52fbc5f09c0b4623727df084dcf": "(z_1 - T)^{-1} - (z_2 - T)^{-1} = (z_1 - T)^{-1} (z_2 - z_1) (z_2 - T)^{-1}.\\,", "d54ff0490ef406c3ea1efa46c4b27f45": "P = S cos(\\phi)", "d5502c3f549cd8ef2aff78132b0bfa99": "P_1(V)", "d5504836605c1a15ae4eb77668eee7df": "\\hat{a},\\hat{a}^\\dagger", "d55096053886efb19970b390bb143a96": "\\sum_{i=1}^{n} \\frac{x_i}{x_{i+1}+x_{i+2}} < \\frac{n}{2}", "d5509b7cbc547969742b268dd8cc393e": " S(t) \\le 1 + \\Delta, \\, ", "d550adc9941521b20771254141588585": "\\mathcal P(E),", "d550cdedd3e8bcae77de78f4fdfc0a09": "r = b + a \\cos \\theta \\ .", "d550dcb359a9bad4f4ab7034944075c1": " T_0 \\supset T_1 \\supset T_2 \\supset \\ldots \\supset T_k \\supset \\ldots ", "d550e2664dd10f09ffda0ad4e48ca3c5": "\\vec{v}=\\dot{\\vec x}", "d550eb1417cef447acb212a9c86cfcc1": "ds^2 = -dx_0^2 + \\sum_{i=1}^n dx_i^2.", "d551038b3f2623a8037a169f64a41ab8": "f (x) \\geq x", "d55138ce2a61d15946057f2d74bdc901": " (*) \\quad f(x) = f(a) + \\frac{f'(a)}{1!}(x - a) + \\cdots + \\frac{f^{(k)}(a)}{k!}(x - a)^k + \\int_a^x \\frac{f^{(k+1)} (t)}{k!} (x - t)^k \\, dt. ", "d5514289dddb72e39b602aac6bd343d7": "\ny_c = \\begin{cases}\\frac{p}{m^2} \\left( 2m \\frac{x}{c} \\right) - \\left( \\frac{x}{c} \\right)^2, & 0a) ", "d569f4e15f3e62be8ffcee25ccff10d6": "T = S \\cdot X = { 8 \\over {5^{5/4}} } \\cdot {5^{7/4} \\over 16} = {5^{2/4} \\over 2} = {\\sqrt{5} \\over 2}.", "d56a757717feaf6310318f2d61eae579": "R = \\frac{MW}{mn/2}-1", "d56abbb93ada8202fd0916f218fc5966": "\\scriptstyle H_\\mathrm{norm} \\approx 1", "d56adc5ef97b0d6b9faf586dacb0edfc": "r(k)\\sim k^{-d} L(k) ", "d56af9c33b71824cd62ba981bc00fa52": "X_{tsc} = {{1 \\over{2 \\pi f C_{tsc}}}-2 \\pi f L_{tsc} }", "d56b375ea7d03e133ea020672f946268": " \\mathbb C^m", "d56b8173bafb4f6b7d0cfcffb48950dc": "\\sum_{k=1}^K \\rho_k(x) = 1, \\quad \\forall x \\in \\Omega_x, \\quad ", "d56b9afdb3644dbb305b513d62414f5b": "\\Phi^X_{t}", "d56bb008dc01c8039dad42b035802744": "[X+f,Y+g]=[X,Y]+Xg-Yf", "d56bbe8394467c92e275abdb56f03bce": "r(x)\\,", "d56c15d5b13117a7d081ccabbae9a1f8": "(a + b\\mathbf{i} + c\\mathbf{j} + d\\mathbf{k}) (e + f\\mathbf{i} + g\\mathbf{j} + h\\mathbf{k}) =", "d56c69de3f523a79032d607324ae7622": " System Noise Figure = F_1\n+ \\frac{ F_2 - 1 }{ 1 } \n+ \\frac{ F_3 - 1 }{ 1 \\times G_2 } \n+ \\frac{ F_4 - 1 }{ 1 \\times G_2 \\times G_3} \n", "d56cf024593fbb5d5599fbb1f9645374": "\\Pr(X = x) = \\prod_i \\Pr(X_i = x_i|X_j = x_j\\ \\mathrm{for\\ all\\ } j\\ \\mathrm{for\\ which}\\ \\lbrace X_i,X_j \\rbrace \\in E).", "d56cf7b7f7202bda2046adf7335c6338": "f=\\{u,v\\}", "d56d20aab41e9e8b54a47c3bf7ac672f": "c^T X", "d56d3f38efa451dc8385f483610abd85": "\\vec P\\!", "d56d68dbd01898273f68eb585c39adc1": "\\|u_\\lambda\\|_{L^q(R^n)}^q=\\int_{R^n}|u(\\lambda x)|^qdx=\\frac{1}{\\lambda^n}\\int_{R^n}|u(y)|^qdy=\\lambda^{-n}\\|u\\|_{L^q(R^n)}^q", "d56d6f8f579604a624c43d2e3bcebed9": " \\mathcal{A} ", "d56d70fea618ebcf11b7431e6a2b3567": "S=\\frac{C_p} {C_f}", "d56da9811e7ca7ba1bdfcacdb5065b32": "x=c_0", "d56deb8f916265c91536fc6ade55326d": "Q = \\bold{v} \\cdot \\bold{A} ", "d56dffb216d1ca2cc26e52a60ad95f6e": "\\gamma \\leq \\tfrac{1}{n}\\,", "d56e1c8ccb917ddd686073c1ae6b60c8": "Z_2^{12}", "d56e525d7bcab8cab12ac1bc87941682": "\\tau_{12}^y, \\tau_{23}^y, \\tau_{31}^y", "d56eaad3181662388bf82c6ff0d1705a": "Y_W", "d56f160d20dca78e35af5c9f29400a73": "\\{ F_1, \\cdots, F_n \\}", "d56f29ff29a8acd3bb13256a053bf1b9": "n\\geq 2k", "d56f51751c6facc4085329d5da4c5934": "f*K(t) := \\frac{1}{2\\pi} \\int_0^{2\\pi} f(s) K(t-s) ds", "d56fb08010d1eea62327ca037c1e8497": "\\xi=f(\\xi_1,\\xi_2,\\ldots,\\xi_m)", "d56fd22938c092a922a724c573bc2290": "u \\approx \\,", "d5701fbd4eedaf8d378a9c8ae971d840": " -n(n-1)~r^{n-2}~\\sin(n\\theta) \\,", "d570ca817bea1a835157edd3a9f69b1c": "P(\\text{positive}|\\text{well})=1%,\\text{ and } P(\\text{negative}|\\text{well})=99%.", "d570d04b14f571631c91052f1a3c1afc": "\\sigma = \\frac{k\\alpha_{abs}}{\\omega\\mu}", "d5713697a5aa242e3688dbe57c4c3f0b": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 56\\cdot 4.49)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 54\\cdot R_{\\bigodot}\n\\end{align}", "d571533dd95ae7c50a69be575b8d4a89": "s\\in \\partial D", "d571717013c5ee2f73ebfd6a74555958": "\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}{X(e^{i \\vartheta}) \\cdot Y(e^{i (\\omega-\\vartheta)}) d\\vartheta} \\!", "d5719c019d850174052d7b83ea585d71": "\n\\begin{align}\n& {}\\quad \\pi^{-s/2}\\Gamma(s/2)\\zeta(s) \\\\[6pt]\n& = \\frac{1}{s-1}-\\frac{1}{s} +\\frac{1}{2} \\int_0^1 \\left(\\theta(it)-t^{-1/2}\\right)t^{s/2-1}\\,dt + \\frac{1}{2}\\int_1^\\infty (\\theta(it)-1)t^{s/2-1}\\,dt.\n\\end{align}\n", "d571a267dd6341f35f238c1c79b329ab": "\\partial_t f_t(z) = v(f_t(z)),\\,\\,\\, f_t(z)=0", "d571ba7774b7f469cc75d9c699e6b109": "lift(A \\Rightarrow 0) = \\frac{P(0|A)}{P(0)} = \\frac{P(A \\and 0)}{P(A)P(0)}", "d572956b265c891bdb3bacbcca08e1fd": "p ", "d572da40133c08b2860573309219ca91": "X_k = \\sum_{n=0}^{N-1} x_n e^{-\\frac{2\\pi i}{N} nk }\n\\qquad\nk = 0,\\dots,N-1. ", "d572f3489fe0a97b189c1b71b8fb00f8": "\\{ H_n(\\theta) \\}", "d57357d45f140c36b5e5cb02f48f3657": "\\sigma_\\mathrm{n} = \\sigma_{ij}n_in_j=\\sigma_1n_1^2 + \\sigma_2n_2^2 +\\sigma_3n_3^2\\,\\!", "d57362b2a738d38c97094a7dc0dc5c2e": "\\Delta H_{rxn}", "d5736887870462d9d366e8bf6becda1b": "\\mathfrak{su}_6\\oplus\\mathfrak{sp}_1", "d573c03099582b173d2d6d1e7f1c97df": "= (1 \\text{eV})/c^2", "d573d39865ab4afe59e79d3de2c8952d": "\\Delta IC", "d573fb122845b5f179feef0f4b611668": " 4\\pi \\left(\\chi_\\text{e}, \\chi_\\text{m}\\right) ", "d57422375178bc30ef0eee185f6f6391": " \\Delta t_+ = \\frac{2 \\pi r_0}{1 + \\omega \\, r_0}, \\; \\; \\Delta t_- = \\frac{2 \\pi r_0}{1 - \\omega \\, r_0} ", "d57426297d5fa7ce52a46c3015b3da84": "f(z) = a_0 z^n + a_1 z^{n-1} + \\cdots + a_n \\, (a_0 > 0) ", "d5742827d055e40cd568fd71271f4681": "B\\,", "d5745f497d6e55d2c2f05c15976e623c": "O(A_1:A_2) \\triangleq \\frac{P(A_1)}{P(A_2)}", "d574c1e532f0226b10c0189b8750bdf2": "\\{e\\} \\times G'", "d575a956a877fd68294442ee72c7889a": "N_z", "d575c5797072f5d6ec6e57809c4a8666": "\\lambda(t+1)", "d57601c8c6fb22bdc1ae834f9f3bda0f": "\\varphi(\\mathbb{E}\\{X|\\mathfrak{G}\\})\\leq \\varphi(X)-(D\\varphi)(\\mathbb{E}\\{X|\\mathfrak{G}\\})\\cdot (X-\\mathbb{E}\\{X|\\mathfrak{G}\\}).", "d5761890243a0c0901929da5479dde9e": "\\begin{align}\n & n^\\alpha(\\hat\\theta_n^H - \\theta) \\ \\xrightarrow{d}\\ 0, \\qquad\\text{when } \\theta = 0, \\\\\n &\\sqrt{n}(\\hat\\theta_n^H - \\theta)\\ \\xrightarrow{d}\\ L_\\theta, \\quad \\text{when } \\theta\\neq 0,\n \\end{align}", "d5763244a2440fce7460a3f1f45b9322": " \\{T_a, T_b\\} = \\frac{1}{3}\\delta_{ab} + \\sum_{c=1}^8{d_{abc} T_c} \\,", "d5766c35dae3d4926c88cb905df01fe2": "\\hat{\\mathbf{w}}_i = \\mathbf{w}_i + \\mathbf{e}_i", "d57671ec4d8b363979223e0e43441a57": "F_{ab} \\, = \\, \\nabla_a A_b \\, - \\, \\nabla_b A_a .", "d5767520e67de515a5d799372b39e34b": "\\binom{d}{i}", "d5768c23398269877a93b1760af9b7b4": "\ny - y_0= -c\\ \\frac{R_{12} (X-X_0)+ R_{22}(Y-Y_0) + R_{32} (Z-Z_0)}\n{R_{13}(X-X_0) + R_{23} (Y-Y_0) + R_{33} (Z-Z_0)}\n", "d57692c8a407a71a5d5392e0bccb3547": "g'(x) = k \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} \\quad \\mbox{(*)}", "d57705f1e3d66f26a38122108d7fa6d0": "=\\mathbf{P}(n-1)\\mathbf{x}(n)\\left\\{\\lambda+\\mathbf{x}^{T}(n)\\mathbf{P}(n-1)\\mathbf{x}(n)\\right\\}^{-1}", "d577218dcee3b3d02b1eb2aa54d737f6": "(x-5)^2 - 7 = 0,\\,\\!", "d577a904a64fd6ffaae7ba757a1656a6": "P(x) \\land (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\land Q(y))", "d5781de8477323225f184dcb913a0922": "\\phi(f) \\, ", "d5789133f71125c89046d8086f112b24": "\\rho > 0", "d578a4c65b89094b0f0aab704b9638f9": "x=\\cos^3 (2\\pi t)+ \\cos(2\\pi nt)-1,", "d578f810957df8c02010c4db9ff7f860": "\\mathcal{Z}_2(_R R)", "d5791d4e1ece822dfb8b77b3732990c9": "\\tilde D_{n+1} \\to \\tilde B_n", "d579288f144b81adc480a37b87439cef": "\\xi \\rightarrow \\acute{\\xi} = q(\\xi,\\tau),", "d5792dd271d053a486a6c17447c0989f": "f(x) = \\frac{1}{2\\pi i} \\oint_C {\\frac{f(z)}{z-x}}\\, \\mathrm{d}z. ", "d579684be7ede7be26f98bbe70357412": "D_4 \\bar R", "d5799d9feead09ccc4a5980d7bc9b03c": "\\begin{vmatrix} x_1 & x_2 & x_3\\\\x_1^q & x_2^q & x_3^q\\\\x_1^{q^2} & x_2^{q^2} & x_3^{q^2} \\end{vmatrix}", "d579b85b4056bc67be4d48d3b2e213eb": "\\top \\equiv \\bot \\rightarrow \\bot.", "d579d9674223d7ff8f8b42942e1fda91": "\\Gamma_k^\\text{gf}", "d57a221c1e684822f908396194e640ec": "X,\\ Y", "d57a2eb2ef33928f3054e05dfd0a2da0": "T_5 \\left[ 1+2+3+\\cdots + (n-1) + n + (n + 1) \\right] - T_5 =\\left[ \\frac{1}{2} (n^2 + n) \\right] T_5 + (n + 1)T_5 - T_5 = T_5 \\left[ \\frac{1}{2} (n^2 + n) \\right] + n T_5 = \\left[ \\frac{1}{2} (n^2 + 3n) \\right] T_5", "d57a38c8507c1dc27208ef75f5478ffe": "v_\\mathrm\nN", "d57a5260cde8511ebed82165135f7f8d": "{\\pm 0} \\times {\\pm \\infty} = \\mbox{NaN}\\,\\!", "d57a7d373a8f8ddf3c0673c5509aa0f2": "\\eta_{Carnot}", "d57ab2401f1d3b692ce37b11ea8f08a5": "\\mathbf{K} = k(\\mathbf{X},\\mathbf{X})", "d57ad13310b3e766d05a595171c48a8c": "i_s=i_1\\sin(2\\pi\\frac{2eV}{h}t).", "d57b0737f0ed9badd343c97909f8907e": "d^{448} = d^{558} = d^{668} = d^{778} = -\\frac{1}{2\\sqrt{3}} \\,", "d57baa498675e3502467606b86e62518": "I=aJ", "d57bdeb06ecf8bd086b6bfa3ad9fe492": "S(\\rho^{123})+S(\\rho^2)\\leq S(\\rho^{12})+S(\\rho^{23})", "d57beaf6e30870a691d6b11db6dc2849": "g:=r_{i-1};", "d57bf08b72e4c281b00f8bfd3cdc853d": " d = \\frac{\\ln 4}{\\ln \\sqrt 5}\\approx 1.7227", "d57bf12cabfc66b5578edf823a868a87": "\\scriptstyle f\\colon A \\to {\\Bbb R}", "d57c1741eae6938530c492f26878b683": " \\mathrm{SML}: E(R_i)= R_f+\\beta_i (E(R_M) - R_f).~ ", "d57c2e9888c35dd0d2535b2471eefa99": "\\omega_i = \n\\begin{cases} \n 4/9 & i = 0 \\\\\n 1/9 & i = 1,2,3,4 \\\\\n 1/36 & i = 5,6,7,8 \\\\\n\\end{cases}", "d57c36b7a583461a68541a46a1fb16ad": "b_2 = B / (1 + (o_2/o_1) + (o_2/o_3) )", "d57c475f728d71f8847306358e9a7374": " x^{(4)} = f^{\\prime\\prime\\prime}f^3 + 3 f^{\\prime\\prime}f^{\\prime} f^2 + f^{\\prime}f^{\\prime\\prime} f^2 +(f^\\prime)^3 f,", "d57ca486acdc60142d79ce6a7b1f941d": "HU = 1000\\times\\frac{\\mu_X - \\mu_{water}}{\\mu_{water}}", "d57cc9a0d65a4a0e9b00b664a52c3412": "\\sum_{m=0}^n P(3m+2)=P(3n+4)-1", "d57ccaa3398055a17319012943b79293": "L = \\frac{\\mathrm{d}^2 \\Phi}{\\mathrm{d}A\\,\\mathrm{d}{\\Omega} \\cos \\theta} \\approx \\frac{\\Phi}{\\Omega A \\cos \\theta}", "d57cff2ebc76509061147a348664d5d5": "\n\\begin{array}{llrl}\n \\text{Context} & \\Gamma & = & \\epsilon\\ \\mathtt{(empty)}\\\\\n & & \\vert& \\Gamma,\\ x : \\sigma\\\\\n \\text{Typing} & & = & \\Gamma \\vdash e : \\sigma\\\\\n\\\\\n\\end{array}\n", "d57d27cc492b9e3381f2f3a4433b5da3": "{10}^{\\,\\! 4 \\cdot 2^{60}}", "d57d32f7075342bce4b0b339031c2628": "\\dot{H} + H^2 = \\frac{\\ddot{a}}{a} = -\\frac{4 \\pi G}{3}\\left(\\rho+\\frac{3p}{c^2}\\right) + \\frac{\\Lambda c^2}{3}", "d57d3406c77acfafa1d049329b41478f": "\\operatorname{depth} M \\le \\operatorname{dim} R", "d57d886d11eba0b3bceea8f6f80385ae": "a_1 \\leq a_2 \\leq \\cdots \\leq a_n", "d57d9d356ff6bffc3f98f6e37400e518": "J_3\\ =\\ -2.619\\ 10^{11}\\text{ km}^6/s^2 \\, ", "d57da5498aee751d19e71d8e81a1ce0f": "F(c,\\ X)=X^3-cX^2+c^pX-1\\in GF(p^2)[X]", "d57e2d6dfba97d0dc610c056ce7b7ff0": "2t<1", "d57e65a2710084da5e0a985e38cf9f50": " \\frac{1}{\\lambda} = R \\left(\\frac{1}{m^2} - \\frac{1}{n^2}\\right),", "d57e706f259f9925ede4da6dd656c794": " \\begin{pmatrix}1&1\\\\0&1\\\\ \\end{pmatrix}, \\begin{pmatrix}1&0\\\\ \\rho&1\\\\ \\end{pmatrix}", "d57e84810a4a58f09a88b1f0f8923b65": "[\\mathfrak{g}, \\mathfrak{g} ]", "d57e920ebc76aeb9bc22b0594753e070": "\\mathbb{HP}^n", "d57e9ab81fc2273b9eed8b40e851bb8f": "V_{OCi}", "d57eb57466b4f72a3f85810344640984": "\\textbf{y} + \\hat{\\textbf{x}}", "d57ebf83cec6e8d44b299106be3aeb74": "\\tau(C) = \\mathrm{diam} (C)^s,\\,", "d57faa725cc4b4803f355b2f6efcf5cb": "\\lfloor \\varphi^n+1/2 \\rfloor", "d57fac5bd84d189e6ee3c53ece5a3965": "\\mathfrak{g}_\\alpha", "d57fafa64ef5c1d38045c802fd09615c": "\\int_M K\\;dA+\\int_{\\partial M}k_g\\;ds=2\\pi\\chi(M), \\, ", "d5800bbfe152093da8557e8ec0f89ff2": " 1 - e^{-b {\\mathrm{Log}[x]}^2} x^{-1-a} \\left(1+\\tfrac{2b \\mathrm{Log}[x]}{a}\\right) ", "d5801e02be0d97f914cd8a4f7c9c3cd2": " \\varphi_{} = \\frac{T \\ell}{G J_T}. ", "d580cab50b5472f9c17eda7927c59952": "\\frac{V_r}{V_t}\\ =\\ \\frac{e_g\\ \\sin u\\ -\\ e_h\\ \\cos u}{\\frac{p}{r}}", "d581215a94277a312354bf461cea928e": "[g_{ij}]", "d5813a7e10e7fadb5884f678dc3b46d1": "\\mathrm{CH}", "d5813b59eba0a3584e00644cd289f707": "\\mathbb{R}^0", "d581491a62a64666656dacd7195d7bee": "\\frac{\\partial u}{\\partial t} = \\frac{1}{2}\\frac{\\partial^2 u}{\\partial x^2}.", "d5814baf4d601c4cfc4b82a95929183f": "S_i-S_j \\cong 1", "d5815b3abb872e2d3f64072e075de3c5": "\\lVert x_n \\rVert \\to \\lVert x \\rVert", "d581956c63d336e4b0ece6767be4f906": "\\gamma:B \\times A \\rightarrow A \\times B", "d581c5e5f3d210f4a4b104ef570e9b6d": "v_s(t)", "d581cb617f887fc1f89d205b37c5ad23": "T_{skew}", "d5824bb7255b7249ab5dcf99b06c755b": "\\min_{c \\in C,\\ c \\neq c_0}d(c,c_0)=\\min_{c \\in C, c \\neq c_0}d(c-c_0, 0)=\\min_{c \\in C, c \\neq 0}d(c, 0)=d.", "d582c0bf80bbe693acf11165d566d625": "H_i(2, 2) = 4 ", "d582d18816d302e6425e411b0356bb7c": "P^{-1}(a') = \\left\\{ \\begin{matrix}\nMN - 1 & \\mbox{if } a' = MN - 1, \\\\\nMa' \\mod MN - 1 & \\mbox{otherwise}.\n\\end{matrix} \\right.\n", "d58314331704ceb0fce7c2959331eb09": "T_{10}", "d583197da49caccc468e1c96603c5fc9": "\\tau=\\bigoplus_{i=1}^{K}\\sigma_i", "d583c39fffda7697501632c041776c78": " \\operatorname{build-param-lists}[g, D, V, T_5] \\and \\operatorname{build-param-lists}[m, D, V, K_5] ", "d583d500fa123d7ae60d81597972e824": "\\sum_{r\\in R_{\\nu}^-}w_r=\\sum_{r\\in R_{\\nu}^+}w_r", "d58422a3c09df9119d59c8c8e5b2c79d": "Y_0=\\sqrt{\\frac{G+j\\omega C}{R+j\\omega L}}", "d5849a6b3ccc300bd70d9ebbe814eaf5": "(\\partial_\\mu\\Gamma^\\rho{}_{\\sigma\\nu}\n - \\partial_\\sigma\\Gamma^\\rho{}_{\\mu\\nu}\n + \\Gamma^\\alpha{}_{\\sigma\\nu}\\Gamma^\\rho{}_{\\alpha\\mu}\n - \\Gamma^\\alpha{}_{\\mu\\nu}\\Gamma^\\rho{}_{\\alpha\\sigma})V_\\rho", "d5849f6f30f956130e1abcc6527344d0": "\\sum_{i0 .", "d59aa3cd8bf82cb217c4e5fe998db991": "I(z) = \\frac{z}{1 - z}.", "d59ac533525f1eee0a1d79683a9c95d5": "x^0", "d59acd6043cb30f3c613eb2eb4a09d54": "\\ L/D >> 1 ", "d59ad8e03a6670f2c677836819489948": " \\frac {df}{dE} = \\frac {1}{m_w^*} \\frac {dk}{dE} \\cot(\\frac {k l_w} {2}) - \\frac {k} {m_w^*} \\csc^2(\\frac {k l_w} {2}) \\times \\frac {l_w} {2} \\frac {dk} {dE} + \\frac {1}{m_b^*} \\frac {d \\kappa} {dE} \\quad \\quad (9-2)", "d59adf4652659a4a63ee30443ef38c7f": "a_2=40,692", "d59afce572a2fe6bbcf458734ab53457": "number \\equiv_{(base-1)} \\sum_{i=0}^n{digits_i} \\times 1", "d59b2d98ee90641a2a3dbc608e7e14d3": "F[x,y,z]=\\frac{\\sqrt{(z''y'-z'y'')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}", "d59b6d19c020dfaa17980942bcd3cfd0": "\\tau = \\operatorname{arsinh}(2 a z/Q)", "d59b850cd7db1e1f8fd00aa0dbd82a96": "sin({\\alpha})", "d59b88209ac28f5d0cec4452974df7f1": "(c+n-1)", "d59bda90976a27424fd25b0b6599ad9e": "r_\\mathrm{s} = \\frac{2Gm}{c^2},", "d59be207a886828204269ff7eba837c2": "+s", "d59bebbd7b9703b7616317f758b2414e": "\\| \\Phi \\otimes I_n \\|", "d59bf82e646cd325e06e6b1e9068e929": " P =\\sum_{|\\alpha|\\le m} a_\\alpha(x) D^\\alpha,", "d59c110376413344c39a9eb87142c706": "I = 2evn_{s}A", "d59c12f4e5f024e1281a74bb39654c67": "u = e^{\\omega_p/kT} - 1", "d59c4b731ded42dd1f7be0858ed957eb": "\n\\dot{V}=-\\kappa V\n", "d59c53d583d413297c3d6f65a1040c4c": "\\vartriangle =r_2 - r_1", "d59c684894c924ce2c9fd5a4f1b7531b": "\\boldsymbol{P} = \\boldsymbol{N}^T", "d59c6ad29222ef7785da3967d036c753": "p = H \\sin{(at+\\Phi)}+p_m", "d59c82cfd45bae946a071cb9e1e2a482": "\\sum F = F_{bottom} + F_{top} + F_{weight} = P_{bottom} \\cdot A - P_{top} \\cdot A - \\rho \\cdot g \\cdot A \\cdot h.", "d59ce2e905f778068576479b3e1176f0": " I_{SL} ", "d59d4bfd1003c4edb4b1df1abb934a85": "\\mathbf{RP}^\\infty := \\lim_n \\mathbf{RP}^n.", "d59d802fc98018a1fa478c56be674e36": " \\frac{ z^{-1} \\sin(\\omega_0)}{ 1-2z^{-1}\\cos(\\omega_0)+ z^{-2} }", "d59d8fce620a2055e4a85679480fe0aa": "\\nu:x \\mapsto (x,x^2,x^3)", "d59da0f7bb8249414c9a4439af1e781c": "{P(N)_{t}}", "d59dae66709082a9815fbfdcf8e672b7": "AB=-1", "d59db6c9aa021b4c65b1e74ed2b2f424": "S^5 = \\left\\{ x \\in \\mathbb{R}^6 : \\|x\\| = r\\right\\}.", "d59de1131bedd4f1d1ba35d6b473dc80": "\\sin_k(i) \\equiv \\sin_r(t). \\, ", "d59e1fe2c566f76da0eda1fe4696a9ad": "B_{\\nu,\\text{max}}(T) = \\frac{ 2k_\\mathrm{B}^3T^3(3+W(-3\\exp(-3))^3}{h^2c^2} \\frac{1}{e^{3+W(-3\\exp(-3))} - 1}\\approx 1.896\\times 10^{-19}\\text{watt m}^{-2}\\text{Hz}^{-1}\\text{kelvin}^{-3}T^3", "d59e2fe181a9c3bb98988cbd53f38d82": "a x^2+b x+c", "d59e306c11540cda4b046758f2ee922e": " \\kappa \\approx 2 - \\eta / 6 = 2 - 0.77 = 1.23. ..........(46) ", "d59e54c4228d7bf5b7e1abff2adbd56b": "C = \\frac{C_s + C_u}{2} ", "d59f18ef31d316eb8ae8af8c8368a97a": "\\chi_{\\nu,k}\\begin{pmatrix}z& 0\\\\ c& z^{-1}\\end{pmatrix}=r^{i\\nu} e^{ik\\theta},", "d59f53a226072d392d5c6c363d7e105f": "2s = - \\left( \\frac {1} {\\tau_1} + \\frac {1} {\\tau_2} \\right) ", "d59fbb7daf0474af328de7ac890df277": "\\frac{1}{2R}(x_2 + x_1)", "d5a0a8114ddc8acf70eb13048787790d": "N(\\varepsilon, \\mathcal{F}, ||\\cdot||)", "d5a0b3f70fe7f7cb402b04233de26b45": "K_{sp^{ }}", "d5a0d2caa1acd95acb79b7c32a36b5af": "b\\mapsto \\|b\\|", "d5a0e12abec6e63a8698343df8d1909b": "y_i= \\sum {\\gamma_{ijk}a_j \\acute {a}_k} + \\sum {\\lambda_{ijk} \\acute{a}_j \\acute{a}_k} + \\sum{ \\xi_{ij}a_j} + \\sum{\\acute{\\xi}_{ij}\\acute{a}_j} + \\delta_i", "d5a115eb3e1e2708dc34b6f3cbcc373f": "f_3(x)\\,", "d5a1228f9b23eff9fe3bb10e1233b001": "k=\\frac{C^{SL}}{C^{LS}}", "d5a15a41cc98da7750066b2fccfca3a2": "\\sigma(X) = \\sqrt{\\operatorname{Var}(X)}:", "d5a1c682c76e339f26ad092a8b45813a": " \\langle \\psi_1 | A | \\psi_2 \\rangle = 0", "d5a1e09271bd8c44420ea9c0c4f39c56": " \\tau_{ki} ", "d5a24ab300ebfdfaa22a486b4c89e1d4": "s_\\lambda s_\\mu=\\sum_\\nu c_{\\lambda,\\mu}^\\nu s_\\nu.", "d5a2a24c90efef62fbc93d14c92f6e4e": "(W_1,\\ldots,W_n)", "d5a2d69b31399ca5ff5e0098bf4365f6": " \\Pi^+ ", "d5a3217416c720979f371f0b5fc55c91": "0<\\frac{1}{b}\\leq\\left|A_{n}+B_{n}\\zeta(3)\\right|\\leq 4\\left(\\frac{4}{5}\\right)^{n},", "d5a3c8f6e4575e4f9f8d04d990b6ffad": "\\frac{s \\cdot a}{2}", "d5a45b9b501c2e4cc111c5265d10b3cd": "F_{\\mathrm{Anchor}} =\\frac {\\mathrm{Weight}} {2 \\cos (45 ^{\\circ} + \\frac 1 4 {\\theta_{\\mathrm{Bottom}}} )} \\approx\\mathrm{Weight}\\times 0.707 + O(\\theta_{\\mathrm{Bottom}})", "d5a4cea691060b257665acbee85c371c": "\\mathbf \\gamma(0)= \\mathbf x_0 ", "d5a5340383bd2d05e6eb468e43ec9198": "\\dot{n}_i = n'_i-n_i", "d5a57bab7c9b8ea325e75d53e35d8963": "\n \\cfrac{\\lambda \\bar{a}^2}{2}\\left[(m^2-2)\\sec^{-1} m + \\sqrt{m^2-1}\\right] + \\cfrac{4\\lambda\\bar{a}}{3}\\left[\\sqrt{m^2-1}\\sec^{-1} m - m + 1\\right] = 1\n ", "d5a5c7f05cfb89e93091d7076fcf85b0": "0.\\overline{18}", "d5a61eece7d8636b0e390a0e600b3941": "\\left | C_G(g) \\right |", "d5a65febbcccf7cc29afe7f080bf0e2b": "\\left |\\{Mf > \\lambda\\} \\right |< \\frac{C_d}{\\lambda} \\Vert f\\Vert_{L^1 (\\mathbf{R}^d)}.", "d5a66630d9fde7eac98fe0dae5c9aa16": "Tf(x)", "d5a67e9958317c136deb5f5d1aacecd5": "Q \\subset D", "d5a6a69b19b58a7d312f49b96c51bfdb": "c = 9.90 \\times 10^{-8}", "d5a76586bca62db9e04f41fbabc66588": "\\int\\tanh^2 ax\\,dx = x - \\frac{\\tanh ax}{a}+C\\,", "d5a78f9c7c7414981a5b1da1e0799290": "\\scriptstyle\\sup_t |G_n(t)|", "d5a83483b3adeb6d05bc5f493f06de16": "S_{22} = \\frac{b_2}{a_2} = \\frac{V_2^-}{V_2^+}\\,", "d5a88c6cf7c458676905838149f571a4": "\\boldsymbol\\theta", "d5a895f1b7730fd429e63cbf0217f770": " \\sum_{k=0}^{n}\\binom{n}{k} \\frac{B_{k}}{n-k+2} = \\frac{B_{n+1}}{n+1} ", "d5a8c2455f87736aad2ed9522933384f": "\\mathbf{\\mathit{Tr}}(\\mathbf{\\mathit{\\rho}}_{S}(t)) = 1", "d5a8ccfb0a687fe0c4821a9d27378c77": "\\mathbf{J}_i=-\\sum_j D_{ij}[c_j \\nabla c_i - c_i \\nabla c_j]", "d5a9467ac6d9b6801237f63d8ecfbba7": "\\pi (Y_{\\ell\\,m}) =(-1)^{\\ell}", "d5a96d3abc4e087d9edb48905e3b630f": "\\frac{1}{m}", "d5a98c8c5ae51d2e48916aa1cfbdff9c": "^hp(x,y,z)=0,", "d5a99808a1dd6609c90e57aa13ae4240": "x \\in A\\cap B", "d5a9adb022d9548872b2bee0ddf53daa": "\\pi \\int_a^b ([h-R_O(x)]^2 - [h-R_I(x)]^2)\\, \\mathrm{d}x.", "d5a9b9a65afc208646f832419317bd4c": "\\displaystyle \\nabla^2u+e^{\\lambda u}=0", "d5a9bf2a3558d8d359865dbc196e74bc": "F_\\mathrm{Poisson}(x;\\lambda) \\approx F_\\mathrm{normal}(x;\\mu=\\lambda,\\sigma^2=\\lambda)\\,", "d5a9c2f6f2875f03181fafe1824fbcf7": "du = -bu\\,dt + \\sigma_2\\,dW_2(t)", "d5a9f19dacf3ee53121a19f62acdd829": "\\sigma_m\\,\\!", "d5aa3d16b269a2dc0af22a118a81171a": "R = T(A, R),", "d5aa489f1be8d6c2fd0034009d9da973": "\\Re s > 0", "d5aa871aedc71cb6a7a4585617701a5d": "sa+tb=c\\ ", "d5aa95fe8edd67e7614ef3e01bf632cb": "c+NL+SL=W", "d5aab8990695f12ded0d401d38d1ee97": "\\rho_w = \\sqrt{\\frac{L_w}{C_w}} = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} = \\rho_0. \\ ", "d5ab2a23c2d9ef058e93898056c2c91f": " \\cot \\theta =\\frac{\\mathrm{adjacent}}{\\mathrm{opposite}}\n= \\frac { \\left( \\frac{\\mathrm{adjacent}}{\\mathrm{adjacent}} \\right) } { \\left( \\frac {\\mathrm{opposite}}{\\mathrm{adjacent}} \\right) } = \\frac {1}{\\tan \\theta}.", "d5abf2cad0fc3d15e976f7a928c5c642": "\\langle Q\\rangle", "d5ac2ffc2b907daf433a2343b2a8f4c3": "\\hat A", "d5ac70482cf07fba0340d36155aa60d0": "~\\sigma_{\\rm as} ~", "d5ac7bb716d9dda141aae19fd782088b": " \\pi_G(x) = \\frac1{|G|}\\sum_{g\\in G} g\\cdot \\pi(g^{-1}\\cdot x).", "d5acfcd1ca30ff324bb5bb5e418a1d39": " \\frac{|(I_t-\\mu_t)|}{\\sigma_t} > k \\longrightarrow \\mathit{ Foreground }", "d5ad0463e8f90a12ec761f92956b429e": "\\begin{align} 2\\cdot R_*\n & = \\frac{(28.23\\cdot 3.72\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 22.6\\cdot R_{\\bigodot}\n\\end{align}", "d5ad1d7c71ac1aa80c34f7ebdcd31c68": "\n|\\tilde n(\\mathbf R)\\rangle=e^{-i\\beta(\\mathbf R)}|n(\\mathbf R)\\rangle\n", "d5ad8f4d603bae3ddf5bf0a2559abf68": "F[\\varphi(x)]", "d5adfd1b779ef06ec1395f5b769edf7a": "A = \\begin{bmatrix} \n1 & 0 & 0 & 0 & 0 \\\\\n3 & 1 & 0 & 0 & 0 \\\\\n6 & 3 & 2 & 0 & 0 \\\\\n10 & 6 & 3 & 2 & 0 \\\\\n15 & 10 & 6 & 3 & 2\n\\end{bmatrix}", "d5ae4762a94ae4f8509b8ff587f3edd3": "\\theta = 2\\,", "d5aeb2c7bf52b37eea8e5dcd12363b5f": "T' (N')^* = (N')^*T'. \\,", "d5aeb9e9c82c81406b04e5db0e591d6d": "f_p(x)=\\sum_{n=0}^\\infty f(p^n)x^n.", "d5aecce8b1ca601bf1c6c4213bad9c39": "\\chi_m = \\mu_r - 1.", "d5b0648379e6b50d446e3ea0ddab0c86": "\\operatorname{L}(V \\otimes W)", "d5b07dcb14491b8bcf21b02c246098ab": "\\operatorname{E}(T) = k\\sigma^2 + \\sum_{i=1}^k n_i(\\mu + T_i)^2", "d5b092c7c244156f8d65a3b1be7cd3d9": "\\frac{\\sqrt{15}}{2}\\cos(2\\theta)\\cos^2(\\phi)", "d5b0ddaa4c9c22c058fd708d4de645e4": "N - \\varepsilon N = (1-\\varepsilon)N", "d5b120d24c16819584b79a221a927f5b": "Q = \\frac{\\omega L}{R}", "d5b13868bbbaa56f0f533a7a2a64d0b0": "\\frac{\\partial U}{\\partial x_i}-\\lambda\\cdot p_i =0~~\\forall i", "d5b1449c3a68a82d50f66d9dda43fb6e": " Diffusion Rate = -\\pi \\cdot (ravg)^2 \\cdot Dc \\cdot \\frac{dC}{dx}", "d5b144f26aacca526b3ef267b2ad61c4": "A \\rightarrow B: \\{N_B\\}_{K_{PB}}", "d5b1de7f1a2667cb260fd3f9521e7b1b": "\n\\dot{r}=-\\frac{\\kappa}{2}r\n", "d5b1e4cbda1975ea0fefd6b56696c098": "X \\sqcup Y = \\{ (x, \\xi)| x \\in X\\} \\cup \\{ (\\xi, y)| y \\in Y \\} .", "d5b1f87050ae5e13b2ba262d7bca5f2a": "x \\rightarrow bx", "d5b23bc0807a080aac8849134b3f4fd0": " \\displaystyle Mf(a)=\\int_U f(ua^2) \\,du.", "d5b25305d28c80ccbe6cd3c0a5f5f6ed": "\\mathit \\Gamma := \\frac{V_\\mathrm r}{V_\\mathrm i}", "d5b2a795abcf7bab5efe3714d3393ec7": "\\sum_{i=1}^n X_i\\,", "d5b2e46cde1943f7a7ea0337ee441ed9": "\\sqrt{2} = \\frac{p}{q}", "d5b35ae1ae2b12c266824f7aa9a0dcbc": "d_0=3G_4", "d5b36157d9f35e6d822158ae799cbc73": "{\\nabla^2 A \\over A } = -k^2 ", "d5b39adef9934e8370c7ff3230043dc6": "X_3 = D^2-G-2H-a_3E = 48-8\\sqrt{39}", "d5b3b7511d756735735ebb1709cceb4f": "\\widehat K=\\widehat{GL}(n|2m; \\Lambda) ", "d5b3ffca3db71f53fa0ae2448c5091c7": "\\mathbf{A}(\\mathbf{x},t) = \\frac{\\mu_0}{4 \\pi} e^{-i \\omega t} \\int d^3\\mathbf{x'}\\mathbf{J}(\\mathbf{x'})\\frac{e^{i k \\|\\mathbf{x}-\\mathbf{x'}\\|_2}}{\\|\\mathbf{x}-\\mathbf{x'}\\|_2}", "d5b462619306bf13e8d0fa1bf177e667": "\\mathcal{P}X", "d5b486fac11b9b89ab95ab0287ceea0e": "q_{or}", "d5b4ba6e842a8a74f1e03f65c6337f2e": "y_c = \\left\\{\\begin{array}{ll}\n\\displaystyle{m\\, \\frac{x }{ p ^2} \\left( 2\\, p\\, - \\frac{x}{c} \\right)}, & 0 \\leq x \\leq pc \\\\\n\\\\\n\\displaystyle{m\\, \\frac{c-x}{(1 - p)^2} \\left( 1 + \\frac{x}{c} - 2\\, p \\right)}, & pc \\leq x \\leq c\n\\end{array} \\right. ", "d5b4e029dea48f67a369ee9a16d91048": "m \\in [q^k]^K", "d5b4ff2faa0b4853121e1fab52a65a11": "(x-h)(y-k) = m \\, \\, \\, ", "d5b550866cdd60555610a13efce8b67e": "u_{xy}=\\exp u \\,\\!", "d5b57e841d3509323fae09d6d262cbe4": "\\gamma \\colon R \\to \\mathcal{P}(\\Lambda \\times R)", "d5b58e002c625ab3e469cd8724d019c6": "\n \\lambda_J^2\\phi_{xx}-\\omega_p^{-2}\\phi_{tt}-\\sin(\\phi)\n =\\omega_c^{-1}\\phi_t - j/j_c,\n", "d5b5a728b8519a1be5e461e0a5c9e1df": "\\Delta/J_{k-1}\\subseteq J_k/J_{k-1}", "d5b65a48478bafeb4e49878dda07fab9": "q = p_1^{e_1} ... p_s^{e_s}.", "d5b70d9cc837281219e2a1b83810b548": "\\beta(T)-\\beta(0)=2\\pi m", "d5b7627c5e3c9143a07c0d6c693d7171": "f(x;k,\\theta) = \\frac{x^{k-1}e^{-\\frac{x}{\\theta}}}{\\theta^k\\Gamma(k)} \\quad \\text{ for } x > 0 \\text{ and } k, \\theta > 0.", "d5b766e34c455af55f28d9344feff071": "\\mathrm{im}(d^{n - 1}) = B^n(X)", "d5b7e21ea2d830faffd69b5f2d92221c": "\n \\hat\\mu(x) = \\sum_{i=1}^n \\hat\\mu(x_i) f_i(x)\n", "d5b81f696bd2aff869eb610b1e05e567": "dF=0", "d5b88dfbf7b571dbed7bf02107105c80": "F(\\mathbf{x}^{(1)}) \\le F(\\mathbf{x}^{(0)})", "d5b895376231aa239544ea53d0cd1549": "{w_{\\mathrm{abs}}} = {{N_{\\mathrm{after}}} \\over {N_{\\mathrm{before}}}}", "d5b89844da71a2b9a41e66b2466822c2": "(r, \\theta, \\phi, t) \\rightarrow (r, \\theta, -\\phi, t)", "d5b8db09c6a6b22398bbb0f5c0fa5dd0": "V \\ ", "d5b8e6978607828dbe73821ec487a1ec": "v_T=\\frac{|\\boldsymbol{r}\\times\\boldsymbol{v}|}{|\\boldsymbol{r}|}=\\omega|\\boldsymbol{r}|", "d5b918ee93673d3fb037ca0c76ba353d": "\\scriptstyle{|\\psi(t)|^2}", "d5b92d78a09efa8dbf40ca760adaa435": "\\mathbf{P}^{(1)}", "d5b9f9c3ba760868d1a3dce1b25d042b": "dm_{fuel}", "d5ba06301ba115bd6e51d97c7445f16d": "1 + \\sin(\\theta)", "d5ba18747edc77f48ea4da7c75552b3b": " \\Rightarrow RT\\ln \\frac{f}\n{{f^\\circ }} - RT\\ln \\frac{P}\n{{P^\\circ }} = \\int_{P^\\circ }^P {\\Phi dP}", "d5ba9242936c6590745e7e33e3ab4dc0": "B+W = 7,460,514k + 10,366,482k = (2^2)(3)(11)(29)(4657)k \\, ", "d5ba9f20e5e994a372d7ecf4ef2abfc1": "z_k - z^*", "d5babb2253fcb85621cdc4b8fbfc9566": " \\textrm{Re} \\left [ {}_1F_1(\\alpha; \\alpha+\\beta; it) \\right ] = \\textrm{Re} \\left [ {}_1F_1(\\alpha; \\alpha+\\beta; - it) \\right ] ", "d5babdad6deb7f182e8ee4ee3173cbe1": "(X-k)/\\sqrt{2k}", "d5bb01369e765e95766469697b30a42f": " H = H_0 + \\lambda V ", "d5bb17bc9fc992034d7526a64ea2513f": "p_i \\mid n - 1", "d5bb5b6472b1d3b0d54f20a1965b4e5f": "\\hat{f}=r_c \\cdot \\hat{r} + \\cos \\alpha \\cdot ( \\hat{c} - r_c \\cdot \\hat{r})+\n \\sin \\alpha \\cdot \\hat{r} \\times \\hat{c}.", "d5bba4830f1917b8750dc813ced8d51c": "\\bar{v} =\\nabla \\phi", "d5bcc68395d963b9961bacd96d6898ef": "\n\\mu'_j = \\int_{-\\infty}^{+\\infty} x^j f(x)\\,dx.\n", "d5bd04883d31799879b9ff0889a238ee": "G\\rightarrow G'", "d5bd4f9b8137afc8fb0fb9ba645cc908": "P^{\\mu}\\!", "d5bdef0e5b3961e2ea479abef50a309c": "(a \\oplus b)", "d5bdf836b4c5a0807850a6e2f86e4890": "\n\\lim_{T\\rightarrow\\infty} \\frac{1}{T} \\log\\left(\\frac{Z_\\eta(t)}{Z_\\mu(t)}\\right)\n= \\lambda^{(m, m+1)} \\mathbb{E}\n\\left(\n\\frac{M_{(1)}}{M_{(1)} + \\cdots + M_{(m)}}\n+ \\frac{M_{(m+1)}}{M_{(m+1)} + \\cdots + M_{(n)}}\n\\right) > 0.\n", "d5bdfef0428e7e2f1940d0e4174635e5": " \\begin{align}\n& \\operatorname{E}\\left[\\frac{X}{1-X}\\right] =\\frac{\\alpha}{\\beta - 1 } \\text{ if }\\beta > 1\\\\\n& \\operatorname{E}\\left[\\frac{1-X}{X}\\right] =\\frac{\\beta}{\\alpha- 1 }\\text{ if }\\alpha > 1\n\\end{align} ", "d5be3a4f9d65d068063df100b27de08a": "P \\cup Z", "d5be5e75ff8a4be32b0445cea9d9cdac": "d^\\prime = \\cos{\\theta_\\mathrm{c}} d + \\sin{\\theta_\\mathrm{c}} s; ", "d5be9f42c0fb1f11693144bd1ec64465": "\\tilde f|_T=\\tilde g\\,", "d5bee1ddb3262b6e34ffdb69b8f068ba": "\\textrm{Hav.\\, MZD} = {\\textrm{Hav. TZD\\, -\\, Hav.H\\, Cos.L\\, Cos.D\\, }}\\, ", "d5bf0e2281df5526290c3f6992c47813": "\\scriptstyle\\hat\\chi", "d5bf4b18109914970c5759047b4c4756": "\\begin{pmatrix} p & q \\\\ q & p \\end{pmatrix} ", "d5bf500fb656942b45804bc53df9bd71": "u\\in W^{1,2}_0(\\Omega)", "d5bf7cf80da9aed45e939400d6742bfd": "P(\\text{reject }H_0 | H_0 \\text{ is valid}) = P(X \\ge 10|p=\\tfrac 14) =\\sum_{k=10}^{25}P(X=k|p=\\tfrac 14)\\approx 0{.}07.", "d5bfc46acf923e81e18386d731256406": "y=L_0(x)", "d5c08598d6eb91e36b48445abf25f937": "\\lim_{p\\to-\\infty}{\\left(\\sum_{i=1}^n |x_i-y_i|^p\\right)^\\frac{1}{p}} = \\min_{i=1}^n |x_i-y_i|. \\,", "d5c0c4b9cb81bc11b1af068334b5fba1": "\\underline{P}(A)=0", "d5c0c9712a300ed09b71fc7a46b92f77": " e^{\\frac{1}{12}-\\zeta^\\prime(-1)} =\n e^{\\frac{1}{8}-\\frac{1}{2}\\sum\\limits_{n=0}^\\infty \\frac{1}{n+1} \\sum\\limits_{k=0}^n \\left(-1\\right)^k \\binom{n}{k} \\left(k+1\\right)^2 \\ln(k+1)}", "d5c0cdbf791e96598ce9246331c5cca8": "\\sum_{n=0}^{\\infty} \\frac{1}{(n+2)n!} = \\frac{1}{2}+\\frac{1}{3}+\\frac{1}{8}+\\frac{1}{30}+\\frac{1}{144}\\ldots=1\\,.", "d5c0d5582ff75f30b314e8f632cd2d74": "P(\\mathbf{E}|M) = \\prod_k{P(e_k|M)}.", "d5c10f75cbb4a0b86edbac4b691acf91": "m_\\text{red} = \\frac{m_\\mathrm{e} m_\\mathrm{p}}{m_\\mathrm{e} + m_\\mathrm{p}} = m_\\mathrm{e} \\frac{1}{1+m_\\mathrm{e}/m_\\mathrm{p}}", "d5c12a65d9eb452146379ea027cf2dcc": "A\\triangle B\\subseteq A\\cup B", "d5c182c195dc214f1ef18a8a971a01b6": "(\\nu, \\hat{\\nu})", "d5c19f89ac27d38d12ffc201d2c18cd0": " \\alpha(t) = \\frac{d\\omega}{dt} = \\frac{d^2\\theta}{dt^2}.", "d5c1bb3e116a3b2501600ef37fb08d46": " E = (e_{i} - e_{a})/Pr", "d5c1f76b519dc90a58e35860b2a05b47": "g_{0i}=-\\textstyle\\frac12(4\\gamma+3+\\alpha_1-\\alpha_2+\\zeta_1-2\\xi)V_i-\\textstyle\\frac12(1+\\alpha_2-\\zeta_1+2\\xi)W_i\n-\\textstyle\\frac12(\\alpha_1-2\\alpha_2)w^iU-\\alpha_2w^jU_{ij}+O(\\epsilon^{\\frac52})\\;", "d5c20217dd2c21057628f845ac18c53b": "R_T \\subseteq T", "d5c211313f00ee392f5d3eacbed9f5d0": "\\rho = \\frac{r_p}{r_s} = \\tan ( \\Psi ) e^{i \\Delta}", "d5c230123a6de4752039ffcaffbff110": "(A-\\tilde\\lambda_\\star I)", "d5c2443b57ff072b719865306c67f28d": "\\displaystyle \\ u\\ ", "d5c24b47c465e1a6e7773e95a23c5418": "a_i < m_i", "d5c27edad7c41d35c1b41e80be1c7986": "n\\in\\omega", "d5c2d410d74b14a5dce222dccf9dc12b": " \\chi \\neq 0 ", "d5c311d756a076316cbab72e562bcf53": "Fe(x) = \\frac{\\varphi^x - \\varphi^{-x}}{\\sqrt{5}}", "d5c33cff84c662e48a9c27f2ffd019c9": " \\lim_{z \\rightarrow b^-} \\int_a^z f(x) \\, dx", "d5c34c757b341ff5d1eb8967d33f37f2": " H_1 ", "d5c37d158032d66a3a43cec561da3889": "\\tilde{\\kappa}_{(\\ell)}", "d5c39063192e0d9fd208984e960efad8": "1/r^{\\ell+2}", "d5c3b87a369ab6ce8fff5ec227e172bf": "K_D", "d5c3f08e16220fd0bcdf73c5d3fdbd79": "\n\\int^{R_e}_0 I(R) r dr = \\frac{1}{2} \\int^{\\infty}_0 I(R) r dr .\n", "d5c42a2316da8c20cfafda16bded8041": "M = [357.5291 + 0.98560028 \\times (J^{\\star} - 2451545)] \\mod 360", "d5c45fb02d9c3f3ce6887f7b91573599": "\\lim_{y \\to \\pm \\pi / 2} u_{n} \\left( x, y \\right) = + n \\text{ for } - \\frac{\\pi}{2} < x < + \\frac{\\pi}{2},", "d5c4861265d9206d1e54487c866ff5a4": "\\exp(tX) = \\gamma(t)", "d5c4d7a2751334a18b9dc436d3ba5aba": "\n\\frac{\\partial A_2}{\\partial z}=A_1 \\chi e^{i \\Delta k z},\n", "d5c52c259fbe6b03153550cfc27df832": "E \\subseteq V", "d5c56899b5c50e991e5d8d897ff8321c": "\\displaystyle\\varphi_i(\\mathbf r)=0,\\quad i=1,\\,\\ldots,\\,K", "d5c57b310204d0493168acf1ab36ff0d": "(\\mathbf{J},\\nu)", "d5c5ad6dbf748160ab3df05680dd31f8": "\\{\\,M_\\alpha\\mid \\alpha<\\gamma\\,\\}\\subseteq K", "d5c5babdfe7524d31adae00ce7931bb6": "\\mbox{E}^{o}_{cell}={\\mbox{RT} \\over \\mbox{nF}} \\mbox{ln K}\\,", "d5c5e56873d83952e45b33003565c950": "(1+i)/2 \\not\\in\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left({A}\\right)", "d5c67717a7e9fd803b1f7f772caca4cc": "{\\rm blanc}(x)", "d5c67aae749497a7aceaabc3bcb9e074": "RTb(1-\\varphi) = G_{ex} - b \\frac{dG_{ex}}{db}", "d5c6b3f7e93fd31114ee75d977a840d1": "0.187859\\ldots = \\sum_{k=1}^{\\infty} (-1)^k (k^{1/k} - 1) = \\sum_{k=1}^{\\infty} \\left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\\right).", "d5c6c633d05064fe7cf80c5e4e5003c3": "\n\\sum_{i = 1}^n {\\sum_{h = 1}^m {s_{ih} x_{ih}^k } } \\leq L^k \n", "d5c740565f3b1790b267b81aaacffd55": "P = \\frac{{V^2 }}{R}", "d5c74dbfa496b09be14d0a8887bfb65c": "\\left [ \\hat{a}, \\hat{a}^\\dagger \\right ] = 1", "d5c78841350a4522603aa669a3385cc5": "\nH^2(P,Q) \\leq \\delta(P,Q) \\leq \\sqrt 2 H(P,Q)\\,.\n", "d5c7e4c006665993625c72778bb5c71c": "\\operatorname{E}_Y \\left( \\operatorname{E}_{X\\mid Y} (X \\mid Y) \\right) ", "d5c82ba6d6b3d93a72d8b38f3e2394e5": "\\eta_{s+1}", "d5c84652d282b01865275b67e8ebca69": "\n2 \\int \\sqrt{2E - {l^2\\over r^2} + { 2\\over r}}\\ dr= k h\n", "d5c8a1ab7970b7a8bfdeb0bd3fa85d82": "\\hat{a}_i^{\\dagger\\bullet}\\, \\hat{a}_j^\\bullet = \\hat{a}_i^\\dagger\\, \\hat{a}_j \\,- \\mathopen{:}\\,\\hat{a}_i^\\dagger \\,\\hat{a}_j\\, \\mathclose{:}\\,= 0", "d5c8d66e30df9d26f718764394cc9956": "c_{ii0}=v_{i}", "d5c8fe4be4c0880c8210cab484307593": " 4\\pi \\times 10^{-7} \\text{ T}\\cdot\\text{m/A} = 1.257 \\times 10^{-6} \\text{ T}\\cdot\\text{m/A}", "d5c939d35faf42415b9f0fd0c4c762db": " B_n=\\langle \\sigma_1,\\ldots,\\sigma_{n-1}|\n\\sigma_i\\sigma_{i+1}\\sigma_i=\\sigma_{i+1}\\sigma_i\\sigma_{i+1}, \n\\sigma_i\\sigma_j=\\sigma_j\\sigma_i \\rangle, ", "d5c97fc48d6a559200c0d5e7f87019fc": "\\mathbf{\\hat{U}}", "d5ca3680a5e857b573d31a96ecff0a7b": " E = \\frac{(20\\frac{ft^2}{s})^2}{2(32.2 \\frac{ft}{s^2})(1 ft)^2} + 1 ft = 7.2 ft", "d5ca4853074cb9dbd4062d0498f8a860": "\\boldsymbol{\\tau}=[0,1,0,0]", "d5cabe28f36414871c2334dc0c7002cc": "\\hat{\\mathcal{P}}_{\\overline{S}_l^k}\\hat{\\mathcal{H}}\\hat{\\mathcal{P}}_{\\overline{S}_l^k} \\left| \\Psi_{l\\mu}^{k} \\right\\rangle =\nE_{l,\\mu}^{k} \\left| \\Psi_{l\\mu}^{k} \\right\\rangle", "d5cacab81eb860f94402f9396d71a4a7": "\\Beta(\\tfrac{1}{2}, \\tfrac{1}{2}) \\sim\\frac{1}{\\sqrt{\\theta(1-\\theta)}}", "d5cb243e620d8a57d807f9c320f05827": "U =\\frac{1}{2} \\iiint (\\sigma_x\\,\\epsilon_x\\,", "d5cb2caa8cc07bc70389d524937e638b": "F(3)", "d5cb9b659df90777f1be98b8efb99561": "M(n) > n/4", "d5cb9fd3c4270efd6642c4c3b41e1ef6": "\\bar{\\psi}", "d5cbd564e073faaa8982c75f3277f057": "\\inf \\emptyset", "d5cc2ec0c25517eeade0e1be16999ac5": "(1,2,3,\\dots, n)", "d5cc63cdd0a3099bcda10c501a107dd8": "\\tfrac{dM}{dT} = B - \\delta MS - \\mu M", "d5ccea3035e63661d1dbfc683b609d39": "I_z = \\iiint_V \\rho r^2\\, dV.", "d5cd3e2e37dd0ee47dc82b6b414d756a": "C(i,j)", "d5cd51364d1ad3569a156da20107aaa4": "f=F\\cdot(1+m)", "d5cd53324f4620b9ffd018682ca50f2c": "I=12log_2(h_2/2^n)=12log_2(h_2) - 12n", "d5cd8e4e1baaf3f5a73e3051d01fac52": "\\tilde f(p,q)=f(q)", "d5cde26a1390538189a5f7f6016d6ff1": "I_n = -\\frac{\\sin{ax}}{(n-1)x^{n-1}}+\\frac{a}{n-1}J_{n-1}\\,\\!", "d5cdfe1a5dd967c49cd62a0a9999a4bb": "\\Omega(\\log n/\\log\\log n)", "d5ce2ffa6248b8fe297300559072a097": "\\Big(\\frac{R}{M} \\Big)^j P_j\\,.", "d5ce311d32fdd219af0f9562eaac0396": " \\frac{q^2}{4}+\\frac{p^3}{27}\\, ", "d5ce6b66e737eceefb8536fed1f773da": "\\displaystyle{e_t(z)=\\exp t{z+1\\over z-1}.}", "d5cec0ce7a6b090fa00db2fad1c437e0": "\\sigma \\sqsubseteq \\sigma'", "d5cf0b6e8c7699f52e5409a19586a7ff": "S = \\iint_{\\mathrm{source}} B(\\theta,\\phi)\\mathrm{d}\\Omega", "d5cf1a2227bc909fb55cdfdd1dd7c31a": "MMH^*_{32}=\\big\\{g_x (\\big\\{0,1\\big\\}^{32} )^k \\big\\} \\to F_p, ", "d5cf562f8eede5df929141d34b71de68": "\\ p\\alpha,", "d5cfa92f54b27d873e1c2cadf9bfd3de": "h^a", "d5cfacc69109ec392110053366b55f29": "\n \\cfrac{\\Gamma \\vdash A, \\Delta \\qquad \\Sigma, B \\vdash \\Pi}{\\Gamma, \\Sigma, A\\rightarrow B \\vdash \\Delta, \\Pi} \\quad ({\\rightarrow }L)\n ", "d5cfec1dd45d154a3e07458427eafa8a": "\\nu_{i-1}+\\delta=s_{\\gamma_i}(\\nu_i+\\delta)", "d5d07270f84605af0ec9fd235b661f13": "A(t) = k \\cdot a(t)", "d5d074d81b11b296e8c9c5c3f336d9c5": "p_M=\\frac {n h \\nu}{c}", "d5d0a0d306f7aba669f2acade2a76109": "(s_1, s_2, \\cdots, s_n)", "d5d1668abb4e9111f914b23ff475e9d3": " L\\, ", "d5d20f293f091036ebfc4481821ceffc": "S_{6}", "d5d20ff847d5e9c0a2839ada8cc03c22": " \\sin \\theta\\ ", "d5d220b9079d14f82548d445eb69e777": "P_1 = (x_1, y_1)", "d5d235d73d36c12fdb7d5c31c94edee3": " \\mathbb{R} \\,", "d5d30b8ed488ac2fd4c21459781d69ff": " r_{\\mathrm{A}} = \\frac{dC_{\\mathrm{A}}}{dt} ", "d5d33183aa27c7fcdefd1dddb68e8d23": " p = \\hbar k ", "d5d336e5e304aab19ee1a8d72825ef76": "\\mathcal{E}(f_{t}^{\\mathbf{z}}) - \\mathcal{E}(f_{\\rho}) = \\left[ \\mathcal{E}(f_{t}^{\\mathbf{z}}) - \\mathcal{E}(f_{t})\\right] + \\left[ \\mathcal{E}(f_{t}) - \\mathcal{E}(f_{\\rho})\\right]", "d5d3681bf22f775cfc3ab3b8862f8201": "f(x)=ax^2+bx+c \\quad \\Rightarrow \\quad f'(x)=2ax+b \\,\\!,", "d5d39279a1ab75e20aa3d71787e054ff": "\\neg p == l", "d5d39c9df29875376ee2cfd7202da0df": "\n\\mbox{If }k \\ge 4,\\;\\;\\; r_k(n) > 0.", "d5d445eddc484cf01b8e7a3920aec0ae": "f(x) \\equiv 0 \\pmod N", "d5d4c3d0ae25edd20b20bb54864db58c": "w(\\sigma)", "d5d4f43489ed68d344434e89f1cbed38": "R_n(x,h)= (n+1) \\sum_{|\\alpha| =n+1}\\frac{h^\\alpha}{\\alpha !}\\int_0^1(1-t)^n\\partial^\\alpha f(x+th)\\,dt.", "d5d524d743800be3d0182ec1899679d6": "C(j)", "d5d647c7a6077884a68278cd5ed89458": "\\beta \\ge 5", "d5d6555c6d958ff8ccf3d12a98742cb1": "\\begin{align}\n\\frac{x_1}{\\alpha} - 1 + \\frac{x_2}{\\alpha} - 1 + \\cdots + \\frac{x_n}{\\alpha} - 1 & = \\frac{x_1 + x_2 + \\cdots + x_n}{\\alpha} - n \\\\\n& = n - n \\\\\n& = 0.\n\\end{align}", "d5d65d70799fa178b3d6d63c0fdd60e3": "G=CG_1C^{-1},", "d5d679990ea5787332d92473d6b57170": "\\pi^*F = \\mathrm{d}A", "d5d71ef9324eebc1e25210dd4331885e": "\n \\sin\\, \\psi = \\operatorname{sn} \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m \\end{array} \\right).\n", "d5d7668b35f4c5ce0a96f2c946e95927": "\\eta^{\\nu \\rho}=0", "d5d7918736c30a6f81f03acdab1262bd": "2x.", "d5d7ce987d60854f45a4dd8a0f4b280f": "X \\supset E_0 \\supset E_1 \\supset \\cdots", "d5d7ed430bc9904c6c7b4761d38bacef": "\n\\Delta = \\left[ \\hat{s} ~\\vdots~ \\hat{m} ~\\vdots~ \\hat{s} \\times \\hat{m} \\right].\n", "d5d82dae2ef40734da1c58e2a26790a6": "B\\preceq A", "d5d85280568eac958fef04a53143cdc1": "T(|A|) = |A|", "d5d86d712b9871e01d3a8ec699221864": "C>0", "d5d8900ec20bca6cd2a58165bd07e50e": " z^4+z^3+z^2+z+1 ", "d5d8947212bd41888029ffc2ec27f7a2": "N^{1/3} a_H \\simeq 0.2", "d5d8d0251305b49c9265618dc1fed2b0": "\\delta^*\\,\\!", "d5d8f68ec8b5f76e977fc10b782d3bed": "T^\\mu_\\mu=\\operatorname{Tr} \\begin{pmatrix} \\rho_0 c^2 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 0 \\end{pmatrix}=\\rho_0 c^2~", "d5d9082a89ea2314ac6c5e6b6d6f859b": "q_1:=1+m_1+m_2\\,\\!", "d5d94a8e5c89ca794c9300988328ff04": "\\mathbb{R}^{n^2}", "d5d968e01c11b9dc1f869df2cdddf896": "E\\left(t\\right)", "d5d9871a9518a8e2a822e3a6c4a8cd7f": "\\gamma = { 1 \\over \\sqrt{1 - {v^2 \\over c^2}} }", "d5d990db4804c1b1320935960d18fb5a": "\\ \\langle \\varepsilon\\rangle =\\frac{\\sigma^2 - \\langle I\\rangle}{\\langle I\\rangle} = \\sum_i f_i \\varepsilon_i", "d5d9cb70000776ecb43c2c196a62e3e0": "p_{1} =\\frac{p_{1}\\cdot P}{P^{2}}P+p\\,,", "d5da1ac64f85b0fc79c26026918ce118": "M_{\\ell m}'=\\int d^3\\mathbf{x'} r'^\\ell Y_{\\ell m}^*(\\theta', \\phi')\\mathbf{\\nabla}\\cdot\\mathbf{M}(\\mathbf{x'})", "d5da4956aac26a353e73b09ebb0cabe5": " d^{n+1} \\circ d^n = 0 ", "d5daa1690d89896c4f15574183218562": "\\ R(r) = A(l,\\alpha) r^l e^{-\\alpha r}, ", "d5db01263fb9a7351f727597b021ae13": "A_M: M \\to J_1(M),", "d5db137ad6196c157e964e46bd41cbd1": "e = (1, -1)", "d5dba503dce8cd53db45bce64e77e211": "\\scriptstyle{|1\\rangle}", "d5dbdbdcabb3e45bfa9a01d931c7f76e": "v(P) = J_{\\varphi_0}(\\varphi_0^{-1}(P))\\cdot {\\bold v}_0(\\varphi_0^{-1}(P))\\qquad(1) ", "d5dc08f7289ba2c7e7c942b541636ba5": "q(n) = \\begin{vmatrix} ~1& ~ & ~&~&~&~&~&~&~1~\\\\\n -1& ~1& ~ & ~&~&~&~&~&~0~\\\\\n -1& -1& ~1& ~ & ~&~&~&~&-1~\\\\\n ~0& -1& -1& ~1 & ~ & ~&~&~&~0~\\\\\n ~0 & ~0 & -1& -1&~1 & ~&~&~&-1~\\\\\n ~1& ~0 & ~0& -1& -1&~1&~&~& ~0~\\\\\n ~0 & ~1& ~0 & ~0& -1& -1&~1 & ~ &~0~\\\\\n ~1 & ~0& ~1 & ~0&~0& -1& -1 &~&~0~\\\\ \n ~ \\vdots & ~&~&~&~&~& ~& \\ddots & ~\\vdots~ \\end{vmatrix}_{(n+1) \\times (n+1)} ,", "d5dca021d4ef58c2f64603c6c8a3a24a": "\nz\\,\\, = \\,\\,a\\,\\,x_1^\\alpha \\,x_2^\\beta \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left[ {x_1 \\,\\,x_2 } \\right]\\,\\, \\sim \\,\\,BVN\\left( {\\mu _1 ,\\,\\,\\mu _2 ,\\,\\,\\sigma _1^2 ,\\,\\,\\sigma _2^2 ,\\,\\,\\sigma _{1,2} } \\right)\\,\\,\\,\\,\\,\\,\\,\\alpha ,\\beta \\,\\,{\\rm constants}", "d5dd56323d2e38ff84c1510de6f2a969": "a,\\ ar,\\ ar^2,\\ ar^3,\\ ar^4,\\ \\ldots", "d5dd61f36cf4b7bc639431750b069259": "\n\\mathbf{v} = \\sum_{k} \\langle \\mathbf{v} \\mid \\mathbf{e}_{k} \\rangle \\tilde{\\mathbf{e}}_{k}\n", "d5dda3032672ccc217c3bd05a023ce0e": "\\int \\coth ax\\,dx = \\frac{1}{a}\\ln|\\sinh ax|+C\\,", "d5ddb2c4d0e60f8d437f1cf9d0d9bd73": " sup_{i} Cr \\lbrace A_{i} \\rbrace < 0.5", "d5ddb8d8725e5f6eb051ce2402d30ec6": " H(X) = \\log \\mathrm{B}(\\alpha) + (\\alpha_0-K)\\psi(\\alpha_0) - \\sum_{j=1}^K (\\alpha_j-1)\\psi(\\alpha_j) ", "d5dddc44c33ef28a06f84250e170642f": "n h/s", "d5de09c29cbd971b45e17d080e5e02bb": " \\beta=\\frac{Z_2}{Z_1+Z_2} ", "d5de4eb9c06f346943839f63a3429dcb": " U_{(\\pi_{s}, \\pi_{o})} = a*\\pi_{s} + b*\\pi_{o}", "d5de5d04ed7cb6bdfe97c6eb8440f5be": " O(h) ", "d5de743bef9e4d2ca2faf85b3ed357bf": "\\chi_\\text{mol} = M\\chi_v/\\rho", "d5dea3e41a60637840b7e4d7fe1e3ac7": "\\sqrt{\\frac{2}{7}}\\!\\,", "d5dec4fd0dae3840e599d633bbed656c": "g_{uc}(\\langle a \\rangle) = \\langle A \\rangle", "d5ded37c4122f67c8c2629b154563356": "H = {\\epsilon_{ijk} F_{ab}^k \\tilde{E}_i^a \\tilde{E}_j^b \\over \\sqrt{det (q)}} + 2 {\\beta^2 + 1 \\over \\beta^2} {(\\tilde{E}_i^a \\tilde{E}_j^b - \\tilde{E}_j^a \\tilde{E}_i^b) \\over \\sqrt{det (q)}} (A_a^i - \\Gamma_a^i) (A_b^j - \\Gamma_b^j) = H_E + H'", "d5dee72dcf755a5938ab788b7f1ee7a7": "\\{\\neg a\\}", "d5def2a57714ed2c0c5a4f5bc8bf5331": " \\textstyle v_0 ", "d5df04a948c55d19f8d85b49392c5ef8": "H_{p+q-d}(M)", "d5df1e24a6f507ea58416aa4e3dc160c": "\\theta\\mathbf{e}_{12} = i\\theta,", "d5df5431a0340f64ae7ad7b75037e29a": "\\tfrac{F_v}{F_m}", "d5df7eb8385f44909e769ab87af8565d": "\\kappa \\ll 1", "d5e043e6bed536de6c8aeb559667d796": "f \\in C(\\Omega)", "d5e064857ee831a331d96a29f7b7ff4d": "a_l", "d5e06516cfe9afefbdb99df69f219a0a": "\\mathbf{M} = \\mathbf{U}_n \\boldsymbol{\\Sigma_n} \\mathbf{V}^*", "d5e068283239cfdfe31789945f0571d2": "d(\\mathbf{v}, \\mathbf{w})= v_1w_1 + v_2w_2 + v_3w_3.", "d5e1285e12e23320b60a30509a59436e": "J_1(ka \\sin \\theta)=0", "d5e1547b32da699cf511f9f4de75b6f4": "2.11 \\times 10^{34} \\mathrm{cm}^{-2} \\mathrm{s}^{-1}", "d5e165b4df4e1716f8246591396dd2aa": "\\left(a \\rightarrow P\\right) \\Box \\left(b \\rightarrow Q\\right)", "d5e168d8886b23680219f332e0e63c1c": "f(\\cos\\theta)", "d5e190ceda87927d332d4b27c26c6bae": "d(C_n)=\\lceil n/2\\rceil", "d5e28780d4405b2c232c19f78f19420c": "c_8 = -8.14971 \\times 10^{-4}, \\,\\!", "d5e2ad18dc650feb58ff95c792c681f1": "SL(n, \\mathbb R)", "d5e2ad692ae00d11289195537c6ebf6f": "Z \\to D", "d5e2eba83ff081d23cbcfbdb93205cee": "\\mathfrak{a}_i + \\mathfrak{a}_j = (1)", "d5e305093bf26537e3bb37c0409dd8cd": " \nI_1 \\left( mr \\right)K_1 \\left( mr \\right)\n\\rightarrow\n{1\\over 2}\\;\\left( {1\\over mr}\\right)\n . ", "d5e389844e1720ee66696413746f5f73": "S=\\{v_1,v_2,\\ldots,v_k\\}", "d5e397a8598907d926c0493b48558ff4": " \\operatorname{E}(f(\\mathbf{Aa})) = 2pq, \\!", "d5e3cea7555f3c5e963ef3848e20ca57": " \\phi_{va2} (r) = \\frac{2 r}{r^2 + 1} ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{va2} (r) = 0", "d5e3f0db3d4d5a40aa01f0dff5bfd814": "\\scriptstyle\\overline{y}", "d5e438e6781979b6c47ef126f878fadc": "\\mathrm {DOF} \\approx \\frac {2 N c (1 + m/P)}{m^2}\\,.", "d5e46d32ee25092be3cca8bed72bc28e": "\n \\sigma_1^2 + \\cfrac{F+R_0 G}{G(1+R_0)}~\\sigma_2^2 - \\cfrac{2R_0}{1+R_0}~\\sigma_1\\sigma_2 = \\cfrac{1}{(1+R_0)G}~.\n", "d5e4949779154633983efe55b75d23cb": "~-3 \\le x \\le 3~", "d5e4d5706d65d9a7d81d52913db6c5f4": "\\mathrm{d}s^2={1\\over k^2 y^2}(\\mathrm{d}y^2+\\eta_{\\mu\\nu}\\,\\mathrm{d}x^\\mu\\, \\mathrm{d}x^\\nu)", "d5e4e8a65c09602fbbc7574d2ff5c3db": "\\tfrac{26}{11} \\simeq 2.36", "d5e4e9805973b7c86e24f6d99ef38828": "\\frac{R}{G} \\gg \\frac{L}{C}", "d5e5594c22f00b5baee611498036a866": " \\textstyle x ", "d5e5c1d2f183e4c4a13420c31507c988": "f_n(x) > 0 \\text{ for } x = \\cos \\frac{2k\\pi}{n} \\text{ where } 0 \\le 2k \\le n", "d5e5e3d98e8e914ae209b90299292fa0": "VMPP_L", "d5e616274181dfe18003ad37beb3cee1": "d^\\prime = V_{ud} d + V_{us} s, ", "d5e64fa5b0b1292419973bbd98430d85": "W^{A}(0,0)=0=W^{B}(0,0)", "d5e67d7c5ff16737ed63531d4f37e332": " \\int_S F \\, dS ", "d5e6d862e3420884d42b6953b5bf243b": "Y = a_n (X_1 + X_2 + \\cdots + X_n) + b_n", "d5e6e77a3fb1b3a4708d27f4555764c7": "{d\\tau}^2 = - g_{\\mu \\nu} \\, dx^\\mu \\, dx^\\nu \\,", "d5e701d6279aa433076a8d5422c0bf9e": "F = \\frac{a^2}{L\\lambda} \\ll 1", "d5e73f231b36fd58e3156b08bdf9711e": "ct'=\\gamma \\left(ct - \\frac{v}{c} x\\right), ", "d5e757ee527bfacad2f77befc00861c0": "m>n\\!", "d5e764e87becbeaae42e12626c55b020": "B = -12", "d5e76d8165fc226b73814fd88f9d78dc": "F=\\frac{\\left(\\frac{\\text{RSS}_1 - \\text{RSS}_2 }{p_2 - p_1}\\right)}{\\left(\\frac{\\text{RSS}_2}{n - p_2}\\right)} ,", "d5e77b3c9d52b85439ee98cb761e2294": "Pot(G)", "d5e81955d9081c26d93800761b4891bf": "Eq \\; 2", "d5e82b71e11a742eb378a3a868d77696": "\\displaystyle (1-mxt+t^m)^{-\\lambda}=\\sum^\\infty\n_{n=0}\\pi^\\lambda_{n,m}(x)t^n", "d5e8d324f497af313cb76ad4093d7c49": "\\boldsymbol{\\mathbf{P}} = m_0 \\boldsymbol{\\mathbf{U}} = (E/c, \\mathbf{p})", "d5e933451601dd50896b7027bdc9c2ab": "\n\\begin{align}\n\\sigma_{1}(12) & = 1^1 + 2^1 + 3^1 + 4^1 + 6^1 + 12^1 \\\\\n& = 1 + 2 + 3 + 4 + 6 + 12 = 28,\n\\end{align}\n", "d5e93554f6cf180c28cd9c01565e1878": "D(\\omega )=\\frac{1}{1-j\\omega \\tau }", "d5e9a92f7f96a6527e2f23b897d26fb7": "\\operatorname{Q}_X(p)=\\frac{1}{\\operatorname{Q}_Y(1-p)}", "d5e9e730754105236c07af228d8361b3": "[f,\\alpha g]=\\overline{\\alpha}[f,g] \\quad \\forall \\alpha\\in\\mathbb{C},\\ \\forall f,g\\in V,", "d5eaa77248d9ed5d911180b207a9c928": "\\sigma^2 dt ", "d5ead5459b25e020b241429bbe9e0846": "\\mathcal{B} (X,\\Delta )", "d5eb02bf6d25fff3b7183808ecb278b4": "\\scriptstyle p = x^2 + y^2", "d5eb3b0746d18ee878b17489eec4c2a8": "\\frac{L_{2}}{L_{1}}=\\frac{L'_{2}}{\\phi}\\left/\\frac{L'_{1}}{\\gamma\\phi}\\right.=\\gamma", "d5ebd641d76e87f0e473cffcafbbc608": "X\\oplus X", "d5ebe70001f9d48aaadbf8db9d90a631": " \\| \\mathbf{v} \\times \\mathbf{u} \\| = \\|v\\| \\cdot \\| u \\| \\cdot \\sin{(\\mathbf{v} , \\mathbf{u})} ", "d5ebf344e329bcd6223db9ff192c9ae0": " \\frac{d}{d\\theta} \\tan\\theta = \\sec^2\\theta ", "d5ec2c0dbf966e82c4a6a65ba6b4224a": "I-V", "d5ec3c414a73c82f3cd380732f081a85": "= (0,1,1,1)^{\\otimes |U|} \\left(\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}^{\\otimes 3} + \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}^{\\otimes 3}\\right)^{\\otimes |V|}", "d5ec8648159fd9d34904440f32fecb5d": "\\frac{d\\,\\operatorname{arsinh}\\, x}{dx} = \\frac{d \\theta}{d \\sinh \\theta} = \\frac{1} {\\cosh \\theta} = \\frac{1} {\\sqrt{1+\\sinh^2 \\theta}} = \\frac{1}{\\sqrt{1+x^2}}", "d5eca3786427da4f80b1b5b26109866f": "T \\ge \\left| S \\right| - \\varepsilon N \\ge (1-R)N - \\varepsilon N = (1-R-\\varepsilon)N", "d5ecb57971ff2a53b3263722c4a76a1c": "a_z = b_x c_y - b_y c_x. \\, ", "d5ecc82caea60995a3d106a11ed0c811": "u\\big|_{x=0}=u\\big|_{x=L}=0 ", "d5eceaf928c08183884ab46b4037d8d1": "t_{sound} = \\frac{R}{c_s} \\simeq (5 \\times 10^5 \\mbox{ yr}) \\left(\\frac{R}{0.1 \\mbox{ pc}}\\right) \\left(\\frac{c_s}{0.2 \\mbox{ km s}^{-1}}\\right)^{-1}", "d5ed0157e2af74089f58bd68471333a8": "M+L+H \\rightleftharpoons MLH:\\log \\beta_{111} =\\log \\left(\\frac{[MLH]}{[M][L][H]} \\right)", "d5ed7e18b33c07a484251cea4d95de50": "T = \\frac{|\\vec J_\\mathrm{incid}|}{|\\vec J_\\mathrm{trans}|} ", "d5edb9c63d68fab605550a59a50049b5": " \\text{cat}(M - {p})=\\text{cat}(M) -1 , ", "d5edbcbb8c01a681e5e96eacd475bbcc": "A_n(x)=\\frac{x^{2n+1}}{2^nn!}\\int_0^1(1-z^2)^n\\cos(xz)\\,dz.", "d5edc6f81f4b359f12bed0313f193a73": "E_1=\\frac{\\pi^2\\hbar^2}{2mL^2}", "d5edf45e3b57ad85d1fd07970b676394": "(1 + x)e^x", "d5edf98ae7c0db8d942a1a8dbfa75a90": "\\mathrm{M} (A + E)", "d5ee0c98f46866438fe6e074f2496c13": "R[T],", "d5ee3876ca83988e57103fe189af4499": "\\Bigg\\lbrack \\frac{T_2}{T_1} \\Bigg\\rbrack", "d5eea2300dc8a5a18c839c747946d3ce": "\\mbox{Order}=1-{C_O\\over C_I}.\\,", "d5eec8de836d2951c0a1ee6e1a34a26c": "\\frac{1}{k}\\sum_{j=1}^{k} \\left|1-\\frac{1+\\alpha}{j}\\right|^2=\n1+\\frac{1}{k}\\left(- 2(1+\\mathrm{Re}\\,\\alpha) \\sum_{j=1}^{k}\\frac{1}{j}+|1+\\alpha|^2\\sum_{j=1}^{k}\\frac{1}{j^2}\\right)\\ .", "d5ef03bcc6580a047dd29cdecd1c7923": "y_p \\cdot p + y_q \\cdot q = 1", "d5ef14b54ecddbec6c2bd7724395d515": "M_{}^{}", "d5ef4e02400b4ce77fc5351596bf4b66": "z_2 = -1.4455692\\ldots + i 0.6992608\\ldots", "d5ef67a69f7babb8336459c47c5f7d38": "\\left|\\arg {zf^\\prime(z)\\over f(z)}\\right| \\le {\\pi\\over 2}.", "d5ef69ec014e9d4b4d786b25f9e72896": "I_e = \\frac {m h^2}{3}+\\frac {m w^2}{12}\\,\\!", "d5ef743aa5ee9aa01a37ac51f1864630": "\\overline{\\mathcal{M}}_{g}", "d5efafe84d6dba91664231d9a9af8caa": "q_1(F_S(s))=q_2(F_S(s-\\ell)", "d5efe00608e23f2535cecb169d0f17a0": " \\le 1 ", "d5f0345786707b4dbafbf81bc30d50b1": "K=\\frac{pq}{2}", "d5f03e1c75d019eed6d54673a445f4d9": "X \\perp\\!\\!\\!\\perp Y \\mid K \\Rightarrow Y \\perp\\!\\!\\!\\perp X \\mid K", "d5f05ca55d6da763f8163cdec264f158": "{\\frac {|BD|} {|CD|}}= {\\frac {|BB_1|}{|CC_1|}}=\\frac {|AB|\\sin \\angle BAD}{|AC|\\sin \\angle CAD}.", "d5f0f0a16765c6d11ebeb0945beb5dc2": "x\\ f = \\lambda y.f\\ (y\\ y) ", "d5f0f480c880dd61eb2149b908907e49": " \\mathbf{D} = \\varepsilon_0 \\mathbf{E} + \\chi \\varepsilon_0 \\mathbf{E} \n= \\varepsilon_0 (1 + \\chi) \\mathbf{E} = \\varepsilon \\mathbf{E} ", "d5f165fe13ebb9e9bb6004fb6a323483": " G(z) = z^n\n\\frac{n-1}{n-z}\n\\frac{n-2}{n-2z}\n\\frac{n-3}{n-3z}\n\\cdots\n\\frac{n-(n-1)}{n-(n-1)z}\n", "d5f19f817a69ed3abb41f686f172d9a7": "X\\subseteq \\bar{X}", "d5f1cf9919a76dfbadff4647d0c544ed": "\\mu_{k,i+1}", "d5f2625bb5679c6aba2a3b28adebc02f": "H\\mathbb{Q}", "d5f2669c1ca820609a5c82ac977a12b5": "\\sum_{\\sigma}\\sum_{n_1 > n_2 > \\cdots > n_k \\geq1} \\frac{1}{{n^{i_1}}_{\\sigma(1)}{n^{i_2}}_{\\sigma(2)} \\cdots {n^{i_k}}_{\\sigma(k)} }", "d5f27f6ad7c1dffe6032b783e885d1c1": "I_i = {g_i}(V_m - V_i) \\;", "d5f3022c1a6d0a7cd7ab33aa7061206d": "\\Delta(C_{in}^i(c_i^1),C_{in}^i(c_i^2)) \\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k", "d5f3205d8db630fb41d5ac5cd5613d1c": "(1+k)^n", "d5f38fc80598d804c3146f35bc47a05b": "r(0,0)", "d5f3e694c009f127bbb52dc543a0e5f7": "\\left(\\bigcup_{\\alpha\\in\\mathcal{I}}A_\\alpha\\right)\\triangle\\left(\\bigcup_{\\alpha\\in\\mathcal{I}}B_\\alpha\\right)\\subseteq\\bigcup_{\\alpha\\in\\mathcal{I}}\\left(A_\\alpha\\triangle B_\\alpha\\right)", "d5f42de4dd214d4741489de099c276e0": "T'=( t_1, \\ldots, t_k ) \\in T^k", "d5f4583bce1dc74674e22c132e8a1150": "\\boldsymbol \\sigma", "d5f4933ef705f9caa2385333c6a76777": "\\frac{H_n}{H_{n-1}}\\, ", "d5f4a0fd44e83b6e1fc6fa26bbd21638": " c - \\varepsilon < \\frac{a_n}{b_n} < c + \\varepsilon ", "d5f4c00690261ea5ae24ba94c3711803": "\\hat{\\mathbf{X}}", "d5f5224453b4f7d93298e764bd9e25ba": " \\mathbf{a} \\cdot \\mathbf{b} = \\overline{\\mathbf{b} \\cdot \\mathbf{a}}. ", "d5f5f7fd30a9499e4d96689e1ce26fe2": "\nP_{\\nu_b\\rightarrow\\nu_a}\\simeq L^2|(h_\\text{eff})_{ab}|^2,\\quad a\\neq b.\n", "d5f616cf1b6c50f36dde230bd20f6b60": "V = \\{x \\in \\{0, 1\\} : (x = 1) \\vee P\\}.", "d5f62393131296219d9c89d89b065726": "\\frac{1}{\\frac{1}{r}-\\frac{1}{R}}", "d5f63275acc2df058dd9bff52238a2f9": "F>0", "d5f6dda017b06088d3ba297c3e6182ac": "\n T(h,a)=\\frac{1}{2\\pi}\\int_{0}^{a} \\frac{e^{-\\frac{1}{2} h^2 (1+x^2)}}{1+x^2} dx \\quad \\left(-\\infty < h, a < +\\infty\\right).\n", "d5f6dff8261a6af710d06e8a64007274": "u_x^+", "d5f6e701f0601f0c83079c089133c207": "x_l", "d5f71b11bed7000eb7a99191fe577afa": "\\eta= \\eta_p \\eta_c", "d5f739754a27efc6c698d1b50028c2c8": "M^{0.97}", "d5f73cf23d561571ed4539f1bebe42f0": "\\phi(y_1,y_2,\\ldots,y_m)", "d5f7d7c3c951692a1df647c7a355cae0": "{\\mathcal N}", "d5f7fe28017df21daaab6808d653b8e9": "a \\cdot b := \\frac{1}{2}(ab + ba) = \\frac{1}{2}((a+b)^2 - a^2 - b^2) ,", "d5f8bf3ec26e1927bbbf5e223f85976f": "{\\mathcal O}_P", "d5f90729ee2e4b82f2456a832fab2e4a": "\\Delta m_1", "d5f93c1dd477e2000ed33755a3b42517": " d_H(C) \\le 1 + k^2.", "d5f970369906bbb48380b8f06a52c72b": "\\ h^\\mathrm{H} R_v h = 1 ", "d5f9af8b79f933140ccf8017631f6b68": "g^{\\alpha \\beta}g_{\\beta 0} + g^{\\alpha 0}g_{00} = 0,\\,", "d5f9c7b57bcfaf3b854ef628dd5686d1": "\\operatorname{Re}\\,(z) = \\tfrac{1}{2}(z+\\bar{z}), \\,", "d5f9ca3f8482c820e2e046cd020fa161": " M = m - 5 ((\\log_{10}{D_L}) - 1)\\!\\,", "d5fa1619012fc14a1d3bbebf5347d0db": "\\nabla^2\\phi(x)=-\\rho(x)", "d5fa3c892e5b11fe77b3fa7a019d5b57": " R = f_s \\log_2(M). \\, ", "d5fa4bcbf4cdc8faf05a677ef54c7583": "T^{\\mathrm {rotational}} = \\sum_{i=1}^{d^{\\mathrm {rotational}}} \\frac{\\hat{L}_i^2}{2\\Theta_{ii}}", "d5fa59af4742c1583050166bb5351098": "\\overrightarrow{G_y}", "d5fa8677116d41f70cb5950bee1f490a": "L_4(2) \\cong A_8.", "d5faaece197292902eb101e1b968851f": "{{du} \\over {dt}}=u-v+H(u-\\theta),", "d5fad28c63e7352671bd47cd12df740a": " \\alpha_P(S)=\\sum_{T\\subseteq S}\\beta_P(T). ", "d5fad53ebf4662d6b53174c41b703c74": "G_{\\mu\\nu}=8\\pi\\,GT_{\\mu\\nu} \\, ", "d5fb827a34a2fd1e42757a7f1afe3ace": "\\alpha U = 4 s_w^2 \\left[ \\Pi_{WW}^{\\prime}(0) - c_w^2 \\Pi_{ZZ}^{\\prime}(0) - 2 s_w c_w \\Pi_{Z \\gamma}^{\\prime}(0) - s_w^2 \\Pi_{\\gamma\\gamma}^{\\prime}(0) \\right]", "d5fba62cf5ce69207996a12819f52e0b": "X=Y+1", "d5fbd2ab5c2e8e9f1a729fdc56190519": "\\alpha(t) = (\\alpha_1(t), \\cdots, \\alpha_n(t))'", "d5fc1fccacc7dfb78c9cfa007bdc8d66": "\\rho_{dc}", "d5fc76b3eaba6cde0ee42c4951e6fa80": "W(n)=0", "d5fc7dc7430bea87fcc7dafd4c9a98bf": "C[W^{-1}]", "d5fcf4313b0c990f246a27377dc9b509": "\\gamma_{\\mathrm{rad}} = N \\gamma_{\\mathrm{rad},0}", "d5fd2053a37d4702c6b15c705ec37d6c": "\\ell_{i,j}:=\n\\begin{cases}\n1 & \\mbox{if}\\ i = j\\ \\mbox{and}\\ \\deg(v_i) \\neq 0\\\\\n-\\frac{1}{\\sqrt{\\deg(v_i)\\deg(v_j)}} & \\mbox{if}\\ i \\neq j\\ \\mbox{and}\\ v_i \\mbox{ is adjacent to } v_j \\\\\n0 & \\mbox{otherwise}.\n\\end{cases}\n", "d5fd96bc3303605fd3ecffde0198643f": "v_1 = v", "d5fde97cebba2a43f3ebe604169c9fe4": "x,y \\in \\mathbb{R}", "d5fe81e274c7ba68512a291181bea144": "A = Q \\Lambda Q^{\\top} ", "d5fe83b9e6bbf4a97b3acd7d644c8958": "s=s'\\left[{1-\\frac{1}{2s'}[2(xx'+yy')+(x'^2+y'^2)]+ \\frac{1}{2s'}[2(xx'+yy')+(x'^2+y'^2)]^2+ \\cdots}\\right]", "d5fe9205ea2f303d75d3b73f80f42dde": "\\nabla^2_{L}\\Phi_{N}=4\\pi\\varrho_{N}", "d5feab54d097911b3f9018862939e575": "L_1 \\le x \\le L", "d5fede3c1b116b8ce666486c64264e26": "\\scriptstyle \\frac{\\Delta y}{\\Delta x}", "d5ff2e2d5230abe44697f63bc545617b": "\\mathrm{A_{1}}", "d5ff836dc0e80a1325ad35929ebde38f": " z=x+iy ", "d5ffbe6bd76e587ba99877721ad0b57f": " i \\in [n]", "d6001bf14ff9a00a97445179a00f8528": "c_2 = -\\pm 1 \\times c_1\\,", "d6002524fb47b58258ed1ba5e6e7782e": "\\frac{1}{z - T} = \\frac{1}{z} \\cdot \\frac{1}{1 - \\frac{T}{z}}", "d6005d2e29461fc56d095a7a163ca740": "\n \\cfrac{\\Gamma \\vdash \\Delta_1, A, B, \\Delta_2}{\\Gamma \\vdash \\Delta_1, B, A, \\Delta_2} \\quad (\\mathit{PR})\n ", "d600682a9f59e4e76324715aacb0c896": "(c_{i-1,0}/c_{i,0})\\mu", "d600e6f67df6bfc6a5ddd1969cb205dd": "(x,y,z,\\dots)", "d6015a4479a33174e71d930726b1b005": "m = l+1", "d60164311702623153064d6cdd621799": "D\\{\\mathcal J\\}=\\{\\emptyset,\\{1\\},\\{2,3,4\\},\\Omega\\}", "d60178125a9e833d234cc23350b6dc20": " g = (1\\ 2\\ 5)(3\\ 4)=\\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\\\ 2 & 5 & 4 & 3 & 1\\end{pmatrix}.", "d6020ac736be310f904acb18d20b7a83": "\\Psi_P=L_Pi_P-Mi_S", "d602662a35cb325a2eac9227234795de": "s * m' = h + xr", "d6033127fccd1bcaebec915a715a7a6c": "a \\neq b", "d6033b3eb044f2626b4bcb8fb48871e6": " \\frac {a} {b} ", "d603d124f054db6fee055d631bc69348": "au+bv=g\\quad\\text{(Bézout's identity)}", "d60401795cec604063c340deb9d8b4ce": "\\Vert S \\Vert = \\Vert R \\Vert.", "d60447f1432b936bc6dc5a509ba6ef44": "\\vec{H}=-\\nabla \\Phi_{M}", "d604576b1ff60bdef511a4082ebdce80": "{\\Pi}", "d604ce176073be89156ff08b9fd5c04b": "\\Gamma = \\frac{q^2}{4 \\pi\\epsilon_0 k_B T a},", "d604e145bbeb9fd4a8064bfb72820303": "\\nabla\\left(\\kappa\\nabla T\\right)+g=\\rho C\\frac{\\partial T}{\\partial t}", "d604e9a1df429b97e570c89371e82ed1": "T = \\mathrm{log}(1 + e^{-x})", "d604fb13a913bd502d701499f2184781": "\\frac{\\overline{e^c}\\overline{u^c}qq}{\\Lambda^2}", "d6054b15cf4fb0b1cb87e17fa669a1e0": "\\Delta v = v_\\text{r} - v_\\text{s} \\,", "d605c70c2cf75a87dc41054fcf1ade4a": "1 \\times 1 \\times 2", "d605ecedaeac265f5edb28147f53c0d1": "\\left( x^{\\ast\n}(t),y^{\\ast }\\left( t\\right) \\right) ", "d60623c27baaadd692fd56548b61c6fe": "\\Bbb{R}^{\\Bbb{N}}/{\\mathbf M}", "d6063a87cd553bd461dc35e38261060f": "\\pi s^2 \\over 4", "d6068292d7a799c4a76d9292318382fc": "\\det\\begin{bmatrix}1 & 1 & 2 & 5 \\\\ 1 & 2 & 5 & 14 \\\\ 2 & 5 & 14 & 42 \\\\ 5 & 14 & 42 & 132\\end{bmatrix} = 1.", "d60686f4106fba69e66e797694669b22": "\nf(n) = f_0 + n\\,f_r\n", "d6069e08330a86a164b76dc909d3718c": " \\text{area}(\\Delta)={{bc-ad}\\over 2} \\ .", "d606ca5099766730270d26a675b04a1e": " \\begin{align}\ny_2 &= y_1 + hf(y_1) = 2 + 1 \\cdot 2 = 4, \\\\\ny_3 &= y_2 + hf(y_2) = 4 + 1 \\cdot 4 = 8, \\\\\ny_4 &= y_3 + hf(y_3) = 8 + 1 \\cdot 8 = 16.\n\\end{align} ", "d606e421d9d305b86b4327e3c9b212bb": " S(r') = S_0 \\exp \\left [ -2 \\left( \\frac{r'}{R} \\right )^2 \\right]. \\qquad(5)", "d606f1d7e6b4087917726bcced7caaa1": "\n a_{\\rm T} = \\frac{\\eta_{\\rm T}}{\\eta_{\\rm{T0}}}\n ", "d60740fa2fe703655f215e92bbe0c8b6": "N^{}_{}", "d6075243a320b5c713266611ce508b3c": "E_d = \\frac{\\frac{P_1 + P_2}{2}}{\\frac{Q_{d_1} + Q_{d_2}}{2}}\\times\\frac{\\Delta Q_d}{\\Delta P} = \\frac{P_1 + P_2}{Q_{d_1} + Q_{d_2}}\\times\\frac{\\Delta Q_d}{\\Delta P}", "d6077a3e5aa130fa97835b716b06cac6": " X_k = \\sum_{n_2=0}^{N/2-1} x_{2n_2} \\omega_{N/2}^{n_2 k} \n+ \\omega_N^k \\sum_{n_4=0}^{N/4-1} x_{4n_4+1} \\omega_{N/4}^{n_4 k}\n+ \\omega_N^{3k} \\sum_{n_4=0}^{N/4-1} x_{4n_4+3} \\omega_{N/4}^{n_4 k}\n", "d607e3e2bb4ec4a7900aa9c916ad1081": "\\mathbf{P} \\left[ \\sup_{0 \\leq t \\leq T} X_{t} \\geq C \\right] \\leq \\frac{\\mathbf{E} \\left[ X_{T}^{p} \\right]}{C^{p}}.", "d6081ccaad4ad0a7dde76e4100c26ab7": "\\lambda_2 \\,\\!", "d60821c32115b5b623ad6cfb52a3941c": "\nR_\\beta=\\frac{\\operatorname{var}[\\beta]}{\\operatorname{var}[\\beta]+\\operatorname{var}[\\epsilon_\\beta]}=\\frac{\\operatorname{var}[\\hat{\\beta}]-\\operatorname{var}[\\epsilon_\\beta]}{\\operatorname{var}[\\hat{\\beta}]},\n", "d60836b74478d5d7b5df546f2e9186f7": "T_1^{(\\mathbf n)}=\\sigma_1n_1, \\qquad T_2^{(\\mathbf n)}=\\sigma_2n_2, \\qquad T_3^{(\\mathbf n)}=\\sigma_3n_3\\,\\!", "d608650a18cb1d3d43c7788d30387b15": " r_1", "d608fc162871af039ed2ca4a94b4ee97": " \\cdots \\rightarrow f^{-1}(g^{-1}(a)) \\rightarrow g^{-1}(a) \\rightarrow a \\rightarrow f(a) \\rightarrow g(f(a)) \\rightarrow \\cdots ", "d60998be2a9c79223144033c0625ee93": "\\sigma^{2}_x", "d609db0cfcea78c49a475b2746180350": "U_f = \\inf\\{U_{f,P} \\colon P \\text{ is a partition of } [a,b]\\} . \\,\\!", "d60a6f9eace72cb722c0a7b841ad07e6": "u_{mf} = \\left [\\frac{(\\rho_p - \\rho)gd_p^2 s^3}{1.75\\rho}\\right ]^\\frac{1}{2}", "d60a83bc21ed7e4b13ac3585992600ba": "\\cos A : \\cos B : \\cos C", "d60ab4b146f8774dc29d88759855c570": "\\tilde{\\nu}_X \\colon X \\rightarrow BO", "d60abfdf5d486c2212dce9f416f5b1ee": "N \\approx \\frac f v \\frac { v_{\\mathrm N} - v_{\\mathrm F} } { 2 c } \\,.", "d60adf3ddfce3080a0ffa77eeae3e2fc": "t_{su}", "d60b1de27afacbe66bda8b00ed8116e2": "U_+(F)(w)=\\frac{1}{\\pi} \\iint_{\\mathbf C} F(z) e^{\\frac{1}{2} w\\overline{z}^2} e^{-|z|^2} \\, dx dy,", "d60b395645b16b8b64e19d9a741fc6d9": "N_s", "d60b8e5d0a5a579b5386cd275efaaffe": " c = \\sum_{e\\in E} w(e) e\\ ", "d60b975167041ddf3ffc7938add3eff7": "Z(x_1)", "d60b9ac96f488629b0f56711c115a3d1": "\\epsilon_\\mathrm{irreversible} = t_1 \\frac{\\sigma_0} \\eta. ", "d60bb544a90ef56df6b906198acbe942": "\nP(u)du=\\frac{1}{\\pi}\\,\\frac{du}{\\sqrt{1-u^2}}\n", "d60c11b12acc70e1ca6c4576174cf463": "=2^2\\cdot521\\cdot829", "d60c5617d41b0a87b2661df266a7e5b0": "\\begin{align}\\nabla(FG) &= e^i\\partial_i(FG) \\\\\n&= e^i((\\partial_iF)G+F(\\partial_iG)) \\\\\n&= e^i(\\partial_iF)G+e^iF(\\partial_iG) \\end{align}", "d60cb730789608ab3fcddb130f74ec25": "\\ln \\ R", "d60cbf86e10b3f9fb70a69e72999113b": "C_{theo} = f(\\sigma_\\bar{C}, \\cdot) = $2.0004 \\,", "d60d26290af6485b671e2c9a22b3b461": "\\frac{v_{\\text{in}}-v_{\\text{out}} \\left( \\frac{Z_2}{Z_4}+1 \\right)}{Z_1}=\\frac{v_{\\text{out}} \\left( \\frac{Z_2}{Z_4}+1 \\right)-v_{\\text{out}}}{Z_3}+\\frac{v_{\\text{out}} \\left( \\frac{Z_2}{Z_4}+1 \\right)-v_{\\text{out}}}{Z_2}.", "d60d3e48f65249aa286e3ebbeec090b7": "\\int _{\\mathbf{R}^d}\\,\\nabla_x^2\\mathbf{1}_{x\\in D}\\,f(x)\\;dx+ \\int_{\\mathbf{R}^d}\\mathbf{1}_{x\\in D}\\,\\nabla_x^2 f(x)\\;dx =-2 \\int _{\\mathbf{R}^d} \\nabla_x \\mathbf{1}_{x\\in D}\\cdot \\nabla_x f(x)\\;dx.", "d60da28ba87ce5e2a7a0429917fbe7ab": " {1 \\over 4\\pi\\varepsilon_0}\\frac{(2.1 \\times 10^{8} C)^2}{(1 m)^2} = 4.1 \\times 10^{26} N.", "d60dfd549421716dead4653f8dccaf96": " s \\in \\mathcal{R}^n ", "d60e28a0bf349689b8e54c5295c5ebfd": "\\boldsymbol{\\tau}=\\frac{\\mathrm{d}\\mathbf{L}}{\\mathrm{d}t}", "d60e4ef14f1acb209f4ab7f7e7d915a9": "n=5, t=2", "d60e5a0d94239ddbc1dd22f7c7a3cf6b": "|\\mathcal V|_S=F", "d60e6fbe53f766ebb4db249783f90d77": "\\displaystyle p_n(x;a,b;q) = {}_2\\phi_1(q^{-n},abq^{n+1};aq;q,xq) ", "d60eb2b403fb148db55faf8bbf84d089": "\\ - \\log_{10}(\\gamma) = \\frac{A|z_+z_-|\\sqrt{\\mu}}{1 + Ba\\sqrt{\\mu}} \\,", "d60ee41ebc48942e31cc705f4ae9a6ad": " 0.8 < M < 1.2 ", "d60ee4d871402131fee3061e7fb7ee1f": "n^{5/2}", "d60efb35eb8a33669014adc9d23e5ad1": "3^{7{,}625{,}597{,}484{,}987}", "d60efc041745ade281a1298c4a55ee34": "1/(1-c^2).", "d60f08b1a759a2521b16516e3c5786fa": "100+80+50=230", "d60f0d1b9471d0df75288e885467d694": "\\|x + y\\| \\le \\|x\\|", "d60f1846aae3de2496135976f9a7d138": " r_0, r_1, r_2, r_3, \\ldots , r_n ", "d60f28f93ea436e0f013daeef9729f29": "\\mathbf{\\bar{n} \\tfrac{2}{m}}", "d60f42b24eaf177089e5e495c2f3caa1": "\\lambda^{'}_{\\beta}|{\\Phi^{'[{\\it{JC}}]}_{\\beta}}\\rangle", "d60f7258bf3ffc152882dc1aeb8370c8": " \\mathrm{d} \\mathbf{A} = \\mathbf{\\hat{n}}\\mathrm{d}A \\,\\!", "d60f8362d9c710feaa78c54d3d4aa0fa": "\\mathbf{M}'", "d6100f9ec596a6640014cc791c336f50": " {\\rm {}}_{}^{2}\\Sigma_{\\rm u}^{+}", "d610745e58f6dd6b406a3aa9246f9d92": " \\left(\\frac{N_1}{N_2}\\right)^3", "d610949b58de7a350b98ab9b3309f5f3": "\\ell k\\,", "d610e03406b0a19b3d4a4650e29c27e0": "z_M = \\frac{N}{I_M} = \\frac{N_m}{I_{Mm}}", "d6110e9f51c0543e1a55a68295e4d0dd": "\\hat{a}_i^\\bullet \\,\\hat{a}_j^{\\dagger\\bullet}= \\hat{a}_i\\, \\hat{a}_j^\\dagger \\,- \\mathopen{:}\\,\\hat{a}_i\\,\\hat{a}_j^\\dagger \\,\\mathclose{:}\\, = \\delta_{ij}", "d6110efd35eb15fb7b4207d5d68512cb": "g_{ij}=e_i \\cdot e_j", "d61112d179fe34d72e271cb50d2d5c03": "A_1, A_2, \\ldots", "d6114cc547067cff44d69f36c85398bc": "\\sum _x \\sin^2 ax = \\frac{x}{2} + \\frac{1}{4} \\csc (a) \\sin (a-2 a x) + C \\, \\,,\\,\\,a\\ne \\frac{n\\pi}2", "d6115cac965d7132b8067ef76540f636": "PPxy \\leftrightarrow (Pxy \\and \\lnot Pyx).", "d611b54d56333cc9a8671295124c6cae": "y=\\dfrac{1}{u}", "d6121998780e26c0a71834d1cccc8b1f": "{r-1 \\choose 4}", "d61244c2892809c27621cebc702f27ad": "U(0,b,z)=1", "d612660af4f524ac6ac0aab80e101009": "ds^2=\\lambda^2(z,\\overline{z})\\, dz\\,d\\overline{z}", "d6127b079652678a8231c8bde3d8898f": "h^{(i)}(x)=f^{(i)}(x), \\qquad i=0,1.", "d61294ca4a2f30234deb3c6965d1636a": " u_0 < 0 ", "d6134c92f231140bbc395ada6921ab86": "\\lambda \\lambda 1 ((\\lambda 1 1) (\\lambda \\lambda \\lambda \\lambda 2 (4 4) (\\lambda \\lambda 3 2 (3 2 (2 (5 1 (2 1))))))) (\\lambda \\lambda 1) (\\lambda \\lambda \\lambda 1 (\\lambda 4 (\\lambda 4 (\\lambda 1 3 2)))) (\\lambda \\lambda \\lambda 1 (3 (\\lambda \\lambda 1)) 2) (\\lambda 1) 2", "d6134cf71a8fa71040cd9034efa1bd85": "\\scriptstyle P_S", "d61352a78bf188950a91c92d35b83697": "d \\,", "d6135935c443d67c175a9eeda2b2211e": " \\mathbb{R}^n, \\,", "d613992b6202940b6275c1e63a5bccc4": "\\frac b d-\\frac{\\lambda_1 a+\\lambda_2 b}{\\lambda_1 c+\\lambda_2 d }=\\lambda_1 {{bc-ad}\\over{d(\\lambda_1 c+\\lambda_2 d )}} ", "d61407b8237701493d886e1db71c207e": "F_{100\\%} = \\left(\\frac{1 + \\sqrt{5}}{2}\\right)^{0} = 1 \\,", "d6146533aa4f6edaba619e999b25f5fc": "v_{Bullet}", "d614a4b7ce23b27ef91ff259d6ca25f4": "F_V(t,T) = V(t)/P(t,S)", "d614df61912532dbb5c25df879ed564a": "\\frac{1}{\\pi (k - j)}", "d61520f9317008f77df420007139ddfc": "\n X^{VG}(t; \\sigma, \\nu, \\theta) \\;:=\\; \\Gamma(t; \\mu_p, \\mu_p^2\\,\\nu) - \\Gamma(t; \\mu_q, \\mu_q^2\\,\\nu)\n", "d61553a1ea1f78f218ab07ee4fc7be30": " K= \n\\begin{bmatrix}\nk_1 & k_2 & k_3 \\\\\nk_2 & k_4 & k_5 \\\\\nk_3 & k_5 & 1 \\\\\n\\end{bmatrix}\n", "d61585451223a349cc9aa9ff734bd6b6": "U=\\{(a,v)\\colon\\|a\\|=1, \\,\\|v\\|=1,\\, a\\cdot v=0\\}, ", "d615892ae70f35e621a1e238f61db9cd": "Y_{9}^{6}(\\theta,\\varphi)={1\\over 128}\\sqrt{40755\\over \\pi}\\cdot e^{6i\\varphi}\\cdot\\sin^{6}\\theta\\cdot(17\\cos^{3}\\theta-3\\cos\\theta)", "d615994994b7f1d7e76411ad9e5ca2b8": "\\frac{dr}{dt}", "d615d8d13bb312c0949b1926d0e69e31": "\\rho(\\vec{r})\\!", "d6164f14ed21453507c092899c7bdad2": "D_3 \\overline{MR}", "d6165650f5a8d09c3d67d59d7f639ba1": "\\pi/4\\sqrt{N}", "d6166528af95c1faea60046495628e17": "\\scriptstyle Q(t)", "d61690f5fa926628a97d1296aa05450c": " | \\Psi \\rangle = \\sum_{s_z} \\int\\limits_R d^3 \\, \\mathbf{r} \\Psi(\\mathbf{r},s_z) | \\mathbf{r}, s_z \\rangle ", "d616a500d710f4aea41169a219ab97e5": "\\forall x \\forall x_1 \\forall z_0 [Rxx_1 \\land Rxz_0 \\rightarrow \\exists z_1 (Rx_1z_1 \\land Rz_0z_1)]", "d616b33d342973626b2c24dae6091fc0": "g = s_1^{k_1} \\cdots s_m^{k_m},", "d6174da1baee99561a1a19edf4fa1607": "a = \\frac{-1}{\\rho} \\frac{dP}{dz} ", "d617848445f683126432e42806be8857": "\\omega^2>\\omega", "d6178a25f850f5b202dc456e411d61ee": "F(t)=\\sum_{n=1}^\\infty f(n) e^{-tn}.", "d617b71df5f57dad29fc2556b8617e8b": "P_{AA}(\\nu,\\kappa,\\pi) = \\left[\\pi_A\\left(\\pi_A + \\pi_G + (\\pi_C + \\pi_T)e^{-\\beta\\nu}\\right) + \\pi_G e^{-(1 + (\\pi_A + \\pi_G)(\\kappa - 1.0))\\beta\\nu}\\right]/(\\pi_A + \\pi_G) ", "d617c591b6c0610e9fb3a27702636ed8": "{x \\choose y}= \\frac{\\sin (y \\pi)}{\\sin(x \\pi)} {-y-1 \\choose -x-1}= \\frac{\\sin((x-y) \\pi)}{\\sin (x \\pi)} {y-x-1 \\choose y};", "d618310dbf4d1f2da7018eaccc46680a": "\\omega_0 = \\frac{1}{\\sqrt{LC}}", "d61839c3ae1b7417f093f82f84ac804f": "G = (g_{\\mu\\nu}) = \\begin{bmatrix} (1-\\frac{2GM}{rc^2}) & 0 & 0 & 0\\\\ 0 & -(1-\\frac{2GM}{r c^2})^{-1} & 0 & 0 \\\\ 0 & 0 & -r^2 & 0 \\\\ 0 & 0 & 0 & -r^2 \\sin^2 \\theta \\end{bmatrix}\\,", "d6185789ec9c074b94b1ed154831755a": "\\frac{\\kappa_0\\boldsymbol\\mu_0+n\\mathbf{\\bar{x}}}{\\kappa_0+n} ,\\, \\kappa_0+n,\\, \\nu_0+n ,\\,", "d618e728ad4f633d7442df33542a3aba": "L_{\\omega_1^{\\mathrm{CK}}}", "d618e786748e85f90ccfe27b85f35f89": "(a_0, a_1, a_2, \\ldots )", "d6194e303546ae1333181438a8af1d35": "P \\times \\sum_{i=1}^{N} 1_{\\text{index}(i) \\in \\text{Range}} \\times \\frac{1}{N}", "d61959af5d2a6055a99e8b93d486aaed": " \\vec{r} ", "d61983dc0c9a47f15cf87e30a3b88377": "X_i, X_j", "d61a516c5e3454c0b5c88c7c02c1a382": " d^2=R (R-2r) \\,", "d61a5419d2f9d9d3a422a0d854681aeb": "\\mathcal{F}_k(\\pi)", "d61a62cf510dc6d37678c6201174e751": " h(x) h(y) = \\int_K h(xky) \\,dk \\,\\,(x,y\\in G).", "d61abbaa3da2fee5f8a0aff99b37346c": "\\alpha[\\mathbf{f}A] = \\alpha[\\mathbf{f}]A.", "d61ac9a70564e1ca395e31265f28a773": " m \\times n ", "d61b076cb1236e3a032da58d69c305ce": "L \\supseteq K \\subseteq R", "d61b341fb215d2e75f1a5e5c55149288": "\\mathbf v = \\nabla \\times \\vec \\psi", "d61b8aecc0edf06e4a144f6df1ff1ae4": "\\bar{x} = \\frac{1}{N} \\sum_{n=1}^N x_n", "d61bd44b9b77b7edcb0dbbfd43a01c29": "\\int_a^b f(x)\\,dx \\le \\int_a^b g(x)\\,dx.", "d61c43c48d9ba7be5015537d728e2e94": "t\\left\\{\\begin{array}{l}p\\\\q,r\\end{array}\\right\\}", "d61c4a2037bf04281bf5741a96a6b363": "\nN=N_1+\\cdots+N_k", "d61d19cd5c01a28974b20735269cb752": "G(\\alpha',\\beta')", "d61d32cc4d5c5a09e54765ff08f6e1de": "||\\mu(A)||\\le |\\mu|(A)", "d61d7df8a0f1880157c8283cb0d6b31f": "\ni^{-s} \\,\\operatorname{Li}_s(e^{2\\pi i x}) + i^s \\,\\operatorname{Li}_s(e^{-2\\pi i x}) = {(2\\pi)^s \\over \\Gamma(s)} \\,\\zeta(1 \\!-\\! s, \\,x) \\,,\n", "d61dd30d2a6c7dfcd3e06fc11f66336b": "\\Psi^{n,K} (r,R) = \\Psi_K(R)\\psi_n(r)", "d61df2f07415268d6c65f8e38a7d487d": "X^+ = \\max(0,X)", "d61dfd5ebc25a813ce2f68c20df4872a": "e^{-w/t}", "d61e91c7e257f655ea0d40406ddddb31": "\\nabla^2\\phi-\\Lambda\\phi=4\\pi\\rho\\;", "d61e940636f6ffa20f725bc10aa9cbbf": "\\tilde x_i", "d61efdd41366af13229e55d8de456fb0": "\\left|\\psi_{AB}\\right\\rang =\\frac{1}{\\sqrt{2}} \\bigg(\\left|\\uparrow\\right\\rang_A \\left|\\downarrow\\right\\rang_B -\n\\left|\\downarrow\\right\\rang_A \\left|\\uparrow\\right\\rang_B \\bigg) ", "d61f64f8313343fe74cc321a6ec6d88f": "\\Leftarrow", "d61fe0a7c230031dbfbe462ce24c3cf1": "k(x,y) = \\overline{k(y,x)}", "d61fe9c1fb1647edc7d7f9fd4a494716": "L_{0, 1} = S_0 e^{rn/365} - X", "d61ffe9d05fd52150404cffc3394628b": "\\mathbf{u} = \\mathbf{x} - \\alpha\\mathbf{e}_1,", "d6201e38298800f418336e10edb9edd0": "\\hat{\\beta}\\approx \\tfrac{1}{2} + \\frac{\\hat{G}_{(1-X)}}{2(1-\\hat{G}_X-\\hat{G}_{(1-X)})} \\text{ if } \\hat{\\beta} > 1", "d6201ee4e622c5fbd7ecf2d25da0ed12": "\\dot{Q_t}=p(t).e^{At},\\dot{R_t}=p(t).e^{Bt}", "d62036578aeaa05db98dee6f10a0f011": "0 = g'(c) = f (c) - d. ", "d621b37b44bd86dac09e96302dd6f66d": "\n \\begin{align}\n _{(x)}\\Gamma_{ijk} & := \\frac{1}{2}\\left(\\frac{\\partial G_{ik}}{\\partial x^j} + \\frac{\\partial G_{jk}}{\\partial x^i} - \\frac{\\partial G_{ij}}{\\partial x^k}\\right) \\\\\n _{(X)}\\Gamma_{\\alpha\\beta\\gamma} & := \\frac{1}{2}\\left(\\frac{\\partial g_{\\alpha\\gamma}}{\\partial X^\\beta} + \\frac{\\partial g_{\\beta\\gamma}}{\\partial X^\\alpha} - \\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma}\\right) \\\\\n \\end{align}\n", "d621cf21c6f85d5b056dca3b5abe947b": "N_f=\\frac {2ln[a_f/a_0]} {\\beta D}", "d621e3c397f33c2beeac8c8c6a158e46": "\\partial_1 x_1= x_1\\partial_1+1,\\partial_1 x_2= x_2\\partial_1, x_1x_2=x_2x_1 \\, ", "d6223ea8862085b1f77e77ed6776539a": " e^{-\\frac{x}{2}} ", "d62244c61a8e51447e7f78552daa2d07": "\\sum_n \\mathbf{N}_n = \\sum_n m_n \\left(\\mathbf{x}_n - t \\mathbf{u}_n \\right) = \\left(\\mathbf{x}_\\mathrm{com}\\sum_n m_n - t \\sum_n m_n \\mathbf{u}_n \\right) ", "d6228b77dea0e332fe0b73183da87938": "L(a) \\neq \\mathit{out}", "d62335b59c0bbfee02d616dfc9fc3f8e": "c<110", "d623487f146cf20aba4a0faae90b7d0d": "\\mathfrak{JKLMNOPQR} \\!", "d6237820cc9bc086246ef46766d14fbc": "f(X)=X", "d6238ac020f9ce717fc0a1c095daed28": "ax \\in \\left( n\\pi - \\frac{\\pi}{2}, n\\pi + \\frac{\\pi}{2} \\right) \\,", "d623b4c905c3c5896db4397105599a0b": "C_{ob}", "d623cda016b0ffa9d6cf388cc48d2262": "\n \\begin{bmatrix} D_{11} & D_{12} & D_{13} \\\\ D_{21} & D_{22} & D_{23} \\\\\n D_{31} & D_{32} & D_{33} \\end{bmatrix}\n = \\cfrac{2h^3}{3(1-\\nu_{12}\\nu_{21})}\n \\begin{bmatrix} E_1 & \\nu_{12}E_2 & 0 \\\\\n \\nu_{21}E_1 & E_2 & 0 \\\\\n 0 & 0 & 2G_{12}(1-\\nu_{12}\\nu_{21}) \\end{bmatrix}\n \\,.\n ", "d62416827713bf6a8bb8bbdfab3e41c8": " X_H", "d624400166c3888960129cb8a83f9622": "S_L > S_B \\,", "d6246c862559a1f3789b9e0acd2dffea": "V = R \\left( \\frac{Tn}{P} \\right)", "d6246df485775c76a1b427c96e64bce4": "d^{2^{c n}}", "d6253e650dbb7b972f9b0d6128b46378": "f a b c", "d625e472051dc04705a5d5461c5ab11c": " V = \\begin{bmatrix} 1 & e^{-j\\omega\\Delta t} & \\cdots & e^{-j\\omega(M-1)\\Delta t} \\end{bmatrix}^T ", "d625e761be8b2dcfa479bb209577df32": "C(\\alpha)_0 = \\{0,1,\\omega,\\Omega\\}", "d62607bf7acf7aac78193a6382bacfa9": "\\mu \\mathbf{F}", "d626992d4f6e0e0ada456afd2d83ae97": "z'_i", "d6269cc0272e792fcc9ff1f4c0ec5690": "a=\\pi^{e_a}p_1^{e_1}p_2^{e_2}\\cdots p_n^{e_n}", "d626ca549a9ec5e1c3916ad47898e8f8": "\\binom{m}{k}\\binom{n}{k}", "d626cc09218a145eb55c30b48291c1b2": " V(x)=A\\int_{-\\infty}^{\\infty} (g(k)+\\overline{g(k)}-E_{k}^{0})\\,R(x,k)\\,dk ", "d626e915c64acc1824b088fa24070983": "m_a^2+m_b^2=5m_c^2.", "d627419a2d59751ce17ba6c4852479d7": "\\scriptstyle f^{64}(4)", "d627cd6fc95ccd32ba68c7a6aa1cc61d": "P_H\\,", "d627e04ecceb3969a210b001a4f7312d": "\\Delta^1_{\\rm LONG}", "d62828ace679269cbf56849793f9c99f": "\\,\\beta_i's", "d6287a230a7bce76c4be68c26845117d": "(2-X)^A P(X)+X^A\\,P(2-X)=2^A", "d6289ad398492aa222354106a3c46d50": "\\begin{align}\n \\tilde{Z}_C &= {1 \\over \\omega C}e^{j(-{\\pi \\over 2})} = -j\\left({ \\frac{1}{\\omega C}}\\right) = -jX_C \\\\\n \\tilde{Z}_L &= \\omega Le^{j{\\pi \\over 2}} = j\\omega L = jX_L\\quad\n\\end{align}", "d6289f8ca656b4b2cea15fe42b199da9": "x, y, z", "d628b44cddc4573047bd7d28a9d5e5a0": " M(t,0) = (-t)^{\\frac{d-q}{p}} M(-1,0),", "d628bb6698c9abf96bc7fe022b2e6c98": "(x_0,y_0)\\in X\\times Y", "d6291d3a710a182661473a6ed408dd34": "M_{\\alpha\\beta}", "d62941f424a511f8d1e51f44a8869a76": "x^2-xy+y^2 = -a(x+y)", "d6294dd511b5ef24054da30ebd78f8e7": "y(t_0+\\tau) = E(\\tau,t_0,y(t_0))\\ y(t_0)", "d62952815130d05b2d862b2d9da827f5": "\\tilde{E}_i^a", "d629dc63176ce273f700466f0c081321": " W_{ADK}(A^{i+})", "d62a0f135903cae0370e3335f24616a3": "R_{r}", "d62ab84ed189df47382621bcdaaba0ce": "x_{i} = x_{i-1}", "d62b66be0fdb6e575cef6adcb18aeeb8": " - { 1 \\over s}\\, \\left[ \\ln(s)+\\gamma \\right] ", "d62b7dd6168cadfc8e28421f92a83ba1": "N^K", "d62b82566089262ba79ec32da406c0a1": "{\\mathbf{}}\\tau(t).", "d62c275c9e20a238a50ae67077bd8eaa": "\\sim A", "d62c4b427a924bfc694b49277bc08896": "g=f*1", "d62c5349d9cf1d95a7bc73be6e7b641e": "\\textstyle P-Q_iA_i", "d62cb08649c4434f7daa99a32fe133dc": "0^{256}", "d62ce0c45a37359b7c5f7f2a314c3d8d": "G_2(4).", "d62d3549b220f337cd526b2e1895e5dc": "\n e^{i\\mathbf{k}\\cdot\\mathbf{r}} = 4\\pi\\sum_{l=0}^\\infty\\sum_{m=-l}^l i^l j_l(kr)\n Y_{lm}(\\theta_r,\\phi_r)Y^\\ast_{lm}(\\theta_k,\\phi_k),\n", "d62d866228b48ca9eadef3152d6e0881": "T=a \\land (b \\lor c)", "d62dad4e1c241853459e5f802fc1f66f": "X=\\{X_i\\,;\\; i\\in I\\}", "d62dc7cead067bad53cd5ca6f946247d": " \\varphi(0) = x(1)", "d62dcf187055f6171d8cd74c0c7a1324": " \\bigl( L^1(R, \\Sigma, \\mu), L^\\infty(R, \\Sigma, \\mu) \\bigr)_{\\theta, q} = L^{p, q}(R, \\Sigma, \\mu), \\ \\ \\text{where} \\ \\ 1/p = 1 - \\theta,", "d62ed14843bf448d41d9efd2f86598d7": "\\,e + ig", "d62fe9665d304d6f979d934f89d847b6": "A_i=\\sum\\epsilon_{pq\\cdots} c_{pq\\cdots,i}", "d6303acbb80a316baae7539b40944076": "\\omega(x) = \\inf\\left\\{\\mathrm{diam}(f(U))\\mid U\\mathrm{\\ is\\ a\\ neighborhood\\ of\\ }x\\right\\}", "d63059872cf6ee372b9ac97b3c93c5fe": "h=| \\mathbf{h} |=\\sqrt{a(1-e^2)}", "d6308cf8848df0291ef71de870dbcfa4": "\n\\frac{\\partial L}{\\partial t} = [ L, A ] + BL,\n", "d63204cd6f5d1ab01026db763c0e02a8": "t_{n+1,n+1}", "d63219e4034933a7dadac1a6d6b02395": "\\Lambda = c .", "d6321a4028d26fcb4fda61c5eee53f1b": "P_\\ell(\\cos(\\theta))", "d6321a8b41004c896c17390884d0b6e3": "\\psi\\in L^2(\\mathbb{R})", "d6322af7671e1706b3dc4fb846cee90c": "q(x) = ax^2\\quad \\textrm{(unary)} ", "d63255c94453416a71a6b4dd923cb5aa": "2000 \\cdot I_e", "d632670f93e673d8d0f7d2baadf2cf04": "\\scriptstyle (x_n) \\,\\mapsto\\, \\sum_n{2^{-n} |x_n|/(1 \\,+\\, |x_n| )}", "d63274d4555f2b97992490e993049366": "\\xi_{1,2}", "d63293a5ebc6ccf443e59400cbe8a0e9": "A^*A", "d632c5fc59ef5692545c695cdcfb39d7": "\\int_U \\left( \\psi \\nabla^{2} \\varphi + \\nabla \\varphi \\cdot \\nabla \\psi\\right)\\, dV = \\oint_{\\partial U} \\psi \\left( \\nabla \\varphi \\cdot \\bold{n} \\right)\\, dS ", "d6331ee6b2c0a72b1dfea5e8b47fc03e": " \\begin{align} \\nabla \\Psi & = \\bold{e}_x\\frac{\\partial \\Psi}{\\partial x} + \\bold{e}_y\\frac{\\partial \\Psi}{\\partial y} + \\bold{e}_z\\frac{\\partial \\Psi}{\\partial z} \\\\ \n& = i k_x\\Psi\\bold{e}_x + i k_y\\Psi\\bold{e}_y+ i k_z\\Psi\\bold{e}_z \\\\\n& = \\frac{i}{\\hbar} \\left ( p_x\\bold{e}_x + p_y\\bold{e}_y+ p_z\\bold{e}_z \\right)\\Psi \\\\\n& = \\frac{i}{\\hbar} \\bold{\\hat{p}}\\Psi \\\\\n\\end{align} \\,\\!", "d633201cf12aee48145f148b7bfd0134": "U = H(k) / D", "d63334839a142622ae70558eb6fe4a01": "A = P^{-1} L U", "d63349e74a4884f131ea9daf2b35bb7f": "\nS_w[p] =\n\\begin{bmatrix}\n\\sum_r w[r] (I_x[p-r])^2 & \\sum_r w[r] I_x[p-r]I_y[p-r] \\\\[10pt]\n\\sum_r w[r] I_x[p-r]I_y[p-r] & \\sum_r w[r] (I_y[p-r])^2\n\\end{bmatrix}\n", "d6336309b8f95596ebe81b9962904935": "\\frac{d\\mu}{dt}=\\frac{Y}{mU} + \\frac{g}{U}\\phi", "d633aa24a81b38ab595a9849b79aaab4": " -[ \\frac{y+11(y \\bmod 2)}{2} + 11 (\\frac{y+11(y \\bmod 2)}{2}\\bmod 2) ] \\bmod 7 .", "d6347594d4aa079ab19779fb936afe9a": " y'(x) = q_0(x) + q_1(x) \\, y(x) + q_2(x) \\, y^2(x) ", "d634d9ad25046b0a10ed6faa9f212705": " \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_m \\end{bmatrix} \\text{ or } \\begin{bmatrix} x_1 \\; x_2 \\; \\dots \\; x_m \\end{bmatrix}^{\\rm T} ", "d634f074035a4eb437bd7da3c03181a0": "K=\\mathbb{Q}", "d635fe2df521ea966ed25d1ef28f68c0": "\n \\begin{align}\n \\varepsilon_{\\alpha\\beta} & = \\frac{1}{2}(u^0_{\\alpha,\\beta}+u^0_{\\beta,\\alpha})\n - \\frac{x_3}{2}~(\\varphi_{\\alpha,\\beta} + \\varphi_{\\beta,\\alpha})\\\\\n \\varepsilon_{\\alpha 3} & = \\cfrac{1}{2}\\left(w^0_{,\\alpha}- \\varphi_\\alpha\\right) \\\\\n \\varepsilon_{33} & = 0\n \\end{align}\n", "d636fe93953c4a3f49b3c116a55c6846": "f_X(x) = \\begin{cases}\\frac{1}{2}, &x \\in \\{0, 1\\},\\\\0, &x \\notin \\{0, 1\\}.\\end{cases}", "d637095099f1b9e25a621f379052aa3a": " \\forall P \\in \\mathcal{P} ", "d6373a08512dac8a026e024865817c0b": "u_k=\\left[ e^{ \\frac{2\\pi i}{N} kn} \\;|\\; n=0,1,\\ldots,N-1 \\right]^T", "d6373cae8c5f65314b72e208fd6f494c": "\\mathcal{N}(A, \\sigma^2)", "d6376f80059932742d703e596ab50d1c": " E(F(X)^2) < \\infty", "d6378660546788d3f67decb7c9ff9667": "\\scriptstyle\\mathbb{Z}\\left[\\,\\sqrt{-5}\\,\\right]", "d637b1388b399f7ec416d08a79afed11": " \\delta = \\frac{ N^2 - 1 }{ 2 } d_{ max } ", "d637d62c28a50cc8ae8efcb7372a3a68": "\\frac{}{}I", "d637d92de16eecb09493a319bfd95382": "\\alpha = \\rho \\frac{\\sigma_x}{\\sigma_y},", "d637e0691f5ade25679af2e00e596705": "A f", "d63815e9d59dbf8ba0b77b838c2ed79f": " \\mathbf{v}' = \\mathbf{v}e^{i\\theta} = e^{-i\\theta}\\mathbf{v} = e^{\\frac{-i\\theta}{2}} \\mathbf{v}e^{\\frac{i\\theta}{2}}.", "d6385d05a0909058b3f39f404bc1fee9": "f(x)=\\operatorname{sech} \\left( \\frac{x}{X} \\right).", "d63883516dfd7e1234323612d8e91c45": "T:S\\rightarrow Y", "d638b05a81d764caa3ca40d42f67619a": " 3600 - 80L = 30 ", "d638ccbb2d3b8a44857896e6b8ca087a": "\\mathcal{L}(\\vec{x},t) = - \\rho(\\vec{x},t) \\phi(\\vec{x},t) - {1 \\over 8 \\pi G} (\\nabla \\phi(\\vec{x},t))^2 ", "d638e27cc12e58c2f1dd2eed6f2ee370": "q(x, t) = \\mu \\frac{\\partial^2 w}{\\partial t^2}\\,", "d6391bd84e5cf48b9d282d63414456a7": "g' \\in G'", "d6392c20fc76c693c85be8fab6bb139d": "N_p= \\{m\\in tM\\mid \\exists i, mp^i=0\\}", "d63945eb983abdcd79c41ccd9da6a40c": "\\Gamma(0.5 - 0.5i) \\approx 0.8181639995 + 0.7633138287 i", "d63991a1da00dfd1938f2c11e599bd5f": "\\textstyle P(\\mathcal{A})=p", "d639b3753681d06d3b760034d7349f27": "\\begin{align}\n\\mathbf{A}' & = \\mathbf{A} - \\dfrac{\\gamma \\varphi}{c^2}\\mathbf{v} + (\\gamma-1) (\\mathbf{A}\\cdot\\mathbf{\\hat{v}})\\mathbf{\\hat{v}} \\\\ \n{\\varphi}' & =\\gamma \\left( \\varphi - \\mathbf{A}\\cdot \\mathbf{v} \\right) \n\\end{align}", "d639ba9ff8c2c14ced5241882b0956c1": "A = A_1 \\times \\dotsb \\times A_N", "d63a1fa5f2348b640ce4584d573e93da": "r_{ij}\\!", "d63a52806b93a5550f2d2d460ee7b799": "K \\otimes_\\Q \\mathbb{R} \\cong \\mathbb{R}^d ", "d63b35b2c2fe5921abcf839597efe6c3": "f_i(x) < 0, i = k+1,\\ldots,m,", "d63b4b2ca0a6d3cd358759664c7440e9": "\\frac{\\sqrt 8}{3}", "d63b51dd644da7052095e58176cbe198": "\\mathbf{a} = r \\left( \\frac {\\mathrm{d}\\omega}{\\mathrm{d}t} \\mathbf{u}_\\mathrm{\\theta} - \\omega^2 \\mathbf{u}_\\mathrm{r} \\right) \\ . ", "d63b8f983ed323123dc1eee904b6cf6e": "\\ W_0(z)", "d63ba323fe0b081eccd82994ea19ab30": " X_i = U U^T X_i V V^T = U (U^T X_i V) V^T = U M_i V^T ", "d63bc76adbf011c7b29b05d65379a5b9": "E_I = vi\\,\\Delta t = 1 \\left(1/Z\\right)\\Delta t = \\Delta t/Z", "d63c598fa08afea45e4fca7b3f856e4b": "\nq \\overset{\\alpha}{\\rightarrow} q' \n ", "d63ca181eab26d939d7def75bce07316": "y = \\; ?(x)", "d63d08069be4e6df6b2685db5fab1c90": "m/p_iP \\neq P_\\infty", "d63d4d818b54d4612b836da886df1c18": "\\mathrm{LCR} = \\sqrt{2\\pi}f_d\\rho e^{-\\rho^2}", "d63d9ebd50a22610c0ab770be4943b43": "r_i = c + \\sum_{j+k=i} a_j b_k,", "d63dd404f1984614ce7e628cdbc28d12": "s_{X_2}^2", "d63de22af0b65860c77c552861ae08e7": "E=\\frac{hc}{\\lambda}, \\,", "d63eae608e7ccc9e8e699a96294e1e27": "J_{\\text{x},\\text{y},\\text{z}}", "d63ed4a434f1ae8d6127f40983c2fc95": "\\omega=\\gamma_\\mu B", "d63ee576d47c2bc930341d064d06cb53": "\\int_0^1\\cos(xz)\\,dz=\\frac{\\sin(x)}x=U_0(x)", "d63f18e78ee9b14693109ac7a837ced8": "\\lambda(n)= \\phi(n) \\mbox{ if and only if }n=\\begin{cases}1,2, 4\\\\\n3,5,7,9,11, \\ldots \\mbox{ i.e. } p^k \\mbox{ where }p\\mbox{ is an odd prime}\\\\\n6,10,14,18,\\ldots \\mbox{ i.e. } 2p^k\\mbox{ where }p\\mbox{ is an odd prime}\n\\end{cases}\n", "d63f6e38684c573762705d92463a21f5": "\\scriptstyle p_{CO_2}", "d63f84f656b6c5352dc27e37e7b28704": "x^*_t = \\pi_0'z_t + \\sigma_0 \\zeta_t,", "d63fbbb653ce5279bd2563162fe54c41": "\\mu [R(Q) - C(Q) - G_{min}] = 0.", "d63ffb39f3b19574522060f43f881d51": "x+i\\,y\\in\\mathbb{C}", "d64044fa23071ea98ecd9bc1e54f7b68": "\\alpha+\\beta+C = k \\pi, (k=1,2,\\cdots)", "d6406839aa2cea089ea796813843750c": "(\\tfrac{\\pi}{2}-\\psi) + (\\tfrac{\\pi}{2}-\\theta) + (\\tfrac{\\pi}{2}-\\phi) = \\tfrac{3\\pi}{2} - (\\psi+\\theta+\\phi) = \\tfrac{3\\pi}{2} - \\tfrac{\\pi}{2} = \\pi\\, ", "d6411cfc056a1db44d0526d879ed38cb": "\\limsup_{n\\to\\infty} \\left| \\frac{f_n}{\\Psi_n} \\right|^{1/n} = \\tau.", "d64168faece7b30f4fc847181bc7b532": "E=(4.161\\times{10^{-5}})\\cdot u^{0.75}\\cdot T_F\\cdot M\\cdot (P_S / P_H)", "d6416b42a0abd08000b5240784537ac8": "\\hat{I}=\\frac{1}{\\pi} \\int |{\\alpha}\\rangle \\langle {\\alpha}|\\, d^{2}\\alpha ,", "d64193cd19bc713b2e017ebebd82c5d8": "f(X)=\\prod_{i=1}^k (X-\\alpha_i)^{m_i}", "d641cf603bcd879a0fd896cdc7622606": "x^{i}", "d641e6e370240c3a961ad30dcf23062a": "\\gamma: [0,1]\\to D", "d641edbfa0d684aed64158b9ea119d8c": "r =\n\\begin{cases}\n\\frac{\\sqrt{5}}{3} S, & \\mbox{if}~T=0 \\\\\nx \\cdot g + \\frac{1}{3}, & \\mbox{if}~T \\ne 0 \\\\\n\\end{cases}\n", "d641fa5e6cd7deb01b0c056151a29c75": "\\, R_{abcd}", "d642913da8d5adac3b71b03e75e9cf6c": " \\boldsymbol{P}^T=\\boldsymbol{N}", "d64295b355724688ba0154250f5d7e5a": "G=\\xi\\partial^\\mu A_\\mu", "d642afb3159f9354e34646e71c959042": "1 \\over 10^7", "d642b50ef86c65b9bb9a85a0a9a362c7": "z=x+iy \\,", "d642cfb9aaca7a38987169bd39ace811": "\\{0^n1^n \\mid n \\ge 0 \\}", "d642f18404f62d4e007b69d8a6f0581b": " arg(\\Phi_c(z)) \\,", "d64312ea872c32aeab786dc500caa986": " 0<2n-\\wp(p-1) F_e/2", "d67b3750986a36631f3ba6eb7853a61e": "\n\\; C_\\Phi = B^* B.\n", "d67b48e5c6c9545ea713a5e6951066e8": "T^{ij} = F^{ik}F^{jl}g_{kl} \n- \\frac{1}{4}g^{ij} \\vert F \\vert^2,", "d67b5b3c6de4844ca373deeb06655bb5": "M_\\text{man}", "d67b788a00c438809a64c93676e8f807": "\\mathbf{p} = \\left(p_x,p_y,p_z \\right). ", "d67bb78bc5b25c540d9604f76e6e71f8": "\\frac{1}{2}\\, \\sqrt{\\frac{g}{k}}\\, =\\, \\frac{1}{2}\\, \\frac{g}{\\sigma}\\,", "d67bc8f1ea958f4029163b5391a86b7e": "\\mathbf{g}(n)", "d67c27eab4f0356ea7af7816a44d6b0d": " \\hat\\Delta\\colon \\hat T(V) \\to \\hat T(V) \\hat\\otimes \\hat T(V) ", "d67c566651d444e692610da0d8a883c9": "\\scriptstyle U_{DC}", "d67d3ceb6788df7650e92154b16b914b": " \\mathfrak{g} > [\\mathfrak{g},\\mathfrak{g}] > [[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]] > [[[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]],[[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]]] > \\cdots", "d67d848d9fa072207ba208b2862e10d4": " Y(T) - Y(t) = \\int_t^T e^{- \\int_t^s V(X_\\tau)\\, d\\tau}\\sigma(X,s)\\frac{\\partial u}{\\partial X}\\,dW.", "d67e134e389e03e6ebc82569a8344ae8": "\\ \\lambda_0 \\,", "d67e2bb911d61a6f7dcb20fe8cbd5a44": "C_{12} = G_8 + G_4 \\cdot P_8 + G_0 \\cdot P_4 \\cdot P_8 + C_0 \\cdot P_0 \\cdot P_4 \\cdot P_8", "d67f58abfc15d36a879ab0022ee764da": "\\textbf{f}_{dyn} \\approx C \\frac{G^2 M^2 \\rho}{v^2_M}", "d67f864f7c6421f3bee5562ff04cb965": " r = d\\sin\\alpha_{crit}\\,\\!", "d67f98a3f5aeefd323e848de8797f5df": "T_i=T_e", "d67fb59d3686de104f8c045eb3e76e52": "\\frac{1}{\\sqrt{1 + (\\omega RC)^2}}\\cdot e^{-i \\phi(\\omega)},\\text{ where }\\phi(\\omega) = \\arctan(\\omega RC).\\,", "d67ff0673b97dd8f83b6a6eb2ee10f2f": "\\psi_{j,j}=\\psi_{4,4}=0", "d67ff31d5cf4dffded63766a89e520fd": "\\sum_{j=1}^n\\left(\\varphi\\delta_{ij} - a_{ij}\\right)x_j = 0.", "d6801e1dcf3eda659e69d0c019395236": "(1,3,1)_{2}\\oplus(1,1,3)_{2}.", "d6802e713eb673b4db82528108ca5274": "u(x) = (x-1)(x-3) = x^2-4x+3", "d68089b4a7568c9c4bd64a31cda999cb": "H_k=(H_0)_k - \\frac{\\gamma_k}{4\\pi} (M_0)_k, \\qquad k = x,y,z.", "d6808c1a8ff79b38d6f46280b94183fb": "T_p(A)=\\underset{\\longleftarrow}{\\lim} A[p^n]", "d680d54ccecdd40182885851f11cf18f": "\n \\displaystyle \n (\\delta)\n \\longleftrightarrow\n S(1,2)\n =\n \\left\\{ \n\t (1),\n\t (2) \n \\right\\}\n", "d680d74f77970b3effea02b09e123fba": "1-\\left[\\frac{15}{16}\\right]^{15} \\,=\\, 62.02%", "d680e62938640a0718b20e3789c632b0": "\\Delta(u,v,w)=\\frac{(u-1)(v-1)(w-1)(vw+1)^2}{vw\\sqrt{uvw}}, \\, ", "d68145bcaed8a1bc69c5a8f1682717f7": "{\\rm st}(f(x_{i_0}))\\geq {\\rm st}(f(x_i))", "d681b80107e3197e23f20938cc2feb94": "f(\\xi )=Tr[\\hat{B}(\\xi )\\hat{f}],", "d681bff567ac07a6d2ef7d582064b200": "D(G, K)", "d681d9d9c7136901cd4648587604cbbb": "p_ie^t + (1-p_i)", "d6821dd0c9dfdf60db67eda0887291d8": "2H_2O \\rightleftharpoons H_3O^+(aq) + OH^-(aq)", "d6823efedf3638b365baeb980b474baa": "Initiates", "d682520425a3bec985986d9f9cc62382": " \\Bbb{R}^2 ", "d6825c1a9f8dd703d0fce8105c04e498": " \\max(0, const - x) ", "d6829394faa8a77128fe36c6142df8b2": "\\sigma \\psi = \\tau", "d682c40defc4077125e4e981298b8922": "\\frac{\\partial \\ln |\\mathbf{X}^{\\rm T}\\mathbf{X}|}{\\partial \\mathbf{X}^{+}} =", "d682c41e3f783015ba2fd31420b7b451": " \\frac{x^2}{(1/\\sqrt{I_1})^2} + \\frac{y^2}{(1/\\sqrt{I_2})^2} + \\frac{z^2}{(1/\\sqrt{I_3})^2} = 1,", "d6830418ef62a4f40fa9cc061f550a66": "q^{r+1}", "d6831a5a62d3a635d45da80461377ca4": "1 \\le i \\le n", "d68330a8809dcec8cd240b9fb9c088b4": "\\begin{align} \n&x_1, \\cdots, x_{K-1} > 0 \\\\\n&x_1 + \\cdots + x_{K-1} < 1 \\\\ \n&x_K = 1 - x_1 - \\cdots - x_{K-1} \n\\end{align}", "d68351d604710484bb17a046f228cf66": "\\tfrac{OD}{OE}", "d683abaab62a22fa4774d8b8123cfa09": "229852x^{10}-653966x^9+1537363x^8-3008720x^7+4904386x^6-", "d683b6ee6dd5e8e18b35ec0e5d4267ba": " W3(A)", "d683ed8059ae9f4be27c2fc06ea4eb2c": "\\lim_{n \\to \\infty}x_n", "d6840b2efc4c47e21652b5fbae79d495": "\\scriptstyle f(1, 2, 3)", "d6846e9317bc0313d802c28a4c3cd561": "O\\left( \\min( n , |\\mathcal{Z}|^{2k} )\n\\right)", "d684b9c04c5cae40d4fb2ccc3d5d5d52": " C(k_1,....,k_n) ", "d684dfd6fa81fffd99b26aee474dce83": "e^{-t^2}=-(2t)^{-1}(e^{-t^2})'", "d684e5ca267aff2af78a831a4aa99a79": "\\begin{align}\nx(t) & = (\\sin \\phi) \\mathcal{L}^{-1}\\left\\{\\frac{s}{s^2 + \\omega^2} \\right\\} + (\\cos \\phi) \\mathcal{L}^{-1}\\left\\{\\frac{\\omega}{s^2 + \\omega^2} \\right\\} \\\\\n& =(\\sin \\phi)(\\cos \\omega t) + (\\sin \\omega t)(\\cos \\phi).\n\\end{align}", "d684f8322b7404cd384bf1e425d9356e": "Fe", "d6850490fa6844bd0f3a22dc6861fa9a": "\\frac{\\mbox{Non-Interest expense}}{\\mbox{Revenue}}", "d6857b3678b0d2afc773a09cad6a7aea": "\n\\Gamma_p(a) = \\pi^{(p-1)/2} \\Gamma(a) \\Gamma_{p-1}(a-\\tfrac{1}{2}) = \\pi^{(p-1)/2} \\Gamma_{p-1}(a) \\Gamma[a+(1-p)/2] .\n", "d68590d11fc14d054ba8aece9c84bdf0": "M \\subseteq V", "d6859bc5ad4d69951b8e2caad54af1b1": "\\,S_c", "d685f44941c67a364ba20db3eeca5365": "b = 2.37 \\times 10^{-4}", "d6861f811ef4999f6b0d77909b2c3e8d": "S^1 = \\{ ( \\cos{\\theta}, \\sin{\\theta} ) \\, | \\, 0 \\leq \\theta < 2\\pi \\}.", "d68636b36909c71484974936ea97c667": "\\frac{V_2^2}{2(h_1-h_2)}", "d68646adac9e5c48efbcd70615631139": "f_A(w_1x_1 + w_2x_2 + \\ldots + w_nx_n)", "d6865d2a27b2c736302435d7cd892af0": "\\int g(\\nu ){\\rm d}\\nu \\equiv N\\,", "d6867bbb0d627c8d2bf608faee8b171a": " (a\\cdot d+b\\cdot c)", "d686fb6b2b90bae0dba90aaff498a972": " f(x)=e^{f(x-1)} \\; \\; \\text{for all} \\; \\; x > -1, \\; f(0)=1,", "d6876d2f730fd36b60ade333282677df": "\\frac{\\ln(1 - p\\,\\exp(i\\,t))}{\\ln(1-p)}\\text{ for }t\\in\\mathbb{R}\\!", "d687ba9e83924db1e7de412e293611ae": "f(\\boldsymbol{x}) = \\sum_{j=1}^{J} \\alpha_j h_j(\\boldsymbol{x}),", "d68897281db0abe9cdf68eb4900e344a": "\\sum_{k=0}^\\infty p_k = 1.\\,", "d6889a89a0bb45b31e300ca0972e66f9": "Pu", "d689330acb46d1e3672d2d842f5d5119": " 0.5 \\lambda \\ge 1.645", "d6a019bba346734083fa0656c0ee9fab": "\\Omega(t)=\\sum_{k=1}^{\\infty}\\Omega_{k}(t),", "d6a066d3619951ca3e5a9fec304dda90": "M_R = \\begin{pmatrix} (R\\mathbf{e}_1R^{-1}) \\cdot \\mathbf{e}_1 & (R\\mathbf{e}_2R^{-1}) \\cdot \\mathbf{e}_1 & (R\\mathbf{e}_3R^{-1}) \\cdot \\mathbf{e}_1 \\\\ (R\\mathbf{e}_1R^{-1}) \\cdot \\mathbf{e}_2 & (R\\mathbf{e}_2R^{-1}) \\cdot \\mathbf{e}_2 & (R\\mathbf{e}_3R^{-1}) \\cdot \\mathbf{e}_2 \\\\ (R\\mathbf{e}_1R^{-1}) \\cdot \\mathbf{e}_3 & (R\\mathbf{e}_2R^{-1}) \\cdot \\mathbf{e}_3 & (R\\mathbf{e}_3R^{-1}) \\cdot \\mathbf{e}_3 \\end{pmatrix}.", "d6a0a39793a5975e6d4c488582ccf829": "p_0= p", "d6a10a47eb14bdd8a5989a706f1a2fed": "L=10\\log \\left({1+\\omega^2}\\right) \\ \\mathrm{dB}", "d6a19e797b7746985c209b7c329d4908": "155.8*10^{-6}", "d6a1a0e23c313c7358af1855045a4af3": " {AB \\over DE} = {BC \\over EF} ", "d6a1b4355a0959b0905af7a900c2dd56": "[T]", "d6a1e0ba1878bf000867b6b7220b9f47": "\\{x | x \\in \\mathrm{U} \\land \\Phi(x)\\}", "d6a1f1e4e32477cad2b83740905a80f2": "\\begin{align}f_1^\\prime(x)&=f_2(x)\\\\\nf_2^\\prime(x)&=-f_1(x),\\end{align}", "d6a1f63a0ae2d059258426dd841db25c": "B = 2 - \\sqrt{3}.", "d6a22eca662a2e26d8d5e47277ee8853": "p_1 = p_2 = P(q_1+q_2)", "d6a279a358a44df3fefd0c62b0466678": "A = \\frac{A_{n+1}\\, A_{n-1}\\, -\\, A_n^2}{A_{n+1}-2A_n+A_{n-1}}.", "d6a2b24a461f9e50502ae5590f2f2d6d": " \\Psi = S_1\\,S_2\\,\\Gamma ", "d6a2cc927724bfcdd68679e8e4feb1c8": "\\vdash X \\rightarrow \\varnothing", "d6a303899a557a8a95978b00de8ed4bf": "V^{\\otimes n}", "d6a307aecb0c6f1cf6729cfdff06b5d9": " \\max_{x \\in J} |p(x)| \\leq e^{\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J} \\left( \\frac{C \\,\\, \\textrm{mes} J}{\\textrm{mes} E} \\right)^{n-1} \\sup_{x \\in E} |p(x)|~, ", "d6a34f56953f1426519baffe31243a3c": "B(x,\\delta)\\;:=\\{y\\in M\\;:d(x,y)<\\delta\\}", "d6a3890537177f627b2d9ed0bf17c382": " \\mathbf{E} = -\\nabla \\Phi ", "d6a3952347728b8a636c1e7908a056e8": "\\frac{2}{\\alpha} \\ ", "d6a3a7383bff3973d2ea27b5d779e497": "f'(z) = \\frac{1}{z}-1,\\,", "d6a3be635c6e80abc639373f2a1755d2": "I_1 = 3", "d6a3d5124dd6b521de7643f95672a76f": "k_{\\rm B} ", "d6a45dc431dda6c84e6f032b9b261110": " r = \\frac{1}{h_a^{-1}+h_b^{-1}+h_c^{-1}}.", "d6a46c875704a931a3f7a16eb17e06ee": "[S_gX_p,Y_p] = g_p(X_p,Y_p)\\,", "d6a4a5182cb219c9bcf4c2849c258f00": "V (r) = \\left(\\frac{V_{0}R_{0}}{r}\\right)+const.", "d6a4a58b4647b4f00190b72b3bf00a94": "E \\subset \\omega \\times \\omega", "d6a4e5f397aee2437717b618f3e0a2a6": "\\hat{a}_{v,{\\mathbf k}}", "d6a57f3321b35ed68cc5ef88b268c854": "( \\cdot , \\cdot ) ", "d6a5c28befc41d91368e140a58773095": "\n\\begin{bmatrix} x \\\\ y \\end{bmatrix}\n=\n\\frac{1}{{e}_{13} + {e}_{23}}\n\\begin{bmatrix}\n{-{e}_{13} - {e}_{23}} \\\\\n{2{e}_{13} +2{e}_{23}} \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix} -1 \\\\ 2 \\end{bmatrix}.\n", "d6a5e943dc14a6d8cccab26fa83cb51c": "ax+b=c\\,", "d6a5f183181053ca58c7d417ebd7ba81": "({31.62\\,\\mathrm{V}}/{1\\,\\mathrm{V}})^2 \\approx {1\\,\\mathrm{kW}}/{1\\,\\mathrm{W}}", "d6a5f6865fa9ac30a68488094b8fb710": "x=\\mathrm{sn}(\\xi, k)", "d6a60d7c36cdeceff1b2cc48b0aab253": "w + hg + l", "d6a612ea59f28bbb3c0a3008572ec16d": "O(1/\\epsilon^{5/2})", "d6a626ede55f496dc8f2a703937b2aa4": "\\Lambda_{\\mathrm{UV}}", "d6a62786289c6a63d7e434db0372ad38": "\\mathcal{L}(H_n, H_A) \\quad \\mbox{such that} \\quad A(i,j) = B^*(i)B(j) \\quad \\mbox{and} \\quad H_A = \\bigvee_{n \\in {\\mathbb Z}} B(n) H_n \\;.", "d6a663842a31d6b13dbbf622087707c3": "Z_0^{p,q} \\supe Z_1^{p,q} \\supe Z_2^{p,q}\\supe\\cdots\\supe B_2^{p,q} \\supe B_1^{p,q} \\supe B_0^{p,q}", "d6a67a483a32782155c770be7ab34d8b": "R = \\sqrt \\frac{L} {4 \\pi \\sigma T_{\\rm eff} ^4}", "d6a6aae07df7d650ef0d439fd07dc38e": "M^{-0.034}", "d6a7161a8f099ca59d1943c9ee955153": "\\xi\\leq t", "d6a727378c4409e69bd2496ece83c7cd": "R = \\rho\\frac{\\ell}{A} \\,", "d6a72fc99e4efef53dc8969ba2f042d2": "k'", "d6a7532dac416d24169a4d8ad6432ae5": "\n\\lim_{t\\rightarrow\\infty}P[\\eta_t(x)\\neq\\eta_t(y)]=C\\lim_{t\\rightarrow\\infty}P[X_t\\neq Y_t]>0\n", "d6a7f7fcabf4148b21b26c6894c1c705": " P \\cdot P = n(X)^2. \\,", "d6a810cd8108f32705c2d108a23a07e1": "\\langle \\mathrm{d} \\omega | S \\rangle = \\langle \\omega | \\partial S \\rangle.", "d6a8303da4426c44b01c30759a983712": "\\sin_k^2(i)+\\cos_k^2(i)\\equiv 1. \\, ", "d6a8331a43bbbcc15eaa8646f6a769c0": "\\frac{k}{m+n}=\\frac{k}{m} - \\frac{kn}{m^2} + \\frac{kn^2}{m^3} - \\cdots.", "d6a84abffe341e91e527f7b81b499f3e": "\\Delta t < \\frac{\\sigma^2}{(r-q)^2}", "d6a854a90ddc09a0f8530fd14378b73f": " f(A) = f(5) A_1 + f(-2) A_2. \\, ", "d6a8753552d0fa5e5ee5304c0001ad20": "\\sigma (t)=Gx(t)-Gx(0)+\\int_{0}^{t}[GBu_{0}(\\tau )+Gf(x(\\tau ))]d\\tau", "d6a895a5d87f4afe0e7c8706f1c2c775": "\\ \\mathcal{F}", "d6a89e0f7526174f1c784f1fdb396314": "Y_j= \\mu + x_j^T\\beta +\\xi_j + \\varepsilon_j, \\, ", "d6a93160b47a09ed66b295b9c5ada973": "E[X]=\\Gamma(1-\\tfrac{1}{\\alpha})", "d6a93511f8722750c606d6f00ca615c2": "y_1=y_0+h(\\tfrac14k_1 + \\tfrac34k_2)=\\underline{1.066869388}", "d6a93b90d393d6e3e27b77703271407c": "D_n(x)=\\sum_{k=-n}^n\ne^{ikx}=1+2\\sum_{k=1}^n\\cos(kx)=\\frac{\\sin\\left(\\left(n +1/2\\right) x \\right)}{\\sin(x/2)}.", "d6a94ad7736c275993a17c5b14f203f0": "pq=x^2+5y^2.", "d6a9f94ace3f7dc7e9f9f58a030956f7": "M(Q,\\Sigma,T)", "d6aa0b759cabbab10ab206bf524196cb": "T = \\frac{1}{2}I_1\\left(\\omega_1^2+\\omega_2^2\\right) + \\frac{1}{2}I_3\\omega_3^2.", "d6aa35b3ea57f6069849415d15116912": "\\frac{\\partial\\rho}{\\partial t}+\\sum_{i=1}^n\\left(\\frac{\\partial(\\rho\\dot{q}_i)}{\\partial q_i}+\\frac{\\partial(\\rho\\dot{p}_i)}{\\partial p_i}\\right)=0.", "d6aa76741743dab97e9cd9dd9fa8622f": "a \\mid c", "d6aa824a88918ff8240c999f73416609": " \\langle {\\Phi_0}\\vert e^{-T}He^{T} \\vert{\\Phi_0}\\rangle = E \\langle {\\Phi_{0}}\\vert {\\Phi_0}\\rangle = E ", "d6aadf2eb12e05e05b5a4c6f0da96aea": "Z_{dr}", "d6ab4f2ef5f9be0ff6cd5968f549eed3": "x, y \\in G", "d6ab81f5c67087c49d1a012c9b3eaa14": "A_1A_2", "d6ab82ca90d36a1f1711cfb41fbf1bf9": " F_t: X\\to X, \\quad t\\in\\mathbb{Z}, \\mathbb{R}. ", "d6ab892d4b591102ca54dd896fa8d89e": "O(|V|*|E|)", "d6ab8b762d3bdc9e1048d37c570fbd87": "\\ell(n)", "d6ab8f047463e20a55dc3c7a3c040b7c": "m\\!", "d6abe91212fa249c47a892603058603c": "(A\\equiv B)\\equiv((A\\equiv C)\\equiv(C\\equiv B))", "d6ac0f11e1268d445ddaa94c2d653163": "C: y^p - y = f(x)", "d6ac1e8a17d15d542b47de39c78c0642": "\\frac{7\\pi}6\\!", "d6ac56fce746ac86ac885e18af1ad00b": "P=(a,b)", "d6ac97ecdaa01a27c2f1b12f35c3aa57": "\\frac{2\\rho\\sin(\\alpha)}{3\\alpha}", "d6acb278d7f2c2cc6bbccc469665ea04": "a_{10}-b_{11}", "d6ad1f1edfebe345930efd48c13214b1": " J_v ", "d6ad2b85abe265ba828c8de42cb3dd4c": " 16\\pi (T_{bd} - T_{ac} \\eta^{ac} \\eta_{bd}/2) \\, = \\, - h_{bd, rs} \\eta ^{rs} \\,.", "d6ad5366199d83a49c000b824c86dc30": "(\\boldsymbol{x}_n,y_n)", "d6adaa4c91c81c0b81828a3dbf859fb0": "f(x)=\\frac{1}{x^2+1}\\,", "d6adb53b78dc33a5cd906a2237432ea5": " \\mathbf{y}_{k} ", "d6adce2b7f216e4cdf2a091c7d9e2184": "B = \\{a^m b^n c^n \\mid m,n \\geq 0\\}", "d6ae496b149cd601e2dbb6ad57733a69": "\\sin^2 \\phi_{\\Gamma} = 1 - \\left|\\tau(\\omega)\\right|^2", "d6ae98ea1450f1fbe5e45af76fa85fb6": " f(\\sigma_1,\\sigma_2,\\sigma_3) = 0 \\,", "d6aeb4bba4e97729e0264d2c167de0a7": "P = \\frac{F}{L^2} = \\frac{Nm\\overline{v^2}}{3V}", "d6aeb7fe9592d40fea12c0903b2b5557": "\\gamma_{\\mathrm{SL}}+\\gamma_{\\mathrm{LV}}\\cos{\\theta_\\mathrm{c}}=\\gamma_{\\mathrm{SV}}\\,", "d6af28d13ddf5f937fe946692596be23": " \\forall z ( (\\phi \\lor \\exists x \\psi) \\rightarrow \\rho )", "d6af38618278e6cd5ce978635e5feb0a": "\\sin x \\, ", "d6af615933cb9c862b25cccf5f4fc34f": "\\epsilon_1 = 1", "d6af64cce599efc384fbd9d62ca85ac9": "\\begin{align}\n \\rho \\, d\\phi \\, dz &\\hat{\\boldsymbol\\rho} \\\\\n+ d\\rho \\, dz &\\hat{\\boldsymbol\\phi} \\\\\n+ \\rho \\, d\\rho \\, d\\phi &\\hat{\\mathbf z}\n\\end{align}", "d6af90534567cb018f130464c4c094bb": "\\tfrac{OD}{OA} = \\tfrac{3}{2}", "d6af95f9b957641d5d035ae7883562a2": "n! := n \\times (n-1)!", "d6af96910eeafb062baf73813481351b": "\n \\langle j_1 m_1 j_2 m_2 | j_3 m_3 \\rangle =\n (-1)^{j_1-j_2+m_3}\\sqrt{2j_3+1}\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & -m_3\n\\end{pmatrix}.\n", "d6afae72ae4b5ebd69508c9d5cfbb861": "{\\rm equivalent\\ weight} = \\frac{m_{\\rm acid}}{c({\\rm NaOH})V_{\\rm eq}} = 52.0\\pm 0.1\\ {\\rm g}", "d6afc569cd3c2953eeaf2ba9eabcf0e5": "{\\mathcal L}^1_{xx}: z_1(x,y)={\\mathcal E}_1(x,y)F_1(\\varphi_1)", "d6b0007ff1dec704d471f2164ce08182": "G = \\frac{2 \\pi^2 L r^2}{M T^2} \\theta\\,", "d6b0407ce166c86177724c9f8403caf4": "\\hat{\\mathbf r}", "d6b06506ca75d9d64fb92f4ecdec12ed": "\\tfrac{5}{16}", "d6b0780be09382e60e2e15f6aaacfc57": "y-x^2=0;", "d6b123635fff889db13375a6d4e6bce9": "m=\\inf_{\\|x\\|=1}\\langle Tx, x\\rangle,\\quad M=\\sup_{\\|x\\|=1}\\langle Tx, x\\rangle.", "d6b21801697001070a231d42b78a3ee4": "C_{D,i}=\\frac{4.822 \\times 10^4}{A \\epsilon \\sigma^2 V^4} (W/S)^2", "d6b236d2267287221af731c6d924fee3": " + \\frac{2C_1R_1+C_1R_2+C_3R_3}{C_2} + \\frac{2C_1R_1+2C_2R_1+C_1R_2+C_2R_2+C_2R_3}{C_3}", "d6b238a673909200586e03c367a1e06c": "\\vec\\Psi : \\R^d \\supset U \\rightarrow \\R^n", "d6b267cb02241af300cc433ccd51ac20": "E_f \\equiv \\mu", "d6b292674136ca2ab176d864d9ec5f95": "[W_1, W_2] = \\tfrac{ih}{2\\pi} m^2 S_3.", "d6b2b60f122b375aa0a087640265d49b": "\\ln \\begin{bmatrix}1 & 1\\\\ 0 & 1\\end{bmatrix}\n=\\begin{bmatrix}0 & 1\\\\ 0 & 0\\end{bmatrix}.", "d6b3462339f3fa4203f00765b0fafd66": "T_{q}^{(k)} = \\sum_q \\langle k_1 , k_2 , q_1 , q_2 | k , q , q_1 , q_2 \\rangle A_{q_1}^{(k_1)} B_{q_2}^{(k_2)} ", "d6b3c8c35d787a3977787cfeb2446319": "Z=j\\omega M \\,", "d6b457d1e4339e81fb2917e14b401693": "L_{p} = 0, L_{pp} \\leq 0, |L_{pp}| \\geq |L_{qq}|.", "d6b4708a93a2bbef4f370ee7eb292bb5": "F:X\\to[0,+\\infty)", "d6b4a838b54e31d4ef9ac55cf94edd4f": "\\Delta_0 u = - 2K e^{u}\\quad\\Longleftrightarrow\\quad \\frac{\\partial^2 u}{{\\partial z}{\\partial \\bar z}} = - \\frac{K}{2} e^{u}.", "d6b4b0d285119e19b7c7fb67f644673b": "\\left(T_u/T_{u-1}\\right)\\cdot J\\left(R\\right)=0", "d6b4c1e8e84022ceb8a483dc31fb788a": "S_4 \\twoheadrightarrow S_3", "d6b4c37edbed150c21ee9a80fe67aeaf": "\\frac{\\partial c}{\\partial t} = D \\nabla^2 c - \\vec{v} \\cdot \\nabla c. ", "d6b4d0bceb43222f3b7b9995d02901b8": " \\text{diffusion vector angle between }B_Y\\text{ and }B_Z = \\arctan \\frac{B_Y}{B_Z} ", "d6b4efb334fab04f93fda2398dad5c9c": "\n\\|Y - DX\\|^2_F = \\left| Y - \\sum_{j = 1}^K d_j x^j_T\\right|^2_F = \\left| \\left(Y - \\sum_{j \\ne k} d_j x^j_T \\right) - d_k x^k_T) \\right|^2_F = \\| E_k - d_k x^k_T\\|^2_F\n", "d6b5136e110c91808078f6c8d9afaced": "\n\\langle \\mathbf{x}|\\psi_{\\{n\\}}\\rangle\n=\\prod_{i=1}^N\\langle x_i|\\psi_{n_i}\\rangle\n", "d6b558f4f92fed348c4656d39c739040": " a_P = {a_P}^0 + a_W + a_E - S_P ", "d6b57b083e4e038b6e2ce07cbd696365": " (I - A)x = d ", "d6b58e42178bb02bdc4a3cdbd5c61eb2": "D(E) = D_0\\frac{E}{\\sqrt{E^2-(E_g/2)^2}}\\Theta(E-E_g/2)", "d6b61201d01cc14f1901815372666175": " \\cos \\theta \\approx 1 - { \\theta^2 \\over 2 } \\ .", "d6b68f8a150441e0a70d4144ccf05dd9": "\\theta:W(\\tilde{\\mathbf{E}}^+) \\to \\mathcal{O}_{\\mathbf{C}_p}", "d6b75f9844c762082d64b29fb5b0c404": "\ne'_i = \\begin{cases}\ne_i & \\text{if } B(e_i,e_i)=0 \\\\\n\\frac{e_i}{\\sqrt{B(e_i,e_i)}} & \\text{if } B(e_i,e_i) >0\\\\\n\\frac{e_i}{\\sqrt{-B(e_i,e_i)}}& \\text{if } B(e_i,e_i) <0\n\\end{cases}\n", "d6b7e79a5de47ff852fea8e711696cd5": "\\partial (u + v) = \\partial u + \\partial v\\ .", "d6b83eca4584489b550543bcb5caa2c1": "A_{\\perp}", "d6b84375963363f984710fab0d3fcdfa": "G, q, g_1, g_2", "d6b886e17fa595dffa9b21b395154cc0": " = 1 + j \\omega ( {\\tau}_1+{\\tau}_2) ) +(j \\omega )^2 \\tau_1 \\tau_2 \\ , \\ ", "d6b9721cfe3ace45d6f925d2b2b0525c": "v_p = \\sqrt{\\frac{T_s}{T_o}\\frac{C_k}{C_d}(CAPE^*_s-CAPE_b)|_m}", "d6b97e48003fb3dbe32168616b3d7d34": "\\tfrac{E-M+S}{4M}", "d6b98460dedbd9e482537e97607cbcc7": "\\frac{X}{\\log X}.", "d6b9bae3c581428b645547337adfde5f": "\\begin{align}\n \\bold{p}^T I \\bold{p}&{}= (\\bold{p}^T Q^T) (Q \\bold{p}) \\\\\n &{}= \\bold{p}^T (Q^T Q) \\bold{p} .\n\\end{align}", "d6b9bfd1e67e94fd2b6c233afb183b42": "\\mathbf{JKLMNOPQR} \\!", "d6b9d21e1515fb55ba3e09ad5f918c9b": " \nP_{n1}={exp(\\beta z_{n1}) \\over (exp(\\beta z_{n1})+exp(\\beta z_{n2}))} \n", "d6b9e7ac527c4e74a3e894f860523ae7": "\\varphi = \\frac{1 + \\sqrt{5}}{2}", "d6ba2fe7a05f6f674d0641b6e378ecb8": "\\frac{T(s)}{r}=n.", "d6ba442dcc681b88b951a3608679e481": "\\eta=2\\cos (2\\pi/7)", "d6ba5194d7d715833086388511651df5": "\\Lambda(x)", "d6ba9c0416ffb977ae70e2dfc14e4087": "\\lambda_n \\leftarrow \\frac{1}{\\ell},\\quad n=1,\\dots,\\ell", "d6babb60573bed2e220095bfd246ee05": "N_l", "d6bac549858a75c4f0cd361c0bccb2a4": "\\Delta_{ki}", "d6bb7007f1ee9e371144f2eb149c79db": "k_n=q_n'(x_n)", "d6bb7909643881a04ef84ff0e55f0bfe": "i\\hbar \\frac{\\partial \\psi (\\mathbf{r},t)}{\\partial t}=D_\\alpha (-\\hbar^2\\Delta )^{\\alpha /2}\\psi (\\mathbf{r},t)+V(\\mathbf{r},t)\\psi (\\mathbf{r},t).", "d6bbd21a9d3f068d9258378ec77ddc5b": "u_\\mathrm{r}(n_\\mathrm{A})^2 \\propto \\frac{(R_\\mathrm{A}-R_\\mathrm{B})^2}{(R_\\mathrm{A}-R_\\mathrm{AB})^2(R_\\mathrm{AB}-R_\\mathrm{B})^2} u(R_\\mathrm{AB})^2", "d6bbd594c3b67548e4f911aebd1a2c7b": " \\phi(\\xi \\eta) = {1\\over 4 \\pi} \\int _0^{2\\pi}\n {R^2 - \\rho^2\\over R^2 + \\rho^2 - 2R \\rho \\cos (\\psi - \\chi) } \\phi\n (\\chi)\\, d \\chi \\; ", "d6bbfa21a93940a7774d9d3f83d742aa": "H^1(E)=0", "d6bc7dc43080cb3c1896591ca2b63924": "O(h^5)", "d6bd07c3681bde46b8179a87eefc7129": "g \\geq 2", "d6bd1565a7722ef784122dc030c4363a": "B_{k} = (1 + r)^k B_{0} - \\frac{(1+r)^k - 1}{r} p", "d6bd98e941dee09aa0e090695334739c": "\n\\sum_A \\vec{R}^0_A\\times\\vec{q}^{\\,A}_r = \\vec{0}.\n", "d6be6a45d2f3cfe2b5919093ecfa3f46": "f(1/u) = a_r = { -GM \\over r^2 } = -GM u^2 ", "d6befc9fe308de813cc46017f6f923f6": " \\operatorname{E}[Z^2]^{1/2} \\operatorname{P}( Z \\ge \\theta\\operatorname{E}[Z])^{1/2} ", "d6bf112fa6f444f00f38ddad6c81e289": "\\omega_0=1", "d6bf1bcf43635be197652bd18cf0d9eb": " Q^{(4)} ", "d6bf3737f534ccda2c0c6616c07ff078": "\\begin{align}\nM & \\longrightarrow & P\\times_G EG \\\\\nx & \\longrightarrow & \\lbrack \\Phi(x,u),u\\rbrack.\n\\end{align}", "d6bf4c23142635acbd65b9a5d17a812d": "\\alpha_p(f) = {N(p^r)\\over p^{rn(n-1)/2}} = {p^{s(n+1)/2}\\over m_p(f)}", "d6bf75749a454fe97233949549ad9cf7": "\\! B = Q^{-1} A P", "d6c039b892954a9c96166d5dc5313a26": "\n-N_j=\\left(\\frac{\\partial U[\\mu_j]}{\\partial \\mu_j}\\right)_{S,V,\\{N_{i\\ne j}\\}}\n", "d6c03f49335e076aa8c410441a73191f": " \\frac{\\ell(\\partial(D_j))}{A(D_j)}\\to 0,\\; j\\to\\infty. ", "d6c08cb77d174fb494c8b165d193f73d": "e^a{\\!}_\\mu = \\partial_{\\mu}x^a + B^a{\\!}_\\mu", "d6c098eab65d985ab498a6b0c19db8d4": "y^{\\prime} \\leftarrow \\tilde{y}^2 rem N", "d6c0ae87e4cc771729b071bd8bc2f53a": " \\left \\langle i \\right \\rangle \\ ", "d6c0ceb0155bacb603712b927c40fb04": "\\mathcal{}L_*", "d6c0d609cdc090f89ad2a462a99711ca": "p \\, \\sim \\, q", "d6c1188af4de1e1add8001138b11b8bf": "\\begin{cases} u_{t}=ku_{xx}+f & (x, t) \\in \\mathbf{R} \\times (0, \\infty) \\\\ u(x,0)=g(x) & IC\\end{cases} ", "d6c27ab676ca3a922aade317dbdfa442": " \\mathbf{r}^o_R ", "d6c297831a7f7024f4823bd39ca31bc8": "q+pz\\,", "d6c2b1b84dbede2ad2d3554df57508ae": "k=k_0 T \\text{ (metal at low temperature)} ", "d6c2c2717e53b541fa5d900caaa84923": "f(x_0+2,y_0+1/2) - f(x_0+1,y_0+1/2) = A = \\Delta y", "d6c2c4eac163b83f8e690cb0a87d203c": "\\int\\frac{f'(x)}{f(x)} \\, dx= \\ln\\left|f(x)\\right| + C", "d6c2d1183f61d3336093fe40f4d600e2": "\\int x^2 \\cosh ax\\,dx = -\\frac{2x \\cosh ax}{a^2} + \\left(\\frac{x^2}{a}+\\frac{2}{a^3}\\right) \\sinh ax+C\\,", "d6c317fe62a9a879db41416c67a8ede2": "M\\{B\\}>0", "d6c3352ab686816986f067bfeba791ed": " E[V:=G] = f\\ (x\\ x)[x := x\\ f] = f\\ ((x\\ f)\\ (x\\ f)) ", "d6c3f0013749757382f204c74088bf7f": "0.693/\\lambda", "d6c3f8122a8c81b0ffb4bdb122c72d7c": " h(u^{-1}) = h(u)^{-1}. \\,", "d6c40c3096e7d46f0cbbea63f4996c19": "\\left|\\Psi\\right\\rang = \\left|1,V\\right\\rang \\left|2,H\\right\\rang ", "d6c41032167f9ee9f540566d968a4251": "|h(x)|", "d6c4280a140de7e2d35041c471be84dc": "(\\alpha \\mathbf{a}) \\mathbf{b} =\\mathbf{a} (\\alpha \\mathbf{b}) = \\alpha (\\mathbf{a} \\mathbf{b})", "d6c430b620376764b3b4546e3877f9b6": " \\sigma(I) ", "d6c4333177bf306a31d2d555e4a31a7a": " Q = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix} ", "d6c43b99f7266d06025017c819f8a0b1": "\\frac{dy}{dx} = \\frac{\\dot{y}}{\\dot{x}} = \\frac{3}{8t},", "d6c44d31881365fcd1aadcef28369341": "a + b + c\\,", "d6c49c42fa5fd71043f378b57083d3b3": "\\overline{A_i}", "d6c4ed3fb6263f09f9775bd6785ed5dc": "k = \\frac {K \\rho_w g} {\\mu_w}", "d6c5068cee0c8db0848e6895076cb733": "P_1(\\sigma_k)", "d6c5353019263ca51429a635b6695b24": "\n\\left| \\frac{\\partial Y}{\\partial X_i} \\right |_{\\textbf {x}^0 }\n", "d6c53cd0b16a941cd6177384bc111f09": " E_B = I_B / I_{B( max )} ", "d6c5d03dff832e301c604a6f7e42498a": "\\langle E \\rangle = \\sum_s E_s P_s = \\frac{1}{Z} \\sum_s E_s\ne^{- \\beta E_s} = - \\frac{1}{Z} \\frac{\\partial}{\\partial \\beta}\nZ(\\beta, E_1, E_2, \\cdots) = - \\frac{\\partial \\ln Z}{\\partial \\beta}\n", "d6c5fc0e3e67d19212dd048ac04c6ae2": "\\tau^{-1}", "d6c631232045dc6f9666783b2c3ad679": "\\operatorname{E}[\\,\\textstyle\\sup_{\\theta\\in\\Theta} \\lVert g(Y,\\theta)\\rVert\\,]<\\infty.", "d6c651d99c5ad339c5d22bcf43b3c906": "P_1^1,P_2^2,P_3^0", "d6c674160b7fa68d20d1f25f025348c3": "R/N", "d6c6c88bf12a6b7f2ea9b47632191ef0": "R_P = R_1R_2+R_2R_3+R_3R_1", "d6c7055090b009404863d53365dff020": " n : S \\rightarrow \\mathbb{N}", "d6c71372e9b4a65e839f70ca148999f4": "V \\to V^*", "d6c71ec5d7e95c9e36fdb8ba6e917f9f": "\\mathrm{P}(u,v)= \\mathrm{A}(u,v)\\cdot\\mathrm{exp}(i\\,\\mathrm{\\Theta}(u,v))", "d6c7330e2a4c97454011b3990b2a3ca0": "\\sigma L=\\tau L+\\mathbf{Grz}.", "d6c7681d656aa98697ff9bba96710be1": "V_h", "d6c7aaa37de05277bdeacdb707c6e106": "\\vDash \\!\\,", "d6c81bc6868eb82315178664a4849ab9": "k_0 = |\\mathbf{k}_0|.", "d6c8734eb508f422f60892dd647930d2": "1 - Nx^2 = y^2", "d6c87401ed2f8589600d6db807438139": "j^{th}", "d6c8a425b970584da975882a61a924ec": " 1 \\to \\{\\pm 1\\} \\to \\mbox{Spin}_V(K) \\to \\mbox{SO}_V(K) \\to K^*/K^{*2}.\\,", "d6c8b0a98cd82d6194fd87f722e21ff6": "f(t_2)/f(t_1)", "d6c8b53b98f4b51ff7efff3fd7d560a8": "\\Delta_n", "d6c8c124f944b299ccf6575d55924571": "p+q=f(\\mathbf{AA})+f(\\mathbf{aa})+f(\\mathbf{Aa})=1", "d6c91b6714c195a02972f93cb56116c7": "\\boldsymbol{v}_R", "d6c92131e01b39ef52691a40a32186df": "\\widehat{\\sigma^2}\\approx \\!\\,\\frac{1}{2N}\\sum_{i=1}^N x_i^2", "d6c95f5d691da401bbfdd0e7f72726c3": "\\phi_0 = U + V\\cos \\Omega t .\\qquad\\qquad (7) \\!", "d6c9fc2188decadd8bc67d1993d017af": "D_2=1P_1+ 5P_2- 3D_{\\infty_1} -3D_{\\infty_2} ", "d6ca304b17f571e5858310eb5ef9adbc": "A_i \\rightarrow \\delta_1\\gamma | \\ldots | \\delta_k\\gamma", "d6ca8e0ab0e5ce204634d70bb2bc71dc": "W_\\text{max}=\\frac{1}{2}\\cdot C_\\text{total} \\cdot V_\\text{loaded}^2", "d6ca95b1b34c41c690496c59465c6ae2": "N_{\\rm packing}(\\varepsilon)", "d6caaa2647b6d1f90bbfe9556156fafa": "(x,f) = \\sum_{g\\in G} a_g f(g),", "d6cad8940ab7f9e3ba2329a6d2464716": "{\\mathbf e}_i\\,", "d6cae1f7242d14dda9f7d8b062f37baf": "\n \\frac{\\partial }{\\partial \\boldsymbol{A}} \\left(\\boldsymbol{A}^{-1}\\right) : \\boldsymbol{T} = - \\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T}\\cdot\\boldsymbol{A}^{-1}\n", "d6caea6d79f1f6fa5f7d90ad58b06c4c": " H(s) = C(sI-A)^{-1}B+D", "d6cafc4dab1b7bfa9f7bb6f4a2cac565": "I_{tcr-max} = {V_{svc}\\over{2 \\pi f L_{tcr}}}", "d6cb53066620ed27a3e64d7b45f2ccf8": "\n p = \\cfrac{2}{\\lambda^4}~\\cfrac{\\partial W}{\\partial I_1} ~.\n ", "d6cb745e85de8155d1c227fe2c400da4": "\\mathfrak{so}(2)\\cong \\mathfrak{u}(1)", "d6cbe1c46f9a7ffdcc7a50b089babbd4": "R_n^{(l)}(\\rho)", "d6cbff8309e1bf5832ad6611fbd3af25": "V_i \\otimes V_j = \\bigoplus_\\ell H_{i,j}^\\ell V_\\ell", "d6cc1157b5fa3e9a7cf332c808a48565": "\\ E_s(f) = |X(f)|^2 ", "d6cc73eb20e0ecc2c0feddbe39a7c93d": "\n\\begin{bmatrix}\n \\lambda_1 & 1 & 0 & 0 & 0 \\\\\n 0 & \\lambda_1 & 1 & 0 & 0 \\\\\n 0 & 0 & \\lambda_1 & 0 & 0 \\\\ \n 0 & 0 & 0 & \\lambda_2 & 1 \\\\\n 0 & 0 & 0 & 0 & \\lambda_2\n\\end{bmatrix}^n\n=\\begin{bmatrix}\n \\lambda_1^n & \\tbinom{n}{1}\\lambda_1^{n-1} & \\tbinom{n}{2}\\lambda_1^{n-2} & 0 & 0 \\\\\n 0 & \\lambda_1^n & \\tbinom{n}{1}\\lambda_1^{n-1} & 0 & 0 \\\\\n 0 & 0 & \\lambda_1^n & 0 & 0 \\\\ \n 0 & 0 & 0 & \\lambda_2^n & \\tbinom{n}{1}\\lambda_2^{n-1} \\\\\n 0 & 0 & 0 & 0 & \\lambda_2^n\n\\end{bmatrix}.", "d6ccaf3e35171f7232cc01133d7652d0": "= \\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}", "d6ccb7a0fbb03b7f0a4d9581155791a1": " \\vec{J_M} = \\nabla \\times \\vec{M}. ", "d6ccbf24215a9ef4a80370e2bbd1392d": "\nE^{(t-1)} ~\\ge~ \\min( E^{(t)}_0, E^{(t)}_1 ).\n", "d6ccce9489a1192ce2288d4898af764a": "G = 1 - \\frac{\\Sigma_{i=1}^n \\; f(y_i)(S_{i-1}+S_i)}{S_n}", "d6ccd8653c07cc022d8a082293445a8c": "\\hat{H} = -\\frac{\\hbar^2}{2m}\\nabla^2 ", "d6ccebda20bc9a67fa9253567e995ffa": " a*r(t) = \\frac{\\mathrm{d}P}{\\mathrm{d}t} ", "d6cd67c12f2ada998146105c19e3f502": "\\operatorname{Hom}_\\mathfrak{g}(V, W) = \\operatorname{Hom}(V, W)^\\mathfrak{g}", "d6cd9e0543c462e481f1369367cbf04e": " \\textbf{P}_{k\\mid k} = \\textbf{P}_{k\\mid k-1} - \\textbf{K}_k \\textbf{H}_k \\textbf{P}_{k\\mid k-1} - \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^T \\textbf{K}_k^T + \\textbf{K}_k \\textbf{S}_k \\textbf{K}_k^T", "d6cdb9fd0fbf9400d1cfb64af9cdf863": "r = \\frac{\\left\\|\\mathbf{a}\\right\\|\\left\\|\\mathbf{b}\\right\\|\\left\\|\\mathbf{a}-\\mathbf{b}\\right\\|}\n {2 \\left\\|\\mathbf{a}\\times\\mathbf{b}\\right\\|}\n = \\frac{\\left\\|\\mathbf{a}-\\mathbf{b}\\right\\|}{2 \\sin\\theta}\n = \\frac{\\left\\|\\mathbf{A}-\\mathbf{B}\\right\\|}{2 \\sin\\theta},", "d6cdc2dc86a25ee91e31d2fbac5b759c": "\\frac{r_1 + r_2}{2} \\,\\!", "d6cde21c2a03a2faa986ff2575dd2a3d": "\n\\forall y\\in Y \\qquad \\sup_{x\\in B}|\\langle x, y\\rangle|<\\infty.\n", "d6ce28bf43a5555f7709a18073c672ed": "(\\mathbb{N}^{|P|},\\le)", "d6ce3b2d10e1b973977ceb2566e78670": "\\rho~c~\\frac{\\partial \\theta}{\\partial t}~+ ~\\nabla \\cdot \\mathbf{q}~=~ 0 ,", "d6ce68c1c7cf3e1a0d47d917caeb6c21": "\\phi(\\boldsymbol{x},t)", "d6cec090bce3cedafebcf4b143656976": "t\\begin{Bmatrix} p , 2 \\end{Bmatrix}", "d6cee1dd21645e06c6958efcdb22e963": "\\text{Gain}=20 \\log \\left( {\\frac{V_\\mathrm{out}}{V_\\mathrm{in}}} \\right)\\ \\mathrm{dB}", "d6cf0b3622808d59d745c8959c0710af": " C_2 = Tr[(XY)^2] - \\dfrac{1}{4} Tr[XY]^2 +\\frac{1}{96}\\epsilon_{ijklmnop}\\left( X^{ij}X^{kl}X^{mn}X^{op} + Y^{ij}Y^{kl}Y^{mn}Y^{op} \\right)", "d6cf27c146426b65c02d2dbfa389e2a7": "i\\in\\{1,2\\}", "d6cf613f194f2ec8fbb6b60f048c44cf": "p \\wedge \\neg p", "d6cf6e60d822332890c87e3b3ff30470": "(X_{4}=0,Y_{4}=0,Z_{4}=R)^{T}", "d6cff9f906f522cf34fc12d02cbdf2b7": "x = a \\sec(\\theta)", "d6d034023e34ebff04e68a3c5d139099": "\\phi_{c_{start},c_{accept},T}", "d6d035958ca7aa50c48d2c9014eaeba3": "M \\circ N = \\sum_{ij} \\mu_i \\nu_j (m_i m_i^T) \\circ (n_j n_j^T) = \\sum_{ij} \\mu_i \\nu_j (m_i \\circ n_j) (m_i \\circ n_j)^T", "d6d03ba72a8a7acbd210fe45879c1653": "C^{\\log(1/\\delta)^{k-1}} \\leq N(k,\\delta) \\leq 2^{2^{\\delta^{-2^{2^{k+9}}}}}", "d6d04d05fb06370fb996e8080f536ccf": "M \\leftarrow \\empty", "d6d063dc16ec49cedd010a15c7308acd": "r,ar,br,abr", "d6d066e5713582bfb2611fa65fb236ee": " t( x, w ) = \\frac{ 1 }{ \\sqrt{ 2 \\mu } } \\left. (\\frac{2}{3} x^{3/2} + \\frac{1}{5} w x^{5/2} + \\frac{3}{28} w^2 x^{7/2} + \\frac{5}{72} w^3 x^{9/2} + \\frac{35 }{704 } w^4 x^{11/2} \\cdots ) \\right|_{ -1 B) \\cdot P(A > B | A=a).\n\\end{align}", "d6dec39b28ae3ddd249138e5f59596df": "\\Sigma_k", "d6dece7d8c25f51db599559cb92864fe": "Q(\\mathbf{y},.)", "d6ded6d42f76d97c84da41a6979b0613": "H_2/H_1 \\cong ({\\mathbf Z}/p{\\mathbf Z})^k", "d6defec3b4f4ce29cb506565b9039de2": " \\kappa(A) ", "d6df119f865181a65d3d4c3e622ef453": "|\\textbf{x}| = \\sqrt{3^2 + (-4)^2} = 5", "d6df5e4858e11e624e3b7f85e5cbfcc5": "P(A|B)\\propto P(A) P(B|A)", "d6df6d9e76b25dec65f73a5973b2958b": "\\psi_E", "d6dfac2422fd95248fd4a3db24a32e1d": "\\tan(15\\pi/16)", "d6e00500bfc7d4949a38c1371952c593": "R^m_{\\ell}(\\mathbf{r})", "d6e00eab5b7bf50b0c3292d4dc3dbc0e": "\\sum\\limits_{n}{{{x}^{2}}(t)}=\\frac{\\sum\\limits_{n}{{{\\left| X(f) \\right|}^{2}}}}{n}", "d6e07a328bdc4519d910675b2c89e280": "[p] + [q] = [p \\oplus q]", "d6e0932fcdc70f4c5c82f42ce664de0d": "\\| g \\|_2 = \\sqrt{{1 \\over 2\\pi} \\int_{-\\pi}^{\\pi} |g(x)|^2 \\, dx}.", "d6e10c332bb67fd019271fbc1d62facc": "\\frac{\\pi^3}{6} R^6", "d6e1337bfcaf5373221c2a83c9f1539a": "\\ell=0,\\quad m=0", "d6e137d75bdc24fc2af9b9c9e5c72ded": "O(k^4)", "d6e161a740b54fcb6a6c9f0fbe44a132": "V(\\mathbf{k})=-g^2\\frac{4\\pi}{k^2+m^2}.", "d6e1b8204cc7d8d1dacfd4ef5afb4063": "g(x) = \\frac{1}{\\sqrt{2\\pi}\\cdot\\sigma}\\cdot e^{-\\frac{x^2}{2\\sigma^2}}", "d6e1c90a1c099d4f88953f318ca117e9": "\\mathrm D_{\\mathsf C}\\leq \\mathrm D_{\\mathsf VC}+1", "d6e20bf9092427e2fb496bd35b6a6cbc": "F_3(a, b) = a^b", "d6e22a8638550ac70425a0151ae95901": "z_0=c", "d6e3838da5348b9b28ad738cd0b84b74": " \\frac{g}{r_s} \\left( \\frac{r}{c} \\right)^2 = \\frac{1}{2} ", "d6e3af948a34fd5f432cb9d377a98ef0": "f(t)", "d6e3d751409fa58ba48f177b596994e9": " \\alpha \\in \\mathbb{Z}^{n}_{q} ", "d6e3ede8ceb3f37498dc0be07b9d37a6": " {d X^\\mu \\over dT}={d x^\\nu \\over dT} {\\partial X^\\mu \\over \\partial x^\\nu} ", "d6e45d43353a2b6a4614155bade8df58": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 \\\\\n 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2\n\\end{smallmatrix}\\right ]", "d6e4ef4c858eff7ba83588366b8b117f": " P = \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v \\cdot (v_1^2 - v_2^2) ", "d6e570a0fcce94f451495febe57ce6cd": "(E-H_{mm})a_m = \\sum^{A}_{n\\neq m}H_{mn}a_{n} + \\sum^{B}_{\\alpha \\neq m}H_{m\\alpha}a_{\\alpha}", "d6e592cce611455d9a9dc67606bb00a8": "z_{n+1}=4z_{n}(1-z_{n})", "d6e5a638d478490a011065bec34accf1": "\\ 0", "d6e5ea3da10942e8c3b6ad366e890835": " t \\ln t \\, , -\\ln t", "d6e632bebb2c96d4adb6d2052de082ff": "|\\Psi^-\\rangle ", "d6e638c2f08cbaeb86b96e65d6bc5047": "B=\\mu_0\\cdot M", "d6e6596ee1fa3098c224802ce4b8e8b7": "B = \\frac{\\mu_0 \\mu_r I N}{l}", "d6e65ecc0ac47a7ce6b34c82a84b9597": "z=\\infty\\,", "d6e6820a490114fe19710bb139b690b0": "x^1(t),...,x^n(t)\\,", "d6e6b9139f74fd42ea768582519bb7a9": "Y_m", "d6e7d7c70b85086e36706d01d9732009": "\\frac{355}{113}-\\frac{911}{2\\,630\\,555\\,928}<\\pi<\\frac{355}{113}-\\frac{911}{5\\,261\\,111\\,856}\\,.", "d6e810848751a36ed9d9739110e66dd7": "{(n+1)(n+2)\\over 2}", "d6e8548ca9472805870ac159e4d79e48": "E_3", "d6e86dab55aa2b7a42e891ab8fa00eed": " X(\\omega) = \\int_{-\\infty}^{\\infty} x(t) e^{-j \\omega t} \\, dt. ", "d6e88bddca085ba6c8a56956e338d551": "\\frac{dC}{dt}", "d6e892905048774de0e12f8ae7f28c41": " f(D+h)v-F\\cos \\left(\\frac{\\pi y}{b}\\right)-Ru-g(D+h)\\frac{\\partial h}{\\partial x}=0\\qquad (1) ", "d6e8fa499c1120e2a75f5b2494c1f544": "S^{\\alpha\\beta\\mu}(\\mathbf{x})\\ \\stackrel{\\mathrm{def}}{=}\\ M^{\\alpha\\beta\\mu}_x(\\mathbf{x})=M^{\\alpha\\beta\\mu}_0(\\mathbf{x})+x^\\alpha T^{\\beta\\mu}(\\mathbf{x})-x^\\beta T^{\\alpha\\mu}(\\mathbf{x})", "d6e91e6895dc4fe02f11ac4196543fc2": " g_F = g_J\\frac{F(F+1) + J(J+1) - I(I+1)}{2F(F+1)} + g_I\\frac{F(F+1) - J(J+1) + I(I+1)}{2F(F+1)}", "d6e92a90f5f6c4c65a88f2ba1f57a4bc": "a_f^{n=0, \\ldots, N-1}", "d6ea0ff4e0578499f7aa5bf633920f96": "\\mbox{isqrt}( n ) = \\lfloor \\sqrt n \\rfloor.", "d6ea278b83bb4af140bdd8b29dc99c96": "p^2+(1-p)^2", "d6ea410c011e74d57b790f4a2ba8baa9": "f(x)=-x^2", "d6ea4ac095669e047a3bd62867d6f144": "H^*(G) \\rightarrow H^*(G\\times G) \\cong H^*(G)\\otimes H^*(G)", "d6ea82b3ff3d07d5a57dedd8e006bcb3": "Y^{-1}(B)\\in \\mathcal A", "d6eabe8c8a74f0ba993fe8322b475056": " d^k \\frac {\\alpha^k} {k!} w^{k-2}.", "d6eb48dc8f6ea5a3e6e7fb1030fd098e": "\\frac{V(t)}{V_0}-1=-e^{-\\frac{t}{\\tau}}", "d6ebf84ffb0d5a094ae5ffb5e847a320": " \\eta_{ij} = 1 \\,\\!", "d6ec5b45df0242c34adaf5d99e53918c": "x \\in (A\\cap B)^c", "d6ec789f7c5a00dc61112c49266e8ae7": "x = \\frac{-b \\pm \\sqrt {b^2-4ac}}{2a}.", "d6ed012f8d354f2a245cb4f779ec0157": "\\mathop {\\lim }_{P^\\circ \\to 0} f^\\circ = P^\\circ", "d6ed9891f30b38c16d1f220a6f4c967a": "\\|\\mathcal{L}f\\|_{H^2} = \\sqrt{2\\pi} \\|f\\|_{L^2}.", "d6eded0c78521a2cccea765788e9f646": "\\mathcal{B}_{\\epsilon}(X^*_{b}, Y^*_{b}; Z)", "d6ee12a0cad513f674b1923f1d02cb41": "p_p\\,", "d6ee18023ea6ea64d55c9d0da3e0b47d": "M_{k}=mk\\left( 1-\\frac{p^{2}}{6n^{2}}\\right)", "d6ee2f1e2d9af8ec29edca6a5d3221e6": "CA_0", "d6ee35bd7e51a5618b0eba4ce303b6ee": "B\\geq 0, \\epsilon>0", "d6ee8c1b334269e7015645b5956ca733": " -k_B T \\ln \\mathcal{Z} = \\Phi_{\\rm G} = \\langle E \\rangle - TS - \\mu \\langle N\\rangle. ", "d6eeaff2e68b6bb4f9367f275fee71c2": "(\\mathcal{L}_{\\!X} f)(p) \\triangleq \\operatorname{d}f_p\\, (X_p)", "d6eef453f694389171109ede07bf2b3f": "\\scriptstyle 2/\\pi\\,", "d6efe47796a20636a224c3f6f68c7915": " F : |K| \\to |L| ", "d6f005ab2be1ab851ce92ec2fd7dfb37": " x^2 - y^2 = 0 \\pmod{b} ", "d6f05930e698d9a030b47cd12a2674c5": "P_X(q,p)=p(X_q)", "d6f0641d9a365efceffd79e6590517d2": "(p-k)!", "d6f066f6feed2495a35310759d097093": "E(m, \\Theta) = \\sum \\phi_x(D|m_x) + \\phi_x(m_x|\\Theta) + \\sum \\Psi_{xy}(m_x, m_y) + \\phi(D|m_x, m_y)", "d6f088d47aed9b9f47e0165297f806c1": "x_1(t) > 0", "d6f08fefb5d3b7adf5f598ada991c8e5": "\\frac{\\cos (nx) + i \\sin (nx)}{(\\sin x)^n} = \\frac{(\\cos x + i\\sin x)^n}{(\\sin x)^n} = \\left(\\frac{\\cos x + i \\sin x}{\\sin x}\\right)^n = (\\cot x + i)^n.", "d6f0cd9491eeb14aedf6a78b5576b369": "-i \\epsilon^{\\sigma 0 1 2} \\gamma_\\sigma \\gamma^5 = -i\\epsilon^{3 0 1 2}\n(-\\gamma^3) (i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3) = -\\epsilon^{3 0 1 2} \\gamma^0 \\gamma^1 \\gamma^2 = \\epsilon^{0 1 2 3} \\gamma^0 \\gamma^1 \\gamma^2", "d6f0e1c702e6dbc900031b17c8e012b2": "\\displaystyle \\omega", "d6f12283f441e37d4aa4b214c97e231b": " s\\cdot x = s(x)\\text{ for }s\\in S, x\\in X.", "d6f13b4951e97b3804a9c593e44c4be8": "\n \\cfrac{\\tilde{p}}{\\tilde{\\rho}} = \\gamma~\\cfrac{\\langle p \\rangle}{\\langle \\rho \\rangle}\n = c_0^2 \\qquad \\implies \\qquad\n \\cfrac{\\partial\\tilde{p}}{\\partial t} = c_0^2 \\cfrac{\\partial\\tilde{\\rho}}{\\partial t}\n ", "d6f14e63a956dbe42398c1b8f473ccbb": "\\Delta_2 f_\\lambda= (\\lambda^2 +{1\\over 4})f_\\lambda.", "d6f15eb90fe216174b2aa5533d28a01c": "\\begin{bmatrix}\n\\alpha^{ck_1} & \\alpha^{ck_2} & \\cdots & \\alpha^{ck_{d-1}} \\\\\n\\alpha^{(c+1)k_1} & \\alpha^{(c+1)k_2} & \\cdots & \\alpha^{(c+1)k_{d-1}} \\\\\n\\vdots & \\vdots && \\vdots \\\\\n\\alpha^{(c+d-2)k_1} & \\alpha^{(c+d-2)k_2} & \\cdots & \\alpha^{(c+d-2)k_{d-1}} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nb_1 \\\\ b_2 \\\\ \\vdots \\\\ b_{d-1}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0 \\\\ 0 \\\\ \\vdots \\\\ 0\n\\end{bmatrix}.\n", "d6f1617ca48b830f8ec65347bf3dc4ee": "A = \\pi R^2 - \\pi r^2 = \\pi(R^2 - r^2)\\,.", "d6f1c5ec613f3eb15c29e180de431b00": "[a_i,a_{i+1}]", "d6f1c8d44972a1a6c13bc86137a52ee7": "155.76\\times 10^{-6}", "d6f1f0fe847459a9a67bd3b6758c618c": "Y_{left}", "d6f23b9dc8b3a814b45bff493e5fcdb7": "\\textstyle \\bar{m}=\\frac{1}{k} \\sum_{i=1}^{k} m_i", "d6f244f700007ac6ec39f23cd8436eb6": "\\psi(\\omega)", "d6f26aa26201f100f1d8dc7275d4ed03": "P\\propto x^{-4}\\,,", "d6f27f03c0c9d026c61ac07adf76aa27": "\\psi=\\begin{pmatrix} \\chi \\\\ \\eta \\end{pmatrix}", "d6f3703016be9dea4e5e725573189518": "(P \\land Q) \\to Q", "d6f3dc596550e3544e8d8ee7361864ae": " p_n < n \\log n + n \\log \\log n \\quad\\mbox{for } n \\ge 6", "d6f40c42586c7493971a34b0c7c78968": "H = (X,E)", "d6f4309f46016dfedf0a63c9a6470be7": "d_i = c/\\omega_{pi}", "d6f47b0a8f619847a59901aba9176497": "f^{(k)}(s)=(-1)^k\\sum_{n=1}^{\\infty}a_n\\lambda_n^k e^{-\\lambda_n s}.", "d6f47d32dd8fbbb4865fb1a5c213beec": " \\alpha >1 ", "d6f5388e38004fc9875f7acb3db8df51": "L_1(x) = 1 - x\\,", "d6f54fbc9fc5f9f2ce2edb2ca60a69d1": "{d{I_{1z}} \\over dt}=-R_z^1(I_{1z}^0-I_{1z}^0)-\\sigma_{12}(-I_{2z}^0-I_{2z}^0)=2\\sigma_{12}I_{2z}^0", "d6f5db1a80385cf8d28491b8fd13549e": "365 = 10^2 + 11^2 + 12^2", "d6f6243deac9a12ea0154b6ab1af07b1": "\\mathbf{a}\\, ,\\mathbf{b}\\, ,\\mathbf{c}", "d6f6416b0c4123817d4e64396eb34564": "S_B = \\begin{matrix}\\frac{4}{10}\\end{matrix}", "d6f68c0c7f14d9f37b7df946ea37ab4b": "\\begin{bmatrix}r\\\\0\\\\c\\end{bmatrix} , ", "d6f6b72f29f494d15ed74390aa59f8a6": " \\deg P'=\\deg P", "d6f6d117b9c9d339408143cd6c0dad86": "\\phi(\\mathbf{r}-\\mathbf{R}) := \\phi_{\\mathbf{R}}(\\mathbf{r})", "d6f734c38e72d759ee93aac40958cb9d": "\n\te_2 = 37.1014913651276582\n", "d6f7a3be9a49b686a4764d56e5bd2e11": "(a_k)_{k \\in \\mathbf{N}}", "d6f81c56fe7a3129122604426390ebda": "B,", "d6f830e6003c5a9881670c880f52de88": "\\vec{p}_{f1},\\vec{p}_{f2}", "d6f835d4f670b948cc79008208c9429b": "\nu(\\varphi) = u_{1} + (u_{2} - u_{1}) \\sin^{2} \\left( \\frac{\\varphi - \\varphi_{0}}{2} \\right) \n", "d6f84c48c4618d9533e0f907282ff142": "\\vartriangle^1_n", "d6f858796e21ad5074cde8e01ba3ee4c": "\\cos\\frac{\\pi}{30}=\\cos 6^\\circ=\\tfrac{1}{8} \\left[\\sqrt{2(5-\\sqrt5)}+\\sqrt3(\\sqrt5+1)\\right]\\,", "d6f8621b183a6907935b456cf647e1a4": "M = \\mu(\\mathbf{x}, \\sigma_{\\mathit{I}}, \\sigma_{\\mathit{D}})", "d6f89120f9be81e433fdb5464113e3d8": "p_{orig}(x)", "d6f89c76fb9533fb1492780384648c31": "\\Sigma_{i}^{\\rm P}", "d6f8a526dc547f63a1b8b585a1f14b03": "\\mathrm{^{235}_{\\ 92}U\\ +\\ ^{11}_{\\ 5}B\\ \\longrightarrow \\ ^{242}_{\\ 97}Bk\\ +\\ 4\\ ^{1}_{0}n \\quad ; \\quad ^{232}_{\\ 90}Th\\ +\\ ^{14}_{\\ 7}N\\ \\longrightarrow \\ ^{242}_{\\ 97}Bk\\ +\\ 4\\ ^{1}_{0}n}", "d6f8f54809f5df089df9092e49dba1a6": "\nv_{\\text{rms}} = \\sqrt{\\langle v^2 \\rangle} = \\sqrt{\\frac{3 k_B T}{m}} = \\sqrt{\\frac{3 R T}{M}},\n", "d6f90a67020f1792bec914d5edb22132": "L_{\\mathrm{MI}} \\ll L_{\\mathrm{fiss}}", "d6f9a7746d477d171124bf8310bd472b": "\\frac{1}{2} \\phi\\cdot R_k \\cdot \\phi", "d6f9aae515ead32514486adcaf10af3a": " conf_{i} ", "d6fa33190e767d770a9f0e5743d6a8d6": " ModD = \\frac{MacD}{(1+y/k)} = \\frac{1.777}{(1+.04/2)} = 1.742% ", "d6fa695e07f4e18eb34ec8ddcde7283e": "\\alpha= \\frac{\\pi}{4n}\\left(2n-1\\right)", "d6faba0b75964a75e36e628ea38d73d5": "\\lambda_1(A),\\lambda_2(A),\\dots", "d6facfdaf5773b1e7996d4521161a057": "\\hat{H} = \\frac{\\hat{p}^2}{2 m l^2} + m g l (1-\\cos(\\phi))", "d6fad5522ed6cd168f1528b4f501a19b": "\\Omega^{1/2}", "d6fb0d251f51bed8c5f9e30964347039": "\\boldsymbol{v} =\\frac{d\\mathbf{r}}{dt} = \\dot r\\hat{\\boldsymbol{r}} + r\\dot\\theta\\hat{\\boldsymbol\\theta},", "d6fb0e617a3640b48b6a3b91333e0e40": "H^r(X, \\mathbf{Z}) \\to H^r(X, \\mathbf{Z}/2).", "d6fb46e3d327843fa2e9baeeda3c7999": " h\\,\\ell\\le \\int_0^h\\int_0^1\\rho(i\\,y+w\\,t)\\,w\\,dt\\,dy", "d6fb624380107b797a4ccf93f5c6aa09": "\\sqrt{4 k_B\\cdot T\\cdot B/R}", "d6fb75b4aa643c241151a357fd3bfca9": "M(t) = \\sum_d n_d t^d.", "d6fba444406884cf6b210211a53c4c14": " {Q}\\ =\\ \\frac{{B_0}^2 d^2}{\\mu_0 \\rho \\nu \\lambda} ", "d6fc015dc5c13a6770417c9685861a76": "\\mathbf{B} = \\mathbf{\\nabla} \\times \\mathbf{A}. ", "d6fcf131dd151f43cf7c61fadf00cf9f": " \\omega = 1 - m_x \\operatorname{cov}( x, y ) \\, ", "d6fcfa5324bffde2e092817edb6191e5": "A_2 x \\le b_2", "d6fd0924e324f50669ae0295adf59567": "mc", "d6fd0e7c73cea3d927081de4b7d50f3f": "\\mathbf{I}_O :=\n\\begin{cases}\n1 &\\text{if } x > median, \\\\\n0 &\\text{else }\n\\end{cases}\n", "d6fd156726f07d19db5f8766bd2a479a": "\\mu_0 \\epsilon_0=1/c^2", "d6fd5fc5ab713279f0457f555ec55507": "(f\\star g)\\star(f\\star g)=(f\\star f)\\star (g\\star g)", "d6fd79f72cd88af839678618f8bcce9a": " (-1)^{n+m} n! \\; [z^n] g_m(z) = s(n,m) ", "d6fda63297c09d361d80d7ab987bf7e3": "Y/2 ", "d6fdad8d0abfd435531ff1a7f16114b7": " \\det \\, X = t^2 - x^2 - y^2 - z^2. ", "d6fdb8e048d336e93d264751d91158a9": "\\textstyle w_i", "d6fdbd4209587e5a5eb3c793017935cb": "\\psi(\\zeta_0+1)", "d6fdbdc5f9e52a005be48999c7d7de5f": "1-c_i", "d6fe09493559ea12bcd8fe79a0dd9050": "p_c=\\tfrac{1}{\\langle k\\rangle}", "d6fe2f42f5f8a0e70eb44e5c206466f2": " \\tilde{q}\\tilde{\\bar{q}} \\rightarrow q \\tilde{N}^0_2 \\bar{q} \\tilde{N}^0_1 \\rightarrow q \\tilde{N}^0_1 \\ell \\bar{\\ell} \\bar{q} \\tilde{N}^0_1 \\rightarrow", "d6fe4a788c1c17ad2bde73d024705e23": "money\\ ratio = { positive\\ money\\ flow \\over negative\\ money\\ flow }", "d6fe4ef8dac7579202bcdb0d0880d76f": "V_\\mu ^\\prime =\\left( O_\\mu ^\\nu V_\\nu +P_\\tau \\right) \\left( \\delta _{\\tau\n\\mu} + Q_{\\tau \\mu }^\\nu V_\\nu \\right) ^{-1}. \\, ", "d6fe4f8db9a9cdf2f618256de2f6895a": " UP+VQ = D ", "d6fe80f991abc97a0e35cc42b0a42721": "{}_pF_q(a_1,\\dots,a_p;c_1,\\dots,c_q;z)\n= \\sum_{n=0}^\\infty\n\\frac{(a_1)_n\\cdots(a_p)_n}{(c_1)_n\\cdots(c_q)_n}\n\\frac{z^n}{n!}", "d6fed375d064f53e31ed32cf9ccb30a0": "({c^{(\\dagger)}}_{\\nu})", "d6ff084a1d119b0151c82039905c15b1": "(0 \\le \\xi \\le L)", "d6ffcc8ddacfe7539a83b11023968a5f": "\\begin{cases}\nQ_1 = 15 \\\\\nQ_2 = 37.5 \\\\\nQ_3 = 40\n\\end{cases} ", "d700ebe0b8a0e13c54fbfd0bc3e14471": "c_{ijk\\ell} = c_{k\\ell ij}\\,", "d700ed9d365e5fdabe992a44ab33eee6": "\\log_{2}\\left(n\\right)", "d70103bf31f9c8e595fce79532b8416c": " \\delta_{kl}\\,\\!", "d7014aa1ba2484a3deb8d8c32e99c58a": "n, r_i", "d7015f8438c6e4b194af1b8b84fd1099": "z_{\\rm{bias}}", "d7017b716dde1b3c762d010fa1ed89b6": "i_i=0", "d7018200fdc2b3cfbcf1e888984938a5": "\\left\\langle\\mathbf{P},\\mathbf{P}\\right\\rangle = P^\\alpha\\eta_{\\alpha\\beta}P^\\beta\n= \\begin{pmatrix}\nE/c & p_x & p_y & p_z\n\\end{pmatrix}\n\\begin{pmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & -1 & 0 & 0 \\\\\n0 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & -1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nE/c \\\\ p_x \\\\ p_y \\\\ p_z \n\\end{pmatrix}\n = \\left(\\frac{E}{c}\\right)^2 - p^2\\,,", "d7018d8d9a8b2ea162af6021db0d9d56": "\\Omega= \\sum_{ij} g_{ij} \\; dq^i \\wedge d\\dot q^j +\n\\sum_{ijk} \\frac{\\partial g_{ij}}{\\partial q^k} \\; \n\\dot q^i\\, dq^j \\wedge dq^k", "d7020c0c67fb703c71572a036585177b": "m_{\\mathrm{e}} \\,", "d7023de41f8327f645c11b79bac47dc6": "p(a)", "d7025a3092f6b8d61a3b4b004ddcc45a": " GDP = f(T) \\,\\! ", "d7026b6cfb6a30e8599f7b4e75ad5ffc": "2(-2)^n\\,\\frac{\\Gamma(n+3/2)}{(n+1)\\,\\sqrt{\\pi}}\\,", "d702762382ebf8fd7c3eecfefdc48455": "\\omega^p = -\\omega", "d7027930e6002a04fd7eb4b52c67bec7": "e^{-\\frac{1}{2}\\|y\\|_{H_0}^2}", "d7027d6bb709762374eb158104255832": "t\\in R^n", "d7029543fe37a0bee2f4bf51c5758da3": "g(x) = (x-\\alpha)(x-\\alpha^2)\\cdots(x-\\alpha^{n-k}) = g_0 + g_1x + \\cdots + g_{n-k-1}x^{n-k-1} + x^{n-k}\\,.", "d702b5047231d56ce416b505009b6971": "(x_1^2-x_0)^2 + (2x_1)^{2}x_0 = (x_1^2+x_0)^2 ", "d702d7675149ac55a8a60551eabe231d": "\\nabla_{(a}X_{b)} -\\frac{1}{n}g_{ab}\\nabla_{c}X^{c}=0,", "d702f515b32d50ec075582bac058ecb9": "\\displaystyle{S=\\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix},\\qquad R=\\begin{pmatrix} 1 & 0 \\\\ 1 & 1\\end{pmatrix}.}", "d70312bbc076fc693187d80bee922db7": "(i_X\\omega) (X_1, \\ldots, X_k) = \\omega (X,X_1, \\ldots, X_k)\\,", "d70439a309579be70a4d0be64df2a6e0": "\\mathcal{E} = -\\frac{\\mathrm{d}\\Phi_B}{\\mathrm{d}t}", "d704b0ce90a256eaa4dd14fd8571d130": " Oxy \\leftrightarrow \\exist z[Pzx \\and\\ Pzy].", "d704be5be04e75cbf6bf537459db9655": "\\pm\\frac{\\sqrt{1 + \\tan^2 \\theta}}{\\tan \\theta}\\! ", "d704dc9ce3f287dfe843702cdcb0620a": " \\left(\\begin{bmatrix} 1 & 1\\\\ 0 & 1 \\end{bmatrix}- \\begin{bmatrix} 1 & 0\\\\ 0 & 1 \\end{bmatrix}\\right)\\begin{bmatrix}v_{21} \\\\v_{22} \\end{bmatrix} = \\begin{bmatrix}1 \\\\0 \\end{bmatrix}.", "d70505d8d98fa96120660622d85683d8": " 100MCalories(feed) * 10% (ECI)(meet/feed)=10Mcalories(meat) ", "d7051229466fda0f7f6f9b8297ba1897": "\\mathrm{O}^\\times_\\mathrm{surface}", "d705279dcab77af246a6b6e9253de003": "\n\\begin{align}\n\\frac{d \\phi_i}{d t} & = -k \\sum_j A_{ij} (\\phi_i - \\phi_j) \\\\\n& = -k \\phi_i \\sum_j A_{ij} + k \\sum_j A_{ij} \\phi_j \\\\\n& = - k \\phi_i \\ deg(v_i) + k \\sum_j A_{ij} \\phi_j \\\\\n& = - k \\sum_j (\\delta_{ij} \\ deg(v_i) - A_{ij} ) \\phi_j \\\\\n& = -k \\sum_j (\\ell_{ij} ) \\phi_j.\n\\end{align}\n", "d7053ef41bfba3a25e568f9d27473c1c": "(1-x)^{\\alpha+1}(1+x)^{\\beta+1}\\,", "d7053f7afc8310199f83e1b47c8117cb": "\\mu\\in\\mathbb{R}", "d705d98b896b5b23b566d00ea0492820": " \\int^T_0 R_X(t,s)\\Phi_i(s)ds= \\lambda_i \\Phi_i(t)", "d705da72f273042a718b7fb89d3df038": "\\lambda(t) = \\lim_{\\Delta t \\to 0} \\frac{P(t \\le T < t + \\Delta t | T \\ge t)}{\\Delta t} ", "d7062aa6862ae98dbcc2c22f6b18d423": "W(A) > W(B) > W(C)", "d70633f20f87ebec5a9a0ffb645c94e8": "x_1 = 3", "d70643b2278e74c8897d0b2e614d96c6": "{\\left( \\frac{d\\varphi}{d\\tau} \\right)}^2 = \\frac{L^2}{m^2 r^4}", "d70660ad18feb033ce83318f65c6c8a3": "P^{\\pi}", "d70666bda51eb0f9c44e4108a1e34161": "\n\\frac {\\partial f} {{\\partial t} {\\partial s}} = j \\circ \\frac {\\partial f} {{\\partial s} {\\partial t}}\n", "d706776d2810e90d358ee42486cb6173": "s(a)", "d706867ebae886c163fc121dbfc95ae2": "n_1-n_2", "d70699eb825b7fbf52c5ff742bea22ce": " \\beta_\\mathrm{F}", "d706ce4904e201469c46add7e65984d0": " ~\\tau_t~ ", "d706def23ed518b032bd0d9985e36bcf": "\\vec r_i", "d70711a5b963a8f3a4500d9d32a6182f": "\\operatorname{H}^k(U) = \\operatorname{H}^k(pt)", "d70736d7f747c0fdae0c56b726b53fa4": " y_{n+1} = u (x_n \\sin t_n + y_n \\cos t_n), \\, ", "d70757de1bc67f6013936d12aaaa080d": "3^{\\frac{M_p-1}{2}} \\equiv -1 \\pmod{M_p}.\\,", "d7075a216d52d1dd19fe8b23f2ce6eb2": "I(f) = \\int_a^b f(x)\\, dx", "d7078bb4183ffbf198c91885c94e1a81": "\\text{Had} : \\{0,1\\}^k\\to\\{0,1\\}^{2^k}", "d707ac8167ae1124d91a23306d0c9217": "R_k(x)=B(x)e^{kx}-A_k(x).\\,", "d707d02bfc37ce42b6fc6cfbd24673e0": "n^2(n+1)(n+2)=", "d70811314753e632667d2384475043db": "\\varphi(x^a \\cdot x^b) = \\varphi(x^{a+b}) = a+b = \\varphi(x^a)+_n \\varphi(x^b)", "d70824ec4f2907bb275b61c8db97ca90": "\\bar{z} f(z)+g(z)", "d7088c09b40615d6c318e9b9ec380d1d": " \\overline{v'_i v'_j v'_k} ", "d708902aa8f1d6e9c5b3366376760172": "f:X \\rightarrow Y\\,\\!", "d708ded346a208aa89270bec958eedd6": "\\scriptstyle \\frac{\\theta}{2 \\pi} T", "d708fb2d1388b74970914049699eddf3": "= [\\,1 - \\rho(x,u,u_{1})\\,]dx + u_{1}(\\theta + u_{1}dx) \\,", "d7091e6bd6d6742357386238371ad0cd": "3 \\eta_0", "d709405ec27480d7a5f455f00ae3c7ac": "{v_1, v_2,\\ldots,v_n}", "d7095bf5a807745ecce1b07c84b49459": "y\\in \\mathcal{F}^n", "d709a15519856b212878d121bbcbc7df": " \\frac{n_1}{\\tau_{12}} = \\frac{n_2}{\\tau_{23}} = 0 ", "d709cbb969811618b7cd4bb057ab8476": "\\top_{\\mathrm{min}}(a, b) = \\min \\{a, b\\},", "d709ce00c6a2178391f2c12f152ec82e": "\\frac{\\partial h}{\\partial t} = \\alpha \\left[ \\frac{\\partial^2 h}{\\partial r^2} + \\frac{1}{r} \\frac{\\partial h}{\\partial r} + \\frac{1}{r^2} \\frac{\\partial^2 h}{\\partial \\theta^2} +\\frac{\\partial^2 h}{\\partial z^2} \\right] - G. ", "d70a3dd4a137131e83ba5f31bf5005dd": "M_{M_5} = M_{31} = 2147483647 ", "d70a62219619afd578208f58a73c1a7f": "\\begin{cases}\n\nx \\equiv & 1 \\ \\bmod \\ 11 \\\\\nx \\equiv & 12 \\ \\bmod \\ 13 \\\\\nx \\equiv & 2 \\ \\bmod \\ 17 \\\\\n\n\\end{cases}", "d70a69bbc4cf277b23816a46877b4664": "C_{abcd}^{}=-C_{bacd}=-C_{abdc}", "d70a6b17e79348011caf7b92704cffbf": "\\alpha = \\beta = 1 ", "d70b3c8c252f1c179bebcfcd77dfa127": "\\mathcal{C}_{n}=\\{(i_1,i_2,\\dots,i_k)\\,:\\ 1 \\le k \\le n,\\ i_1+i_2+ \\cdots + i_k=n\\}", "d70b3e313cdec1f9fb93134ed703be72": "f(t) = {a_0 \\over 2} + \\sum_{n=1}^{\\infty}{ a_n \\cos ( \\omega n t ) + b_n \\sin ( \\omega n t ) } ", "d70b3fb6db79724efb9236575bab617b": "\\textstyle Y = C + E", "d70bab64272ecadb54afa36f388ee24e": "n = \\alpha d^2", "d70bc22a0f995ddd67d05572fc20e44d": "\\mu \\nabla^2 v", "d70cd7409660dcdb34498cc629d6a933": "0\\to \\operatorname{Hom}(X,X)", "d70cf2ba33f5bf33a211dee91ec79c27": "\nP'(x_i - y_i|x,y) = \\frac{1}{|\\mathcal{S}|} \\sum_{z} \\sum_{1 \\leq k \\leq |z|} P(x_i \\sim z_i|x,z) \\cdot P(z_i \\sim y_i|z,y)\n", "d70d063335aeda4622c115db6127931d": "\\scriptstyle N \\rightarrow \\infty", "d70d38848a598dcd2dad0466ab29ad16": "x_1=3", "d70db281a10e6d581b6de3d663b9c1d5": "\\hat{a}^\\dagger_{c,{\\mathbf k}}", "d70dccc541c53deacab6112d07814e9a": "-k\\frac{2\\pi}{3}", "d70dfa0daf2e9f08efc75fae347d0d66": " \\mathbf{Hx} + \\mathbf{He}_i = \\mathbf{0} + \\mathbf{He}_i = \\mathbf{He}_i", "d70e2f7aaedeab2c884ceed7a6a4c099": "\\delta (x) = \\int e^{ikx} dk", "d70e4aebc218f8177971709b87b891b0": " \\sum_{x,y \\in C} d(x,y) ", "d70e5bfef7bfa05c5f4b84b024a3e8a7": " c \\in \\mathcal{C}", "d70ead79325701d453f0a0f20d996895": "(A,i)", "d70f5187915e27be831af0b1edd88c81": "\\frac{\\partial^2 F}{\\partial^2 p}\\cdot\\frac{\\partial^2 F}{\\partial^2 q} - \\left(\\frac{\\partial^2 F}{{\\partial p}{\\partial q}}\\right)^2 > 0", "d70f5f720e371f2ed97348e3c2056969": "1/y", "d70f688d0672fc0e84c6abd3579dd4e7": "y=\\pm\\frac{\\sqrt{b^2-4ac}}{2a}", "d70f7ca99e399a80ec5b32bb978b65eb": "d\\theta^i(E_j,E_k) = -\\theta^i([E_j,E_k]) = -\\sum_r c_{jk}^r\\theta^i(E_r),", "d710117ae98853599f1296fab4501326": " \\Gamma =dx^\\lambda\\otimes[\\partial_\\lambda + \\Gamma_\\lambda{}^\\mu{}_\\nu(x^\\alpha)\\dot x^\\nu\\dot\\partial_\\mu] \\qquad\\qquad (4)", "d710b06f55c8b9ee9307cfe0beaf324b": "\\langle \\gamma\\rangle", "d710bf38db5953c556ac02c274137a03": "h \\nu = g_\\mathrm{e} \\mu_\\mathrm{B} B_\\mathrm{eff} ", "d71127a62ec1ab3c64fb76fbf49b3956": "\nT = \\frac{1}{2} \\frac{d\\mathbf{q}}{dt} \\cdot \\mathbf{M} \\cdot \\frac{d\\mathbf{q}}{dt}\n", "d7114068601dea6f9603bed5eed8b19f": "2^t", "d7116139b30e33b0af2115301c004c30": "\\forall m \\forall X ((\\forall n (\\varphi(n) \\leftrightarrow \\psi(n))) \\rightarrow \\exists Z \\forall n (n\\in Z \\leftrightarrow \\varphi(n)))", "d71170fa8781320f62e96fa1a124feee": "Y_i=Y_0 + Y_1 \\cdot i + Y_2 \\cdot i^2 + \\dots + Y_{k-1} \\cdot i^{k-1}\\,,", "d711944c871b77cb3edafb38a9eb61f3": "M_i=1", "d7129cabd9b41e24e6ee7801feaacbd1": "u=c_1 x+c_2 x^3\\,.", "d712e0be09545e1442e0814bf21d406b": "\\rho = \\sum_i \\rho_i | \\psi_i \\rangle \\langle \\psi_i |", "d712f9f9f27fe978d66637f48af1862e": "\\phi_{X}:\\,\\mathbb{R}^{m}\\rightarrow\\mathbb{R}^{d}", "d7134ab5be980f7de86eb88de44531f6": "\\Phi=\\Psi", "d7135c68ba5ca3d95147fcf8b6267b66": " \\langle \\chi | \\psi \\rangle = \\int\\limits_{R_N} d^3\\mathbf{r}_N \\cdots \\int\\limits_{R_2} d^3\\mathbf{r}_2 \\int\\limits_{R_1} d^3\\mathbf{r}_1 \\chi(\\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N)^{*} \\psi(\\mathbf{r}_1, \\mathbf{r}_2, \\ldots, \\mathbf{r}_N) ", "d71371bda30fa597d95b369d14b0b67e": "[g] = \\left\\{\\left.\\lambda^2g\\right| \\lambda>0\\right\\}.\\,", "d713b68c410351ef3223a4782802197b": "\\overline{F}_\\mathrm{max}", "d713ed4d1519cf7577c45721f1cb8275": "\\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i^2 b_j^2 \n+ \\sum_{j=1}^{n-1} \\sum_{i=j+1}^n a_i^2 b_j^2 - 2\\sum_{i=1}^{n-1} \\sum_{j=i+1}^n a_i b_i a_j b_j ", "d713fd0bac94cbe35cac9fa0a39be398": "put(k, data)", "d71423ffa280434a68c484b7d4b8f694": "\\triangle_{S}=-\\triangle_{LB}+\\tfrac{1}{4}n(n-2)", "d71434790b6c97d854b5df2d74b038cf": "\n\\begin{align}\n& \\sum_{i=1}^n \\sum_{j=1}^{n_i} \\widehat\\varepsilon_{ij}^{\\,2}\n= \\sum_{i=1}^n \\sum_{j=1}^{n_i} \\left( Y_{ij} - \\widehat Y_i \\right)^2 \\\\\n& = \\underbrace{ \\sum_{i=1}^n \\sum_{j=1}^{n_i} \\left(Y_{ij} - \\overline Y_{i\\bullet}\\right)^2 }_\\text{(sum of squares due to pure error)}\n+ \\underbrace{ \\sum_{i=1}^n n_i \\left( \\overline Y_{i\\bullet} - \\widehat Y_i \\right)^2. }_\\text{(sum of squares due to lack of fit)}\n\\end{align}\n", "d714406ece88fcdce18f59d158eba575": "\\scriptstyle\\overline{pq}", "d7147dce9674e9957bcfe037a74c54de": " f(\\theta,\\phi) \\frac{e^{i k r} }{r}.", "d714b3ed3407ef09328a3a712a48f70d": "\n\\frac{du}{d\\theta} = \\frac{-1}{r^{2}} \\frac{dr}{d\\theta} \n", "d714c851bb92785336f796f842b8a04c": "\\Lambda = \\frac{f_Y|H_1}{f_Y|H_0} = Ce^{-\\sum^{\\infty}_{i=1}\\frac{y_i^2}{2} \\frac{\\lambda_i}{\\frac{N_0}{2}(\\frac{N_0}{2} + \\lambda_i)}}", "d714dbbac84d10192c570cd058f72b37": "\\frac{T}{W}=\\frac{3,820\\ \\mathrm{kN}}{(5,307\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=0.07340\\ \\frac{\\mathrm{kN}}{\\mathrm{N}}=73.40\\ \\frac{\\mathrm{N}}{\\mathrm{N}}=73.40", "d714f71aab1b9ff7369d8d0e3ac94a96": "\\displaystyle\\gamma^\\mu\\gamma^\\nu\\gamma_\\mu=-2\\gamma^\\nu", "d715208f0426999be4a37984261d6e80": "f(w) = f(v)", "d71521ab0969386f53ac06b2e10bf1c7": "\n f(z) = \\tfrac{1}{\\pi^k\\det(\\Gamma)}\\, e^{ -\\overline{z}'\\; \\Gamma^{-1}\\; z }.\n ", "d7152f02dde145a8820f9798d2de634c": "\\delta(x)\\in \\mathcal{A}\\,\\!", "d71567bf0b45600a281813ed208bcb62": "\\geq d' \\left\\lfloor \\frac{w_\\min - 1}{a_\\max} \\right\\rfloor ", "d715af2214249ca9b0d5e5e8152de463": "\\sum_{y\\in A} \\langle x,y\\rangle \\, y", "d715cc1df9d742c03818af1135e1f03e": "F(t, x, y) = {\\partial F \\over \\partial t}(t, x, y) = 0", "d715ec4cb92f5032481c6d7467f9fa1a": "\\nabla \\cdot \\mathbf{q} = -\\rho c_p \\frac{\\partial T}{\\partial t} + \\sum_{i,j} \\dot s_{i-j},", "d716043591444b26e0ae473db34fa65d": "\\mbox{V} = \\dfrac{\\mbox{kg} \\cdot \\mbox{m}^2}{\\mbox{A} \\cdot \\mbox{s}^{3}}. ", "d7164ee3273154096fcc02ca7f446093": "V \\rtimes \\mathrm{GL}(V)", "d716b45e1bff87586263fd9da784ddba": "\\nabla_{\\mathbf v}(\\varphi+\\psi)=\\nabla_{\\mathbf v}\\varphi+\\nabla_{\\mathbf v}\\psi.", "d716bc4bb7be4bfd84218afab7bbe5e4": "\\Delta\\mathbf{\\mathfrak{T}} = \\nabla^2 \\mathbf{\\mathfrak{T}} = (\\nabla \\cdot \\nabla) \\mathbf{\\mathfrak{T}}", "d716c401b70ba92621f823a0979aa9e7": "L: \\Theta \\times \\mathcal{A} \\rightarrow \\mathbb{R}", "d7172295e3fdae6e7316b09f6fa78b85": "\\boldsymbol\\Psi", "d7172d41116622462560e0cc15c165f9": "\n \\cfrac{\\partial\\sigma_{ij}}{\\partial\\epsilon_{k\\ell}} = \\text{constant} = c_{ijk\\ell} \\,.\n ", "d7173acf1e6d5da1272f2b0179d0b88d": " F(x) = \\sum_{k=1}^{\\infty}\\frac{k^{\\overline{k+1}}x^{k}}{B_{2k}}. \\ ", "d71781a884656ef3024482dbe0dfdc2c": "\\widehat{\\mathcal{F}}", "d7178d5f14be844ca3e7df3f5e751172": "\\; >\\beth_{\\omega_{1}}", "d7179628eb60ea02c3e694bf215476b1": "\\hat{\\xi} \\stackrel{\\hat{U}}\\longrightarrow \\acute{\\hat{\\xi}},", "d717bc136320bfb5eecf5f1e6d7ab0a8": "\\mathbb{N}_1=\\left\\{1,2,3,\\dots\\right\\}", "d71816e3465977122383a44cdb3e3b49": "\\frac{\\overline{\\partial f}}{\\partial z_i}= \\frac{\\partial \\bar{f}}{\\partial \\bar{z}_i},\\quad \\frac{\\overline{\\partial f}}{\\partial \\bar{z}_i}= \\frac{\\partial \\bar{f}}{\\partial z_i}", "d71821ce556dd2b57de9ecaac242c7c6": "\\alpha: x \\to x', \\beta: y \\to y'", "d7182219df4f3696c19113603106d908": "\\alpha^2:=\\frac{2mE_k}{\\hbar^2}", "d7182a2f31a6201502787244b5d05e49": "G_s(E,P)= \\frac{\n\\{ f\\in E^{\\mathbb N}\\mid\\forall p\\in P,\\exists m\\in\\mathbb Z:p(f_n)=o(n^m)\\}\n}{\n\\{ f\\in E^{\\mathbb N}\\mid\\forall p\\in P,\\forall m\\in\\mathbb Z:p(f_n)=o(n^m)\\}\n}.", "d718461bacd973e6104a240a023f616f": " I_{xx} =\\int\\int x^2 \\, dm ", "d71849bbc733eea1068fe266ba9f43c0": "\\frac {\\mathrm{d}f}{\\mathrm{d}\\varphi}=0", "d71860d3596200971295b049a2b7c406": "0\\leq k\\leq m", "d718727dc5bf24da3d2ae2beb49fceab": "X^*(z^*)", "d7188375df449bc1da645332e5b9327e": "\\Omega_1 = \\{ \\lnot \\} \\,", "d718b023be77e7d39a1d940e69007b91": "w\\div a", "d718d4320b6133cc2f98210f7d6df99b": "M_{biomass} (0)", "d719814fc3095f709b681edb7785df8a": "E=hc\\omega", "d719a8afda00ccc9819e8a51d0203cc8": "\\vec{e} = \\Delta\\vec{F}-\\mathbf{A}\\,\\Delta \\vec{p}\\!", "d719e715d1daaf500e5ea6d3ce32364b": "\\sum_{i=1}^k a_i b_i^p = 1", "d719e81a4fa35164fcc742719852f59b": "\\log \\cos z = \\sum_{k=0}^{\\infty} \\log\\left(1 - \\frac{z^2}{(k + \\frac{1}{2})^2\\pi^2}\\right),", "d71a5503898028d06989c89d986fc67f": " v(S) - \\sum_{ i \\in S } x_i ", "d71aad01ddbfabb28356c919cdb3881d": "a \\div b.", "d71acecc15206278c9cde91daa9b7414": "m_2(\\hat{x}) \\geq |h_3(x(t))|", "d71acfef675292dd4386ee489b107759": "\\mathbf{t}_b = \\frac{\\mathbf{w} - \\mathbf{u} \\cos(b)}{\\sin(b)}.", "d71ae776172b8af147b9a6d1a320c9cd": "\\mathbf x", "d71b3d1562135cb84e131e53ad24994e": "\\;\\deg(G)=\\deg_x(N(G))", "d71bc6c1f537e9cc45cac2e8ef3afb33": "h_1, h_2", "d71bddf90938abebe9f836b1497a38ea": "\\begin{align}\n\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{r}} \\right) - \\frac{\\partial L}{\\partial r} &= 0 \\qquad \\Rightarrow \\qquad \\ddot{r} - r\\dot{\\varphi}^2 &= 0 \\\\\n\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\varphi}} \\right) - \\frac{\\partial L}{\\partial \\varphi} &= 0 \\qquad \\Rightarrow \\qquad \\ddot{\\varphi} + \\frac{2}{r}\\dot{r}\\dot{\\varphi} &= 0\n\\end{align}", "d71c248ecea74c092c544a717b1ba348": "\\underline{c} \\ ", "d71c4383c4233c988c7fc86dc3d58a1f": "{\\partial u \\over \\partial x} = {\\partial v \\over \\partial y} = e^x \\sin y\\,", "d71c5cf45cc24325444a79e631cf5f3a": "\\mathbb{P}(X \\subset K).", "d71c76f75c0b6e42b25dcc7b5bf6e493": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}_k} \\right) = \\frac{dp_k}{dt} = 0~,", "d71c801d8b264969871c79ce5a248b66": "u_{n} =\\frac{p(n)}{q(n)}=\\sum_{k=1}^{m}\\frac{a_{k}}{n+b_{k}}.", "d71ca0e8545d7f76dd5d27a3874a8c5e": "{{Q}_{y}}=-kA\\frac{\\partial }{\\partial y}{{\\left. \\left( T-{{T}_{s}} \\right) \\right|}_{y=0}}", "d71cac920cc8ce018c5326947be66b3a": "\\mathbf{F}\\cdot (\\nabla\\times \\mathbf{F}) = 0.", "d71cbc078d865d5d67a770d11e167eb9": "\\sum_{k=-\\infty}^{-1} a_k (z-a)^k", "d71d0354c9ede3a94dd0ad191c0c4ba0": "\\dot x=f(x,\\lambda)\\quad f\\colon\\mathbb{R}^n\\times\\mathbb{R}\\rightarrow\\mathbb{R}^n.", "d71d080e09a87b9c8863cba3811c1c7f": " f(x) = x^{1 \\over x}. ", "d71d3bd287a8794b411f28930cde8f09": "\\begin{align}\n \\quad AB + AB' + A'B - A'B' &= A(B+B') + A'(B-B')\\\\\n &\\leq 2\n\\end{align}", "d71e09b1238cd7dca7f3e9457509a9ea": "R_{i,j}", "d71e4efc9d53f875266bca06655b5d4b": "s_n = 0", "d71e915cbc18a0f2e4686bab1e7393d7": "b(\\mu) = \\theta = \\mathbf{X}\\boldsymbol{\\beta}", "d71e9d207c1136e0654d16bce8f3cbf8": "y^* \\,", "d71eb4fb2482c76bcfd439de48bb8042": " \\phi \\wedge \\chi \\vdash \\phi ", "d71ef26f56993818f9b334303f0c26ce": "A_n, D_n, E_n", "d71f28f54231f934fefc00a6940b1034": "\\hat{\\bold{H}}_{\\operatorname{SCV}} = \\operatorname{argmin}_{\\bold{H} \\in F} \\, \\operatorname{SCV} (\\bold{H})", "d71f3d105c22c258dc432e32a9ccbc8a": "y = e^{- \\int^x P(\\lambda) \\, d\\lambda}\\left[\\int^x e^{\\int^\\lambda P(\\epsilon) \\, d\\epsilon}Q(\\lambda) \\, {d\\lambda} +C \\right]", "d71f52e6ffb424afa43db1cde1fcecda": "\\frac{[A_{ad}]}{p_A\\,[S]} = K^A_{eq} \\propto \\mathrm{e}^{-\\Delta G_{ad}/RT} = \\mathrm{e}^{\\Delta S_{ad}/R}\\,\\mathrm{e}^{-\\Delta H_{ad}/RT}", "d71f8b56c57f3221758d0abbdeaaf86a": " = \\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\bigg(-\\frac{\\partial}{{\\partial x_i'}}\\left(\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_j(\\vec{r}')\\right) + \\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\frac{\\partial}{{\\partial x_i'}}F_j(\\vec{r}') + \\frac{\\partial}{{\\partial x_j'}}\\left(\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_i(\\vec{r}')\\right) - \\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}\\frac{\\partial}{{\\partial x_j'}}F_i(\\vec{r}')\\bigg)d\\tau'}", "d71fb3ca7341a67284be2ca0a80622f3": "\\langle W(C)\\rangle \\propto e^{-\\alpha L}\\;,", "d71fd1f7b4844b32cc68338c994e5619": "\n\\frac{1}{B}\\frac{dB}{dt}= \\frac{(1 - \\bar{\\lambda})(\\nu+ \\mu)}{\\bar{\\lambda}} +\n \\frac{1}{\\xi}\\frac{d\\xi}{dt} \n", "d7201d5b92ad0683c9bf66b9df2d9dd2": "\\tan(z) = \\sum_{k=0}^{\\infty} \\frac{-2z}{z^2 - (k + \\frac{1}{2})^2\\pi^2}", "d72066b7c7c03faa3be3b4595748f90c": "{}_2F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2F_1 (c-a, c-b;c ; z).", "d72078acda48d601062624d84e3d1fb8": "r = 0.5L_x|N_x|+0.5L_y|N_y|+0.5L_z|N_z|\\,", "d7208530c0fb5404dd149a92cd8a3a7a": "\\{\\cdot, \\cdot\\}", "d7208b294f768ad98b6169e590458ded": "x=y\\phi(x')+\\psi(x')", "d7214175c87d8fc5c2add4b460a1f19e": "{\\Omega^\\hat{m}}_\\hat{n} = {R^\\hat{m}}_{\\hat{n}|\\hat{i}\\hat{j}|} \\, \\sigma^\\hat{i} \\wedge \\sigma^\\hat{j} ", "d721aa931cbfa8f373ae1c80e3951399": "M_J = \\left(\\frac{4\\pi}{3}\\right) \\rho R_J^3 = \\left(\\frac{\\pi}{6}\\right) \\frac{c_s^3}{G^{3/2} \\rho^{1/2}} \\simeq (2 \\mbox{ M}_{\\odot}) \\left(\\frac{c_s}{0.2 \\mbox{ km s}^{-1}}\\right)^3 \\left(\\frac{n}{10^3 \\mbox{ cm}^{-3}}\\right)^{-1/2}.", "d721bcad9050c279c36e197e541a2943": "f:{\\mathbb R}\\rightarrow M", "d7224524c4914bdd456a134367237b3e": " [A(t)]^T[A(t)]=I.\\!", "d722ae27afc57692895df65bcc3b6c9b": " 1/N ", "d722d66e9a3f5c88562c716d0fd407be": "\n \\mathsf{s} = - \\tfrac{\\lambda}{2\\mu(3\\lambda+2\\mu)}~\\mathbf{I}\\otimes\\mathbf{I} + \\tfrac{1}{2\\mu}~\\mathsf{I}\n = \\left(\\tfrac{1}{9K} - \\tfrac{1}{6G}\\right)~\\mathbf{I}\\otimes\\mathbf{I} + \\tfrac{1}{2G}~\\mathsf{I}\n ", "d72310ca3b84b3d43d85318168bcc01a": "\\frac{77076}{(1+0.10)^8}", "d723de428c8364e99ab10637a5a53740": "E < -C_{PL} \\alpha^2 ", "d723e384453808ebdce82ef71f5b3f70": " e^{(A+B)} = \\lim_{n\\rightarrow\\infty}(e^{\\frac{{A}}{n}}e^{\\frac{{B}}{n}})^n", "d72451419f4c9d22e6a0a29de684bbaa": "|f\\rangle", "d7246f1eaf9e48f95d419a53417925ae": " \\mu^-(E)=-\\mu(N\\cap E)", "d7248ed030b47dcd1ee2810169055333": "\\theta_{n+1} = \\theta_n + p_n + K \\sin(\\theta_n)", "d72499c033836822395994a90108125f": " \\lim_k f_k (x) \\geq 1 ", "d724cd28b4a11217152a1b2c9de7cbf7": "\\frac{a^2}{2L}", "d724e4c0a80c8686b632c7b4883e59f3": "\n\\begin{align}\n\\sin(\\pi - \\theta) &= +\\sin \\theta \\\\\n\\cos(\\pi - \\theta) &= -\\cos \\theta \\\\\n\\tan(\\pi - \\theta) &= -\\tan \\theta \\\\\n\\csc(\\pi - \\theta) &= +\\csc \\theta \\\\\n\\sec(\\pi - \\theta) &= -\\sec \\theta \\\\\n\\cot(\\pi - \\theta) &= -\\cot \\theta \\\\\n\\end{align}\n", "d725597ae8f1cb064ea09b7d9dc6bc2b": "l_{x}", "d725f98ac3da9da879bf2dce246c9826": "\\scriptstyle - \\frac{1}{3}", "d726a512ee9c090eaf3e09ad991dc9f6": "\\sin\\frac{13\\pi}{60}=\\sin 39^\\circ=\\tfrac1{16}[2(1-\\sqrt3)\\sqrt{5-\\sqrt5}+\\sqrt2(\\sqrt3+1)(\\sqrt5+1)]\\,", "d726b4b0bd15d2bad55acae74ce9ee72": "\n\\mbox{det} (\\lambda I_2- \\mathfrak{H})\n=\\lambda^2-\\mbox{tr} \\mathfrak{H}\\,\\lambda+\n\\mbox{det} \\mathfrak{H}\n=\\lambda^2-(a+d)\\lambda+(ad-bc)\n", "d726c8e1172f38f69613b41fc7e4686c": "\\omega^2 = a^2-n", "d726f5a4c21570df5b593702dff0373a": "a^1\\,", "d7270dc5b4cb9505af588b07210764aa": "\\hat{\\Psi}_{\\sigma}(\\omega)", "d72742b4d59b0423bc3e3533d5596b6f": "f(z)=\\sum_{n=0}^\\infty c_n(z-a)^n", "d72772c19329d7cd468cfc6195a83fe6": "\\mathbb{Q}(\\sqrt{2})", "d727772949fda59724b9c65e7e4be77b": " \\mu_c ", "d727a0bb8701dcde2e7f6d022cf64141": "n-i", "d727c4a2d1d17933f7f8c0711bd2efb6": "0\\to F(I^0)\\to F(I^1) \\to F(I^2) \\to\\cdots", "d727d523f3c8414c7f9305e93d9e790e": " h \\nu = g \\mu_\\mathrm{B} B_\\mathrm{0} \\,", "d727ea49df8b217e578585d5a63939d0": "dU = \\delta Q - P dV \\,", "d727f78310fc87c0f132705ca039f275": " r^{-n}~\\cos(n\\theta) \\,", "d72806a2ca861effb2b6e0f69f5ed045": "\\pi_n S", "d72838ad4cac4cc5737b30fcb3a4da4a": "\\alpha \\in \\omega^G", "d72849013e19341150b2792e679c8ba8": "h\\ :=\\ f(h,\\ m_n)", "d728675a0f1347210d5f6d54c1cc07ae": "C(id) \\to 0", "d72896ff7ca6070525291d79b7d9de81": "W=XY", "d728b01ef479f2a22f0c79419d2b1193": "\\left(\\frac{p^2}{2m}+V\\right)\\psi = E\\psi", "d728b0ca8d1f67dee771437ef1052629": "\\phi\\colon A \\to B", "d728b6ca6d352232078fc27150d0e36e": "\\tau=\\tau(s)", "d728ca9ebb131069448fdb3edf531457": "H^\\prime_\\max = - \\sum_{i=1}^S {1\\over S} \\ln {1\\over S} = \\ln S.", "d728ebc71006e3b3fd0a1010e2857c9d": "\n \\langle \\hat{N} \\rangle \\equiv \\langle \\hat{B}^\\dagger_1 \\cdots \\hat{B}^\\dagger_K \\ \n \\hat{a}^\\dagger_1 \\cdots \\hat{a}^\\dagger_{N_{\\hat{a}}}\n \\hat{a}_{N_{\\hat{a}}} \\cdots \\hat{a}_{1} \\ \n \\hat{B}_{J} \\cdots \\hat{B}_1 \n \\rangle\n", "d72912227034dc8ae7e82ed96dd0b2a0": "\n |((j_1j_2)J_{12}j_3)JM\\rangle = \\sum_{M_{12}=-J_{12}}^{J_{12}} \\sum_{m_3=-j_3}^{j_3}\n |(j_1j_2)J_{12}M_{12}\\rangle |j_3m_3\\rangle \\langle J_{12}M_{12}j_3m_3|JM\\rangle\n", "d7295876674a25a2770555b59c910799": "P \\cup \\Delta", "d7296a624d30ac9e3c1581b73920672b": "\nf(\\boldsymbol{x}) = f(T_{\\theta}\\boldsymbol{x}), \\quad \\forall \\boldsymbol{x}, \\theta .", "d729724161167c2ef66a2c9cc45f3d81": "|I (i \\omega)|", "d729c08aba10455e2b8ece115acafcb1": "bP^{n+1} = \\mathrm{ker} (\\eta \\colon \\mathcal{S}^{DIFF} (S^n) \\to \\mathcal{N}^{DIFF} (S^n)) = \\mathrm{coker} (\\theta \\colon \\mathcal{N}^{DIFF}_\\partial (S^n \\times I) \\to L_{n+1} (1))", "d729d762ce151fc484a74be5f18673c5": "\\begin{bmatrix}1 & 1 & 1 & 1\\\\1 & -1 & -1 & 1\\\\1 & 1 & -1 & -1\\\\1 & -1 & 1 & -1\\end{bmatrix}\\rightarrow\\left[\\begin{array}{c|ccc}1 & 1 & 1 & 1\\\\\\hline0 & -2 & -2 & 0\\\\0 & 0 & -2 & -2\\\\0 & -2 & 0 & -2\\end{array}\\right]\\rightarrow\\begin{bmatrix}-2 & -2 & 0\\\\0 & -2 & -2\\\\-2 & 0 & -2\\end{bmatrix}\\rightarrow\\begin{bmatrix}1 & 1 & 0\\\\0 & 1 & 1\\\\1 & 0 & 1\\end{bmatrix}", "d72a2b13b77302b6b00859e9f23cf106": "\n\\alpha^*_k = 1/\\alpha_k \\quad\\Longleftrightarrow \\alpha^*_k\\alpha_k = |\\alpha_k|^2 = 1,\\qquad k=1,\\ldots,m.\n", "d72a48a8ebd238e1497a911e938f46e8": "ARC", "d72a7a8e92e97dd5f2cd501dc4aea058": " h_0 ", "d72b33ad10a3364377452230548991c7": "f(x+\\delta)-f(x) = \\frac{[g(x+\\delta+y)-g(x+\\delta)]-[g(x+y)-g(x)]}{g(y)}", "d72c2bed3ad8f311416283459ca137a8": "x^2 = 4\\,", "d72c6f29aeba2926b2a2da131ca0f19b": "\\phi(\\mathbf{R}) = \\frac{1}{4 \\pi \\varepsilon _0} \\frac {q\\mathbf{d}\\cdot\\hat{\\mathbf{R}}}{R^2} + O\\left(\\frac{d^2}{R^2}\\right) \\approx \\frac {1}{4 \\pi \\varepsilon _0} \\frac {\\mathbf{p}\\cdot\\hat{\\mathbf{R}}}{R^2} \\ , ", "d72cd715496e3b43d850df266625b2e8": "h_F^{(2)}(z)", "d72cf70951f33233384c3701899f017c": "A, B : G \\to \\mathit{GL}(V)", "d72d0af88187ea42c8b899f79bf2de24": "L=\\dot{x}_1 p_1+\\dot{x}_2 p_2+p_3", "d72d19f5562da5d8232a14f24d96a282": "w=usu^{-1} w',\\,", "d72d1a815554db8737198b94307894d8": "\\vec{v}_{i} \\rightarrow \\vec{v}_{\\mathrm{CMS}} + \\hat{\\mathbf{R}} ( \\vec{v}_{i} - \\vec{v}_{\\mathrm{CMS}} )", "d72d3521fffcc85b7acc4a7d826f62be": " \\tau = RC\\ ", "d72d46297b09e7457ac25ff8907d40dd": "\nM(X) = \\left( {\\begin{array}{*{20}c}\n { - \\bar \\mu \\bar \\Sigma ^{ - 1} } \\\\\n { - \\bar \\Sigma ^{ - 1} } \\\\\n\\end{array}} \\right)\n", "d72ddc8c42d404c84b96dfc362f28e0f": " \\mathbf{h} = \\mathbf{r} \\times m\\mathbf{v} = h\\mathbf{k}, ", "d72e2c25238e20f677531b3baafa9ce7": "42y^4+21xy-14x^3+42xy^2-42y^2+6=0.", "d72e3c8053058367967e2fddeca6b29e": "T = S^{-1} = \\left(\\sum_{k=0}^\\infty {c_k \\over k!} D^k\\right)^{-1} = \\sum_{k=1}^\\infty {a_k \\over k!} D^k", "d72e5c3e74bc16b026f4af15724d1443": "V_{\\rm m} = \\frac{V}{n} = \\frac{RT}{p} = \\frac{(8.314 \\mathrm{ J} \\mathrm{ mol}^{-1} \\mathrm{ K}^{-1})(273.15 \\mathrm{ K})}{101 325 \\mathrm{ Pa}} = 22.41 \\mathrm{ dm}^3 \\mathrm{ mol}^{-1}", "d72e7b9a7afe4e999ef661cc7ae72e0b": "\\mbox{vec}(\\mathbf{H}_{\\textrm{MMSE-estimate}}) = \\left(\\mathbf{R}^{-1} + (\\mathbf{P}^T \\, \\otimes\\, \\mathbf{I})^H \\mathbf{S}^{-1} (\\mathbf{P}^T \\, \\otimes\\, \\mathbf{I}) \\right)^{-1} (\\mathbf{P}^T \\, \\otimes\\, \\mathbf{I})^H \\mathbf{S}^{-1} \\mbox{vec}(\\mathbf{Y}) ", "d72e998fa51e2db63dc7ff0b8d3a74af": "\\rho(\\mathbf{B},\\boldsymbol\\Sigma_{\\epsilon}) = \\rho(\\boldsymbol\\Sigma_{\\epsilon})\\rho(\\mathbf{B}|\\boldsymbol\\Sigma_{\\epsilon}),", "d72e9d06e3c18c47b0ab5059836283f0": "e:1{\\to}\\tau_1", "d72ea1a24b11c69184bfcf6a0d5bf05f": "= \\left( \\sum_{e \\in E} a_e(f_e^{*})^2 + f_e^{*}b_e \\right) + \\sum_{e \\in E} a_{e}(f_e)^{2}/4", "d72edd23c0f686e8bbf57d76d50f4163": "a_j^-", "d72ef86b825c1e2a4858ec9cec3d05b3": "\\chi N", "d72f1829a554386d77669d124f6029be": " 5 \\cdot \\log_{10}(\\frac{D_1}{D_0}) ", "d72f5121d55eaa34e52cfcc577909f53": " \\psi = \\psi(\\bold{r}_1,\\bold{r}_2) ", "d72f7abda3b73d0b81ea3db7e5d4148e": "\\delta_0 ( \\{ 0 \\} ) = 1", "d72fcb756cb8a1a04684fbb0c6d76c41": "g^{op}x = xg", "d730782481a7320beec0f27dec51c9d5": "O\\left(N_x\\log_2 N\\right)", "d7307c7c7130523a28cb7dce9e0f1630": " P^0=w,\\vec P=\\vec 0", "d7308ac710e72dc7709c928b4eb27e38": "\\frac{1}{\\mu} = \\frac{1}{m_A}+\\frac{1}{m_B}.", "d730aca06337efb9d78c27a04870bdec": "f_z = m \\gamma a_z = m_T a_z. \\,", "d730d522ded19eea61a6d45204a1ef98": "\\ln W = \\log_e W = \\frac{\\mathrm{Log}(W)}{\\log e}.", "d730f299a791e3e76585868dc5b996c9": "H \\star W = E \\cdot W,", "d7319847646716fb7c4512d47501fb56": "\\scriptstyle e={{r_a-r_p}\\over{r_a+r_p}}", "d731d7ee4434e4481fe5b88b89131ebf": "L = \\mathbb{C}TS^{2n+1} \\cap T^{1,0}\\mathbb{C}^{n+1}", "d731fb128de073f16185b32df7a4d4d2": " S((\\mathcal{D}_{\\eta} \\otimes1_{anc}) (\\Phi)) = H_2 (\\left(1 + \\sqrt{(1- 2\\,(1-\\eta)\\, p)^2 + 4\\,(1-\\eta)\\, |\\gamma|^2} \\right)/2) ", "d7327848a254a229356fd62f549a11bd": "|A|<|P(A)|.", "d732796481adbb9ceb32d8d59cba74aa": " a_r = 2 \\Omega u \\cos \\phi + \\frac{u^2 + v^2}{R} ", "d732f32dcb3004e032800b66e3e04488": "\\boldsymbol{\\mu}_A", "d73341829b3bf2f35343da10ccd1ef72": "\\exp^z", "d7336df24abae26ea3d938070adf3a5e": "\\mu_\\,", "d733d571caaae3df6aa14d298535a192": "\\,\\![a, b]", "d733d6c1156639de73139377d906cd48": "W(e^{1+z}) = 1 + \\frac{z}{2} + \\frac{z^2}{16}\n- \\frac{z^3}{192}\n- \\frac{z^4}{3072}\n+ \\frac{13 z^5}{61440}\n- \\frac{47 z^6}{1474560}\n- \\frac{73 z^7}{41287680}\n+ \\frac{2447 z^8}{1321205760} + O(z^9).", "d733fd8595bd7313c92223e51e3ce3a8": "n-m", "d7345ad6186b334e2dfe8318b523f2db": "p(\\Delta X)\\propto exp\\left\\{ -\\frac{\\gamma}{2 k_B T} \\Delta X^T\\Gamma \\Delta X \\right\\}=exp\\left\\{ -\\frac{1}{2} \\left(\\Delta X^T\\left( \\frac{k_B T}{\\gamma} \\Gamma^{-1} \\right)^{-1} \\Delta X \\right) \\right\\}", "d734628e4855559b39ec76cbbfda86fc": "| \\phi(g^*f) |^2 \\leq \\phi(f^*f) \\phi(g^*g), \\, ", "d7346882f64d0112b74b992aae9366f0": "\nD(A,B) = \\frac{1}{2}\\| A-B\\|_{\\rm tr} \\, .\n", "d734980879a6df1a88793b5123afd2e9": "MK_i\\ (i=0,\\ 1,\\ 2,\\ 3)", "d7352cb660729e901b1d43b26b22640d": " \\frac{n!}{l_1!l_2!\\cdots l_k!} \\sum_{w\\in S_n} \\sgn(w)\\left[(l_1)(l_1-1)\\cdots(l_1-w(1)+2)\\right] \\left[(l_2)(l_2-1)\\cdots(l_2-w(2)+2)\\right]\\left[(l_k)(l_k-1)\\cdots(l_k-w(k)+2)\\right] ", "d7356d20677cd7949b92ae77480fe9fe": "a\\leq b", "d735b130647dbc1b032917bc0f2d4f8e": " n=\\frac{2\\epsilon_0 m\\omega(\\Omega_c-\\omega)}{q^2}. ", "d73607e1e3d9ca9b75fca427b0d1fb2e": "\n\\Delta F_{\\textrm{T}}=\\sum_{i 0", "d74f237fd1fdf71dfde2782043c91ef0": "B_{ii}", "d74f87265d6e823777c2e54b316a46df": "\\text{Gain}(i,\\cdot)", "d74f8b123a5143d8193af61a9c55f0f7": " O((b-a)^4) ", "d74fc4e224d7e5b02c4e9d0c5f5b8b67": "(M_\\odot ~\\textrm{ yr}^{-1} \\textrm{ pc}^{-2})", "d74fc7dd1ccfa99f31f1565aa1e710f0": "\\hat{H}_{\\text{JC}} = \\hbar \\omega \\hat{a}^{\\dagger}\\hat{a}\n+\\hbar \\omega \\hat{b}^{\\dagger}\\hat{b}\n+\\frac{\\hbar \\Omega}{2} \\left(\\hat{a}\\hat{b}^{\\dagger}\n+\\hat{a}^{\\dagger}\\hat{b}\n+\\hat{a}\\hat{a}\n+\\hat{a}^{\\dagger}\\hat{b}^{\\dagger} \\right),", "d75007c0f229748f0f78a036d6dfd75c": "I_{rms}^2 \\times R_{DC} ", "d7501c1035b3b48f13079eec78c98055": "\\bar{f}(0) = \\langle\\rangle", "d7503dc0399add520c84c5977e34ebd6": "\\displaystyle{V_t f_y=f_{y+t}=V_y f_t,}", "d7503febfc1a2c842cf18cac8d0ba407": "\\frac{1}{2}K_{24}\\nabla\\cdot((\\mathbf{\\hat{n}}\\cdot\\nabla)\\mathbf{\\hat{n}}-\\mathbf{\\mathbf{\\hat{n}}}(\\nabla\\cdot \\mathbf{\\hat{n}}))", "d750605962ae337bccc4ddcaeb2e2a19": "(X,\\; \\Sigma ,\\; \\mu\\,)", "d750d0da5a8c1d08a6dbc04b7e8e6a6a": " f(t;a,b) = -1 + 2 H_a( t \\mod (a+b) ) ", "d751013afaaebf83bed7634d4b4e4891": "\\xi=\\frac{\\left(1+i \\sqrt3\\right)\\left(\\frac1{2 \\tau}+\\sqrt{\\frac{\\tau^{-2}}4-\\frac8{27}}\\right)^\\frac13+\n\\left(1-i \\sqrt3\\right)\\left(\\frac1{2 \\tau}-\\sqrt{\\frac{\\tau^{-2}}4-\\frac8{27}}\\right)^\\frac13}2", "d7513e389b1fa8b2fbfcacbe078e3d46": ">\\delta D = 8.0*\\delta 18O + 10 per mille ", "d7518f62b20a1f90e91933cd0d75510c": "l-", "d751c22982386778761684727071d711": "x = a_0 + {1 \\over a_1 + {}} {1 \\over a_2 + {}} {1 \\over a_3 + {}}.", "d751fea9034ffd189f8009af3bba3bf2": "E(S_n^2)=\\sum_{j=1}^n E(Z_j^2)=n.", "d7522ca276587ced34b11bf333f36635": " L(\\theta,\\widehat{\\theta}) = \\begin{cases}\n 0, & \\mbox{for }|\\theta-\\widehat{\\theta}| < K \\\\\n L, & \\mbox{for }|\\theta-\\widehat{\\theta}| \\ge K.\n \\end{cases}\n", "d7523b2243d4dd14973b06e5abcbd926": " \\mathbf{Set} ", "d752a1031fe5ba7ac6ba721b8bdb45b8": "F[\\rho(x)+\\varepsilon f(x)]", "d752aad2c99d29b40c9d28eeb7c9190b": "\\vdash \\Box \\Psi \\rightarrow \\Box P", "d752d4e4d548aef187006ebc19208893": " \\left(\\gamma^2 + (\\dfrac{\\lambda}{d})\\gamma \\left(1-\\gamma\\right)\\right)", "d752e3fcd5dc6ad3c18eade068850b54": "Pc = 2 \\tfrac{6}{25} = \\tfrac{12}{25}", "d753104a354747fe64e290bcfc038796": "\\displaystyle{T_{a,b} v=(v,b)a,}", "d753856e0a3c8cf6301a699e3bc99460": "M_\\lambda.", "d753bc6af0d84ac2e55113eb6520fa4c": "\\phi_x(D|m_x) + \\phi_x(m_x|\\Theta)", "d753de2c1b952633a74e548878ec09ed": "\\log 3", "d754598045a270e5c4febf7a383449ea": "Y \\mathbf{\\operatorname{si}} X", "d75460ca14a3cdb895d2116706378f09": "\\,f\\,=\\,0.3164\\,/\\,Re^{0.25}\\,=\\,f_0\\,W^{-0.25}", "d754917a64121b7757fcebf056c3b5f5": "\\textstyle{\\sqrt 2}", "d754e7b42281a545ad501f8c125ed8d4": "\\int_{t-r}^{t}E(t')dt'", "d75539f01782cce05c22ae140c58a3ef": "\\ s", "d7556975d1896c85da5ee3a31f0cafb9": "e^{i0}=1", "d755be8fe49318c5d6a2f86646be909f": "\\frac{M_1^\\mathrm{pass}}{m_1^\\mathrm{inert}} = \\frac{M_2^\\mathrm{pass}}{m_2^\\mathrm{inert}}", "d755ec76ea6ddb0779a0edcf1e043aa6": "\\Bbb{H}_\\mbox{curl}\\ ", "d757643f5877a03e73b287b923f576bc": "\\operatorname{Inn}(G)\\cong \\operatorname{Im}(g\\mapsto \\lambda(g)\\rho(g))", "d7576e4ac1915c03689336e474af5142": "\\sqrt[p]{\\sum_{i=1}^nw_ix_i^p}\\geq \\sqrt[q]{\\sum_{i=1}^nw_ix_i^q}", "d757a0c658e331a9e504aee2541aea8e": " \\mathcal{S}' \\, ", "d757e97bb3d7391a06229b9d72d4eb22": " H = h_0 + h_1 \\mathbf{e}_2 \\mathbf{e}_3 + h_2 \\mathbf{e}_3 \\mathbf{e}_1 + h_3 \\mathbf{e}_1 \\mathbf{e}_2 + h_4 \\mathbf{e}_4 \\mathbf{e}_1 + h_5 \\mathbf{e}_4 \\mathbf{e}_2 + h_6 \\mathbf{e}_4 \\mathbf{e}_3 + h_7 \\mathbf{e}_1 \\mathbf{e}_2\\mathbf{e}_3 \\mathbf{e}_4. \\!", "d757fbd878a2529f082ee2c042b3848b": "S_{a_1} (S_{a_2} f) = (S_{a_1} \\circ S_{a_2}) f = S_{a_1+a_2} f", "d7581faec53f16f4eef810f552e5aabf": " q_k \\leftarrow q_k - h_{j,k-1} q_j \\, ", "d758256c29f5a0371bb90096109a17c0": "\\part y/\\part x", "d7583d1fb53558e3fb639c6bbdf31314": "f(x|a < X \\leq b) = \\frac{g(x)}{F(b)-F(a)} = Tr(x)", "d758449ca1fdefaa3ad66ab396b94f70": "f(S_5) = f(S_5, S_1)", "d758491a1c19f4f1d1583adb84e96c48": "\\frac{dW}{dt} = P(t) = \\mathbf{F}\\cdot \\mathbf{v} .", "d758661db1a6b224f702179aba0214ca": "\\bar{x}= \\frac{1}{n}\\sum_{i=1}^n x_i", "d758a65219677934b09055f7fd97891e": "f:\\bar\\Omega\\to\\R^n", "d7590b6970785a6c65d8c4b8c1c2778b": "\\cos\\left(\\frac{1}{\\sqrt{\\lambda}}\\right)=0", "d75927563fceb1defc4504495c792607": "\n\\begin{align}\n{\\mathbf v}\\wedge {\\mathbf w} & = (a{\\mathbf e}_1 + b{\\mathbf e}_2)\\wedge (c{\\mathbf e}_1 + d{\\mathbf e}_2) \\\\\n& = ac{\\mathbf e}_1\\wedge{\\mathbf e}_1+ ad{\\mathbf e}_1\\wedge {\\mathbf e}_2+bc{\\mathbf e}_2\\wedge {\\mathbf e}_1+bd{\\mathbf e}_2\\wedge {\\mathbf e}_2 \\\\\n& =(ad-bc){\\mathbf e}_1\\wedge{\\mathbf e}_2\n\\end{align}\n", "d759c42335bbbf4b83fc10ec8827c542": " X^{\\prime}(t) = \\alpha \\nu \\left( 1 - \\left(\\frac{X(t)}{K}\\right)^{\\frac{1}{\\nu}} \\right) X(t) ", "d759c7d9d4cf26135f616fe2fb59041f": " K_P", "d759fb5b7649e454f7a4a03b85fa1b8e": "\\scriptstyle Ma\\,", "d75a287e3ac70adcfc929649fe4fb987": "x^3, x^5, ... ,x^{2^k-1}", "d75a2d07b3d6e88c9f78162f44fca381": "dU = \\delta Q - P dV \\, .", "d75a4901b52fe2ef1fe3100a90e99859": "\\|\\cdot\\|_U", "d75a7682d52c57af7dedea3da3c7254c": "\\phi_{SP} = \\phi_1", "d75a881cd9542925d723521b909c6e87": "\\,R= N_\\mathrm{A} k", "d75a990b4aedc4b394e6960d6a2cc103": "\\Omega^k(E) \\to \\Omega^{k-m}(B)", "d75ac2f36962f68316ef36c076af0afa": "\ng^\\prime(x)=-g(x)^2.\n", "d75add80faeade750817844d60da9641": "M_{-\\infty}", "d75b0f98192cc973d703ec34e99b8da1": "r \\in \\{0, \\ldots, m-1\\}", "d75b2674f0954bd6d3fa7792ad8169d3": "F:\\mathcal C\\to\\mathcal D", "d75b58fb10d835a2df26789a0df3aaf7": " \\Box_{c} u\\left(x,t\\right) \\equiv u_{tt} - c^2u_{xx} = 0, \\, ", "d75b883e21e4b2fc31b47bb2986bc78e": "Q = \\left. \\frac{\\partial V_c}{\\partial r}\\right|_{r_T,t_f}", "d75be52f631500585c82c3135928fba4": "f^{0}(x)", "d75c829a03af77ad3294bb5f216096e9": "b > 0", "d75cb08cc0e78023cda7a3fac05642ce": "\\textrm{Ratio}_1 = \\frac{2 \\cdot \\textrm{Uploaded \\,\\, Total}}{\\textrm{Downloaded \\,\\, Total}}", "d75ce72fef506454941e9e383142144a": "\\hat{b}^\\dagger \\hat{b}", "d75cff1146d241a2cde120b4217f4e6d": "f^0 ~ \\stackrel{\\mathrm{def}}{=} ~ \\operatorname{id}_X\\,", "d75d5d8290e91d947207fa38785e09ff": " k[\\Delta]=k[x_1,\\ldots,x_n] ", "d75d6e0be66765f3224fd9e4119197fb": "e_1, \\dots, e_s", "d75dc2d11bbf1ee1e573c677858eafd1": "\\textstyle (n-1)", "d75dd5144aa26f03840d57f84a120e74": "\\mathrm{DPH}(\\boldsymbol{\\tau},{T})", "d75dfd7ce2f18c983d931f9c42d30a0c": "Pr(K=k)", "d75e5a492890b979ad825e57eaf25221": " \\nu = \\arccos { {\\mathbf{e} \\cdot \\mathbf{r}} \\over { \\mathbf{\\left |e \\right |} \\mathbf{\\left |r \\right |} }}", "d75eeb8981059c62fec2b094c4fef17b": "\\oint_{{\\Gamma}_{1}} \\mathbf{F} d\\Gamma +\\oint_{\\Gamma_3} \\mathbf{F} d\\Gamma =0", "d75f146df3fe2a400a1c38f80d4a6a1e": "1-f + f(1+b) = 1+fb", "d75f26a02ff11622ff79d69bc04e9351": "(S,g,\\nabla,\\nabla^*)", "d75fc89c14dccf8ce6506dbaedb5cf19": "e=\\cos\\alpha\\,", "d75ffa444332b1a743c281298f4f219f": "\\exists d>0, 1-F(x)=O(e^{-dx}),x\\to+\\infty;", "d76043da919c97c11a9c79f7e8fd04e5": "j\\in\\N_+^k", "d7605eef31c83c567ae8ef515f88d647": "x \\in S \\setminus T", "d7608f7926a6f505807a5087a774ab42": " \\langle f/g \\rangle_\\rho - \\langle f/1 \\rangle_\\rho\\times \\langle g/1\\rangle_\\rho = \\langle T_\\rho(f)/T_\\rho (g) \\rangle_\\mu.", "d760b7fe28714076972445303abc8c59": "\\, (N+1)", "d760fed7c6d4fcf497af68fcd81d6807": "Y_{0,3} = F_e", "d7612899df4dfa2c5b95269146e871a8": "\\rho = \\rho_m(a) = \\frac{\\rho_{m_{0}}}{a^3},", "d761f711e47c2525f5a25df607002d2e": "\\begin{bmatrix}\\gamma_1 \\\\ \\gamma_2 \\\\ \\gamma_3 \\\\ \\vdots \\\\ \\end{bmatrix} =\n\\begin{bmatrix}\n\\gamma_0 & \\gamma_{-1} & \\gamma_{-2} & \\dots \\\\\n\\gamma_1 & \\gamma_0 & \\gamma_{-1} & \\dots \\\\\n\\gamma_2 & \\gamma_{1} & \\gamma_{0} & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\end{bmatrix} \n\\begin{bmatrix}\n\\varphi_{1} \\\\\n\\varphi_{2} \\\\\n\\varphi_{3} \\\\\n \\vdots \\\\\n\\end{bmatrix}", "d76200db5456e9f16edeab8ddcd9cbba": "\\prod_{k=1}^n f(k)", "d7624512308b9223e73063c3f8b96c88": " e_n(x) = \\sqrt{\\frac{2}{L}}\\sin \\left(\\frac{n\\pi x}{L}\\right)", "d76247c0075a76641840453f34d7e66a": "\\langle P \\rangle = \\mu v \\omega^2 x_m^2/2\\,\\!", "d7625a12a005aa089e3bd92093e4b84b": "F\\colon M \\rightarrow \\mathbb{R} \\quad \\mbox{or} \\quad F\\colon M \\rightarrow \\mathbb{C} \\, ,", "d7628f21315779a85c96a4db02b855a8": "{d}(t) = {IRF}(t) \\otimes {F}(t)", "d7629c1c82af449300560b5a383b2691": "C^*(m) = (C_{in}^1(c_1),C_{in}^2(c_2),..,C_{in}^N(c_N))", "d762cbc2aeb1da1c62f67e62471b0193": "\\mu \\circ (\\mathrm{id}_H \\otimes S) \\circ \\Delta = (\\varepsilon \\otimes \\mathrm{id}_H) \\circ (\\mu \\otimes \\mathrm{id}_H) \\circ (\\mathrm{id}_H \\otimes \\sigma_{H, H}) \\circ (\\Delta \\otimes \\mathrm{id}_H) \\circ (\\eta \\otimes \\mathrm{id}_H)", "d762ce8f91c3d3b144c402ca70ef911b": "\\mathcal{P}_B (A) = (A \\cdot B^{-1}) B", "d762d7f424c2160e90f5a236989456c4": " L(n,k) = {n-1 \\choose k-1} \\frac{n!}{k!}.", "d762f757a58cb8e1df8e38a8799361f8": "\\zeta(s,\\alpha)= \\sum_{n=0}^\\infty \\frac{1}{(n+\\alpha)^{s}} = \n\\frac{\\alpha^{1-s}}{s-1} + \\frac 1{2\\alpha^s} + 2\\int_0^\\infty\\frac{\\sin\\left(s \\arctan \\frac t \\alpha\\right)}{(\\alpha^2+t^2)^\\frac s 2}\\frac{dt}{e^{2\\pi t}-1}.", "d76317f2922d5b676792d9235b17eb51": "a_{0},\\ a_{1},\\ a_{2},...,\\ a_{n}", "d7632954f236347d9b7cc9094170ef97": " u(t,x) = (- 3 \\alpha (x + 4 \\alpha^2 t )^{2/3} . ", "d7633a598ecf21088ee993ffe50a4c61": "\\mathbb{F}^N", "d76371f083ffa0560d297e51098961ac": "\\begin{align}\n\\sum_{k=0}^n f_k g_k &= f_0 \\sum_{k=0}^n g_k+ \\sum_{j=0}^{n-1} (f_{j+1}-f_j) \\sum_{k=j+1}^n g_k\\\\\n&= f_n \\sum_{k=0}^n g_k - \\sum_{j=0}^{n-1} \\left( f_{j+1}- f_j\\right) \\sum_{k=0}^j g_k, \n\\end{align}", "d763727e4ab65026fb4d67ef78dbe393": "\\alpha = H(u_1, u_2, e) \\,", "d763840d28550c811a1c902b368b9923": "(F*G)(k) = \\sum_{m=0}^k F(m)G(k-m)", "d7639b1fd03bc1e4e03ce275e668cd7f": "\\psi^{(-2)}(2)=\\ln(2\\pi)-1", "d763af354de884610dbd15719fbcfcbb": " \\mathbf{Q}_{M \\times 1} ", "d763e00b8ac4256bf0db2c961a7e6e07": "ce(ab)<_sa", "d763e5ab26428b45d4e931c1b9ce63d6": "\\scriptstyle \\Psi=e^{k z}", "d76491f8e8353584d4d34d11f21a54ac": "\\frac{dr}{dt_r}=\\plusmn 1\\,\\!", "d764922474f40598eedd1eb46bd5dbd9": "s_z = m_s \\, \\hbar", "d764c2b6299cedfde968c65f6bf9f893": "\\mu(-A)<0", "d764ccee86664b90ecbf2048185abc72": "S = \\int L \\sqrt{-g} \\, \\mathrm{d}^4x ", "d76547d85c3b688d3ed452a895d04e14": "\\mathrm{d}(\\mathbf{p},\\mathbf{q})=\\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2}.", "d7654b1b47ebeb56905c6a36058d6a68": "Q=V\\cdot \\rho_0.", "d76588ecf0b09540860b38863536812b": "g = \\frac{m^b(x)k_G}{m^b(x)(k_R+k_G+k_B)} = \\frac{k_G}{k_R+k_G+k_B}", "d7659511d7c83d23476e011e19b16e6b": " I_{ion}^{sat} = en_e\\sqrt{k_BT_e/m_i}\\,A_{eff} ", "d765c40268c7531f29895e39080e6773": "\\begin{align}\n \\varphi_2(g_1,g_2) &= \\phi_3(1, g_1,g_1g_2) \\\\\n\\varphi_3(g_1,g_2,g_3) &= \\phi_4(1, g_1,g_1g_2, g_1g_2g_3),\n\\end{align}", "d765d88530bfb4b0f91452852e48111c": "x+x^{-n}=2", "d765f681064e65afdd14080ff9fed957": "\\hat g(f)= e^{-\\frac{\\pi^2f^2}{a}}", "d766092e346691683153a448e1df1197": "2*y+4*x+2*z-w=0", "d7664559ef0f6d3338095a2e0cb7606f": "A \\cong Sym^\\cdot V^*", "d7664855022ef1b7ae75f6db2551c8d5": "m_V = 0.12", "d76673f8f8b53b3beca39ca49ecd0439": " U", "d767436eba8ae9ade4547ae7401092fb": "\\rho(q,t)", "d767818041cd0a499d83499d89d0ec6f": "E_\\lambda:=\\{(y,s)\\,:\\,\\Phi(y,s)>\\lambda\\},", "d767a2ac825d5af8cce3af7a26b5e867": "W_q^{(n)}=\\max(W_q^{(n-1)}+U^{(n-1)},0)", "d767ebf41eec81dfe3ebfaaf8c025923": " z = \\frac{ \\rho_f} {(\\rho_s-\\rho_f)} h", "d767f7622e2e96b34a690167cfa0e761": " A = \\sqrt{{c_1}^2 + {c_2}^2}, ", "d7681a8258911e6345ab017599ad2369": "\nf(t) = \\frac {\\gamma} {\\pi i} \\int_{c - i \\infty}^{c + i \\infty} (ts)^{\\gamma - \\rho - 1/2} \\; G_{p,\\,q+1}^{\\,1,\\,p} \\!\\left( \\left. \\begin{matrix} -\\mathbf{a_p} \\\\ 0, -\\mathbf{b_q} \\end{matrix} \\; \\right| \\, -(ts)^{2 \\gamma} \\right) g(s) \\; ds,\n", "d768a061620b7d05dc1a7dd8140d9758": "B\\backslash A", "d768d21e6d7692f3c2256206a2c76c60": "(X,d |X^2)", "d76970e1fdfa7192ae14733c37c74992": " \\mathbf{H} = a\\cdot\\sigma_0 + c\\cdot\\sigma_1 + d\\cdot\\sigma_2 + b\\cdot\\sigma_3 ;", "d769b18c0606db7c50c493e20bc70346": "{H_0: \\lbrace \\rho_i = 0 \\text{ for all } i \\rbrace }", "d769cf77ece5a9f553d8a0568c5dc9b2": "F_i(x)", "d76a201351a0017527e599ce0467622b": "\\langle A \\rangle_\\sigma", "d76a7dee97b228fb1b757ad5d9133a7c": " \\| r_{k} \\| \\rightarrow 0", "d76ae42024218e672b82e506583af4c9": " = 0.2 + 0.02 \\times \\log_{10}(10,000)\\,", "d76aeeece1ba44a8e650d5eb85459c46": "G_{a/b}", "d76afbe567e690fde430ef5fe05f0809": "(z_2,z_3)", "d76b2077a29f8cc0f4e7f8954ff6dfae": "\\frac{2\\pi}{T}", "d76b3d9c14ee7faac7f4af2262fa96a4": "L_e(\\mathbf x,\\, \\omega_{\\text{o}},\\, \\lambda,\\, t)", "d76b3fe2b3420abb22df589b481ec259": "\\sum_{(i, j) \\in E} c_{ij} d_{ij} ", "d76b4265f582543539f6e118c3a2995f": "R_1R_2 = \\frac{R_aR_bR_c^2}{R_T^2}", "d76b448a3b4cf54190382df396d73f44": " \\epsilon_{[t_l, t_u]}", "d76b986a48a5691af985f6516968c9a5": " P(\\partial_t,\\xi) H(t,\\xi) = \\delta(t) ", "d76bc7dc3aa3fabe186cf0fef64691b2": "{n2}", "d76be5070c5d62fd5a962b6a14c5b994": "c=\\lambda/T=\\omega/k\\,\\!", "d76bedcb0cab87a1df44813486d6d8a1": "\nc^2 {d \\tau}^{2} = \\frac{(1-\\frac{r_s}{4r_1})^{2}}{(1+\\frac{r_s}{4r_1})^{2}} \\, c^2 {d t}^2 - \\left(1+\\frac{r_s}{4r_1}\\right)^{4}(dx^2+dy^2+dz^2)\n\\,.", "d76c37641bb116dbb7567b789806632d": "X=c^2/\\alpha", "d76c54dc96bc8a18ad655ca9077cf167": "g_{0i}, g_{0e}", "d76ce71535a58b4c03baf658c8dde9ec": "\\mathcal{O}_X \\xrightarrow{F^a_{X/S}} \\mathcal{O}_X \\to k(x)", "d76d67102a78542f00e1380c2bd9e139": "V_{seg}", "d76d8e5ef1936ba866332553430a2400": "\\omega_{2k-1}", "d76d8ec39e33b093f69f708921bf3eb8": "\n\\begin{bmatrix}\n\\boldsymbol{G}_j^{(t)}\\\\\n\\boldsymbol{G}_j'\\\\\n\\boldsymbol{G}_j^{(b)}\\\\\n\\end{bmatrix}.\n", "d76da6cc4a845255073f3a6070781d13": "R(P)_i = S(R)P_iS(R)^{-1}", "d76dd6238deb81796691d212da01b25f": "n_r^2 = 16", "d76e6ee51597b56ae0f140a9dea85835": "v_i \\equiv s_i^{2} \\pmod{n}", "d76eb499e1c79e16173ae085305cdfac": "\\frac{\\partial f}{\\partial x}(X,Y) \\cdot (x-X)+\\frac{\\partial f}{\\partial y}(X,Y) \\cdot (y-Y)=0.", "d76ee50829a7e414660da51145671fd3": "C\\mapsto |K_C|^*", "d76f07f37b21f9fb6c005b3ff51180cd": "G \\cong \\Z[b_1, b_2,\\dots]", "d76f0b1dd91c403b4f3142809116e20f": "(y, x]", "d76f2c4d6bdf142af5106c3f36e9e970": "F(x)", "d76f53ed2ad9dd632362fbdc65374ac1": "D_{jk} = \\frac{\\frac{E_k I_k}{L_k}}{\\sum_{i=1}^{i=n} \\frac{E_i I_i}{L_i}}", "d76f5e357ed237cf3f63a34ca905f2d6": "\n\\delta = \\dfrac{\\partial}{\\partial t} + \\sum_{i=1}^{k} f_i(Z) \\dfrac{\\partial}{\\partial x_i}\\cdot\n", "d76f78ab55d0e314f5ba4ee49c33f923": "C_6H_6", "d76fc8b381d6d63be6673607720d462b": "h^{\\alpha\\beta}=g^{\\alpha\\beta}-u^\\alpha u^\\beta", "d76ff42f282f171686a6ca7e584512c5": " \\theta_4 ", "d7703cf6b05825327bd6a489405dc563": " \\operatorname{lambda-named}[(\\lambda F.M)\\ N] = \\operatorname{lambda-free}[M] \\and \\operatorname{lambda-anon}[N] ", "d770aac63962c0f914fa892421f8382d": "\\rho_{ij}", "d770bb09d216b893c6acac33acc32070": "v\\mapsto v^{-1}", "d770c4d67e7da28ecc8fecc0044e9ce1": " h = \\frac{1}{2 g} \\omega^2 r^2", "d770ed7061d0265e8cac05d7418a76e6": "\\eta \\,", "d770ef56cea634d6aabd49d1c913bf2c": "f_{a_1\\cdots a_k}(z)", "d7710970714cbb60210a9dd133242782": "\\frac{\\rm d}{{\\rm d}t}x(t) = ax(t) + bx(\\lambda t),", "d771371bc939af752504186ba31bfa06": "r_{ij}\\in\\mathcal{Z}", "d7714b14a07d03ab9050e79829b96491": "99 - 98 + 0.98x = x", "d7714fbe2533d4a8d655f31680abaa8a": " \\left\\{ \\frac{p^2}{2 m_0 } + \\left( 1 + \\frac{\\epsilon_0 }{2m_0 c^2 } \\right) (V - \\epsilon_0 ) \\right\\} \\Psi_1^0 \\equiv 0 . ", "d7716bdce4b150949140497f57564cbd": "\\rho(x,y,z,t)=|\\psi(x,y,z,t)|^2", "d771d2d581dfe469e3e38db987706ef1": "a_n = \\frac{2}{P}\\int_{x_0}^{x_0+P} s(x)\\cdot \\cos(\\tfrac{2\\pi nx}{P})\\ dx", "d77215bc36ce1ca4cea8d1021df0e016": "(T^{-1})_{ij} = \\begin{cases}\n(-1)^{i+j}b_i \\cdots b_{j-1} \\theta_{i-1} \\phi_{j+1}/\\theta_n & \\text{ if } i \\leq j\\\\\n(-1)^{i+j}c_j \\cdots c_{i-1} \\theta_{j-1} \\phi_{i+1}/\\theta_n & \\text{ if } i > j\\\\\n\\end{cases}", "d772173908721ffc84c596f12e2ab8eb": " {FE}_{T=2}= \\left[\\sum_{i=1}^{N} \\dfrac{x_{i1}-x_{i2}}{2} \\dfrac{x_{i1}-x_{i2}}{2} ' + \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{x_{i2}-x_{i1}}{2} ' \\right]^{-1} \\left[\\sum_{i=1}^{N} \\dfrac{x_{i1}-x_{i2}}{2} \\dfrac{y_{i1}-y_{i2}}{2} + \\dfrac{x_{i2}-x_{i1}}{2} \\dfrac{y_{i2}-y_{i1}}{2} \\right]", "d7722738b8885608c707e93f3d95a1eb": " \n\\begin{bmatrix}\n\\mathbf A^T \\\\ \\mathbf B^T \n\\end{bmatrix}\n\\mathbf x \n= \n\\begin{bmatrix}\n\\mathbf e \\\\ \\mathbf f \n\\end{bmatrix}, \n\\qquad \\mathbf e \\in \\reals^{n\\times 1},\n\\qquad \\mathbf f \\in \\reals^{p\\times 1}.", "d772420d5fd118b509c520cb33d99237": "\\int \\tan (x) \\,dx = \\int {\\sin (x) \\over \\cos (x)} \\,dx", "d77255b6f15d5051846ae1901e53b53d": "y(x) = ce^{3x} \\, ", "d77269ee2f4a01e3f5ddacafeb6c2252": " \\operatorname{gl}\\,\\dim R[X_1,\\ldots,X_n] = \\operatorname{gl}\\, \\dim R + n.", "d77291dfbf863e2e93fd60fcc582cc49": "L \\leftarrow N+1-L", "d772f855def863a90462ed7a92853081": "\nM' = (I_2 \\otimes \\phi)(M) =\n\\begin{bmatrix}\n\\phi(f^*f) & \\phi(f^* g) \\\\\n\\phi(g^*f) & \\phi(g^*g)\n\\end{bmatrix}\n", "d772fa8c01439fee9d1f9b283f6ebf25": "L_v^2 L_{vv} = L_x^2 L_{xx} + 2 L_x L_y L_{xy} + L_y^2 L_{yy} ", "d772fdc134c245c082d3393f1eeaadb7": "G \\wr S_n", "d77351347d48676bff2f946fb3edc086": "\\forall i < n \\; (a_i < m)", "d77393bfe79d97431569179807cb4b97": " H_s(r)=1-e^{-\\Lambda(b(o,r))}, ", "d7741403731cc9a444fdb32257ece41a": "M,N \\in \\mathcal{M}(S)", "d7742c092776c863fbe2f4bb694ecc77": " = D(\\rho||\\sigma) - \\lim_{n\\rightarrow\\infty}\\frac{1}{n}\\left( \\log\\frac{1}{\\epsilon} \\right) ", "d774366b244748b356e3dbceb402afcc": "2^5=32", "d77459cbce561adada7a8763c970597e": "\\ln_q(x) = \\ln(x) \\text{ if } q = 1 ", "d7745e49b0332129888f59866e2a6535": "GT = \\frac{TS}{\\sum_{k=1}^N (T-T_\\text{crit})}", "d7747ebe2f7bc5267026e012604fd249": "\\phi = f - g", "d774e5a6fe0d1277c5cade2dcfb9abe4": "I_{EH}", "d77534f3db999cf3cc6394341de42f82": " \\widehat{A} = \\widehat{A}^\\dagger ", "d775a109917a3972470a7a7b20fe2093": "\\exp : \\mathbf{R} \\to \\mathbf{R}^+ : x \\mapsto \\mathrm{e}^x", "d775a135ed082b62fa31256d195d4f9e": "d(v)=H[i+1]", "d775b54d3484b16f6f929bad821092be": " S_AS_BS_C = S^2(S_\\omega-4R^2)\\quad\\quad S_\\omega=s^2-r^2-4rR \\, ", "d7760c24f0a3792cac9d56c8b7c46368": " x^{(0)} =\n \\begin{bmatrix}\n 1.1 \\\\\n 2.3 \\\\\n \\end{bmatrix}.", "d7767ceef0cfbca636847c7062a89f31": " v_2 (x) ", "d776955ba8b1e9bf0b8fb793c1247126": "\\pi_i p_{ij} = \\pi_j p_{ji}.", "d776a45caeb0e4331c0860cef2d28257": "1 \\leq i \\leq N", "d776bc78da95edcdf6f43489237b54b8": "\\lim_{t\\to\\infty}f(t)", "d7776b4465c1beaa333e6627f4be31ae": "(A + UCV)X = I", "d7777a9581950cacb2d76d66d8fd3d29": "L_{e}", "d7778738123c285825d3816c98c6740f": " s_{\\hat\\alpha} = s_{\\hat\\beta}\\sqrt{\\tfrac{1}{n}\\textstyle\\sum_{i=1}^n x_i^2}\n = \\sqrt{\\tfrac{1}{n(n-2)}\\left(\\textstyle\\sum_{j=1}^n \\hat{\\varepsilon}_j^{\\,2} \\right)\n \\frac{\\sum_{i=1}^n x_i^2} {\\sum_{i=1}^n (x_i -\\bar{x})^2} }\n ", "d7778a89e6503de88fa015377aeb7712": " f_n(y) = \\frac{f(x + \\tfrac{1}{n}, y) -f (x, y)}{\\tfrac{1}{n}}.", "d777b69e81862777c11e99d8c6529a62": " P^{(B,k)}", "d777da51d4291558e7ad89bce01552dc": " M_\\mathrm{max}(B) = -21.726 + 2.698 \\Delta m_{15} (B). ", "d777f1228b68f86332f5d692f34c6648": "\n\\begin{align}\n D_2 & = \\frac{3}{2}n-\\frac{27}{32}n^3, &\n D_4 & = \\frac{21}{16}n^2-\\frac{55}{32}n^4,\\\\[8pt]\n D_6 & = \\frac{151}{96}n^3, &\n D_8 & = \\frac{1097}{512}n^4.\n\\end{align}\n", "d77804b8694d66185c71e77bf9bdc4cf": "e^{\\frac{-iat}{\\hbar}}", "d7781a0750a1dfed4674379acbceadf0": "w = z^{1-b}u,", "d778767cac1b835ab75a896efd14f8df": "\\leq_K", "d7788daf5b077fd24bb737299c35ba31": "\\mathbf{d}_i = \\frac{(1-t)(1+b)(1+c)}{2}(\\mathbf{p}_i-\\mathbf{p}_{i-1}) + \\frac{(1-t)(1-b)(1-c)}{2}(\\mathbf{p}_{i+1}-\\mathbf{p}_i)\n", "d778927878ada54bc4608614746ba0e9": "a^{2^r\\cdot d} \\equiv -1\\pmod{n}", "d778b72775f1af48caadceb8c6d5e3ec": " \\pi_j(\\theta) = \\pi(\\theta)^{1/[1+\\lambda(j-1)]}, \\ \\ \\lambda > 0, ", "d7792874010108ae09514d45bacd6e82": "2^5 \\cdot 9^2 = 2592", "d7798bec6a26013b6686b9650bd80588": "f(\\gamma(i))=0", "d77992003db7c824f03b55c0177aa957": "\\begin{bmatrix}\n3 & 11 & 61 \\\\\n11 & 119 & 653 \\\\\n61 & 653 & 3589\n\\end{bmatrix}.", "d779bd3d971e301a18966ebf447e09cc": "G(\\eta)", "d77a1aaef5182ce2e23f89c86655fe29": " (l_1,l_2,\\cdots, l_k) ", "d77a2cccdf21f5c59bcdcd9f6c8cf957": "n. ", "d77a8389c682c7414fbfbee89dad70a1": "\\circ : \\operatorname{Hom}(C) \\times \\operatorname{Hom}(C) \\to \\operatorname{Hom}(C)", "d77aeac7cf670f1a1504fbc5eeebf2ea": "\nH_T = H + \\sum_k U_k\\phi_k + \\sum_{a, k} v_a V^a_k \\phi_k\n", "d77af4242e656548daa7b4116cffab27": "\\nu',\\lambda',\\lambda", "d77b270288219f01eb10e44952a9150a": "\\begin{array} {rcl}\n\\alpha_i[\\mathbf{f}A] & = & \\alpha(Y_i) \\\\\n& = & \\alpha\\left(\\sum_j a^j_i X_j\\right) \\\\\n& = & \\sum_j a^j_i \\alpha(X_j) \\\\\n& = & \\sum_j a^j_i \\alpha_j[\\mathbf{f}]\n\\end{array}.", "d77b3d6732da3f634f60b1a09619188e": "F = 6 \\pi r \\eta v \\,", "d77b4c81b84ea0af514f1a9b37afa259": "f \\left( T^k x \\right)", "d77b8823f789cd973e60ae1c371f11bc": "C(x,y):=D^{-1}*\\delta_{y}, \\qquad x \\neq y \\in M,", "d77bb833c2b4908a9aabe5d819a89954": "\\mathbb{T} = \\mathbb{Z}", "d77cbde1ef34fdb267ff1d3adc14838f": "Q(x|y)", "d77cfd3de6156780d9abdc18b647806a": "\\mathrm{Nu}_L=-\\nabla'T'", "d77d1cce601bf22156fd39762a8dd597": "\\mathbf{F} = \\mathbf{J} \\times \\mathbf{B}", "d77d348ecbaffbd99354dc050c9c311d": " f(x) \\sim \\lim_{n\\to\\infty}\\sum_{k=0}^n\\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k . ", "d77d3a061c98e0e438715b1ca1c487c9": "t_{1/2}=11.43", "d77d49c947089d7bed56e2726df81223": "\\beta_0=0.5", "d77d84c494f563612bb915e95df8ef02": "f(n)=1+\\frac{n(n+1)}{2}=\\frac{n^2+n+2}{2}.", "d77d9b2894ec076dc3e365c30fddc8af": "\\hat{x}(y)", "d77dd18cfd2a41440ac6ea4c959e46fa": " {dx \\over dt} = (\\alpha + \\beta)xy - \\beta x ", "d77e01d7271db898c25729e0f97f4be3": " \\vec{\\kappa} = \\vec{b}\\cdot \\nabla\\vec{b}", "d77eb31546b9f917cb61e4a08d52056b": "\n\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n 1 & \\lambda z \\\\\n 0 & 1\n\\end{bmatrix}.", "d77ec623b8344b74fcc9909992ce061a": " 0 = \\text {time at the start of the deal} ", "d77ecb7db67b155324ae26ab14a2f3a8": "\\varphi(d).\\ ", "d77ed0ed5c54a68c4bc138a15aa1a6c9": "\\rho_s = -n e x", "d77f3fe3df43b48b391b3017cd8a1b8c": "PSL(2,p)", "d77f7fa107ec3eeed186c4d94d37b960": "{\\cos^2\\alpha_1} = 1 ", "d77f8ee77ae974e2e3ad36b6173dec1b": "M(\\vec{r}) = \\sum_{i=1}^N \\sigma_i", "d7804e48db3a648964511040254e6c7b": "\nh(x)=\\sum_{i=1}^k g_i(x)^2 .\n", "d7810636a4b700265774029b190834b3": "\\eta^{\\gamma\\mu}", "d781731a13b6b08eb968f0375fe5e6b5": "s_i = IV_i", "d7819f2a87874deed3b1eee67f41b202": "\\left|\\rho_{in}\\right| = 1\\,", "d781c6b18505d7123585e2490e5b5e3f": " \\mathbf{p}(0), \\quad \\mathbf{\\dot{p}}(0).", "d7821fcf12708cc5f5c7cb5f56a88ec0": "\\hat{H} = -\\frac{\\hbar^2}{2\\mu}\\nabla^2 - \\frac{Ze^2}{r} ", "d7823ec3db02b80a5ad7f36d5ef8523d": "w_x,w_y , \\epsilon Z^+ ", "d78262c24b87c5d156e417ed356936aa": "\\left\\{\\frac{L(\\theta_1|X)}{L(\\theta_2|X)}\\right\\}", "d7826fc1e2445b238f752b1a2bcbc46c": "P(T)", "d7827bff5d973c60d8c64a2742514227": "\\,f(1)=1\\,", "d78289ceae2059ae749ab0a0dc3f1605": "-\\sqrt{\\frac{25}{84}}\\!\\,", "d7829edc383dd30dad3f2a99bc70578c": "I^{lm}", "d782acaf544abe98ed5058b1d38c9950": "\\frac{-d[COOH]}{dt}=k[COOH]^2", "d782cfa0d66bd53574c9bec06eb5e4ed": "f_x(t) = \\frac{x_1}{\\sqrt 2} + x_2 \\sin(t) + x_3 \\cos(t) + x_4 \\sin(2t) + x_5 \\cos(2t) + \\ldots", "d782ef5acdf7d07df4f19ba144ea327b": "\\bigcup\\nolimits_{n \\in \\mathbf{N}} X_n = X \\neq \\varnothing.", "d782fc4def5aa4b213738b52ee381e63": "\\mathcal{Z}(R_R)\\subsetneq R\\,", "d7832a29f3d25dcd9c378f38f2071721": "dy=f'(x)dx", "d783d8824beb232b096c329851c4f8dd": "1\\Rightarrow 3", "d783f1e817074734341b2d9427921b9d": "f(x) = a^x", "d783f8f9dc8fb7b5ba76fbea8a7f8ac0": "\\scriptstyle\\mathcal{I}^{-1}", "d7842607239cfa52aafb3af442b0ecb8": "\n\\frac{V_\\mathrm{out}}{V_\\mathrm{in}} = \\frac{R_2}{R_1+R_2} = \\frac{6}{9} = \\frac{2}{3}\n", "d784493050f5377e1d525412a8dcb287": "-385\\pm 70", "d78495a44cffdb7ffb774263c547afc3": " \\cdots \\overset{\\partial}{\\to} C_n(T)\\overset{\\partial}{\\to} C_{n-1}(T)\\overset{\\partial}{\\to} \\cdots \\overset{\\partial}{\\to} C_0(T)\\overset{\\varepsilon}{\\to} \\mathbf Z, ", "d784a2795c625439c4091c426cff6577": " E_c=\\frac {1}{2} k y^2", "d784ad2e8ebd4cb5a868c54a9bfcce34": "\\ rate = k_{obs}[I]^{3/2} \\,", "d784d11b1951a487bf33eba07a133fbc": "h=\\frac{g(\\Delta t^2)}{2}", "d785013a79b0fa68b282e817aaee662d": "\n \\sigma = \\sqrt{\\cfrac{2 E \\gamma}{\\pi a}}\n ", "d7850d62c546d1542e2d11ba8bddaa86": "K_pm \\subset K_{\\pm0}", "d7852900ab3a702fdfbd67dbb935f609": "M>4", "d7859544937bbd7b8fe346d4cc5ef3ff": "[1:0:0]", "d7859e7c5a354ecd2970f8c3c044439f": "\nx = 1+\\cfrac{1} {2+\\cfrac{1} {2+\\cfrac{1} {2+\\cfrac{1} {2+\\cfrac{1} {2+\\ddots}}}}} = \\sqrt{2}.\\,\n", "d785cbf81ed5ed8be48d90132bdee932": "(Tx_n)", "d785da866e7bf589726bad5f6a453371": "d_\\text{match}(\\ell_1, \\ell_2)=\\max\\{\\delta(r,r'),\\delta(b,a'),\\delta(a,\\Delta)\\}=4", "d785dffc7c2b33fb99957bb871633b88": "x\\frac{du}{dx} = f(u) - u\\,; ", "d785e276a708dc95e26ee2c07228ff3d": "\\nabla \\cdot \\mathbf{E} = \\frac{\\rho_{\\mathrm e}}{\\epsilon_0}", "d785ed9af0281c00be6a4c7b8b92ee40": "\\scriptstyle{V=} \\tfrac{1}{3}\\scriptstyle{bh}", "d785fd566b085fc31d124a48aff47b86": "d_0 = b_1 - \\frac{b_0}{c_0}c_1 = \\frac{c_0b_1 - b_1c_0}{c_0}; d_1 = b_2 - \\frac{b_0}{c_0}c_2 = \\frac{c_0b_2 - b_0c_2}{c_0};\\ldots \\quad (34)\\,", "d7861c46853a599cc1fc7948cce4761b": "t=0,\\dots,T-1", "d7864b55669e00fe310c1fc0793ba043": "DR_{S}^{T}", "d78686ad71c63f160db1dcb35e3bf8ba": "~\\ell~", "d78695e3ca89acd2346db4846c6542a0": "\\mathbb Z_n \\times \\mathbb Z_n", "d786a6d4b02b63cce8be098f4fca45d8": "\\frac{\\mbox{EBIT}}{\\mbox{Capital Employed}}", "d786d2f4f1e69146abd20685d3e8ab89": "L_1 = \\{ a^n b^n c^n \\ \\ |\\ \\ n\\ge 0 \\}", "d786f3a2e826b8d38b2f71decc3a2059": " \\sigma_+^2 = \\sigma_-^2 = \\frac{ 1 } { 2 } \\sigma^2 ", "d7870daba6043294d452890b51d600c4": "\\!n", "d78778595d3fb55ecf7a140f4ea4bc71": "M_{k}", "d787a044927ca29eaf4f99bd3ef7bfff": "|\\boldsymbol{E}|={1\\over4\\pi\\varepsilon_0}{|q|\\over r^2}", "d787fcfb27811294957bff0541317b79": "P_{n} \\rightharpoonup P", "d78821aa6a9c3830aa90a324ec692c6c": "\\gamma: {\\rm Map}(8,M)\\to LM", "d7889a736092dfa38eff392077840411": "C_{te}", "d788a6e7d0b248208a57fce319adf78a": "\\sum_{}^{} a_n t^n", "d788ad5f2b4844a48d7313038e130d07": "e^{i\\mathbf{K}\\cdot\\mathbf{r}}=\\cos {(\\mathbf{K}\\cdot\\mathbf{r})} +i\\sin {(\\mathbf{K}\\cdot\\mathbf{r})}", "d788e0f3b7eb97a501f97543e0b7b838": "b \\nrightarrow a", "d78948c1d8e6427a3de36b61fe2d0a3f": "x_3 = \\frac{y_2x_1z_2 + z_1x_2y_1}{(y_2^2 + (z_1x_2)^2)}", "d7894ed2951c62d80d84d727a278068a": "K = (A/g^{x_2 x_4 s})^ {x_4} = g^{(x_1 + x_3) x_2 x_4 s}", "d78992cea1f279c0973c5252b7b9c198": "d_d", "d789c3b4fc23e5ad6e6a1aee60c57f72": " L_u=2L_v=\\frac{h}{(0.177+0.000823h)^{1.2}} ", "d789c6ca957388223412909ed3d48778": "\\left(\\sigma \\mu_S^\\ominus+\\tau \\mu_T^\\ominus -\\alpha \\mu_A^\\ominus- \\beta \\mu_B^\\ominus \\right)", "d78ae26f62af0df46dfdb2fdf799900d": "-0.026838601\\ldots", "d78afa2bbc4ed2186bfb3bdb3f600071": "\n\\begin{align}\n\\frac{\\partial u}{\\partial t} - f v& = -g \\frac{\\partial h}{\\partial x} - b u,\\\\[3pt]\n\\frac{\\partial v}{\\partial t} + f u& = -g \\frac{\\partial h}{\\partial y} - b v,\\\\[3pt]\n\\frac{\\partial h}{\\partial t}& = - H \\Bigl( \\frac{\\partial u}{\\partial x} + \\frac{\\partial v}{\\partial y} \\Bigr)\n\\end{align}\n", "d78b7b66540de3f959e3cf7da4630dd2": "\nm \\ddot{r} = F(r)\n", "d78bc1d5072faf6b16bf02e4a1ead61b": "y_{text} = {y_{marker\\_orig} \\over y_{orig}} \\cdot y_{scaled} - {marker\\_size \\over 2} + y_{adjust} + y_{textadjust}", "d78bf8c47df62837dc0058b4acb633c8": "G_i/G_{i+1}", "d78c24cbbb9be603a75cdb3c85f64016": "d_j S(t)", "d78c5a136b381010ca71786790cc792e": " \\frac{d^2 u}{dx^2} -u =0, ", "d78c8bdf1e734bfdd77666505f91fab6": "\\,T", "d78c93ff8c58d51c10164291ded10a93": "\\frac{13}{20}=\\frac{10}{20}+\\frac{2}{20}+\\frac{1}{20}=\\frac12+\\frac1{10}+\\frac1{20}.", "d78ce12b821cca0cd5cd52614cc306a2": "I\\, =\\, (UU),\\; K\\, =\\, (U(U(UU))), \\;S\\, =\\, (U(U(U(UU))))", "d78ceaa067a8cdc2457aaa1147ddffd9": "P(M)_t", "d78cfcdfa053d45e8cd1899726b907c9": "\\boldsymbol{Ax}=\\boldsymbol{b}", "d78d16c689b570aeec79b22e45f56295": "v_2 = v_1 = 0", "d78d6aff2056e0235b087e5f4ffa0bcd": "i\\hbar\\frac{\\partial\\psi(x,t)}{\\partial t} = - \\frac{\\hbar^2}{2m} \\frac{\\partial^2\\psi(x,t)}{\\partial x^2}+V(x)\\psi", "d78d9d81e0cd022ca5889de4683f931a": "v = q/2 - \\sqrt{(q/2)^2-(p/3)^3}", "d78de4905a97f76e8d59e75d73087649": "\\frac{P_l P_m-P_{l+1}P_{m+1}}{r^{l+m+2}}\\,.", "d78e2a8fd6020719068cfaa3a2c8e8eb": "\n \\Gamma_{12}^2 = \\Gamma_{21}^2 = \\cfrac{1}{r} ~;~~ \\Gamma_{22}^1 = -r\n ", "d78e98c1e312399a6e91df8b5d77c74e": "\\scriptstyle \\mathcal {H} ", "d78ed26d948cfcfa4ee5dc3f53c12348": " \\operatorname{merge-let}[\\operatorname{let} V : E \\operatorname{in} \\operatorname{let} W : F \\operatorname{in} G] = \\operatorname{merge-let}[\\operatorname{let} V, W : E \\and F \\operatorname{in} G] ", "d78f54c05fbf0d38f3e0e9a2b9d2cb2e": "f_2(1270)", "d78ff9613e5285a1eef4dfbaef3b65e3": "\n\\operatorname{E} \\left[\\left. \\frac{\\partial}{\\partial\\theta} \\log f(X;\\theta)\\right|\\theta \\right]\n=\n\\operatorname{E} \\left[\\left. \\frac{\\frac{\\partial}{\\partial\\theta} f(X;\\theta)}{f(X; \\theta)}\\right|\\theta \\right]\n=\n\\int \\frac{\\frac{\\partial}{\\partial\\theta} f(x;\\theta)}{f(x; \\theta)} f(x;\\theta)\\; dx\n=\n", "d7903a396ddf1699af7543b4d42fd4b2": "\\scriptstyle \\frac{\\pi}{2}", "d79088bdb7db1c7e2a859ea9735ef9a6": "\\operatorname{cov}(X) = \\operatorname{E}\\left[\\left(X-\\operatorname{E}[X])(X-\\operatorname{E}[X]\\right)^\\mathrm{T}\\right]", "d790f9b23174bde1298fd05bea0bfee6": "\\mathbb{P}(A) = \\frac {1}{100}\\ , \\ \\mathbb{P}(B) = \\frac {99}{100}", "d7911dc41fc16b2609a64a8483a87dac": "s(1)=12,\\ s(2)=13,\\ s(3)=1 \\, ", "d7912df11e0460b59615136834776a46": "E_0 \\supset E_1 \\supset E_2 \\supset \\cdots", "d79133f81ba9213f498feec13c638a73": " Re ", "d79162e57ad0eda5e3db81741bcfad81": "y_{t+h}", "d791df729f9dab86d10ca2c5c6ca8ac0": "V(\\mathbf r) = -\\int_C \\mathbf{F}(\\mathbf{r})\\cdot\\,d\\mathbf{r} = -\\int_a^b \\mathbf{F}(\\mathbf{r}(t))\\cdot\\mathbf{r}'(t)\\,dt,", "d79221d924c14509fe346642aae85c06": "\n\\omega _0 = \\gamma B_0 = \\frac{{2.79 \\times 5.05 \\times 10^{ - 27}\\,{\\rm{J/T}} }}{{6.62 \\times 10^{ - 34}\\,{\\rm{Js}} \\times \\left( {1/2} \\right)}} \\times 1\\,{\\rm{T}} = 42.5\\,{\\rm{MHz}}\n\\,", "d792898578f0fb8ac919f42057028659": " y(t) = \\int_0^t H(s) \\,\\mathrm{d}s = \\begin{cases} 0, & \\text{if } t \\le 0; \\\\ t, & \\text{if } t > 0 \\end{cases} ", "d7928d8a19449fa79b8f666f0c756617": "W_{in} = F_{in} d_{in} \\, ", "d792ccc6c4a45a015e7ed862797cc755": "F(x,y,u,u_x,u_y) = u_x^2+u_y^2-1.", "d792dbd0d4e23d206af10d977cb36d44": "i_i", "d792eb0a64a95c4eb2e006b6a6ed132a": "\n\\begin{align}\n\\hat{P}_{ij} \\Psi\\big(1,2,\\ldots,i, \\ldots,j,\\ldots, N\\big)& \\equiv \\Psi\\big(\\pi(1),\\pi(2),\\ldots,\\pi(i), \\ldots,\\pi(j),\\ldots, \\pi(N)\\big) \\\\\n&\\equiv \\Psi(1,2,\\ldots,j, \\ldots,i,\\ldots, N) \\\\\n&= - \\Psi(1,2,\\ldots,i, \\ldots,j,\\ldots, N).\n\\end{align}\n", "d7933938366125f36ffd1ae45a4f844c": "\\scriptstyle \\prod g^{c_i y_i}", "d79353c7800732f8cedfc3feca5e604b": "{\\pi}_n\\,\\!", "d79380f44a9e49251ea39abb31efc5e2": "\\varphi = 2\\tan^{-1}\\tanh\\tfrac{1}{2}\\theta \\equiv \\mathrm{gd}\\,\\theta.", "d7938d60d1103d92fdf3286690ee0b60": "\\mathrm{kg\\,m^2}\\,", "d79396d5a41854bee651e248aaacbd8c": " (\\kappa-n-1)~r^{n+1}~\\sin(n\\theta) \\,", "d793996db854ad2bc258dfdb8bf4b87d": "\\frac {v- v_\\mathrm F} {v_\\mathrm F} = \\frac {Nc} f", "d793b4682d76fd50db9a35fc799c33dc": "\\begin{bmatrix} 10 & 14 \\\\ 11 & 15 \\\\ 12 & 16 \\\\ 13 & 17\\end{bmatrix}", "d793ee3f3f753b97c36dda89e08f31b7": "\\sin (x^2)", "d7946c3bcc3e442467d91011492bf972": "dS = \\left(\\frac{\\partial S}{\\partial T}\\right)_{V}dT + \\left(\\frac{\\partial S}{\\partial V}\\right)_{T}dV \\,", "d7948038b48d696a8b6515e4786db6a2": "\n u = \\gamma(\\mathbf{v})(1 + \\mathbf{v})\n", "d79497e4b15bd30507a80ee6b7df943e": "{ a + {\\underbrace{1 + 1 + 1 + \\cdots + 1}_{b}} }", "d794a7fbda3a905bf5fd32b577eab7b1": "(\\epsilon_0)\\,", "d794e7879975e45b7bbeae18f2d60161": "\\operatorname{CH}^q(X) = \\operatorname{H}^q(X, K_q(\\mathcal{O}_X))", "d794e8bdbbeb29b3502cce686706edcd": "\\Psi_{xy}(m_x, m_y)", "d79507115ba8584dd45768621ceeecae": " B_i\\subset X", "d7952871519f4680f60c778d1960f13e": "\\frac{V_w}{V_s} = \\frac{\\alpha}{2\\pi}", "d795407f738c1c3958f7a2b4dffb7cbb": " P( | X - \\mu | \\ge k \\sigma ) \\le 1 - \\frac{ k }{ \\sqrt{ 3 } } \\quad \\text{if} \\quad k \\le \\frac{ 2 }{ \\sqrt { 3 } }", "d7955b557b94fbf48889bf308cd7422f": "f(x,y,z) = x^2 + y^2 + z", "d7958b61fd508d4f16b39c34c3f99842": "\\Delta x < 3.4 \\lambda_D,", "d795aa98e61d6a2c25ba66c410fa9ba5": " i = 1 \\ldots t ", "d795e87e4837a827a7a07b6ef25b1804": "p(f_{ik}|c_j)\\ ", "d795f327c1813dc3210998f3055f7733": "F(v) \\leq F(u),", "d795fb74b0f2cd9b7830f7afa6fd66bd": "\\sum_{i=1}^N d_i= \\sum_{i=1}^N \\lambda_i.", "d795ff6348704b3d4aee889d989fecc1": " \\sum_{k=0}^{\\infty} c_k ", "d796311b933700e07bb8084f57804763": "n^\\mu", "d796457d3863ba6b60b544b12711bfee": "F = 1- \\frac{ \\mathrm{obs}(Aa) } { n2pq } = 1- {138 \\over 1612*2(0.954)(0.046)} = 0.023", "d796578e9b9cd8984770cc9bd5b1397e": "\\frac{8388611}{25165824}\\,", "d796597fa5b5821a1f27312cc756a09a": "D^2 f(\\mathbf{a})\\!", "d7965c1c22a4d067494acd57efa4bf9e": "|\\partial u/\\partial z|<\\gamma_0", "d7967250090374d792034ccb862c62e6": "p(I | \\theta, O_{bg})", "d796d2d3404b30192376c201dfffcd87": "\nT^{2k - 1} = \n\\begin{pmatrix}\n0 & 1/2 & 1/2 \\\\\n1 & 0 & 0 \\\\\n1 & 0 & 0 \\\\\n\\end{pmatrix} \n", "d7973f9979da5978507cbf0898640204": " y(t) = (h*x)(t) \\ \\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^{\\infty} x(u) h(t-u) \\, du ", "d797460d7e733e02b11c67f5fc59aa08": "{\\mathrm{d} X_{t}} = a(X_{t}) \\, \\mathrm{d} t + b(X_{t}) \\, \\mathrm{d} W_{t},", "d79781381ead6dbf82f610a1a1b2c840": "a^b \\pmod {N}", "d7978d19edde6af7fb0705fa876a72c5": "\\forall z\\in E", "d797a08841fa9e6acd2968fc4a83077e": "K(a,b)=\\left(1-b \\bar{a} \\right)^{-1}.\\,", "d797b50017e4129a958392f45f05f3e4": " T = {I_{1}\\over I_{0}} = e^{-\\int\\alpha'\\, dz} = e^{-\\sigma\\int N dz} ", "d7981d06c597444ccec9ebda27aa30e6": "\\mathrm{Stk} = \\frac{\\tau\\,U_o}{d_c}", "d7982703436d4e2f79d6da9acfb72b58": "y_1 = f_1(x)\\,,\\quad y_2 = f_2(x)\\,,\\ldots, y_n = f_n(x) ", "d7983501c552f3286a20a1bd3d967afd": "\\mathbf{F} = \\mathbf{F}^e\\mathbf{F}^g", "d798350bee6d749d37421e8d4acb954e": " fd ", "d798447c6f5d9068d7430481404a9496": "\\displaystyle (1-e^{2\\pi i\\alpha})(1-e^{2\\pi i\\beta})\\Beta(\\alpha,\\beta) =\\int_C t^{\\alpha-1}(1-t)^{\\beta-1} \\, \\mathrm{d}t.", "d7985dbeb1a2849394e3a1d6a282b5be": "A_i, B_j", "d79876484c81dbcde757aa8d992e2134": "\\{X_i,Y_j\\}", "d79885a0c5a8d1cf626981759067f4ec": "\\frac{dx}{dt}=k'(p'+x)^{a'}(q'+x)^{b'}", "d798930041132e18093910e43917a311": " \\bar \\nu _{v^\\prime,v^{\\prime\\prime}}", "d798c090e030ba35d7af04efd5f37888": "\nh_{\\lambda} = \\frac{1}{2} \\sqrt{\\frac{\\left( \\mu - \\lambda \\right) \\left( \\nu - \\lambda \\right)}{ \\left( A - \\lambda \\right) \\left( B - \\lambda \\right)}}\n", "d798d377a03334c4b05c0be25395798f": " | \\phi_{\\beta/\\alpha} \\rangle := \\alpha s |0\\rangle + \\frac{\\beta}{\\sqrt{2}}(2^T-2s)|1\\rangle", "d798daff70aace073742a3f86eae8c08": "k_{i,i+1}=\\frac{f_2-f_1}{\\sqrt{f_1f_2g_ig_{i+1}}},", "d7997d4767cf7d17801b927ad89531a8": "\\omega=\\omega^i_{\\ j}", "d79a1550f71235fa5731ce7167032453": "y=c_1y_1+c_2y_2+c_3y_3+c_4y_4", "d79a6204b214f6af9817f56671f19e2c": "|\\alpha\\rangle =e^{-{|\\alpha|^2\\over2}}\\sum_{n=0}^{\\infty}{\\alpha^n\\over\\sqrt{n!}}|n\\rangle =e^{-{|\\alpha|^2\\over2}}e^{\\alpha\\hat a^\\dagger}|0\\rangle\n", "d79a76238e573e0786b0890474081bad": "F_i(k_{xy})=H_i(j_{xy}).\\ ", "d79a81d6f6c675374dfb04a33d06ef05": "(\\mathbf{g}'(\\mathbf{X}))^{\\rm T}", "d79a88f502220f2b5e1f0db43de32678": " G = (P, \\mathbf{C}, \\mathbf{U}) ", "d79b14da08b79c6cc6b7c49f648d8593": "U^* W_u U = V^* W_v V = I.", "d79b9f09e80ca1155f98654c77c42a95": " v_0 h_0 = v_1 h_1 = q", "d79bee470a436e769c0a8f43020f3e98": "\\mathrm{Spin}(12)\\cdot\\mathrm{Sp}(1)", "d79c0339005637eda038fc44f170c076": " \\vdash \\forall x \\, (P(x) \\rightarrow Q(x)) \\rightarrow (\\forall x \\, P(x) \\rightarrow \\forall x \\, Q(x)) ", "d79c0786d37c93c9ef7f37e4b145740a": "\\hat{\\Psi}(\\omega)", "d79c3cb9f0676856f60e4bc1ea4b2872": " 0 \\bold x = {\\bold 0} ", "d79cacc15083145477a644f2100ad02e": " \\lambda=\\frac {E}{\\eta} ", "d79ce89013ce8e67501675dbd935fac6": " \\mu = \\frac{m_1m_2}{m_1+m_2} \\,\\!", "d79d88aea6e8a0ff9b32ee78b4f2cea8": "z = -1 \\quad (L_3)", "d79e7c710e3a26f7b9a92fa0aac02200": " (B \\or C) ", "d79ec1bcc81332bba8a250bfcd9a95c8": "f'(x)-f'(a)\\quad", "d79f2a43f65d655ae63af430f84f5b7a": " f(t, y(t), y'(t))=p(t)y'(t)+q(t)y(t)+r(t). \\, ", "d79f6062088bf9d424c26d1354a4d7a6": "x_1 = \\frac{b(bx_0 - ay_0)-ac}{a^2 + b^2} \\text{ and } y_1 = \\frac{a(-bx_0 + ay_0) - bc}{a^2+b^2}.", "d79f6da65565bb21b2e39eae890c2c5c": "\\neg \\Box A", "d79fa20ec401324e2f883e8c9f7489c3": "\\scriptstyle U_n", "d79fcc77a83d6b069d0e38a418592931": "(i\\gamma^\\mu(\\partial_\\mu + ieA_\\mu) - m) \\psi = 0\\,", "d7a00a4e742f7e1ae0f550986aed2d2d": "X_{hG}\\sim X.\\ ", "d7a00c9cf3706c615117c3b5f01453ad": "\\Delta V = \\sum_{i=1}^n \\omega_i\\, \\Delta r_i\n+ \\sum_{1 \\leq i,j \\leq n} O(\\Delta r_i\\, \\Delta r_j), ", "d7a02b9824ef762e98e2ad759dea93bb": "\\Lambda = (\\lambda/", "d7a09a9732d492ebccb860a040ccc146": " M \\propto R_g^d ", "d7a0e2c2bee578779bdaeb4af09da5c8": "C=(\\frac{1}{2}\\sqrt{5+2\\sqrt{2}})a\\approx1.39897...a", "d7a102af9ea0a34b4516f43656c848fe": "\\operatorname{Lomax}(\\tilde{x}|\\beta',\\alpha')", "d7a1750cb07376f9d4b87d3aa9a875b3": "\\Theta=d\\theta + \\omega\\wedge\\theta = D\\theta,", "d7a1c840f4f5700d399197812b452109": "\\begin{cases}\n\\gamma = \\left( \\lambda + \\mu \\right) /2 \\\\\nb = \\left( \\lambda - \\mu \\right) /2 ,\\,\n\\end{cases}", "d7a1ec01bb8c289acedf3e959b8569f9": "|\\bold B| = |\\boldsymbol {\\sigma_i}|", "d7a24bd00484f665efefdd91088a31d9": " \\mathbf{b}_i\\cdot\\mathbf{b}_j = \n \\begin{cases} g_{ii} & \\text{if } i = j \\\\\n 0 & \\text{if } i \\ne j,\n \\end{cases}\n ", "d7a2dd3423f14ec5b9e18f3ea988d29d": "{df \\over dx} = v {du \\over dx} + u {dv \\over dx}.\\, ", "d7a2e357ac74932e3da69b9f9954a257": " \\begin{align}\n\\mathbf{r} & =\\mathbf{r}\\left ( t \\right ) = r \\mathbf{\\hat{e}}_r\\\\\n\\mathbf{v} & = v \\mathbf{\\hat{e}}_r + r\\,\\frac{{\\rm d}\\theta}{{\\rm d}t}\\mathbf{\\hat{e}}_\\theta + r\\,\\frac{{\\rm d}\\phi}{{\\rm d}t}\\,\\sin\\theta \\mathbf{\\hat{e}}_\\phi \\\\\n\\mathbf{a} & = \\left( a - r\\left(\\frac{{\\rm d}\\theta}{{\\rm d}t}\\right)^2 - r\\left(\\frac{{\\rm d}\\phi}{{\\rm d}t}\\right)^2\\sin^2\\theta \\right)\\mathbf{\\hat{e}}_r \\\\\n & + \\left( r \\frac{{\\rm d}^2 \\theta}{{\\rm d}t^2 } + 2v\\frac{{\\rm d}\\theta}{{\\rm d}t} - r\\left(\\frac{{\\rm d}\\phi}{{\\rm d}t}\\right)^2\\sin\\theta\\cos\\theta \\right) \\mathbf{\\hat{e}}_\\theta \\\\\n & + \\left( r\\frac{{\\rm d}^2 \\phi}{{\\rm d}t^2 }\\,\\sin\\theta + 2v\\,\\frac{{\\rm d}\\phi}{{\\rm d}t}\\,\\sin\\theta + 2 r\\,\\frac{{\\rm d}\\theta}{{\\rm d}t}\\,\\frac{{\\rm d}\\phi}{{\\rm d}t}\\,\\cos\\theta \\right) \\mathbf{\\hat{e}}_\\phi\n\\end{align} \\,\\!", "d7a2f1ad940f23ab6798488e03c2db8c": "4*x", "d7a3759a2a4deeec9e376bbc5ed250fa": " n^2 ", "d7a38e683cec3ccf8dc1c655a4445685": "y_c = \\biggl(\\frac{q^2}{g}\\biggr)^\\frac{1}{3}", "d7a39f6f83d7d667bcacd8fe2f50c362": "\n\\langle T \\rangle_\\tau = -\\frac{1}{2} \\langle V_\\text{TOT} \\rangle_\\tau.\n", "d7a3a85e6f7932f158c83cb4f3f69056": "\\widehat{a}", "d7a42028575c166c85024a2ab01d68a4": "\\scriptstyle Y \\to Z ", "d7a428b1dcd2ae79260e4c5fbc12695e": "{dY}/{dL}=MPL=w", "d7a432f4dff1b11e7a65e6870b69996b": " x_{n+1} = r x_n (1-x_n)", "d7a45c82b5e9ffb73b1b66d27d8a4027": "g (\\gamma^2-1)", "d7a45fe93fa01a70698c498152232481": "X_+:=X\\coprod \\{*\\}", "d7a48ee9c3373b7d10ebb34421912a6d": "\\left( \\frac{3}{\\sqrt{10}},\\ \\frac{-5}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ 0 \\right)", "d7a4b781948d620c690a22c2fb923f49": " \\frac{1}{2 \\pi i} \\oint_C \\frac{x^s}{s \\zeta (s)} \\, ds = \\sum_\\rho \\frac{x^\\rho}{\\rho \\zeta'(\\rho)} - 2+\\sum_{n=1}^\\infty \\frac{ (-1)^{n-1} (2\\pi )^{2n}}{(2n)! n \\zeta(2n+1)x^{2n}}. ", "d7a4d3209d7003b504a276c47c560c2c": "H = \\phi^{-1}(\\sigma)", "d7a4dea708371e2bee94c76fc7599115": " u'\\frac{\\part L}{\\part u} =u'\\frac{d}{dx} \\frac{\\part L}{\\part u'} \\, . ", "d7a4e3e5b2f3f8597256cd69fd9d592b": "\\mathbb{P}^2_k", "d7a53fea76eb769d1f7da885869f0ba8": "\\left ( -x,-y,-z \\right )", "d7a58c5ac703925585a5b96b58ed0823": "\\gamma(s, x) = \\int_0^x t^{s-1} e^{-t} \\operatorname{d}t = \\int_0^x \\sum_{k=0}^\\infty (-1)^k\\,\\frac{t^{s+k-1}}{k!}\\operatorname{d}t = \\sum_{k=0}^\\infty (-1)^k\\,\\frac{x^{s+k}}{k!(s+k)} = x^s\\,\\sum_{k=0}^\\infty \\frac{(-x)^k}{k!(s+k)}", "d7a60dbf086241d1eb3b542093f27f95": " m_\\mathrm{e} v r = n \\hbar ", "d7a63dc42bcd3d6c34bd0a7e00760685": "c_3 = 10.14333127,\\,\\!", "d7a65ed112f0e87314c93617397b3268": "[M]=[N] \\in \\mathfrak{N}_n", "d7a6e9e9db5f20e7558c3d7e590c089b": "x_{n-1}\\,\\!", "d7a77f10e1735ad2a18d996bf94fceb4": "\\sum_{i=0}^n i^3 = \\left(\\frac{n(n + 1)}{2}\\right)^2", "d7a7c234bdc0144300ea58efd45d8965": "x=\\pm\\infty\\,", "d7a7c359025f764589a880eed9098ef1": " L(x,\\dot x , t)=p\\cdot \\dot x - H(x,p,t)\\,,", "d7a81f083d631503e78305a3f211fd57": " \\displaystyle{B(z,t)= \\exp \\,[-z^2 -t^2/2 +zt].}", "d7a865360510018951342227596609f1": "\\Phi=-\\frac{2}{\\pi}\\log|t|.\\,", "d7a8a36c9fb5cde24661c516680a1bb3": "X = -T^{-1/2} + \\frac{1}{2} - \\frac{1}{8}T^{1/2} + \\frac{1}{128}T^{3/2} + \\cdots", "d7a8c0db592ec712333c39ccf766bd9d": "p(x|M)", "d7a91fba1fe6df7e25fb635e1f3e4df7": "\\triangle \\, = \\frac{n_1 - n_2}{n_1} \\ll \\ 1", "d7a9459ffdf3035b95e5e185f8413040": "PGL < P\\Gamma L,", "d7a9bd6672f4e2ac58d8b42d82ab5b58": "I(2\\omega,l) + I(\\omega,l) = I(\\omega,0)", "d7aa34674db2e2391933d937c4c425bf": "\\begin{array}{lcl}\n?\\left(\\frac{x}{x+1}\\right) &=& \\frac{?(x)}{2} \\\\\n?(1-x) &=& 1-?(x)\\,.\n\\end{array}", "d7aa62ff863da3148ee3791ceea184e3": "\\phi(1)=1", "d7aa69b2cfdc4c1929000aa136d716ad": "n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + \\ldots + 1^2 = \\frac{n(n+1)(2n+1)}{6}.", "d7aafc2c8b633209bb3745695532dbbc": "(0,\\ \\pm\\sqrt{2},\\ \\pm\\sqrt{2},\\ \\pm\\sqrt{2})", "d7ab667c142044adcf38c60c9758934c": "|\\rho|=0", "d7ab6dbe5f6c52244f7aeb29274008d3": "f_{alt} = f_R - f_T = 0.1\\; \\mathrm{Hz}", "d7ab7466bd99b839e95578ba59610a1f": "\\lambda_k\\rho_k", "d7abaa3da06b382f1b4e678e12a959e8": "p_{1} \\in A_{1}, p_{2} \\in A_{2}, p_{3} \\in A_{3}, p_{4} \\in A_{4}, ", "d7abb15cae509bd41ba5247246b66f1e": "j_{\\lambda}", "d7abc50ff77fd5b464bc17c7313b958c": "w_i=w", "d7abd236e676921d9c7ded28037ac02c": "\\{\\gamma \\cdot x : \\gamma \\in \\Gamma\\}", "d7abfd76b14e811cd198efd8b2d1f2a0": "p_i \\equiv 2^i - 1 \\pmod{2^i}", "d7ac22b4aec5ec245279d93710f1b997": "\\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix} ", "d7ac5e558c3890d8c6866c497d079c30": "\\vec{r} \\, \\rightarrow - \\vec{r}", "d7acde37ce1169e74ab14449da315e40": "v\\in[0,2\\pi]", "d7ad740e0cb72d2a2d11c5e5084b420e": "\\begin{bmatrix} I & 0 \\\\ -VA^{-1} & I \\end{bmatrix} \\begin{bmatrix} A & U \\\\ V & C \\end{bmatrix} \\begin{bmatrix} I & -A^{-1}U \\\\ 0 & I \\end{bmatrix} \n= \\begin{bmatrix} A & 0 \\\\ 0 & C-VA^{-1}U \\end{bmatrix}\n", "d7adb5f0b33579dbdaee4b0cf3e0fab3": "~ \\theta", "d7adc3d0c4a6fe95ce8e388eafe6746c": " \\scriptstyle{\\mathcal{P}(i)} ", "d7ae88795d946c81014cff5dc9eccd74": "\\mathrm{SO}_4^{2-}\\ ", "d7ae8ff842b0c5364b98eb6fa541ca14": " \\frac{-1}{\\log_{2}p}", "d7aecf858a7cd04b8c0ef3e9ebae4fed": " F_{Y|X=x} (y) = \\mathbb{P} ( Y \\le y | X = x ) = \\frac12 + \\frac1\\pi \\arcsin \\frac{y}{\\sqrt{1-x^2}} ", "d7af1eb0247296b2ff8d2b6a531156ed": " \\sum_{s=0}^r {d \\choose s}. ", "d7af2ce2fdf865ce6cfc95f0d8be8e55": "(G,n)", "d7af558f2ad1219f89ffeb2e5e0a3f34": "\\varphi_{ij}^2:\\mathbb{R}^p\\to\\mathbb{R}^{p}.", "d7afba1dd1ce8cfccf1383fdc0621b07": "\\left(\\pm1/2,\\ \\pm1/2,\\ 0,\\ \\pm\\sqrt{1/2},\\ 0\\right)", "d7afc144fb75f9a9db6dfac191753a76": "w=C(x)", "d7b068a03723c82a4fd9128951c4c641": "\\vec{F} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_1 q_2 }{|\\vec{r}|^3} \\vec{r} = \n\\frac{1}{4 \\pi \\epsilon_0 } \\frac{q_1 q_2}{|\\vec{r}|^2} \\hat{r}", "d7b0994ab74b992cfb8f14bf1c885dda": "\\displaystyle X^\\lambda=\\bigcap_{\\alpha<\\lambda}X^\\alpha", "d7b0d12c56de6b681b78852be0827886": "\\left(\\mathbf{A} + \\mathbf{UBV}\\right) \\mathbf{A}^{-1}\\mathbf{UB} = \\mathbf{UB} + \\mathbf{UBVA}^{-1}\\mathbf{UB} = \\mathbf{U} \\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right).", "d7b0f3f55e504bc9cf8deab7ed3933f1": "\\left(\\bold{r}_{uu}\\right)_v=\\left(\\bold{r}_{uv}\\right)_u", "d7b190ca4d7b0b94eda282037891a472": "8+ 2/3 + 1/10 + 1/2190", "d7b1a41117be4f2aef0ae6175a12b22b": "t = k \\sqrt{\\frac{m}{q}}\\,", "d7b1cf24bfa538b4fc12a86817bddd18": "P'_{n-1}(x)", "d7b23e119081c550a75381532d00d02d": " f(X)", "d7b286c3cf20e1ee0ef7842e7e94a5b8": "(7.e)\\quad \\nabla^2\\Phi =\\,2\\nabla\\psi \\nabla\\Phi\\,,", "d7b2994f27686739e0c0ab35df6af1c8": "y_t = k \\cdot t + u_t", "d7b2cd9216f733f1a1568ba6fcf18822": "X^\\triangledown", "d7b2d9d802184af90d83087781078ab4": "\nA^+ = 26\\left[1+y\\frac{dP/dx}{\\rho u_\\tau^2}\\right]^{-1/2}\n", "d7b355498c752a9f7408f89bce7dd183": "f\\in I^mM", "d7b37094c60221ade5a0165c983a932e": "E(|X|I_{|X|\\geq K})\\le\\epsilon\\ \\text{ for all X} \\in \\mathcal{C}", "d7b3b0723a407d9a561c8e7527b92544": "\\mathbf{r} = \\mathbf{K}^{-1} (\\mathbf{R}-\\mathbf{R}^o ) \\qquad \\qquad \\qquad \\mathrm{(3)}", "d7b4553e760be13cc1d0323bd6352468": "1/(z-a)", "d7b46e67371b8c7160adbd641d48483b": "\\ \\mathbf{B}", "d7b473bfef3a2588526d1cf7fe71969d": "\\left(x!\\approx x^x\\,e^{-x}\\,\\sqrt{2\\pi x}\\right)", "d7b51c01ef7cea24052134848d3044a9": "M=\\mathcal{T}=[~]_{n^\\searrow \\;\\|\\;n^\\nwarrow}\\equiv_{m} [~]_{n^\\nwarrow \\;\\|\\;n^\\searrow}=N", "d7b5830da95c31b5edf7a84a85109e5a": "\\oint_C(N_xdx + N_ydy) = \\iint_S\\left(\\frac{\\partial N_y}{\\partial x}-\\frac{\\partial N_x}{\\partial y}\\right)dxdy = ", "d7b5db3b11203720759748211c0e0f8e": "S, T \\subseteq \\Omega", "d7b68a1a62cdd8802da89c4acd078e5c": "Fe^{3+} + e^- \\rightleftharpoons Fe^{2+} ", "d7b6c9e5cd82e6a2ed7a5dd1d80eb1cd": " b > 0, 0 \\leq c \\leq 1, \\frac {df} {du} < 0 ", "d7b6e8d22a80545ae07114ae44a0fe42": " S^{-1}(p) = m + \\frac ia \\sum_\\mu \\gamma_\\mu \\sin(p_\\mu a) \\;. ", "d7b74f406c12eb1a983c1987da9b8cac": "\\lambda_{CWL}(M)=0", "d7b78fc8da3bf7c73c3d0125829b18a8": " \\int_V f_{V}(g)\\, d \\mu(g) =1.", "d7b7e3525072d9d114b6d623e0d606b0": " I_a(y) = (y/2)^a \\sum_{j=0}^\\infty \\frac{ (y^2/4)^j}{j! \\Gamma(a+j+1)} .", "d7b7fb6a0356ca826a531a105bca9e74": "0 \\to M'_\\mathfrak{m} \\to M_\\mathfrak{m} \\to M''_\\mathfrak{m} \\to 0", "d7b85c5314722b9ec7540bad20c77858": "\\{\\lambda_1, \\lambda_2, \\lambda_3\\}", "d7b89f725185c1fd9984123cbaa89612": "x\\geq 0", "d7b8ee9db57512446c178ae885e29245": "\\alpha=3V\\left[\\frac{\\epsilon_p/\\epsilon_m-1}{\\epsilon_p/\\epsilon_m+2}\\right]", "d7b91f49cd9c490254066dc65ed401b9": "V_B", "d7b9e2a79943d28cc9557f17645ef40b": "G=u \\rho (u)\\,", "d7ba4dd68beb63247807075bd30bc3af": "\n\\begin{bmatrix} a & b & c\\end{bmatrix}\n\\begin{bmatrix}x \\\\ y \\\\ z\\end{bmatrix} = d\n", "d7bae3c358d46098f88b7a29b5db7540": "x\\Leftrightarrow y = (x\\Rightarrow y)*(y\\Rightarrow x).", "d7bafb2562989a27615da6de4cff9876": "\\sum_{u~v}", "d7bb0616470b6d1dcfb5f6681e82f21a": "f_r = f_t \\left( \\frac{1+v/{c'}}{1-v/c'} \\right)", "d7bb7f1fa9adb1b000bc49bcd226a622": " s i n \\alpha", "d7bbb3e4eba754d10b469a7b164c6fe9": " P^{-1}A", "d7bbced9949830b833a5e83b2b57dc0e": "=A \\cup (A \\cap A^C)\\,\\!", "d7bbff3013f03c4dd8c3defa786c9e69": "\\big\\{|0\\rangle_{1}\\otimes|1\\rangle_{2}, |1\\rangle_{1}\\otimes|0\\rangle_{2}\\big\\}", "d7bc0fdc4657cf9f4721e49962035141": "\\mathfrak{p}\\subset \\mathfrak{q}", "d7bc23068201f105f2e7112358753a1e": "\\displaystyle\nH(\\Gamma)/([u]H(\\Gamma)+H(\\Gamma)[w]).\n", "d7bc41c57ce7af0e298cb62c9408d831": "\\gamma_m(x) \\gamma_n(x) = ((m, n)) \\gamma_{m+n}(x)", "d7bc8ee8576fcbf7e20152d087e01657": "\\hat{p}_{nm} = p_{(i-n)(j-m)}, 0\\le n,m \\le i,j ", "d7bcae7410b497b53bb7616bc3efdfbe": " r_n = \\tilde{H}_n y_n - \\beta e_1. ", "d7bcbaa08610b18a31d36216e6f8ad95": "\\displaystyle{f(z) = CT^{-1}h(z) + z.}", "d7bd697bacfe91002e3fe2d4b9d2665d": " p > 1/3 ", "d7bd8e5e5ac75aeb48a226a0c02fcbca": "|j, m_{min}\\rangle", "d7bdae7d943a3e7efe836ea6d3e2f80a": "\\beta(t) = \\omega_{0} b", "d7be06ceaeddfaca575a2e6476eaef08": "E = \\tfrac{1}{2}mv^2 + \\tfrac{1}{2}mr^2 \\cdot \\frac{v^2}{r^2}", "d7be138a923b77d1c8937af9266db770": "\n \\xi(\\xi_0,t)\\approx\\xi_0+(\\hat{u}/\\omega)\\cos(k\\xi_0-\\omega t)+(\\tfrac12 k\\hat{u}^2/\\omega^2)\\sin2(k\\xi_0-\\omega t)+k\\hat{u}^2t/2\\omega\\ .\n", "d7be1d0d7fb28aa6eaca214818102a38": "\\mathbf{p}_n \\cdot \\mathbf{p}_k = |\\mathbf{p}_n||\\mathbf{p}_k|\\cos\\theta_{nk}\\,,\\quad |\\mathbf{p}_n| = \\frac{1}{c}\\sqrt{E_n^2 - (m_n c^2)^2}\\,,\\quad |\\mathbf{p}_k| = \\frac{1}{c}\\sqrt{E_k^2 - (m_k c^2)^2} \\,,", "d7be4eebd028a988da9b0251d0d840e3": "a\\land(a\\lor b)=a", "d7be618209dcc32e27053bc857c31c94": " a\\, ", "d7be7b8eb1a3c920ffd571ce574faee6": "\\begin{matrix} {4 \\choose 1}{3 \\choose 2}{3 \\choose 2}{3 \\choose 1}^2 \\end{matrix}", "d7bed10ce7b2371201cc7be2b929c5fa": "\\log n! \\approx n\\log n - n + \\frac {\\log(n(1+4n(1+2n)))} {6} + \\frac {\\log(\\pi)} {2}", "d7bf3c3b181290b27d2c1f8c9c569db8": "f^{-1}\\colon O(Y)\\to O(X).", "d7bf5a198f79653830d1ec65ae9d4800": "L_2 - L_1 = 0.177\\lambda - 0.098\\lambda = 0.079\\lambda\\,", "d7bf5b6e97abd09ab84b771f67f5121a": "r = r_c", "d7bf6dd7dccd601ee8f569e1d2cc90c2": "\\scriptstyle\\hat\\theta^H_n", "d7bf966248ff0f8d7d9449ec58a21629": "A = 1 - 3X + 5X^2 - 7X^3 + 9X^4 - 11X^5 + \\cdots.", "d7bf98c124664e1fc97d8f52d7650d46": "V_{out} = -\\dfrac{\\dfrac{N V_{in}t_{\\Delta}}{R_{i}} + \\dfrac{N_{p}V_{ref}t_{\\Delta}}{R_{p}} - \\dfrac{N_{n}V_{ref}t_{\\Delta}}{R_{n}}}{C}", "d7bfe652fe879b5554178dc5eefa54f9": "R_{\\text{e}}", "d7bff8a3db67069e9015215a96df4b03": " T^2 = \\frac{4\\pi^2 a^3}{G(m_{1}+m_{2})} ", "d7c0219f632f9cb7eb34d1646878382b": "\\scriptstyle x \\,", "d7c062861d5e2f8c465cdf080aacb65c": " (a \\lor \\lnot b) \\rightarrow c ", "d7c0866407f2fcc6f04c4b7f9aad804f": "(\\rho-\\theta)", "d7c0a132c10f4ab2d8459f2a1ef27961": "\\sigma_{i\\,j} = - \\sum_{k=1}^3 \\sum_{\\ell=1}^3 c_{i\\,j\\,k\\,\\ell} \\epsilon_{k\\,\\ell}", "d7c11dfa942f76604c996f5606cbdf5b": "\\{1,2,\\ldots,p-1\\}", "d7c1415601e7787706053f726d1823e2": "\\Phi^{-1}(p)", "d7c1904a52599396312aad9e8994737a": "P=(x,y) \\in E(\\mathbb{F}_q)", "d7c1d0a8c537daf4bf7fef591c459a6e": "\n V_2 = R_a -x \\quad \\text{and} \\quad \n M_2 = -50 + R_a (x-10) - \\frac{x^2}{2} \\,.\n ", "d7c1f2f2e8a39b4acafcb92357436832": "\\begin{align}\n (m|k)\n &=(m\\mid k)\\cdot 1 \\\\\n &=(m\\mid k){\\sum_{n=0}^\\infty(n\\mid m, k)} \\\\\n &=(m\\mid k){\\sum_{n=0}^\\infty(m\\mid n, k)\\frac {(n\\mid k)}{(m\\mid k)}} \\\\\n &=\\sum_{n=0}^\\infty(m\\mid n, k)(n\\mid k)\n\\end{align}", "d7c23a06d93d0538af4646d0faf5afb3": "\\omega_0 \\approx \\frac{\\sqrt{6}}{\\pi} \\sqrt{\\frac{1-(1/2)^{1/2k}}{m^2-(1/2)^{1/2k}}}", "d7c2408132e8de04316d15e357d3ae37": "\\ \\mathbf{0}: x \\mapsto 0", "d7c26129bf33b068dc927382684248a0": "\\! \\Lambda", "d7c2958273faeadc2e2120111fbe86b5": "\\sqrt{s}", "d7c37c77b8d036566e33c9cca27d7b2f": " \\left( \\textbf{f}_1+\\textbf{f}_2 \\right) \\cdot \\textbf{h} = \\textbf{g} \\pmod q ", "d7c37cd486543934e3b1ff2f746bcf2c": "\nE = E^0 + \\frac{kT}{e} \\ln \\frac{[\\mathrm{Ox}]}{[\\mathrm{Red}]}\n= E^0 - \\frac{RT}{F} \\ln \\frac{[\\mathrm{Red}]}{[\\mathrm{Ox}]}.\n", "d7c3992439f73c886a76145b88690ce5": "\\, \\frac{e^{tb} - e^{ta}}{t(b-a)}", "d7c3b434359db9a133973a4ef92983be": "f,g:X\\rightarrow Y", "d7c425f55efa86eb57f65d71505bbb02": "B\\cup N_x", "d7c492d852dd08b9bd963946832e3f9a": "\\frac{\\partial}{\\partial x} \\left( \\frac{f(x)}{g(x)} \\right) \\geq 0", "d7c4a9896f168cbb60439b58b9bf53f8": "(B y + \\beta)^n - B^n y^n", "d7c4e855474e4bda0e7c97af0b621665": "\\frac{\\text{d} [{_2^0}P]}{\\text{d}t} = \\text{k}_{3(1)} C_1", "d7c4f874449424068f41e791054632eb": "\\Psi_1 = \\sqrt{n_s} e^{i\\theta_1}", "d7c5230545636befae97b719f7d21b1a": " \\sum_{i=0}^{N-1} a_i \\cdot 2^i ", "d7c58195966ce35afc4785aceb4b0e21": "v = \\frac{c}{\\sqrt3},\\ \\gamma = \\sqrt{\\frac32}", "d7c5d6e8156cca992bb52cebe85140e2": "\n \\begin{matrix}\n P & = & \\begin{bmatrix}\n 0.9 & 0.1 \\\\\n 0.5 & 0.5\n \\end{bmatrix}\n \\\\\n \\mathbf{q} P & = & \\mathbf{q}\n & \\mbox{(} \\mathbf{q} \\mbox{ is unchanged by } P \\mbox{.)}\n \\\\\n & = & \\mathbf{q}I \n \\\\\n \\mathbf{q} (P - I) & = & \\mathbf{0} \\\\\n & = & \\mathbf{q} \\left( \\begin{bmatrix}\n 0.9 & 0.1 \\\\\n 0.5 & 0.5\n \\end{bmatrix}\n -\n \\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 1\n \\end{bmatrix}\n \\right) \n \\\\\n & = & \\mathbf{q} \\begin{bmatrix}\n -0.1 & 0.1 \\\\\n 0.5 & -0.5\n \\end{bmatrix} \n \\end{matrix}\n", "d7c5db8b9d2b057b98f952ceaf4ed42e": " \\beta=0.92; 0.57 ", "d7c5e17c4e8b0b2f5e0d7e3d2c43013f": "W_x(t,f)", "d7c651810528e9dcff6f80f14c204ae1": "q'(x_2)=k_2", "d7c65a12412b6eb88d340ef94bf68da7": "Z_{\\text{eff}} = \\sqrt[2.94]{0.2 \\times 1^{2.94} + 0.8 \\times 8^{2.94}} = 7.42\n", "d7c664eda811cc4eeda18e132c45bb55": "K^{\\ddagger\\ominus} = \\frac{[AB]^\\ddagger}{[A][B]}", "d7c6976133b291ddbb01268293c8fea7": "(T_s-q_s^{1/2})(T_s+q_s^{-1/2})=0.", "d7c6e84e3de108b036320e1ab6106659": "I=nAvQ \\, ,", "d7c70f43aad914c1a104cbb064112fb8": " f(x) = a x^2 + b x + c \\,", "d7c72d25ca369bbef4ab5e5e039f287d": "p>\\tfrac{(1+\\epsilon) \\ln n}{n}", "d7c73141e8d0a3f7c697abe7059647cb": " \\sum_{i=1}^m{a_{ij} x_i} + e_j t_j \\ge g_j", "d7c7615b5c5914c5b670f86c235a234d": "f : D^4 \\to \\mathbb R", "d7c78d1776180e38698da7df8f6949e0": "L/2", "d7c7ffadca0062480df14d35faef261a": "im(\\underset{=}{A})=Y", "d7c84c9a447db6a3828a83693a6fa2aa": " P_{IN} = V_{IN} \\times I_{OUT}", "d7c9007a0edcc455b5e5487ddf112437": "\nP(a\\alpha", "d7ce35f468bf858140b3e74fdbcedd0e": " v_o = \\frac{2\\pi a}{T}\\left[1-\\frac{1}{4}e^2-\\frac{3}{64}e^4 -\\frac{5}{256}e^6 -\\frac{175}{16384}e^8 - \\dots \\right] ", "d7ce46701fb6a2151431733c1e6ea633": "\\sigma = \\frac{3FL}{2bd^2}", "d7ce8245722c1576f7251fa04fb56188": "|\\psi\\rangle = {3\\over 5} i |\\uparrow\\rangle + {4\\over 5} |\\downarrow\\rangle.", "d7ce93d0838078030bcc9d038f0885d0": "\\ \n\\left\\{\n\\begin{array}{l}\n\\frac{1}{3} \\rho l_1^{2} \\left(l_1 + 3 l_2\\right)\\ddot{\\alpha}_1 + \\frac{1}{2} \\rho l_1 l_2^{2} \\ddot{\\alpha}_2 + (k_1 + k_2)\\alpha_1 - k_2\\alpha_2 - l_1 P (\\alpha_1 - \\alpha_2) = 0 , \\\\ [5mm]\n\\frac{1}{2} \\rho l_1 l_2^{2} \\ddot{\\alpha}_1 + \\frac{1}{3} \\rho l_2^{3}\\ddot{\\alpha}_2 - k_2(\\alpha_1 - \\alpha_2) = 0 .\n\\end{array}\n\\right.\n", "d7ce9f14a5c4e23eaa11b71126d424c8": "\\{f^n(x): n\\in \\mathbb{N}\\}", "d7ced69ec013e9bd836434bae2f9a9f8": "X \\to \\mathbf{P}^n_S", "d7cf334e9a426faeae34fa03b0528bbf": "w_i(\\gamma) = \\gamma+(-1)^{k(k+1)/2}\\langle \\gamma,\\delta_i\\rangle \\delta_i", "d7cf36e96858a7fb122e9411ddc720be": "{m1,m2,m3,k}", "d7cf46ff9a3389bd0d61711b70adf58b": "x\\sim y\\iff x\\leq y \\land y\\leq x", "d7cf628a754a49a14512a1e820554c95": " x_t e^{\\theta t} = x_0 + \\int_0^t e^{\\theta s}\\theta \\mu \\, ds + \\int_0^t \\sigma e^{\\theta s}\\, dW_s \\, ", "d7cf8060b940bc65ac0143fd2285db89": "dy/dt(t)", "d7cfb23450291969e5f005706414a6ab": "\\scriptstyle P", "d7cfbaa8b2465c1ecdd07f8779c76d31": "\\mathbf{R}_O = R_O\\mathbf{e}_y", "d7d0318ef85d7bb201c79a5ee748c242": "\\mu_1+\\mu_2\\,", "d7d052c3d2ee322abbe96029531f4517": "f(x,\\tau|\\mu,\\lambda,\\alpha,\\beta) = \\frac{\\beta^\\alpha \\sqrt{\\lambda}}{\\Gamma(\\alpha)\\sqrt{2\\pi}} \\, \\tau^{\\alpha-\\frac{1}{2}}\\,e^{-\\beta\\tau}\\,e^{ -\\frac{ \\lambda \\tau (x- \\mu)^2}{2}}", "d7d0549f3917018880d6b5e1bebbe159": "x y-1=0", "d7d0bf898bdfc3b3933a87496c82263c": "\\zeta_{V,p}\\left(p^{-s}\\right).", "d7d0e09e7502cad3f3791dd38ee984b0": "\n\\int_y K(x-y;t)K(y-z;t') = K(x-z;t+t')\n\\, ,", "d7d10cafa1e4f439cac827222272af10": "\\operatorname{E}(T(T-1)) =\n\\left. \\left( \\frac{\\mathrm{d}}{\\mathrm{d}z} \\right)^2 G(z) \\right|_{z=1}.", "d7d16aba9a0f9abca3841beb03c67229": "y^2 = x^3 + 7", "d7d193829674a6a7aa35f45db8e4127d": "b = p^\\beta v", "d7d2d1f4ed84df5ed4393bddd7f55b6d": "M_{\\kappa,\\mu}\\left(z\\right) = \\exp\\left(-z/2\\right)z^{\\mu+\\tfrac{1}{2}}M\\left(\\mu-\\kappa+\\frac{1}{2}, 1+2\\mu; z\\right)", "d7d30e9cdbb5ad9c18711334490043a1": "\\displaystyle{D(f,g)=(f_x,g_x) + (f_y,g_y).}", "d7d35293f1fb421c0dd823122f1de7a3": "\\left |\\xi-\\frac{m}{n}\\right |<\\frac{1}{\\sqrt{5}\\, n^2},", "d7d4bb0ce8c586253c82fa6355dc9aaa": "\\lambda^-_i = \\sum_j \\rho_j \\mu_j p^-_{ji} + \\lambda_i. \\,", "d7d519163c4575a78d51549c1b5b50e4": "\\frac{2}{\\Gamma(\\frac{\\nu}{2})}\\left(\\frac{-i\\tau^2\\nu t}{2}\\right)^{\\!\\!\\frac{\\nu}{4}}\\!\\!K_{\\frac{\\nu}{2}}\\left(\\sqrt{-2i\\tau^2\\nu t}\\right) ,", "d7d54f0f7602c91f52f177f53e789246": "{}\\quad \\stackrel{\\mathrm{def}}{=} \\ \\int_{-\\infty}^{\\infty} x(t-\\tau)\\cdot h(\\tau) \\, \\operatorname{d}\\tau", "d7d5d2295627e0cd9c0a7eea847eb4bb": "g(u)", "d7d65252e71df9f14b51528ff16edae5": "X_\\infty=\\mathcal A/{\\sim}", "d7d65baf0daa0f35cc341242989824aa": "((P \\and Q) \\to R)) \\to (P \\to (Q \\to R)))", "d7d687ff7d654df3a2d524bedbe868a4": " \\mathbf{R} = (\\mathbf{R}-\\mathbf{S}) + \\mathbf{S} = \\mathbf{d} + \\mathbf{S},", "d7d6bba3a43d07d96c0f2a2dfc911b26": "A^T A \\vec\\rho = A^T \\vec{p}", "d7d6ef637ef25b74efdf257fdca77160": "D_\\min = 50", "d7d70593dc6174a985663e14fd220273": "\\frac{dy}{dx},", "d7d72e4fec2c2517e888806c1479d30e": "\\operatorname{sign}", "d7d73471b6e3469d3cc44a99dad91fe3": "0 < abs(\\lambda) < 1 \\,", "d7d776a18a011dc580221cf3ddd519a3": "\\|T(t_0)\\|<1", "d7d7f54164122e89755d4d752d285675": "\\begin{align}\n\\frac{N_1}{d} + \\frac{N_2}{d} &= \\frac{N_1+N_2}{d}\\\\\n\\frac{N_1}{d} - \\frac{N_2}{d} &= \\frac{N_1-N_2}{d}\\\\\n\\left(\\frac{N_1}{d} \\times \\frac{N_2}{d}\\right) \\times d &= \\frac{N_1\\times N_2}{d}\\\\\n\\left(\\frac{N_1}{d} / \\frac{N_2}{d}\\right)/d &= \\frac{N_1/N_2}{d}\n\\end{align}", "d7d800a4f64548b9227d293b5a5d86b1": "I(barked : N^r \\cdot S) = [x] \\cdot barked(x) : E^r \\cdot T", "d7d804272b383ddb01b6695a6e921c3f": "\\Gamma = n \\langle \\sigma v \\rangle \\sim G_F^2 T^5", "d7d854e5619ef083bfb90cbc958e6726": "f \\in L^1(\\mu)", "d7d8614fe8a778c1fe09a1a8f9753eba": "\\vec{F} = -\\nabla V(\\vec{x})", "d7d8a4e353c861257e411d2432ee2013": "\\beta_p", "d7d90db1b86c4a89b5bfa6db46a8090d": "\\tfrac{AE}{CE} ", "d7d92afcc0a3f8e7de1ef4e76b3a5f25": "\\ k_B \\,", "d7d942c497eb3737f9bfb510ecba708c": "[\\Epsilon_c(k_e) + \\Epsilon_h(k_h)+V(r_e-r_h)-\\epsilon]A^{n,K}_{c,V}(k_e,k_h) =0(*)", "d7d96f05ef2bbbf421a1971e3e9f4063": " c\\in \\mathbb{N}, z\\in cP\\cap L\\implies \\exists x_1,...,x_c\\in P\\cap L", "d7da4d640463e716718c2a2dd3dae02d": "\n\\begin{align}\n\\operatorname{E}[\\tau\\mid a_N, b_N] &= \\frac{a_N}{b_N} \\\\\n\\operatorname{E}[\\mu\\mid\\mu_N,\\lambda_N^{-1}] &= \\mu_N \\\\\n\\operatorname{E}\\left[X^2 \\right] &= \\operatorname{Var}(X) + (\\operatorname{E}[X])^2 \\\\\n\\operatorname{E}[\\mu^2\\mid\\mu_N,\\lambda_N^{-1}] &= \\lambda_N^{-1} + \\mu_N^2\n\\end{align}\n", "d7da6169fa02ef60b223c42d22c359c8": "h = 1.5 \\cdot d", "d7da7965d28536f5d0a82e6d48383134": " D = \\partial +\\nabla+\\omega_2+\\omega_3+\\cdots. \\,", "d7daba00b6cab2efd9410f10bdb529e5": "M = R(1-cos( \\frac {28.65S} {R} ))", "d7dae3c8a67650c9fd047d34d7dfb763": "~~~~~S,V,\\{N_{i\\ne j}\\},\\mu_j\\,", "d7db1dd4fef372a581f541ddb8e66cda": "\\tilde\\gamma_n(t)=\\gamma_n(t)+\\beta(t)-\\beta(0)", "d7db38bde999b0bedebbca71d961462b": "-15\\le x \\le -5", "d7db4d67192ecc4bb20844c9617b69d5": "\\begin{align}\np\\\\\np \\rightarrow q\\\\\n\\therefore \\overline{q \\quad \\quad \\quad} \\\\\n\\end{align}", "d7dba1a4b4516558f724a9268e2e4711": "f \\sim g \\; (n \\to a \\in \\R)", "d7dbbbcc6589a4f211848ecb9d2f40cd": "\\beta = \\frac{\\partial f}{\\partial y} = \\frac{1}{a} \\frac{d}{d\\phi} (2 \\omega \\sin\\phi) = \\frac{2\\omega \\cos\\phi}{a}", "d7dbf04e6a39db2b6ff8823294b8ac2e": "r = \\frac{P - L}{L}", "d7dbfb52b7bf6e66775861b2b6c26b7d": "2^2 + 2^1", "d7dc1f14f0872e0336d9a2bbc3377eeb": "f_c(\\infty)=\\infty\\,", "d7dc20951056344f7bdbea2a44597574": "a\\frac{\\partial u}{\\partial \\mathbf{X}}", "d7dcb509d5bb35c4cbb62ffde8fa4d49": "0 \\leq u_1 \\leq u_2 \\leq 1", "d7dcd3f92cb24b5d8fabb0f671e90028": "t \\mapsto t^{\\alpha} \\ ", "d7dcd9903a90c694406c415b6556f3d8": "\\mathcal{L}_h = |D_\\mu h|^2 - \\lambda \\left(|h|^2 - \\frac{v^2}{2}\\right)^2\\,\\!", "d7dd4ae66e435dfd6b90cdd2b64914e8": " ( [P_c - P_i] - \\sigma[\\pi_c - \\pi_i] )", "d7dd54d92ecab33246fa3e02a73a5971": "\\frac{E_b}{N_0} > \\ln(2) ", "d7dd77d7f5c864fd39757faf048a1749": "\\boldsymbol\\Sigma_i", "d7dd8ddb4fec99a794b42783b07a2ec6": "\\mathbf{C}= (C_1, C_2, \\ldots, C_n) ", "d7ddf0ea55cad0a2e013d3e42702193c": "\\sigma=\\langle\\psi | \\cdot \\, \\psi \\rangle", "d7ddfc2ec0f2335a70412fc4149299c4": "\\frac{1}{\\Lambda_{GUT}^2}", "d7de1078bf30615b8123e16e196ca6a4": "S_{ab}=R_{ab}-\\frac{1}{4}Rg_{ab}.", "d7de63678c6e24146574f336b6d61cde": "\\begin{align}\n\\left\\langle (\\mathbf{F}\\circ \\Gamma(t))\\bigg|(J\\psi)_{\\gamma(t)}\\frac{d\\gamma}{dt}(t) \\right\\rangle &= \n\\left\\langle (\\mathbf{F}\\circ \\Gamma (t))\\bigg|(J\\psi)_{\\gamma(t)} \\bigg|\\frac{d\\gamma}{dt}(t) \\right\\rangle \\\\\n&= \\left\\langle ({}^{t}\\mathbf{F}\\circ \\Gamma (t))\\cdot(J\\psi)_{\\gamma(t)}\\ \\bigg|\\ \\frac{d\\gamma}{dt}(t) \\right\\rangle \\\\\n&= \\left\\langle \\left( \\left\\langle (\\mathbf{F}(\\psi(\\gamma(t))))\\bigg|\\frac{\\partial\\psi}{\\partial u}(\\gamma(t)) \\right\\rangle , \\left\\langle (\\mathbf{F}(\\psi(\\gamma(t))))\\bigg |\\frac{\\partial\\psi}{\\partial v}(\\gamma(t)) \\right\\rangle \\right) \\bigg|\\frac{d\\gamma}{dt}(t)\\right\\rangle \\\\\n&= \\left\\langle (P_1(u,v) , P_2(u,v))\\bigg|\\frac{d\\gamma}{dt}(t)\\right\\rangle\\\\\n&= \\left\\langle \\mathbf{P}(u,v)\\ \\bigg|\\frac{d\\gamma}{dt}(t)\\right\\rangle\n\\end{align}", "d7df06a8b55de47ac590acabd26d6d91": "\nD_C(x) = \\min_{y \\in C} \\|x - y\\|\n", "d7df184ec5a6c5bb331cae0f7f639d9d": "\\text{markup} = \\frac{\\text{gross margin}}{1 - \\text{gross margin}}", "d7df186ee74683f4516b952b49b39a46": " \\mathcal{ N}_k(k=1, . . . ,k_{max}) ", "d7df2a8349bc0e178f1d5530ddbf3ba9": "x_N", "d7df591a7a7edc191ca4571335d22fd8": " \\langle r^k \\rangle = a^k \\Gamma(1 + \\frac{k}{3})\\,.", "d7df6bd54910d95515964187b5c101e4": "e(N_{M_1} V) = -e(N_{M_2} V).", "d7dfedf50a7bedfff456c96412c6d5b1": " \\vec u(\\vec r)", "d7e0294b3df903563a44cfe5cd074932": "n =0", "d7e036eaa073e1d1ebbd38a3751cd213": "\nG = \\{ D \\in J_C(\\mathbb{F}_{q^n})~|~\\text{Tr}(D) = \\textbf{\\textit{0}} \\}, ~~~(\\textbf{\\textit{0}} \\text{ neutral element in } J_C(\\mathbb{F}_{q^n})\n", "d7e0e5434f5ff7b1fb4537b33277c192": "-0.75 < \\lambda < -0.5", "d7e11bfb010b62994d5c24856c54f980": "\\int_0^1 x^xdx = \\int_0^1 \\sum_{n=0}^\\infty \\frac{x^n(\\log x)^n}{n!} \\, dx. ", "d7e12fe8587a5fc0e35977c6631eb2bb": "K_+ = Ran(A+i)^{\\perp}", "d7e14f91b1d50501420fe209670129d9": "(f(x))^k = \\sum_{j=0}^{\\infty} B[f]_{jk} x^j ~,", "d7e18604726d4de035b21731e5cb3d9a": "\\mathbf{J} = \\sigma \\mathbf{E} ", "d7e1894e8d141e660905e9659c74ecbc": "\\mathrm{LMMSE} = \\mathrm{tr}\\{C_e\\} ", "d7e1c1188a16a80be39451bb56342ded": "\\ log(k) = s(N + E)", "d7e1ca8be9cc6076198474a217a0aba9": "\\alpha_{p} = \\frac{1}{V}\\left(\\frac{\\partial V}{\\partial T}\\right)_p", "d7e1cd99a7c5b5ea97e7b466e60e4df0": "\\gamma = \\sqrt{1 + \\left(\\frac{p}{m_0 c}\\right)^2}", "d7e20d19d30f74678b4dc9a84510b1ac": "2\\delta", "d7e225580b9de075e42a0a2be3d38824": "2\\phi(x)=\\phi(x)+\\phi(Sx)=1", "d7e235e032f5747a4cbe687dec1108fd": "\\frac{1}{k_{eq}} = \\frac{1}{k_1} + \\frac{1}{k_2} ", "d7e26838ae1f2e996b2b76a2f368ba7d": "h(x)= \\exp({b x})", "d7e2876ed9990cc287088918211f321c": "e^{i\\mathbf{k}\\cdot\\mathbf{r}}=\\cos {(\\mathbf{k}\\cdot\\mathbf{r})} +i\\sin {(\\mathbf{k}\\cdot\\mathbf{r})}", "d7e2a6c60d38ef1fe0eeab3452c6518c": "\\begin{align}\n\\hat{\\alpha}({r_{\\rm c}}) &= \\max_{q \\in \\mathcal{Q}} {\\hat{\\alpha}}(q, {r_{\\rm c}})\\\\\n{\\hat{q}_{{\\rm c}}}({r_{\\rm c}}) &= \\arg \\max_{q \\in \\mathcal{Q}} {\\hat{\\alpha}}(q, {r_{\\rm c}})\n\\end{align}", "d7e2db513366c7c9835fdc2578dda81f": "\n\\mathrm{Residuals} + \\hat{\\beta}_iX_i \\mathrm{\\ versus\\ } X_i \n", "d7e33e2f4b0d5923a0b074a93fb547d1": "z = (x^ox^m)\\,", "d7e34f0cc7b9f7d983661527c29a1dc7": " mv \\equiv Ft ", "d7e36b9e265eede3a238196272eb2107": "i^{4n+3} = -i.\\,", "d7e3702b297cc949fd78f52d0149b075": "t M = 0", "d7e3c937518fdd7781d1b938dcea4690": "\\partial_{tt} \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_{tt} \\psi )} \\right) + \\partial_{xx} \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_{xx} \\psi )} \\right) - \\partial_t \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_t \\psi )} \\right) - \\partial_x \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_x \\psi )} \\right) + \\frac{\\partial \\mathcal{L}}{\\partial \\psi} = 0 . \\quad \\quad (3) \\,", "d7e3cb946e455eda33e136d822082717": "10^{8 \\times 10^{16}}", "d7e432d8a91a8d16123c8d2e409ce01d": "M_2^* + M_2 \\xrightarrow{k_{22}} M_2M_2^* \\,", "d7e489c9bd21eb7274ea7acd2b4f6b5b": "\\alpha=0.5", "d7e48b45493e7fe65288f381d1b6c69a": "-k_1 x_1 = -k_2 x_2. \\,", "d7e5908420ddd97e89bd316fe547dc14": "\\Delta W = \\int_{\\theta_1}^{\\theta_2} \\left | \\boldsymbol{\\tau} \\right | \\mathrm{d}\\theta", "d7e5a3c3c5d897ff6354eae5ad1c4540": "\\eta= \\sigma / (2 \\Omega)", "d7e601618dfd6f8c9055e6fb7d6a17f1": "\nQ = \\frac{q^N}{N!}\\quad \\hbox{with}\\quad q = \\frac{V}{\\Lambda^3}\n", "d7e60bcbc4b28e75a1b6a0225f1ed64b": " A^*\\cup B^* = -N \\ln \\left[ 1 - A \\cup B / N \\right]/k", "d7e6979ea745476ed595cea9f028ab0e": "\\oint_A \\mathbf{D} \\cdot \\mathrm{d}\\mathbf{A} = Q_\\text{free}", "d7e6be4b30546dba6ffd0e0e2bced213": "2n\\,", "d7e6cad997e07ad1e509d8358698b50c": "h_0\\,", "d7e72af7649af20ef97dd327d46a645b": " g_{2,1} ", "d7e78ef5c26c525f39f62b99c85390a0": "\\mathrm{HA + B \\longrightarrow A + HB}", "d7e7dc43bd46c910bca7a305b56c430d": "H(H(u)) = -u", "d7e7e4eb16b091bcf7ae30eec0b11b24": "\\int_{v_0}^{v(t)} v\\,dv = -\\int_{r_0}^{r(t)}\\frac{GM}{r^2}\\,dr\\,", "d7e81008a0bf43d8820b688b510ab8c1": "c_{i \\neq k}", "d7e814c93b3a85d7164311c9e9e0dc7e": "s = \\frac{a+b+c+d}{2},", "d7e819812772effdf9c848d6f9a3a4d5": "Notional_\\text{floating leg}(t) = N \\times \\prod_{i=1}^{n-1} \\left( 1 + r_i \\right)^ {(\\frac{1}{252})} ", "d7e86bdff778b6bedd0fdd5d7d186a3f": " \\Delta'=\\Delta", "d7e87f5f5f6696ebfc81686480456398": "R(\\omega)=\\frac{E_{\\text{out}}}{E_{\\text{in}}}=\\frac{-r_1+(r_1^2+t_1^2)r_2e^{i2\\alpha}}{1-r_1r_2e^{i2\\alpha}},", "d7e8ca3e0e71d573b6cfffe8cf9907a9": "x \\cdot \\sin(x)", "d7e922f045f41163dce8d139466ab316": "a^2 - c\\,b^2)/(a^2 - c\\,b^2)", "d7e932cb44f8f380bb42be41d417e40f": " V \\Lambda V^{T} ", "d7e967120cb6e0512cc79bfcdb73ce67": "\\partial_t \\phi + \\partial^3_x \\phi + 6\\, \\phi\\, \\partial_x \\phi =0,\\,", "d7ea214746cfe8204730c169009aef2c": "-\\frac{v_0^2}{2} = -\\frac{GM}{r_0}\\,", "d7ea42ed1d8b25cc5f5c4baa67b405a2": "d_k=\\mathrm{max}\\{r|e_r = e_k^{\\,}\\}", "d7ea4a2a3287fa28745f5f6589101a73": "\\pi \\int_a^b (\\left[R_O(x)\\right]^2 - \\left[R_I(x)\\right]^2) \\mathrm{d}x", "d7ea6b6089ecb30dd1b4c57f2ee1e0f2": "d_{OS}\\!\\,", "d7eaa1f94d91af037e48af56ef9b3d61": "\\gamma_\\mathrm{ls} - \\gamma_\\mathrm{sa}\\ =\\ -\\gamma_\\mathrm{la} \\cos \\theta", "d7eb2350c61bc62a455bd869397bdd88": "SO(3,1)", "d7eb3ee6a5f7bef4463b6110d6b72885": "\n\\dot{f} \\approx \\{f, H_T\\}_{PB}.\n", "d7eb49cd540917b74d904cbcd1342218": "561 = 3 \\cdot 11 \\cdot 17", "d7ec5a7dd69b808897af30dea111e87e": "f(\\boldsymbol{x}) = \\frac{\\sum_{i=1}^{n} x_{i}^{4} - 16x_{i}^{2} + 5x_{i}}{2}.\\quad", "d7ec6ed67adf9d90b2399ee31dac11fa": "\\cup_{n \\in N} \\scriptstyle S_n", "d7eca9c3834831765cc278bb764642f2": "\\overline{M}", "d7ecdb5dac42e40df324a2d369acf26d": " \\begin{align}\\hat{H} &= \\sum_{n=1}^{N}\\frac{\\hat{p}_n^2}{2m_n} + V(x_1,x_2,\\cdots x_N) \\\\ \n& = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\frac{\\partial^2}{\\partial x_n^2} + V(x_1,x_2,\\cdots x_N) \n\\end{align}", "d7ed0167c3b6da4d0ba985c3c5e7e5eb": "\\mathbb{E}[c] = c", "d7ee0d6619022b4ef0534a7b6ee89ce6": "H=\\sum_{k_+} E_k a^\\dagger _k a_k + \\sum_{k_-} |E(k)|b^\\dagger_k b_k + E_0,\\,", "d7ee93fe0218dc9cc98019df9e36294b": "{{If} f < f_{st}, f=\\frac{-m_{ox}+m_{ox, 0}}{{sm}_{fu, 1}+m_{ox, 0}}}", "d7eea7f56a034e076229e30dba3e453a": "v_{i,j} = A_{ij}", "d7ef12e239369e8792ce4e8d6b194efc": "V_{tn}", "d7ef13c1c61e78c8c084c3ca4f4aa7f5": "\\hbar m", "d7ef3fb0f79e0d2836a19ff8bca0ccfc": "\\bigcap_{n=1}^{\\infty}L_n=\\{0\\}", "d7ef5be4ed1508410f9618d59e1f600f": " \\begin{cases} \\frac{dx}{dt} = -y \\\\ \\frac{dy}{dt} = x + ay \\end{cases} ", "d7ef5e700c7c11b15009a9deba1404c7": " r = \\infty ", "d7efb38fd887fc723f2e2e47bc484bca": "\\deg(P - Q) \\leq \\max(\\deg(P),\\deg(Q))", "d7f007ae874a7439330760902eb0f9f1": "\nF = A \\sigma\n", "d7f04a0947a41c0b052c7ded68422737": "\\Delta \\mathbf F/\\Delta S\\,\\!", "d7f077a12ab0312453df4b15f4f18923": "{}_1F_1(a,b,x) = \\frac{\\Gamma(b)}{\\Gamma(a) \\Gamma(b-a)}\\,e^x\\int_0^1 e^{-xs} (1-s)^{a-1} s^{b-a-1}\\,ds,", "d7f0ec006f48459b5c6351077600edb7": "A(d) = A_0e^{-\\alpha d}", "d7f116997176d81a3bbd4e6dfc6fe6b0": "g_2", "d7f1cf24e00e06935739c8bfc4ca56d3": " \\operatorname{F}_{u, t} (\\operatorname{F}_{t, s} (x)) = \\operatorname{F}_{u, s}(x). \\quad u \\geq t \\geq s. ", "d7f1f44c2d6d193eea28b2dc38d32662": "\\omega = {2 \\pi \\over T}", "d7f21a5465f6860a9b54b551ba824d74": "(\\textrm{g}~\\textrm{pc}^{-2})", "d7f21dcceae2f65257a9555af9221549": "\\mathbf{\\bar{a}}", "d7f23831c9a14f1c05cd19ce765494ce": "\\omega_{pe}^{-1}", "d7f23a02574c13a4129ca96e8b3b8cb5": "\\{x \\in \\mathrm{U} | \\Phi(x)\\}", "d7f25f920077c183a1f0adf6806d091c": "\\det(A)=\\epsilon^{i_1\\cdots i_n}{a}_{1i_1}\\cdots {a}_{ni_n},", "d7f295cb8499408dbee77e8cd68ae9af": "H_n", "d7f2e836f98e9324e73448e9f090ac02": "P(R_t) = \\int_0^\\infty P(R_t|\\lambda) p(\\lambda) d\\lambda = \\frac{1}{1 + k t}\\,", "d7f30ffc8874638bb227974d82df0beb": "\\widehat{\\mathbf{p}} = -i \\hbar \\nabla", "d7f37628d10cf084f425e37cd3ab6966": "k_1, k_2, k_3", "d7f3e12bfb815ee44f79e8edc6fac52f": "p_k = P(N = k)", "d7f3e683f957e873ac9598f2d02c0fc7": "5 + 5 = 10,", "d7f3e7753484678a6fc0b882a284c3b3": "V=Z(\\zeta)\\,\\Xi(\\xi)\\,\\Phi(\\phi)\\,", "d7f3f7c2bd76cc98e4f02f5c578aaf16": "\\Iota \\, \\iota \\,", "d7f42354ffc09699a7cb0a205ab236b4": "\\omega^{A}_{x\\backslash y}=\\omega^{A}_{x}-\\omega^{A}_{y}\\,\\!", "d7f44cf7ddb88c4c22a8e7678bdf1f50": " d\\mathbf{A} = (dA_0, dA_1, dA_2, dA_3) ", "d7f4a752955bc5b92f2dee65673a3b39": "D = [P] + [Q] + [R] + [S] + [T] + [U] - 6 [O]", "d7f4e2ac4becffc2c6662943ab1b5581": " \\langle f\\rangle_i = \\frac{\\sum _{x\\in X}f(x)\\mu_i(x)}{\\sum_{x\\in X}\\mu_i(x)} ", "d7f512ce36e42476042a0dc3920fd40f": "[J,X]=X \\quad\\quad [J,Y] = -Y \\quad\\quad [X,Y] = 2J", "d7f5c32963c0fe059a999003f1fcbd61": "\nc \\,F_3(a_1,a_2,b_1,b_2,c; x,y) - c \\,F_3(a_1,a_2+1,b_1,b_2,c; x,y) + b_2 y \\,F_3(a_1,a_2+1,b_1,b_2+1,c+1; x,y) = 0 ~,\n", "d7f5ce6a9dc138bdf821637a8138d50e": "1 \\ \\overset{\\underset{\\mathrm{def}}{}}{=} \\ \\hat{\\alpha}\\{(\\exists x) . \\alpha = \\iota \\jmath x\\}", "d7f607b5b289c282dbe6ac0560b91c73": " \\mathrm{Ar} = \\frac{g L^3 \\rho_\\ell (\\rho - \\rho_\\ell)}{\\mu^2}", "d7f6aacc1bc05929f22175031adb399a": " W = f_H - f_L = 108 \\ \\mathrm{MHz} - 88 \\ \\mathrm{MHz} = 20 \\ \\mathrm{MHz}", "d7f6c59f4b2e63938e5e265f0c22edd3": "G_U=G_T \\oplus G_\\infty\\;", "d7f7076d35a009e55961f3c50c6cde22": "\\mathbb{F}_{q^n} \\to \\mathbb{F}_{q^m} ", "d7f7bfd06abba02472fff687c11fb3c4": " A / B := \\{ a / b : a \\in A \\and b \\in ( \\textbf{Q} \\setminus B ) \\}", "d7f7db8bcf9115f08f79810f5533cce5": "l_n = kl_n \\cdot \\frac{b_n}{b_a} \\cdot \\frac{s_n}{s_a}", "d7f8bb33070ec7587b04173a3403beba": "S_Y", "d7f8cf384dc557efd5bfe527adcd643b": "{{r}_{O4}}", "d7f915a1fa33b7b8e42bdd0bcf9429fe": "f(x) = x^2 \\,", "d7f997bb864fc0abf19218c447a571bc": "\\int_{\\partial V} \\mathbf{E}\\cdot \\mathbf{n}\\, dS = \\int_{\\partial V} \\nabla\\varphi\\cdot \\mathbf{n}\\, dS = \\int_V q\\,dV,", "d7f9b39c37a420b48977ec9326fc459c": " \\beta_{FGLS1}", "d7fa454b12e63552f4fd5003260e024d": "[A_N(z),B_N(z)]", "d7fa78f5d4e22ad61321a5a066750c94": "\\text{sample mean(Y)}=\\bar{y} = \\frac{1}{N}\\sum_{i=1}^N Y_i", "d7fb224c7cf4963d4813f69e59c05b79": "= \\pm\\sqrt{1 - \\sin^2\\theta}", "d7fb62c1fb5ce20b00dbfb2a5c3ae1c3": "f=K\\left[{L+S_f+h_0\\over L}\\right]", "d7fb95a565d5b96f24fd2a30a7231729": "DG", "d7fbbaab0334f803958918753a79264f": " F_x = \\frac{ee'}{r^2} \\left(1+\\frac{ra'_r}{c^2}\\right) \\left[A cos(rx) \\left(1-\\frac{3ra'_r}{2c^2}\\right) + A\\left(\\frac{ra'_x}{2c^2}\\right)-B\\left(\\frac{u_x u_r}{c^2}\\right)-C\\left(\\frac{ra'_x}{c^2}\\right)\\right]", "d7fbd814d6c7b5cf708157b9b3d6405d": "e_i f_i", "d7fbd82f33d762f294e870ae68d0ec47": "v_i(X_i) \\ge v_i(X_j)", "d7fbe6898ea7370efdd4304a1190e3cb": "V_n^{(2)} = \\left \\{ij \\ : \\ 1 \\leq j \\leq n, i \\neq j \\right \\} \\subset V^{(2)}, \\qquad i=1, \\cdots, n.", "d7fc116f98a724b25df9a84996dbe343": "(\\mu, \\nu)", "d7fc13686525e0391d66db9d6514edbf": "\\textstyle\\frac{Y}{1-X}", "d7fc233b420b79dc559f0d7b80290a0a": "\\begin{matrix} \\frac{1}{3} \\end{matrix}", "d7fc35867a2f613cfb3338fe6ec775a9": "x_n = x_{n-1} - f(x_{n-1})\\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}", "d7fc4727e11cf48de7c61a5f9e74b052": "K^2 \\times S^1", "d7fc773d2eb6dc969614ecbddd01ff55": "\\mathbf{K}=-Q_{\\mathbf{u}\\mathbf{u}}^{-1}Q_{\\mathbf{u}\\mathbf{x}}", "d7fce2866f6ac014304f7905c7cd3623": "\\sigma=\\sqrt{\\left(\\frac{s_i^2 +\\frac{a^2}{12}}{N}\\right)\\cdot\\left(\\frac{16}{9}+\\frac{8\\pi s_i^2 b^2}{a^2 N}\\right)}", "d7fcf63db9043c76698909c260e58883": "0.2 \\times std", "d7fd2d3699ee2cf0114f27d8c5c5593b": "\\Delta_x \\gamma(z,x)=0", "d7fd399a048849f5afb49ee9c6db0512": " = \\sum^\\infty_{k = 0} ({2\\pi}{\\overline{a_k} r^{2k}}){a_k} = {2\\pi} \\sum^\\infty_{k = 0} {|a_k|^2 r^{2k}}", "d7fd7a85ecc478062ebd20c01162cd9c": "r_j{\\left(0\\right)}=0, j=2,3,\\dots,n", "d7fd7a8989de0a4a807aa249ad71ffe0": "\\int_0^\\infty\\left|\\frac{\\sin(x)}{x}\\right|\\,\\mathrm{d}x=\\infty.", "d7fdb6e24c17cbf15aaa9c2349020d5e": "\\mathbf{x}'(t)=\\mathbf{0}", "d7fe1264440b2c31289e2529397cc232": "N_1+N_2-2k", "d7fe944eb344559964253eb41ff93bfb": "\\mathrm{ord}_{s=n-1} \\zeta_X(s) = rk \\mathcal O_X^\\times(X) - rk \\mathrm{Pic}(X)", "d7fea8ef2362bc2b855913eb53076d58": "E = hc/ \\lambda", "d7ffcabd308129d4c37635549be20543": "\\,v = {(1+i)}^{-1}\\approx 1-i+i^2", "d7ffec385fdd54b7f7251bf29446915d": "r(X)=n-c", "d8000501ac4a3d4f4e8d303c1dad0cf5": "\\mathbf{J}=\\mathbf{j}_1+\\mathbf{j}_2", "d8001f88164e5a6f86243e3d48db2f3d": " 0 \\le k \\le [n/2]", "d8003e5893ec8291ba99eabd573fa5e1": " {\\Delta P} = \\frac{L}{T\\,\\Delta v} {\\Delta T} ", "d8004a12e92cccd0dcda66871ec76582": "\\langle\\widehat{a}_j^{\\dagger m}\\widehat{a}_k^n\\rangle = \\frac{\\partial^{m+n}}{\\partial(iz_j^*)^m\\partial(iz_k^*)^n}\\chi_P(\\mathbf{z},\\mathbf{z}^*)\\Big|_{\\mathbf{z}=\\mathbf{z}^*=0}", "d8004d6c00db70161478d5fd74356865": "r_{it}", "d8006df3e71f86b82c8a14f39815a9d4": " \\frac{D}{L} = \\frac{t}{t-d} \\quad ", "d8008f3f88321f570bf4cd7135937700": "f_n(x_1,\\ldots, x_n) = \\operatorname{ave} f(x_{\\varphi(1)},\\ldots, x_{\\varphi(r)})", "d8009fa9f097e64b39d03d092bfc6524": "x\\in [x]", "d800a7e4a5f83d9bbd6d740b427aa52a": "f_{\\text{r}}(\\lambda_{\\text{i}},\\,\\omega_{\\text{i}},\\,\\lambda_{\\text{r}},\\,\\omega_{\\text{r}})", "d800b98d3f25dcd8cfbc709d524feb59": "R\\left(x,w\\right)", "d800c6a9dac2634118e99ba205111a1f": "I = \\ln (s_0/S)", "d800c74950f5888f4fc12e73039c7cf4": " \\cos \\theta = \\frac{v}{n c} = v\\sqrt{\\epsilon_0\\mu_0} \\,\\!", "d800d7524c351c899693029f616a39c3": " \\langle\\alpha|\\widehat{q}|\\alpha\\rangle = 2^{-1/2}(\\langle\\alpha|\\widehat{a}^{\\dagger}|\\alpha\\rangle + \\langle\\alpha|\\widehat{a}|\\alpha\\rangle) = 2^{-1/2}(\\alpha^{*}\\langle\\alpha|\\alpha\\rangle + \\alpha\\langle\\alpha|\\alpha\\rangle) ", "d800e2c5abc82ba78859a14ebed1155a": "\\, m \\in \\mathbb{Z} \\ge 1\\, ", "d8013d8c6e0686cbeb8831e975b867e0": " B(x_1,\\ldots,x_{2N})=(b_1,\\ldots,b_{2N}) ", "d80157d875c422019fab5a662418bdd9": "\\frac{c}{a} = \\frac{a}{r} \\ .", "d8017ca503cb5c36af9627dd3fb8a37a": "g\\ \\varphi\\bar\\psi\\psi", "d80198e894bc3a3de8209076eef58761": "\\psi_{1,4}=1", "d801e0de9cbdedb367e1be7bf31b4843": "x^3-Tr(g)\\!\\ x^2 + Tr(g)^p x -1", "d8023c434a8e254dec66e596a66cac33": "\\mathbf{P}(A): \\mathbf{h} \\rightarrow \\mathbf{h}; \\quad X \\rightarrow A^\\dagger XA, \\quad X \\in \\mathbf{h}, A \\in SL(2,C).", "d80247aa60763b90d004bbfd82778eec": "l_{ij} = \\mathrm{tf}_{ij}", "d802522f3ec458929d8f55d0bb2fde7e": "0 = \\partial_c F_{ab} + \\partial_b F_{ca} + \\partial_a F_{bc} = D_c F_{ab} + D_b F_{ca} + D_a F_{bc}", "d80285ba89ecf975f25897cbdf816686": " a \\triangleright c = b \\iff \nc = b \\triangleleft a ", "d8029ca5ac2c4ae812691e7f01e5b148": "e^{\\frac{\\boldsymbol{\\Omega}\\theta}{2}} = \\cosh\\left(\\frac{\\theta}{2}\\right) + \\boldsymbol{\\Omega}\\sinh\\left(\\frac{\\theta}{2}\\right). ", "d8029e02bfe230dff6cf5ba0b4657840": "x_1,x_2,\\ldots", "d8039b6684e4633626814e3535673561": "\\Omega_2 (E_2) \\frac{d}{d E_1} \\Omega_1 (E_1) - \\Omega_1 (E_1) \\frac{d}{d E_2} \\Omega_2 (E_2) = 0\n", "d803b37c902b2e59674f4a8b3f748d43": "y = r \\sin \\phi", "d803bf5ce080f60f22059157151108e1": "C_D = \\frac{550 \\eta P}{\\frac{1}{2} \\rho_0 [\\sigma S (1.47V)^3]}", "d803ed250249a8fc67d23ecbf763cfc6": "\\scriptstyle{\\hat{H}_\\text{D}}", "d804fdc637399b01f00da2c7c4a6dd8c": "T^\\circ", "d8050656f8926df745241e44e9b14294": "\\mathbf{z}=(z_1,z_2,\\ldots,z_M)^T", "d8052566f029017e9dbfab0ca306eecf": "~\\Phi_{16}(x) = x^8 + 1", "d8054e930f4ed278eae9a5e7cb1fe3cf": "\n\\sum_A \\vec{q}^{\\,A}_r = 0,\n", "d8055c7999fa2b5843fdc0a0ac43b82d": "t_1,\\ldots, t_n \\in T, \\, n \\in \\mathbb N", "d80570d304575e672038601418191e6c": " T_1(z)=\\frac{24(22-K)}{11} (1+z) ", "d8057c421885d85f82c36b4cea72344a": "ds^2 = \\Omega(z)^2\\left| \\, dz + \\mu(z) \\, d\\bar{z}\\right|^2,", "d8059e89c0adbe907634a35bc0f8520f": "\nj_{\\mathrm{A}} = -D_{\\mathrm{A}} \n\\left( \\frac{d\\left[ \\mathrm{A}\\right]}{dz} - \\frac{n_{\\mathrm{A}}F}{RT} \\frac{E_{m}}{L} \\left[ \\mathrm{A}\\right] \\right)\n", "d805c9a172849a6e5bb4ffda66f33ff1": " u_G(x)", "d805df410ff3fd91ddc3fd62283c066b": " \\mathbf{c}_i ", "d805fd6fa6b8242b36ec03d428e16663": "PA=A=AQ\\,\\!", "d8067b2d7a3efc23e9ebc07970db9e5c": "{\\mathbf{}}\\hat{P}_{i+1}=1/2(\\tau_{i+1}\\Psi_i^1+\\Psi_i^1\\tau'_{i+1}),\\hat{P}_0=E({\\mathbf{x}}(0))E({\\mathbf{x}}(0))', rank(\\hat{P}_i)=n_r", "d806b4f5a81d5a6776dce8a33098ba2a": "w=U/u.", "d806d4691f8218335a4c8cd0d5e42462": "p(r) - P = \\frac {2 \\gamma\\, \\rho\\, _{vapor} } {(\\rho\\,_{liquid} - \\rho\\,_{vapor}) r}", "d806d9d494b2b232ada9b2fcabed72f3": " X = U \\Sigma V^{\\rm T}. \\ ", "d807411766d554419a3d1b761b0fb8dc": "\\textstyle{{\\rm monthly}\\atop{\\rm payment}} = \\frac{r}{1-(1+r)^{-N}}{\\scriptstyle{{Who}\\times\\rm Principal}}", "d8077fc97a9bf7d7dbb71d48e62519d8": "\\psi_1, \\psi_2 \\,\\!", "d8079d3e748cedcefbdbe31a3af0d848": "\n\\tilde{\\rho}_\\text{TOT}(\\mathbf{k}) = \\tilde{L}(\\mathbf{k}) \\tilde{\\rho}_{uc}(\\mathbf{k})\n", "d807dee5787d82975352377dc1a31076": "\n\\text{bias}_I(X) \\leq \\epsilon\n", "d807f3127470d9062512936514bc671b": "J_\\alpha(z)=\\sqrt{\\frac{2}{\\pi z}}\\left(\\cos \\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)+e^{|\\operatorname{Im}(z)|}O(|z|^{-1})\\right)\\text{ for }|\\arg z|<\\pi", "d807fcc33b62062c24eec8df8bc73834": "g(\\boldsymbol{r} \\rightarrow \\boldsymbol{r}')", "d808285e67db2de29ccdc7067d1818ba": " M \\ ", "d8083966917929fc1828c0ce7694da3e": "(vw)", "d808cfb63dd448d9bb14b2291da8c25b": "u_c > u_{rest}", "d8090049de3c65aaa823b8d9a2c41e49": "f(0) \\ge f(0+y) - f(y) = 0", "d809343bf79ace6842e91f0a41239637": "dz = h(x,t) dt + dv", "d809ac5d826bf7673868df663950eb1c": "V_n = 0", "d80a11f8da15b54fb9bc450c7caffbbf": "T=GF", "d80a3599286f7ca5a00e675ad8fc64f5": "J_{\\nu \\beta} \\propto \\sum_{\\mathbf{k}} \\phi^\\star_\\nu({\\mathbf{k}}) {\\mathbf{k}} \\cdot {\\mathbf{k}}_{\\rm THz} \\phi_\\beta({\\mathbf{k}})", "d80a488ee7b2cec8611144c73afba966": "E^{\\mathrm {pot}}_{12}", "d80a5fcc4af228962c5ecbec8841b1db": "\\sum_{n=0}^{\\infty} a_n = a_0 + a_1 + a_2 + \\cdots.", "d80a6cdcf0c5effc2315df70b507c88b": " \\begin{bmatrix} I_1 \\\\ V_2 \\end{bmatrix} = \\begin{bmatrix} g_{11} & g_{12} \\\\ g_{21} & g_{22} \\end{bmatrix} \\begin{bmatrix} V_1 \\\\ I_2 \\end{bmatrix} ", "d80a9f038ccccb0a086f4b9086c6b9b3": " I = \\int_0^\\frac{\\pi}{2} \\frac{1}{1 + \\sin(t)^2}\\, dt,", "d80aa2abff65ad84f2d003cd531f71f2": " \\mbox{ch}(f_{\\mbox{!}}{\\mathcal F}^\\bull) = f_* (\\mbox{ch}({\\mathcal F}^\\bull) \\mbox{td}(T_f) ),", "d80ae5d150fc79fd117e4af6e9b64938": "+ C", "d80b6d8cabfa039cf9ca54d48a9b3d65": "m=\\sum a_j", "d80b72d01b1c9a255d6f27d803f53006": "v_p(\\omega)", "d80b83e752d59a164dc627e35aeead8c": "i=1,\\ldots n", "d80b8cda40b70e8788e7148ef5ee9d88": "D_k[f](x):=\\sum_{n\\in\\Z} d^{(k)}_n\\,\\psi(2^kx-n)", "d80b8e8c481f3c93b4527b370a858c62": " m_i ", "d80c57b28ca44a0ca1d17da4d8c738ac": " \\text{(**)} \\qquad \\|f(x+h)-f(x)\\| \\leq M\\|h\\|.", "d80cd3eef62faeaaefef7c1d6e8992fa": "10\\uparrow\\uparrow\\uparrow\\uparrow n", "d80ce4658f6fac83f8e1e2ea3a54ff5c": "\\chi_0/2K_u", "d80ce934a2fd421de6bee1abccb54295": " P=\\frac{B_s}{k_BT}\\, ", "d80d4986bddb2abd03ed88f97bf6e0cd": "\\mbox{angle correction} = -\\frac{-9.5 \\mbox{ cm}}{282 \\mbox{ meters}}= 0.00094 \\mbox{ radians} = 3.2'\\mbox{ (minutes of angle)} ", "d80d4d355907b7928363aa4d931e6bce": "c^T\\left( x^\\ast - \\frac{\\epsilon}{2} \\frac{c}{||c||}\\right) = c^T x^\\ast - \\frac{\\epsilon}{2} \\frac{c^T c}{||c||} = c^T x^\\ast - \\frac{\\epsilon}{2} ||c|| < c^T x^\\ast.", "d80d56416ffd8e9dcf9ab364c1384c77": " \\mathcal{I} ", "d80d8553efe6642ffcb5b00319f9ac9f": "f,g: \\mathbb N\\to [0,\\infty) ", "d80dabe569899227e322c3ea9076a6a3": "J_\\mu(x_1,\\ldots,x_n;q,t) =\\sum_\\lambda K_{\\lambda\\mu}(q,t)s_\\lambda(x_1,\\ldots,x_n)\\ ", "d80dcd03f8c0aa36a1ef6ac1663f6580": "\\hat e_z", "d80dda1e0ac5b72559d1360d067af81a": "E_n(x)=E^{x}_{n-1}(2)", "d80e63cfad886007d19a1cbb72c6e580": "\\ \\delta_{p}", "d80e6ef061c84feb0b9ba9bc599b0b8e": " f = \\; F/F_{\\phi} = (e^3 / 4 \\pi \\epsilon_0) (F/ {\\phi}^2) = (1.439964 \\; {\\mathrm{eV}}^2 \\; {\\mathrm{V}}^{-1} \\; \\mathrm{nm}) (F/ {\\phi}^2). ..........(30d) ", "d80e7550683c184ab5c4a56eeb6c4e33": "E_{ij} = x_i \\frac{\\partial}{\\partial x_j}; ~~~~~\nE_{ij} = \\sum_{a=1}^n x_{ia}\\frac{\\partial}{\\partial x_{ja}}; ~~~~ E_{ij} = \\psi_{i}\\frac{\\partial}{\\partial \\psi_{j}}. ", "d80e83641f4c42d3cea81800dad046d5": "\\mathcal{I} \\models \\mathcal{A}", "d80ea1829234008a23bb5859a44d8e26": "\\left[\\mathcal{S}_n(f)\\right](x)", "d80f59901f9132668fdfc1ab3525ed24": "\\frac{\\partial F}{\\partial n_3}= \\left(\\sigma_3^2-2\\sigma_3\\sigma_\\mathrm{n}+\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2\\right) n_3= 0\\,\\!", "d80f626fa7bf6e753535d901d4959e23": " \\bar{n}_i \\ ", "d80f9a003a1280d5a1179568b1298fa1": " f^{\\prime \\prime} ", "d80fcd746c089abe82085af6f61fdb86": "\\scriptstyle W^{1,p}", "d80ffd07c88ce24c286d050532f170a1": "F\\subset \\mathbb{R}^2", "d810b30e6f7ffd8af0310b4ba243bcd9": "I \\left [ 1-\\left (\\frac{td}{i+d}\\right )\\left (\\frac{1+\\frac{1}{2}i}{1+i}\\right ) \\right ]", "d811444a35cc3e4497f01a8622090f2d": "\\ \\sgn(x) \\approx \\frac{x}{\\sqrt{x^2 + \\epsilon^2}} \\,.", "d811706099a86ff2e2a5c54b8f649530": "CTR=100%*Clicks/Impressions", "d81179c83bebeca32de067532a5a4c3f": "|\\vec r_s|", "d8119f60969c986ed5b628221571778e": "\\scriptstyle \\leq10^{-20}", "d81215b9dccd1cdd239f4c0728554ead": "X\\prec Y", "d81231cb0c2c2ebbee31b2f22c939d95": " f(x) = \\lim \\limits_{\\epsilon \\rightarrow 0^+} \\int e^{i \\phi(x,\\xi)}\\, a(x,\\xi) e^{-\\epsilon |\\xi|^2/2} \\, \\mathrm{d} \\xi ", "d8126019a0ca6c83c9d21463293b5679": "\n\\begin{alignat}{2}\n\\|x\\|_2^2 &= \\|z\\|_2^2 + 2z^*(x-z) + \\|x-z\\|_2^2 \\\\\n&= \\|z\\|_2^2 + \\|x-z\\|_2^2 \\\\\n&\\ge \\|z\\|_2^2\n\\end{alignat}\n", "d812703bf959c11209e19d85956bb4e3": "F(x_1,\\dots,x_n,u,p_1,\\dots,p_n)=0", "d812b1fb98caee016e1643f27134ceb2": "\n(x_1+x_2)\\cdot x_3 \\in [1,2].\n", "d812e10871871f0b107d5b256c315ca3": " 0 \\to VP \\to TP \\xrightarrow{d\\pi} \\pi^* TM\\to 0", "d8133da3b4a33415dbe8b718392a1bff": "\\mu(E)=0", "d81359389d6d9e4f304d938799e19f17": "-\\frac{Nc}{4}(\\delta_1 + \\delta_2)", "d81368672da33e573c94973fdd856dd8": "W_{ij} = M S_{ij}", "d8139d393705742c3fde2c15c0c773e5": "\\displaystyle{(f_1,f_2)= {1\\over \\pi^n} \\int_{{\\mathbf C}^n} f_1(z)\\overline{f_2(z)} e^{-|z|^2} \\, dx\\cdot dy.}", "d813c24b29c7075373f8967e68b12109": "\\dot{Q}_f", "d813c6a6119ab7736fc5768d1f0c4ca1": "(x \\,\\bmod\\, y) \\equiv x \\pmod{y}.", "d8141df7e5212223eff8f4928e8fd74e": "Y_n=(q/p)^{X_n}.", "d81460e0af353c3659a567890891618b": "\n\\boldsymbol{\\sigma}=\\boldsymbol{P}\\cdot\\boldsymbol{F}^T=\n -p~\\boldsymbol{\\mathit{1}} + \\frac{\\partial W}{\\partial \\boldsymbol{F}}\\cdot\\boldsymbol{F}^T\n = -p~\\boldsymbol{\\mathit{1}} + \\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{E}}\\cdot\\boldsymbol{F}^T\n = -p~\\boldsymbol{\\mathit{1}} + 2~\\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{C}}\\cdot\\boldsymbol{F}^T ~.\n", "d814707a258819f66ec0c1b9a9468f02": " f'(0) = \\alpha.", "d814b3706a6743cc91b219e57687e50f": "= \\min_{\\{F_i\\}} \\sum _i \\sqrt{\\mathrm{Tr}[ \\rho F_i ], \\mathrm{Tr}[ \\sigma F_i ]}", "d814cbff19f33029646ab27ee633ffc5": " N(k_2 + \\cdots + k_r, k_2, \\ldots, k_r) = \\frac{(k_2 + \\cdots + k_r + 1)!}{k_2! \\cdots k_r!}. ", "d814f82658d0227c3d293455d4287573": "a_1' = a_0 \\oplus a_1 \\oplus a_5 \\oplus a_6 \\oplus a_7 \\oplus 1 = 0 \\oplus 1 \\oplus 0 \\oplus 1 \\oplus 1 \\oplus 1 = 0", "d815380aa24551b11c254fe22dcadddb": "N=(S,T,W,M_0)\\!", "d81559c3cff8664cbc75af8bd1629777": "\\Psi_c:\\mathbb{\\hat{C}}\\setminus \\overline{\\mathbb{D}}\\to\\mathbb{\\hat{C}}\\setminus Kc", "d81562ed409ea7412b3fafeb10e005ce": "\\lambda\\frac{k-1}{k}^{\\frac{1}{k}}\\,", "d8156bb13476bb314be45c5a6ee612f6": " \\alpha, \\beta, \\gamma ", "d81584f09e174eec259979ae8f92eb09": "t_0", "d815da332cbb51c41f10f350e5b01355": "\\rho_{In} = 0.07517*(1-{0.0035666*E \\over 528})^{5.2553}*({528 \\over T_{In}+460})\\,\\!", "d815e4007fe16013c9e4a51587cefc98": "\\mathcal{C} \\rightarrow \\mathcal{C} \\times \\mathcal{C}", "d8168894d0a027c5fef8ecce7043fec3": "( -200 + 200.00000015 ) / 2 = 0.000000075.", "d816c56d5cd9b745738f8649d67a000f": "\\displaystyle \\alpha(x^2 + y^2) - 2\\beta x - 2\\gamma y + \\delta = 0,", "d816d93e0c96f9b45c176559755b350f": "\n \\sqrt{n}(\\hat\\beta - \\beta)\\ \\xrightarrow{d}\\ \\mathcal{N}\\!\\left(0,\\,(X'\\,\\Omega^{-1}X)^{-1}\\right).\n ", "d8171d75df4c1f1b7d18b784943f47c6": "2<\\alpha<3", "d81740af34c2bb5f112018d174454aff": "\\frac{V}{\\sqrt{2E}} = \\left [ \\frac{ 1 + \\left ( 1 + 2 \\frac{M}{C} \\right )^{3}}{6 \\left ( 1 + \\frac{M}{C} \\right ) } + \\frac{M}{C} \\right ] ^{-1/2}", "d8183b6573725ec7c97b19a3727fc710": "S \\Rightarrow_{r_1} X X \\Rightarrow_{r_2} Y X \\Rightarrow_{r_2} Y Y \\Rightarrow_{r_3} S Y \\Rightarrow_{r_3} S S", "d8183e948e6d05a3b3fc6e36a4b97d1a": "(8)", "d8184954af032bd0e666efeb469221ec": "\\{O_4,O_5,O_6,O_8,O_9\\}", "d818d6d6097a92aaac8a89cac69e5cec": "\n\\left(\\begin{array}{c}\nX_{2}\\\\\nY_{2}\\\\\nZ_{2}\n\\end{array}\\right)=\\left(\\begin{array}{ccc}\n\\cos\\Omega t & \\sin\\Omega t & 0\\\\\n-\\sin\\Omega t & \\cos\\Omega t & 0\\\\\n0 & 0 & 1\n\\end{array}\\right)\\left(\\begin{array}{c}\nX_{1}\\\\\nY_{1}\\\\\nZ_{1}\n\\end{array}\\right),\n", "d818ddfc0dcbb8fe8be412fce11416d9": "\\displaystyle\\mathbf F_\\parallel=\\sum^n_{i=1}\\frac{\\partial\\mathbf r}{\\partial q^i}\\,F^i", "d818e981edc007841ab688ab0a980129": "\\int \\cos ax\\, e^{bx}\\, dx = \\frac{e^{bx}}{a^2+b^2}\\left( a\\sin ax + b\\cos ax \\right) + C", "d818ec88ccacccb754d0504cc40b2536": "H=1/2", "d8191a01af4614110273756bd367bff8": "M_1 = M \\oplus P = ( M \\setminus P ) \\cup ( P \\setminus M )", "d81934eae5b1bd9865bbb0445f435124": "\\scriptstyle\\overline{A}_k", "d819acb9a49e1a8d09645e35461eeb82": "|1| = 1", "d819ae3f7e997cfd52929048e56fdb50": "\n\\langle N_i \\rangle = \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}\\mp 1}.\n", "d819cc14e002b5ed25452e33f85818ec": "|\\tau(p)| \\leq 2p^{11/2},", "d819fe978d7e447457325682479e243c": "f : S(\\alpha)\\rightarrow T(\\beta)", "d81a24e43a817a2ed067e895ba8eb2c8": "e = x - \\hat{x} \\, ", "d81a5fa62ed41645e0da1a682720e609": "j = \\ell_1 + \\ell_2b (\\mod n)", "d81a94e310d09f7008bf3c56228d0b20": "(q/2)^2-(p/3)^3", "d81aa56e7a6996a7ad69f10296849630": "\\Gamma \\, \\gamma \\,", "d81aed439c1d87d45874a7fb8e86adc9": "\\ b=a\\sqrt{1-\\varepsilon^2},", "d81b1c85531f170313de0a72f10870ee": " \\mathbf{x} ", "d81b2f60fb68639fedeb555ce0295f58": "{d\\over dt}{d\\theta\\over dt} = {d\\over dt}\\sqrt{{2g\\over \\ell}\\left(\\cos\\theta-\\cos\\theta_0\\right)}", "d81b7b7384bc9dc7691f1c63d02d1cff": "N_f ", "d81c52b9a3a020321d403606e045862a": "\\scriptstyle{2\\mapsto13}", "d81d0a87873ea5bfbd9994f043bb6a51": "T_{m+n}(x)", "d81d223db886d8fde194c21ca3d8e2f1": "\\vartheta_{s}", "d81d287cf1a4d163123df8b9fc0650ee": "{{1 \\over f_o}{\\nabla^2 \\chi} = {-\\overrightarrow{V_g} \\cdot \\nabla ({{1 \\over f_o}{\\nabla^2 \\chi} + f})} + {f_o {\\partial \\omega \\over \\partial p}}}", "d81d380a1a64cd24de0c4043c86c7db4": "C^{m-1}\\;", "d81d54fcc5d06e824635218920353694": "P(M)_{t}", "d81da883ced1a04ddae509a7607922ae": "\\scriptstyle b^2", "d81da92bf81ab5ee788213dec272291d": " \\mu^{(k)}= d_\\lambda (f,\\xi^{(k)}_\\lambda),", "d81e0080f73ca757c4ca043df0d83581": " r \\rightarrow \\infty ", "d81e240163430e999756fb9b24a610ef": "\\tilde A_{2n-1} \\to \\tilde C_n", "d81e3284b25fcc32175f24112a57e091": "\\mathbf{E} \\left[ \\sup_{t \\in T} X_{t} \\right] \\leq 24 \\int_0^{+\\infty} \\sqrt{\\log N(T, d_{X}; \\varepsilon)} \\, \\mathrm{d} \\varepsilon.", "d81e458a0734451ff5683b716bb3872e": "w\\rightarrow w^3+w_0", "d81e5a9aabc40fdf29eb5dd508dee366": "\\int\\ln x\\;dx = x\\;(\\ln x - 1) +C_{0} ", "d81e89a467fa0da0f276fc1935d37c43": " \\rightarrow z ", "d81e90a1ffaffa11bcab1228f89c7a13": " \\langle S \\rangle = \\text{``limit at infinity''} \\langle S e_n , e_n \\rangle ", "d81ea3ee376bcd466ab3c3b178d40945": "P=(a_1 \\cdot a_{n+1})^{\\frac{n+1}{2}}", "d81eb2cbaf1a34a526358d0549c98c20": "v_A \\approx (2.18\\times10^{11}\\,\\mbox{cm/s})\\,(m_i/m_p)^{-1/2}\\,(n_i/{\\rm cm}^{-3})^{-1/2}\\,(B/{\\rm gauss})", "d81eb5e0c40dd47760efaf6a0cc8bbc9": "\\,\nS = C_v N \\log T + N \\log V - N \\log N = N \\log (T^{C_v} V/N)\n", "d81f4d947c4c02cd2772e375b4c58956": " \\phi ", "d81f874345fe242132e4eb62dea01336": "\n\\begin{array}{lcll}\ndx/dt &= &V_{X} &\\cdots(Eq.1)\\\\\nW/g \\times \\left(dV_{X}/dt \\right) &= &- D \\cos \\varphi - L \\sin \\varphi &\\cdots(Eq.2)\\\\\ndz/dt &= &V_{Z} &\\ldots (Eq.3)\\\\\nW/g \\times \\left(dV_{Z}/dt\\right) &= &D \\sin \\varphi - L \\cos \\varphi + W &\\cdots(Eq.4)\\\\\nd\\theta/dt &= &\\omega &\\cdots(Eq.5)\\\\\nI \\times \\left(d\\omega/dt\\right) &= &57.3M - bV\\omega &\\cdots(Eq.6)\n\\end{array}\n", "d81fba5c6e1584d50b11807736ebd601": "\n\\mathbf{R}(\\alpha,\\beta,\\gamma)=\n\\begin{pmatrix}\n\\cos\\alpha & -\\sin\\alpha & 0 \\\\\n\\sin\\alpha & \\cos\\alpha & 0 \\\\\n 0 & 0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\cos\\beta & 0 & \\sin\\beta \\\\\n 0 & 1 & 0 \\\\ \n-\\sin\\beta & 0 & \\cos\\beta \\\\\n \\end{pmatrix}\n\\begin{pmatrix}\n\\cos\\gamma & -\\sin\\gamma & 0 \\\\\n\\sin\\gamma & \\cos\\gamma & 0 \\\\\n 0 & 0 & 1\n\\end{pmatrix}\n", "d81fc7b6312f60d5b61c07577b74a46f": "\nI=I_0 \\exp \\left (\\sum_i \\beta_i \\sigma_i \\right ) = \nI_0 \\prod_i e^{\\beta_i \\sigma_i}\n", "d81fd64c3e61126fe11ff0ee2209049d": "Tf(\\gamma )=\\left\\langle f,{{\\phi }_{\\gamma }} \\right\\rangle ", "d8208c5d16fded1ebf56157f9133ff36": " (\\gamma f)(v) = \\sum_{(u,v) \\in E(G)} f(u) ", "d820987c91110400ef41782f40463402": "Q = f\\,(L, K),", "d82118b70261f370b5299b9fd09bd320": "D = 2-H = 2 - p/2 ", "d821285c605782984eac62be3c8d5c3a": "L[y]\\in\\mathcal{G}", "d8213f6b0d740c1fb4f2c4d5a3f17af2": "URR = \\frac{U_{pre}-U_{post}}{U_{pre}} \\times 100\\% ", "d8215d0e21d6ca649544ce29bf6a560b": "(2) \\qquad \\frac{\\partial\\Phi}{\\partial z}\\, =\\, 0 \\quad \\text{ at } z\\, =\\, -h.", "d8221fad04cc81c605ec5507264f01a2": "\\omega=\\frac{d\\theta}{dt} = \\frac{v}{r},", "d822383a515b36e4d77797d33d9de017": "r = \\oplus r_i", "d8226af73e3714430ad2e6070044eead": "(3 - \\sqrt{5})/2 ", "d8227b13c522b428cbd6ba251f0ef44c": "\\frac{\\text{d}[{^1_2}S^\\gamma]}{\\text{d}t} \\simeq - \\frac{ \\text{k}_{3(2)} E_0 {^1_2}S^\\gamma }{ ^1_2S^\\gamma + K_2 \\left( 1+ \\dfrac{ {^0_2}S }{K_1} + \\dfrac{ {^1_2}S^\\beta }{ K_2} \\right) }", "d82309a6cff4c0f1bece51061ba9d2ab": "{\\mathbf n}", "d82319a20b0b52b9d23299a63fd9275c": "b_i \\in B", "d823a92287e9195795e693f6a8b1df80": "0 \\le T < RN", "d823eaabcfe5c67fd0e126906cc1ca3a": "\n\\underline{\\underline{\\mathsf{C}}} = \n \\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\\\\nC_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\\\\nC_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & C_{44} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & C_{55} & 0\\\\\n0 & 0 & 0 & 0 & 0 & C_{66} \\end{bmatrix} \n ", "d8241f272e0bcc9a37839d35f594d36a": "I = \\int _0^{\\frac{\\pi}{2}}\\frac{1}{\\sqrt{a^2 \\cos^2(\\theta) + b^2 \\sin^2(\\theta)}} \\, d \\theta = \\int _0^\\infty \\frac{1}{\\sqrt{(x^2 + a^2) (x^2 + b^2)}} \\, dx", "d82427fd57f92e07ed5e1760e7232491": "\\rho = \\sum_j p_j |\\psi_j \\rang \\lang \\psi_j| .", "d824440b7c7edbaca3adac3d63a00fab": "A_{C} = 2 \\pi r^2 + 2 \\pi r h = 2 \\pi r ( r + h ).\\,", "d8245680329f4a91fee6f297e6d1fdd6": "ax^2 + b x +c = 0 \\ .", "d8245a7cbe4175e76554da0b3126e293": "=\\frac{(xyz - 1)^2}{(xz + x + 1)(yx + y + 1)(zy + z + 1)}.", "d8245d415e6a2facc9b7c1407f0e5bf4": "\n \\sigma_{11} = 2C_1\\left(\\lambda^2 - \\cfrac{1}{\\lambda^4}\\right)\n ", "d8246b088e96d7847204694ace75b0a4": "I \\subseteq J \\Rightarrow q(I) \\subseteq q(J)", "d8251a7b65b9887ad39f9b8445df86b4": "\\displaystyle V_o I_o = \\eta V_i I_i ", "d8253bb5ab06210afccb424a1a42f68d": "^{13}\\text{C}", "d82596165efc29c3c5898afb9541b841": "x_T^k", "d825a4a92c54a70ccb780ebfdf3a524c": " G(\\mathbf{x}) = \\begin{bmatrix}\n3x_1-\\cos(x_2x_3)-\\tfrac{3}{2} \\\\\n4x_1^2-625x_2^2+2x_2-1 \\\\\n\\exp(-x_1x_2)+20x_3+\\tfrac{10\\pi-3}{3} \\\\\n\\end{bmatrix} ", "d825d6e109f6ba17ff973e0af7746b7c": "j0", "d82b07377f7366565b93dd2cd578b382": "W_\\varepsilon = \\left\\{ f : \\lambda \\left(\\left\\{ x : |f(x)| > \\varepsilon \\ \\text{and} \\ |x| < \\frac{1}{\\varepsilon}\\right\\} \\right) < \\varepsilon \\right\\}", "d82b39243bf438935c749f7120baaf72": "f(x_1, ..., x_m) \\to \\alpha\\ y \\text{:} g(z_1, ... z_n)\\ \\beta", "d82b7de68d66041facca96df4a95cde1": "R_d", "d82b93e277927c5387af7693be8ade66": " \\approx 1 ", "d82bb00181a93a814757f8b76fd471e7": "R = \\frac{x}{k \\cdot A}", "d82be0f5ebf45192b2c96c9c885aa86f": "\\mathbf{AB} = \\mathbf{I} \\ ", "d82c1408eadac81a28d7092a3e37a075": "[1 + \\zeta n(\\xi_\\mathbf{k})]\\mathcal{Z}", "d82c48c778d08a9e1b9ba310a7a40d73": "x \\vee y = y \\vee x", "d82c64c12d83ef709487f0d791b736fe": "p_n(x)=\\sum_{k=1}^n B_{n,k}(a_1,\\dots,a_{n-k+1}) x^k.", "d82ca939f673f53333023959bedff7c1": "\\mathcal{P}_n(z) = (1+z)^n - 1 -z^n.\\,", "d82cde2d17ca54033b7912f38de96260": "{\\mathcal S}'({\\mathbb R}^n)", "d82d10e4025a996bc7820e6ca2fcc6ac": " c_p - c_v\\, ", "d82d17f356328e9bb5372cc4e540c584": "F(T,V)\\,", "d82d2cc02a7c1e09c60246bc3432ffbb": "2^{n} - 1", "d82d401d2c8b243a2f52c946bb69accd": "\\mathrm{adiabatic},\\,O\\to A", "d82db4ec9ffc74e0d2c500452bfcf6b4": " a > b ", "d82dd183538abcb84a0c184092a98722": " f(x) = \\sum_{i=0}^{k+1} { a_i x^i } ", "d82dda9920a7fd5af5a4c30f5441dfa0": "X_i \\equiv \\left( \\sum_{j=1}^k (-1)^{j-1}Y_{i,j} \\right)\\pmod{(m_1 - 1)}", "d82dde2945be7cebd1ec2c0cde3e688d": "h_4\\;", "d82de8e43c9c0060dc36068c4b7d6de3": "_{s.8\\ s.11\\,}\\!", "d82e8bdffc8420c23b6b1730314b0ea4": "\\mathbf{v}(x,t)\\in\\left[C^\\infty(\\mathbb{R}^3\\times[0,\\infty))\\right]^3\\,,\\qquad p(x,t)\\in C^\\infty(\\mathbb{R}^3\\times[0,\\infty))", "d82ebd9fd1b4e2a0f386e8248c4ea774": "\\forall x \\exists y.\\ [D(x) \\wedge \\neg D(y)]\\, ", "d82ec23c75f8825a560e3e7c3da79264": "\\qquad+j \\cosh\\left(\\frac{1}{n}\\mathrm{arsinh}\\left(\\frac{1}{\\varepsilon}\\right)\\right)\\cos(\\theta_m)\n", "d82edb85e7a4bfa6d7f5b24dbc9e59c6": " \\left[ X^m \\right] f(X) ", "d82f21130dd03b98ba3e22665ba092f5": "(d-1-k)/2,", "d82f2d6895c2d486efe03906a38807e6": " \\log( s^2 ) = \\log( a ) + b \\log( p ) + c \\log( 1 - p ) .", "d82f37764bb1c0dd3fe80756453ac860": "h\\approx 10^{-18}", "d82fb9de061f6ae17c25adf43091f73e": "\\begin{align}\n \\cosh (x + y) &= \\sinh x \\sinh y + \\cosh x \\cosh y \\\\\n \\sinh (x + y) &= \\cosh x \\sinh y + \\sinh x \\cosh y\n\\end{align}", "d82fdb338ae1feb9477997517176e7cd": "g^{efghab}", "d82fde2c3f9342d9548a74f6986c352b": " {d \\over dt} \\left[ \\left( 1 + {1\\over 2} { v_1^2\\over c^2 } \\right)m_1\\mathbf v_1 +{q_1\\over c}\\mathbf A\\left( \\mathbf r_1 \\right) \\right]=\n-\\nabla {q_1 q_2 \\over r}\n+\\nabla \\left[ {q_1q_2 \\over r }{1\\over 2c^2}\n\\mathbf v_1\\cdot \n\\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right]\n\\cdot\\mathbf v_2 \\right] ", "d83006b156bb5bd5ca682abfc54c921b": "[\\ ,\\ ) \\!\\,", "d8306dfa22297f1602eb5e73a698e957": "D = \\frac{A}{N} \\,", "d8307d9b8f99f829d3e6768d31fb555e": "i_\\mathrm{c}", "d830ad5ee3ad4e3651d3c7b9f1719af8": "\\begin{align}\n \\sum_{n=1}^\\infty \\frac1{ p_n}\n &\\ge \\sum_{n=6}^\\infty \\frac1{ p_n} \\\\\n &\\ge \\sum_{n=6}^\\infty \\frac1{ n \\log n + n \\log \\log n} \\\\\n &\\ge \\sum_{n=6}^\\infty \\frac1{2n \\log n} \\\\\n &= \\infty\n\\end{align}", "d83153a85764b3b080bd600559592dbb": "\\mu = \\{p_1,\\ldots,p_N\\}^\\mathbb{Z}", "d8318f4f9cdc305b9f3a4d16f47902a3": "xf\\in \\phi(d,c), gx\\in\\phi(d',c')", "d8323c9ff5404682940105b6d622908b": " \\int_{X(r)} dxdy = {1\\over 2i} \\int_{\\partial X(r)}\\overline{z}\\,dz = -{1\\over 2i}\\int_{|z|=r}\\overline{g}\\,dg={1\\over 2\\pi r^2} -{1\\over 2\\pi}\\sum n|b_n|^2 r^{2n}.", "d832dd960f21985be14fb6d0c78f3aa9": "\\ T2' - T2 = - \\tilde{o} + d", "d833052e321e57d7b5c10ef22cf5f4ec": "\\underbrace{\\tfrac{1}{T}\\ S_{1/T}(f)\\ \\stackrel{\\text{def}}{=}\\ \\sum_{n=-\\infty}^{\\infty} s(nT)\\cdot e^{-i 2\\pi f nT}}_{\\text{Poisson summation formula (DTFT)}}\\,", "d8334995e492c4643482377252f99232": "\\mathrm{enc}(x_1,x_2,x_3,\\dots,x_n) = 2^{x_1}\\cdot 3^{x_2}\\cdot 5^{x_3}\\cdots p_n^{x_n}.\\,", "d833adf0f4ea779a569ea6a2d25ab037": "\\langle\\mathbb{Q},<\\rangle", "d833dd25d3dd2b369ffbc9c49228204d": " -\\frac{\\hbar^2}{2m}\\nabla^2 \\psi = E\\psi ", "d834069b778650538fea239afc2d5006": "\\int |\\psi|^2~da=A+2~\\mathrm{Re}\\left(\\frac{f(0)}{z}\\frac{i2z\\pi}{k}\\right)", "d8345faf26a15fc4968ef9e9a64aa898": "\\hat G_T ", "d83523c86248f219e4340162b2be5305": "\\color{RoyalBlue}\\text{RoyalBlue}", "d835271ae12ec42b234923d48ce50d52": "(Df,f) \\ge (f,f).", "d8352ea930057872502747e55491692e": "a^n \\cos(\\omega_0 n) u[n]", "d83538de3f67c9dcdc5b115f2de3c665": " d\\ln(r) = [\\theta_t + \\frac{\\sigma'_t}{\\sigma_t}\\ln(r)]dt + \\sigma_t\\, dW_t ", "d835bbad5bf622d42906975b865c86a0": "n=\\sup \\{k : \\text{There exists a }Z[G]\\text{ module }M\\text{ with }H^{k}(G,M)\\neq 0\\}. ", "d835c2f6df59caddc83ed74ceffeb0ea": " X = R \\cos \\lambda ", "d835c85ee08b11f84d994782e87e2022": "h*g", "d835c9a8ebc5c0c73762336d3f870e22": "C \\subseteq \\mathbb{R}^n", "d8363388d58428c07656253bd2f5b62d": "\\hat{P}_{ij}", "d836683b8c79ef0b6ec00e15f22df5c1": "\\begin{align}\nR &= \\begin{bmatrix}\n\\cos \\alpha & -\\sin \\alpha & 0 \\\\\n\\sin \\alpha & \\cos \\alpha & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos \\beta & -\\sin \\beta \\\\\n0 & \\sin \\beta & \\cos \\beta \\end{bmatrix} \\begin{bmatrix}\n\\cos \\gamma & -\\sin \\gamma & 0 \\\\\n\\sin \\gamma & \\cos \\gamma & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\end{align}\n", "d8367563c1e536b8eeab40402de2459b": " \\theta_1 = 2 \\psi_1 ", "d83679e9c148ce084b915f590a09a63b": "\\sqrt{\\mathrm{sat}(T_i)}", "d836914627545f4be8dc1470fd557ea7": "\\frac{1}{14} + \\frac{1}{21} + \\frac{1}{42} = \\frac{1}{7}", "d836b35f812916aa9ad53738171f33a2": "C = D^{-1}b", "d836bdc6fbdfd383440ec80c092af5cd": "V - V_p", "d836cb60a3cd93d8d005654bb39be8d6": "\\widehat{\\Omega} = {U}^\\dagger \\widehat{\\Omega} U ", "d83730dec023c172b709fd2f6aadca7d": " -2 p_2 \\cdot p_4 \\,", "d8375270826613e9ac2d96dd8dce77ea": "\\left|\\frac {P(A\\cap V)}{P(V)}-L\\right| < \\epsilon,", "d8376443f2f3f436221a77b06391a617": "\\textstyle\\frac{\\partial f}{\\partial x}", "d8377f31816d1f708e4f1cfe320dfb6a": "X_3 = (XX_1-1)^2 = 9 \\, ", "d8379002889223753f2c0689787300bc": " \n\\begin{align}\nq &= p + 2D - \\frac{6D}{p(p-1)+2}, \\\\\n\n\\nu &= \\frac{p(p+2)}{3D}, \\\\\n\\end{align}\n", "d83792a0b545e0ad6f5dc064b08ba5bf": "\\int_0^{2\\pi}f^2(x)dx=\\sum_{n=1}^\\infty(a_n^2+b_n^2)", "d837a9f1ae55daa2cb0ee63156c555a7": "\n1 = \\frac{1}{\\mathcal{Z}} \\sum_{\\alpha,\\alpha'} \\mathrm{e}^{-\\beta E_\\alpha} \\left(\n\\langle\\alpha |\\psi_\\mathbf{k} |\\alpha' \\rangle\\langle\\alpha' | \\psi_\\mathbf{k}^\\dagger|\\alpha \\rangle - \\zeta \\langle\\alpha |\\psi_\\mathbf{k}^\\dagger |\\alpha' \\rangle\\langle\\alpha' | \\psi_\\mathbf{k}|\\alpha \\rangle\n\\right).\n", "d837aa46e33ba40975a5a8c7748448d7": " \\nu=\\rho \\phi_0/|B^*|.", "d838901a624c354394fa57b658929dd8": "Net\\ Cash\\ Flows\\ from\\ Operating\\ Activities\\ = Net\\ Income\\ + Rule\\ Items\\,", "d838ce3dfbf899792f67a4068d3d8835": " (U^{-1}g)(x) = \\int_1^\\infty g(\\lambda) \\Phi(x,\\lambda) \\, d\\rho(\\lambda).", "d8396c21245c20086e96a9bd56f1b1d7": "e^{-i\\frac{{\\delta}}{2}F^{[l]}}", "d8397b4b4e9ddc743a2886b5e9bd35c7": "Te = 0", "d8397be1afb3113069a4fc0af5a6ed65": " X = eQ ", "d8397cb958110e9b7b2967d11b12c3f8": "\\zeta(s) = \\sum_{n=1}^N\\frac{1}{n^s} + \\gamma(1-s)\\sum_{n=1}^M\\frac{1}{n^{1-s}} + R(s) ", "d839ba79d38f03e63bb9336c7c78108b": " \\mathbf{P}(t) = x_P(t)\\vec{i} + y_P(t)\\vec{j} +z_P(t) \\vec{k}, ", "d839d0c571bb018d7b7ff17dc1227bf4": "1^5 + 2^5 + 3^5 + \\cdots + n^5 = {4a^3 - a^2 \\over 3};", "d839e95d6ce014546f3a8badf20149ad": "\\frac{}{\\star:\\!\\!-~~\\alpha ~\\vdash~ \\top}", "d83a8e9f87c209f7d20961626cb305b0": "[P,x,x]=1", "d83ace0420493aa3ac3a2afa1b2a567d": "\\phi(x)=\\mu\\left(\\frac{2}{\\lambda}\\right)^\\frac{1}{4}{\\rm sn}\\left(p\\cdot x+\\theta,-1\\right)", "d83b1b15b07536c1271b036e1129a0e2": " \\begin{pmatrix} y'_1 \\\\ y'_2 \\\\ 1 \\end{pmatrix} = \\frac{1}{x'_3} \\begin{pmatrix} x'_1 \\\\ x'_2 \\\\ x'_{3} \\end{pmatrix} ", "d83bb704fa9b5116f5c59570a4f56ae4": "\\{1,1\\} \\cup \\{1,2\\} = \\{1,1,2\\}\\, ", "d83bc30853bb35e26591002a1bccb11c": "J=\\tfrac{1}{2} \\int_{0}^{\\infty}[\\,\\textbf{x}^{\\text{T}}(t)\\textbf{Q}\\textbf{x}(t) + \\textbf{u}^{\\text{T}}(t)\\textbf{R}\\textbf{u}(t)\\,]\\, \\operatorname{d}t", "d83c0dc1a957c4e4af9c8a7e4ec392f1": "\\frac{HD}{AD} + \\frac{HE}{BE} + \\frac{HF}{CF} = 1.", "d83c71639b430833c586aeddbdfe0ba0": "1/t ,) P(saw|,I) P(the|I,saw) P(red|saw,the) P(house|the,red) P(|red,house)\n", "d855e5c5af20723987090916f406da37": "\\tilde{b}", "d85629f8b0da5e5ef317e6ed326aad43": "p_{n}=p({\\mathbf{X}},z_{n+1}-z_{n})", "d8563f6a8620f0abade08807373f0c58": "T(F)\\in\\Theta", "d856475ee7280f843a9edfe98e672f57": "\\mathop{\\rm det}_{j,k} \\frac{\\partial^2 S}{\\partial x_j \\partial y_k}(x,y)\\ne 0", "d8566e5cccf785b23d07c8bc0fc6d526": "s>(N^{1/4}+1)^2", "d8568c712d2e94df2b4dfd4871ff93ee": "\\left \\langle N,e\\right \\rangle", "d8570518dc7ac2fac2c933c054123d39": " \\mathbf{F} = \\frac{q_1 q_2} {4\\pi \\epsilon_0 r^2} \\left[\\left[1 - \\left(\\frac{v}{c}\\right)^2 + 4.5 \\left(\\frac{\\mathbf{v\\cdot r}}{c^2}\\right)^2 - \\frac{r}{2c^2} (\\mathbf{a\\cdot r}) \\right] \\frac{\\mathbf{r}}{r} - \\frac{4}{c^2} (\\mathbf{v\\cdot r})\\mathbf{v} - \\frac{r}{c^2} (\\mathbf{a})\\right] ", "d8579478d368cf4e2802e608a4cf74f8": "\\nabla\\times\\mathbf{B}=\\alpha \\mathbf{B} ", "d8583a0918c0eb9f476eb949c0f0e543": "\\displaystyle D_q (f(x)/g(x)) = \\frac{g(x)D_q f(x) - f(x)D_q g(x)}{g(qx)g(x)},\\quad g(x)g(qx)\\neq 0. ", "d8584311108d7851f1594414a743063e": "\\mathcal{V}", "d85866ded3cac5e0b18ba58701659285": "\\gcd \\left(\\sum_{x=1}^{n-1} x^{n-1}, n\\right)=1", "d8587b2da0c999cf14cbe4417aa0e84e": "d(f^n(x),f^n(y))\\geq \\varepsilon_0", "d858ddc8ed961bea7ff8cd200e76e25b": "x = \\begin{vmatrix} \\color{red}{e} & b \\\\ \\color{red}{f} & d \\end{vmatrix}/\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = { {\\color{red}e}d - b{\\color{red}f} \\over ad - bc}", "d858f8ff9342d108431e407d1cf8589d": "\\eta_A\n", "d8592b048bef728a8f29e50fbd8c7b37": "\n W(r) = 0 \\quad \\text{and} \\quad \\cfrac{d W}{d r} = 0 \\quad \\text{at} \\quad r = a \\,.\n", "d859bd93e5e7adbe7cfb81ba2b434bad": "\\scriptstyle 0\\cdot\\infty", "d85a10109afb5b49e59dddae77d01a4b": "S=\\begin{pmatrix} \na_0 & a_1 & a_2 & a_3 & a_4 & 0 & 0 \\\\\n0 & a_0 & a_1 & a_2 & a_3 & a_4 & 0 \\\\\n0 & 0 & a_0 & a_1 & a_2 & a_3 & a_4 \\\\\nb_0 & b_1 & b_2 & b_3 & 0 & 0 & 0 \\\\\n0 & b_0 & b_1 & b_2 & b_3 & 0 & 0 \\\\\n0 & 0 & b_0 & b_1 & b_2 & b_3 & 0 \\\\\n0 & 0 & 0 & b_0 & b_1 & b_2 & b_3 \\\\\n\\end{pmatrix}.", "d85a11dccd888d77343545939bd3db9d": "\\pi \\ ", "d85a1d815d4f08fc2fced79e93ccd9de": "X \\sim \\mathrm{GH}(?, ?, ?, ?, ?)\\,", "d85a48ba1dbcbae9048ca23866126ee2": "\n \\frac{\\partial {\\bold U}}{\\partial t} + \\frac{\\partial {\\bold F}}{\\partial x} + \\frac{\\partial {\\bold G}}{\\partial y} +\n \\frac{\\partial {\\bold H}}{\\partial z} = 0\n", "d85a58b1d0c3391c973d4b55b5cd076a": "H=\\int d^3x\\mathcal{H},", "d85a89f729816604983201e8fc79e0bf": "\\rho\\frac{D\\mathbf{v}}{D t} = -\\nabla p + \\nabla\\cdot\\mathbf{\\tau} + \\rho\\mathbf{f}", "d85b1b0316bd1a8248f014f9b14afd19": "\nx_3 =\\frac{b}{a} \n", "d85b20167e33f2356508e28762f202f8": "f\\in L^{m+n}(\\underbrace{V,V,\\dots,V}_m,\\underbrace{V^*,V^*,\\dots,V^*}_n;W)", "d85b3918b5c3fa0b6db9a6347a9a86d2": " \\displaystyle{f_t=\\lim_{y\\rightarrow 0} f_{y+t}=\\lim_{y\\rightarrow 0} V_t f_y = V_t f.}", "d85b4e76535ff08b8d44f5336efed2ca": "\\begin{align}\n B_1(x) &= x - \\frac{1}{2} \\\\\n B_2(x) &= x^2 - x + \\frac{1}{6} \\\\\n B_3(x) &= x^3 - \\frac{3}{2}x^2 + \\frac{1}{2}x \\\\\n B_4(x) &= x^4 - 2x^3 + x^2 - \\frac{1}{30} \\\\\n & \\vdots\n\\end{align}", "d85b510baa0d92ea5011782098881b17": "\\frac{\\vec{\\Omega }\\cdot \\vec{A}}{\\lambda L}", "d85be1eb964e68b27a7c79a5d37cd6dd": "U^\\dagger(i\\gamma^\\mu\\partial_\\mu^\\prime - m)U \\psi(x^\\prime,t^\\prime) = 0.", "d85c6702266607080f166fc6c1dec560": "\\lambda\\ll r", "d85c6d57c531159b31de03b01348cc86": "t\\cdot 6 \\equiv -1\\,\\bmod{7}", "d85c7027c29a09e137dc76534730ac1b": "u,v\\in\\left( \\mathbb{Z}_{2}\\right) ^{2}", "d85c90c07749d2b97ea38bdcbd689037": "F \\circ h", "d85cb2abbb2c5a9c80bd1d7c43cf20fa": "\\mathbf{F} = q (\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}).", "d85cb8cfb0080f4272ab4abf84e90a45": "P(0) = 1/2", "d85cbe02cc02e5eb719efacb6e2ed3a2": " p", "d85ccc89ec1e980e905aa87dcb01a71c": " \\operatorname{Support}(f * g) \\subseteq \\operatorname{Support}(f) \\cdot \\operatorname{Support}(g) ", "d85d71a65bac367e3ea1b0d908c41a2a": "\n \\vdash A , A \\or \\lnot A\n ", "d85da11a91886f72a9ed2976c2898f7b": "\\csc(\\theta) - 1", "d85dbb15ab078d243d6e8dfe7111648e": "\\mathcal{O}_U", "d85e08ad74d5fc30cacbd391c63f50d3": " \\det A_{33} > 0 ", "d85e583ad144a83dfc3da92f5eb08255": "\\kappa_{\\mathit{ri}}", "d85e70822137cb3cee74948b52b1a040": " U = \\begin{bmatrix} x_{00} & x_{01} \\\\ x_{10} & x_{11} \\end{bmatrix} ", "d85e755798ab7c2219656031354a0352": "a=2, p=z^3-1", "d85f70bee82ab74729f90c2244ac5fb1": "\\mathbf{q}^{-1} (\\mathbf{q} \\vec{v} \\mathbf{q}^{-1}) \\mathbf{q} = \\vec{v}", "d85f8dc2ecc32e41c74041460d434f35": "\\mathrm{d}x = \\rho~\\mathrm{d}\\theta", "d85fb0f9c23129001438a66ac8419bd7": "V_{ind} = \\frac{\\partial \\Phi}{\\partial t} = -L\\frac{\\partial I}{\\partial t}, \\ ", "d85fe00097b8aa63585cc0baa279929d": "\n\\log\\left(\\frac{Z_\\zeta(t)}{Z_\\mu(t)}\\right) \n= \\log\\left(\\frac{\\mu_{(1)}(T) + \\cdots + \\mu_{(m)}(T)}\n{\\mu_{(1)}(0) + \\cdots + \\mu_{(m)}(0)} \\right) \n- \\frac{1}{2}\\int_0^T \\frac{\\mu_{(m)}(t)}{\\mu_{(1)}(t) + \\cdots + \\mu_{(m)}(t)}\n\\,d\\Lambda^{(m, m+1)}(t).\n", "d85ff049422f91766287841b6ff50762": "I_{sp} = c \\cdot \\sqrt{2 \\eta - \\eta^2}.", "d86007d116882b8849296a4d391bb7ad": "A_i\\,", "d8601cb082dbca6a3cbb7d19bfccc15c": "R(X) = D(X) - \\mathbb{E}[X]", "d8603282ea949574fd5e604362084c80": "Ix=gm.(V+ - V-)\\,", "d8609be9da7cf968ca8c7487aa708342": "\\|Df^{-n}v\\| \\le c\\lambda^n \\|v\\|\\text{ for all }v\\in E^u\\text{ and }n>0,", "d860a194946f410a417d31f5ee61d54e": "\n \\vdash \\left( \\left( A \\rightarrow \\left( B \\or C \\right) \\right) \\rightarrow \\left( \\left( \\left( B \\rightarrow \\lnot A \\right) \\and \\lnot C \\right) \\rightarrow \\lnot A \\right) \\right)\n ", "d8614e786a7ce30e416664807d17c36d": "\\ A_r=\\frac{G\\lambda^2}{4\\pi}", "d861933510deb54a5886d0ec514fbf9b": "a(u,v) = a(v,u).", "d861d3900eda89c562d90ba7bd7fa341": "u'", "d861d637239e86fddd074d9c3754d406": "g>0", "d861d8b92d64ba50347d7aedd396f6fe": "\\mathbf{\\ddot{r}} = \\mathbf{a}_{\\text{per}} - {\\mu \\over r^3} \\mathbf{r}", "d86226be3cae9880f0aeb3a840f44d4f": "\\mathfrak{g}\\oplus\\mathfrak{g'}", "d86227deaeb07172737d5a9e1ef2446a": " c = C m \\,,", "d8622ded452547147dee6d43b18a869a": "\\angle CAD=90^\\circ -\\angle DBC=\\angle BCD", "d8627dd4133cc9433c6d0fcd3fe286ad": " \n{1 \\over 4}\n\\begin{pmatrix} \n1 & 0 & 0 & 0 \\\\ \n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 1\n\\end{pmatrix}\n\\quad \n", "d862a19ea3e749b04c9c0a89a1fb90f6": "C_z=C_L ", "d862b64e208044c2ef3e397e0966373e": "\\delta (\\phi R) = R \\delta \\phi + \\phi R_{mn} \\delta g^{mn} + \\phi \\nabla_s (g^{mn} \\delta\\Gamma^s_{nm} - g^{ms}\\delta\\Gamma^r_{rm} )", "d862b657c641ec074cb610cc3f12e669": "x^\\ast = \\sum_{i=1}^t \\lambda_i x_i", "d862c50733a9bd6fbe688f0e536e65b3": "\\eta = \\beta_0 + f_1(x_1) + f_2(x_2) + \\ldots \\,\\!", "d862dc9b04d90defd90cd6e7e2395cb7": "\\mathbf{x}_0\\equiv\\mathbf{x}", "d863cbe5d8e4ae53d35641251f87ba86": " r^{(g)} = 1 - \\frac{rank(a'^{(g+1)})}{\\mu} \\mbox{ if } a'^{(g+1)} \\in Q^{(g+1)} ", "d863f056e62caf3b1def326075d8e46d": " e = - N\\times S\\frac{\\mathrm dB}{\\mathrm dt}", "d86426e3a00985a8a21d37f4abec93ef": "\\pi_{j+1}", "d86431d4e43a18d4fd6fe1afd8bfd51b": "\n\\frac{ \\partial v_x }{ \\partial x} + \\frac{ \\partial v_y }{ \\partial y} + \\frac{\\partial v_z}{\\partial z} = 0\n", "d864580cc065ab9192790b410c5b281c": "\\frac{x_0x}{a^2}+\\frac{y_0y}{b^2}=1", "d864675d62c7905c57953077118ca433": "m x=-\\frac{g}{2v^2{\\cos}^2 \\theta}x^2 + \\frac{\\sin \\theta}{\\cos \\theta} x", "d86512fd284c02b21d1b073b07ee98ab": "f^{\\mathrm{D}}_p(x) = \\left(\\frac{1-x}{x}\\right)^p.", "d865345ed61fa2938cba8eda9d0de0ad": "\\Pr(X_i \\in [a_i, b_i]) = 1. \\!", "d8659fe975acb1db0b1d7820e5ae4c3b": "\\mathcal Z = \\sum_k e^{ ( \\mu N_k - E_k) /k_B T}", "d865b4b12e8d2dc674ce28818eff696c": "\\hat{g}_N(\\cdot)", "d865c03ff89c74ec2d379d358f94a139": " \\Delta ", "d865cb8e0ad37315fc1d6ce37b3fa14f": "\\alpha\\approx\\frac{1}{d}\\left(\\frac{I_0}{I}-1\\right)(1-R)", "d865d5c0087aa972ac63410726d84a97": "\n \\hat\\beta = {\\rm arg}\\min_\\beta\\,\\lVert y - X\\beta \\rVert,\n ", "d86610f593f39f8ffa3bfdb527596000": "M_o = D_o", "d866300748876698cb58ed80de1777cc": "v_{gu} = \\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} {1000 \\cdot m_{gu}}", "d866b3610c7da5243d963aee3c650eec": "x^2 - 2\\ ", "d866ceda6bf67207c79b780e80bf71e9": "\\int\\frac{u\\;dx}{x} = u-a\\ln\\left|\\frac{a+u}{x}\\right| \\qquad\\mbox{(}|x|\\leq|a|\\mbox{)}", "d866f3038e3bc455be42f5765eb61d0c": "u_{xx}", "d86702c005b8f0b51a77e8f44cd6deed": "0 < r < 2GM", "d867050c69be46ac0afda9cf21befcd1": "H(X_2) \\ge 0", "d86715a1d8fdd5fee96a00533cecb397": "\\frac{d B}{dz} =-i \\gamma_{\\perp}|A|^2A", "d8672171c1c17c30f6b569a17ee66c1d": "T=400 N", "d8674abdeb0a2bfe3056e783f1f6df7b": "S = (\\{\\omega_{0}\\}, r, \\xi)", "d867af563099f679f91e0ac122a37e1c": "p_i(s)=g_i", "d867b2bdf6046b296e14bc3b2619a068": "\\begin{align}\\bigwedge \\{ \\bigvee Y \\mid Y\\in S\\} = \\bigvee\\{ \\bigwedge Z \\mid Z\\in S^\\# \\}\\end{align}", "d867c7e279d2d950a62dea3de1d5c98c": "-\\|\\mathbf{P}\\|^2 = - P^\\mu P_\\mu = - \\eta_{\\mu\\nu} P^\\mu P^\\nu = {E^2 \\over c^2} - |\\mathbf p|^2 = m^2c^2 ", "d867e6cdb521d3b1bbebe103a8f724ad": " \\psi(x) = \\sum_{\\hat{p}^k\\le x}\\log \\hat{p}=\\sum_{n \\leq x} \\Lambda(n) ", "d86813eb77efb2fb6244fd64c5f3765a": "\\langle \\mu, \\xi \\rangle(x) = \\langle \\mu(x), \\xi \\rangle", "d8684678a8a99589a4588c2058d86ac3": "(a_1\\otimes b_1)(a_2\\otimes b_2) = a_1a_2\\otimes b_1b_2", "d8687d4831068c5ad57798c6d8483bc8": "|A \\cup B|", "d8688b4a118a24687300477e39a6c1b9": "Dx + Ey + F = 0", "d868985f0dc80900b437cfc418761678": "\\scriptstyle\\gamma(P_n)\\, =\\, 2^{n}-1", "d868f92b4fa71421851e477e0c77e45e": "q_\\text{P} = \\sqrt{4 \\pi \\varepsilon_0 \\hbar c} ", "d86907b87ab7cce2415bf86f6898a1a0": "\\phi_{i}", "d8695d5abc869151f11470de6acef7be": "(f_0, f_1, ..., f_{d-1})", "d8697f8478e112741dd78af651d4d61a": "{\\rm E}[s]\\,\\,\\, \\ne \\,\\,\\sqrt {\\,{\\rm E}\\left[ {s^2 } \\right]} \\,\\,\\, \\ne \\,\\,\\,\\sigma \\,\\sqrt {\\,\\gamma _1 }\n\n", "d869a6ed1dd347cbb4313872f6d47011": "(c)'=0", "d869de8a418cd4c3fd02636fd7162c72": "H= U + pV \\,\\!", "d869df99f1c3f18f1741a26212678acd": " e^{\\Phi} \\to e^{Q_B \\Lambda} e^{\\Phi} e^{\\eta_0 \\Lambda'} .", "d869f18f859b6e39fb2f4f894906ddba": "\\int_X \\|f-f_n\\|_B\\,d\\mu \\to 0", "d86a5d427813ac9754f1287e05fc6d1e": "\\mathrm{Points}\\ \\mathrm{percentage} = \\frac{\\mathrm{Points}}{\\mathrm{Total \\ possible \\ points}}\n= \\frac{ \\mathrm{Overtime \\ Losses + (2 \\times Wins)}}{\\mathrm{ 2 \\times Games \\ Played }}", "d86a71b1a4f038ab18795ea35ca75e08": "\\epsilon^2\\left[\\frac{1}{\\delta^2}\\left(\\sum_{n=0}^\\infty \\delta^nS_n'\\right)^2 + \\frac{1}{\\delta}\\sum_{n=0}^{\\infty}\\delta^nS_n''\\right] = Q(x).", "d86a951ae1fd2b1a47cfe764a67271ef": "Q^* = \\sqrt{\\frac{2DK}{h}} ", "d86add876c6aab0dfe62fe33134d0d1d": "3x \\equiv 1 \\pmod{11}", "d86b27ba6ebe78e26e828077297e7720": "E(\\alpha_i) = 0", "d86b2a929cfdc884367b4adbaec9cfee": "~ A_0=\\frac{ND}{\\sigma_{\\rm as}+\\sigma_{\\rm es}}~", "d86b879f577cb5c68b2d5df9000d8144": "\\dot{Q} = \\left( \\Pi_\\mathrm{A} - \\Pi_\\mathrm{B} \\right) I", "d86b8d9495a805cfeb2dd2d75552500a": "\\scriptstyle X_L \\;\\gg\\; (-X_C)\\,", "d86bab545b349a2b0dc22ace5bc2bac5": "\\operatorname{E}(\\theta|y)", "d86bca46dfa08a1bcd9e7b3d70e55bc7": " \\mathrm{atomic \\ ratio} \\ (\\mathrm{i:j}) = \\mathrm{atomic \\ percent} \\ (\\mathrm{i}) : \\mathrm{atomic \\ percent} \\ (\\mathrm{j}) \\ .", "d86c02df3f8618a573b30150424e102c": "\\sum_{t^{'}=1}^{T} \\frac{\\sin2\\pi W(t-t^{'})} {\\pi(t-t^{'})}w_{t^{'}} = \\lambda w_{t}.", "d86c06c364c26866d105cc780ba346d0": "PSL_2(p)", "d86c5eb985112e16c6f2e1d02df15235": "x_1 = \\alpha \\cosh(t/\\alpha) - r^2 e^{t/\\alpha}/2\\alpha,", "d86c6256187f8ec8b22d6e8877e50879": "A\\overline{AB} = A\\overline{B}", "d86cfe428339aead23d9a8c3e48d8bee": " c,\\,0 \\le c < m", "d86d00e10fab1b5b547e40bac8a3dab7": "E = (E_i - E_f)-E_B", "d86d0b52d5acd84f7b7f492c582c870f": "\\kappa^{\\sigma}", "d86d19ef435746ebd2fb7bbbb6d66efe": "{\\mathbb R}^N", "d86d1b729b1d7d58741f3f249f630be1": " x = \\rho \\cos (\\alpha \\theta), \\ ", "d86daf5b48bc527c58cd913e79a91c5f": "\\overline{x}\\langle R \\rangle.P | x(Y).Q \\rightarrow P | Q[R/Y] ", "d86e0c72d31d946f968ba5e8ab6ffab9": "\n1 - (p_1 p_2 r_3 + p_2 p_3 r_1 + p_1 p_3 r_2) - (p_1 p_2 r_1 r_2 \n+ p_1 p_3 r_1 r_3 + p_2 p_3 r_2 r_3) + p_1 p_2 p_3 ( r_1 r_2\n+ r_1 r_3 + r_2 r_3) + \n", "d86e12f4fa6dda20da61cb06be94f77c": "f(x)=1-[\\frac{1}{1+e^{-a(x-c)}}]", "d86e1e2e38efd9ce47c4401062a5992d": "f(x+e)", "d86e6829805d92aa5ee7640cc56970cb": " {\\pi\\over 4} = 2 \\arctan \\left({1\\over 3}\\right) + \\arctan \\left({1\\over 7}\\right) \\; .", "d86e69cb5ff0f4a1cc7bf224c053241b": "\\Omega^{n/2+2}P(g)\\phi = P(\\Omega^2g)\\Omega^{n/2-2}\\phi.\\,", "d86ec5806315f24d17eba8602a5e2277": "F(t) = t^6 + t^5 + 2 t^4 + 3 t^3 + 2 t^2 + t + 1.", "d86ee5d1ea8b96b106f8375678e698f2": "\\{E_{a^{n}}:a^{n}\\in T_{\\delta}^{\\mathbf{p}^{n}}\\}", "d86ee92927cc1cb86cb16d22fab36589": "J_i=\\frac{\\partial f(x_i,\\boldsymbol\\beta)}{\\partial \\boldsymbol\\beta}", "d86ef01a836af5ffb4f623899211937d": "H^i_\\mathrm{cell}(C_\\phi) = \\begin{cases} \\mathbb{Z} & i=0,n,2n, \\\\ 0 & \\mbox{otherwise}. \\end{cases}", "d86f157780f141acd863a5f8bcd79cee": "k_{\\rm A}=\\frac{k_{\\rm C}}{c^2}", "d86f77ebdfae50f49075c55732304a06": "\\bar x_B - \\bar x_A", "d86fd29ff48fb37a3232f25c81b1be68": "\\scriptstyle{\\ll}", "d8700183af656b0d1056a921975a1797": "f(x) = x + 5", "d8706baa69501526b1a102b2ffbd7508": " \\mathcal{E} = \\{A,\\ G, \\ C, \\ T\\}", "d8706c87caab39ee1885c648e20490ef": "n^*\\ =\\ N \\exp \\left( \\frac{-\\Delta G^*}{k_BT} \\right)", "d8707bb0cd8018406d4915df82a307c2": "wx^2 = \\frac{1}{5}x^5 + C", "d87095ee63ceac08aaaf0a8029018bcb": "S = F e^{-rT}", "d8709d4b92a12763f55761f3fb11c5d2": "x^n-1.", "d870ad902932946fe06442f754a5709c": "\\operatorname{core}(A) = \\operatorname{core}(\\operatorname{core}(A))", "d870c4f2f8f2a02a38876d8ebdcf8191": "~\\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1", "d870d2b0769b64c77662fcf8f08db68d": "\\iiint\\limits_D f(\\rho,\\theta,\\phi)\\,\\rho^2 \\sin\\phi \\,d\\rho \\,d\\theta\\, d\\phi .", "d870dfe4a99bec0e595b83fd0f9a57ae": "\\ \\nu (A) \\equiv \\int_{A} w(x) \\, \\mathrm{d} \\mu (x), \\ \\ \\ A \\in \\Sigma", "d870f5fa00fa65cc74981171ac331a7e": "\\,\\!\\omega_0", "d870fa39149555fe61b0a466f060a7f3": "\\frac{35}{18}", "d8711d05b42ead08a91d08525c2da1e6": "\\mathbf{b}_{x,y}", "d871a27060c5ce42b3146ad4430f57cf": "f_1(x,y)= \\frac{1}{\\sqrt{2}}\\begin{pmatrix} \\cos 45^o & -\\sin 45^o \\\\ \\sin 45^0 & \\cos 45^o \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix}", "d871cc5b0455ef9a798334b6735ab492": "t/\\epsilon", "d871d76c5f427f7cde3744a57a2bd67c": "\\omega '", "d87200ff0b5098c7c680ca2f5ec8b2ce": "\\operatorname{wnchypg}(x;n-1,m_1,m_2-1,\\omega) \\frac{m_2}{m_1\\omega+m_2}\\,.", "d87203d47f33c0b3553a4bc37ffda418": " \\forall x(Fx \\rightarrow Gx) \\land \\exists xFx \\rightarrow \\exists x(Fx \\land Gx)", "d8722b529a183461167d81fd61b186f3": "U(z)=w(v-B(z))=F(z|v)(v-B(z))", "d87298ceeb4e242ad8c2d63f8d34f100": " Z_{min} = E_{2c} -E_{1} = 6.40 -4.00 = 2.40 \\text{ ft}", "d873449306f74d6b34d2c5593ce31c2f": "sin(2\\theta{})", "d8734eb7c21db136c60f0af43e46296a": "p=\\mathrm N(\\alpha)=a_0^2+a_1^2+a_2^2+a_3^2.", "d873e28e0a182d6539ed3ff582d9129b": "p_N = 0 ", "d873efffa51a08b7bd192f3226daefd1": "\\mathcal{H}\\, =\\, \n \\iint \\left\\{ \n \\int_{-h(\\boldsymbol{x})}^{\\eta(\\boldsymbol{x},t)} \n \\frac12\\, \\rho\\, \\left[\n \\left| \\boldsymbol{\\nabla}\\Phi \\right|^2\\,\n +\\, \\left( \\frac{\\partial\\Phi}{\\partial z} \\right)^2\n \\right]\\, \\text{d}z\\, \n +\\, \\frac12\\, \\rho\\, g\\, \\eta^2\n \\right\\}\\; \\text{d}\\boldsymbol{x}.\n", "d87418c16ddd2cda3354aa5ecfbe3858": "\\nabla_{\\bold{v}}{f}(\\bold{x}) \\sim \\frac{\\partial{f(\\bold{x})}}{\\partial{v}} \\sim f'_\\mathbf{v}(\\bold{x}) \\sim D_\\bold{v}f(\\bold{x}) \\sim \\mathbf{v}\\cdot{\\nabla f(\\bold{x})} \\sim \\bold{v}\\cdot \\frac{\\partial f(\\bold{x})}{\\partial\\bold{x}} ", "d87430edbc8af1a11fe9bc6565d4f32d": "a^{\\dagger}", "d8744be0761a27f83782a43a739b9ecd": " A_{sc} = \\frac{J_{ex}S^2}{a}", "d8747c0a21f1c8c12576aef811f411da": "10.30 \\%", "d87495e00d190adb509e70e4d73eb368": "\\scriptstyle{I_\\circ}", "d874c094f671943a900f7f737384d116": "O' \\in \\mathcal{B}", "d874e31ffeb47760323065d93f08fc93": "f=\\prod_{i=1}^{\\deg(f)} f_i^i", "d87526212d7aa45457e0037ece0057d3": "\\Delta^{it}M\\Delta^{-it} = M", "d8753d3a7539c1aa382f0661e531d704": "\\displaystyle{\\Lambda(f)=\\left|g_b\\left( -{a_{-1}\\over \\overline{a_{1}}}\\right)\\right|}", "d87544155337b8993e3d0ee87420edfa": "X^{(0)} \\gets X", "d875570d3d6885841437b57c58cda859": "L(s,\\pi,r), \\ L(s,\\tilde{\\pi}, r) \\ ", "d87578ae335afdacbf4b4c1de6783f6c": "2\\omega_{n} \\sqrt{1 - \\frac{f_{0}}{2}}", "d875863a6bb0a4c8a09fbe026895b927": "\\bigoplus, \\bigotimes, \\bigodot \\!", "d87588dc21174d76517590fcc2ce2897": "x_3=x_4", "d8758c815c940b38e3e7f67bf0ea99e9": "g = G \\left[ \\frac{1+k\\sin^2 \\phi}{\\sqrt{1-e^2 \\sin^2 \\phi }} \\right] \\ , ", "d8759a9163303e16a4de720854005f94": "\n\\phi_{ij} =\n\\begin{cases}\n1, & \\mathrm{\\quad if\\;} ( \\mid X_i - x_n \\mid + \\mid Y_j - y_n \\mid ) > B \\quad \\\\\n0, & \\mathrm{\\quad else}\n\\end{cases}\n", "d875c0fe7d3f53c83c1ee54ece0ef44d": " \\tfrac{1}{X} \\sim \\textrm{Cauchy}(0,\\tfrac{1}{\\gamma})\\,", "d875c5fed965b78800c7c1a9d4a2cc34": "\n\\begin{align}\ny_0(x) & = 1 \\\\\ny_1(x) & = x + 1 \\\\\ny_2(x) & = 3x^2+ 3x + 1 \\\\\ny_3(x) & = 15x^3+ 15x^2+ 6x + 1 \\\\\ny_4(x) & = 105x^4+105x^3+ 45x^2+ 10x + 1 \\\\\ny_5(x) & = 945x^5+945x^4+420x^3+105x^2+15x+1\n\\end{align}\n", "d875e6a9c8c337f9b33bebdecf379d4a": "\\pi\\in\\Pi", "d87602dbd690753b2463e86742edbe2e": "\\frac{100-95}{100} = 5.00\\%", "d8761d890b1c494071815c4ad4b2e391": "\\vec{\\hat{\\alpha}}= \n\\frac{4 G}{c^2} \\int \\frac{(\\vec{\\xi}-\\vec{\\xi}^{\\prime})\\Sigma(\\vec{\\xi}^{\\prime})}{|\\vec{\\xi}-\\vec{\\xi}^{\\prime}|^2}d^2 \\xi^{\\prime 2}", "d87637d35caa952dffb9ea131607f83d": " \\Delta n = \\lambda_0 K |\\mathbf{E}_0|^2, ", "d876581f55f2322ba361aa9707c787f4": "R = \\frac{P}{I^2} \\,", "d8769401c06cf3a9af2f704c75d50c5c": "\\{|n\\rangle\\}", "d8769d56c8d407d5fec92de8cd955475": "\\mathbf E\\;\\cdot\\mathrm{d}\\mathbf A = \\frac{Q(V)}{\\varepsilon_0}.\\,\\!", "d876b7c0db632069b52fcffad98631e3": "e ^ { \\varepsilon{}_n \\alpha } = \\cosh ~\\alpha + \\varepsilon{}_n ( \\sinh ~\\alpha )", "d8772266c4e8a14d2aff337d1faf1cb4": "\\tau_s ", "d8777aad40e22697a95df0572c90ea7a": "\\dim_FD_B(V)\\leq\\dim_EV", "d877c7372941a6b80070c7189cb37877": " \\|x\\|_{\\operatorname{C}^*} = \\sup_f \\sqrt{f(x^* x)} ", "d877e8a8fd321461e16f38790fcb1401": "q(st)=(qs)t", "d877eaac432355e98aeb83b0ba502307": "\\varphi_{ij}:U_{ij}\\rightarrow U_{ji}", "d87805a8351bff272232815ff0d7fec1": "\n\\mathbf{e}^{(1)} \\equiv \\frac{-1}{\\sqrt{2}}(\\mathbf{e}_x + i \\mathbf{e}_y)\\quad\\hbox{and}\\quad\\mathbf{e}^{(-1)} \\equiv \\frac{1}{\\sqrt{2}}(\\mathbf{e}_x - i \\mathbf{e}_y)\\quad\n\\hbox{with}\\quad \\mathbf{e}_x\\cdot\\mathbf{k} = \\mathbf{e}_y\\cdot\\mathbf{k} = 0.\n", "d8783e3e7c33803b030f6d82546c1adc": "\\sqrt{\\frac{20}{63}}\\!\\,", "d878414aa7561bca744bf59a8f429aa0": " \\Sigma\\ ^i ", "d8789a3af3a301f88bcefb94f4f02405": " i \\hbar {d \\over dt} U(t) | \\psi (0) \\rangle = H U(t)| \\psi (0)\\rangle.", "d878c7e7c0a4415435bd0dfd08fdaf18": "\\mathrm{vec}(\\mathbf{M})", "d878f7baa83a423d698e78b74e5fd743": "p_\\gamma", "d8792166b625a1a2f8d080c6e195408f": "L(K)", "d87922a765f12f89c3c4acbaaf5e468c": "\\frac{\\Delta y}{\\Delta x}", "d87935816de27673c65794f63d7f5e8a": "V \\subseteq Y", "d8793ffeee4b7bc8594891f5afb20189": "\\textstyle{{{4 \\over 3} \\over \\left ({9 \\over 8} \\right )^2} = {256 \\over 243} }\\,\\!", "d879511db8c7d3b5cd38641468d67e02": "CAGR \\approx AR - \\tfrac{1}{2}\\sigma^2", "d8797af3554773f4e748e48d54b0854d": "\\Gamma = \\frac{V_{rev}}{V_{fwd}}", "d879b9bd9d300567c740fb9ab0382c68": "\\bar{x} = (1/d_2 - 1/d_1)/2", "d879f8443481ca65ad848bccf6f18e2c": "T_k^{(n)} =\\sigma_{mk}n_m\\,\\!", "d87a02b519d917971931c5b610f97bfc": "\\delta N \\rightarrow 0", "d87a1237e31ba734f891ca0008a22cd3": "F\\, =\\, C_m\\, \\rho\\, \\frac{\\pi}{4} D^2\\, \\dot{u}\\, +\\, C_d\\, \\frac12\\, \\rho\\, D\\, u\\, |u|.", "d87a1a0942a12e2a4d9691ffd38df856": "||A||", "d87aa243cd5d995cc5200c8597b11393": " 4 x^3 + 2 x y^2 = (2 a^2 y - 2 x^2 y ) \\frac{dy}{dx} ", "d87aa9aa437dc5307aada1dd03127ac9": "\ng_2(\\omega_1,\\omega_2)=\n\\frac{\\pi^4}{12\\omega_1^4}\n\\left(\n \\theta_2(0,q)^8-\\theta_3(0,q)^4\\theta_2(0,q)^4+\\theta_3(0,q)^8\n\\right)\n", "d87b1dffd2cb7b3f937950759f086859": "\\begin{align}\n y_1 &= b_0 \\left(\\frac{(\\alpha)_{1 - \\gamma} (\\beta)_{1 - \\gamma}}{(1)_{1 - \\gamma} (\\gamma)_{-\\gamma}} x^{1 - \\gamma} + \\frac{(\\alpha)_{2 - \\gamma} (\\beta)_{2 - \\gamma}}{(1)_{2 - \\gamma} (\\gamma)_{-\\gamma}(1)} x^{2 - \\gamma} + \\frac{(\\alpha)_{3 - \\gamma} (\\beta)_{3 - \\gamma}}{(1)_{3 - \\gamma} (\\gamma)_{-\\gamma}(1)(2)} x^{3 - \\gamma} + \\cdots \\right) \\\\ \n&= \\frac{b_0}{(\\gamma)_{-\\gamma}} \\sum_{r = 1 - \\gamma}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r (1)_{r + \\gamma - 1}} x^r.\n\\end{align}", "d87b6201283d397ba69feec3c8a9b417": " U_{ex} = \\sin^{2}\\theta_{ex}\\mid 1+r^{TE}_{10}\\textit{e}^{i\\Phi_{in}}\\mid^{2} + \\sin^{2}\\theta_{ex}\\cos^{2}\\theta^{in}_{1}\\mid 1-r^{TM}_{10}\\textit{e}^{i\\Phi_{in}}\\mid^{2}+2\\cos^{2}\\theta_{ex}\\sin^{2}\\theta^{in}_{1}\\mid 1+r^{TM}_{10}\\textit{e}^{i\\Phi_{in}}\\mid^{2}\n", "d87c146d57960c7505ee791def8f8af1": " \\gamma_{ij}(x,t) = \\sum_{k=1}^N\\sigma_{ik}(x,t)\\sigma_{jk}(x,t),", "d87c62129cfcc15bf4fa0bab2d4464a4": " ~T ", "d87c7817962ccb1426a93d87928eda63": "||\\Delta x_i\\|", "d87c8f8a6bbc2067376a857c268d3002": "r^\\mu", "d87c9e919d4ece058c7c2ba79278a15e": "q = \\left(\\cos\\left(\\frac{\\theta}{2}\\right), \\omega \\sin\\left(\\frac{\\theta}{2}\\right)\\right)", "d87cc8c499a8cc60069dac19b0de8da6": "\\sum_i \\sigma_i(B)", "d87d3b576aa71a3948f695c556d4d315": "\\displaystyle{K(u,v)={\\dot{z}(t)\\over z(t) - z(s)} - {ie^{it}\\over e^{it}-e^{is}}= \\partial_t \\log\\left( {z(t)-z(s)\\over e^{it} - e^{is}}\\right).}", "d87d61382201595cfc145af90ee0e52e": "\\eta:I\\to X\\otimes X^*", "d87d871aa2b5c378b582252f3a6c02a7": "M \\circ (\\mbox {Id} \\times M) = M \\circ (M \\times \\mbox {Id})", "d87d97608a5945ff059a6b076c343038": "[a(f),a^\\dagger(g)]=\\langle f|g \\rangle", "d87dcb75b7f3b89648674d23665b80e5": " \\Psi_1,\\dots, \\Psi_n ", "d87e3786eaece3b65165eff0adb264ee": "K\\in C^1(\\mathbf{R}^n\\setminus\\{0\\})", "d87e5509babfff7304b665cf278e5022": " F_{e} \\phi_{e}-F_{w} \\phi_{w}\\,= D_{e}(\\phi_{E}-\\phi_{P})-D_{w}(\\phi_{P}-\\phi_{W})", "d87eb0a2320d5f41bb577c8d067b1237": "\\eta_{\\alpha\\beta} = \\mathrm{diag}(1,-1,-1,-1)", "d87eb1b3b8d8cb83e5ca042a1cdfdde3": "{d \\over dt}\\left\\{ Y \\right\\} = \\left\\{B \\right\\} \\left\\{X \\right\\} - \\left\\{ X \\right\\}^2 \\left\\{Y \\right\\} \\,", "d87f106c0c5636ce89c24c7271bdc194": "-\\infty < x < \\infty.", "d87f43597b536cfc0cd05c1cee5b853c": "\\kappa < 0", "d87f741028ef6ff65d490789ab1bf85f": "A \\times B = \\{(a,b) \\mid a \\in A \\wedge b \\in B\\}", "d87fa1504dae96adb3490d375fedb2e4": "\\frac{1}{\\left(b-1\\right)^2} = 0.\\overline{012\\cdots(b-4)(b-3)(b-1)},", "d8803390c800a4f9d11fe29d2097abef": "\n \\sum M_A = 10x - R_a (x-10) + (1)(x-10)\\frac{(x- 10)}{2} + M_2 = 0 \\,.\n ", "d8812b99183edd8b6cb326a55f97731f": "\nSE(log(RR)) = \\sqrt { [1/a + 1/c] - [1/(a+b) + 1/(c+d)] }\n", "d88155618cab254e81d9b855eec3fe0b": "\\bar \\alpha", "d881572d2f2090061928e358f678b20a": "\\det\\left(\\mathbf{I}_n + \\frac{\\beta}{2}\\boldsymbol\\Sigma^{-1}(\\mathbf{X} - \\mathbf{M})\\boldsymbol\\Omega^{-1}(\\mathbf{X}-\\mathbf{M})^{\\rm T}\\right) =", "d88161c4af108a7d40255f1e39c8b681": "\\frac{\\partial u}{\\partial t}", "d88165183f9d1b1c7f61e185ae702e76": "\\sum_P", "d88179d6ccb9cb9b9f684f0e3e892831": "P(x_i) = {Q(x_i) \\over E(x_i)} = y_i", "d881c1f314aeaa86e138b3dbe04c80e2": " D = E[(x-Q(x))^2] = \\int_{-\\infty}^{\\infty} (x-Q(x))^2f(x)dx = \\sum_{k=1}^{M} \\int_{b_{k-1}}^{b_k} (x-y_k)^2 f(x)dx ", "d881c7777d3ca442c5a4972da1d11a28": "(12)\\qquad \\tilde{\\omega}_{ab}=\\frac{1}{2}\\,\\Big(\\rho-\\bar\\rho \\Big)\\,\\Big(m_a \\bar m_b-\\bar m_a m_b \\Big)=\\text{Im}(\\rho)\\cdot\\Big(m_a \\bar m_b-\\bar m_a m_b \\Big)\\,,", "d881ec15215572783883aebc4d1ed239": " T(z) = \\sum_{p\\ge -1} t_{ij}^{(p)} z^{-p+1}", "d882794756f639bb5688d4fcdbd70e34": "a=\\frac{1}{4f},", "d88297aed89f89131e0eac12d3bb82f9": "\n2\\mu = \\sum_{p=1}^{N} \\mu_p \\alpha_{p}.\n", "d8829c302fb85b62892fe38c49ce1520": " \\kappa(A) \\ge 1 .\\,", "d882e8db05b8ea0150ea097d7310af34": "\\left\\{\\frac{1+x_2}{x_1},x_2,\\frac{1+x_2}{x_3} \\right\\},", "d883017da3cd43f1fad36da8719ee19a": "\\rho_p\\;", "d8832d86cd319f5dc3396c29f1e2def5": "2.42\\overline{314}_5 = 2.42314314314314314\\dots_5", "d883500ca51d005f7b03197de51e5704": "g : \\alpha \\rightarrow \\alpha'", "d8836bac267d360d6c0be2b2ff9fbf3f": "\\nabla g\\left( p \\right)", "d8837018d94228de7a8c29c8765272c1": "a,\\overline{a},b\\! \\in\\! V, u,v,x,y \\in V^*, ux,vy \\in V^+", "d884303d6de66b110d52edde6a95d5f9": "A^{*}=\\alpha_{m-1}P^{*}A_{d}^{*}\\left(P^{*}\\right)^{T},", "d884499dc29b6dce8bdffe077847ddf1": " z^{-1} = z^{*} / \\lVert z \\rVert .", "d884a186cb54ba3467b36abad5e6aede": "T_\\text{P} = \\frac{m_\\text{P} c^2}{k_\\text{B}} = \\sqrt{\\frac{\\hbar c^5}{G k_\\text{B}^2}}", "d884b88e9d73b1c117108b757a283d34": " \\psi_{m,n}(t)=a^{-m/2}\\psi(a^{-m}t-nb). \\, ", "d8850932e5c96b0f96f03f341db1c7b8": "\\mathcal{C} = \\begin{bmatrix}B & AB & A^{2}B & \\cdots & A^{n-1}B\\end{bmatrix}", "d88549ba799af3a6eaa4cd048c48f7f2": "Q=\\int\\limits_L \\lambda_q(\\bold{r}) \\,d\\ell", "d8857b806b7069491fa2569bd53c19d5": "\\mathbf{F}=(F_1, F_2, \\dots, F_n),", "d88594922218710a21a572cc753dc84b": "P = -\\frac{2}{3}\\frac{q^2}{m^2c^3}\\frac{dp_{\\mu}}{d\\tau}\\frac{dp^{\\mu}}{d\\tau}.", "d8859c7c82477168cb8d3af69d4e5c96": "B_{k+1}^{-1} = \\left (I-\\frac { s_k y_k^T} {y_k^T s_k} \\right ) B_{k}^{-1} \\left (I-\\frac { y_k s_k^T} {y_k^T s_k} \\right )+\\frac \n{s_k s_k^T} {y_k^T \\, s_k}.", "d885bb6498f28f1896c043d6dfeb1b62": "\\mathbf{x_2}, \\mathbf{y_2}", "d885d669917c4dc1a26a8f49b0175919": "E[X-\\mu_X]=0", "d885eeb0794ef2ff9da00dc3dbc3dd2c": "\\frac{du_2}{dt}=\\gamma(T_1-T_2)", "d885ef3710e74e2c5568414f76949d76": "{\\xi \\over \\mathit{\\Delta}} = \\cos \\delta \\cos \\alpha", "d8866edf5faa118a669b6732d2afb9b9": " x=x_1, \\dotsc, x_n ", "d887203cb031a65ab8e8e83a564e516c": "\\int_E f \\,d\\mu = \\int_E f^+ \\,d\\mu - \\int_E f^- \\,d\\mu", "d887f8d0f01be13e41c5dae6ebf6cefe": "T(\\Delta V) \\approx \\sum_{n = 0}^N a_n (\\Delta V - \\Delta V_{\\rm comp})^n, \\quad \\Delta V_{\\rm comp} = E(T_{\\rm known}) - E(T_{\\rm ref}). ", "d8882664f8e91eb110ce1247a7101335": "u(x) = u*\\chi_r(x)\\;", "d8883c4fc54e83e83d4e82ed057e2746": "10^{-4.5}", "d8884e7bbb2bc5e254e3ba150e74127c": "\\Gamma = \\left\\{ (e, e') \\in E \\oplus E' \\; | \\; g(e) = g'(e')\\right\\}.", "d8888c9eb26f75e00c7f92f18903d91d": "(2,0)", "d888afbeeba19927b1b7a6eb748ba7ea": " -1 ", "d888b2fab3361870dc650d568856dfbc": "{X^a}_{;b}", "d888c35df62366fd2afc98b7c7c011f1": "\\Delta C^\\circ", "d88943e47c7c46ea380f2b73254e7a7b": "Q=aP^{c}", "d889748151b56f1dd1e94480d5165f18": "M_{xy}(t) = M_{xy} (0) e^{-i \\gamma B_{0} t}", "d8897a94bf57cfe437ad8d936faa9ff4": "\\hat e ^T \\hat e", "d8897f5191a1674c4cf49b46e99d4877": "x_0,\\dots,x_n", "d889bffefe613ec52c08175dbf2288ea": "\\left\\{\\,k 2^n - 1 : n \\in\\mathbb{N}\\,\\right\\}", "d88a2192181a4051a7b50312a62bbfa9": "12c3", "d88a5a3c268089b88086ccd64c7cb062": " | \\phi \\rangle \\,", "d88a7cb6b93ffaf6e6e6563f2fe56230": "\\hat{\\sigma}^{2}_w = \\frac{1}{N} \\sum_{k=1}^N {(\\hat{x}_{k+1}-\\hat{F}\\hat{{x}}_{k})}^{2}", "d88a8128f64ad466b5f0abf7912474b4": " \\Phi_A : \\mathrm{Hom}(A,A) \\to F(A) ", "d88a8667a90e05d7f9fecde993fa66b1": "2 \\pi \\exp(v \\tan v) = n \\frac{\\cos(v)}{v}", "d88b8f97ff8ee3cf14cd03de68312c3e": "t\\,\\!", "d88bc87a5e68d121a7344ba79abbcc87": " \\phi_h(g) = H(g) = H(G)=\\langle h|(G) = \\langle h|(\n|g\\rangle). ", "d88be2bdec0a511c72284c9b7f96f17b": "V = M / \\rho\\,", "d88c7b5489ae7a8a6216ee38130d3512": "\\mathrm{C_nH_m + \\frac{2n+m}{4} \\ O_2 \\rightarrow n \\ CO + \\frac{m}{2} \\ H_2O}", "d88d22c85d6c82cd2ae2f90e8fe16c33": "\\cos(3\\theta)=\\frac{3q}{2p}\\sqrt{\\frac{-3}{p}}", "d88d81a0e531e27f6ff724769c31590f": "(A \\wedge b \\wedge A) \\cap (C \\vee b \\vee C ) = (C \\vee a \\vee C ) \\wedge b \\wedge (C \\vee a \\vee C ) = (A \\wedge c \\wedge A) \\vee b \\vee (A \\wedge c \\wedge A)", "d88da74346a29f3606a7627cc286951b": "G(\\textbf{r}, \\textbf{r}^{\\prime}) = \\frac{e^{-j k |\\textbf{r} - \\textbf{r}^{\\prime}|}}{|\\textbf{r} - \\textbf{r}^{\\prime}|}\\,", "d88dfee2c7eca67b37ce676558fddece": " -\\left(\\frac{\\partial S}{\\partial P}\\right)_T = \\left(\\frac{\\partial V}{\\partial T}\\right)_P = \\frac{\\partial^2 G }{\\partial T \\partial P} ", "d88e0f78bccf77145d9b49ce871b30c1": "\\textstyle\\deg A_i<\\nu_i", "d88e272547a3b6ba711b1019c97ba664": "{n \\choose p}", "d88e47238eac51cd73060fa2f4f6f06d": "\n\tE_{tps}(f) = \\sum_{i=1}^K \\|y_i - f(x_i) \\|^2 + \\lambda \\iint\\left[\\left(\\frac{\\partial^2 f}{\\partial x_1^2}\\right)^2 + 2\\left(\\frac{\\partial^2 f}{\\partial x_1 \\partial x_2}\\right)^2 + \\left(\\frac{\\partial^2 f}{\\partial x_2^2}\\right)^2 \\right] \\textrm{d} x_1 \\, \\textrm{d}x_2\n", "d88e9e3278879db3b897b6e0587756cf": "|\\{1,\\ldots,n\\}| = T^2(n)", "d88eac84a24fc97c21e46c6b7c4a6e58": "\nN = A^2+2B^2+3C^2+4D^2+5E^2+....\n", "d88eb852e887176425688521e18b46a8": "\\operatorname{sgn}(\\mathord{\\cdot})", "d88ec2cb379779739ade87ec4f02edf2": "D = (-25x^2 +176x -15 , -\\frac{72}{125}(17x-5))", "d88ec9f555282a2634546335d42e5db1": "M^{\\textit{d}}=\\textit{k} \\cdot P\\cdot Y", "d88f017bdd67fcc96e101bb890ac99c9": "(\\lambda K + I)u=f \\, ", "d88f6e996b0c10d3effaa9809db825d5": "\\sum |b_n|", "d88fd81a5d8d828dceff38edf2e09fec": " {\\prod_i}' G_i\\, ", "d88fda088e592f12dbb3df7a5947db35": "\\frac{dy}{y} = -f(t)\\, dt", "d88ff66e41578cc5e9addb2f00adca34": " \\zeta = \\frac {\\alpha}{\\omega_0} ", "d8910ebb0d9f97c51dba2f0b0e563f90": "u = \\frac{1}{2}\\left(\\mathbf{E}\\cdot\\mathbf{D} + \\mathbf{B}\\cdot\\mathbf{H}\\right).", "d891478cdb3de9adf8ef756eec74c03a": "r_{nk}", "d8914f650a2ee895a3893db3db1692c6": "X[n] = T^n X", "d8915130bc7ae8136a1b81aebff09c48": "Y[x, y] = y - \\frac{y'}{\\sqrt{x'^2 + y'^2}} \\int_a^t \\sqrt{x'^2 + y'^2} \\operatorname{d}t", "d891a8b191f0c16d81f0a5bc39eab534": "S(d_1)=d_1", "d891e124f0b2372ff4c70447d3f49031": "f:\\mathbb{F}_2^n\\rightarrow\\mathbb{F}_2", "d89227e72aca99bad45b7d79c3a5764c": "\\vec \\psi_C", "d892479c25146b9e677f1ce2e36dca57": "(\\Sigma, S, s_0, \\delta, F)", "d8925f9b46bc490a7ef3122f079a865f": "G_\\theta", "d892846186c08b761850f5ebb510159c": "\n\\left[ \\eta \\right] = \n\\left( \\frac{4}{15} \\right) (J + K - L) + \n\\left( \\frac{2}{3} \\right) L + \n\\left( \\frac{1}{3} \\right) M + \n\\left( \\frac{1}{15} \\right) N\n", "d8928f732d619254526b1e5eadd840ec": "T = 2 \\pi \\sqrt{\\frac{a^3}{\\mu}}", "d8929068f239cc2984d214ec4717cc5f": "Combo", "d8929e206878ac323635a80ef3e54c42": "\\{1(\\mathrm{mod}\\ {2}),\\ 2(\\mathrm{mod}\\ {4}),\\ 4(\\mathrm{mod}\\ {8}),\\ 0(\\mathrm{mod}\\ {8})\\},", "d892be6e6b4b29dc7eab4c388cc9691f": "x_1, x_2,\\ldots,x_m\\in\\Gamma^{*}", "d892da58e497a44a80c9e5998e102731": "\\frac{\\beta_{n+1}}{\\beta_n} = \\frac{A(n)}{B(n)}", "d892ec60a8dc2662771190993dc9fa21": "\\{g_1, g_2, g_3\\}", "d8931e4835f952df4aadf08eb4b59ce8": "y(t) = C(t)x(t)", "d8932933f53c5ec0c0ee46e38437a26a": "(f * g)(x,y):=\\sum_{x \\leq z \\leq y} f(x,z) g(z,y).", "d89413df4be5cf34a6f9f3ca3b283b42": "{\\tilde{B}}_{n+1}", "d89446fa2e17eb509ed17ef63718ed44": "u_0\\propto V_0 L^2 ", "d8945c96ec466f86b588b1d2b9956d09": "A \\rightarrow B[n]", "d894676027939f76d2c7f43214f69810": "x^2-a^2y^2=1,", "d894772f40c3640ebdc421e8118ab7c2": "\\rho (\\mathbf{x}(t))", "d894a2b2dec7428407adac3bbbe02cbc": "\\forall n \\in \\mathbb{N} : \\nu \\circ f(n,t(n)) = \\nu \\circ t(n).", "d894a6b60e6a26b7230c35fb062ccb74": "\\frac{\\lambda-1}{\\lambda}", "d894ffee78a9c151f6b2745157543a05": "\\operatorname{cov}\\left[\\epsilon^T\\Lambda_1\\epsilon,\\epsilon^T\\Lambda_2\\epsilon\\right]=2\\operatorname{tr}\\left[\\Lambda _1\\Sigma\\Lambda_2 \\Sigma\\right] + 4\\mu^T\\Lambda_1\\Sigma\\Lambda_2\\mu", "d8952081a84a442ebbfbc0efbca529c3": "\\operatorname{E}[X]", "d895317b7addae2f58c77b8efba12a0a": "r_s=(3/(4\\pi n))^{1/3}", "d8959a226334f20062a86ba440b9a23d": " \\partial_0 = \\frac{1}{c} \\frac{\\partial}{\\partial t}, \\quad \\partial_k = \\frac{\\partial}{\\partial {x^k}} ", "d895aa001f1c63cc9127aa300403c98a": "\n\\frac{Eh^3}{12(1-\\nu^2)}\\Delta^2 w-h\\left(\\frac{\\partial^2\\varphi}{\\partial x_2^2}\\frac{\\partial^2 w}{\\partial x_1^2}+\\frac{\\partial^2\\varphi}{\\partial x_1^2}\\frac{\\partial^2 w}{\\partial x_2^2}-2\\frac{\\partial^2\\varphi}{\\partial x_1 \\, \\partial x_2}\\frac{\\partial^2 w}{\\partial x_1 \\, \\partial x_2}\\right)=P\n", "d895cbcb55da3d8ef871e525553bd319": "h(t) = h_{0} - [ (h_{0}-h_{\\infty})(1 - e^{-t/\\tau_h})]\\, ", "d895f253cac90090286259593c2a46a5": "d\\neq 4", "d896102549bbd67521634b493e123b22": "\\delta \\in R^d", "d896a79bd13565fb8c9b6aaea1988d3c": "(\\mathbf{S})\\int_{t_0}^tg(t^\\prime)\\mathrm{d}B(t^\\prime)=\\lim_{n\\to\\infty}\\sum_{i=1}^n\\frac{g(t_i)+g(t_{i+1})}{2}\\left(B(t_{i+1},t_0)-B(t_i,t_0)\\right)\\,,", "d896e290f6f7d73fa786e44ad4e28fc6": " \\sum_{i=1}^n p_i x_i \\leq M.", "d897acda036dd75ecb34c9202111d272": "b_{7}", "d897cff626373219460b12c0c3e9be91": " \\aleph_1", "d897d0d44a3ba16a02c1513b6c626cfb": "\\widehat{\\sigma}^2 = \\frac{1}{n}\\sum ( y_i-\\overline{y} )^2", "d897f13a15e012bbaddf10cd9d9c57b5": "do(a,s)", "d8982abe5d20bcf8ac92a003dbe3723b": "\\deg f(x) < \\deg g(x)", "d89853866a42256ce401f2edd2c89cb8": " V_t \\times F_e = V_a \\times F_a ", "d89881785c3ae8d31a3888310f90e002": "\\mathbf{Q} = {1 \\over {n-1}}\\sum_{i=1}^n (x_i-\\overline{x})(x_i-\\overline{x})^\\mathrm{T},", "d8991b3ca671cbf52cc35667821eff2e": "\\scriptstyle Z_B", "d8995146b686514040b614a5363ac800": "\\delta_{xy}=1", "d89969fa96f66ce117a5e31c64baf7ef": "(\\tfrac{B}{b}) = (\\tfrac{b}{B}) = -1.", "d8997a0d6885470eccf2275fb2520f96": "\\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots, X_0=x_0)=\\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1})", "d8998daac9a8444a5af0cb92a3b04c56": " \\frac{1}{\\sqrt{a}} \\int_{-\\infty}^{\\infty} \\varphi_{a1,b1}(t)\\varphi\\left(\\frac{t-b}{a}\\right) \\, dt", "d899987890073052ca02682d49308d58": "d_i>0", "d899c291bacb233f95bd660120730478": "|a_i|^2", "d89a17462d860256a9d9230f76ce4220": " M(t) = \\hat \\phi(-\\mathbf{i}t) = e^{ \\mu t} e^{\\frac12 \\sigma^2 t^2 }", "d89ac1cc027e21b0ac94db4f25c1f418": "x\\;\\psi_n(x) = \\sqrt{\\frac{n}{2}}\\psi_{n-1}(x) + \\sqrt{\\frac{n+1}{2}}\\psi_{n+1}(x).\\,\\!", "d89ae4e6ebbb8cde21ca113f1cb1faf8": "\\nabla S=0", "d89b00bd05b69c4226df69b6bc983e7c": "a,i=1,2,3", "d89b5607cf34cc9227b11e8619caf154": "R_\\mathrm{L}", "d89b786dd8a1f1edf830f18285b03801": "\\mathcal{O}_{[gh]}", "d89b79e1abc3928b95968875b46aac1f": "\\alpha = \\sqrt {RG}", "d89b95a7ac464f9c8b59ee87e2dca7bd": "\\mathrm{var} (Z)\\propto \\mathrm{E}(Z)^p", "d89b9925cff7572f51aff39e341b73b2": "{\\mathcal L}_{xy}^1: L=kl;", "d89b9eb83473f299efb174df07d88b59": "(\\frac{b}{2})^2 + h^2 = b^2", "d89bce0f44daba848807a9fc02f6a2c8": "\\Psi_{MC} = C_1\\Phi_1 + C_2\\Phi_2,", "d89bd220edecafcf7fed42836fead745": "\n\\mathrm{DR_{dB}} = 6.02 \\cdot 2^m\n", "d89c6721c15ff64884d8746b8a43f274": "F(x,\\lambda\\xi) = \\lambda F(x,\\xi), \\quad \\lambda>0", "d89c6c5d6cf77113e58444f3aabb81cb": " T \\gg 0. ", "d89c6ffefc64984c08ffe7955ed58a6e": "\\vec v_{B|A}=\\vec v_{B}-\\vec v_{A}", "d89c95f30de8bdb86b9c3f36638eb81b": " + P_{iN})", "d89d89f56dcef248615f3d7096d9cbbc": "F(X) \\to \\operatorname{holim} F(H_n)", "d89dbba597a8771155fdf9c60900c03e": "\\widehat{a}(\\widehat{D}(-\\alpha)|\\alpha\\rangle)=0", "d89dd8e6cabc90b2b09378dc729d6aeb": "\\;ord_P(G)=r+s", "d89e0bb1ef4aeffcac0455d65d5f221b": "\\sec^2\\left(\\frac{A}{2}\\right) : \\sec^2 \\left(\\frac{B}{2}\\right) : \\sec^2\\left(\\frac{C}{2}\\right)", "d89e41ab64e5ed3b14bd9924dcbd6d1b": "O(N\\,\\log N)", "d89e610d6dea77a20e03bec9f2ce7488": "f_1(k) = 1 \\quad\\text{and for}\\quad d|k,\\; d2 \\\\\n \\text{Indeterminate} & \\text{otherwise}\\ \\end{cases}", "d8ad9d508e0f870d9263629fdaadd601": " f(x)=x^{-\\alpha} ", "d8adba910014513f94558f3dd04ef4b1": "\\bar{g}(\\theta)", "d8ae026480eb3cfc40b5801197cbc7de": "\\scriptstyle f_\\mathrm{red}\\,", "d8aec1d07cb0987d2c66fb1aa4ca8a35": "\\mathrm{resultant}(h, T) = \n\\mathrm{resultant}(\\cdots(\\mathrm{resultant}(h, t_s),\\ldots, t_i)\\cdots)\\neq 0", "d8af3ce57631dcbd84f4362afcf5c511": "T(e_i) = \\sum_{k=1}^{r+s}f_k\\,{T^k}_i.", "d8af655efb9142cc98d5d1b1211b4ff5": "\\scriptstyle{Z_{ij}}", "d8af8e9b0c7251c4ee6e4202d5c143f7": "\\frac{dy}{dx} = e^{\\sin {x^2}}\\cdot\\cos{x^2}\\cdot 2x.", "d8af933054a2ad515e09e05b002624e4": "R_M = R_\\infty\\times(1-m_{\\text{e}}/M),", "d8afd9b4062307e71f2197e7bc1e5fe7": "n_{s} = \\sum y_i", "d8b01518a3434c61a02a006428e4169e": "c_\\mathrm f\\,", "d8b05162e02becc213e52adc80ff32b3": "(1+x)^{3}p(\\frac{x}{1+x}) = x^3+7x^2+14x+7", "d8b06d8a795f7d0fa12a4cdeadf020a9": "n \\equiv 2^a \\not\\equiv 0 \\pmod 3.\\,", "d8b072422fa6c44fa4716c65caa4af4c": "d^2 + h^2 = a^2.\\,", "d8b0756b251f2b6e25f88517b7dda7fb": "N_{E}/N_{NE}\\approx H_{E}/H_{NE},", "d8b09d44906b5f64e45c74f4183f28be": "\n\\beta = i\\gamma^0 = \\biggl(\\begin{matrix}\n0 & I\\\\ \nI & 0\\\\\n\\end{matrix}\\biggr),\n", "d8b0a189600db750ad996224e3ed4cda": "1 - 1/74 = 99\\% ", "d8b0a86188e5aecaccb34cab8a5047b9": "f(A)=C^{-1}f(J)C=C^{-1}\\left(\\bigoplus_{k=1}^N f\\left(J_{\\lambda_k ,m_k}\\right)\\right)C", "d8b0d65a81f4c0eb44801d627d2e18f5": "\\left(\\nabla_XY\\right)^k=X^i\\left(\\nabla_iY\\right)^k=X^i(\\partial_iY^k+Y^j\\Gamma_{ij}^k)", "d8b121ef7a4ac043a3b1daa872d2f9a7": " <_p", "d8b15a52fc26a35b3f40806b883da854": "\\tilde{\\theta}", "d8b1e689cc9766ad55355436e4f73604": "\\exp_{10}^3(1.09902)", "d8b21b3e6f50e63e66d6663f577783d3": "f(x)=g(x)^{h(x)}\\,\\!", "d8b226afdb21b44a07419b8e966dcf31": "\n\\begin{bmatrix}\nU'_0 \\\\ U'_1 \\\\ U'_2 \\\\ U'_3\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\gamma&-\\beta \\gamma&0&0\\\\\n-\\beta \\gamma&\\gamma&0&0\\\\\n0&0&1&0\\\\\n0&0&0&1\\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nU_0 \\\\ U_1 \\\\ U_2 \\\\ U_3\n\\end{bmatrix}\\ \n", "d8b22cbe639a254e499f00aeb1180982": "b_0=\\frac{{n_0}}{{n_0 M_0}}=M_0^{-1}.", "d8b2cce72d2ecc8be949888bb9bd14ce": "F_2= mg. \\tan \\theta_2 \\,\\!", "d8b3283ff4f6d4e3a0d67b104155a65f": "s=y\\left(1+\\frac{T}{E}\\right)p,\\,", "d8b36c1bd9df6649945767292214fae0": "y(\\sigma)=\\sum_{k=0}^\\infty \\sigma_{-k-1} 2^{-(k+1)}.", "d8b460a7c76bf2f30987647810ebdd8c": " p_i^{(\\infty)}(n) ", "d8b4ea06316bebe935337694ac95f204": "\\mathrm{Sh}_D", "d8b4f17d806451e8d2eaa1d72aca53c8": "\\begin{align}\ny'&=\\sum\\limits_{r=0}^{\\infty }{a_{r}(r+c)s^{r+c-1}}\\\\\ny''&=\\sum\\limits_{r=0}^{\\infty }{a_{r}(r+c)(r+c-1)s^{r+c-2}}\n\\end{align}", "d8b503974b9152703c4f370d92e2d7e8": "\\mathrm{diam}(Q_j) \\le \\text{dist}(Q_j, \\partial A)\\le 4\\,\\text{diam}(Q_j). \\, ", "d8b5369ec798e917b148dbda35c52429": "\\Delta X_{t}=\\mu+\\Phi D_{t}-\\Gamma_{p-1}\\Delta X_{t-p+1}-\\cdots-\\Gamma_{1}\\Delta X_{t-1}+\\Pi X_{t-1}+\\varepsilon_{t},\\quad t=1,\\cdots,T", "d8b549654d1be42f3207da298cb17de8": "t^2-xt-1=0.\\,", "d8b5a0a2055f2024f7d8e482d94dae05": "g(S(t),t)", "d8b5aa18bee9fa82ff46cd0988638087": "\\mathbf{\\Theta} = \\operatorname{atan2}\\left({ \\mathbf{G}_y , \\mathbf{G}_x }\\right)", "d8b60e9622e094bfe8561244a0ce6df2": "\\Psi(z_{u_1}...z_{u_i})", "d8b64d6b8274144d237a266d191f61c9": " (q-1,q+1)", "d8b6a0116a4bf7b02bb89c03ff5973f9": "\\gamma >1", "d8b6ba4a604d1a5b4d8a7aeeaa94f93f": "u_{in},u_{out} ", "d8b6f05533cc21ea33adea67e146d95a": "\\zeta(2)=\\pi^2/6.", "d8b7000c0d4c30e91b6f9e2d4d644acc": "(10\\uparrow)^{255}", "d8b718dbe246e81a319f9111834dd3ef": "N\\,\\! ", "d8b7e20376955bbdc78ed02323930c6b": "\\underline{\\underline{\\boldsymbol{\\sigma}}} = \\begin{bmatrix}\n\\sigma_{11} & \\sigma_{12} & 0 \\\\\n\\sigma_{21} & \\sigma_{22} & 0 \\\\\n 0 & 0 & \\sigma_{33}\\end{bmatrix}\\,\\!", "d8b8315f84106db6405745ae37accbf0": "\\mathbf{E}\\cdot d\\mathbf{S} = \\frac{Q}{\\epsilon_0}\\,\\!", "d8b8a95f854dc0290426a383903fb8f1": "f \\colon B \\to C", "d8b8dc20b4c6dcda5fc8e90d2a17b617": "\\alpha \\twoheadrightarrow \\beta", "d8b93fa1e4683cbc33449a9cdaae6626": "\\cos nx = 2 \\cdot \\cos x \\cdot \\cos ((n-1) x) - \\cos ((n-2) x) \\,", "d8ba0b86d2a0d602b2d4af1e3a9e1824": "\\left( \\theta^{G}\\right)_K = \\sum_{ t \\in T} \\left([\\theta^{t}]_{t^{-1}Ht \\cap K}\\right)^{K},", "d8ba0cb587bef40f8a421d06d9a1d51c": "\\displaystyle \\sin ( a x^2 ) \\,", "d8ba196769fbe176f78a40e7e89e3323": "A \\in \\mathcal{A}", "d8ba25f616133596659efc87d64f45e4": "\\vdash \\varphi\\quad\\&\\quad \\vdash ( \\varphi \\rightarrow \\psi )\\quad\\Rightarrow\\quad \\vdash \\psi", "d8ba2f2913465ed93f5a19b57b2a034e": " R(2\\pi)\\phi(-x) R(\\pi)\\phi(x) \\,", "d8ba4358df3bff93e52b5def4ca895f3": "q\\,{(4a)_{+/+}}=\\frac{{\\Omega }^{10} - \n 2\\,{\\Omega }^4\\,\n {\\mho }^2 + {\\mho }^3}\n {{\\mho }^2\\,\n \\left( -{\\Omega }^4 + \n \\mho \\right) }", "d8ba4afe808679595d07bfca1e43afc6": "y'", "d8baf44ddd2f3f847f8bd321c217c9dc": "\\phi\\ (r) \\ge 0 ", "d8bb3a8d2ed4954fed1bc8eb71c88bdb": "\\nabla u(x,y) \\cdot \\nabla^\\perp u(x,y)=0 , \\rVert \\nabla u\\rVert =\\rVert \\nabla^\\perp u\\rVert", "d8bbc7c3dc6e8099f389000b9f150d54": "\\int_S \\phi(x')\\nabla' G(x,x')\\cdot d\\hat\\sigma' = \\langle\\phi\\rangle_S ", "d8bc1152828a5b9a7ae2c4e33d5810b1": "dy=\\frac{dy}{ds}ds+\\frac{dy}{dt}dt", "d8bcf5877dffa88bcfe363540585eef9": "f_0(z) = \\,_1F_1(a;b;z)", "d8bd55193406b434d6b06a43ac969d5c": "\\mathbf{z}=[x_1\\ q_1\\ x_2\\ 0\\ 1\\ 0\\ 1\\ 0]^T", "d8bd95fdefe801fffd0cd34456134d31": "D(X + r) = D(X)", "d8be1f0b29e9f32fffcc80a273181193": "g(q)=\\wedge", "d8be210fc5ec9a78ab5c271cb3f7392d": "\\varepsilon{}_n", "d8bebdd7ddc31cb686dd2a1385ad4471": "\\frac{d^c u^c u^c e^c}{\\Lambda^2}", "d8bedb11e2e6565719e8d7ba37521f65": "d\\alpha=d\\beta=0", "d8bedf34220adc0d0d73e02ed7497b19": "M_1...M_{p+1}=id", "d8bf0f987205c404539c03f5742d66f4": " \\Omega^{\\bullet}_{X-D} ", "d8bf26327f21a5e8278c2ecc11199dac": " c_{2j}=\\frac{ 1 \\cdot 3 \\cdot 5 \\cdots (2j-1)}{2^{j+1}}=\\frac{(2j)\\,!}{j!\\, 2^{2j+1}} \\ . ", "d8bf9084ac96a98c504ce020f9d45502": "E = \\frac{3}{5} \\left( \\frac{1}{4 \\pi \\epsilon_{0}} \\right) \\frac{Q^{2}}{R} = \\frac{3}{5} \\left( \\frac{1}{4 \\pi \\epsilon_{0}} \\right) \\frac{(Ze)^{2}}{(r_0 A^{\\frac{1}{3}})} = \\frac{3 e^2 Z^2}{20 \\pi \\epsilon_{0} r_0 A^{\\frac{1}{3}}} \\approx \\frac{3 e^2 Z(Z - 1)}{20 \\pi \\epsilon_{0} r_0 A^{\\frac{1}{3}}} = a_{C} \\frac{Z(Z-1)}{A^{1/3}}", "d8bff1a6b04148ad8b3b57101d6748a8": " \\frac{N}{K}", "d8bff3e365722df77e64633fc14c5035": "Fr=\\frac{v}{\\sqrt{gy}}", "d8c003139176b3a503876b2abf58a3b4": "N_1+N_2 = N \\,", "d8c034f2e7365a6c020ce672c6b3367e": "\\begin{align}R_{\\frac{\\lambda}{2}}\n&= \\frac{Z_0}{2\\pi} \\left[\\ln(2\\pi\\gamma)-\\operatorname{Ci}(2\\pi)\\right] = \\frac{Z_0}{4 \\pi} \\operatorname{Cin}(2\\pi) = 29.9792458 \\int_{0}^{2\\pi} \\frac{ 1-\\cos(\\theta)}{\\theta} d \\theta,\\\\\n&\\approx 73.0790102 \\ \\Omega;\n\\end{align}\\,\\!", "d8c04b9805f62f81c5cf38845c6a6e39": "0.99 \\times 0.01", "d8c04fb8329bd1aa506311d6fabb2385": "\\log(p)", "d8c065098603b94ef35a3ed6c83f20f3": "C^N(L) = \\{0\\}", "d8c091b9aa6118ffbe488b5633cc8d00": " {dx \\over dt} = x \\alpha y - \\beta x^2 - \\beta x(1-x-y) ", "d8c0f4d2745dead861333b4d7b92fc63": "v_1^2+v_2^2+v_3^2", "d8c104b7341a9d0e49decdb567766975": "\\Phi_\\Lambda", "d8c1109d681afcadc115eb00f788f7cf": "\\max(3,d)^{\\min(n,s)}.", "d8c1f33cfd398ff860c05dc5cecb3161": " \\begin{align} \n\\frac{\\partial f}{\\partial \\mathbf{v}}(\\mathbf{x}) &= \\lim_{t \\to 0} \\frac{f(\\mathbf{x} + t\\mathbf{v}) - f(\\mathbf{x})}{t} \\\\\n&= \\lim_{t \\to 0} \\frac{\\int_{\\gamma[\\mathbf{a}, \\mathbf{x} + t\\mathbf{v}]} \\mathbf{F}(\\mathbf{u}) \\cdot d\\mathbf{u} - \\int_{\\gamma[\\mathbf{a}, \\mathbf{x}]} \\mathbf{F}(\\mathbf{u}) \\cdot d\\mathbf{u}}{t} \\\\\n&= \\lim_{t \\to 0} \\frac{1}{t} \\int_{\\gamma[\\mathbf{x}, \\mathbf{x} + t\\mathbf{v}]} \\mathbf{F}(\\mathbf{u}) \\cdot d\\mathbf{u}\n\\end{align}", "d8c2103518d6aa4fbf693973eb8b3881": "10 \\div x", "d8c226e759cf0fe11772bc0261ec344a": "s(F)", "d8c24963b37de56b49fcfdb47b5ec231": "\\!\\,SUBCLU(DB, eps, MinPts)", "d8c27b7b1d33a55ae6672956c750c9bc": "\\begin{array}{lll}\n\\hat m_{pq} & = & \\alpha^p+q+2\\int_{-1}^{1}\\int_{-1}^{1}[(x-x^c)cos(\\theta)+(y-y^c)sin(\\theta)]^p\\\\\n & = & \\times [-(x-x^c)sin(\\theta)+(y-y^c)cos(\\theta)]^q\\\\\n & = & \\times f(x,y)dxdy,\\\\\n\\end{array}\n", "d8c3432efa8a404ad53fa989d8622b76": "\n\\begin{align} \n \\phi &= \\operatorname{arctan2}(A_{31}, A_{32})\\\\\n \\theta &= \\arccos(A_{33})\\\\\n \\psi &= -\\operatorname{arctan2}(A_{13}, A_{23})\n\\end{align}\n", "d8c38212f7229bad6a84ed6990662d77": "g(x) = m_1(x) = x^4+x+1.\\,", "d8c39226db6cd06dba5976dcb36aff9d": "x = y\\epsilon^{-\\nu}", "d8c3d71966e545fbea522752f5e5486b": "mr { \\dot{\\theta}}^2 \\ . ", "d8c43c58ccc479670095603c5e824284": "I_x(a,b)", "d8c45e7cb4834d9b1fec68bb0588fa35": "\\scriptstyle\\mathbb{Z}", "d8c4b0c2dd3bba62f1dcc77325001852": " \\Delta \\theta = \\frac{\\Delta A}{|d\\langle A\\rangle / d\\theta|} = \\frac{1}{N}. ", "d8c4cbcc77cc639d4e800b6cfd7b0b22": "\\pi_{n,a}(x)", "d8c4d3dbaf503eefa0e78001be45d002": "Ca^{2+}", "d8c4fa16b9b347b5ea32f231276bb1f5": "(N + 1)/p = \\mathit{p}^{k-1}", "d8c51dad1c7e49c92ad49f05b3fd2101": "\\langle x_1,\\ldots,x_n \\rangle \\in R", "d8c51fbe455374328629fe1a836d79f1": "x \\in [0; +\\infty)\\,", "d8c558e150c8d69c5839304d0252d7dc": "\\frac{\\mathrm{d}^2 y}{\\mathrm{d}x^2} + (1-2k^2) y + 2 k^2 y^3 = 0", "d8c5eb75d7efa6f73ed0d5c438ada5c1": " \\delta>0 ", "d8c6191759faad6f08c8278524f4641b": "P=(-1)^{\\sum_i l_i}\\ ,\\!", "d8c64dffafafc207cacc1e92e9b2bb7e": "V = (V_0, V_1,....., V_{n-1})", "d8c695eae90273f9c8aaa40f90be287d": " H(x) = \\mathbf{1}_{(0,\\infty)}(x).\\,", "d8c715e0826ebdfd3b592028ea0daa26": "G = k A / x \\,\\!", "d8c744808e91e4d0b483d39e1ff8274b": "\nz=2\\lambda e^{-\\left(x-x_e\\right)}\n\\text{; }\nN_n=n!\\left[\\frac{a\\left(2\\lambda-2n-1\\right)}{\\Gamma (n+1)\\Gamma (2\\lambda -n)}\\right]^{\\frac{1}{2}}\n", "d8c74841dfaf1e644a0731c128ccd95e": "t_{jj'}", "d8c7a10f3b2b573e41ae6a25cda7cc24": "\\sigma_{ij} = \\begin{pmatrix}\n\\sigma_{xx} & \\tau_{xy} & \\tau_{xz} \\\\\n\\tau_{yx} & \\sigma_{yy} & \\tau_{yz} \\\\\n\\tau_{zx} & \\tau_{zy} & \\sigma_{zz}\n\\end{pmatrix}", "d8c7a13162dc009fff0905852d2dec07": "C_\\text{sr}=((155.4\\,\\text{ES})^5-(30.4\\,\\text{ES})^4-(43.3\\,\\text{ES})^3+(46.3\\,\\text{ES})^2+19.5\\,\\text{ES} + 3.6)\\,\\text{EM}", "d8c7c1efe83dc378c7b82f1cb3697f70": " \\begin{align} \\Delta E & = \\zeta (L,S) \\{ \\mathbf{L}\\cdot\\mathbf{S} \\} \\\\ \n\\ & = \\ (1/2) \\zeta (L,S) \\{ J(J+1)-L(L+1)-S(S+1) \\} \\end{align} ", "d8c7e527cb48c8ef347b218b57313232": "\\frac{\\partial C_i (q_i)}{\\partial q_j}=0, j \\ne i", "d8c81c36157c010ae896df674cb80ef4": "0\\leq x\\cdot y", "d8c82d55987298b9a8aab64f3adc71d4": "\\qquad \\sum_{j \\in J} v_j\\,x_j \\ \\ge v_i", "d8c8644d14f1cf7c836fc0f96a5fdc29": "rowgroups", "d8c8685dc8e36ea6786ea83e93ac5194": "\\psi\\ {\\sim}\\ \\psi + 4\\pi.", "d8c899f57be03b4226f12979702784b1": "A(w)=\\sum_{n=0}^\\infty a_n w^n \\mbox{ with } a_0 \\ne 0. ", "d8c8ba936acfcc6118eceb399c5c67dc": "\\sin 36^\\circ = \\cos 54^\\circ = \\frac{\\sqrt{10 - 2\\sqrt5}}{4}", "d8c8d35a1c71c1d12d7f0572aed1ecd9": "\\frac{ds^2}{dx^2}=1+\\frac{dy^2}{dx^2}", "d8c948ea807434e50018e63b7f0e872d": "V_\\mathrm{pp}", "d8c975c6186564373272f445e2e5840d": "\\mathrm{NO}_2", "d8c984f01e863587cea6ac15a428dd01": "[M:K] = [M:L] \\cdot [L:K].", "d8c990fc8a9256c7a92a2841836e3b73": "~\\Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1", "d8c99219ba6d34973ed90332e2a10f00": " 0.9890559955", "d8c99e81449bcd43620355e0d4d4b2fe": " f_k(x) = x^{k} \\left [(x-1)\\cdots(x-n) \\right ]^{k+1},", "d8c9b3b1f9b1449d9e66f0195baa6551": "\\ln{[A]}", "d8c9e59bda117a401dd1d18e40b3dfa9": "c \\equiv b^{e'} \\pmod{m}", "d8c9e716c2655de9d3d1933e45e31a6b": "x^\\prime = k\\ell\\left(x + \\varepsilon t\\right), \\qquad y^\\prime = \\ell y, \\qquad z^\\prime = \\ell z, \\qquad t^\\prime = k\\ell\\left(t + \\varepsilon x\\right)", "d8c9f508c83fddf8f0d5d76fd41eac28": "E[\\tilde{x}^2] = \\frac{x_{max}^2}{3\\times4^\\nu}", "d8caa6ec4d842ed27716ddc6aa444d0f": "S, T \\in Q", "d8cbce4319d02f54e8f40b623e537a00": "|\\psi(t)\\rang = c_+ |+\\rang + c_- e^{\\frac{-i(E_{-}-E_{+})t}{\\hbar}} |-\\rang,", "d8cbfa5f1660cbb839b521e32d92d47c": "d = 1 \\textrm{\\ AU} \\cdot 180 \\cdot \\frac {3600} {\\pi} \\approx 206,265 \\textrm{\\ AU} \\approx 3.2616 \\textrm{\\ ly} \\equiv 1 \\textrm{\\ parsec} .", "d8cbfe202e7319d9b81f0655134a23da": "\\psi'(t)+t\\psi(t) = 0", "d8cc0ed370e99ea753d8a9d6e32704b4": "t^\\alpha = \\frac{dx^\\alpha}{d\\tau}.", "d8cc3070b622e03d4131c33224ab5767": "\\sum_{i=1}^n \\log(i)^c \\in \\Theta(n \\cdot \\log(n)^{c})", "d8cc76909929ed060859869d5e6bd1c1": "\n \\int (d+e\\,x)^m (A \\left(b\\,c\\,d\\,e (2 p-m+2)+b^2 e^2 (m+p+2)-2 c^2 d^2 (3+2 p)-2 a\\,c\\,e^2 (m+2 p+3)\\right)-\n", "d8ccd0c54e8e49872386ac710c988cf7": "r^2 = x^2+y^2", "d8ccea2839b469a3819cd61f01d1fdba": "d_p \\rightarrow \\infty", "d8ccfd5b6c101c6fa088f6f7cec064ed": " z'_2 = \\frac{\\left(z'_1+1\\right)}{2}", "d8cd257b8eb6142cf88515e6a6c5ed30": "\\varphi(p)=q.", "d8cd31879453c5d47cd766fb12d816e4": "E(v_m+1) - E(v_m)", "d8cd687479f56a3d18184fc61afe518d": "\\nu:x \\mapsto (x,x^2, \\ldots ,x^n).", "d8cd7ec79694544bf55e756132a63e43": "\\left( z_{1},\\ldots\n,z_{n}\\right) ", "d8cdb5e37ffeece8fc183a8737e0ffc0": "U_\\mathrm{E} = \\frac{1}{2}\\sum_{i=1}^N q_i \\Phi(\\mathbf{r}_i)", "d8cdb8fb6b71e43b613dbdd6e718fe60": "T_{min} = t_r + \\frac{kV}{2} \\left(\\frac{1}{a_f} - \\frac{1}{a_l} \\right)", "d8cdf53b33f4860343f05d8dc5721562": "\n E\\{H_i x(n) G_j x(n)\\} = 0;\\qquad i < j\n", "d8cdffa7e5362dd8e399a92fc7533412": "2. \\ \\exists K_0, K_1 \\text{ such that } \\left| \\frac{\\partial u / \\partial x_k}{\\partial u / \\partial t} \\right| \\leq K_0 + K_1 |t|", "d8ce1b9f5c45cc7584fc0a70527f5b3b": " y'(t_0) \\approx \\frac{y(t_0+h) - y(t_0)}{h} ", "d8ce45ff535c7e6bae31f1cb7c9a02e5": "*_a : H \\otimes H \\to H", "d8cee84220666bbdc24ab3a8e0daa0dd": "a\\in\\mathcal{A}", "d8cfe5ebe9d0dcf29844a9d520021582": "\\nabla(z) = z^4 + z^2 + 1, \\, ", "d8d0741ace43a563d7f8f8e7a2c1cec5": "-0.25 \\le \\lambda < 0", "d8d0c37cf0aa95c8d0a34e2370801795": "\\sin(\\alpha \\pm \\beta) = \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\,", "d8d0dba762fd45f110659941ca86eef1": "{P \\over Q} = {{Y + \\sigma (l_2 + \\alpha )} \\over {X + \\sigma (100 - l_2 + \\beta)}}", "d8d10cbec3ab7e47eca441bbbb630fd5": "T_{\\mu \\nu} g^{\\mu \\nu} \\, = \\, 0 \\,", "d8d11da7a069542279e81cbb89efdfd7": "f(+\\infty)=0 \\therefore A = \\arctan (+\\infty) = \\frac{\\pi}{2} + m\\pi,", "d8d1411df8abb1f971804b6fb657fde7": "8_{S-} \\times 8_{S-} \\,", "d8d1fbe749634190f03cb14efb2720b0": "A_{n-1} \\cong S_n", "d8d2710404edc4d54756f28f05e9ea8d": "\\sgn\\tau\\,= (-1)^{2n-(i+j)} \\sgn\\sigma'\\,= (-1)^{i+j} \\sgn\\sigma.", "d8d2a20982114f1c669636c00138acd6": "S_s\\, \\dot{S}_t", "d8d304e789f545b39970fa2f639c3e41": "\\displaystyle\\underbrace{x\\diamondsuit\\cdots\\diamondsuit x}_{k\\ \\mathrm{factors}}.\\,", "d8d314922156e86701d466ffca6c6da5": "T F \\square \\square = 1 ", "d8d3433ce2a3480964fbe54579cfbc87": "(10^8)^{(10^8)}", "d8d3c4404850974c4f7dc52e3b0e782a": "n_{d,s}", "d8d3f7d3410939285a7ade48d3908871": "nH_0(s) + o(|s|\\log \\sigma)", "d8d41ada5b3e0ae3c5dc24a3d1657372": " P = 370 + \\left( {21.6 \\cdot LBM} \\right)", "d8d45a25a496182c81bd55133fc03622": "\nF_A^{(2)} \\equiv F_2 ,\\quad F_B^{(2)} \\equiv F_3 ,\\quad F_C^{(2)} \\equiv F_4 ,\\quad F_D^{(2)} \\equiv F_1.\n", "d8d461c8f95ed15f894e8f057e2121e7": "\np_{dyn} = \\rho \\cdot g \\cdot H/2\n", "d8d46990862256338c77fb5ba7f698ba": "\\textstyle f(x, y)", "d8d48c81df83e10ee90bee1259f7d6bb": "z^5 + 3z^3 + 7", "d8d4a055117bc20d1f4a2b5358b4f069": "{5 \\choose 1}{4 \\choose 3}{4 \\choose 1}^4 = 5,120", "d8d4d4714a4ef66dc129bc0c72c9d420": "3*10^8", "d8d4d691fd6cad95185da6c5850461a1": "E = V \\to \\operatorname{sink}[(\\lambda E.V)\\ Y, X] = Y ", "d8d4e3a982810e0ae7e0591f9c66a6f0": "Z_M = \\frac{\\dot N}{\\dot {I}_M} = \\frac{\\dot {N}_m}{\\dot {I}_Mm} = z_M e^{j\\phi}", "d8d4fec799e64395c4a300b9fd6242b3": "\\frac{\\theta \\vdash \\psi \\quad \\phi \\vdash \\psi}{\\theta \\vee \\phi \\vdash \\psi}", "d8d50263bf45623a5a5cbb90ad16dd45": "\\left(\\int_{-\\infty}^\\infty x^2 |f(x)|^2\\,dx\\right)\\left(\\int_{-\\infty}^\\infty \\xi^2 |\\hat{f}(\\xi)|^2\\,d\\xi\\right)\\ge \\frac{\\|f\\|_2^4}{16\\pi^2}.", "d8d5190bd275c970944c56ad8d4d3cd1": "\\mathbf{E}=\\sum_{l=0}^\\infty\\sum_{m=-l}^l\\left(E^r_{lm}(r)\\mathbf{Y}_{lm}+E^{(1)}_{lm}(r)\\mathbf{\\Psi}_{lm}+E^{(2)}_{lm}(r)\\mathbf{\\Phi}_{lm}\\right)", "d8d55f88c850c5db599c68f9f3633526": " \\int_{-\\infty}^{\\infty} x^4 e^{-{1 \\over 2} a x^2}\\,dx = \\left ( -2{d\\over da} \\right) \\left ( -2{d\\over da} \\right) \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} a x^2}\\,dx = \\left ( -2{d\\over da} \\right) \\left ( -2{d\\over da} \\right) \\left ( {2\\pi \\over a } \\right ) ^{1\\over 2} = \\left ( {2\\pi \\over a } \\right ) ^{1\\over 2} {3\\over a^2}", "d8d5a37de839a2f3239a2ec494924d48": "\nF_D^{(3)}(a,b_1,b_2,b_3,c;x_1,x_2,x_3) = \n\\sum_{i_1,i_2,i_3=0}^{\\infty} \\frac{(a)_{i_1+i_2+i_3} (b_1)_{i_1} (b_2)_{i_2} (b_3)_{i_3}} {(c)_{i_1+i_2+i_3} \\,i_1! \\,i_2! \\,i_3!} \\,x_1^{i_1}x_2^{i_2}x_3^{i_3}\n", "d8d5c40c61475439515bc813ce339547": "\\bigvee A=a_1\\lor\\cdots\\lor a_n", "d8d6c93844eaf93aff1270df9b200754": "\\Psi=A \\frac{e^{i\\left( k (\\frac{x^2}{2 L}+L)-\\omega t +\\phi\\right)}}{L}\\sum_{n=0}^{N-1} e^{-i k \\frac{x n a}{L}}", "d8d6fb9b2a8d8cae85fd332358649c16": " \\theta = \\frac{ s^2 }{ m } ", "d8d700da3fdd230f6e04442c65c84943": "\\frac{1}{2\\pi fX_C} = C = 24.78\\ pF", "d8d72b997729507cbf971c79de5819d5": "{\\frac{U_y}{U_{max}}} = \\left [ 1 - \\frac{Y}{R} \\right ]^{1/n}", "d8d77d2960f9677702fba344199f5ef1": "p(\\boldsymbol\\theta|\\mathbf{D})", "d8d7bf4d546b0f5593be89c3af7582b8": "g_{00} = 1, \\quad g_{0 \\alpha} = 0,\\,", "d8d7e2bfff870a9422cb5ab7749c43ca": "\\gamma_\\mathrm{L} ", "d8d7e5e64ec3bdf64f44f749d4845130": "\\delta t = \\sum_r \\epsilon_r T_r \\!", "d8d8128d01eba06c184ffb5a97cc7669": " c = f(\\Phi,u,w,\\theta) \\text{ all mapped into the unit square.} \\, ", "d8d814357635c6766e65606d65f216f9": "U_x", "d8d86f0717585ee12e52db11ba85bad5": "\\log{(\\operatorname{E}(Y|x))} = \\log{(\\text{exposure})} + \\theta' x", "d8d8d0c148836896508dd88622981054": "A \\mathrm{e}^{\\mathrm{i}\\mathbf{k}_{\\mathrm{in}} \\cdot \\mathbf{r}}", "d8d8d2d870ae1369eae9a7e62445df35": "\\dots\\to H^n(A,\\mathfrak{g}(A))\\to H^n_{def}(A)\\to H^{n-1}(A,\\nu(A))\\to H^{n-1}(A,\\mathfrak{g}(A))\\to\\dots", "d8d8d54d8045a06125dac97bb3979079": "\n \\begin{align}\n I_1 =\\ & \\eta_{20} + \\eta_{02} \\\\\n I_2 =\\ & (\\eta_{20} - \\eta_{02})^2 + 4\\eta_{11}^2 \\\\\n I_3 =\\ & (\\eta_{30} - 3\\eta_{12})^2 + (3\\eta_{21} - \\eta_{03})^2 \\\\\n I_4 =\\ & (\\eta_{30} + \\eta_{12})^2 + (\\eta_{21} + \\eta_{03})^2 \\\\\n I_5 =\\ & (\\eta_{30} - 3\\eta_{12}) (\\eta_{30} + \\eta_{12})[ (\\eta_{30} + \\eta_{12})^2 - 3 (\\eta_{21} + \\eta_{03})^2] + \\\\\n \\ & (3\\eta_{21} - \\eta_{03}) (\\eta_{21} + \\eta_{03})[ 3(\\eta_{30} + \\eta_{12})^2 - (\\eta_{21} + \\eta_{03})^2] \\\\\n I_6 =\\ & (\\eta_{20} - \\eta_{02})[(\\eta_{30} + \\eta_{12})^2 - (\\eta_{21} + \\eta_{03})^2] + 4\\eta_{11}(\\eta_{30} + \\eta_{12})(\\eta_{21} + \\eta_{03}) \\\\\n I_7 =\\ & (3\\eta_{21} - \\eta_{03})(\\eta_{30} + \\eta_{12})[(\\eta_{30} + \\eta_{12})^2 - 3(\\eta_{21} + \\eta_{03})^2] - \\\\\n \\ & (\\eta_{30} - 3\\eta_{12})(\\eta_{21} + \\eta_{03})[3(\\eta_{30} + \\eta_{12})^2 - (\\eta_{21} + \\eta_{03})^2].\n \\end{align}\n", "d8d9359676bc0baf2eada7e05e3a68b3": " p_n = n (\\ln (n \\ln n) - 1) + \\frac {n (\\ln \\ln n - 2)} {\\ln n} + \nO\\left( \\frac {n (\\ln \\ln n)^2} {(\\ln n)^2}\\right).\n", "d8d937f6b31256e9403a4f007a7b3d01": "P = Q + E + \\Delta S", "d8d971db473048add1ffd0d006a4af32": "\\sum_{i=1}^n\\rho(x_i, \\theta),\\,\\!", "d8d9ca78df2bb51934725df16633cb6a": "\\lceil x \\rceil ", "d8d9d88e8e74e8eaf363261ea58ce0b5": "R_X+R_Y=10", "d8da26d3cae55e4e941edadcd749f0bd": "k(T) = a(T) \\cdot c_P(T) \\cdot \\rho(T)", "d8da70494ba6fd7e8441fe6943ad2217": "\\overrightarrow{ab} = \\vec{v} ", "d8da835d01d6f5a313de3972f3463796": "\\langle P\\circ h|Q\\circ h\\rangle = \\langle P|Q\\rangle", "d8da8fd0b941b5944bba5fce99976a40": " \\mathrm\\varphi (\\mathbf{r}) = \\frac{1}{4\\pi\\epsilon_0}\\int \\frac{\\rho (\\mathbf r' )}{|\\mathbf r - \\mathbf r'|}\\, \\mathrm{d}^3\\mathbf r'", "d8daeee8703becbb1d09668747e447f8": "\\textstyle {N}_1,{N}_2\\dots", "d8db682d06ca29345eeeb3f55e288760": "(\\mathrm{Con_c}\\,L_i,\\mathrm{Con_c}\\,f_i^j\\mid i\\leq j\\text{ in }I)", "d8db82bf95deb8931c1c42f65c49c521": "\\int x\\arccot(a\\,x)\\,dx=\n \\frac{x^2\\arccot(a\\,x)}{2}+\n \\frac{\\arccot(a\\,x)}{2\\,a^2}+\\frac{x}{2\\,a}+C", "d8db9bc4f8a7cb00fb2c08a00c8ed6cd": "(\\hat{g_n}(u_n))_i = \\frac{J(u_n+c_n\\Delta_n)-J(u_n-c_n\\Delta_n)}{2c_n(\\Delta_n)_i}.", "d8dbb1f326bd922f61292e3c3a9cfdaf": "\\left( 0, n+\\alpha+ (n-1) \\sqrt{n+\\alpha} \\right].", "d8dbfb2c34418ce56dfc0a79020e8d96": "w^i := Tr^i,\\,", "d8dc0b9f6bbc5ac5a4e3f13768bffd88": "\n \\gamma_1 = \\operatorname{E}\\Big[\\big(\\tfrac{X-\\mu}{\\sigma}\\big)^{\\!3}\\, \\Big] \n = \\frac{\\mu_3}{\\sigma^3} \n = \\frac{\\operatorname{E}\\big[(X-\\mu)^3\\big]}{\\ \\ \\ ( \\operatorname{E}\\big[ (X-\\mu)^2 \\big] )^{3/2}}\n = \\frac{\\kappa_3}{\\kappa_2^{3/2}}\\ , \n ", "d8dc37c13ea5428c474b78415670e9ed": "\n\\mbox{OCS}=r(\\lambda) \\frac{D^4}{1.4876 \\lambda^2}\n", "d8dc740966316928ce06cb143d6fcc4e": "dm = M(\\varphi) \\, d\\varphi", "d8dc7eea5f264f1239d4f11be3939a80": "(ab)", "d8dcd7ee17022fdbd0b9632893d65dc8": "B_\\nu(T) = \\frac{u_\\nu(T)\\,c}{4\\pi},", "d8dcdbb08c2db4ad8efaacaa021e9a4b": "\\phi^2=Im{\\phi}", "d8dce77478ef419ac01d951fea6628df": "s=\\left(\\sum_{i=1}^N |k_i|^\\alpha\\right)^{1/\\alpha}.\\,", "d8dcf69be74b039ab8fe59a8580de4dc": "t(d,n) \\leq \\mathcal{O}(d^2 \\log^2{n}) ", "d8dd7d0f3eb7145ca41c711457b7eb8f": "a_{i}", "d8ddb40cbcccd127ae6b83aeb7da44d7": "(N - Z)", "d8dde65116dded023efe22a5d5310963": "\\vec \\omega = (\\dot\\alpha\\sin\\beta\\sin\\gamma+\\dot\\beta\\cos\\gamma){\\bold I}\n\n +(\\dot\\alpha\\sin\\beta\\cos\\gamma-\\dot\\beta\\sin\\gamma){\\bold J}\n +(\\dot\\alpha\\cos\\beta+\\dot\\gamma){\\bold K},", "d8dde97b6044808082a2d5788228ac39": " \\Delta w_i = \\alpha (t - y) x_i ", "d8ddfa021b8fa15e5335b7256dd36193": "Rx=\\{z \\in X | zRx\\}", "d8de55f72b43aadd3019a01634efc634": "x_i,y_i", "d8de6bee5e16a8a7e9b680c63cfe5421": "T = \\frac{-\\ln(U)}{\\lambda}.", "d8de9591deaa8399139f3389f57e02d9": "(1+r)^NP - (1+(1+r)+(1+r)^2+ \\cdots +(1+r)^{N-1})c", "d8dea4c168d47abfbb15b08b6018bc30": "X \\vdash a\\mbox{ iff }X \\vdash a \\mbox{ in the propositional calculus}.", "d8debe04ce02f8537f8ba450e41a8b64": "X_g = 0", "d8def6c070990b4a814ea9e493db85ce": " -n(n-1)~r^{n-2}~\\cos(n\\theta) \\,", "d8df05c8f516cd4c478b8fb21a9d374d": "\\tfrac{p}{q} \\not= \\alpha = \\tfrac{a}{b}\\,,", "d8df157930fa48a7963b10c941870b5c": "B\\, ", "d8df2cd10bcd07b4f05d7fa07d4b7907": "\\mbox{for all }\\epsilon>0,\\quad d(n)=o(n^\\epsilon).", "d8df437337f6a2553595bf972a14bd9f": "P_{A_{O_2}}", "d8df5d582f90606ce21ea00bb2a395af": "\\Theta(n\\, \\log\\, n)", "d8df719370e29aa83f172301c68cb15b": "\\eta = 2", "d8df72511d4eb0f40a76caa7d4ae4e0b": "{\\rm Data\\;Rate\\;Savings} = 1 - \\frac{\\rm Compressed\\;Data\\;Rate}{\\rm Uncompressed\\;Data\\;Rate}", "d8df9f3954aeda13d0cf2715dfe6aba4": "F_{s}", "d8dfa0be523e3d885c1ec511ed233338": " \\operatorname{S}(U)(x - Ux)= i(x + U x) \\quad x \\in \\operatorname{dom}(U). ", "d8dfa1cd73d15422bacac70c1280a667": "\nSS_A \\equiv n \\sum_{i} (\\bar{Y}_{i \\cdot}-\\bar{Y}_{\\cdot\\cdot})^2 \n", "d8dfb43a721a3b37710c4d89e17e5cdd": "E_S = \\frac{m_e^2 c^3}{q_e \\hbar} \\simeq 1.3 \\times 10^{18} \\, \\mathrm{V} / \\mathrm{m},", "d8dfea5779b60d477918a3540e85901c": "\\dot{R} = \\partial R / \\partial t", "d8e00732f8217a120452a550866f233a": "x_1 \\succ x_2", "d8e00dab2c2092bee340a9e12e277d39": "\\vert", "d8e0a8fa5a8b9ba112b44b8e21842564": " \\equiv C_f = \\dfrac{f}{q}", "d8e0cd24514c5d16dea5cffee933421a": "\\simeq 10^{\\left ( n\\times\\log_{10}\\left ( \\frac{n}{e} \\right ) \\right )} ", "d8e0cd4f780f9e868eb6a6d84eb9b096": "t_\\frac{1}{2} = \\frac{{\\ln 2}\\cdot{V_D}}{CL} \\,", "d8e0df444b4f0f1da1a3fe6c8322ba92": "H=1/2\\sum a_{ij} q^i q^j, a_{ij}=a_{ji} ", "d8e0eb1dbcc913e1befd43009fc46dde": "\\nabla \\cdot \\mathbf{E} = \\frac{1}{\\epsilon_0} \\rho", "d8e19d609f2d890599684f91122d4538": "\\frac{d w}{d p} = \\frac{1}{f(w)} ", "d8e1a255fd6c1f66f8d6690b504eb4a1": "[S_0] = \\frac {[A_{ad}]}{K_{eq}^A\\,p_A} + [A_{ad}] = \\frac{1+K_{eq}^A\\,p_A}{K_{eq}^A\\,p_A}\\,[A_{ad}]", "d8e1a96f2a8f09d0a87f74f1271c3423": "k_E = -\\frac{1}{RR'}", "d8e271bd3ea683228ce17b94ccc4f867": "\\mathbf A = \\begin{bmatrix}\n0 & -1 & -2 & -3\\\\\n1 & 0 & -1 & -2\\\\\n2 & 1 & 0 & -1\n\\end{bmatrix}", "d8e29de5f2a951dee4df40bb55f41ffb": "\\frac{\\partial g(\\mathbf{U})}{\\partial x} =", "d8e2b262456011f222e32201ad649af4": "\\Delta H = \\Delta U + p \\Delta V \\,\\!", "d8e2da32575ca72575d8b5c501e1f223": "u_1(z)", "d8e306090f337ae5326be2d0f700f995": " C \\ ", "d8e3594abd930f6b2cd387a0d04968d4": "\\}|\\,", "d8e38ede776c9d8ccc065a601e2d8234": "\\kappa = \\left\\|\\frac{d\\mathbf{T}}{ds}\\right\\|.", "d8e3d585ea7f4bb334fdbd34ae006657": "\n y'^T \\mathbf{E} y = 0\n", "d8e3dfc20f4821db59d8fdbe7c173f85": "C=(P,L,I).\\,", "d8e3e1d9f1e7618b4ab9ab5b16af4f94": "\\gamma=(\\gamma\\div\\beta)\\times(\\beta\\div\\alpha)\\times\\alpha", "d8e40f69afcafa992a506bae01ffb63b": " S \\approx \\frac{ 1 - ( 1 - \\frac{ 2 }{ k } )^3 }{ 2 } ", "d8e424d62241abdf31b20572f9a45a7f": "\\langle\\phi_n|\\phi_m\\rangle = \\delta_{nm}", "d8e4651d81db260aaa2f408cd2098e51": "\\mu^'", "d8e491646308fdb45c4e5ce38c1b06d3": "\\!\\rho_w g D sin(\\alpha + \\beta)", "d8e4a1e0c69ecdc99d534f652cd7c06d": "z_1,z_2,\\ldots,z_k", "d8e57947ddbad5f90765b15fe7ee07be": "\\begin{align}\n\\lim_{h \\to 0} \\frac{\\sgn(0+h) - 2\\sgn(0) + \\sgn(0-h)}{h^2} &= \\lim_{h \\to 0} \\frac{1 - 2\\cdot 0 + (-1)}{h^2} \\\\\n&= \\lim_{h \\to 0} \\frac{0}{h^2} \\\\\n&= 0 \\end{align}", "d8e61818e1939392041f18ffadc00440": "\\langle x,y \\mid xyx^{-1}=y^{-1}, x^{2m}=y^n\\rangle", "d8e618a60838a9a828dadc78266ea142": "g_{q+1}(p) = g_q^p(1)", "d8e66b94af3fb44cae251f87cd0991d4": "w(v)+w(u)=w(v|u)\\geq d", "d8e6750014eea8851684c0af17285291": "(\\varphi,\\varphi)", "d8e675d01065f3dca95fb7031475618f": "\\ddot q=kq", "d8e67d1b77fd943263632f76f4236109": "T_1, T_2", "d8e682a122112203bf18013bc9638e21": "\\dot{\\vec x}(t_0)=\\vec v_0", "d8e6ea52ee2d8015af2cfbbf1f7703dc": "|A| \\leq |B| \\wedge |B| \\leq |A| \\rightarrow |A| = |B|", "d8e6f35e769c03013e7aaf5700882268": " \\tilde{G}(s, t) = \\sum_{\\lambda_k > 0} \\hat{\\lambda}_k \\hat{\\varphi}_k(s) \\hat{\\varphi}_k(s). ", "d8e79b69650c228d8b29a1ed8c4b3b1a": "TX\\to X", "d8e814a1a3a3b019de19d1ff12089913": " d_{\\mathrm{F}} ", "d8e85932c1dbee138fce5aeac2be3c7a": "O(\\log^{1.5} n)", "d8e87024f0bd10027e66c61999d58e68": "\\frac{LVEDP \\cdot LVEDR}{2h}", "d8e890f7c440f0587d66942bf33027f1": "\\scriptstyle j \\;\\ne\\; i", "d8e8a0b22c6c6c1c010962dd1472839c": "\\frac{dx}{dt}=k(p-x)^a(q-x)^b", "d8e8b3b880a47c9884645d4a5d51fbab": "n'\\geq N(k-1,d/2)", "d8e8e6d264147426b69dbe6ee3a079e4": " T^{-1}Q = \\{T^{-1}Q_1,\\ldots,T^{-1}Q_k\\}.\\,", "d8e909d1d30742067df8e92fae0a1b84": "g(0, m) = 1 \\text{ for }m=1,2,\\cdots,M", "d8e918f039ea4f26b73d32c1a3d9ddf1": " {}+13803759753640704000\n x^2-8752948036761600000\n x \\,\\!", "d8e91d5ed922a9ca2e7418af7b83f8f6": "u\\in H^1(\\Omega)", "d8ea31e1b5948eb0d3058cab5698ef5f": "A^\\alpha B_\\beta{}^\\gamma C_{\\gamma\\delta} + D^\\alpha{}_\\beta{} E_\\delta \\rightarrow A^\\lambda B_\\beta{}^\\mu C_{\\mu\\delta} + D^\\lambda{}_\\beta{} E_\\delta ", "d8ea7ed7a5ebfa1708a69924d5e2d650": "x_0 = u+v", "d8ea920879a0b56c70e90d6b188a6aa4": " \\forall \\alpha\\, \\varphi\\!", "d8eaa9efbefa1958ac59e3c3161be1e8": " \\displaystyle\\chi_\\lambda(e^X)\\equiv {\\rm Tr} \\, \\pi(z) = A_{\\lambda+\\rho}(e^X)/A_{\\rho}(e^X),", "d8ebe850464769905c3eb158fc025939": "-\\frac{\\eta}{2\\xi^2}(c_\\eta(0,\\xi)+ n_\\eta^\\prime(\\xi))", "d8ec282bddbe3e188c1353644f1a2193": " \n[Vc_n + \\sum_{i=1}^KQ_i(t)a_{in}]x_n(t) \n", "d8ec5b28d52e85671817d3977f2e6754": "\\ge 2", "d8ec7851c6d21051a434ea0dc5d6c974": "Y=1-\\exp(-Kt^n) \\,\\!", "d8ec88174ba845492a74869a51e9edbd": "P_n'(x) = nP_{n-1}(x),\\,", "d8ec8fbe3c0d89380ae81fa8e744e360": "\\kappa\\times i\\colon {\\mathrm {Spin}}(n)\\times {\\mathrm U}(1)\\to {\\mathrm U}(N).", "d8ecb75ff9f37ff55b5fcacae390307f": "\nB(x,y)=\\frac{\\Gamma(x)\\Gamma(y)}{\\Gamma(x+y)}.\n", "d8ecb816d439da5f1546f7bee8e60f53": "\\begin{matrix}{5 \\choose 4}\\end{matrix}", "d8ece74e977e3b23cee70352738505bc": "\\,P_t", "d8ecfa94df4c432998b7941f7135be3c": "L^{4k+1},", "d8ed05fa01e3d9924c595a7959f061c2": "\np_{\\theta_1} = \\frac{\\partial L}{\\partial {\\dot \\theta_1}} = \\frac{1}{6} m \\ell^2 \\left [ 8 {\\dot \\theta_1} + 3 {\\dot \\theta_2} \\cos (\\theta_1-\\theta_2) \\right ]\n", "d8ed269ddddbfd08298970dc70dd4504": " r_1^2 =(x+a e)^2 + y^2 = x^2 + 2 x a e + a^2 e^2 + (x^2-a^2)(e^2-1)=\n(e x + a)^2", "d8ed744c28f4461069991912dae60789": "\\Gamma(n+\\tfrac13) = \\Gamma(\\tfrac13) \\frac{(3n-2)!^{(3)}}{3^n}", "d8ee3dac593ed48568003ebeadc07804": "\\Delta(x) = \\left( \\pi_0(x) - \\operatorname{R}(x) + \\frac1{\\ln x} - \\frac1{\\pi}\\arctan\\frac{\\pi}{\\ln x} \\right) \\frac{\\ln x}{\\sqrt x}.", "d8ee3e88db2957a1c1303a69db1c5749": "\\otimes_R M", "d8ef072b786159ec2150f42774c7b958": "\\textstyle j-i", "d8ef2c82567eb2cd924b694af2089c3b": "\\frac{t \\cdot a}{2}", "d8ef5767dd62793a447f297a97e4d059": "\\frac{\\pi}{3\\sqrt{2}} = 0.740480189\\ldots", "d8ef8fd9f6a6a662c61ed1a174a79fa7": "\\Pr", "d8efbee72e837dad2024c8bbf7fdab66": " \\left| \\frac {V_\\mathrm{out}} {V_\\mathrm{in}} \\right| = \\frac {1} {\\sqrt { 1 + ( \\omega R C )^2 } } \\ . ", "d8efd951c9119144ef03f5d8407a34b9": "X_{k+N} = - X_k", "d8eff00557f7baf4c475991c745e42c0": "y_2= \\frac{y_c}{y_2''}", "d8f05f6dcf7786d4a11c3a90e4cc4c5e": " mpk=ak^{(a-1)} ", "d8f0a607369954cec655c845e9d66c19": "KIE=\\frac{k_L}{k_H}", "d8f0a8742cfd970a15d7f8eb782282ac": "g'(x)\\neq 0 \\text{ for all } x \\in I \\setminus \\{c\\}, \\text{ and}", "d8f0aa7bc15c8251eb7491cc826ae7d4": " \\mbox {Diameter in inches} = T \\times A / 1270 + W ", "d8f0fa3c344f97f0a19d0e304de9d7de": " a_{\\gamma} ", "d8f132735efe2d994c232b155ff56322": "\\pi_{a_1, \\ldots,a_n}( R )", "d8f14f3ca822e2e1e0a747981bf63a9a": "n\\in\\mathbb{N}", "d8f1c49598b49257efa6e9e121198710": " S\\Bigl(\\sum_k \\lambda_k\\rho_k||\\sum_k\\lambda_k \\sigma_k \\Bigr)\\leq \\sum_k\\lambda_k S(\\rho_k||\\sigma_k),", "d8f25e05bcf74021dd13cedc116aff72": "f\\left(\\mathbf{A}\\right)=\\mathbf{Q}f\\left(\\mathbf{\\Lambda}\\right)\\mathbf{Q}^{-1}", "d8f26db3ff8e0d3625f5ce8866c63995": "K^\\ominus = \\frac{\\left\\{\\mathrm{{C}_{12}{H}_{22}{O}_{11}}(aq)\\right\\}}{ \\left \\{\\mathrm{{C}_{12}{H}_{22}{O}_{11}}(s)\\right\\}}", "d8f2821b729750665cc31903dc536c08": "\\frac{d}{dt} \\int_{\\Sigma(t)} \\mathbf{B}(t_0) \\cdot d\\mathbf{A}", "d8f2871f193ec1c2746daceabc9bd399": " X \\sim LL(\\alpha,\\beta)\\,", "d8f2c5ff721a4562d9c5c6d0c91a20e5": "L_{m-1}", "d8f30c61cdc4c6599ec1d7fd62ee47de": "\\operatorname{D}_2 (z) = \\operatorname{D}_2 \\left(1-\\frac{1}{z}\\right) = \\operatorname{D}_2 \\left(\\frac{1}{1-z}\\right) = - \\operatorname{D}_2 \\left(\\frac{1}{z}\\right) = -\\operatorname{D}_2 (1-z) = -\\operatorname{D}_2 \\left(\\frac{-z}{1-z}\\right).", "d8f3471ef0522dc14012f0dc5d01d570": "u_1(x)", "d8f34adc2d385c2c4662fcc3e1d44948": "y_t = B_0^{-1}c_0 + B_0^{-1} B_1 y_{t-1} + B_0^{-1} B_2 y_{t-2} + \\cdots + B_0^{-1} B_p y_{t-p} + B_0^{-1}\\epsilon_t,", "d8f3bb8a9c9e054d196a4563fc9a8a53": " \\oint_{\\partial \\Sigma} \\mathbf{E} \\cdot d\\boldsymbol{\\ell} = - \\int_{\\Sigma} \\frac{\\partial \\mathbf{B}}{\\partial t} \\cdot d\\mathbf{A} ", "d8f41d2c616e77e44342613b3e6e6c5b": "\\left( \\frac{\\partial ( \\Delta G^\\ominus/T ) } {\\partial T} \\right)_p = - \\frac {\\Delta H} {T^2}", "d8f41e09b3a5511bc4954d7ebddb31b3": "0 \\leq {\\rm JSD}( P \\parallel Q ) \\leq 1", "d8f426540300a52c9600c093e6ef132e": "r(x) = s(x) + e(x)", "d8f45e6dfbff0a146f69dffd22722075": " \\int_B\\psi \\, dx=1. ", "d8f4d3176a27da92e5fdc17ecf07a970": "\\rho = \\rho_f + \\rho_b\\,.", "d8f542767af4d2e5e6b89ff44de22252": "B_r(p) = \\{ x \\in X \\mid d(x,p) < r \\}", "d8f5439ae7890ce136699d3af7b8c1d9": "T_n(\\lambda+\\mu)=\\sum_{k=0}^n {n \\choose k} T_k(\\lambda) T_{n-k}(\\mu).", "d8f54b7afb62966b003afeb819a80f82": "\n \\mathbf{t} = n_i~\\sigma_{ij}~\\mathbf{e}_j ~~~\\text{(repeated indices indicate summation)}\n ", "d8f56d940b4d69abba0c235d601dc068": "\\psi(\\Omega^\\omega)", "d8f61422a1b551b6a395ba00afdf47e1": "I_{sp} = v_e", "d8f61a1c0dcbbbaba8347234d87302c9": "|\\psi \\rangle = | j,m \\rangle", "d8f6596d03b6988b45312da530942316": "0 \\xi^2 \\, .\n", "d900b3eea8b85373a27cf1194ddf6c7f": " |u| = 1+2 \\phi +O(v^3)", "d900bcf444be7d8a7fe643b1c2b8c173": "\\tau_n=T_n-T_{n-1}", "d9011d83227a5d4d9f711a7861683da8": " \\|T\\|<\\infty ", "d90148af1b89bef4d6a67adead5b9e5d": "f(a_1 \\mathbf{x}_1+\\cdots+a_m \\mathbf{x}_m) = a_1 f(\\mathbf{x}_1)+\\cdots+a_m f(\\mathbf{x}_m). \\!", "d90191a4e0ecefbc5953ab8faa69ab9f": "dW_t", "d901dabbe7e9f2cbdfd45dfcb7807f14": "\\scriptstyle{\\psi(t_0)}", "d90212d46f739360911cce5f86861813": " \\underset{L/K}{\\operatorname{inv}} : \\operatorname{Br}(L/K) \\rightarrow \\mathbb{Q}/\\mathbb{Z} . ", "d902877b3b96423401100b1e1f9e03b4": "y^{(n)}(x) + \\sum_{i=0}^{n-1} a_i(x) y^{(i)}(x) = b(x).\\quad\\quad {\\rm (i)}", "d90326e19a22c4328b8598ae0cd07d50": " \\left(\\frac{\\alpha}{\\beta }\\right)_m =1 ", "d903c5b8b71d27e28e1a98b6db6f1a43": " RBM_{Q(0)}(\\theta,\\Gamma;R).", "d903f2a9b2663c5cef950668d1b88f66": " \\partial_\\mu \\partial^\\mu A^\\nu = 0 ", "d9043005918d88fbcd1a60784c97eb6b": " s_i = \\left[ \\frac { p_e ( 1 - p_e ) } { n } \\right]^{1/2}, ", "d9044b6c9ce01ff770369a33b1bfcfd2": " A^- = \\langle A \\rangle _1 + \\langle A \\rangle _3 + \\langle A \\rangle _5 + \\cdots ", "d9048b09222fe13730ab41e356424bca": "Z[J]\\propto\\prod_i{\\sum^\\infty_{n_i=0}{\\frac{C_i^{n_i}}{n_i!}}}=\\exp{\\sum_i{C_i}}\\propto \\exp{W[J]}.", "d904a910f9b7e59f817b6d8eccc41685": " e^+e^- \\to 2\\pi,~~ 3\\pi,~~ 4\\pi,~~ 5\\pi ", "d904dbba02e14604f6d5884b5e33eb41": "\\hat{\\mathbf{b}}", "d904ee2e9ce07babc8d9466842119260": "p{\\mathcal O}_K", "d905371f192fca5d9539e26cb9783e7c": "V(p)=0", "d9057b66b48134585531e196f049fcd1": "\\begin{align}\n x_i &= x_{i-1} + v_{i-1/2}\\, \\Delta t , \\\\[0.4em]\n a_i &= F(x_i) \\\\[0.4em]\n v_{i+1/2} &= v_{i-1/2} + a_{i}\\, \\Delta t ,\n\\end{align}", "d9063336108f39dce4ad509861c6d9f6": " \\lim_{n \\to \\infty} {H}_{2n+1} = \n\\textstyle \\left(\\frac{1}{2}\\right)\n^{\\left(\\frac{1}{3}\\right)\n^{\\left(\\frac{1}{4}\\right)\n^{\\cdot^{\\cdot^{\\left(\\frac{1}{2n+1}\\right)}}}}}\n = {2}^{-3^{-4^{\\cdot^{\\cdot^{{-2n-1}}}}}} ", "d9063d97f514fc4130eb6a3000308ffd": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\end{smallmatrix}", "d906869752b3b704977070aef993d6c1": " T_{ab} = T \\left( G_{ab} - \\frac12 h_{ab} h^{cd} G_{cd} \\right) = 0 ", "d906f03a8c8d4668f3e0ba4352263b31": "(-1)^{i+j}", "d9073d412f0acc2acb923e41b4c62cc8": "\\sum_{m=-\\infty}^\\infty a_{cm+d}\\cdot x^{cm+d} = \\tfrac{1}{c}\\cdot \\sum_{k=0}^{c-1} w^{-kd}\\cdot F(w^k\\cdot x),", "d907f3084c9d86d6c7822d3d4cc53100": "GL(6,\\mathbb{R})=36", "d908468210ef1da457eeb1ce65292bd0": "\\left\\Vert\\mathbf{s}+t\\mathbf{d}-\\mathbf{c}\\right\\Vert^{2}=r^2.", "d908e8bf1b49fe08491386315efebee6": "\\left|x - a\\right| + \\left|y - b\\right| = r.", "d9097df7c59ba1e19d9c863388550a27": "\\, \\! V_-=A", "d909885462d9435f93910e67eda8ace4": "TA = \\frac{TB+HBP+BB+SB}{AB-H+CS+GIDP}", "d9098960d24181f2ea168c95bc837785": "\\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^{s}} = 1 - \\sum_{a=2}^\\infty \\frac{1}{a^{s}} + \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\frac{1}{(a \\cdot b)^{s}} - \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\frac{1}{(a \\cdot b \\cdot c)^{s}} + \\sum_{a=2}^\\infty \\sum_{b=2}^\\infty \\sum_{c=2}^\\infty \\sum_{d=2}^\\infty \\frac{1}{(a \\cdot b \\cdot c \\cdot d)^{s}} - \\cdots.", "d9098e7edb77f14cfb086b80d279bc3f": "\\alpha_V(T)", "d909911065deef2d2d85558476ef7170": "h(x)=x^p", "d909a7972c6e0d5c8183008743d5edb8": "\\left( \\frac{3}{2} \\right) ^6 \\times \\left( \\frac{1}{2} \\right) ^3", "d90a0225ffc761b64c4d9941e02471d5": "x_{t+1} = x_t \\frac{2 - p + \\lambda_{t+1}}{2}", "d90a2acfb31bced722cf779d8889790c": "x = c_0 + \\sum_{i=1}^m k_i x_i,", "d90a3f0007737ac41a23e27690ec1472": "S_{12}\\,", "d90adcd266171760746a1b39dc626e07": " [\\mathbf{k}]_\\times = \n\\left[\\begin{array}{ccc}\n0 & -k_3 & k_2 \\\\\nk_3 & 0 & -k_1 \\\\\n-k_2 & k_1 & 0\n\\end{array}\\right]\n", "d90b09c805e07ce65a2449d1648a8697": "\\scriptstyle R_\\mathrm L > R_0", "d90b3df66cb03c05364d33609a5b34aa": "a \\uparrow^{3}b", "d90b6c1e92f7e4bfb4c250852e48ca99": "{{P}_{CPU}}=\\left[ \\propto ,\\beta ,\\ldots \\gamma \\right]*Vecor\\text{ }of\\text{ }CPU\\text{ }performance\\text{ }Counters+{{\\lambda }_{constantCPU}}", "d90b8d23bf0555a134d1f9ff614944fe": "\\scriptstyle\\lambda", "d90b965fcf312ce93982adccf2cd95c5": "\n\\delta \\mathcal{S} = \n\\int_{t_1}^{t_2}\\; \n\\left(\n\\varepsilon{\\partial L\\over \\partial x} -\n\\varepsilon{d\\over dt }{\\partial L\\over\\partial \\dot x} \n\\right)\\,dt.\n", "d90bd166d73a180e747de1a205534f2a": "s=h^y\\,", "d90bd6bb624ed8f840af04695e62b50e": "{AC}:{BC} = {AD}:{DB} \\, ", "d90c9b79a2a359376fe4113323762c61": "I : D \\subset \\mathbb{Z}^2 \\to S", "d90cbd6bd58a6835a7af63f0dc33911f": "\nL_{\\overline{S},\\overline{S}} x_{\\overline{S}} = - L_{\\overline{S},S} x_{S},\n", "d90ccffa01275a7a7795fb6d1274e75f": "mass", "d90ce4bed11b81542dc38ffb60da18ba": " y \\cdot ( u \\wedge v) = 0", "d90d01c0b35fb93ec66d6345d8d9e28d": "\\vec{x}\\in P", "d90d1b3f8f2704d2f746b51ac2c75d25": "\\mathrm{H}(p, q) = \\mathrm{E}_p[l_i] = \\mathrm{E}_p[\\log \\frac{1}{q(x_i)}]", "d90d2898a11bcf08d304ac52663a1d8b": "B_{oi}\\ ", "d90d976fd3300fbef28e71f297191ebb": "E\\to((A\\to B)\\to(((D\\to A)\\to(B\\to C))\\to(A\\to C)))", "d90da38c3eff54eb41bcad06ae1a69ec": "\n\\begin{align}\n\\operatorname{Var}(T) & = \\;\nn^2 H_n^2 - 1 + n^2 H_{n-1}^{(2)} - 2 n H_{n-1} + n H_{n-1} + 1 - n^2 H_n^2 \\\\ & = \\;\nn^2 H_{n-1}^{(2)} - n H_{n-1} < \\frac{\\pi^2}{6} n^2, \\quad \\text{QED.}\n\\end{align}", "d90ecca369935be8f1b15f43f758ed8e": "\\langle X_i, f_{ij}\\rangle", "d90ed7e9a24ff23ba2ec468430b4909b": " a( \\cdot ,\\cdot )", "d90ee8fa7983d65853323b379d5307ad": "N = L^2", "d90ef1374e765186cd45676a18348710": "\\mathrm{GF}(q)", "d90f238343305418f23ca7186022775c": "|n^{(3)}\\rangle =\\Bigg[-\\frac{V_{k_1 k_2}V_{k_2 k_3}V_{k_3 n}}{E_{k_1 n}E_{n k_2}E_{n k_3}}+\\frac{V_{nn}V_{k_1 k_2}V_{k_2 n}}{E_{k_1 n}E_{n k_2}}\\left(\\frac{1}{E_{n k_1}}-\\frac{1}{E_{n k_2}}\\right)+\\frac{|V_{nn}|^2V_{k_1 n}}{E_{k_1 n}^3}\\Bigg]|k_1^{(0)}\\rangle", "d90f300fda90bcc8a316988fa95e0a94": "\\boldsymbol{v} = \\lim_{\\Delta t \\to 0}{{\\boldsymbol{x}(t+\\Delta t)-\\boldsymbol{x}(t)} \\over \\Delta t}={\\mathrm{d} \\boldsymbol{x} \\over \\mathrm{d}t}.", "d90f307427d0cc3fb44b3b881b0d256f": "\\Gamma(z) \\approx \\sqrt{\\frac{2 \\pi}{z} } \\left( \\frac{z}{e} \\sqrt{ z \\sinh \\frac{1}{z} + \\frac{1}{810z^6} } \\right)^{z},", "d90f977dc33519e4086372656555dd8c": "\n(a_na_{n-1}\\cdots a_1a_0.c_1 c_2 c_3\\cdots)_b =\n\\sum_{k=0}^n a_kb^k + \\sum_{k=1}^\\infty c_kb^{-k}.\n", "d90fc25b8e1ce7659713141fc46432ae": "Q_{Y}(\\tau)=F_{Y}^{-1}(\\tau)=\\inf\\left\\{ y:F_{Y}(y)\\geq\\tau\\right\\}", "d90fca17e371b8cfae08080ba24b6bb1": "\\phi_1(v)=(1) \\!\\left(\\frac{1}{6}\\right)=\\frac{1}{6}\\,\\!", "d90fefb248a5b08a491389f90b07c49e": " C'(s) = \\Delta\\,", "d9105b87e4c47958cd278e51958284c6": "G=-H", "d9108cda4482e68dce5486746cc9e51e": "\n \\mathbf{v}\\cdot\\mathbf{b}^i = v_k~\\mathbf{b}^k\\cdot\\mathbf{b}^i = g^{ki}~v_k\n ", "d910f242f0dbb8d80110578f3b7362ab": "x, y, z \\in X", "d91110193491dbaa8aa209c7239d35c9": "L_n^{(\\alpha-n)}(x)\\approx e^x\\cdot {\\alpha\\choose n}", "d9113b12c4d8540199a406518ef1b58b": "\\mathbb{C}\\smallsetminus\\{0\\} = \\mathbb{R}^+ \\times \\mathbb{S}^1", "d9115058e98181d2ed75a75897aa6d5c": "\\scriptstyle \\varphi(0) = \\varphi(0+) =\\lim_{x\\rightarrow 0_+}\\varphi(x) = 0", "d911a9bf985596b254e8d51e6a774c2b": "\\mathrm{Hom}(\\mathrm{Gr},-)", "d911ae9d798c36fd1baac381171a80af": "f_{z^3}", "d911b228c22959ddf611d25fe718a717": " \n\\liminf_{n\\rightarrow \\infty} \\int_{E} f_n \\, d\\mu_k \\geq \\int_E \\phi \\, d\\mu,\n", "d911e153768f3b8ac905ddf9dab2de54": "S = \\frac{s^2 * m^2}{C_M\\alpha \\div \\sin(a) * t * d * v^2}", "d9120436cd995bf241cc4f882914494e": "O(1,n-1) < GL(n)", "d9122d2c6ab9a1b19414286b65678454": "e = H(M || r)", "d912733359b5fc14f256717de010945a": " b=r_1,", "d9128cd2203934c8fc1e9674286ebb9b": " f = \\frac{4}{3} \\pi\\ \\dot NG^3 \\int_{0}^{t} (t-t_0)^3\\, dt = \\frac{ \\pi\\ }{3}\\dot NG^3t^4 \\,\\! ", "d9129b06ffef0982391e6155109ef7e2": " S_1 \\geq \\sqrt{S_2} \\geq \\sqrt[3]{S_3} \\geq \\cdots \\geq \\sqrt[n]{S_n}", "d912c104238e17fbf6e0d5412ff6e8f2": "\\begin{align} v_1 &= u_2\\\\\nv_2 &= u_1\\,. \\end{align}", "d912f86c7caaf1fa70ecfe8295a522a1": "\\scriptstyle{e/m}", "d9131eaae86dab429d4649931c351c0e": " E_p(x) = \\exp\\left(x + \\frac{x^p}{p} + \\frac{x^{p^2}}{p^2} + \\frac{x^{p^3}}{p^3} +\\cdots\\right).", "d9134779d697cf90a333b00603e960a3": "\\dot\\sigma=\\mathbb{C}\\dot\\varepsilon,\\qquad {(1)}", "d9135f994581a9dc49db445444d7d88d": "y=\\int^x \\frac{1}{\\sqrt{g(v)}} \\, dv, ", "d9135fa3f2162987504d9448ca50a5bf": "z=\\begin{pmatrix}\n 1 & 0 & 1\\\\\n 0 & 1 & 0\\\\\n 0 & 0 & 1\\\\\n\\end{pmatrix}", "d91401abd7708910a893f9dc506e9b9c": "\nf(r)=-\\frac{1}{\\pi}\\int_r^\\infty\\frac{d F}{dy}\\,\\frac{dy}{\\sqrt{y^2-r^2}}.\n", "d9146790693de85d684045fb4633c501": "(n = 1)", "d914752b77359097f6c0817a36392294": "v^0=1", "d9148cf43a846adcc324be33dbfbd620": "\\operatorname{second} \\equiv \\lambda p.p\\ (\\lambda x.\\lambda y.y) ", "d91492b4d0b791b9fa8e7c0d473b777e": "\\boldsymbol{F}_1", "d914a66599aa7ace6c9d0968664e9c97": "\n \\begin{bmatrix}N_{11} \\\\ N_{22} \\\\ N_{12} \\end{bmatrix} = \n \\cfrac{2hE}{(1-\\nu^2)}~\\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} u^0_{1,1} \\\\ u^0_{2,2} \\\\ \\frac{1}{2}~(u^0_{1,2}+u^0_{2,1}) \\end{bmatrix}\n", "d914a86bd2aebac3c5ffe59960fa7e2f": "\\vec{v}_\\mathrm{B rel A}", "d915112cc01f9206fc8e75c57c6d121a": "(k,s,M,m) \\in W^8 \\times W^4 \\times Z \\times B^{\\ast} \\,", "d915cb245c5189b2ffdf4324fece25e9": " \n\\leq B + C + Vp^* \n", "d91679d0662fd61d247b8f4b803c29ab": "\\Omega_R", "d9172e672debbbb2d07a8fafcadb6e12": " \\begin{align} & \\left ( \\frac{E}{c} \\right )^2 - \\left ( p_1^2 + p_2^2 + p_3^2 \\right ) \\\\\n& = \\left ( \\frac{E}{c} \\right )^2 - p^2 \\\\\n& = \\left ( mc \\right )^2 \\end{align} \\,\\!", "d91791bcf761369b33c0c1e743218d8a": "Y_{8}^{-4}(\\theta,\\varphi)={3\\over 128}\\sqrt{1309\\over 2\\pi}\\cdot e^{-4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(65\\cos^{4}\\theta-26\\cos^{2}\\theta+1)", "d917c633ff3ef527512a3a1403e51d9d": "\\scriptstyle \\geq10^{15}", "d91834e856a5064754ede3f053a50592": "\\phi_2(y_2) = \\!\\int_{-\\infty}^\\infty \\psi(y_1,y_2) \\phi_1^*(y_1) dy_1", "d918b3c4247f30da77075923fa9ac4ba": "\\pi_1(S^1)\\rightarrow\\pi_1(M)\\rightarrow\\pi_1(B)\\rightarrow1", "d918e613184c31484bdf4723cec9db3f": "\n\\Phi _\\alpha \\left( {A_\\alpha ^1 , \\ldots ,A_\\alpha ^n } \\right)_ {+},", "d918fe789747a6871dc87041982224cc": "m_{\\mathit{eff}} = 9 \\times 10^{-35} (Wcm^{3}s^{-2})", "d91966739a70ab4604cc134604475746": " T_m f = a_m f, \\quad a_m a_n = \\sum_{r>0, r|(m,n)}r^{k-1}a_{mn/r^2},\\ m,n\\geq 1. ", "d9196eae00b0ba4a274e56a98c90bafe": " T_{ij}^{(1)}=s_{ij} ", "d9197139fde456751bb5281097a3b7c5": " \\theta\\!_2 ", "d919b002c0633770ba6c031c98cbba2b": "P(r,t\\mid r_0)", "d919b865e0e116163b9348daaf34fb22": " \n \\hat\\theta = \\underset{\\theta\\in\\Theta}{\\operatorname{arg\\;max}}\\ \\widehat{Q}_n(\\theta),\n ", "d919dd67de093d178c55ff33b412129e": "d = e / n^2", "d919e7e6e120de973d48c4e28b30116c": "P \\cdot Y", "d919f01d901f0c667c1265f08fbd995b": "F = G \\frac{m_1 m_2}{r^2} + B \\frac{m_1 m_2}{r^3} \\ ", "d91a1d90326466737807554e431b6368": "\\displaystyle \\cos (a x)", "d91a20c1b3731b4c6559c783bb3ece89": "\\lbrace \\exp(j t) = \\cosh(t) + j \\sinh(t) : t \\in R \\rbrace,", "d91a5a895e0a2e1786c1b9f4f49f51e6": "w^{-\\gamma}L(w)", "d91a83359238e72122eaa9912c557c79": "x'_s\\leftarrow 1", "d91aa0269386864ee9ef43dd8ab7fa4a": "a(i)", "d91ac3791fd61a84d70aefbdaeb0ce22": "\n\\begin{align}\n\\left(\\frac{s+2r}{n}\\right) &= \\left(\\frac{t + at^{-1} +2r}{n}\\right) = \\left(\\frac{t\\left(1+at^{-2} +2rt^{-1}\\right)}{n}\\right) \\\\\n &= \\left(\\frac{t\\left(1+r^2t^{-2} +2rt^{-1}\\right)}{n}\\right) = \\left(\\frac{t\\left(1+rt^{-1}\\right)^2}{n}\\right) \\\\\n &= \\left(\\frac{t}{n}\\right) \\left(\\frac{1+rt^{-1}}{n}\\right)^2 = \\left(\\frac{t}{n}\\right)\\left(\\pm 1\\right)^2 = \\left(\\frac{t}{n}\\right) \\\\\n\\end{align}\n", "d91acc7c58560015ebe14fe3943c53a1": "\n+ P(female) \\, p(height | female) \\, p(weight | female) \\, p(foot size | female)\n", "d91b2721145bae4dda63d2d32f1a86f9": "\\mbox{eGFR} = \\mbox{163}\\ \\times \\ \\mbox{(SCr/0.9)}^{-0.411} \\ \\times \\ \\mbox{0.993}^{Age} \\ ", "d91b29071e7a89c71eff1e16e8300ff2": "\\mbox{k-distance}", "d91b62d6d937b845a566487c1fc31a46": "V^* \\to V : f \\mapsto f^* ", "d91b6938fa67c0b494befc0cc17493af": "A < B \\to 2^A \\le 2^B.", "d91bd0a63ec7c53183a7326c1d8961a2": "(x,w)\\sim(a x,a w)\\,", "d91c01dad4c3b54cdd8e6d71530df265": "* \\rightarrow * \\rightarrow *", "d91c671636b4d5e15c74c67fd7699cd3": "\\psi(\\omega^\\omega)", "d91d14822bee24f1ecb1878a9dd8a012": "\\begin{align}\n\\int_0^{2\\pi} d\\phi \\int_0^{3a} \\rho^3 d\\rho \\int_{-\\sqrt{9a^2 - \\rho^2}}^{\\sqrt{9 a^2 - \\rho^2}}\\, dz &= 2 \\pi \\int_0^{3a} 2 \\rho^3 \\sqrt{9 a^2 - \\rho^2} \\, d\\rho \\\\\n&= -2 \\pi \\int_{9 a^2}^0 (9 a^2 - t) \\sqrt{t}\\, dt && t = 9 a^2 - \\rho^2 \\\\\n&= 2 \\pi \\int_0^{9 a^2} \\left ( 9 a^2 \\sqrt{t} - t \\sqrt{t} \\right ) \\, dt \\\\\n&= 2 \\pi \\left[ \\int_0^{9 a^2} 9 a^2 \\sqrt{t} \\, dt - \\int_0^{9 a^2} t \\sqrt{t} \\, dt\\right] \\\\\n&= 2 \\pi \\left[9 a^2 \\frac{2}{3} t^{ \\frac{3}{2} } - \\frac{2}{5} t^{ \\frac{5}{2}} \\right]_0^{9 a^2} \\\\\n&= 2 \\cdot 27 \\pi a^5 \\left ( 6 - \\frac{18}{5} \\right ) \\\\\n&= \\frac{648 \\pi}{5} a^5.\n\\end{align}", "d91d4555da75a267838654f2fe149235": "c_n = {\\langle f, P_n \\rangle_w\\over \\|P_n\\|_w^2}", "d91e076c6d76962dca0779b2d278536d": "\n \\sigma^{*}_{\\rm fracture} = B~(p^*)^m~\\left[1 + C~\\ln\\left(\\cfrac{d\\epsilon_p}{dt}\\right)\\right]\n ", "d91e0782e3daa4d0505a3fb357c11cc5": "\\{X_n\\} = \\{ \\{50\\}, \\{20\\}, \\{-100\\}, \\{-25\\}, \\{0\\},\\{1\\},\\{0\\},\\{1\\},\\{0\\},\\{1\\},\\dots \\}.", "d91e13d2b34356b4f9fa80b84f6e8918": "l(f)", "d91e494571dd711f94be8e4961b0c02f": "\\hat{t}=-\\sin(u)\\ \\hat{k}\\ +\\ \\cos(u)\\ \\hat{l}\\,", "d91e4f304c84a7181bef110a238b40a3": "\\Box A", "d91e60c20efc5c0cb2b8f18eae9aa3ea": "U(P) = \\frac {1}{4 \\pi} \\left[\\int_{A_1} + \\int_{A_2} + \\int_{A_3} \\left( U \\frac {\\partial}{\\partial n} \\left( \\frac {e^{iks}}{s} \\right) - \\frac {e^{iks}}{s} \\frac {\\partial U}{\\partial n} \\right)\\right] dS ", "d91e742156b0dafc2006714b5e69db8e": "S_{i}^c", "d91e8ed4d0e8c2c9ec78b4cd340bb308": "\\sqrt[1/34]{2.1}-1=2.2%", "d91eb1aeda4ecccf4cc8b92e86826320": "\\mathrm{Hom}_n(-,-)", "d91ed4bcc204a092348b606a79658c4a": " \\nabla_{\\vec{e}_{0}} \\, \\vec{e}_0 = 0", "d91ee61ae0e7eedba579fe6cebaa6a5b": "=1+\\frac{f(\\theta)}{z}e^{ik(x^2+y^2)/2z}+\\frac{f^*(\\theta)}{z}e^{-ik(x^2+y^2)/2z}+\\frac{|f(\\theta)|^2}{z^2}.", "d91f29a3a1150484706c55a0e1a54eb3": "\\mathbf J_a=\\mathbf L+\\mathbf S", "d920030babcb53c089f4df6c2bfa35e7": "\n\\begin{align}\n L(E)&= I'/I.\n\\end{align}\n", "d92019a1a17b0eee353c336c63846f6c": "\\tau_{g}(\\omega) = -\\begin{matrix}\\frac{d\\phi(\\omega)}{d\\omega}\\end{matrix}", "d9203c4ae6197e4381b2cf9fdd4bf606": " \\tan(20^\\circ) \\cdot \\tan(40^\\circ) \\cdot \\tan(80^\\circ)=\\sqrt 3 = \\tan(60^\\circ). \\, ", "d92071c99c78a9d060016a1eaa53c1a8": "\\scriptstyle a_i \\,=\\, \\bar{a_i} \\,\\pm\\, \\epsilon_i", "d920d06d0542f190feeaffd15b987bc4": "L_{oc}^{pri}=L_P=L_P^\\sigma+L_M", "d920de8bb143783b970a601d0543883e": " \\operatorname{E}[X] = np , ", "d920ff504c984a4b117e9ca27f8bf24c": "\\scriptstyle (1+\\mathit \\Gamma) \\cosh(\\gamma x)", "d9210410e3f34ae655ec031f45e8f9b9": "\\operatorname{Hom}(X,Y):=d(X,Y)\\in \\operatorname{Ob}(R^*)", "d9212159ce566192b18fc130d51d4adb": "\\widehat{L}_z = i\\hbar\\frac{\\partial}{\\partial \\theta}", "d92147a97a2bd4b3f4d5699829616a2b": "y(0)=0, \\ y(\\pi/2)=2.", "d9215d8a0d9ec991c86dc558532df3ec": "| \\psi \\rangle = (| \\psi_1 \\rangle + | \\psi_2 \\rangle)/\\sqrt{2} ", "d92179be7bcb9c3a9ed15cd7a4459b73": "I_{n} = \\int x^n e^{ax} dx\\,\\!", "d921a8f52dd8e14bdff316b0ca544a8c": "Q_{\\ell m}", "d922004745f9bfce9ef97a29a87eddaa": "V(x) = \\frac{W}{m} = \\frac{1}{m} \\int\\limits_{\\infty}^{x} F \\ dx = \\frac{1}{m} \\int\\limits_{\\infty}^{x} \\frac{G m M}{x^2} dx = -\\frac{G M}{x},", "d922069c6371c3627a188ba8fad75aa7": "\\mathcal{A}(i_{U,\\bar{U}})", "d92219d48a2823576e6acdd9fa931eb6": "\\tilde I(\\omega)", "d92247d513f440c1ab5c16842264b118": "\n\\mathcal{L}=-\\frac{1}{2}(\\partial S)^{2}-\\frac{1}{2}(\\partial P)^{2}-\\frac{1}{2}\\bar{\\psi} \\partial\\!\\!\\!/ \\psi\n\n", "d92252b6399db5366650eb14cafedaf0": "P(\\mathbf{X}|\\mathbf{\\Psi},\\nu) = \\int P(\\mathbf{X}|\\mathbf{\\Sigma})P(\\mathbf{\\Sigma}|\\mathbf{\\Psi},\\nu) d\\mathbf{\\Sigma} = \\frac{|\\mathbf{\\Psi}|^{\\frac{\\nu}{2}}\\Gamma_p\\left(\\frac{\\nu+n}{2}\\right)}{\\pi^{\\frac{np}{2}}|\\mathbf{\\Psi}+\\mathbf{A}|^{\\frac{\\nu+n}{2}}\\Gamma_p(\\frac{\\nu}{2})}", "d922abc39bd1b2086d908e2a5a833d7f": "x \\div 11", "d923109fc2c094745f6cb828a567c196": "(k_{AXU},genState) \\leftarrow GenWords((5L_b(\\left\\lceil m/64 \\right\\rceil)+24),genState).", "d923869404d9238da6cad7ff38a63852": "EE(G)=\\sum_{j=1}^n e^{\\lambda _j}", "d923afe5b32f34462cd03a8b6d5cc7ed": "\\Omega_0", "d923bd24f8ab016ed2029970255cdcdd": "H_n(X,\\mathbb{Z})=H_n(X)", "d9245b7daa566263dc06a8cbd7977c29": "=\\frac{df_x}{dt}\\hat{\\boldsymbol{\\imath}}+\\frac{df_y}{dt}\\hat{\\boldsymbol{\\jmath}}+\\frac{df_z}{dt}\\hat{\\boldsymbol{k}}+[\\boldsymbol{\\Omega \\times} (f_x \\hat{\\boldsymbol{\\imath}} + f_y \\hat{\\boldsymbol{\\jmath}}+f_z \\hat{\\boldsymbol{k}})]", "d9246bffb3ab4efa86265af8af017b43": "\\mathbf{H_{1}} = \\mathbf{H_{2}} ", "d92476167f2f223dcd921a4fe0131549": "\n \\mathcal{E}_{ijk} = \\left[\\mathbf{b}_i,\\mathbf{b}_j,\\mathbf{b}_k\\right] =(\\mathbf{b}_i\\times\\mathbf{b}_j)\\cdot\\mathbf{b}_k ~;~~\n \\mathcal{E}^{ijk} = \\left[\\mathbf{b}^i,\\mathbf{b}^j,\\mathbf{b}^k\\right]\n", "d9247c111206f290ae53449632da76da": "s(b,c)=\\frac{1}{4c}\\sum_{n=1}^{c-1} \n\\cot \\left( \\frac{\\pi n}{c} \\right)\n\\cot \\left( \\frac{\\pi nb}{c} \\right).\n", "d9247e7f796c241cf34be86cee7c3893": "\\operatorname{havercosin}(\\theta)", "d9248613f0c7dc27451b241a23188df9": "p(v) ", "d924cef311d791bdd688f6e56a4d6e9a": "O(1/\\sqrt{n})", "d924ee356a02eae77038ed21fec62e1b": "BV_{CEO}", "d92561a442d934553c691b53e152b0ae": "\\max(0,c-x)", "d925dc70033da5403f36f9c89138231e": "\\Gamma\\left (\\frac{1}{2}+n\\right ) = \\left (-\\frac{1}{2}+n\\right )! = \\Pi\\left (-\\frac{1}{2}+n\\right ) = \\sqrt{\\pi} \\prod_{k=1}^n {2k - 1 \\over 2} = {(2n)! \\over 4^n n!} \\sqrt{\\pi} = {(2n-1)! \\over 2^{2n-1}(n-1)!} \\sqrt{\\pi}.", "d925ea80214f3fc033748be1a8507d8f": " i\\pm\\frac{1}{2} ", "d925fc68e32c430af34dfb85a53b0907": "\nf_i(\\sigma)(a) = \\frac{g_i(\\sigma)(a)}{\\sum_{b \\in A_i} g_i(\\sigma)(b)}\n", "d9263b9a41e116a83194858a38ed90e0": "B(u_{f}, v) = \\langle f, v \\rangle \\mbox{ for all } v \\in V.", "d92648ea7d99bc4f2d2539fe26725460": "P_{\\mathrm{s, max}} = 2 \\frac {k \\Delta T}{q} \\frac{L^2}{h}\\,", "d926da0847a42bc546609f46a07a656e": " \\Phi^{n}(a_{\\pi(1)},\\ldots,a_{\\pi(n)}) = (-1)^{\\left|a_{\\pi}\\right|}\\Phi^{n}(a_{1},\\ldots, a_{n}) ", "d927840ecbba0cb783f6aa9fa7090bec": "\\alpha,\\beta\\in\\mathbb{N}^n_0", "d92795689195953993f18cbfc71cb3b5": " B \\cdot A = \\left( \\sum_{j=1}^r a_{i,j} \\cdot b_{j,k} \\right)_{i=1\\ldots s;k=1\\ldots t} \\;\\in\\R^{s\\times t} ", "d927c4649277e28c720b4871d50a547a": "r(\\theta) = x_0 \\cos \\theta + y_0 \\sin \\theta", "d927d6f99111c8d7b08c62a0b4d03216": "F_U = (N_1 - N_2)I_1 = (1-\\frac{1}{a})N_1I_1", "d927eccd9d31e883c4a1bfc973eacb18": "\\mathbf{y}=(y_1,y_2,y_4,y_4)", "d928139affc00249640997286944bfca": "R(p,n) \\leq \\log_p(2n),\\ ", "d9284afed9363ff0c9b4a47844236a1b": "M_V = 0.12 - 5 \\cdot (\\log_{10} \\frac{860}{3.2616} - 1) = -7.02.", "d928cc1f8a6326dd8ff809d6d813ba3f": "10^n", "d928d7ccdd4d37dbfe8fc0c3b6d60896": "_{\\not\\subset}\\!", "d928f449ea0fe5ae14b4dfbb2675dfda": "p(x) = x^3+2x^2-x-1", "d92903b470fbb4254aa5c4a2fb722b1d": "\\ldots\\,\\!", "d9292d85deb6cd4d9d8ad60860f30fd4": "\\cos(\\omega t) = \\begin{matrix}\\frac{1}{2}\\end{matrix}(e^{j \\omega t}+e^{-j \\omega t})", "d92967e6496f6d2e208cf8318a506030": "P_D\\left( \\frac{y \\sum_j^t \\alpha_j h_j (x)}{\\sum |\\alpha_j|} \\leq 0\\right) \\leq P_S\\left(\\frac{y \\sum_j^t \\alpha_j h_j (x)}{\\sum |\\alpha_j|} \\leq \\theta\\right) + O\\left(\\frac{1}{\\sqrt{m}} \\sqrt{d\\log^2(m/d)/ \\theta^2 + \\log(1/\\delta)}\\right)", "d929e64d23207aedd560462936bc609d": "\\alpha_{m,l}", "d92a39a76384dff3547cd6d6ff040cc2": "\\langle \\exp(i\\theta[\\sigma]) \\rangle_p \\propto \\exp(-f V/T)", "d92a3a4e9c9bc9316db88acbb32202f0": "E = KV \\left(\\alpha^2\\beta^2+\\beta^2\\gamma^2+\\gamma^2\\alpha^2\\right).", "d92aa79f7067a522ef55750320c0dde9": "\\left| \\alpha - a \\right| < \\frac{r}{2\\left|b\\right|} ", "d92ab6286f889038bbea5bf7a147847e": " x^{(1)} = \n \\begin{bmatrix}\n 0 & -1.500 \\\\\n 0 & 1.071 \\\\\n \\end{bmatrix}\n \\times\n \\begin{bmatrix}\n 1.1 \\\\\n 2.3 \\\\\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 5.500 \\\\\n -2.071 \\\\\n \\end{bmatrix} \n =\n \\begin{bmatrix}\n 2.050 \\\\\n 0.393 \\\\\n \\end{bmatrix}. ", "d92afd3da34ab37ea896cd00d8979f68": "x(t)=\\frac{1}{P}.[\\int_0^t{\\bar{p(\\tau)}.e^{A\\tau}d\\tau}.e^{-At}-\\int_0^t{\\bar{p(\\tau)}.e^{B\\tau}d\\tau}.e^{-Bt}]", "d92b18bc940731e280d1f3806693798f": "\\operatorname{ht}(\\mathfrak{p} R[x]) = \\operatorname{ht}(\\mathfrak{p})", "d92b88a10aeb56c592457c45a6287581": "\\|x\\|_\\infty = \\sup_i |x_i|", "d92bb615ca809c2d497ce90baa0e6ba7": "x \\neq a", "d92bba6dcb5b90e5be8dba4aad8183d3": "\\tan\\alpha = t", "d92be4096901cc961011fb30ce18209e": " H(x,t)=-\\frac{1}{2}\\frac{\\partial^2 }{\\partial x^2}+K\\cos(x)\\sum_{n=-\\infty}^{\\infty}\\delta (t-n) ", "d92bfacddc8dd04b6ff380bbca87c475": "\\scriptstyle F:M\\,\\longrightarrow\\, N", "d92c86f8e52aa42c2af55a9786eb8f1c": "a^2 = a", "d92ca9e2b9b32aab62eb30652abd3343": "z_{k+1} = f(z_k)", "d92d2d8510ddb577408da8b2b1c7fca5": " A \\subset \\bigcup_{k} I_k ", "d92db72e1b046c139599b20b300b3db5": "V \\subset {\\mathbb{C}}^n", "d92de51caff69686ad28df5a82ba0b1a": "P(\\boldsymbol{r}) = 1/\\rho(E(\\boldsymbol{r})) = \\exp (-S(E(\\boldsymbol{r})))", "d92df65bb94a22117d725b86636cf24a": "W_{1},W_{2},\\dots,W_{b}", "d92e05fb2f17e67663cc48af95e25816": "\\kappa_{(\\ell)}", "d92e68338574554a9f99c57dc103c581": "W_0 = V, W_1 = \\{\\}", "d92e98aa45e9251d9e615322ea40f470": "\\Psi^{n,K}(r_e,r_h) = \\sum_{c,k_e,v,k_h}A^{n,K}_{cv}(k_e,k_h) \\psi_{ck_e}(r_e) \\psi_{vk_h} (r_h)", "d92ee7e672a27310acbddc8382f96d04": "P^{(n)}_e \\rightarrow 0", "d92f58f69490a543a84eb63379665163": "G_{ex}", "d93082868e441bd4654e8b0b108360eb": "\\Delta_{\\mathrm{LB}}\\log\\; f = -K.", "d930de643402a82ef6a8f1277eaca253": "RAB_t = RAB_{t-1} +Capex_t - Dep_t\\,\\!", "d9312a97871ca4efda5e6099ce74fa3c": "S_{v \\times v} \\Lambda_{v \\times 1} = -C_{v \\times 1\\,} .", "d931570a0ef5be154f33563114ecd97b": "\\frac{\\operatorname{d}e_k}{\\operatorname{d}t} = \\frac F m \\cdot v = a \\cdot v", "d931ead991d7bea28c58e1ec7229e78b": "\nA = \\begin{bmatrix} a_{1,1} & a_{1,2} & \\dots & a_{1,n} \\\\\na_{2,1} & a_{2,2} & \\dots & a_{2,n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\na_{n,1} & a_{n,2} & \\dots & a_{n,n} \\end{bmatrix}.\\,", "d9327f33e6aa4e63f05977fada84ed7e": "u = \\frac{u_1u_2}{d^2} = u_1u_2= x^4 -10x^3 +32x^2 -38x +15", "d9329af07d98b8da400aeff16b2af959": " I(n, \\Lambda) = \\frac{n}{2}I(n-1, \\Lambda) + \\zeta(-n) - \\sum_{r=1}^{\\infty}\\frac{B_{2r}}{(2r)!} a_{n,r}(n-2r+1) I(n-2r, \\Lambda),", "d932b349494c3e92ab8bc52f6ed95e07": "\\left[K\\right]\\left\\{d\\right\\} = \\left\\{-f\\right\\}", "d932c66b9ea192bbe8998be95f3c28c4": "\\, (x,y)=(a/c,b/c) ", "d932fb5dba669c24405d4a381a41cc8e": "\\rho < 0", "d932ff55c61e6037ce92822462395e96": "\\mu(x) = \\frac{1}{P(x)} e^{\\int \\frac{Q(x)}{P(x)} \\mathrm{d}x},", "d933120ecf778d7fcfbc0c26e75a8c72": "\n\\sigma_2 = \\sigma_y =\n\\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix}\n", "d9331539f11460633e58eb2f354b72a6": "\\ddot{u}_{\\beta} = (1 - 2\\beta)\\ddot{u}_n + 2\\beta\\ddot{u}_{n+1}~~~~0\\leq 2\\beta\\leq 1", "d93358c7ac697be1bdc72dd753b79d8c": "\\begin{matrix}{r \\choose 1}{r - 1 \\choose 2}\\end{matrix}", "d933df149c62be04ea54d3a9bfb0372c": "nat", "d93425c83f0121805391059b09ae0f2f": "f(c+\\Delta c)-f(c) \\approx -(\\Delta c)f(c)^{Q+1} ", "d9348169b9abad774afe16238cbe0476": "\\R^{d_{mm'}}", "d934ce4a9408b5b22c32820b4fc68ad4": "f: X \\to \\mathbb{R} \\cup \\{+\\infty\\}", "d934e32e5f67f58afdeecce493e9a032": "A\\textbf{x}=\\textbf{b}\\qquad \\text{and}\\qquad A\\textbf{x}=\\textbf{0}\\text{.}", "d934f65baf5ef67cc9af29ddf5b43c68": "n_\\text{piv} \\in n_\\text{clause}^\\text{left}", "d93528ea4e7c7c1b0600a4893cbc9674": "P_m=[m]P(X_m:Z_m)", "d93538422faf30843af1287c7fdd9aeb": " r_{\\mathrm{ess},k}(T) = \\max \\{ |\\lambda| : \\lambda\\in\\sigma_{\\mathrm{ess},k}(T) \\}. ", "d9357f205f9045167050a8cc793bf3f0": "\\frac{d [B]}{d t} = -k_1[A][B]", "d9358282f665075ae09e22562c5cd7b3": " \\mathbf{p} \\and \\mathbf{C} = 0. ", "d935dd09704af426914c025e4813bdb4": " + {e^2 \\over \\pi^2 \\hbar} \\left [ ln \\left ( {B_{SO} + B_e \\over B}\\right ) - \\psi \\left ({1 \\over 2} + {B_{SO} + B_e \\over B} \\right ) \\right] ", "d93605c751cd87bdc48ed6f346f1dc98": "X_{out}", "d93633c43d63e1480528d937e1999007": "\\lfloor \\log_k n\\rfloor +1 \\ (n > 0)", "d93634d8b3815eb1dba6ebb25f9e3abe": "\\epsilon_{c} = \\frac{1}{2}\\left(\\frac{g_{0}}{r_{s}} + \\frac{g_{1}}{r_{s}^{3/2}} + \\dots\\right)\\ ,", "d9363d531d7ee5e536de2c4dca049132": "w(u)+w(v)\\ge S", "d93752577c61c63f2a8b0c9ddfd196a0": "J=j_\\mathrm{ion}^\\mathrm{sat}", "d9376da46d74139999fb90a9abc108fa": "B_t=W_t-tW_1", "d9379474a2ade8641cb10f2d25f40141": "d_r^{p,q} : E_r^{p,q} \\rightarrow E_r^{p+r,q-r+1}", "d937bdf53cf1a799ce3406ceb7159b38": "\\begin{align}\n\\mathrm{P}(A \\cap B) = \\mathrm{P}(A)\\mathrm{P}(B) &\\Leftrightarrow \\mathrm{P}(A) = \\frac{\\mathrm{P}(A \\cap B)}{\\mathrm{P}(B)} \\\\\n&\\Leftrightarrow \\mathrm{P}(A) = \\mathrm{P}(A\\mid B)\n\\end{align}", "d938864fd0f182a10e17502a089ff0ad": "F_l = \\{x| W'x=\\gamma \\}", "d938b3275e198099db51945bc1226c7b": " \\hat{H} = \\sum_{n=1}^{N}\\frac{\\hat{p}_n^2}{2m_n} + V(x_1,x_2,\\cdots x_N,t) \\,,\\quad \\hat{p}_n = -i\\hbar \\frac{\\partial}{\\partial x_n} ", "d938d46639e7055d5ec3bfd747d032e4": "\\epsilon( t, \\omega )", "d938e485d778ca9dcf58cd10a1f24ef2": "-A^{-1}(A+\\delta A)\\mathbf{v}=-\\mu A^{-1}\\mathbf{v}", "d938e59b9756c42f4ca0872b2185e29e": "A \\succsim\\! B", "d939117b4421b3909cfb1e1b81086150": "(R\\mathbf{v_1})\\times(R\\mathbf{v_2}) = (\\det R)(R(\\mathbf{v_1}\\times\\mathbf{v_2}))", "d9395e5518db2cb7eb6b65d425064947": "D_\\alpha", "d9398fc2da5673fb6c47f76c55abbc2c": "M(x,M(y,z))=M(M(x,y),M(x,z))", "d9399709874d742d13442ed0dff38620": " E_n = - \\frac{me^4}{2(4\\pi\\varepsilon_0\\hbar)^2}\\,\\frac{1}{n^2} = - \\frac{13.6\\,\\text{eV}}{n^2}. ", "d939af3c02344ba388fc91124f64d517": "P_m\\equiv\\sum_{\\alpha=1}^m e_\\alpha", "d939e95eb4cbfa6d56d71e3d96b5f3fd": "E(X^r;p,\\beta)=-r!\\frac{\\operatorname{Li}_{r+1}(1-p) }{\\beta^r\\ln p},", "d93a177066512aecd64daf61889dab70": "= \\frac{\\textrm{MYS}}{13.2}", "d93a39bc1b3b56a6f0247e8f7f0c2200": "h\\}", "d93a480a9c53f456b07a3825ee33ca10": "\n \tE[Y|Z=z, X=1]> E[Y|Z=z,X=0] \\text{for all } z, \\text{ while } \tE[Y|X=1]< E[Y|X=0]\n", "d93a5a30c1e75d5c1105ce8408fcac67": "\\frac{\\mathrm{d}\\mathbf{A}^{-1}\\mathbf{A}}{\\mathrm{d}t}\n=\\frac{\\mathrm{d}\\mathbf{A}^{-1}}{\\mathrm{d}t}\\mathbf{A}\n+\\mathbf{A}^{-1}\\frac{\\mathrm{d}\\mathbf{A}}{\\mathrm{d}t}\n=\\frac{\\mathrm{d}\\mathbf{I}}{\\mathrm{d}t}\n=\\mathbf{0}.", "d93b107b1dcd8eb6c737d7ded32441a4": "f' \\colon M' \\to X", "d93b40b24ad7a0062e91aed2994b2521": "S(-1)=f^{-1}(x), ", "d93b436f70c06837a496ecf62918a8fa": "\\Delta x = \\Delta y = \\Delta z = \\Delta l", "d93b6ba47812b85cf009343b0de17bb7": "\\displaystyle{\\|R\\|\\le \\left(\\max_i \\sum_j \\|R_i^*R_j\\|^{1\\over 2}\\right)\\left(\\max_i \\sum_j \\|R_iR_j^*\\|^{1\\over 2}\\right),}", "d93b9c4aa7e98819998a5d1fce4e5858": "m = \\frac{y_2 - y_1}{x_2 - x_1}.", "d93bda6f04bc98048a10ceeea9fe9b1e": "\\theta\\in[-U,U]", "d93bf50e72bc663792642bcce8d2064c": " l, j , m_\\text{j}", "d93c38402660a79969a0234e5b789b81": "[0,\\infty)\\,", "d93c5e124d413e53755d9e47ca43e7ff": "E = mc^2 \\,\\!", "d93c7b09d52749ac2778e172aeb4b7bf": "A_{\\ell m}^{(1)}", "d93cbacf6cad0d5042175f814cc9e0ef": "\n\\int_1^\\infty \\cos(ax)\\frac{\\ln x}{x} \\, dx =\n-\\frac{\\pi^2}{24}+\\gamma\\left(\\frac{\\gamma}{2}+\\ln a\\right)+\\frac{\\ln^2a}{2}\n+\\sum_{n\\ge 1}\\frac{(-a^2)^n}{(2n)!(2n)^2},\n", "d93cee9c404536f02e89ce4b5098e096": "\\frac{2\\pi}{\\tau}", "d93d0510d1f73efe94fdbde622a15f78": "A\\mapsto A\\otimes B", "d93d42d08d887761d46ea317a5b7ab54": "0.2_6\\,", "d93d6f1ff306457b9438582d26efdf20": "\\frac{\\sin A}{20} = \\frac{\\sin 40^\\circ}{24}.", "d93e1018314db4920f7398ebae716686": "(\\ ,\\ ) \\!\\,", "d93e203fd78898fc426a53a063613005": "f_c'(z_0) = \\frac{d}{dz}f_c(z_0) = 2z_0 ", "d93e264771d8bfe8b3c16b5c52e9bfe1": "\\Delta p_{\\text{B}}(0)", "d93e428f77c24b3aaaae6f2d1fdb8927": "\\varphi\\mapsto 2\\cdot D_{1/2} (h * \\varphi)", "d93e5e2a975e46efa06277cdb1e69450": " \\mathbb{G}_{a} \\to \\mathbb{G}_{a}", "d93e89154122844b90ad71c9a0ec3d35": "\\varphi(x) = x^{-1}", "d93e95ffd391510f8a9696fa40dfb973": "\\frac{\\partial\\theta}{\\partial t}~=~\\alpha~\\nabla^2\\theta ,", "d93ee314f1a277096a43adf09aa5a3e5": "\\mu_g(E) = \\inf\\left\\{\\sum_i \\mu_g(I_i) \\right\\vert \\left. E\\subset \\bigcup_i I_i \\right\\}", "d93f29244e61ae1dffe3ca877f9937fb": "y = kx.\\,", "d93f57d24bbe3378bf1116d752877d4f": "A_{k}", "d93f78d95e83fa23c0a0d2f9d94a1ef1": "\\mathbf{e_3}=\\mathbf{e_2}+\\mathbf{e_1}", "d93fba774ab46fa4e11bc62c7e955d4a": "\\binom xn", "d93fd21d6bcc52eb0521ab46bd0c81c9": "(6)~~~~~\n \\left(\\frac{\\partial z}{\\partial u}\\right)_y\n =\n \\left(\\frac{\\partial z}{\\partial x}\\right)_y\n \\left(\\frac{\\partial x}{\\partial u}\\right)_y\n", "d93fdfb9cf841e3b377eb419a3e613f6": "\\sigma_v = \\sqrt{\\tfrac{1}{2}[(\\sigma_{11} - \\sigma_{22})^2 + (\\sigma_{22} - \\sigma_{33})^2 + (\\sigma_{33} - \\sigma_{11})^2 + 6(\\sigma_{12}^2 + \\sigma_{23}^2 + \\sigma_{31}^2)]}", "d93ff3fafabe44a9e7bd438b110559c8": "\\lambda \\,^{-0.7}", "d94007dc870dc14c400aa73e353493e9": "d_{n} = p_{n+1} - p_{n}", "d940141ea2c8676b1f81a1783ab4a36b": "g : Y \\to GX", "d9401bb9ad34c21dfb5a1cdc826c716c": "\\varepsilon \\left[ M \\right]=\\left\\{ {{\\left\\| f-{{f}_{M}} \\right\\|}^{2}} \\right\\}=\\sum\\limits_{m\\notin {{I}_{M}}}^{N-1}{\\left\\{ {{\\left| \\left\\langle f,{{g}_{m}} \\right\\rangle \\right|}^{2}} \\right\\}}", "d940558a683c175cad08fb9b510e3d33": "(p,T)\\ ", "d9405dc183fc452a766b83f2e4b6a4fd": "\n\\cfrac{ax}{1+axT_H}\n", "d94082335ee81c33a8324db8b2a91ccd": "\\eta = \\eta_c \\eta_p", "d9409266ef5ba89f638d090bf11a817a": "m*", "d9411350d03272611cf58de5ab69ac85": "\\varphi = \\tanh^{-1} \\frac{| \\mathbf p | c}{E}= \\frac{1}{2} \\ln \\frac{E + | \\mathbf p | c}{E - | \\mathbf p | c} ", "d94141d34136984fd3a0da7071e49b55": "e^{(2+i)x} = e^{2x} e^{ix} = e^{2x} (\\cos x + i \\sin x)", "d941a12d37c6be77afe3b640659d42f6": "\\displaystyle 2u_{tx}+u_xu_{xx}-u_{yy}", "d941c83e9286c229fcf1db1e7c23360f": "\nV(\\mathbf{R}) =\\frac{1}{4\\pi \\varepsilon_0} \\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^{\\ell}\n(-1)^m I^{-m}_\\ell(\\mathbf{R}) Q^m_\\ell\n", "d9421dcd59a85278d48af1c009518f6b": " t=t_0", "d9425db43da0314ccd0644b434d0ab7c": "2.6459", "d94262552a3b681ab7beb5e1bd3bf783": "\\Omega \\times G(n,m)", "d9427b1326209c920d70a3e43a5ea290": "P_{r}=\\left\\langle r\\left|\\frac{\\exp\\left[-\\beta H\\right]}{Z}\\right|r\\right\\rangle\\,", "d9428c553318bbc29bd7481578d99aea": "W = X/Y", "d942f62d0bc14d1037d75fa001fab2ef": "\\textstyle x_{i+} = \\sum_j x_{ij}\\,", "d94325184fc5366d0b2ecb970e46de06": "\\int_0^\\alpha \\int_0^{r_\\theta}", "d9432b3100499a2ce63020d45b3fbf56": "\\sin^2(X) + \\cos^2(X) = 1,", "d9433795515b9ca1e6e851d22bb6186d": "\\operatorname{Li}_2(-z)-\\operatorname{Li}_2(1-z)+\\frac{1}{2}\\operatorname{Li}_2(1-z^2)=-\\frac {{\\pi}^2}{12}-\\ln z \\cdot \\ln(z+1)", "d943d81b97e041527e112abf17f4f76b": "\\N\\,", "d943ed2b1d0a6206f1327e291d0b2e81": " \\tau_{yx} = k \\left | \\frac{du}{dy} \\right | ^{n-1} \\frac{du}{dy} = \\eta \\frac{du}{dy} ", "d943f00deb0d87280e4ff4a322aad4e8": "u_n \\in H_n(S^n)", "d944267ac25276f12cb03fc698810d94": "0=1", "d9442c2c65be12c5f8b3a784e8d4a153": "\\theta(0)=0", "d94442159e9c858adcc53daeabcbb9a1": " \\Box p ", "d9445c2acc922092c41489fc130b7937": "\\#(n)\\ge a^n", "d9449e61c7e61b26a4d0691464f37d31": "\\int\\ \\frac{dx}{\\sqrt{x^2+c}}", "d945113a7f3bbcc171365f2725432738": "\\begin{array}{ll}\nSS_\\text{between}/\\sigma^{2}\n& = \\frac{SS\\left(M_{i}\\left(X_{i,j}\\right);i=1,2,\\dots,K,\\; j=1,2,\\dots,n_{i}\\right)}{\\sigma^{2}}\\\\\n& = SS\\left(\\frac{M_{i}\\left(X_{i,j}-\\mu_{i}\\right)}{\\sigma}+\\frac{\\mu_{i}}{\\sigma};i=1,2,\\dots,K,\\; j=1,2,\\dots,n_{i}\\right)\\\\\n& \\sim \\chi^{2}\\left(df=K-1,\\; ncp=SS\\left(\\frac{\\mu_i\\left(X_{i,j}\\right)}{\\sigma};i=1,2,\\dots,K,\\; j=1,2,\\dots,n_{i}\\right)\\right)\\end{array}", "d94522570bc15a09d4ff7214a2132366": "\\scriptstyle \\rho_n", "d9453295a786b99f2310f5f22030e4d1": " \\int_{-\\infty}^\\infty \\mathrm{Ai}(t+x) \\mathrm{Ai}(t+y) dt = \\delta(x-y)", "d9454e755699ea24a6ef442ca1acd0f0": " E' \\to {\\mbox{Pic}}^0(E')\\to {\\mbox{Pic}}^0(E)\\to E\\,", "d9458cc7bccfcc23ccbc8c6ee55a09cd": "\\mathcal{X} \\times \\mathcal{X}", "d945c5fa668c3c6ad7881dc86a98c20c": "a< \\frac{\\lambda _{o}}{2}", "d945da4e7780680b506c12ca22623826": "R_{0,0} = 1", "d945ffd36dd93d9ac258bfd757995cd4": "\\frac{1 + {\\scriptstyle\\frac{3}{4}}z + {\\scriptstyle\\frac{1}{4}}z^2 + {\\scriptstyle\\frac{1}{24}}z^3}\n{1 - {\\scriptstyle\\frac{1}{4}}z}", "d94626244babd94d0a0a535837fc0ea2": " D = \\frac{ N - \\sqrt{ \\sum_{ i = 1 }^K n_i } }{ N - \\sqrt{ N } } ", "d94644e7f0d6a553256f0bf4bd32be4d": "H\\left(A,C\\right) = \\left\\{0,10,11\\right\\}", "d946578780aa215de5ea347742186fb4": "\\psi(\\Omega^{\\omega^\\omega})", "d946713126831d6fdf5ea6a1c0583509": "\\forall i~\\sum_{j=1}^N \\mu_{ij} = 1", "d9471eed942053d8d6369123d93ee2e6": "\\phi^x_t \\to \\exists x \\, \\phi", "d947291b6ce17f469c486330e683c114": "{{\\mathbf{k}}}^n", "d9481affcdb4872ec21a642c90069e8e": "w = 2^{x_1} + 2^{x_2} + \\cdots + 2^{x_r}", "d94853219aa2bf88c9bc901eaa3f45fb": " \\operatorname{drop-params}[g\\ q, D, V, [F_5, S_5, A_5]::[F_4, S_4, A_4]::\\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "d948859f85d08cf914626b63d6eee2ed": "0=N_F(K) - P_F(K) = N_{f}(K) - N_g(K),\\,", "d948b061273756c35e72bc3845dd280e": "2 d\\sin\\theta = n\\lambda\\,", "d948b5b49fec74d13d6c088537138aa1": "g^{x_2}", "d9494324f7d72b9e711159abbb1379d9": "\\Phi \\in H^0(End(F)\\otimes K),", "d94961974f5fc876545e06ec66537ab0": "x=r||s\\,\\!", "d949c32bfa516be3c119b9e3174866c4": "\\Pi(k)", "d94a191858037b3e95cb12e9a644694d": "\\mathbf{J}_{\\text{D}} = \\frac {\\partial}{\\partial t} \\mathbf D (\\mathbf r , \\ t) \\ , ", "d94a2a7cb0f5af4ef6d93143c7f404ab": "\\operatorname{Ric} \\ge 0", "d94a30ac6f5a47e10aa7f27aede7a440": "\\sigma=0.0450461875791687011756", "d94a4afb91f916c9161547c03e681371": "\\delta r(t)\\cong \\delta r(0)e^{-i\\nu t}+c.c.", "d94aa42e38e4ef613c5cb9a398883c3e": "P_4", "d94ac949779cfbf6968c92258fb53484": "\\left\\{\\sin \\frac{n\\pi x}{L}\\right\\}_{n=1}^{\\infty}", "d94aef9df6d986e56593bc7ce51b49f2": "E\\left(r\\right) = \\rho - \\frac{\\rho \\left(1 - \\rho^2\\right)}{2 \\left(n - 1\\right)} + \\cdots ", "d94aeff296ad22b02b9cefa7e2e1750b": " 0 \\rightarrow \\ker T~\\overset{Id}{\\rightarrow}~V~\\overset{T}{\\rightarrow}~\\operatorname{im} T \\rightarrow 0", "d94af5366c87f3c9d7e909ed2ebba46f": "\\int_0^T e^{-x t}\\phi(t)\\, dt \\sim\\ \\sum_{n=0}^\\infty \\frac{g^{(n)}(0)\\ \\Gamma(\\lambda+n+1)}{n!\\ x^{\\lambda+n+1}},\\ \\ (x>0,\\ x\\rightarrow \\infty).", "d94b0b84bd25713588c03517cc1ceb94": "\\frac{\\partial y}{\\partial \\mathbf{n}}(\\mathbf{x})=\\nabla y(\\mathbf{x})\\cdot \\mathbf{n}(\\mathbf{x})", "d94b142277762914a4680d3994f177d2": "h_{AB}", "d94b3469204f8a4cf128b6990b92fb93": "dV = \\rho^2\\sin\\phi\\,d\\rho\\,d\\theta\\,d\\phi.", "d94b7ce8b45ac6cdc8521105c3431a7f": "\\left|\\alpha-\\frac{p}{q}\\right| < \\frac{1}{\\sqrt{5}q^2}\\,.", "d94b7db5dcf7b5a76d8923f86ccd3a8f": "\n\\begin{bmatrix}\n\\left\\langle x,p_{1}\\right\\rangle \\\\\n\\left\\langle x,p_{2}\\right\\rangle \\\\\n\\vdots\\\\\n\\left\\langle x,p_{n}\\right\\rangle \\end{bmatrix}\n=\n\\begin{bmatrix}\n\\left\\langle p_{1},p_{1}\\right\\rangle & \\left\\langle p_{2},p_{1}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{1}\\right\\rangle \\\\\n\\left\\langle p_{1},p_{2}\\right\\rangle & \\left\\langle p_{2},p_{2}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{2}\\right\\rangle \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\left\\langle p_{1},p_{n}\\right\\rangle & \\left\\langle p_{2},p_{n}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{n}\\right\\rangle \\end{bmatrix}\n\\begin{bmatrix}\nc_{1}\\\\\nc_{2}\\\\\n\\vdots\\\\\nc_{n}\\end{bmatrix}.", "d94c494675a6ced11e7f454edac82713": "\\vec F_{n} = \\mathbf{A}^n \\vec F_{0},", "d94c4f2c80403926949061b4d139e547": "(\\mathbb{Z}/20\\mathbb{Z})^\\times", "d94d026521bc2a3cc11c545c7c156872": "\\left(X, \\lbrace e_i | i\\in J \\rbrace \\right).", "d94d4e6791cc969d4764bcba97bbfa7d": "(b) \\qquad |K(x,y) - K(x',y)| \\leq \\frac{C|x-x'|^\\delta}{\\bigl(|x-y|+|x'-y|\\bigr)^{n+\\delta}}\\text{ whenever }|x-x'| \\leq \\frac{1}{2}\\max\\bigl(|x-y|,|x'-y|\\bigr)", "d94d50b0e64e1a8edc2361644976c0c6": "\\frac{\\partial c_i}{\\partial t}=- \\mathrm{div}\\mathbf{J}_i=D_i[z \\Delta c_i - c_i \\Delta z] \\, .", "d94d5f253e11a895face1ad20112ceb2": "\\left[f\\left(\\mathbf{\\Lambda}\\right)\\right]_{ii}=f\\left(\\lambda_i\\right)", "d94d72e066fd43c12fb143f7bd8776ff": "\\qquad x_{n+1} = r x_n (1-x_n) ", "d94da6445b2bf76ea31ff28132592417": "\nE_{m} = \\frac{RT}{F} \\ln{ \\left( \\frac{ P_{\\mathrm{K}}[\\mathrm{K}^{+}]_\\mathrm{out} + P_{\\mathrm{Na}}[\\mathrm{Na}^{+}]_\\mathrm{out} + P_{\\mathrm{Cl}}[\\mathrm{Cl}^{-}]_\\mathrm{in}}{ P_{\\mathrm{K}}[\\mathrm{K}^{+}]_\\mathrm{in} + P_{\\mathrm{Na}}[\\mathrm{Na}^{+}]_\\mathrm{in} + P_{\\mathrm{Cl}}[\\mathrm{Cl}^{-}]_\\mathrm{out}} \\right) }\n", "d94dfaa007873b6dd546187f82a2cf35": "a_i(\\mathbf{x})", "d94e696727c30a6521de6952873de3ce": "d = s - C_h", "d94e6d3c3ab44230037ea02fb1c9ab0a": " r = a\\ \\operatorname{csch} (n\\theta) ", "d94e7fed2ea4643c203f484a4e43d073": "12 \\otimes 3", "d94ea01dc98f9a71e2bb0b1a309e4675": "\\beta \\mapsto \\omega^\\beta", "d94ecd36b6d12d96eb528df459da2d0c": "\\sigma_a(\\zeta) = \\zeta^a", "d94edcc9be0a454646acb7f64a25df76": "\\varepsilon=\\mathrm{sgn~\\det}\\partial x\\left( y,0\\right) /\\partial y", "d94efaa00cccf525755ec85f2c2e1e33": "V_t(x,t) = A \\sqrt {4\\rho\\cos^2 kx+(1-\\rho)^2} \\cos(\\omega t + \\phi),\\,", "d94fa5be45dea87ca8d2c027ce306c46": "V(x,y)=-V_0(\\cos 2\\pi x/a+\\cos 2\\pi y/a), V_0>0", "d94fb7dd8e609556ab9d38d5c36322be": "T(t,r) = \\frac{Q}{4 \\pi k} \\mathrm{Ei} \\left( \\frac{r^2}{4 a t} \\right)", "d950221abe34c56ee23146d834e2a9c4": "ik_1(B_re^{iak_1}-B_le^{-iak_1})=ik_0(C_re^{iak_0}-C_le^{-iak_0})", "d9507971c2c36249d03ea7dc2dfa28c0": " T_s = 289.5 ~\\mathrm{K} ", "d9508439757341356af7e6cd49d23af1": "M_{A,B} \\rightarrow E_{a,d}", "d950a2594b53400941f4627fed78cfd5": "(e^{-x^2}\\,y')' + e^{-x^2}\\,\\lambda\\,y = 0.\\,", "d950d859f3c2c19c2996a1e29ef85d5a": "I_{x_4}r=I_{x_5}r=-1", "d9510f72a9ffc3b13acc9480eb57609a": "O_{ij} = ", "d9512fa9cc76bbd4a0c5b36cbb9615e4": " Q_1 ", "d951770f115ade392ad0d78a647b1fb8": "\nO\\left(M(m)\\log^2 m\\right) = O\\left(M(n)\\log n\\right). \\, \n", "d951b3ba91c30f476940efa87c4682af": "\nz\\rightarrow \nz^9-36 z^7 (x^2+y^2)+126 z^5 (x^2+y^2)^2-84 z^3 (x^2+y^2)^3+9 z (x^2+y^2)^4 + z_0\n", "d951da96a29a6419818c349e68eee68a": "\\sin{(\\chi_{nk}/2)}", "d95225c171b3d5247ec592be8e362882": "\\delta_\\omega g = \\omega^2 g", "d9525c9e0524617e86908b0a849e3d17": "0.7777\\ldots \\;=\\; \\frac{7}{10} \\,+\\, \\frac{7}{100} \\,+\\, \\frac{7}{1000} \\,+\\, \\frac{7}{10000} \\,+\\, \\cdots.", "d95318f927683a3babe1f7c182cf5e81": "a\\equiv b\\mod\\begin{cases}4n,&n\\equiv2\\pmod 4,\\\\n&\\text{otherwise.}\\end{cases}", "d95352e6a8184de1027314b9ebb90d64": "C_e = C_X - C_{\\hat{X}} = C_X - C_X A^T(AC_XA^T + C_Z)^{-1}AC_X .", "d95370625bfc78ad809541e9041cdbc7": " \\Delta L = R \\omega t_1. \\, ", "d953c27d6bf97977b3e79992697d142b": "P(a) = I-Q(a)", "d953f8ef5be3008dbdbb832bd381c04d": "|g\\rangle", "d9542820ee8691618f3f2c7e45aa4315": "\\leftrightarrow, \\nleftrightarrow, \\longleftrightarrow \\!", "d95460edda9b82b1875826d61a6872d3": " (gL_1 \\times gL_2) \\cdot gL_3 = 0 \\qquad \\qquad (3)", "d954caec8034ff50a5b21a53968c8d31": "F(x) = \\begin{cases}1, &x \\geq c,\\\\0, &x < c.\\end{cases}", "d954fc98bb709c04e4c127aaf311f58a": "\\alpha_{k}", "d954fd00b23198afb3e74a9def8eb041": "\\frac {\\psi '(t)}{\\alpha \\psi (t)}= - \\lambda = \\frac {\\phi '' (x,y)}{\\phi (x,y)} ", "d9550ce3c2c8a6ebd9576ffc1f42c26e": " \n\\sum_{n\\le \\lambda} \\left(1-\\frac{n}{\\lambda}\\right)^\\delta \\Lambda(n)\n= - \\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i\\infty} \n\\frac{\\Gamma(1+\\delta)\\Gamma(s)}{\\Gamma(1+\\delta+s)} \n\\frac{\\zeta^\\prime(s)}{\\zeta(s)} \\lambda^s \\, ds\n= \\frac{\\lambda}{1+\\delta} + \n\\sum_\\rho \\frac {\\Gamma(1+\\delta)\\Gamma(\\rho)}{\\Gamma(1+\\delta+\\rho)}\n+\\sum_n c_n \\lambda^{-n}.\n", "d9550d4fb957ffa0a88349708ec3b039": "\\hat\\mathcal{O} = \\hat x_{free} + \\delta\\hat\\mathcal{O} + \\delta\\hat x_{BA}[\\hat\\mathcal{F}]\\,,", "d95523c0fbe3943614f0f33ad71af0db": "f\\in K[T,X]", "d95578f9f241273ab5491601f91c1b25": "P(D_-) = (x^{-1/2}-x^{1/2})P(D_0) + P(D_+)", "d955c7b43f85ea08a90fe4210feb9c37": "G_{0}^{(i)}(x_{1},...,x_{n}) = \\sum_{k} p_{k}^{(i)}x^{k}", "d955ea1f9e93961962312b31f3d43469": "C(X_1, X_2, \\ldots, X_n) \\equiv \\operatorname{D_{KL}}\\left[ p(X_1, \\ldots, X_n) \\| p(X_1)p(X_2)\\cdots p(X_n)\\right] \\; .", "d955f3c9194ed8d7bdea151a82967c97": "\n {k+r-1 \\choose k} = \\frac{(k+r-1)(k+r-2)\\cdots(r)}{k!} = \\frac{\\Gamma(k+r)}{k!\\,\\Gamma(r)}.\n ", "d9561625c1aaa61b8b6cc6e271b795c7": "tangent = \\frac{16}{z} \\,", "d95676cca9cf23e0ddf2361af5248679": "\\scriptstyle \\mathbf{A}_\\text{total} \\;=\\; \\mathbf{A}_2\\mathbf{A}_1", "d95679752134a2d9eb61dbd7b91c4bcc": "o", "d9567a86cd78914ce35fcd98c2e6b935": "u(x,y,z)=A(x,y,z) \\exp(i \\phi(x,y,z))", "d956abf7962207524536dac300ec7712": "(x_U,y_U)", "d956cdb7499333e9019ab551b9d8d803": "\\frac13", "d956d573eb9c6a27eb06e8fb1655e2cc": "\\mathcal{G}=\\mathbf{E}\\cdot\\mathbf{B}", "d957504ea2945050b6779b8cbaf63e31": " M^n \\to N^q ", "d95762f007bba332972d1fb4d5d4bbeb": "E_s^j(z)= \\frac{1}{\\Gamma(j+1)}\\int_1^\\infty (\\log t)^j \\frac{e^{-zt}}{t^s}\\,dt", "d957b08c421e342e965b8032be03bc23": "\\mathcal{O}(n \\log n)", "d957d047ab24c25e1954c3028acd381e": "X_n = (\\mathfrak{X}, \\mathcal{O}_\\mathfrak{X}/I^{n+1}), S_n = (\\mathfrak{X}, \\mathcal{O}_\\mathfrak{X}/K^{n+1})", "d957e37645bc590286f5f5dc3931ade3": "\\phi \\left(t,\\tau \\right) = \\begin{cases} \\frac{1}{\\tau} \\exp \\left(-2\\pi \\alpha \\tau^2 \\right), & |\\tau | \\ge 2|t|, \\\\ 0, & \\mbox{otherwise}. \\end{cases} ", "d9582259baed7dfeaaf5c637c271daaa": "\\overline{\\lambda_{\\epsilon,g}}=\\frac{\\sum_{\\epsilon=1}^\\Epsilon \\lambda_{\\epsilon,g}}{\\Epsilon}", "d9582c09c9bc32be5317b4311e1b82f4": "(n - 1)^{th}", "d9585f127200825c693599460c6f1748": "\\mathrm{su}(3) \\,", "d95882f246d2956dc938f94682235add": "\n\\begin{array}{lcl}\n \\mathbf{b}_x= (\\mathbf{d}_x \\mathbf{s}_x ) / (\\mathbf{d}_z \\mathbf{r}_x) \\mathbf{r}_z\\\\\n \\mathbf{b}_y= (\\mathbf{d}_y \\mathbf{s}_y ) / (\\mathbf{d}_z \\mathbf{r}_y) \\mathbf{r}_z\\\\\n\\end{array}.\n", "d959cfe927e74351e8c46f890b49f8a1": "dP = - \\rho \\cdot g \\cdot dh.", "d959e036bcb9d81a5f8c46e1051545e3": "\\hat{H}=-\\frac{\\hbar\\Delta}{2}\\hat{\\sigma_z}+\\frac{\\hbar\\Omega_{\\perp}}{2}\\hat{\\sigma_x}", "d959faa9378b5dfd240a5469b0fddceb": "v_e\\,", "d959ffc1f4904b7c1e401813cf0bce91": "f:M \\to \\Bbb R", "d95a06f033fbc0a13fff522c9b7d13a9": " \\frac{\\partial u}{\\partial t}=- \\nabla \\cdot \\mathbf{q} ", "d95a50bca2438bcb4be77e2cb523779b": "L = - m c^2 \\sqrt {1 - \\frac{v^2}{c^2}}", "d95a6f1fad029487f9dd2a902d6e41e8": "k>v,", "d95a8a14cec571415a61d956f13c29ab": "\\delta(t-kT)", "d95aa848b57bff598e3e0d6ff765a61d": "\\frac{\\mathrm d}{\\mathrm d t} = \\frac{\\partial}{\\partial t}+\\mathbf u\\cdot\\nabla", "d95adb386591ff452b16dd87abba31ff": " \\frac{\\mathrm{d}\\mathbf{p}}{\\mathrm{d}t} = -{\\partial \\over \\partial \\mathbf{r}}\\left ( \\sqrt{(\\mathbf{P}-e\\mathbf{A})^2 + (mc^2)^2} + e\\phi \\right ) \\,\\!", "d95b2afe9acf681c40243d4d61086445": "R^{i+1} = R \\circ R^i", "d95b382ed0239faeccbb338b737f2c31": " a'_{k\\ell} = a'_{\\ell k} = (c^2-s^2)a_{k\\ell} + sc (a_{kk} - a_{\\ell\\ell}) = 0 \\,\\! ", "d95b98e42f8c9b88c70a4f92a7336837": "U'=\\{\\omega:|g(\\omega)|=+\\infty\\}", "d95bca60d987fff5510aa9f4c7949db7": "y''-4y'+5y=\\sin(kx)", "d95c063f73ebcf46fd4beb4e32f070ca": " 0 \\le \\rho \\le \\min\\left\\{ \\frac{ \\theta_1 }{ \\theta_2 }, \\frac{ \\theta_2 }{ \\theta_1 } \\right\\}", "d95c078b5b2a0bf4a28206ae088b4004": "\\sigma_f=E\\epsilon_f", "d95c7dc60f3b709a84b19427048b1d9e": " \\boldsymbol{\\sigma} (\\boldsymbol{\\sigma} \\cdot \\mathbf{q}) = \\mathbf{q} + i \\mathbf{q} \\times \\boldsymbol{\\sigma} ", "d95ca4a74f400a86fd018a7053e72451": "\\,\\mbox{R}(z, t) = \\exp\\left(- \\frac{i}{h}\\ t\\ l_z\\right)", "d95ce6af8e793545684a7a78d933db74": "\\Delta x \\Delta p_x \\approx \\left( \\frac{\\lambda}{\\sin \\varepsilon/2} \\right)\\left( 2\\frac{h}{\\lambda}\\sin\\varepsilon/2 \\right) = 2h", "d95d413000b0e487872aa87580b69405": "\n \\begin{matrix}\n & \\underbrace{a_{}^{a^{{}^{.\\,^{.\\,^{.\\,^a}}}}}} & \n\\\\ \n& b\\mbox{ copies of }a\n \\end{matrix}\n ", "d95d614f1e4658aa6ff8b5ba2566ba5f": "c_p=\\omega/k", "d95d73977b6889023802b5662ee2d467": "\\mathrm{(Fe,Mg)_2SiO_4 + nH_2O + CO_2 \\rarr Mg_3Si_2O_5(OH)_4 + Fe_3O_4 + MgCO_3 + SiO_2}", "d95e343056c5fb62015c4f2d9195752d": "\\psi_\\alpha", "d95e3f23833e523c6232af9c8805b81e": "A\\! \\succsim\\! A\\!", "d95e430960dddb06080a21387d567dfd": "\\lfloor\\pi\\rfloor=3", "d95e6553266efe91d52e50449ffe8053": " \\lambda = \\langle y , y \\rangle^{-1} \\langle x, y\\rangle", "d95f17bb027f32a87483f8e618ce3fad": "R_{\\alpha\\beta}^{\\ \\ \\ IJ}", "d95f3401e7cb33e1fe67178f7b4bfd86": "\\mu m", "d95f551458502686b5c53b49c80ffe29": "g:\nX \\to Q", "d95f6d89dc6630e18e43121f05eb4cb9": "\n\\omega_{2} = \\frac{d\\theta_{2}}{dt} = k \\frac{d\\theta_{1}}{dt} = k \\omega_{1}.\n", "d95f82dd42019d3cc082c7aecbca3682": "A_1 , A_2, \\ldots \\in \\mathbb{B}_b(S)", "d95f857253bdc6807a19d5aecb7b0b38": "S_*", "d95fd1519e587418ebe3da8fb081701f": "\\tau \\,", "d95fe7d09c11aeccb3e95897edbe511e": "\\mathbf{R}^n", "d960056d47bc0fd3539d034e53d25ce4": "\\mathcal{Z}_{\\rm gr} ", "d96021cafd7db2a2018b040efdf895d3": "\\frac{14}{11}", "d960905c4886139d0641358aa4c47763": "\\pi_j : \\prod_{i \\in I} X_i \\to X_j \\mathrm{ , } \\quad \\pi_j((x_i)_{i \\in I}) := x_j", "d960bb9e5104db2e3072503f325518dc": " \\tau : E \\rightarrow 2^{\\Gamma^{*}} ", "d960d1eec3c335f92da5340647f43db0": "\\sum_{k=0}^n k \\tbinom n k = n 2^{n-1}", "d9611154f4e44f0af810f0c370285a7f": "~\\mathrm{{}_{~94}^{239}{}Pu+{}_2^4{}He \\longrightarrow {}_{~96}^{242}{}Cm+{}_0^1{}n}", "d961282d8c6ccc3498305401d63a0e72": " \\lim_{n \\to \\infty} \\operatorname{Var}(\\overline{X}) = \\rho.", "d9612e0d12c9e9045ce71dfdd018f6de": " \\zeta(k) = \\lim_{\\varepsilon \\to 0} \\frac{\\zeta(k+\\varepsilon)+\\zeta(k-\\varepsilon)}{2},", "d961464744d190e8a914c7fca4f1fc5e": " + \\frac{E_\\mathrm{LO}^2}{2}(1+\\cos(2\\omega_\\mathrm{LO}t)) ", "d9618a64222f607a28043e07b3089e1c": " -i\\hbar\\frac{\\partial }{\\partial x} \\psi = p \\psi.", "d962565bb57371a702f4b568686222c7": "f(x) = \\frac{1}{x^3 - 1}", "d962d52753f568e01d2d2c8b25dbae02": "\\upsilon \\triangleleft S", "d9634e202c37ceab03b83ca7db684ae6": "d(\\varphi,\\beta)", "d963c48a9760e4baca2d01323b784ba1": "c_{12}^2-(b_{12}-a_{12})^2", "d9645ac2053977e7f53568aee464dfbd": "TD = MD - pitch ", "d96477459d6dd9008d123682afb7c8aa": "dy = \\frac{\\partial y}{\\partial x_1} \\Delta x_1 + \\cdots + \\frac{\\partial y}{\\partial x_n} \\Delta x_n.", "d964913cbd9b28cc2a0f4e0f514c733f": "\\phi_0=hc/e", "d9652f0e8c9aea3077e2e634e0f227fa": "\\epsilon N", "d9653fed7a8879151f6fdc73ba52603c": " \\bigl[ \\begin{smallmatrix} 1 & 1 \\\\ -1 & 1 \\end{smallmatrix} \\bigr] ", "d96567b32ad6f386a69e01b862a21fa3": "L(x_i) = y_i", "d9659f3c7c99201444a249ce203aad7f": "Sum_n=22", "d9666d6ce540ec895f9c0911e446dfab": "\\lambda_1 = \\alpha,\\quad \\lambda_2 = -\\gamma.\\,", "d967237960f3b9b2fadd839e70ed2a29": "\\Delta (l,p) = \\begin{cases}\np^{\\left (\\tfrac{1-\\ \\gamma\\,\\!}{2}\\right )}l^{\\left (\\tfrac{-1}{2}\\right )}\\log^{2}p &\\text{, if } l \\geqslant p^{\\gamma\\,\\!}\\\\\np^{\\left (\\tfrac{1}{2}\\right )}l^{-1}\\log^{2}p &\\text{, if } p^{\\gamma\\,\\!} > l \\geqslant p^{\\left (\\tfrac{2}{3}\\right )} \\\\\np^{\\left (\\tfrac{1}{4}\\right )}l^{\\left (\\tfrac{-5}{8}\\right )}\\log^{2}p &\\text{, if } p^{\\left (\\tfrac{2}{3}\\right )} > l \\geqslant p^{\\left (\\tfrac{1}{2}\\right )} \\\\\np^{\\left (\\tfrac{1}{8}\\right )}l^{\\left (\\tfrac{-3}{8}\\right )}\\log^{2}p &\\text{, if } p^{\\left (\\tfrac{1}{2}\\right )} > l \\geqslant p^{\\left (\\tfrac{1}{3}\\right )} \\\\\n\\end{cases}", "d9673c0053d6c98c1ceedd0692ef99b3": "=\\sum_{i=0}^{n-1}\\lambda^{n-i}d(i)\\mathbf{x}(i)+\\lambda^{0}d(n)\\mathbf{x}(n)", "d9675215835ba04244e28c4eaf3a368e": "\\sigma(X,Y)", "d9677611604e082771ca0fb150e65d8e": "| \\varphi \\rangle", "d968d7fdb86a28d4bea44e486cf8c211": "\\scriptstyle \\mathcal{H}^m(x)", "d968fafb8958d0b5031bbd098d385c1d": "S(T(r/s)) = \\langle r + s \\rangle", "d96903aebaa5973ee8cb4736ea3e4b47": "\n\\mathbf{H}_{SB}=\\mathbf{H}_B+\\mathbf{H}_U+\\sum_{}\\mathbf{S}_{x_i}\\mathbf{S}_{x_j}+\\mathbf{S}_{y_i}\\mathbf{S}_{y_j}+\\mathbf{S}_{z_i}\\mathbf{S}_{z_j}\n", "d969352a407a5f78adc07919bc59c667": "-\\mathbf{Q} \\cdot \\mathbf{P}", "d969c6fc421d3a542fc65f3448fa07fd": "\\scriptstyle v_\\mathrm i i_\\mathrm i", "d96aa8b78e5f3f815148a086d6733c09": "\\{(y,t) \\ : \\ |x-y|1.", "d9716341dba662779ca339766f6785a4": "\\tilde g = e^{2\\varphi}g ", "d9726612551961cf66d962e3d0038c37": "\\{X(t), t\\ge0\\,\\}\\,", "d9726aec38f50f07610f0b5a5ff7b423": "1-2+3-4+\\cdots=\\frac{1}{4}.", "d97289f9458050093a2eaf6a1a586802": "\\gamma(t) = e^{it} \\quad t \\in \\left[0,2\\pi\\right]", "d972901d21ca85c6b8858a6ba5c25fde": "K_m^\\text{app} < K_m", "d972a4e3f0d6aacf66c65636249f00d9": "\\dot{f}", "d972c24c2a433747f6e699dcc232c1f5": "{Du \\over Dt} = -{1 \\over \\rho}{\\partial P \\over \\partial x} + f \\cdot v", "d972d394e4e6bbacb192a92898bf9907": "C_*(X \\times Y) ", "d972e7b56fd56e005a57e5925184fa5e": "{32\\over 8} * {32\\over 7} * {32\\over 6} * {32\\over 5} * {32\\over 4} * {32\\over 3} * {32\\over 2}* {32\\over 1}={109951162776\\over 40320}", "d972f320add0d34d6a28b62a1b33a76a": "\\dfrac{\\alpha : X}{\\alpha : T\\backslash (T/X)}T_<", "d97304cbafc40d74d35ee27c96d0c58f": "I_{max}", "d9731b5d30bba7b08cdacb1574bd7dc1": "\n\tf(x) = \\sum_{i = 1}^K c_{i}\\varphi(\\left\\| x - w_{i}\\right\\|)\n", "d9733f47008a1cde3d9dbb1a9c2bcbb6": "\\epsilon = \\pm \\frac{a\\mathbf{\\hat{i}} + b\\mathbf{\\hat{j}} + c\\mathbf{\\hat{k}}}{\\sqrt{a^2 + b^2 + c^2}}", "d9734757e2e9d7363d18a5d238f419b4": "tol", "d9735dcb5606ed8e34e3f020b09f7653": "f(X_\\tau)", "d973c928361810cf07c117939c1553f4": " e(m,m) = 0 ", "d973dcba2ec4748480ef7c1750fc2364": " \\frac{\\text{d } {_2^1}P}{\\text{d}t} = - \\frac{\\text{d}[{^1_2}S^\\beta]}{\\text{d}t} - \\frac{\\text{d}[{^1_2}S^\\gamma]}{\\text{d}t}", "d97446f1d0af787d9932516e0f4179e9": "A=10", "d97469849883b3ee65358aaa13b6731d": "x = x'\\cos\\left(\\Omega t\\right) - y'\\sin\\left(\\Omega t\\right)", "d974798d86bfc49d02a176aa6d5c374b": "10; ", "d9a366c912318834905d1da06a236f3a": "C^{(4)}_{abcd}", "d9a464ae91c37147611582a76d05303d": "\\varepsilon_{\\rm{p}}", "d9a4b151136c1c40edb86adecdd8e045": "\n\\Omega = \\frac{r_{s} \\alpha c}{r^{3} + \\alpha^{2} r + r_{s} \\alpha^{2}}\n", "d9a4c8b77b1c560ac7dcd44857186b88": "a=n^2-m^2", "d9a5643679b43419b550abd8f4fc0d4c": "D(\\mathbf{X}, \\mathbf{Y}) =\\int_{\\Omega} \\int_{\\Omega} d(\\mathbf{x}, \\mathbf{y})F(\\mathbf{x}, \\mathbf{y}) \\, d\\Omega_x \\, d \\Omega_y.", "d9a580c029ee357a57ca29ce8a6889f4": "A^l", "d9a58c34b7cef342ba6406e752b1a0fc": "y \\le x", "d9a58dc2da8401287fcc230523cd20a4": "M=\\max\\ X_k", "d9a6a60812d052a132aa72edf44b54e4": "\\frac{6 4}{1 6} = \\frac{/\\!\\!\\!{6}\\;{4}}{{1}\\;/\\!\\!\\!{6}} = \\frac{4}{1} = 4", "d9a71fc62e116c310c9c2174db394beb": "\\overline{-x}", "d9a7a804004c0e6dd347449fa181f295": "f_{\\varphi}", "d9a7d11962d00abfe69260d41fe5bb15": "W_{2\\,p}=\\frac{(2\\,p)!}{2^{2\\,p}\\, (p\\,!)^2}\\, \\frac{\\pi}{2} \\sim \\frac{C\\, \\left(\\frac{2\\, p}{\\mathrm{e}}\\right)^{2p}\\, \\sqrt{2\\, p}}{2^{2p}\\, C^2\\, \\left(\\frac{p}{\\mathrm{e}}\\right)^{2p}\\, \\left(\\sqrt{p}\\right)^2}\\, \\frac{\\pi}{2} ", "d9a8515c4481232ccdc8e93ad63bf63d": "\\zeta(s) = \\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}}= \\frac{1}{1-2^{-s}}\\cdot\\frac{1}{1-3^{-s}}\\cdot\\frac{1}{1-5^{-s}}\\cdot\\frac{1}{1-7^{-s}} \\cdots \\frac{1}{1-p^{-s}} \\cdots", "d9a94a37ad1759a84f76d4b0c5e49d0c": "\\log_i e = \\frac{\\log e}{\\log i} = \\frac{1}{\\pi i/2} = -\\frac{2i}{\\pi}.", "d9a96b5976936d3608037d87afb5536a": " \\iota: H \\rightarrow \\widehat{G} ", "d9a9ce3734464ec8b2fa333bb4a87500": "\\neg B", "d9a9fabebd3acce217a73d873376055f": "\\text{P}_h", "d9aa46abfe45c26e1f3bd54c152d35d1": "A(i\\omega)=\\sqrt{\\frac{Z_{I2}}{Z_{I1}}}e^{-\\gamma}", "d9aa50ac7f706ecaf733cec6b60eece5": "\\overline{V 'R}", "d9aaf39e94ac847c9312dec8c8fb750c": "\\{ 1,~i_1,~i_2,~i_3 \\}", "d9ab187472140a413804a289e77f23b7": "\\begin{align}\nV_{12} & = V_1 - V_2 = (V_{LN}\\angle 0^\\circ) - (V_{LN}\\angle -120^\\circ) \\\\\n&= \\sqrt{3}V_{LN}\\angle 30^\\circ = \\sqrt{3}V_{1}\\angle (phase_{V_1}+30^\\circ) \\\\\\\\\n\nV_{23} & = V_2 - V_3 = (V_{LN}\\angle -120^\\circ) - (V_{LN}\\angle 120^\\circ) \\\\\n& = \\sqrt{3}V_{LN}\\angle -90^\\circ = \\sqrt{3}V_{2}\\angle (phase_{V_2}+30^\\circ) \\\\\\\\\n\nV_{31} & = V_3 - V_1 = (V_{LN}\\angle 120^\\circ) - (V_{LN}\\angle 0^\\circ) \\\\\n& = \\sqrt{3}V_{LN}\\angle 150^\\circ = \\sqrt{3}V_{3}\\angle (phase_{V_3}+30^\\circ) \\\\\\\\\n\\end{align}", "d9ab1cf359df5cb47e92390d834dff4a": "RP = R+RBI-HR", "d9ab29e8507125cba0c84ba4fea8a1a2": "= (4 + 289) \\cdot (3364 + 49) \\, ", "d9ab889f736526736fe0e1ac42821cc9": "E^2_{\\ast,\\ast}=\\text{Cotor}^{H_\\ast(B)}_{\\ast,\\ast}(H_\\ast(X),H_\\ast(E))\\Rightarrow H_\\ast(E_f).", "d9ab9ba3b739686612b6622d8f745083": " \\frac{Gmu}{r^2} = \\frac{2GMur}{d^3}", "d9aba546e64f66e7df6ceea00b9a9790": "R^f \\oplus \\bigoplus_i R/(d_i) = R^f \\oplus R/(d_1)\\oplus R/(d_2)\\oplus\\cdots\\oplus R/(d_{n-f})", "d9abc03f995bd3d6eef190f8622e4799": "\\sum_{u:\\,\\,(u,v)\\in E} f_{uv} = \\sum_{u:\\,\\,(v,u)\\in E} f_{vu}", "d9abf5c8f035aed1b297ea8d22cd9dbe": "\\eta=-2\\omega\\;", "d9abfaffff255b5440f17cee32b35e31": "\\Gamma_S = - \\frac{1}{RT} \\left( \\frac{\\partial \\gamma}{\\partial \\ln C} \\right)_{T,p}\\,.", "d9ac02b5932459d84893f6d5f08cec09": "n\\ge1.", "d9ac14e2bd2d44e95213270ebe7a7163": "M \\,", "d9ac4027dd5c179b47dcbb555d9bc722": "1+a(p)p^{-s} + a(p^2)p^{-2s} + \\cdots .", "d9ac4114a0432ef6d2ac9460bf347728": " y=\\frac{\\omega}{\\omega_\\text{c}} ", "d9ac471721842d6c5f93b0e60e08473f": "\\mathrm{d}U = T\\mathrm{d}S - P\\mathrm{d}V + \\sum_i \\mu_i \\mathrm{d}N_i\\,,", "d9ac5e20087284f0eea4283ff9610c40": "\\lambda_L=\\left(\\frac{m}{\\mu_0 n q^2}\\right)^{\\frac{1}{2}}", "d9ac8a803337b1d494df1664c13a9f40": "0=F(-\\infty)\\leq F(b) \\leq F(\\infty)=1", "d9aca7ee34fb9d99c5ad2adf0ba0c3ae": "H(s)=\\frac{K \\frac{\\omega_{0}}{Q}s}{s^{2}+\\frac{\\omega_{0}}{Q}s+\\omega^{2}_{0}}.", "d9accf9beb5c57597129302bcdf4b498": "|\\mathbf F|=k_e{|q_1q_2|\\over r^2}\\qquad", "d9ad360f5c5894436e5824dc6b95eb61": "(n_1h_1)(n_2h_2) = n_1 h_1 n_2 h_1^{-1}h_1h_2 = (n_1\\varphi_{h_1}(n_2))(h_1h_2)", "d9ad58608156c259ca9e671b822e68cf": " v = v( x_1, x_2, t ), ", "d9ad72a07782195e9e4ad7cb16f4e10b": "n e \\mu_e \\mathbf{E}", "d9adac491a90b8d81e6ecda3be5f8cc5": "\\varepsilon = +i,\\quad a = -2mn,\\quad b = m^2 - n^2;", "d9adcd69954f394fd5131490cb47cc2b": "\\mathrm{N}_\\mathrm{P}\\,", "d9adcf73a05d229f6b54754d7d2ef9f7": "\\dot{y} = b \\cdot \\cosh E \\cdot \\dot{E}", "d9add849334f0897a56c36f046528626": "R_T(w^\\ast) = \\sum_{t = 1}^TV(\\langle w_t, x_t \\rangle, y_t) - \\sum_{t = 1}^TV(\\langle w^\\ast, x_t \\rangle, y_t) \\ .", "d9ade80ce3ea81f7942ed3fbffb48175": "(\\mathbf{u}\\times\\mathbf{v})\\cdot\\mathbf{w}=\\mathbf{u}\\cdot(\\mathbf{v}\\times\\mathbf{w})=(\\mathbf{w}\\times\\mathbf{u})\\cdot\\mathbf{v}=\\det(\\mathbf{u}\\mathbf{v}\\mathbf{w}).", "d9ae052260ce0dd6d439f0d1cbd505f7": " \\left( {Q\\over T} = \\Delta S = 0 \\right) \\,", "d9ae229a5544da187a892ec119b2811a": "= 3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow (\\cdots (3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow 1))\\cdots ))\\, ", "d9ae50eb5e41414a92f405e077bf926e": " \\frac{dS}{dt} = - \\beta S I + \\gamma I ", "d9ae6111dd2e17d1ea5140e6e6bf83ae": "\\scriptstyle{A}", "d9ae8b6d676e5afc16b9aa5837fdcd4e": "v(\\{1,2,3\\}) - v(\\{2,3\\}) = 1 - 1 = 0\\,\\!", "d9ae97201a2b0931e149c81b741583fb": " K = \\sqrt {(s-a)(s-b)(s-c)(s-d) - abcd \\cdot \\cos^2 \\left(\\frac{\\alpha + \\gamma}{2}\\right)}.", "d9aebc64f748838a4832326165146441": "t_0\\ge t_1\\ge t_2\\,.", "d9aed1e3ecfc9c5d3951871b7291f5c3": " \\scriptstyle \\Omega ", "d9aeef58b838c00b8b96e8d4c087ba89": "L_\\mathrm{final}", "d9af65877e3b876e44a54d929ea8dcb8": "x,y \\in B", "d9af7bdfe3da2579d8abbd7b8b2b90e4": "s t", "d9af8cd28d47a8407b736460ec1ffe97": "{}^*\\mathbb{R}=\\mathbb{R}^{\\mathbb{N}}/\\mathcal{F}", "d9afacb15f7f8c689e8b1384a79ba0c3": "\n\\mathrm{Maximize: } \\sum_{a=1}^N\\sum_{b=1}^N\\mu_{ab}(t)W_{ab}(t) \n", "d9afca31d5840cecdeab5bbf1af63dd0": "\n- \\Delta T = 2 \\text{ in } \\Omega, \\text{ } T=0 \\text{ on } \\Gamma, \\text{ } \\partial_{n}T=0 \\text{ on } \\partial \\Omega \\setminus \\Gamma\n", "d9b005923a166f6dce92f73de124e338": "p \\notin r.Clause", "d9b00ba7d09c42f66acc0915fd36b203": "\n \\Delta x_{\\mathrm{meas}}\\Delta p_{\\mathrm{b.a.}} \\ge \\frac{\\hbar}{2} \\,.\n", "d9b06b0105f19ed78969edc6fb510991": "\\frac {\\cos(\\varepsilon - (\\alpha/2))} {\\cos (\\varepsilon + (\\alpha/2))}", "d9b0c3a5fdb436804c7a00f5571443b2": "f(n)\\in O(g(n))", "d9b0fb3906822afd632323751fc62682": " R= \\phi \\tan{\\beta_m}.", "d9b1b0375da7dc48c7f08ff1bbe01d08": "R_{n+1}", "d9b1ef83a06d7484e06c184c81da4679": "\\left\\{ S, S \\right\\} = \\left\\{ \\overline{S}, \\overline{S} \\right\\} = 0", "d9b1f9d4e2384f6fd6a6d54227e26395": "c_{i,0}", "d9b225d6df8fd71585d0d65a629ecaad": "p(C \\vert F_1,\\dots,F_n)\\,", "d9b26f255fc9f3a1a15e7a0c6a49ce7c": " g= S(\\tilde{f}),", "d9b2e121409db40f7f6768a92fa0abad": " H |\\psi\\rangle = (2^T |0\\rangle + (2^T - 2s)|1 \\rangle)/\\sqrt{2} ", "d9b3038801d300832a12311b6a845182": "\nq_i = \\frac{1}{N - 1} \\sum_{j \\neq i} \\frac{\\sqrt{\\langle r_{ij}^2\\rangle - \\langle r_{ij} \\rangle^2 }}{\\langle r_{ij} \\rangle}\n", "d9b31fd0404a106565aa0190e8088d8d": " y \\notin F(S)", "d9b32843a8c7755730a821fc44e00c5c": "M_{1} \\equiv \\int d\\zeta \\ \\frac{\\lambda(\\zeta)}{\\zeta^{2}}", "d9b37eececd1b342cbed06b97a0431f0": "\\Delta G_{mix} = \\Delta H_{mix} - T\\Delta S_{mix}\\,", "d9b39cf49bb8068959cbdeaba0ec2a22": "\\varphi\\rightarrow\\psi", "d9b3c33cf2b2f74a54c54f6064b69ecd": "{CE}_{9}", "d9b439a3882ba0b47538c825f0ae7182": "S^{1}(n),S^{2}(n),S^{3}(n),\\dotsb", "d9b450db53ebe0612beb82ad5957f734": "A(S,i)", "d9b471d1ab6472bb1e16a50ba1076015": "^{\\;}\\{\\xi^{k},\\xi^{l}\\}=-I^{kl}.", "d9b4a43932d3f688a9432e7efd61f91e": "\\ ln(1-x)= -x -\\frac {x^2}{2} - \\frac {x^3}{3}-...", "d9b4b087349109f034d9e359efbdb31d": "\\mathbf{n} \\cdot (\\mathbf{r}-\\mathbf{r}_0)=0.", "d9b4bbb65e6263a9ebcc96bf361d3418": " f:x\\mapsto e^x", "d9b564a7f712e9e4f3ae75cf097dfce5": "z=x+iy\\,", "d9b60bc6ca67628ce1cedf889e902cb9": "\\mathbf{x} \\leftrightarrow \\mathbf{x'}", "d9b61b8f245873d41ec20f71fefb4995": "\\underset{T\\to\\infty}{L^2\\!-\\!\\lim} \\frac{1}{T}\\int_0^T f(T_tw)\\,dt = \\int_{\\Omega_E} f(y)\\,d\\mu(y).", "d9b62db3f3c74df53ab08b697949fa19": "alive(3)", "d9b6cf551f4adf6e8c21c0f5dd65da1f": " y_i \\in \\{0,1\\},", "d9b74154b4ad2ba9b24c7b7cef5140e6": " y' = f(t,y) ", "d9b77b21431623613e439746d07afa85": "\\sigma \\in \\operatorname{Gal}(K/k).", "d9b7ac9fe9fb8d264bc16fb4e4d02a2b": " Rate =(\\Delta t/T)*c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1) ", "d9b7be1b67c967c8918e6c57dd15940c": " ds^2 = -dT^2 + dZ^2 + dR^2 + R^2 \\, d\\Phi^2, \\; \\; -\\infty < T, \\, Z < \\infty, \\; 0 < R < \\infty, \\; -\\pi < \\Phi < \\pi ", "d9b7f965dcc444c9ea23612df422e0cb": "Position \\ge 0.6", "d9b868fe1d88d72a06ffec05bce8c3cb": "\\Psi=\\begin{pmatrix}\ng_{n,k}(r)r^{-1}\\Omega_{k,m}(\\theta,\\phi)\\\\\nif_{n,k}(r)r^{-1}\\Omega_{-k,m}(\\theta,\\phi)\n\\end{pmatrix}=\\begin{pmatrix}\ng_{n,k}(r)r^{-1}\\sqrt{(k+\\tfrac 1 2 -m)/(2k+1)}Y_{k,m-1/2}(\\theta,\\phi)\\\\\n-g_{n,k}(r)r^{-1}\\sgn k\\sqrt{(k+\\tfrac 1 2 +m)/(2k+1)}Y_{k,m+1/2}(\\theta,\\phi)\\\\\nif_{n,k}(r)r^{-1}\\sqrt{(-k+\\tfrac 1 2 -m)/(-2k+1)}Y_{-k,m-1/2}(\\theta,\\phi)\\\\\n-if_{n,k}(r)r^{-1}\\sgn k\\sqrt{(-k+\\tfrac 1 2 -m)/(-2k+1)}Y_{-k,m+1/2}(\\theta,\\phi)\n\\end{pmatrix}", "d9b87798fdb339b91a8db6c1d6397180": "S=T=1", "d9b8f1724805d93a7cce5c14b97f6e88": "n_T = 2n_0", "d9b90befe9b5eff9274509115f916f62": "\\operatorname{id} \\to g'_* \\circ g'^*", "d9b94ea4402405a74eb32ce7e7b6bf7d": "\\text{RCA}_{c,i}=\\frac{{x(c,i)}/{\\sum_i x(c,i)}}\n{{\\sum_c x(c, i)}/{\\sum_{c,i}x(c,i)}}", "d9b96186d24d93e347ea662b6539eb80": "\\text{Cl}_2(\\theta) = \\mathcal{L}s_2^{0}(\\theta) ", "d9b9faa95cb195d63c9c0f24b86b3c59": "a \\times b \\div d^n\\pmod r", "d9b9fe8c548caf3cfcee7cb10a5fe244": " \\left| +z \\right\\rangle \\otimes \\left| \\phi\\right\\rangle \\quad \\phi \\in V ", "d9ba1807d2c933c5ed0e8750fb930b5e": "x={-{a \\over 2}}", "d9ba29ddc2266db7e91ce3e0d922aab8": "(u_\\alpha)_{\\alpha<\\kappa}", "d9ba3392242aa31aa73cfb5e3c619e0c": "\\mathcal{B}(X^*_{\\sigma(X^*, X)}, Y^*_{\\sigma(X^*, X)}; Z)", "d9ba598bab96f3a5783d2820551ff90d": "\\theta = \\arctan \\sqrt{2}", "d9ba63ae9e91e2ac4991a3dca931382c": "f(c_n)", "d9ba6fe4d185d53c4c1c062e5ac7a4d9": "\\dot{P}", "d9ba783ecf0c42e5e7bc3b1ec729a951": " \\Omega_x ", "d9baab23fff2fb5e946f198021a8ccec": "\\arctan (\\Im j / \\Re j)", "d9badedd05fe8868726a6814058826ec": "k=O(\\log n)", "d9bb76ca609e3e2b5484209311356860": "g(n) = \\sum_{1 \\le m \\le n}f\\left(\\left\\lfloor \\frac{n}{m}\\right\\rfloor\\right)\\quad\\mbox{ for all } n\\ge 1.", "d9bba149947c4298a110f5cfa7295800": "v_x=v_{\\text{out}} \\left( \\frac{Z_2}{Z_4}+1 \\right).", "d9bbb0d9b644aa77bb0ccb5788699228": "\\hat{\\mu}_{MAP} \\to \\hat{\\mu}_{ML}.", "d9bbb9bf6e23cf56256fa5fed4a071bc": "R_\\mathrm{linear}", "d9bbed12e9a2f73932ffa64eb4d5bac1": "\\mathrm{Gz} = {D_H \\over L} \\mathrm{Re}\\, \\mathrm{Pr}", "d9bc760300c82b7581052a14afec214d": "AB=BC=\\frac{d}{\\sin\\theta}\\,", "d9bca399710210767729ecf248377f45": "(q_i) \\neq R", "d9bd5067b209817adbd784b8d872e740": "y_i=-1", "d9bd5767b014b23577515289228939da": "area \\simeq \\frac12 \\theta r^2 \\, ", "d9bd9c478e4b5b8fe8b3b7f4e63b7583": " \\operatorname{Odds}(I|E) \\ge \\operatorname{Odds}(I)\\cdot P(E|I) ", "d9bde335836a2b5ec7a61d5027796da0": "\\operatorname{Spec} A/I", "d9be06397dd4cedc86f5e4a3d7195dd6": "\n w^0 = w^K + \\frac{\\mathcal{M}^K}{\\kappa G h}\\left(1 - \\frac{\\mathcal{B} c^2}{2}\\right)\n - \\Phi + \\Psi\n", "d9be14c66752826b8c3e1705214fcb62": "{W} = {U * \\Delta V_w} = {U*(2*V_1\\cos\\alpha_1-U)}", "d9be781a4ce562c6f74f05dd70c8c0c0": "\\mathcal M_n(\\R) ", "d9bea17f762c3b599d2371e7caf49d54": "P\\times\\mathfrak g", "d9bea33a729193f162b4f5c47302a6d2": " \\mathbf{B} = \\mu_0 \\left(\\mathbf{H} + \\mathbf{M}\\right)\\,.", "d9beb43fd759681b0e9bb04f4b546d5b": "X \\leftarrow Z \\rightarrow Y", "d9bee8c36850f89df944b6410a49dcab": "\n{n \\choose m} = \\frac{1}{m!}\\prod_{k=0}^{m-1} (n-k) = \\frac{1}{m!}\\prod_{k=1}^{m} (n-k+1)\n", "d9bf664a64ba14d1f1e8158298320a42": "p_2(x_n)", "d9bf7e61ede33d0147a0a47e80ede376": "\\boldsymbol{\\dot q}", "d9bf98dc985678dc1a2d0657f99ed74f": "c_n \\approx \\mu^n n^{11/32}", "d9bff2107cb3ca5ddd4c142f5e83cf22": "\n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 4 & 1 & 0 \\\\\n0 & 2 & 0 & 1 \\\\\n\\end{bmatrix}\n", "d9c00082aac31700ec5060cc247355a5": "\\omega_{\\operatorname{c}} = qB/m . \\,\\!", "d9c01091484311f52a805fd93359393c": "\\chi = \\frac{n \\mu_0 \\mu^2}{3k_BT}", "d9c02f2751f3875d64b3a025fff20fcd": "\\beta_{1,1}>\\beta", "d9c05a932a74ccd70a257912be5dc0bb": "C_n = \\int_0^{\\infty}\\frac{dx}{(1+x)^n\\left(\\ln^2 x + \\pi^2\\right)},\\quad n=1,2,\\dots.", "d9c061bce29b5d9e5796a31f0c5a7d72": "\\int e^x \\cos (x) \\,dx = {e^x ( \\sin (x) + \\cos (x) ) \\over 2} + C'\\!", "d9c0ece3f74c8c9b77e588fe588ecc5a": "Tr_{theta}", "d9c10875bab7bb0fb5163671ad570b60": "\n\\mbox{E}\\left[F\\right]=\n\\begin{cases}\n\\frac{\\nu_2(\\nu_1+\\lambda)}{\\nu_1(\\nu_2-2)}\n&\\nu_2>2\\\\\n\\mbox{Does not exist}\n&\\nu_2\\le2\\\\\n\\end{cases}\n", "d9c12277a4a365a11377d81c6b9b2c66": " U(\\mathfrak{g}) ", "d9c12c35e3ecf95d985aef6060a7a86f": "\\sum_{i\\in\\N}X^iY", "d9c15f7fc52214cf053ab4270241448a": "\\displaystyle{a^2 a^{m-1} =a^{m-1}(a^2)=L(a^{m-1})L(a)a=L(a)L(a^{m-1})a=L(a)a^m=a^{m+1}.}", "d9c1e8746d92b481d0ddb99c34e96e22": "\\frac{\\langle{y,Ly}\\rangle}{\\langle{y,y}\\rangle} = \\frac{\\int_a^b{y(x)\\left(-\\frac{d}{dx}\\left[p(x)\\frac{dy}{dx}\\right] + q(x)y(x)\\right)}dx}{\\int_a^b{w(x)y(x)^2}dx}.", "d9c1edad64192fc963a283938725f7ae": "V^{T}CV\\left\\{ z\\right\\} '+V^{T}KV\\left\\{ z\\right\\} =V^{T}F\\left\\{ u\\right\\}", "d9c228c762bd5ce66023f8d6ffd94c8e": "\n\\frac{\\Delta M}{\\Delta \\theta} = \\frac{\\mathrm{d} M}{\\mathrm{d} \\theta} = FL \\cos \\theta - k_\\theta < 0\n", "d9c29791dd3b792c7702ed2b7cf5ac40": "\\bar{\\alpha}", "d9c3738ad2d0326cd6a1e450f29b6987": " \\Pr(X_{n}=j | X_0=i) = p_{ij}^{(n_{ij})} > 0.\\, ", "d9c3751b140b97daa6ce602e0b2aea14": "Z_{10}", "d9c38280db835be571ad54c42a521195": "T=\\frac{1}{2}bh", "d9c3adc1524df72f61b21f901454a7a6": "\\sup_{y^* \\in Y^*} -F^*(0,y^*) \\le \\inf_{x \\in X} F(x,0), \\,", "d9c3dbd7325b8be694b91293fd4e9866": "2^{(n+3)} - 3", "d9c40a14be462dd0e7b15c9a1ea815cd": "\\phi \\,.", "d9c40d84cd8e537dac14b34675b3f275": " b^{c-d} = \\tfrac{b^c}{b^d} ", "d9c40fd5af431498ad7ba35e4e0e108c": "{u_n}", "d9c421a88d32dae55439bdfc572d77ea": "e^{\\gamma} ", "d9c46ac4ba41ac650c0e039c3cb4adf9": "\n \\cos\\, \\psi = \\operatorname{cn} \\left( \\begin{array}{c|c} \\displaystyle \\frac{\\xi}{\\Delta} & m \\end{array} \\right)\n", "d9c4b266316bf7104dad228e1decc9d3": "(1-\\zeta_l)", "d9c4d2384265d65a93e402f4c5e12d8e": "S_{\\rm B} = \\frac{A k_B}{4 l_{\\rm P}^2},", "d9c4d6a5d3e30a0cc7b132a8f8fa5dd1": " \\Delta G_{ad}", "d9c523073b51a3ae71ca1bcd2bab0971": "(8)\\qquad \\Psi_0\\,\\hat{=}\\,0\\,,\\quad \\Psi_1\\,\\hat{=}\\,0\\,,", "d9c539f323b4dfb30ad9a0753696513d": " x = A^{-1} y", "d9c65ecd3395e3c609a3ad96660334d7": "\\omega_s=\\omega_i=\\omega_p/2", "d9c67850d4fe0180a03746ac09068dbb": "\\frac{1}{3}\\left(\\text{base} \\times \\text{height}\\right)", "d9c6a790bc54a10444c9c322cbb2a277": "{\\hat{v}}", "d9c6eb13212bc1128fa2bacf21bfdc1e": "\\pi_1 (\\mathbb{C} P^n)=1", "d9c776e7293e28d71f0485d48431ea84": " \nv = \\left ( 1,1,2 \\right )^T \\in \\Gamma_2^*\n", "d9c7d347e38779eda600a713f72e9cee": "(Sh_K(G,X))_K", "d9c80fa69689853aad1af028d51c8887": "{\\mathbf E}(t)", "d9c81605ca4039a3b73bfd8ec7562d10": "v = \\sum_{i =1} ^m \\alpha _i u_i \\otimes v_i.", "d9c816276ec2509e89f1c51bb2c5f85a": " \\Delta \\boldsymbol{x} = \\frac {( \\boldsymbol{u} + \\boldsymbol{v} )}{2}\\Delta t.", "d9c81a7926c5ac0fba7e93e9fb97287b": "I : X \\to X", "d9c860e93aa720982bff8ace3bc3a9b6": " x_1, x_2, \\dots ", "d9c863932b122af94bda688e04cdf7e1": "P \\cap H_r", "d9c8736f8a1f2d71cad0102f15595790": " 1-\\alpha= \\frac{P+B}{Y} \\,\\ ", "d9c87eee5fbc340da8094ccca028aa68": "\\mathbf{A}(\\mathbf{r},t) = \\frac{\\mu_0c}{4 \\pi} \\left(\\frac{q \\boldsymbol{\\beta}_s}{(1 - \\mathbf{n} \\cdot \\boldsymbol{\\beta}_s)|\\mathbf{r} - \\mathbf{r}_s|} \\right)_{t_r} = \\frac{\\boldsymbol{\\beta}_s(t_r)}{c} \\varphi(\\mathbf{r}, t)", "d9c8bdb1c6db03992d6f1055ae5dc71f": "width=\\frac{W}{H} \\times height", "d9c8c9f5cf10b782e6fd6a18ab40a546": "\nx = k_{x1} \\cdot \\frac{I_4 - I_3}{I_0 - 1.02(I_2-I_1)} \\cdot \\frac{0.7(I_2+I_1) + I_0}{I_0 + 1.02(I_2-I_1)} \n", "d9c91c52179282b7c1e6b313763b65fd": "Radius\\ of\\ turn\\ in\\ metres = \\frac{TAS(m/s) }{rate\\ of\\ turn\\ (in\\ degrees/s)\\times 20 \\times \\pi}", "d9c9a1abcc5062bb976b736a64fcb7bb": "{\\ p = \\rho (\\gamma-1)U}", "d9c9d1c0759546bc68c0c8c61b4a426b": " i \n", "d9c9e6286e7b308e396eebbb1e83154e": " \\rho_1 = M\\;\\frac{P_1}{Z\\;R\\;T_1}", "d9c9ece2d72edc7a5cded383ddadf32f": " \\begin{align}\n\\hat{p}_x \\psi & = -i\\hbar \\frac{\\partial}{\\partial x} \\psi = p_x \\psi \\\\\n\\hat{p}_y \\psi & = -i\\hbar \\frac{\\partial}{\\partial y} \\psi = p_y \\psi \\\\\n\\hat{p}_z \\psi & = -i\\hbar \\frac{\\partial}{\\partial z} \\psi = p_z \\psi \\\\\n\\end{align} \\,\\!", "d9ca1a08dc0124ddb91350d33a57652f": " \\chi = \\frac{R^2+{R^\\prime}^2+(z-z^\\prime)^2}{2RR^\\prime}", "d9ca9b7d7b27fbb9f9f6f48fb0b3f521": "\\mu_{i1} = \\mu_{i2} \\text{ or } \\mu_{i1}^2 = \\mu_{i2} ", "d9caa8fd278706a921033aaaadb442ed": " C_p - C_V = nR \\,\\!", "d9cab7d98d11b12e6f46f123b70e069d": "M_p \\to N \\to B(Homeo(F))", "d9cb2213c616567148ad80a2c7d724ad": "\\{ x \\mapsto a, \\; y \\mapsto a+1, \\; z \\mapsto a+2 \\}", "d9cb6d5e25f593ac99fd41dd51af0eeb": "\\mathcal{H}^{B^R_j}\\otimes\\mathcal{H}^C", "d9cb7340fff4a6693c29aab5e88e77dd": " \\epsilon_1=\\epsilon_2, \\quad \\epsilon_3=\\epsilon_4, \\quad \\epsilon_5=\\epsilon_6", "d9cb741476305f454667e117d9e4f7ba": "3.", "d9cb9fc3d918cc11761204809c8b75c2": "P_c = (\\frac{k\\rho_c C}{\\mu H})^\\gamma", "d9cc160a662ee4689d42a221c77232ef": "k={\\log n \\over \\log 1/P_2}", "d9cc2c62eb941dde7110e42d93f81ba6": " \\lambda_T = (\\pi k_{\\mathrm{B}} T / d_{\\mathrm{F}}) / \\mathrm{sin}(\\pi k_{\\mathrm{B}} T/ d_{\\mathrm{F}}) \\approx 1 + (1/6) {(\\pi k_{\\mathrm{B}} T/ d_{\\mathrm{F}})}^2, ..........(28) ", "d9cc3ad3d099ea290753f81e15c66a1c": "V_\\text{P} = \\frac{E_\\text{P}}{q_\\text{P}} = \\frac{\\hbar}{t_\\text{P} q_\\text{P}} = \\sqrt{\\frac{c^4}{4 \\pi \\epsilon_0 G} } ", "d9cc57cd2a7b1d2c4aa21bf5db0315e6": " \\mathbf{A}\\cdot\\mathbf{U} = A^\\mu U_\\mu = \\frac{dU^\\mu}{d\\tau} U_\\mu = \\frac{1}{2} \\, \\frac{d}{d\\tau} (U^\\mu U_\\mu) = 0 \\,", "d9cc968de132278e78588e27cd23eaef": "u = -\\frac{1}{2} \\ln \\left( \\frac{y}{x} \\right)", "d9ccb1f45ce16b57e73fd6eb43704d2b": " R_{\\lambda\\mu}^i = \\partial_\\lambda\\Gamma_\\mu^i - \\partial_\\mu\\Gamma_\\lambda^i + \\Gamma_\\lambda^j\\partial_j\n\\Gamma_\\mu^i - \\Gamma_\\mu^j\\partial_j \\Gamma_\\lambda^i. ", "d9ccb2acc1eb24f2d76a7c7ce91b2d46": "\\varepsilon_{i_1,\\cdots,i_n}=0", "d9cd7cfc155ab5e3ecb994ceb5581c9c": "\\scriptstyle T_{ab} \\;=\\; T_{ba}", "d9cde43c131756743e5ef91eaae3c4ee": "dy/dt = f(y),", "d9cde67f4a6aafa5248d7afcf3b8e051": "\\int_0^{2\\pi} \\sin(n x)\\cdot g(x)\\,dx.", "d9ce1028fdc0542e2b30bdb07d1e0971": "f_\\mathrm{T}", "d9ce6cb55592fa4523587ddf8f165490": "\n\\sqrt{\\frac{1}{M N}\\sum_{i=0}^{N-1}\\sum_{j=0}^{M-1}(I_{ij}-\\bar{I})^2},\n", "d9cf47f538093f3587ce1b979ee2544c": "\\prod_{i=1}^m K[x]/p_i(x)", "d9cf5436938bf484c051ae0b026fc1c6": " a + L s_1 +L s_2 ... +L s_{D-p}", "d9cf859a45865f4458f7984ec4961102": "A= \\pi a^2", "d9cf87b7da06a338461524adec76c68a": "f/g", "d9cf8a40459434a979dd0dbdeee39c87": "kl_n", "d9cf91f0d9b6ec20b32e0bc926e0aee1": " n \\times p", "d9cfac4215818ecd89dc331798ae71b1": "\n\\psi^{(0)}(z)\n", "d9d0000189bb905f67ceec38d3c8b2bd": "\nRR_{total} = {\\frac{V^2}{2 C_p}} e ", "d9d0471f04ec4723d47692ee32bbb7bf": "E_n X = \\pi_n (E \\wedge X) = [\\Sigma^n S, E \\wedge X]", "d9d096151aec3032558c7a810da9ef5b": "\\sum_{a \\in A} f(a) w(a).", "d9d12f037ae78ea717c4848913376630": "G = g\\frac{R_\\text{earth}^2}{M_\\text{earth}} = \\frac{3g}{4\\pi R_\\text{earth}\\rho_\\text{earth}}\\,", "d9d13ff146da8c824bbbf7e27f78a4b1": "\\frac{\\mathrm{d}S}{\\mathrm{d}t} - \\frac {\\dot Q}{T} \\geq 0.", "d9d153ce26e95ada43fe47824ce88a5f": "t=\\frac{\\overline{d}-d_0} { ( s_d / \\sqrt{n} ) } ,", "d9d1c15e8adba8a36b66f80d66794dce": "\\mathfrak m\\otimes\\mathbb C= \\mathfrak m_{-}\\oplus\\mathfrak m_{+}", "d9d1d192b8d835cc63d0e02eac93005b": "\\rm \\ VF_4 \\xrightarrow{873K,N_2} VF_3 + VF_5", "d9d1d8b64fb5bbb726e00a0e32562f72": "Ex'(t) + Ax(t) = f,", "d9d205872695b6b815c122e0c7f4e720": "\\mathrm{DK}", "d9d21a601249af6d4cf3d914a3bbef96": "\\|x\\|_p = \\left(|x_1|^p + |x_2|^p + \\dotsb+|x_n|^p + |x_{n+1}|^p + \\dotsb\\right)^{\\frac{1}{p}}", "d9d234e715380b8ad95028a080f1707b": "r={{h^2}\\over{\\mu}}{{1}\\over{1+\\cos\\nu}}", "d9d24ea125615fdeaf65cedefcd3ae76": "\\hat{g}\\,", "d9d27ffee66760d275f0b2bc1d6a9aef": "|z| < 2", "d9d291f7e9285fd651d2db89b4d936b0": "\\mathfrak{g}_1 = \\mathfrak{g}/\\mathfrak{n}", "d9d29a3994c24b90c57e75328f0402b6": " a_0 ", "d9d33f4ebce7a61766050ddc5230e269": "\n\\begin{align}\n x\\left(\\theta\\right) &= {|\\cos \\theta|}^{\\frac{2}{m}} \\cdot a \\sgn(\\cos \\theta) \\\\\n y\\left(\\theta\\right) &= {|\\sin \\theta|}^{\\frac{2}{n}} \\cdot b \\sgn(\\sin \\theta).\n\\end{align}\n", "d9d35bba26d50399ec05ae883313ad42": "b\\subset U", "d9d37a4b40003657260fcffbecd23000": "\\mathrm{Im}(\\gamma) \\approx \\omega \\sqrt{LC}. \\, ", "d9d40c5f60998dbe34ab3ced3770f86e": "X_1=\\{-1, 0, 1\\}", "d9d469cb49fff0ff084eb89413351114": "C_{L}=\\frac{F_L}{{1}/{2}\\;\\rho AW^{2}}\\text{ };\\text{ }C_{D}=\\frac{D}{{1}/{2}\\;\\rho AW^{2}}\\text{ };\\text{ }C_{T}=\\frac{T}{{1}/{2}\\;\\rho AU^{2}}\\text{ };\\text{ }C_{N}=\\frac{N}{{1}/{2}\\;\\rho AU^{2}}", "d9d4f495e875a2e075a1a4a6e1b9770f": "46", "d9d509ad5b937af6a0e73f9c45bad59c": "\\log_e(3)=1.0986\\ldots", "d9d52ff9c4cdc275e7c6fa331ac584ca": "\\|x+y\\| \\le \\max \\left\\{ \\|x\\|, \\|y\\|\\right\\}", "d9d595f02000428452267f5276c37f4f": "\\overline{\\operatorname{Sp}}(E)", "d9d5c22bcb03bceff7ff72ac5a2096aa": "\\textstyle \\delta = \\deg(d(x))", "d9d5d6e410c9b44220ca0a95789a56bc": "a=(z_1,\\dots,z_n) = (x_1+iy_1, \\dots, x_n+iy_n) = (x_1,\\dots,x_n) + i(y_1,\\dots,y_n)=x+iy.", "d9d5f6f3f52331e8606690b0501df4f7": "\\mathbf{h}\\,", "d9d64374cb0b195fcc63a00b1e860120": "\\mathbf{j} = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}", "d9d65b524c7f1728587eac1b14f5aa28": "D : \\mathbf{Set} \\rightarrow \\mathbf{Set}", "d9d65e708619200408bbca25a232922b": "\\scriptstyle\\mathbf{n}", "d9d66cf866d9d6d668448fd5614150f2": "(Q_{\\geq\\mathbb{N}}x)\\phi (x)\\equiv\\exists a(Q_Hx_1,x_2,y_1,y_2)[\\phi a\\land (x_1=x_2 \\leftrightarrow y_1=y_2) \\land (\\phi (x_1)\\rightarrow (\\phi (y_1)\\land y_1\\neq a))]", "d9d698b363f3a13cdf18485eb5ff51d7": "\\frac{L'}{L_{0}}=\\frac{T'_{0}v}{Tv}=1/\\gamma", "d9d698bd0a501baa42c0f55aecf4fda4": " {\\rm int} V_i\\cap {\\rm int} V_j=\\varnothing", "d9d703ebe614a95f3afc420fea96130d": "{\\partial \\det(A) \\over \\partial A_{ij}} = \\sum_k \\mathrm{adj}^{\\rm T}(A)_{ik} \\delta_{jk} = \\mathrm{adj}^{\\rm T}(A)_{ij}.", "d9d70f70f6963378bc10445035439542": "\\frac{d}{dt} \\left( \\frac{\\vec \\omega}{\\rho} \\right) = \\left( \\left( \\frac{\\vec\\omega}{\\rho} \\right) \\cdot \\vec \\nabla \\right) \\vec v ", "d9d71b55a52b98babf5f58787765f623": "\\inf_{S_{k-1}} \\max_{x \\in S_{k-1}^{\\perp}, \\|x\\|=1} (Ax, x) \\ge \\lambda_k.", "d9d7268ea91a13cb5cdd5b870153e6ef": "\\mathbf{j}_f \\in \\mathbb{R}^3", "d9d729a2fc731c33ea1360682d73aad5": "FN", "d9d72b7f21bc3808d77b9e387473878e": "f=(b_0+mb_1)x+(c_0+2mc_1+c_2m^2)x^2+\\dots.\\,", "d9d733152398d65e3cc3ee7fe18b9995": "\\langle E \\rangle = \\sum_{i=1}^N \\langle E_i \\rangle.", "d9d73b94b002795abe96d37292162c79": "l + L = T\\,\\!", "d9d74de3577a145573e9666b32c767ae": "\n\\bar{Y} = \\frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4} \n", "d9d765ad6a3bc760e989036076177d14": "\n\\delta^{\\mu_1 \\dots \\mu_p }_{\\nu_1 \\dots \\nu_p} =\n\\begin{cases}\n+1 & \\quad \\text{if } \\nu_1 \\dots \\nu_p \\text{ are distinct integers and are an even permutation of } \\mu_1 \\dots \\mu_p \\\\\n-1 & \\quad \\text{if } \\nu_1 \\dots \\nu_p \\text{ are distinct integers and are an odd permutation of } \\mu_1 \\dots \\mu_p \\\\\n\\;\\;0 & \\quad \\text{in all other cases}.\\end{cases}\n", "d9d7a8cbdedd260ce48a80d667641f95": "\\forall \\lambda \\in L^{\\infty}_t, \\lambda \\geq 0: \\rho_t(\\lambda X) = \\lambda \\rho_t(X)", "d9d7cb49d614bd3695a8336e75f8e1a9": "f_{\\Gamma, R}^{*}", "d9d7d38232a799b416bbf23a7a233cc5": "\\scriptstyle\\sigma_X^2", "d9d7e99fc2fc353787b2eb802513e375": " \\ U ", "d9d80c730413ad1a91a24a47c673bea4": "E_k = \\frac{\\hbar^2 k^2}{2 m}", "d9d887417ac6830edf361301cf07b2ef": "MS\\subseteq S", "d9d88ca5c4eae8531c88ef87fad195f2": "\\scriptstyle \\varphi\\,-\\, \\frac{\\pi}{2}\\,", "d9d8c32b165a0d7643ec3895e560cf50": "\\sigma_{readout}", "d9d92ca2b3332c4e29ab436002b40cb1": " \\mathbb{Z}/p", "d9da225f79566baf22a33d810d09b58c": "R_R = 80\\pi^2 \\left ( \\frac {h}{\\lambda} \\right )^2 \\,", "d9da69e0e905dfe427a6e85829ebfa90": "y_i>x_i", "d9da9b611092ae410359f70bf285b869": " p\\Vdash\\phi", "d9daef8ffc1ee8b0f1cb9b6a6a09fff2": "v = 0.95 c", "d9db408cb0ab3807711af4297a8782f1": " (1-g)(1+g+\\ldots+g^{n-1})=1-g^n,", "d9db4349193d279771b225e06d6b63b8": "\\Omega_M \\simeq 0.27", "d9db70559175123600e9ac0df80753cb": "f_2(x_1,x_2) = 0, \\,", "d9db72f95aa01ed5befc78db9bdeaaf2": "\nv_{n} = \\sqrt{ 4 k_B T R \\Delta f }\n", "d9dbfaf7a6159c363e64f8e3601a3890": "y_{n+1}=\\sum_{i=0}^s a_i y_{n-i}+h\\sum_{j=-1}^s b_j f(t_{n-j},y_{n-j}).", "d9dc1a7bff0a45cdf98c2e223f8c325d": " \n \\begin{bmatrix}\n 1\\\\ 0\n\\end{bmatrix} \n \n ", "d9dc3560f68b5a75f64617f9cd6772fd": "F= 10 \\ \\log_{10} (f)", "d9dd33d463a435a478344ac204ced81e": "\\lim_{k\\rightarrow\\infty}|Dq(x_{2k})|\\ge C", "d9dd3a432a60bddff054af326c4e2487": "\\frac{dr}{dt_r}=0\\,\\!", "d9dd671828cf8b839d19d7a6b25ef17b": "\n\\left(\\frac{\\partial T}{\\partial V}\\right)_{S,\\{N_i\\}} =\n-\\left(\\frac{\\partial p}{\\partial S}\\right)_{V,\\{N_i\\}}\n", "d9dd767c715a7966b3ad3d2ef57faddb": "\\mathbb{C}_{\\sigma_c}=\\{s\\in\\mathbb{C}: \\text{Re}(s)>\\sigma_c\\}.", "d9dd76eb7c78c44af0b5356aa8f7cbdc": "\\deg(v_i)", "d9ddafa3c35259aa717ab3917eb47a3e": "[x:=x+1](x \\ge 4)\\,\\!", "d9ddc73fd875e2514c8faebc50358c3f": "\\mathrm{dim}_H(A)= \\sup\\{s\\ge 0:C_s(A)>0\\}.", "d9ddcb1fac8939775953f328125338ac": " Z[J]\\propto\\sum_k{D_k}", "d9de177a2a1e75070fa706d689a43126": "\\frac{d}{dz}p_n(z) = p_{n-1}(z).", "d9de1cf6bc8a4a51ed91f8ff68179102": "\\Rightarrow_{Y \\to S}\\ SYYY \\ \\Rightarrow_{Y \\to S}\\ SSYY \\ \\Rightarrow_{Y \\to S}\\ SSSY \\ \\Rightarrow_{Y \\to S}\\ SSSS", "d9de53e81b3616aa893aac012342f248": "\\begin{matrix}\np \\oplus q & = & (p \\land \\lnot q) & \\lor & (\\lnot p \\land q) \\\\\n& = & ((p \\land \\lnot q) \\lor \\lnot p) & \\and & ((p \\land \\lnot q) \\lor q) \\\\\n& = & ((p \\lor \\lnot p) \\land (\\lnot q \\lor \\lnot p)) & \\land & ((p \\lor q) \\land (\\lnot q \\lor q)) \\\\\n& = & (\\lnot p \\lor \\lnot q) & \\land & (p \\lor q) \\\\\n& = & \\lnot (p \\land q) & \\land & (p \\lor q)\n\\end{matrix}", "d9de8777b6dde782d07457fb0ec2043d": "dP = - \\rho(P) \\cdot g(h) \\cdot dh.", "d9dea426eb5ed95ffca6d6bc71404e38": "\\{f_i,f_j\\}", "d9dea8b40bed368eabd394877cc9c2e2": "\\mathcal{R}_\\mathcal{R}(C^{knn}_{n}) - \\mathcal{R}_{\\mathcal{R}}(C^{Bayes}) = \\left\\{B_1\\frac 1 k + B_2 \\left(\\frac k n\\right)^{4/d}\\right\\} \\{1+o(1)\\},", "d9df54fc75819bcbbd871c7479e1e5f7": "\\mathbf e_3", "d9df89cdf7c324e9a0e380cce3d2ad8f": " {N}_c=\\bigcup_{x\\in{N}}N^x. ", "d9dfe7ae2fd147da95455e01c1d3ed24": "$2.75 + $1.25 =", "d9e01dc48a41ed2c850d639c0e34e686": "\\frac{\\sqrt{2}}{\\sigma} F \\left(\\frac{p-\\mu}{\\sqrt{2}\\sigma}\\right)", "d9e054eea4c68c49cafc382075c0b1be": "\\frac{9}{7}", "d9e05babaa3aeec4af7c1803e3692f0f": "\\nu_n(t_m) ", "d9e0a4d56274c4a7e09e9a4f4b2e1502": "L=-{a_0^2\\over 8\\pi G}\\tilde f(l_0^2 g^{\\mu\\nu}\\,\\partial_\\mu\\phi\\, \\partial_\\nu\\phi)\\;", "d9e0e16a4a37a07a4c7cb2f05efd18e9": "\\{ \\infty \\} = Y \\setminus c(X)", "d9e118ad987fcf33af2f325c60983d01": "V(\\phi)", "d9e11e87f65b212ee148c75b676697a1": " \\Delta = u_x v_y w_z + u_y v_z w_x + u_z v_x w_y - u_x v_z w_y - u_y v_x w_z - u_z v_y w_x ", "d9e1b81a079b6d47a4c74b19df1ccd06": "f(z)={\\sin{z} \\over z^2-z}", "d9e1e80e5172a5feca10ac9803426d26": "= 10^{\\left ( 1.18\\times 10^{98}\\times97.6 \\right )} = 10^{\\left ( 1.15\\times10^{100} \\right )} ", "d9e1ebe1feb0533dd70c2e5e5937d156": "D = \\frac{k_\\mathrm{B} T}{6\\pi\\,\\eta\\,r}", "d9e23d6e65cb1092ca6146fb0fc0bde2": "|\\Psi^+\\rangle_{AC}", "d9e276a54ad26bedbc5792546d66fde6": "X \\sim \\chi_k(x)", "d9e30f53879f7ab6807fbf6551ac2692": "\\Delta = (Z_{11} + Z_0) (Z_{22} + Z_0) - Z_{12} Z_{21} \\, ", "d9e331dc38847d8990f24bfe096ecbb5": "\\mathbf n_2", "d9e33da4bc3a44b427c433ef9a19aa5d": " \\mathrm{rad}(B_j) ", "d9e3b885fc73724a6d15481057e82e75": " \\frac{n}{2}(p\\ln 2 + \\ln|\\mathbf{V}|) + \\ln\\Gamma_p\\left(\\frac{n}{2}\\right)", "d9e429b24fb06026ff1d4ebbacf11789": "r=a\\tan\\theta", "d9e445eede4c773561a13ddfef4bc6a8": " (f\\otimes g)^\\dagger=f^\\dagger\\otimes g^\\dagger:B\\otimes D\\rightarrow A\\otimes C ", "d9e457daa9b7351ef8a7184bdebf49db": "j_1 = 1/2", "d9e467339b68cc07ef04039787f6ec47": "\\begin{align}\nf(x) &=\n \\begin{cases}\n 0, & \\text{if }x\\text{ is rational} \\\\\n 1, & \\text{if }x\\text{ is irrational}\n \\end{cases}\n\\end{align}", "d9e47ca8326708e8e3b5a6025602586d": "f_\\mathrm{detected} = f_\\mathrm{rest}{\\left(1 - \\frac{v}{c} \\cos\\phi\\right)/\\sqrt{1 - {v^2}/{c^2}} }", "d9e4869a7b3af393093b242a43e71861": "\\iota_w\\omega", "d9e495aea921e47fe48698052cd10fb8": "\\pi.", "d9e498a02858f25a7ac3ef9a021e5fd7": "m^* = \\xi m_0 ", "d9e49fe67a0ce7d9feabde30eff4045a": " e^{-c x} = a_o (x-r) ~~\\quad\\qquad\\qquad\\qquad\\qquad(1)", "d9e4acc6f4d15be956b705e19850ff2e": "\\mathrm{ker}(A)", "d9e5165a2fe53616dc9b62463e189bfe": "(::=)\\subseteq V \\times (V \\cup \\Sigma)^*", "d9e5830c2587d99e8a98d5eb39acc0eb": "\n\\delta \\boldsymbol B = \\boldsymbol {\\ {(B\\cdot\\nabla)\\xi}},", "d9e586f64a766eced22bc04e7993bcae": "\nNT = K_2 \\times V_c \\times (\\tfrac {4d}{3D})^2 + K_3 \\times ( N_1 + \\frac {N_2}{10} )\n", "d9e5ac5393bef62b5b6caa8e5149d8a2": "\\ \\displaystyle g(d,s) \\ ", "d9e5d2febc1bef47dd75f8a652ea1d8e": "\\lim_{k\\rightarrow\\infty}\\mathbf{P}^k=\\mathbf{1}\\pi", "d9e626d37aea61f6d2833290cd7fa9f4": "\\begin{bmatrix} x \\\\ y \\\\ z\\end{bmatrix} = c \\begin{bmatrix} -1/16 \\\\ -13/8 \\\\ 1\\end{bmatrix}.", "d9e64d56a6d22175570c8d0797852f91": "c_{k,0}", "d9e6c2b548ee88df71766833907c16d9": " N_{+} = \\operatorname{ker}(A^* - i) ", "d9e6f29652dd1740574338ba936fb924": "(1-z^2)\\,y'' -2zy' + \\left(\\lambda[\\lambda+1] - \\frac{\\mu^2}{1-z^2}\\right)\\,y = 0.\\,", "d9e74824571471b0c30e1d6f8de7e2a2": "\n\\begin{align}\nSPI = {EV\\over PV}\n\\end{align}\n", "d9e766c2db97308e0a4125a42411c462": "\\mathbf{v} = V (\\cos\\theta, \\sin\\theta).", "d9e7b98272c0daa9903765798b63640a": "x_{K}", "d9e7bb3dee89191f47e7de0a91e8ee75": " Q = \\sum_{i=1}^n (\\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}} + \\mathbf{T}_i\\cdot\\frac{\\partial \\vec{\\omega}_i}{\\partial \\dot{q}}),", "d9e7bc2cb9d047c115c797f9a2a7b5bb": "[\\mathbf{a}]_{\\times} \\, \\mathbf{a} = \\mathbf{0}", "d9e7d8abe5a80b51c8a270797a844fca": "\\sin \\alpha=\\frac{|\\bar{r}|}{2}", "d9e7dc6c35de034f07d05be5d734c18b": "{A^2}_2 = {R^2}_2 + {S^2}_2 = (C)-(E)-(G)-(H)", "d9e7eefe248400e299690045785c7423": " x^{24} + x^{23} + x^{18} + x^{17} + x^{14} + x^{11} + x^{10} + x^7 + x^6 + x^5 + x^4 + x^3 + x + 1 ", "d9e7f5ca50dc2a2538fbb9452a71891d": "\\boldsymbol{D(t)}=\\varepsilon_0\\int_{-\\infty}^t \\ \\varepsilon_r (t-t') \\boldsymbol E (t')\\ dt' , ", "d9e87a36e8ca3aa511be890f8af7d25d": "{ \\partial^2 u \\over \\partial t^2 } = c^2 { \\nabla^2 u}~,", "d9e8824345e5de686aa34b87aa16f18f": "d_{AB}", "d9e8b01f7e25162a755acd8a9d19fb29": "\\bold{a}_2=-{m_1 \\over m_2} \\bold{a}_1. \\!\\,", "d9e8d7bde3730b58af7ece6b9495f959": "\\max_{v \\in V} \\min_{s \\in S} d(v,s)", "d9e8e1477c5a61586a5af4bd297cb642": "\\vdots \\,", "d9e8f5094d9d8ccc2fd495cb1793a04d": " \\nu(N) ", "d9e9e5d83ed0a3332c42d4348f817ae4": " F_X(t) < F_Y(t) ", "d9ea3f28c89cb3550b218d8dcfc99396": "f^i(\\bot) \\sqsubseteq k", "d9ea41fbdfb373aa761bdb52109dd4fe": "\\Delta p_{\\mathrm{b.a.}}", "d9eaa0e042998ac78dfa5102fdc0dd34": "c(x) = 3x^3 + x^2 + x + 2", "d9eae6b559f52257639084ce60703b03": "\\delta /\\delta t", "d9eb6bfea4ca7833248b266c92665426": "R_{zz}=8\\pi T_{zz}", "d9eb831c1d479ce9eff8e5454c2395d5": "\\Delta_3(f_i) = k_i^{-\\frac{1}{2}} \\otimes f_i + f_i \\otimes k_i^{\\frac{1}{2}}", "d9eb94e84375ea108f1f38465a99c2e4": "2\\epsilon=C_3=v_{\\infty}^2\\,\\!.", "d9ebb98689c4ea160434ebeb52c07a28": "A \\rightarrow a", "d9ebbd4ec76c6edebe09841e851219da": "\n\\begin{array}{lcl}\nK,N &=& \\text{as above} \\\\\n\\phi_{i=1 \\dots K}, \\boldsymbol\\phi &=& \\text{as above} \\\\\nz_{i=1 \\dots N}, x_{i=1 \\dots N} &=& \\text{as above} \\\\\n\\mu_{i=1 \\dots K} &=& \\text{mean of component } i \\\\\n\\sigma^2_{i=1 \\dots K} &=& \\text{variance of component } i \\\\\n\\mu_0, \\lambda, \\nu, \\sigma_0^2 &=& \\text{shared hyperparameters} \\\\\n\\mu_{i=1 \\dots K} &\\sim& \\mathcal{N}(\\mu_0, \\lambda\\sigma_i^2) \\\\\n\\sigma_{i=1 \\dots K}^2 &\\sim& \\operatorname{Inverse-Gamma}(\\nu, \\sigma_0^2) \\\\\n\\boldsymbol\\phi &\\sim& \\operatorname{Symmetric-Dirichlet}_K(\\beta) \\\\\nz_{i=1 \\dots N} &\\sim& \\operatorname{Categorical}(\\boldsymbol\\phi) \\\\\nx_{i=1 \\dots N} &\\sim& \\mathcal{N}(\\mu_{z_i}, \\sigma^2_{z_i})\n\\end{array}\n", "d9ebcb629ce40eb773111c08eab0613f": "C_0=A\\setminus g[B],\\qquad C_{n+1}=g[f[C_n]]\\quad \\mbox{ for all }n\\ge 0,", "d9ebccc57bd55be9bd2871c1458ca2aa": "O(NQn^{O(1)} + NT_\\text{out})", "d9ebdf4e9739274be094c3043631aa5a": "(re^{i\\varphi})^n=r^ne^{in\\varphi} \\,", "d9ec0daf0717b761d19bcc485aabc2a4": "[x,y,z] = 0", "d9ec174787f45251f532888cf5c6ee64": "\\ Y=cK \\Rightarrow d\\log(Y)=d\\log(c)+d\\log(K).", "d9ec4850722c6d8b0356e7961de198a5": "\nH = \\frac{E}{\\rho} + \\frac{p}{\\rho},\n", "d9ec69f8fdd49fb15e3aa63e6ced6c12": "\\mathbf S(c_1)=\\{x_1,x_2\\}, \\mathbf S(c_2)=\\{x_1\\}, \\mathbf S(c_3)=\\{x_2\\}, \\mathbf S(c_4)=\\emptyset", "d9eca75d8652d9f5eb5d63df3ac67219": "K^M_2(\\mathbb{R})", "d9ecaa79f08d04b7ea3d593868c295cd": "\\nabla_{\\partial_i} \\partial_j = \\sum_l \\Gamma^l_{ij} \\partial _l.", "d9ecb216419c922eaf3cb6f2f5b906bb": "RSS = ||y - {\\hat y}||_2^2 ", "d9ed2c1edc1eb1cb62c255d1694709f4": "\\frac{f(1-F)}{F}\\,", "d9ed3135f8cff758fc137dd7c118c28b": " R_{\\rm specific} = c_{\\rm p} - c_{\\rm v}\\ ", "d9ed5e3d595ff2f35b9a035c2cb43225": "F = A\\overline{C} + A\\overline{B} + BC\\overline{D}.", "d9ee48c250df8e4dc5a3ed26b53a21b3": "g = \\frac{2J+1}{(2s_a+1)(2s_b+1)}", "d9ee5b44e3b5bffc4ffa854cd16b9383": "s^2_n=\\frac{Q_n}{n-1}", "d9ee6431b253b71edcc44f18a681343b": "O(n\\cdot\\log n) \\subset O(n^2)", "d9eedaf53b80c2b774cae3b6f7518f45": "| b, a\\rangle", "d9ef61686650da8e72ee7903b797498a": " \\cos \\phi = \\frac { \\rho} { | s | } , \\sin \\phi = \\frac { \\mu} { | s | }.", "d9ef713e7ca35b224f7ca1c387c9255d": "f(x) = x^2 \\!", "d9ef7e87e2bdc58c98c89897a3f048a5": "\\frac{\\sigma_f}{E_f} = \\epsilon_f = \\epsilon_m = \\frac{\\sigma_m}{E_m}", "d9ef9fe94f9dd859696d0e0260994dd0": " \\alpha_1^{\\lambda_1}\\cdots\\alpha_{n}^{\\lambda_{n}}", "d9efa624fec5ced8ec34c5fddfca7de5": " \\frac{\\frac{d[D]}{dt}}{\\frac{d[C]}{dt}}\n = \\frac{k_2K[A]}{k_1[A]}\n = \\frac{k_2K}{k_1}\n", "d9efeb1d13572b48cc9dcaec42e2f993": "K = L_\\gamma", "d9f0232e53bebdbed8d1895435f963c8": "i\\hbar \\frac{\\partial\\psi}{\\partial t} = H \\psi .\\,", "d9f026d028bdc1f7483aafbca9fc9268": " (E(\\mu)f)(x)= \\sum_{i.j}\\int_0^\\mu \\int_0^\\infty\\psi^{(i)}_\\lambda(x)\\psi^{(j)}_\\lambda(y) f(y)\\, dy \\,d\\sigma_{ij}(\\lambda) = \\int_0^\\mu \\int_0^\\infty\\Phi_\\lambda(x) \\Phi_\\lambda(y) f(y)\\, dy \\, d\\rho(\\lambda).", "d9f0e893dc980645dec1c4fff8100442": "c^2 = (b+a)^2 - 2ab = a^2 + b^2.\\,", "d9f17e2ad69bc45e08d0639ac19da17d": "P(t), 0 \\leq t \\leq T", "d9f1b6f8500286ff69e05ba0203cc694": "\\displaystyle{c=2c +Q(a)D_c(a^{-1}),}", "d9f2180c7167cb9682c29cf9c15c4dd8": " {\\mathbb Q}", "d9f22dd3c6bac42badc0a9e3305a6409": "\\nabla = \\mathbf{\\hat{x}} {\\partial \\over \\partial x} + \\mathbf{\\hat{y}} {\\partial \\over \\partial y} + \\mathbf{\\hat{z}} {\\partial \\over \\partial z}.", "d9f22ff6222200955991aedab32e012b": " \\psi(x)", "d9f2395e43f13c5c58d820535e5434ef": "\\mathbb{F}, l, d", "d9f2489c575722e924e1d310d54887ef": "\\scriptstyle r_S = \\frac{2GM}{c^2}", "d9f258eed5d563a736f5948420db4b1c": "\\sup u \\le C ( \\inf u + ||f||)", "d9f2a5aaacd12f59ef451873beb0b0a0": "m=\\tan \\alpha", "d9f347d91d6a70afd04cd9e203199f33": "c_0=c_0^{p^2}\\equiv(a_0+b_0)^{p^2}", "d9f3e5c41259a4ca05eecc43ad8f043f": " a_n>0,a_{n-2}>0, \\ldots,\\, \\Delta_{4}>0,\\Delta_2>0", "d9f464c8b8084e34fb5b681afc4f2673": "k_0>0", "d9f4815d13b2c231f5b8241baa45de26": "\\scriptstyle{s/\\sqrt{n}}", "d9f49f101256c35a08428de11c5bd4ac": "0^{(\\delta)}= 0", "d9f4fac5381c9c5fffcb04c9c12b8ed7": "4k,", "d9f5131e14e02b0688c552ef18080e9a": "w^k(y)=v^i(x)\\frac{\\partial y^k}{\\partial x^i}(x).", "d9f51e864a6151f57e727294da7ac28c": "\\beta_2", "d9f53b16f9e1ca69a56218a6b4b46d3a": "\\Delta E_{\\rm c} = \\chi_{2} - \\chi_{1}\\,", "d9f5755b39f3792a6db56de21cd25971": " c_{n+m} \\leq c_n c_m ", "d9f60b365cecaf0d97e250eb89d2068d": "\\scriptstyle\\|a_0-a_j\\|\\leq\\|x_0-x_j\\|", "d9f6228e09b914f28925ae064b985bcf": "X^2-\\Delta", "d9f64c8ba6766dd195aec8123fc2e39d": "\\sigma_m = \\frac{1}{2} \\left(f_y +\\sigma_e\\left(1+\\theta\\right) - \\sqrt{ \\left(f_y + \\sigma_e\\left(1+\\theta\\right)\\right)^2 - 4f_y\\sigma_e} \\right) ", "d9f66edbc71ee83c08ea500128cad3f3": "\\pi=\\frac{2\\sqrt{11}}{11Z} \\!", "d9f68fa6000fed2de38a87969cb591c0": "X_{t-s}. \\,", "d9f6a264ba21e8db5354f7fa901657fa": "v^2=Q(v) \\in C\\ell(V,Q)", "d9f6ab9f21fb4f0e30ea1752bbe591b0": "x\\le_+ y", "d9f6b4366659c759564fe590e9e15cd0": "P\\left (n \\right ) = \\frac{\\left\n (n-m+1\\right )^{m-1}}{\\left (m-1 \\right )!\\left\n (g-1+\\frac{1}{m}\\right )^{m}}\\exp \\left ( -\n \\frac{n-m+1}{g-1+\\frac{1}{m}}\\right )", "d9f70a86fc6c6b2dda3ff6c105fb55fd": "f^{-1}(k)", "d9f71c27f99d78bb341de3bd01ba9837": "\n\\lim\\sup \\frac{\\varphi(n+1)}{\\varphi(n)}= \\infty.\n", "d9f766080d3bdb428201b6e939625210": "\\max \\ \\{\\rho\\ge 0: p\\in P(s), \\forall p\\in B(\\rho,\\hat{p})\\} \\equiv \n\n\\max_{\\rho\\ge 0}\\min_{p\\in B(\\rho,\\hat{p})} f(\\rho,p)", "d9f77e329bfeb3622838b90015429a0b": "\\text{prod}(\\{\\}) = 1 \\qquad \\text{prod}(\\{a_i\\}_{i \\le n}) = \\text{prod}(\\{a_i\\}_{i \\le n-1}) \\times a_n.", "d9f7c5a43018871ab103447d5be38b87": "\\mathbf{\\hat{n}} = \\frac{\\Big(\\begin{smallmatrix}0\\\\K\\\\0\\end{smallmatrix}\\Big) - \\Big(\\begin{smallmatrix}0\\\\0\\\\K\\end{smallmatrix}\\Big)}{\\Big|\\Big|\\Big(\\begin{smallmatrix}0\\\\K\\\\0\\end{smallmatrix}\\Big) - \\Big(\\begin{smallmatrix}0\\\\0\\\\K\\end{smallmatrix}\\Big)\\Big|\\Big|} = \\frac{\\Big(\\begin{smallmatrix}0\\\\K\\\\-K\\end{smallmatrix}\\Big)}{\\sqrt{0^2+K^2+(-K)^2}} = \\begin{pmatrix}0\\\\\\;\\;1/\\sqrt{2}\\\\-1/\\sqrt{2}\\end{pmatrix}", "d9f7ccb87cfa07e3d6b4abd09374aa7c": "\n(p \\quad <\\!+\\!> \\quad q)(j) = p(j) \\cup q(j)\n", "d9f7d1a0e30b0862f298339bcc251ad7": "\\{w | w \\in L_1 \\land w \\in R\\}, R \\text{ regular}", "d9f7d4cb5896592a704a1943b17c8e4e": "\\Lambda(x)=\\frac{\\sup\\{\\,L(\\theta\\mid x):\\theta\\in\\Theta_0\\,\\}}{\\sup\\{\\,L(\\theta\\mid x):\\theta\\in\\Theta\\,\\}}.", "d9f7fa451a9ced04b01d04f2c977e69b": "\\frac{V}{T}=k", "d9f82bc4bdacb4a82fc08f0649cc08be": "\\begin{align}\n\\operatorname{Var}(X) &= \\operatorname{E}\\left[X^2 - 2X\\operatorname{E}[X] + (\\operatorname{E}[X])^2\\right] \\\\\n&= \\operatorname{E}\\left[X^2\\right] - 2\\operatorname{E}[X]\\operatorname{E}[X] + (\\operatorname{E}[X])^2 \\\\\n&= \\operatorname{E}\\left[X^2 \\right] - (\\operatorname{E}[X])^2\n\\end{align}", "d9f83d4a1ee5bea1b439de28d7fb0d62": " \\lfloor x\\rfloor", "d9f86c36f9c227990d523a5395cf25b1": "\\lambda \\Pi 2", "d9f889ed43bcfee4fbb9eb5dbec9812e": " [2^r-1, 2^r-r-1,3]_2", "d9f8b79e56f197e9006f85a23cf72e1b": "(\\mathbf{x},z_1)=(\\mathbf{0},0)", "d9f8bca8e79c2664421947ac794b1417": "E_{\\Psi}", "d9f8f818207bf854697bbfa4ff8aeeb7": "\\mu = M/m = 3", "d9f9311392cb554ef738990217716a69": "P(E=\\bar G \\bar D|C=c) = (0.99 - 0.16(c-11))(0.5 + 0.09(c-11))", "d9f942bcf790076a963214d006c79871": "J_{i,k}", "d9f961cbcea8352f953e4e55629d33a9": "\\tfrac{(c+a)}{b}=\\tfrac{b}{(c-a)}", "d9f96eae9cb1354c89bcd15efe678a17": " a_2 = 0, \\; \\; a_1^3 + 4 a_3 = 0, \\; \\; a_1^4 + 16 a_4 = 0 ", "d9f9e291943eedac67daf0ec8395b723": "\\alpha+\\beta", "d9fa039e8ad3ceecc2aa3d95aa021e62": "f_i(\\sigma^*) = \\sigma^*_i", "d9fa07e64232de26367b430357eac434": " f = \\frac{ - m_{ox} + m_{ox, 0}}{sm_{fu, 1} + m_{ox ,0}} ", "d9fa42c3fb06db069f2f27b8a9fa116d": "[\\hat{x},\\hat{p}] | \\psi \\rangle=i \\hbar | \\psi \\rangle \\ne 0.", "d9fa467d440352e723834be6860a0168": "V={a}{v_x}", "d9fa5c16a5e5cf87a23331ba16198922": " \\bar E_l = {E_l \\over hc} = \\bar {B}l \\left (l+1\\right ) - \\bar {D}l^2 \\left (l+1\\right )^2", "d9fa86d58f510f3abf426dcfd8a70e8a": " \\mathrm{Pr}(X=0) = 1-p ", "d9faa707faeccd4002c9cf1240619917": "h_5\\;", "d9fab2e07be7c2b63a15451ada1933ac": " \\eta = -\\theta,", "d9fb10855856c0b98aed469e4c5fcc0d": "\\mathbf{F}_p", "d9fb364db92f9721f91e50ad16ef2007": "\\scriptstyle\\ x,\\ y", "d9fbca4257e256799655cf70e468412f": "f: \\mathfrak{X} \\to \\mathfrak{Y}", "d9fc40bea0dc3a8b0950cff90e28ccbb": "U (\\mathbf r,t) = A_1(\\mathbf r) e^{i [\\varphi_1 (\\mathbf r) - \\omega t]}+A_2(\\mathbf r) e^{i [\\varphi_2 (\\mathbf r) - \\omega t]}", "d9fc86e763a04f9d4678681b17481e0d": "i \\ge 3", "d9fc87106620618e6b0670afa921a998": "\nS = k_{B} \\ln \\Omega + S_{0}\\,\n", "d9fd1d58a48c4b90eb2d70147d2606ab": "\\{\\lambda_n(A)\\}_{n=1}^N,", "d9fd53242d06fccead295338833773b4": "\\mathrm{NPV}", "d9fd9a81f14da193255b09cbb9f2ca20": "\\{x_i \\}^n_{i=1}", "d9fdfca78fbe8ba0c467060191efb555": "x=\\frac{a}{c},\\quad y=\\frac{b}{c}", "d9fdfe965335b3cb056af083b870c17a": "f:U\\rightarrow \\mathbb{R}^m", "d9fe4d8a7e3ef34437c665fc4e129932": "\\mathsf{cap}(\\mathbb{Z})", "d9fef37d3313ddb798e7bc63875c69f0": "\\,f_2'(x)=\\cos (x)", "d9ff00671fc3867eb4901403f8746cdb": "\\lim_{(x_0,\\dots,x_n)\\to(\\xi,\\dots,\\xi)} f[x_0,\\dots,x_n] = \\frac{f^{(n)}(\\xi)}{n!}", "d9ff6d7543b014a17a10c512de6a87dc": "v = \\frac{dx}{dt}", "d9ffb49d9a2cd04aae2d3ed8b161e1e8": "v\\in\\R^n", "d9ffcabe9fb4a67e735aec8f9940e89c": " \\Bigg\\uparrow ", "da001027ff3a5c96dd7806d32aedad96": "\\begin{align}\n \\operatorname{erfc}(x) & = 1-\\operatorname{erf}(x) \\\\\n & = \\frac{2}{\\sqrt{\\pi}} \\int_x^{\\infty} e^{-t^2}\\,\\mathrm dt = e^{-x^2} \\operatorname{erfcx}(x),\n \\end{align} ", "da006fb2ddd91d464353374b44518f28": "\\mathcal{S}_{\\alpha}\\to \\max", "da009006a467db3c928800f6f058c121": "\\lambda \\in U(1)", "da00bfdb2c6eb55ee194cab96455d137": "\n{\\mu_t}_\\text{outer} = \\alpha \\rho U_e \\delta_v^* F_K\n", "da014b31e7fb28d91d9f935ce77331f5": " \n\\ddot{\\rho}+\\Omega^{2}\\rho=\\rho^{-3}.", "da017a75a25a025c2625da52f61200e1": "\nP\\left(X\\mid A\\right)=\\frac{P\\left(A\\mid\nX\\right)P\\left(X\\right)}{P\\left(A\\right)}\\propto P\\left(A\\mid\nX\\right)P\\left(X\\right)\n", "da02139353601e1517a570587ab677f9": "\\varphi^4", "da024e4f2a1a8d06e02d88e4c1e8c2f0": "\\scriptstyle x(u-\\tau)", "da02f00c9530d7c0b891684608217aa0": "\\scriptstyle{1/4}", "da02f2f7b7aa3b753537bffb6748a963": "g_\\lambda(u)=-\\log u/\\lambda", "da0335a95bc2fc8e0b757263d54b88ce": "m_1 > m_2", "da03d02336d454191bfcc575fd93109b": "\\frac{80}{3}", "da041b97bc85667938a2d6597371acfc": "\\beta^i=\\gamma^{ij}\\beta_j", "da04231ee06d9411173ff1b8e111ea7a": "r^2 \\equiv n \\pmod p ", "da043d377b8d8177b67a2633d58bbde3": " \\varepsilon_i - \\varepsilon_j", "da04926cff23265d25f70c123c04aa8a": "\\left [\n\\begin{smallmatrix}\n 2 & -1 & 0 & 0 & 0 \\\\\n-1 & 2 & -1& 0 & 0 \\\\\n 0 & -1 & 2 & -1 & -1 \\\\\n 0 & 0 & -1 & 2 & 0 \\\\\n 0 & 0 & -1 & 0 & 2 \n\\end{smallmatrix}\\right ]", "da05071e39ae62a34cacb1c6457baaeb": "? = \\sqrt{ax+(n+a)^2 +x\\sqrt{a(x+n)+(n+a)^2+(x+n) \\sqrt{\\mathrm{\\cdots}}}} \\, ", "da050db0cf19d43a60872f65a44c05a5": "M = \\frac{m-m_{od}}{m_{od}}", "da0557f2eb2cbc992f155e6fd9dc56c1": "\\alpha=1,2,3", "da057b2e5cf30e8d4968425ca7d2dc48": "\\mathrm{St} = {\\omega L\\over v} ", "da05ee903a2691cdb8730592dfae44f8": "P_{\\pi\\circ\\sigma}", "da068e101fa69d8fbf7819b4ae1a248e": "\\underline{\\Lambda}_n(T) = \\min_{-1 \\le x \\le 1} \\lambda_n(T; x)", "da06fe5d6c687ece4571ccaa116ec6dc": "2(k+2\\lambda)\\,", "da073a7fb7044295517b968b06800d5e": "1 + \\operatorname{dim} R \\ge 1 + \\operatorname{dim} R_\\mathfrak{p} \\ge \\operatorname{dim} R[x]_\\mathfrak{q}", "da076b07450164ac7e7ffff562673456": " H^{n}(M,{ R}) \\otimes { C} = H^{n,0}(M)\\oplus\\cdots\\oplus H^{0,n}(M). \\, ", "da07732b61d4a1c919e491dbf4f88091": "ab = a \\cdot b + a \\wedge b", "da07d691602c91c18d97c7e027818aae": " \\sum_{k} t_k.\\,", "da07fdf93b655ab274bb0ddd2f7c4db8": "\\begin{align}\n\\ln\\Omega_{E,\\ell} &\\approx n\\ln\\ell + n \\ln\\sqrt{\\frac E n} + const.\\\\\n&= \\underbrace{n\\ln\\frac{\\ell}{n} + n \\ln\\sqrt{\\frac E n}}_{extensive} + \\,n\\ln n + const.\\\\\n\\end{align}", "da0805c28298654ca7fe687175bfc77d": "\\bigwedge(\\mathbf{v}_1,\\dots,\\mathbf{v}_{n-1})=\n\\begin{vmatrix}\nv_1{}^1 &\\cdots &v_1{}^{n}\\\\\n\\vdots &\\ddots &\\vdots\\\\\nv_{n-1}{}^1 & \\cdots &v_{n-1}{}^{n}\\\\\n\\mathbf{e}_1 &\\cdots &\\mathbf{e}_{n}\n\\end{vmatrix}.", "da083e79c87d6bd085ba3ccf0ba7b16d": "\\varphi_{\\beta\\alpha}", "da0840b6192097139d4742429482d379": "[1..t] = abdcabdab", "da086c7feb8479a99d35c756e3fb10e3": "\\boldsymbol{\\nabla} \\rho \\times \\boldsymbol{\\nabla} p=0", "da08b9ad596c96818c6a15401884d033": "(x_1 + x_2 + \\cdots + x_m)^n \n = \\sum_{k_1+k_2+\\cdots+k_m=n} {n \\choose k_1, k_2, \\ldots, k_m}\n \\prod_{1\\le t\\le m}x_{t}^{k_{t}}\\,,", "da08e4716fa988d5794e521ea9e85f31": " F(a_1, \\ldots , a_n) = \\sum_{j=1}^n w_j b_j", "da0913f6ac52667fa61faaa9c584fea2": "e_{i,j}", "da0976063ab8fcf33c19ccc37265b4e5": "\n=\n\\frac{1}{\\theta^k \\Gamma(k)}\n\\left(\n \\frac{1}{y}\n\\right)^{k+1}\n\\exp\n \\left(\n \\frac{-1}{\\theta y}\n \\right)\n", "da099eb11484aecf512d5088093ee437": " j > i", "da09a6200bd6176c80d14ac77470887b": "(m - r)", "da09b77345d643613d8ec59e9e9eaa04": "x = \\frac{1}{a} \\int_0^{L'} \\cos {s}^2 ds", "da0a1e4708fd8e4923f6a36ca7eae098": " \\mathbb{E}[p_i \\mid \\mathbb{X},\\boldsymbol\\alpha] = \\frac{c_i+\\alpha_i}{N+\\sum_k\\alpha_k}", "da0a86e7d2df08ef3874838599e83729": "Y_{3}^{-1}(\\theta,\\varphi)\n={1\\over 8}\\sqrt{21\\over \\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(5\\cos^{2}\\theta-1)\\quad\n={1\\over 8}\\sqrt{21\\over \\pi}\\cdot{(x - iy)(4z^2- x^2 - y^2)\\over r^{3}}", "da0aa4803e22e86844ac1002fe7d8a96": "\\phi_{ind}\\left(\\omega\\right)", "da0ab82445b832e2929f56f7815f30cb": "\\,\\phi(v_i)=\\exp\\left(-\\frac{\\|v_i-c_i\\|^2}{2\\sigma^2}\\right)", "da0adb242af841a38bd651bd24cfcdbf": " y_U", "da0aff0517876dd5187071c5813c014c": "\\bigcup_{i=1}^\\infty A_i \\in \\mathbb{B}_b(S)", "da0b068a2815ad4efeef7ec7526efc95": "M, u \\models \\varphi", "da0b186e509831fe831189a4858964e7": "\n\\mathbf{a}_{B} =\n\\mathbf{a}_A ", "da0b5e862aa499ae403eea1bb6012e9d": "-\\frac{3}{5}", "da0b68c36aa47e778b40a64e7bd9d7e3": "\\,J+iK = [i\\rho_1-\\rho_2]+ [J_1+iK_1]\\mathbf{i}+ [J_2+iK_2]\\mathbf{j}+ [J_3+iK_3]\\mathbf{k}\\quad ", "da0b6acddc66d7f5935b4f6df012453b": "f:X\\rightarrow Y,", "da0bbd4201370e26f01789139a50f075": "X_j(n)", "da0bc11a9c704e98c6895a68139bd22a": " \\begin{align}\n\\dot{q}_i &= \\frac{\\partial\\mathcal{H}}{\\partial p_i}\\\\\n-\\dot{p}_i &= \\frac{\\partial\\mathcal{H}}{\\partial q_i}\\\\\n-\\frac{\\partial \\mathcal{L}}{\\partial t} &= \\frac{d \\mathcal{H}}{d t}\\,.\n\\end{align}", "da0c1c032541cb7f3979bb4e437b3234": "(q_1, p_1)", "da0c27118f9c32a90f2ce952720a22bb": "\n\\begin{align}\na &= b \\\\\na^2 &= b^2 \\\\\nab - b^2 &= a^2 - b^2 \\\\\nb(a - b) &= (a + b)(a - b) \\\\\nb &= a + b \\\\\nb &= b + b \\\\\nb &= 2b \\\\\n1 &= 2\n\\end{align}\n", "da0c2754d58f25ccdcddfdbeb72a0ff2": " \\sum_{1 \\leq q \\leq x^\\theta} E(x;q) \\leq \\frac{C x}{\\log^A x}", "da0c87ebe4b94d68fe3af570f4aa5dde": " E_{ZP}(V) = \\frac{1}{N} \\sum_{\\mathbf{k}, i} \\frac{1}{2} h \\nu_{\\mathbf{k},i}(V) ", "da0cb42eba3e6f9a1d234bc7c343930b": "v[\\mathbf{f}] = G^{-1}[\\mathbf{f}]a[\\mathbf{f}]^\\mathrm{T}", "da0cdbf6eea2b3306e626f24d3be49d5": "p(r) = \\frac{r^2}{2\\pi^2}\\int_0^\\infty I(q)\\frac{\\sin qr}{qr}q^2dq.", "da0ce26084a4324f3423dbc096a10981": "M(G,2)", "da0ce6de8e7349c933954af67c584c84": "\\begin{align}\n p_k &= {1 \\over {2k + 1}} \\prod_{n=1}^{k} \\frac{(2n)^4}{[(2n)(2n - 1)]^2} \\\\\n &= {1 \\over {2k + 1}} \\cdot {{2^{4k}\\,(k!)^4} \\over {[(2k)!]^2}}\n\\end{align}", "da0d1f3fe1c11534e94f6a741e545185": " \\exists i", "da0d2146265d3888a8a7123346024d4d": "U''\\subset X", "da0d444b3b5fcbf1a4059e92eeeb3103": "||\\phi||_1", "da0dd913436f5fb48c8a8a5079b91f8c": " X_0 \\cap X_1 \\ \\ \\text{and} \\ \\ X_0 + X_1 = \\{ z \\in Z : z = x_0 + x_1, \\ x_0 \\in X_0, \\, x_1 \\in X_1 \\}.", "da0e0f3fc9d0df5457c3f3a01cc0b479": "y = \\phi\\,", "da0e6bf569702a438b1ffe9d65ec7898": " \\hat p = X/n", "da0eb2c60efbf3858606dce65755179f": "n=7", "da0ee77c5e0b0b1163aeae0fb43fbaad": "S[i,j]", "da0f148ff3480a4e3fbedc23f2860889": "S(2,n)", "da0f2dc8dc3876cd2590e3550af7a014": " A\\mathbf{x} \\leq \\mathbf b ", "da0fb85cd7f4b53778cbb80ff252839a": "f_o = \\gamma\\left(1-\\frac{v\\cos\\theta_s}{c}\\right)f_s.", "da0fe1d8c22556dc550a90941bfd961e": "50^\\text{g} = 50 \\cdot \\frac {\\pi} {200^\\text{g}} \\approx 0.7854 \\text{ rad}", "da0fff999908bdd0801b68a65093dee0": "\\ E_{nonbonded} = E_{electrostatic} + E_{van der Waals} ", "da1031bef0ff32159a39e8a15ff76448": "2^{bh(v')-1}-1", "da104838417366c2679218013ceb41bc": "{x \\choose y}= \\frac{\\Gamma(x+1)}{\\Gamma(y+1) \\Gamma(x-y+1)}= \\frac{1}{(x+1) \\Beta(x-y+1,y+1)}.", "da106ab83268749db22655f558c29d2d": "-\\infty < x < \\infty\\,", "da108bf82e3e54874e4ec1f6e3d1a1be": " (2)\\quad Y = X^b. ", "da1094c77254248096b261850bb049ed": "\\theta = \n \\begin{cases}\n 0 & \\mbox{if } x = 0 \\mbox{ and } y = 0\\\\\n \\arcsin(\\frac{y}{r}) & \\mbox{if } x \\geq 0 \\\\\n -\\arcsin(\\frac{y}{r}) + \\pi & \\mbox{if } x < 0\\\\\n \\end{cases}\n", "da10a52abe42f7304a3c5b9459d3837e": "\n\\dfrac{I_2}{I_1} = \\dfrac{{r_1}^2}{{r_2}^2} \\,\n", "da1165e3858a1c58d0c59a612d8dbebf": "A_n \\to A_0\\alpha_{n+1} \\mid \\ldots", "da122c4cde0d678930d7d6183801ab56": "(\\rho v^2)_{sw}\\approx \\left( \\frac{4 B(r)^2}{2\\mu_0} \\right) _m", "da122d620fbd904161c8195404ee4af3": " \\begin{align}\n X & = \\left( N(\\phi) + h\\right)\\cos{\\phi}\\cos{\\lambda} \\\\\n Y & = \\left( N(\\phi) + h\\right)\\cos{\\phi}\\sin{\\lambda} \\\\\n Z & = \\left( N(\\phi) (1-e^2) + h\\right)\\sin{\\phi}\n \\end{align}\n", "da1246d66a81de7fd2a1bf75287c875b": "f_n = \\frac{k-1}{kN} + \\frac{k(N-1)}{kN}f_{n-1}", "da12492316aefd5c87c82e46be128e3d": "\\Psi_{n,\\mathbf{k}} (\\mathbf{r}) = \\sum_{\\mathbf{R}} e^{-i\\mathbf{k}\\cdot(\\mathbf{R-r})}a_n(\\mathbf{r-R})", "da12807feefbfdfbeb5c9d0c1a9ebc8d": "q_i\\,", "da129e78835d09c24a7c1d9ca5031133": "\\alpha_n = q^{n^2+n}\\sum_{j=-n}^n(-1)^jq^{-j^2}, \\quad \\beta_n = \\frac{(-q)^n}{(q^2;q^2)_n}. ", "da12a0eaf2754d67c8d13af3926b2801": "\\beta=v_p/c", "da12a93c0d2aa5ac1c7e33b0235f2e67": " \\mathrm{tr}(\\mathbf{A}_1\\mathbf{A}_2\\mathbf{A}_3\\ldots\\mathbf{A}_{n-2}\\mathbf{A}_{n-1}\\mathbf{A}_n) = \\mathrm{tr}(\\mathbf{A}_2\\mathbf{A}_3\\mathbf{A}_4\\ldots\\mathbf{A}_{n-1}\\mathbf{A}_n\\mathbf{A}_1) = \\mathrm{tr}(\\mathbf{A}_3\\mathbf{A}_4\\mathbf{A}_5\\ldots\\mathbf{A}_n\\mathbf{A}_1\\mathbf{A}_2) = \\ldots ", "da12e7112a3e3e57a99555e74121a130": "{v^2 \\over 4g}", "da132a23db59c1705ddacc3be701593e": " W^+ \\, / \\, W^-", "da140938f34caebb448e2f52d7745238": "\n\\epsilon_\\mu^j(p) \\cdot \\epsilon_\\mu^{j*}(p) = 1, ", "da144a489bbadcfc75dc174921a8f0d5": "\\mathbf S(c_1)=\\{x_3\\}, \\mathbf S(c_2)=\\{x_1\\}, \\mathbf S(c_3)=\\{x_2\\}, \\mathbf S(c_4)=\\emptyset", "da1457003102ab49e8536a6a8d04de3e": "\n\\begin{align}V &= \\begin{bmatrix} Y(0) & Y(1) & Y(2) & Y(3) & Y(4) & Y(5) & \\cdots & Y(99) \\end{bmatrix} \\\\\n & =\\begin{bmatrix} 29 & 278 & 529 & 782 & 1037 & 1294 & \\cdots & 34382 \\end{bmatrix}\\end{align}", "da146874e00f3817717e8d8ac3975511": " \\frac{| \\nu - \\mu |}{ \\sigma } \\le 1.", "da147612f474ce1bbf3666f71e00fb8c": " [0, 2/\\pi]", "da149f34b32993fdf9d2153a934b1ecb": "{\\Psi}_{n,\\mathbf{k}} (\\mathbf{r}) = e^{i \\mathbf{k}\\cdot\\mathbf{r}} u_n(\\mathbf{r}) ", "da16491d1526801f31685266e23f4de8": "S \\rightarrow AbA \\mid B", "da16bbce250e819f597db95400e8cdc8": " = f\\, \\Delta h ", "da16bdafbfd15584736dd285f45b1e7a": "(1+x)^{m+n} = \\sum_{r=0}^{m+n} {m+n \\choose r}x^r. ", "da16c3185989bfb78a292dc44e4060d6": "\\mathbf{k}_i", "da1719623b72f4e5829b650da5d15de3": "W_\\gamma [A]", "da179092769f82d8c135e237e1eb4890": "A \\cdot B", "da1812472bbcad84973845f80eadaa1a": "b^n\\pm1\\,", "da182cf861a7911486f89210766179d3": "\\scriptstyle t \\;<\\; \\lambda\\,", "da1850316d3a2cec520d295086f636af": "(R/a)^2", "da1878b0096c1fd68553231304b4e824": "v_i < \\max_{j\\neq i} b_j < b_i ", "da18901775032060a2194cd6491cf9f7": "\\frac{dy}{dt}=\\frac{10\\times0-6\\times3}{y}=-\\frac{18}{y}.", "da18b06824b3d8b228cd1023088cd267": "x_i\\geq x_\\min", "da18b6e23beafc016445c57aa23485a3": "\n2\\xi p_{\\xi}^{2} - mk - mE\\xi = -\\Gamma\n", "da1947eb4cb83ad276efc250636cbd1c": "m_p(t)", "da196a25860d01b5e0e62bcb36f3b810": "L(G) = \\{ w\\in\\Sigma^{*} : S\\stackrel{*}{\\Rightarrow} w\\}", "da199118f15e2cd30a0cc83e62fc1932": "P=100 \\times \\frac{W_{i}-W_{f}}{W_{i}}", "da19fa760639f3c9861948baf5879ae1": "Homeo(F)", "da1a26d1cb37c3aae21330c01ce256e0": "\\psi(x) \\propto \\sum_{n} A_n e^{i p_n x/\\hbar}, ", "da1a2b2db55aa62140d56d957ceb5819": "Q \\approx \\int_a^bf(x)\\,\\mbox{d}x ,", "da1a699df701aff3fe30309e22c014cc": " E=\\prod_{k \\in G} S_k ", "da1a9fa198f1730d442ece9cb6ffff5b": "=\\frac{\\text{100 * kva base}}{%\\text{ X}* \\sqrt{3} *\\text{kv}_{L-L}}", "da1aadbf9d3cb9b9de2b9b56157f5e3c": "\\forall x\\in\\mathbb{R}\\text{ and }\\exists x\\in\\mathbb{R}.\\,", "da1ae88bf3634e4d864eead483b8825b": "T_2=22^{\\circ}C\\!", "da1b0b96f6a8155dec4923e115c290d7": "(\\ln U)_t", "da1b371aaa62066bbceb82b3a3be0393": "K_{-1}=1.74540566240\\dots", "da1b99d90776bf89b4734cbec0794870": " (c,b_c,a_{bc^b}) ", "da1ba3c202b5343c4c8ab09989410ba9": "\\scriptstyle M_0", "da1c07e226df70b885c7c80bfd7dd7f2": "v_p=\\frac{\\lambda}{T}=\\frac{f}{\\tilde{\\nu}}=\\frac{\\omega}{\\beta}", "da1ce75ebac3d2a1be21bf8c4cc757f5": "(x_1,\\dots,x_{D+1})", "da1cf5cf60c8db9e697fb4f7e0dee83d": "T(s,\\mathbf{x})=\\frac{N(s,\\mathbf{x})}{D(s,\\mathbf{x})}", "da1d4bd14d58ba5d3fdeea57de906dca": "\\begin{align}\n \\sigma^2_Z &= \\log\\!\\left[ \\frac{\\sum e^{2\\mu_j+\\sigma_j^2}(e^{\\sigma_j^2}-1)}{(\\sum e^{\\mu_j+\\sigma_j^2/2})^2} + 1\\right], \\\\\n \\mu_Z &= \\log\\!\\left[ \\sum e^{\\mu_j+\\sigma_j^2/2} \\right] - \\frac{\\sigma^2_Z}{2}.\n \\end{align}", "da1d6c7db9e736eb4964828efd096e33": "S=G-T+NX+I", "da1d98613370ce5c46ee21b4e23436f4": "\\xi_m = V_{3m - 1}^{1 / \\delta}, \\ \\eta_m = V_{3m} \\xi_m^{\\delta - 1}", "da1d9edf30d622751b9abe194371ee99": "n \\geq 3.", "da1dd4b731931ba407b9acae546a1ba5": "\\mathbb{P}^x", "da1e0bfeea19711ab25b4f374c8607cb": "\n\\varphi(\\theta) = c^1c^2 + \\frac{1}{2}\\sum\\limits_{l=1}^{\\infty}\\bigg((a^1_la^2_l + b^1_lb^2_l)\\cos(l\\theta) + (a^1_lb^2_l - b^1_la^2_l)\\sin(l\\theta)\\bigg).\n", "da1e19c374a718e610aa88dd832246ff": "{d^2 h^{ij} \\over dt^2} = \\left( {1 \\over m^i} + {1 \\over m^j} \\right) p^{ij} + \\sum_{k\\neq j} C(ij,ik) \\psi(\\alpha_{ij,ik}) {1 \\over m^i} p^{ik} + \\sum_{l \\neq i} C(ij,jl) \\psi(\\alpha_{ij,jl}) {1 \\over m^j} p^{jl}", "da1e23711dea5fa077b660bd205fdc0f": "f^1=f", "da1e26631919d0123034201fc9ce059e": "\\rho_{H}", "da1e3b9e73dc5da707e21fe99cffe25c": "g_J= g_L\\frac{J(J+1)-S(S+1)+L(L+1)}{2J(J+1)}+g_S\\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.", "da1e4bf4f111e6375bcf28861f2e3669": "P_W=I_{1,2}\\,I_{2,3}\\,I_{3,4}\\,I_{4,1}", "da1e661046e7d0c3a43284588a06e4c6": "dT = dN = 0", "da1e9da71598c4bb35e8a7fc407c5f6c": "AB+BC=AD+DC\\quad\\Leftrightarrow\\quad AE+EC=AF+FC.", "da1f241266f0adf72319c555ca2982e2": "A_{n} = a_0 \\left( 1 + \\frac{3}{5} \\left( 1 - \\left(\\frac{4}{9}\\right)^{n} \\right) \\right) = \\frac{a_0}{5} \\left( 8 - 3 \\left(\\frac{4}{9}\\right)^{n} \\right)\\, .", "da1f2cda108b48a099d82c654ba17df3": "m^2+n^2\\leq r^2.\\,", "da1f55a1912e881c5d6471840ec7d725": " E_\\text{B} = -\\dfrac{Ry}{n^2}", "da1f668e6662777de3be33a49056f5b4": " I_n ", "da1fa41c314703e72f05c8e5a1690224": "r_o \\rightarrow r_{ps}.", "da1fe37330396358c6ee59939b7175f1": "(a-\\overline{b+c})", "da1ff5d501868f21abaaadc74681c0f4": "I=\\left\\langle p_0x_1-p_1, \\ldots, p_0x_n-p_n\\right\\rangle.", "da204c5a46881c23e1beb45e70402c9d": "\\begin{bmatrix} {1/\\sqrt{2}} & -{1/\\sqrt{2}} \\\\ {1/\\sqrt{2}} & {1/\\sqrt{2}} \\end{bmatrix}", "da2082cc42b63b75a9772dd2de85b8ea": "z = 0.80 + j1.40\\,", "da2086134dd460bae21efb093f43aae4": "[-1/4 , 2]", "da20f949ff03c34fb2e7a3636b056c99": " \\frac{dN}{dt} = 0", "da21178b9b5c80ece034b1df494fb761": "T_i/P_i", "da21560803bbbbd61535081bc9a78be2": "O(\\log^2 q)", "da2197fac4ccb13cf05ac6c0fb424cb7": " f_\\gamma v^\\gamma = f_1 v^1 + f_2 v^2 + \\cdots + f_n v^n ", "da21c5ba4fff2d4dd2ec0bc1bcf115c1": "\\omega_3", "da21d138ca68c9dced92bdd38c95dd4e": "r_d = \\frac {s_d} {t -2}", "da21d9ee3e9c035956116182c631ea9a": " \\dot m = \\rho*Q ", "da22b87ee3a29d5f2efe6a2c70fdaa85": "\n\\begin{align}\n\\varphi: K[X] &\\rightarrow L\\\\\np(X) &\\mapsto p(\\theta)\\,.\n\\end{align}\n", "da22bb708ffa48f6ae6e76399dae0455": "\\int_0^z \\tan(w) dw = \\log \\sec z", "da22bcbe1021be77294a2bcecd19bff5": "T_\\mathrm{X}^{\\mathbf{k},\\mathbf{k'}}", "da22dd6408fbb3039867375aefb1d550": "L(gh) \\leq L(g)+L(h)", "da231155381df286e4faf6ee55f99243": " e_2 \\leq ub_2 = (k-1) \\sum_{i \\in S_p}\\max_{j\\in\\Theta,j \\neq i}\\left[ \\min_{l\\in{1,\\ldots,H} } \\Pr\\left\\{ \\tilde{J}_{jl} \\leq \\tilde{J}_{il} \\right\\} \\right],", "da233fa808395d62ecabf9e8f2c198c0": "B2^B", "da236af9b83f2e8aa94eb9b168681ec5": "C \\wedge D := \\sum_{r,s}\\langle \\langle C \\rangle_r \\langle D \\rangle_s \\rangle_{r+s} ", "da237ecfa2baa542d6832a56e55bb8da": "\\frac{h}{2 e}", "da239bd1707e28182fdcd698ed289472": "4\\pi r^2\\ \\text{or}\\ \\pi d^2\\,\\!", "da23ad157cff58707b9ec97b93e52e73": "\\sqrt{\\frac{1}{10}}\\!\\,", "da23b88be29a0aac05c6a5675466a587": "\\psi^{(0)}(z) = \\psi(z) = \\frac{\\Gamma'(z)}{\\Gamma(z)}", "da23d930edf0bbc45f969c6453415725": "d=3,", "da23e1fb3a4fdeb500bec90a1d37d9ba": "\\sum_{e_j \\in E} y_j", "da23e5eeab023722dc1b8c825be1feed": "Q^\\pi", "da2406ea09a0ea84e415d5b9af00a744": "G_2 \\supset \\mathit{USp}(2)", "da24092fc7758be06ea566cf71b18e9e": "\\Delta=-2", "da241c2f3844c34cf33670ef2f6d3630": "W^{-1}_\\mathrm{Theil-T} = \\overline{\\text{Income}} \\cdot \\mathrm{e}^{T_T}", "da243a414b1db1c59a1429c9f26065cb": " \\mathcal{C}_{Y \\mid X}", "da24b91f677848f436435b39caeef572": "\\scriptstyle A_\\bot", "da24b9a2eb8d6dc0240559ae6b3002cf": "w(x) = \\dfrac{1}{\\left \\vert G'(x)-F'(x) \\right \\vert}.", "da24bf0a0e6061e79147597da7d728ed": "\\mathbf x_k", "da2514338224b5c96c78ce583ecb048a": "\\hat{W} = \\bigg(\\frac{1}{T}\\sum_{t=1}^T g(Y_t,\\hat\\theta_{(1)})g(Y_t,\\hat\\theta_{(1)})'\\bigg)^{-1},", "da2538e2f7fdac1f738f85bc22a5a5fd": "\\pi=\\frac{85\\sqrt{85}}{18\\sqrt{3}Z}\\!", "da25fc177fd4c53a2c3399c25685dd4c": "\\left \\langle \\psi \\right |", "da26399188fc7830c88ce22bbe8dcd00": "B_B", "da26b4a00c8bcd8492a425313e69ed2f": " A_1\\dots A_t", "da26ffa791208b6f6e87a77b24bdfa97": "\\epsilon(q,0) = 1 - V_q \\sum_{k,i}{\\frac{-q_i \\frac{\\partial f}{\\partial k_i} }{ -\\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m} }}\n = 1 - V_q \\sum_{k,i}{\\frac{q_i \\frac{\\partial f}{\\partial k_i} }{\\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m} }}", "da2705c68892d150657ac5c38159f1b3": "X_1 = \\{1,3\\}", "da272ae9a7f5783f2035b201d82f4038": "p^{th}", "da2737a2b58b02e934d35395e0e46242": "y_b=y(x_b)", "da2774ac97fc7099555daef3b593740d": "\\mathfrak{m}=(\\mathbb R/\\mathbb Z)\\setminus\\mathfrak{M}", "da2775c69c8bd2b40568df70b4204881": "V (Y, P)", "da27a779fcba426b55fa48fa7ad1c558": "\\cdots\\to\\pi_n(F)\\to\\pi_n(E)\\to\\pi_n(B)\\to\\pi_{n-1}(F)\\to\\cdots\n", "da282d0f7295536f3ce87dfc9b247feb": "I_{0,Airy} = (P_0 A)/(\\lambda^2 f^2)", "da2830dd0570010c74080133bafc56e9": "\\scriptstyle{|\\phi_2(t)\\rangle}", "da284afdf21c640b0ae4a63ea4bbdd80": "R_\\mathrm{ac} = \\frac{R_aR_b + R_bR_c + R_cR_a}{R_b}", "da286263611bb15ebf68c8481611caf9": "\\phi\\colon A_i \\to A_i", "da28825bc776c495163e79ce702efb92": "\\delta s", "da288499829f082d8fcd0def76e48b6e": "\\underline{\\underline{\\boldsymbol{A}}}", "da28a09f3a906cc350633a2a572de083": "Gr_p = \\frac{D_p V_s \\rho}{(1-\\epsilon)\\mu}", "da28a966e0b9eaeb28f763837f47ef36": "\\hat{a}|\\alpha\\rangle=\\alpha|\\alpha\\rangle", "da28d578b4a01c92e65484112143124a": "\\widehat{f}(n)=\\frac{1}{2\\pi}\\int_0^{2\\pi}f(t)e^{-int}\\,dt, \\quad n \\in \\mathbf{Z}.", "da2904e38fae48d1f07ebe5e724bc3d1": "-\\pi < \\gamma_1 < \\pi", "da290c1291ec47f1fd75fec740b4946a": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=6,\\cdots ,11", "da29139947ba3ddb1301cf183635fcc7": "x := x + e = \\frac{S+x^2}{2x} = \\frac{x+\\frac{S}{x}}{2}", "da29213490a887c8b32a5563bbe95138": "X \\leq_A Y", "da2958f885f9f53a20e8c802877a1f0a": "\\sigma(x_1,x_2)= x_1 + x_2 = x + \\dot{x}", "da29640e2677ca1247f64d3b0dc8a055": "\ndV = \\frac{\\left( \\lambda - \\mu \\right) \\left( \\lambda - \\nu \\right) \\left( \\mu - \\nu\\right)}{8\\sqrt{-S(\\lambda) S(\\mu) S(\\nu)}} \\ d\\lambda d\\mu d\\nu\n", "da2a32078c2d6d7b614d7f19cdfcb244": "\\mathbf s+t\\mathbf d", "da2acf09c160e1c6c48f607207fb5fab": "a_0 > 0", "da2b398081cbead6603939000cb85383": "\nI\n=-E\\left(\\frac{\\partial V}{\\partial\\sigma^2}\\right)\n=-E\\left(-\\frac{(X-\\mu)^2}{(\\sigma^2)^3}+\\frac{1}{2(\\sigma^2)^2}\\right)\n=\\frac{\\sigma^2}{(\\sigma^2)^3}-\\frac{1}{2(\\sigma^2)^2}\n=\\frac{1}{2(\\sigma^2)^2}.", "da2b3a97d04bf64201f65f5fb04228c2": "\n \\hat\\mu \\ \\sim\\ \\mathcal{N}(\\mu,\\,\\,\\sigma^2\\!\\!\\;/n).\n ", "da2b7e0cc3fa3d7af1dbe276c6f71639": "k_n \\otimes_k k_m \\simeq k_{nm}", "da2bb3b4fa467ddd0a179b0e9b0846d8": "\\scriptstyle \\,\\frac{a}{2^b}", "da2bb7a04b60d8baac156d61fdb211e0": "\nT_{\\delta}^{\\mathbf{p}^{n}}\\equiv\\left\\{ a^{n}:\\left\\vert -\\frac{1}{n}\n\\log_{2}\\left( \\Pr\\left\\{ E_{a^{n}}\\right\\} \\right) -H\\left(\n\\mathbf{p}\\right) \\right\\vert \\leq\\delta\\right\\} ,\n", "da2bbddfb4d7f7d5d2a522825d374a8b": "M(V)", "da2be8cd66e7887ab99184eb222fb515": "b+1", "da2c32d54c1e83ec8905d890269f575a": "M_s = \\max_{0 \\leq \\omega < \\infty} \\left| S(j \\omega) \\right| = \\max_{0 \\leq \\omega < \\infty} \\left| \\frac{1}{1 + G(j \\omega)C(j \\omega)} \\right|", "da2c9e63dab0c30c70737bcd91a1455f": "b = \\frac{1}{\\delta + \\tau}", "da2ccb10c1cb28b49eec32f1b2b5c4df": "C_{\\alpha \\beta}\\,\\!", "da2d29b432d3bdf7ef0d6db3f94598fd": "j_n(x) = (-x)^n \\left(\\frac{1}{x}\\frac{d}{dx}\\right)^n\\,\\frac{\\sin(x)}{x} ,", "da2da9dc7e822c3f85f2d99d8c0b39d5": "m=1 0 1 0 0 1", "da2db1a73d38c976581d6e9e020cbe30": "\n u(1 - p_1 p_2 - p_3 p_4 + p_1 p_2 p_3 p_4) = 0\n", "da2dc11349e300eb7b1bd041f3917391": "2mxP_{\\ell}^{m}(x)=-\\sqrt{1-x^2}\\left[P_{\\ell}^{m+1}(x)+(\\ell+m)(\\ell-m+1)P_{\\ell}^{m-1}(x)\\right]", "da2de309461a9ecfd41878ce8c9c58f6": "\\begin{align}\nx&\\mapsto ax+by + \\alpha\\\\\ny&\\mapsto cx+dy + \\beta,\n\\end{align}\n", "da2e1046a4aa1362289d145f32077301": "\\lnot\\phi\\,", "da2e12caa56e09b919f135e40e1f23fb": "var(qx+(1-q)y)=q^{2}\\sigma^{2}_x+(1-q)^{2}\\sigma^{2}_y", "da2e2832b97e023d30afb5387d54db9b": "\\cos \\,(\\theta(\\bold{\\hat{n}}))", "da2e2c8309979ea2efdecc0bb735295b": "\\int_{-\\infty}^\\infty f(x)\\,dx = 1.", "da2e497046cc543ed13a8d7c72d2d5bd": "s_s", "da2ea49642ecc764e4b7b281ad09aff9": "\\text{TIME}(t(n))", "da2ec8e633c791f072fec98886cbd7c5": "(\\phi \\or \\psi) \\in \\Phi", "da3015600a451dfb619465acde3e460c": "(x -\n\\{w \\}) \\cup \\{z\\} = x ", "da3054b88016e79c192511f81bc8e17b": "P(k) = 126392k^5 + 412708k^4 + 531578k^3 + 336367k^2 + 104000k + 12463.\\,", "da30c4d3ecc0913ee0592764c702a048": "\\kappa(X) = \\operatorname{dim}\\ X.", "da30cfd00db66ceb348fa165e8cbf39d": " \\delta W = \\left(\\sum_{j=1}^m \\mathbf{M}_j\\cdot \\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_1}\\right) \\delta q_1 + \\ldots + \\left(\\sum_{j=1}^m \\mathbf{M}_j\\cdot\\frac{\\partial\\vec{\\omega}_j}{\\partial\\dot{q}_n} \\right) \\delta q_n .", "da30e6ea6c52dee7b8260086e4a09a5d": "{(1.1)_2}\\times 2^{-2} ", "da30ee97ee6fd138d59d6a2983deec00": "a, b, c, d = -31764, 7590, 27385, 48150", "da310f6bd62a7df787bf1796cface09f": "R_{2}\\ge R_{1} * \\text{Prob}( D_{1}>x ) ", "da31ac559be79b1dc08989c2fa6244d7": "dU = \\plusmn dV\\,", "da32032cd2413186568d9105217ec477": " \\tau_k = \\min \\{ t : X_t = -k \\}. ", "da326f7200e158a864695985b2e2f095": "q_i", "da32c91dc697ff7b9571b54c65f2a0d1": "\\epsilon_G", "da32f77280402f66a1cd01fc601a28df": " f(k;n,p) = \\Pr(X = k) = {n\\choose k}p^k(1-p)^{n-k}", "da332f3d7a2457948675bd44fd6e65b4": " 2(PH^2-PE^2) = PD^2-PB^2.", "da339be2f8b07937b8a47a4a38480196": "\n\\sigma=\\frac{2\\pi^5 k^4}{15c^2h^3}= 5.670 400 \\times 10^{-8}\\, \\mathrm{J\\, s^{-1}m^{-2}K^{-4}},\n", "da33aa408c95c2286c586cf5a305f79e": "\n\\frac{d u}{d t} = - \\frac{\\nabla p}{\\rho} - \\nabla \\Phi = 0 \n", "da33c74a839a0840d17fc029b52f9886": "\\pi_n^S/J\\,", "da33d1e8ca3f566fcff1caf48de12628": "\\sigma_i\\,\\!", "da3437d8891f0d6c6fb5e895d7913e71": "H_3 B \\rightleftharpoons\\ H ^ + + H_2 B ^ - \\qquad K_1 = {[H ^ +] \\cdot [H_2 B ^ -] \\over [H_3 B]} \\qquad pK_1 = - \\log K_1 ", "da3451bea665d2fd5dcf7a7b5cf2ef44": "\\tilde{V}_N", "da34c0a4caff0269be2cbe0c92a1c06e": "\n\\int_{-\\infty}^a f(x) \\, dx = \\int_0^1 f\\left(a - \\frac{1-t}{t}\\right) \\frac{dt}{t^2}", "da34c6f6e6d56f2d8c79441ef60ef057": " \\varinjlim (G_k, G_k^+),", "da34ca1171e9cb986d8837d62cc35dcb": "\\textbf{Q}_k", "da34e0b127bca636e3da1b670d380163": "\\mu_{k,i}(w_1\\ldots w_{P(k)})=\\left(\\sum_{s\\in S_k, s_1\\ldots s_i=w_1\\ldots w_i}\\mu_k(s)\\right)\\left(\\frac{1}{2}\\right)^{P(k)-i}", "da35275d6f47ec67904727b9d0e858c2": "L(x_j) = y_j \\qquad j=0,\\ldots,k", "da356b7a7d4e934787af241ecc1c8b8b": "C = \\sqrt{2\\, \\pi}", "da357dcffc8df3904446da2d58015334": "|V_H| = O(\\log V)", "da35910661bd80f202bc43156013e074": " |\\psi\\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ \\pm i \\end{pmatrix} \\exp \\left ( i \\alpha_x \\right ) ", "da3596c863ae77f73bbe6c43ebdeec98": "\\omega = e^{2 \\pi i/N}", "da35e42c1397a8dfc5e0ead77d5d45bf": "\\displaystyle{Q(a)R(b,a)=R(a,b)Q(a).}", "da360a8fcdaff39deb71d362893230f7": " \\log |S|-K(x|S) ", "da36122ef5dc1440988e7cb0eaaac3cf": "\\Delta r_{ij}", "da369d6ac19b0a415ceb547aa7f61435": " \\partial_\\gamma T^{\\beta\\gamma} = 0 ", "da36bcd8ffc429d69fb8945d834ab600": "\\eta = {\\text{energy provided to the load} \\over \\text{heat energy absorbed at hot junction}}.", "da36c9b415a6d8cb1936b6d554ee96af": "\\sigma\\, a\\, \\frac{\\sinh\\, \\bigl( k\\, (z+h) \\bigr)}{\\sinh\\, (k\\, h)}\\, \\sin\\, \\theta\\,", "da36f9c7a42a1683ca39f26f1ccf7463": "a(b+c)", "da3710f91e385c61a3d90522dfa3cfbe": "Tr (T^i) = 0", "da3729571fa13fcb08abb7c3f2ce2107": "\\gamma = 1/2", "da373148ee7e1880a23c60d5cf6a5da3": "\\mathbf{w}/\\alpha \\,", "da37d1938914200bf2b6f89ff7a6d977": "\\delta_{t}=\\frac{a'(t)}{a(t)}\\,", "da380362060367c606462cb9dc374530": "\\lambda_{\\bold{k}} = \\epsilon", "da383dbb670de59731575d9935f3f084": "x^4+480*x^3-270000*x^2+15552000*x+1866240000=0", "da38ccc31893e39d23a12e0f7a5e2225": "(X^*,d^*)", "da38f9d4b84e3a21c5864579e30a2cd2": " R = \\frac{ \\left( \\int{\\overrightarrow{E} \\cdot dl} \\right)^2}{P_d} = \\frac{V^2}{P_d} ", "da38fac34cdc9a0b6730e38fa9a86205": "\n\\frac{1 - X}{X^2} = 3.84 \\eta \\left(\\frac{k_\\text{B} T}{m_\\text{e} c^2}\\right)^{3/2}\\exp\\left(\\frac{Q}{k_\\text{B} T}\\right).\n", "da3931c6cdccbc071c36cf2000f201d9": "v+1/2", "da398db99fac5f6e2e1f1ce30fa61b66": "|x_1 - x_2| < \\delta", "da39d27158615c55b13fc93659db6602": "\\sum_{n=0}^\\infty f_n(x)", "da39d7a01331c8f11df953b4ca587003": "1 + i_t = (1 + r_{t+1})(1 + \\pi_{t+1})", "da3a0c49402268d43eb8e10132c2f910": "\\sin\\sin\\sin\\frac1x^2", "da3a30399c174012f545e0b15def77e1": "\\frac{\\ddot{Z}}{Z}=-m^2", "da3ab569a39aac99d4ce80b4a84bf5f1": "\\phi'(x) = e^{ie \\lambda(x)}\\phi(x)\\quad\\textrm{and}\\quad A_\\mu'(x)=A_\\mu(x)+\\partial_\\mu \\lambda(x)\\ .", "da3ae0828773e4ebd0f0e5160e4308cf": "\\{\\boldsymbol\\mu+\\boldsymbol{\\Sigma ^{1/2}}\\mathbf{v} : \\mathbf{v} \\in \\mathbb{R}^k \\}", "da3b7271fc3472267f9345bf12c918e6": "\n\\mathrm{C} \\rightleftharpoons \\mathrm{P} + \\mathrm{L}\n", "da3be4d82d5d478030fa438ffa93763b": " j \\ge 0", "da3c1ade7272968634b087ffdc9b577f": " ds^2 = E \\, dx^2 + 2F \\, dx \\, dy + G \\, dy^2 ", "da3c2663610150d454c366253f0d7344": "\n\\forall x \\ne 0, \\exists u \\qquad \\dot{V}(x,u) < 0.\n", "da3c2ae94e776977317bebbedd652b44": "\n\\mathbf{P} = \\mathbf{d}\\, \\sum_{\\mathbf{k}} P_{\\mathbf{k}} + \\operatorname{c. c.}\\;,\n", "da3c7d7fd4419fa8e62755e151be13fa": "f_\\mathrm{C}=2N{\\omega}u.", "da3cf30ed9357075219e0db4b0bc8891": "e_1 = -\\Omega(x_1)/\\Lambda'(x_1) = -649/54 = 280 \\times 843 = 74\\,", "da3cf9ffb8134660f73ec13d4cf47fed": "\\mid \\mathbf{E} \\mid", "da3d5eba3ebff0f991bf393f1b97d4ec": "HV = \\frac{F}{A} \\approx \\frac{1.8544 F}{d^2}", "da3d6e410faaf7fd5279f77607e8e679": "\\beta \\in \\Omega^k(P,V)", "da3d7f49cdb87e6d1def4329a570ec22": "T(t)\\,", "da3d86857afbd707895c0fd90e942bc9": " T(h,a) = \\phi(h)\\int_0^a \\frac{\\phi(hx)}{1+x^2} \\, dx", "da3db63e213950e98d6574745ff00bdf": "\\sec \\theta = \\frac{2}{e^{i\\theta} + e^{-i\\theta}} \\,", "da3ddcf9188ef513d8445fa4f905e6e3": "q = G Q^{2}/(4 \\pi \\epsilon_{0} c^{4})", "da3e19199dce673466e57fea9f2b6a70": "\\Psi\\propto\\begin{pmatrix}\n\\rho^{\\gamma-1} e^{-\\rho/2}\\left(Z\\alpha\\rho+(\\gamma+1)\\frac{\\gamma\\mu c^2+E}{\\hbar cC}(-\\rho+2\\gamma)\\right)\\\\\n0\\\\\ni\\rho^{\\gamma-1}e^{-\\rho/2}\\left((\\gamma+1)\\rho+Z\\alpha\\frac{\\gamma\\mu c^2+E}{\\hbar cC}(-\\rho+2\\gamma)\\right)z/r\\\\\ni\\rho^{\\gamma-1}e^{-\\rho/2}\\left((\\gamma+1)\\rho+Z\\alpha\\frac{\\gamma\\mu c^2+E}{\\hbar cC}(-\\rho+2\\gamma)\\right)(x+iy)/r\n\\end{pmatrix}", "da3e433e175261da15628d4c2e466c84": "\n\\omega_{r} = \\omega_{\\varphi} \\left( 1 - \\frac{3r_{s}^{2}}{4a^{2}} + \\cdots \\right)\n", "da3eca892a552252ed6d02402667a544": "L_{X^{(1)}}\\theta_1 \\equiv 0 \\pmod{\\theta_1}", "da3ecfd42504a6484773d9e7d6266207": "a>c", "da3f04acf56e83dd026b7689596bfb8b": "\\begin{bmatrix}\n|U_{e 1}|^2 & |U_{e 2}|^2 & |U_{e 3}|^2 \\\\\n|U_{\\mu 1}|^2 & |U_{\\mu 2}|^2 & |U_{\\mu 3}|^2 \\\\ \n|U_{\\tau 1}|^2 & |U_{\\tau 2}|^2 & |U_{\\tau 3}|^2 \n\\end{bmatrix}\n= \n\\begin{bmatrix}\n\\frac{2}{3} & \\frac{1}{3} & 0 \\\\\n\\frac{1}{6} & \\frac{1}{3} & \\frac{1}{2} \\\\ \n\\frac{1}{6} & \\frac{1}{3} & \\frac{1}{2} \n\\end{bmatrix}.\n", "da3f2277578893ce586a26f89912d927": "{ }^\\dagger\\,", "da3f3df937ea46baa481a81a54590833": "\\sum_{i=1}^k f_i(x)=g(x)+f_k(x)", "da3f42e25585ca4483d325d528c8c7c1": " n_\\alpha~M_{\\alpha\\beta,\\beta}", "da3fa2bfa7c542c7e0a0c2eb3128da39": "\\scriptscriptstyle\\hat\\theta", "da3fb077e0c1bbae1529bb65532a808c": "q_j,", "da3fd800b078fc9d3b265b0d569e4da0": "(3)\\quad F r_1^2=\\frac{q^2}{g y_1^3}=\\frac{(10)^2}{32.2*(0.24^3)}=224.65", "da3feeae85c2419b060af7e598327249": "{s} = {A} {x} - {b}\\,", "da3ff230d1acc6c0dbbdf6809d35fc9e": "D(f)=20\\log_{10}\\left(R_D(f)\\right),", "da400ada08263d1f2751cefe0545515e": "\\gamma = (1 - v^2/c^2)^{-1/2}\\!", "da4011deaada048da68941a9045490c3": "Fr={v \\over \\sqrt{gy_0}}", "da4015fb43a855da48fa3456e88014a1": "E_1=0", "da403596be5fd8c7530407805c9c7853": " u_{tt} = c^2 \\left( u_{xx} + u_{yy} \\right). \\,", "da4081e0d3dfff3320ed6d92ea5ec95d": " X(\\omega) = \\frac{1}{\\sqrt{2\\pi}} \\int_{0}^T x(t) e^{- i\\omega t}\\,dt ", "da40bccdfd416f70863e9b5527124614": "\\left|\\psi_1\\psi_2\\right\\rangle = \\pm\\left|\\psi_2\\psi_1\\right\\rangle", "da40db82a462e190abed75ab9bdd53ce": "\nWr=\\frac{1}{4\\pi}\\int_{C}\\int_{C}d\\mathbf{r}_{1}\\times d\\mathbf{r}_{2}\\cdot\\frac{\\mathbf{r}_{1}-\\mathbf{r}_{2}}{\\left|\\mathbf{r}_{1}-\\mathbf{r}_{2}\\right|^{3}}\n", "da40ecdeec17a03423d53e89be64c60c": "\n\\begin{bmatrix}\nc t' \\\\ x' \\\\ y' \\\\ z'\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\cosh\\phi &-\\sinh\\phi & 0 & 0 \\\\\n-\\sinh\\phi & \\cosh\\phi & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nc t \\\\ x \\\\ y \\\\ z\n\\end{bmatrix}\\ .\n", "da40f71b6c4050a336ba2a0f8e2fd412": "N^{\\prime}-1", "da412199ab7635cea130b2b24e1eb301": "y = 2x + 10", "da412b2487e24d3447161a0f2f6baef0": "Pe = \\frac{F}{D}=\\frac{\\rho u}{\\Gamma/\\delta x}", "da4137d24d0dbdea22a49d9a8be470da": "X_{(1)},\\ldots,X_{(n)}", "da414f6339242c43191efe2615a57a1a": "S^k \\mathbb{C}^n", "da4153814bf17358bf72231646a761ae": "\\pi(k_1,k_2) := \\frac{1}{2}(k_1 + k_2)(k_1 + k_2 + 1)+k_2.", "da41a953fbf94a1eb1ca61454c27a95d": " \\boldsymbol{w} ", "da41ecb3fbf385e148e0e5be3696ab8d": "480-x", "da41f08ffed6b980a3d931aab689d265": "\\gamma \\neq 0", "da4212a136305409709df2885e51fee0": "|\\log(z)|<2 \\pi;s\\neq 1,2,3,\\dots; a\\neq 0,-1,-2,\\dots", "da423095d76c957c0fea234d6dded85e": "m_{A,B}=\\varepsilon_{FA\\bullet FB}\\circ Fn_{FA,FB}\\circ F(\\eta_A\\otimes \\eta_B):F(A\\otimes B)\\to FA\\bullet FB", "da428a04d4f4d50d060778abd0e0829e": " y = 2 A \\sin \\left ( \\langle k \\rangle x - \\langle \\omega \\rangle t \\right ) \\cos \\left ( \\frac{\\Delta k}{2} x - \\frac{\\Delta \\omega}{2} t \\right ) \\,\\!", "da429736bcee12e2b04cc55f1eceaa13": "D_a \\tilde{E}^a_i = 0", "da42fa6f957c7c63d07ce3ad323c08fe": "\\underline{\\alpha}", "da4309927fe0cb00317cff34eec8fd74": "r_c = \\frac{2K}{b-c+a} = \\sqrt{\\frac{s (s-a)(s-b)}{s-c}}.", "da430b928e29a9242a876337528d3d3d": "I = 1.1 \\times I_\\mathrm{o} \\times [ (1 - h/7.1) 0.7^{(AM)^{0.678})} + h/7.1 ]\n\\,", "da443c5a1662b03cdf9ae1c2b02ee163": "\\langle \\hat{S} \\rangle", "da447a6af4c28a489d7b387458af4a92": "F_D\\, =\\, \\tfrac12\\, \\rho\\, v^2\\, C_D\\, A", "da449413f8a8fad5185c88f3f703a145": "\n\\operatorname{Arg}(x + iy) = \n\\begin{cases}\n2 \\arctan \\left( \\frac{y}{\\sqrt{x^2+y^2}+x} \\right) & \\qquad x > 0 \\text{ or } y \\ne 0 \\\\\n\\pi & \\qquad x < 0 \\text{ and } y = 0 \\\\\n\\text{undefined} & \\qquad x = 0 \\text{ and } y = 0\n\\end{cases}\n", "da44aac94b82135888c080a06999d25a": "\\Delta<0", "da450365f1af225eb7b96eeb7519937b": "RED_{\\mathbf{Cl}}(P)", "da456a537e47f8ad249cc422cee68ed2": "\\left(\\sum_n \\mathbf{p}_n \\right)^2 = \\left(\\sum_n \\mathbf{p}_n \\right)\\cdot\\left(\\sum_k \\mathbf{p}_k \\right) = \\sum_{n,k} \\mathbf{p}_n \\cdot \\mathbf{p}_k = 2\\sum_{nm\\ge0\\mbox{)}", "da45929d62ee36173bbb99a42f28dea9": "T^{k+1}p=T(T^kp)", "da459b82022da8f9a1cc5fcf862f9548": "a(m\\otimes n):=\\Delta(a)(m \\otimes n)=(a_1\\otimes a_2)(m\\otimes n)=(a_1 m \\otimes a_2 n)", "da4682343482548c8530a1a3c4fd03c0": "\nn \\int\\limits_{-\\infty}^\\infty \\frac{(F_n(x) - F(x))^2 }\n{[F(x)\\; (1-F(x))]} \\, dF(x),\n", "da46a57784ec4d869a49dacc6f71e6d6": "a_i = \\frac{1}{2}\\mathrm{tr}\\,\\sigma^i M", "da471af709fb34c7841ea2b69f85b8c7": "\\hat{H}_{\\lambda}", "da473daf75359f0ea725e72deac43b61": " \\textstyle 2\\pi ", "da47536f6d620a702d80082a51160efd": "\\mathbf{M}^{-1}(\\mathbf{A x}-\\mathbf{b}) = 0", "da4760886b448bf1f9ecd7e0fd994bce": "u\\in V", "da4789c6e7a5694ce3984ad16348b672": "H \\otimes K", "da47e6dbe751985e060abef693f9e92f": "\\Phi_\\alpha(z)=U_\\alpha z^{\\alpha_0} E_-(\\alpha,z)E_+(\\alpha,z),", "da485cb26f6e37c697c28ef265552833": "h \\circ \\gamma \\circ h^{-1} = \\rho(\\gamma)", "da48a817b18226ea2411181b019b187a": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ 0.08\\end{smallmatrix}", "da48ac63d0e554d2124e8bf62b29df71": "P = 8ac - 3b^2", "da4990aacbd02d89812c0cac435048e3": "\nn_i = \\frac{g_i}{e^{\\alpha+\\beta \\epsilon_i}+1}.\n", "da4995b96bd636b7376c1e6a9257d7bd": " \\int \\frac{dx}{a+bx+cx^2} = \\frac{1}{ \\sqrt{q}} \\ln \\left( \\frac{2cx + b - \\sqrt{q}}{2cx+b+ \\sqrt{q}} \\right) + \\text{constant, where } q = b^2 - 4ac. ", "da4a694579f52f1461e5226f9df4f263": "\\lim_{x\\to a} h_k(x) =0. \\, ", "da4a854075c7bfd4947da242fb5fb746": "p_n,\\ p_{n+1},\\ p_{n+2}", "da4b86a8fe2321ae8376e32bd1d1c2a7": "\\hat{\\textbf{y}}_{k\\mid k} = \\hat{\\textbf{y}}_{k\\mid k-1} + \\textbf{i}_{k}", "da4b9e0fca5a3233fdae7bf8e53e9ae1": "[NO_3]=K_1\\frac{{[NO_2]^2}} {{[NO]}}", "da4c02d707e8a0d08cda834024b21d19": " \\left| +z \\right\\rangle \\otimes \\left| -z \\right\\rangle ", "da4c169c08b2ad8eba4c6dcf88bab891": " \\eta_{0} ", "da4c444a170f2dfb8640e7c226899e4f": "\\zeta(s)=\\overline{\\zeta(\\overline{s})}", "da4c9130192c6408077fdf3898a9fc43": " x_{n+1} = 1 + u (x_n \\cos t_n - y_n \\sin t_n), \\, ", "da4e09ab863b7b0047650b01ee277cfd": "\\frac{x:\\alpha \\in \\Gamma}{\\Gamma \\vdash x:\\alpha}", "da4e29661b4e91a52b4351070344898c": "p(x + a)", "da4e70ac9a4684569760d1c3f2ad4871": "y.\\ Q \\rightarrow x:=y", "da4e780830f1e986077cae93d0f4138c": "\\sum_{n=2}^\\infty \\frac{1}{2^n}[\\zeta(n)-1] = \\ln 2 -\\frac{1}{2}.", "da4e914e615a503723ac283d7c686acf": "\\begin{matrix}J\\!D & = & J\\!D\\!N + \\frac{\\text{hour} - 12}{24} + \\frac{\\text{minute}}{1440} + \\frac{\\text{second}}{86400}\\end{matrix}", "da4f38d6baa2e71475e415f2b9e856a9": " Y = (I_1)^{a_1} * (I_2)^{a_2} \\cdots ", "da4f5a2410a07accbd14cab56baca699": " \\operatorname{Pr}(\\lambda)= \\langle \\operatorname{E}(\\lambda) \\psi \\mid \\psi \\rangle ", "da4f77fb9da3ad8307bee2629fda05dc": "A_0 A= A A_0", "da4fb5c6e93e74d3df8527599fa62642": "120", "da4fe5c9cf708d3a9f4c0855721480e3": "\\gamma^\\mu \\gamma^\\nu + \\gamma^\\nu \\gamma^\\mu = 2g^{\\mu\\nu}I.", "da5017a3d117d6db4199c4d512d256ad": "T_{m,n}=\\emptyset", "da503b411eacdd691ac902e2b3454b07": "V_i(t=0)=0\\frac{}{}", "da5067a6e5a240a77b68cc2bbf059f40": "A \\rightarrow S: \\left . A,B,N_A,\\{A,\\mathbf{N_B'}\\}_{K_{BS}} \\right .", "da50cce763c3a5767d9e2e5dd5f3b7b9": " Z(P, Q, s) = \\sum_{n=1}^{\\infty} \\frac{f_n(P)f_n(Q)}{ \\lambda_{n}^s}", "da5101638e6780c426e6eba0ff8279b9": " \\int_0^1H_{x}\\,dx = \\gamma\\, , ", "da511a7e73ef6c46a271385b05cca9ae": "(\\bar{\\mathbf{3}},\\mathbf{1},-\\textstyle\\frac{4}{3})", "da511fb83d24081e9a770e4b5063edc9": "\\left(\\frac{x}{N}\\right)", "da5177f072504469b213acdaab4f1d99": "z(x,y)\\geq 4", "da51f84154fd08b62c6f47ee73bb5fcb": "C(\\rho)", "da521981c9dca4af944edecf08187e7f": "a_i\\in{\\mathbb Q(x)}", "da5245fc8e013d7b677ca7fa04a26613": "sim(q_{or},d_j)=\\sqrt[p]{\\frac{w_1^p+w_2^p+....+w_t^p}{t}}", "da524b0671730e1505bbdf0ed209094b": "L\\subset M", "da526e787a84d935a11672147e67a629": "\\nabla \\cdot \\mathbf{E} = \\frac {\\rho} {\\epsilon_0}", "da526e7c73c57f4d48bb5e5ea27d5b9d": "\\lim_{||x|| \\to \\infty}\\rho(\\left \\Vert \\mathbf{x} - \\mathbf{c}_i \\right \\Vert) = 0", "da52700c88298121870a77037f06c5da": "(\\;5)\\quad \\quad \\frac{p}{\\rho^\\gamma} = \\text{constant}.", "da5385fc579833ee59e652da5667b421": " \\chi^2 = {(44 - 50)^2 \\over 50} + {(56 - 50)^2 \\over 50} = 1.44", "da53a06472bde4e464a34de4754d569c": "\\mathrm{Defensive Rating} = \\frac{\\mathrm{Points Allowed*100}}{\\mathrm{Possessions}}", "da5407f087c613e0a15e914f0f74777b": "P = P^\\text{H}, ", "da5490d7f5053223bc4c83c9cea305e2": "W=\\int_{t_1}^{t_2}\\boldsymbol{F}\\cdot\\boldsymbol{v}dt = \\int_{t_1}^{t_2}F_z v_z dt = F_z\\Delta z. ", "da54bd9838b2c02a72f84c65bb46424e": " I_{\\mathcal{A}}(X)", "da54e184c6adb675eb810cfa49c65e25": "u_\\star = \\sqrt{\\frac{\\tau_w}{\\rho}},", "da54e55235b219f470936bb7e528f33a": "k_c = \\int_{\\omega}s(\\lambda,x)\\rho^c(\\lambda)d\\lambda", "da554e9a077e718e1434c104d818ee80": "PV = \\frac{Itd}{i+d}", "da5583177553f69ecdd5e09c53b760b7": "f(w)=1", "da55ac6a2602eb7cf540a92586de1805": "\\frac{\\pi r^2}{h^2} \\int_{0}^h (h-x)^2 dx", "da55b7b9f43cf516748b588a4409d4e5": "\\partial_{xx} \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_{xx} \\psi )} \\right) = \\partial_{xx} \\left( -\\partial_{xx} \\psi \\right) \\,", "da5617e4353779d7eaca836fcfaf4e69": "J_{top}^\\mu =\\epsilon^{\\mu\\nu\\rho} F_{\\nu\\rho}\\ .", "da563e52e2f4fb5500551bb1656d5b79": "\\mathbf{T} = \\begin{pmatrix}\n 0.7 & 0.3 \\\\\n 0.3 & 0.7\n\\end{pmatrix}\n", "da564a5791a6665c8cc92488bd35e35b": "\\#X(\\mathbb F_{p^n}) = \\mathrm{Tr} F^n|_{H^0(X)} -\\mathrm{Tr} F^n|_{H^1(X)} +\\mathrm{Tr} F^n|_{H^2(X)}.", "da56eedd57cb570eacd881608028d50b": "X_\\text{et}", "da57203f5ac1b685ec929eb656b2ad6b": "d(v_0) q _0", "da57262a0a0c58d0efa852f963da3925": "(\\nabla I)(\\nabla I)^T", "da57762aafad4ec5da2bf47b8fd795bd": "5 \\pi / 6", "da581910058d3ed2c36a38422d1d1a51": "H(x_1,\\dots,x_2)=\\frac{h(u_2,\\dots,u_n|u_1)}{|J*|}", "da5881f2563975425dcf6e6d176149c7": "d=2^{k-2}", "da5892b063cf584f1b1c01137a5ad72d": "\nm_{1}\\ddot{\\mathbf{x}}_1 + m_2 \\ddot{\\mathbf{x}}_2 = (m_1 + m_2)\\ddot{\\mathbf{R}} = \\mathbf{F}_{12} + \\mathbf{F}_{21} = 0\n", "da58c9b405d4a8ba56de59599adba4a0": "x^DP(x^{-1})\\ne P(x)", "da58e3a3a311c74e42b98fdcb18b0d09": "P(\\exist n_0\\ge0, \\forall n\\ge n_0, |X_n| \\ge A) = 1", "da591029205d7e3a2f47f80e64cec4ff": "\\Sigma_i = \\Sigma_j,\\ \\forall i,j", "da593f18882afa31cc49d1b48c17929a": "\\operatorname{Cl}(S_n)", "da59d3e50f6ff99ac1112a26eedf894c": " I_{SN} = I_{S} \\times \\frac {V} {V_{S}}", "da5a1bd215205ff8547053d716c7d191": "\\alpha_d={{\\pi}\\sqrt{\\varepsilon_r}\\over{\\lambda}}{\\tan \\delta}", "da5a5828efd8f97e7cbe45b826f6de9f": "( G2 + 1 ) / 2", "da5a5a54368288c660952dd7027681ac": "T_0(q) = \\sum_{n\\ge 0} {q^{(n+1)(n+2)} (-q^2;q^2)_n \\over (-q;q^2)_{n+1}}", "da5a8431d3821ae1a3f9273e61c154d4": "\\zeta^r", "da5a87b94c81f6c65dcf303b41083d04": "\\hat{n}_b", "da5aca1a37d003d099e65c65182def4e": " B = \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}. ", "da5ace130c921528cd5b83c02cfd4f96": "p_{n+1}-p_n = O(\\sqrt{p_n}\\,\\log p_n)", "da5b0e44ce2af0be4c27e09177e2a8d2": "\\scriptstyle \\,{}^{0}i \\;=\\; 1", "da5b38b7845fdda822f3755c40f792b0": "\\ p'=\\frac{1}{3}(\\sigma_1'+\\sigma_2'+\\sigma_3')", "da5b5dbc24903ed9e720a10e49537ef5": "\\beta = \\left ( \\mathrm{dB} \\right ) 10 \\log \\left | \\frac{I}{I_0} \\right | \\,\\!", "da5b9dd39b5f8ae174f88f84492adf20": "[L]^T[L]=[I], ", "da5bf7afe6569e11fe35c79b9a4fa8f8": "F\\,=\\,eN_{A}", "da5c010d65d7a2036b26757ea9863895": "R = \\frac{n(n-1)}{\\alpha^2} = \\frac{2n}{n-2}\\Lambda.", "da5cf0b1527ac862a5004b671475e240": "0 = -\\frac{\\partial \\phi}{\\partial p} - \\frac{R T}{p}", "da5d0b55424f9b240537f193eb83aba3": "\nI_0 = \\;10^{-12} \\, \\mathrm{W/{m}^{2}} \\,\n", "da5d8bceb1f1162ec1f8a59beb8c8c8b": "d_i|d_{i+1}", "da5d8dff960fe1a3fc0c4f95c788424e": "\\Delta P = \\frac{1}{2}\\rho f L \\frac{S}{A}v^2", "da5d986308e011f5fb90ce5340e210ac": "\\scriptstyle \\theta_c ", "da5dd3387c28cb68d5a53fda61f569b8": "\\displaystyle iu_t+((1+|u|^2)^{-1/2}u)_{xx}=0", "da5de85e74fe58f1f5f017931001b911": "1/\\epsilon_{0}", "da5e0cc19f6ce04765b340b1440a2806": "62745 = 3 \\cdot 5 \\cdot 47 \\cdot 89\\,", "da5e423189c7d40dc4c38b07efb99018": " \\scriptstyle\\Omega\\setminus\\{x_0\\}", "da5e7368811be4e9cffb20b8b5d1ffda": "\\frac{Z_{i 1}}{Y_{i 2}} = \\frac{Z}{Y}", "da5f1327c14464afa6cf9d6390189d16": "f_o.", "da5f48507944c69a6d4654436a6b0d71": "\\scriptstyle cu\\in BV(\\Omega)", "da5f69c939df3b3e26efc34201d9c430": "\\left[Uf\\right](x,y) = (f\\circ S^{-1}) (x,y).", "da5fb72c905d0234494cbb4bfa36f8da": "[\\mathrm{zero},\\mathrm{succ}] : 1+\\mathbb{N} \\to \\mathbb{N}", "da5ff62593fa825430a54e7bcf6ba8f1": "\\tau = \\inf\\left\\{t \\geq 0: W(t) = a\\right\\}", "da5ff8bb1d79ffa0df7b03e7fd7d4a21": " \\Delta S_{mix} =-k_B\\sum_{i=1}^r N_i\\ln(N_i/N) = -N k_B\\sum_{i=1}^r x_i\\ln x_i = -n R\\sum_{i=1}^r x_i\\ln x_i\\,\\!", "da60012cc5a08022d6e117767c2a5a09": "t.", "da604c4e80535cb376b2510ff1b147f8": "G(r) \\approx { e^{-\\sqrt t r} \\over r^{d-2}}", "da609d1f26360b29c40da34c6b0f8123": "f \\sim g", "da60a5624dd1a57ab723b1697e3b72c5": "V[] \\to d ~|~ dV[]", "da60a9d40ed99a192c656a8b672554d2": "x = \\frac{2 \\sqrt 2}{\\pi} \\left( \\lambda - \\lambda_{0} \\right) \\cos \\left( \\theta \\right),", "da60cee190ba64f336c1f4f88c072dbb": "P|\\psi\\rangle", "da60eba77db4aadac85e916f343ff4ed": "w=T_0 \\frac{d}{ds}\\frac{dy}{dx} = \\frac{T_0 \\frac{d^2y}{dx^2}}{\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}}.\\,", "da610f62a014da43c0592035a1d7db3a": "s(a, b) = \\frac{C}{\\left|I(a)\\right| \\left|I(b)\\right|}\n \\sum_{i=1}^{\\left|I(a)\\right|}\\sum_{j=1}^{\\left|I(b)\\right|}\n s(I_i(a), I_j(b))", "da6193e1c81b898063286239aafbe5d1": "f:X\\otimes U\\to Y\\otimes U", "da61abdfbcd5df31148beeb8ab7b9138": "\\deg y < \\deg p", "da61cf837f2486ff79fae55d773596cb": "\\Pi = D\\sum_{j=1}^n \\oint_{S_j} w_j \\left(\\lambda \\sqrt{I_1} - \\sqrt{I_2}\\right)^2 \\operatorname{d}s", "da62ade2712b858b76deb2e65ea30aa0": "\n\\| {d\\over dt} |\\theta\\rangle \\| = \\| H |\\theta\\rangle \\| \\ge m_0 \\| \\; \n|\\theta\\rangle \\| ~.", "da62c69139cc8c25cab9da66feaccdb9": "S(\\mathbf{Q},\\omega)", "da62dbfb6c570472f1d95d738343361a": "\\mathfrak{R}=\\operatorname{Hom}(\\pi,G).", "da6354938498a9a94bf4e9dd3bcccb50": "\\hbar\\equiv\\frac{h}{2\\pi}", "da636b3dd3c1d6a92c198b07429db043": "\\mathbf{g}=\\mathbf{g}_{-1}\\oplus\\mathbf{g}_0\\oplus\\mathbf{g}_1", "da641431308ee21ac4cc1f5869209924": " \\Sigma \\alpha_i", "da644ce4bdd73f472501cfc5a52ef459": "\\overline{\\mathsf{f}}(a) = \\nabla_b \\left\\langle a\\underline{\\mathsf{f}}(b) \\right\\rangle", "da647630f411ef60e8b3a261900e7571": "x \\ne y ", "da648e7753d875a2b16fece960cf5052": " \\xi < \\xi_\\mathrm{cutoff} ", "da649a489656e899c9f173aa7ba44296": "p_{j+1}(x)=(-1)^{\\deg p}(e(x)^2-x\\,o(x)^2)", "da64a279849876126cda735070b5868c": "\\operatorname{Mode}(X) = \\mu+\\frac{\\sigma}{\\xi}[(1+\\xi)^{-\\xi}-1] .", "da64c73bdc1efeda249f7d5d00e36eed": "y_{it}", "da6507cb6bdfc0a3dde877f9a069f014": "{{Tonnage}} = \\frac {({Length}- ({{Beam}\\times\\frac{3} {5}})) \\times {Beam} \\times \\frac {Beam}{2}} {94}", "da655102aa4afa98cb4866ffad7d98b5": " TTKG = {\\frac{urine_K}{plasma_K}} \\div {\\frac{urine_{osm}}{plasma_{osm}}} ", "da6557ed4ff258c05dbc8d637881d025": "S={1\\over 16\\pi G}\\int d^4 x\\sqrt{-g}L_\\phi+S_m\\;", "da659e56d9b54a4dac0e1511d0d1dca7": "Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, \\,", "da65aa1b85bc8338316b76b31b9e173f": " \\log n ", "da65b8e7f26a3dc09473ef6dd3f667ac": " \\mathbf{x}'^{\\top} \\mathbf{F x} = 0. ", "da65bbc2d2d152a28a1047a73d88abf8": " E=mc^2 ", "da661f36d88529756b444c0427ed6435": "\\overrightarrow{P_1 P_2} = (x_2-x_1 ,\\, y_2-y_1)", "da662f722587fc5c6cea418bf4e13a0a": " \\cdots ", "da66946ee801c2ca93fd0c3973c177db": "2\\sin\\left(54^\\circ\\right) = \\phi ", "da66cf7c4d727a1e03b8ee3aa4642686": " Q - X_1 - X_2 - P(M_3,3) - P(M_4,4) - P(M_5, 5)", "da670bc6ab219b5c56ff9320af968a0e": "\\tau_{\\rm MS}", "da67526ec1b9b423ca5489d6e8312669": "c_i m(i)", "da675431d90204355afb80cb5ea4b3d6": "\n(7.1)\\quad\nd_W(\\mathcal{L}(W),N(0,1)) \\leq \\frac{5 E|X_1|^3}{n^{1/2}}.\n", "da677668a73c5229c56c4f2738290c1c": "\\nu = \\frac{3-q}{q-1},\\text{ but only if }\\beta = \\frac{1}{3-q}.", "da67f2cc71f35b98eaeb29b6d145eca6": "\\left\\lfloor\\binom{n}{2}\\Big/\\binom{3}{2}\\right\\rfloor=\\left\\lfloor\\frac{n^2-n}{6}\\right\\rfloor.", "da680cffbecaf0c20be5d86fc3dafd6d": " T(t) = T_{\\mathrm{env}} + (T(0) - T_{\\mathrm{env}}) \\ e^{-r t}. \\quad ", "da6834ea306c993ae190d8ac693a25f0": "c_t", "da686f5d648c3d33dedaca83a2b438c2": "Q_{Y|X}=X\\beta_{\\tau}", "da691c47548f86dfde73daff02561ee2": "\nS_{X_1X_2} \\approx 0.084 \\, \n", "da6927b645cb798da8b59e64490a4960": "\\begin{align}\n f(C) &= \\frac {9}{5} C + 32 \\\\\n f^{-1}(F) &= \\frac {5}{9} (F - 32)\n\\end{align}", "da692b4af9962f4840e3283a4f4d542d": "\\Re(s)=\\tfrac{1}{2}", "da697731b6df9aa41bd330febf3ed379": "O\\left(\\frac{y\\log^2 y}{\\log\\log y}\\right)", "da697a998925d6bae1a491a59fe490a7": "g>2.3", "da69a59d739404136199adac4bbc9f7d": "f(x)=(1+\\sin(1/x))x^2", "da6a08fcd623f64167f3e436e808ffc9": "\\ g_b(x,y) \\ \\stackrel{\\mathrm{def}}{=}\\ g_a((x - \\Delta x) \\bmod M, (y - \\Delta y) \\bmod N)", "da6a34bf26f6d2f113205615479852bc": "J^\\alpha f=\\mathcal L^{-1}\\left\\{s^{-\\alpha}(\\mathcal L\\{f\\})(s)\\right\\}", "da6a86aefe7454c42c43d33a84e25f0a": " \\mathbf{s}(x) = \\sum_{i=0}^{p-1} \\mathbf{d}_i N_i^n(x) , ", "da6ae254d0b6ca1abf2d7f8e430e01fd": "\\boldsymbol{p}", "da6b556115419acdea7018a8dd0908e2": "B_{ij} = \\frac{\\mbox{Energy absorbed at }A_{j}\\mbox{ originating as emission at } A_{i}}{\\mbox{Total radiation emitted from }A_{i}}", "da6b7169ce0803ff94ea52584d418806": " {\\vec{x}}_{i}(t) ", "da6b71ba06b30acbf0a3f35ca93f2b6c": "\\Delta_\\infty u = 0", "da6b7370512b49296c30813803b1ea0d": "\\langle {\\Phi_0}\\vert e^{-(T_1+T_2)}He^{(T_1+T_2)} \\vert{\\Phi_0}\\rangle = E ", "da6bad4be0dfeb98a9f11bd3ad9bfa86": "\\delta_{ij}=1", "da6bccfbc9701773231096a0303e9516": " -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2} \\, ", "da6c0916262aa26eb8897a194afce310": "2^{\\frac{13-1}{2}} = -1 \\pmod {13}", "da6c1d2f0a86d09d312f2f7b70a30a96": "[a;b]p \\equiv [a][b]p\\,\\!", "da6c1fa1808cbe608bf41d1c8904bd84": "v\\in T_g G ", "da6c4916b8d57d1a58677c6f0771ad21": "C=\\frac{1}{N}\\sum_{i=1}^N \\mathbf{x}_i\\mathbf{x}_i^\\top", "da6c6336b2940a38d3cbac1e73b018ce": "T(n) = \\Theta\\left( n^{\\log_b a} \\log^{k+1} n\\right) = \\Theta\\left( n^{1} \\log^{1} n\\right) = \\Theta\\left(n \\log n\\right)", "da6c6a6201eb8c8559ac978a86f33a74": "\\displaystyle x_{n-1}x_{n+1}=1+x_n", "da6c731820fb3a595253a055985bedfc": "AX +XB = C", "da6cb39da009dbf146fb70ec90ce46af": "v_A = \\frac{B}{\\sqrt{\\mu_0 n_i m_i}}~~", "da6cea1585cf298d6dd1e507bf8ee083": " a_n =\\frac{(X_n (x), s(x))}{(X_n(x),X_n (x))}", "da6cf714a48e8ee452343c595fba8a37": "\\mu(t|m)", "da6cf7528bf8d085b5d33918761cc02a": "\\textstyle F_{2,B}(x,Q^2)\\rightarrow F_2(x,Q^2) = f(x) \\log{ Q^2 / \\Lambda^2 } ", "da6cfb2faeb620c40ff590e58a9a50f6": "\\sqrt[193]{\\frac{10^{100}}{11222.11122}} = 3.14159\\ 26536^+", "da6d033051bfb26b43340799102ec8b6": "\\therefore u_n =\\sum_{m=0}^{N-1} a_m (-1)^m\\cos\\left(\\frac{m\\pi}{N}(n+\\frac{1}{2}) \\right)", "da6d837af7a3a5c351553ff9a525f09d": "\\frac{1}{x-x_{i}} = \\frac{1-\\left(\\frac{x}{x_{i}}\\right)^{k}}{x-x_{i}} + \\left(\\frac{x}{x_{i}}\\right)^{k} \\frac{1}{x-x_{i}}", "da6daa23a729bc6ab604d3d53fbfe525": " m(b - a) \\leq \\int_a^b f(x) \\, dx \\leq M(b - a). ", "da6dbac3afbed119a3627ed0cc9d4d8c": "c_i=0", "da6dfc80a924b1307060b7caddc9074f": "\\pi-\\psi = \\angle AKP_2", "da6e2fbcfe6e1a52dd6ba5089333856b": "b_2=\\frac{8,00 - 32\\times 41}{31}=\\frac{-512}{31}=-17\\mbox{ with remainder }15", "da6e47b9f044a5593e63ee8f2b6b3aab": "{\\partial h\\over\\partial t} -c{\\partial h\\over\\partial t} = \\frac{-\\nabla \\times \\vec{\\tau}}{\\rho_0f_0}", "da6e696f594d960e4ffe62a0046b35da": "y_{n_1,k_2}", "da6e7216ffd86df3b8b723ad5eccb226": "Ax \\le b", "da6e921c5f04d9f63ce6100555a5bf3f": "v_{\\rm xc}(\\mathbf r)\\equiv{\\delta E_{\\rm xc}[\\rho]\\over\\delta\\rho(\\mathbf r)}", "da6eb7845a7640771f8fa7276a0ce6a5": "x-a", "da6f0b9c1a1e48c613a437de77b04d74": "R_\\mathrm{spatial}(\\hat{n},\\phi) = \\exp(-i \\phi L_{\\hat{n}}/\\hbar),", "da6f24aa00d10b972ccfa62e1757c929": "\\begin{matrix}2&2\\end{matrix}", "da6f79f899cfe7f428f68677c61049ca": "1.1861", "da6f88cfa3cfe1b1f3f51f50ab35979e": "\\mathbb{H}^3", "da6fc0ffee51f1a088b6caa787cef495": "\\Delta{}E = W", "da6fcae739bf05f08627de8d3be950f5": "ds^2 = \\gamma_{ab} \\xi^{(a)}_i \\xi^{(b)}_k dx^i dx^k", "da7063912863053e24e37bad34ccb822": "\\scriptstyle{r_1 > r_0}", "da709eed35f385ce551ebf79fe920ad4": "v = \\frac{L^2}{8r}", "da70d42918100329ec12f964dd76853a": "\\lambda(f(N))=0", "da70e62c38a162a3c3283ec3d4dbbd16": "p(t)I_n=(I_nt-A)B", "da71313dc814f56c0e1ea92c3a6cd51d": "\\begin{smallmatrix}\\frac{L_{\\rm S}}{L_{\\odot}} = {\\left ( {\\frac{42.3}{1}} \\right )}^2 {\\left ( {\\frac{4,530}{5,778}} \\right )}^4 = 676 L_{\\odot}\\end{smallmatrix}", "da7135230c951494bb3b00e08b268c81": "\n\\phi _m^{\\mathrm{even}}(x)=\\frac 1{\\sqrt{a}}\\cos \\left[ (m-\\frac 12)\\frac{\n\\pi x}a\\right] ,\n", "da713746e01705a1136054437ffe2b66": "z^{-1} = {z^* / |z|^2}.\\,", "da71a8c545bf1c523aae3399c39cd021": "\\displaystyle \\sum_{n=1}^\\infty p_n 10^{-\\left(n + \\sum_{k=1}^n \\lfloor \\log_{10}{p_k} \\rfloor \\right)}", "da72b0974127c255a40b61d15b7b65c3": "H^1(G_K, GL_n(\\mathbb{Z}))", "da72c7d1778960306b02140d100cccf7": "x\\rightarrow 0", "da72f07f9dc98ab3ae360add1c2f4c94": " E_{external}=E_{image}+E_{con}", "da72ff17db5adb998602b89651f37328": "\\lambda( \\beta^\\vee ) = 0", "da7387654402190e1c74fb5ef981f7e7": "\\vec{\\alpha}(x_i) = 2\\vec{b}(x_i)/u+2\\vec{a}(x_i)\\,\\!\n", "da73cd6a246ff698c34dfd815a49ed3e": " f_X(x|\\boldsymbol \\eta) = h(x) g(\\boldsymbol \\eta) \\exp\\Big(\\boldsymbol\\eta \\cdot \\mathbf{T}(x)\\Big)", "da74041282df7cd5d594feb246dda379": "s_V(\\mathcal{R}) \\stackrel{(I)}{=} s_{V_1}(\\mathcal{R}) + s_{V_2}(\\mathcal{R}) \\stackrel{(II)}{>} s_{V_1}(\\mathcal{R}') + s_{V_2}(\\mathcal{R}) \\stackrel{(III)}{>} s_{V_1}(\\mathcal{R}') + s_{V_2}(\\mathcal{R}') \\stackrel{(I)}{=} s_V(\\mathcal{R}') \\quad q.e.d.", "da7407b23f732d8713e61eb16ab867e2": "\\mathbf{g}(n)+\\mathbf{g}(n)\\mathbf{x}^{T}(n)\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)", "da748141236a35dccfc405b4465af49f": "\\textstyle\\{(3,4),(4,3)\\}", "da7543d2c35db551c91b26eeb74d27ae": "\\operatorname{dim}_R M := \\operatorname{dim}( R/\\operatorname{Ann}_R(M))", "da75d01de881fc858eaf5dc0fd8560b5": "\\frac{r\\sqrt{2}}{R}\\le \\frac{1}{2}\\left(\\sin{\\frac{A}{2}}\\cos{\\frac{B}{2}}+\\sin{\\frac{B}{2}}\\cos{\\frac{C}{2}}+\\sin{\\frac{C}{2}}\\cos{\\frac{D}{2}}+\\sin{\\frac{D}{2}}\\cos{\\frac{A}{2}}\\right)\\le 1", "da76116a838f38f8a2f4bcc644771559": "u(t) = U_0 \\cdot \\mathrm{e}^{-t/\\tau_\\mathrm{s}},", "da761ece4cc118c377232d26b157891c": " = \\left((1 + x)\\cdots (1 + q^{m - 1}x)\\right) \\cdot \\left((1 + (q^m x))(1 + q(q^m x)) \\cdots (1 + q^{n - 1}(q^m x))\\right)", "da7675f506cbd394642d7e32fa3e6876": "[E\\; F] = -U_Y\\Sigma_Y \\begin{bmatrix}V_{XY}\\\\V_{YY}\\end{bmatrix}^*= -[X\\; Y] \\begin{bmatrix}V_{XY}\\\\V_{YY}\\end{bmatrix}\\begin{bmatrix}V_{XY}\\\\ V_{YY}\\end{bmatrix}^*", "da768f50f04ec603b32ef369910104be": "O_F / \\mathfrak p", "da769847049d11947a0cbed706254d23": "u_x(\\mathbf{0}) = 0", "da76a07f1a9836a87b72fe5bda3b7c58": "P_{it}=c_{0t}+\\sum_{j=1}^{k}c_{it}z_{jit}+\\xi_{it} ,", "da76a6b3d7db4a7758df379cc1fdac75": "\\psi_\\alpha^\\dagger|n \\rangle", "da76c620de4523e46f0fc2766da420f2": "\\frac{d y}{d t} = f(t, y)\\,", "da77495d1220dac4a855428b0dbdf555": "{\\frac{\\partial g}{\\partial \\bar{z}}(z_0)}\\overset{\\mathrm{def}}{=}\\lim_{r \\to 0}\\frac{1}{2\\pi i r^2}{\\oint_{\\Gamma(z_0,r)}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g(z){\\mathrm{d}z}}", "da77915e74ce76c1b050aafcbd1b439c": "CE = 3AE", "da77da2f85cc99ba9f7aad72246d9826": "\\mathbf{B}(\\mathbf{x},t)", "da78582354723993181fc473eeb821d2": "{P_t}", "da7870e835d1ec5459f6539e37474d26": "\\alpha \\neq 0", "da7881a24452c046a493700d541004ff": "\n \\phi(v) \\phi(u)=\\phi(u +v -vu)\n", "da78ab3bf460a4628957bbae877ff688": "{T_1}", "da78b1fbbd412a0b812e4bd2f80b56a3": "1_\\gamma", "da78b6182555d3fb280aef2473518408": "\\frac{\\partial \\tau_{xz}}{\\partial x} + \\frac{\\partial \\tau_{yz}}{\\partial y} + \\frac{\\partial \\sigma_z}{\\partial z} + F_z = 0\\,\\!", "da78c3ec2e51f3aaa04ed734c07328fd": "\\operatorname{Disc}_y(f)", "da78d0e4f749deb97f348563d0c66b4e": "d\\mathbf F_C= \\mathbf T^{(\\mathbf n)}\\,dS", "da78e62859bac6cf6e227212010761f9": "\\scriptstyle a", "da79a481749dd31295b85ab8e16b3611": "B \\subseteq \\widehat{\\mathbb{R}}", "da79a9003b379094ed86ee0860adb268": "\\alpha_{eff}", "da7a38e326e0505746e9d53da35b8dc3": "p_{t-1} = -q_{t} + 2p_{t} \\mod N", "da7a5ad6a7dc882fb7562b07e677c5f4": "\nJ=\\left(\\begin{matrix} 1/2 &0\\\\ 0&-1/2\\\\ \\end{matrix}\\right) \\quad \\quad\nX=\\left(\\begin{matrix}0&1\\\\ 0&0\\\\ \\end{matrix}\\right) \\quad \\quad\nY=\\left(\\begin{matrix}0&0\\\\ 1&0\\\\ \\end{matrix}\\right)\n", "da7a6c4d8167e51503b17a7d06a6c28a": "\\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta) = \\frac{\\part\\, \\ln G_X}{\\partial \\alpha} > 0", "da7ab8e11a3735d858b1e9af8aa6a9b6": "\\varphi(z) = \\lim_{k\\to\\infty} \\frac{\\log|z_k|}{2^k},", "da7abf83456965dc1ae983922c9868da": "d(x,y)\\leq \\lambda", "da7b54f8752e2d02c2b29890adbcdbe6": " \\Delta = g^{-1/2} \\partial _{\\mu} (g^{1/2} g^{\\mu \\nu} \\partial_{\\nu} ) ", "da7b884ecc84d44afd6dbd7afbf2de7e": "((C \\lor D) \\land (C \\to E)) \\lor (\\lnot (C \\lor D) )\\lor (\\lnot (C \\to E))", "da7bcd62208074dcb814612acea99e63": " \\nu_\\text{max} ", "da7bfe7b333376f070dc38db0962b1a2": " |z - a| < \\frac{1}{\\lim_{n\\to\\infty} \\frac{|c_{n+1}|}{|c_n|}} = \\lim_{n\\to\\infty} \\left|\\frac{c_n}{c_{n+1}}\\right|. ", "da7c05bb09e7e186ee8fa5624e1a1cd7": " \\operatorname{build-list}[\\lambda y.o\\ x\\ y, D, V, L_2] \\and D[g] = [x, \\_, \\_]::[o, \\_, \\_]::L_2 ", "da7c509dd29d59867f929d6803bb8c99": "\\sin 84^\\circ = \\cos 6^\\circ =\\frac{\\sqrt{10 - 2\\sqrt5} + \\sqrt{15} + \\sqrt3}{8}", "da7cdf40db2588ea109ff0e81a37e772": "h(a_i) = \\log_2{1 \\over w_i}. ", "da7d41db58db842431bb9a6bdaf564a6": "f|_{A} = g|_{A}", "da7d5e7ec3317d9ebe3bf573aa71e3e3": "A\\subseteq\\Sigma^*", "da7d80f57a3e0fb5081fc6fe1766d340": "=\\operatorname{st}\\left(2x + dx\\right)", "da7dae1d4733043778627fc37e179312": "\\phi_a (\\mathbf{r})=\\mbox{Re}[\\mbox{FT}[R(\\mathbf{k})K(\\mathbf{k})]]", "da7ddc3f0ac1f4097117f973e642d03c": "\\mathbf{B}'(t) = n \\cdot \\sum_{i=0}^{n-1} (\\mathbf{P}_{i+1} - \\mathbf{P}_i) \\cdot \\mathbf{b}_{i,n-1}(t)", "da7e6b595eb6bd1035199b48f6485949": "\\Delta(y_n,y)", "da7e74f839943a42143d845d7c8acd80": "\\frac{1}{r^4} P^2_3(\\sin\\theta) \\cos 2\\varphi = \\frac{1}{r^4} 15 \\sin\\theta \\cos^2 \\theta \\cos 2\\varphi", "da7e8b8fd10ac984be205892f832feea": "H_{el} = D_{ac} \\frac{\\delta V}{V} = D_{ac} \\, div \\, u(r)", "da7f1554f1c16fbe1bfb0241df9d77ff": "Y_d", "da7f371657dfc6a8eb10985cbd95d5f3": "\\gamma(k,0) = 1\\,\\!", "da7f3b622cffaed08e8b7b5948ff12cc": "\\Psi(x)", "da7f56cb01aa3d59948d1dc187247f82": "\n \\forall A\\in {\\mathcal A}\\qquad \\forall\\lambda\\in{\\mathbb F}\\qquad \\lambda\\cdot A\\in {\\mathcal A}.\n ", "da7f62d9e05695692323c27ef1c4f4a2": " R_{sp} ", "da7f7522db539c58e03f633835a98e11": "\\sum_{k=1}^n B_{n,k}(1,1,1,\\dots) = \\sum_{k=1}^n \\left\\{{n\\atop k}\\right\\}", "da7f7985e218278d775665b174ec8b0f": "\n\\begin{align}\n& \\cos\\left( \\frac{2\\pi}{21}\\right)\n + \\cos\\left(2\\cdot\\frac{2\\pi}{21}\\right) \n + \\cos\\left(4\\cdot\\frac{2\\pi}{21}\\right) \\\\[10pt]\n& {} \\qquad {} + \\cos\\left( 5\\cdot\\frac{2\\pi}{21}\\right)\n + \\cos\\left( 8\\cdot\\frac{2\\pi}{21}\\right)\n + \\cos\\left(10\\cdot\\frac{2\\pi}{21}\\right)=\\frac{1}{2}.\n\\end{align}\n", "da7fef5462e68230fd7e11249aa53780": "\\displaystyle\\lambda", "da80182e390d5926afd8633afd2f7dff": "f=(a-b)/a", "da802b58c001efc3035a58e6e6c426a9": " \\mathbf{Z} = { \\mathbf{B} \\over \\mathbf{h} \\cdot \\mathbf{s}^{T} } ", "da80c03d6610e198c5e394329da3ed23": "\\begin{align}\nB\\left(\\frac{R}{2}\\right) &= 2 B_1(R/2) \\\\\n &= \\frac{2\\mu_0 n I R^2}{2(R^2+(R/2)^2)^{3/2}}\n = \\frac{\\mu_0 n I R^2}{(R^2+(R/2)^2)^{3/2}} \\\\ \n &= \\frac{\\mu_0 n I R^2}{(R^2+\\frac{1}{4}R^2)^{3/2}} \n = \\frac{\\mu_0 n I R^2}{(\\frac{5}{4}R^2)^{3/2}} \\\\\n &= {\\left ( \\frac{4}{5} \\right )}^{3/2} \\frac{\\mu_0 n I}{R} \\\\\n &= {\\left ( \\frac{8}{5\\sqrt{5}} \\right )} \\frac{\\mu_0 n I}{R} \\\\\n\\end{align}", "da80f285e97d5783a72fca2a826bf985": "B\\cup\\{x\\}", "da819e6f3f33a8f683adc64771a32096": "\\int_0^\\infty\\frac{\\sin(x)}{x}\\,\\mathrm{d}x", "da81ad1cab3f1e00d3b02457be702682": " \\ \\ x^2 - Ny^2 = k_2, ", "da82103e34341695b96de5e4ac002052": "\\mathrm{Sp}(n)\\,", "da8218612da19e522c39ede23cfdef5c": "W = [w_1, w_2, w_3]", "da82206ea66071f41e7b436cf0d0f961": "\nds^{2} = F(x, y, z) dx^{2} + G(x, y, z) dy^{2} + H(x, y, z)dz^{2} \\,\\!\n", "da828f3438cc7787c815ae25ffa08fa3": "\\frac{k_z}{k} = \\cos \\theta \\cong 1 - \\frac{\\theta^2}{2} ", "da82a71b005666eb567d5931588765e5": "\\frac{Q} {2} F(1-x)", "da82c1b569e383cc4936f85b45231b89": " T = \\frac{ts}{te} ", "da82c82a4e11e5262c898e18ad52e437": "\\mathbf{x} (s,t)=( x(s,t), y(s,t), z(s,t))\\!", "da82cdd217dc6116c904a0260bcf3318": "n_{ij}, a_i, b_j", "da82dd92ec7e4cffd4a4186d50d13fd0": " \\int_{-\\infty}^\\infty f(x)\\, d \\lambda(x) = \\int_{-\\infty}^\\infty f(x+\\varepsilon)\\, d \\lambda(x) .", "da82ee27e04a4b755eaece9a25ec0c97": " \\boldsymbol{\\hat\\beta} =( X ^TX)^{-1}X^{T}\\boldsymbol y.", "da83915e5a0779f7b0b92b781e269e3c": "G = \\bigcup_{k=1}^\\infty G_k.", "da84dd3011433d4f7f62fdd958e48d28": "\\partial u/\\partial n", "da84e6d52d047b700de50c65dde32c34": "n_0 \\ldots n_{N-1}", "da85271d6cddddba45e6fb9457df9a94": "\nM_{i}(r_{k}^{A}) = E_{k}(r_{k}^{A}) + \\sum_{x=1}^{N} \\sum_{y=1}^{p} P_{i-1}(r_{x}^{y}) E_{xy}(r_{k}^{A}, r_{x}^{y})\n", "da859582427c2a4f9db77f024fbdb328": "H_{abcd} = g_{ac} \\, g_{db} - g_{ad} \\, g_{cb} = 2g_{a[c} \\, g_{d]b}.", "da85c89b780d827800cfa4b10c05f705": "\\det U = e^{i\\theta}", "da85d03d4f773b009aa2268864355f83": "\nS_{ij}(\\omega_k) = Coh_{ij}(\\omega_k)\\sqrt{S_{ii}(\\omega_k)S_{jj}(\\omega_k)}\n", "da86190bf89808f5fc87a21d969627fa": "\\theta= \\int_{f(\\Delta)} K", "da862cdff93ecce389b5bf8b9edcccd2": "{\\color{Blue}~2.9}", "da865b89998fa8f1bc69bf2ce5721fe3": " \\sum_{r} n_r = N \\;", "da866c4682111140c7c4d788791c46b9": "\\begin{align} f\\colon \\mathbb{R}^3 &\\to \\mathbb{R} \\end{align}", "da868bffd8d45a7edb0fdd89fa5ab771": "E[(\\hat{g}_n)_i]", "da86982de631cbbabfc147d09f47437f": "\\hat{\\tilde{E_i^a}} \\Psi (A) = - i {\\delta \\Psi (A) \\over \\delta A_a^i}", "da872f4736e708770e4a82e7f297ea28": "\\neg \\neg \\exists n\\;P(n)", "da876425c4a3e86f8cb9f93669d6bb47": "P(\\{1,2\\})/2", "da879268cf06dfd4cf0a42c59dd5b121": "U(S) \\geq d(1-\\varepsilon)|S| - d\\varepsilon|S| = d(1-2\\varepsilon)|S|\\,", "da87cb4d89fdd2988fb0fee8812e65d5": " (a+b)/2n", "da87ebe6a4ea6e5e271a1f1ee826cff3": " \\lim_{t \\rightarrow \\infty}e^{-t} \\sum_{n=0}^\\infty \\frac{1 -z^{n+1}}{1-z} \\frac{t^n}{n!} = \\lim_{t \\rightarrow \\infty} \\frac{e^{-t}}{1-z} \\big( e^t - z e^{tz} \\big) = \\frac{1}{1-z}, ", "da8838d13bdb11afdb5dec5af7af631f": "(x^3+x)+(x^2+1)=x^3+x^2+x+1", "da885148cc3216e05c8ccf71c8fed8c2": "\\psi_\\mathbf{k}^\\dagger|\\alpha' \\rangle", "da889a01e9217a5e47b8cce46747d067": "\\mathbb{Z}[\\sqrt{-5}]", "da88b210b51420b4c75c72a91ec8b9a2": "P>0", "da88ca6ac5462985d6d01880c7d53b36": "\\mathcal{N}(\\mu,\\,\\sigma^2)", "da8936548bdd89766f218d7122b86b25": "\n\\frac{\\partial \\mathcal{H}}{\\partial q_j} = - \\dot{p}_j, \\qquad\n\\frac{\\partial \\mathcal{H}}{\\partial p_j} = \\dot{q}_j, \\qquad\n\\frac{\\partial \\mathcal{H}}{\\partial t } = - {\\partial \\mathcal{L} \\over \\partial t}.\n", "da896e39037cfacf15c231efe98a05e6": "[\\omega]", "da896fefe06e84687299deef321f54f3": "\\displaystyle \\phi", "da897f138d26264540dd514d5465a2f6": "\\mathbf{p}(t)\\rightarrow \\mathbf{p}(t+t_0)", "da89b361c0f0b670a72e57c99ca4af57": " \\sigma < \\sigma_0 ", "da89b4b1358ddc5cef3596b9b4bddd20": "R+2\\pi i", "da89f8c92ec19d4e63dc00391ffb46ac": " xX''+(x+1+m)X'+(n+\\tfrac{1}{2}m+1)X.\n", "da8a0b16195ad35153e308f8d10010e7": "P / {\\rm hp} = {{2025 \\times 5 } \\over 375} = 27", "da8afc6c295e41268406233b9c420756": " \\mathbb{F}_{q^n} ", "da8b2a004834fef87d39e9f513876369": "\\mathbf{F} = m\\mathbf{a},", "da8b4c77de0b408ec552e248dc4e4d41": "m^2(t) = 1", "da8b7c59823f8d796f23400c8027a8bd": "F=-\\frac{GMm}{r^{2}}\\left(1+(2+2\\gamma-\\beta)\\left(\\frac{h}{rc}\\right)^{2}\\right)", "da8bb0be5b8dbb98c37c8b1faf751fbe": " P(Z_{(m,n)}|\\boldsymbol{Z_{-(m,n)}},\n\\boldsymbol{W};\\alpha,\\beta)=\\frac{P(Z_{(m,n)},\n\\boldsymbol{Z_{-(m,n)}},\\boldsymbol{W};\\alpha,\\beta)}\n{P(\\boldsymbol{Z_{-(m,n)}}, \\boldsymbol{W};\\alpha,\\beta)} ,", "da8c3c195bbac90cdd5d7a4e5b5c256e": " d_G=1.375 ", "da8c5c397651eeda8269fd5c43b24555": "V\\otimes V", "da8cafb81082de0c79574e15530b58cb": "(1,3)_{0}", "da8cba0496ec40e0521d09d4e0414ed3": "r = 2 f \\tan(\\theta / 2)", "da8ce2bb31f36bd297b2d031177fef30": "\\hat G", "da8cf00256e7b57171c5039b4640760e": "q_i \\circ q_j^{-1}(u_j)(h_j)=h_i", "da8cf84d8751d4ab6b3dd235122b5850": "\\langle x_i, x_j\\rangle = x_i^* x_j", "da8d71c309a007cd8014e4d2f8b862d5": "\\prod_{\\sigma\\in G_F}\\sigma(J^{a_\\sigma})", "da8d7d5e2fe915ff05436712546b7eeb": "\\cos A + \\cos B \\cos C : \\cos B + \\cos C \\cos A : \\cos C + \\cos A \\cos B", "da8dcf9ea94e63b7472e66072e2ac7aa": "H_1(M)", "da8dfae451975d171fbce5e2dd184be0": "\\lambda H(z)=\\lambda h(k^{-1} (z)) = h(f(k^{-1}(z))=h(k^{-1}(\\lambda z)= H(\\lambda z).", "da8e9e91bafd554ccb0761cee0e319d8": "\n\\mathrm{A}^* + \\mathrm{Q} \\rightarrow \\mathrm{A} + \\mathrm{Q}\n", "da8eaf7ecbf0312c3cfe91d09f851d89": " k X \\sim \\mbox{Scale-inv-}\\chi^2(\\nu, k \\tau^2)\\, ", "da8ed5586ce9c9a511f4f6da278e30ae": "\\alpha \\gamma = \\mu \\gamma \\alpha, ", "da8f0bd21e67b7079c649ef1bd9d9819": "\\displaystyle q(s) \\sim \\textrm{Ai}(s), s\\rightarrow\\infty.", "da8f84f34a01d51d853ce4148c9103d2": "W^k{}_{ij}=\\omega^k(\\nabla_{\\mathrm{X}_i} \\mathrm{X}_j)\\equiv0 \\, ,", "da8f965bf103227696322dfe8e15dcb5": "J_+ |j,m \\rangle = \\hbar \\sqrt{j(j+1) - m(m+1)}|j,m+1\\rangle", "da8fbb79f890bbff8c590a15f7912237": "(A,p)", "da90045e30c1092cf4ae64634a641733": "x = \\frac{\\alpha -1 +\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "da9027b3e4cdc43dd367d95bd17b580e": "b^{-n} = 1/b^n ", "da90541a4a46d0c2cff9bfacb63ecbdb": "\\overline X = \\frac{1}{n}(X_1 + \\cdots + X_n).", "da9080bcf63e90b899e294b5323df8cb": " \\alpha = \\left( \\frac{\\varepsilon_i-1}{\\varepsilon_i+2} \\right) a^{3}", "da90898d1d7178f6b2e845b4dbd3b616": "\\mbox{ for all } X,Y \\in \\mathcal{C} \\mbox{ if } \\underline{X} \\subseteq \\underline{Y}, \\mbox{ then } X=Y \\mbox{ or } X = -Y", "da90b18522791dc4d45e11dfaf40dadc": "f(x)\\mathbf{e}", "da90b5a706c985ce78533cc54048fc3b": "R(\\theta,\\delta^*)\\le R(\\theta,\\delta)", "da90c8f45c1db452eb974af86769bd29": " F \\rightarrow E \\rightarrow B \\! ", "da90c99385f4ad5db69ef81ad073ae2a": "\\dim_{\\rm box}(S) := \\lim_{\\varepsilon \\to 0} \\frac {\\log N(\\varepsilon)}{\\log (1/\\varepsilon)}.", "da91c74c1ba413b471c1852d36b07fa7": "\\ \\sin w' - \\sin w", "da91d0976c514cc4122a0c4ce9956c45": " in_b = (out_b - kill_b) \\cup gen_b ", "da92394314ee12494cb8b15c93b184b1": "\nB= \\begin{bmatrix}\na & 1-a \\\\\n1-a & a \\end{bmatrix}\n", "da932b5bbdceedc545c11142b6574c97": "\\frac{\\partial f}{\\partial y}(X,Y) = \\frac{\\partial f}{\\partial x}(X,Y) =0, ", "da941925a728403c29b624fa87960270": " c(n) := \\sum_{ij = n} a(i)b(j) = \\sum_{i|n}a(i)b\\left(\\frac{n}{i}\\right) , ", "da949510048733315735a62838678c40": " dA_{\\bold{x}}^2 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1-d\\theta^2 & -2d\\theta \\\\ 0 & 2d\\theta & 1-d\\theta^2 \\end{bmatrix}~. ", "da94a759eda06ec73320f3a55164bf0e": "B_{ab}", "da94f12656a28dbb619df57ee01955de": "L(x, y, t_2) = g(x, y, t_2 - t_1) * L(x, y, t_1)", "da95a5ef57450902d9515189725059b5": " \\Gamma(x) ", "da95a84fb71e86d0fb2c4dd3074dc7e9": "\\dot{r}", "da9612ebd9fc6669029c3ece9f8d65eb": " K=\\left\\{ \\begin{pmatrix}\n\\zeta & 0 \\\\\n0 & \\overline{\\zeta}\n\\end{pmatrix} : \\zeta\\in\\mathbf{C},\\,|\\zeta| =1 \\right\\}.", "da96423e7b435396e46361196a8bc1c1": "\\left( m\\mid n\\right) ", "da9646b39b6b7f93db5d6920d16b05f2": "\\scriptstyle\\mathcal{R}_m", "da96a6648f6a6c8d4728c8fb663b69ff": " { {P} \\over {B}} \\, = \\, k_B \\, T ", "da96b2eb98b85a29fcc3135c96c8442f": "v_f^2 = (v_i + at)^2 = v_i^2 + 2av_it + a^2t^2\\,\\!", "da96e1ded02c18b6d836bb4741ed03d3": "f^*:\n\\Omega^q_{\\mathrm c}(X) \\to \\Omega^q_{\\mathrm c}(Y)\n\\sum_I g_I \\, dx_{i_1} \\wedge \\ldots \\wedge dx_{i_q} \\mapsto\n\\sum_I(g_I \\circ f) \\, d(x_{i_1} \\circ f) \\wedge \\ldots \\wedge d(x_{i_q} \\circ f)", "da97331afc9b0392045faea1d01993a8": "g_0^+", "da97357cd2473dfd718a3de93f372930": " \\mathcal{F} \\,", "da973e842f747a36cfbb11b413c3d26a": " k_{i,||} = k_{s,||} + q_{||} + G ", "da976d6533aadd49a5631ea6dc870aac": "\\lim_{x\\to a^+}f(x)\\ ", "da979019adc6afddb6679b82981afca8": "G=\\{R\\}", "da9853f817edf0d4b2f9fa7b7702f718": " \\sum\\limits_{ p \\, : \\, p + 2 \\in \\mathbb{P} } {\\left( {\\frac{1}{p} + \\frac{1}{{p + 2}}} \\right)} = \\left( {\\frac{1}{3} + \\frac{1}{5}} \\right) + \\left( {\\frac{1}{5} + \\frac{1}{7}} \\right) + \\left( {\\frac{1}{{11}} + \\frac{1}{{13}}} \\right) + \\cdots ", "da9873cf317dddb2ee828db92da97fc7": "C_{i+1}\\ ", "da98b284d85fb370724ed0f9ffb036e7": "x=\\sqrt{1-y^2}", "da98d6440d4611b2087f33e3dcb54fa3": "|q_j\\rangle", "da993a7698b0218c103978f8e322910c": "\\nabla_\\sigma ( g^{\\mu\\nu} \\delta\\Gamma^\\sigma_{\\nu\\mu} - g^{\\mu\\sigma}\\delta\\Gamma^\\rho_{\\rho\\mu} ) ", "da99504c2449074f4c19863bf9c7c43e": "kh_k=\\sum_{i=1}^kp_ih_{k-i}\\quad\\mbox{for all }k\\geq0.", "da9950d34527d016ac10f0dbaabf840e": "\\int \\coth^n ax\\,dx = -\\frac{1}{a(n-1)}\\coth^{n-1} ax+\\int\\coth^{n-2} ax\\,dx \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "da995a6dbc22af4c682b43af03aeb7ae": "\\frac{\\widehat{P}_{\\eta}(z^{t+1})}{\\widehat{P}_{\\eta}(z^{t})},", "da9981b346601b64fe979c421be0bcf2": "\\mu=\\begin{cases} (1-\\lambda) \\delta_0 + \\lambda \\nu,& \\text{if } 0\\leq \\lambda \\leq 1 \\\\\n\\nu, & \\text{if }\\lambda >1,\n\\end{cases}\n", "da9a0c4126e2d3066b5cd9dc4a3983bd": "\\langle n_i, n_j \\rangle \\in E", "da9a68a33578f5b9e926dfe4528e5bec": " K=K(p) ", "da9a91f329d8c1b533468688b2178d8f": "T_P^*(X)", "da9a9c96f819c596425d91a08c910137": "\\sum_{i=1}^n a_i L_i^2", "da9abfcdcd98ec949ba3e1a30364bc50": "V(y,t)", "da9ac2f4a6d3e200f8d73128084c1fb3": "\\textstyle \\frac{P(E|H)}{P(E)}", "da9bc00f41ce5be0b474f48f9759fc41": "M\\cdot V_T = P\\cdot T", "da9bc6a11ac2eb046005550d0f76422a": " u{\\partial u \\over \\partial x}+\\upsilon{\\partial u \\over \\partial y}=-{1\\over \\rho} {\\partial p \\over \\partial x}+{\\nu}{\\partial^2 u\\over \\partial y^2} ", "da9c445be077548d9d6e17ca220e6ea3": "\\hat{A}_1 = -p_1/24", "da9c55a22fff55a9c5f282a39ab4ac0d": "\n\\begin{align}\n{\\mathrm{d} \\over \\mathrm{d}x} \\sin x & = \\cos x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arcsin x & = {1 \\over \\sqrt{1 - x^2}} \\\\ \\\\\n{\\mathrm{d} \\over \\mathrm{d}x} \\cos x & = -\\sin x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arccos x & = {-1 \\over \\sqrt{1 - x^2}} \\\\ \\\\\n{\\mathrm{d} \\over \\mathrm{d}x} \\tan x & = \\sec^2 x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arctan x & = { 1 \\over 1 + x^2} \\\\ \\\\\n{\\mathrm{d} \\over \\mathrm{d}x} \\cot x & = -\\csc^2 x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arccot x & = {-1 \\over 1 + x^2} \\\\ \\\\\n{\\mathrm{d} \\over \\mathrm{d}x} \\sec x & = \\tan x \\sec x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arcsec x & = { 1 \\over |x|\\sqrt{x^2 - 1}} \\\\ \\\\\n{\\mathrm{d} \\over \\mathrm{d}x} \\csc x & = -\\csc x \\cot x, & {\\mathrm{d} \\over \\mathrm{d}x} \\arccsc x & = {-1 \\over |x|\\sqrt{x^2 - 1}}\n\\end{align}\n", "da9c611667b059f29e78145dbf7bcfc8": "P(x_1,\\ldots,x_n)=\\prod_{i \\frac{m \\pi}{\\Delta k}", "daa32ef0bbe472d9dedc0d43d6b501e7": "\\begin{align}\n\\pi_0 B_{00} + \\pi_1 B_{10} &= 0\\\\\n\\pi_0 B_{01} + \\pi_1 A_1 + \\pi_2 A_0 &= 0\\\\\n\\pi_1 A_2 + \\pi_2 A_1 + \\pi_3 A_0 &= 0 \\\\\n& \\vdots \\\\\n\\pi_{i-1} A_2 + \\pi_i A_1 + \\pi_{i+1} A_0 &= 0\\\\\n& \\vdots \\\\\n\\end{align}", "daa37833f4cde3311fdd556d03df8c5f": " r \\times c", "daa379ae6d08730dd5052a339d73f471": "e\\ ", "daa3fb18e27ae5ed74f30a9694be161d": "\\Lambda^2\\mathbb C^{2n}", "daa4156669f487393b19dbf8f8ad3dee": "M/\\!\\!/G", "daa4cf521085764fa5f5ba84af54eed4": " I(X;Y) = \\sum_{y \\in Y} \\sum_{x \\in X} w(x,y) p(x,y) \\log \\frac{p(x,y)}{p(x)\\,p(y)}, ", "daa4db89434fbe87cb6f1870e1074b86": "\\,p_x = i h \\frac{d\\mbox{T}(0)}{da}", "daa50ff62a3515dfb2ea84eab03414c7": " \\frac{d}{dx} \\left(\\sum_{1 \\le i \\le n} f_i(x)\\right) = \\frac{d}{dx}\\left(f_1(x) + f_2(x) + \\cdots + f_n(x)\\right) = \\frac{d}{dx}f_1(x) + \\frac{d}{dx}f_2(x) + \\cdots + \\frac{d}{dx}f_n(x) ", "daa55236b1117b28dbdb3239fd5f8942": " \\boldsymbol{\\sigma} = 2C_1\\mathrm{dev}(\\boldsymbol{B}) ", "daa57bdfeea8dfa2c011fa7c5c6b1998": "\n \\nu_{\\chi'} \\approx \\frac{\\displaystyle\\left(\\sum_{i=1}^n k_i s_i^2\\right)^2}\n {\\displaystyle\\sum_{i=1}^n \\frac{(k_i s_i^2)^2}\n {\\nu_i}\n }\n", "daa5c18c626a4368568dd3c49bf7c8f5": " \\scriptstyle \\nu ", "daa5f91197fbe972e1ce4114d6063fb1": " E(s)", "daa63e39aa2945c52059f421d4faa6f2": "F_e/D_e < 2", "daa65a3b39e56a390962e7ea590ebefa": "\n\\frac{d}{dq} \\left[ - \\frac{r^{2}}{c} \\frac{d\\varphi}{d\\tau} \\right] = 0\n\\,.", "daa6abf2ec161debd5d76e721a75fc89": "W_{2}^{II}(x,x)\\geq 0", "daa6e84edd910fcbab1485c1beca3c3a": "> 10^9 ", "daa725f57458146eac99b5d8787ea20d": " \\left| \\psi (t) \\right\\rangle = e^{-iHt/\\hbar} \\left| \\psi (0) \\right\\rangle.", "daa7406d2ac8946e3b28d1f032900bbd": " \\Phi^{2}(\\Phi^{0},a)+\\Phi^{1}\\left(\\Phi^{1}(a)\\right)", "daa749fc83045bbc610d44799396b5e3": "0 < d < 1", "daa7506acc56766b4938320ea060437a": "\\mathbf{NL} \\subseteq \\mathbf{P} \\subseteq \\mathbf{NP} \\subseteq \\mathbf{PH} \\subseteq \\mathbf{PSPACE}", "daa7684d01af9465d6c71fa1a5e501b8": "\\displaystyle\\sum\\limits_{i=1}^n i", "daa7c68840d3a840b5afff0a7a6e8783": " p_\\theta, \\theta \\in \\Omega", "daa7de95c92334b78f2453eda2acd127": "\\sin A = \\frac {\\textrm{opposite side}}{\\textrm{hypotenuse}} = \\frac {a}{h}\\,.", "daa7e531acfba78d2276b80f41266b36": "\\frac{8}{77}=\\frac{1}{10}+\\frac{1}{257}+\\frac{1}{197890}=\\frac{1}{11}+\\frac{1}{77},", "daa7fc5ae44d5a57f82da4933358d4a5": "\\displaystyle{F=\\sum_{m\\ge 0} F_m}", "daa8191cb345ad8ee248e54404eada47": "G_{Eq} V = G_1 V + G_2 V\\ \\,", "daa894d4f8550a3aab61b9789323b587": "w=-a_{N-1} 2^{N-1} + \\sum_{i=0}^{N-2} a_i 2^i", "daa92116298664ee0dee4a8880f29356": "\\dot{\\varepsilon}, \\dot{\\varepsilon_{\\rm{0}}}", "daa931be7caddb07d446043d6e9ef588": "\n\\boldsymbol{S_x} = (\\boldsymbol{A}^T \\boldsymbol{S_y^{-1}} \\boldsymbol{A} +\n\t\\boldsymbol{S_{x_a}^{-1}})^{-1}\n", "daa974770f3b6c707c161047c6a0ca49": "B_{n}^{(1)}", "daa997d00dfb73788f5b2774d4bd36a1": "\\mathrm{Ba}=\\frac{\\rho d^2 \\lambda^{1/2} \\gamma}{\\mu}", "daa9d1b452da7fe99ba5038269ba79bd": "x \\neq y \\in S^2", "daaa0c9f43d00dff1cde9550ced8955f": " U^{A}_{jj'} ", "daaa0ec873dfb99cbf47df913744c33a": "d(T(x),T(y)) \\le q d(x,y)", "daaa1e8d7a6f8b71f68434bcaa752986": "\\Theta_1", "daaa314144bbef70220f25bca7a505b1": "\\hat{B}^\\dagger_\\omega\\,\\hat{B}_\\omega", "daaa563d3022a373f908a4da141407ce": "a_2x + b_2y= 0", "daaa92dd0d6e8928e9dad992b873d4db": "\n\\sum_{n=-\\infty}^{+\\infty}e^{-\\pi n^2\\tau}=\\frac{1}{\\sqrt{\\tau}}\n\\sum_{n=-\\infty}^{+\\infty}e^{-\\pi n^2/\\tau}\n", "daaa9d184d4ec03e1151c8df655f9ce0": "Y' := Y \\times_X X'", "daaab8f870de4da303b5fa2545918eb2": "dx < dy", "daaaebc01166d635bb4b5ad5072eb71b": "\\text{If } Y \\sim \\mbox{Pareto}(x_m = \\lambda, \\alpha), \\text{ then } Y - x_m \\sim \\mbox{Lomax}(\\lambda,\\alpha).", "daab239389af5fc5f58686dfc8493a29": "P' = (y_0,\\ldots,y_m) \\,\\!", "daabb42945a328289ff92db28b0033e8": "q=e^{\\pi {\\mathrm{i}}\\tau}\\,", "daabfdd634112f3e3300cf18abea5e04": "\nF_2(r) - F_1(r) = m \\frac{h_1^2 - h_2^2}{r^3}\n", "daac45da4acef121dfa1882130c12e7b": "\\{1(\\mathrm{mod}\\ {2});\\ 0(\\mathrm{mod}\\ {3});\\ 2(\\mathrm{mod}\\ {6});\\ 0,4,6,8(\\mathrm{mod}\\ {10});\n", "daac464aadcfac77b426e0918ba0e89e": "n=x", "daac5a974b8ccadf2cbe056368ab8f6d": "c \\in \\mathbb R", "daac70a03d76f440d60e90306715bcc7": "y^2 = x^3 +3(x+1)^2", "daacd7e9b73f914902b56b8fe5b62fd1": "BP=\\frac{RL}{L}\\bigg|_{L\\approx0}", "daacfae157b66a6613e22be3ffec962b": "~(\\Delta n)^2={\\rm Var}\\left(\\hat a^\\dagger \\hat a\\right)=\n|\\alpha|^2~", "daad026ddace4991f6eb79bdf2994b4c": "S\\cdot S\\subseteq S", "daad074bdb3179c3b423c36a43661a89": " t= |1-\\sqrt{1-4c}|,", "daad1fe844114cc99324b8a47ba83d5d": " -\\frac{\\hbar^2}{2m}\\frac{ \\partial^2\\psi}{ \\partial x^2} + V\\psi =E\\psi", "daad2fa6a8f88a0e3c8e0b4529678add": " PV_\\text{annuity due} = PV_\\text{annuity immediate}(1+i) \\,\\!", "daad872c88ed3a0d2cde7e41b0eae868": "\\tau = \\frac{X_o^2 + A X_o}{B} ", "daadd39c93308a651f5672dd9734dfbf": "\n N_{\\alpha\\beta} := \\int_{-h/2}^{h/2} \\sigma_{\\alpha\\beta}\\, d x_3 ~,~~\n M_{\\alpha\\beta} := \\int_{-h/2}^{h/2} x_3\\,\\sigma_{\\alpha\\beta}\\, d x_3 \\,.\n ", "daae31ebb540d85d500ff439da406829": "g() \\,", "daae6489f547bb9ee57266636f15b378": " x_k(t) = x_k(0) + c^2 p_k H^{-1} t + {1 \\over 2 } i \\hbar c H^{-1} ( \\alpha_k (0) - c p_k H^{-1} ) ( e^{-2 i H t / \\hbar } - 1 ) \\,\\!", "daae76bd83bfb1bc52bc7db370b6c19a": "\\|Tf\\|_{q,w} \\le N_q\\|f\\|_q,", "daae91259769b38564bdc675fe78e02a": "(x_2 - x_1)(y - y_1)=(y_2 - y_1)(x - x_1).\\,", "daaeaa40ef1ae834b924fbe327ce92bf": "\\operatorname{logit}\\left(\\mathbb{E}\\left[\\left.\\frac{Y_i}{n_{i}}\\,\\right|\\,\\mathbf{X}_i \\right]\\right) = \\operatorname{logit}(p_i)=\\ln\\left(\\frac{p_i}{1-p_i}\\right) = \\boldsymbol\\beta \\cdot \\mathbf{X}_i,", "daaeec4c269593a4bb18b61b4d139071": "\nT = \\frac{ma^{2}}{2} \\left( \\cosh^{2} \\xi - \\cos^{2} \\eta \\right) \\left( \\dot{\\xi}^{2} + \\dot{\\eta}^{2} \\right).\n", "daaf19d7d759bcb65e154bc47032e330": "z \\mapsto z^n,", "daaf4a3404a05ea851462c661f451a64": "\\text{duplicate} \\circ \\text{duplicate} = \\text{fmap} \\, \\text{duplicate} \\circ \\text{duplicate}", "daaf5b17ccbf054d7ca0477019ad6e80": "A = \\frac{1}{2},\\, B = -1,\\, C = \\frac{1}{2}.\\,", "daafb1e74f1b318279ccf089cca02d8f": "T_{12}M_{12}", "daafbeaa8cfb38089c6196049f77663d": "\\frac{dV(x(t))}{dt} \\le u(t) \\cdot y(t)", "daafc13951b2a43b0a924207484c4f7b": "X_3=0", "daafd5da34afbe2576bca8eb56315849": "g_1,g_2,c,d,h_1,h_2\\,", "dab040832ac8dec433412428f4ee245e": "= \\int\\limits_{-\\infty}^\\infty f(\\tau) \\delta(\\tau-(t-T)) \\, d\\tau", "dab0faee3d941937987aad5a05c333d6": "v = \\sum _{k=1} ^m \\alpha_k u_k \\otimes v_k ,", "dab11662a18f8b8e51db3260822482af": "\\langle M^2 \\rangle", "dab14bcaf699525bc922f10339038e1e": "E= {1\\over 2m} \\sum_k p_{k1}^2 + p_{k2}^2 + p_{k3}^2", "dab19f9a03d667e3f2ad8946528c54cb": "y_2(t) = \\,\\!y(t + \\delta) = (t + \\delta) x(t + \\delta)", "dab1abb0f5ba6c72b451a70a47b8db2c": "t(x|\\mu,\\nu,\\sigma^2)", "dab1f5f0a07787ef90a8dabf729863d8": "\\hat{\\theta }(\\tau ,\\gamma )=\\int_{-\\infty }^{\\infty }{\\int_{-\\infty }^{\\infty }{\\theta (u,\\xi )}}.{{e}^{-i(u\\gamma +\\xi \\tau )}}du.d\\xi ", "dab2253848dd558ff6b7e0eed0f57d90": " (m,l, t, \\epsilon) ", "dab2346b24e0fcedaf528f9fa85fbc41": "V(\\mathbf{r}_1,\\mathbf{r}_2)=\\left\\{ \\begin{matrix}0 & \\mbox{if}\\quad |\\mathbf{r}_1-\\mathbf{r}_2| \\geq \\sigma \\\\ \\infty & \\mbox{if}\\quad|\\mathbf{r}_1-\\mathbf{r}_2| < \\sigma \\end{matrix} \\right. ", "dab2ae304c12ec6e2518d21743aca447": "\\mathcal{P}(\\mathbb{R})", "dab2db60adbf12bfb04373901291c62e": "V_n(P,Q)\\, ", "dab31ec839567e1c67f1f8cd26d31927": "f^* = \\frac{1}{2\\pi}\\cos^{-1}\\left(\\frac{\\varphi_1(\\varphi_2-1)}{4\\varphi_2}\\right)", "dab3db2014564cc160f8c51d2990626a": "\\rho - \\sigma", "dab43fba6721dc122c40d4df61558a68": "f^{(0)}(c) = f(c)", "dab5293d21276239acc15b28d4c522ff": "e = \\,", "dab53fd27a2fb4290d87fa88f4fd4b74": "\\mathbf{A}^{-1}\\mathbf{A}=\\mathbf{I}", "dab5429101b284c3fc8cb162c881673d": "\\{(3,6,1), (4,4,2), (4,1,5)\\}", "dab569f6a6d0d21cc23038fecf256764": "B_{ij}=e_i\\wedge e_j", "dab572becbdb18ef9384ed3dc2fae10a": "TdS = dU+PdV", "dab584617b9835b55e336b851126aaf8": "\\alpha=1", "dab59cc844b6448e1d66a3a9a29aa7b0": "TripleEMA_0 = (1-f)^3 (p_0 + 3fp_1 + 6f^2p_2 + 10f^3p_3 + \\dots)", "dab5c2be2306f6fe67d6e6e178b610d8": "\\{f_{n_1}\\}\\supseteq \\{f_{n_2}\\}\\supseteq \\cdots", "dab5c55894d92a0879d0947d5300dc8e": "\\delta = s(a,c) - \\frac{a+d}{12c} - s(a,k) + \\frac{a+d}{12k}", "dab5cd64b89fa03f7f2a3ca79fc101dc": "\\alpha(f(f(x)))=\\alpha(x)+2 ~,", "dab647621ae2ce0966796aa005ccc91c": " 0 \\rightarrow A~\\overset{f}{\\rightarrow}~B~\\overset{g}{\\rightarrow}~C \\rightarrow 0", "dab68d223065ef39e2a6268bf96ae12c": " V(s) := \\max_a \\left\\{ \\sum_{s'} P_a(s,s') \\left( R_a(s,s') + \\gamma V(s') \\right) \\right\\}. ", "dab6a592f307ef90737b181b89a5b8d6": "\\psi(\\alpha)=\\delta", "dab748e2a6f51a0a9ad5b103b6ccb0a1": "x' = \\gamma \\left( x - vt \\right ) ", "dab8179e8b467dabe0514974582f7bab": " X_{1} ", "dab818287f9aabfcddbb04f67ace5f07": "E_{tgu} = 70.188 / 4.54 = \\,", "dab8650bc6aec11ca5dfb32f4e411b8f": "\\left(\\tfrac{a}{p}\\right)_n", "dab876c377f895b72f00e425c91fce02": " \\dfrac{\\partial \\mathbf{p}}{\\partial t} = -\\dfrac{\\partial H}{\\partial \\mathbf{q}} ", "dab8a9095710d1e8a4ed6da524869758": "\\psi_0 \\rightarrow \\psi_1 \\ldots \\psi_m", "dab902bd1db94cedad05c025388d076f": "\\vec{v}_A=d\\vec{x}_A/dt", "dab93faf2ab6f8da202a8db25a64e982": "\\zeta(s).", "dab9697dc57e412647fddac768048b71": "_{=\\,}\\!", "dab9d80fcbd8de0377ab7427e7cd9f9f": "\\scriptstyle \\sigma^2", "daba621242129f32c364751bd3bc3879": "\n\\begin{align}\nT_n(x) & = \\frac{(x-\\sqrt{x^2-1})^n+(x+\\sqrt{x^2-1})^n}{2} \\\\\n& = \\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{n}{2k} (x^2-1)^k x^{n-2k} \\\\\n& = x^n \\sum_{k=0}^{\\lfloor n/2\\rfloor} \\binom{n}{2k} (1 - x^{-2})^k \\\\\n& = \\tfrac{n}{2}\\sum_{k=0}^{\\lfloor n/2\\rfloor}(-1)^k \\frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} \\quad (n>0) \\\\\n& = n \\sum_{k=0}^{n}(-2)^{k} \\frac{(n+k-1)!} {(n-k)!(2k)!}(1 - x)^k \\quad (n>0)\\\\\n& = \\, _2F_1\\left(-n,n;\\frac 1 2; \\frac{1-x} 2 \\right) \\\\\n\\end{align}\n", "dabaf9cb5708ea9b58d1e2f5028c4d9e": " \\alpha = \\alpha (n) \\in (0, 1) ", "dabb2aca4f472176be7b3c836fdbba40": "y=\\arcsin x\\,\\!", "dabb58eea2838b4581655d999cc53cce": "\\langle x, y\\rangle = 0", "dabbb2626f9759ebfc7c426f0403dacd": "\\textstyle x^{j-1}+1", "dabbc8913ffdeed60b08d79b571a6d09": "\\mathbf{k}_{||}", "dabc10a5d67ace8c0cd99a016c36ee0f": "u_P=0", "dabc15ed7ca641192905d829d60b92a5": "2\\, \\operatorname{Cl}_2(\\theta) -2\\int_0^{\\theta} \\log\\Bigg| 2 \\cos \\frac{x}{2} \\Bigg| \\,dx", "dabc4b68d631d9e76c4063939d1736d6": "\\| \\Psi \\| = \\sqrt{\\langle \\Psi, \\Psi \\rangle}", "dabcf71b98e8cc0e798c319a8bfdffb5": "\\mu_1^{'}= \\sigma \\sqrt{\\pi/2}\\,\\,L_{1/2}(-\\nu^2/2\\sigma^2)", "dabd0b55ca30c3b7c9a0c9a4753b20a6": "E_{internal} ", "dabd1006937f2f193aad78edf9ba7dfc": "\\phi^2 \\,", "dabd16fd15c71d0e83a6f50071f03f93": "\n\\binom{m + n}{k}_{\\!\\!q}\n=\n\\sum_{j} \\binom{m}{k - j}_{\\!\\!q} \\binom{n}{j}_{\\!\\!q} q^{j(m-k+j)}.\n", "dabde3cb52702ed020ed9852bffe3e07": "\n\\frac{1}{2} \\int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \\, dx = \n\\begin{pmatrix}\n l & l_1 & l_2 \\\\\n 0 & 0 & 0\n\\end{pmatrix} ^2\n", "dabe14b3aa70ad88b65a777c5709919f": "|H(\\omega)|=e^{-\\frac{\\omega^2}{\\sigma^2}} ", "dabe341224b06a2de0d461e117895117": "{{f}_{M}}=\\sum\\limits_{k=1}^{M}{\\left\\langle f,{{g}_{{{m}_{k}}}} \\right\\rangle {{g}_{{{m}_{k}}}}}", "dabe5e8b1a99648f7dc9ae35e4c3b32e": "\\gamma = 0", "dabed0a77f45a84751bd6cb248950dca": "C[c, d], \\,", "dabeec48377cb43c3270bd8821c4efe3": "\\{1,1\\} \\subseteq \\{1,1,1,2\\}\\, ", "dabf428f8ec966ca1064dfd7b37e1aad": "\\scriptstyle \\sum_{u:(u, v) \\in E} f_{uv} = \\sum_{u:(v, u) \\in E} f_{vu}", "dabf73f31caa68ced00a610e69d9e24c": "t/2 ", "dac013ab2307679b1c88999b434106e3": "\\left( r+1 \\right)^2/r", "dac01627e7a06e08497e9c92176535fa": "\n\\frac{\\partial u}{\\partial t} = r u - (1+\\nabla^2)^2u + N(u)\n", "dac07c6f55eba5d4846e03482340f467": "+22639.55''\\sin(l) +4586.45''\\sin(2D-l)", "dac18db8033cc097d3034dca2f5ca847": "=\\frac{1}{2}((-1)^{f(0)}|0\\rangle(|0\\rangle - |1\\rangle) + (-1)^{f(1)}|1\\rangle(|0\\rangle - |1\\rangle))", "dac1d199ed4b56d1729d64e238d6aba2": "\\textbf{V}_P=[\\dot{T}(t)][T(t)]^{-1}\\textbf{P}(t) = [S]\\textbf{P},", "dac1e2e3b034adc4c77797090adee5f6": " R^a_{\\ bcd} = e^a((\\nabla_c\\nabla_d - \\nabla_d\\nabla_c)e_b) ~~~~~~~~~\\text{(wrong!)}", "dac2088359726b775f582d8a77576c00": "f_{clock}", "dac20ed83881e7ca2c18a7c905797811": "z \\in \\mathbb{R}^n", "dac226ae9dae9836a2eba31da333db59": "G_{yy} = |H(f)|^2G_{xx}(f)", "dac24008bbdbc153e18f19c79d51352c": "-\\sigma_c = \\sigma_1", "dac24e6b3c91f0cff9f6c7c604a23282": "\\rm{PGL}_{n+1} K", "dac28830e4341f5363a3b9b66ce7a8d6": "L_1 / L_2 = \\{w \\ | \\ \\exists x ((x \\in L_2) \\land (wx \\in L_1)) \\}", "dac28a13bf7eac8fd83b750cbbb2b8b3": "\\Delta T = (T_\\mathrm{final} - T_\\mathrm{initial})", "dac2ba2cc02fb27193003ea68c6d4ec9": "L = L_{0}\\sqrt{1-v^{2}/c^{2}} = L_{0}/{\\gamma(v)}", "dac31f843c599f6411093f25ca33986f": "{{}^\\star C^\\star}_{abcd} = -C_{abcd}", "dac32c9b688534007dd704ccbfcbcab8": " E(e^{\\theta\\Sigma})=\\textrm{exp} \\left(\\int_{\\textbf{R}^d} [e^{f(x)}-1]\\Lambda (dx)\\right), ", "dac352189820622f5b364ff6e995e61a": "\\mathbf{\\left(J^TJ\\right)\\Delta \\boldsymbol \\beta=J^T\\Delta y}.\\,", "dac35e20354a1f821456f11ec5651830": "~\\exp(g)~(1-\\beta-\\theta)=1~", "dac3bbd0caf8370389cce23922be1be0": "c\\in C", "dac3e4684fc05061f952a0a7b9efed45": "K = 800 / (N_e + m)\\, ", "dac440979327971efa4c43c6d9ab8d2d": "I_1\\,\\!", "dac45352a81a22175b45aed889c6c1f7": "a\\in\\mathfrak{g},b\\in[\\mathfrak{g},\\mathfrak{g}].", "dac45d02c0511944769c76144022898b": "\\mathrm{d} X_{t} = b(X_{t}) \\, \\mathrm{d} t + \\sigma (X_{t}) \\, \\mathrm{d} B_{t},", "dac494a4863844cd986dfa3f14666a81": "\nH = -\\frac{\\hbar^2}{2M_\\textrm{tot}} \\nabla^2_{\\mathbf{X}} + H'\n\\quad\\text{with }\\quad H'=\n-\\frac{\\hbar^2}{2} \\sum_{i=1}^{N_\\textrm{tot} -1 } \\frac{1}{m_i} \\nabla^2_{i}\n+\\frac{\\hbar^2}{2 M_\\textrm{tot}}\\sum_{i,j=1}^{N_\\textrm{tot} -1 } \\nabla_{i} \\cdot \\nabla_{j} +V(\\mathbf{t}).\n", "dac49a3dacbd1b6a3581cdfc2c8cfb65": "{\\bold{v}}_{\\bold{p}}(f)", "dac4d29dcbdfbb964b47809a9cd13ef6": "G^{\\mu \\nu}\\,", "dac4e153bab1d191ccd221ce7d279b4e": "s_n^2 = \\sum_{i=1}^n \\sigma_i^2", "dac50fad1bea73226cf0d947b19bc050": "\\operatorname{Out}(\\mathfrak{g})", "dac51b91cb57ad0a364e8e08a251f868": "m^{\\rm{th}}", "dac535805570c3e30b845b5a5d22ed90": " \\frac{\\partial H(x)}{\\partial x} B(x)", "dac5395281fafa57ce5e95f09e6d1cde": "\\triangle ADB", "dac56cd57723d7868f438407644d9e51": "x_4 = \\frac{1}{2} \\left(x_3 + \\frac{S}{x_3}\\right) = \\frac{1}{2} \\left(354.059 + \\frac{125348}{354.059}\\right) = 354.045.", "dac58885b42dda89f2cecff1e86d3dfc": "|a|", "dac5fb9e7756bb5fb2b06fe7fc46fee9": "~F[\\Pi(r/a)] = \\frac {2 \\pi J_1 (qa)} {q} ", "dac7026e377b89e22efae19fff9d6bbf": "U = A+TS = \\frac{3}{2}\\,NkT-\\frac{a'N^2}{V}.", "dac76be1fd04c1841b81f2492b6322c7": "\n \\Phi(x) = \\frac12 + \\phi(x)\\left( x + \\frac{x^3}{3} + \\frac{x^5}{3\\cdot5} + \\frac{x^7}{3\\cdot5\\cdot7} + \\frac{x^9}{3\\cdot5\\cdot7\\cdot9} + \\cdots \\right)\n ", "dac77b18d4df3eb770097dbc85f13ca7": "0 \\leq x_0 < L_x ", "dac7bcd57df14e551edc2a70f5e8e0fa": "(p, q) \\in R", "dac7e311834e5a3610a1b0d6621c2c96": "||a^2|| = ||a||^2", "dac7eaa8765d72ab50ae610ae1f58eb0": "(b^2 + {{a^2}\\over 2})\\arccos {b \\over a} - {3\\over 2} b \\sqrt {{a^2} - {b^2}}", "dac7fe46035802568a5f16a05197d232": "\\theta \\neq \\pi", "dac81d793917df74ad8f893de0716e33": "\\psi=0", "dac82776a5c18d7273fe64494c710604": "S(\\mathbf{x}) = \\mathbf{x} \\cdot \\mathbf{d} - |\\mathbf{d}| |\\mathbf{x}| \\cos \\theta", "dac86736063de2e002fe10ce1a1edc37": "X_2^2+\\ldots+X_k^2", "dac8e6ba47181b2414812a2f31896cc2": "w(z' - z) = \\langle w_i z_i \\rangle - wz", "dac937178765ee2607b98fc621a51fc7": "\\boldsymbol{\\theta}^*", "dac967e1406aae87ae518c6c759e36cc": " |w|^{-1} \\le |a_2| + |a_2 + w^{-1}|\\le 4, ", "dac969a6528d4bc23c1e432d9e4a7595": " u(0,x) = f(x),", "dac9fb33b742b72d4e5afa9b1d0e85c0": "A \\cap U = \\phi^{-1}(\\mathbb{R}^m)", "daca3fe257cc99f8e901bbc2556dbae6": "(\\lambda x.K\\ (x\\ x))\\ (\\lambda x.K\\ (x\\ x))", "daca9cf9e2fe8106c46da85a588cef76": "\\displaystyle{F(w)={1\\over 2\\pi i}\\int_{\\partial\\Omega} {f(z)\\over z-w}\\, dz}", "dacaf75a081f42bdeaa21cb039c7f496": "(s,u_1,u_2,v,e) \\leftarrow CDecode(L_B(P),\\psi) \\in W^4 \\times Z \\times Z \\times Z \\times B^{\\ast}", "dacb823e2d90aabe006ba73bc9a08616": "p(x_1,\\ldots,x_n)", "dacb9a7232875b7c74a19311330a2d7a": "L = {{w_\\max - \\bar w}\\over w_\\max}~~~~~~~~~~(1)", "dacbfb423a07165d802e547811f5cbc5": " \n\\mathbf{a}_{\\mathrm{r}} = \n\\mathbf{a}_{\\mathrm{i}} - \n2 \\boldsymbol\\Omega \\times \\mathbf{v}_{\\mathrm{r}} - \n\\boldsymbol\\Omega \\times (\\boldsymbol\\Omega \\times \\mathbf{r}) - \n\\frac{d\\boldsymbol\\Omega}{dt} \\times \\mathbf{r}\n", "dacc5bf960324665542f860763fe71de": "S(\\mathbb{A}^2)\\to\\mathbb{PA}^1", "dacd7b5e89228f7e7aa6be829031aed3": "\\color{Black}\\tfrac{\\infty}{m}m", "dacd823b8c11300771614cbef105b178": "\\psi_1(0) = \\varepsilon_{\\Omega+1}", "dacdb16a8bf1920118a6cc956b04db75": " \\psi(n) = H_{n-1} - \\gamma.", "dacdb55b61e4aba4633040cf5a5c6a3c": "\\mathbb{C}_{/\\sim_{\\mathcal{B}}} = \\{o\\in O\\mid o\\sim_{\\mathcal{B}}c\\,\\forall\\,c\\in\\mathbb{C}_{/\\sim_{\\mathcal{B}}}\\}.", "dacdd7eeef99dfcecd1d9f033c4a686b": "\\vec{L}=0", "dacdfb0e9737cfbdfef544df5e2ee5f9": "L = nl\\, \\cos(\\theta/2)", "dace38f9fa2571bc8cea294a979b801d": "{\\mathbf x}_1,\\dots,{\\mathbf x}_{n_x}\\sim N_p(\\boldsymbol{\\mu},{\\mathbf V})", "dace3ca586873d84b0d565b0500fc884": "\\beta(f) := \\sup \\left. \\left\\{ \\int_{X} f \\, \\mathrm{d} \\nu \\right| \\nu \\in \\mathrm{Inv}(T) \\right\\}.", "dacea54a9f4f73d00e82b34f0803708c": "\\lambda_8 = \\frac{1}{\\sqrt{3}} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2 \\end{pmatrix}", "daceb882cd34950ab5f4c20d088bd216": "d_{C(f)} = \\begin{pmatrix} d_{A[1]} & 0 \\\\ f[1] & d_B \\end{pmatrix}", "dacec947337587e3f68c644aceac92c9": "J=Div^0/P", "dacf12347a13c801dd6eb7757ea84d2e": "\\ell(\\alpha, x_\\mathrm{m}) = n \\ln \\alpha + n\\alpha \\ln x_\\mathrm{m} - (\\alpha + 1) \\sum _{i=1} ^n \\ln x_i.", "dacf6b2d5823ca9872aaf120537328ee": "\\left [ \\mathbf z \\right ] = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{12} & z_{11} \\end{bmatrix} ", "dacfd2383239f676ebea16c14b8fffa4": "\\gamma = \\gamma^\\prime = 1\\,", "dad0255e61f1481959cf3029726e62a9": "\\displaystyle (d)", "dad04e95022cd26cfddd4fb7b6714342": "{|C|Vol_q(0,e)} \\over {q^n}", "dad05d2487628d91bcbd13c541ac062c": "OLD(T_i) = []", "dad061e5600311967b6aac6eaaee72af": "\\mathbf{Gr}(r, \\mathcal E)", "dad12fe5084dec9dc988f0bdab21a57c": "x=(x_1, \\cdots, x_K)", "dad1533ece96bb22f10da231df9680a5": "x=7", "dad1b6ebdb2a4d5ac2476918083dda7e": "C_P=\\frac{(\\partial H)_P}{(\\partial T)_P}", "dad2105fdc9a1af349571037f9d33525": "K = \\sqrt{(S-p)(S-q)(S-r)(S-s)}.", "dad2c81d5bd4b7d7b27ff4ea61467484": "\\mathfrak{P}^{67}", "dad2d77f3188fa166bdddc93a1dd8ab5": "I \\rightarrow 0^+", "dad2d797d37ff2fb972734955fa1cc3c": "\nh_{\\phi} = a \\sqrt{\\left( \\sigma^{2} - 1 \\right) \\left( 1 - \\tau^{2} \\right)}\n", "dad30c53d6c21f66f9e4799086b94534": "\\kappa \\, = \\, - { 8 \\, \\pi \\, G \\over c^2 }~", "dad498991a72fb818effca2a6cab94f2": "C_\\nu(x) = \\sum_{k=0}^\\infty\n\\frac {\\cos((2k+1)\\pi x)} {(2k+1)^\\nu}", "dad4e157060178179c4e9368ed924c4c": "M_{-\\infty} (x_1,\\dots,x_n) = \\frac{1}{M_\\infty (1/x_1,\\dots,1/x_n)}.", "dad4e6984d1943944e6c57e5ef5b22f7": "F(Y) = \\varinjlim_{i \\in I} \\operatorname{Hom}_C(X_i, Y)", "dad4f36f824a26e6e85b7d3b4253b169": "n-j", "dad52858ed7476ae8229b70b1e9dc7a1": "V_{th}", "dad52aeb961dac7cdf2c142c8a424981": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{F}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "dad56bfde41d42d2793ce57265618f12": "(g^a)^b = g^{ab}", "dad5797dcb482f9006c46f8b9a35e045": "\\Delta x \\Delta p \\geq \\frac{\\hbar}{2}, \\quad \\Delta E \\Delta t \\geq \\frac{\\hbar}{2}\\,,", "dad58f567c9226b2e7c65e2f39db017a": " U(t) = \\begin{pmatrix} e^{\\frac{-iE_{+}t}{\\hbar}} & 0 \\\\ 0 & e^{\\frac{-iE{-}t}{\\hbar}}\\end{pmatrix}", "dad59af16cb938782895fd48f7e87919": "t= \\,", "dad5d28dab26f200a4792d1837e59f7e": "T(x_1,x_2,\\dots) = (x_1,x_1+x_2,x_1+x_2+x_3,\\dots).", "dad5d63733967283f74d1c54a86defff": "H^1(\\mathfrak{g}; M) =Der(\\mathfrak{g}, M)/Ider(\\mathfrak{g}, M)", "dad66958977a5d11bab442b3ae5db0c3": "F \\simeq 1/3 ", "dadfa5325951486e394d8a4c4456ed5f": " \\beta\\;", "dadfe10fc1c7e74d8cbfe83d3fe7e5f7": "W\\approx\\prod_i \\frac{(N_i+g_i)^{N_i+g_i}}{N_i^{N_i}g_i^{g_i}}\\approx\\prod_i \\frac{g_i^{N_i}(1+N_i/g_i)^{g_i}}{N_i^{N_i}}", "dadff51750baf6be9214103900658f46": "\n\\operatorname{Li}_{n}(e^{2\\pi i x}) + (-1)^n \\,\\operatorname{Li}_{n}(e^{-2\\pi i x}) = -{(2\\pi i)^n \\over n!} \\,B_n(x) \\,,\n", "dadffa7e730db35c2c16642d2f3f2be7": "\\chi^{2}(p;n)", "dae00a267fb4db69dd077b8fa79f1c57": "\\mathcal{O}_X/\\mathcal{I}", "dae0476d0b310c379b568f9ad497322b": "I_p(P,Q) = I_p(Q,P).\\,", "dae04c304a789f6fb61662ce3ddcf116": "V(x) = \\frac{1}{g^2} \\operatorname{Tr} [\\phi , \\bar{\\phi} ]^2 \\,", "dae063bc4c78a01cc0f7d2ac94cb09f7": "\\mathit{\\Omega}", "dae069f13a5691d240e13d7a2393fe24": " \\langle x, y \\rangle =\\rho(x^*y).", "dae07951b05749c27c4d380d2d2dab0c": "f=\\operatorname E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-m(\\vartheta)\\right )^2\\right ]\\rightarrow max", "dae0c795e549c47c9c46dfe5b1717490": "\\! w := w - \\alpha \\nabla Q_i(w).", "dae1d27977ccf03a595a581fa3e90987": "\nLy = -1 - y^{2},\n", "dae22151afb92759273559a91e5a0a14": "\\Omega_{p}", "dae24ead86105b3c3e5cfabb8863341f": "\\zeta(1-s,x)=\\frac{1}{2s}\\left[e^{-i\\pi s/2}\\beta(x;s) + e^{i\\pi s/2} \\beta(1-x;s) \\right]", "dae283ce2b0a2e5f5b8d7f0fb7fc5f29": "(\\Omega, \\mathcal F, \\mathrm{P})", "dae2d6913e300c87b26f67cbe2278a55": "(6)\\quad B_{ab}=\\frac{1}{3}\\theta h_{ab} +\\sigma_{ab}+\\omega_{ab}\\;,\\quad \\theta=g^{ab}\\theta_{ab}=g^{ab}B_{(ab)}\\;,\\quad \\sigma_{ab}=\\theta_{ab}-\\frac{1}{3}\\theta h_{ab}\\;,\\quad \\omega_{ab}=B_{[ab]}\\;.", "dae2e43c91075d0b3d1708979b2d591e": "\\scriptstyle \\frac {17}{12}", "dae3a9f7ae1a5cd1b69bc9c69c7d271c": "\\,S(t) = \\exp(-\\Lambda(t))", "dae3b54065d043abc92c3aedbd2da1a7": "f(x) = b_2\\,x^2 + b_1\\,x + b_0. \\qquad (2)\\!", "dae3d39370e11e3b51687e845d4cc621": "\\mathbb{PT}^+", "dae3e6c5aebe81e5d0315483cb903c9d": "\\int \\left(\\frac{du}{dx} + \\frac{dv}{dx}\\right) \\,dx = \\int \\frac{du}{dx} \\,dx + \\int \\frac{dv}{dx} \\,dx", "dae402a21c0542aa841beb32545ac707": "c\\in(1,2)", "dae429b58553a252762cd47920ba9977": "\\scriptstyle y_q", "dae476a8c6d6366b29cf0aaf0ed566e7": "u\\rightarrow 0", "dae483a00ecac9c8cf048f22b1e83f23": "s = 1000 \\log_{10}{\\frac{f_2}{f_1}}", "dae4f14bd031d0c0f53cb347c6da238a": "g_{ab}=-l_a n_b - n_a l_b +m_a \\bar{m}_b +\\bar{m}_a m_b\\,, \\;\\; g^{ab}=-l^a n^b - n^a l^b +m^a \\bar{m}^b +\\bar{m}^a m^b\\,.", "dae527cde27db3a89c8cc1e79950124b": "QH", "dae5607dba23679cda574386ffc72733": "\\pi_1(SO(2)) \\cong \\mathbb{Z}", "dae575e7219662398bb990ac2f750adb": " z = B_{(n,n)} ", "dae59947f4c19dfac06f828dcba357b3": "H_{\\alpha ,\\beta }", "dae59e93f2e542bf11e5e7e481c1ca3c": "{2n\\choose n}-{2n\\choose n+1}", "dae5b4e7091c40070f3610e477456c53": "Q= (X_2:Y_2:Z_2)", "dae6645314924f81225772b6e7916ab4": "K=\\tfrac{1}{2}pq", "dae72b7a8bb35a5ea95c9eaba466cafa": "m_u", "dae746abcfc8a6026e9fb5cb190d231b": "j=1,\\dots, J", "dae841e72407ae04b6622f9d012a18c0": "\\ln \\frac{S_t}{S_0} = \\left(\\mu -\\frac{\\sigma^2}{2}\\,\\right) t + \\sigma W_t\\,.", "dae844499055f79d26144f8a35f926da": " (\\ln \\text{GDP})_t ", "dae8856ca560d1d9da08050f770617e5": "x^2-2b_{2}x+((b_{2}-b_{14})^2-b_{14}^2=0", "daeabe1351e97356eafb82fb55f316c0": "S=S^{-1}", "daead24b3fbd36d2e2fc7fd8cbc7a43c": "\\vec{v}_{\\mathrm{CMS}}", "daeadfdd7f064630a7a353c875146d8b": "= \\mathbf{I}_p + \\mathbf{UBVA}^{-1} - \\mathbf{U} \\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right) \\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right)^{-1}\\mathbf{BVA}^{-1} ", "daeb10e3aa056d80c78d3294aeb90f09": "v_{\\mathrm N}", "daeb2848aceea644b15dd56be8c52ae6": "J(\\rho)\\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt\\rho R(r)", "daeb9d0f8f18f9f9db09c755bd623ff1": "\\iiint_V \\varphi(x,y,z) dV = \\iiint_V \\chi(q_1,q_2,q_3) Jdq_1dq_2dq_3 ", "daebca60f37ed6cfef17a609379f70c0": " \\phi = 0 ", "daebde381e403aa1902f7e679d90c4ea": " Z \\equiv \n\\langle 0 | \\exp\\left ( -i \\hat H T \\right ) |0 \\rangle\n= \\exp\\left ( -i E T \\right )\n= \\int D\\varphi \\; \\exp\\left ( i \\mathcal{S} [\\varphi] \\right )\\; \n= \\exp\\left ( i W \\right )\n", "daebf5985ed8ca0a75de2a6746e5afb0": "\\begin{align}\n y' &= \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} \\\\\n y'' &= \\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1) x^{r + c - 2}.\n\\end{align}", "daec309d7511904b05f86dbf881232f1": " \\left(\\frac{\\partial U}{\\partial V}\\right)_{T} = u ", "daec59eb0d2e204a3c5773fedba244eb": " 1 - \\left (\\frac{R} {2} \\right)", "daeca20f5baf6b1551efa33a1a575d5a": "n_j^0", "daecab326c51a34ea4d565be2b4cffdd": "\\int_{z_0}^z R\\left(x,w\\right)dx,", "daed106e78b6637ec5d8d4820ff63801": " 1/\\sqrt N ", "daed80e785912c34a6112851f8310248": "\\frac{\\partial \\phi}{\\partial t} + c^2 \\left ( \\frac{\\partial v}{\\partial y} + \\frac{\\partial u}{\\partial x} \\right ) = 0", "daed86af1f10f13b90829da23751301f": "\\{\\psi_n\\}_n", "daed8960e263eca125fe07a3336119b0": "\\mathcal S^2", "daedb8b26e6218dea4b545ce9a4ab926": "(a_1,a_2, \\dots, a_N)", "daedda438f92a040191a9124a6659381": " v( S \\cup \\{ i \\} ) = v( S \\cup \\{ j \\} ), \\forall~ S \\subseteq N \\setminus \\{ i, j \\} ", "daedf467d8972c04757f504433711ba1": "71.6\\pm 3", "daedf9b7b38a91701144a83f8876070e": "S=\\frac{4U}{3T}", "daeeb8c7390d287929f33c39c2838008": "\\|\\mathbf{x}-\\mathbf{p}\\|", "daef3a9dc26a87f7b04c5cd825218025": "D^2f \\;", "daef4210c54390528d7d53037bfd675e": "{\\mathcal H}^2", "daef42f89b4620d85ee3b6d5d44a2268": "\\alpha = 1 - \\frac{D_sP_n}{D_nP_s} ", "daef5d877c639bf25d68cfa7fb8eb20d": "\\int_{-\\pi}^{\\pi} \\cos(mx)\\, \\cos(nx)\\, dx = \\pi \\delta_{mn}, \\quad m, n \\ge 1, \\, ", "daefd834bf12d9ab28cff46a5267de76": "h(n) \\, \\forall \\, n \\, \\in \\mathbb{Z}", "daefe1b1ca74a4606f6d836a3a298987": "\\Delta = -1/2", "daeff9f50fd99594f95ad1f36cba270f": "p^{-n}", "daeffd3846476be7c8c93262a71cc4e1": "\\dot{\\hat x}=f(\\hat x,u)+C", "daefffb93e6c3be7136ba40edae4f2f1": "srt", "daf044d99125f679ca16d6ad206f6daf": "\\frac{dp_\\alpha}{d\\tau} = q_{\\mathrm e} F_{\\alpha\\beta}v^\\beta + \\frac{q_{\\mathrm m}}{\\mu_0} {\\star F_{\\alpha\\beta}v^\\beta} ", "daf0543da87e7d97c1513d02a48158be": "{6\\choose 3}{43\\choose 3}\\over {49\\choose 6}", "daf05bb666da57b885de704b1a807313": "b(X, X^*)", "daf06f70bb8f010424cb6ee4ad03fba2": "\\begin{bmatrix} d+1 \\\\ n_1\\ n_2\\ \\dots\\ n_p \\end{bmatrix} q^{s_2(n_1,\\ldots,n_p)},", "daf07af67a195cbc05ebf25aad48f23e": "\\alpha \\approx 2.50290,", "daf0eacefb543162bc22c4e2212568ed": "s_1 \\leq t_1", "daf152ca84f7db31280eaa64cb518b95": " \\sinh^{-1}(y')=B-\\frac{V_t}{V_d}\\ \\ln(A_x-x) ", "daf177dd1e1bfabaf0b2f76ad57e5365": " \\mathbf{F}_i = - m\\ddot{\\mathbf{r}}_c", "daf1c7e3552ab367a63739fe0f9aab94": "\\eta_3=\\sqrt{2}", "daf1dd07eb168dae9c12148e2ec24aee": "\\mathcal{H}_N = \\mathcal{H}_K \\otimes \\mathcal{H}_M", "daf24c18894d46cb59707ec98e326487": "\\rho_{ij} = \\left(\\Sigma^{\\mathrm{(diag)}}\\right)^{-\\frac{1}{2}} \\, \\Sigma \\, \\left(\\Sigma^{\\mathrm{(diag)}}\\right)^{-\\frac{1}{2}}", "daf290d7901405adbf53c19b12828ed4": "\\overline{2} \\cdot \\overline{3} = \\overline{2}", "daf291540624999e7cc702ff8956d3e5": "r_1{\\left(0\\right)}=1", "daf2f4220f5211705882f09928a6bd4f": "-\\nabla \\times \\nabla \\times \\mathbf{E}= \\mu_0\n\\frac{\\part^2 \\mathbf{D} }{\\partial t^2}", "daf322f760a544e7878287d86cd6fb22": "QH_*(M)=H_*(M)\\otimes\\Lambda", "daf326f65bd429dd1195738f692db049": "\\Sigma^1_2\\mbox{-}\\mathsf{AC} + \\mathsf{BI}", "daf40f5f924e28261fb1f0e5666e4aaf": " y'\\in \\mathbb{F}_{q^n} ", "daf42e8cdfa494e6f5f82044ccbdc832": " V_A ", "daf441a38e088fb9b15045292a708065": "p_0 =1 - \\frac{1}{d^{m+n}} \\sum_{i=1}^m \\sum_{j=1}^n S_2(m,i) S_2(n,j) \\prod_{k=0}^{i+j-1} d - k", "daf4800ba9ea4032875c242d6a8a52c0": "c_R=0.5 c_B (d_{BL})^{0.5}\\,", "daf4d7e439274f7b1816298bb013af95": " D_\\alpha = \\max_x | P_\\mathrm{emp}(x) - P_\\alpha(x) | ", "daf4e47adc4bf441e51161f4710d2354": "1 \\rightarrow C \\otimes C^*, C \\otimes C^* \\rightarrow 1", "daf4f8a1d4b7e8ebc056c387716459fb": "\n\\begin{align}\n\\dot X&=f(X,U)+M(X) w \\\\\nY&=h(X,U) +N(X)v.\n\\end{align}\n", "daf4feb1148386d1d569b321344f800c": "\\sum_{n=-\\infty}^\\infty a_n x^n.", "daf53cfcb6753c21be5079d3d900736c": "i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \\phi_x,\\,", "daf5c0b908dae73edeb17610c9c078ce": "g_{oo}T \\and g_{oo}F = \\forall x_o \\centerdot g_{oo}x_o", "daf5d42f8a044a3183b448da69aeb642": "F_g", "daf62106be48433cd81754195774faa1": "\n=\\int_{-\\infty}^{+\\infty}\n\\frac{1}{f(x; \\theta)}\\frac{\\partial f(x; \\theta)}{\\partial \\theta}f(x; \\theta)\\, dx\n= \\int_{-\\infty}^{+\\infty} \\frac{\\partial f(x; \\theta)}{\\partial \\theta} \\, dx\n", "daf657a86c9943fa65c16eb3cbdb6dc3": " \\alpha_k, \\beta_k = k \\delta, \\quad \\delta \\in R^+", "daf66caae00545ea2e88e3a47644c5b3": "|\\Gamma(A)| < 2^{\\frac{k}{48}}", "daf69992363f9bf8e358d220286eca4f": "\\scriptstyle \\{i,\\, j,\\, k\\}", "daf6bcdb48aec7493fe5353ca064ae47": "i, j \\in \\{1, 2, \\dots, n\\}", "daf7128341add585a6878963e6090e76": "k_1= 6.34 + 7.75\\times 10^{-4}\\,T - 9.35\\times 10^{-5}\\,T^2 ", "daf7132e27e6a598de912af148690e1d": "n^2 = 25", "daf7409865ea5c8e358a02134d8b70ed": "P(\\alpha,\\alpha^*)=\\frac{e^{|\\alpha|^2}}{n!} \\frac{\\partial^{2n}}{\\partial\\alpha^{*n}\\partial\\alpha^n} \\delta^2(\\alpha).", "daf74a8812d680bd73763b60d9ef9aca": "\\mathbf{p}, \\mathbf{q}", "daf789f322465236d5f5c33e1cf22379": "\\langle k|H'|k' \\rangle", "daf79d8346187076080e335cfa01ae13": " \\vec{\\mu}_S = -\\frac{g \\mu_B \\vec{\\sigma}}{2}", "daf7a18c67cfdea3fd205a32c518adc0": " E_{2} =\\varepsilon _{2} \\cosh \\mathcal{G}-\\varepsilon _{1}\\sinh \\mathcal{G\n}.\n", "daf7c8fe42e08b686f7463cf4ea6abf9": "y\\in G", "daf81a7447e88f3c1010ca4a0f0087ce": "f(x_0^-)", "daf875114e8664df011220f1b3b666e5": " y_{R}(x) = e^{r_{1}x}(c_{1} + c_{2}x + \\cdots + c_{k}x^{k-1}) ", "daf9361d1ddb3427505d45cfdbeca21a": " \\overline{\\Gamma_4^*} ", "daf94f2eb782445b8e34163c32194795": "x\\ne0", "daf9957fcec5eea224851a0d8d728c33": "O(n^{2.375477})", "dafa07fd075588cd24bbc49dcc14dc6e": "R_2 = 2\\ \\mathrm{k \\Omega}", "dafa3ea730388ecc6d194de28a588c10": "\\scriptstyle \\begin{align}\n \\scriptstyle k &\\scriptstyle \\,+\\, \\ln\\theta \\,+\\, \\ln[\\Gamma(k)]\\\\\n \\scriptstyle &\\scriptstyle \\,+\\, (1 \\,-\\, k)\\psi(k)\n \\end{align}", "dafa3f0e9a9fb2654c8ac490b1af0fdd": " A_{N} ", "dafa401a4b9491e0f9bd586927b809b2": " \\frac {dR}{dz}= \\frac {2iw}{ Q} K - \\gamma (i-K^2) \\quad (2.9)", "dafa78b808ced94b54417f86c439c49f": "\nP(w_1,\\ldots,w_m) = \\prod^m_{i=1} P(w_i|w_1,\\ldots,w_{i-1})\n \\approx \\prod^m_{i=1} P(w_i|w_{i-(n-1)},\\ldots,w_{i-1})\n", "dafab35ef994b2339339fe2ca0b4bf77": " \\epsilon = \\epsilon' - j \\epsilon'' ", "dafab59c4d9e389ba6128b8b0952fc2d": "1,2,3, \\ldots", "dafae519c3eba3c3d394d671293c326c": " |\\phi\\rangle ", "dafb00f664e3faca394fb2310256434b": "nana$", "dafb61e96e9cc467772a6370e944f973": "f\\mapsto f\\cdot +i\\hbar^{1/2}\\mathcal{L}_{X_f}", "dafc749670ba9f395949c585ca8f4e33": "r^* = -\\frac{2 \\sigma}{\\Delta G_v}", "dafcd38eb6a2b963dca8a34bc45e8aa4": " {\\epsilon_0 }", "dafce0177ff4ea0e16ca39907b8ebe36": "\\,f(v,w) = \\sum_i f_i(v,w)", "dafce4105c571b84061dd3c1e29d7f83": "Y_{0,2} = D_e", "dafd23ebe9117bd8def62f946b6d7caf": "e^z = 1 + \\cfrac{z}{1 + \\cfrac{-z}{2 + \\cfrac{z}{3 + \\cfrac{-2z}{4 + \\cfrac{2z}{5 + {}\\ddots}}}}}", "dafd5929eaad3021e2873a9680227b20": " x \\oplus y = \\min\\{\\, x, y \\,\\},\\,", "dafd5b528514f0ab344d23e03ca81a23": " {{documentation}}", "dafd87ee641dbaca819235543b4719fe": "\\mu_{\\epsilon}=\\sum_{i=1}^{B} {m_{i,\\epsilon} p_{i,\\epsilon}}", "dafd978d8af5b3f4448c43318b246f07": " x=ka\\sin(\\theta) \\approx \\frac {\\pi R}{\\lambda N}", "dafdb349403ce5495be154a3f39fc0fd": "\n \\begin{bmatrix}\n 1 & 0 & 5 & 7 & 4 & 0 & 0 & 25 \\\\ \n 0 & 1 & 2 & 3 & 4 & 0 & 0 & 0 \\\\ \n 0 & 0 & 3 & 2 & 1 & 1 & 0 & 10 \\\\\n 0 & 0 & 2 & 5 & 3 & 0 & 1 & 15\n \\end{bmatrix}\n", "dafde7e70a156876d069ffb056990945": "Z = n \\times [Z] = n [Z].", "dafdf82a16a0eea87fedc6355bf1820e": "\n\\begin{align}\n\\mathbf{A} & = {1 \\over 2}|3 \\times 11 + 5 \\times 8 + 12 \\times 5 + 9 \\times 6 + 5 \\times 4 \\\\\n& {} \\qquad {} - 4 \\times 5 - 11 \\times 12 - 8 \\times 9 - 5 \\times 5 - 6 \\times 3| \\\\[10pt]\n& = {60 \\over 2} = 30\n\\end{align}\n", "dafe6a0fd72a1de253dd9cc711eee40d": "\\tau = \\frac{m}{F_{out}+L+D}", "dafe754e1bdab0f031c4191c7526194b": "\\!H_{i+1}=f(H_{i}, m_{i})", "dafec39644658f0dfa7695d68bdd44af": "\\frac{1}{R-d} + \\frac{1}{R+d} = \\frac{1}{r},", "dafec67651be9044069b625129b1707b": "1/\\sqrt{3}(\\pm 1,\\pm 1,\\pm 1)", "dafee31c362c29349b9838d7ffe49519": "\\ R_q = \\frac {L} {A_q},", "daff7c18efafeb78000fdaa83a023d1c": "\n |\\psi_1\\rangle = \\frac{1}{\\sqrt{2}}\\left(|0\\rangle_A\\otimes|0\\rangle_B + |1\\rangle_A\\otimes|1\\rangle_B\\right)\n", "daff8251e8a1901abf6584b4a13b023c": "\\chi_0 = \\frac{c}{H_0}", "daffb139c9e456469f21564158ad0ded": "g_2(\\tau)=g_2(1, \\omega_2/\\omega_1)", "daffb4f37744698c87d9694e9c3b1e24": "\\left(x, y\\right)", "db001a1c0d7d3b76f9f5d8f75f5b47ff": " \\frac {\\textrm''{static \\,\\, enthaply \\,\\, drop\\,\\, in \\,\\, rotor}''}{\\textrm''{stagnation \\,\\, enthalpy \\,\\, drop \\,\\,in \\,\\,stage}''}", "db00212d6591af6f7a6ae0f595cc6533": "\\mathrm{P}(A) = \\frac{|A|}{|\\Omega|}\\,\\ ( \\text{alternatively:}\\ \\Pr(A) = \\frac{|A|}{|\\Omega|})", "db00428f1e5c5bb2ce0f4f7d58486265": "A_m(p,p)=A_{m+1}(p,1)", "db007d6a923c2909d42c4292bffca5f0": "w_1", "db0081086dcd9a53ef573a2ee1198fc9": "x_4=\\tfrac{3}{2}", "db00a81ccfdf7ac0badc5f882d71bc95": "\\{x_3,x_4,x_5\\}", "db00f01f3f4e5b6ca41b858a9c366658": "y=\\tan(\\theta) \\cdot x-\\frac{g}{2v^2_{0}\\cos^2 \\theta} \\cdot x^2", "db0137cf882d98dc383111dd220a6a4b": " (0.25)_{10} =(1.0)_2 \\times 2^{-2} ", "db013bb87bed87fd912c1aa47badf4e0": "L\\in F", "db015ab4e506b838f7e095f369373b70": "\\mbox{Equity Ratio} = \\frac{\\mbox{Total Shareholder's Equity}}{\\mbox{Total Assets}}", "db019ee2a4af96f0817806f7e1ceb4dd": "\\sin\\left(\\frac{\\pi}{2} - A\\right) = \\cos(A)", "db01a371e9aebc0e81a3113f811f034b": " -\\cos(\\phi) \\, \\partial_\\theta + \\cot(\\theta) \\, \\sin(\\phi) \\, \\partial_\\phi", "db01f0ad60c4a4a601ba56e0c699a0b8": "\\vert\\downarrow\\rangle", "db02055ec66b1dd94c59d367714e30c3": " d(x) = [[x]](x)+1", "db02104c154e30ad47a03cb668773e64": "\\operatorname{E}[S_N-T_N] = \\operatorname{E}[M_0] = 0.", "db0220d9243aa66200dc11faebc9234e": "\\gamma^5 \\,", "db023243af729186cc1e30862d3fdd2e": "\\begin{align}\n0\\leq [D_{1}]_{i,j}&<\\infty \\\\\n0\\leq [D_{0}]_{i,j}&<\\infty \\quad i\\neq j \\\\\n\\, [D_{0}]_{i,i}&<0 \\\\\n(D_{0}+D_{1})\\boldsymbol{1} &= \\boldsymbol{0}\n\\end{align}", "db02695b1cd17fba4a7792dd118007cd": "\\mathbf{\\Delta B} = \\mathbf{P_\\pm} \\left(1, 0, 0 \\right)", "db026fbdaeca7c0cc33691bc96173ac2": "\\beta(\\alpha_s)=-\\alpha_s^2\\frac{11N}{12\\pi}\\frac{1}{1-\\frac{17N}{11}\\frac{\\alpha_s}{2\\pi}}.", "db028a45610818473e3c55752b749739": "\\le \\|x\\|^2 + 2\\|x\\|\\|y\\| + \\|y\\|^2", "db0296f8a7636fc8420d23e321dadcfe": "\\{S\\}", "db02c374cd6ad266ce7dab3223454152": "\\nwarrow", "db0351e928952a000c9eddd5e7a14e51": "x_{ij}\\,", "db036fd769c4b20d85b6f7a399598998": "\\displaystyle a^2+b^2=c^2\\quad (\\text{Pythagoras})", "db038f01db1d6ee8dea5f25727619cc1": "q_t", "db03ef6fda8646c897157ca83992b869": "\\Gamma_{m}\\,\\!", "db048818a8285a929e341a36ac17bd64": " z = x - \\frac{\\sigma(x,y)}{\\sigma^2(y)} y.", "db04c6904ce532c9dfc7ac7c3c7aef08": " A_{1 1} A_{2 2} - A_{2 1} A_{1 2} = -1.\\,", "db04cdd3120325520c3046fd9bbd6692": "\\partial_{k+1}h_{\\beta}^{k+1}=\\pm h_{\\alpha}^k", "db0501f8135c13a9912ec3f26ac85eb8": "\\{p: f(x)=0 \\in p\\}", "db0537b24dafaef51097de00b055cbf3": " u = x / (\\sqrt 2 \\sigma ) ", "db0559f3b290cb1a8e8dbdc6e7759e5d": "\\frac{\\pi^6}{945} \\! = \\! \\prod_{n=1}^\\infty \\! \\underset{p_n: \\text{ prime}} \\frac{1}{{1-p_n}^{-6}} = \\frac{1}{1 \\! -\\! 2^{-6}} \\! \\cdot \\! \\frac{1}{1 \\! - \\! 3^{-6}} \\! \\cdot \\! \\frac{1}{1 \\! - \\! 5^{-6}} \\cdots", "db057215df70be1462461d7488409614": "\\mathbf{w}(i-1)", "db057c847774000a2b2e1436d8c56bf9": "\\Delta(v_{ij}) = \\sum_{k=1}^n v_{ik} \\otimes v_{kj}", "db05b0098dfc4e291502925f8cbc3597": "\\lang n | n \\rang = 1", "db05cd39082a36d9008632d39ff5c101": "T \\times N", "db0654cb860481923a03a3c19530b99a": "m_B", "db065cc9fddd9626bb4df05a5bfc0907": "\\zeta\\;", "db06e2db97f32b49acd2d1a812e10225": "2^{k-1}-k", "db0701af54f7c11f6e78fba49fb6b5a2": "y^* =", "db07218181c72b187bca6d08f8e38895": "k[x_1, \\dots, x_n]", "db07414e59c9178383b4672755f1b1d3": "\n\\begin{array}{lcl}\nP(0, 0) & = & 1, \\\\ [6pt]\nP(m, n+1) & = & P(m,n)\\dfrac{n + 1/2}{m + n + 1}, \\\\ [12pt]\nP(m+1, n) & = & P(m,n)\\dfrac{m + 1/2}{m + n + 1}.\n\\end{array}\n", "db0763d2aabc2e12c987233d57394946": "Z_{TP} ", "db077e38e2d99ec6b267b97b78218998": " X = - \\frac {1}{\\omega C} ", "db07810babccf57bd9eef58d58b189f2": "\\mathbf{B}_\\text{g} = \\frac{G }{2 c^2} \\frac{\\mathbf{L} - 3(\\mathbf{L} \\cdot \\mathbf{r}/r) \\mathbf{r}/r}{r^3},", "db078cf9efadae17e7b093e1a3f80473": "b_{k\\prime} = a^\\dagger_k\\ \\mathrm{and}\\ b^\\dagger_{k\\prime}=a_k,\\,", "db07ab6f9708ccc81665a6896de44def": " P_z = \\varepsilon_0 \\chi_{zz} E_z", "db07bb0f384dfc5f2a3f6d27f0d4cbc3": " c^{\\dagger}_{i,\\sigma}", "db07fe5f3314d2ec1701416d373e0ec2": "\\mathcal{GW}^{-1}", "db0854dc1367e863f03169cdd4df4779": " \\frac{\\partial}{\\partial t}\\left( \\nabla \\times \\mathbf{j}_s + \\frac{n_s e^2}{m c} \\mathbf{B} \\right) = 0.", "db08a3e4077d17ad32b98c049e9f8d9f": "F = T\\,", "db09846ef8a7fbf6c5ba716a8f438aab": "\\chi_{\\text{e}}\\ = N \\alpha", "db09a13eef4cd183ce179ae54ed39fa2": "\\mathrm{D}_{H} F", "db09cfd0bc43ecbc74b49a2d34b27bea": "{13 \\choose 5} - 10 = 1,277", "db09e07fecc145408e3453d6acfb8167": "\\rho(\\hat{D},D)", "db09f6590de066625543474e9a3b4382": "\\operatorname{pred}(n) = \\begin{cases} 0 & \\mbox{if }n=0, \\\\ n-1 & \\mbox{otherwise}\\end{cases}", "db0a0af85956475594d2e3efe82ef7a6": "\nP(E,\\Omega) = \\lim_{\\lambda\\to 0}\\int_\\Omega \\vert DW_\\lambda\\chi_E(x)\\vert\\mathrm{d}x\n", "db0a2462d0c1141f4a95da5720497b9a": "\\Phi \\left(\\eta,\\tau \\right) = \\exp \\left[-\\alpha \\left(\\eta \\tau \\right)^2 \\right].", "db0a7ccc4bc832d3e37ee76a266c4158": "=\\frac{5}{13}", "db0b22b4090a9efef1fd33f9d581bd7d": "\\tan{\\theta'} = \\frac{\\sin{\\theta}}{\\gamma \\left(\\beta / \\beta' + \\cos{\\theta} \\right)}", "db0b684bf89ce14abd42d003a8f976a1": "k', k =1,2", "db0b891b7e85126be8db7a394f5fe86c": "B_{1,2}=\\frac{p(D|M_1)}{p(D|M_2)}=\\frac{p(D|S(D),M_1)}{p(D|S(D),M_2)} \\frac{p(S(D)|M_1)}{p(S(D)|M_2)}=\\frac{p(D|S(D),M_1)}{p(D|S(D),M_2)} B_{1,2}^s", "db0bd6cb95cd555c140092463c3cd3ae": "\\operatorname{PRESS} =\\sum_{i=1}^n (y_i - \\hat{y}_{i, -i})^2 ", "db0bda07b5d7496d18b098efad2d60dc": "\\Pr \\left(\\max_{1\\leq k\\leq n} | S_k |\\geq\\lambda\\right)\\leq \\frac{1}{\\lambda^2} \\operatorname{Var} [S_n] \\equiv \\frac{1}{\\lambda^2}\\sum_{k=1}^n \\operatorname{Var}[X_k], ", "db0c03d04f6f9d593b2bd919702bdba8": "d_i \\vert d_{i+1}", "db0c23b6079e207e37fdc74a60029691": " \\partial_\\mu F^{\\mu \\nu} = 0 ", "db0c46f5effba7fe0e8fc8d2a228ab2a": "\\mathrm{Aut}(A)", "db0c7e610e55c7ef376567f77f767e33": "P_n(p)=\\frac{2 \\pi L}{\\hbar}\\, \\frac{n^2 \\left(1-(-1)^n \\cos (k L)\\right)}{\\left(k^2 L^2-\\pi ^2 n^2\\right)^2}", "db0c8e12b9fc2b4a5546f1aea1900083": "z = \\frac{f_{\\mathrm{emit}} - f_{\\mathrm{obsv}}}{f_{\\mathrm{obsv}}}", "db0cab4e1da99b5f5a699b694dee942d": "C_{abs}", "db0d27c312fb980a0862d9105b7e5547": " \\gamma = \\lim_{n \\to \\infty}{\\left(H_n - \\ln\\left(n+{1 \\over 2}\\right)\\right)} ", "db0d5225df27a1a98bdc5ebc1ca6d82d": "|r| = 1", "db0d5d59e5a45956a6409d424991cfed": "\\chi = {1 \\over \\sqrt{5}} \\begin{bmatrix}\n 1\\\\\n 2\\\\\n \\end{bmatrix}\n", "db0d61e88af0e08c5bc4bcf6c8c36f01": " X = \\mathbb{F}_2^3 = \\{ (0,0,0), (0,0,1), \\ldots, (1,1,1) \\}, ", "db0da847da6b233e1f837a5a27921aea": " Q=\\int_0^VC(V) \\, dV\\ ", "db0dd92d325615f4d9b25615d74e234d": "\\frac{-a_1 + \\sqrt{{a_1}^2 - 4 a_2 a_0}}{2 a_2} ", "db0e1438701f546ca69aa8dc92207889": "\\mathbf r = \\mathbf d - 2(\\mathbf n \\cdot \\mathbf d ) \\mathbf n.", "db0e22124ddb5e7b00305b2dc934fa0c": "=4*x^2+644*x+11760=d", "db0e38e0f73f5eafc48bc08c9857353e": "E-V_0", "db0e8759911ea186a0ccf80ec6302874": "T \\times D", "db0e8dbeede99db0ed09e9dbae92161a": "A_M", "db0ea2e1de1a785bb6dce7e2bb82be47": "\\vdash_M", "db0eae86574e8d6f86ceab289c6fb44d": " s = \\int_{\\lambda_1}^{\\lambda_2} d\\lambda \\sqrt{ g_{ij}\\frac{dq^i}{d\\lambda}\\frac{dq^j}{d\\lambda}} ", "db0f0471db93285b7654ddd81ff20eed": "p*", "db0f4234b0f92b16b9d67815061b40e9": "M = \\rho(\\psi(\\delta)\\psi(\\gamma)^{-a})", "db0f7504eac3fde8eb63d2275f45e66e": "\\Gamma \\backslash \\mathbb{H},", "db0f91145d263d9d2c762c3a87ff26f0": "\\Sigma _{YY} ^{-1} \\Sigma _{YX} \\Sigma _{XX} ^{-1} \\Sigma _{XY}", "db0fa4ed0219e6c8f730021f33385e1c": "\\mathbf{W} = \\mathbf{A}^{*}\\mathbf{uA}", "db0fbd27cb58af54b924787a1f825365": "P_n^t(x)=\\frac{tP_n(x)+(1-t)(x-c_1)Q_n(x)}{\\sqrt{t}}", "db0fc32ea3573e0d45578805def994ec": "g = f\\left[\\frac{a \\cdot x + b}{c \\cdot x + 1}\\right]", "db109c78e6d09883b09f2b038e60728a": "-\\bar{t}", "db10b87ea4effa3d04d575486422c215": "a \\to c\\;", "db10c8f56c9bcdba394656ef642a3430": "(\\tfrac{v}{c})^2", "db11e2d8d2def088ef681a62f1395f0f": " ( c \\vec{\\alpha} \\cdot\\vec{p} + \\beta m_o c^2 + I V ) \\Phi = E \\Phi ", "db128be7e6fbb1e35df03bdc14f6bc27": "0.143814381438\\ldots \\;=\\; \\frac{1438}{9999}.", "db12a0043d187e06501f1389638e247e": "(i) \\ne (j)", "db1343cb91e14c01770b958d9a3bb9d0": " 1 - 2 \\Phi \\leq \\lambda_2 \\leq 1 - \\frac{\\Phi^2}{2}. ", "db1371b586c19356464a4ea0e63a5d2e": "T_{ij}^r = \\widehat{\\overline{u_i u_j}} - \\hat{\\bar{u}}_i \\hat{\\bar{u}}_j", "db1392282fb00d372c4b08f5447bac89": "\\scriptstyle a_c = -2(v\\times\\Omega)", "db13ab567034a8b83aa2430505faf3cf": "\\Delta = \\,b^2-4ac.", "db13beceaf3c0d56bae7f773772b6eb7": "\\operatorname{sech}(t)\\!", "db1444b5e29911c5285fa648b68caab9": "W(S,T)\\geq\\gamma'|S||T|", "db1477e4ae4de5f99debb35bc7263684": "\\int_S{ \\mathbf{J} \\cdot \\mathrm{d}\\mathbf{A}} = \\int_V{\\mathbf{\\nabla} \\cdot \\mathbf{J}\\; \\mathrm{d}V}", "db147a61f2dfd7b7c06b152a86f36ef9": "A=\\Gamma +\\theta, \\qquad \nA_\\lambda^\\mu=\\Gamma_\\lambda{}^\\mu{}_\\nu \\dot x^\\nu +\\delta^\\mu_\\lambda, ", "db14d6fbbe213a3be2c25d748fd472d0": "X=n", "db1513096b1d1a35dc1ec859570ff35a": "\\sqrt{q^*}.", "db151d024a26fabc9863adc84751c874": "ay\\bmod 2^w", "db156193d5e71f02068331111af68fb6": "\\scriptstyle \\{0,1\\}^{Z^d}", "db1561f84d7ab4fa7764fd1180d28b83": "(h^{r-s}h^{ab(x-y)})^{\\alpha\\beta}=h^{\\alpha\\beta(r-s)}h^{\\alpha\\beta ab(x-y)}=th^{\\alpha\\beta ab(x-y)}", "db1588c57f55d31c1e7cd6db73d7d7ea": "x^t", "db1595e9f67b08d744e03981eebbba47": "I=\\sum \\phi c ", "db15c7f893471358e0bffd227eaf7739": "k_\\mathrm{PE}", "db15e90771d994c7c24ae18b2aa4f975": " \nc_\\pm = c_{\\rm B} e^{ \\mp \\beta q \\psi}\n", "db161cfd4798da0bfd9d3b754b1ece03": " A\\subset X ", "db164a86329f373dafff67e262370ded": "\\mathbf{L}= I \\boldsymbol{\\omega} ", "db164f577364a0370d97f4e0dabf955c": "p_1=\\textstyle \\frac{1}{4}\\ ,", "db1691a66c8f0b0b3a52031fc290caf8": "A^s x_1=\\sum_{r=0}^{m-1}\\binom{s}{r}\\lambda^{s-r}x_{r+1}", "db170c9561feca0fce2d227ffb2a3de7": "L \\;", "db172391c5903afd2171015a5eb773f4": " \n(Eq. 6) \\text{ } \\text{Minimize: } P(x) \n", "db1724bca86987593980ca69667e6b21": "\\{0.4 ; 0.35 ; 0.2 ; 0.05 \\}", "db174fc7f252c2d2a2c3b0b50b917762": "\\xi(m) = k m^{-\\alpha} ", "db17b22e81b91b5d1c680e4ae44ea22f": "|f(x)|<\\infty", "db18736444fc1804d2506f843723b0aa": "\\mathbb{E}[f(X_t)|\\mathcal{F}_s]=\\mathbb{E}[f(X_t)|\\sigma(X_s)]", "db18a5204086265437293193b2d94fdf": "Q -\\!\\!\\ast\\, \\_", "db18ed2d48f78866905dd449aa2e51e4": "I/V", "db18f30e1b7c6ea35a1c2e26d4351521": "\\lambda = \\int_0^\\infty \\frac{\\rho(t)}{t+2} dt ", "db18f45dc386059b9b8272c3fff3ba77": "\\ \\beta = 70 ", "db19c41aa891fbeee41bd6d61c692ff1": "\\overset{\\ }{\\leftarrow}", "db19ed81cd034d0392160fa5d947da7c": "\n\\left. - \\left[Q_R^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{1*}(\\mathbf{p})\n+ Q_L^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{2*}(\\mathbf{p}),\n\\right]e^{-i p x} \\right\\}.\n", "db19edd0838b8c83a6e997dc605aa7e2": "\ndx = {\\partial G \\over \\partial p} ds = \\{ G,X \\} ds\n\\,", "db19fb643a194a957e675463616d2a37": " J_{ij} \\sim |i-j|^{-2} ", "db1a1c29ac65f0cda97f113c9f0ce28e": "\\frac{d^{2}u}{d\\theta^{2}} + u = -\\frac{m}{L^{2}} \\frac{d}{du} V(1/u) = \\frac{km}{L^{2}}", "db1a88a660c6f5b0de4eee964b7aba41": "\\{\\gamma_{j,i} | 0\\leq j \\leq t\\}", "db1aa36ec4170c9c7519472732557a70": " I = \\iint \\mathbf{J} \\cdot \\mathrm{d}\\mathbf{S}", "db1b07a80ba0045958b9add22bdc3dab": " -k ", "db1b1b84fab8e812935c540b6ab62b74": "\\beta_i^{rev}", "db1b88cf18d51a66ba47f3fff9d1fba1": "\\sqrt[20]{3/2} ", "db1b8c180e5d8f9398cf9ff710094d43": "\\frac{3\\varphi^{n+1} - 4\\varphi^{n}+\\varphi^{n-1}}{2\\Delta t} = F(\\varphi),", "db1ba0579b7c0992a006324dff725c46": "\\hat{s}'", "db1bfb9bd91d177162f2e2a9fd9b5849": "\\,\\frac{d m_j(t)}{d t} = \\phi(\\textbf{c}(t))d_j(t)-\\epsilon m_j(t),", "db1c436e3fc5afd36217e5eed061c80a": "s_i = \\pm 1 ", "db1caf22475de5dbccb7056170df282a": "X_n", "db1cb712475cac7e8e141924be54df55": "\\gcd(f(x),a(x)-1) = \\prod_{i \\in C} p_i(x).", "db1cda7a62d0f933eecdea380b1c6d11": "\\sigma_{23}^y, \\sigma_{31}^y, \\sigma_{12}^y", "db1d7f6e5a26d31b068b4524a622fe69": "{\\color{white}.}\\qquad\n\\displaystyle\n \\frac sb = E(\\sigma, ik),\n", "db1d9f273bc273e4221579efc375900f": "NaN = NaN \\Rightarrow False ", "db1db04089770c29d8d75c35427ec923": "a \\geq b \\geq c", "db1dd4a6c9b1796051aee002b2135803": "A_2, \\, A_1, A_0 \\in \\mathbb{C}^{n \\times n}", "db1e839e26493d4bc4c51c42cdb7fad1": "\\lambda = b-a-1", "db1edad24af12ad0eb41d0353059e2d1": "M=\\{(\\varphi(t)\\cos \\theta, \\varphi(t)\\sin \\theta,\\psi(t))\\colon t\\in (a,b), \\theta\\in [0,2\\pi)\\}.", "db1f1ae5495ef558cc8a3a7bffd9788b": "\\mathrm{lift}: \\mathrm{M} \\, A \\rarr \\mathrm{M}(W \\times A) = m \\mapsto \\mathrm{bind} \\, m \\, (a \\mapsto \\mathrm{return} \\, (\\varepsilon, a))", "db1f50944c05e4d6a180a936d0d9c34d": "x=x_1 - x_2", "db1fb2964a82eaa6acb2a7c7996be39e": "(\\mathcal{C}){:}", "db1fedafc03a9cc8b2bbbd7870c959f4": "R = -4\\pi \\mathrm E^2 h \\ ; ", "db1ffe6da696b54fa9e8fede01f11aef": "\\hat{v}(x,z,t)", "db2000c1b56ae61040acef26995a4c96": "Z(x_0)", "db207a0e49d2df776cce773fd9dd4101": "fs: \\Lambda_k^n \\to Y", "db20ace4f88f6cdb5c68427989beee35": "f(x) = x^2 + bx |_{b=\\{-1,-2,-3,-4\\}} \\!", "db2107c9f688fc760f86fedf9dadec2e": "\\langle x,y \\mid (xy)^2=x^3=y^5\\rangle.", "db210ec18200681b58262c0693dc7cb6": " ax^2 + by^2 + c , \\,\\!", "db211f3ddb5ff8488b23f32df63eac9b": "e^{\\pi} = \\left( e^{i \\pi} \\right)^{-i} = (-1)^{-i} = 23.14069263 \\ldots ", "db2120ff1cffed4a8a7d6b57b62a6e23": "\\displaystyle x_0 \\in (a,b)", "db2128a3c93232c811080169eb15e746": " r\\le 1/2 ", "db2138186a127c9e4164c7e47b40a3ed": "\\langle \\mathrm{d} \\omega | M \\rangle = \\langle \\omega | \\partial M \\rangle.", "db214f0f8643aa1d36d6100edfa32107": "F : \\mathcal{C} \\rightarrow \\mathcal{D}", "db21c5bc3527f038f8e6f9a1d9a6507f": "X =\\textbf R^2_+", "db21f415f1de15b2b2e672c4e9457c05": "reject \\leftarrow 0 \\,", "db2245d71e4c6810c9cd51e27884f652": " P_d'+P_m'=1 ", "db229029f8c293f368310f0bbbbcb842": "\\nabla_\\rho g_{\\alpha \\beta} = 0", "db22b8051df2338684cd6a228b7b2d8c": "|pq| = |p| |q|.\\,", "db2305e13015aed7197139e34cbafffe": "\\mu F_n", "db230f2f232f5bfc13fc5faa35ba5e9e": "S^{*} = \\{ (o_{i}, o_{j}) | o_{i} \\in X_{k_{1}}, o_{j} \\in X_{k_{2}}, o_{i} \\in Y_{l_{1}}, o_{j} \\in Y_{l_{2}}\\}", "db2315ecf70d3adb46aa141c49a78995": "H_{\\mathrm{Tot}}=\\bigoplus_{i}H^{i}", "db236db8fe5a8bbaa2505002ed56099e": "P_i(t=0)\\frac{}{}", "db23a8a5f8fc6ff5655119762b477d6c": "\n\\begin{align}\n\\Phi_3(-z^2) \n&=z^4-z^2+1 \\\\\n&= (z^2+3z+1)^2 - 6z(z+1)^2\n\\end{align}\n", "db240569226b5d1c43a4501eacbbb1c0": "p , e , \\theta", "db24329327ae74a2d0200e7931025ebf": "\\displaystyle{Q_y(a)R_y(b,a)=Q(a)Q(y)R(b,Q(y)a)=Q(y^{-1}) Q(Q(y)a)R(b,Q(y)a)= Q(y)^{-1} R(Q(y)a,b)Q(Q(y)a)=R_y(a,b) Q_y(a).}", "db24e17fdc4fc75815c1258b6a1caf1e": "Y = \\beta_0 + \\beta_1 X_1 + \\beta_2 X_2", "db250852bfbabe532534738d101f5149": " s(A) = s(BAB^{-1}).", "db25c4bc84894440ee90ed23f2f0469f": "\n \\mathrm{J}_\\pm|(j_1j_2)JM\\rangle = \\hbar C_\\pm(J,M) |(j_1j_2)JM\\pm 1\\rangle =\n \\hbar C_\\pm(J,M)\\sum_{m_1m_2}|j_1m_1\\rangle|j_2m_2\\rangle \\langle j_1 m_1 j_2 m_2|J M\\pm 1\\rangle.\n", "db25f55ff7541e5e95f0183c43068eba": " x_{i_0} ", "db2604b0e7f9ed12a3d0b8a512db8bb5": "\\nu_2: \\subseteq \\mathbb{N} \\to S_2", "db261af767976d56375853f136dcfedc": "\\pi=\\mathrm{10.011T111T000T011T1101T111111}...", "db26a65a6fd5fa71899a4b39f32f5200": "\\mathit{alg}^{\\prime}(A^{\\prime},B)", "db26f12cc058d47c20e5b63022efeed7": "\n \\bar{p}p = m^2\n", "db26f700538f340244d4e92058c9f49f": " (t+X_1) (t+X_2) \\cdots (t+X_n)= \\sum_{\\sigma \\in S_n} \\sigma t^{\\text{number of cycles of }\\sigma}.", "db2743f1ae689715c4f90a86eda84938": "3/8 = 0.375", "db276d5a7c3acc1ed36c85cd296f5d2d": " a_w \\equiv p / p_0 ", "db276e38ac49eeff90c5802fd7955856": "g(z) = \\frac{1}{f(z) - \\mu}.", "db27ac4206b1ce13e12a926c8d86cfb5": "z < 0", "db27ca7c8c3076db4966aa0f23fb91c1": "\\ m, n", "db280172e97e5cc0efc3c20e001641b8": "\\langle 0|T_{00} |0\\rangle = \\sum_n \\frac{\\hbar |\\omega_n|}{2}", "db2836344b269cd42e29d374a02dc303": "\\sigma_{ij}\\,\\!", "db2894abd9cf2f4b15bdbe7ffbee7d04": "|\\psi_{+}\\rangle", "db2897a56883984e2f8ecfc48a551f3e": " \\boldsymbol{r} ", "db28a8e8d00630d4f053d0ace7cc98d0": " r=\\frac{|n-a^2|}{2a\\pm r}", "db28f2ec74757ae279d06977bc63b850": "\\sum_{L=1}^P w^{(L)}_k = 1", "db2931bbb3ec5a04250621f313cc270c": "Z^{(\\ell)}(\\theta,\\phi) = P_\\ell(\\cos\\theta)", "db295293bedd71332271b121ee55ff0d": "N = 167", "db297cd14db2a18fc7c68cd1a95782eb": "S = - k_B \\sum p_i \\ln p_i \\,", "db2a02e295381b418d306a6f7a6abcfa": " \n\\int_0^{\\infty} {k\\; dk \\over k^2 +m^2} J_1^2 \\left( kr \\right)\n=\nI_1 \\left( mr \\right)K_1 \\left( mr \\right)\n . ", "db2a13d31adaeec199bbc6c7e79cda17": "\\omega_{abc} = \\eta_{ad}\\omega^d{}_{bc}\\,", "db2a17354c20e715ba726ebf7e9c126a": "f(x_1,\\ldots,x_n \\mid \\mu,\\sigma^2) = \\left( \\frac{1}{2\\pi\\sigma^2} \\right)^{n/2} \\exp\\left(-\\frac{ \\sum_{i=1}^{n}(x_i-\\bar{x})^2+n(\\bar{x}-\\mu)^2}{2\\sigma^2}\\right),", "db2a31e07a9bd6afe78f6200de705244": "\\textbf{x}_{k} = f(\\textbf{x}_{k-1}, \\textbf{u}_{k}) + \\textbf{w}_{k}", "db2a4b74bba22540bdb107452e8c2afd": "\\sinh x = x \\prod_{n = 1}^\\infty\\left(1 + \\frac{x^2}{\\pi^2 n^2}\\right)", "db2a6a76d74e5a06419ab3a92a948210": "Y = b_0 P^0(z) + b_1 P^1(z) \\cdots + b_k P^k(z).", "db2a9ce97a6a04a885d7abd4a66874c3": " \\Delta", "db2aeb9304fd55b7e1ea10ff374038e2": " v=\\sqrt{\\frac{mg}{k}} \\tanh \\left( \\sqrt{\\frac{k}{mg}} g t\\right)", "db2b0e2e2507c312b28837d6699e5b1f": "q_\\tau^*(\\tau)", "db2b2a912d4200e61d15bc64faf6b500": " h_f = x_f h_g+(1-x_f)h_h.", "db2bdcdd3d91c71724e56bf926fe4ae2": "{\\partial \\over \\partial y_1} = X", "db2bde606032fdff61edcac4173ed6bc": " \\left( \\frac{\\partial \\mu}{\\partial p} \\right)_{T,n} = \\left( \\frac{\\partial V}{\\partial n} \\right)_{T,p}", "db2bf7e8d26b8d15f899ed371aeed414": "\\lang \\psi | \\psi\\rang=1", "db2c483dd437346aac81b5956675ed3f": "\\ln\\left(\\frac{x_\\mathrm{m}}{\\alpha}\\right) + \\frac{1}{\\alpha} + 1", "db2cad3e0e17c461a2446b2aa95bd963": "A=\\{a_1 ,..,a_n\\}", "db2cfd575262217415c795ea1539233c": "I(X;X)", "db2d482c24bf0b2800420443b797b90c": "n=1,2,3,4, \\dots\\,", "db2d648ca5a268b2538dc766a50915d0": "a = d \\ne b = c, \\alpha = \\beta = \\gamma = \\delta = \\epsilon = \\zeta = 90 ^\\circ", "db2d880f6a1d72b82474e08513ec631e": "\\scriptstyle\\boldsymbol \\nabla V = -S\\boldsymbol \\nabla T", "db2da5c6f333640a3c3d7719bef469b1": "\\tan (\\alpha - \\beta) = \\frac{\\tan \\alpha - \\tan \\beta}{1 + \\tan \\alpha \\tan \\beta}\\,", "db2dc1318b97d0c6f005e32f2708e7a0": " \\Pr( \\sum_i X_i a_i > t || a_i ||_2 ) \\le e^{ - \\frac{ t^2 }{ 2 } } ", "db2de4e23d2732ef69834bd0a23d5152": "r(t,k)", "db2e5b2706bb021ac2e405a2291aba51": "p \\mathrel{:} \\{1, 2, \\dots, N\\} \\to \\{1, 2, \\dots, N\\}", "db2ea48f5bc5d06bf3adc9c463c68d6b": "\\sqrt{n}\\frac{r-\\rho}{1-\\rho^2} \\Rightarrow N(0,1)", "db2eedc8041576edc852785006a73fb0": "\\mathbf{E}_1 \\times \\mathbf{H}_2 = \\mathbf{E}_1 \\times \\hat{\\mathbf{r}} \\times \\mathbf{E}_2 / Z", "db2f03efd6db64de762c7fede4193edb": "\\displaystyle{B(a,b)^{-1}(a-Q(a)b)= Q(a^b)Q(a^{-1})(a-Q(a)b)=Q(a^b)(a^b)^{-1}=a^b.}", "db2f0bf9b32f2c4a6f601f74ec0adb3a": "\\Sigma \\, \\Omega + \\Omega \\, \\Sigma", "db2f1780fea0090711f3fbddc58e6758": "F_{I}(\\omega)", "db2fabc2206a3d493232945314c1f870": "~A \\oplus B \\oplus C", "db2faf572e98593a8cf5f229ac027761": " \\chi (e) = q _{v} + q _{w} - q _{v \\cap w} ", "db2fcab1485284417f917ede73874d57": "N=\\mathbb N^2", "db301703af055686945391ad700915c1": "\\forall s\\in S: |s^\\bullet|=|{}^\\bullet s|=1", "db3032d81f77b3514fe5a2f39462a712": "1/k_{\\text{off}} >> r^2/D", "db305ba89c3f824b7b36cf0fe6182009": " \\begin{align}\nP_{0,0}&=1\\\\\nP_{i,i-1} &= \\frac{N-i}{r \\cdot i + N-i} \\cdot \\frac{i}{N}\\\\\nP_{i,i} &= 1- P_{i,i-1} - P_{i,i+1}\\\\\nP_{i,i+1} &= \\frac{r \\cdot i}{r \\cdot i + N-i} \\cdot \\frac{N-i}{N}\\\\\nP_{N,N}&=1.\n\\end{align}", "db3113a5784cccf8641827df1cfaffb5": "\\{A_{j_1\\cdots j_n}^k \\mid 1\\leq j_i\\leq d_i, 1 \\leq k \\leq d\\}", "db3157d147cd04402662b96329ce8f07": "\\lceil x \\rceil", "db31724383f710f7f53eb5eaeeaec86c": "\n\\hat{Y}_{ij} = \\hat{\\mu} + \\hat{\\alpha}_i + \\hat{\\beta}_j + \\hat{\\gamma}_{ij} \\equiv Y_{ij}\n", "db319c922c9eb4452e69db7d358ec62c": "\\pi^*\\colon K_G^*(X)\\hat{_I}\\to K_G^*(X\\times EG)", "db31ae0163cf3419fa3da26f6ef59dc9": "\\mathbf{E}_{\\text{Electric quadrupole}}(\\mathbf{x},t)=Z_0(\\mathbf{H}_{\\text{Electric quadrupole}}\\times\\mathbf{n})", "db31d283e866118d2a64e410ad40956e": "h(x)=2x_1^4-2.5x_1^3x_2+x_1^2x_2x_3-2x_1x_3^3+5x_2^4+x_3^4", "db321e0b40a98385de680dfed6de6d79": "\\sqrt{\\frac{2}{\\pi}}", "db32548b952f2d710a235db3c2dd9ef8": "\\frac{d}{d\\theta}\\operatorname{Cl}_{2m+2}(\\theta) = \\frac{d}{d\\theta}\\sum_{k=1}^\\infty \\frac{\\sin k\\theta }{k^{2m+2}}=\\sum_{k=1}^\\infty \\frac{\\cos k\\theta }{k^{2m+1}}=\\operatorname{Cl}_{2m+1}(\\theta)", "db32d7f4ff670934aec41eb2385d4660": "{(2n+1)}^{-1}", "db3336f9788bf9c8aeacd7c351deea34": "y = \\sum_{i=1}^{n}w_ix_i", "db33d329748daf02f341be2ff6fad7d1": " N_{2} = (Q_{2},\\ \\Sigma ,\\ T_{2},\\ q_{2},\\ A_{2})", "db33e8a56bfb9a77341434945a81508a": "\\mu_b", "db34192187b5cb5652492d6e15ded8ce": "g_{\\mu 4}'=\\frac{\\part x^{\\alpha}}{\\part x^{'\\mu}} \\frac{\\part x^{\\beta}}{\\part x^{'4}} g_{\\alpha \\beta}= -g_{\\mu 4}", "db343a7163f685b480a12623335682d5": " M^{(n)}(B_1\\times,\\dots,\\times B_n)=E [{N}^{(n)}(B_1\\times,\\dots,\\times B_n)], ", "db345558dd152dc3d6ea555329da495a": " |2,0\\rangle + |0,2\\rangle", "db3477f7a3f2d6a77ae4e82a64e0a2fc": " \\mathcal L^\\infty(X,\\Sigma,\\mu)", "db349c577059b3116533096f19df638b": "t^{n} e^{-\\alpha t} \\cdot u(t) ", "db35349e6acd6f25e4b0a9cbd67dda3d": " V_1 \\subset V_2 \\subset \\ldots ", "db35d70c78cf97923c1d5d67c5e222de": "z=\\Phi(w)", "db35f187aa91988e7211123f916282e9": "\\scriptstyle A_2=a_2", "db361114aa2a099d4b1658559e9b6a63": "\n\\frac{r^2}{\\phi}\\left(\\frac{\\partial^2 \\phi}{\\partial x^2}+\\frac{\\partial^2 \\phi}{\\partial y^2}+\\frac{\\partial^2 \\phi}{\\partial z^2}\\right)\\ =\n\\frac{1}{R}\\frac{d}{dr}\\left(r^2\\frac{dR}{dr}\\right)+\\frac{1}{\\Theta\\cos\\theta}\\frac{d}{d\\theta}\\left(\\cos\\theta \\frac{d\\Theta}{d\\theta}\\right)+\\frac{1}{\\Phi\\cos^2\\theta}\\frac{d^2\\Phi}{d\\varphi^2} \n", "db36360e7ca5ffe7ae0108abb4cf8877": "\\mathit l=\\mathit l^{\\prime}", "db3645ea4ea1b8a242256f79a268b6d1": "\\overrightarrow{ab} + \\overrightarrow{bc} = \\overrightarrow{ac} . ", "db36c7351034f659301783409fc92f78": "\\scriptstyle\\mathbf{F}", "db36e9db23668cc9423c5c8d11a2dc80": "\\|f\\|_\\Phi = \\sup\\left\\{\\|fg\\|_1\\mid \\int \\Psi\\circ |g|\\, d\\mu \\le 1\\right\\}.", "db37d52054187762fe761e3aa5c85ee1": " Q_r = K_{eq}~", "db37fb490204c7be0ddf01d8f3a3450d": "V^{\\vee}", "db38585eaf0bdc336a19c0da4ca95ab0": "2^{65\\,536}\\approx 2.0 \\times 10^{19\\,728}", "db386137e00a88aefcdc8a76eb860c9c": "\\scriptstyle \\sqrt{1 - \\frac{v^2}{c^2}}", "db397649f4127385418550d53db90bb2": "\\mathcal{F}=\\frac{1}{2}\\left(\\mathbf{B}^2 - \\mathbf{E}^2\\right)", "db398220052683209605b02e33cd068d": "\\ln[W_0] + \\Sigma_{t=1}^T \\text{E}\\ln [w_{1t}r_{1t}+w_{2t}r_{2t}+\\cdots + w_{nt}r_{nt}].", "db3994f33f430d4ed8a97f6c367ba02b": " A = A_o (k_1) e^ {i \\alpha (k_1)} \\ , ", "db39d023f20f6217e089225f716b81b8": "\\hbox{I}(\\hbox{V}(J))=\\sqrt{J}", "db39faaf9921a604bed7dea03ffe5200": "\n\\Gamma_p(a)=\n\\pi^{p(p-1)/4}\\prod_{j=1}^p\n\\Gamma\\left[ a+(1-j)/2\\right].\n", "db3a175ca34d6b5d6aa1c4342cd714e3": " \\mu{ \\left( \\frac{a}{a_0}\\right)} = \\frac{a}{a_0} ", "db3a249735272b04334b76259f6d3ed5": "\\left(\\frac{dy}{dx}\\right)^2 = \\frac{2r}{y} - 1.", "db3a35253a4bae99a8630cd83f7f06c4": "\\begin{align}\n \\frac{d}{d t} \\operatorname{OE}[a](t) &= a(t) \\operatorname{OE}[a](t) \\text{,} \\\\\n \\operatorname{OE}[a](0) &= 1 \\text{.}\n \\end{align}", "db3a9eeb5d2e57c2fa35c4fe23586d7e": " p^+ \\, / \\, p^-", "db3ac641e74c23209c06a48e49fc636e": "M_1 \\times \\{1\\}.", "db3ac88fe92c1cf6d23c83a5af5bfb7d": "= \\frac{32!}{2}\\cdot 60^{32}\\cdot \\frac{80!}{2}\\cdot \\frac{24^{80}}{2}\\cdot \\frac{40!\\cdot 80!}{2}\\cdot \\frac{6^{80}}{2}\\cdot \\frac{2^{40}}{2}", "db3ae3af6bc7a02cfdf6127d9ab5bfe6": "a_j=\\prod_{k=0,k\\neq j}^m(\\lambda_j-\\lambda_k)^{-1}.", "db3b162db3b499387d2cc5c6d295c8fa": "\\psi _\\mu (\\tau )", "db3b34906044713b26dca603b1e11b6a": "\\operatorname{Ind}_\\mathfrak{h}^\\mathfrak{g} \\simeq \\operatorname{Ind}_\\mathfrak{h'}^\\mathfrak{g} \\circ \\operatorname{Ind}_\\mathfrak{h}^\\mathfrak{h'}", "db3b8a54e9fe1a634820e66966e7e916": "Y = a_{00} + a_{10}v + a_{01}w + a_{20}v^2 + a_{11}vw+a_{02}w^2 + a_{30}v^3 + a_{21}v^2w + a_{12}vw^2 + a_{03}w^3", "db3be50aa27d2356c606f1989ad92905": "\n \\mathbf{e}_3 \\rightarrow \\mathbf{e}_3^\\prime = R_0 \\mathbf{e}_3 R_0^\\dagger\n", "db3bf6af5b0c446c423927c8073e77ee": "R \\colon \\mathbf B \\to \\mathbf A", "db3c2ce0506115e843a078b1b3777627": "\\mu'_{20} - \\mu'_{02} \\ne 0", "db3c3c07ef3ce25647f8950c1f5de377": "L_1 \\vee \\cdots \\vee L_n", "db3cbc658d2b704f4d95a29bbbec8e98": "A\\times B\\subset U''\\times V''\\subset N", "db3cfa59ffabc3d0ad8429ad9b16290e": "p \\vdash \\neg \\neg p", "db3da305b93360ac3c8776e312eb89ac": "\\begin{bmatrix} y_{11} & y_{12} \\\\ y_{21} & y_{22} \\end{bmatrix}", "db3e1643f530b1b4215dd785b685761b": " \n\\begin{align}\n(R_{pq})_{m,n} & = \\delta_{m,n} & \\qquad m,n \\ne p,q, \\\\[10pt]\n(R_{pq})_{p,p} & = \\frac{+1}{\\sqrt{2}} e^{-i\\theta}, \\\\[10pt]\n(R_{pq})_{q,p} & = \\frac{+1}{\\sqrt{2}} e^{-i\\theta}, \\\\[10pt]\n(R_{pq})_{p,q} & = \\frac{-1}{\\sqrt{2}} e^{+i\\theta}, \\\\[10pt]\n(R_{pq})_{q,q} & = \\frac{+1}{\\sqrt{2}} e^{+i\\theta}\n\\end{align}\n", "db3e3c257fcdf84ff48f909133c8cafc": "n_i=3(1+\\omega_i)", "db3e4a7f6bc2282cbbf02d4655664b08": "E_{zx,x^2-y^2} = \\frac{3}{2} n l (l^2 - m^2) V_{dd\\sigma} +\nn l [1 - 2(l^2 - m^2)] V_{dd\\pi} - n l [1 - (l^2 - m^2) / 2] V_{dd\\delta}", "db3ec6e3ac3c8645439f4d34f98f31c7": "\\int_0^{\\frac{\\pi}{2}} \\sin^{n}(x)\\,dx = \\int_0^{\\frac{\\pi}{2}} \\sin^{n-2}(x)\\,dx - \\int_0^{\\frac{\\pi}{2}} \\sin^{n-2}(x) \\cos^2(x)\\,dx", "db3ee47187fd430ce48d2bb522439c12": "\\mathbf{E} (X_{n+1}\\mid X_1,\\ldots,X_n)=X_n.", "db3f4c01703ae0d19a807cd189ef3712": "\\,\\sigma_s = \\frac{u_2}{c_0} = [2 ( 1 + \\phi_2 \\cot{\\beta_2}]^{- 1 \\over{2}}", "db3f9ce37f187c092582a3ed27b350b2": " k\\nabla^{2}\\theta+\\frac{k}{\\kappa_{s}}g=\\frac{\\partial\\theta}{\\partial t} ", "db3fa614bd10312c93f9787bc42c8d88": " f \\mapsto \\widehat{f}(\\chi).", "db3fa984fe9e7ed6df79ba24180c3884": "P_{L1}=\\frac{V_P I_P}{2}\\left[\\cos\\varphi-\\cos\\left(2\\theta-\\varphi\\right)\\right]", "db3fb815a031d3fd89073d840476e6f1": "\\scriptstyle{d}", "db404fbf2a22d5322e243cd488076f0e": "\\partial_\\nu j^\\nu = 0", "db409233cf4c644577899d2645bd2d2c": "G = \\int_{0}^{\\pi/4} \\frac{t}{\\sin t \\cos t} \\;dt \\!", "db4093feb5312c5b1897332eefb760aa": "\\textstyle N_1=\\frac{C_1}{C}", "db40c0d68fc13a025c09246a548a2be7": "\\tilde{W}", "db40f1c8251703c1616181aef1850436": "a.P_1", "db40f4b3a5aef078398bdfe6f9030e0d": "\\mathbf{S}(0) = \\sum_{k,k',\\sigma,\\sigma'} c^{\\dagger}_{\\mathbf{k}\\sigma} \\mathbf{\\sigma}_{\\sigma,\\sigma'}c_{\\mathbf{k'}\\sigma'}", "db4142d1f303c9432904006c028b512b": "|\\mathbf{AXB}|(\\mathbf{X}^{-1})^{\\rm T}", "db414653a8fede659dde291729a2f2ff": "k_{\\infty} = \\eta f p \\varepsilon", "db414fd3a327656589465fe61987e193": "|\\Psi_E\\rangle", "db41b4f2a5aaba31c4ec996f5195ae2c": "\\frac{du_1}{dt}=\\gamma(T_2-T_1)", "db421a6a56eb7593c4f6ee0252b439b0": " q(a \\otimes_B a') = (a \\gamma_1(a'),\\cdots,a\\gamma_n(a')) ", "db4233ec870a70f74f3f0161f379c620": "\\textstyle \\delta_\\psi", "db4277554b8232b402aa6e8071b57132": "\\mathbf{x}_w^{(k-1)}", "db4296dcd950d4ee4d1a676d8316ea8a": "\nf(\\lambda A + (1-\\lambda)B) \\leq \\lambda f(A) + (1 -\\lambda)f(B) .\n", "db42ddab4f50fe6629a02f6f785d4782": " (W(t): t \\geq 0) ", "db42f5774b12742c8a9d0a0b138bd4a4": "\\langle\\mathcal{S}_{i_n = p}, \\mathcal{S}_{i_n = q}\\rangle = 0", "db43022b83ec99f89f78f505e028c77c": " ds^2_R = \\frac{1}{T} ds^2_W \\, ", "db431286613f96ca50e6fb222cb76ad1": "\\begin{alignat}{5}\n x &&\\; - \\;&& 2y &&\\; = \\;&& -1 & \\\\\n 3x &&\\; + \\;&& 5y &&\\; = \\;&& 8 & \\\\\n 4x &&\\; + \\;&& 3y &&\\; = \\;&& 7 &\n\\end{alignat}", "db431365d55e0a60e6742caa27cc3ddf": "\\sigma:X\\to X", "db434248ff916950d40034e2d6304cff": "\\mathcal{L}_{EW} = \\mathcal{L}_K + \\mathcal{L}_N + \\mathcal{L}_C + \\mathcal{L}_H + \\mathcal{L}_{HV} + \\mathcal{L}_{WWV} + \\mathcal{L}_{WWVV} + \\mathcal{L}_Y", "db435a7e73958d923aedff7f482f540d": "v\\in T^{1,0}M", "db4374f12ae57ba5d8da898191c58017": "=\\frac{1}{3}\\cdot\\left(\\frac{7}{4}\\right)", "db4376b089220c110b6eafa432438926": "{GE} = {{GE}_p + {GE}_f + {GE}_{cho}} \\,", "db439cd0244d31b2a4b4c0f7538307a4": "r \\times s", "db43f9fbf72c70fee82b69acfda3db13": " H(j \\omega) \\ ", "db440bb0fcd5fff45d5abf446479fa95": "G^\\psi (R,S,T,m)=\\left( \\frac \\partial {\\partial T}+\\frac{T}{2R}\\frac \\partial {\\partial\nR}-mS\\right) G^F(R,m).", "db442d6a549dbba112b11d7e2c414abf": " (x, y) = (\\{0\\} \\times s(x)) \\cup (\\{1\\} \\times s(y))", "db4459f9e65a381278ef513706dbd984": "\n\\phi = \\frac{1+\\sqrt{5}}{2}= 1.618\\dots\n", "db4462f38370812ca2ffbd5ac57e9434": "E[g_i y_i]=E[g_i(f_i+\\epsilon)] = E[g_i f_i + g_i \\epsilon] = E[g_i f_i] + 0", "db4481b1205a51ba4e77199577a9b3bd": "\\begin{align} \n\\ln \\,\\operatorname{var_{GX}} &= \\operatorname{E} \\left [(\\ln X - \\ln G_X)^2 \\right ] \\\\\n&= \\operatorname{E}[(\\ln X - \\operatorname{E}\\left [\\ln X])^2 \\right] \\\\\n&= \\operatorname{E}\\left[(\\ln X)^2 \\right] - (\\operatorname{E}[\\ln X])^2\\\\\n&= \\operatorname{var}[\\ln X]\n\\end{align}", "db448443b03977cb4d57a341f226c065": "\\tau_0=1", "db44989cb3b83bb387aedb1f12738a00": "S_0", "db44fe0773ebd30940c2e0345ac625dc": "P = {2 \\over 27} \\rho_a A C_L({C_L \\over C_D})^2 V^3 ", "db4501d2c93c3edc023723b80fd57ec7": "\\left[E_k(\\mathbf{R})+\\mathcal{T}_\\mathrm{k}(\\mathbf{R})\\right]", "db45271b9a3f73ca496dd7d99e61bf95": "n\\times\\log_2(n)", "db452d0b8306b2c2bdbb6033ab8bf38b": "\\Tau \\, \\tau \\,", "db454df8ebe44fe65d355bae9f7d6b5b": " \\zeta(k) \\!", "db4559eb6d944f92a83ae0fe0fa5c7fc": "n_2=1.5", "db45c89243f0851eb207b16f6b8d4861": " \\frac{0.25n^2}{n(n-1)}. ", "db45cb8adaec5e1dfe1d4677808cb40e": "\\oint {\\chi(\\omega') \\over \\omega'-\\omega}\\,d\\omega' = \\mathcal{P} \\!\\!\\!\\int \\limits_{-\\infty}^\\infty {\\chi(\\omega') \\over \\omega'-\\omega}\\,d\\omega' - i \\pi \\chi(\\omega) = 0.", "db45cf894a0e340923840ef4a56d0559": " \\mathfrak{g} \\geq [\\mathfrak{g},\\mathfrak{g}] \\geq [[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]] \\geq [[[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]],[[\\mathfrak{g},\\mathfrak{g}],[\\mathfrak{g},\\mathfrak{g}]]] \\geq ...", "db45ec8486cbe8b6a8df1a6601d459d3": " {\\rm Compression\\;Ratio} = \\frac{\\rm Uncompressed\\;Data\\;Rate}{\\rm Compressed\\;Data\\;Rate}", "db460b0278dfa6d5345052330464fac2": "L^1(\\mu_1)", "db4623821744cdfe8bf9e3af98e5e116": "O(\\log\nN)", "db462473195ee8c5ca87f79c45b5f379": "M_X^c", "db462620fa253a6ca753bb97a92b973c": "p > 2^x", "db4663e7fee821c999a3f221dce71205": "\\{p_2,p_4\\}", "db4681a405b3d0ef0aabb80e34d53f17": "e = 33", "db46e4f08cf3b46a164e11e37114b260": "0.2/0.2=1.0", "db470095a987c362aa59c5a1a6f408f5": "\\operatorname{Var}\\left(\\sum_{i=1}^N X_i\\right)=\\sum_{i=1}^N\\operatorname{Var}(X_i).", "db47018a4dfcfbe22a527f04a830b892": "z_{i,j}=f(y_{i,j})", "db4745d51bb4a5304cf81eb7d9814982": "D_s", "db4763d1295de5aba49569523a0bfbb2": "A(\\rho') = \\exp { \\left [ -\\frac {\\rho'} {\\sigma} \\right]^2}", "db47b7e3b80f2f06455663bdc41a1594": "\\textstyle a=\\frac {GA} {M}=\\frac {G} {W_S} ", "db47cbe04220f261f11e243a0abf4afb": "\\Delta v = v_\\text{e} \\ln \\frac {m_0} {m_1}", "db47e53bd2a91bb0d699915a869fe7cd": "\\left(\\textstyle{\\binom{n}{2}}=\\frac{n^2-n}{2}\\right)", "db47e8a7d535354a6613893b68548e75": "L = 20\\ \\log_{10}\\left(\\frac{4\\pi d}{\\lambda}\\right) ", "db47fc71fa00fd079b8131c5742a5aed": "1_N = [1, \\ldots ,1]^T", "db48c05fc359a0b601b2145659fb68aa": " y(t_0) = y_0. ", "db48cdd6fd9d6d477f6ceef91a6d78ba": " W(\\mathbf{E}, \\omega)", "db490ada8165ed8474dc0e30eefe350b": " (1-{1 \\over q})n, |C| \\le {qd \\over {qd -(q-1)n}} ", "db491de1e1abcae5ad7bea87edad4733": "~\\Phi_{2p}(x) = 1-x+x^2-\\cdots+x^{p-1}=\\sum_{i=0}^{p-1} (-x)^i.", "db494004f77447fc55a0c885ec9fc97d": "[X_\\mu,Y_\\lambda] = \\delta_{\\mu\\lambda}H_\\mu,", "db494df94e0dfdc0ae11657561f75358": "[p(t_{1}), p(t_{2})]=i\\hbar m\\omega\\sin(\\omega t_{2}-\\omega t_{1}) ", "db49860c890ffd29a3ac7f2030f87a61": "\\tau=3.03", "db498b7243ae6313688aa2e09cbdfb90": "[1,7077888]", "db49a7de632dac0ffc0ad4417a5ce782": "\\phi_{h,v} = \\eta'_{h,v} - \\frac{1}{2}\\beta'_{h,v}\\eta_{h,v}/\\beta_{h,v}", "db49bdd5adada659cdc343209b61fb89": "P(x)=a_nx^n + a_{n-1}x^{n-1} +\\cdots + a_1 x+ a_0 \\, ", "db4a5bdf786722bbc158129f075284f5": "\\theta\\, ", "db4a633162d725818b10c2739a5c7e02": "Y_r", "db4a7b8d3069e5305965247f00e6c0f6": "\\textstyle \\leq n-1", "db4b066d8fe4020979f1b6201f491e19": "\\bar\\omega", "db4b8c931dd3fe14f441fcb162a6f3fd": "\\mu-\\beta\\log \\left(\\frac{e^{-X}}{1-e^{-X}} \\right) \\sim \\mathrm{Logistic}(\\mu,\\beta). ", "db4c1deb2c791cc62a5947c9db680a41": "\n \\operatorname{E}[S_t] = (1-\\alpha_t) S_0 + \\alpha_t\\left[\\delta_t S_B + (1 - \\delta_t) S_G\\right] \\;.\n ", "db4c4e5627b5c5c734a43f66ff8b21c5": "r_2 = \\frac{k_{22}}{k_{21}} \\,", "db4c781576bb39235c00caa218239706": "x^2 - 4x + 1 = 0.\\ ", "db4c79af1c17cb5d644dab2ccc4dc57a": "Y' = 0.299 \\times R + 0.587 \\times G + 0.114 \\times B", "db4cbf1d787c88dbd3bfcc3c3421130e": "s_w<\\frac{I}{Y} 0, \\alpha", "db57d27652715f467aa804f44cd7e65b": "\\{ v_1 \\} \\subset", "db57f54938d22b63cc26ce09b5add251": "T_n(-1) = (-1)^n\\,", "db580f5a6e02ebaa9a82b1e8c0a5c2b2": "|\\nu_b\\rangle", "db58990a4929832c0ef1ad9b04ea33b2": "\\sum_{n=0}^\\infty z^n", "db58b1144e25d552ab7e78bc546a9266": "T \\hat{\\mathbf{x}} T^\\dagger = \\hat{\\mathbf{x}} ", "db58f49bca9927e6a548c91c5978222b": "n_\\mathrm{s}", "db5966565b121b95c27dcd53149be442": "x_1^{\\alpha_1}x_2^{\\alpha_2} \\cdots x_k^{\\alpha_k}", "db59686310f19faa544c31d26a90b258": "m \\ddot{y}(t) = u(t) - k_1\\dot{y}(t) - k_2 y(t)", "db598b473aefc4cb392578e9bdad8d1d": "\\kappa, \\tilde{\\kappa}", "db5a1ec6a9fe8548e2fef23354060c3c": "(f - g)' = f' - g'.\\,", "db5acf694afee787abaea7727d9a314f": "\\varphi(y_1)\\ldots\\varphi(y_n)", "db5ad23c37a5fbfc4bac7c1632886554": "\\Rightarrow ad=2\\varphi ac", "db5ae835839b3957535e0393ec7263ae": "\\dot{\\theta} = 2[ \\cos^2(\\theta/2)\\tan^2(\\theta/2) + \\cos^2(\\theta/2)\\Delta I ] = 2[ \\sin^2(\\theta/2) + \\cos^2(\\theta/2)\\Delta I ]", "db5b03412096cc319683894f69bca2d4": " y", "db5b41be1cb2c3c29675c3b16c8b3bb7": " \\boldsymbol{\\nabla} = \\sigma^k \\partial_k", "db5b447c21c6eb3749c9a1f0b010e8c1": "\n \\hat\\delta = \\Big(Z'(I-\\kappa M)Z\\Big)^{\\!-1}Z'(I-\\kappa M)y,\n ", "db5bb25d9613224096142626ddcb7b4f": "\\frac {I}{I_S}+1=e^{V_D/(nV_T)}", "db5bb32c543a6121ec61a989bda2ae6e": "\n \\alpha_1 := \\tfrac{1}{2} ~;~~ \\alpha_2 := \\tfrac{1}{20} ~;~~ \\alpha_3 := \\tfrac{11}{1050} ~;~~ \\alpha_4 := \\tfrac{19}{7000} ~;~~ \\alpha_5 := \\tfrac{519}{673750}.\n ", "db5bb6a5d899a261aacfc11fd17f3f08": "P_{\\aleph_1}", "db5c53445427603c6186800214029a61": "r=\\frac{L^2}{8v}", "db5ce4ef0d6eb99aa01f854ee5f5bce8": "\\sum\\limits_{i=1}^\\infty a_{\\sigma(i)}=A", "db5d1eca3382078904beea1325202c87": "\\mathrm{sign}(M) = \\int_M L(p_1,\\ldots,p_l) ", "db5d4960f5822232f5b7f7cb781d7964": "d \\Xi = d \\Phi - \\frac{T (P d V + V d P) - P V d T}{T^2}", "db5d63d91a19acabed7035b3a37b9444": " -1<\\gamma<\\alpha", "db5d90967648787037b0f9b792454179": "\n \\begin{align} \n N_{\\alpha\\beta,\\alpha} & = 0 \\\\\n M_{\\alpha\\beta,\\alpha\\beta} - q & = 0 \n \\end{align} \n", "db5daf7ff8c8469e599c741ab4aa4a16": "\n\\Psi = \\psi_{11} P_3 - \\psi_{12} P_3 \\mathbf{e}_1 + \\psi_{21} \\mathbf{e}_1 P_3 +\n \\psi_{22} \\bar{P}_3, \n", "db5de93753e2d57b4aedfa57ebf6391d": "\n\\,Y = \\cosh^{2} \\xi - \\cos^{2} \\eta\n", "db5e5d77c8cfc671288f961511611fa3": "Q^*_j = \\sum_{i=1}^n \\mathbf {F}^*_{i} \\cdot \\frac {\\partial \\mathbf {V}_i} {\\partial \\dot{q}_j},\\quad j=1,\\ldots, m.", "db5eb29ad9185df882180802a4c668e9": "{}^1\\!D= \\exp\\left(-\\sum_{i=1}^R p_i \\ln p_i\\right) =\\exp(H')", "db5ee3f0ff1dd29d0ee8947d906f572d": "(\\sum\\limits_{i=1}^{n} \\epsilon_i)", "db5eef1e6b583a43260c43493c574b64": "m \\equiv c^d \\pmod{n}.", "db5ef9b20be996010a9d0c38f1945800": "\n\\frac{d^2 W}{d\\Omega }=\\int_{0}^{\\infty}\\frac{d^3W}{d\\omega d\\Omega }d\\omega\n=\\frac{7e^2 \\gamma^5}{64\\pi\\varepsilon_0\\rho}\\frac{1}{(1+\\gamma^2\\theta^2)^{5/2}}\\left [1+\\frac{5}{7}\\frac{\\gamma^2\\theta^2}{1+\\gamma^2\\theta^2} \\right ] \\qquad (12) ", "db5f10fad97fd2313624204327686d86": " p = \\frac{RT}{V_\\mathrm{m}-b}-\\frac{a}{V_\\mathrm{m}^2} \\Rightarrow\n\\left(p + \\frac{a}{V_\\mathrm{m}^2}\\right)(V_\\mathrm{m}-b) = RT.\n", "db5f16953a147e54bf4ec8a4783ffb64": "\\nabla \\cdot \\mathbf{\\xi} = 0", "db5fb951d147405f0af3ea872afd6b67": "\n\\begin{align}\n\\mathcal{F} \\{ \\exp (-x^2/2 + 2xt-t^2)\\}(k) & {} =\n\\frac{1}{\\sqrt{2 \\pi}}\\int_{-\\infty}^\\infty \\exp (-ixk)\\exp (-x^2/2 + 2xt-t^2)\\, \\mathrm{d}x \\\\\n & {} = \\exp (-k^2/2 - 2kit+t^2) \\\\\n & {} = \\sum_{n=0}^\\infty \\exp (-k^2/2) H_n(k) \\frac {(-it)^n}{n!}.\n\\end{align}\n", "db6016f7d81024074f71d1d0e721b2ab": "\\mathrm{\\ \\xrightarrow{MGMT} }", "db6048484db54d6f9592f7e29fac77f7": "\\vec{r}_u\\times\\vec{r}_v", "db60e9bd01442676d86b1ad95b73d18b": " \\widehat{\\boldsymbol{\\beta}}_{L^{*}} ", "db610ff5e41db3e31a3741c6303674dc": "k \\log n", "db6129879d9231ee6d7eb402c1c2faed": "|\\psi(t)\\rangle = \\sum_i \\, | \\varepsilon_i \\rangle \\langle \\varepsilon_i | \\psi(t)\\rangle = \\sum_i c_i(t) | \\varepsilon \\rangle ", "db612e9ec9ce75fafbd0877f767212a8": "\\{ \\varepsilon_\\beta, \\beta < \\alpha \\}", "db61f48d89348c8cd4659c8d08d71a35": "H_\\mbox{b}", "db62626cd8c9a6f0c3968d33980769e4": "\nM_c = \\lambda_1 \\lambda_2 - \\kappa \\, (\\lambda_1 + \\lambda_2)^2\n= \\operatorname{det}(A) - \\kappa \\, \\operatorname{trace}^2(A)\n", "db629ac711f263ae78bf5029ed371548": " \\tau_k \\to \\infty ", "db630bd381f57c37e2d5affe260b1164": "=\\quad - (0.6 \\log 0.6 + 0.4 \\log 0.4)", "db633c5f3d5c0241b90d5e36b51518f9": "0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A.", "db63aa50dd0b552b436e0c34074d916c": " r = \\sqrt{x^2 + y^2} \\, ", "db63d6b07cb30e0430b95700f3f5776a": "2 \\lfloor \\log_2(x) \\rfloor + 1", "db640dd4afc82b0e304782aaf8f3aa5b": "\\scriptstyle 1+\\lceil n/2\\rceil", "db6411238f250f4e52c4e2b95daa7ce2": "K(x,x_i ) = \\exp \\left( { - \\left\\| {x - x_i } \\right\\|^2 /\\sigma ^2 } \\right),", "db6452550a9985fa49400d3344070676": "\\mathbb{Z}^m \\rightarrow \\mathbb{Z}^n", "db64c4ed681f3c8fd11f22e006377fbf": " dA_{\\bold{x}}^T \\, dA_{\\bold{x}} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1+d\\theta^2 & 0 \\\\ 0 & 0 & 1+d\\theta^2 \\end{bmatrix} , ", "db65127f743296005874b34ab7a4bec0": "j=k=\\ell=0", "db65155e8d03000f01fcd25e664a7d14": " \\cos \\theta =\\!", "db652611856ffff598bd4d183bd75ac1": "\\frac{3 k_B T}{L_c b}", "db66c581a23b4856a999637ad9be3766": "\\frac{\\Gamma(n+\\alpha+1)}{n!}\\,", "db670c6fb24db40eb7aec2d1fa2ddd34": " \\frac{(u/v)'}{u/v} = \\frac{(u'v - uv')/v^{2}}{u/v} = \\frac{u'}{u} - \\frac{v'}{v} ,\\! ", "db67cf6734971eaa35ac8c18e89e08e6": "\\textstyle E = E_i + E_p - E_t", "db68045ac641af891b273fe2482fc2d2": " \\xi_\\rightarrow \\circ \\pi_i = \\mathcal{P}(\\Lambda \\times \\pi_i) \\circ \\gamma ", "db685d5f4999626d9858d990b31aa50d": "a(y_0 -n) - b(x_0 - m) = 0,", "db686d4b8603c222345b51e326f27062": "p_1(x)=x;", "db68e0bee47aa56a50a7a19c10be74e4": " \\chi_w(k,n)= 2^{-1/2} \\bigl( \\chi_w(2k,n+1)+\\chi_w(2k+1,n+1) \\bigr)", "db69421b916ee3f66bed3584fcfa1b58": "N/M", "db69691b3a4a8d4f4924046770f230ca": "|x|\\leq (E/q^{2})^{1/\\beta }", "db69888a7c97fb0e0aee82debfd04f3c": "\\tilde{x}_1, ..., \\tilde{x}_n", "db699aadd228cdfd93c2c83c56f75b18": " V_t = \\sqrt{\\frac {\\mu}{p}} \\cdot (1 + e \\cdot \\cos \\theta).", "db69ed281cc358e3063e964db46d2549": " x_1, x_2, x_3 ", "db6a463cd3aa99734ee9c772993a7f5e": "L=\\Sigma \\partial_x+\\alpha I\\partial_y,\\qquad (3a)", "db6a7b7e50759382532662b184d170b7": " \\frac{E_{t+n}y_{t+n}} {1-L^{-1}R^{-1}} = \\sum_{j=n}^{\\infty}\\left( \\frac{1}{R}\\right) ^{j-n} E_{t+n} y_{t+j} ", "db6aac55a1922dee3731cd31cf4c5b74": "\\chi_\\mathrm{red}^2 = 1", "db6add363730183e91bcf452f75ce482": "A\\left(x^\\prime\\cos \\theta\\ -\\ y^\\prime\\sin \\theta\\right)^2\\ +\\ B\\left(x^\\prime\\cos \\theta\\ -\\ y^\\prime\\sin \\theta\\right)\\left(x^\\prime\\sin \\theta\\ + y^\\prime\\cos \\theta\\right)\\ +\\ C\\left(x^\\prime\\sin \\theta\\ +\\ y^\\prime\\cos \\theta\\right)^2", "db6b3ea6baa4cfc1a45a794a54263a52": "R>0\\!", "db6b53504dbdd142bc4d8fcfb7cd552a": "C=\\left(\\frac{\\mathrm{NA}_R}{\\mathrm{NA}_I}\\right)^2", "db6ba72e858a0e8e95fef1cc7dc9207e": " \\lim_{x\\rightarrow x_{0^-}}\\!\\!\\!u(x) \\neq \\!\\!\\!\\lim_{x\\rightarrow x_{0^+}}\\!\\!\\!u(x) ", "db6bad4725ddcfb9e69b2cd41303171e": "\nEI~\\cfrac{\\mathrm{d}^4 w(x)}{\\mathrm{d} x^4} = q(x)\n", "db6bb46dc6202d6a64a19281d3fb0422": "\n \\boldsymbol{\\xi}(\\boldsymbol{\\alpha},t_0)\\, =\\, \\boldsymbol{\\alpha}.\n", "db6bd47285de7fcdeec2125e5ea24ccb": "\\text{fmap} \\colon (A \\to B) \\to (A^{*} \\to B^{*}) = f \\mapsto l \\mapsto \\begin{cases} \\text{nil} & \\text{if} \\ l = \\text{nil}\\\\ \\text{cons} \\, (f \\, a) (\\text{fmap} f \\, l') & \\text{if} \\ l = \\text{cons} \\, a \\, l' \\end{cases}", "db6c37ddfe2ea577d98f40b27e83c482": " w = \\frac{1}{x_0} - \\frac{ v_0^2 }{ 2 \\mu } \\quad \\text{and} \\quad p = \\left( \\frac{9}{2} \\mu t^2 \\right)^{ \\frac{1}{3} } ", "db6c586d61891e55b4861593038745d5": "\\mathit{q}_{[i,j]}", "db6c9cf3be6d5f929022f9ba903cad25": " R_\\mathrm{out} = g_{22}= \\begin{matrix} \\frac{v_{out}}{i_{out}}\\end{matrix} \\Big|_{v_{in}=0} ", "db6cf64a87eb8df67765605401ab3edb": "\\frac{1}{c^2} \\frac{\\partial^2 \\varphi}{\\partial t^2}-\\nabla^2 \\varphi = 0,", "db6d03aeb94c323707d8b55ea3aa0334": "\\operatorname{Hom}(\\mathcal{T},\\mathcal{F})=0", "db6d2098130ab050955ac8efa7b4cade": "E^\\circ", "db6d8a63b554b5c2a2d4983e7c02e8be": "w(e), W(u,v), D(u,v)", "db6da05811d410a7b62d003a70eaf72e": "f_0:H_0(K)\\to H_0(L)", "db6ddbc41ae3083a39cfd44b97700d3b": "\nL\\left( q, \\dot{q} \\right)\n= {1\\over 2} m {\\dot{q}}^2 - V \\left( q \\right)\n", "db6dfdfe21707066f35e1feba9d0819d": "\\rho = \\frac{(G_s+Se)\\rho_w}{1+e}", "db6e18191eef268eaa9402a6baeacb50": "R=\n \\begin{matrix}\n \\texttt{name} & \\texttt{age} \\\\\n \\texttt{A} & \\texttt{34} \\\\\n \\texttt{B} & \\texttt{47} \\\\\n \\end{matrix}\\qquad\n S=\n \\begin{matrix}\n \\texttt{name} & \\texttt{income} \\\\\n \\texttt{A} & \\texttt{20'000} \\\\\n \\texttt{B} & \\texttt{32'000} \\\\\n \\end{matrix}\\,", "db6e24b1cc8639a90ecf44c31c11792e": "\\mathcal{F}_{\\omega}", "db6e4e457fd67041b8cf15ebbe0dc473": "x_e \\in Q", "db6e597dee4584abb8f090aff5e2f058": "\\inf_{t\\in\\R}\\{t:\\text{Pr}(X\\leq t)=1\\} ", "db6ecbd6a99ce5850f71469bf6287a31": "m_{\\nu}", "db6eda0a45766120e01976fd168423dd": " NRh = \\rho r_h ", "db6f094dab94f6548bde8d90135aaf37": "00} na_n a_{-n},", "db6fcb66ea359fbd4c55a880fa8f5617": "\\phi(0)<\\infty", "db7010d005a6f4cabc1bbed241d32c60": "Z_{21} = {V_2 \\over I_1 } \\bigg|_{I_2 = 0} \\qquad Z_{22} = {V_2 \\over I_2 } \\bigg|_{I_1 = 0}", "db7031feb525b2c1874de7641824e05e": "H^{0,0} \\oplus H^{0,0} \\cong H^{1,1} \\oplus H^{1,1}", "db704a619cd227c36cbd92c2d846bf3c": "\\text{Length} = \\int^{y_2}_{y_1} \\sqrt{ g_{\\alpha \\beta} \\frac{d x^{\\alpha}}{d y} \\frac{d x^{\\beta}}{d y} } \\, d y \\,", "db7064574cdffb19022043c610c6c2ea": "(X, T(X))", "db706c571bbd75bdbe11e0438d498e1e": "\n | F( x ) - f ( x ) | < \\varepsilon\n", "db7070387a3b9d85c5e6ab3c1ac36a0e": "z\\mapsto z-\\tfrac{p(z)}{p'(z)}", "db709b51681e2e420feed7ca5ef32172": "\\arg{\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)}", "db70facee9bef6666231e8e936413fc3": "\\|\\mathbf{A + B}\\| \\le \\| \\mathbf{A}\\| + \\|\\mathbf{B}\\| ", "db7196f30b3ab9761b8f856ff7954769": " \\frac{C_V}{Nk} \\sim 9 \\left({T\\over T_D}\\right)^3\\int_0^{T_D/T} {x^4 \\over x^2}\\, dx ", "db71c86153290d91439936bfd949c55a": "((ab\\,\\bmod\\,n)\\,(b^{-1}\\,\\bmod\\,n))\\,\\bmod\\,n = a\\,\\bmod\\,n", "db71c9e1eee84cb6804128d7463b9f1b": "b0.", "db7b303cd508a1110dcfc20d7cdc82b0": "2^{30}", "db7b347a637248fb9484ca4acfe118e6": "\\mathbf{L}=\\sum_i ( \\mathbf{r}_i\\times m_i \\mathbf{v}_i )", "db7b57ceb4fef7805bf10a04046a1a9a": "e^{ix}", "db7b63b2a2d9fc1519ce2d7f6bfc7975": "f_\\star:\\mathcal{P}(X)\\rightarrow\\mathcal{P}(Y)", "db7b660106977968c1ae68101cccd111": " f(r) \\not\\equiv 0 \\,\\bmod{p^{k+1}},", "db7b6b081e8b12dda071cfa911bd1603": "\\begin{matrix} {13 \\choose 1}{12 \\choose 2}{2 \\choose 1} \\end{matrix}", "db7b921f6f4db6a98ca3595821d82590": "\n\\frac {N^2} {t} = \\frac {L \\cdot S} {K} \\,,\n", "db7c18341cfec86932ead53bf3f34dcd": " \\phi(\\tilde{\\nu}- \\tilde{\\nu}_{0}) ", "db7c1970c141005693a3c8663d9f06d9": " x\\ f\\ y = f\\ (y\\ y) \\and q\\ x\\ f = f\\ ((x\\ f)\\ (x\\ f)) ", "db7c41c9abc6579047d161dfe86a1f00": "\\tau_q(r) = \\sum_{a^{q-1}=1}a^{-r}\\zeta_\\pi^{\\text{Tr}(a)}", "db7c4401f1e6569402f82e8fb8179ba3": "\\vec{B} = B_{\\theta}(r)\\hat{\\theta}", "db7cc6daeb8d834255c187c3a10726bb": " \\|V\\|_{F} ", "db7cf708833161c7fb144b135a26e3aa": "\\left ( \\frac{Number\\ of\\ employees\\ who\\ left\\ during\\ the\\ year} {(Number\\ of\\ employees\\ at\\ the\\ beginning\\ of\\ the\\ year + Number\\ of\\ employees\\ at\\ the\\ end\\ of\\ the\\ year)/2} \\right ) \\times 100", "db7cfb7e907239f0a4782686bff6d0c3": " U U^\\dagger = I=U^\\dagger U", "db7d1fce6a07ba92dc4d1f3cb93efdc1": " p_n \\# ", "db7d7a39084ee3015f67c07f20adfce5": "(a+b)+c = a+(b+c)", "db7d9bff94f7613192171eea57e7a78d": "2\\pi\\,", "db7da25c519ab571f8d1e7e983680e56": "a = (8\\cdot7+67\\cdot1)/3 = 41, b = (8 + 1\\cdot7)/3 = 5, k = (7^2-67)/(-3) = 6", "db7db0bed99e348e7a7c7a22a49e5226": " M = \\sum_{i=1}^j\\ f_i .", "db7dba6f5e61923d3af3b987ed429196": " f(x)=x_1^{n_1}x_2^{n_2}\\cdots x_r^{n_r}", "db7e518b8896b40dd5813a7ee346288c": "\\mathbf{y} = \\mathbf{w}^T \\mathbf{x}", "db7ea2ffe1f3caa18ece22a798b576db": "\\frac{s^3}{12}(3\\sqrt2 - 49\\pi + 162\\tan^{-1}\\sqrt2)\\approx 0.422s^3", "db7eca34fdca35b08182379f4cc14435": "\ng_0 = 0 \\quad g_1 = {\\textstyle\\frac{1}{4}} \\quad g_2 = {\\textstyle\\frac{1}{3}} \\quad \ng_3 = {\\textstyle\\frac{3}{8}} \\;\\dots\n", "db7ece94ecc6b641899a3005abf4fe7b": "\\hat{H}_{I}", "db7ecfcf9f3165546069fd62b4e845bf": "D_r = \\lbrace t + x r : t, x \\in R \\rbrace ", "db7f0700214360f076d8c530205dae86": "\n \\det(\\lambda~\\boldsymbol{\\mathit{1}} + \\boldsymbol{A}) = \n \\lambda^3 + I_1(\\boldsymbol{A})~\\lambda^2 + I_2(\\boldsymbol{A})~\\lambda + I_3(\\boldsymbol{A}) ~.\n", "db7f133b104d6bb93aa9f096a1be53df": "\\hat{P}^{i_1\\ldots i_q}_{\\,j_1\\ldots j_p} =\n(-1)^A A^{i_1} {}_{k_1}\\cdots A^{i_q} {}_{k_q}\nB^{l_1} {}_{j_1}\\cdots B^{l_p} {}_{j_p}\nP^{k_1\\ldots k_q}_{l_1\\ldots l_p}", "db7f62729fb4177113d4c4b9c36bfaf9": "\\Omega^k(\\mathbf{R}^n)", "db7f6953127af41d9aa7fb9ae6be3831": "\\delta \\psi = v_r ( r \\delta \\theta ),\\,", "db800154117b534e8af7696c35855d26": "\\lambda \\in V", "db8011687a47b5e6e0611f159acaa5ef": "N_j^n", "db8039fbac802880d1d4f4e37111850a": "A_{\\delta} \\subseteq A", "db80862940110bfba8967caf278698ba": "|a_{ii}| \\geq \\sum_{j\\neq i} |a_{ij}| \\quad\\text{for all } i, \\,", "db80e85b89a70d088df1a90b83d44563": "P_\\mathrm{error} \\le \\exp(-n(E_o(\\rho,Q) - \\rho R)). ", "db811b4c8ad6f76e7b728b6590e5cd33": "M(0,0,c) \\to e^{ihc}", "db8125a0cef203870f998f87119ee48c": "\\displaystyle H(Y) = - \\sum_{ij} \\mu_i P_{ij} \\log P_{ij}", "db8139c123afb73150da2f710d4edb8e": "\\frac{\\mathrm{d}}{\\mathrm{d}t}(mr^2\\dot{\\theta}) -mr^2\\sin\\theta\\cos\\theta \\, \\dot{\\varphi}^2=0,", "db81582de6c3c55dcbaa0b68065e905d": "s \\leftarrow p,\\ \\hbox{not } q", "db817fe0d906346a2669ca867b3c2d80": "\nh_{\\mu} = h_{\\nu} = a\\sqrt{\\frac{1}{2} (\\cosh2\\mu - \\cos2\\nu}).\n", "db81c4ae3c9009481b5d84c37de6bdae": "\\partial_i(ab)=a\\partial_i(b)+\\partial_i(a)(g_i.b) ", "db81e4bb24b1845678c7b9dbb19508d6": "1-e^{- T}", "db8204fcc6716afff7ddd4e339e0df74": "\\scriptstyle\\frac{b\\,\\,a}{d\\,\\,c} = \\frac{a}{c} + \\frac{b}{cd}", "db825dad8140e6e5d0c4c1d1137d6203": "x_{n} = \\frac{2}{2-pf_{av}}", "db829263f0e40b8f98ac0976115658c3": " \n\\frac{1}{\\log (R/r)} \\log \\frac{M}{|f(z_0)|}.\n", "db82b9a99b106f1f1e34a17790201327": "T(n) = \\sum_{k=1}^n \\frac{\\lambda(k)}{k}.", "db82fb5d0d407d4f3d515aeb11f87b2c": "\\operatorname{Im}\\left(\\tau\\right)>0", "db83064cf18c980153a1008a7ce497d6": "i\\ll_{m} j", "db8350f1b7821fc7eaa7d1c71e0695db": "SR=\\frac{Annual\\ Portfolio\\ Return}{Average\\ Largest\\ Drawdown + 10%}", "db83ae3a246d90cbfac5420a2e5885d2": "1/(1-1/n) = n/(n-1),", "db83c9179a9d2824b9783a97e6a1a847": "\\rho_{[x,y]} = [\\rho_x,\\rho_y] = \\rho_x\\rho_y - \\rho_y\\rho_x\\,", "db83d2d1588676c7642dfade2a3debbf": "u(x,0) = f(x) \\quad \\forall x \\in [0,L]", "db83d8a4d94d73da40ec2c7644657e50": "\\delta_{x_0}(A)=\\begin{cases}\n1 &\\rm{if\\ }x_0\\in A\\\\\n0 &\\rm{if\\ }x_0\\notin A\n\\end{cases}", "db841d27d36ffa7375adc176388fcc71": "\\mathrm{Rot}_G (v,i)=(w,j)", "db844563fd808ee135c03504ea84a905": "g \\;", "db8448a621d8f89ba81b70bbf96b9b8f": " E_{lh} ", "db846dc9699167e7db6ff18eafb17df2": "\n\\alpha_i = \\frac{4 \\pi \\nu}{c} \\mathrm{Imag}\\, n_i\n", "db8486bb915f12e4e2f3e2911b7fffbd": "f(x_2)=27\\,", "db849b6f6ddb8f8d677820829890540d": " f'(t) \\ ", "db84a2fd6d10bac443ae21d2ef4d27f4": "A'(x) + A(x)^2 - B(x)^2 = \\frac{2m}{\\hbar^2} \\left( V(x) - E \\right),", "db84c45145d510949d9d1cfc37089cb2": " p(x)=-1\\ ", "db84ddc5293fad0f9369d0fc106ce727": "\\frac{g}{2\\pi} T^2", "db857e669b39dc01d6a92404e683692d": "L(t)=F(t)+xF^\\prime(t). \\, ", "db85aa8250fcf8056001853461f37046": "\\rho (\\mathbf{x},t) = |\\psi(\\mathbf{x},t)|^2", "db85cac780c3400956d093c9bfb66235": "\\lang xa, y\\rang = \\lang x, y a^t\\rang.", "db85cf9d07e72aa17453c712511ef6b6": "\\gamma_l^{(t+1)} = \\frac{e'A(l)W_l^{(t+1)}}{m}", "db862dfaf5ac4e0f923890e50a712554": "6(2\\pi^2)^{2/3}", "db86a820185492a0a21dd4936ae63ba4": " \\int_{-\\infty}^\\infty G_x(t,f) e^{j2\\pi tf}\\,df = x(t) ", "db86cd968a55f1c0c21a1d0ce58a97bc": "\\begin{align}\nr\\cos\\varphi &= a t + b \\\\\nr\\sin\\varphi &= c t + d\n\\end{align}", "db8712cd2215d21612d68fbcf8a5e307": "\\begin{align}\n \\mathbf{\\nabla\\cdot E} &= \\frac{\\rho}\\epsilon_0\\\\\n \\mathbf{\\nabla\\times E} &= 0\n\\end{align} ", "db8732a7458e218701453a1ec5d1efcf": "\n\\begin{bmatrix}\nX\\\\\nY \\\\\nZ\n\\end{bmatrix}\n\n=\n\n\\begin{bmatrix}\n0.9525523959 & 0.0000000000 & 0.0000936786\\\\\n0.3439664498 & 0.7281660966 & -0.0721325464\\\\\n0.0000000000 & 0.0000000000 & 1.0088251844\n\\end{bmatrix}\n\\begin{bmatrix}\nR\\\\\nG\\\\\nB\n\\end{bmatrix}\n", "db874cb58286011b2fe316b8f0b7f2d2": "{48 \\choose 4} = 194,580", "db8757154e9e5e0ffbb1291a434f3493": "H = \\log S^n \\,", "db876a02b1206380b2fc5f745dc28157": "r_c=\\left(\\frac{2\\pi D_\\odot ^2}{\\mu_0 v \\dot{M}}\\right)^{1 \\over 4}", "db8777ded596852b5888ee47a420c91c": "^{e}\\log", "db878897ca7885acd2d988107583dca9": "\\textstyle Real\\,Net\\,Output\\,Ratio = \\frac{internal\\,production}{total\\,production\\,value} = \\frac{internal\\,production}{internal\\,production\\,+\\,externally\\,produced\\,goods\\,+\\,externally\\,produced\\,services}", "db87bcc53116f8d250a07e13fbbc4174": "\\partial_\\mu(\\partial^\\mu A^\\nu - \\partial^\\nu A^\\mu)+\\left(\\frac{mc}{\\hbar}\\right)^2 A^\\nu=0", "db87debb1542513cfe270072d90f7405": "\\sum_{i\\in I} x_i", "db883ea9684edc2158c57ddbb3667176": "\n \\begin{bmatrix}\n 1 & -\\mathbf{c}^T & 0 \\\\\n 0 & \\mathbf{A} & \\mathbf{b}\n \\end{bmatrix}\n", "db88b7fae670bfc1b396103a456e70b7": "\\tilde{f}_i:S_i \\times \\mathbb{R} \\to \\mathbb{R} ", "db88cdc7de8e6baa4dcba6735a3bd985": "(-k, k)", "db89348e5ea0e71d7e471c1f3b950210": "s(P,E_i)=\\prod_ks(P,E_{ik})", "db8945453e956047e603e395233f7f61": "\\varepsilon_i(x)\\equiv\\exp{(-\\int a_i dx)}", "db89916baa9d796e1efe484f00fa5b4c": "Z_{i 1} = Z + \\frac{1}{2Y+\\frac{1}{Z+Z_{i 1}}}", "db89e62084c3917c1efe2fa17babafb8": "\\; \\Sigma_3 \\Sigma _1 = \\omega \\Sigma_1 \\Sigma _3 = e^{2 \\pi i / d} \\Sigma_1 \\Sigma _3 ,", "db89f0e70c8cede9ca3bd82c58204ec3": "(W_{ijk})_{i \\in o_{jk}, j \\in n_k, k \\in m}", "db89fe938d6d6b34768ff272edc73328": "(\\cdot,\\cdot)", "db8a85543cd85ad0348a6639379ad4a9": "t = h/v_l", "db8aa84a01e0797b0739c2ea916d925f": "H_{E}", "db8aaeebb52b84b34facdd0c2c5d2a9f": "\\frac{5}{12}+\\frac{11}{18}=\\frac{90}{216}+\\frac{132}{216}=\\frac{222}{216}", "db8afc6ff34733e0672f004ee6e74e7b": "p(\\boldsymbol\\Sigma) \\sim \\mathcal{W}^{-1}(\\boldsymbol\\Psi,n_0).", "db8b248bcf2ff1433562b9e119b1b065": "\\varepsilon^* = \\varepsilon + \\frac{i\\sigma}{\\omega}", "db8b2f7a736756fe3b0cbd5607df4ba1": " L = \\frac{1}{2} mv^2= \\frac{1}{2}m \\left( \\dot{x}^2 + \\dot{y}^2 \\right)", "db8b513187eeb00d498ed1ee1cfabdc7": "\\frac{Y \\subseteq X}{X \\rightarrow Y}, \\frac{X \\rightarrow Y}{XZ \\rightarrow YZ}, \\frac{X \\rightarrow Y, Y \\rightarrow Z }{X \\rightarrow Z}", "db8b57a3efc1b0630b85ae9ed63d03c1": "\\Psi(t)= \\sum_{n=0}^\\infty \\Psi_n t^n \\quad", "db8b7be14fddd6fb5270b5e2f4cdfc25": "a \\vert a", "db8b9b36690557d0db96f47347425d6e": "f^{*} g'", "db8bc47411ff1b6b8be3773d2ef4ddcf": "( \\lambda x . xx)( \\lambda x . xx )", "db8bfe624bfa3816ab488117a15909e9": "\nS= \\int_t \\sum_i 2\\psi_r{d\\psi_i\\over dt} - E_i(\\psi_r^2 + \\psi_i^2)\n", "db8c1fbb671574b7b8616d9cacfad7f1": "P_{escape}", "db8c4dc23d6c6efafa57174f68710244": "L\\cap D", "db8c597252258f0136e1b0f2c518969f": "\\lambda_{N+k}=0", "db8c79b6a0f680318b76b6bf9be6cc58": "\\int\\frac{1}{x(ax + b)} \\, dx = -\\frac{1}{b}\\ln\\left|\\frac{ax+b}{x}\\right| + C", "db8c8846e79ed57eeb30774413ddd1af": "G = \\langle N \\rangle \\mu", "db8d87cd62f883c7d6a6b9403847d67a": "Q_E(f)", "db8db9c5447103be48ca4819550851ef": "l^\\mu", "db8dd4310bca0e3d4f48759433338bf4": " \\mathfrak{p} \\in P ", "db8dd85043671900c3be81ca4d8e225b": "w_\\ell=1 - \\frac{\\ell}{L+1}", "db8df14586ed895bc5b6f1463f49d0fa": "\\operatorname{cost}(\\mathcal{S}, \\mathcal{M}, \\theta) = -N^2 \\int_x P_{\\mathcal{M}}\\cdot P_{\\mathcal{S}} ~ dx", "db8e21412f3b2778e4719a2438e15e4f": "\\varphi_{sr}(\\mathbf{r})", "db8f4fd9f22f902aae5910220b9312f4": "616 \\times 10^3", "db8faf4340a07191ea59b7cb43707db7": " \\left(\\sigma \\overrightarrow{\\nabla^2} + f_{\\circ}^2 \\frac{\\partial ^2}{\\partial p^2} \\right) \\omega = -2 \\vec{\\nabla} \\cdot \\vec{Q} + f_{\\circ} \\beta \\frac{\\partial v_g}{\\partial p} - \\frac{\\kappa}{p} \\overrightarrow{\\nabla^2} J ", "db90179663e5a5aaa22dfa5dfbca44af": "\\mathrm{d}(t^2\\,W^3)=3 t^2 W^2\\circ\\mathrm{d}W + 2t W^3\\,\\mathrm{d}t.", "db9032296ff4cd5c23f962688a039b76": "a^{\\dagger}_j = e^{+i\\pi \\sum_{k=1}^{j-1}a^{\\dagger}_k a_k} \\sigma_j^+", "db9055d6993047627e485a3839ab53d7": "\\mathbf{w}_{(k)} = (w_1, \\dots, w_p)_{(k)} ", "db9056b3d0ea04d2b53317e3f56bfb60": "\\mathbb T = \\{ z \\in \\mathbb C : |z| = 1 \\}.", "db908e81a9c1b72a8d7be8cdb5d70b97": "a_{8}+b_{7}=c_{12}", "db91485969bb75705c769a67d37fdd2d": "H (N) = \\int d^3 x N \\{ A_c^k , V \\} F_{ab}^k \\epsilon^{abc}", "db92114e569a4b0574d42df2b8a738c8": "Y_j=\\sum_i a^i_jX_i.", "db92cde644307d5c6d1900fff90e4d1e": " \\frac{\\partial P}{\\partial \\rho} = \\frac{\\partial \\Phi}{\\partial \\theta},\\ \\ \\ \\ \\ \\ \\frac{\\partial P}{\\partial \\theta} = -\\frac{\\partial \\Phi}{\\partial \\rho}", "db92e2e409060516b810e8ce1132da1a": "\\tau = M \\tau_0. \\, ", "db933704e179d7277e50ae8af40d4e66": "U_a(b)=\\{b+na \\in \\mathbf{Z}^+\\, |\\, n \\in \\mathbf{Z} \\}", "db933fb23c91414e8c5e2abbe5b409ce": "\\left | \\psi(t)\\right\\rangle", "db939d103b7b1469d263305ea7157ae8": "R_S=R_{Zero} \\sec(\\alpha)=\\left(R_H \\cos(\\alpha)\\right) \\sec(\\alpha)=R_H", "db93c4bd2cb40ab933d0949308f2ee31": "f_i^{(\\alpha)}=\\int_0^\\infty \\frac{L_i^{(\\alpha)}(x)}{{i+ \\alpha \\choose i}} \\cdot \\frac{x^\\alpha e^{-x}}{\\Gamma(\\alpha+1)} \\cdot f(x) \\,dx .", "db93ce5d82db6cd8c0e1c2d57ef7899b": "F = qvB", "db93d1c933db6d7ec77b1e60ee48355f": "H(x_1,\\dots,x_d)=C(F_1(x_1),\\dots,F_d(x_d))", "db93d563a70a1d061bb80896799916b9": "(xy)^{\\rho} = y^{\\rho}x^{\\rho}", "db93d9beb905ed2ddfa3e5faa84c5eef": "\n\\begin{align}\n\\sum_{i=1}^n f(x_i^*)(x_i-x_{i-1}) & = \\sum_{i=1}^n [F(x_i)-F(x_{i-1})] \\\\\n& = F(x_n)-F(x_0) = F(b)-F(a)\n\\end{align}\n", "db93fd099f87e8a54ac062953e2207d6": " \\gamma_{jk} ", "db93fdf38da33cea7de0135bc92cc57a": "H_A=F_h", "db9403b94295f3a4525e78d4a3d706dc": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) .\n", "db947b1bf02b9fa4575d8e30bd7afbdb": "\\beta : \\Omega(G) \\longrightarrow R(G) ", "db9495cb8c252f4ce9f9f671ef46f785": "f(at)", "db94a28ce543e3d0ea28de94a1e44c5c": "\n J = J_{(1)} + J^{+} - J_{(2)} - J^{-} = 0\n ", "db94c6c2e3621067a5eb6132c5a99ec4": " \\sum^{n}_{k=1}\\bigl(\\begin{smallmatrix} n+1\\ k \\end{smallmatrix}\\bigr)\\sum^{r}_{i=1}i^{k} = (r+1)((r+1)^{n}-1). ", "db9536f85d64f52b8e0cf10fbe3f77f6": "u_Q=\\frac{1}{|Q|}\\int_{Q} u(y)\\,\\mathrm{d}y", "db953a72ce363fb04031c2fd52dc6b98": "=\\|(\\Lambda-\\mu I)^{-1}\\|_p\\ \\kappa_p(V)\\|\\delta A\\|_p", "db95617e1224d06f5f9d2dd9049d97f2": "\nw_k = v_{k+1} - v_k, \\ k = 0,...,n\n\\,\\!", "db9584427999af62a67eda40f1f5bcbb": "u \\in U", "db95d769ea7ddbf468ced3b7311070cf": "xy=1", "db961f001f656a2aabb42cc716d8480b": "\\delta_i", "db961fa0e62eb8eac75bc089f0a27392": "C_x(p_*)", "db965b558804cb87b86bf0593dda014f": "n a^n u[n]", "db9663818b045a01dfa39274f640b34b": "Ly", "db966a7028826d298ad42cc5c0029bdd": "a_1, a_2, b_1 \\,\\!", "db96933eb9a2ce9d02d9a9e8be55b3f4": "W:f\\mapsto \\tilde{f},\\,\\,\\,\\ L^2(K\\backslash G /K) \\rightarrow L^2(\\mathfrak{a}_+^*, |c(\\lambda)|^{-2}\\, d\\lambda)", "db96d8a0c81eae8bfabf7976572adcfa": "{1 \\over \\sqrt{3}} \\approx 0.577", "db96de015db86a608a6e47aa0c3108a8": "\\Lambda_{v \\times 1} = \\begin{bmatrix}\\lambda_{v}\\\\\n\\lambda_{v-1}\\\\\n\\vdots\\\\\n\\lambda_{1}\\end{bmatrix}.\n", "db972a0f02b276c38e87c234961b64fe": "\\vec{e}_3", "db972f1ed27cb13c532bfcfd54e4f2ca": "s(k_x)=F(k_x,0)\n=\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty f(x,y)\\,e^{-2\\pi ixk_x}\\,dxdy\n", "db9773aaf7fd420136380dfa226f4e0d": " \\langle s \\rangle=\\frac{}{M}=\\frac{\\partial{\\ln{\\xi(T)}}}{\\partial{\\beta \\mu}}=\\lambda\\frac{\\partial\\ln{\\xi(T)}}{\\partial\\lambda}", "db979f8b529b463d88f31c7a8370fd94": "f:X\\times\\Theta\\to\\mathbb{R}", "db97c03266fce4da9659860b35693070": "\\log_b(|X|+|Y|)=x+s_b(z)", "db9825a6fcbe20d3a8298b1670bd7de0": "D\\phi_1 -\\bar{\\delta}\\phi_0=(\\pi-2\\alpha)\\phi_0+2\\rho\\phi_1-\\kappa\\phi_2\\,, ", "db982ba68bcea3d00081e9e72be872d6": "\n \\begin{matrix}\n 3\\uparrow\\uparrow\\uparrow3 = 3\\uparrow\\uparrow3\\uparrow\\uparrow3 = 3\\uparrow\\uparrow(3\\uparrow3\\uparrow3) = &\n \\underbrace{3_{}\\uparrow 3\\uparrow\\dots\\uparrow 3} \\\\\n & 3\\uparrow3\\uparrow3\\mbox{ multiplied copies of }3\n \\end{matrix}\n \\begin{matrix}\n = & \\underbrace{3_{}\\uparrow 3\\uparrow\\dots\\uparrow 3} \\\\\n & \\mbox{7,625,597,484,987 multiplied copies of 3}\n \\end{matrix}=\\underbrace{3^{3^{3^{3^{\\cdot^{\\cdot^{\\cdot^{\\cdot^{3}}}}}}}}}_{7,625,597,484,987}\n ", "db9919b0a98af18c2b1fdc7b95429bdd": " \\mathbf{F} = \\frac {\\mathrm{d}\\mathbf{p}} {\\mathrm{d}t}", "db996ea3f1c4a7648b9ab6bc0eaee404": "e^{i \\theta} A", "db99a2b07c7bffc73e90a9ed26f41464": " \\left ( {\\partial T\\over \\partial S} \\right )_P = { T \\over C_P } ", "db99be2fc38bf7516f99dfec9511ea35": "F = 1 + \\frac{(L-1)T}{T_0}", "db99c38c4a6f755ec779b9e1f92d4357": "\\epsilon_e^* = \\frac{l^*-l_0}{l_0}", "db9a0cb35bf4ac6c71df055076a4ac04": "(x_1, y_1) = (x, 1 - ax^2 + y)\\,", "db9a4772c67f9ab1f47cce1200e49a0c": "(x)_n=\\frac{\\Gamma(x+1)}{\\Gamma(x-n+1)}.", "db9a6367b1869291d8de8421833ffe60": "c_n = {\\langle f, \\varphi_n \\rangle_w\\over \\|\\varphi_n\\|_w^2}.", "db9a8ecae8a841f36c191d376ecdaa12": "V \\times \\{0\\} \\ \\text{in} \\ Y = X \\times \\mathbf{C} \\ \\text{or} \\ X \\times \\mathbf{P}^1", "db9b02d57c5a5c9312a3cf445d1b1a5c": "E_{KIN}=m l^2 \\dot \\varphi^2 /2", "db9b07480264783545cb9210caac55b0": "\\sqrt{L_{WL}}", "db9b6192df0802cecca1599d99bbccff": "\\begin{align}L_{\\rm Edd}&=\\frac{4\\pi G M m_{\\rm p} c} {\\sigma_{\\rm T}}\\\\\n&\\cong 1.26\\times10^{31}\\left(\\frac{M}{M_\\bigodot}\\right){\\rm W}\n= 3.2\\times10^4\\left(\\frac{M}{M_\\bigodot}\\right) L_\\bigodot \n\\end{align}\n", "db9b6ced7a62d5e606b5cd23e3c25e26": "\\varepsilon\\,\\!", "db9ba905b7d31310890f0f3642369043": " [n, k, d]_q", "db9c27465163bd4cda5b31603195d1ae": "\\textstyle\\binom nk = \\frac{n^{\\underline k}}{k!}", "db9c7011db6273fd0370df3c148ae3b5": "\\lim_{n\\to\\infty}\\Pr\\left[\\,\\left|-\\frac{1}{n} \\log p(X_1, X_2, ..., X_n) - \\overline{H}_n(X)\\right|< \\epsilon\\right]=1\\qquad \\forall \\epsilon>0", "db9cf4b19a33234af9c1c9031bb49a39": "R_1=R_2", "db9d1e56fd023d04f91127e1c37d9af8": "\\int_0^\\infty x^{m}e^{-ax^2}\\ dx=\\frac{\\Gamma [(m+1)/2]}{2a^{(m+1)/2}}", "db9d2743677d17241598e2e696df278e": "\\begin{pmatrix}y'\\\\y''\\\\\\vdots\\\\y^{(n-1)}\\\\y^{(n)}\\end{pmatrix}\n=\\begin{pmatrix}0&1&0&\\cdots&0\\\\\n0&0&1&\\cdots&0\\\\\n\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\\n0&0&0&\\cdots&1\\\\\n-p_0(x)&-p_1(x)&-p_2(x)&\\cdots&-p_{n-1}(x)\\end{pmatrix}\n\\begin{pmatrix}y\\\\y'\\\\\\vdots\\\\y^{(n-2)}\\\\y^{(n-1)}\\end{pmatrix}.", "db9d69444f32d605445521d97f3fe4b1": "( m + z )/ z", "db9d95ddbf804981c85ed4ae400d8851": " \\beta_{ij} = \\frac{B}{A} = \\frac\n{\\left( \\begin{array}{c} i \\\\ m-j \\end{array} \\right)}\n{\\left( \\begin{array}{c} m \\\\ j \\end{array} \\right)}\n = \\frac{i!j!}{(i+j-m)!m!}", "db9e11d7a8e95af2d9da633d273cb001": " \\frac43 n^3 ", "db9e18c85de372134ed02b32b096c7ba": "\\boldsymbol\\Lambda|\\mathbf{W},\\nu \\sim \\mathcal{W}(\\boldsymbol\\Lambda|\\mathbf{W},\\nu)", "db9e245fb92c7681d8f40eb29fc09b1a": "L_{0}^{'}=L\\cdot\\gamma. \\qquad \\qquad \\text{(1)},", "db9e6b7be3227e9ebb32f84dd5d04f69": "f_{i-1}(r_1,\\dots,r_{i-1}) = \\begin{cases} f_{i}(r_1,\\dots,r_{i-1},0)\\cdot f_{i}(r_1,\\dots, r_{i-1},1) & S = \\forall \\\\\nf_{i}(r_1,\\dots,r_{i-1},0) * f_i(r_1, \\dots,r_{i-1},1) & S = \\exists.\n\\end{cases}", "db9e767bb847e597de7a7358dda5b603": "A_0 = {4 \\pi m k^2 e \\over h^3} = 1.20173 \\times 10^6\\,\\mathrm{A\\,m^{-2}\\,K^{-2}}", "db9e7ff481c7383db44eca6a72a480a0": " w ( u \\wedge v)\n= \\sum_{i,jM", "dbae7abd3123f43656eb4aa375292435": "\\tilde{H}_i(X)", "dbaebd9d6d27b55910acec408ca021ef": "\\hat{\\boldsymbol{u}}", "dbaec4673d0e3572381faf4a38a4613a": "A_2 = 2\\pi", "dbaec4b64e184dc038341431c1927184": "\\int\\frac{\\mathrm{d}x}{\\sin ax\\cos ax} = \\frac{1}{a}\\ln\\left|\\tan ax\\right|+C", "dbaecb1f5cebdd3cfaf906e66b849dca": "f_j = \\sum_\\beta f_{j\\beta} X^\\beta,", "dbaf20f8db855ef2c85179614db643f2": "\\Re(b_1+\\cdots b_n -a_1-\\cdots - a_n) >1. ", "dbaf6ba1a20df1f21e877492acbda7c0": "H(s) = \\frac{ 6 }{s + 2},", "dbaf7de8d2b403d7637f9a90829c965a": "\\lambda_{2}\\,", "dbafb4c890817a560f436f7262192d94": "\\begin{smallmatrix}\\left[\\frac{M}{H}\\right]\\ =\\ -0.10\\end{smallmatrix}", "dbb083ecf81b632fe20f1b04ceeb72c7": " \\begin{align}\n\\mathrm{E}[\\mathrm{TMRCA}] & = \\mathrm{E}[ T_n ] + \\mathrm{E}[ T_{n-1} ] + \\cdots + \\mathrm{E}[ T_2 ] \\\\\n&= 1/\\lambda_n + 1/\\lambda_{n-1} + \\cdots + 1/\\lambda_2 \\\\\n&= 2N(1 - \\frac{1}{n}).\n\\end{align}", "dbb0a407bc284d3efdb4931f2863fe3c": "I_i = 0^+ \\rightarrow I_f = 0^+ \\Rightarrow \\Delta I = 0", "dbb0cfc8e6a96292e0fbdc1f9f5e18c1": "\\gamma_0(1) = -\\psi(1) = \\gamma_0 = \\gamma", "dbb0ecfd6969220a3da187314196aef6": "\\frac{-1}{\\ln(1-p)} \\; \\frac{\\;p^k}{k}\\!", "dbb0ee3c5823928f0833b040bff85116": "\\scriptstyle f_\\text{curried}(1)", "dbb15c621184934b5787687a37624931": "g = \\frac{1}{2}(p-1)(q-1).", "dbb206c26e197d193bc5091505927e5c": "\\gcd{(a^{(N-1)/q} - 1 , N)} = 1", "dbb210981abfe7101f62bd922bb449d1": "\\scriptstyle\\forall\\alpha.\\alpha \\to \\alpha \\to \\alpha", "dbb26011a2f3fa9b330fc3407e303ba5": "\n K = \\left(\\frac{\\beta\\,\\pi}{\\alpha\\,I}\\right)^{\\frac{1}{n+1}}\\,(\\sigma_{\\text{far}})^{\\frac{2}{n+1}}\n ", "dbb2ef230aeca001c819c56052128aae": "|\\beta_0| \\gg |\\Delta \\beta (\\omega)|", "dbb31cd90acad68dd3d33a6838e30680": "\\Sigma^1_n", "dbb32243a9e294c818e44dd8669679b8": "\n\\psi(y;t+\\epsilon) \\approx \\int \\psi(x;t) e^{-{\\rm i}\\epsilon V(x)} e^{{\\rm i}(x-y)^2 \\over 2\\epsilon} dx\n\\,.", "dbb3b84cea7e7015fb1a591964fbd918": "f_{i}", "dbb3ff035971ab345efca67c859b6f22": "\nB = \\dot{x}(0)+\\omega_0x(0) \\,\n", "dbb4539e70cecedc99fb149821eed1ab": "p^{R(p,n)}\\le 2n", "dbb483f6f9aa9a60a04369ca2b3bfec6": "(s+1)(s^2+0.4450s+1)(s^2+1.2470s+1)(s^2+1.8019s+1)", "dbb4b9be3f598b0cd8b595a1fafaf42b": "E = V(x)", "dbb4f724831c7d3d2ed863b2fd33c2fc": "\\frac{\\partial u'}{\\partial t} = - \\frac{1}{\\rho}\\frac{\\partial p'}{\\partial x}\\,", "dbb504fb068626adfa9339cd6dcde563": "7{{x}^{2}}+4x-10", "dbb522a626a7a6d95e2a4b9cde8edeb6": "B \\subseteq \\{1,\\dots, n\\}", "dbb5240051fccb5b5e9100d51f100898": "(ct)^2 - x^2 = [(Cx)^2 + (Dct)^2 + 2CDcxt] - [(Ax)^2 + (Bct)^2 + 2ABcxt] ", "dbb58ef46519b3c7cbc5376bac9bc182": "t - \\ ", "dbb5c2f3f492a7eb5a31a1520775ace5": "\\scriptstyle f_j", "dbb5d4f67fff45f37c4b202210bcc2bc": "\\mbox{CAIDI} = \\frac{\\sum{U_i N_i}}{\\sum{\\lambda_i N_i}}", "dbb5dc13a08029a4471accc90703fbc0": "C_{\\psi}", "dbb5e2b87eea11581376be32787560e9": "C_x = \\frac{1}{6 A} \\sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i)\\,", "dbb5e383665849a00e16dd1b05969968": "0^0=0^{\\underline 0}=0!=\\binom00=\\binom{-1}0=\\left\\{{0\\atop0}\\right\\}=p_0(0)=1", "dbb610bd2bdff4a2a0ce257a2c818326": "R(u,v)w=\\nabla_u\\nabla_v w - \\nabla_v \\nabla_u w .", "dbb6322a64d9fb1e0142f1d718bdbdb7": "a_n x^n + a_{n-1}x^{n-1} + \\dotsb + a_2 x^2 + a_1 x + a_0 = 0", "dbb6477458a847ab764622fcd37871de": "\\|f_r-f\\|_2 \\rightarrow 0.", "dbb68bf429c298a84b39cc98d0b72a14": "\\psi_(r) = \\frac{1}{\\sqrt{\\pi}} Z^{\\frac{3}{2}}\\ e^{-{\\textstyle Zr}}\\; ", "dbb68c992e60115a2dd22ff7900bc8e4": "(1 - H(p + \\epsilon))n", "dbb698b121efff5ae56d95d4f24c9c92": "K_{\\mathrm{eff}} = \\int_0^L\\frac{1}{2}u^2\\left(\\frac{dy}{L}\\right)m\\!", "dbb6d287bbeff3271e380d4686077909": "\\varepsilon_{r}", "dbb74a937bfb6e40b2be8febca67261f": "P_\\mu (n,\\tau )", "dbb75de9248c6eb28a439c7245055bc6": " v_1 \\ldots v_n ", "dbb7a08a83cc355679c6759531cf4bc1": "0 < \\beta < 1", "dbb7bc6c6173484546344e0815dc8c4f": "B=\\frac{g_0}{\\eta_0h_{PR}\\left(1-\\phi_e\\right)\\frac{C_L}{C_D}}", "dbb8030571a087da4d490c6f6283fe88": "a\\mathrm{[H_3 O^+]}", "dbb885ab3e906cfd5ea91948eeb1aea8": " \\begin{align}\n+\\left(\\frac{\\partial T}{\\partial V}\\right)_S &=& -\\left(\\frac{\\partial P}{\\partial S}\\right)_V &=& \\frac{\\partial^2 U }{\\partial S \\partial V}\\\\\n\n+\\left(\\frac{\\partial T}{\\partial P}\\right)_S &=& +\\left(\\frac{\\partial V}{\\partial S}\\right)_P &=& \\frac{\\partial^2 H }{\\partial S \\partial P}\\\\\n+\\left(\\frac{\\partial S}{\\partial V}\\right)_T &=& +\\left(\\frac{\\partial P}{\\partial T}\\right)_V &=& -\\frac{\\partial^2 A }{\\partial T \\partial V}\\\\\n\n-\\left(\\frac{\\partial S}{\\partial P}\\right)_T &=& +\\left(\\frac{\\partial V}{\\partial T}\\right)_P &=& \\frac{\\partial^2 G }{\\partial T \\partial P}\n\\end{align}\\,\\!", "dbb8c29ddd2f8d00af5685f8c56dce80": "Y(t)=K(t)^{\\alpha}(A(t)L(t))^{1-\\alpha}\\,", "dbb8e2056bc346f21543b53b6dc5f62e": "\\phi\\colon V\\to R", "dbb972d442530ced7d6b08f4fb774aba": "\\{ V_I \\colon I \\text{ is an ideal of } R \\}.", "dbb981ba352a8174d77d00a262bda65a": "E_a = \\frac{m^2 e^5}{(4\\pi \\epsilon_0)^3 \\hbar^4} ", "dbb9b9613c35a348208563ecd66e2cd5": "\\sigma (E)=\\sigma_{ax}[1+r_m(E_p,\\alpha)]", "dbb9cca80586f0e4b71392c02116adb0": " \\Delta\\lambda \\approx \\Delta \\theta \\left(M {\\partial\\theta\\over\\partial\\lambda}\\right)^{-1}", "dbb9e6cd39002769118a288703dbfbc1": "f(x)=\\sum_{n=0}^\\infty {a_n} x^n", "dbba42aac78a355e08ac5152474519dd": "\\alpha =3", "dbba651d4fda0782a8cb03dd1beb3ef4": "\\,\\,\\dot{\\sigma}_{ij} = H_{ijkl}(\\sigma_{mn})\\,\\dot{\\varepsilon}_{kl}\\,\\,", "dbba8c5f4d3b2948e2745216d2d62b7f": " f(x+h)-f(x) = \\int_x^{x+h} f'(u)du = \\left(\\int_0^1 f'(x+th)\\,dt\\right)\\cdot h.", "dbba9f9c7ff2cab4fe253766eeb77f55": "\\tilde{h_2} \\leftarrow u_1^{z_2} rem P", "dbbaad276af3796c1da6061650041af7": "\n\\left\\{\\frac{\\varphi(n+1)}{\\varphi(n)},\\;\\;n = 1,2,\\cdots\\right\\}\n", "dbbb20769587d1aa22d7b37ba5d275c8": "\\widehat{\\neg \\alpha} = \\mathbb{I} \\otimes \\mathbb{I} - \\hat{\\alpha}_1 \\otimes \\hat{\\alpha}_2", "dbbb4e7a9bf12256fc38b2e2c2b688c3": "\\delta (y)=\\frac{2\\pi}{\\lambda}[n_2-n_1]\\left (2y-a\\right )", "dbbbd60e5fa838615e372819ff1d0e4a": "({V}_{1}/{V}_{2})", "dbbc6775a67de9ddb198eef70297f1e7": "2^{-n\\left[ H\\left( Y|X\\right) +\\delta\\right] }\\ \\Pi_{\\rho_{x^{n}}\n,\\delta} \\leq\\Pi_{\\rho_{x^{n}},\\delta}\\ \\rho_{x^{n}}\\ \\Pi_{\\rho_{x^{n}\n},\\delta} \\leq2^{-n\\left[ H\\left( Y|X\\right) -\\delta\\right] }\\ \\Pi\n_{\\rho_{x^{n}},\\delta},\n", "dbbcc870c2c86a6872fd74fa26f228d0": "\\lfloor\\cdots\\rfloor", "dbbce738f40d0dfdd011e23378f06af6": "\\mbox{curl}\\,(\\mbox{grad}\\,f ) = \\nabla \\times (\\nabla f)", "dbbd16052a0f59cb52f1f6369f6c3947": "df=0", "dbbd7946d5c0e3d752b179c2a933581a": "\\frac{1}{|X|} \\sum_{U \\in X} f(U) = \\int_{U(d)}f(U) dU", "dbbda7db3e44a6ceaabef0a9b9e44f98": "4 \\ \\mathrm{N} \\times \\frac{7}{4}= 7 \\ \\mathrm{N}", "dbbdcac51735ba35ea0ac74f3fce6537": "m(r)\\le\\Phi(\\overline x)", "dbbde4a6e8612cadf98ee4e2f76fedb2": "\\binom{n}{k} p^k (1-p)^{n-k} ", "dbbe2c74e05dd5552411117e393673b7": " a \\lambda^{2} + b \\lambda + c = 0 \\;", "dbbe731f5604db87fed18f9c244f7c67": "a = \\tfrac 5 6. ", "dbbea004ca140b1fa866e981e87fd7d9": "\\tan 40^\\circ\\cdot\\tan 30^\\circ\\cdot\\tan 20^\\circ=\\tan 10^\\circ.", "dbbea129c5c252249a82fe90798fec13": "d(af+bg) = a\\,df + b\\,dg.", "dbbee5ed15f0e0103942dfe4a5c24a0f": "-\\sqrt{\\frac{1}{60}}\\!\\,", "dbbf0745dfbb9f115f9b1a21e053d27d": "\\Delta p_x", "dbbf1b65c79efc8194ed54ca27f083ee": "\\pi'(b',e) = b'.\\,", "dbbf1f7fd1b395993bc20b6877eb4b3d": "\\frac{\\partial v_i}{\\partial x_{i_k}}=\\frac{\\partial^{p_1+\\cdots+p_l}v_j}\n {\\partial x_{j_1}^{p_1}\\cdots \\partial x_{j_l}^{p_l}}", "dbbf53021b1758e642e8d5d8c3f78da8": "J = \\frac{\\pi D^4}{32}", "dbbf62f5248e7f6fb089c82c2dfb095c": "R_n(\\xi,x)\\equiv \\mathrm{cd}\\left(n\\frac{K(1/L_n)}{K(1/\\xi)}\\,\\mathrm{cd}^{-1}(x,1/\\xi),1/L_n\\right)", "dbbfa7085e43545bbf9a3fcf0428ebc1": "\\mathcal{F}_{d}", "dbbfc11568ea528a8fa34349fda89eec": "\\mathcal{D}(\\nu,\\mathcal{D}(\\hat{\\nu},Z))=\\mathcal{D}(\\nu+\\hat{\\nu},Z)", "dbc006cf8faec6ee1cc3223615980949": "\\langle x:=? \\rangle p\\,\\!", "dbc039805bed056fd0611ac0e8ba1843": "\\sum_{k=1}^N \\lambda_k", "dbc09597b77b069a1c6bb5f870585405": "E_{\\mathrm t} (T)", "dbc154e3546b149e9fc7abf44d331766": " p_{i} \\leq \\frac{\\alpha}{m} ", "dbc1669c2b6246f49c58696e02fe9c21": "a \\in B", "dbc191cfc5815e0db9cc0a067885300c": "\\phi_f=U^{-1}(\\infty)\\phi_i U(\\infty)=S^{-1}\\phi_i S.", "dbc220f3e0dd5b4ecfe3e5affce3e655": "f = \\left ( \\frac {c}{c + v_\\text{s}} \\right ) f_0", "dbc263cbaf838c6e5cb38577ea7c89a9": "\\theta \\in Q", "dbc2cf9d536a390398dfd71311eaea4b": "\\textbf R^L_+", "dbc2dcbe99dbf97650817448ad7f74ad": "i(e,n) \\in A \\Leftrightarrow (e,n) \\in B", "dbc30291b89af88dadd5c34f58002da1": "FDR_{-1}", "dbc32ea20c3066d6602da21a770ce530": "\\mathcal O(n^2)", "dbc3c7255accd04cc742fbf100e2d15b": "R_b = \\frac{T_v}{T_e}", "dbc3c8b472102c11b258bbc891b3430e": " \\overline{r}_j = \\frac{\\sum_{k=1}^j r_k \\cdot D_k}{\\sum_{k=1}^j D_k}", "dbc40f3d8aff9b02d7d1edf217e45e31": "A=A\\Gamma +\\sigma \\qquad\\qquad (5) ", "dbc485925a3f958f5bd624423ba9ab6a": "\\frac{(n-1)(n-2)}{2} + 1", "dbc4982883ea6922c7c794cfdd611ac9": "e ^{ix}", "dbc4b99dd3167789e780321cbc83c1c9": "S = (s_1, s_2, s_3, \\ldots s_T)", "dbc5b9dd72b4b73a28683770c34e51e3": " P(cancer~in~population) = 0.5~years * \\frac{1}{250~per~year} = \\frac{1}{500}", "dbc5bc726c395ab33a326138c849a13f": "(x_n),(y_n)\\in 2^\\mathbb{N}", "dbc5ce1c8c0c38f0ec8ac4a711400c73": "\n\\zeta(0) = \\frac{1}{\\pi} \\lim_{s \\rightarrow 0} \\ \\sin\\left(\\frac{\\pi s}{2}\\right)\\ \\zeta(1-s) = \\frac{1}{\\pi} \\lim_{s \\rightarrow 0} \\ \\left( \\frac{\\pi s}{2} - \\frac{\\pi^3 s^3}{48} + ... \\right)\\ \\left( -\\frac{1}{s} + ... \\right) = -\\frac{1}{2}\n\\!", "dbc62222e940503bc9bd86d4be75d0bc": "{1 - R}", "dbc62f88c0008f0bd2b293ba14088f8f": "{r_{\\rm w}}.", "dbc64a14baa079a762db396e4814c38e": "D = 8 \\times 10^{-7}", "dbc66d7bfe788c96014f77f299d64475": "\\gamma_w", "dbc6960326f7b2b65d1674572a14a22f": "\\nabla(O) = 1", "dbc6e1cad2121064c1324e85b9e561b8": "\\int \\frac\\partial{\\partial\\theta}f(\\theta)\\,d\\theta = 0.", "dbc7b01e7de065b271c7038a2231a8c3": "L\\subseteq\\Sigma^*\\times\\N", "dbc7eafdd2e84757d2dba5d9b5a4c7b1": " K(x,s) = <\\varphi(x), \\varphi(s)> ", "dbc7fddd0637df10a9fea296ea9c7456": "S_{21}=S_{31}", "dbc8026947fd6f323256c968c643cbf9": " \\begin{bmatrix} \\ln x \\\\ x \\end{bmatrix} ", "dbc8604de55751939c742de36af420c1": "Q^{min}", "dbc8672fcc000598869ee1ad249d3743": "\\begin{array}{lcl}\n x'=x_0+v_{2x} t,\\ y'=y_0+v_{2y} t\\\\\n\\therefore x = x_0+r_1\\cos(\\omega_1 t+\\phi_1)+v_{2x} t,\\ y = y_0+r_1 \\sin(\\omega_1 t+\\phi_1)+v_{2y} t,\\\\\n\\end{array}", "dbc89977130904efdaaeedc1905d57e1": "a^x\\,", "dbc8e40fb3c20f02a02b0000069d3718": "x_{k+1}:=x_k-\\frac{f(x_k)}{(x_k-q)(x_k-r)(x_k-s)},", "dbc9011a370bca098d4752346ba71d5c": "\\theta_0", "dbc910fe94ae733ac41001d3957d1324": " S\\left( q, \\dot q \\right) ", "dbc971a34145f9c0a646f19f1b1e89db": "E(T)=\\sigma^2", "dbc99cd745bc4c66c5736fbf549d4fb3": "\\displaystyle{\\sum_{n\\ge 0} s^n H_n(x)H_n(y)= {1\\over \\sqrt{\\pi(1-s^2)}} \\exp {4xys - (1+s^2)(x^2+y^2)\\over 2(1-s^2)}.}", "dbc9fa34cbed3ab86d1a09c49fe7f69c": "\\begin{align}\np(\\sigma^2|\\mathbf{X}) &\\propto p(\\mathbf{X}|\\sigma^2) p(\\sigma^2) \\\\\n&= \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{\\frac{n}{2}} \\exp\\left[-\\frac{S}{2\\sigma^2}\\right] \\frac{(\\sigma_0^2\\frac{\\nu_0}{2})^{\\frac{\\nu_0}{2}}}{\\Gamma\\left(\\frac{\\nu_0}{2} \\right)}~\\frac{\\exp\\left[ \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right]}{(\\sigma^2)^{1+\\frac{\\nu_0}{2}}} \\\\\n&\\propto \\left(\\frac{1}{\\sigma^2}\\right)^{\\frac{n}{2}} \\frac{1}{(\\sigma^2)^{1+\\frac{\\nu_0}{2}}} \\exp\\left[-\\frac{S}{2\\sigma^2} + \\frac{-\\nu_0 \\sigma_0^2}{2 \\sigma^2}\\right] \\\\\n&= \\frac{1}{(\\sigma^2)^{1+\\frac{\\nu_0+n}{2}}} \\exp\\left[-\\frac{\\nu_0 \\sigma_0^2 + S}{2\\sigma^2}\\right]\n\\end{align}", "dbca4796d9e0db981cfad1900eee29ce": " \\left [ A \\left \\{ \\frac { \\left ( \\frac {M}{C} + \\frac { \\beta + 3 } {6 \\left ( \\beta + 1 \\right ) } \\right ) } { A } + A \\left ( \\frac{N}{C} + \\frac{3 \\beta + 1}{6 \\left ( \\beta + 1 \\right ) } \\right ) - 1/3 \\right \\} \\right ] ^{-1/2} ", "dbca642962c1fb52dcdd52e22aff7d26": "a\\in \\Bbb{R}^n", "dbca8f05d42d39881e85ab4ee9a1a41f": "\\mathcal F f", "dbcabdcbf220ac1233deee633fcfc97d": "\\scriptstyle [0,\\, 1] \\;\\times\\; [0,\\, 1]", "dbcb1c44fbf725f0cb35f730dd2d5f42": "O\\left(\\sum_i {e_i(\\log n+\\sqrt p_i)}\\right)", "dbcb1ec5e0bf818a1d253c3bd17efd17": "F_\\nu \\propto \\begin{cases} {\\nu^{2}}, & \\nu<\\nu_a \\\\\n {\\nu^{1/3}}, & \\nu_a<\\nu<\\nu_m \\\\\n {\\nu^{-(p-1)/2}}, & \\nu_m<\\nu<\\nu_c \\\\\n {\\nu^{-p/2}}, & \\nu_c<\\nu\n\\end{cases}", "dbcb24000bb6f391626a38eaaa6bb55e": "\\mathcal{L}_X : \\mathcal{F}(M) \\rightarrow \\mathcal{F}(M)", "dbcb5ad2874ee570462b406fab8a7cf3": "e_i.(v \\otimes w) = k_i.v \\otimes e_i.w + e_i.v \\otimes w", "dbcb68de757c0c420e59ea0d350a87aa": "u_\\gamma=1-u_\\beta", "dbcb8505d72f4b16642ca29df2bf0366": " \\vec{e}_1 = \\frac{1}{g(r)} \\, \\partial_r ", "dbcb8bf8bea9b870ec418f1af99ebae1": "\\mathrm{E} \\left [|\\log[[p(X i)]]|^r \\right ]", "dbcbb2d5d29cca471b94f15574d6bb7d": "t_{ox}", "dbcbc1341f972bb8cca46ed802fa4696": "\\begin{align}\n y &= G \\left \\{ \\frac{1}{(\\alpha + 1 - \\beta)_{\\beta - \\alpha - 1}} \\sum_{r = \\beta - \\alpha}^\\infty \\frac{(\\alpha)_r (\\alpha + 1 - \\gamma)_r}{(1)_r (1)_{r + \\alpha - \\beta}} x^{-r} \\right \\} + \\\\\n& \\quad + H \\left \\{ x^{-\\beta} \\sum_{r = 0}^\\infty \\frac{(\\beta - \\alpha) (\\beta)_r (\\beta + 1 - \\gamma)_r}{(1)_r (\\beta + 1 - \\alpha)_r} \\left (\\ln \\left (x^{-1} \\right ) + \\frac{1}{\\beta - \\alpha } + \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{\\beta + k} + \\frac{1}{\\beta + 1 + k - \\gamma} - \\frac{1}{1 + k} - \\frac{1}{\\beta + 1 + k - \\alpha} \\right) \\right ) x^{-r} \\right \\}\n\\end{align}", "dbcc3fddf8101970e3061250b153c6b0": " \\left.\\frac{1}{n K(N-K)(N-n)(N-2)(N-3)}\\cdot\\right.", "dbcc54bb1a0b11e41a3cae6cdf4673eb": "X=x_n", "dbccb12f1d3595c11985e8d8fdef7a0c": "\\mathcal{O}((m+k)p)", "dbccc50f043e6b5ae19289bb9312f12b": "\\nabla \\cdot \\mathbf{B} = \\mu_0\\rho_{\\mathrm m}", "dbcccac8c600d3278f9f967606846114": " g_n < 2\\sqrt{p_n} + 1.\\, ", "dbccd5086b0711790dc1f231d91e9bde": "1.9314", "dbccd9b4f72dc4090a75ad215959cbb1": "\\bar{S}_{2t}=\\{r:q(d,r)\\leq OPT-2t\\}\\,\\!", "dbcced7f128d4671b79ec31e9883139b": "\\text{75 m/s velocity} = \\frac{500 \\times C}{2 \\times 10^9}", "dbcd0668289102df11d74341e28a216e": "P(\\sigma_k = s|\\sigma_j,\\, j\\ne k) = \nP(\\sigma_k = s|\\sigma_j,\\, j\\isin N_k)\n", "dbcd27066e54548d4522e2a54a4a5191": "\\left[\\hat{f}^\\dagger, \\hat{f}^\\dagger \\right]_+ = 0", "dbcd46aafff10d296f1855abb9cff70e": "\\frac {1} {2 \\pi C_i (R_A//R_i)}", "dbcd4a985280ea3fea4af951eb0bb0e7": "c|xy|_X\\le|f(x)f(y)|_Y\\le C|xy|_X", "dbce4be29cc7d1210f383d52c64aae7b": "\nM = \\frac{1}{16} \\begin{bmatrix}1 & 2 & 1 \\\\\n2 & 4 & 2 \\\\\n1 & 2 & 1 \\\\\n\\end{bmatrix}\n", "dbceefd08131770551a6a80e8937021e": "|H(f)|^2 = (f_z/f_p)^2 \\; {{1 + (f/f_z)^2} \\over {1 + (f/f_p)^2}}", "dbcef08d8adbfc149db59790893c924c": "\\tilde{a}=\\frac{a}{a_0}, \\;\\rho_c=\\frac{3H_0^2}{8\\pi G},\\;\n\\Omega=\\frac{\\rho}{\\rho_c},\\; t=\\frac{\\tilde{t}}{H_0},\\;\n\\Omega_c=-\\frac{kc^2}{H_0^2 a_0^2}\\;", "dbcef9c168971985a486529351e3a4ea": "(\\kappa, \\gamma)", "dbcf034a870681d97ba08383ac6f17a3": "i,j> 1", "dbcf912634bfcf1dbc840d567c58cd63": " V \\ne W \\to \\operatorname{sink}[(\\lambda V.\\lambda W.E)\\ Y, X]", "dbd0119e4242f1d6466182bccfdc2deb": "fg(s) = \\begin{cases}f(2s) & 0\\leq s \\leq \\frac{1}{2} \\\\ g(2s-1) & \\frac{1}{2} \\leq s \\leq 1.\\end{cases}", "dbd016faff78ca8fab54976f7b051707": "\\displaystyle{(Lf,g)=(f_x,g_x)+(f_y,g_y)= (\\Delta f,g)_\\Omega - (\\partial_n f,g)_{\\partial\\Omega}}", "dbd0281b6a71cd30ac02d69f1462d285": "\\forall{Y}{\\in}{p}:{x}{\\in}{Y}.", "dbd0a28523774574bb3bb091d42ea04c": "\\hat J_j(1)", "dbd1846e100027d219d97c620d008664": "1^p+ 2^p + \\cdots + n^p \\approx \\frac{n^{p+1}}{p+1}", "dbd1858d7ec8b409cdb40d2b3ef6b026": "\nX=\\begin{vmatrix} \nx_{11} & x_{12} & x_{13} &\\cdots & x_{1n} \\\\ \nx_{12} & x_{22} & x_{23} &\\cdots & x_{2n} \\\\\nx_{13} & x_{23} & x_{33} &\\cdots & x_{3n} \\\\\n\\vdots& \\vdots & \\vdots &\\ddots & \\vdots \\\\\nx_{1n} & x_{2n} & x_{3n} &\\cdots & x_{nn}\n\\end{vmatrix},\nD=\\begin{vmatrix} \n2 \\frac{\\partial} { \\partial x_{11} } & \\frac{\\partial} {\\partial x_{12}} & \\frac{\\partial} { \\partial x_{13}} &\\cdots & \\frac{\\partial}{\\partial x_{1n} } \\\\[6pt]\n\\frac{\\partial} {\\partial x_{12} } & 2 \\frac{\\partial} {\\partial x_{22}} & \\frac{\\partial} { \\partial x_{23}} &\\cdots & \\frac{\\partial}{\\partial x_{2n} } \\\\[6pt]\n\\frac{\\partial} {\\partial x_{13} } & \\frac{\\partial} {\\partial x_{23}} & 2\\frac{\\partial} { \\partial x_{33}} &\\cdots & \\frac{\\partial}{\\partial x_{3n} } \\\\[6pt]\n\\vdots& \\vdots & \\vdots &\\ddots & \\vdots \\\\\n\\frac{\\partial} {\\partial x_{1n} } & \\frac{\\partial} {\\partial x_{2n}} & \\frac{\\partial} { \\partial x_{3n}} &\\cdots & 2 \\frac{\\partial}{\\partial x_{nn} }\n\\end{vmatrix}\n", "dbd18d6cdf2e31c1e838de70fc99a06e": "m = \\frac{\\ln(F/K)}{\\sigma\\sqrt{\\tau}} = \\tfrac{1}{2}\\left(d_- + d_+\\right),", "dbd1eaf4e28c004ec7efb05c946e8e36": "\\int^x B(x')dx'\\,\\!.", "dbd277c536da9ff041a2c226a2bcc545": "|\\mathbf{x}|' = \\hat{\\mathbf{x}} \\cdot \\mathbf{v}", "dbd29a1d090a2077343b32011324831c": "\\le 1 + P_e^{(n)}nR + nC", "dbd2b920b69ce607100ca1e645ab0708": "\\Gamma(\\omega) = \\frac{- Y(\\omega)}{1 + Y(\\omega)}", "dbd2e891c3efa24195810acf35805907": "\\phi=\\bigwedge_r \\phi_r", "dbd30e9e302798072751dab1ea61400d": "a \\not\\in T", "dbd3a80df6cf543169bb537296da2fe6": "0 \\rightarrow \\operatorname{Hom}_R(A,I^0) \\rightarrow \\operatorname{Hom}_R(A,I^1) \\rightarrow \\dots.", "dbd43458d27c8c5ce67c5f5940cf7442": "\n \\begin{align} \n Y_{1}^{-1}(\\theta,\\varphi) & = {1\\over 2}\\sqrt{3\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\quad \n = {1\\over 2}\\sqrt{3\\over 2\\pi}\\cdot{(x-iy)\\over r} \\\\\n Y_{1}^{0}(\\theta,\\varphi) & = {1\\over 2}\\sqrt{3\\over \\pi}\\cdot\\cos\\theta\\quad \\quad \n = {1\\over 2}\\sqrt{3\\over \\pi}\\cdot{z\\over r} \\\\\n Y_{1}^{1}(\\theta,\\varphi) & = {-1\\over 2}\\sqrt{3\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\quad \n = {-1\\over 2}\\sqrt{3\\over 2\\pi}\\cdot{(x+iy)\\over r}\n \\end{align} ", "dbd45bb9a2aef3d7de2feec199acae01": "\\big. \\frac{\\partial q}{\\partial t} = \\nabla \\cdot D \\nabla q - \\vec{v} \\cdot \\nabla q. ", "dbd4989ee81c9209a9d7ba2f3b414900": "\\{a_i^\\dagger, a_j\\}=\\delta_{i,j},\\{a_i^\\dagger, a_j^\\dagger\\}=0, \\{a_i, a_j\\}=0.", "dbd4cce19a0f534b39f03bb5f57e35fc": "p=(p_x,p_y,p_w)", "dbd4f146bb4079ed36467a4b7c1588a9": "\nS = [s_{ij}] = [\\alpha_i + \\alpha_j]; \\,\n", "dbd5121f3e84a22a35f0f9fcf04a43df": "C(\\mathbf{h})=\\frac{1}{N(\\mathbf{h})}\\sum^{N(\\mathbf{h})}_{i=1}\\left(Z(x_i)Z(x_i+\\mathbf{h})\\right)-m(x_i)m(x_i+\\mathbf{h})", "dbd5352fc207defb5a10a400d8c009bc": "E(x,y) \\leq D(x,y)", "dbd585f8a8add6be266374b294c3bbff": "\\left(\\begin{smallmatrix}1/2 & 1/2\\\\ 1/2 & -1/2\\end{smallmatrix}\\right).", "dbd58a33f89dae315ebce3ed9a153ab1": "(A \\land B)", "dbd5d14cfedd673b39480deb02627eef": "K\\subseteq X^\\star", "dbd61904181fe60d98ece8d66090e081": "\\Theta = (\\theta_1,\\dots,\\theta_l)", "dbd65ceae191944847306aa298a118c4": "t_{avg} \\,", "dbd661c8a3ab7490d52193f42a209d5f": "\\sum_{n \\in \\mathbb{N}} a_{n}", "dbd663d6d323c9cd9707f77badf10602": "C^{(\\beta)}_D(\\{x\\})", "dbd6800561e5ffef3455322a171f7841": "\\exp(-(e'-e)/T)", "dbd69ee9ae289a85ea34dbef8435d7c1": "xz", "dbd6b1dcd27590757de288ac88dfa986": "f:X\\rightarrow Y\\,\\!", "dbd6c07096ef4946dffc1bc02fbc24f1": "P = \\nabla \\psi, ", "dbd7207fd630681b52321cfa5a939e4e": "\\frac{dP}{P} = -\\frac{dV}{V}\\gamma.", "dbd7724333827d8c912f4490d6ba1962": "\\delta_{pitman}=\\sum_{k=1}^n{x_k\\left[\\frac{Re\\{w_k\\}}{\\sum_{m=1}^{n}{Re\\{w_k\\}}}\\right]}, \\qquad n>1,", "dbd7b2bbcae9c9bb7de5a9f889f7bf95": "\\vec{p}_2", "dbd7b679f7c75d2ddb7a8b5261004741": "\\int_{1}^{y} \\frac{dt}{t} = x.", "dbd7b979267c5151bfce28f90081e37d": " t = \\frac{t_\\frac{1}{2}}{\\ln(2)} \\ln\\left(\\frac{K_f + \\frac{Ar_f}{0.109}}{K_f}\\right)", "dbd7c5e79eff7fdcc1bf9acd4200cc2f": "F: X \\to \\mathbb{R}", "dbd7c90a3b066f558b0c41fdf3cde346": "S(U) = \\bigoplus_{i \\geq 0} S_i(U)", "dbd7eeffec8fa2bb51a3d2cac8fc6fca": " \\hbar\\omega ", "dbd84994f3a8379cb31a06a93c464fd8": "\\frac{125}{72}", "dbd87e61db289161263959263c241b2d": "\\ h = (h_{n,m})", "dbd8a409e44220d442596c9a549ef8ee": "P(\\phi_r-\\phi_l)=(\\phi_R-2\\phi_P+\\phi_L) ", "dbd9500bfd163a28ce82b6df4386bc90": "j = a,b;", "dbd9678f5efaac8c2774b1a65f96a308": "(xy)z=x(yz) = xyz \\qquad\\mbox{for all }x,y,z\\in S.", "dbd972b2897bc3140bd8e52b8d9d13db": "|v|,", "dbda14257a569bf4133a91f0e29e6663": "x_n=1/n", "dbda53d847bce18903087a1395ffc68e": "S(H_1\\oplus H_2)=S(H_1)\\otimes S(H_2),\\qquad e^{x_1\\oplus x_2}=e^{x_1}\\otimes e^{x_2}.", "dbda6dd297e1c4ca500596c2c1efee1b": " r_2 ", "dbda77d367899a4e1ac3e80ff12bca69": "(A,a) \\to (B,b)", "dbda98cc389e8d9923134444991e4a13": "D = \\begin{bmatrix} a_{11} & 0 & \\cdots & 0 \\\\ 0 & a_{22} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\cdots & a_{nn} \\end{bmatrix}, \\quad L = \\begin{bmatrix} 0 & 0 & \\cdots & 0 \\\\ a_{21} & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\a_{n1} & a_{n2} & \\cdots & 0 \\end{bmatrix}, \\quad U = \\begin{bmatrix} 0 & a_{12} & \\cdots & a_{1n} \\\\ 0 & 0 & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\cdots & 0 \\end{bmatrix}. ", "dbdaa36ece2625e33d43b51cd568f095": "\\mathcal{H} = \\frac{p^2}{2m} + V(x) + \\mathcal{H}_{int} + \\mathcal{H}_{bath}", "dbdaba67f3c39f352a925e99765b28a4": "\n\\widehat{R}_x \\equiv \\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_x) = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos\\Delta\\theta & -\\sin\\Delta\\theta \\\\\n0 & \\sin\\Delta\\theta & \\cos\\Delta\\theta \\\\\n\\end{pmatrix} \\,,\n", "dbdacc0e2df20851609d07b8854cedcc": " \\textbf{P}_{k-1\\mid k-1}^{a} = \\begin{bmatrix} & \\textbf{P}_{k-1\\mid k-1} & & 0 & \\\\ & 0 & &\\textbf{Q}_{k} & \\end{bmatrix} ", "dbdae82aedd50f879ce123a8decbb910": "H=(n-1)(n-2)", "dbdb009853cbcfc05dbe46e19790966e": "\\underline{\\and}", "dbdb53e62f78e1e314414fa03283827b": "\n\\begin{align}\n\\ell_L (\\mu,\\sigma | x_1, x_2, \\dots, x_n)\n & {} = - \\sum _k \\ln x_k + \\ell_N (\\mu, \\sigma | \\ln x_1, \\ln x_2, \\dots, \\ln x_n) \\\\\n& {} = \\operatorname {constant} + \\ell_N (\\mu, \\sigma | \\ln x_1, \\ln x_2, \\dots, \\ln x_n).\n\\end{align}\n", "dbdbaa4948438da150e6c00b4bec03f6": "q_x^0=", "dbdc21f0d5517a0c412676ddf3640e55": "\\textstyle \\mathbb{R}^\\infty ", "dbdc3eccde7a0ccf7ec7ae3a2830fc44": " \\frac{a + b}{a} = \\frac{a}{b}. ", "dbdc8cea91203c0f095eae011d56db1a": "\\begin{align}\nF(b) - F(a)\n&= F(x_n) + [-F(x_{n-1}) + F(x_{n-1})] + \\cdots + [-F(x_1) + F(x_1)] - F(x_0) \\\\\n&= [F(x_n) - F(x_{n-1})] + [F(x_{n-1}) + \\cdots - F(x_1)] + [F(x_1) - F(x_0)].\n\\end{align}", "dbdca08b714dbaee9c13abe0ed82a129": " K_2 = \\frac{k_2}{k_{-2}} = \\frac{[\\textrm{H}^+]_{eq}[\\textrm{CO}_3^{2-}]_{eq}}{[\\textrm{HCO}_3^-]_{eq}}. ", "dbdca30bc0e68b000b79b22e6555201f": "\\int_{\\mathbf{R}^n} \\bar{\\psi}_k f\\,\\mathrm d\\mu \\to \\int_{\\mathbf{R}^n} \\bar{\\psi}f\\, \\mathrm d\\mu", "dbdcbe8c522a976db1e25bf16734b42c": "\\upsilon(-I)=e^{-i\\pi k}", "dbdcdc9bdeef2e42cec294d090d16325": "\\,t_{ij}^2=\\overline{O_iO_j}^2-(R_i-R_j)^2.", "dbdd10dd967e3745f10c4797c9ca8eb4": "S_p(-n)=(-1)^{p+1}\\left(S_p(n)-n^p\\right)-\\delta_p.", "dbdd1d438ef9013853139d59a9c14de1": "D_1(n) = \\frac{1}{2} n! \\sum_{k=0}^n \\frac{(-1)^k}{k!} - \\frac{1}{2} (1-n).", "dbdd202a33a73fab9f8f828f962bcba8": "\\mu, \\lambda", "dbdd354f6ca5ea35b33271244cf749ab": "|0\\rangle^{\\otimes n}", "dbdd38c3e9ec3cb50860117d8ef7a80d": "F_{\\lambda}=\\frac{\\partial L }{\\partial\nx^{\\lambda}}=\\frac{1}{2v^{0}v_{0}}\\frac{\\partial g_{ij}}{\\partial\nx^{\\lambda}}v^{i}v^{j}.", "dbdd45447659dd279b1deee0f4654cb3": " {N}^n(B_1\\times,\\dots,\\times B_n)= \\prod_{i=1}^n{N}(B_i) ", "dbdd7a8ed6f7ea407192c20d1ecd2bad": "\\left(\\frac{300\\times 300}{2}\\right)^{30}", "dbddd3f57edb63c573360a384c6adcdf": "r=\\frac{\\cos\\theta}{\\sin^2\\theta}.", "dbddff4de9763dad26a13a33d705732e": "\\frac{P}{P_0} = \\sum_{i=1}^{11}P_i e^{-\\lambda t}.", "dbde250c02d3ce2e9147a4aef89a29fb": "\\Delta_n^{(1)}=\\left[\\begin{matrix}\nm_1 & m_2 & m_3 & \\cdots & m_{n+1} \\\\\nm_2 & m_3 & m_4 & \\cdots & m_{n+2} \\\\\nm_3 & m_4 & m_5 & \\cdots & m_{n+3} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nm_{n+1} & m_{n+2} & m_{n+3} & \\cdots & m_{2n+1}\n\\end{matrix}\\right].", "dbde6a8c1322f2e3c2eecfdeff33350b": "-B t = 2 A t", "dbdec5a0c34d114f6bd453623897ba00": "\n\\vert E_{\\pi/2}(r) \\vert = { I_\\circ \\over 2\\pi\\varepsilon_\\circ c\\, r }\\, . \n", "dbdf61b5fdc85df4ce76f250fa267e10": "2n_{\\rm coating}d\\cos\\big(\\theta_2)=m\\lambda", "dbdf7a0edc528a917c06fd1bd4300ecf": "\\sum_{n=m}^\\infty \\frac 1 {\\binom n k} = \\frac k{k - 1}\\frac 1 {\\binom{m - 1}{k - 1}}", "dbdfac5541b4f388eda6a84657bd73b1": " k_{m,n} = \\frac{1}{a} \\rho_{m,n}. \\,", "dbdfce610c9711779ae49abac9bb4056": "B \\in G", "dbe00110a2c6d06788a84ad92d840cd3": "v 2 ", "dbf9e12ea4df19d3ae1348d085f9382b": "\\nabla \\nabla^{\\rm T} ", "dbfa0c3261adb7dae655a82bb2a61d96": "\\mathbf{e}_0 ", "dbfa39ede35221db0df29004077de122": " p_k=\\frac{\\partial L}{\\partial \\dot x_k}", "dbfa5b2f917e338816b19f2d31bd914e": "\\displaystyle v_e = g_n I_{sp}", "dbfa619e269741cd108d46744d0cd85d": "{\\sin \\beta}", "dbfa98fd2ad084b532f8ab0f6d09d2bd": "GR_T", "dbfac3e404b91c01dda3e64b641c5af1": " \\sigma_{ij}\n=K\\delta_{ij}\\varepsilon_{kk}+2\\mu(\\varepsilon_{ij}-\\textstyle{\\frac{1}{3}}\\delta_{ij}\\varepsilon_{kk}).\n\\,\\!", "dbface0ccc30fcfb2e3401a17de19db2": "I \\subset K : J", "dbfb139699be4c17624fff751f4bab2c": "{{P}_{core}}=\\left\\{ \\begin{matrix}\n {{F}_{1}}\\left( {{g}_{1}}\\left( {{r}_{1}} \\right),\\ldots \\ldots ,{{g}_{n}}\\left( {{r}_{n}} \\right) \\right),if\\text{ }condition \\\\\n {{F}_{2}}\\left( {{g}_{1}}\\left( {{r}_{1}} \\right),\\ldots \\ldots ,{{g}_{n}}\\left( {{R}_{n}} \\right) \\right),else \\\\\n\\end{matrix} \\right.", "dbfb246fddc1fe209d01962776f87ba3": "\\int\\frac{x^2}{(ax + b)^2} \\, dx= \\frac{1}{a^3}\\left(ax - 2b\\ln\\left|ax + b\\right| - \\frac{b^2}{ax + b}\\right) + C", "dbfb35e3a2042aeb019d082b7decd278": "T^{\\alpha \\beta} \\,", "dbfb402fed70ea2cf87f6a6aafce5a07": "{\\omega^2}_3 = -\\cos(\\theta) \\, d\\phi", "dbfb68abfbd9d36c5d37cd8c6bf9daac": "\\delta = \\frac{2\\pi}{\\kappa} = 2\\pi\\, \\sqrt{\\frac{2\\nu}{\\Omega}}", "dbfb6f897d1c3fe5ca3bc166951d971b": "~\\mathrm{rect}(x/W)", "dbfb743695819aea4892a791dde25626": " \\vec u(t_0) = \\vec u(t_1) ", "dbfbea648b167a39d5800108a0d92a1f": "E=Z^2", "dbfbfc1f614895bd56ef8704f672aeb1": "\\lfloor\\operatorname{lb}(x)\\rfloor", "dbfc08cc857e59a9daa7a645df09387a": "\\beta_4 = p = ( s - l )", "dbfc2d4aee38538eba35c2f5cfdbe536": "B_t=1", "dbfc7bf2e5a2943d205bcace22e8ae45": "\\pagecolor{Black}\\color{White}\\text{White}", "dbfc7d818ab3f2a67789952e92e982d5": "(\\bigcup_{i \\in I} A_i)^\\circ = \\bigcap_{i \\in I}A_i^\\circ", "dbfc87269ecff106b8d6c93ee6bf930d": "\n\\frac {A^2} {T} = \\frac {B S_x} {K} \\,,\n", "dbfcad588f9c7a1b7930481daf5091c8": "w(0,\\xi)=w_0(\\xi),", "dbfcae0ba82899ce0a6043133ece1323": " R = R_{\\perp} = {GM \\over {c^2 r^3}} = {4 \\pi G \\over {3 c^2 } }\\rho (r) ", "dbfcbbff7d6d811f8b7422c653cf2d48": "\\Re\\, t > 0", "dbfcbc58e5886be40fc8aa9d1cd6d4df": "\\textstyle k \\ge p", "dbfcebc5aa8c225c18beb7332e329ded": " \\frac{dS}{dt} = \\mu N (1-P) - \\mu S - \\beta \\frac{I}{N} S ", "dbfd3aa3a7b9b78db66f2b370607d22f": "C_{4,3} = 13", "dbfd44b6440f1b096994eb4c734d58e6": "(X_0, X_1)_\\theta = \\{ x \\in X_0 + X_1 : x = f(\\theta), \\; f \\in \\mathcal{F}(X_0, X_1) \\}.", "dbfdb079e3fff01022119009c62ce0b4": "D_{192}=0.0016817478", "dbfdc8dd99094cd1518b29482dc3d7f3": "\\sum_{(T,\\theta)\\in \\kappa, \\bmod G^F} {R_T^\\theta\\over (R_T^\\theta,R_T^\\theta)}", "dbfe1efeb31a412334a9d7121f84bbf3": "X_1,X_2,X_3,X_4", "dbfe3741ada8364be34a132911b0b8be": "(a,b)\\in I", "dbfe4b7b112df2231814ed7d80b42fff": "\\scriptstyle B\\, \\cup_A\\, C", "dbfee0795d45667a77c8ffdade3b3565": "\\scriptstyle p=\\frac{P}{D}", "dbfee3ad4381876e7163a4a2706ab98b": "H_{pg}=\\mathrm d(R_g)_p(H_{p})", "dbffbeb50e6a3fdffb579d31c19f3767": " \\{e_k\\}_{1 \\leq k \\leq \\ell} ", "dbfffa5fed00f38f15f2275486fcc36f": "gd_{\\mathbb{Q}}S_3=1", "dc003cbf5df628b6e82754f9c85b0e10": "x^2u''-3xu'+3u=0\\,,", "dc0049455bc3762067574bd8b5b84ab3": " R <1/16", "dc0092bc7fa0a499463c8e0a80c19638": " Q^{j+1}_i = \\left(1-C\\right) Q^j_i + C\\left(Q^j_{i-1}\\right) ", "dc00a7be7282db22346bf61eb29ca2e1": "\nS_{B} = -\\frac{1}{16 \\pi G\\kappa l^2} \\int d^4x \\sqrt{- g}\\, {L}(\\frac{l^2}{4} B_{\\mu\\nu} B^{\\mu\\nu})\n", "dc00fb5a073249c9927642373171a3f9": "\\gamma_{eff}", "dc00fe0acf59e4f2d2affd4f6162219d": "\\{\\alpha_0+i\\alpha_1+j\\alpha_2+k\\alpha_3\\mid \\alpha_i\\in\\mathbb{R}\\}", "dc010f4c1832bfb34256e3b7c50a5201": "\\zeta=1/2Q", "dc0166ed9e2f48263fddaae77b9b5744": "F^{\\mu\\nu} = \\begin{pmatrix}\n0 & -E_x/c & -E_y/c & -E_z/c\\\\ \nE_x/c & 0 & -B_z & B_y\\\\ \nE_y/c & B_z & 0 & -B_x\\\\ \nE_z/c & -B_y & B_x & 0\n\\end{pmatrix}", "dc021d92950d56c8d3447705c258d4fa": "G(m)", "dc029a0ef49bd3b9ead15f291095cfee": " u' = q_1 \\, u + 2 \\, q_2 \\, y_1 \\, u + q_2 \\, u^2 ", "dc0383bdfaf392161482849c4b2e9f1f": " dx_1^2 + dx_2^2 = c^2 dt^2, ", "dc0386a850e67c9553d3973d8b2fe5d0": "(h+dk, dk+\\frac{(dk)^2}{2h}, h+dk+\\frac{(dk)^2}{2h}).", "dc03a6b6b14ecf1ce73593cad211bb55": "e = 2+\\cfrac{1}{1+\\cfrac{2}{5+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\ddots\\,}}}}} = 1+\\cfrac{2}{1+\\cfrac{1}{6+\\cfrac{1}{10+\\cfrac{1}{14+\\cfrac{1}{18+\\ddots\\,}}}}}", "dc04b1ea6c1e23657674358a880ef32a": " g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}\\, ", "dc04fe616c479782920e321b6e21f2f0": "\\text{Hom}", "dc04ff46274db4cc311d02e7a22f87c1": "\\displaystyle{[L(x),R(c,d)]= R(xc,d) - R(c,xd).}", "dc051827dbf2fdbb05a2ae2243de3f2e": "\\mathbf{A = L D L}^{*} = \\mathbf L \\mathbf D^{\\frac 1 2} \\mathbf D^{{\\frac 1 2}{*}} \\mathbf L^{*} =\n\\mathbf L \\mathbf D^{\\frac 1 2} (\\mathbf L \\mathbf D^{\\frac 1 2})^{*}", "dc05863b4eed896717b3c559cd9dcb67": "\\begin{align}\n x &= r[\\theta(t) - \\sin \\theta (t)] \\\\\n y &= r[\\cos \\theta (t) - 1].\n\\end{align}", "dc063756db54c84baaa85337ce94f5bc": "\\sup_{x\\in\\mathbb R}\\left|F_n(x) - \\Phi(x)\\right| \\le {0.3328 (\\rho+0.429\\sigma^3)\\over \\sigma^3\\,\\sqrt{n}},", "dc064b86bea8d493c80d1a14febd8e14": " K_{\\mu;\\nu}+K_{\\nu;\\mu}=K_{\\mu,\\nu}+K_{\\nu,\\mu}-2\\Gamma^{\\rho}_{\\mu\\nu}K_{\\rho} = g_{\\mu 0,\\nu}+g_{\\nu 0,\\mu}-g^{\\rho\\sigma}(g_{\\sigma\\mu,\\nu}+g_{\\sigma\\nu,\\mu}-g_{\\mu\\nu,\\sigma})g_{\\rho 0} \\,", "dc066b8295bfc02815f09d8e3991c864": "\\frac{\\Phi_n(10)}{\\gcd(\\Phi_n(10),n)} = p^c", "dc06d3b9ebd192e75b19c56bfdf24af2": "f_R(x)=(2+x)/3", "dc06e9716e7af876326bb80c57725b16": "{r \\over n}", "dc06f88726059599fc1c1343c8771379": " \\gamma = 180 - \\alpha - \\mu ", "dc073705b7829a9f01b2be8f271fba9e": " \\exp\\left( { -c \\sqrt{\\log\\log X} } \\right) ", "dc0742d8cb1625eed68d208aad6960b4": "\\Leftrightarrow \\neg", "dc074e2a2314a560383276dbcdb902c9": "X^c", "dc076e9bfb86b6fa0592ec8b66ec479f": "X^2+X+d", "dc07a6395280bcde958b5174f1abcfd1": "(\\mathbf y-X\\hat{\\boldsymbol{\\beta}})\\cdot X \\mathbf v", "dc07b5e1cdeeb60afda0831ff0362013": "\\phi(\\omega)=\\Phi(\\omega)+ f(\\omega)", "dc07fb2b7c11d3ea889995319215ce30": "\\sqrt{x} \\approx \\sqrt{x-1} + \\frac{1}{2 \\cdot \\sqrt{x-1}}", "dc082151895017fc17bb10e9d250d9c3": "\\omega = {i\\over 2}h_{\\alpha\\bar\\beta}\\,dz^\\alpha\\wedge d\\bar z^\\beta.", "dc08263316f2edabda12436255111098": "\\mathcal{O}_m ", "dc082ccaa5f635dd1e5087c6735925ed": "\\phi_0 = h / 2 e \\,", "dc08ad62f5ce8bcef51abab8391aa990": "f(x) = \\begin{cases} 1, & \\mathrm{if}~|x| < \\tfrac 12, \\\\ 0, & \\mathrm{otherwise.} \\end{cases}", "dc08d9467cd2299b808551eb6e5912a2": "\\dot V_A/\\dot Q_c ", "dc093a771b7cedf6b57bff58daf71121": "\n\\begin{pmatrix}-(\\alpha^4+\\alpha^{-5}x)&\\alpha^{-3}+\\alpha^{5}x+\\alpha^{7}x^2\\\\\n\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2&-(\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3)\\end{pmatrix}\n\\begin{pmatrix}\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3&\\alpha^{-3}+\\alpha^{5}x+\\alpha^{7}x^2\\\\\n\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2&\\alpha^4+\\alpha^{-5}x\\end{pmatrix}\n=\\begin{pmatrix}1&0\\\\ 0&1\\end{pmatrix},\n", "dc09544fdb33e535b6ba5350e797ac7e": "\\Omega\\,", "dc098a56e5243bca042cb35e56f05d11": " Z_h = \\int e^{-h(x)} \\, d\\mu(x).", "dc0a2932a11109d8192327fc3c2fd5dd": "n! = \\sqrt {2\\pi n} {\\left(\\frac{n}{e}\\right)}^n \\left[1 + O\\left(\\frac{1}{n}\\right) \\right]", "dc0a3ef1c0a22d00f983ff8888e99539": "v_f", "dc0a74e87bdc49fb0f95c76ed0318d62": "V(\\mathbf{x}) = \\sum_{i=1}^n -\\frac{Gm_i}{|\\mathbf{x} - \\mathbf{x_i}|}.", "dc0aa61a0d03c4615d17599bd15bd244": "\\Pi_f=1-\\exp\\left[-\\frac{g_0r_0\\left(1-\\frac{1}{2}\\frac{r_0}{r}\\right)}{\\eta_0h_{PR}\\left(1-\\frac{D+D_e}{F}\\right)}\\right]", "dc0aa8f9ab43805164ec528358413358": "\\mathbf{E} = -\\nabla\\phi-\\frac{\\partial \\mathbf{A}}{\\partial t}", "dc0b2ea4df104935194b639797052690": "e^\\gamma = \\lim_{n \\to \\infty} \\frac {1} {\\ln p_n} \\prod_{i=1}^n \\frac {p_i} {p_i - 1}.", "dc0b380bec5d3222c5275f0263f413ed": "(q)_{\\infty}=\\prod_{n=1}^{\\infty} (1-q^n)", "dc0b5326c0e148cb458bdd1ee648b2d6": " c_{ xy } = \\frac{ s_{xy} }{ m_x m_y } ", "dc0b61e816207b4e9460acb5e474b782": "\\lambda'=y_1-y_2", "dc0bfd2e3092c9de0a61457592dfb675": " f_{(i)}(x) ", "dc0c026f50319c5b6a9794d42b154202": "\nG(z) = \\frac{ z + \\alpha - \\lambda \\alpha - \\sqrt{ (z-\\alpha (1+\\lambda))^2 - 4 \\lambda \\alpha^2}}{2\\alpha z}\n", "dc0c077574a55e81c87b8969a271b285": "\\mathbf{}i", "dc0c0b967005594b5575858a0aadb90e": "g \\in G \\land p\\in P \\Rightarrow \\exists h \\in G, \\; h^p=g\\;", "dc0ca616c10b555ee5b207500c408cc5": "U = \\cap U_i", "dc0cf3a278a6ab9338d5252d9f887c25": "f, g, h_j, j=1, \\ldots, m", "dc0d14d0d7e0846844478f2ee301094a": "D_{\\mathrm{out}}=K_d\\frac{d}{dt}e(t)", "dc0d1ab626a58624fa1184be77179f1b": " (2u^2 \\mp 2)^2 - d(2uv)^2 = 4 \\, ", "dc0d67bedb119bbc286d2396ffdb13f3": " X^\\#_p = \\left. \\frac{d}{dt} \\right|_{t=0} A\\left( \\exp(tX), p \\right)", "dc0d8709e90f8d6b04972dadb417aed1": "v^i[\\mathbf{f}'] = \\sum_{j=1}^n \\frac{\\partial y^i}{\\partial x^j}v^j[\\mathbf{f}].", "dc0d90663479fc44eca6f2450a8040bd": "H^i(\\mathbf{P}^n, F(r-i))=0 \\, ", "dc0dfff65de72197698a699eb54f40a3": "\\eta = ", "dc0e31c8715df548d57adfed48cc4fca": "H_{s}", "dc0e5929210c4c88fe36837d9af1505e": "\\sum_{i=0}^n\\binom{n}{j}(q-1)^i\\mathcal{K}_r(i; n)\\mathcal{K}_s(i; n) = q^n(q-1)^r\\binom{n}{r}\\delta_{r,s}. ", "dc0e8c9b46769786235a3d2c389657e0": "{T_{2i}}", "dc0eeeb6e67f22ac17cd7588722107d9": "v\\,=u_{1}\\beta u_{2}", "dc0f2c841f7e532c5ba44c0fd9b714bc": "\\beta_2=1.4", "dc0f34bbdf9eb37396f858e3311c1f68": "\\boldsymbol{P}^T\\cdot\\boldsymbol{R}", "dc0fe0fd60b7914822397fbdaba52fe3": " q = \\frac{V_y Q_x}{I_x}", "dc0ff38c6d5275d1f7b1c0267aa7a188": " - {\\hbar^2 \\over 2m_0 r} {d^2 \\over dr^2}\\left(r R(r)\\right) +{\\hbar^2 l(l+1)\\over 2m_0r^2}R(r)+V(r) R(r)=ER(r),", "dc10072a7a288bc61a050b20aea4609d": "\\scriptstyle \\sin(2\\pi ft+\\pi/2)\\sin(\\phi),", "dc100eb7b03bf08ad02240095c81db57": "u \\not\\leq v", "dc103410d9cdce626b8ad887594f11ef": "\nE_{\\alpha} = \\frac{3}{2}NkT_{\\alpha},\n", "dc104992a7174a02d5dd61026a3d015e": "\\scriptstyle F_x + i F_y", "dc105891e2a74a303817f51bdc7d3411": "(~x_1,~f(x_1)~)", "dc106b4290572b4c16423f35abbecbb1": "\\scriptstyle{p_{ij}^{+}}", "dc10c420bf623d2326fee2155bb61a6e": " S_j=e(\\alpha^j)", "dc10da82c14856e5d9b4de6d9e09c9b6": "y = \\frac{2x - 1}{1 - \\vert {2x - 1} \\vert}", "dc1105ada1db7c3d034ff0a3c79db8cb": " (f \\cdot e^g)' = (f^\\prime + f\\cdot g^\\prime)\\cdot e^g, \\, ", "dc111726961afcd055783fec3376d5f1": "x < x_0", "dc111e09472a6f13301b067a8b70a76b": " |0 \\rangle ", "dc113105e84137b9450d4081034ad0fd": "\\delta (\\lambda) \\approx [ n (\\lambda) - 1 ] \\alpha ", "dc11b294b1c652cae91a461719ac3a2e": "\\overline{\\mathbf{GT}_{1-2}}", "dc120c741ff22e3509956a9eb76681e2": " x D_p y\\,\\!", "dc130e822b0495efe695b036e86185ac": "\nX^{\\{0\\}} = \\emptyset, \n", "dc132727714803398f73d06c1c5903f8": " f_X(x|\\boldsymbol \\theta) = h(x) g(\\boldsymbol \\theta) \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(x)\\Big)", "dc1335ab3989a582aade5176f82d0d4b": "\\ddot{\\mathbf{x}}=\\frac{D^2\\mathbf{x}}{Dt}", "dc13780e95a6f2fa980577c37bbfd377": "\\log N", "dc137b626269bff9b6231811373b7a8a": "G(\\vec{r},t)", "dc138d5e7e7db80186039c3e992cccd0": "I/2 - I_s", "dc14cca7d419dfbe3d3d5466032ffc44": "\\scriptstyle P_B(\\lambda)", "dc152987d9263b7f9261f2825a9a6ab6": " \\nabla^2 ", "dc157141614b0f8c7bd878691c36b90d": "\\mathfrak{sp}_3", "dc159166cff6c82ad67878b0e8667bf0": "\\int_{|\\xi| \\leq R} (1 - \\frac{|\\xi|^2}{R^2})^\\delta e^{2\\pi i x\\cdot\\xi}\\,d\\xi", "dc1595fcff5ab7cc1ef6c1e23f6d17e6": "\\frac{14}{9}", "dc159a35c77e8713f122c89ee1742e0a": "S(x_1,x_2,\\ldots,x_n)", "dc15af0aa515a455c7b09550cdde4363": "\\nu \\subseteq \\mathbb{F}_q^k", "dc15b99b6416783b7b86e8cde41ebca4": "\n \\lambda_1 = \\lambda_2 = \\lambda ~;~~ \\lambda_3 = \\tfrac{J}{\\lambda^2} ~;~~ I_1 = 2\\lambda^2 + \\tfrac{J^2}{\\lambda^4}\n ", "dc15bdc6b49cbc3adb74f1b94ce131a5": "\\frac{W_q^{(H)}-W_q}{W_q} \\rightarrow 0. ", "dc1652f121bed77bdf8a82974a980d68": "H_q: \\quad (x-1)(y-1)=q,", "dc16bf07b7f271e5d13aea24c496f194": "E_{x^2-y^2,x^2-y^2} = \\frac{3}{4} (l^2 - m^2)^2 V_{dd\\sigma} +\n[l^2 + m^2 - (l^2 - m^2)^2] V_{dd\\pi} + [n^2 + (l^2 - m^2)^2 / 4] V_{dd\\delta}", "dc16dd18cb75035c7a838a7ad4d1fc27": "\\mathbb{R}^{p+4}", "dc1724c7329ccee5557a9152c2084b52": "3^i5^j7^k", "dc1760c87bb859a4f163f6134b7fa632": "\\mu_2=\\mu_2^+-\\mu_2^-", "dc1773b664b777cca01afc7df27521aa": "f(n) = a x^2 + bx + c\\, ", "dc17f0f8ccd2da026b7fd6751c4f5170": "\\frac{d^2 x}{d t^2} = \\left(\\frac{d x}{d t}\\right)^n f(x)", "dc180dfc24ea8f151cd2e2d37baeddb0": "\\Delta N_{21} = \\left( N_2 - {g_2 \\over g_1} N_1 \\right)", "dc189eab971688c00a818e0523b9a659": "F \\in C(\\mathbb{R}^m)", "dc18e484721a9ea7986bb246fc6c04cb": "V^M_N", "dc18edcf5434105654782634af25e3d1": "R = 7/6 \\, R_\\mathrm E \\,,", "dc191e8fbf80e33540c911ca9249dfe8": "f(k;\\lambda)=\\frac{e^{-\\lambda} \\lambda^k}{k!}.", "dc19a9adb9078581ced13945c17ce289": "De = B . Da\\,", "dc19c4324227433289b02f214fc285bc": "c(1)\\left (e^{\\gamma(1)_1}+\\cdots+ e^{\\gamma(1)_{m(1)}} \\right ) + \\cdots + c(r) \\left (e^{\\gamma(r)_1}+\\cdots+ e^{\\gamma(r)_{m(r)}} \\right) \\ne 0.", "dc19c48bb46b10ee0b3f8362564663a1": "|\\Sigma_2|=a", "dc1a0fb6061a58fd96723db1a3726858": "\\varphi_X(t) = M_{iX}(t) = M_X(it):", "dc1a25f2cf3f3def94e985f314d22a31": "(x, y)=(1, 1)", "dc1a3f16093536f764482634fd32e324": "Diag", "dc1aa85f1b7804a11ce33b6451fb1242": "\\begin{align}\ny_{1} &= a_{0}\\sum_{r=0}^{\\infty }{\\frac{(\\alpha )_{r}(\\alpha +1-\\gamma )_{r}}{(1)_{r}(1)_{r}}s^{r+\\alpha } }=a_0 s^{\\alpha} \\ {}_{2}F_{1}(\\alpha, \\alpha +1-\\gamma ; 1; s) \\\\ \ny_{2} &= \\left. \\frac{\\partial y}{\\partial c}\\right |_{c=\\alpha} \n\\end{align}", "dc1ab691bb9d1f086e8951676cea0e35": "\\mathbf{U}_1 = \\mathbf{M} \\mathbf{V}_1 \\mathbf{D}^{-\\frac{1}{2}}", "dc1ac1545acc8c0f5197f3c9dfdaeadc": "(3^k + 1)/(2^k - 1)\\le [1.5^k] + 1", "dc1adacb42268a1e9b429c6ada2422e7": "S(z;x)=\\sin( 2^z\\arcsin (x)).", "dc1b04f06f816f197d231d592d55ad5d": "\\lambda = \\lambda^\\prime \\sqrt{\\frac{1+\\beta}{1-\\beta}},", "dc1b37f029734e0896cc5f739d77ec45": "l + (T-l) = T\\,\\!", "dc1b586523fff4acd692e9cbe9750883": "\n\\begin{pmatrix}\n A & B \\\\\n C & D \n\\end{pmatrix}\n=\nLU,\n", "dc1b64974273c774439f4be9595ab7fb": "L(s,\\tau,r)", "dc1b9e1996e67cdcbe045402b6b37143": "\\scriptstyle \\leq3.4\\times10^{-26}", "dc1bcbf3b49e80052c527d2c58917110": "I_1 = 3, \\lambda_i = \\lambda_j = 1", "dc1c6023b42777386b4138a0cbcf1497": "GF\\left( {q^N} \\right)", "dc1ce08955742bce01c77e54c0ed4eb1": "\\ F(aK,aL)=A(aK)^{b}(aL)^{c}=Aa^{b}a^{c}K^{b}L^{c}=a^{b+c}AK^{b}L^{c}=a^{b+c}F(K,L),", "dc1d152045a1598b9e682d603359bb0e": " \\mathfrak{I}^{-1} = \\mathfrak{I}^{*}\\,", "dc1d3a753deb09480625ee8cc178a62f": "Lu=f\\left( x,y \\right),\\ \\ \\left( x,y \\right)\\in \\Omega, ", "dc1d59ea11a082e8296610f91d161b61": "H_0^1,", "dc1d6ea9fedc4e3920995673b25e0312": "\\scriptstyle \\tfrac{1}{2} (r(t))^2 \\dot \\theta(t)", "dc1e4cbf89271b6df6a9755d6860c276": " D_r ", "dc1e4de4f8dd9015c50db25780fc0656": "f:M \\to G", "dc1e59222ff9a0cc97c5fbc15332d1b5": "\\, R_s \\ge \\,1 ", "dc1ecebb4e7963a0cdf761771c610d30": "f_k=|f|^{\\theta_k},\\quad k\\in\\{1,\\ldots,n\\}.", "dc1f395671ba5df8887e84382ae957f3": "O(n \\log |G| + tn)", "dc1f7778fa1e79f5c7e08f585a5c91de": "ad = bc \\,", "dc1fb12473c798faffc3d04f4a88e0f3": " t_{avg} \\cdot \\ln 2 ", "dc1fd61f4e84cffa94af679552bd0119": "B_2 = \\{ | b_{j} \\rangle _{j=1}^{d} \\}", "dc1fd7a2bd1f31f6cb05bb8f3b8093bf": " |1,1\\rangle = \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}", "dc2028440c56df389448abafd0f48d30": "B = \\mu_0 \\frac{N I}{l}.", "dc202e0354434a03bc40f3e52127d860": "N=30\\cdot 12=360", "dc20f9a23b845f87a2f0390e51f779a0": " {C_x \\over C_2} = {R_4 \\over R_3} - {R_2 \\over R_x} \\, .", "dc2106cb92f38d5c2246cd7d68e616bf": "enc \\leftarrow \\Bigl[M\\Bigr]_i^{i+r} \\oplus mask_e", "dc218704a88b99d703f219868466a9cd": "\\lnot P", "dc21a6164df6b63fcb190a1c0eb6f92c": "\\langle x \\rangle = L/2", "dc21fd289e028b78122d1af29e09c942": "P_f=\\int\\frac{k^2}{3(k^2+m^2)^{1/2}}\\frac{2d^3k}{(2\\pi)^3},\\quad 0a\\}", "dc229a73b3afe54f23e86082708f348e": "H_{n+1}^2-2P_{n+1}^2=(H_n+2P_n)^2-2(H_n+P_n)^2=-(H_n^2-2P_n^2) \\,", "dc229caad2cce74d9035d7ebad676cf0": "H=U(1)\\times\\mathbb{R}\\times\\mathbb{R}", "dc22e031555505ee843faa9e47174da1": "{I}_{2}=D\\left(b\\right)f\\left(b\\right){b}^{2}", "dc22ec780fb00a36f0774d511affd3c3": "\\tilde{g}_{22}=r_{0}^2", "dc23198deedbd6c570223a1257b7f948": "X_{t-1} = \\lbrace x_{t-1}^{[1]}, x_{t-1}^{[2]}, \\ldots, x_{t-1}^{[M]} \\rbrace", "dc236a5c803ecca68210e6eea392cfc7": "\\boldsymbol{\\mu}_d", "dc23875ba9bf7fa2d2cfa02529a6e6c9": "H= \\int \\cdots \\int f_i \\ln f_i \\,d q_1 \\cdots dp_r", "dc23b10a2f45f53633a211cd1da854ff": "y^2+by-a^7 = 0\\,", "dc23cb62d710724bf2451a8adb195946": "\\bar{r_i} = \\Delta E - \\Delta C + \\Delta I \\,", "dc23dcdfcdcbcfed943b2dfa9d3898a0": "\\sum_{i=1}^n \\rho(x_i)", "dc23eba9efc91fe93eabeb964ca59c3f": " (ax + by + c) dx + (ex + fy + g) dy = 0\\, , ", "dc2449504a8a4918f158c4a1485e110e": "2\\,\\frac{J_1(R\\,t)}{R\\,t}", "dc245bf919c0fbb7f0bce67e6ab402c8": " y(t=0)=0 ", "dc247599a228800994ee630d5c81b6a3": " p(S) \\frac{\\part u}{\\part n} + \\sigma(S) u =0,", "dc25066e2b353094b77bb3f373aa6b81": "t[n+1]", "dc254c0babd1b5d3bc6d7a99d4adc408": "{\\mathbf A}_\\mu = A_\\mu^a \\sigma_a\\,.", "dc2572ce610ad7c7f5ec131d866ce364": "\n \\lambda_1 = \\lambda_2 = \\lambda_3 = \\lambda ~:~~ J = \\lambda^3 ~;~~ I_1 = 3\\lambda^2\n ", "dc262a5e707fde40f96bc0dd233409ac": "\\exp\\!\\left(-t^{1/\\theta}\\right)", "dc263a7909f55e8ef86b6f10e5fe67ef": "\\frac{dC_V}{dC_W}", "dc263ccbdfb9ede14d6e71bf54fb28c8": " \\ G(t) = t-a", "dc266cc586fced784ca1317ce5db0454": "[L_x, L^2] = 0, [L_y, L^2] = 0, [L_z, L^2] = 0", "dc26c48d4a65e23d978930f0681b497b": "c(a_i)", "dc26d43d3c448592059b2965925e0163": "T^\\nu", "dc271f86b51b3c20d6510f470ed8adbe": "y_c=\\left(\\frac{q^2}{g}\\right)^\\frac{1}{3}", "dc2781728b0694a6b8850c4fbb1e8981": "\\eta^{\\mu\\nu}\\partial_\\mu\\partial_\\nu\\phi=4\\pi\\rho^*k(\\phi)\\,", "dc27ac25bd60363bf711610cdcafd76a": "\\mu_a(\\lambda_1) = \\ln(10)\\varepsilon_{HbO2}(\\lambda_1)C_{HbO2}+\\ln(10)\\varepsilon_{Hb}(\\lambda_1)C_{Hb} \\,", "dc286848239003549ca4e7120f1f989b": "\\zeta(a,\\bar{b})+\\zeta(\\bar{b},a)=\\zeta(a)\\phi(b)-\\phi(a+b)", "dc2951c36e46ef4901e66973ef683844": " \\frac{\\operatorname{d}}{\\operatorname{d}t} x(t-\\tau) = x'(t-\\tau) ", "dc2964455effd9278ee214060231f738": "\\frac {Lf \\cdot \\sqrt{S}} {\\sqrt[3]{D}} \\le 2,8", "dc29d17cf16f03a33df25224df2ad1a9": "\\lim_{i\\to\\infty}\\frac{\\mu(gF_i\\,\\triangle\\,F_i)}{\\mu(F_i)} = 0", "dc2a0485de662e81e9e04a539876e14e": " x \\cdot 0 = 0", "dc2a0c01506e96e4b557d27d25cb19f2": "S':=\\ker(T\\upharpoonright S-\\alpha)", "dc2a15a5dbfff09e4cbe2432f5a3641c": "G_{0} = 1", "dc2a2ced5b5d1d26fd680be3b683f871": " \\mathbf{A F} = \\mathbf{0} ", "dc2a3fe517305472fdcfeee4bd0507ab": "|\\uparrow\\rangle\\langle\\uparrow|, |\\downarrow\\rangle\\langle\\downarrow|", "dc2a541bc4aea4537bb3b1187f5be31d": "a < b, a < c, b < d, b < e, c < d, c < e, d < f, e < f", "dc2a691b61085192e392a8b384d0c6b6": "\\chi(k)=k^{-1}|f(k,\\pi)|\\sum_j\\ W_j\\sin[2kR_j+\\alpha(k)]\\exp(-\\gamma R_j-2\\sigma_j^2k^2),", "dc2a71e3a62190e34b5b7b5538e3fe10": "R = k[x_0, ..., x_n]/P", "dc2a7898aad41106b27ba18406be9db9": " \n\\begin{align}\n & \\gcd(3,4) & \\leftarrow \\\\\n= & \\gcd(3,1) & \\rightarrow \\\\\n= & \\gcd(2,1) & \\rightarrow \\\\\n= & \\gcd(1,1)\n\\end{align}\n", "dc2a9b3a3e75843ddb005aa0d456ac87": "\\oint \\vec{v}_s\\cdot\\vec{\\mathrm{d}s} =2\\pi v_sr.", "dc2aacf3e2e5796b3e34512156f4a220": "3 \\times x", "dc2ac9c3b6761b0979140f8a68d2d7f4": " \\left \\{\\pi(f) \\xi : f \\in \\operatorname{C}_c(G), \\xi \\in H_\\pi \\right \\} ", "dc2aca446bfd1e7d04673fe73ce09766": "\\Phi_B = \\oint_{\\partial S} \\mathbf{A} \\cdot d\\boldsymbol{\\ell},", "dc2adba4a67c6dc8184fac39ca99eee5": " p=e^{-\\beta(E_{\\text{new}}-E_{\\text{old}}))},", "dc2ae5b4ea23e2e444582d6dd9462b33": " H = \\bar{H_0} + \\bar{H'} ", "dc2b004e791e4ceb6cd8662349f2f54f": " T_n = \\frac{n(n+1)}{2} = 1 + 2 + \\cdots + n. \\,\\! ", "dc2b0eb75146cf7c3335aa887593e235": "\\dot{\\varepsilon_{\\rm{p0}}}", "dc2b701d26640010a32ea75b5d5bcdec": "u = \\arctan(x) \\Rightarrow du = \\frac{dx}{1 + x^2}", "dc2bcf83aa39f860ec3d3be3be480302": "w=Az^2", "dc2c1c55d1fd18bc0084c64f61dfbbb4": "a^{2}+b^{2}=c^{2}", "dc2c20dc4e35c2f086d591c3d253da1d": "y = \\frac{1}{\\infty - x} = \\frac{1}{\\infty} = 0", "dc2c852441340e2d5bec9d3368fca3d5": "\\begin{align}\n \\frac{\\operatorname{d} \\theta}{\\operatorname{d}x} &= \\frac{\\operatorname{d}^2 y}{\\operatorname{d}x^2} = -\\left( \\frac{2k}{rd} \\right) y \\\\\n \\frac{\\operatorname{d}^2 \\theta}{\\operatorname{d}x^2} &= -\\left(\\frac{2k}{rd}\\right) \\frac{\\operatorname{d}y}{\\operatorname{d}x} = -\\left(\\frac{2k}{rd}\\right) \\theta\n\\end{align}", "dc2c9bf745f79fb77274b6b315b9e871": "F=\\gamma ma", "dc2ccd93f217a7af798898be896a6b84": "\\left\\{\\frac{x_1+(1+x_2)x_3}{x_1x_2},\\frac{x_1 + x_3}{x_2},x_3 \\right\\},", "dc2cd6377f2a318c2dadc9428afa2105": "(9) \\,", "dc2ce070bd334ba9cb26342888085b16": "e \\in W", "dc2cf26fa80379cb0e9d58082f2d13b9": "1(u{\\Delta}^{*})=u_0", "dc2cf74e971e5291da39519a97b044db": "f(1)=1", "dc2d35e8f7c1be611f2152f2b3a436d8": "\\begin{matrix}4&4&7\\\\6\\end{matrix}", "dc2d4df529b8c71ab57b9617fe903bac": "\\operatorname{Perm}(A') = 2 \\cdot 1 + 15 \\cdot 15 = 227", "dc2d770df5cd62bd3d92adf5f136090f": "T_{1}(q,x) = T(q,x)\\ \\forall q\\in Q_{1}\\forall x\\in\\Sigma", "dc2d89d70f05805cdf6a2d421449b36f": "\\tau_{x}", "dc2dacb7922be0c666d4976739f91bcc": "\\ \\theta \\,", "dc2dc94468e190079f70831b8ef788b7": "\\theta \\phi(x) = \\phi(x) \\theta", "dc2df03face260cca349a1f67b3ee05d": "A rem f\\,", "dc2e2058360480ca2437af730e4fd8ff": "\\frac{3}{2}(\\rho-\\tau)=r_g", "dc2e22f6d3a07795de2f9660353377fb": "i \\gamma^\\mu \\partial_\\mu \\psi - e \\gamma_\\mu (A^\\mu+B^\\mu) \\psi - m \\psi = 0. \\,", "dc2e2f8366fbf6e40bf5a6454daf6a81": "\\tilde{\\Omega}:=\\hbar^{-1}\\vec{d}_\\text{eg}\\cdot\\vec{E}_0^*", "dc2e3df9145cff38488bdab646093b59": "F \\land E", "dc2ed560b9e9c786412f5d45f654610d": "\\ell'(m')", "dc2ed7d57956db21aafc34054258edb0": " \\Box (B(x,p) \\land B(y,p) \\rightarrow \\exists q \\ne p (B(x,q) \\land B(y,q)) ", "dc2edc16ce39c1bf405226f2f21c0dc8": " 2 \\pi", "dc2f3d1c7bc637b3a9a5fc60f10d2c5f": "(-1)^{2j}", "dc2f9cd511b677094a1b3bfb839d36a5": "\\ k=g \\ ", "dc2fe7f4b26d94ac3d54196dd2e09766": "x^{-{1\\over2}}", "dc30749d702da0ce1534da7e34479337": "\\rho : (a, b) \\rightarrow (c, c)", "dc30941cc4c2f6855903c46de0c69a1c": "\nW = \\int_{\\mathbf{r}_{1}}^{\\mathbf{r}_{2}} \\mathbf{F} \\cdot d\\mathbf{r} = \\int_{\\mathbf{r}_{1}}^{\\mathbf{r}_{2}} F(r) \\hat{\\mathbf{r}} \\cdot d\\mathbf{r} = \\int_{r_{1}}^{r_{2}} F dr = U(r_{1}) - U(r_{2})\n", "dc30d39bb7b661c74ad2462fc74e297e": "h_{z}=1", "dc313c8f7baa93a8828dcc6666313224": "\\frac{dy}{y} = 0.85dt", "dc3176dbdc0313edbe7758c6ac589b34": "G = \\{U\\setminus S\\mid S\\in F\\}", "dc31c66f9860207fe9a32b1e2e0a1db6": "\\tfrac1{2\\alpha}", "dc31ed63ed42b93a590f1143f6f62db7": " \\|\\vec x\\|_1", "dc327ef244932ae548c0e753c9e32a52": "\\Delta r \\approx \\frac{\\Delta}{\\sqrt{1+I_\\max/I_s}}", "dc3291fa474c1f78161d1f0e4b8b09aa": "t_{l}=0", "dc32b47f148a7835de98f4aaffd67473": "\\begin{align}\n\\Delta p &= \\gamma \\nabla \\cdot \\hat n \\\\\n&= 2 \\gamma H \\\\\n&= \\gamma \\left(\\frac{1}{R_1} + \\frac{1}{R_2}\\right)\n\\end{align}", "dc32cc9051639ab5eb1f31f0d6097580": "\\varepsilon_r''", "dc32e3f69afbd663b18c373a03942c0b": "|\\Phi^+\\rangle_{AB} = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_{B} + |1\\rangle_A \\otimes |1\\rangle_{B})", "dc32ea69bc83b437f8aa762ed5f7125e": "a(f)|0\\rangle=0", "dc32eb29d25d263fdb85bd650b055dc0": "\\dot S'_{\\mathrm{gen},\\, \\Delta T}", "dc33066c3993e0d50896e533fd692ce0": "NE", "dc33505a8b9feacb2d9e380c166da94e": "\\nabla^2 u_3 = \\cfrac{\\partial^2 u_3}{\\partial x_1^2} + \\cfrac{\\partial^2 u_3}{\\partial x_2^2}", "dc3355554e17d3fabea970a1a2d99fc4": "\n\\mathbf{v}_i = \\sum_{j=1}^N \\mathbf{H}_{ji}\\mathbf{w}_j\n", "dc337d3039fa4d0272c6db8877016781": "180^o", "dc33862ca129df54235cbb73ae6ef821": "\\dot{q} \\ne 0", "dc3388c570b80cdcae5b0ce7db1ffc16": "\n\\begin{align}\nf_Y(y) & = 2 \\frac{d}{dy} F_X(\\sqrt{y}) - 0 = 2 \\frac{d}{dy} \\left( \\int_{-\\infty}^\\sqrt{y} \\frac{1}{\\sqrt{2\\pi}} e^{\\frac{-t^2}{2}} dt \\right) \\\\\n& = 2 \\frac{1}{\\sqrt{2 \\pi}} e^{-\\frac{y}{2}} (\\sqrt{y})'_y = 2 \\frac{1}{\\sqrt{2}\\sqrt{\\pi}} e^{-\\frac{y}{2}} \\left( \\frac{1}{2} y^{-\\frac{1}{2}} \\right) = \n\\frac{1}{2^{\\frac{1}{2}} \\Gamma(\\frac{1}{2})}y^{\\frac{1}{2}-1}e^{-\\frac{y}{2}}\n\\end{align}\n", "dc33a752b565cc703fc2f5478ef3f175": "\\textit{true}", "dc33d4d420f796e304cacc9fe9c1d3ee": "\\lim_{h \\to 0}\\frac{f(x+h) - f(x)}{h}", "dc346fab17e39c08f030bc2a9a442ef7": "\n d\\mathbf{f} = \\mathbf{t}~d\\Gamma = \\boldsymbol{\\sigma}^T\\cdot\\mathbf{n}~d\\Gamma\n", "dc34887a3123c3742fb07089f80705ec": " \\left\\langle \\vec L \\cdot \\vec S \\right\\rangle = \\frac {\\hbar^2} {2} ( j(j+1) - l(l+1) - s(s+1) )", "dc34a4ed57c6a63832a5dbb8c36088ad": "\\approx 1.3293403881791370205\\,,", "dc34ae4c29404e372c0e565a8ddb579c": "a \\uparrow \\uparrow \\uparrow b = \n \\left. \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{\\vdots}_{a} }} \\right\\} b", "dc34e220774c7b94d5553d8afab1dc9b": "v = 0", "dc34f8fe0921537999d360b13af600d2": "P^{(N)}_{0}(\\xi_{1},\\xi_{2},...,\\xi_{N})=\\frac{1}{Z^{(N)}_{0}}e^{-\\beta \\mathcal{H}_{0}(\\xi_{1},\\xi_{2},...,\\xi_{N})}=\\prod_{i=1}^{N}\\frac{1}{Z_{0}}e^{-\\beta h_{i}\\left( \\xi_{i}\\right)}\n\\ \\stackrel{\\mathrm{def}}{=}\\ \\prod_{i=1}^{N} P^{(i)}_{0}(\\xi_{i})", "dc3553a3dbaf93f6b006b151b61730ab": "\\textrm{Var}\\left(t\\right)", "dc359355fffdcb4015bb43f8b48599d0": " \\mathbf{\\hat T} (\\lambda) \\mathbf{\\hat T} (\\mu)= \\mathbf{\\hat T} (\\lambda+\\mu) ", "dc35c8cbf8589a8c822509c4dce87073": "\\nabla \\cdot \\mathbf{v} = \\nabla \\cdot (\\nabla \\times \\mathbf{A}) = 0,", "dc35ca1b803fe212efa31e20cd892651": "\ni \\hbar \\frac{\\partial}{\\partial t}|\\Psi (t)\\rangle = \\hat H |\\Psi(t) \\rangle,\n", "dc369125e0bcada71de9ca6b216e9737": "(-e^{\\frac{i\\pi}{N}})^k\\cdot W_0(k).", "dc36d3f5eb2f7185504a538e0c5fb8a2": "B_i(x)", "dc37348f090efab712bf9e3266fcca84": "f(x)=ax^4+bx^3+cx^2+dx+e,", "dc379845173f4e079980696447ec656f": "H\\left(\\frac{du}{dt}\\right) = \\frac{d}{dt}H(u)", "dc37cd4c70d93462ada3f9164c74c8f4": " \\liminf_{n \\to \\infty} \\frac{f(n)}{m(n)} = 1 ", "dc37d6e14edb32febcbe23e9afb2b739": "Z^{(\\ell)}_{\\mathbf{x}}({\\mathbf{y}}) = \\sum_{j=1}^{\\dim(\\mathbf{H}_\\ell)}\\overline{Y_j({\\mathbf{x}})}\\,Y_j({\\mathbf{y}})", "dc37dbf50372f1f837f9998276735ece": "\n\\partial_t E+\\partial_i((E+p)u_i)=0.\\,\n", "dc37ef80321d5a553dfa47f45b9c01b9": " H = G \\chi_{[0,\\infty)} ", "dc3804831a5d6ee5b69f26493d2f92f3": " \\mathcal{F} = \\Phi_B \\mathcal{R} = NI ", "dc38149bd5498c9ae2a7eb0c6cfdf5d9": "T(x, y) = \\begin{cases}\n a_i + (b_i - a_i) \\cdot T_i\\left(\\frac{x - a_i}{b_i - a_i}, \\frac{y - a_i}{b_i - a_i}\\right)\n & \\text{if } x, y \\in [a_i, b_i]^2 \\\\\n \\min(x, y) & \\text{otherwise}\n\\end{cases}", "dc381d65a02daf0a882309acdf4e9781": "[M + nH]^{n+} + e^- \\to \\bigg[ [M + nH]^{(n-1)+} \\bigg]^* \\to fragments", "dc3868cef618456208d85f52d54d7f8c": "e_1,e_2,e_3,\\ldots", "dc38961531ab540d27c81b280b5f8867": "\\scriptstyle \\Delta \\lambda_B", "dc3897b780d2b11749c3978d75362bd9": "\\xi_{inf}(\\alpha)=\\mbox{inf}\\{r|M\\{\\xi\\leq r\\}\\geq\\alpha\\}", "dc38afcb92c02e03e43d1cbd193066d6": "\\{p_n ; n = \\dots, -1, 0, 1, \\dots \\} ", "dc38be08460acc7920ababcd9ea7e686": "r_0=a/2", "dc39b0b105f78039adaa53c0c27cc062": "A = \\frac{R T (n^2 - 1)}{3p}", "dc39c22ed626b04a63bbcfef928f1233": "s_N(f)(x) = D_N*f(x) = \\sum_{n=-N}^N a_n e^{inx}", "dc39e7a3e8ac2fd6a63ade3ec48074e5": " J_y \\equiv J_2 = i\\left.\\frac{\\partial \\widehat{R}(\\theta,\\hat{\\mathbf{e}}_y)}{\\partial \\theta}\\right|_{\\theta = 0} = i\\begin{pmatrix}\n0 & 0 & 1 \\\\\n0 & 0 & 0 \\\\\n-1 & 0 & 0 \\\\\n\\end{pmatrix} \\,, ", "dc3a03197c370ea030fe74e1ba594286": "\\ell'", "dc3a031ca7d8c6333cf475d4a58a463d": " J_{2z}", "dc3a4df5e9538b0a2dd26099a89a966d": "4V^2 \\over \\pi^2", "dc3a8d27ed00e6211ad974e1ae2965a3": "f^*(x^*) = \\frac{|x|^{n-2}}{R^{2n-4}}f(x) = \\frac{1}{|x^*|^{n-2}}f(x)=\\frac{1}{|x^*|^{n-2}}f\\left(\\frac{R^2}{|x^*|^2} x^*\\right).", "dc3a917c44fea04202cb1aa60a215817": " (f \\otimes g)(x,y) = f(x) g(y) ", "dc3a98991789247b35d9c0d0d78ee3c3": "p(\\hat{x_0} + \\sigma) \\approx p(\\hat{x_0}) + J_p(\\hat{x_0})\\sigma", "dc3aa6b72a15cbb380bc49e73dc37f30": "A = \\frac{1}{4} (\\frac{256}{81}) d^2 ", "dc3b2ed5984dafadf37ac174d8231fa2": "f_{\\#}", "dc3bfc0f021b41aeb1b64b535f580f64": "\\Delta U = Q\\, - \\, W\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\mathrm{(sign\\,convention\\,of\\,Clausius\\,and\\,generally\\,in\\,this\\,article)}}\\, ,", "dc3c4dec0dc10e6207d5909f9d97764f": "\\{x_n\\}_{n\\in \\mathbb{N}}", "dc3c654c2b9c649e28d120309b0e9b6d": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \n-\\mathbf{r}_{\\mathrm{mean}} \\cdot \\mathbf{r}_{\\mathrm{mean}} + \n\\frac{1}{N} \\sum_{k=1}^{N} \\left( \\mathbf{r}_{k} \\cdot \\mathbf{r}_{k} \\right).\n", "dc3c96a5240fb05418378dde7a232caa": "K[A_1,\\ldots,A_k].", "dc3d18f18056f8180273cd95d0c412f5": "Y = \\mathbb{R}", "dc3d56e97bd793afea420b8a542c4c1c": " D = \\sqrt{ \\frac{18 \\eta \\, ln(R_f/R_i)}{( \\rho_p - \\rho_f) \\omega^2 t} } ", "dc3d5f28303b3a0ef71af463b31964f2": "\\nu_{\\varepsilon} (S) = \\frac{J_{\\varepsilon} (S)}{J_{\\varepsilon} (X)}.", "dc3d863f15bd1672b967f596cb2c5007": "k(x_i,x_j) = \\phi^T(x_i)\\phi(x_j)", "dc3d98bf070ec6252afa550c7f7e7914": "M_{24}", "dc3dbf0ab274b38b0935f666a75c5128": "R_{\\beta'} = \\cos \\frac{\\beta}{2} - i R_\\alpha \\hat x R_\\alpha^\\dagger \\sin \\frac{\\beta}{2} = R_\\alpha R_\\beta R_\\alpha^\\dagger", "dc3dda4e165b3fba7dbcb7224615633a": "K(w) \\geq |w| - c ", "dc3e079053faad8975a3001d8a4b9dd2": " g(x,y)= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} h_M(x-Mx', y-My') ~ f(x',y') ~ dx' dy' ~~~~~~(4.2) ~", "dc3e28561c689327774e77a3edff5f46": " \\rho c \\frac { \\Delta x } { \\Delta t } > \\frac {2K} { \\Delta x } ", "dc3ebfebf51560bbe8ba6771c8c4728a": "\\left( \\Pi\\Pi^\\top \\right)^{-1}\\Pi", "dc3edf7ce18706ef1e176fdc61e09ca6": "\\omega_3=\\omega_1+\\omega_2", "dc3ee444dff2678579c3326561ae5d81": " L_1=\\partial_y+\\lambda\\partial_x-u_x\\partial_{\\lambda},\\qquad (3a)", "dc3f1204d056d2c0a90d446038d9ca03": " M = (b-a) f \\left( \\frac{a+b}{2} \\right) ", "dc3f2fa2fb69c10fd3a8003f461ef76f": "|z|\\log|z|", "dc3f5fa0fc55d6690a614561d9f15724": "\\frac{x}{\\ln(x)},", "dc3f6fb823e2257a23de85781c3ab203": "\\vec v_P", "dc3f759f1ce5d08a561d6a52232e90c2": "SFP ={{\\sum P} \\over q_v}", "dc3f7ec89912aa55617b7b8243066311": "\\text{Throughput accounting Ratio} = (\\text{Return per factory hour}) / \\text{Cost per factory hour}", "dc3faec9797d42560acf01f2242374b2": " C = 65m+\\frac{230m}{2}=180m ", "dc3faf9ba321e9411a237dc3736b6f3c": "\n \\lim_{kh \\downarrow 0} \\omega^2 = \n k^2\\, gh\\, \\left\\{ \n 1 \n + \\frac98\\, \\frac{\\left( ka \\right)^2}{\\left( kh \\right)^4} \n \\right\\}\n + \\mathcal{O}\\left( (ka)^4 \\right).\n", "dc3fc60b8fa57258a29d95c2651e850a": "\\eta_Y : Y \\to \\operatorname{Hom}_S (X, Y \\otimes_R X)", "dc3fcaedb1e527d23f2935c48108e32a": "n, m \\in \\mathbf{Z} \\,\\!", "dc3ffed93b84d4705621e42c4c90c45c": "\\int \\frac{dp}{A(p)^n}=\\frac{1}{\\Gamma(n)}\\int dp \\int^\\infty_0 du \\, u^{n-1}e^{-uA(p)}=\\frac{1}{\\Gamma(n)}\\int^\\infty_0 du \\, u^{n-1} \\int dp \\, e^{-uA(p)},", "dc40140c6b681a54cb18242cece20242": "M_1=\\frac{Q^2}{gA_1}+\\overline y_1A_1=\\frac{Q^2}{gA_2}+\\overline y_2A_2=M_2", "dc40147b2a1e45b7adc7a7edd18c1b28": "R_i= \\frac{\\beta(\\mu_{\\uparrow\\downarrow}-\\mu_{\\uparrow\\uparrow})}{2ej} = \\frac{\\beta^2l_{sN}\\rho_N}{1+(1-\\beta^2)l_{sN}\\rho_N/(l_{sF}\\rho_F)},", "dc404e4e12ddb97429692ebc02ef6888": "\n{d\\over{dR_\\mathrm{L}}} \\left( {R_\\mathrm{S}^2 / R_\\mathrm{L} + 2R_\\mathrm{S} + R_\\mathrm{L}} \\right) = -R_\\mathrm{S}^2 / R_\\mathrm{L}^2+1.\n\\,\\!", "dc40502bdee0d11cb739ab44c4600d72": "\\forall A\\, \\exist B\\, \\forall c\\, (c \\in B \\iff \\exist D\\, (c \\in D \\and D \\in A)\\,)", "dc40fb09ac0a1beb0e0d1eebf0d1e385": "\\mathfrak A_P", "dc4154c57187f92a4708d4b2a72a21d4": "\n W_{n+m}W_{n-m}W_r^2 = W_{n+r}W_{n-r}W_m^2 - W_{m+r}W_{m-r}W_n^2\n \\quad\\text{for all}\\quad n > m > r.\n", "dc4162289e14f993bc927125a2d61482": "\\overline{f}(x)= \\sum_{n=0}^\\infty a_n \\frac{x^n}{n!}", "dc41a741dec164c7bed1e8b6ccbd1d9a": "{\\rm kg/m}^3", "dc41c00b6ac83772846546fe0943cb9b": "\n\\begin{align}\nq_\\tau^*(\\tau) &\\sim \\operatorname{Gamma}(\\tau\\mid a_N, b_N) \\\\\na_N &= a_0 + \\frac{N+1}{2} \\\\\nb_N &= b_0 + \\frac{1}{2} \\operatorname{E}_\\mu \\left[\\sum_{n=1}^N (x_n-\\mu)^2 + \\lambda_0(\\mu - \\mu_0)^2\\right]\n\\end{align}\n", "dc41ded8e26bcd8505c5bddcea0fed55": "\\bar{\\mathsf{h}}(a,x) \\mapsto \\bar{\\mathsf{h}}'(a,x)=\\bar{\\mathsf{h}}(f^{-1}(a),f(x)),", "dc422c9a14a82ca742c2beacdca20383": "E= \\left \\{(a,b)\\in \\mathbf{F}_q\\times\\mathbf{F}_q \\ : \\ a-b\\in (\\mathbf{F}_q^{\\times})^2 \\right \\}", "dc423595e6d3ec6c6ed4a3853b15b2ef": " \\begin{align}\n\\mathbf{0} &= (\\mathbf{a} \\times \\mathbf{b}) - (\\mathbf{a} \\times \\mathbf{c})\\\\\n&= \\mathbf{a} \\times (\\mathbf{b} - \\mathbf{c}).\\\\\n\\end{align}", "dc4250efc38758a7fa25cc3c285951ad": "\\{\\tau \\leq t\\} \\in \\mathcal{F}_t", "dc4267488b9b26224dc67f1b6188a7f3": "\nk(\\mu,\\lambda)=\\int_{y=0}^1\\int_{x=0}^1 k(y,x)\\,d\\mu(x)\\,d\\lambda(y)=\n\\int_{x=0}^1\\int_{y=0}^1 k(x,y)\\,d\\lambda(y)\\,d\\mu(x).\n", "dc426b975232f94a4876c14311708406": "^*K_{A}", "dc42978c67f1b8fc875820171cb2c937": "|n\\rangle", "dc42de96a780f936e7ec6dde2243f5fe": "\\|\\mathcal Ff\\|_q \\le \\left(p^{1/p}/q^{1/q}\\right)^{n/2} \\|f\\|_p.", "dc42ee53bf46fbc110c46abc4a906075": "c_4 = -0.22475541, \\,\\!", "dc43081491d259c85b8e1b3d556adba1": "\\mathbf{MTF(\\xi,\\eta)} = | \\mathbf{OTF(\\xi,\\eta)} | ", "dc430eb2e875180a82bcd6575e29a0e7": "\\vec{v} = (v^0,v^1,v^2,v^3) = \\left( \\frac{dx^0}{d\\tau}\\;,\\frac{dx^1}{d\\tau}\\;, \\frac{dx^2}{d\\tau}\\;, \\frac{dx^3}{d\\tau} \\right)", "dc4368384292580ee1e399ac3f76a2bc": "R_{CF} \\gg R_{P}", "dc439e9ee963565217e61e1feb223c91": "\np(\\mbox{weight} | \\mbox{male}) = 5.9881e-06\n", "dc43afa294e0a49af9210fdda1d11ffe": "y \\cdot r^e", "dc44180d7398c2ecb87da8a0c69a9a69": "\\phi=\\omega \\circ \\alpha^{-1}:H\\to K", "dc442609fed3f7dfbd72279671283c68": "\\begin{align}\nh_1&=h_2=a\\sqrt{\\sinh^2u+\\sin^2v} \\\\\nh_3&=1\n\\end{align}", "dc44581097acd0cd6bd01406588073ba": "\ne_1=4^{-1/3}e^{(2/3)\\pi i},\\qquad\ne_2=4^{-1/3},\\qquad\ne_3=4^{-1/3}e^{-(2/3)\\pi i}.\n", "dc44a5476f5bc75c41f5d3faf5a35dd4": "n^2-1=2^{r}s", "dc44aa3cfe7c2057da58f556f956ff30": "P_{\\rm p}", "dc44e9ac05094470c91d400cc34b0498": "x^2 - 2xy + y^2 -2kx - 2ky + k^2 = 0\\,", "dc44f97fcf28ca1940c5a6732adb0566": "C^i{}_k = A^i{}_j \\, B^j{}_k ", "dc453ba2f74d857604c9f8746dbdf72f": "u_h,", "dc457770817f0a7cc3f0bdf0a662eb23": "F=\\frac{T}{r}", "dc457d2c8ebf3dafac0c79981bdbf23d": "|T_N|=\\sum_{i=1}^\\infty|T_i|\\,1_{\\{N=i\\}}.", "dc458758ebb3d869d7227bfc3c86b1bc": " M^{-1} = \\frac{1}{{\\det}^{col}(M)} M^{adj}, ", "dc463b01165dcce57c92e7eacc73ec77": "\ng(t) = at^{n-1} e^{-2\\pi bt} \\cos(2\\pi ft + \\phi), \\,\n", "dc467721acc09bbb9a7a8fc72c969c4c": "(A^{-1})_{ji} = (-1)^{i+j} \\frac{\\det A^{ij}}{\\det A}", "dc469facc31cfb3953b6df7dd7ee2a9b": "\\{1,\\dots,m\\}.", "dc46cf775786b26111c50c517e48df34": "Tr (T^i T^j) - Tr (T^j T^i) = 2 i \\epsilon^{ijk} Tr (T^k)", "dc4717efb83a493d379ccc3c837c42cd": "o(\\lambda^{d/2})", "dc473634f51b029644dc7801c2fa587d": "v_c = \\sqrt{\\mu/r}\\,", "dc4781949b939325a21be3f73f4ad5db": "[\\nabla]", "dc47b1accd16ddce42b680c9874475ab": "\\int_X \\phi(x) dx = If - Ig\\,", "dc47c8edf69e05dc354b603b74ca36e8": "\\mathbf{r}_{ij}", "dc484821107e028c0a3afd7d88847cd6": "\\Gamma_{4}(s)", "dc48c1f5aefb57df19f3fa5374cb428f": "X^2+1", "dc48eb616831fa11f960467a8a5424da": "y(u, v) = \\left(R + r\\cos{v}\\right)\\sin{u}, ", "dc48f158fbeafb985465e0b8cf78efcf": "\\phi \\approx \\textbf{A}x", "dc490e5f9ba6ab1ed7c166f618625e77": "\\lambda(x,y)=(2x,2y)", "dc49ada67de6b82e400f3046a24415f4": "L_{a} \\geqslant \\begin{cases}\nl^{1-\\ \\delta\\,\\!} &\\text{, if } \\gamma\\,\\! \\geqslant 2\\\\\nl^{\\left (\\tfrac{\\ \\gamma\\,\\!}{2-\\ \\delta\\,\\!}\\right )} &\\text{, if } \\gamma\\,\\! < 2\n\\end{cases}", "dc49f03f0b834962ca58e777ab90e33d": "\\frac{\\partial f}{\\partial t} + \\frac{1}{2}\\sigma^2S^2\\frac{\\partial^2 f}{\\partial S^2} + rS\\frac{\\partial f}{\\partial S}-rf = 0.", "dc49f5497709eb3200431200cbca1b00": "C^{\\mu}_{N}", "dc4a18e031857420b9803030c8f448f4": "F_2=W(S)/E", "dc4a8c3f937ea4458476cf1f86d49bec": "\\ X_{ik}", "dc4a9b41ed7d6f04f77e4159ca6553a3": " r = \\frac{\\lambda}{2 n \\sin{\\theta}} = \\frac{\\lambda}{2 NA} ", "dc4b40d012076a1a6673ac21fae5c6ea": "\\mu_j \\neq 0 \\Rightarrow \\mu_j g_j(x_k)>0", "dc4b4b4c20db67196bc3418504b40c89": "\\sum_{n=0}^{\\infty} \\frac{1}{F_{2^n}} = \\frac{7 - \\sqrt{5}}{2}", "dc4b756b76d06ec6aa4f62c36fb40952": "\\Delta \\mathcal A", "dc4b86976f3e42e0fa9a0bb3c86946a5": "\\bar{X}_1 = \\sum x_i^2/x_i = \\sum x_i H", "dc4b97121a5d867a88bd9a160e22ab63": "\\int\\limits_X |\\psi(x)|^2 d\\mu(x) = 1.", "dc4bbd1bfea351f824f2159a3c09043b": "A \\rightarrow\\, \\alpha", "dc4bc55010a00cd0ba1d1381619db03f": "\\{A_1, \\ldots, A_n\\}", "dc4bda3e5b35f82de373e897bf649ff5": "(m/Q)\\mathbf{a} = \\mathbf{E}+ \\mathbf{v} \\times \\mathbf{B}.", "dc4c5cc0fa36b436e278c26ae719fc60": "S = (1+r)^{N-1} + (1+r)^{N-2} .... + 1", "dc4c60699193233f754ea7ad623c9067": " \\left(\\sigma, S, C\\right) ", "dc4cafb6221d077e35fac269a2ddf0d6": "\\cup_{i=1}^m T_i=\\Omega", "dc4ddf29e1f486a871c9f210d4d11170": "I_{\\mathrm{ref}}", "dc4e53579e7dcf624ab07d6b5f668433": "Re = {{\\rho {\\mathrm v} d} \\over \\mu}", "dc4e5a440b801745d0402884202f2136": "T(n) = T(0) \\cdot C^{n}", "dc4e7d7dc79eb2c87eca893a2927e4b2": " E_{1ss} = E_{2ss} -Z = 6.40 -2.00 = 4.40 \\text{ ft} ", "dc4e88ee6650d101c92ee0e235a3761e": "(f \\cdot g)^{(n)}=\\sum_{k=0}^n {n \\choose k} f^{(k)} g^{(n-k)}", "dc4ebf047a510599633f5fd058d49caf": "{dw^\\prime\\over dt} = {1\\over\\bar\\rho}{\\partial p^\\prime\\over\\partial z} + B - g\\ell", "dc4ed9a2044f6e45aa140f34b31234b4": "\\eta_{pump}", "dc4ee90c8ad2700d593689dad8b96314": "\\operatorname{Var}(\\overline{X}) = \\frac {\\sigma^2} {n} + \\frac {n-1} {n} \\rho \\sigma^2.", "dc4f7fc1e66821e19a3f2a15bc90e443": "y_{t}=\\gamma_{0}+\\gamma_{1}y_{t-1}+\\gamma_{2}y_{t-2}+...+\\gamma_{p}y_{t-p}+\\epsilon_{t}.\\,", "dc500d5981cb8f6442ff1441aa4a21a0": " \\mathbf{A} \\times \\mathbf{B} = \\frac{1}{2}(\\mathbf{AB} - \\mathbf{BA}),", "dc5031481b2a332c9705e155beef88d9": "j = {c_4^3 \\over \\Delta}", "dc509668eadf7f4e57fe399e02d036f8": "\\operatorname{nec}(U) = 1", "dc510b9150beeef5b4be3fed5ef46849": "(\\mathbf{m}_2-\\mathbf{m}_1)", "dc510eb6b856112e3d64498a1ba605e1": "\\mathfrak{T}^{\\mu \\dots}_{\\nu \\dots ; \\alpha} = \\sqrt{-g}\\;^W T^{\\mu \\dots}_{\\nu \\dots ; \\alpha} = \\sqrt{-g}\\;^W (\\sqrt{-g}\\;^{-W} \\mathfrak{T}^{\\mu \\dots}_{\\nu \\dots})_{;\\alpha} \\,.", "dc5126fce8776b952100efc6c3967d71": "\\mathbf{A}\\otimes\\mathbf{B} = \\begin{bmatrix} a_{11} \\mathbf{B} & \\cdots & a_{1n}\\mathbf{B} \\\\ \\vdots & \\ddots & \\vdots \\\\ a_{m1} \\mathbf{B} & \\cdots & a_{mn} \\mathbf{B} \\end{bmatrix}, ", "dc51a917a01ca7a7554e8ae10e16ebc9": "\\Box P \\leftrightarrow \\lnot \\Diamond \\lnot P. \\;\\!", "dc51b6ef87b19951dae519402b292c12": " \n\\bold{J} = \\rho dx \\wedge dy \\wedge dz - j_x dt \\wedge dy \\wedge dz - j_y dt \\wedge dz \\wedge dx - j_z dt \\wedge dx \\wedge dy\n", "dc51d0204252b7a02a9323f279f35236": "\\left \\vert N+1-\\varphi (N)-1 \\right \\vert < 3 \\sqrt{N}", "dc51dcde40f2cdb6b17a12d83aff65e5": " \\int_U \\left( \\psi \\nabla^2 \\varphi - \\varphi \\nabla^2 \\psi\\right)\\, dV = \\oint_{\\partial U} \\left( \\psi {\\partial \\varphi \\over \\partial n} - \\varphi {\\partial \\psi \\over \\partial n}\\right)\\, dS. ", "dc51e967343a0683f6d79d034aaad317": "\n \\left. \\frac{\\partial^2 S(z)}{\\partial z_i \\partial z_j} \\right|_{z=0} = 2h_{ij}(0);\n", "dc5214d97df5900e7bc52b69bf2c4ca4": "|A_n-A|\\le\\frac{\\varepsilon/3}{|B|+1}\\,.", "dc522e04aae73d0c6edb220695091174": "\\begin{bmatrix}0 & 1 & 0\\\\1 & -6 & 1\\\\0 & 1 & 0\\end{bmatrix}", "dc525feb43fb2a6580e4338334cd02fd": "\\dfrac{h_\\mathrm{c}}{k' c_\\mathrm{s}} = 1", "dc527aee787819649bcaf262ea8027bb": "\\rho = \\frac{\\partial V}{\\partial r}", "dc5289ab0e16c55235216ca9ce38c87a": "\\zeta(s) = \\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}}", "dc52a20989964063de7b834ed8ffc594": " \\mathrm{d} f ", "dc52a44a576e8606dc5fdb07a53aab08": "\\overline{\\overline{(\\overline{C} \\vee (\\overline{A} \\wedge \\overline{B}))}} \\wedge (C \\vee A \\vee B))", "dc52f334ad2e8d5700b786dc35a727ab": "[f,g+h]\\ne [f,g]+[f,h]. \\, ", "dc538e7673f2c235a3efa56c86d41cbb": " E_{kin,+/-} ", "dc53a4717d3cfb36c97a1560daf20da5": "\\scriptstyle \\dot{x}", "dc53b634bbdce553c6feb4d7192bcd0f": "\nE\\simeq 25000 M_\\odot^{1/2},\n", "dc549d47182df610acd585d0dfa76086": "{48 \\choose 3} = 17,296", "dc551d76aa9bfd4e7032f782a22b584d": "\\textstyle f: \\left\\{0,1\\right\\}^* \\rightarrow \\mathbb{Z}_n", "dc556b4c4ff46f9cec12b8da8adbb53b": "\\scriptstyle a_2", "dc55933e3b6b064d00add520cc8ef803": "1's", "dc559fa30b16d27552e3400fdc2b4aae": "m/n > 1", "dc560e24d6a9b7470f415e161dc35fdd": "F_{Y}(q_{\\tau})=\\tau.", "dc566dd5a82e654be699408c56981e14": "=\\sum_{k=1}^{d} \\sum_{i=1}^{d} {\\Gamma^k}_{ij}\\dot q_i \\boldsymbol{e_k}\\ ,", "dc5674eaf7822f8597700121484b2e49": "v=\\sqrt{\\mu\\left({2\\over{r}}-{1\\over{a}}\\right)}", "dc56805d6079a6699cc585c79a8de6b2": "(X,\\mu)", "dc56cebbcdcf104b0f898bfaea807ca5": "(n,p)", "dc56cfa74514f57001c7f639db96ca64": "q_k \\ ", "dc578ab694b960eeee1fdacaf10bb68c": "\n \\mathrm{E}\\big[\\, x_i(y_i - x_i'\\beta) \\,\\big] = 0.\n ", "dc57b2394ea916dd06290e52e1d354c5": "E_n=(n+3/2)\\hbar\\omega", "dc57c26410092a27558e25ea849d1ff7": "\n\\mathbf{P}_\\textrm{EM} =\n\\epsilon_0 \\iiint_V \\mathbf{E}(\\mathbf{r},t)\\times \\mathbf{B}(\\mathbf{r},t)\\, \\textrm{d}^3\\mathbf{r},\n", "dc580b31b517ed7b4e3bb853eaed5a2e": "\\scriptstyle f(x)=0", "dc58569f92e8540072864f19b467d765": "\\mathbb{Z}_q = \\mathbb{Z}/q\\mathbb{Z}", "dc5874f35bf0c52b7a3dc9b6b4f9b5e9": "\\displaystyle f'(x_0) = 0", "dc587ed01aa47127af7fc30a0f7c1a0d": "\\forall p((p \\land \\Diamond Kp) \\Rightarrow Kxp)", "dc589286b912593ec187b5adb5985722": "\\bar{Z_2}", "dc58af5bf90de06dd5b7b0e26df5404a": "\\begin{align}R_{J}(x,y,z,p) & = R_{J}(A (1 - \\Delta x),A (1 - \\Delta y),A (1 - \\Delta z),A (1 - \\Delta p)) \\\\\n & = \\frac{1}{A^{\\frac{3}{2}}} R_{J}(1 - \\Delta x,1 - \\Delta y,1 - \\Delta z,1 - \\Delta p) \\end{align}", "dc58b55eaf288f164f6549331351581a": "\\frac{1}{\\lambda} = R_\\infty\\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)=\\frac{m_\\text{e} e^4}{8 \\varepsilon_0^2 h^3 c} \\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right) ", "dc58fc300ede91ec5a3d505b6e2a6d12": "\\sqrt{\\det(B^T B)}", "dc59531d428b99aae2d6007956f388cb": "p^{*} := \\tfrac{2}{3}(C_1~I_1 - C_2~I_2). \\, ", "dc596474a4e0300fc5c26f695f72349e": "\\csc(y) = x \\ \\Leftrightarrow\\ y = \\arccsc(x) + 2k\\pi \\text{ or } y = \\pi - \\arccsc(x) + 2k\\pi", "dc59d8ec8a43738af266cc48b60a4194": "{E}", "dc59dd600fedbfb3e5e8240d4c3d2088": "\\scriptstyle i\\in \\{ 1,2,...,n\\}", "dc5a35df9dd57b773e7b6262559f64db": "\\beta^2u_{i,kkmm}=0\\,\\!", "dc5aa8d9593257d09ba2219d5b7bebfc": "\\scriptstyle{\\phi}\\,\\!", "dc5ab3beeafd25efc653b1aaa0a10363": "f: x \\mapsto a^x", "dc5ae57d839814bc56a503de09c176a4": " \\nabla^2 U = \\frac{1}{c^2} \\frac{\\partial^2 U}{\\partial t^2},", "dc5aed55d41e1405b80985e4d7490971": " d_{Y}(f(x),f(y)) \\leq d_{X}(x,y) . \\! ", "dc5b7aa832ea726525e34e5c15b72cd6": " F = 0 ", "dc5bc518afc3f67b99114f2043435256": "\n \\int_{0}^{2\\pi} \\int_{0}^{\\pi} \\int_{0}^{2\\pi} D^{\\ell}(\\alpha \\beta\\gamma)^*_{nm} \\; D^{\\ell'}(\\alpha \\beta\\gamma)_{n'm'}\\; \\sin\\!\\beta\\, d\\alpha\\, d\\beta\\, d\\gamma = \\delta_{\\ell\\ell'}\\delta_{nn'}\\delta_{mm'} \\frac{8\\pi^2}{2\\ell+1}. \n", "dc5bd75c5af3448d8f4b0d6f4fe66278": "r^{\\ell_n}", "dc5bff94b120446047db5ffce06df257": "\\alpha_{min}=\\frac{1}{d}\\left(\\frac{\\Delta I_{min}}{I_{0}}\\right)(1-R)", "dc5c561941c5d7bbff897a735d8746e0": "\\mathrm{S}(\\mathrm{U}(p) \\times \\mathrm{U}(2))", "dc5c5a51b61406a544f55df9478a1ade": "\n\\Delta E=\\frac{1}{2}\\alpha_0\\left(T-T_0\\right)P_x^2+\\frac{1}{4}\\alpha_{11}P_x^4+\\frac{1}{6}\\alpha_{111}P_x^6 - E_x P_x\n", "dc5c7986daef50c1e02ab09b442ee34f": "001", "dc5cc087137f1ac5ce1156822ffe4b29": "\\frac{\\sigma(n)}{e^\\gamma n\\log\\log n} < 1", "dc5cdf2192267375c250567e00307925": "\\mathbf{r=\\Delta y-J\\ \\Delta\\boldsymbol\\beta}.", "dc5ce4049ed1a44d9b9627505d183cbe": "\\operatorname{div} \\, \\operatorname{grad} f \\equiv \\nabla \\cdot \\nabla f = \\nabla^2 f \\equiv \\Delta f", "dc5ce4c1f6cfd42321f3d5632c3c2f73": "I_x\\,", "dc5cf2fd1e9cce5d54ee8d6b1b2fb59c": "\\nabla^2 \\mathbf{E} - \\frac{n^2}{c^2}\\frac{\\partial^2}{\\partial t^2}\\mathbf{E}= 0.", "dc5d19eec4aae0d601755eadb2b5499d": "\\displaystyle{T=R_1(I+L)^{-1} R_0,}", "dc5d536c7ba96a034ea957dc9dfbecf7": " W_{\\bullet}, F^{\\bullet}", "dc5da37828f1d9bdb9d1ad164d63de9a": "\\mathit{W}", "dc5dba5d8c26f457de08433b01dcc357": " 0 < \\# v \\Lambda^n < \\infty", "dc5dddf41c716a6599c4e890aba6767c": "x = a\\cosh\\frac{y}{a}", "dc5df539ba354142ab226bcba2a01e5d": "\\Delta x\\Delta p\\gtrsim h.", "dc5e008d5c266caf6dcb148d1d344aff": "P(x(t)\\ |\\ y(1),\\dots,y(t))", "dc5e2fb6c87fcedf4237871b53c3e0a1": "\\delta_S(\\mathbf{x})=-\\mathbf{n}_x\\cdot\\nabla_x\\mathbf{1}_{\\mathbf{x}\\in D}", "dc5e3b7ad49c204565cdc9eae1277034": "\\zeta^2 s_1", "dc5e4c0d030d0e3601af7c314553ff65": "F(x) = \\sum_{1 \\le n \\le x}\\mu(n)G(x/n)\\quad\\mbox{ for all }x\\ge 1.", "dc5e942d26a64d7fe16a44b9809613f9": "\\displaystyle{a=\\sum \\lambda_i e_i,}", "dc5ec1da9cb290919796a024315859d6": "E_{y,x^2-y^2} = \\frac{\\sqrt{3}}{2} m(l^2 - m^2) V_{pd\\sigma} -\nm (1 + l^2 - m ^2) V_{pd\\pi}", "dc5f09e44ccc255073f35c89e1d89b66": "\\mathcal{S}'", "dc5f499e83aef4f6811faa4081631740": "At", "dc5f5ea739f3fb0b3b432d813e9eac5d": "c_{\\mathrm{1L}} \\,", "dc5f6d1acccad7152daa5f56e7ac9d56": "cone(\\phi)", "dc5f82d2f85ac452d591c8efa798aa02": "\\lim_{x \\to 0^{+}}{1\\over x} = \\lim_{x \\to 0^{-}}{1\\over x} = \\lim_{x \\to 0}{1\\over x} = \\infty.", "dc5fe60305fe566b99518c106dc74e8f": "R/\\mathfrak{m}_R", "dc5feecee3eef8cf18767a92d9aaf4c1": "\\hat{x}_{k+1} \\leftarrow \\hat{x}_k + \\alpha_k \\cdot \\hat{p}_k\\,", "dc600de3dd4a3eb0c3b6cfd148659a99": "S_{p}=\\frac{n_{p}(t=0)-n_{p}(t)}{n_{k}(t=0)+\\int_0^t\\dot{n}_{k,\\text{in}}(\\tau)d\\tau-n_k(t)}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "dc602fdefc0079647dcab20acc8eb2a4": " \\ell = L \\tan \\theta \\approx L \\sin \\theta ", "dc603dabc25ca9d7ef2089b9a4bf2658": "z\\mapsto z\\,|z|^{s}", "dc6051ff81f517c047d967d138a6ee87": "R_i , R_k", "dc60bae968735e0b9d051bab7cef1622": "P = - n e x", "dc60e17be577ab5e44f4c587e7136647": "j = 0,\\ldots,i", "dc60e9c4473ddda6a8bf896317348425": "G(x,x') = \\langle s(x),s(x') \\rangle. \\, ", "dc619ca7d32baaff2d1ca16683ba3ab2": "\\lambda\\sin ^2(\\theta) + \\frac{\\sin(\\theta)}{\\Theta(\\theta)} \\frac{d}{d\\theta} \\left [ \\sin(\\theta) \\frac{d\\Theta}{d\\theta} \\right ] = m^2", "dc61ac44d722140e6d2ce921009517b6": "[n_0,n_1,n_2,n_3]=[1,0,0,0]", "dc623c5acc444b7a753e4f98b82ab19b": "Q^\\dagger=(p+iW^\\dagger)b^\\dagger", "dc625c6159819942893db823872e020b": "X/\\Gamma", "dc6268f414fcc7c4a5d2812904c85ce0": "\\mathbf{\\hat b} = \\frac {\\mathbf{b}} {|\\mathbf{b}|}\\,", "dc627a1fd41d1d775af75af93e51e5f1": "\\scriptstyle P_1 \\,\\oplus\\, IV_1 \\;=\\; P_2 \\,\\oplus\\, IV_2", "dc629c5d04ec506a537ab6175e706924": " 0.1\\overline{\\mathrm{110TT0} } =\\tfrac{\\mathrm{1110TT0-1} }{\\mathrm{100000T0} }=\\tfrac{\\mathrm{1110TTT} }{\\mathrm{100000T0} } =\\tfrac{\\mathrm{111\\times 1000T} }{\\mathrm{111111\\times 1T0} } =\\tfrac{\\mathrm{1111\\times 1T}}{\\mathrm{1001\\times 1T0}} =\\tfrac{1111}{10010}=\\tfrac{\\mathrm{1T1T}}{\\mathrm{1TTT0}} =\\tfrac{101}{\\mathrm{1T10} } ", "dc62adc75ad528fe6311b11681525391": "\nhttp://!anon!/ \\ options \\ name_1 \\ name_2 \\ ... \\ name_n \n", "dc62ccd7d1ac037caa8f4a4157d47152": "d=2,3", "dc630fb280652ad2661ef0252868b833": "F^{p+1}/F^p = \\Lambda^{k-p}(U) \\otimes \\Lambda^p(W)", "dc6338a9b5b4735e89fb258b4d0aaec0": "\\tfrac{1}{2} < \\text{median} < \\tfrac{1}{\\sqrt{2}}", "dc63453cb0a65947f95a6d38fa2b1930": "H\\propto nvcA\\Delta T.", "dc634e2072827fe0b5be9a2063390544": "br", "dc635127d802ec3cf69c9b229e5abc1e": "\\bar{M}_n=\\frac{\\sum_i N_iM_i}{\\sum_i N_i}", "dc6357a18bd2dcdfa1e82ecedb79e28c": "\n\\langle \\theta \\rangle=\\mathrm{Arg}\\langle z \\rangle = \\arctan(1/\\lambda) ,\n", "dc63cd002a61dd218a4f9f50ae363b88": "C \\subset \\mathbb{R}^n", "dc63ff05a2375e9e6a3c6f86d5d550dd": "\\mathbf M", "dc64034ab3c7c772b0315743616ddcf6": "H^2(M)", "dc6423694de0eb639bceaeee10045509": "Y_i ", "dc643acf38d30018195a785fe4e718ff": "A=dx^\\lambda\\otimes (\\partial_\\lambda + a^m_\\lambda {\\mathrm e}_m), ", "dc6461c338347674372fdfd026eb4c3e": "\\theta(x)=C_1e^{mx}+C_2e^{-mx},", "dc64c869f59087d796cabff0df53b32f": "V = \\{v_1, v_2, ~\\ldots, ~ v_n\\}", "dc64eb87ceab802dad62130c2a6557d0": "f_{i,j}", "dc65132b218c3c40675c0afaaee2d8a6": "O(\\log n/\\log (\\log n))", "dc657249eec6db2a13c12364243f31e5": "\\Phi_a(r_1)", "dc663086ce889a03238f062f2cf777ae": "-\\frac{\\hbar^2}{2m} \\nabla^{2}\\psi_{1} + (\\tilde{u}_{1}- E)\\psi_{1} - \\frac{\\hbar^2}{2m} [2\\mathbf{\\tau}_{12}\\nabla + \\nabla \\mathbf{\\tau}_{12}]\\psi_{2} = 0 ", "dc66baca06261c4a04dc39f9f51dbdc2": " C_D = C_{D0} + C_{Di} \\begin{cases} C_{D0} = (C_D)_{C_L = 0} \\\\ C_{Di} \\end{cases} ", "dc66d28555ef53eea244d8daf0616075": "\\exp \\left\\{-\\frac{1}{2} \\sum_{n=0}^{M-1}X_n^T R^{-1} X_n \\right\\} \\exp \\left\\{\\theta m^T R^{-1} \\sum_{n=0}^{M-1}X_n \\right\\}", "dc66f185208df2da4860dbb207c15c89": "\\delta W = p \\mathrm{d}V", "dc676c3ba12115765adb98682069605a": " \\rho' = \\frac{a^2}{\\rho}. \\,", "dc6793213437eb7ee207db8ae78594b4": "s^{\\overline{2m-1}}", "dc67ccdb4d06c61eb468c1c1a7a3eee2": "H_2(z)=z^{-1}+z^{-2},\\,", "dc683f66b69888090e74b3291404d235": "\\beta={v_1,v_2,\\ldots,v_n}", "dc6896d2ea3fed3bbcf9c473e59e0903": "A = - \\ln\\mathcal{T}\\ = - \\ln\\left({I\\over I_{0}}\\right)", "dc68d80c4129096094f26eff5399e310": "\\mbox{2-EXPTIME} = \\bigcup_{c \\in \\mathbb{N}} \\mbox{DTIME}(2^{2^{n^c}})", "dc69377883d9f7c3f5dc85075986cfb0": "\\exp(-\\psi(1/y))", "dc69638748f55307b735c361c04ac50e": "\\frac{3}{8}(35\\sin^4(\\phi)-30\\sin^2(\\phi)+3)", "dc6999a6cba130d351eb833d7282c7e0": "f'' (0) \\approx 0.332", "dc69a573eed2b19b6ff01502521ff2ec": "\\begin{bmatrix}\\Psi\\end{bmatrix}^{T}\\begin{bmatrix}K\\end{bmatrix}\\begin{bmatrix}\\Psi\\end{bmatrix}=\\begin{bmatrix} ^\\diagdown k_{r\\diagdown} \\end{bmatrix}.", "dc69af3c984e1e9d38f83ec5e5af88fb": "\n\\sigma^x = \n\\begin{pmatrix}\n0&1\\\\\n1&0\n\\end{pmatrix}\n", "dc69c64fcf1029be6b5e91b50de08fb8": "x^{u}", "dc69d99fdaf3ecb2395f582603593005": "\nx = \\left( \\begin{array}{ccc}\n0&1&0\\\\\n0&0&0\\\\\n0&0&0\n\\end{array}\\right),\\quad\ny = \\left( \\begin{array}{ccc}\n0&0&0\\\\\n0&0&1\\\\\n0&0&0\n\\end{array}\\right),\\quad\nz = \\left( \\begin{array}{ccc}\n0&0&1\\\\\n0&0&0\\\\\n0&0&0\n\\end{array}\\right)~.\\quad\n", "dc6a39015078c7730165e051787349e7": "\\{g_\\theta(Z_1),\\ldots,g_\\theta(Z_m)\\}", "dc6a7e6d32c949375857c05b0464057c": "x_t = Ax_{t-1} + b \\, ", "dc6ae01ec94c57ac96510ee919f0dff7": "N = \\frac{V_s \\times c \\times r_s}{n_r} \\times \\frac{V_v}{V_v} ", "dc6b167c4b27cdff3f98068741ca4963": " I\\omega^2 \\equiv L\\omega \\equiv L^2/I \\,\\!", "dc6b4a86cf944f07489bf5de355a2bec": "\n\\begin{align}\nf(x,y,z)\n&\\ge 6 \\cdot \\sqrt[6]{ \\frac{x}{y} \\cdot \\frac{1}{2} \\sqrt{\\frac{y}{z}} \\cdot \\frac{1}{2} \\sqrt{\\frac{y}{z}} \\cdot \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} \\cdot \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} \\cdot \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}} }\\\\\n&= 6 \\cdot \\sqrt[6]{ \\frac{1}{2 \\cdot 2 \\cdot 3 \\cdot 3 \\cdot 3} \\frac{x}{y} \\frac{y}{z} \\frac{z}{x} }\\\\\n&= 2^{2/3} \\cdot 3^{1/2}.\n\\end{align}", "dc6b4f6bfccf28398543d20b3d9cd2d2": " A = {{K}^{1/3}}{{L}^{2/3}} ", "dc6ba68b2e7209c7a302aeafd3509509": "V_{jfe} =\\,", "dc6c072297bb2bc5c3f0f5c8e04c759d": "(X_n)_{n = 1}^\\infty", "dc6cccb59cf1b0f8daa5af7f79642664": "\n\\sum_{c=1}^N\\mu_{ab}^{(c)}(t)[Q_a^{(c)}(t) - Q_b^{(c)}(t)] = \\mu_{ab}(t)W_{ab}(t)\n", "dc6cea90cdd1c0cdb8f174660909b1dc": "B_{i,k}(x) := \\frac{x - t_i}{t_{i+k-1} - t_i} B_{i,k-1}(x) + \\frac{t_{i+k} - x}{t_{i+k} - t_{i+1}} B_{i+1,k-1}(x).", "dc6cebd2f1069f7b657a90797aeb38a3": "\\frac{V_o}{V_i}=\\frac{-D}{\\frac{R_{\\text{L}}}{R(1-D)}+1-D}", "dc6d166a4615163f6ca9ce09ee1755f8": "c_i=\\sum_{k=0}^ia_kb_{i-k}\\,.", "dc6d4771daa15ffe6ef7da6800156611": "V(\\mathbf{S}) = \\bigcup_{n \\in \\mathbb{N}} V_{n}(\\mathbf{S}).", "dc6dec601134e46958eecf0417c4ad92": "y(x) = 2floor(x) - floor(2x) + 1", "dc6e336e48ded4293c7ff0fc068bdea1": " \\text{Minimize }\\; f_0(x) ", "dc6e4d511bf8cc49c7f74476181d8054": "[y_\\nu] := y_{\\nu},\\qquad \\nu \\in \\{ 0,\\ldots,k\\}", "dc6e92ce2e7ca5c2c22eebf6e70ef38e": "\n\\Gamma_{21} = {Z_1 - Z_2 \\over Z_1 + Z_2}\n", "dc6ecb5290f713af8e1e004f35c5c467": "p \\sigma^2\\,\\!", "dc6f020dac5efc170de63875c71f3ec1": "s_i =\\frac{x_i}{\\sum_{r\\in{FB_i}}y_r(t)}", "dc6f02cd7139d7b513108f444cbffec4": " 2B + Ph \\, ", "dc6f4693a08524131af8e7a9eca625bd": "I_{tsm}", "dc6f61fc27b4d79a1279b3b755554f86": "(\\varphi_t)^*", "dc6fc247df49bfb42cbc173d67575874": "\\mathrm{sn}\\,(x)", "dc6fcf61d19178bce9dee8de6f472648": " \\gamma \\left[ \\alpha ( a ), \\beta ( b ) \\right] = \\{ \\gamma^{1} \\left[ \\alpha ( a ) , \\beta ( b ) \\right] ,...,\t\\gamma^{K} \\left[ \\alpha ( a ), \\beta ( b ) \\right] \\} ", "dc6fe95d4495451b3bd132262eea31b8": "\\pi:\\mathbb{R}^{m+n}\\rightarrow\\mathbb{R}^n\\subset\\mathbb{R}^{m+n}", "dc6febf2dbce2b1f92f4b28953305d59": "S(T) = \\int_0^T {\\mu(T') \\over T'} dT'", "dc6ff82de9e390e9197b0bb9a5341c7f": "R_{S}(t)= \\frac{\\displaystyle \\sum_j \\sum_{b_j\\neq0} \\sum_{\\beta_j} \\frac{b_j q}{ ^{b_j}M_{S_j}} \\ {^{b_j}_{a_j}}S^{\\beta_j}_j (t)} { \\displaystyle\\sum_j\n\\sum_{b_j\\neq a_j} \\sum_{\\beta_j} \\frac{(a_j-b_j) p}{^{b_j}M_{S_j}} \\ {^{b_j}_{a_j}}S^{\\beta_j}_j (t) }, ", "dc7032c6da477b548beea468e2c6b38b": "a \\cdot (b \\cdot c)", "dc7036ee9854cce64c819176c46409a9": "P*\\left( {{t}_{1}}-{{t}_{2}} \\right)=E*(SOD\\left( {{V}_{1}} \\right)-SOD\\left( {{V}_{2}} \\right))", "dc7043bdb80890625055406c9c547ba9": "n=n_\\text{G}\\sin\\theta", "dc70b1443a626c56d08a04443cd7d733": "q_{2}^{T}", "dc70ce64d6e8e13fe813e0f9c8f86493": "D = {r-1 \\choose 4}", "dc70cfef985b2ba49a8e6e8d551606c8": "X(z) = \\mathcal{Z}\\{x[n]\\} = \\sum_{n} x[n] z^{n}.", "dc70f3dd270c67b5837ad5bce5ae5748": "\\implies \\,N\\left(\\frac{mx + y}{k}\\right)^2 + \\frac{m^2 - N}{k} = \\left(\\frac{my + Nx}{k}\\right)^2.", "dc70fed63d22082c68e3a0590e06bdf0": "\n\\left(c' \\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1/2} \\right) d \\leq \\left(c' \\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1/2} \\Sigma _{YY} ^{-1/2} \\Sigma _{YX} \\Sigma _{XX} ^{-1/2} c \\right)^{1/2} \\left(d' d \\right)^{1/2},\n", "dc71677534962cf0a0cdd77e9f86ce30": "\\mu(T)", "dc71aa1cbaaae0b5f010aebb4ed64c4d": "t\\in\\left[0,1\\right]", "dc71b255a77f91c4430e70ae6da11b60": " \\omega \\in A,", "dc71bff69244032b888fc180a89f0224": "\\limsup_{n\\to\\infty}x_n<\\Lambda", "dc71c9d68eaa66721b6f8ff847597766": "\nds^2 = - (1- \\Lambda r^2) \\, dt^2 + {1\\over 1-\\Lambda r^2} \\, dr^2 + r^2 \\, d\\Omega^2.\n", "dc71d7cb4eeaa5f7a53cafbea29d2490": "\nD = A_{xx} \\xi + A_{xy} \\eta + B_{x}\\,\n", "dc720d1321bc115046bf46374ae2233b": "\\mathbf{v}'_2 = \\mathbf{v}_2 - \\mathbf{v}_0", "dc725a1c9ebaa4924caa1dd225d778c4": " a_1 ", "dc726778dff7ae4d1f57e4e8fa903e32": "(\\gamma_{0,q},\\mu_{q},\\sigma^2_{q},\\alpha_{3,q},\\alpha_{4,q})\n\\quad ", "dc72768088d9d5f519d627f9e729e040": " x_{n+1} = \\frac{f_{n-1}f_n}{(f_{n-2}-f_{n-1})(f_{n-2}-f_n)} x_{n-2} + \\frac{f_{n-2}f_n}{(f_{n-1}-f_{n-2})(f_{n-1}-f_n)} x_{n-1} ", "dc7281dd7498df2562e5b94c0291365a": "O(|v|)", "dc72f1b354a45a903118222074fe84fc": "\\int_{-\\infty}^{\\infty} e^{-x^2} dx = 2 \\int_0^\\infty e^{-x^2} dx", "dc7325ba9ab2f08b5dc212edf94a44f5": "R_{\\alpha_{i},\\beta_{i}}", "dc7385e55b3572a7b52289ea1b5f40bc": "\\cos\\frac{\\pi}{60}=\\cos 3^\\circ=\\tfrac{1}{16} \\left[2(1+\\sqrt3)\\sqrt{5+\\sqrt5}+\\sqrt2(\\sqrt5-1)(\\sqrt3-1)\\right]\\,", "dc73b57508f4a7a2504062f3af80f898": "\\mu_p = \\frac{\\sigma_p V_{Hp}t}{IB}", "dc73d32fe0c2b30ff5cb5b2748b2acac": "R[f^{-1}] = R[t]/(tf - 1).", "dc73f71bd4f956f63717b59ae5623cac": "NP/N\\triangleleft\\text{the}", "dc73fd5f95c4373c2aa2537a028ae4bd": "\\displaystyle \\phi_{tt}-\\phi_{xx}-\\phi+\\phi^3=0", "dc74155f9123c0ea21e0fcdd48c3f8f0": "-1/2\\,", "dc74156aced1a50f111bf462a682e7ef": "\\scriptstyle A\\,=\\,3ad", "dc741e841d45d9171f4b67917cf99353": "a^2+b^2=c^2\\,", "dc7437de09f0a7027b86e0198fc6da13": "\\mathbf{X}_{\\ell m}(\\theta,\\phi)=\\mathbf{L}Y_{\\ell m}(\\theta,\\phi)/\\sqrt{\\ell(\\ell+1)}", "dc748f39f289352eb61935d4bec65a1b": "f : M \\to BG", "dc74e1ec26e73245fb57420967898be6": "\\Delta = r^{2} - r_{s} r + \\alpha^{2}", "dc75081809c96ef0b47deafc07af13bb": "B(L(S)),B(R(S))", "dc750873be8c028993cdd34e04a1e758": "(x_u,1)", "dc752b3caed1f9add80937eefaf7a6e8": "J_{rise} = J_{transit} - \\left( J_{set} - J_{transit} \\right)", "dc753cfcbdc8f377e6bf04868e4aa4d5": "A^* \\colon W^* \\to V^*", "dc7557f49868bc603f7e13ccf46a3cbd": "\\cos(30^\\circ+6^\\circ)=\\sin(60^\\circ-6^\\circ)=\\sin(60^\\circ)\\cos(6^\\circ)-\\cos(60^\\circ)\\sin(6^\\circ).", "dc756e26e577ff77715a9a7522bda96e": "\n\\liminf_n \\frac{S_n}{\\sqrt{n}}=-\\infty\n", "dc75d1ad6df9c695ce9e6c1be12b23fc": "\\mu_0r \\cdot \\left( \\ln\\left(\\frac {8 r}{a}\\right) - 2 + \\frac{Y}{2} +O\\left(a^2/r^2\\right)\\right) ", "dc75daa432b55383445c59ca39283952": "\\delta(t) * x(t) = \\int_{-\\infty}^{\\infty} \\delta(t - \\tau) x(\\tau) d \\tau = x(t)", "dc75f90e005835fc879fa177f79a21e8": "\\alpha (f(0), f'(0)) = 1,", "dc763b253727fe9cd962f423747a121b": "\nF(r) = \\frac{\\mu}{r^3}\n", "dc763c027c01b549dbd752ff9efd35cb": "\\langle x^2\\rangle", "dc76cf206a775b89fa95c9c436b9ebb9": "q_r", "dc76de71e85c401666785c33fac39b68": "t \\in T_r ", "dc7711db3d10128749fef6c211ebbd0f": "\\ddot u_i={\\left(\\frac{c}{\\Delta x} \\right)}^2 \\left(u_{i+1} + u_{i-1}\\ -\\ 2u_i\\right)", "dc772b3aa3659776db7fd5687b479496": "{dw\\over dt} = -{1\\over\\rho} {\\partial p\\over\\partial z}-g", "dc7733ca656e3d99549050bcde3a9ad7": "3.07 \\, z", "dc7776400633d8d04be59e82dcad4383": "\n\\gamma_r = \\sum_{(x) \\mid r}\\exp(-\\sum_i x_{ni}\\delta_i)\n", "dc77ed394d0c6c0750ce6b0ab5078032": " E_{z+fs} = E_{z} + \\frac{\\alpha^2}{2 n^3} \\left[ \\frac{3}{4n} - \\left( \\frac{l(l+1) - m_l m_s}{l(l+1/2)(l+1) } \\right)\\right] ", "dc785b3f71c1f77e7a50c5036926f3b2": "\\operatorname{Symmetric-Dirichlet}_N(\\gamma)", "dc787d8d0bb562f6c0202e7abe8c5ecc": "P_\\mu P^\\mu \\vert \\phi \\rangle = \\mathcal{M}^2 \\vert \\phi \\rangle", "dc78cf5b39c326ced6c579703416a644": "\\displaystyle x_{t+1}", "dc78d3e2175edb12d2d9f1baa051cc38": " \\delta W\\, ", "dc78dcd5a3b25372aad270f538a69efa": "f:X\\rightarrow S_R", "dc78e9b8fcb6e718d0d3952a173f5e90": " \\left[ \\mathbb{H}_\\mathrm{n}(\\mathbf{R}) + \\mathbb{H}_\\mathrm{e}(\\mathbf{R}) \\right]\n \\; \\boldsymbol{\\phi}(\\mathbf{R}) = E\\; \\boldsymbol{\\phi}(\\mathbf{R}).\n", "dc79346559181610fb9bc6007bc5f4b3": "\nf_a(z) = {1\\over z-a}\n", "dc79629be0f4409191052352b086951c": "\\lambda_{2}=\\frac{3-\\sqrt{5}}{2}<1", "dc7980aa7fbe3ad76c08ce806fbfea97": "f(x)=\\frac{1}{2 \\pi i} \\int_{\\nu-i \\infty}^{\\nu+i \\infty} x^{-s} \\varphi(s)\\,ds.", "dc79a4b0dd37e1d5a8a2c96763820b9c": " \\mathrm{DR_{ADC}} = 20 \\times \\log_{10} \\left(\\frac{2^Q}{1}\\right) = \\left ( 6.02 \\cdot Q \\right )\\ \\mathrm{dB} \\,\\!", "dc79c2e80d07e9f7ca4f51aabc35ca29": "f_{X_{(k)}}(x) = \\frac{(2k-1)!}{(k-1)!^2}f(x)\\Big(F(x)(1-F(x))\\Big)^{k-1}", "dc79e87e8e8a8a321101fd15944932a3": "\\hat\\nabla", "dc7a1c4f4e9a4cd2325e1d6028bf81a8": "R_+M", "dc7afeb7acbf85c926ae610e194ef387": " R = \\frac{(\\rho^2 - \\rho '^2)^2(\\sin (2\\rho))^2}{4\\rho^2\\rho '^2 + (\\rho^2 - \\rho '^2)^2(\\sin (2\\rho))^2} ", "dc7b0e36d0b587d18e6c5f349e368b90": "0, 1, 2, ...", "dc7b2e098690da3f6b380f300cc1f277": "\\mathcal{L}=\\partial^\\mu \\phi^* \\partial_\\mu \\phi -m^2 \\phi^* \\phi -\\frac{\\lambda}{4}(\\phi^* \\phi)^2.", "dc7bc1cd36c978eae63f10ec2570982b": "\\phi_n(a)=\\sum_{m\\bmod p}\\binom{m(m^n+a)}{p}", "dc7bcb2bf15baa1449a3a374d31b864b": "A_2+2 Z \\rightleftharpoons 2 AZ", "dc7bdb91ebdc3564e4b5a58e3fac01a9": "\\log_2{(y)}=-\\frac{1}{2}\\log_2{(x)}", "dc7c86460ac041a3a629fbf69a6043a8": "E_{\\rm x} > E_{\\rm y}", "dc7cad9a1277553abe98065cf73f036c": " f(n x) = f(x + x + \\cdots + x) ", "dc7d2ce30d8ede39c87e67030cbd3163": "x^\\top y + c > 1", "dc7d3c5cccf8eed7596b7a0f6b95d20f": "f'(x)=0\\!", "dc7d45d5dda8d494f29064bdfc4cbee6": "|n_{k_0}\\rangle\\otimes|n_{k_1}\\rangle\\otimes\\dots\\otimes|n_{k_n}\\rangle\\dots", "dc7d6a55a4621acdab67e5bcf6472924": "\\Omega=\\frac{2dz\\wedge d\\bar{z}}{(1+|z|^2)^2}.", "dc7d6bbe676bea47465f8eae5ef3e9b5": "\\| \\cdot \\|_{b}", "dc7db3aa666b0953d09d9d738338a5c9": "x \\in A \\cap C", "dc7dda58600b170e92218970d9c445a9": "|\\det(N)|\\le \\prod_{i=1}^n \\|v_i\\|,", "dc7e58575999b1a8ad8822f782c08c72": "\n\\mathbf x\n= \n[(\\mathbf A^T\\mathbf P_B^{\\perp})^{+}, \n\\quad (\\mathbf B^T\\mathbf P_A^{\\perp})^{+} ]\n\\begin{bmatrix}\n\\mathbf e \\\\ \\mathbf f \n\\end{bmatrix}\n= \n(\\mathbf A^T\\mathbf P_B^{\\perp})^{+}\\,\\mathbf e\n+\n(\\mathbf B^T\\mathbf P_A^{\\perp} )^{+}\\,\\mathbf f \n.\n", "dc7e63e90d1cc523d003058740856c5a": " \\int_{H^3}F \\, dV = \\int_{H^2} (1+b^2 +x^2)^{1/2} QF \\, dA.", "dc7f1d65632009262e1deef1f577db7e": "A\\bold{h}=\\bold{q}", "dc7f494266fbf4190095c1ac2f50e723": "\n\\Delta \\hat{z}\\ =\\ \\frac{1}{\\mu p}\\left[\\hat{g}\\int\\limits_{0}^{2\\pi}F_z r^3 \\cos u \\ du\n+\\ \\hat{h}\\int\\limits_{0}^{2\\pi}F_z r^3 \\sin u \\ du \\right]\\quad \\times \\ \\hat{z}\n", "dc7f783d9dd3d1888d52b4f9ca9e193a": "10^{-62}", "dc7f7fcbbf43104b9863bc811dfb3897": "n_{e0}m_e\\partial_t\\vec v_{e1} = -n_{e0}e\\vec E_1 - \\gamma_eT_e\\nabla n_{e1} ", "dc7f99aeb9a2cead16d15764b5f552ec": " H_n(X) := \\ker(\\partial_n) / \\mathrm{im}(\\partial_{n+1}), \\, ", "dc800bc951dd6ccb1fba7322cc8ae6ea": "V_t = (1/\\sqrt c) W_{ct}", "dc80101539bab14b3665a0759ad6bef7": "R = \\mathbb{Z}_p[x]/\\langle f \\rangle ", "dc8011b1c99fd7d20963f2fa732dd480": " f, g: \\mathbb{R} \\rightarrow \\mathbb{R} ", "dc80126dd86956d8a91444faf7284d87": "\\scriptstyle 2 \\,\\times\\, 2 \\,\\times\\, \\cdots \\,\\times\\, 2 \\,\\times\\, 2", "dc803ad303319062f6ac3710a2484ddc": "\\{p\\in P\\}\\;", "dc80653dcb7186832cd21761856a8068": " c_2= t_2 - at_2^{-1}", "dc8078da9c8e3d8c6bc4a74ff84df033": "H^{out}", "dc809f896bd21ca3c9a6345e1eb841bb": "P(\\{x_0-\\epsilon < X < x_0+\\epsilon\\})=0.", "dc80a38f4ce8032d9b0631da5789cadf": "\\left\\{x : \\exists y \\in \\left(xRy\\right)\\right\\}", "dc80cf056aebfe771e73032cc5e657d8": "23_{11} \\ ", "dc80d880dba711bdac372b02a0f49a30": " F_{X'_n}={[F(x)]}^n \\, ", "dc811247e2f0c32b6e1b4eb7e0202544": "H(X)=-\\int p(x)\\log\\frac{p(x)}{m(x)}\\,dx.", "dc818b529c44764745f90b018437aa47": " V = L_1 \\oplus L_2 \\oplus \\cdots \\oplus L_m \\oplus W, ", "dc821f34a8b7da7828ecbcd4d018ce97": "(e_1,\\ldots,e_n)", "dc825db6108f98d0c9aa2bb5d813f706": "x^{n-1} = \\left(x^{n-1} - \\frac{p_{n-1}(x)}{a_{n-1}}\\right) + \\frac{p_{n-1}(x)}{a_{n-1}}", "dc827d8e977024aa60ceb8ac76749b8a": "\\mu_\\text{sig}", "dc82b8276d69f9db53ec9df72019f115": "\\text{Pressure} = \\frac{\\text{weight density} \\times \\!\\, \\text{(area} \\times \\!\\, \\text{depth)}}{\\text{area}}", "dc82d74b56e081146e9eb551f4117c8d": "x_{i:\\lambda} \\sim \\mathcal N(m_k,\\sigma_k^2 C_k)", "dc8329489b623c72e2156e74d1a56223": "(-1)^s\\cdot \\tau(\\chi,\\psi)^s=-\\tau(\\chi',\\psi').", "dc8371912090f9b39a9ad4d28dcceb8b": "\\mathbf{\\hat{x}}, \\mathbf{\\hat{y}}, \\mathbf{\\hat{z}}", "dc837e1023ec869943a91a71436aad4e": " {s} = ({A} {Q}^{-1} {A}^{T}) {\\lambda} + (- {A} {Q}^{-1} {c} - {b} )\\,", "dc837e2da9ab9af3b8d09a1cd60508ac": "X dx", "dc838b23b70abefa28dbedd4ed105432": "\\psi(\\Omega^\\Omega 2+\\Omega^{\\psi(\\Omega^\\Omega 2+\\Omega^{\\psi(0)})})", "dc839f5a2ad846d4c125ea31372e9eb2": "\\mathrm{Ei}(x) = - \\gamma - \\ln (x) + O(x^2) ", "dc83a905660e35ac2babf3efad99dd15": "O(n^{3 - \\varepsilon/3})", "dc83ad1d7efbf42152b90aaef387cb0b": " \\|f_1\\|_\\infty + \\|f_2\\|_\\infty", "dc84159f245bfecc64c88aadc6c762cf": "B_r(p) = B(p;r) = \\{ x \\in X \\mid d(x,p) < r \\}", "dc8418c84a7100aebeb2bcbce30503c0": "B(r)=-c^2 \\left(1-\\frac{2Gm}{c^2 r}\\right)", "dc841b5bc6832cd761014d572c07eb74": "= \\sum_{k=-\\infty}^{\\infty} x[k]\\cdot h[n-k]\\,", "dc843936dff6846f75e0b1631c847b04": "\\omega_0=\\sqrt{\\omega_1\\omega_2}", "dc844b029653aa52d54c6487e26513f7": "A_m(1,1) = 1,1,1,1,1,1,1,1,1,1,\\ldots", "dc846dc395a2df4f7d007e0efc56abfc": " E^ \\nu (q, \\mathbf{k}) \\approx E^ \\nu(q, \\mathbf{0}) + \\langle \\phi _{q, \\mathbf{k}} \\mid \\frac{\\hbar^2 \\mathbf{k}^2}{2m_c (z)} \\mid \\phi _{q, \\mathbf{k}} \\rangle ", "dc8493f94dfcccdd8019cc653f8fcb51": "x\\ge 1", "dc84b8814d22e1d9aef0b5bbacad97e8": "\\mathtt{string}", "dc84f4da84df65062f8f42ec1e1bfcdd": "\nk_x = \\frac{2\\pi n_x}{L},\\quad k_y = \\frac{2\\pi n_y}{L},\\quad k_z = \\frac{2\\pi n_z}{L},\\qquad\nn_x,\\;n_y,\\;n_z = 0,\\, \\pm1,\\, \\pm2,\\, \\ldots\n", "dc84fcf54d7336804f1dbd88cad18715": "E[h(y)] \\approx \\frac{1}{\\sqrt{\\pi}} \\sum_{i=1}^n w_i h(\\sqrt{2} \\sigma x_i + \\mu)", "dc85010518c24f84179739954c2c6659": "\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}=a \\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\n+b\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\n+c\\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}\n+d\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}\\,,", "dc8510f0976922bdf4247bac9f81cac7": "p\\in {\\mathbb R}^n", "dc8552cc74c8f6558bd9e15862ead14f": "{n\\over 2n+1}={1\\over 3},\\, {2\\over 5},\\, {3\\over 7},\\, {4\\over 9},\\,{5\\over 11},\\cdots ", "dc85646e73835b6a6a85a580d394be8c": " S = \\frac{\\partial M}{\\partial \\log t}.", "dc856e5da732caae67dcc3b17155ed18": "1\\ f\\ x = f\\ x ", "dc858f449443548db82c917d2cfd22b0": " S \\approx \\frac{ \\sqrt{ 2 } - 1.5382 \\Gamma( \\frac{ 3 }{ 2 } ) }{ \\sqrt{ 2 ( \\Gamma( \\frac{ 5 }{ 2 } ) - \\Gamma( \\frac{ 3 }{ 2 } ) ) } } \\approx 0.0854 ", "dc85b9d2c94dd51d13cc7b2fe056b2a0": "1\\hbar=L=r\\cdot p=r\\cdot E/c=T \\cdot E/(2\\pi)=\\lambda E/(2 \\pi c)", "dc869d1a725b0d90481a8caa55177885": "\\Gamma = K_{eq}", "dc86cd52e973bde5453cd7759b32a9fe": "p(X_s = x_s|X_b = x_b) = \\frac{1}{Z} \\prod_{c\\in C(G)}\\Phi_c (x_s^c,x_b^c)", "dc86d4368a7bd038cc1eb57bdb4f1f4e": "x[T \\circ S]z", "dc86f38e16839539ab309aa1063dd7e1": "h\\left(C_1e^{mL}+C_2e^{-mL}\\right)=km\\left(C_2e^{-mL}-C_1e^{mL}\\right).", "dc872abb352116c42b8ae546dfc35f31": "p=-(\\alpha+\\beta)", "dc878206c6fd6dca61ad086a3558a804": "\\,\\mathbb N_0=\\aleph_0=\\omega", "dc8793b61dc9b800e95548ddecdc6c19": "z_i * x_i = z_i * c_i = z_i * y_i", "dc87b06742a275973f6d0207d58017a1": "V = \\frac{4}{3}\\pi(2.0 \\mbox{ cm})^3 = 33.47 \\mbox{ cm}^{3}.", "dc88165321fb81f8f7d09245d1ba023d": "C_{rq}", "dc881a4fe7934b2b295b9696f5de4123": "\\frac{1+2x}{(1+x)(1-2x)}", "dc8848476d5c8c3904acaed10b562689": "\\star \\mathrm{d}x \\wedge\\mathrm{d}y = \\mathrm{d}t\\wedge \\mathrm{d}z", "dc88518d09fd6af948a49acd8c817778": "\\lim_{n\\rightarrow\\infty}\\|f_n-f\\|_\\infty=0.\\,", "dc887642b03df65374abafbf33b392a4": "M=\\frac{1000000}{T}", "dc88a92ab6fa867244eb6ea42309f71e": "l=l_0 \\cdot \\sqrt{1- v^2 / c^2}", "dc88c2f52fb18e81fcaca93574268ef9": "x < x_{\\min}.", "dc88c866f19f2e085a9e73fa7eba0167": "\\begin{align}\np(\\mu|D, I) = & \\int p(\\mu,\\sigma^2|D, I) \\; d \\sigma^2 \\\\\n= & \\int p(\\mu|D, \\sigma^2, I) \\; p(\\sigma^2|D, I) \\; d \\sigma^2\n\\end{align}", "dc88ca6b72035df4bf9bae1e35f67c9c": "\\lambda\\gamma\\left(\\sum_{g=1}^G \\|w_g\\|_2\\right) ", "dc88fc39967363d5a63aeb1773dc2699": "S[t^{-n}\\varphi\\circ\\mu_t] = t^mS[\\varphi]", "dc89b1d72a705de052a4fe692d09e302": "(1,2,2,3) \\neq (1,2,3)", "dc89b386ac2f5f755a535f77831990c4": "v(x,\\,y)", "dc89c3d7a529a1a3b93683b96254d6b8": "BU(1)= PU(\\mathcal{H}),", "dc8a4e81aafc37e37f34d2ed59cf110d": "y_j=1", "dc8ab34d233cc147f3cd3e8df7713fe3": "\\Delta(\\gamma)", "dc8ab5425d1a1008688ad58413c770a8": "J_{z} = J_{z, r_o} - J_{z, r_i} = \\frac{\\pi}{2}r_o^4 - \\frac{\\pi}{2}r_i^4 = \\frac{\\pi}{2}({r_o} ^4 - {r_i} ^4)", "dc8af40ef4ec4304e61efcff1007c007": "C \\left |{\\frac {f^{\\prime\\prime} (\\alpha)}{f^\\prime(\\alpha)}}\\right |\\epsilon_n<1, \\text{ for }n\\in \\Zeta ^+\\cup\\{0\\} \\text{ and }C \\text{ satisfying condition (b) }.\\, ", "dc8b6bd9600800f3a2943ff632e4aedf": "Z=(X=x)", "dc8b7be9a3ac33577dbba858b620ce3d": "\\mathcal{O}_S", "dc8ba1cd257a16b198448fd5220c1cfb": "\\phi^2 = \\operatorname{id}_V\\,", "dc8bbaf98b7570a8803d277420ce6a9e": " \\frac{1}{(2 \\pi)^n} \\int_{\\mathbb{R}^n} e^{i x \\cdot \\xi} \\, \\mathrm{d} \\xi. ", "dc8c0111c24f52a7b118303554f3842c": "\\begin{align}\n x &= 0.000000100000010000001\\ldots \\\\\n 10^7x &= 1.000000100000010000001\\ldots \\\\\n (10^7-1)x=9999999x &= 1 \\\\\n x &= {1 \\over 10^7-1} = {1 \\over9999999}\n\\end{align}", "dc8c1038013938cc284396beefe44271": " Z(\\mathbb{C} [ S_1]), Z(\\mathbb{C} [ S_2]), \\ldots, Z(\\mathbb{C} [ S_{n-1}]), Z(\\mathbb{C} [S_n]) ", "dc8c478bc3151636de9e71ac499f2ba5": "t_{1}, \\dots, t_{k} \\in T", "dc8c4b8b17101c3b91e22d692f9045e6": "a_{C} = \\frac{3 e^2}{20 \\pi \\epsilon_{0} r_0}", "dc8c71393a26890fdebd7a769d09998a": "\\rho(X;U) := |\\mathop{det}(A)|^{-1} \\in \\mathbf{R}^{>0}", "dc8cc1b4e9ca754928e575a8ef580448": "\n\\frac1{2\\pi\\,i}\\oint_C \\frac{p'(z)}{p(z)}z^m\\,dz\n=\\sum_{z\\in G:\\,p(z)=0}\\frac{p'(z)z^m}{p'(z)}\n=\\sum_{z\\in G:\\,p(z)=0}z^m.\n", "dc8ce4a4d8e29d8d2bf905c90ed2af36": "J_\\alpha(z)\\sim\\frac{\\exp\\left( i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)}{\\sqrt{2\\pi z}}\\text{ for }-\\pi<\\arg z<0", "dc8d48b490ca6a987ddd7a5c339fb894": "U = n C_V T", "dc8d4f2d87cd610ca39a997642bec11d": "\\mu (r){\\stackrel{\\rm def}{=}}\\min _{\\alpha >0 ,k+\\alpha 0.", "dc8d5290fd3fbc78d8bc2de12b62d2aa": "\\|x\\|^2=\\sum_{b\\in B}|\\langle x,b\\rangle |^2.", "dc8da4a0908c981ad38a9df95761fd57": " d = 1.26\\; R_M\\left( \\frac {\\rho_M} {\\rho_m} \\right)^{\\frac{1}{3}} ", "dc8edf30188dca9ae6e817da6fbe99f6": "\\frac{1}{z} \\exp(\\tfrac{\\pi i}{4}) \\left( 1 - {\\frac{1}{4} \\choose 1} 3z + {\\frac{1}{4} \\choose 2} 3^2 z^2 - {\\frac{1}{4} \\choose 3} 3^3 z^3 + \\cdots \\right). ", "dc8f081e80d1431d7e2a5d08991211ab": "P_{absorb}", "dc8f30ec1fe57ec733f5cb290c262af8": " u_x(t,L) + b u(t,L) = 0,\\,", "dc8f847ad0ad7a784f2c7a983e21f3ee": "\\Delta x^k = - g^{ij}\\Gamma_{ij}^k.", "dc8f8ab69db533b675aed6be60ee95b7": "\n \\boldsymbol{s} = \\boldsymbol{\\sigma}-\\tfrac{I_1}{3}\\,\\boldsymbol{I}\n", "dc90174889c0394c388ca69ed1729920": "N=2k(a\\phi(front)-b\\phi(rear))=2k(a-b)(\\theta-\\psi)-2k\\frac{(a^2+b^2)}{V}\\frac{d\\theta}{dt}", "dc90174a5ac70245615b158ea9d340ec": "\\alpha=\\frac {(y_i/x_i)}{(y_j/x_j)} = K_i/K_j", "dc901ccd7133c211ec744678fe8968f6": "\\mu\\boxplus\\nu", "dc90551b9101aa4077bdc09bf9a5bcf9": "E_2(\\omega) = \\frac {\\omega E^2\\eta} {\\eta^2 \\omega^2 + E^2} ", "dc90664c60301b8d226c0cc200a1de1d": "B = \\mu_0 \\frac{Ni}{l},", "dc90e2857d91e87d272d1af7c58c21ef": " \nr=\\frac{r_0}{1+\\tau/\\phi}\n", "dc90f69b50a4b6ca5ef59d2d8a4386b6": "\\mathcal{H} (x,t,\\lambda;p,p_t) = \\lambda \\Big( p_t + {p^2 \\over 2m} + {1 \\over 2} m \\omega^2 x^2 \\Big).", "dc91174afcebf4c3bcef85275a38f59d": "\\tilde{u}_k(K)", "dc913413cab5289368f4c88d836269d2": "\\mathbf{B}_{l,m}^{(M)} = -\\frac{i}{k} \\nabla \\times \\mathbf{E}_{l,m}^{(M)}", "dc91359df5342a5b38587ab8000c04f4": "\\scriptstyle\\pi^+ \\rightarrow \\mu^+ + \\nu", "dc9143caffcbc2c90c1c5754cfaf1e05": " R^T T \\wedge = R S ", "dc918314b1b82912ae5f60de6aa6bd06": "Q(\\mathbf{y},\\mathbf{x})>0", "dc91cceae8ff263ac39753d6cd8846a4": "\\forall a \\in Y, X \\vdash a.", "dc9203416496845e03d4d4b696db93c9": "X=\\left(X_1~X_2~\\ldots~X_N\\right)^T", "dc92ee7400a1d8f69c4cb2c32c4e6d33": "\\Phi \\cup \\phi", "dc9339ebf6b3e2568cb4f31f038022c1": "\n\\frac{1}{p} = \\frac{j}{n} + \\left( \\frac{1}{r} - \\frac{m}{n} \\right) \\alpha + \\frac{1 - \\alpha}{q}\n", "dc93a0c5ed672fbf1037df2d65a34952": "c/n", "dc941e040efbe9501c050ea43c65a80e": "\\langle X, ( R_i )_I \\rangle ", "dc9456065e3d4b99945851415b7bbdf8": "[-{\\hbar^2 \\over 2\\mu}{d^2 \\over {dr_i^2}}-eE_ir_i-\\epsilon_i]\\psi(r_i) = 0", "dc9461222fc4600a279658b33375fd49": "\\frac{1}{T_2}=\\frac{K}{2}[3\\tau_c+\\frac{5\\tau_c}{1+\\omega_0^2\\tau_c^2}+\\frac{2\\tau_c}{1+4\\omega_0^2\\tau_c^2}]", "dc946b310ef1b8f4270e7c0c75a5c49b": "{\\hat \\Psi} = D \\circ \\Phi \\circ E : \\mathcal{B}_2 \\rightarrow \\mathcal{A}_2 ", "dc947614f42f41825b86d13653f8dda0": "n \\in X \\Leftrightarrow \\mathbb{N} \\models \\phi(\\underline n),", "dc947d7aeac8ff0ce43d658beebc374c": "1 \\leq i \\leq n.", "dc94a012e74c5e7aee5451831d9c6bd0": "\\{1,2,2,3\\} = \\{1,2,3\\}", "dc94c2abeabefdf5b12236028c15fc87": "L \\cap \\mathcal{P}(S) \\subseteq L_{\\alpha+ 1}", "dc94d0b3336e62cabb8b5c1f1485165e": "L[y]=y^{(n)} \\ ", "dc94d999b6fb0b51c112f1d34c390cc4": "A = \\frac {1} {2}|\\frac {\\pi} {\\lambda - \\lambda_0} - \\frac {\\lambda - \\lambda_0} {\\pi}|", "dc94f5fe646f45b743d3fc3745f349e1": "p(x|\\mu,\\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}", "dc95044d2bbe96edac80ca868499246c": " \\varphi\\left(\\bigcup_{i=1}^\\infty A_i\\right) = \\sum_{i=1}^\\infty \\varphi(A_i).", "dc957188fc37fdfcd39244dbe97b6ba3": "\\ \\displaystyle \\varepsilon > 0\\ ", "dc95c0b06b905811b68d3ffa1c568612": "\\pi _{2}^{2}-p^{2} =-\\left( \\varepsilon _{2}-\\mathcal{A}_{2}\\right)^{2}=-\\varepsilon _{2}^{2}+2\\varepsilon _{w}A-A^{2}, ", "dc95e65d1e8d7ff9a9e156e072aed098": "\\mathfrak{sl}_6(\\mathbf K)", "dc9653af26842ee42c4ef57d3bf451bd": "\\|Tf\\|_r\\le \\gamma N_p^\\delta N_q^{1-\\delta}\\|f\\|_r", "dc967a0e7a01f89060c6929126626a32": "P_\\text{cr}=\\frac{\\pi^2 EI}{(KL)^2}", "dc9697c317a58c3dfcfd646456e80ae1": "\n \\mathbf{e}_p\\times\\mathbf{e}_q = \\varepsilon_{ipq}~\\mathbf{e}_i\n", "dc96c9381a2bc3b26a3e2e9acae70bd4": "s_it_{i+1}-t_is_{i+1}=(-1)^i.", "dc96e427385dc8e9b039b8bb0e5544e4": "\n\\rho = \\sum_i ( \\psi_{1,i} \\otimes \\cdots \\otimes \\psi_{n,i} ) ( \\psi_{1,i} ^* \\otimes \\cdots \\otimes \\psi_{n,i} ^* ).\n", "dc970c396583d58133f8052c199551c6": "A \\leq_1 B \\, \\mathrm{and} \\, B \\leq_1 A", "dc971ca6df011d7c5db1fd9dd7875add": "\\mathfrak{g} \\to C^{\\infty}(M)", "dc975a00682482aedbd5ee6b2fbfa378": "\n X\\ \\sim\\ \\mathcal{N}(\\log (OR),\\,\\sigma^2). \\,\n ", "dc9810cf1b19399045b528a0719adad8": "P(\\rho,\\sigma) = \\sqrt{1 - F(\\rho,\\sigma)^2}", "dc9880f8a8682a5fd4689b7e093d2b49": "g(\\boldsymbol{x}) = \\frac{1}{1+f(\\boldsymbol{x})}", "dc98955995146aae17dcfbcafaf4cb09": "(a)_n", "dc98f1e082b9ace4c25bb9563c11bd5f": "\\lnot F[\\text{false}] \\land G[\\text{true}]", "dc9947a8e4c62524732ad131ed6c8564": "\nT_{ij} = e^{ - \\lambda _i - \\lambda _j - \\beta C_{ij} } \n", "dc995811f10668baa7d12c3d9faf646a": "1/p + 1/q = 1/2", "dc9a18d8570cf21c06a699e43c3b8af0": "-\\pi\\chi(X(p)) = |G|\\cdot D,", "dc9a436634b6f8982a57c88eedb7063b": "x:=x+1\\,\\!", "dc9a6adae0495b1ebdd96df42df0d8ac": "\\displaystyle u_t=\\Delta(u^\\gamma)", "dc9a9b26661281f144f549fdbe81107f": "\\left(\\sqrt{\\frac{2}{5}},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-4}{\\sqrt{3}},\\ \\pm2\\right)", "dc9acc40dcc9dcaf4a2221f4b32686e9": "u(x+iy)=\\frac{1}{\\pi}\\int_{-\\infty}^\\infty\nP_y(x-t)f(t) dt\n", "dc9b07a1c3d5e6ebcf6605035bbf2da0": "Q_s=-Q_r=Q=V\\sin\\left(\\frac{\\delta}{2}\\right) \\cdot \\frac{2V\\sin\\left(\\frac{\\delta}{2}\\right)}{X}=\\frac{V^2}{X}(1-\\cos \\delta)", "dc9b0e3761c6f48d9ea616ea7ec1a736": " -\\nabla^2 u = f\\text{ in }N,\\quad u = g", "dc9b0f1d3610f916610b36bf49f791e9": "\\text{E}\\left(e^{-t X}\\right)=\\eta e^{\\eta}\\text{E}_{t/b}\\left(\\eta\\right)", "dc9b7bae57cc4fdf8892875fca4646a4": "=\\frac{32{e^4}}{(k-k')^4}\\left( (k' \\cdot p') (k \\cdot p) + (k' \\cdot p) (k \\cdot p') -m^2 p' \\cdot p - m^2 k' \\cdot k + 2m^4 \\right) \\,", "dc9bf661ee88c4dd8fd5d4f53db750c3": "y_1 = x_0 - x_1 \\omega^k, \\, ", "dc9c4a8c071ef81dbbe3b8d753bbc9bf": "f(1) = v", "dc9c9e4fe637f4b64ff8dc17de010f64": "\ndT = {1 \\over N C_v} dE\n", "dc9cbcc2a14d947644ec5ef005d65369": " M^{p,q}_m(\\mathbb{R}^d) ", "dc9cc02451f6f1bc7cb5f935a2d93263": " (W\\mathbf{r})\\cdot \\mathbf{s} = L^* \\cdot (\\mathbf{r}\\wedge \\mathbf{s}) ", "dc9e3b0a2c0619c7267ca283f689431c": " \\delta \\sqrt{-h} = -\\frac12 \\sqrt{-h} h_{ab} \\delta h^{ab} ", "dc9e5a131fec44ccbcbc8df174c1ee04": " f \\colon V(G) \\to {\\mathbb R} ", "dc9e6dc57358ddd5bb4c86ea6af6a8c0": "\\mathrm{S} = \\Omega^{-1} = \\dfrac{\\mathrm{A}}{\\mathrm{V}}", "dc9ec681e0a0dc289257ccbf8ec35886": "\\gamma = x^2_{\\xi} + y^2_{\\xi} ", "dc9ed9161e410efcfeba034e87d01f77": "\\int_{1}^{\\infty} f(x)\\, dx = \\lim_{t \\to \\infty} \\int_{1}^{t} f(x)\\, dx < \\infty,", "dc9f5ec9cac31668688f37e2569830ac": "\\phi(t)\\rightarrow x_0\\quad \\mathrm{as}\\quad t\\rightarrow-\\infty", "dc9ff8c9edfde60dab59cbc5a1a14c8d": "f:R\\to A", "dca000c9d6c131d644b170b577566b58": "f(x)=\\begin{cases}\n \\sin\\left(\\frac{1}{x^2}\\right)\\text{ if }x \\ne 0\\\\\n 0\\text{ if }x = 0\n\\end{cases}", "dca015660df66d96c6d2c7fb0e85aac6": "R < 0.07", "dca041becd70589122a4b5f01bce34aa": "\n\\frac{1}{\\sqrt{\\lambda}} = 0.8686 \\ln[\\frac{0.4587Re}{(S-0.31)^{\\frac{S}{(S+1)}}}]\n", "dca05790cee4699deb67461fe6670536": "s \\in V", "dca0bf3fc8a85fba9257e442e0c16b48": "|\\mathbf{L}| = \\hbar\\sqrt{\\ell(\\ell+1)}\\,\\!", "dca0e1f14a4afd2e281404d9a878cce3": "s_4=\\alpha^{1},", "dca11e6ebf75b467c4492e126c79dbe4": "\\alpha: \\Delta\\otimes_R i(\\Delta) \\to J", "dca13da58a9c50f948269f4af12a6db4": "\\chi_1 = 0", "dca1719cfb042237a284723b8aafa4c5": "c'(t)\\cdot n(t)=0", "dca17ee170cbcc05b9c692f428a8f871": "(x)=2/(1+e^{-ax})-1", "dca18e2d98a4c7cbcd3561f29987c4bb": "(5)\\quad M_1=\\frac{(0.24)^2}{2}+\\frac{q^2}{g*(0.24)}=13\\;ft^2", "dca1bfc68629526cc6ce214e5d214001": "\\operatorname{Out}(S_6)=C_2", "dca1c1bcccf69eef785e9be3aa5bce97": "g_i(t)", "dca1e50eafd1b1ae558c77b56f78871c": "C_{slope} = \\pm \\dfrac {V_{ref} T_{slope}}{R_{slope} f_{clk}}", "dca1f6e2204e2d1d0e9961c8256f1df8": "J= \\det\\boldsymbol{F}", "dca21fd1e0605fcc13e00b75f8757264": "B_X", "dca25f5f8a40b49cde255f3171595483": "\\det(\\bar{g}_{\\kappa\\lambda}) = \\det(\\eta_{\\kappa\\lambda}) =", "dca28b9a0d1e603cf7d9a828f213c081": " = C_c", "dca29b4b5df819fa04084dbd918bc3e5": "F(\\mathbf u(s),\\lambda(s))=0", "dca31af0b41e533e5224594916810644": "| SA |:| SB | = | SC | :| SD | =| AC | : | BD | ", "dca367ad0a8cab71ffe83e63bfd1603b": "k\\;=\\;\\dfrac{\\delta x}{a\\cos\\phi\\,\\delta\\lambda\\,}=\\,\\sec\\phi.", "dca37e99b33c87e45cb9a2889bb102f0": "x = r \\cos \\varphi \\,", "dca3aa9df277aeaa54d6f7520986cfb5": "(1)\\qquad D\\rho=\\rho^2+\\sigma\\bar{\\sigma}+\\frac{1}{2}R_{ab}l^a l^b\\,\\hat{=}\\,0\\,,", "dca3ad143742ce6c85cac5b6e9ca83dc": "\n\\Pr \\left\\{ \\lambda_{\\text{max}} \\left( \\sum_k \\mathbf{X}_k \\right) \\geq t \\right\\} \\leq d \\cdot e^{-t^2/8\\sigma^2}\n", "dca3ebb7e66c70f06d072bf6b7b98bc7": "\\operatorname{cov}\\left(\\frac{w_i}{w}, z_i\\right) = \\left(b - a\\right)\\operatorname{var}(z_i)", "dca441e18f15525e1c3f76ba9c76ceca": "\\int_0^1 \\frac{1}{\\sqrt{x}}\\,\\mathrm{d}x=\\lim_{a\\to 0^+}\\int_a^1\\frac{1}{\\sqrt{x}}\\, \\mathrm{d}x = \\lim_{a\\to 0^+}(2\\sqrt{1}-2\\sqrt{a})=2.", "dca45874acfa80723c89857b6ff4a688": "\\frac{d}{d\\xi }\\left[(1-\\xi^2 )\\frac{d\\Xi}{d\\xi }\\right]-\\frac{m^2\\Xi}{1-\\xi^2 }+n(n+1)\\Xi=0", "dca485219b5190ff1e1749b79dc16296": "M_o \\simeq N.k + M_e", "dca4966b41be57f745193df42d40676e": "\\mathit{GF(p)}", "dca582b1bf4dda7dbee8087cedc18b82": "N_R = \\sum_{n=2}^{\\infty}N_n ", "dca59f8cf929d40c4a450291689f81cf": "z = c(x^*) \\geq c_R(x^*)\\geq z_R", "dca5e92819e138631e5e3730b384fa2d": "\\frac{Wkd}{2} = U^2 \\frac{2k}{rd}\\left(C + \\frac{md^2}{4}\\right)", "dca6213e21fef0fc8de0d66962ed0416": " p_{\\tau(w)} ", "dca65dac084cbbee77fb4eca7c1d6545": "\\scriptstyle \\eta(x)=\\zeta(x)=0 ", "dca6f01a0c178c84e96703322d960326": "\\scriptstyle{\\sum_{k=1}^n c_k^s = 1}", "dca72165e480144b8dd7e788531abf86": "\\hat{G}_{(1-X)} = \\prod_{i=1}^{N} \\left (\\frac{\\hat{c} - Y_i}{\\hat{c}-\\hat{a}} \\right )^{\\frac{1}{N}}", "dca7398183f3f03a2888f9caabc42908": "\\omega_c", "dca790e775bf3be888e52e9dd3fa73ad": "f'(x_0) > 0", "dca796c0b434c5ec84c91275700f3625": "\\sum_{i=0}^n {n \\choose i}a^{(n-i)} b^i=(a + b)^n", "dca7992443c31d82a3902316822f9af8": "\\mu_{a, r} (A) = \\frac1{r^{n - 1}} \\mu(a + r A).", "dca7ba330519614b61100b1e5a14ac25": "\\int_{-1}^1 \\sqrt{(3t^2)^2 + (5t^4)^2}\\,dt = \\int_{-1}^1 \\sqrt{9t^4 + 25t^8}\\,dt.", "dca7de0f2438f0f9e83fc40222110323": "\\mathrm{R}\\,\\!", "dca7e14f7a13aa0fedd45ac434753963": " A_y = 2 \\pi \\int_a^b x(t) \\ \\sqrt{\\left({dx \\over dt}\\right)^2 + \\left({dy \\over dt}\\right)^2} \\, dt, ", "dca86eb05068e2d60eb46a3208330b3e": "x' = \\gamma \\left ( x - v t \\right )", "dca87e12de9070e7e88025f4aebd7889": "y_1=x_1^{2^k},\\,y_2=x_2^{2^k},\\,\\dots,\\,y_n=x_n^{2^k}", "dca8b55f902d3996ea807dec68fed533": "\nZ (n,V,\\beta) = Z_0\n\\int D w \\exp \\left[ - \\frac{1}{2 \\beta V^2} \\int d \\mathbf{r} \nd \\mathbf{r}' w (\\mathbf{r}) \\bar{\\Phi}^{-1} (\\mathbf{r}-\\mathbf{r}') \nw (\\mathbf{r}') \\right] Q^n [ i w ], \\qquad (6)\n", "dca8d66f6b3058fcba9983266266e4bf": "E_{xc}^{\\rm B3LYP} = E_{x}^{\\rm LDA} + a_0 (E_x^{\\rm HF} - E_x^{\\rm LDA}) + a_x (E_x^{\\rm GGA} - E_x^{\\rm LDA}) + a_c (E_c^{\\rm GGA} - E_c^{\\rm LDA}),", "dca90348d03e2bdb52ad4cc669a23ebd": "-r", "dca91d214913954ca60d9b5e118508cc": " \\dot{Q}_{1 \\rightarrow 2} = A_{1}E_{b1}F_{1 \\rightarrow 2} - A_{2}E_{b2}F_{2 \\rightarrow 1}", "dca96d47b5915f8fec74daa14256d060": "\n\\begin{bmatrix}\na_{11} & a_{12} & 0 \\\\\na_{21} & a_{22} & a_{23} \\\\\n0 & a_{32} & a_{33} \\\\\n\\end{bmatrix}\n\\begin{bmatrix}\nk_0 \\\\\nk_1 \\\\\nk_2 \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\nb_1 \\\\\nb_2 \\\\\nb_3 \\\\\n\\end{bmatrix}\n", "dca99cbf06c165517cbf2ac856ceee3e": "P_f = \\frac{{13 - s \\choose 3}}{{48 \\choose 3}}", "dca9f0882456fce3f6e86a6725478a07": "\\lim_{t\\to\\infty}H(t)=\\infty", "dcaa2549d570fddd9f65af326d7a27e6": "RR\\cdot M_{frac}", "dcab378b21489b7b461effd426a3c956": "\\nabla^2 f = \\nabla \\cdot \\nabla f = \n\\frac{1}{\\cos(\\phi)^2}\\left(\\frac{\\partial^2 f}{\\partial x^2} + \\frac{\\partial^2 f}{\\partial z^2} - 2 \\sin(\\phi) \\frac{\\partial^2 f}{\\partial x \\partial z}\\right) + \\frac{\\partial^2 f}{\\partial y^2}", "dcab469abf46512a65cf13533c4f6fcb": "\\Delta E = - \\vec{\\mu} \\cdot \\vec{B}", "dcab82832ea3d61cbe73719fa78395e2": "m\\{x:\\, |Tb(x)| \\ge \\lambda\\}\\le m\\{x:\\, x\\notin \\cup J_n^*,\\,\\,\\, |Tb(x)| \\ge \\lambda\\} +m(\\cup J_n^*).", "dcaced68a4b7838b6084ccc03c4eb318": "m_{\\lambda} = \\sum M_{\\alpha},", "dcacf1fb7f81a4c1bcc730c40e537311": " v ( S \\cup T ) \\geq v (S) + v (T) ", "dcad130a391b55c25c0b35d9c7b2da48": "p = hf/c = h/\\lambda\\,\\!", "dcad23fbcca516fbbe132b9c9446ddd2": " w = f[x_{k-1},x_{k-2}] + f[x_{k-1},x_{k-3}] - f[x_{k-2},x_{k-3}]. \\, ", "dcad619e72ec96b1cac919e9b7bbe33b": "\\sigma(R)", "dcade1a2a8bdfb8aeb8aede0befc678c": " x_2", "dcadf4cdefa0b6d681f42a4e6e3f1b98": "r_n=\\frac {(1-r)r}{2[n^2(1-r)^2+r]}", "dcadf80120027e2f955fa8d517fe40e2": "0 \\to (\\mathbb{Z}/2\\mathbb{Z}) \\oplus \\mathbb{Z} \\to (\\mathbb{Z}/2\\mathbb{Z})^2 \\oplus \\mathbb{Z} \\xrightarrow{(0,1,0)} \\mathbb{Z}/2\\mathbb{Z} \\to 0,", "dcae43ec87efb0abc182df91f0ec9aff": " \\frac {dy} {dt} = - y.", "dcae55230b45ea7468c966b64f75508c": "F_n^2 - F_{n+1}F_{n-1} = (-1)^{n-1}", "dcae9ccf19c2b889a46a1630ea7e7402": "\nx_n \\approx (n+\\tfrac12)\\pi - \\frac1{(n+\\frac12)\\pi}\n", "dcaeb5f1c8fa0ec97d475e117917c572": " dA_{\\bold{y}} = \\begin{bmatrix} 1 & 0 & d\\phi \\\\ 0 & 1 & 0 \\\\ -d\\phi & 0 & 1 \\end{bmatrix} . ", "dcaf2395c498eedc6b8d1fef2259fc96": " \\approx 6 \\times 10^{15,151,335} ", "dcaf3cf73fbb808770c52bad0a10dbe5": " C^T_{(+)}=-C_{(+)};~~~C^2_{(+)}=-1 ", "dcaf3d996a8fb8dddbbaf82654b4b493": "\\vdash (A_1 \\land \\cdots \\land A_n) \\rightarrow B", "dcaf743167221b2d354b11629baba6b5": "{\\partial f \\over \\partial \\bar z} = 0,", "dcaf993833e8825e80c88a3a9eef9ed3": "R_2", "dcafb3c775e2897e681eb182d8b8ad6f": "\n[ H_i(z), H_j(w) ]= 0, ~~~~~~~~\n", "dcafca73827f762dbf1a0b4f4c73820d": " \\begin{cases}\ne^{-{\\pi}a^2} \\ge 0.00001; & \\left| a \\right| \\le 1.9143 \\\\\ne^{-{\\pi}a^2} < 0.00001; & \\left| a \\right| > 1.9143\n\\end{cases}", "dcafce8e0dfa43ed6a6170a38d64f7ab": "N_{2,1}", "dcb00afc99e6cbe6f610a7cf944035a7": "4 \\times 2^2 - (2 + 2^2)", "dcb05762bee746313318e59ba511131f": "\\{E_i\\}_{i=1}^W", "dcb08867528dbe8c93d036cc21d396e1": "f(z_j)=p_n(z_j), j=0, 1, \\ldots, n,", "dcb0999600dcbe7e1238d158afb91954": "\\displaystyle{\\varphi=(\\psi^2)^{1/4}}", "dcb0a4f92e7fdb4526e90d2b1425e3f1": "f:\\mathbb{R}^n \\to \\mathbb{R} \\cup \\{- \\infty, + \\infty\\}", "dcb0c2069def78a221b7bab6758e02cb": "z_{\\alpha}\\sigma/\\sqrt{n}", "dcb0d20e7c65323d304ae97b580de907": "\\pmod{2}", "dcb104d470ac54684da8f7cadaeabd67": "S = S(q_1,q_2\\cdots q_N, t)", "dcb14ee93c97a11e5e83a2b43196055e": "s^2 = x", "dcb15b382a7936909a9140e6b0e64969": "r = \\alpha", "dcb16d6fbda0fc2656f77b34160f902f": "\n\\begin{align}\n& {\\mathbf \\nabla} \\cdot {\\mathbf D} \\left({\\mathbf r} , t \\right)\n= \n\\rho\\,, \\\\\n& {\\mathbf \\nabla} \\times {\\mathbf H} \\left({\\mathbf r} , t \\right)\n- \\frac{\\partial }{\\partial t} \n{\\mathbf D} \\left({\\mathbf r} , t \\right)\n= \n{\\mathbf J}\\,, \\\\\n& {\\mathbf \\nabla} \\times {\\mathbf E} \\left({\\mathbf r} , t \\right)\n+ \n\\frac{\\partial }{\\partial t} \n{\\mathbf B} \\left({\\mathbf r} , t \\right)\n= 0\\,, \\\\\n& {\\mathbf \\nabla} \\cdot {\\mathbf B} \\left({\\mathbf r} , t \\right)\n= 0\\,. \n\\end{align}\n", "dcb19a531ba5c69a2f93ea8b42a66046": "m_{em}=E_{em}/c^2\\,.", "dcb1bd30774f16bed8978fc65662ab88": "\\{U_{j}\\}\\subseteq \\mathcal{V}", "dcb1ca0f44976f6cd5c1da336f31ff7d": "\\scriptstyle x\\in U_2", "dcb1d204cd90cf8fbc83a8deada031c1": "\\vdash \\subseteq (Con \\setminus \\lbrace \\emptyset \\rbrace)\\times T", "dcb21110a261bee9dc2702d76af2e5aa": "A_{11} = -B_{11}^* = A_{1,-1}^* = -B_{1,-1} = -\\frac{1}{2} A e^{-i\\xi_0}, ", "dcb229a96d6ba96144d78c19bcda0e1c": "B_\\delta([x,y])", "dcb23fdced07e759a0c2f57e2d9f908b": "\\Delta h=r(\\frac{\\sqrt{2}}{2}-{3}\\frac{\\sqrt{3}}{8})\\approx 0.0576r", "dcb244d819a67fe5a0097756435deacb": "\\frac{AF}{FB} = \\frac{AF'}{F'B}", "dcb24caeb98d1ee6380abaa34b016286": "\\sin \\theta < \\theta < \\tan \\theta\\,", "dcb2592b13a18c5689c185ba7ed69bfb": "f_n = 1 - (1- \\frac{1}{N})^n.", "dcb2a66c51a985cee9aee08b2e02bd57": " ii ", "dcb34e9194ef60d7570fc2e069da323e": "a(n) = \\sum_{i=1}^{n}\\binom{n}{i}a(n-i).", "dcb3d4ed506f9030a22ef22da10fa063": "(E) = \\hbar^2k^2/{2m}", "dcb41c6a1355a25eef13c526d070832e": "A x = b,\\,", "dcb494f8d3dc2b100c97e47df0112f8f": "{{v}_{GS}}", "dcb49546511e5eb80bedaa62d8259bb4": "e^{-2\\lambda}-e^{\\lambda(1/e^2-1)}. \\, ", "dcb4b11103c790ee2a644f948a572a40": "f(x, y)", "dcb51ff34f1faf49a8db21d59c6e3b9c": " X \\hookrightarrow X' \\to Y", "dcb5636fd94e80297258aac92f67aeaa": "\\{\\mathbf{u}_1, \\ldots, \\mathbf{u}_m\\}", "dcb573f7b199204d9f4a86028ef2d535": "|z|=L", "dcb5d859e84edb01f023f58399ef13b8": "\\displaystyle{\\alpha(z,v)\\ge \\|v\\|/R,}", "dcb617351cf42b848c657ffe2a78a61e": "F(z)\\rightarrow 0", "dcb65ccfd0fb7b265b71a2ed3bced1e9": "\n\\overbrace{\\rho \\Big(\n\\underbrace{\\frac{\\partial \\mathbf{v}}{\\partial t}}_{\n\\begin{smallmatrix}\n \\text{Unsteady}\\\\\n \\text{acceleration}\n\\end{smallmatrix}} +\n\\underbrace{\\mathbf{v} \\cdot \\nabla \\mathbf{v}}_{\n\\begin{smallmatrix}\n \\text{Convective} \\\\\n \\text{acceleration}\n\\end{smallmatrix}}\\Big)}^{\\text{Inertia (per volume)}} =\n\\overbrace{\\underbrace{-\\nabla p}_{\n\\begin{smallmatrix}\n \\text{Pressure} \\\\\n \\text{gradient}\n\\end{smallmatrix}} +\n\\underbrace{\\mu \\nabla^2 \\mathbf{v}}_{\\text{Viscosity}}}^{\\text{Divergence of stress}} +\n\\underbrace{\\mathbf{f}.}_{\n\\begin{smallmatrix}\n \\text{Other} \\\\\n \\text{body} \\\\\n \\text{forces}\n\\end{smallmatrix}}\n", "dcb6d93a56f78a9531875f6e05fd83a5": "Q(\\mathbf{x},.)", "dcb7be3f8ded7760b9555adfeeca6c15": "(A\\otimes B)^*\\cong\\mathrm B^*\\otimes A^* ", "dcb7f34ce17d8a173026a12dc048ea48": "v_{\\tau}+v v_{\\xi}-\\delta^2v_{\\xi\\xi\\xi}=0.", "dcb82ecdc5fca56c36cccecc54e92f5f": "f = 2 \\pi^2 \\frac{\\mu N}{P} \\frac{r}{c}", "dcb8556953091f3a2871b268fdd484a3": "S^2_v", "dcb8774f37114b774f6f7067bff9f260": "E \\cap c = O \\cap c", "dcb8bd01e4a011f0e3c1c62eada636c2": "\\bar{\\nu}", "dcb8e0770098dffe8c6dad9ae0844946": "\n\\begin{align}\nT_A & = \\sum p\\beta_{pq\\cdots}[A]^p[B]^q\\cdots \\\\[8pt]\nT_B & = \\sum q\\beta_{pq\\cdots}[A]^p[B]^q\\cdots \\\\ \n& {}\\ \\ \\vdots\n\\end{align}\n", "dcb92126297a947ff0b6d6cd80689ea7": " d\\nu_t = \\theta(\\omega - \\nu_t)dt + \\xi \\nu_t\\,dB_t \\,", "dcb926ded50d81dbbddd1a6999df2122": "0 = 0\\,b = n\\,b", "dcb92d6d0837d01b1c836934bd7e45b9": "\\propto m^2\\varphi", "dcb96692f13dc99a69c1bdae3e22963e": "c+\\varphi X_{t-2}+\\varepsilon_{t-1}", "dcb967658b3781ef6dece09e5c3008c4": " I\\subset \\bigoplus _{n=2}^\\infty T^nV,", "dcb9ab929d2c081b0f0eab86005ef7f7": "\n\\int (d+e\\,x)^m (A+B\\,x) \\left(a+b\\,x+c\\,x^2\\right)^pdx=\n -\\frac{(B\\,d-A\\,e) (d+e\\,x)^{m+1} \\left(a+b\\,x+c\\,x^2\\right)^{p+1}}{(m+1)\\left(c\\,d^2-b\\,d\\,e+a\\,e^2\\right)}\\,+\\,\n \\frac{1}{(m+1)\\left(c\\,d^2-b\\,d\\,e+a\\,e^2\\right)}\\,\\cdot\n", "dcb9b0a08c49ddc5ea2210439b264c69": "s \\cdot (s' \\cdot r) = (s \\cdot s') \\cdot r", "dcb9b8603fc81b5041bca17c3291f22d": "\\sum_{n=-\\infty}^{\\infty} (in)^s a_n e^{in\\theta}", "dcb9cf07c41368fd0750eb5d26c630b8": "\\textstyle \\Phi = \\sum_e \\sum_{i=1}^{n_e} \\frac{c_e}{i}", "dcb9d978fad0614e11b4cdf775a7ef45": "x_4 = x_3 + (x_3 - x_1)\\frac{\\operatorname{sign}[f(x_1) - f(x_2)]f(x_3)}{\\sqrt{f(x_3)^2 - f(x_1)f(x_2)}} .", "dcba16bb1e98da5411440fb942cbb061": "f(n)/g(n)\\rightarrow 1", "dcba27e068fc93471dedb3f8087c3cc2": "\n\\begin{align}\nU(\\theta, \\phi) \n&\\propto a\\sum_{n=1}^N e^{ \\frac {-i 2 \\pi nS \\sin \\theta} {\\lambda}}\\int_ {-W/2}^{W/2} e^{ {-2 \\pi ixx'}/(\\lambda z)} dx' \\\\\n&\\propto a\\mathrm{sinc}\\left(\\frac{ W \\sin\\theta}{\\lambda}\\right)\\frac {1-e^{ -i 2 \\pi NS \\sin \\theta/\\lambda}} {1-e^{-i 2 \\pi D \\sin \\theta / \\lambda}}\n\\end{align}\n", "dcba3b8c1f7e0ad9aa17eeaa7eb7e198": " w = ", "dcba5765fd1b28034beea53c2362b467": "\\sqrt{T'}=\\sqrt{T}=\\frac{2\\sqrt{k_1k_2}}{k_1+k_2}", "dcba69e42160df97063e441f7f892674": " R = P + Q;", "dcba7c8f0780b4495895fc349c5fe320": "\\mathbf{x} = \\begin{bmatrix} x[0] \\\\ x[1] \\\\ \\vdots \\\\ x[N-1] \\end{bmatrix}.", "dcbacaf91f540ccb39cead356105eec5": " \\begin{bmatrix}\n 1/4 & 0 \\\\\n 0 & 1 \\\\\n \\end{bmatrix} ", "dcbaefdf71ab31610a26062f79145f1d": "\\frac{\\mathrm{d}\\,T_2}{\\mathrm{d}\\,z}=k_b (T_2(z)-T_1(z))=k_b\\,\\Delta T(z)", "dcbb47b0ecc3e08ec82ba789f232a7f3": " \\displaystyle{|\\partial_z F_f(0)|^2 - |\\partial_{\\overline{z}} F_f(0)|^2 = |a_{1}|^2 - |a_{-1}|^2.}", "dcbb866d02c483ede44555160f74cd44": "((\\operatorname{id}\\otimes \\Delta)\\Delta h)(v_{(1)}\\otimes v_{(2)}\\otimes v_{(3)})=h_{(1)}v_{(1)}\\otimes h_{(2)(1)}v_{(2)}\\otimes h_{(2)(2)}v_{(3)}=hv=((\\Delta\\otimes \\operatorname{id}) \\Delta h) (v_{(1)}\\otimes v_{(2)}\\otimes v_{(3)}).", "dcbba07d5de6dd17b17021ff9bed535f": "\n\\alpha_L=\\frac{1}{L}\\,\\frac{dL}{dT}\n", "dcbbdb2624a1d4647879d1562e12a2e4": "\\begin{align}\n \\alpha &= \\textstyle{\\frac{1}{2}}(2R - G - B) \\\\\n \\beta &= \\textstyle{\\frac{\\sqrt{3}}{2}}(G - B) \\\\\n H_2 &= \\operatorname{atan2}(\\beta, \\alpha) \\\\\n C_2 &= \\sqrt{\\alpha^2 + \\beta^2}\n\\end{align}", "dcbbff85512ef1754048f3a950e6c0bf": "\\scriptstyle{f(pi r^{2)}}", "dcbc192fb56079f705ccd6687da01a06": "V(S) = (K-P(S,T))^+.\\,", "dcbc276f800c920f64c8f65e4dd07009": "|V|^2 / 8", "dcbc757ffb26865d10e1c389bb52ba5a": " N_B = \\frac{N_{A0}\\lambda_A}{\\lambda_B - \\lambda_A} \\left ( e^{-\\lambda_A t} - e^{-\\lambda_B t}\\right ) . ", "dcbc7b1fb7e5f14b50b5a75d1871284a": "\\mathrm{H}(p,q)", "dcbcb2302897c731e4eb67428a051fc4": "N=\\bigcup_{n=0}^\\infty A_n.", "dcbcb5b2c3d0fbd3fead229a9c9cbdd6": "2p_nq_n", "dcbcc813e4402682583df175d7188494": "\nx_{1}^{2} - x_{2}^{2} = r_{1}^{2} - r_{2}^{2}\n", "dcbd0ac97c0e5170d4d1b6f6576a7a45": "\\mathbf{E} = \\mathbf{F} / q", "dcbd4388bf6f5ec6568d026a7ca5f514": "P(N)", "dcbd971665ffa3d40fb74650c328946a": "l^{(e_i)}_{j_i}", "dcbdbd9bf827f6498a9d136f3b40ad3d": "{v^2\\over{2}}-{\\mu\\over{r}}=-{\\mu\\over{2a}}=\\epsilon<0", "dcbe22d77f19e117b0bd458268f56df4": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 1\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 0 & 0\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n1 & 0 & 0 & 0\\\\\n0 & 1 & 0 & 0\\\\\n1 & 1 & 1 & 0\\\\\n1 & 1 & 1 & 1\n\\end{array}\n\\right] .\n", "dcbe5b2e51dc003adc7565dc0002f02d": "\\log(d+1) - \\log(d),\\,", "dcbe660285b9417b2f085784b60bb53b": "\\theta = \\tan \\theta - (1/3) \\tan^3 \\theta + (1/5) \\tan^5 \\theta - \\ldots", "dcbe9282c38823d230faa0e4ea29f588": "\\lambda_3 = 1/\\lambda^2", "dcbeae9d6a1d268e5ac00a7a69f81a28": "op_B", "dcbf2e546fb1bc00907b2f629e0ebc4e": "\n\\begin{align}\n \\mathbf{u} &= \\mathbf{U} & & \\text{at the object surface},\n \\\\\n \\mathbf{u} &\\to 0 & & \\text{and} \\quad p \\to p_{\\infty} \\quad \\text{for} \\quad r \\to \\infty,\n\\end{align}\n", "dcbf8ad14c94288abadde794345519cf": "(p-1)/k", "dcbfe047713805f753738aca41c4ff59": "\n\\frac{\\partial u_i}{\\partial t}\n+u_j\\nabla_ju_i\n=\n\\begin{array}{l}\n-2\\epsilon_{ijk}\\Omega_j\\left(u^s_k+u_k\\right)\n-\\nabla_i\\left(\\frac{P}{\\rho_0}+\\frac{1}{2}u^s_ju^s_j+u^s_ju_j\\right)\\\\\n+\\epsilon_{ijk}u^s_j\\epsilon_{klm}\\nabla_lu_m\n+g_i\\frac{\\rho}{\\rho_0}\n+\\nabla_j\\nu\\nabla_ju_i\n\\end{array}\n", "dcbfea1e484600672f2836a59ed3bc75": " t \\ge (n+d+1)/2", "dcc0084e2849bf1cbb0e2ea002e23288": " Z_2 = -jX_{\\mathrm{C}} =\\frac {1} {j \\omega C} ", "dcc07bc3dea02e8ea0a6033ad6620428": "\\scriptstyle 2,6^\\prime \\cdot \\frac{2 \\pi \\cdot 1\\,a}{230000000\\,a}", "dcc0a14341dddb92a6f7ea3693a5660b": " M \\models ACF_0 ", "dcc130360ffde48fef62b39fbaeed3e4": "F(t)=\\Gamma_{\\varphi}t", "dcc1326c546914a6b961739e1d9e4672": "\\sum_{n\\ge 1}\\frac{\\mathrm{core}_t(n)}{n^s}\n= \\frac{\\zeta(ts)\\zeta(s-1)}{\\zeta(ts-t)}", "dcc14f51622de30718b1643c9884d885": "\\mathbb{T}_r", "dcc184c059caa30ce227078f3cccdbfd": "\\begin{align}\n\\Sigma_{I,R} = \\sigma_I M_R^{-1} \\\\\n\\Sigma_{D,R} = \\sigma_D M_R^{-1}\n\\end{align}\n", "dcc190775b6e2de901fca55d5a8a574c": "\n\\begin{align}\n SS_{{\\text{total}}} = \\Vert {{\\mathbf{y}} - \\bar y{\\mathbf{1}}} \\Vert^2 & = \\Vert {{\\mathbf{y}} - \\bar y{\\mathbf{1}} + {\\mathbf{\\hat y}} - {\\mathbf{\\hat y}}} \\Vert^2 , \\\\\n & = \\Vert {\\left( {{\\mathbf{\\hat y}} - \\bar y{\\mathbf{1}}} \\right) + \\left( {{\\mathbf{y}} - {\\mathbf{\\hat y}}} \\right)} \\Vert^2 , \\\\\n & = \\Vert {{\\mathbf{\\hat y}} - \\bar y{\\mathbf{1}}} \\Vert^2 + \\Vert{\\hat \\varepsilon }\\Vert^2 + 2{\\hat \\varepsilon }^T \\left( {{\\mathbf{\\hat y}} - \\bar y{\\mathbf{1}}} \\right) , \\\\\n & = SS_{{\\text{regression}}} + SS_{{\\text{error}}} + 2{\\hat \\varepsilon }^T \\left( {X{\\hat \\beta } - \\bar y{\\mathbf{1}}} \\right) ,\\\\\n & = SS_{{\\text{regression}}} + SS_{{\\text{error}}} + 2\\left( {\\hat \\varepsilon ^T X} \\right){\\hat \\beta - }2\\bar y{\\hat \\varepsilon }^T { \\mathbf{1} } , \\\\ \n & = SS_{{\\text{regression}}} + SS_{{\\text{error}}} .\\\\ \n\\end{align}\n", "dcc191f88574106483fa8a45c33cdba4": "\\Rightarrow_{r_4} A S S S \\Rightarrow_{r_4} A A S S \\Rightarrow_{r_4} A A A S \\Rightarrow_{r_4} A A A A", "dcc19a249b12e5f8fa7c3656a40dd18d": "\\eta \\lim_{\\Omega\\rightarrow\\infty}\n\\int_{-i\\Omega}^{i\\Omega}\\frac{\\mathrm{d}(i\\omega)}{2\\pi} \\left(\\frac{1}{-i\\omega+\\xi}-\\frac{\\pi}{2\\Omega}\\right)\n=\\left\\{ \n\\begin{array}{cc}\n 0 & \\xi\\geq0 \\\\\n -\\eta & \\xi<0\n\\end{array}\n\\right.,\n", "dcc1bc0f4931939fc8df49276ee8b792": "x_1 \\ ", "dcc1f9715835cca779eaa058a7ae8955": "x \\mapsto -x", "dcc22047b93d05061615217b3d08a217": "\\sum_{j=0}^\\infty n^j\\sum_{k=j}^\\infty\\binom{k+1}{j}a_{p,k}=\\sum_{j=0}^\\infty\\binom{p}{j}n^j.", "dcc27d2438d1300e3a9eedaf285c2488": "\nf(z) - f_n(z) = \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{a_i(z)}{b_i(z)}\n- \\underset{i=1}{\\overset{n}{\\mathrm K}} \\frac{a_i(z)}{b_i(z)}\\,\n", "dcc27f2362b44fedad5ba12fc9e39db3": "\\scriptstyle (m\\mid n,\\, k) \\;=\\; \\Pr(M\\;=\\;m\\mid N\\;=\\;n,\\, K\\;=\\;k)", "dcc2910328c56b12806f924222ecb40b": "R'(1) = 0 = a(1-r) ...............(2)", "dcc2b574c4725602cdb66d84168869b1": "i\\hbar\\dot{\\underline{c}}^A(t) = \\mathbf{H}_A(t)\\underline{c}^A(t)", "dcc2d7478c14f8448ab67db2367b1ad1": "E=\\int\nd\\vec{r}\\left[\\frac{\\hbar^2}{2m}|\\nabla\\psi(\\vec{r})|^2+V(\\vec{r})|\\psi(\\vec{r})|^2+\\frac{1}{2}U_0|\\psi(\\vec{r})|^4\\right]", "dcc346e9462c3f81625eada95508899a": "p(x)=d_0(p(2x))", "dcc34afa8d94c89f8ae62a655a98b729": "[T,M]=[T,D]=[T,P]=[T,K]=0", "dcc3577f623e5799b52d125b139ccaf3": "((a,b),[c])\\in I \\Longleftrightarrow a=c", "dcc367dcf7e68192317c3175171e2f71": "G_1(x),\\ldots,G_k(x)", "dcc3a290064c626ebc90f4a6160b614d": "\\vec r_A=r_{Ai}+\\vec v_A t", "dcc3b10303cf1bf8a15c9913ae787001": " n>1 ", "dcc3c58f6c216adabe967569f52be8ee": " Lu \\geq 0, \\qquad \\textrm{in~} \\Omega. ", "dcc3d83d97157e5f78ddc8c4d0bd0230": "r_1,r_2,\\cdots,r_N", "dcc4390d60c33bcca0db1085083ef70d": "\\ \\rho ", "dcc44d1d858a45f89b39818c4092ed89": "q_1,\\ldots q_k", "dcc452798e602ada667273ae2b5e5cda": "m \\neq \\mathrm{SF}", "dcc497a466fd5b10d843e0401bc1fcd6": "\\text{Ohms on basis of voltage}_1 =(\\frac{\\text{voltage}_1}{\\text{voltage}_2})^2 \\text{ * ohms on basis of voltage}_2", "dcc4c08e137f92214efee77705bfd8e2": "\\phi\\colon I\\to \\R", "dcc54e78bc3eeaee3d97055bc2b43610": "m,q", "dcc576e36e7bd24318616cc080a53422": "l\\equiv \\partial_x+a", "dcc58ea9d216a7a13ed4fb5597c0a1fe": "{C_d}", "dcc5adb397904e74d55243892fe80340": "A\\land B", "dcc62af08b2c8306e2683a10afda00d9": "M_\\mathrm{v} = -14.18-2.5 \\log(E_\\mathrm{v})", "dcc642ed3e03be3b687e3778bd3aad87": "r^{th}", "dcc67b59f1a15199293a9964a7c6d388": " \\operatorname{Tr}_W(R \\otimes S) = R \\, \\operatorname{Tr}(S) \\quad \\forall R \\in \\operatorname{L}(V) \\quad \\forall S \\in \\operatorname{L}(W). ", "dcc6b2d53cca2d107825b5a57754fd4f": "\\ R(i) = E\\{x(n)x(n-i)\\}\\,", "dcc6fe54d03291acac100cc7d66aff58": "\\left(\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{2}}{2}\\right).", "dcc745f300f891283a73fb11a93c11a5": "\\mathrm{MD} = \\Sigma_{i=1}^n \\Sigma_{j=1}^n f(y_i) f(y_j) | y_i - y_j | .", "dcc74e02085f1fab7b310d9d35b9fe8d": " |\\mathbf{x}| = \\frac{1}{\\sqrt{I_n}}.", "dcc7928554e460b4efa351788eae6dff": "~f(y)~", "dcc7b52a42fc614c9d81354aaf692aa3": "A_{g}", "dcc7b8556c07fa5b09a25cbcbd64edaf": "= \\sum_k p_k \\log p_k - \\sum_{i,j} (p_i \\log q_j) | v_i ^* w_j |^2", "dcc7da312380d96b19e5443875cdd512": "\\begin{align}\n x &= \\frac{1-s^2}{1+s^2} \\\\\n y &= \\frac{2s}{1+s^2}\n\\end{align}", "dcc7f20f05dd04edef7f8e821d2e15b9": "\\mathit \\Gamma = {V_\\mathrm r \\over V_\\mathrm i}", "dcc8207dd6aff40e861029f7c94e7d9a": " \\kappa(A) \\geq \\frac{\\max_i(|a_{ii}|)}{\\min_i(|a_{ii}|)} .", "dcc88304aa61e3433d57fc04700c88f7": "\\hat{x}_i = \\left(\\frac{1/\\sigma_x^2}{1/\\sigma_x^2 + 1/\\sigma_d^2} \\mu + \\frac{1/\\sigma_d^2}{1/\\sigma_x^2 + 1/\\sigma_d^2} d \\right)+ \\left(\\frac{1/\\sigma_x^2}{1/\\sigma_x^2 + 1/\\sigma_d^2} \\xi_i + \\frac{1/\\sigma_d^2}{1/\\sigma_x^2 + 1/\\sigma_d^2} \\epsilon_i \\right) ", "dcc894a95ed264d476eca0cec8d178c7": "\\sum_i Z_i \\in Div(X),", "dcc8ae5fc210642a8023b425de8705c0": "|x - a| < \\delta", "dcc91397fe42f86f4c3f5fc9336af579": "\n(1+x)^n = \\sum_{k=0}^\\infty {n \\choose k} x^k \\quad |x| < 1\n", "dcc92675cc0801fd945f07bad473ff8f": " a_1(S,H) = \\frac{\\ln(S/H) + (r+\\frac12\\sigma^2)\\tau}{\\sigma\\sqrt{\\tau}}", "dcc97287e4ecdba3307ebf064a42f2ba": " E^{(e)} \\in [\\underline E^{(e)},\\overline E^{(e)}] ", "dcc99e93aeb592bf42ff864e0786150d": "F_1()\\,", "dcc9d189ef5e25df4b0244c2b00fdb02": "\\alpha , \\beta, \\gamma", "dcc9f2f4391419a8cc77ad078593874e": "z'=f\\left(z\\right)=\\left\\lfloor \\left(2^d-1\\right) \\cdot \\left(\n\\frac{\\mathit{far}+\\mathit{near}}{2 \\cdot \\left( \\mathit{far}-\\mathit{near} \\right) } +\n\\frac{1}{z} \\left(\\frac{-\\mathit{far} \\cdot \\mathit{near}}{\\mathit{far}-\\mathit{near}}\\right) + \n\\frac{1}{2} \\right) \\right\\rfloor\n", "dcca04df092b015132484de795f44001": "n \\mathbf{V}", "dcca10157b0d87b35e3404e57772ebd5": "\\textstyle x^p -1 = (x-1)(1 + x + \\ldots + x^{p-1})", "dcca31bdb21382b47168947effdf2adc": "\\vdash\\ \\textbf{let}\\, id = \\lambda x . x\\ \\textbf{in}\\ id\\, :\\, \\forall\\alpha.\\alpha\\rightarrow\\alpha", "dcca3d8a92023532ab0775e49fcfd203": "F_n\\equiv2\\pmod3", "dcca619c24a4da72ea7e5a4c5610e7dc": "\nx_3 = \\frac{(-{x_1}^3+(x_2-3a){x_1}^2+({x_2}^2+6ax_2)x_1+({y_1}^2-2{y_2}{y_1}+(-{x_2}^3-3a{x_2}^2+{y_2}^2)))}{({x_1}^2-2{x_2}{x_1}+{x_2}^2)}\n", "dccab589af82bbcb90d1eb2232111c28": "\\omega_{s}-\\omega_{I} ", "dccae57b2dcbdae5da9c853270d56fdb": "S\\to M", "dccb55501a8d5ebd6cbd664a03c13d93": "\\Delta G_{fus} = \\Delta H_{fus} - T_f \\times \\Delta S_{fus} = 0", "dccbde28ecae2fb8de5947639debf12f": " p(t_{n+i}) = f(t_{n+i}, y_{n+i}), \\qquad \\text{for } i=0,\\ldots,s-1. ", "dccbe48ac31a9d5e1dc3e745a861f221": "\\mathbf{E} = k_\\text{e} \\sum_{i=1}^N \\frac{Q_i}{r_i^2} \\mathbf{\\hat{r}}_i", "dccbe82acdcc354f33f95d6a26d04202": " \\begin{align} \n\\|A\\|_{op} &= \\inf\\{c \\ge 0 : \\|Av\\| \\le c\\|v\\| \\mbox{ for all } v\\in V\\} \\\\\n&= \\sup\\{\\|Av\\| : v\\in V \\mbox{ with }\\|v\\| \\le 1\\} \\\\\n&= \\sup\\{\\|Av\\| : v\\in V \\mbox{ with }\\|v\\| < 1\\} \\\\\n&= \\sup\\{\\|Av\\| : v\\in V \\mbox{ with }\\|v\\| = 1\\} \\\\\n&= \\sup\\left\\{\\frac{\\|Av\\|}{\\|v\\|} : v\\in V \\mbox{ with }v\\ne 0\\right\\}.\n\\end{align} ", "dccbf22b7749789f9aa5cd950d751806": "Q_c/W", "dccc99b1d8f618a69ab73a4063913246": "\\theta_\\alpha(\\beta)", "dcccbb94f21931480abf7f769dac0bc1": "\\alpha_g=", "dcccbf67aab3dfa70aa1a7d983e212ae": " X_{ij} ", "dcccbff4f7d9fae5ef2a819e6eb25ef9": " \n x + y = INT\\_MAX \n", "dccd0e822db491659da08f2444e30c9a": "\\frac{\\varphi^{n+1} - \\varphi^{n}}{\\Delta t} = F(\\varphi^n),", "dccd4056b86147af219716b6883923c9": "p=2, 3, 5, 11, \\text{ and } 17", "dccdcf7315ea21b17d41cbfa6bdb3c88": "(M_{(1)} , \\cdots, M_{(n)})", "dccde2b81822f744a83b76c3a60329df": "\\begin{align}\n \\lambda &= \\lambda_0 + \\frac{x}{R}, \\qquad\n \\phi &= 2\\tan^{-1}\\left[\\exp\\left(\\frac{y}{R}\\right)\\right] - \\frac{\\pi}{2} \\,.\n\\end{align}", "dccdf641740d1e309ae1d99cbfcaa896": "\nH = {\\frac{1}{2}} \\sum_{i=1}^N \\color{Blue} \\left| \\color{Black} {\\frac{{E}_i}{{E}_\\text{total}}} - {\\frac{{A}_i}{{A}_\\text{total}}} \\color{Blue} \\right| \\color{Black}.\n", "dccdfe60536654b8988354c607737c4d": "\\mathsf{L}(A;x)=\\langle F(x) A I^{-1} \\rangle", "dcce31d7eeac342f103b790fce605f4e": "(e^{2\\pi i})^i=1^i=1\\neq e^{-2\\pi}=e^{2\\pi i\\cdot i}", "dcce433e50062f42dad5eb3bc8306835": "\\{x\\} = \\begin{cases} 0, & x < 0 \\\\ x, & x \\ge 0. \\end{cases}", "dccf0ac01529ef951cac730036f2f002": "Y_{6}^{3}(\\theta,\\varphi)={-1\\over 32}\\sqrt{1365\\over \\pi}\\cdot e^{3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(11\\cos^{3}\\theta-3\\cos\\theta)", "dccf1d7a452e80676d9414b01db51956": "VAG(x^3 -7x + 7,(0,2)) \\cup VAG(x^3 -7x + 7,(2,4)", "dccf46da3fb3de80deb53e2cf55eb4ef": "112233,\\; 122133,\\; 112332,\\; 123321,\\; 133122,\\; 122331. ", "dccf601ef8641dcda9076c14b4d676d7": "[\\hat{A},\\hat{B}]=\\hat{A}\\hat{B}-\\hat{B}\\hat{A},", "dccf9b7cbec95a7ee1f67e48318ebd8e": "03_8+27_8+45_8=77_8. \\, ", "dccfabfab75688432ecbac2b9bc79974": "\\tan(\\phi) = \\frac{u_y'}{u_x'} = \\frac{u_y}{(u_x-v)} = \\frac{\\sin(\\theta)}{(\\cos(\\theta)-v/c)}", "dccff4091682bbbd6bb4cc86568f6f83": "(\\cdot)^\\mathrm{T}", "dccffb64a59b27abf2c646858f9c3033": "V_{012} = \\begin{bmatrix} V_0 \\\\ V_1 \\\\ V_2 \\end{bmatrix}, \\textbf{A} = \\begin{bmatrix}1 & 1 & 1 \\\\ 1 & \\alpha^2 & \\alpha \\\\ 1 & \\alpha & \\alpha^2 \\end{bmatrix}", "dcd00329ec553b7788a82eb1ea4ee92e": "\\Psi (x,0)=x^i", "dcd007ffe46aa667d98e24570323db97": "\\,(z_1, \\ldots,z_4)", "dcd0188c319a286b20bdac451f43c6bb": "\\lambda^2+4\\lambda+4=(\\lambda+2)^2=0\\;\\!", "dcd01e61503a63deb844026629203bc8": " H_{ij} = \\int_{0}^{1} x^{i+j} \\, dx, ", "dcd05b92add1d66ab237344469b22183": "\\overline{Q}^{\\dot\\alpha}_i", "dcd061c44ebfa6756df79774efe3c92b": "y = (1 - x^2)^1", "dcd08db48c4759694c057d99b5910fc1": "t/\\lambda", "dcd1021aba9148f701dd89b15cbcf36a": "f\\cdot O(g) \\subset O(f g)", "dcd16c0308909a0a0de3dddaa9e02f40": " \\mathbf{F} = \\sum_{i=1}^N m_i\\mathbf{A}_i,\\quad \\mathbf{T} = \\sum_{i=1}^N (\\mathbf{r}_i-\\mathbf{R})\\times (m_i\\mathbf{A}_i), ", "dcd17147fa79d907cd449c3e82ef5fa5": " f(p) > 0", "dcd176e7fab18e37eab34f2ed69e2d0b": "ncp=\\sqrt{\\frac{n_1 n_2}{n_1+n_2}}\\frac{\\mu_1-\\mu_2}{\\sigma}", "dcd1ac59dd41e63f6cf62d5e36f7c75e": "\\sum_{k=1}^n f(k)", "dcd1e12a346c87de8c3c792cfa8a8cbb": " j^{\\star} = \\int_{0}^{\\infty} \\left( {dj^{\\star}\\over d\\lambda} \\right) d\\lambda ", "dcd23e51b37ba948b5d1f722a1c2ba31": "P_{\\mathbf{v}}(x|spike) = \\langle \\delta(x - \\mathbf{s} \\cdot \\mathbf{v}) |spike \\rangle_{\\mathbf{s}}", "dcd2a7eed0e3bbaa94d1f31b3a00c662": " b = \\sqrt{r_{max}r_{min}}.", "dcd2cae7be42ea272875c77373fd931d": "\\varphi_1 < 0", "dcd3022dd21bb0d7010825ec7726fabb": "R = \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2} \\quad \\text{and} \\quad \\sigma_\\mathrm{avg} = \\tfrac{1}{2} ( \\sigma_x + \\sigma_y )", "dcd308aca930e277749c7c01e2690f92": "\\phi_x \\simeq \\phi_y", "dcd333ec47510be2687ecfd814d7ced2": "G = \\frac {\\tau^*} {\\gamma^*}=( \\frac {\\tau''}{\\gamma}+i \\frac{\\tau'}{\\gamma})", "dcd3e17409fb8befb5a53e6c42e5bc92": "p(k,\\theta | p, q, r, s) = \\frac{1}{Z} \\frac{p^{k-1} e^{-\\theta^{-1} q}}{\\Gamma(k)^r \\theta^{k s}},", "dcd3f9771222d015c1ea772283372b67": "D(0) = 0", "dcd414f187fe166b396a9181e8be80cf": "\n\\overline{\\phi^{\\prime}} = G \\star (1-G) \\star \\phi \\neq 0 .\n", "dcd41fa71a7439743a4b18eaf5f5e9ad": "\\left\\{4,{3\\atop4}\\right\\}", "dcd46eefb4847989dc21c567eb38c167": "\n \\operatorname{Var}[\\, \\hat\\beta \\,| X \\,] = \\sigma^2(X'X)^{-1}.\n ", "dcd494b57a2df983d5b50d4e5add42b2": "i = 0, \\ldots, 4", "dcd4a4d88e34887d359da9aa6a1843c7": "\\scriptstyle \\langle S\\rangle", "dcd530bb51003fc340efb57a3ab7f645": "\\mathbf{\\rho}=(u,v,w)", "dcd568f02befe2d9fd59ba44a166f259": " \\alpha = \\frac{2 \\eta\\omega^2}{3\\rho V^3}", "dcd614c2bc19028907c60711878604f6": "P_\\phi=\\frac{\\partial L}{\\partial \\dot \\phi} = mr^2\\dot \\phi \\sin^2 \\theta", "dcd61e626fd34e9ca65e63a6f9a03ade": " v = \\sum_{n \\in \\N} \\alpha_n b_n \\,", "dcd6542ece6488e2af42c770a1ae12cf": "\\mu = A_LT \\cdot e^{Q_H/RT}", "dcd70572ee558d33c30be7bd65b101cb": "\\left [ -\\infin < t < \\infin\\right ] \\longleftrightarrow \\left [-\\infin < E < \\infin \\right ] ", "dcd70e5fbec3c938c58f3b14d32c68e7": "(x,x^2,\\dots,x^n)", "dcd7176504966ffee17786facfd04bce": "\\mathbf{M} = \\mathbf{U}\\mathbf{D}\\mathbf{U}^*", "dcd71e6ce6901a8951d6ff1835150586": "m_{1} = d_{0} \\cdot a_{0} - m_{0} = 1 \\cdot 10 - 0 = 10 \\,.", "dcd72e75ab47b0d77f537b26cb275789": "2.\\ \\mathrm{2 HCrO_4^- \\rightleftharpoons Cr_2O_7^{2-}+H_2O; K_D=\\frac{[Cr_2O_7^{2-}]}{[HCrO_4^-]^2} }", "dcd736e2bb05e047141c462e844ce41c": " \\mathbf{W}^{-1} = \\mathbf{W}^{T} =\\begin{pmatrix} 0 & 1 & 0 \\\\ -1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} ", "dcd750ca3809748e061ec3568c291694": "(f)", "dcd775ac39008be9eb41c8726af9d847": "A^{\\alpha}{}_{;\\beta} = A^{\\alpha}{}_{,\\beta} + \\Gamma^{\\alpha} {}_{\\gamma\\beta}A^\\gamma", "dcd79408472dd526ad1e50da80ab901c": "k \\in \\mathbb{Z}", "dcd7ac7dc4549f867af67d3d4ab6c798": "\\partial : k^*\\rightarrow H^1(k,\\mu_\\ell) ", "dcd7fe171b2f64373ab38fe9f7fe8c93": "\nX \\mid p \\sim \\mathrm{NB}(n,p),\n", "dcd80072f4508ac737df427939d1af73": "\\sum_\\alpha n_\\alpha a_\\alpha = 0, \\quad n_\\alpha\\in\\mathbb{Z},", "dcd82e748e2060ebecd410ac4ba92e54": "U\\subset X_\\alpha", "dcd851681d76b174d4c858d645a7fba0": "\\sigma_y(\\tau) = \\sqrt{2\\ln(2)}\\sqrt{h_{-1}}", "dcd86503062647c54ec875d70592b75f": "\\ell = \\mu^{-1} = ( (\\mu/\\rho) \\rho)^{-1},", "dcd8751aacd27a57e67e7327b4d9856f": "\\vec R + \\vec {dR}", "dcd8a029a1a37af2f5cb5ad87451f90e": "P^\\prime", "dcd8c27488a0eb262c86f689cc53202c": "\\begin{align}\n& \\sum_{k=1}^\\nu ( Y_k X_k^{j+\\nu} + \\Lambda_1 Y_k X_k^{j+\\nu-1} + \\Lambda_2 Y_k X_k^{j+\\nu -2} + \\cdots + \\Lambda_{\\nu} Y_k X_k^{j} ) = 0 \\\\\n& \\sum_{k=1}^\\nu ( Y_k X_k^{j+\\nu} ) + \\Lambda_1 \\sum_{k=1}^\\nu (Y_k X_k^{j+\\nu-1}) + \\Lambda_2 \\sum_{k=1}^\\nu (Y_k X_k^{j+\\nu -2}) + \\cdots + \\Lambda_\\nu \\sum_{k=1}^\\nu ( Y_k X_k^j ) = 0\n\\end{align}", "dcd8c8856730ae7e213defb24ab1798c": "s_0:=1;\\quad s_1:=0;", "dcd8d4c29f187b294540cdc2e337d6fb": "EN_{ija} = 30 \\times (en_{lig} - en_{c.a.}) \\times ss", "dcd8db047b323d1573395da6dac7846c": "ds^2=\\left(1-\\frac{2Gm}{c^2 r}\\right)^{-1}dr^2+r^2(d \\theta^2 +\\sin^2 \\theta d \\phi^2)-c^2 \\left(1-\\frac{2Gm}{c^2 r}\\right)dt^2", "dcd8f70c771ccd629cf661082a222c07": "AE_.", "dcd8fadb52d3702bb9b343dbe68f1e58": " \\langle T_K \\varphi, \\psi \\rangle = \\int_{X \\times X} K(y,x) \\varphi(y) \\psi(x) \\,d[\\mu \\otimes \\mu](y,x). ", "dcd901d443fa998111e7764361cc1a3a": "n+a^2/4n", "dcd90b2cf53537404531a088be1a67b7": "\\sin 20^\\circ\\cdot\\sin 40^\\circ\\cdot\\sin 80^\\circ=\\frac{\\sqrt{3}}{8}.", "dcd91d0e4e79fce11fa08349421b0130": " R(\\bar{\\beta}) = - M\\beta^2 \\left( \\frac{\\partial^2 f}{\\partial c^2} + 2\\eta^2 Y + 2K\\beta^2 \\right)", "dcd91e63b5cfbea0e04e9269466ff3b9": "V_{ref}", "dcd973f6e108c104271579c8060daffc": "A \\neq \\emptyset", "dcd98172d165ae16bd13688a925a9c52": "s=\\sinh \\varphi", "dcd98e5ca0a1a1cd9302d75498982b55": " \\pi_\\text{Within} ", "dcd9aa6b3106f772e00f6d27d2fbf0b5": "X_G", "dcd9d657c333f748717528c8b6b2b437": "v(P) = J_{\\varphi_1}(\\varphi_1^{-1}(P))\\cdot {\\bold v}_1(\\varphi_1^{-1}(P)). \\qquad (2) ", "dcda8739dccc06dd8f15965dcf61e395": " \\mathit{g^{(1)}}", "dcda9ddab5d8b9d4a4741a43502d11d3": "\n\\mathbf{M} = \\chi_v \\mathbf{H}\n", "dcdb3c249d550d26c4a1a59c19a98616": "\\frac{1}{x}R", "dcdb8851a2e2593c9126cb58958cf1b4": "F_\\mathrm{Binomial}(k;n, p) \\approx F_\\mathrm{Poisson}(k;\\lambda=np)\\,", "dcdbbae73afbd90de33fdcdd33347b0a": "h_F^{(2)}(z)=\\frac{-\\beta}{1+e^{\\beta z}}=-\\beta n_F(z)", "dcdbe4599adc39d77c9f6962852402db": "(X^2+aY^2)Z^2= X^2Y^2+dZ^4", "dcdbfe23d55eee0e238f0ca14a7d63ec": " \\Chi_w(k,n)= 2^{-1/2} \\bigl( \\chi_w(2k,n+1)-\\chi_w(2k+1,n+1) \\bigr).", "dcdc2e5aa188d19a86d1339257209ab6": "4^n-1=\\left(2^n+1\\right)\\left(2^n-1\\right)", "dcdc2ef7e0fc10e518808775c6469c08": "((a_{\\omega}), h) \\cdot (\\lambda,\\omega') := (a_{h(\\omega')}\\lambda, h\\omega')", "dcdc2f8ca1ee433a84c904232110b728": "f(\\vec{x},g(\\vec{y}),\\vec{z})\\rightarrow h(\\vec{x},\\vec{y},\\vec{z})", "dcdc455fa0086959f5dabff1eb628c53": "(n-1)I_n = - \\frac{\\cos{ax}}{a\\sin^{n-1}{ax}}+ (n-2)I_{n-2}\\,\\!", "dcdc83024716d83b303918719aa3e907": "K(t+T) = K(t).\\,", "dcdcf42d18ba47619ef3beb2ac4abe14": "u(r) = r R(r)\\,", "dcdd0ac9918693bb80d6b077fb44166b": "\n\\ddot{\\vec x} = - \\nabla U\n", "dcdd2c007786c2d6b5bbb7bcc8b2095b": "\\sum_{i=0}^3 \\sigma^i_{\\alpha\\beta}\\sigma^i_{\\gamma\\delta} = 2 \\delta_{\\alpha\\delta} \\delta_{\\beta\\gamma}\\,", "dcdddcf50b144a70f6e1a2f65d64b0d8": "a=\\frac{s}{2\\tan(\\pi/n)}=R\\cos(\\pi/n).", "dcde33e2a3a88f6f478b7587e9681f5f": "\\{y_1,y_2\\}", "dcde3dcf95a50e41ceb31dc981be0242": "DFS(a)= \\left(n^*(a),0,0\\right).", "dcde769112bb0607b548a543cd37da92": "y_k = \\mathbf{h}_k^T\\mathbf{x}+n_k, \\quad k=1,2, \\ldots, K", "dcdef6ff4050e6a3b139b806c0848903": "\\forall x,y \\in H: \\qquad \\lnot(x \\wedge y)= \\lnot \\lnot (\\lnot x \\vee \\lnot y).", "dcdf2264bd56574ec2fa9ae4473f8b5c": "\\mathfrak{R}'", "dcdf31f2159728a90a7c33817c37ced7": "G_\\infty=\\tfrac{\\pi d^{3/2}}{2\\sqrt{R_i R_m}}", "dcdf32f2ef162a3af17face6a59d77b4": "\\bar{u}_S\\,", "dcdf6e8c9ab5b20d16be8e2e1a8fa29b": " \\operatorname{let} y\\ f = f\\ (y\\ f) \\operatorname{in} y ", "dcdf771afc41b5c7dd7bb63907ac8ccb": "q=W^T X^T s,", "dcdfb9346a9f2a0d541b8ad53fa83435": " \\vec \\sigma(u,v) = \\left(u, v, {u^2 \\over a^2} + {v^2 \\over b^2}\\right) ", "dcdfcfe4357263fb83ecafc97a7e29d1": "x\\in\\R\\,", "dcdfee3fdfcd104bb3e14e6f9bf7156c": "yx = (-1)^{|x| |y|}xy.\\,", "dce05e7b25ac8458f8994460c23e5529": "F(z,t) = \\sum_{n=0}^\\infty f_n(z) t^n = \\sum_{n=0}^\\infty \\frac{g(z)^n}{n!} t^n = e^{g(z)t}", "dce07cd893bf40df089cc0a4716ba932": " \\sum_{|\\alpha| = m} a_\\alpha(x)\\xi^\\alpha \\neq 0.\\,", "dce08db97befc410858bbcce450386c8": "Pmo = 1 - \\dfrac{3}{5} = \\tfrac{2}{5}", "dce0a98d98c29ba9bd42a2e889834592": "\n(ADC_x + ADC_y + ADC_z)/3 = ADC_i\n", "dce0be66076680a01cd6754fbc23cf8c": " T \\in V ", "dce11fb04adca0edb4c65eb03fe112f2": "D_1\\,\\!", "dce1277efe397e44d817971ebd44124a": " \\mu_{pq} = \\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{-\\infty}^{\\infty} (x - \\bar{x})^p(y - \\bar{y})^q f(x,y) \\, dx \\, dy ", "dce12b25427a4ddaf34dbc2d7895748f": " \\frac{H\\Psi(X,a)}{\\Psi(X,a)} ", "dce1d5c24f37ad202c61f2f36934f73a": "=\\frac{\\text{kv}_{L-L} \\text{ * 1000}}{\\sqrt{3} * \\text{ohms reactance}}", "dce1e0a981cce07b2705e60626ec5bd3": "n \\in \\mathbb{Q}", "dce1f13d5064d4dd27a0243da65026f7": "\nz_{k^*j}^{new} = z_{k^*j}^{old} - \\eta (x_{ij}^p-z_{k^*j}^{old})\n", "dce1f6e7e694b66dda2a050c1b694360": "h_{fw} = \\psi \\frac{\\rho}{\\rho_{fw}} + z", "dce2120bbff3b6d7f89ccfe33ebcba38": "\\textstyle\\mathbf{x}", "dce23205bdc82d5f521b8a79349bb48e": "e \\in A,", "dce27bd3ac789188ef0e87e2d2cd979c": "\\vdash \\Box \\Psi \\rightarrow \\Box\\Box\\Psi \\rightarrow \\Box P", "dce2ac28b0cc238310a1aad3d75f0e9b": "Cz = Cz' * ({{\\lambda } \\over {\\lambda + 2}}) ", "dce2addf71fcf6687ef7ae1b9df8c4a1": "M-s_i", "dce2d603f80a7e6f5b3e8f515ac040dd": "\\Phi(y)", "dce2f3eb7904d996489cdd39ec18f1ac": "|xy|=|x||y|", "dce343d12cbdeb2fe7e90b07d42a64b2": "2^{shamt} ", "dce3520244f1864513c242e921673b92": "w_i \\geq 0 ", "dce3c30b230c5eccb06352ef5c3aa288": "a_u\\,", "dce3c834c7131468f24c1a4aa6bb4bb2": "c=\\cosh \\varphi", "dce402992a1e4a9d377084f2dc5dd75a": " 2 \\sin t = (-i ) e^{i t} + ( i ) e^{-i t}. \\,", "dce51773355e63cf1edac7a7de11aa3a": "\\boldsymbol{\\hat{\\varphi}}", "dce53dc6ad2223bd19138b35c70f778f": "_2^2\\text{S} = ^{15}\\text{N}_2\\text{O}", "dce5a936859b4c3de131eb455a8740a2": "V=V_0 e^{-1} \\approx 0.37 V_0 ", "dce5b484d976a08fcfa29c0bc63ed935": "G_n(z) = z \\prod _{k=1}^n \\left( 1+g_k \\left( G_{k-1}(z) \\right) \\right).", "dce5c13dd3551aaefbe601edee0ea944": "\\frac{y_2}{y_1}=\\frac{1}{2}\\left(\\sqrt{1+8 F r_1^2} - 1\\right)", "dce655befaa632420955c3f06d54bd64": "\\overline{\\Gamma}_{\\alpha \\beta}^\\gamma", "dce70f3bbe52e0252c696cc05cd622d5": " \\frac {\\bold p}{V} = {3}\\varepsilon_0 \\left(\\frac {\\kappa-1}{\\kappa+2}\\right) \\bold{E_{\\infty}} \\ .", "dce712adff5de79efd862ee64b87c417": "\\hat{c}_V=T\\left(\\frac{\\partial S}{\\partial T}\\right)_V=\\left(\\frac{\\partial U}{\\partial T}\\right)_V ", "dce74a6b956db49a7fdfb2b163263c90": "K=\\frac{\\phi_{\\text{N}_2\\text{O}_4} p_{\\text{N}_2\\text{O}_4}/{p^\\ominus}}{\\left(\\phi_{\\text{NO}_2}p_{\\text{NO}_2}/{p^\\ominus}\\right)^2}", "dce7995c48e9858ee9637ebe4036ecee": "\\min_{\\rho}\\; \\int_{\\Omega} \\phi(\\rho) \\, \\mathrm{d}\\Omega", "dce7e8be33589fe8339b5b736f47b54d": "DP_{T/D}^{S/V}", "dce7f3198a16e5ae105389668efcabbd": "\\begin{align}\n\\left( T^{(n)} \\right)^2 &= \\sigma_{ij}\\sigma_{ik}n_jn_k \\\\\n\\sigma_\\mathrm{n}^2 + \\tau_\\mathrm{n}^2 &= \\sigma_1^2 n_1^2 + \\sigma_2^2 n_2^2 + \\sigma_3^2 n_3^2 \\end{align}", "dce822cf440cd3c8d3babc31c05faebd": "b\\backsim 2x+e.", "dce8a5551a6a6af1e0674f044071fe5c": "S(f) \\propto \\frac{1}{f^\\alpha}", "dce8aead43bd64472b1689e28c38982e": "~f(x,y,z)~", "dce8bbd7ec968c9a4c9dcb50bf12b683": "\n \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\left(\\frac{\\partial \\mathbf{f}_1}{\\partial \\mathbf{v}} + \\frac{\\partial \\mathbf{f}_2}{\\partial \\mathbf{v}}\\right)\\cdot\\mathbf{u} \n ", "dce933922f67db3113b4632574b7bb5c": "\\frac{\\mathrm{d}^{n}}{\\mathrm{d}\\lambda^{n}} u(\\lambda)\\mid_{\\lambda=0} = n! u_{n}", "dcea1a181d1e2357bfb7604f3aafebdb": "\\{92, 19, \\mathbf{101}, 58, \\mathbf{153}, 91, 26, 78, 10, 13, \\mathbf{-40}, \\mathbf{101}, 86, 85, 15, 89, 89, 25, \\mathbf{2}, 41\\} \\qquad (N = 20)", "dcea1b62d8c585749fb53df5fba79d3c": " \\left (\\frac{T_2}{T_1} \\right )^\\frac {\\gamma}{\\gamma-1}", "dcea2963e20a817b5aba104edf023304": "\ns_5 = y_1y_2y_3y_4y_5.\n", "dcea3203cdf9f926e77a3c89045caa10": "\\textbf{S}(t)", "dcea530838bb05461a38f65cff608253": "O(1/k)", "dcea8ab14d9c3b07a55f28d4f0c0c297": "q_{kj}x^kf(x)^j", "dceaf459686bb6e30a6a337b69d3490e": "I \\hookrightarrow R", "dceb08cc06569d6169b6043c6aece76f": "S_3(k_{\\lambda}) = k_{-\\lambda},\\ S_3(e_i) = - q_i e_i,\\ S_3(f_i) = - q_i^{-1} f_i.", "dceb6c97393120fa779f05a817619cc9": "\\sum_{k=0}^{\\infty}|\\beta_k|^2 \\le \n\\exp\\left(\\sum_{k=1}^\\infty k|\\alpha_k|^2\\right),", "dceb77bd0d9614e031c8314c36a84c07": " \\sum_{i=1}^n{\\frac{1}{i}} \\le \\prod_{p \\le n}{\\left(1 + \\frac{1}{p}\\right)}\\sum_{k=1}^n{\\frac{1}{k^2}}", "dcebfde65fe312981eac95fc46ba6dd2": "\\mathbf C_d = \\mathbf C ", "dcec53c3ed7d9968ba6756323d62f3e4": "\n\\theta_\\text{t} - \\theta_\\text{r} + \\theta_\\text{t'} - \\theta_\\text{r'} = \\pm \\pi\n", "dcecdcdf55d58c79c2cb9e34df8c4a42": "g_h=g_0\\left(\\frac{r_e}{r_e+h}\\right)^2", "dced48bc173776102c0298afa1b8045b": "100\\uparrow\\uparrow\\uparrow n=(10\\uparrow\\uparrow)^{n-2}(10\\uparrow)^{98} (2 \\times 10^ {200})=(10\\uparrow\\uparrow)^{n-2}(10\\uparrow)^{100} 2.3<10\\uparrow\\uparrow\\uparrow (n+1)", "dced7152769c2b9021723e751b638d63": " \\{ f_1,\\cdots , ~f_{N-1},~ \\{ g_1,\\cdots,~ g_N\\}\\} = \\{ \\{ f_1, \\cdots, ~ f_{N-1},~g_1\\},~g_2,\\cdots,~g_N\\}+\\{g_1, \\{f_1,\\cdots,f_{N-1}, ~g_2\\},\\cdots,g_N\\}+\\dots ", "dced930f2ee6beb04d4adec96bb5eb22": "n_i \\times n_{i+1}", "dced9de1e87e7f4c7ed0436d1fb46d0f": "D_0 e^{-i \\omega t} = \\widehat{\\varepsilon}(\\omega) E_0 e^{-i \\omega t},", "dcedec55c839c4f8c0f7ef9a3977461d": "\\mathrm{mRMR}= \\max_{S}\n\\left[\\frac{1}{|S|}\\sum_{f_{i}\\in S}I(f_{i};c) - \n\\frac{1}{|S|^{2}}\\sum_{f_{i},f_{j}\\in S}I(f_{i};f_{j})\\right].", "dcedf48f645300e97581158f6ddd054d": "\\qquad\\sigma_{ik}n_k-g_i=0\\qquad\\text{for } i=1,2,3", "dcee27d553b700a6e879d3f1c0903d60": "\nPV(i,n,R) = R \\times a_{\\overline{n}|i}\n", "dcee4b620d4b2490db4c7faf836d6803": "\\displaystyle{Ch(a)-F(a)= Ch(a) - C_nh_n(a)=C(h-h_n)(a) + [(C-C_n)h_n](a).}", "dcee503d70e4c44727548f23e9939f78": "e_I^\\mu", "dcee512af1a9f7d92f9f7429c5b72d37": "\\sqrt{\\,\\,}", "dcee51612f0640cef6e1385978867af2": "2\\pi/m", "dcee52c6f77b8dc6d49eee57878690f4": "\\mathbf{B}_i", "dcee89d21a92ef75118e4183d7031c57": "f\\left(X_1,X_2,\\ldots,X_n\\right)=f_0+\\sum_{j=1}^nf_j\\left(X_j\\right)+\\sum_{j=1}^{n-1}\\sum_{k=j+1}^n f_{jk}\\left(X_j,X_k\\right)+ \\cdots +f_{12 \\dots n}", "dceedfcc1b42ddf7aeb62c4e99672a23": "=\\left({1\\over12}+{2\\over12}+{3\\over12}\\right)\\times 12 \\ \\mathrm{m}=\\left({6\\over12}\\right)\\times12 \\ \\mathrm{m}=\\left({1\\over2}\\right)\\times12 \\ \\mathrm{m}=6 \\ \\mathrm{m}", "dcef1b3d80c12fa5e5dafb666c140cd5": "RDA = EAR + 2SD(EAR)", "dcef4a772ff013a29d13291ac5ed462c": "Gx = \\left\\{ g\\cdot x \\mid g \\in G \\right\\}.", "dcef60dc866416d05dffbcdc915b09d1": "V \\otimes \\dots \\otimes V", "dcef64b1bebde15e61f1a6e4dc6f98fa": "[x]_{\\mathrm{RED}} \\neq [x]_P", "dcef80dbb44569be6b9cce9e481ac973": "J^{\\mathrm{op}}", "dcef984fd9dc56b8126b2ded4f0b31e2": " \\operatorname{cov}(X_1, X_2) = \\frac{\\theta_1 \\theta_2}{(a-1)^2 (a-2)}, \\text{ and }\n \\operatorname{cor}(X_1, X_2) = \\frac{1}{a}.\n", "dcefa608a7196b701c6144ecf1675a64": "{\\lambda} = \\frac{ln2}{T_{1/2}}", "dcefc91968f01bf3b3f60e380d814505": "i,j\\in\\mathbb{N},\\ 1 \\le i \\le j", "dcf059c4f2304b3c322373fa5e774064": "\\mathbf{p}_n", "dcf0722367872526055a66987d03d039": "\\,(u,v)", "dcf0f6e209b107faa69e6e343be66191": "A_5", "dcf0f7c093cc94fe9a34ea745beb4d2a": "=\\frac{V_{nk_5}V_{k_5k_4}V_{k_4k_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}E_{nk_4}E_{nk_5}}-2E_n^{(2)}\\frac{V_{nk_5}V_{k_5k_4}V_{k_4n}}{E_{nk_4}^2E_{nk_5}}-\\frac{|V_{nk_5}|^2}{E_{nk_5}^2}\\frac{V_{nk_3}V_{k_3k_2}V_{k_2n}}{E_{nk_2}E_{nk_3}}", "dcf11de45ce5b93aeaecb3033b7ce5e3": " \n\\frac{ \\| \\lambda \\cdot \\vec{a} \\| }{ \\| \\vec{a} \\|}\n=\\frac{\\|\\lambda\\cdot\\vec{b}\\|}{\\|\\vec{b}\\|}\n=\\frac{\\|\\lambda\\cdot(\\vec{a}+\\vec{b}) \\|}{\\|\\vec{a}+\\vec{b}\\|}\n=|\\lambda|\n", "dcf138260f16ca8683a9e086e6f0c3dc": " h'= f'-g'>0.", "dcf188acba62d6921a22512887807201": "R=1/2 DpAv^2", "dcf1a1e2b4b24a1964a93dc201918df1": "k^2=\\frac{Z}{Y}", "dcf1effd0f0da9b481ea414a83fe6915": "d(u,k)", "dcf2022f40798e11261abec38440d14d": "\\scriptstyle dx", "dcf2088fdbf730eecb9a31bdd9da9927": "g_{i j} = \\left \\langle {\\partial}_i, {\\partial}_j \\right \\rangle.", "dcf268c2149afca0c05103a7a680b45e": "R(\\textbf{x})", "dcf2a9b6dff9ba45b3a056ed5f90c8e3": "\\frac{ALR - 0} {100 - 0}", "dcf2c66b4c21f5e2e11c6b43082e7783": "E_{i+1} = E_i \\cup E_B(v_i,V)", "dcf2dbcccf27141f04f8ec0056a82403": "\\gamma=\\psi(\\alpha)", "dcf307c8c62499885931c95628f949e5": "\\delta_{x_0}[\\phi] = \\phi(x_0)", "dcf31668e7dfaa55c0a194943cd92f66": "I_2=\\frac{E}{R+r}", "dcf32a5430cc0ce411a5cd3cd0db3a8d": " \nP\\left(\\frac{\\rho}{r}\\leq \\lim \\inf_{t\\rightarrow\\infty}\\frac{T_t}{t}\\leq\\lim\\sup_{t\\rightarrow\\infty}\\frac{T_t}{t}\\leq \\rho r\\right)=1; \\quad\\forall r>1\n ", "dcf3796290b1a97d7d3100addd233f19": "I_{\\mathrm{RMS}} = I_\\mathrm{p}\\sqrt {{1 \\over {T_2-T_1}} \\left [ {{t \\over 2} -{ \\sin(2\\omega t) \\over 4\\omega}} \\right ]_{T_1}^{T_2} }", "dcf41514d65bd91afe75774f3b364a67": " \\int { d^N k \\over \\left ( 2 \\pi \\right ) ^N } \\; {\\exp \\left ( ik \\left ( x-y \\right) \\right ) = \\delta^N \\left ( x-y \\right ) } ", "dcf442812e3a85bc23dd3de3192b09e4": "\\ sA + tB \\rightleftharpoons uX + vY", "dcf45424db5c6b63faf006dae918874e": "\n\\gamma = 2 + {k_0 + a\\over m}.\n", "dcf4882d996c239d09008965ec24d08f": "\\gamma (1 + \\cos \\theta_\\mathrm{C} )= \\Delta W_\\mathrm{SLV} \\,", "dcf4f8f79e09f2bb2749380b520df7b9": " W1(A)", "dcf50a5be8ceec48c114080c911a3a70": "G^{\\phi(u)} = G_u", "dcf512867c5691ff2db74f44b193eb5b": "\\theta^{i\\alpha}", "dcf526cd60dafa6b0f446438fd24e9d5": " |z| \\ge 1 ", "dcf528f9edd561c54ef4c0276c0cc2cc": "(x\\vee y)", "dcf5a9d1d0921e88bbbb7524c84d99af": "\\mathcal{C}", "dcf5b94f05eacccf2005a1a8b86b5158": "a_{n}r^{n} + a_{n-1}r^{n-1} + \\cdots + a_{1}r + a_{0} = 0", "dcf5b979f16c29c98ffbaee12a6665e6": "\\left\\vert1\\right\\rangle", "dcf5ca9b579835eeb90b7c938ad14746": "C = \\frac{C_d}{\\sqrt{1-\\beta^4}}", "dcf674d5abc94d7d02feb6699c0bc229": "\\mathit{T}", "dcf67c7b5daef28286c6444a2b3bfa9d": "G(\\cdot \\mid x)", "dcf709eef9414e27c2127fcd00587e87": "\\Gamma(g,L)=\\left\\{ c \\in \\{0,1\\}^n | \\sum_{i=0}^{n-1} \\frac{c_i}{x-L_i} \\equiv 0 \\mod g(x) \\right\\}", "dcf76ad1358d7bbc14dcac5f0210c2e8": "\\!(x_1, y_1), \\ldots, (x_n, y_n)", "dcf771b6d118a55617c55544f54a0797": "\\displaystyle{\\Omega(g,h)=\\exp -{\\pi i\\over 4}\\tau(u_0,gu_0,ghu_0)}", "dcf77f19d6a0e6ddd3d9e1a9119ddc42": "\nT=\\frac{1}{2}\\sum^n_{i=1}\\sum^n_{j=1}g_{ij}\\,w^i\\,w^j", "dcf811b5b1fc493c7c3297e6afa80fba": "19601+13860\\sqrt{2}=39201.99997\\ldots", "dcf83133081760ca36c62ccf07de6f5a": "\\vec B\\,,", "dcf843804e1ae19b3a50af5e5b82aa72": "f(t) = \\frac{\\Gamma(\\frac{\\nu+1}{2})} {\\sqrt{\\nu\\pi}\\,\\Gamma(\\frac{\\nu}{2})} \\left(1+\\frac{t^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}},\\!", "dcf84a8642938da42bbb624d47fe5a0c": "\n\\underline{P}(Cl_t^{\\geq}) \\subseteq Cl_t^{\\geq} \\subseteq \\overline{P}(Cl_t^{\\geq})\n", "dcf8e013195447c6e83ace3c4a467adc": "\\hat{a}_i^\\bullet \\,\\hat{a}_j^\\bullet = \\hat{a}_i \\,\\hat{a}_j \\,- \\mathopen{:}\\,\\hat{a}_i\\, \\hat{a}_j\\,\\mathclose{:}\\, = 0", "dcf92f5ecf1eb9245b7a5452c887e63a": "A = \\big\\{a, b, c, \\ldots \\big\\}", "dcf9d02c0c48e9b4c5a7a8c6569ee364": "g_sh^*_t = h^*_{t\\exp(-s)}g_s", "dcfa7ce2d39cbb90d68e8c965d07b403": " x = r + s \\,,\\,y = r + t \\,,\\, z = r + s + t. ", "dcfab01b5e2803969e0d3c9444cd2a0d": "m_0 - \\ ", "dcfacb3bc71ffe0de13cedd7760c98d7": "o\\left(\\frac{n^2}{\\log {n^2}}\\right).", "dcfadc14cc6a294811c8b4a6c3fb6c0c": " --~~~~AE = C+I+G+NX ", "dcfb0d29ce4e51ea5d297d78224d5eef": "N = m_1 + m_2", "dcfb4a0a43725ad3add5b6c5dd523bfe": "\\scriptstyle \\lim_{k \\to \\infty}\\frac{1}{\\sigma_k}\\left(\\mathrm{Erlang}(k,\\, \\lambda) \\,-\\, \\mu_k\\right) \\;\\xrightarrow{d}\\; N(0,\\, 1) \\,", "dcfb68bc6761a6f1a5e19b2a682b3053": "a_v: s(v) \\rightarrow A(H)", "dcfb7579fe3d7bacb0e1ab1705b1e87f": "\\mathrm{ess }\\sup (fg) \\le (\\mathrm{ess }\\sup f)(\\mathrm{ess }\\sup g)", "dcfb8a786b5a051d3fe71f3b4b0da257": "\\Delta N = n(\\vec{r}) \\ \\Delta V ", "dcfbbed07363e3a86dead423106e518f": "(E,\\pi,B)\\, ,", "dcfc2d064c18b7c948588bd3883b7444": " \\nabla_{\\dot\\gamma} \\dot\\gamma= 0", "dcfc302db863d22ab7e9526749111ff6": " \\langle \\sigma_A\\sigma_B\\rangle \\geq \n\\langle \\sigma_A\\rangle \\langle \\sigma_B\\rangle ", "dcfc37d1bc5271ab771c4f732aed9754": "P_\\mu = \\left(\\frac{E}{c},-\\bold{p}\\right) \\,\\!", "dcfc808bb38199992ba40954fe74cf00": "\\bar x = x_0 - x_1\\,i - x_2\\,j - x_3\\,k - x_4\\,\\ell - x_5\\,\\ell i - x_6\\,\\ell j - x_7\\,\\ell k", "dcfcc9bba99ff5c5a0d90d003e401b74": "P_3 = \\operatorname{calculate}(P_3,s,s,,)", "dcfccd019879a33eea81925a879aeb41": "\\displaystyle{(f,g)=(f,P_kg).}", "dcfce3ce2f0fa8576e2a734fa9280e1a": "\\mathcal{D} \\wedge \\bar{\\mathsf{h}}(a) = \\kappa \\mathcal{S} \\cdot \\bar{\\mathsf{h}}(a),", "dcfcfe21de731cbe3164a4e5ed656b32": "\\,P_{i,i+1}=p=1-P_{i,i-1}.", "dcfd114520390c290a417801f2f66687": "\\sum_{i = 1}^{m} c_{i} u_{i} = 0", "dcfd57656bfb7ca2ba38ff5855192a81": "\n \\frac{p}{\\cos \\phi} - \\frac{Z}{\\sin \\phi} - e^2 N(\\phi) = 0,\n", "dcfda1e9dedf03350512d1996aaacf4a": "i{ \\partial u \\over \\partial t } = -\\frac{1}{2} { \\nabla^2 u },", "dcfdad2f5a4856d43d555a0b2ef0de10": "\\{ P_t : t \\in [0,\\infty) \\}", "dcfe0b3ef8eb8156a5e83949c2e2f7b6": "3%", "dcfe9fd5f4f5dd0efcc51587f4865223": "l(x^2)=2", "dcfec77429c82d72bacad4efa7be5949": "\\gcd(n^a-1,n^b-1)=n^{\\gcd(a,b)}-1 \\, ", "dcfecf8f67da17150a4afbc44e97b461": "1.32x", "dcfede5763c2d9d86aa2b99c83b6fbe5": "\\frac{1}{p^n}", "dcff23a6faadacad6cf6b17e7676f98a": "\\det(\\mathbf{A})=\\mathbf{x_0}\\cdot(\\mathbf{x_1}\\times\\mathbf{x_2}).", "dcff3ddb526defcbc3fb6b35f41a92fa": "{\\mathbf t}_1, \\dots , {\\mathbf t}_{N-1}\\in [0,1)", "dcffc4d77dfa60a5b5e375dba15aaf97": " -r^{-1}~\\cos\\theta \\,", "dcffd111c22b3446729e2864681acce7": " [[a,b,c],d,e] = [a,[d,c,b],e] = [a,b,[c,d,e]] \\ \\forall \\ a,b,c,d,e \\in H", "dd0023eede1d23abe1ae060f55a17c44": "P(H|D)=P(D|H)", "dd002533a8fd2223867ad809d659f585": "x \\prec z \\prec y", "dd005534084f84558544c2fdacfc5d4e": " r_{\\mathrm{A}} = a ( r_{-1} - r_1 ) ", "dd007861a9bcbed66c9abfb9df1df092": "x_1 x_2 x_3 + x_2 x_3 x_4 + x_3 x_4 x_5 + x_4 x_5 x_1 + x_5 x_1 x_2 = 0. \\, ", "dd00a08e577d900e3b44568ae88309c3": "\\dot x = A x", "dd015011559b79563774a7bdbfd33eb2": "\\varepsilon_{klm}", "dd01e32b57173b05b719b89e09cb1aa0": " E[c^*_n] = \\frac{\\log n + \\log 2 + \\gamma}{n} + O\\left(\\frac{(\\log n)^2}{n^{7/5}}\\right)", "dd0202ceed36ace433212d720bd02568": "a_1=2+\\frac{2}{x}-\\frac{1}{x+\\frac{3}{2}}", "dd024795764e9182e4eabcc8eaa18945": "\\scriptstyle \\Omega_{\\text{E}} (t_i \\,-\\, t_{\\text{rec}}) \\;=\\; \\Omega_{\\text{E}} (\\delta t_{\\text{clock,rec}}) \\Omega_{\\text{E}} (-\\tilde{r}_i/c \\,-\\, \\delta t_{\\text{clock},i}) ", "dd0257fae838ec8b852497ab87694179": "var[x_1+x_2+\\dots +x_n] = \\sigma^{2}_{x} + \\sigma^{2}_{x}+ \\dots + \\sigma^{2}_{x} = n\\sigma^{2}_{x},", "dd028d851626563957f802dc721c53a5": " \\vec{H}(z,t) = \\begin{bmatrix} h_{x} \\\\ h_{y} \\\\ 0 \\end{bmatrix} \\; e^{i(kz - 2 \\pi f t)} ", "dd0295a0629a5aa27aa2491f225c875f": "\n\\begin{align}\n\\tan(\\lambda - \\lambda_0) &= \\frac\n{\\sin\\alpha_0\\sin\\sigma}{\\cos\\sigma},\\\\\n\\tan\\alpha &= \\frac\n{\\tan\\alpha_0}{\\cos\\sigma}.\n\\end{align}\n", "dd030b612f906902c6e4cfa23e2c122f": "\n \\partial f : \\mathbb{R}^N \\rightarrow 2^{\\mathbb{R}^N} : x \\mapsto \\{ u \\in \\mathbb{R}^N | \\forall y \\in \\mathbb{R}^N, (y-x)^\\mathrm{T}u+f(x) \\leq f(y)).\\}\n", "dd0336e1d979c3de0543a87b60b7f9f8": "\\lim_{k\\to\\infty}\\frac{Q}{k!}=0", "dd036145fd2ee2fbd1cc0b8e2dbdbca2": " \\nabla\\cdot(\\phi\\nabla\\psi)=\\phi\\nabla^{2}\\psi + \\nabla\\phi\\cdot\\nabla\\psi ", "dd03a5c3700aa9945824b0ac9f5f896d": "2^2P_{3/2}", "dd03b21b239c306f8990e31072bf517c": "p_1=2", "dd043e6f63ad330ab16c2062ec49914f": "101101 = 7 \\cdot 11 \\cdot 13 \\cdot 101\\,", "dd047007602c9a72a3d99cfe915c00ab": "A_1, \\ldots , A_n", "dd047c5228d04ecb1a521df42fe954ce": "E=\\operatorname{Hom}(S_3,S_3)", "dd05529769d05eeb113684b56533b62c": "a_1,a_2,\\dots,a_n", "dd059c909dc9461355bb92a215b59e6e": "y(n1,n2)=\\sum_{k_1=0}^{M_1-1}\\sum_{k_2=0}^{M_2 -1}{h(k_1,k_2) x(n_1 -k_1, n_2 -k_2) }", "dd0645e2c99e7f1a577ab14103cb650d": "\\|T(f)\\|_{L^2} \\leq C\\|f\\|_{L^2},", "dd06a586140ca5cdf6733d1a51db7ee7": "\\frac{P_\\lambda(\\dots,q^{\\mu_i}t^{\\rho_i},\\dots)}{P_\\lambda(t^\\rho)}\n=\n\\frac{P_\\mu(\\dots,q^{\\lambda_i}t^{\\rho_i},\\dots)}{P_\\mu(t^\\rho)}.", "dd0795f64f6891f6e9e16dba4d69de49": " x = y", "dd079f47d09e7e0b53449277c09cf78b": " G_iG_j=G_jG_i, \\text{if } \\left\\vert i-j \\right\\vert \\geqslant 2, \\text{ and } G_i G_{i+1} G_i=G_{i+1} G_i G_{i+1}, \\,\\!", "dd081b7263d9f3b35ef8aa96d8991894": " 0.90 * 100\\% = 90\\%", "dd08888897efa84ce7399f188c8ff7b6": "\\Omega^{+},\\Omega^{-}", "dd089beb0254317e4c620b8c91b99620": "b_t", "dd08d4cc21c065e9369e7cc84d8f1469": "\\mu_m", "dd08e1808d7acc2e62ab088942e8445a": "T_3 = (T_1Y_2)^2 + (Z_1X_2)^2", "dd08e288357252fdd2aca330cfc77770": "I(q) \\sim Sq^{-4}", "dd0952c77aa751b391a44ac444f771c3": "\\hat{\\boldsymbol{\\imath}}", "dd0952e0856cdc8a018154b6ed5641b6": "2^{\\aleph_\\omega}<\\aleph_{\\omega_1}", "dd0976225303df1c1686059c54fa9c3b": "H_{XX} = - \\frac{\\hbar^2}{2 m_e^*} ({\\nabla_1}^2 + {\\nabla_2}^2) - \\frac{\\hbar^2}{2 m_h^*} ({\\nabla_a}^2 + {\\nabla_b}^2) + V", "dd09c8c8daaea838c57c374f89c9d2c7": " P_{\\rm 1} \\ne P_{\\rm 2}", "dd0a3702c934901ac4dc9d12539ea09a": "\n\\begin{align}\nI(X;Y) & {} = H(X) - H(X|Y) \\\\ \n& {} = H(Y) - H(Y|X) \\\\ \n& {} = H(X) + H(Y) - H(X,Y) \\\\\n& {} = H(X,Y) - H(X|Y) - H(Y|X)\n\\end{align}\n", "dd0a38bf90ccabb2fc096d468529e5f7": "3 \\cdot 2^{x - 1} + 1 = 10", "dd0a5ea34f6d7a2b4ef9cd263c1efe51": " \\iint_S K \\, dA \\le 2\\pi\\chi(S), ", "dd0a60f624cc1069d5c21372c21b1cb8": "\\psi(r, \\theta, \\phi) = R_{nl}(r)Y_{lm}(\\theta,\\phi)\\,", "dd0a9c4a8a5f1265c1b0aad78730526d": "g_{\\mu \\nu }\\rightarrow \\tilde{g}_{\\mu \\nu }=\\Omega ^2g_{\\mu \\nu }.", "dd0ad5788cc624acc3f54088703ced2a": " \\epsilon_f^n \\ = \\ \\sum_{i=1}^{n-1} \\ M_{ni} \\ f_{i}^{n-1} \\ = \\ \\sum_{i=1}^{n-1} \\ t_{n-i} \\ f_{i}^{n-1}. ", "dd0b34fd23130fed765ba0904725c442": "(t,q^i)", "dd0b70d27e8ca100600babf33d183147": "\\Omega(nlog(n))", "dd0b7a211b88badaf3ca6a9265a3f712": "V(t)|\\psi_0\\rangle = i\\hbar\\frac{\\partial|\\psi_0\\rangle}{\\partial\\tau}", "dd0b8a9a7e5851a19348e649a4c9ab08": "f_{a,i}", "dd0b8ae0609a5197fe79310ff0200f64": "\\mathcal{N}_i", "dd0b98bd5b75cbf576282f964484c763": "\\forall x (x \\in V \\rightarrow G(x,y_1\\dots, y_n))", "dd0b9c15467ad16f793cda167cd39fa2": "ds^2 = -\\rho^2 d\\sigma^2 + d\\rho^2,", "dd0b9deb68d94762f457230f70f671a4": " H^p (\\operatorname{Tot}(K), D ) = C^{\\infty}(M_0)^g = C^{\\infty}(\\widetilde M)", "dd0be404e9743259687ec646ab5bc6ac": "\\mathbb Z + {1\\over 2}.", "dd0bec65e9abc482d6c7f1022c7c5631": "K^{\\Dagger } = \\frac{k_BT}{h\\nu} K^{\\Dagger '} ", "dd0c1b11c274a1504e391c4a33fe3476": "{ k}_{ f}", "dd0c92abf99e92e4d2d0af841f43458b": "\\ \\ t_2 ", "dd0c99c3b1c6033f5d7b67ed2b76afa2": " s^{signed}_{ij}", "dd0cd551fa4265ca9ffb3b565d9519eb": "f(x,y) = \\sin \\left(x+y\\right) + \\left(x-y\\right)^{2} - 1.5x + 2.5y + 1.\\quad", "dd0ce8ffdb7f4fb600e8aae3b8cdd6e6": "\\frac{\\partial \\theta_e}{\\partial z} < 0", "dd0d241734aefd69fde7ae2886589fa9": "L = 10\\ n\\ \\log_{10}(d)+C", "dd0d6cb55858c4c97da57d417365207b": "\\mathcal{L}_{1}=\\frac{1}{2}\\, I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta\\cos\\alpha)^{2}+\\frac{1}{2}\\, I_{2}\\,(\\dot{\\alpha}^{2}+\\Omega^{2}\\sin^{2}\\alpha\\sin^{2}\\delta)\\,,", "dd0dd6697862651c44c0cc26e11fc526": "c_x = d_{b_1} \\circ d_{b_2} \\circ \\cdots \\circ d_{b_k} \\circ \\cdots,", "dd0dda7bd0243fa469e10acc9552fb38": "G_{ik}", "dd0dde6a4c641be2744b4278db07f773": "\\text{similarity} = \\cos(\\theta) = {A \\cdot B \\over \\|A\\| \\|B\\|} = \\frac{ \\sum\\limits_{i=1}^{n}{A_i \\times B_i} }{ \\sqrt{\\sum\\limits_{i=1}^{n}{(A_i)^2}} \\times \\sqrt{\\sum\\limits_{i=1}^{n}{(B_i)^2}} }", "dd0de97379a3da28b7d849df8cd1853d": "m\\leq i-1", "dd0dfecd033a3cfa8f71f733e97ab0dc": "I_{o_{\\text{lim}}}=\\bar{I_D}=\\frac{I_{L_{\\text{max}}}}{2}\\left(1-D\\right)", "dd0e199b7101f80de0b6a03f91a5e737": "\\mathbf{F} = d\\mathbf{A} + \\mathbf{A} \\wedge \\mathbf{A}", "dd0e586a720b953c6669f1c417dca3d0": " \\int_{a}^{\\infty}x^{m-s}dx ", "dd0e769f42712f6f6d237e3c69b1b8b9": "\n\\begin{align}\n\\sec x & {} = \\sum_{n=0}^\\infty \\frac{U_{2n} x^{2n}}{(2n)!}\n= \\sum_{n=0}^\\infty \\frac{(-1)^n E_{2n} x^{2n}}{(2n)!} \\\\\n& {} = 1 + \\frac{1}{2}x^2 + \\frac{5}{24}x^4 + \\frac{61}{720}x^6 + \\cdots, \\qquad \\text{for } |x| < \\frac{\\pi}{2}.\n\\end{align}\n", "dd0eeadc8a259338b0759bc0aee93ff6": "\n\\beta \\,\\, \\approx \\,\\,\\, - \\,\\,\\,\\frac{a}{{2\\,n}}\\left( {\\frac{\\sigma }{\\mu }} \\right)^2\n", "dd1002143b6be31668da918484b335f6": "\\lambda_k\\sigma_k", "dd10b0676b3d3aad290e39e93e212566": "u_\\alpha", "dd11108810f45e2301f560a1bf06e3e1": "\\displaystyle \\nabla^2s=(|\\mathbf a|^2+1)s", "dd1142ca8d5a05de90bb309fa4edcb42": "x + y + a = 0 \\,", "dd1147cd39819a3ebd994276bfc664b0": "X = ( \\sigma - \\sin \\sigma) \\frac {\\sin^2 P \\cos^2 Q}{ \\cos^2 \\frac { \\sigma}{2}} \\qquad \\qquad Y = ( \\sigma + \\sin \\sigma) \\frac {\\cos^2 P \\sin^2 Q}{ \\sin^2 \\frac { \\sigma}{2}}", "dd116045e15c408840d85ee2909e55d2": "x^2+y^2=1", "dd119903d1f43f000d67a7b46277f4bd": " \\frac {2 \\pi k}{N} ", "dd119a63460546d71edbd4e6e0b670a0": "W^{s}(p)", "dd11c3c9b6f21f520743271ae6a8104c": "\\mathbf{v}=\\mathbf{Q}\\mathbf{u}", "dd11ec75d51a042f839591dab95317d4": " d\\rho^2 = \\frac{1}{-g_{00}}\\left(g_{jk} \\, dx^j \\, dx^k\\right)", "dd1214bc3cf7c2b3fe3ef2151bb5afa5": " U = u(r_{12})+ u(r_{13})+u(r_{23}).", "dd121f754fc62190380aa610fbf8e1dd": "B(\\mathbf{V},n)", "dd12386c8f0dbe73374291833c0e2012": "Z=N + P=\\text{(number of times the Nyquist plot encircles -1/k clockwise)}+\\text{(number of poles of G(s) in ORHP)}", "dd1246ee3efec380d034c1a7eba5ac03": " \\quad 3", "dd12512987f943b44991ec3d941efb5e": "X(t)=\n \\begin{cases}\n 1, & \\mbox{sys functions at time } t\\\\\n 0, & \\mbox{otherwise}\n \\end{cases}\n", "dd126b67c173024cfc576f075ea1cfae": "A_i'\\in\\mathcal{A}'", "dd126fc28d48dea0667af1ddaad4f0d9": "{\\mathbf P}_1+ {\\mathbf P}_2+{\\mathbf P_3}+ {\\mathbf P_4} = {\\mathbf I}", "dd12ab437b01c37d055f4de5f0a86c01": "|\\epsilon_j\\rang = |\\epsilon'\\rang ", "dd12f527af5b4d962f07a034d32e855f": "1 \\le k \\le n", "dd130a6a5eebd28ad8f2b9b93d31ff92": " \\frac{466}{885}\\cdot 2^n - \\frac{1}{3} - \\frac{3}{5}\\cdot \\left(\\frac{1}{3}\\right)^n +\n\\left(\\frac{12}{29} + \\frac{18}{1003}\\sqrt{17}\\right)\\left(\\frac{5+\\sqrt{17}}{18}\\right)^n +\n\\left(\\frac{12}{29} - \\frac{18}{1003}\\sqrt{17}\\right)\\left(\\frac{5-\\sqrt{17}}{18}\\right)^n.", "dd1312db252d1989bc9508760207cd2a": "(1+r)((1+r)P-c)-c = (1+r)^2P - (1+(1+r))c", "dd132eae4ae91f1b3cc66c18a4f8926b": "\\sqrt{2}-\\sqrt{3}\\,", "dd1358a87e2a24cf063d1f20e4b8ba72": " \\upsilon _M \\,", "dd1377a7036975a01736966c2505c3e4": "Im(x) = \\pm \\sqrt{3}Re(x)", "dd1397db80d40083e9365632e4922587": "D_{0,0} = 1, \\!", "dd13d0b6cb668cd720f89293993210be": " {}^\\phi\\tilde{V}_i \\,", "dd14243a58cdd1e7968f3b7feff18f1d": "\\scriptstyle{0.4\\,\\lambda}\\,\\!", "dd143f5a7a01683583ab1444c6535988": "\\oint_{C} M\\, dy = \\iint_{D} \\left(\\frac{\\partial M}{\\partial x}\\right)\\, dA\\qquad\\mathrm{(2)}", "dd144f12b951f8bacddbdc7b124b8a6d": "\\begin{align}\n \\int_7^{10} (471 + 12y) \\ dy & = (471y + 6y^2)\\big |_{y=7}^{y=10} \\\\\n & = 471(10)+ 6(10)^2 - 471(7) - 6(7)^2 \\\\\n &= 1719\n\\end{align}", "dd14d987ab61f9b346e5d645ffd79af8": "\nG^i(x,v) := \\frac{g^{ij}(x,v)}{4}\\left(2\\frac{\\partial g_{jk}}{\\partial x^\\ell}(x,v) - \\frac{\\partial g_{k\\ell}}{\\partial x^j}(x,v)\\right)v^k v^\\ell.\n", "dd14efc62507b51cbf82cfe4e1965272": "[\\omega, \\eta]", "dd14f7aa13b205dfffb42915a4904dba": "\\hat\\sigma^2", "dd152496153d94ce66e751c90e3c7055": "\\tilde\\Phi=\\Phi-\\Phi_0", "dd156ae5566ed6b72cc28f122b2b3832": "\\mathbf{ABCDEFGHI} \\!", "dd156b05e447dc2e0ae293c5e7e81328": " H_a(j \\omega_a) \\ ", "dd15af433a3b22bb4934656e41b9f884": " \\rho : x \\mapsto \\begin{bmatrix} 0 & 1 \\\\ -1 & 0 \\end{bmatrix}. ", "dd15c3c1c5628074707440d39610e188": "\\scriptstyle\\mathbf{P}", "dd160248c31365acd7a70ba9d7d98544": " \\frac{d\\vec{P}}{dt}= \\int^t_0 \\mathbf{A}(t- \\tau )\\vec{P}( \\tau )d \\tau . ", "dd16a2e831bdc635685e67517aaaaa1f": "\\frac{\\beta^2}{(\\alpha-1)^2(\\alpha-2)}\\!", "dd16d61274528f8a1d2cd9c8ec9805df": " \\scriptstyle 2\\ell+1", "dd16ef218676579d4bd691ed6549f787": "\n\\begin{align}\nds^2 &=\n\\begin{bmatrix}\ndu&dv\n\\end{bmatrix}\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n\\begin{bmatrix}\ndu\\\\dv\n\\end{bmatrix}\\\\\n&=\\begin{bmatrix}\ndu'&dv'\n\\end{bmatrix}\n\\begin{bmatrix}\n \\frac{\\partial u}{\\partial u'} & \\frac{\\partial u}{\\partial v'}\\\\\n\\frac{\\partial v}{\\partial u'} & \\frac{\\partial v}{\\partial v'}\n\\end{bmatrix}^\\mathrm{T}\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n\\begin{bmatrix}\n \\frac{\\partial u}{\\partial u'} & \\frac{\\partial u}{\\partial v'}\\\\\n\\frac{\\partial v}{\\partial u'} & \\frac{\\partial v}{\\partial v'}\n\\end{bmatrix}\n\\begin{bmatrix}\ndu'\\\\dv'\n\\end{bmatrix}\\\\\n&=\n\\begin{bmatrix}\ndu'&dv'\n\\end{bmatrix}\n\\begin{bmatrix}\nE'&F'\\\\\nF'&G'\n\\end{bmatrix}\n\\begin{bmatrix}\ndu'\\\\dv'\n\\end{bmatrix}\\\\\n&=(ds')^2.\n\\end{align}\n", "dd178088cfd24b64f1bce0d383e458f7": "f(x, y) = a_{00} + a_{01} y + a_{10} x + a_{11} x y", "dd179106167de7f7df834873221cd89b": "\\mathbf{a},\\mathbf{b}\\in\\R^n", "dd17a176c52c442b9796c18ff4ef763d": "\\prec_P", "dd17a3da7fd8afc29e3dec6c0a8a8126": "\n\\vec{v}(t)\n=\n\\frac{\\vec{x}(t + \\Delta t) - \\vec{x}(t - \\Delta t)}{2\\Delta t}\n+ \\mathcal{O}(\\Delta t^2).\n", "dd17de5d68d0b061571de56d52ef9319": " y \\in X ", "dd17fbb00a64507c9bd7d0060f106deb": "\\frac{}{}|e\\rangle", "dd181ad473bbe5ccbb2f225e7163ff94": "x_1[n] * x_2[n]", "dd185cdf73f0a341933853c5f552f6d4": " \\mathcal{L}=\\partial_x-\\omega\\partial_y+p", "dd186ec982d01ff42ee748c968bdba33": " k_{Ds} r_{12} << 1", "dd18d0fa82e136142bfb0d4eac2a9f59": " \\frac{V}{R} ", "dd18de4f860542db9ff18c98187fe6b6": " |b_n| \\le 2 \\int_0^{2\\pi} \\Re g(e^{i\\theta}) \\,d\\theta =2. ", "dd193a5defd021b833b5204aba4be1f9": "\\ln I = \\ln I_0 - m \\tau,", "dd1940217f861946f12e7679f41ad143": "\\displaystyle-17.83~\\mbox{dB}", "dd199933be6a6b73830c450f15a48f6b": "\\ddot{\\theta} = a/R", "dd19f6ac36d509e66bcc092b28dcf862": "\\alpha(x,f)=\\omega(x,f^{-1})", "dd19f7e464396afdc493182c5c93318d": "C(s)", "dd1ab8b8f9734c3d4e007ae4a67cd3a0": " \\frac{width \\ of \\ n \\ intervals }{width \\ of \\ n \\ random \\ variables}=\\frac{n\\Delta p}{\\sqrt{n}\\Delta p} = \\sqrt{n} ", "dd1b294682559df0cad29ecd95746417": "\\ \\sum_{i=1}^n \\frac{\\overline{z}-\\overline{a_i} } {\\vert z-a_i\\vert^2}=0. ", "dd1b3e3cdf44dec1a40d89ec7cb978d1": "-\\rho QV_1 + \\rho QV_2 = \\overline{P}_1A_1 - \\overline{P}_2A_2", "dd1b440723758c1c7f88822247544a73": "\\ell\\sim\\frac{\\ln N}{\\ln \\ln N}.", "dd1b4e33e5b8af40b803b97b43ad7137": "\\bar n", "dd1b5dd6d7eb762d1d872ff07b5c1f77": "P(x)(f)=\\sum Sq^{2i}(f)x^i", "dd1b8627abe8e4af0b7afec80e7550d1": "\\gamma^*[1] \\ldots \\gamma^*[L] = \\operatorname*{arg\\,max}\\limits_{\\gamma[t] \\in T(\\sigma[t])} p(\\gamma[1] \\ldots \\gamma[L]) p(\\sigma[1] \\ldots \\sigma[L] \\gamma[1] \\ldots \\gamma[L])", "dd1bc22545939c8037caa9fa27042880": "{\\ell_{OA}}^2 - {\\ell_{AD}}^2 = y^2 + 2xy = \\ell_{OB} \\cdot \\ell_{OD}", "dd1bd03abdeb5de62bd42a721bf0f386": " G = G_m^2 ", "dd1c32728dba43f4f36c5e4050724a80": "\\kappa = \\frac{0.378 - 0.213}{1 - 0.213} = 0.210", "dd1c61e9c900a48bf659865c1b58e699": "x = \\frac{2 \\sqrt 2 \\cos(\\phi)\\sin\\left(\\frac\\lambda 2\\right)}{\\sqrt{1 + \\cos(\\phi)\\cos\\left(\\frac\\lambda 2\\right)}}", "dd1c6b052b3651ef1b73293393dfb162": "\\alpha = \\frac{P(\\lambda > \\lambda_I)}{\\lambda_e} = \\frac{1}{\\lambda_e}\\exp\\left(\\mbox{-}\\frac{\\lambda_{I}}{\\lambda_{e}}\\right) = \\frac{1}{\\lambda_e}\\exp\\left(\\mbox{-}\\frac{E_{I}}{E_{e}}\\right)\\qquad\\qquad(12)", "dd1cc5efb20d025f6413d1a0c0fa6eb0": "\\scriptstyle d/d\\tau", "dd1ce58ee6b747798e2ae3f42fb418e4": "\\mathrm{Div}(C)", "dd1d038098676841eecdbc7ffe2e3c20": "l_t'", "dd1d10b118808e14ec3d23b444252adb": "\\mathbf{X} \\sim {\\rm T}_{n,p}(\\alpha,\\beta,\\mathbf{M},\\boldsymbol\\Sigma, \\boldsymbol\\Omega)", "dd1d410cb634acc166d879c3836e1bb3": "\\epsilon^* = \\epsilon + \\frac{\\sigma}{j\\omega} = \\epsilon - \\frac{i\\sigma}{\\omega}", "dd1d5cd6516d4bdda388e090a81e448d": "Z=\\frac{1}{1+\\frac{\\sigma^2}{v^2m}}", "dd1d65a997eeeba2e80f6fd1a7371290": "O_{5}", "dd1d81e0affd46ba4574a04108d5b4b7": "\\ \\sigma", "dd1d94d8fac36e2ecd2db5bf227c0aa1": "B_1 = (a_1,b_1,\\dots,b_{t-1}, b_{t+1}, ... b_r)", "dd1d9edda6b61c614aa6ec1869a80ed7": "2^{-80}", "dd1de557c92aa8027050cb1afd9267a9": "F_\\infty", "dd1df36e052fdf4ead14c55e12e9c929": "\\Lambda(\\varphi,\\hat{\\mathbf{a}}, \\theta,\\hat{\\mathbf{n}}) = \\exp\\left(-\\frac{i}{4}\\omega_{\\alpha\\beta}M^{\\alpha\\beta}\\right) = \\exp \\left[-\\frac{i}{2}\\left(\\varphi \\hat{\\mathbf{a}} \\cdot \\mathbf{K} + \\theta \\hat{\\mathbf{n}} \\cdot \\mathbf{J}\\right)\\right]", "dd1e6bf4c85dfceb15abd3143ea66505": "U = \\left\\{ z \\in H: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\tfrac{1}{2} \\right\\}.", "dd1e6edf95f3a7ec5538b54e7cc5fc45": "\\Omega (R)=\\Omega_{gp} + \\kappa /m", "dd1e73c54ba0219e8ef83ed4994b3466": "\\mathcal{L}(\\theta |x) = P(x | \\theta)", "dd1ed1ef7a3ea5bfe4077be90e9084b5": "F \\to T_kF", "dd1ed853fdc7f1c7aa1f8eb05c196eb6": "\\psi(g) = \\sum_{x\\in G \\setminus H} \\widehat{\\chi}(x^{-1}gx) ~,", "dd1ee62b5a0a60a84f644f89ad90c112": "A[f] = \\int_0^1 f(x)\\,d\\alpha(x),", "dd1eeddf7ef7565465d2796e90a96af5": "\\operatorname{cl}(S) = \\{ x\\in E\\mid x\\le\\vee S \\}", "dd1efa9bf55c4768d0b8068855240782": "\\mathrm{Tr}^U_{X,Y}(f)g=\\mathrm{Tr}^U_{X',Y}(f(g\\otimes U))", "dd1f1163e06c17f50c9b170147bb0594": "g_3 = \\frac{1}{27} (\\lambda + 1)(2\\lambda^2 - 5\\lambda + 2)", "dd1f4c403434977379557dbbd8942554": "{\\textbf{x}}_1", "dd1f6fcfd2ed573b571e2d6e8949c3ab": "L_\\text{C} = -\\frac{q_1 q_2}{r}, ", "dd1f93a361167050d131f51044cf8fd8": "\\displaystyle{\\|\\mu_n\\|_\\infty \\le \\|\\mu\\|_\\infty.}", "dd1fcb4f8765800e728104cc90a23ad4": " \\Gamma ", "dd2011d078a6569c5b603bdb2baef34e": " C(t) ", "dd203af75da67a259b2efc5c712574d3": "v_{\\mathrm N} \\, - \\, v_{\\mathrm F}", "dd2042d94ba0af7933f78cec22f06a9f": "{u}", "dd2096fcf09fce6422a0013360e5a8f5": "d_{1,1}^{1} = \\frac{1+\\cos \\theta}{2}", "dd20ddce3345106bc627ef1ed3d678e8": "2d(x,y) \\ge 0", "dd20f23ee0f5a05a50c48c881172a818": "m_1 < m_2", "dd212270229767191a8e42b95f2e9c33": "\\mu(n)", "dd2127001c4a7a728131c1017a3e2a29": "_{q\\tilde{\\leftarrow}p=p\\tilde{\\leftarrow}q\\ \\Leftrightarrow\\ q'p=qp'\\,}\\!", "dd215c5aafe12db161cf57ce6e864149": "P(x) \\nrightarrow (\\exists{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\nrightarrow Q(y)),~\\mathrm{provided~that}~\\mathbf{Y}\\neq \\emptyset", "dd216b98dc3d7fd268717eb1e5d489b1": "e = 1.6709\\cdot 10^{-2}-4.193\\cdot 10^{-5}\\left(\\frac{D}{36525}\\right)-1.26\\cdot 10^{-7}\\left(\\frac{D}{36525}\\right)^2", "dd21bc8f394df686d9b06476424418ae": "A_{1}, ..., A_{n}", "dd21f22ec848ae03888c51c5596b5edd": " \\frac{[K_i : Q]}{\\log|D_i|} \\to 0\\text{ as }i \\to\\infty. ", "dd225a00bd5507f34e327f6821bfe323": "\\land\\,(p\\lor q\\lor r)\\land\\neg(p\\land q\\land r).", "dd22b0a9fddb29b839206751a552026d": "f \\cdot v = {1 \\over \\rho}{\\partial P \\over \\partial x}", "dd22d6c746244423ef2765973e2459dc": "x^{128} + x^7 + x^2 + x + 1", "dd2324a21fc936ec7e16105cdee0f3ab": " E' = y' + \\frac{1}{2y'^2}", "dd234f5a2b19b49dff965ddf8286476a": " \\dot{\\tilde{\\rho}}= - \\int^t_0 dt' \\operatorname{tr}_R\\{[\\tilde{H}_{BS}(t),[\\tilde{H}_{BS}(t'),\\tilde{\\chi}(t')]]\\} ", "dd23cc89c6e6f899aa0a1a2b871e5153": "\\textstyle m = \\left(\\frac{t_1}{n}\\right)", "dd2492a94b6fa26914db9d9881efeceb": "\\displaystyle{(X,Y)_\\sigma=-B(X,\\sigma(Y))}", "dd2522118bd742babfe0048f0e66ec21": "R_2=K_2+3K_3", "dd25403004ec2aad07ed9787d2dbfb2a": "p_r", "dd254bf9935ddedcdb3573dd0e8c36c9": "\\mathrm{AB \\longrightarrow A + B}", "dd25680a9e2ff8b8251261aa80d7411a": "xy(x+y)=880", "dd259e8c4ed38ce0009ff2acf0f3df72": "[M] \\in H_{2n}(M,\\partial M)", "dd25b29c12c0ac332f356d49da486a09": "g_{\\kappa\\lambda} = \\eta_{\\kappa\\lambda}", "dd25b2ef76516fb9aeecbe201236bb39": "O(\\sqrt p\\log p)", "dd25bc2d3021f0addbcf1ec73b50a84d": "\\ln G_X = \\operatorname{E}[\\ln X]", "dd264b64423765c5fecfed8201fd8b14": "\\scriptstyle F(n) \\;=\\; 2\\uparrow^n 3", "dd2655ebc625fa1fa4841c5b966a5d7b": "\\|A+B\\| \\le \\|A\\|+\\|B\\|", "dd26aa290d7d5468972ab75f153cbfba": "(W_{ij})", "dd26abf40078784cd2a7da5e19ce9d5e": "(F \\circ G)(x) = F(G(x))", "dd26e6d423f95ebde58524412fbeae0c": "S^{n}=\\{[x]\\in\\mathbb{RP}_{n+1}\\; :\\; g(x,x)=0 \\}.", "dd26fd473d1ad3c2692d13d17defad7e": "|M| = c^{\\mbox{dim }M}", "dd270aa33b0877ebdf93c5b26f4b256f": "a_{\\mathrm{Na}^ + } \\cdot a_{\\mathrm{Cl}^ - } = K_{sp}", "dd2743b681b32995d678b543b7357bf6": "-j \\sqrt \\frac{19}{25}", "dd2797c8bb4a8435d7121a92491b28d3": "\\overline{\\{0\\}} = \\{0\\},\\qquad\\overline{\\{1\\}} = \\{0,1\\}.", "dd279825583bde58e1049a91bf2d75be": "\\reals^2", "dd27b2c033eba36918b99d8358df4e35": " \\,\\,\\, = \\sum_i E_i dp_i / T ", "dd284c63d6caf97ac146c004cc6d723f": "f\\left( x,t\\right)", "dd2860d45096ff84a84f963fae3ee530": "M_{ii}=\\sigma^2_{y,i}+\\left(\\frac{dy}{dx}\\right)^2_i \\sigma^2_{x,i}", "dd28c9339973ed203ec9201e4b6385ac": "2^5 + 2 + 1", "dd2905e0f3e877385e3e8f59b980e196": "MSE = (1/N)\\sum_{i=1}^N (y_i - g(x_i|D))^2", "dd2933b78436a1cdf1660e2648b1f79b": "\\mathbf{H}(\\mathbf{x},t)=\\Re\\left(\\sum_{\\ell=0}^\\infty \\sum_{m=-\\ell}^\\ell \\left[a_{\\ell m}^{(E)} h_\\ell^{(1)}(kr) \\mathbf{X}_{\\ell m}(\\theta, \\phi)-\\frac{i}{kZ_0}a_{\\ell m}^{(M)}\\mathbf{\\nabla}\\times(h_\\ell^{(1)}(kr)\\mathbf{X}_{\\ell m}(\\theta, \\phi))\\right]e^{-i\\omega t}\\right)", "dd293cd42e2ea15c960979b9153e6726": " \\frac{du(0)}{dx}EA=P ", "dd295a566e3690d77fd093765c67ff8c": " w^* = X\\beta + u\\, ", "dd295f13a3866bc67a2baba998f9dcd2": " E \\cap F", "dd29ad63d452e16419ead86c494d1528": "t = t^{i_0i_1\\dots i_{r-1}}", "dd29adce88bd2fed85d47aeb3ca2f6e9": "\\Epsilon \\, \\epsilon \\, \\varepsilon \\,", "dd2a6bea2b00e9e6d0b3d49924ab58b3": " 1 = (a^\\perp \\cup b^\\perp)^\\perp \\cup (a \\cup b)^\\perp ", "dd2a7051774dabaebe7888f713ac7577": "\n\\eta_{y} = \\mathrm{sn}\\, \\chi \\ \\mathrm{dn}\\, \\psi \\ \\sin \\phi\n", "dd2ab6d0d56df03a19186756717ce8eb": "\\beta_{w}", "dd2aef44f299301b2621e662dc84994c": " (P \\rightarrow Q) \\And (R \\rightarrow S) ", "dd2b0095c2cdf093f6446dbd91ac897e": "\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C} = 2R,", "dd2b4e851922e548256509d3736a8235": "{q\\over A} = -k \\left({\\partial T \\over \\partial y} \\right)_{y=0} = h_x(T_S-T_\\infty)", "dd2b8e7de9feed89c521bceca3e167f7": " v = \\sqrt{2\\cdot g\\cdot h}. ", "dd2ba25df9260b19afc38ef641d16b54": "G_2\\,", "dd2ba76d648b515d0b4d95c74f161c8a": "\\alpha^2", "dd2bc95e0626584b9fe372772f537271": "B_{p+n}\\equiv B_n+B_{n+1} \\pmod{p}", "dd2cb61dc11cf63f03fa778db7c177bd": "\\int g(z,\\varphi) \\, dz", "dd2cbd6ec9e134b310db7930b4b0040c": "\\rm \\ 4ClONO_2 + TiCl_4 \\rightarrow Ti(NO_3)_4 + 4Cl_2", "dd2d1ba8f939743bb4146aaa8a84223f": "2L-1", "dd2d1e7e07a493a4e41512070910eb7b": "d(a,b)=-\\frac{1}{a}\\sum_{k=1}^{a-1}\\cot\\left(\\frac{\\pi k}{a}\\right)\\cot\\left(\\frac{\\pi bk}{a}\\right)", "dd2d2460e375f25db3dd3fc79be59799": "Z_\\mathrm{R} = R\\,,", "dd2d2ea8d9b251d96b3fa5bef8fd9c89": "\\left(\\!\\!\\!\\binom{n}{k}\\!\\!\\!\\right),", "dd2d4988675d411a3775af00e7cd5309": " \\int dm = M ", "dd2d5c91a6e6819ebee92bd126a036b2": "t_{1/2} = t/\\log_2(N_0/N(t)) = t/(\\log_2(N_0)-\\log_2(N(t))) = (\\log_{2^t}(N_0/N(t)))^{-1} = t\\ln(2)/\\ln(N_0/N(t))", "dd2d87f6cbe936c8aa706ae25c84f018": "\\mathbf{x}_j = 0", "dd2e6b8e6efe3d4ff53a81b0ed05bbae": "{\\Delta c_V = - T \\cdot \\Delta \\left( {\\left( {{{\\partial P} \\over {\\partial T}}} \\right)_v } \\right) \\cdot {{dv} \\over {dT}}}", "dd2e7c5d436f238b749f1722767487ad": "P(B|A) = \\frac{P(A \\cap B)}{P(A)}", "dd2e9b56c56cba4e2dd9066465ad2185": "\\rho_1(x_0, x_1) = \\frac{x_0 - x_1}{f(x_0) - f(x_1)}", "dd2eb43eb67d209af17f78284c7a6bfe": "\n\\rho A\\frac{\\partial^{2}w}{\\partial t^{2}} - q(x,t) = \\frac{\\partial}{\\partial x}\\left[ \\kappa AG \\left(\\frac{\\partial w}{\\partial x}-\\varphi\\right)\\right]\n", "dd2ebf126f7dd303b4c3178548b7fc90": "\\scriptstyle c_v", "dd2ec5127b11385c9fc349cf86659ac1": "\\sqrt{3}\\sin(\\theta)\\cos(\\phi)", "dd2ed0eb6a9f58e25b2fa2d76dcdede0": "\\sigma_a^2(f)", "dd2efa3c21743b482286454edcc4f9a4": "\\exists y", "dd2f1851cd9c1d7a9290f6b4a1b3cf61": "\n G_n=2 \\,(1-2^n) \\,B_n.\n", "dd2f21bcb5fc1f8a836463967797fd54": " r_n(B) = \\sum_{t=0}^n (-1)^t (m-t)_{n-t}\\; r_t(B'). ", "dd2f4015adf7467e56f17428ef17338c": "\\ell = \\iiint_T f(x,y,z)\\, dx\\, dy\\, dz", "dd2f446e8a5bea42a979a923a6282d90": "(1-z)^{-b} F \\left(c-a,b;c; \\tfrac{z}{z-1} \\right )", "dd2fa4a8beb6346ce91fe215903a2666": "\\textstyle K_m = s \\in \\mathbb{Z}_q^*", "dd3089fa3d086caf915554d35eff5cb5": "\\Delta \\theta \\hat{\\mathbf{n}} = \\Delta\\theta(n_1, n_2, n_3)", "dd309a646eae3867c7c429a53525cfb8": "ds=\\frac{1}{y}\\sqrt{dx^2+dy^2}.", "dd309a6b3901348358de8e93297005e1": "V = \\frac{4}{3}\\pi R^3 \\rightarrow dV \\approx 4\\pi R^2 dR", "dd309da18be48daf875c70cd968d23fa": "\\bar b_i - \\lfloor \\bar b_i \\rfloor - \\sum ( \\bar a_{i,j} -\\lfloor \\bar a_{i,j} \\rfloor) x_j = \\bar b_i - \\lfloor \\bar b_i \\rfloor > 0.", "dd30aefc4c40475c946937fa9c23e88a": " |\\overline{PR}| = \\frac{|Ax_0 + By_0 + C|}{\\sqrt{A^2 + B^2}}.", "dd30eea906007034c395bdaa73759d87": "\\mathrm{Sp}(p+q)\\,", "dd31261947173970d244e06124f33d50": "f_y=-Ky", "dd316846052cf6e4747d11d26294de0a": "\\operatorname{Res}(\\Gamma,-n)=\\frac{(-1)^n}{n!}.", "dd316d0a08faf4b698d83612fe354602": "z\\mapsto e^z", "dd31709855563932eec6cf02cb9e07c5": "{{h}_{rad}}=\\frac{\\varepsilon \\sigma \\left( T_{s}^{4}-T_{sat}^{4} \\right)}{\\left( {{T}_{s}}-{{T}_{sat}} \\right)}", "dd318c13fe26a6ff0d663183a7cde71d": "T \\models \\alpha", "dd320eb3c70221d8e99dab41da4a7fac": "\\mathbf{R}^n,", "dd322167adf23f8e793137a088851c66": "\\bar\\psi(\\mathbf{x},t) = [\\psi(\\mathbf{x},t)]^\\dagger", "dd32531b41ead300d14d9a7ebd19095f": "y_1(x) = e^{-\\frac{b}{2 a} x}.", "dd329003a94f69e185c2d509baadeb12": "\n\\frac{x^{2}}{\\lambda - A} + \\frac{y^{2}}{\\lambda - B} = 2z + \\lambda\n", "dd32dc8ff9929585896b80d37233dc66": "\n\\begin{align}\n \\mathrm{Ai}(x) &{}\\sim \\frac{e^{-\\frac23x^{3/2}}}{2\\sqrt\\pi\\,x^{1/4}} \\\\\n \\mathrm{Ai}(-x) &{}\\sim \\frac{\\sin(\\frac23x^{3/2}+\\frac14\\pi)}{\\sqrt\\pi\\,x^{1/4}} \\\\\n\\end{align}\n", "dd330c994630349f3616160eba94b9d8": "\\bold{\\Sigma}^0_n", "dd33196b107ef0a6d90f99a0dd695315": "\n|\\mathbf{v}\\mathbf{w}|^2 = (\\mathbf{v}\\cdot\\mathbf{w})^2 + |\\mathbf{v}\\times\\mathbf{w}|^2.\n", "dd332e91f60499fcb745446a3549a5b6": "R_J\\left(\\kappa x,\\kappa y,\\kappa z,\\kappa p\\right)=\\kappa^{-3/2}R_J(x,y,z,p)", "dd336271e0ed4a8eb8315228d04b2e26": " \\alpha \\colon \\mathcal{M} \\to \\mathcal{O}_X ", "dd346e5e2d67ce433ac63b5c3ef6bd90": "Y_{3}^{-3}(\\theta,\\varphi)\n= {1\\over 8}\\sqrt{35\\over \\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\quad\n= {1\\over 8}\\sqrt{35\\over \\pi}\\cdot{(x - iy)^{3}\\over r^{3}}", "dd34afa4c114c2104d51c44ca89fc5e7": "\\Pr(B_n) = 0 \\,", "dd34d70b2e79518128f6f2709425d8ea": " \\mathcal{E}^n \\subseteq RP ", "dd34d8ce54270e5fc9cfc383c3145e6f": "p(c_i | x_j) \\,", "dd35048dd7e5a971d42c9ea41dfbe43f": "{U(R)}^\\dagger \\widehat{T}^{(2)}_q U(R) = \\sum_{q'} {D(R)}^{(2)}_{qq'} \\widehat{T}_{q'}^{(2)}", "dd3525342289f3591cc94b2c6432a97d": "\\text{median} \\approx \\frac{\\alpha - \\tfrac{1}{3}}{\\alpha + \\beta - \\tfrac{2}{3}} \\text{ for } \\alpha, \\beta \\ge 1.", "dd352dc6e7e42505bbd62029f96e230c": "\\int_{\\mathbb{R}^n} e^{2\\pi i x\\cdot\\xi}(\\mathcal{F}f)(\\xi)\\,d\\xi = f(x). \\qquad\\square", "dd353eaf9412c52e0695cbde88140a55": "u_i^{n+1/2}", "dd355f5186eb6e9fe41f1cf0f3181e31": "b^{n+1} = b^n \\cdot b", "dd3598b870d81c24a3c9834365ee7db4": "\\scriptstyle y\\,", "dd35bc6f2bbfaf5a4389fd8ab3b9d1cc": "d^l_{mn}", "dd362e9ba398d7a18e55acab527de8de": "P(n|N)= \\frac{[n \\le N]}{N}", "dd3659569c350381f262d12c222ed99f": " f_t(\\psi) ", "dd3675f25cdb7ebba1ad657b95e0586a": "V_{TO}", "dd36c346c58933a32279a8bed79dfe3b": "p(x+a)", "dd373fbf4724163b97fae66ec5ead385": "A\\lambda_1+...+N\\lambda_{14}=\n\\begin{bmatrix}\n 0 & C &-B & E &-D &-G &-F+M \\\\\n-C & 0 & A & F &-G+N&D-K&E+L \\\\\n B &-A & 0 &-N & M & L & K \\\\\n-E &-F & N & 0 &-A+H&-B+I&-C+J\\\\\n D &G-N &-M &A-H& 0 & J &-I \\\\\n G &K-D& -L&B-I&-J & 0 & H \\\\\nF-M&-E-L& -K &C-J& I & -H & 0 \\\\\n\\end{bmatrix}", "dd374a40ea61886f9d71c0870343d2f7": "\\langle x, y \\rangle \\langle y, x \\rangle \\leq \\Vert \\langle x, x \\rangle \\Vert \\langle y, y \\rangle", "dd377ffe442cdcd29041a3f8dfc16a47": "\\phi_t(N) \\subset \\mathrm{int}(N)", "dd37a98086555ddcc57456d20943b580": "\\operatorname{rank}(C) = n", "dd38228f932f74f9d0ddcf6aa5258957": "S_{\\rm prolate} = 2\\pi a^2\\left(1+\\frac{c}{ae}\\sin^{-1}e\\right)\n\\qquad\\mbox{where}\\qquad e^2=1-\\frac{a^2}{c^2}. ", "dd38703b4fd8f76c1104b34730298d27": "\\frac{\\partial T}{\\partial t} = \\kappa\\frac{\\partial^2 T}{\\partial^2 z}", "dd387c0ce3e41d87fefb6f18fc8df93c": "\\tilde{k} \\in \\tilde{K}", "dd3886647bcb99ad4c0e229ebda66bca": "r=0, \\dot{r}=v, \\theta=\\theta_0, \\dot{\\theta}=0", "dd389d26bc2a297ff0c0d25db3eab3c5": "\\langle x_{1 1}, x_{2 2}, x_{1 2}\\rangle", "dd390d9dac8e2785310d0d16e799efe4": "R \\times C", "dd394b46a2a43b3ecb70d1abd189e1a4": "\\mathrm{H_2 + OH \\longrightarrow H + H_2O}", "dd394d068fc8a5c12ddd2516c63464fa": "f\\left(\\frac{X}{X^2+Y^2},\\ \\frac{Y}{X^2+Y^2}\\right)=0.", "dd39a9d02e57c58bd4a4f9c277c92b57": "f+e(1-b)^{-1}c,", "dd39bf81385437d4e21b74b0dd03abf1": "\\gamma(r,\\theta)", "dd3a2c6e7022f68bb6983064dabf92f0": "\\eta_{th} = \\frac{\\text{Desired Output}}{\\text{Required Input}} = \\frac{W'_{out}}{Q'_{in}}", "dd3a8fd43883c425916ce7bc7587afdf": "\\alpha \\circ \\beta' = \\beta \\circ \\alpha' ", "dd3ae17a718e788487da0877446918f3": "\nf(n) = \\sum_{d|n}\\mu\\left(\\frac{n}{d}\\right)g(d).\n", "dd3b00b5abe44f59062a5d73e83fb072": "|S| \\leq \\gamma n\\,", "dd3b0e2878b85e4a725f2752a9c0216e": "\\operatorname{cl}_D(A)", "dd3b17843bfb47ef7861b0d507fd599a": "\\frac{d}{dt}|\\Psi(x,t)|^2 = 0", "dd3b1c92b058d80ab696e80614781519": "N = Z - P", "dd3b9650f50d969e37bda360f6374ab3": "g: g^{-1}(V_i) \\to V_i", "dd3baad47b0f3062cc22a6ea7e307ed9": "\\hat{H}_0 ", "dd3c80eb9e2d4c941295e264d29a9780": "\\partial_u = \\sin \\alpha \\partial_x - \\cos \\alpha \\partial_y, \\partial_v = \\cos \\alpha \\partial_x + \\sin \\alpha \\partial_y ", "dd3cd40bee2ba242719614c2e4b377d6": "\\mathbf{F}\\,", "dd3d4000a187d8ae9d29b8cb00b3aa8e": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 59.4\\cdot 16.71)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 213.5\\cdot R_{\\bigodot}\n\\end{align}", "dd3d44f63d3e211364feb489f0530f15": "\\begin{align}&\\lambda = \\frac {h}{\\gamma m_0v} = \\frac {h}{m_0v} \\sqrt{1 - \\frac{v^2}{c^2}}\\\\\n& f = \\frac{\\gamma\\,m_0c^2}{h} = \\frac {m_0c^2}{h\\sqrt{1 - \\frac{v^2}{c^2}}}\n\\end{align}", "dd3dce2d294a42179b741cf36a56e0c8": " \\gamma {\\beta}(1) \\! + \\! {\\beta}'(1) \\! = \\pi \\! \\left(-\\!\\ln \\Gamma(\\tfrac14)+\\tfrac34 \\pi+\\tfrac12 \\ln 2+\\tfrac12 \\gamma \\right) ", "dd3dcec784a007d4efd7a0b2533afb19": "C^\\perp = \\{x \\in \\mathbb{F}_2^n \\,\\mid\\, \\langle x,c\\rangle = 0 \\mbox{ }\\forall c \\in C \\} ", "dd3dffe019a2623ca888b2c05baf3233": "\\mathrm{Pe} = \\frac{du\\rho c_p}{k} = \\mathrm{Re}\\, \\mathrm{Pr}", "dd3e17ba7323c913d63b3399e2134d03": "{\\scriptstyle\\partial \\Omega }", "dd3e649fdf47c88b4fb41e7b3e16c356": " \\sum_{ i = 1 }^n b_i^2 ", "dd3ebe17bdd2933c9db573f04f5c924e": "E\\left[e^{it\\log X}\\right] = \\lambda^{it}\\Gamma\\left(\\frac{it}{k}+1\\right).", "dd3ec595ec3d85c572c7802acf48d8c6": "L(\\alpha)", "dd3eca90408a1d779e21aa03f8f4da2e": "\\mathrm{On}", "dd3f21256433ddd3e84f64bd4103c2f2": "\n\\begin{align}\nEAC = AC + {(BAC-EV)\\over CPI} = {BAC \\over CPI}\n\\end{align}\n", "dd3f42a762e0eb3c4e9629dddbaa2c4c": "z^2u''+p(z)zu'+q(z)u=0", "dd3f48ceca67766e4c54397035b1c79e": "\\mathcal{A}(x)", "dd3f71989a1f48bc45437d9766f278e0": "i^n", "dd3f918b1cf681480d9953bae883cc90": "R_2R_3 = \\frac{R_a^2R_bR_c}{R_T^2}", "dd3fa02d9612a3bb2ad99c4537b60db3": "\\sqrt[4]{-\\frac{p}{5}}\\operatorname{BR}\\left(-\\frac{\\sqrt[4]{\\frac{5^5}{-p^5}}q}{4}\\right)", "dd3fb7f4848fac29efa90e6f943e9303": "W \\equiv \\oint_C \\vec{F} \\cdot \\mathrm{d}\\vec r = 0.\\,", "dd3fe03f011f8bd408828c83d26b2bda": " \\tilde{ST}_x(t,f) = \\int_{-\\infty}^\\infty x(t+\\tau) w^*(\\tau) e^{-j2\\pi f\\tau} \\, d\\tau \\quad = ST_x(t,f)\\,e^{j2\\pi f t}", "dd3feca9720e59fb4b1957922828e8ce": "=\\frac{\\text{kv}_{L-L}^2 \\text{ * 1000}}{\\text{ohms reactance}}", "dd3fef4623bf50057ba906ae04b6062f": "\nQ = 2 \\pi \\times \\frac{\\mbox{Energy Stored}}{\\mbox{Energy dissipated per cycle}} = 2 \\pi f_r \\times \\frac{\\mbox{Energy Stored}}{\\mbox{Power Loss}}. \\,\n", "dd3ff1227158ba37f81a06ccedfe64f7": "g_{\\mu\\nu}(x^\\alpha)=\\eta_{\\mu\\nu}-2\\int_{\\Sigma^-}{y_\\mu^- y_\\nu^-\\over(w^-)^3} [\\sqrt{-g}\\rho u^\\alpha d\\Sigma_\\alpha]^-", "dd3ff49dc908271487fd4e4c8310ea9f": "\\vec{P} = A \\times (B \\vec{r}) - C\\vec{r}", "dd40003492cc6f377364ca0daa91ed00": "\\int_0^{2\\pi} \\sqrt{ E\\left(\\frac{du}{dt}\\right)^2 + 2F\\frac{du}{dt}\\frac{dv}{dt} + G\\left(\\frac{dv}{dt}\\right)^2 } \\,dt = \\int_0^{2\\pi} \\sin v \\,dt = 2\\pi \\sin \\frac{\\pi}{2} = 2\\pi", "dd406c4ffcbd308ec22b51185db20d77": "\\textbf{let}\\ f = \\lambda x . x\\, \\textbf{in}\\, (f\\, \\textrm{true}, f\\, \\textrm{0})", "dd407fd599ca5971666602927de1d709": " a_1 v_1 + a_2 v_2 + \\cdots + a_k v_k,", "dd40d979ee78ed7e54aaaf4b95ea3419": " \\mathcal{L} \\, = \\, \\frac12 \\left(\\epsilon_{0} E^2 - \\frac{1}{\\mu_{0}} B^2\\right) - \\phi \\, \\rho_{\\text{free}} + \\bold{A} \\cdot \\bold{J}_{\\text{free}} + \\bold{E} \\cdot \\bold{P} + \\bold{B} \\cdot \\bold{M} \\,.", "dd415a5fb1f27e158dd83fcbdb5fe163": "f:Z\\rightarrow X", "dd41ff7d37170734ec1d9c5ec720e4ed": "r_{t_1,t_2} = \\left(\\frac{(1+r_2)^{d_2}}{(1+r_1)^{d_1}}\\right)^{\\frac{1}{d_2-d_1}} - 1 ", "dd420afac520e8223501ee2c9d5eac89": "2^\\mathbb{N}", "dd42414a4712a4a72879d3ce272d1c92": "\\frac{t n^T}{d} P_i = t", "dd429983379bfce971707b6fcdb7330b": "y' = y-y_0", "dd42c0b2bc70c223c3797eae8df48a50": "\\pi(1-a-b)", "dd434ee39819760271b964b57fa3bf58": "\n\\begin{align}\nF(a,z) & = 1 + 2z^{-2} + a^{-2}z^{-2} + a^{2}z^{-2} - 2a^{-1}z^{-1} - az^{-1} - 20z^2 + 2a^{-4}z^2 \\\\[6pt]\n& {} - 8a^{-2}z^2 - 8a^2 z^2 + 2a^4 z^4 + 2a^4 z^2 - 2a^{-5} z^3 + 4a^{-3} z^3 + 6a^{-1} z^3 \\\\[6pt]\n& {} + 6az^3 +4a^3 z^3 - 2a^5 z^3 +42z^4 - 7a^{-4} z^4 +14a^{-2} z^{4} \\\\[6pt]\n& {} + 14a^2 z^4 -7a^4 z^4 + a^{-5} z^{5} - 9a^{-3}z^5 - 2a^{-1} z^5 - 2az^5 - 9a^3 z^5 \\\\[6pt]\n& {} + a^5 z^5 - 28z^6 +3a^{-4} z^6 -11a^{-2} z^6 -11a^2 z^6 + 3a^4 z^6 + 4a^{-3} z^7 \\\\[6pt]\n& {} - 2a^{-1} z^7 - 2az^7 + 4a^3 z^7 + 8z^8 + 4a^{-2} z^8 + 4a^2 z^8 + 2a^{-1} z^9 + 2az^9.\n\\end{align}\n", "dd4381c3dcb3c305272367de2d6d292b": " \\langle\\psi_p\\psi_q | \\hat{v}| \\psi_r \\psi_s\\rangle\n = \\int \\psi^*_p(1) \\psi^*_q(2) \\frac{1}{r_{12}} \\psi_r(1) \\psi_s(2) \\, d \\tau_1 \\, d \\tau_2,", "dd438c310fdd611181d2d78eeca09d6f": "\\iota \\kappa \\lambda \\mu \\nu \\xi \\pi \\rho \\!", "dd43bccea7538547265b6f45df2535f7": "u \\beta y = -\\frac{\\partial \\phi}{\\partial y}.", "dd43c3a441c974f4027cb15e6311f694": " c\\neq 0", "dd44702c7d3b6e28f5526374b887343f": "100-40 = 60\\ \\mathrm{dB} ", "dd4492930304ba7e031cf22f2f619606": "\\ x(t)", "dd44dab794f41015f792bd1620aafa60": "2 \\Omega \\sin(\\phi)", "dd44e646de28cb9dde4038e814afcee7": " d(uv_i) = d u_i \\,.", "dd454d78deae3ea75736ac035c71599c": "\\widehat{T}(\\theta, t) = e^{-t(1 - \\cos \\theta)} \\approx {1 -t(1 - \\cos \\theta)} = F_1(\\theta, t),", "dd4579173e7431003a16bacf7a41cc14": "\\mathbf{UCV}^T", "dd457b6cbea2a4c77c00694360ad8e25": " \\Phi ^m (r) = (N-m+1)^{-1} \\sum_{i=1}^{N-m+1}log(C_i^m (r))", "dd45a4762b70fa1fdc2b033d8904cbc7": "G_{c,v,w} = T_w \\circ F_{c,v},", "dd460025bcc6eb4bbeb0766e9f472abe": "\\omega+dd'\\phi", "dd464ecbc63c74698ce318677eb7df18": "\n\\frac{d\\varphi}{d\\tau} = \\frac{L}{m \\, r^2}\n\\,", "dd46528d7784e2dcd6963830ebc3543c": "\\Delta _r H=\\left (\\frac{\\partial H}{\\partial \\xi}\\right )_{p,T}", "dd468d25c4e4e1111a59964e9e122029": "\\tfrac{1}{2}(0 - \\tfrac{6}{4})= -\\tfrac{3}{4}", "dd469bdcc150f814c74d6e6d58e680d9": "A \\in \\C", "dd46bbaf8a52f7c97fc4f61d038126fc": " PV \\ = \\ \\frac{FV}{(1+i)^n} ", "dd47196c050933c8703e86a583e33d69": "0 \\leq \\beta \\leq 1", "dd47957301a45530b7eaf3c393bc93d7": "g\\circ_T f = \\mu_Z \\circ Tg \\circ f", "dd47f359a23075cb4f872ed414b46780": "w=\\sigma", "dd4822c946562eab8c870acf5859389b": "u*\\chi_r = u*\\chi_s\\;", "dd482f3534e64ddfc0dc80bb552a6e31": "(|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\\ = (a_1+a_2, a_2+a_3, ..., a_n + a_1)\\ mod 2", "dd488166f3fa162e17366da8ce829d7d": "\\!\\lnot (\\lnot \\phi \\vee \\lnot \\psi)", "dd489a03cba998993be3cbbe4aa428b5": "\\mathbf{V}_i = \\vec{\\omega}\\times(\\mathbf{R}_i-\\mathbf{R}) + \\mathbf{V},", "dd48bcb6747c4c8cc2763939f700ee69": "\\tau \n= \\int \\sqrt {\\left (\\frac{dt}{d\\lambda}\\right)^2 - \\frac{1}{c^2} \\left [ \\left (\\frac{dx}{d\\lambda}\\right)^2 + \\left (\\frac{dy}{d\\lambda}\\right)^2 + \\left ( \\frac{dz}{d\\lambda}\\right)^2 \\right] } \\,d\\lambda.", "dd48d152de32b6ca4cd0a83d5559d692": "(y^\\mu)^-(y_\\mu)^-=0,\\;", "dd48f120fcafecb9906029dda1b8cd5f": "\\overline{x}=\\frac{\\sum_{i=1}^{m}{x_i}}{m}", "dd48f438cbd65e7f0195e7db31239f5f": "\\ddot{a}_{\\overline{n|}i}", "dd49184265aed7bec35c3249c36621e5": "\\displaystyle c", "dd492764809e68c1a59fce1cd99ada7b": " d = d - (d/b)b = d - d = 0. \\ ", "dd4951c6083024432d1577eadf7ca1b4": "53\\,", "dd57e8af4e90ed772863c9553f9199b8": "ed - k \\varphi (N) = 1", "dd57e8e04b7471231e201c8175b60aef": "\\forall a\\in\\varnothing : a \\le x", "dd57eae7d88039c0489addd5c3c7188a": "{n\\choose0}_2=\\sum_{k=0}^n\\frac{n(n-1)\\cdots(n-2k+1)}{(k!)^2}=\\sum_{k=0}^n{n\\choose 2k}{2k\\choose k}.", "dd5861d5ec8eeda96ff2d22f73d093d1": "x = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{a_3}}}", "dd5863de7f9a91a9937fc16a6fd056b4": "\\begin{align}\n \\Gamma^m{}_{ij} &= g^{mk}\\Gamma_{kij}\\\\\n &=\\frac12\\, g^{mk} \\left(\n \\frac{\\partial}{\\partial x^j} g_{ki}\n +\\frac{\\partial}{\\partial x^i} g_{kj}\n -\\frac{\\partial}{\\partial x^k} g_{ij}\n \\right)\n =\\frac12\\, g^{mk} \\left( g_{ki,j} + g_{kj,i} - g_{ij,k} \\right) \\,.\n \\end{align}\n", "dd58cec3cd2d5426624333d521aaa8d3": " \\displaystyle L_p(1-n, \\chi) = (1-\\chi\\omega^{-n}(p)p^{n-1})L(1-n, \\chi\\omega^{-n})", "dd59084fd3d8df43f42791360f94303c": "a^{\\rm Ins}_{s'} (s_1, s_2, s_3, \\ldots s_T) = (s_1, s_2, s_3, \\ldots, s', \\ldots s_T) ", "dd5936a009c556241682549517979aac": "\\displaystyle \n\\begin{align}\n\\frac{\\partial \\eta }{\\partial t} + \\frac{\\partial (\\eta u)}{\\partial x} + \\frac{\\partial (\\eta v)}{\\partial y} = 0\\\\[3pt]\n\\frac{\\partial (\\eta u)}{\\partial t}+ \\frac{\\partial}{\\partial x}\\left( \\eta u^2 + \\frac{1}{2}g \\eta^2 \\right) + \\frac{\\partial (\\eta u v)}{\\partial y} = 0\\\\[3pt]\n\\frac{\\partial (\\eta v)}{\\partial t} + \\frac{\\partial (\\eta uv)}{\\partial x} + \\frac{\\partial}{\\partial y}\\left(\\eta v^2 + \\frac{1}{2}g \\eta ^2\\right) = 0\n\\end{align}\n", "dd598f7c6ede130d58f2daa2c4438bc2": "E_f = \\frac{L^3 m}{4 b d^3}", "dd59abbe12dfa547e1cde4790130e4b1": "\\psi(x) \\rightarrow \\psi(x+y) \\Rightarrow P(x,p) \\rightarrow P(x+y,p)", "dd59eda8fcad5fac99d572dc40d3462a": "\\phi : \\hat{\\mathbf Z} \\to \\operatorname{Gal}(K_s/K),", "dd5a0cc11cd1739f86f5ff055b7db24c": "1 =\\sum_{k=1}^m (-1)^{k-1}\\sum_{\\scriptstyle I\\subset\\{1,\\ldots,m\\}\\atop\\scriptstyle|I|=k} 1.", "dd5a1da085dcfe9df7b088d41b2be794": "\n \\begin{align}\n \\sigma_{xx}^{\\mathrm{f}} & \\approx \\cfrac{z M_x}{\\frac{2}{3}f^3 +2fh(f+h)} ~;~~ &\n \\sigma_{xx}^{\\mathrm{c}} & \\approx 0 \\\\\n \\tau_{xz}^{\\mathrm{f}} & \\approx \\cfrac{Q_x}{\\frac{4}{3}f^3+4fh(f+h)}\\left[(h+f)^2-z^2\\right] ~;~~ &\n \\tau_{xz}^{\\mathrm{c}} & \\approx \\cfrac{Q_x(f+2h)}{\\frac{2}{3}f^2+h(f+h)}\n \\end{align}\n", "dd5a1de8496ed08c90e398204971071a": "K(k) = \\frac {\\pi /2}{\\mathrm{agm}(1-k,1+k)}.", "dd5a2be558a966717e942aa00fc8a7d0": "\\scriptstyle t = 2\\sqrt{2} ", "dd5a5d475641d78fc9cd12f31e5acad4": "R = \\lim\\sup \\left|\\frac{a_{n+1}}{a_n}\\right|", "dd5a8a6a0e866b8b1889c956c390e69a": "\\frac{f(t)}{1-F(t)} = p + \\frac{q}{m} N(t)", "dd5aa2ceaaffef5ceaaccdedcead7fbf": "\\scriptstyle e", "dd5aceeb6d46645e59152ee6ae8455f7": "\\sum_{i=1}^k \\frac {n_i} {N_i} = C ", "dd5b0d52ece7c2bebe65808a8ee116f4": "k\\neq 0", "dd5b2764daf378a7c26a63a791335378": "\\sigma_{\\mathrm{max}}", "dd5b4b2ff230b6a5840ec6ec431a94df": "\\mathbf{F}_{ext}", "dd5bc7961a02947bf449dfcb2150c3aa": " \\chi_U(g) = \\frac{1}{|G_0|} \\sum_{\\{ x \\in G : {x}^{-1} \\,\ng \\, x \\in G_0\\}} \\chi_V({x}^{-1} \\ g \\ x), \\quad \\forall g \\in G. ", "dd5bd653409c7f375be1c7d3a96a5363": "\\delta \\not\\in C(\\alpha)", "dd5be4cbba6fd29d5a9e05598622c36c": "\\mathbf J = 0", "dd5bfe3d716615e8182b7ac447f573ed": "A\\cap \\mathrm{dom} \\, f", "dd5c20f7d7c1674480c6b2181c16542a": "\\mathrm{MTF}(\\nu)\\equiv 1", "dd5c3d95c496a0c923f99614f0fa723a": "\\, \\frac{e^{ait} - e^{(b+1)it}}{(b-a+1)(1-e^{it})}", "dd5c93666794c3abfacaebc4f1626b8c": "\\frac{1}{1-\\delta} \\epsilon.", "dd5caf372b2aff72d471f6ea1500a5ca": "0.50778218\\ldots", "dd5d2532ef1e98b45ceb2bae1cb2ba51": "\\eta^{\\alpha \\beta} \\,", "dd5d28e25c86bf980829b0f50e262fe6": " x=\\epsilon/d_{\\mathrm{F}} ", "dd5d48954b1cf3e87c6adc39fca05b8b": "X^1 = X", "dd5d5852d245b5a331a0f0a7e6fd8d53": "C_{\\text{CB}} (1-A_{\\text{v}})\\,", "dd5da0f19b148435c1f7bacffb0e7b71": "\\! J", "dd5dada12d48780200229047b458f6db": "\\operatorname{erf}(x) = \\frac{2}{\\sqrt{\\pi}}\\int_{0}^x e^{-t^2}\\,\\mathrm dt.", "dd5df5ce0b8fa777ea7fc4173262c04b": "^{(h)}", "dd5e46bc13b58e815086b05bec5fe0e1": "\\ell_1=\\frac{x-x_0}{x_1-x_0}\\cdot\\frac{x-x_2}{x_1-x_2}=\\frac{x-2}{4-2}\\cdot\\frac{x-5}{4-5}=-\\frac{1}{2}x^2+\\frac{7}{2}x-5\\,\\!", "dd5e4f06a16b0fd1eb7ec378c163604d": "L(\\epsilon)\\sim F\\epsilon^{1-D}\\,", "dd5e6f1a8dd114232b07126a24819c17": "\\displaystyle{h(z)=(az+b)(cz+d)^{-1}.}", "dd5eb99f781dc0c66ab5b7be03e9af5d": "a \\div 0", "dd5ec9f18a8d1fd02ddb9a47b9e528e4": "=\\frac {5-\\sqrt 5}{2} \\ .", "dd5ed7b6a47683e32c15799dcc0936d4": "\n (P_{0, \\epsilon}) \\quad \\min \\limits _x \\|x\\|_0 \\qquad \\text{subject to } \\|y - Dx\\|_2 \\le \\epsilon\n", "dd5eee01b81d721299619638e0e86a22": "K_k=K_ef \\sum_{i=1}^N K_iG_i", "dd5f52faf083b41d46a9922e045b56c5": "N=N_A n", "dd5f9d3cb057f9917b441f952f959917": "\\tfrac{X}{X+Y} \\sim \\Beta(\\alpha, \\beta)\\,", "dd5fa273f0717b5daf556008d19e350b": "v_i \\in B", "dd5fb9a35e8a9e561cd92e62be7458b4": "\\widehat{\\sigma}_\\pi^{2}=\\widehat{\\sigma}_m^{2}-K. ", "dd6003efce4139f78c9d60245c0b9af5": "I_{src}^{dst}: src \\rightarrow dst", "dd6015a922e7b23ce786ea90ea7caf79": "\n= {1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega \\over (1\\,\\mathrm{k}\\Omega + 1\\,\\mathrm{k}\\Omega) + 2\\,\\mathrm{k}\\Omega} \\cdot 15 \\,\\mathrm{V}\n", "dd6089a76e4b735b6b599416baf15ae8": " s_{(2,+)}=+1,~~~ s_{(2,-)}=-1", "dd60fec997be0fb36f651d2a832215b3": "a-a=0", "dd6111a98b0419264182d90952d00952": "\n\\Delta f = \\frac{\\partial^2 f}{\\partial x^2} + \\frac{\\partial^2 f}{\\partial y^2} + \\frac{\\partial^2 f}{\\partial z^2}.\n", "dd614ab60f6d3a5e674a6f7fc4ad6ba2": "\\frac{1}{\\Gamma\\left(\\frac{k}{2}\\right)}\\;\\gamma\\left(\\frac{k}{2},\\,\\frac{x}{2}\\right)", "dd61928c8e9fbe7b4f7d399920728949": "\\,Q\\left(x;\\;\\Gamma(x),\\;\\Gamma'(x),\\;\\ldots\\;,\\;\\Gamma^{(n)}(x)\\right)=0\\,", "dd61a8eebc21d9cacb18cf40a3486ad4": "\\mathrm P(S,R|do(G=T)) = P(S|R) P(R)", "dd61e2995a002f6396a35bc6d6abff6b": "\\vec{X} \\approx \\partial_t - q \\tan(q u/\\sqrt{2}) \\, \\left( x \\partial_x + y \\partial_y \\right) ", "dd621b52f0676a718be66003f0ba4bca": "\\scriptstyle \\| z \\| \\;=\\; 1,", "dd62602c9d90a25fc85e0df88c0220f3": " T_{o}\\, ", "dd62a4f58c909c19753f284f018d02bb": "L'= L", "dd62cd5d10c4d8bc9aa2b6d35d6845e7": "f^{high} = \\ ", "dd6305d74898cf8dd4ac240168d6f967": "\\Delta L=\\alpha_L L \\Delta T", "dd63264b62cee4bb049a087090ed8646": " \\mathbf{N} = {\\frac{d\\mathbf{T}}{ds} \\over \\left\\| \\frac{d\\mathbf{T}}{ds} \\right\\|}. \\qquad \\qquad (2) ", "dd636dcee6a9b21b4a8772807da3c245": "((a^{-1} \\cdot a) \\cdot (b \\cdot b^{-1}))^{-1} \\stackrel{R2}{\\rightsquigarrow} (1 \\cdot (b \\cdot b^{-1}))^{-1} \\stackrel{R13}{\\rightsquigarrow} (1 \\cdot 1)^{-1} \\stackrel{R1}{\\rightsquigarrow} 1 ^{-1} \\stackrel{R8}{\\rightsquigarrow} 1", "dd63b1a2871b1a1c082b4995c5554238": "\\cos(m\\phi)", "dd63c73c56feafec2c3c8b9f9b4ef500": "q_2 = q_1 + \\frac{t_2 - t_1}{t_1 - t_0} \\left( q_1 - q_0 \\right) - \\frac{\\left( t_2 - t_0 \\right) \\left( t_2 - t_1 \\right)}{2m} \\frac{d}{dq_1}V\\left(q_1\\right)", "dd64369bc0daca3ae2ccf8fc458b2493": "X^{\\vartriangle\\triangledown}", "dd643e516e8474460e008d5dd19929ba": "(AB)_{jk} = \\sum_i A_{ji} B_{ik}.\\,", "dd645b70ad204847ed70b554bfa5550f": " \\frac{F}{S} = \\frac{mv^2}{rS} < \\frac{mc_s^2}{rS} \\approx \\frac{mG}{rS \\rho} \\approx G ", "dd652b4595ad592f2ab7b9673ac1634a": " \\mathcal{P}_b (a) = (a \\cdot b^{-1})b ", "dd654cd17e5a407a5707965d04a23163": " \\alpha= \\frac{1}{1+k \\cdot (1+j)} \\,\\ ", "dd65bd550d13832ba2aad11fc9e606de": "\\rho_\\text{e}", "dd669e9d15e7af7daa5b4c2f3c4040be": "x_1,x_2,\\ldots ,x_N", "dd6702325871e0dcf3191d24e3b6a487": "f(x,d)\\,\\!", "dd676d617efcc09c65b9af7aba1f9dfa": "\\displaystyle \\gamma y", "dd67995d7a221efaeda88c425098e264": " \\theta^{*} ", "dd67b2032bb13a05c4a9db3ea7def4e2": "\n\\psi(\\mathbf{x} ,\\tau) = \\varphi_\\alpha(\\mathbf{x} ) \\psi_\\alpha(\\tau),\n", "dd67bdbaa381c06b072a76134aeaa268": "\\psi_C(a)=\\psi_R(a)", "dd6824ed63a6dde2ae7259aea1e5f8c8": "W=\\int_{1}^{2} F\\,ds.", "dd682e88ee323c935658065f42d5dda0": " \\mathbf{v}(t) = v \\mathbf{u}_\\mathrm{t}(s)\\ ; ", "dd6857f181cea5535f89a0e5d7085c40": "\\bar{g}(-y). \\, ", "dd68b2e09439270321aa5ca8a16e82d3": "\\textrm{mes} E_\\lambda\\leq C \\,\\, \\textrm{mes} J\\left(\n\\frac{\\lambda e^{\\max_k |\\Re \\lambda_k| \\, \\mathrm{mes} J}}{\\max_{x \\in J} |p(x)|}\\right)^{1/(n-1)}\n", "dd68cfc0a3feebfb753301511a7fb706": "S_2=C_2", "dd68d69d3fd1959370ea5f3a3144f429": " I(n, \\Lambda )= \\int_0^{\\Lambda }dp\\,p^n \\sim 1+2^n+3^n+\\cdots+ \\Lambda^n \\rightarrow \\zeta(-n)", "dd68e53a5b6c594e58b286dd69e4778b": " \\mathbf{E}(\\mathbf{r}) = -\\nabla V(\\mathbf{r}).", "dd68ef8ed826f33747b675c8541cae3e": "M_2(x_1,\\dots,x_n) = \\sqrt{\\frac{x_1^2 + \\dots + x_n^2}{n}}", "dd68fba3b827cc0fce6c388ee196515b": "\\frac {g_l} {\\sqrt{2}}=(-1)^l \\frac {A_{|2l-3|}/2} {ce_{\\nu}(0,q)}", "dd6948293b16e46508bb50b30c24f977": "0\\leq [f]", "dd6971775003f0ddb843798ac3119d88": "\\min(A, i,j)", "dd699d305a3d06dea90290c89fbc8f02": " \\hat{A}^* = (A^*, B^*) = A^* + \\epsilon B^*. \\!", "dd69d896eeab196af76a39bbfb4dc7c4": "\\mathbb{E} \\left[ \\ln(1+r) + \\sum\\limits_{k=1}^{n} \\frac{u_k(r_k - r)}{1+r} -\n\\frac{1}{2}\\sum\\limits_{k=1}^{n}\\sum\\limits_{j=1}^{n} u_k u_j \\frac{(r_k\n-r)(r_j - r)}{(1+r)^2} \\right]", "dd6a10ed76e00c63428e1e489cdbcb9d": "\\chi^2(0.05) \\le \\frac{(d.f.)s^2}{\\sigma^2} \\le \\chi^2(0.95)", "dd6a1f0395a008c270facde60b527698": "B\\subseteq X", "dd6a2bbf451fe92983342dd1d8fc3a65": " \\lim_{n\\to\\infty}u_n(z)", "dd6a5c7cf8f7e28c15b7fda57d07f6ba": " \\frac{\\rho DRT}{2} = \\rho\\epsilon = \\int\\ g \\, d\\vec{e} ", "dd6a9b636022ac88f5508c3938116065": "h^0(\\mathcal{O}(D)) - h^1(\\mathcal{O}(D)) = c_1(\\mathcal{O}(D)) +c_1(T(X))/2\\ \\ \\ ", "dd6ab6999652f2a61849ef59dd43a6be": "\\int_{E}\\phi\\, d\\mu=\\infty", "dd6aca07510ad5949cb7708a5d20bbd6": "\n l_k := \\max (l_{k - 1}, f(\\mathbf{x}_k) + (\\mathbf{s}_k - \\mathbf{x})^T \\nabla f(\\mathbf{x}_k))\n", "dd6acf5a53a7b33454e75754b02228c6": " {V} = {E}_0 - {iR}_\\omega - {\\eta}_{cathode} - {\\eta}_{anode} ", "dd6aeb6b892f5664def6de206d0a2e67": "m_{body} = C \\cdot V \\qquad (3)", "dd6b949f5fb0f7ba791e5aa687249b76": "C = p_{\\theta}^{2} + \\left(\\frac{L}{\\sin\\theta}\\right)^{2}", "dd6baf9846321468b49a445d0dd923b7": "\\forall \\alpha \\in \\mathbb{F} : \\vert \\alpha \\vert \\ge r \\Rightarrow x \\in \\alpha S", "dd6bcbe720590ffa2497e6d39a6a0607": "\\chi_{T}(G)\\leq k =\\chi(G)", "dd6c1a443b717b02c5b6d7c02cffc7fe": "i\\in \\{1...n\\}", "dd6c2df1257423b7a142c143ce867f77": "h(f_i) = g_i", "dd6c441526b2b994c335b9697bf5b1c0": "\\lim_{n \\to \\infty} \\prod_{n=1}^n N_k(n)= \\frac {k^n} {n!} 1(1+X)(1+X+X^2)\\cdots(1+X+X^2+\\cdots+X^{n-1}),", "dd6c4fc53f21cdfddedd9373fec32910": "M_A l + 2 M_B (l+l') +M_C l' = \\frac{6 a_1 x_1}{l} + \\frac{6 a_2 x_2}{l'}", "dd6c5d828c044361ce6f921576dada52": "\\begin{pmatrix} 1 \\\\ -1 \\\\ 0 \\\\ 0\\end{pmatrix}", "dd6c8ba9031125d60e555259bc3e1710": "L(\\lambda, \\alpha, s) = \\sum_{n=0}^\\infty\n\\frac { \\exp (2\\pi i\\lambda n)} {(n+\\alpha)^s}.", "dd6ca8d28ac8d29d5bd1a3181ac4ecc6": "\\mbox{THD}_i", "dd6cc46535caf65ba3f8940e6ec690bf": "\n\\begin{array}{lcl}\n\\left [ (x_i^2 - 2 x_i + 1) + (y_i^2 + 2 y_i + 1) - r^2 \\right ]^2 & < & \\left [ x_i^2 + (y_i^2 + 2 y_i + 1) - r^2 \\right ]^2 \\\\\n\\left [ (x_i^2 + y_i^2 - r^2 + 2 y_i + 1) + (1 - 2 x_i) \\right ]^2 & < & \\left [ x_i^2 + y_i^2 - r^2 + 2 y_i + 1 \\right ]^2 \\\\\n\\left ( x_i^2 + y_i^2 - r^2 + 2 y_i + 1 \\right )^2 + 2 (1 - 2 x_i) (x_i^2 + y_i^2 - r^2 + 2 y_i + 1) + (1 - 2 x_i)^2 & < & \\left [ x_i^2 + y_i^2 - r^2 + 2 y_i + 1 \\right ]^2 \\\\\n2 (1 - 2 x_i) (x_i^2 + y_i^2 - r^2 + 2 y_i + 1) + (1 - 2 x_i)^2 & < & 0 \\\\\n\\end{array}\n", "dd6cf8a0bc6e1cb3422ef65848cedb2f": "z \\in \\mathbb{\\hat{C}}\\setminus K_c", "dd6d06ec3546e422e5410ea6a19c7d0a": "X_1,\\dots,X_k.", "dd6d66d3c79c98f524e001082a174b98": "\\sum _{i = 1} ^{nm} V_i ( \\cdot ) V_i ^*", "dd6db066be4017893e9cb8bb4cab01b1": "\\scriptstyle{E=mc^2}", "dd6dc0757bb7d3d27d57b33512954e5f": "\\sqrt{3} = 2 \\sin 60^\\circ = 2 \\cos 30^\\circ", "dd6de4ee00b070d47471c59e88c73c78": "\n \\cfrac{\\Gamma \\vdash \\Delta}{\\Gamma, A \\vdash \\Delta} \\quad (\\mathit{WL})\n ", "dd6e16175f36615dd0e7425123571c24": " C_{(+)} ", "dd6e636b28f0f9cdbb458a0524508b1e": " \\boldsymbol{\\omega}_\\mathbf{T} = {1\\over 2} \\mathbf{T} \\times \\mathbf{T'} = {1\\over 2}\\kappa \\mathbf{T} \\times \\mathbf{N} = {1\\over 2}\\kappa \\mathbf{B} ", "dd6e992053f654bc734646263719e9c7": "T = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix}", "dd6effaa5db1942916d9ed6e04930adc": "\\mathbf{e}_2 = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix}", "dd6f3d0ea7dac9092ce94284c79b6920": " u_p ", "dd6f5c5ffb17cc62c02276b4c089bb87": "E = {E_s \\over Z} = { 1 \\over Z } \\int_{-\\infty}^{\\infty}{|x(t)|^2}dt ", "dd70bd39b1628da34f4a878eefe37529": "\\int d^2 \\theta\\; \\lambda_3\\; L E^cL ", "dd70e682b4e349efa49af34e9eccc7ff": "p_\\mathrm{v}", "dd713ed1faf135e5b454816065dc93b0": "\\mathbf{r}_1", "dd714dc3e3ec2cb582e2cb8482b49c5c": "{n_\\mathrm{D}}^\\mathrm{rand\\_degree}", "dd717393c9ea00e82579582d4835e0ad": "F_{A0} - F_A + \\int_{0}^{V} v\\, dV = \\frac{dN_A}{dt}", "dd717840c53427c33cb3d17043e4fc56": " CV=w-e(p_1,u_0)", "dd71c3e852e6b410eeb321896476861b": "\n\\lim_{x \\to 0^+} x^x\n= \\lim_{x \\to 0^+} e^{\\ln x^x}\n= \\lim_{x \\to 0^+} e^{x \\ln x}\n= e^{\\lim_{x \\to 0^+} (x \\ln x)}.\n", "dd71e99347939a75348d1e363db8380e": "\n|\\psi\\rangle=\\sum_i\\sqrt{\\lambda_i}|i_A\\rangle\\otimes|i_B\\rangle\n", "dd71ee89a3d64e9a2cbcba47b63e9f48": "\n\\lim\\inf\\frac{\\varphi(n)}{n}= 0.\n", "dd721d29245cf97fbccba1fc389b9b22": "P(g)= \\int dk P(k) \\delta (g-k^\\eta)", "dd7240341640b2a5b62c444dd4ea99f3": " = \\int_{-\\infty}^t G(t-\\tau)F(\\tau)\\, d\\tau ", "dd7240a6796cd6aafa7c49a1e5a810a5": "\\lim_{N\\to\\infty} \\frac{2\\pi\\log |\\langle K\\rangle_N|}{N} = \\operatorname{vol}(K), \\, ", "dd72948c30bf03285d2cc82ef37bda8e": "\\langle R(u,v)w,z \\rangle=-\\langle R(u,v)z,w \\rangle^{}_{}", "dd729b27b080de2418f12dea4c6107c2": " S = L ^ {-2}", "dd72a423821aa7c9c7ebdf4b23f63b51": " C_P", "dd72ba7f9c3429409d3e39615b4b4488": "\\frac{8}{9} \\sqrt[4]{8}", "dd72e2d14fc97813d092aaffb5c8489f": "E_1 = E_2 = E_3", "dd730f75694986c5b43d2684c59934da": " P(\\omega) = \\frac{24 \\pi k_BT\\eta R}{36\\pi^2 R^2 \\eta^2 \\omega^2 + \\left(\\frac{F}{l}\\right)^2} ", "dd734242efb5a75bd0b0c6917781a169": "\\mathcal{B}_r = ( \\lfloor n(r+1) \\rfloor)_{n\\geq 1}", "dd73f73fccd575c3702becdd8bf9469b": "*,\\star", "dd73fc4381f5a744d97909f4ceba6821": "V\\subset E\\,", "dd73ff0d883be42b94142c8acc768c83": "\\vec{b}\\,\\!", "dd7413bea8d39bff95ca373d25c686ef": "\\begin{align}\nL_{f,P} &=\\sum_{k = 1}^{n}(x_{k} - x_{k-1})\\inf_{x \\in [x_{k-1},x_{k}]}f = 0\\\\\nU_{f,P} &=\\sum_{k = 1}^{n}(x_{k} - x_{k-1}) \\sup_{x \\in [x_{k-1},x_{k}]}f = 1\n\\end{align}", "dd74264e6e69240cba0712f4f9b73856": "R_{c,\\theta}(p) = R_{0,\\theta}p + v.\\,\\!", "dd7432ea0b8649dbe77b04c0e48ab174": "X_n = S^n \\wedge X", "dd74438850068bf0c6d679600d56aca3": "\\left\\lfloor \\dfrac{i - 1}{2} \\right\\rfloor", "dd74cdc7c6823778984c828ac44d8fbf": "\\varepsilon_o", "dd7536794b63bf90eccfd37f9b147d7f": "I", "dd75a6d9ab48adc36ef2494179cb0884": "|y|<\\left |\\frac{a}{d}\\right |.", "dd75ea512109323e1128cb61783773cd": "_0(x)", "dd75f32fde35ed398142320d0c77fbf8": " L_t=q S_t\\left(\\frac{\\partial C_l}{\\partial \\alpha}\\left(\\alpha-\\frac{\\partial \\epsilon}{\\partial \\alpha}\\alpha\\right)+\\frac{\\partial C_l}{\\partial \\eta}\\eta\\right)", "dd763ebc58b1ce2ce83808890d895cae": "DBH", "dd7645c8394442512acc069ad552f795": "\\displaystyle{\\|\\mu\\|_\\infty < 1, \\,\\,\\, \\mu(z)=F_{\\overline{z}}/F_z.}", "dd76ecdfaef5d98fb79f1982cede9f2b": "\\mathrm{E}(e^2) = R_s(0) - 2\\int\\limits_{-\\infty}^{\\infty}{g(\\tau)R_{xs}(\\tau + \\alpha)\\,d\\tau} + \\iint\\limits^{[\\infty, \\infty]}_{[-\\infty, -\\infty]}{g(\\tau)g(\\theta)R_x(\\tau - \\theta)\\,d\\tau\\,d\\theta},", "dd77105f1487c4aeda38e4e20edf830e": "\n\\begin{align}\n\\left.\\frac{\\partial^{k+2}}{\\partial t^{k+2}}(\\rho(\\mathbf r,t)-\\rho'(\\mathbf r,t))\\right|_{t=t_0}&=-\\nabla\\cdot\\left.\\frac{\\partial^{k+1}}{\\partial t^{k+1}}\\big(\\mathbf j(\\mathbf r,t)-\\mathbf j'(\\mathbf r,t)\\big)\\right|_{t=t_0},\\\\\n&=-\\nabla\\cdot[\\rho(\\mathbf r,t_0)\\nabla\\left.\\frac{\\partial^k}{\\partial t^k}\\big(v(\\mathbf{r},t_0)-v'(\\mathbf{r},t_0)\\big)\\right|_{t=t_0}],\\\\\n&=-\\nabla\\cdot[\\rho(\\mathbf r,t_0)\\nabla u_k(\\mathbf r)].\n\\end{align}\n", "dd7739b3930e38613b097e92fe279e85": "W^{12}=Sp(3)/Sp(1)^3", "dd774ad345237c6c9e07ce0dd3e67eaf": "\nK_{d} = \\frac{[A]^x \\times [B]^y}{[A_x B_y]}\n", "dd77731d26eee36705a3ef8afc40b50d": "\\begin{matrix}\n {} & \\left[ \\text{5},12,13 \\right] & {} \\\\\n A & B & C \\\\\n \\left[ 45,28,53 \\right] & \\left[ \\text{55,48,73} \\right] & \\left[ \\text{7,24,25} \\right]\n\\end{matrix}\\quad \\quad \\quad \\quad \\quad \\quad \\begin{matrix}\n {} & \\left[ \\text{5},12,13 \\right] & {} \\\\\n {{A}'} & {{B}'} & {{C}'} \\\\\n \\left[ 9,40,41 \\right] & \\left[ \\text{35,12,37} \\right] & \\left[ \\text{11,60,61} \\right]\n\\end{matrix}", "dd780dd964a10b23cf7526bd8477765d": "\\hat{\\mathbf k}", "dd78837131b4fa3265fd75a6bc882537": "C^k(X,Y) \\subset \\mbox{Hom}(X,Y)\\,", "dd78b16516c90be546c30d74a6e6b440": "\\displaystyle \\partial_t u + \\partial_x (\\partial_x^2 u - 2\\, \\eta\\, u^3 - 3\\, u\\, (\\partial_x u)^2/2(\\eta+u^2)) = 0", "dd794f3e4d1d642ec7fe41e200491206": "\\pi_2: G \\times H \\to H\\quad \\text{by} \\quad \\pi_2(g, h) = h", "dd798983954d66eb4c96494cc14083d6": "\\exist t\\in \\mathbb{R}\\quad \\forall d\\in D\\quad ||td|| \\geq \\frac{1}{k},", "dd79af4cd59225aba7de2fbf20a24f4e": "\\begin{align}\n(2x-3)(3x-4) &= (2x)(3x)+(2x)(-4)+(-3)(3x)+(-3)(-4) \\\\\n&= 6x^2 - 8x - 9x + 12 \\\\\n&= 6x^2 - 17x + 12\n\\end{align}", "dd79d9ea161e43a45e2f701504c2af04": "\\delta W = \\boldsymbol{\\mathcal{Q}}\\cdot\\delta\\mathbf{q} = 0 \\,,", "dd79da6268d282087b018c51994cdb23": "\\rho_{t}(x)=\\frac{t\\rho(x)}{\\left (\\tfrac{1}{2}(t-1)(x-c_1)\\varphi(x)-t\\right )^2+\\pi^2\\rho^2(x)(t-1)^2(x-c_1)^2},", "dd79dcbb9c7ec7dcde034158cbbe9968": " \\mathbf{x} \\in \\mathfrak{g}^{*} ", "dd7a2810c48195a7d46645eff882b94e": "Y = c + ax_1 + bx_2 + \\text{error}\\,", "dd7a4c65a1447497bc97248ad3f1565a": "\\langle A_1,B_1,\\ldots,A_n,B_n|A_1B_1A_1^{-1}B_1^{-1}\\ldots A_nB_nA_n^{-1}B_n^{-1}\\rangle.", "dd7a6829dcd03187f82f2756b9f18f4f": " \\aleph_0", "dd7a787a5f60357813be32e5424109dc": "V_0(P,Q)=2, \\,", "dd7aa9903789433c16b8cbf70c3b84e0": "\\mathbf{q}_i, \\mathbf{p}_i", "dd7ad2118bf580300d19992366bbb65a": "\\lim_{\\delta x \\to 0}({Z_0}^2 - Z_0 \\delta Z) = {Z_0}^2 = \\frac {dZ}{dY}", "dd7aec87c7708fe00f4168f47f6783e4": "\\Phi_{00}=0", "dd7b635f9fd9eea00a4455a3a36d41a4": "x \\preceq y ", "dd7ba11dcaa6701c7d0884c79da619f8": " \\left( \\limsup_{n \\to \\infty} \\frac{|a_n|}{\\alpha^n} \\right) \\leq \\kappa \\, \\left( \\limsup_{n \\to \\infty} \\frac{|b_n|}{\\beta^n} \\right)^{\\frac{1}{2}} \\left( \\limsup_{n \\to \\infty} \\frac{|c_n|}{\\beta^n} \\right)^{\\frac{1}{2}}.", "dd7c0ddf9fcbc34ac11fda893b6569f7": "\nCoh_{ijk} = \\exp^{-2C\\Delta r_{ij} \\omega_k/(u_i + u_j)}\n", "dd7c6ceff7feb998ba4b5e58fd958230": " z = \\frac { \\, | p_o - p_e | - \\frac{1}{2n} \\, } { s_i } ", "dd7c97d33c890185922296777d57ef01": " k_{||} ", "dd7c990d1f1abf3d6d0cbffa3e14d6fa": "\\left\\{\\begin{array}{c}\\exp(a D_T) = 1 + a D_T\\\\\n\\exp(a D_V) = 1 + a D_V\n\\end{array}\\right.. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (8)\n", "dd7cdef15912974be755d600f8271a9e": "(X, \\mathcal{A}, \\mu, T)", "dd7d0bb6f6be24edd994cda2e1ed3f23": "(a_{11} x_1 p_1 + a_{21} w x_1 p_2) (1+r) = x_1 p_1", "dd7d2107d8ed81402a1b3d6f372deaa8": "\\mathbf C=\\mathbf F^T\\mathbf F=\\mathbf U^2 \\qquad \\text{or} \\qquad C_{IJ}=F_{kI}~F_{kJ} = \\frac {\\partial x_k} {\\partial X_I} \\frac {\\partial x_k} {\\partial X_J}.\\,\\!", "dd7d636f9be64147dc5df46b0550480d": "d(k) = x(k-i-1)\\,\\!", "dd7dc1cbec069d0fcdf0dd9cfc9315bc": " (\\alpha,\\beta)=d \\Psi(\\alpha,\\beta). \\,", "dd7e1f276818c882960abb0835663566": "A = \\frac{1}{2} \\log(u^2+v^2) - \\frac{u}{u^2+v^2}", "dd7e9546a09dd021722d8d1caa14fa0d": "\\int\\frac{x\\;dx}{s^5} = -\\frac{1}{3s^3}", "dd7ee34d997c4fccdeb93a2db931da79": "\\left[\\frac{\\alpha}{\\beta}\\right]\\left[\\frac{\\beta}{\\alpha}\\right]^{-1}=\n(-1)^{\\frac{bd}{4}}", "dd7ee4ec03bc4b3093133b42c3bfde14": "\\neg \\textit{on}(0)", "dd7eeec8486c4f4748d315fd799aa5db": "C \\leftarrow \\alpha A B + \\beta C \\!", "dd7f029eda4414558bb89c60bf7f9ac3": "\n\\left(\\frac{x^2}{p}\\right) =\n\\begin{cases} 1&\\mbox{if }p\\nmid x\\\\0&\\mbox{if }p\\mid x.\n\\end{cases}\n", "dd7f3944498235f1181ed1bd54c8bf09": "s_D^2", "dd7f44be2cbe877f2ee505f6d9f2151e": "2:2\\ ", "dd7f9886a2dec9bd1f4eb78d0ab728eb": "\\displaystyle -\\frac{1}{2} \\frac{1}{\\left| \\xi \\right|} - \\gamma \\delta \\left( \\xi \\right) ", "dd7fbcac20f579c81f89a08aaae6c8f1": "\\ |a^\\mathrm{H}b|^2 \\leq (a^\\mathrm{H}a)(b^\\mathrm{H}b),\\, ", "dd80c39eceb31bbe17740b67eb9862b0": "f(x_1,\\ldots,x_k) = n\\,", "dd80df869afecf12ff0c2198eda5e704": "\\hat{L}", "dd80e6f8c8206c9e6e59c772195b75b8": "[H,R]=0", "dd80f4da61c27e9b3fec7ce93ed06f1d": "\nG(\\boldsymbol{x} - \\boldsymbol{r}) = \\begin{cases}\n\\frac{1}{\\Delta}, & \\text{if} \\left| \\boldsymbol{x} - \\boldsymbol{r} \\right| \\leq \\frac{ \\Delta }{ 2 }, \\\\\n0, & \\text{otherwise}.\n\\end{cases}\n", "dd812d6500e649e586bcd1947c47c5f6": "\\rho=\\frac{1}{2}(\\zeta_1\\wedge\\zeta_2\\wedge\\zeta_3+\\bar{\\zeta_1}\\wedge\\bar{\\zeta_2}\\wedge\\bar{\\zeta_3})", "dd81321a923e304ea5d5bf80cfd594dd": "V = \\|v_1 \\wedge \\cdots \\wedge v_n\\|.", "dd817bb9914072aee708b457ad6e1955": "{P}^{6}-6{P}^{4}Q+9{P}^{2}{Q}^{2}-2{Q}^{3}\\, ", "dd819e14c3d8e372416250a97abdd879": " H_2: \\quad (x-1)(y-1)=2,", "dd81b9437dff23a8fb2e196e11e50ac5": "P \\not\\in \\operatorname{Ass}(Q)", "dd82160e1866231e2edb75d4b24adba7": "\\dot{m} = \\rho \\bar{u} A ", "dd825713eaac9b06e7d7c13ea946e92d": "\\textstyle p(N-1)", "dd82a3bd62fcbf0943fa0f11a7fab6dd": "A = \\left\\{a,b,c\\right\\}", "dd82cd9e4071c483ff9672f323dbd3eb": " \\Delta_x ", "dd82db66d240e6eb23eb919b3a4cf713": "\n L(x) = \\tfrac{1}{3} x - \\tfrac{1}{45} x^3 + \\tfrac{2}{945} x^5 - \\tfrac{1}{4725} x^7 + \\dots\n ", "dd831136b8ae5ceba42cc012ecb8d9a0": "V_{b,optimum} = \\frac{V_{a1}\\cos\\theta_{1}}{2n}", "dd832ade1e1b3253a8ef838806ea8485": "\\textstyle \\mathbb{F}_2^{8}", "dd833dc576ac1fb7abc9be593b6a94d5": "C_i^j = \\nabla_{k} \\left( R_{li} - \\frac{1}{4} Rg_{li}\\right)\\epsilon^{klj},", "dd83783beb45613e4473935a3ca50b25": "\\Pi(A, \\omega, \\alpha, x) := A^{G}_{\\omega,\\alpha_{G}}A^{G-1}_{\\alpha_{G},\\alpha_{G-1}}\\dotsb A^2_{\\alpha_3,\\alpha_2} A^1_{\\alpha_2,\\alpha_1} x_{\\alpha_1}", "dd83a882c0506f02d3947fcf09973d08": "\\lim_{N \\to \\infty} \\left \\Vert S_N f - f \\right \\|_2 = 0,", "dd83d41833afdadfe74e9f5cf62c1cbc": "\\frac{p}{Q^2} = - \\frac{\\partial F_3}{\\partial Q}", "dd8407ed1d19ab9a2f2464f0539d99cd": "\\mathbf{B} = \\mathbf{H}+4\\pi\\mathbf{M}", "dd842bf079c5cc8701d570488f1426d7": "b\\in\\mathbf{N},>1", "dd84342622009e01ac064c4432b816df": "m\\times p", "dd848177891f00fb7ee06a580459db30": "y_{k+1} = y_k + m", "dd84ec41c7d7bbb017c5d440f76cfe31": "\\arcsin x= -i \\ln(\\cos (\\arcsin x) + i \\sin(\\arcsin x))", "dd8570d9f11ef053eebea3598e01cc2d": "\\phi_1={(1.7-1) \\over -0.174\\ \\mathrm{m}}=-4.02\\ \\mathrm{dpt}", "dd85bbc7497dc339d61c90eb336f9f72": "r^2 + R^2 = 1", "dd864a00e464b3880a8032487ec723d3": "Vol({\\it M,g})", "dd865790764ad591bd883faf0072e009": "F \\times F", "dd865c37799ad5fe028d4f75e358438b": "p\\ ", "dd86ee61b7c275f7254257d0d5482417": "i \\rightarrow i\\pm 5", "dd87104809a382d7c50ce16dd3cc38e4": "\n\\widehat{B}_z \\equiv \\widehat{B}(\\varphi,\\hat{\\mathbf{e}}_z) = \\begin{pmatrix}\n\\cosh\\varphi & 0 & 0 & \\sinh\\varphi \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\\n\\sinh\\varphi & 0 & 0 & \\cosh\\varphi \\\\\n\\end{pmatrix} \\,,\n", "dd8742a1717a9abd9c2e2ab846c7c1c0": "\\left|\\frac{a_{n+1}}{a_n}\\right|", "dd87df1587f305c5e0f46563b6bec525": "\\phi^{\\Rightarrow x} = \\phi\\,", "dd88509795ab3582d881a841dd46292f": "\nr(h)=\\prod_{i=1}^s\\max\\{1,|h_i|\\}\\quad\\mbox{for}\\quad h=(h_1,\\ldots,h_s)\\in\\Z^s.\n", "dd886160a8d392df59b1abb5fc0ef3ea": "\\begin{bmatrix}\nc\\phi_{1}c\\phi_{2}-s\\phi_{1}s\\phi_{2}c\\Phi & s\\phi_{1}c\\phi_{2}+c\\phi_{1}s\\phi_{2}c\\Phi & s\\phi_{2}s\\Phi\\\\ \n-c\\phi_{1}s\\phi_{2}-s\\phi_{1}c\\phi_{2}c\\Phi & -s\\phi_{1}s\\phi_{2}+c\\phi_{1}c\\phi_{2}c\\Phi & c\\phi_{2}s\\Phi\\\\ \ns\\phi_{1}s\\Phi & -c\\phi_{1}s\\Phi & c\\Phi \n\\end{bmatrix}", "dd88708490f541d12a7f6330d8deb530": "2\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) = \\int_0^\\infty \\theta(it)t^{s/2-1}\\,dt,", "dd88943e9c8bcc02f3137889b48ff758": "s_{\\nu/\\lambda} =\\sum _\\mu c_{\\lambda\\mu}^\\nu s_\\mu.", "dd88a057f076a7a619e6bcdf7d6ce5b8": "\\hat{a}= c- ab -a_x \\quad \\text{and} \\quad \\hat{b}=c- ab -b_y.", "dd8904c3890e4aa5fec3a09cbf5029b8": "u(c_t)=\\frac{c_t^{1-\\sigma}}{1-\\sigma}.", "dd890a8e56c515b62e1fde0e04bf4106": "\\tilde{G}_j^{(M)}:= \\sum_{k=0}^j {j-k+M-1 \\choose M-1} g_k.", "dd890f861b83061c0c2068e2922ec18f": "S_6 \\rtimes C_2", "dd89113d71033c2eb1351e40c007b7c6": " \\Delta S_{mix} = -N k_B\\sum_{i=1}^r x_i\\ln x_i \\,\\!", "dd89a4c64867f9453049cb9c81dfa914": "R = A[x] / (x^n-1)", "dd89c87091a18ba028bb27b68d79d0a6": "\n\\mathcal{I}(A)\n=\n\\mathrm{E}\n\\left(\n \\left[\n \\frac{\\partial}{\\partial A} \\ln p(\\mathbf{x}; A)\n \\right]^2\n\\right)\n=\n-\\mathrm{E}\n\\left[\n \\frac{\\partial^2}{\\partial A^2} \\ln p(\\mathbf{x}; A)\n\\right]\n", "dd89c963115c1091025b204625d36daa": "\\lim_{b\\to c^-}\\int_a^b f(x)\\,\\mathrm{d}x\\,", "dd89dad241a74e2088727ce7bb30b72a": "dt' = \\gamma \\left ( dt - \\frac{v dx}{c^2} \\right )", "dd89ea67f6617d6354797f1114d9c216": "L_4(2) \\cong A_8,", "dd8ae0e641b77000714fbc2348c6e214": "\\frac{\\text{number of control-flow branches executed by the test data}}{\\text{total number of control-flow branches}}", "dd8aefd4ad5f8e9cebd58068ccd7b02c": "\\dot{u}=0", "dd8b2428de7117f3ee4737251c56b4d2": "\\overline m_a", "dd8b2d0a8b621a47368ca58fd9a9abd4": "A_{i} J=J A_{i}=v_{i} J. \\qquad(2)", "dd8b66abb72f5615d66d79a2f49a29c4": "y' e^{\\int_{s_0}^{x} P(s) ds} + P(x) y e^{\\int_{s_0}^{x} P(s) ds} = \\frac{d}{dx}(y e^{\\int_{s_0}^{x} P(s) ds})", "dd8be815dd54c07ea137aca251ebebc5": "\\log\\left ( \\frac{k_s}{k_{\\text{CH3}}} \\right )= \\rho^*\\sigma^* + \\delta E_s", "dd8bee34d4b4b74fee30f3dc7d0922a8": "\\rho(\\mathbf{x}) = \\frac{1}{4\\pi G}\\Delta V(\\mathbf{x}).", "dd8bf292060a7c91184dffa2768486be": "f(x,\\alpha, \\beta,c,\\mu)", "dd8c3c1c5ec671118edc48e948c54934": "(\\mathbf{r})", "dd8c54f3a97f9a2fab3dc07ce2cf3747": "X \\to BG", "dd8c81c5d14a70064f754617504a1e3e": "j_2(q) = q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \\cdots = ((\\eta(q)/\\eta(q^2))^{24}", "dd8c8d59c3da582345842b5b097ffdb2": "\\mathbf{F}=(L,M,0)", "dd8d0346fc2dfd0442b98f639e6bb1f6": "0 \\le t_{e1} \\le 40, 0 \\le t_{e2} \\le 40, -20 \\le t_{e1}-t_{21} \\le 0", "dd8d2c4a0d2e91adb9021d718fcf2fd3": "G = f_1 ( x, y, \\xi + z ) + f_2 ( x, y, \\xi - z ).", "dd8d30b0430acdea04043d430d207067": "\\mathbf{x}_t", "dd8d3ef884ccd05144c934cd6359fc66": "(x_1, y_1) = k \\times G", "dd8d5d27d32deaa577870dc025e7f6dc": "K_{\\sigma \\delta}", "dd8d606fc6ae20a2e8c0b7e542684e27": "\n\\begin{align}\n \\delta_1 & = \\operatorname{E}(\\operatorname{1}_{\\{X_1=0\\}} \\mid \\sum_{i=1}^n X_{i} = s_n)=P(X_{1}=0 \\mid \\sum_{i=1}^n X_{i} = s_n) \\\\\n& = \\frac{P(X_{1}=0, \\sum_{i=2}^n X_{i} = s_n)}{P(\\sum_{i=1}^n X_{i} = s_n)} = e^{-\\lambda}\\frac{\\left((n-1)\\lambda\\right)^{s_n}e^{-(n-1)\\lambda}}{s_n!} \\Bigg / \\frac{(n\\lambda)^{s_n}e^{-n\\lambda}}{s_n!} = \\left(1-{1 \\over n}\\right)^{s_n}.\n\\end{align}\n", "dd8d7fd3bfe11a5d80db076e5adf3d60": "|\\operatorname{tr}|/2 > 1,", "dd8d8fd272bb143b5efb60c3ffe1bb28": "L(x, y) = L(x) \\cdot L(y).", "dd8da4b3038dcc6b0aced42c67e897c0": "\\bar H^{(\\lambda+1)}(z)", "dd8da76097c9963663d84ac894060be6": "S\\! c=\\sum_{i,j}\\langle R(e_i,e_j)e_j,e_i\\rangle=\\sum_{i}\\langle \\text{Ric}(e_i),e_i\\rangle, ", "dd8dc0211de50cebdc2af7afb8f1ac65": "\nf(x;\\mu,\\sigma,a,b) = \\frac{\\frac{1}{\\sigma}\\phi(\\frac{x - \\mu}{\\sigma})}{\\Phi(\\frac{b - \\mu}{\\sigma}) - \\Phi(\\frac{a - \\mu}{\\sigma}) }", "dd8e3eeefc545b58d66e54df54b2039e": "\\frac{dI/dV}{I/V}=\\frac{\\rho_s\\left(r,eV\\right)\\rho_t\\left(r,0\\right)+\\int_0^{eV}\\frac{\\rho_s\\left(r,E\\right)\\rho_t\\left(r,E-eV\\right)}{T\\left(eV,eV,r\\right)}\\frac{dT\\left(E,eV,r\\right)}{dV}\\,dE}{\\frac{1}{eV}\\int_0^{eV}\\rho_s\\left(r,E\\right)\\rho_t\\left(r,E-eV\\right)\\frac{T\\left(E,eV,r\\right)}{T\\left(eV,eV,r\\right)}\\,dE}\\ .\\qquad\\qquad (8)", "dd8e7c8313166e9af0ffff6d83ff1e6f": "= {\\pi \\over 4}", "dd8ec601ceaefc85c49e48c47a4d2fd9": "\\scriptstyle t \\;=\\; t_0,\\; x \\;=\\; x_0,\\; y \\;=\\; y_0,\\; z \\;=\\; z_0", "dd8ee6ccf1db44324e89761d103e7d07": "\\operatorname{cofactor}(X)^{\\rm T} = |\\mathbf{X}|\\mathbf{X}^{-1}", "dd8ef0b29025c9c3d49c1ebd18bffe2f": "q^{m-1}(x)", "dd8f0401c79cd45933c2cbf0a2ca4929": "a y_1^{\\alpha_1 + r \\alpha_2 + ... + \\alpha_m r^{m-1}}.", "dd8f09be25cfa9d3f86d30e6a0e33190": "\\Delta C_\\mathrm d = 10^{-4} \\dfrac{2 F_\\mathrm d}{\\rho v^2 A}\\, ,", "dd8f12da02ff677f6d0d844af13d6774": " B_{r}(a)", "dd8f19730710d6b10753eaf41b9bc399": "r_1 = a \\cdot {m_2 \\over m_1 + m_2} = {a \\over 1 + m_1/m_2}", "dd905c704312bbbe2cf5b4096f7044aa": "\\Sigma^p", "dd90ae29ad9fc885b622d96652aa3d19": " \\frac{\\mathrm{d}}{\\mathrm{d}z} \\frac{n-k}{n-kz} = \\frac{k(n-k)}{(n-kz)^2}", "dd90ce9f387d82e89a6bcd1bcfc9ae7c": "\\{f:[0,1]\\to X : \\ f(0)=x_0=f(1)\\} / h", "dd910baebdc1464ae8e8843e2c6d7e89": "L_{\\mathrm{MI}}", "dd91474b0daa1cfd07585bfe21a1ae50": "f(\\epsilon)=e^{-\\beta(\\epsilon-\\mu)}", "dd916e014caaeef94666206164216020": " (\\neg Q \\vee \\neg S) ", "dd919c24fe60adc05490940e5c29befd": "\\operatorname{dim}(\\mathcal{U}):=k\\leq \\operatorname{dim}(\\mathcal{W}):=l", "dd91e4413a9f3748c45e1088db29c4d2": "\\mathcal{B}_\\mathbf{C}", "dd925ee6c9e9a9c3fbfce3d10bb10bb7": "\\frac{X*R}{K}", "dd928f8b71d0fcce62112830350ca9ed": "\\begin{align}\n\\alpha &= \\textstyle{\\frac{1}{2}}\\left(r_1+r_2\\right)\\\\\n\\beta &= \\textstyle{\\frac{1}{2}}\\left(r_1-r_2\\right).\n\\end{align}", "dd92ada7af4cf006dc7d1551aca5ea10": "n \\in\\, \\mathbb{N}", "dd932b5f3fa292c7f5563a48bb5cb5b8": "h : a_n \\rarr n\\cdot a_n", "dd933d593312cff49198c83044d996e4": "\\mathbf{\\delta}(A) = \\lim_{x \\rightarrow \\infty} \\frac{1}{\\log x} \\sum_{n \\in A, n \\le x} \\frac{1}{n} \\ . ", "dd93ab03361af5359f13064f63026c9e": " 1 \\,", "dd93abb8e1e383e44fd55f70ce948a5a": "x \\text{:}f(x_1, \\ldots, x_n)", "dd93b9af98fa6e8372260b80fd0c6afc": "0.006", "dd93c604852a8df5cd90c3ff4156c8e3": "\\theta_C : C \\rightarrow C", "dd94192718ab202bb95ca23d63eb6785": "S^{\\alpha}_{\\beta\\gamma}", "dd946a91f8a7a1535f438f05d16b6f46": "\\displaystyle{\\partial_s e^{-Ms}\\|f|_{\\partial \\Omega_s}\\|^2 \\le 0,}", "dd94cd7a0ddd8565f4de935498ab2c46": "Y_{1}^{-1}(\\theta,\\varphi)={1\\over 2}\\sqrt{3\\over 2\\pi} \\, \\sin\\theta \\, e^{-i\\varphi}", "dd951e71eab551b80ee211837d8b4563": "(N/2+1)\\times(N/2+1)", "dd953753abd26d4e13695d38b43f0bd4": "x_0 = \\mathrm{B}", "dd9553e77a99e9bdfd5fae1fc2026db3": "\\textstyle\\theta", "dd9561b1ac430855e464928dd2cbd97c": " \\mathbf{x} =\\begin{bmatrix}\n x_1 \\\\\n x_2 \\\\\n x_3 \\\\\n\\end{bmatrix}", "dd9567c5d3dc5bae9c51bbc1ef729c88": " E_\\text{avg}= 4 f N a B_\\text{peak} \\!", "dd9588ed2df0acf65a0de77068fa71cb": "x_v\\in C_o", "dd95ac40368985b51615d769bb6ca0de": "g(z)^n", "dd95c919ec46049f36fddff1147757a9": "\\overline{\\operatorname{Sp}}(E)=\\{u\\in X | \\forall\\epsilon>0\\,\\exists x\\in\\operatorname{Sp}(E) : \\|x-u\\|<\\epsilon\\}.", "dd9623e37ec3eb00e5ab16131fe41085": " \\mathcal{C}_{Y \\mid X} = \\mathcal{C}_{YX} \\mathcal{C}_{XX}^{-1} ", "dd9624903f98a10f831738c2efe7b8fa": " V\\in {}^H_H\\mathcal{YD}", "dd965ffc2131f56df25d8ddfb8ac4e55": "\\begin{align}\n f^+(x)&=\\max(\\{f(x),0\\}) &=&\\begin{cases}\n f(x), & \\text{if } f(x) > 0, \\\\\n 0, & \\text{otherwise}\n \\end{cases}\\\\\n f^-(x) &=\\max(\\{-f(x),0\\})&=& \\begin{cases}\n -f(x), & \\text{if } f(x) < 0, \\\\\n 0, & \\text{otherwise.}\n \\end{cases}\n\\end{align}", "dd966a5f50999730ba80ab9a0b37dc0d": "|\\mathbf{v}-\\mathbf{u}|^2 = r^2", "dd97044b5cc7cfcc044fa1fc4cce08ec": "\n\\begin{pmatrix}x' \\\\ y' \\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 \\\\\n\\lambda & 1\n\\end{pmatrix}\n\\begin{pmatrix} x \\\\ y \\end{pmatrix}.\n", "dd9777b028b21fc1eedd707f9fe774ea": " \\rho_A = \\rho \\cdot l", "dd9867fed8efb96f98bd4b8c1f31b1dd": "\nY^\\prime=0.299R^\\prime+0.587G^\\prime+0.114B^\\prime\n", "dd986fdc71253091c3867b3700f7f507": "\\Box P", "dd98e01ffc4336190977463f3f9e52ca": "X_v \\perp\\!\\!\\!\\perp X_{V\\setminus \\operatorname{cl}(v)} \\mid X_{\\operatorname{ne}(v)}", "dd98f9bee8a6383c2243c9cb5ce43c4f": "f(t)= \\phi(t) - \\lambda \\int_a^bK(t,s)\\phi(s)\\,ds.", "dd99391bdb46d485ea44060e6658545c": "Y^l + Y^r", "dd9979dce9abee3f49a39950fd8fa311": "\\frac{\\partial \\tau_{xy}}{\\partial x} + \\frac{\\partial \\sigma_y}{\\partial y} + \\frac{\\partial \\tau_{zy}}{\\partial z} + F_y = \\rho \\frac{\\partial^2 u_y}{\\partial t^2}\\,\\!", "dd99b92b94fbeb9178ff755d0275baa8": "\\Gamma_G", "dd9a10728789cea4b26fcc2fe13fb35a": " F(\\varphi, \\sin \\alpha) = F(\\varphi \\,|\\, \\sin^2 \\alpha) = F(\\varphi \\setminus \\alpha) = F(\\sin \\varphi ; \\sin \\alpha).", "dd9a1f79dfc9b5d097796c6cfcbbc01f": "\\mathbf{\\theta}_{x,y,z} = \\langle 0,0,0\\rangle", "dd9a21d337294c386404831b9a7011c9": "I_a/S", "dd9a4984cb414be8f1e199c16c27d4bf": "\\dot{x}=A(w(t))x(t)+B(w(t))u(t)", "dd9a51e9f841f5a52a580bd32643e971": "a, b \\in RC", "dd9a95fadc6fab7432df45c9967b2fe9": " \\alpha_k y_{n+k} + \\alpha_{k-1} y_{n+k-1} + \\cdots \n+ \\alpha_0 y_n", "dd9ad287604ae67bd752d2d83971d96b": " C^S_{v_2} = \\frac{-1}{\\varepsilon^{2}_S - \\varepsilon^{1}_S} ", "dd9b1a6bfc5fc34ea548b5e87b09b16b": "F_1\\cdot x_1+F_2\\cdot x_2 + x_4 = F", "dd9b41e84dc73592e8af38d1893f9805": "P_{E_{6}}(x) = (1+x^3)(1+x^{9})(1+x^{11})(1+x^{15})(1+x^{17})(1+x^{23})", "dd9b565e5a74bdf8b8d75c79b336ce37": "(a_1,a_3,a_5,...), \\qquad a_k = k^2", "dd9b954104688f55cd944e7c1f684090": "\\mathrm{Tr} \\, \\pi_{\\mathbf{f}}(U) = {\\mathrm{det}\\, (z_j^{f_i +1/2 +n -i} - z_j^{-f_i -1/2-n +i})\\over \\prod_{i 0", "ddbab18a9819c9a21fbf00030835dc4b": " h_c", "ddbacde256ff92873b169fa406aa02fd": "\\operatorname{Trans}_{Z_{i}}(d_i)\n = \\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & d_i \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix}, \\quad\n\\operatorname{Rot}_{Z_{i}}(\\theta_i)\n = \n\\begin{bmatrix}\n \\cos\\theta_i & -\\sin\\theta_i & 0 & 0 \\\\\n \\sin\\theta_i & \\cos\\theta_i & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix}.\n", "ddbae2a9a67c59a604a933c0a5d01cef": " \\kappa = {\\partial \\psi \\over 2 \\partial \\theta_1 \\partial \\theta_1 } + {\\partial \\psi \\over 2\\partial \\theta_2 \\partial \\theta_2 } , \n~ \\gamma_1 \\equiv {\\partial \\psi \\over 2 \\partial \\theta_1 \\partial \\theta_1 } - {\\partial \\psi \\over 2\\partial \\theta_2 \\partial \\theta_2 } , \n~ \\gamma_2 \\equiv {\\partial \\psi \\over \\partial \\theta_1 \\partial \\theta_2 } ", "ddbae5a102bba83a1596973af90d62ad": "\\mod 7", "ddbaf04deb1070ca2ff753422329c350": "g(T)v", "ddbafcbef7675b99dc8614465fe56049": " \\varphi: M \\to N ", "ddbb2f92086f610701e9ec59a5946e5b": "F(\\mathbf x^*)=0", "ddbb4778288c643f390d1eba54638b80": "\\mathcal G_y", "ddbb60691daa4a7d248df5818c1b3962": "\\begin{cases}x^* \\log(x^*) - x^* & \\text{if }x^* > 0\\\\ 0 & \\text{if }x^* = 0\\end{cases}", "ddbb71ad4d8b5bc38f28d3ff53f5a627": "4 = 4x + 20", "ddbb89a4fd81d70657f033761a6c5a47": "u_i=G_{ik}F_k\\,\\!", "ddbb8a42f1861aaf857b41e986932079": "X = \\prod \\{ 0,1 \\} ,", "ddbbc60502bf46baad51018d4390b745": "P(A|X)(\\omega)=P(A\\mid X=X(\\omega)) .", "ddbbccb0f767ca8b78904db63dde7516": "M \\subseteq E", "ddbbe1534dd38c89084e4c257c27252f": "D_{mono}=\\frac{4\\pi}{\\Omega_{A,mono}}=2D_{dipole}", "ddbbe52c6bf7d27bcd1a1122e98099aa": "\n\\begin{align}\ng^k & \\equiv g^{H(m)w}g^{xrw}\\\\\n & \\equiv g^{H(m)w}y^{rw}\\\\\n & \\equiv g^{u1}y^{u2} \\pmod{p}\n\\end{align}\n", "ddbbeee0d3a3db687d4fcc0a05d57abd": "[p\\cdot g,f] = [p,\\rho(g)f] \\mbox{ for all } g\\in G.", "ddbc04b1a85dc1e9462b2f4d65e005d7": "|\\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1", "ddbc0d576f7103c0d1c276b0bb6e0de9": "Re(z) < 0", "ddbc1822806dc526111ac70b332c46bf": "\n\\frac{\\partial}{\\partial \\rho_{i \\alpha}} = \\frac{\\partial}{\\sqrt{M_i} (\\partial R_{i \\alpha} - \\partial R^0_{i \\alpha})} = \\frac{1}{\\sqrt{M_i}} \\frac{\\partial}{\\partial R_{i \\alpha}} ,\n", "ddbcabc573fd5577ce3ac738d6125695": " \\theta_{m+1}=\\arg\\max_{\\theta}g(\\theta|\\theta_m) ", "ddbcdad209fc599e153fa72b3fc93032": "x_k=a_k", "ddbd9002a03732aa24257e9150e36e4f": "t_1 = t_2", "ddbdaf7f063e763dce2e028e5640f091": "0 < \\varepsilon < \\frac {1} {252n^6}.", "ddbdb315722b6e0d79ddc775b82ac442": " S = \\lambda x.f\\ (x\\ x) ", "ddbdc74de99b2437b94635c2fdce03ab": "\\delta \\in \\mathbb R^d", "ddbddbc7a4324531ac3425ded2f9e702": "\\frac{2m}{m/2} = 4", "ddbddbf852ac0a26cd6ef132202d8aeb": "W(C)=0", "ddbe49d9a0e9607bddcdbc9c3f84aff1": "\\ d\\ ", "ddbe86f4544c4064d29c25e5efbf80cf": "Y_{7}^{0}(\\theta,\\varphi)={1\\over 32}\\sqrt{15\\over \\pi}\\cdot(429\\cos^{7}\\theta-693\\cos^{5}\\theta+315\\cos^{3}\\theta-35\\cos\\theta)", "ddbe93a589dd8597f2fca6f39dd07c9d": "X^*_{b}", "ddbed5ecefb76b24259fdc74ab5d90a8": "f_1: L_1 \\to M", "ddbede01ab53dd0e447f97e49eb68085": " \\operatorname{CNOT}\\ ", "ddbf0b4c474e7a614bee4c0e8485eb70": "\\forall i\\in\\ \\{1,...,m\\}, \\mathbb{E}_{S} [|V(f_S,z_i)-V(f_{S^{|i}},z_i)|]\\leq\\beta.", "ddbf0c68616b3c0278159cce30074aed": "\\Sigma_i \\lambda_i = 1", "ddbf37b6f7fc5d474766da9c74f18b2d": "R \\cdot V", "ddc02855d4f5bec354f72148faaada2b": "\n\\frac{\\partial y}{\\partial \\mathbf{x}} =\n\\left[\n\\frac{\\partial y}{\\partial x_1} \n\\ \\ \\frac{\\partial y}{\\partial x_2} \n\\ \\ \\cdots \n\\ \\ \\frac{\\partial y}{\\partial x_n}\n\\right].\n", "ddc02e06d99533377b73d2cebc4e2417": "t_\\text{in}(k) = 2^{O(k)}", "ddc062f09fe3040d1f0d372bacf2488d": " = \\lim_{h\\rightarrow 0} \\frac 1 {h^2} \\int_{-\\infty}^\\infty \\left[\n \\frac 1 2 \\left( 1 - \\frac{\\mathrm dX_\\theta}{\\mathrm dX_{\\theta+h}} \\right) ^ 2\n \\right]\\mathrm dX_{\\theta+h}\n", "ddc1a3cae2e2b324a8b15a516f02e264": "A\\approx a(n)^{1/3^n}.", "ddc1ab7c3b7e9812a8243724c813cb7c": "\\int_0^1\\int_0^1 f(x,y)\\,dy\\,dx", "ddc1da7b3efdd9c242ae6c6d84248027": "\\scriptstyle f_s/2", "ddc206732c929a8c91fcb78f4ea30940": "p_n = q", "ddc29c52290ef2c031285cacd8c357a5": "a_i \\in R", "ddc2b1f572640f4e52979469eaa652a8": "\\frac{dv}{dt}", "ddc33bde5e7c1a9db12e168b7c204ae2": "\\bar p \\pm 3\\sqrt{\\frac{\\bar p(1-\\bar p)}{n_i}}", "ddc36459949b866365bfe2b63f3a59a5": "\n\\begin{bmatrix} x' \\\\ y' \\end{bmatrix} = \\begin{bmatrix} 1 & k \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix}\n", "ddc37255809dbd9c83c6cf854f21a72a": "F(3) + F(5)", "ddc375a7568aea26d69e5652b755f243": "\\beta = \\frac{1}{k_B} \\frac{d S}{d E}.", "ddc397d63ad171c5a31902c04cb9000f": "\\lambda=\\frac{1}{3}", "ddc3f0b5a6187dcec495a28dd4f59f8e": "X_1 \\le X_2 ", "ddc3fe516ce4ecf527c7b2321549797d": "a\\leq c\\leq1", "ddc402238778376e781f553be6b68814": "=-\\operatorname{tr} (\\gamma^0 \\gamma^0 \\gamma^5)", "ddc467a8c6fd0a5fd3017b04b71604d6": " a^2 + d^2 -2ad \\cos \\alpha = b^2 + c^2 -2bc \\cos \\gamma, \\, ", "ddc52c971744e367861b46f21fd790ba": "e^{a x} \\left(\\left(\\sum_{i=1}^n Q_i x^i\\right) \\cos(b x) + \\left(\\sum_{i=1}^n R_i x^i\\right) \\sin(b x)\\right)", "ddc596fd8717985898af930ad6f741f7": " E = \\sum_{p=1}^{P}H(\\nu(p)), ", "ddc5a97ab8bde325553cbf89698d8f0c": "q = 53", "ddc5c34779cea47a0aa6e68807fa3b3c": "\\scriptstyle e \\, \\equiv \\, e^{1/2}/e^{-1/2}", "ddc5cab76c74fca0b0ead4880f13944b": "\\pi = \\frac{2 l}{t P}", "ddc5d1ae8e909265c128b15e336348d1": "Q(s_f,a)", "ddc63a607cc90aadf2cfe5e0cc49d15a": "p \\cdot x_i = w_i", "ddc6beb80f9fab46bfd9b1cd358f6e05": "X \\leq C + P A", "ddc77e44962b7580ef354f240672c64f": "\\frac{\\mbox{total} \\; \\mbox{votes}}{\\mbox{total} \\; \\mbox{seats}}", "ddc7812e4b118b3302b173911ae859c8": "6^2+y^2=10^2", "ddc78acad9b63775f42e6ff7bd4020b3": "m = \\frac{\\sqrt{E^2 - \\left(p c\\right)^2}}{c^2} ", "ddc78e9c690148c6be10702fbd15759b": "a_{10}+b_{10}", "ddc82c0dde353d983cffb306813c0494": "\\text{number of base pairs} = \\text{mass in pg}\\times0.978\\times10^9", "ddc8bb9bf87c90d802d15f6e943743c8": "\\mbox{Spec} \\; A_i", "ddc93558e5405b0fa05eed9fd26aa1ef": "F_u(y) \\rightarrow G_{k, \\sigma} (y),\\text{ as }u \\rightarrow \\infty", "ddc93fc1490372070130498469b4667b": "\\tbinom{p+q}{q-1}", "ddc98203becab41e24c38f9a055a1847": "N_{(-)} = N_{(+)} = 1", "ddc9fef6b7826fc2831e8b806d977e6b": "O((n/\\sqrt{\\epsilon})\\log n)", "ddca5cc275c39939aa942fa043772f1b": "\\frac{\\partial V}{\\partial t} + \\frac{1}{2}\\sigma^2 S^2 \\frac{\\partial^2 V}{\\partial S^2} + rS\\frac{\\partial V}{\\partial S} - rV = 0", "ddca9c3e1c86cb978e51244fe09e122c": "E_{1}^{\\dagger}E_{2}\\in\\mathcal{S}", "ddcabf2bff120bc632e463364fb0415e": "\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu) = 4\\eta^{\\mu\\nu}", "ddcad3447fe5b763b29f3f9481dba4d3": "Q = u \\cdot u,\\,", "ddcb8a828becbf5afe7c15c641579f4b": "\\sin(x)\\sin(y)=\\frac{\\cos(x-y)-\\cos(x+y)}{2}.", "ddcbbe66dd4c0f6b93b7e14573cf8824": "\\frac{1}{3} \\sqrt {3}", "ddcc61522bcbb526b0ebe5b998f967aa": "\n \\frac{\\partial}{\\partial X_j} = \\frac{\\partial}{\\partial x_j} + u^{(0)}_{k,j}\\frac{\\partial}{\\partial x_k} + \\cdots\n ", "ddccd2c2f6f1a9bcc00042080c07c8b5": "|R|^2 = |S|^2 + |E|^2 + |C|^2.", "ddcce64886a737bb487f4807fc60ad32": " (\\Delta(1) \\otimes 1)(1 \\otimes \\Delta(1)) = (1 \\otimes \\Delta(1))(\\Delta(1) \\otimes 1) = (\\Delta \\otimes \\mbox{Id})\\Delta(1)", "ddcd080812cb41d66fb2eb4cb37001ab": "[a(a^n+b^n)]^n+[b(a^n+b^n)]^n=(a^n+b^n)^{n+1};", "ddcd25d0c4ea6c42a3e61b706de4b5b8": " \\mathbf{f} \\equiv ", "ddcd364b4631ec164fccc70a13a3c0f6": "Ih_n", "ddcd3d69cada66b3f60a24678c43b8d7": "A_{2}^{\\mu } =\\big((\\varepsilon _{2}-E_{2})\\big )\\hat{P}^{\\mu\n}-(1-G)p_{\\perp }^{\\mu }+\\frac{i}{2}\\partial G\\cdot \\gamma _{1}\\gamma\n_{1}^{\\mu },\n", "ddcd4aac3126485cae9582df987017a7": "F_i = \\Sigma_{j=1}^i \\; f(y_j)\\,", "ddcd73d2244af5b4442f2dac25a5d9db": " \\lambda g.\\lambda n.\\operatorname{drop-param}[ n\\ (g\\ m\\ p\\ n), D, \\{p, q, m\\}, [F_1, S_1, A_1]::\\_]\\ \\operatorname{drop-param}[(g\\ q\\ p\\ n), D, \\{p, q, m\\}, \\_] ", "ddcde2f10c708a9dc1de19b6046ef88c": "y_p = -3", "ddce18dc46d1373f8d4d034138d664f8": " Z/pZ \\oplus Z/p^2Z ", "ddce3dc1f0738a76f274fde3bcd93f16": "\nA_{nm} = \\sum_q q \\psi^{*q}_n \\psi^q_m\n", "ddce8069eaf1f71e858cc7253bec3103": " ( \\frac{2}{\\pi}", "ddcebc22318c9b9371658d7e25ab53a8": "C_1(\\dot{u}(t),u(t),z(t),\\beta_1,\\beta_2,\\beta_3) = \\beta_1 \\operatorname{sign}(\\dot{u}(t)z(t)) + \\beta_2 \\operatorname{sign}(u(t)\\dot{u}(t)) + \\beta_3 \\operatorname{sign}(u(t)z(t))", "ddcf003107ab62db5c583231915229cf": "\\text{6KC8}", "ddcf09014d2dd847e4d7b74e4de185d7": " \\forall x ( ( \\phi \\lor \\psi) \\rightarrow \\forall z \\rho )", "ddcf3c96485336d3d84b455a6a98463e": "\\bigcap_{i\\in I}", "ddcfa10518f691bd4da406fb4330c8db": "p'_G-p'_L", "ddcfcc64ec703bbdb959edf617c13cb4": "\\psi(x;q,a)=\\sum_{n\\leq x\\atop n\\equiv a\\pmod q}\\Lambda(n),", "ddd0a0f3be5e287fa824c5c4bc13e22c": "y\\ne0", "ddd0e11a867f68ed58e0f3f22dad1acd": "\\zeta^{(r+1)} = U \\left(n^{-1} U(z, \\zeta^{(r)}), \\, - \\zeta^{(r)} \\right)\\,", "ddd1123a5b3061723ef51e0703632546": "x_{Ai}=202", "ddd12a240e0683564f61b993956d52c7": " s = \\tfrac{1}{2} ", "ddd19b1c5ca41e07ac807e2445ebacd4": "x=Ae^{-bt/2m}\\cos\\left ( \\omega' \\right )\\,\\!", "ddd1b8d7b02cb6a8117be1ba64c52573": "e(P_{\\alpha, i})", "ddd1dae3ceb2cfe451eb028e08b2e117": "R_n(\\xi,\\xi/x)=\\frac{R_n(\\xi,\\xi)}{R_n(\\xi,x)}\\,", "ddd1e58bd3ae6f10f85309dd83891813": "f_{i+1} = f_i g_i f_i \\ ", "ddd2d3ec9a9a6919ee25e9ff764907de": "B = \\begin{cases} B' &\\text{ if } f_\\infty = 0, \\\\ B'\\cup\\{\\infty\\} &\\text{ otherwise.}\\end{cases}", "ddd2da6f1ec0852cb69114e8a20496e0": "\\left(1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ \\sqrt{1/3},\\ \\pm1\\right)", "ddd2f1d6b5faebd0f73bc6b0eae8ab0b": "R_2 = r_1 Q_3 / r_3", "ddd307b7b52355117aa122b4932246d2": "\\scriptstyle{\\boldsymbol{y}}", "ddd33a69685fcc3638ba040be0b70abe": "\\phi(x) = ce^{-h^2 x^2},", "ddd34d5007af3139de325b9403304ab9": "E_{tot} = \\sum_j c_j E_j", "ddd3a38af87f7a1a030648d77aa334d5": " \\mathcal{B} = \\mathfrak{P}(\\mathcal{U} \\; \\mathfrak{P}_{\\ge 1}(\\mathcal{Z})).", "ddd3aaf2fd7a3893b1cae8864c0c7eba": "\\mathbf{F}_\\mathrm{Cor} = - 2 m \\boldsymbol\\Omega \\times \\mathbf{v}_{B} = 2m\\omega_S \\omega_R R \\mathbf{u}_R ", "ddd3ae719d17e9b6c93afcb74c061841": "v_{1,2,3,4}:=v_1 \\otimes v_2 \\otimes v_3 \\otimes v_4", "ddd3bed22dba92655b8f943ddb170e05": " g(s_1,\\ldots,s_n)=\\sum g_i ", "ddd444a39f40198367dd52d12a7f716a": "g_{\\alpha \\beta} \\, u^{\\alpha} \\, u^{\\beta} \\, = \\, - c^2 ", "ddd48afb881ca647cb701c1afca3d25b": "\\theta_c=\\frac{1}{\\gamma}\\left ( \\frac{\\omega_\\text{c}}{\\omega} \\right )^{1/3}", "ddd4967a972bdb86f2536edff0aeaceb": "\\lfloor n\\cdot \\log_{10}2\\rfloor+1", "ddd496bf57d4673adf8de0832fe1baf0": " \\left( E_c(z) - \\frac{\\part }{\\part z} \\frac{\\hbar^2}{2 m_c (z)} \\frac{\\part }{\\part z} + \\frac {\\hbar^2 \\mathbf{k} ^2}{2m_c (z)} \\right) f_k (z) = E f_k (z) ", "ddd4fac42b615b77bb6a856a4cb4182d": "F = m^\\mathrm{inert} a", "ddd586411557e30c4bd71f5b30efd1c1": " \\operatorname{ var }( r ) = r^2 \\left[ \\frac{ 1 }{ n } \\left( \\frac{ 1 }{ m_x } + \\frac{ 1 }{ m_y } - \\frac{ 2 s_{ xy } }{ m_x m_y } \\right) + \\frac{ 1 }{ n^2 } \\left( \\frac{ 6 }{ m_x^2 } + \\frac{ 3 }{ m_x m_y } + s_{ xy }\\left[ \\frac{ 4 }{ m_y^2 } - \\frac{ 8 }{ m_x m_y } - \\frac{ 16 }{ m_x^2 m_y } + \\frac{ 5 s_{ xy } }{ m_x^2 m_y^2 } \\right] + \\frac{ 4 s_{ x^2 y } }{ m_x^2 m_y } - \\frac{ 2 s_{ x y^2 }}{ m_x m_y^2 } \\right) \\right]", "ddd593821b994dec61d04df37dd33480": "Lc=(\\mathrm{Lan}_KT)c=\\int^m\\mathbf{C}(Km,c)\\cdot Tm", "ddd597d94909426b6ed269fe8d90d01e": "\\frac{\\partial \\phi_B} {\\partial t} = 0", "ddd59a9a319a066577a11b5a009e0c4f": "c^p_i=\\frac{1}{\\pi}\\int\\limits_{-\\pi}^{\\pi} f^p(x)dx, p = 1,2\n", "ddd5c10a6c69fd6803e4abf7884b38fe": "\\left \\langle \\boldsymbol{L}\\cdot\\boldsymbol{S} \\right \\rangle={1\\over 2}(\\langle\\boldsymbol{J}^2\\rangle - \\langle\\boldsymbol{L}^2\\rangle - \\langle\\boldsymbol{S}^2\\rangle)", "ddd5f0d31990780c4d45ace173a4a94b": "\\left(n\\right)S", "ddd62ccc3a7b6ce690f6ef10282a59c8": "\\log_b \\colon H \\rightarrow \\mathbf{Z}.", "ddd6ff47dbef9ec8c535a477bcc6bee0": "F_{1} \\cap \\dots \\cap F_{n}", "ddd78495986e9ec19825029b84822b1c": "y=f(x, \\boldsymbol \\beta),", "ddd79a53bca26039c3794bc73a72b93d": "\\ or ", "ddd829bb360824dff26e43e4259c8b53": "(n+1) \\times (n+1)", "ddd8ce7434f73ae65b75c8e87a8b1530": "\\tilde{e}_i", "ddd8d5cb68c2c2e991865feb85bee30a": "\n\\mathbf{w}^{\\text{T}}\\mathbf{S}_W^{\\phi}\\mathbf{w}=\\mathbf{\\alpha}^{\\text{T}}\\mathbf{N}\\mathbf{\\alpha},\n", "ddd9364697189196e620a0d2dc0d576b": "a_q\\,", "ddd97f3a26baad1983ae81e7c0dc5ccb": "\\ker(A+\\sqrt{5}I)", "ddd9a12bee47f0c255a289622dbea188": "R_{a,\\theta} \\in \\mathbb{R}^n ", "ddda0a10f88d4f43bfaeb6968ba59d54": "(a\\cos t,a\\sin t)", "ddda12a487280ec65ebee84076d37bc5": "\\mathbf{\\Psi}_{lm} = r\\nabla Y_{lm}", "ddda1efa104f8d07bca22c0a962f065c": " \\sum_{i=1}^{N}w_i = 1. ", "ddda568295be425a4ede31f59afd1cd6": " c^{p}(\\mu)=\\int\\int\\int c(x,y,z)^{p}d\\mu(x)d\\mu(y)d\\mu(z).", "ddda5fb447c9679ca30b95f1ee2b5f69": "= Z \\, s_p = \\frac {Z}{2 \\, \\sqrt{n}} ", "ddda8b5bc69e2d6de186707e64e9161c": " \\mathbf{\\hat T^\\dagger} (\\lambda) \\mathbf{\\hat T} (\\lambda) = \\mathbf{\\hat T^{-1}} (\\lambda) \\mathbf{\\hat T} (\\lambda) = \\mathbf{I} ", "ddda905b6054b5c2270bbc2e7761ff9d": "J(\\theta) = \\operatorname{E}_\\theta[f(x)] = \\int f(x) \\; \\pi(x \\,|\\, \\theta) \\; dx", "dddadfc73bb8f7b216dc844db56d9eb9": "\\hat q", "dddae6e536d8f9d1ff9907a8a725b646": "\\int\\frac{\\cos ax\\;\\mathrm{d}x}{1+\\cos ax} = x - \\frac{1}{a}\\tan\\frac{ax}{2}+C\\,\\!", "dddaf7f2e8a9a4155374cdbb35142bec": "G(\\bold R,t) = \\bigg( \\frac{m}{2 \\pi i \\hbar t} \\bigg)^{3/2} e^{-\\frac {\\bold R^2 m}{2 i \\hbar t}}.", "dddb7add1ee57e0e1acc440d537f2599": "z(r) = 10-\\sqrt{125-r^2}, \\; 0 < r < 5 ", "dddb7b605b52ef365c08c568c9924646": " \\omega\\lambda", "dddb9f5dd8574f4a3220265912be9a54": "g=u_n+u_{n-1}z+\\dots+u_1 z^{n-1}+u_0 z^n=0.\\,", "dddbd1bab5a2fecc2f2adab0a1e89c01": " \\operatorname{perm}(A)=\\sum_\\sigma \\prod_{i=1}^{n} a_{i,\\sigma(i)}", "dddbe61cef3871c0ccfc057e41a1ad38": "j_e=+\\mu_n n E+D_n \\frac{\\partial n}{\\partial x}", "dddbe8d313c305de7e6ff6464e34b6e2": "x^2 + q = px", "dddc3cb9a41dfcb59499108cbad2a384": "x^+ = \\left\\{\\begin{matrix} 0, & \\mbox{if }x=0;\n \\\\ x^{-1}, & \\mbox{otherwise}. \\end{matrix}\\right. ", "dddc4cf39c3293e6953152fc0d804cc4": "\\frac{(x-X)(y-Y)}{(x-Y)(y-X)}.", "dddc647d6069a79c623a50c267f1058d": "R=\\int_{W_2}^{W_1}\\frac{1}{c_T W} \\sqrt{\\frac{W}{S}\\frac{2}{\\rho}\\frac{C_L}{C_D^2}}dW", "dddc954143964f1c165a6728664fa798": "p = u^2 - N", "dddca453e23d0f99ff3fb65720e79393": "\\int\\frac{\\cos ax\\;\\mathrm{d}x}{1-\\cos ax} = -x-\\frac{1}{a}\\cot\\frac{ax}{2}+C\\,\\!", "dddca46787f9fb35e5852d010cf932b9": "x\\rightarrow \\tilde{x}(x)", "dddd3018da7aa774d6826f82471c846b": "\\oint_{v}\\nabla\\cdot fdv=\\oint_{S}f\\, dS ", "dddd3c609f67dddbc47ac6d4c47b8f78": "H=F(X)*(*_{j\\in J} g_jH_jg_j^{-1}),", "dddd4ea4703f00928ec2c620c9fdcb65": "\\scriptstyle(\\Omega,\\mathcal{F},\\mathbb{P})", "dddd691880d47cce1031c6a565021698": "A \\subseteq B_n", "dddd79c199b5e14d693c10a1ed36c4e5": "f^{**}(x)=\\frac{1}{4c^*}x^2=cx^2\\, ,", "dddd7da8807ad81463e6d0f304f7380a": "\\frac{\\{C\\land I\\}\\;\\mathrm{body}\\;\\{I\\}} {\\{I\\}\\;\\mathbf{while}\\ (C)\\ \\mathrm{body}\\;\\{\\lnot C\\land I\\}}", "dddd97c1d174df09cf614f3b25cc8059": " AA^{-1} = \\{ ab^{-1} \\mid a,b \\in A \\} \\, ", "dddda7a16f0bd1e0f95dbb666e7098a4": "p_x = N_1^c \\frac{x}{r} = -\\frac{1}{\\sqrt{2}} \\left(Y_1^1 - Y_1^{-1}\\right)", "ddde0d6db602125f7f1e2217b8e73150": "\\frac{dy(t)}{dt}=-m1*x(t)-(1/2*(m0-m1))*(|x(t)+1|-|x(t)-1|)-e*y(t)+\\gamma*cos(\\omega*t)", "ddde11811f01cc4ce21dbba2342fe98b": "\n\\mathrm{Fr} = \\frac{v}{\\sqrt{gd}}.\n", "dddf2e5a6feea5ba9a2872f51b96243b": "f: M_\\mu\\to M_\\lambda", "dddf4304243e963e924cc2824ca1160d": "\\rho_J = \\sqrt{\\frac{L_J}{C_J}} = \\frac{V_0}{I_J}\\cdot \\frac{1}{\\sqrt{\\cos \\phi}}. \\ ", "dddfdefe750c6ce2a7573dfd0a3adacb": "\n{dJ \\over dt} = \\int_0^{2\\pi} \\bigg({\\partial p \\over \\partial \\theta} {\\partial x \\over \\partial \\theta} + p {\\partial \\over \\partial \\theta} {\\partial x \\over \\partial \\theta} \\bigg) d\\theta\n\\,", "dde031e95b87a5931735eacfd721cf6b": "|E_l| = O(N_l^{-\\frac{r}{d}})", "dde055ff4d40332e9f9b133aef358648": "c : K \\leftarrow \\mathbb{N}", "dde064d80f856362c21afdfccd80d324": "\\Sigma_2 ", "dde09346e6b0433b8b471a8cb603bf6f": "\\{\\theta : \\mathcal{L}(\\theta | x)/\\mathcal{L}(\\hat \\theta | x) \\ge 0.10\\},", "dde0f0d698d4959e0754389b7e88ea37": "S_\\mathit{wr}", "dde115b9bd8da03c475c3d09f29078a8": "\\alpha=L\\cdot p\\,\\exp\\left(\\mbox{-}\\frac{L\\cdot p\\cdot d\\cdot E_{I}}{eU}\\right)\\qquad\\qquad(14)", "dde140b3429b735eed32c50dfc40affc": "C_{ending} =", "dde1b87badbcbd9f75a02b365e721c0b": "{0.05918 V}/{n}", "dde1c98f26ce0935838427cfc0c74e57": "\\mathsf{ABCDEFGHI} \\!", "dde218b0f9a0ce240a0b08f20f24d32c": "x = X_a(X - X_c) + Y_a(Y - Y_c)", "dde267ba49a1d51f4ff241f029a3befd": "f(x),", "dde28ca70f1b2e85138008b3ab087451": "\\lambda_1,\\lambda_2,\\cdots,\\lambda_a", "dde2e3ff1197dd0af34d95e09de1783a": "\n\\frac{\\partial\\varphi}{\\partial x}=\\frac{\\partial\\psi}{\\partial y},\n\\qquad\n\\frac{\\partial\\varphi}{\\partial y}=-\\frac{\\partial\\psi}{\\partial x}.\n", "dde2f2b0a94f4c1857dfd127e78e4f52": "C_{int}", "dde325fd78cd198d61d5f1977d8db706": "d_\\mathrm{o}", "dde327bfddd90b1ff07afbcec56cc250": "\\operatorname{Hom}(Y \\otimes X, Z) \\cong \\operatorname{Hom}(Y,\\operatorname{Hom}(X,Z)).", "dde3699e0b6edd503e95634582ef5a9e": "\\frac{9(e^{10}-1)}{10}\\approx 19823", "dde41352903a1a01a8c805f1d8b90d21": "i\\cup j", "dde4a876bec0c0bf66b972e98d38d4dd": "\\Phi_V(Q_d,k) = {d\\choose r+1}.", "dde4bcc2e708e04f89114510e30a13fc": "(\\Bbb{C}, \\uparrow)", "dde4e5fae160dfc0c5dbd0957598fa59": "\\nabla_{\\mathbf v}({\\mathbf u}+{\\mathbf w})=\\nabla_{\\mathbf v} {\\mathbf u}+\\nabla_{\\mathbf v} {\\mathbf w}", "dde4ecd3767421f9bcb7633f0447193b": " n=b/a ", "dde4ef1cf028c0de69e322d0061b9847": "O_{n}", "dde504e2f66a223e59414dc87f928a3d": "Z \\cap \\Gamma", "dde528bed10462b47ee6db0f4c6f9afd": " \n\\begin{alignat}{3}\nA' &= \\pi - a , &\\qquad B' &= \\pi - b , &\\qquad C' &= \\pi - c ,\\\\\na' &= \\pi - A , & b' &= \\pi - B , & c' &= \\pi - C .\n\\end{alignat}\n", "dde541ee945b92e3181ac60e4c345542": "\\prod_x f(x) = F(x) \\,", "dde566a4813fb3c3bd55c84c59faa8c6": "{\\mathbb Q}(x,y)", "dde5ad1042f980952ab32afae73f2fa6": "wt(m) = \\Delta(0,m)", "dde5c1e211a5d79a42f30a3f468f7440": "~{\\rm sexp}_b(z)~", "dde631ec29d4ec6c19f74707fc53a4a5": "F_{m} = W_{m} U \\Sigma", "dde6589172773c6cffa8355c6518f12f": "\\begin{matrix} \\frac{49}{25} \\end{matrix}", "dde66eac0b8eb0ee1c9bed617d1cdefb": "\\rho([X,Y])=[\\rho(X),\\rho(Y)] ", "dde683267962981f92a13cecff6973d2": " D_o(r)=1-P({N}(b(o,r))=1\\mid o). ", "dde6b706ea139c404a9c0c411bd687ca": "\\scriptstyle P_1 \\,\\oplus\\, P_2 \\;=\\; IV_1 \\,\\oplus\\, IV_2", "dde7169769065d0612a131d0e5572388": "\\mathfrak{g} \\to T_x M", "dde77744fb033e19528508ec2be3e797": "\\mathrm{VF_5 + KF \\ \\xrightarrow{}\\ K[VF_6] }", "dde7958306f9036410af0f186ebdcda2": "\\sqrt{\\pi A}", "dde7d38fd5327c5f74fdeb38354bb8d8": "{Z}_n^r", "dde821f178e1b1ef237ad2a8fbeffc03": "- \\sum_i \\log P(x_i, y_i),", "dde823eb686dd4daa86e1658b6ca58e3": "g(z, u) = \\exp\\left(\\sum_{d\\mid k} \n\\left(u^d \\frac{z^d}{d} - \\frac{z^d}{d}\\right) + \n\\log \\frac{1}{1-z} \\right)=\n\\frac{1}{1-z} \\exp\\left(\\sum_{d\\mid k} \n\\left(u^d \\frac{z^d}{d} - \\frac{z^d}{d}\\right)\\right).", "dde85eb4a87f9cd7e31d203d41ddd524": "{D}_{6}^{(2)}", "dde88d9a06ab20cf382924b7890c948d": "ds = dz = dr = 0", "dde8a8a89507f04d5cf5b20e9a862064": "\\begin{align}\nx' &= \\gamma \\left( x - v t \\right)\\\\\n\\end{align}", "dde8e8ddcb3f6bfef02172f9d82584a4": "1 < p \\le 2, \\quad \\frac 1 p + \\frac 1 q = 1, \\quad \\text{and} \\quad \\omega = \\sqrt{1-p} = i\\sqrt{p-1}.", "dde8f3ae4889de35bfe3704707777b65": "K\\le \\tfrac{1}{4}(a^2+b^2+c^2+d^2)", "dde928a4a6359fcfc7b2e51c22acdcc5": "\\mathrm{ - (RS)_2Pb + S \\longrightarrow RS - SR + PbS}", "dde94a23a34385df053e468115b104c8": " \\rho(\\mathrm{e}^h) - h\\sigma(\\mathrm{e}^h) = O(h^{p+1}) \\quad \\text{as } h\\to 0. ", "dde94bbe6f56c3cd68d8a35bcd3bf240": "{S}={6n^2(6n^2 + 1)\\over 2}", "dde95f3b835100efaf1c6d8685030607": "\\mathrm{C_{12}H_{24} + 6 \\ O_2 \\rightarrow 12 \\ CO + 12 \\ H_2}", "ddea0762df0ed73e7cd20016b79a1d8b": "s_n=\\frac12+\\frac14+\\frac18+\\frac{1}{16}+\\cdots+\\frac{1}{2^n}", "ddea0b3dae4dcd6a77915d62ecf98342": " z_1, \\dots, z_n \\in V", "ddea1e8e0c04b015c0d4daadb1671cf3": " (\\exists x) A(x)\\ \\equiv \\ (\\tau_x(A)|x)A ", "ddea4ea88fd014c96328a9de3917f89d": "I\\ nat\\ 3\\ 4", "ddea62c8cdfdbc43a7c89459cb23c016": "\\scriptstyle jX_\\mathrm L", "ddea88dccb4ff23f466b7e83bb1dd6e3": "\\Gamma_a^i = {1 \\over 2} \\epsilon^{ijk} \\tilde{E}_k^b [\\tilde{E}^j_{a,b} - \\tilde{E}^j_{b,a} + \\tilde{E}_j^c \\tilde{E}_a^l \\tilde{E}^l_{c,b}] + {1 \\over 4} \\epsilon^{ijk} \\tilde{E}_k^b \\Big[ 2 \\tilde{E}_a^j {(\\det (\\tilde{E}))_{,b} \\over \\det (\\tilde{E})} - \\tilde{E}_b^j {(\\det (\\tilde{E}))_{,a} \\over \\det (\\tilde{E})} \\Big]", "ddea893e9e100dd9c1862cc0dee46291": "\\ln x_A = - \\ln(1 + M_A \\sum_i b_i) \\approx - M_A \\sum_i b_i,", "ddeaa5275a98a2d8a1cc0cee7642cc1f": "\\mathcal{F}\\colon\\mathbb{C}^N \\to \\mathbb{C}^N", "ddeab027dd0b888b47a3db5f00149705": "\\scriptstyle K_h=\\frac{[H_2CO_3]}{[CO_2]}", "ddeaba0d3914e8b3bd6aa56130760925": "\\{f_L,f_R\\}", "ddeaf92b519feb46d98a2eecec4b70eb": " \\omega_{\\rm Z}=\\frac{k_{\\rm B}T}{\\hbar}\n\\ln \\left(\\frac{n_1}{n_2}\\right)_{T} ~~~~~~~~~~~~~~~~.~~{(\\rm oz)} ", "ddeafa3af89a71fc0998ae33eecb618d": "T(X, Y) := \\nabla_X Y - \\nabla_Y X - [X,Y]", "ddeb839a48777c056b1085978ee89ac5": " \\sum_{\\sigma_i = \\pm 1} \\sigma_i^n = \\begin{cases} \n\t0 & \\mbox{for } n \\mbox{ odd} \\\\\n\t2 & \\mbox{for } n \\mbox{ even} \\end{cases}\n ", "ddeb9586ca31f14de1d769a20476e70c": "\\prec, \\nprec, \\preceq, \\npreceq, \\precneqq \\!", "ddeba1cf3b5e65a841b4a35e0ed8c95b": "N(t) = N_0 2^{-t/t_{1/2}}. \\,", "ddebc4c6a94ac8efee3daab4a22db035": "(A+UCV)^{-1} = X = A^{-1}-A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}.", "ddec5a0d4c87cabeb5397d8cdbfb957e": "p^n, q^n, r^n", "ddec5d70a613e7f176a75b700476efac": "G \\approx G_p = 1000 \\,\\, J/m^2", "ddec7ac94460841e8e4a2ed3301ac0e0": "\n\\begin{align}\nJ^2 = j(j+1) \\hbar^2,\\qquad & j = 0, \\tfrac{1}{2}, 1, \\tfrac{3}{2}, \\ldots, \\\\\nJ_z = m \\hbar, \\qquad\\qquad\\quad & m = -j, -j+1, \\ldots, j.\n\\end{align}\n", "ddecc7347ade4b1a75a7d1e1ed223d4c": "Q^{(1)}", "ddecc8aed6e647034971440056e41e04": "\\{k_1, pk_1, p^2k_1, \\ldots, p^{m_1-1}k_1\\},", "ddece431c76bb3c736334e175840bf24": "R= \\int_{t_1}^{t_2} V dt = \\int_{W_1}^{W_2}-\\frac{V}{F}dW=\\int_{W_2}^{W_1}\\frac{V}{F}dW", "ddece8f15270c0695a9c8d2432f6c306": "\\ \\gamma_{j,i+1} = \\gamma_{j,i}\\,\\alpha^j", "dded45f5d1352b9ef71132e341aa487d": "n^{1/3} = \\lambda_{dB} = \\hbar (m_{\\mathit{eff}}kT_{cr})^{-1/2}", "ddeda24a7890a006db9d852658d41370": "(\\phi_x,\\phi_y)=(0.7,0.2)", "ddedd55eba04a2ebec39218092016611": " t / U ", "ddede92608ace8d5bcebc38533cc39f9": "\\displaystyle n = d + l.", "ddedfb4e29c5bf8b36861be5c6831f34": "\\frac{\\pi}{2\\sqrt{2}}", "ddee1bc372090261f7933e15b72c472b": "\\|f\\|_w = \\sqrt{\\langle f, f\\rangle_w}", "ddee6c934e3b9c5e77a85e1590c51dfd": "\nH(\\mathbf{s}) = - J [\\cos(\\theta_1-\\theta_2)+\\cdots+\\cos(\\theta_{L-1}-\\theta_L)]\n", "ddee72a1cbc6b1cc292262f8cf939817": "D{\\mathbf e}_i = \\sum_{j=1}^n {\\mathbf e}_j\\omega_i^j.", "ddeea942e8d074fe1439c220b00147ed": "\\delta R^\\rho{}_{\\sigma\\mu\\nu} = \\nabla_\\mu (\\delta \\Gamma^\\rho_{\\nu\\sigma}) - \\nabla_\\nu (\\delta \\Gamma^\\rho_{\\mu\\sigma}).", "ddeeb097a28a7bff0bd30b7032285e7b": "N(t) =N(0) e^{ - A_{21}t }= N(0) e^{ - \\Gamma_{rad}t }, ", "ddeeeec5f5d87541c94563d1fedbbb6b": "\\epsilon = (-1)^k", "ddef1a034741c5efc061ef52bc20c5ea": " \\frac{dg}{d \\ln \\mu} =\\beta(g)=\\beta_2 g^2+\\beta_3 g^3+\\ldots \\qquad\\qquad\\qquad (4) ", "ddef43827322d6543ed6e7e813039987": "| x | < 1", "ddef77fca22fb79d4d7a17ba489adfa6": "z = (af-be)\\,", "ddef84ff912ea4df41d82bf116eecd73": " [\\psi(x),\\psi^\\dagger(y)] = \\delta(x-y) ", "ddef9b1c9965042855766d9bc5b07950": "f_U", "ddefa407c7e730a03450b341d5be148b": "E [ X_{t+j} | \\Omega_t] = E_t [ X_{t+j} ] \\,.", "ddefaf05dbcd55546c40c52adc9cf865": "\\langle x, y, z \\mid x y x = y x y, x z x = z x z \\rangle. \\, ", "ddefd54878ec852d8b37dd1102e7572f": "(X,f,\\alpha)=0", "ddf01d9cab9ef5e82d2c6e8952002429": "R[x_1,...,x_n]", "ddf0564bb750903cbef6c7b78a22241e": "h(\\mathbf{x})", "ddf09324ce624c94d2b14a73d82401ec": "\\displaystyle{g=\\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix}}", "ddf0c42da6d49e1baa42d966f3986a19": " n \\cdot \\lambda = 2 d \\cdot \\sin(\\theta)", "ddf10b8d643788d200e6db4c6b2e7824": "\\scriptstyle \\tau\\,\\in\\,(0,1).\\,", "ddf1acba89c11ddde1ed61a0079f118b": " \\Pi_1 = ", "ddf2116b5c559965df9925ef00da01cc": "v = \\sqrt{2gh}", "ddf30cc84d95ca112f6182765c3f14b4": "\\mathcal K_D = (\\mathcal K_X + D) = (-(n+1) H + \\mathrm{deg}(D) H)_D", "ddf3647c41baed85402c0cec551637cd": "a \\equiv \\dot{v} = \\dot{\\beta}c", "ddf3b9dd9ff4ece78a342a95a4c72ec7": "\\mathbf{<\\psi_{1s}|\\psi_{1s}>} = 1", "ddf3ba488c85edb7c916e520cb6bb707": "\\frac{1}{p} + \\frac{1}{p'} = 1", "ddf3f6466e037451b75cc629ef503055": "\\Lambda(t)", "ddf42e72a6547853544e34137326a074": " \\Delta S = {\\Delta Q \\over T} ", "ddf4fcf4ab43c4adf34c0181d16fb50f": "y \\,\\!", "ddf534d217a303dae72a773794621235": "(x_1, y_1), (x_2, y_2), \\ldots ", "ddf53b0e179a42ac5dfc1040496d3e6c": " \\mathbf{b}_i ", "ddf584588797792e15675ba6b789e8ee": "W(s,t)", "ddf5d7f5ea9e526166ef05cfa6f08147": "n < 468.\\ ", "ddf5ebdea7bd6a9826bbad7735be0f40": "V_\\text{capinit}", "ddf635626a3cae3eb2ee18507a1fba06": "{\\mathcal L}_{\\rm M}", "ddf693e1a631506838efaa84c1040038": " B = \\sum_{ i = 1 }^K B_i ", "ddf6ab384507d499d5957afba2a7ce48": "e^{\\pm2\\psi}=\\sum_{n=0}^{\\infty} \\frac{(\\pm2\\psi)^n}{n!}", "ddf6b935e8578425d6dd975eadf4e2ff": "\\frac{5}{12}\\,", "ddf6f4baecd2efb9e76bb5854e63002c": "\\operatorname{Var}[Y(t)] = \\operatorname{E}(\\operatorname{Var}[Y(t)\\mid H_{1t},H_{2t},\\ldots,H_{c-1,t}])+ \\sum_{j=2}^{c-1}\\operatorname{E}(\\operatorname{Var}[\\operatorname{E}[Y(t)\\mid H_{1t},H_{2t},\\ldots,H_{jt}]\\mid H_{1t},H_{2t},\\ldots,H_{j-1,t}])+ \\operatorname{Var}(\\operatorname{E}[Y(t)\\mid H_{1t}]).\\,", "ddf74aa445c961ef7e7307bd8f2577d0": "\nA=\n\\begin{bmatrix}\n\\lambda_1I + N_1& & & \\\\\n & \\lambda_2I + N_2 & & \\\\\n & & \\ddots & \\\\\n & & & \\lambda_kI + N_k \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\lambda_1I & & & \\\\\n & \\lambda_2I & & \\\\\n & & \\ddots & \\\\\n & & & \\lambda_kI \\\\\n\\end{bmatrix}\n+\n\\begin{bmatrix}\n N_1& & & \\\\\n & N_2 & & \\\\\n & & \\ddots & \\\\\n & & & N_k \\\\\n\\end{bmatrix}\n=\nD+N\n", "ddf755aaf02baf17ae3bc38491d16d51": " \\lim_{h\\to0+} \\max_{n=0,1,\\dots,\\lfloor t^*/h\\rfloor} \\| y_{n,h} - y(t_n) \\| = 0. ", "ddf779a2db64483150b74f728e9e261e": " \\hbar \\ ", "ddf7b443e8d8596f2edee8914799b5fa": "\\frac{3\\pi}{2}", "ddf7dadfedab3b06840596a1779fb4ad": "(a,1) \\circ (b,1)=(a - b,0)", "ddf81d818a4411368c24eda5a34b9b74": "i\\hbar{d \\over d t}A(t) = [A(t),H_{0}].", "ddf8a083aaf3a4e53fddd1a9d7757f39": "(X,d),(X',d')", "ddf8b100120c070620d6c31e88eb67fc": "| f(z) | \\leq M \\text{ for all } z \\in \\partial \\Omega \\,", "ddf8c34173a178589b8a382948b31f0e": "{}_2F_1(a,b;c;z) = \\sum_{n=0}^\\infty \\frac{(a)_n (b)_n}{(c)_n} \\frac{z^n}{n!}.", "ddf97ce7d4bdd6138b6adb6a2c9f7cf6": "\nA_{u,v} = \\left\\{\n\\begin{matrix}\n1 & \\text{if } \\textrm{\\{u,v\\}} \\text{ } \\epsilon\\text{ } E\\\\\n0 & \\text{otherwise}\n\\end{matrix}\\right.\n", "ddf9c357b32c7b5b6c4cdb5d0d6912da": "\n2 \\uparrow\\uparrow\\uparrow 3\n", "ddf9f88115f53d8e27c559a5bc35f383": "D_x = D_y = \\frac {0.61 \\lambda} {NA}", "ddfa4f01ef5ce9fda13843c079d58625": "\\sum_k (A_{ik}B_{kj}) \\lambda = \\sum_k ( A_{ik} \\lambda ) B_{kj} = \\sum_k A_{ik} ( B_{kj} \\lambda ) ", "ddfa5d40aadc15da4e6949971f9a6963": "\\langle h_n\\rangle_{n=1}^{\\infty}", "ddfa6736342412e68b96cd4d6a12e823": "\\displaystyle{R(a,b)= 2Q(a,b^{-1})Q(b).}", "ddfadffbffc235652b92887a7fb5e681": "E_{m} = \\frac{RT}{F} \\ln{ \\left( \\frac{ \\sum_{i}^{N} P_{M^{}_{i}}[M^{+}_{i}]_\\mathrm{out} + \\sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\\mathrm{in}}{ \\sum_{i}^{N} P_{M^{+}_{i}}[M^{+}_{i}]_\\mathrm{in} + \\sum_{j}^{M} P_{A^{-}_{j}}[A^{-}_{j}]_\\mathrm{out}} \\right) }", "ddfaf2324d5441702637ab0c30e32ece": "\\cfrac{5 Pmf}{3} = 1", "ddfb70de8f4885c081606c102da96bb9": "n_F^{\\prime\\prime}(\\xi)=\\frac{\\beta^2}{4}\\mathrm{sech}^2\\frac{\\beta \\xi}{2}\\mathrm{tanh}\\frac{\\beta \\xi}{2}", "ddfbe7bfca37d0bc5565763ffafa8a3b": "\n\\sum\\limits_{i = 1}^n w_i^\\mathtt{KED} (\\mathbf{s}_0 ) \\cdot q_k (\\mathbf{s}_i ) = q_k (\\mathbf{s}_0 )\n", "ddfc3a3c133f5157d78ec5b7c77baee3": "l_1\\ll l_2,l_3", "ddfc3b396eff9dcda382e99f617c7e5f": "p = t + \\left(\\frac{B \\wedge (q-t)}{B \\wedge v}\\right) v", "ddfc59e656f13b35dd3b0cc5d1bbe222": "e^{i \\int_\\gamma A}", "ddfc731a993af0afaa14e82030d0775c": "\nF_4(a,b,c_1,c_2;x,y) = \\sum_{m,n=0}^\\infty \\frac{(a)_{m+n} (b)_{m+n}} {(c_1)_m (c_2)_n \\,m! \\,n!} \\,x^m y^n ~.\n", "ddfc9593cbfca6b878f9690a5f867b3a": "\\frac{18 mol O_2}{1.5 mol C_6H_6}=12 mol O_2", "ddfd10a531449db7889b03802267e87b": "l \\ne 0", "ddfd134130d7cae181ce88086a6fa2f5": "SE\\text{ of }\\Delta_{ij}=\\frac{1}{2N}\\sqrt{a-4N\\Delta_{ij}\\left (\\frac{B+D}{2\\sqrt{BD}}-\\frac{\\sqrt{BD}}{N}\\right )}=0.00367", "ddfd2cabddcc55c27eb0b25b1bba4165": "\\displaystyle{\\mathbf{SL}(2,k) = \\mathbf{B} \\cup \\mathbf{B} \\cdot J\\cdot\\mathbf{U},}", "ddfd4a24488dac05be910838cbe9595d": "(-1)^n{d^n \\over dt^n} f(t) \\geq 0", "ddfd54a664c72b36a2c5fdce027bdda6": "P(R_A,\\theta) = P(R_{NP} \\cap R_A, \\theta) + P(R_{NP}^c \\cap R_A, \\theta).", "ddfd63aca00694195668b59dde306a75": "t_i, \\ i=1:m", "ddfdd9f37d544b756e2468aa2a5760eb": "p_w(z)", "ddfe02861e297f2362319c7bd116ea46": "L=\\{a^nba^n \\mid n\\in\\mathrm{N} \\}", "ddfe2a15c4d23cca8cb81986b31925ab": "\\lim_{n \\rarr \\infty} p_{ij}^{(n)} = C \\frac{f_{ij}}{M_j}.", "ddfe4d71b5a2e40bebea0acdbe54f898": "\\nabla\\times(\\nabla \\Phi)=0\\ ", "ddfe4e1e067acd98d3e08bef374bbdc1": "f = \\nu = \\frac{m_e q_e^4 }{8 h^3 \\epsilon_{0}^2} \\left( \\frac{3}{4}\\right) (Z-1)^2 = (2.48 * 10^{15} \\ \\mathrm{Hz})(Z-1)^2 \\,", "ddfe7b1b4d9514d0c352fb78f987d313": "\\omega = \\omega_0 + \\varepsilon \\omega_1 + \\cdots.\\,", "ddfeaad369ecb2a71b034a98f7516083": " \\mathrm{d}U \\leq T\\mathrm{d}S - \\delta W\\, ", "ddfeb358745049a7b2fae6496e1b3753": "\\epsilon_0\\,", "ddfec331183b1424767eccae25c51286": "\\| P_n(x) \\|", "ddff02459ac93549182941dcd0151049": " J = \\frac{ 1 }{ n } \\sqrt { \\frac{ \\pi }{ 2 } } \\sum{ | X_i -a | } ", "ddff2ced976fce3d84b0202401fa7e7d": "e_0 ^2 = e_1 ^2 = e_2 ^2 = -1", "ddff2f847a73dce63bfd558393383275": "R_z", "ddff3e2ad5c6d6b476bb6eecf11bdbd8": "\\Delta G_m = RT[\\,n_1\\ln\\phi_1 + n_2\\ln\\phi_2 + n_1\\phi_2\\chi_{12}\\,] \\,", "ddff9a7aba0ac55d0331c64ab77b1980": "a_s", "ddff9c37b62b3a6d8748da61791cce53": "\nI = B_0 \\sigma + \\sum_{j=1}^\\infty B_j \\sin 2j\\sigma\n", "ddffff829a88c819111cf7ef635e325b": " f_{x}(x,y) \\approx \\frac{f(x+h ,y) - f(x-h,y)}{2h} \\ ", "de000a6ae958dacb8be81de2d78171f6": "\\sigma_z\n= \\frac{\\partial^2A}{\\partial y^2}\n+ \\frac{\\partial^2B}{\\partial x^2}", "de001d886863d60469a2f463c5822626": "A_{ij} = a(e_j, e_i), \\quad f_i = f(e_i).", "de00555d60a9999e9251a811c21adb92": "\\delta_l \\approx 10^{-18}", "de008ec8ed13e404df269b3f0729a7b8": "K_0(X)\\,", "de00c2d8bdc8ba03832cff850f9dd8fa": "o_1^{-1} + o_2^{-1} < 1", "de00ce39152351a28771faa0736aec55": "dU = dQ - dW", "de011263535634be3669878edad469bd": "\\operatorname{Li}_{-1}(z)=\\sum_{k=1}^\\infty k z^k=\\frac{z}{(1-z)^2}\\,\\!", "de017c09ca4dc0b118ed36e0dab29e8f": "\n\\begin{pmatrix}\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3&\\alpha^{-3}+\\alpha^{5}x+\\alpha^{7}x^2\\\\\n\\alpha^{3}+\\alpha^{-5}x+\\alpha^{6}x^2&\\alpha^4+\\alpha^{-5}x\\end{pmatrix}\n\\begin{pmatrix}\n\\alpha^{-3}+\\alpha^{-2}x+\\alpha^{0}x^2+\n\\alpha^{-2}x^3+\\alpha^{-6}x^4\\\\\n\\alpha^{-4}+\\alpha^{4}x+\\alpha^{2}x^2+\n\\alpha^{-5}x^3\\end{pmatrix}.\n", "de01b8ed2d187b76dc7346450a08bd57": "\\pi_1(U,w)", "de01bc871295883871a19e798266afc9": "n < 3", "de020dfe4a93b7103318208567f9bcf0": "O(|E| + |V|^2) = O(|V|^2)", "de0221f153e344fb970116ed9773a4ff": "\\lnot(f_1 \\land f_2)", "de02879d268e158dc5836da5ac2569a7": "H=X(X^T X)^{-1}X^T.\\,", "de02e2c972af6471e9aca19920710cef": "B_\\max", "de02ea52f12ef70c980e683395096ddc": "\\pi_1 (X\\times Y) \\cong \\pi_1(X) \\times \\pi_1(Y)", "de03202de219789b187562dce32a2a28": "S \\subseteq \\mathbb{P}^n", "de035e7cb432e46f37c739abfee0b46b": "S_{r_1, r_2}(r_1) = S_{r_1, r_2}(r_2) = 0,", "de0399e6f9ed54c174e239b36e3c35dc": "\\frac\\pi6\\!", "de041523d550fad243a8a2a510ccac17": "\\mathbf v_B = \\frac {\\nabla S}{m}", "de0421d4f87eb8e2e031759079aee4ff": "{v_0,\\ldots,v_n}", "de0443440eeb9fdeed0d910da7478d3a": "A = v_1w_1^\\mathrm{T} + \\cdots + v_kw_k^\\mathrm{T}.", "de04587032d3a720aa3b1df78c50d078": "\\scriptstyle \\lim\\limits_{n\\to\\infty}\\frac{1}{n} = 0", "de04ac04d37b18e66279a307847bafc1": "x_{2}=a_{2}cos(\\theta)", "de04c982fbddf83e6cb28a6353dababd": "\\Sigma^y", "de04f74792ca922f39bed06b3cf2ae45": "f*\\Gamma(x) =\\lambda \\Gamma(x),\\quad \\lambda=\\int_{\\mathbb{R}^d} f(y)\\,dy.", "de04fb90786be4426a082a8d6729ab06": "R = \\rho \\frac{L}{A} = \\rho \\frac{L}{W t}", "de050b946993774fa790b348290ac214": " x_{ij} = 1", "de0537b80f2fafa5629039e14fe86855": " \\operatorname{Q}(\\xi, \\xi) = \\langle \\xi \\mid T \\xi\\rangle + \\langle \\xi \\mid \\xi \\rangle \\geq \\|\\xi\\|^2.", "de0594ad8cf091681198f2f7f1899f6a": "\\beta_E = \\beta =\\left[ \\beta_A - \\beta_D \\left(\\frac {D}{V}\\right) \\right] \\frac {V}{E}", "de05bbe04fa76b5035d504409f16f560": "\\mathbf D_j = \\mathbf A_{jj} - \\sum_{k=1}^{j-1} \\mathbf L_{jk} \\mathbf D_k \\mathbf L_{jk}^\\mathrm T", "de06a0d790211c8dacbc89ad3c40e336": "Ew", "de0711b70aff6334cc7fb4c1a87a8e58": "\\sigma \\cdot dS = g \\cdot \\rho \\cdot S \\cdot dr", "de0737f28801d2156eec5601cae2d325": "x_{42}=\\xi\\sqrt{\\left(1-\\sqrt{t}\\right)\\left(1+t+\\sqrt{t(t+1)}\\right)}\\,", "de07643bf72a1ce91b07e4b9dfe51310": "\nh Q^* = k (T_s - T_w)\n", "de07885464786497b05f41e3eb448909": "\\sigma_1, \\sigma_2, \\sigma_3", "de07d7a8e10aeec65a76712b0390d5e6": "\n\\begin{align}\nd\\Omega_+&=\\sin\\Theta_+\\ d\\Theta_+,\\\\\nd\\Omega_-&=\\sin\\Theta_-\\ d\\Theta_-.\n\\end{align}\n", "de081de4c322af0cce4fd7aa8dd5011d": " \\displaystyle{|b| < |a|^{-1} - |a|.}", "de087e1333a637f819e948d4dcb664ad": "\\xi_y = -\\frac{IB}{nqtW} = +\\frac{R_{Hn}IB}{tW}", "de08d87f3dc2d31ada3e353cc75f0ebd": "p(z) = 0", "de0908d87a200ac43103d236da45abaa": "x \\in A \\leftrightarrow \\phi", "de09ab5ffef5afa4c15561c30b7af288": " \\lang Ax , y \\rang = \\overline{\\lang x , A^* y \\rang} \\quad \\text{for all } x,y\\in H.", "de09e1b279e45f242c6c868c78db1d2e": " M_a ", "de0a3596bf183ac7e09616377776cd67": "q^n-1", "de0a4d68e07eff498ef30fdd07a522c3": "I_\\mathrm{v}(\\lambda)= 683.002 \\cdot \\overline{y}(\\lambda) \\cdot I_\\mathrm{e}(\\lambda)", "de0a607fe02c0953df9c5804bfe34264": "a+b \\ge 0\\,", "de0af64b3d3866648ffd2a487e1c81e0": "(1,1)_0", "de0aff7ad31a1d086002dc3b924b6476": "L^{-1} = \\{(y, x) \\in Y \\times X \\mid (x, y) \\in L \\}", "de0b0dd3b95aa4a222af427e1b555842": "\\arg\\min_{\\mathbf{w},\\mathbf{\\xi}, b } \\max_{\\boldsymbol{\\alpha},\\boldsymbol{\\beta} }\n\\left \\{ \\frac{1}{2}\\|\\mathbf{w}\\|^2\n+C \\sum_{i=1}^n \\xi_i\n- \\sum_{i=1}^{n}{\\alpha_i[y_i(\\mathbf{w}\\cdot \\mathbf{x_i} - b) -1 + \\xi_i]}\n- \\sum_{i=1}^{n} \\beta_i \\xi_i \\right \\}", "de0b850b7782582ea183a3a09561501c": " 3 \\cdot 5^e ", "de0bacd550153506dd9826694bda113c": "\n \\pm EI\\dfrac{d^2w}{dx^2} = M \n", "de0bd35ecf8189b7ef76d243a837127d": "m+n ", "de0c3bd86e3439bd3c0f914c7acafb99": "e = 2+\n\\cfrac{1}\n {1+\\cfrac{1}\n {\\mathbf 2 +\\cfrac{1}\n {1+\\cfrac{1}\n {1+\\cfrac{1}\n {\\mathbf 4 +\\cfrac{1}\n {1+\\cfrac{1}\n {1+\\ddots}\n }\n }\n }\n }\n }\n }\n= 1+\n\\cfrac{1}\n {\\mathbf 0 + \\cfrac{1}\n {1 + \\cfrac{1}\n {1 + \\cfrac{1}\n {\\mathbf 2 + \\cfrac{1}\n {1 + \\cfrac{1}\n {1 + \\cfrac{1}\n {\\mathbf 4 + \\cfrac{1}\n {1 + \\cfrac{1}\n {1 + \\ddots}\n }\n }\n }\n }\n }\n }\n }\n }.\n", "de0c57ed48a4c03417ce58f89471a19c": "ab \\ne 0, 1", "de0c67aaa2e2d30d1ea779c1ab4979dd": "\\tau=\\frac{1}{\\omega_0}", "de0cbc1f817e65b65f7adcc43e58dad6": "h = h_1 + h_2 + \\cdots", "de0d3444ce44fa88d31a0176e50170e6": "\\lambda_1 \\ge \\lambda_2 \\ge \\dots \\ge \\lambda_d", "de0d377f9c186329104a02c11b5c0c1f": "X_{1..i-1}", "de0d423802ccae27aa5adb219a174799": "P = \\frac{ R \\; T } { v - b } - \\frac{ a\\; \\alpha(T) } { v^2 + 2bv - b^2 }", "de0e29542300d8eda9e88ebac15de630": "\\Delta p = \\frac{2 \\gamma \\cos \\theta}{a}.", "de0ed7585c03b28b6a20805998ac2aee": "\n\\begin{align}\n \\sum F_x &= 0,\n \\\\\n \\sum F_y &= 0,\n \\\\\n \\sum T &= 0\n\\end{align}\n", "de0f7fa927c4b73213a8adf9c6be649f": " (\\bar{s}=0 \\to A) \\wedge (\\bar{s} \\neq 0 \\to B) ", "de103242b836222c38744b11cc206232": "\\biggl(\\frac{1-p}{1 - p e^t}\\biggr)^{\\!r} \\text{ for }t<-\\log p", "de104476c74cb4ae114265cd52314e38": " c(E) = \\prod_{i=1}^{k} (1+x_i) = c_0 + c_1 + \\cdots + c_k \\, ", "de10c7d56c50eca5b1b0cc99c83f7768": " \\ln[1+Z_q(V_o,T)]=\\frac{V_o}{2\\pi^2}\\sum_{n=1}^{\\infty}\\frac{1}{n}\\int_0^{\\infty}dm \\int_0^{\\infty}dp \\, p^2 \\rho(n;m)[1+(q-1)\\beta \\sqrt{p^2+m^2}]^{-\\frac{nq}{(q-1)}} \\,.", "de10c971a8dcc72200f4bd13cccaed56": " |z| > |p| ", "de10def573874b247971d1b32f8b1267": "M_f(x_1, \\dots, x_n) = f^{-1}\\left( \\frac{f(x_1)+ \\cdots + f(x_n)}n \\right).", "de10ee0a52b978ba42287d13427b7ccb": "i = 1,\\ldots,N", "de1114fe795dab996cb6dac8221a3293": "a(z)=a_0+\\tbinom{n}{1}a_1 z+\\tbinom{n}{2}a_2 z^2+\\dots+a_n z^n", "de115c033a69837033fa4e8d6ff12679": "w_{ij} \\in \\left\\{-L,...,0,...,+L \\right\\}", "de117e8316267d602172dd0555fc4a47": "C_P", "de119d8fab375d8fcd930a8efd5bdba6": "\\frac{\\alpha}{\\beta}", "de11a68324c6585c11e1ea64797311d1": "\\frac{s}{b^{p-1}} \\times b^e", "de11b1ca331c390ce435eb1acfbbc6bf": " T | jm \\rangle = (-1)^{j-m} | j,-m \\rangle", "de11bbfd96d2bdaa480cd696fbd88595": " \\Sigma_u = PP'", "de11bc57704d6fb219522ac5479b7138": "\\langle f,g\\rangle = \\int_\\Omega f(x)\\bar{g}(x)\\,dx + \\int_\\Omega D f\\cdot D\\bar{g}(x)\\,dx + \\cdots + \\int_\\Omega D^s f(x)\\cdot D^s \\bar{g}(x)\\, dx", "de11dfd227e6ee3f5cf6de8548d0f439": " \\frac{n!}{k!} \\sum_{j=0}^k {k \\choose j} (-1)^{k-j} \\frac{j^n}{n!} =\n\\frac{1}{k!} \\sum_{j=0}^k {k \\choose j} (-1)^{k-j} j^n.", "de11e5d10aecb48aaf84ac0bc55e3588": " 16 \\rightarrow (4,2,1)\\oplus (\\bar 4,1,2). ", "de11e8898846aa98179dbd53549a04fe": "|U_i|^d", "de1231ad662d980ad7b2d5f03cb8b184": "\\sqrt{\\frac{10}{21}}\\!\\,", "de1245e08dbd6dea1e8538a95d1c2b1f": "\\begin{align}\nh_1&=h_2=a\\sqrt{\\sinh^2\\xi+\\sin^2\\eta} \\\\\nh_3&=a\\sinh\\xi\\sin\\eta\n\\end{align}", "de1272786d7394041fed6aae5587a14c": " \\boldsymbol{\\alpha} = \\mathrm{d} \\boldsymbol{\\omega}/\\mathrm{d} t = \\mathbf{\\hat{n}} \\left ( \\mathrm{d}^2 \\theta / \\mathrm{d} t^2 \\right ) \\,\\!", "de12a58807c4c8c33b138c1e4d773713": " E_F(G)\\rightarrow \\{\\cdot\\} ", "de12c883696b1a6fe630921d5d46f431": "P^n", "de1340670d994300f5fd6d9ca86ff4eb": "377730^{+3205}_{-3200}", "de1340aed6323a29c4622d8a1cfdd511": "r_1 + r_2 - 1", "de1371b096abd6745333883b09e81a32": "f'(x) = k\\cos kx. \\,", "de137b273e66d1d16b34072904cb6c94": "E_{ap} \\le |x'(t) \\Delta t| \\le 2A \\pi f_0 \\Delta t", "de143d0020a797f4375483a79e6fdd25": "F_i=G", "de14695b27f88eb76f6a75384045a751": "M_X(\\mathbf{t}) = \\exp\\Big(\\ A(\\boldsymbol\\theta + \\mathbf{t}) - A(\\boldsymbol\\theta)\\ \\Big) \\, .", "de146d3b0317bbf559c2faebed4c000d": "\\neg A \\cdot B \\cdot C", "de1470076a8aa54cd086fe481ddb7ffd": "{\\bar{R}}_4", "de149b273e767d38e3dce00b38b78b81": "L^{2}", "de151e5c58fd99ce469382afa5a8f8df": "CD = k_1 \\cdot\\frac{\\lambda}{NA}", "de154974093d29e8c8ec9db802f29644": " \\operatorname{var}[\\ln (X)] > 0", "de15b55e97fda2f6af9c7ad145d4c7de": "x^4 + x + 1", "de15c026f2d5775f3ba0b34bde71cb2a": "A\\phi = \\neg E \\neg \\phi", "de15cd3c818af4f40313918d4974650d": "F_\\mathrm{crit}=\\frac{4 m^2\\omega_0^2\\gamma^3}{3\\sqrt{3}\\kappa},", "de15db25e79bb7c1b134f5015e54a53c": "\\int\\limits_{-\\infty}^{\\infty} \\frac{\\sin x}{x}\\,dx=\\pi \\!", "de169964d0b23eb5622f8dac397974e8": "\\displaystyle \\partial_t u + \\partial_x^3 u - (\\partial_x u)^3/8 + (\\partial_x u)(Ae^{au}+B+Ce^{-au}) = 0", "de16b116d2666c1a8f80f9268bb80c1f": "\\left(\\frac{dn_2}{dt}\\right)_\\mathrm{abs}=B_{12} n_1 I_\\nu(T)", "de16fc13c6fdb4554ad8521819728d44": "Sf:SX\\rightarrow SY", "de174feb3783bda52b2a4b2736ba29a0": "V_t", "de1776643ee4f22d01fbb7358a1d1253": "\n\\mathcal{L}[u_m(x) - \\chi_m u_{m-1}(x) ] = c_0 \\, R_m[u_0, u_1, \\cdots, u_{m-1}],\n", "de17979faa47d41c583812b295831e45": "0{.}03575\\text{ }50174\\text{ }83924\\text{ }25713 \\ldots ", "de17af768f629ad448b870fa2d90b961": " \\equiv \\lambda m,p,q.(\\lambda g.\\lambda n.n\\ (g\\ m\\ n)\\ (g\\ q\\ n))\\ \\lambda x.\\lambda y.p\\ x\\ y ", "de17b405269e7c0c5f882ecbac122731": "\\scriptstyle \\left|I_o\\right|", "de17d35bfe0d34b837fd08e86e958d1b": "\\chi(\\tau,f)", "de17f30f798745eb18a7106c508f64c8": "\\Omega \\left ( \\mathbf{R} \\right ) = c P\\left ( \\mathbf{R} \\right )", "de180b6ec0c458f52e453e9c23a6f100": "A^+b", "de18209bc167ad1877b17698e6b7768f": "\\mathbf{x} = \\{x_1,\\dots,x_n\\}", "de192eba6efb708b72782788d2ec02fc": "h = \\frac{|E(\\mathbb{F}_p)|}{n}", "de19436c15241ff90d7c9f8ba20f6729": "\\mathbb{D}^q f(x) = \\lim_{h \\to 0} \\frac{1}{h^q}\\sum_{0 \\le m < \\infty}(-1)^m {q \\choose m}f(x-mh).", "de1949fb7745e189e6938f2527dd3ea0": "\\rho:G\\rightarrow PSL(2,\\mathbb{R})", "de1957aeab6ef9a458ab769862e06c7a": "R_l[0 \\dots n/l)", "de1957c2948a6b65005e1c2c03d34bb9": " \\frac{dX}{dt} = -X^3+ \\cdots \\text{ and } \\frac{dY}{dt} = (-1-2X^2+\\cdots)Y", "de1980326e337388a953097e5ff82b01": "R_n^m(1)=1", "de19b6cdc4838bf683ab2979d22e2201": "p_{ij}=n_{ij}/n", "de1aae7b1509fdea5adb0724706d8325": "-0.5 .. 7.4999", "de1b024826159aaf2354c1d611737980": "\\hat{a}_{j}^\\dagger\\hat{a}_{j}", "de1b4b6ee22ff102a151e43b7bf0677d": "(2^\\alpha)", "de1b76b2f64e4f81877a0181e81355ef": "\\mathbf{u}\\,\\!", "de1bb49bbd868b8504f106606f19a608": "1.3108", "de1c20cf194f0a96d96c50778f26ed81": "E_{c,t_0}", "de1c58b72db1043bcbfe64cea560bfbe": "\\mathbf{a} = \\mathbf{A}-\\mathbf{C},", "de1c5fae6c596747115a13e1eb4d4477": "\\beta_A=\\sum w_i \\beta_i", "de1c9037aac2406272be1718b5808498": "k=\\frac{\\kappa}{\\rho C} \\,", "de1d0681af2868b57e67a21d81739ca6": " P(X_{i,i'}) = \\frac{1}{M}\\, ", "de1d50cf4dd270426c0ae4397197d876": "-\\bold{\\nabla_{\\bold {r_0}}\\cdot} \\bold{p} ( \\bold{ r}_0 ) = \\rho_b \\ . ", "de1da16db020bff8e55737455c799053": " Y[f,z] \\equiv :f(\\frac{b(z)}{0!},\\frac{b'(z)}{1!},\\frac{b''(z)}{2!},...): ", "de1daa47620480f2cd87b689f753e1f5": " z_{21} \\,", "de1dd1c58d047cc52602271c2ad294f4": "\\sigma_1 - \\sigma_3 \\ge \\sigma_0", "de1e45c979b62e7b15997b103995135f": "\\mathbf{Z}^*_{n}", "de1e69a539168313542e73f74fb85c67": "Uf(x) = f(Tx) \\, ", "de1e82d644991073bdd98dda9e7322eb": "\\langle S\\mid\\;\\rangle", "de1e8c73abcd581b6c9ad4fbda9b4b2d": "(4 (1 (\\lambda 1 5) 3) (10 (\\lambda 2 (\\lambda 2 (1 6))) 6))) 8) (\\lambda 1 (\\lambda 8 7 (\\lambda 1 6 2)))) (\\lambda 1 (4 3))) (1 1)) (\\lambda \\lambda 2 ((\\lambda 1 1) (\\lambda 1 1))))", "de1ef6e98fe6a2df9c307e3aae7484c5": "R=k[x_1, \\ldots, x_n]/\\langle f_1, \\ldots, f_k\\rangle, ", "de1efe04976576aff657b41a4f0c1ca3": "A \\in K^{n \\times n}, x \\in K^n", "de1f2828739d38762b6af0681deed8ed": "A=1,\\ldots, N", "de1fc0a10c9874c527fcd7f3d617ddf1": "P = I - 2 v v^\\text{H}\\,", "de1fc7cf22dff45cf2b9a96243e0d839": "C(\\alpha)_{n+1} = C(\\alpha)_n \\cup \\{\\beta_1+\\beta_2,\\beta_1\\beta_2,{\\beta_1}^{\\beta_2}: \\beta_1,\\beta_2\\in C(\\alpha)_n\\} \\cup \\{\\psi(\\beta): \\beta\\in C(\\alpha)_n \\land \\beta<\\alpha\\}", "de1fd8dc1de0bd4adf41cb588dd5bdcd": "V^* \\otimes V^*", "de20495fa54f75d45cec3bdb05392c03": " H_n(\\theta) ", "de205efd68dc060a7922b70bde6aea17": "\\mathbf{S}= \\begin{pmatrix} S_{xx} & S_{xy} \\\\ S_{yx} & S_{yy} \\end{pmatrix}.", "de20ef50d9d2b844bb84a9878f5eb5c9": "\\,\\phi_k", "de20f9c02a2de33e275037a123f75625": " (\\Pi_1 * \\Pi_4) = ", "de21446bf096d34264632bbf8e86953a": "B\\in\\mathcal{N}\\,", "de21673cc7434a69c31aff454c6fafef": "\\boldsymbol{\\omega}=\\nabla\\times\\mathbf{v}. ", "de21a6138291604890280461b62c6c23": "\\mathcal{F}(f(x))", "de21d3ca0213585fb4602d156e61a568": "r_{1} = r_0 - \\frac{f(r_0)}{f'(r_0)}.\\,\\!", "de2206f9531d5b41c1a95d555b111c24": "\\mathbf{p} \\in \\mathbb{R}^3", "de220ac21415041db8ce242558bd5a11": "{x'}^i, \\;i=0,1,\\dots", "de224efa59761597178c66671893748d": "\n\\overline {w^'\\theta^'_v}=\\overline {w^'\\theta^'}+0.61\\overline{T}\\; \\overline {w^'q^'}\n", "de22bbb9b20a13b3c239c3fa30377617": "F_d = 6 \\pi\\,\\mu\\,R\\,v\\,", "de22e71c37f9087d840cfe14321aecac": "\\Delta t^2 = \\left[ \\int^{\\Delta\\tau}_0 e^{\\int^{\\bar{\\tau}}_0 a(\\tau')d \\tau'} \\, d \\bar\\tau\\right] \\,\\left[\\int^{\\Delta \\tau}_0 e^{-\\int^{\\bar\\tau}_0 a(\\tau')d \\tau'} \\, d \\bar\\tau \\right], \\ ", "de23208a6c90a38fb82f01a7b9dbff17": "\\sqrt{-\\tilde{g}}=\\Phi^{-d/(d-2)}\\sqrt{-g}", "de2358e517ece675cb1ad30769f9ffd8": "\\phi=p/\\rho_0,\\,", "de241fadf7da7c4660a43316bc5cb319": " { E_1 \\over c^2} = \\gamma_1 \\gamma_2 m_1 - \\gamma_2 \\mathbf{p}_1 \\cdot \\mathbf{u}_2 ", "de24461b9092c04a141df18863810529": "\\langle a,b \\mid a^2, b^3 \\rangle\\,\\!", "de244b3785e693655feab29590c3023f": "\\sqrt{a} + \\sqrt{c} \\le 1. \\ ", "de246a57b781b2b58cf89854dce2fa09": "\\Bbb{Z}_3=\\{ e, (1\\ 2\\ 3), (1\\ 3\\ 2) \\}", "de24b6bf11f203f3306e49632f6200e9": " \\epsilon = h \\nu \\, ", "de24cc2b2d0f436af109ed56b06b9af0": "E_{m}\\, ", "de24fc2bb06d6b6711f06cd097542136": "A_{1} \\subset A_{2} \\subset A_{3} \\subset \\cdots.", "de2511576894afaa7452893b06d0579d": "dy=f(x,y)dx", "de251aa26feb7c050d11b9fd56d1c36d": "\\mathbf{H}\n=\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 1 \\\\\n1 & 0 & 0 & 1 & 1 & 0 \\\\\n\\end{pmatrix}\n\\sim\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 1 \\\\\n0 & 1 & 1 & 0 & 1 & 0 \\\\\n\\end{pmatrix}\n\\sim\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 1 & 1 & 0 & 1 & 0 \\\\\n0 & 0 & 1 & 1 & 0 & 1 \\\\\n\\end{pmatrix}\n\\sim\n\\begin{pmatrix}\n1 & 1 & 1 & 1 & 0 & 0 \\\\\n0 & 1 & 1 & 0 & 1 & 0 \\\\\n1 & 1 & 0 & 0 & 0 & 1 \\\\\n\\end{pmatrix}.\n", "de2534a8aacdfd748ac777709e91df0a": "x=\\cfrac{(m_1 + m_2 + ... + m_{n-1}) - s}{n-2} = \\cfrac{ (\\sum_{i=1}^{n-1} m_i) -s}{n -2}", "de258020027fcd53acd039200c393ea7": "\\mathsf{id}", "de25921c4555e8b84f68cf16d713afa3": "V^e_n", "de25af170787305648be302de0b6f4d8": "316 \\times 12", "de2631c21d4a9da5011ad4fcd443c9da": "A_{ji}=1", "de26768abf7f8362ba84f25096b29644": "(E,\\,G)", "de26ae52ad057996131595944fbae945": " \\omega_0", "de26e4afeafe54bf108cb211cf1fce99": "\\scriptstyle \\sqrt{6}r2/3", "de26e88b2bc47e5a2f1988432bcdcda8": "\\mathrm{stsys}_2{}^n \\leq n! \\;\\mathrm{vol}_{2n}(\\mathbb{CP}^n)", "de279f39e03705c0d3f2ef39d6aadab3": " N_{B}=-D_{BA} \\frac{dP_{B}}{RT}dx=D_{AB} \\frac{1}{RT}\\frac{dP_{A}}{dx}", "de27f6a4c127d09d192db37f83715370": " \\operatorname{Var}(X) = \\frac{1}{n} \\sum_{i=1}^n (x_i - \\mu)^2. ", "de2805a20d6694751e9c0e4d69e3cb47": "x_{k}^{\\left(rr'\\right)}=x_{N-k}^{\\left(rr'\\right)}", "de288fba68e52d210a4c92e4eddf6f3c": "c_{3,1}(\\widehat{a}, \\widehat{b}c, \\widehat{d})", "de28c6a8e2ba30935f5df19bbe3d6762": "e^{-st}", "de2931a18787866251c530d34d701b77": "T=\\frac{dt}{\\sqrt{\\left( 1 - \\frac{2GM}{rc^2} \\right ) dt^2 - \\frac{1}{c^2}\\left ( 1 - \\frac{2GM}{rc^2} \\right )^{-1} dr^2 - \\frac{r^2}{c^2} d\\theta^2 - \\frac{r^2}{c^2} \\sin^2 \\theta \\; d\\phi^2}}", "de294063a7ae879a80dbe93f732a1ffa": "c\\ = n_m + n_{m-1} + \\cdots + n_0", "de296d686b2c91c6b88325497cb268c6": "\\varepsilon_i: \\ \\vec n_i\\cdot\\vec x=d_i, \\ i=1,2,3 ", "de29e07f912b64c7ad5a93a75a432adc": " \\dot{\\tilde{\\rho}}= - \\sum\\int^t_0 dt' \\operatorname{tr}_R\\{[\\alpha_i(t) \\Gamma_i(t),[\\alpha_j(t') \\Gamma_j(t'),\\tilde{\\rho}(t)R_0]]\\} ", "de2a534d9cf005b8075b1cbdf54f3467": " f(x) = \\sum_{i=1}^\\infty a_i K_{x_i} (x) ", "de2a61ca506925a8fb7743235ebb0e57": "s \\in G_u \\Leftrightarrow i_G(s) \\ge u+1.", "de2a70b42c4d25064f7ca63bd70fdd71": " \\sum_{n=1}^\\infty \\frac{\\lambda_i^n}{M_i(n)} < \\infty ", "de2a7d34448d02c84ae1747a23ea9df6": "\\ \\alpha_{i,j,k}\\ and\\ \\beta_{l,m,n} ", "de2a9bbc5229ab4582e71baa1210001e": "P = \\frac{\\pi^2 B_\\text{p}^{\\,2} d^2 f^2 }{6k \\rho D},", "de2b7630043a05d941e3753562d31555": "E\\left[ x_i^3 x_j\\right] = 3\\Sigma _{ii} \\Sigma _{ij}", "de2c1732fbae88596d4db00fba17051d": "\\delta \\sim \\Omega_{\\perp} \\sim \\frac{\\pi}{\\tau} \\sim \\frac{\\pi v}{L}", "de2c2b130b4eab72979953e3ba547530": "\\tan\\alpha\\!", "de2c3452ce4c96b73b5670ee0f35ee4a": " r={\\frac {\\sqrt{ \\left( \\left( {\\it x_2}-{\\it x_1} \\right) ^{2}+ \\left( {\\it y_2}-{\\it y_1} \\right) ^{2} \\right) \\left( \\left( {\\it x_2}-{\\it x_3} \\right) ^{2}+ \\left( {\\it y_2}-{\\it y_3} \\right) ^{2} \\right) \\left( \\left( {\\it x_3}-{\\it x_1} \\right) ^{2}+ \\left( {\\it y_3}-{\\it y_1} \\right) ^{2} \\right)} }{ 2 \\left| {\\it x_1}\\,{\\it y_2}+{\\it x_2}\\,{\\it y_3}+{\\it x_3}\\,{\\it y_1}-{\\it x_1}\\,{\\it y_3}-{\\it x_2}\\,{\\it y_1}-{\\it x_3}\\,{\\it y_2} \\right| }}", "de2c4aaa843892f234d084cbb826ad2e": "g_k = \\frac{k(3k -1)}{2}", "de2c67d76239986401574f2821bbd0b3": "\n\\begin{align}\n\\sigma^2=E(X^2)-(E(X))^2\n&=\\frac{1}{n}\\sum_{i=1}^n i^2-\\left(\\frac{1}{n}\\sum_{i=1}^n i\\right)^2 \\\\\n&=\\tfrac 16 (n+1)(2n+1) - \\tfrac 14 (n+1)^2\\\\\n&=\\frac{ n^2-1 }{12}.\n\\end{align}\n", "de2c9f8f381708c5dc7349b82af73f0f": "\\tan\\frac{\\pi}{20}=\\tan 9^\\circ=\\sqrt5+1-\\sqrt{5+2\\sqrt5}\\,", "de2d7d93160f89f842c814f453f7dbe0": " \\{u_i, u_j\\}, \\quad 1\\leq i,j\\leq 2n ", "de2df9f23d739a8b3f5b1e4d46f73318": "f^{\\star\\prime}(x^\\star) = x(x^\\star):= \\arg\\sup_x {\\langle x, x^\\star\\rangle}-f(x);", "de2e21c678f1c6d2ba096a37b03b6799": "X \\preceq Y \\Leftrightarrow Y - X ", "de2eaaec41b74c77269cfaffaffd2400": " \nQ=\\left[\n\\begin{array}{c}\nQ_0 \\\\\nQ_1 \\\\\n... \\\\\nQ_{Ne} \\\\\n\\end{array}\n\\right]\n", "de2ef435d00d00904278d3602d7b2056": "\\zeta(s,q)", "de2f230ce97f8521f91c66506ada5b1b": " - \\frac{GmM}{|\\mathbf{r}|^2} \\mathbf{\\hat{e}}_r + \\mathbf{R} = m\\frac{{\\rm d}^2 \\mathbf{r}}{{\\rm d} t^2} + 0 \\Rightarrow - \\frac{GM}{|\\mathbf{r}|^2} \\mathbf{\\hat{e}}_r + \\mathbf{A} = \\frac{{\\rm d}^2 \\mathbf{r}}{{\\rm d} t^2} \\,\\!", "de2f36ae522fc82246111dcd8f37683e": " \\frac{r_B}{r_A} = \\frac{N_B}{N_A}.", "de2f39620be18f2aaa5cb8b8c28b208f": " \\bold{ (D-P) = -\\nabla } \\varphi \\ , ", "de2f635ef823660a7fe214a83f6531e1": "\\in \\lceil TC(A)", "de2f7c7d3d2f1f862bb9156d833da5ba": "\\Box \\phi = 0", "de2fca968c808ced098bc64e3af90526": "0 \\epsilon} := \\mu^{-1}((\\epsilon, \\infty))", "de3063d7d186b063923a76c59b75e51d": "\\mathcal{O}(N^2)", "de30beed2e8b911a58c4f5f2df32df5f": " \\phi(\\vec{r}) = \\frac{1}{4\\pi}\\iiint_{\\vec{r}'} \\frac{\\vec{\\nabla}_{\\vec{r}'} \\bullet \\vec{E}(\\vec{r}')}{\\|\\vec{r}-\\vec{r}'\\|}d\\tau' = \\frac{1}{4\\pi}\\left(\\iiint_{\\vec{r}'} \\vec{\\nabla}_{\\vec{r}'} \\bullet \\frac{\\vec{E}(\\vec{r}')}{\\|\\vec{r}-\\vec{r}'\\|}d\\tau' - \\iiint_{\\vec{r}'} \\left(\\vec{E}(\\vec{r}') \\bullet \\vec{\\nabla}_{\\vec{r}'}\\frac{1}{\\|\\vec{r} - \\vec{r}'\\|}\\right)d\\tau'\\right) ", "de30cbb2caa04fac7611373616b91385": "W_B", "de30f09a8d42d7510df36a391e8bf03e": "N = \\left \\{S, B\\right \\}", "de31151a0098cb55dafdcfa19c878a8a": " (f,g)=\\int \\int {f(z) \\overline{g(w)}\\, dz\\, dw\\over |z-w|^{2-t}}.", "de3141d3262ce7d42d8100177fd4d372": " E(r) = O(r)", "de316147e199f951fe92e3f5c6d16fbd": "\\frac{\\mathrm{d}}{\\mathrm{d}x}\\mathrm{coversin}(x) = -\\cos{x}", "de3181b497f57563d3271ccc3e760bd8": " \\Phi = B\\times S ", "de31babe9bcf58aa4d43120d9b170d63": "J_\\nu(z_1\\pm z_2)= \\sum_{k=-\\infty}^\\infty J_{\\nu \\mp k}(z_1)J_k(z_2);", "de32217e63695ba34a6e2947bc42deb6": "R \\to Q", "de3268d13b3e4bdde9cc91a035e38572": " P = C_\\mathrm{p} \\cdot \\begin{matrix} \\frac12 \\end{matrix} \\cdot \\rho \\cdot S \\cdot v_1^3 ", "de329b5c3ea55501d426281483af90f0": "R = r N^x", "de32c94452b485c3949ac6e596fdde0e": "E(|X_n|)=1\\ ,", "de32d582280511d78fdbdd51e37fd38b": "(a \\diamondsuit b)_n = \\sum_{j=0}^n {n \\choose j} a_j b_{n-j}.", "de333a6a54c5b247f5e7d401a1b00630": "\\{\\beta_{i,0}, \\beta_{i,1}, \\ldots, \\beta_{i,m_i-1}\\}", "de3347a250806d7c7ccf596b8b304998": "(M,0,0,0)", "de3363a02b7e002be58f6f5e1cb819a0": "0 = -Q_{i} + \\sum_{k=1}^N |V_i||V_k|(G_{ik}\\sin\\theta_{ik}-B_{ik}\\cos\\theta_{ik})", "de336aa2e49e7b0035be8e94ccf5b689": "\n\\bar{Y}=\\frac{1}{n}\\frac{K_I[X] + 2K_{II}[X]^2 + \\ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \\ldots +K_n[X]^n}\n", "de33a13469878b025b828f9fcc3c120c": " S^T S = S^2 ", "de33cac619508a5652de8e872c6db0fc": "\\mathbb P(n_i = k) = \\frac{X_i^k}{G(N)}[G(N-k) - X_i G(N-k-1)]\\quad\\text{ for }k=0,1,\\ldots,N-1,", "de33fc89a32693ce0ee16b3d24f46488": "dm+a", "de341209deea2266193907b25814155d": "M = 11 \\cdot 13 \\cdot 17", "de34b230fd9980bf163cdfcec503e2ae": "p_5", "de34c1fb7746401680cdf8cbd15848aa": "F(r) + \\frac{L^2}{mr^3} = m \\ddot r ", "de34f117aa9efb41b9f955da78eedd42": " \\gamma(\\tau)=\\langle X(t) X(t+\\tau)\\rangle", "de3514419edaab7ceb6f2995084e907e": "P_{SO(2n+1)_{}}(x) = (1+x^3)(1+x^7)...(1+x^{4n-1})", "de3567c395ef5d330866462bf7326cd3": "T(n) = 1/n + 2/n + 3/n + ... + n/n = (1 + 2 + 3 + ... + n)/n = (n+1)/2", "de356cab3c713cda64dea5eb5e9934e6": "\\Delta p = p^{\\star}_{\\rm A} - p = p^{\\star}_{\\rm A} (1 - x_{\\rm A}) = p^{\\star}_{\\rm A} x_{\\rm B}", "de35e4656723612e91c2a98d817dfe09": "v = \\lambda w", "de35e97735e68debff9fafc1925726e7": "y_b = b_0 x^c \\sum_{r = 0}^\\infty \\frac{(c + \\gamma -1 )(c + \\alpha )_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^r.", "de360274166ba6dbcecce5e4d53c474b": "\\lambda_{mn}(c)", "de3623064853bb526e45bcb43ae91277": "\\Delta [\\psi] (x) = \\frac{1}{x}", "de366cafd2301dfaa4c4cb18c2e97f8f": "\\sum_{i=0}^{n-1} X_i \\omega^i", "de366f8182b6be8456c5beaf8b649254": "Z=\\sum_i e^{-\\varepsilon_i/kT}", "de367c98d44194b3634f092f3a0e2bdc": "(x^2+y^2)^2=dx^2+ey^2+f \\, ", "de371cb7d9530aa944dd9ffa0f08ab86": "\n \\begin{bmatrix}\n 1 & -\\mathbf{c}^T & 0 \\\\\n 0 & \\mathbf{A} & \\mathbf{I}\n \\end{bmatrix}\n \\begin{bmatrix}\n Z \\\\ \\mathbf{x} \\\\ \\mathbf{x}_s\n \\end{bmatrix} =\n \\begin{bmatrix}\n 0 \\\\ \\mathbf{b}\n \\end{bmatrix}\n", "de374192cc8f8a55edad7c0607ac8ca9": "I_{0,\\mathrm{Airy}}", "de37471c8eed62f74c376e4a917443a9": "A, C", "de377d572643bde902a64be42278c489": "\\ \\Delta X_{ik}", "de37a6aed304de774c3ac5104d2dcfe1": "u_i(\\overbrace{\\mathbf{x},z_1,z_2,\\dots,z_i}^{\\triangleq \\, \\mathbf{x}_i})=-\\frac{\\partial V_{i-1}}{\\partial \\mathbf{x}_{i-1} } g_{i-1}(\\mathbf{x}_{i-1}) \\, - \\, k_i(z_i \\, - \\, u_{i-1}(\\mathbf{x}_{i-1})) \\, + \\, \\frac{\\partial u_{i-1}}{\\partial \\mathbf{x}_{i-1}}(f_{i-1}(\\mathbf{x}_{i-1}) \\, + \\, g_{i-1}(\\mathbf{x}_{i-1})z_i)", "de37cc24f9643d576bf73df108c799b3": "|\\Psi\\rangle_\\nu=|\\phi_1,\\phi_2,\\cdots,\\phi_n\\rangle_\\nu = |\\phi_1\\rangle|\\phi_2\\rangle \\cdots |\\phi_n\\rangle", "de383053c063c0f8764f069598f64893": "* \\leq m", "de388676b1ae246820e5d52306039718": "a_b = \\nabla_b Z_1 \\qquad Z_1 = \\ln{gtt}", "de389ea3d3223a95d52542030939de43": "\\,\\psi_{\\theta}^{-1}(t)", "de38e5f88bcf373cd09dc5be6f6562fc": "i=1, 2, \\ldots, n-1", "de38e9cffc7dc73ed9382d8586393489": "\n\\begin{pmatrix}\n l_1 & l_2 & l_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n \\approx (-1)^{l_3+m_3} \\frac{ d^{l_1}_{m_1, l_3-l_2}(\\theta)}{\\sqrt{2l_3+1}}\n", "de38f4de8927f0f22c3024d80c5f68d9": "u\\in C^{\\prime}", "de390a86b14c5fe65de624de4d3bee87": " Assets = Equity + Liabilities ", "de3927c07f6c372da6cabfaf455b6bbb": "\\{m\\{nx\\}\\} = \\{mnx - mk\\} = \\{mnx\\} \\,\\!", "de39b783a98b5a3cd68064eeae0356c0": "2x \\times 4y \\div 6z + 8\n - \\frac {y}{z^2} = 0", "de39ddc94425e9e3a8b31aee90d1be46": " \\langle \\pm |H| \\mp \\rangle = \\frac{1}{2} hA \\sqrt{(I + 1/2)^2 - m_F^2}", "de39e943810734f6b4f842b971f2e66c": " \\vec{s}(C_{-1}^{(6)}) = [+1,-1,+1,-1]. ", "de3a4a3c8a852c55875e5e8ab5e3e477": "Z \\subseteq \\{x: u(x) = -\\infty\\}.", "de3adfc3ab9d882f8b57fc73aed4cd65": " A(s) \\Leftrightarrow A^{\\times}_A + V^{\\bullet\\bullet}_X + 2e^{'} ", "de3ae288c6c45006ea72e1d897e903d4": "(15)\\quad\\quad u_s = u_1 + c_1 \\sqrt{1 + \\frac{\\gamma+1}{2\\gamma} \\left( \\frac{p_2}{p_1} - 1\\right)},", "de3b71082e77871a274ba3e142aca3ed": " \\chi(\\lambda) = (\\lambda + 8 \\pi \\sigma)^3 \\, ( \\lambda - 8 \\pi \\sigma )", "de3b77695abd4975eec58f428a3953f9": " q\\lambda", "de3b8c0eee30ab6002e8f82b0b41eb52": "t=u+v=0 , \\qquad t=\\omega_1u-{p\\over 3\\omega_1u}=\\sqrt{-p} , \\qquad t={u\\over \\omega_1}-{\\omega_1p\\over 3u}=-\\sqrt{-p} ,", "de3bd650c1e6ee5dbe3974a983454aed": "A \\subset \\mathbb{N}", "de3bdcbc445f72e90bfb0b7f9e4907d7": "(a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) = 1", "de3bdde388d5e37a64d06bf07de6e173": "3 + 5 \\equiv 1 \\pmod 7.", "de3beec60a555f9e359d419ddf311860": "\n D\\left(w_{,1111} + 2w_{,1212} + w_{,2222}\\right) = \n -q(x,t) - 2\\rho h\\ddot{w} + \\frac{2}{3}\\rho h^3\\left(\\ddot{w}_{,11}+\\ddot{w}_{,22} + \\ddot{w}_{,33}\\right) \\,.\n", "de3cb7e6a63681bc88e97eb65c8b9d0c": "A = \\frac{nsa}{2} = \\frac{pa}{2}. ", "de3cd02acab767049c3f42bfe956e1bd": "a^2 = b^2 + c^2 - 2bc\\cos\\alpha\\,", "de3cd59ca4e41bcf9afce7a811688513": " l = m - 2 log \\frac {1} {\\epsilon} + O(1) ", "de3d405d97a530bd7cac14489a341b14": "[\\mathcal{L}_n(f)](x) = \\sum_{k=0}^\\infty {(-1)^k \\frac{x^k}{k!} \\phi_n^{(k)}(x) f\\left(\\frac{k}{n}\\right)}", "de3d6399d1d9bc992566b8abe9df6190": "|\\Phi^-\\rangle_{AB} = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_{B} - |1\\rangle_A \\otimes |1\\rangle_{B})", "de3d8152a07d71d4e59c520c3ddbf948": "\\sigma = \\frac {\\widehat \\alpha} {\\sqrt n}. ", "de3dd406200e122295eeadf518dafc8b": " \\Lambda_\\gamma ", "de3de1e1c3a52203e7a5cdc27d9f91bf": "\\int_{M} d (x, x_{0})^{p} \\, \\mathrm{d} \\left( X_{*} (\\mathbf{P}) \\right) (x) \\equiv \\int_{\\Omega} d (X(\\omega), x_{0})^{p} \\, \\mathrm{d} \\mathbf{P} (\\omega),", "de3eaf372c314e095badcfcd73b22e42": "AM \\ge GM \\ge HM", "de3ec4803bf01d07e1cd5811a62fd3a1": "f(S^{j-1} \\times \\{0\\}) \\subset M", "de3ef90e29fb09c04968a25682b5867f": "u(x,y) = f(y),", "de3f665bca1873d4aebd5ac0768c4bda": "\\operatorname{erf}(x)=\\operatorname{sgn}(x) P\\left(\\tfrac12, x^2\\right)={\\operatorname{sgn}(x) \\over \\sqrt{\\pi}}\\gamma\\left(\\tfrac12, x^2\\right).", "de3fca81414ab193449070f33e7737db": "\n \\boldsymbol{B} = \\lambda^2~\\mathbf{n}_1\\otimes\\mathbf{n}_1 + \\lambda^2~\\mathbf{n}_2\\otimes\\mathbf{n}_2+ \\cfrac{1}{\\lambda^4}~\\mathbf{n}_3\\otimes\\mathbf{n}_3 ~.\n ", "de3ff39901de5a38a310629029052b49": "{\\Bbb C} \\times {\\Bbb R}", "de406026d0e10344c5812cb808ae0c6b": "{\\Delta v_x}/{\\Delta y}", "de407cc8c501534295271222b6cce519": "\\delta(\\emptyset)=\\emptyset", "de40b936a11e735730e569c41f42f59b": " p(\\tilde{x}|m) ", "de40def2f053dce3d5a8dbea798a7492": "\\!\\gamma_j'", "de4112e13ce8338489f352477314688d": "\\neg A \\vee \\neg B", "de414babf11b93b72f1bccf3d089173b": "\\sum_{i=1}^{N} V_i = V ; \\qquad \\sum_{i=1}^{N} \\phi_i = 1", "de417c01917903bb295a80f8f72fbfe8": "(x_1,y_1)\\,", "de418cbcfda1eaca440ba8756f98a66a": "{\\rm cov}({\\mathcal K}) \\le {\\rm non}({\\mathcal L})", "de419f17e2aa6420ed5236cfe9d6485e": "\\mathbb{C}^n = \\bigoplus_{i = 1}^l Y_i", "de41cbb0d87f05a8abac20e9eaace1b6": "\\Gamma(W) ", "de41f86e42a74b61a37b3a76b7f5edfb": "n > 0", "de4222afd9d8342a2ade102fc7c4837c": " c_k = \\frac{f(b_k) a_k- \\frac{1}{2}f(a_k) b_k}{f(b_k)-\\frac{1}{2}f(a_k)}", "de422348b1f8fb1ce315a77933233d3f": "\n \\frac{\\partial g_{\\alpha\\beta}}{\\partial x^k} = \\frac{\\partial X^\\gamma}{\\partial x^k}~\\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma}\n", "de437c486d4d1fd22f620d54d122844a": "x\\ne s", "de442f0b8c7223d15c81d7a077dc7974": "\\{(x,y_0) \\in X \\times Y: y_0 \\text{ is constant}\\} = X \\times \\{y_0\\}", "de4445c94ec57bbe667702a945e948f6": "\\varepsilon < \\tfrac{1}{4}\\,", "de44507dbaa2f1a1343bdadc7d8e30d1": "\\frac{g-1}{2}.", "de44543a9574373e788528a766f074c1": " \\|\\vec{E}\\|=\\|\\vec{N}\\|=1", "de448e7d3203f6e463140b52d3ea85dd": " 2a \\,", "de44b348818f8ee65564e1ed0d0daa20": "\\mathbf{v}_0(x)", "de44c582df9d8d29dbbd70aca311c641": "\\leq", "de44ef30c90e9e7e41ad555445b5f944": "(\\vee) \\frac{X \\cup \\{A \\vee B\\}}{X \\cup \\{A\\}|X \\cup \\{B\\}}", "de45024f786751af9750dd611cf2297f": "p_c\\,\\!", "de458cf2cbe60cadda568f662ddbbd5b": "z^\\prime = z/\\gamma", "de45fa60108798bf1b51599174cfe74c": "\\phi_1(\\alpha) = \\varepsilon_\\alpha", "de464c687605138484fcfd3785c9537c": "\\begin{align}\n \\mathrm{Area}(r) &{}= \\int_0^{2\\pi r} \\frac{1}{2} r \\, dt \\\\\n &{}= \\left[ \\frac{1}{2} r t \\right]_{t=0}^{2 \\pi r}\\\\\n &{}= \\pi r^2.\n\\end{align} ", "de4660b754a98a9186e36b4ac9836600": "\\forall x(x\\cdot f(x)=e\\text{ and }f(x)\\cdot x=e)", "de46b0f3480329bd45d2be111edc3467": "M_{200} = M(-C h(\\alpha_1)h(\\alpha_2)\\cdots h(\\alpha_n) \\log \\left (\\max\\{\\vert\\beta_1\\vert,\\ldots,\\vert\\beta_n\\vert\\} \\right ),", "de6efed59ce0a3e9fa27da95302eff31": "\\sum_{n=-\\infty}^{\\infty}x[n]z^{-n} \\to \\infty.", "de6f345f63ac045cdce45f9e807addd3": "0 < \\epsilon < 0.5 ", "de6f4dbbe71bf4d366e913b4657c5df9": "\\begin{align}\nNa^{\\dagger}|n\\rangle&=\\left(a^{\\dagger}N+[N,a^{\\dagger}]\\right)|n\\rangle\\\\&=\\left(a^{\\dagger}N+a^{\\dagger}\\right)|n\\rangle\\\\&=(n+1)a^{\\dagger}|n\\rangle,\n\\end{align} ", "de6fe8e1aeacad82b76b0b243cb1332c": " A \\and B ", "de708c70adf65c2a40bb9e81abffe65c": "k = k_0P^aT^b", "de70a9bcc5854a6ee666be0889d34327": "t=\\sum_{m=0}^\\infty t_m=T\\sum_{m=0}^\\infty R^m\\,e^{im\\delta}", "de70ccc4376c3abdb4af3f00bc182a2c": "\\text{Total}_{M+1} = Total_{M} + p_{M+1} - p_{M-n+1} \\,", "de70d51b47eb4d0dae86cf3e5e2936e7": "16K^2 = 16(s-a)(s-b)(s-c)(s-d) - 16abcd \\cos^2 \\left(\\frac{\\alpha + \\gamma}{2}\\right)", "de71510698c4ef12b2fba77f70c1706b": "U_i = \\frac{X_i-\\mu}{\\sigma}", "de71b72c4f62f20d117414dd2f33ca25": "\\partial S", "de71c14a70e450225d388ba9c1787280": "- \\mathbf X^{\\rm T} \\mathbf y+ (\\mathbf X^{\\rm T} \\mathbf X )\\hat{\\boldsymbol{\\beta}} = 0,", "de725b24c7ac8c4bcea19ddbb3deb018": "\\displaystyle{V=\\ell^2(A),}", "de72735b6ea3b2215c124282a5a5eb41": "F_n(S)", "de72aabb0cf5580071906caff5ffb6d9": "\\hat P_{MU}(e^{j \\omega}) = \\frac{1}{\\sum_{i=p+1}^{M} |\\mathbf{e}^{H} \\mathbf{v}_i|^2},", "de7339f0891b9d0c05324d80bb65e1af": "\n\\frac{c_{s'}}{2\\lambda c\\cdot x^*}\n>\n\\sum_{e\\in s'\\cap\\mathcal U} \\prod_{s\\in \\mathcal S, s\\ni e}(1-p_s)\n", "de734ef7a73449722d48895d6d5a480b": "A_r+A_l=B_1\\,\\!", "de73c3dadb0e9c04a6cdec484cdfee29": "\\,\nH= \\omega J\n\\,", "de73c6a63b2743918c2e2eac3e364058": "O(|V|^2 \\log |V| + |V| |E|)", "de73d55e66e3d3e6fcae23588d726344": "\\ell^{\\ p}(X)", "de742730ff73b8c14ad64468ccd3c3ff": "\\Pi_0(x) = \\lim_{\\varepsilon \\to 0}\\frac{\\Pi(x-\\varepsilon)+\\Pi(x+\\varepsilon)}2.", "de747264443cc58a0b49b7052916cad7": " T = \\beta( 1 + 2X^2 + 2X( 1 + X^2 )^{ 0.5 } ) ", "de747c3ddc01e023402f2775c6f57b55": " x \\oplus y = y \\oplus x,", "de74c192727fb79ce77410248e982849": "\nw_{kj}^U = \\max_i\\{x_{ij}^k\\}\n", "de74c7bcc8dd7dc998675828883bd617": "x*y = [x,e,y]", "de74d4b233a2a8ab15c8e10457a3334a": "S_{RBB}(n)", "de750d800d3bee19ec524bfaf6d97dd1": "xp_N(x)-p_N(x)=x^N-1", "de753557550547a61eaa78fc10535961": " \\sigma_p = \\sqrt {\\sigma_p^2} ", "de753a27d2da60535b46e9278800a443": "k=k_{20} \\theta ^{(T-20)}", "de753bef9f4529116896e18f7815aa7e": " simil(x,y) = \\frac{\\sum\\limits_{i \\in I_{xy}}(r_{x,i}-\\bar{r_x})(r_{y,i}-\\bar{r_y})}{\\sqrt{\\sum\\limits_{i \\in I_{xy}}(r_{x,i}-\\bar{r_x})^2\\sum\\limits_{i \\in I_{xy}}(r_{y,i}-\\bar{r_y})^2}} ", "de755482d7e74a1b71a2da5bb923a812": "m(\\overline\\Psi_R\\Psi_L + \\overline\\Psi_L\\Psi_R)", "de75c46f96269935854a8ecedccbdd07": "p(A(q),t)=p_{0}(A(q))\\rho_{1}(q,t)", "de76c92ecd110e52760ea596ee44e585": "\nV_\\mathrm{L} = { R_2 R_\\mathrm{L} \\over R_1 R_\\mathrm{L} + R_2 R_\\mathrm{L} + R_1 R_2}\\cdot V_s.\n", "de77912674fbea64f46e198fa89d5948": "q\\,\\!", "de7849f2fdebd068a5470740bac7d226": " p_u = \\left(\\frac{e^{(r - q) \\Delta t / 2}- e^{-\\sigma\\sqrt {\\Delta t/2}}}{e^{\\sigma\\sqrt {\\Delta t/2}}- e^{-\\sigma\\sqrt {\\Delta t/2}}}\\right)^2 \\,", "de787ec52d2aa85e2a2b4ab4bd61b3bf": "y = \\frac{\\ln((r \\times s) + 1)}{r}", "de789142180075357ebad31cfd22ee81": "b=\\pi/2", "de7892c64fab097e40505f0c6ce6f571": "a=n^{3}, \\,", "de78ad72e36c6ce7a53709fd6758c133": "\\frac{\\mbox{Dividends}}{\\mbox{Earnings}}", "de78f28295c765c9c401cd3ef72998c9": "\\partial_n\\sigma_n(\\Delta^n)=\\sum_{k=0}^n(-1)^k [p_0,\\cdots,p_{k-1},p_{k+1},\\cdots p_n]", "de78fbccccf3e6e3291b316068a819d2": "A^k = (1)-(2^k)-(3^k)-\\cdots-({Q^k}_k)-(N)", "de7951ce44cef36a37eb609f2d64a71a": "\\mathbf{A}' = \\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_x)\\mathbf{A}", "de7a18033a44fa9b24a78a87535d245e": "R \\ge 1 - H_q (\\delta ) - \\varepsilon ", "de7a89bcec946105d3e86bf105314ed6": "\\frac{\\partial X_M}{\\partial x_n}\\,\\!", "de7ac1bcbc0da305e916c7e9622951c5": "\\bar{S}_{ij} = \\frac{1}{2} \\left( \\frac{\\partial \\bar{u}_i }{\\partial x_j} + \\frac{\\partial \\bar{u}_j}{ \\partial x_i} \\right)", "de7aeae00c5cc373971174f2ab1e046a": "(SAT, \\epsilon-UNSAT) \\in PCP(O(\\log n), O(1))", "de7af81caeedfc6f3681acdd9de314fb": "\\begin{align}\\bigwedge_{j\\in J}\\bigvee_{k\\in K_j} x_{j,k} = \n \\bigvee_{f\\in F}\\bigwedge_{j\\in J} x_{j,f(j)}\\end{align}", "de7b0678824c086e3e32b2706746bc9d": "\\tfrac{BW}{DY}", "de7b2cbb94c61a17c0a5241038f47eeb": " \\langle A \\rangle_\\rho \n= \\frac{ \\int D\\sigma A[\\sigma] \\exp(i\\theta[\\sigma])\\; p[\\sigma]}{\\int D\\sigma \\exp(i\\theta[\\sigma])\\; p[\\sigma]}\n= \\frac{ \\langle A[\\sigma] \\exp(i\\theta[\\sigma]) \\rangle_p}{ \\langle \\exp(i\\theta[\\sigma]) \\rangle_p} ", "de7b3621319ecc9e8b060d15bfbb8ec1": " O(|V|\\cdot|E|) ", "de7bc66986cd84134ad402376055f7c7": "L = \\frac{r^2N^2}{9r + 10l}", "de7c0efe39e490b0f53ca5916c55620c": "\\phi_n= e^{-|n|\\gamma}", "de7c24fa8b75ce487aad62bae023752a": "\\frac{d^2}{dx^2} \\sin x = -\\sin x.", "de7c843d958393f9400d472b8fb113ab": " \\scriptstyle d_j^k = \\begin{cases}\n\\scriptstyle 1, & \\scriptstyle if \\, j=k, \\\\\n\\scriptstyle 0, & \\scriptstyle otherwise\n\\end{cases}", "de7d138bff2cf7c0d6782b26cf5d95c9": "k_C(f)", "de7d98e27207873e1038ababc65aae36": "U(x) = \\frac{1}{cx+1} B\\left(\\frac{ax}{cx+1}\\right)", "de7dc3debc824a7fe41bab9ad720e1e5": "n!/(n-k)!k!", "de7e2adc989f3c43260e67a459abc95b": "\\operatorname{Trans}_{X_i}(a_{i,i+1})\n = \n\\begin{bmatrix}\n 1 & 0 & 0 & a_{i,i+1} \\\\\n 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix},\\quad\n\\operatorname{Rot}_{X_i}(\\alpha_{i,i+1})\n = \n\\begin{bmatrix}\n 1 & 0 & 0 & 0 \\\\\n 0 & \\cos\\alpha_{i,i+1} & -\\sin\\alpha_{i,i+1} & 0 \\\\\n 0 & \\sin\\alpha_{i,i+1} & \\cos\\alpha_{i,i+1} & 0 \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix}.\n", "de7e5aaaac4298daf8af20c2064338e5": "\\Bigg[{\\partial{V}\\over{\\partial{\\mathbf{r}}}}\\Bigg]_0", "de7ec67e17dcec45adf78fcf83b0ed4e": "n = pq. \\, ", "de7f57605d9b8d24b69b190c2fb4a7c5": "a = \\frac{1}{2}\\left(\\frac{u^2}{v}+v\\right), \\ \\ b = \\frac{1}{2}\\left(\\frac{u^2}{w}+w\\right), \\ \\ c = \\frac{1}{2}\\left(\\frac{u^2}{v} - v + \\frac{u^2}{w} - w\\right) ", "de7f8a6915379aec5ef142971ef4b55a": "\\hat \\rho = \\sum_i e^{\\frac{\\Omega + \\mu_1 N_{1,i} + \\ldots + \\mu_s N_{s,i} - E_i}{k T}} |\\psi_i\\rangle \\langle \\psi_i | ", "de7f8ac0633c021125f9fcffc1ad01cc": "\\int_0^{2\\pi} \\sin(m\\varphi)\\sin(m'\\varphi)d\\varphi=(-1)^{m+m'}\\pi\\delta_{|m|,|m'|};\\quad m\\neq 0", "de804716f89aa683b783c21a3a428966": "p_{n,k}'", "de80ced36b89165e42aa763c2486fae2": "\n\\Phi (f) \\approx 2 R k_B T.\n", "de80f797209c941fb640cf0c9d6f4b99": "2L_2 \\rightarrow L_2", "de8165d66ae30262eb6f521345ac2874": "a^2 -c\\,b^2 \\ne 0", "de81d6f11d47bdec803c15033f1b429c": "\\frac {d\\ln K} {dT} = \\frac{{\\Delta H_m}^{\\Theta}} {RT^2}", "de821de491d0cf272e7fd9b601488a98": "\\alpha\\in L", "de82fc85abf9afc4fab82b67db4642ed": "(P|Q)|R \\equiv P|(Q|R)", "de836a94e21031c554a97c1510bdf4fb": "\\delta(C)", "de83f71083c813892a5d99c502d8c026": "\\varepsilon_{33}\\,\\!", "de84377ba3c6d563cfd6af368ddd3439": "\\delta(q_0, a, a) = (q_0, aa)", "de845033f225f1c330a66b8e1032503e": " g_{n,m}", "de845209112313180e92efbba35da1bb": "(k_n)", "de8459aada57ac3749a4e649cd355d92": "(x)_{\\infty}", "de8465d723217241a3096b8763f598fe": "h_1", "de8467c947fc864f319220118b101fcf": " (x*z,\\,y*z)=(x,\\,y) * z", "de8490790d181f8125ac7191c06c41dd": "{x\\in s_i}", "de84ddf973db08666d9131c76f44f430": "\\left(\\frac{p-1}2\\right) \\left(\\frac{q-1}2\\right),", "de851b0bc77c390f50284e9d30a05a1d": " 1/(1-x) ", "de85851313691413df8b8c4d376b2dcb": " D[p] ", "de858d7fb246dd186f71f4fa3846952c": "\\frac{\\alpha\\delta K_1 \\left(\\alpha\\sqrt{\\delta^2 + (x - \\mu)^2}\\right)}{\\pi \\sqrt{\\delta^2 + (x - \\mu)^2}} \\; e^{\\delta \\gamma + \\beta (x - \\mu)}", "de85c37a21e15a17ab2becc8c0239771": "N/2+\\sqrt{N}", "de85e6ff8779952ff596e853fa37c472": " Y. \\,", "de86169d165559a6915a1b1cc51c5ce5": "C_1=C_2=C>0", "de86af2490e23d47d6fa62031648cdbc": "\\frac{\\operatorname{Cl}_2(\\theta)}{\\theta} = \n1-\\log|\\theta| + \n\\sum_{n=1}^\\infty \\frac{\\zeta(2n)}{n(2n+1)} \\left(\\frac{\\theta}{2\\pi}\\right)^{2n}\n", "de86bcf3921a5c47d74c34a09df66c35": "v=e^{in x}", "de871fe26d0339180cf1f7a13a4a4b1b": "a_{21}=\\frac{1}{x_1-x_0}", "de87256eb1064f7a501b43eb2069c027": "abcd\\cos^2\\theta=abcd\\cos^2 \\left(90^\\circ\\right)=abcd\\cdot0=0, \\,", "de8750eb72114896b1e5be2ff60ef861": "p=\\frac{|\\bold{H}|^2}{\\mu}", "de878058aba240a24e00333c83b00aa2": " r_s ", "de87b14497f780cc3c55969b80940da4": " d \\alpha_t = (\\zeta_t-\\alpha_t)\\,dt + \\sqrt{\\alpha_t}\\,\\sigma_t\\, dW_t,", "de880b0b3046f1534018e39c9f33c148": "v_\\mathrm r = -v_\\mathrm i \\,\\!", "de88342c3eb65a9f99f832809114c025": " \\mathit{q} = \\mathit{q}_1\\mathit{q}_2 .... \\mathit{q}_n ", "de8867d005121db509485372929d09ad": "1-z", "de8869ea5efcd8f7f17ef576b350d4d2": "\\text{Bern}\\left(\\frac{1}{2}\\right),", "de88a7c9123649d2c8c21b0c4aac9b39": "\\sum_{j=1}^{m}", "de88df1e28da013de34f9af36bfd56bb": "10\\uparrow\\uparrow\\uparrow 8=(10 \\uparrow \\uparrow)^8 1", "de8912f4ff112e5ce1e065d6ebe44876": "PGL(2,5) \\cong S_5,", "de895b017ff23a783c0c992935ae777d": "\\Gamma \\vdash e \\Leftarrow \\tau", "de89dfe956c29b171f1fb62f2d470b95": " F(\\widehat{\\theta }(x)|X) = \\tfrac{1}{2}. ", "de89eb992f42b3b595350f64fdb0f692": "\\alpha_p = \\frac{1}{\\gamma^{2}}-\\eta", "de89fc2de0ea529dbf46b82e0e12193d": "\n \\frac{d^4 y}{d x^4}\n =\\frac{d^4 y}{du^4} \\left(\\frac{du}{dx}\\right)^4\n + 6 \\, \\frac{d^3 y}{d u^3} \\left(\\frac{du}{dx}\\right)^2 \\frac{d^2 u}{d x^2}\n + \\frac{d^2 y}{d u^2} \\left\\{ 4 \\, \\frac{du}{dx} \\frac{d^3 u}{dx^3}\n + 3 \\, \\left(\\frac{d^2 u}{dx^2}\\right)^2\\right\\}\n + \\frac{dy}{du} \\frac{d^4 u}{dx^4}.\n", "de8a09ec0adab4d0530415455debe85b": "b \\cdot a = ba \\neq F(b) \\cdot a = b^p a.", "de8a4a8fb91dca980928edf2ad68c810": "\\mathrm{~^{14}_{6}C}\\rightarrow\\mathrm{~^{14}_{7}N}+ e^- + \\bar{\\nu}_e", "de8a6aefcd3f8fdc3eeb83c376cc4dd1": "\\mathbf{e}_5 \\times \\mathbf{e}_6 = \\mathbf{e}_1, \\quad \\mathbf{e}_6 \\times \\mathbf{e}_1 = \\mathbf{e}_5, \\quad \\mathbf{e}_1 \\times \\mathbf{e}_5 = \\mathbf{e}_6,", "de8a6c356c20c31f0fb65a37c53506ab": "K\\subseteq\\bigcup_{i=1}^{2^n} s_i K + v_i.", "de8a7c002f255bf443ffbc0b8733c192": "V_\\mathrm{rms} = \\sqrt { \\frac { \\sum t_i \\cdot {V_i}^2 }{\\sum t_i} }", "de8aa89751305321805cfb86d71b9d72": " c = \\frac{1}{\\sqrt{t^2+1}} \\,\\! ", "de8ab62f3266d4fad67a09427e2508f1": "J = \\det\\mathbf{F}\\,\\!", "de8ad4f91dc4aabe2775b531ec2cda77": "\n \\sigma_c < \\sigma_3 < \\sigma_1 < \\sigma_t \\,\n ", "de8b47555722b0120603e67ed15b6bde": "\\frac{dF_{O_2loop}}{dt}=\\frac{((Q_{dump}+V_{O_2})*F_{O_2feed}(t)-V_{O_2}-Q_{dump}*F_{O_2loop}(t))}{V_{loop}}", "de8b6bcf75c6a22425a303900e0fb5fe": "Chow(k) := Chow^{eff}(k)[T]", "de8b73d10cfbcd297bac4a59d4cd6fce": " H(M) = \\mathbb{E} \\{I(M)\\} = \\sum_{m \\in M} p(m) I(m) = -\\sum_{m \\in M} p(m) \\log p(m).", "de8b7648723ec56ec8f9d39edee749dc": "\\mu \\in \\mathcal{P}_p^r (x)", "de8bca68040d8d0c281552dd351754df": "O(k + N)", "de8bd43c92816ecca2535acea759be17": "K = (r \\cos\\theta,\\ r \\sin\\theta)", "de8c0dd4800558cc5d987c3db3a059d2": "\\delta_{int}:S \\rightarrow S ", "de8c7206667978713f06300065c5fc4e": "u+u''=f", "de8cd42cc1d132bd6d2f2d5b80090364": "A=\\,", "de8cfff81cc873866e6ec3e0375dc033": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{7}{\\sqrt{3}},\\ \\pm1\\right)", "de8d7b01b7c40dcae0c1975c7d1f7c6a": "v = x - Px = (I - P)x", "de8db30d02f94593577f1e271fa09132": "S^*=\\{(i,i)\\mid i\\in \\mathbb N\\}", "de8e3f8ab0bbbbbdc4a2073006d92171": "F_{tot} = \\int\\limits_0^{\\pi/2}\\,\\int\\limits_0^{2\\pi}\\cos(\\theta)I_{max}\\,\\sin(\\theta)\\,\\operatorname{d}\\phi\\,\\operatorname{d}\\theta ", "de8e99be3b697f1facb3d46c4f02a53c": "C_n^k", "de8ed0eee223c3ea2c5445fbc7a3185b": "t(x) = \\frac {x-a} {x-b} = -1.", "de8ed8a08d140c043d3bea28816df1c4": " T^n e_k = \\lambda_k^n e_k. \\quad ", "de8eda2b2f0f260adfb8ed46f89b1256": "p_n(x,y)\\leq Cn^{-d/2} \\, ", "de8f0cb37c3560856be466c05e3c8f27": "v_o \\approx {2 \\pi a \\over T}", "de8f5fc9446d996f6d09a27f81445271": "h_{\\alpha\\bar\\beta}", "de8f768c3a7a479c31eac399edbdf694": "\\nabla^2\\left(\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|}\\right) = -4\\pi \\delta(\\mathbf{r}-\\mathbf{r}')", "de8f85a65d42244095fefe653bf126b1": "(p \\leftrightarrow q) \\vdash ((p \\lor \\neg q) \\land (\\neg p \\lor q))", "de8fa0b4b3bdee3986df00119bd46385": "t_m = \\frac{p_m s_0 + p_{m-1}s_1 + \\cdots + p_0 s_m}{p_0+p_1+\\cdots+p_m}", "de8faa1230e961803a13acb2cacbd0fc": "+ \\nabla E_{ext} (\\bar v_i) \\Bigg\\} ", "de8fbecec775c3c372303ef95b0d9cb5": "g: P \\to M", "de9010d6045936be5ec8c1cdbb4e0596": "\\frac{\\partial u}{\\partial t} \\in L^{2} \\left( [0, T]; H^{- 1} (\\Omega) \\right).", "de90d0aa505c5dfcebfdf8c2a7a1be57": "\\kappa\\colon {\\mathrm {Spin}}(n)\\to {\\mathrm U}(\\Delta_n),\\,", "de90e73889f27a4b227a598d70c70e5c": "\n x \\approx D \\times r\n", "de90faff7b1b534b300b83a6911d1142": "K[T]/p(T)", "de90fe4b6859fb61bd38692c7e8c5ac2": "(\\forall{x,y\\in\\mathcal{P}(\\kappa)})\\,", "de913d8935507b6aeec4c257dc89d028": " x \\equiv_{pc} y ", "de91600777c00427071c8f11d5ae09ab": " L_3 = y_{21} R_L y_{12} R_{in} \\, ", "de91761be1082750baaac58c6a20b55f": "\\begin{matrix} \\frac{4}{3} \\end{matrix}", "de91cbaf21fbeff36c4fd159cb7a97f0": " \\sqrt \\phi_{ik} = \\rho (x, y, z, ct) \\; . ", "de91d43f062d9048006472668defc4c1": " ~a ", "de920f7a4085fe28e68c906049f88821": "\n \\boldsymbol{\\sigma} = \\cfrac{2C_1}{J}\\left[\\bar{\\boldsymbol{B}} - \\tfrac{1}{3}\\bar{I}_1\\boldsymbol{\\mathit{1}} -\\boldsymbol{\\mathit{1}} \\right] + 2D_1(J-1)\\boldsymbol{\\mathit{1}} = \\cfrac{2C_1}{J}\\left[\\mathrm{dev}(\\bar{\\boldsymbol{B}})-\\boldsymbol{\\mathit{1}}\\right] + 2D_1(J-1)\\boldsymbol{\\mathit{1}}\n ", "de92391f20b3aca6f6d4537ae26c5257": " \\vec{F}_{12} = \\frac {\\mu_0 I_1 I_2} {4 \\pi} \\int_{L_1} \\int_{L_2} dx_1 dx_2 \\frac {(0,-D,0)} {|(x_1-x_2)^2+D^2|^{3/2}}", "de9279704dd6fd1119dcef70e18090f9": "a^{X}", "de9291ae402b4750b21e30faf3e15983": "y = mx + b,", "de92938cb1b0b83ff0053795ad750d56": "f_{s} \\left(\\vec{x},\\vec{v},t\\right)", "de92bf4294f9d26db88b6185275b2d00": "(ab)'\\;=\\;a'b\\,+\\,ab' \\!", "de9330c09a7680dcb222aa65500c24a8": "\\begin{align} 2\\cdot R_*\n & = \\frac{(48.5\\cdot 0.626\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 6.5\\cdot R_{\\bigodot}\n\\end{align}", "de9332935f8c219c2a0c078befa4b6dc": "\\phi ( I ) = I. \\, ", "de9343dae66043b436aff6e6f0a4c307": " \\textbf{A}_P = \\frac{d}{dt}(R\\dot{\\theta}\\textbf{e}_t) = - R\\dot{\\theta}^2\\textbf{e}_r + R\\ddot{\\theta}\\textbf{e}_t.", "de9346fb1db8ca20271b37cd8118bfc4": "v_2(t) = \\int_{t_0}^{t} i_2(\\tau) d\\tau\\,", "de9361fbf1d3b95f3f08db0500fed43b": "\\|\\beta\\|_1", "de93820d7fc70b3fb8b3310030c05220": "h_*\\to \\pi_*\\circ F", "de9383cd9c462d11423752d88820c3e2": "a \\in I^i/I^{i + 1}", "de9422479158a73678385ad48478a4fc": "Bxyx \\rightarrow x=y.", "de944a050ad8d14d9f72d4d3b6439771": "gm\\,", "de94772a2ae0b14a64160e42728a3dab": "1/2^8 = 1/256 \\approx 0.0039,", "de9492dcb6644b80fe73964ff869a722": " \\delta_n(x)=E[\\theta|x]=\\frac{a+x}{a+b+n}.", "de949bcd3d4944c680af8950a1086322": "1, 2,\\text{ or }\\infty", "de94fab5cb128aa64a18407d0f866f18": "\\sigma_{zx}\n= -\\frac{\\partial^2B}{\\partial z \\partial x}", "de956b67edd17c21d3b0cb087c36c0ac": "U(n,q^2)", "de9572f4fbc51ba2fa56b5723567d648": "\\ln \\left( \\frac{p(z_2)}{p(z_1)} \\right) = \\frac{-g}{R \\cdot \\bar{T}} ( z_2 - z_1 ) ", "de9580c12b8c716adba156a9f1c56214": "1+k4^k", "de958d86d8cd55a5a684262d2c501b9a": "\\mu_{JT} = \\left(\\frac{\\partial T}{\\partial p}\\right)_H", "de959863c00f152f9e64823d65d93772": "T_{cond} = L^2/ \\alpha", "de95d14835bb02a45bbbc0eaea253989": "K(u,v) = \\langle R(u,v)v,u\\rangle.", "de95f9954cac2e1a37d4019f80bb1551": "\\psi_{n2c}(\\bold{r}) = R_{n2}(r) X_{2c}(\\bold{r})", "de9640e1e7b23807493c85ef2c641ca0": "L\\cap\\bar{L}=\\{0\\}", "de964a74f81c9633c7dfafc0231b25d0": "\n \\operatorname{atan2} (y, x)=2 \\arctan \\frac{y}{\\sqrt{x^2+y^2}+x}.\n", "de96542c60bbbd70747d71e3753d9faf": "V_{f1}\\,", "de96ab23917d329b02ccf279773da25f": "K_\\nu=1/3J_\\nu", "de976ff5d1cb3c369115a79c47c2066c": "\\mathbf{AA2} = \\begin{bmatrix}\n(1+2\\lambda+2\\beta) & -(\\lambda-\\alpha) & 0 & 0 \\\\\n-(\\lambda+\\alpha) & (1+2\\lambda+2\\beta) & -(\\lambda-\\alpha) & 0 \\\\\n0 & -(\\lambda+\\alpha) & (1+2\\lambda+2\\beta) & -(\\lambda-\\alpha)\\\\\n0 & 0 & -2\\lambda & (1+2\\lambda+2\\beta) \\end{bmatrix}", "de981376cfa8d8b7d9c07a22d1f803b9": "\\gamma_1=14.13472514...", "de986b1f936a6085eda6ffbd4ecc6e89": ">, \\ngtr, \\gg, \\not\\gg, \\ggg, \\not\\ggg, \\gtrdot \\!", "de987659c65b498a4c2cee00069f4357": "\\left (a \\rightarrow P\\right ) \\left\\vert\\left[ \\left\\{ a, b \\right\\} \\right]\\right\\vert \\left(b \\rightarrow Q\\right)", "de989137e0d79c8f3c085f3e7022c94e": "b_{x+ 1}", "de98a99aa0bb4c38dd9b9662f3c45abe": "\\dfrac{1}{5}=\\dots 121012102_3.", "de990777f95ed4ad5d2b5b4743ee5097": "S_i \\leq a", "de993bbb7c1591a6400fd028f10f5b38": "u_\\text{res}(\\text{Scherzer})=0.6\\lambda^{3/4} C_s^{1/4},", "de9950bf1aafaaa7dfc9c61e08fd952f": "((A\\to B)\\to C)\\to((C\\to A)\\to(D\\to A))", "de99ca35497c7cf0104e5e67c82853ff": "f(x, y) = f(y, x)\\qquad\\mbox{for all }x,y\\in A", "de9a2f848439f5d5726698c1313e2b3c": "{\\rm R}^i F", "de9a4c89fb039141f0c869f278dc2cac": "n^{(1)}", "de9a90739d1780d3de88cf5bdc4c269e": "\\left(\\frac{F_n}3\\right)=-1", "de9acea563e7964c0dcb67a9a62ee329": " = f \\, \\Delta h + \n(\\partial_i f \\, \\partial^i h + \n\\partial_i h \\, \\partial^i f){*\\mathrm{vol}_n} + \nh \\, \\Delta f ", "de9afdf6cc3eef34080dbff9e360c558": "-G", "de9b41b84059465a3718966a10058d85": "\n\\hat{a} = \\frac{2}{N} \\sum\\limits_{i=1}^N R_i \\cos{\\theta_i}\n", "de9b5a93a90cc8c8c92fae8facb888e7": " [I_R] = -\\sum_{i=1}^n m_i[r_i-R][r_i-R],", "de9b5dae69f199902591c1d2d5fc97ed": "y_{0}\\in V\\,", "de9b65fa8c1df01bccdf7d79f0d02db7": "\n \\sigma_\\alpha = \\cfrac{\\sigma_0~\\sigma_{90}}{\\sigma_0~\\sin^2\\alpha + \\sigma_{90}~\\cos^2\\alpha}\n", "de9b877e9dcb2694a59ffb7f6238fe78": "A^+ = (A^*A)^{-1}A^*\\,\\!", "de9b970b2c558d99d3c5d0f3e3d26034": "\\frac1x - \\frac1y = \\frac{y-x}{xy}", "de9bccc7e5558f84e16f30401ee1bffe": "\\sqrt{A}", "de9bcddc38fcfc5c7e611174f39755b6": "\n \\left[\\cfrac{1}{\\sqrt{3}~\\cos\\phi}~\\sin\\left(\\theta+\\cfrac{\\pi}{3}\\right) - \\cfrac{1}{3}\\tan\\phi~\\cos\\left(\\theta+\\cfrac{\\pi}{3}\\right)\\right]q - p~\\tan\\phi = c\n ", "de9bf74d40dff3504b9f0ecae7a288e4": "Q=\\int d^3x \\rho(\\vec{x})", "de9c0e243bb0a769a141e6c3cfc13bc7": "{| \\psi \\rangle}\\in\n\\mathcal{H}_N", "de9c16b8fe13ed807f531ef02dd7aa8b": "|U| \\leq \\frac{1}{2},", "de9c2e923cbbccb5fb96fcfc93984a72": "\n a \\approx b+\\frac{(1/f)^{(d-1)}(b)}{(d-1)!}\\;\\frac{d!}{(1/f)^{(d)}(b)} = \n b+d\\;\\frac{(1/f)^{(d-1)}(b)}{(1/f)^{(d)}(b)}.", "de9c373f6df34516413105815d04e4b9": "\\tfrac{3}{7}", "de9c668fc32c32f39c06656327207ae0": " u \\cdot x = 0", "de9c76fbe6519d650acc7b88c68c8d2b": "\\langle \\phi_n, \\phi_m \\rangle = \\delta_{nm}", "de9cd2ea84427168b2750545561aa8f0": "\\mathcal{D} \\subset \\mathbb{R}^n", "de9cf03455d7d82eb2d64a0e44c448f1": "C=-13.6 eV", "de9d06c6b48b39b1e4dc641c7319d156": "\\forall k > K \\ : \\ \\left|M - \\sum_{n=1}^{k} M_n\\right| < \\varepsilon.", "de9d6a1b69d461e6fa90e99d7be91a24": " T_r = \\frac{T}{T_c + 8}", "de9d82c1340ef6e38e2b6bb941382267": "x=\\theta", "de9dadde775ef50a7df54836c8ead6ad": "Q_t-Q_0", "de9df2c6d65740492da3e496d88df81f": " 12bc\\, s(b,c)=b^2+1 \\mod kc.", "de9e5e6acb3bb30740410906191041b8": "d(1-\\varepsilon)|S|\\,", "de9e60fe0a88bcc797f4e6d27ab9dcd5": "f:V \\rightarrow V'", "de9ede1c5ac1d5993aebfd239fd2a58a": " |\\tilde{\\psi} (\\eta, \\tau)| = \\sqrt{2 \\pi} \\exp \\left( -\\frac{|\\eta|}{2} \\sqrt{1+4 \\tau^2} \\right). ", "de9ef3216dc5699c3fbc9c5805628232": "\\rho(\\mathbf{r}^{\\prime}) d\\mathbf{r}^{\\prime}", "de9f289aa92d42eb7f52ecde259e24f3": "Q_{dm}", "de9f4e58e48720356a877b21d530dddf": " \\overrightarrow{B} ", "de9fad665966248826e2b00091b347dd": "e = a - d.", "dea009d17d27a62b02287c685eeb1d17": " Q=\\frac{ \\pi \\Delta P R^4}{ 8\\mu_p L }[1-(1-\\frac{\\delta}{R})^4(1-\\frac{\\mu_p}{\\mu_c})]", "dea023e740773374e6a32b602a2d5a2c": " \\xi (1/2+iz)= A\\sqrt \\pi (\\lambda)^{-1} \\int_{-\\infty}^\\infty e^{\\frac{-1}{4\\lambda}(x-z)^{2}} H(\\lambda , x) \\, dx ", "dea03dda4827d11b800382ab1a4fb05a": "1 = \\frac{s_0b_1}{\\lambda} + \\frac{s_0s_1b_2}{\\lambda^2} + \\cdots + \\frac{s_0\\cdots s_{\\omega - 1}b_{\\omega}}{\\lambda^{\\omega}}. ", "dea06325f5343c4a30d7348695c1e3ce": "B\\,\\!", "dea0927a139ff372cecffb7fee7a6dfb": "\\Psi_{l}^{k}", "dea0faa1c8346c20cca24ba866ed3ad8": "1 \\over 1000", "dea11cf79c1dd2c2e443d36f223fc7b3": "R_{ab} \\, R^{ab}", "dea12f84075fe116807c588a90f9eae2": "B\\in{\\mathcal A}", "dea17df2c9cce89693374c04ddd972b3": "z (\\rho,\\phi,\\tau) = R (\\rho) * \\Phi(\\phi) * cos (\\omega * \\tau)", "dea241a056ba3d69aa460bc10ff5c0f0": "O(1+mn/L)", "dea290495ded4e2d39b0c9525c2f4b59": " \\left[ a; \\frac{2at}{t^2+1}; \\frac{a(t^2-1)}{t^2+1}\\right] =\\left[ \\frac{20}{5};\\frac{16}{5};\\frac{12}{5} \\right]=\\frac{4}{5} \\left[5;4;3\\right].", "dea29ce35673e3917c346b3894ce4800": "E = 1 - \\frac{1}{\\frac{\\alpha}{P} + \\frac{1-\\alpha}{R}}", "dea2a88dc923e97785e8cffb70255d4f": " p_k = \\frac{\\partial L}{\\partial \\dot{q}_k} ", "dea2e70cd0db642d38fefa96bb842680": "{2a_{14} \\times b_{14} \\over c_{14} - a_{14}}=d", "dea2f575fed8247360f9979e12a46e61": " U^* T U = A \\;", "dea31c18e37272a38dca8280732ed1c3": "\\phi,\\; \\Phi \\vdash \\phi.", "dea31e87ed8b013256079851301ba258": "\nk^{(i)} \\sim P(i=k|z_t) \\propto \\omega^{(i)}_t p( z_t | \\mu^{(i)}_t )\n", "dea343c384d0757626c7616a29fe33eb": "|\\Phi|", "dea348ca58cc1cf13666b0502d5dcded": "\\psi_3(x) = (\\sqrt{3} \\, \\pi^{1/4})^{-1} \\, (2x^3-3x) \\, \\mathrm{e}^{-\\frac{1}{2} x^2}", "dea39074850f05b4a6bfb645c6437183": "\\limsup_{n\\rightarrow\\infty} \\frac{p_{n+1}-p_n}{(\\log p_n)^2} = c,", "dea3bf38446080dd8523e23055002a36": " X \\times I", "dea40c63b6dc0bbdc3f2b00d2b4528e8": "-a_k", "dea46b15403ddce404f326a4325c8c4e": "\\scriptstyle{\\Psi^m_{1,0}}.", "dea51f2bd5c5162448829359e41624c7": "\\scriptstyle \\delta n_0", "dea54f898a76f6016e26a472dcc4aa9d": "\\partial y^{a} / \\partial x^{i} = \\partial H / \\partial p^{i}_a", "dea57611fd0b6af5d8bc3a69f9fddd22": "\\displaystyle{T=\\sum \\lambda_i P_i}", "dea584b4fdf16938f76568103b095174": "\\frac {E(R_i)- R_f}{\\beta_{i}} = E(R_m) - R_f ", "dea5bc78b41c8b4a350e0f3b9f0fb4f9": " \\frac{\\partial u}{\\partial E} = \\frac{-PL}{E^2A} < 0 ", "dea5e98d8a02bee037de8ab5fc2a8868": "j=1,\\dots,k", "dea5fbcaeb788a70a84cf2caba6ccc91": "X\\circ g=g", "dea63aae7b65d70fe88c457448f32eed": "Z= \\int_{380}^{780} I(\\lambda)\\,\\overline{z}(\\lambda)\\,d\\lambda", "dea63e22ee0dfee40749a54601f16bed": "Ax_1, Ax_2,\\ldots, Ax_r", "dea66b3188eeebf0141e9232d859c28e": "k=\\lfloor d/2 \\rfloor", "dea6a2970e90a906f035dccf8895cb3e": "O(\\sqrt{\\log n})", "dea6c1dc59e843a7cdb77fd431d38591": "u(x_i,t_i) = \\frac{f(x_i + c t_i) + f(x_i - c t_i)}{2} + \\frac{1}{2 c}\\int^{x_i + c t_i}_{x_i - c t_i} g(x) dx + \\frac{1}{2 c}\\int^{t_i}_0 \\int^{x_i + c \\left ( t_i - t \\right )}_{x_i - c \\left ( t_i - t \\right )} s(x,t) dx dt.", "dea6c911647fef447b24308476932e94": "W_\\beta = W_{TOT} - W_\\alpha", "dea70cfe348d1c34fee0661d341b3662": "v \\in V\\setminus\\{s,t\\}", "dea7279df6cde2e5043ce6e12024da65": "w(\\xi; s)", "dea757f4e88052f9ce05849ab24c7031": "B_{T}^{x_c} ", "dea7b4199ff86c15e65f6bf05d7e1bf9": "M_0=\\{a\\in M| \\tau(a^*a) <\\infty\\}", "dea85d9b3410592641030b16ebcd03f6": "x(yz)=(xy)z", "dea87c1a0a181b25a0eca741cdd734c1": "\\frac {1}{n}\\sum_{i=1}^n {x_i}", "dea89c4c11c4be83a9abdf0ebb308100": " \\Phi_{(h)}= M_{h,k} \\Phi_{(k)} M_{h,k}^T ", "dea8f112f0f1bc6025d51032d693069e": "V.\\,", "dea97532d33ad7f33e36129b1793bda5": "[f,g] = -[g,f]", "dea9a78f4049c9f361ffaa2f298a9b0c": "\\neg((p\\land q)\\lor(p\\land r)\\lor(q\\land r)).", "dea9d6f1cfed9a93e673d0ed601d3281": "\\beta^2 (\\omega) - \\beta_0^2 = [ \\beta (\\omega) - \\beta_0 ] [ \\beta (\\omega) + \\beta_0 ] =\n[ \\beta_0 + \\Delta \\beta (\\omega) - \\beta_0 ] [2 \\beta_0 + \\Delta \\beta (\\omega) ] \\approx 2 \\beta_0 \\Delta \\beta (\\omega)", "deaa8d4159d0711ca3cc72703de40b27": "\\int\\operatorname{arcosh}(a\\,x)^n\\,dx=\n x\\,\\operatorname{arcosh}(a\\,x)^n\\,-\\,\n \\frac{n\\,\\sqrt{a\\,x+1}\\,\\sqrt{a\\,x-1}\\,\\operatorname{arcosh}(a\\,x)^{n-1}}{a}\\,+\\,\n n\\,(n-1)\\int\\operatorname{arcosh}(a\\,x)^{n-2}\\,dx", "deaabd6c05eca8194f45a8f01e76b169": "\nz=\\pm\\int_x^{x_\\max}\\!\\!\\frac{{\\rm d}a}\n{\\sqrt{2}\\sqrt{E-\\Phi(a)}}\n", "deaae74a5b259bb1557df7d5ab41976c": "(0, p)", "deaaee59f155267a9354e6163725998f": "k_T=\\frac{\\omega n_2}{c}", "deab03ec5be0002d83a4cc9225cb19aa": " h_i = h(h_{i-1},m_i) = f(h_{i_1}\\oplus m_i) \\oplus m_i \\oplus \\theta m_i", "deab1de8767014001fb0f602e511a088": "G \\equiv \\sum_{odd \\ \\ l}(K^{l}_1 + K^{l,l+1}_2) = \\sum_{odd \\ \\ l}G^{[l]}.", "deab31084f742496c7f8ba6ec20e9769": " r =\\sinh t", "deabd5c3880a4d17c4af6692803fd8e9": "k_{B}T", "deac245b41fa5be88605dc1d764c8ee7": "\\sigma \\sqrt{\\frac{\\pi}{2}}", "deac75e2f9996f42fd23d4826c38b568": "\\delta{R^\\rho}_{\\sigma\\mu\\nu} = \\partial_\\mu\\delta\\Gamma^\\rho_{\\nu\\sigma} - \\partial_\\nu\\delta\\Gamma^\\rho_{\\mu\\sigma} + \\delta\\Gamma^\\rho_{\\mu\\lambda} \\Gamma^\\lambda_{\\nu\\sigma} + \\Gamma^\\rho_{\\mu\\lambda} \\delta\\Gamma^\\lambda_{\\nu\\sigma}\n- \\delta\\Gamma^\\rho_{\\nu\\lambda} \\Gamma^\\lambda_{\\mu\\sigma} - \\Gamma^\\rho_{\\nu\\lambda} \\delta\\Gamma^\\lambda_{\\mu\\sigma}.", "deac9cefcf9794f6b49c5067a16f7b5a": "\\succeq ", "deaca4d9c9f121d9f856fe22c37f279d": "|D\\chi_E|", "dead2c304b8a0c9c0bb0fc41a692e10c": "E_j^{rot}=\\frac{\\bold{J}^2}{2I}=\\frac{j(j+1)\\hbar^2}{2I}=j(j+1)\\epsilon.", "dead6f7f1ff2ac9d831efede50102e51": " = \\sqrt{\\int_{0}^{\\infty} 4 \\pi \\left (\\frac{m}{2 \\pi k T} \\right )^\\frac{3}{2} v^4\\ e^{-\\frac{v^{2}m}{2kT}}dv}\\,\\!", "dead8e12594169186b7360f3d68bd57f": "\\mid \\downarrow \\rangle \\to \\frac{3i}{5} \\mid \\uparrow \\rangle + \\frac{4}{5} \\mid \\downarrow \\rangle", "deadcd27674da73debd26f8d07c49f4e": "n_1 = a.\\,", "deadf7f0f2f0f8004fef36225cc4979b": "S_{\\frac13}(q),", "deae2cf29306f286f3eec2360cf1b29f": "p_0=\\frac{1}{61}", "deae305b629111f5980cec181804fadc": " \\alpha(A) \\leq \\mu(A)\\,", "deae55a42bfdb56884d46b727e61ffa8": "\\frac{q_C}{q_H} = f(T_H,T_C) = \\frac{T_C}{T_H}", "deae72e2411bbb3bdebaae6bf33bf2c2": "(b_1^*,b_2^*,\\dots ,b_k^*)", "deae9a6514ba72e868040b78b89df649": "t \\equiv \\bar{t} \\pmod l", "deaeae624fd56bae607935abfa7a61e3": "r \\rightarrow a^{1/4}", "deaefceb42a5514b007953b0e4a7a655": "\\int_X|f(x)-g(x)|^p d\\mu\\,\\!", "deaf0adf12b82f4218d3dec043cb18f7": "\\textbf{H}", "deaf2c246c6d38583463c27dd366ef12": "k = \\frac{\\ln(r_{\\rm{max}} / r_{\\rm{min}})}{\\omega^2} \\times \\frac{10^{-13}}{3600}", "deaf5148561e4f19c82f5cb52c14de87": "\\, \\frac{e^{t\\mu}}{1 - b^2t^2}", "deaf96ffe659efbb3a96caf4ef310f37": "n(n+1)\\,", "deaf98094ed4ef9df3eb322d61e91319": " P(s,b,m,A) = \\sum_{k=0}^{\\infty}\\left[ \\frac{1}{b^k} \\sum_{j=1}^{m}\\frac{a_j}{(mk+j)^s} \\right]", "deafb98c75747ec2336e58ae0c4b4fd0": "\\mathbf{g}^1,\\mathbf{g}^2,\\mathbf{g}^3", "deb0d5edfdacb680efdcc6b50db3a3ca": "\\lnot0=0.", "deb176cc0699adc382eca9695c3c430f": "B[m,p] = X[(p+m)\\Delta_{F}]\\cdot e^{-\\pi \\frac{p^2}{m^2}}", "deb1959f1b55c516d704973a5073bf1d": "F(x;\\alpha,\\beta) = \\dfrac{\\Beta{}(x;\\alpha,\\beta)}{\\Beta{}(\\alpha,\\beta)} = I_x(\\alpha,\\beta)", "deb1c3fcb5fce1bfc10ade04397718bf": "g(\\nu) = { g'(\\nu) \\over g'(\\nu_0) } = { (\\Gamma / 2)^2 \\over (\\nu - \\nu_0)^2 + (\\Gamma /2 )^2 } ", "deb1d2378b4c46ffaed3a9d4d330ae8f": " \\{\\,N(t) : t \\geq 0\\,\\}.\\,", "deb22d88add1f82bd794de6eb16dbc9d": " NPSH_A = \\frac{p_{0}}{\\rho g} - \\frac{p_{v}}{\\rho g} - ( z_i - z_{0} ) - h_f", "deb257c6657dcde579dfb1bfb4e958cc": "T\\leftarrow\\Gamma", "deb28b6f638a2f1254f7ffdbeda69c02": "\\ 0 \\leq v < 8", "deb2b870cd8ca2483e054ae7fbfd8cba": " \\mu((\\lambda_1,\\lambda_2]) = \\lim_{\\delta\\rightarrow0} \\lim_{\\varepsilon\\rightarrow 0} \\frac{1}{\\pi} \\int_{\\lambda_1+\\delta}^{\\lambda_2+\\delta} \\mathrm{Im}(N(\\lambda+i\\varepsilon))d\\lambda.", "deb2bc7cb2b6e355eec0bf9bf7250c10": "ABCD \\doublebarwedge A'B'C'D',", "deb2e6665fcde0abf4ca124274838f79": "b_m = 1", "deb358589dbea32cbdea7d64163e2e48": "\n\\prod_{i=1}^{\\varphi(n)} x_i \\equiv \n\\prod_{i=1}^{\\varphi(n)} ax_i \\equiv \na^{\\varphi(n)}\\prod_{i=1}^{\\varphi(n)} x_i \\pmod{n},\n", "deb35ac698cfcb065d7aab02afd1dcd0": "j=i", "deb3b1991a9322bbeb763eb1b6f49ad8": "\\mathcal{D}^\\star = \\{ \\mathbf{x}^\\star_i | \\mathbf{x}^\\star_i \\in \\mathbb{R}^p\\}_{i=1}^k \\, ", "deb430aa38beddcc7bb5efb3a50d3f47": "\\begin{align}\n\\mathbf E_{(1)}&=\\frac{1}{2}(\\mathbf U^{2}- \\mathbf I) = \\frac{1}{2}(\\mathbf{C}-\\mathbf{I}) & \\qquad \\text{Green-Lagrangian strain tensor}\\\\\n \\mathbf E_{(1/2)}&=(\\mathbf U- \\mathbf I) = \\mathbf{C}^{1/2}-\\mathbf{I}& \\qquad \\text{Biot strain tensor}\\\\\n\\mathbf E_{(0)}&=\\ln \\mathbf U = \\frac{1}{2}\\,\\ln\\mathbf{C} & \\qquad \\text{Logarithmic strain, Natural strain, True strain, or Hencky strain} \\\\\n \\mathbf{E}_{(-1)} & = \\frac{1}{2}\\left[\\mathbf{I}-\\mathbf{U}^{-2}\\right] & \\qquad \\text{Almansi strain}\n\\end{align}\\,\\!", "deb43beb31e82808640c552176cb3bf0": "\\lim_{n \\to \\infty} f(\\frac{1}{n},\\frac{1}{n}) = 1.", "deb46492b3ccdd9947342dba429ceb7f": "{\\mathrm{Div}}^0", "deb46851d15fe47111e312ca9a0d07d3": "\\phi\\colon \\mathfrak{U} \\to [0,+\\infty),", "deb471c0f7eb25323eddbb47417fb88d": "U(\\rho,z) \\propto \\frac{J_1(\\pi W \\rho/ \\lambda z)}{\\pi W \\rho / \\lambda z}", "deb48227ce53851bafe5640f023bad2b": "\\Bbb{H}_1\\ \\xrightarrow{\\mbox{grad}}\\ \\Bbb{H}_\\mbox{curl}\\ \\xrightarrow{\\mbox{curl}}\\ \\Bbb{H}_\\mbox{div}\\ \\xrightarrow{\\mbox{div}}\\ \\Bbb{L}_2", "deb4890bece48214a421738bf6aeb7ac": "\\int_0^{\\frac{a}{2}}\\frac{dx}{\\sqrt{a^2-x^2}}=\\int_0^{\\frac{\\pi}{6}} d\\theta = \\tfrac{\\pi}{6}.", "deb48b15c92ee95b53ce20f31cd23fc2": "M^* + S \\to S^{+\\bullet} + M + e^-\\,", "deb4ff417d7e6c09940d294580386f13": " \\frac{1}{\\sqrt{2\\pi}x^{\\frac{3}{2}}} ", "deb519b076a61eb124161c08eb1593c8": "\\aleph_2", "deb54617a586dac13f5941bb991a4254": "E(X^n)=\\sum_{k=1}^n \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\}\\lambda^k.", "deb55ab6f5a23359dcd90a60ccb18acd": "\\left(\\beta mc^2 + \\sum_{k = 1}^3 \\alpha_k p_k \\, c\\right) \\psi (\\mathbf{x},t) = i \\hbar \\frac{\\partial\\psi(\\mathbf{x},t) }{\\partial t} ", "deb6a72ec80e4b9ce5fba7ce9fd86fb8": "\\,\\{J+iK\\} = e^{i\\theta}\\{J+iK\\}", "deb724c9f303f7e632cdddbe625780f4": "\\Big(M_1\\times \\ldots \\times M_n, N(d_1,\\ldots,d_n)\\Big)", "deb7282322333caa4e5bec4ef5b219b1": "x\\cdot\\infty=\\infty", "deb73e2fafc412fa6e24fef912bcd93d": ":\\quad c / a \\ \\text{arsinh}( a \\ T_a/c )\\,", "deb7a70f855c6983cadcfc7e3c3e297e": "d \\Phi_n (P; p_1, p_2,\\dots, p_n) = (2\\pi)^4 \\delta^4\\left(P - \\sum_{i=1}^n p_i\\right) \\prod_{i=1}^n \\frac{d^3 \\vec{p}_i}{2(2\\pi)^3 E_i}", "deb7b3c02a43e590582ac4b79c79d8c7": " r\\in R,", "deb7c83ef02a55383ecc205d0995a6cd": "K=\\tfrac{1}{4}\\sqrt{(2(a^2+c^2)-4x^2)(2(b^2+d^2)-4x^2)}.", "deb8a4183ce166c68324c6910a4cfe28": "\\Lambda_{LH} = \\sum _{1...p}(\\lambda_{p})", "deb935593af868fd0e4885b54d7e861c": "\\,x_2", "deb985255315a0240bcdb129071c2b37": "E_k = \\hbar \\omega_k", "deb9b1690cd679cc8d281a72083cc5aa": "{\\mathbf{s}}_0", "deb9c2dd63e3fb63a4d8769fb0eb4c07": "\\omega=ck+\\delta", "deba03c703116db23368e039c9ea32c5": "\\lambda_n=\\sqrt{x_ny_n}+\\sqrt{y_nz_n}+\\sqrt{z_nx_n},", "deba4a53528dcc49b94d1ec3c27abbc6": "R_{k,0}=\\{0,1\\}^{P(k)}", "deba8d15b25e514ba246e1847e56d4df": "\\displaystyle R=2r\\quad\\text{(Chapple-Euler)}", "debade86ce864a095b07a9195ff7692c": " \\widehat{U} = \\widehat{U}^\\dagger ", "debaf7eeb207a1ca9ce32b7f004666af": "Q(x) = \\prod_{n\\ge 1}(1-x^n).", "debb2013cefdea7d4b78d19203be0930": "z_1,\\dots,z_N", "debbce086b56f4248952721c92ad97e8": "\\left.\\left(\\frac{\\mu_0}{\\sigma_0^2} + \\frac{\\sum_{i=1}^n x_i}{\\sigma^2}\\right)\\right/\\left(\\frac{1}{\\sigma_0^2} + \\frac{n}{\\sigma^2}\\right),", "debced7537815c037a8543cacea15dd7": "\n(2n-5)!! = \\frac{(2n-5)!}{2^{n-3}(n-3)!} \\,,\\,\\text{for}\\,n \\ge 3\n", "debd09095fa8af85bb9c1acb02b3018c": "T_{e0}=T_{i0}\\equiv T_0", "debd35682116fb5bfc2e3d78b553aab1": "E_\\text{kin}\\, =\\, \\frac14\\, \\rho\\, g\\, a^2.", "debd3b5f3b1972efc96b63defc664dde": "D(X_1,\\ldots,X_n)", "debd4d802718b96f4e93c4e56a188860": " \\partial_t u(x,t) = \\kappa(x) Q(x,t)\\, ", "debd9645373500a7e99cd740619c6f96": " f(x,y) \\approx \\begin{bmatrix}\n1-x & x \\end{bmatrix} \\begin{bmatrix}\nf(0,0) & f(0,1) \\\\\nf(1,0) & f(1,1) \\end{bmatrix} \\begin{bmatrix}\n1-y \\\\\ny \\end{bmatrix}.", "debdc92eac3155a0679b740e6df9385a": "\\cos \\theta_o=\\frac{\\cos \\theta_s-\\frac{v}{c}}{1-\\frac{v}{c} \\cos \\theta_s} \\,.", "debdecc285ed910dc76fd3ecc321117b": "L(1-n, \\chi) = -\\frac{B_{n,\\chi}}{n}", "debe235aeec90f9a417bb47b0c6a5103": "W_{in} = \\epsilon Q_{in}", "debeeca4fd51d21a7e73c11205401db2": " \\mu = \\beta( 1 + \\frac{ \\alpha^2 }{ 2 } )", "debeede3ab89b198c2aa4efd670b1be3": "\ns_{f}(n)=\nO(M(n)\\log^2 n). \\,\n", "debef447bf8143e210a07751b7cafcf2": "f: X\\to X", "debef96ff35ffe8df1221d3ad0222288": "U_T", "debf1082a6b0849b1446819515858684": " t_\\text{max} ", "debf5136ce1935535180b47d95d65f29": "CAS\\,", "dec00813c7d192889342deb23484a00c": "V_7\\,=\\frac{16 \\pi^3}{105}\\,r^7", "dec039c1dbaaa734ddd79270fe99f8e5": " H_{HPF}(f) = 1 - \\mathrm{rect}\\left( \\frac{f}{2B_H} \\right).", "dec040af8d5c2569401a530b32140fc0": " \\varphi(E) = \\inf \\biggl\\{ \\sum_{i=0}^\\infty p(A_i)\\,\\bigg|\\,E\\subseteq\\bigcup_{i=0}^\\infty A_i,\\forall i\\in\\mathbb N , A_i\\in C\\biggr\\}.", "dec06817a8a52b9160c85bf46054f9e4": "\\pi^{-\\frac{z}{2}} \\; \\Gamma\\left(\\frac{z}{2}\\right) \\zeta(z) = \\pi^{-\\frac{1-z}{2}} \\; \\Gamma\\left(\\frac{1-z}{2}\\right) \\; \\zeta(1-z).", "dec081168f8aaea1ae5b929859f00842": "\\scriptstyle \\dots1\\quad 1\\quad 0\\quad 0\\quad 1\\quad 1\\quad 0\\quad 0\\dots ", "dec0bb55f74cb25cdc518c9516f5ede6": "u[3] := 2*atan(\\sqrt(a1^2+b1^2)*tan((1/2)*\\sqrt(a1^2+b1^2)*\\eta)/a1+b1/a1)", "dec0ce9b45f362a38da9166d1bf707ab": "\\begin{matrix}\n [{b^{\\dagger}}_{\\nu_j},{b^{\\dagger}}_{\\nu_k}] = 0 & [b_{\\nu_j},b_{\\nu_k}]=0 & [b_{\\nu_j},{b^{\\dagger}}_{\\nu_k}]=\\delta_{\\nu_j\\nu_k}\\\\\n {b^{\\dagger}}_{\\nu_j}|n_{\\nu_j}\\rang=\\sqrt{n_{\\nu_j}+1}|n_{\\nu_j}+1 \\rang & b_{\\nu_j}|n_{\\nu_j}\\rang=\\sqrt{n_{\\nu_j}}|n_{\\nu_j} -1\\rang & b_{\\nu_j}|0\\rang=0\\\\\n {b^{\\dagger}}_{\\nu_j}b_{\\nu_j}|n_{\\nu_j} \\rang=n_{\\nu_j}|n_{\\nu_j} \\rang &\\left({b^{\\dagger}}_{\\nu_j}\\right)^{n_{\\nu_j}}|0 \\rang=\\sqrt{(n_{\\nu_j})!}|n_{\\nu_j} \\rang & n_{\\nu_j}=0,1,2,\\dots\\\\\n\\end{matrix}", "dec0f08186e67312be893d681c05d681": "\n\\overline{P}(Cl_t^{\\leq}) = \\{x \\in U \\colon D_P^+(x) \\cap Cl_t^{\\leq} \\neq \\emptyset\\}\n", "dec130235bb9231c13e30e8166a1f2b7": "P \\to Q,\\; P\\;\\; \\vdash\\;\\; Q", "dec1587312d29140e798191f0dc33d1b": "\\lambda_k=\\lambda", "dec16259ddd73b956f062584b97f9642": " p \\leftarrow x^3+3x^2-4x+1, M\\leftarrow x+1", "dec1793635f9b61cb5f50ca9a8a32ae0": " \\scriptstyle U_{89} = 500+53\\sqrt{89}", "dec219d8b369e585b65db6c4feb54b90": " N > 1 ", "dec244bf5a94848f8a22d128c9493758": "\\mathcal{S}^h (X)", "dec276d8a7452fbfb101a7adec094f70": " f(\\zeta) = \\int^\\zeta \n \\frac{K}{(w-1)^{1/2}(w+1)^{1/2}} \\,\\mbox{d}w. \\, ", "dec28108b5c76a36cbcbe2540b0c8374": "\\Delta = \\frac{2 A}{L - 1} ", "dec28b3ab851775dac4f136798eac984": "|B_n(G,T)|", "dec291ba871762ff8fc53dd4df5dafe2": "L(n) = an+b", "dec2db9edfda2f9a13877e1ceffdbfc9": " \\sin \\theta = \\frac {\\mathrm{opposite}}{\\mathrm{hypotenuse}} = \\frac {a}{h}", "dec3207f4bf99ce40974a0fe311dd3d0": "(3+4i)^\\ast = 3-4i", "dec33d0a61fa1431a39db8d415d2e7e3": " n- \\nu ", "dec35b0098c06d1f52d0062292352286": " \\mathbf{e}_1 = \\mathbf{u},\\qquad \\mathbf{e}_2 = \\mathbf{v}-\\mathbf{u}.", "dec360434038f89b810d49a86a791ef7": "d(x,c)\\,", "dec3987bfb4c77974587344dd8832bde": "F(x,y) = \\min(x,y) + 2\\cdot\\sgn(y-x)", "dec3b034c75fe95627e3fc245eb0df65": "\\left( \\mathrm{D}_{t} F \\right) (\\sigma) := \\frac{\\partial}{\\partial t} \\left( \\left( \\nabla_{H} F \\right) (\\sigma) \\right).", "dec4073d8fe2b168c5bd7987ef7e9450": "C_\\epsilon", "dec41d47066f2b37e0bdb6662a1bb5b1": "F_{i,kk}=0\\,\\!", "dec4432dd50cf3efb4b2160f2fa55d40": "T_1, T_2, \\ldots T_k", "dec487c462d231378ceaa701bb49b133": "p\\mathrm{d}V\\,", "dec48e0556f687bf728d5c00b87c5f73": "|x_{k+1} - x_{k}| < c\\ ", "dec4b78f34646f2f264f4bcb79302ede": "\n Y_n = \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n \\frac{X_i - \\mu}{\\sigma}.\n ", "dec4d34c0db8a96918a4cd20c5691fa8": "\\frac{-1}{\\sqrt{-g}} \\partial_{\\mu} \\left ( g^{\\mu \\nu} \\sqrt{-g} \\partial_{\\nu} \\psi \\right ) + \\frac {m^2 c^2}{\\hbar^2} \\psi = 0", "dec503b772d048519878f390fac4c101": " a + 2 = 5\\text{ and }b + 2 = 5. \\, ", "dec539c39f7b0990a25341255268de46": "\\int z p(z|x) dz = x", "dec56df952e202e7f29557c6c2f5ad7d": " {\\mathbf A_{11} } \\sim \\mathcal{W}^{-1}({\\mathbf \\Psi_{11} }, \\nu-p_{2}) ", "dec64c16c32c2bb94bfc73df0f2b0a26": "\\phi(x)=\\int_V \\dfrac{\\rho(x')}{|x-x'|} \\, d^3x'.", "dec686c74303a33d1faa6bd045e75877": "\\underset{x}{\\operatorname{argmax}}\\ P(x) = \\frac{1}{Z} \\exp(-1/2x^TAx + b^Tx).", "dec727ef3ad7295e190eb8477490ca0e": "e^{i(nx)} = \\cos (nx) + i\\sin (nx).\\,", "dec7488c8c54f1d9f1c2befba759c9f3": "\\sin^2((2n+1)\\theta)\\,\\!", "dec7773c06daed3a0010e83517e7c2c2": "u^{\\alpha}", "dec78e0a15adae292e71ccf00f42bcce": " \\text{Capture (3D)}\\; \\begin{cases} \\left\\vert X(\\text{new}) - X(s) \\right\\vert < \\text{Separation} \\\\ \\left\\vert Y(\\text{new}) - Y(s) \\right\\vert < \\text{Separation} \\\\ \\left\\vert Z(\\text{new}) - Z(s) \\right\\vert < \\text{Separation} \\end{cases} ", "dec78f95ec55a60374aea2ce1f0c0f29": "\\int_{E} F(x + t i(h)) \\, \\mathrm{d} \\gamma (x) = \\int_{E} F(x) \\exp \\left( t \\langle h, x \\rangle^{\\sim} - \\tfrac{1}{2} t^2 \\| h \\|_{H}^{2} \\right) \\, \\mathrm{d} \\gamma (x)", "dec7ae7271e9804572d3a48f00d8985e": "\\mathcal{L}=\\frac{1}{2}\\eta^{\\mu\\nu}\\partial_\\mu\\phi\\cdot\\partial_\\nu\\phi -V(\\phi\\cdot\\phi)", "dec81f9749b0f4a5d22bb0e8184697c8": "= \\mathcal{L}_{V^{2}}du_{1} - (\\mathcal{L}_{V^{2}}u_{2})dx - u_{2}(\\mathcal{L}_{V^{2}}dx) \\,", "dec8269f1939c9beec59287dc7a1e1a8": "\\omega \\in \\mathbb{R}", "dec8cba544022167684eeec9ebf00073": "\n\\begin{align}\n\\hat{x} &= (1^T \\frac{1}{\\sigma_Z^2}I 1 + \\frac{1}{\\sigma_X^2})^{-1} 1^T \\frac{1}{\\sigma_Z^2} I y \\\\\n &= \\frac{1}{\\sigma_Z^2}( \\frac{N}{\\sigma_Z^2} + \\frac{1}{\\sigma_X^2})^{-1} 1^T y \\\\\n &= \\frac{\\sigma_X^2}{\\sigma_X^2 + \\sigma_Z^2/N} \\bar{y},\n\\end{align}\n", "dec8dd902775deeb713ae4581830eebc": "\n\\begin{align}\nR_w[m] =& E\\{w[n]w[n+m]\\} \\\\\nR_{ws}[m] =& E\\{w[n]s[n+m]\\} .\n\\end{align}\n", "dec8e93e56ea1aede90d73d85847ddd4": "\\textstyle\\binom mn", "dec8f73d92c109f5c588f4bc1c6deb60": " \\tau_2 = \\frac {\\tau_1 \\tau_2 } { \\tau_1} \\approx \\frac {\\tau_1 \\tau_2 } { \\tau_1 + \\tau_2 } \\ . ", "dec93020b4601bf5788aa8c21810d469": "D^\\alpha \\partial_\\alpha u = 0 \\, ", "dec971b15601b6915a1d72ebfe907745": "Z \\in \\mathbb C^N", "dec9d2811e9a201e5ee9f891781e1658": "\\frac{1}{2}(2mv^2)(n_{s}lA)=\\frac{1}{2}L_KI^2", "dec9f07379019cb9f4db310e242c292d": "\nR=|\\langle z \\rangle| = e^{-\\sqrt{c}}\n", "deca698309f1a30f8e9cf3c29d70d7a5": "a_n \\rightarrow 0", "deca92369386507b5ab70e23c2000ccc": "1\\le i\\le p", "decaa4445d804962997326566d85410d": "(H_n)_{ij} = m_{i+j}\\,,\\,\\!", "decaad24fa1e5fe695a49c03a55e63a2": "H_0 = P(0,T) \\operatorname{E}_P\\left(\\frac{dQ}{dP}H_T\\right)", "decaaf6e34b6b8042b2a7baf2fa62763": "S = k + \n{c_{o} \\cdot r^{j} \\cdot a_c \\over 100} + \n{v_{o} \\cdot w^{j} \\cdot a_v \\over 100} =\n{s \\cdot v_{o} \\cdot w^j \\over 100}\n", "decb1c716de33015b9ac1482aca547d7": "\n\\Theta = \\{\\mu, \\Sigma, p(b), M\\}\\,\n", "decb4dda06fb9cc4db146a26535c5473": "W_0(a)", "decb9dbc651cb692d35cca32f90ad85d": "\\ell_1,\\ell_2,\\ldots,\\ell_n.\\,", "decbb19d56b886b43bc459c6fdc7a872": "R, G, B, Y \\in \\left[ 0, 1 \\right], \\quad I \\in \\left[-0.5957, 0.5957\\right], \\quad Q \\in \\left[-0.5226, 0.5226\\right]", "decbfff4cc8b9b916d33343166a5041d": " \\int \\left| f - \\sum_{k=0}^{n} u_k \\right|^p \\, \\mathrm{d}\\mu \\le \\int \\left( \\sum_{\\ell > n} |u_\\ell| \\right)^p \\, \\mathrm{d}\\mu \\rightarrow 0 \\text{ as } n \\rightarrow \\infty.", "deccac2b76f2c5a8b40e32d50661826d": "\n f := J_2^3 - \\alpha~J_3^2 - k^2 \\le 0\n ", "deccc1b9a39a537c3714f4c1181a3d76": "\\frac{\\delta(FG)}{\\delta \\rho(x)} = \\frac{\\delta F}{\\delta \\rho(x)} G + F \\frac{\\delta G}{\\delta \\rho(x)} \\, , ", "decd99f7a6f42e7b4eff63ba77c94a17": " 4^1 - 1 = 3 ", "decd9a48d018d5c5e9848ea3fa9b3401": "\n\nREAD THE ABOVE TEXT!!!", "decda10b10a6f5d3e07b49e49e9d41a8": "\\pm\\frac{1}{\\sqrt{1 - \\cos^2 \\theta}}\\! ", "decdb419d890d4d867646b952d518206": "P(t)=P^t.\\,", "decdb4f6824823f388d09b788a727c0e": " {\\dot{m}_O} ", "decdc026d2f4fac719ac17d8e542cd96": "{\\mathbf{}}K_r(t)=H(t)P(t)C'(t)W^{-1}(t),", "decdd45fbb099841b42810b2d5dbb62e": " \\langle R \\rangle ^2 = \\frac {64 \\gamma c_{\\infty} v^2 k_s} {81 R_g T} t ", "decdef0adee2337af50bef36442ae77c": "C_{nm} ", "dece28150a1c43c66c1c2302ae88be76": "\\ln(2)L/R", "dece8befceee4ced9e5358a3b2c6017d": "\\tfrac{OD}{OA}", "deceb6df4585b4e959fdbd07eeb64bb8": "t_1\\ldots t_{n-1}", "decefb19f62a3e455be4f2b79e0a9dbd": "\\theta'", "decf22ceaf4e2ce8d2a8695a29ea56d1": " \\bold E(x,y,z) ~ = ~ \\sum_{mnp} ~ \\bold E(\\alpha_m,\\beta_n, \\gamma_p) ~ e^{j(\\alpha_m x + \\beta_n y + \\gamma_p z)} ~~~~(2.1b) ", "decf3ee4318382135f42b487c484135c": "\\Delta E_{pot} = -\\Delta E_{kin}", "ded01e79d6bb0521312fd1b8f462e2c6": " Q(x) = \\int_\\Omega P(x \\mid y) \\mathrm{d} \\pi(y) ", "ded0409d440cf5331eee07f5b772b0a0": "\\gamma_k=\\beta_k-1", "ded062f8372388ad9898d12cadbe8027": "\\mathbf{A p}_k", "ded0675bb602e07bb6c7afa58dc0e5f6": " X_t = \\mu t + \\sigma W_t", "ded0804cf804b6d26e37953dc2dbc505": "pn", "ded1177069859f8c9b4f85481667ec02": "\\epsilon^{\\mu\\nu\\alpha\\beta}", "ded13365e3de50897fb55e0d8e0a5718": "\\mathbf{O}(\\mathbf{n}\\log{\\mathbf{n}})", "ded1b617b48a0c5734c35a3a57396a12": "H_{trt}=\\frac{2GM}{3r^2}", "ded1cb567853c5ddd1b09f28a1e244b4": "H_E= 0", "ded1e8bcb95feb2f42fb93387fbaf387": "\\frac{\\part^2 f}{\\part x^2}, \\; \\frac{\\part^2 f}{\\part y^2}, \\text{ and }\\frac{\\part^2 f}{\\part z^2}", "ded22837a863c327849cad3c11883f68": "x(t) = A \\cos t + B \\sin t", "ded22c21debefedf3ee06c8a7824c161": " d\\mu^X = \\iota_{X^\\#}\\omega, ", "ded29c49b78af285227df1219363c2cd": "h = \\left(q + \\left\\lfloor\\frac{(m+1)26}{10}\\right\\rfloor + Y + \\left\\lfloor\\frac{Y}{4}\\right\\rfloor + 6\\left\\lfloor\\frac{Y}{100}\\right\\rfloor + \\left\\lfloor\\frac{Y}{400}\\right\\rfloor\\right) \\mod 7,", "ded2adf77564c1e348ba78b80494adfa": "\nJ_z = \\hbar\n\\begin{pmatrix}\n1&0&0\\\\\n0&0&0\\\\\n0&0&-1\n\\end{pmatrix}\n", "ded2beb6fe372aa82ccc4292b25d9162": "\\scriptstyle I=[0,1]", "ded30af843bb3bb35bffb7a64b55631d": " \\dot{F}(t) = \\lim_{h\\rightarrow 0} {F(t+h) - F(t)\\over h}", "ded32af9a8b69b369eebacdf216f7a20": "\\theta (t) = \\theta_0 \\cos( 2 \\pi t/T)\\, .", "ded401c7ff5fae604b724f4c65a36445": "\\text{bias}_{\\emptyset} (X) = 1", "ded4542ce374c22d5d81139add595979": "\\lambda_x > 0", "ded4c81d6f771a7de32ce81ffcec8663": "(\\tfrac{29}{5}) = +1: \\qquad \\tfrac{1}{2}\\left (5(\\tfrac{29}{5})-5 \\right )=0, \\quad \\tfrac{1}{2}\\left (5(\\tfrac{29}{5})+5 \\right )=5.", "ded4c93b0d1b9dc28df63721fb7a505c": "P^{n}", "ded4e6f8121102542018872979f44781": "FV = PV \\cdot (1+rt)", "ded525ee46a68c68d6bcb744685bba2e": "f:\\mathbb{N}\\rightarrow\\mathbb{N}", "ded59e3d064919442e4cc1010b27388b": "T_2^{*}(K)", "ded5c74953cd2b37f282461b01fa1eb3": "S_{mn}(-i c,\\eta)", "ded625b7a01501be6b7332993996528e": "s_t^i", "ded6758b9e053a7acd57c24de081e1d9": "\\mathbf{q}_{M \\times 1} ", "ded681eaa02d11064c9a469dd1b3e04c": "a=0", "ded7216b226e7850d6c0410a32b40f66": "A = \\partial_{x}^3 + \\frac{3}{4}( v( x, t ) \\partial_{x} + \\partial_{x}v( x, t ) )", "ded76c2d455134318f296a87e27c6b02": "\\Delta \\mathbf{r}_i = \\mathbf{r}_i - \\mathbf{R}, \\quad \\mathbf{v}_i = \\boldsymbol\\omega\\times(\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{V},", "ded7749a496069a0b9a81addfb972449": "(gate2\\vee \\overline{gate7}\\vee gate4)\\wedge (gate8\\vee \\overline{gate6})\\wedge (gate8\\vee \\overline{gate7})\\wedge ", "ded7ae1d06cedd8ba79fc29b4cb22945": "y_{2,t} = c_{2} + A_{2,1}y_{1,t-1} + A_{2,2}y_{2,t-1} + e_{2,t}.\\,", "ded7d5a649e642d25e2935382d7f0d0b": "|\\mathbf{R}(t+1) - \\mathbf{R}(t)| < \\epsilon", "ded7d87de6e637700a5aa8573423a003": "y_2(t)=v(t)y_1(t)\\, ", "ded7f8b75cb6fef8fb544eec470935dd": " 0 < u < N_\\text{ZC}\\,\\and \\, \\text{gcd}(N_\\text{ZC},u)=1, ", "ded829dc473638c10409f8898b5273fb": "x_1=2\\,", "ded8de2cc1ebea202420fc1a5d967041": "w:\\mathbb{R}\\rightarrow\\mathbb{R}^q", "ded8ee808360477f68ff07e1669926bb": "\\,\\Lambda(t) = -\\log S(t)", "ded93ca89124ae7e095cc0b9da29a550": "B\\in\\mathcal B", "ded940ee6fa76d71fc5266bdcac4ac22": "\\| u \\|_{L^{\\infty} (T; X)} := \\mathrm{ess\\,sup}_{t \\in T} \\| u(t) \\|_{X} < + \\infty.", "ded9425a7b340a14ec981e8e1dc463f8": "m_{n}", "ded95f411835463f139ffe86529830b3": "\n _{(X)}\\Gamma_{\\alpha\\beta\\gamma} = \\frac{1}{2}\\left(\\frac{\\partial g_{\\alpha\\gamma}}{\\partial X^\\beta} + \\frac{\\partial g_{\\beta\\gamma}}{\\partial X^\\alpha} - \\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma}\\right) ~;~~\n _{(X)}\\Gamma^\\nu_{\\alpha\\beta} = g^{\\nu\\gamma} \\,_{(X)}\\Gamma_{\\alpha\\beta\\gamma} ~;~~\n g_{\\alpha\\beta} = C_{\\alpha\\beta} ~;~~ g^{\\alpha\\beta} = C^{\\alpha\\beta}\n", "ded96e6c087000efcfcdd4839b90a1d1": "s_{\\bar d}^2", "ded97dc7c585d6c0e2ed7b9a6c920953": " [R_j , P_0] = 0, \\; [R_j , P_k] = i \\varepsilon_{jkl} P_l, \\; [R_j , N_k] = i \\varepsilon_{jkl} N_l, \\; [R_j , R_k] = i \\varepsilon_{jkl} R_l\\,", "ded99ac1ef36009528f8e93ba3072a39": " \\int\\!\\!\\!\\!\\int_{S} \\mathbf{j}_1(\\mathbf{r},t) \\cdot d\\mathbf{S} = \\int\\!\\!\\!\\!\\int_{S} \\mathbf{j}_2(\\mathbf{r},t) \\cdot d\\mathbf{S} ", "ded9a55f42b3d4745bb5d739c33dc16b": "O(A)=O(A:\\neg A)", "ded9db03101d935ec16ffa55d428abf0": "{a\\oplus_M {z}}= \\frac{a+z}{1+a\\bar{z}}", "ded9faa31fae1b66ebe902eb376cc5b1": "\\frac{d\\theta}{dt}=\\omega", "deda09336263e528bca1263d4da4236c": "\\scriptstyle\\operatorname{\\phi}:\\; A \\,\\times\\, A \\;\\to\\; V,\\; \\left(a,\\, b\\right) \\,\\mapsto\\, b \\,-\\, a \\;\\equiv\\; \\overrightarrow{ab}", "deda7818906f55fa56ff20dadf5b5fbb": "-2a^4 \\sqrt {2} a^3 x-2a^2\\left(x^2-5y^2\\right)+\\left(2x^2+y^2\\right)^2+2 \\sqrt {2} ax\\left(2x^2-3y^2\\right)+2a^2\\left(y^2-x^2\\right)=0", "dedabdc607dd2ca954b1f8056259e45f": "N(\\mu,\\sigma^2),", "dedaf29919693bd7c843cd0e04739e2c": "\\sum_{i=1}^3 W_i = 1", "dedb10563d67afe581aaee8b3ff33ca6": "\\| u_{f} \\| \\leq \\frac{1}{c} \\| f \\|.", "dedb357fa537cada7dadc36cdee4dbee": "Tr(g)\\in GF(p^2)\\ \\forall p\\in GF(p^6)^*", "dedb44592f55af0741f3b5564c4deb1c": "\\mathcal G(2,0)", "dedb6867311766bc5b0c39a283cc61eb": " \\beta(g) ", "dedc07dc9e388b81e4e8989e4a6c6c79": "\\mathbf{A} = \\begin{pmatrix} \na & b\n\\end{pmatrix}\\,, \\quad \\mathbf{B} = \\begin{pmatrix} \nx \\\\\ny \n\\end{pmatrix}\\,,", "dedc1c2cf6d2fe7faddb7d19720dc73a": "\\nu=n/(2pn\\pm 1) ", "dedc557ddb07d2f96667c53b2510c109": "s_n=\\prod_{k=0}^{n}\\binom{n}{k} = \\prod_{k=0}^{n}\\frac{n!}{k!(n-k)!}~, ~ n \\geq 0.", "dedc56a8d789b3db2dd23d2332c05c34": "P = K \\rho^{\\gamma} \\rightarrow T \\sim \\rho^{\\gamma-1}.", "dedc63526e4df61b478fdeecadd90f39": "\\mathbb Z[\\eta]", "dedc8f8c2c18ec07f6b7810a536d7f8f": "\\Pi(k_i) \\sim a_\\infty k_i", "dedd40b07f6fd84a9d28883e3a4f0491": "\\frac{8k_BT}{3\\eta\\,}", "dedd9be9863a2d4d7c2ff1ad8346c415": "\n\\begin{align}\nT(x) & \\in \\Theta \\left( x^p\\left( 1+\\int_1^x \\frac{g(u)}{u^{p+1}}\\,du \\right)\\right) \\\\\n& = \\Theta \\left( x^2 \\left( 1+\\int_1^x \\frac{u^2}{u^3}\\,du \\right)\\right) \\\\\n& = \\Theta(x^2(1 + \\ln x)) \\\\\n& = \\Theta(x^2 \\log x).\n\\end{align}\n", "dedda70cbc3733a2239496ab8174e04d": "G := \\left[\\cfrac{\\partial U}{\\partial a}\\right]_P = -\\left[\\cfrac{\\partial U}{\\partial a}\\right]_u", "deddcead195bd393dd09013f2a7d89cf": "7758\\ ", "dede1dcb13a4dac8c32622daa03b868b": "\n\\frac{r_1}{A} = \\frac{0.49q^{2/3}}{0.6q^{2/3} + \\ln(1 + q^{1/3})}\n", "dede90bfa3eb4236aa296d750e5c463b": "e^{\\frac{(i \\pm j \\pm k)}{\\sqrt 3}\\pi} + 1 = 0. \\,", "dede95839a3b06a2c5c943de09274fe0": "(f,b) \\in G \\subset \\bar{G}", "dedec0d9d0fa17c1eb043497a64c48e6": "(F \\land E) \\lor (G) \\lor (D)", "dedece8e5c74609c8d884483b8cd6ca5": "\\,\\phi(k),\\phi(-k)", "deded1104ff7c720d128773b826dd8fa": "D:\\{0,1\\}^n \\rightarrow \\{0,1\\}^k", "dedf4103e26c7da0c0eb933aa3e29bd3": "\\alpha^{i_\\beta}", "dedf82e4fed86621ecc367621baf0145": "m_m", "dedfaea2651e5cad0e95b3f458d9f2a5": " {8! \\times 3^7 \\times (12!/2) \\times 2^{11}} = 43,252,003,274,489,856,000", "dedfe86e85f1976fb2c532f46be0faeb": "E \\otimes_R -", "dedff78114adb528b33734b9c17a1ce5": "\\sum_j \\sum_i y_{ij}", "dee0543653a5ca075600e0e850065f5c": " \\alpha \\psi + \\beta |\\psi|^2 \\psi + \\frac{1}{2m} \\left(-i\\hbar\\nabla - 2e\\mathbf{A} \\right)^2 \\psi = 0 ", "dee0f7030348baa436a95ed1231d0734": "\\begin{matrix} {4 \\choose 1}{3 \\choose 1}{9 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "dee1d34afb783daf03ea4cdad9527f9b": "T \\to U(\\mathfrak{g})", "dee2025ea61363fa020735646275753a": "\\left(1-\\frac{n}{n+2}\\right)\\sigma^2=\\frac{2\\sigma^2}{n+2}", "dee289d3ea476881d8589048d2880632": "d(x,y)+d(y,z)\\le d(x,z)", "dee2c2397231f66a545652662e7467e1": "\\begin{align}\n Z_L &= \\omega Le^{j\\frac{\\pi}{2}} \\\\\n Z_C &= \\frac{1}{\\omega C}e^{j\\left(-\\frac{\\pi}{2}\\right)}\n\\end{align}", "dee2c8a9ba94a619eadcd8063dfc98a5": "\\mathbf Q_d = \\int_{\\tau=0}^{T} e^{\\mathbf A \\tau} \\mathbf Q e^{\\mathbf A^T \\tau} d\\tau ", "dee30774e738f870cd8033eedfb1f776": "E[(x_u-x_v)^2]", "dee309ae81907a0fcba3ec07047075b8": "s-t = s'-t'", "dee3173c097dcc49d8eb85b8fa0910a8": "e^{i\\gamma Pz}", "dee45f2f1c6357e8086229159683cbb9": "V_{OUT}", "dee478126418be50b0997eb84853cca5": " \n\\frac{4\\sqrt{S(\\mu)}}{\\left( \\mu - \\lambda \\right) \\left( \\mu - \\nu\\right)}\n\\frac{\\partial}{\\partial \\mu} \\left[ \\sqrt{S(\\mu)} \\frac{\\partial \\Phi}{\\partial \\mu} \\right] \\ + \\ \n\\frac{4\\sqrt{S(\\nu)}}{\\left( \\nu - \\lambda \\right) \\left( \\nu - \\mu\\right)}\n\\frac{\\partial}{\\partial \\nu} \\left[ \\sqrt{S(\\nu)} \\frac{\\partial \\Phi}{\\partial \\nu} \\right]\n", "dee494713f3e89052dcd2535dc9a45c4": " \\frac{x}{\\ln x}\\left(1+\\frac{1}{\\ln x}\\right) < \\pi(x) < \\frac{x}{\\ln x}\\left(1+\\frac{1}{\\ln x}+\\frac{2.51}{(\\ln x)^2}\\right). ", "dee4f17d1beca95447df51418bd5215b": " \\frac{b-a}{3} (2 f_1 - f_2 + 2 f_3) ", "dee51e3fc833a7f1f7ef51c4bb364f59": "\\Phi:\\mathrm{hom}_C(F-,-) \\to \\mathrm{hom}_D(-,G-)", "dee53dd0e8a61fda769580cc76ce0515": "y = \\frac{P}{\\omega g} \\cosh \\left(\\frac{x \\omega g}{P}\\right)", "dee54887a0cb9be345709efe9fdc7905": "(R_3 - C_0)^2 = D_0", "dee5f06b97a15c73e9ae2bd89c21bc18": "x \\in R^n", "dee5f4b52720a8c9e1b7104b5ba67eef": "u_x^-", "dee61ef25895ec8f5cc6d15b97fde797": "A\\neq B", "dee62fd5ec0206f0ecd8d8e93bf9cd75": " Gr= \\frac{g \\beta \\Delta T L^3}{\\nu^2} ", "dee6387d563c3f665f44939e2a8e2d48": "\\langle x,x \\rangle", "dee6f916fcfe1007343f65e2a2d0ea8e": "\\hat D=- {{\\hbar}^2 \\over {2m}} {\\part^2 \\over \\part x^2}", "dee7141541b0575f29b52d84bb7580f6": "n^{th}", "dee79086cb21deb3e234d5204b634928": "p^2\\mid2^{m\\lambda}-1", "dee7ae315846c8115892302d239450e0": "\\scriptstyle y_R \\in Y_R", "dee7b2e4ec01390267ee2223c5572c26": "p(\\lambda v) = | \\lambda |^k p(v)", "dee7b94967ac37741d765e82285192d5": "\\Delta U = U_{1}(s^{N}) - U_{0}(s^{N})", "dee80045dbc40c42f7338341fb93837d": "\\theta_o \\,", "dee87b458302d8985c76deb1be17ede3": "(x-4) (x-2)^4 x^6 (x+2)^4 (x+4)", "dee8d44b2268dc6e493051592f070cb2": " \\tilde{\\chi} ", "dee924b340601624c3affa94d796d773": "\\mathbb Q/ \\mathbb Z", "dee967286870809893b5290475cae1f7": "\\begin{align}\n \\nabla \\times \\left( \\nabla \\times \\mathbf{E} \\right) \\;&=\\; -\\frac{\\partial } {\\partial t} \\nabla \\times \\mathbf{B} = -\\mu_0 \\varepsilon_0 \\frac{\\partial^2 \\mathbf{E} } {\\partial t^2}\\\\\n \\nabla \\times \\left( \\nabla \\times \\mathbf{B} \\right) \\;&=\\; \\mu_0 \\varepsilon_0 \\frac{\\partial } {\\partial t} \\nabla \\times \\mathbf{E} = -\\mu_o \\varepsilon_o \\frac{\\partial^2 \\mathbf{B}}{\\partial t^2}\n\\end{align}", "dee9d3539d5223a23678b5f73fd680a4": "\\vec D\\!", "dee9fbfb5e2d8c3a42107144e9e13f79": " [g,h] := g^{-1}h^{-1}gh ", "deea91404d5bf7eb7ea29f506d0b8187": "\\lim_{\\underset{h\\in\\mathbf{C}}{h\\to 0}} \\frac{f(z_0+h)-f(z_0)}{h} = f'(z_0)", "deeae2272a9bd6f1fd5c67b4173468f7": "{{\\frac{q}{A_{max}}}}=C{{h}_{fg}}{{\\rho }_{v}}{{\\left[ \\frac{\\sigma g\\left( {{\\rho }_{L}}-{{\\rho }_{v}} \\right)}{{{\\rho }_{v}}^{2}} \\right]}^{{}^{1}\\!\\!\\diagup\\!\\!{}_{4}\\;}}", "deeb1b106fc874dc2c953d8f0f6647ec": "\\frac{0.11}{1} + \\frac{0.18}{5} + \\frac{0.23}{20} + \\frac{0.48}{1.6} = 0.4575.", "deeb8f130128ce0d350e1afb9fef737d": "\\omega=v_sk", "deebc5001f8a94fff3f5e174ccbc3aaa": "E_{t-1}", "deebf0f3df277dd50dd03137c96f833a": "V_{max}^\\text{app} < V_{max}", "deec33233d684cfbae38d5f9029510b5": "\n P_e \\geq 1- \\frac{4A}{n(R-C)^2} - e^{-n(R-C)}\n", "deec449816a595ec5aabb5cefd30070a": " a_2^\\dagger | N_1, N_2, N_3, \\dots \\rang = \\sqrt{N_2 + 1} \\mid N_1, (N_2 + 1), N_3, \\dots \\rang.", "deec69a8caa16a8fd9f1b27907c1ad2c": "\\left[D,a\\right]:\\mathrm{Dom}\\left(D\\right)\\to\\mathcal{H}", "deed50f55fd5e5ef3575eb606781c7f1": " \\left|\\arg {zf^\\prime(z)\\over f(z)}\\right|\\le \\log {1+|z|\\over 1-|z|}.", "deed946a0272077de27b4a42fb598149": "d(\\mathbf{X}^{\\rm H}) =", "deeda62364555b291c729eddcdc0eddd": "P_2(n)", "deedc88ed7a69315adf8be0fd8124ee2": "\\cfrac{\\cfrac{\\beta \\qquad \\gamma}{\\delta} \\, p \\qquad \\alpha}{\\eta} \\, q", "deee771f6bbed50d9601b0235da1cb04": "g_\\mathrm{rev} = 20\\log_{10}\\left|S_{12}\\right|\\,", "deee9f24e5b092f1dac80e6b90311b22": "P_{i,1}", "deeea289b2b8859d4b42479f3851fdc0": "d'([x],[y]) = \\inf\\{d(p_1,q_1)+d(p_2,q_2)+\\dotsb+d(p_{n},q_{n})\\}", "deeec50d4a232cd4ffe032e3587ef14b": "t\\begin{Bmatrix} p , q \\end{Bmatrix}", "deeec820f074acc64eae886a0afc04ed": "Q_n(x)=\\int_I \\frac{P_n (t)-P_n (x)}{t-x}\\rho (t)\\,dt.", "deeed1eeffddbecc4d4f33c5a4435a33": " S _C(n) ", "deeed382c3c72e665cf4184f1dc89d72": "a,b \\geq 0", "deeeeb17544cff81c470f49359f1f651": "(XY)^{st} = (-1)^{|X||Y|}Y^{st}X^{st}.\\,", "deef1d5639228d46566d18b2b61131be": " h ", "deef8654edb465e2f538d35fd5bbe14a": "vme=m'e'u", "deef92a14e87f9595236c060e0d40aa1": "\\frac{1}{\\Delta t}\\int_0^t P dt = \\frac{1}{\\Delta t}\\int_0^t \\textbf{F} \\cdot \\textbf{v}\\,dt.", "deef960baa57f54245d16f1b6fc6243a": "i^{\\ast}_{Q}(TE) = \\{(q,v) \\in Q \\times TE \\mid i(q) = \\tau_{E}(v)\\}\\subset Q\\times TE,\\,", "deef97a67ae090ee32eeaf324e57d439": "\\text{B} \\in \\text{PTAS} \\implies \\text{A} \\in \\text{PTAS}", "deef9a4de1a1de0a1f0c2ad60e394c3a": "~\\arccos(x)~", "deefbeec4e063764beae7f9dad4b39dc": "\\{(+,-,-,-); l^a n_a=1\\,,m^a \\bar{m}_a=-1\\}", "deefd42c56e2519d841c003f6ae38f98": "\\widehat{\\boldsymbol{\\mu}}_S = - \\frac{g\\mu_B}{\\hbar}\\widehat{\\mathbf{S}}\\,,\\quad \\left|\\boldsymbol{\\mu}_S\\right| = - g\\mu_B \\sigma\\,,", "def0022a8ea0c4660f5c7f8ebdcad105": " AB = 1.\\ ", "def045723f33972f5b7d16a599955ee1": "\n1-\\kappa^{\\prime} = \\lambda(1-\\kappa)\n", "def087fbb2f0fe823655ce6d05417d2f": "p^M = \\frac{a + cb}{2b}", "def0943484a4779c2c2e74ea8f6b97c3": "\\inf \\theta \\le 33/100.", "def0a98fc0c5fc8e753594487fa5e0f6": " \\widehat f(\\chi) = \\int_G f(x) \\overline{\\chi(x)}\\;d\\mu(x),", "def0e9ebd4dab75b168185b1527eb2cd": "\\alpha |0\\rangle|\\psi\\rangle+\\beta |1\\rangle|H\\psi\\rangle", "def0fceff3b277a685bdb2936e614835": "\\Delta V", "def127bf5aecb7b335f946eda0f17ba7": "{\\frac{F}{E A} L } = \\alpha_L \\Delta T \\ L", "def13b32b0c8fda36dff281c03997469": "Pc = Pmf Pwf + Pmo Pwo", "def15dd6d6425860db213f7feb2b3718": " Pz= z-\\langle z-x, Z\\rangle Z.", "def19d2ea286af5b7ceee57ce6a47bd3": "\\mathcal{F}^j=-\\partial \\mathcal{H}/\\partial X_j", "def19e0a4eb4b19ade478e82eda34dd7": "S=(A*B)+C+D", "def1ce597b3b8817cf4e8a78c04393e0": "\\vec a(t)=A(\\vec x(t))", "def1d8ce3104c0fc616a3ae4fd550f49": "\\nabla : \\Gamma(X,E)\\to \\Gamma(X, \\Omega_X^1\\otimes E)", "def1f38031e816ffe75dd76a440b057a": "\\frac{\\partial V}{\\partial x} = -i_l r_l\\ ", "def207cb2c7891bc9a72b3aac997f74c": " j_m = \\frac{1000 \\times (1.5 \\times 10^{-3})}{ \\pi \\times (2 \\times 10^{-2})^2 \\times 2} = \\frac{3}{16\\pi}\\times 10^4 ", "def22ed8146edf9607f29d511c0b1904": "\\displaystyle{c_0(X+iY)=\\sigma(X)- i\\sigma(Y)}", "def2b202857cdc1e602abafd5d32f83f": "Ix+=gm.Vz\\,", "def2fc8ad6b82fae5033dd59bb114eda": "\\tfrac{4}{3} \\pi r^3 \\rho_\\text{air}", "def37e1e009218cb3330ee04bade33d8": "\\Gamma^\\mu = -{1\\over e}{\\delta^3 S_{\\mathrm{eff}}\\over \\delta \\bar{\\psi} \\delta \\psi \\delta A_\\mu}", "def3b3f073403dea6ee6df3d080aec94": "A = \\frac{ns}{4} \\sqrt{4-s^{2}}.", "def3c39e30d837907c620537dad64017": " \\dot {\\bold{H}} = \\frac{d}{dt}\\left(\\bold{r} \\times {\\dot{\\bold{r}}}\\right) = \\dot{\\bold{r}} \\times {\\dot{\\bold{r}}} + \\bold{r} \\times {\\ddot{\\bold{r}}} =\\bold{0} + \\bold{0} = \\bold{0}", "def3d7aaf565e3dc3e36b57355ae87ee": "\n\\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} + v\\frac{\\partial u}{\\partial y} =\n-\\frac{1}{\\rho} \\frac{\\partial p}{\\partial x} + \\nu \\left( \\frac{\\partial^2 u}{\\partial x^2} +\n\\frac{\\partial^2 u}{\\partial y^2} \\right),\n", "def412f4005d6f3a5de475c423e45238": "\\int_0^1 \\tilde{B}_m(x) B_n(x)\\, dx = \\delta_{mn}", "def42325074a0e78d445972603aff0e0": " \\frac{d}{d \\tau} R = \\frac{1}{2} (\\Omega - \\omega) R ", "def427efe88c12f6ff4ec5c36039dc6a": "p \\vdash (p \\lor p)", "def47f9adefee20f5d837c75085b5fc2": "\\phi_\\omega(x)e^{-i\\omega t}\\,", "def4bb960bc1e2e671bd66da347e8f34": "\n D^{\\ell}_{0,0}(\\alpha,\\beta,\\gamma) = d^{\\ell}_{0,0}(\\beta) = P_{\\ell}(\\cos\\beta).\n", "def55a97d383325692504eb6e8f72c64": "c_4=\\sqrt[3]{199-3\\sqrt{33}}", "def5654d277e855ba6c50e180e0b00bd": "v_{i+1}", "def56b1891313031bc5bf902628c3218": "B \\subset \\mathbb{P}^1(k)", "def58808ca623ea4d11fd5df1094cb54": "h \\in \\Gamma(E\\otimes\\bar E)^*", "def5a2b2119e08c2808052ed3e5030fb": "(X, \\mathcal{F}, \\mu)", "def5b11504c8a2d548fbe14c739bc3ed": "\\overline{A + B} \\equiv \\overline {A} \\cdot \\overline {B}", "def5cf801f1eccba6e0d56182f3d417b": "g(\\theta) = \\theta, \\quad g(\\omega) = \\omega^2,", "def5fe54174a34dd4b98d05e4731631d": "(P,Q)", "def6293b737928878699c90ec0ff5a56": "S \\Rightarrow aSb \\Rightarrow aaSbb \\Rightarrow aababb", "def65f177a003cc40dea193c4657852d": "\\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1} \\Sigma _{YX} \\Sigma _{XX} ^{-1/2}", "def65f882d797d9b80373f9f49746a5b": " df = \\left(a\\frac{\\partial f}{\\partial x} + \\frac{\\partial f}{\\partial t} + \\frac{1}{2}b^2\\frac{\\partial^2 f}{\\partial x^2}\\right)dt + b\\frac{\\partial f}{\\partial x}\\,dB ", "def6742df4b90cab17a17daa34f49bcc": " \\sum_{g \\in G} \\chi([g]) |g|^{-s} ", "def679a29cda43a955e60a73cd317e1c": "(\\alpha e^{-i\\theta}, \\widehat{U}|\\alpha\\rangle) ", "def685661862a028c7c7596e69c83609": "\\textstyle{i_{1}}", "def698af2f32b60b8fccb63d54603305": "2\\log n", "def7830f5332588efb1728e08d72e383": "4\\mathbb{Z}", "def812cc51a095b85292d505ccdd69fd": " {d\\over dt} (g^{-1}Fg) = -g^{-1} \\dot{g} g^{-1} Fg + g^{-1}\\dot{F} g + g^{-1}F\\dot{g} = g^{-1}(-\\dot{g}g^{-1}F + \\dot{F} +F \\dot{g}g^{-1})g=0.", "def832d9e7395a2a98562df295b0bdaf": "j^2=+1.\\,", "def83995dc7401d6ece6c655b9a02bfe": "M(256,4,3)\\approx10^{\\,\\!10^{10^{1.99\\times 10^{619}}}}", "def88c525b73817490deb4a19fee5e4c": "{\\bar{R}} ", "def8cc3f306b671f0c903de926c77a40": "\\,\\mathrm{slog}_b(z) > -2", "def920753c333e3709a430320bbc156e": "\\delta S = \\frac{C(T,X) \\delta T}{T}.", "def966a81b43fe912d3a2507f1d503c4": "\\frac{\\partial}{\\partial x^\\mu} \\left[\\frac{\\partial\\mathcal{L}}{\\partial(\\partial\\phi/\\partial x^\\mu)}\\right] - \\frac{\\partial\\mathcal{L}}{\\partial\\phi} = 0,", "def975ad220bf02e474c0fb88ae57768": "I_3 = J^2", "def9f04aa37f23d1b42d75dcb1a63424": " \\hat H \\Psi = \\left[{\\hat T}+{\\hat V}+{\\hat U}\\right]\\Psi = \\left[\\sum_i^N \\left(-\\frac{\\hbar^2}{2m_i}\\nabla_i^2\\right) + \\sum_i^N V(\\vec r_i) + \\sum_{i-2 \\}", "df0721d08fa78cf5da0b5e844ee02a52": "L'=\\mathrm{AC}=L_{0}/\\gamma=18\\ \\mathrm{cm}.", "df0732b2306b8bdb54a06db50d665e81": "f_{IN}-f_{LO}", "df0751d1fb02bc01ab13616277f4f33a": "\\forall \\varepsilon > 0 \\; \\exists c \\; \\forall x > c :\\; |f(x) - L| < \\varepsilon", "df0763ab79685f46e071778c53801d9d": " \\langle \\theta^{G}, \\chi \\rangle_G = \\langle \\theta,\\chi_H \\rangle_H ", "df076bef4f8b998552afa51ba3a9e9db": "M |z|^n \\le |f(z)|", "df0791f6695e32a4c778837fac8a3b0e": "x \\in N,", "df079471e7b91c417dd950f18272a09e": "\\sigma = \\frac{\\sigma_{SD}}{\\sqrt{P}} .\\,", "df07e11cb1989e081fd42fe9749a8362": "(p_1, p_2, l)", "df0824eb920c2cc9748b3bf89404a52a": " \\begin{align} \n m_{k+1} \n &= m_k - \\underbrace{[\\tilde{\\nabla} \\widehat{E}_\\theta(f)]_{1,\\dots, n}}_{\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n \\text{natural gradient for mean}\n \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n } \n \\\\\n &= m_k + \\sum_{i=1}^\\lambda w_i (x_{i:\\lambda} - m_k) \n \\end{align} ", "df08448f218e483ef56359feb130fee7": " \\mathbf{B} = \\frac{\\mu_0}{4\\pi}\\int_C \\frac{I d\\mathbf{l} \\times \\mathbf{\\hat r}}{|\\mathbf{r}|^2}", "df0881c45f0a0f72c0fd7ef0c5486d4e": "\\Pi_{i+1}^{\\rm P} = \\left( \\Pi_{i}^{\\rm P} \\right)^{K_i^{\\rm c}}", "df091d31e0086e76c31278c273f04233": " x = z ", "df09aea884019cb88a2957126faba316": "\\Rightarrow ", "df09c37fe00ec6eaac563191b2adc57d": "g\\mapsto g^p", "df09d292a02345ee5a579f859d5e2b3b": "\\overline \\partial : \\mathcal E^{p,q}(E) \\to \\mathcal E^{p,q+1}(E) ", "df09d90e67a14b7de6e0f398e2f318f2": "\\varepsilon(t) = \\sigma C_0 + \\sigma C \\int_0^\\infty f(\\tau)(1-\\exp[-t/ \\tau]) \\,d \\tau", "df0a289ce64404c75620eaae6cbc2390": "b_2 = a_1^2 + 4a_2,\\quad b_4 = a_1a_3 + 2a_4", "df0a301d6dc84368730e37da7c703ba8": "\\Phi=\\Phi(E^a)", "df0a36d6c64e77d1363c30291fc1a612": "g(S^2)\\subset f(S^2)\\cup U", "df0a4e32f658401b07bae9276deb4f33": "g_6^2g_1^4g_7=1", "df0a9f196bde01a401cf2d57ec25d548": "\n\\begin{cases} \n\\{O_{1},O_{2}\\} \\\\ \n\\{O_{3},O_{7},O_{10}\\} \\\\ \n\\{O_{4},O_{5},O_{8}\\} \\\\\n\\{O_{6},O_{9}\\}\\end{cases}\n", "df0af164d33a273830458c3aea243a77": "\\delta x^{\\mu} = \\epsilon X^{\\mu}\\!", "df0b16d70e0e87b0b91341c839f1a0ae": "120 = 2^3 3^1 5^1\\,", "df0b80064ee2a2bcba9c3a02e4686bfd": "\\overline{s}", "df0b8283db49642fa791885e8119bf45": "\\ v ", "df0c6769ca5f46f8a5ae8e6ff81c2de3": " \\frac{d^3y}{dx^3} = - \\frac{d^3x}{dy^3}\\,\\cdot\\,\\left(\\frac{dy}{dx}\\right)^4 -\n3 \\frac{d^2x}{dy^2}\\,\\cdot\\,\\frac{d^2y}{dx^2}\\,\\cdot\\,\\left(\\frac{dy}{dx}\\right)^2", "df0c9c3f114d4bdfd90c65b8b43576db": "\\bigl( \\begin{smallmatrix}\\\\ \\pm 1&0\\\\ 0& 1\\end{smallmatrix} \\bigr),", "df0cd22b1181ce6f4bd6a8b5db3d6d02": "\\mathfrak{so}(1,1)\\cong \\mathbb R", "df0cd32b93ed52fb10bf7f71ffd4519b": "u(x,y)=e^x \\sin y. \\,", "df0cd68b9617acdfa513a15dbe2aae7d": "\\text{If }A\\subset B\\text{, then }B^{c}\\subset A^{c}.", "df0d3891da3baf018ab8853ab3d02e47": "f\\left( x^\\prime ,t+\\varepsilon \\right) =\\int_{-\\infty }^\\infty \\, dx\\left(\\left( 1+\\varepsilon \\left[ D_{1}\\left(x,t\\right) \\frac{\\partial }{\\partial x}+D_{2}\\left( x,t\\right) \\frac{\\partial ^{2}}{\\partial x^{2}}\\right]\\right) \\delta \\left( x^\\prime - x\\right) \\right) f\\left( x,t\\right)+O\\left( \\varepsilon ^{2}\\right).", "df0dacd40f78f96ad6c050461551a5a2": "p\\in\\mathbb{R}^{n_p}", "df0e30046acd594beb59741d5a309183": " \\langle A | B \\rangle = \\text{the inner product of ket } | A \\rangle \\text{ with ket } | B \\rangle", "df0ea23ea1c302ab0d0db973fe430c0d": "\\begin{pmatrix} q_n & q_{n-1} & \\cdots & q_1 & q_0 & 0 & \\cdots & 0 \\end{pmatrix}.", "df0ecff2f6c89719812432ce8625a62e": "\nU_5=\\begin{pmatrix}\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix} \\quad\nL_5=\\begin{pmatrix}\n0 & 0 & 0 & 0 & 0 \\\\\n1 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1 & 0\n\\end{pmatrix}.\n", "df0effea8f0b4c5a5a65cbcc83b29cc7": "V(k) = \\frac{4 \\pi Z e}{q^2 + k^2}", "df0f0049ea5b5155bbf97822f68f62ab": "\n \\forall x\\in V\\qquad \\exists A\\in {\\mathcal A}\\qquad x\\in A,\n ", "df0f11a4291b7b687823e7b86e1d21b3": " t = \\frac{\\overline{x} - \\mu_0}{s/\\sqrt{n}} ", "df0f1f43b13a5185a3bb2d2ed51eea3d": "y=(\\,\\cos\\beta,\\sin\\beta\\,)", "df0f43baf0ff2928a7ff90e510c955c5": "f(x)=a-\\frac{1}{x}", "df0f65a7955475582fe0f728ca59a194": "m \\in M^n", "df0f65aaff6be45608fd94ae464eeee6": " \\mathcal{V}_1(S_i) + \\dots + \\mathcal{V}_1(S_p) ", "df1035859adb9da81602c6705c2b66ad": "\\!\\mathcal A \\models_X^+ t_1 = t_2", "df10e99dc690fe3945f619816da09105": "\n f_c = \\frac{k_c}{2}(2H-c_0)^2+\\bar{k}\\,K\n", "df11e3928543ee96086a0139ed5fed45": " c \\equiv a \\times b \\pmod {N}", "df122a0239fae03ca7887853f6688024": "k'[x_1^{1/q}, ..., x_d^{1/q}]", "df12665d6a3f250b737d43b4728cc32d": "d(v)/5", "df128b695ddc67e7983bf27fdcdc46dc": "[x,p_x] = i\\hbar", "df129e64b15aaeacca1715f11757b538": "\\vec{A} = \\frac{ \\sqrt{2} a \\int C\\left( \\frac{u^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right) \\, \\sin(\\omega u) \\, du }{ C\\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right)} \\; \\partial_x ", "df12b8da71920414987d65170042493d": " t = J \\dot{\\omega} + \\omega \\times J \\omega ", "df12fb05ad61947217a250afe18eeca0": "f(z)=i\\frac{z-i}{z+i},", "df13fcf0528b1ba302778cbe73499af5": "S_n = \\sum_{i=1}^n a_i X_i,\\,\\!", "df140ec8e3714733d2f4c9f11f5e3c42": "\n \\mathbf{E} = \\xi \\exp[i(kx - \\omega_j^R t)] \\exp[\\gamma t] \\mathbf{\\hat{x}}\n", "df1425ef9fde6206f6480d00cbffd797": " \\lim_{n \\to \\infty} \\frac{a_n}{b_n}\\ ", "df146cf41bbb1225f995acb1490a4ecf": "\\mathfrak{p} \\subset \\mathfrak{p}'", "df149fd6587d32c729719a0c3b21a962": "\n \\begin{array}{lcl}\n b + p^2 & = & s + q \\\\\n c & = & (s - q) p \\\\\n d & = & sq\n \\end{array}\n ", "df157c197b385b28509a0c0d78351207": "pA+qB...\\rightleftharpoons A_pB_q \\cdots: \\beta_{pq\\cdots}=\\frac{[A_pB_q \\cdots ]} {[A]^p[B]^q \\cdots }", "df15be36f50bec4f7c4bd6066a6b7f55": "[M_{\\mu\\nu},M_{\\rho\\sigma}]=\\eta_{\\nu\\rho}M_{\\mu\\sigma}-\\eta_{\\mu\\rho}M_{\\nu\\sigma}+\\eta_{\\nu\\sigma}M_{\\rho\\mu}-\\eta_{\\mu\\sigma}M_{\\rho\\nu}", "df15c7759cc9a26302f327b329006eec": "\\begin{matrix}\nn \\approx \\sqrt{1 + \\sum_i B_i } \\approx \\sqrt{\\varepsilon_r}\n\\end{matrix},", "df16251cfe6a513218cb82dc9fc29710": "E = \\left[ 3 + W \\left(\\frac{-3}{e^3} \\right) \\right] k_\\mathrm{B}T \\approx 2.821\\ k_\\mathrm{B}T,", "df16bd726d0cacf2dc2e949fd9000ef7": "f(z)=\\sum_{n=0}^\\infty (z-a)^n {1 \\over 2\\pi i}\\int_C {f(w) \\over (w-a)^{n+1}} \\,\\mathrm{d}w.", "df174342a4f9e72a5f64eacef1016cff": "\\mathcal{M}(U)", "df1747301f34aba0e52574b3a5aaea60": "L(s,\\chi) = \\sum_{n=1}^\\infty \\frac {\\chi(n)}{n^s}\n= \\frac {1}{k^s} \\sum_{m=1}^k \\chi(m)\\; \\zeta \\left(s,\\frac{m}{k}\\right).", "df175023d9be9e47111c95805277f568": " L_{m,n} = \\frac{\\mu_0}{4\\pi} \\oint_{C_m}\\oint_{C_n} \\frac{\\mathbf{dx}_m\\cdot\\mathbf{dx}_n}{|\\mathbf{x}_m - \\mathbf{x}_n|} ", "df17986311bb0ccaa3649f4a83fe33a0": "\\mathbf{X} \\in\\mathbb{R}^{n\\times p}", "df17adeb42d3cf8b5d8b9810ae23a242": " z^{-a}\\, _2F_1 \\left (a,1+a-c;1+a-b; z^{-1} \\right)", "df17bca0e55c1237a3ea68926347d9a2": "\\mathbf{Z}\\left(\n\\mathbf{z}\\right) \\mathbf{X}\\left( \\mathbf{x}\\right) ", "df17c7020a90f95bd9844cfe1b2f5c26": "S(A)_\\rho \\ \\stackrel{\\mathrm{def}}{=}\\ S(\\rho^A) = S(\\mathrm{tr}_B\\rho^{AB})", "df17f53ce204b8816d29b572ec354372": "\\frac{1}{n_0} = \\frac{16}{3}\\frac{\\pi\\ell d^2}{4}", "df185bcad4119382268199f5683948a8": "\\gcd(N,D)=1", "df18d6c67c99a38e342cbc93dfbdb085": "x'_{\\lambda}=x_{\\lambda},", "df1910eac0df235d8be9380dfce031ba": "\\displaystyle{\\mathbf{t} =\\dot{\\mathbf{v}},}", "df196e4ce677ca07af1a77b79fbf4e12": " \nP^\\eta(\\eta_t\\equiv 1 \\quad\\text{on }A)=P^A(\\eta(A_t)\\equiv 1),\n", "df199e0bd70f1160fa45f82a0c49cde3": " p = mv = \\hbar k ", "df19f054b713ad9798b044c86054d406": "\\frac{\\Gamma(\\alpha+\\beta)}{k!\\Gamma(\\alpha+\\beta+n+k)}", "df1a352ce4f1e0c072e1ea1db3286395": "\\lim_{c\\rightarrow \\infty} f(c)=0", "df1a43e5cbbc0267d4a017fe4928bc0a": " \\frac{\\sigma_\\delta}{\\gamma} = \\sigma_{x'} = \\sigma_{y'}", "df1a450228a751a92443ac4ab151f270": "\\sigma = \n\\begin{bmatrix}\n\\sigma_{11} & \\sigma_{12} & 0 \\\\\n\\sigma_{21} & \\sigma_{22} & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix} \n\\equiv \n\\begin{bmatrix}\n\\sigma_{x} & \\tau_{xy} & 0 \\\\\n\\tau_{yx} & \\sigma_{y} & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix}", "df1a48d8b36e3e8332d3754d629f0b39": "\\mathbf{L} = \\mathbf{x}\\times \\mathbf{p}", "df1a7fea6678aa6b4360b886d35b257f": "M=\\angle zcy,", "df1ac00ea68e5ce62b564ec79c85cd44": "\\nabla_X\\sigma = X^i(\\partial_i\\,\\sigma^\\alpha + {\\omega_i}^\\alpha\\!{}_\\beta\\sigma^\\beta)e_{\\alpha}.", "df1b9d9504c40ce7199ba2de919cb881": "\n\\langle j_1 m_1 j_2 m_2 | j_3 m_3 \\rangle = (-1)^{-j_1+j_2-m_3}\\sqrt{2j_3+1}\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & -m_3\n\\end{pmatrix}.\n", "df1bb3dec04a07272346f79b9a29f174": "\\int_a^b f(x)g(x)w(x)\\,dx = 0.", "df1c72c03c70fcff49fcb3f91367afb9": "f^{-1}(K) \\subset f^{-1}(\\cup_{i=1}^s V_{k_i}) \\subset \\cup_{\\lambda \\in \\Gamma} U_{\\lambda}", "df1cb7bf1d0548b04cf90b8b3ca5d574": "\\tan[\\theta(x)]\\,", "df1cbfa4edbcdc5cd5c0385e0669fed7": "C_p\\ln\\frac{T_2}{T_1}\\;", "df1cc8c2ccb922826efc9b56d700e960": "Z_\\Lambda^\\Phi(\\bar\\omega) = \\int \\lambda^\\Lambda(\\mathrm{d}\\omega) \\exp(-\\beta H_\\Lambda^\\Phi(\\omega | \\bar\\omega))", "df1cef4bd94ed6e445027939a8cd834a": "dU = T dS - P dV + \\sum_i^n \\mu_i dN_i + \\Phi dQ + v dp", "df1d26529f372718f180ac7a7825ec32": "1, \\ldots, n", "df1d53ac84bf4e259f6c04a07c0c2f87": "A(Y)=A(y_4\\ \\mathcal{k}\\ y_3\\ \\mathcal{k}\\ y_2\\ \\mathcal{k}\\ y_1) = (y_1 \\oplus y_2)\\ \\mathcal{k}\\ y_4\\ \\mathcal{k}\\ y_3\\ \\mathcal{k}\\ y_2", "df1d75802a5f47b89cea2ea8f199625c": "AV = \\frac{1029.4(PO_{bpm})}{ID^2-OD^2}\\,", "df1d7e5665126425b6c4bfdb6662c446": "p_{3/2} ", "df1decf4aaaf254998475a2b6b84b466": "\\frac{1}{2(l+1)}+\\sqrt{k}\\left(l+\\frac{3}{2}\\right)-E_{l} = 0,", "df1e2e3fa99270883228b42ca4de6415": "p(\\boldsymbol{x},z,t)\\,", "df1ef79e6b561a9cdc44997a1353c9b3": "-g = g_{00}\\gamma.\\,", "df1f0bf44b6fdfd62058b007f88d1ecb": "\\dot{c}_m(t) = - \\sum_n c_n\\langle\\psi_m|\\dot{\\psi_n}\\rangle e^{i(\\theta_n-\\theta_m)}", "df1fbb28d33be5c58d0c0d07c589b76d": "\\frac{d^2\\mathbf{r}}{ds^2} + \\alpha\\ \\mathbf{r} = -\\mathbf{P}", "df20248b2a9d562dd594bce7604f1a63": " s_{i} ", "df203260f11aebc905593fe720f20c5c": "S_{i+1}", "df2053099ed77ae734cbe0c008a44ff5": "\\mathfrak a", "df2082c3d78f6ffef705609e7ba3b6bc": "\\boldsymbol{\\tau} = \\bold{p} \\times \\bold{E} \\ , ", "df20a0c8d7d475ac42db347395cbb5af": "b = a - \\left\\lfloor \\frac{a}{n} \\right\\rfloor \\times n - n.", "df20c2a244f7145d55eebe0a78b9bd56": " A = v w^{\\mathrm{T}}. ", "df20e5ec82a971878cd84d8d08b805a3": "[T^{-1}]_\\gamma^\\beta=([T]_\\beta^\\gamma)^{-1}", "df21713fecd9aa4e9ea5850b82bff2c5": "x^{23} + x^{22} + x^{21} + x^{8} + 1", "df21d56b5b341393d0513dada22ff101": "A + B = 4000", "df2237b756809c12c8bfc895a23a37e1": "z=0.", "df227c5949affa4a0d420cf85793d7b8": "\\bar{f}(r,t,h)= (2 r \\cos(t)+ r \\sin(t) + 5) r", "df2296600d8398f8c227ec8fd8ac72a7": "d = \\sum_{i=0}^{N-1}f(x_i)", "df22a4fdf8e076a2047e3b2797adf667": " Incidence Rate = \\frac{events}{persontime} ", "df23574ee7947db4a0d602add68526ab": " \\frac{\\Delta_h[f](x) - \\frac12 \\Delta_h^2[f](x)}{h} = - \\frac{f(x+2h)-4f(x+h)+3f(x)}{2h} ", "df23961f6392f0baa9e5276b102a75b2": "(y \\downarrow k)[n] = y[k n] ", "df23d4dbaa8016ed5aa570f26e9465e2": "g(u)= \\frac{\\partial}{\\partial u}J(u)", "df23d5680e624ab9688458c65dd50dcd": "\\lambda < \\kappa ", "df23d92b6b9250607a0259bd5fdb6a1f": "\\mathbf{AA1} = \\begin{bmatrix}\n(1+2\\lambda+\\beta) & -(\\lambda-\\alpha) & 0 & 0 \\\\\n-(\\lambda+\\alpha) & (1+2\\lambda+\\beta) & -(\\lambda-\\alpha) & 0 \\\\\n0 & -(\\lambda+\\alpha) & (1+2\\lambda+\\beta) & -(\\lambda-\\alpha)\\\\\n0 & 0 & -2\\lambda & (1+2\\lambda+\\beta)\\end{bmatrix}", "df241440b03bf46463a7866a9e8e8fa5": "D_i\\cap C_m\\ne\\emptyset", "df24553d75a768c7cbc0ea2af337c03e": "\\textstyle )", "df247b19b069d523a0b09d0d0ec5ab64": " : \\hat{f}\\,\\hat{f}^\\dagger \\, \\hat{f} \\hat{f}^\\dagger : \\,= \\hat{f}^\\dagger \\,\\hat{f}^\\dagger \\,\\hat{f}\\,\\hat{f} = 0 ", "df2546ab7be376411577dbe32e5c6764": "\\int_a^b \\ldots\\, dx", "df25901920ee2d89340e149986f85ed0": "\\mathbf{\\hat T^ \\dagger} (\\lambda) \\mathbf{\\hat T}(\\lambda)=\\mathbf{I} ", "df25a71976cb34c6ecec2e0bd252ae7a": "\\Delta=\\frac{\\partial_r}{\\partial x^a}\\frac{\\partial_l}{\\partial \\theta_a}", "df25b99da0f0724d6a02bba5f277b8e7": "L = \\int_a^b \\sqrt{ (dx)^2 + (dy)^2}. \\ ", "df266783cc2060f41750a0e7f8f7ce1b": "k_\\mathrm{e}", "df266a6e32576e11ebeecb29a7b14a73": "\\frac{|v\\rangle + i |w\\rangle}{\\sqrt{2}}", "df26b94cb3b0b1158dd823f0da3ac886": "=\\widehat{D}(-\\alpha)(\\widehat{a}|\\alpha\\rangle - \\alpha|\\alpha\\rangle)", "df26c21d9189c79fa452b906d279470c": "\\scriptstyle(-3.1(2.5))\\times10^{-16}", "df26d11ef7a37c99e0b5656b08cf03f9": "N_f(50%)=\\frac{1}{2}\\left[\\frac{2\\epsilon'_f}{\\Delta D}\\right]^{\\frac{-1}{c} }\\quad\\Delta D(\\text{leadless})=\\left[\\frac{F L_D \\Delta(\\alpha \\Delta T)}{h}\\right]", "df26e979292b5761b241167d6fa2ff8d": "D_{N}^{*}(x_1,\\ldots,x_N)\\geq c'_s\\frac{(\\ln N)^{s}}{N}", "df27cae1b95acc2d252d0fbddbda304b": "A/\\mathcal{J}_\\lambda", "df27cc8a5f29430667a882c1cb88ff2f": "p(\\sigma) = \\frac{m}{\\Gamma(m) \\sigma_{av}} \\left ( \\frac{m\\sigma}{\\sigma_{av}} \\right )^{m - 1} e^{-\\frac{m\\sigma}{\\sigma_{av}}}\nI_{[0,\\infty)}(\\sigma)", "df282d4588e56116082e96ee5ca71321": "\\|x\\| \\to \\infty \\Rightarrow f(x) \\to \\infty. \\, ", "df2863e41efdc3ce183764536fb9bbca": "\\,a", "df28e18a7ace309d6a986cfea32bcefc": "4x=\\frac{7+x}{10} \\mbox{ so } x=\\frac{7}{39}.", "df28fd63b1e0c8ab817e48b473d30281": "\\lambda_1,\\lambda_2,\\ldots,\\lambda_k", "df291ad59435b102adbe21096eb79b49": "A \\sim M^{\\frac 7 8}", "df2950c2ed33190d8329ff1147d8e755": "Y_{p}=\\frac{n_{p}(t)-n_{p}(t=0)}{n_{k}(t=0)+\\int_0^t\\dot{n}_{k,\\text{in}}(\\tau)d\\tau\\underbrace{-n_k(t)}_{\\text{only for Definition 1}}}\\left |\\frac{\\mu_k}{\\nu_p}\\right|", "df2992a0d3000ebd1b1068d13034c0ba": "{\\rm Area}(w)\\le f(|w|),", "df29df9a514d13299d454040970f63e8": "n = 1 \\dots N", "df2a11c3c7447e62d1bd39cfa4544234": "p_n^2>p_{(n-i)} \\cdot p_{(n+i)}", "df2a68ababa17c1ab4e30ee5d708d4df": "({\\or}L)", "df2a81d98d083cab103259295f7476ee": "a, b, c \\in \\mathrm{R}", "df2a989d91b1295df276ad490167a055": "H(u)(t) = \\text{p.v.} \\int_{-\\infty}^{\\infty}u(\\tau) h(t-\\tau)\\, d\\tau = \\frac{1}{\\pi} \\ \\text{p.v.} \\int_{-\\infty}^{\\infty} \\frac{u(\\tau)}{t-\\tau}\\, d\\tau", "df2abe036351604b108f60b93dbcefe5": "\\bigcup_{x \\in X}\\ \\bigcap_{t_x > 0}\\ \\bigcup_{t \\in [0,t_x]} \\{x_0 + tx\\} \\subseteq A.", "df2b0170f7eafab7f4dab1af840e2bb5": "\\Gamma\\vdash x : \\text{Int}", "df2b037ba48e11fbd3ea77fcd36edcc5": "{10}^{\\,\\! 4 \\cdot 2^{50}}", "df2b06e9f0db5cf2deeac11d36c025c4": "\n\\begin{align}\n &\\nabla p = \\eta\\, \\nabla^2 \\mathbf{u} = - \\eta\\, \\nabla \\times \\mathbf{\\boldsymbol{\\omega}},\n \\\\\n &\\nabla \\cdot \\mathbf{u} = 0,\n\\end{align}\n", "df2b38fc4d58fc19cf2caab89afaad61": "Y = -X", "df2b535f57ed448578775164157f385a": "z_j=\\chi_{\\psi_{j,j}}(z_j,\\rho_{\\psi_{i,j}}(z_i))", "df2bcb94efb4336700da22aac15d6f5f": "x = \\frac{\\rho \\sin E }{ \\sin \\phi_1} \\,", "df2be761ccde436f9afd317db713699b": " \\pi \\, {G} = 4 \\sqrt{\\tfrac2\\pi}\\,\\Gamma{\\left(\\tfrac54 \\right)^2} = \\tfrac14 \\sqrt{\\tfrac{2}{\\pi}}\\,\\Gamma {\\left(\\tfrac14 \\right)^2} = 4 \\sqrt{\\tfrac2\\pi}\\left(\\tfrac14 !\\right)^2", "df2bfd6ad0fc795097a35e646f1caf2e": " \\sinh x = -\\sqrt{\\pi}i \\; G_{0,2}^{\\,1,0} \\!\\left( \\left. \\begin{matrix} - \\\\ \\frac{1}{2},0 \\end{matrix} \\; \\right| \\, -\\frac{x^2}{4} \\right), \\qquad -\\pi < \\arg x \\leq 0 ", "df2d1ab2534c70efc8dc84f4c473c2a4": " \\tau \\circ d = -d ", "df2d3a2d3ac42482d9101f06e0ab8e00": " \\sum_{0 < k \\leq n} k^{c} = \\sum_{k \\geq 0}\\frac{B_{k}}{k!}c^{\\underline{k-1}}n^{c-k+1}.", "df2d5e559bc28175c62f0e1f888438a1": "V_{\\beta}\\setminus\\bigcup_{\\xi<\\alpha}\\subseteq V_{\\beta}\\setminus V_{\\alpha}\\,", "df2d92d8eb551c3b92c4301a8c80f608": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ -0.30\\end{smallmatrix}", "df2dbb0e6010ce8363e0adf618419e05": "\\vec{e}_0 \\neq \\vec{f}_0, \\; \\vec{e}_1 \\neq \\vec{f}_1", "df2dbd0d82cfd6f238d80ee67cc75f9d": "\\sigma = \\frac{F}{A}", "df2dc4274031d3f96ea653b61e9443da": "\\Phi'(x)", "df2e04cebe41dbc7d2ccca45225286e9": "x\\cdot 0 = 0", "df2e34d3231041da6d269c0e53908a19": "f(S)", "df2e354bdc71ed29050edc9ef5491910": "\\begin{align}\n & \\left( 8\\,\\frac{\\mathrm{bits}}{\\mathrm{channel}} \\times 24\\,\\frac{\\mathrm{channels}}{\\mathrm{frame}} + 1\\,\\frac{\\mathrm{framing\\ bit}}{\\mathrm{frame}} \\right)\n\\times 8\\,000\\,\\frac{\\mathrm{frames}}{\\mathrm{second}} \\\\\n = {} & 1\\,544\\,000\\,\\frac{\\mathrm{bits}}{\\mathrm{second}} \\\\\n = {} & 1.544\\,\\frac{\\mathrm{Mbit}}{\\mathrm{second}}\n\\end{align}", "df2e3c4ebdfad06463b062261039721c": "\\mu_U", "df2e6863bb4b546989f941c79b5847fa": " \\{B_{j}:j\\in J'\\} ", "df2f1e6899687f27754e3925de7c9d00": "\nf(x) = \\exp(-x^TAx+s^Tx) \\;,\n", "df2f6d8f627b6be819014a71e2120ff9": " \\Omega_{n+1}=e^{-\\Omega_n}.\\,", "df2fb73001d6712314a2e4695d486ac4": "C_Z = \\sigma^2 I,", "df2fbcda76a372b2624d583491f35b85": " T_i = K_p/K_i, T_d = K_d/K_p", "df2fe7d4767f4b024a80691758acbdf1": "\\hat{a}_i \\,\\hat{a}_j^\\dagger \\, \\hat{a}_k \\,\\hat{a}_l^\\dagger= (\\hat{a}_j^\\dagger \\,\\hat{a}_i + \\delta_{ij})(\\hat{a}_l^\\dagger\\,\\hat{a}_k + \\delta_{kl})", "df2ff010049b6230f031b24f4d6e16a1": "\\operatorname{erfi}(z)", "df3024287f44038159df14d343f82f39": "(a_1,a_2, ... ,a_n).", "df3077ad9b27b6dac803c09a83aa233b": " \\frac{(b-a)^3}{36}\\,f^{(2)}(\\xi) ", "df307b6040f0f035f0655c545646ce55": "A^n", "df309921a6d0a68ce3079c593973c748": "45:64 \\approx 1:1.4222", "df30e6fb06a77637aae2c017d54e9e05": "{\\displaystyle}x^2+y^2=1", "df318c8f76d8292dad5fe1daf8590f42": "\\begin{align} \\mathbf{a} \\cdot \\mathbf{b} &= a_1b_1 + a_2b_2, \\\\ \\mathbf{a} \\wedge \\mathbf{b} &= (a_1b_2 - a_2b_1)\\mathbf{e}_1\\mathbf{e}_2 = (a_1b_2 - a_2b_1)\\mathbf{e}_{12}. \\end{align}", "df31904fbbbe2813f92a47d0a2849be3": " \\sum_{i=1}^n \\frac{\\gamma^2}{\\gamma^2 + [x_i - x_0]^2} ", "df319a37e6d1ad790bd2fdf2d025301b": "D(X,Y) = 1 - I(X;Y)/H(X,Y)", "df31cb51885692aa228373abde0a1001": " \\frac{1}{\\zeta(s) }= \\prod_{p} (1-p^{-s})= \\sum_{n=1}^{\\infty} \\frac{\\mu(n)}{n^s}", "df31dc548e0fdc72830c87efbc5130db": "[\\phi[f],\\phi[g]]=i\\Delta[f,g] \\,", "df31e3e4e2f5d1ecc980446548e9bd7f": "\\mathrm{Borel}([0, t]) \\otimes \\mathcal{F}_{t}", "df32165ef72c589db401ea6a03406c1b": "E(2)=E_{1,0} -d {\\epsilon} \\frac {}{} ", "df328d3b88cf40b4efab2003f4b49042": "\\overline{[a,b]}", "df32aea37346ffe95502ecdab3881388": "\\mathbf R\\,\\!", "df32b22496f9c21317308db7f94b110a": "-\\nabla\\cdot\\mathbf{M}", "df32b77e9d30a67dbf3ce1074cf7279e": "\\frac{x}{\\log x}\\exp\\left({ (C+o(1))(\\log\\log\\log x)^2 }\\right) \\ ", "df32ce52fb2e08263046d21b3efaf248": "W = -e\\phi - E_{\\rm F}, ", "df32ee78f79fdcb4eb8ad4144d98090f": "\\rho(\\mu^2)", "df331d8ea3f0a08799328ecac9ef7ec1": "=[1]_q \\cdot [2]_q \\cdots [n-1]_q \\cdot [n]_q", "df33648c51c391103ea6a38f7883025a": " {\\lambda_n}^{-} \\sim \\frac{- n^2 \\pi^2}{\\left(\\int_a^b \\sqrt{(w/p)_{-}(x)}\\, dx\\right)^2},\\quad n \\to \\infty. ", "df33cfaf52f5a2c8c4f3744a70765773": " 1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}+ \\ldots = \\frac{2\\cdot 3\\cdot 5\\cdot 7\\cdot 11\\cdot\\ldots}{1\\cdot 2\\cdot 4\\cdot 6\\cdot 10\\cdot\\ldots} ", "df33ed32e5084893183c2c193f8985bd": "\n m(\\varphi)=A_0\\varphi+A_2\\sin 2\\varphi+A_4\\sin4\\varphi\n +A_6\\sin6\\varphi+A_8\\sin8\\varphi+\\cdots,\n", "df340406ed702f8b7c4fade43f3b21ad": "2\\pi - 5\\sin^{-1}\\left({2\\over 3}\\right)", "df3421e6003b2618d11cf89f70886dd9": " C = \\frac { D( n N - 1 ) - n N } { ( 2 N ( n^2 - n ) ) ^{ 1 / 2 } } ", "df343e47745cde5942e7ac2295ec0e25": "\\mu\\boxtimes\\nu", "df3448a84fd9d966e99b1152bee85f42": "\\begin{align}\n c_{2n}^2 &{}= \\left( r + \\frac{1}{2} c_n \\right) 2r \\\\\n c_{2n} &{}= \\frac{s_n}{s_{2n}} .\n\\end{align}", "df34853869e63325665e5f7ff19f3501": "\\Phi:PH\\to QH", "df34f133717bdde45717bd0806939ba5": " \\quad \\sum n^m = \\frac 1{m+1}\\left( B_0n^{m+1}-\\binom{m+1}1B_1n^m+\\binom{m+1} 2B_2n^{m-1}-\\cdots +(-1)^m\\binom{m+1}mB_mn\\right) ", "df34fc83e40c2e6725a3a25cbcab94ed": "\\overline{g(x)}", "df351f21017114b9c2f549b295392819": " \\gamma = \\ln\\pi - 4\\ln\\Gamma(\\tfrac34) + \\frac4{\\pi}\\sum_{k=1}^{\\infty}(-1)^{k+1}\\frac{\\ln(2k+1)}{2k+1}.", "df353a40be0f29d2fb27f300e8c90faf": "\\hbar {\\mathbf{k}}", "df356b04a20ec1ea87be73226e2de849": "\\frac{\\partial E}{\\partial \\hat{h}_i} = -2R_{xs}[i] + 2\\sum_{k=0}^{N-1}\\hat{h}_kR_{ss}[i - k] = 0 ", "df35baa1c5789eebc4190082372a8627": "\\Delta y = y \\otimes 1 + (-1)^F \\otimes y", "df35ea9d965146fd7a5f4cbeff9c9a0d": "M_{l}=\\frac{1-\\phi_{sl}}{\\phi_{sl}}M_{s}", "df36ce79859cb63354fa2316c8371d0b": "0 \\leq r \\leq 1", "df36e7a0fd4476e7972d420065a53250": "X' \\to Y", "df3791aa8db89555544ca963746e7432": "C_{in}", "df3809dff893d9cb86080e48e1f9d642": " \\frac{d T(t)}{d t} = - r (T(t) - T_{\\mathrm{env}}) = - r \\Delta T(t)\\quad ", "df381ae39cd5fa060f977eba63c609e0": "a : 1\\ 2\\ 4", "df3858395ef43bdbd87581eed0232e1f": " (\\tfrac{ 1}{10},\\,\\tfrac{ 3}{10},\\,\\tfrac{ 7}{10},\\,\\tfrac{ 9}{10}),", "df385f31fe6ff458a9dc34bc84fafad4": "|{\\mathcal P}|=n^2+1", "df389e7e610452c2e1c23a600d582aa7": " \\frac{a \\cdot b}{\\|a\\| \\|b\\|} ", "df38b34cd13c605b90a093242925375e": "K_p = \\frac{{p_S}^\\sigma {p_T}^\\tau} {{p_A}^\\alpha {p_B}^\\beta} = \\frac{{X_S}^\\sigma {X_T}^\\tau} {{X_A}^\\alpha {X_B}^\\beta} P^{\\sigma+\\tau-\\alpha-\\beta} = K_X P^{\\sigma+\\tau-\\alpha-\\beta}", "df38f769e4be28bef76f3de66b56ae4e": "\\frac{\\partial \\mathbf{a}}{\\partial \\mathbf{x}} =", "df394fe203a68b9d9d364020e08ea9dc": "{{i}_{D}}\\propto \\frac{W}{L}{{\\left( {{v}_{GS}}-{{V}_{TH}} \\right)}^{2}}", "df39713a38f7c3d0d17512a0cb306874": "2\\uparrow\\uparrow n", "df399813061e4b9528ba917861f798a1": "[E] + [ES] = [E]_0", "df39be3bf865671667f327d312e39849": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, w z \\right) =\nw^{b_1} \\sum_{h=0}^{\\infty} \\frac{(1 - w)^h}{h!} \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ b_1+h, b_2, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right), \\quad m \\geq 1,\n", "df39e1f4c42cdfa1e6cefcb075504cb6": "NL+SL", "df39fe167735f560f81c9866dcd483e6": " \\begin{align}\n\\alpha &= \\mu \\nu,\\\\\n\\beta &= (1 - \\mu) \\nu,\\\\\n &\\text{where }\\nu =\\alpha+\\beta= \\frac{1-F}{F} \\text{ and }0 < F < 1, F\\text{ is (Wright's) genetic distance between two populations}\n\\end{align}", "df3a559a063d1f8542326f5bcf935e20": "\\delta \\mathbf{r} = \\mathbf{r} - \\boldsymbol{\\rho}", "df3a5d1685c6e6d9e9c8fbca4e27d3f7": "\\Delta P = P_{end}-P_{top} < 0 ", "df3a70110c28ad2f9ff4f7e8b9ecf211": "V_\\max = k_\\mathrm{cat} [E]_0", "df3a8909186142d6c4637735509c7d47": "CP^1\\times T", "df3aa4072f548f551215a4e82385140f": "P(a) \\equiv P(b)", "df3b0afa088b42d13ee53f34aa4a44c6": "\\ x, y,", "df3b131b77c57548ed848a40d5734fc0": " \\zeta = \\frac{1}{\\sqrt{1 + (\\frac{2\\pi}{\\delta})^2}}. ", "df3b2be8924492c5f19dda39d7d079b6": "\\chi_{\\text{2}}(2\\omega)=\\frac{Nq^3\\zeta_2}{\\varepsilon_0m^2} \\frac{1}{D(2\\omega)D(\\omega)^2} ", "df3b6e754bfabe2fb1d6c15ea3a1f6d4": "+V_{nn}^2\\frac{V_{nk_5}V_{k_5k_4}V_{k_4n}}{E_{nk_4}^3E_{nk_5}}+V_{nn}^2\\frac{V_{nk_5}V_{k_5k_3}V_{k_3n}}{E_{nk_3}^2E_{nk_5}^2}+V_{nn}^2\\frac{V_{nk_5}V_{k_5k_2}V_{k_2n}}{E_{nk_2}E_{nk_5}^3}-V_{nn}^3\\frac{|V_{nk_5}|^2}{E_{nk_5}^4}", "df3bf4b1fb091e811ce5e6a7be3134dc": "(n_{l-1}\\dots n_0)_s", "df3c6783a66b40ef85847fb35e625f1a": "L \\approx 25", "df3c6ad238405b6b754f2c31a08e94d3": " \\tbinom{n}{k} + \\tbinom{n}{k-1} = \\tbinom{n+1}{k},", "df3c8dd59b90be310dd799c85dfa4933": "P_{base} = 1 pu", "df3cec7ce0ca7502408c59bde5b599d1": "g(x) \\in R", "df3d3cc5d45f7bd2764ebebc0f071aad": "R \\,", "df3e031774f2184f740133fbee6fb93a": "b = a - \\left\\lfloor \\frac{a}{n} \\right\\rfloor \\times n,", "df3e12156886014cbaf20fcfd60c4a96": "(\\tau,q^\\lambda,a^\\lambda_\\tau)", "df3e88c03373899db511facb70c5e57c": "\\! \\chi = \\sin^{-1} r", "df3ed5c8c456392d9b09e924f1984443": "\\{p\\}", "df3ed5ec7a129384216347482bb3af18": "\\int \\left(\\sum_k a_k 1_{S_k}\\right) \\, d \\mu = \\sum_k a_k \\int 1_{S_k} \\, d \\mu = \\sum_k a_k \\, \\mu(S_k). ", "df3f0ed0402a9ff069512e69ce53b539": "({\\rightarrow}L)", "df3f1372e24067c006c8b38848d152d9": "R(r) = J_0\\left(\\frac{\\alpha_{0n}}{a}r\\right).", "df3f5bab53a880bdbbc9283108ad7b24": "\n\\Pr \\left\\{ \\lambda_{\\text{max}}\\left( \\frac{1}{n} \\sum_{k=1}^n \\mathbf{X}_k \\right) \\geq \\alpha \\right\\} \\leq d \\cdot e^{-nD(\\alpha \\Vert \\bar{\\mu}_{\\text{max}})} \\quad \\text{for } \\bar{\\mu}_{\\text{max}} \\leq \\alpha \\leq 1.\n", "df3f7b2a966f46c86afd5e9cbae00c36": " a_i(\\mathrm{mod}\\ {n_i}) = \\{ a_i + n_ix:\\ x \\in \\Z \\} ", "df3fb5e84f42164fefa15275a33a76c7": "x(t + \\Delta t) = (2-f) x(t) -(1-f) x(t - \\Delta t) + a(t)\\Delta t^2,\\,", "df40091e112390006dc6f4e98c36033c": "U\\in\\mathcal{O}\\,", "df403c7b1bdf7b2b87cf7cd06c3660da": "DR*(n) =\n\\begin{cases}\n \\frac{DR(n)}{2}, & \\mbox{if } DR(n) \\mbox{ is even}\\\\\n \\frac{DR(n) + 9}{2}, & \\mbox{if } DR(n) \\mbox{ is odd}\n\\end{cases}\n", "df406e5ff9d1bd834c08639d5b971b16": "-\\sqrt{\\frac{2}{7}}\\!\\,", "df407694afb75ae1e4b6cdb4bcd2cc25": "|\\alpha \\rangle", "df40ddd4e767f177f674b97adc604883": "\n\\frac{ \\partial \\overline{\\phi} }{ \\partial t }\n+ \\frac{\\partial}{\\partial x_j} \\left( \\overline{u}_j \\overline{\\phi} \\right)\n= \\frac{\\partial \\overline{J_{\\phi}} }{\\partial x_j} \n+ \\frac{ \\partial q_{ij} }{ \\partial x_j }\n", "df40debccd0bc7f7f0777d299ca9acbe": "P \\or P", "df40f34e8ed89fbc56a587f82524f605": "\nn_{i+1,j,k} \\leq n_{i,j,k}\\quad,\\quad\nn_{i,j+1,k} \\leq n_{i,j,k}\\quad\\text{and}\\quad\nn_{i,j,k+1} \\leq n_{i,j,k}\\quad,\\quad \\forall\\ i,\\ j \\text{ and } k\\ .\n", "df4113e361a653e8f5a9621a4100b846": " \\subseteq [D \\times D^{'} \\rightarrow D^{''}]", "df41155ec0b068737fcb861a908f3b91": "\\phi_{w} \\,= \\phi_{W} ", "df412cfb076cca896d4158127c3ab5df": "b^{(\\log_b (2)+\\log_b (8))/2} = 4,", "df413a872b1e33f1c04edecfe496e885": "P^{p}(\\Omega_{e_{i}})", "df41acbb0f8f46382b980d43eeb0a713": "Y\\in \\mathfrak{g}", "df421dfc09eb976b7f50f507764620ee": "n_{1\\bullet}", "df42dc990b5f1a39aa9aaab0f594aef4": "q_1 q_2 \\equiv \\pm n^2 \\pmod{p}", "df42e229267a42cd7ad4705cfa02a2b7": "\\pi_0 (map(X, Y)) = Hom_{Ho(M)}(X, Y).", "df4344a8d214cca83c5817f341d32b3d": " \\frac{1}{2}", "df43c785eb67e8d21ed9c2b85f6b0497": "S_{k_i}", "df43d81ae695673a3873e019d89aeb3d": "a = 3p_c \\,V_c^2", "df43e6aff6fc6b16b18cbf6a8a2bd603": "\\lim_{n \\to \\infty}\\frac{A_n-\\ell}{x_n-\\ell}=0.", "df43f97de6982ec1dc158c41f74cc79c": "\n\\begin{align}\n\\theta & = \\arccos \\left( \\frac{r-h}{r} \\right)\\\\\n & = \\arccos \\left( 1 - \\frac{h}{r} \\right)\\\\\n & = \\arccos \\left( 1 - \\frac{1}{2\\pi} \\right) \\approx 0.572 \\,\\text{ rad,} \\mbox{ or } 32.77^\\circ.\n\\end{align}\n", "df441106fe451681453ea5464e8b9fbd": "L_d^p(K_t) = \\{X \\in L_d^p(\\mathcal{F}_T): X \\in K_t \\; P-a.s.\\}", "df44347863ac17dc898a13f44f681d01": "a\\neq 0", "df4439a5a72d1a2c0c1e47bb00d24b54": "\\lim_{n=\\infty}T_i=L", "df443a4c752e9cd69437b2f10cc491b7": "\\lambda\\in\\mathfrak{h}^*\\backslash\\{0\\}", "df4463bf7887585a8b750648824ba01b": "K_{\\rm J} = \\frac{\\nu}{U} = \\frac{2e}{h}\\,", "df44a4f4a412a1e820d2a458e511835d": "\n[[Category:Paralympics infobox templates|{{Template:Infobox National Paralympic Committee}}]]\n", "df44e5bc6b4334ad017f5531cbf40cc0": "\\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{x}} =", "df44e7bc4b0daec448f16c269de6e975": "\\hat z({\\mathbf{s}}_0 )", "df44f73753cd9b702a65772a5ed4efc3": " dT = 0", "df452e76e9f9413b55564b635cd1df19": " \\mathbf{E} = \\frac{1}{4\\pi\\epsilon_0} \\int_V \\frac{\\mathbf{r} \\rho \\mathrm{d}V}{\\left | \\mathbf{r} \\right |^3} \\,\\!", "df453bd3a4e49ed654684872cba93a78": "\n\\begin{align}\n\\int_V \\nabla \\cdot \\left( \\mathbf{u} \\cdot \\boldsymbol {\\sigma} \\right) dV \n &= \\int_V \\frac{\\partial}{\\partial x_j} \\left( u_i \\sigma_{ij} \\right) dV \\\\\n &= \\int_V \\frac{\\partial u_i}{\\partial x_j} \\sigma_{ij} + u_i \\frac{\\partial \\sigma_{ij}}{\\partial x_j} dV\n\\end{align}\n", "df4599f64cac1d97327cd804c4101a08": "\\Gamma = c + jd = \\frac{a^2+b^2-1}{(a+1)^2+b^2} + j \\left(\\frac{2b}{(a+1)^2+b^2}\\right)\\,", "df45d632b52bb306bd9e22c71eedbfcb": "\\to Y", "df46124736c89001ce003a3e36344d67": "f,g\\in G", "df461d7b080561774ad9b37afd8304e7": "\\frac{k (n - k)}{2} \\binom{n}{k}", "df4669cc919d15e2355df42c55e6d6f1": "\\mathbf{A}^{-1}", "df466fc68ae367432d5016a9b71a4c6e": "T= A^{\\alpha -1}", "df4695ce3a53d663354a2e1c9244a715": "2^{\\mathrm{round}[\\log_2 (x)]}", "df46c7fabec26d5783773651f3b9f270": "x=\\frac{r_1\\cdot r_2}{r_1+r_2}=\\frac{r_1\\cdot r_2}{r}", "df46d6b03922edbd8214129df45e3df6": "g' = \\frac{1}{f'\\circ g}.\\,", "df4721d7c2da30112c2942f800cb1168": " S=\\left\\{ | \\phi_k \\rangle : |\\phi_k \\rangle \\in \\mathbb{S}^d \\right\\} ", "df477801bab6dcd0cb5407cd0773c272": "\\Gamma(s) = \\lim_{x \\rightarrow \\infty} \\gamma(s, x)", "df477c9fd00534a8d45ac86e0cec1269": "(f(b)-f(a))g\\,'(c)=(g(b)-g(a))f\\,'(c).\\,", "df47962cb11d5d70368ed6514d1887dc": " \\hat{\\boldsymbol{\\beta}} = \\left(X^\\top \\Sigma^{-1} X \\right)^{-1} X^\\top \\Sigma^{-1}\\,\\mathbf{y}, ", "df47a9e454e061676810ae944c6c9ffa": "\\mathbf{\\Lambda}_{i=1 \\dots K}", "df47c4d42ec738d345f4970c8a03c676": "p_n(x+y)=\\sum_{k=0}^n{n \\choose k}p_k(x)p_{n-k}(y).", "df482fd781a9a57e7b213e5ae5d187ed": "\\begin{pmatrix}\\bar{x}_1\\\\\n\\bar{x}_2\\\\\n\\bar{x}_3\n\\end{pmatrix}=\\begin{pmatrix}\\frac{\\partial\\bar{x}_1}{\\partial x_1} & \\frac{\\partial\\bar{x}_1}{\\partial x_2} & \\frac{\\partial\\bar{x}_1}{\\partial x_3}\\\\\n\\frac{\\partial\\bar{x}_2}{\\partial x_1} & \\frac{\\partial\\bar{x}_2}{\\partial x_2} & \\frac{\\partial\\bar{x}_2}{\\partial x_3}\\\\\n\\frac{\\partial\\bar{x}_3}{\\partial x_1} & \\frac{\\partial\\bar{x}_3}{\\partial x_2} & \\frac{\\partial\\bar{x}_3}{\\partial x_3}\n\\end{pmatrix}\\begin{pmatrix}x_1\\\\\nx_2\\\\\nx_3\n\\end{pmatrix}\n", "df48bb6d349b6f238435e7d944f0823a": "F_1(x_1,\\dotsc,x_n),\\dotsc,F_m(x_1,\\dotsc,x_n)", "df48bbbcb377476d5a10cd7f3043b522": "s,y", "df48d2ebdf113a7cc22d66b97545c8f3": "|i-j|\\le1.", "df493657f8a778e368788b5d3ddc33d0": "E[F_6] = 4", "df4958337e91c67bec9bf15b3f6d3955": " \\vec S_o = \\mathrm M_3 \\big(\\mathrm M_2 (\\mathrm M_1 \\vec S_i) \\big) \\ ", "df4996978b74a4a70dd18167ba16eb0c": " \\theta_{\\Gamma}(\\tau)=1+\\sum_{n=1}^\\infty r_{\\Gamma}(2n) q^{n} = E_4(\\tau), \\quad r_{\\Gamma}(n) = 240\\sigma_3(n) ", "df49cdaf8f844d32a0164fe1f37dece0": "\nr^{TE}_{ij} = \\frac{n_{i}\\cos \\theta_{i} - n_{j}\\cos \\theta_{j}}{n_{i}\\cos \\theta_{i} + n_{j}\\cos \\theta_{j}}\\quad r^{TM}_{ij} = \\frac{n_{j}\\cos \\theta_{i} - n_{i}\\cos \\theta_{j}}{n_{j}\\cos \\theta_{i} + n_{i}\\cos \\theta_{j}}\n", "df4a04ae680638daab562e267345f8e7": " 1-\\gamma ", "df4a7990a998e97037c77c3f233d9644": "\n\\mathbf{e}_{j}(s) = \\frac{\\overline{\\mathbf{e}_{j}}(s)}{\\|\\overline{\\mathbf{e}_{j}}(s) \\|} \n\\mbox{, } \n\\overline{\\mathbf{e}_{j}}(s) = \\mathbf{r}^{(j)}(s) - \\sum_{i=1}^{j-1} \\langle \\mathbf{r}^{(j)}(s), \\mathbf{e}_i(s) \\rangle \\, \\mathbf{e}_i(s).\n", "df4b47490271fe5be79b727bb4757f15": "CPP=MAP-ICP", "df4bb1ba768e37e30716b484ad0bb5e6": "N(t) = N_0 e^{-t/\\tau}. \\,", "df4c1c1190a5bfd597f4169e9983fba4": "\n \\cfrac{\\partial\\mathcal{L}}{\\partial w} - \\frac{\\partial}{\\partial t}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{w}}\\right) + \\frac{\\partial^2}{\\partial x^2}\\left(\\frac{\\partial \\mathcal{L}}{\\partial w_{xx}}\\right) = 0\n ", "df4c826b6d61956eabae3170f43915c0": "B = g^{(x_1 + x_2 + x_3) x_4 s}", "df4d73d95bfce7c32f80cef310e0a2b0": "y = \\! w_1 + w_2 x", "df4d93b51aa3def0411338b9d20aa0e2": "\\left(\\begin{smallmatrix}1 & 1\\\\ 1 & -1\\end{smallmatrix}\\right),", "df4da61721180e749a4b9567cdef674e": "\\mathrm pOH \\approx -log(1.9 \\times 10^{-5}) = 4.7 ", "df4dd52387754d2055016a61fb9c708c": "|\\psi\\rangle=c_1|\\psi_1\\rangle+c_2|\\psi_2\\rangle", "df4de303890be474741deb06b5486a4c": "C_{2k+1} < \\operatorname{SO}(n).", "df4e1f646cab4ff7f0e3c103011e5837": "\\scriptstyle z\\, = \\,0", "df4e2f72328435c84da94d462fdfa79c": "\\ J_v = K_f ( [P_c - P_i] - \\sigma[\\pi_c - \\pi_i] )", "df4e34ab148d38fa235fd9948a2a19df": "\\tbinom{k - 1}{n-1}", "df4e9f07d34de7d1c1fd4e953d701a3a": " {\\Delta}_{\\rho}^{2}(ab) = {\\Delta}_{\\rho}^{2}(a)b+ a{\\Delta}_{\\rho}^{2}(b) ", "df4ee019fbf84c56401bb6b79122f68a": "\\mathfrak{U}", "df4efe9dfaa71f1741b2a0c28a72ad45": "M_y = \\lim_{m,n \\to \\infty}\\,\\sum_{i=1}^{m}\\,\\sum_{j=1}^{n}\\,x{_{ij}}^{*}\\,\\rho\\ (x{_{ij}}^{*},y{_{ij}}^{*})\\,\\Delta\\Alpha = \\iint_{}{} x\\, \\rho\\ (x,y)\\,dx\\,dy", "df4f4365cab58b4a1357e8331dbd458f": " \\nabla \\cdot T' ", "df4f8c2d80772aac5ba26c7e00188600": "F\\subseteq A\\,", "df4f8cfe2be28cc5e72a4f6dca958966": "(-\\lambda\\ + 1)^2 = (\\lambda\\ - 1)^2", "df4f9ca1ce85ac9cd97125fbfd20afc7": "A_{\\alpha_1\\alpha_2\\cdots\\alpha_p\\,,\\,\\alpha_{p+1}\\cdots\\alpha_q} = \\partial_{\\alpha_q}\\cdots\\partial_{\\alpha_{p+2}}\\partial_{\\alpha_{p+1}} A_{\\alpha_1\\alpha_2\\cdots\\alpha_p} = \\dfrac{\\partial^{q-p}}{\\partial x^{\\alpha_q}\\cdots\\partial x^{\\alpha_{p+2}}\\partial x^{\\alpha_{p+1}}} A_{\\alpha_1\\alpha_2\\cdots\\alpha_p}.", "df5011eba4858d3351ce53af4c854b86": "E \\le \\frac{(b-a)\\Delta^2}{24}f''(\\xi) ", "df5020c77c003cc37133157b098c122c": "\\bold p_1 = (x_1,y_1,z_1) ", "df50a760e0643d0d5af1dda87bbccf3c": "\\Omega\\rightarrow\\infty", "df51093364e83f4f75a54ecba31e2af6": "t= \\tau", "df51109c26329a9bb8aea8f53de7c416": "(1,2,\\ldots,n)", "df5166d5ff0c2a9157228b761a3acc07": "z = a + b\\omega ,", "df5171ac7165dab02af85a223d21bb66": "\\mathbf{r}' = \\mathbf{x}_s(t')", "df519b5fdc8848b86a15c7a92dfcab76": "\\beta = x/\\sqrt{2} + \\pi/8", "df51b5325ae56dcc95969965b1256096": "|R_n(t)| \\le \\frac{\\bigl(\\mu(I_{a,t})\\bigr)^n}{n!} \\int_{[a,t)} |u(s)|\\,\\mu(\\mathrm{d}s),\\qquad t\\in I.", "df52127d10da1c487f5f087543c5059b": " \\int_a^\\infty f(x) \\, dx ", "df5249090b05934791c44748d25462a5": "H=-p_3 \\ ", "df525fb4c0f15f11be0ec19ebd579f47": "\\,c > 0", "df52a9ed08aec29413cef2bc9927e7b0": "\\mu=1.176280818\\dots", "df52ffbda8f4d3b2ad20942d612d1e08": "\\frac{1}{2}\\left(\\max x + \\min x\\right)", "df53020bc8861afecdaff99463ee06b8": " \\epsilon_1(\\omega=0,p) = \\frac{\\epsilon_d}{|p-p_c|^s} ", "df532022612bbfbe43c8cb3f308b5df0": "D = \\dfrac{d}{dx},", "df5363e046f56089826f7287aa1ed2bd": "\\mathbf{D} = \\epsilon_0 \\mathbf{E}+\\mathbf{P}", "df5368d2e969f8a9a37546d1bca12e1d": "c=5.7", "df53a754e2911204c0e762595e3f25f3": "O(c^n)", "df5442678aef364fdf0f77062c98ff4d": "\\underline{P}(Cl_t^{\\leq})", "df54713bcf5d6d68d7fdcff03f25aa5e": "\\frac{X}{1-X} \\sim \\beta^{'}(\\alpha,\\beta)\\,", "df54c294e44d87a95d0a0d0b9b8f9ba5": "f: [0,1] \\to \\mathbb{R}", "df55008b15277595645950edc0638c92": " \\Psi \\to \\Psi + Q_B \\Lambda + \\Psi * \\Lambda - \\Lambda * \\Psi \\ , ", "df552dde3de18b9d2edef030f3340290": "\\left [\n\\begin{smallmatrix}\n 1 & 4 & 2 \\\\\n 4 & 1 & 3 \\\\\n 2 & 3 & 1 \n\\end{smallmatrix}\\right ]", "df55393281229356454c27e65fdd3a13": "A_w", "df56361fd187ecc61e7b02dbfbaa51c9": "\\text{Phase 1}: AL (\\text{from figure}) = MP = 0 \\text{ and }AB (\\text{from figure}) = AP \\,", "df565cf9e9df1f36f2dd7bbe9b38ae55": "n\\times r", "df569133dbfc28bd32bd4f4bdf8f41ae": "A_l", "df56b59011e3121a4b5c68ad6525e7c1": "\\mathcal{D}=\\mathcal{E}\\left[\\mathcal{A},\\mu\\right]", "df56fa01df8fd75e08e54090b20d6839": "E \\vdash F ", "df570e9c8f085d1656ba20646480e9c6": "n=3 \\ \\ \\rightarrow l_3=0, l_{31}=0,", "df57421cc0abbaf0ba973ddd3329657a": "\\tilde v_i:= \\left(I-P \\right)^* v_i", "df575fa5c2204434a79a3ff7b084cd8d": "\\psi(\\Omega^\\Omega+\\Omega^\\alpha) = \\phi_{\\Gamma_0+\\alpha}(0)", "df576b23f6a956b0288b07314ff1dad4": "\\langle \\bar{T}T \\bar{T}T\\rangle_{ETC} \\cong \\langle \\bar{T}T\\rangle^2_{ETC}", "df578a06de743e34975fcaa0304dcc43": "50i-10", "df57ba84a090a52b16f946b1a3336ba8": " \\mathbf{\\mu} = \\frac{1}{2c}\\mathbf{r}\\times\\mathbf{j} = \\frac{e}{2c}\\mathbf{r}\\times\\mathbf{v}.", "df57c429e86c895de792b340fcc0689e": "\\{x|0>V_0", "df58437daf34645f4f14a7e477fac629": "\\,\\forall v, u,\\, f_i(u,v) = -f_i(v,u)", "df58ce483274e952e7a0febc6c0dcf25": "y=+1", "df58ed348551ec15dd44cc4e528deaf2": " x = \\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{2+\\cdots}}}} ", "df592a3e5532211d35d4ad9a8580f94f": "(f,f)", "df593cee3da23c6f27db3bf95798f3c7": " (y_{n})_{n \\geqslant 0}=(1,5,13,2,4,7,1,\\dots )", "df593dd020c00f46fd321e3958504341": "\\begin{pmatrix} 1 & -1 \\\\ 1 & -1\\end{pmatrix}", "df596dedbb4792a749d1ed8b0d1c95f2": " |k \\rangle \\mapsto |f_k \\rangle ", "df59cca8a6e1bba4736b79453abdf20e": " s_2 = -\\alpha -\\sqrt {\\alpha^2 - {\\omega_0}^2} ", "df59d7f4a2cab92a64045aa1b774dd76": "\\mathcal{N}=(\\mathcal{N}_L,\\mathcal{N}_R)", "df59eb907799146847410f0a9cec72d0": "t \\mapsto \\begin{pmatrix} p & q \\\\ q & p \\end{pmatrix}, \\quad p = w + xi, \\quad q = y + zi ", "df5a8b1cc3da8b8ca01052fac7e99131": " \\left\\lfloor \\frac{n^2}{4} \\right\\rfloor", "df5ae920f005a329fa5526d604fd5cc3": "(R_{n+1}, L_{n+1})", "df5b5113bfa8ca95f2c0a595ee910f1a": "\\theta=0^\\circ, 60^\\circ", "df5b5892d10bc7b867a3dd135e876f3c": "\\frac{\\partial \\rho}{\\partial t} + \\nabla\\cdot\\mathbf{J} = 0 \\quad \\rightleftharpoons \\quad \\partial_\\mu J^\\mu = 0 \\,, ", "df5bb878c0c689ec8dd899eedce9f85a": "f(x;c,k) = ck\\frac{x^{c-1}}{(1+x^c)^{k+1}}\\!", "df5bd79006b687544fdeb49f2276714b": "n = \\rho/\\mu", "df5c54e2f16b73677a5f6be0295e1826": " \\Psi = e^{-i{E t/\\hbar}}\\prod_{n=1}^N\\psi(\\mathbf{r}_n) \\, , \\quad V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots \\mathbf{r}_N) = \\sum_{n=1}^N V(\\mathbf{r}_n) ", "df5c80f346557c9668170d77bc386b89": "\\alpha(d) \\ge (1.2)^\\sqrt{d}", "df5d0690c374a07f7fd4f93452e95071": "\\text {with }0 <\\text {F}\\left(x^*\\right)<1-e^{-1} = 0.632121, 0<\\eta<1 ", "df5d107b267daa286ad6d9e0a2dbb68a": "\\det(A) = \\sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j} = \\sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}.", "df5d7e7e50aecb223ac18fee940a2dd1": " R \\longleftarrow \\mathbb{Z}_p[x]/\\langle f \\rangle ", "df5d8389f34d52ab5efe4b3b6891f675": " Z_N(K,L) = \\sum_{\\{\\sigma\\}} \\exp \\left( K \\sum_{\\langle ij \\rangle_H} \\sigma_i \\sigma_j + L \\sum_{\\langle ij \\rangle_V} \\sigma_i \\sigma_j \\right). ", "df5dbb006b9338cdf0643fc65985ee64": " g* ", "df5de1b9d10da8d09680250df0145780": "S(-2)=f^{-2}(x)", "df5df7e80da575a7844f8996d37d8a06": " C \\approx 1.44 \\cdot {S \\over N_0}", "df5e20043bb1138d23dbc33dc8f1eb35": "(0\\le t\\le 1)", "df5e3222f48aac526f57fbb8521a7131": " \\cos ^2 a + \\cos ^2 b + \\cos ^2 c = 1\\,.", "df5e38255550b79814e0cce6b0992863": "[K,S]=[K,\\overline{S}]=0", "df5e76fc6e0cf33469544ef57ad8cd17": "(b)", "df5e80caf87d9f6b823ed5d2157bb6b5": "\\mathbf Z \\rightarrow X,", "df5e9c0d8897d6c37a1323bb3fe73b50": "t = 2.792 \\times 10^{-5}\\;\\mathrm{s}", "df5e9db6690080166a7499493ee61184": "\\sqrt a", "df5f6f1376d82822a3850f0f150a813a": "Q(n,L,Z)", "df5f7a53dd0ba19bc34823192004fe4e": "p^s=q^t+d", "df5fbbc4211e92cb6d39f6555db7c25c": "0 \\leq x \\leq 1", "df5fe4a13dfab65b47fbc0f3552ae444": " e_{ij} (t+1) = e_{ij}(t) + \\nu \\big [ y(t) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ] \\frac { v_{ij} \\big ( \\mathbf{x}(t) - \\mathbf{c}_i \\big ) } {\\sum_{i=1}^N \\sum_{j=1}^n v_{ij}^2 \\big ( \\mathbf{x}(t) - \\mathbf{c}_i \\big ) } ", "df60102736734d65136a76616ece1b9d": "\\gamma=\\frac{-e g}{2m}", "df60c648f910f1002ef03502ab1c6f7c": "k\\leq \\delta r^2", "df611445abd2710899988a74127703ff": "AND(\\alpha,\\alpha')=s^TC(u\\otimes v)=\\alpha\\alpha'", "df611bdd57083a92ae9755ce86696c12": "\n \\tan\\beta = \\frac{R\\sec\\phi}{y'(\\phi)} \\tan\\alpha\\,,\\qquad\n k = \\sec\\phi\\,,\\qquad\n h = \\frac{y'(\\phi)}{R}.\n", "df613b0f502601b55093465e83bf5a8a": "M = \\int d^3x {[H (x)]^2 \\over \\sqrt{det q (x)}}", "df619e29085db87bc9e4da40601c3df5": "\\int_{( J_n^*)^c} |Tb_n(x)|\\, dx= \\int_{( J_n^*)^c} \\left|\\int_{J_n} (K(x-y)-K(x-y_n))b_n(y)\\, dy\\right|\\, dx \\le \\int_{J_n} |b_n(y)| \\int_{( J_n^*)^c} |K(x-y)-K(x-y_n)|\\, dxdy \\le A\\|b_n\\|_1.", "df61ddcd6b6b620b10c6194447d514ec": " {\\dot{m}}B ", "df620f86cf02747cca1d49dad8f54437": "\\theta.", "df6238b8f1d243651cf4b1827441595a": "L^S(s,\\pi,r_i) = \\prod_{v \\in S} \\gamma_i(s,\\pi_v,\\psi_v) L^S(1-s,\\tilde{\\pi},r_i).", "df62590fed4125ecc3a088162da83877": "R(s) = \\frac{-\\Gamma(1-s)}{2\\pi i}\\int \\frac{(-x)^{s-1}e^{-Nx}dx}{e^x-1}", "df627118ba4bb96f59ba0f2c48d9a8ea": "\\langle \\psi |\\psi\\rangle", "df6271ecc8a36e496637c1fe5fea9384": "\\textstyle H_2: \\theta = x", "df62cf33ce5ac5f4fba95f78b3925068": "\\frac {F_{out}}{T_{in}} = \\frac {2 \\pi}{l} \\,", "df62da883d3d101bba4a1b9358a969b4": "\nL^{SCC}[g,\\phi ]=\\frac{\\sqrt{-g}}{16\\pi }\\left( \\phi R+\\frac{3}{2\\phi }\ng^{\\mu \\nu }\\nabla _{\\mu }\\phi \\nabla _{\\nu }\\phi \\right)\n+L_{matter}^{SCC}[g,\\phi ], ", "df632de6e488aa478e0eeddb08003c4f": "\\left|\\sum_{k=0}^n u_k \\right|^p \\le \\sum_{k=0}^n |u_k|^p \\text{ when } p<1 ", "df634816a4fc5e7b6a25454bbe8ca4cd": "T^2 = \\frac{n_1 n_2}{n_1+n_2}(\\overline{\\mathbf x}_1-\\overline{\\mathbf x}_2)'{\\mathbf S_\\text{pooled}}^{-1}(\\overline{\\mathbf x}_1-\\overline{\\mathbf x}_2).", "df637c4f5fe662b47eaedd7f25b2a61f": "S_k = S_{1/T}(k/P).\\,", "df644baa826167ce6a9f0c6e4d994a53": "P(X,\\varepsilon)", "df6454e9d38bd1dd6cb83e6df4890c8c": "{\\text{wheel}}\\;\\overset{\\textstyle}{\\underset{\\textstyle\\omega}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\\rightarrow}}\\;\\text{tachometer}", "df646d09e88bc692292c881527d61b0e": " \\int_a^b f'(x) \\, dx = f(b) - f(a). ", "df6486b5b6048205b927137a82ee02ed": "\\frac{1}{2}\\hbar", "df64ca0fbdf953c36cf22005c7a34ba4": "\\int d^4 x T^{\\mu 0}(\\vec{x},t) = P^\\mu", "df6582c548b99dba5b4693e4236b17d5": "d_{11} \\neq 0", "df6589094a9a6aef1959e0d9a07e4427": "\\begin{align}\n e_1(X_1,X_2) &= X_1 + X_2,\\\\ \n e_2(X_1,X_2) &= X_1X_2.\\,\\\\\n\\end{align}", "df65a360ac26786ddffff371fe46a338": "{\\zeta_X(s)}=\\frac{1}{1-q^{-s}}", "df65bb0e3d103854c94bd0994663d568": "\n \\displaystyle \n w(n,g) \n =\n \\sum_{k=0}^{n}\n w(n-k,g-1)\n", "df65c4b69e08ef98bdc483b5febc3e59": "\n\\begin{align}\nC_A & = [A]+\\sum p\\beta_q [A]^p[H^+]^q \\\\\nC_H & = [H^+]+\\sum q\\beta_q [A]^p[H^+]^q -K_w[H]^{-1}\n\\end{align}\n", "df65c8ac61ba7847212428d145fc1834": "\\begin{align} \\hat{H} & = \\sum_{n=1}^N \\hat{T}_n + V \\\\\n & = \\sum_{n=1}^N \\frac{\\bold{\\hat{p}}_n\\cdot\\bold{\\hat{p}}_n}{2m_n}+ V(\\mathbf{r}_1,\\mathbf{r}_2\\cdots\\mathbf{r}_N,t) \\\\\n & = -\\frac{\\hbar^2}{2}\\sum_{n=1}^N \\frac{1}{m_n}\\nabla_n^2 + V(\\mathbf{r}_1,\\mathbf{r}_2\\cdots\\mathbf{r}_N,t)\n\\end{align} ", "df65ecd5f9c2fb0b97f1681ffaee73bd": "\\partial_i = \\frac{\\partial}{\\partial x^i}", "df65f80c35e9dfe7f405dbc175916f8c": " \\sqrt{|x_1|^{2}+\\cdots + |x_N|^{2}}", "df666424ee1cdc6fc4b99d3e663bcd2a": "\\mathbf{A} \\in [\\mathbf{A}]", "df66722c24e1bf2fc43d319bf8f9105c": "\\dot{x} = \\Gamma V(x)", "df667dab32d576e8ecdad3bd24168905": "\\,p\\in(0,1]\\,", "df66ced23bf5be56f7492e3a709f942b": "\\rho M(G \\setminus e)", "df66dfaf8231884e162e0321013e0940": "w_1, \\ldots, w_n", "df6722b7dcb65f8ac902e9e4c1d0d971": "f : \\{0,1\\} \\to X", "df6734e146e0598e828b4bb8eee8c6b7": "a(x), b(x)", "df676b7a5ff1ba75f32b0b37e7fff381": "\\frac{d \\vec{u}}{dt} = -\\nabla \\phi", "df6780c1fbac0aba895c3ffe2e228615": "\\alpha_{QF}", "df6788ef7ea3fc05f9d1ceb8bfa1c38a": " \\{ a^n b^n c^n : n \\ge 1 \\} ", "df67abfac99e8a20d6bdc19fa04e7620": "\\Gamma(1-z) \\Gamma(z) = {\\pi \\over \\sin{(\\pi z)}},", "df67f9574e10ede2d04338d1e010d276": " \\mbox{Handicap differential} = \\frac{ ( \\mbox{Equitable Stroke Control} - \\mbox{course rating} ) \\times 113}{ \\mbox{slope rating}} ", "df6810c7ca29f0cc2886e80a920f0b80": "\\binom{t}{k} = \\sum_{i=0}^k \\frac{s_{k,i}}{k!} t^i.", "df68341738699c2e3e18aa1e004ff7d0": "n < 5 ", "df6840bf9273ff64f27a3103ec34b270": "\\Omega\\left(\\frac{1}{np}\\right)", "df688b7144aaef7da035b38e4d380571": "\\sqrt{\\det (-S_{xx}''(x^0)) }", "df68920417728fa612ac0503b6c0f9cc": "\\scriptstyle b_1", "df68a4b49ea36d26c07c8c94ee389e61": " \\delta^{\\mu_1 \\dots \\mu_s \\, \\mu_{s+1} \\dots \\mu_p}_{\\nu_1 \\dots \\nu_s \\, \\mu_{s+1} \\dots \\mu_p} = \\tfrac{(n-s)!}{(n-p)!} \\delta^{\\mu_1 \\dots \\mu_s}_{\\nu_1 \\dots \\nu_s}.", "df6938e32d01e80b643ef10323100093": "\\mathbf{v}' = R\\mathbf{v}", "df696f9a6255f00127d006e5ea223792": "\\delta T = \\epsilon h_{\\epsilon}^{*}(\\mathcal{L}_{X}T_{\\epsilon}) \\equiv \\epsilon (\\mathcal{L}_{X}T_{\\epsilon})_{0}", "df698608362342d68f57cd7de4d25930": "940 \\pm 60", "df699774c1292ebd596080b583af7cba": " D \\cdot A = T\\cdot I \\,", "df6a4e11a2c29ff351e7e9c2e2ddeea3": "A^{(H)}(t)", "df6a894498a89017fc2c76d677515bc7": "\\sum_{k=1}^n\\frac{k}{\\varphi(k)} = \\frac{315\\zeta(3)}{2\\pi^4}n-\\frac{\\log n}2+\\mathcal{O}\\left((\\log n)^{2/3}\\right)", "df6a974f7a4ddbf624e4b742a061e140": " M_f(x_1,\\dots,x_n) = f^{-1} \\left({\\frac{1}{n}\\cdot\\sum_{i=1}^n{f(x_i)}}\\right) ", "df6a9e21e0f8e1e617e328b028e29acc": "x,y \\in \\Omega", "df6af8fb443dd9c295732b8a3fa282fc": "\\underset{H}{0.\\underbrace{999\\ldots}}\\; = 1\\;-\\;\\frac{1}{10^{H}}.", "df6b290df974bc4c72e8ffbbb210b2fe": "\\frac{\\partial W(q,p,t)}{\\partial t} = -\\{\\{W(q,p,t) , H(q,p )\\}\\}~,", "df6b9233c91933bcc5baefc27c0c24cd": "\\frac{\\sin x}{x}", "df6bb161c1600fe25875809c76680ada": "\\alpha=0.0168", "df6bf5123483836a9cfab9a653a41bbf": "\\left[\n\\begin{array}{cc}\nF_u & F_{\\lambda}\\\\\n\\dot u^* & \\dot \\lambda\\\\\n\\end{array}\n\\right]\\,\n", "df6c022e6e13054bcf8f194e62de916c": "\\displaystyle{G=G_0G_{+1}G_{-1}G_{+1}=G_0G_{-1}G_{+1}G_{-1}.}", "df6c1f8d48818f30fcbb52ac37d4a896": "\\begin{align}\n \\delta_\\Phi\\mathcal{L}\\, =\\, \n &-\\, \\rho\\, \\int_{t_0}^{t_1} \\iint \n \\left\\{ \n \\frac{\\partial}{\\partial t} \\int_{-h(\\boldsymbol{x})}^{\\eta(\\boldsymbol{x},t)} \\delta\\Phi\\; \\text{d}z\\; \n +\\, \\boldsymbol{\\nabla} \\cdot \\int_{-h(\\boldsymbol{x})}^{\\eta(\\boldsymbol{x},t)} \\delta\\Phi\\, \\boldsymbol{\\nabla}\\Phi\\; \\text{d}z\\,\n \\right\\}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t\n \\\\\n &+\\, \\rho\\, \\int_{t_0}^{t_1} \\iint \n \\left\\{ \n \\int_{-h(\\boldsymbol{x})}^{\\eta(\\boldsymbol{x},t)} \\delta\\Phi\\; \n \\left( \\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\nabla}\\Phi\\, +\\, \\frac{\\partial^2\\Phi}{\\partial z^2} \\right)\\; \\text{d}z\\,\n \\right\\}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t\n \\\\\n &+\\, \\rho\\, \\int_{t_0}^{t_1} \\iint \n \\left[ \n \\left( \\frac{\\partial\\eta}{\\partial t}\\, +\\, \\boldsymbol{\\nabla}\\Phi \\cdot \\boldsymbol{\\nabla} \\eta\\, -\\, \\frac{\\partial\\Phi}{\\partial z} \\right)\\, \\delta\\Phi\n \\right]_{z=\\eta(\\boldsymbol{x},t)}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t\n \\\\\n &-\\, \\rho\\, \\int_{t_0}^{t_1} \\iint \n \\left[ \n \\left( \\boldsymbol{\\nabla}\\Phi \\cdot \\boldsymbol{\\nabla} h\\, +\\, \\frac{\\partial\\Phi}{\\partial z} \\right)\\, \\delta\\Phi\n \\right]_{z=-h(\\boldsymbol{x})}\\; \\text{d}\\boldsymbol{x}\\; \\text{d}t\n \\\\\n =\\, &0.\n\\end{align}", "df6c504d93b0e4c15d73f180d4eccab6": "\\phi(10)=4", "df6d1a9fab541bbc7841beaa9d825326": "(\\varepsilon,\\delta)", "df6de95e5c9509f32f67c2b6422647ae": "\\Sigma_2^{\\rm P} \\cap \\Pi_2^{\\rm P}", "df6dea8bea38553528c534f814926a96": "A = B = 1", "df6dfbfc89e65bf1c9635825ad9d6967": "S(S-1) > 8m+20", "df6e4e5ee9b7ff8ceebd6f3ccb8838b6": "f(\\vec x,\\vec x)=2\\rho(\\vec x)", "df6e6a505cf4c963498cbab77413f36b": "\\sup_{x\\in\\mathbb R}\\left|F_n(x) - \\Phi(x)\\right| \\le C_0\\cdot\\psi_0, \\ \\ \\ \\ (3)", "df6e6f43dd3c48b2fd26d4b6f39341f6": "Q(x,p(x))", "df6e76bee903a03b9f54ccea919e6623": " M(\\vec X,\\vec {\\rm E},Y) = \\left[ {\\begin{array}{*{20}c}\n 0 & 0 & b \\\\\n 0 & 0 & A \\\\\n 0 & { - \\Sigma ^{ - 1} } & I \\\\\n {A^T } & I & 0 \\\\\n\\end{array}} \\right]\n", "df6e80da46d55e8b0bbd5e88f0fe611a": "0.\\overline{01} = 1.\\overline{10} = \\tfrac13", "df6f39057fcb788585b34bd994abb5c8": "\\mathrm{Sh}_0", "df6f916b52ba09a00592f0b8ffbf6456": "Y^*", "df6f9aefd37f864e20b56cde76048f62": "\\tau^2 = n/\\sum_{i=1}^n \\frac{1}{x_i}.", "df6fd8209589466879382a77c30fd71c": "A \\rightarrow I: \\{N_A, A\\}_{K_{PI}}", "df6fdbe0cb263098edbef21651d9956b": " \\operatorname{ker} T := \\{\\mathbf{v} \\in V : T\\mathbf{v} = \\mathbf{0}_{W}\\}\\text{.} ", "df70a5b5ae944d57d0351137d7f3e1f7": "\\tan \\theta ", "df70ec88d069e3d5e4ae817ec45a5c79": " Z_q(V_o,T) \\rightarrow \\bigg(\\frac{1}{\\beta - \\beta _o }\\bigg)^{\\alpha}", "df70ed80186da5a64c47d60bbdaf6de1": "\\sup_{0t\\}}(x) \\, dt,", "df941ab0af4753e28987e50f00a23ac9": "m_{0}=\\frac{4}{3}\\frac{E_{em}}{c^{2}}", "df9465ab72fda4ec72ba42ead1eebec0": "r = \\frac{(1/n - n) \\sin k'd}{(n+1/n)\\sin k'd + 2 i \\cos(k'd)}", "df94d9c4d0a5498a17826477aed65f74": "\\sup_{x\\in \\bar{X}}|f_{t}(x,t)|", "df950280842bcd1580751e370355a505": "\\scriptstyle \\sqrt{ 2^N}", "df95590d8976144256825be679155030": "n T -W = 0.", "df963bd4cbca0485186ae2f5342ea069": " h_{t+1} = f(h_t) \\oplus h_t ", "df965a96aad03e535cc6d07a0a32dc26": "\\frac{3x}{x^2+2x-3}", "df965ea719ed3ff38796950697754451": "\n \\langle x - a_i\\rangle = \\begin{cases} 0 & \\mathrm{if}~ x < a_i \\\\ x - a_i & \\mathrm{if}~ x > a_i \\end{cases}\n ", "df96859eeb7c55b93dc73b0ad00e7c91": "S_2 P = S_2^P", "df96bbef2f49e1205fcf7229d2ef6653": "m\\geqslant \\frac{m_{P}}{2}\\left( \\frac{\\pi n_{k}}{k}\\right) ^{\\frac{1}{2}}", "df96c3c02d19de38b6c02bdcb30d4fee": "n \\times a\\,", "df970f62c95e895409a9db8976fc89a6": "f\\circ\\eta = \\eta'", "df9771274cdeff40178374dca4f150c1": "T_{ab} \\, = 0", "df97a9d1ea7b56167a2a000dec5402e5": "\\gamma_{wv}", "df97da85b73dc7d51afd169a9d575e04": " =-\\frac{v^2}{2}(\\partial^\\mu \\theta)(\\partial_\\mu \\theta) + m^2 v^2~.", "df97e264b2820fcbcc9e075a85171e8b": "F = \\frac{\\sum_j m_j F_j}{\\sum_j m_j}", "df98203fbf1d4486621b5075b816622f": "u_v", "df9821626d402adfad3b9aa4fcd5772e": "\\frac{\\partial L(t,q(t),q'(t))}{\\partial x_i}-\\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{\\partial L(t,q(t),q'(t))}{\\partial v_i} = 0\n\\quad \\text{for } i = 1, \\dots, n.", "df985bb1b3ce63454e5261f01665237a": "T_w\\colon Q\\to Q", "df986237c451a19aa679141b31eea1c1": " c = (1-s)f(k)", "df986fbc018c0e7c26fa696977710de7": "\\hat \\theta_s", "df98b533d024e9b240685673289c1267": "2k+(v-1)(\\lambda-\\mu) = 0", "df98f2ed49d7a9e8ad013974185b43fc": "CX = (X \\times I)/(X \\times \\{0\\})\\,", "df9900a6019a8d41a5d955c69a4e9c0e": " : \\hat{b}^\\dagger \\, \\hat{b} \\, \\hat{b} \\, \\hat{b}^\\dagger \\, \\hat{b} \\, \\hat{b}^\\dagger \\, \\hat{b}:\\, = \\hat{b}^\\dagger \\, \\hat{b}^\\dagger \\, \\hat{b}^\\dagger \\, \\hat{b} \\, \\hat{b} \\, \\hat{b} \\, \\hat{b} = (\\hat{b}^\\dagger)^3 \\, \\hat{b}^4.", "df9909a17ae494467ae7213157781302": "\\frac{(k'+t+1)_t}{(p)_t}a_{p,k'+t}=a_{p-t,k'}.", "df992ef7f99dc5db6c55e18337cf9e9a": "a_{14}+b_{14}+c_{14}=c_{1}-a_{1}", "df9934dcb84f1f1b33842bae697fc1da": "F_Y(y; \\mu, \\sigma) = \\int_{-\\mu/\\sigma}^{(y-\\mu)/\\sigma} \\frac{1}{\\sqrt{2\\pi}} \\, \\exp \\left(-\\frac{1}{2}\\left(z + \\frac{2\\mu}{\\sigma}\\right)^2\\right) dz\n+ \\int_{-\\mu/\\sigma}^{(y-\\mu)/\\sigma} \\frac{1}{\\sqrt{2\\pi}} \\, \\exp \\left( -\\frac{z^2}{2} \\right) dz.\n", "df994416de21969ea1f6d90b2d62e494": "\n\\begin{pmatrix} a_1 \\\\ a_3 \\\\\\end{pmatrix}_j=\n{1\\over144}\\begin{pmatrix} 130 & -34\\\\ -34 & 10 \\\\\\end{pmatrix} \n\\begin{pmatrix} -2&-1&0&1&2\\\\-8&-1&0&1&8\\\\\\end{pmatrix}\n\\begin{pmatrix}y_{j-2} \\\\ y_{j-1} \\\\ y_j \\\\ y_{j+1} \\\\ y_{j+2} \\\\\\end{pmatrix}\n", "df9952eb67a6b26facd57f9d37b08b01": "R^m_\\ell(\\mathbf{r})", "df9a1a6a2e57f32b1696e79b2efc581d": "\\sum_{i=1}^{n} X_i =1", "df9b2d533f18ef840550f8f914c01fa6": " f''(x) \\ge 0 ", "df9b2d7f2cc59f73de87e33b94dcb012": "\\begin{pmatrix} 15 & 17 \\\\ 20 & 9 \\end{pmatrix}\\begin{pmatrix} 7 \\\\ 8 \\end{pmatrix} = \\begin{pmatrix} 7 \\\\ 4 \\end{pmatrix},", "df9b3b17cf651a3bac18cc8e69541e8b": "A=ap/2", "df9b4b30f62b524de7cc946ba500f590": "L_{uu}>0", "df9badca0a104f53de71278d2c1889cb": "\\tilde{I}_{26}[f] = A_1 \\sum_{i=1}^6 f(a_i^1) +A_2 \\sum_{i=1}^{12} f(a_i^2) +A_3\\sum_{i=1}^{8}f(a_i^3),", "df9bf35305e8ce550b26e54706e86e19": " \\frac{d(1/f)}{dx} = -\\frac{1}{f^2}\\frac{df}{dx}.\\,", "df9c186186e19bc095385698e3b44f7e": "\\mathsf{2CH_2\\!\\!=\\!\\!CH_2+O_2\\ \\xrightarrow{Ag}\\ 2(CH_2CH_2)O}", "df9c3bceb1a3977382f185d51aee2801": "\\otimes \\!\\,", "df9cc789d573048d3aab01245cd95a76": "F\\colon(\\mathbf{Bord}_{\\langle n-1,n\\rangle},\\sqcup,\\emptyset)\\rightarrow(\\mathbf{kVect},\\otimes_k,k).", "df9d47167ab06bc174a916a80582a592": " \\Psi(\\bold{r})={1\\over \\sqrt {V}} \\sum_{\\bold{k}} e^{i\\bold{k\\cdot r}}a_{\\bold{k}}", "df9d50936f5d7651d31ee8cc1178e372": "j < i \\leq h", "df9d5de6fd056d27099e6c612029cf65": "I(a) = \\emptyset", "df9dec265a7f4488e3bfa2af4e109d01": "y(t) = 10 \\,x(t)", "df9df27bceaa09d04ebe550f93564d11": "xy \\vee \\bar{x}z \\vee yz = xy \\vee \\bar{x}z", "df9e66cbfddcab051f26aa9c70746d0f": " FRINGE_d ", "df9e720df6a9bf2b98752ca4a1b55973": "20n", "df9e87b24043064fd28aa16a3756d33b": "\\displaystyle{\\kappa(\\theta)=y^{\\prime\\prime}(\\theta)x^\\prime(\\theta) - x^{\\prime\\prime}(\\theta) y^\\prime(\\theta)}", "df9e8ede78a721c9cf7a75a001bed24e": "\\frac{\\partial}{\\partial y^i} = \\sum_{k=1}^n\\frac{\\partial x^k}{\\partial y^i}\\frac{\\partial}{\\partial x^k}", "df9e9a66738394f04f46c10c50aa6a2b": "a_n=1+p_n", "df9ecfd177afb11b413501b69ba3a9df": "\\mathbf{E}=\\eta\\mathbf{J}", "df9ed87e836e463cd086106035aef441": "x^{(t)}", "df9edbefd10556281bc6baffcda93b81": "g[\\tau+\\log(\\gamma)]= g(\\tau)", "df9f3b752dae7c85559fcc2f77f84257": "\\lim_{n\\to\\infty}\\frac{F(n+1)}{F(n)}=\\varphi.", "df9f43e3d2f9c67a36d5d995cfb95d21": "\\mathfrak{p}R[x] = R[x]", "df9f4586bd8babf7b5c3a9419b8cb0c9": "T_{00} = \\rho(r) c^2 g_{00} = \\rho(r) e^{\\nu(r)} c^2 \\;", "dfa03d5342e3c3a92b1c4963e742851b": "\\; \\rho = \\rho^2", "dfa05e2df8af98564b5a99e1f3548367": "\\tilde g_{ij} = \\int_\\Omega \\int_\\Omega \\tilde\\kappa(x,y) \\varphi_i(x) \\psi_j(y) \\,dy\\,dx\n = \\sum_{\\nu=1}^k \\int_\\Omega \\kappa(x,\\xi_\\nu) \\varphi_i(x) \\,dx\n \\int_\\Omega \\ell_\\nu(y) \\psi_j(y) \\,dy\n = \\sum_{\\nu=1}^k a_{i\\nu} b_{j\\nu}", "dfa065d7c14b7d70981831cfe169234e": "\\scriptstyle{\\lambda_1(M)}", "dfa1240e3ae62036fc583bb085367923": "\\frac{\\partial \\mathbf{F}}{\\partial \\mathbf{X}_{ij}}", "dfa12e0a7eae8fb29c640941305b1a4e": " {\\partial \\mathcal{L}\\over\\partial I_k} = 0 \\qquad k=1,\\dots,n ", "dfa1bac8494014c9344e047ede7cd83b": " f(x_t, t) = x_t e^{\\theta t} \\, ", "dfa248bbd28a0c40c6b115a01eabfc7a": "x^m(a - x)^n", "dfa29ab2df61c4745072e321a2a0a814": "k=\\frac{\\kappa}{\\rho C}", "dfa32f84dea179f0553fcb730ccc7bb8": " p(k) = - \\log_2 \\left( 1 - \\frac{1}{(1+k)^2}\\right)~.", "dfa35fdb8a9c3a161caaf59ef4b83dce": "(\\hat{n} \\cdot \\vec{\\sigma})^{2n+1} = \\hat{n} \\cdot \\vec{\\sigma} \\, .", "dfa3a5c7ee4d31a3484a9d182a8ce44f": "w(K)", "dfa3e981cac266791898e00a491c7587": "CC(M)", "dfa3f87a979025661b063ea232b22c2c": " F^*(s^*) ", "dfa406a941bf8f4010c18f68dfa0f1e5": "\\int_0^\\infty x^n e^{-ax^b} dx = \\frac{1}{b}\\ a^{-\\frac{n+1}{b}} \\, \\Gamma\\left(\\frac{n+1}{b}\\right)", "dfa44b01aa65aea00f65ed5c9bf6c2d6": "C = \\frac{1}{\\sin^2 \\theta} ", "dfa48d4b9df9c27b223dbcbd61d37b4f": " E\\ ", "dfa4c3528d7ae76b55b28cfd10995842": " \\vec{x}_{P}(t,\\tau_{P}) ", "dfa4d107a894d2e374c2fb5445b028c7": "f(x) = \\sum_{i=0}^\\infty {x \\over 2^i}", "dfa5371ed0522091839fc3634e91ef60": "O^*(2^n)", "dfa54979788e514c28579d303de08c23": "\\sigma^{\\mu \\nu} \\equiv i/2\\left [ \\gamma^\\mu , \\gamma^\\nu \\right ]", "dfa55045225e2a21683aa870f5c18bf5": "x,N\\,", "dfa55c118c0c72a886cfa93e4e992dd0": "+ + +", "dfa59dead61863ab3464374de01def8c": " F_{\\text{viscosity, slow}} = \\eta 2 \\pi (r+dr) \\Delta x \\left . \\frac{dv}{dr} \\right \\vert_{r+dr} ", "dfa5a1a7ab26a98bb9ce7399081e7cd9": "(x^{l(2t - 1)} - 1)b(x) = a(x)(x^{2t-1}- 1)p(x)", "dfa5b9b343c08191f68cf461b7662b0e": "x^3 \\cdot 2^{-1} = \\frac{\\sqrt{5}x}{2}", "dfa5d5a16cc2100668ae7cf40769a60b": "r_g = \\frac{p_{\\perp}}{|q| B}", "dfa5e07c45590964272aaf4ee696f15a": " E_B = \\frac{1}{2}[\\bar{E_V} + \\bar{E_C}] ", "dfa5e8225a70a23f364535e1c7eeaf30": "\\scriptstyle 0.025^{1/5} \\;\\approx\\; 0.48,\\,", "dfa5f856df499baf9a405cfb4ae2387d": "(\\tau+\\sigma m)", "dfa63b48048b41ac08e9910f28f58747": "-\\alpha- \\delta\\frac{x^\\star-\\beta}\\lambda+ \\gamma \\cdot f^\\star \\left(\\frac {x^\\star-\\beta}{\\gamma \\lambda}\\right)\\quad (\\gamma>0)", "dfa6439946f4fbf66166b105c298f88c": "\n\\begin{align}\n\\mathcal{S}[x] & = \\int L[x(t),\\dot{x}(t)] \\, dt \\\\\n& = \\int \\left(\\frac{m}{2}\\sum_{i=1}^3\\dot{x}_i^2-V(x(t))\\right) \\, dt.\n\\end{align}\n", "dfa668602bf4d807a39703719d910c4b": "\\begin{bmatrix} \\mathbf{V}_1^* \\\\ \\mathbf{V}_2^* \\end{bmatrix} \\mathbf{M}^* \\mathbf{M} \\begin{bmatrix} \\mathbf{V}_1 & \\mathbf{V}_2 \\end{bmatrix} =\n \\begin{bmatrix} \\mathbf{V}_1^* \\mathbf{M}^* \\mathbf{M} \\mathbf{V}_1 & \\mathbf{V}_1^* \\mathbf{M}^* \\mathbf{M} \\mathbf{V}_2 \\\\ \\mathbf{V}_2^* \\mathbf{M}^* \\mathbf{M} \\mathbf{V}_1 & \\mathbf{V}_2^* \\mathbf{M}^* \\mathbf{M} \\mathbf{V}_2 \\end{bmatrix} =\n \\begin{bmatrix} \\mathbf{D} & 0 \\\\ 0 & 0 \\end{bmatrix}\n", "dfa67d838a7535fcfdbb401ca3e287a2": "\\langle A(\\mathbf{s})\\rangle", "dfa69fd9732aef207c285b89b0d1cdea": "N > 4\\sqrt{q}", "dfa6cbf0f1eefeaf25ff98c281e58cfd": "|CF|", "dfa70ea159239eb31ae805025a5f5d9d": "\\rho(0) = 0", "dfa724c2b0b50d48193d72ad9f7126bc": "e^{i\\theta}(|\\phi\\rang+|\\psi\\rang)", "dfa73e39093ca17eab8a1b1158476aca": " w \\cup \\{w\\} \\!", "dfa786feb5c9e9803d737e35f03f39f6": "\\{\\lfloor r\\rfloor,\\lfloor r\\rfloor+1\\}", "dfa7bb897aadff657e84ef926b670f6f": "\\scriptstyle \\operatorname{P}(A|\\mathcal{B})", "dfa88447ac4ccd983f687654164a22f1": "\\theta^{1,2}(t)", "dfa8c91a1e2f3601af1c782a42f64b58": "\nQ_n = n \\left( \\frac{2}{n} \\sum_{i=1}^n \\mathbb E \\|x_i - X\\|^\\alpha - \\mathbb E\\|X - X'\\|^\\alpha - \\frac{1}{n^2} \\sum_{i=1}^n \\sum_{j=1}^n \\|x_i - x_j\\|^\\alpha \\right),\n", "dfa9833f4e6820670caf459243fe5300": " q\\mbox{ and }p = \\tfrac14 \\left(L^2+ 27M^2\\right)", "dfa9c2e13793046ad7991cff68538737": "E_1(X) \\ne 0", "dfaa05b4f6a4e7f8b1c4761664e4d468": "\\hat{\\sigma}^2\\,\\!", "dfaa2d6979c930c6869445583ddcc7de": "P_3(x)=x \\,", "dfaa48329edb8812add60ae27d440272": " T_1(x) = x \\,", "dfaa9199f5389db7675992f53f05cbcc": "Q(i)\\,\\!", "dfaa9b703538f8fa5ccbc98114b41300": "x^* * x", "dfaac927fff6e2c03f40402926a346cd": "[OH^-]_{0^{ }}", "dfaaf9f5a0a395a34cf4845e22268b18": "\\Pr(\\tau_{n+1}\\le t|T_0, T_1, \\ldots, T_n)=\\Pr(\\tau_{n+1}\\le t)\\, \\forall n \\ge1, \\forall t\\ge0 ", "dfab15c2a0a2f39d55de3007a0b95d5c": "\\lambda=\\tilde{\\lambda}", "dfab4473f3bab15adca7f8a349981dcf": "\\nabla \\cdot \\left( \\boldsymbol{\\kappa}\\, c_p\\, c_g\\, a^2 \\right)\\, =\\, 0,", "dfab9d7c992a3bd2842e9e22ab521180": " \\mathbb{Q}(\\sqrt{3}, i) ", "dfaba8650e056df497356b5777d2306c": "\n\\begin{align}\nA_1 \\cos(\\omega t + \\theta_1) + A_2 \\cos(\\omega t + \\theta_2)\n&= \\operatorname{Re} \\{A_1 e^{i\\theta_1}e^{i\\omega t}\\} + \\operatorname{Re} \\{A_2 e^{i\\theta_2}e^{i\\omega t}\\} \\\\[8pt]\n&= \\operatorname{Re} \\{A_1 e^{i\\theta_1}e^{i\\omega t} + A_2 e^{i\\theta_2}e^{i\\omega t}\\} \\\\[8pt]\n&= \\operatorname{Re} \\{(A_1 e^{i\\theta_1} + A_2 e^{i\\theta_2})e^{i\\omega t}\\} \\\\[8pt]\n&= \\operatorname{Re} \\{(A_3 e^{i\\theta_3})e^{i\\omega t}\\} \\\\[8pt]\n&= A_3 \\cos(\\omega t + \\theta_3),\n\\end{align}\n", "dfabd3fd2175ff683978f421e7075427": "C_i \\equiv M^3\\,\\bmod\\,N_i", "dfabffda7dc14b8a902efdc95b4b7103": "T_{h*(1,1)}\\cdot (x*(1,-1)) = \\lambda\\cdot (x*(1,-1))", "dfac1066b964beb010ca8e44d66175fb": "\\rho\\,\\mathrm{B}(k, \\rho+1)\\,", "dfac28023d0712acf3715461815efbd0": "\\lim_{x\\rightarrow0} {}^{n}x", "dfac69387ded872ce5e18828f5db5cc2": "x + ax + bx = c", "dfac6e68c37a594b107a48bbde1480ad": "\\overline{x} \\in X(R)", "dfaca991b6502ddc1bf60d4d4d08a349": "\\omega =\\omega^{\\prime} + i\\omega^{\\prime\\prime}", "dfacd982d94fb18757b92de73ab7bc43": "\\gamma_{c,p}(n)", "dfad8a6d7669d486fd543e950d12a78f": "TY", "dfadee831f41b61cfd192e8150c193f4": "d_{e''} \\neq 0", "dfae2023179d64cfdbfa209dbec66d94": "\\Psi_g\\colon G \\to G\\,", "dfae683a07cd7b21eb102ba115b9831d": " 2 \\frac{p - p_\\infty}{\\rho U^2} =2\\frac{R^2}{r^2}\\cos(2\\theta)-\\frac{R^4}{r^4}.", "dfae8159326c8ff0a5e6df4aca2af5d7": "\\pi/5", "dfae9b1291112dd71d58bb4067b915fe": "R(x,y) = 0", "dfaeade9ad04f13a92622722bb80adf4": "h:R\\to K", "dfaf22cc21b8688a8be1fd52c6da0790": "\nc = \\sqrt{\\frac{K}{\\rho}}\\,\n", "dfaf98538d928740290276fabf703d82": " \\sigma = \\frac{ 0.6325 }{ ( n - 1 )^{ \\frac{ 1 }{ 2 } } } ", "dfafa81730a1977461da40ba2f401ec2": "q^s", "dfafd95bb475b0a7155d48c0443edd51": "g(1) - g(0) = g'(c) \\!", "dfafe09741d544a1373805d3869fd02a": "P_1 = P_0+P_0*r-c ", "dfb0604b5f551f13dd63f104ec4f8baa": " \\alpha = M ", "dfb067b89ce6ac6c0cf989626055fe85": "\\mathrm{det}({\\mathbf v}, {\\mathbf w}) = ({\\mathbf v} \\times {\\mathbf w})\\cdot {\\mathbf n},", "dfb06ee677c88519e3a5a6983a88fae8": "y_1, \\ldots , y_k", "dfb0ca9352cb8f08d9514d4b2f37f28f": "\n \\delta U = \\int_{\\Omega} [\\boldsymbol{\\nabla}\\cdot(\\boldsymbol{\\sigma}\\cdot\\delta\\mathbf{u}) + \\mathbf{b}\\cdot\\delta\\mathbf{u}]~ {\\rm dV} ~.\n ", "dfb0d197ba4741dd509549aaafcdaf46": "f(x) = x^p", "dfb10ecec64142031a2c0073e5ca45a2": "\\hat{b}^\\dagger = u^* \\hat{a}^\\dagger + v^* \\hat{a}", "dfb125a9e0a4de0c826644b1e320a5b7": "\\langle A \\rangle = \\sum_s p_s \\langle \\psi_s | A | \\psi_s \\rangle = \\sum_s \\sum_i p_s a_i | \\langle \\alpha_i | \\psi_s \\rangle |^2 = \\operatorname{tr}(\\rho A)", "dfb160f82711091087f833cfcc0725bd": "\\ln (f(x))=\\sum_i\\alpha_i(x)\\cdot \\ln(f_i(x)),", "dfb18a4de48ea44940fa8d11969df5c3": "Q^\\dagger(\\mathbf{p})", "dfb194b9717373b0fadc0bcb5fcd7707": "\\texttt{ZF}\\vdash(\\texttt{AC}_{\\mathcal{P}(\\kappa)}+\\neg\\texttt{AX}_{\\kappa})\\leftrightarrow \\texttt{CH}_{\\kappa}\\,", "dfb1f782cc5cc3e4a054f1ce280c9ad9": "k \\approx \\pi / a ", "dfb2430a2aece98837d8f0dc5a235914": "\\mathrm{P}(1\\times \\mathrm{U}(n)) \\cong \\mathrm{PU}(n).", "dfb26a008e91f790a54152152366f72f": "\\pmod{n}", "dfb27e2a66431dc563a245c11d725998": "\\frac{1}{2} \\left (1\\pm i\\frac{\\zeta}{\\|\\zeta\\|} \\right ).", "dfb2d21eb99c7439d110ea4eb2dc692b": "t^* ", "dfb2db6a2da885a8aadf31166fc5970b": "\\scriptstyle -{1 \\over z^\\star},", "dfb3568b94e1e15505bba3fcbc185a25": "E\\approx \\frac{1}{2}m_1 v_\\text{e} \\Delta v ", "dfb36c502c24bb91baf04102e9eac85f": "\\partial:\\; C_k \\mapsto C_{k-1}", "dfb398320237940795dabcb5f14e1c2c": " \\Delta P_j = p_{j} \\Delta T \\,", "dfb39ebe1575f330087cd171c90420ef": "\\rho^{2^k}", "dfb4389aad572297071aade62c22cdf1": "\\cos \\omega_\\circ = \\dfrac{\\sin a - \\sin \\phi \\times \\sin \\delta}{\\cos \\phi \\times \\cos \\delta }", "dfb44848eb9f802c87b4f63e7f20944b": "\\displaystyle{\\pi_s(g^{-1}) f(z)=|\\overline{\\beta}z+\\overline{\\alpha}|^{1-2s}f\\left({\\alpha z +\\beta\\over\\overline{\\beta}z +\\overline{\\alpha}}\\right).}", "dfb4c56bb4a4f9a1dbe3245748a14a43": "[HA]_0", "dfb4debc5be664164654455afda1dfcd": " P(X_1,\\ldots, X_n) \\in \nA[X_1,\\ldots,X_n]^{S_n}", "dfb583fc3703341895e9ee7c1776c813": "c_H (\\alpha) = \\min \\max \\{ A_H (x_i)\\ :\\ a_i \\neq 0 \\}. ", "dfb58c0e0cbca41cc31a341b712082f7": " N(x)", "dfb5c1efbbf99e4b354d735b014d489d": "y = a_0 \\sum_{r = 0}^\\infty \\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^{r + c}. ", "dfb620ca4b8db680f866309d7c971d4f": "S(g)", "dfb69550cc85d28f28bdff5ebb2f6604": " \\mathbf{A}=\\sum _i \\mathbf{a}_i\\mathbf{b}_i \\quad \\mathbf{B}=\\sum _i \\mathbf{c}_i\\mathbf{d}_i ", "dfb707a0f69659bcd507e51857457f24": " \\dot{x} = 2p ", "dfb71db5a154d6fa1ff0128bfc510865": "z^2y^{\\prime\\prime} + zy^\\prime +(z^2-\\nu^2)y = (z-\\nu)\\sin(\\pi z)/\\pi", "dfb7b990f71110db7a99c9c943a7aef5": "\n1 = \\gamma^2 + v\\delta\\gamma = \\gamma^2 (1 + \\kappa v^2)\n\\,", "dfb7ebb91af2e44d899d8cba76614645": "|\\text{MCC}| = \\sqrt{\\frac{\\chi^2}{n}}", "dfb8ac2199676abc28896d1670325aba": " G = \\frac{p'(x_k)}{p(x_k)}", "dfb8cfc9bee3369db473d63af9dddfc7": " y = - \\left(\\begin{matrix}\\frac{a-b}{2}\\end{matrix}\\right)\\sin (2 \\omega t) ", "dfb8e044005fbcd5af6203167f312ff7": "- \\mathbf{E} = \\nabla V. \\,\\!", "dfb8fc7073b152f52be476863ade7850": "H_X = \\frac{\\alpha-1}{2\\alpha-1},", "dfb97b7bc03fcb1470929fc6bd45f138": "y = c_1 x^m \\ln(x) + c_2 x^m \\,", "dfb98532c4efa2b28018fbbeacc635a2": "\\{ \\alpha_{\\iota} | \\iota < \\gamma \\},\\!", "dfb9fc7f2b1c439253286df4a0944247": "BU(n)=EU(n)/U(n) ", "dfba077999a0f6c7f95a6cd19ece7411": " \\begin{align}\nE_X & \\approx \\frac{.0267 \\ \\mathrm{ V}}{z} \\ln \\frac {X_o}{X_i} \\\\\n & = \\frac{26.7 \\ \\mathrm{ mV}}{z} \\ln \\frac {X_o}{X_i} \\\\\n & \\approx \\frac{61.5 \\ \\mathrm{ mV} }{z} \\log \\frac {X_o}{X_i} & \\text{ since } \\ln 10 \\approx 2.30 \\end{align}", "dfba1af03d79730b8daeb6a0681820f1": "\\mathrm{SO}(2, \\mathbb{R})", "dfba88e19a4fb5fd9f3d678c3c9b6ee2": "\\omega = \\omega_i", "dfbaa67aa50e916530ed68cb98a344bd": " E_00, V\\geq 0, p^*", "dfc1144a277f57b7ea9734c086483dad": "\\, \\Phi_n \\,", "dfc16d19bfb96463f06ea621a1d9338e": "\\twoheadrightarrow.", "dfc176bbbefa0256086dda1c904ac67d": "v=y_{\\xi}", "dfc1d32ebe84ddbd72eebb267d8a066f": "H \\,", "dfc248d3cc690a611e4cdefef2b8930e": " \\lim_{x \\to c} f(x) = 1,\\ \\lim_{x \\to c} g(x) = \\infty \\! ", "dfc25a6f4c86586c073fdc8d8433712f": "\\begin{align}\n p(7k + 5) &\\equiv 0 \\pmod 7\\\\\n p(11k + 6) &\\equiv 0 \\pmod {11}.\n\\end{align}", "dfc260e9cc21798263824e14b33828de": "\\alpha (ST) = (\\alpha S) T = \\beta T = \\gamma", "dfc2af609f548d8d961f20821f8fb1a5": "\\sigma<\\sigma_a", "dfc2c00a6876dab9f29736c37fbec7a3": "2.32727", "dfc2fec0f86115b6ca057b661b0f55d3": " xx^{s}=a^2 +b^2", "dfc31d5be16ce0454a85e5ba972bcce3": "K_t(x,y)\\equiv\\sum_{n\\ge 0} e^{-(2n+1)t}H_n(x)H_n(y)=(4\\pi t)^{-{1\\over 2}} \\left({2t\\over \\sinh 2t}\\right)^{1\\over 2} \\exp \\left(-{1\\over 4t} \\left[{2t\\over \\tanh 2t}(x^2+y^2) - {2t\\over \\sinh 2t}(2xy)\\right]\\right).", "dfc365c28de8ea2c060577317ff613a0": "a^3 (1-e^2)^{3/2}", "dfc36839d4d43aa70c0fb4fff5cc2313": "c\\in (a,b)", "dfc39b012a8afc3f3e8b86a322025452": " \\operatorname{E}(T) = n + n H_{n-1} - (n-1) = n H_{n-1} + 1 = n H_n,\n\\quad \\mbox{QED.}", "dfc3e04518e8a3c01bf42247c01f8e9d": "\\int_{-c}^c \\frac{1}{x}\\,dx = 0,", "dfc3ed293466d13b72cd427729f1a732": "\\|f\\|_{BV} = V_f(I) + \\lim\\nolimits_{x\\to a^+}f(x)", "dfc450814ad91a98535fcea645dea922": "R^{(3)}_{ij} = 2[\\nabla_{i}\\Phi_{A}\\nabla_{j}\\Phi_{A} - (1+ 4 \\Phi^{2})^{-1}\\nabla_{i}\\Phi^{2}\\nabla_{j}\\Phi^{2}], ", "dfc4d958a848e9af85a83b612ab4ea37": "\n\\forall i \\in \\mathrm{N} \\quad \\forall \\tau\\ _i \\in\\ \\Sigma\\ ^i \\quad \n\\pi\\ (\\tau\\ ,\\sigma\\ _{-i} ) \\le \\pi\\ (\\sigma\\ )\n", "dfc4e484a8e46b68fae9582dc5b4adf1": " \\forall (\\alpha,\\beta)\\ \\mathbf T \\left( \\alpha \n \\begin{bmatrix}\n \\vec f \\\\\n \\ \\\\\n 0 \\\\\n \\end{bmatrix} + \\beta\n \\begin{bmatrix}\n 0 \\\\\n \\ \\\\\n \\vec b \\\\\n \\end{bmatrix} \\right ) = \\alpha \n \\begin{bmatrix} 1 \\\\ \n 0 \\\\\n \\vdots \\\\\n 0 \\\\\n \\epsilon_f \\\\\n \\end{bmatrix} + \\beta \n \\begin{bmatrix} \\epsilon_b \\\\\n 0 \\\\ \n \\vdots \\\\\n 0 \\\\\n 1 \\\\\n \\end{bmatrix}.", "dfc4ea0c5b6ec52595197316debc6be7": "\\ln \\left ( i_M (V)/i_M (0)\\right ) = -eV/\\mathcal{E}_p ", "dfc597b9889815e962bcbbd59f63c79c": " F_{0k} = - F_{k0} ", "dfc5a4e4ad9120128273ab4ad34e596e": "|M|>256", "dfc5b2adb5089269fbfe872b354450c5": "\\frac {D}{T} = \\varphi = \\frac {1+ \\sqrt {5} }{2} \\ , ", "dfc5f7050373376d93e490107ecef92f": "\\displaystyle J", "dfc5ffc855b109a2e9c17885dd4bb2e3": "\\mathbf{H}_{ij} = 0, i\\neq j", "dfc60104725c17e2c276c50dda142e77": "\\underset{a_{i0},a_{i1},...,a_{im}}{\\operatorname{arg\\,min}} \\operatorname E\\left [ \\left ( a_{i0}+\\sum_{j=1}^{m}a_{ij}X_{ij}-\\Pi \\right)^2\\right ]", "dfc618130793aeaafae9d400c97771f7": "A_{ij} = \\sqrt{A_{ii} A_{jj}}", "dfc64d74c8f0ba7f81e5bca67d130252": "\\delta \\mathbf{q} = \\sum_r \\epsilon_r \\mathbf{Q}_r ~, ", "dfc65450b09fb73e6786c5a6a6ddc2e8": "\\begin{align}\n q_{n}({\\mathbf{X}}) = \\exp \\{-i\\sigma \\int \\limits_{z_{n}}^{z_{n+1}} V({\\mathbf{X}},z')dz'\\}\n \\end{align}", "dfc66af63c0ef326433baa2f314b27d9": " \\omega_{i} = 0 ", "dfc69844b205fe9e5a824258972d772f": " K = 0.2 + 0.02 \\times \\log_{10}(V)\\,", "dfc69f47051c7cf598032661ff4cef53": "\\tfrac{1+\\sqrt{5}}{2}", "dfc6d4f2c6a12c77a6bf1e8b3554b7c2": " \\text{Var}(\\Sigma)", "dfc6f9cc3d2e77d4b0193f467ed3f201": "\\gamma_{P}(Q)", "dfc7493029f837e28e40af9ea633447f": "\\rho = T \\mu^* - \\sum_{t=1}^T \\widehat{r}_t", "dfc7af2f887d78df3390d06d0aeeda17": "E=2\\mu(1+\\nu)", "dfc7ed8d226d67bd4d99cba1748125d4": "H = \\tfrac1{2} (xp+px) = - i \\left( x \\frac{\\mathrm{d}}{\\mathrm{d} x} + \\frac1{2} \\right).", "dfc807abc9e60a777a47bed5af2b91b6": " f\\mapsto\\int_E f \\,d\\mu, \\,", "dfc8395a15978b5e09d8db07338a364f": "{dx_2 \\over dt} = r_2x_2\\left(1-\\left({x_2+\\alpha_{21}x_1 \\over K_2}\\right) \\right).", "dfc88d95d2dea4af23f8f895880a7560": "{}_1 Y_{10}(\\theta,\\phi) = \\sqrt{\\frac{3}{8\\pi}}\\,\\sin\\theta", "dfc903969f00988dd07b46cb30778bdd": "\\frac{1}{n}x + \\frac{1}{n}h", "dfc9521434efcdd8ed39bb03ac9669fe": "T \\mathbf{x} = \\mathbf{y}", "dfc96001568235d60dc12dca4b32d260": "w_{i}", "dfc98a2d688286df7b2a0d803fd9014c": "R \\gg y_1 - y_2", "dfca183ab2cad27c6404be617ad81daf": "\\mathbf{R} =LD= \\begin{bmatrix}\n1 & 0 & . & 0\\\\\nr_{12}/r_{11} & 1 & . & 0 \\\\\n. & . & . & . \\\\\nr_{1n}/r_{11} & r_{2n}/r_{22} & . & 1 \\end{bmatrix}", "dfca8d24196fa7339b4ade21c8aabd0a": "s_1,\\ldots,s_M>0.", "dfcab92d4b53d5cd959666aab5a32319": "\\Delta _{\\mathcal{G}}(x_{\\perp }) =\\gamma_{1}\\cdot \\gamma_{2}\\frac{\\mathcal{G}(x_{\\perp })\n}{2}\\mathcal{O}_{1},\\text{vector}\\mathrm{,} ", "dfcae036093bf769d7102ceab34184c2": " \\mathbf{P}(\\mathcal G)\\to \\mathbf{P}(\\mathcal E_{\\mathbf{Gr}(r, \\mathcal E)})=\\mathbf P({\\mathcal E})\\times_S {\\mathbf{Gr}}(r, \\mathcal E). ", "dfcb5b49ec546ecf225ffc7d8bc40014": "(A\\to B)\\to((\\neg A\\to B)\\to B)", "dfcbdb10609ad2e02b566b0f9722784c": "f\\star g=f\\circ g+\\frac{i\\hbar}{2} f\\wedge g.", "dfcc0132f5ea4c10a0050c7d53968313": "\\displaystyle{x^4, y^5}", "dfcc254cab6ff9fd1bdeb7bb0f372a1a": " |1, 2, 0, 0, 0, \\dots \\rangle.", "dfcc8124f54e0d79d20de894c5b39ac1": "f(x; \\sigma) = \\frac{1}{2\\pi\\sigma^2} \\int_{-\\infty}^\\infty du \\, \\int_{-\\infty}^\\infty dv \\, e^{-u^2/2\\sigma^2} e^{-v^2/2\\sigma^2} \\delta(x-\\sqrt{u^2+v^2}).", "dfcc92637e9462725e6a9502ae267b78": "L,", "dfccc1cfbe601bfc72e92f98bff7d279": "H^r(X, \\mathcal{F})", "dfccd73ba6c6b855fad625537e6056bb": " d = \\frac{1AU}{tan(p)} ", "dfccf5a8f70cbfee59f76ca4c4f1ed5d": "MRS_{xy}=-m_\\mathrm{indif}=-(dy/dx) \\,", "dfcd6b54de3f89fb584174302986f31c": "\\alpha(f(x))=\\alpha(x)+1\\!", "dfcd952447a088738e8fcb25dfdff728": "AB \\rightleftharpoons A^+ + B^-", "dfcdb143e54efab3277f684af232e04c": "\\textstyle\n (1/f)'''(x)=-\\frac{f'''(x)}{f(x)^2}+6\\frac{f'(x)\\,f''(x)}{f(x)^3}-6\\frac{f'(x)^3}{f(x)^4}\n", "dfce2769d5dfad112445781651592526": "{\\pi\\over 3}\\ {\\pi\\over 3}\\ {2\\pi\\over 3}", "dfcebb213865dd96ca29e13a2a184efc": "F(\\mathbf{q}) = \\int f(\\mathbf{r}) \\mathrm{e}^{-\\mathrm{i}\\mathbf{q}\\cdot\\mathbf{r}}\\mathrm{d}\\mathbf{r}", "dfcee5fd5f65288914731e8595c21a87": "\\frac{\\partial\\rho(\\mathbf{x},t)}{\\partial t} + \\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x},t)=\\left(-i\\omega\\rho(\\mathbf{x}) + \\mathbf{\\nabla}\\cdot\\mathbf{J}(\\mathbf{x})\\right)e^{-i \\omega t}=0", "dfcf005d5b9096d1ca78534550ad0565": " \\mathbf{B} = \\mathbf{B_0} + \\mathbf{b} ", "dfcf145fa9cc4052d78c16d0b3b869f2": "\\varphi (N) ", "dfcf28d0734569a6a693bc8194de62bf": "G", "dfcf2a34dec3d92607038e2c58cb6595": "M = N\\left\\langle\\mu\\right\\rangle = N \\mu \\tanh\\left({\\mu B\\over k T}\\right)", "dfcf34cf8f543cb2e548346acd37c48b": "\\Omega_{\\lambda} = \\sum_{U (p:z) = \\langle x,z \\rangle,\\,x\\in NF} 2^{-\\ell(p)}", "dfcf711f257289d41d623783ffe53226": " \\varphi (L)", "dfcf79e3d5443ac2ec1e8739062d54e2": "\\Delta f", "dfcfbea8cedac524bbe6dacba5cae6d1": "p(\\mathbf{X},\\mu,\\tau) = p(\\mathbf{X}\\mid \\mu,\\tau) p(\\mu\\mid \\tau) p(\\tau)", "dfcfc2f7818ea0e6bbbed792c208116f": " ~{\\mathcal U}^{(t)}_{1...t} ", "dfcff7604be75e69d9b190736aa4ab24": "\\text{d}(P,Q)", "dfd005bc2b7d84c6c50c7d4367350e0d": "\\frac{\\partial u}{\\partial y} = \\frac{\\partial u}{\\partial r}\\frac{\\partial r}{\\partial y} + \\frac{\\partial u}{\\partial \\varphi}\\frac{\\partial \\varphi}{\\partial y},", "dfd0fff8974cc168aa14c2b0464881f2": " {\\hat x} ", "dfd157b0a702a1053171ba67b07402d8": "(0|0|0|0)", "dfd1dfae7e9c8d8718af7f967cb95fa9": "(\\mathcal{A},\\phi)", "dfd22a236088e7ea345aa6daf782158d": "\n\\phi_R = \\angle H_R(j \\omega) = \\tan^{-1}\\left(\\frac{1}{\\omega RC}\\right)\n", "dfd26afa6091c61f9775456ee723a190": "\\mathbb{R}^4", "dfd27717377911c9307cb202317e798e": "r(\\alpha)_i = (r\\alpha)_i", "dfd27f2edf4c2555a55f922b2e2cc181": "\\limsup B = \\inf\\{ \\sup B_0 : B_0 \\in B \\}", "dfd29d416a2320c94fa96adb007579c3": "\\partial^\\mu \\partial_\\mu \\phi+m^2 \\phi^2=0", "dfd2d7dcbfef8c03c7cf014d8991bf69": "\\parallel z_i\\parallel\\approx 0", "dfd2e1798734a13cf3351bfcaff8694a": "\\mu + \\frac{1}{2} \\ln(2\\pi e \\sigma^2)", "dfd2ef145de2fe479f496fcf6391ead6": "\\underline{\\lnot \\varphi \\quad \\quad}\\,\\!", "dfd2f30836d2fc52abfbedb6bf43da6d": "\\operatorname{recc}(A) = \\{y \\in X: \\forall x \\in A: x + y \\in A\\}.", "dfd2fe159859b987dda71f8252fa665c": "y'(x)=-2y+e^{-x}\\,", "dfd3335dc9bfe6fb05257ab7c5ec2ff3": " f^*(t) ", "dfd38e2e7f814c02037324ab817c0d29": " \\begin{pmatrix} 1 & 0 \\\\ -\\frac{2}{R} & 1 \\end{pmatrix} ", "dfd3e631f04f9a9ca64886e342ad1540": "U_{k}(\\beta) ", "dfd3f5fc4bef54ebd732b479edb51bea": "{\\color{Red}\\bar{\\infty}} = \\tfrac{\\infty}{m}", "dfd422ca58f9059a622dcb4d201d278d": " \\Delta (f g) = f \\,\\Delta g + g \\,\\Delta f + \\Delta f \\,\\Delta g ", "dfd48c4d503fa2d44ba804d0a79f8f7b": "f(x) = \n \\left\\{\\begin{matrix}\n \\textbf{c}_Ne^{\\textbf{A}_Nx}\\textbf{b}_N & \\text{if }x < 0\n \\\\[8pt]\n \\textbf{c}_Pe^{\\textbf{A}_Px}\\textbf{b}_P & \\text{if }x \\geq 0\n \\end{matrix}\\right.\n ", "dfd4ad4c530a836514320013cdc19629": " \\gamma_r^\\mu = \\underbrace{I_4 \\otimes I_4 \\otimes \\cdots}_{r-1\\,\\text{matrices}} \\gamma^\\mu \\cdots \\otimes I_4 ", "dfd4d31bac14e59e52933e6883c15751": "I_{1k}", "dfd52bd0c7bbbe215ad7e0f85ca11c7a": "s(F)\\ne\\infty", "dfd530dac647fc3dd5f245814ded9066": "\n\nf_{p}(\\mathbf{x}; \\mu, \\kappa)=C_{p}(\\kappa)\\exp \\left( {\\kappa \\mu^T \\mathbf{x} } \\right)\n\n", "dfd54f29174ad73da84766575a85d4ce": "J_\\mu^3", "dfd5a3a9870f5ee3476cf402dac4700a": "T_1(q) = \\sum_{n\\ge 0} {q^{n(n+1)} (-q^2;q^2)_n \\over (-q;q^2)_{n+1}}", "dfd5b7384fed1fee6356274c9c3eb638": "N_y = my - p_yt = \\gamma(u)m_0(y - u_y t) ", "dfd5d79dc125ce5e2fe9175dd0298bbc": "\\det \\boldsymbol{\\varphi}'_w(0) = +1", "dfd5eddd5484c94633fd1b5d3c01b10c": "E_n(mx)= m^n \\sum_{k=0}^{m-1}\n(-1)^k E_n \\left(x+\\frac{k}{m}\\right)\n\\quad \\mbox{ for } m=1,3,\\dots", "dfd5fba3a0829c9da009efbf0e298b76": "uv \\in E ", "dfd606def88854c060754549bdc252e7": "-(g^{\\mu \\alpha }g_{\\beta \\sigma }(\\sqrt{-g}g^{\\nu \\sigma }),_{\\rho }(\\sqrt{-g}g^{\\beta \\rho }),_{\\alpha }+g^{\\nu \\alpha }g_{\\beta \\sigma}(\\sqrt{-g}g^{\\mu \\sigma }),_{\\rho }(\\sqrt{-g}g^{\\beta \\rho }),_{\\alpha })+", "dfd620172eda5597666709163e395670": "H_u^L", "dfd63460e884d50ef974aa42b9ec9095": "s\\neq 1", "dfd6513d16ceaabca22f76fa7dc9950a": "\n \\frac{\\partial\\tilde{\\rho}}{\\partial t} +\n \\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\nabla \\cdot\\langle\\mathbf{v}\\rangle+\n \\langle\\rho\\rangle\\nabla\\cdot\\tilde{\\mathbf{v}} +\n \\nabla\\tilde{\\rho}\\cdot\\langle\\mathbf{v}\\rangle\n = 0\n ", "dfd6832263d8b4fee56ff298b1daa696": " N_1^{(e)}(x)=1-\\frac{1-x_{0}^{(e)}}{x_{1}^{(e)}-x_{0}^{(e)}}, \\ \\ N_2^{(e)}(x)=\\frac{1-x_{0}^{(e)}}{x_{1}^{(e)}-x_{0}^{(e)}}. ", "dfd6a07427dfdc5011263d23148aa092": "\n\\begin{pmatrix}\n1 & 0 & 1 \\\\\n\\end{pmatrix} \n\\cdot\n\\begin{pmatrix}\n1 & 0 & 0 & 1 & 0 & 1 \\\\\n0 & 1 & 0 & 1 & 1 & 1 \\\\\n0 & 0 & 1 & 1 & 1 & 0 \\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1 & 0 & 1 & 0 & 1 & 1 \\\\\n\\end{pmatrix}.\n", "dfd758c7d69f08ecc1ee602667cbc715": "A \\oplus B = \\N.", "dfd780758779013fd583f7fecb6fc2f7": " r^2(a^2+b^2)-c^2=0 ", "dfd781d9521b1e755e34cb7e5256d13b": "B_z = \\{b+z|b\\in B\\}", "dfd794a84c84a0a4f7ffe4b651b411bc": "\\sum_{i=1}^n a_i>\\tfrac{B-1}{2}V", "dfd7ac18598f3db5ce8850a4b38c8cd3": "\n\\begin{align}\n \\mbox{Change in NFA} & = \\mbox{Current Account} \\\\\n\\end{align}\n", "dfd7cbef30e0d8be27d56442c8d3d07d": "2P_{\\frac{3}{2}}", "dfd7f8a91d95e0d711ebaf950fa7eef1": "S(\\omega_j)", "dfd81830e9606da4d5a33c64a5f017c1": "\\displaystyle{\\nabla_z N(z-w)=-\\nabla_wN(z-w) =-\\partial_{n,w}N(z-w)\\mathbf{n}_w -\\partial_tN(z-w)\\mathbf{t}_w,}", "dfd8825b51a6231a4cd9e28d6d052956": "P = [0 : \\dots : 0 : 1]", "dfd8da02a5c06c74a3732924bbcbaabc": "G_1 \\ \\stackrel{\\mathrm{def}}{=}\\ \\{\\pm I,\\pm iI,\\pm X,\\pm iX,\\pm Y,\\pm iY,\\pm Z,\\pm iZ\\} \\equiv \\langle X, Y, Z \\rangle", "dfd92e87dbaa02990fd90e9df7887934": "g^2_\\ast <1", "dfd941a2c924d8dbc26ab034209f5fcf": "\\sqrt{2+\\sqrt{2}}", "dfd99198ef4a5f58daf004d119b271e8": "\\langle T x, y' \\rangle = \\langle x, T' y' \\rangle", "dfd995c6b4300b6493af724de36e0292": "a_n=\\sum_{k=1}^n s(n,k) b_k,", "dfd9ba36cd5df549d27b2aeb020f555e": " K\\subset \\mathbb{R} ", "dfd9f762c80e7f9d4b5212804a3f2a26": "=\\frac{d}{dt}\\boldsymbol {r} =\\sum_{k=1}^{d} \\dot q_k \\ \\boldsymbol{e_k} + \\sum_{k=1}^{d} q_k \\ \\dot{\\boldsymbol{e_k}} \\, ", "dfdada414e5cb03669df4a808423e016": "\\sum_j w_j \\approx \\int_{r_0}^R \\frac{2\\pi r\\rho dr}{r^p} = 2\\pi\\rho\\int_{r_0}^R r^{1-p} dr,", "dfdbafeac6c08e357e6d52c5cbb9ede7": "\\left| \\int_a^b f(x) \\, dx \\right| \\leq \\int_a^b | f(x) | \\, dx. ", "dfdbcbf64e2a990c0456261d0af9d581": "{\\mathbf\\Sigma}", "dfdbceb6d94832d63c3603ceea14d525": "(s_{m})", "dfdbd5a660a6b793ff33cbc8caf17260": "\\scriptstyle [0.5/L,\\ 1-0.5/L]", "dfdc4fbfe1ba33598c0d3dd0b3129526": "end\\,for", "dfdcc5a21125f21f98b85aa1179ad57c": "a^m + b^n = c^k\\quad", "dfdcd795b7d8cced3e686507c8398caa": "k_B T/2", "dfdd2d1ff930999b7629387badbdb7bb": "\\Lambda_n(z)=\\int_0^z \\frac{\\log^{n-1}|t|}{1+t}\\;dt.", "dfdd49da836ee683e8902004bb68c89b": "z = \\frac{M - \\mu}{\\mathrm{SE}} = \\frac{96 - 100}{1.62} = -2.47 \\,\\!", "dfdd589456ee03cd7c8be386e1b8b0bf": "d = gh + p", "dfdd89291a6a1c7754c0c702e7b76044": " C_y=\\beta_{xy} \\times \\rho_{xy} \\times \\epsilon_{f}", "dfddb86982a7c5f8def0539e0ee4c426": " \\mathbf{F} = (\\mathbf{T}')^{T} \\, \\mathbf{\\bar F} \\, \\mathbf{T} ", "dfddd81bf77a16d2c0b261686eff819f": "x = 2Ae^{t} + Be^{-5t} \\,\\!", "dfddd996473d5a71dd6d190bab09be72": "T_n={n+2\\choose3}.", "dfdde6296de21b9e967089622fe78182": " U x = y ", "dfde11e197c21790b1ae9c48352a3ddc": "P(0)", "dfde61a1dcb68e1a930467c6b2608dd7": "B_{3}", "dfdebea9f2d593d9062b9f2ec4da83eb": "\n\\nabla T = \\begin{pmatrix}\n{\\frac{\\partial T}{\\partial x}}, \n{\\frac{\\partial T}{\\partial y}}, \n{\\frac{\\partial T}{\\partial z}}\n\\end{pmatrix}", "dfdf05f9d38d424a01c6814e0acc337c": "b_0, b_1...b_n", "dfdf078108764cabe7f510c4b392d939": "Z_{r}", "dfdf96dc9cea46c275b542c90fd80350": "X(x_0)", "dfdfb965c2641916b160500565a3573b": "\\mathcal{M}_{ij} \n= 2 \\overline{\\Delta}^2 \\left( \n\\overline{ \\left| \\hat{S} \\right| \\hat{S}_{ij} } \n- \\alpha^2 \\left| \\overline{\\hat{S}} \\right| \\overline{\\hat{S}}_{ij}\n\\right)", "dfe0322bcdcc129bd518e1926dd2073b": "\\mathbf{v}'_2", "dfe052d3cffa1ba6c816c15dc9f0a613": "\\frac{d\\lambda}{dt}=0.", "dfe065593b9706046290287673d61181": "(\\mathbf B^T \\mathbf P_A^{\\perp} \\mathbf B)^{-1}", "dfe06a63a90937f52d3488a5d792c32c": "t=64", "dfe0908563d620edb981c7ce042e2b7e": "G_{\\rm ij}\\,", "dfe159d909d5ac130c8526ada6868f95": "X=\\frac{1}{2}AB\\cos(\\boldsymbol\\theta)", "dfe19bdfac59a2e77c03d0ec412962d1": "|S| \\le N\\,", "dfe19d6aaf7129709f3c54602ba458b3": "E(U)=\\sum_{k=1}^\\infty\\frac{(\\ln(w+2^{k-1}-c) - \\ln(w))}{2^k} < \\infty \\,.", "dfe1eb4da83fc28075da2c306c73a9ea": "(NH_{2})_{2}CO + H_{2}O \\stackrel{urease}{\\rightarrow} NH_{3} + H_{2}NCOOH \\rightarrow2NH_{3(gas)} + CO_{2(gas)}", "dfe25ac24c5204b36335c310a00bea22": "\\displaystyle \\frac{1}{2\\pi\\cdot|ab|} e^{\\frac{-\\left(\\omega_x^2/a^2 + \\omega_y^2/b^2\\right)}{4\\pi}}", "dfe26c4d38cba456e691188943fe7349": "\n\\begin{pmatrix}\n l_1 & l_2 & l_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n \\approx (-1)^{l_3+m_3} \\frac{ d^{l_1}_{m_1, l_3-l_2}(\\theta)}{\\sqrt{l_2+l_3+1}}\n", "dfe28f0d1e031809209c86c99ec2b791": "HL \\rightleftharpoons L+H:pK_3=-\\log \\left(\\frac{[L][H]} {[HL]} \\right)", "dfe313fae8d021742d4f67e19cd51c55": "\\varepsilon_{t}", "dfe3334646239b3e9799bfedcdd90f31": "ad_g", "dfe363d8a58a89341874b1451ed78bbb": "\\left.\\right. H_2", "dfe366a5a3adeb855df228f8da09059e": "F^\\prime ", "dfe3d97e57544fd3743ee6281b98f760": "{\\lambda}=M + \\varpi ", "dfe3f441e0f79fbcb93a4b08cc934851": "v = \\sqrt{\\frac{M}{r-2M}}", "dfe48eef99b1b2a981d5cae7b1e3ecbd": "\\scriptstyle \\mathcal{A}_q^n", "dfe4eb22533e9e41d1f17ba04bb202dd": "P = \\{L_1,L_2,\\dots | R_1, R_2,\\dots\\}.", "dfe51cbf3897a6af4903b6279acd21d8": "\\begin{array}{rcl}\nwp(x := x - 5, x > 10) & = & x - 5 > 10 \\\\\n & \\Leftrightarrow & x > 15 \n\\end{array}", "dfe52023e77eeb05a48a454a4406e85a": "\\mathbb{C}^g \\cong \\mathbb{R}^{2g}", "dfe520f756120732175bf419d8c53ed5": "\\ln \\left ( \\frac{I_L}{I_O} \\right ) = \\ln \\left ( 1 - \\frac{V_S}{V_O} \\right )^{k_O}", "dfe5406219e42499841e958bf59362fe": "\\left| \\psi \\right\\rangle = \\left|\\psi_{\\text{test}}\\right\\rangle - \\left\\langle\\psi_{\\mathrm{gr}} \\mid \\psi_{\\text{test}}\\right\\rangle \\left|\\psi_{\\text{gr}}\\right\\rangle ", "dfe542c6af00a9cd8266623608e22451": "\n\\bar{\\kappa}_{bf} = 4.34 \\times 10^{25} \\frac{g_{bf}}{t}Z(1+X) \\frac{\\rho}{\\rm g/cm^3} \\left(\\frac{T}{\\rm K}\\right)^{-7/2} {\\rm \\, cm^2 \\, g^{-1}},\n", "dfe552eb0a805db35c15ca105ea3166c": "w_{cv} = {2 \\pi \\over \\hbar}^2 ||^2\\delta[\\Epsilon _c (k') - \\Epsilon _v (k) - \\hbar \\omega]", "dfe5739ada51fd473a8510b1e2318554": "\\bar{r_u}", "dfe58dc478d3bcdca7bc5a809a92b2e8": "v' = 2", "dfe5aff266fabb901efa6166c6b72c2b": "f \\ll 2h", "dfe5d21cb42e873ee21634d82f82dbb7": "R_{P_t}", "dfe5f6b1725d189d948468bbeb47634d": "\\# \\!\\,", "dfe5fa154c3cdd1a91681be1cbad2bef": "h(z)=\\frac{a}{z+b}+c.", "dfe60204397cc717fe19047b680692f7": "y(i)+y(j) \\leq c(i, j)", "dfe632152bdfbca5a220d1fe8ac9d8f0": "p_n=n", "dfe662983936f31fafd3395c2a7ab5b0": "|A\\cap C_n|<\\omega,\\forall n<\\omega", "dfe6f751873ce031a4e08ff3fc471a7b": " \\textstyle \\sigma=\\{\\sigma_j\\}_{j \\in \\Lambda}", "dfe76a6917c50e4cc33e4fa050b41346": "F=\\mathrm{d}p/\\mathrm{d}t", "dfe7751263c218a30ad51501fa56e83b": "bf{A} = 0", "dfe776f0068d3751f5b4c1b68c02f827": "Q_m=\\frac{ K_u -I_u\\omega^2 }{2Kd^2}\\tau", "dfe78cf0185baee59730d40edccecb57": " \\frac{1}{R_\\mathrm{E}} = \\frac{1}{R_\\mathrm{L}} + \\frac{1}{r_\\mathrm{O}} ", "dfe7ac7d88401c9e305107f36b29b2c3": "\nS \\ \\stackrel{\\mathrm{def}}{=}\\ \n \t\\left\\{ \\begin{matrix}\n a_{11} + a_{12}x + a_{13}y & = & 0 \\\\\n a_{21} + a_{22}x + a_{23}y & = & 0\n\t\\end{matrix} \\right.\n\\ \\stackrel{\\mathrm{def}}{=}\\ \n \\left\\{\\begin{matrix}\n D/2 + Ax + (B/2)y & = & 0 \\\\\n E/2 + (B/2)x + Cy & = & 0\n \\end{matrix} \\right.\n", "dfe7b5ca4225f673db19e9a5ce7cb800": "(X^*_{b})^* = X^{**}", "dfe7cee54d6101d4e1ffa171ef9efe62": "\\Bigg(\\frac{a}{n}\\Bigg)", "dfe81406c7b3cb2ce4c2639cda51c3c3": "\\mathbf{r}", "dfe814f13e2ea69c439629b27a7792e7": "U(x,z)=W(x,z)=0,\\,", "dfe825b4ea2ca198f0870f91a8729ac4": "(f \\cdot g)(x) = f(x) \\cdot g(x)", "dfe838bb8e5bb25dd41ab8c6eebf0817": "P(x) = 1 + 2x", "dfe8b0424e3c205c22c13a94b9bb745b": "(X,\\mathcal{F})", "dfe8d780bcacbeef965c340a86b6cc97": "\\mathbf{M^r=\\left(I-H \\right) M^y \\left(I-H \\right)}", "dfe8e8e2433d89c718ad89988bbfae4e": "q^\\prime\\in \\delta(q,a)", "dfe8ee931c46b482a8a26b8f3ce2ccc6": "e^{3100}\\approx 2 \\times 10^{1346}", "dfe96356d8b6e281735cbdb9fa6628c9": "\\varphi(\\mathbf{q};V)", "dfe9729cff79713d384e29d448d32ffc": "\\alpha_3 = {{3\\alpha_0 + 2\\alpha_1} \\over 5}", "dfe9857a210e3823f4c8d3a17bd51beb": "\\kappa_1 = \\kappa_2 = r^{-1} \\,", "dfe9c95d5704a8e3a45e7c0839cb7a0c": "Z_c", "dfe9ca9ac320978d9c874941696f6c47": "\\omega(X,\\mathbb{D})(E)=\\int_E \\frac{1-|X|^2}{|X-Q|^2}\\frac{dH^1(Q)}{2\\pi}", "dfe9e2003ef364cc0fe9dce2a5ee9248": "\\textit{NOUN}", "dfe9ea55ebffdf69042e8bae9f0300ac": "E(\\ \\tau,t_0,y(t_0)\\ )", "dfea10bcc7b05a37980b86429acd54ec": "c \\in A", "dfea224415e763bcf41bbefb6ca8a996": " \\displaystyle \\sum_{n=0}^\\infty B_{n,\\chi}\\frac{t^n}{n!} = \\sum_{a=1}^f\\frac{\\chi(a)te^{at}}{e^{ft}-1}", "dfea415e85b4aec6177edfe46a24e468": "\\mathbf{T} = \\rho A {v}^2, \\mathbf{P} = \\frac{1}{2} \\rho A {v}^3", "dfea645280848291047867c77a879b72": "\\ddot{\\theta}_1 m_2 L_1 l_2 \\cos(\\theta_2)\n+\\ddot{\\theta}_2 \\hat{J_2}\n- 1/2 \\dot{\\theta}_1^2 \\sin(2 \\theta_2) \\hat{J_2} \n+b_2 \\dot{\\theta}_2\n+g m_2 l_2 \\sin(\\theta_2)\n= \\tau_2", "dfeabb39ed54dfc88ee0844442940ba6": " \\displaystyle{Tu_n=\\lambda_n u_n.}", "dfeb049e24cfda15011d86b663e2b095": " \\| \\mathbf{x} \\| = 1", "dfeb0f2ba7819fb8f325b8a8abcc31a2": "\\phi_{A,B}:FA\\bullet FB\\to F(A\\otimes B)", "dfeb120bbf33728ef0f507d6c4648b5b": "Employed", "dfeb839fc39bc822f9b4c0dec0d8cfa7": "D = \\cap_{i = 0}^m \\operatorname{dom}(f_i)", "dfec20084568de0d20662b0ea4a069e8": "xe^xE_1(x) \\sim \\sum_{n=0}^\\infty \\frac{(-1)^nn!}{x^n} \\ (x \\rightarrow \\infty) ", "dfec274a7a1738b077e089c2629d204b": "Pr[x^{(n)} \\in A_\\epsilon^{(n)}] \\geq 1 - \\epsilon ", "dfec8688c508fd654b14aa6214bbef9c": "\\ P _{ik}", "dfeca44fc77e1f9dbc63a3744eddee10": " W = \\begin{bmatrix}\n 0 & 0 & 0\\\\\n1 & 0 & 0\\\\\n1 & 1 & 0\\\\\n1 & 1 & 1\\\\\n0 & 1 & 1\\\\\n0 & 0 & 1\n\\end{bmatrix}.\n", "dfed6bf9ad67a07ed05e727c965c526c": "\n\\Phi(r, \\theta) \\propto\n\\frac{1}{r} \\sum_{k=0}^{\\infty} \\left( \\frac{a}{r} \\right)^{k} \nP_{k}(\\cos \\theta)\n", "dfed88fae7f3b4e8c5c9adf3a01e086c": " \\pi \\ \\sim \\ 2 \\left(2^{2n} - 4^{2n} \\right) \\frac{B_{2n}}{E_{2n}}. ", "dfed90eed9b37e5c25c6cdb8f7d48731": "\n\\frac{d \\vec x}{dt} = \\vec v(\\vec x, ~t)\n", "dfed913346ac9b879900a54e4bd58320": " Z = 1.69 ", "dfeda9e45f253466921a195292ba34be": "Q=\\int\\limits_S \\sigma_q(\\bold{r}) \\,dS", "dfeddec154755fb61f788fbea3e021d1": " v_{k+0.5} = 2 \\beta v_{k} - v_{k-0.5}, \\text{ for some }\\beta \\,\\!", "dfede4408721d2c0e8ed487baacf9aa8": "= \\sum_{e}(a_{e}f_{e}f_{e}^{*}) + \\sum_{e \\in E}f_e^{*}b_e.", "dfee5dbf969a089f8c474ffe6510b525": "2 \\pi", "dfee72ba64183038e1dadd26acf69900": "\\textstyle \\sum_{f_i \\in F}", "dfee75cee251cdd8467e36f733563274": "\\sigma_{3t}=10", "dfeea995ea131dd3071e3d77ff7c98db": "A=(\\Sigma_1,\\Sigma_2,\\ldots,\\Sigma_n)", "dfeebfdb5775d1ebaf0385b8e8952d70": "f_{X_{(1)},\\ldots,X_{(n)}}(x_1,\\ldots,x_n)=n!f_X(x_1)\\cdots f_X(x_n)", "dfeed5559c9b9bdb469828d2874d9f0a": "C \\approx \\frac{\\varepsilon A}{d}", "dfeefbabe76eb18d9468f6ce14fb82a4": " M_{ij} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{\\Phi_{i}}{I_j} = \\frac{\\mu_0}{4\\pi} \\oint_{C_i}\\oint_{C_j} \\frac{\\mathbf{ds}_i\\cdot\\mathbf{ds}_j}{|\\mathbf{R}_{ij}|} ", "dfef5a982b615e022a909d8fd0765b01": "x_1> x_2> \\dots > x_n", "dfef5cc242e659faa0f83b11a728d1d8": "\np(z) = \\int p(z|x) p(x) ~ dx\n", "dfefc05da6cfa7d3db53199825e08c66": "j \\equiv -960^3 \\pmod{167} \\equiv 107 \\pmod{167}. \\, ", "dff01255b075a502d2ba91eb9260ae0b": "\\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} = \\left(\\frac{\\partial \\mathbf{f}_1}{\\partial \\mathbf{v}}\\cdot\\mathbf{u}\\right)\\times\\mathbf{f}_2(\\mathbf{v}) + \\mathbf{f}_1(\\mathbf{v})\\times\\left(\\frac{\\partial \\mathbf{f}_2}{\\partial \\mathbf{v}}\\cdot\\mathbf{u} \\right)", "dff02d13dd1d853050303b36cebfd47d": "\nF_\\nu(k) = \\int_0^\\infty f(r)J_\\nu(kr)\\,r\\operatorname{d}\\!r\n", "dff042b57495b784255ba084d06856c4": "f: (-1, 1) \\to (-1, 1) \\, ", "dff06b5d5167a2d6c0430948ef6afb34": "P(X|\\theta)", "dff087a4684518eacba3ff02d9821d2a": "\n\\mathbf{G} \\equiv\n\\mathbf{B}^\\mathrm{int} \\mathbf{M}^{-1} (\\mathbf{B}^\\mathrm{int})^\\mathrm{T}.\n", "dff0b451eb4ce8623685afab56eb7b1b": "\\mathbf{R}^{*T} \\mathbf{r} = \\mathbf{R}^{*T} \\mathbf{B}^{T} \\mathbf{q} ", "dff0daeeb4c350d7c6853eec676811ea": "\\sqrt{a}\\sqrt{b}=\\sqrt{ab}", "dff0e49e14855958152c73430d692ef6": "t:\\tilde{\\mathbf{E}}^+\\to \\mathcal{O}_{\\mathbf{C}_p}", "dff0f23c29978b45f47353a965c48baa": "\\hat a_i=\\min_j count[j,h_j(i)]", "dff0f8daf2e8fd42cba149a3ef21f374": "{N-K-K+B\\over N-K}{K\\choose B}{N-K\\choose K-B}\\over {N\\choose K}", "dff0ff2936fe737008126094a6a99a46": "S_1,\\ldots, S_n", "dff12dd8b9cba757b0b803af4eb6feac": " c\\rho_c = (h_w\\rho_w)+(b_2\\rho_m)+[(c-h_w-b_2)\\rho_c] ", "dff1743412f784508b8c300e246ce169": "r_b=\\infty", "dff1e683559bf13871634c642b81d640": "(e^{i\\theta_1} z_1, \\dots, e^{i\\theta_n} z_n) \\in G", "dff1e6bb4130043f2728f0e02be04a94": " \\mathcal{L}_e ", "dff1e8d631d70805abc8b39e3200eb2b": "\\left(\\frac{m}{n}\\right) \n= \\left(\\frac{n}{m}\\right)(-1)^{\\tfrac{m-1}{2}\\tfrac{n-1}{2}} =\n\\begin{cases}\n \\;\\;\\;\\left(\\frac{n}{m}\\right) & \\text{if }n \\equiv 1 \\pmod 4 \\text{ or } m \\equiv 1 \\pmod 4 \\\\\n -\\left(\\frac{n}{m}\\right) & \\text{if }n\\equiv m \\equiv 3 \\pmod 4\n\\end{cases}\n", "dff20a8d732f12dbd822bc7e6cdd375a": "\\ Cz' = 2 * \\pi * (\\alpha + \\alpha 0) ", "dff21d5d0b4fcd59d0338bd96ad5e1da": "g\\left((a+b\\sqrt{2})+(c+d\\sqrt{2})\\sqrt{3}\\right)=(a+b\\sqrt{2})-(c+d\\sqrt{2})\\sqrt{3}=a+b\\sqrt{2}-c\\sqrt{3}-d\\sqrt{6}.", "dff220e5282ac9af7695f16d4cc66ef6": "u(\\mathbf{y}^1)=u(\\mathbf{y}^2)", "dff2264d5a350753cc3f458cc8e8f979": "\n\\begin{align}\n\\mathrm{E}(Y) & {} =\\sum_{k=0}^\\infty (1-p)^k p\\cdot k \\\\\n& {} =p\\sum_{k=0}^\\infty(1-p)^k k \\\\\n& {} = p\\left[\\frac{d}{dp}\\left(-\\sum_{k=0}^\\infty (1-p)^k\\right)\\right](1-p) \\\\\n& {} =-p(1-p)\\frac{d}{dp}\\frac{1}{p}=\\frac{1-p}{p}.\n\\end{align}\n", "dff247e4ed3ca00c1ce030e5019570bd": "\\dot y(t)\\in HY", "dff26a150647678d1ee1a90f9cfe0c98": "C=4\\pi\\varepsilon_0R \\,", "dff27a7a3e94f78c8af4d155f8bfa7b1": "f\\left(\\frac{az+b}{cz+d}\\right) = \\chi(d)(cz+d)^k f(z)", "dff2b6f2859163735762bc12241e75cd": "\\Pi^0(f_1,f_2)=f_1f_2", "dff2cbb9db009e99257d8b51ae610948": "R_\\text{a} = \\frac{1}{n} \\sum_{i=1}^{n} \\left | y_i \\right |", "dff345981c6ce921d73c1d563a822269": "\\{\\{a\\},\\{b,c\\}\\}", "dff3aa0773b63af0d34ab3c5ce9a113e": "\nK(a,b; m) = 2 \\left(\\frac{\\ell}{m}\\right) \\sqrt{m}\\cdot \\text{Re}\\left(\\varepsilon_{m} e^{2\\pi i \\frac{2\\ell}{m}} \\right).\n", "dff3c86c7e691fe176f9a5b44eacbbbd": "\n G = K_{\\rm II}^2\\left(\\frac{1-\\nu^2}{E}\\right) \\quad \\text{or} \\quad\n G = K_{\\rm II}^2\\left(\\frac{1}{E}\\right) \\,.\n ", "dff40c0b84f9a7d8f574fcb372894a05": " \\bar r_1\\ ", "dff4459099ad0e790f2699b232b938cd": "\\Gamma^{(\\lambda)} (R)_{nm}\\,", "dff468758dfdc9e700898e6eccf64dd9": "\n\\sqrt{{s_1^2 \\over n_1} + {s_2^2 \\over n_2}} \\approx 0.0485\n", "dff4af5c6f3562ec5bd246085ecfb5bb": "\\mathbf{F}_G", "dff4beccbd5be9dcb104027b291047ce": "\n\\begin{cases} \n\\{O_{1},O_{2},O_{3},O_{7},O_{10}\\} \\\\ \n\\{O_{4},O_{5},O_{6},O_{8},O_{9}\\} \\end{cases}\n", "dff4d2f5c9bffb1859957fcc15ffc096": "S_{\\lambda'/\\mu'} = (e_{\\lambda_i - \\mu_j -i + j}), 1\\leq i,j \\leq l(\\lambda)", "dff4e4515d6d08ed0a1fd402ea680f80": "\\operatorname{arg}\\, x = 2 \\pi / 3", "dff55335db7e2ffe69a04158420fa8fc": " \\tfrac{D\\rho}{Dt} = \\tfrac{\\partial \\rho}{\\partial t} + \\mathbf u \\cdot \\nabla \\rho = 0 \\Rightarrow \\nabla \\cdot \\mathbf u = 0 ", "dff5b3f7cd42661db1ed138226461c30": "\\mathrm {Sh}(X, \\mathcal C)", "dff6120548c0af1ad572d02babb0ef6e": "\\sqrt{\\log n}/n.", "dff6e05ec00b53d46e366975136d0d2e": "t \\in [0,1]", "dff6e0c26f3533c258f9322d525fa91c": "D = 7.5\\;\\mathrm{m} \\cdot \\frac {0.33} {0.33 + 0.66} = 2.5\\;\\mathrm{m}", "dff7905d2a3d10b2c5f09bcb7fdde412": "(x^2)^2 + (18)^2, \\,\\!", "dff7e1bbd1a2176dcade724878cd4fe5": "f(z)=g(\\bar{z},z)", "dff7f3221711c0b3efb80b651c1ebb54": "\\eta_0=\\sqrt{\\frac{\\mu_0}{\\epsilon_0}}", "dff8066f65a1e8d321dff76edc062532": "\\mathcal{H} = -\\sum_{i,j} \\mathcal{J}_{ij} \\vec{s}_i \\cdot \\vec{s}_j\\quad (2)", "dff8162ca2d4458f0158111066d39c51": "(16)\\quad \\Psi_2=-\\frac{M(v)}{r^3}\\qquad \\Phi_{00}=\\frac{M(v)_{\\,,\\,v}}{r^2}\\;.", "dff84fd6b211dda8056ca3c8ab5fddce": "q_1 \\equiv \\pm q_2^{\\pm 1} \\pmod{p}", "dff860d53c96b1657b5bb161b155a87b": "\\forall x\\,P(x)", "dff86260ca3ea2c75b72045763e3d9eb": "\\frac{1}{2 k_0 n} \\frac{\\partial^2 a}{\\partial x^2} + i \\frac{\\partial a}{\\partial z} + \\frac{k_0 n n_2 |A_m|^2}{2 \\eta_0} |a|^2 a = 0", "dff8682852bccbf2370bedf1ada1e597": "x = a {\\cos t \\over t}, \\qquad y = a {\\sin t \\over t},", "dff8d5a15a48497854cdf2a3b4a958ea": "N_l^k = \\left\\langle\\Psi_l^k\\left.\\right| \\Psi_l^k\\right\\rangle = \\left\\langle\\Psi_m^{(0)}\\left| \\left(V_l^k\\right)^+ V_l^k \\right| \\Psi_m^{(0)} \\right\\rangle", "dff8d7db60a9354774324dd691654a14": " : \\hat{b} \\, \\hat{b}^\\dagger : \\,= \\hat{b}^\\dagger \\, \\hat{b}. ", "dff8e06f0a3b9133d22396ff900bb446": "\\mathbb{Z}[\\zeta_n]", "dff94f91ac487969d254685d73b09c79": "\\scriptstyle \\mathit \\Gamma", "dff96554fbd6018bef550c141b4affd9": "\\bigwedge^{n-k} V", "dff96d915ae42e12678b83f58e85e026": "\\theta^\\prime", "dff971391660f158738618affdd98ed5": "5(n^2-n)+1 \\, ", "dff97597cc7ff5bbf878d92c1e1bc3ea": "2 < \\alpha < 3", "dff97fe795f91e00be935470de48fb20": "B_<", "dffa61aa0b19d906c582f27d26bf83f2": "Fe_2O_3(s) + 2 Al(s) \\rightarrow 2 Fe(l) + Al_2O_3(s)\\,", "dffaa38d7bfef856a3bba157488581af": "- \\frac{\\hbar^2}{2\\mu R^2}[ \\frac{\\partial}{\\partial R} (R^2 \\frac{\\partial}{\\partial R})- J(J+1)]F_s(\\mathbf R)+[{E'}_s(R)-E]F_s(\\mathbf R) = 0 ", "dffab4e7bc1fbb031a2f76bb228e7999": "X_{lc''}(\\bold{r}) = \\frac{1}{i^{n_{c''}}\\sqrt{2}}\\left(Y_l^m + Y_l^{-m}\\right)", "dffad635c92c43c34989477974797710": "\\int_{T_1}^{T_2} d\\tau = (T_2 - T_1) \\sqrt{ 1 - \\left ( \\frac{r\\omega}{c} \\right )^2}.", "dffae05e102297ba35d7f96fe90ff86b": "\\mathbf{T}(\\mathbf{X}) = \\langle X, \\tau\\rangle", "dffafe360708d2a4acfa2d76a90bf7f4": "|q|^2", "dffb91329d3417169e4ebcba1e025ecc": "D_4 \\to B_3", "dffba1c638e2b5c1a8e02e989b3f20b4": "I(1) \\leq I(2) \\leq \\cdots \\leq I(k)", "dffbfa995163e0c84be25c8247356099": " K = \\lim_{n \\to \\infty}K(n) \\approx 0.6072529350088812561694 ", "dffc05b77d3ac7b80bb539204118dba5": "A\\mid B\\Leftrightarrow B\\mid A,", "dffc8a647e90f12c318ddde114bc243a": "y=b \\left(a^2 d_a^{\\,'} d_b^{\\,'} - b^2 d_b d_b^{\\,'} + c^2 d_b d_c \\right)", "dffc9a0bc5deb39e3f5e164857d7535c": "\\doteq, \\doteqdot, \\overset{\\underset{\\mathrm{def}}{}}{=}, := \\!", "dffcd60368bdf9055d99390417148e65": "\\chi\\!\\,", "dffceb2fa68eb4e39f9031bb8e7a6ddc": "r=\\sqrt{\\xi_1^2+\\xi_2^2}, \\omega=\\arctan \\frac{\\xi_1}{\\xi_2}", "dffd2aacbdef904014140874ee86ad80": "\\begin{matrix} {13 \\choose 4}{4 \\choose 2} \\div 2! \\end{matrix}", "dffd464c38bf0970066ce31f6b86b5df": "\\{fg,h\\} = f\\{g,h\\} + g\\{f,h\\}", "dffd99bde441082757e9f82c415942a3": "C(\\alpha^{j}) = 0.", "dffddbfe2c5fc7f1d0501e08751a30dd": "O(\\log_bn)", "dffdee2c95f1e550d8c6b564216e376c": "{A}_{1+}", "dffdf7ced51e109bce501d12aface961": " \\Vert M \\Vert_2 \\leq \\gamma ", "dffe41ed8a4685ed4427fc2d7fc05674": "D_{2D} = \\frac{m^*}{\\pi \\hbar^2} \\ ", "dffeae588fbf377a383b09b173bed055": "v_{em}=c/n", "dffec94cb51c7626fafa3769585e9a44": "M\\left(|\\downarrow \\rangle \\otimes |O_{\\uparrow} \\rangle \\right) = 0", "dffef38fc86f7c50f20c2f78556d6806": "X\\,\\sim\\,\\textrm{Levy}(\\mu,c)", "dfff654710e6b66557b85f961caf0afe": "f_n(x) = \\begin{cases} 2n(1-nx), & \\mbox{if } 0 \\le x \\le \\frac{1}{n} \\\\ 0, & \\mbox{if } \\frac{1}{n} < x \\le 1\\end{cases}", "dfffae1b8aaf2fd8844e6c7df354332c": "\\frac{m}{100} = 1 - e^{-\\frac{n}{100}}", "dfffc0e97e834bb1a6e4ba97c4e27069": " \\sum_n S(n)^{-\\alpha} {S(n)!}^{-1/2} <\\infty\\, (\\alpha>1). ", "dfffc4ec20dbab5db71ad7bad00a4b75": "f: X \\times \\Omega \\rightarrow \\mathbb{R} ", "e0000bf0f3beab5abeb0fb5216c6bbb7": "f(\\mathbf{X}_i;\\mathbf{u}) = f(\\mathbf{X}_i;\\mathbf{v}^{(t-1)})", "e00021673dbd5259773e0fb0b932dd05": "O(nd)+O(\\frac{d}{\\epsilon^2})", "e00046cd5fade80c74dfac72c93e9198": "\\left(\\beta\\right)", "e00059eba0ebdadf4615eefcfa9e04b3": "\\frac{dN_B}{dt} = \\lambda_A N_A - \\lambda_B N_B", "e00072d25ab3dd5ba05634a92e334d37": "p_{1} \\otimes p_{2} \\otimes", "e0008778bcc7061bdaf6a665198cf075": "\\sqrt{xy} = \\sqrt{x}\\sqrt{y}", "e000b04ffc3662e85231319f7a91b183": "\\mathrm{coim}(f) = \\mathrm{coker}(\\ker f)", "e000e45a59a2de99c0b0e855428a6ee8": "y=\\infty", "e00138a199c47cd886c51ce923bfa6a8": "x^5 + d_1x + d_0 = 0\\,", "e001a4a4a91bd074796f22c2ed6e5c65": "\\scriptstyle{\\varkappa_{\\alpha}^{\\beta}}", "e001a87554bdec82dcd453079d434522": "B_\\alpha=\\left\\{\\psi\\in BV([0,1]);\\Vert \\chi_\\alpha - \\psi \\Vert_{BV}\\leq 1\\right\\}", "e001ad21616bdbdf2ad24746e9360fa6": "\\mathbf{R}_{x}^{-1}(n)", "e002118514c82fa06c3cf11c6b854e52": "\\text{WMA}_{M} = { n p_{M} + (n-1) p_{M-1} + \\cdots + 2 p_{(M-n+2)} + p_{(M-n+1)} \\over n + (n-1) + \\cdots + 2 + 1}", "e00232139cb6d86d61819124282cdbd7": "U = a' X", "e0024641df802a5fa71cbe95cba81d11": "(b_{14}*c_{14})*(a_{15}+c_{15}) ", "e00285e8b24e9f9fd1b5684e2f6939fa": "\\operatorname{MSE}(\\hat{\\theta})=\\operatorname{E}\\big[(\\hat{\\theta}-\\theta)^2\\big].", "e00297b2793c53608d0f3cec16766480": "X=X(t)=\\frac{x(t)}{x(t)^2 + y(t)^2},\\ Y=Y(t)=\\frac{y(t)}{x(t)^2 + y(t)^2}.", "e002dd740599461c4ee3f3b9818d2460": "\\delta_i=\\prod_{k\\neq i}(l\\alpha_i-l\\alpha_k)", "e0035a31be8c5b212bcf3d664d4ed749": "100/.50 = 200", "e00368e765d42c76b5f3e9786d1e1036": " {d^2 x^n \\over dt^2} =- \\Gamma^n {}_{00}.", "e00372be984df3855aa98279f4185796": "\\mathcal{L}_{V_H}(\\omega) = d\\omega(V_H) = 0", "e004154270585fd8ef80f2a5ebd2bdd0": "y_{R}\\left(t\\right) - y_{L}\\left(t\\right)", "e0043352777526dd090b1526563694a9": "t_{LL}^{\\mu \\nu} + \\frac{c^4\\Lambda g^{\\mu \\nu}}{8\\pi G}= \\frac{c^4}{16\\pi G}((2\\Gamma^{ \\sigma }_{\\alpha \\beta }\\Gamma^{\\rho }_{ \\sigma \\rho }-\\Gamma^{ \\sigma }_{\\alpha \\rho }\\Gamma^{\\rho }_{\\beta \\sigma }-\\Gamma^{ \\sigma }_{\\alpha \\sigma }\\Gamma^{\\rho }_{\\beta \\rho})(g^{\\mu \\alpha }g^{\\nu \\beta }-g^{\\mu \\nu}g^{\\alpha \\beta })+", "e0045c18b4282209236f97c12d37c5ad": "{2^{2^{2^{2^{2}}}}}-3", "e0048a4ffb61dec961b76bb715e1cc52": "\n\\log(1+x) = \\frac{x^1}{1} - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\cdots \n=\\cfrac{x} {1-0x+\\cfrac{1^2x} {2-1x+\\cfrac{2^2x} {3-2x+\\cfrac{3^2x} {4-3x+\\ddots}}}}\n", "e004aa7e818d21beff28912f99d62e72": " W_n=\\sum_{d|n}dX_d^{n/d}.", "e004d5295ba309ee32e67be1154c8a07": "q_\\alpha([x]) = \\inf_{x\\in [x]} p_\\alpha(x).", "e005158ecbe97955fe5c92057c21cd98": "x=c+2v\\,\\!", "e0053291005365b1537dfcdc73a18c14": "D_0(n) = n! [z^n] q(z) =\n\\frac{1}{2} n! \\sum_{k=0}^n \\frac{(-1)^k}{k!}\n+ \\frac{1}{2} n! \\frac{1}{n!} - \\frac{1}{2} n! \\frac{1}{(n-1)!} ", "e00571e61bcc598084583aa7f94ef0ed": "\\forall\\varphi\\in\\mathcal{C}, \\ \\ \\mathcal{S}[\\varphi]\\equiv\\int_M \\mathrm{d}^nx \\mathcal{L} \\big( \\varphi(x),\\partial\\varphi(x),\\partial\\partial\\varphi(x), ...,x \\big).", "e00571f3b8b882bbcf5424b2bd604f71": " F(k) = 1 + \\frac{\\Beta(p; k+1,0)}{\\ln(1-p)}", "e005d6bc7b4ec7610faf493d44f6715b": "\\hat{\\mathcal O}_{Y,y} \\to \\hat{\\mathcal O}_{X,x}", "e0064b9f3e8d2b81aec9ac6023b9ed5d": "\\scriptstyle \\ a/h = h/b ", "e00651955a1a3ac024ace87e76b10870": "A=\\{A_i\\}_{i\\in I}", "e0067ea8fed5882d830dbcda1122f776": "\\neq", "e006ddf7da853d7ce821fc07ffa68ab0": "\\mu_{\\operatorname{eff}} = { {F \\left( \\frac {\\partial u} {\\partial y} \\right)}\\bigg/{\\frac {\\partial u} {\\partial y}}} ", "e007ebef1125c04533ad1d650bc6b47c": "\\psi_i(r)=\\int_0^\\infty\\phi_i(r)\\frac{\\sin(qr)}{qr}", "e007f5c99eaeffa37c625ec43f7ad81b": "x=\\{x_1,\\dots,x_n\\}", "e00828967bfbf29f0e76d71a378ed1eb": "\\left\\{\\frac{1+x_2}{x_1},\\frac{x_1+(1+x_2)x_3}{x_1 x_2},\\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3}\\right\\},", "e00828e4efc386f4ba74a62ba44a19ce": " \\omega \\, ", "e0083d210605f6100a080571d0a78b08": " \\log_b\\!\\left(\\!\\sqrt[y]{x}\\right) = \\begin{matrix}\\frac{\\log_b(x)}{y}\\end{matrix} ", "e0089a4dd39577f8380a035950a0ba99": "(n-1)! = \\frac{n!}{n}.", "e008c9f7206cf329d28ae80442cec7f7": "k_i=\\begin{cases} 1 \\ \\ \\qquad \\text{if the} \\ i^{th} \\ \\text{coefficient of} \\ \\textbf{f} \\ \\text{and} \\ \\textbf{g} \\ \\text{is} \\ 1 \\\\ -1 \\qquad \\text{if the} \\ i^{th} \\ \\text{coefficient of} \\ \\textbf{f} \\ \\text{and} \\ \\textbf{g} \\ \\text{is} \\ -1\\\\ 0 \\ \\ \\qquad \\text{Otherwise}\\end{cases}", "e008dc00cd514ce204d0999cb1014979": "\n\\Delta g\\ =\\ \\int\\limits_{t_1}^{t_2}\\left(\\frac{\\partial g }{\\partial v_1}\\ h_1\\ + \\ \\frac{\\partial g }{\\partial v_2}\\ h_2\\ + \\ \\frac{\\partial g }{\\partial v_3}\\ h_3 \\right)dt\n", "e009341616390e4288b5b1d6a5b64d19": "a \\mapsto -a", "e0095a3c0b7da7940c6a483bdf65fe8f": "T_{11} = P(r) g_{11} = - P(r) e^{\\lambda(r)} \\;", "e0098039890a6435c2defae3494e4535": "\\begin{align}\n \\sinh(x + y) &= \\sinh (x) \\cosh (y) + \\cosh (x) \\sinh (y) \\\\\n \\cosh(x + y) &= \\cosh (x) \\cosh (y) + \\sinh (x) \\sinh (y) \\\\\n \\tanh(x + y) &= \\frac{\\tanh (x) + \\tanh (y)}{1 + \\tanh (x) \\tanh (y)}\n\\end{align}", "e009a1b285d33d05cf7a4fc852a0dd2c": "\\frac{1}{|B|} \\int_{B} \\omega^q \\leq \\left(\\frac{c}{|B|} \\int_{B} \\omega \\right)^q", "e009a4cde8d40d0b4f3f154ccf24dd1f": "n = p \\cdot q \\cdot r", "e009d0388b7811107d92d0946862a335": "\\cos \\omega_{p}t", "e00a1cf3e30f9cdfe98fa86474f38b9f": "j1 ", "e0144a4f1ff1a6df23fb645f1037d11c": " \n- \\pi /2 + \\epsilon < \\arg ( f_i ) < \\pi / 2 - \\epsilon, i \\geq 1. \n", "e01484fedfc8fe183fa8ee55c1c616bf": "m = 2a_y + 10", "e0148c8962572e05f0cfcdc1ed8c54a1": "3\\theta", "e014bb372bfe8a3b3c1ab67de5fe1036": "\\frac {v^2}{2}+ gz+\\left(\\frac {\\gamma}{\\gamma-1}\\right)\\frac {p}{\\rho} = \\text{constant}", "e0153ccf7979b6f82bdab6a811805daa": "2l", "e01568bca82bb1cadd38a49b02b2670f": "0 \\leq A \\leq B < \\infty", "e01584141eecf501e8cffa2a83dd0caa": "\\sum_{n=0}^\\infty\\frac{u^n}{n!}\\Delta^na_i = e^{-u}\\sum_{j=0}^\\infty\\frac{u^j}{j!}a_{i+j}.", "e0166e47b0efe039b3dd8357a7b9040c": "{i}^2 = -1", "e01670f12091fc199e214d04e884d31d": "\n\\oint_{C(\\varepsilon, R)}\\frac{\\exp\\left(ikz\\right)}{\\exp\\left(z\\right)-1} \\, dz = 0.\n", "e016746938d3759cbecb043701ad96de": "\\dot{x} = v(x), ", "e01684f5c5c1411a1c69d0b009a93510": "u_{2,1}^{0} = \\delta_{2}^{0}(1) = 3", "e016be761d28a4fe918c99b85dc38b29": "t + C_2 = C_1 \\int e^{-\\int f(x) dx} dx", "e016d347dc6b8e50dfc539332918ea29": "M: \\mathbb{R}^n \\to \\mathbb{R}", "e0176ae412d5627a2d5854a1bb02f719": "x_1,\\ldots,x_n\\,\\!", "e01778ca4681d9bab2e61337fa3508f2": "\\operatorname{Var}(X) = \\sum_{i=1}^n p_i\\cdot(x_i - \\mu)^2 = \\sum_{i=1}^n (p_i\\cdot x_i^2) - \\mu^2", "e017790122d3103ccfb97b7e64481f9b": "(Z_1/X_1, Z_1/Y_1)", "e017b345dd6ae6d71fb48c40259e6da0": "\\mathrm{On}(\\mathrm{box},\\mathrm{table},t)", "e017c480c0ee78db9b1374f074c79173": "E = \\frac{1}{2} \\rho A v^3 t", "e017ce3c7f94a12ec0dc880d2cc5eebc": "n-l\\ge 0", "e018005f2c8aa13a01d453529b903959": " M = \\frac{r_+^2 + r_-^2}{l^2},~~~~~J = \\frac{2r_+ r_-}{l} ", "e0180a30da7e9874987644920dd7216f": "\\bigvee P = \\bigvee \\{ x \\in L \\mid x \\le f(x) \\}", "e01892fe008e63ac76d32dcfe987a857": "x \\in S \\Rightarrow T(x) \\in S.", "e018b6d88003cbcec117020404fd4706": "r(Q) +r(O_{r})", "e01956a656b506f7fb52bc8d48f58d4b": " \\mathrm d\\varphi_x\\colon T_xM \\to T_{\\varphi(x)}N.", "e0198cfa66098a948e30864a8d185239": "C^\\perp ", "e01990442452a3e9abe3e330e42de884": "x(t) = 4t^2 \\,", "e019e761c6ef0620b5f2140e292ade52": "a^2\\equiv b^2\\pmod p", "e01a0e9203828cee5b264900c1cc69a1": "\\mathbf{B} = \\mathbf{P}\\mathbf{A}\\mathbf{P}^{-1}", "e01a2b82258a3e84d0029669bc050b5d": "M\\equiv C^{d} \\bmod\\ N ", "e01a7b33425eec94b43c3b6b2eb462bd": "E_{xc}^{\\mathrm{LDA}}[\\rho] = \\int \\rho(\\mathbf{r})\\epsilon_{xc}(\\rho)\\ \\mathrm{d}\\mathbf{r}\\ ,", "e01a8a45ce7930deb2f9835f2d4cfcf5": "\\mathfrak{p}\\,", "e01a8a91467e7ae520574c9899b7aeb6": " \\ell\\le \\int_0^1 \\rho(i\\,y+w\\,t)\\,w\\,dt .", "e01aa05c9ccb224bbf1d5c159a06f43d": "\nI_n = \\begin{bmatrix}\n1 & 0 & \\cdots & 0 \\\\\n0 & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & 1 \\end{bmatrix}.", "e01ae7c5e9cbc311e3832343a27a6e1b": "\\bar \\partial_{j, J}", "e01af34d347d342380e3a424f21c1bd4": "\n\\Phi_{G} = - k_{B} T \\ln(\\Xi) = - k_{B} T Z_{1} e^{\\beta \\mu}\n", "e01b18f82cf354e0d8081d0863c77f3b": " max(0, S-K) ", "e01b4b9355187347c900bb9668830dca": "\\varphi(N)", "e01b69b1fd246c0c66e133bfbfcf1485": "i\\hbar\\frac{\\partial}{\\partial t} |\\Psi\\rangle = E_{\\Psi}|\\Psi\\rangle", "e01b7aa4075fac841a119e226e95ba70": "p_0 \\leftarrow r_0\\,", "e01bbc1f5edc791591e54a3b87268e44": "X^{a}-Y^{a} = (\\xi^{\\mu},0)", "e01c294f65bc4ffd02c4995e6dfecadb": " [\\mathfrak f,\\mathfrak f]\\subset \\mathfrak h, \\qquad [\\mathfrak f,\\mathfrak h\n]\\subset \\mathfrak f. ", "e01c779a9cae7729c99e2cf1841ffdb4": "A:=\\,\\!", "e01c77f4706ccb104f865709ed775380": "\\forall X_0,X_1\\in Ob(C). \\forall Y_0,Y_1\\in Ob(D). \\forall x:X_0\\to X_1. \\forall y:Y_1\\to Y_0.", "e01c7ec0b84025a00e066aa5d812309f": " \\cos \\theta = 1 - \\frac{\\theta^2}{2!} + \\frac{\\theta^4}{4!} - \\frac{\\theta^6}{6!} + \\cdots", "e01c8420d22a6af9624658a1306b9734": " {\\mathbf{u}}(t)= -L(t) \\hat{\\mathbf{x}}(t).", "e01cac4541d5d64e2498f6dd28c1e413": " a(t)=a_0\\,t^{\\frac{2}{3(w+1)}} ", "e01cfd4b273a1339e507241f11542e8a": "f_x = s_xb_{x+1}.", "e01d48c6b1d22a3a7a2e99e3f0e55539": "P_0=\\frac{K}{1+Ae^0}", "e01d661ada9faf32d116956a355d0ca4": "\\textit{animal} \\subseteq \\textit{creature}", "e01da06ab897e5746c1a9cad276cfb42": "x \\wedge y \\wedge z \\wedge x = x \\wedge z \\wedge y \\wedge x. (N ) ", "e01db94e80aaebc6abf4d0822064cda6": "\\operatorname{Pr}_Y", "e01dbf326cfc24e74993c5f94ef67176": " e^{it} = \\frac{1+iu}{1-iu},", "e01dc7cdf3f9030c9af5bd52cd30412b": "\\frac{dS}{dt} \\ge 0", "e01dd0cdf962887c60dc786d12adc070": "\\mathbf{K}=\\begin{bmatrix} k_1 & k_2\\end{bmatrix}", "e01dd1f90b2025b89ff2733857434efd": "\\textrm{ad}_x([y,z]) = [ \\textrm{ad}_x(y) , z] + [y , \\textrm{ad}_x(z)]", "e01df8a2693572ac1282ad2854582163": "\\alpha = \\frac{k}{\\rho c_p}", "e01e10ead55833d54f1bcbb3617f66b3": "\\textstyle \\mathbf Z^d", "e01e3d6e1ab3d0b8116efcd572fdab2a": "rK+wL\\,", "e01e4b585eb9cc179edccd09076cfe61": "\n\\nabla^2 \\psi +\\alpha\\left(\\frac{\\partial \\psi}{\\partial x}\\right)=\\gamma \\sin \\left(\\frac{\\pi y}{b}\\right)\\qquad (5)\n", "e01ea81e89f8808658ea6128a3415c56": "\n a_\\text{map} = \\frac{256 \\times 2^{\\text{zoom Level}}}{2 \\pi} \\ \\text{pixels} .\n", "e01eb72a5964b4bb76d2411a62bbc201": "\\ \\sin(2x) = 2 \\sin(x) \\cos(x) ", "e01ed3735329ab9a6a73f5c6a0140ecb": "\\scriptstyle P_1,\\, P_2", "e01f151d01cf567def4a32181b667573": "{\\rm tr}(A)={\\rm tr}(A^T).\\,", "e01f184704d281026980fef4b5edd3f7": "\\text{price}_{\\text{yesterday}}", "e01f2ca97e4743549c5a298079b47662": " \\and (S_6 \\implies (\\operatorname{equate}[A_6, n] \\and V[y] = n)) \\and D[y] = D[n]) ", "e01fd45bf10305e41d37459223778301": "stx", "e0200e87dff5eb195850667ca8bb61ea": "\n\nF^{-1}(y) = \\sgn(y) \\begin{cases} {|y| (1 + \\ln(A)) \\over A}, & |y| < {1 \\over 1 + \\ln(A)} \\\\\n{\\exp(|y| (1 + \\ln(A)) - 1) \\over A}, & {1 \\over 1 + \\ln(A)} \\leq |y| < 1. \\end{cases}\n", "e02014c899472587b4909e9f168d6b03": "Y \\to Z \\to X' \\to", "e0207b9335ee97bc9ce3b66107cf9231": "\n \\dot{u}_i = \\frac{\\partial u_i}{\\partial t} ~;~~ \\ddot{u}_i = \\frac{\\partial^2 u_i}{\\partial t^2} ~;~~\n u_{i,\\alpha} = \\frac{\\partial u_i}{\\partial x_\\alpha} ~;~~ u_{i,\\alpha\\beta} = \\frac{\\partial^2 u_i}{\\partial x_\\alpha \\partial x_\\beta}\n", "e020cb94eba1171f1ea0a9fe4811b4ce": "\\frac{1}{22} + \\frac{1}{33} + \\frac{1}{66} = \\frac{1}{11}", "e0214c19f1e10e28cee668c6cd193583": " \\dot x(t) = u(t), \\quad x(t_1) = x^1, \\quad t_1 \\le t \\le t_2 ", "e0218ad3799f7938df5c75bf99175901": " \\mathbf{v} = \\nabla \\times \\mathbf{A}. ", "e021946e709666b6621cbf9acaef080c": "C = \\pm e^B", "e0219b366e7a521815b78355a7d71e4c": "\\mathcal{B}\\subseteq \\mathbb{F}", "e02220b42fbef907c905f6aeb56464ba": "(4,2,1)\\oplus(\\bar{4},1,2)", "e022f69122d1e83ce458622e3bece249": " {r}_{k} ", "e02315935db7078e4020886b7dad0c6d": " \\iint K(x,y)g(x)g(y)\\,dx dy \\geq 0. ", "e023f7f50dba987ca1d11780f9a57ba0": "G_x = \\{g \\in G \\mid g\\cdot x = x\\}.", "e0242df84dca4a4bc4a222055a9806db": "\\mathrm{d}\\omega(V_0,...,V_k) = \\sum_i(-1)^{i} V_i\\left(\\omega(V_0, \\ldots, \\hat V_i, \\ldots,V_k)\\right)", "e02477e0de01238fbf2a528006e73a3e": "\\ C_4^3 (3)=\\frac{17}{49}\\ \\ldots", "e02513d4873474d7333b7e9e399559b3": "d(v_1, v_2)", "e025580e2d4d92a0864bcce20cd7eba5": "T_{coh}", "e0255bfd8869b58052f559c0fbe7161c": "\n\\begin{align}\nW & = N!\\prod_{i=a,b,c,...}^k \\frac{1}{N_i!}\n\\end{align}\n", "e025ae6c804ca1dc0c6f932866041730": "-\\Lambda", "e025d8a9c0000181609af165bd8ed950": " \\psi^{(\\operatorname{Sha}) }(t) = \\operatorname{sinc} \\left( \\frac {t} {2}\\right)\\cdot \\cos \\left( \\frac {3 \\pi t} {2}\\right)", "e025ffb8724ba352b5641581d960def6": "(\\tau \\succ_C \\sigma \\iff \\sigma \\supset \\tau", "e0264647690a22748a18a4addebb430d": "f : (\\mathbf{R}, \\mathcal{L}) \\to (\\mathbf{C}, \\mathcal{B}_\\mathbf{C})", "e02666d7c948846a9537dd1baab73f75": "\n\\overline{m}_n=\\frac{1}{N}\\sum_{i=1}^N z_i^n.\n", "e0267c0f4b0796449ef519addee38df7": "k^1", "e026a131d94f08e453a9302e14bcb9d4": "\\mathfrak P_n(K)", "e026b77c377d9b3f0efc93e3252d7064": "\\frac{\\mathbf{\\hat{p}}^2}{2m} \\psi = \\hat{E}\\psi", "e026d3f6a93d66f75aa46e2520527134": "\nT = \\frac{dq_\\mathrm{rev}}{dS}\n", "e0270ff19ac93d4813852ae7855a4443": "F(x; a,b)=1-(1-x^a)^b.\\ ", "e0272740a423e5511c068401c36bcbaa": " z=D_i .", "e02728fcc21e12e8fff3158378dc5f88": "q_{and}", "e027b3c2b31b7272b9dd85d6011ed502": "y_2'=v'(t)y_1(t)+v(t)y_1'(t)\\,", "e027e0c41837d35b071c050f1127978e": "\\frac{\\int_{-\\infty}^{\\infty}{L(\\delta(x)-\\theta)f(x_1-\\theta,\\dots,x_n-\\theta)d\\theta}}{\\int_{-\\infty}^{\\infty}{f(x_1-\\theta,\\dots,x_n-\\theta)d\\theta}},", "e027f64b1c9407b26f71824e29e2104c": "I_\\mathrm{C}", "e0287d09b6a47a35d3424e86b5fb8eb5": "A=(V,R^A_1,\\ldots,R^A_n)", "e028c4ef19b04d57e26b39af281ab506": "w \\leq w'", "e028d917fa7c1656694bcd3ad493bd2c": "\\langle \\mathbf{s}_i\\cdot \\mathbf{s}_j\\rangle_{J,2\\beta}\n\\le \\langle \\sigma_i\\sigma_j\\rangle_{J,\\beta}", "e02a127f67994e6e98911b5667451890": "\\exists x (P \\land Q(x))", "e02a3aee58616b52c90fc0691d1ece02": "(g,e')N = (g,e')(p_1(N) \\times \\{e'\\}) = gp_1(N) \\times \\{e'\\} = p_1(N)g \\times \\{e'\\} = (p_1(N) \\times \\{e'\\})(g,e')=N(g,e')", "e02aa1940a4d5176f62e5f2679a0d171": "p_{X^{n}}\\left( x^{n}\\right) ", "e02ac127b985fb78c404444ac275e725": " K^M_*(F) := T^*F^\\times/(a\\otimes (1-a)), \\, ", "e02b14c81d4ca63fc552894c7761c60b": " (\\mathbf{y}')^{T} \\, \\mathbf{F} \\, \\mathbf{y} = 0", "e02b4e67ef15ef75019913d07c58a85c": "\\int \\cos^n x \\, dx = \\frac{\\cos^{n-1} {x} \\sin {x}}{n} + \\frac{n-1}{n} \\int \\cos^{n-2}{x} \\, dx", "e02bc517e96628c02f7154b5517172e0": "\\sigma^2 = \\sigma_{mn} \\, \\sigma^{mn}, \\; \\omega^2 = \\omega_{mn} \\, \\omega^{mn}", "e02bd8ebfd12998291f84a23c5280b7e": "V=\\sqrt{ 2 \\mu g d_{skid} } ", "e02bfdbe27ae7c4bd308ce031dfde77b": "-13.6 \\,\\text{eV} = -\\frac{m_{\\text{e}} e^4}{8 h^2 \\varepsilon_0^2},", "e02c45d363f220b173e0dee3250c6fc7": "\\mathcal{J}", "e02c61f00d3d1ebf3c0c075dd63e6694": "\\mathcal{S}_1", "e02c659e06a09d25d7537892fb3b5515": "\\boldsymbol\\beta^{k+1}=\\boldsymbol\\beta^k+\\Delta \\boldsymbol\\beta", "e02c6c02cff0241876fe2b10f98771e3": "\\begin{align}L_n^{(\\alpha)}(x)- \\sum_{j=0}^{\\Delta-1} {n+\\alpha \\choose n-j} (-1)^j \\frac{x^j}{j!}&= (-1)^\\Delta\\frac{x^\\Delta}{(\\Delta-1)!} \\sum_{i=0}^{n-\\Delta} \\frac{{n+\\alpha \\choose n-\\Delta-i}}{(n-i){n \\choose i}}L_i^{(\\alpha+\\Delta)}(x)\\\\[6pt]\n\n&=(-1)^\\Delta\\frac{x^\\Delta}{(\\Delta-1)!} \\sum_{i=0}^{n-\\Delta} \\frac{{n+\\alpha-i-1 \\choose n-\\Delta-i}}{(n-i){n \\choose i}}L_i^{(n+\\alpha+\\Delta-i)}(x).\\end{align}", "e02cb3f39d4d91304e2a2d69f4c5fd11": "\\pi/4=\\left(\\prod_{p\\equiv 1\\pmod 4}\\frac{p}{p-1}\\right)\\cdot\\left( \\prod_{p\\equiv 3\\pmod 4}\\frac{p}{p+1}\\right)=\\frac{3}{4} \\cdot \\frac{5}{4} \\cdot \\frac{7}{8} \\cdot \\frac{11}{12} \\cdot \\frac{13}{12} \\cdots,", "e02cb6b1bb2f1f77b5eb225941e6928a": "Z = Z_{L} + Z_{C}", "e02d67a4a95cc663c2792e4735ba8da3": " \\int_V \\mathcal D \\phi \\; e^{-\\langle \\phi|S|\\phi\\rangle} = \\prod_i \\frac1{2\\sqrt{\\pi\\lambda_i}} = \\frac N{\\sqrt{\\prod_i\\lambda_i}} ", "e02d83f5c2f0d10c5d3a93589e13cf27": "\\neg p?;b\\,\\!", "e02dd28953fef75a2d2e7b0c3eb4d390": " F_{\\mathbf{g}}=\\sum_{i} f_i e^{-2\\pi i\\mathbf{g} \\cdot \\mathbf{r}_i}", "e02e069bbc1de89c88b60a887109ee0c": "h(x_1,\\dots,x_n)=f(g_1(x_1,\\dots,x_n),\\dots,g_m(x_1,\\dots,x_n)),", "e02e232aa38603127b07386663360274": "p_k^* = v_k^* A \\left(1- P_k \\right) A^{-1}", "e02e31d85ccb1cd032251c041e580a73": "[L_A]_\\beta^\\gamma=A", "e02e899d84c147e9db312a2e6cd0f3b8": "\\pi(\\delta_{A})\\leq \\pi(\\delta_{B})", "e02ee47a7e548486b35777daa19c0ee9": " \\operatorname{E}\\left[\\epsilon^T\\Lambda\\epsilon\\right] = \\operatorname{tr}(\\operatorname{E}[\\epsilon^T\\Lambda\\epsilon])", "e02ee47b9d2b68b1a22b1b5ddecf77a8": "e(M) \\in H^4(M;\\mathbf{Z}_2)", "e02ef6bda75454fa0620bce3674b4385": "P(\\text{negative}|\\text{ill})=1%\\text{ and }P(\\text{positive}|\\text{ill}) = 99%. ", "e02f0fff7f4764c27639187e185aacac": "\n\\mathcal{L}= \\overline{\\psi}(i\\partial\\!\\!\\!/-m)\\psi -\\frac{g}{2}\\left(\\overline{\\psi}\\gamma^\\mu\\psi\\right) \\left(\\overline{\\psi}\\gamma_\\mu \\psi\\right),", "e02f11661282f153f187036dca8e3809": "{\\mathbf z}^T\\mathbf{X}{\\mathbf z}\\sim\\sigma_z^2\\chi_m^2", "e02f44e9402391c00b7c8572e88fb762": " r = {k_1 [A]^s[B]^t} - {k_2 [X]^u[Y]^v}\\,", "e02fc2a46b4cdbe8227129846754d1c8": "Q = 2{\\pi}\\left ( \\frac{E}{{\\delta}E} \\right )", "e02fe04213072f4651d662666b719b39": "H(x) = \\prod_{n\\ge 1}\\frac{1}{(1-x^{5n-3})(1-x^{5n-2})}", "e0301a39525bb078270e9dc16ce73a65": "\\Delta\\otimes\\bar{\\Delta} \\cong \\sigma_-\\Gamma_0\\oplus\\sigma_+\\Gamma_1\\oplus\\dots\\oplus\\sigma_\\pm\\Gamma_k", "e030d6ceee3bb64f69875882b9e79453": " t_{ik} ", "e0311c6077ae54c39e9888746d8172ae": "-1 \\leq \\frac{x^2}{x^2+y^2} \\leq 1", "e0312ee142fd99997a3f2a8584181d22": "c_\\eta(a,b)=\\frac{\\mathrm{sinh}\\beta b}{2b(\\mathrm{cosh}\\beta a-\\eta\\,\\mathrm{cosh}\\beta b)}", "e031910cffaadfcda0ee9ef4b7c4233e": " \\mathbb{N}, x_{1} ... x_{n}", "e03216fdf457bf96d3fd2bf578c9d949": "v^2 = v_0^2 + 2 a s \\,", "e0323214a79c98cb4c6de53bd9667c10": " \\succeq 0 \\; \\forall \\; 1 \\leq k \\leq p", "e03280b9bb1244248ecf4e05a3c08844": "(A, \\| \\cdot \\|)", "e032b6c6f94078cfdab4c4833ff4e414": " F_i=\\frac{1}{d} \\Pi_i,", "e032d3a9fcf6119d8f06d84ae893cb97": "y_O=e^{1-t}\\,", "e0330d7a2b5bccb280b186e7c899ac43": "\\sum^{\\infty}_{i=1} f_i g_i = \\int^T_0 g(t)f(t)dt", "e0332654dbfdf7b364340dfa9645ae66": "E_v = A_v + T_v \\,,", "e033447cbf19010fef024ecd16ac65bb": " Z = i \\omega L \\,", "e0338952dbb3cd3333323d879653328c": " V|n^{(0)}\\rangle = \\Big( \\sum_{k\\ne n} |k^{(0)}\\rangle\\langle k^{(0)}| \\Big) V|n^{(0)}\\rangle + \\left(|n^{(0)}\\rangle\\, \\langle n^{(0)}|\\right) V|n^{(0)}\\rangle \n", "e033a85b9ed39e6190d492ef02225386": "(A^+A)^* = (V\\Sigma^+U^*U\\Sigma V^*)^* = (V\\Sigma^+\\Sigma V^*)^* = V(\\Sigma^+\\Sigma)^*V^* = V(\\Sigma^+\\Sigma)V^* = V\\Sigma^+U^*U\\Sigma V^* = A^+A", "e033fde6b6a2a93f2cc6a33c0014b758": "\\tau_{R^\\omega}(x)=lim_{n\\rightarrow\\omega}\\tau(x_n+I_\\omega)", "e03413d126a759d185795003238e00ae": "s \\models_K \\mathcal{P}_{\\sim\\lambda}(\\square f)", "e0345290be73993800db8fb119617799": "\\frac{dF(P)}{dP}=\\frac{F(P_1)-F(P_0)}{dP}=F'(P)=G(P).\\,\\!", "e03462e07d8b9557e4d1d0c2e893efc0": "\\varphi = \\arcsin \\left[\\frac{\\theta + \\sin \\theta \\cos \\theta + 2 \\sin \\theta}{2 + \\frac \\pi 2}\\right]", "e034bc0810d923be240c3a778daeb0d8": "\nb =\n\\left[\\begin{array}{l}\n -\\Delta x^2 g_{22} + u_{12} + u_{21} \\\\\n -\\Delta x^2 g_{32} + u_{31} ~~~~~~~~ \\\\\n -\\Delta x^2 g_{42} + u_{52} + u_{41} \\\\\n -\\Delta x^2 g_{23} + u_{13} ~~~~~~~~ \\\\\n -\\Delta x^2 g_{33} ~~~~~~~~~~~~~~~~ \\\\\n -\\Delta x^2 g_{43} + u_{53} ~~~~~~~~ \\\\\n -\\Delta x^2 g_{24} + u_{14} + u_{25} \\\\\n -\\Delta x^2 g_{34} + u_{35} ~~~~~~~~ \\\\\n -\\Delta x^2 g_{44} + u_{54} + u_{45}\n\\end{array}\\right].\n", "e0350c281b958f67599923f79ba24dbc": "\\pmod k", "e035a5e77fc6398626c05716bcd2aa65": " 01(01)^* \\cup 100^* ", "e035bafeb2deedd236c480d85c7b2124": "\\scriptstyle \\otimes", "e03604ebbb448e965f93ea40f167a3be": "\ndA = \\left( \\sigma^{2} + \\tau^{2} \\right) d\\sigma d\\tau\n", "e0368f56036f46e12636c53407b8f6d2": "\\pm 2w \\pmod l", "e037507af24a3602d0a009295a20029b": "R < C", "e03768231199c6a8bf3902077c460207": "\\scriptstyle R_s", "e03812655ef1c7cac348c0dcf83eb2d1": "\\textstyle{\\sum_{i=1}^k p_i = 1}", "e0387f0ad232d328cffbc294847ca93d": "T= \\mathbf{Gr}(r, \\mathcal E)", "e038a5b9e6e88a257b6bece9184d5e80": " pq = uw + vz + (uz + vw) \\omega .\\!", "e038c2180f0efe450f058640acf18cfb": "\\bold k\\perp \\bold B_0", "e0399e0ebc2068f80e6fba21a087d4ec": "\\,E(Z)", "e03a0ed4eb2f23f3b53688fb937ee289": "P(\\mathbb{N})", "e03a55ffc35e473b100c5b1d368cc799": "\\lambda_B \\approx 0.7 \\mbox{nm}", "e03a6652f695539aa92bca3722a85168": "\\begin{align}\n\\sigma_\\mathrm{oct} &= T^{(n)}_in_i \\\\\n&=\\sigma_{ij}n_in_j \\\\\n&=\\sigma_1n_1n_1+\\sigma_2n_2n_2+\\sigma_3n_3n_3 \\\\\n&=\\tfrac{1}{3}(\\sigma_1+\\sigma_2+\\sigma_3)=\\tfrac{1}{3}I_1\n\\end{align}\n\\,\\!", "e03aab0a27a65cf466e5c3e41a25578e": "O_{bef}", "e03b5af981be71c0249c98351cd63773": "1-2\\,", "e03b7b294f7fdab436cb8fa5e4503b43": "\\gamma=\\frac{1}{\\sqrt{ 1 - (v/c)^2 }}", "e03b849651b7c7abbe98fbdc41045a87": "\\binom nk=\\binom{n-1}{k-1}+\\binom{n-1}k,\\text{ for }0i}_n = \\wedge^{m>i+1}_n", "e03c6afe84d5f43350892f1310b2ba55": "\\prod_{p \\text{ prime}} \\frac{1}{1-p^{-s}}.", "e03c746d4878ecef07314a60b5e9188b": "\\sum_{i=1}^{n-1} \\left\\lfloor \\frac{im}{n} \\right\\rfloor = \\frac{1}{2}(m - 1)(n - 1).", "e03c82a813e23e9023165cdf4bc48e1f": "\\updownarrow \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\, \\updownarrow", "e03c9f89af141dc6ec837061d47ff2ee": " (A-\\alpha I)^{-1}x = \\mu x", "e03cd9e25a8a7af3f44721329fd4d411": " {^{15}}\\text{N}^{14}\\text{N}\\text{O} \\rightarrow {^{14}}\\text{N}{^{15}}\\text{N}, ", "e03ce054a8d682c7999b80224bba987a": "\\Phi\\in C(X,\\mathbf R)", "e03cf44e50bebca4cc54bdda24757b1f": "\\begin{align}\nV_n \n& = \\int_0^1 \\int_0^{2\\pi} V_{n-2}(\\sqrt{1-r^2})^{n-2} \\, r \\, d\\theta \\, dr \\\\[6pt]\n& = \\int_0^1 \\int_0^{2\\pi} V_{n-2} (1-r^2)^{n/2-1}\\, r \\, d\\theta \\, dr \\\\[6pt]\n& = 2 \\pi V_{n-2} \\int_{0}^{1} (1-r^2)^{n/2-1}\\, r \\, dr \\\\[6pt]\n& = 2 \\pi V_{n-2} \\left[ -\\frac{1}{n}(1-r^2)^{n/2} \\right]^{r=1}_{r=0} \\\\[6pt]\n& = 2 \\pi V_{n-2} \\frac{1}{n} = \\frac{2 \\pi}{n} V_{n-2}.\n\\end{align}", "e03cf771d5070871f78ed07668d3ac88": "x_{\\alpha} = \\inf\\{x \\in \\mathbb{R}: P(X \\leq x) \\geq \\alpha\\}", "e03d0ba035ff20f10b6bff977d63d14a": "\\mathbf{F}\\cdot\\mathbf{n} \\, dS", "e03d0fe705b618159468aa145e22bb4c": "\\mathcal{U}^N := \\left\\{ u : u(x,t)=\\sum_{k=-N/2}^{N/2-1} \\hat{u}_{k}(t) e^{i k x}\\right\\}", "e03d2805b2c72912e586b6fdcae4334f": "\\phi =", "e03d654c8e65800864f9f58e934d6a96": "p\\wedge(e-p)=0", "e03d66272b074684c9e774103bf924fe": "k[[x,y]]", "e03de97597ad88aedd12ef3d480da988": "F^m_{\\gamma}", "e03e472ece109a565352786f3490704f": "\\left\\{ :\\mathop{{}}_{{}}^{{}}\\circ \\mathop{{}}_{{}}^{{}}\\circ \\mathop{{}}_{{}}^{{}}\\circ \\mathop{{}}_{{}}^{{}}: \\right\\}", "e03e57b17a3b60fae2d957be03aae264": "TdS = dH - dP", "e03e8d807308be963b4687e0195a6349": "\\frac{2r^d\\pi^{d/2}}{d\\Gamma(d/2)}", "e03e8ffa966194e7372fb09cb8b6f8a7": "f \\circ g_1 = f \\circ g_2 \\Rightarrow lfg_1 = lfg_2 \\Rightarrow g_1 = g_2.", "e03e96300527ae5b8be6c41eaa103ccd": "M_\\lambda = \\mathcal{U}(\\mathfrak{g}) \\otimes_{\\mathcal{U}(\\mathfrak{b})} F_\\lambda", "e03ef7351173218600175ef984b9a830": "S_0 = \\sqrt[(x+y)]{\\frac{K_{sp}}{x^x y^y}}\\,", "e03f1a7080544a94a5755143b7fc359b": "\\Phi : L(H_A) \\rightarrow L(H_B)", "e03f38c2d5747c29120b312dcd714d90": " WF_A(Pf) \\subseteq WF_A(f)", "e03f39764b764b246da9e6d6e1be9056": "\n \\int ((m+1) (A (a\\,d\\,f-b(c\\,f+d\\,e))+B\\,b\\,c\\,e)-(A\\,b-a\\,B) (d\\,e(n+1)+c\\,f(p+1))-d\\,f(m+n+p+3) (A\\,b-a\\,B)x)(a+b\\,x)^{m+1} (c+d\\,x)^n(e+f\\,x)^p dx\n", "e03f4521cbf730a8637dbea8c7b4249e": "m = j", "e03f47d2c06e361de371de35f750ea4e": " Q(x,f(x)) \\equiv 0", "e03f6aebe341529792c26e7691471dd3": "\n \\sigma_{jk}\\cfrac{\\partial\\epsilon_{jk}}{\\partial x_1} = \n \\tfrac{1}{2}\\left(\\sigma_{jk}\\cfrac{\\partial^2 u_k}{\\partial x_1 \\partial x_j} + \\sigma_{jk}\\cfrac{\\partial^2 u_j}{\\partial x_1 \\partial x_k}\\right)\n ", "e03fa6158be18b85160b52744609b4ee": "WW^{T}=wI", "e03fb76b49d1c4acfafc6e2eacbf3e32": " (\\partial f)(v) = \\partial(f(v)) - (-1)^{|f|} f(\\partial(v)) ", "e03ff440b8dd32783792a8a47b191553": "\n\\begin{matrix}\n g_{10}\\ $_{11}\\ g_{11}\\ $_{12}\\ g_{12} \\ \\dots \\ $_{1m_1}\\ g_{1m_1} \\\\\n g_{20}\\ $_{21}\\ g_{21}\\ $_{22}\\ g_{22} \\ \\dots \\ $_{2m_2}\\ g_{2m_2} \\\\\n \\dots \\ \\dots \\ \\dots \\ \\dots \\ \\dots \\ \\dots \\ \\dots \\ \\dots \\\\\n g_{k0}\\ $_{k1}\\ g_{k1}\\ $_{k2}\\ g_{k2} \\ \\dots \\ $_{km_k}\\ g_{km_k} \\\\\n \\\\\n \\downarrow \\\\\n \\\\\n h_0 \\ $'_1 \\ h_1 \\ $'_2 \\ h_2 \\ \\dots \\ $'_n \\ h_n \\\\\n\\end{matrix}\n", "e0402ab5ac86e8715e385ce51363db20": " 1 mg/l ", "e040a740a0552d0eaab23123ea117139": "K=\\frac{[B]_{eq}}{[A]_{eq}}", "e04123485c128291eb154df7335e6e97": "||| f |||_{L^{p,\\infty}}=\\sup_{0<\\mu(E)<\\infty} \\mu(E)^{-\\frac{1}{r}+\\frac{1}{p}}\\left(\\int_E |f|^r\\,d\\mu\\right)^{\\frac{1}{r}}", "e0412dbb6e50de76a383b8f6e7d81dce": "PV=nRT", "e04147ce3f016abbb1f84a38fcfc8fb1": "D=P^2-4Q", "e04153ac525eced9b0b4077e7ba26c2c": "x \\in \\Omega \\backslash Y", "e04170fb56065f9bf1cdbfc622d1125d": " \\mathbf{e } ", "e041bb7008e22f4b28ef25a6435772a9": " s_k = \\frac{P_{2k}}{2}, \\quad t_k = \\frac{P_{2k} + P_{2k-1} -1}{2}, \\quad N_k = \\left( \\frac{P_{2k}}{2} \\right)^2.", "e041d21ba27901c5fb3eb97deabb6ef7": "|S\\cup T| \\le |S| + |T|.", "e041dffbcf5503f70173aa6134a900f0": "i = 1, 2, 3, \\ldots, 32", "e0421a1512eeaf020045437c78e20ff7": "R_{WB}\n", "e0421d96eca75ba7cde932d6654c5198": "\nD(\\theta'||\\theta) = \\frac{1}{2}(\\theta'-\\theta)^2\\underbrace{\\left(\\frac{d^2}{d\\theta'_i d\\theta'_j}D(\\theta'|\\theta)\\right)_{\\theta=\\theta'}}_{\\text{Fisher info.}}+...\n", "e0424a461a0ff9a7f72f9c5907d03810": " l \\mid \\varphi(a^n-1)", "e0425c6982246db0e89e5b75b9045d06": "f : S^{j-1} \\times D^{m-j} \\to \\partial M", "e0427b1a932652f084526b0ccb721105": "=-\\frac{1}{2\\sqrt{g}}\\bigl(\\epsilon^{mij}D_i R^n_j+\\epsilon^{nij}D_i R^m_j).", "e042afa214826595b2f17f5719ece440": "\\frac{2x}{2} = \\frac{8}{2}", "e04328db5d10094d8e67fae5f563bd53": "\\operatorname{Var}(c^T X) = c^T \\Sigma c .", "e04328fba1e2a81cb308510d745e3de8": "(x_1 i + y_1 j + z_1 k) + h (x_2 i + y_2 j + z_2 k)", "e04337b37cc42bc91d54983a08d2ef5e": "\\sigma=1-\\frac{M^2}{L_PL_S}=1-k^2=\\frac{L_{sc}}{L_{oc}}=\\frac{L_{sc}^{sec}}{L_S}=\\frac{L_{sc}^{pri}}{L_P}=\\frac{i_{oc}}{i_{sc}} ", "e043d9e4dac9f977a9dda15a0a000ac0": "X\\!", "e044c31aa01579376de522b40068a63c": " X = \\mu - \\sigma \\ln(U) \\sim \\mbox{GPD}(\\mu,\\sigma,\\xi =0).", "e04502a3c40fd6139c7237a6a6a64e02": "s_r", "e0450a2dde6249f29cf1687adf3f528f": "A_{\\mu}(x)", "e0453610e190d45908386bfaaff18e23": "U = -\\mathbf{M}\\cdot\\mathbf{B} = -k{{\\mathbf{B}\\cdot\\mathbf{B}} \\over |\\mathbf{B}|} = -k{|\\mathbf{B}|^2 \\over |\\mathbf{B}|} = -k\\left (B_x^2 + B_y^2 + B_z^2 \\right )^{\\frac{1}{2}};", "e045635af82f8f48986a6e931cb70b8f": "a_j", "e0456a583d02b5fef390d21f1f9c153e": " S = k_B \\ln W \\,", "e045a64511e6de71c2f838a362470676": "\\text{Peak intensity } (\\mathrm{W}/\\mathrm{cm}^2) = \\frac{\\text{peak power } (\\mathrm{W})}{\\text{focal spot area } (\\mathrm{cm}^2)}", "e0460f94489cb29e1a95be9c67a9c60d": "(x_1+x_2)^p=x_1^p+x_2^p.", "e04683536e0969658b1b88f229a21163": " \n\\frac{1}{t}\\sum_{\\tau=0}^{t-1} E[p(\\tau)] \\leq p^* + (B+C)/V\n", "e046a79caff44961d4f37021ff6c715a": "t'=it", "e046d39e3b75cf79fc5762568eabae13": "\\mathbf x \\cdot \\mathbf y = \\sum_i x_i \\hat{\\mathbf e}_i \\cdot \\sum_j y_j \\hat{\\mathbf e}_j = \\sum_i x_i y_i", "e046fa7ba4295036dd3fc8ecdb01c4f3": "\\zeta_2", "e04701cea57bcc453bd91a31fb974c9e": " \n\\begin{bmatrix}\n\\mathbf A, & \\mathbf B \n\\end{bmatrix}\n^{+} = \\left[\\mathbf P_B^\\perp \\mathbf A( \\mathbf A^T \\mathbf P_B^\\perp \\mathbf A)^{-1}, \\quad \\mathbf P_A^\\perp \\mathbf B(\\mathbf B^T \\mathbf P_A^\\perp \\mathbf B)^{-1}\\right]^T, \n", "e047541afcd5f8c287085c4687af89ba": "u_i^{(k)}", "e0478cc3f35363508ec1ca8616b34d69": "k=\\frac {nGI} {L} ", "e047af25e81473fc7a98d30c0f6cfcf1": "p : \\mathbf{R}^k \\to E_x.", "e047cc878f4e1ea816289ba5c4ac1f6c": "\\begin{align}\n \\mathbf{B}_{\\mathbf{P}_0\\mathbf{P}_1\\mathbf{P}_2\\mathbf{P}_3\\mathbf{P}_4\\mathbf{P}_5}(t) = \\mathbf{B}(t)\n = {} & (1 - t)^5\\mathbf{P}_0 + 5t(1 - t)^4\\mathbf{P}_1 + 10t^2(1 - t)^3 \\mathbf{P}_2 \\\\\n {} & + 10t^3 (1-t)^2 \\mathbf{P}_3 + 5t^4(1-t) \\mathbf{P}_4 + t^5 \\mathbf{P}_5,\\quad t \\in [0,1]\n\\end{align}", "e0485ca254130ec7d0638b20b4ee7190": "\\mathbf{I_0}", "e04886836ef7c45f7bc8366ec984030d": "(a,b) \\div (c,d) = (ad,bc)", "e04889cc6e846a99d3d2eecb4e0302e1": "(-1,0.5)\\ ,\\ (0,0)\\ ,\\ (3,3)", "e048a68bd2cb8bbeee62fd0af612700e": " \\beta^* =1/(\\nu^*-1) \\approx 0.6 ", "e048afd1562588b0795b2db3862d7bc8": " p^2-p+1", "e048cab2adf6cb338db07eb6c7edc83c": "\\nabla\\cdot\\mathbf{E} = 0", "e048cb73e2fea6d54360e3724cec3a60": "\\|xy\\| = \\|x\\| \\|y\\|", "e048dbdd825426ff486ae7c18a7f040c": "\\phi(x_1, x_2, \\ldots, x_n, y) = 0 ", "e04953d5dc07458d0b132d0071148c56": "\\frac{a}{\\sqrt{2}}", "e049a43b723920e90f7e06dd133af84f": "S_{\\text{new}} = S_{\\text{old}} - \\frac{ \\left( 5-T \\right) ^2} {2}", "e049b047b8863378d262a8aae2413ba0": "R: L_d^p \\rightarrow \\mathbb{F}_M", "e049b844245a930df1277cee521c4d19": "e_1,e_2,\\ldots, e_k", "e049dc0584788236812c48eb7023a8f5": "\n\\sum_{j m} (2j+1)\n\\begin{pmatrix}\n j_1 & j_2 & j\\\\\n m_1 & m_2 & m\n\\end{pmatrix}\n\\begin{pmatrix}\n j_1 & j_2 & j\\\\\n m_1' & m_2' & m\n\\end{pmatrix}\n=\\delta_{m_1 m_1'}\\delta_{m_2 m_2'}.\n", "e049dfb5727b1a1843fa6eebda4a6011": "2 \\pi f \\ ", "e049e60a1e81485e47fbb8c6a4c6a4ee": "X^*_{\\mathcal{G}}", "e049eac547fa8897125075d8037ff246": "N_{kj} =", "e049f39657617f9ad8d16871d3bd762f": "(q, \\gamma) = \\delta(p, A)", "e04a49ec03f6d43619d0531e481e5cee": "\\pi(\\theta\\mid x)\\,\\!", "e04a6795ed64315cc2c81c78c071018d": "\\mathbf{r}_{io}", "e04a9844a32ab7716f9be5dbb30c8e46": "f(q_1,q_2,\\ldots,q_n)=0\\, ", "e04ad8afa2042c36ff9899880e616f33": "\\mathbf{E} = -\\nabla V", "e04b2a7cdcd4933bad63b581578950d1": " \\left( \\frac{v}{c} \\right)^2 \\left( \\frac{r}{r_s} - 1 \\right) = \\frac{1}{2}.", "e04bbfe79105f434fc101bf15a1d738d": "I \\rightarrow B: \\{N_A, A\\}_{K_{PB}}", "e04c365a99d915d00ec010cf81e97c57": "E_{u,v}(\\tau,s) =\\sum_{(m,n)\\ne (0,0)}e^{2\\pi i (mu+n\\tau)}{y^s\\over|m\\tau+n|^{2s}}", "e04d069c2ae010c1192b385ddf8fb395": " (1+i) = \\left(1+\\frac{i^{4}}{4}\\right)^4 ", "e04d12fc69546ac93ad4361799ca0bf0": "\n \\mathbf{x}\\ \\sim\\ \\mathcal{N}(\\boldsymbol\\mu,\\, \\boldsymbol\\Sigma),\n ", "e04d17fd9ac03de4cdf81f20a5bd7473": "f_j(x_1,\\dots,x_n)=0\\quad\\text{for}\\, j=1,\\ldots, r.", "e04d3f45266126c3a8f8e2360a40fafd": "R_H =\\frac{E_y}{j_xB}", "e04d85de978e3dd06f3e0b1c3db007f9": "Q_L", "e04d98601a9e3138b395a840a511f7f3": "\\scriptstyle BV(\\Omega)", "e04dac2b8af2e345ee476b56374246eb": "AS'C = ABC - \\pi/3", "e04ddd7afa62ea2354ca82a5eae67eb4": "F(\\theta_{x_t})", "e04e31b0e170850ca902c6d0193b3656": "\\log n=2", "e04e4e8b2a2a1a932e78fb877c13825a": "Prelevant", "e04e4fc39711838128b0a40f2fa7a284": "z^{p-1} -1 = z^{4n}-1 = (z^{2n}-1)(z^{2n}+1)", "e04e68aecf393b546bf8103653edc228": " f(n) = \\begin{cases} n/2 &\\text{if } n \\equiv 0 \\pmod{2}\\\\ 3n+1 & \\text{if } n\\equiv 1 \\pmod{2} .\\end{cases} ", "e04e8eba9040ffb3dd91266182894369": "\\ddot{\\vec{x}}[t] = - \\nabla \\Phi [\\vec{x} [t],t] \\,.", "e04ec1264e90c8bc3dcf8e95eb76cbb5": "H=-\\frac{\\hbar^2}{2 m} \\frac{\\partial^2}{\\partial x^2}+\\frac{m \\omega^2 x^2}{2}\n+\\lambda x^4", "e04f95ca02f0a5fb57f5d307e3ceff9e": "x^4-x^3-1", "e050289b829f2735ebc6c8a507254b83": "J\\in[0, \\infty)", "e050759513ea33043bde7e7d24e4035d": "\\mathcal{L}_X", "e050e9395ebd999ac04e2fb83a1d1bd9": "\\frac{d \\Pi}{dx}=(1-2x-y)-x\\frac{dy}{dx}-ax = 0", "e05107624e1e4934c9ed6241058c82fc": "T=\\frac{S_{\\mu}-m\\mu}{\\sqrt{S_{\\sigma^2}}}\\sqrt\\frac{m-1}{m}=\\frac{\\overline X-\\mu}{\\sqrt{S_{\\sigma^2}/(m(m-1))}}", "e0511590a5d007a4f7c76a571b444d7c": "h+k", "e051433367157385dd804c6219cbb44e": "\\displaystyle \\mbox{Hess}(f) \\in \\Gamma(T^*M \\otimes T^*M) ", "e05145d32539e06e84d609b03d8d98e3": " r_i < \\ell_j ", "e0516e91a1770706483961b6a6ce7dd6": "\\frac{D \\eta}{D t} = 0,", "e0518dd72f7054a889bbbf5a9b8318d2": "2^{N_d/2}/2", "e0519c46c607b9980358ee95d6272baf": "(\\mathbb{R}^n_+,\\mathbb{R}^d)", "e051dbfac6243cfca362daac8ebe70a3": "L_n=R_n(\\xi,\\xi)\\,", "e05228d3bdd2ecf0088921bafb7c1525": "\\phi_e(x)", "e05235dc8c8da17505fb08c1aa10e43b": "s_n \\rightarrow s", "e0523ed9bfaed560fea615e4e863890f": "F_{CE}", "e05241120f82e4f893f1ba50fe8875fd": "\\Gamma_{e}(x=d) = \\Gamma_{e}(x=0)\\,\\mathrm{e}^{\\alpha d}\\qquad\\qquad(1)", "e052837af87608ca05fcc330616d13ff": "\\gamma = \\frac{\\partial^2r}{\\partial E \\, \\partial C}", "e0528c76868868a19f937eddb9935f00": "\\gamma = c_p/c_v", "e052afea3bd9d4160805f74d9fe08732": "e^{ \\mu \\phi} \\,", "e052b5d3001ec7e387b5a77b5f8e0b70": " f(x)=Sxe^{-\\alpha|x|}", "e052c2e92dea7d0591cfd19b9465d502": "\\scriptstyle g/\\pi^2\\approx{1}", "e052e0a5c5713c34c836206283392cf1": "d \\vec{\\ell}_1", "e05327498339504b87c844e89536a579": "y = {A - {1 \\over T} \\over 2C},", "e05361381efdddc7dd277c449870a4ed": "F(\\phi_r-\\phi_l)=D(\\phi_R-2\\phi_P+\\phi_L)", "e053a1d72b2718861aea6c902d3bfc55": "S(\\rho^{AB})", "e053ac2238dfdb5d5649983cc57d50a8": "\\frac{\\partial}{\\partial t} \\vartheta(x,it)=\\frac{1}{4\\pi} \\frac{\\partial^2}{\\partial x^2} \\vartheta(x,it).", "e053e3b5b3b790c9981016d808b3dc58": "S_{cw}", "e0546f56cf78db1c8143647354e1a09b": " \\lim_{x \\to 0} \\frac{x}{x} = 1, \\! ~~ (1)", "e055070aa9709be5427b1d93282f1669": "u_{n+1} = u_n - a_n\\hat{g}_n(u_n),", "e0553d8a14abba0685b263c416920d87": "|x|\\to \\infty", "e0556e109c321067820b9d64dd96f6d1": "\\big\\{e_i(0)\\big\\}", "e0557a9b36ae2b7f9542eb41c21615ee": "\\sum_{j}L_{ij} = 0", "e055c67d4622f9bd88a41ae56feb749a": "\\delta z \\propto e^{im\\phi}", "e055d8f4189291cea934a183f8ae7d7a": "1 \\leq i,j \\leq n", "e05638886b2c204b8264a978796871b6": "\\begin{align} \n \\omega_r & = cos(2 \\pi \\frac{K}{N}) \\\\ \n \\omega_i & = sin(2 \\pi \\frac{K}{N}) \\\\\n y(N) & = (\\omega_r\\ s(N-1) - s(N-2)) + i (\\omega_i\\ s(N-1)) \n\\end{align} \n", "e0568e52cc38d9ef9aff6b0d4760df5a": "\\ p_c", "e056a0d2b82f2b6cd1dafe669f67e595": "\\frac{d[W']}{dt} = -a_2[W'][E_2] + d_2[W'E_2] + k1[WE_1]", "e056bd15c15351d08c81329a81da502c": "\\alpha_{ij} = \\begin{bmatrix}1 & \\alpha_1 & 0 & 0 & \\alpha_{-1} \\\\ \\alpha_{-1} & 1 & \\alpha_1 & 0 & 0 \\\\ 0 & \\alpha_{-1} & 1 & \\alpha_1 & 0 \\\\ 0 & 0 & \\alpha_{-1} & 1 & \\alpha_1 \\\\ \\alpha_1 & 0 & 0 & \\alpha_{-1} & 1 \\end{bmatrix}.", "e0578b0c0ce3e6987b28a13f66695a17": "P / {\\rm hp} = {(F / {\\rm lbf}) (v / {\\rm mph}) \\over 375}", "e0578fe60e9f9b35b526574f8380da7d": " (l_A a_B + l_B) l_D + (1 + r) l_A a_D ", "e057ef6251b079fab5e10a6562cfe80c": "(R,I-R)", "e0580572018b2717e4feb268fabbebd7": "\\begin{align}\n0\\,x&=0\\\\\n1\\,x&=x\\\\\nn\\,x&= x+ (n-1)\\,x \\qquad \\text{if} \\quad n>1\\\\\n(-n)\\,x&= -(n\\,x) \\qquad \\text{if} \\quad n<0\n\\end{align}", "e05807f413f172093c4c91990437b1a1": " H(f) = - \\sum_S \\hat{f}^2(S) \\log \\hat{f}^2(S) \\!", "e058191d32425e8a0dd80337382c8095": "\\scriptstyle |\\zeta|^n\\leq \\|a\\|_p \\|(\\zeta^{n-1},\\cdots,\\zeta, 1)\\|_q ", "e058351d9c488300de9670fe4fa56801": "(3 + 5) + 7 = 3 + (5 + 7)", "e0588599c636964426426dd6cd65d2b1": "\\phi(t)=\\sqrt2 \\sum_{n\\in\\Z} h_n\\phi(2t-n),", "e0591c46c33a6149567c8a7d41e63019": " X_t = \\frac{\\theta (L) }{\\varphi (L)}\\varepsilon_t,", "e05944ee94e8f5c07c9ec496ba5d5dcf": "f_3(x)\\neq 0", "e05986d72b1370817c7c96546ecaf7f8": " \\lambda (v-1) = k(k-1). ", "e059b32a37483f66a7d148e0ff4b61b7": "\\alpha = (6.75 \\times 10^{-7}) I^3 - (7.71 \\times 10^{-5}) I^2 + (1.792 \\times 10^{-2}) I + 0.49239", "e05a01388517dc08c6d23ac03fb6a34c": "\\sqrt{f(\\Theta)}:", "e05a30d96800384dd38b22851322a6b5": "\\lambda ", "e05a3e47d0d67f24d248a218ee11eaf4": "\\begin{align}\n\\hat{\\mathbf x} &= \\cos\\phi\\hat{\\boldsymbol\\rho} - \\sin\\phi\\hat{\\boldsymbol\\phi} \\\\\n\\hat{\\mathbf y} &= \\sin\\phi\\hat{\\boldsymbol\\rho} + \\cos\\phi\\hat{\\boldsymbol\\phi} \\\\\n\\hat{\\mathbf z} &= \\hat{\\mathbf z}\n\\end{align}", "e05a80bd7f029581eb1ebddc559d1b3b": " \\Psi = \\sum_{I=0} c_{I} \\Phi_{I}^{SO} = c_0\\Phi_0^{SO} + c_1\\Phi_1^{SO} + {...} ", "e05b2f546b9536f3c511e1d12f84a1cf": "2\\Gamma(\\omega)", "e05b59372a8109718a788f034830f383": "(\\partial_t -\\Delta)u=0", "e05b9d41331a46c9ddda6a8c6cb12682": "\\gamma(t)=(\\zeta(\\sigma+i t),\\zeta'(\\sigma+i t),\\dots,\\zeta^{(n-1)}(\\sigma+i t))", "e05bb4fe802d9bfccc664ce2c96b72a3": "\\begin{align}\n\\tau_\\mathrm{n}^2 + \\left[ \\sigma_\\mathrm{n}- \\tfrac{1}{2} (\\sigma_2 + \\sigma_3) \\right]^2 \\ge \\left( \\tfrac{1}{2}(\\sigma_2 - \\sigma_3) \\right)^2 \\\\\n\\tau_\\mathrm{n}^2 + \\left[ \\sigma_\\mathrm{n}- \\tfrac{1}{2} (\\sigma_1 + \\sigma_3) \\right]^2 \\le \\left( \\tfrac{1}{2}(\\sigma_1 - \\sigma_3) \\right)^2 \\\\\n\\tau_\\mathrm{n}^2 + \\left[ \\sigma_\\mathrm{n}- \\tfrac{1}{2} (\\sigma_1 + \\sigma_2) \\right]^2 \\ge \\left( \\tfrac{1}{2}(\\sigma_1 - \\sigma_2) \\right)^2 \\\\\n\\end{align}", "e05c068f4799ab1a911791acb3e61ae7": "(2\\pi)^{-\\frac{k}{2}}|\\boldsymbol\\Sigma|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(\\mathbf{x}-\\boldsymbol\\mu)'\\boldsymbol\\Sigma^{-1}(\\mathbf{x}-\\boldsymbol\\mu) },", "e05c21bedad7d0a5434a2a9498010c3a": "o(x^k)", "e05c88f9df23e20be83fb2cb33d5f1f2": "E=\\hbar\\omega\\left(n_x+n_y+n_z+3/2\\right)~~~~~~~~n_i=0,1,2,\\ldots", "e05cf60d0830f4cb1c234c99e309e7d6": "-k_{eq}\\left( \\frac{k_2+k_1}{k_2} \\right)x_1 = - k_1 x_1 .\\,", "e05d095bfc26cd1d1650be673bcd0bac": "~E =\\frac{\\omega_{\\rm s}}{\\omega_{\\rm p}} \\frac{1-V/p}{1+U/s}~", "e05d14da01b576bceb42bc93108c34ed": "{Q_c} = {I_c} . {A_c}", "e05d34802f6958e7c9a708f78643f208": "\\frac{\\operatorname{d}^2\\! F_\\nu}{\\operatorname{d}\\!k^2}+\\frac{1}{k}\\frac{\\operatorname{d}\\!F_\\nu}{\\operatorname{d}\\!k}-\\frac{\\nu^2}{k^2}F_\\nu", "e05d63fc4c6786885a2d0a89bcdea917": "\\mathbf Z/2^k", "e05d808ed982168c2df9f9d963f0e340": " \\frac{1}{4} \\left(15 + 7\\sqrt{5}\\right)\\, s^3", "e05d8b852e17e76eee9f8a8645d74bc8": " n = ( t / d_p )^2 p^{ -1 } q ", "e05e089fcd6e0e78035070279d13c8cb": "\\theta = \\theta_0", "e05e2ddf9b2a9285f26e29e077a6fde9": "t_0+\\delta t", "e05e3e783132837fa81692158e91fd7f": "\\psi=\\exists x_1 ... \\exists x_n \\phi", "e05e8735636104e7638933b7b57d7d6d": " \\int_V \\mathcal D \\phi \\; e^{-\\langle \\phi|S|\\phi\\rangle} = \\prod_i \\int_{-\\infty}^{+\\infty} \\frac{dc_i}{2\\pi} e^{-\\lambda_ic_i^2}. ", "e05f0082568f77efdf2f6090054cbcb1": "x=a\\cot(t)", "e05f08b02324baee2ec32197b1b079c7": "B l= \\mu_0 N I,", "e05f10f7d56d7ff0dc1390f032a64da0": "\\mathcal{\\epsilon}", "e05fdf579db136347775580d31a9bd39": "\\scriptstyle 1 - \\frac 1 3 + \\frac 1 5 - \\frac 1 7 + \\cdots = \\frac \\pi 4 = 0.7853981\\ldots ", "e060354440421a38e141e6e43f7b0b39": " \\rho(z_1,z_2)= \\log (z_1,z_2^\\times ; z_2, z_1^\\times).", "e060500b60ae78a87a863317788c517e": "\\lambda_{03}=8.65373", "e0607703814149a49215b2b4d6b4a263": "\\min[]", "e0609989b08a0698584a074012d7459f": " \\operatorname{drop-param}[ \\lambda g.\\lambda n.n\\ (g\\ m\\ p\\ n)\\ (g\\ q\\ p\\ n), D, \\{p, q, m\\}, \\_] ", "e060d0e293e8e1c7652a360866c6f997": "\\mathbb{Z}/2\\mathbb{Z}", "e0610d681c6feb0cc296971f5bc076e7": "\\tilde P_n", "e06126e414d39d8d97a9632ece309138": "\\cdots\\to P_2\\to P_1\\to P_0 \\to X \\to 0", "e0617ca2704d5c51128bd8387cc0f076": "S_t \\,", "e0621f80f5b6a4140981e5d3762d1833": "\\left( 1-e^{-k(n+0.5)/(m-1)} \\right)^k.", "e06227e6aca0c10bf7a3cc8913f40d0f": "I_\\mathrm o", "e062344e4180142d7bbdc8e1da16638d": "\\zeta(3) = \\frac{4}{3} \\sum_{k=0}^\\infty \\frac{(-1)^k}{(k+1)^3}", "e0632763c36983e37f8f09fd0d6e5901": "k = \\frac{j}{1728-j} \\pmod{167} \\equiv 158 \\pmod{167}", "e0632a8d681e57aecfd9b71d8dcbd6e8": "M^{\\mathrm{ET}}(x) = h - eFx \\qquad\\qquad\\qquad\\qquad\\qquad\\;\\;\\; (2)", "e0632b4f2d627e66cc72e702cfb02fdf": "\\arccot x ", "e063c4754207d953f75703a8548c6e07": " \\,P\\left(x;\\;\\Gamma(x),\\;\\Gamma'(x),\\;\\ldots\\;,\\;\\Gamma^{(n)}(x)\\right)\\equiv 0.\\!", "e06477a860cf15cae5d9feaa53721bde": "\n l_\\epsilon u_\\epsilon = g^\\epsilon(x), x \\in \\partial \\Omega\n", "e0647ebcd77b1a895c79b3c92daac3d4": "\\sum_{n=1}^\\infty \\bigg| \\frac{(-1)^{n+1}}{n} \\bigg|", "e0649afed02e1861bab827fd874881b6": "op_x", "e064dcc5164c0aa2098d21a67ec037b4": "\\int x\\,\\operatorname{arcsch}(a\\,x)dx=\n \\frac{x^2\\,\\operatorname{arcsch}(a\\,x)}{2}+\n \\frac{x}{2\\,a}\\sqrt{\\frac{1}{a^2\\,x^2}+1}+C", "e065102c6ba69f778d4796754f2f76bf": "2x - y = 1 \\,", "e0658abf5707b011923d35b9c0c091f9": " F_{G} ", "e06619783e970b7d43a1c80fd290d270": "\\text{If }x \\in A\\text{ and }C\\subset B,\\text{ then } p(x, C)\\geq \\varepsilon \\rho(C).", "e0669c5c9b62197b15d9b3578798d6d1": " G(v)+F_v(J) = \\left[ \\omega_e (v+{1 \\over 2}) + B_v J (J+1) \\right]- \\left[ \\omega_e\\chi_e (v+{1 \\over 2})^2 + D J^2 (J+1)^2 \\right]", "e066afbb2e3094d9432da5a64df34691": "A \\ni x,", "e066e90269453dce427138a7899e50fe": "\\frac{\\partial \\ln \\mathcal{L}(\\alpha,\\beta|X)}{\\partial \\alpha} = \\sum_{i=1}^N \\ln X_i -N\\frac{\\partial \\ln \\Beta(\\alpha,\\beta)}{\\partial \\alpha}=0", "e066ec56c9e7257b3486c55f63e4176f": "\\ -(MRS_{xy})=-(P_x/P_y) ", "e0675f4b59c1a2483f40ff08b959b5f4": "\\mathbf{BB1} = \\begin{bmatrix}\n(1-2\\lambda-\\beta) & (\\lambda-\\alpha) & 0 & 0 \\\\\n(\\lambda+\\alpha) & (1-2\\lambda-\\beta) & (\\lambda-\\alpha) & 0 \\\\\n0 & (\\lambda+\\alpha) & (1-2\\lambda-\\beta) & (\\lambda-\\alpha)\\\\\n0 & 0 & 2\\lambda & (1-2\\lambda-\\beta)\\end{bmatrix}", "e067ae583e665b881df1488a22121aca": "S\\left(t\\right)=\\int E^2\\left(t^{'} \\right)+E^2\\left(t^{'} -A\\left(t\\right)\\right)+2E\\left(t^{'} \\right)E\\left(t^{'} -A\\left(t\\right)\\right)\\, dt^{'}", "e06838817fb6e16ecf6b169eb9983e7d": "\\bar{a} = (a_0, \\dots, a_{k-1})", "e0685fc2413eabf45a74796f167ca415": "\\nabla p = (d p /dA)\\nabla A", "e0686656f277cb1d32db23173a8183cb": "R_{g0}=\\sqrt{N b^2/(6)}", "e068853df4ad178032cb8d773ef17895": "A + B = \\{\\mathbf{a}+\\mathbf{b}\\,|\\,\\mathbf{a}\\in A,\\ \\mathbf{b}\\in B\\}.", "e0688c1e5c5afe5b35d83684af7562b4": "\\begin{bmatrix}\n2&-1&0&0\\\\\n-1&2&-2&0\\\\\n0&-1&2&-1\\\\\n0&0&-1&2\n\\end{bmatrix}", "e068d05ea07e9454b9ace9561555260a": "k\\log k", "e0696878d82fb295ef11c5d285c8a0ab": "Y=y", "e0697b272f80ca577d11e589273ff3fd": "u_i = u_1,...,u_n, u_i\\in[0,1]", "e069a8be695c97cf79277588f974207b": "\\liminf B := \\inf \\bigcap \\{ \\overline{B}_0 : B_0 \\in B \\}", "e069c37f457f03eaeb522ab5ea812f69": "N_i=2(a_{r}+a_{s})", "e069d11b41e6278d016db7c84db293c8": " (X'X)^{-1}X' + D ", "e06a022b86322d6ee502b6760a0ed4ec": "\\pi \\,\\!", "e06a170121199dff148cd21e922daf8e": "\\mathcal{R}^K _{\\theta} = \\Psi_c(\\mathcal{R}_{\\theta})", "e06a62011c18ed69d50ff48088b23615": " \\quad N = \\frac{60Q}{Vol} ", "e06ad1a44207dc60641b554f56849ee6": "|z_1 - z_2|=\\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2},\\,", "e06adf2d21660a0c4fe10bba5a332719": " k_1 \\dots k_n ", "e06afbf5bd411cc41743c995ed1fff51": "\\lambda_2\\lambda_3 = 1", "e06b46e6d824c30e00aaf3c6202b0dce": "\\mathrm{^{238}_{\\ 92}U\\ +\\ ^{2}_{1}D\\ \\longrightarrow \\ ^{238}_{\\ 93}Np\\ +\\ 2\\ ^{1}_{0}n \\quad;\\quad ^{238}_{\\ 93}Np\\ \\xrightarrow[2.117 \\ d]{\\beta^-} \\ ^{238}_{\\ 94}Pu}", "e06b65d1d4e47548c42042e106fc2dcd": "\\ M_{pitch_{max}} > D_{pitch} \\times F_{forward} ", "e06b792bd573514e3dcdb4c4bd9e4abe": "F=A.\\mathrm{dp}", "e06b960c00d9533eba6cc710978c9b84": "V=G\\times_{H}\\mathbb{V}\\;\\text{where}\\;(gh,v)\\sim(g,\\rho(h)v)\\;\\forall\\;g\\in G,\\;h\\in H\\;\\text{and}\\;v\\in\\mathbb{V}.", "e06b99fccccf1764ce7e28e1041c6da0": "\\phi_2 = -\\phi_1 V_2 / V_1", "e06c0a8fc5eaec5ecbc43d0dd4222a24": "(x + y) + z = x + (y + z)", "e06c4392cdb0652604040091dd70c1d0": "{P}_{o}, {P}_{g}", "e06c94cec883623df390ad315ce32ea2": "I_K^\\mathbf{c}/i(K_{\\mathbf{c},1})\\mathrm{Nm}_{L/K}(I_L^\\mathbf{c})\\overset{\\sim}{\\longrightarrow}\\mathrm{Gal}(L/K)", "e06cf4e2de7dfadebf6bb3b5fce9dd5a": "2f(j) - 1", "e06cfc4a872e8ff89bc256ab873252c4": "F_j=\\sum_{i=0}^l\\sum_{s=0}^{m_i-1}a_{ijs}\\left(\\sum_{t=0}^{m_i-1}\\beta_{i,t}^{p^t}f_{p^{t}k_i\\bmod{N}}\\right)", "e06d7dd5b3661ab48a4d96924afa956b": "0\\rightarrow \\mathbf Z(p)\\rightarrow \\Omega^0_X\\rightarrow \\Omega^1_X\\rightarrow\\cdots\\rightarrow \\Omega_X^{p-1} \\rightarrow 0 \\rightarrow \\dots", "e06db42aa6868bd8a5f08b34dd49f703": "m+ld", "e06db867ee633ec15c2986ad17a97d6b": "\\Xi(s) = \\frac{1}{2}\\prod_\\rho \\left(1 - \\frac{s}{\\rho} \\right).\\!", "e06e2f53b48baeab747f901f949ea7e8": " \\int_{-\\infty}^\\infty x^2 | g(x)|^2\\; dx = \\infty \\quad \\textrm{or} \\quad \\int_{-\\infty}^\\infty \\xi^2|\\hat{g}(\\xi)|^2\\; d\\xi = \\infty. ", "e06e5028c136ee0dddaee5f1ad488a0c": "f(x+kp)=(x+kp)^2-n", "e06e864147f551cb3bddc94b27d882b2": "H(X) \\gets \\lg(L)", "e06eccfbee53e09530aa4291d76438e7": "\\mathbf{Cat}", "e06edcfb17d7492b3fdfc2ca7dade4a7": "\\operatorname{ip}(x,y) = |y-x|", "e06ee324a22b43065742d2921feca55d": " I_i \\cap O_j = \\varnothing, \\, ", "e06f3ba42c89003ab63f620ff2f317bd": "\\lim_{n \\to \\infty} c_n = A.", "e06f5cda93f3f59171a5c3aae6db8bc2": "S(P_j) = -e^{\\lambda P_0} P_j\\,", "e06f5fc4203b03aba9ba1bc3de5f3ef4": " |\\psi\\rangle = \\cos\\theta \\exp \\left ( i \\alpha \\right ) |x\\rangle + \\sin\\theta \\exp \\left ( i \\alpha \\right ) |y\\rangle = \\psi_x |x\\rangle + \\psi_y |y\\rangle. ", "e06f97f9d8b3a2808207b160832be086": " g'=((a_\\text{f}^2+g^2))^{0.5} ", "e06ffa923ca1e3b0c195c4504723b810": "C_l", "e0700443071a34a9cfe222eccde86d39": "G = \\langle N, \\Omega, \\langle A_i,u_i,T_i,\\tau_i,p_i,C_i \\rangle_{i\\in N} \\rangle", "e07067a7f9105cf4c4b1afd40660786b": "n \\geq 5.", "e070945e3487ea58a4c46fe835934a52": "G(n)=\\frac{\\sum^{n+1}_{i=1}X^2_i}{T_n^2} ,", "e070b8ad0a97cf5427c73aaf9967b988": " \\mathit{l}_{H} = (-C_{mgH}\\,\\mathit{l}+C_{tHd})/C_{nH}\\,", "e070b94a8b352e5f9b5a65e96a0200f0": "\\begin{bmatrix} \\lambda_{1} \\end{bmatrix} ", "e070f2e3af5d26970693af115a65ada2": "\\dim_x \\cap_{i=1}^n Z_i = 0", "e07117fae220bf82bee7b5f96b02d0b1": "\\sqrt{4.5/6 + 4.5/6} = 1.2", "e0712587239549464b706dc8f3958b6e": "\\lim_{\\kappa\\rightarrow 0}f(x\\mid\\mu,\\kappa)=\\mathrm{U}(x)", "e071834bf99a5ec9300f9ea836d9086c": "\n\\ \\delta E = \\sqrt{k_B T^2 C}\n", "e071b88b0319cf492ab4ba24336487b3": "i:N\\subset M", "e071c9d45d708edeb1a613f9b4f5ba12": "2^h - 1", "e071ccf5761f5d8853081cbbf9505674": "\\theta_{cr}", "e0722741e3c2f4a0b2340d27f61edcff": "\\int_V \\nabla \\cdot \\vec A\\ dV=\\int_S \\vec A \\cdot d\\hat\\sigma.", "e0722756705b3e6cbd2600ce18deb17e": "\\scriptstyle t \\;=\\; N\\tau", "e0722fa5ab229fa62862cce3cbd1bbbb": "P^\\mu = m \\, U^\\mu\\!", "e0723b7befa635cd8e660cd983aada69": "x_i\\in A_i", "e0725121ed0e3e2d49ee32cc1f6e7cf5": "\nA= (a_{n,n})\n", "e072a5a85305fbd100e22f80022f231d": "\\epsilon_n \\rightarrow 0", "e072daafa953da4d0e8cd2625bea22bb": "=\\oint_C \\left({1 \\over\\;z^5}+{1 \\over\\;z^4}+{1 \\over 2!\\;z^3} + {1\\over 3!\\;z^2} + {1 \\over 4!\\;z} + {1\\over\\;5!} + {z \\over 6!} + \\cdots\\right)\\,dz.", "e0736de8649ab710df26cf7b254303e9": "\\frac{\\partial f_i}{\\partial t} + \\vec{v}_i\\cdot\\nabla f_i + e\\Bigl(\\vec{E}+\\frac{1}{c}(\\vec{v}\\times\\vec{B})\\Bigr)\\cdot\\frac{\\partial f_i}{\\partial\\vec{p}} = 0\n", "e07373fffade6b2cc34a163b11c09817": "N-n_1-n_2 \\choose n_3", "e07380e9312de7c0ecfc39fb6f0f3660": "\\bar{x}_{i}", "e073b24885521a2c0f4da7139de0a423": " {e^B A e^{-B}} = A + [B,A] + \\frac{1}{2!} [B,[B,A]] + \\frac{1}{3!}[B,[B,[B,A]]] + \\cdots .", "e073b8468f3aebac0bbe06b0c8d7916c": "\\textstyle{\\frac{1}{2}}(d+1)(d+2) - 1 = \\textstyle{\\frac{1}{2}}(d^2 + 3d)", "e073cdcd0e4945de9a42184beff5fba7": "m \\times (n+1)r", "e073dedcf2738241d78ad5573a92bd6c": " k X + b \\sim \\textrm{Frechet}(\\alpha,k s,k m + b)\\,", "e07493fe4be516b7ef7d289606783cf5": " \\mathbf{\\hat U} (t) ", "e074f58f384c0920aa017603534f9be4": "y^2 + 2y + 1", "e0753cf1a701a7ac11370e2ff989be02": "C_D = \\frac {dQ_D}{dv_D} = \\tau_T \\frac {d i_D}{dv_D} = \\frac{i_D \\tau_T}{V_{th}} \\ . ", "e0754a2f15275979cdb833d979cb8dfa": "\\displaystyle w_t+w_{xxx}+3uw_x=0", "e0757a686cba283c6b11e425990e6e04": "\n\\begin{align}\n\\zeta_0 & = [a_0; a_1, a_2, a_3, \\dots]\\\\\n\\zeta_1 & = [a_1; a_2, a_3, a_4, \\dots]\\\\\n\\zeta_2 & = [a_2; a_3, a_4, a_5, \\dots]\\\\\n\\zeta_k & = [a_k; a_{k+1}, a_{k+2}, a_{k+3}, \\dots]. \\,\n\\end{align}\n", "e0757d6e946608faa9898bff2cc194b1": "1,2,\\ldots,2m, 2m+2,\\ldots, 3m+1", "e075aa502f99c7f1de7038c931692d38": "\\text{data density}=\\frac{\\text{number of entries in data matrix}}{\\text{area of data graphic}}", "e075adadcf1695197266980775e3d3cb": "{p \\choose k_1, k_2, \\ldots, k_a} \\equiv 0 \\pmod p \\,\\!", "e075c2077175ea0637ab1b7cbf6d391c": "\nP_i = \\frac{1}{2^{nR}}\\sum_{w}x^2_i(w)\n\\,\\!", "e075f3d20717050120318d730bfe841a": "U(S)", "e075f4b43d3ad226eb28c0493cbc6588": "T_3/T_2 \\,", "e0760e9e34bc67162b061e293ebe2f75": "\\theta = {\\pi\\over 4 }", "e076587641e73b72c7ed334d1eead033": "V_t \\ = \\ V_g - V_w", "e0766d5847066a49bf6babcf1cfe7669": "\\bold{x}=\\{x_1,x_2,\\dots,x_n\\}", "e0768d738c3d8960a6784b81a3275bc5": "\\psi = \\sum_{k=1}^{\\infty} \\frac{1}{F_k} = 3.359885666243 \\dots", "e076a0d7e3415a5506fb653702a42bd3": "\\beta = {\\frac{1}{\\rho}} {\\frac{d\\rho}{dp}}.", "e076b69e4fbf47f627cb5da24e51b973": "S \\to VP\\ NP_{subj}", "e076ce1bd37a6a49a25e13e26dd58f43": "\\cdots\\rightarrow H^q(B^\\bull) \\rightarrow H^q(C^\\bull) \\rightarrow H^{q+1}(A^\\bull) \\rightarrow H^{q+1}(B^\\bull) \\rightarrow\\cdots", "e076eda3d49893df08146a4431eb9cdb": "q_{int,g}=y_g\\cdot\\sqrt {2g\\bigl(E_{int,us}-y_g\\bigr)}=0.5\\cdot\\sqrt{2\\times 32.2\\times \\left(3.63-0.5\\right)}=7.1 \\text{ ft}^2/\\text{s} ", "e0770d66d8585d56f427c0b7789d78e1": "J_3", "e0772f1e7d3fe1f1f46ed8dc1b8173f7": "b = \\pi - \\arcsin \\left( \\frac{\\sin a\\,\\sin \\beta}{\\sin \\alpha} \\right)", "e077465cdc74922485e8a0fa3f8702e4": " e^x \\approx 1 + x.\\ ", "e0782478c269d258d5700672b64b102f": " \\phi(x') < \\phi(x)", "e079009ad345d15c6ce7be662c1e2946": "\\forall p: \\mathcal{B}p \\to p", "e07906f2e7ceaaf2b858625bff7ff0a7": "\\quad B\\in", "e0790d8cfa76e3013eb652843eec7b8c": "\n \\tilde{\\epsilon}_{\\mathbf{k}} \n = \n E_{\\mathbf{k}} \n - \n \\sum_{\\mathbf{k}'} V_{{\\mathbf{k}}'-{\\mathbf{k}}}\n \\left(f^e_{\\mathbf{k}'} + f^h_{\\mathbf{k}'} \\right)\n\\,,\n", "e079706528dc7219342fdef7e167366c": "\\textstyle \\omega_1, \\ldots, \\omega_n", "e0797467cbd3075453bed3a0dc3fd835": "=(a'b+ab')c+(ab)c'=(ab)'c+(ab)c'=((ab)c)',", "e0797e660f11bc9b61ddd5c918c5caf7": "\\operatorname{E}\\,\\hat\\sigma^2\n = \\tfrac{1}{n}\\operatorname{tr}\\Big(\\operatorname{E}\\big[M\\,\\operatorname{E}[\\varepsilon\\varepsilon'|X]\\big]\\Big)\n = \\tfrac{1}{n}\\operatorname{tr}\\big(\\operatorname{E}[\\sigma^2MI]\\big)\n = \\tfrac{1}{n}\\sigma^2\\operatorname{E}\\big[ \\operatorname{tr}\\,M \\big] ", "e079a43a2b059e3a7a7c355a2bc040f1": "Modulation \\ Spectrum \\ Spread \\approx 2 (\\Beta + 1 ) f_m \\sin (\\delta t ) ", "e079efba4748ee53763173e501722dff": "\\overline{S}^{\\dot\\alpha i}", "e079f7d8a0eab5452a8ae17a4d07baff": "\\forall m_t \\in L^{\\infty}_t: \\; \\rho_t(X + m_t) = \\rho_t(X) - m_t", "e079fd17926a958df111a4b3eedb86de": " [\\textrm{CO}_3^{2-}]_{eq} = \\frac{K_1K_2}{[\\textrm{H}^+]_{eq}^2 + K_1[\\textrm{H}^+]_{eq} + K_1K_2} \\times \\textrm{DIC}, ", "e07a551478c59db44ece5956f64ab1b8": " \\rho \\log_2 \\rho = \\int \\lambda \\log_2 \\lambda d P_{\\lambda} .", "e07a68410f47b6c9a23d84e40c588160": "X_1 \\rightarrow X_2", "e07a719d24b0cb2b2d66c06a96847fb3": "\\cfrac{2}{c + \\cfrac{2}{d + \\cfrac{2}{4}}} = a", "e07b00b0153915052a967f597061fb0e": "M_{score} = \\frac{M(A,B)}{\\min(n_A,n_B)}", "e07b32fcb0433f8cb7f724f788cbca57": "p_k\\sim k^{-(2+1/m)}", "e07b38024ba8ae5bf18e19ff61c0c54b": "\\#C(\\mathbb{F}_q)", "e07b50d8fff0107fdccd29c573e9a52a": "\\textstyle x\\in\\Omega", "e07b70ed7e8fb1f80dc5b6370582487f": "A_1,\\ldots,A_n", "e07b93597804e6a9345228a662cda73f": "\\alpha_k \\le \\frac{k-1}{k+4}\\quad(26\\le k\\le 50)\\ ,", "e07bc844fbb03aff4a909fb11ce43532": "\\displaystyle\n\\begin{array}{lcl}\nb_{k} &=& \\frac{A^{k}b_{0}}{\\| A^{k} b_{0} \\|} \\\\\n &=& \\frac{\\left( VJV^{-1} \\right)^{k} b_{0}}{\\|\\left( VJV^{-1} \\right)^{k}b_{0}\\|} \\\\\n &=& \\frac{ VJ^{k}V^{-1} b_{0}}{\\| V J^{k} V^{-1} b_{0}\\|} \\\\\n &=& \\frac{ VJ^{k}V^{-1} \\left( c_{1}v_{1} + c_{2}v_{2} + \\cdots + c_{n}v_{n} \\right)}\n {\\| V J^{k} V^{-1} \\left( c_{1}v_{1} + c_{2}v_{2} + \\cdots + c_{n}v_{n} \\right)\\|} \\\\\n &=& \\frac{ VJ^{k}\\left( c_{1}e_{1} + c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right)}\n {\\| V J^{k} \\left( c_{1}e_{1} + c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right) \\|} \\\\\n &=& \\left( \\frac{\\lambda_{1}}{|\\lambda_{1}|} \\right)^{k} \\frac{c_{1}}{|c_{1}|}\n \\frac{ v_{1} + \\frac{1}{c_{1}} V \\left( \\frac{1}{\\lambda_1} J \\right)^{k} \n \\left( c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right)}\n {\\| v_{1} + \\frac{1}{c_{1}} V \\left( \\frac{1}{\\lambda_1} J \\right)^{k} \n \\left( c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right) \\| }\n \n\\end{array}\n", "e07c4cef54862886f8411cc782cbe311": " \\Delta G_{micelle} = RT* ln (CMC) ", "e07c749b4ac775f9ee6b9a8de4bb8f5c": "\n\\langle\\Delta i_1\\Delta i_2\\rangle = \\langle(i_1-\\langle i_1\\rangle)(i_2-\\langle i_2\\rangle)\\rangle =\\langle i_1i_2\\rangle-\\langle i_1\\langle i_2\\rangle\\rangle -\\langle i_2\\langle i_1\\rangle\\rangle +\\langle i_1\\rangle \\langle i_2\\rangle\n", "e07ca09bb3123e63fa20f06274cabf9a": "\\ v = \\frac{[M]_0-[M]}{[I]_0}", "e07cd9ad0bb7284e57cc5e88487210f5": "\\left\\{{ n \\atop k-1 }\\right\\}", "e07d10581add95a5b4ef55e87cb9d0fa": "h>0", "e07d297563d43de5c9e8233d53c926b2": "MacD = \\sum_{i=1}^{n}t_i\\frac{{CF_i \\cdot e^{-y \\cdot t_i}}} {V} ", "e07d45fb55a136228bce86f0df262391": " {{\\lambda^2 \\alpha} \\over {(\\alpha-1)^2(\\alpha-2)}} \\text{ for } \\alpha > 2 ", "e07d9b338f4f7e95ecf1cebbd439ad2a": ">x ) ", "e07daed7e3f6c33f8dde4528d41f85be": "\\Gamma_1(\\cdot)", "e07dbf0c3b00f6990f789e96023e0bf7": "B'\\subseteq B\\,", "e07dbf515708428e520f1fe7e90057c4": " 0 I_0", "e0ba9410e1672b6fc7e507cef21cba1a": "2^{\\aleph_{\\omega_1}}<\\aleph_{(2^{\\aleph_1})^+}.", "e0baaab01e15adb6024c7cb6c01af7d4": "t_{\\rm ff} < t_{sound}. ", "e0baab99cd06112e6bd73f3f85a94749": "\\int_{-1}^{1} P_m(x) P_n(x)\\,dx = {2 \\over {2n + 1}} \\delta_{mn}", "e0bab8f046feba0d12b91916dc2c9d42": "\\left( {1 \\over 2}\\times1 \\right) + \\left( {1 \\over 2}\\times{1 \\over 3} \\right) = {2 \\over 3}", "e0babe0657f2aff2e7fcc6d735937e04": "\\mu_{3,1}", "e0bad1932a2405eb9703841d652d30c0": "\\mathrm{\\bot_{max}}(a, b) \\le \\bot(a, b) \\le \\bot_{\\mathrm{D}}(a, b)", "e0bb4287232583d6b122b079c119a1fd": "p(x)=\\sum_{n=-N}^N p_n e^{inx}.", "e0bb50e104dfb45e892efc2a807b857f": "O(V^{1/2}E)", "e0bc256622da08f3d27db074ddf988b1": "ui < u_{i+1} za i \\in N, i < p", "e0bc33d0ae257c081d7bf5ae6b959eec": "\\{\\mathbf{X}_k\\}", "e0bd36938f1e8f142ad6871985ee485e": "\\langle\\left(\\nabla_X Y\\right)(m),m\\rangle = 0\\qquad (1).", "e0bdcc62a325a6bab28b0636939d79b4": "{{{6}}}", "e0bdcf4ce51130ff9a81cc21a637587e": "\n \\begin{align}\n & W(1,1,1) = 0 \\\\\n & \\cfrac{\\partial W}{\\partial \\lambda_i}(1,1,1) = \\cfrac{\\partial W}{\\partial \\lambda_j}(1,1,1) ~;~~\n \\cfrac{\\partial^2 W}{\\partial \\lambda_i^2}(1,1,1) = \\cfrac{\\partial^2 W}{\\partial \\lambda_j^2}(1,1,1) \\\\\n & \\cfrac{\\partial^2 W}{\\partial \\lambda_i \\partial \\lambda_j}(1,1,1) = \\mathrm{independent of}~i,j\\ne i \\\\\n & \\cfrac{\\partial^2 W}{\\partial \\lambda_i^2}(1,1,1) - \\cfrac{\\partial^2 W}{\\partial \\lambda_i \\partial \\lambda_j}(1,1,1) + \\cfrac{\\partial W}{\\partial \\lambda_i}(1,1,1) = 2\\mu ~~(i \\ne j)\n \\end{align}\n ", "e0bddb06ec5d2167bc571c8c093c5190": "v_1\\sin\\theta_2\\ = v_2\\sin\\theta_1", "e0bdf4222c7c249e9777a2011414540a": "\\begin{Bmatrix} x & y \\\\ z & v\n\\end{Bmatrix}", "e0bdfd459352e34948f96925db25f6b8": " \\mathbf{a} \\succ^w \\mathbf{b} ", "e0bdfde5284b494df438a954b2dca0aa": "|s_k|<\\frac{b}{GCD(a,b)}\\quad \\text{and} \\quad |t_k|<\\frac{a}{GCD(a,b)}.", "e0be4d6258414ad18eace12e2100c6bc": "0 \\leq x \\leq \\frac{L}{2}", "e0be5a9a3e17780c028c2841a6dc7695": "\n\\frac {\\Delta P} {L} = \\frac {4 K} {D} \\left( \\frac {8 V} {D} \\right) ^ n \\left( \\frac {3 n + 1} {4 n} \\right) ^ n \\frac {1} {1 - X} \\left( \\frac {1} {1 - a X - b X^2 - cX^3} \\right) ^ n\n", "e0be6b1af981fbb043cee04a66a3dbe3": " \\int \\sec^n x \\tan^m x\\, dx ", "e0bea049aa5bec0e304ca1e96c7f5913": "S(\\rho^{12}|\\rho^1)", "e0bf675ff857607b02223f9561d5570f": " \\displaystyle{{d\\over ds} U(s)f= iP U(s)f,\\qquad {d\\over dt} V(t)f=iQV(t)f }", "e0bf77db24f88ffa7944e553eee0ec41": " v\\left( L \\right) ", "e0bf98d0907fb2e98e5f309ae19c070e": "\\tau_{s,0}", "e0bfa00bc5c5a47d7e052e33201a98d3": "cos(2\\pi\\cdot F_{bfo}\\cdot t)\\,", "e0bfe0257b8704462b80354a93117556": "a {}^{(n)} b\\,\\!", "e0c07ce5b6a584a580c974c7b7b13c15": "\\top_{\\mathrm{Luk}}(a, b) = \\max \\{0, a+b-1\\}.", "e0c0b41a3dc794a923f242dc26baf3ae": "\\Pi_\\alpha\\,", "e0c0f29dd648d714ce0da3e29b998ea2": "C = \\sum_{i=1}^n a^{-s_i} \\leq 1", "e0c12d615090d3574f32ebeab63f5601": "I_x", "e0c153c0f6eec92e95c8a8dff96f1272": "(X, \\tau)'", "e0c1bb309aaa0bb64ede6c03c1755a82": "\\mathbf{A} = \\mathbf{a} \\wedge \\mathbf{b} = -\\mathbf{b} \\wedge \\mathbf{a} \\ ,", "e0c1d9ada103b38fa9dbfcf360fa0000": " [\\forall \\text{ internal } A\\subseteq{^*\\mathbb{R}}\\dots] \\,", "e0c1dc4712c9c995c0ca1810bb6840bf": "m_1(x) = m_2(x) = m_4(x) = x^4+x+1,\\,", "e0c2293d25ce627e714c608f04d5ee76": "|FF_j|", "e0c2b5b736d3af49ac76ba3efd734430": "\\textit{occludeopen}(1)", "e0c2d01ce0268b001f1f467c1168ec39": "\\frac{1}{\\sqrt{2}}(1,1,0)", "e0c2ee790eed48bf2effede586a1ad25": "((\\mathbf{a}\\times \\mathbf{b})\\cdot \\mathbf{c})\\;((\\mathbf{d}\\times \\mathbf{e})\\cdot \\mathbf{f}) = \\det\\left[ \\begin{pmatrix}\n \\mathbf{a} \\\\\n \\mathbf{b} \\\\\n \\mathbf{c}\n\\end{pmatrix}\\cdot \\begin{pmatrix}\n \\mathbf{d} & \\mathbf{e} & \\mathbf{f}\n\\end{pmatrix}\\right] = \\det \\begin{bmatrix}\n \\mathbf{a}\\cdot \\mathbf{d} & \\mathbf{a}\\cdot \\mathbf{e} & \\mathbf{a}\\cdot \\mathbf{f} \\\\\n \\mathbf{b}\\cdot \\mathbf{d} & \\mathbf{b}\\cdot \\mathbf{e} & \\mathbf{b}\\cdot \\mathbf{f} \\\\\n \\mathbf{c}\\cdot \\mathbf{d} & \\mathbf{c}\\cdot \\mathbf{e} & \\mathbf{c}\\cdot \\mathbf{f}\n\\end{bmatrix}", "e0c32525ddb2a61d735053061e5cdbaf": "e^{t A} = Q_t(A)", "e0c354566b2f078d1fc1f7c76a089a64": " {\\left( a - \\lambda_{\\pm} \\right)\\over c } = -{\\left( b - \\lambda_{\\mp} \\right) \\over c}. ", "e0c3bd8ba3757c333a9f63cae42a5373": "f\\in [H^1_0(\\Omega)]'", "e0c3cc9a63da79f2b90e9102bf42a564": "\\left(\\frac{1}{|B|} \\int_B \\omega(x) \\, dx \\right)\\left( \\frac{1}{|B|} \\int_B \\omega(x)^\\frac{-p'}{p} \\, dx \\right)^\\frac{p}{p'} \\leq C < \\infty,", "e0c41bcdb433dd1f72add8728d1de6d7": "(B_1,B_2)", "e0c478c81544dbf184050af8da7320d6": "\\ p_{ij\\ldots}(\\mathbf x,t)", "e0c49e621a81093a48d44b9d07c48511": "\\partial_\\alpha {\\star F^{\\alpha\\beta}} = \\frac{1}{c} J^\\beta_{\\mathrm m}", "e0c4a36a80fe571e662a5764a1058dc4": "e^{i z \\cos \\theta}=\\sum_{n=-\\infty}^{\\infty} i^n\\, J_n(z)\\, e^{i n \\theta}", "e0c4d29d9b2a4834ed21cb08ea37a25f": "\\alpha_N", "e0c51e9a0d8eac70ff150a5daaea7166": "V(a,x)=\\frac{\\Gamma(\\tfrac12+a)}{\\pi}[\\sin( \\pi a) D_{-a-\\tfrac12}(x)+D_{-a-\\tfrac12}(-x)] .", "e0c55c8b66d512508e0a058a15cfd9fa": "\\text{FA} = \\sqrt{\\frac{1}{2}} \\frac{\\sqrt{(\\lambda_1 - \\lambda_2)^2 + (\\lambda_2 - \\lambda_3)^2 + (\\lambda_3 - \\lambda_1)^2}}{\\sqrt{\\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2}}", "e0c564c80f74d020a3d231ef9732cb54": "\\beta =\\frac{d \\ln \\Omega}{ d E}.", "e0c59d3dab7c7bb4983b79d1dfe11a68": "x^{i+1} := x^i + \\alpha^i w^i,\\,", "e0c5de3ed0206bae1b8622eeabc86e5f": "\\nRightarrow", "e0c5df3160560ce09fb53c739e289f4c": " d_g(A,B) := \\| \\log(A^\\top B)\\|_F ", "e0c5f3887ca9158997e6b2cf1c44d1f2": "B = -13.1", "e0c604d61dd26a5c60ca63d6d91913fc": "|\\mathbf{a}|\\,|\\mathbf{b}|\\,|\\sin\\theta|,", "e0c6de17e13ee61d0c6da535a46dee5a": "\\displaystyle X_t = \\begin{cases}\n W_{\\min(\\frac{t}{1-t},T)} &\\text{for } 0 \\le t < 1,\\\\\n -1 &\\text{for } 1 \\le t < \\infty.\n \\end{cases} ", "e0c732bf137bd83bf2afcd40837b9195": "f(a,i) = a_i", "e0c7e45fa0909478d0e2f17b08cb27f2": " GG(y;a,\\beta,p) = \\lim_{q \\to \\infty}\nGB1(y;a,b=q^{1/a}\\beta,p,q) ,", "e0c8776d74f96e2977fd54f6af2769a5": "\\begin{pmatrix} 7 \\\\ 8 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 19 \\end{pmatrix} \\to \\begin{pmatrix} H \\\\ I \\end{pmatrix}, \\begin{pmatrix} A \\\\ T \\end{pmatrix}", "e0c8a0321767b6fe53b7e5e226a6c5a9": " \\boldsymbol{\\mathcal{E}} = \\epsilon_{ijk}\\mathbf{e}^i\\otimes\\mathbf{e}^j\\otimes\\mathbf{e}^k ", "e0c8b4d3ced786ae7f5afc075539389f": "\\Delta\\tilde{\\nu}_{D} = \\frac{V}{c}\\tilde{\\nu}_{0}\\cos\\theta ", "e0c8ccabd2706e31d21dd8cc359f7256": "\\dot{q}_j", "e0c8d2441fd03dbfd2227bac5455be15": "\\Pr \\left[ X > x \\right]=x^{-2}\\mbox{ for }x>1,\\ \\Pr[X<1]=0", "e0c91d94315a9b5c2669b47da343c5f8": "0.667 \\times 0.417 = 0.278", "e0c946fb028b82763aba6a7d1c66418a": "\\mathrm{Pic}(A)", "e0c9517d81ad89b406c5d019cbb08c9a": "c_{f,\\mathrm{id}_T} = c_{\\mathrm{id}_S,f} = \\mathrm{id}_{f^*}", "e0c962682555b99fcd2cf56f5ba44948": "h_\\ell^{(2)}", "e0c9aac9cdfcfc509046c222405cbf85": "\\frac{\\overline{d^c}\\overline{u^c}ql}{\\Lambda^2}", "e0c9d5273ea4da9833cb0ec7242bc5a5": "(\\cos x + i\\sin x)^n = \\left(e^{ix}\\right)^n = e^{inx} = \\cos (nx) + i \\sin (nx),", "e0ca3f84eaede1b1a08c46c208499f35": "x - 2\\langle v,x\\rangle v = x - 2 v (v^\\text{H} x), ", "e0ca81e81d8568a7427e71db44f65cfb": " \\Omega_n(t) = \\sum_{j=1}^{n-1} \\frac{B_j}{j!} \\,\n \\sum_{\n k_1 + \\cdots + k_j = n-1 \\atop\n k_1 \\ge 1, \\ldots, k_j \\ge 1}\n \\, \\int_0^t \\,\n \\mathrm{ad}_{\\Omega_{k_1}(\\tau )} \\, \\mathrm{ad}_{\\Omega_{k_2}(\\tau )} \\cdots\n \\, \\mathrm{ad}_{\\Omega_{k_j}(\\tau )} A(\\tau ) \\, d\\tau \\qquad n \\ge 2,", "e0caa8d581f0b8df05bd857a63a2cb9f": "(G, v)", "e0caaf1db93dba4d1a9efff0186bca50": "\\frac{5d}{3}+O(1)", "e0cacb2534b776736a0b05050d071fea": "\\delta \\mathbf{r}_i = \\sum_{j=1}^m \\frac {\\partial \\mathbf {r}_i} {\\partial q_j} \\delta q_j,", "e0cacb2a047e446c5c05c1b643fa2bd8": "{x^3 + 7x^2 + 8x + 2 \\over x + 1} = x^2 + 6x + 2", "e0cb24084bf813f665c1d389b51449ce": "({x}',{y}')", "e0cb8a7eb099ba7e5db4d49b536aea54": "W_N(x)\\,", "e0cb994439dab0f4294f91b8081c40f2": "\\psi\\in\\mathcal{H}", "e0cbc50f56b7a16e8c8bcf9d4e3cd607": "\\delta \\int d\\tau = 0\\,", "e0cc4deee372e25a292280cce23a095a": " {\\mathbb E}\\bigl[|XY|\\bigr] \\le \\bigl({\\mathbb E}\\bigl[|X|^p\\bigr]\\bigr)^{1/p}\\; \\bigl( {\\mathbb E}\\bigl[|Y|^q\\bigr]\\bigr)^{1/q}.", "e0ccfde9fdeca7928efedf17293be708": " D \\cap \\partial M = \\beta ", "e0cd0f26322ba3076d18664b24df3051": " \\{ z \\in \\mathbf{C} \\mid |z+1| \\le 1 \\}, ", "e0cd1d82c04e447377dea1daa80df8c7": "\n\\int_1^\\infty e^{iax}\\frac{\\ln x}{x^2}dx\n=1+ia[-\\frac{\\pi^2}{24}+\\gamma\\left(\\frac{\\gamma}{2}+\\ln a-1\\right)+\\frac{\\ln^2 a}{2}-\\ln a+1\n-\\frac{i\\pi}{2}(\\gamma+\\ln a-1)]+\\sum_{n\\ge 1}\\frac{(ia)^{n+1}}{(n+1)!n^2}.\n", "e0cd4c8c202ea3592abacce9c7aad4cc": "I_{n} = \\int x^{ax} \\cos^n{bx} dx\\,\\!", "e0cd876c3cd0d04cada971dc2e3e7068": "\\left( e\\left( \\rho \\right)=\\sigma /\\rho \\right)", "e0cdae50b71a7d6de1d1657ded231613": "\\mathcal B([0,\\infty))\\otimes \\mathcal E^* ", "e0cdc2e194c4424c35ae930ff0f4ffb7": "(10)\\quad ds^2\\approx-\\Big(1+2\\psi(\\rho,z)\\Big)\\,dt^2+\\Big(1-2\\psi(\\rho,z)\\Big)\\Big[e^{2\\gamma}(d\\rho^2+dz^2)+\\rho^2 d\\phi^2\\Big]\\,.", "e0cec141c3a80499734d68bf756548db": "w\\Vdash(A\\to B)[e]", "e0cecc48689a9f7e6a9a726be8994503": " -21 = 7 \\times (-3)", "e0ced8d496b692cad1976fec2214d652": "T^*(\\eta_{T(V)}(T(v))) = \\eta_{V}(v)", "e0cf10bf58c225de3a9a5a7e6ef873d3": "\\varphi(\\varphi(u))=\\exp(u)", "e0cf142862a317fe3ce1309f7be728a3": " \\begin{align} a_{n+1} & = \\frac{\\sqrt{a_n} + 1/\\sqrt{a_n}}{2} \\\\\n b_{n+1} & = \\frac{(1 + b_n) \\sqrt{a_n}}{a_n + b_n} \\\\\n p_{n+1} & = \\frac{(1 + a_{n+1})\\, p_n b_{n+1}}{1 + b_{n+1}}\n \\end{align}\n", "e0cf1487c95d62516683004e72ca336d": "\nx = a \\ \\frac{\\sinh \\tau}{\\cosh \\tau - \\cos \\sigma} \\cos \\phi\n", "e0cf174b4790e7b03231cca74ffbf349": "\\wedge^3", "e0cfa3f7b19ea334d2c3a8c024606d8d": "d= \\frac{N}{P_d \\cos \\psi}", "e0cfcc317ed36668ca0eb17953a85d7b": "\\frac{\\pi_5}{\\pi_2 \\pi_3} = \\frac{z_0 q^2}{4 \\pi a \\varepsilon_r \\varepsilon_0 k_b T} = Z_0", "e0cfdadf8df1ada7c4dcbc368a0abf36": " \\Phi = \\min_{S \\subset X, \\pi(S) \\leq \\frac{1}{2}} \\frac{Q (S \\times S^c)}{\\pi(S)}. ", "e0cff5fe162b34a29b2124bb7bf93fa5": "\\hat d_\\infty(\\mathbf x, \\mathbf y)=0.", "e0d03ccdf693ee6b34cb1dc577c1eb99": "RRR", "e0d09e8d52c2fa24dbd598b5a5cce80f": "\\tau_s", "e0d1011c1f001081cb9dbde73bcd94c4": "\\mbox{ for all } X,Y \\in \\mathcal{C}, X \\neq -Y, \\mbox{ and } e \\in X^+ \\cap Y^- \\mbox{ there is a } Z \\in \\mathcal{C} \\mbox{ such that }", "e0d17c2039354f48f608c271ec70590b": "d=1 \\;mod\\; n^s", "e0d2038bfa6a660a16d4005225ec37bc": "v_1 = \\left( {\\sqrt 3 \\over 2} , {1 \\over 2} \\right) \\,", "e0d20426027f855f5cbb109bd790e135": "\\scriptstyle \\vec{r}_u=\\tfrac{\\partial \\vec{r}}{\\partial u}", "e0d2045f248f706669a28f3fd68b0235": "\nV^\\ast(j,p) = p\\int^{\\infty}_0 V(j,t) e^{-pt} \\, dt\n", "e0d22bd42367fe98508182ab6ca5b890": "f(\\lambda x)=\\lambda^{\\Delta}f(x)", "e0d23995db5326bdb56afede78f2751a": "dW_{input} = e ~ i ~ dt \\;", "e0d24eb096dc4fc7b9af96d65ace3d31": "p_w", "e0d2bd935f0f3ef7691be53694acae17": "\\text{As } n = 12\\text{, }n^2=144 \\text{ and } n^3 = 1728. \\text{ Also}", "e0d2ee2bbd1982d42cb6c79bc6245ef4": "\\alpha^{\\mathrm{N} \\mathfrak{p} -1}\\equiv 1 \\pmod{\\mathfrak{p} }. \n", "e0d330236be1219cd4e69fa32b749204": "|\\lambda_j/\\lambda_1| < 1", "e0d3f6d00c7629d7bf828906e2249778": "u''=0", "e0d3fe8bbe543ab5fb72deba57594073": "\\mathbb{H} := \\{ (x, y) \\in \\mathbb{R}^2 | x^2 + y^2 < 1 \\}", "e0d43646a069b744c730fb276e36f7e7": "u_{i}=\\left( z_{i}|x_{i}\\right) ", "e0d43ae8f44f80869d3e49612fcfbe17": "\\sum_{P \\in C}{\\mathrm{ord}_{P}(f)} = 0", "e0d456933b6dae630375ca93840eac84": "d_p=\\int_{0}^{t_0} \\frac{c}{a(t)}dt\\ .", "e0d4b5baf876bfbfcf1a220d6b2be8f3": "{\\sqrt{a^2+DC^2}}\\,", "e0d4c078cfc9f5a35928027a3b4941e6": "\\psi(\\mathbf{r}) \\equiv \\lang \\mathbf{r} | \\psi \\rang. ", "e0d4ce8cd45f164d47dd5f3fd65bcb6d": "\\eta = \\phi + \\pi ", "e0d51331fa6701957c192990a0e58c35": "C(10^n)", "e0d521b10746e43a1fb2b243f7cfea71": "\n\\int \\left(\\hat\\theta-\\theta\\right) f \\, \\frac{\\partial \\log f}{\\partial\\theta} \\, dx = 1.\n", "e0d5cdde67c331825b643c3713fd3351": " time = \\frac{u_n N_d N_t}{2}", "e0d6133c9630d0543a47fcaaf6d3426a": "x=r\\cos\\theta", "e0d6408a8b5415146a30057977f4c423": "\\mathbb{J}(\\mathbf{r})", "e0d6d99508e7ba8402e1112e2c0099c6": "\\tfrac{m}{n}<2", "e0d6df13211cd361c356c79f6d2d438f": "|A|\\cdot|B|", "e0d72fa068f09feefd095dd339498d02": "O(n^{d-1} \\log n)", "e0d7b08aa3dc4bbc78883342dcd4d376": " \\sqrt{\\pi}x e^{x^2}{\\rm erfc}(x) = 1+\\sum_{n=1}^\\infty (-1)^n \\frac{(2n)!}{n!(2x)^{2n}}.", "e0d7dd08c6ebabbd732ff53a27d3bfcb": " F(x) \\approx \\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\left[-\\frac{(x-\\mu)^2}{2\\sigma^2}\\right]\\left[1+\\frac{\\kappa_3}{3!\\sigma^3}H_3\\left(\\frac{x-\\mu}{\\sigma}\\right)+\\frac{\\kappa_4}{4!\\sigma^4}H_4\\left(\\frac{x-\\mu}{\\sigma}\\right)\\right]\\,,", "e0d816a1f6f8758d89c63f435fc79833": " \\tilde{f} \\colon X \\to Y^I", "e0d82897decea4418d8cf2a11e25c19a": "R_s/R=(1+\\tan \\theta \\tan \\alpha)\\sec \\alpha ", "e0d8483d89777e7bf9b6c36b9d0d9014": "E_d / E_k ={sin^2(\\alpha)}", "e0d8bfdaf0013dc5430a560f27f4ac53": "E\\mapsto \\int_E f\\, d\\mu", "e0d8fbe5c17355e14126ef53e771678b": "||x_i - x_k||_2^2", "e0d9181c4c4abe297452787dbe3fba7c": "\n\\frac{d\\mathbf M}{dt} = -\\gamma\\mathbf M \\times\\mathbf H - \\frac{\\lambda\\mathbf M \\times\\mathbf M \\times\\mathbf H}{M^2}\n", "e0d95850449f6681e6ed1706ea8ec400": "1 \\div +0 = +\\infty", "e0d95f16add3a459a5fe0e4c898f0a49": "\\rho_{liquid}=\\frac{m_{fullliquid}-m_{evactube}}{V}", "e0d97ea228baa3ae6059d47fa9d60fea": "\n\\sqrt{ {\\rm var} (T) {\\rm var} (V)} \\geq \\left| {\\rm cov}(V,T) \\right| = \\left | \\psi^\\prime (\\theta)\n\\right |", "e0d9ecb649f13892a311a8ee8a2b3abc": "\\| v_{n} - u_{0} \\| \\to 0 \\mbox{ as } n \\to \\infty.", "e0d9fa2bc0660828334b493044b607a6": " R", "e0da00a3dbdec4694e94989efd24064f": "x(t)^{1/n} = t+\\cdots", "e0da0752208c84d085d1522d7333bb66": "\\ 4 e_1 := e_1 + e_1 + e_1 + e_1 ", "e0da2bda0049d867625b1ee3629caab9": "(x_0,x_1,x_2,x_3) = (ct,x,y,z)", "e0da7319098127e8f6e1749ebfa90801": " \\left | \\int_S{f\\,d\\mu} - \\int_S{f_n\\,d\\mu} \\right|= \\left| \\int_S{(f-f_n)\\,d\\mu} \\right|\\le \\int_S{|f-f_n|\\,d\\mu}.", "e0dab0713a3fa99a395a4a3e372f8a65": "\\left({\\mathit{He}}_n^{[\\alpha]}\\circ {\\mathit{He}}^{[\\beta]}\\right)(x)={\\mathit{He}}_n^{[\\alpha+\\beta]}(x)\\,\\!", "e0db2f5d9468a546ba33f557b59ebb8c": "[\\gamma^i,\\gamma^j]_{+} = 2\\eta^{ij}", "e0dba033be6507796e34b794caf5a684": " \\frac{v_a^2}{2} - \\frac{v_p^2}{2} = \\frac{GM}{r_a} - \\frac{GM}{r_p} ", "e0dbbc12ff7e036ba97d5eec7e728bbd": "\n\\mathbf{F}_{\\mathrm{Euler}} = \n-m\\frac{d\\boldsymbol\\Omega}{dt} \\times \\mathbf{r}\n", "e0dbc48d9d5e9e92d4bb02359105362f": "\\left(H^\\dagger W^{\\mu\\nu}H\\right)\\left(H^\\dagger W_{\\mu\\nu}H\\right)/\\Lambda^4", "e0dbc74ac3959813d9af5b3a3cb3986a": "G_1A \\approx \\begin{pmatrix}\n12 & -51 & 4 \\\\\n7.21110 & 125.6396 & -33.83671 \\\\\n0 & 112.6041 & -71.83368\n\\end{pmatrix}", "e0dc0d9d916f6143c47199c93c21fc9f": "\\epsilon_t", "e0dce255f2c978248bfbcdd934853156": " {3 \\over 112} (b-a)^4 ", "e0dcf2a004aa67cdbebfb2064df9952c": "d_j = \\frac{1}{3}\\left(\\frac{4}{6} + \\frac{4}{5} + \\frac{4-0}{4}\\right) = 0.822", "e0dd01ea72b7dd2366a53f0668e29902": " (\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta)^{\\rm T}(\\mathbf{y}- \\mathbf{X} \\boldsymbol\\beta) + (\\boldsymbol\\beta - \\boldsymbol\\mu_0)^{\\rm T}\\boldsymbol\\Lambda_0(\\boldsymbol\\beta - \\boldsymbol\\mu_0) = (\\boldsymbol\\beta-\\boldsymbol\\mu_n)^{\\rm T}(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\boldsymbol\\Lambda_0)(\\boldsymbol\\beta-\\boldsymbol\\mu_n)+\\mathbf{y}^{\\rm T}\\mathbf{y}-\\boldsymbol\\mu_n^{\\rm T}(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\boldsymbol\\Lambda_0)\\boldsymbol\\mu_n+\\boldsymbol\\mu_0^{\\rm T}\\boldsymbol\\Lambda_0\\boldsymbol\\mu_0 .", "e0dd163665dd09a27ddfee200002385a": "\\psi_0=\\left( \\frac{\\alpha}{\\pi}\\right)^\\frac{1}{4}e^{-\\alpha x^2/2}", "e0dd7679b4b96d87298af6133e0c64d2": "dz = -\\frac{A}{2 \\sigma^4} d \\sigma^2,", "e0dde16626a346a75f77e296c6539baf": " T ", "e0de1e6901e3411f3d7413e5bbf9bbb0": "N'_{\\rm covering}(\\varepsilon)", "e0de783582c53e5738fedf687becddd5": "\\langle\\partial T,\\alpha\\rangle = \\langle T, d\\alpha\\rangle", "e0de961c90353ef27ddf04d6e3dba693": "\n= \\frac{1}{\\sqrt2}\\begin{vmatrix} \\chi_1(\\mathbf{x}_1) & \\chi_2(\\mathbf{x}_1) \\\\ \\chi_1(\\mathbf{x}_2) & \\chi_2(\\mathbf{x}_2) \\end{vmatrix}\n", "e0deac9c73fbf2e7b31ec2ad62a74115": "P_{A}dA", "e0def8b241a7468607d052ce32f6b3c5": "\\mathbf{n} \\times (\\mathbf{E}_{scat} + \\mathbf{E}_{inc}) = \\mathbf{E}_{int}", "e0deff3fbc99a6d781fe6eaa5582eb31": "\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n =\n (-1)^S\n \\begin{Bmatrix}\n j_4 & j_5 & j_6\\\\\n j_1 & j_2 & j_3\\\\\n j_7 & j_8 & j_9\n \\end{Bmatrix}\n =\n (-1)^S\n \\begin{Bmatrix}\n j_2 & j_1 & j_3\\\\\n j_5 & j_4 & j_6\\\\\n j_8 & j_7 & j_9\n \\end{Bmatrix}.\n", "e0df6f1bb27f2501dc54315b96d0abf7": "h\\left(k\\right)=k\\mod 11", "e0df6fcc6eca8f0589ff7a708632e3fb": "w(\\mathbf{e_3}) \\leq 2", "e0df9ba8faaf5bbfded2ec8b78384666": "(S^n).", "e0df9cd7445b47178b8c4bac9eb669e1": "(x,\\,y)", "e0dfcbe1353a7e5dad29ce2c443cac60": " | \\psi_e(t) \\rangle = e^{-iHt / \\hbar} | \\psi_e(0) \\rangle ", "e0e03192cd6c445de063c7c99a9ed422": "\\Delta g_{AB}=g_{copper}g_{S3}^{-1}=\\begin{bmatrix}\n0.970 & 0.149 & -0.194 \\\\\n-0.099 & 0.965 & 0.244 \\\\\n0.224 & -0.218 & 0.950 \\\\\n\\end{bmatrix}", "e0e0496fb694151500eff9c75630c998": "IV(Ex,a)= -\\sum_{v\\in values(a)} \\frac{|\\{x\\in Ex|value(x,a)=v\\}|}{|Ex|} * \\log_2\\left(\\frac{|\\{x\\in Ex|value(x,a)=v\\}|}{|Ex|}\\right)", "e0e07da860973d8897979f2391c6fb66": "\\zeta_G(u) = \\frac{1}{\\det (I-Tu)}~,", "e0e0a37af16fba09911539853a8d1014": "\\alpha = xi + yj + zk\\,", "e0e0b18a1888e9fd58a2aaa39766052d": " J_{(h)}= M_{h,k} J_{(k)} M_{h,k}^T ", "e0e0d993992e68e84be17ee8a853fbe8": "Decrease.waste.using.com.lib=waste.unit*(num.users.individual.cons(scalar)-num.max.rent.units.stocked(scalar)) ", "e0e0fb40068ddc8943fc67ea03d23e75": "{m+n \\choose m,n}", "e0e1392b5e268ab54650519f2f9906ca": "f\\left(t,T\\right)", "e0e17929666b7eeb9134d942bfabf41d": "t_i \\,", "e0e18e34390160a3ed1ccb2cc87f41e7": "\\theta = \\sin^{-1}(1) = \\begin{matrix}\\frac{\\pi}{2}\\end{matrix} = 90^{\\circ}", "e0e1c1bcb6eaacbb4acd3c0ef48d02e2": " P(X_1^n|A_m) ", "e0e1f013b03d08fca957983963d81901": "(a,\\infty)=\\{x\\,|\\,x>a\\}", "e0e2234d8d2e121b7b2313f43667859e": "0=-\\dot Q + \\dot m h_1 - \\dot m h_2 + P.", "e0e22bac469fe0a352f5aaadb3156df6": " (s\\alpha)^\\vee= s(\\alpha^\\vee).", "e0e25346fa31e961a41148e8b57d360e": "\\mathbb{C}-\\mathbb{R}_{\\ge 0}", "e0e25851cff045ae5188e0b83c50429d": "u, v\\in (V\\cup\\Sigma)^{*}, ", "e0e2c8e6acbf22b8acbafd13243be4b7": " r^2\\frac{d}{dr}\\left(\\frac{1}{\\rho}\\frac{dP}{dr}\\right)+\\frac{2r}{\\rho}\\frac{dP}{dr}\n==\\frac{d}{dr}\\left(\\frac{r^2}{\\rho}\\frac{dP}{dr}\\right)==-4\\pi Gr^2\\rho ", "e0e3460ffe73980992b8233d041e561e": "\\scriptstyle(x_t,\\,y_t)_{t=1}^T", "e0e34a80050950605634a61edd871a76": "S_n + E_n", "e0e37ad5cb67048f62e6c2f7be1576da": "\nE^{(1)}_\\mathrm{electrostatic} = \\langle \\Phi_0^A \\Phi_0^B| V^{AB}| \\Phi_0^A \\Phi_0^B \\rangle .\n", "e0e3b7a7eddbd4d06b9d236885eb80c0": "{p\\choose 3} = \\tfrac16p(p-1)(p-2)", "e0e3e9fd8bcb08670790b6e31dca2dbb": "\\varepsilon_\\alpha", "e0e3fe05bb31da2288a0565031485410": "a\\,e \\ ", "e0e40d47c6dbd6a0e782ccc5b7b83e78": "\\Delta x = vt", "e0e43b15327bd385a14428853ff33bb2": "q_{0i}=(s_{0i},ta_i(s_{0i}),0)", "e0e4d6001b7d33174b8487f797f4f768": "\ni=1,...,N_p\n", "e0e502e7356675f547604bf1d73e88df": "\n\\sigma(n,m)=\\binom{n+m-1}{m}\n", "e0e581b6a6fe58ee2c686cf6bbff99fb": "\\int_M A\\wedge dA.", "e0e5c039ca155b84ca400d683c1bcc37": "z=r e^{\\phi i}\\,", "e0e66ce658615e19a79a2a14a47f8b09": "\\omega_0\\tau", "e0e6bceff0b3e784af476c1a1f80ce1e": "S_n^\\wedge", "e0e6c3d38c4ab7a8b0a15389c99638fb": "\\left(\\frac{1}{\\sqrt{10}},\\ \\frac{-7}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "e0e6e27b718db97467b07be2587ab0c5": "F(X) \\xrightarrow{F(u)} F(Y) \\xrightarrow{F(v)} F(Z) \\xrightarrow {\\eta_X F(w)} F(X)[1]", "e0e72dd9ad7755508fdeb015700336d1": "\\lambda\\geq 1/2", "e0e7890039a0ad1401a6732d11f06401": " \\phi: A \\times B \\to A \\otimes_R B ", "e0e795e021e5447ecf956fee87d76ff9": "P(x)=x^3+2x^2-x-2.\\,\\!", "e0e7dc10f9d3f844d7ec95285c5606b0": "E = \\frac{p^2}{2m} = \\frac{\\hbar^2 k^2}{2 m}", "e0e7e05f53c763a38fb97ad479c1bca0": " \\lim_{\\omega \\to \\pm \\omega_0 } Z(\\omega) = \\infty ", "e0e7faeaa79fa168208f25151fc27263": "-h(x)\\leq 0", "e0e7fbbc58fb5a5f891f280f04e86f0d": "K_0(R) \\cong \\mathbb{Z} \\oplus Cl(R)", "e0e8026d0b6e63126a2586a1f1fffd59": "\\underline{x}_0", "e0e86afa587720f08d8e55b2705a0d2e": "q_{c,t_n}=q_c=q_{c,t_0}\\, \\forall c", "e0e8df1bb589167f4821658def3ef372": "(-1,0)", "e0e9a3b45d64ee063d1dcd6155b81e9a": "\\left| \\frac{fl(x) - x}{x} \\right| \\le \\Epsilon_\\text{mach}.", "e0e9dab3826a3a757cc745904086602f": "\\hat\\alpha = \\alpha", "e0ea2acf262d0cd8c5288d7caec000fc": " (\\Box + m^2) \\psi = 0. \\, ", "e0ea61948a91e2deb56d3d78ebe81e75": "\\mathcal{L}^{-1} \\left\\{ \\frac{s\\cos\\phi - \\omega \\sin\\phi}{s^2+\\omega^2} \\right\\} = \\cos{(\\omega t+\\phi)}. \\ ", "e0ea79af8d8e9643c7d2eed51b22e061": " f(X;\\theta) = g(T(X), \\theta) h(X) \\!", "e0eab75530b8c21dd17fe833e89dd584": " \\|x\\|^2 = \\|x^* x\\| = \\sup\\{|\\lambda| : x^* x - \\lambda \\,1 \\text{ is not invertible} \\}.", "e0eae0a4536ef3450f8d71a408487cfc": "x+3 = -2 \\quad\\text{or}\\quad x+3 = 2,", "e0eb1ee04b847ed9452ef78c686f06e6": "\\beta_{n+1}", "e0eb3765928a114d9b90fae391633c1f": "a^x", "e0eb4f97fded441aae4a8b5766c218ad": "\\mathbf{x}_0 \\in N,", "e0ec31f3d38a5ffba2be9bac7a5abde0": "(L_i',R_i') = \\mathrm H^{-1}(L_{i+1}' - T_i, R_{i+1}' - T_i)", "e0eca907c201593213da66f59ba1bbfc": " \\lambda_d(E)\\leq \\lambda_d \\Bigl( \\bigcup_{j\\in J}B_{j} \\Bigr) \\leq \\lambda_d \\Bigl( \\bigcup_{j\\in J'}5B_{j} \\Bigr)\\leq \\sum_{j\\in J'} \\lambda_d(5 B_{j}).", "e0ecc3d7038da0f45340bc100e36f904": "[X,Y]_x = x\\cdot[X,Y]", "e0ecd2c350065dfa1b08945634139154": "\\Delta V^{\\ddagger} = \\bar{V}_{\\ddagger} - \\bar{V}_A - \\bar{V}_B ", "e0ed5c850dd2fb71e76bad9ba312c8a1": "1) A \\rightarrow KDC : ID_A || ID_B ", "e0ed5df7b7decb0120dd5cc2307c2f65": "E(x_s^{i_p},x_b^{j_q})", "e0edb9e586e97034e47c1de2d14f19b9": "16*x^4+5152*x^3+508816*x^2+15146880*x+138297600", "e0eddacefa3e910737616e5aa86a68ef": "H+\\vec{v}", "e0ee1560f493ef7abb823fa3bba42f89": "\\qquad\\mathbf F_1=k_e\\frac{q_1q_2}{{|\\mathbf r_{21}|}^2} \\mathbf{\\hat{r}}_{21},\\qquad", "e0ee29040e4787a42c912892c8ed8af9": "\\frac{ P^m_n(\\sin\\theta) \\cos m\\varphi}{r^{n+1}}\\,, \\quad 1 \\le m \\le n \\quad n=1,2,\\dots", "e0ee86a6e0446425ac66e35bc988fe36": "\\arctan(x)", "e0eefc87a6f35c674841f00176d7453f": " {1\\over \\alpha }\\left(\\frac{P(0,t)}{P(0,T)} - 1\\right)", "e0ef832dc63a59ea4367f6c4234b07f2": "E(u)=E_s(u)E_c(u)E_d(u)E_v(u)E_D(u),\\,", "e0ef9aeb9bfe306822adf00f9cf99e26": "J_+|j\\,j\\rangle = 0, \\,", "e0efdd16a09e02704f761555b9099227": " (x_1,x_2)", "e0f0ce44e31c271a77143b10c66167c6": "KM = KB + BM", "e0f14103abdaee64fd8b432e419d2dbd": "~k=\\sin(\\theta)\\frac{mv}{\\hbar}~", "e0f16a44c4e618908705d58e6adbac9e": "\\psi_k", "e0f1ec5c1e9b379dff872f31f1464bae": "M^{*m}(A) = \\limsup_{\\varepsilon \\to 0} \\frac{\\mu(A_\\varepsilon) - \\mu(A)}{\\alpha_{n-m}\\varepsilon^{n-m}}.", "e0f205155645e345b84968240f7b39c2": "\\frac{G}{N} = \\mu = \\mu^\\circ + kT\\ln \\frac{p}{{p^\\circ }}.", "e0f21b1c52f06d70078ed95a248e06c8": "r^2=b^2\\cos^2\\alpha=b^2-b^2\\sin^2\\alpha=b^2-\\left[a\\sin\\theta\\pm\\sqrt{c^2-a^2\\cos^2\\theta}\\right]^2.,\\,", "e0f22d76ee068d6c0834f1456f1ef099": "\n S_L = 1 + \\frac{0.015 \\left( \\bar{L} - 50 \\right)^2}{\\sqrt{20 + {\\left(\\bar{L} - 50 \\right)}^2} } \\quad\n S_C = 1+0.045 \\bar{C}^\\prime \\quad\n S_H = 1+0.015 \\bar{C}^\\prime T\n", "e0f235f25b973ed0f3e9e9c2b50aa2d4": "\\forall m_\\bullet \\forall X_\\bullet \\exists Z \\forall n (n\\in Z \\leftrightarrow \\varphi(n))", "e0f261d3243d12414e8bd16ea41385df": " {d \\mathbf{p}_1\\over dt} \n= {q_1 q_2 \\over r^2}{\\hat{\\mathbf r}}\n+{q_1 q_2 \\over r^2}{1\\over 2c^2} \n\\left\\{ \\mathbf v_1 \\left( { {\\hat{\\mathbf r}}\\cdot \\mathbf v_2} \\right) \n+ \\mathbf v_2 \\left( { {\\hat{\\mathbf r}}\\cdot \\mathbf v_1}\\right)\n- {\\hat{\\mathbf r}} \\left[ \\mathbf v_1 \\cdot \\left( \\mathbf 1 +3 {\\hat{\\mathbf r}}{\\hat{\\mathbf r}}\\right)\\cdot \\mathbf v_2\\right]\n \\right\\} ", "e0f26f9cec3e34b8854086879aeda7b1": "C=\\left(\\begin{matrix}0&1\\\\-1&0\\end{matrix}\\right),", "e0f3b3dd1fcf49b3858ce38c9fbb4d4a": "\\left(x,\\ y\\right)", "e0f3dba3248a6ccb26950955635d93e2": "NM", "e0f3e52fcf00b5d8507e5ec4e66bf400": "u_1,\\,u_2", "e0f3ed485a62afbfe77bd0f85f6c301c": "\\! h_i(x)", "e0f3f2a1c09247f350edaa1a6cddc686": "\\lim_{k \\to \\infty} R_k(a, b) = s(a, b)", "e0f49000fae0611ed8432d9bd18d9d7e": "[Y], [Z]", "e0f49afdfee8f6f53943bd18be972d01": "V = V \\vert_{K_1} \\oplus V\\vert_{K_2} = (\\oplus_{\\alpha \\in A} S) \\oplus U,", "e0f4baad3a5c6e4b9a24fe93f815757c": "C':y'^2+\\bar{h}(x')y'=\\bar{f}(x')", "e0f4cc211aca27df4aaf8284c8cc7f82": "\n0\\sigma}t_{ijj'} c^{\\dagger}_{ij\\sigma}c_{ij'\\sigma} + \\sum_{ij,\\sigma}(V_j f^{\\dagger}_{i\\sigma}c_{ij\\sigma} + V_j^* c^{\\dagger}_{ij\\sigma}f_{i\\sigma}) + \\sum_{i\\sigma,i'\\sigma '} \\frac{U}{2}n_{i\\sigma}n_{i'\\sigma '}", "e0f6b041de64ceb13816535eecd66fea": "\\ D_{heel} = D_{pitch} ", "e0f70dfaa4254b9439196468cf3018fc": "n^2 + n + p \\, ", "e0f71c19ee796df9cd083449910ccd88": "\\mathbf{M}^3,\\mathbf{N}^3", "e0f74078e4c565b6aa4c5b61105ca668": "p_{\\mathbf{s}}=(1-\\tanh(\\mathbf{s})^2)", "e0f74f1551b36c62968994dfea9098f5": "2^{2^{0}} + 1 = 2^{1} + 1 = 3,", "e0f760449af0e3fd27bbb7a8cb6d9e16": "\\ \\Delta R(\\theta, c)= R_A(\\theta) - R_0(\\theta)", "e0f7628be8e20bdb62c5510915ae6772": "G = \\frac {\\omega} {resistance \\times reluctance} = \\frac {\\omega \\mu \\sigma A_m A_e} {l_m l_e}", "e0f791c6958cd4b876bc2482a73acad4": "|v-c|/c", "e0f7d4fdb96d87fe3aaacbfae83fc88e": "\\begin{cases}\n \\text{any value in } [0,1] & \\text{for } n=1 \\\\\n \\frac{n}{2} & \\text{otherwise}\n \\end{cases}", "e0f8007e645c7d5d28913299e518746e": "\\mathbf E_{1s} = <\\psi_{1s}|\\mathbf - \\frac{\\nabla^2}{2}|\\psi_{1s}>+<\\psi_{1s}| - \\frac{\\mathbf Z}{r}|\\psi_{1s}>", "e0f8e490f747615e49c169b434076c36": "\n \\mathcal{M} := D \\left(\\frac{\\partial \\varphi_1}{\\partial x_1} + \\frac{\\partial \\varphi_2}{\\partial x_2}\\right)\n", "e0f96f71d9198ce075aa53a8e86a87d8": "A^j:=\n\\begin{pmatrix} \\frac{\\partial f_1^j}{\\partial u_1} & \\cdots & \\frac{\\partial f_1^j}{\\partial u_s} \\\\ \n\\vdots & \\ddots & \\vdots \\\\ \n\\frac{\\partial f_s^j}{\\partial u_1} & \\cdots &\n\\frac{\\partial f_s^j}{\\partial u_s}\n\\end{pmatrix}\n,\\text{ for }j = 1, \\ldots, d.", "e0f9e3e1f909c1e2c9926d64d94b4641": "|w|< 1", "e0fa0c6717bbb46d7926c8f184a258e7": "\\frac{\\partial^2\\rho}{\\partial t^2} - \\nabla^2 p + \\nabla\\cdot\\nabla\\cdot\\sigma = \\nabla\\cdot\\nabla\\cdot(\\rho\\mathbf{v}\\otimes\\mathbf{v}).", "e0fa147b4bf11a8f8c38d1521b73067e": "d_n = e^ {-1.12436 - 0.11594n}\\ \\mathrm{inch} = e^ {2.1104 - 0.11594n}\\ \\mathrm{mm} ", "e0fa1d8b3153d0ea997f6b8559a2813a": "[X, G/O] \\to L_n(\\mathbf{Z}[\\pi_1(X)]).", "e0fa3b096fa853e6a4e762ae4124d939": "F-P", "e0fa3d331753774cae0a0c2e085813a8": "I_{str}=-\\frac{\\epsilon_{rs}\\epsilon_0 a^2 \\pi}{\\eta} \\frac{\\Delta P}{L} \\zeta ", "e0fac8db03a3d5b3d303d98048cf67d9": "Q(z) = \\sum_{d|k} \\mu(k/d) \\times Q_d(z)\n= \\sum_{d|k} \\mu(k/d) \\exp\\left(\\sum_{m|d} \\frac{z^m}{m}\\right).", "e0fb0cf7523f93b5614167419c539f4b": "B(V)", "e0fb44faf09f26d48386b5b22ab46938": "h(f) = H(f)'", "e0fb5a5b04eb90e0f4965967f44ac34a": "\\int_M d\\omega = \\oint_{\\partial M} \\omega.\\!\\,", "e0fbc862c9bf01aef45b77337d894697": "\\begin{matrix} ((g_m+g_{mb}) r_\\mathrm{O}+1) \\frac {R_L} {r_O + R_L} \\end{matrix}", "e0fc1c22308f9a4416b807efad33f0c9": "E_1\\ ,\\ E_2\\ ,\\ E_3\\ ,\\ E_4", "e0fc708a849c12e023366a85af13dadf": "{\\rm div\\,\\,}\\vec A\\stackrel{!}{=}0", "e0fc743e5f958ede9fbea2a2b85d1757": "\\beta_1,\\ldots,\\beta_k", "e0fc74f87e6d699e1050c2d07401083f": "h(x)=\\sum_{n=1}^\\infty {a_n \\over n!} x^n", "e0fc8cc236573be2f718aa53c43012b8": " \\phi= \\zeta * e^{\\kappa h} ", "e0fc93c2ef5c2cfdf97764ebbae1ea5c": "D_{p}", "e0fca2e5df59ba74d3a35e73214753fd": "K\\in\\{1,\\ldots,N\\}", "e0fce2fe34415234798d3c38ff9eeafc": " C_{D_0} = c_{d_0}", "e0fcf55878cb203ffdc2b165153bf0c0": "\\mathcal{L}[x] = + J^{\\gamma}[x] A_{\\gamma}[x] \n - {1 \\over 4\\mu_0} F_{\\mu \\nu}[x] F_{\\alpha \\beta}[x] g^{\\mu\\alpha}[x] g^{\\nu\\beta}[x] \\sqrt{\\frac{-1}{c^2} \\mathrm{det} [g[x]]}. ", "e0fcfc00befb502fb78c2b1e870b1a59": "\\pi_n(X) \\,", "e0fcfcf1bc30612af09825187fbbad7d": "f:A\\to B", "e0fd029eb0b4be249a1dd7e374f30430": " c = \\frac{1}{\\alpha} \\ ", "e0fd3e81e27041d222c071c2f0f41f08": "\\left(0,\\ 0,\\ \\pm1,\\ \\pm1,\\ \\pm1\\right)", "e0fd6fd5da26e68429b2900a75590d7f": "W_L = \\frac{\\Phi_L^2}{2L} = 2\\pi \\alpha W_{LC}. \\ ", "e0fda777d1ff2b4afe618bd2a435bdb1": "\\operatorname{Aut}(S_2)=\\operatorname{Out}(S_2)=\\operatorname{Aut}(A_2)=\\operatorname{Out}(A_2)=1", "e0fe2eef7b94ab6737890db132c3f77b": "\\int_{\\partial \\Omega}\\omega=\\int_{\\Omega}\\mathrm {d}\\omega", "e0fe3cc1110398095eed2dd4d7995e9e": "\\exists x \\in y", "e0fe67b7e8907dc5bfd52fff5b7cb414": "B(A(P))", "e0fe706e4acf54949019bf6d6c2ded1e": "R^M_L", "e0fe8d53ae7d129284b013d6dd09c0b2": "\\mathbf{I}_O :=\n\\begin{cases}\n1 &\\text{if } x >\\mu, \\\\\n0 &\\text{else }\n\\end{cases}\n", "e0febc0f67a3ed567d3c235fcf6f0047": "\\{it:t\\in\\mathbb R\\}.", "e0fecd10c38b4a2e76b7dd6d4b310345": "{T'}^{a \\dots}_{b \\dots} = J^W {\\partial x^{'a} \\over \\partial x^c} \\dots {\\partial x^d \\over \\partial x^{'b}} T^{c \\dots}_{d \\dots}", "e0fed9e858e352277de6e98b12bac86e": " (v_x, v_y, v_z) ", "e0ff0a8df8f20c22dbcbc15218f47172": "\\mathbb P(\\mathcal E)|_{p^{-1}(U)} \\simeq \\operatorname{Proj}\\, A[x_0, \\dots, x_n] = \\mathbb{P}^n_A = \\mathbb{P}^n_U,", "e0ff3fee6f68f2b004386fef691a8296": "b(y,z)", "e0ff5353aed1f4158102d93c83a02044": "(10,000-9,500)/10,000 = 5\\%", "e0ff62fecee2f1b8553b111caa77dcbd": " a = - 4 \\pi^2 f^2 x\\; . ", "e0ff7196e4d6fdd1cfb1f40e4c6c07cd": "e_1 = (1, 0, \\ldots, 0) \\,", "e0ff8fa6b221d83c85d54e7cfcd581b1": " \\begin{pmatrix} \\alpha & 0 \\\\ \\beta & \\alpha^{-1}\\end{pmatrix}.", "e1008fadaa5946bb01cc2425da9c71e0": "n/p_k^{\\alpha_k}", "e1009c8ee3fd9cf1e454e3a153ca23b6": "(Nkr * Nwp / Tkn) * 100", "e10107aebece5dd909801f0843d00ee2": " \\sin(\\alpha) \\pm \\sin(\\beta) = 2 \\sin\\left( \\frac{\\alpha \\pm \\beta}{2} \\right) \\cos\\left( \\frac{\\alpha \\mp \\beta}{2} \\right), \\;", "e101094f0cb96fc87af06df13f013828": "C_{\\lambda \\pm k}=c_{1} a_{\\lambda k} + c_{2} a_{\\lambda -k} + c_{3} a_{\\lambda k}^+ + c_{4} a_{\\lambda - k}^++c_{5} B_{\\lambda k} + c_{6} B_{\\lambda -k} + c_{7} B_{\\lambda k}^+ + c_{8} B_{\\lambda -k}^+", "e1010b62d4648391ba5f683a7398900f": " \\cos \\theta _1 \\cos \\theta _2 + \\sin \\theta _1 \\sin \\theta _2 = 0", "e10141cbf2fc5e0b377dab9ed52d67ad": "G = S_{21}\\,", "e101483eb90b9f69bc8b77a5d1ccc879": "\n\\vec{x} = \\mathbb{A} \\vec{w}\n", "e1016dd1e1b0315ec44c0ad480ec220e": "{\\mathbf{f}}=-\\frac{\\mu}{R^2}\\mathbf{\\hat{r}}+\\sum_{n=2}^\\infty \\sum_{m=0}^n {\\mathbf{f}}_{n,m}", "e1018eccfe82117563b43ee42d18f203": "\n\\begin{align}\n\\ln q_\\tau^*(\\tau) &= \\operatorname{E}_{\\mu}[\\ln p(\\mathbf{X}\\mid \\mu,\\tau) + \\ln p(\\mu\\mid \\tau)] + \\ln p(\\tau) + \\text{constant} \\\\\n &= (a_0 - 1) \\ln \\tau - b_0 \\tau + \\frac{1}{2} \\ln \\tau + \\frac{N}{2} \\ln \\tau - \\frac{\\tau}{2} \\operatorname{E}_\\mu [\\sum_{n=1}^N (x_n-\\mu)^2 + \\lambda_0(\\mu - \\mu_0)^2] + \\text{constant}\n\\end{align}\n", "e101b237482ff099d81954c60b0bbbfa": "\\mbox{System internal virtual work} = \\sum_{e} \\delta\\ \\mathbf{r}^T \\big( \\mathbf{k}^e \\mathbf{r} + \\mathbf{Q}^{oe} \\big) = \\delta\\ \\mathbf{r}^T \\big( \\sum_{e} \\mathbf{k}^e \\big)\\mathbf{r} + \\delta\\ \\mathbf{r}^T \\sum_{e} \\mathbf{Q}^{oe} \\qquad \\mathrm{(15)}", "e101c8f0d09a2e23ad802162976bd200": "D^\\prime ", "e102821a3b75d5ba6903fa91c0e48ba2": "f_{os} ", "e102916a253ea6c3c77e26091a467409": "X = \\mathrm{AGTA} + \\mathrm{CGCA}", "e102a130c89909f947de113387e55bf4": "\\left\\lceil \\frac{k-1}{n-2k} \\right\\rceil + 1", "e103280e9cdab0d2e505ee64f9d393cf": "y=e^{-a(x)}\\left(\\int g(x) e^{a(x)}\\, dx + \\kappa\\right)", "e1038463285483e08f0fef177bbad86b": "\\tau = n\\tau_o", "e103c64fd024fb9ac41e7dd3540b4507": "y_{n+1} = y_{n} - \\frac{f(y_n)}{f'(y_n)}", "e103d4a56505f1a07a89efa011b5cbd7": " T = \\begin{bmatrix} \\frac{1}{2 (\\log_2 2)^2 }& 0 & \\cdots & 0 & \\cdots \\\\ 0 & \\frac{1}{3 (\\log_2 3)^2 } & \\cdots & 0 & \\cdots\\\\ & & \\cdots & \\\\ 0 & 0 & \\cdots & \\frac{1}{n (\\log_2 n)^2 } & \\cdots \\\\ & & \\cdots & \\cdots \\end{bmatrix} ", "e103e1a21913f1f613806c86ae065520": "\n \\cfrac{d}{dt}\\left(\\int_{\\Omega} \\rho~\\eta~\\text{dV}\\right) \\ge\n \\int_{\\partial \\Omega} \\rho~\\eta~(u_n - \\mathbf{v}\\cdot\\mathbf{n}) ~\\text{dA} + \n \\int_{\\partial \\Omega} \\bar{q}~\\text{dA} + \\int_{\\Omega} \\rho~r~\\text{dV}.\n ", "e103feeb1d0a29560cd0bb251e419244": "\\lim_{a\\to -\\infty}\\int_a^cf(x)\\, \\mathrm{d}x + \\lim_{b\\to \\infty} \\int_c^b f(x) \\, \\mathrm{d}x", "e1041124971479cd322dc0687d345b30": "t_r=\\cot^2 x_r\\,", "e1043dd26d986fb2b6f28524e214a4dc": "O(n\\cdot k)", "e104594e4193053616814001b334f66f": "EM_{12}(endo, exo, fendo, fexo)= RE", "e1048f9617121b84a59502e40b06df0f": "P_0 + P_1 x_2 + P_2 x_2^2 + P_3 x_2^3 + \\dots + P_N x_2^N - f(x_2) = - \\varepsilon\\,", "e104a6230032b63799ecc58b26506c18": "p=(x,x')\\in J^m(\\mathbf R^n)", "e104be7e792dc5a7590077465c6aa5f2": "\\overline{c} = N/m", "e105a45ae0f7aaeb3b3c7314fb462dc0": "\\frac{\\Gamma,C,C\\vdash B}{\\Gamma,C\\vdash B}", "e105ee66e993092d02ee0e068125e3d2": "S=X^T X \\,\\!", "e1065daf3bb1ee73a41c72b2a4fa9e47": " (1.x_1x_2...x_{23})_2 \\times 2^{e}", "e106bedadbbdaf4c429423554c97514c": " K_\\text{C}(x,x') = C ", "e106da479797a76c5f4b6130fc52c8ed": "\\lambda_1,\\lambda_2,\\lambda_3", "e1071ae99b04bacb15ddd82253ea9dd3": "\\varnothing = \\{u \\in w \\mid (u \\in u) \\land \\lnot (u \\in u) \\}", "e107255aad56e4ebc8c6e8ab6c132d87": "f(t)=(t-1)^2+(t^3-1)^2-10", "e107606cb2e46c4530a5350e35b9395a": "\\gamma\\ ", "e107cd1903761ce53568d7bb2d894df4": "v(x) \\equiv \\sqrt{s(x)^{-1}-x} \\mod g(x)", "e10836e768ba7832b84efea059232b91": "5\\, ", "e1083814c1a4fde411f8a837df97e14f": "P(\\Delta,v,L,\\Omega_{\\perp})=\\frac{1}{1+\\left(\\frac{\\Delta}{\\Omega_{\\perp}}\\right)^2}\\sin^2\\left(\\frac{L}{2v}\\sqrt{\\Omega_{\\perp}^2+\\Delta^2}\\right)", "e1085cffe6c275da08284b091d83fa02": "\\sigma_1(W)\\geq\\gamma^3\\sqrt{pq}", "e108621b122f2d78a3fec022f25189c2": "\\alpha + 2\\beta - 3 \\le 0", "e108a4527c9fefedfb0fc5e6b28fa292": "\\mathbf E", "e1093103e5a05fcac0edb0e87ccf76a1": " \\partial |w_i| = \\begin{cases}\n1,&w_i>0\\\\\n-1,&w_i<0\\\\\n\\left[-1,1\\right],&w_i = 0.\n\\end{cases}", "e10956b161463b666aabbfc1630e5a8a": "\\scriptstyle p^{\\nu}", "e1097c377b3b814d816d3eb340c15e4e": "\\sin(\\theta)", "e1097c802517f4a48c6b0de914e8b0c4": " GB1(y;a,b,p,q) = GB(y;a,b,c=0,p,q). ", "e109b7236bfa09f2a79914a19907e7ef": "2^X \\,\\!", "e109fe953001068f9f90395f24533ff1": "\\rho =\\rm const", "e10a2785c0192be2b1f9119d31226452": "m=\\frac{1}{4}A+B", "e10a86cd4aff41f7772eb28e53d82ec1": "a(b-r_t)", "e10ae1cc17bdac9b680c87f20a4502ec": "Q = \\sqrt{\\frac{a_1^2 + a_2^2 + \\cdots + a_n^2}{n}}", "e10ae5694a5edf10f6df9b9f474796eb": "\n \\dot{\\varepsilon}_{\\mathrm{vp}} = \\left\\langle \\frac{f}{f_0} \\right\\rangle^n (\\boldsymbol{\\sigma}-\\boldsymbol{\\chi})\n ", "e10b00b12b54c8d68d20a898b9b797c4": "\\Psi_k(X)=\\bigcup_{i=1}^{N} X\\circ (B^{(i)})_k", "e10b772a4fecc82047d03c516f3aad1b": "C_t\\,", "e10b78ba299ad15e6e8c99036ab0ba84": "n = 0.4", "e10c7c0d4ab85f29cb096d9e49701011": "\\displaystyle \\Phi([t_1,\\dots, t_n])=\\int {\\Phi(t_1)\\cdots \\Phi(t_n) \\over |y|^2 + q_\\mu^2} (|y|^2)^{-z({c\\over 2} -1)} \\, d^D y,", "e10cc756b3ddd408ec9fd8f6aab82a1b": "sTSHI = (TSHI - 2.7)/0.676", "e10d1ca4c0756d7eddcd039654d833d6": "GL_n(O_F)", "e10ddb63c32290b0301a96b48681721f": "y_i = f(i) + \\varepsilon_i = \\boldsymbol\\beta^{\\mathrm T}\\mathbf{x}_i\\ + \\varepsilon_i,", "e10de9fe5abd807438db2d09d6c1bb08": "\\phi_s", "e10e1db60c05f3db85433b93b47e5f20": "\nQ", "e10e91042e0a78cd795438a8a74782b5": " b \\, ", "e10eadd4da39ac83cc9da999293115dc": "\\psi(\\Omega+\\zeta_0) = \\varepsilon_{\\zeta_0 2}", "e10ef702a794449d364821a5a023abe4": "Z^{\\alpha}=(\\omega^{A},\\pi_{A'})", "e10f0848fb1929406911ad3b7f1dee6b": "\\scriptstyle x_2 \\;=\\; kB \\,-\\, A \\;=\\; \\frac{1}{A}\\left(B^2 \\,-\\, k\\right)", "e10f4c489a6aec47b0888a26cb786c52": "S\\in\\mathrm{RAT}(N)", "e10fb7a68dfca67560e09f6eb36e6767": " \\langle\\Omega| \\phi_1 \\,\\phi_2\\, \\phi_3\\, \\phi_4 |\\Omega\\rangle ", "e10fc1dc6e32092853cbf37f57804f58": "\\int_{-\\infty}^\\infty {1 \\over (x^2+1)^2}dx,", "e10fcc7693b89ac5faf5749e79dd23a3": "\\frac{\\mathrm{d}S}{\\mathrm{d}t} =\\sum_i\\frac{\\partial S}{\\partial q_i}\\dot{q}_i+\\frac{\\partial S}{\\partial t} =\\sum_ip_i\\dot{q}_i-H = L. ", "e1106307057dff348db3bd996b762eee": "K^{n \\times n}.", "e110e680e819ed38a070a8d96859c23a": "\\scriptstyle (x,0)\\,", "e111e7cc4f3dd81a75bd96f53f3cf7df": "S_i,\\ i=1,2,\\ldots\\ ", "e112322d920704d51e7212f4dd9b523c": "K_{h_\\lambda}(X_0 ,X)", "e1123909c6f7d83e925fc111d3610f80": "A_{loaded} =\\frac {i_L} {i_S} = \\frac {R_S} {R_S+R_{in}}", "e1127678737d4ca800dd4fd2d4547711": "c_1 \\sqrt{n+1} \\le | p_n(z) | \\le c_2 \\sqrt{n+1} . \\, ", "e112ab41493238cf91798de68a077940": "F^{[l']}", "e1130d111f04aaaf4d498b9219243476": "g^{ij}g_{jk}=g_{kj}g^{ji}=\\delta^i{}_k=\\delta_k{}^i", "e113370ae4ec552d1cbd46a51717cf0c": " = L_y (-i \\hbar L_z) + (-i \\hbar L_z) L_y + L_z (i \\hbar L_y) + (i \\hbar L_y) L_z ", "e11344ef85508bc3f2bd54bdea15d10d": " u\\,", "e113a011eecd1515ae9e33de953bb4ed": " M6 = K[ 1 - \\frac{ \\sum_{ i = 1 }^K | p_i - \\frac{ 1 }{ N } | }{ 2 } ] ", "e113ae2fb6663a269d3e980634162d6c": "\\tilde{ f} (k) = \\sigma e^{-\\sigma^2 k^2 / 2} \\ . ", "e113d22a9ea4b7588a2d3d3eccd3fc89": "y_{b}=b_{0}\\sum_{r=0}^{\\infty} \\frac{(c-\\beta )(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}s^{r+c}", "e113d53890838bb0d77c66fcf62e7124": "a\\frac{\\partial u}{\\partial \\mathbf{x}}", "e114239ca771f41abba34eccf146094c": "A \\in P, B \\not\\in P", "e11437d6c8d54966486aabe7e9182c92": "c_{7}", "e11569c282c45e7e953852ccd80fcf92": "\\Sigma_1(E) = (E\\!\\ggg\\!6) \\oplus (E\\!\\ggg\\!11) \\oplus (E\\!\\ggg\\!25)", "e115892a0f3927ee8d3b7db8156c0832": " \\mathbf{e }.\\,", "e11590e506cdefca009bef744ef58fdb": "(1+A_v)", "e1159d62df6a32e77d56577009d1b7af": "x_{1} = x(t_{1})", "e115aba8c44f22d694dce3c51c71d61c": "|h|+|k| \\ne 0", "e115d08de96dc010065916d346c541ac": "{1\\over 970864271032320000} + {1\\over 685597979049984000} = {691\\over 277667181515243520000 }.", "e1166d618f3e58c7c0a7c7230693fac2": "\\frac{1}{A} d_2(f(x),f(y))-B\\leq d_1(x,y)\\leq A d_2(f(x),f(y))+B \\text{ for all } x,y\\in M_1", "e1166f61d9eb0c0880b2160a79273d0a": " g\\left(Z_{i}\\right) ", "e1167004d0b4ab7d0210810ccab9f38c": "R_s=\\frac{2v^2\\cos^2\\theta}{g} \\left(\\frac{\\sin \\theta}{\\cos \\theta}-\\tan \\alpha\\right) \\sqrt{1+\\tan^2 \\alpha}", "e116c7a795078ea9cb74ffbdec48d0fb": "a \\mapsto a \\otimes 1", "e11708e8bca13d0e283e0b3aca78f87d": "(x_1, x_2, \\dotsc, x_n)", "e11729b0b65ecade3fc272548a3883fc": "x=0", "e11744f3b2350f87410f4e5abc24a61c": "\ng(\\lambda A_1 + (1-\\lambda)A_2,\\lambda B_1 + (1-\\lambda)B_2 ) \\leq \\lambda g(A_1, B_1) + (1 -\\lambda)g(A_2, B_2).\n", "e1176053148050dde348a907243fd67e": "\\frac 1 {z} \\cos \\left (\\sqrt{z^2 - 2zt} \\right )= \\sum_{n=0}^\\infty \\frac{t^n}{n!} j_{n-1}(z), ", "e117ba72da0fc629e402b5cadf8cd277": "a(1+e)", "e117e4ff0b683a8894191781706cbb4a": "\\tilde{x}_0", "e1186e41e311b53a94aa25eedfc152be": "\\operatorname{Ric}_{ab} = Kg_{ab}. \\, ", "e11884cd102558d8528ded0339497773": " p = - \\rho c^{2} ", "e1189044a081bb2dd3c3fa5ca55cb631": "E_{internal}=(\\alpha\\,\\!(s)\\left | \\mathbf{v}_s(s) \\right \\vert^2 + \\beta\\,\\!(s)\\left | \\mathbf{v}_{ss}(s) \\right \\vert ^2)/2 ", "e118a12ccc48242659809074a6fb54ba": "\nQ(x) = a_4x^4+a_2x^2+a_0\\,\\!\n", "e118dfa839b3046c7a135926da5c4084": "\\mathbf{f}^I", "e1193b87f8e42499aea39de17c9a91cd": "\n c \\in [-\\infty ,0].\n", "e119509ef0df4470b20f4f16906cbdb7": " -\\infty < T < \\infty, \\; 0 < r < \\infty, \\; 0 < \\theta < \\pi, \\; -\\pi < \\phi < \\pi", "e1199764f225a6602b579ada270b195b": " [(a^2 - 1)l^2 + 1 - m^2]^2 - ", "e119f8a80a48bc19d1dd6a863e8ccdc0": "h_{s} \\equiv f_{s 1} + \\frac{Z_{s} e \\phi}{T_{s}} f_{s 0}", "e11a1b59d6f9e180e50aff58b1f1f834": "\\Pr(\\text{RCA}_i \\geq 1 \\mid \\text{RCA}_j \\geq 1)", "e11a264b849086a895fcbb87916294a6": "\nE_{i,f}=E_{kin,i/f}+m_e c^2=\\sqrt{m_e^2 c^4+\\mathbf{p}_{i,f}^2 c^2},\n", "e11a685f9225a10ce85cb204a550460d": "M^d=P \\frac {Y} {V} \\,", "e11a789c4cfadf63b86b834f95ee1a01": "\n\\begin{align}\n \\hat{\\mathbf{e}} &= \\frac{\\check{\\mathbf{q}}}{\\|\\check{\\mathbf{q}}\\|} \\\\\n \\theta &= 2\\arccos(q_4)\n\\end{align}\n", "e11a84d17c490f4e9f7e102e7558a985": "\\left[ 1,\\dots,2\\lambda \\right]", "e11a926945b6eaf090abae26a6a4424c": "F_{\\omega} = \\frac{dC}{dz} [V_{DC} - V_{CPD}] V_{AC} \\sin(\\omega t)", "e11aaa369b7856f0f7f7fe86ed85db98": " L= \\frac{1}{2} L_1 \\frac{dQ_1}{dt} ^2 + \\frac{1}{2} L_2 \\frac{dQ_2}{dt} ^2+ m \\frac{dQ_1}{dt} \\frac{dQ_2}{dt} - \\frac{Q_1^2}{2C_1} - \\frac{Q_2^2}{2C_2} ", "e11aec3909a0ea63b363122e972ce372": "\\sigma_X^2", "e11b0f7f93bd4dd3c7eddecd68f6a6cd": "X_n \\, \\xrightarrow{P} \\, X\\,.", "e11b25ab19c27158015a4324d42f786f": "n - m \\sim n' - m' \\leftrightarrow n + m' = n'+ m", "e11b54083c469092873a113b49ac9efa": "e^{i\\varphi} = \\cos \\varphi + i\\sin \\varphi. \\,", "e11b81205fc0914e910f59d5f5d9932b": "(x,\\sigma^2) ", "e11b8e1a6256bb066d5d2adea17aa664": "\nf = t+ \\sum_{n=2}^\\infty a_n t^n\n", "e11b93ac67f8d0c0192a107e9297cc4b": "ND(v)", "e11b97f052c221fc8a9c9ffd748e28b1": "(-1)^{x+1}\\Gamma(x)(!(-x))\\,", "e11ba37d5d784af689e175bec8a2f284": "g_1", "e11baea6205c13434a10f24c2f8efe93": "f(y, x_1, \\ldots, x_k)\\,", "e11be49d027643a49311e6d919e1f7dc": "Y_{n,1}, Y_{n,2}, \\dots, Y_{n,n} \\, ", "e11c10ddb354581eab0d982330bb56b8": " \\left \\{ \\sigma_1^2, \\sigma_2^2, \\sigma_3^2, \\ldots \\right \\} ", "e11cace8ea7f00c43d876ed49666eb04": "price =x ", "e11cb3970177191c14161795a9d7cf68": "\\left(\n\\begin{matrix}\n\\lambda&v^i\\\\\nw_j&a_j^i\n\\end{matrix}\n\\right),\\quad \n(v^i)\\in {\\mathbb R}^{1\\times n}, (w_j)\\in {\\mathbb R}^{n\\times 1}, (a_j^i)\\in {\\mathbb R}^{n\\times n}, \\lambda = -\\sum_i a_i^i\n", "e11cca07e83ab35c8d124eb1eccbea31": "\\cos c= \\cos a \\cos b + \\sin a \\sin b \\cos C, \\!", "e11d20b015c2f9d04494335ea56692f2": "\\textstyle{\\frac {\\log(7)} {\\log(3)}}", "e11d46150777ef45105541eec7e6d61a": "{\\mathrm{S}\\widetilde{\\mathrm{L}_2(}\\mathbf{R})}", "e11dce2be6483d819af052cfa7e6e376": "g(X)", "e11dceac46865cc80fa474097a557de6": "K_E=\\frac{V_E}{V_{O_2}}", "e11def33b2cc8483f9cf5d4ffc65514f": "C=100\\times(1-\\frac{\\rho_B}{\\rho_T})", "e11e00dfc4564a28287a2cb44328f74c": "\n\\begin{pmatrix} & h& \\\\[-0.9ex] v & & v'\\\\[-0.9ex]& h'& \\end{pmatrix} \\circ_1\n\\begin{pmatrix} & h'& \\\\[-0.9ex] w & & w'\\\\[-0.9ex]& h''& \\end{pmatrix} =\n\\begin{pmatrix} & h& \\\\[-0.9ex] vw & & v'w'\\\\[-0.9ex]& h''& \\end{pmatrix}\n", "e11e034ba9a37e463af090f79f02c5ff": "\\psi)", "e11e4608f9b7bafa536288f6b33404db": " = d_0(m + 2 \\frac{d_1}{d_0} (m + 2 \\frac{d_2}{d_1} (m + 2 \\frac{d_3}{d_2} (m)))). ", "e11e5c7a20e3101e9f3243735a5fd147": "W\\subseteq T^{*}", "e11e7dc346a4f3f8196cc186c06f11d1": "i, j, k", "e11e9cb2d49f14b221ad2b0164d1c53e": "n/2^n < n/n^3", "e11ef21e80ef221565bde97c8c7b160c": "\n \\oint_{C_1} \\frac{\\left(\\frac{z^2}{z-z_2}\\right)}{z-z_1}\\,dz\n =2\\pi i\\frac{z_1^2}{z_1-z_2}.\n", "e11f01d032a40a7ac1823d191a1b44b1": " \\sin(\\alpha) = \\frac{v_\\text{sound}}{v_\\text{object}} ", "e11f219f3fe2043f4dae06284797e3cb": "\\hat{\\tilde{H}} |\\psi \\rangle = 0", "e11f263cac5ac29f003ae0fe3eced82c": "w: U' \\to V_i", "e11f8652d3cb4dce9a1437bae80f70f6": "\\mathrm{A}_3 \\twoheadrightarrow \\mathrm{C}_3", "e11f91079bf8bd099a171d42f1b86dba": "S_1=1 \\, ", "e11fe4e70ad3c179453aef2e5fc967eb": "\\{\\langle x,y\\rangle; x\\in A\\}", "e1203f54f61784e64fd68d663a256aec": "M = F/f_1 = (F - B)/D", "e12041a64fd1217a02f9e5219f433fdd": "L(s,\\pi,r_i) = \\epsilon(s,\\pi,r_i) L(1-s,\\tilde{\\pi},r_i),", "e1212c7400f6aca470c411d53bf2117c": " \\theta = 2 \\pi \\frac{t}{T} = \\omega t\\,", "e121775839de240d1170c06b7d496a14": "\\mathcal{H}_{Kin}", "e121f7e889b19b16a4e360c85d4d9058": "\\frac{a}{2^b}-\\frac{c}{2^d}=\\frac{a-2^{b-d}c}{2^b} \\quad (d< b)", "e12201f79ab0d62cc3060cbfb446aa7f": "u = \\mathbf{K}(s) \\, v", "e1226b60739b0e97198454558189b458": "\nA =\n\\left(\\begin{matrix}\nm_0 & m_1 & m_2 & \\cdots \\\\\nm_1 & m_2 & m_3 & \\cdots \\\\\nm_2 & m_3 & m_4 & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\end{matrix}\\right)", "e1228375b192f5c1d19e61c5472b2440": "{\\bar{P}}_7", "e12283e820ef380267d52bce31c6a3b8": "\\mathit{SS}_{i,j,k}\\,", "e122bd52ef3eec0de532b1ecb18f6ab0": "\\bigwedge^i C^n", "e122d3d7a17d8e160aa49b842c88d0b3": "ax+b", "e122e3570875c5439f5e1f202339142f": "\\boldsymbol{A}(\\mathbf{X})", "e123459ca8543254184075dc05b5cde9": "P(0) = 0", "e12347d5cbb09d789ee55abdedc7cdef": "\n\\begin{matrix}\n\\phi_1 = (p_{11} - p_{12})Q \\\\\n\\phi_2 = (p_{21} - p_{22})Q\n\\end{matrix},", "e12357029ebab390cead8b4a9989f086": " \\tau = it \\,, \\!", "e1235798e41c3a9e505f25e9d60f1e7c": "\\mathrm{}\\ \\Pr(X>40 \\mid X > 30)=\\Pr(X>40)\\,", "e1235e9f2f6903fc336f85387a4755c3": "Y^{2}", "e1237ac05de233dc295c8efcdfb167ab": "H(u)(t) = \\text{p.v.} \\int_{-\\infty}^\\infty u(\\tau)\\left\\{h(t - \\tau)- h_0(-\\tau)\\right\\}\\,d\\tau", "e12414bcfaecef3ad43e90523a57effe": "\\displaystyle Wg(1,d) = \\frac{1}{d}", "e124534d0178cd1ffe3b19b2d7f49efb": "N(x)=(c+o(1))\\frac{x}{\\log x}", "e124cec14d80303c6d0509b6e9589687": " \n\\lim_{t\\rightarrow \\infty}P[\\eta_t(x)\\neq\\eta_t(y)]=0\n", "e1254ccd51c1c577e3eec541b529ed10": "{n} \\cdot \\lambda\\,", "e1255a3b8d68fc5792386dd693a35409": "V = 2 \\arctan \\left( \\tan \\left({H \\over 2}\\right) \\times {h \\over w} \\right)", "e1255f5309beacf91e18c2998b50fca0": "U_t = U_{t-1} + e_t,", "e12574d5c75f8d6d5fc49b3dc69fc05e": "\\frac{\\partial E}{\\partial \\hat{h}_i} = \\frac{\\partial}{\\partial \\hat{h}_i} \\sum_{n=-\\infty}^{\\infty}[x[n]^2 - 2x[n]\\hat{x}[n] + \\hat{x}[n]^2 ]", "e1259c27990847af5b20cedabf65c269": " N_S := \\frac{1}{2} n (n - 1) - \\sum_{\\mu = 1}^{m} \\frac{1}{2} \\nu_{\\mu} (\\nu_{\\mu} - 1) \\le N ", "e1259f3f8dcf14a916ef9d105db7767e": "A=2(6+\\sqrt{3})a^2\\approx15.4641...a^2", "e125c608650eafd01b06adf1245bd713": "\\frac{\\mathrm{d}h}{\\mathrm{d}t}", "e12629cc4f4e4f7da80caebcf99cd8b7": "a_{13}*b_{13}", "e126398adf65948f65d56b3ec4b508a1": " - \\varepsilon < g(x) - L\\leq f(x) - L\\leq h(x) - L\\ < \\varepsilon, ", "e12650de03bf2b089d35f5c62200e254": "\\tfrac{O_1O_2}{BE}", "e126621dd1e653553949b75a715a3edf": "L^{SCC}[\\tilde{g},\\tilde{\\phi }]=\\frac{\\sqrt{-\\tilde{g}}}{16\\pi\nG_{N}}\\tilde{R}+\\tilde{L}_{matter}^{SCC}[\\tilde{g}].", "e1268c0557412431a64b3fa38321656f": "t_0 = 1", "e126e0149b23b34b7a14ecdb285fd17d": "-y-z+x-x^2+xz=0", "e127194a104c7d88221cef819a274646": "x\\in U", "e12735f9d2172e1e350098e4059279a5": "(q^0, q^i)", "e1276233837c568291bef0e5880dd3e7": "\\varepsilon_{ij}= \\varepsilon'_{ij} + \\varepsilon_M\\delta_{ij}\\,\\!", "e12781516425f043184177e7d670e66e": "f(i) \\in A_i", "e1278409682a13f70b666f933fe66570": "\\;J_2 (w,b,e) = \\mu E_W + \\zeta E_D", "e1279247f55af012ed01710a3e9c5d05": "\\operatorname{E}((X - \\mu)(X - \\mu)^{\\operatorname{T}})", "e128158876a74438ffbef3d871bcb7fa": "E_3 \\cong A_1 \\times A_2", "e128487c584d0d572ab4950919a2f11b": "O(1.3289^n)", "e12874c66dc8b48123cd71c15e6efffa": "B(p, w) = \\{x \\in \\mathbb{R}^L_+ : \\langle p , x \\rangle \\leq w\\} \\ ,", "e1288c68d2dcd0897f8c205b3f2d4cc4": " t \\leq 1", "e12895d0c9ed190f8b19f1ec8336030a": "\\{1,i,j,k,-1,-i,-j,-k\\}", "e128a1380b71c0ed680dedd957e877fe": "0 = \\Gamma^{\\alpha}_{\\beta \\gamma} g^{\\beta \\gamma} \\,.", "e128a5ba17a0d28a6882494a6ac20b25": "\\forall C [\\lnot MC \\leftrightarrow \\exist F ( \\forall x [Mx \\rightarrow \\exist! s (s \\in C \\and \\langle x, s \\rangle \\in F)] \\and ", "e12988c9b5703bb0aabbcc56cdfa2751": "U_g(t)", "e129a78f5a1d9de458ad3bd3cba0ef39": "k' = \\frac {C_x}{p_x}", "e129b5a7283bcc817a70dcf75e09e570": " G=\\sum_i X_i^T X_i ", "e129d2bae71ce7bf0f92739d7891be40": "T_{opt}^5-(0.75T^0)T_\\mathrm{opt}^4-\\frac{T^0IC}{4\\sigma} = 0 ", "e12a04d2988437467c7ee7185eff640b": "p^2=\\frac{4}{3}(r^2-a^2)", "e12a26b06dbda8c739c4807465168faf": "N_r = |\\{ x | R_x = r \\}|", "e12a3440d1a6c9e5ab510513e40fd216": "[5,12[", "e12a3cacda800d2e1c72e769d4003447": "\\phi = \\left( \\frac{1}{\\tanh{\\gamma}} - \\frac{1}{\\gamma} \\right) \\ \\frac{x_j}{\\gamma} \\sigma_i ", "e12a49be7e63aac325034c70caec5a0d": "\\arcsin\\alpha \\pm \\arcsin\\beta = \\arcsin\\left(\\alpha\\sqrt{1-\\beta^2} \\pm \\beta\\sqrt{1-\\alpha^2}\\right)", "e12afc1530ac34ebdd268370752f6853": " E\\left[ e^{- u \\int_0^t V(x(\\tau))\\, d\\tau} \\right] = \\int_{-\\infty}^{\\infty} w(x,t)\\, dx ", "e12b20fc5f9c0547eb4c85b6b0eb4139": "V_{i}(t) = 2V \\cdot \\cos (\\omega \\cdot t)", "e12b2c8ae811a7a0fa36df107f98f258": "x = \\frac{y-\\mu}{\\sqrt{2} \\sigma} \\Leftrightarrow y = \\sqrt{2} \\sigma x + \\mu", "e12b4d3a07e2197407a3adb0b24b4e0e": "G(\\omega) = \\sqrt{\\frac{1}{1+{\\omega}^{2n}}},", "e12bc663c337a66bb7d040d359cc410f": "\nx = \\underset{i=1}{\\overset{\\infty}{\\mathrm K}} \\frac{1}{b_i}\\,\n", "e12bd934d7c7dfc6f1d2d85c7d7879d1": "\tq_{1}\\stackrel{\\epsilon , T_{1}}{\\rightarrow}r_{0}\\stackrel{x_{1} , T_{1}}{\\rightarrow}r_{1}\\stackrel{x_{2} , T_{1}}{\\rightarrow}r_{2}\\cdots r_{m-1}\\stackrel{x_{m} , T_{1}}{\\rightarrow}r_{m}, r_{m}\\in A_{1}\n", "e12be441898e0bc1bfa4c775a672f80d": "U_g\\otimes \\lambda(g)", "e12bf37a6272b95c117ef0d9ebfeafe5": "S_w[p]", "e12c0e1f8f90c80afb30f07b960747f5": "C_s = \\frac {{\\omega}^2 L^3}{24P^2}", "e12c34826ba11a21fe338ab9ceda5163": "\\frac{c}{\\sqrt{g\\, h}} = 1.0376,", "e12c3e8cad7503d41ec94f3907e448fa": "\\partial_1", "e12cadb0181873ec7d658c337a0d70b4": "k = k + 1", "e12cec3d873511a7db651aaf1856f210": "W^{(\\gamma)}(t)=W(t)+\\gamma t", "e12d0ad33314ed3076a160c79cbcdc29": "\\Delta P_i", "e12d0e7738da237149f68524c7e3bb27": "\\mathcal{F}=\\mathcal{BG}", "e12d110bc3106bc450dc053a639d6b37": "\\lambda X", "e12d1e4d64011d7e1f98756b326b3b28": "L\\subset U", "e12d2b21c8b318c52ffd814e4c81ae67": "\n \\boldsymbol{F} = \\boldsymbol{\\mathit{1}} + \\gamma\\mathbf{e}_1\\otimes\\mathbf{e}_2.\n ", "e12d2cd71d0734720f72a1dbc904eff5": "m = H + 2.5 \\log_{10}{\\left(\\frac{d_{BS}^2 d_{BO}^2}{p(\\chi) d_0^4}\\right)}\\!\\,", "e12d791b0eb003b551ee14c190693e10": " f_n ", "e12d7cc43e9e8c2f3777d71d5dcdb48d": " R = (-1) \\int_m^n f^{(2p)}(x) {P_{2p}(x) \\over (2p)!}\\,dx ", "e12d7e827364eb37b18089e9269dc793": "\n \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p + (2\\mu + \\lambda)~\\nabla(\\nabla\\cdot\\mathbf{v}) + \\rho\\mathbf{b}\n ", "e12d9f7346ce542f97618d6c4f09d8c0": "w(e) = w(u,v) ", "e12db21033ae798479471f613ba12ccf": "f\\mapsto P_n f=\\int_S f \\, dP_n=\\frac{1}{n}\\sum_{i=1}^n f(X_i)", "e12e1b27a8c5552d2a4b155890e30a6a": "V(x) = -\\int w(x)\\, dx", "e12e272654fd0e51b08b241669474d49": "dr=(b-ar)\\,dt+\\sigma \\,dW", "e12e39fd19e9c2771917c18d744fa02b": "\\begin{array}{rcl}\n\\displaystyle B+C+\\cdots+Z+\\frac{B}{3}+\\frac{C}{3}+\\cdots+\\frac{Z}{3} & = &\\displaystyle \\frac{4B}{3}+\\frac{4C}{3}+\\cdots+\\frac{4Z}{3} \\\\[1em]\n & = &\\displaystyle \\frac13(A+B+\\cdots+Y).\n\\end{array}", "e12e46a514225a904ed1ee09a7586f88": " \\{ {{|}} x + C_2 \\rangle \\mid x \\in C_1 \\} ", "e12e991b9c11ec8f5072ec7f459e8e5c": "|\\mathbf{\\tau}|=|\\mathbf{r} \\times \\mathbf{F}|=|\\mathbf{r}||\\mathbf{F}|\\sin\\theta ", "e12ec2a73aee5a17b0c2042024f7c534": "\n\\dot{x}(t) = f\\left(x(t), u(t)\\right), x(0) = x_{0},\n", "e12ee72942bcabe605a994d235886f44": "^{\\;}f(\\xi,\\tau)", "e12efcc8658d80c8f0ea4caa4fea77ae": " f(x,t) = a x", "e12f09c83ad57b4376db62cab395acfe": "B_T = 110", "e12fd3b491ca520114a0e84870aa1117": "\n\\chi(q) = \\sum_{n\\ge 0} {q^{n^2}\\over \\prod_{1\\le i\\le n}(1-q^i+q^{2i})} \n", "e12fd942f583de48527803485b62a1bb": "e_3=-\\Omega(\\alpha^{10})/\\Xi'(\\alpha^{10})=(\\alpha^{-4}+\\alpha^{-1}+\\alpha^{7}+\\alpha^{-5})/\\alpha^{7}=\\alpha^{7}/\\alpha^{7}=1,", "e130087ec70acccaeaa9f73117eb56eb": "\\delta = -A^2 - A^{-2}", "e1302a0ae98614d169e6f3a7411bdb94": " \\frac{\\mu^2}{2\\sigma^2} + \\ln \\sigma", "e1303c39cc8811c80342fad6758729d4": " r \\approx z", "e130a1649dc388a0ddf8b554583dc349": "\\Phi_q", "e130a8961cbbb6c521b593b9eda63943": "\\sum_{n=0}^{\\infty}\\tbinom{n+2}2 x^n={1\\over(1-x)^3}.", "e131131850edc9f127b84f0b062c1292": "\n\\begin{bmatrix}\n {*} & & & \\cdots & & & * \\\\\n & \\ddots & & & & & \\\\\n & & a_{kk} & \\cdots & a_{k\\ell} & & \\\\\n \\vdots & & \\vdots & \\ddots & \\vdots & & \\vdots \\\\\n & & a_{\\ell k} & \\cdots & a_{\\ell\\ell} & & \\\\\n & & & & & \\ddots & \\\\\n {*} & & & \\cdots & & & *\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n {*} & & & \\cdots & & & * \\\\\n & \\ddots & & & & & \\\\\n & & a'_{kk} & \\cdots & 0 & & \\\\\n \\vdots & & \\vdots & \\ddots & \\vdots & & \\vdots \\\\\n & & 0 & \\cdots & a'_{\\ell\\ell} & & \\\\\n & & & & & \\ddots & \\\\\n {*} & & & \\cdots & & & *\n\\end{bmatrix}.\n", "e13120663e4c6d625de0579777567ec2": "\\mathrm{Mo} = \\frac{\\mathrm{We}^3}{\\mathrm{Fr}\\, \\mathrm{Re}^4}.", "e13123adebee3303bc2bff13a25de6f7": "\\epsilon_1,\\ldots,\\epsilon_6", "e13165b722b745b6111bd7457b0ca2e3": "\nQ_p = \\left( \\sum E_p + P V_p \\right) - \\left( \\sum E_r + P V_r \\right)\n", "e131d8009c5ac0a80c4569be3e4984a8": " A = \\frac{ E \\pi d^4 }{ 64 }", "e13208d13e633f979ecea29d8af08661": "x = -\\frac{p}{2} \\pm \\sqrt{\\left(\\frac{p}{2}\\right)^2 - q}", "e1321da24923b86f1a1f442c0ec5fa47": " \\{f_i\\}_{i=1}^m ", "e132bea15d5c3c1331bee28085ac49a7": " = x_k - \\frac{x_k^n - A}{n x_k^{n-1}}", "e132ce66f3d2b53fdba574b44b90fd0f": "a \\rightarrow b \\rightarrow c", "e133592bae422d7515222d6bc546e2d8": "L_{\\omega , \\omega}", "e134a39e59a5c3cbfedb30e1b4575808": "\\displaystyle 3N", "e13542ed322f2d828fb75625056d3f2c": "\\phi_{sl,v}=\\frac{1}{1+\\frac{M_{l}SG_{s}}{\\phi_{sl,m}M_{s}}\\frac{M_{s}}{M_{s}+M_{l}}}", "e1354cc842cc323c307b3424ed3dfa81": "A.", "e1354e7a0dc185efad2cdd8aef9d61c6": "V_{oc} = \\frac{(V_{out+})+(V_{out-})}{2}", "e1355f6268b15c142a5d6c9e2208026b": "f\\left( \\left( y,z\\right)\n,t\\right) =ty-z", "e1360038f24c8a303e394f48817e1b5c": "P_a + \\frac{1}{2} {\\rho C_s^2} = P_2 + \\frac{1}{2} {\\rho C^2}", "e13614c3ee1a16585d9734e8f379aa39": "\nx_1=x_3-x_2 \\Rightarrow x_1 \\in [-\\infty ,5]\\cap ([6,\\infty]-[-\\infty ,4]) =[-\\infty ,5] \\cap [2,\\infty ]=[2,5]. \n", "e1361962aad05e9994994f431bce4c6e": "D = a + bS", "e136291c336d8dfb9a3ead28cfb628ce": "\\Delta P = \\rho\\,g\\,h_{\\mathrm{f}}", "e1366e33bb2f177caf8a910b2b9e15e8": "\\mathbf\\varphi ", "e136a577580bbc0d52cae129ed959147": "\\frac{d\\mu}{dn} > 0", "e136a83d8974315fd6dc95d44a2ea8c3": "T^*u = \\sum_{k=0}^n (-1)^k D^k [\\overline{a_k(x)}u].\\,", "e136aaefb69f3f6a25198d6a1b4218d1": "\\varphi(h)(n)=(-1)^h n", "e136dba8a270c4bab07c30239dade2a8": "\\rho = \\nabla \\cdot(\\mathcal{E}_0\\mathbf{E})", "e136e19bf098007bb348503b4d978fb4": "\\Box = \\partial_\\alpha\\partial^\\alpha ", "e13797ac364f0e661b246f4136debc14": "\\mu_1 \\times \\mu_2", "e13810d5e57e13c77959e52f909872d2": "\\{\\ X\\}", "e138231d459eb3b143aaadfa8ec868da": "|0\\rangle , |1\\rangle", "e1383083a9fcacfc7210a1d5661ce22d": "\\{W|P|Q\\}(\\mathbf{\\alpha},\\mathbf{\\alpha}^*)=\\frac{1}{\\pi^{2N}}\\int \\chi_{\\{W|P|Q\\}}(\\mathbf{z},\\mathbf{z}^*)e^{-i\\mathbf{z}^*\\cdot\\mathbf{\\alpha}^*}e^{-i\\mathbf{z}\\cdot\\mathbf{\\alpha}} \\, d^{2N}\\mathbf{z}.", "e13858c6bee3eacc8a9a8fb4c4db2bea": " \\theta = k \\frac{s}{r}. ", "e1389212a79ef5fe4405a82914641a6c": "\n\\forall \\alpha \\in \\Lambda. \\, \\forall p' \\in S. \\,\np \\overset{\\alpha}{\\rightarrow} p' \\, \\Rightarrow \\, \n\\exists q' \\in S. \\, q \\overset{\\alpha}{\\rightarrow} q' \\,\\textrm{ and }\\, (p',q') \\in R\n", "e13892b659c623960319ea980ae6bbff": "t = 2 \\pi \\sqrt {\\frac {r} {g \\tan \\theta}}", "e138c3449315ebe84fb76c573f4616e5": "\nC_x = c_l\\cos\\phi + c_d\\sin\\phi\n", "e138d87d693e0a31f1cc9abff57c7840": "y[n] = (h*x)[n] = \\sum_{m=-\\infty}^\\infty h[n-m] x[m] = \\mathcal{Z}^{-1}\\{H(z)X(z)\\}.", "e138ea53a07b333f05d6783565b79b0d": "pKa\\,", "e138ed889e4d6e823a46c3585d2e134f": "\\frac{d^2y}{dt^2} +\\frac{1}{2}\\left(\\frac{1}{t-e_1}+\\frac{1}{t-e_2}+\\frac{1}{t-e_3}\\right)\\frac{dy}{dt} = \\frac{A+Bt}{4(t-e_1)(t-e_2)(t-e_3)}y", "e1392b97f19d658f9756ffe159007fac": "\\mathbf{D} = \\mathcal{E} \\mathbf{E} ", "e139a48c9781339a13361082f983b3b6": "\\ln |y| -\\ln |1-y|=x+C", "e139f9f60b6774375998e6b81b85d394": "n_{1}=\\frac{p_{1} V_{ref}}{z_{f1} R T_{ref}}", "e13a13a3b498caa1070f1c936c227531": "\nL = {MR^2\\over 2} \\dot\\theta^2 + {MR^2\\over 2} (\\sin(\\theta)\\dot\\phi)^2\n\\,", "e13a7ab7c0dc59775d56686be4482a0c": "\\{x_i\\}_{i=1}^n", "e13a8ba16efbe561f44c649afdb15645": "O_1=\\{e_1,\\ldots,e_{k-1}\\}", "e13aade25a5553259d581beb9715965b": "\n\\begin{pmatrix}(1+\\alpha^{-4})+(\\alpha^{1}+\\alpha^{2})x+\\alpha^{7}x^2&\\alpha^{7}+\\alpha^{-3}x\n\\\\\n\\alpha^4+\\alpha^{-5}x&1\\end{pmatrix}\n\\begin{pmatrix}\n\\alpha^{-7}+\\alpha^{4}x+\\alpha^{-1}x^2+\n\\alpha^{6}x^3+\\alpha^{-1}x^4+\\alpha^{5}x^5\\\\\n\\alpha^{-3}+\\alpha^{-2}x+\\alpha^{0}x^2+\n\\alpha^{-2}x^3+\\alpha^{-6}x^4\\end{pmatrix}=\n", "e13ae9793494c59b9753fbda7275ea51": "C_7", "e13b4da2159203d8142e71d183fa86fd": "{H}=\\hbar \\omega \\left({P}^{2}+{X}^{2} \\right)\\text{,}\n\\qquad\\text{with}\\qquad\n\\left[ {X},{P} \\right]\\equiv {XP}-{PX}=\\frac{i}{2}\\,{I}.", "e13b50739a259c5ff5dd3a845135a4fc": "H_\\varepsilon^h f(e^{i\\varphi})=\\frac{1}{\\pi}\\int_{|e^{ih(\\theta)} -e^{ih(\\varphi)}|\\ge \\varepsilon} \\frac{f(e^{i\\theta})}{e^{i\\theta}-e^{i\\varphi}}e^{i\\theta}\\, d\\theta,", "e13b66a08d192bbec0eb7513071a847d": " P = \\frac{n m \\overline{v^2}}{3}.", "e13b8bb50c1efb652eb3dc96d4d4b51d": "R_2 =2.3397", "e13c1418f2d7b18516a932a3ae28cb4f": " w,v ", "e13c3aa4d8551ae4619eb3806389732a": "H(p) \\,", "e13c40d4c2053d7c75ef63dd3715df08": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{ x^{m+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{m+2 n\\,p+1}\\,+\\,\n \\frac{n\\,p\\,x^{m+1} \\left(2 a+b\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}}{(m+1)(m+2 n\\,p+1)}\\,-\\,\n \\frac{b\\,n^2 p (2 p-1)}{(m+1)(m+2 n\\,p+1)} \\int x^{m+n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}dx\n", "e13cdddbd3ca603799c0e4e17ce32ac8": "\\phi_{A,B}^{0}", "e13d657d5576646fed43ef33b5df2a55": " 1.2,\\,\\,\\lambda\\,=\\,0,", "e13dda788af9098408e86a73bf689c83": "\\varphi\\colon\\Delta\\to X", "e13df3847e6c1c62541e6ebb6726e1b3": "a\\!\\!\\!/b\\!\\!\\!/+b\\!\\!\\!/a\\!\\!\\!/ = 2 a \\cdot b \\,", "e13dff611a6d4c52ae17f594be7dbd93": " u_{i - 1/2} ", "e13e0daaf2797de2dfc58bd17471ad78": "\\mathrm{Ad}_G \\cong G/C_G(G_0).", "e13e27e67fb758f9033afab13a54fae0": "\\sup\\nolimits_{T \\in F} \\|T\\|_{B(X,Y)} < \\infty.", "e13e2c517bd3b9ae98fd5b2f5f8bc899": "M_2' = M_2 + \\delta^2 \\frac{ n-1}{n}", "e13e54612e18b803c60fd4fb459d4c4a": "\\tau_r", "e13e6a4ce89691568a505f8f34b61afd": " \\hat{H}_1 = \\frac{1}{\\alpha^2} \\hat{H'}_1 = \\frac{1}{2 \\alpha^2} (I - \\beta) V ", "e13e7971643aad025095bd535e78a0aa": "f(x,y,z)=0 ,\\ g(x,y,z)=0", "e13ea291fcbeda692a65bc8fa84e2870": "\\alpha V K_T = \\left[ \\frac{1}{V} \\left(\\frac{\\partial V}{ \\partial T}\\right)_P \\right] V \\left[-V \\left(\\frac{\\partial P}{\\partial V}\\right)_T\\right] = - V \\left( \\frac{\\partial V}{\\partial T} \\right)_P \\left(\\frac{\\partial P}{\\partial V}\\right)_T", "e13ea6696e2c03191df247f2588897cd": " k_{\\perp} ", "e13ed6d3e60b42d59acc91d9dc85fdcc": "X_t=W^{(\\beta)}(A_t)", "e13eed8a0f07cce404f5329b62ee2ed6": "[E,P_i]=0", "e13efe04e07bfdd074345e253a72257b": "D_2= (u_2,v_2)", "e13f27038da71f27fdb9abbceee602cc": " y_1 ", "e13f3648e17355ac61259707cb80eb6d": "C(X) > X^{2/7}.", "e13f6efad9a902dfdc4b216201a5eff1": "{S_{Th}}^{(2)} \\geq {S_{Th}}^{(1)} ", "e13f869b1f814f59e05d717fb36176fd": "b^2(f_1,f_2) = \\frac{\\langle F_n(f_1)F_n(f_2)F_n^*(f_1+f_2) \\rangle}{ \\langle |F_n(f_1)F_n(f_2)|^2 \\rangle\\langle |F_n^*(f_1+f_2)|^2\\rangle} ", "e13fae7730301ddc54b9bb1fe15a6a90": "\\alpha : X \\longrightarrow X\\times A+1 = FX", "e13ffcc42a6411b5f3cfb3a39befe94b": "\n (\\lnot R)\n ", "e13ffdb4b4929f16ec5f992c42fb1e46": "\\binom{n}{k}_F = \\frac{F_nF_{n-1}\\cdots F_{n-k+1}}{F_kF_{k-1}\\cdots F_1} = \\frac{n!_F}{k!_F (n-k)!_F}", "e14016e9ba56d1980cc298acac6c2158": "\\frac{1}{p}+\\frac{1}{q}=1.", "e14033c243dded7b1f7bc903710e9873": "\\ \\xi' = kz", "e1406843da61b1754e723fa093bf7ce3": " e_k (X_1 , \\ldots ,X_n ) ", "e1409d814e21ccd3c0d031fbf3c35a2e": "\\mathrm{Tor}_i^R", "e140a52f014e1acfded09c7e71d7f980": "F:D(X) \\subset \\Bbb{R} \\times \\Omega \\longrightarrow M", "e140effa0535ba49697ab161e9df2e36": " u^L_{i + \\frac{1}{2}} = u_{i} + \\frac{\\phi \\left( r_{i} \\right)}{4} \\left[ \n\\left( 1 - \\kappa \\right) \\delta u_{i - \\frac{1}{2} } + \n\\left( 1 + \\kappa \\right) \\delta u_{i + \\frac{1}{2} } \n\\right],", "e1411726e089acbe0158c61e9e44e2c8": "\\, \\Re(z) = 1/2. \\, ", "e14181e6d130ce861cf7b8fd3c47e695": "D,", "e141c81c4cfc667a2d02dec00b924442": "[f]([\\mathbf{x}])", "e141e4f397f14c83cde3ae5fc0e5bbad": "\\theta_j\\in (-\\pi,\\pi] ", "e141e6449c67ef5a08317e0eb851a119": "\\frac{(\\log n)^d}{n},", "e1421494b04a0cb62f681e04ca0cc8f1": "f_{n}< 10^{-8} ", "e1422890d3639545bde61046c495b560": "\n\\begin{align}\nu(0) &= 0 \\\\\nu^{\\prime}(0) &= 0 \\\\\nu^{\\prime}(x) &\\to 1, \\qquad x \\to \\infty \n\\end{align}\n", "e1422e46f79a17ddc16b2650fedeb2cb": "\\;1-\\lambda\\,", "e1423244da57f73ce2beed5512b6d4fe": " (\\alpha, \\beta) = (\\lambda \\alpha, \\lambda \\beta)", "e1423d7ddf487b26937c1e99d9101c3b": "\\tilde{v}_{t+1}", "e142a25ac481a40e8e8312b4697fa075": "\\Delta G^\\circ = -RT\\ln\\frac{[A][B]}{[AB]}", "e142b04d5e0d86fddc18698d621df293": "0_{A^n}\\stackrel{\\mathrm{def}}{=}(0_A,...,0_A) \\in A^n", "e143561e0bddc2ec34ec783b34e59d3b": "(\\Sigma, I)", "e143a4a3257ef5f0aa19a71329e19420": "(Z/X)/(Y/X) = Z/Y.", "e143a4e84cf5e777c94f1ff0ad73aad5": "\nA(z) \\propto z^{l(E^2)}\n", "e143b495383ea250a7dc3dffd3f5f8e9": " w_A=\\begin{pmatrix}2 & 2 & 3 & 4 & 5 & 6 & 6 & 8 \\\\ 4 & 6 & 4 & 7 & 5 & 3 & 4 & 1 \\end{pmatrix}. ", "e143d5a4d9c6eed15b3736b57ff45b0f": "\\mathcal{} L_{n+1} (\\pi_1 (X))", "e144a7eb0780c953124219174351f577": "L^{(h)} = \\frac{\\beta_h}{\\epsilon_h}\\begin{pmatrix}\n1 & -\\gamma\\phi_h & 0\\\\\n-\\gamma\\phi_h & \\frac{\\gamma^2 {\\mathcal H}_h}{\\beta_h} & 0\\\\\n0 & 0 & 0\\end{pmatrix}", "e144e3e00da87c3d7dad4c9c5f3286ce": "\\nabla_{\\perp}^2 A + 2ik\\frac{\\partial A}{\\partial z} = 0,", "e1458597d6ed05f0ecd7d511f9cd02c9": "X\\to M_p(n)", "e1460b33f6f425dea18ae7872d388c19": "p(y) \\propto \\left(\\frac{a_1-a_2}{a_1}\\;y\\right)^{(-a_1+a)\\nu} \\left(\\frac{a_2-a_1}{a_2}\\;(1-y)\\right)^{(a_2-a)\\nu}.", "e1467c7525eb3e4834e808af76d11be9": "\\frac{f(y)}{g(x)}", "e146a116ef55ae09f11b628c842cc484": "ba(\\Sigma)", "e1472d0f634325bddd0a5ae85ba60bd3": "\\varphi_0(\\gamma+1) = \\omega ^{\\gamma+1} = \\omega^ \\gamma \\cdot \\omega \\,,", "e1473af1b1bc96470153e0152b546a99": "P^{(N)}_{0}(\\xi_{1},\\xi_{2},...,\\xi_{N})", "e14754fef329bb4daf4f8e6a99e2b6ee": " s\\in S ", "e147b0a69a357322822abbb1f45e2225": "\\mathbf{Set}", "e14828ab99c0e06df8fd68fcda5c3ef0": "s_1 = ffghkll", "e1485ff6eb43e49283780fdca8b01951": "L = I \\omega ", "e14870c22762c18e3021dd8ccaf2d6c4": "\\varepsilon_{\\text{s}} = \\lim_{\\omega \\rightarrow 0} \\widehat{\\varepsilon}(\\omega).", "e148722931653f5fac12f24c11035ae9": "y = c_1 e^x + xe^x.", "e148fc3b567646c41dbe57f03239eca2": "\\mathbf{A}=\\begin{pmatrix}\n A_{11} & A_{12} & \\cdots & A_{1m} \\\\\n A_{21} & A_{22} & \\cdots & A_{2m} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{n1} & A_{n2} & \\cdots & A_{nm} \\\\\n\\end{pmatrix},\\quad\\mathbf{B}=\\begin{pmatrix}\n B_{11} & B_{12} & \\cdots & B_{1p} \\\\\n B_{21} & B_{22} & \\cdots & B_{2p} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n B_{m1} & B_{m2} & \\cdots & B_{mp} \\\\\n\\end{pmatrix}", "e14932504123a91dd3e44694b752607c": "z^3-z^2-z-1=0", "e14932f8b7a229d5b591c7034ee5c879": "(x + i0)^{-k} - (x- i0)^{-k} = 2\\pi i \\frac{\\delta^{(k-1)}}{(k-1)!},", "e1494bd12379b6bab5be69071864566f": "K/2", "e1495ae3d866b87824408756a5418f62": "\\scriptstyle D_F(8\\rightarrow 4)= 4(1)-2+0-1=1", "e149928aedb607fcd08de66bc64fdeb6": "L^{p_0}+L^{p_1}", "e1499f15f80b470d028374ffd1dee750": " S = \\sqrt { \\frac{ 2 }{ 4 - \\pi } } [ ( \\frac{ \\pi }{ 2 } )^{ 0.5 } - \\log_e( 4 ) ] \\approx 0.1251 ", "e149f25b346036fcdadcc3e12a6bbcf3": "\\vert e, n-1 \\rangle \\rightarrow \\vert g, n-1 \\rangle.", "e14a1ea75498cf45d3885adf23e39324": "\\psi\\colon D\\nrightarrow E", "e14a7b9f67daf0b2dc2711e0f51515e5": "a_1^{a_2^{a_3^{\\cdot^{\\cdot^\\cdot}}}}", "e14a7d9225127eae638fe04fee13b944": "y^\\dagger x", "e14ac76730cb2177fad44bd5ce2f1116": "\\Gamma ^{-1} ", "e14ada872c4ad7f22b4e222357d6b700": "r^2/4", "e14aea91fe28a89b8ca1cf5c9b92b859": "\nQ = \\sum_{i=0}^5 P(M_i,i) + X_1 + X_2\n", "e14b572ad85579ba543f78708b79ed48": "f(j) M(i,j) = \\frac{\\lambda}{N} A(i,j) = \\frac{\\lambda}{N} A(j,i) = f(i) M(j,i)", "e14b5bafe5095b1bafb68f93ea87716f": " H( X ) = \\sum \\frac{ x_{ ij } }{ X } log \\frac{ X }{ x_{ ij } }", "e14b6e4c3752d7e80af71a6d54a61fbb": "q(x_1,\\ldots,x_n)>0\\,", "e14ba2cedbaf59a3617aa0bfb3a3e908": "\\Omega \\wedge * \\Omega < 0", "e14be76465026f243df3ba83c77d66c7": " e^+ ", "e14c12d3fe5c01341e14b7ee2a97fef7": "q = r_2", "e14c13471af1c9fcebfb9a39d95dd91a": "A^6,A^5B,A^4B^2,A^3B^3,A^2B^4,AB^5,B^6\\,", "e14c685875f481c005cf31530917fabc": "xy + yx = 0\\,", "e14d0b03c3b2aa34b63dd7803137f263": "a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{ \\ddots + \\cfrac{1}{a_n} }}},", "e14d18bf34320755d39fca5f1cc30262": " \\Delta E_{F=I\\pm1/2} = -\\frac{h \\Delta W }{2(2I+1)} + \\mu_B g_I m_F B \\pm \\frac{h \\Delta W}{2}\\sqrt{1 + \\frac{2m_F x }{I+1/2}+ x^2 }", "e14d198a2466103ac11e6c72dde80f89": " \\scriptstyle FV = PV \\cdot (1+i)^n", "e14d1cfbe38cd7bad77875b4175f2a46": "\\hat{f} \\rightarrow \\acute{\\hat{f}}=\\hat{U}^{+}\\hat{f}\\hat{U}", "e14da4ad678150e0bf9503ad515ae26d": "H^\\infty(\\mathbf{D})", "e14dceb294ae917e34d01ca1ff12d43c": "\\pi: M \\times [0, 1] \\to M", "e14deb67a0d05f51b943b43f14fbe0b6": "\\alpha_{Ji}\\,\\!", "e14e1776aa9dded081f4efc1d94a8e73": "\\mathcal{S}_2", "e14e2a2c9afe3ffc54cca738db8ca5da": "A \\; = \\; 10 \\; - \\; 20 \\; C_N ", "e14e64e8a1be6f79ed7e94d8a2d5d6f8": " G_0 = b_0 \\, ", "e14ee6ed87646cb2cd65f372e8681e62": "\\left\\{(x,y)\\mid y\\ge \\frac{1}{1+x^2}\\right\\}", "e14ef005ba3d5532ab4d4a79d89bfb7b": "V_1(\\mathbf{x},z_1) = V_x(\\mathbf{x}) + \\frac{1}{2}( z_1 - u_x(\\mathbf{x}) )^2", "e14f0b08e3e8c9f3e4ebff978d530b1f": "\\vec{F} = q(\\vec{v} \\times \\vec{B})", "e14f415f34cc679748e25481db8292c2": "F(x) := \\int_0^x f(\\xi) d\\xi", "e14f57b7067ab66e86c32ad8e8cdbcab": "R = e^{\\frac{\\mathbf{A}}{2}}.", "e14f66e13b18bd4c553c7fb6b9a18bc5": "\nG(\\mathbf{k}) = \\iiint \\mathrm{d}^3r \\; G(\\mathbf{r}) e^{-i \\mathbf{k} \\cdot \\mathbf{r}}\n", "e14f745ff1ac80fe4e7a6fe95a5cafa0": "P_n^{(\\alpha,\\beta)}(x)= \\sum_s {n+\\alpha\\choose s}{n+\\beta \\choose n-s} \\left(\\frac{x-1}{2}\\right)^{n-s} \\left(\\frac{x+1}{2}\\right)^{s} ", "e15007aaa75136f1b6e9f68fbc1cb325": "\n\\overline{\\overset{\\{q\\} }{\\bigcup }X_i}=\\bigcap^{\\{q\\}}\\overline{X_i}.\n", "e1505d0ba0ef7db21d5568f171051e42": "\nJ(\\mathbf{\\alpha}) = \\frac{\\mathbf{\\alpha}^{\\text{T}}\\mathbf{M}\\mathbf{\\alpha}}{\\mathbf{\\alpha}^{\\text{T}}\\mathbf{N}\\mathbf{\\alpha}}.\n", "e150a9fac01f9152fdca4f5ac32fd469": "d = (a^2 + b^2 + p^2)/(2p)", "e150dfa7d23a54faca975084d8166ac8": "\\mathbf{A}\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}", "e150e7b1f4bc82ad91c6e6d66bab6f90": "X\\supseteq Y", "e15152aad8c82b2413ca96ca94e9ed42": "\\scriptstyle\\flat ", "e15157909544316a51f7a9f95d55d11c": "\\Gamma[\\phi]=-\\langle J,\\phi\\rangle-E[J] \\, ", "e1515e0a481442adc2ae994044909411": "\\frac{\\mathrm{m}^3}{\\mathrm{N}\\cdot \\mathrm{s}^2}", "e1516b068facbb79c9a411270da2e42d": "P(0) = 36.79%", "e1518f791886149f64508f36a8553531": "\\mathit{x}", "e151e10934767cff7b3de987a18515f4": "U_\\mathrm{E} = \\frac{1}{2} \\frac{1}{4\\pi\\varepsilon_0} ( \\frac{Q_1 Q_2}{r_{12}} + \\frac{Q_1 Q_3}{r_{13}} + \\frac{Q_2 Q_1}{r_{21}} + \\frac{Q_2 Q_3}{r_{23}} + \\frac{Q_3 Q_1}{r_{31}} + \\frac{Q_3 Q_2}{r_{32}})", "e15266ce01f572120ec7a2c7844ddc0f": "\\rho_{a / b}(\\rho_{c / d}(R)) = \\rho_{c / d}(\\rho_{a / b}(R))\\,\\!", "e152edc32eec357486d830d15019a3c1": " \\mathbf{v}_1, \\mathbf{v}_2, \\dots, \\mathbf{v}_p ", "e1533844b395e39c4fd5a4796f86053c": "\\left \\{ J(q_1), J'(q_1), J''(q_1), \\ldots, J(q_n), J'(q_n), J''(q_n) \\right \\}", "e15339624a352ad3f424bc02bf9f5e58": "6601 = 7 \\cdot 23 \\cdot 41 \\qquad (6 \\mid 6600;\\quad 22 \\mid 6600;\\quad 40 \\mid 6600)", "e1535e14eea3787aa6d32fa82346e923": "B\\Gamma", "e1536c2a290676ea3625e52b59a593f3": "\\{T,F\\}", "e1536e4b4a45add2709920424244e62f": "H(S)", "e1537368eb687a476ab6748e4a0fd65d": "\\sum_{n=-\\infty}^\\infty a_n ( z - c )^n", "e153ceb2c0d3f65ce6b60f2d16fa995b": "\\sigma = 1", "e1543a063c611e60e7d79f68a74cfcd1": " \\Gamma^l{}_{jk} ", "e15470cd75697519321bf984180afb08": "\\delta{t}", "e1547a62ef78046b336dedf5fbb23902": " \\rho = \\int \\lambda d P_{\\lambda}", "e155132790d696109608602ba3304d7d": "\\mathcal{I}_{a, a}", "e155285c657d0f3b1e6b728680f68979": " \\gamma(t)=x(t)+iy(t).", "e15551c22a407aececa1f7d0a749dd3c": "\\beta_2 = \\frac{-1 + i\\sqrt{3}}{2}", "e15555509377e165dad679262b1cde6c": " \\Omega \\times (0,T) ", "e1555f98c6e0d66ec103f740341c31cc": "\\sqrt{(x-c)^2+y^2} = -{1 \\over 4a} (x^2 + 2xc + c^2 -4a^2 -x^2 +2xc -c^2)", "e1557565e09aefc5ba09a700b8bd1913": "\\theta = \\theta^*", "e15596e7e18c54cf6b33ea52e3ac01b4": "-\\frac{\\xi}{2 t} \\frac{\\partial c}{\\partial \\xi} = \\frac{1}{2 \\sqrt{t}} \\frac{\\partial}{\\partial x} \\left[ D(c) \\frac{\\partial c}{\\partial \\xi} \\right]", "e155d7ecc4a75b0953e760330df6634c": "\\mathrm{d}\\mathbf{\\Sigma}^2 = \\frac{\\mathrm{d}r^2}{1-k r^2} + r^2 \\mathrm{d}\\mathbf{\\Omega}^2, \\quad \\text{where } \\mathrm{d}\\mathbf{\\Omega}^2 = \\mathrm{d}\\theta^2 + \\sin^2 \\theta \\, \\mathrm{d}\\phi^2.", "e1562ec09928580cb342bb60831404b9": "x=r(\\varphi)\\cos\\varphi \\,", "e156325c7d96143c5ba5fc4348caaa8e": "\\int_0^\\pi \\int_2^3 \\rho^2 \\cos \\phi \\, d \\rho \\, d \\phi = \\int_0^\\pi \\cos \\phi \\ d \\phi \\left[ \\frac{\\rho^3}{3} \\right]_2^3 = \\left[ \\sin \\phi \\right]_0^\\pi \\ \\left(9 - \\frac{8}{3} \\right) = 0.", "e15665180a6ed6c9c9c6dfcdd4545760": "\nP\\left(A\\mid\nX\\right)\\approx\\prod_{i 0", "e15d2735a57c59502876ae9d09527550": "\\frac{v^{2}}{2 g}+z+\\frac{p}{\\rho g}=C", "e15ecaa3c36be5a57bc65e3006ff0697": "B=(b_{ij})", "e15f1ca8079ac7135a8ca92a8ac85d9e": "\\mu(AB)<\\mu(A'B').\\,", "e15f21cd18c0900b58c2b79c41a2a07e": "E\\left(2\\,\\sqrt{-4-3\\,\\sqrt2})\\right)=\\frac{\\left(2+\\sqrt2\\right)\\left(\\pi ^2+4\\,\\Gamma\\left(\\frac34\\right)^4\\right)}{4\\,\\sqrt\\pi\\,\\Gamma\\left(\\frac34\\right)^2}", "e15f473536bc8951c8131d49ffa2f710": "\\frac{\\zeta(s-1)}{\\zeta(s)}=\\sum_{n=1}^{\\infty} \\frac{\\varphi(n)}{n^s}", "e15f58d6b82c86cbf14855e28e423113": "X\\times I.", "e15fec42cb52e7bf07caa321db545cfe": "f''(x\\otimes y)=\\sum_{g\\in G}\\langle x,\\rho(g)[y]\\rangle g", "e15ffa4fc49b445d7bcc9ab61d7fa171": "\\mathbf{w_n}", "e1601c461c72773dc295845b5001fe96": "-\\int_{-\\infty}^\\infty |f(x)|^2 \\log |f(x)|^2 \\,dx -\\int_{-\\infty}^\\infty |g(y)|^2 \\log |g(y)|^2 \\,dy \\ge 0.", "e1603e17780b1225fcb2bc077610f7cd": "127_{11} \\ ", "e16073c9f5f75be119d01205b4af5957": "K: D \\times D \\rightarrow \\mathbb{R}", "e160c3d4876b1b714837c26d972c20f5": "P_{\\mathrm{s, max}} = 3 \\frac{R_{\\mathrm{T}}}{q} \\frac{L^2}{h},\\,", "e1611cccf8fe2d0c897a02277f2e8565": "2n-3", "e16149f5d631c62313957d1c2dba86f1": "\\forall x ( e \\leq x \\to \\exists z \\leq x [ \\mathrm{Proof}_T (z,\\mathrm{neg}(\\#\\rho))],", "e161687e0ab475295e13aa103b2ee1ab": "\n\\left(\\mathbf{U}^\\dagger\\mathbf{A} \\mathbf{U}\\right)^\\dagger = \\mathrm{diag}(\\alpha^*_1,\\ldots,\\alpha^*_m) =\n\\mathbf{U}^\\dagger\\mathbf{A}^{-1} \\mathbf{U} = \\mathrm{diag}(1/\\alpha_1,\\ldots,1/\\alpha_m)\n", "e161a47b049df35b25dbcddb38c8160d": "I_J", "e161b075d8e430de878f1f010e1771d5": "{\\delta}_{[k,j,c]}:\\mathbb{R}^{k}\\to \\mathbb{R}^{k+1}", "e161c65622da42a9de60a807384a9708": "V_{j(\\alpha)}", "e162380878bba2e3271a84640ecbc253": "U(x,y)=\\left(\\alpha x^\\rho +(1-\\alpha)y^\\rho\\right)^{1/\\rho}", "e16249990beb431611cb2f801e68c5fc": "H = \\frac {G^2} {A}", "e1626377b2fd98fd95c657bfc7f3d76e": "\\left| h_{\\mu \\nu} \\right| \\ll 1", "e162a3c5b3a2d98cc7c0d0354a5a1030": "\\int_{\\mathbb{R}} |(F/E)(\\lambda)|^2 d\\lambda < \\infty ", "e162e07d5bbaaaafa876fa0286e7f11d": "\\int_{\\mathbf{R}^n} |\\psi_k-\\psi |^2\\,{\\rm d}\\mu\\, \\to 0\\,", "e162e4967a0623f755626d5cf91488c4": "Q^{(d)}", "e16316b962cbaf4d7c6c1b05125ff528": "B_{2}(T)", "e1633c5ad15170baa5e7d314fc193f54": "\\gamma /\\theta <<1", "e1635e4f2f8fedd24e26efeaca5a78ab": "\n(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1} = \\mathbf{A}^{-1}+\\mathbf{A}^{-1}\\mathbf{B}(\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\mathbf{CA}^{-1}\\,\n", "e163ea2d167da714c32402d0e6bd50ea": "w = \\frac{1}{y}", "e16407f6c3c105224d4dc7fb6fc03760": " 144 = 2 \\times 2 \\times 2 \\times 2 \\times 3 \\times 3 = 2^4 \\times 3^2.", "e1641e9fd9d2a2e72e8a48c5fa41f411": "\\forall xyz\\, [ C(x;yz)\\rightarrow\\neg C(y;xz) ],", "e164269f3301a5510e755e07e36cb9b0": "\\lim_{n\\to\\infty} \\frac{s'_n-\\ell}{s_n-\\ell} = 0", "e164c58f868517d4d6e8a3e00b0abf33": "N_i = \\left ( f_i\\left ( x \\right ) \\right )^{e_i}-C_i \\bmod\\ N_i", "e164c72702eaccb55d9d204f8316c302": "\\phi_a(\\mathbf{r})", "e164ef0a23470cb3c276af06cc94e949": "0 = \\operatorname{Tr}(A^*A) = \\sum_{j=1}^n (A^*A)_{jj} = \\sum_{j=1}^n \\sum_{i=1}^m (A^*)_{ji}A_{ij}=\\sum_{i=1}^m \\sum_{j=1}^n |A_{ij}|^2", "e16506a1177a51269ccbfef7bf30828c": "\\textstyle F = G \\frac{m_1 m_2}{r^2}\\ ", "e16534233072f6563e1cc43f6e92475c": "\\mod p", "e16552f72c5ea5b2c8e8e37f368404db": " N_l = im_c - t( i ( a + 1 ) m_c + ( b - 1 ) m_c^2 )^{ 1 / 2 } ", "e1655a6e8aea1ce3fcc454888399e0bf": "\\left (\\textit{choc} \\rightarrow \\textit{STOP}\\right ) \\sqcap \\textit{STOP}", "e165830055b68e06bb09953f93016403": "\\frac{\\partial}{\\partial q_2} \\boldsymbol {e_1} = -\\boldsymbol{e}_2 \\frac{1}{h_2}\\frac{\\partial h_1}{\\partial q_2} -\\boldsymbol{e}_3 \\frac{1}{h_3}\\frac{\\partial h_1}{\\partial q_3} \\ ,", "e1661baa4a64ddd1790a6d9429b843b3": " \\tilde{N}^0 \\rightarrow \\tilde{\\ell}^+ \\ell^-", "e1662170535109095a71663b4d22a563": "\\omega_c < \\omega", "e1663cecc58d84e0a2858c8339bd795d": "a = 2r, b = r, c = r\\,\\!", "e166942b9428402b39440a01dc0cfe26": "= b \\int_0^A \\frac{ds}{W(s)}", "e166b8cdac08fa795dfec71dc678660e": "g(\\lambda) = \\inf_{x} L(x,\\lambda) ", "e1671797c52e15f763380b45e841ec32": "e", "e1675ccab6dbd7d56462b2735bc8ef5f": "\\scriptstyle x\\in U_1", "e1676e688525443ba132d0389fae0ebf": "(\\mathrm{id}_C \\otimes \\epsilon) \\circ \\Delta = \\mathrm{id}_C = (\\epsilon \\otimes \\mathrm{id}_C) \\circ \\Delta", "e1677d04afa04ceb74ee2ef36df2464e": "GF(p^2)", "e167aae6c202be6459cf6575b6f70813": "d_1|d_2|\\ldots|d_k", "e167b1d29a73e6d9ea780148991267c8": "f \\circ \\phi:S^p \\to X", "e16800ef65a85bd79e3e269cc62a1557": "F_\\beta", "e16874daed15f2c5d2b1f51daa67fb12": "\\ S_i = \\alpha_i \\times S_c ", "e168b09897570b3a8ff785c299795cad": "O\\left(|V|/\\sqrt{\\log |V|}\\right)", "e168f20a4e0a94d1259e38d6a07dfb02": "\\scriptstyle \\Delta \\phi(nT) \\ > \\ \\pi.", "e169f106237c0216d478cbf8e18a96ca": "\\{\\psi_{j,k}\\}_{j,k\\in Z}", "e16a30424a38d338923c6d00a18d1e88": "[G_o]", "e16a4d13ec7687dfc1662eb9090e7cf4": "\\sin\\theta=y/r", "e16a5dadd2af2a7af9c4d42325283629": "GL(n)", "e16a72d03d893d082530c55c74998436": "H(v)", "e16ab8dd35703ed8f1801d9e52f9aff7": "\\bold{x} = (x^0,x^1,x^2,x^3) = (ct,\\bold{r}) \\,\\Rightarrow,\\, d\\bold{x} = (cdt,d\\bold{r})", "e16ac16ec9aa52c8f2c149f34168913b": "R_1+R_3=R_2+R_4.", "e16af572039533a36589f4f971d1f43f": "\\frac{2}{(\\nu-2)^2 (\\nu-4)}\\!", "e16b834d6949986fd110ffd65c180e80": " O(n^{-1/2})", "e16ba092946c3d566f843c3482ef41f2": "A^c=U \\smallsetminus A", "e16bd208d92a322a1028de8c28dd1b6c": " 1/4 ^{th} ", "e16bf9021663aa7719cfe4c01fe68246": "\\det(XD+(n-i)\\delta_{ij}) = \\det(X) \\det(D) \\, ", "e16c6d6cfa2242892688a7a153667c56": "\\{\\, a \\mapsto 1, b \\mapsto 011, c\\mapsto 01110, d\\mapsto 1110, e\\mapsto 10011\\,\\}", "e16cb2f37ad196553dc88e93fbd64e22": "M(H) \\approx n \\mu \\tanh\\left(\\frac{\\mu_0 H \\mu}{k_B T}\\right)", "e16cf37e05dd76d68885a5f7f2fd0541": "(t_{11}, \\dotsc, t_{1k}), \\dotsc, (t_{m1}, \\dotsc, t_{mk})", "e16d02abe2ff4d643aab4a40c46e527e": " p_i = \\begin{bmatrix} 0 & \\operatorname{e}_i^{\\mathrm{T}} & 0 \\\\ 0 & 0_n & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}, ", "e16d1dce74f82fc7043f03bfaba19d4d": " g_{22} = \\left. \\frac{V_2}{I_2} \\right|_{V_1=0}", "e16d55bf2858da160f5931cf18cc97a7": "\\alpha=\\alpha_0 \\delta^t", "e16d71878047624c9f4bde5fff9bdcaa": " \\begin{align}\n\n R = N \\cdot u \\cdot \\rho = u \\quad \\text{if} \\quad \\rho = \\frac{1}{N}.\n\\end{align}", "e16d9c02743806218f781bc1f23b738f": "y(e) = \\sum_{1 \\le i \\le k}(g_i(e)v_i)", "e16d9de7a4dea68f6c22fed1c763b1e2": "H_\\phi= i \\frac{I_0\\delta \\ell\\, k}{4\\pi r} e^{i(\\omega t-k\\,r)}\\,\\sin(\\theta)", "e16dcce73fbd64b3295d9a48a07a414e": " \\mathbf{I}\\cdot\\mathbf{a}=\\mathbf{a}\\cdot\\mathbf{I}= \\mathbf{a} ", "e16e0fa6f211c8a395b2edeecc505f00": "\\scriptstyle B(t)", "e16e65944645f239af568c00ee3f50e5": "N = 100", "e16e6afaa340a96ffd842f4ce3117f29": "\n\\frac{\\partial}{\\partial x^{\\sigma}} \\left( \\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\right) =\n\\frac{\\partial L}{\\partial \\phi^A}\n", "e16e820a9e6d9b1e17f71ad2eba6f2c7": "\\delta^{\\beta_k}(\\gamma_k-1) + \\delta^{\\beta_k}", "e16ea2ea8056483662e774ba58e476da": " \\mathbf{E} = - \\nabla \\varphi - \\frac { \\partial \\mathbf{A} } { \\partial t }", "e16eb1eb7f8fb72c697453bfe10239c5": "\\mathrm{e}^{'}_{}", "e16f068234c6a9c9b30b1c9e950802c1": "m + i", "e16fd340f2f58a575dd6f966e1d72208": "\\overbrace{\\int_{y_1}^{y_2}x(y)dy}^{A_1}+\\overbrace{\\int_{x_1}^{x_2}y(x)dx}^{A_2}=\\biggl.x_iy_i\\biggl|_{i=1}^{i=2}", "e17008420cff012475ac7ccec2b1a6d3": "\\mu_i=RT\\ln c_i+\\mu^{\\ominus}_i", "e17054240e006ae7c319adfd12d287cc": "\\mathrm{Zn_n(Cu) + CH_2I_2 \\longrightarrow IZnCH_2I + Zn_{n-1}(Cu)}", "e1707a111e97f8d1dd19792b3c60ec58": "(z_1 - T)^{-1} (z_2 - T)^{-1} = \\frac{(z_1 - T)^{-1} - (z_2 - T)^{-1} }{(z_2 - z_1)}.", "e170879248f42957f7dfc8100a4c9061": "\\vec{\\beta} \\cdot \\dot{\\vec{\\beta}} = 0 ", "e170b7cc968e86e210cc34a868b02f61": "A \\to A\\alpha \\mid \\beta", "e170df3f7c2d2397e6363bb4612a4880": "\\mathbf{m}=-\\gamma \\mathbf{L}", "e1710bfd5c4f23efcccc9f675b28eca2": " \\tan \\theta\\! ", "e1712349dba90bc9088c469600e6ae4a": "\\mathbf{\\tau}(t)\\;", "e171681e2b6322eb92959a3635244013": "\\mathrm{d}S=\\left(\\frac{\\part S}{\\part T}\\right)_P\\mathrm{d}T+\\left(\\frac{\\part S}{\\part P}\\right)_T\\mathrm{d}P.", "e1719056f444e54388721838d52c41bd": "X : I \\times \\Omega \\to \\mathbb{X}", "e171d5073e51caa3b7da66633c48d322": "T : H \\otimes H \\to H \\otimes H", "e17248166a98ae5ae6fbb0770b9e45b6": " U_0(q) - 2U_1(q) = \\mu(q)", "e172815bad46814e594b691115c331ac": "\\mathbf u \\cdot \\nabla \\rho", "e1729c2440a7d1784740c5e1a1b0d6f9": "X = \\bigcup_{i=1}^n U_i", "e1729e9df42786077caf165ade1f5287": "g = EP / (e_{i} - e_{a})", "e172e3e53f2a05f0fe4eaeb40102d969": "dr\\,d\\theta", "e172f1202fd9d4fd9db719318575eb2b": "\nf(w_2,w_1,\\beta_2)=\\sum_{i=1}^{\\beta_2}(f^\\text{pmi}(X_i^{w_2},w_1))^\\gamma ", "e17303b445bb8e02e796233159c780b8": "\n\\frac{ \\partial \\bar{u_i} }{ \\partial t }\n+ \\frac{ \\partial }{ \\partial x_j } \\left( \\overline{u}_i \\overline{u}_j \\right) \n= - \\frac{1}{\\rho} \\frac{ \\partial \\overline{p} }{ \\partial x_i } \n+ 2 \\nu \\frac{\\partial}{\\partial x_j} \\bar{S}_{ij}\n- \\frac{ \\partial \\tau_{ij}^{r} }{ \\partial x_j }\n", "e1730732c5f714b18bd6fa3c5ed169b4": " \\sum_{i=1}^n x_i y_i = 1_A", "e17360baac0b1330f39c89ab17b3f485": "U_{DF} = E_F-qV_{SC}-qV_{DS}.", "e173847b10b19d7f3015b42d5af9bd48": "\\lim_{h \\to 0}{f(a+h) - f(a)\\over{h}}.", "e173b1bd3aa0030e3fd2ee6e8104d4b2": "", "e173c3557af461b9d53087107b085c8e": "M^\\ast", "e173c80421fcd62e1ac5ac9246371643": "\\log_{10}(10 x) = \\log_{10}(10) + \\log_{10}(x) = 1 + \\log_{10}(x).\\ ", "e174115ad5705f4b34748c350d9b8f04": "\n\\begin{matrix}\nb_{k} &=& \\left( \\frac{\\lambda_{1}}{|\\lambda_{1}|} \\right)^{k} \\frac{c_{1}}{|c_{1}|}\n \\frac{ v_{1} + \\frac{1}{c_{1}} V \\left( \\frac{1}{\\lambda_1} J \\right)^{k} \n \\left( c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right)}\n {\\| v_{1} + \\frac{1}{c_{1}} V \\left( \\frac{1}{\\lambda_1} J \\right)^{k} \n \\left( c_{2}e_{2} + \\cdots + c_{n}e_{n} \\right) \\| }\n &=& e^{i \\phi_{k}} \\frac{c_{1}}{|c_{1}|} v_{1} + r_{k}\n\\end{matrix}\n", "e17415a46cb374e9fbf198e192233279": "\\hbox{Prob(radius}\\ge R) = R^{-x}", "e1743b4ebe2fa9fcf900e9be82882437": " E(f) = \\frac{1}{2} \\int_U \\Vert \\nabla f \\Vert^2 \\,dx.", "e1743fbbea17d30a567143b1146ea9e8": "\\int 1_S \\, d \\mu = \\mu (S).", "e174671cd3d55a724ff5329d5d31fa31": "n = (-1)^s \\times\n (1+m*2^{-23})\\times\n 2^{x - 127}", "e1746c7d8c84152d957c4df58b0ad18a": "\\scriptstyle n\\geq 3", "e174c974dbf7994f30dce42e8ecaa26f": "T a^{-3}", "e1756c8ac6cf93afd46c0ce2b8d323bf": "K=\\tfrac{1}{2}pq\\sin{\\theta}\\le \\tfrac{1}{2}pq,", "e1759e95c8c61e97ac711bab69f189ac": "U = \\pi_{r_1,\\ldots,r_n,c_1,\\ldots,c_m,s_1,\\ldots,s_k}(P)", "e175a2852391ae4548c6782617a2d07a": " \\scriptstyle \\sigma^2 S ", "e175fbeb9f039a56c010b5311d5f9451": " \ni \\gamma^{\\mu} \\partial_{\\mu} \\psi = m \\psi, \n", "e176347a0cd9269f20c16aa70fc6b4b2": "V(r)=\\frac{2\\pi\\hbar^2}{m}b\\,\\delta(r)", "e17647f19a37e62162916832829465d1": "U( x )", "e17652bbd7fdb8ad1e7c0195e7a59c00": "f = 1/T \\,\\!", "e176798d9e7f8dfe1e31f29dc839c14f": "P^{i} = -2 \\pi^{ij}{}_{;j}", "e1768150d513f47d56133750ee708d01": "\\mathrm{H_2CO_3 \\xrightarrow{Carbonic\\ anhydrase}\nCO_2 + H_2O}", "e177751cea78072332cbcde1ef6dd9b0": "q'(x_1)=k_1", "e1778e0359c8114c497c27dc1fc3b332": "I/I^2 \\to f^*\\Omega_{Y/Z} \\to \\Omega_{X/Z} \\to 0.", "e177c71d2e2b8bfc7e7fc1262b76bd88": "\\xi^{i} \\rightarrow \\acute{\\xi}^{i} = q^{i}(\\xi,\\tau) = Tr[\\hat{B}(\\xi ) \\hat{U}^{+} \\hat{\\xi}^{i} \\hat{U}],", "e177ccad6a0fdd925552d3504cb5d2f2": "d \\circ d = 0", "e177e171d9fd84e145764266c9fdca37": "y=-1/4 \\,", "e17841e6011d0b92541185ed9b25d9f0": "\\binom{n}{1}_F = \\binom{n}{n-1}_F = F_n", "e1785830309b99a43fe530c092a6258c": "\\mathbf{E} \\ ", "e1787c0420e8f2023e2e394e0d874451": "a = 2^\\alpha u", "e178a576d8949218b331709d20079e15": "p_k = \\frac{\\partial L}{\\partial \\dot{q_k}}.", "e1790244301e58c27d152d9035478594": "Y_0, y, Y_1", "e179b9f56a49c6fca9e0403107643775": "f = \\frac{\\omega}{2\\pi} \\,\n", "e17a29ff72309c9897841cd3d22bc15c": "f(x) = C", "e17a7364c3fc65dad57f1b528441abe0": "Tr(\\Phi^k)", "e17a88767d65b3915b179676b9fd23d4": "K_{bellows}", "e17aa703021f7499c46f420ebf7f8520": "\\varepsilon(t>t_1)=\\varepsilon(t_1)e^{-\\lambda (t-t_1)}. ", "e17ac3cd9c4f988411b5719a3fe884e5": "\np = \\rho c 2 \\pi f \\xi = \\rho c \\omega \\xi = Z \\omega \\xi = { 2 \\pi f \\xi Z} = \\frac{a Z}{\\omega} = Z v = c \\sqrt{\\rho E} = \\sqrt{\\frac{P_{ac} Z}{A}} \\,\n", "e17ae397fdefee230dab138c7c444e99": "p^i", "e17b4a011ea48b1673a0b3b26798b0c3": "x^2-y^2.", "e17b74fe802f2e2ee51bb30e31567f43": "d(||\\Omega ||_\\omega \\omega^2)=0", "e17c0f5b289cc3804e3d53f932e26705": "\\{q\\in\\mathbb{Q}:a_q\\neq 0\\},", "e17c216a6b276715eafc3022c1cadf2b": "{I_1 \\choose I_2} = \\begin{pmatrix} Y_{11} & Y_{12} \\\\ Y_{21} & Y_{22} \\end{pmatrix}{V_1 \\choose V_2} ", "e17c2c1be56684997e4fc95d3e92cb92": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 2}{4 \\choose 1}^2 \\end{matrix}", "e17c5d7c76dd0cc034267bba20e21590": "\\sum_{k=1}^{N}k^n=\\sum_{m=0}^{n-1}A(n,m)\\binom{N+1+m}{n+1}.", "e17cc69818c0fefffa04c38f5632c0a1": "u(x,0)", "e17cdd3bee87da7ea0082dd735df5f9d": " P(W_n|W_{n-1}) = { P(W_{n-1},W_n) \\over P(W_{n-1}) } ", "e17d162da2a9166e9830b087a48c84ce": "\n \\Delta\\psi\\, +\\, k_c^2\\, \\psi\\, =\\, 0\n \\qquad \\text{with} \\qquad k_c^2\\, =\\, k^2\\, -\\, \\frac{\\Delta\\left(\\sqrt{c_p\\,c_g}\\right)}{\\sqrt{c_p\\,c_g}},\n", "e17d3cbea20c63d245a9799b6f6c6f47": "[x:=x+1](x \\ge 4) \\equiv (x+1) \\ge 4\\,\\!", "e17d90774dab9069c93d867ddd5ff6e9": "\\frac{\\partial^{2} C}{\\partial S^{2}}", "e17dd99b4f3e0e5dd8bb04336c311e25": " \\cos \\theta_c =\\mathit{f}_1 \\cos \\theta_1 + \\mathit{f}_2 \\cos \\theta_2 \\,", "e17e1ac84f4a47fe1b0cd77a630891cf": " \\Delta f \n= {1 \\over \\rho} {\\partial \\over \\partial \\rho}\n \\left(\\rho {\\partial f \\over \\partial \\rho} \\right) \n+ {1 \\over \\rho^2} {\\partial^2 f \\over \\partial \\varphi^2}\n+ {\\partial^2 f \\over \\partial z^2 }. \n", "e17e3ecc5c35b401e5e972039533be4c": " \\frac{e^{-\\frac{x^2}{2\\sigma^2}}}{\\sqrt{2\\pi}\\sigma} ", "e17efb8c77354ab6d6479ff95a64c088": "x_i = \\frac{1}{\\lambda} A^T_{i,j}x_j", "e17efd9317dce960048099a602211a61": "Q_{k}", "e17fc5240b04e317835d7a0be9098fc3": "{h S}=1.65(D)\\tau_c*", "e17fef1f3627c40dfd1ec216c5674295": "x_0=\\sqrt{\\hbar/m\\omega_0}", "e180031a7e32627b7a693cdc2506ba8a": " T^r_s(V) = \\underbrace{ V\\otimes \\dots \\otimes V}_{r} \\otimes \\underbrace{ V^*\\otimes \\dots \\otimes V^*}_{s} = V^{\\otimes r}\\otimes V^{*\\otimes s}.", "e1800d30b4b8b97541dd8ea95246c9c1": "\\biggl|\\int_\\Gamma \\frac{1}{(z^2+1)^2}\\,dz\\biggr| \\le \\frac{\\pi a}{(a^2-1)^2}. ", "e1802b3c7f4c9aa9ea20d2023301390d": "\\boldsymbol{\\alpha}(tI-\\Theta)^{-1}\\Theta\\mathbf{1}", "e1814d7aa16f31207d82c3845d82f3da": "\\nu_1: \\subseteq \\mathbb{N} \\to S_1", "e181de48753836a9f7cbc0129dde2ee2": "\\frac {1}{0}=\\infty,\\qquad \\frac {1}{\\infty}=0,", "e18253448448635abbd8cdb84318dba8": "\\nabla_\\mu \\omega_\\nu - \\nabla_\\nu \\omega_\\mu = 0,\\,", "e182541e74cf4d2b0a61b41f270a691d": "\n \\exists y \\left( \\forall x \\left( p(x,y) \\right) \\right) \\vdash \\exists y \\left( p(x,y) \\right)\n ", "e1825dec6bcc75ab9241895e2c6fd50c": "Uf(\\lambda)=\\int_1^\\infty f(x)\\, P_{-1/2 +i\\sqrt{\\lambda}}(x) \\, dx", "e1828147358f2df68368db5dac605af2": "=A \\cap U\\,\\!", "e182921e1055f4dd589ae8b4e8ea7874": "R(a) = J_0(\\lambda a) = 0.", "e182b80b5e19e688e322cb4969f1d51b": "\\overline{\\mathbf x}=\\frac{1}{n_x}\\sum_{i=1}^{n_x} \\mathbf{x}_i \\qquad \\overline{\\mathbf y}=\\frac{1}{n_y}\\sum_{i=1}^{n_y} \\mathbf{y}_i", "e182ebbc166d73366e7986813a7fc5f1": "AD", "e1831c6eec1ecefa788e160fdcb0cfec": "\\ \\bar{A} = \\dot{A}-wA+Aw. ", "e18332d13b159dc9dffb2b047307f808": "y(t) = A\\sin(2 \\pi f t + \\phi) = A\\sin(\\omega t + \\phi)", "e18361f329bc3163f9245ef96d4b78f7": " A(m, n)", "e1838ff45540f822f2a34c8dc51708c5": "r_2\\,", "e183e29975c8bc71035466e2a2946467": " (\\lambda p.\\lambda f.(p\\ f)\\ (p\\ f))\\ (\\lambda f.\\lambda x.f\\ (x\\ x)) ", "e183f63289a040deeed67102dd243d0e": "AC \\ = \\ v\\delta t \\cos\\theta", "e1841405c37c65db037be00f24adc290": "x = x_0e_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7,\\,", "e18432b398d50236b16df65c97b2564e": "b^{th}", "e18461d0f4bd3bddffd1024fcc1263b0": "\\log \\colon \\mathbb{R}^\\times_{>0} \\to \\mathbb{R}", "e18496e3c7d453edcb8e6b0f32159494": "\\bar{x} = \\sum_{i=1}^n w'_i x_i= \\sum_{i=1}^n \\frac{w_i}{\\sum_{j=1}^n w_j} x_i = \\frac{ \\sum_{i=1}^n w_i x_i}{\\sum_{j=1}^n w_j} = \\frac{ \\sum_{i=1}^n w_i x_i}{\\sum_{i=1}^n w_i}.\n", "e1857da78828d27072f5a5610498873a": "{c\\over v} = {P\\over{2\\pi\\tau}}", "e1858cf76338b6ac39ae380862664c11": "\\int g F_j \\frac{\\partial f}{\\partial p_j}\\,d^3p=-nF_j\\left\\langle \\frac{\\partial g}{\\partial p_j}\\right\\rangle", "e185c4564717db8191a8b88654d8e017": "ab/R", "e185f05d9df540f6eb1f33aafcb62a97": "(n-k)\\times(n-k)", "e18631017fb9c54b9b6093254b324400": "\\displaystyle Y(s)", "e186c2cde408cb0eaa3d312c913c0bc6": "\\frac{dC_\\text{V}(x)}{dx} = - b_\\text{ext} C_\\text{V}(x)", "e1877d18d8f66a1d064260d67bbed420": "k = k+1 ", "e1879e96ccebb7c5dd87dccf7d984bd7": "|\\alpha|^2+|\\beta|^2=1", "e187a158da9789b04de618309017679f": "\\theta=dx^\\mu\\otimes \\dot\\partial_\\mu ", "e187b91ad653e5aaedef6aa6b7cc2017": "\n\\left[1 + (\\gamma+1) \\frac{\\phi_x}{V_\\infty} \\right] M_\\infty^2 < 1\n", "e188035b14d5391badfc0f9ced16e28b": "i \\leq n", "e18816729f860aaddd66cd448aebedf4": "((ab)c)d", "e1884f28eb91d788d1ca4744ad1bf4a7": " q_{(a,b,c)} = \\frac{1}{\\sqrt{2(1+c)}}(1+c-\\bold{i}b+\\bold{j}a) ", "e1885f1430d08c77b0974321e868c89a": "k = \\frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}.", "e1888719bcc69dcfc480884d52752456": "\\textstyle P(E|H)", "e188c7c159631461381b0c5f7f8a4f2c": "\\{y_i,\\, x_{i1}, \\ldots, x_{ip}\\}_{i=1}^n", "e18906c74e08a7298a48ecf03a4139e1": "\\varphi_f(x,y) \\simeq \\varphi_{F}(f,x,y)", "e1897796c19b9d3d9af5e8ea6ae20b9c": "\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\varphi}} \\right)-\\frac{\\partial L}{\\partial \\varphi} = 0 \\qquad\n\\Rightarrow \\qquad \\ddot{\\varphi} + \\frac{2}{r}\\dot{r}\\dot{\\varphi} = 0.", "e189880aac6398b57e0d1183fde06732": " \\mathbf{x}^* \\in \\mathcal{R}^n", "e18990d1dadbaaff08b0ba7715bdf59b": " F_x +F_u p + F_p p_x + F_q p_y =0, \\,", "e1899691c54a4f2dfdd660eba81f4359": "\\pi(y)\\,", "e189a9285348329ea074a105ff9669c8": "1+\\sqrt{6}", "e189cf6cb35b7e341e1f4a184a9c704a": "\\mathbb{L_A}", "e189dec5975d5fe3f464f514abe70826": "Y_f=Y\\cos(\\psi)+X\\sin(\\psi)", "e18a20275c679cddbd2631ac110298b1": "F_\\text{nodes} = -A^\\top K A \\mathbf{x} - A^\\top K L = 0", "e18aca81314d9719b02cada3e81fccd1": "T_n(1) = 1\\,", "e18accb61147fb9f5bc5d376fc21e105": "Y_{11} = {Z_{22} \\over \\Delta_Z} \\,", "e18b4d75c4beb32aefa263a01f6c7728": "\\operatorname{E}", "e18bbdb5ccd329305649961a76208792": "e_i \\le e_{i+1}", "e18bd5056bc32525e1028da3393b5590": "f(z)=e^z", "e18c13727c77b22a6e2aa5e78996d820": "\\{a(n)\\}", "e18c41c35ae780c419f47ae7aa54d037": "H_1: \\theta > \\theta_0", "e18ca88da53d464de3f23f2ff29fadef": "\\mathbf{C} = \\frac{\\mathbf{x}_1+\\mathbf{x}_2+\\cdots+\\mathbf{x}_k}{k} ", "e18cd3888e66c4dfbc7973b22488969c": "\n\\Phi(\\vec{\\xi}) = - \\int \\frac{d^3\\xi^{\\prime} \\rho(\\vec{\\xi}^{\\prime})}{|\\vec{\\xi}-\\vec{\\xi}^{\\prime}|} \n", "e18d2c7888416cf8a72e966ab71b92e7": "\\mathbf{\\omega}(t)\\;", "e18d398857e7cc235bc89fe1d36b5661": " a_{s1} ", "e18d8500104ec27e3f46dbd42afd7eb2": "x = r \\cos \\left ( \\theta + \\omega t \\right )", "e18d8dad912110118d3315082a648f76": " D_\\mu \\mapsto \\partial_\\mu - i e A_\\mu - i (\\partial_\\mu \\Lambda) ", "e18e13f62bd24ebce493683b0af87651": "q_j = P_{j,j+1}=\\frac{j}{N} \\frac{N-j}{N}", "e18e341bd0a9276233460dc95794eeb9": "(\\tau_i,f_i)_{i\\in I}", "e18e4c578456836988c3a85f3578708f": "\\alpha(t)", "e18eb6ce4799829b019b569cd08229ab": "W-FDR", "e18ee3f277738af6f217327942d2cd55": "\\delta_\\alpha", "e18ee87988e8648f20b6fd9d213f0dea": " S: f(x,y,z)=0\\ .", "e18f21e1aa0ca4c1289ff27dcedad54d": "S =\\ln \\Omega_{E,\\ell,N}", "e18f375a8ea465c0986cf1779c8451c2": "\n\\cos\\frac{\\hat{\\gamma}}{2}+\\sin\\frac{\\hat{\\gamma}}{2} \\mathsf{C} = \n\\Big(\\cos\\frac{\\hat{\\beta}}{2}\\cos\\frac{\\hat{\\alpha}}{2} - \n\\sin\\frac{\\hat{\\beta}}{2}\\sin\\frac{\\hat{\\alpha}}{2} \\mathsf{B}\\cdot \\mathsf{A}\\Big) + \\Big(\\sin\\frac{\\hat{\\beta}}{2}\\cos\\frac{\\hat{\\alpha}}{2} \\mathsf{B} + \n\\sin\\frac{\\hat{\\alpha}}{2}\\cos\\frac{\\hat{\\beta}}{2} \\mathsf{A} + \n\\sin\\frac{\\hat{\\beta}}{2}\\sin\\frac{\\hat{\\alpha}}{2} \\mathsf{B}\\times \\mathsf{A}\\Big).\n", "e18f4f9a2404c47153c94f0a2c3f6f15": " (T_a)_{jk} = -if_{ajk}.", "e18f59c4cb52f897acc34599dc66b183": "\\{ \\omega : X(\\omega) \\le r \\} \\in \\mathcal{F} \\qquad \\forall r \\in \\mathbb{R}.", "e18f7ff02ef880c0e6bcef78053f2117": "\\sqrt{\\Delta U^2 +\\Delta V^2} = 13", "e18fa12906db27f71caa63679e88c999": "\\hat{H}_\\text{int} = \\begin{cases} \\hat{1} \\hat{\\tau} & \\text{Fermi decay} \\\\ \\hat{\\sigma} \\hat{\\tau} & \\text{Gamow–Teller Decay} \\end{cases}", "e18fa50b8aaf96dccc034e5f8e360fe6": "\\begin{align}\n R &= \\begin{bmatrix}\n\\cos \\alpha & -\\sin \\alpha & 0 \\\\\n\\sin \\alpha & \\cos \\alpha & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix}\n\\cos \\gamma & -\\sin \\gamma & 0 \\\\\n\\sin \\gamma & \\cos \\gamma & 0 \\\\\n0 & 0 & 1 \\end{bmatrix} \\end{align}\n", "e18fa93e90bfe8384ac7f2a9f6a8cedb": "\n\\frac{1}{r} = A \\theta_2 + \\varepsilon\n", "e1902c442afe029b3d6163c114449701": "\\scriptstyle 1.5m", "e190446ae4435e8152ba9acc769c99a2": "f_s/2", "e19084302542a1ffabf7f98873df78c7": "R^m_{ijk}", "e190a5d1822b1da3b37ffb4aa397207c": "\\Delta\\varphi = \\frac{q\\Phi_B}{\\hbar}.", "e190c36274fc0912b87f2ef217194b27": "P(s,f|q=x) = {s+f \\choose s} x^s(1-x)^f, ", "e190f4a8e64792878726608a2f4e2be6": "\\sigma_i^' \\in \\Delta_i, i \\in N.", "e1914b4a5a809083d171b33777b76085": "|\\tilde{\\phi}_i\\rangle", "e1915ac7f2d1f0c0ebdb15279792339b": "\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^q", "e191b77e163a9eb9b977b8c65b729650": "\\mathrm{root} \\simeq a - \\frac{a^2 - x}{2a}\\,\\!", "e1921303863a0fd30c42d9ab3a86b1f5": " (a_1, \\dots, a_n),", "e1922d50864de32b5fb81631f631bbdb": "S_{2m+1}=\\sum_{n=1}^{2m+1} (-1)^{n-1} a_n", "e1927c3bff8a547565f31ce63588dcc3": "\\mathcal{L}(s,f|x=p) = {s+f \\choose s} x^s(1-x)^f = {n \\choose s} x^s(1-x)^{n - s}. ", "e192c79b6901adacefe02e0728390d92": "|x|^{O(1)}", "e192eae88ab394343a3b0bb7b8f54c5a": "\\Diamond \\varphi\\implies\\Box\\Diamond \\varphi", "e1936b7326fc79783df4ad07cb156b7b": "\\mathcal{C}_m=\\mathcal{C}_1\\mathcal{C}_{m-1}-\\mathcal{S}_1\\mathcal{S}_{m-1}, \\mathcal{S}_m=\\mathcal{S}_1\\mathcal{C}_{m-1}+\\mathcal{C}_1\\mathcal{S}_{m-1}, \\mathcal{S}_0=0, \\mathcal{S}_1=\\mathbf{R}\\cdot\\mathbf{\\hat{e}}_2, \\mathcal{C}_{0}=1, \\mathcal{C}_1=\\mathbf{R}\\cdot\\mathbf{\\hat{e}}_1", "e19398986d98db08f7ed832d3c3fe9bf": "\n(L_1,L_2,L_3) = l_1 \\mathbf{\\hat e}^1 +l_2\\mathbf{\\hat e}^2+ l_3 \\mathbf{\\hat e}^3.\n", "e193a0673e300b15ffc4b4fa7a38c41b": " \\langle x| a^{\\dagger} |0\\rangle =\\psi_1 ~,", "e193adb6b39cdbb91c9d823ee66fbca4": " \\pi^{-s/2}\\Gamma(s/2)\\int_{0\\swarrow 1}\\frac{x^{-s}e^{\\pi i x^2}}{e^{\\pi i x}-e^{-\\pi i x}}\\,dx\n+\\pi^{-(1-s)/2}\\Gamma((1-s)/2)\\int_{0\\searrow 1}\\frac{x^{s-1}e^{-\\pi i x^2}}{e^{\\pi i x}-e^{-\\pi i x}}\\,dx\n", "e193defd47e8b0f094d1e3ee45dddb02": " \\lim_{n \\to \\infty} \\left \\| v - \\sum_{k=0}^n \\alpha_k b_k \\right\\|_V = 0.", "e193e2ef2c5fe696f93e9ff6a493e094": " \\ \\rho ", "e193f7f10ac00689d033270c8c203c89": "\\int_r^s f(t) \\Delta t = \\int_{[r,s)} f(t) d\\mu^\\Delta(t)", "e1940c9b996f08963494b8198070ae2a": "g(r) = \\frac{4\\pi}{3} G \\rho_0 r - \\pi G \\left(\\rho_0-\\rho_1\\right) \\frac{r^2}{r_e}.", "e19452d42a4cf08a83b75024db0e917c": " \\phi_X: \\mathbb R \\times M \\to M ", "e194b9ad6152b12726a8b1df70c4c229": "A\\mathbf{x} = \\mathbf{y}", "e194f123266dbdcfda0ac33f370a1833": "S^{\\mathrm WZ}(\\gamma) = S^{\\mathrm WZ}(\\gamma')+n ~,", "e1950b542cf13bbbf9a807bed59843fb": " [ w_a ( \\operatorname{E}(R_a) - R_f ) ] / [2 w_m w_a \\rho_{am} \\sigma_a \\sigma_m] = [ w_a ( \\operatorname{E}(R_m) - R_f ) ] / [2 w_m w_a \\sigma_m \\sigma_m ] ", "e19517f653bbf085c6107dceb00ae081": "\\mathbf{r}(t) =R(t)\\mathbf{u}_R = \\begin{bmatrix} x(t) \\\\ y(t) \\end{bmatrix} = \\begin{bmatrix} R(t)\\cos (\\omega t + \\pi/4) \\\\ R(t)\\sin (\\omega t + \\pi/4) \\end{bmatrix}, ", "e195669fa9ac4d3c99455fbce26d9ee3": "n(1 - e^{-n'/n})", "e195c80d3f38322ab71bf6609d12ff1a": "m_\\text{red} = \\frac{m_1 m_2}{m_1 + m_2} ", "e196291402c8d6f3b1707b47b1cdb75e": "F_{max}", "e19643d6a046ce5cc736205d3cc79eef": "T, V", "e19654578b8eb488cb214543b88340e9": "\\frac{f^{(j)}(x_i)}{j!}.", "e19697dbe7a173ec1cf586a642541486": "R(x)^{-1} = R(x^{\\rho})", "e196d71233c82dcb7552d64594d71457": "HELP \\to \\begin{pmatrix} H \\\\ E \\end{pmatrix} , \\begin{pmatrix} L \\\\ P \\end{pmatrix} \\to \\begin{pmatrix} 7 \\\\ 4 \\end{pmatrix} , \\begin{pmatrix} 11 \\\\ 15 \\end{pmatrix}", "e196e81395c68685a46687b68ee7dbae": "a_n=\\frac 1 {2 \\pi i} \\oint_{|z|=c'} \\frac{\\Gamma(\\alpha+1)}{\\left(\\frac z 2\\right)^\\alpha}f(z) O_n^{(\\alpha)}(z)\\mathrm d z,", "e196eddb2c0baaec48bd929bb72194d2": "\\text{Cl}_{2m}\\left( \\frac{q\\pi}{p}\\right)= \\sum_{j=1}^{p} \\Bigg\\{ \\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+j)\\frac{q\\pi}{p}\\right]}{(kp+j)^{2m}} \\Bigg\\} =", "e196fb40f3f4bc025653cb4139e74ffc": "E_{xy,x^2-y^2} = \\frac{3}{2} l m (l^2 - m^2) V_{dd\\sigma} +\n2 l m (m^2 - l^2) V_{dd\\pi} + l m (l^2 - m^2) / 2 V_{dd\\delta}", "e1974ea64071c6f15f92bc52d0f70c10": " \\Delta \\sigma = B \\sin \\sigma \\Big\\{ \\cos(2 \\sigma_m) + \\tfrac{1}{4} B \\big[ \\cos \\sigma \\big(-1+2 \\cos^2(2 \\sigma_m) \\big) - \\tfrac{1}{6} B \\cos(2 \\sigma_m) (-3+4 \\sin^2 \\sigma) \\big(-3+4 \\cos^2 (2 \\sigma_m)\\big) \\big] \\Big\\} ", "e19761f5a294005e8dc89c511538b2bd": " c_{\\infty} ", "e19763bde5768c59aed85967c481630d": "\\hat{x}(t) = j\\cdot e^{-j \\omega_0 t}\\,", "e197d09a88cc895ba667e3c0152c74c9": " \\neg\\;(\\neg\\;(x \\le y) \\;\\wedge\\; \\neg\\;(y \\le x)) .", "e197d6e562fd2e93c5e5b937ec2bee3b": "\\frac{x}{\\sigma^2}\\exp\\left(\\frac{-(x^2+\\nu^2)}\n{2\\sigma^2}\\right)I_0\\left(\\frac{x\\nu}{\\sigma^2}\\right)", "e197f607e911a38f72abc265199c556c": "c_{ab}^{opt}(t)", "e197f8bb4e51d761357ca8fd239afce5": "\n \\boldsymbol{B} = \\boldsymbol{F}\\cdot\\boldsymbol{F}^T = \\begin{bmatrix} 1+\\gamma^2 & \\gamma & 0 \\\\ \\gamma & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\n ", "e1981204a46a1e39a758ad34c295a408": "(G, \\cdot)", "e1981400f31dc5291020fc868b4a9ed2": "(\\Phi'_{i,1},\\dots,\\Phi'_{i,n})\n=\\sum_{j=1}^n a_{i,j}(\\Phi_{j,1},\\ldots,\\Phi_{j,n}), \\qquad i\\in\\{1,\\ldots,n\\}.", "e19846c7fc67863c9fa68fd550306891": "\\scriptstyle \\tilde{t}_i", "e1987d7cbadecd3a8a04bc15593f6ae5": "\\omega=\\delta\\alpha", "e1989f284e98008083c55112b6090e8b": "109.0\\pm 1", "e198a715a5de00fd0358969a76ebe665": "\\nabla^2 \\mathbf{V} = \\nabla \\cdot \\left( \\nabla \\mathbf{V} \\right)", "e198b60006e5c5e2009e54c4887bdf74": "s\\left\\{\\begin{array}{l}p\\\\q\\\\r\\end{array}\\right\\}", "e198d33bbf7e3eace5c412e52aa4f3e3": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{1}&\\mathrm{*}&\\mathrm{T}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "e198e904906fc11da27c73d8b2e3edbf": "v (\\Delta)", "e19958338222a006c7539d5a3a1e3384": "\\omega_0 = \\frac {\\pi v}{2 l}", "e199786fb1d271177639e431bc61a09d": "\nj^{\\sigma} = \n\\left[\\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\mathcal{L}_X \\phi^A - L \\, X^{\\sigma}\\right]\n- \\left(\\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\right) \\Psi^A\\,.\n", "e199a0b6e3743700c741d9d34fa7b0da": " \\mathrm{P} = V_\\mathrm{rms}I_\\mathrm{rms}\n", "e199ab4d50d158a20efe02d549ba7a9d": "\\langle f,g\\rangle_{L^2(\\Omega_E,\\mu)} = \\int_E f\\bar{g}\\,d\\mu.", "e199d9c215ebe277ddd7dad1b5acf919": "\\omega_{0a} = - \\omega_{a0} = \\varphi a_a \\,,\\quad \\omega_{ab} = \\theta \\varepsilon_{abc} n_c \\,,", "e199e2794e24035c05fadbe515104372": " \\langle e_n, e_m \\rangle = \\int_0^L e_n(x) e^*_m(x) dx = \\delta_{mn}", "e199e8ec296c18ca8fac8c9cfd13b988": "\\mathbf{I}\n\\!\\!\\begin{array}{c}\n _\\times \\\\\n ^\\cdot\n\\end{array}\\!\\!\\!\n\\left(\\mathbf{ab}\\right)=\\mathbf{b}\\times\\mathbf{a} ", "e19a0006e4a43521bcfe22479cb1c1a0": "\\mathbf{X}= \\langle X, \\mathcal{F} \\rangle", "e19ada0a2cd0cf7b1eb10e7fcd699910": "I_p", "e19af8d9e158fc019574930d9ac6cb2b": "\n\\frac{\\partial\\psi}{\\partial t}\n+{\\bold u}\\cdot \\nabla\\psi\n+{\\bold u}({\\bold u}\\cdot D\\nabla^2\\psi)\n+(D\\nabla^2\\psi-{\\bold u}({\\bold u}\\cdot D\\nabla^2\\psi))\n=0\n", "e19b10713e8a1290edaba9fa4af17424": "(x^\\mu,\\widetilde x^\\mu=\\dot x^\\mu +a^\\mu(x^\\alpha)), \\qquad \\widetilde x'^\\mu=\\frac{\\partial x'^\\mu}{\\partial x^\\nu}\\widetilde x^\\nu + b^\\mu(x^\\alpha). \\qquad\\qquad (2) ", "e19b8ac42ef1ad15903dba6b0768d60d": "p,q_1,\\ldots,q_n", "e19bdf06dab3cc2dbaf79fc034815bc9": "-D u_l^2 \\equiv N^l", "e19bea2b34c774d881256aaa727381aa": "Y_{2}^{1}(\\theta,\\varphi)={-1\\over 2}\\sqrt{15\\over 2\\pi}\\, \\sin\\theta\\,\\cos\\theta\\, e^{i\\varphi}", "e19bf34127243c74eb4ac738eb1b0111": "{\\alpha \\choose \\beta} = | \\{ B \\subseteq A : |B| = \\beta \\} |", "e19c19bf35c78648c92de6ad57b90493": "(\\mathcal{L}\\Phi)(x) = \\sum_{y\\in f^{-1}(x)} g(y) \\Phi(y)", "e19c25987858f24b595c97bb19ac4807": " \\operatorname{H}(S) = - \\sum_i \\lambda_i \\log_2 \\lambda_i. ", "e19c305d30423ef7cbd83d8587cf2afb": "a \\in F_q", "e19c3ef76279bb6fdd9024dab8eb09ae": "\\mathbb{C}^+", "e19c7e7c1b4615e2b5a8b0a8ca63a539": "\\left(\\sum_{i=1}^n x_i y_i\\right)^2\\leq \\left(\\sum_{i=1}^n x_i^2\\right) \\left(\\sum_{i=1}^n y_i^2\\right).", "e19cdd9fead2dd8a930a1aec62f4c2d6": "\\csc A = {\\cot A \\over \\cos A} ", "e19d6a3f106040578192c0bd771725fd": " \\tan(\\beta - \\alpha) = \\frac{\\tan\\beta - \\tan\\alpha}{1 + \\tan\\beta\\tan\\alpha} = \\frac{\\frac{b}{x} - \\frac{a}{x}}{1 + \\frac{b}{x}\\cdot\\frac{a}{x}} = (b-a)\\frac{x}{x^2 + ab}. ", "e19d7b7db71bfdf06be76631df9a94ad": " 8\\pi T_{bd} \\, = R_{bd} - R_{ac} \\eta^{ac} \\eta_{bd} / 2 ", "e19db409d404b2365c1106037eaf286c": "\n\\begin{array}{rll}\n\\min & \\operatorname{Tr}(P_1) \\\\\n\\text{subject to} & \\Omega_b(A_n)\\leq Q, \\\\\n&\\operatorname{Tr}_{\\mathcal C_k}(A_k) = A_{k-1}\\otimes I_{\\mathcal A_k} & (2\\leq k\\leq n),\\\\\n&\\operatorname{Tr}_{\\mathcal C_1}(A_1) = I_{\\mathcal A_1},\\\\\n&Q_k = P_k\\otimes I_{\\mathcal D_k}&(1\\leq k\\leq n),\\\\\n&\\operatorname{Tr}_{\\mathcal B_k}(P_k) = Q_{k-1}&(2\\leq k\\leq n),\\\\\n&A_k\\in\\operatorname{Pos}(\\mathcal C_{1\\cdots k}\\otimes A_{1\\cdots k}) & (1\\leq k\\leq n),\\\\\n&Q_k\\in\\operatorname{Pos}(\\mathcal D_{1\\cdots k}\\otimes B_{1\\cdots k}) & (1\\leq k\\leq n),\\\\\n&P_k\\in\\operatorname{Pos}(\\mathcal D_{1\\cdots k}\\otimes B_{1\\cdots k}) & (1\\leq k\\leq n),\\\\\n\\end{array}\n", "e19e3c077c872cca5a5ddeccf55383d3": "\\displaystyle ax^3+bx^2+cx+d", "e19e442d997a57c243490ab549c71770": "P(\\cdot)", "e19e6212b7b861152b9367fee23cee02": "C(-,A)", "e19e7664e7f2953fc29ab7f1956e01b6": " \n \\partial^2 \\varphi + m^2 \\varphi =0 \n", "e19e7f24c2c7c7e964d26808d61992e4": "10^{10^b}", "e19ea60c00d6d392b405fc57f0ebd061": "G(R) = \\{g \\in R[\\![t]\\!] | g(t) = b_0 t + b_1t^2 \\dots, b_0 \\in R^\\times \\}", "e19edad589431b0ec4bc567b1e57e7b7": "\n\\delta_{2s}(n)=\n\\frac{\\pi^s n^{s-1}}{(s-1)!}\n\\left(\n\\frac{c_1(n)}{1^s}-\n\\frac{c_4(n)}{2^s}-\n\\frac{c_3(n)}{3^s}- \n\\frac{c_8(n)}{4^s}+\n\\frac{c_5(n)}{5^s}-\n\\frac{c_{12}(n)}{6^s}-\n\\frac{c_7(n)}{7^s}-\n\\frac{c_{16}(n)}{8^s}+\n\\dots\n\\right)\n", "e19ef85efb90f3e31db99cff26feba91": "W_i \\in W", "e19f20bf00c708529b92a4f2c5ca7a87": "\n\\frac{a^2x^2}{x^2+y^2+z^2-a^2w^2} + \\frac{b^2y^2}{x^2+y^2+z^2-b^2w^2} + \\frac{c^2z^2}{x^2+y^2+z^2-c^2w^2} =0\n", "e19f409c8f38599b568fd94e40909989": "\\tau _1 \\wedge \\tau _2", "e19f87c1c19beefa5c75b782288e1e14": "FDR_{+1} = pFDR = E \\left[ \\left. {\\frac{V}{R}} \\right| R>0 \\right] ", "e19fc1709d09345548f03775044079c9": "X_{0} = Z;", "e19fccb7dc876aeb1c4124c3dd41e271": "\\mathbb{C}^2\\otimes\\mathbb{C}^n", "e19fe1a891ebead944c216a422714005": " M ", "e1a01c9c2877c097fbe5a16a7f8e815e": "R = ( r_{ij} )_{m \\times n}", "e1a05cab61b5a77359df9c7e58584320": "\\Delta t_{e y}=-2e\\sin M+y\\sin(2M+2\\lambda_p)=[-7.659\\sin M+9.863\\sin(2M+3.5932)]\\mbox{min}", "e1a08a44e4d281033de82e30be533d61": "sL \\,", "e1a08f6f315c342ede86860ecbffd0fd": "\\mathrm{Re}\\,\\zeta\\bigl(\\tfrac{1}{2}+it\\bigr)", "e1a0bd9cb30d223008c634b5ac206fb3": "\n\\frac{\\partial^2 \\vec\\Psi}{\\partial x^i \\, \\partial x^j} = \\Gamma^k{}_{ij} \\frac{\\partial\\vec\\Psi}{\\partial x^k} + \\vec n\n", "e1a10371dca5d2fe85e9426099a6c301": "\\pm\\frac{\\cot \\theta}{\\sqrt{1 + \\cot^2 \\theta}}\\! ", "e1a1084e480f0cfad255f0292fa79ca0": "+\\sqrt{-r}", "e1a10ce0504233de72396627240868c8": "BE = \\frac{0.60\\,\\beta}{1 - \\frac{\\beta}{2}}", "e1a11b86e3994137e1e83e097d38a016": "\\ell^q", "e1a1a3ea8150df8687d1d0628479f1f4": "|1\\rangle \\otimes |0\\rangle = \\frac{1}{\\sqrt{2}} (|\\Psi^+\\rangle - |\\Psi^-\\rangle),", "e1a1a4341d8b3e77cfce8e6c2dbfd098": "\\mathrm{X}_i\\equiv \\mathbf{u}_i", "e1a1c0e8022d78a35d166c8f982ffdf3": " \\lambda = \\text{Overall Death Rate},~\\lambda^*=\\text{Expected death rate},~\\nu=\\text{Disease-specific death rate}", "e1a1da4a886590c307602763f9f28593": "p_t \\approx 1", "e1a20bfb6ff8509bdb171c3281f34dfd": "\\pm\\frac{\\cos \\theta}{\\sqrt{1 - \\cos^2 \\theta}}\\! ", "e1a212724012a53ae2ca39be60d4b53b": "r = R_\\mathrm E / y_\\mathrm{atm}", "e1a2e0b6aa9060e3644b93916113d928": "k\\cdot S", "e1a2ee3c7760b4fa970aba45b7cfe893": " P(x) = \\sum_{k=1}^\\infty \\underset{p_k\\text{ prime}}{p_k x^k} = 1+2x+3x^2+5x^3+\\cdots", "e1a2f85103fa58beb218e3a344a060cc": "\nPr\\begin{cases}\nDs\\begin{cases}\nSp(\\pi)\\begin{cases}\nVa:\\\\\nS^{0},\\cdots,S^{T},O^{0},\\cdots,O^{T}\\\\\nDc:\\\\\n\\begin{cases}\n & P\\left(S^{0}\\wedge\\cdots\\wedge O^{T}|\\pi\\right)\\\\\n= & \\left[\\begin{array}{c}\nP\\left(S^{0}\\wedge O^{0}|\\pi\\right)\\\\\n\\prod_{t=1}^{T}\\left[P\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)\\times P\\left(O^{t}|S^{t}\\wedge\\pi\\right)\\right]\\end{array}\\right]\\end{cases}\\\\\nFo:\\\\\n\\begin{cases}\nP\\left(S^{t}|S^{t-1}\\wedge\\pi\\right)\\equiv G\\left(S^{t},A\\bullet S^{t-1},Q\\right)\\\\\nP\\left(O^{t}|S^{t}\\wedge\\pi\\right)\\equiv G\\left(O^{t},H\\bullet S^{t},R\\right)\\end{cases}\\end{cases}\\\\\nId\\end{cases}\\\\\nQu:\\\\\nP\\left(S^{T}|O^{0}\\wedge\\cdots\\wedge O^{T}\\wedge\\pi\\right)\\end{cases}\n", "e1a3051256b96eb0e105692be69dbc3e": "y_1(t)", "e1a319bea7834b4a1ee41210d64fbe7d": "A \\leq_m B", "e1a34b2946caf0205e6aa98a405843b7": "\\,(\\eta_1,\\ldots,\\eta_k)", "e1a37529adaeac5448d9e4796e2ae6b7": "\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)) ", "e1a3ae6e7f966a372050d0fc28a7634d": "I_\\mbox{linear}", "e1a3e137235906be930013a11a73b37e": "f + O(g) \\in O(f + g)", "e1a40bec648aea54c52a00c5615c4eac": "i : \\mathbf{R} \\to \\mathbf{R}^{2} : x \\mapsto (x, 0)", "e1a416734b32fb16a3c4dfb7af0cdb47": "\\begin{bmatrix}\nA & B \\\\\nC & D\n\\end{bmatrix}\n.", "e1a45d613710cf15102319364b40ea0e": "\\widehat{\\Omega}", "e1a4f5ee05ba932fb8d04ce321f70eb6": "\n\\begin{array}\n[c]{c}\nX\\\\\nZ\\\\\nI\\\\\nI\n\\end{array}\n\\left\\vert\n\\begin{array}\n[c]{cccc}\nZ & X & Z & I\\\\\nZ & Z & I & Z\\\\\nY & X & X & Z\\\\\nZ & Y & Y & X\n\\end{array}\n\\right.\n", "e1a53e59d4e406201db09bb9c1ac377a": "v'(t)", "e1a54f0506c233e3ad31b552df16aa48": "\n J_n = \\frac{2^n - (-1)^n}\n 3.\n", "e1a5fd158138e1d79db93d3fc26438da": "\\hat{D}(\\alpha)|0\\rangle=|\\alpha\\rangle", "e1a5ffa6e577bd1559dd5e7666d148af": " z^{\\mathrm{T}}I z = \\begin{bmatrix} a & b\\end{bmatrix} \\begin{bmatrix} a \\\\ b\\end{bmatrix}= a^2 + b^2", "e1a608b4343d4db50eece1d7161804b3": "\n\\mathbf r=\\begin{Vmatrix}\n\\mathbf r_1\\\\ \\vdots\\\\ \\mathbf r_N\\end{Vmatrix},\\qquad\\qquad\n\\mathbf v=\\begin{Vmatrix}\n\\mathbf v_1\\\\ \\vdots\\\\ \\mathbf v_N\\end{Vmatrix}.\n", "e1a6a5d6558c2e783cf6661f68f9f6b6": "\\dot{x} = x^2 - y^2,\\; \\dot{y} = 2 \\, x y", "e1a6bdc09ce658f1bf0e37c4a13d4f62": "r_{SOI} = D\\left(\\frac{m_s}{m_c}\\right)^{2/5},", "e1a6de647c217291dfb072bf7c6af03d": "\\Delta G^\\ominus = -RT \\ln K.", "e1a73624af562a936a4655f8cbe02aeb": "\\min\\{m,n\\}", "e1a79b217685b7cf1d1841a7abb71922": "\\sum E", "e1a7d17c1080f2aac246a28e65d98ccb": "L_x<0", "e1a7d4c24bda0a5133d23405a0a37be9": "\\psi(\\mathbf{r}+\\mathbf{a}_i) = C_i \\psi(\\mathbf{r})", "e1a7dde2e3f3e1f0d39c788242248aac": "(q_i)", "e1a7f896e82eafe157097f131bb2e28e": " |\\langle x,y\\rangle| \\leq \\|x\\| \\cdot \\|y\\|.\\, ", "e1a80fe7ac801abfbd9ba898fd8d74bc": "S^3 \\subset \\mathbb{C}^2", "e1a8a9978b669b716d10b5f8eab2f745": "d\\Omega^2 = d\\theta^2 + \\sin^2\\theta\\,d\\phi^2", "e1a8e31a4b4218f86285a1dbf3dfaef0": "\n(u,v) \\mapsto (x,y) = \\left(\\frac{u}{v},\\frac{u-1}{u+1}\\right),\na=\\frac{A+2}{B}, d=\\frac{A-2}{B}\n", "e1a94676cbd4a863220eafb131c7ade5": "\\sigma = E \\varepsilon", "e1a97890780264bf7caf9de7b30a0110": "\\mu\\ = m_H / m_T", "e1a97e50038e5c40f39339e3f558be06": "r_c(t)=\\sqrt{4\\nu t}", "e1a991b7b1e9c02a33ad20a96b9d08d5": "K'_v\\cap K'_u\\ne \\varnothing", "e1a9fdbccf4cab02092fbfc346df9b5e": "p^n a_{p^n} = a\\;", "e1aa38feb79e93aa92ba16802c9ef959": "V=\\frac{e^{2}}{r}", "e1aabd379031b6b973b65f3ccbfc16c3": " d(A,B)=\\log\\left(\\frac{|YA|}{|YB|}\\frac{|XB|}{|XA|}\\right). ", "e1aafccbcfb4cab07b39f970b31bf485": "H(T',J')", "e1ab24349a752f0ee1d3a61f16e1d157": "B = B_0 \\left(1 - \\frac{b}{v} \\right)", "e1ab397ccca5f8ef62ceba4d68e18b11": "\\zeta_m, \\zeta_m^2,\\dots\\zeta_m^m=1", "e1ab41280854f10825bb6a55a2101ff5": "H: \\{0,1\\}^* \\rightarrow \\mathbb{Z}_q", "e1ab669817e81b35855b49a962977da9": "\\frac{3^3 2^0 + 3^2 2^1 + 3^1 2^4 + 3^0 2^5}{2^8 - 3^4} = \\frac{125}{175}", "e1ab7e44639a7e71fa0e6b01ad580ae9": " [ G : H ] = { |G| \\over |H| } ", "e1abaeadaa331f615893ce4917749a52": "F^{\\alpha\\beta}{}_{;\\beta} \\, = 0", "e1abd26b09a160cbfe64091f517c3e0c": "C(s,t) = \\operatorname{corr} ( X(s), X(t) ),", "e1ac39030c6afec621553c8d81c698a5": "\\mu_\\alpha", "e1ac8b637a22b197ee3840eb6acaf69c": "z(U):= \\max_{x\\in X}\\ \\{f(x): g(x,u)\\le b, \\forall u\\in U\\}", "e1ac8d01d720dde2c6b7271171263a43": "|z^2+1|\\ge |z|^2 - 1 = a^2 - 1>0", "e1aca5810e70aba5920b96a1770cd7cb": "\\iota < \\rho,\\!", "e1ace2a40c5ed91e330319dded350cb3": "n_{kmax} = n^h", "e1ad077227d1793e36a3169286549c83": "v = \\varphi(n\\tau)\\,", "e1ad08986586435873cc23d2611bb3b0": "r_\\max = \\frac{n_0 + 1}{2n_0},", "e1adce9694ddcdf1f51ce9a341e31a67": "F_k(\\mathbf{a}) = \\sum_{i=1}^n\na_i^k", "e1ae007face8b75f2a847cf953a259df": "2^5\\cdot L_5(2)", "e1ae2944a91dd8b67d17ab57f7da8785": "0.1\\, 0", "e1b9015779dcf7dd1daba64de3b5f3ae": "\\frac{a-b}{a}", "e1b927ee7f5545f8fc25729aceda1b08": "(a\\cos\\phi)\\delta\\lambda", "e1b93e68a4dbf946313d1e81f6d4cc3e": "\\displaystyle m_f/m_0", "e1b94038c5d837a290be5d3b34d26515": "P(T|h_i)", "e1b94237975af85f83cb0e29477e0baa": "\\frac{1}{T} \\int_0^T Z(t)^2 dt = \\log T + (2\\gamma - 2 \\log(2 \\pi) -1) + O(T^{-15/22})", "e1b945188e6f4b6923d52dcfe933db43": "\\begin{align}u_{i,j}^{n+1} &= u_{i,j}^n + \\frac{1}{2} \\frac{a \\Delta t}{(\\Delta x)^2} \\big[(u_{i+1,j}^{n+1} + u_{i-1,j}^{n+1} + u_{i,j+1}^{n+1} + u_{i,j-1}^{n+1} - 4u_{i,j}^{n+1}) \\\\ & \\qquad {} + (u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n} - 4u_{i,j}^{n})\\big]\\end{align}", "e1b983fbbd8ed41e8d0b55ae26db97a8": "{}^{x}a", "e1b99c5be830edc7a14a57b82f85e882": "A_{o}^{D} = \\left( \\frac{ Total \\ Time }{Total \\ Time + Diagnostic \\ Down \\ Time } \\right) ", "e1b9cbe8acb1fe43bfd4e5c562b0031a": "\\mathbf{\\Pi}^0_\\alpha", "e1ba03ff39c83356c4de5066af214ba2": " \\forall m \\in \\mathcal{M}, ID \\in \\left\\{0,1\\right\\}^*: Decrypt\\left(Extract\\left(\\mathcal{P}, K_m, ID\\right), \\mathcal{P}, Encrypt\\left(\\mathcal{P}, m, ID \\right) \\right) = m ", "e1ba08ff2d902129f1111a6d14aab5e6": "=\\lambda\\mathbf{R}_{x}(n-1)", "e1ba4537c9cceb348a032082d5e3b8f6": "w'_{ij} = w_{ij} - y_j + y_i", "e1ba820a5731f700b38f062e8271b8ca": "\\sum_{i=1}^3 \\frac{\\partial L}{\\partial \\dot{x}_i}\\dot{x_i}-L", "e1bb114d67cf7c5e01504e4d9d9373c2": " \\phi_{crit} '\\ ", "e1bb385f27659aae564cc666b40fc2af": "\\langle x,x\\rangle \\ge 0", "e1bb39a12543c7666509b26a376027df": " \\varphi\\left(\\bigcup_{i=1}^\\infty S_i\\right) = \\sum_{i=1}^\\infty \\varphi(S_i). ", "e1bba51182e384da7023b13811bbf632": "N=C\\frac{d^2\\psi}{dt^2}", "e1bbba518795bc4d36fdfed39b30b610": " q_j = -\\frac{\\kappa}{\\mu} \\frac{\\partial P}{\\partial x_j} ", "e1bbc3760b2115291f812d6b2c203072": "\n \\begin{align}\n & \\nabla^2 \\left(\\mathcal{M} - \\frac{\\mathcal{B}}{1+\\nu}\\,q\\right) = -q \\\\\n & \\kappa G h\\left(\\nabla^2 w^0 + \\frac{\\mathcal{M}}{D}\\right) = \n -\\left(1 - \\cfrac{\\mathcal{B} c^2}{1+\\nu}\\right)q \\\\\n & \\nabla^2 \\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) \n = c^2\\left(\\frac{\\partial \\varphi_1}{\\partial x_2} - \\frac{\\partial \\varphi_2}{\\partial x_1}\\right) \n \\end{align}\n", "e1bc26320874e75094782e6979be0b11": "E_{em}^{v}=E_{em}\\left[\\frac{1}{\\beta}\\ln\\frac{1+\\beta}{1-\\beta}-1\\right],\\qquad\\beta=\\frac{v}{c},", "e1bc269b7f74b31d023f5b75873d700e": "P(X=x|\\theta)", "e1bc31fab3eefb058e2066fce7f248a4": "m = m_0 + \\delta_m", "e1bc3957126b933953cc1d68168aef94": "\\int x^2\\,\\operatorname{arsech}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{arsech}(a\\,x)}{3}\\,-\\,\n \\frac{1}{3\\,a^3}\\,\\operatorname{arctan}\\sqrt{\\frac{1-a\\,x}{1+a\\,x}}\\,-\\,\n \\frac{x(1+a\\,x)}{6\\,a^2}\\sqrt{\\frac{1-a\\,x}{1+a\\,x}}\\,+\\,C", "e1bc39791da7451a8b4ccf851b8ece96": "\\textrm{ID}_A \\,", "e1bc4c6fda771618494355daea1ad197": "p_{j_1 j_2} p_{j_2 j_3} \\cdots p_{j_{n-1} j_n} p_{j_n j_1} = p_{j_1 j_n} p_{j_n j_{n-1}} \\cdots p_{j_3 j_2} p_{j_2 j_1}", "e1bc72d8dd69b91845b23853602b8979": "\\alpha, \\;\\;\\; \\beta", "e1bc845a0bf3f0b04742c6d31d23733f": "x\\in\\{x|0 n \\cdot(\\ln n + \\ln(\\ln n) - 1). ", "e1c7c7ccc760a34ccccc056f3c3434a4": "\n\\begin{matrix}\nD_1 \\quad D_2 \\cdots D_n \\\\\n\\vdots \\\\\nJ\n\\end{matrix}\n", "e1c80021de97c332da97a06fb25b1bfe": " \\boldsymbol{\\Omega} = \\frac{wr}{I\\boldsymbol{\\omega}} ", "e1c850fb5e7f480040aa4bf8722077b0": "\\phi([x,y]) = [ \\phi(x), \\phi(y) ] \\, \\forall x,y \\in \\mathfrak{g}_1.", "e1c8557ffaa90214aafbcf0483977029": "\\hat{f}(\\xi_2) = \\delta(\\xi_1-\\xi_2)", "e1c866a8073a99f716c265d03af136f5": "\\frac{2}{\\sqrt{3}} \\approx 1.15470054", "e1c896393030cd3c72b4917887c5f5a7": "\\theta(x,y) = \\operatorname{atan2}(y,x)", "e1c8d75bba475473febbbb2536ade98a": "v_\\min", "e1c94f31393410a7de4e5087d2813801": "E\\{z\\}=0", "e1c9a65812d1194477aca6597748e7a2": "\\mathit{USp}(2n) \\supset \\mathit{USp}(2n-2)", "e1c9cde719682d211b13a4b9432f3254": " E(T_{j}) = \\frac{ \\partial A(\\eta) }{ \\partial \\eta_{j} } ", "e1c9df97b921ff2507070ac7f3d4b085": "\\ BPP \\subseteq \\Sigma_2 \\cap \\Pi_2 ", "e1c9ffa9ac15e49b11c9c62903d35276": "\\operatorname{pos}(\\varnothing) = 0", "e1ca107f20fbd24db24ccd878643828f": "E_\\text{K} = \\frac{1}{2}\\boldsymbol\\omega\\cdot[I_C]\\boldsymbol\\omega + \\frac{1}{2}M\\mathbf{V}_C^2.", "e1ca1ae15dd7cff76b677aa7bac64327": "H^{+_{ }}", "e1ca212adc4f1719b1aaaf3b4a67975b": " = \\frac{\\sum{M_n\\bar{u}_n}}{\\sum{M_n\\bar{u}_n^2}}", "e1ca4b8e4f8c490e7d0201186ec5ac0e": "\\mathbf{u\\times v}=\n\\begin{vmatrix}\nu_2&u_3\\\\\nv_2&v_3\n\\end{vmatrix}\\mathbf{i}\n-\\begin{vmatrix}\nu_1&u_3\\\\\nv_1&v_3\n\\end{vmatrix}\\mathbf{j}\n+\\begin{vmatrix}\nu_1&u_2\\\\\nv_1&v_2\n\\end{vmatrix}\\mathbf{k}\n", "e1cac17901747d113da0461431879865": "\\delta_\\epsilon \\Omega^{(d+1)}=d\\Omega^{(d)}( \\epsilon ).", "e1cad97f45453a2d3465b3ca9582d055": " 1 \\rightarrow K^* \\rightarrow \\Gamma \\rightarrow \\mbox{O}_V(K) \\rightarrow 1,\\,", "e1caed09c8c48bfaaa9b4fb3402af96f": "\\displaystyle \\sqrt{\\frac{2}{\\pi}} \\cdot \\frac{\\operatorname{rect}\\left( \\displaystyle \\frac{\\omega}{2} \\right)}{\\sqrt{1 - \\omega^2}} ", "e1cafd6f1a97c0cf291c5036e3bd91e7": "H_{2k+1}=\\langle a_1,b_1,\\dots, a_k,b_k,t| [a_i,b_i]=t, [a_i,t]=[b_i,t]=1, i=1,\\dots, k, [a_i,b_j]=1, i\\ne j\\rangle", "e1cb4fe3d9524c5edc5b3eda0bb4d05f": "5=n=(a^2+1)/2", "e1cb73a60217b38076e02793cb66f43f": "V \\subset A", "e1cbc8080544e166c48a7aa50f17f4a6": "\\frac{q_{\\mathrm e} q_{\\mathrm m}}{2 \\pi \\epsilon_0 \\hbar c^2} \\in \\mathbb{Z}", "e1cc1a7ba137bc6cf857a77bb9b61880": " f_{ijk} = f_{i} + f_{j} - f_{k}, \\mathrm{where}\\, i, j \\neq k ", "e1cc25999dd4a8d23c0ecb321b3497d8": "J_{m_i}(\\lambda_i)=\\begin{bmatrix}\n\\lambda_i & 1 & 0 & \\cdots & 0 \\\\\n0 & \\lambda_i & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\lambda_i & 1 \\\\\n0 & 0 & \\cdots & 0 & \\lambda_i\n\\end{bmatrix}\\in \\mathbb{C}^{m_i,m_i}, 1\\leq i\\leq s.", "e1ccbe7a77639b22db7d9d73b60be3c5": "\n{\\rm Var}[z]\\,\\,\\, \\approx \\,\\,\\,\\sum\\limits_{i = 1}^p {\\,\\sum\\limits_{j = 1}^p {\\,\\left( {\\frac{{\\partial z}}{{\\partial x_i }}} \\right)_{\\bar x_i } } } \\left( {\\frac{{\\partial z}}{{\\partial x_j }}} \\right)_{\\bar x_j } {\\sigma _{i,j} \\over n_{i,j}}", "e1cd792d82876b8e646997dfa0afd710": "y = \\mathbf{w} \\cdot \\mathbf{x}", "e1cd94528840158b1423f44676086992": "n (n - 2)", "e1cdc0a8ac68b9c6742cc4b5ede0d79c": "I(\\omega) = \\frac{1}{2} \\int_0^T | \\dot{\\omega}_t - b(\\omega_t) |^2 \\, \\mathrm{d} t", "e1ce566bee30c389f19a2e5437482306": "e x > a\\,\\!", "e1ce80af061f0ca037436a8b31e737ba": " \\dot{\\rho}_{{\\rm rad}}+4H\\rho_{{\\rm rad}}=0.", "e1ce83cffef6340efd2af1b25b58508f": " \\frac{1}{c^2}T'' +k^2 T=0,", "e1ceb95944b18899e7ba7fba7a5c3dc0": "\\boldsymbol{\\tau} = \\frac{\\mathrm{d}\\mathbf{L}}{\\mathrm{d}t}", "e1cee06319d5db83ba40a301db5e2052": "\\frac{C (A + B)}{2 \\pi} \\cdot \\operatorname{sinc} \\left[ \\frac{A - B}{2\\pi} n \\right] \\cdot \\operatorname{sinc} \\left[ \\frac{A + B}{2\\pi} n \\right]", "e1cf71be9ead24f2602893885abc98cb": " \\dot{y} = f(t, y), \\quad y(t_0) = y_0. ", "e1cfbd7698099f56c5d4cf3918be13c4": "\n\\mathcal{A}_3=(p_1\\partial_x+p_2\\partial_y+p_3)(p_4 \\partial_x^2 +p_5 \\partial_x\\partial_y + p_6 \\partial_y^2 + p_7\n\\partial_x + p_8 \\partial_y + p_9).\n", "e1cfd2263454da315b28246633c61227": " \\xi_i = s_i - \\frac {u_i + P v_i} {T}", "e1d019e93718724d82ed927f0d6ab2e3": " \\text{(4)} \\qquad d U = \\alpha n R \\, dT\n = \\alpha \\, d (P V)\n = \\alpha (P \\, dV + V \\, dP). ", "e1d02da097e8cfc0b1741596c849de81": "\\ U(h) = U(0) h ^ \\zeta\n", "e1d03e1ae15fb3e808e30e62ef10297a": "f \\circ eq = g \\circ eq", "e1d05039943632bf0c5bb9a6f886ed4f": "dA", "e1d06a151a28b2c9f1eea601ad72065d": "x(\\tan(x)+\\cot(x))", "e1d07a8e14fc6008767aee35186bdc54": " \\sigma(n) \\le H_n + e^{H_n}\\log H_n", "e1d139fdd661c9cf12393afc9baa9d66": "\\lim_{n\\rightarrow\\infty}\\sqrt[n]{|a_n|},", "e1d14065b6b7e2176e25251987e14f96": "\\scriptstyle \\mathbf{u}=(u,v,w)", "e1d17e89b7699daa41a07afed7a31a99": " a_n =2\\int_0^{2\\pi} e^{-in\\theta}\\, d\\mu(\\theta).", "e1d188b0d85643f9d7e4fe513d9ec4ab": "(S, \\otimes)", "e1d1b1d6f568b5cbe04bcae96ff286a8": " \\sqrt{\\gamma_i} A_i \\to \\sqrt{\\gamma_i'} A_i' = \\sum_{j = 1}^{N^2-1} v_{j,i} \\sqrt{\\delta_i} A_j ,", "e1d1b2f9918609eb2c868f50c5669964": "\n \\Beta(x,y) \\cdot \\Beta(x+y,1-y) =\n \\dfrac{\\pi}{x \\sin(\\pi y)},\n\\!", "e1d1ebf1d22a47b0393a4aa0008e26a8": "L \\subseteq X \\times Y", "e1d24a7494d8fc77ced205e2a9ca19e4": "\\text{EVaR}_{1-\\alpha}(X)=\\mu+\\sqrt{-2\\ln\\alpha}\\sigma.\\,", "e1d25af816941d1f9f5e156330ff479c": "x\\wedge y \\wedge x = y \\wedge x", "e1d284c27796b3e6d8eefc596b039c61": "f(k;\\rho)", "e1d2c60e8d29faf8d958c9702275471d": "\nF_{n} = F_{n-1} + 2^{2^{n-1}}F_{0} \\cdots F_{n-2}\\!", "e1d33fd458533bd19d968dcb39000809": "H^i(X, \\mathcal O_X)", "e1d344814c20de21398d885c74fb7aa6": "C(\\varepsilon) \\sim \\varepsilon^\\nu \\, ", "e1d35633d4cb3cfe717c5922f3691f70": "T = \\{a, b, c\\}", "e1d386a1708795c82f15f6de6368d739": "\\mathbb{R}^{d}", "e1d3a47ab353f33780dc92fcf87e0e90": "\\eta_i=\\xi_i", "e1d3aa7e5d1c423de5b396183ac92af6": "\\begin{matrix} {3 \\choose 1}{11 \\choose 4}{4 \\choose 1}^4 \\end{matrix}", "e1d3d1023c78d287b3ac785c6d96c47f": "\\{F_\\alpha\\}", "e1d3e8394390d95994dbea3919430572": "R(M, c x) = R(M,x)", "e1d3fd27642fa435835381363f223225": "\\scriptstyle{\\pi/2}", "e1d4015a5db423660320aedf8642f7a1": "{n \\choose p} = \\frac{n!}{p!(n-p)!}", "e1d45316f245426a7a3e0ea12d317c8f": "p_\\ell", "e1d4723343907a82f69e381cb0c5c7f4": " r_{xx} ", "e1d5005a4af6faa54ab97f0501baa60a": "p_\\textrm{kin} = p - qA \\,\\!", "e1d563a306317613c4c8c4240c8b0f76": "\\sqrt{x}(1-o(1))", "e1d5c4b15afff9fbc5ee69f570286302": "L_t", "e1d62043f497a413c55dc1f85521c209": "\\varphi_Y(t) = \\textrm{e}^{\\lambda(\\varphi_X(t) - 1)}.\\,", "e1d626952efa2e84a47860b7d4fad23f": "{\\sqrt{\\beta} \\over C_q} e_q^{-\\beta x^2} ", "e1d639a7cd7ef639638750166a3d49ff": "(G_1, P_1) \\ ", "e1d675fefacf688bcc74151b1714a624": "\n \\sum_{i\\neq m}\\mathbb{E}_{X^{n}}\\left\\{ \\text{Tr}\\left\\{ \\Pi_{\\rho\n_{X^{n}\\left( i\\right) },\\delta}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho_{X^{n}\\left(\nm\\right) }\\ \\Pi_{\\rho,\\delta}^{n}\\right\\} \\right\\} ", "e1d71d7382dcd20c80a649071a9a1d30": "I=(0,1)", "e1d74b3bc06c51274d4e586684d25684": "\\sigma = (1-\\bar{\\eta})\\tau + \\eta", "e1d788910c94fb49def647353ca4d353": "E_{ext} (\\bar v_i) ", "e1d7cbc7ac5acbf68b140bb7b2913f47": "\\mathcal{P}=\\{\\mathbb{P}_{\\theta} : \\theta \\in \\Theta\\}", "e1d802c27b9bded08e0c72fbe6277dea": "\\prod_{i=-\\infty}^\\infty x_i = \\left(\\lim_{m\\to-\\infty}\\prod_{i=m}^0 x_i\\right) \\cdot \\left(\\lim_{n\\to\\infty}\\prod_{i=1}^n x_i\\right),", "e1d873d17d04578e0f62c804ffbbc1d5": "\\mu_n=E[X^n]", "e1d8ca5c382617e223a8ec35805c1413": "S^d_{\\langle T_i: i \\in d \\rangle} = \\bigcup_{n \\in \\omega} \\left(\\prod_{i0", "e1e4d023ab4961efb44c44d7f098bc91": "\nC \\frac{d^{2}V}{dt^{2}} + \\epsilon \\left( V^{2} - 1 \\right) \\frac{dV}{dt} + \\frac{V}{L} = 0.\n", "e1e502790fbb6d992ec31c16f0369787": "R\\approx \n{{L \\rho} \\over {\\pi (D-\\delta) \\delta}}\n\\approx\n{{L \\rho} \\over {\\pi D \\delta}}\n ", "e1e518d29e951381d88e48bb6ec88b03": "k_{z}=\\beta -j\\alpha", "e1e523ed63a5d0c2a2d06043d1280a95": " \\quad \\quad \\theta_{23}^{PMNS}+\\theta_{23}^{CKM}\\simeq 45^\\circ \\ .", "e1e5710300b06fa44ec1b6296d135cba": "f(x)^2 = \\sum_{k=0}^{\\infty} M[f]_{2,k} x^k ~,", "e1e5afbf367380a247b9281980c3d5d6": "\\varphi = \\gamma \\,", "e1e5cef4341ade39581327f8149bbe5b": "A\\in\\mathcal{I}", "e1e5e20643b4e5e0486e1a783bdab4e1": " \\Psi(\\bold{r})=\\sum_{\\nu} \\psi_{\\nu} \\left( \\bold{r} \\right) a_{\\nu}", "e1e60058f23e7e522f08d8592e8bd1e1": "W^s_{\\mathrm{loc}}(f,p,U) = \\{q\\in U: f^n(q)\\in U \\mbox{ for each } n\\geq 0\\} ", "e1e605e0304c344710d62ac242d2276e": " - \\mathbb{N} ", "e1e633851e9121f465141d0d1a444b18": "\n\\Psi (f \\otimes A) = \\begin{bmatrix} f_1 \\Psi_1(A) \\\\ \\vdots \\\\ f_n \\Psi_n(A)\\end{bmatrix}\n", "e1e6536f145a896c6bcbc811217e5cdf": "\\operatorname{sech}\\,x = \\sec { {\\rm{i}} x} \\!", "e1e65c5a0adf410ffe500fd0a8856796": "K_r=k_r^+/k_r^-", "e1e664e715a4e09e2fd693171b9624d6": "I = \\sum_i \\langle \\cdot, u_i \\rangle u_i .", "e1e7091475f2137597281443915aa8b0": "n|\\{ x \\in C | c_{n-1,x} \\neq 0\\}|", "e1e70cbf5e765ac7a8c39c4a0c115437": " P = F_{in}V_{in}= F_{out}V_{out}. ", "e1e724ab7532bc6f9fe777fe7fb3986d": "\n\\xi_{\\pm} = (l_1\\pm i l_2 )^2- 2 mgh I(n_1\\pm i n_2),\n", "e1e743fe9b4003d69c68898dbc860a45": "\nL(\\beta) = \\prod_j \\frac{\\prod_{i\\in H_j}\\theta_i}{\\prod_{\\ell=0}^{m-1}[\\sum_{i:Y_i\\ge t_j}\\theta_i - \\frac{\\ell}{m}\\sum_{i\\in H_j}\\theta_i]\n}.\n", "e1e74db9e66aa97cb002e37db36d2712": "f: \\{0,1\\}^n \\rightarrow \\{0,1\\}", "e1e7eb2b5afddccaf2ce3e05820ab492": "I(t) = \\frac{\\mathrm{d}Q(t)}{\\mathrm{d}t} = C\\frac{\\mathrm{d}V(t)}{\\mathrm{d}t}", "e1e810428d5d1a0c339c0eb306adf080": "d_{ij} \\neq 0", "e1e8923f57c17201969b656eb014b1dd": "E^2 - \\mathbf{p}\\cdot\\mathbf{p} c^2 = m_0^2 c^4", "e1e8b963714dbedb92738bcd8ee3d190": "\\operatorname{Mut}(f,r,s,t)", "e1e8bcf058923d43b517b0f79b565b53": "\\Delta T = K {dq\\over dt} = K C_p\\, \\beta", "e1e8d9e67d5a2973da3dd85655a9b7de": "\\beta=0, \\delta=\\sigma^2\\alpha,", "e1e95c4e65dbf4788cf092e3f6a9d164": "\\nabla^2 f = \\frac{\\part^2 f}{\\part x^2}+\\frac{\\part^2 f}{\\part y^2}+\\frac{\\part^2 f}{\\part z^2}.", "e1ea0638f054e3e1bec9bd23944e1eb9": "Y' = Y \\oplus N", "e1ea30f87b0ac8cf540038b3059bf6d0": " q=\\int q_\\nu \\mathrm{d} \\nu ", "e1ea5606e9a5cb1e3ce4bf44db3f291f": "\\theta + \\sin \\theta \\cos \\theta + 2 \\sin \\theta = \\left(2 + \\frac \\pi 2\\right) \\sin \\varphi", "e1eab336cb49da987f32c6061a1d4f4e": "R \\le 1 - H_q( J_q(\\delta))+o(1)\\text{,} ", "e1eabf7f122c6ebbf94d20520f75c28e": "\\langle e_1,e_2,e_3,\\ldots | e_1=e_2^{p_1},e_2=e_3^{p_2},e_3=e_4^{p_3},\\ldots\\rangle_P", "e1ead6a01a6d7b3efaf0d9f06b892057": "B(U)", "e1eaff6d2f0bb54d2f66921b107e61f3": " x \\in C_1 : {{|}} x + C_2 \\rangle := ", "e1eb14a37287405e49a6598dd59b957b": "\n\\begin{align}\nI & = [0, \\ldots, 99]^2 \\\\\nS & = \\R \\\\\nS_0 &: \\Z^2 \\to \\R \\\\\nS_0((x, y)) & = \\begin{cases}\n1, & x < 0 \\\\\n0, & 0 \\le x < 100 \\\\\n1, & x \\ge 100\n \\end{cases}\\\\\ns & = ((0, -1), (-1, 0), (1, 0), (0, 1)) \\\\\nT &\\colon \\R^4 \\to \\R \\\\\nT((x_1, x_2, x_3, x_4)) & = 0.25 \\cdot (x_1 + x_2 + x_3 + x_4)\n\\end{align}\n", "e1eb218bc4a58fe06f15f86534e181bd": "\n\\ | H(e^{j \\omega}) | = \\frac{1}{\\sqrt{(1 + \\alpha^2) - 2 \\alpha \\cos(\\omega K)}} \\,\n", "e1eb30089e36a7d96aaebe259547835b": "E=-\\vec{\\mu}\\cdot\\vec{B}.", "e1eb6c21ccb0ac8001c41f6c6077a43f": "\\begin{bmatrix}\\begin{bmatrix}M\\end{bmatrix}^{-1}\\begin{bmatrix}K\\end{bmatrix}-\\omega^2\\begin{bmatrix}M\\end{bmatrix}^{-1}\\begin{bmatrix}M\\end{bmatrix}\\end{bmatrix}\\begin{Bmatrix}X\\end{Bmatrix}=0", "e1eb7c8a006a88391f4409769db88a2c": "\\Phi_1", "e1ebb40970a183ae2e032242bbd64b87": " = [1+j \\omega (C_L+C_C) (R_o//R_L)] \\,\\!", "e1ebe7acc103566e023182c73689f602": "L\\subset\\Sigma^*", "e1ec1f55526910e2b68ddc0fbbb117c7": "Y = J", "e1ec311721786d93c665e7c36b32b2af": "\\mathbb{E}(X_t^n) = \\lambda^{-n}\\Gamma(\\gamma t+n)/\\Gamma(\\gamma t),\\ \\quad n\\geq 0 ,", "e1ec3211e809e83664513cef06c04d47": "\\mathcal{A}\\cap\\mathbb{C}", "e1ec7b43f7c81481eb237aa23ade9192": "\\left\\{\\begin{matrix}\n1+1 & = & 2 \\\\\n1+2 & = & 3 \\\\\n1+3 & = & 4 \\\\\n2+1 & = & 3 \\\\\n2+2 & = & 4 \\\\\n2+3 & = & 5 \\\\\n3+1 & = & 4 \\\\\n3+2 & = & 5 \\\\\n3+3 & = & 6\n\\end{matrix}\\right\\}\n=\\left\\{\\begin{matrix}\n2 & \\mbox{with}\\ \\mbox{probability}\\ 1/9 \\\\\n3 & \\mbox{with}\\ \\mbox{probability}\\ 2/9 \\\\\n4 & \\mbox{with}\\ \\mbox{probability}\\ 3/9 \\\\\n5 & \\mbox{with}\\ \\mbox{probability}\\ 2/9 \\\\\n6 & \\mbox{with}\\ \\mbox{probability}\\ 1/9\n\\end{matrix}\\right\\}\n", "e1ec918f8e33cbc79386880010b0ad09": " SD", "e1ecba702d5df9fd8f07796f8a938433": "a+b = (a+(b-1))+1", "e1ecbe112590e6f5f680cecef3e368e9": "P_n^{[r]} = aP_{n+1}^{[r+1]} + bP_n^{[r+1]} + cP_{n-1}^{[r+1]}", "e1ecd95b7ca60a06298b9a002977e44e": " : \\hat{f}_1^\\dagger \\,\\hat{f}_2 : \\,= \\hat{f}_1^\\dagger \\,\\hat{f}_2 ", "e1ecdbb3561aabbe79d162336f20e4f0": "\\int\\frac{\\tan ax\\;\\mathrm{d}x}{\\tan ax - 1} = \\frac{x}{2} + \\frac{1}{2a}\\ln|\\sin ax - \\cos ax|+C\\,\\!", "e1eceb1748c2a506a83171cffe396191": "m_i = p_{i,1} p_{i,2} ... p_{i,n_i}", "e1ecef21ee5b830853553cd1f77f4784": " k_A \\ \\overset{\\underset{\\mathrm{def}}{}}{=}\\ \\frac {\\mu_0}{ 4\\pi} \\ ", "e1ed19181b448d309852df1e08be3d09": "\\nabla (z)=4z^4+4z^6+z^8, \\, ", "e1ed47b599aeee45e07c1948006bdeab": "\\frac{dN}{dx}= -N n \\sigma", "e1ed7e79a311144134af2621a36eff4b": "g_{jk}(\\theta_0) = \\left.\\frac{\\partial^2}{\\partial \\theta^j\\partial \\theta^k}\\right|_{\\theta = \\theta_0} D_{KL}(P(\\theta)\\|P(\\theta_0))", "e1edbfa3fb7a6ef20fae3784db626341": "((A\\equiv(B\\equiv C))\\equiv B)\\equiv(C\\equiv A)", "e1ee4065a0ab3ed47c89cb0fda3dfe09": "\\Sigma_1^T \\Sigma_1 + \\Sigma_2^T \\Sigma_2 = I_r", "e1ee7340623173e7b54128f2d1b32b99": "\\bold{r}(t)\\rightarrow \\bold{r}(t+t_0)", "e1ee969c32f1004d66cef2c49265a21e": "\\mathbf{D}=\\gamma_1 \\left \\langle\n\\begin{bmatrix}\n \\left | \\hat{\\vec B_1} \\right |^2 & \\hat{\\vec B_1} \\cdot \\hat{\\vec B_2} \\\\\n \\hat{\\vec B_2} \\cdot \\hat{\\vec B_1} & \\left | \\hat{\\vec B_2} \\right |^2 \n\\end{bmatrix}\n\\right \\rangle_1 + \\gamma_2 \\left \\langle\n\\begin{bmatrix}\n \\left | \\hat{\\vec B_1} \\right |^2 & \\hat{\\vec B_1} \\cdot \\hat{\\vec B_2} \\\\\n \\hat{\\vec B_2} \\cdot \\hat{\\vec B_1} & \\left | \\hat{\\vec B_2} \\right |^2 \n\\end{bmatrix}\n\\right \\rangle_2 ", "e1eedc69eb98513b9df0d4e33ffc27d2": "\\varepsilon_k \\sim \\operatorname{EV}_1(0,1),", "e1ef53d46647fc704c1563bb257a1d6e": "Y=Xe^{hX}/m_X(h)", "e1ef7ec21130f1e9a48c1353389e41fa": "x_1 < x_3 < \\cdots < x_{2n-1} < x_2 < x_4 < \\cdots < x_{2n}", "e1eff0f79b959a72d9e1d7c618d86506": "Z_c={1 \\over 3}", "e1f08c6d8b7d961e5f39537475f3638a": "{\\rm E}[u(A)] \\le {\\rm E}[u(B)]", "e1f0a435a5e38498c5fa4fd7efbe22e5": "P(S_0, t) = e^{-r(T - t)}[KN(-d_2) - FN(-d_1)]\\,", "e1f0bacba7f4ed4d019c371301b183ea": "B:A\\to\\mathcal{U}", "e1f0cb5d24e08b859073ae3957fbe466": "(0.4 \\cdot 1 \\cdot 1.77 + 0.1 \\cdot 2 \\cdot 1 \\cdot 1) \\cdot (1+0.1030) = 1 \\cdot 1", "e1f13582906ab97642058de0769ef37a": "\n Y^1_3 = - \\frac{1}{r^3} \\left[\\tfrac{7}{4\\pi}\\cdot \\tfrac{3}{16} \\right]^{1/2} (5z^2-r^2)(x+iy) =\n- \\left[\\tfrac{7}{4\\pi}\\cdot \\tfrac{3}{16}\\right]^{1/2} (5\\cos^2\\theta-1) (\\sin\\theta e^{i\\varphi})\n", "e1f196b8b2c55bc6dfdbe8c84d6eaf80": "\\begin{align}\n\\text{TR} &= \\text{TC}\\\\\n\\text{P}\\times \\text{X} &= \\text{TFC} + \\text{V} \\times \\text{X}\\\\\n\\text{P}\\times \\text{X} - \\text{V} \\times \\text{X} &= \\text{TFC}\\\\\n\\left(\\text{P} - \\text{V}\\right)\\times \\text{X} &= \\text{TFC}\\\\\n\\text{X} &= \\frac{\\text{TFC}}{\\text{P} - \\text{V}}\n\\end{align}", "e1f1dc870272a0a14691cf342a8254b7": "\n\\begin{align}\n& {} \\qquad (M_{i_1} \\oplus \\dots \\oplus M_{i_d}) \\oplus\n(M_{i_1} \\oplus \\dots \\oplus M_{i_{k-1}} \\oplus M_{i_{k+1}} \\oplus \\dots \\oplus M_{i_d}) \\\\\n& = M_{i_1} \\oplus M_{i_1} \\oplus \\dots \\oplus M_{i_{k-1}} \\oplus M_{i_{k-1}} \\oplus M_{i_k} \\oplus\nM_{i_{k+1}} \\oplus M_{i_{k+1}} \\oplus \\dots \\oplus M_{i_d} \\oplus M_{i_d} \\\\\n& = 0 \\oplus \\dots \\oplus 0 \\oplus M_{i_k} \\oplus 0 \\oplus \\dots \\oplus 0 \\\\\n& = M_{i_k} \\,\n\\end{align}\n", "e1f26ca6bb9962fd1828f7e8a6551a10": "\\Delta t = \\frac{0.441}{N \\Delta\\nu}.", "e1f275251aac64abf0ea0cd5a9efe627": "{}_2F_1(a,b;c;z) =\\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(b)} \\frac{1}{2\\pi i} \\int_{-i\\infty}^{i\\infty} \\frac{\\Gamma(a+s)\\Gamma(b+s)\\Gamma(-s)}{\\Gamma(c+s)}(-z)^s\\,ds.", "e1f29c3352a4c5b03fb0266dd4a6a11b": "\\oint_{\\gamma}\\sum_{n=-\\infty}^{\\infty}a_{n}\\left(z-c\\right)^{n-k-1}\\mathrm{d}z=\\oint_{\\gamma}\\sum_{n=-\\infty}^{\\infty}b_{n}\\left(z-c\\right)^{n-k-1}\\mathrm{d}z.", "e1f2dd1ce1b9d6f0eec99d146465d060": "\\theta = \\lambda 1_V", "e1f3027ea8e477d22b5b1602ab2ccaf8": "R[t_1, \\cdots, t_n]", "e1f320cc2cbce907e04d2aaabdbe1039": "r \\ne \\pm r^\\prime \\mod n", "e1f359d564c918bb446b6f9e05a89a2e": "m_{(2,1)}(X_1,X_2) = X_1^2 X_2 + X_1 X_2^2", "e1f36d2b0228a00eac6c99951210a231": "A^+AA^+ = V\\Sigma^+U^*U\\Sigma V^*V\\Sigma^+U^* = V\\Sigma^+\\Sigma\\Sigma^+U^* = V\\Sigma^+U^* = A^+", "e1f3836cb65c406a737eaeba1bb791f8": " i\\in\\{1,\\dots,k\\}", "e1f3f788712411ed973fc87aadcbf30d": "R_{HOxI,-19} = R'_{HoxI}\\left ( \\frac {1 + \\frac {-19} {1000}} {1 + \\frac {\\delta^{13}C_{HoXI}} {1000}}\\right )^2", "e1f414f8efcad55586dc65e41251f412": "|\\langle Tx,Ty\\rangle|=|\\langle x,y\\rangle|", "e1f42e3676db2c0c4c6689f4f146379d": "\\{m \\setminus L \\,\\vert\\; m\\in M\\}", "e1f4d44ad5658e78d4e110d518e96108": "\n \\boldsymbol{\\nabla} \\mathbf{v} = \\sum_{i,j = 1}^3 \\frac{\\partial v_i}{\\partial x_j}\\mathbf{e}_i\\otimes\\mathbf{e}_j = \n v_{i,j}\\mathbf{e}_i\\otimes\\mathbf{e}_j ~;~~\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\sum_{i=1}^3 \\frac{\\partial v_i}{\\partial x_i} = v_{i,i} ~;~~\n \\boldsymbol{\\nabla} \\cdot \\boldsymbol{S} = \\sum_{i,j=1}^3 \\frac{\\partial S_{ij}}{\\partial x_j}~\\mathbf{e}_i \n = \\sigma_{ij,j}~\\mathbf{e}_i ~.\n ", "e1f56be6669d0bd121746182c45088b6": "(P \\uparrow P)", "e1f5e42254751132910aae10be8fad19": "AC=1", "e1f6315d5ae8d6f09c7921cf1cea7d94": "\n \\pi r^2.\\, ", "e1f66aaf4f93eae27059dc3572965f38": "s_n=\\sum_k C_k \\rho_k^n e^{i 2\\pi \\omega_k n}", "e1f6728aae81df70744ae98465b7ec43": "D_N", "e1f688ffc7c27bd3d025cc958552ff32": "p_j = (w^{(j)}_{k_j},\\ldots, w^{(j)}_2, w^{(j)}_1).", "e1f692e8e74ed3990d5463ff1076a562": "\\left(\\tfrac{D}{n}\\right)=-1", "e1f69edcf67547d1e6a7d7d517c11ec3": " \\Delta(l,m,n) = \\langle a,b,c \\mid a^{2} = b^{2} = c^{2} = (ab)^{l} = (bc)^{n} = (ca)^{m} = 1 \\rangle. ", "e1f6a1cfb2bbde049c63406592af9811": " |\\Omega_Q(B) \\psi |^2 = | \\chi _B \\psi |^2 = \\int _B |\\psi|^2 d \\mu ,", "e1f6a50c443c0a656e8c225f12200b99": "8^2+7^2+4^2", "e1f6b36cb4ed91dd82396012cc498274": "\\mathbf{c}=S(k)\\hat{\\mathbf{c}}", "e1f6f6782a3a99638372f9a6426f4146": "\n \\begin{cases}\n H_0: d=1 \\\\\n H_1: d> 1\n \\end{cases}\n ", "e1f6fec2a38f3ac4ac7368defd2de180": "\\frac{f(k)}{k}=A \\,", "e1f712980cc13603ca5d684242aa0e16": " m-2r < -1 \\qquad \\int_{a}^{\\infty}dxx^{m-2r}= -\\frac{a^{m-2r+1}}{m-2r+1} ", "e1f72ea661dd89f3fa0bdc422b7be1fe": "x/a = \\cos(t)\\,", "e1f752b08b1522982250dbd364e2aa15": "\nf(1) = 1 + 1 + 1 + \\cdots\\,\n", "e1f8874bf17752d34cc70b4dc7cf3457": "i=1,\\dotsc,I", "e1f90900d3aad8b5b176a836850ae7f9": "LQ = K + E_1 \\times LP + E_2 \\times LP^2 ", "e1f90dd16a737c7de53323229833aa22": "\\chi_{{\\scriptscriptstyle \\rm{Alt}^2} \\rho}(g) = \\frac{1}{2} \\left[ \\left(\\chi_\\rho (g) \\right)^2 - \\chi_\\rho (g^2) \\right]", "e1f920493c043197099f6abc30314d48": "\\mathbb F_3", "e1f9471038142fd6b6dddbae27ff0421": "(\\mathbf{Q}, \\mathbf{P}, K)", "e1f9d837c0a669a0ec9ea4cb903b453f": " \\sigma: \\mathcal{A \\otimes A} \\rightarrow \\mathcal{A \\otimes A} ", "e1f9ddf0906da705fea08cae2a4b7a3c": "\\left [\\hat{a}_i, \\hat{a}_j^\\dagger \\right ] = \\delta_{ij} ", "e1f9decc92cb7b19a5528cbf1f85ca91": "\\scriptstyle \\eta \\equiv 1 ", "e1fa0cd705887a65433677f9645fe13a": "10-10=0", "e1faaf5654e31d33e33d5f6270a5875f": "\\sum_{d\\,|\\,n}\\phi(d) = n.", "e1fabbfaacf8d95b26e2ac20d4266556": "f(x;k,\\lambda; \\alpha) = \n\\alpha \n \\frac{k}{\\lambda}\n\\left[\\frac{x}{\\lambda}\\right]^{k-1}\n\\left[1- e^{-(x/\\lambda)^k} \\right]^{\\alpha-1}\ne^{-(x/\\lambda)^k}\n\n \\,", "e1fad842216cac1b42fa06fd0e432d38": "1\\le i\\le k", "e1faf4b2fc1fe00e54cde4e799cd4190": "{d (\\rho u \\phi ) \\over d x} = \\frac{d}{d x}\\left(r \\frac{d \\phi}{d x}\\right) .", "e1fb1c96760da309e68629ee176d4a34": "\\{P_3,P_4,P_5\\}", "e1fb3ce19e4f837d1ec8c2a71de057b7": "\n \\begin{bmatrix} \n a_{1,1} & a_{1,2} \\\\ \n a_{2,1} & a_{2,2} \\\\ \n \\end{bmatrix}\n\\otimes\n \\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix}\n=\n \\begin{bmatrix} \n a_{1,1} \\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix} & a_{1,2} \\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix} \\\\ \n & \\\\\n a_{2,1} \\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix} & a_{2,2} \\begin{bmatrix} \n b_{1,1} & b_{1,2} \\\\ \n b_{2,1} & b_{2,2} \\\\ \n \\end{bmatrix} \\\\ \n \\end{bmatrix}\n=\n \\begin{bmatrix} \n a_{1,1} b_{1,1} & a_{1,1} b_{1,2} & a_{1,2} b_{1,1} & a_{1,2} b_{1,2} \\\\ \n a_{1,1} b_{2,1} & a_{1,1} b_{2,2} & a_{1,2} b_{2,1} & a_{1,2} b_{2,2} \\\\ \n a_{2,1} b_{1,1} & a_{2,1} b_{1,2} & a_{2,2} b_{1,1} & a_{2,2} b_{1,2} \\\\ \n a_{2,1} b_{2,1} & a_{2,1} b_{2,2} & a_{2,2} b_{2,1} & a_{2,2} b_{2,2} \\\\ \n \\end{bmatrix}.\n", "e1fb46a5587c855d0bf7c932c304d812": "2^{160} \\le e \\le 2^{161}", "e1fb5ec6acb4fe4811dd21f87f2d6a44": "f_i:\\mathbb{R}^m \\to \\mathbb{R}^n", "e1fbd9679f4676aa85ee604df2237aa0": "\\mathbf{P}=\\mathbf{F}\\frac{d}{t}", "e1fc059439506153bfc02093ed9ea838": " \\gamma_\\text{chir} ", "e1fcbbf4d4a776065b1d713362c62c99": "A\\bullet B \\subseteq C\\bullet B", "e1fd1d8d4d7d5ca00eb998fce087de10": "n \\le n'", "e1fd57d055dfb1b60d2efc94e4bd291d": "h \\mapsto h^t", "e1fd601dbae82a538d518550acb1af19": "\\mathbf{R}", "e1fd6bdf739a200e750706971ec9f9e7": "\\phi=12.92", "e1fd7ed70439d8223d14df17c0f62fb4": "g(z) = \\frac{G m_e}{(r_e + z)^2} ", "e1fd8af2cefc4bee5502f7727b05969b": " \n\\mathcal{Q}=\n \\begin{pmatrix}\n \\begin{matrix}\n 0&a\\\\\n -a&0\n \\end{matrix}\n & \\begin{pmatrix}\t\t \n\t\t\t \\leftarrow w^t\\rightarrow \\\\\n\t\t\t \\cdots 0\\cdots\\\\\n \\end{pmatrix}\\\\\n \\begin{pmatrix}\t \n \\uparrow & \\vdots\\\\\n w & 0\\\\\n \\downarrow & \\vdots \n \\end{pmatrix} & 0\n \\end{pmatrix}.\n", "e1fe6841751d0fd2041c14533582c7d4": "O(n!)\\,", "e1fee2dd39c0769a3112b91823822037": "\\{a, b, c\\}", "e1ff5cff85e5879be4fafea3a596b876": "f_{\\omega^\\omega}(n)", "e2006636da425ce8b122e37da0eb7ebf": "(h {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} k)(x,y,z,w) = h(x,z)k(y,w)+h(y,w)k(x,z) -h(x,w)k(y,z)-h(y,z)k(x,w)", "e2006aaae409ac5b1e681290c2e1c321": "{c}= 38.945 \\sqrt{K}", "e2007680a909ec883b675d60a47d3567": "P = (\\coprod_U U\\times H)/\\sim", "e2015c346696ea2e8d278e0bb5cb35af": "\\ \\psi", "e20176a41c2b563b3b9f8b57473526f4": "a_y = \\frac{1}{m}(-kv_y - mg) = \\frac{-kv_y}{m} - g = \\frac{dv_y}{dt}", "e20182ef95986eef63db63c19a682b45": "\\Pr \\left( P=p \\right)=\\frac{1}{N}for,0\\le p e^{(a+b)/2}.", "e20decd986ff0a8b3425b0a754df2382": "\\frac{1}{\\sigma \\sqrt{2 \\pi}}\\, e^{-\\frac{(\\operatorname{logit}(x) - \\mu)^2}{2\\sigma^2}}\\frac{1}{x (1-x)}", "e20e181c9f9683e55f9bcc403aee4e1c": "(4)~~~~~\n dz = \n \\left[\n \\left(\\frac{\\partial z}{\\partial x}\\right)_y\n \\left(\\frac{\\partial x}{\\partial u}\\right)_v\n +\n \\left(\\frac{\\partial z}{\\partial y}\\right)_x\n \\left(\\frac{\\partial y}{\\partial u}\\right)_v\n \\right]du\n", "e20e437d9912295acf48d12097fceaa8": "G'\\colon X\\times I \\rightarrow Y", "e20e6bf96cef9a63b310252bdfe4a50f": "\\cos (0) = 1", "e20e820b78c173474523159d60c7ebf6": "x_0 ", "e20e85547f6b8c5a057389d5fd0f8696": "d^*_1 \\geq d^*_2 \\geq \\ldots \\geq d^*_\\ell", "e20ea76a0e1590d873ac14ff5e7a11b9": "L _ {0, 1}", "e20eae17c143d3c4a845c26848846e0c": "GDGT ratio-2=\\left(\\tfrac{[GDGT-2]+[GDGT-3]+[GDGT-4']}{[GDGT-1]+[GDGT-2]+[GDGT-3]+[GDGT-4']}\\right)", "e20ed77c736b24a6d6307bb918339eae": "\\zeta_{\\mathrm{Ai}}(2)=\\sum_{i=1}^{\\infty} \\frac{1}{a_i^2}=\\frac{3^{5/3}\\Gamma^4(\\frac23)}{4\\pi^2},", "e20ee98742abb262e22bee15b57bc35a": " (x_1,y_1,\\ldots)(x_2,y_2,\\ldots)\\ldots(x_{\\underline{\\ell}},y_{\\underline{\\ell}},\\ldots)", "e20ee9af8adbe5906e8faad1629667e7": " \\tan(\\varphi) = p/x ", "e20eed3bcb96902919dcff0bf07e1dfb": " \\arccos ( -\\frac{4}{5} ) ", "e20eff2033d8708279d737d2b5ad9c8c": "C_r = C_m\\cdot\\frac{(20.9-r)}{(20.9-m)}", "e20f14c694000a5a653057c0195a5de5": "X= 2\\lambda ap,\\,Y=2\\lambda bq,\\,Z=2\\lambda cr,\\,Xp+Yq+Zr=0.", "e20f5fa904fd15bae665d44e9a5f860c": "d_{x^2-y^2} = N_2^c \\frac{x^2 - y^2}{2r^2} = \\frac{1}{\\sqrt{2}} \\left(Y_2^2 + Y_2^{-2}\\right)", "e20f62ac7bdad337526989eb4428a80e": " p^* = \\pm p \\equiv 1 \\pmod 4 \\text{ if } p \\text{ is odd} ; ", "e20f6632da276f26986e2acad51d9ed6": "[V_\\mathrm{Mg}''] = e^{-\\frac{\\Delta H_F}{2k_BT}+\\frac{\\Delta S}{2k_B}} = Ae^{-\\frac{\\Delta H_F}{2k_BT}}", "e20f9fe5e13809357a9708d42d50fdec": "\\overline{X}=\\frac{1}{n}\\sum_{i=1}^nX_i, \\qquad S^2=\\frac{1}{n}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2,", "e20fe5553f95f7c9f2b987c40f79b11d": "8\\, ", "e2102390e4e1feb1f2dbc49569b25023": "\\Lambda \\alpha . \\lambda x^\\alpha . \\lambda f^{\\alpha\\to\\alpha} . f (f x)", "e2103961da1ec546da061271209061eb": " \\gamma^5 = \\frac{i}{4!} \\varepsilon_{\\mu \\nu \\alpha \\beta} \\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\alpha} \\gamma^{\\beta} ", "e21068da467a972cb978664c4661bce2": " E_y^0 = \\mid \\mathbf{E} \\mid \\sin \\theta ", "e2109d8b1a81e56795ef98db40a714bd": "\\delta^{d-2} \\pi^{d/2}2t/\\Gamma((d-2)/2)", "e210eb863f888850e20fe6f6b0519552": "\\displaystyle{(T_c \\Phi_-)|_{\\Omega^c}= (\\lambda + {1\\over 2})\\Phi_+,\\,\\,\\,(T_c \\Phi_+)|_{\\Omega}=(\\lambda-{1\\over 2})\\Phi_-.}", "e2114aade5c9c8e1efe5c1cb5554eb23": "R(d_k)=e_k\\,", "e211826065d0319792cc52f8607527b1": "k_r=k_1 + k_3 ", "e211b99d1619e781398b8ae0586a768f": " Z(\\beta) =\\sum_{\\sigma_1,\\ldots, \\sigma_L} e^{\\beta J\\sigma_1\\sigma_2}\\; e^{\\beta J\\sigma_2\\sigma_3}\\; \\cdots e^{\\beta J\\sigma_{L-1}\\sigma_L}= 2\\prod_{j=2}^L \\sum_{\\sigma'_j} e^{\\beta J\\sigma'_j} =2\\left[ e^{\\beta J}+e^{-\\beta J}\\right]^{L-1}. ", "e211c79b919756228e73210fdca32282": "r = \\lim_{n \\to \\infty} \\frac{1}{n} H(X_1, X_2, \\dots X_n);", "e21255d88dbc384d445574eb8f5eff43": "k[\\![t_1, \\ldots, t_n]\\!]", "e212732f536eb12938b99d4b136e9cbe": "\\Psi(t)=\\Psi(t+T)", "e212afca20c6c612b3bd12a6387ab652": "\\sigma_3=0\\!", "e212bd68da286cccea62385982fb164e": "\\mathbf{E}(\\mathbf{r}) = \\frac{1}{4\\pi\\mathcal{E}_0} \\int \\frac{\\rho(\\mathbf{s})(\\mathbf{r}-\\mathbf{s})}{|\\mathbf{r}-\\mathbf{s}|^3} \\, d^3 \\mathbf{s}", "e212ebbe41ea2517be680e637379eba4": "t_1,\\ldots,t_n", "e213012d96f453091fd27303bab5acaf": "L(x) \\approx x/3", "e213659c252a04ece8e26833e447c929": "u(t_k)", "e213892dad5202313a943c2a7e6bd255": " (\\partial H)_U=-(\\partial U)_H=-VC_P+PV\\left(\\frac{\\partial V}{\\partial T}\\right)_P-PC_P\\left(\\frac{\\partial V}{\\partial P}\\right)_T-PT\\left(\\frac{\\partial V}{\\partial T}\\right)_P^2", "e213b126e75860facf0a91652e22acc7": "90 \\cdot b_n", "e213bcdab1e68ca87c80560e339516f0": " \\theta = \\alpha(1-\\theta)p", "e21417853caa731b772df776c8adfd93": " \\mathbf{X}=(X_i)_{i=1}^J", "e2142e6844be3725b2f45ee52d2961c3": "\\bar{\\kappa}", "e214d38187bbb9692750dc71b78ecd86": "F_k^n(x)=\\frac{1}{n} \\sum_{i=1}^n \\mathbf{1}(X_k^i\\leq x)", "e21552c22a4cbe8e8d3c40ecf6bd8156": "O(N^{5/3})", "e21576e3ab97e8d82fb3117437395709": " \\mathrm M_3 \\mathrm M_2 \\mathrm M_1 \\vec S_i \\ne \\mathrm M_1 \\mathrm M_2 \\mathrm M_3 \\vec S_i \\ .", "e21596ad2f11f639c2690517a8f3d2b6": " \\sum_{j=1}^n w_j K(s_i,t_j)u(t_j)=f(s_i) ", "e2164b13990287d87db8da75559c9835": "\\ v = \\sqrt 2\\sqrt{\\frac {GM} {r}\\ } = \\sqrt{\\frac {2GM} {r}\\ }.", "e216bdf6bc130d1d2684b7c4de04f771": "g(t) = \\mathbb{E}[Y_t].\\, ", "e216cd7a46c0a85eaead636cfc323670": "\\Phi_{\\rm G}", "e216d3d608ef72230cbe016f98ccc76f": "g=\\frac12 (d-1)(d-2) . \\,", "e216e3f9c1316848cdb60edbeb2dce3e": "|\\psi(x,y,z,t)|^2", "e2174c4050c2256386e78a83cae4040f": " (-t)^{-\\alpha'} \\geq \\mathrm{constant}\\cdot(-t)^{\\gamma'}(-t)^{2(\\beta-1)} ", "e217dcb6c38085e1bdac3170e598831c": " \\scriptstyle \\omega_0 = \\frac {1}{\\sqrt {LC}} ", "e217e4adcbe577046ad94758792cfe51": "2 \\leq p < \\infty ", "e217f21da70a9f6473a18041cb249e21": "\\int \\frac{dp}{A(p)B(p)}=\\int dp \\int^1_0 \\frac{du}{\\left[uA(p)+(1-u)B(p)\\right]^2}=\\int^1_0 du \\int \\frac{dp}{\\left[uA(p)+(1-u)B(p)\\right]^2}.", "e217fcd04d1315d0cecb430857a9ded5": "\\mathbf{d} = \\mathbf{a} - \\mathbf{c}.", "e21836bf409b1e085f6811c3c2795bfb": "\\frac{d^{2}y}{dt^{2}} + \\omega_{0}^{2} y = 0", "e21843728b062583960f8a658bc8f1f5": "\\mathbf{r}_{dx}(n-1)", "e2184881f442700d73fcb2db44424c1a": "\\mathbf{H}_{ba} = \\mathbf{H}_{ab}^{-1}.", "e218625f42c8193e468fbff62acf6061": "N(m) dm", "e2187f240bce27a93d1ceb7e59eca216": "F_X(x)=1-\\left(\\frac{k}{x}\\right)^a", "e218b0f9305f766fa4ecea7dc463f1e3": "A + B = \\{a+b : a \\in A, b \\in B\\}", "e2197c65696cb3ced2c46340db4e15d9": "p(n,1/\\epsilon)\\,", "e2198b8292460cf864575fb944e09e82": "F(1/x_1,\\dots,1/x_d)=(-1)^d F_{\\rm int}(x_1,\\dots,x_d).", "e219910800665b15c5c4cee98e089c79": "g_{\\mu \\nu}=g_{\\mu} \\cdot g_{\\nu}.", "e21a148e93e0679634217d3c4c5cd124": " D\\cdot A = 0 ", "e21a7b4a2646dc274a7e2b0e5807796f": "\\hat{m}_x(t)_{T}", "e21ae7b1e321344f0fd05b84687a2f66": "\\tau_c \\Delta f \\approx 1", "e21b03b6e52c9a379118a48840a35a27": " \\hat\\mathfrak r_i = \\vec\\mathfrak r_i / \\left \\|\\vec\\mathfrak r_i \\right \\|", "e21b75e98214f0c05f487cceedfa63d3": "P_c^p(z)", "e21b86a67faf32c69521adaed7d9494d": "m \\in \\mathbb{Z}", "e21b9882214869460baa54609c3d5778": "y_i = \\overbrace{\\frac{RC}{RC + \\Delta_T} y_{i-1}}^{\\text{Decaying contribution from prior inputs}} + \\overbrace{\\frac{RC}{RC + \\Delta_T} \\left( x_i - x_{i-1} \\right)}^{\\text{Contribution from change in input}} ", "e21c3d615b78a3b7c0b444b4052550e7": "\\hat{\\mathbf{n}}\\cdot\\widehat{\\mathbf{L}}", "e21c4b7d9c4f7f874fc2c84676eb6be1": "\\cos^2\\theta + \\sin^2\\theta = 1\\!", "e21c5ae5146a8867c946223bc3ba3df2": "\\vec{F} = q \\vec{E}.\\,", "e21c9534ce25a74ed076fa8eeb9b3b18": "1/(x^3 + 4x)", "e21c968ab1b95e1160bd449f28e772dc": "\\displaystyle n=4", "e21d1898db753ff65651b491785038ec": " u(-b)=u(a-b) \\qquad u'(-b)=u'(a-b). \\,\\! ", "e21d6f00d303f5ed7cd495ef34609fd2": " s = \\theta r ", "e21ddc1700a258a7b566e33e4ecdebde": "\\forall x \\in t\\ (\\phi) \\Leftrightarrow \\forall x ( x \\in t \\rightarrow \\phi)", "e21df1015b725563648c2dc6724a17af": "T(v_1,v_2,\\dots,v_r) = T(v_{\\sigma 1},v_{\\sigma 2},\\dots,v_{\\sigma r})", "e21e1571c3e27d8c18372e97a7120eab": "A = \\sqrt{1 + e'^2\\cos^4 \\phi_1}, \\quad B = \\sqrt{1 + e'^2\\cos^2 \\phi_1},", "e21e1795c06c0f3837fba7343bdf250b": "v(x_1)", "e21e2cd63acf5c00da23fcb6328e478b": "TR^2", "e21e9148e375b781fe760b80d4b1b00a": "\\mathbf{q}(s)", "e21f4d2e9830b9542c82b783dca372ed": "\\left[ A \\right]=\\left[ A \\right]_{0}\\frac{1}{k_{f}+k_{b}}\\left( k_{b}+k_{f}e^{-\\left( k_{f}+k_{b} \\right)t} \\right)+\\left[ B \\right]_{0}\\frac{k_{b}}{k_{f}+k_{b}}\\left( 1-e^{-\\left( k_{f}+k_{b} \\right)t} \\right)", "e21faa2cc52e8b00acdb4b060405caa4": "p(F_i\\vert C=c)", "e21fdb780488973d5c2fbe9d94a79bf4": "C\\left\\{ x\\right\\} '+K\\left\\{ x\\right\\} =F\\left\\{ u\\right\\}", "e21feb9c0f4c2692c78ebf3c05df19ae": "Poss", "e21ffb0ec033824e4e8e8f78bf8d935b": "\\lim_{x\\rightarrow \\infty}f(x;\\mu,c) =\\sqrt{\\frac{c}{2\\pi}}~\\frac{1}{x^{3/2}}.", "e21ffca333fee64c7c91941abe485923": "\\mathcal{O}_2", "e2202cdfe1de9979a59f5b9b988df1f5": "i = 0, 1, \\dots, N-1", "e2206474c9a019fb35ad1b32441bb530": "h'(\\alpha_k)=0", "e220f1e5d3a1a8fa183ba6d6f91ae564": "\\frac{}{\\pi_1:\\!\\!-~~ \\alpha \\times \\beta ~\\vdash~ \\alpha}\\qquad\\frac{}{\\pi_2:\\!\\!-~~ \\alpha \\times \\beta ~\\vdash~ \\beta}", "e221739078b7e47bd714d3eb04a54d79": "1- (1.25*3.9)/(1.25+3.9)= 5.34%", "e22196a124b858471a5d6627a6e2eccc": "P_1 < P_2", "e221acd9ac0b8902abd821e615365bf6": " MP_L = 90 - 2L ", "e221ef7abe5459a7ab6bf0c20cfd8520": "v(x,t,\\theta)", "e221fb5b53e8d406810a92d6312af877": " \\hat{a} = (\\text{sample mean}) - \\left(\\frac{\\hat{\\alpha}}{\\hat{\\nu}}\\right)(\\hat{c}-\\hat{a}) ", "e22274eae9b8c6cc2ed3d6063d7b433f": "\\alpha > 4", "e22284cac08da35dff4f0ffce2c45f0f": "b^2=d^2+h^2", "e222886886b8c67bf32f0a5cd6323b6d": "\\lim_{x \\to a} \\cos x = \\cos a", "e222953d85c55d1b3b2c59608d66aa5a": "\nH_E= - E_0 z\n", "e222a1020033024dd1b1bac8af91b968": " I = I_0 \\frac{ 1+\\cos^2 \\theta }{2 R^2} \\left( \\frac{ 2 \\pi }{ \\lambda } \\right)^4 \\left( \\frac{ n^2-1}{ n^2+2 } \\right)^2 \\left( \\frac{d}{2} \\right)^6", "e222a8f50167725c1ffb0f0fce036224": "\\mathbf{Q}^{om} ", "e222b62431e44ace9266becca21ff09a": "B_1\\lor\\cdots\\lor B_l\\lor\\lnot C", "e222e5839a13ad9eb21fa64fcf38cbc6": "x>s", "e22312837d71c5baa0b5de607f80ccb1": "p \\in (1,\\infty)", "e22312b8c3a34b03ccf48b815b937bec": "O^j_1 \\in \\{O^i_1\\}", "e223409219fc33f8c198058dfd498431": "A_0 \\to \\ldots \\to A_{i-1} \\to A_{i+1} \\to \\ldots \\to A_k", "e2236a37ab1fcc007276bc7521a7e732": "\\forall \\!\\,", "e223f87a26fd3eea787dfec8453ff243": "\nx\\rightarrow x^2-y^2-z^2+x_0\n", "e223ffb40ab6a6830ac799c838f14f69": " \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}", "e22428ccf96cda9674a939c209ad1000": "sr", "e224970cc13e54c80a4c43ab591eceff": "H^q(X,\\varprojlim F_i) \\to \\varprojlim H^q(X, F_i)", "e224d93281daa9f8171d13517da01767": "\\mathbf{a} = a_\\text{x}\\mathbf{e}_\\text{x} + a_\\text{y}\\mathbf{e}_\\text{y} + a_\\text{z}\\mathbf{e}_\\text{z} ", "e2254109b9c47b042f2f6c8e39a6cd84": "L^{\\infty}_+(P)", "e225b87ad7c0eefc0a010321fa60d166": " g_0\\approx 1 ", "e2260293dbd59b4e249395985893a67b": "f(x) - f(\\bar{x}) \\not\\in -\\operatorname{int} C", "e2263518af9f4f6d145e1deb6fc94ae9": "\\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} = 2 \\partial_{[\\alpha} \\omega_{\\beta]}^{\\;\\; IJ} + \\omega_{\\alpha I}^{\\;\\;\\; K} \\omega_{\\beta K}^{\\;\\;\\;\\;\\; J} - \\omega_{\\beta I}^{\\;\\;\\; K} \\omega_{\\alpha K}^{\\;\\;\\;\\;\\; J}\n", "e22685435a6b0f88238e0a423332b009": "f(-t)", "e2277f824cd27613805afc2f147382f0": " \\lambda f.((\\lambda q.f\\ (q\\ q))\\ (\\lambda q.f\\ (q\\ q)) ", "e227c0223dd629481484c926113b727b": "f(z_1,\\ldots,z_n).", "e227d5a43b680c006052be5e7b64d2c7": "3=2^{m/2n}", "e227d8c80728746a337c7305bdffd7e5": "A_i B_i = A_i E_i + \\rho_i \\sum_{j=1}^n A_j B_j F_{ji}", "e22834f2dcf1eb58472ee6208723bc15": "O(n^2 (n+\\log q))", "e22855954c9ceff0db8cafd138525704": "a + b = b + a", "e22869a2fa61c64b2326f1f4e21ef0ef": "T = \\Gamma \\oplus U", "e228a96f31772aed503e2d09f9df7e8b": "\\begin{align}\nE_{\\text{i}} &= E_0 e^{i( \\omega t+\\beta\\sin(\\omega_\\mathrm{m} t))} \\\\\n &\\approx E_0 e^{i\\omega t} [1+i \\beta \\sin(\\omega_\\mathrm{m} t)] \\\\\n &= E_0 e^{i \\omega t}\\left[1+\\frac{\\beta}{2}e^{i\\omega_\\mathrm{m} t}-\\frac{\\beta}{2}e^{-i \\omega_\\mathrm{m} t}\\right].\n\\end{align}", "e228ce370aac018ba9bdce8ccad45ede": " {}_RQ_P = 0 ", "e228f32567a915bd757856332cc07c2a": "(28)\\quad r^2\\psi_{,\\,rr}+2r\\,\\psi_{,\\,r}+\\psi_{,\\,\\theta\\theta}+\\cot\\theta\\cdot\\psi_{,\\,\\theta}\\,=\\,0\\,.", "e2290b643fa9cdedd765443acfe5e24b": "\\mathbf{r}_2 \\to (0,1,0,0)", "e2292cf7108df967ec9c692442ba28b1": "p,q \\in Q", "e2297134051a307d8813c8433e81e0c2": "\\mathbf{f} = \\epsilon_0\\left[ (\\boldsymbol{\\nabla}\\cdot \\mathbf{E} )\\mathbf{E} + (\\mathbf{E}\\cdot\\boldsymbol{\\nabla}) \\mathbf{E} \\right] + \\frac{1}{\\mu_0} \\left[(\\boldsymbol{\\nabla}\\cdot \\mathbf{B} )\\mathbf{B} + (\\mathbf{B}\\cdot\\boldsymbol{\\nabla}) \\mathbf{B} \\right] - \\frac{1}{2} \\boldsymbol{\\nabla}\\left(\\epsilon_0 E^2 + \\frac{1}{\\mu_0} B^2 \\right)\n- \\epsilon_0\\frac{\\partial}{\\partial t}\\left( \\mathbf{E}\\times \\mathbf{B}\\right)\\,", "e2298204d87ffe8cb048924da64277bc": "T_1, T_2 \\in [n]", "e229b03c7f65d533c82f09367068175a": "|\\cdot|", "e229beccb13a37b3cbeb0c2b9654faf0": "y\\csc\\varphi=\\frac{y}{\\tfrac{dy}{dx}}\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}.", "e22a0efefc3390782cee169831c00890": "p(\\theta \\mid D, I) \\propto p(\\theta \\mid I) p(D \\mid \\theta, I)", "e22a88efbe322386ac1c731f1ee48619": " \\int_0^\\infty x^{s-1} \\left(\\frac{e^{-ax}}{1-e^{-x}}-\\frac{1}{x}\\right) \\, dx = \\Gamma(s)\\zeta(s,a) \\!", "e22a8aa836865e941486751aae59f09c": "E=\\{0,1,2,...\\}", "e22a8dbfe3355fe8e2d7944cf2ddca39": "p_x = \\| p \\| \\cos \\theta", "e22ae49b5809f754ed464674051f662c": "n^5 + 10n^3 + 10n^2 + 10n + 3", "e22af77d4d8b6f9260fc273528284016": "-\\frac{\\hbar^2}{2m}\\nabla ^2 \\delta\\psi^*+V\\delta\\psi^*+g(2|\\psi_0|^2\\delta\\psi^*+\\psi^2\\delta\\psi) = i\\hbar\\frac{\\partial\\delta\\psi^*}{\\partial t}", "e22b20480510ada0ec3e0a8f8b7ebf0b": "0 = \\int_{\\partial N} T^{\\mu \\nu} \\mathrm{d}^3 s_{\\nu} \\!", "e22b463ae5c84356aa7b8ed794bc8b89": "k(f,\\epsilon)", "e22b70153ee609daf7eadbb13fceb526": "{\\Bbb H}", "e22b86ce3b43e9e9cba2b3cb1338d918": " f(x) = a_2 x^2 + a_1 x + a_0 \\qquad \\, ", "e22b89ed754ff09874c1a043a2d71d01": "\\Lambda(x)=\\sum_{i=0}^v\\lambda_ix^i=\\lambda_0\\cdot\\prod_{k=0}^{v} (\\alpha^{-i_k}x-1).", "e22bad07812d4851c36bf332c4e70f1c": "f(x)=o\\left( \\vert x-x_0 \\vert^{2-n}\\right),\\qquad\\mathrm{as\\ }x\\to x_0,", "e22c78ae25a4dc12d5dc11f9311f88d7": "p \\cdot x_i \\geq p \\cdot x_i^*", "e22c98123efeae66a4ef7e6640fc3b6c": "\\mathbf{v}_\\parallel = \\mathbf{v} - \\mathbf{v}_\\perp", "e22ca3609f6bad32f9fdd81e1e63f551": " \\| u(t) \\| \\le \\| u(0) \\| e^{\\lambda t}", "e22d14a95c7b477ec84102ef584787e1": "y = ix + az", "e22d52309e9eb3b0b602d4481d1e9b8f": "\\mathrm{MSE} = \\mathrm{tr} \\left\\{ E\\{(\\hat{x} - x)(\\hat{x} - x)^T \\}\\right\\}", "e22dd1f6addb0d2005d1a3b837639355": "\\tfrac{X}{Y} \\sim \\mathrm{Pareto}(1,n)", "e22de88376adb3e31deef5e2397571b3": "M_X(t) = (1-2t)^{-K/2}", "e22deb259f8b785560b497112749dc88": " x_i = x_{\\text{low}} + \\left( i - 1/2 \\right) \\Delta x ", "e22dee867ba87a4db75ecc6cae6f209e": "\\theta_{max}", "e22dfc56a17b11fb68babe05d3483554": "( \\mu_a - \\mu_b ) =\\frac{\\partial f}{\\partial c} - 2 K \\nabla^2 c ", "e22e21ea932efe7a4b0dfa7d5e485c05": "\n=\\rho \\sqrt{R_\\beta R_\\theta},\n", "e22e250ab89ad1ee14d5510c6f5f4e2f": " \\sqrt f = k_1 \\cdot \\left(Z - k_2\\right) ", "e22e52e8405ff5006d7260ff868c25ea": "S_{j} \\subset \\{1, ...\\,,n\\}", "e22eb450db971ca14a1f840ef29a864e": "X_n\\xrightarrow{P} X", "e22f47635527569e84c38ec1ad4da2cd": "p(\\mu|I) \\propto 1 ", "e22f5e4a74ca651f3674115ad9fea019": " s_{j} ", "e22f6045a3d1cc1ef2e56fdaef85e5d6": "A = \\sqrt{W^2 + V^2 +2WV\\cos{\\alpha}}", "e22fa8df081b7ddf11b8d607f2cfe033": "\n\\varphi(\\mathbf{r}, t) = \\frac{1}{4\\pi \\epsilon_0} \\iint \\frac{q\\delta^3(\\mathbf{r'} - \\mathbf{r}_s(t'))}{|\\mathbf{r} - \\mathbf{r}'|} \\delta(t' - t_r') \\, dt' \\, d^3\\mathbf{r}'\n", "e2300bf8bb602d02008ddc40426cefd6": "\\lambda = \\pi/2 - \\theta", "e2302d26fe104d3c40d64f1c79412668": " v\\left(\\bigcup_{i\\in I}U_i\\right) = \\sup_{i\\in I} v(U_i).", "e2305ed176c690163fefc2dbb34585fe": "D = \\inf \\left\\{ \\left. \\frac{\\det \\left( \\sum_{i = 1}^{m} c_{i} B_{i}^{*} A_{i} B_{i} \\right)}{\\prod_{i = 1}^{m} ( \\det A_{i} )^{c_{i}}} \\right| A_{i} \\mbox{ is a positive-definite } n_{i} \\times n_{i} \\mbox{ matrix} \\right\\}.", "e23088546422b897400915ed5ff92408": "\\text{Symmetrical short circuit current:}", "e231869e2a1798839e3b23ffb4fe5af7": "\\mathbf{F} \\mathbf{C} = \\mathbf{S} \\mathbf{C} \\mathbf{\\epsilon}", "e2318cf8b9baef7600e6891a83868390": "\\sqrt{a} = 2^{-n}\\sqrt{4^n a},", "e231c47640ebe16bce09ac194b22e9e2": "\\mathrm{\\tfrac{u\\bar{u} + d\\bar{d} + s\\bar{s}}{\\sqrt{3}}}", "e2320e6da72a43a3e2c8f31f5941001d": "\\{x_n=\\cos(n)\\}_{n\\ge1}", "e2325adb4a8724fa7316cc25f808c7a6": "\\mathfrak{H} =\n\\begin{pmatrix}\na & b \\\\\nc & d\n\\end{pmatrix}", "e23271bf11b167862336947dbb3a05a3": "k_c", "e232da8a94e017e5a19b4f4391a38a57": " \\psi = f \\circ \\phi ", "e233a331d508857c071c660bfc86358d": "\\textstyle{1 \\over x-1}", "e233ffd9b2e7d099aa3593b8249e2306": " ATEE =0.9 \\cdot TEE + 200 ", "e23422199dd92e1c88fbdc012fd1df6b": "m_\\mathrm{PbO} = \\left(\\frac{200.0 \\mbox{ g }\\mathrm{PbS}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }\\mathrm{PbS}}{239.27 \\mbox{ g }\\mathrm{PbS}}\\right)\\left(\\frac{2 \\mbox{ mol }\\mathrm{PbO}}{2 \\mbox{ mol }\\mathrm{PbS}}\\right)\\left(\\frac{223.2 \\mbox{ g }\\mathrm{PbO}}{1 \\mbox{ mol }\\mathrm{PbO}}\\right) = 186.6 \\mbox{ g}", "e23435798d7e51f0bed2281bdd83e9cd": "T = (p\\cdot n_1 + q\\cdot n_2)\\cdot T_0", "e2348c52141cd72dd4d9c6e655f38305": "\\frac{8192}{6561} \\sqrt[3]{2}", "e234a9c0b34faac554012f29a14613bf": "a \\uparrow^3 b", "e234e4f56ac9e20e84eafd020ea50f00": "\\hat{B}_\\omega", "e234f483a671bc8c5c57d0edaab2b86c": "\\mathit{\\bar{G}}=\\mathit{C}", "e2350fde9df30a4d917545b3ecbe07c1": "(-1)^n = \\pm 1", "e2352a9d88a32660c9d12b315e4f3312": "I_\\alpha(z) \\sim \\frac{e^z}{\\sqrt{2\\pi z}} \\left(1 - \\frac{4 \\alpha^{2} - 1}{8 z} + \\frac{(4 \\alpha^{2} - 1) (4 \\alpha^{2} - 9)}{2! (8 z)^{2}} - \\frac{(4 \\alpha^{2} - 1) (4 \\alpha^{2} - 9) (4 \\alpha^{2} - 25)}{3! (8 z)^{3}} + \\cdots \\right)\\text{ for }|\\arg z|<\\pi/2 ,", "e2355d2667e3bd5dbdeeff5cb1b49195": "F_X(x)=\\left(1-\\frac{k}{x}^a\\right) I_{[k,\\infty)}(x),", "e2357327a56e2af6536c18de1747e21c": "O(n \\log{n})", "e235900271ebce7e675cc5f3bd724021": "V^T", "e235e6387e12732a92f2642d621166c3": "\\tau=\\frac{2Gb}{x}", "e235f3f0afb63f571a191afdf759e605": "\\bar{X}, \\bar{Y}, \\bar{Z}", "e236488eb7a726e16e01359ff01d2337": "L = T - V.", "e23697009ea7d033156d4377951ca050": "tr~~~", "e236fd8f283a4620dd1a36a6218d7a97": "\nX(x)\\equiv (x^3+Ax+B)((x^3+Ax+B)^{\\frac{q^{2}-1}{2}}-\\theta(x))\\bmod \\psi_l(x).\n", "e23757064cac60cc51753351150e00ec": "K_{m,2}", "e2378fdc1d949098783d1eb20ae67fbe": "E > V_0", "e237aab10e9b3ff3545683ec579f076f": "f(x,y)=\\frac{\\partial f}{\\partial y}(x,y)=0.", "e237f96a44637d52011899008447cc33": "f_{\\langle X | R\\cup \\{w\\} \\rangle}(x) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ x=1\\ \\mbox{in}\\ S_w\\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ x\\neq 1\\ \\mbox{in}\\ S_w.\n\\end{matrix}\\right.", "e2388d1bdd909b165755616490f8f1fa": " \\delta \\boldsymbol{\\epsilon} = \\mathbf{B} \\delta \\mathbf{q} \\qquad \\qquad \\qquad \\qquad \\mathrm{(9)}", "e238a7c9d3ce9bf21da786d7fc9806db": "\\int_{-\\infty}^{\\infty} e^{-ax^2} e^{-2bx}\\,\\mathrm{d}x=\\sqrt{\\frac{\\pi}{a}}e^{\\frac{b^2}{a}} \\quad (a>0)", "e2390731bc64a4999f4ebdee84c8c44a": " \\mathrm{S} = \\left( \\frac{r}{c} \\right)^2 \\frac {\\mu N}{P}", "e239112edf34da7a8a0f34766b1e5d17": "\\mathit{y}", "e239123eb64eed85a4753023b0734d1e": "\\int_0^\\infty \\cos ax^n=\\frac{1}{na^{1/n}}\\Gamma(1/n)\\cos\\frac{\\pi}{2n}\\quad ,n>1", "e2395085410dea95bef096238677d423": "\\displaystyle a_n = F_{2n-1}", "e23a2396446622c4a29557a963378af1": "\\displaystyle{\\ell(\\gamma)=\\int_0^1 \\alpha(\\gamma(t),\\dot{\\gamma}(t))\\, dt.}", "e23a5a6fe01bbeecc86028fa90594a2d": "\\scriptstyle(1.8\\pm5.3)\\times10^{-30}", "e23a932cfdaccecc523f085d5f367c78": "\n\\begin{align}\n V_\\text{g} &= \\frac{1}{4} (\\rho-\\rho') g a^2 \\lambda,\n \\\\\n V_\\text{st} &= \\frac{1}{4} \\sigma k^2 a^2 \\lambda,\n \\\\\n T &= \\frac{1}{4} (\\rho+\\rho') \\frac{\\omega^2}{|k|} a^2 \\lambda.\n\\end{align}\n", "e23ada974b87004187762f4c18caf2bf": "0\\leq n\\leq 7", "e23af8386e56886b8d6f028d86e9c05e": "\\boldsymbol\\Sigma_{YY}", "e23b2f0f9f8b30fd5da647e1046ecd53": "([\\,]\\; \\lambda x.(x\\;x))", "e23b54f98c838729a21cdd63d49002be": "GL(V) \\times \\mathfrak{S}_n", "e23b9dfff579ecbcd1ec3b1a6a62dccd": "i : Y \\to X", "e23bc42f0efb040fa46d6a85c4e72ed6": "d=(x+yc)", "e23bc449f91f5ada249a334266962b34": "0 \\le \\lambda \\le 1", "e23be3169198a821fe5ee282363fc9b1": " \\mathfrak{G} ", "e23c5f8886b62f141bd43c0a54e5637b": "\\nabla v", "e23c8d690bc4027d2cc106e55bdd9c88": "Q_3", "e23cc3c6c263977526bb986f2cd274f1": "(\\phi \\to \\chi) \\to ((\\chi \\to \\phi) \\to (\\phi \\leftrightarrow \\chi))", "e23d1bd4be52e3946e18be4e66b994ae": "v,w\\in\\Sigma^*", "e23d378109cc504234c9e658ce9b2ba0": "n + \\sqrt{n} + 1", "e23d68be74dd3ef234367cdb9931109c": " \\mathbf{F} = - \\frac{m \\left ( \\mathbf{v}\\cdot{\\mathbf{v}} \\right ) \\mathbf{\\hat{r}} }{\\left | \\mathbf{r} \\right |} = q \\left ( \\mathbf{v}\\times \\mathbf{B}\\right ),\\,\\!", "e23d78b2dd38c892219441012eca1827": "s(t).", "e23dc475d75328be579f2f7c80a42b73": "\\chi (-1) = -1", "e23dcbc6bfc4c57b0dbfb0bb1401f8e5": "\\Bbb Q", "e23dfe69ddc6c4f17aa265d16307cde5": "x=\\lim_\\omega x_n", "e23e4f2a1563d6186498c30069cb42fc": "P=\\vec{F}\\cdot\\vec{v}", "e23e5c6178fcede4de9a9673bf51cfd7": "b_k", "e23e7a596c1b832d3b099030a3f8f3ac": "\\int_0^\\pi P_\\ell^{m}(\\cos\\theta) P_\\ell^{n}(\\cos\\theta) \\csc\\theta\\,d\\theta = \\begin{cases} 0 & \\text{if } m\\neq n \\\\ \\frac{(\\ell+m)!}{m(\\ell-m)!} & \\text{if } m=n\\neq0 \\\\ \\infty & \\text{if } m=n=0\\end{cases}", "e23ecb868c6d9b086f4eaa5599745a2a": "P_a: \\bigoplus_{n=0}^\\infty H^{\\otimes n} \\to F_a(H). \\, ", "e23ede2fb37f2c2a556b16792e1d1671": "\\bar{\\psi}\\gamma^\\mu\\psi", "e23ef94cdada12cc763371eb74fd8f8a": "f(0) := 0,", "e23f2a6e8fe6884a6cf5b78320885f48": "E_\\mathrm{dihedrals} = \\frac {V_1} {2} \\left [ 1 + \\cos (\\phi-\\phi_0) \\right ] \n + \\frac {V_2} {2} \\left [ 1 - \\cos 2(\\phi-\\phi_0) \\right ] \n + \\frac {V_3} {2} \\left [ 1 + \\cos 3(\\phi-\\phi_0) \\right ] \n + \\frac {V_4} {2} \\left [ 1 - \\cos 4(\\phi-\\phi_0) \\right ]", "e23fd1ef1f25d9ad998f411be684cc90": "\nm_p = m(\\pi/2)\\,\n", "e23ff85ad746f6bdd3dd88f133b51797": "\\scriptstyle X \\;\\sim\\; \\mathrm{Erlang}(k,\\, \\lambda)\\,", "e24008a58e065764fe8bdc0bbf5a4dcd": "e_{k-1} = \\sum_i w_i \\log \\left( m_{k-1}^{-1} q_i \\right)", "e24020538b3f18078c1a6b9c72d84011": "\\begin{align}\n\\Psi^{+} ({\\mathbf r} , t)\n& =\n\\left[\n\\begin{array}{c}\n- F_x^{+} + {\\rm i} F_y^{+} \\\\\nF_z^{+} \\\\\nF_z^{+} \\\\\nF_x^{+} + {\\rm i} F_y^{+} \n\\end{array}\n\\right]\\,, \\quad \n\\Psi^{-} ({\\mathbf r} , t)\n=\n\\left[\n\\begin{array}{c}\n- F_x^{-} - {\\rm i} F_y^{-} \\\\\nF_z^{-} \\\\\nF_z^{-} \\\\\nF_x^{-} - {\\rm i} F_y^{-} \n\\end{array}\n\\right]\\,. \n\\end{align}", "e2403a4e45f18f1c12a1bfffb3463059": "n = \\frac{\\ln(\\cos \\phi_1 \\sec \\phi_2)}{\\ln [\\tan (\\frac14 \\pi + \\frac12 \\phi_2) \\cot (\\frac14 \\pi + \\frac12\\phi_1)]}", "e24044197a33f8f0f3cb98bb8a0d4fa8": "\\frac{{}_{(1)0}\\partial x^{-1}}{\\partial x}=-x^{-1}\\,\\!", "e2404df31e443e1b9349763742daad2e": " \\sum_{i=1}^n (x_i - \\mu) = 0,", "e2418626933b1131c6dde63e51496e2f": "P_{\\rm s}", "e241d83e9020b800c3eea89c432f0057": "\\ \\displaystyle \\hat{\\beta}(q,r_{c}) \\ ", "e241eee71cc3d8ba55a73b49fb8c9eae": " A_t = \\#\\{c \\in C \\mid w(c) = t \\} ", "e24205a3b382451cb4230bf396018548": "\\Delta\n> c", "e242a62ca1ffedd18461c6367454e898": " r_1=u+\\int \\frac{c}{\\rho}d\\rho, ", "e242d9f3a50aa99c38c52dd84418b970": " \\and T_5 = [F_5, S_5, A_5]::\\_ ", "e242f5edda96fe428a0a06b321cebd00": "\n e_{ABC}~\\cfrac{\\partial F_{iB}}{\\partial X_A} = 0 \n ", "e2434d0a503d0227177c6d450a4f39c0": "\\alpha^{-1}(M_{Z^0})", "e24366f95cd4cd491ddcc479641f1ef9": "x_1 \\geq 0", "e243679f057a79ae40821175f0cf7e97": "X_i^'", "e243b8724ee0860a2b24f48b7c5c2360": "\\gamma >0", "e2447caf8d0dc6cd97a5a93d38cfed5b": " T_{0} = T", "e2448ac2e20f54d56a050a6c1ccfe113": "f_{\\rm{best}} \\ ", "e2449fcc9a374c05a281b8a5d1be852f": " A \\subset \\mathbb{R}^n", "e244f57d7ee85083ec1d428fe1f65be7": "\\pi(x;q,a)\\le{(2+o(1))x\\over\\varphi(q)\\ln(x/q^{3/8})}", "e244f60f5d06baded712940ceb4e5e21": "\\scriptstyle 2\\pi/T_n", "e24579ea0379fcead44bad39e482cccd": "d(z, m)^2 + {d(x, y)^2 \\over 4} \\le {d(z, x)^2 + d(z, y)^2 \\over 2}.", "e245c35b0d6172aa911e70bc7cb4e411": "(m,n)(n,p)=(m,p)", "e2466dd9478f59e3e84e6731d47a2877": "\\mathcal{X} = \\left\\lbrace{x\\in X \\vert g_1(x)\\le0, \\ldots, g_m(x)\\le0}\\right\\rbrace.", "e24691522eafc06f9fd71a694401d16f": "a_6", "e246a123dded338d35fc8b9f8c9d4a07": "\n\\min \\{ \\max \\{ R_A(x) \\mid x \\in U \\text{ and } x \\neq 0 \\} \\mid \\dim(U)=n-k+1 \\} \\geq \\lambda_k\n", "e246b724fcd42d4f45933bb0687f2615": "E_F/k_B", "e246f1e7e9aea8e1e1746ac80938193c": "\\Delta{z_i} \\;\\stackrel{\\mathrm{def}}{=}\\; z_i' - z_i", "e24735461f75b9c10c15385ad6434090": "\\frac{2\\,d_2^2\\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\\!", "e2475374e267a1ee9ed2fd190037f73a": "\\left(R, B\\right)", "e24884c95bcf5bd006c3d5062313ef52": "\\hat{C}\\pm\\,s_\\hat{C}\\sqrt{\\left(r-1\\right)F_{\\alpha;r-1;N-r}} ", "e2490a13a2343bc4a6677515d18de6fc": "P_{n+1}/P_n", "e249527a7a364032c95799a0c692ba57": " = \\operatorname{E}_\\theta \\left[ f(x) \\; \\nabla_{\\theta} \\log\\pi(x \\,|\\, \\theta)\\right]", "e2497c07c6023471316fbe7fec8c6522": "\\boldsymbol\\epsilon_{i}", "e249c68fdd3eda16b1ce268b2866ac1e": "\nE(Y\\mid X) = E(Y) + r\\sigma_y\\frac{X-E(X)}{\\sigma_x},\n", "e249fec223e8210e535f674b19444d23": " \\sum_{n = -\\infty}^\\infty a_n e^{inx} = \\frac{1}{2}(f(x+) + f(x-)) ", "e24a0453407b5c2596163e552946249e": "\\Delta\\varphi = -\\Delta a f(\\xi_1, \\alpha + \\Delta\\alpha) + \\int_a^b [f(x, \\alpha + \\Delta\\alpha) - f(x,\\alpha)]\\,dx + \\Delta b f(\\xi_2, \\alpha + \\Delta\\alpha)", "e24a254c7150d07dc9c8d98e44dcc158": "\\sigma_x^2(k) = \\lambda\\sigma_x^2(k-1) + x^2(k)\\,\\!", "e24a5601777c447da6dcdab4efb64f53": "p \\lor q", "e24a641cfe94ebf138116cbda514f6fe": "f_s(K)", "e24a6bd6a61f7e157b63e55138f36715": "\\deg(G)-(n-d) \\geq 0", "e24a9867295847530add00ff809e8ed9": "g(\\nabla f, X) = \\partial_X f, \\qquad \\text{i.e.,}\\quad g_x((\\nabla f)_x, X_x ) = (\\partial_X f) (x)", "e24aebafc4d8cd5167887660f111daa2": "x=R(y,\\phi)=\\frac{1-y}{2+a+\\phi}", "e24afaefd7f073122c542413d7d8f526": "a:1{\\to}A", "e24b217a6913ffc49d7a1742ee9a732c": "\n Z = \\int \\operatorname{tr} \\left( \\mathrm{e}^{-\\beta\\hat{H}} |x,p\\rangle\\,\\langle x,p| \\right)\n \\frac{ dx\\, dp}{h}\n = \\int\\langle x,p| \\mathrm{e} ^{-\\beta\\hat{H}}|x,p\\rangle ~\\frac{ dx\\, dp}{h}\n", "e24b29e66eba3c7545af7b96b03676ee": "\\mathbf{a_{\\mathrm{Cptl}}} = -\\omega^2 \\mathbf{r_B}(t) \\ , ", "e24b6ef587b828ec0f46c005ef2f2825": "\\mathbf{T} = \\begin{pmatrix}\n0 & - a_\\text{z} & a_\\text{y} \\\\\na_\\text{z} & 0 & - a_\\text{x} \\\\\n- a_\\text{y} & a_\\text{x} & 0 \\\\\n\\end{pmatrix}", "e24b95843a2d9734ed50e1f65631dd12": "\nCU(C,F) = \\tfrac{1}{p} \\sum_{c_j \\in C} p(c_j) \\left [\\sum_{f_i \\in F} \\sum_{k=1}^m p(f_{ik}|c_j)^2 - \\sum_{f_i \\in F} \\sum_{k=1}^m p(f_{ik})^2\\right ]\n", "e24c4ea21db34954b75631e806807fef": "C_0\\,", "e24c64ada28506db75f196bc742e4907": "F_n^{(0)}=F_n", "e24c7bf1993d3472494a17335af726ed": "\\binom{(n+1)^2+1}{2}-\\binom{n^2+1}{2} = (n^2+1)+(n^2+2)+\\cdots+(n+1)^2.", "e24cbf59316fbbe36af6649e0efc5d85": "\\mathrm{F} = \\left\\{ \\tau \\in \\mathrm{H} : \\left| \\operatorname{Re}\\tau \\right| \\leq \\frac{1}{2}, \n\\left| \\tau \\right| \\geq 1 \\right\\}", "e24cfe4d1d54f8e8d060d865cd5341a1": "\\beta=\\tfrac{1}{3}(2 e^{i\\theta}+e^{-2 i\\theta})", "e24d0366146ffb16b9b8bcc19582f0b0": "\\rho(\\gamma(t))", "e24d0f8a5197f87acaf80c25019d988c": " H[(X,Y)]\\leq H(X)+H(Y).", "e24d3ce092fdca80d5cf0b3005dd6764": "E = E_0 + \\langle\\Psi_0 | H_\\epsilon-H_0 | \\Psi_\\epsilon\\rangle", "e24d759802467522860ff0f568a0b393": " E = p\\ f, F = (\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x)) ", "e24daa0aec6bede3776c6759eb6c5fb1": "\\left|\\psi_1\\psi_2\\right\\rangle = (-1)^{2\\,s}\\left|\\psi_2\\psi_1\\right\\rangle.", "e24dbbbfb4040b88e2418e096adaca63": "\\varphi(H) = H", "e24dcfb924e63cad698ca86164993f97": "\n \\langle \\hat{x}(t) \\rangle = \\sqrt{\\frac{2\\hbar}{m\\omega}}\\Re[\\alpha(t)] \\qquad \\qquad \n \\langle \\hat{p}(t) \\rangle = \\sqrt{2m\\hbar\\omega}\\Im[\\alpha(t)]\n", "e24e4dfdde5bea3a954bdb8d7640c788": "v^{sup}(k),v^{sub}(k)", "e24e6e487325a1859ffc98fcc2c07677": "P(\\emptyset) = 0", "e24e778c508f6f876b137ae4aa5096cc": "\n\\pi = \\cfrac{4}{1 + \\cfrac{1^2}{3 + \\cfrac{2^2}{5 + \\cfrac{3^2}{7 + \\cfrac{4^2}{9 + \\ddots}}}}}\n", "e24ebe2ff6ca6615006aa2c1eb373c69": "\\sum_{k\\ge 0} c_k<\\infty.", "e24ef83291f74655d33777c54b66264d": " \\begin{align}\n y(t) &= \\frac{a_0}{2} + \\sum_{k=1}^{\\infty} \\left[ a_k \\cos(2 \\pi k f_0 t ) - b_k \\sin(2 \\pi k f_0 t ) \\right] \\\\\n &= \\frac{a_0}{2} + \\sum_{k=1}^{\\infty} r_k \\cos\\left(2 \\pi k f_0 t + \\phi_k \\right) \\\\\n \\end{align} ", "e24f8d818ca0c25fe12b1514d2b3ee02": "f(x) = ax^2 + bx + c", "e24fa43b7fafbcdd418c2a5037a2dd64": "\\det(A^{-1}) = \\frac{1}{\\det(A)}=\\det(A)^{-1}.", "e24fc7d5e436c59e19449f91426ce914": "C_L=\\frac{L}{\\frac{1}{2}\\rho A W^2}", "e25009306f4c4060d403008a7fcda449": "\\displaystyle b_n = 2 F_n F_{n-1}", "e2502b2ed48c198e12100633d3c2895f": "e^{-1/2}", "e2508653681c6c93dbc7ac0564d90224": " x_{n+1} = x_{n} - \\frac {f(x_n)}{f^\\prime(x_n)} \\,,", "e250c3f744c07857fcb2c9c98978671b": " (q,q^2)", "e250c42e39e34e6c2c3c89bae4b6f911": "\nE(S)=a_1M\\,\n", "e2510adf9ddf7100059f060a57b5b045": "\\Gamma_\\tau = \\{ U_\\tau(1,a,b) \\in H_\\tau : a,b \\in \\mathbb{Z} \\}.", "e25188c9b82fa0f41ba9e7274919a926": "\\det(\\bar{T}_{\\kappa\\lambda})", "e2519d33e8314ce1b62c1624f92d7a01": " \\langle \\delta_k \\delta_{k'} \\rangle = \\frac{2 \\pi^2}{k^3} \\, \\delta(k-k') \\, \\mathcal{P}(k).", "e2520120bddc4b2832f480f60243c4ec": "\\log(e/2),", "e2521e1bdc4d2794ef7d2326cc1bceba": " \\leftarrow ", "e252451b294f760446c2d142543e1d27": " \\text{Capture (2D)}\\; \\begin{cases} \\left\\vert X(\\text{new}) - X(s) \\right\\vert < \\text{Separation} \\\\ \\left\\vert Y(\\text{new}) - Y(s) \\right\\vert < \\text{Separation} \\end{cases} ", "e252beca82ce457121477c51b66ed224": "\n(-\\hat{E} + mc )\\psi_{1,2} = (-\\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}})\\psi_{3,4}\n", "e252c2bc6286df853e40a17570679c7e": "V^+=V^-", "e252f5360fffe2be50c90f6d75e70b2b": "\nr_{xy}=\\frac{\\sum x_iy_i-n \\bar{x} \\bar{y}}{(n-1) s_x s_y}=\\frac{n\\sum x_iy_i-\\sum x_i\\sum y_i}\n{\\sqrt{n\\sum x_i^2-(\\sum x_i)^2}~\\sqrt{n\\sum y_i^2-(\\sum y_i)^2}}.\n", "e2534cefc2d6258851b771e24814a73c": "\\Omega (R)=\\Omega_{gp} - \\kappa /m", "e2534ecfba51778f76317087779d74e5": "\\mathcal{A} \\star \\mathcal{B}", "e25356666f7a61f3468bf4809dbf4cfe": "\\left(\\sum_{n=0}^\\infty a_n\\right) \\left(\\sum_{m=0}^\\infty b_m\\right)", "e253945e5b250ac4a8444e8ca9a09f05": "\\mu = A \\cdot e^{Q/RT},", "e253ac90eb8265364180bb8b2edb62ea": "A \\subseteq C\\,\\!", "e253f1a9aea9846880af734d249c60bd": "\\kappa_{\\sigma}", "e254c0a0dde702406b8c347f56efbb4f": "\n\\vec T_b = \\vec T_{b0} + \\sum_{i=1}^n(\\vec T_{bi} - \\vec T_{b0})C_i\n", "e254d869b7e434fb7d68b9370002b9e9": "d \\in {L,R}", "e254dfe0abd5ffed9eb0e26b99b46d13": "\\delta(k,i) = \\lambda\\delta(k-1,i) + \\frac{e_b(k-1,i)e_f(k,i)}{\\gamma(k-1,i)}", "e254edbe1775baeefaa382e7ae23016c": " \\forall a \\in \\mathrm{A}, \\; \\exist \\sigma\\ _n \\in \\Sigma\\ ^n \\; s.t. \\; \\forall \\sigma\\ _{-n} \\in \\Sigma\\ ^{-n}: \\; \\Gamma\\ (\\sigma\\ _{-n},\\sigma\\ _n) = a ", "e255297e6aef54bdc6178a6414a7c248": "Q =-\\frac{|s_{pm}|}{2\\mathrm{Re} (s_{pm})} = -\\frac{1}{2\\cos(\\arg(s_{pm}))}", "e25565d5ca1e0fb3bd877eec18f8fc0b": "\\operatorname{vers}(\\theta)", "e2559aac7807f5c79776f02178ca976a": "\\mathop{\\mathrm{vol}}(B_n) \\, = \\, \\exp\\left( - (n-1)^2\\ln n + n^2 - (n - \\frac{1}{2})\\ln(2\\pi) + \\frac{1}{3} + o(1) \\right) .", "e2560757c3856e61d25f459fe1b2fda1": "\\int _a^x B_n (u) ~du = \\frac{B_{n+1}(x) - B_{n+1}(a)}{n+1} ~.", "e256076c3e9371e2b54459ca5abb0c6b": " \\pi_t(o_1, a_1, ..., o_t,a_t) ", "e256203cd97c535281604ce3124e4598": "\\hat{g}.", "e2562cd7ba145fed017286df371cc211": "dU =C_{V}dT +\\left[T\\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - p\\right]dV\\,\\,\\text{ (1)}.\\,", "e2565d83861d36841c8105512369a9f9": "\\hat{\\mathbf{x}}_1", "e256691ee63c8d0da2f3c37bcc651bf4": "\\sqrt[p]{\\sum_{i=1}^n\\frac{w_i}{x_i^p}}\\leq \\sqrt[q]{\\sum_{i=1}^n\\frac{w_i}{x_i^q}}", "e2567428380f6678de0171818afadc99": "\\mathbf{P}_{X_i|z^n}", "e2567c768c9f5d197fe20dac9f7e55c4": "\\pi u", "e256a8b9a0ce646a8e44cd470444703b": "\\Psi_k\\Psi_m(X)=\\Psi_m\\Psi_k(X)=\\Psi_{\\max(k,m)}(X)", "e256aecf330617b8a766e500a6dcab8d": "\\textstyle A=1", "e256b4dab08f0b99e0924b84a478e061": "V=2\\times \\frac{2}{3}\\pi r^3=\\frac{4}{3}\\pi {r}^{3}", "e256d1540c68a27d2835f38f87d073be": " F_{ST} = \\frac{f_0-\\bar{f}}{1-\\bar{f}}", "e256f88cf081ba916d48d0eb5c10c701": "x_{n-1}^\\ast = x_{n-1}(t)-v_{n-1}(t)^2/2b_{n-1}", "e256ff0f38214ff7c35ff3b180545fed": "\nr'_2(n)=\n\\frac{\\pi}{4}\n\\left(\n\\frac{c_1(4n+1)}{1}-\n\\frac{c_3(4n+1)}{3}+\n\\frac{c_5(4n+1)}{5}-\n\\frac{c_7(4n+1)}{7}+\n\\dots\n\\right)\n", "e257189db3190bd5f889829dd78aa34b": " U = k{x}^{2}/ 2 . ", "e2572457a1f04ed2dce3862a9d308afa": "{(\\eta_b)_{max}}", "e257c144cc7c37a23127124d893ae65d": "f = AB(C); \\left(\\frac{\\sigma_f}{f}\\right)^2 \\approx \\left(\\frac{\\sigma_A}{A}\\right)^2 + \\left(\\frac{\\sigma_B}{B}\\right)^2+ \\left(\\frac{\\sigma_C}{C}\\right)^2.", "e2583c600e4e8b2682143892923767b8": "p=\\rho gy", "e2584560b8d77220b07dc5b8699a5c85": "\\|u\\|_{(2)} \\le C(\\|\\Delta v\\| +\\|\\Delta w\\| + \\|v\\|_{(1)} + \\|w\\|_{(1)})\\le C^\\prime(\\|\\psi\\Delta u\\| + \\|(1-\\psi)\\Delta u\\| + 2\\|[\\Delta,\\psi]u\\| +\\|u\\|_{(1)}) \\le C^{\\prime\\prime} (\\|\\Delta u\\| + \\|u\\|_{(1)}).", "e2585c41a38bb6cee7f61cc0019253b7": "\\hat k (C)", "e258bb5b94763d9b9fd86faafa49aea8": "v_\\mathrm{p} > c, \\,", "e25970827b6338a45778c1b1a75108e1": "W_i = W_i'', W_{1-i} = W_{1-i}'' \\cup B", "e25996063538395f6f9b32caade76c48": "\n\\ \\ <^\\text{colex}\n", "e259977ede054cf75958663b5ef750da": "\\scriptstyle T", "e259fe620a666f7fc722e219660af490": "K_n \\sim \\frac{n}{\\log n},", "e25a0b5459051102f0126a67a0db5e27": "\\partial L / \\partial \\dot{q_i} ", "e25a178a731b6c5264c1e5ad2619181a": "I(\\mathbf{q}) \\sim \\left | \\phi(\\mathbf{q}) \\right |^2 = \\left | f(\\mathbf{q}) \\right |^2 \\times \\left ( \\sum_{j=1}^{N} \\mathrm{e}^{-i \\mathbf{q} \\mathbf{R}_{j}} \\right ) \\times \\left ( \\sum_{k=1}^{N} \\mathrm{e}^{i \\mathbf{q} \\mathbf{R}_{k}} \\right )= \\left | f(\\mathbf{q}) \\right |^2 \\sum_{jk} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_j - \\mathbf{R}_k)}", "e25a2812c90b4760c04acbc62309a3f0": " T^{ii}", "e25a3f84150a4a59ce2a34ff320733a2": "8\\pi", "e25a542daafc89dcaba571777c627618": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = x_{1} \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = g\\left(\\boldsymbol{x}\\right) h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) \\\\\n g\\left(\\boldsymbol{x}\\right) & = 1 + \\frac{9}{29} \\sum_{i=2}^{30} x_{i} \\\\\n h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) & = 1 - \\sqrt{\\frac{f_{1}\\left(\\boldsymbol{x}\\right)}{g\\left(\\boldsymbol{x}\\right)}} \\\\\n\\end{cases}\n", "e25a7f0346b911be444a4cf67e2f2291": "g\\,'", "e25a9b438f5aa18783fabddd463ccd31": " \\frac{[D]}{[C]} = e^{ - \\frac{\\Delta \\Delta G^{\\ddagger}}{RT} } ", "e25a9ed7a5b522d46c98a8813634ab5d": "s^2", "e25ac973e5a3000372a1bd633e2a8dab": "(\\textbf{q}_i)", "e25ae63fbfc4805ad2d862a2627c6484": "\n\\tan 2\\Phi = \\frac{2A_{xy}}{A_{xx} - A_{yy}}\n", "e25b53d6f8e3cbe43ff34eff2c967688": " \\frac{d}{dx}\\left( EA\\frac{du}{dx} \\right)+n=0 ", "e25bf085288bcbeb9eaea8cf59f01e97": "\ny=-\\frac{(1+i\\sqrt{3})x}{2^{2/3}\\sqrt[3]{729-108x^3}}-\\frac{(1-i\\sqrt{3})\\sqrt[3]{-27+\\sqrt{729-108x^3}}}{6\\sqrt[3]{2}}.\n", "e25c1f5abf4fd8e1386aa72bbefba87c": "\nW(r)=-kT\\log g(r)=-kT\\log\\frac{P(r)}{Q_{R}(r)}\n", "e25c5d7287c0e0308b0ec48e4ea9c930": " N = \\bigg\\{\\begin{bmatrix} 1 & 0 & t \\\\0 & 1 & s \\\\ 0 & 0 & 1\\end{bmatrix}: s,t \\in \\mathbb{R} \\bigg\\}. ", "e25c888b4fb40980cb8f822411339a6e": "\\sum_{n=1}^\\infty \\frac{1}{n^2}", "e25c9a357839820dc439ce21dac62434": "\np_w(\\theta)=\\sum_{k=-\\infty}^\\infty {p(\\theta+2\\pi k)}.\n", "e25d01b1924f75fa98ad99862922391c": "\nA_{DFA} = \\{\\langle B,w \\rangle \\mid B \\text{ is a DFA that accepts input string } w \\}\n", "e25d2d05eb2d6ff86d6d93dedacd51ea": "n! = P(x)", "e25d56fe2c7159dcbd009861760c169f": "E/2", "e25d6897a638d0581141eace4a15bdf1": "0 \\to N \\to M \\to M/N \\to 0.", "e25d70407331c8d458712a096768799e": "0 \\le y,y^{\\prime}<256^l", "e25da5ef4e5e563b7c1f4a85a22eac6b": "(l=0)", "e25dedcd10760e22a4d5713261645a06": "\\psi_{2}", "e25e72446c8f56341764ec2ed3d145c7": "\n\\boldsymbol{\\Tau}_{\\boldsymbol{n}}(z) = \\frac{zA_{n-1} + A_n}{zB_{n-1} + B_n}\n= \\frac{A_n}{B_n} \\left(\\frac{z\\frac{A_{n-1}}{A_n} + 1}{z\\frac{B_{n-1}}{B_n} + 1}\\right)\n\\approx \\frac{A_n}{B_n} \\left(\\frac{zk + 1}{zk + 1}\\right) = \\frac{A_n}{B_n}\\,\n", "e25ebe177db18c22fe906d3dc5b431fa": "\\scriptstyle f_n", "e25ebeb6610e660437e61d4e3728441b": " I_t \\, ", "e25eccc87b4ee0828c0e01234477bedb": "\n S_D = 100 - \\cfrac{20(-78.188 + \\sqrt{6113.36 + 781.88 E})}{E}\n ", "e25ed382148b5d237011065d94de9a3f": " N = \\sum{N_h} ", "e25f08ba5da74ef91ac2d44c9e544847": " \\forall xA \\rightarrow A(r/x)", "e25f0f461f6b57a0c15afffcc275e0ad": "f(x,y)= 2x^3+4x^2y+xy^5+y^2-7.\\,", "e25f6691447fb799637708cf95c47c4c": "\\; \\mathrm{Ker}(A - \\lambda I) / Q.", "e25f7d42f8eae3a397010ee758e01b74": "\\frac{1}{c^2}\\frac{\\partial^2\\mathbf A}{\\partial t^2} - \\nabla^2{\\mathbf A} = \\mu_0 \\mathbf{J}", "e25f9cd6dbdd725419a9d7435ac86459": " a_w \\equiv l_w x_w ", "e25fcfc7a10c2b8dbe921d248ac1d52c": "\\frac{\\alpha^{-1}_3 - \\alpha^{-1}_2}{\\alpha^{-1}_2-\\alpha^{-1}_1} = \\frac{b_{0\\,3} - b_{0\\,2}}{b_{0\\,2} -b_{0\\,1}}", "e25fd9a48be58d2b5a92e94fadd2cf7f": "{\\mathbb P}(X_1\\le x_1,\\ldots,X_n\\le x_n)\\le\\min_{i\\in\\{1,\\ldots,n\\}}{\\mathbb P}(X_i\\le x_i),\\qquad (x_1,\\ldots,x_n)\\in{\\mathbb R}^n,", "e26059c56d8ff90d9813e8e36a0975af": "(2-DX_i)", "e260c07bca4895184a8c6b97b8ea23ea": " \\Rightarrow \\frac{1}{2}\\bigg[\\bigg(|0 \\rangle + |1 \\rangle \\bigg) |f_k \\rangle |f_k' \\rangle + \\bigg(|0 \\rangle - |1 \\rangle \\bigg)|f_k' \\rangle |f_k \\rangle \\bigg]", "e260c9218ae2e960179e95e29cea4ed7": "z_0\\in D_r", "e2611f36cf0c239e14e95a6829f5ae03": "d(x,y) = 1", "e2613bc41bb190787c9fcfd07a985a37": "n_{01}", "e261a9fc1ba377983a34fef225b009e0": "q_m^0=", "e261e9336b68cb3d9de1e86b74f38bbd": " a_j^* = z_j ", "e262223f0aba8ba8e5629727e2582889": "\\mathcal{E}_\\lnot(z)", "e2623217441736ca6ad3689bf68db409": "|\\phi_i\\rangle=\\mathcal{T}|\\tilde{\\phi}_i\\rangle", "e26298e76d65416f864b93de34dbb4a5": "\\log_r y = a \\log_r x + b \\quad\\Rightarrow\\quad y = r^b\\cdot x^a", "e263357423e183f3402822c20be03a84": " ds^2 = -8 \\pi m r_0^2 \\left( 1 + 2 \\log(r/r_0) \\right) \\, du^2\n- 2 \\, du \\, dv + dr^2 + r^2 \\, d\\theta^2 ", "e2634926780c0c433fa38bb427c50c59": "e_{(\\mathbf N)}=\\frac{dx-dX}{dX}=\\Lambda_{(\\mathbf N)}-1\\,\\!", "e26355e1154f53d68319aad78fd8c74f": " L = N \\frac{\\mathrm{d}\\Phi_B}{\\mathrm{d} I} ,\\,\\!", "e263d128b50284ddb9622311be1ee3a9": "E = [\\vec e_1 \\vec e_2 \\ldots \\vec e_n]", "e26415d9152def53ba732c0cce71f7e5": "P + Q = x + 5xy + 4y^2 + 6 ", "e2652fd33d0e575de640506b97976394": "\\int_0^\\infty x^{n}e^{-ax}\\ dx=\\frac{\\Gamma (n+1)}{a^{n+1}}", "e2654f6c48ea5eadc8bcdcb3c473d9bf": "\\frac{\\pi}{4} = \\arctan \\frac{1}{2} + \\arctan \\frac{1}{5} + \\arctan \\frac{1}{8}.", "e265dec59f81dd8e27433e718c87eb4a": "\nE = \\begin{cases}\nx < y \\\\\ny > x \\\\\nx \\ngeq y \\\\\ny \\nleq x\n\\end{cases}", "e265e4b9ce2524ede244326c55c092aa": "\\Pr(X - \\mu \\geq \\sigma) \\leq \\frac{ 1 }{ 2 }. ", "e2660196668251006e232b3dcbd5e614": "N_{}", "e266241f1ca7d753dd3be35ccd3c2845": "\\operatorname{let} p : p\\ f = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) \\operatorname{in} p ", "e2666486abcdd456a75bb4ad60cd1996": "p(f_i|\\bar{c})", "e266a5c37884d2695f420f6c4af6845f": " S=\\sum_I (-1)^{|I|}e^{\\Sigma I} \\, ", "e266f488f260fb777e5c856d63f935f4": "\\int_M u\\nabla^{2} v\\, dV+\\int_M\\langle\\operatorname{grad}\\ u, \\operatorname{grad}\\ v\\rangle\\, dV = \\int_{\\partial M} u N v d\\tilde{V}", "e267168cfe25856bc464ef028d51fdb3": "\\mathbf{N}", "e2677298b9a995803e82c416d3b87bad": "r_x(\\tau) = E[(x(t)-m_x(t)) (x(t+\\tau)-m_x(t+\\tau))]", "e267822546448d627285d3146f55f852": " R(x_1,x_2) = \\frac{f(x_1) - f(x_2)}{x_1 - x_2}", "e267a0022f5623982ab3469c0c711efc": "1/2^n", "e267d9069be3ed195b520c3c98817862": "\\nu({0})=0", "e267dd7cc8b46e03fc31f7afe714bf13": "A_{S}", "e267ffffc5aea78a66c308b59310e7fb": "C = p_{\\theta}^{2} + \\cos^{2}\\theta \\Bigg( a^{2}(m^{2} - E^{2}) + \\left(\\frac{L}{\\sin\\theta} \\right)^{2} \\Bigg)", "e2683e7667ee09e086a778e14ed09f9e": "v \\otimes f \\mapsto f(v).", "e2686bea0b81a95f427214cf3fc01b6a": "\\frac{{\\dot a}^2}{2} - \\frac{G \\frac{4 \\pi a^3}{3} \\rho}{a} = - \\frac{k c^2}{2} \\,.", "e2689195fc6fbfbd2e3f956f3534c47c": "m_0\\,", "e2689d39248fa366594e07306704d481": "Mp(2,\\mathbf R)= \\left \\{ \\left ( \\begin{pmatrix} a & b \\\\ c & d\\end{pmatrix},G \\right ) \\ : \\ G(\\tau)^2=c\\tau +d, \\tau \\in \\mathbf{H} \\right \\},", "e2691670012d3cb3ac0f0995bb6e2e28": "AB =\\begin{pmatrix}4&6\\\\6&2\\end{pmatrix}", "e2693a2e33f79b0fcb14d5ed0fd4a8d2": "L(f) = \\sup_{x\\in D} \\|f'(x)\\| ", "e2698dcd5bd382fbf25483b8877c7109": " \\Phi_P(r) = -\\frac{G M}{\\sqrt{r^2+a^2}}\\,,", "e269d6e64be77a4a4f93a9f5a22ac18f": "i+j+k-1,", "e269dce030d2bb38a01193b7ad218567": " h_0 (X_1, X_2, \\dots,X_n) = 1,", "e26a0041f4a953445420e080f40a0562": " g^{ij} ", "e26a4532eaff867913fe6444c41cd0d6": "\\, G_{adv}(x-y) = \\Delta(x-y) \\Theta(y_0-x_0) ", "e26a562b3b3aad441bce2cd174ad088e": "Literature \\leq bad", "e26aa610907a715a3e51545022ac3838": "i,j\\ne1,n", "e26ac9f25ec011002e1f2eb5927da066": "c'\\equiv cs \\pmod{q}.", "e26ad1babde109565af77f44a21e9166": " V = \\left| f \\right| R ", "e26b0c9ae068a3e485db057a592683ba": "{BE}_{10}", "e26bad72377541dee32ea7be63991198": " \\ C_{\\text{L3D}} ", "e26bdc90e31104a895939e05fac6cc45": " A = A_0 e^{-i \\gamma_{||} |A|^2L} ", "e26be6f2b01aa425e43c79cd183b88f5": "\\Gamma\\,'", "e26bec6217bb681a5ad09302140d5fcc": "\\mathrm{id}_S", "e26c15cf9546421861271218d1b5c4c2": "(B, \\pi_B)", "e26c17f40cfab86b73e0c700933f2a33": "\\operatorname{TVaR}_{\\alpha}(X) = \\operatorname{E} [-X|X \\leq -\\operatorname{VaR}_{\\alpha}(X)] = \\operatorname{E} [-X | X \\leq x^{\\alpha}] ,", "e26c23c5813682b712dc2d51f3063fb7": "\\mbox{Tr} \\mathcal {L} = \\sum_{\\{i\\}} \\rho_i", "e26c869d73b75c388f0d284783c410be": "d^2(p, q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+...+(p_i - q_i)^2+...+(p_n - q_n)^2.", "e26d033c87a9e250dce1bfc78ee4b049": "R_\\text{z}", "e26d161f4dbe1bc7b528ad53a852409f": "\\operatorname{recc}", "e26dee6db87444f67530132f900527d1": "S_{11} = S_{22}", "e26e125da688f7948b7e0b31702c28a3": "\\text{period of } \\tfrac{1}{p^{m+c}} = p^{c} \\cdot \\text{ period of } \\tfrac{1}{p}", "e26e56d8e2756e8008c4249ab1672732": " T = \\frac{K \\cdot m}{k_B} \\rho_c^{1/n} \\theta_n ", "e26eaf1e5303699ebcdae101532c55b9": "E_30", "e28d0d978f23da9f18c9ddf1e8af7a4a": "u[9] := 2*atan(\\sqrt((a0+b1)/(b1-a0))*sinh(\\sqrt(b1^2-a0^2)*\\eta)/(cosh(\\sqrt(b1^2-a0^2)*\\eta)+1))", "e28d2b0ee2fae8a9567a1d3f0710df8c": "y = f(\\mathbf{z}) \\,", "e28d3367424b54cec4ef60925b4a2a9c": "B_{n+1}=\\sum_{k=0}^n {n\\choose k} B_k", "e28d3bd9b08d28616cdf7fc94c308adb": "W^{II}(0,0)=0=W^{I}(0,0)", "e28d3d6b1cc818501bdd9041ab49f27d": "1+ee=\\frac{1-c+cee''}{1-c}", "e28d56be2642e118f1469a1953fa7fa3": "|c_{22}| \\ge |c_{21}| + |c_{23}|", "e28defcbdefa1a8c32496f500741384d": " \\left\\{(-\\infty, a] \\colon a \\in \\mathbb R \\right\\}", "e28e12568d9d2b44a30a47cd7f855e82": " \\mathbf{n} \\equiv ", "e28e4a7a696eb600b22afb350ee13065": "\\log N", "e28e910e0690fbf1e1f193a4ebce39b3": "\\|f\\|_U=\\sup_{x\\in U}{|f(x)|}", "e28eb0e298945bf4285b1974a5ded8f5": "\\left(\\begin{smallmatrix}1 & 1\\\\0 & 1\\end{smallmatrix}\\right)", "e28f58fe1da833a74562581ca682676c": "2V a_0 + (L\\varphi)a_0 = 0 ", "e28fd2296509b917de9de80cccb8e229": "\\tau = \\lambda_2 / \\lambda_1, ", "e290338f6b01a3a8b979253faff63d92": "\\frac{s+0.5}{n+1}", "e290560bc97a820cae60aabbb04babc4": " r_2 = r_1 = r", "e2905fab5d831aae37ce88da44a18bad": "y=(y_n)", "e29069e520e6c2701bc70b96ae872fc4": "[y_1-\\Delta/2,~y_M+\\Delta/2)", "e290758247359036718e6bd4ce3b07e2": " X_c ", "e2907c0db307d285c8e577a051672356": " \\mathbf{u} \\cdot \\nabla \\mathbf{u} ", "e2909669dcc1e093df178a5419988c5b": "H_\\Lambda^\\Phi(\\omega | \\bar\\omega) = H_\\Lambda^\\Phi(\\omega_\\Lambda\\bar\\omega_{\\Lambda^c})", "e290a0c34aceadcc5c69b4689830a9cf": " 1 \\le n \\le \\left\\lfloor \\frac{f_H}{f_H-f_L} \\right\\rfloor", "e291453dc0ccd8aad8a57a0da3b7a873": "M_\\mu\\rightarrow M_\\lambda", "e291a30480453403ea441a3265ef0b40": "{n \\choose 6} {6 \\choose 2, 2, 2} \\frac{1}{6}", "e291e072c6161974068782ed9848be5b": "ji=-\\!k", "e29251aeff34ce1402cfbdf9dc579d84": "0 = (x^\\alpha)_{; \\beta ; \\gamma} g^{\\beta \\gamma} = ((x^\\alpha)_{, \\beta , \\gamma} - (x^\\alpha)_{, \\sigma} \\Gamma^{\\sigma}_{\\beta \\gamma}) g^{\\beta \\gamma} \\,.", "e2925d8c818e08eacf8cc5fb9139be70": "H(x,y) = u_x(x,y) = -u_y(x,y)\\, ", "e29269b38d19fb4bebaef4cac65fb4dc": "\\int_0^{\\infty} \\sup_{Q} \\sqrt{\\log N(\\varepsilon ||F||_{Q,2}, \\mathcal{F}, L_2(Q))}d \\varepsilon < \\infty", "e2926ae47d83c895b53309f7363a1541": "\\sin\\frac{2\\pi}{15}=\\sin 24^\\circ=\\tfrac{1}{8}\\left[\\sqrt3(\\sqrt5+1)-\\sqrt2\\sqrt{5-\\sqrt5}\\right]\\,", "e29291d2b031febe7b76648d475392fc": "Ih_n \\to 0", "e292fc504c4cad195eb633bd634f8f24": " N=\\binom{n_t}{t}+\\binom{n_{t-1}}{t-1}+\\ldots+\\binom{n_j}{j},\\quad\nn_t > n_{t-1} > \\ldots > n_j \\geq j\\geq 1. ", "e2932e388ab88801dd270ad03a2706f0": "\\mathcal{M}\\models p(\\boldsymbol{b})", "e29340a96545aac3da6d133c3b924c0f": " N_10 = (1)2^5 + (0)2^4+(0)2^3+(1)2^2+(1)2^1+(1)2^0", "e2935eaa524ca3df8d102db768362920": "\\omega=\\omega(\\mathbf{k}),", "e2935fc8a9169b59ebd23ad71288326c": "\\underline{V}", "e29361b727d2a4b6c78c10ac157141ed": "\\Omega\\omega", "e293970e587c9ceb8469e2e50bfe6a5a": "\\mathbb N", "e293b1d4870d771a80e2ac6980757515": "z_{ij} = 0", "e2945ff8fdc42f0386edccb7dcdd1170": " M = 2", "e294a2dd708ab47f89f93185b6bf78a0": "{Q}=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\,", "e294adc9934f568c10333e2cbd350192": " \n\\begin{align}\n\\left( \\pm \\frac{\\sqrt{2}}2 (1 + i) \\right)^2 \\ & = \\left( \\pm \\frac{\\sqrt{2}}2 \\right)^2 (1 + i)^2 \\ \\\\\n & = \\frac{1}{2} (1 + 2i + i^2) \\\\\n & = \\frac{1}{2} (1 + 2i - 1) \\ \\\\\n & = i. \\ \\\\\n\\end{align}\n", "e294b97594175b1c41232fa35c42449b": " \\left( \\sqrt{2}\\,rs^3 \\right)^{1/4} ", "e2950b5a17e3e21e09c6b1c14c200a28": "\\lim_{p\\to -\\infty} S_p(x,y)", "e2954f0a3d2e8bb017bbb0926cc33436": "\\iint_D \\left( \\frac{\\partial u}{\\partial x}-\\frac{\\partial v}{\\partial y} \\right )\\,dx\\,dy = \\iint_D \\left( \\frac{\\partial u}{\\partial x}-\\frac{\\partial u}{\\partial x} \\right ) \\, dx \\, dy = 0", "e2959a426758ee7a655bc30b28e9b755": "f_s = {\\mu}_s f_r", "e2959e4efa178d8c605db61908ac9256": "\\Omega=2\\pi", "e29628de1c944b6994fff534f3e51dd1": "a^2J_n=ax^n \\sin{ax} + nx^{n-1} \\cos{ax} - n(n-1) J_{n-2} \\,\\!", "e2965152230314a378974c2e7af694ee": " \n\\exists \\sigma\\ _{-i} \\in\\ \\Sigma\\ ^{-i} \\quad s.t. \\quad \n\\pi\\ (\\sigma\\ _i ,\\sigma\\ _{-i} ) < \\pi\\ (\\tau\\ _i ,\\sigma\\ _{-i} )\n", "e296af611d3504e42165fcf8de8321af": "g''(t)=g'(t)\\cdot(1+e^t\\cdot (\\varepsilon^{-1}-1))^{-1} \\,", "e296bdf42f410d425a75413792d86152": "f_i\\,\\!", "e297145b96f958a33904bb90c0cab7d4": "4 \\times 10^{10}", "e297387ac29dbd534df870c9b2807b0c": "T_a\\ :\\ Y^2 = X^3 + 3aZ^2(X+Z^2)^2.", "e2977776e061ebf72ebaf857451469a0": "d \\cdot|S| \\cdot |T| / n", "e297b4e38b79540a334361560c7a3243": " \\mathrm{card} (T(\\aleph_0)) = \\aleph", "e2986ad18259ba8a23265582b2b1165f": "\\boldsymbol{\\gamma}_{ij}", "e29871fe1ab290146e9a725b69909c89": "\\gamma_i=1", "e298bf89772e76ebd9e0089b524a5134": "(\\Omega,\\mathcal{F},(\\mathcal{F}_n)_{n \\in \\mathbb{N}},\\mathbb{P})", "e298ccb52cedeaf4e0788515f021e9e6": "\\mathbf{f}(x)", "e298ec574573f91c1ce1ec268b112611": "P = 227\\,\\bmod\\,17 = 6", "e2997f9e2b1cde1db48259979d602ffb": "\\sigma = -\\pi I + \\mathbb{T}", "e299a62baf311bebf76470f3a9fdc2cd": "\\omega(x,D)(E)=0", "e299d5db9873eabb09bfa9aa37a37efe": " f = \\frac{sm_{fu} + m_{ox, 0}}{sm_{fu, 1} + m_{ox ,0}} ", "e29a177691ba9e35ad94bc1ad3652095": "rB > C ", "e29a39c7f2b8c835fbafdc963972387b": "\\mathbf F_2=-\\mathbf F_1", "e29a3ee2063ab0e80f483f5e9a3c77c9": "\\overline{\\Lambda}_n(T) \\le \\frac{2}{\\pi} \\log(n + 1) + 1", "e29a4597b267167662dd89d11a4500d9": "L = {\\phi \\over i} \\,", "e29a78a60a8f925a6e5cbe6131dd0637": "y_\\text{t} = \\sqrt{2} m_\\text{t}/v \\simeq 1", "e29a7f70793170fabebc06258903440d": "h_f = r Q^{n}", "e29b0a2ec9c692d129140d351f4c1f57": "\\frac{c}{x}S_0'=\\frac{c}{x}=o(S_0')\\,", "e29bac9d7227d40cacb6a87a972ee737": "R={V_s-V_f \\over I}", "e29c2695827017df2b6c593f63a61b2d": "{x}\\,", "e29c42535bdb8901e854cdc964e3b08f": "k_{\\rm C} / k_{\\rm A} = c^2", "e29c5cc6702483562f2a6adb3eaf564e": "\\frac12 + \\frac12 \\ln(2\\pi\\sigma^2) + \\mu", "e29c6bfae721ffde818cd6c6dc413a56": "\\delta ^{i}_{j}", "e29cc39fded37cad91da9f108c418e90": "(u, v, z)\\in[0,\\infty)\\times[0,2\\pi)\\times(-\\infty,\\infty)", "e29cd7a265623c3a8201f26ca1ded026": "t_\\nu(\\boldsymbol\\mu,\\boldsymbol\\Sigma)", "e29ce010780cc55014b81d58534f5eb2": " D = \\{y\\in\\mathcal{S}: \\phi(y) > M(y) \\} ", "e29ce4ca872062f3c704e1324a06b1c6": "\\theta (f,b)", "e29cf62c83aae6a86a638b3c6db1e239": "l(q)=\\lfloor log_b(q) \\rfloor+1", "e29d0c07e82ad990b90eb378274d3d8d": "T_{lower} = \\frac{F d_m}{2} \\left( \\frac{\\pi \\mu d_m - l}{\\pi d_m + \\mu l} \\right) = \\frac{F d_m}{2} \\tan{\\left(\\phi - \\lambda\\right)}", "e29d1340a6f05da525d18bad2f788bdc": "f^'(E)", "e29d3c159cdc788c5cece0f34798df12": "\\frac{x^3 + x^2 + x}{x+1} = (x^2 + 1) - \\frac{1}{x+1}", "e29d68198afbe9633bf42637341e38a7": "a^m \\equiv 1 \\pmod{n}", "e29d809323ffa5eb94ea1cbce12182f3": "\\eta_\\varepsilon * \\eta_\\delta = \\eta_{\\varepsilon+\\delta}", "e29d8d4ad3e4947f75c6470a518de080": "I:=\\int^b_a\\varphi(t) \\, dt", "e29db8fbf9c91a08be4e71c5abef64bf": "\\left(\\tfrac\\cdot n\\right)=\\left(\\tfrac{n^*}\\cdot\\right)", "e29dbc0ac651b73fabd28b2292f9352e": "66{2\\over 3}^g", "e29df8b61f1dbbefdd92ac4af8305ea8": "{n0}", "e29e20db89eaab706880719811fab797": "\\mathfrak{g}_{\\pm \\alpha}", "e29e3938e10311aea77de7ea86eb395c": "\\tau=t_1-t_2", "e29e42c4297f46accd3ccbd52eb753f3": "L(r, c) = \\frac{1}{r {r-1 \\choose c-1}}", "e29e43414fdf8da8d3259fbc9471f89e": "\\vec{F} = q\\vec{E}", "e29e5adf41aa7166d263c86a5aa73eba": " \\gamma = {1 \\over \\sqrt{1-{v^2 \\over c^2} } } ", "e29e7fbd0e7c53810999f2dad76bff96": "k_nh_{n-1}-k_{n-1}h_n=(-1)^n.", "e29e87f7cd0dda20bb1d2458ea53c8b5": " \\frac {\\mathrm{DOF}_2} {\\mathrm{DOF}_1} \\approx \\frac {N_2 \\, c_2} {N_1 \\, c_1} \\left ( \\frac {m_1} {m_2} \\right )^2 \\,.", "e29e9674b06566f5fb390340e97ac452": "T_b^{-1} = \\frac{k}{h\\nu}\\, \\text{ln}\\left[1 + \\frac{e^{\\frac{h\\nu}{kT}}-1}{\\epsilon}\\right]", "e29eec091d70d81390272f8e1f7f71a6": "I^n M \\cap M' = I^{n-k} ((I^k M) \\cap M') \\subset I^{n-k} M'.", "e29ef2c1a6cddf7f0a2dd2273fe5fc6a": " \\textstyle v=(D_2-D_1)\\frac{\\partial N_2}{\\partial x}", "e29efde8f04f6ce5bd350e579a25ac1a": "\\left(\\begin{smallmatrix}-1 & 0 \\\\ 0 & -1\\end{smallmatrix}\\right) = \\left(\\begin{smallmatrix}4 & 0 \\\\ 0 & 4\\end{smallmatrix}\\right)", "e29f47da54a1ee43d29a1df7488f83bc": "\\frac{1}{\\Lambda_m}=\\frac{1}{\\Lambda_m^0}+\\frac{\\Lambda_m c}{K_a(\\Lambda_m^0)^2}", "e29f4ec4709f5763b1e0d2e6e49f45fa": "\n\\begin{align}\n\\frac{\\partial \\eta }{\\partial t} + \\frac{\\partial (\\eta u)}{\\partial x} + \\frac{\\partial (\\eta v)}{\\partial y} & = 0\\\\[3pt]\n\\frac{\\partial (\\eta u)}{\\partial t}+ \\frac{\\partial}{\\partial x}\\left( \\eta u^2 + \\frac{1}{2}g \\eta^2 \\right) + \\frac{\\partial (\\eta u v)}{\\partial y} & = -g\\eta\\frac{\\partial H}{\\partial x}\\\\[3pt]\n\\frac{\\partial (\\eta v)}{\\partial t} + \\frac{\\partial (\\eta uv)}{\\partial x} + \\frac{\\partial}{\\partial y}\\left(\\eta v^2 + \\frac{1}{2}g \\eta ^2\\right) & = -g\\eta\\frac{\\partial H}{\\partial y}.\n\\end{align}\n", "e29f94e9e8b552bb4249cf33950cc25c": "\\textbf{P}_{k\\mid k} = \\textrm{cov}(\\textbf{x}_{k} - \\hat{\\textbf{x}}_{k\\mid k})", "e29fa6653a0caaada06c723b951e4787": "p^0>0", "e29fbfae0803edf34c2a94277a6df611": " C = \\bigcap_{m=1}^\\infty \\bigcap_{k=0}^{3^{m-1}-1} \\left(\\left[0,\\frac{3k+1}{3^m}\\right] \\cup \\left[\\frac{3k+2}{3^m},1\\right]\\right).", "e29ff8369a19f2236ac6e5a101d78a9a": "g(x)=\\sum_{n=0}^\\infty {b_n \\over n!} x^n", "e2a0050046f55f809090fd494ffa3b76": "\\gcd(y_i,N) = 1", "e2a04ff7dc5fbad732da4f8ac91f8aae": "\nE (A_+) = \\frac{1}{2} \\sqrt{\\frac{\\pi}{2}}, \\ \\ E(A_+^2) = \\frac{5}{12} \\approx .416666 \\ldots, \\ \\ Var(A_+) = \\frac{5}{12} - \\frac{\\pi}{8} \\approx .0239675 \\ldots \\ .\n", "e2a061a5ee974f36bf4280bac3962260": "f(0)=0", "e2a09be9a18e6ad44bcd86d9bbc6c055": "\nS_z | \\mathbf{k}, \\mu \\rangle = \\mu | \\mathbf{k}, \\mu \\rangle,\\quad \\mu=1,-1.\n", "e2a0da46f2ecc1059d54f1b120a10f34": " Vol_q(\\boldsymbol{y}, \\rho n) = |B_q(\\boldsymbol{y}, \\rho n)| ", "e2a0f4ed694dcfc5cdf9e50c9a8faf87": "E_{B}", "e2a12a9667fed0ff66ae18a5bf75429e": "p_{\\mathbf{y}}", "e2a133198a70e280f74b12275606d31b": "H_{rot}=B\\mathbf R^2=B(\\mathbf N-\\mathbf L)^2", "e2a165050174b5156f3f8b8631fb8a0c": " \\pi_i(x) \\xi \\rightarrow \\pi(x) \\xi \\quad \\mbox{ normwise } \\forall \\xi \\in H_n \\ x \\in A. ", "e2a16b620891bedcd972decb764e935f": "\\langle m \\rangle = (-J)\\times P(-J) + \\cdots + J\\times P(J) = \\left(\\sum_{m=-J}^J m e^{xm/J}\\right)/ \\left(\\sum_{m=-J}^J e^{xm/J}\\right)", "e2a198b222e9e04d536901458ef6bbbf": "DIV = \\frac{1}{L_\\max}", "e2a19af7e27381c26ce237e42bdc1426": "\\tfrac{1}{2}Q_4", "e2a1ee3d4165ea062ef2c9fa3623c4a5": "(\\alpha) \\frac{\\alpha}{\\begin{array}{c}\\alpha_1\\\\ \\alpha_2\\end{array}}", "e2a21f08a7710daf7c0c8fb3780898a1": "Processing~Speed \\cdot \\log_2(n)", "e2a23163e185961c3134d4a443f283a3": " {d (ab) \\over dx} = \\frac{\\partial(ab)}{\\partial a}\\frac{da}{dx}+\\frac{\\partial (ab)}{\\partial b}\\frac{db}{dx} = b \\frac{da}{dx} + a \\frac{db}{dx}. \\, ", "e2a2910c72ee67c0cebe53bd2a08d0bd": "\\mathcal{E}(g) = \\frac{\\int_M R_g \\, dV_g}{\\left(\\int_M \\, dV_g\\right)^{\\frac{n-2}{n}}},", "e2a291bb29a1f2943fa977f7348df3f6": " R_{max} ", "e2a30f3bea8879a42c2b5132c2452c36": "N \\oplus P = M", "e2a35b901b0ca3898df78eee3fbcbdaa": " \\overrightarrow{v} = (\\alpha, \\beta, \\gamma) \\,", "e2a35cc54e39e0fa181cd226bed3a659": "\\mathbb{R}\\setminus\\mathbb{A}", "e2a38941b5edb599c26c5d393f159c08": "A\\equiv B", "e2a3db2ab8c3773e11fae0403a2d815c": " \\; \\overline{X} = (X_1,\\dots,X_k) ", "e2a429418c589a27c426da86b737a22d": " P = 1 - a e^{ b m } ", "e2a46322e63bfa56830b49a7bfd3241a": " \\chi_{v}", "e2a4a97db79efdd34c7bbc19ea4c3f03": "\\begin{align}\n r(t) &= s(t) \\cos (2 \\pi f_0 t) \\\\\n &= I(t) \\cos (2 \\pi f_0 t)\\cos (2 \\pi f_0 t) - Q(t) \\sin (2 \\pi f_0 t)\\cos (2 \\pi f_0 t)\n\\end{align}", "e2a4bcfab519e21e3d4f8a8e5734f6f5": "\\{ f_n \\}", "e2a4d24915c200242f70fd74db02fb54": "T^n(\\Omega) < \\alpha", "e2a4d436a86a44cd0af27dac1b10cf51": " (u,v) = (-9D^2 + \\varepsilon x, 3\\varepsilon(y - 1)) \\, ", "e2a4e3f51d3ba19b186239f412a5fbed": "p= -\\pi +2\\pi n/L", "e2a527c04e915c448f4a5c9152aae4b8": "t = \\arcsin(1/\\sqrt{N})", "e2a535a2ae97ce5f5d2aceceada62c62": "h_{j}(x_{i})=1\\,\\!", "e2a54b9c01d79101716d607a0ae57c5b": "v = (a_1 + \\frac{3}{4}a_3) sin(\\omega t) - \\frac{1}{4}a_3 sin(3\\omega t)", "e2a553004ed801c89092ce0aab06fb6d": "q_\\ell", "e2a5649c19e2506f9a69ec7e504fefb2": " x = r + \\frac{1}{c} W\\!\\left( \\frac{c\\,e^{-c r}}{a_o } \\right)\\, ", "e2a572cb07d1ac55b50f2300323208bd": "\\prod_{i=1}^{p-1} i \\equiv -1 \\pmod p", "e2a5d1fa73a34f8c1f064ce51168c24b": "f^n = \\underbrace{f \\circ f\\circ \\cdots \\circ f}_n .", "e2a5d30f007c707a882d2089e275713a": "\\displaystyle{\\|R\\|^{2m} =\\|(R^*R)^m\\|\\le \\sum \\|R_{i_1}^* R_{i_2} R_{i_3}^* R_{i_4} \\cdots R_{i_{2m}}\\| \\le \\sum \\left(\\|R_{i_1}^*\\|\\|R_{i_1}^*R_{i_2}\\|\\|R_{i_2}R_{i_3}^*\\|\\cdots \\|R_{i_{2m-1}}^* R_{i_{2m}}\\|\\|R_{i_{2m}}\\|\\right)^{1\\over 2}.}", "e2a5e69a7d09ad535e51dbdc0c254246": "\\left(p,q,Tr(g)\\right)", "e2a5fc392fa45a5064b5d4e6a911dbbb": "\\scriptstyle{r'}", "e2a63359e8ea687d37a111c80fad92cf": "\\mathbb{E} \\Phi \\left( \\left|\\left| \\dfrac{1}{n}\\sum_{i = 1}^n e_i[f(X_i) - f(Y_i)] \\right|\\right|_{\\mathcal{F}} \\right) ", "e2a670b0ab6487ae4300eef1b9c44869": "xM", "e2a6757b9572753dd9046e353ded595c": "z\\in T", "e2a690516ff4489696bc4b31216a2b59": "\\mathbf{A} \\mathbf{u} = \\lambda \\mathbf{u}", "e2a6bc66436c15f87c272d1d991856b8": "x = \\frac{X}{X+Y+Z}", "e2a6e8781c1a79a386f6fa88012f3f30": "a_k=\\int_{-1}^1\\varphi(y)\\cos(2k+1)\\frac{\\pi y}{2}\\,dy.", "e2a731da91135a2c75e53d6630ce7047": "(A - \\mu I)", "e2a73f3d76253203f03431d846d29e37": "P+UU^\\dagger=1. \\, ", "e2a7d2ea52fb14e9dbd41aa45b24a7d0": "|\\psi\\rangle_A \\otimes |\\phi\\rangle_B", "e2a82db8be72349c06779bc7897e85e6": "M*share_i", "e2a83d62c738a04c461f66b8c69e14ca": "\\rho_{T_0}", "e2a86499ac4b14bdebfcb8849e9f89eb": "[q_0,q_1,\\ldots,q_n]", "e2a868180e440299deb6eaf782747778": "\n\\lim_{\\mu \\rightarrow 0} \\Gamma(1 \\!-\\! s) \\,(-\\mu)^{s-1} = 0 \\qquad (\\textrm{Re}(s) > 1) \\,.\n", "e2a8a695628eca6e107134dcdc8b6a34": "(X, \\mathcal{B}, T, \\mu)", "e2a8a6bb0a4d59c0a19c9562715d5266": "q(x)=a_1 x_1^2 + a_2 x_2^2+ \\ldots +a_n x_n^2.", "e2a8f0d9d1335819ce4aa0a8d0834dd5": "O(N^K \\, T)", "e2a8fa57fd79db35ce067cb510112522": "i=0,\\ldots,n", "e2a92df68d740be1f999a74b7167d145": " f_+(z)={1\\over2i\\pi} \\int_\\Gamma{\\phi(t)-\\phi(z)\\over{t-z}}\\, dt\n+ \\phi(z) ", "e2a99a11d1223255e1b2dc5c87148ddb": " \\bar{f}: \\Gamma(G',S')\\to \\Gamma(G,S),\\quad", "e2a9ce4972126e382dd8645c82cd9b28": "K_m(S1, A)", "e2a9ed1abb8972d4692f63648bd749b6": "L\\ll\nS_0", "e2aa131155d14be8b231a53a4ec9c5c5": "\\text{RAC h.p.}=\\frac{D^2\\times n}{2.5}", "e2aa241b9eaaaef928354e81b00c92a8": "\\kappa = \\bigcup_{i \\in \\kappa} \\{i\\}", "e2aa511c059121c4720989170aaae6bb": "L(r,Q)=\\frac{Q}{V(r)}", "e2aa6b37694c5c848bdbb9a72019ba6f": "f_t: U \\to U, x \\mapsto tx", "e2aa6fda20c585fa2bc29e664358ab4b": "a=-0.5", "e2aa82e7cefd269e4c1b17adb0afa979": "\nT_\\mu{}^\\nu =\n\\sum_{\\sigma} \\left[ \\left( \\frac{\\partial L}{\\partial \\boldsymbol\\phi_{,\\nu}} \\right) \\cdot \\boldsymbol\\phi_{,\\sigma} - L\\,\\delta^\\nu_\\sigma \\right] \\delta_\\mu^\\sigma = \n\\left( \\frac{\\partial L}{\\partial \\boldsymbol\\phi_{,\\nu}} \\right) \\cdot \\boldsymbol\\phi_{,\\mu} - L\\,\\delta_\\mu^\\nu\n", "e2aa9fb305f88ea24c404991990d0e50": "(6)\\qquad \\nabla^2\\Psi \\,=\\,e^{- 2 \\Psi} \\,\\nabla\\Phi\\, \\nabla\\Phi", "e2aac7b3379df42c472957820ca66740": "W = \\int_0^V\\ dV\\ \\int_0^V \\ dV' \\ C(V') = \\int_0^V \\ dV' \\ \\int_{V'}^V \\ dV \\ C(V') = \\int_0^V\\ dV' \\left(V-V'\\right) C(V') \\ , ", "e2ab0f98f2067fe2992825b16f6d71e3": "e_a = e_s - \\gamma * \\left( T_{dry} - T_{wet} \\right) ", "e2ab3e8d493b0d92811e2c6d7d8bf49d": "\n\\begin{array}{cl}\n & P\\left(Searched|Known\\wedge\\delta\\wedge\\pi\\right)\\\\\n= & \\sum_{Free}\\left[P\\left(Searched\\wedge Free|Known\\wedge\\delta\\wedge\\pi\\right)\\right]\\\\\n= & \\frac{\\displaystyle \\sum_{Free}\\left[P\\left(Searched\\wedge Free\\wedge Known|\\delta\\wedge\\pi\\right)\\right]}{\\displaystyle P\\left(Known|\\delta\\wedge\\pi\\right)}\\\\\n= & \\frac{\\displaystyle \\sum_{Free}\\left[P\\left(Searched\\wedge Free\\wedge Known|\\delta\\wedge\\pi\\right)\\right]}{\\displaystyle \\sum_{Free\\wedge Searched}\\left[P\\left(Searched\\wedge Free\\wedge Known|\\delta\\wedge\\pi\\right)\\right]}\\\\\n= & \\frac{1}{Z}\\times\\sum_{Free}\\left[P\\left(Searched\\wedge Free \\wedge Known | \\delta\\wedge\\pi\\right)\\right]\\end{array}\n", "e2ab47054222d5d9585b05ec42ce7d1e": " f( x ) = p g_1( x ) + ( 1 - p ) g_2( x ) ", "e2ab5dbcba87374b09d88823b85f822c": "\\begin{align} \\frac{\\Delta s}{R} \n& = ln \\bigg[ \\bigg( \\frac{p_2}{p_1} \\bigg)^{1/(\\gamma-1)} \\bigg( \\frac{\\rho_2}{\\rho_1} \\bigg)^{-\\gamma/(\\gamma-1)} \\bigg] \\\\\n& \\approx \\frac{\\gamma+1}{12\\gamma^2} \\bigg( \\frac{p_2 - p_1}{p_1} \\bigg)^3 \\\\\n& \\approx \\frac{\\gamma+1}{12\\gamma^2} \\bigg[ \\frac{\\rho_1 w_1^2}{p_1} \\bigg( 1-\\frac{w_2}{w_1}\\bigg) \\bigg]^3\n\\end{align}", "e2ab69f6af57e86b8d30e607ce7ab3fe": "\\frac{d\\mathbf{Y}}{dx} = \\mathbf{A}", "e2ab9dd8421b107edf742e7fd1cca277": "Q_{i,j}^{(d)}", "e2abb16f9e12453c36db239cee244395": "\\operatorname{P}(n) = (k-1)k^{-n}\\,", "e2abbe1409115a86201a27cb196e88ce": " c_{t+n} = (1 - R^{-1}) \\left[A_{t+n} + \\frac{E_{t+n}y_{t+n}} {1-L^{-1}R^{-1}} \\right]", "e2abe4402f63b60310af9f739305aacb": "\\begin{align}\nv & = at+v_0 \\quad [1]\\\\\nr & = r_0 + v_0 t + \\frac{{a}t^2}{2} \\quad [2]\\\\\nr & = r_0 + \\left( \\frac{v+v_0}{2} \\right )t \\quad [3]\\\\\nv^2 & = v_0^2 + 2a\\left( r - r_0 \\right) \\quad [4]\\\\\nr & = r_0 + vt - \\frac{{a}t^2}{2} \\quad [5]\\\\\n\\end{align}", "e2ac1b34641ac6671fc6ddd93c30c618": "[2^m,k,2^{m-r}]_2", "e2ac339b0b5e961961ed2ace44ed7aca": "a_n = a_{n+1} = a_{n+2} = \\cdots.", "e2ac963cc8388563a9ede7bd69a7848a": "N_A\\,", "e2acc2046f4e45bab3589fc3b4d9fbf3": "{50 \\choose 4} = 230,300", "e2ad34b025d39401fdfcf4822456c58b": "\\{p,\\ \\bot\\rightarrow \\bot,\\ p \\land \\neg\\bot \\rightarrow s\\}.", "e2ad646514ecc93722eace49fdda3903": "= \\operatorname{E}_X\\left[\\operatorname{E}[Y^2\\mid X]\\right] - \\operatorname[{E}_X\\left[\\operatorname{E}[Y\\mid X]\\right]]^2", "e2ad757c6c46531f28f15fa5a186bbca": "\\Delta S_{ad}", "e2adac3e41b9c54b545a720b148a8578": "M_{i,j} = f_j(\\alpha_i)", "e2adb3fa3719bca5421880d2e937573d": "n_A sin\\theta_A=n_B sin\\theta_B \\ ", "e2adebf00edb1ae72096a1d8a07327e2": " \\sigma _c ", "e2adeeddf9ac88436cdf388462d3e06e": "\\gamma_1>0", "e2ae0574c2e44044276aa3676def0d4c": "\\begin{align}R_{F}(x,y,z) & = \\frac{1}{2 \\sqrt{A}} \\int _{0}^{\\infty}\\frac{1}{\\sqrt{(t + 1)^{3} - (t + 1)^{2} E_{1} + (t + 1) E_{2} - E_{3}}} dt \\\\\n & = \\frac{1}{2 \\sqrt{A}} \\int _{0}^{\\infty}\\left( \\frac{1}{(t + 1)^{\\frac{3}{2}}} - \\frac{E_{2}}{2 (t + 1)^{\\frac{7}{2}}} + \\frac{E_{3}}{2 (t + 1)^{\\frac{9}{2}}} + \\frac{3 E_{2}^{2}}{8 (t + 1)^{\\frac{11}{2}}} - \\frac{3 E_{2} E_{3}}{4 (t + 1)^{\\frac{13}{2}}} + O(E_{1}) + O(\\Delta^{6})\\right) dt \\\\\n & = \\frac{1}{\\sqrt{A}} \\left( 1 - \\frac{1}{10} E_{2} + \\frac{1}{14} E_{3} + \\frac{1}{24} E_{2}^{2} - \\frac{3}{44} E_{2} E_{3} + O(E_{1}) + O(\\Delta^{6})\\right) \\end{align}", "e2ae3f39011de95824ff15e9a33fcc39": "\\psi(x) \\rightarrow \\psi(-x) \\Rightarrow P(x,p) \\rightarrow P(-x,-p)", "e2ae5d9e3129131d79c1493b802434f8": "\n \\begin{align}\n c & = \\cfrac{\\sigma_0}{\\sigma_{90}} + \\cfrac{\\sigma_{90}}{\\sigma_0} - \\cfrac{\\sigma_0\\sigma_{90}}{\\sigma_b^2} \\\\\n \\left(\\cfrac{1}{\\sigma_0}+\\cfrac{1}{\\sigma_{90}}-\\cfrac{1}{\\sigma_b}\\right)~p & = \n \\cfrac{2 R_0 (\\sigma_b-\\sigma_{90})}{(1+R_0)\\sigma_0^2} - \\cfrac{2 R_{90} \\sigma_b}{(1+R_{90})\\sigma_{90}^2} + \\cfrac{c}{\\sigma_0} \\\\\n \\left(\\cfrac{1}{\\sigma_0}+\\cfrac{1}{\\sigma_{90}}-\\cfrac{1}{\\sigma_b}\\right)~q & = \n \\cfrac{2 R_{90} (\\sigma_b-\\sigma_{0})}{(1+R_{90})\\sigma_{90}^2} - \\cfrac{2 R_{0} \\sigma_b}{(1+R_{0})\\sigma_{0}^2} + \\cfrac{c}{\\sigma_{90}}\n \\end{align}\n ", "e2ae5f9dc278c1f277b9deeafd82d20d": " f(\\phi,\\psi) = \\int\\psi(x)\\phi(x) dx", "e2aeb35cc2a5d2fa3fa64f28d609ee3c": "\\mathbb{Z}^n \\oplus \\mathbb{Z}_{q_1} \\oplus \\cdots \\oplus \\mathbb{Z}_{q_t},", "e2aecf10148fa226ded4bb5211fcc8b6": "H = \\otimes _k H_k", "e2af259df4e753f9fbfc46b42110188c": " r \\ = \\ (1+i/n)^n - 1", "e2af419451e1054ea5ed295cf666a7c1": "\\mathcal{L}(\\psi;\\mathbb{X}) = p(\\mathbb{X}|\\psi) = \\int_\\lambda p(\\mathbb{X}|\\psi,\\lambda) \\, p(\\lambda|\\psi) \\ \\operatorname{d}\\!\\lambda ", "e2af7bbb871f85fa0d057f2ac6478ff7": "\\inf_{x \\in S_k, \\|x\\| = 1}(Ax,x) \\le \\lambda_k", "e2af8dfac010e4914d178cee3e9ffa19": "{J^{\\alpha}}_{,\\alpha} \\, \\stackrel{\\mathrm{def}}{=} \\, \\partial_{\\alpha} J^{\\alpha} \\, = \\, 0 \\,.", "e2afd96980327040cb4f64f10d96c819": "H(s)=\\frac{15}{s^3+6s^2+15s+15}.", "e2b0838dd4b3f1bf61e54deb10c08a7b": "MC = \\frac{I_{c,p}-I_{c,ap}}{I_{c,ap}}", "e2b0b09018afeeec7f9e1753f3539c51": " L = \\Phi \\, dt \\, dV ", "e2b0c11ead48f33d4612aaeea864e492": "\\tau_{ij}= \\epsilon_{ij}-\\epsilon_{ji}", "e2b1174f83004fe99b45062dd809ba88": "F(x) = \\int_a^x f(t)\\, dt", "e2b13b3b1400bbbb0b069b91b2c67188": "-p_1", "e2b18b41d93b8cad163801b12748de19": " \\lambda = \\frac{ k_1( 1 + \\rho ) + \\sqrt{ ( 1 - \\rho^2 )( k_1^2 + \\rho ) } }{ 2k_1 - 1 + \\rho } ", "e2b1f1ecc4b1f070193d68259d0e4592": "\\{\\vec{p}^{2}+\\Phi (\\vec{x})\\}\\Psi =b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\\Psi \\,,", "e2b24a39a04d3287221c7d3750dce128": "= \\operatorname{E}_X\\left[\\operatorname{Var}[Y\\mid X]\\right] + \\operatorname{Var}_X\\left[\\operatorname{E}[Y\\mid X]\\right]", "e2b2b95e389fa6c84e2163b0db292221": " W = W(I_1) ", "e2b345c62a81033ba21739c94a85a9fa": "MR_i = \\big | x_i - x_{i - 1} \\big | ", "e2b388a8759117a2d5a23b59e99d53b0": "f(x)-a^2(x)", "e2b39d6a96c9532d8bcae79fe733d066": "{n\\choose k_1,k_2}={n\\choose k_1, n-k_1}={n\\choose k_1}= {n\\choose k_2}.", "e2b412db778f966ca75b197247cd754c": "p_1 \\ne p_2", "e2b44f2723cf8d8832d7a5166d1fcc2d": " {\\rm tr}(t_at_b)= 2g_{ab}", "e2b4d2c9d7df874e1071bdd34cf7f2b6": "\\int_0^\\infty \\left[\\vartheta (z,it) -1 \\right] t^{s/2} \\frac{dt}{t}=\n\\pi^{-(1-s)/2} \\Gamma \\left( \\frac {1-s}{2} \\right)\n\\left[ \\zeta(1-s,z) + \\zeta(1-s,1-z) \\right]", "e2b5149a02566866805f56ffedc6aeb9": "\\partial_\\alpha \\partial^{[\\beta} A^{\\alpha]} = \\mu_0 J^\\beta", "e2b5572d50908e2154e8176cd00a0834": "\\tfrac{d y}{d t}", "e2b58095cb97a2d68b561c19b7f85864": "b_2 = \\frac{2(n^{2}+n+3)}{9n(n-1)}", "e2b5bc0ee5746e87e5f18ad8820f12d1": "(\\operatorname{adj}(A))_{i,j} = (-1)^{i+j} M_{j,i}.\\, ", "e2b5d6aa9dcc66eec217396ca4b4d591": "U \\subset \\mathbb{R}^n", "e2b6153e52f80909fcdefe991848bd65": "v \\approx \\frac { v_{\\mathrm N} + v_{\\mathrm F} } {2}\n= v_{\\mathrm F} + \\frac { v_{\\mathrm N} - v_{\\mathrm F} } {2} \\,.\n", "e2b615dd68586f2c8a18ae484e4fee5e": " \\frac{d}{dx} B_n(x) = nB_{n-1}(x).", "e2b62c0ca862a9081a4ed16518345bcb": "R_O", "e2b66b63f40a47d55d87ec5725b0fdb2": "Payout = Claim \\times \\frac {Sum\\ Insured} {Current\\ Value} = $3 \\mbox{M} \\times \\frac {$5 \\mbox{M}} {$10 \\mbox{M}} = $1.5 \\mbox{M} \\!", "e2b685687c7b2a3ac0f0dce7d8754c2c": "\\mathbf{H}_{\\text{Electric dipole}}(\\mathbf{x},t)=\\frac{c k^2}{4 \\pi}(\\mathbf{n}\\times\\mathbf{p})\\frac{e^{i k r - i \\omega t}}{r}", "e2b69ba4b9e49f2e1e5cce7f36a1a716": "K_0(\\mathrm{Vect}_{\\mathrm{fin}})", "e2b6a4c1c8854b69cd314027b92f0ebc": "(\\mathbb Z, +)", "e2b6a5418369bdda58e4b9223da28b43": "\\,\\Pi\\,", "e2b6b6532cdd0096b806a5f7a360a4b4": "x = \\sqrt[n]{a}\\cdot e^{i 2\\pi k/n},\\quad k=0,1,\\dots,n-1.\\ ", "e2b75529a7bc42a4e648b25b626f106c": "\\partial \\psi_i(\\boldsymbol{\\theta})/\\partial \\theta_j", "e2b768902073453d636d85146185ccb9": "adx+bdy+cdz \\mapsto F(x,y,z,-a/c,-b/c).", "e2b7bd074f9bf79c7157dca5775c3654": "R = 25 - 3 \\cdot \\frac{25}{3} = 0", "e2b7bff52c706158d5acfeed59881bcd": "T_{2B}", "e2b7f89324ef2bf6ce0e47f6be91f7e2": "\\ F \\ll 1 ", "e2b836e78baf10c0202aea2f73a44d43": "{\\zeta_X(s)} = \\prod_{x} ({1 - N(x) ^{-s} })^{-1}", "e2b851172c4ac3fd0c227f049e6fd72b": "z \\mapsto k z \\, ", "e2b8587c3156fd8f12a80f9edfc80a40": "[M + nH]^{n+} + e^- \\to \\bigg[ [M + (n-1)H]^{(n-1)+} \\bigg]^* \\to fragments", "e2b952886f1a9bd303900b9293b3bf9c": "\n\\tilde{\\boldsymbol{q}}_{\\boldsymbol{k}}(\\boldsymbol{x},t) =\n\\left(\\begin{array}{c}\n\\tilde{u}(t)\\\\\\tilde{v}(t)\\end{array}\\right) e^{i\n\\boldsymbol{k} \\cdot \\boldsymbol{x}} ", "e2b9648c42beda108547b9b564f07037": "\\overline{\\phi}: R[t] \\to S", "e2b971f8f05f3ee32b66fc8039217e10": "\\begin{align}\n\\sigma & = \\sqrt{\\operatorname E[(X - \\mu)^2]}\\\\\n& =\\sqrt{\\operatorname E[X^2] + \\operatorname E[(-2 \\mu X)] + \\operatorname E[\\mu^2]}\n=\\sqrt{\\operatorname E[X^2] -2 \\mu \\operatorname E[X] + \\mu^2}\\\\\n&=\\sqrt{\\operatorname E[X^2] -2 \\mu^2 + \\mu^2}\n=\\sqrt{\\operatorname E[X^2] - \\mu^2}\\\\\n& =\\sqrt{\\operatorname E[X^2]-(\\operatorname E[X])^2}\n\\end{align}", "e2b99dbe5e5b659c32821bd4f044228e": "d\\!", "e2b9cd212b9e254f65c863ba027f4fd1": "\\frac{\\partial F}{\\partial n_2}= \\sigma_2^2-2\\sigma_2\\sigma_\\mathrm{n}+\\sigma_\\mathrm{n}^2-\\tau_\\mathrm{n}^2 = 0 \\,\\!", "e2b9e359977439280141ea005b962652": "\n\\begin{align}\n&\\Delta \\bar{e}\\ =\\frac {J_3}{\\mu\\ p^3}\\ \\sin i\\ \\cdot \\\\ \n&\\int\\limits_{0}^{2\\pi}\\left(-\\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ 2 \\ \\sin u\\,\\ \\left(5\\sin^2 i \\ \\sin^2 u\\ -\\ 3\\right)\\ - \\ \\left(2\\ \\hat{r}-\\frac{V_r}{V_t}\\ \\hat{t}\\right)\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\frac{3}{2}\\ \\left(5\\ \\sin^2 i \\ \\sin^2 u\\ -1\\right) \\ \\cos u\\right)du = \\\\\n&2\\pi\\ \\frac {J_3}{\\mu\\ p^3}\\ \\sin i\\ \\frac{3}{2}\\ \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)\\left((1-{e_g}^2\\ +\\ 4\\ {e_h}^2)\\hat{g}\\ -\\ 5\\ e_g\\ e_h\\ \\hat{h}\\right)\n\\end{align}\n", "e2b9ebebf3ae4992e1611573a66ff00f": "EG\\rightarrow BG", "e2ba0988b86c8551cc93b8b6c48688bc": "\\begin{matrix} {12 \\choose 1}{4 \\choose 4} \\end{matrix}", "e2ba0ee5f6f2c33321a73b8c2f86da47": " f^*=f, \\, ", "e2ba1b68ce44f2bb523e4c1f58ca0c42": "\\eta\\,", "e2ba2092aaaec1ccdb535801d24bc60e": "T_{\\rm c}", "e2ba3b4dfc263b741e72ea3609ced62e": "\n \\cfrac{\\partial\\lambda_i}{\\partial\\mathbf{C}} =\n \\cfrac{1}{2\\lambda_i}~\\mathbf{N}_i\\otimes\\mathbf{N}_i =\n \\cfrac{1}{2\\lambda_i}~\\mathbf{R}^T~(\\mathbf{n}_i\\otimes\\mathbf{n}_i)~\\mathbf{R} ~;~~ i=1,2,3\n \\,\\!", "e2ba5e1a3bafcbc4f2a904fa9fff9c7a": "L.L' = 1", "e2ba8a17855620fbe286c1495415e898": "N(H)", "e2ba912de7075df1c5050d506070abd6": " X + G_1^{-1} G_2 X= G_1^{-1} Y ", "e2bac8c9a1c380dbebc575dc127f5e25": "(W_i,W_j)", "e2bb0c96320701a0ed5070cfdc67cb17": "\\boldsymbol \\tfrac{1}{\\lambda} \\boldsymbol\\Sigma", "e2bb11758b766a3ac9e02790656fc584": "\\left[ \\begin{array}{cccc} 0&\\color{red}{\\mathbf{2}}&1&-1 \\\\ 0&0&\\color{red}{\\mathbf{3}}&1 \\\\ 0&0&0&0 \\end{array} \\right]", "e2bbbaf24ed879b680c5f6c1d6f5c2e7": "\\hat{d}=r_a \\cdot \\hat{r} + \\cos \\alpha \\cdot ( \\hat{a} - r_a \\cdot \\hat{r})+\n \\sin \\alpha \\cdot \\hat{r} \\times \\hat{a}", "e2bbe315f6556ef67c00aaa58c39d946": "\\cos\\phi_0 = -b/r", "e2bbe8fb3f85f196430f1f94d63b184e": "P=\\frac{2 K e^2}{3 c^2} a^2", "e2bc007c6c6f83c64fb6aa4616a814ce": "\n\\left(\\frac{d\\mathbf{L}}{dt}\\right)_\\mathrm{rot}+\n\\boldsymbol\\omega\\times\\mathbf{L}=\\mathbf{M}\n", "e2bc5a2cd0cf40eacf6a04a2678008ec": "\n\\left(\\frac{\\alpha}{\\mathfrak{p} }\\right)_n= 1 \\mbox{ if and only if there is an } \\eta \\in\\mathcal{O}_k\\;\\;\\mbox{ such that } \\;\\;\\alpha\\equiv\\eta^n\\pmod{\\mathfrak{p}}.", "e2bc60960ebac3e512f7d6b8c7f9dfb3": "\\bar{F}", "e2bca8784d4743357f2c218e6a9e3660": "e+n=k", "e2bcf8f56c3391eb853284e7650b9f4d": "g_{\\mu \\nu} ", "e2bd04b8b30fbc8d63d3712c1a451d0a": "\\frac{32}{27} \\sqrt[3]{2}", "e2bd23a4c32aa06dceefa1eb992f7216": "p_5(x)=x^2-2\\,=(x-\\sqrt{2})(x+\\sqrt{2})", "e2bd2420911924d223b24f3b5afe34d2": "G:[0,\\infty) \\rightarrow [0,\\infty)", "e2bd77c9896ac16f4aa40ef66cff182f": "\\textstyle {23 \\choose 2} = \\frac{23 \\cdot 22}{2} = 253", "e2bda64f63d9daa93db29944a9116a7a": "\\begin{align}\n\\int_{L_2} \\left ( - c^2 u_x(x,t) dt - u_t(x,t) dx \\right ) &= - \\int_{L_2} \\left ( c u_x(x,t) dx + c u_t(x,t) dt \\right )\\\\\n&= - c \\int_{L_2} d u(x,t) \\\\\n&= c u(x_i,t_i) - c f(x_i - c t_i).\n\\end{align}", "e2bda699629c13da9a5bf81ef55c4ac5": "W = \\int_C \\mathbf{F} \\cdot d\\mathbf{x} = Fd\\cos\\theta.", "e2bdfc1e1a14d6b3cc996701050ceec1": "H\\left(x,y\\right)=\\begin{cases}\n1 & \\text{if }x^{2}+y^{2}\\leq1\\\\\n0 & \\text{else}\n\\end{cases}", "e2be069b10594a09b07ab51a436818cb": "A=BC\\,\\!", "e2be2fd47d0f077a38ace46688951c62": "e_{i+1} = 0", "e2be4a89db8be8c82067585d13f339a1": "\\scriptstyle p \\,\\to\\, \\infty", "e2be5f3bd6825526c2970bcf50da4b3d": "k_{col}", "e2be76d32d86dea63430941f6a914001": "j_A : I \\to \\left[A\\ A\\right].\\, ", "e2bedbf72ebb46f04edafa1440b6c84b": "p \\rightarrow e + \\pi^0", "e2bf02a70dc3bea22f15d9a7a6fbe92a": " S_E=V_E/N_E ", "e2bf38b0417438d8cb662ba6ec1c23fd": "\\frac{\\partial \\mathcal{G}}{\\partial N_j}=0", "e2bf50e21cfc29eed8ca255ba0028e6d": "d^\\circ(P)", "e2bfe9c92f744a3177ccfc1f9fd60748": "C_xH_y + zO_2 \\to aCO_2 + bCO + cH_2O + dH_2", "e2c0127195be8a3984715cb427f9a91a": "\\mathbf{x}_{ub}", "e2c0187423a6734610852b44bd364e51": "x^{(k)}.", "e2c02876d4e86d503ff71821f3460cd4": "\nc \\,F_1(a,b_1,b_2,c; x,y) + c(y-1) F_1(a,b_1,b_2+1,c; x,y) - (c-a)y \\,F_1(a,b_1,b_2+1,c+1; x,y) = 0 ~.\n", "e2c07452fcd2161747afb73513589716": "\\frac{P(x\\rightarrow x')}{P(x'\\rightarrow x)} = \\frac{P(x')}{P(x)}", "e2c0b9c5700adadfbe4f41e4ce259af3": "E_0=mc^2", "e2c0c027792cf846f4a28cb11da2dd8d": "X(t)=c_{on}", "e2c0dc0434311eab9290a7de65223770": "\\left( k_1 + k_2 + k_3 + k_4 \\right)^2 = 2\\left( k_1^2 + k_2^2 + k_3^2 + k_4^2 \\right),", "e2c0f70b5589184d258eb04c64f92748": " \\mathbf{v_i} ", "e2c13492ea6fe532f1068fd58e5bbb7e": "\\begin{bmatrix}\n2 & 0\\\\\n0 & -2 \\\\\n\\end{bmatrix}\n", "e2c17396f30408ba973361cfd559d932": "(\\mathbb{Z}_2 \\times \\mathbb{Z}_3, +)", "e2c176e1172b217d8b9da3085f0452ce": "I_{2lm}", "e2c1bd89c4bb8023e280b259b62a92e3": " 16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2 ", "e2c1cb20d4ca2bc68d5d7afc398359d7": "\\delta(-1,i) = \\delta_D(-1,i) = 0\\,\\!", "e2c1f30f10f805c1ecd1913596541c03": "\\begin{align}\n\\begin{bmatrix}v_1\\\\ v_2\\end{bmatrix} &= R^{-1}\\begin{bmatrix}v^1\\\\ v^2\\end{bmatrix} \\\\\n&= \\begin{bmatrix}4&\\sqrt{2}\\\\ \\sqrt{2}&1\\end{bmatrix}\\begin{bmatrix}v^1\\\\ v^2\\end{bmatrix} = \\begin{bmatrix}6+2\\sqrt{2}\\\\2+3/\\sqrt{2}\\end{bmatrix}\\end{align}.", "e2c1fa6c4af83d4735f5d0173c5b7eef": "\\Bigl|\\int_S fg\\,\\mathrm{d}\\mu\\Bigl|\\le\\int_S|fg|\\,\\mathrm{d}\\mu\\le\\|f\\|_p\\,,", "e2c2808b61007d5df60292b46cb93577": "l(b - 1)", "e2c2a1e871a7edf7671a1fed2d3f6dbd": "\\scriptstyle \\boldsymbol\\varphi", "e2c2aee4747574ad34af5d8e2840607c": "\\omega_1", "e2c2ce40aad962a5253c933f1b080544": "\\scriptstyle\\sqrt{8/k}\\,", "e2c2ddee42042da7e457d378016cf313": "Y=\\frac{1}{2}AB\\cos(\\boldsymbol\\theta +\\frac{\\pi}{2})=\\frac{1}{2}AB\\sin(\\boldsymbol\\theta) ", "e2c331efabad6e3a1eff58e5499288a3": "p \\ll k", "e2c33764b268dbf384fdede79e3cc43a": " I", "e2c345a3e7dc3581b9d29ad0730b72bf": "1-q ", "e2c3c89826bab26d68fe67ee421be0c0": "|\\psi(0)\\rang", "e2c3d4787bd96254d66f497d79454610": "\n \\color{Gray}\n \\Bigl( \\frac{\\partial}{\\partial t}\n \\color{Gray}\n + \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\Bigr)\\,\n \\left( \\frac{\\partial\\Phi}{\\partial t} + \\tfrac12\\, |\\mathbf{u}|^2 + g\\, \\eta \\right) \n = 0\n", "e2c3d58adbfbd8883fb5a8306921564d": "\\sigma_{max} = \\sigma\\left(1+2\\cfrac{a}{b}\\right) = \\sigma\\left(1+2\\sqrt{\\cfrac{a}{\\rho}}\\right) ", "e2c3e3d7a29ae49a2cd1beadc1be166c": "(27)\\quad \\theta_{(\\ell)}=\\hat{h}^{ba}\\nabla_a l_b=m^b\\bar m^a\\nabla_a l_b+\\bar m^b m^a\\nabla_a l_b =m^b\\bar \\delta l_b+\\bar m^b \\delta l_b=-(\\rho+\\bar\\rho)\\,,", "e2c463ea2e8d757099c2bba2be5f7af1": "k_r^+>0", "e2c4f6394a6b2ff9385ba054f032d7f0": "get(k)", "e2c54e4196da55b059010db68377ce47": "\\ g_m R_L ", "e2c5723768e641f88fc9dc8200dc7aa8": "O_{X,x}", "e2c5b5122ef6ff432c3a358444dda49f": " \\delta(\\bold r)", "e2c5bbae7057c1fe9c262e56ac3edce1": "\\; = -\\sum_{x,y} p(x,y) \\log p(x) p(y) .", "e2c5bfb324d0057368b628db948c0a6d": "\\frac{V_{\\,d}}{V_{\\,t}} = \\frac {P_{\\,a\\,CO_2} - P_{\\,e\\,CO_2}} {P_{\\,a\\,CO_2}}", "e2c5e37d429f057b15ac5a2c14b38100": "\\sqrt{2-\\sqrt 3}", "e2c61f8cb59ae4670958a05b0fde76c8": "(\\dot{a})", "e2c63736f2afc2c5a80e6669f47e8588": "1 + \\cot^2 A = \\csc^2 A \\ ", "e2c6611a4f2bbf3151cae504e56ceb04": " k_{\\mathrm{H,cc}} = \\frac{c_{\\mathrm{aq}}}{c_{\\mathrm{gas}}} ", "e2c672b00ee5deeb2a93ab0730527b69": "V({\\mathbf{X'}},z)", "e2c67a8335da764b734ed4df03030369": "f''(x)=-\\lambda f(x)", "e2c6d1117dba2c9ea79ad6afda31e1f0": "\\ \\mathcal{F}\\{g\\}", "e2c71632c3bc8f0a51946fcdab4e45da": "\\displaystyle u_t = \\alpha u_{xx}", "e2c7420a617705f374dec08c0b290ab6": " f(t)= \\sum_{n=0}^{\\infty}\\frac{a_{n}}{M(n+1)}x^{n} ", "e2c76f23c045b3a4e63ee9fa3aa4f062": " \\gamma (v \\otimes w) = w \\otimes v ", "e2c7b3f88770490ba8908f597607f040": "\\displaystyle L_2 \\phi = L_3( | u |^2)", "e2c8d0d0cefa42259c55666984c3a9b7": "S_n(R) = S_n R^n", "e2c8edd97850b979156a3ba602e9bbc6": "-1.4110", "e2c911895e53efa701d3769365a98591": " Q = \\frac {4 \\pi \\sin \\left ( \\theta \\right )}{\\lambda} ", "e2c97395fed352658dd22114257e0de3": "\n f(y;\\alpha, \\lambda, y_0 ) = \\frac{n}{\\cosh[\\alpha(y - y_0)] + \\lambda}, \\qquad -\\infty < y < \\infty,\n", "e2c9b57a943ac4ba55349165df27803e": "\\mathbf{u} \\oplus_U \\mathbf{v}=\\mathbf{u}+\\mathbf{v}+\\left\\{ {\\frac{\\beta_\\mathbf{u}}{1+\\beta_\\mathbf{u}}}{\\frac{\\mathbf{u}\\cdot\\mathbf{v}}{c^2}} + {\\frac{1 - \\beta_\\mathbf{v}}{\\beta_\\mathbf{v}}} \\right\\} \\mathbf{u} ", "e2c9c3e22805a45e5ea8387d57e7a10a": "S_{n} \\rightarrow S, \\partial S_{n} \\rightarrow \\Gamma", "e2ca2177c788e358cbd74f3bdfddc9da": " X \\times Y \\subseteq \\mathcal{P}(\\mathcal{P}(X \\cup Y)). ", "e2ca57c4cd6287e8bb84af0f3530b042": "Z^'", "e2cabe0346b3461171d1837b2147c051": "\\sqrt{\\sigma_L^P}\\over \\sqrt{\\sigma_L^D}", "e2cb43033e76f7e2856b07affdf5d37e": "\\begin{align}\n\\overbrace{\\int_{-\\infty}^{\\infty} h(\\tau) \\, A e^{s (t - \\tau)} \\, \\operatorname{d} \\tau}^{\\mathcal{H} f}\n&= \\int_{-\\infty}^{\\infty} h(\\tau) \\, A e^{s t} e^{-s \\tau} \\, \\operatorname{d} \\tau\n&= A e^{s t} \\int_{-\\infty}^{\\infty} h(\\tau) \\, e^{-s \\tau} \\, \\operatorname{d} \\tau\\\\\n&= \\overbrace{\\underbrace{A e^{s t}}_{\\text{Input}}}^{f} \\overbrace{\\underbrace{H(s)}_{\\text{Scalar}}}^{\\lambda},\n\\end{align}", "e2cb55d84ae3a25f45517f9aed2bf833": "\\mathcal{H}_{S}", "e2ccb998e576e42ee43283589704b698": "\\sqrt{2+a}=2\\cos\\tfrac\\phi2", "e2ccd1bd4d16de95253ff3880a41ebe3": "1 - L / G", "e2ccd26a10669b36fc66aa6b2d4fc47a": "\\theta(z;q)", "e2cd37b851dfba7b283a52c0da939815": "k = {2 \\pi}/{\\lambda}", "e2cd41eaafde190b6dbaf6de3f741912": "G(n + 1) = f(G(n))", "e2cd46f08b954b94442f5b9c62cbc0b2": "\\xi_1<\\sqrt{2}\\lambda<\\xi_2", "e2cd55b0ce0738d298914c999d54e753": "\\alpha_{crit} \\,\\!", "e2cd791906e64a5b1a062b1152ce0773": " Y_i = X'_i \\beta + g\\left(Z_i \\right) + u_i, \\, \\quad i = 1,\\ldots,n, \\, ", "e2cdce335860e5d4e11f95311eb11f68": "\\ W_{n + 1} \\sim W_n", "e2ce10d4d1a70cbb186512ce57205129": "P(x) \\lor (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\forall{y}{\\in}\\mathbf{Y}\\, (P(x) \\lor Q(y))", "e2ce1badb468ce8c7a7b0cf2676a3d0e": "m_{em} =(4/3)E_{em} /c^2", "e2ce4bb0d88879e4c89fecbc6c1df55d": " c_1 ... c_K ", "e2ce53ae5c69d64f2931534cab90c71d": "\n\\int x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n -\\frac{(m-3 n-2 n\\,p+1) x^{m-2n+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{2 c\\,n^2(p+1)(2p+1)}\\,-\\,\n \\frac{ x^{m-2n+1} \\left(2 a+b\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{2 c\\,n(2p+1)}\\,+\\,\n \\frac{(m-n+1)(m-2n+1)}{2 c\\,n^2(p+1)(2p+1)} \\int x^{m-2n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}dx\n", "e2ce9e43658a1bd9dba60936b731cdbc": "ps_i = a_i \\oplus b_i \\oplus c_i", "e2cef0cef1c9b6179691b3edc513b1ec": "Q=N_k.", "e2cefe4337beb01a394e4d2b37917b3b": "h(t) \\sim h_0 - 350 \\sqrt{t}", "e2cf11ba75d20976b2f0ca8a8e8aeed9": "R_i\\,R_j = R_{i+j},\\;\\;\\;\\;R_i\\,S_j = S_{i+j},\\;\\;\\;\\;S_i\\,R_j = S_{i-j},\\;\\;\\;\\;S_i\\,S_j = R_{i-j}.", "e2cf5130a5a81bb644dc38afe2530eea": "\\varepsilon(\\phi \\otimes x) = \\phi(x).", "e2cfd0f4b1527a197d7a8177b19e29f0": " \\mathbf{v} = q^i \\mathbf{e}_i = q^1 \\mathbf{e}_1 + q^2 \\mathbf{e}_2 + q^3 \\mathbf{e}_3. \\, ", "e2d00a4fe18380c61d2d8e8d084598c4": " \\tau_{\\alpha, \\beta}", "e2d029658e781de5139821b215d86f91": "\n V_1 = -10 \\quad \\text{and} \\quad M_1 = -10x \\,.\n ", "e2d04ba1cddcc246758883752b748ead": "\\tfrac{dy}{dx}", "e2d06062c71feae1a1cf418d1fb7ffa3": "(d_1, d_2, 1- (1 - p_1)^r, 1 - (1 - p_2)^r)", "e2d08c9e5ec120697688f2001c531aa3": "{\\mathit{He}}_{10}(x)=x^{10}-45x^8+630x^6-3150x^4+4725x^2-945\\,", "e2d0b668bf2411425dc9400cd85e5e37": "D_2 = \\lim_{\\epsilon \\rightarrow 0, M \\rightarrow \\infty} \\frac{\\log (g_\\epsilon / M^2)}{\\log \\epsilon}", "e2d0c6d1adabef63db9f0dcad5079c87": "(\\mathrm{Tor}(G))_p = \\oplus_{i \\in I_p} \\mathbb Z[p^\\infty] = \\mathbb Z[p^\\infty]^{(I_p)},", "e2d0e9b336bb522be1b86c6daced9ee5": " S_i^-", "e2d0f80864e9802cfe9f84b25fd368c9": "\\hat{h}_E(Q) \\ge \\frac{C(E/K)}{D} \\left(\\frac{\\log\\log D}{\\log D}\\right)^3 ,", "e2d118f5691f91630ef93cdc07248df9": "H^{k}_{\\mathrm{dR}}(M)", "e2d11c2c76ec3a744cd4ddf929af55ca": "a(0) = 1", "e2d11df7d3a4030e09cbaa83d8a88d74": "\\sum|M_i|=\\sum |L_i|(1+\\frac12+\\frac14+\\cdots)\\leq 2n=O(n),", "e2d15cf5eb34cdcb430a784f21de6be6": "u_{xx} =xu_{yy}.", "e2d1744dc6fae650b7ed66f562e147a2": " \\left(\\frac{Q_1}{Q_2}\\right) = ", "e2d17f8d8d4fa9587ba00640ef6fd650": "X \\rightarrow Z \\leftarrow Y", "e2d1e5e8181455794220433cd495f0eb": "t \\mapsto t^p", "e2d26e319058cc5914908ab18971c081": "-\\frac{\\partial^{2}S_{l}(u)}{\\partial u^{2}}+\\left(\\frac{l(l+1)}{u^{2}}+\\frac{1}{4}ku^{2}+\\frac{1}{u}\\right)S_{l}(u) = E_{l}S_{l}(u).", "e2d274dec61b95447337e44c7fe8a8d9": "r_{444}", "e2d2b63831d8db63f96e41d272004379": "1, 2, 3, \\dots, \\frac{p-1}{2}.", "e2d2c7aa72f494dba29d8904f438035a": "\\mbox{cond}(A)=\\|A\\| \\|A^+\\|.\\ ", "e2d2ea768eb2fbb06e99ed03ccfdc7c6": " \\nu(\\varepsilon) = \\nu_0 + \\delta_\\nu \\varepsilon(t) ", "e2d30c1cc6d0db6d016c4bd9aa065155": "y = \\frac{x^2}{\\frac{1}{5}x^5 + C}", "e2d34e64f1619c13dedca836e96f6356": "x_0=\\infty", "e2d35b3c6c8cad7394d1f1d5f982ea39": "(2-\\eta)^3= 7(\\eta-1)^2,\\ ", "e2d370333f97eebd1c8494771ed3652d": " 5 > \\beta \\ge 3", "e2d37aaaafadb3bcfd84b604e1410063": "g(x) \\equiv f(\\sqrt{x}) \\,\\;", "e2d39c445e35d114d0317d1223dae178": "\\lang 2,0 \\rang", "e2d3bab8399e8838d7cf9e15084a71e6": "\\frac{p}{p_c} < \\frac{T}{2T_c}.", "e2d4953482ed3714730e407e6d7674f6": "\\mathbf{r}_3", "e2d5163048ed10496950d943ba3f2b39": "\\sqrt{\\frac{g}{\\gamma}}", "e2d556b925826f819ce8a4e1b873788d": " R_s ", "e2d58bf191f4792638a92134012ee8b6": "a_n=4n+1\\, .", "e2d5bd67aa34e8e686d78818dd091085": "2 \\tau", "e2d5c8d0fb11638fb52b93c6187e7460": " \\mathbf {a}", "e2d627aec390436845d63f54e36129c0": "S_{a/b}", "e2d69e403c714ce173fe050914b48a21": "x_3 = \\frac{1}{2} \\left(x_2 + \\frac{S}{x_2}\\right) = \\frac{1}{2} \\left(357.187 + \\frac{125348}{357.187}\\right) = 354.059.", "e2d749cdcd48e2acdbc13b17a2ca0ece": "\\mathbf{r}\\rightarrow -\\mathbf{r}", "e2d75604774d1d6dcad508e9d15fa7a2": " I = I_\\mathrm{cm} + mr^2.", "e2d82e2cb2916df07c7ddd399a9b83c6": "S_k(r) =\n\\begin{cases}\n\\sqrt{k}^{\\,-1} \\sin (r \\sqrt{k}), &k > 0 \\\\\nr, &k = 0 \\\\\n\\sqrt{|k|}^{\\,-1} \\sinh (r \\sqrt{|k|}), &k < 0.\n\\end{cases}\n", "e2d84ef3a57e3a2211f5bd09b0447e8f": "\\det\\begin{pmatrix}\n\\Phi_{1,1}&\\Phi_{1,2}&\\cdots&\\Phi_{1,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi'_{i,1}&\\Phi'_{i,2}&\\cdots&\\Phi'_{i,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi_{n,1}&\\Phi_{n,2}&\\cdots&\\Phi_{n,n}\n\\end{pmatrix}\n=\\det\\begin{pmatrix}\n\\Phi_{1,1}&\\Phi_{1,2}&\\cdots&\\Phi_{1,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\na_{i,i}\\Phi_{i,1}&a_{i,i}\\Phi_{i,2}&\\cdots&a_{i,i}\\Phi_{i,n}\\\\\n\\vdots&\\vdots&&\\vdots\\\\\n\\Phi_{n,1}&\\Phi_{n,2}&\\cdots&\\Phi_{n,n}\n\\end{pmatrix}\n=a_{i,i}\\det\\Phi", "e2d8ac1fb4a3de09db12df10d4f8b12f": "\\partial_l", "e2d8f97086a038f423f996b32f2c53d9": "= 2 \\eta^{\\nu \\sigma} \\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\rho \\right) - \\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\sigma \\gamma^\\nu \\gamma^\\rho \\right) \\quad \\quad (2) \\,", "e2d91cf95aae8fc682c76b5f90496c46": "z = x + iy,", "e2d9235101f694f00abc4f1910c7d3db": "L = \\psi_{,\\nu} \\psi^{*}_{,\\mu} \\eta^{\\nu \\mu} + m^2 \\psi \\psi^{*}.", "e2d946d194ffa10a87c447d350642cbb": "\\bar{t}\\pmod l", "e2d96531ce1650967d16270a8f2f1f71": " w_{ij}=\\frac{1}{n}\\sum_{\\mu=1}^{n}\\epsilon_{i}^\\mu \\epsilon_{j}^\\mu ", "e2d9de092509f12748d30babe03d61e5": "\\chi(\\omega) = R(\\omega)R^*(\\omega+\\omega_\\mathrm{m})-R^*(\\omega)R(\\omega-\\omega_\\mathrm{m}) .", "e2da168e632c562f83dd038d653493f9": " \\mathcal{W} = \\int_S H^2 \\, dA - \\int_S K \\, dA", "e2da5850400017ec6f62f02585aa4310": "\\Delta := \n\\begin{pmatrix}\n\\delta_{1,1} & \\delta_{1,2} & \\cdots & \\delta_{1,I} \\\\\n\\delta_{2,1} & \\delta_{2,2} & \\cdots & \\delta_{2,I} \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n\\delta_{I,1} & \\delta_{I,2} & \\cdots & \\delta_{I,I}\n\\end{pmatrix}.\n", "e2dac462019439d571710f3a2b85d039": "N_{BOC}=2m/n", "e2db5b4ed68857958dbc1e7ec060ea11": "\\sigma(\\mathbf{x})=\\mathbf{0}", "e2db7255ac61937cd62f7111fd30c610": "\nH_K= \\frac{(l_1)^2+(l_2)^2+ 2 (l_3)^2}{2I}+ mgh n_1.\n", "e2dbb47d77a6b745dcfd1acb9b331d40": "n = 1\\,", "e2dbc60e27b13ba0248733da74d98280": "\\operatorname{rank}[\\lambda \\mathbf{I}-\\mathbf{A},\\mathbf{B}]=n", "e2dbd4b26d758137070f5b0edfc107d6": "n\\geq 0", "e2dbeadfbc4d33d972f612f7dd215407": "\\sum_1^k \\left(\\frac{X_i}{\\sigma_i}\\right)^2", "e2dccf91c02f916bd4cb1eaba43afc5d": "r_{0} \\,", "e2dd2c4bd51f49509033f14ae2de3889": "\\sum_{i=1}^m", "e2dd2f090da84831008988768a0d60bc": "\\displaystyle{J_f(a)=|\\partial_zF_f(a)|^2 - |\\partial_{\\overline{z}} F_f(a)|^2.}", "e2dd73805e7f2c75e25d93ee9a8fa643": " \\sum_{j=0}^n (-1)^j\\tbinom n j P(j) = 0.", "e2dd7ee848a704b1fcfcc2e82fe5cf2c": "\\zeta(-n,x)=-{B_{n+1}(x) \\over n+1}.", "e2ddbba54bdc11a48f8a5a45623d4e2c": " 2 \\cdot 4^2 + 2 \\cdot 4 + 1 ", "e2dde8bb4f033474ec68d86031343d07": "\\int_K f(gk\\cdot y)\\,dk", "e2de039328f4c717918c1f5d2739a0fc": "y = u. \\,", "e2de6c2bac4038e65dceed2848e97675": " + \\frac {l_w} {2} ", "e2de7b95775dc63b5d49fecf8208f122": "\\mathfrak{u}(n)", "e2de81452abf87cba35f6290e0e00d28": "\\frac12 \\log_2 \\big( 2\\pi e\\, np(1-p) \\big) + O \\left( \\frac{1}{n} \\right)", "e2deac3ff29d1b93a210919ea152b7e8": "\\lambda_1 = 1,\\quad \\lambda_2 = 9", "e2deb02cf10b5b23da2b22eb6be2d4c4": "I(\\theta ) = \\frac{{I_0 N\\Delta V}} {{(kr)^2}}i(\\theta )", "e2deca1d20531bcbedb54a35a7e3a25f": "X_0 + Y_0 C_c = X_1 + Y_1 C_c", "e2df09ad0b5bd1a3baa7d1cc86a53d6f": "\\displaystyle{B(c+b,a)=B(c,a^b)B(b,a)}", "e2df0cbc4a624d0992c9713de4808cbe": "\\mu=0.122565-0.744864i", "e2df21441d6980ad9e8476628b94167a": "\\sum_{k=-a}^a(-1)^k{a+b\\choose a+k} {b+c\\choose b+k}{c+a\\choose c+k} = \\frac{(a+b+c)!}{a!b!c!}", "e2df5dd7e171f175543db2805a897536": "\\delta^{(3)}", "e2dfa6cf946b28536bc15c8e7d83da00": " T \\equiv {T^{\\alpha}}_{\\alpha} = -u_{\\alpha} u^{\\alpha} T = -u^{\\alpha} T \\eta_{\\alpha \\beta} u^{\\beta} \\rightarrow - T_{00}", "e2dffd9da72da221313d9e67fe84daa6": "r = \\lim_{n \\to \\infty} H(X_n|X_{n-1},X_{n-2},X_{n-3}, \\ldots);", "e2e01954299ca020cbfa5a921d1c5138": "g(a_1, a_2) = a_1a_2-a_1-a_2", "e2e0240d23280bc2cc219180b55e1b34": "F= F_1(q, Q, t) \\,\\!", "e2e040b9c69b7c3ee4c4dc69759fe2d4": "w * \\lambda = \\lambda", "e2e0b194db8a0b981fa0926dc71b3eab": "\\overline{z-w} = \\overline{z} - \\overline{w} \\!\\ ", "e2e0ccc84e4582434af8f6a219854373": "c_p/c_v", "e2e0f7a4da872b4d5bc801a7b111b4e2": "\\phi_\\epsilon", "e2e115bb31975e87271843218fea32b3": " \\lambda_1,\\lambda_2 ", "e2e116f09f9b020a151194ea2911f329": "R_\\text{q}", "e2e15cf61384601f3d36107409d483bc": "\n\\begin{array}{lcl}\n\\sigma_{\\mathrm{f}}&=&M_{\\mathrm{f}}\\bar{\\sigma} + A_{\\mathrm{f}}\\Delta T\\\\\n\\sigma_{\\mathrm{m}}&=&M_{\\mathrm{m}}\\bar{\\sigma} + A_{\\mathrm{m}}\\Delta T\\\\\nt_{\\mathrm{i}}&=&M_{\\mathrm{i}}\\bar{\\sigma} + A_{\\mathrm{i}}\\Delta T\n\\end{array} ", "e2e165e436eb19de2a099948f8bc99f2": "y \\in \\Sigma^{n}", "e2e17fa0bffeed85fefaa87f947e88fb": "\n\\left\\langle \\Psi _m|x|\\Psi _n\\right\\rangle =\\frac{2(-1)^{m-n+1}}{(m-n)(2N-n-m)} \\sqrt{\\frac{(N-n)(N-m)\\Gamma (2N-m+1)m!}{\\Gamma (2N-n+1)n!}}.\n", "e2e1a8c4b7dcaf61ffbb773d8b91d739": "0\\to \\Lambda^1(U)\\wedge\\Lambda(V) \\to \\Lambda(V)\\rightarrow \\Lambda(W)\\rightarrow 0", "e2e1e3985bb85dc3041bf5c411740080": "\\frac{V_1/k_1}{V_2/k_2} \\sim F'_{k_1,k_2}(\\lambda)", "e2e1ed375f32afb97b318df3a87ed1aa": "R(x) = e^{\\int \\frac{L(x)}{Q(x)}\\,dx}\\,", "e2e2285564768a6ece35baa6260cea93": "\\sum_{k=1}^n\\varphi(k) = \\frac{1}{2}\\left(1+ \\sum_{k=1}^n \\mu(k)\\left\\lfloor\\frac{n}{k}\\right\\rfloor^2\\right)", "e2e2b63f4bc28d693d731ef3f87c918e": "{\\color{Blue}~2.17}", "e2e2e4e3c0463c91f35fd4e3e8e0960e": "V = \\frac{h}{3}(B_1+\\sqrt{B_1 B_2}+B_2)", "e2e31ecb8b2ede28524e8e502de64af9": "\\mathbf{e}_{31}", "e2e35ccdd6bbe4922347bed5a5564446": "\n\\mathcal{I}_{m,n}\n=\n\\frac{\\partial \\mu^\\mathrm{T}}{\\partial \\theta_m}\n\\Sigma^{-1}\n\\frac{\\partial \\mu}{\\partial \\theta_n}\n+\n\\frac{1}{2}\n\\operatorname{tr}\n\\left(\n \\Sigma^{-1}\n \\frac{\\partial \\Sigma}{\\partial \\theta_m}\n \\Sigma^{-1}\n \\frac{\\partial \\Sigma}{\\partial \\theta_n}\n\\right),\n", "e2e35e65178eee8c1453ba2ed0c42b0b": "4\\sqrt 3", "e2e37e3fdc1ee794f44a00866e4582bc": "\n \\cfrac{\\partial \\lambda_i}{\\partial\\boldsymbol{C}} = \\cfrac{1}{2\\lambda_i}~\\boldsymbol{R}^T\\cdot(\\mathbf{n}_i\\otimes\\mathbf{n}_i)\\cdot\\boldsymbol{R}~;~~\n i = 1,2,3 ~.\n ", "e2e43898a8b5ec77238f1a8820ef9a68": "\\gamma = 1/\\sqrt{1 - v^2/c^2}", "e2e446ac4c51f25c0e5570bd8060dbef": "\\frac{AF}{FB} \\cdot \\frac{BD}{DC} \\cdot \\frac{CE}{EA} = 1,", "e2e472868a816e449472195e8e168704": "(\\;7) \\quad\\quad \\frac{d}{dt}\\left(\\int_{x_1}^{x_s(t)}w \\, dx + \\int_{x_s(t)}^{x_2} w\\,dx\\right) = -\\int_{x_1}^{x_2}\\frac{\\partial}{\\partial x}f\\left(w\\right)\\,dx", "e2e48f95e800b0418fccfd82d531fcb4": "\\tan^2\\frac{x}{2}=\\frac{1-\\cos x}{1+\\cos x}.", "e2e4a33d7016acb182107efeb2638ffb": "C = 2\\pi a \\left[1 - \\sum_{n=1}^\\infty \\left(\\frac{(2n - 1)!!}{2^n n!}\\right)^2 \\frac{e^{2n}}{2n - 1}\\right],", "e2e546e1d3f423f95eace25902e9c8fd": "B_2(T)", "e2e55e0664adf058b191ba57ea42a653": "S(x) \\oplus S(x\\oplus a) = b ", "e2e56717d1593d38fee5240d22fb97ae": " [2n-1][2n-3]\\cdots[1]= [2n-1]!!. \\, ", "e2e5674ab455ab4ac1c3878a48b28d63": "w=\\begin{cases}w_{min},&\\mbox{if }w_{min}\\ge\\;w(L)\\\\w(L), &\\mbox{if }w_{min}\\le\\;w(L)\\end{cases}\\,\\!", "e2e57382adf7bbfee5597adf4aaeef83": " (a+b)/2 ", "e2e5cae7b63b2ba898d575294730b5c0": "\\scriptstyle c( ", "e2e61e94c99ec37fa848bdda31126f36": "f_0(x)=0", "e2e643399f285b0efc0310e52afa3112": "v_2", "e2e64ab72a0ca6a4da673e372f6ab5e4": "f_2'(x)>g_2'(x)", "e2e69cf075ef1dc41eb3c77e53423bf7": "1RM = w \\cdot (1 + 0.025 \\cdot r)", "e2e6bb7cfdfac392a3fb5a96789e405d": "P\\,Tr\\, \\exp i T_\\mu\\oint d\\sigma A_\\sigma^{(\\mu)}(\\sigma)", "e2e6e27387ea4859b97fecea710769a1": " \\gcd(x,y) =\n \\begin{cases}\n x & \\mbox{if } y = 0 \\\\\n \\gcd(y, \\operatorname{remainder}(x,y)) & \\mbox{if } y > 0 \\\\\n \\end{cases}\n", "e2e73a010fda62698e31965bfab25def": "f(s)=\\sum_{n=1}^{\\infty}a_n e^{-\\lambda_n s},", "e2e75979e4645e407a36dac38d0f3bc5": "E^{\\alpha \\beta \\gamma \\delta} E^{\\rho \\sigma \\mu \\nu} = - g^{\\alpha \\zeta} g^{\\beta \\eta} g^{\\gamma \\theta} g^{\\delta \\iota} \\delta^{\\rho \\sigma \\mu \\nu}_{\\zeta \\eta \\theta \\iota} \\,", "e2e759b3e11e9e70134e3116b5452f43": "T\\colon V \\to V", "e2e7e12b557ee100a75f2337a505e523": "\\frac{\\mbox{number of new cases in the population at risk}}{\\mbox{number of persons at risk in the population}}=\\mbox{Rate}", "e2e82f901403d3227859c6c384625d25": "V_\\mathrm{+} = \\frac{R_1}{R_1+R_2} \\cdot V_\\mathrm{s}", "e2e854fdc5f11ed40800b27b38237796": "\n\\sigma_N = B f'_t \\left(1 + \\frac D {D_0} \\right)^{-1/2}\n", "e2e85da2fc7c1e3c46f7309805eaba1a": "\\rho >0", "e2e8996c0a5764df25f2030e1490bc93": "\\sum e^{\\frac{2 \\pi i r^2}{p}}", "e2e8a872203aa89317282b0a6d3fd306": "B^7=SO(5)/SO(3)", "e2e8e045433d5584698c1702a69f19c0": " 1/11 = 0.090909\\dots \\,", "e2e9951d1b5002ea327f0748c70cc71d": "g(0) = 0", "e2e9985573b7e94c5422ee0b1dceda4b": "n \\geq m \\geq N", "e2e9ace112fac80f2ef4ecc0a5325978": "\\displaystyle{(\\int_G \\Phi(x)\\, dx\\, f,g) =\\int_G (\\Phi(x)f,g)\\, dx}", "e2e9bedd1a4d9fd774787f24c5435982": "U=e^{i\\hat{p}\\sigma_z}H", "e2e9ee58c06ed385d31328bcf375452d": "U(t)V(t) = U(0)V(0) + \\int_{(0,t]} U(s-)\\,dV(s)+\\int_{(0,t]} V(s-)\\,dU(s)+\\sum_{u\\in (0,t]} \\Delta U_u \\Delta V_u,", "e2e9fd5bebe6ee4c450af436317786d3": "R(t)=\\int_0^{a_M}{r(t,a)da}", "e2ea1b37642129dae6d045030b88d691": "\\, L\\prec M\\, ", "e2ea5203b78bcfd0dd8c86cfaa66a444": "b = \\frac {fm_\\mathrm s} {N} \\frac { \\left| D - s \\right | } { D }\\,.", "e2eaad4b473870ce32caa71382c6dede": "(1,2,3,\\dots)", "e2eb765daecf67123776c67e0de86f4e": "y_n(x)=\\sum_{k=0}^n\\frac{(n+k)!}{(n-k)!k!}\\,\\left(\\frac{x}{2}\\right)^k", "e2ebf3653debfb6c4707e2a9437ab08b": "\n\\left [ Z+Z^{*} \\right ]\n", "e2ec09cf8e53ec1db807fafa1043fe7b": "dI = (C_O - k_O C_L) \\, dx\\;", "e2ec584729b4ab1084f474ac625c5ea2": "\\mathbf{\\hat{d}}_\\mathrm{s} = 2 \\left(\\mathbf{\\hat{d}}_\\mathrm{n} \\cdot \\mathbf{\\hat{d}}_\\mathrm{i}\\right) \\mathbf{\\hat{d}}_\\mathrm{n} - \\mathbf{\\hat{d}}_\\mathrm{i},", "e2ec5da3df85f88adb727b75d245ca23": "g(\\gamma)", "e2ecadebd82a7ab1d6f7129bc3c4e26c": "\n X_n\\ \\xrightarrow{d}\\ X,\\ \\ Y_n\\ \\xrightarrow{d}\\ c\\ \\quad\\Rightarrow\\quad (X_n,Y_n)\\ \\xrightarrow{d}\\ (X,c)\n ", "e2ecb3ae64e76e07814ced68d05588a3": "p'", "e2ecbf368b3668049ee1c0c3cb7dc5db": "0 < \\operatorname{var}(X) < \\tfrac{-11+5 \\sqrt{5}}{2}, ", "e2ecd5dc0ff4bda65f4f9e055efca404": "\\begin{align}\n\\dot{y} = \\frac{\\operatorname{d}h(x)}{\\operatorname{d}t} &=\\frac{\\operatorname{d}h(x)}{\\operatorname{d}x}\\dot{x}\\\\\n&= \\frac{\\operatorname{d}h(x)}{\\operatorname{d}x}f(x) + \\frac{\\operatorname{d}h(x)}{\\operatorname{d}x}g(x)u\n\\end{align}", "e2ed60e71b70baa3d6724d6cd8b9e16a": "\\lim_{N\\to\\infty} \\int_0^{2\\pi}\\int_0^\\pi \\left|f(\\theta,\\varphi)-\\sum_{\\ell=0}^N \\sum_{m=- \\ell}^\\ell f_\\ell^m Y_\\ell^m(\\theta,\\varphi)\\right|^2\\sin\\theta\\, d\\theta \\,d\\phi = 0.", "e2ed731db0dc80eee6534bef727f46bd": "\\textrm{Effort} = \\left[ \\frac {\\textrm{Size}} {\\textrm{Productivity} \\cdot \\textrm{Time}^{4/3}}\\right]^3 \\cdot B ", "e2edbe6b0dd4a9429711c59ebcde2718": "\\zeta_r", "e2edd41ca11d6ce01834ec6183493565": "f_n^{(k)}", "e2ede447460bed3d1b2e6aafa42cf957": "\n a^3 = \\cfrac{3R}{4E^*}\\left(F + 4\\gamma\\pi R\\right)\n ", "e2ee3fba2695b4b931cd33d17e1a6a7a": " q_{all} = \\frac{q_{ult}}{FS} ", "e2eeab043488af8eee38ca7264e3e7b6": "b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \\dots {x_2}^{(1)}x_1", "e2eec3a40bbb4e97fe3598c65165783f": "x^2-y^2=0", "e2eef02f41bd10a42c9e66a3cd6979c5": "k=0,1,\\ldots,5", "e2ef013c4b3b46971330c25efb6f83a4": " \nI(X_3,X_4)=I(X4,X_2) +I(X_1:X_3,X_4)+3I(X_3:X_4|X_1)+I(X_3:X_4| X_2)\n", "e2ef7cff0d4468854cd5fc97cbcde4e2": " \\Bbb{Z}\\subset \\Bbb{Z}[i] ", "e2ef961d82826cf503e362ed91e967a3": " \\left| \\int_{C_L} \\frac{f(z)}{5-z} dz \\right| \\le 2 \\pi \\rho \\frac{\\rho^{\\frac{3}{4}} (3+\\frac{1}{1000})^{\\frac{1}{4}}}{5-\\frac{1}{1000}} \\in \\mathcal{O} \\left( \\rho^{\\frac{7}{4}} \\right) \\to 0.", "e2efaf15359e91d14a375d63cb90ca37": "\\mathbb{RP}^n \\to \\mathbb{RP}^{\\infty}", "e2f00767d8f4e79833d9408581bd5ed8": "T = Ia", "e2f01e0e313002ceb5924d911afeb33d": "\\mathbb{Z}[\\alpha]", "e2f0328b3e1cc31ea02fad39f0fb8b5f": "\\sqrt{\\frac{m_e}{m_i}}", "e2f07275f0897a90d5b0f57346894a82": " mv^2 \\equiv pv \\equiv p^2/m \\,\\!", "e2f0948a737a52303ccb41a8beb57a7a": "2 \\over 25", "e2f0993716fcc5655b26f8eabc0fb364": "e<0", "e2f09ebf7252c841af602b4a3ac30236": " \\Chi(q)=\\sum_{n\\ge 0}{(-1)^nq^{n^2}\\over (-q;q)_{2n}}", "e2f0a0a44514fd291d68c06d427f3eb6": "dV = dA(\\mathbf{v} \\cdot \\mathbf{n}) dt", "e2f0f5760c223b926347aaccafaf1a8d": "0\\leq y_i\\leq a_i", "e2f0fecd3b500adbd2596f7e7aff58e2": " u^3v^3=-p^3/27", "e2f127437d5467517a834702e4f17709": "H_0=\\hbar\\omega_1|1\\rangle\\langle 1|+\\hbar\\omega_2|2\\rangle\\langle 2|+\\hbar\\omega_3|3\\rangle\\langle 3|,", "e2f170a643c8ba82638fac513fc64f1d": "f(x)=\\sum_{k=0}{\\frac{x-a}h \\choose k} \\sum_{j=0}^k (-1)^{k-j}{k\\choose j}f(a+j h).", "e2f1d8b987aff11ad59b9df45d76214f": "\\mathbf{F} = \\frac{d\\mathbf{p}}{dt}.", "e2f23b222365a1a25fbfbf69ddbfbef8": "\\sum_{i=1}^N\\sum_{j=1}^N w_{i}\\gamma(x_i,x_j)w_j \\leq 0", "e2f25b2a6aae054ea631997ce71cbaa8": "\\partial_\\mu \\gamma_{\\mu-} =0,", "e2f28d7cec626c088b6d6189ecfba06d": "\\prod _x f(x)g(x) = \\prod _x f(x)\\prod _x g(x) \\,", "e2f2ed0879c246af4fc8b42e9511095d": "\\mathcal{X} := \\bigcup_{i \\mathrm{\\ even}} \\mathcal{P}(n-g_i)", "e2f3b697b012a29f50cc3b9bd639a185": " A[C] = \\int_{t=t_0}^{t_1} n(X) \\sqrt{ \\dot X \\cdot \\dot X} dt. \\,", "e2f3c6d94e3ab3c2bd99ef175ce6d662": "\n u^H(c) = \\cfrac{1}{\\pi R} \\left[a^2(2 - m^2)\\sin^{-1}\\left(\\cfrac{1}{m}\\right) + a^2\\sqrt{m^2-1}\\right]\n ", "e2f4c9b0500bbb01e4f8d1e3cdaa16dc": "\\bigcirc, \\triangle \\bigtriangleup, \\bigtriangledown \\!", "e2f507f8cbca698ea82eb7b1678b57d9": "k = F_{2i} F_{2i+3} - 1,\\,", "e2f510420a98eaae6be9d39cdced58cb": "\\textstyle \\rho _\\max", "e2f5282733d5660ea7f2e86b4e3624e5": "T_1 \\sub T_2\\sub ...", "e2f53369fc5e6c767ccadcfb6cb7ad21": "\\Re s>1", "e2f5438e26e56de7b3155e89cb1ce90d": "(t, x_i) = \\frac {\\left( 1+\\sum{y_i^2},\\, 2 y_i \\right)} {1-\\sum{y_i^2}}", "e2f5d8c119148fec0553c0a0bed8eab2": "\\sigma_4", "e2f625ac4c6d3e655d831db20de35873": " M(\\beta):=|\\langle \\mathbf{s}_i\\rangle|>0", "e2f6dd9effc3846a14c873583b804996": "0 \\le x \\le 4", "e2f786990bc567a05f17fbffc17e5eb3": "(A\\equiv B)\\equiv(C\\equiv((C\\equiv B)\\equiv A))", "e2f7c9249fe60c32e1429e532285f223": "\\mathbf{v}=(\\cos \\alpha_X,\\cos \\alpha_Y,\\cos \\alpha_Z)", "e2f7d2886be333f1ec7359e59c6c5260": "\\sqrt{2} \\sqrt{ 2 + \\sqrt{2} }", "e2f7fa6636ef306b29912f51212606d8": "\\mathbf{a\\times b} = -\\mathbf{b\\times a}", "e2f84b8034027f6fa4fbf2f93ee0804b": "\\scriptstyle\\rho", "e2f856207b5cc19c539fe3bcc9883ea0": "\\epsilon^{IJKL} = - \\epsilon_{IJKL} .", "e2f897854f1828e49a7ac3b20f381df7": "f_{k+1}(x) \\neq 0\\,", "e2f8aee943b1cbd24ba285459ac4b7ca": "A_{i j}=a_{i j}", "e2f8b3699c4bb8e00be367c9f7b9e19a": "\\text{Cl}_2\\left(\\frac{\\pi}{3}+2m\\pi \\right) =1.01494160 \\cdots ", "e2f8e04bf7e45d8f0600e61e987d2c0b": "O(\\sqrt{E})", "e2f92a4c3652768ee6eeb629c95744e0": " j \\notin T ", "e2f942885491ba8f2d668de2b75bcffe": "{{u}^{*}}\\left( x,y \\right)=\\sum\\limits_{i=1}^N \\alpha_i\\phi \\left( r_i \\right)", "e2f9b777f89876e2930032f1b672da9d": "\\arccos\\left(\\sqrt\\frac{3}{5}\\right) \\approx 39.2^{o}", "e2f9bea177d0383eb0f6d338d9b2d771": "R_S=\\frac{v_{Bullet}^2}{g}\\,\\, 2 \\left(\\sin(\\theta)\\cos(\\theta)-\\frac{\\cos(\\theta)^2 \\sin(\\alpha)}{\\cos(\\alpha)}\\right)\\sec(\\alpha)\\,", "e2f9d909d7ee0693863e257c075e6e80": "\n\\frac{\\beta }{z}\\,\\,\\, \\approx \\,\\,\\, - \\,\\,\\frac{1}{{2\\,n\\,\\,\\ln (b\\mu )}}\\left( {\\frac{\\sigma }{\\mu }} \\right)^2", "e2f9dccb50c3df306e406472cbbbfd2a": " \\frac{(T_{m-1}^i - 2T_{m}^n +T_{m}^i )}{\\Delta {x}^2} + \\frac {e}{k} = 0 ", "e2fa00da81f4219b42a7792fe27d9d88": "g = \\begin{pmatrix}r^2\\sin^2u_2 & 0 \\\\ 0 & r^2\\end{pmatrix},", "e2fa6d78a97b289d3661bfccc3ca2430": "P\\to X", "e2faa15fb7eb03050509b9a7b815053f": " {} + \nh \\, \\Delta f", "e2fae4750ff8b9bc91f0b17209d3314a": " \\begin{align} 4K^2 + \\frac{(a^2 + d^2 - b^2 - c^2)^2}{4} &= (ad)^2 + (bc)^2 - 2abcd \\cos (\\alpha + \\gamma) \\\\\n&= (ad + bc)^2 - 4abcd \\cos^2 \\left(\\frac{\\alpha + \\gamma}{2}\\right). \n\\end{align} ", "e2fb2966b496a32b54e58515e547f083": "p(\\tfrac{x+1}{2})", "e2fb7eade8ef5feaa5f28208062cea82": "70^2", "e2fb8d4fb6f6e83144db09fd5a527ef5": "\\Xi_{b}^{-}", "e2fbb5e6a6161bdec0a0a8d06c8a2cc5": "a_2=2a", "e2fbb97fd0e310f1102ccc8c4c9cd5e8": "R \\ge 500", "e2fbcf48122034d51d3654f0cddad90a": "f(x) = x^2 + x", "e2fc0ac401c06f7916d624804d799edb": "\\overline{A} = X", "e2fc13fb18c0b7269d2688601a13cf0a": "0 \\leq m < n", "e2fc1696af24e9ea874bc204c2e375a4": "\\phi\\colon \\{*\\}\\to\\mathbb Z", "e2fc239838c1b4fbc1532399f64a6891": " L = v^{K(x)+1} - P\\ddot{a}_{\\overline{K(x)+1}|}", "e2fc2c49f2a1d61e133c77c9d64d8584": "|\\partial\\Omega|", "e2fc51b651a2c7f9454a8d3cd6107c49": "\\frac{r^4\\delta^2}{16n^4\\pi^4}", "e2fc5478dd1862c798f18491e2f16c22": "\\rho_k", "e2fca8135c2fadca093abd79a6b1c0d2": "DP", "e2fcbd8faf26237d5c413715898bd3ef": " \\sigma_p^2 = \\sum_i \\sum_j w_i w_j \\sigma_i \\sigma_j \\rho_{ij} ", "e2fcddaaf688cbb8bff2f3f8c1273247": "\\theta(t) = -\\frac{\\gamma + \\log \\pi}{2}t - \\arctan 2t \n+ \\sum_{n=1}^\\infty \\left(\\frac{t}{2n} \n- \\arctan\\left(\\frac{2t}{4n+1}\\right)\\right).", "e2fcde4a9d3ac3af01f98bfa9918de81": "A + B \\longrightarrow AB", "e2fce1bdfc4dd990fe7ad22ce3590b5e": "C(k) = k^{-1}. \\, ", "e2fcf38648c344e113a129a2a823bdaf": "PSO(2k) \\neq PO(2k)", "e2fd2cf45c1c567d4472215b3a27efdb": "\\sum_{m=0}^{N-1}U_{km}U_{mn}^*=\\delta_{kn}", "e2fd4529eb63eaa93dd7f4af4c976d1e": "c_{1'1'} = c_{11} + 2(2c_{44} - c_{11} + c_{12}) (l^2m^2 + m^2n^2 + l^2n^2)", "e2fd61494769468b6460a38eba60b9f5": "\n\\begin{align}\n\\sin(\\theta + \\pi) &= -\\sin \\theta \\\\\n\\cos(\\theta + \\pi) &= -\\cos \\theta \\\\\n\\tan(\\theta + \\pi) &= +\\tan \\theta \\\\\n\\csc(\\theta + \\pi) &= -\\csc \\theta \\\\\n\\sec(\\theta + \\pi) &= -\\sec \\theta \\\\\n\\cot(\\theta + \\pi) &= +\\cot \\theta \\\\\n\\end{align}\n", "e2fd623543776faa1cf4651e4bf5ed2e": " N \\times 6 ", "e2fdaaa6aea8099ef7c3d959e7b157fd": "\\lambda_{02}=5.52008", "e2fdd5f3f268ae499fef396baa368c0f": " c_1^2(X) = 2 e(X) + 3\\sigma(X) ", "e2fdd6111809bc478d64e8fe87a980c1": "E(X_i) = \\frac{n K_i}{N}", "e2fde2cffb5fa5918817bfad173fa1fe": " a = (1 - \\left|z\\right|^{t})^{1/t}", "e2fdf5c2f79c68b1f18fc464991f734c": "\\varphi: U\\times V \\to U\\otimes V.\\ ", "e2fe000b78b21b10680ef3b47bf8c039": "R = O\\left((x-x_0)^3\\right).", "e2fe5d1f8fb522c12c645e0bda7d0b6c": " -f(x) ", "e2fe7298c1f504859992654cd027b909": " X^{(1)}, X^{(2)}, \\ldots , X^{(r)}", "e2fea3e8a048cd64685e873c3a1906d9": " \\frac {1} {2M-r}", "e2fefb16b838d971f0d36515554e2599": "d_Y(f(y), f(x)) < \\epsilon \\,", "e2ff7de0e116f3a41eea314e61ecb317": " ax^2 + by^2 + c = |z|^2 + c . \\,\\!", "e2ff84f0adc07ecce8aa4d4935b740ad": "e^{X}=Pe^{J}P^{-1}.\\,", "e2ffe08e79acbf638f833b8054091a50": "\\lambda = \\lambda^{(0)}_n + \\epsilon \\int f^{(0)}_n(x) D^{(1)} f^{(0)}_n(x)\\,dx ", "e3001e50a11774a77deed6b8032879b1": "\\sqrt[3\n]{E}/\\rho", "e300b509935a7cae78577b41a015beb4": "\\mathbf{n}_{t+1} = \\mathbf{L}\\mathbf{n}_t", "e300bfc342c0b2cb44f062fed4978196": " \\tilde A = {h\\over{8\\pi^2cI_A}} ", "e300caf51f9a4e2aa7de9b04f1379c86": " f(10) = 0.1003290624. \\, ", "e300dddd07969d49349c0f042846e4ca": "\\mod l^n ", "e30113c18214cb5deddd8241bb00573a": "w(\\mathfrak{D}_{L/K}) = \\sum_{s \\ne 1} i_G(s) = \\sum_0^\\infty (|G_i| - 1).", "e301149f28298b7754aa7cee999f7eb5": "x_{\\mathrm{ZOH}}(t)\\,", "e301374a4f1885ed36c8df3c02923544": "(g\\cdot f)(t)=[1-t - \\sin (2\\pi t)] +i[t-1+\\cos(2\\pi t)]\n", "e3014e86e51b740240d9f4bf4668c633": "\\forall x \\in (A\\cap B)^c, x \\in A^c \\cup B^c", "e30185449758c19134159f3f0ac268f6": "\\mathbf e=\\frac{1}{2}(\\mathbf I - \\mathbf c)\n\\qquad \\text{or} \\qquad\ne_{rs}=\\frac{1}{2}\\left(\\delta_{rs} - \\frac{\\partial X_M}{\\partial x_r}\\frac{\\partial X_M}{\\partial x_s} \\right)\\,\\!", "e301b31864ff9db07139cc7a2bad3999": " f_t(x) = f(x+t)~.", "e301b858ad29df0b2122b1109658cbfe": "u_n", "e301d27426dfd063df514c878518cec4": "\\mathrm{wt}(c)=\\mathrm{wt}_1(c)+\\mathrm{wt}_2(c)=3", "e30227743c9b7105c1e0b89c170566c9": "\\bold{v}", "e302679aa4d71cfe8e57010b0c7822dc": "H^0 (X, \\omega_X \\otimes \\mathcal L(D)^\\vee)", "e302941bccd3bd998559cbc90b38884a": "\n\\begin{align}\n\\text{Minimize}\\quad &g\\\\\n\\text{s.t} \\quad & g-\\sum_{j \\in S}q(j|i,a)h(j)\\geq R(i,a)\\,\\,\n\\forall i\\in S,\\,a\\in A(i)\n\\end{align}\n", "e30335071e739e737f1cb2e6a1a41ec4": "G = ((V,E),c,s,t)", "e30386ec552d53d765200a302b3d257b": "-1.0327", "e3038e06efe0e436e58a3f533195379e": " B (x; a, b) ", "e303b07f039e8b7224e02d15838be2b0": "\n \\delta W = \\delta U = \\int_{\\partial\\Omega} (\\mathbf{n}\\cdot\\boldsymbol{\\sigma})\\cdot\\delta\\mathbf{u}~{\\rm dS} + \\int_{\\Omega} \\mathbf{b}\\cdot\\delta\\mathbf{u}~{\\rm dV}\n ", "e303dfe33c143365f36bdd42b82c03c9": "x_{2}^{1} = x_{2}^{0} + u_{1,2}^{0} + u_{2,2}^{0} = 10 + 4 + 6 = 9 \\in Z_{11}", "e3043fdd2d029084ff5b0c96640dc385": "(\\mathbb{Z}/4\\mathbb{Z})^\\times \\cong \\mathrm{C}_2,", "e30473e00e4680ee8e82be808ced6359": "\\oint_{C} {f'(z) \\over f(z)}\\, dz=2\\pi i (N-P)", "e305318afde3042909f471dd92342750": "f(a,b,c) = g(h(P^3_3(a,b,c),P^3_1(a,b,c)),h(P^3_1(a,b,c),P^3_2(a,b,c))).", "e3055b7d09c1dcac19cdad90da0cdb4f": " \\forall t \\in \\mathbb{R} \\quad \\sum_{j=1}^d |a_j-t| \\geq \\sum_{j=1}^d |b_j-t|", "e3058c55f47c126947a26f8275504fee": "\\mathbf{F} = Q (\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B})", "e305b29f0c2c494e512465b89526797a": "\\scriptstyle P(z) \\;=\\; P(-z) ", "e3061a28ae998ba5da793b035ae79e44": "\\left[ J_\\pm , \\widehat{T}_{k}^{q} \\right] = \\hbar \\sqrt{( k \\mp q )(k \\pm q + 1)} \\widehat{T}_{k}^{q \\pm 1} ", "e3067125cdea758b6199d8e8a46c665e": "\\forall a \\in A,\\; V \\to A\\colon v \\mapsto v + a\\quad", "e306792179a248d6edc78f85d882b0af": "\\mathbf{F}_\\text{dipole}=\\left(\\mathbf{m}\\cdot \\nabla \\right) \\mathbf{B}", "e30688a966f4db6370f1b96fc6b40106": "\\langle a, b \\mid a^n = b a^m b^{-1} \\rangle\\,\\!", "e30721504db531ec21d5875b4c9b0b35": "\\vec{F}=-\\vec{\\nabla} U.", "e3073a8393e6f92fc519e9a7e0023564": "A' \\subseteq B'", "e3073e8dce8a765d128d1edccc9c5798": "G_0 = \\cdots = G_{i_0} = G = G^0 = \\cdots = G^{i_0}", "e307509b6c26898f10516d796e4e71cf": "Z \\to D \\subseteq \\mathbb{C}", "e307a5452bbdcba2a7d92313d0f11868": "10 \\times \\log_{10} \\mathrm{year}", "e307dc22b1976a05f2f63a52d5696c89": "a^f b^{n-f}\\,", "e3083f1d4dfe7306a069b8274d6e4230": "P_Z=Z(Z^\\mathrm{T} Z)^{-1}Z^\\mathrm{T}", "e308a394a9a01a22eb0a4e7d45df8c10": "\\pi_{1}", "e30944edb703b14214046d2eb1b6a755": "\\displaystyle{U_i^2=-I,\\,\\,\\, U_iU_j=-U_jU_i \\,\\, (i\\ne j).}", "e309541137e3ec4e2aa13260de739bc6": "0 = Z^3-Z^2+Z\\left(A-B-B^2\\right)-AB \\;,", "e309bae0d94f1f75e9be668fbcfd0da5": "U\\cap I'_x", "e30a363591f4420296cb65a97ddec516": " e + \\mathfrak{g}_f ", "e30a38421421345d93677b64b2dd9c89": "\\varlimsup_{n\\to\\infty}x_n:=\\limsup_{n\\to\\infty}x_n", "e30a520430ff061d01d6131fc1046b1e": "\\sqrt{\\frac{4}{7}}R ", "e30a78296df446d3e19e15c791cdad68": "\\begin{align}\n \\int_a^c f(x) \\, dx &{}= \\int_a^b f(x) \\, dx - \\int_c^b f(x) \\, dx \\\\\n &{} = \\int_a^b f(x) \\, dx + \\int_b^c f(x) \\, dx\n\\end{align}", "e30aa52ab5871f00676b4b4a9d2f23f6": "\\langle\\hat{\\rho}(\\mathbf{k},t)\\hat{\\rho}(\\mathbf{k}',t)\\rangle=k^{n_s-1}\\delta^{(3)}(\\mathbf{k}-\\mathbf{k'})", "e30af751921547612930fe513e06ea4c": " \\alpha_i\\ge 0", "e30b5b60d404bca6ac833d81523a2a62": "M\\equiv M_p \\bmod\\ p ", "e30b5c496d59507701a6530e5d0efe0e": "1^m + 2^3 = 3^2", "e30bc11bed8c7723defad46d75953af4": "\nW = \\pi t 2^{(3/2)\\left(n-1\\right)}.\n", "e30bca90182dd4a1869e9c9507b8e1de": "X\\to x+\\tilde{x}", "e30bf993bd9799a167cfa081b327d308": "\\left [ -\\cos^{-1}\\left(-\\frac{1}{e}\\right) < \\theta < \\cos^{-1}\\left(-\\frac{1}{e}\\right)\\right ] \\longleftrightarrow \\left [-\\infin < E < \\infin \\right ] ", "e30c4e7dc974136e993cb0fd7e3f7a8b": "X \\overset{\\text{id}}{\\to} X \\to 0 \\to \\cdot", "e30c597692824e0e983b36f0501c55a8": "R \\in \\mathrm{ker }\\ d\\alpha, \\ \\alpha (R) = 1 ", "e30c6dd5f6a3bdef0972f1fc8494fa64": " b_k = \\frac{A^kb_0}{\\|A^kb_0\\|}. ", "e30c89b78f180cb0e3b3d36f958f3c9a": "E_{X}", "e30cf163a7dbe7965debde83703f52cb": "-p_1 p_2 p_3 p_4 ( \\delta_1 + \\delta_2 + \\delta_3 + \\delta_4 ) \\frac{Nc}{4}", "e30d0339503e311e7e8d1ef2e79c09fc": "\\sigma^{2}_{r}", "e30d0ae0f593d5266277e4adcc9262e8": " (px+(1-p))^{n} \\,", "e30d0d0909e4e76ebbf5400900b8e459": " ze ", "e30d338ff525d1d5db2df0d492840dbf": " \\|f\\|_{C^*_r} := \\sup \\left \\{ \\|f*g\\|_2: \\|g\\|_2 = 1 \\right \\},", "e30d56b9bf3ac38915e6d322231e0aeb": "A = \\bigoplus_{p,q} H^q(X_p)", "e30d6287e61c0d3156b554fee09ad4d2": "\\frac{1}{k'}", "e30d8b8d179ca001b6cd00b6f5ded8de": "t = t_1 + t_2 \\;", "e30dc583791fea9cba62294e155e329b": "\\scriptstyle \\left\\{ \\overline{B}(x_\\alpha,\\, r_\\alpha) \\right\\}", "e30dedb565cc4cc8cca5fc9b32725e6f": "N\\left(T\\right)", "e30ebddfd33a5bab10313b14b26e97f5": "\\nabla . \\mathbf{F} = 0", "e30f2d86248651951d2e129940a98b2a": "\\nabla_{\\dot{x}_0} Y = \\lim_{h\\to 0} \\frac{1}{h}\\left(Y_{x_0}-\\tau^{-1}_{x_h}(Y_{x_h})\\right) = \\left.\\frac{d}{dt}(\\tau_{x_t}Y)\\right|_{t=0}", "e30f7fb7ddcb659883aa306c5ceac77e": "y^\\prime_i=\\sum_{j=1}^n f_{ij}(x)y_j\\quad(1\\leq i\\leq n)", "e30fa107557d2a24b6735e4ab8dbcdb2": "(h_1\\ ,\\ h_2\\ ,\\ h_3)\\,", "e30fde78847396c96a5e42e5dff6586a": "K:=Quot(R)", "e30fefca6bf13c871a60ddec2e1be573": "\\vec{\\nabla}\\cdot\\vec{E} = {\\rho\\over\\varepsilon_0}.", "e3103a9ff06702d5be60b9c183de77a1": "c_{n+1} = c_n^2 (c_n - 3) / 4 \\,\\!", "e31083339cea76f3e9396aa58005d4f3": "X \\otimes Y", "e310d280bb1c2fe6fb776974b04a081a": " T_0 = [-a, a]^n", "e3110d86a16fd765fe1f7bb0b3a85f08": "x_\\mathrm{sawtooth}(t) = \\frac{A}{2}-\\frac {A}{\\pi}\\sum_{k=1}^{\\infty}\\frac {\\sin (2\\pi kft)}{k} ", "e3115ab94aa0d94ccf99452acb707f37": "\\frac{\\partial^2 H}{\\partial u^2} > 0", "e31187d6b56226b049799c8216ad86e4": "Y_n=\\prod_{i=1}^n\\frac{g(X_i)}{f(X_i)}", "e3119aca85bec7d42d45b085866472a2": "C^{(3)}_{abcd}", "e311c5cfc7cc0ed340bc2c47294bb297": "\n\\nu = 2\\arctan(B - 1/B)\n", "e311ce0dec20a43d484c678582df07df": "\\mathfrak{p} = R \\cap \\mathfrak{q}", "e311dbefb4641880b7e07e9944afc19d": "X=e^{e^{e^{...^{e^{f}}}}}", "e3122347b32eb79d248cb1e5b963f10a": "4\\cdot n_{\\rm Te} - 2\\cdot n_{\\rm Mi} - \\dot\\Omega_{\\rm Te}- \\dot\\Omega_{\\rm Mi}= 0", "e3124006b62a41938c5e0915aaa60eff": "T_1,T_2,\\dots, T_m", "e3124cf3d5b5f951707fab51669f978b": "W^+ = \\{ W_t^+, t \\in [0,1] \\}", "e312866450703691942e8b44e6e7da1a": "(1+x)^n = {n \\choose 0}x^0 + {n \\choose 1}x^1 + {n \\choose 2}x^2 + \\cdots + {n \\choose {n-1}}x^{n-1} + {n \\choose n}x^n,", "e312dab22beb6b33a62ee5eb5dc3e006": " [z^{2n}][u] g(z, u) < \\log 2\n\\quad \\mbox{and} \\quad 1 - [z^{2n}][u] g(z, u) > 1 - \\log 2 = 0.30685281,", "e3130cfe419782d91120e767035e6268": "\\psi(x) = \\sum_{p\\le x} \\log p \\left\\lfloor \\frac{\\log x}{\\log p} \\right\\rfloor \\le \\sum_{p\\le x} \\log x = \\pi(x)\\log x", "e31318dc49a6f1b9f44102f37bbb6a52": " x \\in |K| ", "e31323801989a1b61ee570fbe19e8de6": "X_n + Y_n \\ \\xrightarrow{d}\\ X + c ;", "e313269466e233883b5596ada86a01c1": "{25 \\over 2}^2\\pi", "e31369e9d59745951c6d5e1fe15e4dde": "f'(x) = \\frac{1}{2 \\sqrt{x}}", "e313dc332209dcc92768bda85556ca3b": "D \\in \\mathbb{R}^{n \\times K}", "e314627d3d7c7de719cccc62ee1b3e7a": "A = \\tfrac14ns^2 \\cot \\frac{\\pi}{n}", "e314ab019c8602e6440ba1df509c309c": "0.524", "e314abb61c6742449f29352913b629b2": "I = C_m\\frac{{\\mathrm d} V_m}{{\\mathrm d} t} + g_K(V_m - V_K) + g_{Na}(V_m - V_{Na}) + g_l(V_m - V_l),", "e314adb3cb9e51d496361ca53ab33fee": "\\mathrm{B}(\\alpha) = \\frac{\\prod_{i=1}^K \\Gamma(\\alpha_i)}{\\Gamma\\left(\\sum_{i=1}^K \\alpha_i\\right)},\\qquad\\alpha=(\\alpha_1,\\cdots,\\alpha_K).", "e315549a84b47efee9f380fb643fcfb5": "\\int \\exp\\left[\\theta^TA\\eta +\\theta^T J + K^T \\eta \\right] \\,d\\theta\\,d\\eta = \\det A \\,\\,\\exp[-K^T A^{-1} J ] ", "e3155724dc3b7a9c5d80c6c891651ff1": "\\text{4,500 m/s} = \\frac{30,000 \\times C}{2 \\times 10^9}", "e315818791dae9dc0653aa23ee86291d": " p \\vee q", "e3159ed813ff9798e1547b8a94fa78b2": "r\\ ", "e315c0ad51b2d8512eed80b028b98623": "\\alpha_{m}\\sim\\kappa^{m\\left(m+1\\right)/2},", "e315c7ecaf9d069e35af60277214b41e": "q\\equiv1\\ \\text{mod}\\ 3", "e315cef192416f3708b0d0dda2d0745f": "\\kappa = q^2 / (8 \\pi^2 m_e \\epsilon_0) = c^2 r_e / (2\\pi)\n", "e315e52470a26bc31af409ff69008733": "n_v^{k,(-n)}", "e316315d355545b79c48d6d9a8d9dc68": "\\mu (\\phi^{-1}B) = \\nu(B)", "e316cdb2d395661c56608796d7b1aa7d": "\\scriptstyle{{\\lambda\\over 2}}", "e317015ad841223a45e2b1678de8ef4d": "\\langle x, y\\rangle={\\|x+y\\|^2-\\|x-y\\|^2\\over 4},\\,", "e3177d50086cfbcc219693d7a56e8ce8": "x^2+2x=0", "e317d5cd5c2c18b1fe9b37fe021f7770": "f:M\\to R", "e31833d6aca05bfcbe5b39406a5cbee9": "f(x)=\\frac{g(x)}{h(x)}", "e3183472f9026bee10bd925325d30eb4": "\\varepsilon(h_{(1)})h_{(2)} = h = h_{(1)}\\varepsilon(h_{(2)})", "e318f5ccbd8b4423739ce0c723b951c2": "F_\\nu(H)", "e3190108e6dd53053fae2a2d62c6d298": "\\varpi\\,\\!", "e3190fd499419c5ce15a4cb8d2cc34b0": "H_0 = \\frac{p^2}{2m}. \\,", "e3192b88404559dc1268af6a165d19a6": "t, s", "e3196968090eed3948b4f20ae72b7a6b": "i \\frac{\\partial a}{\\partial z} + i \\beta_1 \\frac{\\partial a}{\\partial t} - \\frac{\\beta_2}{2} \\frac{\\partial^2 a}{\\partial t^2} + \\frac{1}{L_{nl}} |a|^2 a = 0", "e31984b019a75fd79051bbc980349ec5": "\\delta \\it{\\phi}=\\rm{Im}(\\it{\\delta r/r})", "e319f47c86f95d1e68aabbb18558fd0f": " [B_0,B_1] = 0 ", "e31a024537d1652e3ec2221a0149ea2a": "[f(x_1), \\dots , f(x_n)]", "e31a0f05f494181406aaed3d169b056e": " b , v_2, \\ldots, v_n ", "e31a19b10b6d864c5c9f64e67023341b": "L_{1},\\cdots,L_{K}", "e31a2814700fc2d8c9ee34de51500a3d": "\\sum_{n=1}^{\\infty} M_n", "e31a94b9b9b4897a48a70396a60843d9": " e^{\\langle X \\rangle} \\leq \\left\\langle e^X \\right\\rangle, ", "e31ab47e966887a54e01a8aab1508fd5": "A\\geq B", "e31aeb46004f6cfc700ce85946e26967": "L_p", "e31aee587a16580d562f79bf9a236fe7": "P(\\Omega) = 1.", "e31b18ede43223d49a5b3e0a46f664ec": "(\\log n)^2", "e31b1c5335990e0393acbf0f33f34f87": "\\mathcal{E}(b_1)\\cdot \\mathcal{E}(b_2) = x^{b_1} r_1^2 x^{b_2} r_2^2 = x^{b_1+b_2} (r_1r_2)^2 = \\mathcal{E}(b_1 \\oplus b_2)", "e31b458b48dd58470b662e66b9742071": "R^2", "e31b4b478b353127f8be1144be060cb6": "T_0 = 2\\pi\\sqrt{\\ell\\over g}\\quad\\quad\\quad\\quad |\\theta_0| \\ll 1.", "e31b9d8145339b0834a0389923ec832b": "N + \\delta N", "e31bc2cec3684d4a0bd58079afdc30bb": "\\displaystyle{(ac,d^*b)=(bc^*,da).}", "e31c21caa54ffc99638345d8c58ac0a8": "E = \\frac {(\\lambda - \\lambda_0) \\cos \\varphi} {\\rho}", "e31c30fcf45829f34a83169282deab50": "\nT_v\\exp_p(w_N)=T_{\\alpha(0,1)}\\exp_p\\left(\\frac{\\partial \\alpha}{\\partial s}(0,1)\\right)=\\frac{\\partial}{\\partial s}\\Bigl(\\exp_p\\circ\\alpha(s,t)\\Bigr)\\Big\\vert_{t=1,s=0}=\\frac{\\partial f}{\\partial s}(0,1).\n", "e31cdaf897ec25037daa6c4f1e517885": " \\frac{dE_{\\lambda}}{d\\lambda}=\\frac{\\partial E[\\psi_{\\lambda},\\lambda]}{\\partial\\lambda}+\\int\\frac{\\delta E[\\psi,\\lambda]}{\\delta\\psi(x)}\\frac{d\\psi_{\\lambda}(x)}{d\\lambda}dx. ", "e31d208012147ff1108ddd820ac80e5c": "(A_1,\\le_1)", "e31dbbba6a43b3847f3a25c921dcd72b": "\\left(\\!\\!{n\\choose d}\\!\\!\\right) = \\binom{n+d-1}{d} = \\binom{d+(n-1)}{n-1}\n = \\frac{(d+1)\\times(d+2)\\times\\cdots\\times(d+n-1)}{1\\times2\\times\\cdots\\times(n-1)} = \\frac{1}{(n-1)!}(d+1)^{\\overline{n-1}}.", "e31dd5d61dc7aa09cd7074dea27e25c5": "f^{-1} (x)", "e31dede2f5f04be7de51cddff4e8728d": "f\\in m_P\\subset\\mathcal{O}_{C,P}", "e31e32edb1fc68abb22ff6829db18a41": "|A|=|B|", "e31efb0a107f582f5bb480800452aa20": "\\bar x = 1", "e31f6a4e8743d9b51680d93bf1a697f8": "\\beta >> H ", "e31f723a5b6def6cdc310f2580b8dc7c": "f(c)=0", "e31fa2aa69349052e1a5030d855e8fe7": "L_{i} = R_{i+1} \\oplus {\\rm F}(L_{i+1}, K_{i})", "e31fed5b831e7ebd8c1bed42afc78eb9": "\\langle A\\otimes B,R_a\\rangle = \\langle B,\\Omega_a(A)\\rangle", "e32007fde44249b949d308b4173613ac": " \\langle X,Y\\rangle_{\\beta H}=\\frac{\\partial^2}{\\partial t\\partial s}{\\rm Tr}\\,{\\rm e}^{\\beta H+tX+sY} \\bigg\\vert_{t=s=0} ", "e3201166405d6c4274ee6f56f9df268a": " \\frac{d}{dt}\\langle p\\rangle = \\frac{1}{i\\hbar}\\langle [p,H]\\rangle + \\left\\langle \\frac{\\partial p}{\\partial t}\\right\\rangle = \\frac{1}{i\\hbar}\\langle [p,V(x,t)]\\rangle ,", "e3201f905703d72b1720defa5825c13c": "4^3 - 4 = 60", "e320283fd2cf40333ee74f156cd3f0c1": " z=f(d) \\text{ where }d = \\frac{x-\\text{origin}}{\\text{range}} ", "e32066b09428ae6e88720cf311237c5f": "\\mathop{\\uarr}x = \\{y\\in X: x\\leq y\\} = \\bigcap \\mathcal{N}_x.", "e320a149c59d0bf0124b747a280cd82b": "H^{2 \\mathrm{dim}(X) - 2d}(X, {\\Bbb Q}).", "e320ce34c3b825ac177a18b9a3b10a1c": "\\,z_i\\,", "e32132f549180d1a5e6980037bda3dc9": " E_\\text{r} = \\int \\frac{v^2 dm}{2} = \\int \\frac{(r \\omega)^2 dm}{2} = \\frac{\\omega^2}{2} \\int{r^2}dm = \\frac{\\omega^2}{2} I = \\begin{matrix} \\frac{1}{2} \\end{matrix} I \\omega^2 ", "e3213d36b6be0ddf343e13130a7df8d6": "\\omega_1/\\omega_2", "e32197abf09a1ac2e80f909b39806b29": "\\det(M) \\det(M_{1,k}^{1,k}) = \\det(M_1^1)\\det(M_k^k) - \\det(M_1^k) \\det(M_k^1). ", "e321991c23b25a3f4861fb945030f869": " 2 \\uparrow^n 2 ", "e3219accfd05d9e8ec9139c847f88edb": "2ad = 0.", "e321de9f88bb0b3e414fcf9f2abf7749": "( \\text{World Biocapacity} / \\text{World Ecological Footprint} ) \\times 365 = \\text{Ecological Debt Day}", "e321efb26608670e61a466a49cae996a": "1/B", "e3222deafd3d9e4c7f5be4adbe6e4aed": "\\mathbb{E}[\\ln x]", "e3223f7bb24c02f6901d4693c275f16c": " -2^n \\leq \\hat{f}(a) \\leq 2^n. ", "e32247c76b8abda37d09ab97f34b4140": "{1 \\over \\pi}\\iint |h^\\prime|^2 \\, dx dy = \\left[\\sum_{n\\ge 1} {1\\over n} |\\lambda_{-n}|^2 - \\sum_{n\\ge 1} |\\alpha_n|^2 r^{2n}\\right] +\\left[\\sum_{n\\ge 1}{1\\over n}|\\lambda_n|^2 - \\sum_{n\\ge 1} |\\beta_n|^2 R^{-2n}\\right].", "e3225cd4f32716182288c5da07eee1bb": "\\binom{k}{n}_{(1)_{a\\in A}}=[x^n](\\sum_{a\\in A} x^a)^k", "e32262155f4c57563f6bf8c2b5cfd35e": "x \\mapsto x + by, \\quad y \\mapsto y, \\quad b \\ne 0,", "e3227cc3037cde52c631bfa3794878dd": "\\boldsymbol\\Lambda_n=(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\boldsymbol\\Lambda_0) ,", "e322e7c3a375e44c27afc0afb1cddf1e": "\\forall\\hat{S}_{i}\\in\\mathcal{O}_{SB}(\\mathcal{H}_{SB})", "e32355cfbf6c1aba3f8873af5839ba86": "v_e = \\sqrt{\\;\\frac{T\\;R}{M}\\cdot\\frac{2\\;\\gamma}{\\gamma-1}\\cdot\\bigg[ 1-(p_e/p)^{(\\gamma-1)/\\gamma}\\bigg]} ", "e32361030fd183cf3f48c4ddd2b0abca": " \\operatorname{FV}(S) = \\{f\\}", "e323626f9412d0a8e13dd0b3994be867": "0 \\leq s, t \\leq T", "e323717a038a008fca4f3208af6eb6d8": " det \\left( \\Gamma \\right) ", "e3239987527994cf1ec6daefc28d8e71": " \\phi \\to \\forall x \\left( \\phi \\right) ", "e32399f48c5554e5a8f2d26eeacbc234": "\\displaystyle{f(z)=\\overline{f(\\overline{z}^{-1})}^{-1},}", "e323bb775849afa1d2a68ea23bb30e29": "(X,\\le)", "e323c3b1f6a205a8c0872775e36c3c2c": "n_i\\,", "e323dfe793a30a20b24a90a6ef779c57": " \\begin{bmatrix} V \\\\ F \\end{bmatrix} = \\begin{bmatrix} z_{11} & z_{12} \\\\ z_{21} & z_{22} \\end{bmatrix} \\begin{bmatrix} I \\\\ v \\end{bmatrix} ", "e323fe72417acda37de0d9c0ae8fc9bb": "1 + z = \\gamma \\left(1 + \\frac{v_{\\parallel}}{c}\\right)", "e324028a1dbd31f793cef8f485bcde2d": "_k\\mathbf{a}_{l,m,n}", "e3242697ec416448cb3d8bdad7adc144": " \\nabla \\cdot v = \\partial_1 v_1 + \\partial_2 v_2 + \\partial_3 v_3;", "e3242737cc75325d01c3a2afa2ebe283": "x \\mapsto \\left[F_i\\left(x\\right)\\right]_{i \\in I}", "e3243f63e34db8a2238a51b5977b06c3": " \\phi = \\arctan\\left(\\frac{2\\omega \\omega_0\\zeta}{\\omega_0^2-\\omega^2}\\right)", "e32464ad8e66caf4bf1fdd131a2b2dbc": "\n e = e_0 - \\lambda \\ln\\left[\\frac{p_c}{p_{c0}}\\right]\n ", "e325269ab80ed0dfbd4350c5650dfcb8": "\nS_{N_A}\\otimes S_{N_B} \\subset S_{N_A+B_B} \\Longrightarrow \\forall \\pi \\in S_{N_A+B_B}:\\quad \\pi = \\tau \\pi_A \\pi_B, \\quad \n\\pi_A\\in S_{N_A}, \\;\\; \\pi_B \\in S_{N_B},\n", "e3254f874a8988f2f00f50260ce04025": "\\tfrac{r}{N} \\Big / \\text{average} \\left( \\tfrac{r-1}{N},\\tfrac{r-2}{N}, ...\\ ,\\tfrac{1}{N}\\right) \\ = \\ \\tfrac{2r}{r-1}", "e3255e3dbcef2455194184119947b474": "\n\\begin{align}\n\\bar a & \\equiv aR \\pmod {N} \\\\\n\\bar b & \\equiv bR \\pmod {N} \\\\\n\\end{align}\n", "e3261a39ee4894983307e49a19813ea6": "\n\\mathrm{GCI}_{i\\rightarrow j}(t) = \\ln \\left( \\frac{V_{i,n}(t)}{V_{i,n-1}(t)}\\right)\n", "e3262c794a09ca8544ee4cdec8630571": "C_{pi}^*=\\bar{C_{pi}}", "e326aff8388e30e8b48c79e94f4ca4bb": "x_s(t)", "e326c9a40f3ce0cc91cad9aa5cedf4c9": "\nH_{B-l}=H_B\\otimes\\mathbb{I}+S_{x_B}\\otimes S_{x_l}+S_{y_B}\\otimes S_{y_l}+S_{z_B}\\otimes S_{z_l}\n", "e326cb35b49b2779c0eb2a436401b5da": "\\, mgh", "e326d5a94c487fbedb57e06ca6ba7d3a": " r = \\begin{pmatrix} \np_\\text{x} & \np_\\text{y} &\np_\\text{z}\n\\end{pmatrix}\\begin{pmatrix} \nT_\\text{xx} & T_\\text{xy} & T_\\text{xz} \\\\ \nT_\\text{yx} & T_\\text{yy} & T_\\text{yz} \\\\\nT_\\text{zx} & T_\\text{zy} & T_\\text{zz}\n\\end{pmatrix}\\begin{pmatrix} \nq_\\text{x} \\\\ \nq_\\text{y} \\\\\nq_\\text{z}\n\\end{pmatrix}\\,,\\quad r = p_i T_{ij} q_j ", "e326d64dee4ea0812d8fc7409d4190b3": "\\delta l^a-D m^a=(\\bar{\\alpha}+\\beta-\\bar{\\pi})l^a+\\kappa n^a-(\\bar{\\rho}+\\varepsilon-\\bar{\\varepsilon}) m^a-\\sigma\\bar{m}^a\\,,", "e326fe27baa7d15aaa8026d767ba4994": "G=[[G_{nm}]]", "e327acf8b16924c10fb68fba8a25dd9d": "i=n", "e327efa50f0f0146d10d5b04477ff798": "(\\omega\\wedge\\eta)(v_1,\\cdots,v_{p+q}) = \\frac{1}{p!q!}\\sum_{\\pi\\in S_{p+q}}\\sgn(\\pi)\\omega(v_{\\pi(1)},\\cdots,v_{\\pi(p)})\\otimes \\eta(v_{\\pi(p+1)},\\cdots,v_{\\pi(p+q)}).", "e3281c4f91ad4541a147d3c68e5b3b7d": "\\langle n \\pm 1|n\\rangle= S \\ .", "e3283dd5411bec2dc687ae3a4150b880": " -{\\partial p}/{\\partial n}", "e3285c0925b05f6366057a4cf1f5e164": "f\\left(a_0,\\dots, a_{n-1}, s\\right) = 0 \\leftrightarrow \\forall i < n \\; \\left(\\beta(s,i) = a_i\\right)", "e3289b9e49528c26a273a36117f50f18": "p\\ge 1", "e328b85e6c1b089ebcc14fca8e0624cf": "D\\Theta = \\Omega\\wedge\\theta", "e329153e0852eb86cfac5af3d21be239": "V_{GEO}", "e329a59b164d7fafc14d0e9f6463d9d2": "\\mathcal{L}_X g_{ab}=\\phi g_{ab}", "e329b678b474b8bdc1333870be827a48": "\n\\mbox{PF} = \\mbox{DPF} {I_{\\mbox{1, rms}} \\over I_{\\mbox{rms}}}\n", "e32a01dc7800c03a4b78c9befbb0da8d": "\\xi^{L,R}", "e32a1c865646fea4d7bd6769bfe2f005": "\\varepsilon_{\\textrm{r}}", "e32ab0979f29df03983e2fd6dbdcef4b": "\\begin{matrix}{5 \\choose 3}\\end{matrix}", "e32abea2894806a85a5647103a3db6ba": " f(x,y) = u(x,y) + iv(x,y)", "e32b5d87a1bedddd31eca3f1b6273245": " y' = y\\ ,", "e32beb2d3a21c861e558fbcf6b751320": "\\lambda x.x", "e32c0c17dfc217033ec82c86c2d16cad": "i_1, i_2,\\ldots", "e32c3f0ee0b1cb8b434803837cdf3a86": "|\\gamma|\\leqslant \\sqrt{(1-p)p}", "e32c5b6b54ce1398ae1134e7688631ad": "w_{i,j}", "e32c83d80f9baaa8f5287def5bda2e9a": "y = mG", "e32c989d591ea3a56b4e0753fb98b877": "{\\frac{tan\\delta}{\\mu_i}}*10^6", "e32ccf7c21e967562c3e37532109ddba": "y=\\frac{1-t^2}{1+t^2}\\,,", "e32cdbb413e16dc1fc0e53195aef887d": " i=0 ", "e32d298fc79c609b507091612e81ac2d": " \\psi(x) = \\sum a^\\dagger(k) e^{ikx} + a(k)e^{-ikx} ", "e32d444f4158e6ee8c694f115fdd6988": "\\circleddash, \\circledcirc, \\circledast \\!", "e32d5701976f988946b8acfcdbc559c2": "S = \\int d^4x d^2\\Theta 2\\mathcal{E}\\left[ \\frac{3}{8} \\left( \\overline{D}^2 - 8R \\right) e^{-K(\\overline{X},X)/3} + W(X) \\right] + c.c.", "e32dbea839e6a9cf8c731eb12fcd78c0": "\\bar{K}", "e32dd5231e1d9666fc087d875145e19a": "\\frac{S}{a_1}", "e32dd9401b69b863d5321939930c8068": " \\delta \\phi = \\frac{4 \\, m}{R}", "e32dfae25686f4460c6088bdec24a657": "S_E=\\int_{\\tau_a}^{\\tau_b}\\left(\\frac{1}{2}m\\left(\\frac{dx}{d\\tau}\\right)^2+V(x)\\right) d\\tau.", "e32e619030eb58183774b16b937e7d01": "\\frac{x_2}{x_1} = \\frac{1-a}{a}\\theta ", "e32e7dfca3ace36f27c529837dfc74f6": " \\varepsilon_r ", "e32ed9ee61840cff6fb2a91088292239": " \\left( \\ker A^* \\right)^\\bot = \\overline{\\operatorname{im}\\ A}", "e32f0007e8fb35c768fc14098a3dca0e": "\\mathbb Q(\\eta)", "e32f3f5b66f72b2340a1e444a18a409f": "|\\{v\\colon v>0,v\\in S\\}|-|\\{v\\colon v<0,v\\not\\in S\\}|.", "e32f595216b3bd199dfcbc6214eb6ce8": "\\vartheta(0;\\tau)", "e32f674f218784123f2d097bdc713915": "\\delta S = S - S_0 = k_B ln(\\Omega) = \\frac{ \\delta Q}{T} ", "e32fc06fed6cf6ebd9a36453d3cb8bd8": " \\zeta(k)=\\frac{2^k}{2^k-1}+\\sum_{r=2}^\\infty\\frac{(p_{r-1}\\#)^k}{J_k(p_r\\#)},\\quad k=2,3,\\dots ", "e33014df865bb8bdd479b7e4ec2fd0b2": "S^2_{n-1} = \\frac{1}{n-1}\\sum_{i=1}^n\\left(X_i-\\overline{X}\\,\\right)^2", "e33018dbe062085fbee952b2f1db1dc2": "(\\lambda _{1}+\\lambda _{2})\\left\\{ p^{2}+\\Phi (x_{\\perp\n})-b^{2}(w^{2},m_{1}^{2},m_{2}^{2})\\right\\} \\Psi =0\\,,", "e3301a2f46597893c31e289c07e7d019": "H_R(f)\\cdot H_T(f) = H(f)", "e330444c9b27ed6e48d41af68d3cc302": "{a^2}\\cos^2\\theta\\pm{b^2}\\sin^2\\theta = r^2", "e3306fa5b304b831deaad53086acd8f9": "\\rho(X,t) = |\\psi(X,t)|^2", "e33092c9144dc11152ef33b2739be50b": " H = X(X'X)^{-1}X'. \\, ", "e330a9ca93f6d282ffc77765336ba10d": "\n\\min \\{ R_A(x) \\mid x \\in U \\} \\leq \\lambda_k\n", "e330cdb52ee547b03d42fa0f628f26e1": "\\textstyle M \\times N", "e331365b473237c4d84ed0ec66ace90b": "L(x) = L_1(x) \\otimes L_2(x).", "e331634969f0553cecbc5f45526b9691": "AB\\cdot BC=IB^2+\\frac{IA\\cdot IB\\cdot IC}{ID}.", "e331d71cc8226ef7af1cdc0defd36c13": "\\cos(\\alpha \\pm \\beta) = \\cos \\alpha \\cos \\beta \\mp \\sin \\alpha \\sin \\beta\\,", "e331f5a972d3f3ca147bb53af1b412ca": "\\tfrac{\\mathrm{d}F}{\\mathrm{d}x}=f(x)", "e3320b19875321154e636d2e8b32318b": "C_{v}", "e3322bb2f836cbfeb81d87098dc25fea": " h_1(x) = \\lim_{\\varepsilon\\to0+} \\mathbb{E} ( Y | x-\\varepsilon \\le X \\le x+\\varepsilon ),", "e33293fd3ee73a6612e4231d043f10d7": "\n\\left. + \\left[Q_R^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{1*}(\\mathbf{p})\n- Q_L^\\dagger(\\mathbf{p}) \\epsilon_\\mu^{2*}(\\mathbf{p}),\n\\right]e^{-i p x} \\right\\}.\n", "e332d4e4ffaccf69ad8131b5b0454e03": " {G^a}_b \\, {G^b}_c \\, {G^c}_a = R^3/4 ", "e332faee297287751e35495d7e5273b1": " \\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\} =\nn! [u^k] [z^n] B(z, u) =\nn! [z^n] \\frac{(\\exp z - 1)^k}{k!}", "e33306a4a885e74f55071d0f017e7eae": "g(s) = \\operatorname{Re}\\left[\\frac{-s^k\\log(-is)}{k!(2\\pi i)^n}\\right]\n=\\begin{cases}\n\\frac{|s|^k}{4k!(2\\pi i)^{n-1}}&n \\text{ odd}\\\\\n&\\\\\n-\\frac{|s|^k\\log|s|}{k!(2\\pi i)^{n}}&n \\text{ even.}\n\\end{cases}", "e3334000b631387fcd6c00c56c7617ad": "Y_{8}^{4}(\\theta,\\varphi)={3\\over 128}\\sqrt{1309\\over 2\\pi}\\cdot e^{4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(65\\cos^{4}\\theta-26\\cos^{2}\\theta+1)", "e3336aad11adabae93dc8ca4fb2a9d18": "T=q", "e33376d47a9894793b753be6f3124536": "Z_u", "e33387cd9d1414ee7124c7537cd645c8": "\\widehat{\\text{MMD}}(P,Q) = \\left| \\left| \\frac{1}{n}\\sum_{i=1}^n \\phi(x_i) - \\frac{1}{m}\\sum_{i=1}^m \\phi(y_i) \\right| \\right|_{\\mathcal{H}}^2 = \\frac{1}{nm} \\sum_{i=1}^n\\sum_{j=1}^m \\left[ k(x_i, x_j) + k(y_i, y_j) - 2 k(x_i, y_j) \\right] ", "e333952e606baedee5d525286d16fdfb": " \\nu^*_0 = nV(b)/c^2 \\, ,", "e333b8d310c0508f64ebc7c4d7761bde": "P_2 \\cdot p_2 = X_2 \\cdot p_2-N_2 \\cdot w \\,", "e333d5c4edffdd5bfdb36f5b73a44a0a": "\\displaystyle V_{{LJ}_{trunc}}", "e333d87b4a41d9ab978cf70ede38c53a": "\\theta\\in[0,2\\pi]", "e333f5aa43ccf406bcfd9afa8c9f2b27": "(0 \\le \\boldsymbol{\\varepsilon} \\le \\boldsymbol{\\varepsilon}_1 )", "e333fb654809ba4275523963d5a64fe5": "q_s = k_d S \\,\\!", "e3346192547796769535a897e5b2613a": "\\lim_{\\alpha\\to 0^+} (I^\\alpha f)(x) = f(x).", "e334bd8005a58575a6f74dc1dd43630d": "\n\\begin{align}\n& \\int_{\\mathbb{R}^n} e^{- x^T A' x + s'^T x} \\left( -\n\\frac{\\partial}{\\partial x} \\Lambda \\frac{\\partial}{\\partial x} \\right) e^{-\nx^T A x + s^T x} \\, dx = \\\\\n& = \\left( 2 {\\rm tr} (A' \\Lambda A B^{- 1}) + 4 u^T A' \\Lambda A u - 2 u^T\n(A' \\Lambda s + A \\Lambda s') + s'^T \\Lambda s \\right) \\cdot \\mathcal{M}\\;,\n\\\\ & {\\rm where} \\;\nu = \\frac{1}{2} B^{- 1} v, v = s + s', B = A + A' \\;.\n\\end{align}\n", "e33501d340aa326541d9542957dcb597": "\\lfloor \\mu \\rfloor", "e335344a1da88c2e14901c63e2c8a263": "h_p\\left(t\\right)", "e33561095b0e57101625509538809958": " \\frac{6}{\\pi^2} = \\prod_{n = 0}^\\infty \\underset{p_n: \\text{ prime}}{\\! \\left(\\! 1- \\frac{1}{{p_n}^2} \\! \\right)} \\! = \\! \\textstyle \\left(1 \\! - \\! \\frac{1}{2^2}\\right) \\! \\left(1 \\! - \\! \\frac{1}{3^2}\\right) \\! \\left(1 \\! - \\! \\frac{1}{5^2}\\right)\\cdots", "e335621e9128d9cc46b589158b6905f0": "|\\psi(x)-x|=O(x^{1/2+\\varepsilon})", "e335bed1ae4790f32485c1b19896b7ab": "\\int_\\Omega \\nabla u\\cdot\\nabla v = \\int_\\Omega gv.", "e335da924f58d5203faf9231ee86709a": "\\left\\{0, 1, 2, \\dots, \\omega_0, \\omega_1\\right\\}", "e33663bc5c407607eba42bc9e933f4a0": "\\gcd(|r-s|,n) = p", "e3370b7918865fa3028de30e6bb33aa2": "y-y_0=-\\lambda (y_P-y_0)", "e3370f14d299b2fc3a55839c5454f558": " \\begin{alignat}{2}{52 \\choose 5} &= \\frac{52!}{5!47!} \\\\\n&= \\frac{52\\times51\\times50\\times49\\times48\\times\\cancel{47!}}{5\\times4\\times3\\times2\\times\\cancel{1}\\times\\cancel{47!}} \\\\\n&= \\frac{52\\times51\\times50\\times49\\times48}{5\\times4\\times3\\times2} \\\\\n&= \\frac{(26\\times\\cancel{2})\\times(17\\times\\cancel{3})\\times(10\\times\\cancel{5})\\times49\\times(12\\times\\cancel{4})}{\\cancel{5}\\times\\cancel{4}\\times\\cancel{3}\\times\\cancel{2}} \\\\\n&= {26\\times17\\times10\\times49\\times12} \\\\&= 2,598,960.\\end{alignat}", "e33761b1cc1d4995826b4623d93a475e": " \\sup_A |P(A) - Q(A)| \\, ", "e337a7864462348cf0c98719fcc496ea": " h^{ab} \\rightarrow \\tilde{h}^{ab} = h^{cd} \\frac{\\partial \\tilde{\\sigma}^a}{\\partial \\sigma^c} \\frac{\\partial \n\\tilde{\\sigma}^b}{\\partial \\sigma^d} ", "e337aa9cdc928f3a435275a4afe84d45": " \\Delta_1 = \\begin{vmatrix} kaI_0 \\left ( ka \\right ) - I_1 \\left ( ka \\right ) & K_1 \\left ( ka \\right ) & -kaK_0 \\left ( ka \\right ) - K_1 \\left ( ka \\right ) \\\\ I_0 \\left ( ka \\right ) + kaI_1 \\left ( ka \\right ) & -K_0 \\left ( ka \\right ) & -K_0 \\left ( ka \\right )+ kaK_1 \\left ( ka \\right ) \\\\ \\frac{\\mu_A}{\\mu_B}kaI_0\\left ( ka \\right ) & K_1 \\left ( ka \\right ) & -kaK_0 \\left ( ka \\right ) \\end{vmatrix} ", "e337d81a6f22abaeee109b0f27d5514f": "\\hat G_T = {{\\beta _T (D_T + [TSH])[TT_4 ]} \\over {\\alpha _T [TSH]}}", "e3384b503cde2b67f5c39b03102a50c8": "N +1\\,.", "e338651e9306d447473bb1ec8bb2e30f": "G = \\bigcup_{s\\in W} B s B,", "e33876fe0db2c58dabdf3a57692ab17a": "ln(Y_d)=\\psi+\\eta^T\\Z+\\beta^T{X}+d\\ln(D)+\\epsilon\\,\\!", "e3389787f9dc939dc4da1dbe5ea64dcf": " G=(E,V) ", "e338a0fbb2ce9e5b21ebf02ac21d0cae": "5.67\\times10^{-5} \\times 2.34\\times10^2 \\approx 13.3\\times10^{-3} = 1.33\\times10^{-2} ", "e338b511fdf805ebe8b36b1d016733e9": " L_\\delta (a) = (1/2){a^2} \\qquad \\qquad \\text{ for } |a| \\le \\delta ,", "e338cf58fd45b7a84b19d2f0b5aee3b7": "EG\\longrightarrow BG", "e33907c83ea4ec02f46a7ce1d05e5166": "\\exists k>0 \\; \\exists n_0 \\; \\forall n>n_0 \\; g(n)\\cdot k \\leq f(n)", "e33945e6127bbdaafa62c800b55b7e25": "h=\\frac{\\sqrt{3}}{2} a", "e3397d4fa3a009bcb23f56566f6017ea": "e^{g z}", "e3399d9b147627fb67c55d3c1a91b64b": "\\{C_n\\}", "e339c8b5e1ab71140bc59fab1694f223": "M_{2} := \\{ \\delta_{1 / n} | n \\in \\mathbb{N} \\}", "e33a235a31b7d3f15ec7fc1898617100": " v = \\frac{1}{ u}( u \\cdot v + u \\wedge v)", "e33a38afb7b10ac57f05d7429838f61d": "~\\tau~", "e33a5fce6bac736e60019f82f6528ea7": "|\\alpha_x|^p", "e33a5fd4d71ff5ae4c8d95e44ac6ab05": " T=\n\\begin{bmatrix}\na & * & * & *& * & * & * & * & * \\\\\n0 & 0 & b & * & * & * & * & * & * \\\\\n0 & 0 & 0 & c & * & * & * & * & * \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & * & * \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}\n", "e33a66550b017bcd3541949aedff3b95": " f_s(f_t(z))=f_{t+s}(z)", "e33a9f2252b2e2476af6fb9dc431d4d4": "\\mathbb{V}_R(F)=\\{a\\in R^n|\\ f(a)=0 \\text{ for all } f\\in F\\}.", "e33aa732bfc0b13b20d40ffc46d1b1f0": "\\,I^+(x) ,I^-(x), J^+(x), J^-(x)", "e33abf1f58bc43104587a820ee9b3153": "\\operatorname{ch}(V) = \\frac{\\sum_{w\\in W} \\varepsilon(w) \\xi_{w(\\lambda+\\rho)}}{\\xi_{\\rho}\\prod_{\\alpha \\in \\Delta^{+}}(1-\\xi_{-\\alpha})} = \\frac{\\sum_{w\\in W} \\varepsilon(w) \\xi_{w(\\lambda+\\rho)}}{\\sum_{w\\in W} \\varepsilon(w) \\xi_{w(\\rho)}}", "e33ad5de46f37c281da172f7f94e5b61": "\\hat{\\alpha}^{eff} = \\hat{\\alpha} - 2 n \\pi", "e33b08ea143ee879d535b1e743e93a0c": "\\bar x\\ =x_1\\hat e_1+x_2\\hat e_2+x_3\\hat e_3", "e33b3ac2ff2aec182768d32579eb90cb": " \n\\Pi = kT c_{\\rm B}(e^{+\\beta q \\psi} + e^{-\\beta q \\psi} - 2) - \\frac{\\epsilon_0 \\epsilon}{2} \\left(\\frac{d \\psi}{dz} \\right)^2\n", "e33b46cd8d8a645b97d40de127f327ee": "\\displaystyle{f(z)=\\sum_{n\\ge 0} a_n z^n}", "e33b744e667f4487ff25bb174c03265c": "\n\\begin{array}{lcr}\nC_6H_{12}O_6 + 12 U^{6+} + 6H_2O +24 e^- \\longrightarrow& 6CO_2 + 12 U^{4+} + 24H^+ \\\\\n\\qquad Water\\ Soluble & Water\\ Insoluble \n\\end{array}\n", "e33bb0d71a79ea8eb3756aa7470648d0": "f(\\mathbf{x},\\sigma^2|\\mu,\\mathbf{V}^{-1},\\alpha,\\beta) = |\\mathbf{V}|^{-1/2} {(2\\pi)^{-k/2} } \\, \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} \\, \\left( \\frac{1}{\\sigma^2} \\right)^{k/2 + \\alpha + 1} \\exp \\left( -\\frac { 2\\beta + (\\mathbf{x} - \\boldsymbol{\\mu})' \\mathbf{V}^{-1} (\\mathbf{x} - \\boldsymbol{\\mu})} {2\\sigma^2} \\right). ", "e33bcf24300ab06c6198838ac570e79e": "P_\\alpha", "e33befd156163bdf3a0796ee0544441c": "< \\int [d N] (1 + i \\int d^3 x N (x) \\hat{H} (x) + {i^2 \\over 2!} [\\int d^3 x N (x) \\hat{H} (x)] [\\int d^3 x' N (x') \\hat{H} (x')] + \\dots) s_{int} , s_{fin}>_{Diff}", "e33bf2222d4625dfc2fec71ad42a8f96": "\n \\approx E_J \\sin(\\phi_0)\\delta\\phi\n = \\frac{E_J \\sin(\\phi_0)}{I_c \\cos\\phi_0}\\delta I\n", "e33bf5a0132d171fce4b443abaed1d6f": "\n\\frac{\\partial c}{\\partial t} = \nD \\frac{\\partial^{2}c}{\\partial z^{2}} + \nsg \\frac{\\partial c}{\\partial z}\n", "e33bf905fdcfec20eb5896e2955cf531": "\\frac{P(A_1|B)}{P(A_2|B)} = \\frac{P(B|A_1)}{P(B|A_2)} \\cdot \\frac{P(A_1)}{P(A_2)}.", "e33c28d821a6fd300de1893ffe47b26c": "y_1(t) = \\,\\!y_2(t)", "e33c414ab4f95f78dae9e3fe51c5ba45": "S(\\rho):=-{\\rm Tr}(\\rho\\log \\rho)", "e33c84846b8c7e6c765f4b0d57e0ebc3": "\\operatorname{E}(I - C) = (n - 1)\\sigma^2", "e33d127dd827a49e5a7ea76c968f4c1a": "\\gamma = \\lim_{n \\rightarrow \\infty } \\left(\n\\sum_{k=1}^n \\frac{1}{k} - \\ln(n) \\right)=\\int_1^\\infty\\left({1\\over\\lfloor x\\rfloor}-{1\\over x}\\right)\\,dx.", "e33d813f8235dcf345fd106ab3ba0e67": "A_{3,5}", "e33df0b04bdad278aa4eea8312386f6e": "(x+\\Delta x)^n,", "e33e93d928e3ed254e363a1513c37cec": "(X,Y,\\pi)\\,", "e33f036b656d0d4925d32028710e14d2": " \\delta \\vec x", "e33f0c3098efc9b989d7b28d8c389c47": "\\left(\\frac{a}{p}\\right)=\\prod_{n=1}^{(p-1)/2}\\frac{\\sin{(2\\pi an/p)}}{\\sin{(2\\pi n/p)}},", "e33f1c4110c32e2f6e9dd543129a43a7": " {ample(s)=\\empty} \\iff {enabled(s)=\\empty} ", "e33f250531fc8161ca8e0b44ff725995": "X(s) = \\frac{s\\sin\\phi+\\omega \\cos\\phi}{s^2+\\omega^2}", "e33f341f1fe60533e1a99a0370302bbd": "x_j\\in A_j", "e33f48db6b1215a27c999d9aa953e480": "V =\\frac{RT}{p } \\Rightarrow \\left(\\frac{\\partial V}{\\partial T}\\right)_{p}^2=\\frac{R^2}{p^2} ", "e33fa6eb1afd6c60b16f5eb04b89cc61": "c_e", "e33fa76c757bee48e7281103d0cd6b59": "t,\\,", "e33ff2a8cea3d1c959adc928119ed853": "\\mathbb{D}^q(e^{at})=a^q e^{at}", "e33ff62c71aa49c632bb6c11badb47da": " \\frac{{\\rm d}}{{\\rm d} t} \\left ( \\frac{\\partial L}{\\partial \\mathbf{\\dot{q}} } \\right ) = \\frac{\\partial L}{\\partial \\mathbf{q}} ", "e340732d68c0ef40a9f334b17bb8b87f": " x^{*n} = \\underbrace{x * x * x * \\cdots * x * x}_n,\\quad x^{*0}=\\delta_0 ", "e340973a1757b2ca77440a1cfeffcae3": "a_{ij} + a_{kl} \\le a_{il} + a_{kj}\\,", "e340aed3ea384ab0899f62c9f77c9461": " 0 < R < \\infty ", "e341212d50ddfec3236e343f77ab4cef": "\\sum_{k=1}^{n} f(\\gamma(t_k)) [ \\gamma(t_k) - \\gamma(t_{k-1}) ]\n=\\sum_{k=1}^{n} f(\\gamma_k) \\Delta\\gamma_k.", "e3419f8fadd8623395f221a4d4e61f94": " d_1 ", "e341bc3e767e66125b2e78ad17850555": "O\\left(e^{-\\frac{k n}{\\ln n}}\\right)", "e341d948ac58e13b53af38489733652d": "(1 + x)^r > 1 + rx\\!", "e342287762426ee43bc0734fde9a89ea": "\\operatorname{rank}(R)=n", "e342709395723434d0d7c6b4e2a39658": " \\mbox{diam}(M,g_\\varepsilon)\\le 1 ", "e342c05a1bfadbfd062b1cdae89ecc51": " \\delta_{ext}(q, x)=s'=(\\ldots,(s_i', t_{ei}'), \\ldots) ", "e342e9dface73313c21113a7ec1dacb0": " \\neg \\neg P \\leftrightarrow P ", "e34345513784d49c46933a5ad1ca2977": " \\psi (x) = M\\left( \\frac{x}{2} \\right) \\log(2)+M \\left( \\frac{x}{3} \\right) \\log(3) + M \\left( \\frac{x}{4}\\right )\\log(4) + \\cdots. ", "e343513a1878b1d4cd5e38023ada2c3b": "(\\rho, \\theta)", "e3439ef254f5cb199d6e21df829e9796": "K=2MN\\sqrt{EQ\\cdot FQ}", "e343ba9bdd89c02734ecb700466e0321": "= \\sgn( \\cos (\\theta - \\frac{\\pi}{2})) \\sqrt{1 - \\cos^2\\theta}", "e343d4fea0f908c0f9bc3623ca715d5b": "\\vec{x_i}", "e343f0bd11066734797b7b546684f29d": "\\mathrm{2\\ Na_2O_2\\ + 2\\ H_2O\\longrightarrow\\ 4\\ NaOH\\ +\\ O_2}", "e344424081ff16c5cddcd73d786999be": "f(x-\\theta)", "e3445c282264e9f435574ab0574a8352": "D_1 = \\lim_{\\epsilon \\rightarrow 0} \\frac{-\\langle \\log p_\\epsilon \\rangle}{\\log\\frac{1}{\\epsilon}}", "e3446314962562aa0e25550e7d844494": "\\begin{align}\n\\mathbf{E} \\left [ X_i | X_1, \\dots, X_{i-1} \\right ] &= 0, \\\\\n\\mathbf{E} \\left [ X_i^2 | X_1, \\dots, X_{i-1} \\right ] &\\leq R_i \\mathbf{E} \\left [ X_i^2 \\right ], \\\\\n\\mathbf{E} \\left [ X_i^k | X_1, \\dots, X_{i-1} \\right ] &\\leq \\tfrac{1}{2} \\mathbf{E} \\left[ X_i^2 | X_1, \\dots, X_{i-1} \\right ] L^{k-2} k!\n\\end{align}", "e34475438781e1ef53d5b7196e58ceee": "X \\, ", "e3449070f22dc7799399bfa368e235e8": " E' = \\frac {\\sigma_0} {\\varepsilon_0} \\cos \\delta ", "e34494ee2e52c9acc516eb84d78b2552": "E = \\sum_T W_T \\cdot H_T = \\sum_T W_T \\sum_R W_R \\cdot \\bar{D}_{T,R}", "e3451acae88cbe8e9bb560f510ce7318": "\\frac{dx }{dt }=\\lambda ", "e345343fd71616b7fd3cc615df249b0e": "\\rho \\mathbf{v}", "e34561bd90381bdb9e847e407e5ace31": "x_0 \\gg 0 ", "e345e1b8aecd711bb13b84c60c2540f0": "= -e^{-4x}(2x-1)+2xe^{-4x}= (-2x+1+2x)e^{-4x} = e^{-4x}.\\;\\!", "e34607f750343aed43aef204df1e0859": "\np_\\alpha \\longrightarrow -i \\hbar \\frac{\\partial}{\\partial \\alpha}\n", "e34643501551a80c20578b85f515b146": "\\dim W(\\lambda) = \\prod_{(i,j) \\in Y(\\lambda)} \\frac{r+j-i}{\\mathrm{hook}(i,j)},", "e34674a81d429f5b14b2f9008d3c5f2a": "(T+S)(\\omega):= T(\\omega)+S(\\omega),\\qquad (\\lambda T)(\\omega):=\\lambda T(\\omega).", "e3468741889257d66fb687c0c5327bcf": "\\phi(T) = \n\\begin{cases} \n1 & \\text{if } T > t_0 \\\\\n0 & \\text{if } T < t_0\n\\end{cases}", "e346f6493d7b138932104c5cb966ead2": "\\frac{-a}{-b} = \\frac{a}{b}", "e34700921c53c7a8b92fa7db547ebbf4": " \\models_{\\mathrm P}(\\phi \\rightarrow \\psi)", "e347164d8cc774b623f63491e4f16ebf": "F.", "e3471adbf15b3443c7f51cc3c8dd6b93": "c \\in \\{1, \\ldots, N\\}", "e347484761256aab02b0b5b16f7ae636": "p = \\hbar k,", "e34763c02e58cb98db3737e0bc568c7c": "V = \\frac{ n_d - 1 }{ n_F - n_C }", "e34787d5ffc05a29996c133c56b556cd": "\\frac{d T(A, R)}{d R} = A\\cdot 2\\cdot R.", "e347a52c15a55e0612aaecc889c0cfcb": "\\varphi: \\mathrm{hom}_{\\mathcal{C}}(FY,X) \\cong \\mathrm{hom}_{\\mathcal{D}}(Y,GX)", "e347ef8b4051d49548afc90a463026a0": "H^{2n}(C_\\phi) = \\langle\\beta\\rangle.", "e3482350068e67169cf9be3aa4385668": "(-1)^k\\cdot W_0(k).", "e34836d753ecb182bd730bc1c89dfcfc": "X_tY_t = X_0Y_0+\\int_0^t X_{s-}\\,dY_s + \\int_0^t Y_{s-}\\,dX_s + [X,Y]_t", "e3483cdd561e281bb392687a40d2dfbe": " \\tfrac5{36} ", "e34850782aca7f80c64c3e3eeaf10e82": "\\frac{R}{c}", "e3487804d53f9741b6c34845ad763d23": "\n\\mathbf{B}(t) =\n\\frac{\n\\sum_{i=0}^n b_{i,n}(t) \\mathbf{P}_{i}w_i\n}\n{\n\\sum_{i=0}^n b_{i,n}(t) w_i\n}\n", "e3487c36fd8f0a414a55ecbe4c549162": "\\pi\\delta_{ij}\\,\\!", "e348a027e3a1ee2b9bdb81ae58c91b35": "O(n^2\\log(r)\\log(q))", "e348a703091178aa5334984fb6175efd": "q_{m}", "e34908b44c68a5e1421baa0df9e59ea5": " \\nabla_{(\\alpha}K_{\\beta\\gamma)} = 0 \\,", "e3495f79a221a8b101ce87167c78286b": "\\alpha_1=\\frac{A_{32}-A_{23} }{2} ", "e349619823c8d1be505f1166701e9f2a": "7\\frac{10^2-1}{10-1}", "e3499e15ed8c9a2a6954c09b1afda692": "\\sec \\varphi - \\tan \\varphi = e^{-x/a}.\\,", "e349deefcf20a8c950a2fc7a12133016": "|n,g\\rangle, |n,e\\rangle", "e349eb5949d1614c230ddf282b76d440": " \\gamma _s = \\gamma _0 \\theta (A - \\ln \\theta) \\,\\! ", "e349f3bc64cc75f30fa4f5ac539d7a57": "\\bar{\\sigma\\,\\!}", "e34a595323be7ed74bfce59f66545567": "N_1+N_2=M", "e34aa96d9c590dad2d673233c6d4b311": "S_n = \\sum_{k=0}^n a_k b_k", "e34ac0a54e404b3cec606d204280c2dc": "1\\leq p \\tau", "e35bd4238e622251dd504260f79fa618": "\\overline{F}(x) \\equiv \\Pr[X>x] \\, ", "e35c0f648bf5405856c21894e0158368": " \\rho = \\sum p_i \\rho^A_i \\otimes \\rho^B_i ", "e35c2cb913c0e74327a06290d9db6f70": "\nw_0(n) = \\frac{I_0\\left(\\pi\\alpha \\sqrt{1-(\\frac{2 n}{N-1})^2}\\right)}{I_0(\\pi\\alpha)}", "e35c3556a333b3d26dfcf9cfc9ee9431": "R_s=\\sqrt{x^2+y^2}=\\sqrt{\\left(\\frac{2v^2\\cos^2\\theta}{g}\\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)\\right)^2+\\left(m \\frac{2v^2\\cos^2\\theta}{g} \\left(\\frac{\\sin \\theta}{\\cos \\theta}-m\\right)\\right)^2}", "e35c9342763f403f3bb208f8912ee7d9": "\n\\lim _ {T\\rightarrow\\infty} \\frac 1{2T} \\int_{-T}^T x(t)^2 dt.\n", "e35cb183fd3f61320411396672d847b3": " \\|v\\|_{\\infty}= \\|A^k v\\|_{\\infty} \\ge \\|A^k\\|_{\\infty} \\min_i (v_i), ~~\\Rightarrow~~ \\|A^k\\|_{\\infty} \\le \\|v\\|/\\min_i (v_i) ", "e35cc3d0f8305c1fa08b0ff9b13da143": "\\Phi \\nvdash \\bot", "e35ce663763eb4db5d302f4601e281bf": "\nJ_{\\alpha}^{\\prime} = \n\\int_{0}^{\\infty} \\frac{dx}{\\left( x + b^{2} \\right)^{3} \\sqrt{\\left( x + a^{2} \\right)}}\n", "e35cf0f2970cbb53973fa44ba8db7142": "G \\to \\widehat{G}", "e35d8c229a361ebc1eb4082c6c4c8d65": "P = A - 0.9 K - 2.5 L", "e35df863fc94d6c6fa5b34788e170373": "\\sqrt{b}", "e35e48d79b595b3d5845a24bc2eda7b9": "\\left |E_{n}T -2\\pi k_{n}\\right |<\\delta", "e35e4cbe1b95f729af575bb0b901d680": " \\{ X_t : t \\in T \\}", "e35e4de1d91b8e59e20e21d363df99e7": "\\overline{Y}_j - \\overline{Y}", "e35e77210cd4ea8a02609a8ee7b113f7": "\\sum_{n=0}^\\infty \\frac{\\pi^{n}}{n!} = \\frac{\\pi^{1}}{1} + \\frac{\\pi^{2}}{2!} + \\frac{\\pi^{3}}{3!} + \\frac{\\pi^{4}}{4!}+ \\cdots", "e35e8d0717b26e843f9c449c75a655a1": "S_{RNN(n)} = S_{RNN(n-1)} + 1 + S_{NBR(n-1)}", "e35eb8c05ada620c35f398824f03d8c5": " x_1", "e35ed233e2bf85382c9c2834416d4fbc": "V_r = \\dot{r} = \\frac {H}{p} \\cdot e \\cdot \\sin \\theta = \\sqrt{\\frac {\\mu}{p}} \\cdot e \\cdot \\sin \\theta", "e35ee4b5c77c3ed5aa785153bf7d1347": "\\phi(t) = \\arg \\left\\{ x_\\mathrm{a}(t) \\right\\}.\\,", "e35f23235dc105968d7244015b78203d": " \\nabla\\cdot(\\nabla\\psi)=\\nabla^{2}\\psi ", "e35f486891b32b8a12f8728d33c48cfa": "\na - x_{n+1} = \n- \\frac{2 f'(x_n) f'''(\\xi) - 3 f''(x_n) f''(\\eta)} {12 [f'(x_n)]^2 - 6 f(x_n) f''(x_n)} (a - x_n)^3.\n", "e35f4f965788a1298fc336f107d67d7d": "N\\times M", "e35f73afb36e13d1b5236ffeae85d6a3": "\\{X_n : n\\in[0,\\infty)\\}", "e35f740b14f54239c907478fbaeb9838": "\\,\\mbox{T}(da) = \\mbox{T}(0) + \\frac{d\\mbox{T}(0)}{da} da + ... = 1 - \\frac{i}{h}\\ p_x\\ da", "e35fa317ad63cb9aecc536dace187395": "\\langle \\mathbf R \\rangle = 0", "e35fc3ac23c7b77b0d1d5b55580bc353": "\\,f_i(v,w)", "e35fccc6a1a7b0debd94f8e0a9ea1ed2": " S_{xyz} ", "e36033987ff173cc8aab84547601f52f": " \\lang n_1 n_2 \\cdots n_N; S | n_1 n_2 \\cdots n_N; S\\rang = 1, \\qquad \\lang n_1 n_2 \\cdots n_N; A | n_1 n_2 \\cdots n_N; A\\rang = 1. ", "e3605ab9b28b21ce658fdf1772ae1257": "\\begin{matrix} {11 \\choose 2}{4 \\choose 2}^2{36 \\choose 1} \\end{matrix}", "e360fd7d7e91a646e0c183da40baf09c": " \\tfrac14 - \\tfrac16 \\sqrt3 ", "e36103e72c194f6830f514d8e553cd7e": " f(p) = p ", "e3611864265dbb8ba66d3fb97ef1b2ab": "\\rho =\\left( \\mathcal{P}+\\mathcal{Q} \\right)\\rho ", "e361678cfc672cde37e170eb2c85b9af": "f \\colon R \\to S", "e36212d09bece62cca4da3237f9712ba": "\\theta\\;", "e362176175ce1496f332a2a1da7e119d": "\\sin 2\\theta = 1\\,\\!", "e36255cb56573ef2555858f0599386e6": "q = \\frac{m}{16} + 8\\, \\left( \\frac{m}{16} \\right)^2 + 84\\, \\left( \\frac{m}{16} \\right)^3 + 992\\, \\left( \\frac{m}{16} \\right)^4 + \\cdots.", "e3625a90866343d0cc90fb8cde0b62cd": "\n\\frac{\\partial H}{\\partial P_{m}} = \n\\frac{\\partial H}{\\partial \\mathbf{q}} \\cdot \\frac{\\partial \\mathbf{q}}{\\partial P_{m}} + \n\\frac{\\partial H}{\\partial \\mathbf{p}} \\cdot \\frac{\\partial \\mathbf{p}}{\\partial P_{m}} \n", "e36263860644b77d98037cf9898c5aa0": "c/c'=1\\,", "e362b3f2c6ab47909e9e726671a385f7": "F(b) - F(a) = \\int_a^b f(x)\\,dx,", "e36314e624d2b2ca257e1f1ecb381f93": "2a", "e36357ed1c3f20ec5cc9ef6c7a62e937": "(x-s)^k + n\\cdot(y-s)^k = 1.", "e36374e7c7bb88dcecdbe464b97488a9": "p,q \\in G", "e363aadf98b082bc714796e1dc8122a7": " \\text{Var}(\\widehat{\\boldsymbol{\\beta}}_{ols}) = \\text{MSE} (\\widehat{\\boldsymbol{\\beta}}_{ols})", "e363d89e9b5601e915631ff47332fc21": "r_u = \\frac{a}{2} \\sqrt{2} \\approx 0.7071067 \\cdot a", "e363dc94de285751b7df8e17c2627a52": "\\operatorname{Inverse-Gamma}(\\nu, \\sigma_0^2)", "e364744e25ec11db68b569a830614729": "c_{e'} \\neq 0", "e364adc43e9011c90f8572cfd66aca3d": "\\mathbb RP^2", "e36559358df306434f6b29e315ec3aac": "\\int_0^\\infty \\sin ax^2\\cos 2bx\\ dx=\\frac{1}{2}\\sqrt{\\frac{\\pi}{2a}}(\\cos \\frac{b^2}{a}-\\sin\\frac{b^2}{a})", "e36592787baa06629806aed9235452c5": "|{\\psi'}\\rangle=V|{\\psi}\\rangle", "e365cb5790e8254c05af7ae982779c57": "A \\equiv_m B", "e366322ea002e6ff95680c2bfcc29112": "\\left [ -\\infin < \\theta < \\infin\\right ] \\longleftrightarrow \\left [-\\infin < E < \\infin \\right ] ", "e366802eda17b8f775a9194743eefb96": "\\int_a^b f(x)\\,dg(x) = \\int_a^b f(x)\\,dg_1(x)-\\int_a^b f(x)\\,dg_2(x),", "e366883b227c8f38755c23615f3d4e3c": "K^\\ominus =\\frac{{\\{R\\}} ^\\rho {\\{S\\}}^\\sigma ... } {{\\{A\\}}^\\alpha {\\{B\\}}^\\beta ...}", "e366d8612cac24b6f5583e0f698dec2c": " r = e^\\rho. \\, ", "e366e18368a70fc1a2bed5faa3ce5429": "\\begin{align}\n{}_7F_6 & \\left(\\begin{matrix}a&1+\\frac{a}{2}&b&c&d&e&-m\\\\&\\frac{a}{2}&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\\\ \\end{matrix};1\\right) = \\\\\n&=\\frac{(1+a)_m(1+a-b-c)_m(1+a-c-d)_m(1+a-b-d)_m}{(1+a-b)_m(1+a-c)_m(1+a-d)_m(1+a-b-c-d)_m}\n\\end{align}", "e3670ae140dd779ca79fbc96bad27a5b": " N = ", "e3679ceece568488170009494d99d268": "\nY_{\\bullet[i]}\\mathrm{\\ versus\\ }X_{i\\bullet[i]} \n", "e36806ea6368e47e0995e025954dc6bc": "V^{e}", "e3682e0032a17bcf652578199f06cbc4": "P = 1/W,", "e368959ba6653f8bfb89abec78c9840f": " W_x(t,f)=\\int_{-\\infty}^\\infty x(t+\\tau/2)x^*(t-\\tau/2)e^{-j2\\pi\\tau\\,f} \\, d\\tau", "e368bb98d0c7e59622e9967ce65298d0": " \\rho c \\frac{\\partial T} {\\partial t} = \\frac{\\partial \\frac{ k \\partial T} {\\partial x}} {\\partial x} + S ", "e368fa3e61aedacdf26c80589b5e50a8": "I^pH_{n-i}(X) = I^pH^i(X) = H^{i}_c(IC_p(X))", "e368fdf26054acf11ed79e7269a42f90": "\\left(\\sqrt q \\int_{\\mathbb R} |g(y)|^q \\,dy\\right)^{1/q}\n \\le \\left(\\sqrt p \\int_{\\mathbb R} |f(x)|^p \\,dx\\right)^{1/p}.", "e36989edeca17e28be254aa5f7ce61ad": "\n1~\\mathrm{W} \\times 10^\\frac{25~\\mathrm{dB}}{10} = 316.2...~\\mathrm{W}", "e3699e7be2d33718dc0a9e4e40954a41": "\\tilde{P_n}(x) = \\frac{1}{n!} {d^n \\over dx^n } \\left[ (x^2 -x)^n \\right].\\, ", "e36a2f7d6e790db6008950ae3d2dd693": "\\hat{\\Pi}\\equiv I-\\Pi", "e36a406bfd3c947f6a7a5e37d5756477": "\nf(x,y) = \\frac{x^2}{x^2+y^2}\n", "e36a6d19982124ac5256628d2024157d": "H\\gg S_0", "e36a6fb71be3e2447eb1803fb630108f": "\\alpha_2:=V,", "e36ad4d912ae8182be54b6e9ed497979": "\\frac{2Gm}{c^2}=r_s", "e36b5a6de3965126dd61088e0a283dd4": "Q = [M - (M_d + M_e)]c^2", "e36b62dd584787251481765ea8b878c2": "u,v \\in K^n", "e36bcec2629b58ace2cfbb26f492d4f6": "x \\notin P", "e36be4e3538e06f869df29a03a69d288": " D_{\\mathrm{KL}}(P\\|Q) = \\operatorname{Tr}(P( \\log(P) - \\log(Q))). \\!", "e36c3571e91ef0b2f99b46d97beb8413": "k = \\pi / a", "e36c428043c6a07667478e160337e391": " \\mathrm{Ref}(\\phi) \\, \\mathrm{Rot}(\\theta) = \\mathrm{Ref}(\\phi - \\theta/2). \\ ", "e36c597c44a622a59347f2e79096f492": " \\frac{f/1}{(\\sqrt{2})^3} ", "e36c8e6d243ab68a9dd9a246aec82278": "\\cos\\delta = \\frac{(g-1)\\cdot 1+g\\cdot 1}{\\sqrt{(g-1)^2+g^2} \\sqrt{3}} =\\frac{2g-1}{3} =\\frac{\\surd 5}{3}\\approx 0.74535599.", "e36d57dc2fc466247775a881ed14d5b7": "\\mbox{At x=1}, B_1=B_2=0.5; \\frac{dB_1}{dx}=\\frac{dB_2}{dx}=1", "e36d617844245be03f6756194a7b32f6": " \\beta_{1}y(b)+\\beta_{2}y'(b)=0\\qquad\\qquad\\qquad(\\beta_1^2+\\beta_2^2>0),", "e36dab301b4b90927a973481412368e7": "PC_x", "e36dae998e5348eb6641631c20b50e6d": " Q_j = \\sum_{i=1}^n \\left(\\mathbf{F}_i\\cdot \\frac{\\partial \\mathbf{V}_i}{\\partial \\dot{q}_j} + \\mathbf{T}_i\\cdot\\frac{\\partial \\vec{\\omega}_i}{\\partial \\dot{q}_j}\\right),\\quad j=1, \\ldots, m.", "e36dbaff937d4eb1f4c10abb8397acc0": "\\lambda \\in \\mathcal{P}^*", "e36e0cdbde311d5e1c0622e6fd058c3f": "\\hat{x}=\\mathbf{x} + \\varepsilon \\mathbf{p}\\times\\mathbf{x}\\quad\\text{and}\\quad\\hat{X}=\\mathbf{X} + \\varepsilon \\mathbf{P}\\times\\mathbf{X}.", "e36e1494ecbd92d56b15c1556375489b": " D_{f}= \\frac{L!}{N_{f}!(L-N_{f})!}. ", "e36e41bab0835f74f43b87a7914c273f": "\\gamma \\in GL_2^+(\\mathcal O_F)", "e36e7697b28e1e104097a74485767f4f": "\\Delta(x) = D(x)-x\\log x - x(2\\gamma-1)", "e36e91f47c5cb8d7893cd1a2a0a09f51": "\\zeta^{ki} \\equiv \\cos_k(i)+j\\sin_k(i). \\, ", "e36e957c33d8e9d91fc477ac6aee0013": "\\epsilon (a,b,c,d)=e^{{\\rm{i}}\\pi [\\frac{a+d}{12c} - s(d,c)\n-\\frac{1}{4}]}\\quad(c>0).", "e36ea1c92f4193cbdd66681938aa4b0a": "\\frac{\\ln(5280^3(236674+30303\\sqrt{61})^3+744)}{\\sqrt{427}}", "e36eb184dba1e5888434f5dee6b1cf20": "P = 2 (\\pi r+a)", "e36ece2c97ed25b9b10d0c810b7e5e67": "T_{x,1\\#}\\nu\\in\\mathrm{Tan}(\\mu,a)", "e36edacf3c2086466f3bbf6035326f6a": "\\Delta = e^{-\\varphi} \\left(\\frac{\\partial^2 }{\\partial x^2} + \\frac{\\partial^2 }{\\partial y^2}\\right)", "e36f19471a6318d6ada1119057c697fd": "\\scriptstyle p_2 \\,", "e36f8dc6f2bb610ffacfe7da4a25644b": "G_H", "e36fedfe8a263b25427596d5b4147487": "\\ln(\\operatorname{var_{GX}} (\\Beta(\\alpha, \\beta))) = \\ln(\\operatorname{var_{G(1-X)}}(\\Beta(\\beta, \\alpha))) ", "e37041dd4be9c18d44f487148f5364c9": "G(i)", "e37093beb66ea47e58fc750c5bf161ae": "\\forall^\\mathrm{st}z\\,(z\\text{ is finite}\\to\\exists y\\,\\forall x\\in z\\,\\phi(x,y,u_1,\\dots,u_n))\\leftrightarrow\\exists y\\,\\forall^\\mathrm{st}x\\,\\phi(x,y,u_1,\\dots,u_n).", "e370b2b165b5c130d4bd3333df2da54a": "\n\\begin{array}{rl}\n\\mbox{Ax. 1.} & P(\\varphi) \\land \\Box\\; \\forall x [\\varphi(x) \\rightarrow \\psi(x)] \\rightarrow P(\\psi)\\\\\n\n\\mbox{Ax. 2.} & P(\\neg \\varphi) \\leftrightarrow \\neg P(\\varphi)\\\\\n\n\\mbox{Th. 1.} & P(\\varphi) \\rightarrow \\Diamond\\; \\exists x\\; [\\varphi(x)]\\\\\n\n\\mbox{Df. 1.} & G(x) \\iff \\forall \\varphi[P(\\varphi) \\rightarrow \\varphi(x)]\\\\\n\n\\mbox{Ax. 3.} & P(G)\\\\\n\n\\mbox{Th. 2.} & \\Diamond\\; \\exists x\\; G(x)\\\\\n\n\\mbox{Df. 2.} & \\varphi\\;\\operatorname{ess}\\;x \\iff \\varphi(x) \\land \\forall\\psi\\lbrace\\psi(x) \\rightarrow \\Box\\; \\forall x[\\varphi(x) \\rightarrow \\psi(x)]\\rbrace\\\\\n\n\\mbox{Ax. 4.} & P(\\varphi) \\rightarrow \\Box\\; P(\\varphi)\\\\\n\n\\mbox{Th. 3.} & G(x) \\rightarrow G\\;\\operatorname{ess}\\;x\\\\\n\n\\mbox{Df. 3.} & E(x) \\iff \\forall \\varphi[\\varphi\\;\\operatorname{ess}\\;x \\rightarrow \\Box\\; \\exists x\\; \\varphi(x)]\\\\\n\n\\mbox{Ax. 5.} & P(E)\\\\\n\n\\mbox{Th. 4.} & \\Box\\; \\exists x\\; G(x)\n\\end{array}\n", "e370e029f6fa2b48afd48652e6fccace": "Z=(Z_t)_{t\\geq0}", "e371027b0198df522f70bb8b9f301663": "||h'||\\leq 1", "e371196802369519b8d75c79fb159a2f": "\\pi(v)", "e3711973f639d7909e93eb59e7818d8d": "I_S = 10^{-6} ", "e37145f2a115e1cfc021f17f4e76f9c1": "\\frac{a}{2}[\\text{0 1 1}] \\rightarrow \\frac{a}{6}[\\text{1 1 2}] + \\frac{a}{6}[\\text{-1 2 1}]", "e371527bb6c78a6700d9729171e73b86": "T(\\omega)", "e37181741bd5469f61ea6e5d61efc47b": "\n V=\\kappa\\left(1-\\frac{A^*-B^*}{S_G-S_B}\\right) \\;.\n", "e371cc0db9311b75ba71ecbe570169f8": "{\\mathbf v}", "e3724d733609756334b1769298d91e6f": "\n T_{11} = \\cfrac{\\sigma_{11}}{\\lambda} =\n 2C_1\\left(\\lambda - \\cfrac{1}{\\lambda^3}\\right)\\left[\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1}\\right]~.\n ", "e3729b0943789c4e823ab3a418ba5727": "x = x'Ax'', y = x'wx''", "e3731cc44fdb9cbef6be183dd7ef9d29": "\\sum_{k\\in\\mathbb{Z}}\\exp\\left(-\\pi\\cdot\\left(\\frac{k}{c}\\right)^2\\right) = c\\cdot\\sum_{k\\in\\mathbb{Z}}\\exp(-\\pi\\cdot(kc)^2).", "e37344d797d54c6312c0d5e54432d9cd": "\nx(t+\\epsilon) - x(t) \\approx \\sqrt{\\epsilon}\n\\,", "e373d35a4118c80365724b34b8ca3bd0": "J_{0,3}\\oplus J_{i,2}\\oplus J_{i,2}\\oplus J_{7,3}", "e373dc7e3bab648ef8c972255c4605cb": "W_\\mathrm{Theil-L} = \\overline{\\text{Income}} \\cdot \\mathrm{e}^{-T_L}", "e3742f99f92455557fae0bb1ce8cd23b": "H(2)=1", "e3745300edd209345d42153713e58da4": "x^2\\frac{d^2y_n(x)}{dx^2}+2(x\\!+\\!1)\\frac{dy_n(x)}{dx}-n(n+1)y_n(x)=0", "e374a1b27865ff7e3ed7c2a10778c13a": "O(\\log^i n)", "e374b07b5aa50ba13c273a4a0c95c875": "f(tx+(1-t)y) \\le t f(x)+(1-t)f(y) - \\frac{1}{2} m t(1-t) \\|x-y\\|_2^2 \\,", "e374fff84a44f9d786c317cb4b737426": "\n \\begin{align}\n l_x & = \\int_0^1 \\left| \\cfrac{d \\mathbf{x}}{d s}\\cdot\\cfrac{d \\mathbf{x}}{d s} \\right|~ds\n = \\int_0^1 \\left| \\left(\\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\\cdot\\cfrac{d \\mathbf{X}}{d s}\\right)\\cdot\n \\left(\\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\\cdot\\cfrac{d \\mathbf{X}}{d s}\\right) \\right|~ds \\\\\n & = \\int_0^1 \\left| \\cfrac{d \\mathbf{X}}{d s}\\cdot\\left[\n \\left(\\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\\right)^T\\cdot \\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\\right]\n\\cdot\\cfrac{d \\mathbf{X}}{d s} \\right|~ds\n \\end{align}\n", "e3752d4f40d94bb90cc57caed0e04054": "C = A \\cdot B \\mod M, ", "e375a961658132693a6569fca4f5a77b": "\\alpha = n^2 a^3\\,", "e375be010d8010c4c961299a1c2edcfe": "I(c)=\\log_2 |C|", "e376174c486af787ba890fc746c0045d": "k = k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1 \\mod N,", "e3771f81a028e191f8e336f9bd30b494": "\\frac{a_{n+1}}{a_n} \\le \\frac{b_{n+1}}{b_n}", "e377ed31b3880693214ac1759509c1a5": "g(y_1,\\dots,y_n;\\theta)=g_1(y_1; \\theta) h(y_2, \\dots, y_n | y_1),\\,", "e3780c44fb6ea8f982bf8635eaa5f6c8": "\\widehat{f}: \\widehat{X} \\to \\widehat{S}", "e37817e5a1c166ea8c106d3a090e326c": " \\text{EVaR} ", "e3781ab58fbb818965cd41a561cfacf4": "\n PIN_t=\\frac{\\alpha_t \\mu}{\\alpha_t \\mu + 2\\epsilon}\n", "e37825f0b51b3f28fdcd12efd5c27989": "\\Box (K \\rightarrow Q)", "e37840c0c4ffb51e354bad7342c9227d": "I(at : S^r \\cdot N^{rr} \\cdot N^r \\cdot S \\cdot N^l) = [x] \\cdot y \\cdot [y] \\cdot at(x,z) \\cdot [z] : T^r \\cdot E^{rr} \\cdot E^r \\cdot T \\cdot E^l", "e37865bf8cf8dd477d9368bbb61ccec6": "\\alpha_{t-1}(x_{t-1})", "e3789371022c0f81928cbad43ca81857": "{1 \\over {2 + 3}}(3(\\log 5 - \\log 3) + 2(\\log 5 - \\log 2))", "e378c7a0261fec112288baaf474064b9": "\\alpha=0.1", "e37959d6f0274557440f025b8fe5e153": "f'(r) \\not\\equiv 0 \\pmod{p}", "e37993881a445f93b01cea1306e8d98b": "\\rho M=\\{A\\mid M\\vdash T(A)\\}.", "e379b850f85c2e50f4aeaa31339cc6aa": "1 - a z^{-1}", "e37a4515123e86862ec2e592a9eb2527": "R_jf(x) = c_d\\lim_{\\epsilon\\to 0}\\int_{\\mathbf{R}^d\\backslash B_\\epsilon(0)}\\frac{(t_j-x_j)f(t)}{|x-t|^{d+1}}\\,dt", "e37a4e7483170d451785a5e124870968": "p_1 =a_1 x+b_1y+ \\cdots +h_1z\\ge 0,", "e37a733cc5e5d137709ea3db7a2c664e": "\\int x^2 r\\;dx= \\frac{xr^3}{4}-\\frac{a^2xr}{8}-\\frac{a^4}{8}\\ln\\left(x+r\\right)", "e37a8e3e427592d0bd82cd6c0efa9ce0": "L^{-}=\\lim_{x\\to x_0^{-}} f(x)", "e37b2b23820db9f15fc8e5320e063e33": "\\theta = 0\\,", "e37b77363dfed689d3fcb24135f57f6c": "\\in \\Gamma", "e37bbae678eb5cba637791b398a471fa": "A, B \\in \\mathcal{R}", "e37c2cc52eae0f0df33a95d8130a7be6": "Z_0 = 50 \\ \\Omega", "e37c62f88549faa71c660048616799a5": "\n\\begin{align}\n\\dot{\\hat{\\mathbf{x}}}(t) &= f\\bigl(\\hat{\\mathbf{x}}(t),\\mathbf{u}(t)\\bigr)+\\mathbf{K}(t)\\Bigl(\\mathbf{z}(t)-h\\bigl(\\hat{\\mathbf{x}}(t)\\bigr)\\Bigr)\\\\\n\\dot{\\mathbf{P}}(t) &= \\mathbf{F}(t)\\mathbf{P}(t)+\\mathbf{P}(t)\\mathbf{F}(t)^{\\top}-\\mathbf{K}(t)\\mathbf{H}(t)\\mathbf{P}(t)+\\mathbf{Q}(t)\\\\\n\\mathbf{K}(t) &= \\mathbf{P}(t)\\mathbf{H}(t)^{\\top}\\mathbf{R}(t)^{-1}\\\\\n\\mathbf{F}(t) &= \\left . \\frac{\\partial f}{\\partial \\mathbf{x} } \\right \\vert _{\\hat{\\mathbf{x}}(t),\\mathbf{u}(t)}\\\\\n\\mathbf{H}(t) &= \\left . \\frac{\\partial h}{\\partial \\mathbf{x} } \\right \\vert _{\\hat{\\mathbf{x}}(t)} \n\\end{align}\n", "e37cc3180ef29478fa1d0a379a786890": "e = (1,\\dots, 1)' \\in R^{m \\times 1}", "e37d03976bd2ee8a52bdcf163654156a": "2 I", "e37d1f94814483f550f60227c0fa7469": "x^7+x^5+x", "e37d4dd9cb2187515959d06c9cde45a1": "\\mathrm{W}(x,p)", "e37d86d22a40ad833c7f0bf2d07e4b91": "{V}^{\\bullet\\bullet}_{O}", "e37d93bfadcd79c1388994e1a386da41": "\\scriptstyle 6/ \\pi^2", "e37dc9058d95491bcb6b896cacf18ee5": " a_1 = 1 ", "e37ddb6050d73584c127109ebf0270ab": "\\phi_l = \\frac{180^\\circ + (l - 1)360^\\circ}{P-Z}, l = 1, 2, ..., P - Z", "e37de75d1608a706aece88a73003af08": "\\frac{\\mathrm{mm}}{\\mathrm{hr}} = \\left ( \\frac{10^{(dBZ/10)}}{200} \\right )^{5 \\over 8}", "e37e13ddd569f9f69eb0c175bfa31820": "a'-a^2+\\frac{6}{x^2}=0", "e37e150557fd648ca200c344ee59023e": " \\frac{dx}{dt}=rx+x^3. ", "e37e4fa5eecf006c0d215a5237af7f4b": "X^{<\\omega}", "e37e6a0dee6926c66300501086b13dd2": "\\bar{m}=2m+1, k=1", "e37ecd34931ec1c5285c1ee612b2f8e2": "N'(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{1}{2}x^2} ", "e37eea89ead881aa479afbaf209291de": "\\mathbf{p}(\\mathbf{r},t)=\\mathbf{p}(\\mathbf{r})e^{-i\\omega t} \\, ,", "e37f6990cf492e9bd16b2135c28c7434": "\\lim_{t\\to\\infty} \\widehat{d}(w,t) = d(w).", "e37f75338e2b8d6926e4c09b722af238": "e^{+2 \\pi i \\omega}", "e37f8858c8985e6fa47b4184344f0427": "D_R", "e37f98307be4ff5d498a179c2c9cb56e": "\\mathbf{P}_{\\theta} (A) = \\left( \\int_{A} e^{- \\theta \\varphi(x)} \\, \\mathrm{d} x \\right) \\Big/ \\left( \\int_{\\mathbf{R}^{d}} e^{- \\theta \\varphi(y)} \\, \\mathrm{d} y \\right)", "e37fc81b35bb04ca73604d2639f9082d": "\\scriptstyle r_0", "e37fcc6d0b355f596c47a23bee4302c1": "F^{\\mu \\nu}", "e38080faf665b4b20ef3db06bd915275": "E^\\mathrm{tot}(\\mathbf{x},t)=\n\\sum_{n}\\frac{E_n^\\mathrm{ret}(\\mathbf{x},t)+E_n^\\mathrm{adv}(\\mathbf{x},t)}{2}.\\ ", "e38088ae5fe368b1a4735fa3a35b3e19": "\\scriptstyle \\leq9\\times10^{-10}", "e380d615ef43afb5e8dbbf9186627bef": "w'=(U'u-Uu')/u^2=(U/u)'", "e380db6a66bf6faeb353a8440f40f29e": " \\wedge ... \\wedge (\\forall x_1...x_m, y_1...x_m) [(Eq(x_1, y_1) \\wedge ... \\wedge Eq(x_m, y_m)) \\rightarrow (Z(x_1...x_m) \\equiv Z(y_1...y_m))] ", "e3812de8ecfcf74a2c0b04a16c101741": " p; Inv; inv; \\lnot; I", "e381c3a3f7a1929e8c2e8babe7753aea": "\n f(x) = \\frac{n}{x [1/2(x/x_0)^{-\\alpha} + \\lambda + a/2(x/x_0)^\\alpha ]}, \\qquad x > 0,\n", "e381c5a4c913ffc5798e28f646687a89": "\\varphi^{-1}(t)=\\varphi(St)", "e381dc4a147f9409ed01d7a8401a514d": " = \\lim_{h_1 \\to 0} \\frac{\\lim_{h_2 \\to 0} \\frac{f(x+h_1+h_2)-f(x+h_1)}{h_2}-\\lim_{h_2 \\to 0} \\frac{f(x+h_2)-f(x)}{h_2}}{h_1}", "e381ea3761f27061ae351d3c264a91c8": "\\begin{bmatrix} \\dfrac{a_{22}}{a_{12}} & \\dfrac{-\\Delta \\mathbf{[a]}}{a_{12}} \\\\ \\dfrac{-1}{a_{12}} & \\dfrac{a_{11}}{a_{12}} \\end{bmatrix}", "e3823d0d685328498a843c5c87850a23": "z\\mapsto z+\\omega", "e3824539c1e3a84a34b29423928d3fb7": "Q \\to R", "e3824bb61ef1bcc5aa440c714e174bd4": "u_i:X\\longrightarrow\\mathbb{R}", "e3826c22c3547b33d0d43ae495d02b3a": "\\psi \\land \\varphi \\,\\!", "e3828db21fc02c79b0f5ea390867fc49": "h =\\,", "e38295bf9b3995ad077b2c05e4d8f013": "\\scriptstyle 1/P.", "e382d344d2899040b895a20e08c8738d": "\n\\begin{align}\n& {} \\qquad \\frac{1}{k} \\binom{n-2}{k-2} \\left( \\frac{n-k}{k-1} \\sum_{i=1}^{n}f(x_i) + nf\\left(\\frac1n\\sum_{i=1}^{n}x_i\\right) \\right)\\\\[6pt]\n& \\ge \\sum_{1 \\le i_1 < \\dots < i_k \\le n} f\\left( \\frac1k \\sum_{j=1}^{k} x_{i_j} \\right)\n\\end{align}\n", "e3831841e6f9f5a22c3f72cb17ab5644": "\\partial_{xy}f\\ne\\partial_{yx}f", "e3833d6edf68b31936b746bb791d1926": "\\displaystyle{XY^n-Y^nX=2n Y^{n-1}}", "e3845bba342d109882ba25b98e07498c": " \\mathrm{all} \\ s \\,", "e3847e219e7f23622b70d2dcc7bffd86": "x \\to b", "e3856ffb0fa27301141805559c46ddfb": "x \\vee y = x \\vee z", "e3858aebfa35fbcb3a98269bad85a5eb": "m\\frac{\\mathrm d\\vec{v}}{\\mathrm d t}=q(\\vec{E} + \\vec{v}\\times\\vec{B})", "e385e3e85a0f6e62a9bab14b4e4aa87f": " x= \\varphi(t),\\,\\, z=\\psi(t)", "e3863908c4e43cc2c560944dd4a85d61": "\\sqrt{1-\\rho^2}y_1 = \\alpha\\sqrt{1-\\rho^2}+\\left(\\sqrt{1-\\rho^2}X_1\\right)\\beta + \\sqrt{1-\\rho^2}\\varepsilon_1. \\,", "e38669dff0d5f8c2400cb886bfe78022": "\\langle\\partial_\\mu m| n\\rangle=\\frac{\\langle m|\\partial_\\mu H | n\\rangle}{E_m-E_n},", "e3867919ab8bca1bd9d42b4f3ea3020b": "((ca)b)a=c((ab)a)", "e386a157969a4c7a2e3d7eb67dc7ab80": "\n[M] [\\ddot U] +\n[C] [\\dot U] +\n[K] [U] =\n[F]\n", "e386ab02d6599ce123551eef2e48f11c": "\\sum_{\\Omega(n)=2} \\frac{1}{n(n-1)} \\approx 0.17105", "e3871938da75b0741e57522517197bdd": "\n S(n,g) = \n \\Big\\{ \n \\left( m_1 , m_2 , \\dots , m_n \\right) \n \\Big| \\Big.\n m_i \\ge m_{i-1} ,\n m_i \\in \\left\\{ 1, \\dots, g \\right\\} ,\n \\forall i = 1, \\dots , n \n \\Big\\}.\n", "e38737892525643b04a4e75ae3b29c53": "L(b) = \\mathit{in}", "e387bd5d11c9b628cfd3d82273c7b83e": "\\langle a \\mid a^8 = 1\\rangle.", "e387f113e96241a6ea0e0fcae0f5dced": "\\rho(\\omega_{S,i}, \\omega_E)", "e3880e699b0e37d309036834c0c5346a": " \\mathcal C", "e388406dbe10eb6a400c863d2fb3c0c3": "\\zeta(3) = \\frac{1}{4} \\sum_{k=1}^\\infty (-1)^{k-1}\n\\frac{56k^2-32k+5}{(2k-1)^2} \\frac{(k-1)!^3}{(3k)!}", "e38874576e3328844e90f825f5de7835": "\nx[m] \\equiv \\sum_{k=-\\infty}^{\\infty} x[k]\\cdot \\delta[m-k],\n\\,", "e388cc91ca6d4e6e1495ea1406ca6d2f": "\\frac{1}{s} \\sum_j^s [d_j - y_j(t)] \\,", "e388e0de61910c12e1249cf3c6199699": "\\frac{\\partial u}{\\partial t} = \\alpha \\frac{\\partial^2 u}{\\partial x^2}", "e388e21621d01c02db82481ca6716632": "\nI(s) = \\frac{V_{in}(s) }{R + \\frac{1}{Cs}} = { Cs \\over 1 + RCs } V_{in}(s)\n", "e388e75a1c8ed29f8bbae60b05b9e26a": "\\frac{\\log\\left(GDPpc\\right) - \\log\\left(100\\right)} {\\log\\left(40000\\right) - \\log\\left(100\\right)}", "e388f46ae1201c24cf4f8d87061ef355": "\\epsilon^2\\approx 0", "e388fa5b63fc564c91f47e0e3e8a5d01": "T_m = \\frac{\\Delta H^\\circ}{\\Delta S^\\circ-R\\ln\\frac{[AB]_{initial}}{2}}", "e3890134a664c4c6ca8235b28ad3fe15": "\\lozenge P", "e389ac3bc838fba3f1fa36d75984c566": "|x| \\ll 1", "e38a6e22e3bc21af63b43dd1aab3da3c": "H=(V^\\prime,E^\\prime)", "e38a8c184942fe560b75e6b0903e2d9d": "\nT = \\frac{1}{2} Y F.\n", "e38ad09a205026a65d1361a22a7769c5": "\\pi^-", "e38b3c636aa6d1c6c5e8d81c4f257539": "[K_\\mu,K_\\nu]=0,", "e38b58d2b4ab82d9b34d189bb23f61e4": "\\sqrt{pr + qs}", "e38bd02df7d25524074798d67f29e934": " B_{w} ", "e38bf8d76284fefee1fa439eab953290": "d(1,0)=1", "e38c1b56792c7d7063b7237da47f7ae7": "{R f = \\nabla (-\\Delta)^{-\\frac{1}{2}}f}", "e38c434fdcf3fecb27c96af3b06e269d": "h\\nu_1", "e38c6b78528418c26a5d07dd10748ee7": "P(n)\\ll \\log^\\delta n", "e38c85372ef256b7744a5ebaef12fd4d": "E/V = K_1 \\left(\\alpha^2+\\beta^2\\right) = K_1\\left(1-\\gamma^2\\right). ", "e38c8e2c0a630ebbe9a0f69b489cc0c2": "1 \\in T", "e38c90c22cc1f8d40ae23e878c88b967": "\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;\n\\dot{G}\\dot{Y}\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!|}}\\;\n\\qquad \\text{or} \\qquad\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!|}}\\;\n\\dot{G}\\dot{Y}\n\\;\\overset{\\textstyle}{\\underset{\\textstyle}{|\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!-}}\\;\n", "e38d1e69c883b30a6518be4c71c02d71": "d\\mathbf{r}=\\dfrac{\\partial\\mathbf{r}}{\\partial q_1}dq_1 + \\dfrac{\\partial\\mathbf{r}}{\\partial q_2}dq_2 + \\dfrac{\\partial\\mathbf{r}}{\\partial q_3}dq_3 = h_1 dq_1 \\mathbf{b}_1 + h_2 dq_2 \\mathbf{b}_2 + h_3 dq_3 \\mathbf{b}_3 ", "e38d678d9e5259238356444b4eb7ed09": " (c_1\\mathbf{a}) \\cdot (c_2\\mathbf{b}) = c_1 c_2 (\\mathbf{a} \\cdot \\mathbf{b}) ", "e38d7ef5af03d23bc56e8f34eb278ef8": "\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\frac{\\partial v^i}{\\partial q^i} + \\cfrac{1}{2g}~\\frac{\\partial g}{\\partial g_{mi}}~\\frac{\\partial g_{im}}{\\partial q^\\ell}~v^\\ell = \\frac{\\partial v^i}{\\partial q^i} + \\cfrac{1}{2g}~\\frac{\\partial g}{\\partial q^\\ell}~v^\\ell\n", "e38e113c36ab5a7045b165f99ba9f5d9": "\\left | f^\\prime (p) \\right | ", "e38e28f0bc92f16407c166eee8af1203": "H_z^p(z_1,z_2)=\\sum_{i=1}^NH_z^{(i)}(z_1,z_2) ", "e38e567dc905170a6fb5815a4567c729": "g\\colon A\\rightarrow A'", "e38e71d6ce4cc8b73ac5d09b67ef8013": " \\sqrt{\\frac{\\beta}{\\alpha\\lambda}} (x - \\mu) ", "e38ed6a14fd32de149aab3abded9ad14": "\\Delta F = -k_{B}T ln (Q_{1}/Q_{0}) = - k_{B} T ln (\\frac{\\int ds^{N}exp[-\\beta U_{1}(s^{N})]}{\\int ds^{N}exp[-\\beta U_{0}(s^{N})]})", "e38ee3dcaddd325422529e95595c70d2": "\\dfrac{\\partial g_t(z)}{\\partial t} = \\dfrac{ 2}{g_t(z)-\\zeta(t)}.", "e38f4453157d7d9498ca0f6d460ce02e": "\\rho_{a / b}(R)", "e38f52d2dd8e896e3387c9ee887e561b": "0 + 120 \\angle 0^\\circ + 240 \\angle 120^\\circ = 120\\sqrt{3} \\angle 90^\\circ", "e38f83a6fb231c746d05f7df4fccbcd0": "H = \\gamma^0 \\left[mc^2 + c \\gamma^k \\left(p_k-\\frac{q}{c}A_k\\right) \\right] + qA^0.", "e38fcc6e17ef236e3327d8ee5c03ebbf": "|j_1m_1;j_2m_2\\rangle=|j_1m_1\\rangle \\otimes |j_2m_2\\rangle", "e38fdd1f3aca00846709a833a588947e": "\\frac{\\dot{Q}}{T}", "e38ffe89b3f75c8c55488bdb26f64474": "\\alpha=\\alpha_{i_1i_2\\dots i_p\\bar{j}_1\\bar{j}_2\\dots\\bar{j}_q}", "e3900bf31224f7678a5552479a3631d9": "\\deg (a * b) = \\deg (a) + \\deg (b)", "e39019106e7cfacdc38980daa2bb04bb": " \\rho(x,y) ", "e3902718520331398cb1a44ad13a4a27": "p(c)", "e3902ca1429c7ee28701b05c18d0c09b": "{{\\int_{0}^{t} C_p(\\tau)d\\tau} \\over \\mathrm{ROI}(t)}", "e390ba1c6cf193aa0eaf99a1aa316106": "L = \\frac{1}{l} \\mu_0 K N^2 A", "e39130f91ac09a058548a7f42dbf738e": "CTDI_w=\\frac{1}{3} CTDI_{100}^{central} + \\frac{2}{3} CTDI_{100}^{peripheral}.", "e3913718358eb0c76966c1d18355877c": "\\Pr(N > xn) = \\int_{N=xn}^{N=\\infty} \\Pr(N\\mid n)\\,dN = \\frac{1}{x^{\\alpha}}.", "e39194991c9052d4bd7943b98dfc9dcc": "2 = 2", "e391a80371e03c7a2025b8669a1eed55": "\\rho '", "e391cf3858431df8023521317ce0ddd3": " Q = \\pi_1 \\, ", "e3925e7c32b5a50d20db3505236da405": "\n\\sqrt{114} = \\cfrac{\\sqrt{32^2+2}}{3} = \\cfrac{32}{3}+\\cfrac{64/3} {2050-1-\\cfrac{1} {2050-\\cfrac{1} {2050-\\ddots}}}= \\cfrac{32}{3}+\\cfrac{64} {6150-3-\\cfrac{9} {6150-\\cfrac{9} {6150-\\ddots}}},\n", "e392cd3ed5aaa228d71ef3f6eacbd262": "\\sigma_{12}.", "e392d289bbd6ae416ea953ee90dca0e7": "\\int_0^1 G^{-1}(x)dx = E[X]", "e392d96575ab08eb98e52e4d996dea2e": "\\psi^q_n", "e392e902df2b0384d73b97f22cba3a2f": "\\zeta^2 s_2", "e392f4dbba841b3ae17187215f202fcd": "Z(C_4) = \\frac{1}{4}\\left ( a_1^{16} + a_2^8 + 2a_4^4 \\right).", "e393240ed585b52da6674c754e93b1db": "i = 2", "e39380e936f93fd66b2e0d616e7c64b4": "\\scriptstyle \\partial V\\,\\!", "e393ab1ab058b3ae3709e5776c304de3": "\\mathrm{Nu}_D =\\frac{48}{11} \\simeq 4.36", "e394700b516f27fc3314e6c435ef8c1d": "\\frac{dy}{d\\mathbf{x}} = \\mathbf{a}", "e395749c6a6a497d729be52525d5d71d": "\\hbar", "e3958fab0eeb7d982f6f0e597351baa2": "x_1*w_1", "e3960f66691bde12efc787fcd5e07543": "(k=0)", "e39622b9b274dd13bed7ac35a22d1839": "xI", "e3964e6c4dcf35fd420592492102d5bb": "w_C(c)", "e3967757e35a061adad7f8f40fef5f0e": "A(\\rho)= \\int_{D^*} h(|w|)^2\\,dw\\,d\\bar w \\le \\int_0^{2\\pi}\\int_0^\\infty (r\\,\\log (r+2))^{-2} \\,r\\,dr\\,d\\theta<\\infty.", "e39689ee3d4621c402f9722a49523ed4": "\\begin{cases}\n\\dot{\\mathbf{y}} = f_y(\\mathbf{y}) + g_y(\\mathbf{y}) z_2 &\\quad \\text{( where this } \\mathbf{y} \\text{ subsystem is stabilized by } z_2 = u_1(\\mathbf{x},z_1) \\text{ )}\\\\\n\\dot{z}_2 = u_2.\n\\end{cases}", "e396a1ab11c995c5be23febea56a2df4": "E = hf = \\frac{hc}{\\lambda} \\,\\! ", "e396bd9215cada7b1f807bc0ca5b1d56": "[h,h'] = 0\\ ", "e396bfd08f88dd628639d874874aca72": "\\boldsymbol{\\nabla}\\times \\mathbf{F}_l\\left(\\mathbf{r}\\right) = \\mathbf{0}", "e396ca4bf29e1021841f12f93937b232": "\\theta=\\phi \\mod 2\\pi", "e39786064cf3679c31b7df68f786893c": "\\textstyle R[t]/t^{n+j}R[t]", "e398ae793ebf07a44c214566b9d3d421": "\\partial_uf(a)=\\lim_{h\\to 0^+}\\frac{f(a+h\\, u)-f(a)}{h}", "e398eb0aae737d76fc74070a0e5ae1b7": "a_{C} = 0.691 \\; \\text{MeV}", "e399267073c284f2e0134e66d886a0a7": " \\mathcal{A}_f := \\mathcal{O} / J_f. ", "e3994f414fe74022ad7b7eab4c4d6e5f": "\\mathbb{R}[Z]", "e3996f9d75e3191ba5ff3e1d44571f6e": "d_1 | d_2 | \\cdots | d_r", "e3999ae59e7199bc7d5ec438bc1269f4": "\\mathcal{A} (E)", "e39a335c3dc58f2312a46b10e9ffbd44": "YA=\\tfrac{3}{4}", "e39a76790bdac1926b3bea6b6965e478": "\\sum_{k=0}^\\infty p(7k+5)q^k=7\\frac{(q^7)_\\infty^3}{(q)_\\infty^4}+49q\\frac{(q^7)_\\infty^7}{(q)_\\infty^8}.", "e39a7c6862601ec61c0a814b5167c731": " i^i_{ion}", "e39a839405f8f842a3e1a3247f30bd8b": "b_1=f/a_0", "e39a9c621a47f92dc13afea54e4a57fb": "(p\\times n)", "e39aa1b7facee572d6ca215608452430": "(\\omega_{zm})", "e39aafba1a92d8c68334634c6c7e7e3b": "G \\to \\{\\pm 1\\},", "e39ab4c5e7fd63cc0244a55cd599298c": "T_{em} = \\frac{21.21I_r^{'2} R_r^{'}}{n_r s}", "e39ad300eeedf468d671cae9396f84a2": " K_0^M (F)={\\mathbb Z} ", "e39af0e951d7bc081cd8b2ae6a2da6d2": "\\nabla u_1,\\nabla u_2,\\dots,\\nabla u_{n-r}", "e39b6721519def0ad5a1326988ca7718": "\\mathbf{B}_{l,m}^{(M)}", "e39c2f48a22cb9d714b317cead6c83fa": "k_2 = 0.25(\\epsilon_1+\\epsilon_2)/2\\epsilon_1", "e39c3621aba225755628f37266f8cdab": "1131_{2i}", "e39c76b80ddf8a025bb58782d49665f6": "L_n[0, c] = (\\ln n)^{c+o(1)}\\,", "e39cb533c1adb1a2647f0ac3b272ff9b": "\nc_2=d_1/a_1 = \\sum_{i=2}^k (k-1) a_i' a_2 \\cdots a_{i-1} a_{i+1} \\cdots a_k.\n", "e39cc8c9931e695fc07710171a602326": "k=\\varphi + \\sqrt{\\varphi} \\approx 2.89005 \\ ,", "e39d53bc2c8cc942b5d7a30354e60048": "\\mathbf{u}(t) = (u_1(t),\\cdots,u_M(t))^\\mathrm{T}", "e39d842860a55870398c57629c6e0e3a": "I_n=\\frac{2(px+q)^n\\sqrt{ax+b}}{a(2n+1)}+\\frac{2n(aq-bp)}{a(2n+1)}I_{n-1}\\,\\!", "e39db40e3adaf23021d796538c24c2fb": " A = \\frac{\\pi}{4} \\sum_{q\\ge 1} \\frac{1}{q^4}\n\\sum_{ (p, q)=1 \\atop 1 \\le p < q } 1 =\n\\frac{\\pi}{4} \\sum_{q\\ge 1} \\frac{\\varphi(q)}{q^4} =\n\\frac{\\pi}{4} \\frac{\\zeta(3)}{\\zeta(4)},", "e39df69a42e1946fda86a6567f975eb6": "\\theta(z;q)=(z;q)_\\infty (q/z;q)_\\infty", "e39eb20e3d6264bf4984f98eba1c8fb9": "\\Gamma_e(x) = \\Gamma_e(0)\\,\\exp{\\left(-\\frac{x}{\\lambda_e}\\right)}\\qquad\\qquad(10)", "e39eb4429428ebf2feb25f12ee42a0b3": "A f = \\mathcal{A} f \\mbox{ for all } f \\in D_{A}.", "e39ec8983ebc91024ef0b6bdafe0fdcd": "\\exp(c\\mathbf{r}) \\exp(a\\mathbf{s}) = \\exp(b\\mathbf{t}) \\!", "e39eeb9cdc7b94fccdfd43729fcc0bbf": "{\\infty}", "e39eedabb521b795046fbec166952c9e": "\nH_n = \\ln n + \\gamma + \\frac {1} {2n} - \\frac {1} {12n^2} + \\frac {1} {120n^4} - \\varepsilon ", "e39f0eeb5d8baa2601b17e13092064a9": "\\rho_{a / b}(R \\cup P) = \\rho_{a / b}(R) \\cup \\rho_{a / b}(P)", "e39f260272088c55ae922573fa9e9e04": "k_a=\\frac{2\\pi a}{\\lambda}", "e39fc17447a92dabb134fcc19fc8ff0b": "\\delta L=0", "e39fcc9c96c7e331c97f5b6962f94e99": "a_n = 1/n", "e39ff4598d20974e269849cbe83fdb27": "W_C = 0.5CV_C^2 - \\ ", "e3a02b0a3d82a73511591da6a1605f9c": "x^\\frac{\\alpha}{\\alpha+1}+y^\\frac{\\alpha}{\\alpha+1}<1.", "e3a03a4c7f7849d5813b3d697a8384bc": "\\mathcal{B}[\\neg\\exists p: \\neg p \\wedge \\mathcal{B}p]", "e3a03d5458a56b8bcc97740850958fbe": "\\pi(M^{\\mu\\nu})(\\sigma^{\\rho\\sigma}) = [\\sigma^{\\mu\\nu}, \\sigma^{\\rho\\sigma}] = i(\\eta^{\\sigma\\mu}\\sigma^{\\rho\\nu} + \\eta^{\\sigma\\nu}\\sigma^{\\rho\\mu} - \\eta^{\\mu\\rho}\\sigma^{\\nu\\sigma} - \\eta^{\\nu\\rho}\\sigma^{\\mu\\sigma}),", "e3a04e43eda944b291c1be4d13fc5809": "\nr_6(n) = 16 \\sum_{d|n} \\chi\\left(\\frac{n}{d}\\right)d^2 - 4\\sum_{d|n} \\chi(d)d^2.\n", "e3a06c9181aede25bed627d9d43bea2f": "n \\neq 6,", "e3a0d602287745f88d028dfa36f35ffe": "\\mathcal{O}(m + \\log n)", "e3a1439c3569490c437a939755baf2b6": "I_X = \\frac{R_T}{(R_X+R_T)}I_T \\ ", "e3a16c268e23b3548220d86f3f0abb0a": "\\eta(-s) = 2 \\frac{1-2^{-s-1}}{1-2^{-s}} \\pi^{-s-1} s \\sin\\left({\\pi s \\over 2}\\right) \\Gamma(s)\\eta(s+1).", "e3a189cb770ba61cf43f73540d059256": "\\operatorname{erf}(x)\\approx 1-\\frac{1}{(1+a_1x+a_2x^2+\\cdots+a_6x^6)^{16}}", "e3a1afe0c2106678ba97ee1edd06d847": "\\det (\\sigma_i) = -1,", "e3a219601e9af163cdc6e56a41258da3": "Z_1^{p,q} = \\ker d_0^{p,q} : F^p C^{p+q} \\rightarrow C^{p+q+1}/F^{p+1} C^{p+q+1}", "e3a21cad4b9c644e716d3bb918aa289e": " 4AB-E^2 >0 \\,", "e3a228daf814883471cb0dcdf40d7f1f": " \\eta = \\mathbf{X}\\boldsymbol{\\beta}.\\,", "e3a2e92a35b7558ac026a1d18a58f5fb": "\\phi(\\beta)=\\frac{3}{4\\beta^{2}}\\left[\\frac{1}{\\beta}\\lg\\frac{1-\\beta}{1+\\beta}+\\frac{2}{1-\\beta^{2}}\\right],\\;\\beta=\\frac{v}{c}", "e3a3169b3cc3c8966ba47eeb506f100c": "\\int f(x_k)p(x_k|y_0,\\dots,y_k) \\, dx_k\\approx\\frac1P\\sum_{L=1}^Pf(x_k^{(L)})", "e3a31c12a069d89c747d8f464bc8aaf9": "N(x,t)", "e3a32774c77c2d9c7062c453d32ab7d4": "\\alpha_{fuel}", "e3a32bcb25f7c8d2121b3e816f25bfa8": "[CF_i,child_i]", "e3a36b823df5f01e88a28b66337a750a": "-[OH^-]_{0^{ }}", "e3a3adf97f00df111934adbb4c1c963e": " \\ \\textbf{f} = \\textbf{f}_1 + \\textbf{f}_2 ", "e3a3b19fde5677e46f1056c1c291fa3e": "\\left \\{ a_n \\right \\}", "e3a3f1d460bb730f0e536e459621eab9": "\\frac{K \\cdot t}{V} = -ln ((1-URR) - 0.03) + (4-3.5 (1-URR)) \\cdot \\frac {UF}{W}", "e3a40a04a2cfccac5e5424e3309e28b7": "\nH_x(x^*,u^*,\\lambda^*,t)=\\begin{bmatrix} \\frac{\\partial H}{\\partial x_1}|_{x=x^*,u=u^*,\\lambda=\\lambda^*}\n& \\cdots & \\frac{\\partial H}{\\partial x_n}|_{x=x^*,u=u^*,\\lambda=\\lambda^*}\n\\end{bmatrix}\n", "e3a4a1b784c03d4f581b611d1e44e6fe": " { \\theta } = \\frac 1 2 \\sin^{-1} \\left( { {g R} \\over { v^2 } } \\right) ", "e3a4d907adf85a9a74e241e3e00cebe3": "P_{LOSS} = ( V_{IN} - V_{OUT} ) \\times I_{OUT} + ( V_{IN} \\times I_{Q} )", "e3a4d9cb879788b80e741ff502cc58fe": "\\psi(\\Omega+1) = \\varepsilon_0 \\omega", "e3a52e492a795cd0ac007cfccb4d4378": "\\overline{X}_n=\\frac1n{\\sum_{k=1}^n X_k}", "e3a5408673f8a7d541e585172ee0b93c": " \\alpha_x^{ } , \\alpha_y ", "e3a54993384f3a4310ead4a65cd3d6c7": "\\sqrt[3]{2+\\sqrt{3}+\\sqrt[3]{4}\\ }.", "e3a5c2b3d375f4313746dbb857b02ae6": "g_\\ast^2 = \\frac{b_0}{b_1}.", "e3a5d90ad2bfe6d43315cf56e634ee24": " f (t) ", "e3a607b31a413ad0fe0bda6ed62b448c": "f^*(x)-L", "e3a6409a1dbcc8dbba549fc062c840ea": "p'_x(x,y)=p'_y(x,y)=p(x,y)=0.", "e3a671595c7c9e95f466a4fb0e5c5a3e": "\\mathit \\Gamma = { {Z_\\mathrm L - Z_\\mathrm S} \\over {Z_\\mathrm L + Z_\\mathrm S} }", "e3a6786ee94dea76cbf936de8ec8c1ab": "\\int\\frac{e^{-x^2}}{x^2}\\; \\mathrm{d}x = -\\frac{e^{-x^2}}{x} - \\sqrt{\\pi} \\mathrm{erf} (x) ", "e3a69b5448947100f796907f4d127242": "(V,\\tau)", "e3a70294989d494dec770e9b85cb926d": "-1.0930", "e3a7069e2d1bdac159fef885ddc823b6": "Q[^y\\!/\\!_v]", "e3a747148712d57dc2de931928abbec1": "a_0/2\\,", "e3a75eb040aa1ecf9c90b6f427a166ef": "\\begin{align}\n x &\\mapsto f(x)\n\\end{align}", "e3a76389e029592feddea3de7698b17b": "\\underset{x\\in \\mathcal{X}}{\\max}\\ \\bold{E}[c(A,x)] \\geq \\underset{a \\in \\mathcal{A}}{\\min}\\ \\bold{E}[c(a,X)] ", "e3a76e7890f3b72e67229f8e44df5b7d": "\\mathbf{v}_{d_j} = [w_{1,j}, w_{2,j}, \\ldots, w_{i,j}]", "e3a7a1e58c56e4f30c92f49446174945": "\\phi(\\mathbf{k})=\\sum_j \\psi_j \\phi_j(\\mathbf{k})", "e3a7ae68cbdef957bbd43c8e3061fa7d": "i_0\\in I", "e3a8392e5c2ebc39197954b7c546c13f": "\\bar a", "e3a84af176e46c859abaf7c45ad2b596": "\\pi=(\\pi_1,\\ldots,\\pi_n)", "e3a8e062487a05ab9bcdc9c28ba4a552": "\n\\begin{align}\n s(n,n-p) &= \\frac{1}{(n-p-1)!} \\sum_{0 \\leq k_1, \\ldots , k_p : \\sum_2^p mk_m = p} (-1)^K \n \\frac{(n+K-1)!}{k_2! \\cdots k_p! ~ 1!^{k_1} 2!^{k_2} 3!^{k_3} \\cdots p!^{k_p}} ,\n\\end{align}\n", "e3a945a39b0ceeb29568353061c784db": " \\lambda v.(\\lambda h.h\\ v) ", "e3a9548fe8af065ffba0e1b6e10f93b5": "B_0,B_1", "e3a99813a375e7581469dfd5bb131449": "p( x )\\!", "e3aa421439615a52e9c2cd48298050f0": "B_1, \\dots , B_k", "e3aa4afafcaf3032d7c43342d48eeb74": "a(x) \\in R", "e3aa50f395c84fc5efa33294189c83ca": "(x_k, x_{k+1})", "e3aa56fbe85d36f8d908a9279906917a": "\\alpha=(a)", "e3aa89a30feb6e245cfa8bd0141b51ea": "\nH = \\frac{p^2}{2m} + mgz - \\frac{\\lambda}{2}(r^2-R^2) + u_1 p_\\lambda + u_2 (r^2-R^2)\n", "e3aab30332b7867321c467e0a13df277": "\\psi_1=Ae^{ipx}\\left( \\begin{matrix} 1 \\\\ 1\\end{matrix} \\right)+A'e^{-ipx}\\left( \\begin{matrix} -1 \\\\ 1\\end{matrix} \\right) ,\\quad p=E_0 \\,", "e3aab5f4c02e8940ac057d66310ff73a": "A(\\boldsymbol \\theta)", "e3abb8f3f2c7e537c22cf20aef9a7373": "\\left|\\phi\\right\\rang = \\frac{1}{\\sqrt{2}} \\left(\\left|+x\\right\\rang \\otimes \\left|-x\\right\\rang -\n \\left|-x\\right\\rang \\otimes \\left|+x\\right\\rang \\right)", "e3abd3b4b8f8f37d6b13baec6c81faa4": "\\begin{bmatrix} \\mathbf{p}_{n-j+1} & \\cdots & \\mathbf{p}_n \\end{bmatrix}", "e3ac6f55e80951c72ffd04e5552b3f6d": "T^{\\mathrm{SS}}_p", "e3acd0227f65ccb9c9cb704e4f8106d9": "x\\,R\\,y", "e3acf0925ba96b380898f142379968ef": "W_x(t,f) = W_y(t- \\frac{1}{3} f,f) \\, ", "e3ad06343366537cf7472de674b90319": "O_n", "e3ad4ec6ca7b1f6124232e85e844a887": "Q(\\alpha,\\alpha^*)= \\frac{1}{\\pi} \\int P(\\beta,\\beta^*) e^{-|\\alpha-\\beta|^2} \\, d^2\\beta.", "e3adb70d1974b70e105fce3023f8a936": "\\beta \\mathbb{N}", "e3add34090436fcac83c2524322ea666": "A = X_2ZZ_1 = 0", "e3addc179c38767a819f67ac0857f33b": "\\sigma_{ab}=\\frac{i}{2} \\left[\\gamma_{a},\\gamma_{b}\\right] ", "e3ae02f8bec88f9b5b0ba6329d41d7f0": "\\frac{1}{p^k q^\\ell r^m \\cdots }", "e3ae90af8584208d04a7a6e4b7e5aa0c": "C_{OA}\\left(N\\right)=C_{OA}\\left(2^m \\right)", "e3aecd7e19e25cc06b135feb178f1b0e": " \\ PV \\ = \\ {A(1-e^{-rt}) \\over e^r -1}", "e3aede9137d241039c0b29085f1ed6b2": "\\nabla \\cdot \\mathbf{D} = \\rho_\\mathrm{f},", "e3aee9da26b7e12f9c6d0958c9547dc4": "\\nabla f(x) + \\sum_{j=1}^m u_j \\nabla g_j(x)", "e3af6148f1140ddff60a9487f9f5f8f1": "\\left[\\delta_{\\epsilon_1},\\delta_{\\epsilon_2}\\right]S=\\int_{M^{d+1}}\\left[\\delta_{\\epsilon_1}d\\Omega^{(d)}(\\epsilon_2)-\\delta_{\\epsilon_2}d\\Omega^{(d)}(\\epsilon_1)\\right]=\\int_{M^{d+1}}d\\Omega^{(d)}(\\left[\\epsilon_1,\\epsilon_2\\right]).", "e3af6f537e9916cd166976061b015652": "\\displaystyle{\\|A_\\varepsilon\\|_2^2=(R^2(R+\\varepsilon I)^{-2}K, KR^2(R+\\varepsilon I)^{-2})_2\\le \\|K\\|_2^2,}", "e3af7dce41231e1b592556a9f0231d2b": "ba = ca", "e3af995e7d4c4eefc8e19ca442f6dd48": "E = \\frac{V}{d} \\,", "e3afaf987d6602854f4621d35d3a3940": "\\to a T[gg] T[fgg] T[fgg] \n\\to ab T[g] T[fgg] T[fgg] \n\\to abb T[] T[fgg] T[fgg]", "e3afcb3755a36559258e5f44619887df": "\\boldsymbol{\\Gamma} = \\frac{d \\mathbf{M}}{d\\tau} = \\mathbf{X}\\wedge \\mathbf{F}", "e3afdb266bac9e5fde70dbdde2ded9e4": "S_l^k", "e3b0ca6e3a53bdc492e280fb6589e387": "\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in SL_2(\\mathbf{Z})", "e3b117ee1e86e87130078ae2d7ca7c3e": "k_a, k_b, \\ldots, k_N", "e3b1365b0ee5602af412ca4fe12394e5": "\\left( \\tfrac{a-1}{2} \\right)", "e3b15b5129149f16c066dc12350381eb": "v = \\mu \\mathcal{E}", "e3b19a13ebd06dd9c2c162410ae62f28": "\\; (\\pi, V, K)", "e3b1ee0f97efa440dbb4154c20619c53": "L_e(x) = a_e x + b_e", "e3b214184a6cbd790cc49e809e7affe3": "O\\left(\\exp\\left(2 \\sqrt 2 \\sqrt{\\log n \\log \\log n}\\right)\\right)", "e3b22676c1f0a07e6e8604685eb87c1d": "\n\\begin{align}\n\\mathbf x \\cdot \\mathbf y & =\n\\left(x^1 \\mathbf e_1 + x^2 \\mathbf e_2\\right) \\cdot \\left(y_1 \\mathbf e^1 + y_2 \\mathbf e^2\\right) \\\\[10pt]\n& = \\left(x^1 h_1 \\hat{ \\mathbf e}_1 + x^2 h_2 \\hat{ \\mathbf e}_2\\right) \\cdot \\left(y_1 \\frac{\\hat{ \\mathbf e}^1}{h_1} + y_2 \\frac{\\hat{ \\mathbf e}^2}{h_2}\\right) = x^1 y_1 + x ^2 y_2\n\\end{align}\n", "e3b26b6d4e6a5b40df947848acb3e131": "\\mathrm{(SNR)_{C,AM}} = \\frac{A_c^2 (1 + k_a^2 P)} {2 W N_0}\n", "e3b27f97c406fba9e4980b9a8e660f2e": "(\\phi \\to \\psi) \\,", "e3b2dd84a5c5d894162ead2cfeb369bf": "\\tau \\frac{\\partial q}{\\partial t}+q=-K \\nabla h", "e3b2e15ddb2694635c42f9c2012e2338": "dT_{on} = DT = \\frac{D}{f}", "e3b3627ba54e1d34b8a12fe31266d4f5": "dH= TdS+VdP+\\sum_i \\mu_i dn_i,\\,", "e3b3b90e5898953088a880162632e9ed": " (F_n / F_{n-1}) \\times (F_m / F_{m-1}) \\to F_{n+m}/F_{n+m-1}, \\ \\ \\ \\ \\ \\left(x+F_{n-1},y+F_{m-1}\\right) \\mapsto x\\cdot y+F_{n+m-1}", "e3b3d3953e580d0e51a2ea87a9d72106": "\\Phi:Y\\to Y'", "e3b40b36d734a0f24e80524d458b99dd": "\\frac{abXd}{[ffg]}", "e3b411ab87ff5f356d486fc68b16421f": " \\arccos ( -\\frac{24 + 15\\sqrt{5}}{61} ) ", "e3b458f1a6c5c37123d6a45ad195268f": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = J^{-1}~\\boldsymbol{F}\\cdot\n \\left[\\cfrac{d}{dt}\\left(J~\\boldsymbol{F}^{-1}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{F}^{-T}\\right)\\right]\n \\cdot\\boldsymbol{F}^T ~.\n", "e3b476681660794660b3b381c4e73e48": " \\mathit XCF = \\mathit E - \\mathit Div - \\mathit gBV \\,", "e3b4caa1d2edfa2a9b61bfa52c42c119": "\\langle,\\rangle", "e3b4e163af21db71a0ab275eb42d05ef": "O_{i}", "e3b4e330d3a936faff63bf1ff16ee682": "(dG)_{T,P} = -\\mathbb{A}\\, d\\xi \\,.", "e3b52a115fec05c213c6978253818fa5": "{\\mathcal{I}}_{\\alpha, \\beta}= {\\mathcal{I}}_{\\beta, \\alpha}= \\ln \\,\\operatorname{cov_{G{X,(1-X)}}}", "e3b560be7454fb735ce671862f5cc2d4": "mB_y\\not\\subseteq A", "e3b60f39be5b705fe2e13617dd34bb74": "\\begin{smallmatrix}r_{per}\\ =\\ (1\\ -\\ e)\\cdot a\\ \\approx\\ 44\\end{smallmatrix}", "e3b69edfa048ed7e02b8f5f97c37cbcf": "\\dot{u}=0\\quad\\text{and}\\quad \\dot{r}=\\frac{F}{2}\\dot{u}\\;.", "e3b748152d7de33598ae9cca86850d5a": " \n (pq|rs) = \\int\\int \\frac{ \\chi_p( \\mathbf{r}_1 ) \\; \n \\chi_q( \\mathbf{r}_1 ) \\;\n \\chi_r( \\mathbf{r}_2 ) \\;\n \\chi_s( \\mathbf{r}_2 ) \\;\n }\n { \n \\mid \\mathbf{r}_1 - \\mathbf{r}_2 \\mid \n }\n \\;\\; d\\mathbf{r}_1 \\; d\\mathbf{r}_2 \n ", "e3b7571c5df90078cb9832ec1cf0f53e": "\\lim_{x \\to 0^+} \\log_a x = +\\infty \\quad \\mbox{if } a < 1", "e3b7a0deb9df7d24ea3872f4fb6b4110": "\\frac{2ax-x^2}{a^2}", "e3b7a5a0fac12b1bbcf3519eefca961b": "w(n)=a_0 - a_1 \\cos \\left ( \\frac{2 \\pi n}{N-1} \\right)+ a_2 \\cos \\left ( \\frac{4 \\pi n}{N-1} \\right)- a_3 \\cos \\left ( \\frac{6 \\pi n}{N-1} \\right)", "e3b803fa99a440fefee4d18e57cc6642": " S", "e3b82e4c97cf2d524f9bee2124b01592": " \\frac{V_\\mathrm L}{V} = Q ", "e3b89ac5b7e0ae2c464c2a2d43d2ddf3": "S=\\{S_k\\}_k", "e3b8d8b4d894407584eccb4620f0f749": " B_2 = \\begin{pmatrix}\n1 & 0 \\\\\n0 & k\n\\end{pmatrix}", "e3b91ed7004abfd8305a6ac4ade328b7": " O(|V|+|E|) ", "e3b950aab865097ff50bf8c1e5f6f3e5": "\\displaystyle\\varphi^c(n)=2,", "e3b980e40a559d41d334bb2c53a20aeb": "\\partial Y", "e3ba165a9590bd75490df1be59d7d398": "\\zeta(a,b)+\\zeta(a,\\bar{b})+\\zeta(\\bar{a},b)+\\zeta(\\bar{a},\\bar{b})=\\frac{\\zeta(a,b)}{2^{(a+b-2)}}", "e3babe6eb14124c72e626cab38391fc7": "K_\\text{vert}=\\frac{F_\\text{max}}{\\Delta y}", "e3bac660062a3a74fd0c7bafc4ae1717": " MRT = \\frac{1}{N}\\sum_{i=1}^m t_i n_i", "e3bb0a439eebd8caa51ca7fe4d9679af": "W(\\alpha,\\alpha^*)=\\frac{2}{\\pi} \\int \\delta^2(\\beta-\\alpha_0) e^{-2|\\alpha-\\beta|^2} \\, d^2\\beta=\\frac{2}{\\pi}e^{-2|\\alpha-\\alpha_0|^2}", "e3bb1e36cc3286180ccf8c0ab8ab1290": "dw, dv", "e3bb50fbc206b0427e9e73e4194dc12e": "\\rho(\\mathbf r)=\\sum_i^N |\\phi_{i}(\\mathbf r)|^2.", "e3bb6a6d31acf048f8533bb689afabb3": "\\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/4\\mathbb{Z}", "e3bbbbc8ed8698ae76a765206396e1c2": " x = \\cos(6\\pi nt)\\cos(2\\pi nt) - \\cos(6\\pi t)\\cos(2\\pi t), ", "e3bbeae3d77b8512b18898144cbb94d0": "\\rho(X;U)", "e3bc61fe5fa977c58455775c5d999363": "k(t) = \\frac{x_1'(t) \\cdot x_2''(t) - x_1''(t) \\cdot x_2'(t)}{\\Big( x_1'(t)^2+x_2'(t)^2 \\Big)^{\\frac{3}{2}}} \\qquad\\qquad \\qquad\\qquad\\qquad N(t)\\,=\\,\\frac{1}{|| \\gamma'(t)||}\\cdot\\begin{pmatrix} -x_2'(t) \\\\ x_1'(t) \\end{pmatrix}", "e3bc896546747809737a8c1db3c598df": "f_{\\mathbf{k}}^e", "e3bc9b6f47dbc11d2ca617b61f312a7a": "\\Omega(M)=\\bigoplus_{p=0}^{2l}\\Omega^{p}(M)", "e3bd06107affaf0822430fa2570dbe6f": "\\textstyle x_i\\in\\{0,1\\}", "e3bd163a48563d78c3a5018bf2e27bff": " C_3,C_2,C_2.\\ ", "e3bd293e0d59c3774a2577c536911d89": "\\mbox{m}\\,", "e3bd470fa891218c1bcd699ae02d01ee": "C^\\infty(M)", "e3bd4e5aaa7149b32a02565d249990ba": "S = \\int d^4 x \\; e \\;e_I^\\alpha e_J^\\beta \\; \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} [\\omega]", "e3bd63ed7279ff9873a28570fd2d7838": "O^\\prime", "e3bdb7ffcca5d920820b87b8f9b04005": "u^2 = 1", "e3bdf48bd510477dec8ef583475efa3e": "\\mu=3", "e3bdfa4ffd04a8f410d695aef13cea4f": "(z_1,z_2;z_3,z_4) = \\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)}.", "e3be3358ec7e8f7c4230ef1c5db8deda": "T^{\\mathrm{AA}}_p (x,y) = \\begin{cases}\n T_{\\mathrm{D}}(x,y) & \\text{if } p = 0 \\\\\n e^{-\\left(|\\log x|^p + |\\log y|^p\\right)^{1/p}} & \\text{if } 0 < p < +\\infty \\\\\n T_{\\mathrm{min}}(x,y) & \\text{if } p = +\\infty\n\\end{cases}", "e3be9864edfa956050c342939eab4711": "w=(w_e), e\\in E", "e3bea8769d539956dd0cd95e81e507e3": "[f]([\\mathbf{x}]) \\supseteq \\{f(\\mathbf{y}) | \\mathbf{y} \\in [\\mathbf{x}]\\}", "e3bee076727a8aaab102358287edc870": "c_1=c_\\sigma=0", "e3bf29bb71d61ab0a22df36123f02d31": "\\Lambda_m =\\Lambda_m^0-K\\sqrt{c} ", "e3bf32d037e6a4fa65811d3587c7f664": "E(X) = X^{e - \\Delta(y, P(\\alpha_i)_i)} \\prod_{1\\le i \\le n| y_i \\ne P(\\alpha_i)} (X - \\alpha_i)", "e3bf71b4fdfcde6b918772979123640e": "\n\\frac{dA_1}{dt} = \\frac{1}{2} r^2 \\frac{d\\theta_1}{dt} = \\mathrm{constant}\n", "e3bfc2deac37cdd2252251c3c99dfff2": "f(x_1,x_2) = x_1 x_2 + \\sin(x_1)", "e3bfcc78f87c03f30379d0d7c2f3709b": "K_0=0.03928", "e3c01a10c008bfe19d6f9f88df26c369": "A'(z)", "e3c07c729fd828411080942aa3e1ef67": "\\frac{d^2f}{dz^2} - \\left(\\tfrac14z^2+a\\right)f=0", "e3c0cbf91ece024841ff9fdb38694186": " n\\,\\!", "e3c0da4f87bf8eb804f656317a805a78": "S_{2m}=\\sum_{n=1}^{2m} (-1)^{n-1} a_n", "e3c0ea179afecd1a286c576ec82c197f": "-u_0 < u < u_0", "e3c10d3ee2e521ca28adefe9b4e43e40": "g_\\text{s}\\approx 2", "e3c1395c23ac4f21baa1b2389ee21430": "e_1 = R\\, d\\alpha", "e3c173ed340d1ce5c099c42c8241d438": "\\beth_3", "e3c187f9a9fd8bcbaa6bc92e91ff4dfa": "\\sum_{}", "e3c1fe46f6c6f5a1765a9bd920ba0148": "(x=3, \\;y=-2,\\; z=6)", "e3c2213eebb8e788a10299fd10c4f074": "|g\\cap \\mathfrak o|=0", "e3c28d03b4f850257fe2739b0b4153e7": "t_\\text{P} = \\sqrt{\\frac{\\hbar G}{c^5}} ", "e3c296b4b4ef63ca51602f6970b7172d": " x = a \\cos(\\omega t) ", "e3c2b9c6d36cf84613e47b71fc2358f3": "K_M = K \\cap M", "e3c3320ddcd839525b8cbc58ce211244": "h^2 = a(a - 2b\\cos\\gamma).\\,", "e3c361bdaf95a2ec9af4361ccfba3e5d": "~f(\\omega)=\\int f(t) \\exp(i\\omega t) {\\rm d}t ", "e3c3836cce24a85e2733cdd101602dc7": "\\alpha \\succeq \\beta", "e3c38bdefa669ed4914d3639ca922f10": "|D(\\varphi_n)(z_n) - D(\\varphi)(z)| \\le \\int_{|w-z|\\ge\\delta} |K(z_n,w)-K(z,w)||\\varphi(w)|\\,|dw|\n+\\int_{|w-z|\\le\\delta} (|K(z_n,w)|+|K(z,w)|)\\cdot|\\varphi(w)| \\, |dw|", "e3c3b8e9e2e0ef31c8c7e1cc9e40806b": "\\bar{5}", "e3c3c59758897007beb0167595d36dc9": "f_{yz^2} = N_3^c \\frac{y (4 z^2 - x^2 - y^2)}{r^3 \\sqrt{5}} = \\frac{1}{i \\sqrt{2}}\\left(Y_3^1 + Y_3^{-1}\\right)", "e3c419ba14d5c8b7f6134ce4be630f9c": "(a,c)", "e3c44ef1bd6edc3a458afce664924d58": "\\mathrm{TKOF}=\\frac{m_{\\mathrm{bullet}}\\cdot v_{\\mathrm{bullet}}\\cdot d_{\\mathrm{bullet}}}{7000}", "e3c4b7450c028eadc9d13f07fd06469a": "\\int_{0}^{a}\\frac{x^{m}dx}{x^{n}+a^{n}}=\\frac{\\pi a^{m-n+1}}{n \\sin [(m+1)\\pi /n)]}", "e3c4d04f63b02b1e66e768a2b756c328": "L = \\int_a^b \\sqrt{ \\sum_{i,j=1}^n g_{ij}(\\gamma(t))\\left({d\\over dt}x^i\\circ\\gamma(t)\\right)\\left({d\\over dt}x^j\\circ\\gamma(t)\\right)}\\,dt.", "e3c4fc9ae905fd3ae7489d46d3d9951f": "\\sum_{i=0}^{j-1} \\frac{1}{s_i} = \\sum_{i=0}^{j-1} \\left( \\frac{1}{s_i-1}-\\frac{1}{s_{i+1}-1} \\right) = \\frac{1}{s_0-1} - \\frac{1}{s_j-1} = 1 - \\frac{1}{s_j-1}.", "e3c51f2407e07c21e65e63a1fcc8a5fa": "\\sum_{i=0}^{k-1}d_{i}\\cdot(2^{n})^{i}", "e3c52053f79e4d809d3f871c8874eb38": "\\Delta_2", "e3c5f1b1e999d682962d18cf425f9ec9": "\\mathcal{Q} =\n\\mathfrak{P}(\\mathcal{V}\\mathfrak{C}_1(\\mathcal{Z})\n+\\mathcal{U}\\mathcal{V}\\mathfrak{C}_2(\\mathcal{Z}))\n+\\mathcal{U}^2\\mathcal{V}\\mathfrak{C}_3(\\mathcal{Z})\n+\\mathcal{U}^3\\mathcal{V}\\mathfrak{C}_4(\\mathcal{Z})\n+\\mathcal{U}^4\\mathcal{V}\\mathfrak{C}_5(\\mathcal{Z})\n+\\cdots)", "e3c60b9e375dde3eed5ad8830d04a76f": "\n\\frac{1}{2}\\sum^n_{i=1}F^Q_id^Q_i = \\frac{1}{2}\\int_\\Omega \\sigma^Q_{ij}\\epsilon^Q_{ij}\\,d\\Omega\n", "e3c63fe3232954a6961b42ce7c3852d1": "\\phi_1, \\phi_2, \\lambda_2-\\lambda_1", "e3c7454351f5e3ede7f81aabddfb861d": "\\frac{e}{p}=\\frac{w}{w+{\\epsilon}}\\, ,", "e3c74dbddf4e1c8188285d0ecf8e019d": "\\mathbf{j}^2", "e3c7734b83a86595b2470df954a7cfdc": "\\hat{w}_r (\\omega) = e^{-\\mathrm{i} \\omega \\frac{N-1}{2}} \\frac{\\sin(N\\omega/2)}{\\sin(\\omega/2)}", "e3c7a685b1ed36085c26647807cbccbe": "\\mathcal{S}^s (S^n)", "e3c7bb0f618bdc89be58caf9333b28f5": "\nD(P\\parallel P^{\\prime })=-\\sum I(X_{i};X_{j(i)})+\\sum\nH(X_{i})-H(X_{1},X_{2},\\ldots ,X_{n})\n", "e3c83abad8dda85a185a624acdd03a8b": "\\mathbf{AB} = \\begin{pmatrix} \na & b \\\\\nc & d \\\\\n\\end{pmatrix} \\begin{pmatrix} \nx \\\\\ny \\\\\n\\end{pmatrix} =\\begin{pmatrix} \nax + by \\\\\ncx + dy \\\\\n\\end{pmatrix}\\,,\n", "e3c8517a2e5d3657da7fd135e70f9fc2": "W_p {{=}} \\frac{{2p-1 \\choose p-1}-1}{p^3}.", "e3c853089f3a01c7bd3bf3e74b4c59fd": "k=\\sec\\phi'", "e3c8644c0b894eaf69ec6018af4986b6": "\\rho = 1/a", "e3c87978abffdfdcc65b72d6d569fab7": " \\alpha_1 = \\frac{3-k}{4} ", "e3c87e88c0b0eb907758ada2403b4b73": " 17 500= 100\\times 100 + 4800 + 2700 ", "e3c9002104b886506afc30a490638c93": "(t,y,x)", "e3c91568b095ac57c69e35ed3ddb15d5": " r = R ", "e3c923dfff541096ebdcbe4f7362d662": "\n\\begin{align}\n \\epsilon(f) & = \\mathbb{E} \\left| X(f) - G(f)Y(f) \\right|^2 \\\\\n & = \\mathbb{E} \\left| X(f) - G(f) \\left[ H(f)X(f) + V(f) \\right] \\right|^2 \\\\\n & = \\mathbb{E} \\big| \\left[ 1 - G(f)H(f) \\right] X(f) - G(f)V(f) \\big|^2\n\\end{align}\n", "e3c942edf39b65557e001ec54f44e2c8": " s(n,k)\\,", "e3c9793bc819e64351e493d619261d61": "S_i = S(i\\Delta t)", "e3c992227ad28b592b4de58efa089d72": "\\displaystyle{\\partial_s \\|f|_{\\partial \\Omega_s}\\|^2 \\le M \\|f|_{\\partial \\Omega_s}\\|^2,}", "e3c9b7ff556bea49ca0b6a04bea192e5": "p(z)=p(z_0)+c_k(z-z_0)^k+c_{k+1}(z-z_0)^{k+1}+ \\cdots +c_n(z-z_0)^n.", "e3c9f382043f989112e00d453d940e66": "\\alpha=\\frac{a V}{(2 \\pi \\beta)^{3/5}}", "e3ca92426a7e887dd5afe87da7c55c6a": "x+x'*y=y'", "e3cabce3002e2827f34a363a0ca89639": "\\alpha n = \\omega^{\\beta_1} c_1 n + \\omega^{\\beta_2} c_2 + \\cdots + \\omega^{\\beta_k}c_k \\,,", "e3caddccda23afebc248c4b1d6e69ab6": "x(t)=|\\tilde{\\chi}(\\omega)| h_0 \\sin(\\omega t+\\arg\\tilde{\\chi}(\\omega))\\,,", "e3cae9aebe2d70360242c333c52b5094": "A=X^TCX", "e3cb027823fba9841e4e5abc74c53f28": "V = \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty f(x,y)\\,dx dy=2 \\pi A \\sigma_x \\sigma_y.", "e3cb031bda98206d94e92c97185bc5d4": "177147 + 59049 = 226196", "e3cb2ca3e4e18eedf1d0ee43675df655": "\\langle \\hat{A}(t)\\rangle={1\\over Z_0}Tr[\\hat{\\rho}(t)\\hat{A}]={1\\over Z_0}\\sum_n \\langle n(t) | \\hat{A} |n(t) \\rangle e^{-\\beta E_n}", "e3cbbf2e58031b4cec044b01b6f9af31": "f:D(A)\\to H", "e3cbd6e5bd836467f18eb6b9a92d1592": "COP_{cooling}", "e3cbd722615aab5ddecc7169b01b5621": "** T^{IJ} = {1 \\over 4} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} \\epsilon_{MN}^{\\;\\;\\;\\;\\;\\;\\; KL} T^{\\;\\; MN} = - T^{IJ}", "e3cc2d9ca6745d83876122d1c005ef34": "\n T_{11} = \\cfrac{\\sigma_{11}}{\\lambda} = \n 2~\\left(\\lambda - \\cfrac{1}{\\lambda^5}\\right)~\\cfrac{\\partial W}{\\partial I_1} = T_{22}~.\n ", "e3cc3d9b75ce888769d3c95ad2e5d2be": "i,j=1,2,3", "e3cc66187cbfcccc3635e574988768f1": " x^r(x^d - a)^{(p^n - 1)/d}", "e3cc75b3601fea87356f8ae55e76475d": "\n\\begin{align}\n\\mbox{FSPL(dB)}\n &= 10\\log_{10}\\left(\\left(\\frac{4\\pi}{c}df\\right)^2\\right) \\\\\n &= 20\\log_{10}\\left(\\frac{4\\pi}{c}df\\right) \\\\\n &= 20\\log_{10}(d) + 20\\log_{10}(f) + 20\\log_{10}\\left(\\frac{4\\pi}{c}\\right) \\\\\n &= 20\\log_{10}(d) + 20\\log_{10}(f) - 147.55\n\\end{align}\n", "e3cca915bf6ff189114b820a48b0303d": "\\langle E^2\\rangle_c", "e3cccfc23abfcffa0de23cbbfd34fb42": "(\\gcd[6-1,35])\\cdot(\\gcd[6+1,35]) = (5)\\cdot(7) = 35.", "e3cd16836a2f34d6ea9239ed5ddc0db9": "\\mathbf{r} \\cdot \\mathbf{v} < 0", "e3cd2c3765f5b1a8545e6ee7ec8095cd": "C = \\sqrt[3]{\\frac{\\Delta_1 + \\sqrt{\\Delta_1^2 - 4 \\Delta_0^3}}{2}} \\qquad \\qquad {\\color{white}.}", "e3cd4ba9faa3098aad29fcbf6187c795": "3 \\times 0 = 0 + 0 + 0,", "e3cd88ad63d7c08c4e5c790dff2662f8": "r=a_0+\\dfrac{1}{a_1+\\dfrac{1}{a_2+\\dfrac{1}{a_3+\\cdots}}},", "e3cde3f61de01c172fa395c723c68d2f": "f.", "e3ce5561febf9d438c53cfea88f5d031": "\\frac{13}{30} - \\frac{\\pi}{8} =\\sum_{n=1}^\\infty \\frac{1}{4^{2n}}\\left[\\zeta(2n)-1\\right]", "e3ce756c1a06a660adbdaa44e26a3179": "\\begin{align} (\\mathbb{Z} / 5\\mathbb{Z})^\\times & = \\{ [1], [2], [3], [4] \\} \\\\ & \\cong C_4 \\\\ \\end{align}", "e3ce8190ab6c2eb40f806ab3fec302a2": "f+df", "e3ce8d1caf5c4295a5026ef3a447f1be": "f(\\mathbf{x})= \\frac{1}{\\sqrt { (2\\pi)^k|\\boldsymbol \\Sigma| } } \\exp\\left(-{1 \\over 2} (\\mathbf{x}-\\boldsymbol\\mu)^{\\rm T} \\boldsymbol\\Sigma^{-1} ({\\mathbf x}-\\boldsymbol\\mu)\\right)", "e3cea55686aadc50137b0b51f0d69c9a": "\\boldsymbol{F_{12}}=-\\boldsymbol{F_{21}};", "e3cf398f0d3189471ed5d6abbb62911f": "A_1, \\ldots, A_k", "e3cfad510343f12adb780e72c4e7ea7f": "\n \\begin{align}\n \\frac{\\partial M_{11}}{\\partial x_1} + \\frac{\\partial M_{12}}{\\partial x_2} & = Q_1 \\quad\\,,\\quad\n \\frac{\\partial M_{21}}{\\partial x_1} + \\frac{\\partial M_{22}}{\\partial x_2} = Q_2 \\\\ \n \\frac{\\partial Q_1}{\\partial x_1} + \\frac{\\partial Q_2}{\\partial x_2} & = - q \\,.\n \\end{align}\n", "e3cfc203fe87b88faf32931949edf227": "\\kappa=8/3", "e3cfd096a16d654388f41d14ce907bab": " f: \\Omega \\mapsto \\mathbb{R} ", "e3cfd4c9906bf7baeff01051cc3e0cdb": "\\delta_s", "e3cff86a4f6cf13540812ea3df6b8e9b": "H^* (M; \\mathbb R^w)", "e3d002fd3c24977593c48de5e528f948": "n + m = n'", "e3d020b1893214477764439f5caaa09e": "ax^2\\!+\\!bx\\!+\\!c \\;=\\; a(x\\!-\\!\\alpha)(x\\!-\\!\\beta) \\;=\\; (x\\!-\\!\\alpha)^2t^2", "e3d044beda7abf65a605082ba69cf9b8": "cX \\sim \\mathrm{Gamma}( \\alpha, \\beta/c).", "e3d04bc633ad693077ddc0363560acd5": " a_{10} = \\mathcal{L} a_{20} +p_3 a_{20} +p_6, ", "e3d0c727c3fae6a94e607475547ed10a": " H|\\psi\\rangle = E |\\psi \\rangle \\,", "e3d0dacfdaba0cf0591e2577d9f0dc98": "P_k = \\frac{k}{|G|} \\sum_{g \\in G} c_k(g)", "e3d0f39081a11691e7206c2c2eee5dee": " B = \\{ x | Q(x) \\} ", "e3d102734160f17289851c541c1ca162": " c \\in C ", "e3d135a3e159c8c73217839ebe0e4b75": "\\chi(G)=k", "e3d19b54c9099f18506e1d04affc18a9": "\\operatorname{MSE}(S^2_{n})=\\operatorname{E}((S^2_{n}-\\sigma^2)^2)=\\frac{2n - 1}{n^2}\\sigma^4", "e3d1b7f5845c52e268b5db8f82fdd5c0": "\n z_+ := \\cfrac{-\\beta + \\sqrt{\\beta^2 - 4\\alpha\\gamma}}{2\\alpha} ~,~~\n z_-:= \\cfrac{-\\beta - \\sqrt{\\beta^2 - 4\\alpha\\gamma}}{2\\alpha}\n ", "e3d1f61fd4e0c656d0b46c58a134cfe4": "\\operatorname{hacoversin}(\\theta)", "e3d1facd7c27a08d54a16315711aecb9": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 1}{10 \\choose 1}{4 \\choose 2} \\end{matrix}", "e3d21d2037617ddcadc4c31d87b0901e": " \\alpha_i\\wedge \\alpha_j,\\; i\\neq j,", "e3d237d5db2bf27d96e90b07df6cf05b": " \n\\Delta(t) + Vp(t) \\leq B + C + VP(\\alpha^*(t), \\omega(t)) + \\sum_{i=1}^KQ_i(t)Y_i(\\alpha^*(t), \\omega(t)) \n", "e3d247e5b4a80bc0fb46b0576a0c6792": "\\textbf{P}_0'", "e3d2519b4040ca3ce12682a8c506eb45": "a^{(b-1)/2}\\equiv -1 \\pmod b\\;", "e3d26b8c0c1b0eda8f5c938e92f21e2c": "L_1\\preceq L_2 \\; \\mathrm{iff} \\; u(L_1)\\leq u(L_2).", "e3d28f36b25d04f9e81c5141307d6a92": "\\rho \\equiv N/V", "e3d2a6cb36df9fde6a067bee81d1de52": "\nI_{probe} = -I_{e} e^{-e V_{probe}/(k T_{e} )} + I_{ion}^{sat}\n", "e3d2a8faa2cb027bd212037bda1323dd": "\\ln(f_w(\\theta))= c_0 + 2\\sum_{m=1}^\\infty c_m \\cos(m\\theta)", "e3d2dd9dfb14374dba6a8072ddf76916": "\\scriptstyle a \\in \\mathbb{R}", "e3d3e0b642ee336a38144a3a2f4bd9d2": "\\displaystyle{\\pi(g) W(z) \\pi(g)^* = W(g(z)).}", "e3d412ae7d561f9fa3a60173f3afe7fb": "\\frac{1}{4n}p^2\\cdot \\cot(\\pi/n)\\,\\!", "e3d43cc796a64ca5d71748c404a8ec63": "(\\phi \\to \\chi) \\to ((\\phi \\to \\neg \\chi) \\to \\neg \\phi)", "e3d478f0bca91779c68a03d64bb34486": "c < K_\\varepsilon\\, \\operatorname{rad}(abc)^{1 + \\varepsilon}", "e3d4a1e279fd4e937489ccd42f1b15e3": "1.0023^{+0.0056}_{-0.0054}", "e3d4a2b06fd620c7386deeedf0a17393": " P(y_i)= \\frac{e^{\\beta_0+\\beta_1 x_{i1}+\\cdot+\\beta_k x_{ik}}}{1+e^{\\beta_0+\\beta_1 x_{i1}+\\cdot+\\beta_k x_{ik}}} =\\frac{1}{1+e^{-(\\beta_0+\\beta_1 x_{i1}+\\cdot+\\beta_k x_{ik})}} ", "e3d4a9927c9a8ea148a568ac6ca7f413": "\\nabla \\times \\mathbf{H} = \\frac{4\\pi}{c} \\mathbf{J} + \\frac{1}{c}\\frac{d\\mathbf{D}}{dt}", "e3d4e70efd2d41a41872608279b27dca": " \\log G(z+1)=", "e3d55cd354e46fd611f0efd9cfe32595": "M(M - 1)/2", "e3d5a0812795b1a35bd21b90b4f4b29b": "\\mathcal{L}(Q)", "e3d5d7d36d3955d3a787adf57230ad09": "(x,y)=(0,l)", "e3d62c4021c13b88e2f1e756b6b91fca": "\nV(x,y)= {q^2\\over |x-y|}.\n", "e3d646891084c792e3d619205e24763e": "S \\Rightarrow^{ac}_{f} AA \\Rightarrow^{ac}_{g} AA \\Rightarrow^{ac}_{h} SA \\Rightarrow^{ac}_{h} SS \\Rightarrow^{ac}_{h} \\text{failure: h cannot apply, no A to rewrite}", "e3d6f83df9630b907cb7c9c63ccc5ad3": "Rq_i/r_i\\,", "e3d71d43ba95670ac9a5a8ac2a311a19": "f\\circ g:{\\mathbb R}^n\\rightarrow{\\mathbb R}^\\ell", "e3d7a47e6dc40565036117fad0fcfccf": "\\frac{\\partial v}{\\partial t} - u \\beta y = \\frac{\\tau_y}{\\rho h},", "e3d7bcf1824c24863e3734fd3d7b1d4e": "f(k;k_0)=\\left\\{\\begin{matrix} 1, & \\mbox{if }k=k_0 \\\\ 0, & \\mbox{if }k \\ne k_0 \\end{matrix}\\right.", "e3d83b8c47ed599260dd326d3f141b1d": " E[n] = T[n]+ U[n] + \\int V(\\vec r) n(\\vec r){\\rm d}^3r ", "e3d8683b56c960493ce4babe2eeddd4e": "\n \\omega_j = \\omega_j^R + i\\gamma_j\n", "e3d8aed536716179bdeb378a836db4c1": "e^{-2\\pi i k/ N}", "e3d8ebd4a3f929be9c33cce74ae6cf50": "\\alpha^n = 1", "e3d90de498fad1109298d3440ff558d0": "m,n,\\dots", "e3d98698ad766d4221f9159e76d82ae8": "x\\to+\\infty", "e3da1b182914c755c43f86713c3d6c84": "E^\\mathrm{free}(\\mathbf{x},t)=\\sum_{n}\n\\frac{E_n^\\mathrm{ret}(\\mathbf{x},t)-E_n^\\mathrm{adv}(\\mathbf{x},t)}{2}=0 ", "e3da9c0d3e9d91523467262d2ca68baa": "\n\\lambda(t)=-\\frac{\\partial H}{\\partial x}\n", "e3daa32a00df8a965edbb3b014b4c385": " |1/2,+1/2\\rangle\\;|1/2,-1/2\\rangle\\ (\\uparrow\\downarrow)", "e3daa39decd9c0bd811b6a3c6329bb5b": "V_a\\ ", "e3dabd437edacc2847b80a64580215f2": " |P(a)| > F(A,d) ", "e3dae74f61eeae3fd687ad090e2b72e3": "a\\equiv^\\ast\\!b\\,(\\mathrm{mod}\\,\\mathbf{p})\\Leftrightarrow \\frac{a}{b}>0", "e3db75d25732657380126670db0cdb5e": " = \\omega(X_H(\\gamma),X_H(\\gamma)) = 0", "e3dbc2fa91d3d7db63ccb153fde4b521": "\\Phi_{\\xi}", "e3dc052993ef01410cfa6b1dbd9bde39": "2 I \\times C_{7}", "e3dc544a75c02aec3f407d7cd0dfc5ba": "\\mathbf{} v ", "e3dd1c2f726d91dd707437af43ef9d65": "Z = \\frac{L_{\\nu_s}}{\\int_{\\nu_0}^\\infty \\frac{L_\\nu}{h\\nu} d\\nu}\n = h\\nu_{H\\beta} \\frac{\\alpha_{H\\beta}^\\text{eff}}{\\alpha_B} \\frac{F_{\\nu_s}}{F_{H\\beta}}", "e3dd2d686fc1826a92ccbfcc0e3c72f9": "[n]=\\{1,...,n\\}", "e3de4b681a696cabc06e0092a8ec726d": "m,n,l", "e3de71bd1263d5be2003ab28862c8794": "y = a \\cosh \\tfrac{x}{a}\\,", "e3decb13096fd9846d7aa3fb9e8c24fb": "\\Phi_{1}(\\mathbf{r})", "e3dfba4a9ff00905fe397b9f4cd85cfb": " {b_1(\\rho_m-\\rho_c)} = h_1\\rho_c ", "e3dfc9e13de36b8375d643460bcded08": "x^2 + y^2 = 1 + \\frac{121665}{121666}x^2y^2", "e3e056a7b1b20a2b9373fde488474038": "S_B", "e3e062996416c72e36ff0d9c684fe7d3": " \\sqrt{\\mu_0}\\mathbf{H} ", "e3e09265f92ed9b932dc2672062d9a16": " \\phi_h(g) = \\mbox{IP}(h,g) = (h,g) = \\langle h,g \\rangle = \\langle h|g \\rangle ", "e3e0a1918d622e98b1b728bcc3e7ecf2": "\\frac { \\mathrm{P}(see\\ gold \\mid GG)} { \\mathrm{P}(see\\ gold \\mid GG)+\\mathrm{P}(see\\ gold \\mid SS)+\\mathrm{P}(see\\ gold \\mid GS) } =\\frac{1}{1+0+1/2}= \\frac{2}{3}", "e3e0e9809f41a0f4806cf61ebf857365": "\\delta_k(i)", "e3e0ee3a57b520cbbe537c75c7c7d85d": "h_j=\\lambda \\gamma_j", "e3e0feab0bc6d161706e21429a6d7ca1": " \\delta(t-\\tau) \\ ", "e3e16079f5f2d8b4d7f0bddf9d234d19": "\\varepsilon:= -\\frac{1}{2}\\big(n^aDl_a-\\bar{m}^aDm_a \\big)=-\\frac{1}{2}\\big(n^al^b\\nabla_b l_a-\\bar{m}^al^b\\nabla_b m_a \\big)\\,,", "e3e1b9e885743ae738ca74fb94c9c195": "\\frac{1}{|\\mathbf{x}_1 - \\mathbf{x}|} = P_0(\\cos\\gamma)\\frac{1}{r_1} + P_1(\\cos\\gamma)\\frac{r}{r_1^2} + P_2(\\cos\\gamma)\\frac{r^2}{r_1^3}+\\cdots", "e3e296e194dc849aa83f770fbdfd3ad1": "\\sigma_n^2", "e3e2c7bd1f9647055afc8d315abd9267": "\\mathbf{O}(\\mathbf{m}+\\mathbf{n}\\log{\\mathbf{n}})", "e3e2f48da89910709024239a6bbfb385": "k\\in \\Bbb N", "e3e2fcba65c4e7a857af2c743759b0ba": "\\beta = 0.5", "e3e3586aa778c15f4ccd6a5c6c159c96": "\nI(X;Y) \\leq \\frac{1}{2}\\log(2 \\pi e (P+n)) - \\frac {1}{2}\\log(2 \\pi e n)\n\\,\\!", "e3e35915c930bcc73b12214dca976d08": "\\omega^{\\omega^\\omega} + \\omega", "e3e365d782b1660805f004484a2e90b1": "[0,t)", "e3e427967aadd743e401d97a6d465248": "c = \\frac{2}{3} \\sqrt{-m_c^2 + 2m_b^2 + 2m_a^2} = \\sqrt{2(b^2+a^2)-4m_c^2} = \\sqrt{\\frac{b^2}{2} - a^2 + 2m_b^2} = \\sqrt{\\frac{a^2}{2} - b^2 + 2m_a^2}.", "e3e42ce87739ae36482e3db9dfecc756": "k_B T \\ln 2", "e3e496a39e143290c7efec2177b6d8e3": "\\mu(T,V,N)=kT\\left(\\hat{c}_P-\\ln\\left(\\frac{VT^{\\hat{c}_V}}{N\\Phi}\\right)\\right)", "e3e4fc4afa38947defbe8f62c27972bc": "h,m\\!\\in\\!\\mathcal{N}; u\\!\\in\\!V^+,v,v',w,w'\\!\\in\\!V^*", "e3e53cb569bfc9dfaa93619759421c27": " \\qquad a_2 \\ne 0", "e3e58552e95cccbf60c8edb824d879d0": " {\\gamma^2} \\left( x - v t \\right)^2 + y^2 + z^2 = c^2 \\left[ \\gamma t + \\frac{ \\left( 1 - { \\gamma^2} \\right)x}{ \\gamma v} \\right]^2", "e3e60ada0b8823b2a76cbbadcf1ee70a": "\\mathbf{r}=\\mathbf{r}(u,v),", "e3e611f47bc5569f4e61b87a869e08eb": "\\begin{align}\nt &= \\gamma \\left( t' + \\frac{vx'}{c^2} \\right) \\\\ \nx &= \\gamma \\left( x' + v t' \\right)\\\\\ny &= y' \\\\ \nz &= z',\n\\end{align}", "e3e65176180373570b677d4b2c59495d": "F_{max}\\,", "e3e66e20f0628a0a4473ee07bb962ed3": "X \\mapsto \\operatorname{Hom}(-,X)", "e3e66f0646604d79b37b27c4d5771ff3": "\\left| F, p_1\\ldots p_n \\right\\rangle", "e3e6c7c3f41a143f2b510daf3624dc46": "\\overline{\\mathcal C}", "e3e6df628c582b8bb9362996f061ecd1": "\\Omega_e \\propto t^{-\\frac{1}{2}}", "e3e6e3ef55df959ab60ce54d6c3a661c": "S[p_1..p_1+l-1]=S[p_2..p_2+l-1]", "e3e78b4765a42b2df10ea5381d4d4ae9": "\\beta \\mapsto \\alpha^\\beta", "e3e78fd98fcc0f2d80f5d8efebecf038": "fl(x \\cdot y)=\\hat{x} \\cdot \\hat{y}", "e3e79e8f871cd62987755e5b0be69ac8": "d\\omega(E_j,E_k) = -\\sum_i c_{jk}^iE_i(e) = -[E_j(e),E_k(e)]=-[\\omega(E_j),\\omega(E_k)],", "e3e808b38951e31e0aafcd381900a4f7": "y[j+nL] = \\sum_{k=0}^{K/L-1} x[n-k]\\cdot h[j+kL],\\ \\ j = 0,1,...L-1,", "e3e81ca37c836ca36d376bff594a46bf": "\\Phi_{ij}=\n\\begin{bmatrix}\nA&0&0\\\\\n0&B&0\\\\\n0&0&C\n\\end{bmatrix}\n", "e3e8238e9991c4278a21d50c074d8c2a": "\\dfrac{\\alpha : X/Y \\qquad \\beta : Y/Z}{\\alpha \\beta : X/Z}B_>", "e3e8df8d04f718a940415df54165f529": "\n\\,\\!V_R = V_{in} - V_C\n", "e3e93b26d73b3a09ad012bccc1cda77a": "\\varphi(t;\\mu,c)=e^{i\\mu t-|ct|^{1/2}~(1-i~\\textrm{sign}(t))}.", "e3e9607188f9483410d0ea5b8450c38e": " P(S^{t+1} \\mid h^t, o^{t+1}) \\propto P(o^{t+1} \\mid S^{t+1}) P(S^{t+1} \\mid h^t) ", "e3e98daf15f06485c440f41262f799a6": "-1.1641", "e3e9e8437447d08b5d0b3451fee50fe6": "\\{O_1,O_2\\}\\rightarrow \\frac{1}{i}[O_1,O_2]. \\, ", "e3ea440cca52472647e9a4e4e2b7a48d": " f_X(x| \\theta) = h(x)\\ \\exp\\Big(\\ \\theta x - A(\\theta)\\ \\Big) \\,\\! .", "e3ea5b5d8fb4ce6bac2c4570d67ab552": " \\mathbb{C}^n\\otimes\\mathbb{C}^n = S^2\\mathbb{C}^n \\oplus \\Lambda^2\\mathbb{C}^n.", "e3ea95396dd11f5892f7b7bab23061d7": "= \\sgn( \\sin (\\theta+ \\frac{\\pi}{2})) \\frac{\\sin \\theta}{\\sqrt{1 - \\sin^2 \\theta}}", "e3eaa526bffc904c4647133acbe2a975": " \\mu\\left(\\bigcup_{i=1}^\\infty E_i\\right) = \\lim_{i\\to\\infty} \\mu(E_i).", "e3eadc87783f3378e5ca282cac8e8770": " \\min \\Phi_k (\\bold x) = f (\\bold x) + \\frac{\\mu_k}{2} ~ \\sum_{i\\in I} ~ c_i(\\bold x)^2 - \\sum_{i\\in I} ~ \\lambda_i c_i(\\bold x)", "e3eaf9ee17dff54d23284615061e2e57": "(i,i)", "e3eafe18ced16a207783fb89fe55af2a": "\n\\text{(Eq. 2)} \\qquad\n\\text{Subject to: } (\\mu_{ab}(t)) \\in \\Gamma_{S(t)} \n", "e3eb35f5cca77083f8d0741da4a4f47d": "(\\pi/6) A^2C \\approx 0.523\\, A^2C", "e3eb8e1320d1b88a046368092d08fb3e": " R = U_s\\Lambda_s{U_s}{^H} + U_n\\Lambda_n{U_n}^H ......(7) ", "e3eb9beb5dd0ea7e6f97a16a9ec41c83": "\n (X-s_\\lambda)\\cdot H^{(\\lambda+1)}(X)\\equiv H^{(\\lambda)}(X)\\pmod{P(X)}\\ .\n", "e3eb9c0e4e2528230dd44a40bbf7f58e": "M(a,b,z)= \\frac{\\Gamma(b)}{\\Gamma(a)\\Gamma(b-a)}\\int_0^1 e^{zu}u^{a-1}(1-u)^{b-a-1}\\,du\\,\\quad .", "e3eba226e146fb45a38ff70d216175a7": "\\{x_{kb^m}, ..., x_{(k+1)b^m-1}\\}", "e3ebaf221bfa7404a9a2d5b2e423d32f": "z \\mapsto \\frac{az + b}{cz + d}.", "e3ebcc3895b4b70c9b3bdac75eb1e752": "F_{t,T} < S_t e^{r (T-t)}", "e3ebfc3d6170a347bd01b4a168749fcf": "\\nu_{T}", "e3ec0a00abd0011deb0f056344a224e3": "\\scriptstyle 6 = 2 \\cdot 3", "e3ec5a14969a866814f5a8611b19e73e": " = {T \\over 2}\\int \\mathrm{d}^2 \\sigma \\sqrt{-\\tilde{h}} \\tilde{h}^{ab} g_{\\mu \\nu} (X) \\partial_a X^\\mu (\\sigma) \\partial_b X^\\nu(\\sigma) \\, ", "e3ec7970a924ea4fd4475e15c99a1f91": " 2 (d^2 \\psi / dz^2) d\\psi = d (d\\psi / dz)^2 ", "e3ecf9410666b55daa5a4f3c81004237": " Q' = \\frac{Q}{K}", "e3ed32cba997e90bef8d8cea9cee9fbd": "775/1.85=418\\text{ years}", "e3ed756e5d3464bd0339d6dee0c020b4": " \\Gamma_{ij}{}^k", "e3edb5392d36feba4a43df1f8e38e412": "\\,^{z_{13} = x_{13} y_1 - x_{14} y_2 - x_{15} y_3 - x_{16} y_4 + x_9 y_5 + x_{10} y_6 + x_{11} y_7 + x_{12} y_8 + x_5 y_9 - x_6 y_{10} - x_7 y_{11} - x_8 y_{12} + x_1 y_{13} + x_2 y_{14} + x_3 y_{15} + x_4 y_{16}}", "e3ee189591f651d41d3a6278e9772ac5": "2px-p^2=2qx-q^2.", "e3ee4f614e8f641ea359f500e9d24045": "O_X", "e3ee824bb46b19d382ddc03299639867": "\\frac{\\rm d}{{\\rm d}t}x(t)=A_0x(t)+A_1x(t-\\tau_1)+\\dotsb+A_mx(t-\\tau_m)", "e3eef9dde8f87dbc5194554b1d4891ee": "\\sin^4\\theta = \\frac{3 - 4 \\cos 2\\theta + \\cos 4\\theta}{8}\\!", "e3ef283eeed547d8e5cd4695345a0abb": "(T_s \\text{ or } \\phi)", "e3ef58ce8c6ebec45b51ddb9d2030dc2": "i\\le j\\le k", "e3ef7319e443a846928a5258a6ee2677": " y_2(x) = x y_1(x) = x e^{-\\frac{b}{2 a} x}.", "e3ef76905788b1a6216b1a210eb68016": " v_B = v_{B_1}+\\cdots+v_{B_m}. \\, ", "e3ef9e761d5d84f431c1c78c0740b2ab": "=e^{i\\pi/2}", "e3efed2ba2c215ac09f2bb49d8b52bdb": " \\theta = \\frac{1}{2}(\\omega _0 + \\omega _1)t", "e3f0188c7de20055cefdd41742827b19": "r\\rightarrow 1", "e3f04d4a9ffceb49e55d3c476a3d0e0a": "\\forall n\\ \\forall i,j \\ge g(n)\\quad |f(i) - f(j)| \\le {1 \\over n}", "e3f07e861be9bd278b304103dce78f5f": "E(\\nu )\\propto k", "e3f08ba2ba8223012b2773cc5f7c5d22": "F_k\\,\\!", "e3f093e09364b4fd324ca28fa50828f6": "A = \\pi r^2", "e3f0a06abf3e996930b0008548daf07c": "t_u - t_l \\rightarrow\n\\infty", "e3f0b371c8ac732c43ffa1abc4ea9227": "\\begin{align}\n e_1 &= 1p_1,\\\\\n 2e_2 &= e_1p_1-1p_2,\\\\\n 3e_3 &= e_2p_1 - e_1p_2 + 1p_3,\\\\\n \\vdots &= \\vdots \\\\\n ne_n &= e_{n-1}p_1 - e_{n-2} p_2 + \\cdots +(-1)^ne_1p_{n-1}+(-1)^{n-1}p_n\\\\\n\\end{align}", "e3f0fda185283aee6d247881bd1b1497": "\\displaystyle{Q(a,b)= 2L(a)L(b)-\\frac{1}{2}R(a,b).}", "e3f173ae8a94807d4813ddf823969daf": "\\displaystyle S_{\\mu,\\nu}(z) = s_{\\mu,\\nu}(z) -\\frac{2^{\\mu-1}\\Gamma(\\frac{1+\\mu+\\nu}{2})}{\\pi\\Gamma(\\frac{\\nu-\\mu}{2})}\n\\left(J_\\nu(z)-\\cos(\\pi(\\mu-\\nu)/2)Y_\\nu(z)\\right)", "e3f17839b43f6e7f2ed7fae783c67ee3": "n_c = 2", "e3f19bedc9425927df1750dba5f1bece": "S_n(t) = S_n(0)S_0(t)", "e3f1e3bcc5a5450f4fe1f5a409a9b030": "\\hat\\mu(u, v; \\tau) = \\mu(u, v; \\tau) - \\frac{1}{2}R(u - v; \\tau)", "e3f1e55e9b2678194bd88bf239022efe": "E = \\int \\frac{1}{2} \\nabla\\phi_0\\cdot\\nabla\\phi_0 d^2 x + \\int \\frac{1}{2} \\nabla \\sum_{i=1}^n n_i \\arg(z-z_i)\\cdot\\nabla\\sum_{j=1}^n n_j \\arg(z-z_j) d^2 x", "e3f1e6cac3cf261be641781f157744f8": "\\alpha_{B,V}:B\\otimes_FD_B(V)\\longrightarrow B\\otimes_EV", "e3f1e93dc8bdb481841d26a0bdb35e3e": "a_j=0", "e3f1f434505139056b9a62f5e2af8ad3": "R_{ijkl}=(\\mathsf{R}(e_i \\wedge e_j) \\cdot e_k) \\cdot e_l", "e3f21d4067454ed4fd433a852420d048": "i_x", "e3f29579d0778bab592fa78358090033": "\\Sigma_k^{\\rm P}", "e3f29f20780ebbd8a462f340e2b856d8": "{\\color{Salmon}(f^{-1})'}({\\color{Blue}f}(x_0)) = 4~", "e3f31f77023e53086109f24ddde8a7da": "x_1^2 + \\cdots + x_n^2 + 2a_1x_1 + \\cdots + 2a_nx_n + 1 = 0,", "e3f35a5970892eba1cc7937135164c17": "r_m^-=r_m", "e3f36fce897df1272a66d0a62dcbf188": "\\mathit{o(q)}", "e3f37a59c81e3750462a753ee86f6a0e": "\\cos(p\\phi)", "e3f3845e242326eeae72ab3380dba6d8": "\\delta w = -V\\,(-P\\delta_{ij})\\,d\\varepsilon_{ij}=PVd\\varepsilon_{kk}", "e3f393a4c25d8307397fa74665642b1b": "\n\\begin{align}\n k(\\lambda,\\phi)\n&=k_0\\left[1\n+\\frac{\\lambda^2c^2H_2}{2}\n+\\frac{\\lambda^4c^4H_4}{24}\n+\\frac{\\lambda^6c^6H_6}{720}\\right],\\\\\n\\gamma(\\lambda,\\phi)\n&=\\lambda s\n+\\frac{\\lambda^3c^3t H_3}{3}\n+\\frac{\\lambda^5c^5t H_5}{15}\n+\\frac{\\lambda^7c^7tH_7}{315},\\\\\nk(x,y)&=k_0\\left[1\n+\\frac{x^2K_2}{2(k_0\\nu_1)^2}\n+\\frac{x^4K_4}{24(k_0\\nu_1)^4}\n+\\frac{x^6K_6}{720(k_0\\nu_1)^6}\\right]\\\\\n\\gamma(x,y) &=\n\\frac{xt_1}{k_0\\nu_1}\n + \\frac{x^3t_1 K_3}{3(k_0\\nu_1)^3}\n + \\frac{x^5t_1 K_5}{15(k_0\\nu_1)^5}\n + \\frac{x^7t_1 K_7}{315(k_0\\nu_1)^7}.\n\\end{align}\n", "e3f40b43502b1b6d8e002f4cb69f29ff": "\\textstyle\\left\\langle{n\\atop k-1}\\right\\rangle", "e3f4664301c603bbfae5d04fb3e124ee": "(N,v)", "e3f46f8a0c92145576081d260f9fc4d9": " 20 \\log_{10}(|G(s)|) ", "e3f4fca8497443ca0b80581de98524c6": "\\scriptstyle A=D", "e3f5235862c49ff6d6cad8c7154aebd1": "Y_{1,1} = -\\alpha_e", "e3f545976b098d7a35ba8b6a8b80e36d": "\\frac{\\left(\\Gamma_\\mathrm{res}/2\\right)^2}{\\left(\\Gamma_\\mathrm{res}/2\\right)^2 + \\left(E-E_\\mathrm{res}\\right)^2}", "e3f56a2f3fdef92beb9f8e05f84beb2d": "H^2 = \\frac{Var(G)}{Var(P)}", "e3f62c333e9b1db337498a7091a35a88": "H=\\frac{i}{2}(\\bar{\\partial}-\\partial)\\omega.", "e3f6b04c77f86e3c157436e456ab81ac": "P \\not = P'", "e3f6d8ff5e7cf9b2382af57a308a1673": "\\,c = \\sqrt{a^2+b^2-2ab\\cos\\gamma}\\,;", "e3f6df4822408f78c7801909e0227272": "\\sqrt{a+b \\sqrt{c}\\ }", "e3f6fe8957cdbbac0b10bd7c28e28665": "\\chi_r:=\\frac{1}{|B(0,r)|}\\chi_{B(0,r)}=\\frac{1}{\\omega_n r^n}\\chi_{B(0,r)}", "e3f712fc6277a87adc3fc16edc0e9994": "\n (6) \\qquad \\cfrac{\\partial^3\\varphi}{\\partial x \\partial t^2} = \\cfrac{EI}{J}~\\cfrac{\\partial^3 \\varphi}{\\partial x^3} + \\cfrac{\\kappa AG}{J}~\\left(\\frac{\\partial^2 w}{\\partial x^2} - \\frac{\\partial \\varphi}{\\partial x}\\right)\n", "e3f720154d720b58ef5f2672f44c7b0f": "\\rho\\ln\\rho", "e3f72a5c82a765079e518503cd542e54": "\\gamma(K) = \\sup \\{|f'(\\infty)|;\\ f\\in\\mathcal{H}^\\infty(\\mathbf{C}\\setminus K),\\ \\|f\\|_\\infty\\leq 1,\\ f(\\infty)=0\\}", "e3f74c3e0e54e20866a3882876e1784a": "\\eta=(1-R_1)(1-e^{-\\alpha d} )[ \\frac {( 1 +R_2 e^{-\\alpha d} )}{1 - 2 \\sqrt{R_1 R_2} e^{- \\alpha_c d}\\cos(2\\beta L+ \\phi_1 + \\phi_2) + (R_1 R_2) e^{- \\alpha_c d}}] ", "e3f761c9c4de35911b093e9658212903": "1/n^2=0", "e3f76870bf1627f7bd64f98ae26ae702": "\n \\partial = \\mathbf{e}_0 \\partial_0 - \\mathbf{e}_1 \\partial_1 - \\mathbf{e}_2 \\partial_2 - \\mathbf{e}_3 \\partial_3,\n", "e3f775f935b35bd53431c97eb2e0c0c9": "\\mu(A) =\\begin{cases} (1-\\frac{1}{\\lambda}) \\mathbf{1}_{0\\in A} + \\nu(A),& \\text{if } \\lambda >1\\\\\n\\nu(A),& \\text{if } 0\\leq \\lambda \\leq 1,\n\\end{cases}\n", "e3f789b361d76454614e1c481bcec731": "\\mathbf{B}^k", "e3f7d6e76430369c042b639ca2d14a5f": "Z^{I}_{i,j}:", "e3f7f58a0f7dbba509e341714a95548a": "G_{cd}F^{cd}=\\frac{1}{2}\\epsilon_{abcd}F^{ab} F^{cd} = - \\frac{4}{c} \\left( \\vec B \\cdot \\vec E \\right)", "e3f86c9a66d753397e247a9a175ea87e": "\nd = \\frac{2}{3} \\left(\\frac{2e}{m_\\mathrm{i}}\\right)^{1/4} \\frac{|\\varphi_\\mathrm{w}|^{3/4}}{2\\sqrt{\\pi j_\\mathrm{ion}^\\mathrm{sat}}}", "e3f86ee7878af614605f7aadcb4a5206": "\\mathcal{B}=\\textbf{1}", "e3f87063941444b8bfecee0ca1d73d0c": "\\nu = 0,\\pm 1,\\pm 3, \\pm 4", "e3f89c20dbd7ddb0f9023e9c1e0416aa": "\\; \\tau(A : B : C) = \\tau(A : BC) - \\tau(AB) - \\tau(AC) ,", "e3f8e0694222d991e4e31d969aaad828": "\\exists r D(r)=\\mathrm{true}\\;", "e3f9151e7d891153a3f48b8326daeb92": "\nL_{x}<0\\,,", "e3f935947abb2e3b6d7e18774d9fff49": "\nw(n) = \\left\\{ \\begin{matrix}\n\\frac{1}{2} \\left[1+\\cos \\left(\\pi \\left( \\frac{2 n}{\\alpha (N-1)}-1 \\right) \\right) \\right]\n& 0 \\leqslant n \\leqslant \\frac{\\alpha (N-1)}{2} \\\\ \n1 & \\frac{\\alpha (N-1)}{2}\\leqslant n \\leqslant (N-1) (1 - \\frac{\\alpha}{2}) \\\\ \n\\frac{1}{2} \\left[1+\\cos \\left(\\pi \\left( \\frac{2 n}{\\alpha (N-1)}- \\frac{2}{\\alpha} + 1 \\right) \\right) \\right]\n& (N-1) (1 - \\frac{\\alpha}{2}) \\leqslant n \\leqslant (N-1) \\\\\n\\end{matrix} \\right.\n", "e3f9a946a99c6e301b8add29de488681": "F_\\mathrm{Z}=\\frac{F_\\mathrm{L}}{M\\!A}\\quad,\\quad h=\\frac{s}{M\\!A}\\quad.", "e3f9afa292262e95c5890491100847fd": " \\nabla \\cdot (\\mathbf{A} \\times \\mathbf{B}) = \\mathbf{B} \\cdot (\\nabla \\times \\mathbf{A}) - \\mathbf{A} \\cdot (\\nabla \\times \\mathbf{B}) \\ . ", "e3f9b5475d5ce6d004ca565484a3c743": "\\mathrm{lift}: \\mathrm{M} \\, A \\rarr S \\rarr \\mathrm{M}(A \\times S) = m \\mapsto s \\mapsto \\mathrm{bind} \\, m \\, (a \\mapsto \\mathrm{return} \\, (a, s))", "e3f9de1fa47a5bd5c64f0784661814a3": "F_n(u) = \\frac{(-1)^n}{\\pi \\hbar} L_n\\left(4\\frac{u}{\\hbar \\omega}\\right) e^{-2u/\\hbar \\omega} ~,", "e3fa0b66d207d04e74c696d911940284": "{T_{sample} \\ge T_m + 2T_a }", "e3fb5142179054ae411b45e2c22aa34e": "\\, \\frac{e^{itb} - e^{ita}}{it(b-a)}", "e3fb9092c9a66f598f37f365c9c4f1f5": "\\mu^*", "e3fb9a9c35b6f5ee01cba3f1dfd996bb": "e = a^2", "e3fbb75fd1f1cd3e36d3a51f1f8a6c9c": "[X/G]", "e3fc139a5297ff44f4c62b4e6027b015": " L = \\left\\{ \\langle A,k,B,x \\rangle \\in \\mathcal{C} \\times \\mathbb{N} \\times \\mathcal{C} \\times \\{0,1\\}^* \n\\left|\nB \\mbox{ has at most } k \\mbox{ gates, and } A(x)=B(x) \n\\right.\n\\right\\} ", "e3fca50d483464160c53de1c1679ad5d": "n_B^\\prime(\\xi)=-\\frac{\\beta}{4}\\mathrm{csch}^2\\frac{\\beta \\xi}{2}", "e3fcd15c338e36ec29bffdd11c5b9612": "\\left\\langle T_{f}, \\varphi \\right\\rangle = \\int_\\mathbf{R} f(x) \\varphi(x) \\,dx. ", "e3fcdd4fd25b0c8dcdf5361c933918ad": " \\frac{\\partial^2}{\\partial\\theta^2}\n \\left[\n \\int T(x) f(x;\\theta) \\,dx\n \\right]\n =\n \\int T(x)\n \\left[\n \\frac{\\partial^2}{\\partial\\theta^2} f(x;\\theta)\n \\right]\n \\,dx.\n", "e3fd274cfdbb1ea426a1c3606185a24f": "\\textstyle \\overline{y}(\\lambda)", "e3fd396751f7786bfc1513c5c2bd32d4": " q_{n}, n=5,6,\\dots,\\infty ", "e3fd569281fc736052261daded25de4f": "(\\hat{a}^{\\dagger}_{a} =) \\hat{a}^{a}", "e3fe1f1ae45642d609ff49b9bb7745dd": "-Ay", "e3fe79bf98672eef435b628cf608cadb": " \\Omega = U - TS - \\mu N\\,\\!", "e3fe829fcac8b8ec7b4ab96053e6e09d": "\\phi(n)\\;", "e3fe8c119977f257c09713019353741d": "\\phi(x,y)", "e3fee2ff37bb46de83227ca3f3636b62": "I={\\big\\langle\\big\\langle} l_1,\\ldots,l_p{\\big\\rangle\\big\\rangle}", "e3ff4cc595d70d9b40268618bb66188b": "\\mu=25", "e400088f1549055916fef3660e04feea": " \\forall x \\in \\mathbb{R} \\quad \\exists y \\in\\mathbb{R}\\quad x < y. ", "e400319b901a5dd4974a7d9efe26619a": "\n\\arctan z = zF({\\scriptstyle\\frac{1}{2}},1;{\\scriptstyle\\frac{3}{2}};-z^2).\n", "e4007f9f65194b9121d458110426a565": "\\lambda=\\lambda_n", "e400ad1bc8ccb79d9b58a4c8561cd158": "m = (m_1, \\ldots, m_K) \\in [Q]^K", "e400cf9e09b530ddb91d5026b1f5de12": "\\lambda_7 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & -i \\\\ 0 & i & 0 \\end{pmatrix}", "e400db5a97b98d693076726ab657a2cf": "x -\\sqrt{y}", "e400e2d10369c11507e494b26922d310": "SM(levol,endo,exo)", "e400e6b07428a8b126c6d983d8c398d6": "\\mathcal{L}\\big(\\varphi(x),\\partial_\\mu\\varphi(x)\\big) = \\mathcal{L}\\big(T(\\omega)\\varphi,T(\\omega)\\partial_\\mu \\varphi\\big).", "e400e81d511fdbbcb63c049d6d06b72e": "\\begin{matrix} D \n &=& \\{a,b\\}\\times\\{a,b\\} \\quad \\cup \\quad \\{a,c\\}\\times\\{a,c\\} \\\\\n &=& \\{a,b\\}^2 \\cup \\{a,c\\}^2 \\\\\n &=& \\{ (a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)\\} \n\\end{matrix}", "e40148dd0b37872a0fd4a2e44aeb846e": "\\|f\\|_2 = \\lim_{r\\to 1} \\sqrt{M_r(f)}.", "e4016352c979a40bb5e7a96cfb2fdf9f": "G_Y+G_{(Y|X)}=G_X", "e401a2d683e6086c47fe2790fd31ea37": "\n\\begin{align}i & = 32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}+\\frac{1}{2i}d_{-1}+\\frac{1}{-4}d_{-2}+\\frac{1}{-8i}d_{-3}\\\\\n& = i(32d_{5}-8d_{3}+2d_{1}-\\frac{1}{2}d_{-1}+\\frac{1}{8}d_{-3})+16d_{4}-4d_{2}+d_{0}-\\frac{1}{4}d_{-2}\\\\\n\\end{align}\n", "e401ad1f95b78f5725450502f765a929": "a^\\dagger\\psi_n", "e40203005d3716026f314f8a5b2f3291": "\\nabla_2 ( (x_1 \\otimes x_2) \\otimes (y_1 \\otimes y_2) ) = \\nabla(x_1 \\otimes y_1) \\otimes \\nabla(x_2 \\otimes y_2) ", "e402572f007e4e0603c1115789cebe91": "V_i-V_o=L\\frac{dI_L}{dt}", "e40287717450f0f687a387d13bafbe87": "\\sqrt{3} \\cdot v_{e,th}", "e402c5a8e57a7e58569310c98f15fbdc": "\\begin{align}\nB &{}=7,460,514 \\\\\nW &{}=10,366,482 \\\\\nD &{}=7,358,060 \\\\\nY &{}=4,149,387 \\\\\nb &{}=4,893,246 \\\\\nw &{}=7,206,360 \\\\\nd &{}=3,515,820 \\\\\ny &{}=5,439,213\n\\end{align}", "e403246d164a70eaf08da4d7ae91bfbd": "c - c_0 = exp[R\\bar{\\beta}t] cos\\beta \\cdot r ", "e40329e526e2c97addeafe833788255b": "\\overline{|\\mathcal{M}|^2} \\,", "e4032b0a1dbe68fa1bb7754778372dbf": "S < 40 ", "e4035b8a26d011170a3fa514355355c8": "F_r={{V}\\over {\\sqrt{g \\left({A \\over B} \\right)}}}={{V}\\over {\\sqrt {gy}}} ", "e403d3209462f7b653083c980aadd14a": "C = \\frac{0.02 P_1 P_2}{d^2},", "e403eba24840220a05298080aa6079c0": "y_n \\in \\{0,1,\\dots ,q-1\\}^k", "e40444a94ae90f2ae253721b8132f977": "V_{0}", "e40472544a7f713cd4e2ebf5076948a0": "\\mathbf D = \\epsilon \\, \\mathbf E \\, .", "e404a266021b8c3f013e8496d6cf9661": "C \\cong B/q(A)", "e404c0ad91387a23063eb94ea8e60b5f": "\\left[\\hat{b}_i^\\dagger, \\hat{b}_j^\\dagger \\right]_- = 0 ", "e404ce5b46a9c2a36578d4e3d7fb8faa": "\\sum_{k=0}^\\infin \\frac {1}{n^k}=1+\\frac{1}{n-1} \\quad\\mbox{for all} \\quad n\\in\\mathbb R > 1.", "e404d873d7c1b42e87cfb785c7ab93e7": "x \\in L \\Rightarrow \\mathrm{Pr}[A\\,\\mathrm{accepts}\\,x] \\ge 1/2 + 1/2^{f(|x|)}", "e4053d940eea60a02abbd38b8b59e0dd": " E_{in} ", "e4054c64c12e0fd4a738f247fcbc9496": "\\frac{\\partial^2V}{\\partial x^2}+\\frac{\\partial^2V}{\\partial y^2}+\\frac{ \\partial^2V}{\\partial z^2}=0", "e4059338a6002f32c6989808d02e870e": "Deposits = \\sum_{n = 0}^{\\infty}\\left[\\left(1 - \\alpha - \\beta - \\gamma\\right)\\right]^{n} = \\frac{1}{\\alpha + \\beta + \\gamma}", "e405b4ed7bdbb45e209e88e2f2d8a41e": "(\\alpha_m,r_m)", "e405d2d3ca2e1c0461b4543c2c755d49": "E^r_{p,q}", "e4065550721aa70428d1dcdede7dc23c": "z^2 + w^2 = r_2^2, x^2 + y^2 \\leq r_1^2", "e406ac4d7c470823a8619c13dd7101be": "s_i", "e407733d7abfc522f0cda640c538d60f": "D_{\\mathrm{KL}}(p \\| m ) = \\int \\log (f(x)) p(dx) = \\int f(x)\\log (f(x)) m(dx) .", "e408a31f06cb3c9ba7bee772abc6768c": "\\sum_{i=1}^{n}\\sum_{k=1}^{m} X_{ij}W_{ii}X_{ik}\\hat{ \\beta}_k=\\sum_{i=1}^{n} X_{ij}W_{ii}y_i, \\qquad j=1,\\ldots,m\\,.", "e408bed601b9932f92230b52c8365324": "\\! (1-p+pe^{it})^n", "e408d7201752a427d602e79bd75f48f2": "\n\\epsilon = 1 + k \\langle \\mathcal{M}_{\\rm Tot}^2 \\rangle\n", "e40936007e43d3b2a6e93b55ec6ad046": "\\left\\langle -2, \\{0,1,\\mathrm i,1+\\mathrm i\\}\\right\\rangle", "e409c47941db0b7e589c2e376fad0050": "C_\\beta(s) = \\begin{bmatrix}\\beta'(s) & \\beta''(s)\\end{bmatrix}.", "e409ced00ce7756b716cc499a3bf264e": " S \\cdot X^2 \\approx 1.1682 \\approx 269.2 \\ \\hbox{cents} ", "e40aa159332af5204d403b76f49b113a": "\\alpha^j", "e40adb22b8bcfd91c592043d2efcb0e8": "_aC_b", "e40b3b85e7566e436a7053b3dd3de5f2": "N_{\\text{vol}}", "e40b53e14e3d3c13fa7bdea5a55d75e1": "N>0", "e40b5f325566245b3e9d84d939d0e0c2": "\\partial / \\partial z ", "e40b7c9f89a9ba71805990bf1be12d39": "P_3=(x_3,y_3)=P_1+P_2", "e40bc20dec54aac2e357c6491e2c416a": "\\phi_{{\\Omega^\\Omega}}(0)", "e40bd7823726bd797bc40b961b4fbc3f": "\nf_{out} = \\frac{f_{in}}{n}\n", "e40c1dc8570df140a21035257210ccb5": "P_1 \\uparrow MS(X,Y,W)", "e40c637c0a0dc10d90fb2f56ec8b32a5": "\\eta \\colon 1_{C} \\to T", "e40c84bc593bb4807de273383fea177f": "k(x,\\cdot)", "e40c89c89782baa93ab037d7e48a2619": "\\mathcal{O}_{K,\\mathfrak{p}}/\\mathfrak{p}", "e40c9e00b48ef3b03b5360a6b6f63588": "\\mu_a(\\lambda_2) = \\ln(10)\\varepsilon_{HbO2}(\\lambda_2)C_{HbO2}+\\ln(10)\\varepsilon_{Hb}(\\lambda_2)C_{Hb} \\,", "e40cbeab9bc88536b52696c0b239388e": "e^{727.951346801}", "e40cc70c40525bca81b08557ef092ec4": "s_0\\ldots s_i = \\frac{\\ell_1}{\\ell_0}\\frac{\\ell_2}{\\ell_1}\\cdots\\frac{\\ell_{i + 1}}{\\ell_i} = \\ell_{i + 1}. ", "e40d706dffc582b722e3dda8fb706dc3": " \\forall j ", "e40d9e371ec06b852d850633fae04de2": "s,u_1,u_2,v\\,", "e40dd18e65bca91ae82e7e36bd8c6046": "\\varphi_{ij}:\\mathbb{R}^n\\to\\mathbb{R}^n", "e40ddd640f5ed4794a1d2c2edf362752": "\\alpha=\\mathbf{V}p", "e40dec85132562939bd707984e46562b": "\\nabla \\cdot \\mathbf{B} = 4 \\pi \\rho_{\\mathrm m} ", "e40e23f8301fff19a2d13da0e26f4999": "\nP(X \\in A \\mid Y \\in \\cup_i[y_i,y_i+\\delta y_i]) \\approxeq\n\\frac{\\sum_{i} \\int_{x\\in A} f_{X,Y}(x,y_i)\\,dx\\,\\delta y_i}{\\sum_{i}\\int_{x\\in\\Omega} f_{X,Y}(x,y_i) \\,dx\\, \\delta y_i} , ", "e40e2b09727553ea7b599134c245f64d": "z = re^{it}+c", "e40e8a34f14880ab22f711e5491334ef": "\n \\sigma_{11}= 2C_1 \\left(\\lambda^2 - \\cfrac{1}{\\lambda}\\right) \n = 2C_1\\left(\\frac {3\\varepsilon_{11} + 3\\varepsilon_{11}^2 +\\varepsilon_{11}^3} {1+\\varepsilon_{11}}\\right)\n ", "e40e9f2138648da5341cb7b03a0711b0": "K \\sim \\mathrm{Geometric}(\\exp(-W))\\, .", "e40ef013275e8c379514acc85300b556": "x_1 e_1+\\cdots+x_n e_n=0.\\,", "e40f38f3433c6077940575672174339f": "G_{\\nu}( \\omega)", "e40f3deacf6bc77c454105d553e3aff6": "x_{4} = [1,2,3,4]^{T}", "e40f50e5812e2d266c5f36385115bdf6": "W=\\frac{12 S}{m^2(n^3-n)}.", "e40f60242de63b6875a0a6b3ae4a4859": " v_{0y} = v_0\\sin\\theta", "e40f63106ef6855f458b3078d0beb82c": "\\mathbf{t}_{(i)} = (t_1, \\dots, t_p)_{(i)}", "e40f75f52d37164f8a29b0fb4646ff2d": "|b_{11}|", "e40fe0c88e4a2cad40a62a769390382b": "B_z(x) = B_0 e^{-x / \\lambda}. \\,", "e410085a545845482ace3a45846b7bf3": "\\gamma_{\\mathrm{SG}}", "e4105d8ec6ae673360c111d70b8988b8": " a_{i} ,b_{i} \\in \\mathbb{Z}_{p_{i}} ", "e4107b3220fe9a2b04c890949ffa4b54": "w\\nVdash A", "e410ba19127b68ccd9d50c810f252044": "C_{AB}", "e410faa1a0b68a7393ee6e517112f88e": " \\nabla^2(\\phi\\psi)=\\phi\\nabla^2\\psi+2\\nabla\\phi\\cdot\\nabla\\psi+\\psi\\nabla^2\\phi", "e4110c7c1a0f0ad26549185264a9c1bc": "\\Phi''(x) + \\Phi'(x)^2 = \\frac{2m}{\\hbar^2} \\left( V(x) - E \\right).", "e4114d5bfa512b67f1deacf8109c82a2": " I_t ", "e411da69c47733b24100493755fd74c9": "(1, 1)", "e4122518acee3321b79e0287aab57617": "\\rho\\bar{u}_j \\frac{\\partial \\bar{u}_i }{\\partial x_j}\n= \\rho \\bar{f}_i\n+ \\frac{\\partial}{\\partial x_j} \n\\left[ - \\bar{p}\\delta_{ij} \n+ \\mu \\left( \\frac{\\partial \\bar{u}_i}{\\partial x_j} + \\frac{\\partial \\bar{u}_j}{\\partial x_i} \\right)\n- \\rho \\overline{u_i^\\prime u_j^\\prime} \\right ].\n", "e4122d982a7107ceea18dff55038d855": "\n\\begin{align}\n \\mbox{Change in NFA} & = \\mbox{Current Account} +\\mbox{Valuation Effects} \\\\\n\\end{align}\n", "e4125fea23eac71720d0c123886c03fa": " U = \\lambda/\\delta x \\,\\!", "e412a7a48bc49234cd7d2ce362b6b314": "{E_k}^{-1}\\!", "e412cad40b1c9a30c024bd078fe0b24b": "(10 | 13) \\equiv 10^6 \\equiv 1 \\bmod 13.", "e4134716e99dd146e426f35ff1c085ac": "\\begin{align}\n \\hat x' &= \\hat x\\left(\\frac{1}{4} - \\frac{1}{3} \\frac{3}{4}\\right) + \\frac{2}{3} \\hat y \\frac{\\sqrt{3}}{2} \\left(\\frac{\\sqrt{3}}{2} + \\sqrt{3}\\frac{1}{2}\\right) + \\frac{2}{3} \\hat z \\frac{\\sqrt{3}}{2} \\left(\\frac{\\sqrt{3}}{2} - \\sqrt{3}\\frac{1}{2}\\right) \\\\\n &= 0 \\hat x + \\hat y + 0 \\hat z = \\hat y\n\\end{align}", "e4134d260b4cc95ecf2f938972c4e238": "K(k) = \\frac{\\pi}{2} \\sum_{n=0}^\\infty \\left[\\frac{(2n)!}{2^{2 n} (n!)^2}\\right]^2 k^{2n} = \\frac{\\pi}{2} \\sum_{n=0}^\\infty [P_{2 n}(0)]^2 k^{2n},", "e4135c268b2d5b07bbc2b29bc0c9ca78": "{\\bar{M}}_5", "e413835a3256eca2bcbdda1113c65ff3": "\\mathrm{P}(A) = \\mathrm{P}(A \\cap A) = \\mathrm{P}(A) \\cdot \\mathrm{P}(A) \\Rightarrow \\mathrm{P}(A) = 0 \\text{ or } 1", "e413a738379ed7a3aca7b312962347ed": "s(\\lambda,x)", "e414491e4cb08df31265ce3ba5288078": "\nT_qM=\\left\\{X^i\\partial_i\\Big|X\\in \\mathbb{R}^n, \\partial_i=\\frac{\\partial}{\\partial \\xi^i}\\right\\}\n", "e4148a25522b540321e475ddaa9a9b2e": "\\mathbb{E}_m[\\Pr_{e \\in BSC_p}[D^{*}(E^{*}(m) + e)\\neq m]] \\leq 2^{-\\delta n}", "e41493956bf4bf300a068592e17ac9fd": "\\hat h", "e4149e647b7af6ecc20869969a5d4d08": "\\gamma_{SV}", "e414c7222238d00793602b30142f9b41": "YA = p\\times EP + (1 - p)(1 - EP)", "e414e57225f363fcae11d34016f3ad3f": "V_L", "e414fff65fdb9eef2b09339eaabc12ef": "\\Delta (0)=0,", "e41552fdef52568c89e4fbb67a9fc69d": "\\mathrm{Dom}(\\varphi_i) = \\mathrm{Dom}(\\varphi_{f(i)})", "e41558d6f18771ed4447a6412290aac8": "\\left(HJ_i\\right)^2=\\left(HE\\right)^2+x^2=\\left(r_1\\right)^2+\\left(r_2\\right)^2+x^2", "e415b01fa71c83e02f00aaf54f476b0f": "+j\\infty\\,", "e41634aa932c3b168b4758ce8050880a": "s = \\frac {2 D_{\\mathrm N} D_{\\mathrm F} }\n{D_{\\mathrm N} + D_{\\mathrm F} } \\,,\n", "e41649ab09d20c91f627fb6279d941fe": " a*(b + c) ", "e41651838ddfef96890505eed7374c03": "R_{\\rho\\sigma\\mu\\nu}", "e416ad5410133c4cd31030ce265c1025": "\nK'(t) = e^{-Ht}\n\\,", "e416d3ff557b8eba4511ac4f8572c756": "f = \\frac{Nc}{2d}\\qquad\\qquad N \\in \\{1,2,3,\\dots\\}", "e416fde036125486c3605639b8808681": "\\omega(\\mathbb N)=1", "e41704883eeb8e34206f8ccf0e9431cd": "\\mathbf{v} = \\mathbf{u} + \\mathbf{a} \\mathbf{t}\\;\\!", "e418224cce442b6b2a905ca8da785322": "g= ", "e418226b7fb62834cefc3fdbac7bb392": "\\gamma : [a,b] \\to L", "e418652ccac754e4ba62ec4012296ff5": "\\mathbf{f} ", "e418870ccb2925593f863b4519071c10": "\\textstyle i = 1, \\, \\ldots, \\, N", "e4188975d5c185faf787a83190dfce6c": "\n\\begin{align}\n\\int_{H < E} x_{m} \\frac{\\partial ( H - E )}{\\partial x_{n}} \\,d\\Gamma &= \n\\int_{H < E} \\frac{\\partial}{\\partial x_{n}} \\bigl( x_{m} ( H - E ) \\bigr) \\,d\\Gamma - \n\\int_{H < E} \\delta_{mn} ( H - E ) d\\Gamma\\\\\n&= \\delta_{mn} \\int_{H < E} ( E - H ) \\,d\\Gamma,\n\\end{align}\n", "e418aa48aa4ec03eb293489ccd63a9f7": " = -(n-2)\\mathbb{E}_\\theta \\left[\\frac{1}{|\\mathbf{X}|^2}\\right].", "e418ae62fc11a53506e3c16dd06e9774": "m a_i = F_i\\,", "e418e895cfb0ebb63d90eb0e3e7b77c0": "O(Nlog(N))", "e418ecc559bf152a2a8426c27863db3f": "\\sum_{i=1}^n a_{s,i} \\delta q_i = 0~~~~(s = 1, 2, ..., k).", "e41913ab572f40b281054f5ad2f5fe80": "\\log C \\leq 0", "e41925fe5b4053897f251ef1a46f51a6": " \\mathcal{Z}(\\mu, V, T) = \\sum_{i} \\exp((N_i\\mu - E_i)/k_B T). ", "e419554623d47f235b9b897a6cd658e3": "L\\ :=\\ L\\ +\\ \\mathcal{j}\\ m_n\\ \\mathcal{j}", "e4196a5e792caf4fab4baa5771579884": "\\bigcirc \\phi", "e4199b1d159bab812162651cb599ac60": "q > n^{1/2} + 1 +2n^{1/4}", "e419fdad16f9f8bdb83b34d1d65b3c8e": "\\mathbf X = \\mathbf X_2\\otimes\\mathbf X_1,", "e41a07ce8a0b6ce9a813d4b2dea49b86": "gate3", "e41a215471c8d94d0af7527fdf03922a": "co'(ci, x, y) = \\mathrm{AND}(M_3,M_5,M_6,M_7) = \\mathrm{NOR}(m_3,m_5,m_6,m_7).", "e41a30ce1787515ab4550c811b1daab0": "F(\\cdot)", "e41a883ab628a9e3bd6eeb9ce307fd07": "\\displaystyle{\\pi(g,\\gamma)f(w) = {1\\over \\pi}\\iint_{\\mathbf C} K(w,\\overline{z}) f(z) e^{-|z|^2}\\,dxdy,}", "e41a932ece3ff8b40b9f5feb0a8b3f6c": " \\|f\\|_{n} = \\sup_{z \\in \\mathbb{C}} \\exp \\left[-\\left(\\tau + \\frac{1}{n}\\right)|z|\\right]|f(z)| ", "e41aac5f30fa857310f169a611be3208": " \\exists X_1 \\forall X_2 \\exists X_3 \\ldots f", "e41ab6c701cb7b0859a6028629d0584c": "r_{loan} = \\frac{I_{T}}{nB_{0}} = r + \\frac{1}{\\lambda_{n}} - \\frac{1}{n}", "e41abf8fea7b51a927095705301c5723": "d \\ln \\mathcal{L}(\\mu,\\Sigma) = -{n \\over 2} \\operatorname{tr} \\left[ \\Sigma^{-1} \\left\\{ d \\Sigma \\right\\} \\right]", "e41ac66d8a70819b4fbe6efc22337251": "\\mathit\\Delta", "e41aca2e754028707535a57f77dd5077": "AP \\cdot AP = AR \\cdot AS \\, ", "e41af8f191c2e68dea3ba852b8e185be": "\\frac{n + 1}{n + 2} \\leqslant \\frac{W_{n + 1}}{W_n} \\leqslant 1", "e41b49a69914afdb60298deb0e2d98ed": "GE(\\alpha) = 1/2 (\\sigma/\\mu)^2 \\quad \\quad\\quad \\text{ for } \\alpha = 2.", "e41b56d6ec72b85224efa573a1963809": "= C_{\\beta I}^{\\;\\;\\; K} e^\\alpha_K e^I_\\gamma \\delta_\\delta^\\beta - C_{\\beta J}^{\\;\\;\\; K} \\delta_\\gamma^\\beta e^\\alpha_K e^J_\\delta ", "e41b6d1a62a03241e4f165129c6fd6f7": " {S^{\\mu}}_{\\nu\\lambda}", "e41ba35a38842289b3892b1c48f23a8a": "\\sigma_{\\lnot \\varphi}(R) = R - \\sigma_\\varphi(R)", "e41beb1f3d324fae976410d4cfacc7a7": "\\Delta R_{i}^{2}=\\sum_{j=1}^{3}r_{ij}E_{j}", "e41cc9445003ed0b14c993e4d7f39af3": "\\mathfrak{D}^{A\\times B}_w(s) = \\mathfrak{D}^{A}_u(s) \\cdot \\mathfrak{D}^{B}_v(s).", "e41ccd243059599de206a5368bf224b6": "J \\approx \\,2.25 a^4", "e41d5a682c168428b90d0439b4114bc4": "dP/dx>0", "e41d7c23611b267b8c4896bd1e80efda": "\\ln \\left( G \\right) + \\bar{g}(\\nu) { I_{in} \\over I_S } \\left( G - 1 \\right) = \\gamma_0(\\nu) \\cdot z", "e41d8bfe8bbace502fe34ed3a405af0b": "L_n (1) =\\mathbb{Z}, 0, \\mathbb{Z}_2, 0", "e41da00665e1862a98fcc22270643ff5": "\\dot{\\sigma}(\\mathbf{x}) = \\mathbf{0}", "e41db0c83d191f4f1b2762f74a35b255": "\\Sigma_{a \\in H} \\Sigma_{j \\in S(a)} p_j = \\Sigma_{a \\in H} W(a)", "e41e3f9e0e6828385699ceb97cdf85ca": "\\mathit{d_H}^{R}(\\mathcal{E}_w \\geq \\mathit{d_{min}})", "e41e58a1279c72d55cba415d9ccba8f6": "\\int d^3x\\, J^0", "e41ebb1e5c0718e904d3df8f033d8d3f": "e^{\\frac{k}{m}t}v_y = \\frac{m}{k} e^{\\frac{k}{m}t}(-g) + C ", "e41ebf43cbb75b6b75898550d84bdd6d": "\\overrightarrow{\\Gamma_n^*}", "e41edd80eb509407f11738fe2c5a0a5c": "f = \\frac{1 + \\sin(k x)}{2}", "e41ef16c2633443a5ad7c02b955954f3": "b_{10}-(3/4)a_{11}", "e41f676421be865fb427a4b8517e54fc": "\\psi \\circ \\varphi", "e41f69de4ba863eb5983494bf4b06600": "(u(t),v(t))=(t,\\frac{\\pi}{2})", "e41f7d60e8c168a1074c2a9a64743bd4": "\\{ 1, i_1, i_2, i_3,~i_0, \\varepsilon{}_1, \\varepsilon{}_2, \\varepsilon{}_3 \\} ", "e41ff44899c4f1c2a9174efc361bafdb": "2^{p(n)}", "e4202876915eb091a491b87652ec941f": "Q_t", "e4206ad0f91b2be6da31afb7bbabb8ea": " r^2=2 d D t, ", "e420adc2c45597d1ffe49c36ee57b46e": "\\mathbf{G}_2 = \\mathrm{diag} \\left\\{ \\tilde{\\mathbf{G}}_2, \\tilde{\\mathbf{G}}_2, \\dots, \\tilde{\\mathbf{G}}_2 \\right\\} ", "e420bf634fd4e41d0126b22b299eb039": "\\alpha_0 = \\frac{2 \\cos \\frac{\\pi}{8}}{1 + \\cos \\frac{\\pi}{8}} = 0.96043387...", "e420e0f08fa318fd4c3a17f7550b8a26": "C^{cand} := C^{cand} \\cup DBSCAN(cl, cand, eps, MinPts)", "e421496d7533e5af19fbe1bf3631a53d": "\nX \\perp\\!\\!\\!\\perp A,B\n\\quad \\Rightarrow \\quad\n\\text{ and }\n\\begin{cases}\n X \\perp\\!\\!\\!\\perp A \\\\\n X \\perp\\!\\!\\!\\perp B\n\\end{cases}\n", "e42155ed9c8dd4068658ff4bfeb66f3c": "\ny = \\left( X-\\mu\\right)^T \\, V^{-1} \\, \\left( X-\\mu\\right) .\n", "e421ba292e1dbae25369a6afa8f3faf0": "\\mbox{P}^d(p, q, ...) = \\neg P(\\neg p, \\neg q, \\dots).", "e421ce9ad70f4c1304ca135e2353ac25": " Q^m_1 ", "e421dd479302308760fd917b2c9acd5e": "\\frac{1/k^s}{H_{N,s}}", "e421f686870f36697ec5b615b5eb375e": "Con := \\{ X \\in \\mathcal{P}_f(T) \\mid X \\mbox{ has an upper bound} \\}", "e42200bbccdef7a3c49829353ecf8fc6": "\\Theta_1 \\le \\Theta_2.", "e4225386d7f9f799b15d5acad78dec29": "R_{l}(u)=S_{l}(u)/u\\,", "e42268058080d89a2c6fce64ceeaeb81": "\\sqrt[3]{\\frac{1}{2}+\\frac{1}{6}\\sqrt{\\frac{23}{3}}}+", "e42282e81a7a7a02198af319b9911d6a": "\\scriptstyle|\\zeta|\\leq \\|(1,\\|a\\|_p)\\|_q=R_p ", "e4233c01031282b86a87dc4c86fd2355": "\\textstyle\\left\\{2 \\left| -\\frac{1}{2}\\right.\\right\\} = \\frac{3}{4} \\pm \\frac{5}{4}", "e4234d249bc516fee1c7297892010da8": "\\lambda = \\lim_{n\\to\\infty} \\frac1n \\sum_{k=2}^n \\frac{\\log(P_1(k))}{\\log(k)},", "e42367ade6fb313d2c55b7d7c0abde87": "0 < n < 2N", "e423a9d6a6ac963093eeec55bf8ae078": "U_{A}(\\delta_{A}) U_{B}(\\delta_{A})\\leq U_{A}(\\delta_{B}) U_{B}(\\delta_{B})", "e423ba96ac7af9469a3202941a2d2af6": "x^3+y^3+1=0.3xy", "e423bf1b1ed8486c67b9b09e84ea21d5": "\\boldsymbol{\\tau} = I\\boldsymbol{\\alpha},", "e423d2945892db93ef044ad90e4c8621": "K_{E} = R_F + \\beta_E (R_M - R_F)", "e423ede2435668ef35dcb3da89845de3": "\\omega = \\sqrt{\\omega_n^2 - \\alpha^2} = \\sqrt{\\kappa/I - (C/2I)^2}\\,", "e4240ff886dbbfcedd3db989e33b4853": "|X|", "e424632ac09987e01c092290648f9151": "M\\left({u\\atop d}\\right)", "e424d9f5deafca847a12e518fd70cc81": "S \\subseteq H \\subseteq S'.", "e424f98ab3f725242131de4edd3a9ba2": "\\scriptstyle \\{\\eta_t(.) ", "e425025df2ce63bd92d8f7a5791f8dfe": "\\displaystyle K = \\sqrt{abcd}", "e4250d8b91ce03069557161e26d54323": "\ng_{\\pi NN} F_\\pi = G_A M_N \n\\,", "e42538b6e3f3aecd2765ef1eaf8c55ba": "TMDL = WLA + LA + MOS", "e4256bf614844c5b0861dff2f6ca6b22": "\\alpha_1 + \\alpha_2 = \\frac{1-\\sqrt{1-4c}}{2} + \\frac{1+\\sqrt{1-4c}}{2} = \\frac{1+1}{2} = 1", "e425b09bf748a70f82e7c7905ce84492": "0 < c < \\infty", "e425ea5e5f066d5cced3f46add35af48": "\\frac{\\partial E}{ \\partial w_{ji} } = - \\left ( t_j-y_j \\right ) g'(h_j) x_i \\,", "e425f932319a8c12ff071e51eb04a64a": "G_f D", "e42601d529ca13fc8a76e3876b262cab": "(a+b\\sqrt c)(a - b\\sqrt c)/(a^2 - c\\,b^2)", "e426a259c106fa01ff50ac409bbdcb4a": "\\mu = (\\mu/\\rho)_1 \\rho_1 + (\\mu/\\rho)_2 \\rho_2 + \\cdots", "e426d3a67d9816ed4de4e7ae6c7c7148": "e \\ ", "e4270f7f42d17ec68dc8806a69eb4851": "((-1+3i)+1)^2 = (3i)^2 = (3^2)(i^2) = 9(-1) = -9", "e4278c1cb157e8c16373109a060d7c12": "x_B \\overset {d}{=} (x_A + y + z)", "e427dc0c8ea4227f22d69ed8bfbf2130": "C(x,v,t) = x - vt", "e4287ca6f4dcdca8746ed5be65e3ee53": "J_{ij} = \\frac{\\partial \\theta_i}{\\partial \\eta_j}\\,,", "e4289cebaf0989f3a0fed19652d0a791": "{}^{2S+1}\\!\\Lambda^{(+/-)}_{\\Omega,(g/u)}", "e4293bd9fda7554440df32011f05ec06": "\\tfrac{3}{4}\\zeta(4)", "e42964cfef7dbae2e57e39b81a598b39": "s_n(t) = \\sqrt{\\frac{2E_s}{T_s}} \\cos \\left ( 2 \\pi f_c t + (2n -1) \\frac{\\pi}{4}\\right ),\\quad n = 1, 2, 3, 4. ", "e429add47d84db3f8c10645b5eabb0d6": "\\beta < \\varphi_\\gamma(\\delta)", "e429c7d8eb2728e0e2224aa32e85bab2": "\\pi_{+j}=\\sum_{i=1}^r\\pi_{ij}.", "e42a01ac194357b351a6a726361ae0da": "m(t)\\,", "e42a20aef131c7565ce8a842251069aa": "v, v' \\in f'", "e42a3ab07e1a0c8fc579a5558590b1d6": "Q = f(X_1,X_2)", "e42a4f4d7f7c6c166b7df14f047b0964": "r = i - \\pi", "e42a66addadc6f634dc9b765fec90c65": " y =f(x)=x^2", "e42a977369a271dc0aea0f1964fd38a2": "x_{\\mathrm{min}} \\approx 1.46163", "e42b521d5f90297d6b8fbce91e976eff": "\\omega_N = e^{-2 \\pi i/N}\\,", "e42b57bab8418b50d7468c1db45e17f5": "V_{sig}", "e42b785fef30f660d5d406e73d25a882": "n^{\\searrow}.\\sigma\\mid\\sigma_0(P) \\circ\n\\overline{n}^{\\searrow}(\\rho).\\tau\\mid\\tau_0(Q) \\rightarrow_{b}\n\\tau\\mid\\tau_0(\\rho(\\sigma\\mid\\sigma_0(P))\\circ Q)", "e42b86dc4259a60b57bb6003d5169959": " \\sum_{i=1}^N ", "e42bab6352e026cb4caf87e049ba7af4": "\\begin{smallmatrix}{L}\\end{smallmatrix}", "e42bc726ef5e89144628edf9802f1c96": "\\mathbf A\\wedge \\mathbf A=0", "e42bd6e0513e2c3b194a5e8571830eda": "|\\Downarrow\\rangle ", "e42c31cad56617d7a2b89514884a7878": "d\\bar{z}", "e42c3ceb8f2efbffa27abddc7891c085": "-RT\\ln\\gamma_s x_s=\\int_p^{p+\\Pi}\\! V \\, \\mathrm{d}p", "e42c53cf0b9854726d518445985be534": "\\begin{align}\n\\text{PL} &= \\text{TR} - \\text{TC}\\\\\n &= \\left(\\text{C}+\\text{V}\\right)\\times \\text{X}\n - \\left(\\text{TFC} + \\text{V} \\times \\text{X}\\right)\\\\\n &= \\text{C} \\times \\text{X} - \\text{TFC}\n\\end{align}", "e42c564887ad30d0ad45d6be36990680": " \\text{flatness} = \\max \\left ( P_\\text{out} \\right ) - \\min \\left ( P_\\text{out} \\right ) ", "e42c56bdc954019369726aa9256397bf": "v=\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}", "e42c6ab40c9d9cc561d14cf4f584b114": "\\lim_{p\\to\\infty} L_p(x)", "e42c729682ea73ee607eda5fefd45cb2": "A,B \\in \\mathbb{F}_p", "e42c8fb09cccece6bd3a7dc3524b937e": "\\;\\gamma^N", "e42c9bb0f6c07036a05ae31859441399": " f_{X,Y} (x,y) = \\begin{cases}\n \\frac1{2\\pi\\sqrt{1-x^2-y^2}} &\\text{if } x^2+y^2<1,\\\\\n 0 &\\text{otherwise}.\n \\end{cases} ", "e42cad1b22e6b0ea1619970688b62479": "\\chi\\rho_N \\ll \\rho_{F\\pm}", "e42cb0144bcc2813b5f25817d05d8429": "\\begin{pmatrix}x(t)\\\\y(t)\\end{pmatrix} = \\begin{pmatrix}\\Re(e^{i\\omega t}\\psi_x)\\\\ \\Re(e^{i\\omega t}\\psi_y)\\end{pmatrix} = \\Re\\left[e^{i\\omega t}\\begin{pmatrix}\\psi_x\\\\ \\psi_y\\end{pmatrix}\\right] = \\Re\\left(e^{i\\omega t}|\\psi\\rangle\\right).", "e42d159eec76affb0828f3087a71619c": "\\varepsilon_{ab\\ldots n}\\,\\epsilon^{ab\\ldots n}", "e42d276b8925e72f81a5e958c471adaf": " (i \\partial\\!\\!\\!/ - m) \\psi = 0. ", "e42d2ed1604f0fbb93bf704cc8f07c97": "\\gamma=\\tan^{-1}\\left(\\frac{\\tau\\sqrt{1+t^2}+\\sigma t\\tan(\\lambda-\\lambda_0)}{\\sigma\\sqrt{1+t^2}-\\tau t\\tan(\\lambda-\\lambda_0)}\\right).", "e42e05c46c87b81e7c2316dce4aa511b": "p_{i+1}(\\xi)\\ne 0", "e42e40b3f39084caf73c651754fafdec": "\\Pi A = RT.", "e42e4aa1d191a51d18667ce9f9af7094": "f(z)=\\sum_{n=0}^\\infty \\left[ A_n(1-z) f^{(2n)}(0) + A_n(z) f^{(2n)}(1) \\right] + \\sum_{k=1}^N C_k \\sin (k\\pi z).", "e42e62a2cdc0a38f1653fcbc4b78d5d1": " r_{12} = |\\vec{r_1} - \\vec{r_2}| ", "e42e9459f46df4189f826aac6ac15261": "\\frac{1}{C^2}~\\frac{\\partial^2\\theta}{\\partial t^2}~+~\\frac{1}{\\alpha}~\\frac{\\partial\\theta}{\\partial t}~=~\\nabla^2\\theta ,", "e42e9dd1f6d29d13d17da0087fe63de8": "(\\overline{gate6}\\vee gate5)\\wedge (\\overline{gate6}\\vee x3)\\wedge (\\overline{x3}\\vee gate6\\vee \\overline{gate5})", "e42eb8a6c6ace1b4fc6508bf52980825": "\\sin(\\sin(x))", "e42ecec180aacfd906fa6de87f151dbd": "\\!c_w", "e42edd4493e9badf8ce82b529ae75b58": "~|\\psi_\\text{field}(0)\\rangle=\\sum_n{C_n|n\\rangle}~", "e42f3d88b063a90e3cea155c6c86af94": " \\Omega\\left(\\frac{\\log(m)}{\\varepsilon^2\\log (1/\\varepsilon)}\\right)", "e42f92311292b1171edc2cb3d0c49072": "\\begin{align}\n\\int_{\\gamma} y dx+x dy &=\\int_0^{\\pi-{\\tan}^{-1}(\\frac{3}{4})} (5\\sin t(-5 \\sin t)+5 \\cos t (5 \\cos t)) dt \\\\\n&=\\int_0^{\\pi-{\\tan}^{-1}(\\frac{3}{4})} 25(-{\\sin}^2 t+{\\cos}^2 t) dt \\\\\n&=\\int_0^{\\pi-{\\tan}^{-1}(\\frac{3}{4})} 25 \\cos (2 t) dt \\\\\n&=\\left.\\tfrac{25}{2}\\sin(2t)\\right|_0^{\\pi-{\\tan}^{-1}(\\tfrac{3}{4})} \\\\\n&=\\tfrac{25}{2}\\sin(2\\pi-2{\\tan}^{-1}(\\tfrac{3}{4})) \\\\\n&=-\\tfrac{25}{2}\\sin(2{\\tan}^{-1}(\\tfrac{3}{4})) \\\\\n&=-\\frac{25(\\tfrac{3}{4})}{{(\\tfrac{3}{4})}^2+1}=-12. \n\\end{align}", "e42f9d6e5834975e4830db56e461077f": "l_{p}", "e42fa7177e65ef5c946bd098c562a0f9": "X_k^{-1}", "e42fcc0189aea7e14e9958d245e3d70c": "^4 2 = 2^{2^{2^{2}}} = 65,536", "e43013f70d807666efa4fe137e6d579a": "\\displaystyle{g=\\begin{pmatrix} A & B \\\\ C & D\\end{pmatrix}}", "e4303c648610811a5d13486662d68741": "\\textstyle A_i(x)", "e43060a0299c5c64cf17033dded477fa": "P^{\\text{old}}(i,s_j)", "e43063de9e5ac16d7d66d3b42d6df1ec": "\n\\frac{x^{2}}{\\mu^{2}} + \\frac{y^{2}}{\\mu^{2} - b^{2}} + \\frac{z^{2}}{\\mu^{2} - c^{2}} = 0\n", "e43070c2f4de264f670f1c7c3c66aa71": "a+b=c+d", "e4310507c987649f3f91f9cfa0a8cc52": " \\mathbf{W} = \\mathbf{A^{T}} \\mathbf{w} ", "e431f5f1120ccf56f6435294590c26a0": "p = x^2 + y^2 + z^2 + 0", "e432fd3826f4b9831766248c0929d3b9": "\n\\nabla V = \n\\frac{1}{h_{\\zeta}} \\frac{\\partial V}{\\partial \\zeta} \\,\\hat{\\zeta}+\n\\frac{1} {h_{\\xi}} \\frac{\\partial V}{\\partial \\xi} \\,\\hat{\\xi}+\n\\frac{1} {h_{\\phi}} \\frac{\\partial V}{\\partial \\phi} \\,\\hat{\\phi}\n", "e43344321ddcc4ff1081c7a9bce5a4b5": "r = \\frac{\\sum_i{e_{ii}} - \\sum_i{a_i b_i}}{1 - \\sum_i{a_i b_i}}", "e433513806b793f2312273ad14a40a16": "\\left| f(z) \\right|", "e4338ee6fecfd7096442cc466fdc7e2f": " V_p = a (\\bar{ M}) + b \\rho ", "e433e82fcbb05080d6f81aa48c6a8ade": "\\begin{align}\n f(A, B, C, D) = {} &\\overline{A}BC\\overline{D} + A\\overline{B}\\,\\overline{C}\\,\\overline{D} + A\\overline{B}\\,\\overline{C}D + A\\overline{B}C\\overline{D} + {}\\\\\n &A\\overline{B}CD + AB\\overline{C}\\,\\overline{D} + AB\\overline{C}D + ABC\\overline{D}\\\\\n \\Rightarrow f(A, B, C, D) = {} &A\\overline{C} + A\\overline{B} + BC\\overline{D}\n\\end{align}", "e433ecf8997f8211bb14e2fe62414d78": "(fg)' = f'g + g'f. \\!", "e4349f09cc624126d2fbaeb2af534ea4": "F_\\text{P} = \\frac{G m^2}{r_\\text{G}^2} ", "e434c004c18d158c0e8c69ce87bcf41b": "\n\\boldsymbol{\\nabla}\\cdot \\mathbf{A}(\\mathbf{r},t) = 0\n", "e434eda32758d5f6bb4f6bdc0cda4343": "E_{\\text{p}}", "e43591d8a2ad867c4aca14f3e8be71a9": " dh = T \\, ds + v \\, dp, ", "e435e29588d6a706da7c26229c6e64f0": "\\sigma:I\\to \\mathbb{N}", "e435f0305eca10de89b1b7e86c8a936b": "usv\\rightarrow utv", "e43644b4dd21338af38b29bc2284b0ba": " C_0<0,\\quad C_n\\ge 0\\text{ for }n\\ge 1. \\, ", "e436ea47ec5ad8b9e16f27436087849c": "cn^{\\frac 13}", "e436ed6099126525cc57bac476f98f81": "G(x) = \\int_a^x f(t)\\, dt", "e436fc3ecbeded0c2195923a16b89054": "\\mathbf{x_1}, \\mathbf{y_1}", "e4371a28ad1a1401e6bfc95fd437ba07": "(x-3)(x-1)^3(x+1)^3(x+3)(x^2-3)^4.\\ ", "e4380daf57e6a8696793bde8fb83a124": "a_x=0.72\\,\\;", "e4387dcad4c4cfbc3cb5e0a5e4734002": "\\mathbf{e} = e_1^1e_1^2\\cdots e_1^{n_T}e_2^1e_2^2\\cdots e_2^{n_T}\\cdots e_T^1e_T^2\\cdots e_T^{n_T}.", "e439082a22bbf5d1c0fcaa0bed5ca392": "f(x)=\\cos(x)", "e4391bf437f971f528ab602df7f070ea": "H(x,y)", "e4393d68d43df97297cd4224be28be39": "g^iD_i \\oplus g^jD_j = B", "e43947401b8eaa0c398d913ffde7c8a9": "3,294,720 + 123,552 + 54,912 = 3,473,184\\,", "e4394e06d4da850bfeb41fd05722112c": "\\left[M_{\\rm inert}^{-1}\\right]_{ij} = \\hbar^{-2} \\frac{\\partial^2 E}{\\partial k_i \\partial k_j}", "e4395bacf945ef715c763a5c65f853f4": "\\Leftrightarrow x^2 + y^2 + (3c/2) x = 0 ", "e43993811d103e1ab3cd62a2899630ad": "{\\mathcal A}\\,", "e439bb801652ad7abeaa4ff53fa95b02": "\\begin{bmatrix}\\hat{W}\\\\\\mu\\end{bmatrix} = \\begin{bmatrix}\nVar_{x_i}& \\mathbf{1}\\\\\n\\mathbf{1}^T& 0\n\\end{bmatrix}^{-1}\\cdot \\begin{bmatrix}Cov_{x_ix_0}\\\\ 1\\end{bmatrix} = \\begin{bmatrix}\n\\gamma(x_1,x_1) & \\cdots & \\gamma(x_1,x_n) &1 \\\\\n\\vdots & \\ddots & \\vdots & \\vdots \\\\\n\\gamma(x_n,x_1) & \\cdots & \\gamma(x_n,x_n) & 1 \\\\\n1 &\\cdots& 1 & 0 \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\\gamma(x_1,x^*) \\\\ \\vdots \\\\ \\gamma(x_n,x^*) \\\\ 1\\end{bmatrix}\n", "e43a99e948296470221a14b3230f3eb6": "c \\,=\\, 1", "e43ab78fea6ca2f799da771eadf2a145": "S_L - \\ ", "e43abd6c6b3f1c868d9f85bf204a7347": "\\sigma(E_i,T)", "e43b21c614bc8c02f689809eefea4b4f": "NPV = 100\\,(1.05)^{-1} + 200\\,(1.10)^{-1}\\,(1.05)^{-1} = \\frac{100}{(1.05)^{1}} + \\frac{200}{(1.10)^{1}(1.05)^{1}} = $95.24 + $173.16 = $268.40 ", "e43b7bf6b5c027436c0a586ee03e89a4": "\\mathbf{v}_s", "e43bd82beb9aab60ab8c212553aef103": " \\tan\\phi = \\frac{1}{\\sinh a} ", "e43beefb44a3b313b5c3c3976b8b049a": "\\mathcal R = \\frac{l}{\\mu A}", "e43bf86489da8ed8561a5ea5c9d6bbdd": "H(\\varphi,\\eta;\\boldsymbol{x},t)", "e43c7a676fcda66751af1bd573bc08f6": " P(Y_2=0)=\\left(1+\\sum_{n=1}^\\infty \\left(\\frac{3.75}{12}\\right)^n\\right)^{-1}=\\frac{11}{16} ", "e43ca01c4ffab6c2b1b91caf463de23e": "\\theta_\\mathrm{A} - \\theta_\\mathrm{R}", "e43cabac90af3284e025f69d69fe1d64": "\n\\varphi(t;\\mu,c) = \n\\exp\\left[~it\\mu\\!-\\!|c t|^{3/2}~\\right] .\n", "e43cebe8cb24a4795563cdd3b624863f": " G(v) = \\bar \\nu _{electronic} + \\omega_e (v+{1 \\over 2}) - \\omega_e\\chi_e (v+{1 \\over 2})^2\\,", "e43cf315e98c13d77420e3ea95b8dbe9": "\\alpha=2.3", "e43d2bdcb15971e03171a2190b3f7a03": "\\eta = \\sqrt{\\mu \\over \\varepsilon}\\ .", "e43d4240f6475b7f9e3ba2955787e096": " \\frac{\\lVert a_j \\rVert^2}{\\lVert A \\rVert^2} \\qquad\\qquad\\qquad j=1,\\cdots,m ", "e43d4ed83b6aea0b60a41a69517ee48f": "x = \\frac{t^2-c}{b+2t\\sqrt{a}}", "e43da70c79dcb6e00affb1a40a20a265": "-\\lambda'", "e43e0effafb3850c05e050fd3adff8c2": "G={{1\\over 2Z_\\circ} {A^2I^2\\over r^2} \\over {{1\\over 2}R_sI^2 \\over 4\\pi r^2 } } ={A^2 \\over 30 R_s}\\,\\!", "e43e22bce5f3b541e8c2a12b64069e00": " e^{i\\theta} = \\cos{\\theta} + i\\sin{\\theta},\\,", "e43e9d5de2b857f9217b8cf68c05fe97": "\\Delta w=0", "e43ea0caf4871464953cdf4af83ebcfd": "r = b + a \\cos \\theta", "e43ebc9ad8586307409fac69d471b3d9": "\\Omega_X^*", "e43f516bd08fbc66bafad67140113871": "\\delta E", "e43f6a2621b4b64ea6196b7fc31dfd84": "\\textstyle (-1)^{k(k-1)/2}i^{p-q}Q", "e43fb8af5a6adb605b9b403308291b93": "N \\times 2^L", "e43fd061a4bd3b7779478cfb6b383a44": "f_{t+1}", "e43fd26ebd7d7e18faabf5f4fa30b182": "r(s,o)", "e43fe065373220f4448949757ea104fc": "[H_\\lambda,X_\\mu] = (\\lambda,\\mu)X_\\mu,", "e4400a3135dd44a77960d9c11933d003": "\n\\mathrm{pH} = 14 + \\log \\sqrt { \\frac {C_a C_b K_w} {(C_a + C_b) K_a} }\n", "e44036552e6f4f1e45197cfb86aecc49": "dV=dx_1dx_2dp_1dp_2 \\ ", "e4403bf4677e27c38b97af95c2cd0464": "\\left(P ^{2}\\right)_{i,j}.", "e440b1de375112e1e5d2d41ad0370705": "x=1\\wedge x", "e440f3bd6f481c250c058de00af5724c": " f(k) = \n \\begin{cases}\n 1/2, & k = -1 \\\\\n 1/2, & k = 1\n \\end{cases}\n ", "e44104dcca02931e72174ecbf7623e98": "m_adx^a=\\frac{1}{\\sqrt{2}}(\\sqrt{g_{\\theta\\theta}}d\\theta+i\\sqrt{g_{\\phi\\phi}}d\\phi)\\,,", "e44110dba5282c0c94867a876b476038": "\\alpha\\le 1", "e4413b4c3ed853db4d218f3469b21f8d": "\\Bbb{Z} \\;\\overset{2\\cdot}{\\hookrightarrow}\\; \\Bbb{Z} \\twoheadrightarrow \\Bbb{Z}/2\\Bbb{Z}", "e4413b4dbe1cf57ff2834978fe311071": "\\chi_{||}", "e44151928e3eebfc99b0ba0b128113d0": "V = \\sqrt{\\frac{2 (p_t - p_s)}{\\rho}}", "e4418e11db5bae71074e423b3949a589": "y^m = x^n + k\\ ", "e441ac4c2e4383a2984924574e9c8350": "\\frac{Y(s)}{U(s)} = \\frac{1}{s^2}.", "e441c6bd52fc8c06fb60620ec1c1f68a": " \\%A_f ", "e4428017108fe4902ef1ff4b19a31d60": "\\,\\!(r,\\theta,\\varphi)", "e442895781b6a7ba07c5b6de8482df89": "\\ |k_a|+|k_b|+|k_c| = 3", "e4428dc81628a15f2b4a06dd8713e9fa": "\\scriptstyle\\pi/2", "e44290d95ee829947dc35763a45d0164": "-c^2 = g_{\\alpha \\beta} u^{\\alpha} u^{\\beta} = g_{t t} (u^{t})^2 + g_{s s} v^2 (u^{t})^2 \\,.", "e442cf138b7d352bde1deb9b275914f0": "\nK' = \\left( \\frac{\\partial K}{\\partial P} \\right)_T \\qquad (3)\n", "e442fd39d768852da1d252f46588d867": "\\alpha P(L_+) - \\alpha^{-1}P(L_-) = zP(L_0),\\,", "e443315056417d93ec2435f37a192ecc": "r(t) \\sim N\\left(e^{-\\alpha t} r(0) + \\frac{\\theta}{\\alpha} \\left(1- e^{-\\alpha t}\\right), \\frac{\\sigma^2}{2\\alpha} \\left(1-e^{-2\\alpha t}\\right)\\right).", "e4434573eb1c59b5efae56e63c9a3491": "1:(1+\\sqrt{2})", "e443e0926e67c053be1fd948791e4970": "1-(1+\\xi z)^{-1/\\xi} \\,", "e4440592b457afdff84040bd3c9bdf91": "(1-x)^{-1} = 1 + x + x^2 + x^3 \\cdots,", "e444260e38b58a79ff44e7c472372806": "R(A,B; x) := \\frac{x^* A x}{x^* B x}.", "e44478cba27ba120233849b1f23ee5f2": " \\lambda a, b, c.c\\ (\\lambda x.\\lambda a, b, c.b\\ (\\lambda a, b, c.a\\ f)\\ (\\lambda a, b, c.b\\ \\operatorname{mse}[x]\\ \\operatorname{mse}[x]))", "e44491558bda05cbbbb5767bdd856cbd": "T = \\left ( \\frac{\\partial U}{\\partial S} \\right )_{V, N} \\, .\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(2)", "e444e982fccb99e82b41fd2f088607ed": " \\cos \\theta\\! ", "e445281764d970a078197f5d2cf613ff": "\\omega = {1 \\over r} \\sum_{j=1}^{n+1} (-1)^{j-1} x_j \\,dx_1 \\wedge \\cdots \\wedge dx_{j-1} \\wedge dx_{j+1}\\wedge \\cdots \\wedge dx_{n+1} = * dr", "e445488bb68a650b55bafc4455305049": "p \\colon X\\times Y \\to \\text{TRUE/FALSE}", "e4456817fbbbd9c501cf9cfad498b663": " q_{trans} = 45.4 \\text{ ft}^2/s", "e445808e461f27bb0be037bc70870faf": "\\{ F : U \\in F \\}", "e445d83a2d64871679c35df4baab8e81": "\\phi(q)=\\sum_{n=-\\infty}^\\infty (-1)^n q^{(3n^2-n)/2}. ", "e445da4c53268adae8d74c5635914e8c": "a = \\frac{1}{4p}; \\ \\ b = \\frac{-k}{2p}; \\ \\ c = \\frac{k^2}{4p} + h; \\ \\ ", "e445efaf0b2dca8b894417bf7ccfb4df": "x \\in \\mathbb{Z}^*_n", "e445f25fbba5ba8d226b7eb66227d697": "\\mathrm{st}(\\{x\\},\\mathcal U)", "e445fa4af15e678f0a5d3d49d32df269": "\\Gamma = [E_\\min, E_\\max]", "e4466e0ae36ff1af2928f75db913830f": "\\Re(s) < -\\frac{1}{2} ", "e4469f8d8ebf77f1ba735e2feea1c554": "MN/M\\triangleleft G/M", "e446f3b58aab040cf22aa84214933d2f": "{{\\left\\{ {{g}_{m}} \\right\\}}_{0\\le m\\le N}}:", "e4470d90f6792de356d62ab863d2a5d2": "|(x-1)\\cdots(x-n)e^{-x}|", "e447563785869bcfa24e4f681bcf95b9": "\\scriptstyle\\lambda\\,", "e4478117475d15f9425ec3512c570fcd": "\\langle N \\rangle=\\frac{\\partial{\\ln{\\Xi(\\mu,M,T)}}}{\\partial{\\beta \\mu}}=M\\frac{\\partial{\\ln{\\xi(\\mu,M,T)}}}{\\partial{\\beta \\mu}}", "e447b176dbd82e0af06082828ddef59e": "S'_{gen} = \\frac{Q'_{total}}{T_{surr}}", "e447b86d1a6df0de3e14d7414e04090e": "m=(m,\\ldots,m_k )", "e447c37303aa003c6e1b9992045ddfda": "{x_n}", "e44838cdb8d59a8f81524ea6633d7d21": "(m,n_m)", "e4483b119fa4ef549bc20c4922734ef7": "Q_{i*}", "e4483e847a15f2c701f63cf5e60d7f55": "\\{t_c,\\ldots,t_{c+d-3}\\}", "e4489279749568832adeff85b3a9694c": "t_l=0", "e448a70c865dc46ba05f066144907fe3": "x = (r_1+r_2)\\cos t, y = (r_1-r_2)\\sin t\\,\\!", "e448e084175c4d5c724cf7ce109d306c": "J= -D {\\delta C \\over \\delta x} ", "e448e797e5842b1306ee6a379092fd8d": "\\epsilon^i{}_{jk}=\\delta^{il}\\epsilon_{ljk}", "e4491b72a81c851cc9f2d908dd0972eb": "\\begin{matrix} \\frac{m}{s^2} \\end{matrix}", "e4493fa252608fd8329eb814e40ce16b": "u(A) \\preceq u(B)", "e4498229c8ea06cab582dd6a4fe8b10c": "z(x,y)", "e44a1958f95eaecc13938ae0da4de0af": "\n\\begin{pmatrix}\n(3,1)_{-\\frac{1}{3}}\\\\\n(1,2)_{\\frac{1}{2}}\n\\end{pmatrix}\n", "e44a663cdecc8daa7806c0e49af39d47": "1+\\frac{q}{1+\\frac{q^2}{1+\\frac{q^3}{1+\\cdots}}} = \\frac{G(q)}{H(q)}.", "e44a7bfcc1f51a4e17372e0bb39b1476": "u_x=\\delta_x/2", "e44aea85f0d0e8eb57e198d530e0eeca": "x_{B1}=100", "e44b0b9923a128ce7e28bda2e852f2f2": "\\displaystyle (c=2R).", "e44b355437e053d63bf63a1055459952": "R_{lateral}=R_{up}*R_{down}^{*}; R_{axial}=R_{up}*R_{down}", "e44b4b7fa2f22324a4ddf6ae8c64ac90": "(f\\oplus b)(x)=\\sup_{y\\in E}[f(y)+b(x-y)]", "e44b98a96ff0818594c2306d6d18eb91": "\\mathbb C^\\times \\cong \\mathbb R \\oplus (\\mathbb Q / \\mathbb Z)", "e44bb29d47e22a7735a972d59546bf44": " T_s(x) = T(s,x).", "e44bbeb89cc7d8c702f5aab98b39299f": "N_t \\times 1", "e44cc234b0166043eeb5ba02d5bc682d": "~T=T(\\gamma)~", "e44d10eff69a27b8f9cbf5cd54584153": "\\begin{bmatrix}\n\\gamma_1 \\\\\n\\gamma_2 \\\\\n\\gamma_3 \\\\\n\\vdots \\\\\n\\gamma_p \\\\\n\\end{bmatrix}\n\n=\n\n\\begin{bmatrix}\n\\gamma_0 & \\gamma_{-1} & \\gamma_{-2} & \\dots \\\\\n\\gamma_1 & \\gamma_0 & \\gamma_{-1} & \\dots \\\\\n\\gamma_2 & \\gamma_{1} & \\gamma_{0} & \\dots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots \\\\\n\\gamma_{p-1} & \\gamma_{p-2} & \\gamma_{p-3} & \\dots \\\\\n\\end{bmatrix}\n\n\\begin{bmatrix}\n\\varphi_{1} \\\\\n\\varphi_{2} \\\\\n\\varphi_{3} \\\\\n \\vdots \\\\\n\\varphi_{p} \\\\\n\\end{bmatrix}\n\n", "e44d7b0f6dc71d29561fe56971f55487": "A_{\\alpha}", "e44d9bc8442fce804d2fd2f2034eb1f2": " Q = \\Delta H \\,", "e44dd09a9cd15ae986f6c0dba4091aae": "f^\\text{inc}= f", "e44de51380c3a38ca0c2f196bf44d257": " ch(V)=\\sum_{\\mu}\\dim V_{\\mu}e^{\\mu}, ", "e44e1afabaaa755f09061cd30228aa66": " \\frac{ \\sigma^2 } { 1 + \\sigma^2 }", "e44e3de8e41c6d0aacb55ad5c14608cd": "f(x)=1+x^{32}+x^{47}+x^{58}+x^{90}+x^{121}+x^{128}", "e44e50d94fa7fd630b2359cd30b2230f": "\\left|\\mathbf{x}\\right|^2 = x_1^2 - x^2_2 + x_3^2", "e44ebe1fb67a6af17b52837ebd2c384b": "A^+ (x^0,x^1,x^2,x^3) = A^0 (x^0,x^1,x^2,x^3) +A^3 (x^0,x^1,x^2,x^3)", "e44ed02137e5eaf295d991e9a0ebe10c": "F = k \\frac{|q_1 q_2|}{\\epsilon_r r^2}", "e44ed0a21a34f8d24eca887ddb192f54": "x^nG(x^{-1})", "e44f05f566fe1e9a8e5f030b8758d40b": "\\mathbf{e_z} \\times \\mathbf{e_x} = \\mathbf{e_y} ", "e44f6528867359a6253563b8b0829632": "A = \\frac{V_\\mathrm {out}}{V_\\mathrm {in}}", "e44f817d8bff14d56db8fbe451b31614": "\\phi_q =\\frac{V_{\\mathrm{sen}}}{E_{\\mathrm{sen}}}", "e44fbaca38e6a6bf9c2ce62132d2ef35": " =\\lambda_1 \\mathcal{H}_1 + \\lambda_2 \\mathcal{H}_2, \\, ", "e44fc883a94bceeafaee7dcb3e30906b": "\\phi (r) \\,\\!", "e4503e21df85586d71bb9730c585a2d9": " |\\mathbf{a} \\times \\mathbf{b}| = |\\mathbf{a}| |\\mathbf{b}| |\\sin \\theta| , ", "e4504b40190d4afe205a8ac1362753bd": "L = gJdeltaH/U^2", "e4505cb189c65fe7149230cb26fb3c1f": "\\vec{h}_0 = \\frac{1}{\\sqrt{1-3m/r}} \\, \\partial_t + \\frac{\\sqrt{m/r^3}}{\\sqrt{1-3m/r}} \\, \\partial_\\phi ", "e450bc91f08ee2ef717cc5bd196d893f": " \\boldsymbol{\\tau} = \\mathbf{p} \\times \\mathbf{E}", "e4516aaba69c77b96b914c2478c48909": " \\sum_{i=1}^n \\frac{\\gamma^2}{\\gamma^2 + [x_i - x_0]^2} - \\frac{n}{2} = 0", "e4517f62a00970cebc4220e018328cad": "r = s", "e451b6250f9621cefad6a1880b117b59": "\\frac{l}{2}+\\frac{l}{2}", "e451e5f7bea541a4bbbed181c968f74d": "4\\pi/n", "e451e8a920941f58b8624147a2bbd965": "\nf(x; k,\\sigma_1^2,\\ldots,\\sigma_k^2) = \\sum_{i=1}^{k} \\frac{e^{-\\frac{x}{\\sigma_i^2}}}{\\sigma_i^2 \\prod_{j=1, j\\neq\ni}^{k} (1- \\frac{\\sigma_j^2}{\\sigma_i^2})} \\quad\\mbox{for }x\\geq0.\n", "e452099e48fbe35658d53774637d9d26": "\\psi=\\left(\\begin{array}{c}\\psi_L\\\\ \\psi_R\\end{array}\\right)", "e4524e86bddce2fa3759e7ef0619947d": "U_{B,\\varepsilon}(y) = \\cap_{\\alpha\\in B}(p_{\\alpha,y})^{-1}([0,\\varepsilon))", "e4525824fab03906c407610bc4ddc9e0": "\\frac{1}{P\\left(1-\\frac{P}{K}\\right)}=\\frac{K}{P\\left(K-P\\right)}", "e4527b5e6380f7b621f6cf484ec3a286": "\\gamma =\\int_1^\\infty\\left({1\\over\\lfloor x\\rfloor}-{1\\over x}\\right)\\,dx,", "e4527be95d50e2d9f24e9cd79d0bd720": "(D-E)", "e452834097295732e2957e14c9084c65": "{Z_g}(h) = \\frac{\\Phi(h)}{g_{0}}\\, ,", "e452918ef83a0d12757427deae4b586e": "\\sum_u \\Pr(X=u) = 1", "e4529a276c332ecb9615c0138792595c": "x = 4.965114231744276\\ldots ", "e4537a9b4b9130916ad6542942cf64f2": "X \\xrightarrow{\\Delta} X\\otimes X \\xrightarrow{\\phi\\otimes\\psi} X\\otimes X \\xrightarrow{\\nabla} X. \\, ", "e453bb0a2e7af4e5a66a8e17a7d1a5e0": "h_{v} + z_\\text{elevation} + \\psi = C\\,", "e4540405f8cb8abfb922f01a6c4d25ab": "\\Gamma(x)=e^{\\psi(-1,x)+\\frac 12 \\ln(2\\pi)}\\,\\,\\,", "e45453130e0a3308264085ec40e62005": "e\\in I_2\\setminus I_1", "e455025b1485e429864ce3638af18b7a": "\\sin\\sin\\sin\\frac1x^k+v^2-b", "e45575e700d63515f4c00bf9285e1259": "n^{(\\frac{2}{27}+o(1))\\mu^2}.", "e455bae9be931e02911afbdbd42dde0f": "\\mathfrak{sl}_n(F)", "e455c9df2b26dfb7b15b38ce51d9e731": "[T_A^1]~|~[T_A^2]", "e457762792ceaeb8df1d6364f0e31836": "p(\\overline{x}) = 1", "e458221b0faf367ceb854e76ff07b48f": "\\displaystyle S(e^x)", "e4582293c62e0dd8ff3d828df7df250f": "1.00U($1\\text{ M}) - 0.89U($1\\text{ M}) < 0.01U($0\\text{ M}) + 0.1U($5\\text{ M})\\,", "e45830a9fe329caf065a9cc0f451bf01": "[D_1,D_2] = D_1\\circ D_2 - (-1)^{d_1d_2}D_2\\circ D_1.", "e4583704e030fcf9c74a873dec890f81": "{{r_p}\\over{r_a}}={{1-e}\\over{1+e}}={{0.723}\\over{1.277}}=0.566", "e4584fa2620533f78ef24b2517f2f4b9": "(X_0, \\tau_0) \\to (X_1, \\tau_1)", "e45860b6b690726baddaa87f1d2a1211": " (t + \\tau)", "e4587a0d2feabbf510822dfb169d0a17": "\\frac{\\beta}{1-\\beta}=-\\frac{\\alpha}{1-\\alpha}", "e4589703807b6f30f37dba571cc6e356": "\\displaystyle c_\\alpha |\\boldsymbol \\xi|^{-(n - \\alpha)}", "e458af5deaaaa8bbf2be9e5c83d75a0c": "\\displaystyle{a^{b+c}=((a^{-1} - b) -c)^{-1} = ((a^b)^{-1} -c)^{-1}=(a^b)^c.}", "e458c52a783bbec068470782d60dac08": " \\delta v_i/v_i = \\delta E_i/E_i ", "e458d8860567d0b78e468a3e74010d56": "\\begin{cases} \\partial_{t} u(t, x) = \\partial_{x}^{2} u(t, x), & x \\in (0, 1), t > 0; \\\\ u(t, x) = 0, & x \\in \\{ 0, 1 \\}, t > 0; \\\\ u(t, x) = u_{0} (x), & x \\in (0, 1), t = 0. \\end{cases}", "e459390108753d1d745e32b1dbea08b3": "\\frac{f(x)}{f'(x)}=0", "e45986c585ee5732f8926def30a81d79": " R_i = \\frac{1}{\\nu_i} \\frac{\\mathrm{d} \\left [ {\\rm X}_i \\right ]}{\\mathrm{d} t} ", "e459a01a93a7b3edb630643cb8c8560f": "\\Delta{s}\\,", "e459f946b96018e3c7afd8d003509a9e": "\\displaystyle{ W_{\\mathcal F}(z)f(w)=e^{-|z|^2/2} e^{w\\overline{z}} f(w-z).}", "e45a4e184e1ce2b9206c19feeeda6130": "\nH_\\mathrm{e}\\;\\chi_k (\\mathbf{r};\\mathbf{R}) = E_k(\\mathbf{R})\\;\\chi_k (\\mathbf{r};\\mathbf{R}) \\quad\\mathrm{for}\\quad k=1,\\ldots, K.\n", "e45a60c500f0582f6b7de7500eaa3c9b": "\\textbf{Y}_{k\\mid k-1} = \n \\textbf{L}_{k} \\textbf{M}_{k} \\textbf{L}_{k}^{\\text{T}} +\n \\textbf{C}_{k} \\textbf{Q}_{k}^{-1} \\textbf{C}_{k}^{\\text{T}}", "e45aa798030c46ab4f290715aea7510f": "\\displaystyle \\partial_t w = 3\\, (\\partial_x u)\\, w + 6\\, u\\, \\partial_x w - 4\\, \\partial_x^3 w", "e45ab6ceae969b606fdf5ab5144e1ac9": " \\left| \\begin{matrix} x_1 & y_1 & z_1 \\\\ x_2 & y_2 & z_2 \\\\ x_3 & y_3 & z_3 \\end{matrix} \\right| = 0,", "e45b3ba21b6a6d0290577d89339a5e77": "\\Delta t = \\frac{2}{c} \\sqrt{R^2 + r^2 - 2 R r \\cos\\varphi} \\approx253\\,\\mathrm{ms}", "e45b488c1104f8b4e266551ef449ef12": " fv = \\frac{1}{\\rho} \\frac{\\partial p}{\\partial x}", "e45b70f8a2d0cc425814c4022227d7f1": "\nV_{t+1} (N_{i,j}) = \\left\\{\n\\begin{array}{ll}\n1 & \\mbox{if}\\,\\, U_{t+1}(N_{i,j}) > 3\\\\\n0 & \\mbox{if}\\,\\, U_{t+1}(N_{i,j}) < 0\\\\\nV_t(N_{i,j}) & \\mbox{otherwise},\n\\end{array} \\right.\n", "e45b8bb3cfcf35d5b90ce71294ac321d": "\\mathbf{0}", "e45bde72672ac92e181d5ab87379179f": " x_0,\\ x_1=f(x_0),\\ x_2=f(x_1),\\ \\dots,\\ x_i=f(x_{i-1}),\\ \\dots", "e45c0134b6a96e8fa40097a4ca09d9d0": " c^\\dagger_{i\\sigma} , c_{j\\sigma}", "e45c252841509a3bdf4bfc1f4e16f51a": " \\psi(f) = \\int_X f(x) \\, d \\mu(x) \\quad ", "e45c50f5c59791fbd6ab4d6dda0188f1": "\\displaystyle \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = \\nabla \\cdot \\sigma + \\mathbf{f}", "e45c6826e6087d664fbde4788fb3fae0": "{}\\quad = \\int_{-\\infty}^{\\infty} x(\\tau)\\cdot h(t-\\tau) \\,\\operatorname{d}\\tau,", "e45c6e05e9932768c15800224adde164": "\\frac{2}{7}\\sqrt{10-\\sqrt{2}}", "e45c88556d20d7e23c9d3039b6d3c608": "U = \\begin{cases} \\{0,1\\}, & \\mbox{if } P \\\\ \\{0\\}, & \\mbox{if } \\neg P\\end{cases}", "e45cbeb4b4bbca77a7f0ee49fcb62451": "\\left |x-\\tfrac{1}{2}\\left(\\tfrac{a}{c} + \\tfrac{b}{d} \\right ) \\right |=\\tfrac{1}{2}\\left (\\tfrac{b}{d} - \\tfrac{a}{c} \\right )", "e45cd449912d5f150ff8e87cbc2d3bff": " j \\ne 1 ", "e45d3f1cd11df1b95e174326bca49a04": " \\lim_{\\varepsilon \\to 0} \\frac{\\zeta(1+\\varepsilon)+\\zeta(1-\\varepsilon)}{2}", "e45db117ebc23fb78da5261551f28095": "\\textstyle (N-1)(1-p)", "e45dfacb414e955e717c0bd741dc4f57": " \\begin{align}\nf(x) &= f(a)+\\Big(xf'(x)-af'(a)\\Big)-\\int_a^x tf''(t) \\, dt \\\\\n&= f(a) + x\\left(f'(a) + \\int_a^x f''(t) \\,dt \\right) -af'(a)-\\int_a^x tf''(t) \\, dt \\\\\n&= f(a)+(x-a)f'(a)+\\int_a^x \\, (x-t)f''(t) \\, dt,\n\\end{align} ", "e45e0fe63442bb3f51e3f1b9fb908a7c": "F \\left( \\frac { p_{02} } { p_{01} }\\ , \\frac { T_{01} } { T_{02} }\\ , \\frac {{\\dot{m}} {\\sqrt {T_{01}}}} { p_{01} }\\ , \\frac { N } {\\sqrt {T_{01}}}\\ \\right) = 0 ", "e45e2cb5909ca4c52268c79633178c78": "\n\\int Dq(t) =\n\\lim_{N\\to\\infty}\\left( {-i m \\over 2\\pi \\delta t \\hbar } \\right)^{N\\over 2} \n\\left( \\prod_{j=1}^{N-1} \\int dq_j \\right)\n", "e45ef3c67cf4f8c8284af857841c5fa4": "N(n,k) = \\frac{1}{n}{n\\choose k}{n\\choose k-1}.", "e45f13248d88a6123de96e32221dc032": "\\mu_\\varphi=X^k", "e45f473668765c3f8c5d6910d44811c9": "\\left( \\frac{17}{91}, \\frac{78}{85}, \\frac{19}{51}, \\frac{23}{38}, \\frac{29}{33}, \\frac{77}{29}, \\frac{95}{23}, \\frac{77}{19}, \\frac{1}{17}, \\frac{11}{13}, \\frac{13}{11}, \\frac{15}{14}, \\frac{15}{2}, \\frac{55}{1} \\right)", "e45f6f780cbea6d64360932a85a778f0": "C_V = \\frac{3}{2}nR \\;", "e45f93b3ae747726b41336a44c6a8d55": "\\psi(x+N) - \\psi(x) = \\sum_{k=0}^{N-1} \\frac{1}{x+k} ", "e45fd825b3f42c6be7deaa76adb5e100": "\\bar{X}={1 \\over n} \\sum_{i=1}^n X_i", "e45fe74b8e29e7497d137de22ddf85bb": "\\int_{a}^{b} \\, f(t)\\ dt \\ = \\infty", "e4600023dc35f528a05d028b6c3b49d0": "\\lambda\\in\\Lambda_0", "e46009646f13fb2a6912291709113928": " A^2 + B^2 = R^2 \\,", "e46089f12ac8a70fc7afa5c86918d263": "H(s)=\\frac{G_\\mathrm{bpf}\\frac{\\omega_{0}}{Q}s}{s^{2}+\\frac{\\omega_{0}}{Q}s+{\\omega_0}^2}", "e4608b3c5f95b4fa12f162bb617a3512": "U = \\{u_1, u_2, \\dots , u_n \\}", "e460ca03ff498dc666073d556dc10033": "\\displaystyle f^2=de,", "e460eed431b42996727c887deb290918": "\\lambda_1 \\,\\!", "e4610a531e8bce8113ff725edef1898b": "R=[9/[5 \\times 6/2]]-1=-0.4", "e4614b771271bf72ab97a9d6641d4d13": "\\alpha(i)", "e461596b101c75985fbd34a4d095e7a8": "G_{ab} = \\frac{8\\pi}{\\phi}T_{ab}+\\frac{\\omega}{\\phi^2}\n(\\partial_a\\phi\\partial_b\\phi-\\frac{1}{2}g_{ab}\\partial_c\\phi\\partial^c\\phi)\n+\\frac{1}{\\phi}(\\nabla_a\\nabla_b\\phi-g_{ab}\\Box\\phi)", "e46186b3d1ab9c47627c3624f14e0bb2": "\\gamma_n = 1\\cdot 3 \\cdot 5 \\cdot .. \\cdot (n-2)", "e46186b60cff49838738d2baccd0e9ca": " SL(8,\\mathbb C)", "e461b3cf57f61fbd2f8312099f3da0d8": " \\nu_\\mathrm{n}", "e461c07f56eb2ad831b6afc3c76792f1": "(x_1,y_1)\\parallel_+ (x_2,y_2)", "e461c527d4ca1aea0fe1f11360bc82f9": "A=\\mathbb{C}", "e461e6305537d7f9f7e5bb17e8908e07": "\\frac{1-e^{-\\lambda \\theta}}{1-e^{-2\\pi \\lambda}}", "e461f06c3390f33050dbaa5298b4bc31": "HK={{\\textrm{load}(\\mbox{kgf})} \\over {\\textrm{impression\\ area} (\\mbox{mm}^2)}}={P \\over {C_pL^2}}", "e461fec796272c81c1c4f2eb0a88e432": "\\bar F(c) = \\int_c^\\infty f(x)dx ", "e4620f52b2a5f27f166a809d9034a9e2": "B^a{\\!}_\\mu\\rightarrow B^a{\\!}_\\mu - \\partial_{\\mu}\\alpha^a", "e46228ef8afe9de0e1a40fa83c9a1c6b": "\n \\eta\\, =\\, a\\, \\cos\\, \\left( k x - \\omega t \\right),\n", "e462b021ec5c8a338b715e8afc943dd8": "\n (PL)\n ", "e462b7e236c081548166a08ba6b20e24": "C^1", "e462e10727dadda7c23da8b0b2a6eb38": "g-h", "e462f95a6911a181d326ad38c603fb4a": "v_\\mathrm N + v_\\mathrm F = 2v\\,\\!", "e4639487d6663b2b54410b2c1d76540e": " 0\\to VY\\to TY\\to Y\\times_X TX\\to 0, \\qquad\\qquad (1)", "e463acb8cb07c2eaa26994dd6fdc2684": " \\textstyle c \\! \\cdot \\! t / 2", "e464051c0366216799429fa1ed29b37d": "L\\equiv\\partial_{xy}+A_1\\partial_x+A_2\\partial_y+A_3", "e4647f85d704122b18ea09e4fde406f5": " \\frac {a_1 + 8a_2}{(a_1+4a_2)^{3/2}}", "e464998ada33ff4c444ebc34c6804ec2": "\\Delta y = R_W \\sin(\\theta)(T_1+T_2) \\frac{\\pi} {T_R}", "e464f1154eeb356f5f6a7817f20eb34d": "\\sqrt{ax^2+bx+c} \\;=\\; -x\\sqrt{a}+t \\;=\\; \\frac{c-t^2}{b+2t\\sqrt{a}}\\sqrt{a}+t", "e46500c789915fb2fdf0f3db50885062": "x \\wedge y = xy", "e465601851c1a755527c70d9441484af": "vxy = c^j", "e465a0585cc97490baa699f2cc71c453": "\\sigma_{r} = \\frac{MWT}{M_{r}}", "e465caaadf7b61d28cafeb6101715528": "\\mathbf e_2 = (0, 1, \\ldots, 0)", "e465d08c45d14f0406250b5391da08e5": "(b,c)", "e465ee10f145dac3f796f4fd285916a8": "\\ \\displaystyle D ", "e466056848522d116ba9f3f2767fed2a": "\\exp(x + y) = \\exp(x) \\cdot \\exp(y)", "e4660e673eb2c64bb63a07fbd7afc528": "\\textstyle UW", "e466131382383b26c32ffe54889596a0": "\\begin{matrix} {10 \\choose 1}{4 \\choose 3}{2 \\choose 2} \\end{matrix}", "e46699ec1871c15bbd332bc3239857ad": "a^n = g^{\\log_g(a^n)} = g^{n\\log_g(a)} = g^{n\\log_g (a) (mod |g|)}", "e466adc7a7887ce08be7b26df64abdb0": "{4 \\choose 1}{3 \\choose 1} = 12", "e46739e879cefcc52907726dca54163d": "g_{SP}(t,\\omega)", "e46771fad72513c160ae5cea5b7443c8": "Z = (-1)^n \\left(1 \\cdot 2 \\cdot 3 \\cdot \\cdots \\cdot \\frac{p-1}2\\right).", "e46780379288fdf941c3e335f8258482": "\\scriptstyle2", "e4678688924508aab81a283964c9e3ed": "0\\rightarrow B\\rightarrow X_n\\rightarrow\\cdots\\rightarrow X_1\\rightarrow A\\rightarrow0", "e467c405602c7175d11c2b958f24b6a4": "u_t + F_x\\left(u \\right)=0. \\, ", "e468056f0819aac5ee97dee2b5f50de9": "\\partial^\\alpha \\ = \\left(\\frac{1}{c} \\frac{\\partial}{\\partial t}, -\\nabla \\right)", "e46861863093d5ef64b1b6a9ccbcdc03": "X\\rtimes G", "e468636946708fda3d62804beaf23d26": "\\sqrt{2} \\approx 1 + \\frac{1}{3} + \\frac{1}{3 \\cdot 4} - \\frac{1}{3 \\cdot4 \\cdot 34} = \\frac{577}{408} \\approx 1.414216,", "e468648506529685fd793823fde5774f": "\\delta^{\\beta_k}\\gamma_k", "e468b0a3c2227a189a4ae47f074dedab": " \\alpha_2 ", "e468f8f2cff71e07b676e3c77207beb8": "\\begin{align}\n\\mathbf{F}&=q_{\\mathrm e}\\left(\\mathbf{E}+\\mathbf{v}\\times\\mathbf{B}\\right)\\\\\n& + \\frac{q_{\\mathrm m}}{\\mu_0}\\left(\\mathbf{B}-\\mathbf{v}\\times \\frac{\\mathbf{E}}{c^2}\\right)\n\\end{align}", "e46915a8d1349239824726d628499c11": "N \\subset M", "e4691d0fcf4747ee3a84fe6a00795033": "m_\\mbox{linear} = f(12.3, 7.6) = [14.46,0.554]^T", "e46952567948fda7031cef4db698fb00": "{{{11}}}", "e469e296bb5ec7f6ab9781bf2453358a": "\\sum_i (y_{ij})^2", "e46a00371399063a5a2664ec382a19c4": "\\mathbf{n}=\\mathbf{v}/\\|\\mathbf{v}\\|", "e46a0e6923057df8c40b5a1fe241ccc0": "\\text{excess kurtosis} =- \\frac{6}{3+2\\alpha} \\text{ if }\\alpha=\\beta ", "e46a233dc6a79e23a8f8784ac5bb76d3": "\\sqrt{|\\mathbf{p}|^2+|\\mathbf{p}-\\mathbf{q}|^2}", "e46a23d3baa066e8a7458e0b99e35428": " | k_1 \\rangle ", "e46a32de450184d9c5c0aa8f6cd86226": "\\partial{C} = \\partial", "e46a59e66ff448b3da3fd94457166eb5": "\\log_{b} w", "e46a5bc6bd7291dab70f50bbda7779df": "{m+d \\choose m}", "e46a6f6449d942e71ed60de0c391acbd": "\\xi^* = K\\bar{\\xi}.", "e46abad62e57fb36ee1af04838a1a879": "\\dot{r}=-\\frac{G}{2}\\dot{u}", "e46ac64bce9cdf3f108a8a2f96e5431b": "N_U=N_{P_1}", "e46acce2b9a26e1e27e804095f0ed484": "-1<\\rho<1", "e46ae074ea7cff55c8db1f1500079dfd": "\\text{fmap}: (A \\rarr B) \\rarr ((S \\rarr A) \\times S) \\rarr (S \\rarr B) \\times S = f \\mapsto (f', s) \\mapsto (f \\circ f', s)", "e46ae252d49fa8bf38911f5f47b56d44": "\\sqrt[8]{2}", "e46af469512ebc77728687bc93cf8f80": "1 \\leq i \\leq n\\!", "e46b2a760c57eb995b62bd166b835b1e": "]a,b[", "e46b3f9df531f937d28702293355b35b": "1.331^{1/3} - 1 = 10%", "e46b525009b63dc63d6f50b19fa1c906": "\\vec x = [x_1,..., x_n]", "e46b9389a8103855b79c5569e02882c7": " \\bar{c}=2.\\xi.\\omega, \\bar{k}=\\omega^2", "e46ce37d2ca2fb45c0e066b3290c550d": "\\left\\{e_1, ie_1, e_2, ie_2, \\dots, e_n, ie_n\\right\\},", "e46d813a08e1c6f887dc4a3404e18481": "\n\\ Y(z) = (1 + \\alpha z^{-K}) X(z) \\,\n", "e46e47235d766ff441e64416055fd509": "w(n) = \\mathrm{sinc}\\left(\\frac{2n}{N-1}-1\\right)", "e46e6139ee2fc7b08cf9bf9450e961f1": "\\frac {d M_y(t)} {d t} = \\gamma \\left ( M_z (t) B_x (t) - M_x (t) B_z (t) \\right ) - \\frac {M_y(t)} {T_2}", "e46e674a5c64b22e69a9018a7b2f1ab1": "\\sum _{i=1} ^m \\frac{1}{i} \\approx \\ln(m) + \\gamma.", "e46e9ded2d912f3bfa63a8a15babf92f": " G:=(\\mathbb R \\cup \\{\\infty\\}) \\times \\mathbb R,", "e46eb1ed2fa05316f426d29418c2c9a7": "(P)\\psi", "e46ec5128a1236ad13b43414c063c4e8": "\\Gamma \\in Q", "e46ef082fd8aea339aa0362ffd27b191": "\\wedge^m_0", "e46ef1cc3aa498c8c5b09dca6a7f3414": "x_c(\\theta_c(t))", "e46f16009e0d33752a750810c224bb18": " \\alpha(z) \\approx (\\theta - \\beta) {D_d \\over D_{ds}} ", "e46f30560ac534015998fbce2772e575": " v = x_1 e_1 + x_2 e_2 + \\dotsb + x_n e_n .", "e46f5b5387a13c8e62e4c04d8810384d": "0<\\int_{\\partial B}\\omega = \\int_{\\partial B}f^*(\\omega) = \\int_Bdf^*(\\omega)= \\int_Bf^*(d\\omega)=\\int_Bf^*(0) = 0", "e46fc7cfc46b31cbf9510d0dd83b2a73": "\\partial_{x}, \\partial_{y}", "e46ff99ac23524142dcfcbaafc883598": "\\mathfrak{so}(\\mathfrak{g})", "e470632615a4128fba31d8deb0a45788": "\\left(\\!\\!{n\\choose k}\\!\\!\\right) = {n^{\\overline{k}}\\over k!},", "e470fc344b4de6b19343750fbc3632de": " \\alpha < \\beta < \\gamma\\,. ", "e470fe4ba5b2cf2479a0614269e39a7f": "{{\\varepsilon }_{particle}}=1-\\frac{\\omega _{p}^{2}}{{{\\omega }^{2}}}", "e4715373d33edbda4af91373f230f9d3": " \\mathbf \\zeta=1", "e471558a406a6a7c090be67fcd7fbdf8": "n_1 0\\})", "e47bdb775a686078e90db026f0c0cc8b": " M'\\longleftarrow N\\longrightarrow M'',", "e47bf8de381f9e6e15c2dd1fae847c52": "\n \\begin{align}\n \\frac{\\partial \\mathcal{L}}{\\partial f}\n & - \\frac{\\partial}{\\partial x_1}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,1}}\\right)\n - \\frac{\\partial}{\\partial x_2}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,2}}\\right) \n + \\frac{\\partial^2}{\\partial x_1^2}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,11}}\\right)\n + \\frac{\\partial^2}{\\partial x_1\\partial x_2}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,12}}\\right)\n + \\frac{\\partial^2}{\\partial x_2^2}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,22}}\\right) \\\\\n & - \\dots\n + (-1)^n \\frac{\\partial^n}{\\partial x_2^n}\\left(\\frac{\\partial \\mathcal{L}}{\\partial f_{,22\\dots 2}}\\right) = 0\n \\end{align}\n ", "e47c69750c58cac91ed67d5cf944ee29": "\\,q\\overline{q} = -a^2 + b^2+ c^2 +d^2 ", "e47cafb8e939d341f2cb907de151992a": " u(t,x) = T(t) X(x),\\,", "e47d5ddd6492610e3319c87f85db93a1": "{}_Z^A\\!X\\to {}_{Z-1}^A\\!Y+ \\nu + \\beta^+", "e47d81bb3ffbf476ddd8217244baf2b0": " A_{|\\alpha \\beta \\gamma|\\cdots} B^{\\alpha\\beta\\gamma \\cdots} = \\sum_\\alpha \\sum_\\beta \\sum_\\gamma A_{\\alpha \\beta \\gamma\\cdots} B^{\\alpha\\beta\\gamma \\cdots} ", "e47dbe22569505b172edec6beb3840ff": "X=\\prod_{i=1}^\\infty \\mathbb{R} ", "e47e0c2f2c2e3fd676abd0ca3f8586ce": "F_{n+1}^{(1)}=2F_n^{(1)}+F_{n-1}^{(1)}-2F_{n-2}^{(1)}-F_{n-3}^{(1)}.", "e47e0f210267f51515716285479ce356": "\\Rightarrow (b-d)(b+d)=c^2-e^2\\, ,", "e47e4d6fe8c6317b7550aa171e271582": " He_n(x) = n! \\sum_{m=0}^{\\lfloor n/2 \\rfloor} \\frac{(-1)^m}{m!(n - 2m)!} \\frac{x^{n - 2m}}{2^m}. ", "e47e6f8960455f68eaf0628ea7638b94": "W^{\\iota}=\\{(\\{x\\},\\{y\\})\\mid xWy\\}", "e47e85cce5ec2960d4d6198a833176b1": "\\rho - \\rho_0 = \\rho_0\\alpha(T-T_0)\\,\\!", "e47eb99d3626bd626a14675b25ec5754": "\\int_X (g-f)^+\\,d\\mu=0=\\int_X (g-f)^-\\,d\\mu,", "e47f2bba52da3e22b11aa10f9ff9a92d": "\\qquad E_{KL}\\approx e_{rs}\\approx\\varepsilon_{ij}=\\frac{1}{2}\\left(u_{i,j}+u_{j,i}\\right)\\,\\!", "e47f426c58e1b265be6b329817fb458e": "d \\neq 0", "e47f4e0e4ffaceefa6a6d0a1307e2e07": "\\lim_{x\\to\\infty}x/N=\\begin{cases} \\infty, & N > 0 \\\\ \\text{does not exist}, & N = 0 \\\\ -\\infty, & N < 0 \\end{cases}", "e47f5ecc02a8819d991c6bd64b8fdfdd": "\\textstyle{t}\\,", "e47f6243d5b1763b3ca1cfdbcc256d8a": "\\Omega = \\pi / 2", "e47f6ce53282f3846894689c32237aa0": "{m \\choose r}_q = {m-1 \\choose r}_q + q^{m-r}{m-1 \\choose r-1}_q.", "e47f763838ee252bc31f1f84fa1de829": " H(a,b) = {1 \\over {{1 \\over 2}\\cdot {({1 \\over a} + {1 \\over b})}}} = {{2ab} \\over {a+b}}", "e47f7ad86b0c713d7237590da49ff0c9": "T\\in\\mathbb{R}^{+}", "e47f947361bcb23bd6fa7165251c1f4f": "X = Y x / y,\\,", "e47fe10376bc6c5c9a62023df4e9acda": " A \\rightarrow \\; C ", "e47fe7d8c2eadd7cdf207d973c47eeda": "F_X(x) = \\frac{1}{2} - \\frac{1}{\\pi}\\int_0^\\infty \\frac{\\operatorname{Im}[e^{-itx}\\varphi_X(t)]}{t}\\,dt.", "e47fee6613a1317a652c61b86054417d": " e \\ll x ", "e4807c4a35fa43cdddb0af06cf4811e5": " f = f_1 g_1 f_1 g_2 f_1 g_1 f_1 g_3 f_1 \\cdots \\ ", "e4809fb029897d8ffd3bde2c1763dbe2": "(n^3)/4", "e480bed1cdfabaf63a969e3414418f67": "|x+y|\\leq|x|+|y|", "e480e56c8578a924c6da2edd7b6b7f3a": "\\Phi\\left(\\mathbf{r}\\right)\\equiv\\frac{1}{4\\pi}\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\cdot\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\frac{1}{4\\pi}\\oint_{S}\\mathbf{\\hat{n}}'\\cdot\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'", "e4813dd8f85aedf37cb5553b1699f6b3": " \\mathbf{A}\\!\\!\\!\\begin{array}{c}\n _\\cdot \\\\\n ^\\times \n\\end{array}\\!\\!\\!\n\\mathbf{B} = \\sum_j\\sum _i \\left(\\mathbf{a}_i\\cdot\\mathbf{c}_j\\right)\\left(\\mathbf{b}_i\\times\\mathbf{d}_j\\right) ", "e48170d6c1af374e63d233652d020067": "c_1,c_2,\\ldots c_m", "e48193ef7824a62d22f0ae2d77ac930e": " r ", "e481bff891ecbf9e20f515e879f68560": "\\rho_{L^*}", "e481c5deccabd577323e4edb09ad1fc5": " S=\\mathbb{R}^3\\setminus\\{(0,0,z)~|~z\\in\\mathbb{R}\\} ", "e4823eefc2d4156a68cb884778478962": "m_{n}, m_{n-1}, m_{n-2}, ... , m_{1}", "e482452dacd254f2ac53eb4e34039fbd": " \\%A_ F", "e482a0e1a6db2855d9ac8ed7572da9b6": "\\underline{x}_0 \\in \\mathbb{R}^n", "e482aa2528ebb7ce8c6d2abb5974f0ae": "\\frac{1}{i} = i^{-1} = -i\\,", "e482ae364962684a5cc11ccc77b40f0b": "\\operatorname{Ind}^\\mathfrak{g}_\\mathfrak{h} \\circ \\operatorname{Res}_\\mathfrak{h} \\simeq \\operatorname{Res}_\\mathfrak{g} \\circ \\operatorname{Ind}^\\mathfrak{g_1}_\\mathfrak{h_1}", "e482c17c461f28cdbb49a11c8b22faae": "\\textstyle V_\\theta ", "e482e88c75ee20cd162d6a2d970b101f": "\\mathcal{A} = 1", "e482f86e79976573e3cca538abd7c54e": "\\left|\\;\\frac{4}{5}\\;\\right|^2=\\frac{16}{25}", "e482fd258f4076f43101756b56fbfc7a": "\\mathcal{L}_\\text{approx}(\\theta \\mid x \\text{ in interval } j) = f(x_{*}\\mid\\theta) \\Delta_j, \\!", "e48313e4252b1d822fd04666bdf50c8b": "[-2,2]\\cap \\left(0 - \\frac{1}{2\\cdot[-2,2]} (0-2)\\right) = [-2,2]\\cap \\Big([{-\\infty}, {-0.5}]\\cup [{0.5}, {\\infty}] \\Big) = [{-2}, {-0.5}] \\cup [{0.5}, {2}]", "e483174feb31a8e9cc2ab67332b6e739": "\\mathbf{e}_5", "e4833aad5b948325b6dcd5d86a8c0f86": " S_R (s_1) - S_R (s_2) = \\frac{1}{T} (U_R (s_1) - U_R (s_2)) = - \\frac{1}{T} (E(s_1) - E(s_2)),", "e4835f4282cb4e9c9f22bbd7893b0d73": "\nP(\\lambda)= a_0 \\lambda^n + a_1 \\lambda^{n-1} + \\cdots + a_{n-1} \\lambda + a_n\n", "e483e6bef3f396ff1a09b98303cea733": "\\rho_c \\sim 10 - 10^3\\,\\mathrm{\\tfrac{g}{cm^3}} ", "e4846c866c370cf79c62359ad7d0e3ab": "x_{t+1} = \\frac{(d-\\eta c)x_t}{\\eta c+a} + \\frac{c}{\\eta c+a}.", "e484744d8c44532e800faea668bff9c2": "W(\\cdot)", "e484841cc4880cf2a52f7858d2da08af": "(A \\rightarrow ((B \\rightarrow A) \\rightarrow A)) \\rightarrow ((A \\rightarrow (B \\rightarrow A)) \\rightarrow (A \\rightarrow A))", "e484df1d301081071b90eac5668ba4ab": "D(r)=\\mathrm{true}\\;", "e4850a1c034189ebda2cf5b04077fe9e": "b=\\tilde{A}^T\\tilde{b}", "e485546de958cda0a8a41e5326b30d8d": "\n\\left[\\frac{\\lambda}{\\mu}\\right]_2 \\bigg[\\frac{\\mu}{\\lambda}\\bigg]_2 =\n(-1)^{\\frac{\\mathrm{N} \\lambda - 1}{2}\\frac{\\mathrm{N} \\mu-1}{2}},\\;\\;\\;\\;\n\\bigg[\\frac{1-\\omega}{\\lambda}\\bigg]_2 =\\bigg(\\frac{a}{3}\\bigg), \\;\\; \\text{ and }\\;\\;\n\\bigg[\\frac{2}{\\lambda}\\bigg]_2 =\\bigg(\\frac{2}{\\mathrm{N} \\lambda }\\bigg),\n", "e4859115874810cb5c89443a8b60987a": "\\,e^{i\\omega t}", "e485b22e58633f2125bffe4893328b8a": "\\textstyle \\overline{a}_{j.}", "e4861b707ff550ef16e31580a5206c7e": "n\\ ", "e48663ec050a1cc0f7aa7713df7604c6": "P \\vee (P \\to \\bot)", "e4866c1834a506cff1e3d3101a334155": "F_N(x) = \\sum_{n=0}^N D_n(x) = \\frac{1}{N}\\left(\\frac{\\sin \\frac{Nx}{2}}{\\sin \\frac{x}{2}}\\right)^2.", "e4873b2ec60d2d6b6b6c54cdf7176d79": "\\mathbf x \\times \\mathbf y = \\sum_i x^i \\mathbf e_i \\times \\sum_j y^j \\mathbf e_j =\n\\sum_i x^i h_i \\hat{\\mathbf e}_i \\times \\sum_j y^j h_j \\hat{\\mathbf e}_j", "e48748e7f90b587b18bf932030437471": "\\sigma _1 E^2", "e487733b90c95b35cfaa4ad2635e9353": "\\ \\phi^\\prime", "e487908e3032ea60b513ffc569374cec": "-\\lambda^{-1}\\mathbf{R}_{x}^{-1}(n-1)\\mathbf{x}(n)", "e487a72cf0c9bae266825d85eceb0b9a": "\\ U_p = \\,", "e487aab8c3bb559229a3a915ebc41b72": "X = \\{", "e487ae6fb6033dc35f0a040ffddbd228": "dY=F_A dA+MPK dK+MPL dL=F_A dA+r dK+w dL", "e4880bbaf4491dc54a5526896fc18569": "\\mathbf{C} \\cong \\mathbf{R}\\cdot 1 \\oplus \\mathbf{R} \\cdot i = \\mathbf{R}^2", "e48919996a9bdea8d1a585477328def7": "\\mathit{USp}(2n) \\supset \\mathit{U}(n) ", "e48967e122289cc84a7f8831cce0a1a8": "q=f(\\mathbf{aa})+ \\frac{1}{2}f(\\mathbf{Aa})= \\mbox{frequency of a}", "e4898aa266fad088c379980cd3fbdb59": " \nP_{n1} = {1 \\over (1+exp(-\\beta (z_{n1}-z_{n2}))} \n", "e489a48439da93e3d689d2382a561025": "x^2+xy+y^2", "e48a37b093b83cd8c0f1f5380f42e25d": "\\scriptstyle \\| z \\| \\;\\le\\; 1", "e48a3a28e4d010cfb80d4359f2e7ce17": "\\nu+1", "e48a5408f58b10dbc2503e1feb8af34f": "t=\\frac{1}{2 P}\\cdot C\\cdot(U_\\text{charge}^2-U_\\text{min}^2).", "e48a56c9993e391c72010c46b1ecbd02": "\\mathbf{r}^{(i)}", "e48a67547d75abf719b1e3b99516b394": "f^{-1}(x)=\\frac{1}{f(x)}", "e48a9a2aebb27f9c952c49d58b477ea5": "d_1d_2.", "e48b39b07554a9c13746b21dbd7ee957": "|I|=d+1", "e48b5089fb1712ccadebfa8a46d31473": " \\mathbf{x}_i \\quad (i=1,2,\\dots,n)", "e48b5b247dc207a9a7f3d84970375304": "f_Z(z) = \\frac{1}{(z + 1)^2}", "e48b693a8014766ee3a49b94a53891e2": "\\angle POS = m \\text{ arcminutes, } s \\text{ arcseconds, } t \\text{ sixtieths of an arcsecond}", "e48b8d8aeb4bd86b6263625a7f3a559d": "\\pi: F(k[\\epsilon]/(\\epsilon)^2) \\to F(k)", "e48b9aa9720202b7c61fd60776f268bc": "\\frac{1}{h_1^2}+\\frac{1}{h_3^2}=\\frac{1}{h_2^2}+\\frac{1}{h_4^2}", "e48bb91e4028caa0bdfe7a8a72fded2d": " E = -\\boldsymbol{\\mu} \\cdot \\mathbf{B}_0 = -\\mu_\\mathrm{x} B_{0x}-\\mu_\\mathrm{y} B_{0y}-\\mu_\\mathrm{z} B_{0z} .", "e48be0a43a1e0e65a0c7d7c19f270bd3": "\\tfrac{\\partial}{\\partial x^i}", "e48caa8aa8594d1251198ef87acf297e": " \\langle \\sigma_i \\rangle = \\psi^\\dagger\\sigma_i\\psi = R_i", "e48cd3188e49d1bdd20c198bb28cd8a9": " f(0) = 0 \\ ", "e48d062c305696f38ff130700377c7d9": "ds^2 = -dt^2 - 2 h(r) r \\, dt \\, d\\phi + (1-h(r)^2) r^2 \\, d\\phi^2 + \\frac{dz^2 + dr^2}{f(r)^2}", "e48d0acc8de6dfa3f24caef809bf7f94": "{| \\psi\\rangle} =\n\\sum_{k}c_{k}{\\hat{a}_{k}}^{\\dagger}| 0\\rangle ", "e48d7282ed7c49b8bd21b0b5afacdfc6": "v_e = \\sqrt{\\frac{2GM}{r}},", "e48d8d9ae44bb01b3072b38157732779": "\n\\rho = \\frac{1}{2}( \\sigma_0 + x^1 \\sigma_1 + x^2 \\sigma_2 + x^3 \\sigma_3 )\n", "e48e83a9f8b1e5b0eef43a3e09703957": "Y[]", "e48eba56e85eb7e574eef75c528ccfd4": "\\text{request}(ts, i)", "e48f4a6a241355e0e42ac9eb47a7af7a": "\\begin{align}\n271^3 + 2^3 3^5 73^3 = 919^3 &= 776,151,559 \\\\\n3^4 29^3 89^3 + 7^3 11^3 167^3 = 2^7 5^4 353^3 &= 3,518,958,160,000 \\\\\n\\end{align}", "e48f5f5c416f95e205b1a0a1f246b782": " L_u=2L_v=2L_w=2500 \\text{ft}", "e48f5f7df49da5f2eb077338bf9fb8c1": "= \\frac{e^4}{(k-k')^4} \\left( \\bar{v}_{k'} \\gamma^\\mu v_{k} \\right) \\left( \\bar{v}_{k} \\gamma^\\nu v_{k'} \\right) \\left( \\bar{u}_{p} \\gamma_\\mu u_{p'} \\right) \\left( \\bar{u}_{p'} \\gamma_\\nu u_p \\right) \\,", "e48f9e7c16f7366534d685ddc361fca7": " n_1, n_2 ", "e48fd1db3cde7745907a37f248e51409": " \\oint_C x^{\\alpha-1} \\, d\\alpha = 0, ", "e4901309f441f26948a2f7990cdf1ae1": "\\pi = 3 \\tfrac{1}{8}", "e490252d7f90152e1d8e0cd0a5ffd17f": "\\xi\\varphi", "e4910f8bdaa4bae5da617eaae74085c7": "\\displaystyle{\\left(-\\overline{b} z + \\overline{d}\\right)^{1\\over 2}}", "e4912a4c5c7a24d2f7308a61c312f5f8": "\\mathrm{d}s = \\sqrt{g_{\\alpha\\beta} \\mathrm{d} x^\\alpha \\mathrm{d}x^\\beta}", "e491c6775208a72fea74f778759e2e54": "x\\in A\\iff L[S,x]\\models\\phi(S,x)", "e491ec7ddd2731ca4b2e2dfd6873c428": "(\\mu/\\rho)", "e4921cd078c7081f88a6390621c636a1": " \\operatorname{build-list}[\\lambda x.x\\ (q\\ q\\ x), D, L_2] \\and D[p] = [q, \\_, \\_]::L_2", "e49242708afb085d2445b95a8570c594": "\\operatorname{Cov}(X_i, X_j) = \\operatorname{E}(X_iX_j) -\\operatorname{E}(X_i)\\operatorname{E}(X_j)", "e4924408156380c96f17f3c2fbf5d112": "\\mathbf{u} \\oplus_E \\mathbf{v}=\\frac{1}{1+\\frac{\\mathbf{u}\\cdot\\mathbf{v}}{c^2}}\\left\\{\\mathbf{u}+\\frac{1}{\\gamma_\\mathbf{u}}\\mathbf{v}+\\frac{1}{c^2}\\frac{\\gamma_\\mathbf{u}}{1+\\gamma_\\mathbf{u}}(\\mathbf{u}\\cdot\\mathbf{v})\\mathbf{u}\\right\\}", "e492678f7541d604fe50fa99ffa4499a": "\\omega \\to 0", "e492be1c412e5665bc509eabdf22a4ae": "\\tfrac{18-\\sqrt{30}}{36}", "e492c9bc6e7b2e163bb01d5a592cce37": "\\operatorname{Hol}(G)", "e492e5c09d17ebc86264b6f9ad10d50b": "\\mathbb{Z}/27\\mathbb{Z}", "e493550e95679672fd0e96cf42ab8c89": "v=FV=F(I-C)^{-1}Dp=ADp", "e493921590f41795ccc52eacbcdd0725": "B^A", "e493aa7030478b6fb1b48ad8f7a7eae4": "\\Delta M_J = 0, \\pm 1, \\pm2", "e493b9968ba17eb08fdc5e7150db5209": "(v,h)", "e493bf6d8a14a9b332e4e45fdb7983e7": "\\{1\\}\\times[0,h]", "e4940dc1ebd999007cb2df76ab70dfbb": "\\lambda=a e^{-\\lambda}", "e49418e7474e735cb866dfc2cdf46054": " B_2 ", "e4942e58bed8b0a67e299b05862228df": " (a,b) \\mapsto a \\otimes b ", "e49446c6a2339a08f88ca2df2c3414db": "f(t,y) = y", "e494c8def7a885b5ffa6e6da11a54070": " 1 = g_{n_1} g_{n_2} \\cdots g_{n_t} ", "e4950e8aa64356813a66d62e17103fa3": "\\lim_{r\\to\\infty}\\int_{|x|>r}\\left|f\\right|^p=0", "e4957a1408e1404487e59f4151e1d122": " a,\\,0 < a < m", "e495848e2f3530c2b36e75bd24c61875": "A = \\| \\vec{u} \\| \\| \\vec{v} \\| \\sin \\theta = \\sqrt{\\| \\vec{u} \\|^2 \\| \\vec{v} \\|^2 (1 - \\cos^2 \\theta)}", "e4958521aa134cf242c398544e2ca950": "\\mbox{std} \\frac{K \\cdot t}{V} \\ \\stackrel{\\mathrm{def}}{=}\\mbox{ const} \\cdot \\frac {\\dot{m}}{V} \\frac{1}{C_o} \\qquad(8)", "e4958b261125844451f269e498008656": "g: Y \\to Z", "e495b782372ae9b997f00f0bbd7a8a00": "c\\in\\mathbb{C}", "e495c5a4b0ccdeda18b9443a06edca0f": "\\bar{E}", "e495df56ef43da75da0e43f27318a55f": "-\\nabla^2\\mathbf{H}=\\nabla\\times\\mathbf{J}.", "e495e991cb238695d8e59152b7eb76a4": "\\theta(ab)=\\varepsilon(a)\\theta(b) + \\theta(a)\\varepsilon(b),", "e4961de98d5d23f562c6f676a9c315ea": "\\|X - \\mu\\|_\\alpha > 0", "e496616f2eef11574002e1c4f1f5a413": " A W_c + W_c A^T = -BB^T", "e496700d4d6ff9e76d5942886e90c3d0": "t \\rightarrow t+t^3/t_0^2", "e4967a57312c1257eca6dad86a6da163": "\\tau_{ki}", "e496b621c93b65ab3e84bdf22976d899": "\n\\begin{align}\n2(x_1,y_1) & = (x_1,y_1)+(x_1,y_1) \\\\[6pt]\n2(x_1,y_1) & = \\left( \\frac{2x_1y_1}{1+dx_1^2y_1^2}, \\frac{y_1^2-x_1^2}{1-dx_1^2y_1^2} \\right) \\\\[6pt]\n& = \\left( \\frac{2x_1y_1}{x_1^2+y_1^2}, \\frac{y_1^2-x_1^2}{2-x_1^2-y_1^2} \\right)\n\\end{align}\n", "e49736f09a17efd3daec360132426f43": " a ", "e4974751a40b24ac9bf300b666643398": "\n e_t = e + \\frac{u^2 + v^2 + w^2}{2} + gz\n", "e4977140998322825522c666e0bd583b": "h = R(1-\\cos\\frac{\\theta}{2}) = R - \\sqrt{R^2 - \\frac{c^2}{4}} ", "e497881913debd82e89d5812a962fb4b": "d J_S(t)=d_j S(t)-E[d_j S(t)]=S(t)-S(t^-)-(h(S(t^-)) \\int_z z \\eta(S(t^-),z) \\, dz) \\, dt.", "e497898a353fd26b2d180597ba92a45f": "\\mathrm{Lie}_q\\,\\mathcal{F}\\subset T_q{\\mathcal O}_{q_0}", "e49795f07d620ac836a8c869529ab363": "P(R) = 0", "e49808e70bb8404d89dfc0aa541b4cca": "\\lambda_1=\\lambda, ~\\lambda_3=1", "e4980d4473a4f7a4906805cee3df98dd": "\\frac{e}{m} = \\frac{v}{B\\cdot r}", "e498134286def91f9d7f72e0a751f4e6": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\frac{\\partial L}{\\partial \\dot{x}}=\\frac{\\partial L}{\\partial x}", "e4985072230d9b4f61a4b39905e6c3f7": "\\sin^2\\alpha/2\\,", "e498a227f7819719977bc128b7855e85": " W[k,n] = \\alpha - \\left(1 - \\alpha \\right) \\cos \\left( \\frac {2 \\pi n}{N[k]} \\right), \\alpha = 25/46 , 0 \\leqslant n \\leqslant N[k] - 1", "e49901596c4a9c74140aa28afcd0c83f": "\\phi'=\\phi - \\langle \\phi \\rangle", "e499160731c14537eec5b86b2bfc5337": "E_\\text{up}=\\frac{4}{3}\\pi\\sqrt{2}U^{1/4} \\left(-\\frac{m}{g}\\right)N^{3/4},\n", "e499b8bf4e63f2de5134d66cde2c86f3": "= \\frac{\\text{mass}(\\text{lb})}{\\left(\\text{height}(\\text{in})\\right)^2}\\times 703", "e499ea5d0a5ff49bec79fae225c3db5d": "a \\cdot b = 1 \\ast \\phi^{-1}(a) \\phi^{-1}(b)", "e499f3e8cfe88937d835bbf534a73add": "\\mathcal{V}[\\eta ,\\eta ^{+}] =-\\ln Z[G_{0}^{-1} \\eta , G_{0}^{-1}\\eta ^{+}]-\\eta G_{0}^{-1}\\eta ^{+}", "e49a9c7efda0e1d19d88abacf867eacd": "\\sqrt{s(s-a)(s-b)(s-c)}\\,", "e49aa518e6f785dc59e712b4e4be6284": "\\Omega > \\Omega_{gp}", "e49acc5c87ea454cba7559fd64f81592": "f^{-1}(D) \\,", "e49acd145c416754a7696872744b6653": "R\\overset r{\\underset s\\rightrightarrows} X\\to Y", "e49ad34eb145af260eb62c50e3c264ee": "\\frac{d}{\\sqrt{2U}}", "e49b1df15d04f6d41e2ead37807ae7f9": "\\mathbf{B} = \\mu \\mathbf{H},", "e49b531b0a1d02ff5fc69f3f78ba1e4f": "\n\\begin{align}\n\\frac{S}{[\\#]} & \\to \\frac{AD}{[\\#]} \\to \\frac{aAcD}{[\\#]} \\to \\frac{aaBccD}{[\\#]} \\to \\frac{aabBccD}{[\\#f]} \\to \\frac{aabbBccD}{[\\#ff]} \\\\\n & \\to \\frac{aabbbccD}{[\\#fff]} \\to \\frac{aabbbccdD}{[\\#ff]} \\to \\frac{aabbbccddD}{[\\#f]} \\to \\frac{aabbbccddd}{[\\#]} \\\\\n\\end{align}\n", "e49b577fde6b795e7a38fe9d3072d457": "X : [0, T] \\times \\Omega \\to \\mathbb{R}", "e49b7ef192c405cb451d08af763d6798": "\\xi=\\frac{N-N_0}{k_0 A},\\,\\,\\,\\eta=\\frac{E-E_0}{k_0 A},", "e49b80bff75f51bb2956889b39de1699": " |\\Psi\\rangle ", "e49b950f681eb6409ac33bc0d1fc7ead": "\\{e_1\\equiv z_{xy}-\\frac{x^2}{y^2}z_x-\\frac{x-y}{y^2}z=0, e_2\\equiv z_x+\\frac{1}{x}z_y+xz=0\\}", "e49ba494692424f3798247f79d6dec49": "C(SU_{\\mu}(2))", "e49bbfc5762bb61a3b093f7c1d962907": "var_{01}(p)", "e49bd3c99722894416c156be35bbea2e": "(a+b-c)", "e49bd3d87d1ec07e38d179f6a3856f5a": "e^- + e^+ \\longleftrightarrow \\nu_e + \\bar{\\nu}_e", "e49bee45ad51a433670f32db27cebf85": "x_i = x - \\lambda_i", "e49c1f8b3efbe1b415c93a230a8d5cf1": "w_1,\\ldots,w_N>0.", "e49c9791913f7bd92f667db6176e080f": "\\! \\dot{\\phi}", "e49cbc0d5accc4746233c36010ab6459": "M^{-1}", "e49cd1b484c49153a43a3fbb68f5c3a1": "30<\\frac{\\delta}{1-\\delta}35\\Leftrightarrow\\delta>\\frac{6}{13}", "e49d07f536f7408daf6a81456f9936e0": "f(z)=u(z)+iv(z)", "e49d1b75ae50aa11ca7307027895e329": "C_\\mathrm L = {\\frac{L}{\\frac{1}{2}\\rho v^2S}} = {\\frac{2 L}{\\rho v^2S}} = \\frac{L}{q S}", "e49d28afdb287af2b7affab9f821e76f": "\\vec{\\theta} = \\theta \\hat{\\theta}", "e49d2ecb3b59004b07d0993c0279c865": "\\tau(n)\\equiv n^{-610}\\sigma_{1231}(n)\\ \\bmod\\ 3^{6}\\text{ for }n\\equiv 1\\ \\bmod\\ 3", "e49d3152fd28982063d0028808f7f1ef": "T(O_{r}^{*})", "e49d5446a9cdf76884eb8e5224aafa35": "J^\\prime = J^{\\prime\\prime}-1 ", "e49dcc133027ba894644133d258d0492": "r=\\frac{a \\sin \\theta}{\\theta}", "e49dd866daf5e9c779f960e03fb561cd": "\\Pi(t)", "e49df2ee27d9e5b1f192ebedae467947": "f(x) = \\sqrt{x}", "e49df65fb79c09a2c04f21fcf302171f": "\\varphi \\left( {0,x} \\right) = \\left( {{x \\over {\\Delta x}} - {1 \\over 2}} \\right)^2 - {1 \\over 4}, \\quad \\varphi _j^0 = \\left( {j - {1 \\over 2}} \\right)^2 - {1 \\over 4} . \\quad \\quad (12)", "e49e464c0304f76299d63326a773d98f": "DO_{crit} = DO_{sat}-D_{crit}", "e49e4bba63b61f752dd8a89977ba718f": "\\mathbb{T}^1", "e49e5436e63dc10bc7aadb4208eb1e85": "\\left [\\begin{smallmatrix}2&-\\sqrt{2}\\\\-\\sqrt{2}&2\\end{smallmatrix}\\right ]", "e49e882ff5eaa0160af1f6881135da70": "\\frac{\\partial ^2C_i (q_i)}{\\partial q_i^2}=0", "e49e92a8a7f8ecd1d08c29fe00f80d80": "|A + A| < c|A|\\,", "e49f2aadd1fcf7615933c655ad9bb9d6": "\\mathbf{L}^2 = L_x^2+L_y^2+L_z^2", "e49f32bfd52d4a4c87f2dde38681ea9c": "K \\supseteq \\mathbb{R}^d_+", "e49f3b43f2de0d090db7271784afd736": " \\rho(\\boldsymbol\\beta|\\sigma^{2}) \\propto (\\sigma^2)^{-k/2} \\exp\\left(-\\frac{1}{2{\\sigma}^{2}}(\\boldsymbol\\beta - \\boldsymbol\\mu_0)^{\\rm T} \\mathbf{\\Lambda}_0 (\\boldsymbol\\beta - \\boldsymbol\\mu_0)\\right) ).", "e49f9cd2365ce6042081c4abaaacd3ba": "\\Lambda^2\\mathbb C^6", "e4a02cce7cb3e73b37538c7d079b01cc": " U_E(r) = k_e\\frac{qQ}{r}", "e4a04245e4c2538cb6258a9fc01b0a9a": "\\sum_{i=1}^n \\mathrm{Geometric}(p) \\sim \\mathrm{NegativeBinomial}(n,p) \\qquad 0\\kappa} = \\{Y \\subset X \\; | \\; |Y|>\\kappa\\}", "e4a2770d5aa5e2f701172691fc0ca3b7": "a = \\hbar / Z \\alpha m_\\text{e} c ", "e4a2bec1381f73dfa4e5be4adb79495d": "q(D,\\widehat{D})\\geq-\\alpha/2\\,\\!", "e4a2c3e5fded0402618bbd027b3503fc": "T=\\frac {(L- 0.25P) \\cdot P \\cdot \\sqrt{S}} {130}", "e4a326f7ce0c7860e43e4efa84bdb55e": "f\\colon c_1\\rightarrow c_2", "e4a34078ac14120a148dc3af3b29106a": "q \\geq (\\zeta/\\sqrt{n}) \\omega \\sqrt{\\log{n}})", "e4a39919e762775a413381b94a3c36e0": " S(\\Psi) = \\hbar \\sum_{g \\ge 0} (\\hbar g_c)^{g-1} \\sum_{n \\ge 0} \\frac{1}{n!} \\{\\Psi^n \\}_g ", "e4a3ae8fe05c0c4adee4dbbac594bdf7": "f(t) = f_{0} \\sin 2\\omega_{p}t", "e4a42e65de84702747833207c76c120b": " {\\rm sinc}_{\\rm C}(x, y) = {\\rm sinc}(x) {\\rm sinc}(y)", "e4a4b5d86c8e848b5f8f1149e1904fb1": "S_{y_U}", "e4a4c4eeebd8e0b36282abebc46d65e2": "P_{A,B,\\Lambda}", "e4a551b8ce98ab3161dc54b1840f9d98": "\n \\begin{align}\n Q_1 & = \\frac{\\partial \\mathcal{M}}{\\partial x_1} \n + \\frac{D(1-\\nu)}{2}\\left[\\mathcal{A}\\frac{\\partial }{\\partial x_2}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2}\n -\\frac{\\partial \\varphi_2}{\\partial x_1}\\right)\\right] - \\frac{\\mathcal{B}}{1+\\nu}\\frac{\\partial q}{\\partial x_1} \\\\\n Q_2 & = \\frac{\\partial \\mathcal{M}}{\\partial x_2} \n - \\frac{D(1-\\nu)}{2}\\left[\\mathcal{A}\\frac{\\partial }{\\partial x_1}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2}\n -\\frac{\\partial \\varphi_2}{\\partial x_1}\\right)\\right] - \\frac{\\mathcal{B}}{1+\\nu}\\frac{\\partial q}{\\partial x_2}\\,.\n \\end{align}\n", "e4a5d18d3c5c7dfa72afefc068a2ee25": "\\mathbf{R}(t)", "e4a61103316a888c0c48c4d2e449a87b": "u\\sim v", "e4a69d1a7228846f5c125fdd8756b23f": "\\mu\\boldsymbol{\\epsilon}'=2(\\mathbf{r'}\\cdot\\mathbf{F})\\mathbf{r}-(\\mathbf{r}\\cdot\\mathbf{F})\\mathbf{r'}-(\\mathbf{r}\\cdot\\mathbf{r'})\\mathbf{F}", "e4a6a5c7e415464b4f957be8117d0d4f": " \\operatorname{lift-choice}[M\\ N] = \\operatorname{lift-choice}[N] ", "e4a6ba3377309c0881164403b92fd32c": "\\operatorname{cont}_F", "e4a7504747014e65316c8e0dc6015020": "M_p(x_1,\\dots,x_n) = \\exp{\\left( \\ln{\\left[\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)^{1/p}\\right]} \\right) } = \\exp{\\left( \\frac{\\ln{\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)}}{p} \\right) }", "e4a7680b27d31301637e700e0489d805": "E_{N}", "e4a7c7e9c46b222c0296febcb01b427a": "t_{half}\\;=\\;t_2\\;-\\;t_1\\,\\!", "e4a7dc1c7114e54a9ea029007c230485": "s_i =\\sum \\, {s_{k}^{i}}{\\text{for all secrets properly shared}}\\mod n", "e4a84f96d1aa5300c9ac35e77b0a5769": "K \\subset L^G", "e4a86e5a2d32c6bcdb84abd506b6bbfb": "(a g_1 a^{-1}, \\ldots, a g_k a^{-1})", "e4a8924b047b42384e83328a13ed2d7a": "S_m=\\int L(\\tilde g_{\\mu\\nu},f^\\alpha,f^\\alpha_{|\\mu},\\ldots)\\sqrt{-g}\\,d^4x\\;", "e4a896ee9a17e9c9b5474120ccb563a3": "T = \\frac{\\bar{V}}{s_V/\\sqrt{n} } \\sim \\text{noncentral } t(n-1, \\sqrt{n}\\lambda). ", "e4a89a649c726894d7c5047ecf59e2d9": " u \\to u^{h'} ", "e4a8a62d43b7e42df1a8cc02d3699495": " r_1 = \\int_0^\\psi p(\\psi ') \\, d \\psi ' = \\frac{\\psi}{2 \\pi} ", "e4a8b1c8498c823ea58af4c34e276092": "P(Q)", "e4a8c40487338b4f5a4a879f405da4c1": "x=1.", "e4a904dbd0fe7602ab840c27aa8f076e": "i=0, \\dots , \\infty .", "e4a94ae29198ed69e3ffc9216e84e995": " (1-p)^n\\, ", "e4a94bd21096314a69420a2c2a1847b4": "f(x) = \\begin{cases}\n x^2 & \\mbox{ for } x < 1 \\\\\n 0 & \\mbox{ for } x = 1 \\\\\n 2-x & \\mbox{ for } x > 1\n\\end{cases}", "e4a9a257c528c7f2e9c2ac505accbd38": "R_n^2(\\xi,x)= 1", "e4a9b30218486aba4ee2edaafdaf17a7": "k_L\\,", "e4aa55ddc4d7afb0c8039adef2f0cf9a": "\\begin{align}\n\\left[\\sigma_1, \\sigma_2\\right] &= 2i\\sigma_3 \\,,\\\\\n\\left[\\sigma_2, \\sigma_3\\right] &= 2i\\sigma_1 \\,,\\\\\n\\left[\\sigma_2, \\sigma_1\\right] &= -2i\\sigma_3 \\,,\\\\\n\\left[\\sigma_1, \\sigma_1\\right] &= 0\\,,\\\\\n\\left\\{\\sigma_1, \\sigma_1\\right\\} &= 2I\\,,\\\\\n\\left\\{\\sigma_1, \\sigma_2\\right\\} &= 0\\,.\\\\\n\\end{align}", "e4ab3c28b2f3c7793f548cd42414ff0e": "v=\\int\\left[\\int\\left(\\frac{M}{EI}\\right)dx\\right]dx", "e4abe22f405fa433d6b178421442d56d": "p_{01} \\leftarrow x^3+x^2-2x-1, M_{01}\\leftarrow \\frac{2x+3}{x+2}", "e4ac13b365dded002bcdd92b2c09a5d1": " \\and S_8 \\implies A_8 = q ", "e4ac7dcc30e5521672ada87d80478ee4": "H(x)=H_0(x)+fx", "e4aca154c234f48a76312840c7cf6dc2": "\\,f\\,", "e4ad3618f10b94e7ce364e789c6ab974": " f(k)/k ", "e4ad79d953fac51da5f907179d6dd335": "MD(\\Box ( \\Box p \\rightarrow p )) =", "e4ade929cf5503694bbb995e0b04b9a4": "V^p_n(R) = (2\\Gamma(\\textstyle\\frac{1}{p} + 1) R) \\displaystyle\\frac{\\Gamma(\\frac{n-1}{p} + 1)}{\\Gamma(\\frac{n}{p} + 1)} V^p_{n-1}(R).", "e4ae05ab271120c4841b17a694fd2e50": "\\sigma \\circ \\iota_{p, p+1, ..., p + q}", "e4ae08abcd6c52772b260dbb13c6f6c6": "\\mathbf{B}\\in M_{n\\times m}(\\Re)", "e4ae236e6402337b6105ecf5fd0cd457": " = \\operatorname{tr} \\left(\\gamma^\\mu \\gamma^\\nu (2\\eta^{\\rho \\sigma} - \\gamma^\\sigma \\gamma^\\rho ) \\right) \\,", "e4ae66e8e3ebb54403a369f84664bbc0": "\\vartheta(x)=\\sum_{p\\le x} \\log p", "e4aeb37622dae7e596b65100e02f1d2f": "f(n) = \\sum_{1 \\le m \\le n}\\mu(m)g\\left(\\left\\lfloor \\frac{n}{m}\\right\\rfloor\\right)\\quad\\mbox{ for all } n\\ge 1.", "e4aeb66e3236aea5664f28a46864b861": "h(Q_{Y|X}(\\tau))\\equiv Q_{h(Y)|X}(\\tau).", "e4aed0a3157d8f4dccec05e045ed7f78": "\\int e^{cx}\\cos bx\\; \\mathrm{d}x = \\frac{e^{cx}}{c^2+b^2}(c\\cos bx + b\\sin bx)", "e4af59635a5731c6e1d68320f6ffd704": "S_0 ' = 40 - 1.3541 = 38.6459 ", "e4af732c596b62bacc752ae86e9c8b3b": "X = (X_1, \\cdots, X_K)\\sim\\operatorname{Dir}(\\alpha_1,\\cdots,\\alpha_K)", "e4afba6f3c4dfb84546d1ca10b15c306": "\n\\begin{align}\n\\operatorname{var}(\\text{mean})\n &= \\operatorname{var}\\left (\\frac{1}{N} \\sum_{i=1}^N X_i \\right)\n = \\frac{1}{N^2}\\operatorname{var}\\left (\\sum_{i=1}^N X_i \\right ) \\\\\n &= \\frac{1}{N^2}\\sum_{i=1}^N \\operatorname{var}(X_i)\n = \\frac{N}{N^2} \\operatorname{var}(X)\n = \\frac{1}{N} \\operatorname{var} (X).\n\\end{align}\n", "e4aff5d8f2f3cbe6543c78b95cb09037": " g(z, u) = \\exp\\left( - u z + u \\log \\frac{1}{1-z} \\right) =\n\\exp(-uz) \\left( \\frac{1}{1-z} \\right)^u.", "e4b00b4a65a415cf9ebaa9f83719c071": "(2)", "e4b079f6d4d2481de7439cf5b09bcbb8": "\\mathrm{Re}_D \\gtrsim 10\\,000", "e4b0cb272bb6f95887468204ee35fa0b": "(T^n)", "e4b0d5e00d082fe21f07fd457898d95d": "|x|_p = 1\\,", "e4b0dfab5671e037829986364c31e820": "a_n = O(\\alpha^n)", "e4b0e2697bff098a31f454461ec3c5bd": "\\begin{align}\n\\sum_{k=1}^n(X_{t_k}-X_{t_{k-1}})^2&\\le\\max_{k\\le n}|X_{t_k}-X_{t_{k-1}}|\\sum_{k=1}^n|X_{t_k}-X_{t_{k-1}}|\\\\\n&\\le\\max_{|u-v|\\le\\Vert P\\Vert}|X_u-X_v|V_t(X).\n\\end{align}", "e4b118a23baee7c2f090ae5b4536eb55": "\\mathbf{C} = { 1 \\over {n-1} } \\mathbf{B}^{*} \\cdot \\mathbf{B}", "e4b14648802d32b0b47c8632f00a4be3": "\\Delta \\tau \\to 0", "e4b154d4f9211322979b8127e9d46429": "\\cot\\delta'=\\frac{1}{\\tan\\delta'}", "e4b1aa29681ebe666f0daa18cce4681e": "\\xi_n", "e4b2718c46a45ff5b59afbf8c87c4a75": "\\rho_j", "e4b289a41371cc00568cdf0ff478c4d0": "\\cos \\theta_c = \\frac{c}{nv}", "e4b2f28ba3c9f86a289e3c1bb0fe218d": "\nI =\n\\begin{bmatrix}\n \\frac{2}{3} m r^2 & 0 & 0 \\\\\n 0 & \\frac{2}{3} m r^2 & 0 \\\\ \n 0 & 0 & \\frac{2}{3} m r^2\n\\end{bmatrix}\n", "e4b2f73747f3a58d52a0a01d9f472d80": "G = \\langle x_{1}, \\dots, x_{n} | r = 1 \\rangle", "e4b349d43a7bae605a1fc86b77d98148": "J_i=\\psi_i^\\dagger \\sigma_x \\psi_i,\\ i=1,2 \\,", "e4b35277c698cba0e1bce91d740f5460": "U(t)=e^{iHt}", "e4b386cb824db949b21ca93538ef59a3": " M_r ", "e4b39108897b8ca1ef7f77cc6aea16d9": "\\mathbb{R}\\cup\\{\\infty,-\\infty\\}", "e4b3937c6a7a24c1cfc8fc5d89facb35": "(a+b\\sigma_1\\sigma_2)^* = a+b\\sigma_2\\sigma_1\\,", "e4b3c72b72bd32f819c8ecc477bc2c78": "\\lim_{n\\rightarrow\\infty}\\,\\sup\\{\\,\\left|f_n(x)-f(x)\\right|: x\\in\\mbox{the domain}\\,\\}=0.", "e4b4515a7f57353a387053e5adcea871": "\\begin{matrix} S(\\alpha) & \\xrightarrow{S(g)} & S(\\alpha')\\\\ f \\Bigg\\downarrow & & \\Bigg\\downarrow f'\\\\ T(\\beta) & \\xrightarrow[T(h)]{} & T(\\beta') \\end{matrix}", "e4b46950cfa15a6f9ea1b250d0724c9f": "\\boldsymbol{\\Sigma}^1_{n+1}", "e4b48113bc60038c01e31306e7ad69fe": "\\frac{\\partial y}{\\partial x} ", "e4b4968ce8829bc818588455f70ae768": "\n\n\\sigma=\\frac{1}{p(n)} \\frac{\\pi^{\\frac{n}{2}}}{\\Gamma(1+\\frac{n}{2})} \\frac{1}{c^{n-1}} \\frac{n(n-1)}{h^{n}} k^{(n+1)} \\Gamma(n+1) \\zeta(n+1)", "e4b4a4f9e22f15620052c6e1f68f2de6": "\\langle r,f \\mid r^n , f^2 , (rf)^2 \\rangle\\,\\!", "e4b4cf44f3eb4d9ed079787950aab6d4": "\\exists x\\, \\text{Phil}(x)", "e4b4d0a5e7bb33d2ee90bc3c002c23e9": " lf(x,y) = f(x_0,y_0)+(f(x_1,y_1)-f(x_0,y_0))s+(f(x_2,y_2)-f(x_0,y_0))t\\,", "e4b4ee030a8ae471a6805b9789e9b784": "(i \\gamma^\\mu \\nabla_\\mu - m) \\psi = 0", "e4b543fdfd60f3f1cc6ebbdf1fb8ff0c": "\n\\sigma = \\rho K \\frac{\\tilde{F}}{M} \\quad .\n", "e4b54c70ee262ebaeb79477205111c05": "x^5 - 1", "e4b58382be7f5ff9d967d017a65d2693": "I\\;", "e4b5a26f937b3e121ee5d54d46f3960d": " E_F", "e4b5ae7aa7dc13ddff9f94863152e020": "L_t=L_a e^{-k_1 t}", "e4b5b6e4d1b684ee6066724f0adcc3c0": " D(f)=b^2-4ac, \\quad D(f)\\equiv 0,1\\, (\\!\\!\\!\\!\\! \\mod 4). ", "e4b5eec98adb11ecbcd5ff61017b48c6": "x \\in X_{0}", "e4b61596796dd96097b8f17b69bb92af": "\\scriptstyle\\varphi\\colon \\mathbb{R}\\to \\mathbb{R}", "e4b63ff74f1436dc15a48f26bf7203ce": "E(m) = m \\oplus S(k)", "e4b69a8e7eb3ddaafe3f57d1fe87ad6e": "f^{-1}(s)", "e4b69b49680cb2c2e5ba8d5fcaf06793": " a'_{hk} = a'_{kh} = c a_{hk} - s a_{h\\ell} \\,\\! ", "e4b6a9d0e1ef6992a0ef24effbcd7163": "\\sqrt[d]{c} = c^{\\frac 1 d} = b^{\\frac{1}{d} \\log_b (c)}. \\,", "e4b6f1211e728deda9afdbefcb8f42a4": "\\mathbf{C}_j", "e4b718a5de8c48468aa0fe89fb59c500": " \\delta W = \\left(\\sum_{j=1}^m \\mathbf{F}_j\\cdot \\frac{\\partial \\mathbf{r}_j}{\\partial q_1}\\right) \\delta{q}_1 + \\ldots + \\left(\\sum_{j=1}^m \\mathbf{F}_j\\cdot \\frac{\\partial \\mathbf{r}_j}{\\partial q_n}\\right) \\delta{q}_n. ", "e4b766eaeb317aeef81e6828f2d26d46": "\\Gamma(t+1)=t \\Gamma(t).", "e4b7a4248864356d0f933146c0f6189e": "\\nu_p(m\\cdot n)= \\nu_p(m) + \\nu_p(n)~.", "e4b7e9123d78e32e57adf4b1625422e1": " T(\\epsilon) = 1 - {i \\over \\hbar} \\epsilon \\hat{p}", "e4b7f5f68613fa93d48ad2ef3d859840": "x +\\sqrt{y}\\,", "e4b7f873b23c37a4352146a80e82f053": " I = \\int \\left | \\mathbf{r} \\right | ^2 \\mathrm{d} m = \\int \\mathbf{r} \\cdot \\mathrm{d} \\mathbf{m} = \\int \\left | \\mathbf{r} \\right | ^2 \\rho \\mathrm{d}V \\,\\!", "e4b86377abe9cf6529975ca44894a790": "\n\\begin{align}\nF^{(0)}(s) & = 1, \\\\\nF^{(1)}(s) & = \\int^s_0 F^{(0)}(u) \\, du=\\int^s_0 1 \\, du=s, \\\\\nF^{(2)}(s) & = \\int^s_0 F^{(1)}(u)du=\\int^s_0 u \\, du={s^2 \\over 2}, \\\\\n& {} \\ \\ \\vdots \\\\\nF^{(n)}(s) & := \\int^s_0 {u^{n-1}\\over (n-1)!}du={s^n \\over n!}, \\\\\n& {} \\ \\ \\vdots\n\\end{align}\n", "e4b86f5c40542f6fc16bf0dc4b3031b7": "\\begin{align}\n \\operatorname{Pr}\\Big(\\lim_{n\\to\\infty}g(X_n) = g(X)\\Big) \n &\\geq \\operatorname{Pr}\\Big(\\lim_{n\\to\\infty}g(X_n) = g(X),\\ X\\notin D_g\\Big) \\\\\n &\\geq \\operatorname{Pr}\\Big(\\lim_{n\\to\\infty}X_n = X,\\ X\\notin D_g\\Big) \\\\\n &\\geq \\operatorname{Pr}\\Big(\\lim_{n\\to\\infty}X_n = X\\Big) - \\operatorname{Pr}(X\\in D_g) = 1-0 = 1.\n \\end{align}", "e4b876c0cd2642ce1cff4867725057d2": "\\sum_{i=1}^n \\left(\\,X_i-\\overline{X}\\,\\right)^2/\\sigma^2\\sim\\chi^2_{n-1}.", "e4b8cfddb7aa9759b9a5eca49093978a": "\\theta_\\mathrm{R}", "e4b8d4cc5b5f12e6589a50cff7cbeabb": " PV_\\text{perpetuity due} = PV_\\text{perpetuity immediate}(1+i) \\,\\!", "e4b8eee5bb1d449956f1ae0388022061": "\\Pr(\\sigma_i = +1) = \\Pr(\\sigma_i = -1) = 1/2", "e4b92512e100aac03fff17fc8ef7db0e": "w^i", "e4b92d86838b1b525bb2467efb50069c": "\\rho_{o}", "e4b930370a80758b8d6a66755f5186bb": "\\sigma = \n\\begin{bmatrix}\n\\sigma_{11} & 0 & 0 \\\\\n0 & \\sigma_{22} & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix} \n\\equiv \n\\begin{bmatrix}\n\\sigma_{x} & 0 & 0 \\\\\n0 & \\sigma_{y} & 0 \\\\\n0 & 0 & 0\n\\end{bmatrix}", "e4b990ad6b94539cbc87363af93b4b41": "wL=D_L[F(K,L)]*L\\,", "e4b9919773705bdac8170e18fea3375c": "\\sum_{n=0}^{\\infty} q^n = \\frac{1}{1-q}", "e4b9fb9420b4f9c5c079432de4c442cf": " \\begin{bmatrix} p_i, q_j \\end{bmatrix} = \\delta_{ij}z, \\qquad \\qquad\n\\begin{bmatrix} p_i, z \\end{bmatrix} = 0, \\qquad \\begin{bmatrix} q_j, z \\end{bmatrix} = 0~,", "e4ba50d204bb83ebf6b1a6b80b138957": "\\scriptstyle k_0\\,=\\,\\tfrac{1}{\\sqrt{2}}", "e4bb746a509bb88a75062767119cc653": "\n C_{ij}~\\epsilon_i~\\epsilon_j = C_{ij}'~\\epsilon'_i~\\epsilon'_j ~.\n ", "e4bb828c67a0a0116f5a71e7264c1a2f": "C_\\mathrm{max}= \\frac{n^2}{\\sin^2 \\alpha} \\ ", "e4bbfa03317b8a7b4daf200d6e1f0956": "Pr(\\ldots)", "e4bc1a17ff9a4b88dbec83b03f9246ed": "T/(u)", "e4bc36689d774b5fded0e1ea8bb32c38": "\\begin{bmatrix} 1&0&0&0 \\\\ 0&1&0&0 \\\\ 0&0&0&1 \\end{bmatrix} \\begin{bmatrix} a \\\\ c \\\\ b \\\\ d \\end{bmatrix} = \\begin{bmatrix} a \\\\ c \\\\ d \\end{bmatrix}", "e4bca0deaf55ff600f87812ceb55a723": " dv^2 = u^2 \\mp 2 ", "e4bca218d0bb2d9b67e1f7418c0acdfd": "|\\alpha|^2 + |\\beta|^2 = 1\\,.", "e4bcad65a3ee96da1adc781cf2dc982d": "T\\rightarrow 1 ", "e4bcc8af5be8e4bc82535c49a56ec349": "\n \\begin{array}{ccc}\n \\ln((1+r)^t) & = & \\ln 2 \\\\\n t \\cdot \\ln(1+r) & = & \\ln 2 \\\\\n t & = & \\frac{\\ln 2}{\\ln(1+r)}\n \\end{array}\n", "e4bcf78305020b687b7029b5b4d2bf84": "\\scriptstyle\\partial\\Omega", "e4bd28c6ef08af72ff18f26fe874bb0d": "\\Omega = \\Omega_1 (E_1) \\Omega_2 (E_2) = \\Omega_1 (E_1) \\Omega_2 (E-E_1) . \\,", "e4be302ac99708396235cb561d19f92f": "u=-\\mathcal{F}\\boxtimes_{n=1}^N\\mathbf{w}_{n}(p_n(t))\\mathbf{x}(t),", "e4be357be3c8b1e6fd743438f083d38a": "\\mathcal{E}^2", "e4be3bb960b9355c9f9b3312983378aa": "~K~", "e4be429772035f4f95ca99c6fd94be2c": "f^{n-1}(\\bot) \\leq f^n(\\bot)", "e4be5202beaa86f935ec6e898e51cd26": "V(I) \\cap V(J)\\,=\\,V(I + J).", "e4bee9b5c75fa6d29018f1c40a82bd28": "\\eta(s) = -\\frac{1}{d_n} \\sum_{k=0}^{n-1}\\frac{(-1)^k(d_k-d_n)}{(k+1)^s}+\\gamma_n(s),", "e4befbd6da04b76a0de443e323ef3e7f": " \\nabla^2 \\mathbf{A} - \\dfrac{1}{c^2}\\dfrac{\\partial^2 \\mathbf{A}}{\\partial t^2}=- \\mu_0 \\mathbf{J} +\\dfrac{1}{c^2}\\nabla\\left(\\dfrac{\\partial \\varphi}{\\partial t}\\right)\\,,", "e4bf1c71145658aa336a9c81a49f417c": "\\lim_{x\\rightarrow \\infty} F(x)=1\\,.", "e4bf83434bf42aa11156e8f666b8b7d6": "Q^\\text{out}_\\text{hot} = Q < \\frac{\\eta_M}{\\eta_L}Q=Q^\\text{in}_\\text{hot}", "e4bffe10cfffbfb837c6c723be1cae3c": "\\,_1F_1(a,b;z)=M(a,b;z)", "e4c03f1d6659472fbaed6f3fbe2e5dcf": "{f_{n-1} \\choose f_{n}} = M_{n}M_{n-1}\\ldots M_3{f_{1} \\choose f_{2}},", "e4c04fe59376c2fda66571e7107526f4": " Td^*(T {\\Bbb C} P^n) = (\\xi/(1-e^{-\\xi}))^{n+1},", "e4c0b101d6d2a4afeac181a5de42cc26": "0 \\ne w' \\in W", "e4c13baa3defd169a3fdd8b061d09b3f": "\n H_z= \\frac{1}{j\\omega\\mu} k_{xy}^2 \\cos k_x x \\cos k_y y \\sin k_z z\n ", "e4c16f9a50ad2faf5ff2d559a63b05be": "\\lim_{x\\to\\infty}\\frac{\\pi(x)}{x/\\ln(x)}=1,", "e4c194ff58ef86bbd5aff91bd3690e9e": "x < y", "e4c28e31d012dfa7f9bc73a26517e110": "\\mathbf{X}_t", "e4c2c5a68f9b72e95fb19a780d5bf10b": "A \\wedge \\lnot B", "e4c2e8edac362acab7123654b9e73432": "1.0", "e4c2ff90715577d07d8a994ec2740e34": "t_{ig}\\, ", "e4c3293be9eb5a90bd1cfdafe286aa5a": "\\frac{\\partial^2 \\mathbf{x}^{\\rm T}\\mathbf{A}\\mathbf{x}}{\\partial \\mathbf{x}^2} =", "e4c357e9a39eec3d391bdb87612c720c": "\\theta_a ,\\, \\theta_b ,\\, \\theta_c ", "e4c35f7211a4c4ba0364bf2593ee7fa9": "\nc^2=\\frac{\\partial p}{\\partial\\rho}", "e4c38acb3a7aab37fe5eb1718fd8a116": "\nA =\n \\begin{bmatrix}\n 1 & 3 & 2 \\\\\n 2 & 0 & 1 \\\\\n 5 & 2 & 2\n \\end{bmatrix}\n, \\quad\nB =\n \\begin{bmatrix}\n 4 \\\\\n 3 \\\\\n 1\n \\end{bmatrix},\n", "e4c3d0e8912829edf121d0ffecefff6a": " \\mathbf{r}_i = (\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{R}, \\quad \\mathbf{v}_i = \\omega\\times(\\mathbf{r}_i - \\mathbf{R}) + \\mathbf{V},", "e4c4135b88fbee27648b0a4ccbe9a875": "[0,b]", "e4c42a831c9607cfb945bfa9cd8681c8": "ax^2+y^2=1+dx^2y^2", "e4c46b76790f6b55f02bb06d4a0abb39": "\\begin{align} P(A |\\text{not }B) & = \\frac{P(\\text{not }B | A) P(A)}{P(\\text{not }B | A)P(A) + P(\\text{not }B |\\text{not } A) P(\\text{not }A)} \\\\ \\\\\n\n &= \\frac{0.01\\times 0.6}{0.01 \\times 0.6 + 0.95\\times 0.4} \\\\ ~\\\\ &\\approx 0.0155.\\end{align}", "e4c47c8f2a1bd7dee929edbf1bd80837": "\\lim_{\\Delta x\\to 0}\\left ( \\frac{\\Delta f}{\\Delta x}g(x_0) \\right ) = f'(x_0)g(x_0)", "e4c4a5b58e7c817bddb58558cbcea985": "aC_{in}^\\alpha (x) = a(x,\\alpha x) = \\left( {ax,\\alpha \\left( {ax} \\right)} \\right) = C_{in}^\\alpha (ax)", "e4c553f260efac0d002dc0f34587574b": "u_{0n}(r, t) = \\left(A\\cos c\\lambda_{0n} t + B\\sin c\\lambda_{0n} t\\right)J_0\\left(\\lambda_{0n} r\\right)\\text{ for }n=1, 2, \\dots, \\, ", "e4c55854979224c507a6c740069234ec": "\\prod^{\\infty}_{\\begin{smallmatrix} n = 1 \\\\ n \\mbox{ odd} \\end{smallmatrix}}(1-x^n)^{-1} = \\frac{1}{(1-x)(1-x^3)(1-x^5)...} = \\frac{(1-x^2)(1-x^4)...}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)...}", "e4c5a41b3f0329bd28e172a1dcd8ef9f": "C_2 \\times G", "e4c5e3e4e9d2a2f370c25812c9b284fe": "\\Delta x=\\int_{0}^{E_0}\\frac{1}{F}\\, dE", "e4c655e9603e1d4ade3067baf600469e": "p = \\frac{F}{L^2} ", "e4c65cba3eac8081b1f75166b6363b46": "\\psi ( \\cdots \\mathbf r_i,\\sigma_i\\cdots \\mathbf r_j,\\sigma_j\\cdots ) = (-1)^{2s}\\psi ( \\cdots \\mathbf r_j,\\sigma_j\\cdots \\mathbf r_i,\\sigma_i\\cdots ) .", "e4c666f5be85a330f3a13e90ee49a51d": " \\mu_i^* = \\frac{1}{r} \\sum_{j=1}^r \\left| d_i \\left( X^{(j)} \\right) \\right| ", "e4c6a9f883c5b933375659d594426436": "\\sigma(X_2)=X_1", "e4c6b77dd04672fc34b668367ff759e7": "d=\\sqrt{x^2+y^2}.\\!", "e4c712c0a7b2fdd4ccfdc882339ac5a4": " F(z)=f(z) M(a)^{{z-b\\over b-a}}M(b)^{{z-a\\over a-b}}.", "e4c715bfb625dbbbd5a3f1480d09d10b": "5\\ ", "e4c7426091fc1c67b06f655d21dcc02a": "\\mathrm{RM} = \\frac{e^3}{8\\pi^2 \\varepsilon_0 m^2c^3} \\int_0^d n_e(s) B_{||}(s) \\;\\mathrm{d}s \\approx\n (2.62 \\times 10^{-13}\\, T^{-1})\\, \\int_0^d n_e(s) B_{||}(s)\\; \\mathrm{d}s\n", "e4c8293a0f18242d81e212cb5c5d2a0d": "D(\\alpha)=\\exp(\\alpha a^{\\dagger}-\\alpha^* a)", "e4c83ee91c240ba9206dd0a2815a5d98": "A^*A = I_n\\,\\!", "e4c8436b68511fd339e15d7f9bec0258": "\\phi(x=0) = 0", "e4c8855f84930924b23964f5dbb2cd25": "\n \\mathbf{b}_i\\times\\mathbf{b}_j = J~\\varepsilon_{ijp}~\\mathbf{b}^p = \\sqrt{g}~\\varepsilon_{ijp}~\\mathbf{b}^p\n", "e4c8a81a9c6497adf4e799f0d3237ac6": "\\mathbf{M} = \\chi_\\text{m}\\mathbf{H}", "e4c92e2e89fede0bcbc72218eff72204": "\\textbf{S}(t_f) = \\textbf{S}_f", "e4c9e2d7732b4a8a92df5d2016010ea9": "\\left(a, y\\right) > \\left(b, x\\right)", "e4c9fd2fa63564f8b4babd10d1f70f50": " (\\partial U)_P=-(\\partial P)_U=C_P-P\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "e4ca464cd80e8423565c281a6162c178": "\\forall z \\forall x ( ( \\phi \\lor \\psi) \\rightarrow \\rho)", "e4ca4a63eb15158282bd26610b1cbadb": "L\\left(C\\right) = \\sum_{i=1}^{n}{w_{i}\\times\\mathrm{length}\\left(c_{i}\\right)}", "e4ca603766e511a5493c0646a0a1fbe7": "q_\\text{inv} = q^{-1}\\text{ (mod }p\\text{)}", "e4cad415fc3634e43b215edbad38a7c2": "P_1=(2:1)", "e4cb1532c8ca3b81e9d74fe74c10f3cb": "\\sin(\\theta^1(t))\\cos(\\theta^2(t)) = \\frac{1}{2}\\sin(\\theta^1(t) + \\theta^2(t)) + \\frac{1}{2}\\sin(\\theta^1(t) - \\theta^2(t))", "e4cb36e3f1ecc15a7cfc7a12989040d1": "\\alpha_{avg.}", "e4cb3779f3a82e12a90199949ba650a7": "y=x^{p/q}", "e4cb3d3f434dd27f3ab17f9aaa1c91a0": "\n\\text{minimize} \\quad \\text{over } \\widehat D \\quad \\|D - \\widehat D\\|_{\\text{F}}\n\\quad\\text{subject to}\\quad \\operatorname{rank}\\big(\\widehat D\\big) \\leq r\n", "e4cb6b5cf7aef906222214d357f2a4bb": "4 p d \\le 1", "e4cb86e362850cd188170ed88e15519c": "p_0(x)=x^3.", "e4cba1976764ec3b7a8f65e74ae0b713": "K_X + r D", "e4cbdbfd34a09318cc4984a467ec6ad5": "\\mathit{E_p}", "e4cc2ae0b604e377175c3fa7834d850a": "\\sigma_\\mathrm e\\,\\!", "e4cc3e1b5d4447e6af71648f716d75a6": "\\mathbb{R}^{n+1}", "e4cc4e3f41ee4ac70290d99df7c018f9": " \\sin(2 \\alpha) = 2 \\sin(\\alpha) \\cos(\\alpha). \\, ", "e4ccdf84e36ee8167dabf317e4327256": "\\partial_\\mu\\omega", "e4ce7e4bcd910a164cb5918890a372cd": "\\sum_{k=1}^q w_{k}=1", "e4ce92025038f23b4702a03f6fd73f7a": "g:U\\to \\mathbb C", "e4cec09411aa808486eb0336888372fe": "S_n = X_1+\\cdots+X_n \\, ", "e4ced39d2a2ffdfc73935971ddd173ab": "\\rho(\\vec x)=0", "e4cedfb55f73314f9e05eb34fcfa9d10": "\\dot Q_H", "e4cefd1f417cbb8ce6f24d9c55204d1f": " u = \\nu + \\omega ", "e4cf1c98844e8d7b374a41d6f9b6868e": "A \\cong k[[x_1, \\ldots, x_d]]", "e4cf4a178ccae6190ba4c6b45f29cbd5": "\\begin{align}\np(y_i|c_k) & = \\sum_j p(y_i|x_j)p(x_j|c_k) \\\\\n & =\\sum_j p(y_i|x_j)p(x_j, c_k ) \\big / p(c_k) \\\\\n& =\\sum_j p(y_i|x_j)p(c_k | x_j) p(x_j) \\big / p(c_k) \\\\\n\\end{align}", "e4cf51f6a2b8efd5feb4d532b0600275": "\\Pr_x[C(x, y) = f(x)]", "e4cf64dbd7b23224977038bdcc7ace34": "E=<1,0>", "e4cf76be5e6367578038bf53910490bd": "(\\iota_v \\omega)(w)=\\omega(v,w)", "e4cf85821879d9d90fb70804058ccaf3": "\\left ( \\frac{\\partial U}{\\partial V} \\right )_S = -p", "e4d030d34d61d0e91860aa2dcf69a18c": "J(v,w) := (-w,v),", "e4d03b4f453d3f1883523079f7f2643a": " \\frac{1}{\\sqrt{\\sigma^{2} + \\tau^{2}}} {\\partial f \\over \\partial \\sigma}\\hat{\\boldsymbol \\sigma} + \\frac{1}{\\sqrt{\\sigma^{2} + \\tau^{2}}} {\\partial f \\over \\partial \\tau}\\hat{\\boldsymbol \\tau} + {\\partial f \\over \\partial z}\\hat{\\mathbf z}", "e4d04015bfc6575fcb15685459650bd7": "m \\mapsto 1 \\ast m", "e4d0708847d0882d5e82e7b4f3d8a983": "\\textrm{SINR}^{\\mathrm{uplink}}_k = \\frac{q_k|\\mathbf{h}_k^H\\mathbf{v}_k|^2}{1+\\sum_{i \\neq k} q_i |\\mathbf{h}_i^H\\mathbf{v}_k|^2}", "e4d090ef9b554267c856f295ee8d572b": "R_{d}/1000", "e4d0970a4ba62d73c273282499b924dd": "\\sim_e", "e4d171027b6695e7e68be5a5753c4718": "v_j +tv_b \\ge v_i ", "e4d1804a54a375d5a11b4edc4eba68c7": "\\mathbb{R}^d \\times \\{1,2\\}", "e4d1a02ece941ded59712bd7077f9839": "\n\\begin{bmatrix} 1/\\sqrt{2} & 1/\\sqrt{2} \\\\ -1/\\sqrt{2} & 1/\\sqrt{2} \\end{bmatrix}\n\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} + \n\\begin{bmatrix} 100 \\\\ 50 \\end{bmatrix} = \n\\begin{bmatrix} 1/\\sqrt{2}+100 \\\\ -1/\\sqrt{2}+50 \\end{bmatrix}\n", "e4d22b660526dd09e0768ae972c152d6": "\\ h_{0,1}=-(\\gamma/2)(\\bar n+1)", "e4d25d36adb559f339a18ba7aeedb219": "\\psi^{(m)}(z)= (-1)^{m+1}\\int_0^\\infty\n\\frac{t^m e^{-zt}} {1-e^{-t}} dt", "e4d263fd01f4f33d874e8e8657396234": " {m^2}/{s^2}", "e4d2a5714b5262c847fc35d63ea96d13": " e=\\lim_{n \\to \\infty} \\frac{n}{\\sqrt[n]{n!}} ", "e4d2c5565d65f5088f129c1feb5a8495": "\\mathbf{1}_{[0,\\alpha\\sqrt{n}]}(\\|p_\\sigma\\|)", "e4d2dfaeee34a082612565e92f4363b1": " H g_1 K = \\coprod_i H a_i", "e4d314363f28a667a6c52891b77f9baf": "L_\\text{star}", "e4d3311a83d064bf5d969ffb72641e19": "\\lambda_{2}", "e4d355ad375239063176b857d8e51e63": "\\langle p'|\\mathbf{T}(0)|p\\rangle", "e4d3dd1866e2047e21c24363df63d9a0": "\\lbrace h_{t,k} \\rbrace ", "e4d3f4ee6032044f3dd0fcf24a27378b": "\\beta(X,Y)=f(\\left[X,Y\\right]),", "e4d4019f48f0c00eae794417bcad7944": "(\\boldsymbol{\\lambda},\\gamma) \\leftarrow", "e4d41111c8999269b9a9c8d02c1e26f8": " \\phi(r) = \\frac{\\mathrm{Vol} \\, B(p,r)}{\\mathrm{Vol}\\, B(p_k,r)} ", "e4d4824d82f17d0c3b7242f56b78b18e": "\\psi (a, b, p)\\!.", "e4d499b55611e8f91ec307dae3bbe8f7": "A \\ominus B = \\bigcap_{b\\in B} A_{-b}", "e4d4c010abdbce5865762fd9fa380b3b": "x_1 = \\sqrt{\\frac{L^2 - a^2}{3}}", "e4d4e0631a4031894a4065f4edb93463": "\\left\\lfloor \\frac{n}{2} \\right\\rfloor,", "e4d4f395e07a84d1c622d9c07b4f6323": "r=2a\\sin^2\\theta/\\cos\\theta", "e4d50dc2b924909e33e52397cb2b2aa7": " Q_i \\ge 0,g_i \\in R^m ,d_i \\in R. ", "e4d51e76b31fe3aca3da2e63d1762763": " \\varepsilon = \\frac{\\left\\langle \\displaystyle\\sum_{i=1}^N c_i\\Psi_i \\right| \\hat{H} \\left| \\displaystyle\\sum_{i=1}^Nc_i\\Psi_i \\right\\rangle}{\\left\\langle \\left. \\displaystyle\\sum_{i=1}^N c_i\\Psi_i \\right| \\displaystyle\\sum_{i=1}^Nc_i\\Psi_i \\right\\rangle} = \\frac{\\displaystyle\\sum_{i=1}^N\\displaystyle\\sum_{j=1}^Nc_i^*c_jH_{ij}}{\\displaystyle\\sum_{i=1}^N\\displaystyle\\sum_{j=1}^Nc_i^*c_jS_{ij}} \\equiv \\frac{A}{B}. ", "e4d550cdc07813f6e0540bcbe2f4856e": "R_t.", "e4d59611e8cf3ae0383474e4d1d72f42": "B \\to \\tau \\nu", "e4d5b60db7f803d1a9a61e85343c0618": " \\log_b (x) = \\frac{\\log_{10} (x)}{\\log_{10} (b)} = \\frac{\\log_{e} (x)}{\\log_{e} (b)}. \\,", "e4d5e69c4ccdb57ccd684b7ddc3776f4": "G(x,y) = \\frac{1}{2\\pi \\sigma^2} e^{-\\frac{x^2 + y^2}{2 \\sigma^2}}", "e4d5f70bba6a2e025bb19e2893080d93": "\\widetilde{\\mathbb{X}}^{(k)}=(\\widetilde{x}^{(k)}_1,\\ldots,\\widetilde{x}^{(k)}_N)", "e4d673659d73db419aecc575dd7f6eab": "S[\\sigma] \\to T[\\sigma] T[\\sigma] T[\\sigma]", "e4d677e84f7532d3cdf7b949213cf2a6": " \\det(\\mathbf{A}) = \\frac{1}{n!} \\varepsilon_{i_1\\cdots i_n} \\varepsilon_{j_1\\cdots j_n} a_{i_1 j_1} \\cdots a_{i_n j_n},", "e4d6e61c8025870ade93dc08d2671a94": " E = \\lambda h ", "e4d7e4dbe47cfd0730ec56c27c52407b": "\\gamma=", "e4d80dacb30f95ad00b91d6e066513f0": "\ns_f(n) = O(M(n)\\log^2n). \\,\n", "e4d814d9b1fc31dbe0cd6f06cbe08354": " f_2", "e4d8bd391bdb864ae4d64bca7244224f": "LLE-REID=\\frac {LLE}{REID}", "e4d90c1c55060e4e6836518fe51b4d54": "\\Pi(0)\\neq 0", "e4d949c8d746e3c76c2ea9b8ff0b96ef": "H_{2k}(M),", "e4d959bd943c9db7361de10c477eca6e": "X\\mapsto X_+ \\wedge E", "e4d9666b9926c8c9867ed5e447756a3f": " \\nabla\\times\\mathbf{v} = \\frac{1}{h_1h_2h_3} \\mathbf{e}_i \\epsilon_{ijk} h_i \\frac{\\partial (h_k v_k)}{\\partial q^j} ", "e4d9b2eb0bcfb26443791b83db4b1cbf": "A\\to(B\\to(A\\land B))", "e4d9c81116ef699e597439b49ab36d04": " \\text{Maximize }\\; f(x) ", "e4da098dadaac63ad127137d00edcfbf": " F = p, E = \\lambda f.\\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) ", "e4da15463dbc207e3006496a1a0de235": "B_c \\approx {1 \\over \\ D}", "e4da35e8039187a3040b88af9ae7703c": " \n\\vec{R}_i = A \\vec{r}_i \n", "e4da3b7fbbce2345d7772b0674a318d5": "5", "e4dad07c08869e8b5af55319810737c0": "l^a\\partial_a=\\partial_v +U\\partial_r +X^3\\partial_y+X^4 \\partial_{ z }\\, := \\,D \\,,", "e4daea32ef8d4451bdd4c0a0039eb252": " a \\ ", "e4db34e988080365f19b3b7a42b18cbb": " \\Delta \\boldsymbol{x} = \\boldsymbol{u} \\Delta t + \\frac{1}{2}\\boldsymbol{a} \\Delta t^2,", "e4db472c52f9e00d83354b9ff8a224b0": "\\dot{x} = f(x,u)", "e4db4fc1ef9ae443ced49d00f2192445": "11101", "e4dbb87dfa520e532f7a1c25aed5ce7d": " \\langle m \\left ( {\\rm X} \\right ) \\rangle = \\frac{1}{T} \\sum_i^T m \\left ( {\\rm X}_i \\right )", "e4dc50f229412af2faf7e1afd2276a47": "F=k N^{0.5}", "e4ddabfd32426258df2d5b63593dd4ed": "\\int (a\\theta+b)\\, d\\theta = a. ", "e4ddbed6c1f1f23d0f9ca1188bfded8d": " R = \\frac{p(O_{fg}|I,I_t)}{p(O_{bg}|I, I_t)} = \\frac{p(I|I_t, O_{fg})p(O_{fg})}{p(I|I_t, O_{bg})p(O_{bg})}, ", "e4ddc73445f2ef3f0d237bff7b26e69f": " c_i(t) = c_i(0) \\exp(-k \\lambda_i t)", "e4ddcf42b529d5870b945fe7ba8a7550": "\\mathcal{E} =\\oint_{\\part \\Sigma (t)} \\mathrm{d} \\boldsymbol{\\ell} \\cdot \\mathbf{F} / q", "e4ddd754c1435826f561058924d51c5c": "p/q", "e4dded49667d37efc5c1d35af30d172b": "|i\\rangle", "e4de177041827b56c46c9ce9262e68f7": "E_{\\mathit{kin}}=\\frac{1}{2}mv^2", "e4de2e26533a35f0917f4317f39a8d23": "p_j(y,x)=p_j(y|x)p_j(x), ", "e4de4145ee6a0b484cce544121d7beac": "B_t(x)", "e4de730254b3bc4f421f56627701ca80": "\\sigma^2_f \\approx b^2\\sigma^2_a+a^2 \\sigma_b^2+2ab\\,\\text{cov}_{ab}", "e4e00af3129324df5fe593562c4ca4e5": "f = V_{exh}\\ \\rho\\,", "e4e0115009adaf6306a85a9c012bd7d4": "\nQ = \n\\begin{pmatrix}\n\\frac{\\mathbf u_1}{\\|\\mathbf u_1\\|} & \n\\frac{\\mathbf u_2}{\\|\\mathbf u_2\\|} & \n\\frac{\\mathbf u_3}{\\|\\mathbf u_3\\|}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n 6/7 & -69/175 & -58/175 \\\\\n 3/7 & 158/175 & 6/175 \\\\\n -2/7 & 6/35 & -33/35 \n\\end{pmatrix}.\n", "e4e02bd263e1ec158f54139a40feb1db": "pK_{\\mathrm b} \\approx 14 - pK_{\\mathrm a} ", "e4e14b2e964778cb5f2d4edafe84e3f1": " h = \\frac{\\tan \\varphi}{\\tan \\theta} \\ell \\approx \\frac{\\varphi}{\\theta} \\ell \\approx \\frac{\\varphi}{\\theta} L \\sin \\theta ", "e4e16eb431eef2009975dbf84f773a24": "p \\vdash [a]p\\,\\!", "e4e1b79f4653261ab17189ced4db501a": "x \\not\\in [x]", "e4e1bb573b2f93182a21cb3cfa1dcf2f": "\\mathbf{\\omega}", "e4e210b7190c74d3ed503fd1e4003b2a": "2H_2 + O_2 \\rarr 2H_2O", "e4e2527d8e2a70a42bb302a276b25bdd": "\\Bigl| \\nabla^2 E_0 \\Bigr| \\ll \\Bigl| k_0\\, \\nabla E_0 \\Bigr|", "e4e257f9ca3024784a70ff16bdeb1bdb": "V_k =", "e4e25cf830a9b16e90fa47b250caa341": "f = \\sum a_{ij} g_{ij}, \\quad a_{ij} \\in I^{n-j}", "e4e293d23ef2867034c0ff65c7f87db0": " K^0_S \\to \\gamma \\gamma ", "e4e2958b34457041528c0e1cd7cd1952": "\\frac{\\log3}{\\log2} \\approx 1.5849\\dots \\approx \\frac{19}{12}", "e4e2cc26e651d4a99a58ca722a56565f": "0\\to A \\to B \\to C \\to 0", "e4e2ccef05457a988caa5a70dd0d64da": "\\delta(x)=x+K", "e4e2edd9899d0a358239e6e4ffeea743": "(1151, 120, 1)", "e4e2f8db082d62ea6a89db3962766f5a": "D_{dBd} = 10 \\cdot \\log_{10}\\left[\\frac{D}{1.64}\\right]", "e4e308b5ec83e3e774be4e0e23762042": "\\mathfrak u=\\aleph_1", "e4e30a73f2891857267d314bcbe52654": "\\sigma > 0, \\alpha", "e4e34822c51622a5215ea025ece9134e": "1 - \\epsilon", "e4e3640abe91371adecd19e0d5cd9b71": "{\\scriptstyle \\psi^{(1)}(z)}", "e4e397f3fd7c51b2b47b5e1ef4bde6a3": "x+x'=\\sqrt{(h_t+h_r)^2 +d^2}", "e4e39be0bf2ea6db32b0e2c3cd65849a": "ap_a+bp_b+(1-p_a-p_b)\\frac{a+b}{2}", "e4e3a0ecdda0196ace48a7866c78745c": "\\theta(x=0)=T_b-T_\\infty", "e4e3b58c6b0dcdbf88fc839b9ec0c9dc": "\n\\left(\\frac{7}{9907}\\right) \n=-\\left(\\frac{9907}{7}\\right) \n=-\\left(\\frac{2}{7}\\right) \n=-1\n", "e4e40dae19c40392d8ec301bf68dde7b": "r_s=[(x-x_s(t))^2+y^2+z^2]", "e4e46370d0c7c5b5f829806ee9cd35ba": "x \\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1) x^{r + c - 2} - x^2 \\sum_{r = 0}^\\infty a_r(r + c)(r + c - 1) x^{r + c - 2} + \\gamma \\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} - (1 + \\alpha + \\beta) x\\sum_{r = 0}^\\infty a_r(r + c) x^{r + c - 1} -\\alpha \\beta \\sum_{r = 0}^\\infty a_r x^{r + c} = 0", "e4e4860a9080f9ac5c4761fb3807e898": "1/(s-1-e^{-s})", "e4e48cf56c383da540a944e343ed6b3f": "\\{U_1,\\ldots,U_m\\}", "e4e4e0955275f68c7312d6fcd4331435": "S_n := \\frac{X_1+\\cdots+X_n}{n}", "e4e507365545a1b04dbc0d9cd1fd0074": "\\frac{P}{A} = \\frac{2 \\pi h}{c^2} \\int_0^\\infty \\frac{\\nu^3}{ e^{\\frac{h\\nu}{kT}}-1} d\\nu \\,", "e4e53a788bdfdcb2181763f922f842e2": "\\mathbf{v} = \\mathbf{0}\\quad (r=a)", "e4e5465cadfde1c10520b288cbd38c2e": "\\bar{v} = \\sqrt{\\frac {k_B T}{m}}", "e4e5503484e340c19a7d7faba34b8316": "\\begin{array}{rcccl}\n\\boldsymbol\\alpha &=& \\left(\\alpha_1, \\cdots, \\alpha_K \\right) &=& \\text{concentration hyperparameter} \\\\\n\\mathbf{p}\\mid\\boldsymbol\\alpha &=& \\left(p_1, \\cdots, p_K \\right ) &\\sim& \\operatorname{Dir}(K, \\boldsymbol\\alpha) \\\\\n\\mathbb{X}\\mid\\mathbf{p} &=& \\left(\\mathbf{x}_1, \\cdots, \\mathbf{x}_K \\right ) &\\sim& \\operatorname{Cat}(K,\\mathbf{p})\n\\end{array}", "e4e55e306159cea9ad583e2c83510120": "0 = c_1 Ax_1 + c_2 Ax_2 + \\cdots + c_r Ax_r = A(c_1x_1 + c_2x_2 + \\cdots + c_rx_r) = Av, ", "e4e5d90d485ea1d3d259f4d00ab41d80": "\\Uparrow", "e4e606b1e8d041f13aa599167e9ce316": "\\scriptstyle p:\\; G' \\,\\rightarrow\\, G", "e4e62d61c41697224a59ae501e2aaf1f": "\n\\begin{align}\n\\frac{s}{p_1}\n&=p_2 \\cdots p_m \\\\\n&=q_2 \\cdots q_n.\n\\end{align}\n", "e4e66e6783fca1675ed8ea02f344de58": "f: \\mathbb{R} \\rightarrow \\mathbb{R}, f(x) = \\sqrt{x}", "e4e6e70a545d29d40a3802a7b9b77aa9": "y = x + 22", "e4e7323b7129f760ebdc4a9f0d4357f5": "\\ t_ \\frac{1}{2} = \\frac{1}{k[A]_0}", "e4e774378ad31fe987086a5f497db8e1": "x^7+x^4+x^3", "e4e796021d49bd8ae17bd3ef3cf69d5b": "\\frac{df}{d\\tau} = \\sum_{\\alpha=1}^n \\frac{dx^\\alpha}{d\\tau}\\frac{\\partial f}{\\partial x^\\alpha}.", "e4e7b66368ab564a519dc3a804490a5f": "r_1=r_2=r", "e4e8148d388a326271c1b3280a587037": "v (T, p)", "e4e84240d9d55dfe48237563bb8e4860": "\\psi(\\Omega^\\Omega \\psi(\\Omega^\\Omega \\psi(0)))", "e4e8469f921df5e055768af681b409a8": "p:=\\frac{p_n}{p_n-r}", "e4e8d3c53ddf37d92f33c10648802af8": "\\frac{d}{dx}(\\mu y) = \\mu q(x)", "e4e8f75c2d2880cbb81be8e911ddd8dc": "\n\\varphi \\mathbf{(r)} = - \\int_C \\mathbf{E} \\cdot \\mathrm{d}\\mathbf{l}\n", "e4e9444706ef2ba6d016361e82198f5e": "G(t,\\xi) ", "e4e94c5de55d19bd1ec45651465ca84a": "\\begin{bmatrix} 1/\\sqrt{2} & -1/\\sqrt{2} \\\\ 1/\\sqrt{2} & 1/\\sqrt{2} \\end{bmatrix}", "e4e95af32d9e454409fe5044d9ab8156": "5 \\times 10^{14} \\text{J/kg}", "e4e976e72cd8c8b11d6f23b0960ba274": "\n0 \\longrightarrow \\operatorname{B}_2(K) \\longrightarrow A(K) \\stackrel{d}{\\longrightarrow} \\wedge^2 K^* \\longrightarrow \\operatorname{K}_2(K) \\longrightarrow 0\n", "e4e99174877ece8b4a54abaa8dc13d9f": "\\phi_0:= -F_{ab}l^a m^b \\,,\\quad \\phi_1:= -\\frac{1}{2} F_{ab}\\big(l^an^a-m^a\\bar{m}^b \\big)\\,, \\quad \\phi_2 := F_{ab} n^a \\bar{m}^b\\,,", "e4e9c796c570a3e600389a6325e6c456": "\\log_{10} (4.7\\times10^6 / 3.2\\times10^9 ) = -2.83", "e4e9f531308b2edcaf14c2acdf60227e": " y=\\frac{Y}{ZZ} ", "e4ea13db4422574cd29dc2bac9e896ee": " B_k(x)\\!", "e4ea3dd1c74c5f7b5deaf54529cae092": "E_{K_2} = D_{K_1}.", "e4eb400a6ee30f7157f6948313a9611c": "\\scriptstyle <1.7\\times10^{-11}", "e4eb5145694db635d910064246ffa11d": "k=(\\vec{L}\\cdot\\vec{R})^n=[\\vec{L}\\cdot (\\vec{E}-2\\vec{N}(\\vec{N}\\cdot \\vec{E}))]^n,", "e4ebb31080bf87a844924b82aee75aed": "G \\leftarrow G/e", "e4ebb814dfc0a8fc3e3d54e8531f8a7c": " \\frac{x_0 + \\cdots + x_n}{y_0 + \\cdots +y_n}\n= \\frac{x_0}{y_0} \\left(\\frac{1 + \\frac{x_1}{x_0} + \\cdots + \\frac{x_n}{x_0}}{1 + \\frac{y_1}{y_0} + \\cdots + \\frac{y_n}{y_0}}\\right)", "e4ec29b9a0f2674d8c6ace0f48dfc683": "|b| \\leq a \\leq c", "e4ed6b247540cf0388faff38a1e5d06b": "\\partial /\\partial q^i", "e4ed943cdd379a22070847a61e06e889": " d \\sin \\theta = (m + 1/2 )\\lambda \\ . ", "e4ede90a2accd1e01c1adf50602699fc": " x_1^2+x_2^4 ", "e4ee76cfcd9a7199f86da5a47fffbfea": "T \\cdot \\widehat{Var}(\\widehat\\beta_1)<1. \\,", "e4eea429d59015b4d55d646ada2ff2d4": "\n{\\rm Var}[z]\\,\\,\\, \\approx \\,\\,\\,\\sum\\limits_{i = 1}^p {\\sum\\limits_{j = 1}^p {\\left( {\\frac{{\\partial z}}{{\\partial x_i }}} \\right)} } \\left( {\\frac{{\\partial z}}{{\\partial x_j }}} \\right)\\sigma _{i,j}\n", "e4eed9b081420d2692f4d3bb6341af29": "=x(bu'^2-2au'v'-bv'^2)\n-y(av'^2-2bu'v'-au'^2)\n+b(uv'^2-uu'^2-2vu'v')\n+a(-vu'^2+vv'^2+2uu'v')", "e4ef68ce0e136884675beb94c2ba8169": " E(X_1+\\cdots+X_n) = E(X_1)+\\cdots+E(X_n). ", "e4f05a42aa6aa65a0c2efdceecdf9f24": "\\mathrm{lim}F=\\mathrm{Ran}_E F", "e4f0600de19f8bc84eeeb1463d0d8400": "l = \\operatorname{gcd}(a-c,d+b)", "e4f0ac41250acc90015f02725b746bdc": "\\scriptstyle f\\circ\\boldsymbol{u}(\\boldsymbol{x})=f(\\boldsymbol{u}(\\boldsymbol{x}))\\in BV(\\Omega) ", "e4f11314e248030278b1201114bada81": "f(x,y,z)=0", "e4f1be4a59dac756db138d908cefee82": " w = d + [2.6m - 0.2] + 5R(Y,4) + 4R(Y,100) + 6R(Y,400) \\mod 7. ", "e4f1c7fee33d9beb627a5ab2aae42acf": "u(x_s, t_s) = u(x_0, 0)\\,", "e4f1f331e9736ecf1843571062e2194c": "L_y/T^2", "e4f230873191435f75823d2a9d49cca8": "P\\approx P_0 \\frac{Y}{1-e^{-Y}}", "e4f2872e97e972a2d7cded2d021567d5": "J(X)", "e4f2a305c47376559929bfaa3890c4e5": "\\eta = {{\\hbar\\omega - \\epsilon_G} \\over {\\hbar\\theta_x}}", "e4f335e61bed07ecd1e03703b54df29b": "L_\\alpha", "e4f3c89077024720b74727c515da9f5f": " \\dot{m}", "e4f411035b44b06036293d0caea8e85c": "(x_2,\\ y_2)", "e4f44ee3693d8a0e52b190625ba8dcbf": "\\left (\\frac{V_1}{V_2} \\right )^{(\\gamma-1)}", "e4f4a0091d0bf67dbb6766e89277874c": "\n\\; A_i = \\sum _{i = 1} u_{ij} B_j.\n", "e4f4d8339394f07d9b57626d2a21793f": "Z\\,", "e4f4dadd18a46e8db5945a0621106cc5": "|t|=1", "e4f54961797e66e9e2b1ebfc692d4778": "\n-f(x) = f(-x) \\, ,\n", "e4f5d7d4fe894b554b227bbe969afeb5": "H(z, -1) = \\frac{1}{1+z}\n\n\\quad \\mbox{and hence} \\quad\n\nn! [z^n] H(z, -1) = (-1)^n n!.", "e4f63edb47b016a1487c08b7d881e7f6": "P(XY)", "e4f65d65b30d2dfda18ca03ab61ca8de": "X = \\sum_{i = 0}^N P^{-1}(\\boldsymbol{r}_i)", "e4f67943b59128f1f38be178e7fe11bf": " y'(t) \\approx \\frac{y(t+h) - y(t)}{h}, \\qquad\\qquad (2) ", "e4f73ad82429a306d1322604781c506a": "\\widehat{\\theta}_n=\\arg\\max_\\theta h_n(\\theta), h_n(\\theta)=H'_n(\\theta).", "e4f7545b5bb816ae54c409fd762e28d7": " \\eta_{Xe} ", "e4f7629b574dd05d19a2107c87806700": "D_n", "e4f7dd01625f645d1805bb5f6379be82": "(b \\times b) - b - a", "e4f7e9e5b9a9f450f49e585115c08d9b": "K \\leq \\ker f", "e4f7ebc9eaf62d8af276dc3c1b60ec6c": "\\frac{-i}{j}=k", "e4f7fb6ca3b354ba53b5c9dfecd274cf": "y'' + y = 0\\ ", "e4f835fd4e8235a4c0d1536d99e11a7e": "\\sin_k(i+t)\\equiv \\sin_k(i)\\cos_k(t)+\\sin_k(t)\\cos_k(i).", "e4f857a7081ab8c54fbffe14db85f944": " \\{f_i\\}_{i \\in I} ", "e4f88df269b22ff6987b1517ea50ab84": "T_B = \\Bigg(\\frac{\\,R\\,\\ln(P_0)}{\\Delta H_{vap}}+\\frac{1}{T_0}\\Bigg)^{-1}", "e4f89cf3ff0330542f00d82434d962c8": "h_5", "e4f8a1a5c24656a1baad58635bddfae5": "\\left\\langle -\\beta \\tilde{H} - \\log\\left(\\tilde{Z}\\right)\\right\\rangle\\geq \\left\\langle -\\beta H - \\log\\left(Z\\right)\\right\\rangle", "e4f8b617493754b43adc7085cbf57a3c": "\\begin{pmatrix} B_L \\\\ B_S\\end{pmatrix} = \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} A_L \\\\ A_S \\end{pmatrix}\\, .", "e4f9805fa13431a76617a0ea3f73a152": "\\alpha= (1-D) \\bar \\alpha(\\theta_i) + D \\bar{ \\bar \\alpha}", "e4f98ea44815504721c3471bee1ced09": "g\\left( \\theta^{\\prime}\\right) ,\\ ", "e4f9939d39cd8cc6473bc1f037c3eb8b": " \\chi(p) ", "e4f9a254de029ada2ea3db2835263f9c": "\\mu = \\frac{a \\Delta t}{(\\Delta x)^2}.", "e4f9d7435795e7797a6fd6264ae4ad75": "\\frac1a\\frac{ds}{d\\sigma}\n= \\frac{d\\lambda}{d\\omega}\n= \\frac{\\sin\\beta}{\\sin\\phi}.", "e4f9e18cc4635c658095a485578dcab4": "\\tfrac{\\pi}{2}, \\tfrac{\\pi}{3}, \\tfrac{\\pi}{8}", "e4f9fefd627b042ec212deb4b997d449": " a \\wedge x \\le b.", "e4fa22605851641d3334bab8f15fcd44": "P \\uparrow Q", "e4fa316009fd084397cb07b0f22a9b98": "\\scriptstyle{j\\omega M}", "e4fa6b0b11a4b3626487d48ce3313f8e": " u_5 =0.51492 ", "e4fa7720b059184473ce4eb98a336bff": "\\nu >4\\!", "e4fb2ea1e576354da1e9ee4267202729": "\\bar{a}^{(\\mu)}_\\mathbf{k}(t)", "e4fc29fabf9c5b1ed6b6d28b8cd7a14f": "{\\tau_i\\neq \\tau_j,\\, \\forall i,j}", "e4fc30958148981b94ca67b06f54364e": "\\cosh\\frac ak = \\cosh\\frac bk \\cosh\\frac ck - \\sinh\\frac bk \\sinh\\frac ck \\cos\\alpha, \\,", "e4fccef5ab108f19274db9a1bd8453db": "\nS = nR \\ln\n\\left[\\left(\\frac VN\\right) \\left(\\frac{4\\pi m}{3h^2}\\frac UN \\right)^{\\frac 32}\\right]+\n{\\frac 52}nR.", "e4fd8234bc331309f053ffe8388ba291": " \\sim \\, ", "e4fdbad41479e8117da410564962911b": "T = \\sum_{i=1}^{2} \\frac{1}{2} m_i \\dot x_i{}^2", "e4fddcd5cdedc78c62e87aa5ded5f30a": "\\vec{e}\\,^T\\vec{e}\\!", "e4fdf2fc6c4eddc5355edaa6a811ace8": "\\textstyle n = pq", "e4fe026c0353eec99841fd52ab16cca0": "n=\\sqrt{n_x^2+n_y^2+n_z^2}=\\frac{2Lp}{h}", "e4fe4296df35db825c3f31d952764c59": "X \\sim NM(k_0=10, \\{p_1=0.2, p_2=0.1, p_3=0.2 \\})", "e4fe46f7f6fb016e53a4cb50a31c3192": "68 \\cdot 2 = 136", "e4feb84d8f2e3ab6ccca3175fe4f82ed": "(xy + yx)/2", "e4fef4c6272fe0eeb3fd623824ebd81a": "\\xi = {\\mathbf e}\n\\begin{bmatrix}\n\\xi^1(\\mathbf e)\\\\\n\\xi^2(\\mathbf e)\\\\\n\\vdots\\\\\n\\xi^k(\\mathbf e)\n\\end{bmatrix}=\n{\\mathbf e}\\, \\xi(\\mathbf e)\n", "e4ff053b179787368bf4f391b96ae1bd": "X_0 \\cap X_1 \\subset X_0, \\ X_1 \\subset X_0 + X_1.", "e4ff16c60dd1b3ad2cb55d014e488adf": "\n\\begin{align}\nV_j &= \\int_0^1 f_j^2\\left(X_j\\right)dX_j\\\\\n&= \\sum_{ m_j \\neq 0 } \\left| C_{0 \\dots m_j \\dots 0} \\right|^2\\\\\n&= 2\\sum_{m_j=1}^{\\infty} \\left( A_{m_j}^2+B_{m_j}^2 \\right)\n\\end{align}", "e4ff4a0edbe2da54b3919620510f1587": " E_\\mathrm{stored} = {1 \\over 2} C U^2 ", "e4ff6017005b8f9a9a8f2a75f2f155ec": "\\tilde{H}_0(X) = \\ker(\\epsilon) / \\mathrm{im}(\\partial_1)", "e4ff7cf3a59795e0654da47b7f93fee6": "s_P(t)\\ \\stackrel{\\text{def}}{=}\\ \\sum_{k=-\\infty}^{\\infty} s(t-kP),", "e4ff867b996d35e039797e95da26bc74": "\\lceil N / 3 \\rceil", "e4ffaccffc3333125f20d04f0b9144d8": "E_{Can}", "e4fff311e0bf950bbc9eb28c0544af6f": "1/z^2", "e50068bc5bd4bc9c24626ce64f218f03": "\\tau =\\frac{1}{n\\omega }\\frac{{{A}_{n}}}{{{B}_{n}}}", "e50099f8f6020eeeeda17e9a0f2a4b80": "\\Phi'(r,\\theta,\\phi)=\\frac{R}{r}\\Phi\\left(\\frac{R^2}{r},\\theta,\\phi\\right)", "e500d40e8e459aec906db87b8d6faa59": " G(z) =\n\\frac{n}{n} z\n\\frac{1}{1-\\frac{1}{n} z}\n\\frac{n-1}{n} z\n\\frac{1}{1-\\frac{2}{n} z}\n\\frac{n-2}{n} z\n\\frac{1}{1-\\frac{3}{n} z}\n\\frac{n-3}{n} z\n\\cdots\n\\frac{1}{1-\\frac{n-1}{n} z}\n\\frac{n-(n-1)}{n} z.", "e50141cb4b6de9186c45402ee1e7e477": "x^{i} \\to x^{\\prime i} ", "e501ae2ad90dc374410a774da21c5739": "n_2", "e501e2c9f6a9e950236721b165f83a22": "\n|\\mathbf{\\delta{Q}}| = \\delta\\dot{\\mathbf{L}} = \\dot{m}\\delta r = \\rho(2\\pi r\\delta r)U_{\\infty}(1 - a)\\times(2\\Omega a'r^2)= r\\delta F_{\\theta}\n", "e50261d812134d9446ded1ad48b808ad": " \\triangle NOP^\\prime \\sim \\triangle P^{\\prime\\prime}OS \\implies OP^\\prime:ON = OS : OP^{\\prime\\prime} \\implies OP^\\prime \\cdot OP^{\\prime\\prime} = r^2 ", "e502625480d04b49586c387b4d5bc9e2": " K g_2 L", "e502823a69a3e3e51411ac9c3f76eed1": "\n\\vec{S}(u,v)=(1-v)\\vec{c}_1(u)+v\\vec{c}_3(u)+(1-u)\\vec{c}_2(v)+u\\vec{c}_4(v)-\n\\left[\n(1-u)(1-v)\\vec{P}_{1,4}+uv\\vec{P}_{2,3}+u(1-v)\\vec{P}_{1,2}+(1-u)v\\vec{P}_{3,4}\n\\right]\n", "e503516e608dbb0c316caf7803327635": "\\{ \\mathbf{e}_i : 1\\leq i\\leq n\\},", "e503ad0dcb9d03a860c488cb03c29b94": "\\scriptstyle (\\infty,\\, \\infty,\\, 0)", "e503ce00e898beaadeb908e53962f1e2": "\\, C \\bullet_H D := \\sum_{r\\ne0,s\\ne0}\\langle \\langle C\\rangle_r \\langle D \\rangle_{s} \\rangle_{|s-r|} ", "e504178ca986b8de8ce5fae47fc279fd": "\\operatorname{E}\\Big[e^{i\\theta X_t} \\Big] = e^{-t |\\theta |}. ", "e50488d61e90ace7561660e52a354ef5": "i\\hbar{dA \\over dt} = [A,H]= AH - HA", "e504995541c7cf46a2a6c7bc232c2a76": "\\frac{233150}{19773}X-\\frac{102500}{6591},", "e504be89a2e74d8c75dc0f5ff112a7dd": "\n\\mathit{cornerness} = \\det(\\mu(\\mathbf{x}, \\sigma_{\\mathit{I}}, \\sigma_{\\mathit{D}})) - \\alpha \\operatorname{trace}^2(\\mu(\\mathbf{x}, \\sigma_{\\mathit{I}}, \\sigma_{\\mathit{D}}))\n", "e504d1c62aa53a37e4f315636727207c": "S(f) = \\sigma_Z^2.", "e504d3890a58f3793440f9640e1fcdda": "\\mathbf{L}=-i\\hbar(\\mathbf{r}\\times\\nabla)", "e5050afe89654b3a33e996e2c33aab93": "\\begin{align}\n \\frac{\\delta \\rho }{\\delta t} + \\nabla_\\alpha \\left( \\rho V^{\\alpha } \\right) &= \\rho CB^\\alpha_\\alpha \\\\ \n \\\\ \n \\rho \\left( \\frac{\\delta C}{\\delta t}+2V^\\alpha \\nabla_\\alpha C+B_{\\alpha \\beta }V^\\alpha V^\\beta \\right) &= \\sigma B^\\alpha_\\alpha \\\\ \n \\\\ \n \\frac{\\delta V^\\alpha }{\\delta t} + V^\\beta \\nabla_\\beta V^\\alpha - C\\nabla ^\\alpha C - 2V^\\beta B^\\alpha _\\beta & = 0 \n\\end{align}", "e505339c20c6929ef38505c36a47f811": "df_{n}", "e50533bea8c646b6264f5e835bc22045": "M\\setminus x", "e50541d184e591047b736283ceff81af": "{2a_{15} \\times b_{15} \\over c_{15} - b_{15}}=d", "e50628fe754a0b9885a55d554f02aacb": " \\psi_1 \\left(\\tfrac34\\right) = \\pi^2 - 8G.", "e5062c701a20d77493b9863c2c5b2ff4": "\\rho (\\mathbf{x},t) = |\\psi (\\mathbf{x},t)|^2", "e506d26ebadd16c9c31201b196395573": "H\\psi=i\\hbar \\partial\\psi/\\partial t", "e50728072990e35615950b66c8bfe26c": "\\scriptstyle \\pi^*:H^2(M,\\mathbb{Z}) \\to H^2(P,\\mathbb{Z})", "e5072dc81a1e6506c0c22fba19c85f24": "(x,y)=(\\cos \\; t,\\sin \\; t)", "e5073035ca04a523fa8dd38dcd65c26c": " \\sigma^2 =\\sum\\limits_{i=1}^n (1 - {p_i}){p_i}", "e50747b360dfd36477ddd4bfc45d5352": "\\theta =\n\\begin{cases}\n\\pi -2\\arctan\\frac{X_\\mathrm L}{R_\\mathrm 0} & \\mbox{if } {X_\\mathrm L} > 0 \\\\\n-\\pi -2\\arctan\\frac{X_\\mathrm L}{R_\\mathrm 0} & \\mbox{if } {X_\\mathrm L} < 0 \\\\\n\\end{cases}", "e5074b0ce2d4ebc3e675846b7d44c462": "\\scriptstyle c_i", "e507c355de8ea9759c55dc71306a32a8": "\\left[0,\\frac{\\pi^2}{6}-1\\right)\\cup\\left[1,\\frac{\\pi^2}{6}\\right).", "e508d850130c90d51c278c89b15b5890": "a=(1+i)/2", "e509438e5c46cd9b0fb353cbd965a5be": "\\frac{P \\to Q, R \\to Q, P \\or R}{\\therefore Q}", "e50a3b534799b8e261a242fb91e3791d": "\\mathrm{d}U = T\\mathrm{d}S - p \\,\\mathrm{d}V + \\sum_i \\mu_i \\,\\mathrm{d} N_i\\,", "e50a76424e8f5056fa26362fcd4d0579": " \\mathbf{x} \\times \\mathbf{y} = -(\\mathbf{x} \\wedge \\mathbf{y}) ~\\lrcorner~ \\mathbf{v} ", "e50abf3f03ac07031049d181569e9dee": "(n, e)", "e50adb4e0739c5a700e1451ffb155439": "\\limsup_{i \\rightarrow \\infty} A_i = \\bigcap_i \\bigcup_{j \\geq i} A_j", "e50b993d2180763f716f065ea02c2fa0": "\naVF = LL - \\frac{1}{2} (RA + LA) = \\frac 32 (LL - V_W)\n", "e50bc5e8a9d664a0c27ddc849a5966ef": "P+Q=0", "e50be03f4816658e757c1a7e27870157": " |b_n|^2 \\le c_n", "e50c1581aae4079a5d6e22f2be83f141": "\\begin{align}\n S &= k_B \\ln \\Omega \\, \\approx k_B \\ln ( P(r,n) dr ) \\\\\n A &\\approx -TS = -k_B T \\frac{3 r^2}{2 L_c b} \\\\\n F &\\approx \\frac{-dA}{dr} = \\frac{3 k_B T}{L_c b} r\n\\end{align}", "e50c1969a6f116e8955e9b3bc15248db": "H_n\\bigg(\\underbrace{\\frac{1}{n}, \\ldots, \\frac{1}{n}}_{n}\\bigg) = \\log_b(n) < \\log_b (n+1) = H_{n+1}\\bigg(\\underbrace{\\frac{1}{n+1}, \\ldots, \\frac{1}{n+1}}_{n+1}\\bigg).", "e50c2df60c87c71fe5a0c2829ce610e0": "\\sum_{d\\,\\mid\\,12}f(d) = \nf(1) + f(2) + f(3) + f(4) + f(6) + f(12).\n", "e50c731fef440b40f3c5ff4f96c31da5": "\\displaystyle f'(x_{0}) \\approx \\displaystyle \\frac{\\frac{11}{6}f(x_{0}) - 3f(x_{-1}) +\\frac{3}{2}f(x_{-2}) -\\frac{1}{3}f(x_{-3}) }{h_{x}} + O\\left(h_{x}^3 \\right), ", "e50c79e0ab03c8bfffa8749b3a51519d": "m_i \\ge m_{i-1}", "e50ca4202fb314bfabc7e4858fd85540": "f\\colon X \\to \\mathbb{R}", "e50cca57a33f278b73326a5aa76bbf21": "~q=\\hbar\\omega_{\\rm p}-\\hbar\\omega_{\\rm s}~", "e50d0bc72db095bf487b61eb27657de4": "\\Psi_{N}(f)", "e50d5a6440deff3b6921b3417dd47575": " N=\\binom{n_i}{i}+\\binom{n_{i-1}}{i-1}+\\ldots+\\binom{n_j}{j},\\quad\nn_i > n_{i-1} > \\ldots > n_j \\geq j\\geq 1. ", "e50d720e67dd053de9745bc0eeb69e89": "~\\theta_f = \\lambda /d", "e50d7ef979827d3476fbdeb03a86a289": "\\{\\rho\\,,z \\}", "e50dae5d763297de66f1467c9a5f4f46": "\\mu_\\alpha\\,\\!", "e50dcc72f68b206fc92292e79cbae415": "B\\mapsto(B\\Leftarrow A)", "e50dcf749c73b1f95b7eb905802fbf9d": "R_M ", "e50ddb741fed725919defffe71ddc9c7": " |\\sum_{n,m\\ne 0} c_{mn}\\lambda_m\\lambda_n| \\le \\sum_{n\\ne 0} {1\\over |n|} |\\lambda_n|^2.", "e50de50d2a2d58c55e0a1065467775b7": "U=\\bigl(\\mathbb{E}\\bigl[|X|^p\\big|\\,\\mathcal{G}\\bigr]\\bigr)^{1/p},\\qquad V=\\bigl(\\mathbb{E}\\bigl[|Y|^q\\big|\\,\\mathcal{G}\\bigr]\\bigr)^{1/q}", "e50dfd1cf4e7cb54d525e4bdc962fa01": "\\ z^2-z+c=0.", "e50e297480c3c65649d75ec61e3e8e27": "A\\in \\mathcal S_n(\\mathcal A_{1\\cdots n}, \\mathcal C_{1\\cdots n})", "e50e317302b94ab4a8410776c5be561c": "Z = (2\\pi)^{n/2}|\\Sigma|^{1/2}", "e50e8d55c8b6ae61ec4b23561e13ecd0": "(Q(AB) + Q(BC) + Q(AC))^2 = 2(Q(AB)^{2} + Q(BC)^{2} + Q(AC)^{2})\\,", "e50f7f9d08282db8d04580c65bf69c1c": "D*F=0", "e50f8c2bd5f307314cdf3b100b7c69ff": "\\, a=b\\cos\\gamma \\pm \\sqrt{c^2 -b^2\\sin^2\\gamma}\\,.", "e50fa72dca5a58d53077404819a45e81": "{\\nabla \\cdot \\overrightarrow{V} = 0}", "e50fd643c35695332cde4ada4d8c1d94": "\\begin{align}\n\\frac{\\delta S_{\\alpha \\beta }}{\\delta t} & = -2CB_{\\alpha \\beta } \\\\[8pt]\n\\frac{\\delta S^{\\alpha \\beta }}{\\delta t} & = 2CB^{\\alpha \\beta }\n\\end{align}", "e50fefd08c2f333ba28c9b5bb5c290cb": "0 = E \\{ (hx+hw+c-x)(x+w) \\}", "e5108458bff1ebc16e86b8f2567a45dd": "k(x-y)", "e510af7ffd916896dac0e07edb47b726": " \\nabla \\phi \\, ", "e5114e7e5ec52dcf341b412289359271": " \\nabla \\times \\mathbf{F} = \\mathbf{C};", "e5118ef025647f6def11d02193a17756": " \\frac{d \\Pi(\\alpha)}{d \\alpha} = 0 ", "e511b11ba033e860993c2f605279c024": "Z_3 = 1^2 \\cdot 1^2 - 1\\cdot 1^2 \\cdot 2^2 = -3", "e512f498a2dd94412b335730fec4bc7c": "(J, s, h)", "e5132b7cd488b40715eba13edc7175a8": " \\dot{t} = E \\, \\left( 1 + m/r \\right)^2 \\approx E \\, \\left( 1 + 2 m/r \\right)", "e5136acf1a72ff45ac3c267ebfb5f873": "\\begin{align} a_{n+1} & = \\frac{a_n+b_n}{2}, \\\\\n b_{n+1} & = \\sqrt{a_n b_n},\n \\end{align}\n", "e5143b26ae801bf5200728d4099e5f8d": "F(f_{k_j}) \\overset{\\ast}{\\rightharpoonup} \\int_{\\mathbb{R}^m} F(y)d\\nu_\\cdot(y)", "e51485f8e48508b5a81357fd1e5cc3d7": "\\hat{f}(I)", "e51494082a002cda0780ac8cb413c869": "{1\\over (2,q-1)}q^{n^2}\\prod_{i=1}^n(q^{2i}-1)", "e5149d2543792987f8bf0e056711333c": "Y_{1,2} = \\beta_e", "e514c5dae742db395efb96f8c9c5497b": "R(M,x) = \\frac{(\\sum _{j=1} ^n \\alpha _j v_j)' A' A (\\sum _{i=1} ^n \\alpha _i v_i)}{(\\sum _{j=1} ^n \\alpha _j v_j)' (\\sum _{i=1} ^n \\alpha _i v_i)}", "e514f10262a3e4541b086e076e4b49eb": "f^*(\\theta)=\\max\\{f(x,\\theta)|x\\in C(\\theta)\\}", "e5154b15bf8229d48b5cc797c77edb14": "\\begin{matrix} \\frac{9}{8} \\end{matrix}", "e5155a02efd1a45fd0aa79d74571265a": "P(x,y) \\log_2[P(x,y)]", "e515a35d704060184e194c6bfde97815": "\\gamma_p = \\left(\\frac{1}{V} \\frac{dV}{dT}\\right)_p = \\left(\\frac{d(\\ln V)}{d T}\\right)_p = \\frac{d(\\ln T)}{d T} = \\frac{1}{T}.", "e515d76e0a851c4c19f23578a53bcb5f": "\n q(x,y) = q_0 \\sin\\frac{\\pi x}{a}\\sin\\frac{\\pi y}{b} \\,.\n", "e516228bc578445f6971622afed187fb": "S=\\sum_i W_{ii} \\left(y_i-\\sum_jX_{ij}\\beta_j \\right)^2", "e51640d3458bda438c8abbc320b7a5cc": "\\int_c^df^{-1}(y)\\,dy+\\int_a^bf(x)\\,dx=bd-ac.", "e516bc3c948d66dc0573bfc3f78cef37": "\\{(\\mathbf{a_i},\\mathbf{b_i}) + (\\langle \\mathbf{a_i}, \\mathbf{t} \\rangle)/q\\}", "e516c3899878d5daa382af4b9d8d2d86": "\\mathbb{R}(Z)", "e516fbbb4b763d618e1d0a0e35a998b1": "q(x)[S]=\\partial_\\mu j^\\mu (x) \\,", "e51707fa868f5d2794f3b507629d7b04": "{1, \\frac{3}{2}, \\frac{4}{3}, \\frac{7}{5}, \\frac{11}{8}, \\frac{18}{13}, \\frac{29}{21}, \\frac{47}{34}, \\frac{76}{55}, \\frac{123}{89}}, \\dots \\dots [1; 2, 1, 1, 1, 1, 1, 1, 1, \\dots]", "e5172a783feceed7d6ff53620058bb6b": "y=b/2", "e51758c3eb02e1dc88e38dfa63594113": "\\neg B \\to \\neg A", "e5177e1dd65f817b9ebf0cd0dedb12fb": "T =\\frac{1}{2}m \\left(\\sum_i\\ \\frac{\\partial \\mathbf{r}}{\\partial q_i}\\dot{q}_i+\\frac{\\partial \\mathbf{r}}{\\partial t}\\right)^2\\,\\!.", "e517deae3f4f1c8f414aed6740e11e0d": "\\, N", "e517e5daf2808948c34637a27b4b5e80": "{\\Bbb C} \\times H=\n{\\Bbb C} \\times {\\Bbb R} \\times {\\Bbb R}^{>0}", "e5182af7f03d0fc740f6d881db4c2c58": "|\\psi\\rangle = \\int_{-\\infty}^\\infty |y,y\\rangle \\, dy = \\int_{-\\infty}^\\infty |p,-p\\rangle \\, dp", "e518c2968e41067357b413003568d285": "\\textstyle \\binom{n}{2}=\\frac{n(n-1)}{2}", "e518c6ad5ebec0d20d0695c7d5d9d7db": "h(k)=\\begin{cases}\n 1, & \\mbox{if } k=0 \\\\\n -1, & \\mbox{if } k=1 \\\\\n 0, & \\mbox {otherwise}\n\\end{cases}", "e5194ebf46fcb9452fc97816a8546285": "-\\mathbf{e}_{31}", "e5199d8f69f64c9c54cb0e85c0ff8d37": " P_1 = \\left( {m + \\Delta m} \\right)V", "e5199d9434c58cf8c8ef5d6075789325": "B + tC = (1 -t)B + tA", "e519b56e7ec0ebc24784a403bc09d15b": "x = a_0 + a_1t; \\,\\!", "e51a0ad53599fa13468add1da2f88689": "p_{\\tfrac{1}{2}1} \\leftarrow 64x^3+192x^2+80x+8", "e51a2a5597387b7d96b08f97a6313fff": "\n\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}^{n}\\right\\}\n\\right\\} ", "e51a2fe4eec4500500721d4947548d32": "\\hat{y} = y_\\mathrm{obs} / y_\\mathrm{atm}", "e51a48655d1789219f9c39488991cae7": "p_1 = \\{ s, v_4, v_3, v_2, v_1, t \\}", "e51a67a65ff7dda263eac16d1c645f4e": "1/2^{|b|}", "e51ae9423cbfa594a4182db93fbb2b84": "\\exp\\left( \\left(\\sqrt[3]{\\frac{64}{9}} + o(1)\\right)(\\ln n)^{\\frac{1}{3}}(\\ln \\ln n)^{\\frac{2}{3}}\\right) =L_n\\left[\\frac{1}{3},\\sqrt[3]{\\frac{64}{9}}\\right]", "e51b442c812b7362a7b45055df8fd496": "f_C(c|E=e) = \\frac{P(E=e|C=c)}{P(E=e)}f_C(c) = \\frac{P(E=e|C=c)}{\\int_{11}^{16}{P(E=e|C=c)f_C(c)dc}}f_C(c)", "e51b60330e36a98e7ab19faf829b6978": "a^{\\lambda(n)}\\equiv 1 \\pmod{n}", "e51b6833a40816ee945c947ffa352efc": "t_{ig} = \\left ( \\frac{\\pi}{4} \\right ) \\left (k \\rho c \\right )\\left [ \\frac{T_{ig}-T_{o}}{q''} \\right]^2 ", "e51ba0527644978b7a3020226ff3f18e": "(\\cos \\theta, \\sin \\theta)\\,", "e51be0ff0119440bf7fa767f888ed715": "Initiates(open, isopen, t) \\leftarrow HoldsAt(haskey, t)", "e51c922882778e55b19c7541f7b6d552": "H_x = \\sum_{n=1}^{\\infin}(-1)^{n+1}x^n\\zeta(n+1)", "e51cbb740dbc947790a7af1d62c70667": "\n\\alpha = \\alpha' + 2\\pi = -2 \\pi \\sqrt{\\beta}\\Bigg( \\left(1-\\frac{3m}{r} \\right)^{1/2} - 1 \\Bigg).\n", "e51d36e8aaf3ea5b03e799c2c6652941": "D(X) = \\rho(X - \\mathbb{E}[X])", "e51d40f1be808ca82de6dab2091675cd": " Y^\\phi = Y \\cup \\{\\phi\\}", "e51d5bdd28838bf4f1a225daf5c5212e": "N \\times [0,1],", "e51e28d5d2c148795fe33c22b9b2bf67": "-\\mathbf x = (-x_1, -x_2, \\ldots, -x_n).", "e51e2ac4b1d7ce85abcb197dfaa2f887": "h[n] = T \\sum_{k=1}^N{A_ke^{s_knT}} \\left( u[n] - \\frac{1}{2} \\delta[n] \\right) \\,", "e51e4a45e7eb3760aceb47e94a266824": "\\bigtriangleup_{SO,dB} = P_{out,f,dBm}-P_{out,2f,dBm}", "e51e78f52c9107526dee97e5e8ac23b6": "\\bar {u} \\frac{\\partial \\bar{u}}{\\partial x} + \\bar {v} \\frac{\\partial \\bar{u}}{\\partial y} = -\\frac{1}{\\rho} \\frac{d\\bar{P}}{dx} + \\frac{\\partial}{\\partial y} \\left [(\\nu \\frac{\\partial \\bar{u}}{\\partial y} - \\overline{u'v'}) \\right].", "e51e96b50c1a522d6f15e641f61f59db": "\\frac{\\partial L_{ij}}{\\partial t}+\\frac{\\partial (v_kL_{ij})}{\\partial x_k}=\\left(\nx_j\\frac{\\partial P_{ki}}{\\partial x_k}-x_i\\frac{\\partial P_{kj}}{\\partial x_k }\\right)", "e51e9b03cb7d0397209ec93491e66d47": " A_{\\mu \\nu} ", "e51ee0df91e5c9c71721a3dc2d0e9ee9": "\\left(\\mathbf{B} + \\mathbf{BVA}^{-1}\\mathbf{UB}\\right)^{-1}", "e51ef0049a2c45d8bd2b063c424b06d3": "x = \\begin{cases} r \\exp \\bigl(i (\\theta) \\bigr), \\\\ r \\exp \\bigl(i (\\theta + 2\\pi) \\bigr), \\\\ r \\exp \\bigl( i (\\theta - 2\\pi) \\bigr). \\end{cases} ", "e51f4bdddd4625b66c76c443b5aac6f6": "Q(M)=\\sum_{k=1}^{M} \\frac{M!}{(M-k)! M^k}.", "e51f54b0ddd361f2cf798f37a125f2bb": "L_n^{(a)}(z)", "e51f7a98dd43fec78eb52c6b20a36a60": "\\mathrm{SNR} = \\frac{N}{\\sqrt{N}} = \\sqrt{N}. \\, ", "e51f8069c1fc0792c864a9ba761b952b": "\\boldsymbol{\\alpha}'=-\\mathbf{Q}sc_1-\\mu\\boldsymbol{\\epsilon}'s^2c_2-\\alpha'_j\\big[\\boldsymbol{\\alpha}s^2c_2+2\\boldsymbol{\\beta}s^3\\bar{c}_3+\\frac{1}{2}\\boldsymbol{\\delta}s^4c^2_2\\big]", "e51fbb33388a9a3f9637cda16f225e53": "x_{\\{k\\}}", "e51fc133804c49102a9500efb3156829": "\\scriptstyle\\omega", "e51fcf6c4d2802b2ecbf2afa6b027d98": "\\inf \\theta \\le 15/46.", "e51fffd2b3a6305bc22239faacc74c8c": "\\frac{g}{2\\cos \\theta_W}\\overline{d}_{L\\alpha}U_{\\alpha\\beta}\\gamma^\\mu d_{L\\beta}Z_\\mu", "e52018ff69e3ff5e225964aef193b912": "\\theta=\\arctan (12) \\approx 85.2^{\\circ} \\,.", "e52050b7e65168c42338c4341ea33489": "\\,r(e^{ i \\alpha})=-\\cot( \\alpha /2)", "e5209b4ec417234bbd82e831faaf806f": "h\\colon Tx\\to x", "e520aec0df0a5a9bb05acbd666605752": "X_n\\in L^1", "e520b2bb958027f9ed44e997b43e87f1": "i \\in [0,w-1] ", "e520b586037eb2102c2759ef0305de73": "C=C_{o} e^{-k \\tau}", "e520b83fe6fca983f79e720185bf7757": " {\\rm li} (x) = \\int_0^x \\frac{dt}{\\ln t}. \\; ", "e520f4fcd9aca1369e6ce52942bd0595": "T \\square T F F = \\{0 | \\star\\} = \\uparrow ", "e52153b848b546fd830d565c8e7443a1": "f \\circ (g \\circ h) = (f \\circ g) \\circ h", "e521aebf858a30949ebd8d5c7d944638": "x_{i}\\in\\{0,1\\}\\,\\!", "e5227797d9afea16556d5a922f5f60a5": "\\omega({\\mathbf e}\\cdot g) = g^{-1}dg + g^{-1}\\omega g.", "e52298a7cd4629a260f8681d9e41af01": " \\beta = 1/(kT)", "e522c18c051bbdca8d966c1488c14be6": "{\\rho_\\text{substance}} = SG \\times \\rho_{\\rm H_2O}.", "e522cad9d667c5db39a29b76b3e00677": "\n \\mathrm{dev}(\\boldsymbol{B}) = \\boldsymbol{B} - \\tfrac{1}{3}\\mathrm{tr}(\\boldsymbol{B})\\boldsymbol{\\mathit{1}}\n = \\boldsymbol{B} - \\tfrac{1}{3}(3+\\gamma^2)\\boldsymbol{\\mathit{1}} = \n \\begin{bmatrix} \\tfrac{2}{3}\\gamma^2 & \\gamma & 0 \\\\ \\gamma & -\\tfrac{1}{3}\\gamma^2 & 0 \\\\ 0 & 0 & -\\tfrac{1}{3}\\gamma^2 \\end{bmatrix}\n ", "e5236cc4e0c7dfcd42e697772bb07849": " a(u,v) = \\int_\\Omega \\nabla u \\cdot \\nabla v \\,dx ", "e5237d2a6a719a70f2e9f7b688421cc6": "\\mu_0=\\gamma_0^{n-1}+Bn/\\gamma_0", "e523cd70c4d5cc961b4279f22e5950ef": "E=(d,g)", "e5241d4df652abd1d6a206ae26145237": " 0x) = \\begin{cases}\n\\left(\\frac{x_\\mathrm{m}}{x}\\right)^\\alpha & x\\ge x_\\mathrm{m}, \\\\\n1 & x < x_\\mathrm{m}.\n\\end{cases}\n", "e5265786642ef07615a919ecee966e60": "R, F\\in \\mathbf{H}_n", "e5267c71522d51112dcc4e6dcccc9be3": "t_D = 2 \\cdot \\frac D c = 2 \\cdot \\frac {2.5\\;\\mathrm{m}} {300\\;000\\;000\\;\\frac{\\mathrm{m}}{\\mathrm{s}}} = 0.000\\;000\\;016\\;66\\;\\mathrm{s} = 16.66 \\;\\mathrm{ns}", "e5268b4300e0d80de27de1faecf579f7": "X \\sim \\mathrm{Davis}(b=1,n=4,\\mu=0)\\,", "e5270579f08a92a6c424422756e43014": "k_\\rho", "e5272dfe1295de28c794f49467867127": "r(x) = (x-x_0)(x-x_1)\\cdots(x-x_n)", "e527819ad53695d8fa224e563e099b05": "M_a=\\lim_{N\\to\\infty}N\\cdot x(N)=\\lim_{N\\to\\infty}\\frac{P_0\\cdot r}{1 - (1 + \\frac{r}{N})^{-NT}}=\\frac{P_0\\cdot r}{1 - e^{-rT}}. ", "e527c3100c5be134eae76f5c5d4b7762": "\\rho_6(n) =\n\\begin{cases}\n \\varepsilon\\!^{ -1}, & \\text{if }z>\\ln \\varepsilon \\!\\\\\n e^z, & \\text{ otherwise}\n\\end{cases}\n;", "e527fc4036ba7ff15f92dca7a268a7f0": "\n\\sum^n_{i=1}F^Q_id^P_i = \\int_\\Omega D_{ijkl}\\epsilon^Q_{ij}\\epsilon^P_{kl}\\,d\\Omega\n", "e5284ba578dc5dd746e74f096c521462": "X_1 = x\\partial_y, \\quad X_2 = y\\partial_x, \\quad H=x\\partial_x - y\\partial_y.", "e5292027321f305389d90bc69d42ffe8": "G_{\\mu\\nu} = 8\\pi T_{\\mu\\nu}\\,", "e5297da404f60994a1b653a0c046a230": "T_<", "e5297f1ca2d888ae54425cabe9c22117": "h[n] = T \\left( h_c(nT) - \\frac{1}{2} h_c(0)\\delta [n] \\right) \\,", "e529a5568c45933246d1601e1b45f8aa": "\\dot{c}_m(t) = - c_m\\langle\\psi_m|\\dot{\\psi_m}\\rangle - \\sum_{n\\ne m}c_n\\frac{\\langle\\psi_m|\\dot{\\hat H}|\\psi_n\\rangle}{E_n - E_m} e^{i(\\theta_n-\\theta_m)}", "e529b9c0768a257eac4879e8b9e723de": "\\textstyle y \\in Y", "e529fc02f18291123f5d854850bce731": "2 \\cdot a + b,", "e52a0a6548c908057a11305593ae6e9d": " U(z) = \\sum_{p \\mid P(z)} f(p) . ", "e52a3cdf15be4ce912180891b34a7139": "{\\textbf A}_k", "e52aabf5763b6a306109584747ae7eaf": "\n\\ln\\left( r_e \\right) \n = {1\\over L^2} \\oint_\\ell \\oint_\\ell\n \\ln \\vert \\boldsymbol{x} - \\boldsymbol{y} \\vert\n \\; dx \\; dy .\n", "e52ab478040613ce591fb12bcda1dcc5": "P'_R", "e52b493e3180c0ad47217d06600862a4": "s_{\\overline{X}_1 - \\overline{X}_2} = \\sqrt{{s_1^2 \\over n_1} + {s_2^2 \\over n_2}}.\n", "e52cc063dc86fdce03c28b9c1406b41e": "x^2-y^2 = 0", "e52ccc2d7c647e2340f6dc22bfbc346b": "\\mathbf n_{\\perp}= (-\\sin \\theta,\\cos \\theta) ", "e52cd648d84b3ee001e1e141c7870257": "\\theta-\\psi=\\beta", "e52ce913a303a24259c8b313cdc05746": " L_1 \\and \\cdots \\and L_i \\and \\cdots \\and L_n ", "e52cfa25cc99decfa3d0d25ef33f5d5f": "\\epsilon = \\epsilon_r \\epsilon_0", "e52d4ccdc59c668af0ba93e867be1a2d": "x_{1} = x_0 - \\frac{f(x_0)}{f'(x_0)} \\,.", "e52dd0415d480d82ac225450f1b3841a": "ML = 0.936 Md - 0.16 +/- 0.22", "e52dd226102cb5fda9315762bc5a9cf4": " T_3 = (T_1Z_1)^2 + (T_1Y_1)^2 - (Z_1Y_1)^2 = 12", "e52de9d68c96059bb0c050c6e80f3e65": "S^{2n-1-a(n)}", "e52e4cff0440c92b0abcddefbe59c80f": "\\left(1/6,\\ \\sqrt{1/28},\\ \\sqrt{1/21},\\ \\sqrt{1/15},\\ -2\\sqrt{2/5},\\ 0,\\ 0,\\ 0\\right)", "e52e91fc685f4418f36a3106749ab39b": "A(h) = A^\\ast + C h^n + o(h^{n+1})\\;", "e52ebc014174dd4b952398a9a9170f92": "\\mathrm{id}_{X_T} = (\\eta_X)^*.", "e52ed2935d3caf6fcab43120200aa1ff": " x =_E y ", "e52f4f94464c886a0bf28f48b0314dfd": "\\partial_\\alpha F^{\\alpha\\beta} = \\frac{4\\pi}{c}J^\\beta_{\\mathrm e}", "e52f807439cd0a9aa75c7069a9077587": " t_r ", "e53055cc07ebf4a6bcba002e7368ceac": "\\mu^n", "e530591ea5bd1e355209075a275eb876": "\\bullet\\rightarrow\\bullet\\leftarrow\\bullet", "e530d23d909ee913623902abbd260d1e": "x = A+f(A+f(A+f(\\cdots)))", "e530d74e6debab84b5e23a4f5a1602ff": "E[Y(T)|X_t=x] = E[Y(t)|X_t=x] = u(x,t).", "e5313c9d2cd80827b9f41b8984491c79": "= \\frac {3}{4} \\sqrt{\\pi}\\,", "e53141ac3ed4fc44ec9a9aef62f7ab35": "L=n\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}\n=\\dot{x}_1\\frac{n \\dot{x}_1}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}+\\dot{x}_2\\frac{n \\dot{x}_2}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}+\\dot{x}_3\\frac{n \\dot{x}_3}{\\sqrt{\\dot{x}_1^2+\\dot{x}_2^2+\\dot{x}_3^2}}\n", "e5315fafb9642616f5753eb5698babe8": "\\gamma F_y = -q\\gamma vB, \\quad {F_y}' = -q\\gamma v B,", "e5316fb85b084f771ea494800b01f517": "\n \\frac{b}{c}=\\frac{b}{\\frac{a}{2}}=\\frac{2}{\\frac{a}{b}}=\\frac{2}{\\sqrt{2}}=\\sqrt{2}\n", "e531b34efbf2a742110505e069c07e97": "\\left(\\frac{8.16 \\mbox{ g H}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }}{1.008 \\mbox{ g H}}\\right) = 8.095\\ \\text{mol}", "e53274fddf4192cdeaf271be50b762ff": "2 \\pi n i", "e53277c33be00b9175405c2de966bf4b": "a_1^{b_1}\\cdots a_n^{b_n},", "e532df43f5fb281987360db4f11debdb": "C_N = {K \\sqrt{p_{N_2}}}", "e533189a629dae5260c73abcb0370960": "\\int_a^b \\left|\\gamma''(s)\\right|\\sgn \\kappa_{n-1}(s)\\,ds", "e533af026dbeba09e86c9781e5da7b74": "\\sigma \\ ", "e533b31fab9d8e334959f2ca8a6823ad": "A\\equiv((B\\equiv C)\\equiv((A\\equiv C)\\equiv B))", "e533cf13df636a0dde1e9e27df675ffe": "e^{-2\\phi} = \\frac{M}{\\lambda} - \\lambda^2 uv", "e533d6edc1bf7de95b76133a4b25b9a2": " \\textbf{C}_k = \\textbf{P}_{k\\mid k} \\textbf{F}_k^T \\textbf{P}_{k+1\\mid k}^{-1} ", "e53414f64a99ab3e538f0b2c2352f790": " a = z_{\\mathrm{S}} e^2 g^{-2} = e^3 /8 \\pi h_{\\mathrm{P}} \\approx \\; 1.541434 \\times 10^{-6} \\; \\mathrm{A \\; eV} \\; {\\mathrm{V}}^{-2}. ..........(25) ", "e534155e93ca770078c926b96dfbb9d2": " \\frac{g_{\\mu\\nu} - \\frac{k_\\mu k_\\nu}{m^2}}{k^2-m^2+i\\epsilon}+\\frac{\\frac{k_\\mu k_\\nu}{m^2}}{k^2-\\frac{m^2}{\\lambda}+i\\epsilon}.", "e5342f4b384f9f7d8c70d9e22d12b114": "\\det\\begin{bmatrix}a & \\mathbf v\\\\ \\mathbf w & b\\end{bmatrix} = ab - \\mathbf v\\cdot\\mathbf w", "e534648653cdc9b158735226f201a0c2": " c \\sigma \\cdot \\vec{p} ~ \\Psi_2^0 + V \\Psi_1^0 = \\epsilon_0 \\Psi_1^0 ", "e53464e444627ff87a5ef5bf59ad8fea": "D_\\mu D^\\mu \\phi = -(\\partial_t - ie A_0)^2 \\phi + (\\partial_i - ie A_i)^2 \\phi = m^2 \\phi", "e53494345934554ff98274d4bc2568fc": "\\int_0^{\\theta}\\log(1+\\cos x)\\,dx=2\\text{Cl}_2(\\pi-\\theta)-\\theta\\log 2", "e5349e003ea5586ec24f0e4d43ecb8d2": "ad - bc \\neq 0", "e534a5ac5df091f5e699c971e4ec1676": "f^*\\omega_N\\vert_V = \\omega_M\\vert_U", "e534b5b3a4dc802b92490ffd1be1ade6": " \\Gamma,\\mathcal A,\\mathcal A,\\Delta\\vdash\\mathcal C", "e534d79e0e36a1ea5326c9c14f04daa9": "\\alpha\\approx 1/137", "e534e0d048f6004fde994bba89220e7e": "\\|f\\|^2 = \\int_D |f(z)|^2\\,d\\mu(z) < \\infty,", "e534e94d75c069936b26579e886fd881": "D(1+o(1))", "e5355bc51aa82ec5072adadfdcc0336b": "D_F^q(p, q) = \\sum p(i) \\log \\frac{p(i)}{q(i)} - \\sum p(i) + \\sum q(i)", "e5355da4aec1a7cd710d7a59c7a92e12": "\\delta Q=TdS", "e53579c9f12ff1e9d974757b6329f0c4": "\\pi_p(BO, BO_k)=0", "e535c14059b453f327db1fc15f33f108": "ab=ba", "e535f3f0f8ab4c91ac9e3672de43f857": "ds^2 = d\\theta^2 + \\sin^2\\theta\\,d\\phi^2.", "e535fe8c856c70e392775035e2e9dfb1": "\\left(\\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ \\sqrt{1/6},\\ \\sqrt{1/3},\\ \\pm1\\right)", "e536454401d65c622c9530f506913d82": "\\operatorname{Li}_{1}(z)=\\sum_{k=1}^\\infty \\frac{z^k}{k}=-\\ln(1-z)\\!", "e536bf6314ba1080e0d11abcf59753c5": "\\bar{v} <\\bar{x}(1 - \\bar{x}),", "e536e4b4b2b7ea11da9040925b64a3b4": "\\mathbf{u}' ", "e5373cb76828499531f4b45725afad27": "Q = C\\; A\\; \\sqrt {2\\;g\\;H\\;\\frac{T_i - T_e}{T_e}}", "e537a4ce9f54064d52c2aa335b16f44c": "\\det(A - xI) = 0, \\,", "e5385be1023dbd5ca1121e761ec660f6": " x \\otimes y = x + y.\\,", "e538d0b5533646908b61832d88a7009f": "b(x)=2x-\\lfloor 2x\\rfloor", "e53908226619a95abc6058eb902d054d": "\\sum_{n=1}^{\\infty} \\frac{a_n}{n^s},", "e539cf637acfad8344e5eaa4c093a45e": "f(x,y) = x + y", "e539de028ba06c08cd94ddd08ab32e17": "\\begin{align}\n \\frac{\\partial\\mathbf{v}}{\\partial t} &= \\Pi^S\\left(-\\mathbf{v}\\cdot\\nabla\\mathbf{v} + \\nu\\nabla^2\\mathbf{v}\\right) + \\mathbf{f}^S \\\\\n \\rho^{-1}\\nabla p &= \\Pi^I\\left(-\\mathbf{v}\\cdot\\nabla\\mathbf{v} + \\nu\\nabla^2\\mathbf{v}\\right) + \\mathbf{f}^I\n\\end{align}", "e53a2f5962817ecf4494267231cb938c": " \\frac{m_{\\rm C_2H_6}}{m_{\\rm O_2}} = \\frac{1 \\cdot (2\\cdot12+6\\cdot1)}{1 \\cdot (2\\cdot16)} = \\frac{30}{32} = 0.9375", "e53a3e88bcd2738f6c83b7cb89b4ba1e": "A_{12} = -\\cos{(\\alpha)} \\, d\\theta", "e53b0312043f3497c208f70eb1343664": "\\left[0, \\frac{3}{8}\\right] \\cup \\left[\\frac{5}{8}, 1\\right].", "e53b9d43d5c2a5788e4ec786bd114a86": "\n\nW(t) =\n\\begin{pmatrix}\n0 & -\\omega_z(t) & \\omega_y(t) \\\\\n\\omega_z(t) & 0 & -\\omega_x(t) \\\\\n-\\omega_y(t) & \\omega_x(t) & 0 \\\\\n\\end{pmatrix}", "e53baf528fd2b8cb0fc9ce3787f5f0e9": "\\alpha = \\sqrt {RG} ", "e53bb7dad30f2869442c8385a4cd7af6": "\\omega \\rightarrow 1", "e53c0d0f7dd5ac59635e27f473015393": " {f_E \\over f_C} = {5 \\over 4}", "e53c1fddcaa40c70d9bc69b3d9999c65": "P = ( v_1, v_2, \\ldots, v_n )", "e53c206674b8c59dabb316ca6a63bc1b": "\\prod_{k=1}^n a_k^{j_k(g)}", "e53c28186c40181ce4abe27b93fb5d77": "e \\leftarrow \\lambda_B, i \\leftarrow 0", "e53c2b1ecf2f7a847d00c283c2af1131": " \\mathbf{F}(\\mathbf{r}) = kq\\sum_{i=1}^n \\frac{Q_i(\\mathbf{r}-\\mathbf{p}_i)}{|\\mathbf{r}-\\mathbf{p}_i|^3} ", "e53c3341d2ba333950dbd05e3626366a": "\\mathrm{1\\,Bi = 1\\,abampere = 1\\,emu\\; current= 1\\,\\sqrt{dyne}=1\\,g^{1/2} \\cdot cm^{1/2} \\cdot s^{-1}}", "e53c7b8c692a35771f3414fc418ae7e6": " t \\rightarrow \\int_{\\tau=0}^t \\frac{ \\mathrm{d} \\tau }{N(\\tau)} ", "e53c9b5410b471ac042d0b397a47885d": "42l \\equiv - 16 \\pmod{8}", "e53c9e6348db87e7d8421bbe3aa7574e": "= M^* N^* MN = (MN)^* MN. \\,", "e53cf77a0b62bc58faf2194feecec45f": "\\displaystyle{|\\Phi_{w}(0,0)|^2 - |\\Phi_{\\overline{w}}(0,0)|^2=1-\\left|{1\\over 2\\pi}\\int_0^{2\\pi} f(e^{i\\theta})^2 \\, d\\theta\\right|^2 \\ge 0.}", "e53d3a23da234d41bae839936ecdf868": "Y_3 = \\left (Z_1^2 Z_2^2 + eX_1^2 X_2^2 \\right) \\left (Y_1Y_2-2dX_1X_2Z_1Z_2 \\right ) \\ + \\ 2eX_1X_2Z_1Z_2 \\left (X_1^2Z_2^2+Z_1^2X_2^2 \\right )", "e53d5703ceeb4c3fa797f33af3873db1": "m\\ddot{x} = \\lambda(2x),\\quad m\\ddot{y} = -mg + \\lambda(2y),\\quad x^2+y^2 - L^2=0,", "e53d795d5cf19bd135862f84dd1d6729": "x = {a \\over 2} \\frac {\\sin (m + n) p}{\\sin (m - n) p}, \ny = a \\frac {\\sin m p \\sin n p}{\\sin (m - n) p}\\!", "e53da6f73553be7e09ffbc6e5c486e49": "d \\ne 0", "e53dbac6f35da12c8eef11622f96bd05": "S_{-}", "e53de3d2932d15f51a02e6e234e0e08d": "6 \\cdot V =\\begin{vmatrix}\n\\mathbf{a} & \\mathbf{b} & \\mathbf{c}\n\\end{vmatrix}", "e53e3f4ba6d37d2c2ea9d561c1151e19": "\\cos A = {\\cot A \\over \\csc A} ", "e53e9c36e64fc83847f931827eb62bac": "L \\; p \\lor \\lnot L \\; p", "e53f79ceb6ad67d1d116aeebaa39c1d2": "\\ R_v=E\\{ vv^\\mathrm{H} \\}\\,", "e53f8d15d17cf59182749f3cef8431e4": "\\begin{align}\np(\\mathbf{X}|\\mu,\\sigma^2) &= \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{n/2} \\exp\\left[-\\frac{1}{2\\sigma^2} \\left(\\sum_{i=1}^n(x_i-\\bar{x})^2 + n(\\bar{x} -\\mu)^2\\right)\\right] \\\\\n&\\propto {\\sigma^2}^{-n/2} \\exp\\left[-\\frac{1}{2\\sigma^2} \\left(S + n(\\bar{x} -\\mu)^2\\right)\\right]\n\\end{align}", "e53ff0404190418975029296919f5549": "\\frac{n(n-3)}{2}\\, ", "e5400730cc4a14d7fcb10fe818c9054f": "\\mu(E_1) \\leq \\mu(E_2).", "e5400e48d9fd2332ff68026656dd8145": " \\qquad z_n = z_0^{2^n} ", "e5400edb22f027ee999f3fd48c3dcbef": "A=\\Sigma_{model} \\times \\Sigma_{res}^{-1}", "e540510fc5d440b4be7943b0a097ff20": "=\\sqrt{8 \\cdot 1 \\cdot 4 \\cdot 3}=\\sqrt{96}=4\\sqrt{6} \\approx 9.8", "e54062a7edb4afb822e1873ba6fe88e6": " d\\colon \\pi_{2}(X,A,x) \\rightarrow \\pi_{1}(A,x) \\! ", "e5406a542758fe9c69c84d3fbdcbe58b": "\\lfloor n \\rfloor = \\lceil n \\rceil = n.", "e5407c276b9924a066a06cb74ba88e77": "\\frac{1}{P}\\frac{dP}{dT}=\\frac{(n+1)}{T}", "e541238bbd0793254bc6f29a78b9e3a8": "\\begin{align}\n \\int_{-\\infty}^\\infty f(x)\\log(g(x)) dx &= \\int_{-\\infty}^\\infty f(x)\\log\\left( \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}\\right) dx \\\\\n &= \\int_{-\\infty}^\\infty f(x) \\log\\frac{1}{\\sqrt{2\\pi\\sigma^2}} dx + \\log(e)\\int_{-\\infty}^\\infty f(x)\\left( -\\frac{(x-\\mu)^2}{2\\sigma^2}\\right) dx \\\\\n &= -\\tfrac{1}{2}\\log(2\\pi\\sigma^2) - \\log(e)\\frac{\\sigma^2}{2\\sigma^2} \\\\\n &= -\\tfrac{1}{2}\\left(\\log(2\\pi\\sigma^2) + \\log(e)\\right) \\\\\n &= -\\tfrac{1}{2}\\log(2\\pi e \\sigma^2) \\\\\n &= -h(g)\n\\end{align}", "e541564d7d22449fdb62db978d2d2527": "h_B^{(1)}(z)=\\frac{\\beta}{1-e^{-\\beta z}}=-\\beta n_B(-z)=\\beta(1+n_B(z))", "e541e21309162aca27de8f131bd2782b": "R_{ab}=0", "e541e6b37050c6e65d6ddb72d2feee62": "C_x (t,f) = x(t)\\,\\hat{x}^*(f)\\,e^{i 2\\pi\\,t f}", "e5426542b444896d9b544a68b485ae68": "E_0,E_1,\\dots,E_N", "e5426ebaf98e2bfcbae022c619aa3e1f": "d = \\frac{|c_2-c_1|}{\\sqrt {a^2+b^2}}.", "e542b1130498dad79afd3a7961a38506": "\n\\left(\\chi^\\dagger\\gamma^0\\gamma^\\mu\\psi\\right)\\left(\\psi^\\dagger\\gamma^0\\gamma_\\mu \\chi\\right)=\n\\left(\\chi^\\dagger\\gamma^0\\chi\\right)\\left(\\psi^\\dagger\\gamma^0\\psi\\right)-\n\\frac{1}{2}\\left(\\chi^\\dagger\\gamma^0\\gamma^\\mu\\chi\\right)\\left(\\psi^\\dagger\\gamma^0\\gamma_\\mu\\psi\\right)-\n\\frac{1}{2}\\left(\\chi^\\dagger\\gamma^0\\gamma^\\mu\\gamma_5\\chi\\right)\\left(\\psi^\\dagger\\gamma^0\\gamma_\\mu\\gamma_5\\psi\\right)\n-\\left(\\chi^\\dagger\\gamma^0\\gamma^5\\chi\\right)\\left(\\psi^\\dagger\\gamma^0\\gamma_5\\psi\\right).", "e542fccee192859f445ad1d977791581": "(p x_1 + (1-p) x_2)^n", "e54375d2cbc6d407f8382e37e2f1a733": "a_{n} = \\frac{a_{n-1}}{9} = \\frac{a_{0}}{9^n}\\, .", "e543a771dd841959af586f0cb9319828": "\\Delta(x,y)", "e543b3f23f34809a1924f98d076c0b64": "\\sigma_{gt}=\\frac{301655cm^{-1}} {-lg( \\frac{I} {100%}) \\rho_{sample}} ", "e543f89546247c013bae311aa53fc251": "\\beta_{\\tau}=\\underset{\\beta\\in R^{k}}{\\mbox{arg min}}E(\\rho_{\\tau}(Y-X\\beta)).", "e543fe57b2d287f36b88dec62953f5b5": " A = \\varinjlim \\cdots \\rightarrow A_i \\, \\stackrel{\\alpha_i}{\\rightarrow} A_{i+1} \\rightarrow \\cdots ,", "e54411b4ec41da80165ec3fca9131eb9": "Ass(A)", "e544959937206feca2cb04d666c56f9c": " R_\\mathrm{in}=\\begin{matrix} \\frac{1}{ g_{11} } \\end{matrix} = \\begin{matrix} \\frac{v_{in}}{i_{in}}\\end{matrix} \\Big|_{i_{out}=0} ", "e544de692f9edf8235d7d99ffd9a7653": "\n1-\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi\n_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\hat{\\Pi}_{\\rho_{X^{n}\\left(\nm-1\\right) },\\delta}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta\n}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho_{X^{n}\\left( m\\right) }\\ \\Pi_{\\rho,\\delta\n}^{n}\\ \\hat{\\Pi}_{\\rho_{X^{n}\\left( 1\\right) },\\delta}\\cdots\\hat{\\Pi}\n_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\Pi_{\\rho_{X^{n}\\left( m\\right)\n},\\delta}\\right\\} \\right\\} .\n", "e544e48a6209ded52133e6f024bc3cc1": "y^2+a_3y = x^3+a_4x+a_6", "e54523ab67812c067ad4a515aa360826": "\\det\\left[ a\\; b \\right] = a_{1} b_{2} - a_{2} b_{1}.", "e5453e6325a16e380fb34d994fc7bbb0": " \\and (S_5 \\implies (\\operatorname{equate}[A_5, m] \\and V[x] = m)) \\and D[x] = D[m] ", "e545599efbba02d20a21db5469288bbf": "N_{vs} =", "e5456a08dd8c75878fdfcf35116c0d53": "n_0=n_2 \\simeq 1", "e545e68bad76f47688482466d04e94e6": "p(\\theta | y,\\xi)", "e54676f37385c0b94c3604349686b033": "\\partial^{\\mu} A_{\\mu} = 0.", "e5469715eef01c2d3b194f72e3452625": "/\\left(\\tau_b\\right)", "e546a5aeaead18cd8a03f5d4ada96690": "FP(\\sigma, \\sigma, 1, 1, \\alpha) = P(I)(\\sigma, \\alpha)", "e546ad33c244034187df4309bea8ebc3": "\\left . \\frac{A}{B} \\right \\} \\to X", "e546ef98ce9a727c5a06a89c4033276f": "\n S^{\\mathrm{beam}} = Q_x/\\tfrac{\\mathrm{d}w_s}{\\mathrm{d}x}\n ", "e547201b8da8acd9b0c00baec5ea14f4": "H_0 = 1\\,", "e547849e39313c71ed4c9c7b84af4f42": "\\mathbf{c_{s}=c_{s1}+c_{s2}}", "e547cceda3b2d873ae879c18a1d06303": "|\\det(U)|=1", "e5485771f7d667acee4d741785a3a76c": "AG\\phi \\equiv \\phi \\land AX AG \\phi", "e5488df71c7d224bbd9e8f14be76db17": "a + d = b + c. \\,\\!", "e548aa69638e223fbb6adebb3aca2d4b": "\\pi=\\tfrac{355}{113}", "e548be18e8521439b68df243a5d94db3": "^{\\;}c^{i}(\\xi,\\tau)", "e54904a212cb9f1a52cdccc05cfc52db": "P_2= \\frac{(\\frac{q}{p})^{n_1}-(\\frac{q}{p})^{n_1+n_2}}{1-(\\frac{q}{p})^{n_1+n_2}}", "e54906379da36e31ddc9777e4f024080": "p(s|i)", "e549420e9c488000d0e0657c4ee78d84": "\\alpha x_{\\xi\\xi} -2\\beta x_{\\xi\\eta} + \\gamma x_{\\eta\\eta} = -I^2 (Px_{\\xi} + Qx_{\\eta}) ", "e5496d8a1aaf869f27e56322a076a928": "\\begin{matrix} {n \\choose n-x}{4 \\choose 1}{3 \\choose 1}^x \\end{matrix}", "e54987be8cc3a75d055c322e6e734d80": "\\langle \\rho \\rangle", "e54995e4a567324fc088eb7019c8d301": " k^G ", "e549f6468e6eb5df1c86570cfbeed170": "\\rho_{01}(p)=1", "e54a200bde64246d5087ddf72478405a": "i, \\; d", "e54aaa14314c05b3a0784f86c81fdcfb": "\n\\sin\\phi = \\frac{v_{\\infty}(1 - a)}{W}\n", "e54ac3fc1d12e5de336479c5652c309b": "P_{out}=\\frac{W_{out}}{\\Delta t}", "e54ad3a5d3baa9c091c2f4716bb23a9e": "\\displaystyle \\partial_zw=3\\partial_{\\bar z} v", "e54ad4f013b1e1ce4e3a68765e4b7feb": "(n-1)/2 = 2^{m_1}+l_1", "e54b9f3dcfdbfe92d1914b8656c7c9a3": "\\mathcal{L}=\\frac{1}{2}\\left[\\left({\\partial \\phi \\over \\partial t},{\\partial \\phi \\over \\partial t}\\right)-\\left({\\partial \\phi \\over \\partial x}, {\\partial \\phi \\over \\partial x}\\right)\\right ]-{m^2 \\over \\beta^2}\\sum_{i=1}^r n_i e^{\\beta \\alpha_i \\cdot \\phi}.", "e54bf4912914451d6b88eaf1a45b3075": "\\ \\bot", "e54c3e8c78b2cfbd2ee48bc12a6c4667": "\\mathbf{x}\\ \\sim \\mathcal{N}(\\boldsymbol\\mu, \\boldsymbol\\Sigma),", "e54c61eb0da0f112854035822919c65f": "t_f\\rightarrow\\infty", "e54c8ca4ae1d5e3c89b20f64870e98b2": "\\begin{align}\\underline{\\mathsf{f}} (\\alpha x \\wedge z + \\beta y \\wedge z) &= \\underline{\\mathsf{f}}((\\alpha x + \\beta y) \\wedge z)\\\\[6pt]\n&= f(\\alpha x + \\beta y) \\wedge f(z) \\\\[6pt]\n&= (\\alpha f(x) + \\beta f(y)) \\wedge f(z) \\\\[6pt]\n&= \\alpha(f(x) \\wedge f(z)) + \\beta(f(y) \\wedge f(z)) \\\\[6pt]\n&= \\alpha \\, \\underline{\\mathsf{f}}(x \\wedge z) + \\beta \\, \\underline{\\mathsf{f}}(y \\wedge z),\\end{align}", "e54ccd221953732248f1972b33050881": "\\sin \\alpha = \\frac{1}{4A^2}\\,", "e54cf1a1e854b6a351a65b8834f5dfbc": "f(x)=\\frac{dF(x)}{dx}\\,.", "e54d1b1b812183f9045540811024fe6d": "f\\colon X\\rightarrow Y", "e54d28566d6aeecbd8dc3b5422a20563": " v_x {\\partial c_A \\over \\partial x} + v_y {\\partial c_A \\over \\partial y} = D_{AB}{\\partial^2 c_A \\over \\partial y^2} ", "e54decdba62cb61747739b0ba29da91b": "D_\\mathrm{h}", "e54e049bc0217404294b2acc4d006463": "B_{n+h} = \\frac{(n+h)!}{W(n)^{n+h}} \\times \\frac{\\exp(e^{W(n)} - 1)}{(2\\pi B)^{1/2}} \\times \\left( 1 + \\frac{P_0 + hP_1 + h^2P_2}{e^{W(n)}} + \\frac{Q_0 + hQ_1 + h^2Q_2 + h^3Q_3 + h^4Q_4}{e^{2W(n)}} + O(e^{-3W(n)}) \\right)", "e54e6a439e7729392ed3725ffac86cdd": "t_2-t_1", "e54e7b61954e55c729e71ecb6692a5f1": "\\mathcal{L}_{HV}", "e54e9fe8db69aa3e05538562e8b9da33": "\\frac{1}{N} \\sum_{n=1}^N \\mathbb E[\\bar v_n]", "e54eba3a351645e152f07dbae1af4f04": "\\Sigma_k^p", "e54ee6549549b4281208f9e54e93acf7": "APY \\approx e^{i_\\text{nom}} - 1,", "e54f5657b06dd4235c925a5cafbe43a7": "1/(1-(1-p)) = 1/p.", "e54f6876f6ca25364ece8456e0b0a2a6": "= \\pm\\sqrt{1 - \\cos^2 \\theta}", "e54fb46a2ea7b10e9ea7e7dd3ed13d50": "E(0)", "e5501043e34e3c76645b8a1c1c1975c5": "{D}_{12}^{(2)}", "e55044934f5aa6b3be6399f1ac65ee38": " E_2 ", "e5505f35dc235623c1380f1daa31a317": " \\sigma s \\ ", "e550c173d9cf1871e668d7cf235fce35": "\\rho g_{ij}''-\\rho g^{kl}g_{ik}'g_{jl}+\\tfrac12\\rho g^{kl}g_{kl}'g_{ij}'+\\frac{2-n}{2}g_{ij}'-\\tfrac12 g^{kl}g_{kl}'g_{ij}+\\mathrm{Ric}(g)_{ij}=0.", "e550e879dedc7b9b1f782d20b436cd10": "\\chi_c(G) \\le \\chi(G)", "e5516ded69f39c08be90ee78f85ed5ad": "x={-b\\pm\\sqrt{b^2-4ac} \\over 2a}", "e551794d2622cef5a6895352f75a57f7": "\\theta=\\Theta/2", "e551b44c009806aecaeacace6d017317": "\\ \\Pi ", "e552321aa31e29dbe86a33d89ac29920": "M_n(D)", "e5524698d69bebe18ed42994d6bbb048": "2(1-\\varepsilon)\\gamma\\,", "e552b4109e9dc5ede5f53a51088a8d5e": " {2 \\over 3} \\cdot {5 \\over 4} = {10 \\over 12} = {5 \\over 6}. ", "e553a7e484b06b8c5e180cec10a14559": "F\\left(n\\right)=\\frac{\\varphi^n-(1-\\varphi)^n}{\\sqrt 5}", "e553ede932daf7444c927a0448b5af75": "{\\sigma^2}/(2 a)", "e554ab49aff7ca3bf5d8b611926784b4": "\n \\begin{align}\n F_1 & + \\int_{\\alpha}^{\\beta} \\sigma_{rr}(a,\\theta)~a~\\cos\\theta\n ~d\\theta = 0 \\\\\n F_2 & + \\int_{\\alpha}^{\\beta} \\sigma_{rr}(a,\\theta)~a~\\sin\\theta\n ~d\\theta = 0 \n \\end{align}\n ", "e554b0d02be67b4730436d76af4dfdb5": "0 \\leq k \\leq K", "e554df6e9cdbd6a348ab0e5f1fecf3e3": "b^2-d^2=c^2-e^2\\, ,", "e5551c00b435d5e385a00733d4eb87b7": "\\Phi_0 = \\frac{h}{2e}", "e5557e28fe56cd8188b630c03ccac185": " S= \\frac {Q}{T}", "e555a85817f192cdb2752b642e2ebe67": "w_k=-\\frac{\\frac{p(z_k)}{p'(z_k)}}{1-\\frac{p(z_k)}{p'(z_k)}\\cdot \\sum_{j\\ne k}\\frac1{z_k-z_j}},", "e5563b65b8417fcaa03a82e9e62a2a77": "O((\\log n)^2)", "e556ae30768445b1167f5ab314e41073": "\\scriptstyle 8\\times\\log_2(8) = 8\\times3 = 24", "e556eda6888f4b311d91eada8500123e": "k_1=2.813524695", "e556fdb85a3e08a3fba5e6ddeea682e9": "\\operatorname{div}(\\Gamma \\operatorname{grad} \\phi) ", "e5570c7cc736b507e3dc4104e371043b": "ax - 1 = qm.\\,", "e55776a365912d2aec35f03aac6e9289": "\\int_a^b f(x)\\,dx \\approx h \\sum_{n=0}^{N-1} f(x_{n})", "e55784983ddea80bf7ea8dfeade5827b": "\\sigma(X^{k-1})-\\sigma(X^{k})<\\epsilon", "e557b494ae94e80ec9df8e0d7df02bf7": "Y_k^{-1}.", "e557bc85efc961609e67b70dd9a068af": "U_O(o)", "e557c97da6f60d25eed5f709e75e4936": "\\mathbf{P}^n_S = \\mathbf{P}_\\mathbf{Z}^n \\times_{\\operatorname{Spec}\\mathbf{Z}} S.", "e557fac3e3cf2f73ff5d3be0ed2d1acf": "\\tau_1 \\neq \\tau_2", "e5581b8557ba1d3fff7706db0a60ea49": " \\int_{\\mathbf{R}} \\psi_{n, k}(t) \\, d t = 0, \\quad \\|\\psi_{n, k}\\|^2_{L^2(\\mathbf{R})} = \\int_{\\mathbf{R}} \\psi_{n, k}(t)^2 \\, d t = 1.", "e558652e329417b7f6a9dfad13de00e6": "\\sum_{i=1}^kA_i=I_n", "e5586d2a256394527ffe84deb7bf1634": "\\frac{a-b}{a+b} = \\frac{d \\sin \\alpha - d\\sin\\beta}{d\\sin\\alpha + d\\sin\\beta} = \\frac{\\sin \\alpha - \\sin\\beta}{\\sin\\alpha + \\sin\\beta}.", "e558e2093b8bcb20e50a36f9790203df": "p_1\\circ u=q_1", "e558f57b84e0da7e00d2bf6b3beb7543": " Z_i(N-n_i) \\equiv \\ \\sideset{ }{^{(i)}}\\sum_{n_1,n_2,...} e^{-\\beta (n_1\\epsilon_1+n_2\\epsilon_2+\\cdots)} \\;", "e558f864867270afaba02c93215c5c74": "\\underset{x}{\\operatorname{arg\\,min}} \\; x^2 + 1, \\; \\text{subject to:} \\; x\\in(-\\infty,-1].", "e55923abd03d8eacdabefb4f281d8158": "L(z)=L_{o}^{\\dotplus }e^{-jk_{z}z}+L_{o}^{-}e^{jk_{z}z} \\ \\ \\ \\ \\ \\ \\ (16) ", "e559310ae38f72aeef782e0cf05c0cfd": "\\text{excess kurtosis} =\\frac{6}{(3 + \\nu)(2 + \\nu)}\\left(\\frac{1}{\\text{ var }} - 6 - 5 \\nu \\right)\\text{ if }\\text{ var }< \\mu(1-\\mu)", "e559e575a4789bba7ae4e13048aa54aa": "t_1 < t_2 < t_3 ", "e55a22e8457d48933348b6c5dbe29115": "\\prod_{i\\in I}M_i / U", "e55a47104258e4ea87ba246645cd69c3": " \\operatorname{vec}(ABC)=(C^{T} \\otimes A)\\operatorname{vec}(B) ", "e55a4f7ec1e9f82e0046aa5f1359246d": " \\langle \\hat x(\\omega) \\hat x^\\ast(\\omega') \\rangle = 2\\pi\\,f(\\omega)\\,\\delta(\\omega-\\omega')", "e55a58f89053b5d47265bf92b7733a7d": "\\ker(\\partial_n)", "e55a9072e649df912df84ab845d7821f": "k=0,1,\\dots,7", "e55aa6a99b761bb07f6d1edbec0efa56": "(q_1,q_2)", "e55ad87f81625777e2f569f23ad47467": "\\scriptstyle(-5\\pm12)\\times10^{-5}", "e55af5fadbd8774c157e7e2faa7079fa": "\\operatorname{Var}(X) = \\frac{\\sigma^2}{\\xi^2}(g_2-g_1^2) ,", "e55af617cfb0c0a93d6fb55c64369105": "u(\\mathbf{x},z_1,z_2,\\ldots,z_k) = u_k(\\mathbf{x}_k)", "e55b8adfcae373366bc6549ecff3f538": " L_1 : [m_1 : b_1 : 1]_L ", "e55bb1ae59b6a64858a85a2f48c53036": "Sa", "e55be70db45189fbe86a37cc1d64c935": " L \\,", "e55d3e093dabdc1524f09bc4485665a8": "\n \\frac{D}{2\\rho h W}\\left[\\frac{\\partial^4 W}{\\partial x_1^4} + 2\\frac{\\partial^4 W}{\\partial x_1^2 \\partial x_2^2} + \\frac{\\partial^4W}{\\partial x_2^4}\\right] \n = -\\frac{1}{F}\\frac{d^2F}{dt^2} = \\omega^2 \n", "e55d4356e8f792d5cc661baaa406d0fa": "||x||_B \\leq C ||L_t(x)||_V.", "e55d664ee8992277dabee83b51f8e77c": "\\ 2.4(log_2n + m", "e55e328349414752113c4878dc62303f": "f^{-1}", "e55e5bf126ea4e46256ed964a52db4e6": "1 \\le r \\le n ", "e55e78e912af67082f2c13d453188b28": " 1\\over{\\sum_{i=0}^{i=n} N_{i,p}(u)w_{i}}", "e55e7a9839b3e2a83843b086e30eadf8": " E \\in \\Sigma", "e55e9fe5f260c587d5b4d4ff27d5d9d9": " \\lambda x.\\operatorname{drop-formal}[D, (\\lambda y.o\\ x\\ y)[o:=p], F] ", "e55ea56f0a4c8162890cc5400599890c": " \\ldots V_1,V_2,V_3,\\ldots ", "e55f853eb59396da828b5bc4ca8c611f": "\\gamma \\ne 1", "e55fd4129c25fcbc3ed0fc87d736eff2": "\\frac{49}{90}", "e55fe3644d79e6a2f3a85664d763e925": "\\hat{X}^n:\\mathcal{Z}\\to\\mathcal{X}", "e5602508d0734f000bb10dd02c76015a": " \\mathbf{v}\\cdot\\mathbf{b}_i = v^k\\mathbf{b}_k\\cdot\\mathbf{b}_i = g_{ki}v^k ", "e5602d45e1b58eea1f57a8ad60a8b6e6": "0!", "e5605670401dd4f79bebbc4738f706ca": "\\phi_1(x)", "e5607a2d3ab4721df8fcbb9522cadfa7": "\\beta_1 > \\beta_2 > \\cdots > \\beta_k", "e5608c18b4ca43d081cde13df1d0183e": "r_\\infty=1-e^{-({\\alpha +\\beta \\over \\beta})r_\\infty}", "e56095585e36c848c821c7041bfb09f2": "\\hat{\\theta_i}", "e56137918c9da1c574c39d46f1162538": "\n\\mathbf{A}=\\mathbf{B}\\mathbf{P}\\mathbf{D}\\mathbf{P^{-1}} \\qquad \\qquad \n", "e561592c4edee0cf68df878549863a93": " z", "e5615fdf05266c52a4e8bca8b9a6df3b": "V(a) = \\{ p \\in \\operatorname{Proj}\\, S \\mid a \\subseteq p \\},", "e5622799e33a074b330d29bf2d23820a": "\\frac {r}{h-r} = \\tanh\\frac{b}{2}.", "e562a0787cd486f9d5bdfd614585daf9": "A \\otimes_R B := F (A \\times B) / G", "e562d4cb9dcdfa1de2b1565c80bc2811": "\\frac{1}{{z \\choose n}}= \\sum_{i=0}^{n-1} (-1)^{n-1-i} {n \\choose i} \\frac{n-i}{z-i}, ", "e562dfd9776434b51223a12e29373541": " c \\cdot B^{-1} = (m\\cdot B^\\prime +e)B^{-1} = m\\cdot U\\cdot B\\cdot B^{-1} + e\\cdot B^{-1} = m\\cdot U + e\\cdot B^{-1}", "e5633d841ac2e9e356c7e66055807a15": "U(d)", "e563780e9f2888940b39c92eb3a36b55": "\\Delta w_h", "e563adf939464765df23fb649cf7edd6": "\\lambda_{\\bold{k}}", "e564162c8494d48fe0badcb6b4f43071": "\\ \\Delta f = 0.", "e5643aaf3c76d223c708cb5eb24c064e": "\\frac{F({{v}_{2}}|{{v}_{1}}^{\\prime })}{f({{v}_{2}}|{{v}_{1}}^{\\prime })}\\ge \\frac{F({{v}_{2}}|{{v}_{1}})}{f({{v}_{2}}|{{v}_{1}})}", "e564447519e431c62ac6a6ca0d8ecb28": "\\phi_e(y)", "e56451cb5d404bf5e28cea0ab0249202": "x > \\sigma", "e564576878679f56371c162b3815b8c4": " f^\\prime_t(0)=e^t.", "e564a1de83550f4da4acd38075807b25": "\n \\begin{cases}\n \\displaystyle \\frac{(2\\pi)^{n/2}\\,r^{n-1}}{2 \\cdot 4 \\cdots (n-2)}, & \\text{if } n \\text{ is even}; \\\\ \\\\\n \\displaystyle \\frac{2(2\\pi)^{(n-1)/2}\\,r^{n-1}}{1 \\cdot 3 \\cdots (n-2)}, & \\text{if } n \\text{ is odd}.\n \\end{cases}", "e564c419d8261314bfb278b24ada5118": "(\\hat{x}, \\hat{y}) = \\operatorname{argmaxlocal}_{(x, y)}(\\operatorname{det} H L(x, y; t))", "e564d9381f388b23bac2eb957752baf2": "-2\\int_0^{\\theta} \\log\\Bigg| 2 \\sin \\frac{x}{2} \\Bigg| \\,dx -2\\int_0^{\\theta} \\log\\Bigg| 2 \\cos \\frac{x}{2} \\Bigg| \\,dx=", "e56548d7738f1ac3af6a91f21e23eeed": "U = \\int_{0}^{\\infty}\\frac{\\varepsilon}{e^{\\beta\\varepsilon}-1}g(\\varepsilon)\\,d\\varepsilon. \\qquad \\mbox{(2)}", "e5654ba7b71093d3738659a75d9dc733": "\\rho(0)", "e5657af01467c2f29e6a0e325e0c1665": "~[P_\\mu, P_\\nu] = 0\\,", "e5659af8e57403b4649b45cd57282eac": " (1~2~3~4~5~6)^2 = (1~3~5) (2~4~6).", "e565afa55534c2cbe20eec9b16fbb707": "\n\\nabla^{2} \\Phi \n= \\frac{1}{a^{2} \\left( \\sinh^{2}\\mu + \\sin^{2}\\nu \\right)} \n\\left( \\frac{\\partial^{2} \\Phi}{\\partial \\mu^{2}} + \\frac{\\partial^{2} \\Phi}{\\partial \\nu^{2}} \\right)\n= \\frac{1}{a^{2} \\left( \\cosh^{2}\\mu - \\cos^{2}\\nu \\right)} \n\\left( \\frac{\\partial^{2} \\Phi}{\\partial \\mu^{2}} + \\frac{\\partial^{2} \\Phi}{\\partial \\nu^{2}} \\right)\n= \\frac{2}{a^{2} \\left( \\cosh 2 \\mu - \\cos 2 \\nu \\right)} \n\\left( \\frac{\\partial^{2} \\Phi}{\\partial \\mu^{2}} + \\frac{\\partial^{2} \\Phi}{\\partial \\nu^{2}} \\right).\n", "e56601703a79e66ed6dea0a947ae47d6": "\nh_\\phi = \\frac{a \\sin \\sigma}{\\cosh \\tau - \\cos\\sigma}\n", "e5667489ee361ef91c14da79ad239a6a": "{\\it{l}}", "e56689ad7b5b4da7f85835dc7949259f": "C = \\pi\\sqrt\\frac23", "e56736c6a7da423ae8fd786c70e39f47": "l \\in S", "e56762234a5fd260221eb4456197ebf0": "T_{\\mu\\nu}", "e5678cd285c4186cbec11e311c471a96": "AXA = A", "e567c37a26c6a2226e7ce0fa907ec51c": "e^{i \\phi}+e^{-i \\phi}", "e5680463fbd33a3393dc86c363a363de": "p_t = Const.", "e5683317fb3d07e0f685cbee432d5235": "K[X]/\\langle p \\rangle", "e5683a03e86d620fe5492948fc8bf47e": "\\text{release}(i)", "e568534551fc0dd0a2e6a6a5a11c9b6b": "\\cup_i f_i(X_i)", "e5687888198020242af4b3faf64b740e": "E_\\mathrm{rotational} = \\frac{1}{2} I \\omega^2 ", "e568a5f17c5b6b54d5e063f29d757f2b": " E_k = x_2x_6x_{10} \\cdots x_{2N-2}+ x_4x_8x_{12} \\cdots x_{2N}", "e568ddb570fe8f768386228c8b7fd5f0": " v^i = \\frac{dx^i}{d\\lambda} \\;\\mbox{ and }\\;\n {v'}^i = \\frac{d{x'}^i}{d\\lambda} \n", "e56902d2cabe3d0ca68c18cf829fea5f": "\\frac{{\\Delta z}}{z}\\,\\,\\, \\approx \\,\\,\\,b\\,\\Delta x", "e5691461de6b191e9ce08b631c217b51": "3n - 1", "e569b001cd069f1f71658dedb0cf3a14": "1 + e^{-1/x^2}", "e569b308ef7542ad55e82ce1dd47722e": "\\phi: U \\to P = P_1 \\times_S \\cdots \\times_S P_n.", "e56a0d71249134e2a042c416ab8c54d7": "0 0", "e57db32b4b8ad623287a95f7b264fb53": "0 \\to \\mathbf{Z}/2 \\to \\mathbf{Z}/4 \\to \\mathbf{Z}/2 \\to 0.", "e57db6a093a514866d9bdef1f2d56acc": "\\Phi(\\mathbf{r}_2) = \\Phi_1(\\mathbf{r}_2) + \\Phi_3(\\mathbf{r}_2) = \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_1}{r_{21}} + \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_3}{r_{23}}", "e57e04fbceb2826459764c8a8621aa2e": "f(V, Q)=0", "e57e664fae7e7063f985637c544bc043": "\\Big( (\\mathcal{M}, s) \\models AG\\phi \\Big) \\Leftrightarrow \\Big( \\forall \\langle s_1 \\rightarrow s_2 \\rightarrow \\ldots \\rangle (s=s_1) \\forall i \\big( (\\mathcal{M}, s_i) \\models \\phi \\big) \\Big)", "e57ea33eda53c5c2e91ed39599ff4eb1": "Vol_q(r,n)", "e57eb035ae632fbb206c7ceaa992ae2e": "T_{\\rm E} = 254.356\\ \\mathrm{K}", "e57edcbb54425f5b15ee4640d17bb783": "\\sum_{r_j} q_{ij}(r_i,r_j) = q_{i}(r_i), \\forall i, r_i, j ", "e57f24f4811a1b4ae55e6d84589ec077": "\\Phi_{max} = \\pi R^2 \\sqrt{2 \\Delta P / \\rho}", "e57fa423968c1523c644fbc29d451dfc": "y''", "e57faf585ab30b671f9878817897cbbe": "\\frac{n!}{x_1!\\cdots x_k!} p_1^{x_1} \\cdots p_k^{x_k}", "e57fcf1a6c850c99b085114a4c89082e": "\\mathbb{N}=\\{1,2,\\ldots\\}.", "e580062d87014f1e8271e324fd4557e4": "\\{(t,t^2,1)\\mid t\\in GF(q)\\}\\cup \\{(0,1,0)\\}\\cup\\{(1,0,0)\\}.", "e580a40605e58758b18d97a57c0004c3": "\\mathbf{r}_{n+1}\\, \\ldots, \\, \\mathbf{r}_{N}", "e580fb85fd509c154a53797e61c2ac58": "\n \\operatorname{cov}(w_i, z_i) + \\operatorname{E}(w_i\\,\\Delta z_i)\n = \\left[\\operatorname{E}(w_i z_i) - wz\\right] + \\left[\\operatorname{E}(w_iz'_i) - \\operatorname{E}(w_i z_i)\\right]\n = \\operatorname{E}(w_i z'_i) - wz", "e581019a13e418c8e3f43dc420eba9c5": "x(u,v) = \\cos \\theta \\,\\sinh v \\,\\sin u + \\sin \\theta \\,\\cosh v \\,\\cos u", "e5811d4442d9d50cb3c744f2ff4383f2": "dz^A", "e5816a86c01686f8c3c15792833f3525": "R_{5,0} = -6+462 r^5-1260 r^4+1260 r^3-560 r^2+105 r", "e5818bb777818dbed44e870367249571": "\\partial_u+\\partial_v", "e581dd6b82c197dc38a83933316f8821": "k = e^{\\pm \\theta} \\neq 1", "e5821b534995c43dab701dd8747fa34e": "S(\\epsilon)|\\alpha\\rangle", "e582353b535be2c350e71192ae079e21": "{\\sum_{n=1}^\\infty \\ }' \\frac{1}{n}", "e582c566f280d95f81c76324208ef1a4": "\\sum_{k=0}^{n}|\\beta_k|^2 \\le \n(n+1)\\exp\\left(\\frac{1}{n+1}\\sum_{m=1}^{n}\\sum_{k=1}^m(k|\\alpha_k|^2 -1/k)\\right),", "e5836c7b694d50bab88fb3c6a809b24f": "m_a, m_b, m_c", "e583ad2a4d61441a99de9ec90e246c86": "\\approx \\frac{\\zeta(\\alpha)}{\\tau^\\alpha}\n-\\frac{\\zeta^2(\\alpha)}{2^\\alpha\\tau^{2\\alpha}}", "e58407c40fe68fe6e9cdb8abf7295aaf": "O(VE^2)", "e5849345d4a0b21f6048baa75764b6c0": "\\frac{\\partial L}{\\partial x_i} = - \\frac{\\partial V}{\\partial x_i},", "e584e383826415427dd82185ab92b473": " L - u'\\frac{\\part L}{\\part u'} = C \\, , ", "e5856fa5bf6f04d12fa8e8163ee3be1a": "\\neg S", "e585e4b31596180da921759609484099": "r = 9,61 \\%", "e585f3ee91acc0c5c442334fba4d1a8a": "a = a. + jt ", "e5861f5c782716a91ca993b034ec7629": "[x,y,z] = xy^{-1}z ", "e5863c3e27e95bbc108f299196ef79ac": "K^d", "e58645884ab80553271828d672d3451d": "u=(y,z)", "e5866fb786eed4828b5937200749684c": "Y_{6}^{6}(\\theta,\\varphi)={1\\over 64}\\sqrt{3003\\over \\pi}\\cdot e^{6i\\varphi}\\cdot\\sin^{6}\\theta", "e5867833ead522df928f089d629b0538": "\\beta = \\frac{dn}{d(p[H^+])}", "e586e6f86b23348b01a5c43f61eac581": "\\ \\overline{T}", "e58702151fd9f8fe1d4a84367a95767a": "\\Delta\\langle N\\rangle", "e5872799667dcb7d310c4bc18c9c21ac": "\\scriptstyle{I}", "e587355983302d2ad9f93b130f86035b": " C \\approx \\Delta F ", "e58777c3cb0588763d88424e5d4a9047": "q_t(a)", "e587ee0984c98db85ae50de4bbf59097": " g_{p_t}(x) = a_{0,p_t}+ \\sum_{0 p \\\\\n\\end{cases}\n", "e58cb69c09cda2cf67887b5e88925df7": "\\displaystyle{f=\\sum a_n H_n}", "e58cc198b1ad608befe11d3be30dd523": "\\operatorname{var}\\left[\\frac{X}{L};P^{(L)}\\right] < \\operatorname{var}[X;P]", "e58cc51d3b6ea5f351a2f5234b2c23fc": "v\\in H", "e58d0b276644e617a92ef3f69f2b98bf": "Z_{\\Delta^n}", "e58d1ec9f5f871b17894562cf82447b6": "Q^{(g)}", "e58d90256b48170f30dec9d9511ba50c": "1/2,\\ 1/2", "e58df8086e6c552ac6416010584d3289": "P(A|B)=P(A)P(B|A)/P(B)", "e58e557263e9855372f8cbc2a40180dc": "f(\\alpha) = 0", "e58e5965576ca49474d99ab5724ab5a5": "\n(X\\odot Y)^\\star\\cong Y^\\star\\circledast X^\\star,\n", "e58e66e5cec965fb705434a702e8ea2c": "a_1= 10, \\, a_2 = 8 \\cdots a_6=5 ", "e58e8b2f85cd1563c168a03ba6d502d9": "A, B \\subset X", "e58ea54aa6aa6f5fbd68f5b1beb85231": " A_o = 0.9 \\approx 877 \\ hours \\ down \\ time \\ per \\ year", "e58eccc17a94a50f9a70ef0ba1b66ce2": "\\mathrm{2\\ AmF_3\\ +\\ F_2\\ \\longrightarrow\\ 2\\ AmF_4}", "e58ecfea7ae40185a19acc3a9b771c13": "s_C = \\frac{Q(AB)}{Q(BC)} = \\frac{Q(BD)}{Q(AB)} = \\frac{Q(AD)}{Q(AC)}.", "e58ef4ecd62e66ed3b85443a607b5893": "{\\bar{S}}_9", "e58f439e1c75183f582f8b53c78cc531": "\\begin{align}\np(c_i)& =\\sum_j p(c_i , x_j)\n& = \\sum_j p(c_i | x_j) p(x_j)\n\\end{align}", "e58f4e62b329ba620848db41531cf692": "h(K_{1}, K_{2}) := \\max \\left\\{ \\sup_{a \\in K_{1}} \\inf_{b \\in K_{2}} d(a, b), \\sup_{b \\in K_{2}} \\inf_{a \\in K_{1}} d(a, b) \\right\\}.", "e58f589c88d77b4bb0546bd6a8ceb530": " g(\\theta_m|\\theta_m)=f(\\theta_m) ", "e58f8a02cee62a697914ed13e5441a20": "\\begin{cases}\nQ_1 = 25.5 \\\\\nQ_2 = 40 \\\\\nQ_3 =42.5\n\\end{cases} ", "e58f93ddf813432438ba35db3b9f3591": " |\\mathbf{a} \\times \\mathbf{b}|^2 = |\\mathbf{a}|^2 |\\mathbf{b}|^2 - (\\mathbf{a} \\cdot \\mathbf{b})^2.", "e58f94493b03f7e3f1a19d53fdd67859": "\\left(\\tfrac{8}{7} \\times \\tfrac{6}{5} = \\tfrac{48}{35}\\right)", "e58fe3a0219d8ebb50b0ff983cea1eda": " \\mathbb{P}^{n}_X := \\mathbb{P}^n \\times_{\\mathrm{Spec}\\mathbb Z} X", "e59033503f280cb4b1392c9b9b737c30": "\\sin \\theta_1 = -\\frac{(x_0+d) f_y - y_0 f_x}{r_1} \\quad (22)", "e59043413bf2632e2df2143380b19e8d": "x_N\\,", "e590ca4843a9140d50c2c18544788075": "\n\\tau = R C\n", "e590d1bbaec1855a667096aafd21a526": "\\theta_k = (\\theta_k^1,.....,\\theta_k^N)\\in\\mathcal S^1\\times......\\times\\mathcal S^N", "e590d9a849d0066763688077bf9f1ce9": "\\ln \\left( {\\frac{{J_2 }}{{J_1 }}} \\right) = 2\\ln \\left( {\\frac{{T_2 }}{{T_1 }}} \\right) + \\frac{{\\ W }}{k}\\left( {\\frac{1}{{T_1 }} - \\frac{1}{{T_2 }}} \\right)", "e590da16fc0763dd425b566e69d3ab0d": "\\mathbf{D} = \\{ D[k, l] \\} ", "e5911277e0608384df8c50ddbd5bb7ba": "c>a>b", "e591156e47a6d39a8abbafd39da15be8": "T^{K+1}p=0", "e5912259c153310fc772326506bead79": "\\textstyle E_{1} ", "e5914dc1e884a5fed64ce25cdb8b9b89": "2g-2+r>0", "e592020ab52ba56656d2cd92eb062144": "p \\mid b", "e5923378896abb2a99ec9852f0ce5c0b": " \\cos \\theta_c =\\mathit{f}_1 ( \\cos \\theta_1 + 1) -1 \\,", "e592405ab89794c85a5df2ce8c9511e4": "r:= \\lfloor 0.5 + \\mu_{k,l} \\rfloor", "e592a92a7e10d950bd65280459cbd246": " \\mu_g = \\sqrt[n]{ A_1 A_2 \\cdots A_n }.\\, ", "e592c92279ef06a9c9aaf34a72e9f170": "|s_2| = 6", "e592f7d92afcc3a633f0062d8c76996e": "A\\in\\mathbb{R}^{2\\times 2}", "e5934112798dd3b20cee2017a08f21e8": "P_i=\\left(i,v_K(a_i)\\right),", "e593a0e834d9d1e6df6db35501b573fe": "y_1 \\leq y_2", "e5940114e6029a71840fc05b93b468f5": "1-e^{-\\gamma}", "e59407b0213394f61ea5ec3922689cf7": "{\\mathbf Q} \\otimes {\\mathbf Z}_n", "e59446fb86aa58ac25af135b2437dd18": "x^{11} + x^9 + x^8 + x^7 + x^2 + 1", "e5949381c3477b6ea6a430a0616c2cdb": " \\Phi_{I} = ", "e594bae4a0a23eaf623f0c8a261cc063": "r = \\frac {1}{s} = \\frac {p}{1 + e \\cdot \\cos \\theta}", "e59581f55fe91dd8c2dfffcb3591a71f": "r_i(\\boldsymbol \\beta)= y_i - f(x_i, \\boldsymbol \\beta),\\ (i=1, 2, \\dots, m) ", "e595ab76fde013e930ee3d5f08928130": "(X_{n-1}-e_{n-1}^{\\beta})", "e595d5b9e6d50b41bfd70ed42344df74": "K(X)", "e5962e36aadde9808e169a5f4c89ed35": "| R \\rangle ", "e5967f1c5514004cc0b569aaf3b16761": "\\pi=3.14159265358979...", "e596b9e488cd864777ee8018adae580c": "\\!\\mu_3(v_2)", "e596f607f57c1ad55f0d1ba156257a3e": "x_1^{\\min (e_1,1)}\\cdots x_n^{\\min(e_n,1)}.", "e597b142e01ff8e8492cc1e79951ec6d": "\\frac{1}{2} \\left( 1 + \\frac{4\\pi h}{\\lambda}\\frac{1}{\\sinh\\left(\\displaystyle \\frac{4\\pi h}{\\lambda}\\right)} \\right)", "e597d54137fd4277edd27165629fe7ec": "Ak^a-c_{T-j} \\ge 0", "e5982036700fd91ce3cedc16dbade02f": "\n\\left\\{\n\\mathbf{x} \\in \\mathbb{R}^n : \n\\sigma_k(\\mathbf{x}) = 0\n\\right\\}\n", "e598b6f40ed4a39ebdaaa41c73fe2dda": "\\int_{-\\infty}^\\infty e^{-x^2} \\, dx=\\sqrt{\\pi}", "e598cf5f2b28263fdfde326954535b6b": "p(\\mathbf{x})", "e598d56b284a4735bea9d0a151910f86": "[L_m,L_n]=(m-n)L_{m+n}+\\frac{c}{12}(m^3-m)\\delta_{m+n,0}", "e598f4d992a1f67f9c25cc394b05138e": "(\\mathbf{\\lambda}x:A . B)", "e599343e7f9fdfd9d39cbc42a9e4b8ec": " \\int \\mathbf{v}\\mathbf{e_{\\phi}}\\mathrm{d}\\phi=2\\pi ", "e59984d904de8fc22d8b627b44b96de9": " \\mu \\,", "e599d0c25f841b21fa7c5547c04ce182": "u = \\frac{\\partial \\psi}{\\partial y}\\,", "e59a14eb1c04ce0000eaff70397ba1dc": "x/x = 1 + 0x/x\\ ", "e59a3e9bf4f6704352c092706e836bd4": "Z_{in2} = \\frac{{K}{Z}}{K-1}.", "e59a5b12e1c6ea86b7fa63a833e1327d": "{n \\choose k} = \\frac{n(n-1)\\dots(n-k+1)}{k!} \\approx \\frac{(n-k/2)^k }{ k^k e^{-k} \\sqrt{2\\pi k}} = \\frac{(n/k-0.5)^k e^k }{ \\sqrt{2\\pi k}}", "e59b8cc030a99c1009278c29c07e5786": "(\\exists k)(y\\cdot k=x).", "e59bad684a72b87e971857a335375cfe": "G =\\exp \\mathcal{G=}\\exp (\\mathcal{G(}A\\mathcal{))=}\\sqrt{\\frac{1}{(1-2A/w)\n}}.\n", "e59c032f3299c0e1d55ada5b56d0d0e2": "\\mathbf N", "e59c5457c678ca4b9dd85b3dd30d47a7": "e \\ge 0", "e59c995cc3d919cae7c8cc3dd553059f": "\\cfrac{\\mathrm{d}\\boldsymbol{\\varepsilon}}{\\mathrm{d}t} = 0", "e59cf4ae34ea19d032f00b2f01ddd3b0": "L_0 = \\pi_1(X)", "e59d17455cf34c03504ddb9a756ce4cf": "\\pi': X' = X \\times_Y Y' \\to Y'", "e59d1fda8fc31a555203e01b7e2d472f": "\\rho_\\text{ions}(r,t)", "e59d6569b4396ca34bcddf6b654606c5": "\\rho(z_1,z_2)=\\tanh^{-1}\\left|\\frac{z_1-z_2}{1-z_1\\overline{z_2}}\\right|", "e59dc8331a3e4fdcc696872249cfbd56": "~+~", "e59e38b7ba18914a45c73847d856828b": " u, \\bar{u}, \\mbox{ and } u^{\\prime} ", "e59ef1a6bf2a5e29919ca5138a19876d": " S_5 = S_8, A_5 = A_8, S_4 = S_7, A_4 = A_7, S_3 = S_6, A_3 = A_6 ", "e59efa1a144f7615bce2e0af6fa23b70": "\\chi(g)", "e59efe478677fa9850bc1bff1cc58732": "\\mathfrak{P}^{100}", "e59f34c0a8ac44313ead5fe6fe58ebf6": "v, h", "e59f60bb921b75a878115294671b2638": "\\bold\\theta =", "e59f61b1c6dde3d92fac12d5d3d1d494": "L_{\\mathrm{MV}}(x,y,z) = \\sqrt{\\frac{(x-y)\\cdot(y-z)\\cdot(z-x)}{2\\cdot((y-z)\\cdot\\ln x + (z-x)\\cdot\\ln y + (x-y)\\cdot\\ln z)}}", "e59fbb29e74130d05108c0eb4b6ed56c": "\\mu_\\ell", "e5a00d9a13f5c3be6265ea338402077f": "\\check{e}_i", "e5a0245e4258cfce139e2483a609cee3": " Y=\\{y_1,y_2,\\ldots, y_j\\}", "e5a026539305d24fc167282f61c97a0b": "\\frac{\\sin(\\pi x)}{\\pi x} = \\frac{1}{\\Gamma(1+x)\\Gamma(1-x)}.\\,\\!", "e5a048bf5ee3c0a3088100487dc97663": "R_{f}=(\\frac{n-1}{n})R_{p}.", "e5a05a570db574f3304b068eb86006f1": "\n\\begin{align}\nI_1 I_3 \\ddot{\\omega}_{1}&= (I_2-I_3) (I_1-I_2) \\omega_1\\omega_{2}\\\\\n\\text{i.e.}~~~~ \\ddot{\\omega}_1 &= \\text{(positive quantity)} \\times \\omega_1\n\\end{align}\n", "e5a091b47c5d61dc999581eeb657a05a": "\nk^\\epsilon (x)=\\begin{cases} 1, & 0 < x < 1 \\\\ \\frac{1}{\\epsilon^2}, & 1 < x < 2\n \\end{cases}\n", "e5a0c6e05a980f41c1b935d5a47517a0": "p_\\alpha^{-1}(U)", "e5a1132b8a92398b9dd08d836b74acad": "\\Theta(E-E_g/2)", "e5a1327c90b996cfee9e09a2f6d6c345": "z=-\\infty\\,", "e5a1bfb174501ae947987aabd007ea8c": "\\zeta_{a}", "e5a1ecf270d052a7df3e7a08c6cf8d56": "H_{\\nu} (\\omega)", "e5a22dd2cd110cedf00bf7a8bf922f69": "\\frac{\\delta L}{\\delta r_j} + \\sum_{i=1}^e \\lambda_i\\frac{\\partial F_i}{\\partial r_j}=0 ", "e5a22e0a2359937781d39d05207528cf": "\\scriptstyle\\varphi(\\mathbb{E}\\{X\\})", "e5a22e39a07afc76b8f25fd9dfa59eec": "\\operatorname{ncut}(A, B) = \\frac{w(A, B)}{w(A, V)} + \\frac{w(A, B)}{w(B, V)}", "e5a23578cdb4d60badd4ca7cea6cdc55": "\\psi(x)", "e5a28b6e72bd235f8ead70b7f8d401ef": "k^a=l^a", "e5a2b9631ecdbf698f45daf9384baf6a": "\\left(\\frac{a_n}{a_0}\\right)^{\\frac1n},", "e5a2d165167f83a7ea7e38c5f40812a4": "\\delta_j=0", "e5a2ecb3370fcb29d19ab2cb72216301": " \\mathbf{b}=\\mathbf{A}\\mathbf{x}_* = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i.", "e5a305f837d10ce60985466e3c84d16b": " | \\alpha - \\xi | < \\frac{1}{H(\\xi)^\\kappa} ", "e5a3d84e4af0079afe1764fe3bee59de": " \\frac{\\partial M}{\\partial x} \\neq \\frac{\\partial N}{\\partial y} \\, \\! ", "e5a3e971144a9f0ebedb23ed9fd4b332": "j((1+\\sqrt{-d})/2)", "e5a3f8152515d12a138086cb047fa57a": "\\bar{x}_{A}", "e5a40997f3bcf6ed9a42b1c368284231": "U = -\\frac{3M^2G}{5R}", "e5a452eab9bf928fe62d5b97f33af34c": "\\because g{(-x)}=\\frac {f{(-x)}+f{(x)}}{2}=g{(x)}", "e5a48c0002b6b1b2a218087ced1f251b": "\\begin{align}\nx(u,v) &= u - \\cosh(v)\\sin(u)\\\\\ny(u,v) &= 1 - \\cos(u)\\cosh(v)\\\\\nz(u,v) &= 4 \\sin(u/2) \\sinh(v/2)\n\\end{align}", "e5a4e6c24c26458ba859e6b5ba424bba": "f(x) = x^{3}", "e5a4f1d74bd8009e03d9cea9354988b8": "i\\notin\\Lambda", "e5a53dc1645b6ab4c8a5cc92bf391e62": "\\begin{align}\n C(0, t) &= 0\\text{ for all }t \\\\\n C(S, t) &\\rightarrow S\\text{ as }S \\rightarrow \\infty \\\\\n C(S, T) &= \\max\\{S - K, 0\\}\n\\end{align}", "e5a5688c2303465a5345c1f7737325c4": "VAG(x^3 -7x + 7,(0,2)) ", "e5a56fe37f907c6e91ef5499921be343": "\\varepsilon_\\mathrm{trans}", "e5a59637e8991737a83c1a053945a920": " a_{11} = p_2p_4+p_1p_5, ", "e5a5ec53794e9220f4d566f023d24f41": "\nT = {\\rm constant} \\times \\frac{1}{\\sqrt{d}},\n", "e5a6138936c033304ddd3b2c9e9b8040": "\\underline{X}", "e5a61df160c02942b69f9d7f13e88d8c": "\n\\mbox{cas}'(a) = \\frac{\\mbox{d}}{\\mbox{d}a} \\mbox{cas} (a) = \\cos (a) - \\sin (a) = \\mbox{cas}(-a).\n", "e5a61e2b57d9bf0bf03b7cf4bf36d4ff": " \\chi(q)=\\sum_{n\\ge 0}{(-1)^nq^{(n+1)^2}\\over (-q;q)_{2n+1}}", "e5a67b9cffd104fb7ca0db68e658a98d": "\nq_L \\rightarrow e^{i\\theta} q_L \\qquad\nq_R \\rightarrow e^{-i\\theta} q_R ~,\n", "e5a6950911de3c9abdaef2d5f8d203c0": "\\frac 1 {n!} (x_1^1 \\cdot x_2^0 \\cdots x_n^0 + \\cdots + x_1^0 \\cdots x_n^1) (n-1)! \\geq \\frac 1 {n!} (x_1 \\cdot \\cdots \\cdot x_n)^{\\frac 1 n} n! ", "e5a6ad723703c2347b67ca078f023a43": "\\ k ", "e5a6dff0ff0121caff8c2c66ed49f0d4": "\\hat{\\phi}", "e5a6f82aed7c5be5621da514b3e44322": "E_{\\pm}", "e5a7472d780a5a032c7775cc5e3ce901": "s_{i}", "e5a7b44766a1098a7cb7bd3335758caa": "G_s=G_s(h,k)", "e5a80ad36b8e1aa7e3448c0eb6d6ee54": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{4}{\\sqrt{3}},\\ 0\\right)", "e5a877ad9c0219ca4c5f89ec1467371c": "{V_\\mathrm {dc}=V_\\mathrm {av}=\\frac{3V_\\mathrm {LLpeak}}{\\pi} \\cos(\\alpha) } - {6 f L_\\mathrm {c} I_\\mathrm {d} }", "e5a8ca42018b06ead852890e43a6bf04": "l_\\text{m}", "e5a8d405bf46f7c522d473eb0ffb7a70": "h=(b-a)/n.", "e5a9ac6464044fbdf00acd69e29b26d2": "v_g = \\frac{c}{n + \\omega \\frac{\\partial n}{\\partial \\omega}} = \\frac{c}{n - \\lambda_0 \\frac{\\partial n}{\\partial \\lambda_0}} = v_p \\left( 1+\\frac{\\lambda}{n} \\frac{\\partial n}{\\partial \\lambda} \\right) = v_p - \\lambda \\frac{\\partial v_p}{\\partial \\lambda} = v_p + k \\frac{\\partial v_p}{\\partial k}.", "e5aa0528458115afa5f3d8fe261b243c": "\\mathrm{\\Lambda}^{\\!\\otimes}\\! \\left({A}\\right) = \\left\\{ x+yi : 0\\leq x, 0\\leq y, \\sqrt{x}+\\sqrt{y}\\leq 1\\right\\}.", "e5aa293112dd8ba43827ff71bbd6bd96": "8\\pi/3", "e5aa3a8c11f43878bdc4a2a08a128879": "\nw_n\\in\\mathcal{P}, \\text{ that is }: w_n \\text{ is bounded to }\n[-U,U], \\mathbb{E}(w_n)=0\n", "e5aa52c2e9114a67549bb022e374878c": "c_n = \\frac{ \\langle f,(J_\\alpha)_n \\rangle }{ \\langle (J_\\alpha)_n,(J_\\alpha)_n \\rangle } = \\frac{ \\int_0^b x f(x) (J_\\alpha)_n(x) \\mathrm{d}x }{ \\frac12 (b(J_{\\alpha\\pm1})_n(b))^2}", "e5aa9207ec71a600d05fb49733053f71": "0.509\\,\\mbox{mol}^{-1/2} \\mbox{kg}^{1/2}", "e5aa9afdd8c1a9a21d4b131a19b542ae": "z_1,\\dots,z_n", "e5aab23eeb768d6dfe7f657383f59457": "w_i \\leqslant w", "e5aadcb476b5d13944833203f9b8752b": "\\textbf{K}", "e5ab11ef2430ee64ff27952ed243c85d": "\n |j_2 m_2\\rangle,\\quad m_2=-j_2,-j_2+1,\\ldots, j_2.\n", "e5ab3eac6780b9dc14194bc31c87d533": "\\frac{M}{p_i} \\neq O", "e5abcdee474fed8b5fe889ef8c3118a2": " ~P(z,T) ", "e5abdc980f6ec955208ab9f5b148311c": "pV=nRT", "e5ac988d86e689d721745a267756c740": "H'_{abc} = H_{abc}+\\Phi_{[a} g_{b]c}", "e5aca56878d6e8e364036e62a48e5c96": "((y_n),(\\psi_n)) ", "e5acdca1d12acea24864d32adb45ce40": "\\scriptstyle f_{c2}", "e5acf5a16a7b65dcad2aa19ea0022c68": "\\textstyle \\mathbb{F}_{2}", "e5ad8264a716ebd6c5211add1dcba276": "\\varepsilon\\to 0.", "e5ae6f1df3684e9388f5380d8cc610f5": "\\begin{align}\n 0 &= [(0,0)] &= [(1,1)] &= \\cdots & &= [(k,k)] \\\\\n 1 &= [(1,0)] &= [(2,1)] &= \\cdots & &= [(k+1,k)] \\\\\n-1 &= [(0,1)] &= [(1,2)] &= \\cdots & &= [(k,k+1)] \\\\\n 2 &= [(2,0)] &= [(3,1)] &= \\cdots & &= [(k+2,k)] \\\\\n-2 &= [(0,2)] &= [(1,3)] &= \\cdots & &= [(k,k+2)].\n\\end{align}", "e5ae7812279f12bc18d1db6c7370e80a": "\\phi_3=-11.25^\\circ", "e5aec79b2235600dd537c5bfbaf271bf": "\\begin{align}\n\\sigma_{12}' = &a_{11}a_{21}\\sigma_{11}+a_{12}a_{22}\\sigma_{22}+a_{13}a_{23}\\sigma_{33}\\\\\n&+(a_{11}a_{22}+a_{12}a_{21})\\sigma_{12}+(a_{12}a_{23}+a_{13}a_{22})\\sigma_{23}+(a_{11}a_{23}+a_{13}a_{21})\\sigma_{13},\n\\end{align}", "e5af3219e1a72ab715233da7a77df3bb": "R^r \\to R^g", "e5af673754415216c5c1f1c51c9c2cf9": "\\begin{cases} u_{t}=ku_{xx}+f(x,t) & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ u(x,0)=0 & IC \\\\ u_x(0,t)=0 & BC \\end{cases} ", "e5afb03baafa90d4187026c91e303f61": "\n\\lim_{n\\rightarrow\\infty} n^{-3/4}\n \\ln p_3(n) \\rightarrow \\text{a constant}.\n", "e5afb88a631d1f57f5c987e2c401d555": "\\boldsymbol{\\psi}=\\psi\\mathbf{z}", "e5afd2480da99cbb213ddbeca811cf0d": "x \\in C(W,p)", "e5afea7e9d258e7d01bd13af2fca1314": "P \\and (Q \\and R)", "e5b019c1217bb034d278561499814b31": "\\varepsilon_1 = \\psi(1)", "e5b06015102685e093d4d9ed656a12ea": " \\sigma_k = 1.00 ", "e5b0ac10fcda9e4f0ac2ff7036202570": "\\text{MA} = \\frac{F_w}{F_i} = \\frac {1}{\\sin \\theta} \\,", "e5b0feb7437451cbf60a94a83bdfc5c9": "4\\sin^2(\\pi/10)", "e5b13c6e620ae1acfc4ec8c1d357a912": "\\begin{bmatrix}\nu\\\\v\\\\w\n\\end{bmatrix} =\n\\begin{bmatrix}\n\\frac{(\\kappa E_0)^2 + \\delta^2 \\cos \\Omega t}{\\Omega^2} & -\\frac{\\delta}{\\Omega} \\sin{\\Omega t} & -\\frac{\\delta \\kappa E_0}{\\Omega^2} (1-\\cos \\Omega t) \\\\\n\\frac{\\delta}{\\Omega}\\sin\\Omega t & \\cos \\Omega t & \\frac{\\kappa E_0}{\\Omega}\\sin \\Omega t \\\\\n\\frac{\\delta \\kappa E_0}{\\Omega^2} (1-\\cos \\Omega t) & -\\frac{\\kappa E_0}{\\Omega} \\sin{\\Omega t} & \\frac{\\delta^2 + (\\kappa E_0)^2 \\cos \\Omega t}{\\Omega^2}\n\\end{bmatrix}\n\\begin{bmatrix}\nu_0 \\\\ v_0 \\\\ w_0\n\\end{bmatrix}", "e5b18fbc7d87c44f58b7a79d8d6d3df2": "[(E^2 - (mc^2)^2) - (\\hat{\\mathbf{p}}^2)]\\left( (E^2 - (mc^2)^2)^{2j-1} + (E^2 - (mc^2)^2)^{2j-2}\\hat{\\mathbf{p}}^2 + \\cdots (\\hat{\\mathbf{p}}^2)^{2j-1} \\right)\\begin{pmatrix}\n\\psi_{1,2}^{[2j]} \\\\\n\\psi_{3,4}^{[2j]}\n\\end{pmatrix}\n = 0", "e5b1a15875fd3038c927a4315f77bcb8": "I_y=\\frac{bh^3}{12}+Ad^2", "e5b1a5765aa69453c5abed7e2a67d48b": " y = (y_1,\\alpha_1 y_1)", "e5b1be8c24ba288e292c99647564b7a0": "SM_4(gevol,endo,exo)=RE", "e5b1c1a4684015a379a779ff36e85a72": "Z_3 = (Y_1+Z_1)^2-D-ZZ_1", "e5b1d9c483179ccb085d6cee1d7baf6f": "H_\\mbox{b}(p) = H(p, 1-p) = - p \\log p - (1-p)\\log (1-p).\\,", "e5b1eca54dc6cc558bd8ab248f7c1deb": "\\textstyle A=\\{a_1,\\ldots,a_m\\}", "e5b1f9ed81a8e1a8ec7c0ee22237e130": "\\langle (x:=x+1)*\\rangle x=7\\,\\!", "e5b201f4524688be6dad1c27290765af": "\\text{shaves}(x, x) \\wedge \\neg \\text{shaves}(x,x)", "e5b21db82a0b379c177e45d321e76be3": "(1-0.0164)^4-1=-6.4%", "e5b22aba2247b81158c1fac68dd49d52": "\n\\frac{p(S|h,\\Theta)}{p(S|h,\\Theta_{bg})} = \\prod_{p=1}^P G(S(h_p)|t_p,U_p)^{d_p} r^f\n", "e5b2b1c8029cd5df61db5d047ff225eb": "\\mathrm{Res_0}\\big(V(z)^{-k}\\big)=kv_k", "e5b2d750a8f63c6da0393194de2ebac6": " \\mathbb{R}P^n", "e5b2fba388644e3bd40e3970bb53b2c8": "n^2 = n^2", "e5b321471afba5edc66fcb951521d3a5": " \\!\\ \\frac{1}{2}\\left(n+\\sqrt{n^2+4c}\\right) = R ", "e5b334ab88ac9028f4e278630d9a05b0": " \\langle f, f \\rangle < \\infty. \\, ", "e5b336950f5ed734d7c7687039d2129b": "\\gcd(a_1, a_2, \\ldots, a_n) = d", "e5b377adcb9a68e122df5e8c4d83a101": " GT = K \\times V\\,", "e5b3819799b18c542307257031fe8c82": "G(z)", "e5b38c5d35edf6e7566ae4a3d85c3814": "e\\epsilon(t)\\exp\\left(-i\\dfrac{E_{0} - E_{1}}{\\hbar}t\\right)\n\\langle \\psi_{1} |x|\\psi_{0}\\rangle = i\\hbar c_{1}'(t)", "e5b38f835364a39bbd905befe7be0188": "C_x(p_i)", "e5b3bfc1669529b693e25b58527d6034": "W = \\biggl|\\bigcup_{m=1}^nA_m\\biggr|.", "e5b3e96fc890b93b3f46e18730d69db1": "v_n=n\\pi/2", "e5b4a0bcab41ab917c15d3202fb6a33c": "k_{sh}", "e5b4b3bd54c64833d65a89f05fec05a6": "\nn_{1}=\\frac{r_{13}\\times r_{14}}{\\left|r_{13}\\times r_{14}\\right|},\\; n_{2}=\\frac{r_{14}\\times r_{24}}{\\left|r_{14}\\times r_{24}\\right|},\\; n_{3}=\\frac{r_{24}\\times r_{23}}{\\left|r_{24}\\times r_{23}\\right|},\\; n_{4}=\\frac{r_{23}\\times r_{13}}{\\left|r_{23}\\times r_{13}\\right|}\n", "e5b50b755fefadafb7ef230a75586614": "\\mathbf{Q} = \\frac{1}{N-1}( \\mathbf{F} - \\mathbf{\\bar{x}} \\,\\mathbf{1}_N^\\mathrm{T} ) ( \\mathbf{F} - \\mathbf{\\bar{x}} \\,\\mathbf{1}_N^\\mathrm{T} )^\\mathrm{T}", "e5b5270088ae3d49dd03a55eb8d97827": "{ d^2x^j\\over dt^2} = -{\\partial\\phi\\over\\partial x^j\\,}.", "e5b52a3fa5f8cfc91b9a1bb829deb1bf": "\\int \\frac{dx}{S} = \\frac{2S}{a}", "e5b53a148c1ff8536a8de3e422dc9843": "\\overline{ C_1 },\\overline{ S_1 }", "e5b53f6aae0bd836b73cd7d493cbf699": "q = \\kappa_1 / \\kappa", "e5b5a78c71eaab37b5d59e5b7195342c": "\\displaystyle{D(u,v) =(Lu,v).}", "e5b5b8144acd883f81949c9e4836893b": "\\scriptstyle \\left|V_o\\right|=\\frac{V_o}{V_i}", "e5b5bad010002209dfda2fdcac380555": " p_i = {\\partial T\\over {\\partial q_i\\over \\partial t}},", "e5b621768f6a5ddd50f276bf54d8a0e9": " \\left[\\frac{\\left(\\frac{^{207}Pb}{^{204}Pb}\\right)_{P}-\\left(\\frac{^{207}Pb}{^{204}Pb}\\right)_{I}}{\\left(\\frac{^{206}Pb}{^{204}Pb}\\right)_{P}-\\left(\\frac{^{206}Pb}{^{204}Pb}\\right)_{I}}\\right]= {\\left(\\frac{1}{137.88}\\right)}{\\left(\\frac{e^{\\lambda_{235}t}-1}{e^{\\lambda_{238}t}-1}\\right)}", "e5b622870cb76f983fde98f38bc09d54": "E(R_m)-R_f~", "e5b625fb308c16319c8f9cbf4320e75b": "\\frac{p_{2}}{p_{1}}=\\frac{1+\\gamma M_{1}^{2}}{{1+\\gamma M_{2}^{2}}} = \\frac{2\\gamma}{\\gamma+1}M_{1}^2-\\frac{\\gamma-1}{\\gamma+1}", "e5b64e5fa3e5e179a8822725ce0b004b": "=\\left(1/b\\right)\\left[\\beta/\\left(\\beta-1\\right)\\right]\\ln\\left(\\beta\\right),", "e5b6673a0507091d0de6e82fa914465c": "ax+by+cz=0,\\,dx+ey+fz=0,\\,gx+hy+iz=0", "e5b68b6a88566cbd8ff97b536855a0dc": "H^1[0,1]", "e5b68f3bd8385ca80c10e2fbb6d899ca": "A\\times A", "e5b6a989f6ec8ea79014ece17f131420": "P_{(k)} \\leq \\frac{k}{m \\cdot c(m)} \\alpha ", "e5b6bbda68f98699848a44c92a8af554": "\\prod_{m=1}^\\infty \n\\left( 1 - q^{2m}\\right)\n\\left( 1 + (w^{2}+w^{-2})q^{2m-1}+q^{4m-2}\\right),", "e5b76f1e6abd96fb5da4ab29d0024674": "(1+1/n)^n=\\sum_{k=0}^{n}\\binom nk/n^k=\\sum_{k=0}^{n}\\frac1{k!}\\times\\frac nn\\times\\frac{n-1}n\\times\\cdots\\times\\frac{n-k+1}n,", "e5b779594fd8cf60ecc960c793933910": "\\frac{\\Gamma(\\frac{\\nu+1}{2})} {\\sqrt{\\nu\\pi}\\,\\Gamma(\\frac{\\nu}{2})} = \\frac{(\\nu -1)(\\nu -3)\\cdots 4 \\cdot 2} {\\pi \\sqrt{\\nu}(\\nu -2)(\\nu -4)\\cdots 5 \\cdot 3\\,}.\\!", "e5b79a297bde9cf6cf61fb64809f02c6": "C_{yx}\\subseteq \\bigcup_{i \\in D} Y_i \\times \\bigcup_{i \\in D} X_i", "e5b7c9019487859e07a3d45ff1bd72d8": "\\mathrm{P}(X = x | Y = y, Z = z) = \\mathrm{P}(X = x | Z = z)", "e5b848ec663dc064e00a68021a076114": "I(V+k\\sin(\\omega t))\\approx I_0+I'(V+k\\sin(\\omega t))+O(I'')", "e5b88703d1ca70be5c3510ebdb80aa3b": "P(x) = \\frac{bT^4}{3}\\,", "e5b8bf94e36d807239626387afa6bd98": "p(X|\\sigma,I)={1 \\over \\sigma}f({X \\over \\sigma})", "e5b8fece34aaab9a9cdcf795ba965b5a": " \\int_0^2 \\! \\int_{0}^{\\pi/2} \\! \\int_0^2 \\! \\bar{f}(r,t,h) r \\, dh \\, dt \\, dr = 24 + 40 \\pi /3", "e5b974980effa873a008fa4eff14d6e6": "f(x) = \\frac{1}{2}(2\\pi)^{1-n}(-1)^{(n-1)/2}\\int_{S^{n-1}}\\frac{\\partial^{n-1}}{\\partial s^{n-1}}Rf(\\alpha,\\alpha\\cdot x)\\,d\\alpha", "e5b98395fb433f933f1919176b6fc93c": "1-q", "e5b991bfa109ac3f24d617db864b3641": "\nZ(\\mathbf M) = \\oplus _i Z(\\mathbf M) P_i ", "e5ba2845fc262039d7f2a6b0fcd0cba8": "P_\\mathit{leakage} = I_\\mathit{leakage}V", "e5ba284612bd7ea562f8bcac02bbb7d4": "\\sqrt{(x-n a)^2+L^2}\\approx L+ (x-na)^2/2L", "e5ba5f6b6870fa56ec98259da3cf7ef4": "b_2Z_2", "e5ba78ab86fff4c9bf64658371d64547": " F=\\sum_{i,j=1,2}\\frac{1}{2m} |(\\nabla - ie A) \\psi_i|^2 + \\alpha_i |\\psi_i|^2 + \\beta_i|\\psi_i|^4 +\\frac{1}{2}(\\nabla \\times A)^2 ", "e5ba877c113b4d7d00a8aa60cc9a1f2b": "\\frac{dy(t)}{dt}=(c-a)*x(t)-x(t)*z(t)+c*y(t)", "e5bae9763722318e1781d59c0429e465": "\\displaystyle \\int_{-\\infty}^{\\infty}f(x) e^{-2\\pi i x\\xi}\\, dx ", "e5baec818b17c50b9d2f859d3987e14f": "\n\\tilde{\\delta}(t) \\ge \\delta = \\frac{1}{K}\\,\\frac{dK}{dt}, \\quad \n\\tilde{\\nu} (t) \\ge \\nu = \\frac{1}{L}\\,\\frac{dL}{dt}, \\quad \n\\tilde{\\eta}(t) \\ge \\eta = \\frac{1}{P}\\,\\frac{dP}{dt},\n", "e5baf673deace2eab5cd8e391d26d431": "\\kappa.\\,", "e5bb23797bfea314a3db43d07dbd6a74": "gf", "e5bb2fc7d9f565ee1eb00ad1f1c84f26": "(x,\\ y)\\,\\!", "e5bb326917b1f35a6d064a862cc60d48": "J_j: H_j^* \\to H_j", "e5bb51b8a5bc02b0a0bfb3f46c7bc2b1": " \\tan\\left( \\frac{\\alpha+\\beta}{2} \\right)\n= \\frac{\\sin\\alpha + \\sin\\beta}{\\cos\\alpha + \\cos\\beta}\n= -\\,\\frac{\\cos\\alpha - \\cos\\beta}{\\sin\\alpha - \\sin\\beta}", "e5bc3edbe3d14bb8d8623d0048e0284c": "\n\\varepsilon_{FY}\\circ F(\\eta_Y) : \nY \\otimes_R X \\to \n\\operatorname{Hom}_S (X , Y) \\otimes_R X \\to\nY \\otimes_R X\n", "e5bc5727c3ec5c5358ef53c4bbe7a8c7": "C_J=2 \\cdot(\\frac{\\mu_1}{r_1}+\\frac{\\mu_2}{r_2})+ 2n(\\xi \\dot \\eta- \\eta \\dot \\zeta) - (\\dot \\xi ^2+\\dot \\eta ^2+\\dot \\zeta^2)", "e5bcba9087cd7de62fe99f6def934d04": " \\int_A^B e^{iS} D\\phi\\,, ", "e5bcf4dd0c2a6d2072da2d2b911248df": "\n\\begin{align}\n\\lim_{|x| \\to \\infty} x \\sin \\frac{1}{x}\n& = \\lim_{|x| \\to \\infty} \\frac{\\sin \\frac{1}{x}}{1/x} \\\\\n& = \\lim_{|x| \\to \\infty} \\frac{-x^{-2}\\cos\\frac{1}{x}}{-x^{-2}} \\\\\n& = \\lim_{|x| \\to \\infty} \\cos\\frac{1}{x} \\\\\n& = \\cos{\\left(\\lim_{|x| \\to \\infty} \\frac{1}{x} \\right)} \\\\\n& = 1.\n\\end{align}\n", "e5bd45326c1ebe2b756bfa8733efc486": " \\R c ", "e5bdfab61e664b1f90de986984983a89": "\n\\rho({\\mathbf u}\\cdot\\nabla){\\mathbf u}={\\mathbf F}-\\nabla p,", "e5be1b8adcdd29d455156aa88925b45b": " c_0 = a_0^n, \\quad c_m = \\frac{1}{m a_0} \\sum_{k=1}^m (kn - m+k) a_{k} c_{m-k}, ", "e5be1ecc59d017abc9eef8925c553f74": "\\ \\begin{align}\n x & = R\\left(\\cos\\frac{\\theta}{2}\\cos v-\\sin\\frac{\\theta}{2}\\sin 2v\\right) \\\\ y & = R\\left(\\sin\\frac{\\theta}{2}\\cos v+\\cos\\frac{\\theta}{2}\\sin 2v\\right) \\\\ z & = P\\cos\\theta\\left(1+e\\sin v\\right) \\\\ w &= P\\sin\\theta\\left(1+e\\sin v\\right)\n\\end{align}", "e5be21baf997b99ae3b42443f658d8eb": "r_\\mathrm{e} = \\frac{1}{4\\pi\\varepsilon_0}\\frac{e^2}{m_{\\mathrm{e}} c^2} ", "e5be25092862819cf9431a286832acbe": "{}_8C_1 + {}_8C_3 + {}_8C_4", "e5be41dd60ea00198a634d3dc05b593c": "x = A^+ b + [I - A^+ A]w", "e5bea026cc24b93a19e8cb64c37252f2": "dN_B/dt = 0", "e5bf2f3d99d0cc404940bbb12ef2a7a1": "\nH_{\\mathrm{grav}} = - \\frac{3G M^{2}}{5R},\n", "e5bf3539db1881dd6ff1bc13f6541d1a": "{\\mathcal Z}", "e5bf4d3297cf4a2f8328ba0aa6cf6f20": "\nd \\sigma_t = (\\beta_t-\\sigma_t)\\,dt + \\sqrt{\\sigma_t}\\,\\eta_t\\, dW_t", "e5bfc096ce4efde50de3a2cb33838ae1": " e^i \\left (p,u^i \\right ) = f^i(p) + u^i g(p) ", "e5c03e68c4da66047b971e7b906a90d9": "\\delta_u H\\,", "e5c05a9fd204d4e33a2101e5ee6c8f0a": "\\mathbf {F}_i", "e5c0757902564f0f4c00b1954b0b1a2e": "\\mathcal{FL}", "e5c08ddaf76070e82167fdfefc4d6c13": "\n\\mathbf{Q} = \\frac{\\partial G_{2}}{\\partial \\mathbf{P}}\n", "e5c0fd7df7865e9621991ccf55d4564f": "\\begin{pmatrix}\na & b \\\\\nc & d \\end{pmatrix}\n\\quad\\quad\\begin{bmatrix}\na & b \\\\\nc & d \\end{bmatrix}\n", "e5c10437531ce28cfda088d715544872": "(p+A)(V-B)=CT, \\,", "e5c114e863587daac088579bedff4d77": "w_0,\\ldots,w_{k-1}", "e5c119f76a39c376c175ad500408cc3c": "\\psi_R\\rightarrow e^{i\\theta_R}\\psi_R.", "e5c126b2e64889a6606450082daaef07": "\\frac{p}{r}=\\frac{q}{p}.", "e5c16f180f5885ff954ce7233af253d3": "3\\alpha - 4\\beta = \\alpha", "e5c18b3b7c0de062b092299bc8b04bb3": " \\sigma_k (k = 1,2,3)", "e5c18b3fdc65d5a942acbe67e7ae3396": "(-1)^{p_1 r_1}", "e5c191677bb386011af288aa5781765b": " \\vec v = \\frac {d}{dt} \\vec r(t) = \\frac {d R}{dt} \\hat u_R + R\\frac {d \\hat u_R } {dt} \\ . ", "e5c19375e6541390af0f1d762bc04c7f": "d = {(\\mathbf{p_0}-\\mathbf{l_0})\\cdot\\mathbf{n} \\over \\mathbf{l}\\cdot\\mathbf{n}}", "e5c1b983b66620a1b154c824f647c421": "D_\\mu=\\partial_\\mu + \\frac{1}{4}\\omega_{\\nu\\rho\\mu}\\gamma^\\nu \\gamma^\\rho", "e5c1d184e7742967b0d15eaa1cd0feab": "\\scriptstyle 2\\sqrt{1-y^2}", "e5c1d98cc4a4bfe41ba6aed16dcf5b04": "\\mathbf{x}'' = R\\mathbf{x}R^{-1}", "e5c208e982841d4933966788443506ed": "y \\in Z/pZ", "e5c232d4d1d70ed60163a98cc79934a6": "\\frac{x}{t}", "e5c2cd221f73134612a88b60cebe3560": "\\Rightarrow_{r_3} aSSS \\Rightarrow_{r_3} aaSS \\Rightarrow_{r_3} aaaS \\Rightarrow_{r_3} aaaa \\Rightarrow_{r_3} aaaa", "e5c3219ac28137f88ef2f41172152a9b": "\\lim_{\\epsilon\\to 0}\\varphi_\\epsilon(x) = \\lim_{\\epsilon\\to 0}\\epsilon^{-n}\\varphi(x / \\epsilon)=\\delta(x)", "e5c322277623b9d609e7b8f46c48a799": "kj = -i\\,", "e5c3c949edeb6935014612c7c10cadb0": "H^q(V,E)\\cong H^{n-q}(V,K\\otimes E^{\\ast})^{\\ast},", "e5c4211c20dffeec1e760096ec92033e": "dP/dt = acVP-mP", "e5c4d7d1ab8f3cdcee315b460f9aae31": "d\\neq 0", "e5c546afc80a28bf98c07cd348f78316": "q(i|j,a)", "e5c54eec9c8474ab527a795893f3b046": "x^{'}_{i} \\leftarrow x^{int(u(0, 1)*hms)+1}_i", "e5c556c65c8eaa784b3aa3237c949db0": "\\left(\\frac qp\\right) =1 \\quad \\iff \\quad q \\mbox{ splits completely in } \\mathbf Q(\\sqrt{p^*}).", "e5c595b95cca54c9466dc851b62c4854": "f_i = s_ib_{i + 1}", "e5c5dc29dd5a7f44ed8ac0d2b3bc8aab": "\\beta=2", "e5c6150980f0940b56dc4c8acd5837b3": "y=\\Pi x_i", "e5c6269535bc7a24ede6647cc9f705e0": "\\frac{1}{(1-x)(1-2x)}", "e5c68763262a4ec72ce52818ea3bfe83": "\\phi\\rightarrow-\\phi", "e5c6f6cea22bde9b5c3e6ead31fb47e6": "\\langle E \\rangle = - \\frac{\\partial \\ln Z}{\\partial \\beta}.", "e5c70b66331c0298fabfd974dfd7d2b9": "\\dot{\\mathbf{x}}(t) = A \\mathbf{x}(t) + B \\mathbf{u}(t)", "e5c76578f9aff9f97482fc0ead65d0c0": "\\overline{a}_{\\overline{n|}i}= \\frac{1-v^n}{\\delta}", "e5c77d506d50dbfcf62a622222c86080": "a_{15}+b_{14}", "e5c7b0a3c8263a1e8dd047546c58942b": "\\{KgH: g \\in G\\}", "e5c7c960f039ab5acd2407ebeb10a7a7": "\\mu / L", "e5c7e4c5416a7c2db2301e664f2bc701": "M=\\int\\left[\\int w\\,dx\\right]dx", "e5c7e6eda40b26c8ecf0df984f497935": " G(\\alpha) = ([(\\beta,\\gamma) : L(\\alpha,\\beta,\\gamma)], [(\\beta,\\gamma) : R(\\alpha,\\beta,\\gamma)]) \\;", "e5c82f36829a1c723ef12604cb19c1aa": "\\omega = -2, -4, -6, \\ldots", "e5c86082f22f98e257cb927c4ee66778": "\\delta_{Az} = \\frac{\\rho \\lambda}{2 L} = \\frac{H \\lambda}{L sin\\gamma}", "e5c905a29506fa2b44c8a8a652fd7561": "\\begin{align}\n \\sigma^2_Z &= \\log\\!\\left[ (e^{\\sigma^2}-1)\\frac{\\sum e^{2\\mu_j}}{(\\sum e^{\\mu_j})^2} + 1\\right], \\\\\n \\mu_Z &= \\log\\!\\left[ \\sum e^{\\mu_j} \\right] + \\frac{\\sigma^2}{2} - \\frac{\\sigma^2_Z}{2}.\n \\end{align}", "e5c944c8ecbaeddc54bd0e71a7855483": "x^2 - 4x + 7", "e5c9c0210042f13cf5ba7592a8356401": "x = 0\\;", "e5c9ccbf47718c5636bd9bdae5a5ff09": " H = s^{k-1}(h( c_1 )) \\oplus s^{k-2}( h( c_2) ) \\oplus \\ldots \\oplus s( h( c_{k-1}) ) \\oplus h( c_k)", "e5cb16c20d9f01bbbfe8f299e28d1f4b": "\\pi(x)", "e5cb63fec2e196e20fe44c4d5568c95e": "\\alpha=\\frac {|w|}{(2 c_r Q_r)} \\quad (1.5)", "e5cb6cb8b310cd1b230aa4a1f52d2a11": " -(\\kappa-n+1)~r^{-n+1}~\\sin(n\\theta)\\, ", "e5cb7b5fa28861a0b525a8a832cd4369": "\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))\\ K ", "e5cbbd0592ef2cf642bcf15656fcd83e": "a_0z^m + a_1z^{m-1} + \\dots + a_{m-1}z + a_m = 0", "e5cbe1683c172beb22a82395186f117a": "M=\\chi H", "e5cc0848fbfc3bb7a60f74461eddb273": "\n\\int x^m \\left(A+B\\,x^n\\right) \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^pdx=\n -\\frac{x^{m+1} \\left(A\\,b^2-a\\,b\\,B-2 a\\,A\\,c+(A\\,b-2 a\\,B) c\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{a\\,n(p+1) \\left(b^2-4 a\\,c\\right)}\\,+\\,\n \\frac{1}{a\\,n(p+1) \\left(b^2-4 a\\,c\\right)}\\,\\cdot\n", "e5cc13539786eb1b15a7f94b2921de95": "\\mathbf{J} = \\begin{pmatrix}\n0 & - J_\\text{z} & J_\\text{y} \\\\\nJ_\\text{z} & 0 & - J_\\text{x} \\\\\n- J_\\text{y} & J_\\text{x} & 0 \\\\\n\\end{pmatrix} = \\begin{pmatrix}\n0 & - (x p_\\text{y} - y p_\\text{x}) & (z p_\\text{x} - x p_\\text{z}) \\\\\n(x p_\\text{y} - y p_\\text{x}) & 0 & - (y p_\\text{z} - z p_\\text{y}) \\\\\n- (z p_\\text{x} - x p_\\text{z}) & (y p_\\text{z} - z p_\\text{y}) & 0 \\\\\n\\end{pmatrix}", "e5cc4616ed55d8dee425e318acba7348": "L_{\\alpha, \\alpha}", "e5cc484c45be02752163ecce526daf6a": "\\textrm{fuel} + \\textrm{oxygen} + \\textrm{nitrogen} \\to \\textrm{water} + \\textrm{carbon\\ dioxide} + \\textrm{nitrogen}", "e5cc5810e0dc0e8191e1ed0c44aa4067": "\\Omega = \\begin{bmatrix}\n\\begin{matrix}0 & 1\\\\ -1 & 0\\end{matrix} & & 0 \\\\\n & \\ddots & \\\\\n0 & & \\begin{matrix}0 & 1 \\\\ -1 & 0\\end{matrix}\n\\end{bmatrix}.", "e5cc76882276e1c53a55b7086cf1ade3": " \\sigma_1 = \\left[ \\begin{matrix} 0 & 1 \\\\ 1 & 0 \\end{matrix} \\right], \\; \\;\n \\sigma_2 = \\left[ \\begin{matrix} 0 & -i \\\\ i & 0 \\end{matrix} \\right], \\; \\;\n \\sigma_3 = \\left[ \\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix} \\right]. ", "e5cc849cbef41bd9eadd05a1e0a30180": "\\Delta U_{\\text{bath}} + \\Delta U + W = 0\\,", "e5ccbb4d71b43002a35e21bbda3c7ec6": "\\ker D=K", "e5cd028dbd210fb055419e7ef428e668": "(R\\otimes \\mathbf{1})(\\mathbf{1}\\otimes R)(R\\otimes \\mathbf{1}) =(\\mathbf{1}\\otimes R)(R\\otimes \\mathbf{1})(\\mathbf{1}\\otimes R)", "e5cd0e9210bb36f802fb36ed45c31198": "0\\le\\theta<2\\pi", "e5cd15ce5dd3e828183c4361a5265a2c": " s F(s) \\ ", "e5cd18c42ad2df4d1ea1d60e03fbf496": "W(\\mathbf{0})=0", "e5cd4a777e181be834005dca14e380b6": "\\begin{align}\n \\bar H^{(\\lambda+1)}(X)\n &=\\frac1{X-s_\\lambda}\\cdot\n \\left(\n P(X)-\\frac{P(s_\\lambda)}{H^{(\\lambda)}(s_\\lambda)}H^{(\\lambda)}(X) \n \\right)\\\\[1em]\n &=\\frac1{X-s_\\lambda}\\cdot\n \\left(\n P(X)-\\frac{P(s_\\lambda)}{\\bar H^{(\\lambda)}(s_\\lambda)}\\bar H^{(\\lambda)}(X) \n \\right)\\,.\\end{align}\n", "e5cd95e345b326fa667cfdc2c54e200b": "\\,\\!z^\\prime", "e5cd97411f0a4b6f9c9aed79a2d6e805": "(\\lambda y.t)[x := r] = \\lambda y.(t[x := r])", "e5ce0c4ad32e67237894738437827c54": "a\\uparrow^n\\cdots\\uparrow^na", "e5ce72a7640778946fa55493ce397a4c": "\\textstyle m(Z \\cap A) = \\operatorname{mes} (A) ", "e5cf394cb3baed8a0aa89bb92745ee3a": "\\nabla \\times \\left(\\nabla \\times \\mathbf{E} \\right) = \\nabla \\times \\left(-\\frac{\\partial \\mathbf{B}}{\\partial t} \\right) \\qquad \\qquad \\qquad \\quad \\ \\ \\ (5) \\,", "e5cf3b4b01c1e8f792337636f302021c": "{{p_x^c\\, q_x^0}\\over{p_x^0\\, q_x^0}}\n\\left/\n{{p_m^c\\, q_m^0}\\over{p_m^0\\, q_m^0}}\\right.", "e5cf4df374817fca708d89860c009caa": "\\scriptstyle \\lesssim10^{-4}", "e5cf8eefecaacae07022d8082b512756": "\\frac 1 2 R\\le L\\le R.", "e5cfc6203f5c7ce2c749deb291631736": "\n \\operatorname{Var}[X] = \\frac{2}{\\lambda^2}\\bigg(\\frac{1}{1+2\\lambda} - \\frac{\\Gamma(\\lambda+1)^2}{\\Gamma(2\\lambda+2)}\\bigg).\n ", "e5d05c822270407c7d6e8377fd517925": "R \\geq (2^{a-1}-1)", "e5d074cf79c49dcdcc0e18f237ccd89f": "\\bar y=D\\cdot y_{max}", "e5d08d119c5e08da28f5bc9205979852": "F(\\beta) = -\\beta^{-1}\\log Z \\, ", "e5d0afdf4a15700c971adb4f4cc93a94": "I(X)=J(F(X))", "e5d0c33f6f726ad1a3babb74e5ce3ac8": "K(n)=\\prod_{k=1}^{n-1} k^k", "e5d0c5433819de48aee30fa0ac3b982b": "i\\text{ such that }p_i=\\max(p_1, \\ldots, p_k)", "e5d0c8c292fb7920dbd9c97a4899cffe": " \\alpha \\subset n", "e5d1344a78ad8ac2200050aa024ef8bc": "\\textstyle\\prod_pp/(p-1)", "e5d154b2513b75b5e30e2495cd69abfa": "\\begin{bmatrix} CB & CAB & CA^2B & \\cdots & CA^{n-1}B & D\\end{bmatrix}", "e5d16f2b6053123a9c8121faebfadf4d": "\\cos{(0)}=1", "e5d184f5f871b28e887dd0e51eaf2d11": " \\nabla^2 \\Psi = {\\partial ^2 \\Psi\\over \\partial x^2 } +\n {\\partial ^2 \\Psi\\over \\partial y^2 } +\n {\\partial ^2 \\Psi\\over \\partial z^2 } =\n - {\\rho_{e} \\over \\varepsilon \\varepsilon_{0}} \\; .", "e5d2d907f656744a5c4a7710be088f21": "s, h \\cup h' \\models Q", "e5d30aabc1340eece8dfcc14a4484a8c": "C = \\Delta t \\sum_{i=1}^n\\frac{u_{x_i}}{\\Delta x_i} \\leq C_{max}. ", "e5d3203f2840a93f3e4db933c3afc0f9": "F(k) = G(k)", "e5d339a000bd73bee984b18732e6888f": "p \\sim_e q", "e5d33f43696caa29d8e7876ccc2d6279": "S[] \\to aS[f]c \\to aaS[ff]cc \\to aaT[ff]cc \\to aaT[f]bcc \\to aaT[]bbcc \\to aabbcc", "e5d34f6043862f1ddcb8fcb8e7c4f318": "f(s) = 0", "e5d3578616ad576290a12c43459a00ff": "| \\psi \\rang = \\sum_i c_i |{k_i}\\rangle", "e5d37c1e3f40048f713008689c9c15cb": "V=(\\frac{1}{6}(5+4\\sqrt{5}+15\\sqrt{5+2\\sqrt{5}}))a^3\\approx10.0183...a^3", "e5d37d6ca8479dfd8345a122cc0024ea": "r^{-2}\\,", "e5d38f3b8537b96840f5fef212f4594c": "e^{(\\sigma+it)\\log n}\\,", "e5d4181684bb81c7536a5ee094775aaa": "e(x)=\\widehat{\\theta}(x) - \\theta,", "e5d4246a42845cd1093630ba64629d4d": "K \\vee T", "e5d52319f7e8ca9d2eb138517ca6be8b": "4k + 2,", "e5d5268164595060cec5dcd4131f959a": "M \\approx \\mathbf{Z} \\oplus \\mathbf{Z}/2", "e5d5972f4996fc51a8ffc70dd774ad7d": "\\mathbf{y} \\in \\mathbf{R}^n \\mapsto \\left(\\frac{2\\mathbf{y}}{|\\mathbf{y}|^2+1}, \\frac{|\\mathbf{y}|^2-1}{|\\mathbf{y}|^2+1}\\right) \\in S\\sub \\mathbf{R}^{n+1}.", "e5d5cd5a9d1240df5c775337ce8063a6": "\\beta=0.5\\,", "e5d5f6c1cc5161124086198e5c570d43": "\nP_3=-4d A_1 A_2 e^{i(k_1+k_2)z}=4d A_1 A_2 e^{i((k_1+k_2)z+\\pi)}\n", "e5d612af570be5f9c623a03e8a33c370": "P(x_1,x_2)", "e5d65e1b2e320c4a13c4eea625442285": "\\operatorname{E}[\\,\\varepsilon\\,] = 0.", "e5d6831c2a4f72c05f5b73adbf9dbc2d": "\\rho_{s0}^\\alpha/8 \\simeq 4.4\\,\\sigma_{dc}^\\alpha\\, T_c", "e5d68591b35c09d9949f0dcdf79265e2": " \\frac{q}{p} = \\frac{\\nu}{2} \n(d+2-\\eta),~\\frac 1 p=\\nu.", "e5d6b771a84a4ca2289ec5bcb57e1b74": " \\oint_{\\partial S}\\psi d\\boldsymbol{\\ell}=\\iint_{S}\\left(\\hat{\\mathbf{n}}\\times\\nabla\\psi\\right)dS ", "e5d6ca2b664f7d664867e1cff799e644": "a_j\\,", "e5d73408e206840cfea33da88d58be9c": "\\{ | i' \\rangle \\}", "e5d78fbe89dd45a58320eb5659731ebe": "f\\left[ n \\right]=\\delta \\left[ n \\right]-\\delta \\left[ n-1 \\right]", "e5d7ccc23ba4ed26ef58c451f8fbe5e7": "2^{178}", "e5d7d6bf190eb710095ad2c67d2a8acb": "J = \\begin{bmatrix}0 & -I_V \\\\ I_V & 0\\end{bmatrix}", "e5d7e7c776c301efe3b54d732f1b8ef2": "\\frac{\\ddot a}{a}=-\\frac{4}{3}\\pi G\\left(\\rho^\\prime + 3p^\\prime\\right) = -\\frac{4}{3}\\pi G(1+3w^\\prime)\\rho^\\prime", "e5d7f320d84108788e9d2f7fe9540dc4": "\\rho _d", "e5d819aac18b7023fc92d789b3db5618": " u^* \\, ", "e5d967e4625814c05ff2676dedbeccca": " Z_m = \\sqrt{\\left(2\\omega_0\\zeta\\right)^2 + \\frac{1}{\\omega^2}\\left(\\omega_0^2 - \\omega^2\\right)^2}", "e5d97337fdbd305cd303eeb938cfbd1e": "\\frac{d\\sigma}{d\\Omega}=\\left(\\frac{\\alpha}{4E}\\right)\\left[\\csc^{4}\\frac{\\chi}{2}+\\sec^{4}\\frac{\\chi}{2}+\\frac{A\\cos\\left(\\frac{\\alpha}{\\hbar\\nu}\\ln\\tan^{2}\\frac{\\chi}{2}\\right)}{\\sin^{2}\\frac{\\chi}{2}\\cos\\frac{\\chi}{2}}\\right]^{2} ", "e5d983c42233d8895d85b53d8bacd250": "\\alpha = \\frac{n}{|T|}", "e5d9a49c25e68bf6137c8babfa32a28a": " \\operatorname{Tr}_W (T (I_V \\otimes S)) = \\operatorname{Tr}_W ((I_V \\otimes S) T) \\quad \\forall S \\in \\operatorname{L}(W) \\quad \\forall T \\in \\operatorname{L}(V \\otimes W).", "e5d9c50a551f48f2231dfb63ab76aa35": " \\mathfrak{sl_2}", "e5d9dad8ad9ea5e549ac09f2d38cb159": "p\\neq 2", "e5da05f3262be959a97e45dd52544573": "\\int\\mathbf{Y}_{lm}\\cdot \\mathbf{\\Psi}^*_{l'm'}\\,\\mathrm{d}\\Omega = 0", "e5da7132d9deb544bccb22ca87025591": "\\frac{\\partial u}{\\partial r}=\\frac{\\partial u}{\\partial x} \\frac{\\partial x}{\\partial r}+\\frac{\\partial u}{\\partial y} \\frac{\\partial y}{\\partial r} = \\left(2x\\right) \\left(\\sin(t)\\right)+\\left(2\\right) \\left(0\\right)=2r \\sin^2(t)", "e5db2a7f80729cc18ad5b4e15fb488b4": "E \\psi_c = - \\frac{\\hbar^2 (1-\\lambda)}{2 (L_1+L_2)} \\frac{d^2 \\psi_c}{d Q_c^2} ", "e5db8563f82ba3ad460ec614d54a69cd": "p_i = \\frac{f_i}{\\Sigma_{j=1}^{N} f_j}", "e5db8d6a532ac290bb711a3bf577e9cd": " \\frac{1}{K} \\, d_T(g,h) \\le d_S(g,h) \\le K \\, d_T(g,h) ", "e5dbfadb48e732ee1bb34afe0b6c0082": "I_{rim}=M_{rim}R^2", "e5dc03a5d53c1497ff44b43b2c3c67c0": "\\begin{smallmatrix}m=\\frac{b^2-c^2}{b^2\\sin(\\alpha)^2}\\,\\!\\end{smallmatrix}", "e5dc7b28f0839a30213a8910b725df5a": " \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v} \\right) = -\\nabla p + \\nabla \\cdot\\mathbf{T}_\\mathrm{D} + \\mathbf{f} ", "e5dcd74c514591c99ea7ec34370e0c83": "\n\\begin{alignat}{2}\n\\epsilon(q,0) & = \n1 + V_q \\sum_{k,i}{\\frac{ q_i k_i \\frac{\\hbar^2}{m} \\frac{\\partial f_k}{\\partial \\mu} }{\\frac{\\hbar^2 \\vec{k}\\cdot\\vec{q}}{m} }} = \n1 + V_q\\sum_k{\\frac{\\partial f_k}{\\partial \\mu}} = \n1 + \\frac{4 \\pi e^2}{\\epsilon q^2} \\frac{\\partial}{\\partial \\mu} \\frac{1}{L^3} \\sum_k{f_k} \\\\\n& = 1 + \\frac{4 \\pi e^2}{\\epsilon q^2} \\frac{\\partial}{\\partial \\mu} \\frac{N}{L^3} =\n1 + \\frac{4 \\pi e^2}{\\epsilon q^2} \\frac{\\partial n}{\\partial \\mu} \\equiv\n1 + \\frac{\\kappa^2}{q^2}.\n\\end{alignat}\n", "e5dce96fbe46f69759c3582cb9786245": "g \\triangleq\\sum_{i1", "e5e2c0d8926cb121c8c7523238c933ee": "\n\\mathrm{E_n}'(z) = -\\mathrm{E_{n-1}}(z)\n\\qquad (n=1,2,3,\\ldots)\n", "e5e31d267e139d9f3102e0b97dc9e624": "\\nabla, \\bar{\\nabla}", "e5e3b721b7660a2fec46be7f5239b1c2": "\\begin{smallmatrix}\\sqrt{77^2\\ +\\ 38^2\\ +\\ 4^2}\\ =\\ 86\\end{smallmatrix}", "e5e3e7acbff2a98e5355a754c2f32c08": " \\mathbf{p} = \\mathbf{A} \\or \\mathbf{B} = (\\mathbf{A} \\times \\mathbf{B}) J^{-1}.", "e5e4261dba2af39633d09b545caead43": "Q = \\rho V_a^2 D^3 \\times f_q \\left(\\frac{ND}{V_a}\\right)", "e5e4345729183b8088971841d250e61a": "\\lambda_1, \\ldots, \\lambda_k ", "e5e474d8e88d1cc6faf3aa73b34ceeb0": " u(w)= -e^{-aw}", "e5e4a703c3fd3d28ca89e78c1a198c34": "\\left ( \\begin{array}{cc}a & b \\\\ c & d \\end{array} \\right )", "e5e4a9c1bdce89f580d930f78db335d1": "\n \\operatorname{E}[\\,w_th(x^*_t)\\,] = \\frac{1}{2\\pi} \\int_{-\\infty}^\\infty \\varphi_h(-u)\\psi_w(u)du,\n ", "e5e4e87294659b47d3062bf0c4351d92": "f: E \\to F", "e5e54a76d913f57419b54f7cb37552fa": "g(x_1, x_2, x_3)=\\sum_{m_1=-\\infty}^\\infty \\sum_{m_2=-\\infty}^\\infty \\sum_{m_3=-\\infty}^\\infty h^\\mathrm{three}(m_1, m_2, m_3) \\cdot e^{i 2\\pi \\frac{m_1}{a_1} x_1} \\cdot e^{i 2\\pi \\frac{m_2}{a_2} x_2}\\cdot e^{i 2\\pi \\frac{m_3}{a_3} x_3}", "e5e58e54bc77820523ed005534e62e49": "\\phi^{\\prime}", "e5e5f081861fbe82c336a327f420a514": "L(\\psi) < 3L_B(P)+16 \\,", "e5e60c96499e5e3c08d7dddc9a3ffe14": "D = \\dfrac{\\mbox{Medium flow rate}}{\\mbox{Culture volume}} = \\dfrac{\\mbox{F}}{\\mbox{V}}", "e5e65cd292c935e6dd60e9e2429d613a": "\\rho<1", "e5e71230a333966f1354a7a165c5d414": " \\vec n\\cdot \\vec x = \\vec n\\cdot \\vec p", "e5e7342b7199f9d10ecea4a664ec7903": "s = \\dfrac {q^2-q-a_1^2q+a_1q+1} {q-2a_1q+a_1^3-a_1^2+a_1-1} \\bmod{\\ell} ", "e5e74ff9cf5311bffc8f48927e267cdc": "V_{in} = -V_{ref}", "e5e7add7219068ec5bc36a43f9dbcab3": "\\sum_{n=0}^{\\infty}(-1)^nx^n={1\\over1+x}\\,.", "e5e7da08a1c973c1a449b8d41c64c0df": "m=2^L", "e5e824ec161547fa8d004f17baa47ccb": " y[x(t)] \\approx f[x(t)] = x(t+1)- c[x(t),t] ", "e5e89c0d2d1e7bee5a3b4c209f13acd8": "\\begin{align}\nh(x_1^n)= 1,\\,\\,\\,\ng_{(\\alpha \\, , \\, \\beta)}(x_1^n)= \\left({1 \\over \\Gamma(\\alpha) \\beta^{\\alpha}}\\right)^n \\left(\\prod_{i=1}^n x_i\\right)^{\\alpha-1} e^{{-1 \\over \\beta} \\sum_{i=1}^n{x_i}}.\n\\end{align}", "e5e8a7a0ef14efd008f6efb0b20be593": "P(E|H_2) = 20/40 = 0.5", "e5e935b02e1eb7090ecc033016a0c91e": " B", "e5e947bb07998ae090ec1dffcd6ae814": "k = 1,\\ldots,p", "e5e9f58414bc1b9e793cef79a95839d2": "z^2+c", "e5ea54112d03b8c0cd803a385374055a": " \\and (S_7 \\implies (\\operatorname{equate}[A_7, q] \\and V[F_7] = A_7)) \\and D[F_7] = K_7 ", "e5ea5436ac0e1578979c2eadce705b2a": "\\|Df^nv\\| \\le c\\lambda^n\\|v\\|\\text{ for all }v\\in E^s\\text{ and }n> 0,", "e5ea5ea709794a5df2dd5c2dc07c0828": "\n\\delta_v^* = \\int_0^\\delta \\left(1 - \\frac{U}{U_e}\\right)\\,dy\n", "e5ea9c9b3f8c749d2fe84d010302f1a7": "it \\mapsto \\exp(it) = e^{it} = \\cos(t) + i\\sin(t),\\,", "e5eaad2134f8c96951ca6e6eb62c21a7": "\\partial (xy) = (\\partial x) y + x(\\partial y)", "e5eabf5486a557c19a2f79d666afe60f": " a(x) = \\mu x, ~b(x) = \\sigma x ", "e5eb7e97ec6d7e3d5f22d25935428a50": "\n\\begin{align}\n\\Phi &:=\\left(\\phi_1~\\phi_2~\\ldots~\\phi_N\\right)^T\\\\\n\\Phi^T \\Phi &=I\n\\end{align}\n", "e5ec049787dba57db4f39b078c240531": "\\delta'[\\varphi] = -\\delta[\\varphi']=-\\varphi'(0).", "e5ecc6c9803f4dd4760606d657828abd": "x_1, \\ldots, x_m", "e5ece13eb5134be59cec3d5895a7415d": "y(x)=0", "e5ecf73cb0f8f8aa1886b8be0d03d4ba": " P = (s,x) ", "e5ed4277b3aceb77ca656ed7c3fc61c5": "\\begin{bmatrix}\na & b & c & d & e \\\\\nb & c & d & e & f \\\\\nc & d & e & f & g \\\\\nd & e & f & g & h \\\\\ne & f & g & h & i \\\\\n\\end{bmatrix}.", "e5ed628ae6da65cff4df42b7d831918d": "\\scriptstyle f(E)", "e5ed6a92843921a5fd6f597e9afb44b1": "Q_1=\\frac{1}{2}\\left[(p-iW)b+(p+iW)b^\\dagger\\right]", "e5ed85c6165f1e9a113f3e678138f716": "X_{q_i}=-\\frac{\\partial}{\\partial p_i}", "e5ed99341ea346da197dac3755ecf484": " \\frac{w}{4}\\sum_r \\chi(r)\\log \\Gamma\\left( \\frac{r}{D} \\right) = \\frac{h}{2}\\log(4\\pi\\sqrt{|D|})\n+\\sum_\\tau\\log\\left(\\sqrt{\\Im(\\tau)}|\\eta(\\tau)|^2\\right)\n", "e5edf1ae3ab88420d3e9df6811fe9553": "P_1 V_1^\\gamma = P_2 V_2^\\gamma \\Rightarrow\n\\frac{P_2}{P_1}= \\left( \\frac{V_1}{V_2} \\right)^\\gamma ", "e5edf883dca7e81330296982b8c340e9": "\\bigwedge A", "e5ee2e39e4b94c31aa1d4f3a73e5c2a9": "\\|p_\\sigma\\|\\in[0,\\alpha\\sqrt{n}]", "e5ee4c8c3755521cf4efc99eacb2ecbc": "2465 = 5 \\cdot 17 \\cdot 29 \\qquad (4 \\mid 2464;\\quad 16 \\mid 2464;\\quad 28 \\mid 2464)", "e5eea0be53faf84f7687ec90c9e3c91a": "X\\Vdash A\\to B", "e5eec80b79904248d2a83aef5f7cfaf3": "J^-(p)\\cap J^+(q)", "e5eed265e2a58a3934754643307251eb": "W = \\int_C Fds = F\\int_C ds = Fd", "e5ef9aa1500b10c4512500a775c6293c": "\\|v\\|_a=\\sqrt{a(v, v)}", "e5f01aabf446e4f5f31fa773cf9fce6a": "\\hat{\\mathbf{\\nu}}", "e5f04565d7a54e7fcf28995f118e07c3": "\\left [ 0.9 \\left [ \\frac{A+B}{2} + \\frac{A-B}{2}\\sin4\\pi f_pt \\right ] + 0.1\\sin2\\pi f_pt \\right ] \\times 75~\\mathrm{kHz}", "e5f066fbef2534345465a34c8a8813fd": "\\rho^A = \\operatorname{Tr}_B \\; \\rho^{AB}", "e5f07066cfc93469ff0ae3b5bb22a1cb": "\\varphi_1,\\varphi_2\\in\\mathcal{C}(I_{\\alpha}(t_0),B_b(y_0))", "e5f08dcf9d685e1520034d1cd5048c74": "\\scriptstyle a,\\, b \\;\\mapsto\\; a \\,-\\, b \\;:=\\; a \\,+\\, (-b)", "e5f0bee2ae1614c82e1dd5346fba5eb0": "\\mathbb{Q}(\\sqrt{D})", "e5f10b5836a9e4eba7e72c8db0e7e06b": " G_0=F_0,", "e5f182438d39d18c253bd5f175fd7091": "\\Gamma_I", "e5f1c7bc7cb6b02bcb83afccb25dfcb8": " \\sum_{z=1}^{\\infty}f(z), ", "e5f1d196ecbd3663c7701195dda40021": "{\\underbrace{\\partial \\overline{hv} \\over \\partial t}}_{\n\\begin{smallmatrix}\n \\text{Change in}\\\\\n \\text{y mass flux}\\\\\n \\text{over time}\n\\end{smallmatrix}}\n+ \\underbrace{{\\partial \\overline{huv} \\over \\partial x} + {\\partial \\over \\partial y} \\left( \\overline{hv^2}+{1 \\over 2}{k_{ap}g_zh^2}\\right)}_{\n\\begin{smallmatrix}\n \\text{Total spatial variation}\\\\\n \\text{of x,y momentum fluxes}\\\\\n \\text{in y-direction}\n\\end{smallmatrix}}\n= \\underbrace{-hk_{ap} \\sgn \\left({\\partial v \\over \\partial x}\\right){\\partial hg_z \\over \\partial x}\\sin \\phi_{int}}_{\n\\begin{smallmatrix}\n \\text{Dissipative internal}\\\\\n \\text{friction force}\\\\\n \\text{in y-direction}\n\\end{smallmatrix}}\n- \\underbrace{{v \\over \\sqrt{u^2+v^2}}\\left[ g_zh \\left(1+{v \\over r_yg_y}\\right) \\right]\\tan \\phi_{bed}}_{\n\\begin{smallmatrix}\n \\text{Dissipative basal}\\\\\n \\text{friction force}\\\\\n \\text{in y-direction}\n\\end{smallmatrix}}\n + \\underbrace{g_yh}_{\n\\begin{smallmatrix}\n \\text{Driving}\\\\\n \\text{gravitational}\\\\\n \\text{force in}\\\\\n \\text{y-direction}\n\\end{smallmatrix}}\n", "e5f238739567e3d624c35054e192b1a2": "\\theta = \\arg(z),\\ \\theta_1 = \\arg(z-a)", "e5f2387f8984bd491d0c83af447f09fe": " L^p_m ", "e5f23e04589c3f5d47c4d7667684a73e": " m = 1 \\,", "e5f261a0d0364e7292d86ac4fd85b676": " \\Big [ \\mbox{nuclear} \\Big ] \\Big [ \\big [ \\mbox{physic(s)} \\big ] \\big [\\mbox{-ist} \\big ] \\Big ] ", "e5f297667bbdf634df533fbe356cece0": "H_q(x) = x\\log_q(q-1)-x\\log_qx-(1-x)\\log_q(1-x).", "e5f2bc1dec037c53a7a0880a837a8a0e": "\\rho=\\operatorname{tr}_B \\chi ", "e5f2f6e43236bb5379ad4cae22721b86": " T_r=4M\\ln 2 -2M \\approx 0.77M.\\,\\!", "e5f3572738ed31778ced868fc9204f1c": "p(x|y,\\theta)", "e5f376ae3242a17ec7a1bb18e20ca2e3": " \\mathbf{p}=[\\underline p,\\overline p] ", "e5f3aa72ae94dc5915009aafdf765560": "N_{B_1},\\ldots,N_{B_k}", "e5f414b8e3dfb33e889e2ec35b0528dc": "V_s=20 \\sum_{t=1,\\ldots,s^2}\\frac{r_{i,s,t}A_{s,t}}{3}.", "e5f422c5772641bd928f04e0835daace": "c_2=0", "e5f425e545956185e1dbdd8f443beefd": "f = \\sum_{j=1}^n a_j \\mathbf{1}_{A_j}", "e5f446010de82ce9c0dc85dbae7ebb7f": "N\\left( \\mathcal{S}\\right) ", "e5f4667706ba5a8438c01d313886d7de": "\\langle B, \\in\\rangle ", "e5f4824d95a79484df8fe50e8605042c": " G = 6.67384(80) \\times 10^{-11} \\ \\mbox{m}^3 \\ \\mbox{kg}^{-1} \\ \\mbox{s}^{-2} = 6.67384(80) \\times 10^{-11} \\ {\\rm N}\\, {\\rm (m/kg)^2}", "e5f49070a611ecdce80330cccf3e5d2f": "(x_1-x_2)^T(F(x_1)-F(x_2))\\leq C\\Vert x_1-x_2\\Vert^2", "e5f5a22797557f4b2b11a0de016de2e0": "F(\\vec{k},t)", "e5f5b211329a1808ebf2d0e049b14360": "\\mathfrak{m}_s", "e5f5fdc2a9f28bc3fb73cf37ec25c3ce": "F:X \\times S \\rightarrow Y", "e5f644baa6c1bf6a1e45202f99b5dfe5": "f(x) = \\left| x \\right|", "e5f66bbe23a8918d0bab26cdffa14199": "V_1,V_2,\\ldots", "e5f67fc88379c8d0ddea6308d8ad93d1": "L(x)=p(r|x),", "e5f738bdf6035b84962c08ea987284d7": "y=w''/w'", "e5f7504a9f6c643b2a903061e8d76e16": "I(t,V) = g(t,V)\\cdot(V-V_\\mathrm{eq})", "e5f760b94f1669096cad4f9b423debae": " Q(\\lambda \\xi + \\eta) =0,", "e5f80f2837ea23dbed5dd4884722e840": "\\tau _{Auger7_{doped}}(t,x,n) = \\frac{2\\cdot \\tau _{Auger7(t,x)}}{1+(\\frac{n_{i}(t,x)}{n})^{2}}", "e5f82931dc43b670024b39e6064374bb": "f(x,t)=\\frac{1}{2\\pi} \\int \\hat{\\zeta}_0(\\omega) \\exp \\left[-i\\left(k(\\omega)x-\\omega t \\right)\\right] d\\omega ", "e5f84314b8113985200fb98680bf9f70": "\\forall j \\neq i, y_i \\succ_{x} y_j\\,\\!", "e5f84f79660264d267d6ec1833bfe241": "F = x \\cdot F_x + x' \\cdot F_x'", "e5f87fcc0bf607c5ba29267bd25579c2": "\\mbox{lim}_{i\\rightarrow\\infty}|\\xi_i(\\gamma)-\\xi(\\gamma)|=0", "e5f8a5d74e5a079f9aa7721d69d6be55": "\\Lambda_{rot}", "e5f8ac9aa50afeef116939a39f3f954f": " f \\nabla ", "e5f97e799742985a8903e78ef4817c33": "\\widetilde f", "e5f98cce5742dac9ac2c1271669ce9e0": "\\nabla^2f", "e5f9aa5d03e4bad0645ce46ea59d37a8": "\\frac{d v}{dr} = -\\frac{1}{2} \\sqrt{ \\frac{G M}{R^3} }=-\\frac{1}{2} \\frac{v}{r}", "e5f9b741abb6ea4b3c14c9119c43b719": "g_{(a,k)}", "e5f9c393ec6bcf21b756a4d4c2def2be": "\\,P_b = P_{bc}", "e5f9d0435022bd42e52d87e4633ed9fd": "\\scriptstyle1/\\sqrt{n}", "e5f9daaca4bbc85f0e8b647b69154cdc": "x^{12} + x^{11} + x^3 + x^2 + x + 1", "e5fa175c09af934d5c862e196983a686": " \\int_{-\\infty}^{\\infty} x^2 e^{-{1 \\over 2} a x^2}\\,dx = -2{d\\over da} \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} a x^2}\\,dx = -2{d\\over da} \\left ( {2\\pi \\over a } \\right ) ^{1\\over 2} = \\left ( {2\\pi \\over a } \\right ) ^{1\\over 2} {1\\over a}", "e5fa4c6b6e43e5a1f5a3bf25866f6ae2": "(\\varphi_k)", "e5fa95089a50e938cde8f48c88cee91f": "x \\sim y", "e5faf8d9a2ae7650c2d386344617d64b": "C_2=C.\\,", "e5fb033bfdd2e09a4999d51ba49b36de": "i < \\lfloor n/2 \\rfloor", "e5fb241cc2e61f9d8445089bae623721": "\\tau_M\\oplus \\nu_M\n\\colon M \\to BO(n+k)", "e5fb28e7824277e48dad8d3fc0fd6e32": "n=1+D/2", "e5fb3d89519b08f58e491f613fd5e01a": "dX/ds", "e5fb4cbc93feb06bb580d71742915f1e": " G_{{q_i }{q_j}}= \\frac{{\\partial}\\mathbf{F}_{q_i}}{\\partial q_{j}} - \\frac{{\\partial}\\mathbf{F}_{q_j}}{\\partial q_i} - \\left[ \\mathbf{F}_{q_i} , \\mathbf{F}_{q_j} \\right] = 0. ", "e5fbaa1a86c558875b1567b5d3667f3b": "360^\\circ /650 = 0.554^\\circ ", "e5fbc5e443899dc2bddade8d211ff4c4": "f(x,y) = \\frac{y}{x}", "e5fc093d8026a40aa0cb1e3d331f4050": "y=\\frac{1}{1+Be^{-x}}.", "e5fc1c0a7046a5140fbbfde449e3cc86": "m_n = (m, \\ldots, m)", "e5fcb2d781292f9c6f3f81a98d0f2de7": "O(\\sqrt[4]{q})", "e5fcb44b61a30154f02407c60c66e3b4": " = \\frac {c_i}{f_\\mathrm{eq}}", "e5fcba6270d5e10105b2227f9098a8bc": "\\Delta+\\mu", "e5fcc168d061393b82cb28ed7301ee7e": "V(q) = (q^{-1} + q^{-3} - q^{-4})^2 = q^{-2} + 2q^{-4} - 2q^{-5} + q^{-6} - 2q^{-7} + q^{-8}.", "e5fcdfe8f36a33e47844a81f77a1cfe0": "\\mathcal{C}=\\{(-\\infty,x]:x\\in\\mathbb{R}\\}.", "e5fce7c8972fdfaf146b15837bde2a89": " \\mathrm{Oh} = \\frac{ \\mu}{ \\sqrt{\\rho \\sigma L }} = \\frac{\\sqrt{\\mathrm{We}}}{\\mathrm{Re}} \\sim \\frac{\\mbox{viscous forces}}{\\sqrt{{\\mbox{inertia}} \\cdot {\\mbox{surface tension}}}} ", "e5fd81f379d7161e420a19e6edace3cd": "C\\ell(n,0) = C\\ell(0,n) = \\Lambda^* \\mathbf{R}^n", "e5fd8e18abe625488a8f5fcd4a414949": "Q_{\\text{bath}} = \\Delta U_{\\text{bath}} =-\\left(\\Delta U + W\\right) \\,", "e5fd93948276baf9c528abb020885fe7": "\\mbox{curl}\\;\\vec v = \\left( {\\partial v_z \\over \\partial y} - {\\partial v_y \\over \\partial z} \\right) \\mathbf{\\hat{x}} + \\left( {\\partial v_x \\over \\partial z} - {\\partial v_z \\over \\partial x} \\right) \\mathbf{\\hat{y}} + \\left( {\\partial v_y \\over \\partial x} - {\\partial v_x \\over \\partial y} \\right) \\mathbf{\\hat{z}} = \\nabla \\times \\vec v", "e5fda01aa9bc7750ee267c5f2cde92f3": "L\\le s\\le K", "e5fdbd2e0868a68e08c89054f6373104": "dS = \\left(\\frac{\\partial S}{\\partial x}\\right)_y\\!dx +\n \\left(\\frac{\\partial S}{\\partial y}\\right)_x\\!dy", "e5fdc0bafe4fcfa0701c86905919938c": "\nQ(t) = {Q_{{\\rm max}}\\over {1 + ae^{-bt}}}\n", "e5fe504a00fe82208fa7226132931ebb": "\\mathbf{\\hat{d}}_\\mathrm{n} \\cdot \\mathbf{\\hat{d}}_\\mathrm{i}", "e5feb1d2b2593b148b01201d3de42172": "(v(f))(q) = \\mathrm{Re} \\frac{(\\psi, \\frac{i}{\\hbar} [H,\\hat f] \\psi)}{(\\psi,\\psi)}(q)", "e5feb6cf7d091f4d008496c0d73ce4cc": "{\\tilde{B}}_7", "e5ff0d2996b83e929cba8b9967ea7116": "\\,x(t)=(C+m(t))\\cos(\\omega t).", "e5ff4985ea2aa5140ce84806681a6756": "Q(x) = -x^2 + 8x - 15, E(x) = x - 3", "e5ffaa006763444ba3a9e2744700f194": "\\tfrac{\\sqrt{2}}{2}=\\tfrac{1}{\\sqrt{2}}", "e5ffe191d66c297be0d176b6fff1530e": "C^\\prime(a,q,0)=0", "e5fff3411491a4c5cc31be13320c6030": "|\\mathbf{F}_\\mathrm{c}| = m |\\mathbf{a}_\\mathrm{c}| = \\frac{m|\\mathbf{v}|^2}{r} \\ . ", "e600d91a762356f8cd6def3f9f227702": "a=\\begin{bmatrix}-1\\\\ 1 \\\\ 1\\end{bmatrix}.", "e601127cb3f11d65bde531e7eaad104c": "\\begin{matrix}{13 \\choose 3} - 64 = 222\\end{matrix}", "e601469ea50e72adc8b5a9592ef1ee19": "r V_e = b(e)(y+V_u-V_e)+\\frac {dV_e} {dt}", "e6014f65af0d8b11c9a14d5bc6d76247": "Y_{7}^{2}(\\theta,\\varphi)={3\\over 64}\\sqrt{35\\over \\pi}\\cdot e^{2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(143\\cos^{5}\\theta-110\\cos^{3}\\theta+15\\cos\\theta)", "e60151a29494e5692687d1d899276ae7": "G' = G \\setminus A", "e601830616dcf85e830fd38a2e6fd61b": " M_{i,j} \\leqslant M_{i,k} ", "e601b25d263b1159fafe7417b1206833": "\\dot{Q}_j", "e601c40eb20ba74a626b2d6a7a33a793": "L = I \\omega,", "e601efe69614a1360e860789bb5cc08b": "z_1=z_2", "e601fbb172479907113a79cb75446db8": "{{{8}}}", "e6022d9ca0ab3fc4ba1469ec5eca5b91": " r_{t} = \\bar V_{t} - \\gamma \\bar V_{t+1} ", "e60244bc9d67cde32268f63746c046da": "\n\\mathbf{A} \\circ \\mathbf{B} = \n\\left[\n\\begin{array} {c | c}\n\\mathbf{A}_{11} \\circ \\mathbf{B} & \\mathbf{A}_{12} \\circ \\mathbf{B} \\\\\n\\hline\n\\mathbf{A}_{21} \\circ \\mathbf{B} & \\mathbf{A}_{22} \\circ \\mathbf{B} \n\\end{array}\n\\right]\n=\n\\left[\n\\begin{array} {c | c | c | c }\n\\mathbf{A}_{11} \\otimes \\mathbf{B}_{11} & \\mathbf{A}_{11} \\otimes \\mathbf{B}_{12} & \\mathbf{A}_{12} \\otimes \\mathbf{B}_{11} & \\mathbf{A}_{12} \\otimes \\mathbf{B}_{12} \\\\\n\\hline\n\\mathbf{A}_{11} \\otimes \\mathbf{B}_{21} & \\mathbf{A}_{11} \\otimes \\mathbf{B}_{22} & \\mathbf{A}_{12} \\otimes \\mathbf{B}_{21} & \\mathbf{A}_{12} \\otimes \\mathbf{B}_{22} \\\\\n\\hline\n\\mathbf{A}_{21} \\otimes \\mathbf{B}_{11} & \\mathbf{A}_{21} \\otimes \\mathbf{B}_{12} & \\mathbf{A}_{22} \\otimes \\mathbf{B}_{11} & \\mathbf{A}_{22} \\otimes \\mathbf{B}_{12} \\\\\n\\hline\n\\mathbf{A}_{21} \\otimes \\mathbf{B}_{21} & \\mathbf{A}_{21} \\otimes \\mathbf{B}_{22} & \\mathbf{A}_{22} \\otimes \\mathbf{B}_{21} & \\mathbf{A}_{22} \\otimes \\mathbf{B}_{22}\n\\end{array}\n\\right]\n", "e602472814976dd29967bfbbf99418cd": "\nf_{WN}(\\theta;\\mu,\\sigma)=\\frac{1}{\\sigma \\sqrt{2\\pi}} \\sum^{\\infty}_{k=-\\infty} \\exp \\left[\\frac{-(\\theta - \\mu + 2\\pi k)^2}{2 \\sigma^2} \\right]\n", "e6024d2c58b10fd73786885a89cb706f": "\\pi=\\frac{e}{\\sum_i{e_i}}", "e6024f3d91826b5d0f909cc00c75db66": "\n \\mathbf{M}_1 = \\int_A \\left(-x_2\\sigma_{11}\\mathbf{e}_3 + x_2\\sigma_{13}\\mathbf{e}_1 + x_3\\sigma_{11}\\mathbf{e}_2 - x_3\\sigma_{12}\\mathbf{e}_1\\right)dA =: M_{11}\\,\\mathbf{e}_1 + M_{12}\\,\\mathbf{e}_2 + M_{13}\\,\\mathbf{e}_3\\,.\n ", "e6026d37db1345f32d185d04e097869f": " 0 < |i - j|\\mod {(}N-1-{\\frac{K}{2}}{)} \\leq \\frac{K}{2}", "e6028950ae2f84eb79f618dec327f4e8": "\\displaystyle [a_0; a_1, a_2, \\ldots, a_k-1],", "e602ee0a76a082cf2ba153298dd3ffb5": "P_e", "e6032492c41fc794cccbf15dce13f54a": "\\frac{1}{1+f(q\\mid L)/f(q\\mid H)}", "e603401cb84a7fdf14addda025c789b5": "\\begin{align}\n\\dot{\\mathbf{e}}\n&= \\dot{\\hat{\\mathbf{x}}} - \\dot{\\mathbf{x}}\\\\\n&= A \\hat{\\mathbf{x}} + B \\mathbf{u} + L v(\\hat{x}_1 - x_1) \n- A \\mathbf{x} - B \\mathbf{u}\\\\\n&= A (\\hat{\\mathbf{x}} - \\mathbf{x}) + L v(\\hat{x}_1 - x_1)\\\\\n&= A \\mathbf{e} + L v(e_1)\n\\end{align}", "e60363d099c891a38c2a68a0f43d7009": "O(\\left|{V}\\right| + \\left|{E}\\right|)", "e6036593fe22b4f7fe1b04c06cd60b8f": "V_{n+1}(K)", "e60367de41b8ae099bd9da295447bf8f": " L \\,\\!", "e603e77a22ebca48229f9d44f4766b7e": "x + y := \\Psi_{fRep}^{-1}(\\Psi_{fRep}(x) + \\Psi_{fRep}(y))", "e604adc669d1d092f7ee3e80160a9602": " \\Box ", "e6058547a5a5e0ac0a2501ca72be7f26": "\\overline{\\psi_n(x)}", "e6058b0354522d732e13209f11dedc81": "\\delta_{\\perp\\ ij}(\\mathbf x ) = \\frac{1}{(2\\pi )^3}\\int d^3 \\mathbf{k} \\left(\\delta_{ij}-\\hat{u}_i\\hat{u}_j \\right) e^{i\\mathbf{k\\cdot x}} \\ , ", "e605df84c09844139021ecb451de37c9": "r\\mid (x-y),\\;\\;\\;r^{p-1}\\equiv1\\pmod{p^2}.", "e605e0295d8acb84e844283cce89a318": "P(x) \\approx \\frac{n_x}{n_t}.", "e60635611b4ef22ddc639312ee4cea0d": " \\frac{1}{|I|}\\int_{I}|f(y)-f_I|\\,\\mathrm{d}y < C <+\\infty", "e6064849f8fbbc8877724c05244e4bbd": "\\frac{Dv}{Dt} + f u = -\\frac{\\partial \\phi}{\\partial y}", "e6065696fedbd68b8e3d0a6461d4fe26": "V=\\left[\\begin{matrix} A & B \\\\ B^T & C \\end{matrix}\\right],", "e6066b7867f7ead55713e5104021c866": " w = \\frac{2 \\pi}{\\hbar} |M|^2 \\delta (E_{\\psi} - E_{\\chi} ) ", "e606b83e85f27c9a94b1a3fa13263deb": "-V_1+V_2+V_3-V_4 = 2L'\\,\\Delta l\\frac{\\partial{I}}{\\partial{t}}", "e606f1e111cda553ef9dc8dbc6a19b13": "\\omega2", "e606f2b4176190408a2e7967cfcf9ccf": "\\mathbf{Q}_d", "e607261b68b34ff07de100054bfbedf6": "\\epsilon > \\frac{89}{570}", "e6073b947f71731ef7b144442af033b8": "\\nabla_\\alpha ", "e6076a4124e4e68f0cc23cb883f63586": "\\sum_y \\sum_x (-1)^{x.y} \\left| y \\right\\rangle \\left| f(x) \\right\\rangle", "e607714b59ed8f76171ab1e4db3b5946": "\nV = V_0 \\exp(-P/K_0). \\,\n", "e6079b809738fa9034f21e9627bef7b3": "C_\\text{total} = I_\\text{discharge} \\cdot \\frac{t_2-t_1}{V_1-V_2}", "e6079c68c93fd3f04a2caf3b92e1caca": "I_{z} = \\int \\left(x^2 + y^2\\right)\\, dm = \\int x^2\\,dm + \\int y^2\\,dm = I_{y} + I_{x} ", "e607ec6832c166cded958915214dd126": "0 \\equiv (2^{2m\\lambda}-1)/(2^m+1)=(2^m-1)(1+2^{2m}+2^{4m}+...+2^{2(\\lambda-1)m}) \\equiv -2\\lambda \\pmod {2^m+1}.\\ ", "e60831630715ca0ef78a8e018f86fb8d": "\\chi^2_{s-p},", "e6083aa15afd8a0bd79036080f6b27ed": "\\begin{align}\n\\left(\n \\begin{array}{*{3}{r}}\n 4 & 12 & -16 \\\\\n 12 & 37 & -43 \\\\\n -16 & -43 & 98 \\\\\n \\end{array}\n\\right)\n& =\n\\left(\n \\begin{array}{*{3}{r}}\n 2 & & \\\\\n 6 & 1 & \\\\\n -8 & 5 & 3 \\\\\n \\end{array}\n\\right)\n\\left(\n \\begin{array}{*{3}{r}}\n 2 & 6 & -8 \\\\\n & 1 & 5 \\\\\n & & 3 \\\\\n \\end{array}\n\\right)\n\\end{align}", "e60890cc86ede08ef12bcad0770d2402": " \\partial\\mathbf{F} = \\mathbf{J}.", "e60966098229fcfa4f2b4afea153040e": "M_f(x_1,\\dots,x_k,x_{k+1},\\dots,x_n) = M_f(\\underbrace{m,\\dots,m}_{k \\text{ times}},x_{k+1},\\dots,x_n)", "e60974a55ad7a48c1085be8d7d8bf5b0": "\n\\hat{H}=\\hat{H}_{\\textrm{qp}}+\\hat{H}_{\\textrm{ph}}+\\hat{H}_{\\textrm{int}}", "e60991ca7a23cf0e92759c23c98aba9c": "\\lim_\\omega \\gamma := \\bigcap_{n\\in \\mathbb{R}}\\overline{\\{\\varphi(x,t):t>n\\}} ", "e60a59a99c1409d8736f7f823d2d60bf": "L - P", "e60af623ccd7386633f602d4aa4f0191": "Z^2_2", "e60b37f871cf4f6ab5e17a6bc1f6d2bb": "s^* = \\sqrt{\\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}.", "e60b38e1397022eec4858f40e742135b": "A_{i}A_{j}=\\sum_{k=0}^{n} p^k_{ij}A_{k}=A_{j}A_{i}, i,j=0,\\ldots,n", "e60b7491d928a5513dc59af84993f89d": "u_n \\in V_n", "e60bb78c3edb11ca37f30426cf9dc4b6": "\\frac{P(s_1)}{P(s_2)} = \\frac{\\Omega _ R (s_1)}{\\Omega _ R (s_2)}.", "e60bc493da0ec3fc7e3b8414758ee218": "V_k", "e60bf3725b38df6b9508aaee832b8dab": "H'(t)=H + \\delta V^{ext}(t)", "e60c2bd54e437041092adc447432ffdd": " \n\\begin{align}\nw_1 &= \\frac{1/\\sigma_{Z_1}^2}{1/\\sigma_{Z_1}^2 + 1/\\sigma_{Z_2}^2 + 1/\\sigma_X^2}, \\\\\nw_2 &= \\frac{1/\\sigma_{Z_2}^2}{1/\\sigma_{Z_1}^2 + 1/\\sigma_{Z_2}^2 + 1/\\sigma_X^2}.\n\\end{align}\n", "e60c38b291cd7159f0fb31d6b78c7156": " H^2=X(X'X)^{-1}X'X(X'X)^{-1}X'=XI(X'X)^{-1}X'=H ", "e60c6c0415daf19b5e90a4444db57577": "\\operatorname{E} \\exp(u^T X) = \\exp \\left\\{-\\int \\limits_{s \\in \\mathbb S}\\left\\{|u^Ts|^\\alpha + i \\nu (u^Ts, \\alpha) \\right\\} \\, \\Lambda(ds) + i u^T\\delta\\right\\}", "e60c7f29f83355cdf89bded805957ae8": "\\ {w_i}", "e60cec1d4e3ee8fb64be491fd78df26b": " i\\hbar \\frac{\\partial}{\\partial t}\\psi = -\\frac{\\hbar^2}{2m} \\nabla^2 \\psi + V \\psi ", "e60d609564df3f09e98c3681cb862b09": "$400/$500", "e60d6dcca178629237128318e5660da5": "\\bold{g}\\;", "e60dfc72e77dc56ec9f56127674566c6": "\\int \\frac{(\\ln x)^n\\; dx}{x^m} = -\\frac{(\\ln x)^n}{(m-1)x^{m-1}} + \\frac{n}{m-1}\\int\\frac{(\\ln x)^{n-1} dx}{x^m} \\qquad\\mbox{(for }m\\neq 1\\mbox{)}", "e60e18e2d26016ab6dfb952af7d5da6e": "\\bar{X} = K X K\\,", "e60e3de20d57c400947d35bf4c44f9cb": "\\Omega=-\\ln(\\mathcal{Z}) = \\sum_i g_i \\ln\\left(1-ze^{-\\beta\\epsilon_i}\\right).", "e60e4f39431afe1e4724beaf4bbf34f8": "\\begin{matrix} {11 \\choose 1}{4 \\choose 2}{40 \\choose 1} \\end{matrix}", "e60eb58ddfe90e3581920b884abedb1c": "d = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2}", "e60f8f5c4f8aef8e85ee3e8c7ae3e27c": " Y_\\ell^m( \\theta , \\varphi ) = \\sqrt{{(2\\ell+1)\\over 4\\pi}{(\\ell-m)!\\over (\\ell+m)!}} \\, P_\\ell^m ( \\cos{\\theta} ) \\, e^{i m \\varphi } ", "e60fd5db24ce15b302c4521a53e8b51a": " \\frac{L}{t} = \\left( \\frac{\\ell}{t} \\right) \\left( \\frac{180}{\\pi \\theta} \\right) ", "e61032f5f38af3be59c3abff08643c37": "\\sqrt{9+16}=\\sqrt{25}=5", "e611791004cff7d4247e72d1a5c74544": " S_{0.05}=1.33{S_{0.20}}^{1.15}", "e611abd57de78b3187a83a4de3607057": "\\ x = 0", "e611b6f2c2c44fc31795bff683110c18": "~|\\alpha|~", "e611f45b50af7d0af6f369b6c44a0d66": "{G_\\delta}_\\sigma", "e61232de21bf9b50ad74d46540fccb44": "S^3 \\to S^2", "e612cfa405e718245697bf354c3f9717": "A\\parallel_+ C \\parallel_- B", "e612e680c63d35e69e82de7af8101af3": "a_{22} w x_2 p_2.", "e6131d6b2069928bab545b4bc0fd0549": "P\\left(k\\right)\\sim k^{-3} \\, ", "e6136e4fd4396e2a2274d439ea27d1b6": " F : (\\mathbb{C}^n \\times \\mathbb{C}^{\\mu},0) \\to (\\mathbb{C},0) ,", "e613b168bc03a973426926ae9e351bad": "\\ell_{k,j}", "e61402d6468f91617af46e3b4171bb45": " A \\in M(m,n;\\mathbb{K}),~B \\in M(n,p;\\mathbb{K})\\,\\!", "e61443bf549441895c69def46bd8ea8a": " EX=0", "e614945aa0e20372709e7ee6d3887494": "f(0)=0,\\,\\,\\, f^\\prime(0)=1,", "e614c85d1bdfaaf236180f7bb88eed6c": "\\frac{\\partial \\mathbf{Y}}{\\partial x}", "e61516d474c2720345e9de974ac70ba3": "\\lambda_{1},\\dots,\\lambda_{k} > 0\\,", "e6158a44afd6c9193adc9f35201fd0ef": "\\sum a_i^2 = \\sum_{ i \\mathop =1}^n a_i^2.", "e6158e51818bc9313e99281828f962b3": "{\\ \\mathrm{d}H = \\delta Q + V \\mathrm{d}P },", "e61672aad93ad67d0b858224d89d2616": "\\int_0^{2\\pi} \\frac{dx}{a+b\\sin x}=\\frac{2\\pi}{\\sqrt{a^2-b^2}}", "e61677747bc1917796ff1c1b618e39a1": "U(t) = \\frac{\\hbar}{2 e} \\omega ( n + a \\cos( \\omega t) ), \\ \\ \\ I(t) = I_c \\sum_{m \\,=\\, -\\infty}^\\infty J_n (a) \\sin (\\phi_0 + (n + m) \\omega t).", "e61684ab284cf2ad29bb8c9228d7c265": "0 = -P_{i} + \\sum_{k=1}^N |V_i||V_k|(G_{ik}\\cos\\theta_{ik}+B_{ik}\\sin\\theta_{ik})", "e617610498949665fd1fe2464cdd77c3": "\\ell(\\gamma)=\\int_{0}^{1} \\sqrt{ \\pm g(\\gamma'(t),\\gamma '(t)) } \\, dt, ", "e617ad6aab4db72036430358fe76b40d": " 001", "e617e3326f5e82221a3d97da27c73b5b": "\\int \\arctan (x) \\cdot 1 \\,dx. ", "e6186e8248a427f6da557209bbc52f25": "\\tilde C \\to \\tilde D", "e618b3691b6e2a2bf4dca18afbaecfab": "s_r=\\pm 1", "e61940f3117a18cb1a38650e602fa8a0": "S = R\\, \\zeta - \\tfrac12\\, g\\, \\zeta^2 + \\int_0^\\zeta \\tfrac12 \\left[ \\left( \\partial_z \\Psi \\right)^2 - \\left( \\partial_\\xi \\Psi \\right)^2 \\right]\\; \\text{d}z.", "e6198c9eb27ff8495c90a1e77ab040ba": "\\scriptstyle{X^-\\subseteq X}", "e61998130a053481734dc85ab794006f": " R' = R + 2 Ric'(n,n)+ \\|h\\|^2 - H^2 ", "e619e29fc9da06ab9d1dcd505fd321f4": "\\bar{L}(\\alpha) = \\sup_{\\theta: \\alpha = g(\\theta)} L(\\theta). \\, ", "e61a0011650e6f86950d3c4816cfd8d4": "g:Q\\rightarrow\\{\\wedge,\\vee,accept,reject\\}", "e61a1fc802803b23e90837e00c97c123": "P(B) = {\\sum_j P(B|A_j) P(A_j)},", "e61a5de46b072fbeba01dab7b0f455a6": "\\tfrac{13}{2}", "e61a74d6ba8f7a27e8684c590fdcc9d0": "\nH(s) = \\frac{ \\omega_0^2 }{ s^2 + \\underbrace{ \\frac{ \\omega_0 }{Q} }_{2 \\zeta \\omega_0 = 2 \\alpha }s + \\omega_0^2 } \\,\n", "e61a9d4326e3be23df35b68dacb6eabe": " \\boldsymbol{S}_{\\boldsymbol{\\psi}} ", "e61ac24284f415e5b1d6e005304a8981": " \\text{vector area of parallellogram }ABCD = \\vec{r}(t) \\times \\vec{r}(t + \\Delta t). ", "e61ac7fa8d52b3406b3f1dd07fac61c0": "\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}\\mathbf{v} + \\frac{\\partial \\mathbf{v}}{\\partial \\mathbf{x}}\\mathbf{u}", "e61adcfac08359bd0f63dacc27f35ca3": "S \\subseteq \\mathbb{N}", "e61b4660f9d0cf5d4feeb77294653fa1": "1^6 + 2^6 + 3^6 + \\cdots + n^6 = {n(n+1)(2n+1)(3n^4+6n^3-3n+1)\\over 42} = {6n^7 + 21n^6 + 21n^5 -7n^3 + n \\over 42} ", "e61ba502534816d278a41d9aa1358016": "{\\mathcal F}_n", "e61bd0e7b98aef03a0b774da60434a21": "~n_1\\sigma_{\\rm a}(\\omega) v(\\omega)D(\\omega)~", "e61bd715b791bbb209fe62e731af3990": "V_s = V_P e^{i\\theta},\\,", "e61bef3f82db4ac0aac6e315da6a2c87": "f(\\mathbf t) = \\frac{\\Gamma((\\nu+p)/2)\\left|\\mathbf{A}\\right|^{1/2}}{\\sqrt{\\nu^p\\pi^p\\,}\\,\\Gamma(\\nu/2)} (1+\\sum_{i,j=1}^{p,p} A_{ij} t_i t_j/\\nu)^{-(\\nu+p)/2}", "e61ca8be409d6652efac763ed91cbc7a": "r\\geq (a/c)y+(b/c)x", "e61cd36372a99541015a9f871515a358": "-j3", "e61d1f170f3f7eb7a3fd642bf9971011": "*[F,*G]^{IJ} = [*F,*G]^{IJ}", "e61d836f5499047cfcc8208100b59d01": "\\frac{\\partial ^2 y}{\\partial s^2} (\\bar v_i) ", "e61da005487b5672026dcdcb5a84cd3e": "\\frac{\\partial {\\rm tr}(\\sin(\\mathbf{X}))}{\\partial \\mathbf{X}} =", "e61dc2c5b76ccc5ccebfb82fed66d26e": "\\alpha(k) = k", "e61decaba63ac4460429f044d310d199": " \\Box P ", "e61e355153d93550a2f94976cb11acd7": "H(q,p) = T(p)+U(q) = \\frac{1}{2}p^2 - \\cos q, ", "e61e66c7fc3e9ab20c6d34dd5d693819": " z \\neq 0", "e61e807803ae9154b84fa3c43ffbaf9a": "\\frac{\\sqrt{9} + \\sqrt{16} + 9 + 4}{2} =\n\\frac{3 + 4 + 9 + 4}{2} = \\frac{20}{2} = 10", "e61f47993ed6477969f6252cba4c94b1": "h(g(w))=g(w), \\,", "e61fea18bab4efe467a7349252c3a903": "\\sqrt 2 = 1 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\ddots\\,}}}}}.", "e61ff225b424ee09c1a0fb7a43333d68": "\\mbox{ROT}_{13}(\\mbox{ROT}_{13}(x))=x", "e61ff6d50424fa990b58fa1b58e21179": " v^{(1)} = \\begin{bmatrix} v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_n \\end{bmatrix},", "e62027dfab7adebecf0ce351882abf7d": "\\|\\mu*\\nu\\|\\le \\|\\mu\\|\\|\\nu\\| \\, ", "e6206ddbb73bd089dbf2fc4a84dacca6": "g'(x)= \\frac{1-\\ln x}{x^2}.", "e620a14d3b8dc6fd72458bc6b8cf8b7e": " Q_j = 0,\\quad j=1, \\ldots, m.", "e620ab544cc264fe423404e3ec19fef8": "v_{\\infty} +\\varepsilon ", "e620ede1b3908b512be545c18b8dd681": "Q = \\begin{pmatrix} -\\alpha & \\alpha \\\\ \\beta & -\\beta \\end{pmatrix}.", "e620f7418049e0693f0a65c685460101": "s \\in S, \\phi \\in F", "e621288edf66323c0f9624a19da4962e": "j=0,1,\\ldots,n", "e6214b174799a1a4b8a4e42d097fe004": "\\ MSE = \\frac{\\sum_{t=1}^N {E_t^2}}{N} ", "e6218918e6d69889ce189eb932a27070": "\n u_x(x,y,z,t) = u_0(x,t)-z~\\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z) = w(x,t)\n", "e622c5f85065561460bcbdca316a4f3b": "\\sup_{n \\in \\mathbb{N}} \\left( \\left\\| f_{n} \\right\\|_{L^{1} (W)} + \\left\\| \\frac{\\mathrm{d} f_{n}}{\\mathrm{d} t} \\right\\|_{L^{1} (W)} \\right) < + \\infty,", "e6233a597b23695be0e81ed3500c0588": "\\begin{align}\n F_n(x) =\n & \\Phi(x) \\\\\n & - \\frac{1}{n^{1/2}}\\bigg( \\tfrac{1}{6}\\lambda_3\\,\\Phi^{(3)}(x) \\bigg) \\\\\n & + \\frac{1}{n}\\bigg( \\tfrac{1}{24}\\lambda_4\\,\\Phi^{(4)}(x) + \\tfrac{1}{72}\\lambda_3^2\\,\\Phi^{(6)}(x) \\bigg) \\\\\n & - \\frac{1}{n^{3/2}}\\bigg( \\tfrac{1}{120}\\lambda_5\\,\\Phi^{(5)}(x) + \\tfrac{1}{144}\\lambda_3\\lambda_4\\,\\Phi^{(7)}(x) + \\tfrac{1}{1296}\\lambda_3^3\\,\\Phi^{(9)}(x)\\bigg) \\\\\n & + \\frac{1}{n^2}\\bigg( \\tfrac{1}{720}\\lambda_6\\,\\Phi^{(6)}(x) + \\big(\\tfrac{1}{1152}\\lambda_4^2 + \\tfrac{1}{720}\\lambda_3\\lambda_5\\big)\\Phi^{(8)}(x) \\\\\n &\\qquad\\quad + \\tfrac{1}{1728}\\lambda_3^2\\lambda_4\\,\\Phi^{(10)}(x) + \\tfrac{1}{31104}\\lambda_3^4\\,\\Phi^{(12)}(x) \\bigg) \\\\\n & + O(n^{-5/2})\\,.\n \\end{align}\n", "e6234a7c8edc44a05b8ec7d0d3dc1b8a": "\\Pr(w_n=v\\mid\\mathbb{W}^{(-n)},\\mathbb{Z},\\boldsymbol\\alpha)\\ \\propto\\ n_v^{k,(-n)} + \\alpha_v", "e623c422c7235771d7869dd9b4d3fad0": "\\Vert K \\Vert_\\mathrm{HS} = \\Vert k \\Vert_{L^2}.", "e6241ffd21225dfd2df31fd71fa27023": "N_{A}= -D_{AB} \\frac{dC_{A}}{dx}", "e6244eab2797a29e166011663a0f08c9": "Hom_{Ho C}(X, Y) = H^0 Hom_C (X, Y)", "e624c125bdcb522a2b170d60d985d024": "\\alpha_{A,B,C} \\colon (A\\otimes B)\\otimes C \\rightarrow A\\otimes(B\\otimes C)", "e624c4d22a8c9526a71077aff169aea6": "f^{-1}(U_i) \\subset g^{-1}(V_i)", "e624decd8e4ad890d10c7a3c0166a9ed": "k < n.", "e624f525841e0a6162cd0c7846245925": "S^i ={v\\in V^i | E_v \\cap X^{i+1} !=\\emptyset}", "e6254aa162a76adaac85cfd1f10e9672": "A_1 B_1", "e625822fc93c0f663b58a16976dccd4a": "\\frac{{\\rm d}^n}{{\\rm d}x^n}\\,\\Gamma(x) = \\int_0^\\infty t^{x-1} e^{-t} (\\ln t)^{n} \\,{\\rm d}t.", "e625935a1a2d776f86b6a6b5b937dffa": " \\begin{align}\n\\partial_t^+ u_{n,i} &= \\frac{u_{n+1,i} - u_{n,i}}{\\Delta{t}}, & \\partial_x^+ u_{n,i} &= \\frac{u_{n,i+1} - u_{n,i}}{\\Delta{x}}, \\\\[1ex]\n\\partial_t^- u_{n,i} &= \\frac{u_{n,i} - u_{n-1,i}}{\\Delta{t}}, & \\partial_x^- u_{n,i} &= \\frac{u_{n,i} - u_{n,i-1}}{\\Delta{x}}.\n\\end{align} ", "e625d836221c9f33756ea4222aa1d07d": "\\|A\\|_{op} = \\min\\{c\\ge 0 : \\|Av\\| \\le c \\|v\\| \\mbox{ for all } v\\in V\\}", "e625dbe2419f12911b90af14b33b52e4": "\\phi(0) \\wedge (\\forall x \\phi(x) \\rightarrow \\phi(Sx)) \\rightarrow (\\forall x \\phi(x))", "e626ad0b002beef8965853fba81a8ffc": " 2 \\approx 10^{3/10}", "e62710dbd18ffd18550774c2cafc04d5": "\n= \\frac{1}{4\\pi\\varepsilon_0} \\sum_{\\ell=0}^\\infty\n\\left[\\frac{4\\pi}{2\\ell+1}\\right]^{1/2}\\; \\frac{1}{R^{\\ell+1}}\\; \\sum_{m=-\\ell}^{\\ell}\n(-1)^m Y^{-m}_\\ell(\\hat{R}) Q^m_\\ell, \\qquad R > r_{\\mathrm{max}}.\n", "e627c56855704940423c265f9c40b954": "\nU \\sum_{i=0}^n (B_1^*)^i h_i = \\sum_{i=0}^n (B_2^*)^i h_i\n", "e627fafc4f3331b862ab0d3bcb301dab": "f^2(\\theta)=\\cos(\\theta)", "e6284252fbb26c5c78b86644f4d4f732": "E(m) \\oplus t = m \\oplus t \\oplus S(k) = E(m \\oplus t)", "e62853505998b7249cce05f9db105dd8": "\\displaystyle ((x'+u')-(x+u))\\times f(u'-u, x'-x, x)=0.", "e628780b595bd3f39e6a34fab13cfe18": "\\omega _{e}\\approx \\frac{2 \\cdot \\pi}{T_{day}} = 7.27221\\times 10^{-5} \\;\\; \\mathrm{rad/s}", "e6289b59d238e44cf87a8a6e306681c9": "\\hat{r}(x,s)", "e628ab0078f841006c36a97cdcd8f577": "r = \\frac{3}{2}R_s", "e628d6650e76a0496502b3402d8c883d": " \\Lambda(n)=\\lim\\limits_{s \\rightarrow 1} \\zeta(s)\\sum\\limits_{d|n} \\frac{\\mu(d)}{d^{(s-1)}}", "e6290ae8a715f43bba29019288500198": "l=\\frac{\\rho}{r}", "e62920482f5f8e139a5e759fd38f62c6": "P_V^2=P_V.", "e6292bc21dc8ae7e32bd05a981f66478": " \\langle 0| \\phi(-x) \\phi(x) |\\psi\\rangle \\,", "e62949d079995c3009134fb8c5882584": "T(n) = T\\left(\\frac{n}{2}\\right) + O(1)", "e6294c435794b6104188a30e019dd29c": "(x \\in A)", "e6297425017401869651a6796e9704ba": "\\tfrac13\\, Q^2\\, \\left( \\zeta' \\right)^2 \\approx -g\\, \\zeta^3 + 2\\, R\\, \\zeta^2 - 2\\, S\\, \\zeta + Q^2.", "e629c503cddd42f7733020a4bbc8d06c": "\\operatorname{E}[|X - E[X]|] = \\frac{2 \\alpha^{\\alpha} \\beta^{\\beta}}{\\Beta(\\alpha,\\beta)(\\alpha + \\beta)^{\\alpha + \\beta + 1}} ", "e629c7babb71ca469d790beeb40b7554": "e^{\\pi \\sqrt{163}} = 640320^3+743.99999999999925007\\dots\\,", "e62aaa2de3876cced25ac86e9cac69f8": "G(r)", "e62acd7a82b343d6a2eecbb8d42ec05c": " 2H_2 (g) + 2 NO(g) \\rarr N_2 (g) + 2 H_2O (g)", "e62ae2932ab2ddf222817f45f97b7d17": " \\cfrac{\\Gamma[x_1,\\ldots,x_n] \\vdash p[x_1,\\ldots,x_n]}\n{\\Gamma[t_1,\\ldots,t_n] \\vdash p[t_1,\\ldots,t_n]}\n", "e62af2f73d24f16fa9362462fd166040": "\\sin(2\\pi t/23)", "e62b2aa89702da6b2fcc6eb2d76c7e31": "\\angle PAB = \\angle PBC = \\angle PCA.\\,", "e62b3e82097aea400f949a531de86928": "\\! T", "e62b3f86b9a2c8f8640050648ea49e60": " (S_7 \\implies (\\operatorname{equate}[A_7, q] \\and V[F_7] = A_7)) \\and D[F_7] = D[q] ", "e62be666a3c2c094a1db6cb2672f5fbf": "A \\circ (B \\circ C) = (A \\circ B) \\circ C,", "e62c0c0d4440bc690a6aaefeb1591116": " AMI(U,V)= \\frac{MI(U,V)-E\\{MI(U,V)\\}} {\\max{\\{H(U),H(V)\\}}-E\\{MI(U,V)\\}}\n", "e62c22dfa0b28e89202e8c224c46e580": "\n2\\Omega_x > \\Omega_z\\,\n", "e62d09b6636175fcdba9efcf00af0813": "Q(\\psi, P) := -S_\\psi S_P \\,\\!", "e62d0f0560f7e5a477ef0fb579f30b7d": "\\mbox{Cl}(S) = \\{ x \\in X :", "e62d239822831122dd571c0d362408f7": "a = b", "e62d59cf3e24a303b4bd5df50c9ac694": "\\boldsymbol{\\hat \\theta}", "e62db25f6334317b6aa33e1ac5c135df": " \\lim_{n\\to\\infty} \\frac{x_{n +1}-\\alpha}{\\prod^k_{i=0}(x_{n-i}-\\alpha)} = L = \\frac{(-1)^{k+1}} {(k+1)!}\\frac{f^{(k+1)}(\\alpha)}{f'(\\alpha)}, ", "e62dcb48161c08340071c87359b28a81": " T_{NL}^{\\pm} \\approx (k z_k^{\\mp})^{-1} ", "e62e3cdfe80267baf859d66b7c5367d0": "|x|_{0} := \\begin{cases} 0, & \\text{if } x = 0 \\\\ 1, & \\text{if } x \\ne 0. \\end{cases} ", "e62e68240544571c1d5699829472ee8d": "|q|>1", "e62e7effcce2f76ef966864dfad626b9": "\\Psi_3 = -2\\log(\\frac{\\varepsilon}{3.7D} + \\frac{2.51\\Psi_2}{Re})", "e62e88502d92c6af9f842e734585189c": "p(D|S(D),M_1)=p(D|S(D),M_2)", "e62e8ac127192a36f02f7ddd8a1335ff": "\\underline{\\mathbf{R}}", "e62e997bb83a1fd697c15ebeadfcdea0": "s_2 = x_0 + \\zeta^2 x_1 + \\zeta x_2.\\,", "e62fb373ee5d6aaebaf6ea2b0645a871": " w: V^k \\to \\Lambda^k(V) ", "e62fdf7526cff9f9f9f9586d1dcf58c2": "\\bar{\\nu}_e + p \\rightarrow e^{+} + n", "e6303adc0aa83dbd72221ac3a10f99b8": "2\\sigma(r) \\beta + r \\frac{d\\sigma(r)}{dr} \\beta(r) - r \\frac{d\\beta(r)}{dr} \\sigma(r) = 0.", "e6308f4857605113be644436afa87af4": "\\eta_{ab} \\, = \\operatorname{diag}(1, -1, -1, -1)", "e630bcfe4eaa661b67a620bdd3e4cc37": "[\\ ] \\!\\,", "e630dfbb4d04a8cad0b5d09d84977451": "H+\\vec{u}", "e631c1cfa2ac21d7a911ff4f11a43e4e": "\\left [ \\hat{b}, \\hat{b}^\\dagger \\right ]\n = \\left [ u \\hat{a} + v \\hat{a}^\\dagger , u^* \\hat{a}^\\dagger + v^* \\hat{a} \\right ]\n = \\cdots = \\left ( |u|^2 - |v|^2 \\right ) \\left [ \\hat{a}, \\hat{a}^\\dagger \\right ]. ", "e631d2f9f8edad3de07e8befac5790c9": "V_{L1}=V_i-V_C", "e631f7ce9ea74aed4a1b18c80b390618": "\\frac{I_1(\\kappa)^2}{I_0(\\kappa)^2}\\,", "e6320503f39443ede9f989c9d7be9346": "{\\tilde{C}}_3", "e63205843f87830c369d531d4031ac5c": "S_j = \\sum_{k=1}^j D_k", "e6322ca927bc0461dd562ffd3595c74a": " 2^{2^2} + 2 + 1 ", "e6326d3b6c06031deeefe28f3cfa3179": "(Y,\\mathcal{B},\\nu)", "e6327da8635a0412a9fe71baf34799a2": "y=-x'", "e632df1b34368afe01f5bbe2528d00a9": " \\frac{a + b}{c} = \\frac{\\cos\\left(\\frac{\\alpha - \\beta}{2}\\right)}{\\sin\\left(\\frac{\\gamma}{2}\\right)} ", "e633004ea2996053b531d57cc302f7aa": "\nP(n,t)=\\frac{(\\overline{\\nu}t )^n}{n!}\\exp(-\\overline{\\nu} t),\n", "e63345d621eccd1537973f923ff7d12e": "2\\pi\\sin \\vartheta d\\vartheta / 4\\pi", "e63364a56f06a358f871a4f83a206c26": "\\sup_{x^*\\in\\{c\\}}(xx^*-f^*(x^*))=xc,", "e633d2ee0c8c681ef34e5f95987b5ba8": "\\Gamma^{(\\mu)}\\,", "e633f0a66e9cf6b0cb24ffb013cfaee2": "y_t = a + w_0x_t + w_1x_{t-1} + w_2x_{t-2} + ... + w_nx_{t-n} + \\text{error term},", "e634317674f9e57805cdcb0c9a4600cb": "[y_1,y_2,...,y_n]", "e63438687eb3356500aaf22b1b28d94c": "T_p M / T_p N", "e63458d33caca6f7cea5e7757c322387": "[X]^{\\kappa} = \\{Y \\subset X \\; | \\; |Y|=\\kappa\\}", "e6345f70b37931d6b4ea0a02c5353d62": " \\neg L_i ", "e634d30cab8582e6caee0f6f2ed070f6": "= \\theta \\circ j^{2}_{p}\\sigma \\,", "e634f0f7ee8a4ade77684608e77d8da8": "\\boldsymbol{\\sigma}=\n\\left[{\\begin{matrix}\n \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz}\n\\end{matrix}}\\right].\n", "e63564549326a9f963a7d011bcc51cb1": "T_e=1", "e635d535d9883f07e7a33d6e7c5dbcd2": "\\log(1 + x)", "e635da310712cab5dc17a4ddec2296c7": "\\mathbf{A},\\,\\vec{A},\\,\\underline{A}", "e635e3b1ef0e723603c906de5549715a": "R^1\\subseteq G", "e635f222852dabf5210ecacb3592260c": "4k+2.", "e6360b7a408f186ae0cbefb1bc781ab5": " \\sum_i^{N_1}m_i \\mathbf{v}_{\\rm i} = \\sum_j^{N_2} m_i \\mathbf{v}_{\\rm i} .\\,\\!", "e6364ca2955de22f113e28e91fe9398c": "[0,1)^s", "e636cf7eb72f5b7ef0e154d45cedac17": "f(x,y) = \\left( g(x,y)\\cos x , g(x,y)\\sin x , \\sin\\!\\left(\\frac{x}{2}\\right)\\sin y \\right) ", "e636d336c80a538e47dfea29141a9bcc": " \n\\begin{align}\n&\\left(A+UCV \\right) \\left[ A^{-1} - A^{-1}U \\left(C^{-1}+VA^{-1}U \\right)^{-1} VA^{-1} \\right] \\\\\n& \\quad = I + UCVA^{-1} - (U+UCVA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\\\\n& \\quad = I + UCVA^{-1} - UC(C^{-1}+ VA^{-1}U)(C^{-1} + VA^{-1}U)^{-1}VA^{-1} \\\\\n& \\quad = I + UCVA^{-1} - UCVA^{-1} = I \n\\end{align}\n", "e636dd7b75d4fb34955c0214ed718c8f": "t_4 = -299 \\equiv 7 \\pmod {17}\\, u_4=156 \\equiv 3\\pmod {17}", "e637538305f8ec1273ebf9f5a43cf726": " \\int\\limits_0^1\\int\\limits_0^1\\int\\limits_0^1 1 \\,dx\\, dy \\,dz = \\int\\limits_0^1\\int\\limits_0^1 (1 - 0) \\,dy \\,dz = \\int\\limits_0^1 (1 - 0) dz = 1 - 0 = 1", "e6378b4312bc9fa575ab2c1e2a6a2952": " \\textstyle r ", "e638176566882b4719368232f1806276": " P^{-1}(Ax-b)=0", "e63821eccaf640a919761cfe882b1f27": "(z,w)", "e6384b0ee3faddc93a135be3e78067ab": "m<0", "e6387c57e2a6de37ca71f2bef0d7c948": "s=(\\ldots,(s_i, t_{si}, t_{ei}),\\ldots) \\in S ", "e6387f695054c099310eaec79ad26b2b": "1-f_v(x)=t_v(x)", "e638b96725ba5cef54e1b6e551d621ca": "P_t = \\sum_{n = t+1}^{\\infty} m_n B(t,n) = \\frac{m_{t+1}}{1 + r(t,t+1)} + \\frac{1}{1 + r(t,t+1)} \\mathbb{E}[P_{t+1} \\mid \\mathcal{F}_t]", "e638dd14e01b64510fe49a6e98f7105a": " \\delta_n(x)=\\frac{a+b}{a+b+n}E[\\theta]+\\frac{n}{a+b+n}\\delta_{MLE}.", "e639372afedd2bf9861d4672ccedff3b": "\\mu < n", "e63960d7f7d18b682e1cc51261d73476": "x\\ f", "e639743716ecb5fd7c7d4024ffdf5b98": "\\dot{r}=\\frac{F}{2}\\dot{u}", "e63a5e58cbce41efe3faf4e263c2ed37": " = \\frac{2}{\\pi}\\, \\ln\\!\\left[\\tan\\left(\\frac{\\pi}{2}\\,p\\right)\\right] \\! .", "e63b051c7d2ecce5a0321a2268b42894": " -\\frac{\\beta + 1}{\\beta} ", "e63b2abaae07a9f2ce7e6244a8debe98": " f(t)=1 - t + it \\mbox{ and } g(t)= \\cos(2\\pi t)+ i\\sin(2\\pi t)\n", "e63c3e3c02db3603ab87ba27f5e9d349": "\\begin{cases} \n cap~n\\;\\|\\;[~]_{cap~n} & \\mbox{if } A=cap~n\\\\\n cap~n\\;\\|\\;[\\;\\mathcal{T}_1(A_1)\\;]_{cap~n} & \\mbox{if } A=cap~n.\\,A_1\\\\\n \\;[\\;\\mathcal{T}_1(A_1)\\;]_n & \\mbox{if } A=n[\\;A_1\\;]\\\\\n \\;[~]_n & \\mbox{if } A=n[~]\\\\\n \\mathcal{T}_1(A_1)\\;\\|\\;\\mathcal{T}_1(A_2) & \\mbox{if } A = A_1\\,\\mid\\,A_2\\\\\n \\lambda & \\mbox{if } A=0\n\\end{cases}", "e63c40495cd8ea8e6b286d490eae4e56": "< 0.084", "e63c8550f071b9db9c74758dacd87663": "\\psi(u,v)=\\phi(u,v)\\wedge \\forall x, (x=v \\vee \\neg \\psi(u,x))", "e63c8af1c68e957432efa0b56e19d314": " KP(y|x^{\\ast}) \\leq KP(x,y) - KP(x) + O(1)", "e63d174c670dc5a0a850428f32f8e79a": "C_{\\hat{X}} = C_{XY} C^{-1}_Y C_{YX},", "e63d1ea58d45ffeb33e7b33ae200094d": "\\mathit{\\Delta}", "e63d5a2b42843a20d8fa100bca693b4e": "(1-1/2)/2=25\\%", "e63da351a6ad9f7fc8f2dde0d613947e": "\\mathbf{\\Psi} > 0", "e63dea72ec1854ba610536f73e785720": "\\Gamma_n := \\Lambda_n\\ltimes {\\Bbb Z}", "e63decf9ed4f97265dc1e38fb336a66a": "169 \\times {50 \\choose 18} \\times 17!! \\times {32 \\choose 5} \\approx 2.117 \\times 10^{28}", "e63ed716d02b5af3481511e19e42d15a": "H(z) = \\frac{Y(z)}{X(z)} = \\frac{\\sum_{q=0}^{M}z^{-q}\\beta_{q}}{\\sum_{p=0}^{N}z^{-p}\\alpha_{p}} = \\frac{\\beta_0 + z^{-1} \\beta_1 + z^{-2} \\beta_2 + \\cdots + z^{-M} \\beta_M}{\\alpha_0 + z^{-1} \\alpha_1 + z^{-2} \\alpha_2 + \\cdots + z^{-N} \\alpha_N}.", "e63ef2e3e74baadf40768c903116f75b": " \\eta=E-E_{\\text{eq}} ,\\;f=F/(R\\,T),\\;\\alpha_{\\text{o}}+\\alpha_{\\text{r}}=1", "e63ef86040f5da9baf1e7058c2e1c7d5": "\\frac{d \\mathbf{Q}_k}{dt} (t) = \\frac{\\hbar}{m_k} \\operatorname{Im} \\left(\\frac{\\nabla_k \\psi}{\\psi} \\right) (\\mathbf{Q}_1, \\mathbf{Q}_2, \\ldots, \\mathbf{Q}_N, t)", "e63eff7873285a5fafce3488c0dc3734": "(x,y,z,t)", "e6407602d93164f7c926b067f7100a2c": "X^p - X = T^{-1}", "e640d1deb78f5a2daa766d5480391357": "P_{\\rm S\\ emt} = 4 \\pi R_{\\rm S}^2 \\sigma T_{\\rm S}^4 \\qquad \\qquad (1)", "e640df2f7a4bbe689ca8231e5265bf0d": "\\scriptstyle \\Pr(N\\;=\\;n\\mid M\\;=\\;m,\\, K\\;=\\;k)", "e64104157d5158d6d5f2d1747cb137c1": "\\mathbf{a}_{31} = -4", "e641327a582cbecb0602a764a8ef1b3e": " a^7 + b^7 = (a + b) (a^6 - a^5 b + a^4 b^2 - a^3 b^3 + a^2 b^4 - a b^5 + b^6),\\,\\!", "e64143073cac46032a534e4c4b6a3ce5": " b_{s-1} ", "e64157eb5a0f8eac633d557b032e5fa6": "f(x,s,t,\\theta,W)", "e6417b16a767203ee244f1d560811c71": "\\mathbf{J}=- D \\nabla n^m", "e641b816b69335215e26cc121a57d99f": " \\lim_{x \\to \\infty} \\frac{2x-1}{x} = 2. ", "e641c846402058f55146a0afd7e89a2d": "\\mathbf{H} = \\mathbf{H}_0 + \\mathbf{H}_d, \\,", "e641fe448342e2556628868eaa1a0aa5": "\\sigma(x, x) =\\sigma^2(x).", "e6421840e74fbc64c627ccd79c3b6be2": "\\exists\\ X_i\\ D_{**}(X_i, X_i) > 0", "e64247c2921f51e0dbbc79fa94efbfc1": "\\varphi_{X+Y}(t) = \\operatorname{E}\\left(e^{it(X+Y)}\\right)", "e64260f869ad29da76e7ade999c42ac2": "\\mathbf{w}_{n}=\\left[w_{n}(0),\\,w_{n}(1),\\, ...,\\,w_{n}(p)\\right]^{T}", "e642667ee9b16e8b943649245cfc6353": "\\cos (\\alpha +\\beta)=\\cos \\alpha \\cos \\beta - \\sin \\alpha \\sin \\beta", "e642e32c2e1331f17b5d48e018d3f4fd": "h_\\lambda (X_0)=\\lambda = \\text{constant}.", "e64321315d1d9858db0cbc0879ae9b23": "x\\mapsto f(x\\mid\\theta), \\!", "e6432c21347b61124463a1b184a00f55": " (\\bold A - \\lambda \\bold I) ~ \\bold x = 0 ", "e64383030cae5440b1ac183b2b4fe6f9": "{\\Delta V_w} = V_{w1}+V_{w2}", "e643980ecc0d58401af05dd443c7684e": "x\\in A_1", "e643c2d99f6c133fff78888d71c0123e": "N(\\mu,\\sigma^2/n),", "e6444b5275b5e97f8e00a85d3854f97d": " P_\\text{avg} = \\frac{1}{n} \\sum_{k=1}^n V[k]I[k] ", "e644ab2f548694f0653629a06c975563": "m_T = \\gamma m", "e644feffb33c471f74fc864694116636": "o((\\iota \\to \\iota) \\to \\iota) = 2", "e6450d556e852118fbfd09114fabd16e": "HS_S(t)=\\frac{P(t)}{(1-t)^\\delta}\\,,", "e64510756776d415f661bca81ee982a4": "\\bar V = \\frac{{RT}}{P}", "e6456a2d8b3c8506f80d9cdaa8db2c59": "\\frac{T_E}{ T_D} = ?", "e6458cecaa6efb2ad5a2668e0d24fad3": "M_{D}=k\\cdot P\\cdot Q", "e645dc6e5bd4704673185f9cf53ac151": "\\Gamma(\\tfrac32)\\,", "e6460b914748ccc4ef3e19699a1b80c0": " x(t) = x(t + T) \\ ", "e6464225a46b610103735292c61a1111": "(dy, -dx)", "e6467bd3cc11e38332a8587ed285ac72": "P(L_1 \\# L_2)=P(L_1)P(L_2),\\,", "e6469f334fdb5f3a1258730118dfce14": "\\pi_i(X)\\otimes \\mathbb{Q} = 0", "e646c4987140a451c4f050113c2568a7": "P(A_+ > x) \\sim \\frac{6 \\sqrt{6}}{\\sqrt{\\pi}} x e^{- 6x^2} \\ \\ \\mbox{as} \\ \\ x \\rightarrow \\infty.", "e646cd5ae55810388dca4815cfc5a783": " \\lbrace(g\\ \\bmod\\ \\ f) : g \\in \\mathbb{Z}[x] \\rbrace ", "e646deb3c80cb08e4dba2ace32376b51": "\n\\begin{align}\n\\langle \\mathbf{q} \\cdot \\mathbf{F} \\rangle &= \\Bigl\\langle q_x \\frac{dp_x}{dt} \\Bigr\\rangle + \n\\Bigl\\langle q_y \\frac{dp_y}{dt} \\Bigr\\rangle + \n\\Bigl\\langle q_z \\frac{dp_z}{dt} \\Bigr\\rangle\\\\\n&=-\\Bigl\\langle q_x \\frac{\\partial H}{\\partial q_x} \\Bigr\\rangle -\n\\Bigl\\langle q_y \\frac{\\partial H}{\\partial q_y} \\Bigr\\rangle - \n\\Bigl\\langle q_z \\frac{\\partial H}{\\partial q_z} \\Bigr\\rangle = -3k_B T,\n\\end{align}\n", "e64731f233f768dea6f10b65c289966f": "\n\\begin{align}\n\\frac{d^2 x}{d t^2} &= \\frac{d}{d t}\\left(\\frac{d x}{d t}\\right) = \n\\frac{d}{d x}\\left(\\frac{d x}{d t}\\right) \\frac{d x}{d t} = \n\\frac{d}{d x}\\left(\\left(\\frac{d t}{d x}\\right)^{-1}\\right) \\left(\\frac{d t}{d x}\\right)^{-1} = \\\\\n\n& = - \\left(\\frac{d t}{d x}\\right)^{-2} \\frac{d^2 t}{d x^2} \\left(\\frac{d t}{d x}\\right)^{-1} =\n- \\left(\\frac{d t}{d x}\\right)^{-3} \\frac{d^2 t}{d x^2} = \\\\\n\n& = \\frac{d}{d x}\\left(\\frac{1}{2}\\left(\\frac{d t}{d x}\\right)^{-2}\\right)\n\\end{align}\n", "e6475eb5f0e65b5ef1a51c8c82a0088a": "A = \\begin{bmatrix} 1 & 0 & 2 \\\\ 0 & 1 & 0 \\end{bmatrix},", "e64774c459947f148d562954268adc60": "m\\neq j", "e6479e6a4aeefe335fe45a355755f245": "\\, R^2", "e647aa65b6452ce1a2b185a920109fc4": "f^*:\\mathrm{Sub}_{\\widehat{C}}(Y)\\to\\mathrm{Sub}_{\\widehat{C}}(X)", "e647bb2723ef2aa4b3e815eb4c8db51c": "k=\\mathcal{O}/\\mathfrak{m}", "e647c45f9903d3017e900460d956b23e": "\\,\\psi", "e647edd3b71de1dfee66e0f60c131a13": "\\sigma>0", "e6484986b65c169fa149c9933c64a0fa": "\\frac{W_{n + 2}}{W_n} \\leqslant \\frac{W_{n + 1}}{W_n} \\leqslant 1", "e6490a872b50d4cd9855cb99c7969bbe": "J^+(S;T)", "e649ac6251ac6f649586f042e60e0fc3": "\\left(-\\infty\\right)", "e649c57ebbfb98b89fb824244ad28786": " \\int_P \\mathbf{v}\\cdot d\\mathbf{r}=\\varphi(B)-\\varphi(A). ", "e64aa40e511aa03450233dff8b321775": "f' \\circ f \\colon f^{-1}(V \\cap U') \\to f'(V \\cap U')", "e64b10198eb5bafd0d423b828ba6e284": "S = \\{\\{1,2,5\\},\\{2,4\\},\\{1,4\\}, \\{2\\}\\}", "e64b38b67c1ce20f93e84f6904e8489c": "\n\\mathcal{E}(f) = \\int_{X\\times Y}\\left(f(x) - y\\right)^2 d\\rho\n", "e64b6b5075b01429a66e3616da0158b4": "(x,K/x)", "e64b6c492739bd37169b364140e0346b": " c\\log_d b=\\log_d a", "e64b7696ccee52e6f6d8246f72ab6b10": "T=\\{t|t\\; is \\;a\\; sink \\}", "e64bcdd32599e85c535c216a048b7ee9": "\n \\sigma_{11} - \\sigma_{33} = \\sigma_{22} - \\sigma_{33} = \\lambda_1~\\cfrac{\\partial W}{\\partial \\lambda_1} - \\lambda_3~\\cfrac{\\partial W}{\\partial \\lambda_3}\n ", "e64cb9f212c2a411d934b5016c7e0d11": "\n\\frac{\\delta H}{\\delta p(x)} = -1-\\log p(x).\n", "e64d2125031f601ffa319981d5fc54df": "\\left\\{x_1, \\ldots, x_n, x_{n+1}\\right\\} \\overset{\\mathrm{def.}}{=} \\left\\{x_1, \\ldots, x_n\\right\\} \\cup \\left\\{x_{n+1}\\right\\}", "e64d3874eef9069f60583416b524ed38": "\\Delta \\omega\\,", "e64dc046552504b868960adec860e603": " L(\\lambda) = \\prod_{i=1}^n \\lambda \\exp(-\\lambda x_i) = \\lambda^n \\exp \\left(-\\lambda \\sum_{i=1}^n x_i\\right)=\\lambda^n\\exp\\left(-\\lambda n \\overline{x}\\right), ", "e64ddf3c78e8cadb8a1e57b21ae10731": "z=8m\\!", "e64e2aa21b7042e60aaaeb42e08b9915": "\\hat{\\bold{z}} \\times \\bold{B}_\\perp = \\nabla A -(\\bold{\\hat z} \\cdot\\nabla A) \\bold{\\hat z} = \\nabla A", "e64e5b509328eb309f946c7efd16a2d8": "\\frac{d[A]}{dt} = \\frac{k_1k_3[ABCD]}{k_2[D]+k_3}", "e64e5f1e2333329a6a6307293d656f89": "P(A)\\in[0,1]\\,", "e64ea9412782906d850fdeef8682251e": "y \\in \\mathfrak{j}", "e64ec235b4c76a6b66b65f656fdd6724": "1=p^r\\cdot p^{-r}=p^r (1-q)^{-r}=p^r \\sum_{k=0}^\\infty {-r \\choose k} (-q)^k.", "e64ee80e45b6fba9ca6875a1a0b8aae0": "\\left( X_{a:bn}-X_{a:nb} \\right) \\, X^n = E[\\vec{X}]_{ab}", "e64f0905de4a6153e2d596abb6532249": "\\mathbf{Top}", "e64f09cbae124d69e0f1bde9fcee6160": " f(P)=(f_0,f_1,\\ldots,f_{d-1}) ", "e64f185bab92ed6804d7c6b3f4caa55f": "d^{m}", "e64f59ef63099b0719eb22d8dd8a8b0e": "\\gamma k^2", "e64fa76659c8a728ba8a585f5dbfa9ac": "a x^2 + b y^2 = z^2 \\ ", "e64fff4d4d8509a2f6185870a2a86469": "|x-x_0|<\\delta", "e65071dc25a44a3bb1a81fbaec4dd7ed": "\\pi_0(X,*) := [(S^0,*), (X,*)].", "e65075d550f9b5bf9992fa1d71a131be": "XYZ", "e65097e3e5190f4af31781b6fb173295": "j_p^\\infty(\\sigma)", "e650c116296d8b0270f94470ab7c4249": "\\tilde\\kappa", "e650c2115ffe7351f1eda43bf8711af9": "B_\\infty^{p,q} = \\text{im }(C^{p+q-1} \\rightarrow C^{p+q}) \\cap F^p C^{p+q}", "e650ef5f83e0797a259eb430e7d16cdd": "q=\\frac{\\text{value of stock market}}{\\text{corporate net worth}}", "e6512102a449b8d442c00fd6c4e08a7d": "I=\\gamma_0\\gamma_1\\gamma_2\\gamma_3", "e65129a1f4dcf5d5b07113d244bff0dc": "\\scriptstyle{DTFT}\\displaystyle \\{x_N\\}\\cdot \\scriptstyle{DTFT}\\displaystyle \\{y\\} = \\frac{1}{N} \\sum_{k=-\\infty}^{\\infty} \\scriptstyle{DFT}\\displaystyle\\{x_N\\}[k]\\cdot \\underbrace{\\scriptstyle{DTFT}\\displaystyle \\{y\\}(k/N)}_{\\scriptstyle{DFT}\\displaystyle\\{y_N\\}[k]}\\cdot \\delta\\left(f-k/N\\right)", "e6512fe4308391e0bc13980607c847a2": " Reference http://www.parkweb.vic.gov.au/1bays.cfm --!>", "e651ecf50f5982fb1edd28f0665d3dbd": "Z_\\mu(X,t)=\\sum_{n=0}^\\infty\\mu(X^{(n)})t^n", "e651feb028b3dee90541aba91a99f736": " \\displaystyle A(s,\\lambda)F(k)=\\int_{\\sigma(N)\\cap s^{-1}Ns} F(ksn)\\, dn,", "e65212d1f48946643543aeb0d996ce9e": "\\mathbf{v} = (u,v,w)", "e65278ad7eeb29a8c2a583cb7e8e56ef": "\\delta m x \\frac{dv}{dt}=-\\frac{dp}{dt}xz\\delta m + \\frac{dr}{dt}x^2\\delta m ", "e652abb5d318fcb2b77ea15cf5d11b12": " S = -1 ", "e654574b0f8a091f39edc21711315c50": " 0=v_0 \\sin(\\theta) - gt_h ", "e6547b367b1ac67d175d956620095f19": "k_{F}(\\epsilon)\\frac{1-T_{\\text{C}}}{1-T_{\\text{D}}}-1 ", "e66eb05c2e62062d288a512790ca9364": "I_{S} = V_{IN}\\frac{Z_{P}+Z_{1}}{Z_{P}Z_{1}}", "e66f0df0182940fd60e00278fe0b0a80": "F=\\frac{\\sigma_W^2}{\\mu_W},", "e66f42136551af10fb32687cb32b6a80": "LWP=\\int_0^{p=p_0} r_L dp/g ", "e66f6641453d5e15e3c3d9dacec8fffd": "\\hat{X}_{Bayes}(\\mathbf{v}) = \\hat{X}_{Bayes}(\\alpha\\mathbf{v})", "e66f9abb7dafa74aacad8f17c9efe58f": "\\begin{align}\n\\frac{\\partial \\Lambda}{\\partial x} &= 1 + 2 \\lambda x &&= 0, \\\\\n\\frac{\\partial \\Lambda}{\\partial y} &= 1 + 2 \\lambda y &&= 0, \\\\\n\\frac{\\partial \\Lambda}{\\partial \\lambda} &= x^2 + y^2 - 1 &&= 0,\n\\end{align}", "e6700178748dcce7489f79b6ef4ff614": "\\mu + \\frac{\\phi(\\alpha)-\\phi(\\beta)}{Z}\\sigma", "e6703364900afa3cbe8b41bdf50b8f06": "\\bar{\\sigma} = pr_{2} \\circ \\sigma \\in C^{\\infty}(M)\\,", "e6703d7f79c283bb4a486fd586aaa5c0": "E_{tgu} = 0.5 \\cdot [\\tfrac {(m_p \\cdot v_p) + (m_c \\cdot v_c)} { 1000 }]^2 / m_{gu}", "e6703fb70a25f9accce07b6401b69d38": " \\det \\begin{bmatrix} \n 0 & a^2 & p^2 & d^2 & 1 \\\\\n a^2 & 0 & b^2 & q^2 & 1 \\\\\n p^2 & b^2 & 0 & c^2 & 1 \\\\\n d^2 & q^2 & c^2 & 0 & 1 \\\\\n 1 & 1 & 1 & 1 & 0\n\\end{bmatrix} = 0. ", "e670428aa0e3292bb9ffb7b14c326400": "\\epsilon_0 - \\ ", "e6705a3666774e6f31d384dce62f1b42": "\\left\\langle 2e^{\\tfrac{\\pi}3 \\mathrm i},A_4:=\\left\\{0,1,e^{\\tfrac{2 \\pi}3 \\mathrm i},e^{-\\tfrac{2 \\pi}3 \\mathrm i}\\right\\}\\right\\rangle", "e67069a505d432b10c5370ae66fab50d": "z'=z \\,", "e670c7a94bbf14f4a6b7b3068980c3a6": "x\\in F_2^k", "e67101767684659a4b38129e1b133a8c": "\\vec v_s", "e6713711b2f9e557ac73ce941afa3a07": "J^{\\nu \\rho}", "e6719c20c68dddd16c6e075bf6d7b33f": "3 - 6/(n+2)", "e671e9e5d4392f1db3d39a957db2e0b5": "SG_V = SG_A - {\\rho_a \\over \\rho_w }(SG_A-1).", "e67221e0fce64754b45394be207e7431": "\\ell=0,1,2,\\ldots,n-1", "e6728754c810f55e58a80dd456f558a3": "f: U \\rightarrow \\mathbb{C}", "e672c2120f1ce25f345a9943c741f1fd": "\\lambda_6 = \\sqrt{3}.", "e672d798e4a9023489caf2393f748d5b": "\\dot{x} = f(x). \\,", "e6732b8978992cb3f418c4ba0db9385b": "H_4,", "e6732f38be1ba7ad0c2262bd2144d4cd": "M^{0.67}", "e673313ba75e758191629d9777c30bad": "B^{-1}=\\frac{B^{adj}}{\\det(B)}=\\pm B^{adj}", "e673583be8c668229a46aa91ce064a0c": "\\left|\\frac{d}{dt}\\right|^\\alpha \\delta(t)", "e6738df581b249942023738525dc9be3": "\\phi \\in \\Phi\\,", "e673f746da3d7ff0d2deae41b65e517e": " (3) \\frac{\\mathit D} {\\mathit P} = \\frac{\\mathit ROE - \\mathit g} {\\mathit PB} ", "e67423fa3a83de49f1af8f8f89e6ec37": "\n\\eta = {R_\\mathrm{load} \\over {R_\\mathrm{load} + R_\\mathrm{source} } } = { 1 \\over { 1 + { R_\\mathrm{source} \\over R_\\mathrm{load} } } }.\n\\,\\!", "e67424ea78740bc7a9b757478c9c3911": "\\frac{\\text{d}C_2}{\\text{d}t}= \\text{k}_{1(2)} {^1_2}S^\\beta E - (\\text{k}_{2(2)}+ \\text{k}_{3(2)}) C_2", "e6743906a7ac70c0d68f38a1e8ec8b80": "\\sum_{n\\le X}a(n) \\sim \\frac{c}{b} X^b.", "e674626170f61aee2d8ad3f718bd7529": "\\xi \\in (0, L)\\,", "e67492e976c6ec35fff6c88b3c14b8a6": " \\dfrac{M}{Re} \\approx 1", "e674b4bf176386c1337ce131fb311c9a": "b+d=0\\,\\!", "e674ff7ad25d89268f733c60d7f2d47d": "|g(x)-b|>0", "e67518f8cdb6007d306984a947854cef": "\\dot{y} = R y", "e675225d4c14da8b2b853f043178f108": "_{s.14 \\,}\\!", "e6752c7cdde0f279e4dde2df542b32ab": "\\lim_{z\\rightarrow z_0} f(z)=f(z_0)", "e67532514aa5ea66051e1ade331f4927": "\\mathbf{A}\\theta", "e6753e61990bc639ae1869683cb421b7": "m=0", "e67551b619d4f82b58d553d50836c8f3": "\\dot{\\mathbf{x}}(t) = \\left( \\begin{matrix} \\dot{x_1}(t) \\\\ \\dot{x_2}(t) \\end{matrix} \\right) = \\mathbf{f}(t, x(t)) = \\left( \\begin{matrix} x_2(t) \\\\ - \\frac{g}{l}\\sin{x_1}(t) - \\frac{k}{ml}{x_2}(t) \\end{matrix} \\right).", "e675ba396c648588be98c9eaa9055f4d": "\\rho_{2}(q,t)", "e675ee2a020de655edba18fdeec88436": " f(x)=\\min(x) ", "e67622ae95fb44606984ac5459a1e25f": " x_1, x_2, x_3,... ", "e67643843da888659f2549eb11d23a95": "\\mathrm{Rhodopsin \\rightleftharpoons \\ Retinal + Opsin}", "e6766206cf2459e8833de807d5099741": "x=\\log (z)", "e6771287c175d6e5ec6d9d566922f410": "\\frac{u_{mb} }{u_{mf} }=\\frac{4.125 \\times 10^4 \\mu^{0.9} \\rho^{0.1} }{(\\rho_p-\\rho)gd_p}", "e6777797924a815d0a3cb0c62d3b07ca": " E_{m,n} = 2^{n-m}{n \\choose m} ", "e67783ec14d1427a79198ab8207a1dfe": "\\mathcal{L}_X Y", "e677a0d4cc1120fef78a6e999d8e2d9b": " g ", "e677bc850ec032cbfe8e06a4648bb500": "\\sigma^0 = \\begin{pmatrix} 1&0 \\\\ 0&1 \\\\ \\end{pmatrix} ", "e677e22e510acde0106c1ca82408955b": "(\\theta_1,\\ldots \\theta_k)", "e677ecdbaf20cceddec5c507b6562197": "H(x,p) = p^2 / 2 + U(x) ", "e678775c8e136afa0d3f3204550aa616": "F_{drag} = \\frac{1}{2} \\rho V^2 A_s C_D ", "e6787dc9914c4ecd519385ed7a33c09e": "\\hat V", "e678dbac0e353ed13b47ff1c3f4fe251": " \\sum_{k=1}^\\infty 2^k a_{2^{k}} ", "e678e8670ad6b48a954a0575cee08dce": " B_{m_j} ", "e678f151bcf0726dea5e4485fc48498d": "x_{n+1}=2x_n\\,", "e6793622307fcdf4c96a7867f1d5b7b4": "G(x) = G_1(G_2(x))", "e6797b54b17c2c959001e64ead2da6c7": " V \\otimes W", "e679ab98c7b4f81c22e5d9cb03cd2d9b": "P=\\frac{1}{2}\\rho A \\sqrt{U^2+V^2} \\left( C_L UV - C_D U^2 \\right)", "e679c4dcca83035a679dff121bee53fc": "\\ E_a ", "e679d292f8b71dc454c06fac466df967": "\\tilde{G}_{adv}(p) = \\frac{1}{(p_0-i\\epsilon)^2 - \\vec{p}^2 - m^2}", "e67a02673b76aefa638f2369b859836e": "\\beta = (k_B\nT)^{-1}", "e67a82d7fba75f61d6d1e41ad78cb408": "\n(a, b, c)\\circ(a', b', c') = (a+a', b+b', c+c'+ab').\n ", "e67a8e1a466ca08d16b76b061f2b8567": " \\cdots x_{n}.y M_{0} \\cdots M_{m-1}", "e67af61a0cbc8f9b2a3cc8f6361b3fea": "V = \\int d^3 x \\sqrt{det (q)} = {1 \\over 6} \\int d^3 x \\sqrt{|\\tilde{E}_i^a \\tilde{E}_j^b \\tilde{E}_k^c \\epsilon^{ijk} \\epsilon_{abc}|}", "e67b195b6bc8feaa438cc3475c902425": "\\hat{u}_t", "e67b793ec09f4231572a64ee1be48b0c": "\\mathbf{E} = \\mathbf{E}^{(e)} + \\mathbf{E}^{(r)}", "e67b79fc9750512b5955f8f7a8f9559a": " \\int D[x]e^{-\\mathcal{S}[x]/\\hbar}=-A[x]\\sum_{n=0}^{\\infty}(\\hbar)^{n+1}\\delta^{n} e^{-J/\\hbar} \\text{,}", "e67baa0e335ac55235b82984f739083b": "n!/2", "e67bc528fe841a9267a00a960a4281f3": "f_{xz^2} = N_3^c \\frac{x (4 z^2 - x^2 - y^2)}{2 r^3 \\sqrt{5}} = \\frac{1}{\\sqrt{2}}\\left(Y_3^1 - Y_3^{-1}\\right)", "e67bd56d86a516d71b33b38cdefb7ec8": "r_{ij}\\;", "e67c1200dee695f54b5de13010e0f635": "(E_1(X) , Q_1(X)) \\ne (E_2(X) , Q_2(X))", "e67c18850d585957aa8f35b30937e455": "u_4 = \\frac{(x_1^2+x_2^2+x_3^2+ax_4^2)x_8 - 2x_4(x_1 x_5 + x_2 x_6+ x_3 x_7+ bx_4 x_8)}{c}", "e67c1a59ca4736aec2557e8dfa72f586": "T_G(0,-2)", "e67cae1dd749e458c4ea37045a76a1b1": " \\mathbf{P} = \\chi \\varepsilon_0 \\mathbf{E} ", "e67cc021e03fcb76a7a7cca530600bea": "\\ell_1, \\ldots, \\ell_s", "e67cd86712ac84691fe1a8aef473c0c7": "Pr(X \\leq x |H)", "e67cea8792cc6c3feebe2754c80f7b43": "u_s", "e67d2fbfe6634668bd08ff029c2a578b": "\nF(x)=P[X \\le \\ x]=1- \\left(\\frac{k}{x}\\right)^{\\alpha}\n", "e67d5729ac2a88e386d0793693386e88": "T_{\\rm a} =", "e67d6e83066e4bc070af866c01375a4c": "\\beta=0", "e67de88b9472c9d88ddd48e7b0fe1a28": "\\eta = 2 \\sin \\left(\\frac{\\theta}{2} \\right) ", "e67e17918a1455619030b603c5eddc1a": "A \\operatorname{dom}f \\cap \\operatorname{cont}g \\neq \\emptyset", "e67e36f1a391b3e3ae6d76ffb2f177e2": "0 \\to M \\overset{f}\\to M \\to M_1 \\to 0", "e67e6076ef4457c66eb625473b920f1a": "X\\subset O", "e67e7baa970d0b57feccdba2112b4b43": "0 = (g^{\\mu \\nu} \\sqrt {-g})_{, \\nu} \\,.", "e67e90faaf02cacdb41499029abf218a": "\\scriptstyle c(E) = A_{\\rm V} (E_{\\rm V} - E)^{a_{\\rm V}} ", "e67f79b2e23debe99be751fe121a97e5": "Sc_\\sigma(S)", "e67fbdfbf01ffb4093d2d2c511b25ced": " -\\inf_{x \\in E^\\circ} I(x) \\le \\varliminf_N a_N^{-1} \\log\\big(\\mathbb{P}_N(E)\\big) \\le \\varlimsup_N a_N^{-1} \\log\\big(\\mathbb{P}_N(E)\\big) \\le -\\inf_{x \\in \\bar{E}} I(x) ,", "e68004bf18e16551a964b0ecdee84b93": "\\nabla=\\partial_\\rho\\, \\hat{e}_\\rho +\\frac{1}{\\rho}\\partial_\\phi\\, \\hat{e}_\\phi +\\partial_z\\, \\hat{e}_z ", "e6804ad3554bacc6d1ddb27fcb5e2c7a": "R_Y=H(Y)", "e6805657ffc1de2051940ad3e741d65f": "\\exists r \\forall c, \\mbox{fell}(r,c)", "e680655c8e384f32260e2c09a2d39494": " {\\mathbf u}_1(s) ", "e68140f1f25c17f1dbaa6d1afe3fc390": "\\Delta r \\approx R", "e68158061dc619468a6c500b2bbe3a26": "\\sqrt{z} = \\sqrt{r} \\, e^{i \\varphi / 2}.", "e681a57c955d538c197478fc9d8a099f": "\\frac a b = \\frac c d", "e681b1fc44538c00c4d8457ab2398752": "(a,b) \\circ (c,0) = (ac, bc)", "e6824b9917a81e60c37785df0b368fd4": " \\tbinom n{k_1,\\cdots,k_n} ", "e6829ac929ecc50235e8324fbcb34d3b": "\\textstyle B ", "e682b605952725561bf740536656eb27": "H = (N - I_t) N_{\\operatorname{dg}}^{-1}.", "e68362377a734cc4d3d3fbbb0656df64": "\\langle x, y \\rangle \\,", "e6837f85dcad6d89b0724e34d276956a": "{n_e}", "e683eb37a3f89e51e5163d5d4859bafd": "\\partial_t r(t,a) + \\partial_a r(t,a) = -\\mu(a) r(a,t) + \\nu(a)i(a,t) ", "e68409fc7a13685980558394a5a5de20": "\\sigma_1, \\ldots, \\sigma_{n-1}", "e68453c02d8cf98bbf652043f90e36ae": " a+1=\\alpha=\\frac{\\gamma V_o}{2\\pi^2 \\beta^{3/2}} \\,.", "e6847a23c12dcc2b3f1b4fa14f3b387b": "F =\\frac{\\pi^2EI}{L^2}", "e68483b702d4be05c6d7bf1ecc7889f8": "Sp", "e68494b0828e93d9a24033c0651c5f6e": "\\sum_{n=0}^\\infty Q(n)x^n = \\prod_{k=0}^\\infty \\frac{1}{(1-x^{2k-1})}", "e684d4f57f9f9689c5d83d611e61b7bc": "\\| f \\|_{k, \\alpha,\\Omega}", "e684f0c83e1d3a02f5c4c45709c02bb6": "m = \\rho V = \\rho \\frac{4}{3} \\pi abc\\,\\!", "e68541473276c6d67ac60c44ee2db7f4": "\\dot{p}_k=0", "e6858a56444bca5d9538904c7d3b6141": "k = \\prod t_i", "e6858ba0645d88808681f17759d62b24": "\\int_a^\\infty dx \\ \\left(\\int_c^\\infty dy\\ |f(x,\\ y)| \\right )", "e685d168843e7706f0c394ad07a83adf": "\nf(x=i| \\boldsymbol{p} ) = p_i ,\n", "e686427dc1b6bd337e3205890a79851f": "\\mathrm{\\Omega\\ m=kg\\ A^{-2}m^3s^{-3}}", "e68695e80842989c76c9bf745a79b801": "0 = \\tau_0 < \\tau_1 < \\dots < \\tau_N = T\\text{ with }\\tau_n:=n\\Delta t\\text{ and }\\Delta t = \\frac{T}{N};", "e686d95242fba49fdf494123e687dd5a": "P(1)=P(2)=P(3)=1,", "e687644aa4d93b4b0adade82818d0825": "\\,\\!d(x,y)=p(x-y).", "e68788e0d041aefa62e9005cd9941405": "\\omega_n = 2\\pi v_n \\, ", "e687e34286c0af2aa82c5de231764c3e": "\\displaystyle{ \\int_{\\partial\\Omega_s} |f|^2}", "e6884d852833eb80b8c1debf58fc4832": "q(n)", "e68867819db89e578ce06d23d0f42e69": "\\frac{2n + 1}{3}", "e6887967fa95019ce5393e6aa40ee465": "p \\cdot s = s \\cdot p = s", "e6889fb36ba86c9475e4a11efcdd4978": "\\sum_i \\sum_\\alpha X^\\alpha \\otimes a_{i\\alpha} \\mapsto \\sum_i \\sum_\\alpha a_{i\\alpha}X^{p\\alpha}.", "e688ad6eed5d01ca2837b3b5371630a1": " g(\\tau),", "e688b18467643e1929dca9764cba84c6": "f=P", "e688c7596b00b3365ef53fa4e2787392": "{4 \\over 3} Al_{(s)} + O_{2(g)} \\rightarrow {2 \\over 3} Al_2O_3", "e6893d381907edd7b18a7c357cef2b5b": " \\|f\\|_{p_\\theta} \\leq \\|f\\|_{p_0}^{1-\\theta}\\|f\\|_{p_1}^\\theta", "e689424316d9819f7cdd76bedd2e88cb": "\\Pi = \\delta_{(0,0)}", "e6896e869fd909887a00b3eb6b6b1843": "B \\leq_T C", "e68975b031ebbf9ae7ce810ae951c4d1": "(2^{m/n})^n = 3^n\\,", "e689d4ddb0d1c08b60bdb5df049d4804": " x = 0 ", "e68a1a38a31c2d5f61bacebe4493a1ee": "A_{i+1}", "e68a21a94350b6411847f104134ba06e": "N = \\sum_{i=1}^n \\frac{1}{1+(p_1^2/p_i)-p_i} ", "e68a50da0413220cf8a631345d0dc7dc": "\\nabla \\times v", "e68ad743fb740b010b36c393d8d9a2a9": " k \\rightarrow \\infty ", "e68adf834d48813ceeaf79f2a37e855e": "D_r \\!", "e68ae3437d4f43bb00c9b0e4a827007d": "Z(r) = \\{z \\in \\mathbb{R}^{2n} | x_1^2 + y_1^2 < r^2 \\}, ", "e68ae9cfde5e3aa28b2cb134fa34f196": "S * \\operatorname{id}_X = \\operatorname{id}_X * S = \\eta\\circ\\varepsilon.", "e68aef38ec802793356c80b98969c309": "m_{f}", "e68b16685b67c0738287a11c2573b036": "\n{\\mathcal L}_S=-\\frac{1}{G}\\left[\\frac{1}{2}g^{\\mu\\nu}\\left(\\frac{\\nabla_\\mu G\\nabla_\\nu G}{G^2}+\\frac{\\nabla_\\mu\\mu\\nabla_\\nu\\mu}{\\mu^2}-\\nabla_\\mu\\omega\\nabla_\\nu\\omega\\right)+\\frac{V_G(G)}{G^2}+\\frac{V_\\mu(\\mu)}{\\mu^2}+V_\\omega(\\omega)\\right]\\sqrt{-g},\n", "e68bc1fb0d04432e2e72318d55cfeeea": "\\,e^{+{1\\over 2} n\\theta}q {e^{-{1\\over 2} n\\theta}}^{*}", "e68bceb84f46c8ffb7fa6cb6a2f2e0e0": "f(\\hat{\\mathbf{k}})", "e68bd27ad92500934c88eff3b89309ab": " -y \\partial_x + x \\partial_y. ", "e68c27ac8f492260f954d51df44565f6": "P_{reflect} = \\frac{2W}{c R^2} cos^2 \\alpha ", "e68c3165025bc3c5cb772735e98bd791": "w(n)=\\frac{I_0\\left(\\pi\\alpha \\sqrt{1-(\\frac{2 n}{N-1}-1)^2}\\right)}{I_0(\\pi\\alpha)}", "e68cfc1d2b51bfce8707d08c3c77d346": "V(P,Y)", "e68d4ee3836ac7e937160959453d614b": "\\tau''_{ij} \\equiv \\overline{u'_i\\, u'_j},\\,", "e68d7532d5b5f6d344de5ec96f6b8d4b": "\\mathcal{L}(x) = x^T M x -\\lambda (x^Tx - 1), ", "e68d9bd4157dc01268fca25d719c236c": " \\alpha_i, \\beta_i \\ge 0", "e68df205726e3131f36f98687a7131a5": "\\varepsilon=\\sqrt{1-(s^2_a+c^2_a)}.", "e68e3aa4b69b8407492f0b994822acad": "\\alpha > 3", "e68e5f22fbe0686afd96ffae73df2c03": "A\\vdash A,", "e68ece298983e780eec7409b2fa27ce6": "\\{x\\in I:|\\phi(x)|\\le\\epsilon\\}", "e68f01f1b3df45d1d5ad2eeb4524d416": " P_{\\mathbf k} + \\bar{P}_{\\mathbf k} = 1 ", "e68fa2ef8543b669d6f9973fd9bf3b7a": "\\scriptstyle[H^+]=[OH^-]+[HCO_3^-]+2[CO_3^{2-}]", "e68feb5123fd2111db5910512bad4a48": "[T]_{\\Phi_a}^{\\Phi_b}", "e69029eab8e0d87b74288823c8aff8f0": " Q = T_3 + Y_W \\, .", "e6905c34d87daf4512318580892303b9": " p = E/c. \\,\\!", "e69068ebb4caee8792a881e412273e73": "(14)\\quad ds^2=-\\frac{L-M}{L+M}dt^2+\\frac{(L+M)^2}{l_+ l_-}(d\\rho^2+dz^2)+\\frac{L+M}{L-M}\\,\\rho^2 d\\phi^2\\,,", "e69080742d2982197bae23215ecdbc6e": "t _4[R - \\beta] = t _1 [R - \\beta]", "e690ca037eeed8e94af7e547a0401965": "(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t\\ge 0},\\mathbb{P}) .", "e690ffa95954f66d49867f9812ea7033": "e_1 = \\frac {c_1}{a_1}", "e69107d16f6cd04c0f0703ecf6a02158": "\\mathrm{d} * \\mathrm{d} A = \\mu_0 J ", "e69107e94b21aaebf162cec2bb113799": "c_i \\in \\sum_{j\\leq i} \\mathcal{Z}^+ t_j", "e6910cfb8759f22dd444262c5b403d50": "\\mathbf{v}[k] \\sim N(0,\\mathbf R_d)", "e69110b3476134eec80a3ec079079184": "\\displaystyle{|a_{1}|^2 - |a_{-1}|^2={1\\over 4\\pi^2}\\int_0^{2\\pi}\\int_0^{2\\pi} \\Re [f(e^{i\\theta})\\overline{f(e^{i\\varphi})} (e^{i(\\theta-\\varphi)}-e^{-i(\\theta-\\varphi)})]\\,d\\theta\\, d\\varphi.}", "e6914e9141f1cba99881733e2a77605e": "q_{i} = A_{i} \\cdot \\epsilon_i \\cdot \\sigma \\cdot T_{i}^4 - \\sum_{j=1}^{N_s} A_{j} \\cdot \\epsilon_{j} \\cdot \\sigma \\cdot B_{ji} \\cdot T_{j}^4 ", "e69162fee6a28d5d2c496fc78c1cf7b4": "1_Rx = x", "e691776cae6610f2ff071a3190b44ca1": " \\psi^+", "e69188acd7687aebb1dd5b0904e40865": " [I_R] = -\\sum_{i=1}^n m_i[r_i-R]^2.", "e691927a3d8b79a06672d7f4ead69a98": " \n c(x,\\eta)= \\left\\{\n \\begin{array}{l}\n \\lambda \\quad \\text{if}\\quad\\eta(x)=0\\quad \\text{and}|\\{y\\in x+\\mathcal{N}:\\eta(y)=1\\}|\\geq T; \\\\\n 1 \\quad \\text{if}\\quad \\eta(x)=1;\\\\\n 0 \\quad \\text{otherwise}\n\\end{array}\\right.\n ", "e691b74bb4a175ec0d8b8f7d75c59bec": "\\sin x_\\mathrm{deg} = \\sin y_\\mathrm{rad} = \\frac{\\pi}{180} x - \\left (\\frac{\\pi}{180} \\right )^3\\ \\frac{x^3}{3!} + \\left (\\frac{\\pi}{180} \\right )^5\\ \\frac{x^5}{5!} - \\left (\\frac{\\pi}{180} \\right )^7\\ \\frac{x^7}{7!} + \\cdots .", "e691b98bb390831b02bc52185650bf5b": "\\int\\limits_0^\\infty f(v)vdv = \\langle v \\rangle", "e691e01ce4c4d8425dc70aaa2a177632": "[ion]_\\mathrm{in}", "e6927ab1f8a2d00c10b9bb6c9fee534b": " \\int h(x) \\phi(x) d^dx = \\int h(k) \\phi(k) d^dk \\,", "e6927dc8d45e30dc3f4030fd8c0e5cc7": "\\hat{T}^{i_1,\\ldots,i_n}_{i_{n+1},\\ldots,i_m}= (R^{-1})^{i_1}_{j_1}\\cdots(R^{-1})^{i_n}_{j_n} R^{j_{n+1}}_{i_{n+1}}\\cdots R^{j_{m}}_{i_{m}}T^{j_1,\\ldots,j_n}_{j_{n+1},\\ldots,j_m}.", "e6929579615341a43be187a2ee9f1245": "q_\\text{P} = e/\\sqrt{4\\pi\\alpha}", "e692cfe29e679272bd4ee9d594cac061": "\n\\Delta E = E_{-} - E_{+} = \\frac{4}{e} \\, R \\, e^{-R}\n", "e6934bd661845ed227d27634c3b87c78": "{a_n}", "e69399e53c524c62ef55d6eda1609a83": "\\Delta \\tau = 2 T_c \\sqrt{ 1 - V^2/c^2 } + 4 c / a \\ \\text{arsinh}( a \\ T_a/c )", "e693a67b1976a0ca0c7b854b3437a6af": "\\vartriangle^{m-1}_n = \\wedge^m_n", "e693b905640eeb30918ddf81dde087c2": "z_j = q_j/e", "e693d2965c7c606647e3314a9fee1fe6": " \\varepsilon_{ijk} = \\varepsilon_{kij} ", "e69410dc4c23c949aec3357c8c21da07": "k\\operatorname{B}(k-1/c,\\, 1+1/c)", "e6946422a3ad52f04cb0bde8145f4418": "N=2^m", "e694a277e656404c3dd3dd95f13e4a44": "\\operatorname{SU}(1) \\cong \\operatorname{SO}(1) \\cong \\operatorname{Sp}(0) \\cong \\operatorname{SO}(0).", "e694abfc0adf4084834843fb18d7c88c": "\\Leftrightarrow 8(2^b-1) = 4y(y+1)", "e694adfe246447f094b1ac0e9a1ba5eb": "\\boldsymbol{C}(\\mathbf{X})", "e694c7318bd6904cdf9f252d8b1926ea": "\\Delta \\tau = \\sqrt{(10\\text{ years})^2} = 10\\text{ years}", "e694ca60b404ab7143c865e3881ba412": "q_0\\in Q", "e694fa24a0df1fb8ee62b35e660a95e2": "\n\\sec z\\,\n", "e6959e6f3b1b765f1d6cb9dc76ab7ddb": "\\mathcal{G}_{\\alpha\\beta}", "e695c4de219e99554ab2398c2303a706": "\\mathcal{L}=\\bar\\psi(i\\gamma^\\mu D_\\mu-m)\\psi -\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}", "e695d0e46d26fcb5a0f51d1a93413fe4": "\\forall a\\forall b\\;\\lnot a=b\\rightarrow \\exists C\\; aC\\and bC ", "e695e5d318123d3716eeab18e2a8feda": "K_2=\\frac{[\\mathrm{Cr_2O_7^{2-}}]}{[\\mathrm{HCrO_4^-}]^2}", "e69634686eb5bd550d681e810efbbe3a": "\\sin p = \\frac {1 AU} {d} ,", "e6963a7089dc354ae4a3e2be5598fc9c": "\\lambda_0 = 1", "e69660de1070a4ff748514e5e737e272": "[Avg(H(1),L(1),C(1)) + Avg(H(2),L(2),C(2)) + Avg(H(3),L(3),C(3))]/3", "e696649da5d534af1d0d228a5d3faa66": "\\gamma^0 = \\begin{pmatrix} 0 & -I_2 \\\\ -I_2 & 0 \\end{pmatrix},\\quad \\gamma^k = \\begin{pmatrix} 0 & \\sigma^k \\\\ -\\sigma^k & 0 \\end{pmatrix},\\quad \\gamma^5 = \\begin{pmatrix} I_2 & 0 \\\\ 0 & -I_2 \\end{pmatrix}.", "e69673b16c7b54dc50ae1b7d7da7ee36": "= \\sgn( \\sin (\\theta+ \\frac{\\pi}{2})) \\frac{1}{\\sqrt{1 - \\sin^2 \\theta}}", "e696c038314f4f5728f4a6c77d8a8b80": "T : ( \\mathcal{C}^1 ( I ), \\| \\cdot \\|_{\\mathcal{C}^1} ) \\rightarrow ( \\mathcal{C}^0 ( I\n), \\| \\cdot \\|_{\\infty} )", "e696cd0533c3a0e626a3cf67b547cbc6": "H(X_{1},\\ldots,X_{n})", "e697055e7827ff32d6b23abc9b620435": "\\Delta=S^*S=FS", "e69720b9d007b959298971c32cccbd84": "R_S=R_H \\left(1+\\frac{\\sin(\\alpha)\\left(\\cos(\\theta)\\sin(\\alpha)-\\cos(\\alpha)\\sin(\\theta)\\right)}{\\cos(\\alpha)\\cos(\\theta-\\alpha)}\\right)\\sec(\\alpha)\\,", "e697618fdeefdf4265cfafaa236c63f4": " \\textstyle \\epsilon ", "e6976d360cdc3a48f463439e5482d3d0": "\n\\nu(\\phi)=\\frac{a}{\\sqrt{1-e^2\\sin^2\\phi}},\n\\qquad\n\\rho(\\phi)=\\frac{\\nu^3(1-e^2)}{a^2}.\n", "e69784b3823988a1152ad6370556cec4": "\n\\sum_{d\\mid n}\\varphi(d)=n,\n", "e6979bfbcd7b7bb366a406f7088bf408": "P_h = P_0 e^{(-mgh) / (kT)}", "e697a2533440af554da0a63cb0ff3e46": "\n \\Delta p_{\\mathrm{b.a.}} = \\frac{2\\Delta\\mathcal{W}}{c} \\,,\n", "e697ac969d421337827dde075df30330": "\n\\psi(\\mathbf{x},t) = \\mathrm{e}^{\\mathrm{i} K t} \\psi(\\mathbf{x}) \\mathrm{e}^{-\\mathrm{i} K t},\n", "e697d2eecbbfc3c39722bf39149e7c2c": "1 = 1^2", "e69818cf8e028664a3b91d0a9a6a12da": "\\mathbf{A} = \n\\begin{bmatrix}\n\\frac{\\partial F_1(\\vec{q})}{\\partial p_1} & \\frac{\\partial F_1(\\vec{q})}{\\partial p_2} & \\cdots & \\frac{\\partial F_1(\\vec{q})}{\\partial p_n} \\\\\n\\frac{\\partial F_2(\\vec{q})}{\\partial p_1} & \\cdots & \\frac{\\partial F_2(\\vec{q})}{\\partial p_{n-1}} & \\frac{\\partial F_2(\\vec{q})}{\\partial p_n} \\\\\n\\vdots\t& \\frac{\\partial F_j(\\vec{q})}{\\partial p_i} & \\vdots & \\vdots \\\\\n\\frac{\\partial F_m(\\vec{q})}{\\partial p_1} & \\frac{\\partial F_m(\\vec{q})}{\\partial p_2} & \\cdots & \\frac{\\partial F_m(\\vec{q})}{\\partial p_n} \\\\\n\\end{bmatrix}\\!", "e698bdf8647c9303f0098a2be590d106": "C\\in{\\mathcal A}", "e698f4bb3b746df7784691c46747826f": "I, J, K", "e6991c8f16f2caf304ae840b5a39eaa4": "\\frac{\\left|\\tilde{X} - \\mathrm{mode}\\right|}{\\sigma} \\le 3^{1/2}.", "e6997e0b5703c6a3ff901c2b7283decf": "u(t) \\le \\alpha(t) + c\\alpha(t)\\int_a^t \\exp\\bigl(c(t-s)\\bigr)\\,\\mathrm{d}s\n=\\alpha(t)\\exp(c(t-a)),\\qquad t\\in I.", "e6998ea3ada35b5ae38f9487e2d00faf": "E\\to E,\\ e\\mapsto \\overline{e}", "e699abdb55e8bdee01a73e245dafb2c8": "\\log(k) = \\log(k_0) + s_EN", "e699e014add5dd4d01464af56ef73227": "(S_i, C|_{S_i})", "e699eee7dc8fb6bf49e488577433db62": " i\\hbar \\frac{\\partial}{\\partial t} |k(t)\\rang_I=H_{1I}|k(t)\\rang_I ", "e69a217d6755100c80d4e7bfb8fe99f7": "F=ma\\,", "e69a4c358b59554186ff0532eea219e0": "\\overline{g}=p\\overline{u}", "e69a4e502ad0a4d03fcc81bcd63bbf62": "c_i = \\frac{{x_i \\cdot \\rho}}{{\\sum_i x_i M_i}} ", "e69aebac0ff179b36a6b80658dac785b": "\\ln(n)\\,", "e69b381bd6f0bb68a2d3d36a4504d186": "\\Lambda(n) = \\left| \\mathbf{h}(n) - \\hat{\\mathbf{h}}(n) \\right|^2", "e69b49acf2896925fd1b69aca29b8ce8": "(f\\big|\\gamma)=f.", "e69bf8187aae19438102aafc22aa2c00": "a_{i}+b_{i}+c_{i}", "e69c2412a57960ccbde67490264ccb2d": " \\vec E_{\\rm phys}(\\vec x)~=~ \\vec e~ \\hat a~ M(\\vec x)~\\exp(ikz-{\\rm i}\\omega t) ~+~ {\\rm Hermitian~Conjugate}~", "e69c3131c4193f1475163984e586c220": " \\delta \\ge 1-R ", "e69c39203d6951387d594309fe7bec12": "P_L - P_G = \\frac{4 \\sigma \\cos \\theta}{D_P}", "e69c8b8b50a63e9aae8b90d65f8151e4": " \\lambda = \\arctan \\left( \\frac{\\sin \\sigma \\sin \\alpha_1}{\\cos U_1 \\cos \\sigma - \\sin U_1 \\sin \\sigma \\cos \\alpha_1} \\right) \\, ", "e69cb91fea48ac1f1ebc7e3887b03dae": "\\lim_{r \\rightarrow \\infty} \\alpha(r) = \\infty ", "e69cbdb50c86ab4804a9ef04cc05acba": "b+c+\\dotsb < 1", "e69cc3ba00c43d3e514c5eeb3395aef5": "\n \\begin{bmatrix}\n {b_x } \\\\\n {b_y } \\\\\n\\end{bmatrix} = \\begin{bmatrix}\n {s_x } & 0 & 0 \\\\\n 0 & 0 & {s_z } \\\\\n\\end{bmatrix}\\begin{bmatrix}\n {a_x } \\\\\n {a_y } \\\\\n {a_z } \\\\\n\\end{bmatrix} + \\begin{bmatrix}\n {c_x } \\\\\n {c_z } \\\\\n\\end{bmatrix}\n", "e69d07cb235608194edb344ddd50fc7d": "\\vec{q}", "e69d0a832d28d8b2376c1788728c39a1": " \\mathbf{J}_{u} = L_{uu}\\, \\nabla(1/T) - L_{ur}\\, \\nabla(m/T) \\!", "e69d2a2ef59d13291f0743f2ed463773": "\nf_{w3c}(a, b) =\n\\begin{cases}\n a - (1 - 2 b) \\cdot a \\cdot (1 - a)\n & \\text{if } b \\leq 0.5 \\\\\n a + (2 b - 1) \\cdot (g_{w3c}(a) - a)\n & \\text{otherwise}\n\\end{cases}\n", "e69d3d513d1067f8cc7a4022c0fa90d6": " \\textstyle \\|Tf_n\\|_2 = 2\\pi n/\\sqrt2 \\to \\infty. ", "e69d4663ba41b6956be6f9f43883cfe1": "\\Sigma_{i=0}^{n-1} e^{-j2\\pi n^{-1}ik}v_i", "e69ddb2f29e03d5b34febb7da849d216": "\n\\begin{align}\nx_0 & = 1 & & & f(x_0) & = 1 \\\\\nx_1 & = 2 & & & f(x_1) & = 4 \\\\\nx_2 & = 3 & & & f(x_2) & =9.\n\\end{align}\n", "e69e285ddeab0bde95d94553bab8078a": "\\Bigg[\\frac{\\alpha\\beta}{\\pi}\\Bigg]=\\Bigg[\\frac{\\alpha}{\\pi}\\Bigg]\\Bigg[\\frac{\\beta}{\\pi}\\Bigg]", "e69e2a3231ab732f573c1ff456b7c0ea": "e[n]", "e69e597eb1eca6103de436e53a70880e": "\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty |\\psi(\\bold{r})|^2 {\\rm d}^3\\bold{r} = \\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty \\psi(\\bold{r})^*\\psi(\\bold{r}){\\rm d}^3\\bold{r} < \\infty ", "e69e66d4f1ac2108f5d5f6e2f8a0b9cf": "\\frac{P_{\\text{CO}_2}}{[\\text{CO}_2]}\\ =\\ k_\\text{H}", "e69e77e125b19f749c1a7a497931f0f6": "\\quad\\quad\\quad v = \\begin{bmatrix}-20\\\\-60\\\\-80\\end{bmatrix}", "e69e83d74637d407885f574cb489010d": "2\\theta=90^\\circ", "e69ee3eb20d0a96a8f32f24a2d596e09": " \\mu =e^{\\int p(x)dx}=e^{\\int -1 dx}=e^{-x}", "e69f09cc2664808faaa3a52fb71c76fd": "\\frac{a}{1-a} \\frac{x_2}{x_1}=\\theta", "e69f43512ff3c8975509c6c6ad69e1a0": "e^{ar}, \\quad 0 \\le a < \\pi", "e69f43621376c89cfadceb6bc8d6b8d5": " \\Gamma_0^\\dagger= +\\Gamma_0 ~,~ \\Gamma_i^\\dagger= -\\Gamma_i\n~(i=1,\\dots,d-1)\n", "e69fb63ed862283ba070162bd718859b": "\nU(P_1) = \\frac{A e^{\\mathbf{i} k r_0}}{r_0} \\int_S \\frac{e^{\\mathbf{i} k r_1}}{r_1} K(\\chi) dS,\n", "e6a005fd9d0acd5c4b1df153908d4bd1": "E_2=B\\backslash A", "e6a02a1a33bba6accc4264f59abb5044": " d=8 k+2 ", "e6a0504894b45b03b745a41f89257ad5": "2.7\\cdot 10^4", "e6a07676f76a7788223418ad70ccd574": "m_\\mathrm{2D}^*", "e6a0feb499c98b5908e17e5efd971680": "F_D\\, =\\, {\\scriptstyle \\frac12}\\, \\rho\\, C_d\\, A\\, u\\, |u|", "e6a123d9c4ce4a75e333dbc0b37c9240": "W = \\Delta E_\\mathrm{k} = E_\\mathrm{k,2} - E_\\mathrm{k,1} = \\tfrac{1}{2}m\\left(v_2^{\\, 2} - v_1^{\\, 2}\\right) \\, .", "e6a125a322fa861ad79882e6c4d04320": "\\varphi_{sp}(r) = \\varphi_{pc}(r) + B_i = {1 \\over 4 \\pi \\varepsilon_r \\varepsilon_0}{z_i q \\over r} + B_i", "e6a16cea2babdd9936c0250a40e72b7a": "{\\partial}_i = \\frac{\\partial}{\\partial x^i}, \\qquad i=1,\\dots,m. ", "e6a197e2cfdcedbf2fd42fcdb8e43ca1": "\n\\begin{align}\nS_x \\rightarrow U^\\dagger S_x U &{}= e^{i \\theta S_z} S_x e^{-i \\theta S_z} \\\\\n&{} = S_x + (i \\theta) [S_z, S_x] + \\left(\\frac{1}{2!}\\right) (i \\theta)^2 [S_z, [S_z, S_x]] + \\left(\\frac{1}{3!}\\right) (i \\theta)^3 [S_z, [S_z, [S_z, S_x]]] + \\cdots\\\\\n\\end{align}\n", "e6a1c56d7f454b94468bb207faca6620": "\\scriptstyle \\sqrt{\\langle r^2\\rangle}", "e6a1f4428ea96c4d7a413aafcd253661": "\\mathcal A = \\mathcal{C}", "e6a1f72bd47045fe4e64d63e9e1e0b99": " I_x(I_xu+I_yv+I_t) - \\alpha^2 \\Delta u = 0", "e6a21c5156af3aadd05782b628efd1b8": "D\\subseteq \\R^d", "e6a27b608ef45a7563da09b493e7b519": "J_{ij}=1", "e6a29d976bb2bd34149303e91d46de1e": " \\mu_r ", "e6a2a3c6f833016e9ad9fb0df2abb571": " \\mathbf V(x)= -\\int w(x)dx=-10x+C_1 (kN)", "e6a2ffffc698ce23ed3b09d959a1ed88": "T_{(t,x)}(\\Bbb{R}\\times M)", "e6a30a4ac6fd3255c421f6ed451e13b7": "Q(x,y) ", "e6a37989ddebbec3dcf24944345dbbe9": "\\ell_{(M,f)}(x, y)\\ ", "e6a37eb76f7296e8b1bb73e791123d79": "h_v\\leftarrow -Ah_x", "e6a414c7ea01cf2a5e9f2ab2db6df545": "\\rho : G \\to \\mathrm{Aut}_{\\mathbf{C}}(X)", "e6a41c01027a44f0b288151ee04618dc": "\\mathbf{v} = \\mathbf{v}_0 e^{- \\lambda t / m}", "e6a4ac0592869fe71da7ba862c3c81ff": "\\frac{ds}{dp}=1+\\frac{T}{E}.", "e6a4e3b07a1a2e72bd62cea486571336": "\\rho V", "e6a54e01b7527de88ae1d16281180cea": "\n\\Psi(\\mathbf{r},\\mathbf{R}) = \\varphi_1(\\mathbf{r};\\mathbf{R})\\tilde\\Phi_1(\\mathbf{R})+\n\\varphi_2(\\mathbf{r};\\mathbf{R})\\tilde\\Phi_2(\\mathbf{R}).\n", "e6a58081190246d7e3d6b508ad308ad3": "y_1>\\tau", "e6a5c9946da05d1920a1aeabee667191": "Q_{Y^{c}|X}(\\tau)=\\max(0,X\\beta_{\\tau})", "e6a6aa39af25555c34af80c585b60d7e": "\\oint_{\\gamma}\\varphi (\\zeta )d\\zeta =\\lim_{n\\to \\infty}\\frac{c}{n}\\sum_{k=1}^{n}\\varphi^2 \\left (T_{k-1,n}(z) \\right )", "e6a6c70bcf8a4bc8ffae397b1f02e765": " \\boldsymbol{\\gamma}_{SG} = \\boldsymbol{\\gamma}_{SL} + \\boldsymbol{\\gamma}_{LG}~{\\cos {\\boldsymbol{\\theta}_c}} ", "e6a6dd5f80feaef8ea02e3c6de3e2607": "L \\in \\Sigma_4 - \\mathsf{SIZE}(n^k)", "e6a755c6e27160a6a9707c06ed9461ae": "x\\in X^{\\ast }\\left( t\\right) ", "e6a75bf39ca1ff31a0871342735194ae": "PL\\;=P_{Tx_{dBm}}-P_{Rx_{dBm}}\\;=\\;PL_0\\;+\\;10\\gamma\\;\\log_{10} \\frac{d}{d_0}\\;+\\;X_g,", "e6a79adb3626bbbbacdb4611744ecf4f": "\\textstyle l = 5", "e6a81d3c122f30932621061b1109f7ae": "\\tbinom n r / 2^{n+r} \\!", "e6a837e9840292022f77f4cb605c67e4": "{\\frac{w_{abs}}{\\overline{w}_{abs}} = \\frac{w_{rel}}{\\overline{w}_{rel}}}", "e6a85d34ceaa1fec0740787d4c0d78fa": "\\mathbb{Z} / m_0\\mathbb{Z}", "e6a87b169ad638d27d0a7565bc47d0ef": " q_k \\leftarrow \\frac{q_k}{h_{k,k-1}} \\, ", "e6a87f82572b135f198cd8cedfa7481d": "\\displaystyle \\sqrt{2 \\pi}\\cdot\\frac{\\delta(\\omega-a)+\\delta(\\omega+a)}{2}\\,", "e6a8a58927bba7831bf06753f5f61de3": "\\forall x \\gamma_1(x)", "e6a8a63567df37670714d5af90980ac2": "\\begin{align}\n & g\\left( x_k,y_k \\right)=\\sum\\limits_{i=1}^N \\alpha_i\\phi \\left( r_i \\right),\\qquad k=1,\\ldots,m_1 \\\\\n & h\\left( x_k,y_k \\right)=\\sum\\limits_{i=1}^N \\alpha_i \\frac{\\partial \\phi \\left( r_i \\right)}{\\partial n}, \\qquad k=m_1 + 1,\\ldots,m \\\\\n\\end{align}", "e6a9231f6a8d74a62d2b23fd80143512": " x=v\\cos u, y=v\\sin u, z=c\\sin nu.\\,", "e6a929664203e9c7459bcd95d37e60fe": " w=\\frac{ E|X| }{ \\sqrt{E(X^2)} } = \\sqrt{\\frac{2}{\\pi}}. ", "e6a96028f7233c3caa3975558123d3cd": " E ( a \\cdot f(x) ) = E f(x) ", "e6a98e09fbd19c7ef7c486872babba0f": "p(x+y)", "e6a9949c66cdd7d4353d7b55a2320d79": "\\mathcal{E}(X)", "e6a9ac0278d88b9f7297a53e4f1c6558": "h^{-}\\colon \\mathcal C^{\\text{op}} \\to \\mathbf{Set}^\\mathcal C.", "e6aa671cd004319bc58c21b52ae254f5": "\\gamma+\\ln[x]", "e6aa763ac612da09a2c8094b520a6719": "f=f_1-f_2=(n_1-1)(1/r'_1-1/r''_1)+(n2-1)(1/r'_2-1/r''_2)=(n_1-1)k_1+(n_2-1)k_2", "e6aa798074053cc34e695cbbd3a1d601": " y_c = \\sum_{j=1}^n \\left( \\sum_{\\ell=1}^{k_j} C_\\ell x^{\\ell-1}\\right )e^{\\alpha_j x} \\,\\!", "e6aa8a79af68528b7afed98147c34897": "(z_t, x_t^{[m]})", "e6aabf1a453a4bdccbf849ab7e48a2ec": "a \\geq b", "e6aae5177cf4a665a65e6f7960cc6050": "y = x_1 + \\sin (x_1) + q_1 x_1 e^{-x_2}", "e6ab16e2b8f45d515c5dc2c962543bd5": "a,\\alpha,b", "e6ab409ff4add7feaa616c544587126e": "_p = .90 ", "e6ab8feb81f85f1a400f9da79b11e70d": "\\displaystyle r=\\sqrt{\\frac{efg+fgh+ghe+hef}{e+f+g+h}}.", "e6abdaece981daf3148b349ef64d07ae": "x \\in k\\left[M\\right]", "e6abfa77857af964ad3f560eb15958df": "\\alpha\\Vdash\\Box p_j", "e6ac2d96075d73f2e53f99b7bb43aef6": "\n\\frac{\\rho L_{\\mu} + L_{\\mu} \\rho}{2} = \\frac{d \\rho^{\\,}}{d \\theta^{\\mu}}.\n", "e6ac35a273e2f59cdb4f858ac40219f1": "\\frac{16}{11}", "e6ac379576856f47cab20914c2646ada": "L_\\gamma", "e6ac52cc1045868f4fac573e6df51c6b": "\\operatorname{Cl}_{2m}\\left(0\\right)=\\operatorname{Cl}_{2m}\\left(\\pi\\right)=\\operatorname{Cl}_{2m}\\left(2\\pi\\right)=0", "e6ac7bf93afa9ffec6898390d79a9af4": "\n \\omega_p^2 = {4\\pi n e^2 \\over m}\n.", "e6ac82e3a565a22ed989ba9fad3abf0f": "Bu,", "e6ad2f31576999b1a7b6e7cbb5a7ee49": "-1 \\,", "e6ad5e8e3607fb694d7ade045aed9084": "V_t < V_d", "e6ade98d17968fe1fdf6f4e810f5d481": "\\mathbf{S} = \\int d\\mathbf{S}", "e6adf0240356afd2ea1a68a1903b29fd": "\\div", "e6ae1e0d121b8a098ddf6d97b60d7904": " X \\sim e^U ", "e6ae8b83de530b6913d3638daa81bdcb": "\\displaystyle \\sqrt{2\\pi} \\hat{f}(\\omega) \\hat{g}(\\omega)\\,", "e6ae8dc6d0bdb3f9a50f52033122f9a5": " = -k_{eq}\\left( x_1 + \\frac{k_1}{k_2} x_1\\right) \\,", "e6aeb002a9867887af7b61301794fd3f": "\\alpha = \\sqrt{\\nu}, \\!", "e6aebc82f74902079cf031e12a20e92f": "\\mathrm{SU}(6)\\cdot\\mathrm{SU}(2)\\,", "e6aec9065c2807f4f080465b17d40a2f": "\\mathbb{Z}_5", "e6af0dbf335b09945370c915ed45d681": "\\cos x = 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\frac{x^6}{6!} + \\cdots ", "e6af7f6cf82d986ebd692cbf21bb9b15": "\nf(s) = 2^s\\pi^{s-1}\\sin\\left(\\frac{\\pi s}{2}\\right)\\Gamma(1-s)f(1-s)\n", "e6af88c4e7e7670742e264edb9422b78": "Q_{\\bold{y}}(\\theta) = \\begin{bmatrix}\\cos \\theta & 0 & -\\sin \\theta \\\\ 0 & 1 & 0 \\\\ \\sin \\theta & 0 & \\cos \\theta\\end{bmatrix} , ", "e6affeeadd0421f2f90861e246ce17dc": "(-1)^n(\\mathcal L f)^{(n)}(0) = \\mu_n", "e6b029c79563aecab5ad228c1355c731": "p \\mapsto M_p", "e6b02c8281ae43a13ea6b5a34dce230b": " e^+e^- \\to e^+e^- e^+e^- ", "e6b0390a256f208a88d90727539b5d1b": "6p^2q^2", "e6b08d41b7153c696586ededffb2cd07": "\\qquad\\vdots\\qquad\\vdots\\qquad\\vdots\\qquad\\vdots", "e6b0a1a778ddb108fffbc5299d6b7c7a": "G_1, \\cdots, G_c", "e6b139e71eaa78f29733aed3a035bb64": "-(I_3-I_1)/3", "e6b1b18f26fd505402c97caad7dfadae": "\ng(x,n) = \\left\\{\\begin{matrix}\nn & x\\in S_0\\\\\n2n \\ (\\bmod \\ p) & x\\in S_1\\\\\nn+1 \\ (\\bmod \\ p) & x\\in S_2\n\\end{matrix}\\right.\n", "e6b1d3c14a94607428899133c9ae0d1e": "\\ \\sigma_y", "e6b241edf685246042bee7fbc9fc41e2": "a^{N - 1}_p\\equiv 1 \\pmod{N}", "e6b309d35fd315385c63768c3a8ae4ed": " \\sigma^x_i = D {\\psi^\\dagger}_i \\psi_{i+1} + D^* {\\psi^\\dagger}_i \\psi_{i-1} + C\\psi_i \\psi_{i+1} + C^* {\\psi^\\dagger}_i {\\psi^\\dagger}_{i+1}.", "e6b3a6593fda10d140d59cc3f670d32d": "h\\colon\\,x\\mapsto \\sqrt x.", "e6b3ec2c97ed3e12fe72a630daa6eb38": " (x(t),y(t))=(x_3+t(x_4-x_3),y_3+t(y_4-y_3)). ", "e6b4179361292fbc3f6010a9c7dd962e": "\n\\mathbf{r} \\cdot\\left(\\mathbf{p}\\times \\mathbf{L}\\right) = \n\\left(\\mathbf{r} \\times \\mathbf{p}\\right)\\cdot\\mathbf{L} = \n\\mathbf{L}\\cdot\\mathbf{L}=L^2\n", "e6b42b46c0d00e8e99500a7f44f2e019": "\\Gamma_i \\subset U ", "e6b445d809ca00a3357f74bb42e18958": "x\\left(n_1,n_2\\right)", "e6b465d359922d1b6baa7b6f2dce4fbf": "Y = X/\\!\\!\\sim", "e6b471a5b5e3b76cd6876e6b30716ca0": " f(k) ", "e6b4c54f25a29d783cbb3d2f123fa39a": "\\bigcup_{n=1}^\\infty U_n = \\bigcup_{n=1}^\\infty (X\\setminus C_n) = X\\setminus\\bigcap_{n=1}^\\infty C_n", "e6b59580072f649213300917202e194c": "\\varrho(Z + A) = \\varrho(Z) - a ", "e6b5c5cbc09009492f86afe23d968f02": "\n\\det(A) \\int dx_1 dx_2 ... dx_n = \\int dy_1 dy_2 ... dy_n\\,.\n", "e6b721258275a45f9f64057415f1f428": "t=t_a+(n-1)\\cdot t_0", "e6b74fd5a778eb86f7b240d2156454da": " v(t)=0 ", "e6b75d4e2004ba007e02f4534fb5a8b6": "\\mathbf{z}_{\\rm{l}}=\n1+{dt_{ab}+dt_{cd}\\over 2}i+{dt_{ac}-dt_{bd}\\over 2}j+{dt_{ad}+dt_{bc}\\over 2}k\n", "e6b7a5466249914892cb9a763334150b": "\\ell_{j\\ne i}(x_i) = \\prod_{m\\neq j} \\frac{x_i-x_m}{x_j-x_m} = \\frac{(x_i-x_0)}{(x_j-x_0)} \\cdots \\frac{(x_i-x_i)}{(x_j-x_i)} \\cdots \\frac{(x_i-x_k)}{(x_j-x_k)} = 0.", "e6b7b20a91849a6db88a7717939f0b58": "E_{\\text{photon}} = h\\nu", "e6b8d6d170db5638106729755195bbde": "\\left. + \\left[ c_1^\\dagger(\\mathbf{k}) u^{-1}_{-1} (\\mathbf{-k})\n+ c_2^\\dagger(\\mathbf{k}) u^{-1}_{+1}(\\mathbf{-k}) \\right]e^{-i k x} \\right\\},\n", "e6b8e5d5eb8d02220f8c672929e6625d": " Y = (p/2) ", "e6b8f17310a6a02ed0dc3419280d4cf2": "W_{i}", "e6b9460196c7731351f61e8ca0f161da": "(f_{sc}\\;,f_c)", "e6b95a730d00f60f4134a7ede26c915c": " C_\\bullet: \\cdots \\longrightarrow \nC_{n+1} \\stackrel{d_{n+1}}{\\longrightarrow}\nC_n \\stackrel{d_n}{\\longrightarrow}\nC_{n-1} \\stackrel{d_{n-1}}{\\longrightarrow}\n\\cdots, \\quad d_n \\circ d_{n+1}=0.", "e6b9e39992577bf894f1c073d354bd60": "\nr^{2} \\frac{d\\varphi}{d\\tau} = ac,\n", "e6ba1901c237471d0efce4ad33c3e2a8": "a_{p,q}", "e6ba255ada70246738c04c2c65565689": "T_{D(max)} = \\frac{1}{2f_m} ", "e6ba3059d17f48cfd0555b80a060076d": "\\tau = S _0 + \\mu (\\sigma _n - P _f)", "e6ba99318680baf60ea356dc513fd286": "\\bar{f}(r,t,h)= 2 r \\cos(t)+ r \\sin(t) + 5", "e6bada80fb388c1ab677da390f70c66e": "\\varphi=\\sum_{i=1}^4 c_i\\chi_i\\left(z\\right)", "e6badea7267a4f9c84e525d12b27f9be": " -\\textbf{c}_N(Iiu-\\textbf{A}_N)^{-1}\\textbf{b}_N+\\textbf{c}_P(Iiu-\\textbf{A}_P)^{-1}\\textbf{b}_P", "e6bb57328f3dc2601231065b305df30a": " LR- = \\frac{1 - \\text{sensitivity}}{\\text{specificity}} ", "e6bba8c99e39be08b1e4eb0b705da631": "\\gamma_u", "e6bbb0ccfd61a1ce36b89762a4eed8c0": "\\nabla \\cdot \\mathbf{v} = 0", "e6bbbd44227c9cb70cdcbf587949102d": "\\tfrac{(y+1)^3 - y^3}{y + 1 - y}", "e6bbf6e588f7533a4c5e50808888a3a7": "= \\int dt \\Big[ {dx \\over dt} p - {p^2 \\over 2m} + {1 \\over 2} m \\omega^2 x^2 \\Big]", "e6bc01989b43b9dbe7ac0bd7842a931c": "w_{BF}", "e6bc3d0ebbdfb45cc9ad8f270488d98f": "\\{-\\alpha_1, -\\alpha_2, \\ldots, -\\alpha_n\\}", "e6bc63b162004c1814bf882ea4e639f2": "r\\begin{Bmatrix} p \\\\ 2 \\end{Bmatrix}", "e6bcd498374eaa79e07a4ce9ea9a9d7e": "\\phi_n^{k}", "e6bd5d0b45cd0e8d7fad302878190616": "\\scriptstyle\\frac{1}{2 e}h", "e6bebdfd778dc4df505692e5181c7469": "C_n(a_1,\\cdots,a_l)", "e6bed18a38cb38891c74411da4fe8477": "\\langle T_X(a)^q \\rangle \\sim a^{\\zeta(q)}\\ ", "e6bed38882ede01205f0c900c20d2f31": "\n[e]=[f(\\gamma)]=[f(\\gamma_1)]\\ast\\cdots\\ast [f(\\gamma_m)]", "e6bf6d773cbd9acbe1e9ac86f22f1ab1": " a+2 b", "e6bf8d13fafa055675c9320aa0dcb745": "i: (A, C) \\to (X, B)", "e6bfbff40f5dc413b90d0d5e95cee3ea": "d^3 x", "e6bfd514a60f01dc43f387c42a8bfcc7": "\n\\text{Let }a, b, \\text{ and }c \\text{ be integers that satisfy}\n", "e6c0063e26249be6ce362d57de083704": " D^{\\mathrm{face}} \\,", "e6c0229580bd669956198d75e10e5df7": "\\Phi=e^{\\beta(\\epsilon_i-\\mu)}+1", "e6c05fa8b328f2a09ab91a9e7dafb105": "\\Delta f_{\\mathrm{pred}}", "e6c0d8bdb53df212c6d622e515e7968c": "\\scriptstyle \\Re\\{\\cdot\\}", "e6c0f90d9a6d1d723a14575d34f9f104": "[X,Y]=Z,\\quad [Y,Z]=X,\\quad [Z,X]=Y", "e6c1183d3e3f338a44d05ff3285e84f0": "f_c = \\sqrt{2\\ln(c)}\\cdot\\sigma_f ", "e6c156d19b1f8590b005dd909e73c264": " U_\\text{E}(r) = k_\\text{e}\\frac{qQ}{r}", "e6c15c0f2ff432d5c0a292a5eba4538a": "\\Delta G^\\circ = -RT(\\ln K_{eq}) = -2.303RT(\\log K_{eq})", "e6c17ef03821603bd157371b28c00d9a": " m_j < f(t_i) < M_j", "e6c1c6a349becd742bfa7a5494332837": "\\zeta_L(z) = \\exp \\left({ \\sum_{n \\ge 0} s_L(n) \\frac{z^n}{n} }\\right) \\ . ", "e6c1ce8fda2583545ebbd7ebca23bb00": "F^*(x):=\\inf\\left\\{t\\in\\mathbb{R} : x\\leq F(t)\\right\\}", "e6c1f3b227a3ec19049659d219abd96b": " \\partial_{i}f ", "e6c205a3628f8c7c986160fe97a7c68e": "Z_C(t_1,t_2,t_3,t_4) = \\frac{t_1^6 + 6 t_1^2 t_4 + 3 t_1^2 t_2^2 + 8 t_3^2 + 6 t_2^3}{24}.", "e6c236781d22604e327792615a254743": "O(|V|^{2.5})", "e6c240d638a97872d9935d724c99cb95": "s \\models_K f", "e6c248ce261626c10730f5b15c6e946b": " g(x,y)= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} h(x-x', y-y') ~ f(x',y') ~ dx' dy' ~~~~~~(4.1) ~", "e6c2d66db8cf1b0a42105f7e454868a5": "\nk \\rightarrow \\infty\n", "e6c2ed66c4fa3cc94e5cf4da3e926ccc": "s_V^2", "e6c30d179ce807bc9381e00319d89f16": "\n\\begin{align}\n \\lambda_1 &= -3 \\lambda_3 /2 \\\\\n \\lambda_2 &= \\lambda_3/2 \\\\\n\\end{align}\n", "e6c3ac3115bfa6552506ceb6fb9390b7": "\\sum_j(M_j^*)^{-1/j} = \\infty", "e6c3c1b5ecce65fffea17653f5898dce": "M_l", "e6c3d15815e15004e4381a7754e623af": " s^2 F(s) - s f(0) - f'(0) \\ ", "e6c3f727da5291174cbf1e77eca5aabf": "\n\\mathcal{Z}(a_n) = \\{ n \\ge 1 : a_n \\text{ has no primitive prime divisors} \\}.\n", "e6c492f3868f71543b637b2f32ce9830": "\\{{\\mathrm e}_m\\}", "e6c4a5c2c54c10afb58b5d8467d3f69c": "L^2(X, d\\mu)", "e6c4dde7ddc9aaa6ee9e9300bcc29537": " \\overrightarrow{V_g} = {\\hat{k} \\over f} \\times \\nabla_p \\Phi ", "e6c4e0372459d1a92d9746ff7a803a50": "A_{\\mathrm{left}}^{-1} A = I_n", "e6c5250297168673f065429ea4a38954": " \\, j(\\epsilon) = z_{\\mathrm{S}} d_{\\mathrm{F}} D_{\\mathrm{F}} f_{\\mathrm{FD}}(\\epsilon) \\mathrm{exp}(\\epsilon/d_{\\mathrm{F}}) = j_{\\mathrm{F}} f_{\\mathrm{FD}}(\\epsilon) \\mathrm{exp} (\\epsilon / d_{\\mathrm{F}}), ...........(21) ", "e6c56fbd87de891439a3fe37337dfc20": "F = \\{ A \\rightarrow BC \\}", "e6c57682c0e5bc67f4b7fa28896a0eb8": "2\\rho = 2n\\pi + \\delta", "e6c58bd358aba96e7d03cff8c31a65de": "\\operatorname{comb}(\\xi, \\eta)", "e6c5b1b5b7a81f952692110a621dc347": "c^2 = \\frac{1}{\\mu_0\\varepsilon_0} ", "e6c653b3b9dc1beecf06992c17f2707f": "\\log p(v)=E(v)+\\sum\\xi^iF_i(v)-\\psi(\\xi)", "e6c657b7171db8afed008b4b99dfc9e4": "c \\cdot r", "e6c669653c456e27ea6d2cbd998166b5": " \\mathbf{x}^T[\\Lambda]\\mathbf{x}=1,", "e6c6793f934751de6b307e874f04bb37": "M \\subset \\mathbb{C}", "e6c68fc6c6c7a5c9305ae328be9f708b": "p_2 =7 (r =2)", "e6c6aed4f2826eb285b41719bddc27b3": "\\exp(a_1(e^t-1)+a_2(e^{2t}-1))\\,", "e6c795011771a69ffed6c1bf5cc4bbbc": "\\hat{b}_p^\\lambda \\,", "e6c82e7708a62efebc8fd450b279421e": "\\scriptstyle P_\\mathrm r", "e6c831e5809dd63824598d3f1d9ceadb": "\\beta + \\nu", "e6c8a93ed7f44826f037a9139552f60e": "\\left(\\frac{\\partial V}{\\partial T}\\right)_P = \\alpha V", "e6c9041ba8cd1e65331dcbcc67ffe44c": "[0,0,0,1]", "e6c91af7e2487254fc87f67fce2b3ef5": "\\begin{align}\n\\frac{d}{dx}\\left(\\frac{f(x)}{g(x)}\\right)\n&= \\frac{d}{dx}\\left(f(x)\\cdot\\frac{1}{g(x)}\\right) \\\\\n&= f'(x)\\cdot\\frac{1}{g(x)} + f(x)\\cdot\\frac{d}{dx}\\left(\\frac{1}{g(x)}\\right).\n\\end{align}", "e6c9206e8fdebb99de33d7dde9a27bba": "\\mathcal Q_{\\mathrm{Hur}}", "e6c957f7f70fa4225d880e2f464e53d1": "k(\\boldsymbol{x},\\boldsymbol{y}) = e^\\frac{-||\\boldsymbol{x} - \\boldsymbol{y}||^2}{2\\sigma^2},", "e6ca3685cb8b951d9331ce4e4a4eecbb": "T_m(X) = \\sum_{i=1}^n W_{im}(X)Y_i", "e6ca50fc95e4fcefc036a5fdc2ecd523": "\\displaystyle Z_1 = \\{ 0.5 x : x \\in Z \\} \\cup \\{ 0.5 + 0.5 x : x \\in (0,1) \\setminus Z \\} \\, ,", "e6ca69bae5f44cff6bda1a4c82c688b6": "\\scriptstyle\\nu_{13} \\,=\\, \\frac{1}{h}\\left(E_3 - E_1\\right)", "e6ca86356816637fba2bf0dd88c95ac3": "p>\\lambda", "e6caa64176864a7bae6549106cc8c621": "r_i{\\left(t\\right)}", "e6cab14862cf43040cdf8f32e38372c5": "\\langle x, y\\rangle = 0.", "e6cadb1a1a8277c18c66843e26b02e86": "\\Big[(N-1)N^{2}\\Big(N(N+1)-6K(N-K)-6n(N-n)\\Big)+", "e6cadf21c7f2fd975de2ecd63f92dc88": "p_n\\# = e^{(1 + o(1)) n \\log n},", "e6cae2c213d0cb4363b29c7ff8c3e12a": "y_1 y_2 \\dots y_n", "e6cb48c84ffcc96497559b6cd6941360": "\\nabla\\times\\mathbf{B}=0 ", "e6cb5ceae1febf5d46a0ad7a1a8f4fa1": "\n w(x_1,x_2,t) = \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty \\sin\\frac{m\\pi x_1}{a}\\sin\\frac{n\\pi x_2}{b}\n \\left( A_{mn} e^{i\\omega_{mn} t} + B_{mn} e^{-i\\omega_{mn} t}\\right) \\,.\n", "e6cb97ca73dd22435843e6c71979bcae": "4/(3.6450682*10^{-7}", "e6cba3548d818caca53de1651c2b05cc": "\n \\left(\\dot{\\rho} + \\rho~\\boldsymbol{\\nabla} \\cdot \\mathbf{v}\\right)~\\eta +\n \\rho~\\dot{\\eta}\n \\ge -\\boldsymbol{\\nabla} \\cdot \\left(\\cfrac{\\mathbf{q}}{T}\\right) + \n \\cfrac{\\rho~s}{T}.\n ", "e6cbb6f44023f3b3716c5da1b43c9c36": "\\mathbf{1}_A \\colon \\Omega \\rightarrow \\Bbb{R}", "e6cbc771a24b6e1368c72287efb90397": "\\frac{\\partial \\mathbf{Y}}{\\partial x},", "e6cc1d1f74f7ea09b76ea0ba064fb9d3": "G(n)=G(n-1)G(n-2)", "e6cc1e096997cafdab99f02c77232448": "\\frac{\\mathrm{d}T}{\\mathrm{d}t}", "e6cc31fe30e2780c4a5646697df87694": "v=(v_1, v_2)", "e6cc6bb0bd390886032698408186a384": "\\left(x_1 + y_1 \\omega \\right) + \\left(x_2 + y_2 \\omega \\right) = \\left(x_1 + x_2 \\right) + \\left(y_1 + y_2\\right) \\omega", "e6ccc56001164dbf2b8467af70b165ff": "\\lambda\\,\\!_p \\big( \\mathbb{Q} \\big(\\sqrt{D_{0}} \\big) \\big) = 1", "e6ccf56c0d2da138b2693afd23be0032": "\\Pi=f(X)=f(S,\\bar{P},\\bar{E}) \\,", "e6ccfb254ec8bd6494344760bdc42ab5": " \\int_X | x\\rangle\\langle x|\\; d\\nu (x) = I_{\\mathfrak H} ", "e6cd04adfadf222e2be05ecaa280cbb4": "x(t)=\\begin{cases}\n\\cos( \\pi t); & t <10 \\\\\n\\cos(3 \\pi t); & 10 \\le t < 20 \\\\\n\\cos(2 \\pi t); & t > 20\n\\end{cases}", "e6cd69cdbea6a30aa9b0e82bad1c8e61": "{r \\over (1+r)^2}\\le |f(z)|\\le {r\\over (1-r)^2}", "e6cddf5179152f9ba6f5ad72fd8e8b8a": "t'=t^\\alpha\\,,\\quad x'=x^\\beta. ", "e6ce4a1bc1dd10ecdc58aa501b4b416a": "Z' \\xrightarrow{f} Y' \\xrightarrow{g} X' \\xrightarrow {h} ", "e6ce4d925751f43ec7044057b1251a62": " S({\\mathfrak g})", "e6ce52080119c8eb31be9032ffec6e5d": "|C_{n^{*}l^{*}}|^{2}= \\frac{2^{2n^{*}}}{n^{*}\\Gamma(n^{*}+l^{*}+1)\\Gamma(n^{*}l^{*})}", "e6ceca5789f5c7e3ea22b5e9cc5b6872": "\\sigma(s)", "e6cedaae0c2773a87901ffd36ea084d6": "z_{1,2}\\,", "e6cf3d0f42ebb5a3e393677bc665673a": "\\ln(F_{ijk})=\\lambda + \\lambda^A + \\lambda^B +\\lambda^C + \\lambda^{AB} + \\lambda^{AC}+ \\lambda^{BC} + \\lambda^{ABC}, \\,", "e6cf5c88e6c558f37e256d894c517649": "AdS_5", "e6cfe66191866f96cbbd9a6a764b667d": "-|1\\rangle", "e6cfe6f7280cb52d5425b942bec0f0ea": "\\log_{10}(P) = 10{.}3291 - \\frac{1642{.}89}{351{.}47 - 42{.}85} = 5{.}005727378 = \\log_{10}(101328\\ \\mathrm{Pa}).", "e6d041af71226d2a0fcc608094cc80ab": "\nf_\\mathbf{p} (p_x, p_y, p_z) =\n\\frac{c}{Z} \n\\exp \\left[\n-\\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}\n\\right]", "e6d0622744fa7dd99ea25ff5e0cb251b": "\nr =\n\\begin{cases}\n0, & K < K_c \\\\\n\\sqrt{1-(K_c/K)}, & K \\ge K_c\n\\end{cases}\n", "e6d08080442ecbf205d5ad7c2ece779b": "\\mathbf{F}_{1^n}.", "e6d193f5f19173d3d03451d33621ba0b": "(\\partial I)\\times D^{m-1}", "e6d1aaf4fdab76f06df767013288a512": "u=h^{-1}(U)", "e6d20aa17b64e7ad22144db6c94aca7b": " \\Psi = \\pm \\sqrt{\\frac{-r_0 (T - T_c)}{2s}}. ", "e6d21e4425fb12a66c6f9f3e9e28d70f": "w_i(x_0),\\;i=1,\\ldots,N", "e6d29232b6f90999a84d0ee1186a7e4f": "x, y \\in C\\,\\!", "e6d2c43d7c5a9f3d4c9cac2045c67139": " \\pi(x) [\\bigoplus_{f} \\xi_f] = \\bigoplus_{f} \\pi_f(x)\\xi_f.", "e6d2fbafbea5d8c7256313ce462e8357": "5_0", "e6d313daf53064387bc1abec60b80682": "||\\mathbf{P}||^2 = \\mathbf{P}\\cdot\\mathbf{P} = \\gamma^2m_0^2(c^2-v^2) = (m_0c)^2\\,,", "e6d3b4d20d114b56bec4c009e0696e55": " N_x ", "e6d3c389dfedde5b91222b10792dd7b4": "h(a)", "e6d3c955f769b3969e726415f22f5af9": "\\textrm{Ann}(M) := \\{ r \\in R | rm = 0 \\, \\forall m \\in M \\}", "e6d3d50b93d029c41f860ca6a4576018": "A_{geo}(G,H) = \\left(\\prod_{j=0}^{72}A^j(G,H)\\right)^{\\frac{1}{73}}", "e6d3d93e119812475ab7b1e46ceeda2e": "\\textstyle (P_2,L_2)", "e6d3e3acb1ed2dcf796d55023f56305c": "E:= \\{E_n\\}_{n\\in \\mathbb{N}} ", "e6d415928ece94dd380cf95e537ae35b": " 0^0 = \\Phi \\,", "e6d41aa78b13bcd74569c33681827ae4": "\nf \\propto {1 \\over \\sqrt{\\mu}}\n", "e6d421381333dd7aad3e650a86ecf73d": " f = \\frac{\\omega}{2 \\pi}.", "e6d463d5b4218bd18d02a5c94a95c5f2": "\\operatorname{GL}(n, \\mathbf{R})", "e6d498f3738fe5b4aefdebfe0710396b": "\\begin{align}\\sigma_x \\sigma_p&=\\frac{\\hbar}{2}\\sqrt{\\left( \\cos^2{(\\omega t)} + \\frac{\\Omega^2}{\\omega^2} \\sin^2{(\\omega t)} \\right)\\left( \\cos^2{(\\omega t)} + \\frac{\\omega^2}{\\Omega^2} \\sin^2{(\\omega t)} \\right)} \\\\\n&= \\frac{\\hbar}{4}\\sqrt{3+\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-\\left(\\frac{1}{2}\\left(\\frac{\\Omega^2}{\\omega^2}+\\frac{\\omega^2}{\\Omega^2}\\right)-1\\right) \\cos{(4 \\omega t)}}\\end{align}", "e6d4a42a4da7201a204a93c19858c384": "\n \\boldsymbol{S} = 2~\\frac{\\partial W}{\\partial \\boldsymbol{C}} \\qquad \\text{or} \\qquad\n S_{IJ} = 2~\\frac{\\partial W}{\\partial C_{IJ}} ~.\n ", "e6d4a642adc0da75fe8d0a9ee69e1a7f": "\\alpha = {d \\over f} \\times {180 \\over \\pi} \\equiv {180 d \\over \\pi f}", "e6d4a9895a5902d3be8cdb22621d8581": "U_e = \\int {\\frac{E A_0 \\Delta L} {L_0}}\\, d\\Delta L = \\frac {E A_0} {L_0} \\int { \\Delta L }\\, d\\Delta L = \\frac {E A_0 {\\Delta L}^2} {2 L_0}", "e6d4b18ec72cf5a372ce65a56134a31b": "\\mathrm{\\Omega=V\\ A^{-1}=kg\\ m^{2} \\ A^{-2}s^{-3}}", "e6d4b5ddecb2109caf0668df6670609a": "\\beta(1-\\beta)", "e6d531d201877733e0caeae80b467890": "OP\\times OP^{\\prime} =r^2.", "e6d59929a2ce9d146c72153979201fb1": " \\int \\sec x \\, dx = \\ln|\\sec x + \\tan x| ", "e6d59a934340316a069540748c4f60aa": "\\ln n << n", "e6d59febf8759a5e09b43a6b3ff951f2": "N = \\sqrt{- \\frac{g}{\\rho_0} \\frac{\\partial \\rho (z)}{\\partial z}}", "e6d6189210cec4d3515930076a6f28bb": "\\phi(x_n) = y_n", "e6d6269c7600ff6417ccd80bb481f9be": "\\left(\\frac{\\pi}{6\\sqrt{3}}\\right)^{\\frac{1}{3}} \\approx 0.671", "e6d69254cc1129cdb991c5b1acaa3169": "{n_{comp}}", "e6d73d10e50f52d95f2145183b89dec0": " G_x(t-t_0,f)e^{-j2\\pi ft_0}\\,", "e6d7fbd55851147863b71b4516e45a62": " \\|r \\cdot x\\| = |r| \\cdot \\| x\\|.", "e6d8520cf2e4b16a5f44f80fc51d9568": "\\vec{\\nabla}\\times\\vec{E} = 0.", "e6d8a19cf352a4bc661da74e01166577": "=\\mathbf{ \\Omega \\ \\times } \\left( \\mathbf{ \\Omega \\times} (\\mathbf{ X}_{AB}+\\mathbf{x}_B) \\right) + \\mathbf{a}_B + 2\\ \\boldsymbol{\\Omega} \\times\\mathbf{v}_B\\ \\ ,", "e6d8b829c6f6203c6d93d13d427807b9": "r_{i+1}:=\\frac{\\text{prem}(r_{i-1},r_{i})}{\\alpha},", "e6d8c02c347fbb9a5f71d1146d13614f": "d(B,A) = 0", "e6d8f183e049d2c1d76f7d9aaa260c9b": "( -x - 1 ) + ( 2x - 8 ) = x - 9", "e6d8f397b92e0a6d81b751b8d8c63dc1": "|{\\psi_{gr}}\\rangle", "e6d916b154f943ccb7399ea99f40807e": "\\mathbf{E}=\\mathbf{B}=\\mathbf{0},", "e6d91a3adc3cf587b6314197909f4d10": " \\phi_{mg} \\ ", "e6d93cc3a6228bc1950d37522ff6961d": "g=f\\,\\frac{n^3}{6}=f\\,\\frac{(E/\\hbar\\omega)^3}{6}", "e6d94242d969e403d4f0fe5290a0813f": "L(M)=N(M')", "e6d96ff64829e516f3750d13eff7f549": "\\Delta Q_\\text{cool}", "e6d9766586091bc296451bf15113fbf0": "(c-1)", "e6d9bc488dd471e3b1dcf2d77ccf2343": "S=\\{ s_n \\}_{n\\in\\N}", "e6d9d10be1ad1f2ba2901a2c6e41646f": "H(\\kappa) \\!", "e6da1dc03150ed9e876bfe9b5b70df57": "\\text{ Equiangle Skew } =max[\\frac{\\theta_{max} - \\theta_e}{180 - \\theta_e},\\frac{\\theta_e - \\theta_{min}}{\\theta_e}]", "e6da1f3e048ca731320f493c73e46a2f": " dQ_h = T_hdS_h ", "e6da2162cb0e747ff256f855457dc302": "\nh(\\mathbf{X}) = \\frac{\\mathbf{X}}{\\epsilon + |\\mathbf{X}|^2}\n", "e6da2489365ceece5a9a0abfad841abe": "\nu(x) = u_0(x) + \\sum_{m=1}^{\\infty} u_m(x).\n", "e6da514d4b26b737405c9a85619aa50f": "h_B\\;", "e6daadf0489a84556a7d48b42ed28228": "P_\\ell^{(m)}(x)", "e6dae7bde7501927d87291050e69a53f": "tr(\\mathbf{{\\Sigma}}_y)", "e6daf6f7fb1e08e3c01839aeef5a9f26": "p(t) = \\det(t\\delta_{ij}-a_{ij}).", "e6db1270f0e5e7b3f9fc46525c8229e3": "\\sqrt{x^2 + bn}", "e6db4d5f32979ffa303b5ab9a0028be0": "P:=(p_{ij})_{i,j}", "e6db928cef89f058af82805f4008b6af": " D(a,s) + D(b,s) = \\sum_{n=1}^\\infty (a+b)(n) n^{-s} \\ ", "e6dbcfe35d19d6cbb51008ae6a574116": "\\iiint_V \\mathbf{u}(x,y,z) dV = \\iiint_V \\mathbf{v}(q_1,q_2,q_3) Jdq_1dq_2dq_3 ", "e6dbe1f3eded4e244eb843714464e2fc": " \\lambda = \\frac{h}{\\sqrt{2m_0eU}}\\frac{1}{\\sqrt{1+\\frac{eU}{2m_0c^2}}}", "e6dc406577346cf9b2dcc350c05315aa": "\\cos^2\\theta = \\frac{1 + \\cos 2\\theta}{2}\\!", "e6dc7a0fd1a369e20c1954f65061dc3d": " V^{2} = x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + (1 + u_{1}u_{1})\\frac{\\partial}{\\partial u_{1}} + 3u_{1}u_{2}\\frac{\\partial}{\\partial u_{2}} \\, ", "e6dc93097f14d8b87794591dbda5315d": "\\sigma_2 - \\sigma_3 > 0", "e6dc9b2362b7c92ab389bb53fc54e821": "Q \\to P", "e6dd0455de4b5d9a1cdc3546eda788e1": "f(\\vec{X})", "e6dd546f5cc4e53188dd1ebdda1d04e8": "a\\cdot x(t) + b\\cdot y(t)\\,", "e6dd6df2c92712c6b5d407fa452f1fd5": "\\scriptstyle f(x)", "e6dd75f46ff59258a17f33a3e6adc634": "\\sin^2 x + \\cos^2 x = 1 ", "e6de6e9c4a9c6f0575054578d0ffd1e6": "\\! \\rho_m", "e6dea98cc6e35bb3d7de55ab6aebd224": " n_s ", "e6deaaa63d73fb06c394df806d266f3d": "T_u\\cdot J\\left(R\\right)\\subseteq T_{u-1}", "e6dedbaf4ae55de7b0e0e113b6bf16fe": "\\textstyle B_n = {2n\\choose n} = d_n C_n", "e6df4c731960dc85dae1ae5183018997": " G = \\langle B, e| e^{-1}he=\\phi(h), h\\in H\\rangle. \\, ", "e6df773f9b8da64968e3aca7cb816a4e": "\\begin{align}\n \\frac{\\partial y}{\\partial c} &= b_0 x^c \\Bigg ( \\ln(x) \\sum_{r = 0}^\\infty \\frac{(c + \\gamma - 1) (c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r} x^r + \\sum_{r = 0}^\\infty \\frac{(c + \\gamma - 1) (c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r}\\Bigg \\{ \\frac{1}{c + \\gamma - 1} + \\\\\n&\\qquad \\qquad+ \\sum_{k = 0}^{r - 1} \\left(\\frac{1}{c + \\alpha + k} + \\frac{1}{c + \\beta + k} - \\frac{1}{c + 1 + k} - \\frac{1}{c + \\gamma + k} \\right) \\Bigg \\} x^r \\Bigg ) \\\\\n&= b_0 x^c \\sum_{r = 0}^\\infty \\frac{(c + \\gamma - 1) (c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r (c + \\gamma)_r}\\left(\\ln(x) + \\frac{1}{c + \\gamma - 1} + \\sum_{k = 0}^{r - 1} \\left \\{ \\frac{1}{c + \\alpha + k} + \\frac{1}{c +\\beta + k} - \\frac{1}{c + 1 + k} - \\frac{1}{c + \\gamma + k} \\right \\} \\right) x^r.\n\\end{align}", "e6df85e0a3b56a4e5d065eb06fc36298": " \\mathbf{f} = - {4\\pi G \\over {3 c^4}} \\left ( {Mc^2 \\over V }\\right ) \\mathbf{r} ", "e6dfa2fa3ee249c7c5b5b0035c743c98": "\\frac{\\partial \\Lambda}{\\partial x}=2x+2x\\lambda", "e6dfd15de4d1e617ce416fc7716cf2d3": "D_0=I,", "e6dfd45f15468e48023fa7c17feff924": " \\|f\\|_r \\le \\frac{2r}{R-r} \\sup_{|z| \\le R} \\operatorname{Re} f(z) + \\frac{R+r}{R-r} |f(0)|. ", "e6dff5026d5ffe7a81505a9df9ec16c8": "\\displaystyle \\tilde{u}", "e6e03ad61eb9f0d3cc66d4a43144b217": "U=\\{x_1,\\dots,x_n\\},", "e6e051405bb514c9f29b404897dc5800": "(X,U)", "e6e067d9d6e6659a269d92a03bf31504": "\\forall w\\in\\mathbf{C} : \\frac{1}{|f(w)|^{\\frac{1}{n}}}\\,K_g=\\frac{1}{n}\\Delta \\log|f(w)|=\\frac{1}{n}\\Delta \\text{Re}(\\log f(w))=0", "e6e085b0c79f09455639cfab251c65fa": "r(x) = p(x) - q(x)", "e6e0b2bcf4ad5bcd319091d71c1a01f0": "L_\\%(M)", "e6e0b824f093070dbfb492f299c1693c": "\n\\log R_e = 0.36 \\,(\\langle I \\rangle_e / \\mu_B) + 1.4 \\, \\log \\sigma_o\n", "e6e0d011c78c48f7ad60d1a863c16d13": "n=77", "e6e1a2392fee7e77df84f389131a14bc": "\\frac{287+265}{50+ 50}-\\frac{243+267}{50+47.33} = \\frac{552}{100}-\\frac{510}{97.33} \\approx 0.28", "e6e1dea98d57cbcc9805fea247c9a94d": "y = x^2\\sqrt{1-x^{-2}} = x^2 - \\frac12 \\sum_{n \\ge 0} C_n(2x)^{-2n}", "e6e231eaaf13293e83a804e85f350186": "R'=R,~ P'=\\frac{P}{s} ~\\text{and}~t'=\\int^t \\frac{\\mathrm{d}\\tau}{s}", "e6e23821a5e4140aebcd5b3a4987294d": "\\underset{x\\in \\Bbb{R}}{\\operatorname{arg\\,max}}\\, x", "e6e23eb55d09d90b799c1cbc5416ded0": "p_4(x)=-\\tfrac{3}{16}", "e6e25232e83a53d8ef5282f358a26f32": "N_{R_2}", "e6e27bcd99245dbb64df20728a497972": "c=1-\\big([S]-[R]\\big)=1-\\frac{e^{-k_St}+e^{-k_Rt}}{2}", "e6e2a2563fd034053a029093787e5bea": "d(f^n(x),f^n(y)) > c-\\delta,", "e6e2ccada9c27522d7d2c8ea459ff606": "\\left (\\sqrt{2}^{\\sqrt2}\\right )^{\\sqrt2} = \\sqrt{2}^{(\\sqrt{2} \\cdot \\sqrt{2})} = \\sqrt{2}^2 = 2.", "e6e31780fd429f9d56db03eee9e15bb1": "\\dot\\gamma = \\frac{4Q}{\\pi r^3}.", "e6e33af3a6dc359c35a85e6840e507f6": "\\gamma_2=\\frac{2}{\\sigma^2}(1-\\mu\\sigma\\gamma_1-\\sigma^2)", "e6e3521942c469816eaa562fae46db5c": "z_{12} = z_{21}", "e6e35d53d7fa5791280c5e8a95560ed4": "-i \\hbar", "e6e35d86927d4ccc9f1d8ee1102a45fb": "\\scriptstyle \\frac{[CO_2]}{p_{CO_2}}=\\frac{1}{k_\\mathrm{H}}", "e6e381f101331e975c33e8cdb00fd1d5": "C_L \\, ", "e6e3d085f53a71931cc91a3b79385bab": "u = U", "e6e403b417593308ba873545c475d271": " W_{3\\to 4} = \\int_{V_3}^{V_4} P \\, dV, \\, \\, \\text{positive, work done on system} ", "e6e4060818d14f15e77508ab40d5a327": "\n\\mathbf{J} = \\sigma\\mathbf{E}\n", "e6e425d28958e56b6d40bc3d563fbd95": " \\begin{align} \n\\mathbf{r} & =\\mathbf{r}\\left ( r,\\theta, t \\right ) = r \\mathbf{\\hat{e}}_r \\\\\n\\mathbf{v} & = \\mathbf{\\hat{e}}_r \\frac{\\mathrm{d} r}{\\mathrm{d}t} + r \\omega \\mathbf{\\hat{e}}_\\theta \\\\\n\\mathbf{a} & =\\left ( \\frac{\\mathrm{d}^2 r}{\\mathrm{d}t^2} - r\\omega^2\\right )\\mathbf{\\hat{e}}_r + \\left ( r \\alpha + 2 \\omega \\frac{\\mathrm{d}r}{{\\rm d}t} \\right )\\mathbf{\\hat{e}}_\\theta \n\\end{align} \\,\\!", "e6e456d0f55739baf2b177f9bccf5a81": " \\mathbf{u}(\\mathbf{x},t) = \\mathbf{u_{0}}e^{i(\\mathbf{k} \\cdot \\mathbf{x} - wt)} ", "e6e4bb81e086568cb836f37eea969b94": "\\mathbf e_i\\cdot\\mathbf e_i=1", "e6e513797585480ffbe6f21182d6298b": "j^r_p\\sigma ", "e6e5678d51b72b55233be0b2c3d729fe": "G_{dBi} = 10 \\, \\log_{10}(1.698) = 2.30 \\, \\mathrm{dBi}", "e6e5c7ce565c356ad4efa7de0bd658f3": "y_{unk}=\\bar{y}", "e6e5ebf75d47deee328c6a05fb5ec0f6": "\\mathcal{L} = \\overline{u}_L\\,i\\displaystyle{\\not}D \\,u_L + \\overline{u}_R\\,i\\displaystyle{\\not}D \\,u_R + \\overline{d}_L\\,i\\displaystyle{\\not}D \\,d_L + \\overline{d}_R\\,i\\displaystyle{\\not}D \\,d_R + \\mathcal{L}_\\mathrm{gluons} ~.", "e6e5efa89949057054ec91d9027e0daa": "-\\frac{dy}{dx} = \\frac{MU_x}{MU_y}", "e6e628dc946284d779301f352918b25e": "r_{i+1}:=\\text{rem}(r_{i-1},r_{i})", "e6e64d27c483b6ae8700067270601f93": "K_{\\mathrm{J-90}} \\,", "e6e66fcda4d9b0a05096459c959a4c27": "\\pi_T \\ne \\pi_C \\ne \\pi_A \\ne \\pi_G ", "e6e690e44035ab98720dac37cf32945a": "\\oint \\vec{v}_s\\cdot\\vec{\\mathrm{d}s} =\\frac{h}{m_4}n.", "e6e6bf259ee584506726310b0e8919d6": "\\Theta(N * K)", "e6e6e9f99a69ec0f81ff493369a75c8e": "K=\\sqrt[4]{efgh}(e+f+g+h).", "e6e708b30c4fd973a11a7d8a0d2de695": "S = \\sum_{|\\alpha|\\le m} c_\\alpha \\partial^\\alpha\\delta_a.", "e6e746567a93e54cc5ddafedeb9a6955": "m' = s^{-1} * (h + xr)", "e6e79788ecf6b0ae8865bf012e86e08d": "O(h^n)", "e6e7ed94f75e0e98e33a19b20e4948bf": "\n \\begin{align}\n \\lim_{n\\to\\infty} \n \\left[\n L_{\\hat{X}^n_{DUDE}}\\left( x^n,Z^n \\right) - \n \\min_{\\hat{X}^n\\in\\mathcal{D}_{n,k}} L_{\\hat{X}^n}\\left( x^n,Z^n \\right)\n \\right ] =0 \\,,\\,\\text{ almost surely}\\,.\n \\end{align}\n ", "e6e8267a40967f5a7f89c4d2af8a7473": "-\\infty < a < b < \\infty \\,", "e6e8580fa634bc00af1c61a9c8c8ac08": "\\{0\\}\\subset\\dots\\subset O_{0}\\subset O_{1}\\subset O_{2}\\subset\\dots\\subset L^2(\\mathbf{R})", "e6e86b40653f4daae9abbe3506ced4b4": "\n\\left(\\begin{array}{cc} \n A_{11} & A_{12} \\\\\n A_{21} & A_{22} \\end{array}\n \\right)\n", "e6e88b3ad8a2db8d69a44fcf22e5fb03": "\\left(\\frac{p+n+1}{2}-m\\right)\\log \\Lambda(p,m,n) \\sim \\chi^2_{np}.", "e6e89d3377f62a4c4b0b031751fa7954": "-\\frac{\\partial{H_z}}{\\partial{x}} = C'\\frac{\\partial{E_y}}{\\partial{t}}", "e6e8b98d8a7a72e1b0353f7f318a6166": " \\lambda (k) = \\sum_{k'} S(k,k')", "e6e958cfea89d09abad586ad3c2b6722": " \n\\lim_{n\\rightarrow \\infty} \\mu(A-A_n)=0.\n", "e6e95ce1b0aeab5ae056010b1ea4ac28": "\\mathfrak s\\mathfrak o(2n)", "e6e96af5f97317a3ea1fb330dc5b58db": "p = 3 - \\alpha\\,.", "e6e96f741edac24ca9d58688fa91cbfc": " \\mathbf{P}(t) = [A(t)]\\mathbf{p}, ", "e6e989c6d98cda787d5251c99264f576": " J_{\\mathrm{M}} = \\alpha_{\\mathrm{r}} a {\\phi^{-1}} F^2 \\mathrm{exp}[- v(f) \\;b \\phi^{3/2} / F ], ..........(36) ", "e6e9d44ea1095ec8b3cd5a9e8069ec51": "g_m", "e6e9e26598f020b05ca58a18ff5a4db1": "\\mathbb P(W \\leq x) = \\sum_{n=0}^{c-1} P_n \\sum_{k=1}^m \\frac{(-\\lambda(x-kD))^{(k+1)c-1-n}}{((K+1)c-1-n)!}e^{\\lambda(x-kD)}, \\quad mD \\leq x <(m+1)D.", "e6ea03049ec5b52a9bf62eb5d8694f54": "b_1 = S_{11}a_1 +S_{12}a_2\\,", "e6ea520c104f4ade769b4da11a697d0e": "\\mathbf{v} \\cdot \\mathbf{r} = [-y, x] \\cdot [x, y] = -yx + xy = 0\\, . ", "e6ea99e8c0a29609d10d2f377da43e55": "hc\\frac{\\Delta\\lambda}{\\lambda^2} = hc\\frac{\\delta D}{2D\\Delta D} = \\frac{e\\hbar B}{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i}) \\ .", "e6eac15387e1b6f0655d039c60ba07ea": "\\ln K_{eq}", "e6eb30aa1b6941f782928fb854b2e146": "\\tau_b", "e6eb48c865e5f40e9792ec4500bc6aa4": " \\rm p\\cdot\\lambda - \\rm q\\cdot\\lambda_{\\rm N} ", "e6eb4a0e4fee5adef217cb8066ec9908": "\\textstyle{A=\\frac12 \\times (log_2 a + log_2 b)}", "e6eb89b86eb4bffeb723cc8898e87295": "\nE[X_i] = \\frac{a \\theta_i}{a-1}, \\text{ for } a > 1, \\text{ and } \nVar(X_i) = \\frac{a \\theta_i^2}{(a-1)^2 (a-2)}, \\text{ for } a > 2.\n", "e6eb997649ee43d127e2ce0eb71a69ad": "\\sigma(X)", "e6ebe457eeb12352e570e91b576fbb06": "that the meat facility is 600 miles from the city and the farm is 300 miles from the city;", "e6ec09b1a55d6699e0bd9cdf866fff11": "\\mathfrak{m} = \\operatorname{ann}(s)", "e6ec323c302bc89d838fa62676b941c8": "\\vec{v} \\vec{w} = \\vec{v} \\times \\vec{w} - \\vec{v} \\cdot \\vec{w},", "e6ec5694e162ca8c5b609605d241bc1a": "E_{CMI}(\\varrho_{A, B})", "e6ec6c4e367b284e9cbf8fa5dadc1bc8": " M_{\\alpha\\beta} = X_\\alpha\\ P_\\beta - X_\\beta P_\\alpha ", "e6ec7b741fd052ad26823920bc047fd3": "\\mid \\mu_{3,2}\\mid >\\frac{1}{2}", "e6ecb93d0f5e1e5f03afcf2f4b3d2d15": "F(\\tau) = \\sum_NH(N)q^n + y^{-1/2}\\sum_{n\\in Z}\\beta(4\\pi n^2y)q^{-n^2}", "e6ecc1371f82ba7c0b8cd42865ef3dd0": "\nR_H = a \\left(\\frac{M_{\\mathrm{planet}}}{M_\\star}\\right)^{1/3}.\n", "e6ed1db18c72a5c34136454c5a83729c": "\\exp(c_i \\tau D_T)", "e6ed6b73e49a828244f0aade7d14eb01": "\\forall n\\in\\mathbb{N}_0", "e6edc378e37feac2afd38007b028f8dc": "{}^nC_k", "e6edd578382c085d645748990044f784": "\\frac{dG}{d\\omega}=-nG^3\\omega^{2n-1}", "e6ee72e7e2df589526eeb6ffa44e90c8": "\\frac{dx}{dy} = \\begin{cases}\n\\mbox{unbounded} & \\mbox{if } 2x^2 + 2y^2 = a^2 \\\\\n\\pm1 & \\mbox{if } x = 0 \\mbox{ and } y = 0 \\\\\n\\frac{y(a^2 + 2x^2 + 2y^2)}{x(a^2 - 2x^2 - 2y^2)} & \\mbox{else } \n\\end{cases}", "e6ee8a1472381fad1c9b2d28f5c25c6f": "H^q_{\\mathrm c}(X) \\to H^q_{\\mathrm c}(Y)", "e6eeb435f7ef4eb02b9abbe228993a3d": " \\left|q^{i}\\right|+\\left|p_{i}\\right|=1, ", "e6eed16db54ba2cf6f1bdf02ca5db079": "p(A)=i", "e6eedab73a3f9a37500f179fd5dfac7d": "A^{\\ast\\ast}", "e6eefe938c10cf1668671f092d7cf7a2": "\\sigma(n) = (p+1)(q+1) = n + 1 + (p+q), \\, ", "e6ef3af0cfe337a66ccf62a60d6db234": " (ab)^{l}=(bc)^{n}=(ca)^{m}=1. ", "e6ef61b208a7b4e82439ff7f1871eb76": "\\scriptstyle\\|\\cdot\\|", "e6ef6e87a3ec2af974c0edc1974f2f2a": "(2/3)", "e6efa17b7d8d23bbfefb3a9e89a48fda": "P (A | Previous, Personal Signal) = \\frac{pq^a*(1-q)^b}{p*q^a*(1-q)^b + (1-p)(1-q)^a*q^b} ", "e6efa86fae6584a47a3f6fb7d5c1bec2": " \nL_f=\\bigl((\\delta_{f(i),j})_{i\\in[m],j\\in[n]}\\bigr) \\quad\\text{and} \\quad R_g=\\bigl((\\delta_{j,g(k)})_{j\\in[n],k\\in[m]}\\bigr). \n", "e6efe88f24414c3162e52e4ece6d9958": "a'=a/\\sqrt{\\varepsilon},\\text{ }b'=\\sqrt{\\varepsilon}b.", "e6f0195c9189dbbadaf391ae865a0da5": "(X_1, X_2, \\dots, X_N)", "e6f0370a5e5619c3e7ca17171702bac4": " X^* ", "e6f0a12c4f3bdaa7a742d98a01da1f2d": "w=1", "e6f0a63ebf488c46b42bd2e3242d21a4": "r_O", "e6f0d64b62d1b984a92f5ca78d48ebdc": " \\psi(\\bold{k}) = (2\\pi a)^{3/2} e^{- a \\bold{k}\\cdot\\bold{k}/2} ~ .", "e6f13a8ad0b662f57330b3b303986039": "P_0 = \\sum_{(y_0,\\ldots,y_c)\\in \\mathrm{S}}\\prod_{i=1}^{c} \\binom{m_i}{y_i}\\omega_i^{y_i}", "e6f13b63830e3eb4430135847f92b768": "(-x+y-1)^2-4x=(x-y)^2-2(x+y)+1=0.\\,", "e6f154ca92683627cadf5f0d1227fb72": "d_e (x_e)", "e6f167578713fba9eebe5526615403de": "W_r= k_r c_{i_r}^{\\alpha_r}, ", "e6f1786cfba3586b92ae5e67f80e0ddc": "p(\\vec x|c=i)", "e6f1b4335f0899ca9b3cc4a0d1c8f5b3": "r(N)={1\\over 2}G(N)N^2+O\\left(N^2\\log^{-A}N\\right),", "e6f242f0e35d6b4a737174078135a996": "OR=\\frac{D_{E}H_{NE}}{D_{NE}H_{E}}=\\frac{D_{E}/D_{NE}}{H_{E}/H_{NE}}.", "e6f29da08905f1efcf67ad835ca3d7bd": "\\begin{align}\n\\left|A_{1}\\cup A_{2}\\cup A_{3}\\cup\\ldots\\cup A_{n}\\right|= & \\left(\\left|A_{1}\\right|+\\left|A_{2}\\right|+\\left|A_{3}\\right|+\\ldots\\left|A_{n}\\right|\\right)- \\\\\n& \\left(\\left|A_{1}\\cap A_{2}\\right|+\\left|A_{1}\\cap A_{3}\\right|+\\ldots\\left|A_{n-1}\\cap A_{n}\\right|\\right)+ \\\\\n&\\ldots+ \\\\\n&\\left(-1\\right)^{n-1}\\left(\\left|A_{1}\\cap A_{2}\\cap A_{3}\\cap\\ldots\\cap A_{n}\\right|\\right)\n\\end{align}", "e6f2bb169dd478782eb60e25f50ad89e": "x \\in T", "e6f2d85ad5b15bbcf5b01182936d97ef": "A=\\alpha_{m}PA_{d}P^{T},", "e6f2ff5d1ae002c77ae03ee1270f8a76": "\\int\\frac{\\cosh^n ax}{\\sinh^m ax} dx = \\frac{\\cosh^{n-1} ax}{a(n-m)\\sinh^{m-1} ax} + \\frac{n-1}{n-m}\\int\\frac{\\cosh^{n-2} ax}{\\sinh^m ax} dx \\qquad\\mbox{(for }m\\neq n\\mbox{)}\\,", "e6f30ac406e3b166fd2607a84d2c8a2c": "\\mathbf{H} = \\{(a, b, c, d) \\mid a, b, c, d \\in \\mathbf{R}\\}.", "e6f3347b1b9a290239b8e5bd40140ed3": "|\\phi_1\\rang", "e6f3489b3f6690be17a2525115e62665": "z = r \\sin \\phi", "e6f374a82733d2140548aa8f5a92831e": "\\int_{\\Lambda^n}f\\left( \\theta\\right) \\, \\mathrm{d}\\theta=\\int_{\\Lambda^1}\\left( \\cdots \\int_{\\Lambda^1}\\left( \\int_{\\Lambda^1}f\\left(\\theta\\right) \\, \\mathrm{d}\\theta_{1}\\right) \\, \\mathrm{d}\\theta_2 \\cdots \\right)\\mathrm{d}\\theta_n", "e6f3eb54ab69e157109f2f9485318335": "\\Gamma^\\infty (E),\\ \\hbox{and}\\ \\Gamma^\\infty (F)", "e6f4046e3ed1b9e01c761ec2e190caf9": "\\frac{2 \\pi} {a}", "e6f4359d1f902c2b6cd1faf46adbe047": "\\widehat{a}|\\alpha\\rangle = \\alpha|\\alpha\\rangle", "e6f4fae50cb869e776a7e417e6df999d": "h^j\\,", "e6f51c4703b24a30a837a88a50cf5da9": " \\Gamma = \\left[ \\begin{array}{ccc|c} 0 & -\\gamma_z & \\gamma_y & \\rho_x \\\\ \\gamma_z & 0 & -\\gamma_x & \\rho_y \\\\ -\\gamma_y & \\gamma_x & 0 & \\rho_z \\\\ \n\\hline\n-\\rho_x & -\\rho_y & -\\rho_z & 0 \\end{array}\\right]", "e6f541e3e5665d83f21d0c96306d80e7": "i, j\\, ,", "e6f5ad8ff6aee268cf6e82052f026db7": "R_B=\\left( \\frac{Vs}{I} \\right) - R_L", "e6f5ae464651822ca7d9b9d1051eca54": "\\delta=\\frac{V_i D}{V_o-V_i}", "e6f6000e421e97cf5ffe7e130df09548": " \\bold\\R=[\\cos(\\theta/2)-I u \\sin(\\theta/2)] ", "e6f605ed1e9b76f4f1a429b07c51b408": "\\phi_n(\\kappa) =\n0.033C_n^2\\kappa^{-11/3},\\quad\n\\frac{1}{L_0}\\ll\\kappa\\ll\\frac{1}{l_0}", "e6f63ce5e314db4da539b6976190b588": " (S = 1)", "e6f64b5ec7970732d748341f35ad4e75": "\\begin{align}\n\\overline{A}_{i+1} &= \\sum_{j=i+1}^\\infty G^{j-i-1}A_j \\\\\n\\overline{B}_i &= \\sum_{j=i}^\\infty G^{j-i}B_j\n\\end{align}", "e6f6653178a5296d2761c642d4c2101e": " R_{i,j} > R_{i,k} ", "e6f66fb167ef91f22f360961760b1896": "V=K[A]/(p(A))", "e6f6968e38a53531101950f289bdbd5d": "\\textstyle \\frac{\\partial}{\\partial y_j}", "e6f69f42e97f320f150e9770e318a59a": "\\begin{cases} - \\Delta H_{f} (x) = 0, & x \\in D; \\\\ H_{f} (x) = f(x), & x \\in \\partial D. \\end{cases}", "e6f6ef89dc46200ca5389e7d283f4254": "\\vartheta \\left(x^{1/n}\\right) = 0\\text{ for }n>\\log_2 x\\,.", "e6f6fd62e3930a7d80b78e980988f9f0": "V^{1-(r,r)/2}", "e6f7536c3ffa40a00c1e9eadf0b19b4c": "\\alpha = n/m", "e6f7b8b8555739571725492b9aabc9e9": "Pr(\\max_{i > 1} v_i < z)(v-E(\\max_{i>1} v_i~|~v_i\\lambda\\quad\\forall n\\geq n_0.", "e70de24e510af2394c8f36cf9e32cd42": " A^{-1}_{kx} = e^{-ikx} \\,", "e70e69aaac12add857be4860d1387865": "\\gamma (v) \\equiv \\frac{1}{\\sqrt{1-v^2/c^2}}", "e70ed5ea9bf6807f0c5d0ea4ea3a86f5": "e(1-e)=0=(1-e)e", "e70ef4e5c312ac52acf9af6b9851f800": "\n{{\\sigma _z } \\over z}\\,\\,\\,\\,\\, \\approx \\,\\,\\,\\,\\sqrt {{{a^2 \\,r^2 \\,\\mu ^{2r - 2} \\,\\,{{\\sigma ^2 } \\over n}} \\over {a^2 \\,\\mu ^{2r} }}} \\,\\,\\,\\,\\, = \\,\\,\\,\\,\\,{r \\over {\\sqrt n }}\\left( {{\\sigma \\over \\mu }} \\right)", "e70f06ac0c6138185eca8a07234be5f0": "L^{norm} = S S^{*}", "e70f7eef026d4e212d3160b3995147c9": "r(u,v) > 0", "e70f9dfb804ef37ac5bf160e8e21dece": "\\mathrm{lift}: \\mathrm{M} \\, A \\rarr (A \\rarr \\mathrm{M} \\, R) \\rarr \\mathrm{M} \\, R = \\mathrm{bind}", "e70faf5d5c79fcc1d3a496122abfd0b5": "d(x,y)=d(y,z)", "e710421c33dda8e36cfe69dd4c30e02b": "\\frac{\\partial a\\mathbf{u}}{\\partial\\, \\mathbf{x}} =", "e710439f6559e9af5c2afeac94ee07eb": "\\begin{matrix} \\frac{10}{47} \\times \\frac{9}{46} = \\frac{90}{2162} \\approx \\frac{1}{24} \\approx 0.04163 \\end{matrix}", "e710670d2c6395c52ddaa54b4ee46328": "a, b, c, \\theta", "e710e7728e3b32ac5d4451c911d5e5fe": "C_n = C_{n-1} + (x_n - \\bar x_n)(y_n - \\bar y_{n-1}) = C_{n-1} + (y_n - \\bar y_n)(x_n - \\bar x_{n-1})", "e7116e9f4dd28bb1c01451a305d12bc3": "\\displaystyle{2\\pi|T_{1-\\varepsilon} H f(x) - H_\\varepsilon f(x)|\\le \\int_{|y|\\le \\varepsilon} | f(x-y)-f(x)|\\cdot|Q_r(y)|\\, dy + \\int_{|y|\\ge \\varepsilon} |f(x-y)-f(x)|\\cdot |Q_1(y)-Q_r(y)|\\,dy.}", "e7119a5a2698e7e95241f56db3368f6c": " \\phi^4 ", "e711f639ae5122eb8807d97b4f9ab226": "FIP=\\frac{13HR + 3BB - 2K}{IP}+C", "e71234330d97bb8a213d5a11630562e5": "\n(\\mathcal{J}_1 \\pm i \\mathcal{J}_2)\\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* =\n \\sqrt{j(j+1)-m'(m'\\pm 1)} \\, D^j_{m'\\pm 1, m}(\\alpha,\\beta,\\gamma)^* .\n", "e7134285327c5ece326fd1283b4f0783": "(F(x) + C)' = F\\,'(x) + C\\,' = F\\,'(x)", "e71355edb2e85b6d3a8eb38d9781604a": "k=\\frac{2\\pi}{\\lambda} \\ . ", "e71357cb4982f4abd827ff5e9571b228": "\nF\\uparrow =\\epsilon \\sigma T_a^4 + (1-\\epsilon) \\sigma T_s^4\n", "e7136c7e4503bc09cd2309c2f15203b6": "condition_n", "e71375ff3fffb133e156827b7b8bb09f": "\\sqrt{T}\\big(\\hat\\theta - \\theta_0\\big)\\ \\xrightarrow{d}\\ \\mathcal{N}\\big[0, (G'\\,\\Omega^{-1}G)^{-1}\\big]", "e713ec653b80f3b47cde4a52b0c3d1eb": "\\mbox{PSU}(n) = \\mbox{SU}(n)/Z(\\mbox{SU}(n)) \\to \\mbox{PU}(n) = \\mbox{U}(n)/Z(\\mbox{U}(n))", "e71449160c8aed759a0f535b5bde9363": " \\log_{10} \\Delta t_{8-5} = 2.69 CE*", "e714aaf7a1829def0d143c19d8a06037": "Y=G", "e7150c788a63236602bf5f9521c31234": "l_{\\mathrm{sum~of~solid~core~pieces}}", "e71521879790dd1df70ec7046c8aadfd": "M_j, j \\le k", "e7157ddd19a329b505a68ac7ed39a87c": " \\qquad \\frac {p_1V_1}{T_1}= \\frac {p_2V_2}{T_2} ", "e71598d3501a08b160ead67a2b2a6d52": "2^n > k", "e715a3ada1db9c119a0a2cfd089b61f2": "p(\\theta|\\rho(\\hat{D},D)\\le\\epsilon)", "e715a817f6317e34de40c51b7ffcd8eb": "\\langle \\lambda x.x, A\\rangle", "e715cff7da309ec05e391c2fa62e1948": "\\mathrm{I}(X_b;Y_b)", "e715d713c02db67a9be81bf68fa6ee2b": "\\,\\!A=S^2-a^2,", "e715f2f66dab0aec96845db4771ca5a8": "P(Y \\in S) = \\int_S p_y(y)\\,dy, ", "e71666e10c39c693624b5043bb6074ca": " \\{ F_t : t \\in T \\}", "e716681700f02f818498c92765850fa6": "f(x)=\\sup_{i\\in I}f_i(x),\\qquad x\\in X.", "e716b21279662c3595fd529ac1a24f5e": "|c_{1}(t)|^2=\\mathrm{sinc}^2\\left(t\\left(f-\\dfrac{E_{0}-E_{1}}{h}\\right)\\right)", "e716cee43fe3b9d41e131cd2d7e687c6": "p(t)I_n", "e7176fdedd6695e50e51b70791cc1eac": "\n \\|\\hat F_n-F\\|_\\infty \\equiv\n \\sup_{t\\in\\mathbb{R}} \\big|\\hat F_n(t)-F(t)\\big|\\ \\xrightarrow{a.s.}\\ 0.\n ", "e717a28aed658194813a25a6c2289048": "\\{p,q\\}, \\{q,r\\}", "e718123c76d64205d6284348aa254fe3": "r_2=0.154\\times Do", "e718160796bf2f7d9a401c14e189d511": " {\\rho }_i \\left( t \\right) = \\rho \\left( x, t \\right) \\ ", "e71816405ff761f02b918a741bda85f4": "f=f\\left(x,\\theta\\right) \\in\\Lambda^{m\\mid n}", "e7181d9deeb03fd58ea09147a597ba49": "Z^{i}", "e7184239d4090727acc68add6f3205a9": "\\bar{P}_{\\mathbf k}", "e718a4db7f0ab5822fd7f78dc50f476b": "f^{(i)}(0)=c_i=0\\quad\\forall i\\in\\{0,1,2,...\\ n\\}", "e718cec30497b2fb368af05ae08bcacf": "\\theta_{d=1 \\dots M,k=1 \\dots K}", "e719641ee738ab74a3af7ff6ea20bccc": "f_{cr}\\equiv{F_y}-\\frac{F^{2}_{y}}{4\\pi^{2}E}\\left(\\frac{KL}{r^2}\\right)\\qquad (3)", "e7197152d39510288c76c35caa18d20b": "-I_s = \\frac{ V_{\\text{opamp}} - V_s }{ R_3 } = V_s \\frac{ \\frac{R_2}{R_1} }{ R_3 }.", "e7198d5d18e4ece36234e427d1326e4f": "x^3+15x^2+66x-360,\\quad x=3 ", "e719d22a2c663000dd337e45d9b649c8": "\\vec{a}_{acc} = \\vec{a}_{o} - \\vec{a}_{frame}", "e71a01f575ac474c4d2953ea9abb7a5a": "H(U,F)", "e71a1784d934b8be93552c4e8893952f": "R^{\\lambda}_{ij \\sigma}", "e71aa91ad147499c77378e24e3083cec": "x \\in A^{\\ast} L(x)", "e71adfc12904787eb3aac43ac3ffc284": " r\\to \\infty", "e71b054d85061fb851f9c9e54dccd903": "v_{\\text{R}} \\left( t \\right) = {i_{\\text{R}} \\left( t \\right)}R", "e71b97b091a1f18cfc77c47263395145": " \\alpha^2 = \\frac{\\omega^2}{c_l^2} - k^2\n\\quad \\quad \\text{and}\\quad\\quad \\beta^2 = \\frac{\\omega^2}{c_t^2} - k^2. ", "e71b9b5cf026393c80a488cafdad4034": "\\,D = \\frac{P\\mathrm{e}^{-G}}{1 + P\\mathrm{e}^{-G}}, \\qquad\\qquad (6)", "e71c283b78969ac837c2fcf634bf5a95": "K' \\cup J", "e71c340e8157d1c8974fcfb2c20c7dcb": "J(\\mathbf{X},t) = \\det[\\boldsymbol{F}(\\mathbf{X},t)]", "e71c4166e27351a95d321982550016e8": "\n\\begin{align}\n\\beta_k &= \\beta_0 + N_k \\\\\n\\mathbf{m}_k &= \\frac{1}{\\beta_k} (\\beta_0 \\mathbf{\\mu}_0 + N_k {\\bar{\\mathbf{x}}}_k) \\\\\n\\mathbf{W}_k^{-1} &= \\mathbf{W}_0^{-1} + N_k \\mathbf{S}_k + \\frac{\\beta_0 N_k}{\\beta_0 + N_k} ({\\bar{\\mathbf{x}}}_k - \\mathbf{\\mu}_0)({\\bar{\\mathbf{x}}}_k - \\mathbf{\\mu}_0)^{\\rm T} \\\\\n\\nu_k &= \\nu_0 + N_k \\\\\nN_k &= \\sum_{n=1}^N r_{nk} \\\\\n{\\bar{\\mathbf{x}}}_k &= \\frac{1}{N_k} \\sum_{n=1}^N r_{nk} \\mathbf{x}_n \\\\\n\\mathbf{S}_k &= \\frac{1}{N_k} \\sum_{n=1}^N (\\mathbf{x}_n - {\\bar{\\mathbf{x}}}_k) (\\mathbf{x}_n - {\\bar{\\mathbf{x}}}_k)^{\\rm T}\n\\end{align}\n", "e71cb516d52cfc177fad4183e48f7901": "\n \\delta U_0 = \\boldsymbol{\\sigma}:\\delta\\boldsymbol{\\epsilon} ~.\n ", "e71cdfd423dcf02dd58ae2d435062228": " \\mathrm{N} = \\frac {B^2 L_{c} \\sigma}{\\rho U} = \\frac{\\mathrm{Ha}^2}{\\mathrm{Re}} ", "e71ce5f7fa2ae7e3398bf2a6fa6b4c84": "X_k = U_k + \\left( \\omega_N^k Z_k + \\omega_N^{3k} Z'_k \\right),", "e71cf8c719a5a912079484213922ce91": "n^{cd}", "e71d330a491cb26e0c0e45fc7b9d9862": "\\log_e(x)", "e71d3383de67c551db0e04adcb8398a0": "r(i)", "e71d49f7c75223f34208a71766a2629c": "y_1y_2\\cdots y_q", "e71d4d8456f563e6c1233084b7a1742f": "x-5=0", "e71db15bfdc2327053f58cb97cd68971": " \\frac{\\Delta M_{stor}}{\\Delta t} = \\frac{M_{in}}{\\Delta t} - \\frac{M_{out}}{\\Delta t} - \\frac{M_{gen}}{\\Delta t}", "e71e3476b8253d56fa72c7ebd2233021": "a_n \\,", "e71e4db96683664f1452de07dad0cfac": "\\phi_1\\colon\\alpha\\mapsto\\varepsilon_\\alpha", "e71ee387dcee601f09f434aba8760fd5": "n \\rightarrow \\infty", "e71ee56c844cddd4c04d2ae8cfc4eb75": " 1, 2, 3, 6. \\, ", "e71f5e35e855137d75ff14a550dd23c7": "- \\,", "e71f6ada62abeb2386e4620d0d31d319": "\\theta(s)", "e71f7900cd983c09bd4beb532b50a336": "\n\\begin{align}\n\\mathrm{Exp}(\\text{AA}) & = p^2n = 0.954^2 \\times 1612 = 1467.4 \\\\\n\\mathrm{Exp}(\\text{Aa}) & = 2pqn = 2 \\times 0.954 \\times 0.046 \\times 1612 = 141.2 \\\\\n\\mathrm{Exp}(\\text{aa}) & = q^2n = 0.046^2 \\times 1612 = 3.4\n\\end{align}\n", "e71fa68c52b7962bac220e29114366b3": "\\chi^2 = {(|b-c|-0.5)^2 \\over b+c}.", "e72023f39ad625b207b28acc33adfb5f": "\\mathrm{U}(1)\\,", "e7204c948b242cd073af9360009f5e2b": "\n\\sigma_{i0} = \\frac{\\Delta \\sigma_i}\n{\\mathrm{amf}_i(\\theta_2) - \\mathrm{amf}_i(\\theta_1)}\n", "e720541827299ae4bc3f426321097143": "2x^2 + y^2 + 8z^2 = n", "e7205657a4b0666deb78dd1ebc073ea9": "k_D", "e72080a60893f951b0930909ad77ce44": "s = r t", "e7209ef7462a6f90c4e660fb465139fd": "g\\in {\\rm PSL}(2,\\mathbb{R})", "e720d70da5bc3b8f782305e4c8ace6e2": "\n\\mathbf{R}^{3N} = \\mathbf{R}_\\textrm{ext}\\oplus\\mathbf{R}_\\textrm{int}.\n", "e720e34536ecc087814ce6be3417aa19": "\\frac{S}{\\nu^3}", "e7211f46fa235de792f434624ae31d62": "\\int x^3e^{x^2}\\, dx,", "e721464772ed077604fb10ffe2498516": "X_1,X_3,X_5", "e721ab43506854d31b4f09c46fb44086": "I(\\omega) \\propto \\frac{\\left(\\frac{\\Gamma}{2}\\right)^2}{(\\omega - \\Omega)^2 + \\left( \\frac{\\Gamma}{2} \\right)^2 }.", "e722803a2238bf2c6805cf1304aec9e6": "(t I_n - A) \\cdot B = p(t) I_n.", "e7228f193a5a33d72477c6c1746d3e2b": "\\tan \\frac{\\varphi}{2} = -\\frac{\\omega}{\\omega_m}", "e72293cf784462abc7b94903017d5c99": "P_R", "e72300482624b07935b2644588c15d40": "\\begin{smallmatrix}[(11.2+35.6)/2]^3/79.91^2=2.0\\end{smallmatrix}", "e72346a48f87fab86fe93e79e11f8c90": "SVP_\\gamma", "e7235e2473ddb1bf4ad87bb729a620a1": "r = \\frac{\\ell^2}{m^2\\gamma}\\frac{1}{1+e\\cos\\theta}", "e7238cb8e5f46a0fa61fa542051562be": "R \\to Q(R)", "e723c1e75d7026b63f244ac799e6692d": "N\\to\\infty", "e723db4e2f9bed143488739e0a2c6ab5": " log (a_{ij}) = \\beta log (s_{ij}) ", "e723fcf3aa234e61c5d5eed2dedecda2": "(st)(q)", "e7241c495c623bc69b1e80f45811829e": "\\mathbf{v}.", "e7247a396305b239a25986768be4ae25": "\\varphi = \\frac{1}{4\\pi\\epsilon_0}\\frac{(\\bold{r}-\\bold{r}')\\cdot\\bold{d}}{|\\bold{r}-\\bold{r}'|^3}", "e724a4ad13b51ee86ca3c84134123a45": "|\\mathbf{a}|", "e724b73620fa74f6509cb3743dd414ac": "\\left[ \\frac{-2\\pi}{N}, \\frac{2\\pi}{N} \\right]", "e725054afc207527ae29e748b87eaf75": "i_r = i_n - p_e\\,\\!", "e7253ad7c8a95ec47a39401d66d97e13": "B_{p^m+n}\\equiv mB_n+B_{n+1} \\pmod{p}.", "e725607bcdb1e57f551f24a0193f458a": "\\Box(\\Box P\\rightarrow P)\\rightarrow \\Box P,", "e7256faa6f9bc426c3ff11b89f7f7d7c": "\\overrightarrow{\\leftarrow}", "e7259a77db443d708768503386a7aa21": " T_n\\left(\\tfrac{1}{2}\\left[x+x^{-1}\\right]\\right)=\\tfrac{1}{2}\\left(x^n+x^{-n}\\right)", "e7259e4b6e2cc54bd7424ddee79980fa": "J_{gas} = J_{oxide} = J_{reacting} = \\frac {C_g}{\\frac{1}{k_i}+\\frac{x}{D_{ox}}+\\frac{1}{h_g}}", "e725a61b47e327a82bf804450c490249": "F_{y} =0.5(\\rho l(c_{xm})^2)2(\\frac{s}{l})(\\tan\\alpha_{1}+\\tan\\alpha_{2})", "e725b1a22c0f3f0a006e3a0bc6894d40": "f_oG2", "e726517778861b99679cc1832ad965f1": " x_n = Q_n y_n ", "e7267d187bfcf3f3faa01485078a0720": " \n- E[L(0)] + V\\sum_{\\tau=0}^{t-1}E[p(\\tau)] \\leq (B+Vp^*)t \n", "e72680b005af52e6e3f97892ee2bb352": " n \\not \\in FV(E) \\to (\\operatorname{let} n = E \\operatorname{in} L \\equiv (\\lambda n.L)\\ E) ", "e726ba5b7dff853dd2a0701d668cd0a7": "L=L_{0}^{'}/\\gamma. \\qquad \\qquad \\text{(2)}", "e726f56abe97aa8ca006cc41d228e609": "(XA)^{T} = XA", "e72756339efa243a5fc1ce7eaa8fe309": "-\\liminf_{n\\to\\infty} x_n = \\limsup_{n\\to\\infty} (-x_n).", "e727afcc500cfbe9613c698a476258f7": "\nG^{II} = (d\\Phi- \\delta_{ab}I^a d E^b)^2 + \\Lambda\\, (\\xi_{ab} E^a I^b)\\left( \\eta_{cd} dE^c d I^d\\right)\\ ,\\quad\n\\eta_{ab}={\\rm diag}(-1,1,\\ldots,1)\n", "e7280bd558c4dcb5cb8b03870e3682a1": "f(y(x)) \\equiv g(x) \\, ", "e7284dc8ecaffe3cbca596a5931b92a3": "\\text{rank} (M)=\\dim_{R_0} M\\otimes_R R_0", "e7285140f13536dd590a7eec476bd064": " \\mathbb{E} ( |Z| | X=x ) = \\frac2\\pi \\sqrt{1-x^2} ", "e728a58f0aef4b15c6c7b055546a83b5": "r_1 + s_1 = r_2 + s_2 \\quad (5)", "e729348f963cfa7686f32c0f38e00381": "m_{A,B}:TA\\otimes TB\\to T(A\\otimes B)", "e72939a304afcf0ddf6cc5710bc5a438": "\\displaystyle \\sum_{n=0}^{\\infty} H_n(x |q) \\frac{t^n}{(q;q)_n} = \\frac{1}\n{\\left( t e^{i \\theta},t e^{-i \\theta};q \\right)_{\\infty}} ", "e729d4b2fc2468c6393b54325c47d33a": "J(C) := \\mathrm{Div}^0(C) / \\mathrm{Princ}(C)", "e729f60d427cb5eec22fe3079e447927": "\\| x \\|_{Y} \\leq \\varepsilon \\| x \\|_{X} + C(\\varepsilon) \\| x \\|_{Z}", "e72a2a1198823f371454cd9c6a7f13b7": "r = \\frac {L^{<->}}{L}", "e72a3109b5aca930a2ab55b875393b56": "\\displaystyle{V(\\xi_1,\\xi_2,\\xi_3,\\dots)=(T\\xi_1, \\sqrt{I-T^*T}\\xi_1,\\xi_2,\\xi_3,\\dots).}", "e72a5f7bd579bd41feebcd8150e5c66c": " \\varphi \\left ( \\mathbf{x} \\right ) \\ \\stackrel{\\mathrm{def}}{=}\\ E\\left ( y \\mid \\mathbf{x} \\right ) = \\int y \\, P\\left ( y \\mid \\mathbf{x} \\right ) dy ", "e72a682f5730e119e83db21a8588419c": "V_\\mathrm{in} = I\\cdot(Z_1+Z_2)", "e72a68a98f7d8257e1a00c9d1de0e6d0": "\\mathrm{id}_X^{-1} = \\mathrm{id}_X", "e72a8ab275d6556a6880a70786888702": " \\sum_{k=1}^n \\frac{1}{k} = \\log n + 1 - \\int_1^n \\frac{x-\\lfloor x\\rfloor}{x^{2}} dx", "e72b062e7c0a323e848444239b23d8b4": " \\boldsymbol\\beta^{(t+1)} = \\boldsymbol\\beta^{(t)} + \\mathcal{J}^{-1}(\\boldsymbol\\beta^{(t)}) u(\\boldsymbol\\beta^{(t)}), ", "e72b070649097be3b9b9e77c750b4ed1": "P_r", "e72b68e7ebc0d6bd575757203a6d1067": "d\\varphi = {{2\\,dt} \\over {1 + t^2}}.", "e72bfa585aab069682485f56f1b3c3a8": "xp_s", "e72ccd1f8575dee1aeef7e92b20dd18d": "\\operatorname{dVar}_n(X) = 0", "e72cef3c8fda5aef2d2a7388f6df64a4": "\\sigma_{\\rm T}", "e72d2c4f27b1170cd3ab719b4d81d871": "(n/3)^n < n!", "e72d416098be44c2112ef10d20737a9b": "\\int_0^\\infty \\frac {e^{-ax}-e^{-bx}}{x} \\, dx=\\ln \\frac{b}{a}", "e72dd735045cc3f70f4b74875bc3616a": "E = \\frac{1}{2} T \\ ", "e72de7e5104b5743a5ecde5475044431": "\\textstyle \\mathbf{c}_i", "e72ea9592b5b61ba50f7637ffc7eae14": "\\tan(z) = \\sum_{k=0}^{\\infty} \\frac{-2z}{z^2 - (k + \\frac{1}{2})^2\\pi^2} = \\sum_{k=0}^{\\infty} \\frac{-8z}{4z^2 - (2k + 1)^2\\pi^2}.", "e72f1889b05be7522fb6879ae4570709": "b - \\frac{1}{n}x - \\frac{1}{n}l", "e72f4e8a2cdb340a8a037cc682c51b69": "F= F_0 \\sin {(2 \\pi f t)}. \\!", "e72f5522857a04382edb02bf7b02d0f5": "\\tfrac{0.5}{L}", "e72f5d8de961c06cee8c875ca6e29357": "\\forall g \\in G \\quad \\omega_g = \\mathrm{Ad}(h)(R_h^*\\omega_e),\\text{ where }h=g^{-1},", "e72f77005b469e9824052ebe2a8716a5": "p < - \\frac {\\rho c^2} {3}. \\,", "e73029a9b96f6fdcae6097e0c1d48c07": "\\mu=e\\beta D", "e73045e897965f0e40633700edc6715b": "[f,g]=g^f-g, \\, ", "e7307e3d743197a1c75a4a2d30a9f694": "C_{p} = \\frac {C_{p0}} {\\sqrt {|1-{M_{\\infty}}^2|}}", "e730975c1d4209e07b00f35df496b518": "\n\\frac{d \\theta}{dt} = \\frac{u}{r \\cos \\phi}\n", "e730fa04244166462f9d8ea715bd8a88": "{\\rm CO} + {\\rm H}_2 {\\rm O} \\lrarr {\\rm CO}_2 + {\\rm H}_2", "e7311a844c7914839e62c2b9792d6793": " \\varphi(D) = \\deg(D)", "e7311db93ebde9e3bb46314d8f98f46c": " u\\colon Y\\rightarrow\\mathbb{R}", "e73152eba65b2cdc00e60286dbc89625": "\n \\begin{align}\n \\langle j_1, m; 1, 0 | j_1+1, m \\rangle & = \\sqrt{\\frac{(j_1-m+1)(j_1+m+1)}{(2j_1+1)(j_1+1)}},\\\\\n \\langle j_1, m; 1, 0 | j_1, m \\rangle & = \\frac{m}{\\sqrt{j_1(j_1+1)}},\\\\\n \\langle j_1, m; 1, 0 | j_1-1, m \\rangle & = -\\sqrt{\\frac{(j_1-m)(j_1+m)}{j_1(2j_1+1)}}.\n \\end{align}\n", "e731973e8c40443af16f3e5d7674f717": "V[A]\\geq G^{-1}", "e7329530c26607e86fe4aa890bd43246": "\\sigma=\\left\\langle\\cos\\left (\\frac{2\\pi z_i}{d}\\right )\\left (\\frac{3}{2}\\cos^2\\theta_i-\\frac{1}{2}\\right )\\right\\rangle", "e732c1a15d72376dc16888a04b0b19d5": "m=\\varepsilon_{I_{\\mathcal D}}\\circ Fn:FI_{\\mathcal C}\\to I_{\\mathcal D}", "e732fca3b1cbbb2e38024a6e9e8bbc87": "\\forall i,w_i\\geq 0", "e7338781f5e10f1462ba355c782ba1d8": "\\left[0, \\frac{5}{32}\\right] \\cup \\left[\\frac{7}{32}, \\frac{3}{8}\\right] \\cup \\left[\\frac{5}{8}, \\frac{25}{32}\\right] \\cup \\left[\\frac{27}{32}, 1\\right].", "e734148fd9177cbb5b03447da84f9f70": "(gu)(hu) = h^{-1}g.", "e73422c646b1f13c226116758980bb80": " \\sqrt{\\mu_0}\\left(\\mathbf{m}, \\mathbf{M}\\right) ", "e73447561a2c06d9b8aa0fabffa91af4": "T(T^{k-1}M)", "e73483765df29cdc181b57d90171aa80": "m |\\mathbf{g}| \\mathrm{tan}\\ \\theta = \\frac{m|\\mathbf{v}|^2}{r} \\ ,", "e7349a40dda66489ace7a8538e79b7e4": "S_N = a_0 b_0 - a_1 B_0 + a_N B_N + \\sum_{n=1}^{N-1} B_n (a_n - a_{n+1}).", "e7349aad8d5b7ba441acd60ee63d3a35": "U \\approx\\int_0^{\\pi/2}\\int_0^{\\pi/2}\\int_0^R E(n)\\,{3\\over e^{E(n)/kT}-1}n^2 \\sin\\theta\\, dn\\, d\\theta\\, d\\phi\\,,", "e7349b3ab9aeae62969dde57a1238dba": " g = D^{1/2} f", "e7351dfc24afd9c4cac94390a1f563e2": " f = f^+ - f^-, \\quad ", "e73592e1ea5b2342cb0ff9431750fdc5": "\n\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n1&0&0\\\\\n0&\\cos\\alpha&\\sin\\alpha\\\\\n0&-\\sin\\alpha&\\cos\\alpha\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{T}\\\\\n\\mathbf{N}\\\\\n\\mathbf{B}\n\\end{bmatrix}.\n", "e7359c544fbfa5167ece7183fd9fb5d9": "\\overline{\\Bigg[\\frac{\\alpha}{\\pi}\\Bigg]}=\\Bigg[\\frac{\\overline{\\alpha}}{\\overline{\\pi}}\\Bigg]", "e735c6fcd11ac9e3005bd074fa89ca34": "Fm_{ms} = Fm_{uc}\\left ( \\frac {1 + \\frac {-25} {1000}} {1 + \\frac {\\delta13C_s} {1000}} \\right )", "e7362b05177d118c34e846ef4c2ec7f7": " \\operatorname{drop-params}[\\lambda N.S, D, V, R] \\equiv (\\lambda N.\\operatorname{drop-params}[S, D, F, R]) ", "e736864a531c0465a09580624ce948b9": " R(a,b)c=2Q(a,c)b,\\,\\,\\, Q(x,y)=\\frac{1}{2} (Q(x+y)-Q(x)-Q(y)).", "e7369950ced949a7fab48654343aa0b7": "\nK(x-y) = \\int_0^{\\infty} K(x-y,\\Tau) W(\\Tau) d\\Tau\n\\,", "e7369f2630d5dda3728c8c044a43e0e9": "\\mathbf{v} = {1 \\over \\sqrt{14}}(-1, 3, -2)^T", "e736b2904827b28cd2d6a4f7e0251bf5": " \\sqrt (t^2-x^2) = 3 ", "e736df5a8a471a2cd4257cf0c820065d": " h-i-j ", "e73703f564d7467f195123396cbb89de": "a_{12}=(8/15)*b_{12}", "e737043dbd8ae3f918e5b7469373b951": "[g_1\\otimes f_1,g_2 \\otimes f_2]=[g_1,g_2]\\otimes f_1 f_2", "e7372e5da91de930ec5bbdaf2a8c9f3e": "\\int \\frac{\\mathrm{d}u}{u\\sqrt{u^2-a^2}} =\\frac{1}{a}\\sec ^{-1}\\left| \\frac{u}{a} \\right|+C", "e737882894cef85c981076333b457ec1": "H(Y) = \\mu(\\tilde Y),", "e737c1f7b15b895861d48d98a397fa54": "a^\\dagger_i", "e737eaccee5e058668d08b3deaca2db3": "f|_{A} : A \\to Y", "e7380aa01cf63d7503898bf4fb2e1df3": "R_\\infin = \\alpha^2 m_{\\mathrm{e}} c / 2 h \\,", "e7380f33199933b1e265ad98cceaf588": "\\scriptstyle 64\\times\\log_2(64) = 64\\times6 = 384", "e73845758558472b98057607af8b6d74": "v = \\sqrt\\frac{2mg}{\\rho A C_\\mathrm{d}}", "e7385691c1c3a8be67c73125ac627851": " P_X ", "e738aee2b600bd77c0f86f3cdef4b027": "C = \\frac{\\mu_B^2}{3 k_B}N g^2 J(J+1)", "e738c3b15bf41021391c6fc91a5202d2": "z =\\frac{S_c}{\\sqrt{\\operatorname{VAR}(S)}}", "e738cff77be34068359bc32ac8454511": "\\scriptstyle \\varphi_{AB} \\;=\\; -\\varphi_{BA}", "e738fd5db8562b0ef4da94b20ae8d21f": "\\textstyle c>0", "e7392c9c28d168b1ae383cb91c8a5ea6": "\\langle\\psi_n(t)|\\psi_m(t)\\rangle = \\delta_{nm}", "e739929e4728e3b0ea0f15f1110c1c95": " \\mu_{log} = \\ln(5.33) - \\frac12 \\ln\\!\\left(1 + \\!\\left(\\frac{0.42}{5.33}\\right)^2 \\right) = 1.67", "e739ccda719bf85f7304bbcb28cc3f6a": "\\boldsymbol{\\gamma}_A", "e739e05a469c970507b140ae38d28972": "g_{ij} = \\delta_{ij} - \\frac{1}{3}R_{ikj\\ell}x^kx^\\ell + O(|x|^3).", "e73a69503fb82174d86a2717814028b6": "\\cos\\theta,", "e73b1359411923ce2edc6accc8a09e3f": "\\phi(E_i)", "e73c11bf86ebf5a29abb810ac1199291": "\\Theta(|V|^3)", "e73c29ebc704de5bd0890a12a4f594ac": "P^0_i", "e73c4b4377492b39d80a26424aa6642f": " dS_n(t) = S_n(t)\\left[b_n(t)dt + dA(t) + \\sum_{d=1}^D \\sigma_{n,d}(t)dW_d(t)\\right] , \\quad \\forall 0\\leq t \\leq T, \\quad n = 1 \\ldots N. ", "e73c6791602e831fe7c56b9e065b7144": "{R_{mn}(c,\\xi)}", "e73c6d001038666f9d8ffa45d2077f39": "f\\colon X \\to Y", "e73cc9904a64fd7df641a9815775e3be": "U_{k+1}=U_k\\left(I+3\\,U_k^*U_k\\right)^{-1}\\left(3\\,I+U_k^*U_k\\right)", "e73ccf355780614377799b705f920c8a": "\\operatorname{tr} (\\gamma^\\mu\\gamma^\\nu) \\,", "e73d0a1a94b7cb1e5071d3a4a055faca": "m_e = 1", "e73d8048c58f65a2775dad7aa8fe404d": "\\begin{bmatrix}\\boldsymbol{\\hat\\rho} \\\\ \\boldsymbol{\\hat\\theta} \\\\ \\boldsymbol{\\hat\\phi} \\end{bmatrix}\n = \\begin{bmatrix} \\sin\\theta\\cos\\phi & \\sin\\theta\\sin\\phi & \\cos\\theta \\\\\n \\cos\\theta\\cos\\phi & \\cos\\theta\\sin\\phi & -\\sin\\theta \\\\\n -\\sin\\phi & \\cos\\phi & 0 \\end{bmatrix}\n \\begin{bmatrix} \\mathbf{\\hat x} \\\\ \\mathbf{\\hat y} \\\\ \\mathbf{\\hat z} \\end{bmatrix}", "e73dc8b2cb5a8559f07d9d217ea62ec9": "\\Phi(\\ h_n,t_n,y(t_n)\\ )", "e73dd290f006f8193b57db1630b45e76": "~|{\\rm final}\\rangle = U |\\rm initial \\rangle ", "e73ded0d001fe307bb5508bfcde03f52": "\\mathbf y - X \\boldsymbol{\\hat \\beta}", "e73e104d53e92c68a6e8af7655d7721e": " \\mathcal{C}(m_i) \\in B(y, pn)", "e73e99b6408b1f45d26b636c63d7698a": "\\Gamma_8 = \\left\\{(x_i) \\in \\mathbb Z^8 \\cup (\\mathbb Z + \\tfrac{1}{2})^8 : {\\textstyle\\sum_i} x_i \\equiv 0\\;(\\mbox{mod }2)\\right\\}.", "e73ebf4cc7288be53bc65ebc9dec666f": "\n\\mathbf{F}(x,y)=\n\\begin{bmatrix}\n {e^x \\cos y}\\\\\n {e^x \\sin y}\\\\\n\\end{bmatrix}.\n", "e73f097b4826763bb006e5096628b3fd": "^n\\mathbf{C}_k", "e73f201f6daab7e8da8ce2a8ea1a67d9": "\\frac{dx}{dp} = \\frac{T_0}{T}\\frac{ds}{dp} = T_0(\\frac{1}{T}+\\frac{1}{E})=\\frac{a}{\\sqrt{a^2+p^2}}+\\frac{T_0}{E}", "e73f98ae4ca4d4d6a1c773a01d95356a": "Lk = Wr + Tw \\;,", "e73fe35aef727c2678856fea239e04d8": "\n Z\\ \\sim\\ \\mathcal{CN}(\\mu,\\, \\Gamma,\\, C) \\quad\\Rightarrow\\quad AZ+b\\ \\sim\\ \\mathcal{CN}(A\\mu+b,\\, A\\Gamma\\overline{A}',\\, ACA')\n ", "e74011c0f59d1bfbb43639d24cd7036a": "\\displaystyle \\alpha(t)= {|t|!\\over t! |S_t|},", "e740cbd54ef1c219b0605ebc9cc494b7": "g(E)dE = \\iint_{\\partial E}g(\\vec{k})\\,d^3k = \\frac{L^3}{(2\\pi)^3}\\iint_{\\partial E}dk_x\\,dk_y\\,dk_z", "e7418c65f456906d46a8c0ebd80c2ed6": "\\tfrac{|AE|}{|AF|}=\\tfrac{|EF|}{|AE|}=\\Phi ", "e741b43485a1b0dd3cda0d5bd557da8b": "f(n)=\\frac{\\lambda^n e^{-\\lambda}}{n!}", "e741bb9a055f887d263189ef797ea403": "\\mathbf Q^T", "e741d636e2dbe9ae548eae11cb18e780": "\\overline{.}", "e7420057310a20786a71b7529d68808e": " {} + \\frac{f_{n-2}f_{n-1}}{(f_n-f_{n-2})(f_n-f_{n-1})} x_n, ", "e7421bfc2baac71f4d177e29bfdf2b03": "D_{\\mathrm F} = \\frac {s f^2} {f^2 - N c ( s - f ) }\\,.", "e7425c0e8c68732401982e8b426564d0": " -\\|x-y\\| \\leq \\|x\\|-\\|y\\| \\leq \\|x-y\\| \\Rightarrow \\bigg|\\|x\\|-\\|y\\|\\bigg| \\leq \\|x-y\\|.", "e7427c6823f46e0d2e55c462d136d50a": "\\sum_{n_1^2+n_2^2\\leq N^2}", "e74282825c69a032598aed4d0ec63388": "\\hat{\\mathbb Z} := \\varprojlim_n \\mathbb Z / n.", "e7434e57c7d9cd5d2a5e3b14ae207ef4": "\\ 7000\\ forager\\ bees\\ \\times \\frac{10\\ trips\\ in\\ good\\ flying\\ weather}{per\\ day} \\times \\frac{70\\ mg\\ of\\ nectar\\ during\\ honey\\ flow}{per\\ trip\\ and\\ bee} \\times \\frac{1kg}{1,000,000 mg} \\approx 5 kg\\ nectar\\big /day ", "e7436a28dc0ba3d8f0e1e4a5e501fb35": "{{\\left\\| f \\right\\|}_{\\Beta ,p}}<+\\infty ", "e74397f01df8e310f56c0dcabca406fa": "Q = V_1A_1 = V_2A_2", "e743b2d91e7497c290de7716b2bddec9": "\\psi(x;\\theta)", "e7440e67e2198e67935e480f6f534a34": " t_s ", "e7446794fbc135fd3319285de2f995d0": "y[n]\\,", "e74473490c06557290b1426729dc0f5b": "\n\\mathrm{d} \\mathcal{L} = \\sum_i \\left( \\frac{\\partial \\mathcal{L}}{\\partial q_i} \\mathrm{d}q_i + \\mathrm{d}\\left ( p_i {\\dot q_i} \\right ) - {\\dot q_i} \\mathrm{d} p_i \\right) + \\frac{\\partial \\mathcal{L}}{\\partial t}\\mathrm{d}t\n\\,", "e74484b4e7a496a69a6827220eddafd7": "\\int\\arccos(a\\,x)\\,dx=\n x\\arccos(a\\,x)-\n \\frac{\\sqrt{1-a^2\\,x^2}}{a}+C", "e744df9579e8f161d9c2f3ddf6158114": "\\textstyle{\\frac {\\log(5)} {\\log(2)}}", "e744f702f67d3217725b3b10a714fe13": "r_p^3 = r_{d_1}^3 + r_{d_2}^3 + r_{d_3}^3 +...+ r_{d_n}^3", "e7450b13bb4c8dd1f42ad2e4f56f87bf": "\\displaystyle K\\le \\tfrac{1}{2}\\sqrt[3]{(ab+cd)(ac+bd)(ad+bc)}.", "e74550c1572bb7548983a6a8f82ccccb": "h(v_i, v_g)", "e7456c7f0fc7a0a18bbd53c63418a6bc": "\\int y dA = 0 ", "e746486fde35dcb94913b825e114b8a3": "\\varphi\\vdash\\bot", "e7466f36376bc28c2942e910508e7492": "\n\\begin{matrix}\n(x_0, y_0), &(x_1, y_1), &\\ldots, &(x_{n-1}, y_{n-1}), \\\\\n(x_0, y_0'), &(x_1, y_1'), &\\ldots, &(x_{n-1}, y_{n-1}'), \\\\\n\\vdots & \\vdots & &\\vdots \\\\\n(x_0, y_0^{(m)}), &(x_1, y_1^{(m)}), &\\ldots, &(x_{n-1}, y_{n-1}^{(m)})\n\\end{matrix}\n", "e74677280c03b662a8ec3a329d257b57": " x=f(u)+vi(u),\\quad y=g(u)+vj(u),\\quad z=h(u)+vk(u) \\,", "e746bc9b99dc97dc8361e31b2ff12e7a": " \\Omega^{\\bullet}_Y(\\log D)\\rightarrow j_*\\Omega_{X}^{\\bullet} ", "e746cf27de05a5830693ca168de80662": "Q_S\\left(n\\right)", "e747e85afa1e09c01c37e643af0cf07e": "\\frac{\\partial}{\\partial{c}}P_c^p(z_0)", "e747f8a5f2d1658cfa7f6ee5f16cc7e6": "\\frac{d^2 x^\\lambda}{d\\mu ^2}+\\Gamma^\\lambda_{ij}\n\\frac{dx^i}{d\\mu}\\frac{dx^j}{d\\mu}=0,", "e74813e44242dc21b57a43469bfd8cda": "\\bar{v_0} = V_r \\cdot \\hat{r} + V_t \\cdot \\hat{t}", "e7487ca6962cc216d95a1dec9a457f3b": "\\Rrightarrow, \\Lleftarrow \\!", "e748e99b868228ba42cf79fd0ad0ff78": "B = f_H/f_L,", "e748f42c63ef8dc2de596b067586f69d": " \\mbox{cosh}^2(s)-\\mbox{sinh}^2(s)=1 ", "e7495dd31b9ac6ddbc726c4c4943aa1a": "\\Box = \\eta^{\\mu\\nu}\\partial_\\mu \\partial_\\nu = \\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2} - \\frac{\\partial^2}{\\partial x^2} - \\frac{\\partial^2}{\\partial y^2} - \\frac{\\partial^2}{\\partial z^2}", "e7497ae844a5c4ae11c0dbd7b47eb60b": "E_P/E_{ann}", "e749bbca1e2c112e229c54b6e1186677": "f\\left( \\mathrm{Supp} \\Omega_{X/Y} \\right)", "e749becaf5659cc3de0cbbab9dfbebba": "\\frac{\\partial\\phi(\\mathbf{r},t)}{\\partial t} = \\sum_{i=1}^3\\sum_{j=1}^3 \\frac{\\partial}{\\partial x_i}\\left[D_{ij}(\\phi,\\mathbf{r})\\frac{\\partial \\phi(\\mathbf{r},t)}{\\partial x_j}\\right]", "e749dfd7ee4222700304225fcb0e4e90": "\\phi^{-1}(S)", "e749f17a453246b28c9875ae6aa41c8a": "-log(\\theta)", "e74a0dff74f9a3d97328206bec5826aa": " dz_1(t)=\\sqrt{\\rho}dz_2(t)+\\sqrt{1-\\rho}\\, dz_3(t)", "e74a4184e1aca89e328bcbfe07a8278b": "M = \\sqrt{1+\\frac{S}{N}}.", "e74a69cfa2e24f4fc97ee76ae502ee25": "{ \\partial u \\over \\partial y } = -{ \\partial v \\over \\partial x } ", "e74a7a236af0107673c97ada7b11df68": "\\phi (a)", "e74acb97e9a48318cd63d8d5b58e8f41": "\\mathrm{AFD} \\times \\mathrm{LCR} = 1 - e^{-\\rho^2}. ", "e74b1aae5b90a3eb0f8a4c62574a790c": "a_q = x_{g^q}", "e74b25749eef0702f04e224b605048a4": "\\kappa_3(X)=p\\kappa_3(F)+q\\kappa_3(G)\n+3pq(\\kappa_2(F)-\\kappa_2(G))(\\kappa_1(F)-\\kappa_1(G))\n+pq(q-p)(\\kappa_1(F)-\\kappa_1(G))^3.\\,", "e74b3358e61df411a023ab3d6914b577": "\\mathcal{M}_{\\epsilon}", "e74c2dd4e9f3da14d22512866cf7e77c": "\\frac{1}{1}, \\,\\frac{1}{2}, \\,\\frac{2}{3}, \\,\\frac{2}{4}, \\,\\frac{3}{5}, \\,\\frac{3}{6}, \\,\\frac{4}{7}, \\,\\frac{4}{8}, \\,\\ldots,", "e74c7874cd48c42762e4887d779d4c4b": " \\left| \\sum_{i,j} a_{ij} \\langle S_i , T_j \\rangle \\right|\\le k", "e74c8d1d8f4fbbca63f840f43d842ae6": "v_{T} \\equiv \\sqrt{kT\\over{\\mu m_{H}}}", "e74c9613e22b06d39496c426a27781bb": "\\cos\\frac{\\pi}{5}=\\cos 36^\\circ=\\frac{1+\\sqrt5}{4}=\\tfrac{1}{2}\\varphi\\,", "e74cd72f07e778f25a22a41844c36739": "x \\mapsto \\mu_{x} (B)", "e74d63241d57a09684a3ffdefc77cf65": "(x-p)(x-q) = x^2 - (p+q)x + n = x^2 - [(\\sigma(n) - \\varphi(n))/2]x + [(\\sigma(n) + \\varphi(n))/2 - 1] = 0 \\, ", "e74dae53b069dc93ba90ce38813fbd8e": "\\scriptstyle \\gamma^o", "e74db91262f1d0c0a840c416abee6838": "\n{L} {\\Theta}_n + \\varepsilon_n {\\Theta}_n = 0\n", "e74dcfe1f9b351d5e80ef9babc7ed0bd": "1.\\overline{011} = 11.\\overline{101} = 11100.\\overline{110} = \\tfrac{3+\\mathrm i\\sqrt7}4", "e74ddfe65594987e6af3de7f1dcdf0b2": "S'_w", "e74e09caca2a7c8078ce41dc96e56524": "C = M_r \\oplus K_{r+1}", "e74e260f33de85a2706c1f16c500e5b5": "\\pi_1(X,p)", "e74e90827bccae836d92b9af6f45dbbf": "y=-5", "e74eebd4188986c0907a67d848647217": "xp'_x(a,b)+yp'_y(a,b)+p'_\\infty(a,b)=0,", "e74f0d697852eaf917a480523cf63c37": "W_{2}^{A}(x,z)", "e74f14a608e3e66c40fee209f2ef0f35": "\\sin y \\approx \\frac {e} {2 r}", "e74f4f4ea0b2dc84461ca1e3992bddcd": " -\\int_a^b{\\vec{E}\\cdot \\mathrm{d}\\vec \\ell} = \\phi (\\vec b) -\\phi(\\vec a).", "e74f53177cf0fbde73575b1f2d78c7b3": "\\bar{y_R}", "e74f74d69678571ace7f62e358589d71": "\\frac{d}{dx}\\left(\\frac{z_x}{\\sqrt{1+z_x^2+z_y^2}}\\right ) + \\frac{d}{dy}\\left(\\frac{z_y}{\\sqrt{1+z_x^2+z_y^2}}\\right )=0", "e74f9d723e591f600678b7041d4b1b63": "\\frac{\\text{d}[{^{d_h}_{c_h}}P^{\\gamma_h}_h]}{\\text{d}t} =\n\\sum_i u_{\\gamma_{hi}} y_{d_{hi}} \\text{k}_{3(i)} C_i \\qquad \\qquad (3c) ", "e74fc7ed3d1bd49057311369a664b4e5": "\\sinh\\left(2 K\\right)\\sinh\\left(2 L\\right)= 1", "e74fc8c0cdac401276aa0a3d258b3350": " \\lim\\limits_{n \\to \\infty} \\frac{|x_{n+1}-\\alpha|}{|x_n-\\alpha|^{\\psi_k}} = |L|^{(\\psi_k-1)/k} ", "e74fd3fe579077eb961705327ebc5a85": "\\sqrt{2} \\lambda >\\xi ", "e74fe3a8a808b92acb18210fdcd3ff61": "r_{\\mathrm{e}} = e^2 / 4\\pi\\varepsilon_0 m_e c^2\\,", "e7501e762558a63772e3cf928a968e1e": "X_H=\\partial/\\partial q^i; ", "e75039143260ac74a871f1f538209baf": "h:{\\mathcal M}\\to S", "e75071ecc05b07ded1618811f38e90fb": "F_{net} = ma_r\\,", "e75075e8205828018ae5c563817d0e3a": "C_\\mathrm r = C_\\mathrm m\\cdot\\frac {(\\mathrm{reference\\,volume\\,%\\,CO_2})}{(\\mathrm{measured\\,volume\\,%\\,CO_2})}", "e7507bf26bcdf1032169b1e33d8f6976": "36524\\mod 7 = 5", "e75097d8cbf2423943e4cb7a8f6e3348": "\\int_V F dU = \\lim_{n \\rightarrow \\infty} \\sum_{i=1}^n F(x_i) \\Delta U_i(x).", "e750b7cc9c7f6454fb813bbaeea49348": "s \\in [0,1]", "e750e580d4f1b216d5a4c47c4131b032": "\\psi_{SS}", "e7516d52de39030e109e25b806b0454b": "\\Sigma_{i \\in I}A_i", "e751cc9cddbbb1c7ea0494b87ae0edb3": "t\\in\\R", "e751ce678361cbfb7307a7b88c352d6d": "H \\leq Z(G)", "e751eb729d393ad816535073dfbc62cc": "S=C_1\\cup C_2\\cup \\cdots \\cup C_k", "e7524c7a87ee2dcb95271029d289c44b": "\n \\epsilon = - s \\sigma,\n", "e75294eae9f4822278e3320a74858c7d": "M = 100 \\frac{x}{d} \\% = (52 + 65 e^{- \\frac{27}{20} \\frac{w}{h}}) \\%", "e752ca0eaad5e982899ff82b7abacd2b": "[H_j: H_1\\cap H_2]<\\infty\\text{ for }j=1,\\,2.", "e752f5e36c415c608379f9700220c262": "|z_i|", "e7534868d6a46c483523605a116a1f89": "\\{x<10 \\land x\\leq10\\}\\; x := x+1 \\;\\{x\\leq10\\}", "e75404a98770699a708633d40695dca1": "C^{1}[a, b]", "e7542294dd6850b1ecb2914eb51d7a37": "{(S\\gg 1)}", "e7542d276131e03c7233217829bd835e": "O |\\psi_0\\rangle=q_j | \\psi_0\\rangle", "e754474a0fc264f3cda53825359aec4f": "n=|V|", "e754988fe3b690b6e552854b92ed6e58": "\\mathrm{eval}_{A,A\\Rightarrow\\bot}\\circ\\gamma_{A\\Rightarrow\\bot,A} : (A\\Rightarrow\\bot)\\otimes A\\to\\bot", "e7550a5a2b91215e3ffb55bff3be8de6": "rol(K_{i(1+b_i)} \\oplus S_i ,u)", "e7550a85d5ece9853615db573c0ae54d": "N_x \\subset M", "e7557669c6775f120bd0e93dbea02a4b": "y_{1}(t) = y_{2}(t), \\quad \\forall \\ t \\le t_{0}.", "e755b85922d20d9d6845e1959ae0e5f6": "\\psi=-\\frac{A}{r^2}\\sin(2 \\theta).", "e7561392b2fc73f85656ff644a1b5a5c": "I_1=I_2", "e756387a25b8a6249ec5de0550cdb2b5": "p_f\\,", "e7564ab854d9a51444c086457b9330a9": "\\overrightarrow{\\Gamma_n^*}=\\Gamma_n", "e7564def42a56e38393d5e0e23a48a5b": "I_{sp(vac)}", "e7568030bafc652ba2be8446b2fc735e": "O(n(1+\\tfrac{\\log N}{\\log n}))", "e75698ff18f0d23328afdc4aea775faa": "\\mathrm P(X_v=x_v \\mid X_i=x_i ", "e756d31447936e3e5bd54cd8d216df74": "A_{1} \\subseteq A \\subseteq A_{2} \\mbox{ and } \\mu \\big( f^{-1} ( A_{2} \\setminus A_{1} ) \\big) = 0.", "e756d97dd72ca396080f92511b5577dc": "V(x) = A\\cdot\\sum_{n=-\\infty}^{\\infty}\\delta(x-n\\cdot a)", "e756fb21248409d0efd6e8d01c3c2774": "\\rho_{B}=\\mathrm{Tr}_A(\\rho_{AB})", "e75738c025a9c9a119767795ce86b2eb": "\\boldsymbol{V} = \\boldsymbol{I} \\cdot \\boldsymbol{Z}", "e757be13f8060c1d2145f15d9c26db62": "\\mu=\\left(\\frac{\\partial G}{\\partial N}\\right)_{T,P}", "e757c88ec96bd024bdf15020f152105d": "\\Pr(A \\mid \\Sigma) ", "e757d47392030a10f0ecb4441c792499": "2.5753", "e757f3ad07b32adf269d95d77944c9c7": "I(q) = 4\\pi\\sum_{i=1}^N c_i\\psi_i(r),", "e7585f872c5052aa31b4da2b5bd83327": "\\text{reachability-distance}_{\\varepsilon,MinPts}(o,p) = \\begin{cases}\\text{UNDEFINED} & \\text{if } |N_\\varepsilon(p)| < MinPts\\\\ \\max(\\text{core-distance}_{\\varepsilon,MinPts}(p), \\text{distance}(p,o)) & \\text{otherwise}\\end{cases}", "e758c21e6ae8ff57cb4d53a47c13304d": "C \\times D := \\tfrac{1}{2}(CD-DC) ", "e758d0cff387e37702398dabfac0fed2": "\nM_{k}^{\\prime} = M_{k} + l b M_{k-1} + \\ldots + l b^{l-1} M_{1} + b^{l} M_{0}\n", "e758eeb8cc9404ccfa01e372db3ce616": "\\mathbf{v'}_i \\in \\mathbb{R}^3", "e758f3e66c77aa33a89860460f67f613": " H(|x|-1) = G_{1,1}^{\\,0,1} \\!\\left( \\left. \\begin{matrix} 1 \\\\ 0 \\end{matrix} \\; \\right| \\, x \\right), \\qquad \\forall x ", "e7590a82e060dc8013b0c28d8cc1dbea": " \\operatorname{Prob}( D = 1 | Z ) = \\Phi(Z\\gamma),\\, ", "e7592b6b2f01cccd63b2994f196660c3": " y[n] = \\sum_{k=-\\infty}^{\\infty} { h[n,k] x[k] } ", "e75958f698ef6db56f3641ca3c07c41c": "\\lambda^2(z,\\overline{z})\\, dz \\, d\\overline{z}", "e759664903db9fe0f2b8de9dc7c92784": "\n \\Beta(x,y)=\\dfrac{\\Gamma(x)\\,\\Gamma(y)}{\\Gamma(x+y)}\n\\!", "e75974ccc903c2ccb60eb67567f39283": "\\scriptstyle\\mu \\;=\\; \\mu_r\\mu_0", "e759837bf46277f58f463e0b71f8245b": "\\cos(\\theta)/\\cos(\\alpha)", "e7598ab3a5600425f960a4edddb14827": "Q'_{out} = \\epsilon _{s.s.} Q'_0", "e75999dc6a4e7eaa09100fec825164ef": "\\frac{P_{t_1}-P_{t_0}}{t_1 -t_0}", "e75a142129021c45e879b800421ad1d2": "\\gamma_{LG}", "e75a61651e8528df6a83f41ea821049c": "\\mathbf{c}_\\Xi := \\frac{1}{2}\\sum_{n=1}^N \\xi_n .", "e75ac74cbde34911e43e7f2fd78bd6eb": "\\frac{1}{6} \\sqrt{13}", "e75b1dfa4572d28b5102dd6726c2f8ee": "S\\in\\{0,1\\}^\\infty", "e75b2b31de0fcd13b54421aec4c5df1b": " \\{ x_k \\} \\leftarrow \\{ z_k \\} ", "e75b422a7ecd90b277bd013565211efb": "\\sum_{t=0}^{\\infty} \\frac{1}{\\| p_t \\| } < \\infty ", "e75b46b6ca4605216ca02e37ba4c9f20": "\\rho(\\mu^2)=\\sum_{n=1}^NZ_n\\delta(\\mu^2-m_n^2)+\\rho_c(\\mu^2)", "e75b9aa12fab0981a33a313eb61a7965": "S^{IJ} = {1 \\over 2} (T^{IJ} - {i \\over 2} \\epsilon_{KL}^{\\;\\;\\;\\;\\;\\; IJ} T^{KL})", "e75ba3317faf097955fb5cf1e8779a28": "{\\mathfrak a}^*_+", "e75be86992ab7383b3d5b25fa11a8e71": "0 \\le \\ b_n \\le \\ a_n", "e75c288a0c20c9ff4336b52d31c9d8bd": "0=b_d=a_0(1/f)^{(d)}(x)\\frac1{d!}+(1/f)^{(d-1)}(x)\\frac1{(d-1)!}", "e75c345b1edcd1c5bcb34fc2d270a5fd": " f(S_{x,i}, a) ", "e75c559a4e13b6761f77da54774b0aa7": "O(\\sqrt{N})", "e75c91cee8dcd40c8820f60f9ff5e721": "A_m(2,1) = 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\\ldots", "e75cdc22ac94c23ea593124fcb2f204f": " r = r_1 = r_2 ", "e75d588aca126e466731a6084bc89638": " y(t) = \\cos \\left( 2 \\pi ( f_{c} + \\Beta \\cos \\left( 2 \\pi f_{m} (t + \\delta t) \\right) ) (t + \\delta t) \\right)\\,", "e75d654e7d90be71c0683d87978c4b0f": "\\ I(r,d)=\\eta I_bpdS \\left [ \\frac{r}{d} \\tan^{-1} \\left( \\frac {r}{d} \\right)+ \\ln Q \\right ]", "e75d6cf3a8343314ff2391b92c997d03": " \\mathbf{R}^* =\\frac{1}{W} \\vec{k}\\times(\\mathbf{r}_1\\times\\mathbf{F}_1+\\mathbf{r}_2\\times\\mathbf{F}_2+\\mathbf{r}_3\\times\\mathbf{F}_3).", "e75daed2ea75f675f91cdb9d0388a42d": "C \\, ", "e75e3145ff48493d13bd3344095ba174": "\\bar{x}^j", "e75e5a4b40b31d6716dc5fc21829e5ba": "\\ \\mathbf{C} \\mathbf{x} = \\mathbf{b},", "e75e77bcc759a6d70d43f2d55affe50c": "uj > u_{j+1} za j \\in N, p \\leq j < q", "e75e9b6015b7264ff882c42507280f23": "f_i(x) = \\frac {l^{np} (x-\\alpha_1)^p \\cdots (x-\\alpha_n)^p}{(x-\\alpha_i)},", "e75ec627bbd8f7d9f37d85a467371c6c": "o(Av_1,\\ldots,Av_n)=\\operatorname{sign}(\\det A)o(v_1,\\ldots,v_n), \\quad A\\in GL(V)", "e75f1c1f3b6ecc3ee3dc3ac05ad567ca": "\\frac{d h}{d r} = \\frac{m \\omega^2 r}{m g}", "e75f321256cbf0308c0fc5f3652436f6": "s_p=\\sum_{i=1}^m I_{[0,p]}(u_i)", "e75f4f0db24ea9bb19e37770290db53d": "\\Psi_n>0", "e75fd4229fbd52d93d646b89f1dc4617": "S_{\\sigma,\\varepsilon} := \\min\\{ \\sigma(k) | k \\in I_{\\sigma,\\varepsilon} \\} ", "e760b10d2ed85fe9c955be4a13deb9e7": "(z-\\alpha_1)(z-\\alpha_2)(z-\\beta_1)(z-\\beta_2) = 0.\\,", "e760b69e1bc752946c54be0ead544183": "\\textbf{K}_k \\textbf{S}_k \\textbf{K}_k^T = \\textbf{P}_{k\\mid k-1} \\textbf{H}_k^T \\textbf{K}_k^T", "e760bca75296c9925046df5a5f63a70c": "n\\geq2", "e760ea09dcb1c4bb3baa9dc99f30f0cb": "k \\rightarrow \\infty ", "e7613dce7f9c8738e072cb721789bf25": "\n\\begin{array}{l}\n a = 2(22)=44 \\\\ \n b =117 \\\\ \n c = (143 - 18) = (26 + 99)=125 \\\\ \n \\\\ \n \\text{radii} \\to (r_1 = 18,\\quad r_2 = 26, \\quad r_3 = 99 ,\\quad r_4 = 143) \\\\ \n A = (18)(143)=2574 \\\\ \n P = (18 + 26 + 99 + 143) =286\n \\end{array}\n", "e761556aa024ce51210ffefabef20e85": "\\overline{T}_{in,i}", "e761889c2dee2885a80fc6fe80cd11b8": "\\hat{p}_{x^n}", "e76192472430e2e874ed7d07ccff1b87": "(E,\\,\\lambda)", "e761bd719c1d7316bd1fbb73429b19b7": "f^{-1}\\mathcal{G}(U) = \\mathcal{G}(f(U))", "e761c21db88448929a201bac375f79f4": "(0,\\lambda)\\,", "e761c7eed6f320a43ac3617d0e270298": "F_0/F_1", "e761cc696e6ebdf3b09ac2ce33b8730f": "\\left|\\psi_{AB}\\right\\rang = \\left|\\uparrow_{\\mathbf{n}}\\right\\rang_A \\left|\\downarrow_{\\mathbf{n}}\\right\\rang_B ", "e761f0474f3ad9d975f685c806824a32": "\\vec{A}=\\frac{1}{2}\\oint{\\vec{r}\\times}d\\vec{l}", "e7622cc4926843d8709259ace2042e9d": "\\left[\\begin{array}{c}{x}\\\\ {\\mu}\\end{array}\\right] = \\left[\\begin{array}{cc} {Q} & {A}_{eq}^{T}\\\\ -{A}_{eq} & {0}\\end{array}\\right]^{-1} \\left[\\begin{array}{c} {A}^{T}{\\lambda}-{c}\\\\-{b}_{eq}\\end{array}\\right] \\,", "e7622cd786b6d3b9df8d8840050a41ce": "n_E=A", "e762752117a82c40d306145c10d479e1": " c_1 = \\frac{1}{\\sqrt{2}},", "e7629831d99146cdc922d10f0bc59ce7": "F_2[T]\\,", "e762f114e6c633b349ab6320470e1411": "\\displaystyle{\\Delta=\\Delta_1\\times \\cdots\\times\\Delta_s}", "e762fd015bad46dbdfcba4d6565a02d4": "\\ \\mathbf u(\\mathbf X,t) = \\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad \\text{or}\\qquad u_i = x_i - \\delta_{iJ}X_J", "e7632f5e26730db6d28ce36a289f6519": "\\text{PT}^*M = \\text{T}^*M /\\! \\sim \\ \\text{ where, for } \\omega_i \\in \\text{T}^*_pM, \\ \\ \\omega_1 \\sim \\omega_2 \\ \\iff \\ \\exists \\ \\lambda \\neq 0 \\ : \\ \\omega_1 = \\lambda\\omega_2.", "e76355c819584ad54446f8e1f2b7e02e": " \\left| a\\frac{\\Delta t}{\\Delta x} \\right| \\leq 1. ", "e7635707b61fa129265dc313789c0a3c": "|V_1| > 4^{2k}", "e76386137b258e268d35707e038057cf": "t+\\tau \\gg \\frac{A^2}{4B} \\Rightarrow X_o(t) = \\sqrt{B(t+\\tau)}", "e7638e4224671ea7071d7342dd7417f3": "\\mathbf{V}\\,\\!", "e76404df23899f842fba6dc0f9fca303": " \\min \\, \\{\\, d(a,b) : a \\in A,\\, b \\in B \\,\\}. ", "e76486f666bb30c4d7cdd5fe49bec7ba": "\\bar{\\theta}_n=\\int_{-\\infty}^\\infty t \\, dH_n(t)", "e764ab4abece35897b81529994a623d8": "\\mathrm{j}_{+}", "e764c5e2cf287826f8e682289ce08752": "h=\\frac{38\\,MPa}{16\\,kNm^{-3}}", "e76530d5bb91f46ba9fe3f1725c65d41": "I=eN/t", "e76531eb7f5ea2d39390383bfe617e7d": " (\\partial A)_S=-(\\partial S)_A=\\frac{PC_P}{T}\\left(\\frac{\\partial V}{\\partial P}\\right)_T+P\\left(\\frac{\\partial V}{\\partial T}\\right)_P^2+S\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "e7655e7a13467be9d154227f771353f9": "\\, \\delta V ", "e76569e17a65d939568de208b5f07ae8": "\\mathbf{R}\\cdot(\\mathbf{k}-\\mathbf{k^\\prime})=2\\pi m", "e76587eb8c894fd7f86cfc0a337e5f81": "\\xi(s) = \\xi(1 - s).\\!", "e7658c15b95a698ec1ba8bfbd73990ff": "e^{\\bar\\lambda M}T = T e^{\\bar\\lambda N}.", "e76603288bda51e2fc26dead2989d943": "\\mbox{End}\\,G \\to \\mbox{End}\\, G/H", "e766089d4edcce219854690e958b25a9": "mol Fe_2O_3 = \\frac{grams Fe_2O_3}{g/mol Fe_2O_3}\\,", "e7660beef331870d98b7bfeab9471a3e": "\\cos n\\theta +i\\sin n\\theta=(\\cos(\\theta)+i\\sin(\\theta))^n \\,", "e76612618aa77409d197915e835bd835": "\\prod_{j=1}^\\infty\\left({1 \\over 1-\\alpha z^j}\\right)^{M(\\beta,j)}=\\prod_{j=1}^\\infty\\left({1 \\over 1-\\beta z^j}\\right)^{M(\\alpha,j)}", "e7661b81b8a7476e6b0abc19b9a4836c": "\\epsilon_F = \\frac{\\hbar^2 k_F^2}{2m}", "e7665177344eae8b0c9ff114f5fbf792": "2m+2k+\\frac{S(S-1)}{4} +\\frac{1}{2^{S-1}} - 9", "e7665cb8a651713858bb442c7c7a9f07": "\\alpha_0=\\gamma^{-1} \\alpha", "e76684e1ac3369d3941c6483b33191f9": "\\displaystyle{H(e^{i\\theta})=e^{ih(\\theta)},\\,\\,\\, h(\\theta+2\\pi)=h(\\theta)+2\\pi,}", "e766a53c4432ea477b8a28f426335222": "x^3+a_{15}x^2-4a_{15}b_{2}^2=0", "e766b88c626d75ac6026be6821573e79": " G = 2 \\cdot N \\cdot \\sum_{ij}{\\pi_{ij} \\left( \\ln(\\pi_{ij})-\\ln(\\pi_{i.})-\\ln(\\pi_{.j}) \\right)} ,", "e766c0ba5fb6641de27efc099a51a29a": "R^\\rho{}_{\\sigma\\mu\\nu} = dx^\\rho(R(\\partial_{\\mu},\\partial_{\\nu})\\partial_{\\sigma})", "e7670ad27cf6aebaca05ca9298a21210": " x(t) = E^T \\mbox{Diag} (\\exp t e) E a ", "e76712b5f3eaa1f6dcecbcda68def1b3": "\\frac{\\Gamma \\vdash a : X \\circ Y \\qquad \\Delta, b : X, c : Y, \\Delta' \\vdash d : Z}{\\Delta, \\Gamma, \\Delta' \\vdash d[b := a, c := \\epsilon] : Z}[\\circ E_{weak}]", "e767237edf30976578bfc14f2577dea5": " \\mathbf{L} = \\left ( \\mathbf{r} - \\mathbf{r}_0 \\right ) \\times \\mathbf{p} \\,\\!", "e7675f8f3f8b97c429cda2fb987dced7": " e^{- {(\\partial_\\mu A_\\mu)^2\\over 2}}.", "e7678ba7218ec766c4e8b0e152453414": "~y(a)~", "e767a288f983dc05755a99a0f7badda3": "Tr.", "e767cb44e48e19001e1f73faaca68d7b": "\ns(t) = \\cos\\left[2 \\pi f_c t + b_k(t) \\frac{\\pi t}{2 T} + \\phi_k\\right]\n", "e767dd1ee7a038928332d4e009f1b6cc": "\\mathit{\\Sigma\\Tau\\Upsilon\\Phi\\Chi\\Psi\\Omega} \\!", "e768226f79a06c9fdf86c60fe359e8ad": " ( \\mathbf{r} \\times \\mathbf{v} )_z < 0", "e76864aab4199433e6e22f1a25cd4983": "\\Phi(x) = \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\operatorname{d}\\!x", "e768a882be4ab79fb23ec0b20feff9f9": " (g_{k\\lambda}\nv_{0}-g_{0k}v_{\\lambda})\\frac{dv^{k}}{d\\mu}+\\left[\\frac{1}{2v_{0}}\\frac{\\partial\ng_{ij}}{\\partial x^{0}}(g_{00}v^{0}v_{\\lambda}+\ng_{k\\lambda}v^{k}v_{0})-\\frac{1}{2}\\frac{\\partial g_{ij}}{\\partial\nx^{\\lambda}}v_{0} +\\frac{\\partial g_{i\\lambda}}{\\partial\nx^{j}}v_0- \\frac{\\partial g_{0i}}{\\partial\nx^{j}}v_{\\lambda}\\right]v^i v^j=0.", "e768ba5e7b639f314d028a7bca2656e8": "K(x)", "e768c527001596853c408d2befaa4a4f": "E_u = \\frac{Q_{ms}(u)\\;M_u(u)}{Q_{ms}(f)\\;M_u(f)}", "e768c8f1e833a3e79f4fb5ce7deeef60": "\\mathrm{E}(F) = 0", "e768cc3b166411a211c63c5f01df22fb": "\\mathbf X=\\chi^{-1}(\\mathbf x, t)", "e768d8ac4174d26ef1e16da1642585d7": " 1.8 + 0.3(H_s-3) ", "e769102120b1f5cf0647d243920b8bf5": "\n\\begin{matrix}\nX_B=c_x+(1+s\\times10^{-6})\\cdot (X_A-r_z\\cdot Y_A+r_y\\cdot Z_A)\\\\\nY_B=c_y+(1+s\\times10^{-6})\\cdot (r_z\\cdot X_A+Y_A-r_x\\cdot Z_A)\\\\\nZ_B=c_z+(1+s\\times10^{-6})\\cdot (-r_y\\cdot X_A+r_x\\cdot Y_A+Z_A).\\\\\n\\end{matrix}\n", "e76930e4c1ddaae206ed41bbc4a016ba": "f(k;\\rho) \\approx [\\mbox{constant}]/k^{\\rho+1}", "e769f8a1785898acd0787099d46ff985": "Q_3 = \\{O_{4},O_{6},O_{8}\\}", "e769fedbe15e5cee8eabfa0a3e5aec30": "\\frac{\\mbox{365 Days}}{\\mbox{Inventory Turnover}}", "e76a146420465129397f225e8fc3f0a5": " \\int_{-\\pi}^\\pi e^{-i (j-k) t} \\, dt = 0. ", "e76a23a0e3456647f6d67410cb8a967e": "\n\\begin{bmatrix}\n a & b \\\\\n c & d\n\\end{bmatrix}\n = \n\\begin{bmatrix}\n 0 & 1 \\\\\n -1 & 0\n\\end{bmatrix}.\n", "e76a468084478d0df168edd638a66ef0": "{J^{\\nu}}_{\\text{bound}}=\\partial_{\\mu} \\mathcal{M}^{\\mu \\nu} \\,.", "e76a49ffeba0f4d2e8bdefe42c17b3ed": "\\Sigma(R) = C(R)/M(R)", "e76a7a483163891e414de98556dd1de9": "\\vec{p}=\\vec{q}+\\Delta \\vec{p}\\!", "e76abb14fed9437fb539aac39ad7f257": "\\sum_{i=0}^{P(k)}|p_{k,R_{k,i+1}}^{\\mathcal M}-p_{k,R_{k,i}}^{\\mathcal M}|\\geq |p^{\\mathcal M}_{k,R_{k,P(k)}}-p^{\\mathcal M}_{k,R_{k,0}}|=|p_{k,S}^{\\mathcal M}-p_{k,U}^{\\mathcal M}|\\geq\\frac{1}{Q(k)}", "e76acbb94484e5814cb336f8a2275a0a": "\\displaystyle{H=i(2P-I).}", "e76b29bfd74d1acedf0e7b25631c0731": "f(m_1(x_1),\\cdot,m_N(x_N))", "e76b2dfd2ef1515bca6a895a5c1fd602": " \\begin{align}\n\\text{mean}(Y) &= \\text{mean} (X)(c-a) + a = \\left(\\frac{\\alpha}{\\alpha+\\beta}\\right)(c-a) + a = \\frac{\\alpha c+ \\beta a}{\\alpha+\\beta} \\\\\n\\text{mode}(Y) &=\\text{mode}(X)(c-a) + a = \\left(\\frac{\\alpha - 1}{\\alpha+\\beta - 2}\\right)(c-a) + a = \\frac{(\\alpha-1) c+(\\beta-1) a}{\\alpha+\\beta-2}\\ ,\\qquad \\text{ if } \\alpha, \\beta>1 \\\\\n\\text{median}(Y) &= \\text{median}(X)(c-a) + a = \\left (I_{\\frac{1}{2}}^{[-1]}(\\alpha,\\beta) \\right )(c-a)+a \\\\\nG_Y &= G_X(c-a) + a = \\left (e^{\\psi(\\alpha) - \\psi(\\alpha + \\beta)} \\right )(c-a)+a \\\\\nH_Y &= H_X(c-a) + a = \\left (\\frac{\\alpha - 1}{\\alpha + \\beta - 1} \\right)(c-a)+a, \\,\\qquad \\text{if } \\alpha, \\beta > 0 \n\\end{align}", "e76b4150894e2dfd39a1d6669994ab9c": "\\phi(X,n,m)", "e76b5ae8262688ae6742a1366530fcb7": " N = N_A + N_B + N_C ", "e76b5d2529659666842c96147d5eb137": "Q \\propto I^2 R ", "e76b8dfb01f01d89bacb84d674e54f7a": "\nFV_{due}(0.09/12,7\\times 12,$100) = $100 \\times \\ddot{a}_{\\overline{84}|0.0075}\n= $11,730.01.\n", "e76bc4c8aefbd5db8f99312119b736c2": "\\Delta_*", "e76c40f88bcfc8e7e9aec348eb86893e": "C^\\alpha(\\Omega)", "e76c487f3aa4b238683499c9b2615d2a": " \\sum_j \\frac{f_j(x)}{g_j(x)} ", "e76c5fc8363dcdd518f54bb6edebf0f4": "\nd_G(p,q):=\n\\inf \\{ \\ell (\\gamma) \\mid \\text{ all piecewise }C^1\\text{ curves }\\gamma\\text{ such that }\\gamma(0)=p\\text{ and }\\gamma(1)=q \\} .\n", "e76c88bed2ba7e12a64138766a66bbb7": "\\eta_{compound} = \\eta_1 \\eta_2 \\ldots \\eta_n.\\,", "e76cf662c4b094f4ff1f42cd27c75a47": "10\\uparrow\\uparrow\\uparrow 3", "e76cf8a691823107e27238a4976cd98e": "Du", "e76d08741edf77beb5d37fbe72d0ec93": "\\tfrac{1}{24}(q^2 - 13q + 24)", "e76d1691e679b0f720c5caa91fb085b8": "3.5\\leq\\mathcal{A}(D_{1})\\leq3.7\\,\\!", "e76da3b3c6f4f005c0e7b6645f233274": "\\scriptstyle\\Delta uv", "e76da4499e7a729c8e32347079f4b705": "\\mathbb{E}\\left[m\\right]=\\mu", "e76ddbf40850d1d137813ea804cd24f1": "R^2 = 1 - \\frac{\\color{blue}{SS_\\text{res}}}{\\color{red}{SS_\\text{tot}}}", "e76dfd25626445084be7eb87ab946437": "\\int_M (\\operatorname{div} X)\\omega = \\int_{\\partial M} X\\;\\lrcorner\\;\\omega,", "e76e5c4a495d0c0e27b7179018b42387": "D(A)\\subset H", "e76e8b706ac1f60c1d396e41eefdad2a": "g\\colon D^2\\to M \\, ", "e76e96322154ff7a62ddaecffca38a6c": "x \\isin X", "e76eba4d9d09b34e53ce4f56d5d21d67": "A(x,y)=(x+1,y)", "e76eda6aa722e97648162a2e713a4707": " S_x = S_{yz} =\\int_{\\partial \\mathcal{V}} [(y - y_\\text{com}) T^{0z} - (z - z_\\text{com}) T^{0y} ]dxdydz ", "e76f1b2f8b3a271ab0056bb15dd96ca1": "p(z) = \\cosh(z)- 1", "e76f4afe6a09f8d0242ed5fb84192d5f": "\\mu(t,T)=\\xi\\left(t,T\\right)\\left(\\int_t^T\\xi\\left(t,s\\right)\\,ds-\\theta(t)\\right).", "e76f8aea24f1b5323c36fd6ef5ee205c": "3/\\pi", "e76fe098028e82ce966dc176db957b42": "S_1, S_2, \\ldots", "e76ff7dd18039ebceec5e68fc6ecb09e": "(u_1 = x^2-4x+3, v_1 = -4x+12)", "e7701675d1288446516e596be72a818b": "p(x)=a_n x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\\ldots+a_1 x+a_0", "e770233574dce7062565085065acaf48": " r_{\\mathit l \\mathit l^{\\prime}} ", "e77025c76bdc92cbfebfd56ac29a2032": "\\mu(\\sigma) \\leq \\mu(\\tau)", "e77055084665fae894d8fcd0d5924f91": "K_W=[H^+][OH^-]\\,", "e770b455abf79500f74b020321f91e74": " d(\\mathbf{L}'_{1}, \\mathbf{x})^{2} + d(\\mathbf{L}'_{2}, \\mathbf{x})^{2} ", "e770c18a615727296b0f9ad79895617a": "w_\\min = q", "e770cb56ce690f6166d1745a46b88964": "P(r_1, r_2, \\ldots, r_N) = 0.\\ ", "e7710ec5d5880a8b7114bc4471b84da7": "D\\;", "e771708de5d6fad0b30eb0e74e01006f": "p'(b) = a/4,", "e7717c2f356eaadf1973fcf23ff027bc": "y = \\frac{Y}{Z^2}", "e7717df010cbe20186572a5c48193adc": "x^2 = x + 1,\\, x^n = x^{n-1} + x^{n-2},\\,", "e771cf3d644c4c77f297d836506690bb": "b \\wedge a = b' ", "e772a17e0a5c1f05509ea46beec73c8a": "=\\frac{1}{2}\\operatorname{Tr}\\left[\\left(\\frac{\\delta J_k}{\\delta \\phi}\\right)^{-1}\\cdot\\frac{d}{dk}R_k\\right]=\\frac{1}{2}\\operatorname{Tr}\\left[\\left(\\frac{\\delta^2 \\Gamma_k}{\\delta \\phi \\delta \\phi}+R_k\\right)^{-1}\\cdot\\frac{d}{dk}R_k\\right]", "e772b19ee23970cb00ab8df55c62be53": "\\Delta_0 = b^2-3 a c", "e772b7ba6690ab1d9ced2cff1639f701": "(-\\gamma, \\gamma, 0)", "e772e968fd640a18a79ba41d828969af": "\n\\begin{matrix}\nA & = & \\sqrt{\\frac{1}{2}(I_p+|L|)} \\\\\nB & = & \\sqrt{\\frac{1}{2}(I_p-|L|)} \\\\\n\\theta & = & \\frac{1}{2}\\arg(L)\\\\\nh & = & \\sgn(V). \\\\\n\\end{matrix}\n", "e77315219b53f73285b0f26ec22ea9df": "(K_{2k+1}(M;\\mathbb{Z}_2),\\mu)", "e7731832db8daba443c9dc754a0bf663": "\\scriptstyle A <_\\alpha B <_\\alpha C", "e7732bdaaaddf168d36857997069a7af": "f_0 = \\frac{n_0}{n}", "e773349454669bb38bf75f7f8f75c205": "\\frac{\\partial f}{\\partial\\bar{z}} = \\mu(z)\\frac{\\partial f}{\\partial z},", "e773573e6040cdec30ced1daf7944e21": "t ::= f(t_1,t_2,\\dots,t_n)", "e773f2b3de9ed8a75926ef391d3a4c9d": "u(x_1^0,\\dots,x_n^0) = z^0", "e7740364addfbd1548b479ddf20bbd52": " X_{t-1}", "e77436a043ca411d623303068445a614": "y(x)=\\begin{pmatrix}y_1(x)\\\\y_2(x)\\end{pmatrix}", "e774778331cdee17d0decc2fcc549d84": " \\frac {dV_{fb}(t)}{dt} = f \\frac {C_s}{C_{fb}} \\cdot V_s(t)\\, ", "e775277859eb5574500f989628a4178b": "MU\\to BP", "e775696d4a97c3f17581553185a26294": "\n\\begin{bmatrix} A^{-1} & -A^{-1}\\vec{b} \\ \\\\ 0,\\ldots,0 & 1 \\end{bmatrix}\n", "e77572d73f517baf6c2fc65dda3a4f02": "f_{X_{(k)}}(x) =\\frac{n!}{(k-1)!(n-k)!}[F_X(x)]^{k-1}[1-F_X(x)]^{n-k} f_X(x)", "e7757db07b1fd7602ada57faa14fb944": "\\int_a^b [f(x) - g(x)]\\;dx", "e77628ed87707b7ddc1b97ed98628669": "q_j, j = 1,\\ldots,m", "e776406ddb4a2c3db2ecae2cad7dae52": " \\log( \\frac{k_X}{k_0})= \\rho(\\sigma + r(\\sigma^+ - \\sigma))", "e77650894c6feb4eadc0f305fdcff2e0": "\\sqrt[3]{31} = 3.1413^+", "e776584cbb89f608e3093f727a570411": "\\ \\displaystyle D", "e7765c462b7e752003d1defc24c4b5b6": "R_b", "e776b20f19270a1edae1706f44a48abb": " X^* = - \\tfrac{ N \\ln \\left[ 1 - \\tfrac{X}{N} \\right] } { k} ", "e776d0d85f69acc75744971436936dd8": "\nE_f=E_i-\\hbar\\omega,\n", "e776fe11d18965f7718d79bcaa0115ff": "\\Phi_A", "e77710440086590a6c07100c19e6017c": "M_1,M_2,M_3,M_4", "e77742b5de027e6997f016f10aa98360": "d \\ge 0", "e7774a12e248c9961e5f4a4eb04aca66": "\\|\\mathbf{M}\\| = \\langle \\mathbf{M}, \\mathbf{M}\\rangle^\\frac{1}{2} = \\operatorname{trace}(\\mathbf{M}^*\\mathbf{M})^\\frac{1}{2}", "e777e4b648a190796d283447b5c6a78c": "\\int \\cos^n x dx , \\,\\!", "e77816da1a937908b379b1096a28db76": "w'''=0", "e778429d8769714354b1994984a23fe5": "\\epsilon >0", "e778498e9dceb9f08bb9f5bd3d3aff34": "\\bar{n}(x)=n_0 \\exp \\left[-\\frac{e}{\\kappa T} \\Phi_{\\text{P}} (x)\\right]", "e7784d2a98e4d709eb2fc95c943a41ac": "C(Z(x_1),Z(x_2)) = C(Z(x_i),Z(x_i+\\mathbf{h})) = C(\\mathbf{h})", "e778584dae4002bb4cd43f6427f15875": "\\displaystyle{[\\sum c_\\alpha E_\\alpha + \\overline{c_\\alpha}E_{-\\alpha}, E_{\\psi_i} + E_{-\\psi_i}]=0}", "e778e6f92e2a30a94bacd694966124f0": "A\\frac{dp}{dt}-E\\frac{dr}{dt}=L_\\beta \\beta - L_r \\frac{d\\beta}{dt} + L_p p", "e7796d6487e6f51a16ae3aa1aa7ec212": "\n\\begin{align}\ndu \\wedge dv & {} = 0 + ps\\ dx \\wedge dy + qr\\ dy \\wedge dx + 0 \\\\\n& {} = (ps - qr)\\ dx \\wedge dy = (\\det g)\\ dx \\wedge dy.\n\\end{align}", "e77994eb898b194aba3e7dc455d965f3": "f(r | H=h, T=t) = \\frac{1}{\\mathrm{B}(h+1,t+1)} \\; r^h\\,(1-r)^t. \\!", "e7799f62813fc23e7f47f741e9f2fad8": "x' = s_x \\cdot x", "e779f14588ff7bb7e991b9ca095f9cac": "(\\Delta f)(x)=f(x+1)-f(x)\\,", "e77a5e539e2d8f28add78ae5799a2e96": "\\mu_t", "e77abe38fd8393e012b3b4e94f1ffdb5": "E = \\sqrt{|\\vec{q}|^2 + m^2} \\,", "e77b048d2d93d0293638072e50fde851": "\\alpha_\\rho(x,y)", "e77b15c728a4660151aa6ad3250f0952": "\\land, \\lor, \\to, \\leftrightarrow", "e77b2872f03a00e43fb28987b59384bc": " \\frac{1}{2} \\left( \\nabla \\times \\omega \\right), ", "e77b465f47fb8b5b8db63fa4504d29f7": " D_m ^s ", "e77b7397c47bc4f0cbd5d8f605134a3d": "\\nabla \\times V_h", "e77b7985be6cb4c974e60bf2182fc527": "\\frac{N}{k}", "e77b7e0df28d19c995934eb939d927d6": "M v_i = \\lambda _i v_i", "e77b8e3b30259fee2759a88809d3a661": "u_{1}(j^{1}_{p}\\sigma) \\,", "e77c117b9c25b6f52db0d9d62804ff57": "p(m_i\\mid z_{1:t}, x_{1:t})", "e77c617a9988045b5c3006ebc3e1d6e8": " z = 2 \\left( \\sqrt{W_+(p \\wedge c) W_-(p \\wedge c)} + \\sqrt{W_+(p \\wedge \\neg c) W_-(p \\wedge \\neg c)} \\right) +W(\\neg p) ", "e77c7d217a7bf00eb55b43d444ee081c": "\\mathbf{q} = c_1 \\mathbf{v_1} + c_2 e^{\\lambda_2 t} \\mathbf{v_2} + \\cdots + c_n e^{\\lambda_n t} \\mathbf{v_n}", "e77cb8a005e0bfbd6c7537f9cae7892d": "= \\int_{-\\infty}^{\\infty}{\\left|h(t) (1 \\cdot e)^{-j \\omega t} \\right| dt}", "e77d2f88cc688958e38ebf565bd97a1d": "\\begin{pmatrix}\n 1 & -1 \\\\\n 1 & 1 \n\\end{pmatrix}", "e77dde7007bcb1a43e32b0b6a337aa74": "S=\\{\\,2\\cdot x\\mid x \\in \\mathbb{N},\\ x^2>3\\,\\}", "e77de82762e41dfc711a7f2f45e3aeb3": "V_\\mathrm{A}", "e77e05e22b3ba2cdda66e129bf88a296": "x^y = 2^{\\,y\\ \\log_2\\, x}", "e77e18aa1efbc1f91fdfa7568d6f3bca": "p_1^2+p_2^2+p_3^2=n^2", "e77e1d8fe21e01aeb89a7e531e7362c7": "\\frac{\\partial }{{\\partial \\Lambda }}{{V}_\\Lambda }(\\psi ) = -{\\dot \\Delta _{{G_{0,\\Lambda }}}}{{V}_\\Lambda }(\\psi ) + \\Delta _{{{\\dot G}_{0,\\Lambda }}}^{12}\\mathcal {V}_\\Lambda ^{(1)}\\mathcal {V}_\\Lambda ^{(2)}", "e77e279b80c69b7cb55e6261fa00d3a7": " Q = {1 \\over F_\\mathrm b} = \\frac {\\omega_0}{\\Delta \\omega} ", "e77e4a30853e4db6886cf9378f09da0e": "m=\\frac{1}{n} \\sum_{i=1}^{n} X_i \\,", "e77ea85967eaa8ca2ba95d7d79c2aef1": "TV(x) = \\sum_{i=1}^\\infty |x_{i+1}-x_i|.", "e77f6965683c11a18b560d781df1caf4": " \\langle \\mathbf{X}, \\mathbf{Y} \\rangle = \\sum_{k=1}^p X_k Y_k ", "e780192ee1f3747772125f9ea037a538": "\\{p,q,r\\}", "e78071b7320c5247518be17a6a92f2b8": "4f, 6f, 8f,", "e78095fc8860b6553c5d9902dad96372": "\\textbf{P}_{k\\mid k} = \\textrm{cov}(\\textbf{x}_{k} - (\\hat{\\textbf{x}}_{k\\mid k-1} + \\textbf{K}_k\\tilde{\\textbf{y}}_{k}))", "e780d261259a4c13b4267da3aef4b78d": "a\\;", "e7814d6339e6d6b42cfd0c1a1acc030c": "S^{n-2}", "e781afee0003ea16fc5b2cd9c98ee09b": "\n \\tilde{p} \\pm z\n \\sqrt{\\frac{1}{\\tilde{n}}\\tilde{p}\\left(1 - \\tilde{p} \\right)}\n", "e781c5dd05a3bce135e54fae88f93975": "g(\\alpha) \\ge 0", "e781f504f21134fdaa0821509f4718a4": "\\int dV |\\psi|^2=N", "e781fdc3e3eaab12b2de3db58258b70f": "N = {6! \\over {2!\\,4!}}=15", "e782698555dbdd8ed3c9a45cfeee0551": " \\nabla_X(fv)=df(X)v+f \\nabla_X v", "e7826ccb3753b7407e28ac795a81e9fb": "E_{\\mathrm{sen}}", "e782c1445dec7576465989be1b986b80": "\\displaystyle f_t(\\zeta(t)) = \\gamma(t)", "e782f7cac2c4dac94a9216e8bf7ed16f": "4p^3+27q^2=0 \\text{ and } p\\ne 0", "e78311b1a9689580f6cbe82397266fd0": "\n \\oint_{\\mathbf{X}_A}^{\\mathbf{X}_B} (\\boldsymbol{\\nabla} \\times \\boldsymbol{\\epsilon})\\cdot d\\mathbf{X} = \\int_{\\Omega_{AB}} \\mathbf{n}\\cdot(\\boldsymbol{\\nabla} \\times \\boldsymbol{\\nabla}\\times\\boldsymbol{\\epsilon})~da\n = \\boldsymbol{0}\n", "e78338b33d66e39c7695e43f12166d37": "\\overbrace{\\frac{1}{P}\\cdot S\\left(\\frac{k}{P}\\right)}^{S[k]}\\ \\stackrel{\\text{def}}{=}\\ \\frac{1}{P} \\int_{-\\infty}^{\\infty} s(t)\\ e^{-i 2\\pi \\frac{k}{P} t}\\,dt \\equiv \\frac{1}{P} \\int_P s_P(t)\\ e^{-i 2\\pi \\frac{k}{P} t} dt\\,", "e7839fd6274b7367da6897131c4c4d0b": "H_\\alpha^{(2)}(z)\\sim\\sqrt{\\frac{2}{\\pi z}}\\exp\\left(-i\\left(z-\\frac{\\alpha\\pi}{2}-\\frac{\\pi}{4}\\right)\\right)\\text{ for }-2\\pi<\\arg z<\\pi", "e783ba60448d372570d8fad9b2aa72c4": "q^{n-k} = 2^{4-2} = 2^{2} = 4", "e783c8ab3ac3dca55e56df510020fb76": "(f \\cdot \\vec g)' = f\\;'\\cdot \\vec g + f \\cdot \\vec g\\;' \\,", "e78444d50b5e5fabc80a378e0bba9f49": "w_2 \\,", "e7845bbfd4e12ffc2755ce4a927b0fd8": "V_n(P,Q)=V_n(2S,S^2)=2S^n\\,", "e7846cabaa8ec22b90dc7a9147b6cf7e": " \\mathbb{F}_{q^n} \\to \\mathbb{F}_q^n ", "e784adc3316b20e1f9209b9c73d37ce3": "\n\\Psi(\\mathbf{R}, \\mathbf{r}) = \\sum_{k=1}^K \\chi_k(\\mathbf{r};\\mathbf{R}) \\phi_k(\\mathbf{R}),\n", "e784be1875512183876e765318d8169a": " f(x) \\leq b", "e784ccaad5e210cc040a3fb9e6463009": "\\Delta = d \\delta + \\delta d", "e785266abfc2f8603b22bd8740caa89a": "\\begin{align}\n&\\underset{u}{\\operatorname{maximize}}& & \\inf_x \\left(f(x) + \\sum_{j=1}^m u_j g_j(x)\\right) \\\\\n&\\operatorname{subject\\;to}\n& &u_i \\geq 0, \\quad i = 1,\\dots,m\n\\end{align}", "e7856afa58e795795801c3606db04bab": "\\,f^{-1}", "e785978633e6467c87e8c2293a961ac3": "~G~", "e785e4d516b2e574f0506efaacaea579": "\\epsilon^{i}", "e785e7f102343ed32fe74f70ab7d120b": "\\frac 1{\\ln 2}\\ \\frac 1{1+x}", "e785f736268eb66f438cfb5ac1aaf022": "\\prod_{p} P(p, s)\\,", "e78600d1d27eaf2340d4f1adf607d1d5": "E(x) = e\\,x^i,", "e786560a5e5c52264316483db5141acd": "{\\Delta c_P = T \\cdot \\Delta \\left( {\\left( {{{\\partial v} \\over {\\partial T}}} \\right)_P } \\right) \\cdot {{dP} \\over {dT}}}", "e787d875520d1565c7caf4cba9e40882": " u \\delta y + v \\delta x\\ = \\left( u + \\frac{\\partial u}{\\partial x}\\delta x \\right) \\delta y + \\left( v + \\frac{\\partial v}{\\partial y}\\delta y \\right) \\delta x\\,", "e787f90dc36e777ac349dac93ea258a0": "(m + 1)(n + 1) - 1 = mn + m + n.\\ ", "e7881e17cb087622e4c527598ebc4e7f": "\\tan\\alpha = \\frac{a\\sin\\gamma}{b-a\\cos\\gamma}.", "e7885ff05f8286b63552808667cef0ed": "a, 2a, 3a, \\dots, \\frac{p-1}{2}a", "e78904643254c61f61f180839704db4c": "\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\frac{\\partial u(t, x)}{\\partial t} \\varphi (t, x) \\, \\mathrm{d} t \\mathrm{d} x +\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\frac{\\partial u(t, x)}{\\partial x} \\varphi(t,x) \\, \\mathrm{d}t \\mathrm{d} x =0. ", "e7892d285a93149c39a5c7f99a0f91ba": "n \\geq 5", "e7894b3c7d106793d88376c60e54a699": "M\\{B\\} > 0", "e789782e2a009b771e750bc2d8df1dd2": "D = \\sqrt{\\frac{\\pi}{2}} \\left(1 + \\operatorname{erf}\\left(\\frac{\\left| \\alpha \\right|}{\\sqrt 2}\\right)\\right)", "e7899fd1a236417401f9b8f993fc8cd9": " x^2 =n .", "e789dfc5158903f989080a280137c7b7": "\n2 \\pi \\partial_t \\hat{u}_k\n=\n- i \\pi k \\sum_{p+q=k} \\hat{u}_p \\hat{u}_q \n- 2\\pi\\rho{}k^2\\hat{u}_k\n+ 2 \\pi \\hat{f}_k\n\\quad k\\in\\left\\{ -N/2,\\dots,N/2-1 \\right\\}, \\forall t>0.\n", "e789f32d863e247a79a75eb081ac72c2": "\ny\\rightarrow \ny^5 - 10 y^3 (z^2 + A x z + x^2) + 5 y (z^4 + B z^3 x + C z^2 x^2 + B z x^3 + x^4) + D y^2 z x (z+x)+ y_0\n", "e78a00935f908173a40d67d845d814e7": " yxyx = x^2y^2 ", "e78a28c101a53d0c84e88d54844edce6": "\n \\nabla_{\\!\\theta}\\, \\hat\\ell(\\hat\\theta|x) = \\frac1n \\sum_{i=1}^n \\nabla_{\\!\\theta}\\ln f(x_i|\\hat\\theta) = 0.\n ", "e78a6105740516a536eeb8d195dd1f2c": "k:\\mathbb R^n\\times \\mathbb R^n\\to \\mathbb R", "e78a7bb1718206854a529276cb821d34": "(g, h)\\mapsto h", "e78aaa5c531cf8924890951ca9a2f91c": "Z(\\beta) = \\sum_{x_i} \\exp \\left(-\\beta H(x_1,x_2,\\dots) \\right)", "e78aaef1a37bb3fabb847ff7c3d23f42": "E = 0.9465\\left( \\frac{m f^2_f} {b} \\right)\\left( \\frac{L^3} {t^3} \\right)T", "e78acdb6d40eaaab534ed0648ed25b2c": "\n\\tilde{A}_{1}^{\\mu } =\\big((\\varepsilon _{1}-E_{1})\\big )\\hat{P}^{\\mu\n}+(1-G)p_{\\perp }^{\\mu }-\\frac{i}{2}\\partial G\\cdot \\gamma _{2}\\gamma\n_{2}^{\\mu },", "e78ad290474ab4a69d927579337661db": "(X_{t-1}, u_t, z_t)", "e78aff08aee146926860b76a496d1c43": " \\sqrt{-1}\\omega = \\sum_i \\alpha_i dz_i\\wedge d\\bar z_i,", "e78b1a4ef06933eb5bb12ba9304d0831": "C_S=N_x\\left(\\log_2 N_x+1\\right)\\,", "e78b2bffd83ed6aca707d32d97aa107b": "\\lambda_1 \\approx 0", "e78b2e9efdcbfb75f7b77ac170e08e4c": "\\sigma_\\bar{C} = g(\\bar{C}, \\cdot) = 18.7% \\,", "e78b3c4bb909da223d1c2717b7cf11e2": "= \\sum_{n=-\\infty}^{\\infty} x(nT) \\left( \\mathrm{rect} \\left(\\frac{t - nT}{T} - \\frac{1}{2} \\right) - \\mathrm{rect} \\left(\\frac{t - nT}{T} - \\frac{3}{2} \\right) + \\mathrm{tri} \\left(\\frac{t - nT}{T} - 1 \\right) \\right) \\ ", "e78b5a88af426ec9e240a063544c7550": "||(a, b)|| = \\sqrt{a^2 + ab + b^2}.", "e78b65c974e837d94f9a5e6462f633f1": "\\sqsubseteq", "e78b681cdfd995552467569f5caad6c2": "\ng_{\\mu\\nu} = \\left ( \n\\begin{matrix} \n1 & 0 & 0 & 0 \\\\ \n0 & -\\frac{1}{c^2} & 0 & 0 \\\\ \n0 & 0 & -\\frac{1}{c^2} & 0 \\\\ \n0 & 0 & 0 & -\\frac{1}{c^2} \n\\end{matrix} \\right ) .\n", "e78b94a5cb675b2f42590573401d688f": "\\phi \\geq u", "e78b99b382a61faf4e57bb3988a46384": "\\Omega _{1} =\\int_{0}^{t}A (\\tau )d\\tau ", "e78b9c191386314af0c58be93e13478d": "x \\in u^{-1} [ 0 , 1)", "e78bc90d8a060360cab309d2f779b782": " \\frac{a}{r_s} \\left( \\frac{2 \\pi a}{c T} \\right)^2 = \\frac{1}{2} ", "e78bef306a8b6737e7581986cf5cd29f": " \nK=\\left[\n\\begin{array}{ccccc}\n\\frac{E^{(1)}A^{(1)}}{L^{(1)}} & -\\frac{E^{(1)}A^{(1)}}{L^{(1)}} & 0 & ... & 0 \\\\\n-\\frac{E^{(1)}A^{(1)}}{L^{(1)}} & \\frac{E^{(1)}A^{(1)}}{L^{(1)}} + \\frac{E^{(2)}A^{(2)}}{L^{(2)}} & -\\frac{E^{(2)}A^{(2)}}{L^{(2)}} & ... & 0 \\\\\n0 & -\\frac{E^{(2)}A^{(2)}}{L^{(2)}} & \\frac{E^{(2)}A^{(2)}}{L^{(2)}}+ \\frac{E^{(3)}A^{(3)}}{L^{(3)}} & ... & 0 \\\\\n... & ... & ... & ... & ... \\\\\n0 & 0 & ... & \\frac{E^{(Ne-1)}A^{(Ne-1)}}{L^{(Ne-1)}} + \\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} & -\\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} \\\\\n0 & 0 & ... & -\\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} & \\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}}\n\\end{array}\n\\right] \n", "e78c1eadd10041d6ca8be52638f123a3": "R_s = \\sqrt{\\pi f \\mu_c / \\sigma_c}", "e78c30d4ab50b2c5348907a22793b83a": "a < \\infty", "e78c53561c18cae9e0fed0377d99b86a": "\\begin{bmatrix} 1 & 1 & -1 \\\\ 1 & 1 & 0 \\\\ 1 & 0 & 3 \\end{bmatrix}.", "e78c71a557d2a10bd15db6133b943937": "y_{ij} = \\mu + \\alpha_i + \\alpha_j + d_{ij} + e, ", "e78cd7a88137b3de512b94de499e20f3": "A_{i j}", "e78d143e55fae5640f49769c59035154": "B/A\\,\\!", "e78d374269c7ccfa4980237e12ba4378": "\n\\dot{f} \\approx \\{f, H^*\\}_{PB} \\approx \\{f, H\\}_{PB} + \\sum_k u_k\\{f, \\phi_k\\}_{PB},\n", "e78d45546b72e88d8e59eddd6cef8c46": "\\mathcal{L}_X \\psi := X^{a}\\nabla_{a}\\psi\n-\\frac18\\nabla_{[a}X_{b]}\n[\\gamma^{a},\\gamma^{b}]\\psi\\, = \\nabla_X \\psi - \\frac14 (d X^\\flat)\\cdot \\psi\\, ,", "e78d840314fa01705b1d3a3f5dad6069": "x_0,\\ldots,x_n", "e78da1af93ba25cdc40e667b41082d4a": "P(S_1, \\dots, S_n | O_1, \\dots, O_n)", "e78da804ea492e53a0eed510c8721354": "\\varepsilon_{ijk}\\,", "e78dca9f3f2ebedfdf9a02f2d6a69fe9": "b_n = B_n(a_1,a_2,\\dots,a_n),", "e78ddbdfb8d8f1007715831fe47c0cae": " SL(7,\\mathbb C)", "e78e37ead094dc58d35fe4231dcc3e16": "S=\\{ s_0, s_1, \\ldots, s_{k-1}\\}", "e78ec8e03e091f55bff8497d2cd2bc24": "{dy \\over dx}-(1+x^2)=1", "e78ef3ac2a4999efc9b4440bc9e24900": "\\beta_{pqr}=\\mathrm{\\frac{[M_pL_qH_r] } {[M]^p [L]^q [H]^r}}", "e78f2f64861bedb2193d94fe2c337f32": "B\\in\\mathbb{R}^{|V|}", "e78f632038354aae583f795a73d4e6b8": "\\cap ", "e78fce131adae0cfdae0b6a6d0ccead2": "{a}", "e78ff0cbfbea2b2424151900162f6432": "\n \\begin{align}\n & N_{\\alpha\\beta,\\alpha} = 0 \\\\\n & M_{\\alpha\\beta,\\beta}-Q_\\alpha = 0 \\\\\n & Q_{\\alpha,\\alpha}+q = 0\n \\end{align}\n", "e790195a013ef062c1fe8786e76897a3": "\\psi_{4,8}=1", "e79032ec9eb1ae3ffbda059db6e019a4": "\\rho_E[\\varphi_t] = \\exp{[-H[\\varphi_t]/kT]}=\\exp{\\left[-\\frac{1}{kT} \\int\\frac{d^3k}{(2\\pi)^3}\n \\tilde\\varphi_t^*(k){\\scriptstyle\\frac{1}{2}}(|k|^2+m^2)\\;\\tilde \\varphi_t(k)\\right]}.\n", "e7903403db099f29db516440431c3ea8": "{N(x_i)}", "e7908582980bb99f40f232949dbc56e6": "\np(t) = T p(t-1)\n", "e790fb5d90978732ed7162b12f52ea83": "\n \\overset{\\circ}{\\boldsymbol{\\sigma}} = \\boldsymbol{R}\\cdot\\dot{\\boldsymbol{R}^T}\\cdot\\boldsymbol{\\sigma}\\cdot\\boldsymbol{R}\\cdot\\boldsymbol{R}^T +\n \\boldsymbol{R}\\cdot\\boldsymbol{R}^T\\cdot\\dot{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{R}\\cdot\\boldsymbol{R}^T +\n \\boldsymbol{R}\\cdot\\boldsymbol{R}^T\\cdot\\boldsymbol{\\sigma}\\cdot\\dot{\\boldsymbol{R}}\\cdot\\boldsymbol{R}^T \n", "e79121b1c95b374b632e4ea67363d283": " P^{tr}(X) \\neq P^{te}(X)", "e791243cf0d230e7f49105130aeb078f": "j'", "e7913374870fccd35d5d6340133d9da9": "R_T \\cap Z", "e7919450ce2994ce08bea9f15dff2bd6": " e \\in \\mathfrak{g} ", "e7921326b2cfab61280eb226c7fabb25": " S^* + P \\rightleftharpoons SP", "e7922af776fb573e587956cb34ab392f": "\\int \\frac{\\ln x}{x^2}\\,dx = -\\frac{\\ln x}{x} - \\int \\biggl( \\frac{1}{x} \\biggr) \\biggl( -\\frac{1}{x} \\biggr) \\, dx.\\!", "e79241f242da6f6c98f0c17b7659d32a": "TV \\left( u^{n+1}\\right) \\leq TV \\left( u^{n}\\right) .", "e7924a25ba918a3776f5c7144f06417a": "i(-j) = -k\\,", "e792605362921fc2d227a8498b2de11f": "\\mathbf{v}' = e^{\\frac{\\boldsymbol{\\Omega}\\theta}{2}}\\mathbf{v}e^{-\\frac{\\boldsymbol{\\Omega}\\theta}{2}}.", "e7926ed6ef6da5f3c2d8d8c9d06fc223": " t' = \\frac{2h}{\\sqrt{c^2 - v^2}}", "e792806fd161397d1225e1485e2ea170": "r:\\; z \\mapsto i{{(1+z)}\\over{(1-z)}}", "e792c08aeb14d2f64b8026bb73f83d37": "y = \\sqrt[3]{x}", "e792c5d679d78075597b72ebaab15081": "{\\rm Ric} (M) \\ge 0 ", "e792f5341e84b0470d4803df28e00347": " \\left.\\left(\\tau_0 \\mu_0 + \\tau \\sum_{i=1}^n \\ln x_i\\right)\\right/(\\tau_0 + n \\tau),\\, \\tau_0 + n \\tau", "e792f8ac76089d7774939f39c07bbcde": "l_3\\equiv\\partial_{yy}+\\frac{1}{x+y}\\partial_y\\Big\\rangle\n", "e79382ca6f4c9aceebc9d6c4618cd5f0": " \\frac{a+b}{a} = \\frac{a}{b} \\ \\stackrel{\\text{def}}{=}\\ \\varphi,", "e793a05f1b985a93c56ef7fbb8e944a5": "\\cot\\frac{\\gamma}{2} \\cos\\frac{a-b}{2} = \\tan\\frac{\\alpha+\\beta}{2} \\cos\\frac{a+b}{2}", "e793d1218eb5b48d458e20a4b0ffafe6": "\\sum_{k=1}^\\infty{\\frac{1}{k^2}} = \\frac{\\pi^2}6", "e79427ef1b32f2d9b6c0c42c559d50af": " S = - k \\sum_i p_i \\ln(p_i).\\, ", "e79433b0a6cca3af36af59ac7be35ff3": "\\vec v_{n+1/2}=\\tfrac{\\vec x_{n+1}-\\vec x_n}{\\Delta t}", "e794514cc49a428aa338c4b1f0070289": "A = \\sqrt{P}e^{i\\gamma Pz}", "e794549ce88132733b7fc60acb401f6c": "\n L = \\frac{1 }{2} m \\left ( \\dot y^2-2 \\ell \\dot y \\dot \\theta \\sin \\theta + \\ell^2\\dot \\theta ^2 \\right) - m g \\left( y + \\ell \\cos \\theta \\right )\n", "e794659ac1793c6b947d54df2ea12962": "A_0 \\oplus A_1", "e7947291949cf38d5a64669cfd22f52e": "\\! h(x_i)", "e795096015555826c1f0fb723b69f242": "S_o", "e79525222d81a29d575160fe2bb463e1": "{ap \\choose bp} \\equiv {a \\choose b} \\pmod{p^3}.", "e7953206a0476d4e91bae148325106ab": "T\\{f(t)\\}", "e79533ae53fad084ce0ae580323d0752": "\\sin\\frac{11\\pi}{60}=\\sin 33^\\circ=\\tfrac{1}{16}\\left[2(\\sqrt3-1)\\sqrt{5+\\sqrt5}+\\sqrt2(1+\\sqrt3)(\\sqrt5-1)\\right]\\,", "e796bf8ab080239efc771161ea64089d": "P_{3}", "e796ff9d8fc44bd3de3b837e87925562": "r(t) = 1,\\,", "e797110dcf3d39199e3e091d9ef99d04": "R_1+R_2=2n-k", "e79724eae5fcfe60e37f20953840585b": "z_{n+1} = z_n^p ", "e797635205d942be38c98e91e8eeb6dc": "\\stackrel{\\mathrm{def}}{=}\\ \\int_{-\\infty}^\\infty f(\\tau)\\, g(t - \\tau)\\, d\\tau", "e79790138065aaa309e1bb7c2ea0649f": "\\scriptstyle{Z_{ii}}", "e797dffec71ae9791beaa9dc118758ab": "\\|z\\|_2 \\le \\|x\\|_2", "e7983bd000048c94fba04a4e4acc395c": " 0 8 \\sqrt { \\frac{n}{\\pi} } \\left(\\frac{4 n}{ \\pi e}\\right)^{2 n}. ", "e7a0582c661bfb3a487ee399db5e4444": " t=\\sin \\theta, x=\\sin \\varphi ", "e7a0c83d2fc4446a759b90639a33200d": "\\sum_{k = 1}^\\infty \\frac{(-1)^{k + 1}}{k} = \\ln 2.", "e7a1fca257115929e31883f617a2448f": "\nC^+ = \\max_{f(X_1,X_2)} \\min \\{ I(X_1;Y_2,Y_3|X_2), I(X_1,X_2;Y_3)\\} \n", "e7a21d35fb9b62179fdd436cba90e20c": "\\log(N_i+N_e)=23.491 - 7.5\\log(n_m)\\,,", "e7a267c5274fa8bfd588ea8371b55885": "\\frac{d\\mathbf{r}(t)}{dt} = f'(t)\\mathbf{i} + g'(t)\\mathbf{j} + h'(t)\\mathbf{k}.", "e7a2a113d6a8891e1463cb1de0480266": "p_{i-1}(\\xi)\\ne 0", "e7a3465d9a9cbc63fc6d7d7d61afb401": "\\omega(\\zeta,z) = \\frac{(n-1)!}{(2\\pi i)^n}\\frac{1}{|z-\\zeta|^{2n}}\n\\sum_{1\\le j\\le n}(\\overline\\zeta_j-\\overline z_j) \\, d\\overline\\zeta_1 \\and d\\zeta_1 \\and \\cdots \\and d\\zeta_j \\and \\cdots \\and d\\overline\\zeta_n \\and d\\zeta_n", "e7a36f27c93b0907958122a171057ddf": "V(nH)", "e7a3b27358798c095d052a5cfb6b80e8": "\\mathit{c}", "e7a3c3d7163c4a735bf7471c6f54c436": "H^r(E, E \\setminus E_0; \\mathbf{Z}) \\to H^r(E; \\mathbf{Z}) \\to H^r(X; \\mathbf{Z}).", "e7a3d4643bc91ea9c11437c756aff14c": "(D/N) = (-43/167) = 1", "e7a40a33cd7463f05ece80ea0a072a14": "CAPE^*_s", "e7a42bb0486f01fa8bd099f5c636cb8e": " U_g ", "e7a478519cc8a69bbb7b795bbb9a2b1e": "\\displaystyle{\\kappa(t)=\\lambda(t)^{-1}.}", "e7a5400a35270739fe13ee4612220cf2": "(A\\,\\triangle\\,B)\\,\\triangle\\,C = A\\,\\triangle\\,(B\\,\\triangle\\,C).\\,", "e7a5d353701f3051b8ae1f9396fa83d1": "G(n^2;x)=\\sum_{n=0}^{\\infty}n^2x^n=\\frac{x(x+1)}{(1-x)^3}", "e7a62bc63720cc2259ab4351bcc339bc": "a/(2b)\\,", "e7a64e6594e7209bbfacb296dcfc1a7b": "E/(m c^2)= \\alpha", "e7a69a3e677ad3cd6bb4e0e8e6cb2672": "N|n\\rangle = n|n\\rangle, \\,", "e7a6bf17cb5e9e20b01c155886d90bd7": "C^a (x) = 0", "e7a6c2b3c66d901cf335c6aef9952ecc": "\\begin{matrix} \\frac{1 \\;EVU \\;\\times \\;cosine \\;of \\;latitude} {1 \\;EVU \\;\\times \\;cosine \\;of \\;zero} \\end{matrix}", "e7a710332d5e596f08a536fcd84f31aa": " \\dot f = \\alpha f - \\beta f s ", "e7a717566fc730a2f1e9469e869b63c7": "\n\\mathcal{J}^2 \\equiv \\mathcal{J}_1^2+ \\mathcal{J}_2^2 + \\mathcal{J}_3^2 =\n\\mathcal{P}^2 \\equiv \\mathcal{P}_1^2+ \\mathcal{P}_2^2 + \\mathcal{P}_3^2 .\n", "e7a768908124d62c0fc9c5c4d53ad324": "\n-{1\\over 4} \\int d^4x F_{\\mu \\nu}F^{\\mu \\nu}\n= -{1\\over 4}\\int d^4x \\left( \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} \\right)\\left( \\partial^{\\mu} A^{\\nu} - \\partial^{\\nu} A^{\\mu} \\right)\n", "e7a7ebb12571480076ea1aec5cf85748": " \\tilde Ag= e^{-\\varphi}A(e^{\\varphi}g)\\equiv A_\\varphi g. ", "e7a895119ac1705515da1e830756d2e3": " \\sqrt{3}/2", "e7a8a6756c4c7dc5098fcae9c361dd78": "\n\\frac{\\partial^3 f}{\\partial \\eta ^3}+f\\frac{\\partial^2 f}{\\partial \\eta^2}+ \\beta \\left[1-\\left(\\frac{\\mathrm{d}f}{\\mathrm{d}\\eta}\\right)^2 \\right]=0\n", "e7a9156d08dfb6a5b13fb5404779e734": "\\frac{\\partial E}{\\partial t}+\\nabla\\cdot\\left ( \\bold u \\left ( E+p \\right ) \\right ) = 0 \\,\\!", "e7a9199b6894a571dcd8410ebe906bae": "2s=(a+b+c)", "e7a9206fea5d85b6686db598bc13126f": "S(z,\\zeta) = \\sum_{i=1}^\\infty \\phi_i(z)\\overline{\\phi_i(\\zeta)}.", "e7a974b270dd2bfc44591abf4ae491a7": "\\psi(x_1, \\dots, x_n) = \\Phi(x_1, (0, x_2, \\dots, x_n)).", "e7aa3ed7b94fb690e38a64a5ddbae0ea": "I_n \\otimes A + B^T \\otimes I_n = (V\\otimes U)(I_n \\otimes L + M \\otimes I_n)(V \\otimes U)^{-1}.", "e7aa755fea821a3c15f2e22920b4f04a": " \\dot\\theta = \\frac{H}{r^2} ", "e7aacc9cd39fb2d9d2516163dd324877": " t_{e1}, t_{e2} ", "e7aae574de90c6216167571d24341d60": "(\\lambda Y. B \\subseteq Y)(S)", "e7ab2e89bde76fe652088d0a8909cd50": "\\sigma^2_N = \\frac{M_{2,N}}{N}", "e7abc65f6c1650abf79c92bdb33de937": "\\tilde{\\pi}", "e7ac43ec5978d188728f407c7fbe8487": "Z_2 = \\sqrt{-2 \\ln(U_1)} \\sin(2 \\pi U_2) ", "e7ac7cbb9ff0184d499a3a59059f4262": "\nH(\\mathbf{Y})=-\\frac{1}{N}\\sum_{t=1}^N \\ln\\frac{p_\\mathbf{x}(\\mathbf{x}^t)}{|\\mathbf{W}|p_\\mathbf{s}(\\mathbf{y}^t)}\n", "e7acd39307656889417d3f5034fb91e4": " x^{1/2} = \\sqrt{x} ", "e7aceda9df3c21c8f81f274a62cb279c": "H \\; 1000", "e7ad0a1cc939313e13fb2e00fa382a26": "\\mathbf{n} = \\frac{\\mathbf{r}_u\\times\\mathbf{r}_v}{|\\mathbf{r}_u\\times\\mathbf{r}_v|}.", "e7ad5923f05e9166877d46fb67b90188": " \\Re\\, \\overline{w} {h(w)\\over h^\\prime (w)} \\ge 0.", "e7ad59d81783a4d51edf42aeb3c1f0bc": "\\operatorname{E} [k_t/K] = p_t", "e7ad7d93110b0e48884376b09bc6dabd": "\n \\sigma_{jk} = \\cfrac{\\partial W}{\\partial\\epsilon_{jk}}\n ", "e7ae0ef437c357e8e1af4fe763a1c9af": "f(n)=256^n", "e7ae8216e81ce46c56d142435a96f492": "\\mathrm{Am^{3+}\\ _{(aq)} +\\ 3\\ F^-\\ _{(aq)} \\longrightarrow \\ AmF_3\\ _{(s)} \\downarrow}", "e7af660c65fa5aec308c459fb3a39fc4": "B^n x + B^n \\le B^n x + \\alpha\\,", "e7af69dd1b4033890bf8482958767b2e": "r^2 = \\ell^2 + (r-s)^2", "e7af6d939de1d83f92e0e4b2995260f0": "b \\equiv 1 \\pmod{N}", "e7afd79d7ba6a8c1bbf1a9397dbf76d3": " r'(d) = \\frac{1}{\\cos^2(d/R)} ", "e7b05ce60a29d3302337da28a90f5a7c": " \\epsilon > 0", "e7b08f74c9cfa96bd0366781288cc163": "U=\\{z=x+iy:|z|=\\sqrt{x^2+y^2} < 1 \\}", "e7b0e93dba2be7781525edeafcb54ec6": " \\sum_\\ell A_{k\\ell}P_\\ell, ", "e7b0fe013d048acc3ca6e5e2b4ee091b": "C_{SP}(t,\\omega) = \\iiiint G(t^{''},\\omega^{''})C_s(t^',\\omega^')C_h(t^{''}+t^'-t,-\\omega^{''}+\\omega-\\omega^')\\,dt^'\\,dt^{''}\\,d\\omega^\\,d\\omega^{''}", "e7b1215790207aecb77e33a200556457": "\\rho = (Q_1, K_1, Q_2, K_2, \\ldots, Q_M, K_M)", "e7b15a849ced49636b75f4d950b79785": "\\delta = \\theta_0 - \\theta_5", "e7b1638106710e7b1af64ae85532a89f": "\\int\\! x^n \\, dx=\\frac{x^{n+1}}{n+1}+C", "e7b187a19100802abb506815502b5c34": "\\mathbb{S}^n = \\bigcup^{ }_{A \\subset \\mathbb{N}} [A]^n", "e7b1a333a0ca298455e81902a9db4fb3": "x+y>0", "e7b1b57271bd6a4d68ab3c7688e7b2bb": "R-\\Lambda=\\kappa T", "e7b215eac57eb035aca6f8dd8f71efb8": " kR_0 \\;\\simeq\\; 0.697", "e7b21a370577de23941d98eea9c8191d": " b_s = 0 ", "e7b24b112a44fdd9ee93bdf998c6ca0e": "360", "e7b2a195daae63c499153cfe94215bef": "\\Box \\varphi = \\frac{\\rho}{\\varepsilon_0}", "e7b2c482369bfe2c175b244b07d31422": "Attr_i(U) := \\bigcup_{j=0}^\\infty Attr_i(U)^j", "e7b2c73df3fed03fb5242b036ae044b3": "d_{3/2,3/2}^{3/2} = \\frac{1+\\cos \\theta}{2} \\cos \\frac{\\theta}{2}", "e7b2dcfa521cfecd25713a3457ae7320": "\\neg locked(door,s) \\rightarrow \\neg locked(door, result(opens,s))", "e7b2e5c9e9c7e87ee5546e5ee64bb0fd": "M^\\perp = \\{ x' \\in X' : x'(m) = 0, \\ \\forall m \\in M \\}.", "e7b34bc7a5901a664809267721bc20a1": "\nn \\mbox{ is self if }\n[ n - DR*(n) - 9 \\cdot i ] + SOD([ n - DR*(n) - 9 \\cdot i ] ) \\neq n\n\\quad \\forall i \\in 0 \\ldots d(n)\n", "e7b359e6dfd53ddd72aee3541f1f6f4b": "x^{13} \\pm 1", "e7b396fcd8dc5f31a44c745a4da214ff": "(\\tfrac{x-1}{2})^2+(\\tfrac{x+1}{2})^2 = y^2", "e7b398d50e87b946411fccedb13f4614": " \\pi\\mathbf{P} = \\pi.\\,", "e7b46b761b28c506a62f782deb7dad8e": " \\frac{\\partial L}{\\partial u} - \\frac{\\partial}{\\partial x}\\frac{\\partial L}{\\partial u_x} - \\frac{\\partial}{\\partial y}\\frac{\\partial L}{\\partial u_y} = 0", "e7b47ae7a8f30c66bad79790e7242284": "\\int_{(x,y)\\in C} x^3\\, dx + 4y^2\\, dy", "e7b499d67bfea8a0db44a39b055ef3f4": "\\lambda(t)", "e7b4ae126c4904cdba539165fbccf8d1": "\\scriptstyle\\ \\alpha ", "e7b4ff5d32064a38cd8f378f7004a256": "H^r(F, F \\setminus F_0; \\mathbf{Z})", "e7b53ab15f00e295782487b3d0cd4ac2": "\\operatorname{dim} T/(\\mathfrak{p}_1 \\cap T) \\le \\operatorname{dim} T/(u)", "e7b54e052221a9ff4900cbd9a77de466": "U_s", "e7b5a686a8087b4c25360ce9b09d5ff1": "\\mathrm{GL}(n)", "e7b5ad8ad460033e6756589a1d62d04c": "\\delta_v^*", "e7b5b1d56fe39b54548728b7a6d6b736": "\\Omega X", "e7b5b4134d206398082fae4bb3a3efaa": "\n\\mathbf{B}(t) = \\sum_{j = 0}^n {t^j \\mathbf{C}_j}\n", "e7b5bf84ba4a6c78810a201cecb4c369": "\\mathbf{B}(\\mathbf{r},t) = B_\\text{x}(\\mathbf{r},t)\\mathbf{e}_\\text{x} + B_\\text{y}(\\mathbf{r},t)\\mathbf{e}_\\text{y} + B_\\text{z}(\\mathbf{r},t)\\mathbf{e}_\\text{z}", "e7b5d2953505c748139cd20686963cb8": "i^{a+bi} = e^{\\frac{1}{2}{\\pi i} (a+bi)} = e^{-\\frac{1}{2}{\\pi b}} \\left(\\cos{\\frac{\\pi a}{2}} + i \\sin{\\frac{\\pi a}{2}}\\right)", "e7b5d5da1e62a4a06d1befeeb5e58e4c": " \\mathbf{x}_{i}^{k} = V_{k}^{T}\\mathbf{x}_i \\in \\mathbb{R}^{k} ", "e7b5eae07fb02a72002974caadeb87f0": "\\exp(it\\Delta) f(x) = \\frac{1}{(4\\pi it)^{n/2}} \\int_{R^n} e^{i|x-y|^2/4t} f(y) dy", "e7b628fc473be56e4ac1a970937b3572": "E = 1.6067\\left( \\frac{L^3} {d^4} \\right)mf_f^2T'", "e7b665871d636bcbacb426acfa32d3bb": "\\frac{d^2 \\hat u}{d \\hat y^2} = 1 \\quad ; \\quad \\hat u(0) = \\hat u(1) = 0", "e7b6b4bcbd5ffd10877b2e9230e730e5": " \\mathbf{M} = \\chi \\mathbf{H},", "e7b6d67e14843dcc3698b458758d0203": "0 \\leq i \\leq n-1", "e7b74bde5e7ca3380c2f301f3544baec": "\\phi = \\sqrt{5} e^{i\\theta} ", "e7b74dee0483d8f5c5250abc37a00347": "r_e = \\frac{P_1}{P_0}", "e7b773d91b9c77445f6bf8995651f245": "\\ - m \\frac{d \\boldsymbol\\Omega }{dt} \\times \\mathbf{x}_B \\ .\n", "e7b7a4738698f60717c140cc279b0f71": "\\int x^2\\,\\operatorname{arsinh}(a\\,x)dx=\n \\frac{x^3\\,\\operatorname{arsinh}(a\\,x)}{3}-\n \\frac{\\left(a^2\\,x^2-2\\right)\\sqrt{a^2\\,x^2+1}}{9\\,a^3}+C", "e7b7ab0b4a23967b5f62313c510a2b10": " \\Pr (y_j = 0) = \\pi + (1 - \\pi) e^{-\\lambda} ", "e7b7ff43382eab620b3e1ff6b43f3df5": "\\textstyle P=P_1R", "e7b80a9685f93d7f8dae076db167907d": "\\mathbf{e}_0 = \\mathbf{e}_z ", "e7b99f403541e0cbe411efe3b121e07c": "\\frac{\\partial \\rho }{\\partial t}+{\\mathrm{i}\\hat{\\mathbf{L}}}\\rho =0.", "e7ba0ba64c49802c400c0857caef6af8": "\\frac{1}{4} m \\left(U^2 + 2U^2 \\frac{ky}{r} + U^2 \\frac{k^2 y^2}{r^2}\\right)", "e7ba4528eab1a7b60b2746b43906abad": " G(\\mathbf{x}^{(0)}) = \\begin{bmatrix}\n -2.5\\\\\n -1\\\\\n 10.472\n\\end{bmatrix}", "e7ba7a6641b3368f53f44c447f9fd699": "X \\sim N(\\mu_X, \\sigma_X^2)", "e7ba85f4324caee65e8c32b5ff510d39": "m_j=0", "e7baac3cabf59bb740d07466a80311ee": "\\scriptstyle{X^-\\neq X}", "e7bb2d2a2837eeee9ff802965e70c035": "2^{-1} = \\frac 1 {2^1} = \\frac 1 2.", "e7bb8c296b044b760bf7044155e3db68": "\\mathbf{a}(s) = \\frac{d}{dt}\\mathbf{v}(s) ", "e7bb92a1ef3b1bcea5bab7a03516b4f9": " f(u) = \\beta_{1} + \\beta_{2} e^{-u} \\,", "e7bc0f3fbdaaf5a7b7d23c0c0bc6b061": "3^1 + 3^2 + 3^3", "e7bc18662a4da84cca6a3ea7f2bd6525": "X_A := \\sum_{j\\in A} X_j", "e7bd5eef117c089bc48699def9ccaa13": " \\begin{pmatrix} \\hat{\\boldsymbol{\\beta}} \\\\ \\hat{\\boldsymbol{\\gamma}} \\end{pmatrix} = \\begin{bmatrix}X & K\\end{bmatrix}^{-1} \\mathbf{y} = \\begin{bmatrix} (X^{\\rm T} X)^{-1} X^{\\rm T} \\\\ (K^{\\rm T} K)^{-1} K^{\\rm T} \\end{bmatrix} \\mathbf{y} .", "e7bd7c16f4e18d3e2dd2fd4c3baed600": "G[C\\cup D]", "e7bd7e0f8149e8614fd9070e45a88c2e": "k_n = \\frac{n \\pi}{L}, \\quad \\mathrm{where} \\quad n = \\{1,2,3,4,\\ldots\\},", "e7bda069859ad709651a85e16cb88c58": "\nC([x]) =\\bigcap^{\\{q\\}}C_i([x]).\n", "e7bda38f1e14730268ab6329cf55e9da": " dt_\\text{E}^2 = \\left( 1-\\frac{2GM_\\text{i}}{r_\\text{i} c^2} \\right) dt_\\text{c}^2 - \\left( 1-\\frac{2GM_\\text{i}}{r_\\text{i} c^2} \\right)^{-1} \\frac{dx^2+dy^2+dz^2}{c^2} \\,", "e7bdaca857dd188591a2e8e05e50e37f": "\\mathfrak{d} < \\mathfrak{a},", "e7be0fc0feb6ea4a8379a7ee18fd83a6": " e^{\\mathbf{A} \\oplus \\mathbf{B}} = e^\\mathbf{A} \\otimes e^\\mathbf{B}. ", "e7be68a23cb07aea936e14bbaee21338": "\\gamma(t)={ \\mathrm{cov}(X(h),X(h+t))\\over \\sigma^2} ={ E[(X(h)-\\mu)(X(h+t)-\\mu)]\\over \\sigma^2}", "e7bf3638d906cbc92e3b9cfccd19e72c": " \\text{(c)}\\quad\n\\tan\\beta=\\frac{a\\sec\\phi}{y'(\\phi)} \\tan\\alpha.\\,", "e7bf864a0717f43dc3e161ab806e1f7b": "\\rho \\to \\infty ", "e7bfb4cb47c1058a93eb04057541fd9e": "\\displaystyle{ e_\\alpha [\\pi] = [\\pi_r]} ", "e7bfc112babd85026351eda266dc5243": "H = 2a. \\,", "e7bff9e7592c5b9adfb9fa1bbd21c10b": "\\beta=\\gamma/(\\delta-1)", "e7c0491ba9e4e2137d21d4a7efefbccf": " g_n < (\\log p_{n})^2 - \\log p_{n} ", "e7c04f6711aae5c2e539e8cc4b942c3f": "\\sum_{i=1}^K \\|\\mathbf{w}_i\\|^2 \\leq P", "e7c05a60c006b4b26fb2896e72519b9b": "1+\\varepsilon^2T_n^2(-1/js_{pm})=0.", "e7c0b1bc0e56d31816081a2f912dc674": "\\left\\{\\mathcal{B} f\\right\\}(s) = \\left\\{\\mathcal{L} f(t)\\right\\}(s) \n+ \\left\\{\\mathcal{L} f(-t)\\right\\}(-s)", "e7c0cfaf74ce79822b8a0f31b18037c4": "\\frac{y - y_0}{y_1-y_0} = \\frac{x-x_0}{x_1-x_0}", "e7c112275eb8c93e5c71fe4cb2143869": "d_3=\\frac{d_2-r}{d_2}\\,", "e7c178ea20067bf351f0c38f1d0e3261": " \\Sigma ' \\alpha_i", "e7c19dddf8c2c801776f184fb7941ecb": "\\theta=\\tfrac{1}{2}\\pi", "e7c21c3f9d9287e35b6448b546a0f8ba": "\\frac{S_0(z)}{\\hbar}= kz - \\frac{m}{{\\hbar}^2 k} \\int_{-\\infty}^{Z}{V dz'} ", "e7c2508122d75154b3b200818526e8ac": "\\begin{matrix}{k \\choose m}\\end{matrix}", "e7c255d263b31746d23a067a4104912f": "\\varphi = D_{1/2} (2\\cdot (h * \\varphi))", "e7c2a911c6c61006955ae4dd7c35f9c0": "E[x]", "e7c31e0db118882be7f9818474d9aa5d": "y=\\tfrac{a}{x-b}+c, a\\ne0", "e7c331d34ee961618ef117e6a0cd44e7": "H_i (X) \\cong H_{\\text{dR}}^{2n-i}(X)^{\\vee} \\cong H^i(X).", "e7c41893c77dae861097193fb640ed58": "\ny_i(t) \\leq Q_i(t+1) - Q_i(t) \n", "e7c420fbb954eb7753e14089845e0fed": "|c\\rangle", "e7c4e525ae9bd7b4b9e3884840562778": "\\Phi_{ab} = \\frac{m}{(x^2+y^2+z^2)^{5/2}} \\, \\left[ \\begin{matrix} y^2+z^2-2x^2 & -3xy & -3xz \\\\ -3xy & x^2+z^2-2y^2 & -3yz \\\\ -3xz & -3yz & x^2+y^2-2z^2 \\end{matrix} \\right] ", "e7c514596cfcdcc87a7ef91cb3b07f65": "\\cong_{\\mathcal{B},\\epsilon}", "e7c56e72864dce4e5bccfd96adfc94ce": "\\eta_{th} = 1 - \\bigg(\\frac{p_2}{p_1}\\bigg)^\\frac{1-\\gamma}{\\gamma} \\,", "e7c5e0626f65585e844163275b046884": "H=\\frac{\\dot a}{a}", "e7c5f564cacb2f2c37809ced918ce88f": "\n\\begin{bmatrix} x' \\\\ y' \\\\ z' \\end{bmatrix} = \\frac{1}{w_c} \\begin{bmatrix} x_c \\\\ y_c \\\\ z_c \\end{bmatrix}\n", "e7c69c7b44a7a7836fe5d5684541a949": "a\\frac{\\partial^2 u}{\\partial x^2} + c\\frac{\\partial^2 u}{\\partial y^2} + e\\frac{\\partial^2 u}{\\partial z^2} \\text{ + (lower-order terms)}= 0.", "e7c6b7fe422b19985a96e8522249574f": "\\mathbb{S} := \\mathrm{Res}_{\\mathbb{C}/\\mathbb{R}} \\mathbb{G}_m", "e7c6d5da412d45a792ad55b5fb94f16c": " \\gamma_i \\equiv { 1 \\over {\\sqrt {1 - {{\\mathbf{v}_i \\cdot \\mathbf{v}_i } \\over c^2} } } } ", "e7c712ba0cb3b59ee092e895ec6f9c72": "I_{1}", "e7c74639adb19bec44b6d8b275c5caac": "\\ \\begin{array}{rrcl} & \\boldsymbol{\\sigma}^* &=& \\mathcal{H}(\\boldsymbol{B}^*) \\\\\n\\Rightarrow & \\boldsymbol{R}\\cdot \\boldsymbol{\\sigma}\\cdot \\boldsymbol{R}^T &=& \\mathcal{H}(\\boldsymbol{F}^*\\cdot(\\boldsymbol{F}^*)^T) \\\\\n\\Rightarrow & \\boldsymbol{R}\\cdot \\mathcal{H}(\\boldsymbol{B}) \\cdot\\boldsymbol{R}^T &=& \\mathcal{H}(\\boldsymbol{R}\\cdot\\boldsymbol{F}\\cdot\\boldsymbol{F}^T\\cdot\\boldsymbol{R}^T) \\\\\n\\Rightarrow & \\boldsymbol{R}\\cdot \\mathcal{H}(\\boldsymbol{B})\\cdot \\boldsymbol{R}^T &=& \\mathcal{H}(\\boldsymbol{R}\\cdot\\boldsymbol{B}\\cdot\\boldsymbol{R}^T). \\end{array}", "e7c7b9acfd97f6657b31f05e89c52032": "S(X)= S(Y)", "e7c7ca07ca0fc97d9ada5c999e17bddc": "{\\mu}{V}\\over {{\\delta}_1}^2\\,\\!", "e7c89822f8fa5c5752a31285e7ed4f53": "\\scriptstyle\\hat\\beta_j", "e7c8fc70495f62e33001c72aea1833a9": "\\mathcal{C} \\subseteq \\Sigma^{n}", "e7c98ee6f146bee7762bb9737d088309": "s\\in \\C", "e7c9f24f89ecedf871abd362f90bcb2e": "\\mathbf{v} = (q_2-q_1) / \\left|(q_2-q_1)\\right|.", "e7ca1a5d1ad56a1dd4f2dcd0da37ab96": "v(\\{1\\}) - v(\\varnothing) = 0 - 0 = 0\\,\\!", "e7ca88b53e095e5e92638209890c260f": "Tf(x)=\\int_Y K(x,y)f(y)\\,dy", "e7caa5b7ba419086ac7d16b66413fafa": " R_n = \\frac{1}{2 \\sin \\frac{\\pi}{n}} \\quad\\quad \n \\begin{array}{r|ccr|c}\n n & R_n & & n & R_n\\\\\n \\hline\n 2 & 0.50000000 & & 10 & 1.6180340- \\\\\n 3 & 0.5773503- & & 11 & 1.7747328- \\\\\n 4 & 0.7071068- & & 12 & 1.9318517- \\\\\n 5 & 0.8506508+ & & 13 & 2.0892907+ \\\\\n 6 & 1.00000000 & & 14 & 2.2469796+ \\\\\n 7 & 1.1523824+ & & 15 & 2.4048672- \\\\\n 8 & 1.3065630- & & 16 & 2.5629154+ \\\\\n 9 & 1.4619022+ & & 17 & 2.7210956-\n \\end{array}\n", "e7caf2e2974c37b61c3c560f2154dbb8": "(\\text{size}-1)", "e7cb0cad18c18cda140541a4ea127f04": " \\ \\ p0\\mbox{ for }0\\leq t<1,\\;\\alpha>-1.", "e7dfcf27257299a7d30a45dd26874dd4": "N_{1},\\ N_{2}", "e7e053574812c82c732c05d9411c8edc": "C(\\mathbb{C}^m, \\mathbb{C}^{n \\times n}) = 0.", "e7e07b07fe5015e4e5d0701a93ce3f17": "2^p-1", "e7e0b504610fac2eb3eaba34d6edee5b": "(7)", "e7e0b5a38ddd795028774404801236f6": " h(\\mathbf{x})=H\\mathbf{x}. ", "e7e1116b23dddb8c1688842cf504e6e6": "\n\\left(\\frac{x}{c \\cos\\theta}\\right)^2 - \\left(\\frac{y}{c \\sin\\theta}\\right)^2 = 1\n", "e7e1504b04c76ea6dd7752d156a07f53": "F_n(s_0)", "e7e160a68491c1eec5fc3c363dd1293b": "\\textstyle{r}", "e7e16d3c8cc9309b011864327fda7262": "S_1, \\ldots, S_e", "e7e174be372a95c77f3e383596db71d9": " t_g = \\frac {v \\sin \\theta} {g} + \\frac {\\sqrt{v^2 \\sin^2 \\theta + 2 g y_0}} {g} ", "e7e1a7a4bc4ecfc4dbfb68843ebd1a59": "\\mbox{isqrt}(27) = 5", "e7e1f5afbe6c6b47aaa7b643dc16024e": "\\mathbf{Z}[\\sqrt{-5}]", "e7e26ab424a5c7d321e5fb29fceabd81": "\\Pr(X > t + s | X > t) = \\Pr(X > s)\\,", "e7e26c186921b2e3d908b3472d6aaed5": "e_3=e_4=1,", "e7e281d9f1c409086713ecd741678375": "c \\rightarrow \\infty", "e7e359be0b04404bfd8982b1e6f53299": "d\\tilde n/d\\eta", "e7e363a41a0456b3d830bb0def0c4c17": "I_S=\\left(j\\omega (C_J+C_D) + \\frac{1}{r_D} +\\frac{1}{R_S} \\right) V_O \\ , ", "e7e3767a5cf4c71d4efb5ecfe12d951f": "\\! 4 \\leq J \\leq 8", "e7e39505f0fa5f25ee8c91e383d565c9": "A=(3+\\frac{11\\sqrt{3}}{2})a^2\\approx12.5263...a^2", "e7e3b0d9f47e9fb5a48a7e34c2206c53": "M_e \\le N", "e7e3bbc3a185a52fdc36b6dbbd058b43": "j = 1, \\dots, k", "e7e3dfbe13cbcb5421c868f32522c951": "A + B = C", "e7e4378b05d68df32c8dfbb8f7c86dc9": "\n\\left( {j - \\sigma - {1 \\over 2}} \\right)^2 - {1 \\over 4} = \\sum\\limits_m^{M} {\\gamma _m \\left\\{ {\\left( {j + m - {1 \\over 2}} \\right)^2 - {1 \\over 4}} \\right\\}}. \\quad \\quad (15)\n", "e7e473af6eed18352ac1ad94191b45c8": "\\,Cov\\{m_1(y)\\}", "e7e475a1b2e8f1be0a50f2e1234ca61e": "\\phi=\\sqrt{\\frac{\\chi^2}{N}}", "e7e47dcd179422f9025304c137d2d013": "E(t) = g(e(t), c^*) \\, ", "e7e52a435aeedd8182f3980187ecfed7": "g(X,Y) = \\frac{f(Y) - f(X)}{Y - X}", "e7e6128070cb3ee3d3dda01fd3695e18": "\\rho\\mathbf F=\\langle\\rho F,\\le,\\rho V\\rangle", "e7e62e773867fa6f34ca8bf320afbc3e": "AP^{2} + PC^{2} = (Aw^{2} + Az^{2}) + (wB^{2} + zD^{2}) = (wB^{2} + Az^{2}) + (zD^{2} + Aw^{2}) = BP^{2} + PD^{2}.\\,", "e7e6600645d5aa4c9abd993c6b9f3056": "x*y=y*x", "e7e670e9bfb42d2c5262f26f787d0dcb": "g_2^2=g_3^3= (g_2g_3)^7=1.", "e7e691cd140d938ee793398d09fb4aff": "\\displaystyle{ (z_1,z_2;z_3,z_4)={(z_1-z_3)(z_2-z_4)\\over(z_2-z_3)(z_1-z_4)}}", "e7e7095712b1f43f1e54319b4c0f41b9": "3! = 6", "e7e71fd3ef0507f2c781da5bb0b9524f": "\\, r = a\\cdot \\sqrt{1+\\ t^2}", "e7e72c6e0b423c9473a677a72687549d": "\\min\\{v_1, v_2, \\ldots\\}", "e7e751e1be47c4f86f2689c453831aef": "\\textstyle \\mathbf{1}_{B_1} ", "e7e78eac0aa025566d2d9356df668e4c": " \\mathbf{TIME}(f( \\left\\lfloor \\tfrac{m}{2} \\right\\rfloor )) = \\mathbf{TIME}(f( \\left\\lfloor \\tfrac{2n+1}{2} \\right\\rfloor )) = \\mathbf{TIME}(f(n)). ", "e7e9036f730fe632bb391f9a4e9f56e1": "{V_f \\over I} \\;\\Omega", "e7e90bad2abd912cb7ed23212adfdba2": "f^{-1}\\{x\\}", "e7e9256f748d3b164089efca86634599": "\\displaystyle{\\|R\\|^{2m} \\le n \\cdot \\max \\|R_i\\| \\left(\\max_i \\sum_j \\|R_i^*R_j\\|^{1\\over 2}\\right)^{2m}\\left(\\max_i \\sum_j \\|R_iR_j^*\\|^{1\\over 2}\\right)^{2m}.}", "e7e92f37a0909cec86243b38a9080918": "\\ e_{ck} = \\frac{the~number~of~c-k~pairs~in~the~data}{n-1} = \\frac{1}{n-1}\n\\begin{cases}\n n_c(n_k-1) & \\mbox{iff }c\\mbox{ =k} \\\\\n n_cn_k & \\mbox{iff }c\\mbox{ ≠k}\n\\end{cases}\n=e_{kc} ", "e7e94916dafbaa55196c88fe3c424397": "r_a=a(1+e)", "e7e9738c140276cf06302a424f7e613b": "\\Delta t_{i=1}", "e7ea1ec850413bea47b73561f1b02fba": "H(|1\\rangle) = \\frac{1}{\\sqrt{2}}|0\\rangle-\\frac{1}{\\sqrt{2}}|1\\rangle", "e7ea2183f8147d0fb28805b2a32af17a": " {y : \\langle y,a_j \\rangle = b_j} ", "e7ea5942e833702792f0c2ce4fe31db1": "q^n = \\sum_{d\\mid n} d N_d,", "e7ea6b208725ae7d727ba69634e47e7b": "\\varepsilon \\rightarrow \\infty", "e7ea827bc66693a3eff83617a6edf56a": "Z(\\beta) = \\int \\exp \\left(-\\beta H(x_1,x_2,\\dots) \\right) dx_1 dx_2 \\cdots", "e7eaae474a0f1b1c4505d3059b4e1c14": "M \\to M \\quad m \\mapsto r \\cdot m.", "e7eafda1534f590213b691916b762eba": "a.P", "e7eb0313f3787d3a0e66c2c0ac15cb11": " \\omega_m =\\omega'_0\\sqrt{\\frac{-1}{Q^2_C}+\\sqrt{1+\\frac{2} {Q^2_C}}} ", "e7eb648fc26bface14f275eca1bf3cb8": "s_\\mu = 43.37", "e7eb674da08971f83cfe2d4caf86878b": "R/2", "e7eb6c85677ed5c00066ba757e262146": "G = \\left\\{1, f, g, fg\\right\\},", "e7ebc8a1464579976e2234bae8d7a1a4": "a,u\\in[0,1]", "e7ec7a4c37420d76d57e9c6e2f8e8001": "\\ N = \\frac{\\log \\, \\bigg[ \\Big(\\frac{LK_d}{1-LK_d}\\Big)\\Big(\\frac{1-LK_b}{LK_b} \\Big) \\bigg]}{\\log \\, \\alpha_{avg}} ", "e7ec863a71d026f822f32bae8ea2ac68": "\n\\begin{array}{cccccccc}\n1&&&&&&& \\\\ \n3&1&&&&&&\\\\\n5&3&1&&&&&\\\\\n7&5&3&1&&&&\\\\ \n9&7&5&3&1&&&\\\\\n\\vdots&&&&&\\ddots&&\\\\\n\\scriptstyle 2n-3&\\cdots&&&&\\cdots&1& \\\\\n\\scriptstyle 2n-1&\\scriptstyle 2n-3&&&&\\cdots&&1 \\\\ \\hline\n\\scriptstyle =n^2& \\scriptstyle =(n-1)^2&\\cdots& \\scriptstyle =5^2& \\scriptstyle =4^2& \\scriptstyle =3^2& \\scriptstyle =2^2& \\scriptstyle =1^2\n\\end{array} \n", "e7ecab5efd61a5da5c665f20f62bfa71": "\\begin{cases} \\rho = \\log\\sqrt{ x^2 + y^2}, \\\\ \\theta = \\arctan \\frac{y}{x}. \\end{cases}", "e7ecb66b319c8eadba8d24021c1e2802": "\\psi(\\Omega)^\\omega", "e7ecb7cfdf0fc65e6ef5e06e4bb69d04": "\\cos z = \\sin \\phi_1 \\sin \\phi + \\cos \\phi_1 \\cos \\phi \\cos (\\lambda - \\lambda_0)", "e7ed1bf9bf162aa297f8d088c7dd8101": "\\mathbf{a}\\cdot(\\mathbf{b\\times c})= -\\mathbf{b}\\cdot(\\mathbf{a\\times c})", "e7ed1d7d1794326e04aa7267bb48816f": "\\nabla_{\\bold{v}} (f + g) = \\nabla_{\\bold{v}} f + \\nabla_{\\bold{v}} g", "e7ed34403e65c8fc1b6a171471808b50": "r(v)", "e7edf5c917a4125f9a260f53c75be560": "|G_{\\nu}(0)|=0", "e7ee6637473ea1b2a538676cf154c535": "= 101", "e7ee6d8f4ec8306a8de12c8a9509d611": "u(w)=\\log(w)", "e7ee9a601b6d5b6149cf178933881d8b": "V = k/P", "e7eec12d283dbd0a6bf67f22f1d34bd9": "4T^2=4s(s-a)(s-b)(s-c).", "e7ef3bce5fbbdcc3be446537f9cf9f32": " n 2^{-k} ", "e7ef5a79d40b539bec2ea5d383372dc8": "D_1, D_2", "e7ef721ab0f72fd05dc91950dac7908f": "\nds=r\\sqrt{\\theta'^2+\\phi'^{2}\\sin^{2}\\theta}\\, dt.\n", "e7efb5e062ca8b8a92c857137aeb2665": "x = \\lambda", "e7efee91172f79d2bc8c7065f49942e3": "f(x_1,\\ldots,x_n \\mid \\mu,\\sigma^2) = \\prod_{i=1}^{n} f( x_{i}\\mid \\mu, \\sigma^2) = \\left( \\frac{1}{2\\pi\\sigma^2} \\right)^{n/2} \\exp\\left( -\\frac{ \\sum_{i=1}^{n}(x_i-\\mu)^2}{2\\sigma^2}\\right),", "e7effac3ce924e670f81a1dfe7a19b03": "w_i'=\\frac{1}{n}.", "e7f0581a8b095ea4a219c91b89b27122": "= {1 \\over 2} (\\delta^K_O \\delta^I_P \\delta^J_M + \\delta^K_M \\delta^I_O \\delta^J_P + \n\\delta^K_P \\delta^I_M \\delta^J_O - \\delta^K_P \\delta^I_O \\delta^J_M - \\delta^K_M \\delta^I_P \\delta^J_O - \\delta^K_O \\delta^I_M \\delta^J_P) F^M_{\\;\\;\\; K} G^{OP}", "e7f05b6abe5aec6605bba3a04e88c9ad": "\\psi(t)\\,", "e7f06a51a902dd84df83e0da741a1b23": "y = at(3-t^2)", "e7f08be3379cdb7afb7738bd16dae949": "\\pi d_m", "e7f093f494cd6b3c149dea39f540176c": "p_1 \\leq \\cdots \\leq p_c", "e7f0b588b987f1b15ddf574abfda292f": "\\psi^{\\mathrm I}(t,q^{\\mathrm I})=\\psi(t,q^{\\mathrm I},Q^{\\mathrm{II}}(t)). \\,", "e7f0d0559294d1a079370f00b433fe04": "G_1 = 8.873 \\mu", "e7f0f38f9572e3a0dbdb945163dfeed0": "a - b = N/(a + b)", "e7f10bcc73863b96b7636a5196c1d57b": "Mod(R)", "e7f10e8d3d2a86d6a263781d8fff540a": "\\sigma X \\sim HN(\\sigma)\\,", "e7f134d5a8fa8a4aa669f2329edf9a1c": "\\mathrm{Binomial}(1,p)", "e7f14a3941c3ed54442139a92daacd11": "f'(a)\\approx {f(a+h)-f(a)\\over h}.", "e7f1b6272152e40b6022586439df89c2": "\\textrm{SINR}^{\\mathrm{ZF}}_k = | \\mathbf{h}_k^H \\mathbf{w}^{\\mathrm{ZF}}_k |^2.", "e7f1e5f685c2af7f1c2bba9b6cd09621": "\\|A\\|_{*} = \\operatorname{trace} \\left(\\sqrt{A^*A}\\right) = \\sum_{i=1}^{\\min\\{m,\\,n\\}} \\sigma_i.", "e7f212afede0e9c88bad28db5f9db668": "\\triangle ABC \\, ||| \\,\\triangle DEF \\, ", "e7f23efe48975d893f459b1e998d660d": " h = h_{r,s}(c) = {{(pr-qs)^2-(p-q)^2} \\over 4pq}", "e7f252dde526144456dd5c45da12c246": "\t \nj = j_{ion}^{sat} \t \n\\left( -1 + \\,e^{e(V_{pr}-V_{fl})/k_BT_e} \\right)\t \n", "e7f27780554ba787eb81a90089c9b843": "f(t)=e^{ct}, f'(t)=ce^{ct}", "e7f2f9cf2cbdcc2eb8eca8a1916669ff": "(\\phi \\to \\neg \\psi) \\leftrightarrow \\neg (\\phi \\wedge \\psi) ", "e7f305f4d5942f20aff4cc6b69f651b8": "\\frac{dP_{a \\parallel v}}{d\\Omega} = \\frac{q^2a^2}{16\\pi^2 \\varepsilon_0 c^3}\\frac{\\sin^2 \\theta}{(1-\\beta \\cos\\theta)^5}", "e7f348ff9eddd268414bd707625d25fb": "P_c = \\frac {N_1}{N_2} \\cdot P_m - \\frac {\\left( 1,111 \\cdot |\\bar{U_2}| \\right)^2}{R_i} ", "e7f35f1f5eb2bc9cb211f58e45dee300": "q^{n-d+1}", "e7f368a1ac69645daae25fb44f39a501": "(A+Bz)\\frac{d^2w}{dz^2} + (C+Dz)\\frac{dw}{dz} +(E+Fz)w = 0", "e7f37dda86b6c977964d85391ab6b32a": "A \\langle x_1, x_2, \\ldots, x_n \\rangle", "e7f3d0253905a189bf9f25529a16581d": "I_{\\mathrm{sin}} = \\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{q}) \\right|\\sin\\left(\\phi+\\mathbf{q}\\cdot\\mathbf{r}\\right) =\n\\int \\frac{d\\mathbf{q}}{\\left(2\\pi\\right)^{3}} \\left|F(\\mathbf{-q}) \\right|\n\\sin\\left(-\\phi-\\mathbf{q}\\cdot\\mathbf{r}\\right) = -I_{\\mathrm{sin}}", "e7f40df3cbfd139de4add2c07ef2c59d": "a, b\\!", "e7f43a6ee111751958b5f6970483011f": "dS_{y}", "e7f455b8b5d9ce30c70d3eab955f550a": "\\nu_T", "e7f4896b0b18a060642a33035c6b9899": "L_n[1/3, c] = e^{(c+o(1))(\\ln n)^{1/3}(\\ln \\ln n)^{2/3}}", "e7f4e472731ce1210475f97a2c468716": "Q^{2(p+q)-1} = SU(p,q)/P \\subseteq \\mathbb{C}^{p+q}", "e7f59da4a578d676d1faf45864e44762": "( \\mathcal{H}, \\langle \\;,\\; \\rangle_f )", "e7f64b073d01258dc31408fcdaa356be": "\\gamma(x)", "e7f6d7c61299f8bbdce85089f568f8a4": "F_2=F_{load}\\frac{Sin(\\alpha )}{{Cos(\\alpha )Sin(\\beta )+Sin(\\alpha)Cos(\\beta )}} \\,", "e7f6f837e51e921219770a056cc5e9d8": " f(t_{n+s}, y_{n+s}) ", "e7f70f0c721f65442b385a11859a7bfe": "(0, N-y-1)", "e7f7638d51b6805801cf54b37a430f63": "\nH_{2}O_{(l)} + \\underbrace{237.2 \\ \\textrm{kJ / mol}}_{\\textrm{electricity}}+\\underbrace{48.6 \\ \\textrm{kJ / mol}}_{\\textrm{heat}}\\rightarrow H_2+ \\ ^1\\!/_2 O_2\n", "e7f7e742df0c16b9312ee3039bf9b2cb": "\n\\log p(\\mathbf{X}|\\boldsymbol\\theta) - \\log p(\\mathbf{X}|\\boldsymbol\\theta^{(t)})\n\\ge Q(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)}) - Q(\\boldsymbol\\theta^{(t)}|\\boldsymbol\\theta^{(t)}) \\,.\n", "e7f860b288bc11c53a9d629d931986fe": "\\theta=\\pi/4", "e7f8eba817865bf66046fd54a727d9d6": "(\\epsilon/c)", "e7f8f96439fc86e5e998c76768586569": "X_i(t)", "e7f949cd6f2c96905d026db57defc6e7": "\\scriptstyle P_D", "e7f96548cee9df78f31460a0fd486bd7": "j^{\\nu} = i \\left( \\frac{\\partial \\psi}{\\partial x^{\\mu}} \\psi^{*} - \\frac{\\partial \\psi^{*}}{\\partial x^{\\mu}} \\psi \\right) \\eta^{\\nu \\mu}~,", "e7f9a9f84254e34016c275401ffdb646": "\\begin{matrix}{4 \\choose 1}^3{32 \\choose 1}\\end{matrix}", "e7f9b8f0f85c1347dc28a2258960e8b8": "\\mbox{Re}(s) = \\sigma = 0", "e7f9c5d738d97e118d6b4e0dc51523f2": "P\\to P/H", "e7fa25ad03daa94bd48413459489dcd9": "\\bold{r}\\rightarrow \\bold{r}(-t)", "e7fa2f358081c2a1187a97ff5270dffd": "\\zeta(3)=\\sum_{k=1}^\\infty\\frac{1}{k^3}=1+\\frac{1}{2^3} + \\frac{1}{3^3} + \\frac{1}{4^3} + \\frac{1}{5^3} + \\frac{1}{6^3} + \\frac{1}{7^3} + \\frac{1}{8^3} + \\frac{1}{9^3} + \\cdots\\,\\!", "e7fa3fb6173e97f3751ba09b6e7228f8": " \\begin{align} \n\\mathbf{A} & = (A_t, \\, A_r, \\, A_\\theta, \\, A_\\phi) \\\\\n& = A_t \\mathbf{e}_t + A_r \\mathbf{e}_r + A_\\theta \\mathbf{e}_\\theta + A_\\phi \\mathbf{e}_\\phi \\\\\n\\end{align}", "e7fa42039f7b780fba0bd4c7f8b8e47d": "\\textstyle\\ f_{f_{max}}", "e7fa58e942a8d0013054ae6a2347cecd": "\\mathbf{j}(\\mathbf{r},t)\\cdot d\\mathbf{S} = \\Sigma(t)", "e7fa61257c3f88bf5d09adc15fd59995": "\\angle60^\\circ\\,", "e7faa8b09b5df6f5c6e647b655f0e87c": "F(k_x,k_y) = \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} f(x,y) e^{-j(k_x x + k_y y)} dx dy ", "e7fb081e7d6a49314607f263a85eef3c": "A_2", "e7fb4c87a1339379f5e2851dc290f0a2": " \\begin{pmatrix}-\\sqrt 5&2&1\\end{pmatrix}\\begin{pmatrix}\n1&1&0\\\\\n1&1&0\\\\\n1&0&1\\end{pmatrix}\\begin{pmatrix}-\\sqrt 5\\\\2\\\\1\\end{pmatrix}=10-5\\sqrt 5<0 ", "e7fba968fc251145d4c4b110f756781d": "0< \\left |x- \\frac{p}{q} \\right| < \\frac{1}{q^{n}}. ", "e7fbd961814f1d0493b488a0f6787e53": "G=(N,S,u)", "e7fbefa6cf17be0ada127f4f58811d6b": "{\\scriptstyle O(\\sqrt{b-a})}", "e7fc62c6e2e53a21330d95c906893396": "\\langle bb \\rangle", "e7fc66b0ce507b0e69cbd882001305bb": "\\Delta^y", "e7fc94cde321263423619d6db0c74af4": " \n\\vec{B} \\left( \\vec{R}\\right) = \\frac{\\mu _0}{4 \\pi} \\left\\lbrace 3 \\frac{\\left( \\vec{m} \\cdot \\vec{R} \\right) \\vec{R} }{ R^5} - \\frac{\\vec{m}}{R^3} \\right\\rbrace\n", "e7fcbffd521da4c346a3e1a354d88b20": "\\operatorname{cl}(X) = \\bigcup\\left\\{\\operatorname{cl}(Y) : Y\\subseteq X \\text{ and } Y \\text{ finite} \\right\\}.", "e7fcc6b513fc7c50a45c615eb05502c7": "|f_{1i}\\rangle", "e7fd55daeb3df44892ebdcaf22d5cf10": " \\eta_B = \\frac{B_{out}}{B_{in}} = 1 - \\frac{(B_{lost}+B_{destroyed})}{B_{in}} \\qquad \\mbox{(2)} ", "e7fda9c5502d36c2dd32f085720c67ab": "\\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix}.", "e7fe074b0f51e7c3850df428ea277f65": "(\\mathcal{P}(X),\\subseteq)", "e7fe1306ca6f9d4ec9ceb4b411a4f9bc": "\\langle r,f \\mid r^{2n}, r^n=f^2, frf^{-1}=r^{-1} \\rangle\\,\\!", "e7ff17408882c0c0ebc7008e476184f1": "E_{W_{i+1}}[h_i(x_j) y_j]=0", "e7ff3a70ff4acb1e7ffef3b95c4bd0f6": "\\beta=\\beta_1", "e7ff507925ff8f99d90a8bc4198a9f47": " \\tilde{O}(n^2) ", "e7ff7a8b7eba259c40f0d814b0381aa8": "T(e_n)=n\\|e_n\\|\\,", "e7ff8cbc68fdbee50b5db8098bf5bfa2": "[\\mathfrak{g'}, \\mathfrak{g}] \\subseteq \\mathfrak{g'}.", "e8005aa16ee1283fcd7285ed1db042e6": " u_t(0) = u^{2,0} \\mbox{ in } \\Omega", "e8006aaeffc198fe63db939eb059e641": "x=2^{2^i}\\times d+r", "e800f10ab5a5a053eb17737d5085e2f8": "2 T_n(x) = \\frac{1}{n+1}\\; \\frac{d}{dx} T_{n+1}(x) - \\frac{1}{n-1}\\; \\frac{d}{dx} T_{n-1}(x) \\mbox{ , }\\quad n=1,\\ldots", "e800f46e3d1da5fda864076555c56b63": "\\Omega^n = \\Omega", "e800fc6ce297921c442cf385f87881de": "0\\cdot\\infty", "e8011a9c64cdacae28736a3b92a7c2f9": " z^{-n_0}", "e801455a4767c9cc4057059ced44adfb": "M_{y}(t) = \\text {Im} \\left (M_{xy} (t) \\right ) = -M_{xy} (0) \\sin (\\omega _0 t)", "e801c9c16ab203bee11598e414e37e95": "\\tilde{\\phi}(\\vec{k})=\\int d^3x e^{-i\\vec{k}\\cdot\\vec{x}}\\phi(\\vec{x}), ", "e801d2caa34b2b80ba0aecd0b412fb37": "n^x\\,\\bmod\\,n = 0", "e80296408671ca48765083b433f14a20": " \\mu(S) = \\int_S \\frac{1}{|t|} \\, dt ", "e80298df3a4c4574e25999763040f7ab": "n\\cdot (p(t)-p_E)", "e8030ac5ed8b64b85552a7e30f7f6bed": "0 \\chi^2_{k-1,\\alpha}", "e81f91bb37a042f9ef53f80a3dcbc132": "{1\\over 2(1-p^{-2})(1-p^{-4})\\cdots (1-p^{-2t})}", "e8202297593498014e50ce263bd843ed": "\\pi(R_n)\\,", "e820bdd7aa8b4238bb7ed719e1907372": "L_{3} = \\left(1 - \\alpha - \\beta\\right) \\left(1 - \\alpha - \\beta - \\gamma\\right)^2", "e820c8fcfc9f87bce913dcd118eca3b3": " H(I, p, q, \\phi, t) = {1 \\over 2 }I^2 + p^2 + \\epsilon \\cos{( q - 1)} + \\mu\\cos{(q - 1)}(\\sin{\\phi + \\cos t)} ", "e820c925b0b6f89f332b6f683ea5bd0c": "(P_3P_7) ", "e820ee82cc007065f84a7396f4eaa18b": "\\mu~", "e820f27fd3d5a3db6af3fe3a816230f8": "G_{ij}=X_i \\cdot X_j,K_{ij}=Y_i \\cdot Y_j \\,\\!", "e821483b85ea680754a33c1479514760": "\\ F_{tuning}(s) = K_o \\cdot \\ V_{in}(s) ", "e8217630725664a3e29c5eb368f009a4": "\\langle r,s,t \\mid r^2 = s^3 = t^4 = rst \\rangle", "e8218c57f23c12ec7f16c88a4007f9ba": " \\displaystyle \\delta xy", "e821bfe2344e4bdfd8ff0481568ecbe5": "H(x_1,x_2,\\dots) = \\sum_s V(s)\\,", "e821c6207a8135ae719ff76e4a22ad16": "\\begin{align}\ns_1=&u_2v_3-u_3v_2\\\\\ns_2=&u_3v_1-u_1v_3\\\\\ns_3=&u_1v_2-u_2v_1\n\\end{align}", "e821c6e2014bea3046dd205006636cd0": " k = 2\\pi / \\lambda", "e821e06eeca758c767da37115d7fed76": "\n\\vec{\\nabla}\\cdot\\left[\\epsilon(\\vec{r})\\vec{\\nabla}\\Psi(\\vec{r})\\right] = -4\\pi\\rho^{f}(\\vec{r}) - 4\\pi\\sum_{i}c_{i}^{\\infty}z_{i} q \\lambda(\\vec{r}) \\exp \\left[{\\frac{-z_{i}q\\Psi(\\vec{r})}{k_B T}}\\right]\n", "e822711d0d9a1d143e288f9db2811f60": "\\!\\mathcal A \\models_X^- \\exists x \\phi", "e82293904bc096f1b2cea5d8a2dd582d": "\\,H", "e822c59057105b79a98de5b2bcc541a6": "\\vec \\omega(t) \\times \\vec{r}(t) = W(t) \\vec{r}(t) ", "e822f30cd17c815564df66ba9f04f707": " \\forall ", "e823346c6b5e3e5586eb19f3c0e4d7d3": "N=2k(a\\phi(front))=2ka(\\theta-\\psi)-2k\\frac{a^2}{V}\\frac{d\\theta}{dt}", "e8238865fafd01fe952a86465f334187": "\\omega_n = c \\sqrt{{k_x}^2 + {k_y}^2 + \\frac{n^2\\pi^2}{a^2}}", "e823a482c41e0b28d986d5d7160c0d84": "X \\doublebarwedge Y .", "e823a855612985dfa245279b44faf2cc": "x, y \\in X", "e823dc00b0cae164c946cbb63bcf2453": "\n A^{T}A = \\begin{bmatrix}\n 1 & 4\\\\\n 2 & 5\\\\\n 3 & 6\n \\end{bmatrix} \\cdot\n \\begin{bmatrix}\n 1 & 2 & 3 \\\\\n 4 & 5 & 6\n \\end{bmatrix} =\n \\begin{bmatrix}\n 17 & 22 & 27 \\\\\n 22 & 29 & 36\\\\\n 27 & 36 & 45\n \\end{bmatrix}\n", "e823fd7d83218985f490a5c2706db839": " x^{-1} \\in T ", "e82405fc3449f35f8018f1767bbf3bfb": "nk = -(k-n)^2/2 + n^2/2 + k^2/2", "e824210344a7dbfca04bd956b6a09295": "E(s) = R(s) - F(s)Y(s).\\,\\!", "e82421759f0a9c419d90785eeca0fe18": "\\lambda n(n-1)dt", "e8242d92ee6b744ac88196561fa7f136": "\\frac {d M_z(t)} {d t} = i \\frac{\\gamma}{2} \\left ( M'_{xy} (t) \\overline{B'_{xy} (t)} - \n\\overline {M'_{xy}} (t) B'_{xy} (t) \\right )\n- \\frac {M_z - M_0} {T_1}", "e82447b9e6f5407cb74c9bed7ddfe575": "\\sigma^2 ", "e82460cb02f53482659fefc18a6974ba": "S({\\mathbf u})=(x^1({\\mathbf u}),\\dots,x^n({\\mathbf u}))", "e824a53e268915b6f46b2f2b45527532": " \nV^0_{\\textrm{rev}}=\\frac{\\Delta G^0}{n\\cdot F}=\\frac{237 \\ \\textrm{kJ/mol}}{2 \\times 96,485 \\ \\textrm{C/mol}}=1.23 V\n", "e824d2a968021338bc45f7874eb80716": "\\{y\\}", "e824d2e98acf94b3e0d5e5d2e1040895": "\\sum_{k=0}^7 e^\\frac{2i\\pi 3^k }{41}", "e82502ea4d0ee6e73766a0efb48d25fe": "g_{ij}=1", "e8250ea3f830c7e8a0dc666c5393b97b": "P(k'|k)", "e8255aa82c2657787dd2810c1596f628": "\\begin{alignat}{13}\nf(t) &&\\; = \\;&& a_5 t^5 &&\\; + \\;&& a_4 t^4 &&\\; + \\;&& a_3 t^3 &&\\; + \\;&& a_2 t^2 &&\\; + \\;&& a_1 t &&\\; + \\;&& a_0 & \\\\\nf'(t) &&\\; = \\;&& 5 a_5 t^4 &&\\; + \\;&& 4 a_4 t^3 &&\\; + \\;&& 3 a_3 t^2 &&\\; + \\;&& 2 a_2 t &&\\; + \\;&& a_1 & \\\\\nf''(t) &&\\; = \\;&& 20 a_5 t^3 &&\\; + \\;&& 12 a_4 t^2 &&\\; + \\;&& 6 a_3 t &&\\; + \\;&& 2 a_2 &\n\\end{alignat}", "e8256e9ccb9149908eac4c27e3cfe3f2": "f(x_1, \\ldots, x_n, x_{n+1}, \\ldots, x_m) = (x_1, \\ldots, x_n).", "e82570e4e95ef43a95fe6fefb28e03ec": "{}^qH = \\frac{ 1 }{ 1 - q } \\; \\ln\\left ( \\sum_{ i = 1 }^K p_i^q \\right ) ", "e8261ad155e8b0acce6354558c4135a2": "\\operatorname{Aut}(A_3)=\\operatorname{Out}(A_3)=S_3/A_3=C_2", "e8263ebc3771753bd592b07b1455785c": "-2t^3+3t^2", "e826c4963fb48c10a7768dd21e07b67f": "\\mathfrak{P}^{98}", "e827bdf95fc99d6e324b26e101aee4c4": "\\tau_\\max=\\frac{1}{2}\\left|\\sigma_1-\\sigma_3\\right|\\,\\!", "e8280093a989b5c85e0bcdbde56ca024": "75 \\times 0.12 {{=}} 9\\mbox{ units}", "e8280907c8d32d5af0f36e5442e84a34": "LS", "e8284ea194ecadb313b004844693f8de": "{\\mathbb R}^d", "e8284ec9a51c8b843ea34c65f60081ae": "{S = R^{n}}", "e82865151fbe957efa8e39a38b46a68a": "P_\\lambda=\\{ g\\in S_n : g \\text{ preserves each row of } \\lambda \\}", "e828761eda74a0df805dd0f448ae0ba2": "\\lambda_i = 0", "e828b1243cc57db2aad160c47612da9d": " u=u^\\lambda(x^\\mu)\\partial_\\lambda + u^i(x^\\mu,y^j)\\partial_i ", "e829777ab97c93ef78b95b37ec071e96": "\\frac{1}{e}", "e8297ed25f5c91fc1e3cec42c4fd388c": "{\\hat{\\beta}}(q, {r_{\\rm w}}) < {\\hat{\\beta}}(q^\\prime, {r_{\\rm w}}).", "e829f3f3615dbc7b626b75b2edef8990": "|F_{if}-F_{bfo}|\\,", "e82a7fa4ae0857880485af728da768c6": "F(t,z)", "e82afeb8419b03f6229090d8e8eae701": " \\{ x \\in \\mathbf{R} | x^2 = 1 \\} = \\{ x \\in \\mathbf{R} | \\ |x| = 1 \\} ", "e82bdc2cbe62c3207733ce005e88bcd2": "F_{(2)}", "e82bf2e7a0624ec4fc31e93079498f31": "\\psi(x_{\\pi(1)})\\cdots \\psi(x_{\\pi(n)})|\\Omega\\rangle", "e82c515e4b2ae7da4b61bdd7720c24db": "U=\\sum_{all x}\\left\\{ W(x)\\left[y(x) - \\sum_i \\alpha_i B_{i,k,t}(x)\\right] \\right\\}^2", "e82c5433226f7969d8620196d48130dd": "\\sum_{n = -\\infty}^{\\infty}{\\left|h[n]\\right|}\n\n= \\sum_{n = -\\infty}^{\\infty}{\\left|h[n]\\right| \\left| e^{-j \\omega n} \\right|}", "e82c6dfa67fedb90846c563b4aabd662": "\\mathbf{I}^{A}", "e82c7e5591e7cf5106e3b5a4c1cea899": "\\frac{dx_i}{dt} = r_i x_i \\left( 1 - \\sum_{j=1}^N \\alpha_{ij}x_j \\right) ", "e82c8d709305fe02cde8e00a659e043d": "\nR^n+|a_{n-1}|\\,R^{n-1}+\\dots+|a_{1}|\\,R=|a_0|\\,.\n", "e82ce542095df820b8296694260effc5": "M\\times S^1", "e82d069a75db0426e7b8b22e1e19e654": "\\lim_{n\\to\\infty}H^{\\circ n}(A)=S", "e82d1ce31147dbc328c8d56f9890b248": "y^{j+1}\\sum_{i=j}^\\infty {i \\choose j} \\frac{1}{(1+y)^{i+1}}=1.", "e82d9e2b662f1afd3222f7110ef80097": "\\;\\;= \\;\\; 2 + \\tan^2 a + \\tan^2 b - 2 \\sec a \\sec b \\cos c ", "e82e5fc124a1ebe7cb2a2db73bc03eec": "\\gamma:[a,b]\\to M", "e82e61d7f7935bd8d2eb13856489ed8f": "(10) \\ \\alpha_\\ell = \\varphi + \\alpha", "e82eb3df2175ae182190249aaacbd3a7": "\\deg(v)", "e82ec3b96132c4419f5b3b4b6c3bce36": "\\mathbf{F}_c = m\\mathbf{a}_c = -2 m \\boldsymbol{ \\omega \\times v}", "e82efaa5223d0760fdf24d7e1ccca446": "G_1' \\in \\mathcal{G'}", "e82f8ca5e366117b057667450d6d8efb": "(x, f), (x', f') \\in fRep_{red}", "e82fedd962f9871184a6e7dda3a0ca65": "\\mathbf{v}[\\mathbf{f}A] = A^{-1}\\mathbf{v}[\\mathbf{f}],\\quad \\mathbf{v}^\\sharp[\\mathbf{f}A] = A^T\\mathbf{v}^\\sharp[\\mathbf{f}].", "e8301fb6e02a77bcaf9e941eaba0b57b": "C_2=2", "e8302119570eed406f7e6266202510d0": "A \\approx \\frac{d^2}{1.8544},", "e830257866f1bcfb80b18710e596e353": "x_0 \\in \\operatorname{core}(A), y \\in A, 0 < \\lambda \\leq 1", "e8303bfb8221c694634decf41a043c2e": "\\psi\\,:\\,E_{a,d} \\rightarrow M_{A,B}", "e830f8f231bfa519204c9de2a9f4dd26": "\\mathbf \\nabla \\cdot \\mathbf A + \\frac{1}{c^2}\\frac{\\partial \\varphi}{\\partial t} = 0", "e83106502e95f51e707133459ea18441": "\\lim_{x \\to \\infty} \\frac{e^x+e^{-x}}{e^x-e^{-x}} = \\lim_{y \\to \\infty} \\frac{y+y^{-1}}{y-y^{-1}} = \\lim_{y \\to \\infty} \\frac{1-y^{-2}}{1+y^{-2}} = \\frac{1}{1} = 1.", "e8312c3f6269d941322fb6bc88faaa94": " r = |z| = \\sqrt{x^2 + y^2} \\ . ", "e8317651df0fcee753764982f0924d01": "x^2 \\left( {a^2 - c^2 \\over a^2} \\right) + y^2 = a^2 - c^2", "e8319565b873392ff5b5ba3ea79c27af": " J_n = \\sum_{i=1}^n S_i, ", "e831d279eb971da37b20a762b975f611": "\nF_r = P ( \\pi r^2 )\n", "e832016f7923512adad89495055bc1f8": "P(1+kp) = P(1)", "e83205526e0d7b8a0da84dcfb632c89f": "\n-2\\pi\\ \\frac{3}{2} \\left(\\frac{3}{2}\\ \\sin^2 i\\ -\\ 1\\right)\\ \\left(-e_h \\hat{G}\\ +\\ e_g \\hat{H}\\right)\n", "e832e70b5cd213adb09f1e8916a25e69": " \\phi = {{SH*p}\\over {(0.622+0.378*SH) p^*_{(H_2O)}}}\\times 100 ", "e833d1a83635ac53c3804328dd2f91c6": " p_\\text{up or down} = p_\\text{up} + p_\\text{down} = 1 ", "e833dbe5ee508b8cba4edf3f97f2ec07": "\\sigma S={\\left(\\frac{k}{e}\\right)\\sigma_{0}}{I}_{1}", "e833df36762297094a969b1f5a27cc16": "s \\ ", "e83436e61129accfa4104f70a23ed1bb": "( G1 + G2 ) / 2 = ( G2 + G1 ) / 2", "e83446505e31f625b5980ddd65b8cfa2": "\nA_Q = \n\\begin{pmatrix}\n A & B/2 & D/2 \\\\\n B/2 & C & E/2 \\\\\n D/2 & E/2 & F\n\\end{pmatrix}.\n", "e8345cf1f9882b2ce8cd450316ecdc90": "N_{kl}=\\sqrt{\\sqrt{\\frac{2\\nu ^{3}}{\\pi }}\\frac{2^{k+2l+3}\\;k!\\;\\nu ^{l}}{\n(2k+2l+1)!!}}\\,", "e8348b77569801a1180f0a351a726799": "p_{1},\\ p_{2},\\dots,\\ p_{n}", "e8348b935327c13c3e12192db879ddec": "*\\circ * : \\Lambda^k(V) \\to \\Lambda^k(V) = (-1)^{k(n-k) + q}I", "e8348e4e3336714a68aa1af451928c73": "\\langle T_H\\rangle = \\frac{1}{\\Delta S} \\int_{Q_{in}} TdS", "e8349d8f2fc2ee013a7c0da0d28e81ca": "S^{r+q}", "e834ede8aa275254c029c612d765312c": " y = \\frac{8au}{(1+u^2)^2}.", "e834fc7fb2a6c3be620b469e4daac44c": "\\nu(\\gamma, z)=\\upsilon(\\gamma) (cz+d)^k", "e83528f9c4ecfe31e9cf8013a5e49788": " y(t) =\\lambda x(t)+\\int^{\\infty}_0 k(t-s)x(s)ds,\\quad 0\\leq t<\\infty ", "e835574cc8559204c34d1525f43186e9": "Y-C-T=S", "e8359a5a5a5344b623a2b9710d5a58ac": "\\mathrm{^{252}_{\\ 98}Cf\\ \\xrightarrow {(n,\\gamma)} \\ ^{253}_{\\ 98}Cf\\ \\xrightarrow [17.81 \\ d]{\\beta^-} \\ ^{253}_{\\ 99}Es\\ \\xrightarrow {(n,\\gamma)} \\ ^{254}_{\\ 99}Es\\ \\xrightarrow []{\\beta^-} \\ ^{254}_{100}Fm}", "e83608573a3ba6447362260b4a5637b3": "\\{ B_i \\}", "e83640237075ccef39735c794ca365da": "f : R \\times R \\rightarrow S", "e8365cb2f85185c31ad2bf18ebbec384": "< \\tfrac{\\lambda}{10}", "e83669f500913d6cce76b40aa8ceda5d": "G_0 = G \\to G_1 = G_0/Z(G_0) \\to G_2 = G_1/Z(G_1) \\to \\cdots", "e836b82356ffbd10e04d825722d23432": "dV=\\delta W/p", "e836ed68082c8aaf5486ff22925a154f": "\\tau(n)\\equiv n^{-30}\\sigma_{71}(n)\\ \\bmod\\ 5^{3}\\text{ for }n\\not\\equiv 0\\ \\bmod\\ 5", "e83741c3d86122f98a57f31d542cc042": " C_{(-)}= C_{(-)}^* = s_{(d+2,-)} C_{(-)}^T = s_{(d+2,-)} C_{(-)}^{-1} ~~~~ s_{(d+2,-)}=-s_{(d,+)} ", "e83776557d80f0b29f3456db6c0373e4": "\\lVert z \\rVert_\\Psi^2 = \\lVert z_1 \\rVert^2 + \\dots + \\lVert z_p \\rVert^2 - \\lVert z_{p+1} \\rVert^2 - \\dots - \\lVert z_n \\rVert^2", "e8378a60b6aa132013938a1ae8ae355b": "\n\\mathbf{J} = \n\\begin{pmatrix}\n1 & -2 & 4 & -8 \\\\ \n1 & - 1 & 1 & -1 \\\\ \n1 & 0 & 0 & 0 \\\\\n1 & 1 & 1 & 1 \\\\\n1 & 2 & 4 & 8\n\\end{pmatrix}\n", "e8379fb71650fb522bbff9355c2065cf": "ds^2= - c^2 dt^2 + dl^2 + (k^2 + l^2)(d \\theta^2 + \\sin^2 \\theta \\, d\\phi^2).", "e837ad1d71d242ad2bc10867e8ca52d4": "L=N\\left ( \\mathrm{d} \\Phi/\\mathrm{d} I \\right )\\,\\!", "e837b3e4fd2d40cb2ed1e70b5647135b": "L \\in (N \\cup T)^*N(N\\cup T)^*, R \\in (N\\cup T)^*", "e837b77585e3f5c459a51e0640524826": "\\zeta(s,x+y) = \\sum_{k=0}^\\infty \\frac {y^k} {k!}\n\\frac {\\partial^k} {\\partial x^k} \\zeta (s,x) =\n\\sum_{k=0}^\\infty {s+k-1 \\choose s-1} (-y)^k \\zeta (s+k,x).", "e837fedf56b7e887bdf880b9dfd9f7b5": "\\operatorname{Aut}(\\widehat{\\mathbf{C}})\\,", "e8383d21c3e4fb943122623eec88748d": "\\mathbf{P} = \\chi_\\text{e}\\mathbf{E}", "e8385b1c9ee9a8380038d89595071862": " \\Psi_\\mathrm{total} = \\psi_\\mathrm{electronic} \\times \\psi_\\mathrm{nuclear} ", "e838c2e48f634682c69d1f02d3b0e014": "p = 2100", "e8393be12ced9c61144a1b66c3a0e484": " [\\Omega] = \\dot{A}A^T,", "e8395f04f39a000feed67ca9729ebeaf": "f(x) = \\frac{1}{1+25x^2}.\\,", "e83a535898751ac91a44196a8d5dbdaa": "\\textstyle H_0", "e83a7a8d48146999d82bab56701af7dc": "\\begin{bmatrix}1&0\\\\1&0\\end{bmatrix}", "e83aa1c239b8472d27361f4e738bdbfe": "{\\tilde{C}}_{7}", "e83abf63c832eaca06a11c34d9d33232": "C_{112}=2B+2C", "e83ac965d105b951fdc15687495ed2dd": " \\frac{d}{dx}\\tan(x) = \\sec^2(x) = \\frac{1}{\\cos^2(x)} = 1+\\tan^2(x).", "e83afda82a90ef183d91a80322f4ab8c": "m_{\\text{e}}", "e83b00f84bfaf8139ae7d4567d87431d": "\n\\mathrm{round}\n\\left(\n \\frac{-415.37}{16}\n\\right)\n=\n\\mathrm{round}\n\\left(\n -25.96\n\\right)\n=\n-26.\n", "e83b10c0e494a31baa321762566765cd": "y_i^{\\prime\\prime} = ?", "e83b1466d5cd3b2db00973cbf1218b06": " \\pi_A, \\pi_G, \\pi_C, \\pi_T \\ ", "e83b35547eb1b12dbbc4007985357376": "\\{F,G\\}=dG(X_F) = X_F(G)\\,", "e83b49c6dd2c4780e85b2ad2efb7922f": "\n \\begin{align}\n &\\kappa^2\\, =\\, k^2\\, +\\, \\frac{\\nabla(c_p\\, c_g)}{c_p\\, c_g} \\cdot \\frac{\\nabla a}{a}\\, +\\, \\frac{\\Delta a}{a}\n \\quad \\text{and} \\\\\n &c_p\\, c_g\\, \\nabla\\left(a^2\\right) \\cdot \\boldsymbol{\\kappa}\\,\n +\\, c_p\\, c_g\\, \\nabla\\cdot\\boldsymbol{\\kappa}\\, \n +\\, a^2\\, \\boldsymbol{\\kappa} \\cdot \\nabla \\left( c_p\\, c_g \\right)\\,\n =\\, 0.\n \\end{align}\n", "e83b57a9555e461b389607c7a59800a1": " \\mathbf{a} = \\mathrm{d} \\mathbf{v}/\\mathrm{d} t = \\mathrm{d}^2 \\mathbf{r}/\\mathrm{d} t^2 \\,\\!", "e83b76b18add3e1d71c264d63e849363": "\\forall t < T-1: \\rho^{com}_t := \\rho_t(-\\rho^{com}_{t+1})", "e83b8f4fa0149c4eee7635af90fd0f57": "y \\equiv rs_1^{a_1}s_2^{a_2} \\cdots s_k^{a_k}\\pmod{n}", "e83b97023cf122259be110b33e622a8d": "\\scriptstyle\\mapsto\\,", "e83b9881e5d1ee214d7c08b520e86735": "2^{2^{65\\,536}}\\approx 10^{6.0 \\times 10^{19\\,727}}", "e83c0ba26630c79740dd7fc038202923": "{2\\pi^{3N/2}(2mE)^{{3N-1}\\over 2}}\\over {\\Gamma(3N/2)}", "e83c10d15d20cbe07cfafe9298374f1e": "c_\\mathrm a\\, ,", "e83c29b910dd45e9e9a6b745b6e4bdbb": "\\begin{align}A \\sum_{n=0}^{\\infty} \\frac{1}{\\ln F_{n}} &= \\frac{A}{\\ln 2} \\sum_{n=0}^{\\infty} \\frac{1}{\\log_{2}(2^{2^{n}}+1)}\\\\ &< \\frac{A}{\\ln 2} \\sum_{n=0}^{\\infty} 2^{-n} \\\\ &= \\frac{2A}{\\ln 2}.\\end{align}", "e83cb0bab78588b8ca4a585b6d7cc1c6": "\\Delta G = \\Delta H - T \\Delta S \\ ", "e83ceb3225013a9c1d1133ec3a320a70": "d\\mathbf{x}^2 - d\\mathbf{X}^2 = d\\mathbf X \\cdot 2\\mathbf E \\cdot d\\mathbf X \\quad\\text{or}\\quad (dx)^2 - (dX)^2 = 2E_{KL}\\,dX_K\\,dX_L\\,\\!", "e83d272f9af25cb34cfc47b82cc4c695": " \\phi(z) \\to 1 ", "e83d82222b8c48d90f6bdf4d898c5c43": "\ng_3(\\omega_1,\\omega_2)=\n\\frac{\\pi^6}{(2\\omega_1)^6}\\frac{8}{27}\n\\left[\n \\left(\\theta_2(0,q)^{12}+\\theta_3(0,q)^{12}\\right)\\right.\n", "e83db493fc9a14dea3c064b8a36bf83c": "\\varepsilon_{nit}", "e83e7fa6751b6cec386f15a23a205a6b": "f_n = \\begin{vmatrix}\na_1 & b_1 \\\\\nc_1 & a_2 & b_2 \\\\\n& c_2 & \\ddots & \\ddots \\\\\n& & \\ddots & \\ddots & b_{n-1} \\\\\n& & & c_{n-1} & a_n\n\\end{vmatrix}.", "e83eb5e4a91b7a85735b63c4d80eb419": "F_{weight} = -\\rho \\cdot g \\cdot A \\cdot h", "e83ef96c3d3e8e54b49412399153ffc8": "{CE}_{5}", "e83efdc59785cab11f024577d7c8df53": "K=\\mathrm{ \\frac {[H_3N^+CH_2CO_2^-]} {[H_2NCH_2CO_2H]}}", "e83f0f3222c6ddc12a518b1bca44417c": " \n H_{e} = \\sum_{k,s} \\xi(k,s) c_{k,s}^{\\dagger} c_{k,s}\n", "e83f1f39d9573926372d3bb8742b10ac": "S^2|s,m\\rangle=s(s+1)\\hbar^2|s,m\\rangle ", "e83f3c00471daf78f3a16ec324509e48": " W \\in \\operatorname{FV}[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V, W : E \\and F \\operatorname{in} L] \\equiv \\operatorname{de-let}[\\operatorname{let} V: V\\ W = \\operatorname{get-lambda}[V, E][V:=V\\ W] \\operatorname{in} \\operatorname{let} W: F[V:=V\\ W] \\operatorname{in} L[V:=V\\ W]] ", "e83f6a57bb0615d5bfa60746acb9cdbd": "G_{\\mu\\nu}\\equiv R_{\\mu\\nu} - {\\textstyle 1 \\over 2}R\\,g_{\\mu\\nu} = {8 \\pi G \\over c^4} T_{\\mu\\nu}.\\,", "e83fb4bb3fce9c29ffbf2040a9f3de93": "\\left(\\tfrac{3}{2}\\div\\left(\\tfrac{8}{7}\\right)^3 = \\tfrac{1029}{1024}\\right)", "e83fe210b392fad9afb88a579d9d8727": "\\frac{\\partial(x, y, z)}{\\partial(u, v, w)}", "e84031af952968653172b91bdf136910": "s(b,c)=\\frac{-1}{c} \\sum_\\omega\n\\frac{1} { (1-\\omega^b) (1-\\omega ) } \n+\\frac{1}{4} - \\frac{1}{4c},", "e840d603557f1c78d9dab573ee5d8594": "n(\\mathbf{X}^{n-1})^{\\rm T}", "e840db721aaa2a0a1b7c62a12f6f075b": "[\\omega]^\\omega", "e841663af01dc060b6b0a868fe5108ca": "f_{out}(x) = p f_{in}(x) + q f_{in}(x+1)", "e8417cc07e6976b5b215cf8ecd88d999": "P(L) = \\{p(w) \\mid w \\in L\\}", "e8419ed387cf21327b9e2f60c65c4e6d": "\\bigcup_{k\\in\\mathbb{N}} \\mbox{NSPACE}(2^{n^k})", "e841c1914254f4de7c872fc34355c49a": "\nE_0 = 10^{-12} \\mathrm{\\frac{J}{m^3}}\n", "e84202d4af8ccd91460714326d4f3357": "\\nabla' g = \\varphi \\otimes g", "e8424a24ddf0d6359471bc921044e7fc": " = \\frac{V_{CC}-V_{BE1}}{R_1} \\ . ", "e8427eacfb6959afab35ef3f0e8e299f": "LADD = (Contaminant Concentration)(Intake Rate)(Exposure Duration)/(Body Weight)(Average Lifetime)", "e84294cef656f80060d87e374787c790": "\\mathrm{ROOH + QO + ^{\\cdot}OH + RH \\ \\longrightarrow {} \\ ROOH + QO + H_2O + R^{\\cdot} \\ \\longrightarrow {} \\ RO^{\\cdot} + ROH + QO + H_2O }", "e84294dd649c7a79a46d82b3139af0d2": "P = u \\cdot (s-t),\\,", "e842b500fc690b74fda9e377aa5355d8": "X = \\begin{bmatrix}\nc+b & a\\\\\na & c-b\n\\end{bmatrix}.\n", "e842d8aab073c41dcca8264755efcd3d": "ka \\equiv ma \\pmod p, \\,\\!", "e842fbd4ce2ddf854543a0b4fe48930b": "\\vec{e}_0, \\; \\vec{e}_1, \\; \\vec{e}_2, \\; \\vec{e}_3", "e84389b6562a3d3e69f48ba32ab4d15c": "\\pi_0^{S}", "e843c056eb390d7cfb38f24fb7de1bb9": " \\hat{U} \\approx I + i\\hat{H} ", "e843dd47a09dbf35a95c4d39769b21c2": "\n\\eta( \\mathbf{p}) = {i \\over \\sqrt{2}} \\left[ Q_L(\\mathbf{p})\n- Q_R(\\mathbf{p}) \\right]. \\quad\\quad\\quad\\quad (12)\n\n", "e844a4ab46646de73a9a16be89e7817c": "\\frac{X_{n+1}}{s} \\sim T^{n-1}.", "e8455fc80d9cb0e1d9f9f92f602877a7": " E = -1/(n+d)^2", "e845647bb16f8df73b95ecb2e938e067": "W^s(f,p) = W^s(f^k,p)", "e845e27d131f10f2a33baf827dd0700e": " e_j ", "e84655bd674598e577000d9b85c2a9ef": "\\mathrm{Aut}(\\mathfrak g)", "e8467585d706a6fa44dc0bec6d55751c": "\n\\Pr(X = x) = f(x; \\lambda, \\nu) = \\frac{\\lambda^x}{(x!)^\\nu}\\frac{1}{Z(\\lambda,\\nu)}, ", "e84687367168bd1f7e3f0132e08126c8": "\n f(r | H=h, T=t)\n = \\frac{{N \\choose h}\\,r^h\\,(1-r)^t}\n {\\int_0^1 {N \\choose h}\\,r^h\\,(1-r)^t\\,dr}\n = \\frac{r^h\\,(1-r)^t}{\\int_0^1 r^h\\,(1-r)^t\\,dr}\n .\n ", "e846b8c99a8469034fee42a395c484db": "\\left(\\pm\\sqrt{10},\\ \\pm\\sqrt{6},\\ \\pm\\sqrt{3},\\ \\pm1\\right)", "e846bb3c214242f5a20ca962fe8edeb3": "M\\cdot V = P\\cdot Q.", "e846c00732c835902bd6002ec493637d": "u=u_{0}\\cos(\\theta-\\theta_{0})-\\kappa,", "e846e2e89417f26c0ffdbd310e327eec": "\\Delta \\boldsymbol\\beta", "e846e5dc389863e2c7e59e90759a4806": "\\frac{1}{2} c_p \\left( 1 + \\frac{2 k h}{\\sinh\\left(2 k h \\right)} \\right)", "e8471ccdc646afb0eada4e93135ee143": "\n\\begin{align}\na(1)&=a(2)=1, \\\\\na(n)&=a(a(n-1))+a(n-a(n-1)), \\quad n>2.\n\\end{align}\n", "e8477195b5f3703d8cd81f1b9cbc0ff2": "\\scriptstyle\\mathbb{Q}", "e847c7a66ddf353480a6954e31de5c46": "=\\frac{\\mu_0r^2N^2\\pi}{l}\\left[ 1-\\frac{8w}{3\\pi }+\\sum_{n=1}^{\\infty }\n\\frac {\\left( 2n\\right)!^2} {n!^4 \\left(n+1\\right)\\left(2n-1\\right)2^{2n}}\n\\left( -1\\right) ^{n+1}w^{2n}\\right]", "e847d62e027fee7585e03dd59d04da60": "2.7783", "e8482542ce85914c68709f63c15f45aa": " \\Box F_{ab} \\ \\stackrel{\\mathrm{def}}{=}\\ F_{ab;}{}^d{}_d = \\, -2 R_{acbd} F^{cd} + R_{ae}F^e{}_b - R_{be}F^e{}_a + J_{a;b} - J_{b;a} ", "e8482556bbcb942cce9532ddf8d8d5d8": "S:[0,T]\\to\\mathbb{R}", "e848d427018c016c2256c27622e559e4": " V_{\\text{out}} = \\frac{ \\left( R_{\\text{f}} + R_1 \\right) R_{\\text{g}} }{\\left( R_{\\text{g}} + R_2 \\right) R_1} V_2 - \\frac{R_{\\text{f}}}{R_1} V_1 = \\left( \\frac{R_1 + R_{\\text{f}} }{R_1} \\right) \\cdot \\left( \\frac{R_{\\text{g}}}{R_{\\text{g}} + R_2} \\right) V_2 - \\frac{R_{\\text{f}}}{R_1} V_1. ", "e848e29bdb3fa22fd4ccb477a99def01": "\\mathrm{ker}(\\partial_1) = \\{b[v_1,v_2] + b[v_2,v_3] + b[v_3,v_1] | b\\in \\mathbb{Z}\\} \\cong \\mathbb{Z}.", "e8495b4122029586a5a5b60c97866d1f": "\\tau_{(Q)} = \\alpha_{(Q)}Q - f_{\\left(\\alpha_{(Q)}\\right)}", "e849c3c421c8619c94d58c0e5336029f": "f(k;s,N)=\\frac{1}{k^s H_{N,s}}", "e84a3c1c94febecb6c560324a5f89ecd": "\n \\underline{\\underline{\\boldsymbol{\\omega}}} = \\begin{bmatrix} 0 & -w_3 & w_2 \\\\ w_3 & 0 & -w_1 \\\\ -w_2 & w_1 & 0\\end{bmatrix} ~;~~ \\underline{\\underline{\\mathbf{w}}} = \\begin{bmatrix} w_1 \\\\ w_2 \\\\ w_3 \\end{bmatrix}\n ", "e84afaab83ecb301b3d97ce4174d2773": "reply", "e84b23b8284b31d2939609f8d037398f": " \\mu_{A} \\, = \\, \\mu_{g}", "e84b51db5f667fb9755175f9040e62a7": "\\chi=V-E+F \\,\\!", "e84b8f0c51b6a5b09c122428fc4f42c3": "\\scriptstyle -1", "e84be2e8d4396e1a5217a581c69daac1": "I_{im-}", "e84be48e264cc0ab0181426fe92c33cd": "2 \\sqrt{2}", "e84bff6bfe1741089e1af83fac86d987": "\\approx W_x(at,f)", "e84c285264286cf73eccc438fee50c30": "Q = C H^n", "e84cfebcd2624e78f5a3ef5640f0ed81": "x_1 \\ldots x_n", "e84d029471b270941dad20d0b44bb00b": " \\inf_n \\frac{E \\left[ X_n \\right ]^2}{E \\left[ X_n^2 \\right ]}>0\\,,", "e84d2aba83705c628c48b31560c19e57": "\n \\sigma_x = \\frac{\\partial^2\\varphi}{\\partial y^2} ~;~~\n \\sigma_y = \\frac{\\partial^2\\varphi}{\\partial x^2} ~;~~\n \\sigma_{xy} = -\\frac{\\partial^2\\varphi}{\\partial x \\partial y}-(f_{x}y+f_{y}x)\n ", "e84d980f34b1ca7908781350be9a27df": "\n\\frac{\\pi}{4}=\\sum_{n=2}^{\\infty}\\frac{\\zeta(n)-1}{n}\\mathfrak{I}((1+i)^n-(1+i^n))\n", "e84dbbd0058790c5b5de907b4601c22f": "\\varphi = (1+\\sqrt5)/2", "e84de77bec1a9a314759da073382bb29": "( t_k)", "e84e35da625bf60dbffc3b08c9b39c78": "p_2, p_3", "e84e45c8b4b53a8c8eaad5c66996ad2a": "\\forall x \\in t", "e84e4fa8b4d0372dcb2e1b4f251b0ac2": "\\frac{11}{18}", "e84e54c8f293a0a390c5956af1905037": "p=c_s^2 \\rho+c_0h[\\psi(\\vec{x})]^2\\,\\!", "e84e685dd3b0f6a504dea65213836b21": " AgCl(s)+ e^- \\leftrightharpoons Ag(s)+ Cl^-", "e84e9983ffeb1ad5a83522ee59105693": "((P \\to Q) \\and (Q \\to R)) \\to (P \\to R)", "e84ecf291e2c0d0390164b50c2dd8d43": "\\inf_{Q_{Y|X}(y|x)} E[D_Q[X,Y]] \\mbox{subject to}\\ I_Q(Y;X)\\leq R. ", "e84f0d5fcfd9c67a90ce97ffbd8df451": "\\exists x \\forall y, \\mbox{saw}(x,y)", "e84f54c87fc98341cecb9dd9872d713b": " L\\frac{d^2u}{dx^2} + g\\sin u = 0. ", "e84f6b5e1155613d993f8ec09e285b95": "r_{\\rm M}", "e84f7f905e89d4bde51f10d9df55c496": " \\nabla \\times \\mathbf{B} = \\frac{1}{\\epsilon_0 c^2} \\mathbf{J} + \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}} {\\partial t} ", "e84f9c3b559068b04612a35a70be7156": "\\Box^2", "e84fd80db5e2ed836903fcbfb771a7c4": "\\gamma(x,y)\\geq 0", "e84fec1e074026d6fa8e3155482c35c3": "g(x)", "e84ff6c3faf508949b46fa275998201d": "\\forall k > K, \\forall x \\in A \\ : \\ \\left|f(x) - \\sum_{n=1}^{k} f_n(x)\\right| < \\varepsilon.", "e85053589cd13d3ce9757645961f0f19": " \\hat \\sigma = \\sqrt{\\frac 1 n \\sum_{i=1}^n x_i^2}", "e8505882d81b5e2fe4fdbc517ba51020": "H_\\min^\\prime > 0", "e85068f0f068592cbf7acc0c9d710298": "\nu^*(x) = \\arg\\min_u \\nabla V(x,u) \\cdot f(x,u)\n", "e850daf642b2b126389861f5d943aebf": " \\int f(x) D_q g d_q x = (1-q)x\\sum_{k=0}^{\\infty}q^k f(q^k x) D_q g(q^k x) = (1-q)x\\sum_{k=0}^{\\infty}q^k f(q^k x)\\frac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^k x}, ", "e851088198087bdbe6644dc96f75156c": "\n\\rho = \\frac{\\mbox{corr}(\\hat{\\beta},\\hat{\\theta})}{\\sqrt{R_\\beta R_\\theta}}.\n", "e851153d31bd6c05e023116ae2ebf11c": "\nV(T_1, \\ldots, T_n) \n= \\left. \\frac{\\partial^n}{\\partial \\lambda_1 \\cdots \\partial \\lambda_n}\\right|_{\\lambda_1 = \\cdots = \\lambda_n = +0} \n \\mathrm{Vol}_n(\\lambda_1 T_1 + \\cdots + \\lambda_n T_n).", "e8515a5f2702008932d2a58f9366d1c3": " {P}=\\sqrt{\\frac{1}{2\\hbar m\\omega }}\\ {p}\\text{,}\\quad {X}=\\sqrt{\\frac{m\\omega }{2\\hbar }}\\ {x}\\text{,}\\quad \\quad \\text{where }\\omega \\equiv \\sqrt{k/m}.", "e8519fe1b2b62fd53e7aa2d5cdd0c712": "wfe/({\\delta}.\\sqrt{\\theta}) = wfe/[(P/14.696).(\\sqrt{T}/\\sqrt{288.15})]", "e8522544614756ee40ad23b132d0b66f": "\\mathfrak{gl}(n,F)", "e852799b767e672713efaeab7ac5efba": "\\ \\alpha_e ", "e852a57cc6d00c245e21889c361614d5": "p_X(x) = \\begin{cases} 1 \\qquad 0 < x < 1 \\\\ 0 \\qquad \\mbox{otherwise}\\end{cases}", "e85300cf9f8600e60ccc25375ddaf902": "\\scriptstyle{R_{F}}", "e8531ee0758b6c3ce3e82217bb49e5e6": "\n\\begin{align}\n\\sum_{i=1}^n \\left(x_i - \\overline{x} \\right)^2 &= \\sum_{i=1}^n \\left(x_i - \\frac 1 n \\sum_{j=1}^n x_j \\right)^2 \\\\\n&= \\sum_{i=1}^n x_i^2 - n \\left(\\frac 1 n \\sum_{j=1}^n x_j \\right)^2 \\\\\n&= \\sum_{i=1}^n x_i^2 - n \\overline{x}^2\n\\end{align}\n", "e85392a727a54c03e47cae9c26411677": "\\frac{t_{k+1}}{t_k} ", "e854110d2ff960b2cc1134035e093b0f": "\n\\operatorname{Cov}(\\textbf{X},\\textbf{Y})=\n\\operatorname{E}(\\mathbf{X Y^\\top}) - \\operatorname{E}(\\mathbf{X})\\operatorname{E}(\\mathbf{Y})^\\top", "e85472e69e6790fb0bec54751273eb5e": "F_{0j} = d*j", "e854e9b275c1f1b010388b83abb5ada0": "k_1 > 1", "e854fd594c8d59b33a3bfe60b6b30108": "x^2 + px + q = 0 \\, ", "e85508912ebf2a487f852f966f571e34": " f:X \\rightarrow \\mathbb{R} ", "e85521345623d14d93c49e71498aee3c": "\\Phi(x,y) = \\varphi(r), \\quad r = \\sqrt{x^2+y^2}", "e855621e92e3c60a815a0bc45fa52ddb": "L_k", "e8556a2483517f52fc5522d05c79a0b1": "D(a_{k,k},r_k)", "e8561a44f1b390700beef2a958655973": " \\frac{\\mathrm{d}\\mathbf{p}_i}{\\mathrm{d}t} = \\mathbf{F}_{E} + \\sum_{i \\neq j} \\mathbf{F}_{ij} \\,\\!", "e8562f14248980209fb6f5248b4543dd": "\\pi a,\\, \\pi b", "e85645a1099af9e620339c4c26f94b9d": "\\Delta{V} = V_{exh}\\ \\ln(\\frac{m_0}{m_1})\\,", "e85662b75ffb2225de7ac8deb1f977b0": "(1-3p)/4", "e85739ee680bebe52b63aa1e8cdb5230": "\nB = \\bigoplus_{\\ell:\\;\\ell\\not=i\\;\\mathrm{and}\\;\\ell\\not=j}{g^{\\ell}D_\\ell} = \\mathbf{Q} \\;\\oplus\\; g^0\\mathbf{D}_0 \\;\\oplus\\; g^1\\mathbf{D}_1 \\;\\oplus\\; \\dots \\;\\oplus\\; g^{i-1}\\mathbf{D}_{i-1} \\;\\oplus\\; g^{i+1}\\mathbf{D}_{i+1} \\;\\oplus\\; \\dots \\;\\oplus\\; g^{j-1}\\mathbf{D}_{j-1} \\;\\oplus\\; g^{j+1}\\mathbf{D}_{j+1} \\;\\oplus\\; \\dots \\;\\oplus\\; g^{n-1}\\mathbf{D}_{n-1}\n", "e85763833510a68a9e5c2b1bee237c0d": "P_i = \\dot m {\\Delta h_o} ", "e8576a62b096ed848e176ce976608631": "R= \\frac {v}{f}\\,", "e857f0b0b8135f4cb7f9d84eb38a7475": "f(k) \\cdot {|x|}^{O(1)}", "e85832e65aa2ca2a7dc2fcb35bcc2d9f": "\\varepsilon_{\\psi(\\psi_1(0))+1}", "e858ad940ae41ebdde4157b01ef7df53": "2\\lambda^2", "e8593583eb6488875c9cae868e811122": "A=\\lim_{n\\rightarrow\\infty} \\frac{K(n+1)}{n^{n^2/2+n/2+1/12} e^{-n^2/4}}", "e8599d55894a5cde4740d5c11107b735": "[G]", "e859ea39cd200b2fd393b0a7325513cd": "\\boldsymbol{\\Sigma}", "e85a3fe987b12b11ef94e081c03a4b8e": "N>A", "e85a85b365e4367ccfac81a7e510123f": "q_2=\\frac{5000+2c_1-3c_2}{4}", "e85aaa4a7b9b52fdb08522493fb860da": "f_2=-18x^7", "e85b16d9c55094c9918a44a7d3295369": "s^*\\in S_i", "e85b830460e13751cd418682ab66b6d7": "\\frac{s + 10}{s + 5} \\qquad \\text{and} \\qquad \\frac{s - 10}{s + 5}", "e85bb01fb9a27e6fe412b491f004cc75": "\\textstyle\\frac {1}{2-1}=1=", "e85c331f1466a82d2bbcbe74b88ed362": "m\\circ \\omega=\\mathrm{id}_{\\mathbb{F}_p}", "e85c5294b5218d728a4f61a4b77ee24a": "\\phi^{-1}", "e85c8ea7b5f72dc751bedb4b3765f068": "\\delta_{jk}", "e85d061df1fc5456899d4528af63e3d2": " \\psi_{q(\\theta)} = \\dot{q}(\\theta)I^{-1}(\\theta)\\dot\\ell(\\theta)", "e85d9cf6ae03cad213ed906f5c49b826": "3.14159264\\dots\\!", "e85de4d875748fe71a312f8df69351c4": "\\scriptstyle F = x^2 - y^2 - z^2,", "e85deb9f65e16e80bfd7fe0d991c4350": " \\mathbf{p} = \\frac{d}{dt}\\left(\\sum_{i=1}^n m_i (\\mathbf{r}_i - \\mathbf{R})\\right) + \\left(\\sum_{i=1}^n m_i\\right) \\mathbf{v},", "e85e1b953a63ca258824b577510d23e4": "W_{t,d,n}", "e85e51b747c6cf49933b76cbf22d764e": "s_p(n) = \\sum_{j = 1}^\\infty \\left\\lfloor\\frac{n}{p^j}\\right\\rfloor, ", "e85e5a0043896a5d2496ec3d9ace5b39": " \\boldsymbol{\\sigma}\\cdot\\boldsymbol{\\epsilon} = \\sigma_{ij}\\epsilon_{ij} = \\tilde\\sigma \\cdot \\tilde\\epsilon\n", "e85e74867931d290ea66797213250302": " = \\int^{2\\pi}_0 {f(re^{i\\vartheta})}{\\sum^\\infty_{k = 0} \\overline{a_k (re^{i\\vartheta})^k}} \\, \\mathrm{d}\\vartheta", "e85e85e2f0e3308dcaffc8be5f783288": "\\left|\\frac{f(z_1)-f(z_2)}{1-\\overline{f(z_1)}f(z_2)}\\right| \\le \\left|\\frac{z_1-z_2}{1-\\overline{z_1}z_2}\\right|", "e85eb54b2c502199109cd7c5c17bb38d": " D=6 ", "e85eecc9ea77bae166d8b4cfcd8b3692": " \\Delta_{X}=\\{(x,x)|x \\in X\\}.", "e85fe1306444a5f87e657d8ba1ceb75c": "\\mathcal{E} = \\{a \\bar{b} : a, b \\in \\mathcal{B}_w \\}", "e860c0a8c1751d2175bc4a5e63de366e": "\\scriptstyle \\pi = \\frac{4}{1} - \\frac{4}{3} + \\frac{4}{5} - \\frac{4}{7} + \\frac{4}{9} - \\frac{4}{11} + \\frac{4}{13} \\cdots.", "e860f8b98ead905e8dbeb4aa8853e196": "y(t)=\\int_0^1 w(t,\\xi)\\,d\\xi,", "e86135e7c0cae3d1cc4958d7301c0fc3": "\\frac{d\\ v_C(t)}{dt} + \\frac{1}{RC}v_C(t) = \\frac{1}{RC}v_S(t)", "e8615db02abddf465d64e31199bffb8a": "\\mbox{tr}\\,\\mathfrak{H}=a+d", "e861908d8f774e12f3aa383c02c1e140": "M_n = \\frac{2\\zeta(n)n!}{(2\\pi)^n}", "e861b7905f97b95814ed08808df074b3": "f=a+bx+cp", "e861d72d0541da548de3211159bcf3b1": "F_{\\mu \\nu}^a = \\partial_\\mu A_\\nu^a-\\partial_\\nu A_\\mu^a+gf^{abc}A_\\mu^bA_\\nu^c ", "e861edd79faf020025aa55f9f18e6db0": "\\alpha N \\delta^2 (1-\\epsilon)/4", "e862236fa8702c0b242170919e128ff8": "\\gamma_0(q) = -\\psi(q)", "e862365fbd33f4ce1b73d133ff9d6fda": "S^n\\wedge S^m \\to S^{n+m}", "e862d3b0fc61d980cea59e5ee4bd5ef5": "X_i \\sim S(\\alpha, \\beta_{x_i}, \\gamma_{x_i}, \\delta_{x_i}), i=1,\\ldots,n", "e863a010111b73c68ac58b5a3c5fdaec": "qU = \\frac{1}{2}mv^{2}\\,", "e863df6c0cc677657b7efcba31948a87": "m_{1} = {X_1+\\cdots+X_n \\over n} \\,\\!", "e86400f5fc6ceb242055ead9d97c0ab3": "\\sin\\frac{\\pi}{20}=\\sin 9^\\circ=\\tfrac{1}{8} \\left[\\sqrt2(\\sqrt5+1)-2\\sqrt{5-\\sqrt5}\\right]\\,", "e8640387578cc5e9096be4e8a792fea3": "R(t) = \\exp(-\\lambda t).", "e864c03268f1497e59186908fb665251": "g_{ij}=-D[\\partial_i||\\partial_j]=-\\partial'_j\\partial_i\\sum{p(\\log p-\\log;q)}=\\sum\\frac{\\partial_ip\\partial_jq}{q}=[p=q]=\\sum{p\\partial_i\\ell\\partial_j\\ell}", "e864d0dc57bc056e815f5873a5bdf5b8": " \\frac{x}{2} = x ", "e864de52afbe9fd69131fe51c1596155": "\\frac{R_t}{(1+i)^{t}}", "e864f75bdc74e09a8a8888e1f0e6fb38": "x(t) = \\frac{1}{{m\\omega _d }}\\int_0^t {p(\\tau )e^{ - \\varsigma \\omega _n (t - \\tau )} \\sin [\\omega _d (t - \\tau )]d\\tau }", "e864fe298fa73832d6d3552d4c71d02a": "P_2=\\frac{6}{6+6}", "e865a605d42ba8fa98b93f16903a9e0f": " \\lambda_d(E) = 2^{-d} \\alpha_d H^d(E)\\,", "e865dabb8794982b0ffb37be3e8701e3": "\\mathcal{R}_k[\\bar{\\Phi}]", "e8660d2ce2c71ca4d3a8803b8ab6d0fb": "[1].(z_1,\\ldots,z_n):=(e^{2\\pi iq_1/p} \\cdot z_1,\\ldots, e^{2\\pi i q_n/p}\\cdot z_n).", "e86654397d1cf7d76236339605e63727": "\\sigma =\\sqrt{\\frac{hf_{0}^3}{2PQ^2T}}", "e866e33c57bd44a8f8a90a226ca06a8b": "(x,y)=( x_n,\\ f(x_n) )", "e866fb652646eda9e50bc20c297cb67a": "f > f_0", "e86719390f6e931615757b71b84ff2b1": "h_p(x)", "e8672e23017bf7c5ce4a2552368b2896": "\\lim_{p\\rightarrow 0}\\Delta(p)=\\mathrm{constant}", "e8675bb6251f3bcfb08faadf61be442e": "d_{mn}", "e867ddebbe5a79664bd3b71e38b92ca0": " \\Lambda^\\cdot {\\mathfrak g}^*", "e867e260a6d81d1dffdd31d94db0ea71": " M[f] * \\left[ 1,x,x^2,x^3,...\\right]^\\tau = \\left[ 1,f(x),(f(x))^2,(f(x))^3,...\\right]^\\tau.", "e867ff656ab5dbc01891798b75ca0f99": "\\begin{align} \\mathcal{Z}\\{x_1(n)*x_2(n)\\} &= \\mathcal{Z} \\left \\{\\sum_{l=-\\infty}^{\\infty} x_1(l)x_2(n-l) \\right \\} \\\\\n &= \\sum_{n=-\\infty}^{\\infty} \\left [\\sum_{l=-\\infty}^{\\infty} x_1(l)x_2(n-l) \\right ]z^{-n}\\\\\n &=\\sum_{l=-\\infty}^{\\infty} x_1(l) \\left [\\sum_{n=-\\infty}^{\\infty} x_2(n-l)z^{-n} \\right ]\\\\\n &= \\left [\\sum_{l=-\\infty}^{\\infty} x_1(l)z^{-l} \\right ] \\! \\!\\left [\\sum_{n=-\\infty}^{\\infty} x_2(n)z^{-n} \\right ] \\\\\n &=X_1(z)X_2(z)\n\\end{align} ", "e867ff721a3e7326c5cae442d6b637c0": "\\left\\vert \\mathcal{V}\\right\\vert =n", "e868119125c1c84924860086699f0a6e": "\\Pi(\\phi,n,k)=\\sin\\phi R_F\\left(\\cos^2\\phi,1-k^2\\sin^2\\phi,1\\right)+\n\\tfrac{1}{3}n\\sin^3\\phi R_J\\left(\\cos^2\\phi,1-k^2\\sin^2\\phi,1,1-n\\sin^2\\phi\\right)", "e868386f91ae6213604cd097271c47a4": "2^{2^n}\\equiv1\\pmod3", "e8687068457f167229e49d08db2b89ae": "t \\pmod l", "e868de2fa0ea8d4024398ca4aa888afb": "V(\\mathbf{x}) = -\\int_{\\mathbf{R}^3} \\frac{G}{|\\mathbf{x}-\\mathbf{r}|}\\,\\rho(\\mathbf{r})dv(\\mathbf{r}).", "e868f027a7bb4eac1cd32a3711e390f2": "\\scriptstyle M_s", "e8698094049d2b7b75f5938159490489": "C_n=\\sum_{0\\le i\\le n}A_n B_{n-i}", "e869c86d1d212c58525ee34f301a2a1e": " r^3~\\cos\\theta \\,", "e869d88afd8ce609a634bb54b9eab928": "g^R", "e86a68a2d124a09dd2d80d377ed7ed3d": " s\\in S, x\\in X", "e86acecc42fa851fda260f2fb91db779": "C=(u_1 + u_2 d_A) \\times G", "e86b08f42171e26d1cb66934655846b3": " \\tilde D^M=\n\\begin{pmatrix}\n D_{1111} & D_{1122} & D_{1133} & \\sqrt 2 D_{1112} & \\sqrt 2 D_{1123} & \\sqrt 2 D_{1113} \\\\\n D_{2211} & D_{2222} & D_{2233} & \\sqrt 2 D_{2212} & \\sqrt 2 D_{2223} & \\sqrt 2 D_{2213} \\\\\n D_{3311} & D_{3322} & D_{3333} & \\sqrt 2 D_{3312} & \\sqrt 2 D_{3323} & \\sqrt 2 D_{3313} \\\\\n \\sqrt 2 D_{1211} & \\sqrt 2 D_{1222} & \\sqrt 2 D_{1233} & 2 D_{1212} & 2 D_{1223} & 2 D_{1213} \\\\\n \\sqrt 2 D_{2311} & \\sqrt 2 D_{2322} & \\sqrt 2 D_{2333} & 2 D_{2312} & 2 D_{2323} & 2 D_{2313} \\\\\n \\sqrt 2 D_{1311} & \\sqrt 2 D_{1322} & \\sqrt 2 D_{1333} & 2 D_{1312} & 2 D_{1323} & 2 D_{1313} \\\\\n\\end{pmatrix}.\n", "e86b0e9080d4441e795f224f764346ac": " \n \\sigma_{ji,j}+ F_i = 0.\n\\,\\!", "e86b5bbee557102f3348cdfead81e7ec": "4\\ ", "e86c463c08d48a84bdae9e1a5941e0d0": "\\{\\mathbf{k}(i),\\mathbf{K}(i)\\}", "e86c4b144e828fea1eced4a3df7ae865": "eAe", "e86c97abfe437d41a03acc679ddbdf3a": " \\mathcal{A}^f ", "e86ceaeaf18206a404c5c060715cd722": "(x\\in )V\\subseteq W_{U}(\\subseteq\\bar{W_{U}}\\not\\ni x)\\,", "e86d77e3e3f73b0fb7396d7cafd640a7": "\\Phi = \\mu_0 \\frac{NiA}{l}.", "e86ddb2eb1cc132f534bc1e241695f63": "\\Phi = \\frac{\\sum {\\rm d \\ell} }{{\\rm d} V}", "e86e0e084f6964f587413edd26baa1dd": " A_{ij} = \\left\\{ \\begin{array}{ll}\n x_{ij} & (u_i, v_j) \\in E \\\\\n 0 & (u_i, v_j) \\notin E\n \\end{array}\\right.", "e86e11d56aa2cd4b25324115589f68d1": "F_n(a, 0) = 1", "e86e269bb4c272e50edab65e61583c0b": "x + x'\\varepsilon", "e86e45729685111f8f8d7ac0de91656c": "\\langle a,\\;b \\mid a^m=e,\\;b^n = e,\\;aba^{-1}=b^k\\;\\rangle.", "e86e54df2dcd8e65c037b30bf37838ec": "{\\mathcal G}", "e86eeaa8cbc2347175bb8797812b76b3": "\\bigoplus_i R/(d_i) = R/(d_1)\\oplus R/(d_2)\\oplus\\cdots\\oplus R/(d_n)", "e86effb3bd5d6f2ddb3502cbcbd35d52": "f^{*}(x) = \\lim_{h \\to 0}{ \\left({f((1+h)x)\\over{f(x)}}\\right)^{1\\over{h}} } = \\lim_{k \\to 1}{ \\left({f(kx)\\over{f(x)}}\\right)^{1\\over{\\ln(k)}} }", "e86f3a9c507cd2df0613cd80cd39daab": "\\cos{5x}=0\\,", "e86f5fc05a56918388c43d9a289e092a": "\\textbf{a}^*", "e86f74d08f3c0da6380af7e762609a15": " \\mathbf{\\hat y} ", "e86f801914501bae27f78e2d2eb42db8": "\n L^{-1}(x) = 3x \\frac{35-12x^2}{35-33x^2} + O(x^7)\n ", "e86f8068269c3727c13fdcb8754700d3": "\\tilde{G}_{k,n}", "e86f8beeb18dc3e63af271fb1ec4c917": "\\delta\\psi = \\frac{\\partial\\psi}{\\partial x}\\, \\delta x + \\frac{\\partial\\psi}{\\partial y}\\, \\delta y,", "e86fa2e68d00acbb224b8e8b3ec2ad28": "F \\subset {{\\mathbf{k}}}[x_1, \\ldots, x_n]", "e86fc25b081ee2b7c4911e41d01edd75": "A^*(X) = \\bigoplus_{k = 0}^{\\dim{X}} A^k(X).", "e86fdb1afe169420d25b725ebf0ff0c5": "X^j", "e86ff8097c6bf4da84c3360d0576d91d": "\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} I =\n2 {\\mathbf E} \\cdot {\\mathbf d} ( p_{cv} - p_{cv}^\\star ).", "e8700c4cda3961392c276c87414d74d8": "\\frac{M_{Pl}^3}{m_H^2}.", "e870c74669e74ce44a0af3b72829a055": "S = \\{ 1, r, r^2, r^3, \\dots\\}", "e870dfcf719340534eedc3fb3fc952f6": "X_1,X_2,\\dots,X_n", "e870eacbb76c77184d671f85c9faf099": "\\textstyle (0,1) ", "e8711a7a4190e8445e046fe5d424c6ed": "g(x) : \\mathbb{R}^n \\rightarrow \\mathbb{R}", "e8711d93e1752226da266aedd0d42c66": " \\eta_0 \\left(e^{-\\Phi} Q_B e^{\\Phi} \\right) = 0 .", "e87147694d797b820116940b22c8b759": "7^5 = 462*1+252*26+126*66+56*26+21*1 = 16807", "e8719275e9e78a547b88673cc37b8580": " = \\left(\\sum_i q^{i(i - 1)/2} \\binom{m}{i}_{\\!\\!q} x^i \\right) \\cdot \\left(\\sum_i q^{mi + i(i - 1)/2} \\binom{n}{i}_{\\!\\!q} x^i \\right).", "e871941fc5a07a58cce5bd76f8ec8537": "P = a_0 + a_1 x + a_2 x^2 + \\cdots + a_n x^n , ", "e871c2c19562c87cb3bf774ae3009837": "|W| \\le \\Bigl|\\bigcup_{A \\in W} A\\Bigr|.", "e871fa3839360ff1f9d2fb26c3a52fda": "\\varepsilon _{2} =\\frac{w^{2}+m_{2}^{2}-m_{1}^{2}}{2w},", "e872849c7eeb635e2102e6071925a7a3": "\\liminf_{x \\to a} f(x) = \\lim_{\\varepsilon \\to 0} ( \\inf \\{ f(x) : x \\in E \\cap B(a;\\varepsilon) - \\{a\\} \\} ) ", "e872935e981c8ef9349997d4a54df498": " \\mathbf{u}=\\mathbf{u}\\left ( \\mathbf{r},t \\right ) \\,\\!", "e872a3132f4f9e81af43be21337e3ad4": "Q = O", "e872ae15bd3c4f3878ce1f72055c9f1b": "\\psi_1(q) = \\sum_{n\\ge 0} {q^{n(n+1)/2}(-q;q)_{n}}", "e872de54960b6dea5d075ec1f958eaed": "C_1,C_2,\\ldots", "e87308ab3f9b60ecd17690c24f0e495d": "E^f", "e8733e5bb30591c18f5163be47ba0f28": " -\\ln (-1-\\eta) + (1+\\eta) \\ln x_{\\mathrm m}", "e87349480ed41258cef0eeabded1ab78": " D=1; E=-1; F=1; G=-4; ", "e8736f17bf6b5a38cbfe658e2a985193": "M(r)", "e874a774b4df9fc9c62f2378e793e0bb": "\\tbinom{0}{0}=1", "e874c188d1ac5e76230083a680722dc5": " m+ld 0\\\\\n0 & \\text{otherwise}\\end{cases}", "e8763637826a153251ef7994f5c830c0": "\\log n+\\log\\log_2 n+1", "e8765ead28f59bdc3c61611dc0ad2d3a": "\n\\begin{align}\n{\\mathcal L}_B &= \\frac{1}{16\\pi G} (R - 2 \\Lambda)\n+ \\sigma_1 B^\\mu B^\\nu R_{\\mu\\nu}\n+ \\sigma_2 B^\\mu B_\\mu R\n- \\frac{1}{4} \\tau_1 B_{\\mu\\nu} B^{\\mu\\nu}\n\\\\\n&\\quad\n+ \\frac{1}{2} \\tau_2 D_\\mu B_\\nu D^\\mu B^\\nu\n+ \\frac{1}{2} \\tau_3 D_\\mu B^\\mu D_\\nu B^\\nu\n- V(B_\\mu B^\\mu \\mp b^2) + {\\mathcal L}_{\\rm M} .\n\\end{align}\n", "e8766c0c924a4c94ecfdfe8426eab429": "\\mathbb{Q}_2(\\sqrt{7})=\\mathbb{Q}_2(\\sqrt{-1})", "e876b85a3647d19ae3348ba6848f000a": "(x, y) = \\{ \\{ x \\}, \\{ x, y \\} \\} \\in \\mathcal{P}(\\mathcal{P}(X \\cup Y)) ", "e876dded821265a99287e4cf76055aa8": "(\\mathbf{1},\\mathbf\n\n{1},0)", "e876e0d50582f37335cdbeb0ee03f0a1": " v = (1 + \\frac{m_\\textrm{p}}{m_\\textrm{b}}) \\cdot \\sqrt{2\\cdot g\\cdot h}", "e87716a8e6ac8671f8348ce089499ef8": "\\begin{cases} \\dfrac{\\partial \\rho}{\\partial t}(t, x) = A^{*} \\rho (t, x), & t > 0, x \\in \\mathbf{R}^{n}; \\\\ \\rho(0, x) = \\rho_{0} (x), & x \\in \\mathbf{R}^{n}. \\end{cases}", "e8771f68d424a6258f6254c93e23870e": "\\neg a \\vee b", "e87725a3bfb4a0be5877edcf0f81ba63": "S_i := \\{g \\mid L(g) \\leq i\\}", "e8775d361c888bd7d6a0069b163615f4": "\\tau_{\\lambda_0}", "e87770ce7a144831e64ff938b3a6eaa5": "\\theta_0 = \\omega_r \\quad\\text{ if }\\quad P(\\omega_r\\mid f(x_0)) = \\max_{s=1,2,\\ldots,R} P(\\omega_s\\mid f(x_0)) ", "e87797e8e464b899d78e26518446855b": "\\delta(c^p \\smile d^q) = \\delta{c^p} \\smile d^q + (-1)^p(c^p \\smile \\delta{d^q}).", "e8779b1c52345655d23e033d42cf652a": " (A - \\mu I) b_{k+1} = \\frac{b_k}{C_k}, ", "e87802235fe9dee63f62adae59752c3a": "|z||z-a| = c", "e8780c6e18d3d950c259a2039f22be39": " F_{ST} = \\frac{\\sigma^2_S}{\\sigma^2_T} = \\frac{\\sigma^2_S}{\\bar{p}(1-\\bar{p})}", "e878149e116220e7695cf2f7b4355a11": "\\frac{43,416 \\mbox{MWh}}{(366 \\mbox{days}) \\times (24 \\mbox{hours/day}) \\times (20 \\mbox{MW})}=0.2471 \\approx{25%}", "e87824f048ed91e9fcd3603a4aaabf1b": "\\int \\mathcal{D}\\bold{g}\\, \\mathcal{D}\\phi\\, \\exp\\left(\\int d^4x \\sqrt{|\\bold{g}|}(R+\\mathcal{L}_\\mathrm{matter})\\right)", "e87870ee04aba808f9f4412973f80adb": " (\\lambda x.\\operatorname{de-let}[f]\\ \\operatorname{de-let}[x\\ x])\\ \\operatorname{get-lambda}[x, x\\ q = f\\ (q\\ q)] ", "e878965881dc37baa4e3e5f4ffa42403": "= 1 - \\tfrac{1}{6} = \\tfrac{5}{6}", "e878b9505be70425329640807fd018b8": "g(P_1.P_2) = g^{-1}P_1 \\cap g^{-1}P_2", "e878c4bb346b402db1e6a57c54daf9a8": "\\max(normal) = 1", "e878d12c5e7a0f7e61d4e9d96263ca4a": "\n\\hat{\\mu} = \\bar{Y}_{\\cdot\\cdot}\n", "e879729158644b4c1620493dca0d2011": "O(4)\\,", "e8798be911775652315ef3ba37fb7be3": "\\sum_{k\\frac{1}{2}", "e89907df271635e4871489e76eb1ab91": "p(x,t)=p_t(x)", "e8992d54e3f271a613160fa5c4ca8f2b": "\\epsilon_1(p)", "e899500ffed6d213c3f0764195c7e9a6": "\\frac{\\delta K_{\\lambda+1}(\\delta \\gamma)}{\\gamma K_\\lambda(\\delta\\gamma)} + \\frac{\\beta^2\\delta^2}{\\gamma^2}\\left( \\frac{K_{\\lambda+2}(\\delta\\gamma)}{K_{\\lambda}(\\delta\\gamma)} -\n \\frac{K_{\\lambda+1}^2(\\delta\\gamma)}{K_{\\lambda}^2(\\delta\\gamma)} \\right)", "e899578867a55f5d3b14d4507624a108": "\\mu_X", "e89974ec619d7320333f6825282a9837": "\\mathbf{M}(\\mathbf{x},t)", "e899753e5e13de0edb0372281604aa20": " \\alpha = 0.3 ", "e899a3432a7f37e48febb4e3e67cfad0": "x \\odot y", "e899ad7670e81c6427efb50d3d57cf41": "\na_n=\n\\begin{cases}\n-\\log(2) &:n = 0 \\\\\n\\frac{-2(-1)^n}{n} &: n > 0.\n\\end{cases}\n", "e899b57f61a7d99e0701e109e0f142db": "\\hat u_R ", "e899ccf5a99a8e53342e79ed3190489c": "\\mathbf{F}_\\mathrm{B} = \\mathbf{F}_\\mathrm{A} + \\mathbf{F}_{\\mbox{fictitious}},", "e899d63806766fcf22a378e571b86dd0": "L_p=\\frac{J} {\\Delta p}", "e899dd16b1a694a7cdbc1d17158d6ca5": "S_A \\equiv \\left\\{ design \\; h\\in S \\mid \\frac{\\delta^2_{hj_hl^h_{j_h}}}{\\frac{\\hat{\\sigma}^2_{hl^h_{j_h}}}{\\alpha_h}+\\frac{\\hat{\\sigma}^2_{j_hl^h_{j_h}}}{\\alpha_{j_h}}} < \\min_{i\\in \\Theta_h} \\frac{\\delta^2_{ihl^i_h}}{\\frac{\\hat{\\sigma}^2_{il^i_h}}{\\alpha_i}+\\frac{\\hat{\\sigma}^2_{hl^i_h}}{\\alpha_h}} \\right\\},", "e89a2ceedc8d45f4439e85b04dea1d2e": "\\partial M", "e89a518ec702ce61f18b79e6d4efcc0e": "e(k,0) = d(k)\\,\\!", "e89a5b0d2a2a11224399ec8483a01727": "\\frac{P}{\\neg \\neg P}", "e89b28aafadba2f72628479fe63e32f6": "(\\mathbf{X}, \\ln|\\mathbf{X}|).", "e89b2db778417bae1d5b971409d52ede": "\\frac{1}{N} \\cdot \\sum_{i=1}^N (y_{1i}-y_{0i})", "e89b961be195dfe08e27554b074d6fe1": "\n\\sigma_{\\theta\\theta} = \\frac{\\sigma}{2}\\left(1 + \\frac{a^2}{r^2}\\right) - \\frac{\\sigma}{2}\\left(1 + 3\\frac{a^4}{r^4}\\right)\\cos 2\\theta\n", "e89be903156a7237b1b72c11312f7411": "B\\,(V^{-1}X) = UC.", "e89c7416f61dcdcc9b302ef154c8a8f1": " U_0(x,y) = - \\frac{j}{\\lambda} \\frac{e^{jkz}}{z} \\int_{-\\infty}^\\infty \\int_{-\\infty}^{\\infty} e^{j \\frac{k}{2z} [ (x-x_i)^2 +(y-y_i)^2 ] } U_i(x_i,y_i) \\; dx_i\\; dy_i, ", "e89c9b65ccfbc1047f51fe2d330ff7a8": "H = H_{0} + \\Delta H\\,", "e89cc5362535d54d6bdbfc4ff7cee757": "p?;a\\,\\!", "e89d1f819514c18b369de32b171967f7": "(X_1, X_2)", "e89d25bb9278307bcc84e2a97d875918": "\\sigma = \\int c(E) \\Bigg( -\\frac{df(E)}{dE} \\Bigg) \\, dE,", "e89d475918b8198619d96b058e7e5564": "\nj\\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_\\xi + Y^k\\frac{\\partial}{\\partial \\xi^k}\\Big|_\\xi\\Big)\n= \\xi^k\\frac{\\partial}{\\partial x^k}\\Big|_X + Y^k\\frac{\\partial}{\\partial \\xi^k}\\Big|_X.\n", "e89d47bf460f584a35d0035eeead7ae6": "\\begin{align}\n \\sqrt{\\sigma^2 + \\tau^2} \\, d\\tau \\, dz &\\hat{\\boldsymbol\\sigma} \\\\\n+ \\sqrt{\\sigma^2 + \\tau^2} \\, d\\sigma \\, dz &\\hat{\\boldsymbol\\tau} \\\\\n+ \\left(\\sigma^2 + \\tau^2\\right) \\, d\\sigma \\, d\\tau &\\hat{\\mathbf z}\n\\end{align}", "e89da0c0c73b56b18d920816a6df9829": "x=b", "e89e0b9b88d5e2456e8a786314cb27ae": "I_{simp} = \\bigg( \\frac{0.1299}{12} \\cdot $2500 \\bigg) \\cdot 3 = $81.19", "e89e2960c2d87994a8df4866e60dd3bc": "\n{\\left(\\mathcal{I} \\left(\\theta \\right) \\right)}_{i, j}\n=\n- \\operatorname{E}\n\\left[\\left.\n \\frac{\\partial^2}{\\partial\\theta_i \\, \\partial\\theta_j} \\log f(X;\\theta)\n\\right|\\theta\\right]\\,.\n", "e89e5611851d9467cb921ad464c6bc30": "\\mathbf{j}_s = n_s e \\mathbf{v}.", "e89e5d682d035d4a84ea35892eafe1b0": "p = \\frac{1}{n}", "e89eb41d980d68d6b1f22c893dc53514": "WXYZ", "e89f522488163932caa474c86f105aed": "\\mu_k=\\Gamma\\left(1-\\frac{k}{\\alpha}\\right)", "e89f8823987fc4a15a69b4f2530e5cd5": "\\lim_{x \\to 0^+} \\log_a x = -\\infty \\quad \\mbox{if } a > 1", "e89f8c8607e9ecb9211b2e50c8e21a12": "b = \\frac{0.08664\\,R\\,T_c}{P_c}", "e89f95a2629514cc56d83e828c615dde": "ax(t)+by(t)+cz(t)=d\\ ,", "e89fc5ae988482b6620022e63ecb9ccd": "p = k_{\\rm H}\\,x", "e89ff16fc3cbbb77d2a62830d513dd00": "P_s=\\sqrt{-\\frac{\\alpha_0\\left(T-T_0\\right)}{\\alpha_{111}}}", "e89ffc2c75b8642fcccb1fcccbf63bb4": "u \\in \\mathcal{U}", "e8a0073df1677c6349ffebd8f99c239b": "(X,\\partial X)", "e8a00b29dda3e4cbb70a24402c82b999": "d_2=\\|d_1\\|\\,", "e8a043412c848f4de76f2d68db71669d": "\\mathbf{P}_{ins,i}", "e8a096f67a9e4934275cd1d2e353c3c2": "\n\\begin{alignat}{2}\nz^*(x-z) & = (Qz)^*(x-z)\\\\\n&=z^*Q(x-z)\\\\\n&=z^*(A^+ A x - z) \\\\\n&=z^*(A^+ b - z) \\\\\n&=0.\n\\end{alignat}\n", "e8a0b70352917d92faf2a3c4ef7f2c9d": "\n\\frac{1}{2}\\sum^n_{i=1}F^P_id^P_i + \\sum^n_{i=1}F^Q_id^P_i = \\frac{1}{2}\\int_\\Omega \\sigma^P_{ij}\\epsilon^P_{ij}\\,d\\Omega + \\int_\\Omega \\sigma^Q_{ij}\\epsilon^P_{ij}\\,d\\Omega\n", "e8a0c5f80559451a59df4c8d408947ea": " g_{n_i} \\in X' ", "e8a11186eb329edb5beb0aaae312f57a": "H_1=-\\frac \\hbar 2\\left(\\Omega_p e^{-i\\omega_p t}|1\\rangle\\langle 3|+\\Omega_c e^{-i\\omega_c t}|2\\rangle\\langle 3|\\right)+\\mbox{H.c.},", "e8a1308981e0696894f237d5fd399744": "\\operatorname{std}_p(f)= {1\\over 2(1-p^{-2})(1-p^{-4})\\cdots(1-p^{2-n}) (1-{(-1)^{n/2}\\det(f)\\choose p}p^{-n/2})} \\quad", "e8a137fedf204d1d05ac81d03e4c8bfc": "Y_{9}^{-4}(\\theta,\\varphi)={3\\over 128}\\sqrt{95095\\over 2\\pi}\\cdot e^{-4i\\varphi}\\cdot\\sin^{4}\\theta\\cdot(17\\cos^{5}\\theta-10\\cos^{3}\\theta+\\cos\\theta)", "e8a17212515df28c0bd134833153ad0b": "\\begin{bmatrix}a\\\\b\\\\c\\end{bmatrix} . ", "e8a1b99c8dfee925eea8bdae2508c2b9": "d(x_{1+1}, x_1) = d(x_2, x_1) = d(T(x_1), T(x_0)) \\leq qd(x_1, x_0).", "e8a228623e0b6b7cd7a6d542a4c1485b": " x = 1/T = 0", "e8a243119a9098086923b9c03c6af213": "\\textstyle \\textbf{x}_{i}", "e8a2625336b78403e2f3d447cf1e45cf": "R_{earth}\\,", "e8a26bbc55a50d2e462e1c58a43497d1": "v=\\sum_iv^i\\frac{\\partial}{\\partial x^i}", "e8a2d5d25bf0e065e33966060f93acec": "\\sum_{n=0}^{\\infty}T_n(x) t^n = \\frac{1-tx}{1-2tx+t^2}; \\,\\!", "e8a2fa9049b9ec3f6550e1ca1f2636d8": "\na^\\dagger |n \\rangle = |n+1 \\rangle \\sqrt{n+1} \\quad\\hbox{in particular}\\quad\na^\\dagger |0 \\rangle = |1 \\rangle \\quad\\hbox{and}\\quad (a^\\dagger)^n |0\\rangle \\propto |n\\rangle.\n", "e8a35a75a164cbc1676166784f048dee": "S_r(n+1, k)=k\\ S_r(n, k)+\\binom{n}{r-1}S_r(n-r+1, k-1)", "e8a39454d0ac0257152e8a17f3ee47b8": "m(i)", "e8a3fe573f14476fb13f4b6082a1b37b": "A^+ = A^*(AA^*)^+\\,\\!", "e8a4303e12dcb5cf3516e37c928d5cca": "A \\to \\alpha \\and B \\to \\beta \\iff (A <> B \\Rightarrow \\alpha <> \\beta)", "e8a4536e585437814688ae851218a1e1": "-\\mu_e \\mathbf{E}", "e8a4cd9010bf576e8b1a1fd75a86d5ef": "\\mathbf{A} = \\begin{bmatrix}\n -1.3 & 0.6 \\\\\n 20.4 & 5.5 \\\\\n 9.7 & -6.2 \n \\end{bmatrix}. ", "e8a563df264ff4480f502702295bf842": "n^T P_i", "e8a5d21118cc4920d3b5f685429c406b": "s=R\\cos^{-1}\\frac{R}{R+h} \\,.", "e8a64957f6bef686d3523e274762d154": "a + -b + -c", "e8a66821feca0327e1682a68e0648a71": "T(k,l) = C A^{k-l-1} B", "e8a6a42b4db355ad6faea3e02eb56acf": "\\mathbf{NC}^1 \\subseteq \\mathbf{NC}^2 \\subseteq \\cdots \\subseteq \\mathbf{NC}^i \\subseteq \\cdots \\mathbf{NC}", "e8a6b8980a23235df67cd35f374fff52": "P(t)=\\mathcal{P}_{ac}(c_a^*c_c)exp(i\\nu_1t)+\\mathcal{P}_{bc}(c_b^*c_c)exp(i\\nu_2t)+c.c.", "e8a6c406cb96b207931c0a15abd34791": "\\left(\\sum_{n \\in N} a_n X^n\\right) + \\left(\\sum_{n \\in N} b_n X^n\\right) = \\sum_{n \\in N} (a_n+b_n)X^n", "e8a7622760b8b931acc85d2ff716afd2": "\\textit{dau}: \\textit{daughter}", "e8a7887deaac50907bc55c772643f409": "h_k ^d = h ^{d-1} _k \\oplus 0 ~.", "e8a7a4dc98df6c4ad441a0da35eab57d": "\\textstyle \\theta=0.5", "e8a7b55e38403b34193fa087532f9e20": "f : \\mathbb{H} \\to \\mathbb{C}. \\, ", "e8a7e36c64080567c51b0f1a48b66278": "i\\frac{\\partial}{\\partial t}\\psi=-i\\sum_{j=1}^{3}\\alpha^j\\frac{\\partial}{\\partial x^j}\\psi+m\\beta\\psi-g(\\overline{\\psi} \\psi)\\beta\\psi", "e8a85a53446f60561a0790a732545690": "c_{9}", "e8a8a54d9082da56809cb7af7971170d": "M_{1}=q a", "e8a8b4e79733b5093db5bc8f4e5eafa8": "{\\varphi^2 = 1 - \\varphi}", "e8a954f84f6c008bda9881d60f4aa663": "(p,q) \\in R", "e8a96027bd43f28aba5f81cc28a0a898": "n\\log n+\\gamma n+1/2", "e8a981c905a83ee5958e9f07b349a3e1": "\\text{median}=1-\\tfrac {1}{\\sqrt{2}}", "e8a9e15e5fabbe4ee17d95bfb3f6a6bc": "\\frac{d\\mathbf{u}_\\mathrm{t}(s)}{\\mathrm{d}s} = -\\mathbf{u}_\\mathrm{n}(s)\\frac{\\mathrm{d}\\theta}{\\mathrm{d}s} = - \\mathbf{u}_\\mathrm{n}(s)\\frac{1}{\\rho} \\ . ", "e8a9e2c1dd695ce1ca91877edbe48a61": "1 t'", "e8aab6ae1185e2265208ddd5b73c499a": "c_n = (1/T)\\hat f(n/T)", "e8aac2f868951552cafac90c1d4b2616": " j > \\sigma > \\left( j - 1 \\right) \\ ", "e8aae0a601428375abc094e6af3ada74": "\\operatorname{E}(X)=0", "e8aaf87d9a5c35b14cfbc370d3fd7b21": "A_{i}", "e8ab31fb55b43b6ad4fdc9c936485a13": "\n\\alpha_{P}(X) = \\frac{\\left | {\\underline P}X \\right |} {\\left | {\\overline P}X \\right |} \n", "e8ab389f3c411cdc143142d6fd584e5d": "N \\lambda/n_2\\,\\!", "e8ab7b748ad6e75c3b8c8035ba1f08fa": " \\frac{1}{Z_{\\text{GOE}(n)}} e^{- \\frac{n}{4} \\mathrm{tr} H^2} ", "e8ac20115ccfa24d37dcfc0aa6bb0d35": " \\Sigma^n Y ", "e8ac258e1867b22adda6025b5b4d429b": "\n \\boldsymbol{\\sigma} = \\cfrac{2}{J}~\\boldsymbol{F}\\cdot\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}}\\cdot\\boldsymbol{F}^T =\n \\cfrac{2}{J}~(\\boldsymbol{V}\\cdot\\boldsymbol{R})\\cdot\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}}\\cdot(\\boldsymbol{R}^T\\cdot\\boldsymbol{V})\n ", "e8acc6caffa1a3886c6a01cd67c214bf": "CE = \\%C + \\frac{\\%Mn}{6} + \\left(\\frac{%Cr + %Mo + %Zr}{10} \\right) + \\frac{%Ti}{2} + \\frac{%Cb}{3} + \\frac{%V}{7} + \\frac{UTS}{900} + \\frac{h}{20}", "e8acd8d1e1b8dca5c8e7d10c84aaeb0c": "\\ T'", "e8ad012532ae18dce7c05b64ca9cb7d2": "\\sum_{n=1}^{\\infin} 1 = +\\infty \\, ,", "e8ad12fd3106f94aded558a3ea53a348": "g(t)=\\sum_{n=1}^\\infty \\kappa_{n} \\frac{t^{n}}{n!}", "e8ad1cb0a34ad0a6de3fdf6024311b85": " \\Delta Q", "e8adc847caaf2103e848f62e6ab9139c": "\\exp (2xt-t^2) = \\sum_{n=0}^\\infty H_n(x) \\frac {t^n}{n!}\\,\\!", "e8adcdf4060aa5087faf6539eb2e1b18": " \nu\\left(\\omega\\right)=A~w\\left(\\omega\\right)exp\\left[-i\\omega T_0\\right]\\cdot\\hat p\n", "e8ae03b7240d15d02172634375125e31": "\ny = 1 + \\underset{1}{\\overset{\\infty}{\\mathrm K}} \\frac{z}{1}\\qquad \\left(z = \\frac{a}{b^2}\\right)\\,\n", "e8aebd3bd1cc0cc66b5a1c5215daf479": "H(f) = \\begin{bmatrix}\n\\dfrac{\\partial^2 f}{\\partial x_1^2} & \\dfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_1\\,\\partial x_n} \\\\[2.2ex]\n\\dfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_1} & \\dfrac{\\partial^2 f}{\\partial x_2^2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_2\\,\\partial x_n} \\\\[2.2ex]\n\\vdots & \\vdots & \\ddots & \\vdots \\\\[2.2ex]\n\\dfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_1} & \\dfrac{\\partial^2 f}{\\partial x_n\\,\\partial x_2} & \\cdots & \\dfrac{\\partial^2 f}{\\partial x_n^2}\n\\end{bmatrix}.", "e8aefe8135b85b8d959da50e6e812ffe": "e = \\sum_{k=1}^\\infty \\frac{k^5}{52(k!)}", "e8af0fe46d9bfa851c52cf01513abad5": "M_f \\to Y", "e8af3894ee9e34006072f3a3896b7a73": " U(P) = -\\frac{i}{\\lambda} U(r_0) \\int_{S} \\frac {e^{iks}}{s} K(\\chi)\\,dS ", "e8af47e0b232c1942118451aaa18fd4b": "X(e^{i \\omega})", "e8af66df2044b86d67810b33f1300ff5": " \\; {}_0F_1(;a;z) = (1) \\; {}_0F_1(;a;z)", "e8afc5e72d28c7a0542b3603ea6fb1b4": "\n\\begin{array}{ll}\n\\min & g(f_1(x),\\ldots,f_k(x),\\theta)\\\\\n\\text{s.t }x\\in X_\\theta,\n\\end{array}\n", "e8b021f56e457d0c81061e5596daeb26": "\\left|\\frac{a_{n+1}}{a_n}\\right|=1", "e8b02c13e8fd1ae82f9c286f01c0873b": "\\frac{dv_y}{dt} + \\frac{k}{m}v_y = - g ", "e8b0a1536fd320bdbab58561336ba28d": "\\Delta G^\\circ = -R T \\ln K \\,", "e8b0f0336354e1f8bca2aedb3d5ca26c": " 3n(n-1)+1 = \\tfrac{1}{2}n(3n-1)+\\tfrac{1}{2}(1-n)(3(1-n)-1)", "e8b12da9c1bbd2c848d8aca3902e5a54": "\\mathcal{O}(10^3\\hbox{ TeV})", "e8b14d3ff47d9401047f73c8f0208fca": "T=300K", "e8b15a25f0827094a3f1beeb9eef42ab": "R(p,n) = 0.\\ ", "e8b15b1caad6a2640e41cbbee6d5c429": "\n \\begin{bmatrix} V_2 \\\\ V_3 \\end{bmatrix} = \\int_{-b/2}^{b/2} \\int_{-t/2}^{t/2} \n \\begin{bmatrix} \\sigma_{12} \\\\ \\sigma_{13} \\end{bmatrix} \\,dx_3\\,dx_2 \\,.\n ", "e8b16241754a8fdb696732de55bcc374": "\\displaystyle{\\partial_{n+}U={\\lambda-{1\\over 2}\\over \\lambda + {1\\over 2}}\\cdot \\partial_{n_-}U.}", "e8b1651e715ddbcb1663bbfca0a7c5c6": "[\\tfrac{g}{m^2 d}]", "e8b1800a84a083ad3aeedda1208f0bf3": "W(r,k)\\le 2^{2^{r^{2^{2^{k+9}}}}}", "e8b198bfe3e269b7b0d7babb98cfb496": " -j1.49\\,", "e8b198eba72e7cdbc6664e1a38549b00": "= A_{c} \\cos \\left( 2 \\pi \\int_{0}^{t} \\left[ f_{c} + f_{\\Delta} x_{m}(\\tau) \\right] d \\tau \\right)", "e8b1d3303c8ff35f926ba85c4271df50": " f_X(x_1 ,x_2 : t_1, t_2 ) = f_X(x_1 ,x_2 : t_1 + \\Delta, t_2 +\\Delta ), \\, ", "e8b213c52878a1c2ef2080664d1b900f": "q_1\\ ,\\ q_2\\ ,\\ q_3\\ ", "e8b22dabceae81d8fe906cd1111e5d86": "k_\\ell = 0, 1, \\dots, N_\\ell-1", "e8b24e568d6006ed2754b08a494d5ec9": "Lu = \\nabla \\cdot (a \\nabla u) + b^T \\nabla u + c u", "e8b2ac5aa87e86737fdbfb7f7ebb2775": "\\max\\{\\lceil m_S/(n_S-1)\\rceil\\}.", "e8b30a3f0da3f17e93c7bcb1892e72fd": "0 = - \\nabla U(X) - \\gamma M\\dot{X}+ \\sqrt{2 \\gamma k_B T M} R(t)", "e8b350088cd1b7b543efc35cdf2c94c0": " \\mathbf{h} ", "e8b3efc3964484a14fac7bf85c8d0b8b": "\\neg(P\\land Q)\\iff(\\neg P)\\lor(\\neg Q)", "e8b41f8ed7d076c570c62cb11a90c927": " \\psi_x^*\\psi_x + \\psi_y^*\\psi_y = \\langle \\psi | \\psi\\rangle = 1. ", "e8b42a4ef540a659b9a403d390fe3be4": "\\{0,1,2\\}", "e8b4518f1c8306102b1352e5224cd03a": "\\begin{align}\n t_0^{1/3} &= a t_0 + b & \\text{ (match in value)}\\\\\n \\tfrac13 t_0^{-2/3} &= a & \\text{ (match in slope)}\n\\end{align}", "e8b4528f178a790cfa7b49bf20c7e14c": " A(\\Box EFGH)=\\frac{1}{2}A(\\Box ABCD)", "e8b460896ecce6827cdcc257ea397af0": "L = \\frac{4}{5} \\cdot \\frac{r^2N^2}{6r + 9l + 10d}", "e8b46c3490973de3de8e2c89d09b0018": " \\text{Offensive Rebound Rate} = \\dfrac{100 \\times\\text{Offensive Rebounds}\\times \\dfrac{\\text{Team Minutes Played}}{5}}{\\text{Minutes Played}\\times \\left (\\text{Team Offensive Rebounds} + \\text{Opposing Team Defensive Rebounds}\\right )}", "e8b4b2d36bd0adc9d746add3f77a53bf": "Ly=Lclm(D+a_2,D+a_1)y=0;", "e8b58b882153c80ac1f7831ac8fa847c": "\n\\begin{align}\n\\lambda & = \\frac{\\sigma_w T^2}{\\sigma_v} \\\\[2ex]\nb & = \\frac{\\lambda}{2} - 3 \\\\\nc & = \\frac{\\lambda}{2} + 3 \\\\\nd & = -1 \\\\\np & = c - \\frac{b^2}{3} \\\\\nq & = \\frac{2b^3}{27} - \\frac{bc}{3} + d \\\\\nv & = \\sqrt{q^2 + \\frac{4p^3}{27}} \\\\\nz & = -\\sqrt[3]{q + \\frac{v}{2}} \\\\\ns & = z - \\frac{p}{3z} - \\frac{b}{3} \\\\[2ex]\n\\alpha & = 1 - s^2 \\\\\n\\beta & = 2(1 - s)^2 \\\\\n\\gamma & = \\frac{\\beta^2}{2\\alpha}\n\\end{align}\n", "e8b59ec79f97579857ff18c59d81fbf7": "E= \\tau \\theta\\ ", "e8b5aa2663bb7ffdd73c96e525e39606": "T^2_{+1}(q) = -q_{xz} - iq_{yz}", "e8b5b6a00a2d99a149a414f09cffbcb5": "\\mathbf{c}=\\mathbf{a}\\otimes\\mathbf{b}, \\quad c_{ij}=a_ib_j ", "e8b6b239a32aecd13d3dd0e34eb00c12": "(98+2) \\cdot a + b", "e8b74ad8650c2da1a4eff001f33e77bc": "\\textit{open}(t-1) \\leftrightarrow \\textit{open}(t)", "e8b79488e1e25c07125a2d53aae650c7": "\\beta_{K+1}", "e8b7a5beb2231214fb39bbe25a3e552c": "\\langle 0|T\\Phi(x)\\Phi^\\dagger(y)|0\\rangle = \\int_0^\\infty d\\mu^2\\rho(\\mu^2)\\Delta(x-y;\\mu^2)", "e8b7cd08a3ee5ebef13d04ee07c7a5ab": "Q: X \\times \\Xi \\rightarrow \\mathbb{R}", "e8b8098180ba7ab3d10be3bd4e6427d1": "Y=Y^{s}(W/P, \\ \\ P/P^{e}, \\ \\ Z_2)", "e8b85264d3a0d628219935db85eaa371": "_{0 \\nleftarrow p=p}\\!", "e8b883d98a276fdbd3d546d7da9fa2c2": "r_{ij}\\neq 0 \\implies i\\leq j", "e8b8b647be32badcf731abc23e7d7ec3": "\\begin{matrix} {4 \\choose 2} \\times 2! \\end{matrix}", "e8b8f546b0e4e6cad636169ad4bbe95b": " y(0)=0", "e8b93812e2a586f29b6d9c165cce45c9": "1=(1-\\tan \\theta \\tan \\alpha)\\sec \\alpha \\quad \\; ", "e8b98c898b3bf28eba2a80cfaa8b321a": "(A/\\Psi)/(\\Phi/\\Psi)", "e8ba4cea22ac9013de4c5df9eb4ca00b": "\\sum_n V_1^{(n)} I_2^{(n)} = \\sum_n V_2^{(n)} I_1^{(n)} \\!", "e8ba5ab874c548e65fd4755887a875cf": " D \\equiv ", "e8ba83ece6dc58790624468a735f08f0": "p(y_1,\\ldots,y_m|x_1,\\ldots,x_m;\\theta) = \\prod_{i=1}^m \\frac{e^{y_i \\theta' x_i} e^{-e^{\\theta' x_i}}}{y_i!}.", "e8ba8eddf7a2706444db3a34c7c61c78": "\nY_{nk}=y\\in \\{0,1\\}, k \\in\\{0,1,\\dots,m\\} \\,\n", "e8ba9aa1d6118bcef7422e939510c140": "\n \\hat{a} = \\frac{nx_{(1)} - x_{(n)}}{n-1},\\ \\ \\hat{b} = \\frac{nx_{(n)}-x_{(1)}}{n-1}.\n ", "e8baa340d2cea8b5e130ee4b59ef1465": "\n2 T =\n\\begin{pmatrix} \n\\dot{\\alpha} & \\dot{\\beta} & \\dot{\\gamma}\n\\end{pmatrix}\n\\; \\mathbf{g} \\;\n\\begin{pmatrix} \n\\dot{\\alpha} \\\\ \\dot{\\beta} \\\\ \\dot{\\gamma}\\\\\n\\end{pmatrix},\n", "e8bab74fffc03b3db187e716260d40f4": "(x, y)\\in\\mathbb{C}^2", "e8bafa7ee63b1ff0c1ecbc14d0c13bb5": "D^2+\\tfrac{1}{4}[\\gamma^\\mu,\\gamma^\\nu]F_{\\mu\\nu}", "e8bb3b9ee0fffadaba606304138414b5": " \\tilde{C} = C \\; + \\; \\operatorname{grad} \\, \\omega \\; \\lrcorner \\; W,", "e8bb47f19d155ab9dd7fcd948f365e60": "f_{c}", "e8bb5b4808f55b04e10d9949aa4755a4": " K_{a3}=\\frac{[\\mbox{H}^+][\\mbox{PO}_4^{3-}]}{[\\mbox{HPO}_4^{2-}]}\\simeq 2.14\\times10^{-13}", "e8bbb64bd23f372e1765aa12d276c617": "~f(\\omega)=\\frac{1}{\\sqrt{2\\pi}}\\int f(t) \\exp(i\\omega t) {\\rm d}t ", "e8bbb8473e4984af4e8766e65e904cfb": "\\scriptstyle{e}", "e8bbef81e16bfe595358028a537dbdfd": "|\\mathcal P|<\\infty", "e8bc240fb2c37bfebdd556d58777aa04": " d' ", "e8bc389235f58c7a2561e65eac099877": "\\Theta_L", "e8bcc2cf1ed9bd435b4edef78a0d8759": "[X,Y] := XY-YX", "e8bccb9837041c037e1eddff9238eb73": " \\operatorname{let-combine}[Y] ", "e8bcffecd2b51a7c3ceeeef05ce1f707": "T_0(x) = 1\\,\\!", "e8bd54d95fb397de304dd350c05bb834": " p \\mapsto g_p(X(p),Y(p))", "e8bd95557f95a1d2b7ae7b7f04c5b34a": "\nF(x, y)=\n\\begin{cases}\n\\frac{x^3}{x^2+y^2} & \\mbox{ if } (x, y)\\ne (0, 0)\\\\\n0 & \\mbox{ if } (x, y)=(0, 0).\n\\end{cases}", "e8be410e67b813edd36cec671333f36c": " c=1/\\sqrt{\\mu_0 \\epsilon_0}", "e8be8de1eddc02aa3023df5f3cd21292": " = \\sum^\\infty_{k = 0} \\int^{2\\pi}_0 \\frac{{f(re^{i\\vartheta})}\\overline{a_k} (r^k)}{(e^{i\\vartheta})^k} , \\mathrm{d}\\vartheta", "e8bea465ce1910d5daff3b29bcb200b9": "\\{l^a,m^a,\\bar m^a\\}", "e8bec6745a4054cecb541568d4396fb1": "\n\\inf_{x_1, x_2} \\phi(x_1-x_2)=-\\infty\n", "e8bf61880904d45ac8ce28a90bfcbf0d": "\n \\tfrac{1}{\\sqrt{2}}\\rho - \\sqrt{3}~B\\xi = A \n ", "e8bf85fa6fc73036fc51a364ed20e2cb": "x_i=(\\frac{w_i}{\\sum_jw_j})*(B)", "e8c0034f86c7224a339f9ac51609a92a": "U(a,L)U(b,M)= \\pm U((a,L).(b,M))", "e8c01589289ebf112636a38696503a3b": "x_{i}^k", "e8c01a8c3baeca2e121f479ec72135ef": "1^2 + 2^2 + \\cdots + n^2 = {n(n + 1)(2n + 1) \\over 6}", "e8c02db099d187e4f09c1c57da598506": "\n\\begin{align}\n\\frac{d\\mathbf{x}}{dt} & = \\mathbf{f}(\\mathbf{x},\\mathbf{z},t), \\\\\n\\mathbf{z} & = \\varphi(\\mathbf{x},t),\n\\end{align}\n", "e8c10cf79736f62f9035a2026bae4b97": "\\mathrm{P_c} = \\alpha \\, P_1 + \\beta \\, P_2 + \\gamma \\, P_3", "e8c129a910b5c78ea6fe64459475dee9": "\\chi^2(6)\\,", "e8c1d0c05d7e7da3997ea2fdac534a7a": "T [L] = \\bigcup_{t\\in T} L_t", "e8c1d248c7229d85205c538060233242": " \\begin{cases} \\begin{bmatrix} 1 & 0\\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} a \\\\ c \\end{bmatrix} = \\begin{bmatrix} ax \\\\ cx \\end{bmatrix} \\\\ \\begin{bmatrix} 1 & 0\\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} b \\\\ d \\end{bmatrix} = \\begin{bmatrix} by \\\\ dy \\end{bmatrix} \\end{cases} ", "e8c2612c41bee221f487fdf2233e019b": "m+s\\left(\\frac{\\alpha}{1+\\alpha}\\right)^{1/\\alpha}", "e8c27396c7d103eaf6440d122ae23262": "\n | \\phi \\rangle = \\int_X \\Psi (x) | x\\rangle \\; d\\mu (x)\\; , \\qquad\n \\Psi (x) = \\langle x | T^{-1}\\phi \\rangle\\; ,\n", "e8c289dd177484921000b411f96ec5a8": "\\textstyle S = \\sum_{i=1}^T (x_i-\\alpha)^2", "e8c2c3b99b3e03ca76b89e6a77212ddf": "\\frac{d}{dx}\\, \\operatorname{arcoth}\\,x =\\frac{1}{1-x^{2}}", "e8c2fab884fa4774ab6daa470c57b834": "t_e = \\int \\beta_h \\beta_T \\mbox{d} t", "e8c358f87a35a53659e750d45b1bb0eb": "2^{k t}", "e8c36eca761ada7a6c1045fa0a3f58da": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin^2 u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^3\\ \\sin u\\ \\cos u \\ du\\ = \\\\\n&-\\hat{g}\\ \\left(\\int\\limits_{0}^{2\\pi}\\ \\sin^2 u \\ du\\ +\\ \n3\\ {e_g}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^2 u \\ du\\ \\ +\\ \n3\\ {e_h}^2\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^6 u \\ du\\ \\right) \\\\\n&+\\hat{h}\\ 6\\ e_g\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\cos^2 u\\ \\sin^2 u \\ du = \\\\\n&-\\hat{g}\\ \\left(2\\pi \\left(\\frac{1}{2}\\ +\\ \\frac{3}{8}\\ {e_g}^2\\ +\\ \\frac{9}{8}\\ {e_h}^2\\right)\\right)\n+\\hat{h}\\ \\left(2\\pi \\left(\\frac{3}{4}\\ e_g\\ e_h\\right)\\right)\n\\end{align}\n", "e8c3e29a84afbf15308d802e2d73be2b": "W_s = F s=2 \\pi n T", "e8c43536307a598cea3e0bdbe5814183": "\\overline{V}", "e8c4553b526a95c1b61f95f93cf82760": "V_{\\text{T}}", "e8c4c79ce04c833e29ccbeff16f2d758": "(Z_i)_{i=0}^n", "e8c5335a49770730fee3bbf7c3d41807": " M=V_1\\cup V_2\\cup V_3", "e8c563794ec6f2f44b9b16b98e8abf39": "1-p_0=p_1+p_2+p_3=60p_0", "e8c56e5e2d15749e594afc3175198409": "\\triangledown ^{2}F_{z}+k^{2}F_{z}=0 \\ \\ \\ \\ \\ \\ (13)", "e8c6446dbaca093ae515bc17313a7b23": "\\gamma_e=1, \\gamma_i=3", "e8c644b657d3a4434327d3ddad1f0d8c": " X_n\\ \\xrightarrow{d}\\ c \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{p}\\ c,", "e8c68e3963224d956e9a099e63ae1f65": "g^{abcd}", "e8c6a52fff2b098caee59e2971eb8546": " m_f = -log_{2.512} \\left(2.512^{-m_1} + 2.512^{-m_2} \\right) \\!\\ ", "e8c6af998ee92a545d85906e3d33d558": "\\,\\chi_n(z)=2^{-n}z \\Phi (z^2,n,1/2).", "e8c6b17c4e9dbbfa3e79464654a01602": "P_{t+\\Delta t}=P_t(1+r\\Delta t)-M_N\\Delta t\\; ", "e8c6c071fe99bd677f97f7b9aa0c8697": "x = 2 + 3t", "e8c718e52d1770121ac687b56bc93a2a": "S(f) = \\int_{-\\infty}^\\infty R(\\tau) \\cos(2 \\pi f \\tau) \\, {\\rm d}\\tau.", "e8c74dea4e03cdd459c2afa29c838c3e": " W_R(K) =\\text{Tr}_R \\, \\mathcal{P} \\, \\exp{i \\oint_K A}", "e8c787f4706b4dd78ab91439139c3e7b": "\nd_1 =-\\tfrac{1}{24},\\qquad d_2 = \\tfrac{3}{4},\\qquad d_3 = \\tfrac{7}{24}.\n", "e8c87e1869690e0b096d050d8d36b761": "(0.01)^2,", "e8c898540243ba90b7bbadac3e71ebe1": "\\scriptstyle{\\varepsilon_\\circ \\, = \\, 8.85\\times 10^{-12} \\, F/m }", "e8c918afb061b6ac38295b330f80fdf0": "x = L/2", "e8c9422b9a24392006789b394425e22c": " g(x,y) = y^2 - f(x) = 0 ", "e8c9656c3a04dc19b830249df1480608": " p_{\\alpha, \\beta} (\\varphi) < \\infty.", "e8c96853879d1e25e21b7c4866770474": " \\dot{\\omega} ", "e8c98c574592f4b1a0b31a482f0af4c7": "\\frac{n-c}{n}", "e8c9b867d2c78d03c68cd987f8548df9": "\\phi= \\arctan {\\left (\\frac{2 \\zeta r}{1-r^2} \\right)}. ", "e8c9c7b05cf2f7d7e2030255880693cd": "(P \\to Q), (R \\to S), (\\neg Q \\or \\neg S) \\vdash (\\neg P \\or \\neg R)", "e8ca13ccc43f9708b6c7a6e86e5f14ff": "f(x) = \\frac{1}{b-a}", "e8caf2ac54c98df6bc7c461772148ebe": " G(A) \\subseteq G(A^*).", "e8cafd00ca798071e96c439f6703ba00": " f(\\Delta)=(f_{-1},f_0,\\ldots,f_{d-1}).", "e8cb55708e39b3964dec2ca6c8e85a66": " \\lambda_d \\Bigl( E \\setminus \\bigcup_{j}U_{j} \\Bigr) = 0.", "e8cc009841e20b458590575f6960c2d8": "f \\left( t \\right)", "e8ccbe98bf951a040d250499b43b3550": " a \\in W^s ", "e8ccd2696d8c144da4a48b1806ef2ff6": "X\\to \\mathbb R", "e8cce3b30fa9b252209a1bd3bfda7a86": "2^{-54.2}", "e8ccf3c026611f53c6373c7c6ca96f1f": "2.1\\overline{6}", "e8cd1151523c2cedb97a2485ef4d71c4": "r_f T", "e8cd1fb36c8d9b783cdafb9ccf04715d": "\\mathbf{F}^g = \\vartheta^f \\mathbf{f}_0\\otimes\\mathbf{f}_0+\\vartheta^s \\mathbf{s}_0\\otimes\\mathbf{s}_0+\\vartheta^n \\mathbf{n}_0\\otimes\\mathbf{n}_0", "e8cd4455ccbdef821225925cddcb7d45": "2^{31}-1", "e8cd5ce82caaf7bd0f688e86a4edcab6": "\n10.000 \\mbox{ metres} = \\frac{L + 0.25G +2d + \\sqrt{S} - F}{2.5}\n", "e8cdb19c9ffba819dda9d33b38f5664e": "S_N = a_0 b_0 + \\sum_{n=1}^N a_n (B_n - B_{n-1}),", "e8cdd2903dba02066c7f9cc5190f28f8": "a, b, c ", "e8cdfbdd922beb3a9aa8947060e329a4": "\\tau:=\\inf \\{t\\geq 0 \\,|\\, B_t > a\\}", "e8ce909470435169b3b4b3f4510a5265": "\\gamma_{ij}(t,x^k)", "e8ce92c667c3058ab78cd32bc0505d6b": "\\frac {\\mathrm{d} P}{P} = \\frac {L}{R} \\frac {\\mathrm{d}T}{T^2},", "e8cec84df2bc2cc801ccb4e165a576d3": "b^{(B-1)/2}\\equiv -1 \\pmod B\\;", "e8cefe7145e1eb4c1a15a6b62a2d9d22": "r'\\ ", "e8cf2f1b471a07a7fb72f82312766033": " s = \\alpha_1 = -3 + 4j ", "e8cf438d880989bafce63ef929c80160": "V_{n+2}=2\\pi V_n/(n+2)", "e8cfdba0ef3a47f2ff36c8d9b28a5d23": "[\\mathbf{A}]_{ij}=a_i\\cdot a_j", "e8cfe15e364e5d1581abb6be6f8f6d09": " F(\\Delta ,E,E') = \\frac{1}{2}(1 \\pm \\frac{{\\Delta ^2 }}{{EE'}})", "e8d080fdd5a238fa25af0e5f67781140": "(\\Omega,\\mathcal{F},\\{\\mathcal{F}_t\\}_{t=0}^T,P)", "e8d102550f6accb5ef782096729ba3df": "\n\\operatorname{Li}_{-n}(z) = z \\;_{n}F_{n-1} (2,2,\\dots,2; \\,1,1,\\dots,1; \\,z) \\qquad (n = 1,2,3,\\ldots) ~.\n", "e8d16720a9988b546ed772371245b42d": "X \\,\\sim N(\\mu, \\sigma^2) ", "e8d1ca6c7a5fa9f3ad3e7573ce136ef2": "\\nabla J=0", "e8d1fa78fbe2993508163f20d7464126": "h \\ll 1", "e8d20a2449628fca1837a4b156ea6717": "T(W)", "e8d2476641e7af3ac5c026267cc19912": "\\begin{matrix} {9 \\choose 1}{4 \\choose 2}{8 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "e8d2c3468a5a49043656d4684b3cf204": "\\mathbf J=\\boldsymbol\\Omega+\\mathbf R", "e8d2dab0a6e4f625b4500f64d87d1a13": "(10\\uparrow^2)^3 b", "e8d3038e79658de457a1b6cd6c38f55d": "\\mathrm{proj}_{\\mathbf{u}}\\,(\\mathbf{v}) = {\\langle \\mathbf{u}, \\mathbf{v}\\rangle\\over\\langle \\mathbf{u}, \\mathbf{u}\\rangle}\\mathbf{u} , ", "e8d306161b47eddf945a8a4b31abb1ab": "G_4=\\{1\\}", "e8d3089076aafe7358335391365ebd16": "k_y", "e8d3339455d49c83d625c1d8a7cce9a3": "\\delta_{xy}", "e8d34ba90cf5568d3e93345e8544fc39": "\\hat v := 1 + \\varepsilon (v_0 i + v_1 j + v_2 k)", "e8d36fd38970aab957ffde28ee51ee4e": " \\int_{S_u} \\mathbf{u}^T \\delta\\ \\mathbf{T} dS + \\int_{V} \\mathbf{u}^T \\delta\\ \\mathbf{f} dV = \\int_{V} \\boldsymbol{\\epsilon}^T \\delta \\boldsymbol{\\sigma} dV \\qquad \\mathrm{(g)} ", "e8d3bd88e2636ebc1d24ef8412dd265e": "q = \\sgn(y) \\left\\lceil \\left| y \\right| - 0.5 \\right\\rceil = -\\sgn(y) \\left\\lfloor -\\left| y \\right| + 0.5 \\right\\rfloor \\,", "e8d3d4837a22aa118cb8253bea6c98c3": "lb_{computed}>1 ", "e8d3f5a1877f5062d4099f2e75d99de8": "f(x_0 + h) - f(x_0) - \\lambda h", "e8d408717f0252117311587bf4c39efd": "\\frac{\\part f}{\\part x_i}(a_1,\\ldots,a_n) = \\lim_{h \\to 0}\\frac{f(a_1,\\ldots,a_i+h,\\ldots,a_n) - f(a_1,\\ldots,a_i,\\ldots,a_n)}{h}.", "e8d48333436b5d583068382cdff56c59": "\\begin{matrix} {2 \\choose 1}{10 \\choose 1}{4 \\choose 2}{36 \\choose 1} \\end{matrix}", "e8d4cedfa6c537b02023b13f7183b09f": " C_{ijkl}\n= K \\, \\delta_{ij}\\, \\delta_{kl}\n+\\mu\\, (\\delta_{ik}\\delta_{jl}+\\delta_{il}\\delta_{jk}-\\frac{2}{3}\\, \\delta_{ij}\\,\\delta_{kl})\\,\\!", "e8d51765e0200d52b1fb9ca5300932a5": "\\mathcal{F}^{-1}(\\mathcal{F}f)(x)", "e8d55c8bec19da8ff395e2b1f99da00e": "r_{k+1}^* \\leftarrow r_k^*- \\overline{\\alpha_k} \\cdot p_k^*\\, A ", "e8d5b739b781a20247f68515e97d8df0": "3363+2378\\sqrt{2}=6725.99985\\ldots", "e8d67bde0664749b2d049a8235d9be52": "(a\\cdot a_i)^{n-1} \\equiv a^{n-1}\\cdot a_i^{n-1} \\equiv a^{n-1} \\not\\equiv 1\\pmod{n}", "e8d6ef43b324429b506b22916d57774f": "\\mathcal{L}_X(fg)=(\\mathcal{L}_Xf) g + f\\mathcal{L}_Xg", "e8d733bf62337c6528bcc091555efed9": "\nT_+(x) = 576\\int^\\infty_0 \\frac{dt}{t^3}J_0(xt)[J_4(t)]^2\n", "e8d766b25f6fcd04cc6518e737113edc": " A=\n \\begin{bmatrix}\n 2 & 1 \\\\\n 5 & 7 \\\\\n \\end{bmatrix},\n \\ b=\n \\begin{bmatrix}\n 11 \\\\\n 13 \\\\\n \\end{bmatrix}\n\\quad \\text{and} \\quad x^{(0)} =\n \\begin{bmatrix}\n 1 \\\\\n 1 \\\\\n \\end{bmatrix} .", "e8d7782216bbb8fe6af16a2ef759d23e": "\\scriptstyle{k_e}", "e8d79d2a7b822de93ad2ded53a3e0c7e": "\\forall ij~\\mu_{ij} \\in \\lbrace0, 1\\rbrace", "e8d7bc1a67db44e249ed90a160afd0b5": "q.", "e8d8132364db7f4b7f084c163db34ceb": "|\\Psi\\rangle=\\sum_nc_n|\\psi_n\\rangle", "e8d844377b082101d19fd65fe2744ffe": " J(\\chi,\\psi) = \\sum \\chi(a) \\psi(1 - a), \\, ", "e8d88eeeff5516bdd992a4c083865843": "DPW = \\frac{\\displaystyle \\pi d^2}{4S} \\left(1 - \\frac{\\displaystyle 2\\sqrt{S}}{d} \\right)^2", "e8d8d1cee30f2789839e2a679089ca70": "\\beta=\\lim_{n \\to \\infty}2nE_{2n}(f),\\,", "e8d90dbcc6bd68f0625363f2505c64f2": "P_{\\lambda}^{\\mu}(z) = \\frac{1}{\\Gamma(1-\\mu)} \\left[\\frac{1+z}{1-z}\\right]^{\\mu/2} \\,_2F_1 (-\\lambda, \\lambda+1; 1-\\mu; \\frac{1-z}{2})", "e8d929f669ddfe60d1392eef1a2ebe6a": "a = R_t/R_0", "e8d941d1a33744e03d0e4c2f59383ef5": "q^k < \\frac{q^n}{\\sum_{j=0}^{d-2} \\binom{n-1}{j}(q-1)^j}.", "e8d94a3ffbd3bcc67c3b708b71288031": "\\scriptstyle\\mathcal{C}[\\cdot]", "e8d9d2af05f27f0417e406183c24da8d": " B_Q(v,w)= \\tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).", "e8da3a790b746618f81cd25e1d088944": "X_{i+1},X_{i+2},...,X_{n},", "e8da41447bd25495221cbb57054ab2b9": "K^\\times", "e8da46f23d09f2925c004dc8ea4503b1": "(N \\cup \\Sigma, P)", "e8da507b26afa1d951274d4f92086f65": " \\left| \\nu_{i} \\right\\rangle", "e8da6aae46d2a240bad9232befcd8ef6": "\\frac{1}{\\sqrt{|a|}}x(\\frac{t}{a}) \\rightarrow W_x(\\frac{t}{a},af)", "e8dad7b292a77aca67977201a138a616": " \\frac{\\partial q}{\\partial t}= 0.9~10^{-11}~ \\frac{\\partial^2 T^{7/2}}{\\partial x ^2 }", "e8dae1ce9b388eb270ab6f24252f24d3": "r=r_e", "e8daffb471608898fd359c44859939c5": "{n\\choose k}_2={n\\choose-k}_2", "e8db3d77895fffa2817538cb28425485": "H(t)=\\sum_{k\\geq0}h_k(X)t^k=\\prod_{i=1}^\\infty\\left(\\sum_{k\\geq0}(X_it)^k\\right)=\\prod_{i=1}^\\infty\\frac1{1-X_it}.", "e8db995e2bedfe234664166f424d969d": "\\Delta=d\\delta+\\delta d \\,", "e8dbe824fe57f439c65840c69a4e7e90": " n_s=n\\sqrt{P}/H^{5/4} ", "e8dc0c0dacfdba71d4fdf01ba596ca42": "C_P=\\frac{4}{27}C_D", "e8dc14c99b4f238d1bbf1154e8a720c5": "x_0 \\equiv a_i \\pmod{m_i}", "e8dc4aaaaf03a31cf9bd7b1776138087": " D(h,F) \\approx D_{\\mathrm{F}} \\; \\mathrm{exp}(\\epsilon / d_{\\mathrm{F}}) \\; \\mathrm{exp}(-K_{\\mathrm{p}} / d_{\\mathrm{F}}) .......... (19b), ", "e8dc6a1fcf1c149faa14ea83adbfc4de": "=10^7", "e8dcc15efd6dcc0bb69c0395a473cf36": "\\Delta ({\\hat \\Psi}, \\Psi_{id}) = \\inf_{E,D} \\| {\\hat \\Psi} - \\Psi_{id} \\|_{cb}", "e8dd3ae76e6302589fcb8e90cc96bb01": " dl = {dz \\over c \\cos \\alpha(z) } ", "e8dd4e103f61bf0b5878afaca00a350d": " \\alpha_i = 0 ", "e8dd8b29e6ffc1ff7f429633e777600f": "\\neg P \\or \\neg R", "e8ddb24b01ff97c9c0ff52e2559f38f8": "\\bigcup_{n \\in \\omega} \\left(\\prod_{i3", "e8eb38c022ae007c15832b64c99d3686": " \\zeta (1/2+i\\hat H) = 0 ", "e8eb5362f2745efa6472c9d760354c65": "Tr h'_\\gamma = Tr h_\\gamma.", "e8eb5869a6a33ef15f6bf0bff53f10a4": "\\forall a:\\exists b : \\forall c: \\exists d : R(a,b,c,d) ", "e8eb6952f2ae848ed8e7e7888d4da79d": " \\mathrm{Tr} \\, \\pi_{\\mathbf{f}}(U) = {\\mathrm{det}\\, (z_j^{f_i +n -i} + z_j^{-f_i-n +i})\\over \\prod_{i0", "e90040282fc74470d97fdb34e7dc8edf": "x_{\\mu}", "e90051f1e764967b4301adc394e02ef4": "\\left(a, r, u\\right)\\succsim \\left(c, p, u\\right)", "e90054dc1c5bd5d9af8d1dbe49810743": "\n\\int p \\, dx = \\hbar \\int k \\, dx = 2\\pi\\hbar n\n", "e9006a08d8d5ad5e91e611e83c7ff682": "(INF/(C + INF)) * 100%", "e900d82d0da46cc5814939cabd836246": "\\frac{\\partial f(g)}{\\partial g} \\frac{\\partial g(u)}{\\partial u} \\frac{\\partial u}{\\partial \\mathbf{x}} ", "e901e222ebde3f066a7a1f40e8f3f7e4": "\\left( a,\\tfrac{a^2-1}{2},\\tfrac{a^2+1}{2} \\right)", "e90215434589db18a43d4483b1383c27": "\\{f,g\\}(e) = 0", "e902944dd3f1af6a2b0cc0bb5f74f4de": "\n\\frac{dV}{dx_3}=\\int_V \\nabla \\cdot \\mathbf{v}dV\n", "e902ad015cbc9323508591e4fe35dd6d": "(\\xi + \\eta)_{inf}(\\alpha)=\\xi_{inf}(\\alpha)+\\eta_{inf}{\\alpha}", "e902b3f386d17d7f7fe33d0389e9c735": " I(z) = I_0 \\, e^{-\\alpha z} ", "e902e2fce476debca4fa75541647a36d": " \\gamma(\\phi) = \\gamma\\phi", "e902f324692bd466a89f00c7ae224e4d": "\\Phi = \\sum_i X_i \\otimes Y_i \\otimes Z_i ", "e9032bf83c322fd38366fd9fee8572e7": "\\begin{align} 2\\cdot R_*\n & = \\frac{(73.3\\cdot 3.72\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 58.7\\cdot R_{\\bigodot}\n\\end{align}", "e90355f4b2b8435c8e7bcdcff0aff6bd": "F_x = A (P_2 - P_1) = \\frac{1}{2} \\rho A (C_s^2 - C_u^2) ", "e903a4a1c08f75e279eedfbea63ff544": "\\mathcal{L}_X\\omega=0", "e903b24beaf283686407ac1345afcf3c": "n_s ", "e903be465169e449a76a052c01a60239": "p \\times q", "e903d13ca6ea6e77d5ea1f309c1e8f69": "\\scriptstyle T_{input}=u+P_U", "e903d59ade77b2fff99a0ebf3f69beca": " q = \\left(x - \\frac{1}{4}\\right)^2 + y^2 ", "e903f27e6e406aebbbc95cf0efb1f9f9": "V_{\\rm bi}", "e904091b1bb15268d5b5aee7f5aa7b92": "S^* \\subseteq S", "e904831979167e6f69522ceda9a07b35": "(\\mathbb{C}^n,0) \\to (\\mathbb{C},0)", "e9049a2628c16ad27fe484f4e233ad79": "k_3", "e9049ff8e23cd7588a5073f38a7e1efb": "\\langle \\phi_a(v), v'\\rangle >0, \\langle \\psi_b(v), v'\\rangle=0", "e904b1c1bc3826a47a3619e56c2b334c": "P^1_r", "e904b2610684abcd1d6106f68d59023b": "E(l) = \\frac{6}{\\pi^2} \\sum_{l=1}^\\infty \\frac{1}{l} = \\infty . \\,", "e904f6c69d045f0ba32c056f27344a0e": "\n{}_t q_x = \\frac{S(x)-S(x+t)}{S(x)} = \\frac{t}{\\omega-x}.\n", "e9050fa932f50da701193bef6a8ca5db": " \\left(\\frac{1}{c}\\dfrac{\\partial }{\\partial t} + \\boldsymbol{\\nabla}\\right)\\mathbf{F} = \\frac{1}{\\epsilon_0}\\left( \\rho - \\frac{1}{c}\\mathbf{J} \\right).", "e90545bc9a72caa72b81d266d605873a": "\\ 6.3y'^2 - 2y'^3 - 1 = 0 ", "e9055cdc8b5e8bcbfe084a7bdc91efb8": "X+Y", "e905633c78aba20b6edc29a0e79f679c": " \\psi(q+\\varepsilon) = \\psi(q) + i\\frac{\\varepsilon}{\\hbar}\\langle q|\\mathbf{\\hat P}|\\psi\\rangle + O(\\varepsilon^2) ", "e90572b9c349ccbfdf5fddaac1285bae": "\\mu_k/\\sigma^k", "e9057a5fd30c74ad873e2b7c807180aa": "\\vec{F}_\\mathrm{d} = - b \\vec{v} \\,", "e9059d9f4143482d553911a359bd7a4e": "\\bar{\\rho}(E)= \\begin{cases} \\sqrt{2N-E^2}/\\pi\n& \\quad \\left\\vert E \\right\\vert < \\sqrt{2N} \\\\ 0 & \\quad \\left\\vert E \\right\\vert > \\sqrt{2N} \\end{cases} ", "e905b645ef43dfea61b0dd78512e3155": " \\lim_{t \\to 0^+} {t}^{t} = 1, \\quad \\lim_{t \\to 0^+} \\left(e^{-\\frac{1}{t^2}}\\right)^t = 0, \\quad \\lim_{t \\to 0^+} \\left(e^{-\\frac{1}{t^2}}\\right)^{-t} = +\\infty, \\quad \\lim_{t \\to 0^+} \\left(e^{-\\frac{1}{t}}\\right)^{at} = e^{-a}", "e905bed52c01f93d570ecd6355b7e6d3": "\\mathrm{Ox} + ne^- \\rightarrow \\mathrm{Red},", "e905bfd1c6eea9f0291414a32405a3a3": "ExprRest \\rightarrow \\epsilon\\,|\\,+\\,Expr\\,ExprRest", "e905e400599bfa3d6a2c4c34c4092520": "\\mathrm{E}[D]=\\tau", "e905edf01da022ada3db3d24f98e88fe": " \\frac{\\mu(A)}{\\mu(X)} = \\frac 1{\\mu(X)}\\int \\chi_A\\, d\\mu = \\lim_{n\\rightarrow\\infty}\\; \\frac{1}{n} \\sum_{k=0}^{n-1} \\chi_A\\left(T^k x\\right) ", "e905faf3a6f1433a3756d55588c1aafc": " [E] = \\frac{(k_{-1}+k_2)[ES]}{k_1[S]} ", "e90610b1c99ce36d8a2a2251f3eb4f73": "T(\\mathbf{M}) = a\\mathbf{M}\\mathbf{R}^T + \\mathbf{1}\\mathbf{t}^T", "e9062608c1d002d01b6223a3bfc1581d": "\n\\begin{align}\nA_{-1}& = 1& B_{-1}& = 0\\\\\nA_0& = b_0& B_0& = 1\\\\\nA_{n+1}& = b_{n+1} A_n + a_{n+1} A_{n-1}& B_{n+1}& = b_{n+1} B_n + a_{n+1} B_{n-1}\\,\n\\end{align}\n", "e90642b9290f3f7780b2a4b9a502c892": "H'_{ab}{}^b=0.", "e906661c4f71e6974da49bc6d453e645": " (\\mu_{i,j}: i\\in I)", "e9068e80792f97745ffd954e48f7d372": "n G = \\infty", "e90736037df4cb0711c595848b8f448b": " Fr \\approx 1 ", "e90749f758fa44cf24b23c4a8055cbd6": "T\\to 0", "e907b8a7279dcf1c938099a5a9b46cd7": " ln[S(x + t)] - ln[S(x)], ", "e907b8aa721d963e5b03df13c9934a68": "\\frac{\\operatorname{d}N}{\\operatorname{d}T} = B - D = bN - dN = (b - d)N = rN, ", "e9084aebb62121f887ae44e9c80442cd": "[\\mathfrak{k}, \\mathfrak{p}] \\subseteq \\mathfrak{p}", "e908903eb1dcdc3e3068add3ff230162": "\\textbf{A}", "e908c36984bcc8222aaf5d939302619f": "\\Delta H_{vap} ", "e908c632eaa6ac5a25b6b13988434c2f": "\\textit{Re} \\ll 1", "e908d683b66dd9bfe2fc006fa8ae5279": "\\scriptstyle \\not \\Rightarrow", "e908d7d721c4e46cda92fbfe2313cc61": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \\sum_{k=0}^{\\infty} I_{k}\nr^{k} P_{k}(\\cos \\theta )\n", "e908f65b06d5b7387c9447ea8c8fd58e": " PCI = \\frac{ X_t }{ Z } [ 1 - | \\frac{ \\sum_{ i = 1 }^{ r_+ } X_+ }{ X_t } - \\frac{ \\sum _{ i = 1 }^{ r_- } X_-} { X_t } | ] ", "e9093c6cf5c16b045ff3eff26930d905": "\\chi(\\tilde{M}) = k \\cdot \\chi(M).", "e9093d9b6aa80ec50c73ee6041563611": "I_{L_{Max}} = \\frac{V_i-V_o}{L}D T", "e909bd71e095e1a51416f0fcb0fd94fe": "|\\phi_2\\rangle", "e909e2bfe321e0fc27ea447220d5498f": "f=g\\circ T", "e90a0d7ca60c94e7cf95f4e2daa91814": "-1.3,", "e90a180941f8c9a222900b9e9f4b2ba3": "^{\\;}q^{i}(\\xi,0)=\\xi^{i}.", "e90ab95d88bc4c764633fa6ed155acda": " k \\in \\{ 1,\\dots,K \\} ", "e90b19dbeeb39d8392afddc645156dab": "V(\\rho,\\varphi,z)=\\sum_{n=0}^\\infty \\sum_{r=0}^\\infty\\, A_{nr} J_n(k_{nr}\\rho)\\cos(n(\\varphi-\\varphi_0))\\sinh(k_{nr}(L-z))\\,\\,\\,\\,\\,z\\ge z_0", "e90b5f485e0f213109e66f6dcf7e99f3": "G\\subset \\mathbb{C}", "e90b70296487d2b04c8b380b9bec6d38": "P_{L}", "e90b783e4ac15e924e2de4ada4316322": "\\kappa_3=100", "e90b8db9d704132bd60fac44aad0db59": "\\xi'=\\tan^{-1}\\left(\\frac{t}{\\cos(\\lambda-\\lambda_0)}\\right), \\,\\,\\,\\eta'=\\tanh^{-1}\\left(\\frac{\\sin(\\lambda-\\lambda_0)}{\\sqrt{1+t^2}}\\right),", "e90bf808e3a8e5cc136e5c60453dfee2": "WXY + XYZ + YZW + ZWX =0", "e90bfe25ca772786bbf4581ad78c92fa": " \\oint_C \\Gamma(\\alpha)\\,d\\alpha = \\oint_C \\int_0^\\infty x^{\\alpha-1} e^{-x}\\,dx \\,d\\alpha ", "e90c371c60032075c70cfe547afd602c": "\n \\displaystyle \n V_{LJ}\n (r)\n =\n 4 \\varepsilon\n \\left[\n \\left(\n \\frac\n\t {\\sigma}\n\t {r}\n \\right)^{12}\n -\n \\left(\n \\frac\n\t {\\sigma}\n\t {r}\n \\right)^6\n \\right].\n", "e90c3b58ecce0d06036fad48993d90cb": "\\textstyle \\lambda_i\\in\\mathbb{R}", "e90c44446e66d3a1466d1f2f909a099d": "f_y(\\mathbf{y}) \\triangleq \\begin{bmatrix} f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 \\\\ 0 \\end{bmatrix}\\,", "e90c62c6885534faf202e80630d8709f": "a^{a^{\\cdot^{\\cdot^{a^a}}}}", "e90c646ac790f3b55ec00663b2b4ff57": "\n\\begin{align}\n\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T \\right) \n& = \\left( \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\right) \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( \\frac{d}{dt} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\right) \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} \\, T \\\\[6pt]\n& = \\frac{\\partial L}{\\partial \\mathbf{q}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\left( \\frac{\\partial^2 \\phi}{(\\partial \\mathbf{q})^2} \\dot{\\mathbf{q}} \\right) \\dot{\\mathbf{q}} T + \\frac{\\partial L}{\\partial \\dot{\\mathbf{q}}} \\frac{\\partial \\phi}{\\partial \\mathbf{q}} \\ddot{\\mathbf{q}} \\, T.\n\\end{align}\n", "e90c8f8b550100335ed712fe8bb15c92": " A_w = {(p - n) \\over (2.0665 - 1.0665p/100)} = f_{pn}(p - n)", "e90ca015de0aade8262800b0762a78e5": "\\mathfrak f", "e90d4f4fbfe60fadc56553050a8d2392": "e^{-1} = \\sum_{k=0}^\\infty \\frac{(-1)^k}{k!}", "e90d9921fa3e645bef6c436d0945dffa": " S_1 \\to S_0 + h \\nu_{em} + heat ", "e90dbeaea5d9ad664c80333879ace691": "\\frac {dy}{g(y)} = f(x)dx", "e90dc2f51d0087c3dc8a110daa395387": " 0 \\longrightarrow \\mathbb{Z} \\langle e \\rangle \\stackrel{\\partial_1}{\\longrightarrow} \\mathbb{Z} \\langle v \\rangle \\longrightarrow 0, ", "e90ded3a1c42af9d620e8df936144752": "P_n^{(\\lambda)}(x;\\phi) = \\frac{(2\\lambda)_n}{n!}e^{in\\phi}{}_2F_1(-n,\\lambda+ix;2\\lambda;1-e^{-2i\\phi})", "e90ead8de7f38102ae73b8e5dd8803bf": "F(x;k,\\lambda; \\alpha) = \\left[ 1- e^{-(x/\\lambda)^k} \\right]^\\alpha \\,", "e90ed26dbfd3993ba59f1094287bfca1": "h = \\frac{c_0 M_{\\rm u} A_{\\rm r}({\\rm e})\\alpha^2}{R_{\\infty}} \\frac{1}{K_{\\rm J-90} R_{\\rm K-90} F_{90}}", "e90efef01731a7b483dcd8554df2d29a": "A = (a_1, a_2, \\dots , a_m)", "e90f1058e25acdcf21be30d503c4e5c6": "y=R1(x;s)", "e90f8367f908d36b01f350340055f2c7": "\\gamma_{in}", "e90ffa9ad70b1970e3bab9ceda693569": "0 \\leq s < s_1", "e91006552ff6a979413dd37be9581710": " p > 2 \\quad \\mbox{and} \\quad \\left(\\frac {d_K} p\\right) = 1,", "e9105360e5c5005961f7249ddbe8dadb": "\\boldsymbol{\\mathbf{U}} = \\gamma(\\mathbf{v}) (c, \\mathbf{v})", "e910cbf3f0d8837fc50cb18330195b38": " \\log \\left ( \\frac {p(r)} {P} \\right ) \\approx \\frac {p(r) - P} {P} ", "e9113c49bc0ff2cdf5ac48176f961b46": "\n \\begin{align}\n X & = \\{1, 2, \\dots, n\\} \\\\\n Y & = \\{a_1, a_2, \\ldots, a_n\\} \\\\\n F & = \\{(1, a_1), (2, a_2), \\ldots, (n, a_n)\\} \\\\\n \\end{align}\n ", "e911552a5fe0e8908c2d955166e424ae": "y^{-s}", "e91203a5996c4f77bdcce4c66a1b5296": "D=\\sqrt[3]{\\frac{3d^2C^2}{8Lf^2\\pi^2}}", "e9129cab390692d27b3dc3e640ae82a1": "Ab(S)\\,", "e91343649a6d3d9736d34dd7dd352a40": "=\\quad {1 \\over n}(k_1(\\log n - \\log k_1) + k_2(\\log n - \\log k_2))", "e913e41bbad94e5adbd3505679f310e3": "P_{n} \\xrightarrow{\\mathcal{D}} P.", "e9141b4169c35bcdd02da059b85ab909": "L= 92.45 + 20\\ \\log_{10} (d) +20\\ \\log_{10} (f) ", "e9142c1a906f021930c12fa448511656": " \\rho_e", "e91450cd832f2a196bd153d3b21e6a83": "\n\\langle H \\rangle = \\sum_{n=0}^{\\infty} E_{n} P(E_{n}) =\n\\frac{1}{Z} \\sum_{n=0}^{\\infty} nh\\nu \\ e^{-n\\beta h\\nu} = \n-\\frac{1}{Z} \\frac{\\partial Z}{\\partial \\beta} = \n-\\frac{\\partial \\log Z}{\\partial \\beta}.\n", "e91473d7e1dbe91641c82b61077adfe0": "t = a \\cdot \\sqrt{\\frac{a} {\\mu}} (e \\cdot \\sinh E-E)", "e91474943f619d9342203adeffc17916": "\\left\\{\\left[\\begin{smallmatrix}a & b \\\\ 0 & 0\\end{smallmatrix}\\right] : a,b \\in \\mathbb{R}\\right\\}", "e914ad0a2fe548188bfd945d06a09c8c": "\\mathbb{P}(N=k)=\\sum_{n=k}^m(-1)^{n-k}\\binom nk S_n, \\qquad k\\in\\{0,\\ldots,m\\},", "e914b9f3632d19ffe7c00cdfd6d9d155": "\\frac{\\sum_i \\gamma_i c_{V,i}}{\\tilde{C}_V} = \\frac{\\alpha K_T}{C_V \\rho} = \\frac{\\alpha V K_T}{\\tilde{C}_V}", "e914cb5e941dbe8721fb27b0d323e9cd": "h_0,\\; h_L", "e914ef6cda0b0bb3cec5c5a6614814e2": "p(y_i | c_j) \\,", "e914ff1afcb83d43dad1d6ed1cc980fd": "\nJ = L(F_e = 0)F_e. \n\\,", "e91501d9bc84ce051a1b1e904ffcc2cb": "W = \\sqrt{A^2 + V^2 - 2AV\\cos{\\beta}}", "e915adc67d0fa9769f4649649af283c1": "S'=G S G^\\top \\, ", "e915e2a13366e5a5e2d90d3b1e66eeb5": "E[F|x^{(t)}] \\le E[F|S^{(t-1)}]", "e916181fa0a0be4a7fda22b65c5ae408": "\\dot{g}=-e \\cdot w(t).", "e9161ec1b693c9f482712d6794338481": "ncp_F", "e9163b1b05459124bacb38e03b95a802": "f(x) = \\frac{1}{\\pi\\gamma \\left[1 + \\left(\\frac{x - x_0}{\\gamma}\\right)^2\\right]} ", "e9165ebd48e3c826bf908709ffd9516e": "\\frac{x}{1-x}=[0;a_1-1, a_2, a_3,\\cdots]", "e916ae28c61dbba674141ecbf0c297e9": "E(\\gamma)=\\int_0^1 g_{\\gamma(t)}(\\dot\\gamma(t),\\dot\\gamma(t))\\,\\mathrm{d}t.", "e916cf5343451482d340055542e70edd": "CN_{I}", "e91712019a49ec27396e53b85b22b67d": "x \\in \\mathbb{R}^n", "e9176d235e7882e3c94d5443fe67a1b0": "\\mathbf v = (v_x, v_y, v_z),", "e917a74cba68c53ada96df35f257616b": "O\\left({X^{1/2}(\\log X)^{3/4}}\\right)", "e9182e55aef9bdc5aa2ff9ab00ddbf48": "\\sum_{x \\in \\{0,1\\}^T} |x \\rangle |f(x) \\rangle", "e9185ab606ebe5327b4daf811d2b5450": "u=\\frac{vf}{v-f} \\,,", "e9187ccaaf350f080840bbd883a2bbe4": " K^*=\\left(\\frac{\\alpha}{\\beta e}\\right)^\\alpha\\left(\\frac{\\gamma}{\\delta e}\\right)^\\gamma, ", "e9195bcb405a080fd5a36aff7ed5e270": "-x^4", "e919a3d6f3e11747b583143e949f79d7": " T^{\\mu \\nu} = \\frac{1}{\\mu_0} \\left( F^{\\mu \\alpha} g_{\\alpha \\beta} F^{\\nu \\beta} - \\frac{1}{4} g^{\\mu \\nu} F_{\\delta \\gamma} F^{\\delta \\gamma} \\right) ", "e919b81a4be5c5a9717a146b558d70a5": "\\{ [a,b): 0\\leq a< b\\leq 1\\}", "e919dfac9c72ba0e082d1cba90ae68f5": "\\psi(\\Omega^{\\Omega^2 \\omega^3})", "e91a238bb9154fd83e11f3ad39c74e52": "O(\\log^4 n)", "e91a2869c03acb7b6d982ecf083a32a4": "n^2(n^2-1)/12", "e91a3c43778f406e4b6d119e5f89b336": "S(\\alpha)\\to T(\\alpha)", "e91b32135bb98c42aef13152c0ec3843": "ABCD^+ \\to AB^+ + CD\\,", "e91b4bb9fd0dc0a3d105f29dd2af57dd": "\\ C-\\text{vertex} = 1 : 1 : 0", "e91baa26340d5ebce71765fa75eb2ef3": "D>1", "e91c94beda933fc9cb5114c5027a1f06": "\\mathbf{e}_l", "e91cbd34ef3eeae97e752746da7c7cd4": "Y_{8}^{-3}(\\theta,\\varphi)={1\\over 64}\\sqrt{19635\\over 2\\pi}\\cdot e^{-3i\\varphi}\\cdot\\sin^{3}\\theta\\cdot(39\\cos^{5}\\theta-26\\cos^{3}\\theta+3\\cos\\theta)", "e91cf681e3d41b77c34a1bab6d67b70d": "1*x = x", "e91d6204b9b7b43ece82eb9b7f47c19f": "\n|J_C(\\mathbb{F}_{q^n})| = \\prod_{i=1}^{2g} (1 - \\tau_i^n)\n", "e91dc47116fc52967108d6ea95a504be": "ds^2 = c^2 d\\tau^2 = g_{00} \\left ( dx^0 \\right )^2,\\,", "e91dd2c9c87110ff431291d2c023f982": " (S_4 \\implies (\\operatorname{equate}[A_4, p] \\and V[q] = A_4)) \\and D[q] = D[p] ", "e91dda252f72cb35230b02b5d3ca2544": "\\left ( \\frac{\\partial U}{\\partial S} \\right )_V = T", "e91e634a3c9a2fc3a6dca0c9307792a3": "{1\\over k_1} \\times {d[A_1]\\over [A_1]} = {1\\over k_2} \\times {d[A_2] \\over [A_2]}", "e91e66b11b5b49797e9d84c736a1fbd4": "\nr_{12}\n =r_{\\mathit l \\mathit l^{\\prime}} \n= \\sqrt{\\mathit l + \\mathit l^{\\prime}}\\;r_B\n.", "e91ea65a71a7c5ea521c3277e2518425": "\\{1, \\dots,n\\}", "e91ec299ccdeffedfa8ce5ab07a4f653": "(Ry')' + \\frac{R\\,\\lambda}{Q}\\,y = 0.\\,", "e91f3577b88a3863a93a4dd1fcfd891a": "f(x) = x^5 - x + a\\,", "e91f4fdbdc5461e542ef2f13b94bcf1a": "f(x_1,\\ldots, x_{2n}) = x_1x_2 + x_3x_4 + \\cdots + x_{2n-1}x_{2n}.", "e91f93951982bd98407c62ebb8781a37": "a_0^{14} a_1^9 a_2^6 a_3^4 a_4^4 a_5^3 a_6^3 a_7^3 a_8^2 a_9^2 a_{10}^2 a_{11}^2 a_{12}^2 a_{13}^2 a_{14}^2 a_{15}^2 a_{16}^2 a_{17}^2 a_{18}^{2} a_{19} a_{20} a_{21}\\cdots a_{229},", "e91fa449d9c9d7b32402552a4cbec673": "\\scriptstyle k_0", "e91ff664c7f60f00970fda3be4cc7334": "|x-a| = T_{i}\\frac{m_{i}n_{j}}{(m_{i}+s_{ij})(m_{i} + n_{j} + s_{ij})}.", "e9231113d262e214f5780ffec9de541c": "c=\\tfrac{k^2(3s^4-10s^2 r^2+3r^4)}{4} \\, ", "e9236a7c6c8cff3bcdb492eb8653383f": "(r-1)\\times(r-1)", "e9237c3e7cfc5324b881d4c0f2bca7f9": "\\frac{\\partial p}{\\partial z} = \\frac{\\beta p}{\\rho_{0} c_{0}^{3}}\\frac{\\partial p}{\\partial \\tau} + \\frac{\\delta}{2 c_{0}^{3}}\\frac{\\partial^{2} p}{\\partial \\tau^{2}}", "e9237f52a1a012270ac273803e6ff596": "\\gamma(t) = 2i\\sqrt{t}", "e92383ef66d985455b86567688ae67ef": "E(\\omega)", "e923b04bf43f2776debe3eee2b7f285d": "\\mathbf{while}\\;C\\; \\mathbf{do} \\; S", "e923b4494776a63371a93d227e129d6c": "(P_2,T_2)", "e92496a085494cafeedfba88690c4823": "y_1,\\ y_2,\\ y_3,\\ y_4", "e925847634831184bf0a485a451caab5": "h\\colon\\pi_{2n-1}(S^n)\\to\\mathbb{Z}", "e9258cbe0df1a067f9bdb63149866191": "F_\\text{seq}(l)", "e925c5d604a91eae4a961f23980940c3": "{4 \\choose 1}^4 - {3 \\choose 1} = 253\\,", "e926244dfb9328aff468d5c698c0052c": "\\kappa_1 =\\mu_1;", "e926481d379b275a901ae9c13be4088c": " \\ln P_r = f^{(0)} + \\omega \\cdot f^{(1)} ", "e92654edf156c280aaa52ff7ab0b1d89": "\\begin{align}\n\\int_S&\\sum_{k=1}^n|f_k(x)\\,g_k(x)|\\,\\mu(\\mathrm{d}x)\\\\\n&\\le\\biggl(\\int_S\\sum_{k=1}^n|f_k(x)|^p\\,\\mu(\\mathrm{d}x)\\biggr)^{\\!1/p\\;}\\biggl(\\int_S\\sum_{k=1}^n|g_k(x)|^q\\,\\mu(\\mathrm{d}x)\\biggr)^{\\!1/q}.\\end{align}", "e9267748d722abfbc900894c3363ef81": "\\left(\\frac{\\partial U}{\\partial X}\\right)_S = -\\,\\frac{\\left(\\frac{\\partial S}{\\partial X}\\right)_U}{\\left(\\frac{\\partial S}{\\partial U}\\right)_X}\n=-T\\left(\\frac{\\partial S}{\\partial X}\\right)_U = 0", "e9269651175a442f4babd790e4507d0b": "\\displaystyle{R^kP_s(z)={k\\over 2\\pi i^k} {z^k \\over (|z|^2 +s^2)^{k/2+1}}}", "e9269ee9a7677881837a37ed64de2a70": "A\\in R^{m\\times n},", "e9270db242655935a126b30560dc6d65": "V\\setminus\\{0\\}\\to \\mathbf{P}V.", "e9273a0b260b086432da2efae9472c5e": "n/p^j \\bmod 1\\ge 1/2", "e9276ebe6958d9e8c989ef2a5a3c20e7": "\\mu = c^3-c-\\gamma\\nabla^2 c", "e927be30b5e74dc037168ab970a2d44d": " a_1, ... , a_{m/n} ", "e927caca67beb4bc1cbe752593f164d7": "Q_{base} = 1 pu", "e927ddd35b3b7935c4fcad349873c74a": "x_{n+1}=y_n\\,", "e927ecfe6a8df777abc0dbe82ca170cc": "P = \\frac{V^2}{R}\\,", "e927fccd96438e46a8d5bdd643eb7f28": "\nF(\\phi,k) = \\int_0^\\phi \\frac{\\mathrm{d} \\theta} \n{\\sqrt{1 - k^2 \\sin^2 \\theta}} = \\sin \\phi \\,F_1(\\tfrac 1 2, \\tfrac 1 2, \\tfrac 1 2, \\tfrac 3 2; \\sin^2 \\phi, k^2 \\sin^2 \\phi), \\quad |\\real \\,\\phi| < \\frac \\pi 2 ~,\n", "e929505098af5e1843e56d947fe4d730": "x\\in B(x,\\delta)\\subseteq G", "e9296c3a411a76ea209ab8b7de88d846": "C_1 , \\dots , C_{n-1}", "e929cfc10be0c08465d4f62770b343d7": "\\frac{m\\lambda^2}{2\\hbar^2}\\,\\!", "e92a2c9937570d494aa947ec788b54ac": "d: L_i \\to L_{i-1}", "e92a3d74451895e23a1eac941242b65b": "u_t = -Lu,\\ ", "e92a6aea851fd52417b7a201d95da5b5": "f(x,y)=0,", "e92a6ee23f464587360803546e90e1c1": "S_\\varphi", "e92a910658f22a5c2e527f699896f55e": "\\frac{dW_f}{dR}=\\frac{\\frac{dW_f}{dt}}{\\frac{dR}{dt}}=\\frac{F}{V}", "e92aa1c17221c941a243999ae1b0ce88": "Q(R) = (R^{\\times})^{-1}R", "e92aef2450b14a8969dfed03fcab2a76": "F_\\text{thrust}", "e92b217137039ae434c705ab5f5d0b82": "{n\\over 4n+1}={1\\over 5},\\, {2\\over 9},\\, {3\\over 13},\\, {4\\over 17},\\cdots ", "e92b5936a61f960d23e805f9473446df": "\\tilde{a}^i_j", "e92b5ecb9b3b1d2fd30534e7ab5f7611": "x_1, x_2, x_3\\,\\!", "e92b756d9a88edcf4bb82e0da2d27271": "1, \\varepsilon{}_1 , \\varepsilon{}_2 , i_3", "e92be05599dfb6e5d5503efa2dba00ca": "\\mathit{C_{|L|}}\\,", "e92c0a0ad23070c4497096bc290389c3": "\\langle \\psi^{jk} \\vert \\psi_{lm} \\rangle = \\delta_{jl} \\delta_{km}", "e92c22c4d78b74db63fa0c8e9905f938": "-i_R + i_C = 0", "e92c327455d56d81f658e01293320a41": " D_i", "e92c3924fda2adab16d18b919b4d2c36": "j\\colon Y \\to \\operatorname{Aim}(X)", "e92cbde69f10413d3b1fef2bc5702ea9": "10 ^ x\\,", "e92ce5640c6b68bfb977dd5aa4496f7f": "\n\\frac{\\sigma^{pd}}{2\\sigma^{pp}} = \\frac{1}{2}\\left[1+ \\frac{\\bar{d}(x)}{\\bar{u}(x)}\\right] \n", "e92d0657501b29b700bff8a883d241f9": "g^{-1}\\,", "e92d32a6d0f8058dc55771e77faaabb7": "\\Pr(R \\cap B \\mid \\text{not } Y) \\not= \\Pr(R \\mid \\mbox{not } Y)\\Pr(B \\mid \\text{not } Y).\\,", "e92d3b6fe96970b14b3790942f2244a9": "\\displaystyle{[L(a^2),L(c)]+2[L(ac),L(a)]=0.}", "e92d634d041da8001f027ab9d0a33338": "V=(210+90\\sqrt{5})a^3\\, .", "e92d6ab6e25a3a9a0ac5b27b536e2a13": "(0,0,0,x)", "e92d76ca85eb5eb4314d40e1491cd710": "\\int xs\\;dx = \\frac{1}{3}s^3", "e92dbcc04df4b3e227ea5314fab02bd6": "x = \\lambda \\sqrt{1 - 3\\left(\\frac{\\phi}{\\pi}\\right)^2}", "e92e4931589310e331e614c10bf944ee": "A(\\alpha(t))", "e92e4af863c1923827294a9f8938cfa0": "\\vec{u} = (u_x, u_y, u_z) = u_x\\mathbf{i} + u_y\\mathbf{j} + u_z\\mathbf{k}", "e92e92ea1f54db7be0c7998fce9c646e": "\\neg(A\\lor B)\\leftrightarrow(\\neg A\\land\\neg B)", "e92ee59ee4b77d2411dc2e82078ca30a": "\\int_{-\\infty}^\\infty x\\,\\mathrm{d}x", "e92f1356f5f44636b7eec84fe0da8016": "d_i+1", "e92f2dced423398c94df15c713d3d45b": " dE = \\Big(\\frac{\\partial^2 L}{\\partial x^i \\partial \\xi^j}\\xi^j - \\frac{\\partial L}{\\partial x^i}\\Big)dx^i +\n\\xi^j \\frac{\\partial^2 L}{\\partial\\xi^i\\partial x^j} d\\xi^i ", "e92f5f49ae1eac2891dadd501b96ed51": "p(\\textbf{z}_k\\mid \\textbf{Z}_{k-1}) = \\int p(\\textbf{z}_k\\mid \\textbf{x}_k) p(\\textbf{x}_k\\mid \\textbf{Z}_{k-1}) d\\textbf{x}_k", "e92f6456a1c842d498e0bd1f3692b320": "{\\mathbf{}}F_r(t), K_r(t), L_r(t) ", "e92fa001594baf4f934ba19fbdd61e4b": "\n\\begin{Bmatrix}n\\\\k\\end{Bmatrix}\\,\\bmod\\,2 =\n\\begin{cases}\n 0, & \\mathbb{A}\\cap\\mathbb{B}\\ne\\empty\\\\\n 1, & \\mathbb{A}\\cap\\mathbb{B}=\\empty\n\\end{cases}\n", "e92fa10cffd421b134468774a758f31e": "n\\;", "e92fa5f9786313f3bc9a88d9b52bf310": "\\hat x = x", "e92fcdd6d24b476a54a1e798ba2aa952": "\\mathrm{^{249}_{\\ 96}Cm\\ \\xrightarrow [64.15 \\ min]{\\beta^-} \\ ^{249}_{\\ 97}Bk\\ \\xrightarrow [330 \\ d]{\\beta^-} \\ ^{249}_{\\ 98}Cf}", "e92fd4f19eb20d240117facb23a380ae": "\\!\\left(0,a\\right)", "e9304423021c166e1f394254e9cd8a8e": " \\omega(u,v) = \\langle Ku, v \\rangle \\quad\\text{and}\\quad \\kappa(u,v) = \\langle Lu, v \\rangle ", "e9304a3a135e0dfaaf63c49cae80c088": "\\mbox{Average days to sell the inventory}=\\frac{\\mbox{365 days}}{\\mbox{Inventory Turnover Ratio}}", "e93058cef257b2af639312b8014e01a9": "(u, v, z)\\in(-\\infty,\\infty)\\times[0,\\infty)\\times(-\\infty,\\infty)", "e93084b05daa5edee8b027ff76f6494e": "c_v", "e93088d38044f07716776060f0e0d329": " R\\,\\!", "e93089a2967372d88aedbdc13f283c91": "\n\\frac {\\rho_E}{\\rho_M} = \\frac {V_M}{V_E} {\\left( \\frac {r_E}{d} \\right)}^2 \\frac {1}{\\tan \\theta}\n", "e93098bd0e7161ff60af4e5667b9c309": "\n\\hbar \\frac{\\partial}{\\partial t} f^{h}_{\\mathbf{k}}\n=\n2 \\operatorname{Im} \\left[ \\Omega^\\star_{\\mathbf{k}} P_{\\mathbf{k}} \\right] + \\hbar \\left. \\frac{\\partial}{\\partial t} f^{h}_{\\mathbf{k}} \\right|_{\\mathrm{scatter}}\\;.\n", "e930fb13f52cb62f383ddb6ae3dd554f": "[\\mathbf{b}] \\in [\\mathbb{R}]^{n}", "e9310587118b2bd149663045c06c07fe": "\n \\mu^+ \\rightarrow e^+ + \\nu_e + \\bar{\\nu}_{\\mu}~.\n", "e93117f8dba482f03ed8e52636968eb9": "\\sqrt{3}/{2}", "e931ca9947af04ce44ccf4ed0574a9a6": "\\frac{3}{\\sqrt[4]{8}}", "e931e920c69de512928bc71922a1a7f2": " \\mu\\sigma^2 \\propto m^3", "e932549eb777d7b51f6d1a2b33f445b8": "1^3+3^3+\\dots+(2y-1)^3 = (xy)^2", "e93264bf4adc39bdf6664fadf1899bc2": " V_0 := \\{\\} .", "e93277e3585a170461c801af7585b020": "E_h(x,v)=\\tfrac12\\left(v^2+\\omega^2\\,x^2-\\omega^2\\Delta t\\,vx\\right)", "e93299422ba586fd6c17710c9e28f7b1": " \\mathbf{\\bar Y} \\, \\mathbf{\\bar f} ", "e932a48168f4240573ba6354957bb843": "\\sum_{m=0}^{n-1}(A_{m+1} - A_m) = A_n - A_0 = \\sum_{m=0}^{n-1}\\frac{g_m}{\\prod_{k=0}^m f_k}", "e9334b93717b89ae0f2e0fa74cfc3a88": "W = \\mathbf R \\times (\\mathbf R^*)^k \\times S^2( (\\mathbf R^*)^k) \\times \\cdots \\times S^{m} ( (\\mathbf R^*)^k)", "e9336b00f6d98558aa952207b30b7d86": "\n{\\rm E}\\left[ {s^2 } \\right]\\,\\, = \\,\\,\\sigma ^2 \\,\\left[ {1\\,\\,\\, - \\,\\,\\,{2 \\over {n - \\,\\,1}}\\,\\,\\sum\\limits_{k\\, = \\,1}^{n\\, - 1} {\\,\\left( {1\\,\\, - \\,\\,{k \\over n}} \\right)\\rho _k } } \\right]", "e93371a916ba0cadf249dc870f1f48b7": "s_6 = 1001.", "e9337575f552cf691f44809291960bf0": "\\widehat{\\mathrm{CH}}_p(X)", "e9339f86072baf417d014734726e1685": "J^{\\prime}(u_0) = \\frac{2}{u_0} \\left[\\frac{m}{L^2 u_0^2} f(1/u_0)\\right] - \\left[\\frac{m}{L^2 u_0^2} f(1/u_0)\\right]\\frac{1}{f(1/u_0)} \\frac{df}{du} = -2 + \\frac{u_0}{f(1/u_0)} \\frac{df}{du} = 1 - \\beta^2 ", "e933fc86c06bde450961987480b57f65": "\\left(\\sum_n E_n \\right)^2 = (M_0 c^2)^2 \\Rightarrow \\sum_n E_{\\mathrm{COM}\\,n} = E_\\mathrm{COM} = M_0 c^2 \\,,", "e934aae63a659195a41b1502b0de257d": "\\operatorname{E}[S_N]=\\sum_{i=1}^\\infty\\sum_{n=1}^i\\operatorname{E}\\!\\bigl[\\operatorname{E}[X_n]1_{\\{N=i\\}}\\bigr]=\\sum_{i=1}^\\infty\\operatorname{E}[\\underbrace{T_i1_{\\{N=i\\}}}_{=\\,T_N1_{\\{N=i\\}}}].", "e9350c983761073c946ecc6eb6190164": "\\hat{\\pi}_{\\phi^T}= |\\phi^T\\rang\\lang\\phi^T|, ", "e93562e261d96573a3ae7b72473dccaa": "\\sigma(n)=\\min \\left\\{\\,k \\mid n \\le \\binom{k}{\\lfloor k/2 \\rfloor} \\,\\right\\},", "e9359d74aa8d3e356285633ca692d819": "f^{(-n)}(x) = \\int_a^x \\int_a^{\\sigma_1} \\cdots \\int_a^{\\sigma_{n-1}} f(\\sigma_{n}) \\, \\mathrm{d}\\sigma_{n} \\cdots \\, \\mathrm{d}\\sigma_2 \\, \\mathrm{d}\\sigma_1", "e935b3ae63ddbbb9420caa0898def2d2": "\\text{E}(e^{-tx})", "e935df554952d34ad13009a8cdd546fd": "\\{0,1,3,7\\}", "e935e9119bb3096d064e39e4fe6543d8": "(r_d", "e9362b2ab4f1e6abe1b1ba4d8101ee39": "2 \\pi /d ", "e936a6589e4a47987803298ab7c70ac8": "(22201221202001110211210200)", "e936afb064c48257d5d39037999d281e": "T(\\theta)= \\frac{M(\\theta)-C_1}{C_2}", "e936d197e62b59813a395a90612dd6db": "\\displaystyle{M(a)b=\\frac{1}{2} R_y(a,e)b= \\frac{1}{2} R(a,Q(y)e)b= \\frac{1}{2}R(a,y)b=\\{a,y,b\\}=L_y(a)b.}", "e936d571a4cfe5d99381ded556592900": "J=\\frac{1}{2(n-1)}R", "e9371ffca250dfb274cec5c7dd5dfa7d": " {N \\choose n_A} = \\frac{N!}{n_A!n_B!}", "e93751877200d706a68711eec77e18a6": " f(x_2) ", "e9376938b60da82a035571c0445d6766": "P(x):SV[[x]]\\rightarrow SV[[x]]", "e937c68bd8f3fad1677983c9dc78f9b1": "\\eta_s = \\frac{n_B - n_{\\bar B}}{s}", "e937f1b01e26686c2984ef9397c33b05": " \\nu_3 = p [ S_1 \\sigma_1^3 + 3 \\delta_1 \\sigma_1^2 + \\delta_1^3 ] + ( 1 - p )[ S_2 \\sigma_2^3 + 3 \\delta_2 \\sigma_2^2 + \\delta_2^3 ] ", "e93853d8a12c76e3c277d180110672a4": " \\ p\\textbf{r} \\cdot \\textbf{g} \\pmod p =0 ", "e9390d8d1ebea814c5aab5a899216c9f": "\\Phi : \\mathbb{R} \\rightarrow \\mathbb{R}", "e9396cec26b88696631acc722c009960": " S_i(t) ", "e9399df894023d4858dcee6becc9c8f6": "y = e^{- \\int_{s_0}^{x} P(s) ds} \\int_{t_0}^x Q(t) e^{\\int_{s_0}^{t} P(s) ds} dt + Ce^{- \\int_{s_0}^{x} P(s) ds}", "e939e61e949b2925202bc62e08e9e24d": "\\mathbf{L}_\\parallel ' = \\gamma(\\mathbf{V})\\left(\\mathbf{L}_\\parallel + \\mathbf{V} \\times \\mathbf{N} \\right) ", "e939f4c326e318a89ca3ace6079c64b5": "G = (N,T,S,P)", "e93a02c4a0a091579e7ffca7996bfb4d": "\\delta\\,\\!", "e93a1268f126d08f6403b2b7088605b3": "100 \\sqrt{25/100} = 100 \\cdot 1/2", "e93a30b6c0486dc73f9b31afc38baf97": "\n\\langle f, g \\rangle_\\mathcal{D} = \\sum_{x \\in \\{0,1\\}^n} \\mathcal{D}(x)f(x)g(x) = \\mathbb{E}_{\\mathcal{D}} [fg]\n", "e93a44af2771f338a8bb2ebb83534e15": " \\mathrm{tet}_b ", "e93a6ed511918897aa955f275a8b65e9": "t (1 - t)^2", "e93a915523c48bb907e683aa5c1b8ce9": "\\beta=-0.58", "e93ab1d2e399b0030ecf957d613bf01b": "Z_{in} = Z_1 + Z_2 + g_m Z_1 Z_2", "e93abaaab47e34852f05fc0ad0a62118": "f(x_\\varphi)", "e93adf7c930ca4fffe9c76d8d8f94155": " \\mathbf{m} = \\mathbf{r}m \\,\\!", "e93b14f140b1912c888daedf1c6b3bbd": "F_b:=\\left\\{b\\right\\}\\times F, b\\in B", "e93b2585368fbda35f9f4460fb15fe8c": "\\vec{P} = (0,0,\\ldots,0)^T", "e93b9a0c3897ea937b61fa8750e36968": "\\operatorname{ad}\\ \\mathfrak{g}", "e93b9a6d068f334e43e168aa2574918c": "\\, \\mu_e", "e93bd17839f43017110ab1d0b95ae2eb": "\\alpha = 0\\,", "e93c0159c5aecb3d0114058077e73959": "\\frac{\\mu}{\\mu^4+1} \\to \\frac{\\mu^2}{\\mu^4+1} \\to \\frac{\\mu^3}{\\mu^4+1} \\to \\frac{\\mu^4}{\\mu^4+1} \\to \\frac{\\mu}{\\mu^4+1} \\mbox{ appears at } \\mu \\approx 1.8393", "e93c22abfeb8717fdb9504c6c2f1bd43": "p' \\;=\\; 1 \\!", "e93c30a0599eabf5c43609b23306ac79": "\\frac{1}{DNM}\\sum_{j=1}^N \\sum_{i=1}^M \\lVert s_j - m_i \\rVert^2", "e93c37e377738bb1b06ce293893dabdb": "m \\lambda = d \\sin \\theta", "e93c523971ad49e742e6392e4a4de908": " r = \\frac{R_0 - 1}{D} ", "e93cc7683826116ea1f1fd31bfed9225": " Y^*_1 ", "e93d0acc9e53f93737ba3b3cf839fd3f": "\\mathcal{M}\\{\\Gamma\\}=1\\text{ for the universal set }\\Gamma", "e93d4afadc1ef8c72a405e7f6cfe2e92": "{\\mathbf A_{22\\cdot 1}} = {\\mathbf A}_{22} - {\\mathbf A}_{21}{\\mathbf A}_{11}^{-1}{\\mathbf A}_{12}", "e93d80262e0037f47021655fd05f0112": "\n\\phi = \\int_d^{\\infty} u(r) \\frac{N}{V} 4\\pi r^2 dr ,\n", "e93d9bd6ad5d023f1a9a9a179ef434c5": "\\text{Upper endpoint} = \\bar X + 1.96 \\frac{\\sigma}{\\sqrt{n}}.", "e93da6f5777c4cb6bf8e4b4b61a16afc": "R_n/I", "e93de486da8d17307ee09f1eb96aefce": "\\overline{Z}^\\alpha", "e93e140f347cf32d169034f6adde7c99": "K_{13} = K_{31} = K_{23} = K_{32} = 0", "e93ebaa1dc1bbf6c41d1fd01e2278883": " t f(t) \\ ", "e93ee989d6546e73e6dc95c9f5a6002f": "a/r \\ll 1", "e93f1d3a2ff3e7a42f9ae5600c7ae5eb": " \\frac{ \\sqrt{ 6 } [ \\gamma + \\log_e( \\log_e( 2 ) ) ] }{ \\pi } \\approx 0.1643 ", "e93f2dc49230e7792b578f2fa7190915": "\\sum_{k=0}^\\infty a_{p,k}(n+1)^{k+1}=\\sum_{k=0}^\\infty a_{p,k}n^{k+1}+(n+1)^p", "e93f63e124dbb0d884430813c0c9a9eb": " {\\mathfrak g}\\to C^{\\infty}(M) ", "e93f9158591e67cb2594d5bd10d9df2b": "\\omega=\\aleph_0", "e93fb86004483acecbf08b094c0e7dfb": "k[x_1, x_2,\\ldots, x_n]", "e93fbea343d7a72a57e664e9bcc69678": "S \\approx 2.414a", "e93fce012d79f5f33102e6d0a92869e8": " \\frac{p_1}{p_2} =e ^ { {g \\over R \\cdot \\bar{T}} \\cdot ( z_2 - z_1 )}.", "e9406f36b6c84cfc587eb52814ae9fe8": " \\tau \\mapsto \\tau+1 \\ :\\ \\lambda \\mapsto \\frac{\\lambda}{\\lambda-1} \\, ;", "e940a7c7357a6ccdbd3f625b75e15d62": "L = 1", "e940c3d5174d05cc4baeef3e13b5f5df": "a_{(n)}b", "e94177450658b8e82516f3ca91402436": "\\scriptstyle 2\\times 4\\times 12", "e94192b873b94597a30e63e871cca942": "\\delta(\\{0\\})=1.", "e941d984ed7d8390a4a5d1f2d1507d5a": "\\Delta \\langle \\hat{B}_\\omega \\hat{B}_{\\omega'} \\rangle", "e941dcf50fef152b721b70d66e74fba1": "(\\sqrt{t} - 1)^2,\\,2(1-\\sqrt{t})", "e941f070ff325d04d6b2698de8b5d47e": "2^{2^{n-1} - n}", "e941fc64301ee59ba516f828bec087e5": "I_{\\text{D}} = I_{\\text{S}} \\left( e^{\\frac{V_{\\text{D}}}{V_{\\text{T}}}} - 1 \\right)", "e9422779ad875e8bc91c3ebd99d6d3ed": "\\operatorname{Ric}", "e9422ffed434667cbec71e9b8f8fb243": " u \\in K", "e9428d7abf82fde0dc70a376b4295806": " {N}(B) =\\#(B \\cap {N}). ", "e942943e93a02bb38891ca6e756e8771": "\\mu \\ne 2", "e942be67d17381542cc68ad9da7521ab": "\\dot =", "e942f87f3858677489c409c4c6e8503e": "I_k(\\mathbf{y}, t)=D_k(\\mathbf{y}, t)e^{j\\phi_k(\\mathbf{y}, t)}", "e943396ca92e6ebba01c830f53b1388e": "-0.0467", "e9434650ea21a3772903f89a83f81e55": "b=\\pi/\\beta", "e94493696a708c5abb4e55a0c326e81d": "\\Omega_k \\to 0", "e9449af571f257b7742d57c4e4a87a33": " F(x;\\mu,\\sigma) = \\left[\\left(\\frac{e^\\mu}{x}\\right)^{\\pi/(\\sigma \\sqrt{3})} +1\\right]^{-1}.", "e9449f6647983b7365c2da7775ea4627": "K(\\Delta^{1/n}),\\,\\!", "e944db2c7eb3118d6a3e8fd1866e1be2": "\n| \\Psi(t) \\rangle = \\sum_{\\alpha=1}^{N} c_{\\alpha} e^{-i E_{\\alpha}t / \\hbar} | E_{\\alpha} \\rangle.\n", "e944ea270f0ae3db0c70c76a559aa60f": " A \\to X \\to X/A", "e94541b9920ef131f3eacdaa5e17daab": "M+\\text{OH} \\rightleftharpoons M(\\text{OH}):[M(\\text{OH})]=K[M][\\text{OH}]=K K_\\text{W}[M][\\text{H}]^{-1} ", "e945e8d3eb915cb4adb68f5302df8282": "X (\\tau, \\sigma) = (X^0(\\tau,\\sigma), X^1(\\tau,\\sigma), X^2(\\tau,\\sigma), \\ldots, X^d(\\tau,\\sigma)).", "e945ffebbff2976108dc63d4aaf20398": "\\tfrac{p}{q}\\;", "e9461d062776a62b21888337212cc8c6": "\\gamma^o", "e9466002a28833608c9eb87afc2257f8": "I_2(p)", "e9468d3d242201a5dc7f66c79f8ec986": "E\\approx pD", "e94698404fd69bf8e0f2010b99a27c43": "\\bold{u}\\cdot\\dot{\\bold{u}} = \\frac{1}{2}(\\bold{u}\\cdot\\dot{\\bold{u}} + \\dot{\\bold{u}}\\cdot\\bold{u}) = \\frac{1}{2}\\frac{d}{dt}(\\bold{u}\\cdot\\bold{u}) = 0 ", "e947a68b86204f591e232bf3d5604caa": "\\varphi = \\frac{1 + \\sqrt{5}}{2},\\,", "e947cf7a53038b775ec51e28144b5f9a": " \\{u,v,\\{w,x,y\\}_+\\}_+ = \\{w,x,\\{u,v,y\\}_+\\}_+ + \\{w, \\{u,v,x\\}_+,y\\}_+ - \\{\\{v,u,w\\}_-,x,y\\}_+ \\,", "e9481a477158fce8aad4e787d86ccae8": "\\textstyle {4\\choose 4,0,0} \\ {4\\choose 3,0,1} \\ {4\\choose 2,0,2} \\ {4\\choose 1,0,3} \\ {4\\choose 0,0,4}", "e948b867dd59a60e8605a2ef87e19e94": "= [ (r+s)^2 - (p-q)^2 ][ (p+q)^2 - (r-s)^2 ] \\,", "e948c09bfee86a46cb2c1945051fde8f": "s = 1 / \\beta", "e948e858912c40b838e18865f67f6bd8": "\\mathbf r = \\mathbf r_1 - \\mathbf r_2 ", "e948f45feb77c8476c9fbb91a4f2556e": "\\sum_{n\\ge 1}\\frac{\\psi_k(n)}{n^s} = \\frac{\\zeta(s)\\zeta(s-k)}{\\zeta(2s)}", "e948f4a2c83eaf489aad72d05d1f99c4": "\\hat X_n(t) = \\frac{X(nt) - \\mathbb E(X(nt))}{\\sqrt{n}}", "e9492245c4d18e94f4fd7738dcd980e7": " P_\\ell(x) = o(x^\\varepsilon)\\,", "e9497b6c2dd4f928c3a86136fd36e94e": "\\omega = e^{2 \\pi i /Q}", "e949831a20629a8d97c5bd4fd822fbf0": "\\dot{\\mathbf{x}}(t) := \\frac{\\operatorname{d}}{\\operatorname{d}t} \\mathbf{x}(t)", "e949d8e3840877bdd9d0189d5f114260": "F_{0} = \\,\\!", "e94a02e5a59f72875bc6dde965c65b10": "s(N) = 2 \\cos(2 \\pi \\omega) s(N-1) - s(N-2)", "e94a0bdc70a0eed9071142fd9998ee79": "(\\beta+1)i_\\mathrm{b} = i_\\mathrm{x}-\\frac{v_\\mathrm{x}}{R_\\mathrm{E}} \\ , ", "e94a3fc26e4b0ebc79aaaf7c82f52ae3": "x_1 = x", "e94a517ff2aa6d1df56cd9db708135a2": "\\zeta(3)=\\frac{8}{7}-\\frac{8}{7}\\sum_{k=1}^\\infty \\frac{{\\left( -1 \\right) }^k\\,2^{-5 + 12\\,k}\\,k\\,\n \\left( -3 + 9\\,k + 148\\,k^2 - 432\\,k^3 - 2688\\,k^4 + 7168\\,k^5 \\right) \\,\n {k!}^3\\,{\\left( -1 + 2\\,k \\right) !}^6}{{\\left( -1 + 2\\,k \\right) }^3\\,\n \\left( 3\\,k \\right) !\\,{\\left( 1 + 4\\,k \\right) !}^3}", "e94adaa6554a72418daddc34fad161fe": "D = \\Phi/\\Phi_0", "e94b4819aad197a615b6d66dda916096": "\\operatorname df", "e94b5986e6b8abee2d61da045529e383": "\\boldsymbol\\mu_0 ,\\, \\kappa_0 ,\\, \\nu_0 ,\\, \\boldsymbol\\Psi", "e94b6080f687af8e68d4d46f995b36ba": "\\Delta G^\\ominus=-RT \\ln K\\,", "e94b9375af2ac828120aa8eb65d53e47": " \\frac{[\\mbox{H}_2\\mbox{PO}_4^-]}{[\\mbox{H}_3\\mbox{PO}_4]}\\simeq 7.5\\times10^4 \\mbox{ , }\\frac{[\\mbox{HPO}_4^{2-}]}{[\\mbox{H}_2\\mbox{PO}_4^-]}\\simeq 0.62 \\mbox{ , } \\frac{[\\mbox{PO}_4^{3-}]}{[\\mbox{HPO}_4^{2-}]}\\simeq 2.14\\times10^{-6}", "e94bd647faf9b95b5e7ed881792c17d1": "\\sigma = \\frac{\\alpha-1}{\\sqrt{n}}", "e94c29b888210583c98a2c0645bd8a32": "\\ A = \\frac{1}{( 2/3) - (0.420166/k)} \\qquad(12)", "e94c336278d144d9713159a9a41a2f7d": "g'\\,", "e94cc3514a7ce3c62e5ff0e60d791a2f": "(N_C,N_R,N_O)", "e94d104377c0300db3bbb9de5356b0a3": "i \\in S_j", "e94d24a3fc56e82d27c6610a6f7aed55": " \\sigma_-^2 = \\frac { \\sum (m - x)^2 } { n - 1 } ", "e94d90716454a410386959bdc94dcce0": "\\frac {Z_{i1}}{R_0} = \\frac {R_0}{Z_{i2}}", "e94d94a91b52d42e85357a7044f45a93": "0\\to \\mathbf Z_2^\\infty\\oplus\\binom n2\\mathbf Z_2\\to MCG (\\mathbf{T}^n)\\to GL(n,\\mathbf Z)\\to 0", "e94db1e5ee159609954da70f7e5096d8": "\\gamma _m \\ge 0 \\ ", "e94db629666e2bd555ca03c5bd3fb3cb": "\\left\\langle l\\right\\rangle\\sim\\ln{N}", "e94e4eb823ac3dbf2f5e9b03dd5299ca": " w_2 - w_1 ", "e94e974817ef4c04ac444ee2c7e561b2": "k_1,\\ldots,k_n", "e94eee501b1c0bf9c13c76eb5894e17f": "N(q,n)=\\frac{1}{n}\\sum_{d|n} \\mu(d)q^{\\frac{n}{d}}.", "e94f1fe04154499492c9a187834f0fc2": "\\mathbf{\\ddot r}_{Moon} = G{m_{Sun}}{r_{{Moon},{Sun}}^{-2}}\\hat{\\mathbf{r}}_{{Moon},{Sun}}+G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}\\hat{\\mathbf{r}}_{{Moon},{Earth}}", "e94f987333c751d04ea57e0a3c75d7b8": "^{\\;}E_n", "e94fa244e032c99bfa6766f42f38bfa5": " \nK=\\left[\n\\begin{array}{ccccc}\n1 & 0 & 0 & ... & 0 \\\\\n0 & \\frac{E^{(1)}A^{(1)}}{L^{(1)}} + \\frac{E^{(2)}A^{(2)}}{L^{(2)}} & -\\frac{E^{(2)}A^{(2)}}{L^{(2)}} & ... & 0 \\\\\n0 & -\\frac{E^{(2)}A^{(2)}}{L^{(2)}} & \\frac{E^{(2)}A^{(2)}}{L^{(2)}}+ \\frac{E^{(3)}A^{(3)}}{L^{(3)}} & ... & 0 \\\\\n... & ... & ... & ... & ... \\\\\n0 & 0 & ... & \\frac{E^{(e-1)}A^{(e-1)}}{L^{(e-1)}} + \\frac{E^{(e)}A^{(e)}}{L^{(e)}} & -\\frac{E^{(e)}A^{(e)}}{L^{(e)}} \\\\\n0 & 0 & ... & -\\frac{E^{(e)}A^{(e)}}{L^{(e)}} & \\frac{E^{(e)}A^{(e)}}{L^{(e)}}\n\\end{array}\n\\right] = K(E,A)=K(E^{(1)},...,E^{(Ne)},A^{(1)},...,A^{(Ne)})\n", "e94fbc52cb34d83bedbbfac068b811c6": "\\displaystyle k=2{\\alpha}K_BT", "e94fcc37f474b9f2e0768493653d2252": "i(-k) = j\\,", "e9501d98c048a8df16f60ee5ac57dc18": "\\phi(x)=\\int R(\\omega; x,y) f(y)\\,dy.", "e95045c837ff8671dee37d64a123f139": "C_x = \\{x\\}", "e9504be0ef7dae2e057bd47ee597ed16": "|\\phi_1\\rang, |\\phi_2\\rang, ", "e95067566178b2711007d4e89718923f": "\\ \\ln(k/T) ", "e9508ce618562e9d0c078291b3a518c9": "e = (v_C, v)", "e950aa9e7653682511bf2f1ac991ba70": "\\sigma:\\Omega_{4i}^{\\text{SO}} \\to \\mathbf{Z}", "e950c5fecb7d58dc463a3c3fc8db4eb0": "k_\\mathrm{FE}", "e951206bea8f314bbaf215ecb4635bbf": "10 \\cdot y + z,", "e9517322786b23e38582ff9ad099be86": "\\neg \\textit{locked}", "e9518ecb9c0b5aab8978919f8f5cd6df": " \\Sigma = \\mathrm{E}(\\mathbf{X X^{\\rm T}}) - \\boldsymbol{\\mu}\\boldsymbol{\\mu}^{\\rm T} ", "e951b686ce771096353c18893baa0f6b": "\\ \\mathbf{c} \\star \\mathbf{x} = \\mathbf{b},", "e951cc22b72e825e6e7f02f3ec31a275": "2^r-r-1", "e951f698e162be0be9aa674d19d5ea73": "C(X) \\otimes L(H)", "e9520cd50af8837ec514607eab6813e4": "\\cos A + 2 \\cos B \\cos C : \\cos B + 2 \\cos C \\cos A : \\cos C + 2 \\cos A \\cos B", "e95235543c2857ad699b4828843fd6c0": " A=X-E\\left( X\\right) \\mathbf{e}_{1\\times N} =X-\\frac{1}{N}\\left( X\\mathbf{e}_{N\\times1}\\right) \\mathbf{e}_{1\\times N}, ", "e9524e6de250b0b8953a8588ac3f6e0a": " \\text{body fat percentage} = 1.2\\times \\text{BMI} + 0.23 \\times \\text{age} -5.4- 10.8 \\times \\text{gender}", "e952b05d306def7129ea2e1415e48277": "j=1,2,\\ldots", "e953cd04f6fec69360ada9ac03c54aed": " \\textstyle\\sigma = \\sqrt{\\sigma_1^2 + \\sigma_2^2 - 2 \\sigma_1\\sigma_2\\rho}", "e953de73dff7eb0ccbf09ff5c058887c": "\\,\\Delta C_{\\,{\\rm V|\\,phonon}}\\propto T^3", "e95411f4fa4d4525155bd840f231b2c5": "b_{1^{ }}", "e9546c0a410203d11288d819cf222c43": "\\displaystyle\\mathcal{L}_t\\left\\{f(t)\\right\\}", "e954819877df73d465ec9095d6e03047": " \\frac{\\sigma^2_j}{N},", "e954affaf832a6f8a42a54e7f24a9593": "\\tilde{F}_{\\mu\\nu} = \\frac{1}{2}\\epsilon_{\\mu\\nu}^{\\rho\\sigma}F_{\\rho\\sigma}", "e954bb988e06b110b287a9c034b0ad70": "\\sigma(Y^*, Y)", "e954e568ce9f7bc5acfe09a64bef7cc9": "e_2 = 1 - \\Pr\\left\\{ \\bigcap_{i \\in S_p} E_i \\right\\}", "e954e7dc0f57b6c814afa1b25f52aeb5": "\\det(MY+ Q \\,\\mathrm{diag}(n-1, n-2, \\dots , 1,0) ) = \\det(M) \\det(Y). \n ~~~~~~~ ", "e954eae294e619b66c20c0cf78ec8776": "\\{ [M(OH)]^{(z-1)+} \\} = K_{1,1}\\{ M^{z+}\\} \\{OH^-\\} ", "e9551415fbc7adf2afd6e49ddaf2c934": " g(z) = z + b_1 z^{-1} + b_2 z^{-2} + \\cdots.", "e9555e27233fe4ecb5e2865a116e4617": "\nf^n\\left(x\\right) = x + \\frac{(x-a)^2}{2!}(n f''(a))+ \\frac{(x-a)^3}{3!}\\left(\\frac{3}{2}n(n-1) f''(a)^2 + n f'''(a)\\right)+\\cdots\n", "e9562c50e6b6ef4faa3cee7058fb97f1": "= 293 \\cdot 3413 \\, ", "e9563b5c086ec11a1d13bdb4613e3cdd": "FVA = \\frac{C}{i} \\left[ \\left(1+i \\right)^n - 1 \\right]", "e95651981c9227e9e3c24775dda93fc3": "\\begin{align}\\tfrac{a}{b}<1,\\end{align}", "e956722c6eb0e2be7aa07a6ed883e662": "T_\\varepsilon(q)=q", "e956c597abd1694e2f584f9741c232e1": " p: V \\rightarrow D ", "e956c6dbf1bf63bd1f5f3131e4c0f540": "\\langle\\cdots\\rangle_p", "e957030243471702d1a67f4612393329": "3V = kba \\left( \\frac{Tn}{P} \\right)", "e95716b2a5b967d4050622b8efd2a905": " \\begin{align}\n k_i^j \\quad = \\quad\n \\left\\{\n \\begin{array}{lcr}\n \\frac{i}{j} \\cdot k_j^j & \\text{for} & j \\geq i\\\\\n \\frac{N - i}{N-j} \\cdot k_j^j & \\text{for} & j \\leq i\n \\end{array} \\right.\n\\end{align}", "e9572585587a7ebe30c51042d772cd74": "\\frac{Q} {2}", "e957545c2460d41de1c611d6d43121d3": "X_0, X_1, \\ldots, X_{N-1},\\ldots", "e9575a6eb7c349fbb041c3e15214fed1": "W_0', W_1' = solve(G \\setminus A)", "e9576603c6f36ba59b9a2949800a9edd": "\\scriptstyle \\alpha^0\\,>\\,0", "e957827c0a81687784c2bcb45c8bc3ee": "[n+1]^F x = F(x, [n]^F x)", "e95789fad9216249766cb9589ef037a3": "\\! a' = {-a_{\\| m} + a_{\\perp m}} = {-(a \\cdot m)m^{-1} + (a \\wedge m)m^{-1}}\n= {(-m \\cdot a - m \\wedge a)m^{-1}}\n= -mam^{-1} ", "e957c629be2c318c51c75e799eca06b4": "E[W_t] = 0.", "e957d1de3eaeed1b9a07196a8e8489da": " c_1 \\frac{dV_1}{dt} + \\frac{V_1}{r_{M1}} = \\frac {V_2 - V_1}{r_1} + \\frac {I_{electrode}^1}{A_1}", "e957d93e8f90589c3c33cc9e42a6ee47": "MF = \\frac {1}{ \\left [ 1 - \\left ( \\frac{f_d}{f_n} \\right )^2 \\right ] ^2}", "e957f6aa1279647141eb0e504d2a367f": "S^\\alpha_i", "e9582087c08e9e75ce870b4b0e107ec7": " \\vec{e}_1 = \\frac{1}{\\sqrt{2}} \\, \\left( \\partial_v - \\exp(-2f) \\, \\partial_u \\right) ", "e9582742c4d9977391e42cbf40d85d24": " \\partial_i = \\frac{\\partial}{\\partial x^i}", "e9584528036ab184181d54aaa8c901aa": "\\hat{X}(z)=\\hat{X}_{Bayes}\\left( \\mathbf{P}_{X|z} \\right)", "e95858cd7fe5154f7fb30181bb664937": "mP_2 v_2 - mP_1 v_1.", "e959331023d536ec367644b5f743ad98": "\\Gamma_{12} (u, v, 0) = \\iint_{\\textrm{source}} I(l, m) \\frac{e^{i \\omega \\left( \\frac{R_1 - R_2}{c} \\right) }}{R_1 R_2} \\, dS", "e959aaafaee3ecbe188efeffa951ef67": "C= \\cfrac{t_L}{S(S-1)/2}", "e959d1dd9b5c8d0813ff3042d10ccda9": "f(x,y) = \\begin{cases}1 & x^2+y^2<1 \\\\0 & x^2+y^2 \\ge 1 \\end{cases}", "e95a842093311006dc115ba4928738de": "v_g = \\frac{\\partial \\omega}{\\partial k}", "e95ad791e9a3f653d29f43d84030c8b9": "\\mathrm P(X_1=x_1, \\ldots, X_n=x_n) = \\prod_{i=1}^n \\mathrm P(X_i=x_i \\mid X_{i+1}=x_{i+1}, \\ldots, X_n=x_n )", "e95af92b89ca984e1c96efb1994c7452": "T\\in V^{\\otimes k}", "e95b1e729cecfc458f02e8b35db29a38": "\\{x+n \\theta : n \\in \\mathbb{Z}\\},", "e95b2f5707b522af7a647b21c2e52371": "\\vec f^1 = \\vec b^1 = \\begin{bmatrix}{1 \\over M_{11}}\\end{bmatrix} = \\begin{bmatrix}{1 \\over t_0}\\end{bmatrix}.", "e95bdcb7ba4502d96dd73532bcf04d87": "\\scriptstyle \\boldsymbol{\\nabla} p", "e95bf0807e7cb7f4c1bce89d4e44f793": "\\,\\mbox{T}(a) \\mbox{T}(da)|x\\rangle = \\mbox{T}(a)|x + da\\rangle = |x + a + da\\rangle = \\mbox{T}(a + da)|x\\rangle \\Rightarrow", "e95c19c5497f2db79684a6c948a7ae6f": " \\phi = {{e_w} \\over {{e^*}_w}} \\times 100\\% ", "e95c5a5569e998dc43bfcada986bec21": "\\left| \\int f \\, dg \\right| \\leq c ||f||_{p,q}^\\alpha ||g||_{p',q'}^{1-\\alpha}.", "e95d410f70685d7d94dfab6bc50aa6fe": "r_{i,x}", "e95d5879e8e665126aef053f0b683d28": " \\!\\ K_n = 2 K_{n-1} + K_{n-2} ", "e95d76e638954d924e9595afc700507a": "x^y = (b^{\\log_b(x)})^y = b^{y \\log_b(x)} \\Rightarrow \\log_b(x^y) = y \\log_b(x)", "e95db0fed2d3ed0f9b860aff82600f0b": "x^p+y^p+z^p=0\\;\\;", "e95db6a439571f9c663195e796009ab0": "(S^n)_n", "e95e1ca27d0e39aa03eb5a611ce4122f": "0.6", "e95e369f4fc8ba2cc435aee3d08650e2": " \\underline{\\underline{\\mathsf{C}_i}} =\\begin{bmatrix}\na_i & a_i - 2e_i & b_i & 0 & 0 & 0 \\\\\na_i-2e_i & a_i & b_i & 0 & 0 & 0 \\\\\n b_i & b_i & c_i & 0 & 0 & 0 \\\\ \n 0 & 0 & 0 & d_i & 0 & 0\\\\\n 0 & 0 & 0 & 0 & d_i & 0\\\\\n 0 & 0 & 0 & 0 & 0 & e_i\\\\\n\\end{bmatrix}\n", "e95e50591ac6cbf18eb9d9baeb2943a6": "\\Phi_{ij}", "e95f18bf32259b553845df64958aad45": "\\mathbb{R}^n/\\mathbb{Z}^n", "e95f84f03d08f2708fecc032a69dd74a": " f \\,\\!", "e95f9a22de3cfc9c51e17fab62b1f107": "\\int_1^a \\frac{1}{x}\\,dx = \\ln a,", "e95fc7e4551ce91328b3652ec848f8f9": "(x)(y)", "e95fde8e4658433295a38953bba72bd5": "T_b \\, = \\, 198.2 + \\sum {G_i}", "e95ffd6c630c6871408fb68928a3d0e3": "t\\in {\\Bbb C}, \\ (x,y)\\in {\\Bbb C}^n\\backslash 0 \\times {\\Bbb C}^m\\backslash 0\\ \\ |\\ \\ t(x,y)= (e^tx, e^{\\alpha t}y)", "e9600c627fa5046792d72b4fea495c21": "\\nabla\\times\\mathbf{w}^{||}=\\mathbf{0}", "e9602ef7d6c3c0b06ea97ac01ded3932": " y(t+h) \\approx y(t) + hf(t,y(t)). \\qquad\\qquad (3) ", "e9604b03f63341561089b0629d03babd": "{\\psi}_n(x) = (h_n)^{-1/2}\\,e^{-x^2/2}H_n(x).\\,", "e9606bced42a8344bf58ac2c57be19d1": "c=1,", "e96088e26d9f4509770cdd82464345de": "(1 + 2) \\times (3 + 4)", "e960aa18923fae3dbe0714ce9fa70367": "E^{(+)}(\\mathbf {r}, t) = i\\sum_{j}[\\frac{\\hbar\\omega_{j}}{2}]^{1/2}\\hat{a}_{j}\\mathbf{\\varepsilon}_{j}e^{i(\\mathbf {k}_{j}\\cdot\\mathbf {r} - \\omega_{j}t)}", "e960d15bda6a9f4aa9963a1e691c6ee2": "(x,y)\\in E(G)", "e960d36250bdc87538ac58c4f8cc7774": " y(t) = \\sum_{n = -\\infty}^{\\infty} x[n] \\cdot h(t - n T_s) ", "e960eb2d12a4369c23f0f2b2329d4c4a": "\\tfrac{\\lambda(1-\\nu)}{\\nu}", "e9614098acda01e9e9eaaf2da7e67f41": "(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +\\,", "e96196f2a8501dd9d7fe623fcd606a7e": "\\lim_{n\\rightarrow\\infty}f_n=f\\ \\mbox{pointwise},", "e9619dac65e78e0f2c71fee87fbb6ff7": "\\scriptstyle u(x)", "e961a4fc166846b7c9b81136274c7eae": "10^{10^a}", "e961c748b23dee6fafa469f33f25c5d5": "+ Z", "e9621c464684c6963ae93e4dafa6cf1c": " \\gamma = \\frac{7}{5} = 1.4", "e962241465b96533f89411d88c5ed8cf": " O(n \\log n \\log \\log n)", "e9622671e7d7f8ea3b39c4c677bea657": "\\mu=\\mu_1+\\mu_2\\,", "e96242cce22cea504dfd5db42473ece4": "F/L= \\rho V G", "e9624e33b00104c65341e2d88cbc93f8": "V \\subset {{\\mathbf{K}}}^n", "e9628785e820f79a18e5ee1b8fc7f3cc": "\\mathrm{2 \\ H^{+} + SO_4^{2-} + 2(CH_2O) \\longrightarrow 2 \\ CO_2 + H_2S + 2 \\ H_2O}", "e962e996241f15cd9eec7a2989161ce5": "\\begin{align}\n \\sum_{n}\\mathcal{L}(n) &= \\frac{m - 1}{1} \\sum_{n=m}^\\infty \\frac{1}{\\binom n 2} \\\\\n &= \\frac{m - 1}{1} \\cdot \\frac{2}{2 - 1} \\cdot \\frac{1}{\\binom {m - 1}{2 - 1}} \\\\\n &= 2\n\\end{align}", "e9630c1c570ed0e827666c832741da66": "I=\\frac{1}{a}K(\\frac{\\sqrt{(a^2 - b^2)}}{a}) ", "e96394728431f1d608959a24eac9b2c7": " \\mathcal{E}^0 \\subseteq \\mathcal{E}^1 \\subseteq \\mathcal{E}^2 \\subseteq \\cdots ", "e963a1eff657568504a1ee0a8ee6f33f": "\\textbf I\\;=\\;\\int_0^{\\frac{\\pi}{2}}\\,\\frac{1}{\\left(a\\,\\cos^2\\,x+b\\,\\sin^2\\,x\\right)^2}\\;\\mathrm{d}x,\\qquad a,b > 0.", "e963a4c77c9094ffb36de47018596f38": " F_5 = o, S_5 = \\_, A_5 = p ", "e963c0c4cea140446afc4deeb8663668": "\n\\sigma_{ij} = -p\\delta_{ij}+ \\mu\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right) + \\delta_{ij} \\lambda \\nabla \\cdot \\mathbf{v}.\n", "e964032d4a3c2761c610f997ca548e23": "A+B=\\{a+b:a \\in A, b \\in B\\}", "e964042514f40ddc3fa090eae0273567": "1/n^q", "e9641457c23831e379975d334a98eeec": "S\\equiv \\frac {\\overline{D}} {\\sigma_D} ", "e96434cf2db58d656db362ce37040881": "{\\Delta m}_t + {\\Delta v}_t = {\\Delta x}_t = {\\Delta p}_t + {\\Delta q}_t \\,,", "e96495eafd55e6793e10ff85f70d314e": "\\det\\frac{\\partial\\phi_t}{\\partial (q_0,p_0)} = 1", "e964deb6b7399c4d1777656f2c5b8a08": "\\mathbf{F}_{g} = \\sum_{i=1}^{N} -G \\frac{m m_i}{r_{i}^3}\\mathbf{r}_{i}.", "e9659010deb4b0654233fabb6107a498": " \\mathbf{P} = P_{\\nu} := \\left(P_0, P_1, P_2, P_3 \\right) = \\left(E, p_x, p_y, p_z \\right) ", "e965c2f1b33f1b1bcd7341ccd0ee4478": "(a^2-n|p)", "e965e75b09cc5c1d19d26df06a876fbc": " |\\mathcal{F}| = \\sum_{i=0}^{k-1} {\\binom{n}{i}} ", "e9662c630d3573723f14c9d0d16760b5": " \\lambda'_{x}(\\alpha) =\n\\min_{P} \\{\\Lambda(P): P(x)> 0,\\; K(P) \\leq \\alpha\\},\n", "e9663170c8d20666f0556b594b82de51": " E_{\\omega} = \\int d^{3} x \\left[ \\frac{1}{2} |\\dot{\\phi} - i \\omega \\phi|^{2} + \\frac{1}{2} |\\nabla \\phi|^{2} + \\hat{U}_{\\omega}(\\phi, \\phi^{*}) \\right] ", "e9667f36a6c1b52f14fbdc0101ee6576": "\\textstyle {N}", "e966c1dbca9877ef73fc56f694603dbc": "\\frac{\\mathrm{d}}{\\mathrm{d}t} A(t) = - \\frac{i}{\\hbar} [H_\\mathcal{S} , A ] = 0,", "e966ca06376b14db4cd7504a0742a92c": " \\frac{R} {Q_o} = \\frac{V^2}{\\omega U} = \\frac{2 \\left( \\int{\\overrightarrow{E} \\cdot dl} \\right)^2}{ \\omega \\mu_o\\int{|\\overrightarrow{H}|^2 dV} } = \\frac {2k}{\\omega_o}", "e966f8f880642b50e5a5a9cf415a89d7": "\n\n\\begin{align}\n\n\\mathrm{P}(X_1=x_1,\\dots,X_n=x_n) & = \\mathrm{P}(X_1=x_1) \\\\ & \\qquad \\times \\mathrm{P}(X_2=x_2\\mid X_1=x_1) \\\\ & \\quad \\qquad \\times \\mathrm{P}(X_3=x_3\\mid X_1=x_1,X_2=x_2) \\times \\dots \\times P(X_n=x_n\\mid X_1=x_1,X_2=x_2,\\dots,X_{n-1}=x_{n-1})\n\n\\end{align}\n", "e967ae3b2f33346c9f08e7ed30914033": "J^k_0f", "e96821ced5981fbfcefd80598b4de067": " 133 \\rightarrow 55 \\rightarrow 250 \\rightarrow 133 \\rightarrow ... ", "e9685c90bb09dd638dbcc8e5a449c572": "01011001", "e9691a81ec30f7412d7b76dcfab7058c": "\n\\begin{align}\nCPI = {EV\\over AC}\n\\end{align}\n", "e96928918db96dbed1fa8fc885c940ff": "\\scriptstyle \\left(f_i\\right)_{i \\in I}", "e9692efa0f88711aab4eb53a22f68ea3": "\\theta \\colon \\mathcal{N}_\\partial (X \\times I) \\to L_{n+1} (\\pi_1 (X))", "e9695bb8f61eaa68a3eb0f1a1dcb5cfe": "= m\\ddot r\\hat{\\mathbf{r}}+ m r\\ddot\\theta'\\ \\hat{\\boldsymbol\\theta} ", "e969ab48f0edf0ec0a5b343fc47695ea": "v(p)=1", "e969c347e954972abfa4c64cc024c8f8": "\ng(E(y)) = g(\\mu) = \\eta = X \\beta + v \\,\n", "e96a093d28b99fca977d7c9fb392a6a3": "x=2^{1+\\max(U\\cup V)} + \\sum_{u\\in U} 2^u.", "e96a0ef66e3f7955ae6a80f7a29be3e0": "\\hat{\\phi}_{j,0,0}:=2^{\\frac{-3j}{4}}W(2^{-j}r)\\tilde{V}_{N_j}(\\omega), r\\ge 0, \\omega \\in [0,2\\pi), j \\in N_0", "e96a2958a1411d10c0281b037ef2df5c": "t=\\sqrt{1-y^2}/y", "e96a3d3425d19b6e916adbdb1d153004": " Q = 2 - \\frac{4}{p} \\sin{p} + \\frac{4}{p^2} (1-\\cos{p})", "e96ac219e666302e52f038908ffac439": "\\frac{\\partial u}{\\partial \\nu} (x) = \\nabla u (x) \\cdot \\nu (x)", "e96ad267166b1eafbcc0a5709b269fa9": "E = -\\sum_{i=1}^{N}\\left(\\frac{Z-s_{i}}{n^{*}_{i}}\\right)^{2}.", "e96ae1f076a5d07d13ee249c95de384b": "X_{1}, X_{2}, \\ldots, X_{n}", "e96b3641d6be3afce6e3141632692bc2": "p = mv ", "e96b626633edcfc594674f14f804b707": "\\eta _{th} = \\frac{{{}_2W_3 + {}_3W_1 }}\n{{{}_1Q_2 }}", "e96b9c6dd3f1cb90cf5a271524e0bec1": "\\mathrm{lcm}(k_1, ..., k_m)", "e96ba3dde3ff822e99077b147c27875b": "(a_{12} x_2 p_1 + a_{22} w x_2 p_2) (1+r) = x_2 p_2", "e96c90d279d37850be802203bebccc77": "\n\\cfrac{\\cfrac{\\cfrac{\\mathbf{pq} \\qquad \\mathbf{p \\overline{q}}}{\\mathbf{p}}\\, q \\qquad \\cfrac{qr \\qquad \\overline{p} \\overline{q}}{\\overline{p} r}\\, q}{or}\\, p \\qquad \\overline{o} s}{rs}\\, o\n", "e96ccb81dbe1c87036565935db315e31": "U(0,1)", "e96cd8b9843b813384bbaaec7fae6d5d": "f(x)=\\Omega_L(g(x))\\ (x\\rightarrow\\infty)\\;\\Leftrightarrow\\;\\liminf_{x \\to \\infty} \\frac{f(x)}{g(x)}< 0", "e96d2a4c0360c68377a66400d68398ff": "W = \\int \\! p \\,dV \\,", "e96d528efd0b613fee3b69f24a0cb34e": "\\tfrac{1}{8}(b-a)^2", "e96d6d21d8f421656f675cd785ad7e61": "\\|u-\\pi u\\|.\\,", "e96d760576ec511905f92838e439656b": "\np + 2q + r = 0.83943 + 0.15771 + 0.00286 = 1.00000 \\,\n", "e96d8343f8104413c9d9601befbe2966": "\\dot{u}_{n+1}=\\dot{u}_{n}+{\\Delta}t~\\ddot{u}_\\gamma \\,", "e96d836fb0dcd868eb7e5558078ac63b": "\\langle (x_1,x_2), (y_1,y_2)\\rangle_{H_1\\oplus H_2} = \\langle x_1,y_1\\rangle_{H_1} + \\langle x_2,y_2\\rangle_{H_2}.", "e96dc4e091b855f84a0d3afeaad2118d": "F(x)=\\alpha G(x)+\\beta,", "e96dd3eb79e3ebe66e0c5fb405e4c05e": "(1,3,\\bar{3})\\rightarrow2\\,(1,2)_{-\\frac{1}{2}}\\oplus(1,2)_{\\frac{1}{2}}\\oplus2\\,(1,1)_0\\oplus(1,1)_1", "e96df0adfb6c3874fe058ee7f6211769": "\\Delta_1^2-4\\Delta_0^3 = - 27 \\Delta\\ ,", "e96df136114b87a81a85ba1fbd26c334": "x_i \\in A", "e96df6ab1b89683a8a7e6ed8e8b513c9": "E_x^{\\rm HF}", "e96e29c9d19d3f3398a71462c0224717": "V_{\\rm peak} - (-V_{\\rm peak}) = 2 V_{\\rm peak}", "e96e61a887811eecdf4303767adbfea1": "10^{6.7}", "e96ea49b312fb972e4b4365060dfe1b2": "D_d~", "e96eab7c9c019a61318f17ec9438ce40": "({\\mathbf P_1},F_{\\mathbf P_1})", "e96f5e25c55e398a8edf14c60d0a84e0": "\\theta_\\mathrm{p}", "e96fccf98e751ca11001861c4a68867e": "\\ln(b-a) \\,", "e96ffbfbbbea3fd25b1f0f26bbde30db": "{x_1,x_2,\\ldots,x_n}", "e9700e731b52a157f4a049067fd2e94a": "C_H(x)= \\sum_{y \\neq x}\\frac{1}{d(y,x)}", "e9702bd7ea3c21a34bf9aa19d86fc79d": "J_{z} = I_x + I_y = \\frac{b h^3}{12} + \\frac{h b^3}{12} = \\frac{b h}{12}(b^2 + h^2)", "e9704ee4ba5100480446afd898cea9be": "f^{(n)}(x)\\,", "e97094445e46db891c6bd787be4ec091": " R_{22} = \\left. \\frac{V_2}{I_2} \\right|_{I_1=0} ", "e970ecb2076a3ec127c65020bc308693": " s = \\left [ 2 s_0 \\cos \\left ( \\omega' t \\right ) \\right ] \\cos \\left ( \\omega t \\right )\\,\\!", "e971187b1cf60d2eb58b0c22faf53eea": "G(\\omega) = \\frac{|Y|}{|X|} = |H(j \\omega)| \\ ", "e9711c0751e4eb223741fece3ee3ed05": "\n\\delta_n = (\\gamma_{n+1}-\\gamma_{n}) \\frac{ \\log{ \\frac{\\gamma_n}{2 \\pi} }}{2 \\pi}\n.", "e971218a552398fbcce2969b5e487323": "\\frac{f^2}{F}\\,", "e9712a3245e73e3bbf46e65b3034aa35": " \\varepsilon_2 < 2^{-5} < 10^{-1}. \\, ", "e9712de863eb9db124a6cbf7c412a74b": "\\lim_{x\\searrow0} \\frac{f^{(n)}(x) - f^{(n)}(0)}{x-0} = \\lim_{x\\searrow0} \\frac{p_n(x)}{x^{2n+1}}\\,e^{-1/x} = 0.", "e971ddaed635808ce1d5d2ff86850d5a": "P(x_{i} | x_{i-(n-1)}, \\dots, x_{i-1})", "e971e8d27623be68e7e2dbf240555a25": "F_{n} = 2^{(2^n)} + 1", "e9720e3a83462dcba80864273fcfb2fc": "f(x)=x^{\\frac{q}{p}}", "e972724def6ce93ef633f8d9198ce77b": "\\left(\\operatorname{ad}_g\\right)^n(x)=e", "e9727c00e43ea4d721aedac8ff890e79": "\\forall x\\in\\mathbb R:\\;f(x)=g(x+m2^{-k})", "e9728c4330012bb285d12f5b50f4621b": "M_{i}", "e9734b40b42560d6f6a1870217f886af": "\\beta_1=\\alpha", "e973554a66d89975aaadc2dd6f936056": "A_{\\perp} = \\frac{2 \\pi e \\Delta H}{\\hbar c}", "e9739cbe5771a0f7f5dcca879c918442": "I_C=\\frac{M}{1-\\alpha M}(I_{CBO} + \\alpha I_B)\\iff I_C =\\frac{I_{CBO} + \\alpha I_B}{1-\\alpha - \\left(\\frac{V_{CB}}{BV_{CBO}}\\right)^{\\!n} }", "e973b87579064e810d6831009925c670": "(4)\\quad T_{ab}=\\frac{1}{4\\pi}\\,\\Big(\\, F_{ac}F_b^{\\;c} -\\frac{1}{4}g_{ab}F_{cd}F^{cd} \\Big)\\,,", "e973b9ebfd6cb22c0b80c334589226d4": "\\Vert T_n x - T x \\Vert_H", "e973c66710cdfff3f9a31e8bf4669d99": "10^{14}", "e9744f0a3fc443d0865a8af4e46be9dd": "0<\\delta<1/2", "e9753b171d365ed3540db3e8c9732fe3": "(12)\\quad \\Psi_2=-\\frac{M(u)}{r^3}\\qquad \\Phi_{22}=-\\frac{M(u)_{\\,,\\,u}}{r^2}\\;.", "e97569d46fec876bdd6c04f41a08a052": " \n\\begin{align}\n \\pm \\sqrt{-i} & = \\cos(3\\pi/4) + i\\sin(3\\pi/4) \\\\\n & = -\\frac{1}{\\pm \\sqrt{2}} + i\\frac{1}{\\pm \\sqrt{2}}\\\\\n & = \\frac{-1 + i}{\\pm \\sqrt{2}}\\\\\n & = \\pm \\frac{\\sqrt{2}}2 (i - 1).\\\\\n\\end{align}\n", "e975e8bce8b13e84fd8d914033ff18a9": "\\pi_2(PU(\\mathcal H))=\\mathbf Z,", "e975efe886f84e7f77d909db4ca2a09c": "\\psi_{s'}", "e9762866439b311c512b2c9fd7734660": "\\begin{align}\n&& \\max(a(0,1)h_1+a(0,2)h_2+\\cdots+a(0,n)h_n)\\\\\n&&{\\operatorname{s.t.}}\\quad a(1,1)h_1+a(1,2)h_2+\\cdots+a(1,n)h_n\\ge b_1\\\\\n&&\\quad a(2,1)h_1+a(2,2)h_2+\\cdots+a(2,n)h_n\\ge b_2\\\\\n&&\\quad \\vdots\\quad \\vdots\\quad \\vdots\\quad\\vdots \\quad\\\\\n&&\\quad a(m,1)h_1+a(m,2)h_2+\\cdots+a(m,n)h_n\\ge b_m\n\\end{align}\n", "e9767501e1a5ea2cde53ab74de8c9eca": " b, b'\\in B", "e976c33443f3872f52e0496920417294": " 0.50 \\atop m\\text{ odd}}} \\frac{X^m}{m!}\\left[2\\zeta^\\prime(-m)+\\zeta(-m)\\left({1\\over 1}+{1\\over 2}+\\cdots+{1\\over m}\\right)\\right].", "e98a10f46d0601f1ebe321f3f4af314f": "\\varepsilon_2' = -\\frac{\\nu}{E}\\sigma_1", "e98a484cab1ecabcc24c8091d142ad1d": " \\scriptstyle Y = \\mathrm{log}(1 + e^{-X}).", "e98a5222040d97e4dab09786bce62bf0": "\n \\int (d+e\\,x)^{m-1}(B (2 a\\,e\\,m+b\\,d (2 p+3))-A (b\\,e\\,m+2 c\\,d (2 p+3))+e(b\\,B-2 A\\,c) (m+2 p+3) x)\\left(a+b\\,x+c\\,x^2\\right)^{p+1}dx\n", "e98ae5b3859e3b58729925c7e586f636": "(-1)^{m'-m+s}", "e98af1ac6c04f3b9804a70854cbe50cf": "\\epsilon_2 (p)", "e98b00a0befc66ef34fcf6531553b0fe": "I_\\mathrm{rev}\\,", "e98b06aa296ed8ddd4b6560a7d0f9964": "n\\le2.", "e98b1160db296a6761931f627764f39c": " S \\mapsto A(S) ", "e98baea357520f8580ebf678f736535b": "\\displaystyle{\\sum m_i(1-|\\lambda_i|) <\\infty,}", "e98bf67c7ce5d05d86dff2abf009cb49": "\\dot{v} = (F\\cos\\alpha)/m - D/m - g\\cos\\theta\\,", "e98bf734628f09495972eb8b7d9c5ef9": "e_b(b') = \\begin{cases}\n1&\\text{if } b=b'\\\\\n0&\\text{otherwise.}\n\\end{cases}", "e98c508ff7537a8a040d75f7e5900099": "\\dot{v}_4 = {1 \\over C_4} ({{v 1 - v 5} \\over R_6} - {{v 7 - v 3} \\over R_2}) ", "e98c88008e9a19219141bab1442b1d28": "X^\\mu=(ct,x,y,z)\\,.", "e98cda1b5225f00309187785b528591c": "X_a^b", "e98d23cf1872b790009e9a6936bc7842": "l_1 + l_2 > m", "e98d2f8a4d53f8bfdbec5e43fb00005e": "D(\\mathbf{r}) = \\int_{-L/2}^{L/2} k(s)N(\\boldsymbol{\\sigma}_{\\mathbf{r}}(s)) ds", "e98d4dfa0d14a36ef9c7bd24d67f0b25": "\\frac{1}{\\tau_n(N)}= A + BN + CN^2", "e98d930410ebaf4900c291de9b6a9af0": "f^* \\ge g^*", "e98e31ac2e7dc198b56725b45376a537": "(\\tilde{\\nabla}_X \\alpha)(Y,Z) = D_X\\left(\\alpha(Y,Z)\\right) - \\alpha(\\nabla_X Y,Z) - \\alpha(Y,\\nabla_X Z).", "e98e3bd3abce038f48e95f2f78d91c19": "T_1[i,j]", "e98e69a3c05fd00cdebb01f0a9e05079": "F = \\prod_{i=1}^K f_i.", "e98e6ed247b66d2c5d81e2ef5fb26ae5": "\\frac{\\sin\\alpha}{\\sqrt{2-\\sin^2\\alpha}}", "e98e8d4d4fb43935192ccb682ba7472b": "< r ", "e98eec89490f97d52d0dfc8eed246f23": "\\operatorname{E}_{\\mathbf{\\mu}_k,\\mathbf{\\Lambda}_k} [(\\mathbf{x}_n - \\mathbf{\\mu}_k)^{\\rm T} \\mathbf{\\Lambda}_k (\\mathbf{x}_n - \\mathbf{\\mu}_k)]", "e98f2a45282c3fbee5494910f77a84ad": "n \\leq m", "e98f3128c6e112134dce2b9d9e7145c0": "u_X:A \\otimes X \\rightarrow X \\otimes A", "e98f71c97bee616d740616990660f272": "C_k h", "e98fcdcc97dd0e589266eef5cf4a90c6": "-\\ell\\,", "e98fe4807695708489ea149bd94afe4b": "\\det ( A) A^{-1} = \\sum_{s=0}^{n-1}A^{s}\\sum_{k_1,k_2,\\ldots ,k_{n-1}}\\prod_{l=1}^{n-1} \\frac{(-1)^{k_l+1}}{l^{k_l}k_{l}!}\\mathrm{tr}(A^l)^{k_l},", "e98fe8802ae6bd4bcb5799870707481b": "I_j \\subset R_j", "e98ff62ca32150b04508a3fefbb14234": "x = a \\cdot (e - \\cosh E)", "e99062b6d9ef3c90bb2beb89cf830f28": "\\overline{\\mathbb{F}_{q}}", "e9906bf88782ef654c8b849d06bc36f8": "\\begin{alignat}{2}\ne_1 & = (ab)^* \\\\\ne_2 & = \\left(aa(ab)^*bb(ab)^*\\right)^* \\\\\ne_3 & = \\left(aaaa \\left(aa(ab)^*bb(ab)^*\\right)^* bbbb \\left(aa(ab)^*bb(ab)^*\\right)^*\\right)^* \\\\\n\\, & \\cdots \\\\\ne_{n+1} & = (\\,\\underbrace{a\\cdots a}_{2^n}\\, \\cdot \\, e_n\\, \\cdot\\, \\underbrace{b\\cdots b}_{2^n}\\, \\cdot\\, e_n \\,)^*\n\\end{alignat}\n", "e99083be1ac09a1ea110e4ace225a1fe": "(\\varphi, \\varphi)", "e990876198026a0240ff0806f9ecd761": "\\%R = { high_{Ndays} - close_{today} \\over high_{Ndays} - low_{Ndays} } \\times -100", "e990ab718bb2e9321a1ce132e14b65d9": "{\\rm MCG(N_3)} = {\\rm GL}(2, {\\mathbf Z}). ", "e9910ccf7cbe670b7b64c24047fa560a": " P = \\sum_ { i_1 = S_1, i_2 = S_2, .. i_n = S_n} K_{i_1 i_2 .. i_n} ", "e991afdc39256b54c1401be503e4b60d": " E_y = V_{in} B_{in} ", "e99218939bd36a2753792215f7da646e": "R = r_0 A^{1/3} \\,", "e9921e1f26d0831cecd4de202a465562": "\n\\begin{array}{l}\nv(x) = \\mathrm{ord}_t(t) = 1 \\\\\nv(x^6-y^2)=\\mathrm{ord}_t(t^6-t^6-2t^7-3t^8-\\cdots)=\\mathrm{ord}_t (-2t^7-3t^8-\\cdots)=7 \\\\\nv\\left(\\frac{x^6 - y^2}{x}\\right)= \\mathrm{ord}_t (-2t^7-3t^8-\\cdots) - \\mathrm{ord}_t(t) = 7 - 1 = 6\n\\end{array}\n", "e99285503506a62b54574864603de9f9": "\\alpha M \\subseteq M", "e992883ab9e8e4abc4183d1800d2865d": "{\\mathcal P} = {\\mathcal P}({\\mathcal O}) = \\{A_1, \\ldots A_n\\}", "e9932240ae74a4e1fcd4d0acfa61460f": "X_{2}^\\mathrm{opt} = \\alpha W + \\beta \\mu_2.", "e99374e10bc2743951361402eacde577": "\\sqrt{3} + 4\\,\\!", "e9939c8ee7c5951d56ac23a495012e9b": "\\frac{dh}{dx}=S=\\tan\\Theta", "e993a695021468cbbf107739e704a20e": "T < T_1", "e993c502442bad3f0d4234f2d24e941e": "\\frac{\\partial L}{\\partial x}= -q\\frac{\\partial \\phi}{\\partial x}+ q\\left(\\frac{\\partial A_x}{\\partial x}\\dot{x}+\\frac{\\partial A_y}{\\partial x}\\dot{y}+\\frac{\\partial A_z}{\\partial x}\\dot{z}\\right)", "e993dd2e084295ff6d592b60f6571751": "E = \\tfrac{1}{2}(t - y)^2", "e99410f8cb6e0991ca54615382f5c106": " l_3", "e9945b259545e5cf351ff9faca68cc71": "S_{t} = \\alpha \\times (Y_{t-1} + (1-\\alpha) \\times Y_{t-2} + (1-\\alpha)^2 \\times Y_{t-3} + \\cdots + (1-\\alpha)^k \\times Y_{t-(k+1)}) + (1-\\alpha)^{k+1} \\times S_{t-(k+1)}", "e9947631a8dfff7e3d903f9027bf2c68": "\\chi_{T}(G)=\\chi(G)", "e9947b5881708f103cf0264ec84d1cc1": "\\chi(X)=\\sup \\; \\{\\chi(x,X) : x\\in X\\}.", "e9956d9eff7ba53e3c70c4ebe46322bf": "\\mathbf{P} \\left ( \\sum_{i=1}^n X_i \\geq 2 t \\sqrt{\\sum \\mathbf{E} \\left [ X_i^2 \\right ]} \\right ) < \\exp (-t^2), \\qquad \\text{for } 0 < t \\leq \\tfrac{1}{2L}\\sqrt{\\sum \\mathbf{E} \\left[X_j^2\\right ]}. ", "e995c49285dcfb08e4b0b02a4eacb51f": "|x| + |y| = 1", "e995ea11b6db2d3dd52c4378707d48b7": "B_{2m}", "e9960f6a2af6e5e2c6e0cca12586e639": "\\vec v = [v_1,...,v_n]\\,.", "e99653761806bc6a0f5d6751476702e1": "D^+ (S)", "e996f6dfba16371a77e987fafa5b921d": "p(c_j|f_i)-p(c_j)\\ ", "e99709d801f6c73bd9a967d18f2f32e6": "\\mbox{newspaper } \\mathbf{\\{ \\operatorname{d}, \\operatorname{s}, \\operatorname{f} \\}} \\mbox{ dinner}", "e99712c932eff2dee49569253bbde8d4": "\\sum 1/n^2 ", "e9974d8163a4f40e134a48d3b7bec748": " \\varphi := \\sum_{k=1}^\\infty \\xi_k \\psi_k,", "e997fb05708ef8b1f6439c318ec8f2e2": "\\textbf{w}_{k} \\sim N(0, \\textbf{Q}_k) ", "e998030d1585873cdb84b42d4c7505b6": "M[\\xi] =\n \\frac{1}{37} \\sum_{\\omega \\in \\Omega} \\xi(\\omega) =\n \\frac{1}{37} \\left(\\xi(\\omega^\\prime) +\n \\sum_{\\omega \\ne \\omega^\\prime} \\xi(\\omega)\\right) =\n \\frac{1}{37} \\left(35 \\cdot r - 36 \\cdot r \\right) = -\\frac{r}{37}\n \\approx -0.027r.\n", "e9983d628ba752951200e6b567000bb2": "\\begin{align} E_B = & a_v A - a_s A^{2/3} - a_c Z(Z-1)A^{-1/3} \\\\\n& -a_a (N-Z)^2 A^{-1} + 12\\delta(N,Z)A^{-1/2} \\\\\n\\end{align}", "e99860c7ddea57336e4c997032d5e2e5": "E_{ij} ", "e998bf8e25dbad9bea56c53828fb021e": " U_x: H_x \\rightarrow K_{\\eta(x)} ", "e998e6e2b92b4caaffc5a62ec11566ac": "{}+ (a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2)k.", "e998ff00db11a8cdfa1744b14876ded4": "F_\\text{P} = \\frac{E_\\text{P}}{l_\\text{P}} = \\frac{\\hbar}{l_\\text{P} t_\\text{P}} = \\frac{c^4}{G} ", "e999097148205df316ad9ef6dd1bbd5f": "R_{45}", "e999560d0d229c54e132f88ab0569527": "x,\\;", "e9996521b2725b2e7de3d90cf9658c8e": " \\langle\\phi| = \\langle \\psi | A^\\dagger", "e999705d3462c5c8f0040a7138cb2c21": "\\omega + 1 = \\left\\{0, 1, 2, \\dots, \\omega\\right\\}", "e999ca00f5b5268a4dca1d3d4b2fcb2f": "\\alpha = \\frac{1}{\\gamma} = \\sqrt{1- v^2/c^2} \\ , ", "e999e686d65a9362655e7a13f816dabd": "\n\\begin{align}\n\\mathbf{L}(\\mathbf{X} | \\tau, \\mu) & \\propto \\prod_{i=1}^n \\tau^{1/2} \\exp[\\frac{-\\tau}{2}(x_i-\\mu)^2] \\\\ \n & \\propto \\tau^{n/2} \\exp[\\frac{-\\tau}{2}\\sum_{i=1}^n(x_i-\\mu)^2] \\\\\n & \\propto \\tau^{n/2} \\exp[\\frac{-\\tau}{2}\\sum_{i=1}^n(x_i-\\bar{x} +\\bar{x} -\\mu)^2] \\\\\n & \\propto \\tau^{n/2} \\exp[\\frac{-\\tau}{2}\\sum_{i=1}^n\\left((x_i-\\bar{x})^2 + (\\bar{x} -\\mu)^2\\right)] \\\\\n& \\propto \\tau^{n/2} \\exp[\\frac{-\\tau}{2}\\left(n s + n(\\bar{x} -\\mu)^2\\right)] ,\n\\end{align}\n", "e999ec2e43a0df3a0000049a40db51c5": "\n\\frac{\\ }{\\top\\hbox{ true}}\\ \\top_I\n", "e999fe30d28aa67378c9a5c38b7da3f0": "| x_{n+1} - a | \\le K \\cdot {| x_n - a |}^3,\\text{ for some }K > 0.\\!", "e99a30fc0876a45125b3ad2ee6cd470b": "\\cos(a\\sqrt z)", "e99b5446570973433972b4199b45416d": "S_i \\cdot S_j \\subset S_{i+j}", "e99bb8f91f54d5e0579ef7e5844d7880": "\\epsilon_{ij} = \\epsilon_{ij}^{\\text{v}}+\\epsilon_{ij}^{\\text{s}}", "e99bd60aea016184db95d8c1db1701e2": "s : U \\to E_{|U}", "e99c19dec2b574bc5d4990504f6cf550": "AH", "e99c36b04d61129ba8bcce41217a3a45": "|\\Delta|\\ll\\Omega_{\\perp}", "e99c3d5962cbe1a252b97277d1a3a24b": "C_W \\,\\ ", "e99c529a944e5c02f3f78c1fc390b2bc": " I = m r^2", "e99c8f87312c324e293c2af5589fa05d": "E = \\alpha \\mp \\beta", "e99cfc1413bba2de537d30bec9f974a3": "\\alpha = \\operatorname{atan2}(Z_1 , -Z_2),", "e99d0106d75106632fbfa41ad21c3c30": " A^{\\times}_A + X^{\\times}_X \\Leftrightarrow A(s) + X^{''}_i + 2h^{\\bullet} ", "e99d0afdfce2336612ff516079145bec": "M'(\\theta)", "e99d5bcad28e5bb395a19b6704de8a1f": "(k,l)", "e99d8557ea0afb23ab90c854c0c4802a": "\\longrightarrow_R", "e99df632d28967b5871da566ca209079": "\\ln 2 = \\frac{2}{3}+\\sum_{k\\ge 1}\\left(\\frac{1}{2k}+\\frac{1}{4k+1}+\\frac{1}{8k+4}+\\frac{1}{16k+12}\\right)\\frac{1}{16^k}.", "e99df6a010094cc407fcb508e0edda22": "e^{-\\frac{x^2}{2}}\\cdot H_n(x) \\sim \\frac{2^n}{\\sqrt \\pi}\\Gamma\\left(\\frac{n+1}2\\right) \\cos \\left(x \\sqrt{2 n}- n\\frac \\pi 2 \\right)", "e99e17513cb1f6f952482c5382e83fe3": "\\displaystyle I(P) = \\min_{Y_r} I(X_r \\land Y_r)", "e99e1e34f9413fe98395ded5b4d9018b": "p_1, p_2, \\ldots, p_j", "e99e2fa9720ded4cafb470e6c17bf009": "2^{-{\\epsilon^2}n}", "e99e408d195c0a1a9840159870c1a763": "\nr_e = \\exp\\left\\{\n {1\\over L^2} \\oint_\\ell \\oint_\\ell\n \\ln \\vert \\boldsymbol{x} - \\boldsymbol{y} \\vert\n \\; dx \\; dy \n \\right\\}\n", "e99e7d16d0c112e9aa0d0a8941bf6565": "\\psi_m\\mathbf{(k,r)}=\\frac{1}{\\sqrt{N}}\\sum_{n}{a_m\\mathbf{(R_n,r)}} e^{\\mathbf{ik\\cdot R_n}}\\ ,", "e99eb93ff68d2f1851ec4a4918fa56e8": "\\vec \\psi_P = \\frac{\\partial \\vec v_P}{\\partial t}", "e99ecc5ba73946e9e3423e48b5735cbd": "\\sum_{d,e\\ge 1}\\frac{\\frac{d e}{ [d,e]}\\sum_{n\\ge 1} (X_d^{ [d,e]/d } Y_e^{ [d,e]/e } t^{ [d,e] })^n}{n}", "e99ee5b28c608c1945fb0e7335547bbb": " a: G \\to G' ", "e99f03344ab9722020e7382af4e0a200": "\\cdot \\!\\,", "e99f0781988cdfb35294ee4fa3c54da5": " \\begin{align}\n \\zeta(s) & =\\sum_{k=0}^m \\frac{B_k}{k!} s^{\\overline{k-1}} + R(s,m) \\\\\n & = \\frac{B_0}{0!}s^{\\overline{-1}} + \\frac{B_1}{1!} s^{\\overline{0}} + \\frac{B_2}{2!} s^{\\overline{1}} +\\cdots+R(s,m) \\\\\n & = \\frac{1}{s-1} + \\frac{1}{2} + \\frac{1}{12}s + \\cdots + R(s,m).\n\\end{align} ", "e99f1c6e6ed9f2ea0b8928f9de603271": "\\,I^+(x)", "e99f6e17362166ca80d34b2a167e95ec": "H:=\\{\\vec{u}\\in (L^2(\\Omega))^n| \\operatorname{div}\\,\\vec{u}=0 \\text{ and }\\gamma(\\vec{u})=0\\}", "e99f6f3064b9fe4485bb5905adb7d29f": "T := \\{ \\phi \\mid \\phi \\mbox{ is satisfiable} \\}", "e99f6f57711e00bec8f3d001498b0439": "a_\\text{r}\\in\\Sigma_\\text{r}", "e99f9322dfecfbb857f8f64388740af4": "r(\\theta) = {e \\over 1+e \\cos \\theta}", "e99ff53b3b0f26d80359a2c3204ff886": "j_! \\leftrightarrows j^!=j^* \\leftrightarrows j_*", "e9a0349291677387107bf13999edb0b1": "\nw^{(L)}_k = \\frac{\\hat{w}^{(L)}_k}{\\sum_{J=1}^P \\hat{w}^{(J)}_k}\n", "e9a0a57e3689faeb741a47d94f4ed2b1": "\\mathbf{F} = Q (\\mathbf{E} + \\mathbf{v} \\times \\mathbf{B}),", "e9a0c6c3ad09423ea6d64225a9cc70c9": "P_n(x) = \\sum_{k=0}^n (-1)^k \\begin{pmatrix} n \\\\ k \\end{pmatrix}^2 \\left( \\frac{1+x}{2} \\right)^{n-k} \\left( \\frac{1-x}{2} \\right)^k.", "e9a0ce83c20b37282a612897e6fd1680": "\\triangle P_1KA", "e9a0fa1d69c5a4310f77dbf23978ac32": "\n\\langle x \\, p| \\hat{\\lambda}_x | \\Psi \\rangle = -i \\frac{\\partial}{\\partial x} \\langle x \\, p | \\Psi \\rangle, \\qquad\n\\langle x \\, p| \\hat{\\lambda}_p | \\Psi \\rangle = -i \\frac{\\partial}{\\partial p} \\langle x \\, p | \\Psi \\rangle.\n", "e9a1715c6f69bf78aa5cf3112cabcae5": "X_{i}-Y_{i}-B_{i}", "e9a1a5f2fcf6e30348e40b3923239a1f": "N-L", "e9a1dcb7af4470ed7815cf5a69643e0c": "\\mathbb Z\\to\\mathbb Z", "e9a264b8ff0746cd1b01b1d7dcf78b46": "y'=y", "e9a28ee79ab19c925b404f96d843eb4b": "\\mathbb Q[p_1,\\ldots,p_n].", "e9a297ae9a591959c2353d81bcb6bcf0": "\\langle \\sigma',C'\\rangle\\leq\\langle\\sigma,\nC\\rangle", "e9a2d3bf9228795461bdb9ed33680339": "N-P", "e9a2ff90ffc001198cd3b9e8898dd491": "f(x)(y)", "e9a30d179a782fcc3ad391ca8e4c64e6": "\\displaystyle{ B(z_1,z_2)=x_1y_2-y_1x_2=\\Im\\, z_1\\overline {z_2}.}", "e9a3132f2a64eccc86805dbfe3f73c83": "D[k,l] = \\lambda_k \\qquad \\text{for } k = l = j", "e9a3244887db6c80e67d8b69c7f70e9c": "6 \\times \\tfrac{3}{4} = \\tfrac{6}{1} \\times \\tfrac{3}{4} = \\tfrac{18}{4}", "e9a3304fc2c691cb243d63832d23fcc6": "\\mathbf{E}(\\mathbf{r},t)=\\frac{q}{4\\pi\\varepsilon_0}\\left[\\frac{\\hat{\\mathbf{n}}-\\vec{\\beta}}{\\gamma^2R^2(1-\\vec{\\beta}\\mathbf{\\cdot}\\hat{\\mathbf{n}})^3}+\\frac{\\hat{\\mathbf{n}}\\times[(\\hat{\\mathbf{n}}-\\vec{\\beta})\\times\\dot{\\vec{\\beta}}\\,]}{c\\,R\\,(1-\\vec{\\beta}\\mathbf{\\cdot}\\hat{\\mathbf{n}})^3}\\right]_{\\mathrm{retarded}} \\qquad \\qquad (2)", "e9a33b8098629f332ff75bec52144c55": "J = J_x = J_y \\neq J_z = \\Delta", "e9a35b6bfd35e858b0efeb29116e48e9": " {\\dagger} ", "e9a3ea7021f61857b080c8b0fa9ff882": "R_{1,1} = r", "e9a41d0a937020538efb57779ef000de": "y = Q^{-1}(t) x", "e9a440b7eeac6ec80df2584cdb438402": " Q = \\Delta U ", "e9a4552a26d056dc82c4271a95e46c1f": "T_Y", "e9a4757e12618f1fe6e009fec36bc822": "\n\\mathbf{f_{0:t}}(i) = \\mathbf{P}(o_1, o_2, \\dots, o_t, X_t=x_i | \\mathbf{\\pi} )\n", "e9a47e0e155f457737ab32e9b2f40174": "(A,\\mu,\\eta,\\delta,\\varepsilon)", "e9a4d2bc2cb22d0ed47a2287e24d7612": "\\delta^\\dagger \\circ \\delta = 1_A", "e9a4d42632bc9d0e6b095360ed944f61": " Q_n = \\operatorname{tr} [ e^{- H(1,2,\\ldots,n)/(k_B T)} ]. ", "e9a4ff10b2d3fd490591dae115f58593": "\nf(\\zeta) = \\int^\\zeta \\frac{K}{(w-a)^{1-(\\alpha/\\pi)}(w-b)^{1-(\\beta/\\pi)}(w-c)^{1-(\\gamma/\\pi)} \\cdots} \\,\\mbox{d}w \n", "e9a5136cdb7eeb589fc9d75c310bfe1d": "_2\\pi_*^S", "e9a52f248fbe548056b9a704981bf0e9": "x\\pm iy = -a", "e9a58576452a3c609929efa3585190a0": "\\cos A=\\frac{\\textrm{tanh(adjacent)}}{\\textrm{tanh(hypotenuse)}}=\\frac{\\tanh b}{\\,\\tanh c\\,}.\\,", "e9a59f8c7d08cbc166693ab02959ee43": "s = \\frac {1} {2}(v_0+v) t ", "e9a5ae14ed66d55f0a8eabeff560232c": "(-f)\\in C(E)", "e9a5affbcc794cdff989983157e6dc53": "\\nabla \\gamma_{23} = - (F_{23} \\cos\\gamma_{12} + F_{13} \\sin\\gamma_{12})", "e9a5c9b748a492884e28b704cb824684": "N h", "e9a5f19946548c665fc7a5260208eed3": "P = VI", "e9a5f8df76ca3808a15685e15c5ba95b": "_{s.8 \\,}\\!", "e9a636fa75f0292a9d26f1ca071238e8": "\\textstyle\\frac{4}{3} \\pi r^3", "e9a64cf228a528603d944fd8cf1c3125": "\\ N_i = n_i N = n_i T_j n_j = n_i n_k \\tau_{jk} n_j.", "e9a66e8f1ee37e8b63216ce6d56b71ee": "y = -(x+1)^{1/2} + (x+1)^{3/2}", "e9a6a1bc5ed09bf4ed0879df6de6e21c": "a=e^\\alpha,\\ b=e^\\beta,\\ c=e^\\gamma,", "e9a6afb535495f586ee05b55be485d88": " \\lambda_n(t) \\approx {n \\choose 2} \\frac{2 \\beta}{I(t)}", "e9a6d33d7c630916cff7b572942ccbdd": "\n\\lambda = \\pi_1^{\\alpha_1}\\pi_2^{\\alpha_2}\\pi_3^{\\alpha_3} \\dots", "e9a6e1dd91063fa81b2f94aaa735527b": " d_m = \\frac { CI } { 2m } ", "e9a70977a00dfa8456f4d21ef3c3ce97": "b_0 = 2a_0 + 3a_1 + 1a_2 + 1a_3", "e9a7871c5d3274b0c8ef4bc4f8a6227c": "m^{e^f} \\equiv m^{k(n-1)}m \\neq m \\pmod{n}", "e9a7ca8e2f7ba5112681fcc58977b382": "n = (1-\\gamma)K + \\gamma", "e9a7ce7b7be583deb120c494d53d5e1d": "\\displaystyle{I-E^*=JUEU^*J.}", "e9a7e175d43ead6b4fe383f86872116d": "Dis: \\{0,1\\}^{n}\\times \\{0,1\\}^{d}\\rightarrow \\{0,1\\}^{m}", "e9a8eef9f86c830d5830dc20c3166919": "X \\in U", "e9a8f95fd02d74296cfa1b01b09f5025": "Q_{n+m}", "e9a90bb778297406c3a1cfc6e3870568": "m:\\textit{Mary}", "e9a939dbf3991ea000414cbfc001d2dd": " R_0 \\int_{0}^{T}{S^*(t)dt} < 1 ", "e9aa5134dd0362339dcfc0cc929b643e": "v=\\sqrt{T\\over\\mu},", "e9aa595227731858ec8aaeafaef360bb": "b \\Leftarrow a;", "e9aa602f9d9f9d5c0bfda872a48ea69a": "R=\\begin{bmatrix} R_1 \\\\ 0\\end{bmatrix}", "e9aa78f39713b8d5993e9558f9e40334": " \\eta_{E} = \\frac{\\dot{W}_{net}}{\\dot{m}_{fuel} \\Delta H^{0}_{T}} \\qquad \\mbox{(4)} ", "e9aabe40a928518332d41449ce12c6b2": "\\psi(\\Omega^2 3 + \\psi(0))", "e9aae19a372352367528cb8f0646c6fe": "L_{p} = 0, L_{pp} \\leq 0, \\partial_t(R) = 0, \\partial_{tt}(R) \\leq 0,", "e9ab364e4b06eb3af08d3297c3da8c09": "S = \\frac{E[R-R_b]}{\\sqrt{\\mathrm{var}[R-R_b]}}.", "e9ab46d59f1edb3a4e2caf2a7643ef08": "\\theta=\\Omega / m ", "e9abcbe3f0220807477dad82626c575e": "\\mathrm{Kn} = \\frac {\\lambda}{L}", "e9ac5deeb7662f5565a80cbc0d80f863": " f_i(s)=\\tilde{f}_i \\left( s_i,\\sum_{j=1}^n s_j \\right) ", "e9ac77c6cdd5727500c4d3bfe0ca7bf9": "|x_m|\\ge \\rho|x_{m+1}|", "e9ac7a6b7e3f2e8a3f334c614bd33b8f": "T_\\lambda u(x)=\\int_{\\mathbf{R}^n}e^{i\\lambda S(x,y)}a(x,y)u(y)\\,dy, \\qquad x\\in\\mathbf{R}^m, \\quad y\\in\\mathbf{R}^n,", "e9ad73d89555bfe93d0108940bbb6100": " \\operatorname{E}(Y_{i})=\\frac{1}{\\mu_i}\\big[\\ln(\\lambda_i/\\delta)+\\digamma(\\nu)\\big], ", "e9ad73f25f83a56bdd549dc493420105": "\\rho \\frac{D h}{D t} = \\frac{D p}{D t} + \\nabla \\cdot (k \\nabla T) + \\Phi", "e9ada128188a2476c8e932edb67b0291": " \\mathbf{x}^{(k+1)} = (1-\\omega)\\mathbf{x}^{(k)} + \\omega L_*^{-1} (\\mathbf{b} - U\\mathbf{x}^{(k)}). ", "e9adb8061347446f0b9251b2ad021287": "\\bar W", "e9ae540ae41c90d6b63b9946153e60a6": "\\bar{z} = \\overline{x+iy} = {x-iy}", "e9ae9179dee557cff2386d90d6b4e808": "\\frac{\\partial \\theta}{\\partial t}= \\frac{\\partial}{\\partial z} \n\\left[ K(\\theta) \\left (\\frac{\\partial \\psi}{\\partial z} + 1 \\right) \\right]\\ \n", "e9af1aef0c8a0856808a3325a01434a8": "z^2+x_1^2+x_2^2+\\cdots+x_n^2.\\,", "e9af3119f20a44c64226077cc7b68e32": "u(t) = A\\cdot \\cos(\\omega t + \\phi_m(t))", "e9af343acabd3ee319e7d1cae3b3d618": "a, b, c \\in X", "e9af37bfda225410aca39f30876bc42d": "\\mu(AB)>1.\\,", "e9af641219a93840a61dbabee7e482b7": "\\sqrt{s(s-a)(s-b)(s-c)}", "e9afd454a180c22fe7fc1e910f08eb15": " L_1(\\widehat{\\mathit{G}}) ", "e9afed3328785dc1ddb87a7c7610d5b7": "\\Psi = \\sum_n c_n\\psi_n \\,,", "e9b05de400cf300219f9837bf0ac3430": " 3\\rightarrow 3\\rightarrow 64\\rightarrow 2 < G < 3\\rightarrow 3\\rightarrow 65\\rightarrow 2.\\, ", "e9b0dbbf7fe3c58bd29d6f0041db1e61": "G =(V;A )", "e9b0dfc31b5d8b8a31c6ed82bf0fac06": "w (\\mathbf{r}) = \\sum\\nolimits_{\\mathbf{G}} w (\\mathbf{G})\\exp \\left[ i \\mathbf{G} \\mathbf{r} \\right]", "e9b0e47777a072642be3b4087fe4beb6": "\\sec(\\alpha)\\ge 1 \\,", "e9b0f17a97ec394555bc8def5e6bf1d8": "A=P(1+r/n)^{nt}", "e9b15202c0fe9c533d3fa87153a102ee": "\\frac{z}{\\sqrt{a^2 + b^2}}", "e9b19943f154c966601110d227b43cd0": "{\\rho} \\left (\\frac{{\\partial}u}{{\\partial}t}+u\\frac{{\\partial}u}{{\\partial}x}+v\\frac{{\\partial}u}{{\\partial}y}+w\\frac{{\\partial}u}{{\\partial}z}\\right )= {\\rho}g-\\frac{{\\partial}P}{{\\partial}x}+{\\mu}\\left( \\frac{{\\partial^2}u}{{\\partial}x^2}+\\frac{{\\partial^2}v}{{\\partial}y^2}+\\frac{{\\partial^2}w}{{\\partial}z^2}\\right)\\,\\!", "e9b1b11de37d9052db5d8df3d5407d3d": "[{\\rm L}]", "e9b20ecbcdaf550a061e520e6a824137": "\\lambda = \\frac{|\\Delta \\mathrm{Price}_t|}{\\mathrm{Volume}_t}", "e9b2398716c7e79537f308db11182574": "2 \\leq i \\leq r", "e9b2477666bba742b4c6506b58cfb059": "\\tfrac{N-b}{N} \\Big / \\text{average} \\left( \\tfrac{N-b-1}{N},\\tfrac{N-b-2}{N}, ...\\ ,\\max(0,\\tfrac{N-b-r}{N})\\right)", "e9b2ad38d27bd872dc34e07c50907a1a": " \\begin{bmatrix} a_1 \\\\ b_1 \\end{bmatrix} = \\begin{bmatrix} T_{11} & T_{12} \\\\ T_{21} & T_{22} \\end{bmatrix} \\begin{bmatrix} b_2 \\\\ a_2 \\end{bmatrix} ", "e9b2b60428b45bb7324bdbbd7dac8b33": "\\epsilon_k\\,\\!", "e9b2c13ace8168f5fd6d6d6cd1a57dae": "\\psi_2(x)= \\frac{1}{\\sqrt{k_2}} \\left(B_\\rightarrow e^{i k_2 x} + B_\\leftarrow e^{-ik_2x}\\right)\\quad x>0", "e9b2db0a716fe4db6f29bf3ac680db5d": "Y_c(u)", "e9b36cbf59e19d7cef9de8e2a65b4371": "\\mathcal{A}\\cong R[X_{0},X_{1},\\ldots,X_{n-1}]/\\langle p_{0},\\ldots,p_{N-1}\\rangle", "e9b36d4bb3771ed6b880764e3cb7b302": "\\text {Work}=2\\times0.1\\times980\\times0.57 =112\\text{erg}\\,\\!", "e9b392fc4640801db4196095c3050f27": "{X \\leftarrow \\Gamma}", "e9b4020eeb76a75200b47a978d97f432": " \\sigma_t = \\rho r^2 \\omega^2 \\ ", "e9b433973a50958723b2e3399943eb45": "\\{0, 1, 2, \\ldots, N-1\\}", "e9b44b74be9607893f86a385d8c9adbe": "\\Delta E_0 = IP - EA", "e9b45e236442e4962b8ded15c80c96c1": " \\alpha < \\gamma ", "e9b45f31f618f5d629f957a5b639e970": "\\scriptstyle X(t)-Y(t) ", "e9b4a7a754682788dcc26789662dc405": "0 < k \\le n < P", "e9b53fd98f609f4946da44c726e26e5e": "[\\psi]", "e9b55870a3f3ae34ce9242e178e80e22": " y(t) ", "e9b5fbf9ed2243088830b07a71aee7ec": "\\tan\\varphi = \\frac{2t}{1 - t^2},", "e9b660e8c57f83ffdf1451b338144035": "{d^2\\theta\\over dt^2} + {g\\over \\ell}\\sin\\theta = 0,", "e9b66e4d10e2704b065c3053baa97f54": "\\scriptstyle x - x \\times x \\equiv (1-x)\\times x", "e9b6f78b64f210c6c549c070716db0c6": "h(x) = \\int_0^x f(t) dt", "e9b75bd033796a63b2b65400f16133b6": "i,j^{th}", "e9b764ed2e4f2208d3ac3a0e7f14f3ad": "\\mathrm{sing}(X)", "e9b7822db240a296c08fbd2419c07576": "T=a + b \\int_{C} \\frac{ds}{W(s)}", "e9b839624bc453454e7b6909dfbd92e1": "\\Omega =2\\pi - \\arccos { {n_x} \\over { \\mathbf{\\left |n \\right |}}}\\ \\ (n_y<0).", "e9b89cc9c05c7aee5935c83800c9a495": "\\Gamma(\\tfrac14) = \\sqrt{\\sqrt{2 \\pi} S},", "e9b8ce70046876a6260dc6b83a7483b0": "\n\\mathbb{E}\\,\\mathbf{Z}_k = \\mathbf{0} \\quad \\text{and} \\quad \\Vert \\mathbf{Z}_k \\Vert \\leq R\n", "e9b8e253b6338bbdfe1343f250b8f110": "\\int\\!\\!\\!\\int\\!\\!\\!\\int_V \\, dV ", "e9b9f5c69b14692cfb034ead2761ba5c": "a^3\\;", "e9ba2d707f864363d433aa9675ca8e8e": "\\liminf_{n\\to\\infty}x_n\\leq\\limsup_{n\\to\\infty}x_n.", "e9ba7116a9c019c14d934e743da41785": "\\frac{E_b}{N_0} =\\frac{E_s}{\\rho N_0}", "e9ba775003ff05453819f6cec26abd6a": " \\frac{dS}{dt} = \\mu N - \\mu S - \\beta \\frac{I}{N} S, S(n T^+) = (1-p) S(n T^-) n=0,1,2,\\dots ", "e9baa9190003c9a1d7d05fdec92e3b4d": "(1,-g,0), (g,0,1), (0,-1,g),", "e9baf5f761862b5033718efba07718be": "R(L)", "e9baf7a3f28d61bb8f48eb58635a2f8c": "student \\leq bad", "e9bb712699424c43d27362a206833548": "C_{Ace}=-\\left(\\frac{1}{r_{Ace}\\omega_\\beta}-C_{ob}\\right)", "e9bb8494b7ae8bbae4674baa5dec66d2": "\\{M_{a}\\}", "e9bba564920fa61cfcabcb069cd14ae0": "\\mathfrak{so}(2,1)\\cong \\mathfrak{sl}(2,\\mathbb R)", "e9bbb48585e80bd9bc8b3398752662ea": "~t~", "e9bbcca060b77a342c14b9f02442933c": "\\scriptstyle \\leq1.2\\times10^{-5}", "e9bc00f22ade70d5d6520e8bcbdb4b32": " V \\in \\operatorname{FV}[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} L] \\equiv \\operatorname{de-let}[\\operatorname{let} V : V\\ V = \\operatorname{get-lambda}[V, E][V:=V\\ V] \\operatorname{in} L[V:=V\\ V]] ", "e9bc7ca75b2ed3b39f99fcfb846719f0": "(a,b,c,d)", "e9bcb6dd8534562c7d57c114e3d00561": "\\left \\lceil r \\right \\rceil", "e9bcd49ae09f0fa5155643dc364adbb5": "O_{r}", "e9bcecbd98c2f540394b878a0124477f": "\n\\mathbf{A}(\\mathbf{r}, t) = \\frac{\\mu_0}{4\\pi} \\left(\\frac{q\\mathbf{v}}{|\\mathbf{r}-\\mathbf{r}_s| (1 - \\boldsymbol{\\beta}_s \\cdot (\\mathbf{r}-\\mathbf{r}_s)/|\\mathbf{r}-\\mathbf{r}_s|)}\\right)_{t_r} = \\frac{\\mu_0 c}{4\\pi} \\left(\\frac{q\\boldsymbol{\\beta}_s}{(1-\\mathbf{n}\\cdot \\boldsymbol{\\beta}_s)|\\mathbf{r}-\\mathbf{r}_s|}\\right)_{t_r}\n", "e9bd06031487245defe155f24f9a83da": "\nP(\\lambda) = \\lambda^2 - \\mu \\lambda + 1.\n", "e9bd576d53b07e7da883ef05ef1bf9c7": "\\mathbb{OP}^2", "e9bdc7edc24da07f7a313a8a32ea87e5": "\\sigma_n\\,\\!", "e9be0970b4af13fcf0dddf914669e3d5": "\\underbrace{_\\,3x^2}_{\\begin{smallmatrix}\\mathrm{term}\\\\\\mathrm{1}\\end{smallmatrix}} \\underbrace{-_\\,5x}_{\\begin{smallmatrix}\\mathrm{term}\\\\\\mathrm{2}\\end{smallmatrix}} \\underbrace{+_\\,4}_{\\begin{smallmatrix}\\mathrm{term}\\\\\\mathrm{3}\\end{smallmatrix}}. ", "e9be3360c61801a4a81c28b01901cc15": "\\mathbf{A}\\left(\\mathbf{r}\\right)\\equiv\\frac{1}{4\\pi}\\int_{V}\\frac{\\boldsymbol{\\nabla}'\\times\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}V'\n-\\frac{1}{4\\pi}\\oint_{S}\\mathbf{\\hat{n}}'\\times\\frac{\\mathbf{F}\\left(\\mathbf{r}'\\right)}{\\left|\\mathbf{r}-\\mathbf{r}'\\right|}\\mathrm{d}S'", "e9beee28f52c0f07e1285b8b585479fc": "\\begin{matrix}g_{00} = -1+2U-2\\beta U^2-2\\xi\\Phi_W+(2\\gamma+2+\\alpha_3+\\zeta_1-2\\xi)\\Phi_1 +2(3\\gamma-2\\beta+1+\\zeta_2+\\xi)\\Phi_2 \\\\ \\ +2(1+\\zeta_3)\\Phi_3+2(3\\gamma+3\\zeta_4-2\\xi)\\Phi_4-(\\zeta_1-2\\xi)A-(\\alpha_1-\\alpha_2-\\alpha_3)w^2U \\\\ \\ -\\alpha_2w^iw^jU_{ij}+(2\\alpha_3-\\alpha_1)w^iV_i+O(\\epsilon^3) \\end{matrix}", "e9bf0608e454848d1fb921e19d185156": "\\operatorname{var}(X\\mid Y) = A-BC^{-1}B^T.", "e9bf279d054105c00aa837547df23747": "\\textit{dog} \\subseteq \\textit{carnivore}", "e9bf3a5e1f578e38fb2119200a3e67e2": "\\mathrm{comp}_{x,y,z}(f,g)", "e9bf4a9efc8d3cd004a3e9a0fa2c1a4e": "f(c^-)", "e9bf5f43e5d03f18a31407dc7a66ca85": "\\ln (a\\times 10^n) = \\ln a + n \\ln 10.", "e9bf7aa9ca5769731ac05cf8f636c434": "f(z) = \\frac{z-4}{e^z-1}", "e9bf907c17a69869850b73050ad86557": "\\alpha, t", "e9bf920ee8e85db466c482078a9bc6c0": "\\rho_{\\text{realized }} := \\frac{\\sum_{i\\neq j}{w_i w_j \\rho_{i,j}}}{\\sum_{i\\neq j}{w_i w_j}}", "e9bfd52195ab2f1fd311e81e2f8da160": "V_\\text{out}(t) = f(V_\\text{in}(t))", "e9bfeb5877f560d58378716d2f375c21": "\n\\bar{Y} = \\frac{1}{4}\\cdot{}\\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}\n", "e9c07dceaa8b4de9bad010a0a621d032": "V_{n+1}", "e9c0a07df66ba5eb16e47bc99b511e10": "F_f = {d_r \\over L}mg - {h_{cm} \\over L}ma", "e9c0c5d5f04be6750d66f62a794ccb92": "~E_{\\rm R} =\n\\left(\\frac{\\hbar\\omega}{2\\epsilon_0 V}\n\\right)^{1/2} \\!\\!\\!\\cos(\\theta) X~", "e9c12027a141313f72dcfacaf5ce045a": "f(x) = a + mx", "e9c23e8b61c531b0e1ed931a67ee7cce": "\\psi(\\mathbf{r}, t + t_0) = \\widehat{U}(t - t_0) \\psi(\\mathbf{r},t_0) ", "e9c2cab335434624b546cde6e9ed4261": " V(A,h) = \\frac{1}{3}A h", "e9c2cc77e4030cb00f993edf362d3ab8": "\\textit{SMM} = \\text{Median}( p_M, p_{M-1}, \\ldots, p_{M-n+1} )", "e9c30d55beda06e6e06a96fc1d63f281": "F_a \\propto \\vec{v}^2", "e9c359532579a425aeac3f39b24d9c71": "dv = e^x \\, dx \\Rightarrow v = \\int e^x \\,dx = e^x", "e9c360b14cc9e82b0ed8951f54390f94": "C \\ll \\sqrt{\\frac{\\kappa}{I}}\\,", "e9c3e481cff1a623a79aa3c0f0b71dc5": "L = \\textstyle{\\frac{1}{2}}(M + m)\\,\\!", "e9c3f9cc1a87e90cd98533a24138d69a": "~a~", "e9c4144b9c42e3b9cd4adda08883e110": " \\mathbf{a} \\cdot (r\\mathbf{b} + \\mathbf{c})\n = r(\\mathbf{a} \\cdot \\mathbf{b}) + (\\mathbf{a} \\cdot \\mathbf{c}).\n", "e9c4199f5c97956f14eb4bdaa3ae2af0": " v^2d\\bar{z}=|v|^2dz, \\,", "e9c44a98d9f83e014970ae037e361854": "f\\colon [0,1] \\to M", "e9c47e3bc5aecd720bea34ad7856255b": "x_{ij}=1", "e9c487423c2dad65f1f7648dc7264d4f": "|{\\Psi}\\rangle", "e9c4bfc6565083962947d8eda636fbf8": "g(x^2) = 2ax + [g(x)]^2", "e9c4c59fc1502a0942797105c2e303ca": "f_1(x,v,0) = g(v)\\exp(ikx)", "e9c4c9fb17f4e21aa762f39a7f5a8e36": "\\Box(\\varphi \\implies \\psi) \\implies (\\Box \\varphi \\implies \\Box \\psi)", "e9c4cb750ea4e34de0b96ec4a2e3873e": "\\phi::=\\Phi \\mid (\\neg\\phi) \\mid (\\phi\\and\\phi) \\mid (\\phi\\or\\phi) \\mid \n(\\phi\\Rightarrow\\phi) \\mid (\\phi\\Leftrightarrow\\phi) \\mid X\\phi \\mid F\\phi \\mid G\\phi \\mid [\\phi U \\phi]\n", "e9c4d034d38ec86d200468268da4dd77": "\\Epsilon_c(k_e) = {{\\hbar^2k_e^2} \\over {2m_e}}+\\Epsilon_G, \\Epsilon_h(k_h)={{\\hbar^2k_h^2} \\over {2m_h}}", "e9c4d16888e294a3c6259954a848cd81": "\\textstyle g(x) = (x^{2l-1}+1)p(x)", "e9c55065546516ea1ecc4f34c9b75f9b": " \\epsilon_{} ", "e9c56a0cc4fc49978c1d46092e342b9c": " \\begin{align}\n\\tilde{y}_{i+1} &= y_i + h f(t_i,y_i), \\\\\ny_{i+1} &= y_i + \\tfrac12 h \\bigl( f(t_i, y_i) + f(t_{i+1},\\tilde{y}_{i+1}) \\bigr). \n\\end{align} ", "e9c586e0dbd206d2854188b804f4c240": "\\frac{N}{N_0} = 10^{\\left(-\\frac{t}{D}\\right)}", "e9c59f2e5fb371bf27af2a4bbfcb35ea": "(G; c", "e9c5b9d8093d059895bdb542be5f3b3d": "{B}_{5}^{(1)}", "e9c5e7e5b1109c33cf871ac4a17d42a1": "\\left(\\nabla\\phi\\right)^2", "e9c632e184f8497f32c8cab5290aedff": " \\frac{d}{dt}=0 ", "e9c66658a28315d419ad3125e10177cd": "\\hat{p} \\,", "e9c69f08b37f37ef04893bf7a1d74b9b": "Sq^{2^i}", "e9c6f6db4322632894a23bfb26459895": "\\frac{1}{\\lambda_{\\mathrm{vac}}} = R\\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)", "e9c71d08772237a583414dfbf27e0dec": "T = \\frac{1}{2} \\begin{pmatrix} \\omega_\\text{x} & \\omega_\\text{y} & \\omega_\\text{z} \\end{pmatrix} \\begin{pmatrix} \nI_\\text{xx} & I_\\text{xy} & I_\\text{xz} \\\\ \nI_\\text{yx} & I_\\text{yy} & I_\\text{yz} \\\\\nI_\\text{zx} & I_\\text{zy} & I_\\text{zz}\n\\end{pmatrix} \\begin{pmatrix} \\omega_\\text{x} \\\\ \\omega_\\text{y} \\\\ \\omega_\\text{z} \\end{pmatrix} \\,.", "e9c73b8a6cade2ce5e70cc2d638d699c": "1\\cdot 2\\cdots (p-1)\\ \\equiv\\ -1\\ \\pmod{p}", "e9c76fd9d3e098d59e2b9fddad0b9a47": "(\\mathcal{L}_X g_{ab})_{;c}=0\n", "e9c77a8efa8679bdec1350373e1f7d40": "\\text{Holant}(G, f_u, f_v).", "e9c77fafe2630a833006566d8467c4a1": "\\mathrm{L} \\subseteq \\mathrm{SL} \\subseteq \\mathrm{NL}", "e9c7a4cc8313ec6bde05f985a4621538": "\n\\sum_j {T_{ij} = T_i } ,\\sum_i {T_{ij} = T_j } \n", "e9c81b0974e986a15ec3491ef042ec9a": "= \\begin{vmatrix} \\boldsymbol{i}&\\boldsymbol{j}&\\boldsymbol{k} \\\\ 0 & 0 & \\omega \\\\ v t \\cos \\alpha & vt \\sin \\alpha & 0 \\end{vmatrix}\\ ", "e9c8656abe02a22d325e9a20587698dc": "-3x\\,", "e9c877428579973ea6225eda0163b2fa": "0\\leq\\rho,\\sigma,\\Lambda\\leq I", "e9c8aa00842354c5f15ee88552b764a2": "\\mathbf{P}_{i}", "e9c8d5976d0f22c483c1fbe8b5fc33ea": "\\frac{M(\\lambda x, \\lambda y)}{N(\\lambda x, \\lambda y)} = \\frac{M(x,y)}{N(x,y)}\\,. ", "e9c8d6ba3d6223298da471c9c37c347a": "\\textstyle L^{p}", "e9c95ad321ecc16c889efdc9863fa68c": "h \\lbrack X \\rbrack = \\lbrace h(x) \\mid x \\in X \\rbrace. \\, ", "e9c96b6a2d3111c9363a5b2ba215b517": "\\hat{b} = \\frac{1}{N} \\sum_{i = 1}^{N} |x_i - \\hat{\\mu}|", "e9c97fea34db6a7662c03edff352baeb": "V D V^{-1} ", "e9c9a43946f54c54a6706da775648ff7": "L = \\{ x \\in \\mathbf{Q}|x^2 \\le 2 \\vee x < 0\\}.", "e9c9ccef0ec686f824809241f7343c13": "[SFP] \\equiv {kW \\over m^3/s} \\equiv {W \\over l/s} \\equiv {kJ \\over m^3} \\equiv {kPa}", "e9caaf0163a83a848dab0cc439517d87": " 1 \\to \\{\\pm 1\\} \\to \\mbox{Pin}_V(K) \\to \\mbox{O}_V(K) \\to K^*/K^{*2},\\,", "e9cb86b4416c0bced28906c868973572": " v = \\pm u\\sqrt{\\frac{3\\sqrt{2} - 2u}{6u + 3\\sqrt{2}}} ", "e9cb947b6fded17a63ca5b6464d91642": " |\\Psi\\rangle = \\int d^{26} p \\left (T(p) c_1 e^{i p\\cdot X} |0\\rangle \n+ A_\\mu (p) \\partial X^\\mu c_1 e^{i p \\cdot X} |0 \\rangle + \\chi (p) c_0 e^{i p \\cdot X}|0\\rangle + \\ldots\n\\right),", "e9cc121d24d0d02624149887d4f14340": "\\mathbf{P} -e\\mathbf{A} = \\frac{m\\dot{\\mathbf{r}}}{\\sqrt{1-\\left(\\frac{\\dot{\\mathbf{r}}}{c}\\right)^2}}\\,", "e9cc2ef41713580a370fa132dbc23fe3": "\\scriptstyle x[n-i]", "e9cc8e4ceb96d56c7b648275fc8cd6bd": "{12\\over2}=6", "e9ccb6afbf00d12dcb883324bfc32b31": "\\beta^{(h+1)} = n \\left( \\sum_{i=1}^n\\frac{x_i}{1-(1-p^{(h)})e^{-\\beta^{(h)}x_i}} \\right)^{-1},", "e9cce9d549dd050e5b47946aea385e20": "\\zeta(3)", "e9cd51d320b51cdba0335cf4ba84df5b": "b_4=a_1a_3+2a_4^{}", "e9cd54afbf23a3e174a5081c405aad4c": "\\mathbf{u}_{n+1}", "e9cda83c0a04851f619d976e5c228baa": " \n(\\mathbf{u} + \\mathbf{w})^2\n= \\mathbf{u} \\mathbf{u} +\n\\mathbf{u} \\mathbf{w} + \\mathbf{w} \\mathbf{u} +\n\\mathbf{w} \\mathbf{w}, \n ", "e9cdc95c2e45273e581e3314a0883278": "\\gamma = \\frac{1}{\\sqrt{1-v^2/c^2}}", "e9cdcfa0146620ab520a79e9be16e0af": "\\hat{X}^n_{DUDE}:\\mathcal{Z}^n\\to\\mathcal{X}^n", "e9ce159fdb09fb2a6bbf939e415136f8": "f^{-1}(\\mathrm{cl}'(A')) \\supseteq \\mathrm{cl}(f^{-1}(A')).", "e9ce4cf30058cc48fa03b1dbeb68cc48": "F_{net} = ma\\,", "e9ce60a0c67e6c548fa110f3c7f6e76e": "l=0\\,", "e9ce73432a1f0872a76e17df8f0db42b": "\\frac{\\varphi^{n+1} - \\varphi^{n}}{\\Delta t} = F(\\varphi),", "e9cec17b4e49f8fe48ae0ec823358680": "\\cos \\pi = -1 \\, \\! ", "e9ceeb8058726ba0a6fb37e1363a2cb3": "[t,t+\\Delta t]", "e9cefee261af6775b8ca094683c1be44": "0\\,i_0+1\\,i_1+\\cdots+n\\,i_n=n(n-1)", "e9cf38a02df4958c460bb792ed60b54c": "577 \\times 10^3", "e9cf400560ed2ac266fd1d37f7252949": "\\mathcal{D}\\big\\{x^{*n}\\big\\} = (\\mathcal{D}x) * x^{*(n-1)} = x * \\mathcal{D}\\big\\{x^{*(n-1)}\\big\\}", "e9cf784e2d918d780cccaec6b90d9d66": "S[g]= \\int {1 \\over 2\\kappa} f(R) \\sqrt{-g} \\, \\mathrm{d}^4x ", "e9cfb92b6633e4c7d73bf60989cf3e89": "\\displaystyle u_t=6uu_x-u_{xxx}+3ww_{xx}", "e9d095aca66298cce9fe72bf51099300": "\\mbox{div}\\,(\\mbox{curl}\\,\\vec v ) = \\nabla \\cdot \\nabla \\times \\vec{v} = 0", "e9d16a4a2931931309d9c2bb6f44ca32": " \\left| x(t) - x \\left( t + T(t) \\right) \\right| < \\varepsilon \\ ", "e9d17f3c0edeec7c5609d169ad6f0df4": "\\Delta G_B^\\circ ", "e9d1bd28eb2eb5d0cbd92939784de305": " d\\, ", "e9d1d3f9a908326c8e34106d72d72011": "M/X=(M^\\ast\\setminus x)^\\ast", "e9d227c5de353767826d0700ab34c5c7": "Y_{mn}(c,\\eta)", "e9d237483b4608417f3ff9185bcfcacd": "\\rightarrow\\!\\leftarrow", "e9d23779d292c4aad142035f0ce00b90": "\\mu_{n}", "e9d239c5dcf366780edc6ea1a678dfc9": "\\mathbf{r} =(r\\cos\\theta,\\ r\\sin\\theta)\\ ,", "e9d240e6c69462e2c4bbfa70c5aac5df": "v_\\parallel", "e9d266f8b13b4c312e9dd6e06ef20057": "\n \\bold{A} \\oplus \\bold{B} =\n \\begin{bmatrix} \\bold{A} & \\boldsymbol{0} \\\\ \\boldsymbol{0} & \\bold{B} \\end{bmatrix} =\n \\begin{bmatrix}\n a_{11} & \\cdots & a_{1n} & 0 & \\cdots & 0 \\\\\n \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n a_{m 1} & \\cdots & a_{mn} & 0 & \\cdots & 0 \\\\\n 0 & \\cdots & 0 & b_{11} & \\cdots & b_{1q} \\\\\n \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & \\cdots & 0 & b_{p1} & \\cdots & b_{pq}\n \\end{bmatrix}\n", "e9d2dfce12ba54b018218ba85466677e": "(\\overline{\\xi},\\overline{\\zeta}) = (0,0)", "e9d365d8d2cdd7bb09695d0d1fc85594": "g=f\\oplus h", "e9d3671f689e69cd888f5771f363c305": "p(x) = p_{orig}(ub*x)", "e9d36fcba3b7c674c8b1a1b2c559056a": "\\bar{m}^a\\partial_a=\\bar{\\Omega}\\partial_r +\\bar{\\xi}^3\\partial_{ y}+\\bar{\\xi}^4\\partial_{ z } \\,.", "e9d3979c7a438b0ca34aaf1dd033c548": "(1-\\epsilon) P(a) \\leq x \\leq (1+\\epsilon) P(a)", "e9d3aa47c8f81da6b754e296ecbe2222": "Q_1 \\cap \\dots \\cap \\widehat{Q_i} \\cap \\dots \\cap Q_n \\nsubseteq Q_i", "e9d43b97cf2d2039e56c835dbafd8005": "H\\left( Y|X\\right) ", "e9d499dee9407a759d71a5dabe4d41c7": "\\scriptstyle{\\mathbf{n}-\\mathbf{e}_i}", "e9d4b41b177516e65036f7ee84c60110": " F(x) = {F_0}\\left(\\frac{x}{x_0}\\right)^\\frac {\\log (F_1/F_0)}{\\log(x_1/x_0)}, ", "e9d4c33f3f547cc57f31007fd7654d44": "E_k = m c^2 \\left( \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} -1 \\right)", "e9d4d5ab8c37af2f52a049c35d20a7f7": "\nW =\n\\frac{1}{\\sqrt{d}} \n\\begin{bmatrix}\n1 & 1 & 1 & \\cdots & 1\\\\\n1 & \\omega^{d-1} & \\omega^{2(d-1)} & \\cdots & \\omega^{(d-1)^2}\\\\\n1 & \\omega^{d-2} & \\omega^{2(d-2)} & \\cdots & \\omega^{(d-1)(d-2)}\\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n1 &\\omega &\\omega ^2 & \\cdots & \\omega^{d-1}\n\n\\end{bmatrix}~.\n", "e9d5213f0379d9549078287ea2fabab6": "\\frac {H^2} {r^4} \\cdot \\left ( \\frac{d^2 r} {d\\theta ^2} - 2 \\cdot \\frac{\\left (\\frac {dr} {d\\theta} \\right ) ^2}\n{r} - r\\right )= - \\frac {\\mu} {r^2}", "e9d546ecce99d24d78deeb11ee5bf22c": "\\frac{\\delta(t-\\frac{r}{c})}{4 \\pi r}", "e9d549525024715a9ec8b2c1035bf5fe": "n_+", "e9d56b36467403a9f9a8055d4a19930b": " c_t=(1-R^{-2} b^{-1}) A_t - \\frac{u_1}{u_2} \\frac{(R^{-1} b^{-1} L^{-1})} {1 - R} + \\frac{(1-R^{-2} b^{-1})} {1-L^{-1}R^{-1}} E_t y_t ", "e9d5b53bef41dac682627d388a4cac33": "F_G(G/H, X) = X^H", "e9d5b794f5e7104f7c02b525a40da4f8": " \\frac{dV_r}{dt} + \\frac{9}{2} \\frac{\\mu}{\\rho_p r_p^2}V_r - \\left(1-\\frac{\\rho_f}{\\rho_p}\\right) \\frac{V_t^2}{r} = 0", "e9d5defb71d8a5ffff4e49578c133af7": "p(x;q)", "e9d63d9f2b20ea38057509e169f42377": "\\text{Gradient} = \\frac{N}{m^2}", "e9d64753bbbb0ad1b60bb57a821aaae4": "\\textstyle \\mathbb{Z}^n", "e9d67388aff4dd9dd8a5ff1f219e9a41": " \\int_0^{2\\pi} \\frac{P(\\sin(t),\\sin(2t),\\ldots,\\cos(t),\\cos(2t),\\ldots)}{Q(\\sin(t),\\sin(2t),\\ldots,\\cos(t),\\cos(2t),\\ldots)}\\, dt", "e9d680fe6f5a68c00300425e0df23a63": "\\textstyle n\\geq1", "e9d6d020eef3ea18b3250fe54be1ba83": "V_{L3-N}=\\sin \\left(\\theta-\\frac{4}{3} \\pi\\right) * V_P = \\sin \\left(\\theta+\\frac{2}{3} \\pi\\right) * V_P", "e9d7397e9772df2e11071d57657f4b22": "AFX", "e9d76b9f02282e343c0bcae85e800178": "\\Delta f=-\\frac{f_f}\\pi \\left( \\arctan \\frac{Z_{\\mathrm{F}}}{Z_q}\\tan \\left( \n\\frac{2\\pi f}{Z_{\\mathrm{F}}}m_{\\mathrm{F}}\\right) \\right)", "e9d7a49b944bb7dc7e19f68000a5c82e": "S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \\cdots)", "e9d7eaa7e30e43ae18e70c88e9ed4e09": "\\frac{-(2k-1) z^2}{k (2k+1)}", "e9d86dee15ad90b1c287945da524d67b": " \\int_k {1\\over (k^2 + m^2)} {1\\over ((k+p)^2 + m^2)} \\,.", "e9d89be5ce92234359747b6479df8773": "y = \\operatorname{Im}\\,(z) = \\dfrac{z - \\overline{z}}{2i}", "e9d8f4958f19b25d106e9d9f018ecd01": "f(X_1, X_2, \\dots , X_n) = X_1 \\cdot f(1, X_2, \\dots , X_n) + X_1' \\cdot f(0, X_2, \\dots , X_n)", "e9d90205a3ba2c420716857b46fee588": "\\partial^* E", "e9d912333186bd7c63bd09cc608b4fb0": "\\lim_{x \\to \\infty} \\frac{\\pi(x)}{x/\\log x} = 1.", "e9d957b709b3d2336ed15457b309ef1c": "P(t,T) = A(t,T) \\exp(-B(t,T) r_t)\\!", "e9d9a35111665393e21bd535c4ce38fd": "\\mathrm{sum} = 2 \\times C_{out} + S", "e9d9a8cef904b49d33e22711f1e47bea": "~ \\alpha(t) = e^{-i\\omega t}\\alpha(0)~", "e9d9ba6eedfb1ad2008aa2ee4e6be506": "\\frac{12}{5}", "e9d9c3089cfd1b679c68c922a0084b87": "\n\\lambda_p(x)=\\lim_{\\delta \\to 0}\\frac{1}{|B_\\delta (x)|}{P}\\{\\text{One event occurs in } \\,B_\\delta(x)\\mid \\sigma[N \\setminus(B_\\delta(x))] \\} ,\n", "e9da38be1791ff361acd35437fb2298c": "\\chi = 0. ", "e9da4b40cb350d44f46311efc2ddcab6": "\\operatorname{Ker}(A)", "e9da7713a0acfd1970a66059c74a14fa": "MCH = \\frac{Hb}{RBC}", "e9daffb620ceaf415fa41ea3ecb3ba2f": "t_{2} \\equiv 0 \\pmod 2", "e9db11b71e7a8c7aef40c2b42e48da0c": "Q_2 = \\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 7/25 & -24/25 \\\\\n0 & -24/25 & -7/25 \\end{pmatrix}", "e9db1d6e90961472bc527c0d7e0ff889": "0 \\in \\operatorname{core}(\\operatorname{Pr}_Y(\\operatorname{dom}F))", "e9db8ff190ea3337539b7b92a91de198": "\\sum_{n=0}^{\\infty} p_{ii}^{(n)} = \\infty.", "e9dbb9ca6a270d353b1a9ad50a4b137e": " \n \\gamma_2 = \\frac{(\\alpha + \\beta)^2 (1+\\alpha+\\beta)}{n \\alpha \\beta( \\alpha + \\beta + 2)(\\alpha + \\beta + 3)(\\alpha + \\beta + n) } \\left[ (\\alpha + \\beta)(\\alpha + \\beta - 1 + 6n) + 3 \\alpha\\beta(n - 2) + 6n^2 -\\frac{3\\alpha\\beta n(6-n)}{\\alpha + \\beta} - \\frac{18\\alpha\\beta n^{2}}{(\\alpha+\\beta)^2} \\right].\n", "e9dc0430387f82c077a3d86a0fa0bea8": "H L", "e9dcb900d60e04a1dfa5a79cd80b10f0": "\\tbinom81", "e9dcbfc39e25f58c5cb2d56a73628c83": "v_d", "e9dd313dcc4e94fd9d1fcbe172b9eafc": "\n \\begin{align}\n E_{11} & = \\frac{\\partial u_1}{\\partial x_1}\n + \\frac{1}{2}\\left[\\left(\\frac{\\partial u_1}{\\partial x_1}\\right)^2\n + \\left(\\frac{\\partial u_2}{\\partial x_1}\\right)^2\n + \\left(\\frac{\\partial u_3}{\\partial x_1}\\right)^2\\right]\\\\\n &= -x_3\\,\\frac{\\partial^2 w}{\\partial x_1^2} \n + \\frac{1}{2}\\left[x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_1^2}\\right)^2\n + x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2}\\right)^2\n + \\left(\\frac{\\partial w}{\\partial x_1}\\right)^2\\right]\\\\ \n E_{22} & = \\frac{\\partial u_2}{\\partial x_2} \n + \\frac{1}{2}\\left[\\left(\\frac{\\partial u_1}{\\partial x_2}\\right)^2\n + \\left(\\frac{\\partial u_2}{\\partial x_2}\\right)^2\n + \\left(\\frac{\\partial u_3}{\\partial x_2}\\right)^2\\right]\\\\\n &= -x_3\\,\\frac{\\partial^2 w}{\\partial x_2^2} \n + \\frac{1}{2}\\left[x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2}\\right)^2\n + x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_2^2}\\right)^2\n + \\left(\\frac{\\partial w}{\\partial x_2}\\right)^2\\right]\\\\ \n E_{33} & = \\frac{\\partial u_3}{\\partial x_3}\n + \\frac{1}{2}\\left[\\left(\\frac{\\partial u_1}{\\partial x_3}\\right)^2\n + \\left(\\frac{\\partial u_2}{\\partial x_3}\\right)^2\n + \\left(\\frac{\\partial u_3}{\\partial x_3}\\right)^2\\right]\\\\\n &= \\frac{1}{2}\\left[\\left(\\frac{\\partial w}{\\partial x_1}\\right)^2\n + \\left(\\frac{\\partial w}{\\partial x_2}\\right)^2\n \\right]\\\\\n E_{12} & = \\frac{1}{2}\\left[\\frac{\\partial u_1}{\\partial x_2} + \\frac{\\partial u_2}{\\partial x_1} \n + \\frac{\\partial u_1}{\\partial x_1}\\,\\frac{\\partial u_1}{\\partial x_2}\n + \\frac{\\partial u_2}{\\partial x_1}\\,\\frac{\\partial u_2}{\\partial x_2}\n + \\frac{\\partial u_3}{\\partial x_1}\\,\\frac{\\partial u_3}{\\partial x_2}\\right]\\\\\n & = -x_3\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2} \n + \\frac{1}{2}\\left[x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_1^2}\\right)\\left(\\frac{\\partial^2 w}{\\partial x_1\\partial x_2}\\right)\n + x_3^2\\left(\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2}\\right)\\left(\\frac{\\partial^2 w}{\\partial x_2^2}\\right)\n + \\frac{\\partial w}{\\partial x_1}\\,\\frac{\\partial w}{\\partial x_2}\\right]\\\\\n E_{23} & = \\frac{1}{2}\\left[\\frac{\\partial u_2}{\\partial x_3} + \\frac{\\partial u_3}{\\partial x_2} \n + \\frac{\\partial u_1}{\\partial x_2}\\,\\frac{\\partial u_1}{\\partial x_3}\n + \\frac{\\partial u_2}{\\partial x_2}\\,\\frac{\\partial u_2}{\\partial x_3}\n + \\frac{\\partial u_3}{\\partial x_2}\\,\\frac{\\partial u_3}{\\partial x_3}\\right]\\\\\n & = \\frac{1}{2}\\left[x_3\\left(\\frac{\\partial^2 w}{\\partial x_1\\partial x_2}\\right)\\left(\\frac{\\partial w}{\\partial x_1}\\right)\n + x_3\\left(\\frac{\\partial^2 w}{\\partial x_2^2}\\right)\\left(\\frac{\\partial w}{\\partial x_2}\\right)\n \\right]\\\\\n E_{31} & = \\frac{1}{2}\\left[\\frac{\\partial u_3}{\\partial x_1} + \\frac{\\partial u_1}{\\partial x_3} \n + \\frac{\\partial u_1}{\\partial x_3}\\,\\frac{\\partial u_1}{\\partial x_1}\n + \\frac{\\partial u_2}{\\partial x_3}\\,\\frac{\\partial u_2}{\\partial x_1}\n + \\frac{\\partial u_3}{\\partial x_3}\\,\\frac{\\partial u_3}{\\partial x_1}\\right] \\\\\n & = \\frac{1}{2}\\left[x_3\\left(\\frac{\\partial w}{\\partial x_1}\\right)\\left(\\frac{\\partial^2 w}{\\partial x_1^2}\\right)\n + x_3\\left(\\frac{\\partial w}{\\partial x_2}\\right)\\left(\\frac{\\partial^2 w}{\\partial x_1 \\partial x_2}\\right)\n \\right]\n \\end{align}\n ", "e9dd9013ec300ceba41484dfc2c9a876": "\\otimes ", "e9dd9946a402d289aa3f2bc2f02b968e": "x\\mathbb{Z}_n", "e9dd9a279f3ffe5b8bf415aea6322018": "\\nabla \\cdot (\\mathbf{a} \\otimes \\hat{\\mathbf{\\mathfrak{T}}}) = \\hat{\\mathbf{\\mathfrak{T}}}(\\nabla \\cdot \\mathbf{a})+(\\mathbf{a}\\cdot \\nabla) \\hat{\\mathbf{\\mathfrak{T}}}", "e9dd9d09a1e8eb205cda364edd84ab63": "\\Delta S_{system} = S_{in} - S_{out} + S_{gen}", "e9ddef220c1b5a01c3f48996cddd167d": "C^{n \\times n} \\otimes C^m", "e9de2328533cf5f6ae1d35d019035caa": "(F,B) \\colon (W,M,M') \\to (X \\times I, X \\times 0, X \\times 1)", "e9de424f090b7930fd65e08ebdffebe4": "2Na_{(s)} + Cl_{2(g)} \\longrightarrow 2NaCl_{(s)}", "e9de4e185c0e97d2e9ee8542a2c8da2e": "F_1 = 1,\\; F_2 = 1.", "e9de51399c2b20eaff5f3227acca7a55": "W_\\mu\\approx-2g^{,\\nu}_{[\\mu\\nu]}\\;", "e9de93a3eb205ad198f042feeaf6625c": "X_i^e = 0", "e9dead734bc3ef022eaa985fa204ffeb": "S = \\frac{2}{\\rho_1\\rho_2}\\,", "e9dee919920c31649edc25663bf65e4a": "\\beta \\ll 1", "e9df3c1516fedccc7400eefd829a632e": "Z[J] =\\int \\mathcal{D}\\phi e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{g\\over 4!}\\phi^4+J\\phi\\right)} = Z[0] \\sum_{n=0}^{\\infty} \\frac{i^n J(x_1) \\cdots J(x_n)}{n!} \\langle 0|\\mathcal{T}\\{{\\phi}(x_1)\\cdots {\\phi}(x_n)\\}|0\\rangle.", "e9df6f9e9ecf5eb905884cef3fe3d79c": "a_{13}*b_{1} ", "e9dfa3b8836fe2b1f7c5245e584ec3ea": "\\frac{2}{3}s \\;=\\; \\frac{2}{3} \\,+\\, \\frac{4}{9} \\,+\\, \\frac{8}{27} \\,+\\, \\frac{16}{81} \\,+\\, \\cdots", "e9dfaa22db2ed39e185e77480c724ed8": " \\mathrm{chord}\\ \\theta = 120 \\, \\sin\\left(\\frac{\\theta^\\circ}{2}\\right) = 60 \\cdot \\left( 2 \\, \\sin\\left(\\frac{\\pi\\theta}{360}\\right) \\right).", "e9dfaecd72c9eacc8d380f774025c72f": "\\frac{n}{n+1}\\,", "e9dfb00eed7618c9e5e5d540c14b14a3": "A_\\mathrm{e} = \\frac{3 \\lambda ^2 }{8 \\pi} = \\frac{G \\lambda ^2 }{4 \\pi},", "e9dfb52137d532fb3f703c2de05599af": " \\!\\ \\sum_{n=1}^{\\infty} \\frac{6}{n(n+1)(n+2)} = \\frac{3}{2}.", "e9dfd31cc505d51fc26975250750deab": "Spam", "e9dfd90c58489233689988ed6ed05c4f": "p_{n}^{1/n} > p_{n+1}^{1/(n+1)}", "e9e034de6cf425134003867b3038e820": "R'_{std} = R_{std} - R_{mb}", "e9e03c2b3c4cb6bb93b0576a19a97bc3": " \\mathbf{x} ", "e9e04003437039bcd13599889b8e9ccb": "r^e\\bmod N", "e9e05d8f960c6da9691e1f114ac69d32": "\\sum_{p=1}^{m}b_{ip}*2^{m-p}", "e9e08948083e80c34c1a9ca29dbc9f8b": "a_j \\ne 0", "e9e0d7d351ca57bb598d19f9376cd167": "\\lambda-\\mu", "e9e0fe35c55938a5d0824556d319a28c": "L^2 (G, d\\mu)", "e9e1094d1468e4d23e581e750de71f87": " L(y,y') = \\sqrt{ {1+y'^{\\, 2}} \\over y } ", "e9e13c1f4bca82622f3a04cef0b85d42": "f(x)=x^6-x", "e9e144b500eb02ff3094d1b9e73981d2": " \\mathbf{J}_p = \\frac{\\partial \\mathbf{P}}{\\partial t} ", "e9e186be9b7bf8ef58799d126041a5d1": "\\displaystyle{{1-(r/R)\\over [1+(r/R)]^{n-1}}f(x_0)\\le f(x) \\le {1+(r/R)\\over [1-(r/R)]^{n-1}} f(x_0).}", "e9e18bdb1934c8effa7680f74987e923": "x_1 > x_2 > ... > x_N", "e9e18f5c8d4dfbdc2260f88bf2950fd3": " \\int_C T_{ab} \\, k^a \\, k^b \\, d\\lambda \\ge 0.", "e9e19a6521ce57f7cf55dcd3173c1e02": "\\phi_e = (\\phi_E + \\phi_P)/2 , \\phi_w = (\\phi_P + \\phi_W)/2", "e9e1e0a50ae7ec7ccc32eb8c28320a97": " [n,R,R] = [R,n,R] = [R,R,n] = \\{0\\} \\ . ", "e9e2c3f5ee931b0af51a3a6b603dfd17": "\\psi^{(2k)}", "e9e2d0e7f31469e64f6434cd932d5861": "2 \\times 2", "e9e2f6a2982939dc8ea5edd4158048cb": "\\mathrm{2S_2F_2 \\ \\xrightarrow{180^oC}\\ SF_4 + 3S }", "e9e305863bbd335d6f149bd12422cdb6": "\\emptyset\\in\\mathcal{I}", "e9e345c34de829c54fc7763283f434fa": "\\displaystyle{g(z)=f(z^{-1})^{-1},}", "e9e359f750b915de38e402d91fab1db4": "P^{*}", "e9e3dd5d902e4ffa7860a45cb71e1c30": "D(x) \\hookrightarrow D(y)", "e9e40b999b75c9c6baea8f141ab62043": "(2n-1)(4ac-b^2)I_{n+\\frac{1}{2}} = \\frac{2(2ax+b)}{(ax^2+bx+c)^{n-\\frac{1}{2}}}+{8a(n-1)}I_{n-\\frac{1}{2}}\\,\\!", "e9e4449a266359d42db9116719d777d5": "u\\rho'(u) + \\rho(u-1) = 0\\,", "e9e4679b13d557c3510e0e363ecaaa0d": " s = \\begin{cases} b_k - \\frac{b_k-b_{k-1}}{f(b_k)-f(b_{k-1})} f(b_k), & \\mbox{if } f(b_k)\\neq f(b_{k-1}) \\\\ m & \\mbox{otherwise } \\end{cases} ", "e9e4d9283dda1115f77a25f1ae4f0dcb": "\\forall x,x'\\in X: d_Y(f(x),f(x'))\\leq\\omega(d_X(x,x')).", "e9e4e5077313dc415040bd99e3c1ac5c": "\n\tE^{\\pm}(k) = K^{\\pm} (\\Pi^{\\pm})^{4/3} (\\Pi^{\\mp})^{-2/3} k^{-5/3}\n", "e9e4ffc80a677b22b1ea86b85d494a36": "S \\Rightarrow_{r_1} AA \\Rightarrow_{r_1} AA \\Rightarrow_{r_2} SA \\Rightarrow_{r_2} SS \\Rightarrow_{r_2} SS", "e9e51613b4101b260eb6bc4b97e7e044": " L_*=\n \\begin{bmatrix}\n 16 & 0 \\\\\n 7 & -11 \\\\\n \\end{bmatrix}\n", "e9e52f5facb24bd81458933d14a10627": "\\textstyle |p(z)| \\le \\left(\\sum_{k=0}^n|a_k|\\right) |z|^n", "e9e54e68244d0bc630539caa779adb03": "\n\n100,000 \\,A_{\\stackrel 1 x :{\\overline 3|}} = 100,000 \\sum_{t=1}^{3} v^{t} Pr[T(G,x) = t]\n\n", "e9e557ff982e48a64bbc3ae6390ee0b2": "R_\\text{a}", "e9e5a6e7083a917eea0dabe27a19c24f": "\\frac{N_0}{2}", "e9e6384df4e49a96d89968137fead1d5": "V_x = V\\sin\\theta", "e9e63bef5b38623e8f7637e296f8b811": " w''(t) = \\frac{3}{2} w^2, \\quad w(0) = 4, \\quad w'(0) = s", "e9e63f6c5e124fda34f0e24ee5db788d": "\n \\sum_{j_3} (2j_3+1)\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6\n \\end{Bmatrix}\n \\begin{Bmatrix}\n j_1 & j_2 & j_3\\\\\n j_4 & j_5 & j_6'\n \\end{Bmatrix}\n = \\frac{\\delta_{j_6^{}j_6'}}{2j_6+1} \\{j_1,j_5,j_6\\} \\{j_4,j_2,j_6\\}.\n", "e9e643642117ec6cd1df0b68f640c9c6": "\n0<\\mu<\\frac{2}{\\mathrm{tr}\\left[R\\right]},\n", "e9e67473388c33da8ea91deaef7bc2b8": "|\\psi\\rangle = \\frac{1}{\\sqrt{N}} \\sum_{k=0}^{N-1} |k\\rangle", "e9e67574b30ac61a727dc6358d5a84cf": "C_i = E_K(P_i \\oplus C_{i-1}), C_0 = IV", "e9e6903cfae2643a1b5c0277a832c768": "\\rho_b = - \\nabla\\cdot\\mathbf{P}\\,.", "e9e6b6f930386cc31fa73212f8ac0024": "R=S\\left( {\\frac{{E{t}}^{2}}{\\rho}} \\right)^{\\frac {1} {5}}", "e9e6b7e657c34861355b4fde71a4d182": "\\begin{align}\n \\hat{f}(t) &= \\sum_{i=1}^{N} A_i e^{\\sigma_i t} \\cos(2\\pi f_i t + \\phi_i) \\\\\n &= \\sum_{i=1}^{N} \\frac{1}{2} A_i e^{\\pm j\\phi_i}e^{\\lambda_i t}\n\\end{align}", "e9e6c60c9cd2a633c5cf812a5999b7ad": "\\lambda_r", "e9e714a8c5097eb1f89597e7cadc8e41": " \\min\\left(\\int f^+ \\, d \\mu, \\int f^- \\, d \\mu\\right) < \\infty. ", "e9e7236860a880edb6dccb6c0c7bb2a3": "h_{01}", "e9e75f59512afbd0508f3deb3cb46d54": "b^{\\downarrow}_i", "e9e785478fdc879138a827b033838fc1": "\\mathfrak{U},", "e9e7a9fd53b6dd5a98e7b5b244117206": "A + B \\overset{M}{\\to} C", "e9e85e9b79c3c8b2c0db43b19a69981f": "\\bar{f}(n)", "e9e87d34c99cfd330115c08b5a2524da": "\\tfrac{3}{2} r_0", "e9e8b665f4d4085638650b781818d24c": " \\bowtie ", "e9e940cb78faa948d7cfba8549b23d32": "\\displaystyle N", "e9e9544d52c4e5231b3ee5976a032588": "|x_n-x| 2\n\\end{align}", "e9f162b513bc2506d3ee113112ef3cb0": " k_1^{d_1} \\cdots k_n^{d_n} ", "e9f170857a1f6e4930a58d31c668cbd9": "r.\\ ", "e9f1fb23f74c891cc01b2dd5285928e5": "\\left(\\nabla^2-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial{t}^2}\\right)u(\\mathbf{r},t)=0.", "e9f1fb520dba48018ab725f7592a0822": "E(Q)", "e9f2082f1b4de183042d425d3931ba7e": "U_E = q\\frac{1}{4 \\pi \\varepsilon_0} \\left(\\frac{Q_1}{r_1} + \\frac{Q_2}{r_2} \\right) ", "e9f265002781cf8b2874f9b25c9906e0": "\\tau_{\\alpha,\\beta}", "e9f27ddd2b1df5358afac8ee71ea3a40": "\\dot{x_i}=x_i\\left(\\left(Ax\\right)_i-x^TAx\\right),", "e9f2a64939b20d9c2d7af4ddd95ea7b7": "\\hat{a}^\\dagger", "e9f3511c6cb784c1692ae48efb8bd406": "|x\\rang|q\\rang \\overset{U_{\\omega}}\\longrightarrow |x\\rang|q\\oplus f(x)\\rang", "e9f396dc66edf87c720ce990f97250b4": "\\tbinom{n+1}{1}", "e9f3a0bfe0f578d49fb39f5bae53d484": "1\\leq\\,j\\leq\\,n", "e9f3ae7663065a4283465e5531c9f624": "e\n\\,", "e9f43f9a79c585c93b2e1f525b2e6dcb": "y_1,", "e9f4882c5d936e884e6caf0e24f1836f": " n \\mid m ", "e9f49df7cdfe1d1db0cd7f63a873e7c2": "2^{-k}\\mathbf{Z}^n= \\left \\{2^{-k}(v_1,\\dots,v_n):v_j \\in \\mathbf{Z} \\right \\}", "e9f4a2832f599548d65bba20075628fe": "\n\\mathbf{C}' = \\int_{\\Delta'} \\mathbf{x}'\\mathbf{x}'^{\\mathrm{T}} \\, dA'\n= \\int_{\\Delta^0} \\mathbf{A}\\mathbf{x}^0\\mathbf{x}^{0\\mathrm{T}}\\mathbf{A}^{\\mathrm{T}} a\\, dA^0\n= a \\mathbf{A} \\mathbf{C}^0 \\mathbf{A}^{\\mathrm{T}}\n", "e9f4ad102474fc9b3761a7126256619c": "{ P }_{ out }=\\left( \\sigma { { T }_{ eq } }^{ 4 } \\right) \\left( 4\\pi { { R }_{ p } }^{ 2 } \\right) ", "e9f4b393016348f79420168383b05ee7": " p(r) \\approx P ", "e9f4cfa51d3855d0437b1e58d3f89d90": "K^M_n(k)", "e9f525416e9f354aa1263b0f3ab0b691": " E_{s} = - \\frac{\\varepsilon\\zeta (\\rho -\\rho _{0})\\phi _{p} }{\\sigma ^{\\infty }\\eta } gH(\\kappa \\alpha )+\\vartheta(\\zeta ^2)", "e9f53ce80875693c637e728572350d5f": "PN^2=\\frac{ON^3}{NA}.", "e9f55f659bec93e84662d80b5ccb9bf3": "\\mathbf{r}' = \\mathbf{r} + \\left(\\frac{\\gamma-1}{v^2}\\mathbf{r}\\cdot\\mathbf{v} - \\gamma t \\right)\\mathbf{v}\\,. ", "e9f587fb32eabb4205559eaec634aa31": "X_{rms}", "e9f592194f5d7182e87e88760daf57f9": "M \\otimes_\\mathbf Z \\mathbf Z/n = M/n.", "e9f5b0e25831acb3c73333bacaccda42": "x = \\omega^2 \\cdot r_0/g_0", "e9f64bd9983c7ea69928337b066a0eb0": "\\int_Y f(y)\\,d\\rho(y) = \\int_X f\\circ\\phi(x)w(x)\\,d\\mu(x).", "e9f652db7acc6574fd2cb2360beb38bf": "0 < \\operatorname{Re\\,} \\alpha, \\operatorname{Re\\,} \\beta < n,\\quad 0 < \\operatorname{Re\\,} (\\alpha+\\beta) < n.", "e9f6639695df1d8f82e0a8133d6c2aae": "(W_R^2 B+WW_R B_R) + (W^2 B - W W_R B_R)= \\left(W_R^2 + W^2\\right)B = B,", "e9f678ed43a129f9228fb05ac6bc5ecf": " 4 \\times 4 ", "e9f6a25e8aab6659022f37cd450944a3": "K = \\{(H \\cdot S)_{\\infty}: H \\text{ admissible}, (H \\cdot S)_{\\infty} = \\lim_{t \\to \\infty} (H \\cdot S)_t \\text{ exists a.s.}\\}", "e9f6c048a17a9b10f0d536bcb8ed13b5": "x_i\\sim\\, x_j", "e9f6cbd9c2ec1a249c099fa2d35b8b96": "S _0", "e9f6fda23744c63862371641b5ad2212": "\\delta_{q+1} \\circ \\delta_q = 0 ", "e9f7030086ba7b26d87ef2e39b60f2ba": "f(\\hat{x},\\theta) > f(x,\\theta)", "e9f7f72abdb951cb6a38d2ec8ed74ba2": "\\frac{\\lambda}{N}", "e9f8256b93f838b7bd03d9f88d048e04": "f(y|x;\\theta)", "e9f832d78a3fffe61abe71249f43e689": "\\ z = \\,", "e9f8b4478acf382ccd256b42817b4f82": "\n\\Phi_2 (\\begin{bmatrix} \\rho(F_1) \\\\ \\vdots \\\\ \\rho(F_n)\\end{bmatrix}) = \\sum _i \\rho (F_i) R_i.\n", "e9f8e66b8070240e6e799d5132400ae4": "P_{r_{dBm}}=P_{t_{dBm}}+ 10 log(G h_t ^2 h_r ^2) - 40 log(d)", "e9f961e44bf2d30f6cc2fdf3262ff36f": "T ': {B_2}^* \\to {B_1}^*", "e9f99d79eff76a49780a0bdafdd92cc6": "\\mathrm{d}n/\\mathrm{d}T", "e9f9ab33548c552b0da891124b02592b": "\\partial_\\mu J^\\mu = \\sigma", "e9f9af70a27f80c14a424163e34d59dc": "m_1=\\frac{m_2|\\boldsymbol{a_2}|}{|\\boldsymbol{a_1}|}\\!.", "e9f9f582cff37acac59c1933ddb6e743": "\\psi=\\begin{pmatrix} \\psi_R \\\\\\psi_L \\end{pmatrix},", "e9f9fa8b8e6621d22b2af7406f16b1a9": "\\sigma_h", "e9fa01108dbb482c37498a05d2acb801": "\\lambda = \\frac{v}{f},", "e9fa31e85d5b04ffaa9222a5d4cfe4f2": " I_P = \\int_V \\rho(\\mathbf{r})\\,\\mathbf{r}^2 \\, dV.", "e9fae02728084b0ced97269595a3a0c5": "\\rho\\in (-1, 1)", "e9faed23177dcc8a31dc81b467ba4476": "\\binom{n}{n_1\\ n_2\\ \\dots\\ n_p},", "e9faf8320c36aca9763b377900b0d920": "\\begin{align}\n(dx)^2 - (dX)^2 &= dx_j\\,dx_j-\\frac{\\partial X_M}{\\partial x_r}\\frac{\\partial X_M}{\\partial x_s}\\,dx_r\\,dx_s \\\\\n&= \\left(\\delta_{rs} - \\frac{\\partial X_M}{\\partial x_r}\\frac{\\partial X_M}{\\partial x_s} \\right)\\,dx_r\\,dx_s \\\\\n&=2e_{rs}\\,dx_r\\,dx_s\n\\end{align}\\,\\!", "e9fb3f07029552788182b49b884a6538": "\\alpha=\\frac{x_1+x_2+\\ldots+x_n}{n},", "e9fba1ada50bb85a51096cf4db6e69db": "\\mu_{a, r} \\xrightarrow[r \\to 0]{*} \\theta H^{k} \\lfloor_{P},", "e9fba5727da468788f78dd8e5327be7b": " S_{2m+1}-S_{2m}=a_{2m+1} \\geq 0 ", "e9fbba356a10fb42215c459d83203a53": "V = |\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c})| = |\\mathbf{b} \\cdot (\\mathbf{c} \\times \\mathbf{a})| = |\\mathbf{c} \\cdot (\\mathbf{a} \\times \\mathbf{b})|", "e9fbcda2ddbb0d6700462585938ff0f4": "1: \\quad term \\quad *= \\quad {a_n}^2", "e9fbd3a3edec852fe5d7c7b7fb51d98c": " D_{ \\Gamma } = (1 - \\Gamma^*\\Gamma )^{\\frac{1}{2}} \\quad \\mbox{and} \\quad D_{\\Gamma^*} = (1 - \\Gamma \\Gamma^*)^{\\frac{1}{2}}.", "e9fc0bca9fe5912b91453ca7a0e682b8": "M_{ij} = \\langle x_i, x_j\\rangle", "e9fc3ea36072e61be762aa8a089a55a7": "{DE}_{10}", "e9fc59b6e16c8d0333c17e4ea33aa87c": "P_{(i)}\\leq\\frac{\\alpha}{m-i+1}", "e9fcc3f462c609621aa2b8ad6b308de0": "\\nabla P=-\\left(\\frac{k}{\\mu}+D_K\\right)^{-1}q", "e9fce2b384cc4409e705acd47974d930": " L(\\mathbf{V})=L(a\\mathbf{v}+b\\mathbf{w})=aL(\\mathbf{v})+bL(\\mathbf{w}).", "e9fd26e080206139073c52db95d32462": "\\mathfrak{p}_1 \\cap A \\ne \\mathfrak{p}_2 \\cap A", "e9fd5992674941beb85ad9850f4ff07b": "u(0,x)= u_0\\ \\left({\\frac{x-x_3}{x_1}}\\right)^2", "e9fd7537bf4851dfb9bd1c3ef1c26939": "K=LGD * \\left[N\\left(\\sqrt{\\frac{1}{1-R}} * G(PD) +\\sqrt{\\frac{R}{1-R}}*G(0.999)\\right) - PD\\right] ", "e9fd82bd2962a33d7362a70b86c80548": "N \\lambda\\mathbf{a} =K\\mathbf{a}", "e9fd8ebdd5b72c1e3d868449cf1b69b9": "s\\geq s'\\leftrightarrow \\theta\\leq \\theta'", "e9fdcea338494e4d3e2117fcfcd7b003": "L(q)=1-24\\sum_{n=1}^\\infty \\frac {nq^n}{1-q^n}=E_2(\\tau)", "e9fe0006ed16206b7c948b24662532fe": "\\frac{d}{dt}A = \\frac{1}{i\\hbar}[A,\\widehat{H}]+\\frac{\\partial}{\\partial t}A\\,,", "e9fe295c38cf48a487562df323d6569f": "b_n", "e9fe32a16f6889c67857db3770b964a8": "k^{1-s} \\operatorname{Li}_s(z^k) =\n\\sum_{n=0}^{k-1}\\operatorname{Li}_s\\left(ze^{i2\\pi n/k}\\right).", "e9fe372f7a304840bb2cc77839c55f92": "\\mathrm{SCl_4 + 2HNO_3 + 2H_2O \\ \\xrightarrow{}\\ H_2SO_4 + 2NO_2\\uparrow + 4HCl }", "e9fe420ef23de7e89090bdd93d75f1a5": " 0 = \\sum_{n=1}^e \\left(\\sum_{m=1}^d b_{m,n} u_m\\right) w_n,", "e9fe4acc5b16f6996332cc0e804713e8": "\\inf \\theta \\le 35/108", "e9fe8998375a51bf43c0ac568349511a": "\\mbox{MIRR}=\\sqrt[n]{\\frac{FV(\\text{positive cash flows, reinvestment rate})}{-PV(\\text{negative cash flows, finance rate})}}-1", "e9fe947d0796f7803ce141ac6116df11": "\\operatorname{dCor}_n(X,Y) = 1", "e9fe9ecc93d99f563d922cb1638b7561": "\n (G + H)~(\\sigma_1^y)^2 = 1 ~;~~ (F + H)~(\\sigma_2^y)^2 = 1 ~;~~ (F + G)~(\\sigma_3^y)^2 = 1 \n ", "e9fea215eeb09ef7ad56f0afb587bf57": " E_3(x,\\alpha) = x^3 - 2x\\alpha \\,", "e9fec4eca25960b11c4dd1f37aa08151": "m v_{\\perp} \\rightarrow p_{\\perp}", "e9ff0e7df2a7d1cd7fd0eb5280b31ee1": "\\frac{{dY}}{{dx}} = \\frac{1}{h}\\left( {a_1 + 2a_2 z + 3a_3 z^2 } \\right) ", "e9ff227570924a1bc3bc94b6d3b027ae": "\\omega_f(t;x):=\\sup\\{ d_Y(f(x),f(x')): x'\\in X,d_X(x,x')= t \\},\\quad\\forall t\\geq0.", "e9ff3d956555d971d3da402c9b937a43": "\\hat S_1\\cdot\\hat S_2\\,,", "e9ff657c9a757070ca76aa1e1a9cf53b": "\\int_{0}^{\\infty }\\frac{x}{\\sinh ax}\\ dx=\\frac{\\pi^{2}}{4a^{2}}", "e9ffd98d9f8f0f749ea7416ef79aacb7": "\\psi = \\theta_1 - \\theta", "ea00b8ca8ae49197302709ee8a2c7a04": "y_i - \\overline{y}", "ea00ddf7503580d927f9aabe434d2f35": "\\alpha_p = \\sum_i g_i(p) (dh_i)_p\\,\\!", "ea0103cfdbd4242f129f21938633345f": "h \\colon M_0 \\rightarrow M_1", "ea011177439724d319a6db2e3d6334a9": "\\det \\begin{pmatrix}x-x_1&y-y_1\\\\x_2-x_1&y_2-y_1\\end{pmatrix} = 0.", "ea01203591cfec2683eaf18c2f6b5607": "\\scriptstyle \\leq8.4\\times10^{-8}", "ea014d78c3dbd16d51d14346961416c8": "v=q^a\\mathbf{e}_a+p_a\\mathbf{f}^a+uE", "ea016d0257c181b5fe27df955b65077c": "n - \\frac{2 (\\lambda,\\alpha_i)}{(\\alpha_i,\\alpha_i)} \\ge 0", "ea018a8fd9378aada30d000721d0aeb0": "n_\\mathrm{A} = n_\\mathrm{B} \\frac{R_\\mathrm{B}-R_\\mathrm{AB}}{R_\\mathrm{AB}-R_\\mathrm{A}} \\times \\frac {x(^{j}\\mathrm{A})_\\mathrm{B}}{x(^{j}\\mathrm{A})_\\mathrm{A}}", "ea021d52603d2a0500ec17f15ef354c6": "f(A) = f(B^{-1}AB) ", "ea023da8d51962b74ff428fb90519f96": " Z=\\frac{nK^2}{4(1+K^2/2)} ", "ea02750e58cc2b8a6bf804a58885acee": " H(\\xi,\\eta) = \\frac{R(\\xi,\\eta)}{ \\left\\vert R(\\xi,\\eta) \\right\\vert}", "ea02a49ef607049695e8b37ad0978c2f": " \\lambda^2 - 2g \\lambda + 1 = 0 ", "ea02f5f09fa635e33c4857ec99404ad9": "2n-1", "ea0301e468558043e80ab4cf67220823": "D = N_1 D_2 + N_2 D_1.", "ea032fa577f74b17e34a95a06774fa9b": "\\int\\limits_{V_f} \\rho(\\vec x, t) \\, d\\Omega = 0", "ea035f5b69d5f534ee3134ae3f3eb248": " (x,y) = (x_0,y_0)-f_0*(x_1-x_0,y_1-y_0)/(f_1-f_0)\\,", "ea0374f0fbdc01658199ed836d17f247": "\\hat{H}_\\text{D} = -\\boldsymbol{\\mu}_\\text{I}\\cdot\\mathbf{B}", "ea03b85346a2f5dc484aa0c34b1491f5": "q_1 = a_2 q_2 + 1", "ea042dab0acde25e8643d0f8889772e3": "\\operatorname{Li}(x) = \\int_0^x\\frac{dt}{\\log(t)}.", "ea048f26c701a29c9d22b98a926798e7": " Y = 2 \\bar Q.", "ea04a4ab7c1ff8f92931f3e19577cb5b": "\\left|\\frac{a_{n+1}}{a_n}\\right|\\ge 1", "ea04b0fb1ed2a1f8e4d268fcead8eaa6": "B \\subseteq X", "ea04b37c4a5dc28665224328196698cd": "j^{\\pm}=j^{1}\\pm i j^{2}", "ea05138e57d3478ed949079adfadd829": " \\frac{1}{|\\mathbf{x} - \\mathbf{x'}|} = \n\\frac{1}{\\pi\\sqrt{RR^\\prime}}\n\\sum_{m=-\\infty}^\\infty e^{im(\\varphi-\\varphi^\\prime)} Q_{m-\\frac{1}{2}}(\\chi)", "ea0525773fc2ffb970a1b91d76e806ba": " \\bar{w}_{1L}(s;L)", "ea055b9b9e516686bf3f6f925afd8d5c": "\\Gamma_\\gamma(s)=0", "ea056cf707aa29e243aa968aaf8a5983": "|\\rangle", "ea05b7a0e3c182a38fc1f0b1d1df424a": "\n\\overline{ \\hat{ \\phi } }(\\boldsymbol{k},\\omega) = \\hat{ \\phi }(\\boldsymbol{k},\\omega) \\hat{G}(\\boldsymbol{k},\\omega)\n", "ea05defa177ca83e2de52f34823e6207": "\\star \\mathrm{K}^{3-4} \\; \\mathrm{C}^{3-4}", "ea05ebceaced3ca76e627a20cd52c2b4": "\\Psi_{jk}^{i} = \\frac{1}{3} \\left( \\frac{\\partial \\Phi_{j}^{i}}{\\partial v^{k}} - \\frac{\\partial \\Phi_{k}^{i}}{\\partial v^{j}} \\right).", "ea0646cb4b4d5212e3b33dcc5ff3912c": " f : \\mathbb{R}^2 \\to \\mathbb{R}", "ea067d6f6daf5cf5be19bb4af24df2cc": "\\displaystyle{T=\\sum \\lambda_i U(g_i).}", "ea06843442afdf518370f9ab77d6b643": "z_1, \\ldots, z_n", "ea0752d8aac1d2f622519e607f5c99ff": "\\mathbb E[N(x)] = M(x)", "ea0756669f33a492c5e0d344e3c25fff": "I=\\{ \\mathsf{a, b, c,} \\ldots\\}", "ea0765dd7c501f47f065cc98a73b3dee": "\\alpha<\\biggl\\{\\frac{an^{*}+bn}{m}\\biggr\\}\\le\\beta", "ea07bc658e59eef816f46cc4961edc49": "P^{(i)}_{0}(\\xi_{i})=\\frac{1}{Z_{0}}e^{-\\beta h_{i}^{MF}(\\xi_{i})}\\qquad i=1,2,..,N", "ea0812853912e7d93b751afc6a462974": "\\mathcal G \\begin{pmatrix} \\pi^+ \\\\ \\pi^0 \\\\ \\pi^- \\end{pmatrix} = \n\\eta_G \\begin{pmatrix} \\pi^+ \\\\ \\pi^0 \\\\ \\pi^- \\end{pmatrix}", "ea084229e600c4f10ff5039bcb9f3e1e": " N(E)\\sim \\frac{\\sqrt{E} }{2\\pi } \\log \\left( \\frac{\\sqrt{E} }{2\\pi e} \\right) ", "ea08b7f7f736b88c1d3e2f3b9ac659e5": "\n\\sigma_{red}=\\sqrt{({\\sigma}^2) + 3({\\tau}^2)}\n", "ea09147ec4acfba1df6b8de7140d49e4": "\\hat e", "ea09325f4d81c3e1c411d8d3c95457c6": "T = 2 \\pi \\sqrt{\\frac{l}{g}}", "ea09352e81254c4d56fb04627f29970d": " \\sqrt{\\frac{2D*K}{h}} ", "ea0989d9a8bc59fcb9f4239836ef945e": "\\nabla_{\\partial_a}\\partial_b=\\Gamma_{ab}^{c}\\partial_c", "ea09e3c09223e0302054df73a16405b8": " t_0, \\ldots ,t_{n-1} ", "ea0a232726b49e1e99b65766dbb91131": "\\mathrm{ACA}", "ea0a34704841156f2d7311063eac3511": " f(x) = \\prod_{i=1}^n x_i ", "ea0b13842ff4c214ab1015651c1b59c8": " \\{\\mathbf{Shape}=square, \\mathbf{Color}=blue, \\mathbf{Size}=small\\} \\Longrightarrow \\; \\{\\mathbf{Class}=good\\} ", "ea0b1639fe5e92a981f8527534259019": " S(1 + \\sum_i \\pi_i) = 0", "ea0b1bdefa982f55d1d1df1c5bc2da31": " (x+1)^{n+1} = (x+1)(x+1)^n = x(x+1)^n + (x+1)^n = \\sum_{i=0}^n a_i x^{i+1} + \\sum_{i=0}^n a_i x^i.", "ea0b81372cfa164336533ab997c73c82": "\\mathbf{C} = [A]\\mathbf{C} + \\mathbf{d} - (\\mathbf{d}\\cdot\\mathbf{S})\\mathbf{S}.", "ea0c57c4de2a96c54d2310fbfeff8a4a": "u=v", "ea0c7e207b62d7ad7b49024cd04a00c4": "\\lambda_j = \n\\left\\{\n\\begin{array}{lr}\n-\\frac{j^2 \\pi^2}{L^2} & \\mbox{j is even.}\\\\\n-\\frac{(j+1)^2 \\pi^2}{L^2} & \\mbox{j is odd.}\n\\end{array}\n\\right.\n", "ea0d98aa002d29974a282e2c713ac40e": "1 x = x 1 = x", "ea0da39cf51a0fc8e8a6bdc454922b98": "\\textstyle \\left( a,b\\right) ", "ea0df1fb5a0a0537eae63754e66e8dc4": "ds^2= {4(dx^2 +dy^2)\\over (1-x^2-y^2)^2}.", "ea0e89a877f8156736011449ece89a7d": "t_a \\mathcal{A}^a_\\alpha = \\mathcal{A}_{\\alpha} ", "ea0e8cae8e9c0bac59de44b2be4f9f95": "\\gamma _1", "ea0efb67507c93d3a78f1d5616b82bca": "\\mbox{BFL} = \\frac{f_2 (d - f_1) } { d - (f_1 +f_2) }.", "ea0f62dd873d9d2afb19facab1a4b3c0": "R_2 = -\\frac{2DB}{F - B - D}", "ea0fc7f60cf86ae9a7135bdbc61124c3": "h(r) = \\frac{\\Omega^2}{2g}r^2 + h(0) \\ , ", "ea0fc9b5719ef0fe9d5f2cf27086a189": "\\Psi(\\mathbf{r}_1, \\mathbf{r}_2 \\cdots \\mathbf{r}_N, s_{z\\,1}, s_{z\\,2} \\cdots s_{z\\,N}, t)", "ea0fcabd51d3205f1a59b2545b85e196": " T^k{}_{ij} := \\Gamma^k{}_{ij} - \\Gamma^k{}_{ji}-\\gamma^k{}_{ij},\\quad i,j,k=1,2,\\ldots,n.", "ea0fccc8563bbee46519e002d8a33112": "\\left( f(p_1, p_2, \\ldots, p_j) \\right)", "ea1021fe79a19b1d92622831432bb2b0": "(x+1)^2R_n(x)=\\frac{1}{n+1}\\frac{d}{dx}\\,R_{n+1}(x)-\\frac{1}{n-1}\\frac{d}{dx}\\,R_{n-1}(x)\n\\quad\\mathrm{for\\,n\\ge 2}", "ea10cae09da943e5492d6b68c48c30b3": "y''+y=0 \\,", "ea10fb928d38496c9bd9930bfd9c2445": "\\Omega=\n[a_1,b_1]\\times[a_2,b_2]\\times\\cdots\\times[a_N,b_N]\\subset\\R^N", "ea1114c4654d806bcba00b20d4ebf7eb": " M^-_\\infty < M^+_\\infty < +\\infty ", "ea112b7bdd7337e31dd6addd8d266591": "\n \\cfrac{\\partial W}{\\partial I_1} = C_1~\\sum_{i=1}^5 i~\\alpha_i~\\beta^{i-1}~I_1^{i-1} \\,.\n", "ea113807432eb950fcbc4463d0fd41d1": "\\mathbf{P}_\\alpha = \\hat{\\mathbf{a}} +\\alpha \\mathbf{e}_\\infty", "ea1164f2b44266b483889a22d5f60e56": "q\\mid p^2-p+1", "ea11be838e2d0382417aa2441364798c": "k=\\Lambda", "ea11ca1fe760929dc220238f65a2f6b0": "S_0(x) = \\pm \\int_{x_0}^x \\sqrt{Q(t)}\\,dt.", "ea126cd9943d53a636dd5e27b5859655": "\\mathfrak h^* = \\mathfrak k \\oplus i\\mathfrak m\\subset\\mathfrak g.", "ea12715d6ec6022e81c0cc4cce245feb": "\\mathbf{M_{B}} = \\begin{bmatrix}\n49.321 \\\\\n20.138 \\\\\n12.881 \\\\\n20.150 \\\\\n5.910 \\\\\n9.150 \\\\\n8.217 \\end{bmatrix}", "ea12b8fafdb1d3224ff710293dd3868b": "t_1,\\ldots,t_m", "ea12bfb8a028b455b311e1d08f83618f": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = 1 - \\exp \\left(-\\sum_{i=1}^{n} \\left(x_{i} - \\frac{1}{\\sqrt{n}} \\right)^{2} \\right) \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = 1 - \\exp \\left(-\\sum_{i=1}^{n} \\left(x_{i} + \\frac{1}{\\sqrt{n}} \\right)^{2} \\right) \\\\\n\\end{cases}\n", "ea131b1920dcd9954239de9fccf026b7": "(a,b) \\in R", "ea1350d41f700619432cfe7c3a862462": "\\Delta_{1}=2\\left(\\frac{L}{1-\\frac{v^{2}}{c^{2}}}-\\frac{L}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}}\\right)", "ea136f99002085bac72fac738e05be8b": "\\omega_1,\\omega_2", "ea13b64804c18f2f61ca06e1f678abf7": "\\rho(f) = \\int_G f(g) \\rho(g) d\\mu(g).", "ea13ef0133ea20d30613e7159e9c665c": "\\partial p / \\partial s = 0", "ea142cf1e69c3de7c4500e2fb7caad1c": "\\{0,0\\}, \\{1,0\\},", "ea1456e1f6917cd8e1539d7671bbc51c": "\\epsilon_3,\\epsilon_4>\\epsilon_F", "ea14d829079f3a3d36e311206ea0ec14": "\\operatorname{\\rho}(T) = S \\alpha^{\\frac{B}{T}}", "ea14e6d7ccf026e03b6cb63c6fa8186c": "\\scriptstyle t=0", "ea14eb92a09af346481a999c310094f2": "\\chi ^2", "ea15a3426a8fc9c73dcc0c3bd52aa58b": "A \\subseteq B \\and B \\subseteq A \\Rightarrow A = B", "ea15e6b8b6bc7a151e69645fe1c1095e": "N = 0", "ea15f66f9f43c0428f653bd2bab0a507": " - (\\alpha + 1) P \\, dV = \\alpha V \\, dP,", "ea15f9c053bfd8f65290dbfc4262fdf1": "\\operatorname{E}[g(X)] = \\int_{\\Omega} g(X)\\, \\mathrm{d}\\mathrm{P} \\neq g(\\operatorname{E}[X]),", "ea16a259e96fcf7e918750d29551d145": "\\partial_{[\\alpha} F_{\\beta\\gamma]}= 0 ", "ea16ac2bb7db756c5134db1f34a922e8": "\\sin^2(x) + \\cos^2(x) = 1\\,", "ea16b041ed8498ce592a0a5cc986fed2": "\\frac{\\partial(x^{i_1},\\dots,x^{i_k})}{\\partial(u^{1},\\dots,u^{k})}", "ea16cd1c43a240cfb4f05108efc17ef6": "P_{\\mathrm{error}\\ 1\\to 2} \\le \\sum_{x_1^n(2)} Q(x_1^n(2)) \\left(\\frac{p(y_1^n|x_2^n(2))}{p(y_1^n|x_1^n(1))}\\right)^s. ", "ea16dc397a115f3663e7749bfe5df153": "S=h_1(X_1,\\ldots,X_m)", "ea16e1a3c94a3556653515e3bf4152fb": "10\\log \\left({\\frac{1}{1+\\omega^2}}\\right)", "ea1766a24c53e488f8269be591517e1c": "g(x)=\\sum_{n=1}^\\infty {b_n \\over n!} x^n", "ea17b9ac052aa9e9083e6a5dc0f5dd96": "\\forall x_v\\in Dom(v),\\; \\mu_{u \\to v} (x_v) = \\sum_{\\mathbf{x}'_u:x'_v = x_v } f_u (\\mathbf{x}'_u) \\prod_{v^* \\in N(u) \\setminus \\{v\\}} \\mu_{v^* \\to u} (x'_{v^*}).", "ea17ed5ecd5b9a9feac05c8f0253b96e": "(\\mu * \\nu) (A) = \\int_{X} \\mu (A x^{-1}) \\, \\mathrm{d} \\nu (x) = \\int_{X} \\nu (x^{-1} A) \\, \\mathrm{d} \\mu (x)", "ea17fe7566a518c3d4fa8b474856ba6a": "\\operatorname{Cov}(X_i,X_j)=0\\ ,\\ \\forall\\ (i\\ne j) ,", "ea183e72ccb1cd1ebc8c5c4e3b9b74f9": "g_{vs}=\\frac{1}{R_{vs}}", "ea186ca916681c590d847d37775ead2a": "Z^{\\infty}_{in} = 2\\|1 +1 = \\frac{5}{3}", "ea1884d864a0787c8397148e074e3e6f": " HETP = A + \\frac{B}{u} + C \\cdot u ", "ea19284081eb34abf1f906f481a6336f": "p(x) = p_1(x)\\;-\\;p_2(x),", "ea1953492f21519296f3fa4f1e11672f": "\\vec{e}_0", "ea1a0953792274b741959a11085a9b0a": "\\sum_{n=1}^\\infty \\left(a_n - a_{n-1}\\right) = - a_{0}.", "ea1a1db7086a2499b31ddaeb0a51da0c": "\\frac{dz}{dt}=z((\\lambda + i ) + b |z|^2), ", "ea1a2f2734a8ade68d698a9eab610c96": "\\langle u_i, v_j \\rangle = \\delta_{ij}", "ea1aaf1542f1721361d0416dedab257c": "\\zeta = \\frac{1}{2 \\omega_n \\tau_1} + \\frac{\\omega_n \\tau_2}{2}", "ea1b090838aab254e5243c1bf4bd0b67": "f_C = \\frac{1}{2\\pi (C_D+C_J)(R_S \\mathit{\\parallel}r_D)} \\ , ", "ea1b17035bb4fcb3e722d6955a360b04": " G^{ab} = R^{ab} - {1 \\over 2} g^{ab} R \\ ", "ea1b4408f208eb4e8a2919c6537a70e3": "\\!t_2", "ea1b6372868f1d5db55dd131e8e5d016": "A_{\\alpha}\\,", "ea1b7ecb82449bb273603a5cd34f3129": " \\varphi(q)", "ea1ba8e59a9d73505a64814032f15306": "G=\\left\\{ \\begin{pmatrix}\n\\alpha & \\beta \\\\\n\\overline{\\beta} & \\overline{\\alpha}\n\\end{pmatrix} : \\alpha,\\beta\\in\\mathbf{C},\\,|\\alpha|^2 -|\\beta|^2=1 \\right\\}", "ea1bfca5f1c7d92e6dc214a6651b6354": "(p_1(x) u')' + q_1(x) u = 0 \\, ", "ea1c0b2c11453b6c014881d8ac473bd8": "\\forall, \\exists, \\nexists \\!", "ea1c79f5fb412705db4399add64ad057": "\\delta^{2}n = n^{2\\gamma}", "ea1c7a41612e51c083816a85dd33b561": "\\sigma\\to\\tau", "ea1c9c361df57fd7b514e4722e6d04b1": "\\Delta v \\ = 3 v_\\text{e} \\ln 5 \\ = 4.83 v_\\text{e} ", "ea1ca23fca760f1d5fccd2dd13620551": "e^{A t}", "ea1cd13e797ca36b77e6e83e6f1526a8": " \\operatorname{sink}[(\\lambda p.(p\\ f)\\ (p\\ f))\\ (\\lambda f.\\lambda x.f\\ (x\\ x)), X] ", "ea1cd8efe96920af589c8d4528ad28be": "0.02260\\pm0.00053", "ea1d2cdf43ca3149a9d5dd8ac4481a50": " \\operatorname{F}_{s,t}(S) = U_{s-t} S U_{s-t}^* ", "ea1d2f971c85f919230a16ca9768af26": "K (\\sqrt[W]{\\varepsilon})/k", "ea1d4447ff33fd3f864657bfa0bb533e": "A=X_1\\cup X_2\\cup X_3\\cup\\cdots", "ea1d6fc004c7b18bf685068edd2081f3": "G(\\chi)\\ /\\ |G(\\chi)|,", "ea1d74aec4f7b6dfa92ad6dfbb950732": "c_m=\\frac{e^{-m\\gamma}}{m}\\qquad \\mathrm{for}\\,m>0", "ea1d8716833004c21cb164b8e9df6aba": "f(z) = \\frac{P(z)}{Q(z)}", "ea1de0f74b6831b7507d66595e5a61be": " f^\\prime = 10 + 12x + 3x^2 ", "ea1e66b66a9508c2ea2e1746f05a76c5": "\\eta\\left(\\frac{3\\xi_1^2+\\xi_2^2}{2\\xi_1^2(\\xi_1^2-\\xi_2^2)}c_\\eta(0,\\xi_1) - \\frac{n_\\eta^\\prime(\\xi_1)}{2\\xi_1^2}\\right)+(1\\leftrightarrow 2)", "ea1e70e18c208d8258055cb44ea35a37": "\\mathbf{A} =LU'= \\begin{bmatrix}\n1 & 0 & . & 0\\\\\nl_{12} & 1 & . & 0 \\\\\n. & . & . & . \\\\\nl_{1n} & l_{2n} & . & 1 \\end{bmatrix}", "ea1e7738acf72c864b93dfb16f7874ab": "\\operatorname{erf}(),", "ea1e8ab2cee98f853c26025c1fc81d33": "T = t g^p", "ea1eb27015805e505bebc700222c9cdd": "\\!\\mathcal A \\models_Y \\phi", "ea1ed0d1902f57adbbe7f80d6f001810": "\n\\begin{align}\nu &{}=&\\arcsin x &\\quad\\quad\\mathrm{d}v = \\mathrm{d}x\\\\\n\\mathrm{d}u &{}=&\\frac{\\mathrm{d}x}{\\sqrt{1-x^2}}&\\quad\\quad{}v = x\n\\end{align}", "ea1edc90bcaaad03dd26ea084167081c": "98 \\cdot a + 2 \\cdot a + b,", "ea1ef749c4b2420564562ccb05511d82": "{\\rm Tr}_{q^{n+2}/q}(x)=x+x^q+\\cdots+x^{q^{n+1}}", "ea1f0142b6de653f01a4804a548ac6bb": "\\frac{d}{dx}f(x)=\\frac{\\frac{dg(x)}{dx}-\\frac{dh(x)}{dx}\\cdot f(x)}{h(x)}", "ea1f0ec71ee2a4a63b00223e463ceffd": "cr(K_{m,n}) \\le \\left\\lfloor\\frac{n}{2}\\right\\rfloor\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor\\left\\lfloor\\frac{m}{2}\\right\\rfloor\\left\\lfloor\\frac{m-1}{2}\\right\\rfloor", "ea1f6845be9b35dd43b2d89ebd7efff7": " \\xi_k = \\int_\\mathcal{T} (X(t) - \\mu(t)) \\varphi_k(t) dt \n", "ea1f96a8ae7d2ecdbd5572202c063024": "\\mathcal{L} \\supset B_{\\mu} h_u h_d + A h_u \\tilde{q} \\tilde{u^c}+ A h_d \\tilde{q} \\tilde{d^c} +A h_d \\tilde{l} \\tilde{e^c} + h.c.", "ea1fba71fa7cd2e9f1bea219c630cd9c": "h(\\mathbf{y},\\tau)", "ea1ff818d55b5e74f16e007c80676246": "\\Re^n", "ea20a043c08f5168d4409ff4144f32e2": "010", "ea20a564cb3b13134ed5af4fcf9a55cd": "f_c - f_d", "ea20c5057604ff723c53aa7f7cb241c7": " \\{ x^2 - y^2 = 0\\} = \\{(x+y)(x-y)=0\\} = \\{x+y=0\\} \\cup \\{x-y=0\\}. \\, ", "ea20d940a1505b8deba6d8d6508b4012": "\\sqrt{2}\\lambda<\\xi", "ea2128e39baf77a628952f46aca5e63b": "f_1(\\xi_1),f_2(\\xi_2),\\ldots,f_m(\\xi_m)", "ea213af9b537b71336de9491225d3a65": "D_C", "ea21a8a70eb14109a31157b9904bdb3d": "h_n = t_{n+1}-t_n", "ea21e47e69b002c7b350f346069d521a": "f(x,y) \\rightarrow f(\\rho \\cos \\phi,\\rho \\sin \\phi ).", "ea223f82ddd2b059cc0b46e041013133": "\n\\frac{T_\\mathrm{total}}{T_{s}}={1+\\frac{\\gamma -1}{2}M_a^2}\n", "ea2251085890dadf6e7dac69996dd30c": "\nP_1 = 10^\\frac{L_\\mathrm{dB}}{10} P_0 \\,\n", "ea22c8f8eb52783d58c3cdb3de621cb7": " x_1 * x_2 = -(x_2 * x_1) \\qquad\\forall(x_1,x_2)\\in A\\times A", "ea22ea33eca23adef83dc385dfa76521": "x^{2^k-1}", "ea22eaa794c916e0b82637b42cecf87f": "(A, 1, \\cdot)", "ea23cae5a0cefa1b5fd957596ff1d7e7": " f(p) = \\frac{(n+1)!}{k!(n-k)!} p^k (1-p)^{n-k}\\text{ for }0\\le p \\le 1 ", "ea23e1918465232454373794c1db7623": "~s~", "ea23f5b9108ae3ae21e9387cc011bde7": "u^{*}_{0}", "ea243cd95bc4aecc68db9b58b4b11a48": "C_{2k}", "ea243f5fbb4b8ee4f801cb635e157227": "V(q) = q^{-2} + q^{-4} - q^{-5} + q^{-6} - q^{-7}.", "ea245ccad283b1c208bc9980d5d4e6cf": "\\sum_{j=1}^{n} X_{ij}\\beta_j = y_i,\\ (i=1, 2, \\dots, m),", "ea252eb83a6c383028616584516638c5": "[\\text{ }a,\\text{ }b,\\text{ }c]", "ea25640493df38509a070e79760611e4": "A^Q \\to \\Omega^2 (TM)\\,", "ea2576d6e65ddeb3736d762ebb4e2dd0": " T_n\\left(\\cos\\theta\\right) = \\cos(n \\theta), ", "ea26044b79d2b1c0ca226d4254afa5ee": "\\mathrm{III}(t)", "ea260c4a57c1213cd40cc7edf85f9377": " \\widehat{\\boldsymbol\\beta}_{ols} = (\\mathbf{X}^{T}\\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{Y} ", "ea264934a2badde093058b0cc7007d53": "0.75c < v_p < c", "ea268079e1ea1edbbf075b7f389423bf": "S(x,t) = S(x+U(S)t)", "ea268a384873bcabd66a862cef2fe22c": "k = \\sum_{i=0}^n \\left( a_i\\times 8^i \\right)", "ea26a587cef7aa88b5a5d0bada5a428f": "m_1^2+m_3^2=m_2^2+m_4^2", "ea26cfaa31b7d5b527989390d8d8cb38": "B_{r}", "ea276c176d98a72c48ae7ceabcdcb6e5": "C_a = \\tilde{E}_i^b F^i_{ab} - A_a^i (\\mathcal{D}_b \\tilde{E}_i^b) = V_a - A_a^i G^i = 0", "ea277d7ceeb907875c7f8c2adfeabdfa": "[Y] \\cdot [Z] = de \\; \\omega^n", "ea279e1da6161b264822170731686c48": "(1 + i_{lt})^n=rp_{n}+((1 + i_{st}^{\\mathrm{year }1})(1 + i_{st}^{\\mathrm{year }2}) \\cdots (1 + i_{st}^{\\mathrm{year }n}))", "ea27e9d3d653f43ff383bf756cbc11d7": "x \\to a", "ea281af5a716ac8441e9ad4219917744": " A(\\varepsilon) = A_0 - \\delta_A \\varepsilon(t) ", "ea283fa9b1ff39b91ba60d22359f34cb": " 3^{a + b} ", "ea284e26b82c894c676230cd82202eb2": "-\\log P(Ba|Ra\\overline{H})", "ea2869d605636cc63f957d7bda528a08": "\\rho^* R_j [(\\rho^{-1})^*f] = \\sum_{k=1}^d \\rho_{jk} R_kf.", "ea28883a2a197480c7607a6162aa2960": "a = b = \\frac{1}{\\sqrt{2}}.", "ea28916863edd0b8c22e35c9b9144f99": "P_\\mathit{SW} = \\frac {VI_o (t_\\mathit{rise} + t_\\mathit{fall})} {6T}", "ea28e7deff369233490d7c0b0ff27efa": "\\scriptstyle f(a)", "ea291366006f7774319b988f67783cc2": "\\scriptstyle\\sqrt{2k-1}", "ea292fc3da2130cd76afc158a0c4850c": "{}^q\\!D_{\\alpha}=\\sqrt[1-q]{\\sum_{j=1}^N w_j ({}^q\\!D_{\\alpha j})^{1-q}}", "ea2934f0f6c7ab55d1f80597e49947e9": "s_k^T y_k=s_k^T B s_k>0. \\, ", "ea296d67cdcf8a2112d605f05ce1d36d": "\\mu_r = 1 + 2.5 \\cdot \\phi", "ea29c3cdff676183e3d1fe2a28f74aa3": "x_N[n]\\ \\stackrel{\\text{def}}{=} \\sum_{m=-\\infty}^{\\infty} x[n-mN].", "ea29d50c452ed2d318bc707275e26655": "P(s_1,s_2)", "ea29f41890b5917c1e2332cc3c5c1ca3": "\\int\\frac{x\\,dx}{r^3} = -\\frac{1}{r}", "ea2a014e445bcbfbabdaf3d7a8124089": "{\\pi\\over 3}\\ {2\\pi\\over 5}\\ {3\\pi\\over 5}", "ea2a0bada2f8048a42f26311ab98980b": "\\frac{\\partial C}{\\partial r}", "ea2a7233be528d653a8da1f0735f3562": "\\sum_{n=0}^\\infty A_n^\\alpha x^n=\\frac{\\displaystyle{\\sum_{n=0}^\\infty a_nx^n}}{(1-x)^{1+\\alpha}},", "ea2ad12890f0b98c8de5435972a49f88": "S_{e2e}(t)", "ea2b068575658454ef8b891e2681eaf0": "u_0=\\frac 4{\\left( n\\pi \\right) ^2}dQU_{\\mathrm{el}}\n", "ea2b0cdd04d7a9d8f0e9476566d3abe4": "f(I)", "ea2b2f9c11212942fdf4154bde8e95dc": "\\textstyle \\zeta(\\alpha)", "ea2b645a59bd6af90a20e9f4987611d4": " E = E^0_{\\text{Hg}_2^{2+}/\\text{Hg}} - \\frac{RT}{2F} \\ln\\frac{1}{a_{\\text{Hg}_2^{2+}}} ", "ea2b67f3b756490d3782a95b08822827": "I_0 = I(0,0)", "ea2b81094028c02578dd7e0903bfd806": "|x-x_0|", "ea2bbf496f1a98465b31872f47a014c7": "c_0(x) = \\cosh {\\sqrt {-x}},\\text{ for }x < 0", "ea2bcca24991b30b8237c438c10f6e01": "K_M^{N}", "ea2c3fceb988a5fcf3cb0620fa354078": " {\\textstyle \\sum} a_kz^k = a(z) \\, (\\boldsymbol{B}) ", "ea2d585fbaa53d25c5f6e50bc9ccd223": "g(t) = \\delta_0 (t)", "ea2d7a72b51604f087bbcb645f8c3a2b": "m\\stackrel{\\text{def}}{=}\\lim_{x\\rightarrow a}f(x)/x", "ea2dc45d8731d1508f93dd302fc9a2c9": "\\mathbf{E} \\big[ \\big| N_{t} - N \\big| \\big] = \\int_{\\Omega} \\big| N_{t} (\\omega) - N (\\omega) \\big| \\, \\mathrm{d} \\mathbf{P} (\\omega) \\to 0 \\mbox{ as } t \\to + \\infty.", "ea2dc84ff59612074d7a48c15b5037f5": "A(z) = 0.5[P(z) + Q(z)]", "ea2dd824425e319841c276d870b4d3de": "\\displaystyle (x-p)^2 + (y-q)^2 = r^2,", "ea2deed027e695e73a6ad1218979944a": "H\\left(e^{j\\omega}\\right) = \\frac{1}{3} + \\frac{1}{3}e^{-j\\omega} + \\frac{1}{3}e^{-j2\\omega}", "ea2e0f5739b1543e72367a1dc1c69c01": "K_*(l^1(G))\\rightarrow K_*(C_r(G))", "ea2e5da52e82ce31f5749fd039328115": " b = 2(mq+np),\\,", "ea2e97dd79b8d5c6d5b4d0a50f1ebebb": "\\text{Ch}_K", "ea2ec3f5fbf81be779fcfe1296867718": "\\langle x,y,z \\rangle", "ea2efe0b207e451fe146f7abe9ef84af": "t^{\\prime }0 \\quad \\text{for}\\ i=0, \\ldots, z-1.\n\\end{align}", "ea30becfaddd0f45d74f1aae2727a097": "\\mathbf{p}^2 c^2 + m^2 c^4 = E^2", "ea30ed2149844e373ab715ca7d1ae94f": " S_C = S \\cot C = ab \\cos C= \\frac {a^2+b^2-c^2} {2}\\,", "ea310648f4e28fee9baaf114d6911f9b": "\\frac{1}{2}\\, B'\\,\\chi^\\alpha \\chi_\\alpha \\, \\text{,}\\quad \\frac{1}{2}\\,B\\,\\eta^\\alpha\\eta_\\alpha \\,\\text{,}\\quad\\text{or} \\quad M\\,\\eta^\\alpha \\chi_\\alpha \\,\\text{,}", "ea31853720e3e167e6c821054b4f614e": "\\iiint\\limits_V \\nabla \\cdot \\mathbf{E} \\ \\mathrm{d}V = \\frac{Q}{\\mathcal{E}_0}", "ea31867deb06f805cbd7b5e589b4c54d": "x\\in R", "ea32165e70c78739c7bb739acb8b5240": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=0,\\cdots ,5 ", "ea32245572a33bdddebe4d2435953ac8": "W\\operatorname{E}[\\nabla_\\theta g(Y_t,\\theta_0)]", "ea32564ceccffd9ed719e62ff0eec733": "\\displaystyle E(\\xi) = \\alpha_\\xi(\\xi) - L(\\xi)", "ea3257c7db23344ae10b16b23905e50e": " r= k[A][B]\\, ", "ea3274946bf91384452937a8e56f6dad": "Q(s_f, a)", "ea3283f4432261e5ce2946d2e3572960": "F_{50\\%} = \\frac{1}{2} = 0.500000 \\,", "ea32c75e3fbe978e5f7ed5608e50a741": "k-FDR= E \\left( \\frac{V}{R}1_{(V>k)} \\right) \\le q", "ea32d4312658c3ea8a6281dda9d458a1": "\\sum_{i=1}^c \\mu_i = n", "ea332d297106b0d5f317d9298b7fd9db": "g_1 = \\frac{f_1}{f_0} = \\cfrac{1}{1 + k_1 z g_2} = \\cfrac{1}{1 + \\cfrac{k_1 z}{1 + k_2 z g_3}}\n = \\cfrac{1}{1 + \\cfrac{k_1 z}{1 + \\cfrac{k_2 z}{1 + k_3 z g_4}}} = \\dots\\ ", "ea3375118c74c1e4b03309b6739ac57f": " D_{KL}(Q||P):=\\int\\frac{dQ}{dP}(\\ln\\frac{dQ}{dP})dP ", "ea3384fb5b64a429c0a54368751ee292": "e^i", "ea33ed242a7257f8250ae93daf45d048": " \\left(\\frac{N D}{a}\\right)", "ea3458a2593050c00e02d80a0e1bda39": "S = Ax^{2} + B\\sqrt{r^{2} + x^{2}}\\ + C", "ea35398e603852f6ffd3a1ab17f6eee8": "\\ell.\\mathrm{Ann}_R(S)\\,", "ea363ed8fec6e52881d9d63b0be5ed62": " \\frac{\\partial u_i}{\\partial p_j} ", "ea3642bf49580cd5b5ed1e23447c2424": " \\sum_{m=0}^{p-1}{(-1)^m{p-1\\choose m} m^{2n}}\\equiv0\\pmod p \\!", "ea366bd5da79d2fce61c90e31b5e6f37": "F_t - S_t = E_t(S_{t + 1} - S_t) + P_t", "ea36c3be0e10d1f1a81b55d382f9b16c": "\\displaystyle \\partial_t u + \\mu\\, \\partial_x^3 u + 2\\, u\\, \\partial_x u -\\nu\\, \\partial_x^2 u = 0", "ea36e15537b5f9ee749d41fbd9369ea9": " \\mbox{EV}|\\mbox{PI} = 0.5\\times1500 + 0.3\\times600 + 0.2\\times500 = 1030", "ea3717683e39ed416face2938feec395": "\\mathbf{k}=\\frac{2 \\pi}{\\lambda}\\mathbf{n}", "ea374dfdf109c079e4b934b9d4f9b9ce": "M,M+1,\\dots,N", "ea375de6c81ebb7c9ae424abd326585f": "(x_1,x_2,...,x_n)", "ea37abcae4f2899185d52e5ada17c95d": "q_2 \\cong \\langle b_1, ... , b_m \\rangle", "ea38675a3d4ffb72c5ce70d946167b59": "{K} = \\frac{{I}}{{I_0}}\\cdot\\frac{{2}}{{\\beta}^3{\\gamma}^3}\\cdot (1-\\gamma^2f_e)", "ea38865bd15fc0591cb3c05029a12f88": "[a_n, a_{n-1}, a_{n-2}, \\dots]", "ea388f3d5d822855a362233d4e7ad2ce": "a \\div b = a \\times (1/b).", "ea3899c3d1c0844b43753d18821ac6c4": "dE=-p_sdV", "ea38ced67956de9da418bf38b817c47a": "\\sqrt{(1-\\sin^2(u))} = \\cos(u)", "ea38ebd6d782789cac39e5ca8a40a36d": "y'_0", "ea38f92c811c27fac5a4163956a6148a": "\\sigma(X)-\\sigma(M)", "ea3915e7cabbe0ecc693a42c38498dc6": "\\langle u \\rangle \\,\\!", "ea3924c5ad74077d6ff2df35d1a0776b": "\\tau(\\omega)", "ea395f8015cdfa577e66265cd3295793": " \\arctan x", "ea396bd1872a36ab02124f32e0cafebf": "\\langle \\rho^*(g) \\alpha, v \\rangle = \\langle \\alpha, \\rho^*(g^{-1}) v \\rangle", "ea39a9f9924f110a93271a8e7eae97f5": "\\frac{p}{\\gamma}", "ea3a277dc3a5cebcfbbb68b11bfaeff2": "\\mathbf{P} = \\{ L : L=L(M) \\text{ for some deterministic polynomial-time Turing machine } M \\}", "ea3a4e6422536ac3f239ae517cfb1bde": "H_2(S(2k+1,n)) = \\begin{cases} 1 & n < 4\\\\\n\\mathbf{Z}/2 & n \\geq 4.\\end{cases}", "ea3ab4eda75ea2a1979c72c6e93a6def": "\\Omega=\\Omega^i_{\\ j}", "ea3abb59fe24e8a37b3aa2a8057282a7": "\\{\\min cx \\mid Ax \\ge b, x \\ge 0\\}", "ea3ac8c77dad4d8943e1c6fb6e88dc04": "\\underset{i}{\\overset{0}{x_j}}(t_0)", "ea3af8c7a2ae52ab9e717fa8db009d69": "\\cos\\,{\\theta_\\text{C}}= \\frac {\\phi-1}{r-\\phi}\\ ", "ea3b446985d0b5eb3b72c159bb8a6f0b": "|\\Psi\\rangle_\\nu = (|\\Psi_0\\rangle_\\nu , |\\Psi_1\\rangle_\\nu , \n|\\Psi_2\\rangle_\\nu, \\ldots)", "ea3bdfcdaeffbfa4c780a168f36356f6": "\\left(\\frac{dr}{ds},\\ r\\frac{d\\theta}{ds}\\right)\\cos n\\theta \\frac{ds}{d\\theta}\n= \\left(-r\\sin n\\theta ,\\ r \\cos n\\theta \\right)\n= r\\left(-\\sin n\\theta ,\\ \\cos n\\theta \\right)", "ea3c90ee9173141fce091b52b544e415": "{\\mathbf{}}A(t), B(t), C(t), Q(t), R(t)", "ea3c96002ff74746574b92bad7c9dddc": "\\mathrm{d}\\Omega=\\frac{\\mathrm{d}S_r}{r^2}=\\sin\\theta\\,\\mathrm{d}\\theta\\,\\mathrm{d}\\varphi.", "ea3ce231d0eccf17b3ac2145829a34ff": "u= \\frac{v + u'}{1+ \\frac{v u'}{c^2}} \\Leftrightarrow u'= \\frac{u -v}{1- \\frac{v u}{c^2}}", "ea3cf474e784950da365b2f424b46789": " \\operatorname{D}_A(U) = \\operatorname{Tr}(\\operatorname{E}_A(U) S). ", "ea3d1900d98dd861855792a335014744": "E(x)=k\\theta,\\,E(\\ln(x))=\\psi(k)+\\ln(\\theta)", "ea3ddb6e0ad1ec8e3e8dee587dbbb57e": "S - 0 = k_B \\ln{N} = 1.38 \\times 10^{-23} * \\ln{3 \\times 10^{22}} =70 \\times 10^{-23}J/K", "ea3decbed4e5fdf75447115061f02063": "Z=\\frac{1}{N! h^{3N}} \\int \\, \\exp[-\\beta H(p_1 \\cdots p_N, x_1\n\\cdots x_N)] \\; d^3p_1 \\cdots d^3p_N \\, d^3x_1 \\cdots d^3x_N ", "ea3f171935ed06881c65ae0aafe81e3b": "\\sigma(\\pi)= 1", "ea3fb07a8d95f10ad8ffdf8506818a5f": "v_{k}(x)=\\begin{cases} {x-x_{k-1} \\over x_k\\,-x_{k-1}} & \\mbox{ if } x \\in [x_{k-1},x_k], \\\\\n{x_{k+1}\\,-x \\over x_{k+1}\\,-x_k} & \\mbox{ if } x \\in [x_k,x_{k+1}], \\\\\n0 & \\mbox{ otherwise},\\end{cases}", "ea3fd411effad7cc30135213199a8de6": "R - I", "ea3fe4395d00b2f99a4edf1ce1d30502": "E\\,'=\\gamma vB", "ea401120712751e7f049705e51ec6704": "p = \\frac{ \\sum_{i=1}^N k_i / N } {r + \\sum_{i=1}^N k_i / N }", "ea4049726d98a5d2731af356d170c0e5": " \\zeta = \\frac{1}{\\sqrt{1 + (\\frac{2\\pi}{\\ln (x_0/x_1)})^2}}, ", "ea406a8ac2abf6cc82c631f0d9c781fb": "\n \\sqrt{R}(\\hat\\beta - \\beta) \\ \\xrightarrow{d}\\ \\mathcal{N}\\Big(\\,0,\\; \\Big(\\tfrac1R X^\\mathsf{T}(\\Sigma^{-1}\\otimes I_R) X \\Big)^{\\!-1}\\,\\Big) .\n ", "ea4073ae89fc9a508a18cc32e113450d": "\\operatorname{tr}(\\gamma_5 a\\!\\!\\!/b\\!\\!\\!/c\\!\\!\\!/d\\!\\!\\!/) = 4 i \\epsilon_{\\mu \\nu \\rho \\sigma} a^\\mu b^\\nu c^\\rho d^\\sigma", "ea40b910fec638b2c86047280037e5a3": " \\ln A' ", "ea40d1e372dd96098fdcce84c201778e": "\n\\bold{p} = {\\partial G_2 \\over \\partial \\bold{q}}, \\quad\n\\bold{Q} = {\\partial G_2 \\over \\partial \\bold{P}}, \\quad\nK(\\bold{Q},\\bold{P},t) = H(\\bold{q},\\bold{p},t) + {\\partial G_2 \\over \\partial t}\n", "ea40dd439f54cb12b9340434be58c516": "\\And", "ea40ee9f2d33003fa3309bdc5ae36fb3": "\\mathbf{v_B} = \\frac{d\\mathbf{r_B}(t)}{dt}=(v \\cos \\alpha + \\omega t \\ v \\sin \\alpha,", "ea41092eb737ba805262ad0f4efea60f": "\\Delta\\mu_1 = -v_s\\Pi", "ea415a98ff921198bb57068a855c6de0": "(10\\uparrow\\uparrow)^2 11", "ea4190dd18070d84aacef4c62980b15b": "\\operatorname{tr}(\\gamma^5)", "ea42081c06d31c567728039ae2ead46e": "t_i + \\tfrac{\\delta}{2}", "ea42733726b5de19a0c1cc56ef3695e5": "X=\\{x_1,x_2\\dots,x_n\\}", "ea427c852238099bd83e03d9fc340d55": "\\mathcal{A}_{\\alpha} = t_a \\mathcal{A}^a_\\alpha \\equiv t_1 \\mathcal{A}^1_\\alpha + t_2 \\mathcal{A}^2_\\alpha + \\cdots t_8 \\mathcal{A}^8_\\alpha ", "ea431f3fbe364c2fea2e24bd33a7abc5": "L'", "ea433709cad4e0828a0914138b7bb9cb": "t = \\frac{\\overline{X}_D - \\mu_0}{s_D/\\sqrt{n}}. ", "ea43779f8d4912a4e26ea699ce9ef61a": "\\nabla = \\gamma^\\mu \\partial_\\mu", "ea43f6d7798abbca496d9d639508d55b": " \\ln (1+e^{\\eta})", "ea444aded051964f1438d626aa21635f": " x^{*} = [I-A]^{-1}b \\, ", "ea4552ddd1983660132777c07633e983": "\n\\{\\,a'\\,\\} = M\\{\\,a\\,\\} \\oplus \\{\\,v\\,\\},\n", "ea4554cedcb950bf0d7e76fa335a63cb": "\\begin{bmatrix} 1 & -sL \\\\ 0 & 1 \\end{bmatrix} ", "ea4569cc9c08a47320e4c18ad2f7c4a5": "f(x) \\propto x^{\\alpha_1-\\alpha_2}_\\text{th}x^{\\alpha_2}\\text{ for } x>x_\\text{th}", "ea456e472b32c7ad76d4c9f706559ca1": "\\begin{align}\n p(t) I_n &= (t I_n - A) \\cdot B \\\\\n &=(t I_n - A) \\cdot\\sum_{i = 0}^{n - 1} t^i B_i \\\\\n &=\\sum_{i = 0}^{n - 1} tI_n\\cdot t^i B_i - \\sum_{i = 0}^{n - 1} A\\cdot t^i B_i \\\\\n &=\\sum_{i = 0}^{n - 1} t^{i + 1} B_i- \\sum_{i = 0}^{n - 1} t^i A\\cdot B_i \\\\\n &=t^n B_{n - 1} + \\sum_{i = 1}^{n - 1} t^i(B_{i - 1} - A\\cdot B_i) - A \\cdot B_0.\n\\end{align}", "ea45bc1a2e87994178cf9632424a9a45": "s' = a s \\,", "ea45cbfba120cedaeabe564135bbfb08": "E_1,E_2,\\ldots,E_m", "ea45ed66452bb73d227a84169c810402": "\np = \\frac{\\left( a+b+c\\right) \\left(ab+bc+ca\\right)}{a^2+b^2+c^2}\n", "ea462f8b180e4a2022d88a1b0c07f3ff": "f: V \\to W", "ea46678a73b1fc02f4ca25dc3432f78f": "|\\langle \\psi | \\phi_j \\rangle|^2", "ea46c48cd2a0870603c98e1dd7e2ad42": "(1 + 2 t) ( 1 - t)^2", "ea47ae2cfba011d900575512c3db8aaf": "u = x \\Rightarrow d u = dx", "ea47e3dabeadb51a986b5bbb33f8b1c1": "u'=u(x',t)", "ea480faa103c452552f9d82203fda8de": "I t / \\psi \\ ", "ea48428a571a87c20e2602bd07cfc1cc": " \\rho_c ", "ea4848a6ca46c2d9e94a48e0a8de444f": "\nJ = \\{x \\in M \\mid x \\mbox{ is a critical point of } f + \\lambda g \\mbox{ or } \\lambda f + g\\}.\n", "ea49070e1ac28105bf378f429750597d": "c(u) \\neq c(v)", "ea4913b68661e5c87aa390edb1ca4e2e": " f(x_1,x_2,...,x_n) ", "ea491600c3ab1a973c22b3696d97528b": "\\cos \\boldsymbol{\\Phi}_s = \\frac{dr_s}{dt_s} =\\pm \\sqrt{1-\\left(1-\\frac{2M}{r}\\right)\\frac{\\mathit{I}^2}{r^2}},\\,\\!", "ea493c5d21dabcb106142eaf93b02f32": " \\mathrm{MFSL} = 10^{7.91 - 8.1 a_w} ", "ea49b66ab2e5b359cfdff343775e3fc7": " M^{1,1}_m(\\mathbb{R}^d) = M^1(\\mathbb{R}^d)", "ea4a24142a8f4ee3d30d8be983032b86": "so (1,3)_\\mathbb{C}^\\pm", "ea4a44a424124c7c2b0fe862e1ea5660": "\\triangledown", "ea4a586ec0a8433c4e4e06024598af6c": "2(2n+1)R_n(x)=(x+1)^2(\\partial_x R_{n+1}(x)-\\partial_x R_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))", "ea4a5e78edb22801ff0f9994c409cb11": "\\Delta_2=\\{amount(lactose,medium), amount(glucose,medium)\\}", "ea4ae96ef9712a4083607c7889688af9": " (f,g) ", "ea4af7acc6061e4e7432e36edef776fc": " P= \\begin{pmatrix} {{1\\over4} + {3\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} \\\\\\\\ {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} + {3\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} \\\\\\\\ {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} + {3\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} \\\\\\\\ {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} - {1\\over4}e^{-t\\mu}} & {{1\\over4} + {3\\over4}e^{-t\\mu}} \\end{pmatrix}", "ea4b07e872793ca16ef63f650ed3edf3": "\\mathfrak{t}^*", "ea4b67fe7635c93cea3591a5f8a94946": " \\mathbb{H}/\\Gamma ", "ea4b85ec2dff43dca06ee7d963ac3b65": "I_s=\\hbar c \\gamma k^{3}/12\\pi", "ea4c9695971ab32140cbd10521b00a39": "(A.2)\\quad \\theta_{(n)}:=h^{ab}\\nabla_a n_b\\;,", "ea4d39efb5f04185b9956174f7775210": "(((P \\to Q) \\and (R \\to S)) \\and (\\neg Q \\or \\neg S)) \\to (\\neg P \\or \\neg R)", "ea4d7ef2c3d9d17bc6457c6b5786fce9": "m c^2=\\hbar \\omega=\\hbar c/R", "ea4d91fc3b21443b9da40a301591849f": "\\left.\\right.\\omega_f(\\delta;0) < \\Omega(\\delta)", "ea4de10ada3237b7df03343d65ff43f8": "\\Gamma_N(w|a_1,...,a_N) = \\exp\\left(\\frac{\\partial}{\\partial s}\\zeta_N(s,w|a_1,...,a_N)|_{s=0}\\right)", "ea4e11123887a175e2e0ebf6ba25c9b5": "\\beta_2 < 0", "ea4e4ad1da1ee9cff8df802310ca56f1": " \\lim_{\\alpha = \\beta \\to \\infty} \\operatorname{excess \\ kurtosis}(X) = 0", "ea4e4fb9226a25d3fb6be90604f75929": "\\rho_n(x)=\\frac{1}{d_0^{n-1}} \\frac{\\rho(x)}{\\left(P_{n-1}(x) \\frac{\\varphi(x)}{2}-Q_{n-1}(x)\\right)^2 + \\pi^2\\rho^2(x) P_{n-1}^2(x)}.", "ea4e69d46ddc9372302b4e8a2f55245c": "\\displaystyle{\\mathcal{U}f(z) ={1\\over \\sqrt{2\\pi}}\\int_{-\\infty}^\\infty e^{-(x^2 +y^2)} e^{-2ixy}f(t+x) e^{-t^2/2} \\, dt ={1\\over \\sqrt{2\\pi}}\\int_{-\\infty}^\\infty B(z,t) f(t)\\, dt,}", "ea4ea967af5272009842c8942030b00c": " F:\\mathbb R \\rightarrow \\mathbb R", "ea4eaaa4bd5a7ce5e49de8882f7b122f": "\n \\mathbf{u}(\\mathbf{X}_B) - \\mathbf{u}(\\mathbf{X}_A) = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} \\boldsymbol{\\nabla} \\mathbf{u}\\cdot d\\mathbf{X}\n = \\int_{\\mathbf{X}_A}^{\\mathbf{X}_B} (\\boldsymbol{\\epsilon} + \\boldsymbol{\\omega})\\cdot d\\mathbf{X}\n", "ea4ec92b5487db4587fc43a3f868520c": "B = Y_2ZZ_1Z_1", "ea4f2cd6e72f70dd0e8200a86ad5cd8d": "\\alpha_0 > 0", "ea4f57c5b3bf49e27797990b15ffb4e0": "\\operatorname{Cl}_{3}(2\\theta) = 4\\operatorname{Cl}_{3}(\\theta) + 4\\operatorname{Cl}_{3}(\\pi-\\theta) ", "ea4f8d31c2b62d78a1e9e3c90801f208": "E = - \\frac{Ze^2}{2r}", "ea4fb44c51b62f4f4f5035d459715dfd": "{E}_{6}^{(2)}", "ea50331620e20d45a791b0bee1088b5e": "f_{ij} = 0.5", "ea5051168a17a8feb927204f4ef5bddf": "X_{hG}", "ea5068ff44e1c87a66751df3ecff79c3": "= \\frac{1}{q_T} \\left( 0, -q_y, q_x, 0 \\right) \\,", "ea50977044de341d713baf79d306062e": "X_{[m]}", "ea50ccd1df2a7bb1c13b08f2abdc40cc": "\\Gamma =dx^\\lambda\\otimes (\\partial_\\lambda + \\Gamma_\\lambda^i\\partial_i), \\qquad \n\\Gamma_\\lambda^i=\\Gamma_\\lambda{}^i{}_j(x^\\nu) y^j + \\sigma_\\lambda^i(x^\\nu). ", "ea50d8a642d920ad558f190b76c52779": "\\zeta(3)=\\frac{\\pi^2}{7}\n\\left[ 1-4\\sum_{k=1}^\\infty \\frac {\\zeta (2k)} {(2k+1)(2k+2) 2^{2k}} \\right]", "ea5163ada96b00ded9a9e52c1e921797": "\\omega = \\sqrt{\\Delta E\\over{m a^2_{0}}}", "ea516d31dbff2cf694655bfe0ea91e0f": " \\left ( \\begin{array}{c|c} \\mathbf{I} & \\mathbf{t} \\\\ \\hline \\mathbf{0} & 1 \\end{array} \\right ) ", "ea51ba2dace89775e21fa1efb9964023": "0 = (g^{\\mu \\nu} \\sqrt {-g})_{; \\rho} = (g^{\\mu \\nu} \\sqrt {-g})_{, \\rho} + g^{\\sigma \\nu} \\Gamma^{\\mu}_{\\sigma \\rho} \\sqrt {-g} + g^{\\mu \\sigma} \\Gamma^{\\nu}_{\\sigma \\rho} \\sqrt {-g} - g^{\\mu \\nu} \\Gamma^{\\sigma}_{\\sigma \\rho} \\sqrt {-g} \\,.", "ea51ef64fdb767fd20af28ee44e7b27d": "x \\in d(X)", "ea51f22c39ea12d656fdd6be5d070757": "v\\left(z\\right)=\\sqrt{v_{i}^{2}-2az}", "ea52828ae50bc769dc05848549d22007": "\\partial \\Omega_D", "ea52b718316bd1d1247df33af7d42a0c": "\\mathbf{a}_{x,y,z}", "ea52d899a3acf9a83682270b0eb89333": "\\sin 2x = 2 \\sin x \\cos x, \\,", "ea52e04c4f215fa257e8c5945790b932": "\\alpha = \\pi D_\\text{p} / \\lambda", "ea535cabff318d99933b0fbf3ee4c26a": "\\mathcal{L}v(x_1,x_2)=0", "ea53b7e33fba21989134bec2e879df6e": "\\omega_D = i^*(\\omega_X \\otimes \\mathcal{O}(D)).", "ea53be318746869622ffe68a4ee37ca7": "\\begin{align}\n \\phi({\\mathbf{r}}) &= 1 - \\frac{m}{2\\pi\\hbar^2}\\int{\\frac{\\exp[ik|{\\mathbf{r-r'}}|-i{\\mathbf{k}}\\cdot({\\mathbf{r-r'}})]}{|{\\mathbf{r-r'}}|}V({\\mathbf{r'})\\phi({\\mathbf{r'}})}dr'}\n \\end{align}", "ea54230cc663f124be4bc592695c0641": "\\vec{d}_{\\text{eg}}:=\\langle\\text{e}|\\vec{d}|\\text{g}\\rangle", "ea54daf9e5614889f66b92c4d9444d72": " f_\\mathrm{3dB}=\\frac {1}{2\\pi R_\\mathrm{A} C_\\mathrm{M}}= \\frac {1}{2\\pi R_\\mathrm{A} [ C_\\mathrm{gd}(1+g_\\mathrm{m} (r_\\mathrm{O} \\| R_\\mathrm{L})]}", "ea5526bf6662e15d70c3d15fceda8f64": "\\rho_1 = \\gamma_1 / \\gamma_0 = \\frac{\\varphi_1}{1-\\varphi_2}", "ea55452f8c2af2103dcebb8781a2d633": "|F(s)| \\approx 1", "ea554955b6757e10fd93eb7a00e4398c": " \\rho \\in [0,\\infty), \\, \\theta \\in [0,2\\pi]", "ea56261ef6f3f04a249bea8eea3f5f67": "\\mu_{i,j}=\\frac{\\langle\\mathbf{b}_i,\\mathbf{b}^*_j\\rangle}{\\langle\\mathbf{b}^*_j,\\mathbf{b}^*_j\\rangle}", "ea5629bd70a64299738406289cc0cb8e": "M_3", "ea562c7d5f0a1db22d739a4e122f1171": " [s_1 ,\\dots ,s_N ]^T = (E\\otimes I)\\cdot \\tilde{s}(t): \\qquad E_{ij} =\\frac{(i-t)^{(j-1)}}{(j-1)!} ", "ea5641e7f3ba29daf553c04194e8d00f": "BS = \\frac{1}{N}\\sum\\limits _{t=1}^{N}(f_t-o_t)^2 \\,\\!", "ea566cc0a07b1301e73c21f7864d7d09": "b = \\sqrt{2\\phi}", "ea56b855831dbcdd49f50ad03bc330fa": "\n\\frac{N}{1-x}\n = \\frac{N\\cdot(1+x)}{1-x^2}\n = \\frac{N\\cdot(1+x)\\cdot(1+x^2)}{1-x^4}\n = ...\n = Q' = \\frac{N' = N\\cdot(1+x)\\cdot(1+x^2)\\cdot\\cdot\\cdot(1+x^{2^{(n-1)}})}{D' = 1-x^{2^n} \\approx 1}\n", "ea5722e2a8812be009503f3eb6e594e8": "\\frac{P_{1r}}{P_{2t}} = \\frac{P_{2r}}{P_{1t}}", "ea576e2f8283d37ee1b5e872ef9bd91e": "\\mathcal{S}^*", "ea578aa42991a42421a69d626a597c75": "d_m", "ea5799e1a5e4b35225e116a4897880aa": "(2n-1)!! = \\sum_{k=0}^{n} \\binom{2n-k-1}{k-1} \\frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!!.", "ea57dff55d489770c566bc8b56786f5b": " \\rho_X(t) \\le C \\, t^p, \\quad t > 0.", "ea580c28b44bf7b64e6207d741245eb1": "I_{o_{lim}} = \\frac{V_i\\left(1 - D\\right)}{2L}DT", "ea581215636a38de7efd847303191e4c": "\\scriptstyle \\alpha \\;\\ll\\; 0.5", "ea5877aa377ced4c233d5b37f17ee2dc": "f^{*}f_*\\mathcal{F} \\rightarrow \\mathcal{F}", "ea58f5af3b9db13ca7ef61ed8281fd1b": "N = \\frac {f^2} c \\frac 1 { 2 D_\\mathrm N - f }\n\\approx \\frac {f^2} c \\frac 1 { 2 D_\\mathrm N }\\,.", "ea5938d93d6e8466b3cc186cacf1e973": "L=L^{'}_{0}/\\gamma", "ea594adbf382e4e7b372afeb6f5052ec": " \\frac{\\partial\\psi}{\\partial t} +\\nabla\\cdot\\left( \\psi{\\bold u}\\right) =0 ", "ea59bf17aa6803a2397c1febc34ed78a": "Q(x'; x^t)=\\mathcal{N}(x^t;I) \\,", "ea5a381e7865b31a0834ea418949c268": "\\mathcal{H} \\approx 0\\rightarrow \\mathcal{H}\\Psi =0", "ea5a63d06d6c84026276f85990d380fc": "\\gamma_\\mathrm T = 2 \\gamma_\\mathrm L = 2 \\sinh^{-1}{\\sqrt{\\frac{R_1}{2R_2}}} \\,", "ea5a82e6b85e474cfbdef3af95f2d7d9": "a^b = \\sum_{x=1}^b \\,\\! P ( a - x + 1 , b + 1 ) K ( b , x )", "ea5ab261fbd0e3e967559ce94a94aeda": "\nT(t) = T_0 + \\int_0^t R(t)\\,dt\\ + \\epsilon(t) = T_0+(R_0t + \\frac{1}{2}At^2+...) + \\int_0^t E_t(t)\\, dt + \\epsilon(t)\n", "ea5abe05f5a9081532d4ea4c47d99bb9": " r_1^2 + r_2^2 + \\cdots = \\infty \\text{ and } \\frac{ r_k^2 }{ r_1^2+\\cdots+r_k^2 } \\to 0, ", "ea5af9e341016b4676aa75eca36ec96c": "t = t_{1}", "ea5b182f074d541f2408b839287adcb3": "\\scriptstyle{\\scriptstyle{\\rm cost}}\\approx \\Big(\\big(\\frac{r}{1-(1+r)^{-N}} +r_{\\rm PT}\\big)", "ea5bc40e91ce1fc235bab7b7a39ee5cc": "C= \\frac{\\mathrm{d}Q}{\\mathrm{d}V}", "ea5bd0e03e8b75c50dacc8bf5e501ae2": "\\mathrm{grad\\,} \\varphi\\,", "ea5c04a2b11a4e960c3a4a221b5b92c5": "\\left(U_{m}\\right)_{\\sigma|\\sigma'}=\\delta\\left(\\sigma_{1},\\sigma_{1}'\\right)\\cdots\\delta\\left(\\sigma_{m-1},\\sigma_{m-1}'\\right)w\\left(\\sigma_{m},+1,\\sigma_{m}',\\sigma_{m-1}\\right),", "ea5c1430f28a651b122d1333e0a6d276": "\\text{Hom}(I, \\text{hom}(-, -)) \\simeq \\text{Hom}(-, -)", "ea5c1fc12d0ee4330ed7da0d062d3c5a": "mg = \\frac{GmM_{earth}}{R_{earth}^2}\\,", "ea5ce790fdfee5969e3485fe36991d4d": "y = r_1\\, \\sin(\\theta)\\, \\sin(\\phi) \\dots,", "ea5d2f1c4608232e07d3aa3d998e5135": "64", "ea5da08f1c9cd52e7ed7b96812bd5ff0": "\\sigma_i^2=1 ", "ea5dce534e2081842c2cea8c730ea263": "\\displaystyle{E(f)(x)=\\sum_{m=1}^\\infty a_m f(-b_mx)\\varphi(-b_mx) \\,\\,\\, (x < 0),}", "ea5e1f3bd4d70e2fbbd3e477c2946fb7": " \\int_0^\\theta \\sec\\zeta\\,d\\zeta = \\ln\\left|\\tan\\left(\\frac{\\theta}{2} + \\frac{\\pi}{4}\\right)\\right|. ", "ea5e32b442e7a9a1ba547c5d3c921cbc": "\\mathcal{F}_{\\mathfrak{H}}", "ea5e9bf2bb53ab9f5986998bb5a37d3b": "b\\in\\text{cl}(C\\cup\\{a\\})\\smallsetminus\\text{cl}(C)\\,", "ea5f3abb3976c89166a804d1f3d59e36": "\\mathbf{e}_\\mathrm{x}\\hat{p}_x + \\mathbf{e}_\\mathrm{y}\\hat{p}_y + \\mathbf{e}_\\mathrm{z}\\hat{p}_z = -i\\hbar\\left ( \\mathbf{e}_\\mathrm{x} \\frac{\\partial }{\\partial x} + \\mathbf{e}_\\mathrm{y} \\frac{\\partial }{\\partial y} + \\mathbf{e}_\\mathrm{z} \\frac{\\partial }{\\partial z} \\right ),", "ea5f62b2862ac6d9a68e1541283c9db7": "k\\equiv 1 \\pmod 4", "ea5fd726dea51cd4c91f5d8bd198f747": " A_{i+1}^{\\mu} \\downarrow^{A_{i+1}}_{A_i} = \\bigoplus_{\\lambda \\in \\hat A_i} g_{\\lambda,\\mu} A_i^{\\lambda}.", "ea604a75e262f3e5d1886e98ecdd7298": "F_e - F_w = 0", "ea6096c2b2ddfd1e1b5f20119fa5b084": "\\rho_i = c_i \\cdot M_i", "ea60a1895ea8d500cc50c6e62f809c96": "y_{12}-y_{22}", "ea60be4e0e453a49d1830094398c8b50": "\\theta = \\arctan \\left( \\frac{\\text{opposite}}{\\text{adjacent}} \\right).", "ea60c1f552078c3dfca078ddd3928205": "\\eta_{\\text{max}} = \\eta_{\\text{Carnot}} = 1 - \\frac{T_C}{T_H}", "ea60c2b51aeab605eeeaf07c8803b91c": "f(q) = q \\ \\forall q \\in \\mathbb{Q}", "ea60fb95886bd83b19229c07aa24875a": "\\operatorname{Area}(B) = \\int_B f(u_1,u_2)\\,du_1\\,du_2.", "ea616391a883793ac8fc3bfd2adda746": "\\mathfrak f_4\\cong \\mathfrak{so}_9\\oplus \\Delta^{16}", "ea616df6c1888973b6bc04fe52845132": "\\mathrm{SU}(p+q)\\,", "ea61c3b50985bf8cd43b54291032a0f5": "\\kappa \\cdot \\kappa = \\kappa", "ea62a4eedd273b3bf7d367f76ac24931": "F(t) = \\frac {C(t)}{C_{0}}", "ea62b58f1b9eaf314230a3ea21313566": "\nr^2 = 4 b (a- b \\sin^{2} \\theta)\\,\n", "ea62bf2c702113efbc59e356181168e2": "{e}'(v)=\\frac{f(v)}{F(v)}(v-e(v))", "ea63343f3dc2cb16c879a8c8eac22c78": " \\mathrm{HOMA} = 1- 257.7/n\\sum^n_i(d_{\\rm opt} - d_i)^2 \\,,", "ea633b182d50dacff94acf269b6c69f7": "Z_0 = R \\cos(\\Theta) =\\sqrt{-2 \\ln U_1} \\cos(2 \\pi U_2)\\,", "ea63c2de8e7f94d39a51a9954366c6fa": "Z = -\\frac{\\mu_x - \\mu_y}{\\sqrt{ s_x^2 + s_y^2}}", "ea63cade65113eb8caf139ef7a813dbd": "\n\\Pi = 2 \\epsilon \\epsilon_0 \\kappa^2 \\psi_{\\rm D}^2 e^{-\\kappa h}\n", "ea63f5c546a800e7b0a97baf291d77b5": "Z_{i\\cdot} = \\frac{1}{N_i} \\sum_{j=1}^{N_i} Z_{ij}", "ea63fa81a4fcea6432935173fa4650c3": "\\mathbf A\\cdot\\mathbf B = \\sum_i B_i(\\mathbf A\\cdot\\mathbf e_i) = \\sum_i B_iA_i", "ea6411eb519d870789184825653face3": "\\left( 5,12,13 \\right)", "ea641f99f6fae2a963e5e237bb40a6dc": "= \\int_{t}^{t + \\tau}C\\, \\operatorname{d}t", "ea6426c5f737ccee258a6d4652b0f109": "C_{out}: B^K \\rightarrow B^N", "ea64971cc9d87436fe8cc8280e4c60aa": "\n\\begin{align}\nE[|B(y, pn)|] & = \\sum E[X_c] \\text{ for every } c \\in \\mathcal{C} \\\\\n& = \\sum \\Pr[X_c = 1]\\text{ for every } c \\in \\mathcal{C} \\\\\n& \\ge \\sum q^{-n(1 - H_q(p) + o(n))} \\\\\n& = \\sum q^{n[R - 1 + H_q(p) + o(1)]} \\\\\n& \\ge q^{\\Omega(n)}\n\\end{align}\n", "ea64c99923024e72c6171bfbd1f34957": "E_b(f)", "ea64dd9383620c29826bd15bc909bfe8": "\\zeta_{2n}(s)\\,", "ea6504ed7567c2b70cc9f1eaf4cd5b30": "hv=\\Delta h(v_{(1)}\\otimes v_{(2)})=h_{(1)}v_{(1)}\\otimes h_{(2)}v_{(2)},", "ea651ac2002fa1b10a702bf10f4f6e12": "0 \\le \\eta_{th} \\le 1", "ea655a03ae728cbed5bc9a9fd25c4551": "{\\mathcal L}^4_{xy}", "ea656cbc36187bb98312da881f98eb7b": "f^* N = N_R", "ea65dbe574efeed6e757d678e6a56c1a": " V_i = \\sum_k D_{i k}p_k + \\sum_j C_{i j}V_j", "ea663ae14a549fc9e602a17b0393d66c": "Y=U/V", "ea663c8d703929f441b031a805234f43": "(A \\to B) \\iff (\\neg B \\to \\neg A)", "ea66669c5433fea2380249860c22d8ad": "\\displaystyle{\\begin{pmatrix}\\overline{a} & -\\overline{c} \\\\ -\\overline{b} & \\overline{d}\\end{pmatrix}.}", "ea66915d6d9a0974ed673464aed3a22a": " {\\mathbf{u}}(t)= -L_r(t) \\hat{\\mathbf{x}}_r(t).", "ea66fe941d1380b437edfb4d2002c3c0": "\n \\beta\\ \\sim\\ \\mathcal{N}(\\beta^*, \\Sigma^*).\n ", "ea6719d6e313cc06c9a0c8876fc62aeb": "\\mathrm{MD} := E[|X - Y|] .", "ea671f07639cca8bae34394bd7c5b71c": "= p(C) \\ p(F_1\\vert C) \\ p(F_2\\vert C, F_1) \\ \\dots p(F_n\\vert C, F_1, F_2, F_3,\\dots,F_{n-1}).", "ea67797d28d0776af9ef74ef0ef2a16c": "\\cos (\\arccot x) = \\frac{x}{\\sqrt{1+x^2}}", "ea67b4af6ab95a0b919367c0f49eb660": "\\int\\frac{x}{R^{2n+1}}\\;dx = -\\frac{1}{(2n-1)aR^{2n-1}}-\\frac{b}{2a}\\int\\frac{dx}{R^{2n+1}}", "ea6800bd576c12683c4f0e28b6fde5fc": "(q,\\epsilon,\\epsilon,q)\\in\\delta^*", "ea68461414395fceb4eccbff710ec1d1": "y_T = \\frac{Y_T}{X_T+Y_T+Z_T}", "ea69099359437b4602c22ec5fcf12d0c": "\\{x\\in\\mathbb{Z}^n : Ax=0,\\ x\\neq 0\\} \\, ", "ea694f9db3563afd83df5395b7ca4d45": "{A}_{2}^{(2)}", "ea69dfb3064cf0189696278dfb9b7db4": "\\delta L", "ea6a64a353827fb0f640af1ced5ed1a0": "\\tfrac{k}{n}", "ea6b26bd311b96a9fae4241c38ec3bb8": "t = t", "ea6b739321b5b6aee4d9a3d224cbba09": "\\delta=(\\delta_1=1,\\delta_2,\\dots,\\delta_m)\\in \\{\\pm1\\}^m", "ea6b85d45c526bb8add70cefaff3b50f": "G \\subset \\mathcal{B}_B", "ea6bdddd6b20f42889feb028adfe3068": "A\\mathbf{x} = \\mathbf{0}.", "ea6bf367bf30ddb5a0ce9e68e5cbdad8": " \\nabla^2 f = 0 ", "ea6d6128b584952f5c329ad135620a9f": "\\Psi\\, ", "ea6d83fa73fb673d237add906af4d432": "(p?;a) \\cup (\\neg p?;b)\\,\\!", "ea6d9204bd29a8e5361983b8e282e3da": " n(n+1) ", "ea6dfbc2c91d7210f34e6863a96a6619": "\n\\frac{1}{|\\mathbf{r}-\\mathbf{r}'|} = \\frac{1}{\\sqrt{r^2 + (r')^2 - 2 r r' \\cos\\gamma}} = \n\\frac{1}{r_{{\\scriptscriptstyle>}} \\sqrt{1 + h^2 - 2 h \\cos\\gamma}} \\quad\\hbox{with}\\quad h \\equiv \\frac{r_{{\\scriptscriptstyle<}}}{r_{{\\scriptscriptstyle>}}} . \n", "ea6e797a33a48b7d64f1a36dcdea8e9e": " \\mathbf{v}_2 = \\mathbf{v}_1 + \\boldsymbol{\\omega}_1\\times\\mathbf{r} = \\mathbf{v}_1 + \\boldsymbol{\\omega}_1\\times (\\mathbf{r}_1 - \\mathbf{r}_2) .", "ea6ec54d4b8d25f56382cd4cde446605": "f_{\\lambda}(\\theta) = \\langle \\theta, \\star \\lambda\\rangle", "ea6ed349727458cd7dc0e7b2558721d7": "\n\\operatorname{E}(X_1 | X_2 < z) = -\\rho { \\phi(z) \\over \\Phi(z) } ,\n", "ea6f31cd4a8f97ab122d94c0395183fe": "\\mbox{CAIDI} = \\frac{\\mbox{sum of all customer interruption durations}}{\\mbox{total number of customer interruptions}} = \\frac{\\mbox{SAIDI}}{\\mbox{SAIFI}}", "ea6f990577858c840f78a1f0f8727c53": "\\frac{ \\frac{10}{100}(-100)+\\frac{10}{100}(-20) }{ \\frac{20}{100}} = -60.", "ea6fe1dba823ee607e15d7c6691634db": "J=\\pi L_{e+r}", "ea7003bbb4563c59f5bbb9356fe90c8e": "X_n \\, \\xrightarrow{\\mathcal D} \\, X\\,.", "ea703369bd28ee95f936e4b0fb71eec4": "ln p_{1}(\\Delta U) = \\beta(\\Delta F -\\Delta U) +ln p_{0}(\\Delta U)", "ea704b0ecd69aee0e5c1fc8ba71db329": " (x_1+x_2+\\cdots+x_k)^n = \\sum_{\\lambda\\vdash n} s_\\lambda f^\\lambda. ", "ea706edd8e147199930884ab42f0c678": "{\\rm Tr}(B^{r/2}A^rB^{r/2})^q\\leq {\\rm Tr}(B^{1/2}AB^{1/2})^{rq},", "ea707df6b737bc2b65404f1ed1d8290c": "n = 2 \\int_{G(\\omega=0,p)>0}\\frac{d^D k}{(2\\pi)^D}", "ea70c4aca4381d222a650af7583b6f13": "\\det (tI - A) = (t - \\cosh(\\phi))^2 - \\sinh^2(\\phi) = t^2 - 2 t \\ \\cosh(\\phi) + 1 = (t - e^\\phi) (t - e^{-\\phi}).", "ea714edbf15aaf55049729f9fa8ae575": " (x_{n}) ", "ea7152adb399e6e4d528adf4814987bc": "\nb \\ \\stackrel{\\mathrm{def}}{=}\\ \\lambda_{z}^{2} - \\frac{1}{2} \\left( \\lambda_{x}^{2} + \\lambda_{y}^{2} \\right) = \\frac{3}{2} \\lambda_{z}^{2} - \\frac{R_{g}^{2}}{2}\n", "ea71d5452126ef1b0962d0c2df0cd9f6": "\\displaystyle w_t=\\epsilon u", "ea720431a22935412ba226ccd6548300": "c=G=1", "ea7239a51f822ab902c514f8c1510fff": "\\overline{\\tau}=\\frac{\\int_0^\\infty\\tau A_c(\\tau)d\\tau}{\\int_0^\\infty A_c(\\tau)d\\tau}", "ea72de69d94a30cd66b3b97c73a661f5": "I \\,\\!", "ea72e5c1d5460ff9d38fa350b5173d70": "\\cosh nx + \\sinh nx", "ea72e9e35e692265ac4ac30c653d409a": " \\Psi = \\psi(\\mathbf{r}) e^{-iEt/\\hbar} ", "ea72f202348f87c0ea42b6b8cf9f9664": "\\scriptstyle{C^\\infty_0(K)\\subset C^\\infty_0(\\Omega)}", "ea7323154f41b23cc8d481c4fdaf1e58": "\\scriptstyle |c_n|^2 ", "ea734387228d7f7c4457cd547758aae9": "de = C_vdT", "ea7381fbbcefc1145e82f952c17a03d7": "\\int_{-\\infty}^{\\infty} x e^{-a(x-b)^2}\\,\\mathrm{d}x= b \\sqrt{\\frac{\\pi}{a}} \\quad (\\operatorname{Re}(a)>0)", "ea739d76eb0595180986534c2f68f391": "\\frac{\\partial \\; \\|\\mathbf{x}-\\mathbf{a}\\|}{\\partial \\; \\mathbf{x}} = ", "ea73ce6da32db4471ccdd3d91962793d": "G:\\mathbb{R}\\rightarrow\\mathbb{R}", "ea742cf0151eccc935c04dc3b717f66d": "T_\\text{P} = \\sqrt{\\frac{\\hbar c^5}{G {k_\\text{B}}^2}}", "ea742d85077a9aa48d5b38a1d3371e13": "\\arctan (1/x) = \\tfrac{1}{2}\\pi - \\arctan x =\\arccot x,\\text{ if }x > 0 \\,", "ea74afdb0f9be6bd93b20529edbffa3c": "\n\\left(\\mathbf{A}+\\mathbf{UBV}\\right)^{-1}=\n\\mathbf{A}^{-1} - \\mathbf{A}^{-1}\\mathbf{UB}\\left(\\mathbf{B}+\\mathbf{BVA}^{-1}\\mathbf{UB}\\right)^{-1}\\mathbf{BVA}^{-1}\n", "ea74cd40be9099bbcae783b838a9a75c": " n(n-1)~r^{-n}~\\cos(n\\theta)\\, ", "ea74e72d65262372af6a9439e0415944": " \\mathbf{q} = 2\\mathbf{k}_F ", "ea751a8f44d329a0b294ee95a21bf6a6": "\\frac{1}{3^s}\\left(1-\\frac{1}{2^s}\\right)\\zeta(s) = \\frac{1}{3^s}+\\frac{1}{9^s}+\\frac{1}{15^s}+\\frac{1}{21^s}+\\frac{1}{27^s}+\\frac{1}{33^s}+ \\ldots ", "ea7572a2effe774ac78f4548a877a796": "\n \\sigma_{31} = 2G\\varepsilon_{31} \\quad \\text{and} \\quad \n \\sigma_{32} = 2G\\varepsilon_{32} \n", "ea757f50b2a3600a5df31a1caa77fb6e": "P_{total} = p_1 +p_2 + \\cdots + p_n", "ea75c55aa591be9a7e1d74fd60b8c4e5": "x \\mathbb{Q}, x \\in A", "ea75e3c17288bb6a29301cf1d85a0ffe": "r'=bc_0+\\gamma sc_1", "ea75f30abd7ae58c3a670f4408f30fd3": "\\bold c_\\mathrm x\\,", "ea7637b78633138d2d267fc0bdf34f58": "Y[\\mathrm{111}] = \\frac{ 6c_{44} ( c_{11} + 2c_{12} )}{c_{11} + 2c_{12} + 4c_{44}} ", "ea7642aa368b07bca0530ed7bf100c80": " \\gamma = \\lim_{s \\to 1^+} \\sum_{n=1}^\\infty \\left ( \\frac{1}{n^s}-\\frac{1}{s^n} \\right ) = \\lim_{s \\to 1} \\left ( \\zeta(s) - \\frac{1}{s-1} \\right ) = \\lim_{s \\to 0} \\frac{\\zeta(1+s)+\\zeta(1-s)}{2}", "ea766f16d7c3e699d479170dbe2810a7": "GL(P)\\times GL(TM/P)", "ea7692c75e5144ac277f3ba8a293b129": "\\mathbf{f_{0:t+1}}", "ea76a7e5784e9b1bf4d3f45b26ef3eed": " S=(E_S,V_S)", "ea76e028a2295e198978ffe28474e4fc": "\\; \\langle N, v\\rangle \\;", "ea7739c8ab6e07d265fe4b9cb2703bb4": "\\partial_{\\xi} = t^{\\gamma/2} \\partial_x\\quad", "ea773e2aa7a8d4983feca35f354883ba": "x \\in {\\mathbb R}^d", "ea77995a73b25a2dc049344ecf166fc7": "\\log_b n + \\log_b m = \\log_b (nm) \\,", "ea779dd11ee2dbbe93468840ef346841": "\\text{Wi} = \\dot{\\gamma} \\lambda.\\,", "ea77a812db69b7a57439dbf9d569902f": "\\eta(\\tau+1) =e^{\\frac{\\pi {\\rm{i}}}{12}}\\eta(\\tau),\\,", "ea77fd795e23269e6b8c201002e5fbe3": "\\begin{smallmatrix}\\frac{b}{a}\\ =\\ \\sqrt{1\\ -\\ e^2}\\end{smallmatrix}", "ea781cb353917dd031c63db166a7a67e": "a = \\frac{\\cos^2\\theta}{2\\sigma_x^2} + \\frac{\\sin^2\\theta}{2\\sigma_y^2}", "ea78551b2462485338616a075d86fd84": "\n0 = \\delta \\int { c \\frac{d\\tau}{dq} dq }\n= \\int { c \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{dt}{d\\tau} \\frac{d \\delta t}{dq} dq }\n\\,.", "ea789afd5b539cd17c9dc54e444bbd5c": "\\sigma_{av}", "ea789b0413e8db6ec4aba81845ee0f9a": "\\begin{pmatrix} x \\\\ y \\end{pmatrix} \\mapsto A \\begin{pmatrix} x \\\\ y \\end{pmatrix}, \\quad \\text{where } A \\text{ is in } SU(2)", "ea78a00bac987edb4fe699c679955769": "c = \\lambda f \\,", "ea7947dc1ea9196239eecddb0e20434b": "U(\\{\\alpha y_i\\})=\\alpha U(\\{y_i\\})\\,", "ea795b0d1cbcb2e06e105925c7f9f967": "\\kappa = \\sqrt{\\frac{8\\pi n e^2}{\\epsilon k T}} ", "ea7974e955e9e5bcc40b9f4dfa5661cd": "\\bold X = \\bold A(\\theta)\\bold S + \\bold N", "ea79b75e8dfd968b3461021891dcc703": "G_\\phi", "ea79ca97444f32e2ebbe6c97a27f45bb": "R\\colon X\\to Y", "ea79e2418ada5672af14a449727812a3": "G_\\infty\\cong\\bigoplus_{i\\in I}F_P\\;", "ea7a893f835059977be9ad7f4858f403": "{\\tilde{B}}_9", "ea7a92d74880d8f4b43f2ea652257039": "(k - 1)", "ea7b29ca4a7bab3cbda57fd37aad8682": "P \\or P \\vdash P \\, ", "ea7b33aaf78ee96c8fe4f7f7f471e997": "\\Pi^1_m", "ea7b672b0f20e40371fa22ef8cd7f000": "\\mathbf{y}(k) = C \\mathbf{x}(k) + D \\mathbf{u}(k)", "ea7c2bd8af07a5e1b0623cda226f5eb8": "H = \\langle R_H \\mid S_H \\rangle", "ea7c2bf458e8c2675105733cac11a8c5": "R_\\infty = \\frac{m_\\text{e} e^4}{8 {\\varepsilon_0}^2 h^3 c} = 1.097\\;373\\;156\\;8539(55) \\times 10^7 \\,\\text{m}^{-1},", "ea7c5b807838d8d0ab2ee29a2a2cfcd3": "d = a - e,", "ea7c9aa46ec1ce38d9663d73b8ae6deb": "O_{3}", "ea7caa9ee4059f4ac7f453d43a6c5f70": " f(0) \\geq 0~, \\quad |f(x)| \\leq f(0) ", "ea7cd756ec83ea7e155a3847502d8939": "T_{\\mu\\nu}=-\\frac{2}{\\sqrt{-g}}\\frac{\\delta(\\sqrt{-g} L_\\mathrm{m})}{\\delta g^{\\mu\\nu}},", "ea7d7c5ddc711712d5992136a649d28f": "p = \\left(\\frac{-1+\\|v\\|^2}{1+\\|v\\|^2}, \\frac{2\\mathbf{v}}{1+\\|v\\|^2}\\right) = \\frac{-1+\\mathbf{v}}{1+\\mathbf{v}}", "ea7dbc1fb45b26f5f3f10bf04a8a5b6a": "e^{it'\\mu} \\Psi(t' \\Sigma t) \\, ", "ea7dcbf82017b8e99eb8ab5f222415bf": " g(\\alpha_1) \\equiv\n\\lambda_{12}(\\alpha_1;\\phi_1,\\phi_2) - \\lambda_{12} = 0,\n", "ea7e118fbdb48e60a2d4dfd93719cb77": "= -\n\\left.\n\\left( \\begin{array}{cccc}\n \\tfrac{\\partial^2}{\\partial \\theta_1^2}\n & \\tfrac{\\partial^2}{\\partial \\theta_1 \\partial \\theta_2}\n & \\cdots\n & \\tfrac{\\partial^2}{\\partial \\theta_1 \\partial \\theta_n} \\\\\n \\tfrac{\\partial^2}{\\partial \\theta_2 \\partial \\theta_1}\n & \\tfrac{\\partial^2}{\\partial \\theta_2^2}\n & \\cdots\n & \\tfrac{\\partial^2}{\\partial \\theta_2 \\partial \\theta_n} \\\\\n \\vdots &\n \\vdots &\n \\ddots &\n \\vdots \\\\\n \\tfrac{\\partial^2}{\\partial \\theta_n \\partial \\theta_1}\n & \\tfrac{\\partial^2}{\\partial \\theta_n \\partial \\theta_2}\n & \\cdots\n & \\tfrac{\\partial^2}{\\partial \\theta_n^2} \\\\\n\\end{array} \\right) \n\\ell(\\theta)\n\\right|_{\\theta = \\theta^*}\n", "ea7e18d8d3dce14ce5341c517603744c": "w^{(k+1)} \\gets {X^{(k+1)}}^T y ", "ea7e3aac9a5619c7042c41f5d6da3eed": "S_{11} = S_{22} = 1 \\,", "ea7e3ede50d40980fcf9faaaf3976486": "\n I_1 = \\lambda_1^2+\\lambda_2^2+\\lambda_3^2 = \\lambda^2 + \\cfrac{1}{\\lambda^2} + 1 ~.\n ", "ea7e7d8fcb7c4e5ec4a484b3ba1ecb8f": " \\operatorname{E}\\left [\\left|X \\right|^\\alpha \\right ] = \\alpha \\int_{0}^{\\infty} t^{\\alpha -1}\\mathrm{P}(\\left|X \\right|>t) \\, \\mathrm{d}t.", "ea7e9a6b47d4e2ab9ff84e423ab2f802": "{\\mu}_d", "ea7ec03111ed6444b419c968ba866df5": "\\leq \\kappa", "ea7f545e38139b62932b418a31966f8f": "f^{(n+1)}(c)<0 \\Rightarrow c", "ea7f8d9a1cab2de4144bc17280bc90ad": "n\\log_2(1/\\epsilon)", "ea7fab683a0e92127878bb6dbc99a610": " L(u) = -\\sum_{i = 1}^d\\partial_i (|\\nabla u|^{p - 2}\\partial_i u).\\,", "ea7fe085f792a0380741101b2c9fb112": "K \\otimes L", "ea7fe74f85571cd4de03ba2c8d573aa8": "\\begin{align}\n PV &= D_1 e^{-(r)(\\frac{\\Delta t_1}{m})} \n\\end{align}", "ea804ab086b0d5dc535a2f43f44bbdfc": " \nv(P/Q) = \n\\begin{cases}\nv(P) - v(Q) & P/Q \\in {\\mathbb{C}(x,y)}^* \\\\ \n\\infty & P \\equiv 0 \\in \\mathbb{C}(x,y) \n\\end{cases}\n", "ea8070742801c8357c4b32e38949aaeb": "\\begin{array}{cc} \\begin{array}{rrrr} \\\\ \\\\ \\\\ \\\\ j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} & & & & qj & & & \\\\ & & & pj & pk & qk & & \\\\ & & oj & ok & ol & pl & ql & \\\\ & nj & nk & nl & nm & om & pm & qm \\\\ a & b & c & d & e & f & g & h \\\\ \\hline a & o_0 & p_0 & q_0 & r & & & \\\\ n & o & p & q & & & & \\\\ \\end{array} \\end{array}", "ea80981b3eae4329f36d866081c3241b": "-W^{\\mathrm{path}\\,P_0,\\, \\mathrm{reversible}}_{A\\to B} + Q^{\\mathrm{path}\\,P_0,\\, \\mathrm{reversible}}_{A\\to B} = \\Delta U\\, .", "ea80b1f8d0292b02d62d8d4c8bd3a292": "\\varphi_P : \\mathbb{R}/(P\\cdot\\mathbb{Z}) \\to \\mathbb{R}", "ea80d683090c0f33ccddf3c844740d44": "(2,6,3)", "ea80e39455a68e1290cc90a515b9aa11": "\\scriptstyle{Rc}", "ea80fadc83d220b10c1bd6a544a20c00": "N = A \\epsilon^{D_B}", "ea818a9ee4e373cf6b6ab744c65c532e": "\\|\\boldsymbol{x}\\| := \\sqrt{\\boldsymbol{x}^* \\boldsymbol{x}},", "ea818ecfcfcf46c6987d5d9c6026ff5d": "E_{n}^{(1)}=-\\frac{1}{8m^{3}c^{2}}\\langle\\psi^{0}\\vert p^{2}p^{2}\\vert\\psi^{0}\\rangle", "ea81e1e2b2e5af04e70931677cf1de13": "P(n), {n\\in\\mathbb{N}}", "ea820acbdf43116434c2a9e628124ef4": "2^{O(t)} = 2^{t'}", "ea820e13f33ab606bcc7b32588e24373": "\n1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0,\n", "ea82586abbad11739447e4333590394a": "P(x) \\nrightarrow (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\nrightarrow Q(y))", "ea828b5b9890d18bec945f6624a4f57f": "W_m ", "ea828dba60b581c31c6332d3b4273e08": "\\left(\\begin{array}{c}\nX_{5}\\\\\nY_{5}\\\\\nZ_{5}\n\\end{array}\\right)=\\left(\\begin{array}{c}\nX_{4}\\\\\nY_{4}\\\\\nZ_{4}\n\\end{array}\\right)-\\left(\\begin{array}{c}\n0\\\\\n0\\\\\nR\n\\end{array}\\right)\\,,", "ea82a09c553d0dbb335af24c4488e8d3": " \\log( s^2 / n^2 ) = a + b \\log( p ( 1 - p ) / n ) ", "ea82b50081a9a02d33a73320663aae2a": "\\tfrac {1}{2} 0 + \\tfrac{1}{2} 100", "ea82c0938741053c77902e92729250b4": " I^\\prime=q^\\prime/t", "ea82c7596d3aa216535df9509a47245c": "\nL(x,v)=\\frac{1}{2}\\left|\\frac{v-b(x)}{\\sigma}\\right|^2 + \\frac{1}{2}\\frac{db}{dx}(x).\n", "ea82ecd772768f535bf462fc3ff8a623": "\ns_2 = y_1y_2 + y_1y_3 + y_1y_4 + y_1y_5 + y_2y_3 + y_2y_4 + y_2y_5 + y_3y_4 + y_3y_5 + y_4y_5\n", "ea830060810f782b3811ccc453e58561": "2 = \\left( \\frac{\\left(3-x\\right) \\times 2}{3-x} \\right)", "ea836a17d9a9a3936df1c83b78d13282": "\\left( -\\ell \\sin \\theta , y + \\ell \\cos \\theta \\right)", "ea8371ff4876048a176c6987356666b5": " C_{Dmin} = C_{DM} \\neq (C_D)_{CL=0} ", "ea8375b900f098c8d7221520b8bcf6bc": "\\Rightarrow \\lambda_j v_j ' v_i = \\lambda _i v_j' v_i", "ea83a9bc4d66363472620b769aed95d6": "\\scriptstyle \\partial S/\\partial q_k ", "ea83e6976a150501900fcc7fabd41e3a": "z\\mapsto \\log(z)", "ea8426b307aab13e1669f9d5159890a0": " b^2 = f(a) ", "ea84598b201bcda951dafa5cc0c34424": "\\Box\\mathbf A+\\mathbf \\nabla \\left ( \\mathbf \\nabla \\cdot \\mathbf A + \\frac{1}{c^2} \\frac{\\partial \\varphi}{\\partial t} \\right ) = \\mu_0 \\mathbf J", "ea84747a5ea64115c1f8219cc71878cf": "\\langle x_i, x_j \\rangle^m", "ea848829a5e738047ed1e9201c264e4c": "x\\text{-index} = \\frac{x - \\min\\left(x\\right)}{\\max\\left(x\\right)-\\min\\left(x\\right)}", "ea849a00936965eefc02cab29ad07426": "\nx = \\sigma \\tau\\,\n", "ea84a0cd695cf857614c875a138cd514": "A_m(1,3) = 1,3,6,10,15,21,28,35,45,55,\\ldots", "ea84d225d04c929f8d87bb788771257f": "{\\mathbb P}(X_1\\le x_1,\\ldots,X_n\\le x_n)\\le{\\mathbb P}(X_i\\le x_i),\\qquad (x_1,\\ldots,x_n)\\in{\\mathbb R}^n,", "ea84ed5deb1da0269166c1678b9c5a5d": "\\epsilon \\ll t \\ll 1", "ea8507bab582740f3b4060ad93a6a5b4": "\\pi_{\\mathbf R}\\circ F_{\\mathbf P}=\\pi_{\\mathbf P}", "ea852a565e43811f9c729b308b8c3f50": "G(y)=\\int_0^y g(z)\\,dz", "ea8533bcf1de475ca1605213d43d49b5": "f(t) = f_0 k^t", "ea85465b99d000306fbfc0a5cfc3f8dd": "\\mathfrak{P}^{43}", "ea859585c232dcfbd6dab31f97753155": "\\quad x_k", "ea85c75222a9eda1e7ba8c06688cca00": "\\underbrace{\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad}_{}", "ea85cadc11ad28f104a56764c4282753": " E_{\\alpha+1}(n) = E_\\alpha^n(2) ", "ea85ecda234dba228545aae2bac9d73b": "s>0\\,", "ea85ff017deb966079145f54b33bd962": " \\Sigma(\\mathbf{x}) = \\sigma(\\mathbf{x},\\mathbf{x}) .", "ea860353cc525f8d4f5152e93788f224": "y=\\sum w_i x_i", "ea8612c2989e6133899412c93ee21e57": "\\langle \\cdot \\rangle", "ea8629effc6a227244ee13d8b5ff4e26": "\\beta> 0", "ea8634d26f0f8929616db0959dd3b775": "r=\\sqrt{\\rho^2+z^2}", "ea866770e1a5645ef086f61fce4ccab3": "G_{13}", "ea86ad59818434ac35de01ca0842e149": "b = \\frac{c.r}{\\delta r + 1/a }", "ea86b10d40da0795a3199cc980b2189b": "\\tau _{Auger7}(t,x) = 10\\cdot \\tau _{Auger1}(t,x)", "ea86c067185f03c5aed90a8eba9c462b": "H_2(\\mathrm{S}_n,\\mathbf{Z}) = \\begin{cases} 0 & n < 4\\\\\n\\mathbf{Z}/2 & n \\geq 4.\\end{cases}", "ea86e02bbba9fc904bde564fcb2b6be8": "\\begin{bmatrix}3 & 7 & 2 \\end{bmatrix}", "ea870aa06069ebc43897b9796d5fe1d4": "\\left\\langle F_{,i} \\right\\rangle = -i \\left\\langle F \\mathcal{S}_{,i} \\right\\rangle", "ea8735863b06229380986948f43a342e": "\n{42.2 J \\over 40 fs} = {42.2 J \\over 40*10^{-15} s} \\approx 1*10^{15} J/s = 1 PW\n", "ea8764fdf887e0e546fa2e66540ab4fa": "\\ln P = \\frac{2-\\gamma}{1-\\gamma}\\ln M + \\frac{3\\gamma - 4}\n{1 - \\gamma}\\ln R - \\gamma \\ln T", "ea8767bdf3c1e927bf8edfe329e02ae6": " S_1(t) \\ldots S_N(t) ", "ea8772ff987854511e22ed284d6905df": "[0,h]", "ea88506e776df5d97f133782377b512d": " 0.024265 \\times W^{0.5378} \\times H^{0.3964}", "ea8866f19218794a98cd8f265bdc7441": "\\Gamma(X, \\mathcal{O}_X) = k", "ea887a09de271279e06a97e6761e1808": "\\lambda\\downarrow0", "ea88e5e647ba7de6fb759841105670cd": "\\{v_i\\}_{1\\leq i \\leq k}", "ea88f484ce0744b85d19a5810771d3f2": "a_{t+1} = k_1 \\frac{a_t}{ \\left(1+k_2 a_t\\right)^c}. ", "ea88fcae110b17d551b4950af70e9e54": "{\\mathfrak H}\\;", "ea89088c61ab50935a2a7c4d9fba44d1": "\nE^2 = \\frac{m_0^2 c^4}{1-v^2/c^2}, \\quad p^2 = \\frac{m_0^2 v^2}{1-v^2/c^2}.\n", "ea891026f238c4483986bf4efffa1ac9": "120^\\circ", "ea8969ba67344f4baae8262a033f894d": " {1\\over C} \\ d_F(x,y) \\leq d_{E}(x,y) \\leq C \\ d_F(x,y). ", "ea8a08df96710910a422a0467b865dea": "~ A/A_0 +G/G_0=1~.\\ ", "ea8a1a99f6c94c275a58dcd78f418c1f": "AE", "ea8a5b2ba6d86eb668a4a9c1b8c4151c": "\\begin{align}|B| & {} = b_{i1} C_{i1} + b_{i2} C_{i2} + \\cdots + b_{in} C_{in} \\\\ & {} = b_{1j} C_{1j} + b_{2j} C_{2j} + \\cdots + b_{nj} C_{nj} \\\\\n& {} = \\sum_{j'=1}^{n} b_{ij'} C_{ij'} = \\sum_{i'=1}^{n} b_{i'j} C_{i'j} . \\end{align}", "ea8a8136c3d9b072d0eb3a978ffd48a1": "\\! w=-1", "ea8a855dec7be39f377259222d8b4f75": "f(x_i,y_i)", "ea8ad3416a9bb654a2653037dc4f3512": "\\begin{align}\n\\vec{x}(t + \\Delta t)\n&= \\vec{x}(t) + \\vec{v}(t)\\Delta t + \\frac{\\vec{a}(t) \\Delta t^2}{2}\n+ \\frac{\\vec{b}(t) \\Delta t^3}{6} + \\mathcal{O}(\\Delta t^4)\\\\[0.7em]\n\\vec{x}(t - \\Delta t)\n&= \\vec{x}(t) - \\vec{v}(t)\\Delta t + \\frac{\\vec{a}(t) \\Delta t^2}{2}\n- \\frac{\\vec{b}(t) \\Delta t^3}{6} + \\mathcal{O}(\\Delta t^4),\\,\n\\end{align}", "ea8adcfc4d67aa5ccdce459537aa8367": "-dU\\,", "ea8af0f6ae7d515a8fa48f421eeda9f0": "T^4 = \\frac{3}{4}T_e^4\\left(\\tau + \\frac{2}{3}\\right)", "ea8b7e95bdb35df184f1caa63787b561": " \\rho_{\\rm R} = \\frac{\\pi^2}{15} \\, T_\\gamma^4 (1+z)^4 \\left[ 1 + \\frac{7}{8} N_{\\rm \\nu} \\left( \\frac{4}{11} \\right)^{4/3} \\right],", "ea8ba613d35d6acd160237129acb0b76": "d_p = d \\bmod\\ (p-1)", "ea8bf18d527137ad97c404a4cfbb9a26": "f_\\epsilon:\\partial V\\longrightarrow\\R", "ea8bf784b1c25c6f2cf42b281d93b61d": " s = c t \\,\\! ", "ea8ca06c10884e41d0eb02d836dddf33": "\\Gamma_i^1 = \\Gamma_i -\\Gamma_1\\,\\left(\\frac{{C_i}^{\\alpha}\\, - {C_i}^{\\beta}\\,}{{C_1}^{\\alpha}\\, - {C_1}^{\\beta}\\,}\\right)\\,.", "ea8ce288db5a3a9d8e350c51816c5c49": "I_{D,Sat} = \\frac{W}{L} \\mu\\, C_{inv}\\frac{(V_{G}-V_{th})^2}{2}", "ea8cf42c5a670b8d1c289e1d9838f34a": "Q(x_0,x_1,x_2,x_3) = x_0^2-x_1^2-x_2^2-x_3^2.", "ea8d61db2f1ba192d4309ec217ed82e0": "\\scriptstyle\\eta\\,\\perp\\,x,", "ea8d767c4ac13037255c7e1b28262d1c": "z_R = \\min \\{c_R(x) : x \\in X_R \\subseteq \\mathbf{R}^{n}\\}", "ea8df6b884d1bead6aa69af7be3dabbd": "e=(u,v)", "ea8e0a053d05cb3bc994ee9b31852667": "v_c\\,\\!", "ea8e2250003d466d6e1fd729476b7d00": "2^{m_2-1} 0.", "ea914b8139109dbcd7b1fcea61796cb0": "U_\\text{pol} = -\\dfrac{e^2\\alpha_\\text{d}}{(4\\pi\\varepsilon_0)^2r^4}", "ea916168e3790ddb94e49fd87e9c0627": "\\hat{x}(t)\\,", "ea916bce970e445716dcd3f675af4b80": " z = {\\ln{q \\over g}} ", "ea91db96cda71e6c448040e592f34811": "U_E=q^2/2C=Q^2\\cos^2(\\omega t + \\phi)/2C\\,\\!", "ea91de0a0f7eeb1f7fb488d354c9ed01": "\\mathrm{C_H = [H] + \\Sigma r \\beta_{pqr}[A]^p[B]^q[H]^r - K_w[H]^{-1}}", "ea91eb9e94011f1ceab33f8412e725a0": "(xy)z = (xz^{-1})(zyz)", "ea91ebecf98f8d8e92da6a74961fd814": "\\scriptstyle 6 = (1 + \\sqrt{-5}) ( 1 - \\sqrt{-5})", "ea91f6f8be818c7409b4b85955e00c96": "T_{m'}", "ea92373cc59110dc4c8a4f44a9b4070f": "\\int_{-\\infty}^{\\infty} f(x)\\, dx = \\lim_{\\Delta \\to 0} \\sum_{i = -\\infty}^{\\infty} f(x_i) \\Delta", "ea9243aee97fe6cdcb482ed6770299d6": "= c^2 (s'-t')^2 - (x'_1-y'_1)^2 - (x'_2-y'_2)^2 - (x'_3-y'_3)^2 ", "ea92605d84697415d5fa188a03a01f72": " \\mathbf{Image(x,y) = }", "ea9281edd750e51c4631c27e50f19d22": "\n\\mbox{distortion power factor} = {1 \\over \\sqrt{ 1 + \\mbox{THD}_i^2}} = {I_{\\mbox{1, rms}} \\over I_{\\mbox{rms}}}\n", "ea9285049631b6073ddc5a345ab0cbdd": "\\frac{\\Delta \\mathit B}{\\mathit B} = \\mathit g ", "ea92883c5f60d14a7ae84be393127d3f": "r = x_1\\,\\bmod\\,n", "ea9288707554a38826b6b473154ceaa9": "a \\times 1 = a,\\,", "ea92cd4c5510848c66142fc3a9cfcf1d": "(ts)[x := r] = (t[x := r])(s[x := r])", "ea9346376f7b9472e98aad904f499eb5": "\\frac{1}{11} = 0.0909\\dots = 0.\\overline{09}.", "ea93daff8e65755063ff3a453d23f3bb": "f_a(k) = a^2 k (1-a)^{k-1} \\, ", "ea93f230166a705c20e1b713d0b46156": "{\\delta \\over \\delta_c} = Sc^{1/3}", "ea945b06dc7e28eccb760ce1760094ec": " \\scriptstyle \\beta_1=0 ", "ea94ad2e4c51971becd736637ce7d134": "z(n;t)=O(n^{2-1/t}).", "ea94bb87dc1c0631198c25a77a33b9b5": "v_{\\theta} = (1 - e^{-r^2/(4\\nu t)})\\Gamma/(2 \\pi r).", "ea950745c486d2187abfc348d6b6f848": "\n\\begin{align} \n& \\Omega=\\frac{v}{r} \\\\\n& \\frac{d\\Omega}{dr}|_{R_{0}}=\\frac{d\\frac{v}{r}}{dr}|_{R_{0}}=-\\frac{V_{0}}{R_{0}^{2}}+\\frac{1}{R_{0}}\\frac{dv}{dr}|_{R_{0}} \\\\\n\\end{align}\n", "ea951291b69416e8fb582a88920848f8": "\\int_0^\\infty e^{-x}e^{-x}\\,dx=\\int_0^\\infty e^{-2x}\\,dx=\\frac12.", "ea9590b721b28731f1780848c25dccd0": "(E \\wedge X)_n = E_n \\wedge X", "ea95d24e774cc3236df297b62927c174": "\\text{Passer Rating}_{\\text{NFL}} = \\left ({mm(a) + mm(b) + mm(c) + mm(d) \\over 6} \\right ) \\times 100", "ea95e1fc2cd1ac5435037aeaaf882060": "\\rho \\frac{D \\mathbf{v}}{D t} = \\boldsymbol{\\nabla} \\cdot \\boldsymbol{\\sigma} + \\mathbf{f}\\,,", "ea9627e1424a7128b77795b039b02ce4": "i=1, \\ldots, n", "ea964755d93e373aaf5b830c1a95baef": "\\operatorname{str}(T_1 T_2) = (-1)^{|T_1||T_2|} \\operatorname{str}(T_2 T_1)", "ea966fd82331f8a9ce17d9c33597c125": "I_{L3}=I_P\\sin\\left(\\theta-\\frac{4}{3}\\pi-\\varphi\\right)", "ea96756b3dadddfe82491da2f94f21c1": "|\\Phi^+\\rangle_{AC}", "ea96f7adc05c503098138cb458e11f1e": "A^{ik}\\ ", "ea978f775de6ae82162fa8454255f8a6": "e^{\\alpha_j x}", "ea98d5f6e783cbaa99f91f274b61b5a1": "S_{\\sigma^2}=\\sum_{i=1}^m (X_i-\\overline X)^2,\\text{ where }\\overline X = \\frac{S_{\\mu}}{m} ", "ea990fc45d6c966b0e4ddc087449c9a5": "D(E) = 0", "ea9916567747a4b6a657a540efc30442": "\\sum _x \\cos ax = \\frac{1}{2} \\cot \\left(\\frac{a}{2}\\right) \\sin ax -\\frac{1}{2} \\cos ax + C \\,,\\,\\,a\\ne n \\pi", "ea99275276484b0bd89c831fa4032c4d": " i \\in S \\Leftrightarrow \\chi_s(i) = 1", "ea993bd0f3fc6b781e207882c672aff1": " \\prod_{p>2} \\Big(1 - \\frac{p+2}{p^3}\\Big) = 0.723648... ", "ea994efb1370ebde4cbfac6f17d7f94e": "\\frac{\\sigma_t}{\\rho} ", "ea9956410c81b9460d63a15c3a1e99c4": "\\overline{\\alpha} = \\frac{\\Delta \\omega}{\\Delta t} = \\frac{\\omega_2 - \\omega_1}{t_2 - t_1}.", "ea997ee52151dc8474147d3bc4d44eb9": "D\\ \\tau\\dots\\tau", "ea9a604bf029a5557d3de4cbf27d5a77": "\\mu = V M_s", "ea9a7da95996b7b1f52e81a1450536ac": "f\\circ g=f", "ea9ad69fbfb3a8e276aed838aa36219b": " (\\nabla \\cdot T)_i = \\partial_j T_{ij}", "ea9adc7859337ac76d2053e44eb85d79": "C_n = \\frac{1}{n+1}{2n\\choose n}.", "ea9b092624e7252b785bebac7d41c51a": "m\\,=\\,n", "ea9b21764ccf2746dcb5ece2113af7e5": "U_{-t} = \\pi \\, U_{t}\\, \\pi", "ea9b245370666faea2f8921ee54caf40": "E_{n,l} = - h c R_{\\infty} \\frac{{Z_{\\rm eff}}^2}{n^2} \\ ", "ea9b7c00b69821e61d82de8a6f3b625d": "\\langle\\mu|\\xi\\rangle = \\bar{\\mu}_1\\xi_1+\\bar{\\mu}_2\\xi_2", "ea9ba6b515f91c22c0275c7e69b36985": "R(\\theta ) =\n\\begin{pmatrix}\nr & t'\\\\\nt & r'\n\\end{pmatrix}", "ea9ba95792831901cbcf088a1c0c59d1": "P_{\\mu }(n)", "ea9bbbd1277f18fe25ecd3fdfea4644f": "{\\color{Blue}BCT} = 5.876 \\times {\\color{Red}ECT} \\times \\sqrt{U \\times d}", "ea9bd46e404afa570ae4358fd8b28d42": "{\\rho}^*_{xx'}=\\frac{N{\\rho}_{xx'}}{1+(N-1){\\rho}_{xx'}}", "ea9bf1c713926bebf17d502e4877e12a": "x_1 \\dots x_k ", "ea9c57d9bfdf107711d2fed4b3f4243a": "\\ln |x|\\,", "ea9c7dafabdc0c43f5e7a57bfdca9818": " P = {{\\mu_0 e^2 a^2} \\over {6 \\pi c}}. ", "ea9cf5d27d0d6fa254d4a2bce6b3fefc": "Z_{F \\cdot G}(x_1, x_2, \\dots) = Z_F(x_1, x_2, \\dots) \\, Z_G(x_1, x_2, \\dots)", "ea9d47c35ea5a4cbcd968485160cb104": "\\mathbb T^2", "ea9d64e263bcbf39e294fe7812cab948": "e_j(n)=d_j(n)-y_j(n)", "ea9d89996bf71aa550255fb01521ba03": "K= \\begin{pmatrix} 3 & 3 \\\\ 2 & 5 \\end{pmatrix}", "ea9d93355948eced55f2366afdfee32c": " I_D \\approx I_{D0}e^{\\begin{matrix}\\frac{V_{GS}-V_{th}}{nV_{T}} \\end{matrix}}, ", "ea9d9b6cb977217934b5234c5d1e75f1": "V\\in\\mathcal{V},\\ni x\\,", "ea9ddfd8cc0dba67aaa92dbe6c8fa1a4": "X/\\!/G", "ea9dfe52c47d60cd696065e641944162": "\n\\mathbf{D}^{-1}\\mathbf{C}(\\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C})^{-1} = (\\mathbf{D}-\\mathbf{CA}^{-1}\\mathbf{B})^{-1}\\mathbf{CA}^{-1}\\,\n", "ea9e60bfcb952846496a4dd029127c18": "17^7+76271^3=21063928^2\\;", "ea9e80b3e443f6d4f0afc75e2c097b66": " {x^2 \\over 2} = {\\sqrt{5} \\over 2}, ", "ea9ee793090bbb829738bc1b968ba7f1": "\\sum_{m=-\\ell}^\\ell Y_{\\ell m}^*(\\theta,\\varphi) \\, Y_{\\ell m}(\\theta,\\varphi) = \\frac{2\\ell + 1}{4\\pi}", "ea9f105a392140db179a3253b767bbd0": "\\mathbf{d} = \\mathbf{r}_+ - \\mathbf{r}_- \\ ,", "ea9fd7d51a3216c21a73c74b6fae7d21": "\\frac{S_l}{S_s} = \\frac{\\alpha}{2\\pi}", "ea9ff63c3bbbc5b1ef1a295354158990": "F_{\\theta_1}(x) \\leq F_{\\theta_0}(x) \\ \\forall x", "eaa06606645c3db54f6f9b8687319946": "\\langle \\langle x,w,y\\rangle ,w,z \\rangle = \\langle x,w, \\langle y,w,z \\rangle\\rangle", "eaa06a51b167f3773c67ff32d5dc58cc": " \\mathfrak{spin}(7,\\mathbb C)", "eaa0825ff17c1a84c367543f86e3224f": "x^2-3x+1", "eaa0905f504b537660cee2ec58f67570": " \\mathbf{E}\\left[e^{\\lambda X}\\right] \\leq e^{\\frac{1}{8}\\lambda^2(b-a)^2}", "eaa0bf4550092e34bf8f2f72a9a8f8ef": " T : (-x, -y, -z) \\rightarrow ((-y) (-z), (-z) (-x), (-x) (-y)) = (y z, z x, x y). ", "eaa0c7b99b8a9631388954c19a3f9659": "R_{DM}=\\frac{k}{1\\cdot n}=\\frac{k}{n}.", "eaa0fcaa02018c7a4138b0072756d1a0": "V_\\max = V_f + V_r = V_f + \\rho V_f = V_f (1 + \\rho).\\,", "eaa0ff0631c009fc2074f512629a29e0": " \\phi_1, \\dots, \\phi_l, \\psi_1, \\dots, \\psi_m:\\; {\\Bbb R}^n \\mapsto ({\\Bbb R}^d)^*", "eaa144234c89d83ffefe840a63380391": "C = \\frac{|S_A-S_B|}{\\sigma_o}", "eaa178a4f51d2d355c560b129dad7762": " \\nabla_{\\mu , \\sigma}V_\\nu ", "eaa19d0c2ac9fde38d591ef06a615d34": "R[t] \\to S, \\quad f \\mapsto f(x)", "eaa1d8abb25364c620129edcb8a19011": "J = (1-r_{\\mathrm{av}}) A_0 T^2 \\mathrm{e}^{-W \\over k T}", "eaa1dc950405632bd8741ad194b6b821": "(b_{ij})", "eaa2002e9663bc43972ca2a21ef5d294": " \\left( \\sum_{n=1}^\\infty |a_n|\\right)^4 \\leq \\pi^2 \\sum_{n=1}^\\infty |a_n|^2 \\, \\sum_{n=1}^\\infty n^2 |a_n|^2~.", "eaa2287b776ec41eb72d62904bb5c66d": "\\frac{4}{3} \\pi r_s^3 = \\frac{V}{N}.", "eaa258676862df4032d0ab28e574fb5e": "a + ib \\equiv \\left(\\begin{matrix} a & -b \\\\ b & a \\end{matrix}\\right). ", "eaa31358e3d47655bf1976c8a7037cd5": "\\scriptstyle\\epsilon \\;\\in\\; \\left\\{-1,\\, 0,\\, 1\\right\\}", "eaa35ac4cf15a0f4f12e8251cd06fd52": "A^+ = V\\Sigma^+ U^*", "eaa35e3d71d54087969fb503b0e3c4c1": "\\dot{p}_i = - \\partial H / \\partial x_i", "eaa3de0971b53d942e2283c34ee207c2": "T = Y + 20", "eaa437a13984c763a7fd54eea4977f3a": "\\mathrm{moser} < f^{3}(4) = f(f(f(4))), \\text{ where } f(n) = 3 \\uparrow^n 3.", "eaa465fca6dae0dd7524ad4b51228e95": "q\\geq 2", "eaa480a9fb9be4ab02732634e9a69c62": "\\mathrm{Ai}(z)\\sim \\frac{e^{-\\frac{2}{3}z^{\\frac{3}{2}}}}{2\\sqrt\\pi\\,z^{\\frac{1}{4}}}", "eaa4ab54ccefc112e06fd32cf15fd5e4": "\\neg (\\neg (\\neg A\\lor B)\\lor (C\\lor (D\\lor E)))\\lor (\\neg (\\neg D\\lor A)\\lor (C\\lor (E\\lor A)))", "eaa4d7d163141e28c38563e45d634557": "\\alpha=\\frac {\\mu_0 c e^2}{2 h } \\ . ", "eaa4edd649bed6ffe6ac59fb8b846bb6": "\\mathcal{E}(u,v)=\\mathcal{E}(v,u)", "eaa52fe3767087ff23dd321100322b25": "1, 5, 19, 41, 109, \\ldots", "eaa56f7f3f63eff2bde9973e5e25e2fc": "\nU = U_{\\rm 2} = U_{\\rm 1}\n", "eaa5c1eda456d3192bdd7616fcd7e802": "\\mathbf{z}=(Z_1,\\ldots,Z_n)", "eaa5cb809c595971321a97dcbb027bff": "F : V \\times V \\rightarrow U ", "eaa5d681d8b51b7293b4172cac256b31": "\\scriptstyle S_i", "eaa5fe392bba02c500ed7ba6b1e8e1d7": "\\mathcal C \\, |\\psi\\rangle = | \\bar{\\psi} \\rangle.", "eaa60a7949f15b5493ca97635801a7ed": "v(d)", "eaa64d899dc6659a8bf0f58dd75342c7": "rx'^2 + 2sx'y' + ty'^2", "eaa66b016987a6502aa0d951af7984c0": "\\frac{ht}{u} \\pmod{p}\\;", "eaa691606057bdaf707957084f232b1b": "\\{\\lambda_n(A)\\}_{n=1}^N", "eaa6d5caf8c0e2e7fe230d0228fc93b1": "SLG = \\frac{TB}{AB}", "eaa723cc86656d67d8304992446b1b3e": "\\scriptstyle 2^n-1", "eaa73dc0d44d0a492d7ddb5f83c52619": "A = \\frac{a\\,\\alpha\\,P}{R^2\\,T^2}", "eaa74f0e6ece7c7287ffcef4eabc298f": "y_{1}^{\\star}", "eaa79c11341f9acad9bff21e6014bf35": "V_{m} = \\frac{RT}{F} \\ln{ \\left( \\frac{ rP_{K}[K]_{o} + P_{Na}[Na]_{o}}{ rP_{K}[K]_{i} + P_{Na}[Na]_{i}} \\right) }", "eaa7b505b05542b5e1b114f52c3562e7": "\\forall_x \\exists_y G(x, y).", "eaa7e6c5168891c297fcaaae55b4c1e9": "\\frac{d\\beta}{dt}", "eaa80d29ac966716f8083fa4345a7ab0": "\\mathbf{A \\cdot B} = \\left( \\begin{matrix}A^0 & A^1 & A^2 & A^3 \\end{matrix} \\right) \n\\left( \\begin{matrix} -1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{matrix} \\right)\n\\left( \\begin{matrix}B^0 \\\\ B^1 \\\\ B^2 \\\\ B^3 \\end{matrix} \\right) ", "eaa81bc5669eae5d5fc247f2861cabd8": "\\mu_\\pi\\,\\!", "eaa828063cc8ef8640855a70c3c436d9": "~m~", "eaa8601e9c5550e0064ffde3bbc3cae4": " g(k)=i \\sum_{n=0}^{\\infty} \\left(\\frac{1}{2}-\\rho_n \\right)\\delta(k-n) ", "eaa862c7b6ba8e76bb2f677d74796cac": "\\frac{\\sqrt{\\pi}}{2}", "eaa8c489288330755033c3bc5459dfd7": "\\mathbf{d} = \\begin{bmatrix}\n(\\lambda+\\alpha)(C_{10}^{j+1}+C_{10}^{j}) \\\\ 0 \\\\ 0 \\\\ 0 \\\\ (\\lambda+\\alpha)(C_{20}^{j+1}+C_{20}^{j}) \\\\ 0 \\\\ 0 \\\\ 0 \\\\ (\\lambda+\\alpha)(C_{30}^{j+1}+C_{30}^{j}) \\\\\n0\\\\\n0\\\\\n0\\end{bmatrix}.", "eaa8ddafc08e01a1c70408b027ae1459": "\\displaystyle{(x_1\\otimes \\cdots \\otimes x_k,y_1\\otimes \\cdots \\otimes y_k) =k!\\cdot \\prod_{i=1}^k (x_i,y_i).}", "eaa8ed9691a6e380afe7b246059a7b38": "\\sqrt{J}", "eaa9d1c336b17c616d4e90f637a59c4c": "(\\beta_2+1)I_{B2} =\\left(1 + 1/\\beta_2 \\right) I_{C2} = \\frac{V_{BE1} - V_{BE2}}{R_2} = \\frac{V_T}{R_2} \\ln \\left(\\frac {I_{C1}I_{S2}}{I_{C2}I_{S1}}\\right)\\ , ", "eaa9eb66bea4bc558a2a7f92b0e61eed": "\\pi_1(\\mathbf{T}^2) = \\pi_1(S^1) \\times \\pi_1(S^1) \\cong \\mathbf{Z} \\times \\mathbf{Z}.", "eaa9ff13812b8aa55ce30ce564639ec8": "i_{0}=m-m_{0}+1", "eaaa5280a3da9f3c76f21c2a91c8d171": " V(z) = \\sum_{\\begin{smallmatrix}d < z \\\\ d \\mid P(z)\\end{smallmatrix}} \\frac{\\mu^2(d)}{g(d)} . ", "eaaa52a711b621ad2cc3a4f7f300a5be": "c\\in(0,\\infty),p\\in\\mathbf{P}", "eaaa534ac59153d0f6da26bdcc0658db": "g(t) = e^{-t} \\sum_{n=0}^\\infty f_n t^n.", "eaaa7e25ae640df0a3c4910726cbaa2d": "\\gcd(a,b):= \\text{if}\\;b=0\\; \\text{then}\\;a\\;\\text{else}\\; \\gcd(b, \\text{rem}(a,b))", "eaaaa42ce20e8a340605da7a5ca0c970": "\\scriptstyle J_z = \\hbar m_j", "eaaaa99e8d5f9ed52db8dbd60e70b99e": "A = \\begin{bmatrix} a_1, & a_2, & \\ldots, & a_n \\end{bmatrix}", "eaab055aec3ba9d9b322c6a02dac7e1d": "J \\to \\infty", "eaab5aa920371d688f7655deb3e50385": "~|n\\rangle~", "eaab6d7e257145dde9160bf92a01a14e": "\\mbox{for }-\\pi \\le u \\le \\pi,\\quad -\\pi \\le v \\le \\pi \\,", "eaab83092687b5f2d02091bc978d102c": "t^{3/2}", "eaaba0593fce410b720a8d15f2aa0b65": "\\mathbf{Z} \\!\\,", "eaabc224d7cf62914ffdb92d7ade6aa3": "\n \\frac{1}{n} \\sum_{\\tau=1}^{n} \\left(V_\\tau^B + V_\\tau^S\\right)=V=\\alpha\\mu+2\\epsilon \\; .\n", "eaabd195ed74495afa3995b5e0cc1b55": "5-3=2", "eaabe882209818bf7011c9f09427087a": "C_\\text{in}(y_i^{\\prime\\prime}) \\ne c_i", "eaabeaab9b5acb5c864a1c2991748b35": "\\beta = \\rho_{a,b}(\\sigma_a/\\sigma_b)", "eaabfe90c8969a76f9e0efd62f968783": "\\mathrm{a\\ A + d\\ D \\longrightarrow b\\ B}", "eaac054e157ee806b72fb401f7ec86fc": "K \\supseteq F", "eaac382ecbc7e01daeeeebe34cc03fee": " \\frac{ -\\log_e(2)} { \\lambda^\\frac{ 1 }{ 2 } } \\le S \\le \\frac{ 1 }{ 3 \\lambda^\\frac{ 1 }{ 2 } }", "eaacb85415b9ede3663b62d1c0c72cc1": " P^*_i ", "eaacff4dda9f7a02f5a0de07fd823d97": "\\begin{bmatrix}\\mathbf{\\hat r} \\\\ \\boldsymbol{\\hat\\theta} \\\\ \\mathbf{\\hat z}\\end{bmatrix}\n = \\begin{bmatrix} \\cos\\theta & \\sin\\theta & 0 \\\\\n -\\sin\\theta & \\cos\\theta & 0 \\\\\n 0 & 0 & 1 \\end{bmatrix}\n \\begin{bmatrix} \\mathbf{\\hat x} \\\\ \\mathbf{\\hat y} \\\\ \\mathbf{\\hat z} \\end{bmatrix}", "eaadbb41958b9dbab905c5cee7025b48": "\n \\mathcal{M} = D\\left[\\mathcal{A}\\left(\\frac{\\partial \\varphi_1}{\\partial x_1} + \\frac{\\partial \\varphi_2}{\\partial x_2}\\right)\n - (1-\\mathcal{A})\\nabla^2 w^0\\right] + \\frac{2q}{1-\\nu^2}\\mathcal{B}\n", "eaadc3ae9d1f612ffba77d5c56cf5d8b": "f(tx) = f(tx+(1-t)\\cdot 0) \\ge t f(x)+(1-t)f(0) \\ge t f(x)", "eaadcfb44f878cf7221da0e0049f6c17": "\\mathfrak{a}\\subseteq R[X_{0},X_{1},\\ldots,X_{n-1}]\\,", "eaae1aca677a43f1aba13c559bf6cf30": " x(t+1)\\ \\stackrel{\\mathrm{def}}{=}\\ f\\left [ x(t)\\right ] = 4 x(t) \\left [ 1-x(t) \\right ] ", "eaae69313e85e708b39328f12c29c9b4": "\\sigma_I^{(k+1)} = t \\sigma_I^k", "eaae848e27c7a66a45f7dbd9b87a94e4": "\\mathrm{MV(t)}=K_p{e(t)} + K_i\\int_{0}^{t}{e(\\tau)}\\,{d\\tau} + K_d\\frac{d}{dt}e(t)", "eaaea0a5c3ac2386b830a38f8922a58d": "H(j\\omega)", "eaaeeecf5548168d648ee52005cc048d": "\\nabla \\times \\mathbf{H} = \\mathbf{J}.", "eaaf79dd88b57950fc9f6b908047b984": "\\begin{array}{cc}\n \\begin{array}{r} \\\\ 3 \\\\ \\\\ \\end{array}\n &\n \\begin{array}{|rrrr} \n 1 & -12 & 0 & -42 \\\\\n & 3 & & \\\\\n \\hline \n 1 & & & \\\\\n \\end{array}\n\\end{array}", "eaafae7dd9e6a84541de1637565ec778": "12a + b", "eaafb875d449c8b124e5c6df5f80f9f7": "\\tau(\\theta)=\\kappa^\\prime(\\theta)=\\mu", "eaafdd775bd9ea33ec886b22b41898e5": "d(\\alpha x, \\alpha y) = |\\alpha| d(x,y)", "eab06ed1bc6bb4601205f9510c376c3b": "z^{2n}-1", "eab139ffcaaacf8dec96f71585404548": "\\mathbf{E} = - \\mathbf{\\nabla} \\phi(\\mathbf{x}) ", "eab15fb77049df2ada82d963f5acb9f6": "1 \\leq d \\leq \\zeta(n)/\\gamma(n)", "eab1720de657a1fe99616a7ea69b88cb": "\\sigma = F/A", "eab1baaccb4e835cf85813b8ef405ac6": "|n| = n", "eab1d95676e1d4e5cd27ea87e3f3d5a1": "\\scriptstyle\\mathbf{y} ", "eab249ec929886840f6645878b726103": " C = \\sum_{n=1}^N 2^{|\\mathcal{S}(n)|} ", "eab327c8f47d40e0a5bee93ac2c2d52a": "x_{Bi}=200", "eab33ab5f052c14970a0bf15aea7d168": "a=\\text{percentage change depicted in graph}", "eab33ca535e39d25716486526b4b3b13": "R_{\\alpha} g (x) = \\mathbf{E}^{x} \\left[ \\int_{0}^{\\infty} e^{- \\alpha t} g(X_{t}) \\, \\mathrm{d} t \\right].", "eab341da0926f3382058988849fdb7ef": "\\sin 3^\\circ = 3\\sin 1^\\circ - 4\\sin^3 1^\\circ", "eab347150701e33291bcfc3ddbc40c9a": "{\\mathrm{f}}", "eab3581a974a8d2c110df2cd7fa49a81": " \\zeta = 1 ", "eab3e595b4453be9ec2f6ba82793dfb1": "dU = C_{V}dT + \\left[ \\frac{n R T}{V} - \\frac{n R T}{V} \\right] dV.", "eab4c6826b6aab62fc2a3b7e90661d22": " \\mathcal{F} T^t = M^t \\mathcal{F}, ", "eab4e4310cd33384510fbfb868c1e4e4": "||f||_\\infty = \\sup_x|f(x)|", "eab5123d19e3311c94c187171d29380f": "Z=|Z|e^{j\\varphi}", "eab551f948473b7ef4c74e4ae4a23954": "\\int_{CV} \\nabla (\\rho\\mathbf{u}\\phi)dV \\,= \\int_{A} \\mathbf{n} \\cdot (\\rho\\mathbf{u}\\phi)dA", "eab55d6f932d3fecdf7b3f6b3afa9be1": "\n \\mathbf{x}\\ \\sim\\ \\mathcal{N}_k(\\boldsymbol\\mu,\\, \\boldsymbol\\Sigma).\n ", "eab592179e674cf2f0ffa9c28aa64a63": "\n(D)_{ij} = \n\\begin{cases} \n \\lambda_i, & \\text{if }i = j\\\\\n 0, & \\text{else}\n\\end{cases}\n", "eab5a6b67f9a8ce2360316789064ab74": "D(f) = O(R_1(f)^2)", "eab5b711dda2f9e4e52a1f0367c3f02b": "100 \\cdot a + b", "eab5cd2a27998db5d1b6d3ac065935be": "C.\\,A", "eab5d6e8423f9aa7226f724dd5afbb73": "a^\\hat{r} = \\frac{m}{r^2 \\sqrt{1-\\frac{2m}{r c^2}}}", "eab603232fafc696f07ffbe850516050": "\\dot{v}_3 = {1 \\over {C_3 R_2}} v_4 - {1 \\over {C_3 R_2}} v_3 ", "eab6845273c53463bc6e253b9386e102": "\\lambda(\\cdot)", "eab6fdf90a7fb79ecfa3b50f9377e5fa": "\\mathbf J_{\\mathrm{f}}", "eab6fe5baa3aa6ef29280ac1430ca9a2": "f_{X_i}(x_i) = \\frac{f_i(x_i)}{\\int f_i(x)\\,dx}.", "eab7164e96be8f529bbb5886914987c2": "W=\\sum_{i=1}^k Y_i", "eab7194ac36b8e118d14342abe544d89": "r_{\\pm}", "eab76444773333225e879751e5c9f7f5": "\\tfrac{46468}{324}=143\\tfrac{136}{324}", "eab76c8515f0d5abcf15d407ed3a9e53": "\\zeta_n\\in\\mathcal{O}_k,", "eab80f416cc6a3bfc351a8c6f62ca00f": "s= \\rho(\\boldsymbol\\theta;z_1,\\ldots,z_m)", "eab80fbcc7803aa19a909ac230437dc2": "P-P_0", "eab84379455833ba66cffb5892438ac0": "\\{y_1, y_2, y_3, y_4\\}", "eab8537f17f8bee2c60328ddbb064052": "L(\\theta|Y=12)=\\begin{pmatrix}11\\\\2\\end{pmatrix}\\;\\theta^3(1-\\theta)^9=55\\;\\theta^3(1-\\theta)^9.", "eab8d66ecb113e740dbc59f2b1b80957": " \\operatorname{cl}(C) = C ", "eab90171e3a910f35aaa5acad7c7824b": "C_2 = D_2", "eab908a8c601bf34e842ccb03d4c5559": "m_g", "eab90a7ff7a4778a5f534691819d1810": "a_j\\in\\,S^{\\pm}\\,\\forall\\,1\\leq\\,j\\leq\\,n.", "eab9377e9fe452827ac7b026ef00d9ca": "Q(x;k)=\\frac{\\gamma(k/2,x/2)}{\\Gamma(k/2)}\\,", "eab9651f13e4c972443502a9a173155b": "a_0=\\frac{4\\pi\\epsilon_0\\hbar^2}{me^2}", "eab98c207f6a0c6b0412432052e8fb13": "(1.059463094359295264561825)^{12} \\approx 1.99999999999988 ", "eaba857a5500418c0a93586724961662": "\\displaystyle (ab)^2=a_1^2b_2^2-2a_1a_2b_1b_2+a_2^2b_1^2.", "eabac3f8210d3d9453903e1c233144be": "\n= \\delta \\int d^4 x \\; e \\; e^{M [\\gamma} e^{\\beta]}_N C_{\\gamma M}^{\\;\\;\\;\\; K} C_{\\beta K}^{\\;\\;\\;\\;\\; N}", "eabb1ebfc1096be5ff59ddc35ae41a22": "\\frac{V_{dd}}{2}=V_{dd}(1-\\frac{3}{2}e^{\\frac{-1}{RC}\\frac{T}{2}})", "eabb3f5729e2b1569daa43095a221115": "C = \\frac{\\dot{m}}{K} + (C_{o}-\\frac{\\dot{m}}{K}) e^{-\\frac{K \\cdot t}{V}} \\qquad (9)", "eabb77d3fb3daafbc710e6252ea9810c": "\\lim_{ \\varepsilon \\to 0^+ }\n \\sup_{ \\Vert \\delta x \\Vert \\leq \\varepsilon } \n \\left[ \\frac{ \\left\\Vert f(x + \\delta x) - f(x)\\right\\Vert }{ \\Vert f(x) \\Vert } \n / \\frac{ \\Vert \\delta x \\Vert }{ \\Vert x \\Vert }\n \\right],", "eabbce89b20364245bb11eba3e39d831": "\\boldsymbol z=\\{z_1,\\ldots,z_m\\}", "eabbdeba1ce1aa2c985b7283ca8673de": "\\phi_3(v) = \\frac{4}{6} = \\frac{2}{3}.\\,", "eabc4524bbf11ecad96cab7a312089b1": "\\mathbf{I}_1", "eabc73c533521cf2377da4d1405b245e": "\\frac {V_r}{V_t} = \\frac {e_g \\cdot \\sin u\\ -\\ e_h \\cdot \\cos u}{\\frac {p}{r}}", "eabcc93a245d7a93934736e1518f4026": "\\begin{pmatrix} \\alpha_1 \\\\ \\alpha_2 \\end{pmatrix} =\n \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\n\\begin{pmatrix} \\omega_1 \\\\ \\omega_2 \\end{pmatrix},", "eabcc9c849be17a54ec3b68e386fc487": "e = \\left [ -\\frac{12}{\\pi^2} \\sum_{k=1}^\\infty \\frac{1}{k^2} \\ \\cos \\left ( \\frac{9}{k\\pi+\\sqrt{k^2\\pi^2-9}} \\right ) \\right ]^{-1/3} ", "eabd3495eb11f03340fe798ba6df0e92": "{\\mathcal O} = \\{z_1, \\ldots z_n\\}", "eabd61b63329d4c8b53978cd35e786b3": " \\sigma_T = \\sigma_S + \\sigma_A ", "eabd6c6070bddafba8dee20ea9ff585e": "\\Sigma\\theta/(\\theta+\\rho A-1)", "eabe0e70e4e23b8551515b9597b039ea": " [\\alpha/Fe] = \\log_{10}{\\left(\\frac{N_{\\alpha}}{N_{Fe}}\\right)_{Star}} - \\log_{10}{\\left(\\frac{N_{\\alpha}}{N_{Fe}}\\right)_{Sun}} ", "eabe2a2f7035c793f48b3885a2ea0009": "y = \\mathbf{w} \\cdot \\mathbf{x} + b", "eabe6ea0e233e12dc161b6e46f4acbd0": "f\\mapsto Pf(z)", "eabeb592ae684898c3c541101686e593": "L_m\\,", "eabedbea5e33bb9679e6bf5cffb401fe": " g\\cdot f(v)=f(g^{-1}v), \\quad g\\in G, v\\in V.", "eabee4715993b152b20169c1d61d1f78": " \\mathbb{P}( (X - \\mathbb{E}(X))^2 \\ge a^2) \\le \\frac{\\operatorname{Var}(X)}{a^2}, ", "eabfe980bf965f350a60d02fe299be81": "f_c(\\beta_1) = \\beta_2", "eabfef93ef10ccc15427b399527c6f9a": "\\psi_{1,i} \\otimes \\cdots \\otimes \\psi_{n,i}", "eabff19f2f0222464c522c4fb6da298b": "f_2: \\mathbb{N} \\longrightarrow\n\\mathbb{N}", "eac0ac4fa8f5eb96ca9ce403d2bcd1cf": "c_{4}", "eac0c0441b38a38b9009725954857d88": "\n F_{11} F_{22} - F_{12} F_{21} = 1 \\quad \\implies \\quad F_{22} = 1\n ", "eac0f37961e13927bcd638d5c974da61": "\\frac{1}{R}\\frac{d}{dr}\\left(r^2\\frac{dR}{dr}\\right) = \\lambda,\\qquad \\frac{1}{Y}\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\left(\\sin\\theta \\frac{\\partial Y}{\\partial\\theta}\\right) + \\frac{1}{Y}\\frac{1}{\\sin^2\\theta}\\frac{\\partial^2Y}{\\partial\\varphi^2} = -\\lambda.", "eac10b7ff007aa29cd78321945f9f73b": "Sq^I = Sq^{i_1} \\ldots Sq^{i_n},", "eac13864a01f83f1aa9e0bc18764a24e": "\\mbox{cosh}(s)", "eac16964481b94f043d960d16ae39b6f": "p(\\nu, \\lambda) = \\mathbb{P}(\\lambda^{(n)}=\\lambda~|~\\lambda^{(n-1)}=\\nu) = \\frac{f^{\\lambda}}{nf^{\\nu}},", "eac19ae2bdcfe4b3217dddf4727ab182": "q_2^* = \\frac{a - b \\cdot \\frac{a + \\frac{\\partial C_2 (q_2)}{\\partial q_2}- 2 \\cdot \\frac{\\partial C_1 (q_1)}{\\partial q_1}}{2b} - \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2b},", "eac2e1c3a123b1e0611bd00648efae4c": "(X+Y)^{[p]} = X^{[p]} + Y^{[p]} + \\sum_{i=1}^{p-1} \\frac{s_i(X,Y)}{i}", "eac327264487e304aa96fbe21d6f00fd": "c_{in}(i) = (0,\\ldots,0,1,0,\\ldots,0)", "eac377e7f92fd88ec10ec0ce3ceeed16": "Square numbers end on 9", "eac37d6aa93e6298bae796d41d2ac6ec": "\\models \\phi", "eac39f6f7d3fe516552fc7ab0eaed0fc": "\\mathcal{G}_3", "eac3c062e654f028416aa2f784cd493e": "\\hat n ", "eac467fceb0111ebff85bad891e094f0": "I = \\frac{\\mathrm{d}Q}{\\mathrm{d}t} \\, .", "eac47bc5951a2a359449daad09dec929": "\\operatorname{cov}(W_s,W_t) = \\min(s,t),", "eac487df22312fa889bd44a213dd8dca": " V_R = I_R R \\,\\!", "eac4ca748956dea4eed9d96afa72e88d": "K_2 = -9K_1/4", "eac50702c55d65dbc3acddd9309d545b": "\\Delta_L", "eac5616946bdedda260e2c394ea7d1f2": "q\\le'r \\to q \\le_Q r", "eac5709ca7cb8bf2f69fd88cebfeeea6": "\\zeta(\\bar{a},\\bar{b})+\\zeta(\\bar{b},\\bar{a})=\\phi(a)\\phi(b)-\\zeta(a+b)", "eac58b9a7e56c168384f044b70839c4d": "\\dot H_k", "eac58e64b85b81e6a907bbefe9187580": "T_qQ = E_q^+ \\oplus E_q^0 \\oplus E_q^-", "eac5b4e1b73941d68f1a5deabca0d9a7": "\\lambda t", "eac5b6111939f306b1ad0c1fde5f76b9": " \\operatorname{get-lambda}[x, x\\ q = f\\ (q\\ q)] ", "eac5e7ba578f657ea96994f34a6877b0": "(c)\\text{ }R_{i}\\bigcap R_{j}=\\varnothing \\text{ for all }i=1,2,...,n.", "eac6896d96b4b5ccd368ac48a93a4422": "|\\mathbf{x} \\times \\mathbf{y}| = |\\mathbf{x}| |\\mathbf{y}| \\sin \\theta, ", "eac6f45445182224c273834bcc89d981": "Q^{n}(c)", "eac6f4d33c4f43e40a07a1a20f39c774": "{\\Gamma \\vdash e_1{:}\\tau_1{\\to}\\tau_2\n \\;\\;\\;\\;\\;\\;\n \\Gamma \\vdash e_2{:}\\tau_2{\\to}\\tau_3\n }\\over{\\Gamma \\vdash e_2\\circ e_1 : \\tau_1{\\to}\\tau_3 }", "eac738d0ac020a93e44dffd94718901c": "\\dim K_{+}", "eac7c1106b3f19a6e32e65e631a5d0bc": "\\scriptstyle D_F", "eac7f16cc26dd7cdfa9ad21eaeabc7cf": "v=wbv'", "eac837c6b893899c212349754942bbbd": "\\frac{d}{ds}u = a \\frac{\\partial u}{\\partial x} + \\frac{\\partial u}{\\partial t} = 0.", "eac84e7f735e6121c03b44a34c6496ca": "D_{Ae}", "eac88dec0af0d856162ee1518deba1ee": "b=V_c-\\frac{RT_c}{4P_c}", "eac896151a22d31a60cceba6deea3cbb": "g(n)", "eac8a1fa0b2886ea138aab89d12c1313": "g(x)=x-\\frac{f(x)}{f'(x)}", "eac8c7f3cae3b19d0bf6a7f599490dfc": "\\frac{CE}{EA}=\\frac{|\\triangle BCO|}{|\\triangle ABO|},", "eac944bb380648bff01c46756d0a5a77": "\\lambda_0\\,", "eaca178063176669aee4adbe44d983b9": "~\\sigma_{\\rm e}~", "eaca1fe5205c95db1dc4249f6f5fa724": "\\vec{e}_1", "eaca6f245304c67e9fa6b7ae01858f5d": "X_i=-\\frac{1}{k}\\frac{\\partial S}{\\partial x_i}", "eacaa8dc2ae3cb5138b5b31c8de9ef34": "\\ \\frac{T_0}{T_0^*} = \\frac{2\\left(\\gamma + 1\\right)M^2}{\\left(1 + \\gamma M^2\\right)^2}\\left(1 + \\frac{\\gamma - 1}{2}M^2\\right) ", "eacac5e8bcd76049d827cf10618f7e2c": " n =m + m ", "eacad005b08ced7e3268f1f24147bdf6": " A / B = -(A / {-B}) = -(-A / B)= -A / {-B} \\, ", "eacb5e6ab7085bcf826e75da4ea2ca00": "V_\\alpha \\subseteq M", "eacb8f559d2e29e119076f2f5f2c2072": "\\scriptstyle \\alpha_i \\,\\in\\, k", "eacbc69a93baed4841abf9961031b311": "\n\\begin{align}\nk(t)&=\\frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/3}}-\\frac{1}{2}\\left[\\frac{1}{(x'y''-x''y')^{2/3}}\\right]''\\\\\n&= \\frac{4(x''y'''-x'''y'')+(x'y''''-x''''y')}{3(x'y''-x''y')^{5/3}} -\\frac{5}{9}\\frac{(x'y'''-x'''y')^2}{(x'y''-x''y')^{8/3}}\n\\end{align}", "eacc475c0c6010c2d61f742507ef1f8a": "x^{12}+y^{12}+z^{12}+22\\left(x^6y^6+y^6z^6+z^6x^6\\right)+220\\left(x^6y^3z^3+y^6z^3x^3+z^6x^3y^3\\right).", "eacc6a194738def66bf7231ce3b94eb3": "\n\\begin{align}\n \\sigma(x, y) &= \\sigma(x, x^2) \\\\\n &= \\operatorname{E}\\!\\left[x \\cdot x^2\\right] - \\operatorname{E}[x] \\cdot \\operatorname{E}\\!\\left[x^2\\right] \\\\\n &= \\operatorname{E}\\!\\left[x^3\\right] - \\operatorname{E}[x]\\operatorname{E}\\!\\left[x^2\\right] \\\\\n &= 0 - 0 \\cdot \\operatorname{E}\\!\\left[x^2\\right] \\\\\n &= 0. \n\\end{align}\n", "eacca985973be6b187950f878df627c0": "f_a(y) = a^2 + ay + y^2.\\,", "eaccd98db0ac9da70c753d7d6a2477e5": "\\Delta P =f \\frac{\\rho V^2}{2}\\frac{l}{d}, ", "eaccdc1240753689148e003a139d6d93": " e^{A}Be^{-A}=B+[A,B]+\\frac{1}{2!}[A,[A,B]]+\\frac{1}{3!}[A,[A,[A,B]]]+\\cdots \\equiv e^{\\operatorname{ad}(A)} B.", "eacce1fed74108bdf6e606b2edf26ddc": "q=10", "eacd201649d34a5c09ed71d8fb265994": "\\{ e \\}", "eacd79d0b20bdf1095bda46d253f0e48": "\\, Y ", "eacd9f2d118d061f634448eb382c9ae6": "S=\\{a,b,c,d,e\\}", "eacdb39c894f7cac9d10f1b3c22c15d0": "s_K", "eacdcde7c52ff52fb403e3c2f87f24a3": "\\ \\|x\\|_p=\\left(|x_1|^p+|x_2|^p+\\cdots+|x_n|^p\\right)^{1/p} \\,.", "eace5c6491efd63fd3637568544477d4": "P\\to P/G\\to X", "eaced23b20bda42aeb0071718daf0e8d": "\\left(\\sum_{i=m}^n i\\right)^2 = \\sum_{i=m}^n ( i^3 - im(m-1) )", "eaced9e9630994bf1702290f832769ee": "{\\mathbf Z} = \\left ( \\begin{matrix}\n-3 & +2 & -1 & -1 & -3 & -1 & -3 \\\\\n-2 & +1 & +1 & +3 & +1 & +3 & +3 \\\\\n-1 & -1 & -3 & -1 & -3 & -3 & +2 \\\\\n-1 & -3 & -1 & -3 & -3 & +2 & -1 \\\\\n-3 & -1 & -3 & -3 & +2 & -1 & -1 \\\\\n+1 & +3 & +3 & -2 & +1 & +1 & +3 \\\\\n+3 & +3 & -2 & +1 & +1 & +3 & +1 \\end{matrix} \\right ).", "eaceef3ef402b970842afe1ddb3d54b5": "|n(x^\\mu_0)\\rangle", "eacf03dd7d353c355350540f06fc64a5": "\\overline{\\mathbf{GQ}}", "eacf18fcf8b3360f8ee6d3c529f7fc35": "\\mathbf{v}^{\\textrm{MMSE}}_k = \\frac{( \\mathbf{I} + \\sum_{i \\neq k} q_i \\mathbf{h}_i \\mathbf{h}_i^H )^{-1} \\mathbf{h}_k}{\\|( \\mathbf{I} + \\sum_{i \\neq k} q_i \\mathbf{h}_i \\mathbf{h}_i^H )^{-1} \\mathbf{h}_k\\|} ", "eacfc190bf1b003fa3165d7344d3bafe": "\nb =\n-\\Delta x^2\\begin{bmatrix} g_{11} , g_{21} , \\ldots , g_{m1} , g_{12} , g_{22} , \\ldots , g_{m2} , \\ldots , g_{mn}\n\\end{bmatrix}^T.\n", "ead001b48ad6e470259a761262fbf932": "F_r = \\mu N_r \\,", "ead01eb1550e82865d98142ad2d8b63d": "x,y\\in[0,\\infty)", "ead0342c2eca5b801a32525b4eaf8455": "R \\triangleq \\left\\{\\mathbf{x} \\in \\mathbb{R}^n: x_i = c_i \\ (\\text{for } i\\in S) \\text{ and } a_i \\leq x_i \\leq b_i \\ (\\text{for } i \\notin S)\\right\\}", "ead046a1641bc7f94ce03d802c29a865": "A \\lor \\lnot A", "ead05812f9bb2808fb6e7785263f8a1f": "\\frac {1}{\\sqrt 2}", "ead07bf3ea757c547b02733ccb146dcf": " \\ln\\mathcal{L}(\\beta) = \\sum_{i=1}^n \\bigg( y_i\\ln\\Phi(x_i'\\beta) + (1-y_i)\\ln\\!\\big(1-\\Phi(x_i'\\beta)\\big) \\bigg)", "ead0a40950b0322bb11334e6b50ef435": "\\begin{align} \\hat{H} & = -\\frac{\\hbar^2}{2}\\sum_{i=1}^{N}\\frac{1}{m_i}\\nabla_i^2 + \\sum_{i=1}^N V_i\\\\\n & = \\sum_{i=1}^{N}\\left(-\\frac{\\hbar^2}{2m_i}\\nabla_i^2 + V_i \\right) \\\\\n & = \\sum_{i=1}^{N}\\hat{H}_i \\\\\n\\end{align}", "ead1015bd066071e6f7cbb323ced3bc5": "\\pi_N:P_N\\to Q_N", "ead12384ed5f1094e5117836ed16a6fa": "\\mathbb{Q}^+", "ead149de4276e53afea51f64086e98c0": " \\ge ", "ead15eadc320cbb4505b8c0f7e57a20b": "f(x) = \\frac{1}{\\sqrt{2\\pi*5^2}} * e^{ -\\frac{(x-0)^2}{2*5^2} }", "ead16a2360795971de62ba74f44c5c7c": "C_{high}", "ead178bc217038b3a67ca53e213eea58": "\n\\frac{\\mathrm{d}}{\\mathrm{d}t}{\\partial{L}\\over \\partial{\\dot x}} - {\\partial{L}\\over \\partial x} = F\n", "ead29bba3fd19f2e671b219a56a68496": " \\mathbf{R_k} ", "ead2c1d1c3602428fd46d67a9055c99c": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log (\\frac{\\varepsilon}{3.7D} + \\frac{5.158log(\\frac{Re}{7})} {Re(1 + \\frac{Re^{0.52}}{29} (\\frac{\\varepsilon}{D})^{0.7} }\n", "ead2f2d0acd5ef3bded5cf52f3b37ff4": "\\hat\\mu(\\xi)=\\int_{\\mathbf{R}^n} \\mathrm{e}^{-2\\pi i x \\cdot \\xi}\\,d\\mu.", "ead30092ce1d7a2f5056a301082bc146": "\\displaystyle l_n(x)=(-1)^nL_n^{(x-n)}(x).", "ead365355f44e37726480c08580889b9": "b = 0.75", "ead37ee3310d16b701a5dc8a3e48217e": " v = 100 * { (1/p + 1/q + 1/t)-1 \\over 1/p + 1/q + 1/t}", "ead3b6829ef4190657f71310597be4eb": "+\\left(\\mathit{u}_3-\\mathit{u}_4\\right)=\\left(9 - 4\\right)=+5", "ead3fd3aa2949ed2449e81d413775478": "E(\\mathbb{F}_p)", "ead4123f7d32088d3892895ebf5828ef": "\n\\begin{array}{rl}\n\\min & p \\\\\n\\text{subject to} & \\Omega_b(A)\\leq pQ, \\\\\n& A\\in \\mathcal S_n(\\mathcal A_{1\\cdots n}, \\mathcal C_{1\\cdots n}),\\\\\n& Q\\in\\text{co-}\\mathcal S_n(\\mathcal B_{1\\cdots n},\\mathcal D_{1\\cdots n})\n\\end{array}\n", "ead41b510c36708fe23b68083dd75f0b": "D \\, \\frac{\\mathrm{d}^{2}u}{\\mathrm{d}\\theta ^{2}}=\\sec \\theta \\tan^2 \\theta + \\sec^3 \\theta = \\sec \\theta (\\sec^2 \\theta - 1) + \\sec^3 \\theta = 2 D^3 u^3-D \\, u.", "ead430a97348b908fc3e5153e406a871": "dS_2 \\times S_2", "ead47a78e068fc7548659b5fc6fb2f6c": "\\Re(\\rho) \\le 1/2", "ead47edf2cde86f73a053215c84e25af": "R_{nl}(r)", "ead4987ce8b1fbe2ae245d146665e154": " \\frac{\\partial r} {\\partial s} = \\cos\\phi, \\frac{\\partial r} {\\partial s'} = -\\cos\\phi' ", "ead4ac7549071d42c5bcfcbba4b9f62b": "F_{weight} = -\\rho \\cdot g \\cdot V.", "ead4b48aef877e1afd7c78cfe7a94e6b": " v = v_L + v_C \\,", "ead4ced109234b2f906f72270171d0ab": "r_{0}", "ead4d3897195eceff0a670e1f97d8ef6": "f^{\\mathfrak T_{\\Phi}} (\\overline {t_0} \\ldots \\overline {t_{n-1}}) := \\overline {f t_0 \\ldots t_{n-1}}", "ead4dffe9eae40c6191d5e3bc91ee7e1": "\\forall x, y, z \\,(xRy \\wedge yRz \\rightarrow xRz)", "ead555af0218f63229fe3b85b9138183": "\\ell_R= \\theta R, \\ell_r=\\alpha r", "ead651c12fb1d16b7c27d819b6c72e46": "\\tfrac{8}{4} + \\tfrac{3}{4}", "ead6a05cbc018b67ad67055daf01ec4f": "|\\alpha_1|<|\\alpha_2|=\\min{}_{m=2,3,\\dots,n}|\\alpha_m|", "ead6baa9f8f9b1bf9d5ccd58d26d22b0": "\\widehat{h} = \\widehat{\\mathbf{S}} \\cdot \\frac{c\\widehat{\\mathbf{p}}}{E} ", "ead6f9abe95b7a1a850c14747c4740ed": "\\mathit{SS}_\\text{total} = \\mathit{SS}_\\text{between} + \\mathit{SS}_\\text{within},", "ead6fa0310a1c45c308b3af59d62f53e": "[z^{2n}][u] g(z, u) =\n[z^{2n}] \\frac{1}{1-z} \\left( \\frac{z^{n+1}}{n+1} + \\frac{z^{n+2}}{n+2} + \\cdots \\right) =\n\\sum_{k=n+1}^{2n} \\frac{1}{k} = H_{2n} - H_n.", "ead71d06751545c76df827be170779ff": "\\chi'(G) = \\Delta(G)", "ead7408ed188f41f069eb2761acddbbb": "a^{(p-1)/2} \\equiv \\pm 1 \\pmod{p}.", "ead7bd47edc87d972b00e140e68ceb42": " (\\lambda p.\\operatorname{de-let}[\\operatorname{let} q : q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p])\\ \\operatorname{get-lambda}[p, p\\ f\\ x = f\\ (x\\ x)] ", "ead7d36891399fce39181a658aba8b15": "{0,1,2}", "ead7e65e04a63fa239967b63a73e4749": "\\begin{align}\n& \\hat{L}^2 = \\hat{L}_x^2 + \\hat{L}_y^2 +\\hat{L}_z^2 \\,\\, \\rightleftharpoons \\,\\, \\bold{L}\\cdot\\bold{L}= L^2 = L_x^2 + L_y^2 + L_z^2,\\\\\n& \\hat{S}^2 = \\hat{S}_x^2 + \\hat{S}_y^2 +\\hat{S}_z^2 \\,\\, \\rightleftharpoons \\,\\, \\bold{S}\\cdot\\bold{S}= S^2 = S_x^2 + S_y^2 + S_z^2,\\\\\n& \\hat{J}^2 = \\hat{J}_x^2 + \\hat{J}_y^2 +\\hat{J}_z^2 \\,\\, \\rightleftharpoons \\,\\, \\bold{J}\\cdot\\bold{J}= J^2 = J_x^2 + J_y^2 + J_z^2,\\\\\n\\end{align} ", "ead82c7c3d773de00ddd8c9dc0cc1f24": "\\tau=-1/4t \\,", "ead850529c5bb4b1776990abb95ef0f4": "n_s-1", "ead8ae04ae6b1d8a2454c7212426bbbd": "(\\Sigma_{SFR})", "ead8c78604e0576240d366cd0a3daec2": " f(n;b,c) \\propto n^{-b} \\exp(-c(\\log n)^2) ,", "ead8c91e49025ec64b969c227b09b702": "\\pi: V - 0 \\to \\mathbf{P}(V)", "ead8f2716b3f3f6d8cc12e110cadd48c": "\\forall \\alpha, \\beta \\in \\langle 0,...,k \\rangle, \\alpha \\oplus \\beta = min(k,\\alpha+\\beta)", "ead90eea41945aa0882b608faad74371": "f = 2^{n/12} \\times 440 \\,\\text{Hz}\\,", "ead92f18ed15f637057088bbf2f4f235": " \\prod_{c\\in g} a_{|c|} = \\prod_{k=1}^n a_k^{j_k(g)}", "ead97f3349dd93d0e76712f9470c8380": "h^{998}(password)", "ead9a20f2ce249b84d3892fe35cb182a": "\\mathcal{I}_{ij}(\\theta)\n= \\int f(X; \\theta) \\frac{\\partial \\log f(X; \\theta)}{\\partial\\theta_i} \\frac{\\partial \\log f(X; \\theta)}{\\partial\\theta_j} \\,dX\n= \\int \\frac{1}{f(X; \\theta)} \\frac{\\partial f(X; \\theta)}{\\partial\\theta_i} \\frac{\\partial f(X; \\theta)}{\\partial\\theta_j} \\,dX\\,.", "eada4da81081a3793b9405b57f136744": "\n \\begin{bmatrix} M_{11} \\\\ M_{12} \\\\M_{13} \\end{bmatrix}\n := \\int_{-b/2}^{b/2} \\int_{-t/2}^{t/2} \\begin{bmatrix} x_2\\sigma_{13} - x_3\\sigma_{12} \\\\ x_3\\sigma_{11} \\\\ -x_2\\sigma_{11} \\end{bmatrix}\\,dx_3\\,dx_2 \\,.\n ", "eadab120981d6ec69e3fdff524758ba9": "\\begin{align}\n\\boldsymbol{V}_i&=\\begin{bmatrix}\n\\boldsymbol{v}_1 & \\boldsymbol{v}_2 & \\cdots & \\boldsymbol{v}_i\n\\end{bmatrix}\\text{,}\\\\\n\\boldsymbol{\\tilde{H}}_i&=\\begin{bmatrix}\nh_{11} & h_{12} & h_{13} & \\cdots & h_{1,i}\\\\\nh_{21} & h_{22} & h_{23} & \\cdots & h_{2,i}\\\\\n& h_{32} & h_{33} & \\cdots & h_{3,i}\\\\\n& & \\ddots & \\ddots & \\vdots\\\\\n& & & h_{i,i-1} & h_{i,i}\\\\\n& & & & h_{i+1,i}\n\\end{bmatrix}=\\begin{bmatrix}\n\\boldsymbol{H}_i\\\\\nh_{i+1,i}\\boldsymbol{e}_i^\\mathrm{T}\n\\end{bmatrix}\n\\end{align}", "eadaf62f9b4734dc0335512d870197f7": "\\lim_{a\\to 0}\\Vert\\tau_a f-f\\Vert_{L^p(\\mathbb{R}^n)} = 0", "eadb0357fda5b7c77c4f5aaf9618cceb": " n=10", "eadb222aea594e57dee1c1dd343dc05f": "t=-(d_0/4)(-d_1/5)^{-5/4}", "eadb996bc281895415babbed06886911": "\\forall z\\in\\{0,1\\}^{q(n)}\\,\\Pr\\nolimits_{y\\in\\{0,1\\}^{p(n)}}(M(x,y,z)=0)\\ge2/3.", "eadbb7212079b71938e1f943c23d6949": "\n\\begin{align}\nR_\\alpha^A & = \\xi^{-1} Q_\\alpha^A + \\xi \\sigma_{\\alpha \\dot{\\beta}}^0 \\bar{Q}^{\\dot{\\beta} B}\\\\\nT_\\alpha^A & = \\xi^{-1} Q_\\alpha^A - \\xi \\sigma_{\\alpha \\dot{\\beta}}^0 \\bar{Q}^{\\dot{\\beta} B}\\\\\n\\end{align}\n", "eadbef3aae20d4d82dca7fcfc07657f9": "\\frac {b}{a}", "eadbf26e3cb9eae366fbb8c8593ac9e9": "\\pi \\,", "eadbfad0a9cc27acc1c8ab98656e0020": "\\{ \\}, \\O \\empty \\emptyset, \\varnothing \\!", "eadc301e453319fa700caa857da8ef4a": "\\mathrm{d} X_{t} = b(X_t) \\, \\mathrm{d} t + \\sigma (X_{t}) \\, \\mathrm{d} B_{t},", "eadc96beac6f8ea825e8d2a9a784ba2c": " \\hat{H} = \\frac{\\hat{p}^2}{2m} + V(x) \\,, \\quad \\hat{p} = -i\\hbar \\frac{d}{d x} ", "eadcb74b9752ea7dfdda94db2f4cdee1": "\\mathfrak{R}''", "eadce7b3da7d59146b2236ec4b0b0b00": "\\int_\\gamma \\rho(h \\circ \\gamma)\\,ds", "eaddb08d4935ffe9ce061174797e3e82": " |\\psi(t)\\rangle = U(t) |\\psi(0)\\rangle.", "eadde12a19c0eb23ea31956c1a07d288": " { {p}_{n} } ", "eaddf300f479d8ffc75aa4770e2bbbda": "\\mathbf{1}_A(x) :=\n\\begin{cases} \n1 &\\text{if } x \\in A, \\\\\n0 &\\text{if } x \\notin A.\n\\end{cases}\n", "eadf37b1f8e741cae43225ed2aae0e23": "\\ \\phi_{i,j,k} = \\phi(r_i, s_j, t_k) ", "eadf51d10f26e6cf6a198631d596c759": "g\\colon Y\\to Y", "eadfdad75560e11598ed7ef21e05837b": "\\|\\vec{p}\\|\\le g = \\mu p_n\\,\\!", "eadff482d1f71e41485f90ffce11e639": "\\mathrm{Games}\\ \\mathrm{behind} = \\frac{(\\mathrm{Team A's}\\ \\mathrm{wins}-\\mathrm{Team B's}\\ \\mathrm{wins}) + (\\mathrm{Team B's}\\ \\mathrm{losses} - \\mathrm{Team A's}\\ \\mathrm{losses})}{\\mathrm{2}}", "eae0130fd3ea008de3b1a9d188c81284": "q^i ", "eae064fe3b2771017f9264c377dd155c": "(f^{-1})^k", "eae0a06a666b0bd5d632ac06a8c59272": "\n\\begin{align}\n\\frac{dy}{dx} & = \\tan \\theta \\\\\n dx & = \\cot \\theta \\, dy \\\\\n\\cot \\theta dy & = - \\frac{g}{k^2} \\cos^2 \\theta \\,d\\theta \\\\\ndy & = - \\frac{g}{k^2} \\sin \\theta \\cos \\theta \\,d\\theta \\\\\n & = - \\frac{g}{2k^2} \\sin 2 \\theta \\,d\\theta \\\\\n y & = \\frac{g}{4k^2} \\cos 2 \\theta + C_y\n\\end{align}\n", "eae0f97fbd7156017102ee5515c28f3a": "\\frac{ d}{ dt}p(t) = -\\frac{\\partial}{\\partial q}\\mathcal{H}", "eae1185126ebdbd278be6d4d5c17810c": "f_\\mathbb{H}^\\prime(\\alpha)", "eae1b49d2bff2e490af7d241f21a6173": "x\\in \\tilde{\\mathbf{E}}^+", "eae1f8b745c1deb7c5a7916bc4276720": "\\omega,\\ \\omega_\\omega,\\ \\omega_{\\omega_\\omega},\\ \\ldots.", "eae1fbf241a011caa0d07699926906b7": "\\gamma_m < \\varphi_{\\beta_m}(\\gamma_m) \\,.", "eae2032faeab1de2cb798574dcf400ba": "\\triangle^{(k)}y_{i} := \\triangle^{(k-1)}y_{i+1} - \\triangle^{(k-1)}y_{i} \\mbox{ , } k \\ge 1.", "eae210c6572156417b4d49455df27d56": "f(\\dots,x,\\dots,x,\\dots) = 0 , ", "eae25926235d11bc72aee2af69cbfc06": "\n\\delta(c_i,c_j) = \\sum_r S_{ir} S_{jr}\n", "eae28cd027de6244147ad890d52ee1cb": "t\\rightarrow \\infty", "eae2a0dc577af8a8fba371f3f9e7e8d5": "m=\\frac1R", "eae2f189bd1416ceeaf6207fcff21118": "f \\star g = g \\star f", "eae31556a07164618af80d5548e2f9f8": "\\overline{F} = \\overline{A}\\,\\overline{B} + \\overline{A}\\,\\overline{C} + \\overline{A}D", "eae3451d23162534bb5a2665c36f7099": "V(x - v^b(k),k) = V(x,0)", "eae34a4b198bb90578851dd7c5855e5b": " PV = k,", "eae3d132b7b70d62a5e8df1663727446": "\\mathsf{ATR}_0", "eae3e3db630f40f5489ed432fa73c10c": "0\\rightarrow A_i\\rightarrow B_i\\rightarrow C_i\\rightarrow0", "eae3f25e52879c59acdae7b463311d82": "20.75 \\mu_B", "eae4034dbbc57b5c15c9cffaed775f17": "K(k)= \\frac \\pi 2 \\theta_3^2(q),", "eae418e1ad338a69a7daeb7c5c231ba1": "N=n_{1}+n_{3}'+n_{4}'-n_{2}-n_{3}-n_{4}", "eae49e8591bf7a2eaf528d963b4a50dc": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{11 \\choose 3}{4 \\choose 1}^3 \\end{matrix}", "eae4c4dbf4177c799a1cfaf7cb3d75f4": " RCM = EVB \\times (H_i - H_m) ", "eae4f8d68707bedbfea5b150609c447b": "f(x) \\le f(y)", "eae51a18389051ee670d8b6ab2eee18e": "x_{(n-1)/2}", "eae542ef833c602cc7fdf3444ebe3620": "v=\\sum_{b \\in B} l_b b", "eae579a609b3965be0577fd21075fcf0": "\\varepsilon(n) = \\begin{cases} 1, & \\mbox{if }n=1 \\\\ 0, & \\mbox{if }n \\neq 1 \\end{cases} ", "eae6085f2b51a2857c4112b08c5b2fb8": "\\mathbf{1}_{\\{\\tau_n>0\\}}X^{\\tau_n}", "eae61b2c36d93183f67573432450a011": "(5\\times7)", "eae61b6899c24807c67e7bf2f2cfe9ac": "nE_b", "eae629abfd4b386c12d4ab710fa6da94": "b^{p^{e-1}} \\equiv a^{(N-1)/p}_p \\not\\equiv 1 \\pmod{v}", "eae6bd752f14df3079fd21f80c8be3f9": " \n{1 \\over 2}\n\\begin{pmatrix} \n1 & -1 & 0 & 0 \\\\ \n-1 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & 0\n\\end{pmatrix}\n\\quad \n", "eae6e61771d6d69c4fd076c6e3f79069": "P,L,I", "eae6ee268eeca89a905895c36d9386a4": "k > 2 d_A.\\ ", "eae72bc07f454da1b80dcf886de5baa1": "vxy = b^j", "eae74116874cbb945fecb01833ee5280": "\\begin{align}\nS_1S_3+S_2S_4 & = 4R^2(\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4)\\\\\n& = 2R^2[\\cos(\\theta_1-\\theta_3)-\\cos(\\theta_1+\\theta_3)-\\cos(\\theta_2+\\theta_4)+\\cos(\\theta_2-\\theta_4)]\\\\\n& = 2R^2[\\cos(\\theta_1-\\theta_3)+\\cos(\\theta_2-\\theta_4)].\n\\end{align}", "eae764c848dfd58f2b350d3329c85729": "\\begin{alignat}{2}\n dg_E & = \\quad \\ \\left(\\frac{Vf}{\\Lambda^3}\\right)\n\\frac{2}{\\sqrt{\\pi}}~\\beta^{3/2}E^{1/2}~dE \\\\\n P_E~dE & = \\frac{1}{N}\\left(\\frac{Vf}{\\Lambda^3}\\right)\n\\frac{2}{\\sqrt{\\pi}}~\\frac{\\beta^{3/2}E^{1/2}}{\\Phi(E)}~dE \\\\\n\\end{alignat}\n", "eae77e3432a5457298b4b63e3aa1d27f": "y' = y' \\,,\\quad p_y' = p_y ", "eae819105e2fc6324ffd9c80bb1157c7": "1_{|\\xi| \\leq R}", "eae86fb75608f73472457ad4e7c3b698": " \\epsilon \\ ", "eae8a2f17f124d000145fbcaee870c12": " R = \\frac{\\omega_A}{\\omega_B} = \\frac{N_B}{N_A}.", "eae973825fcb24fc45adc6610e552f37": " \\boldsymbol\\Sigma' =\n\\begin{bmatrix}\n\\boldsymbol\\Sigma_{11} & \\boldsymbol\\Sigma_{13} \\\\\n\\boldsymbol\\Sigma_{31} & \\boldsymbol\\Sigma_{33}\n\\end{bmatrix}\n", "eae9dfb59bc0016d59d00ead862e61a4": "~\\beta_{\\mathrm{max}}=\\left(\\frac{R^2}{PQ}\\right)^{\\frac{1}{3}}.", "eae9e2afb83fd15f3ea9820593535880": " \\sum_{k=0}^{2n} (-1)^k \\binom{2n}{k}^3\\, = \\, (-1)^{n} \\binom{3n}{n,n,n}.\n", "eae9eccf729ff297f2447b3bf83c861d": "\\int xu\\;dx = -\\frac{1}{3} u^3 \\qquad\\mbox{(}|x|\\leq|a|\\mbox{)}", "eae9edc1f937bd8c824ce9e3d6d077b2": "-\\exp\\left(0.5\\left(\\cos\\left(2\\pi x\\right)+\\cos\\left(2\\pi y\\right)\\right)\\right) + 20 + e.\\quad", "eae9f656f28c4bf0ebc25d29f29e9ff3": "O(N\\ln N)", "eaea20da7b9af1298bd9371302659bbc": " \\frac{1}{p(1-p)} ", "eaea8cbfebd7f5ae48b48e00d61f6d1f": "\\!\\sum_{k=0}B_k z^k,", "eaea8e6dca3e486ecfe854c88ad742ce": "\\boldsymbol{\\mu}", "eaeb21aa58a33c8e53756f73f2aa643d": "= \\frac{1}{2} \\eta_{\\mu \\nu} \\left(2 \\eta^{\\mu \\nu} I_4 \\right) = \\eta_{\\mu \\nu} \\eta^{\\mu \\nu} I_4 = 4 I_4. \\,", "eaeb2dd34b9745cbbbfd9eed8c5d955f": " f^{\\prime\\prime} = 12 + 6x ", "eaeb3b53c20adf7c1bb3ee4319e6afad": "e^P = I + \\sum_{k=1}^{\\infty} \\frac{P^k}{k!}=I+\\left(\\sum_{k=1}^{\\infty} \\frac{1}{k!}\\right)P=I+(e-1)P ~.", "eaeb48a91ce81d0193ab3b9c4dec9892": "\\mathrm{bei}_n(x) = \\left(\\frac{x}{2}\\right)^n \\sum_{k \\geq 0} \\frac{\\sin\\left[\\left(\\frac{3n}{4} + \\frac{k}{2}\\right)\\pi\\right]}{k! \\Gamma(n + k + 1)} \\left(\\frac{x^2}{4}\\right)^k", "eaeb78fa419865762d4f43fa77a4b65a": "D = \\frac{V_o+(V_\\mathit{SYNCSW} + V_L)}{V_i - V_\\mathit{SWITCH} + V_\\mathit{SYNCSW}}", "eaec6331d0e138b9a2cf4f7786de8e02": "(\\mathbf{x},\\sigma)", "eaecdc50f53c65768eec7e3aea313741": "v = {dx \\over dt}", "eaecf2389a088e4d9c664ff08611ee63": "\ng(y) = \\left\\langle y, x \\right\\rangle - f\\left(x\\right), \\,\nx = \\left(Df\\right)^{-1}(y)\n", "eaecf41bd8ffde0d6c23894c3db5ba08": "P_{cr}= \\frac{m_e^2 c^5 \\omega^2}{e^2 \\omega_{p}^2} \\simeq 17 \\bigg(\\frac{\\omega}{\\omega_{p}}\\bigg)^2\\ \\textrm{GW}", "eaecf4bee1dbe94a85e242c939144dcf": "\nT_C=\\frac{8a}{27bR}\n", "eaed1741d42cc7b40be886c1c38b5feb": " \\psi^0_i", "eaed1b76979d34aac0e7c55699bd1eba": " \\begin{matrix}\n \\mathbf1&5&0&6&3&4&2\\\\\n 1&\\mathbf4&0&5&2&3&1\\\\\n 1&4&\\mathbf0&4&2&3&1\\\\\n 1&4&0&\\mathbf3&1&2&0\\\\\n 1&4&0&3&\\mathbf1&2&0\\\\\n 1&4&0&3&1&\\mathbf1&0\\\\\n 1&4&0&3&1&1&\\mathbf0\\\\\n\\end{matrix}\n", "eaed52b75f6c7f7798a962b43a1f7bba": "\\left[r(\\varphi_i)\\right]^2 \\pi \\cdot \\frac{\\Delta \\varphi}{2\\pi} = \\frac{1}{2}\\left[r(\\varphi_i)\\right]^2 \\Delta \\varphi.", "eaed742f0b1ac565f0ae82772858ce22": "\\sin \\pi = 0,\\,\\!", "eaee04ce826230d114df4071fbcf2f65": "n_{\\rm e} \\tau_{\\rm E} \\ge 1.5\\times10^{20} {\\rm s}/\\mbox{m}^3", "eaee2f2ce2fe5587e07da4f0ffc5b37c": "I_{no}", "eaee5525f563daa7442750375f5d620d": "\ndI = -I n \\sigma dx\n", "eaee5d497518de27d6f8614f3819df03": "[H_{\\alpha_i},H_{\\alpha_j}]=0", "eaee932ba046a2abc6c864c373aaca20": " Y(x_i) = R_n(x_i) - \\frac{R_n(x)}{W(x)} \\ W(x_i) = 0 ", "eaeec37922bbbd5cb4f1426288e62d67": "H_\\Delta (N) = \\sum_\\Delta N(v(\\Delta)) \\epsilon^{ijk} Tr \\big( (I + {1 \\over 2} F_{ab} s_i^a s_j^b) (I - A_c s_k^c) \\{ (I + A_d s_k^d) , V \\} \\big)", "eaef3244331ee20cebb51ad7b68f05e2": "m=\\lfloor i\\phi\\rfloor", "eaef66e1bbc24ba3872e02d99cfb222e": "D_\\mu = \\partial_\\mu - \\frac{i}{4} \\omega_{\\mu}^{ab} \\sigma_{ab}", "eaef8859c73c2706a9b348bea7b7c955": " \\sum_{0 < k \\leq n} k^{c} = \\frac{n^{c+1}}{c+1}+\\frac{1}{2}n^c+\\sum_{k \\geq 2}\\frac{B_{k}}{k!}c^{\\underline{k-1}}n^{c-k+1}.", "eaef9ff2af424c4155f2d65ccbf3b180": "|f(x) - P_{n-1}(x)| \\le \\frac{1}{2^{n-1}n!} \\max_{\\xi \\in [-1,1]} |f^{(n)} (\\xi)|.", "eaf01ed28776e1dc856b660a95ccdcd3": "\\mathbf{F} = m \\gamma(\\mathbf{v}) \\left( \\mathbf{E}_\\text{g} + \\mathbf{v} \\times \\mathbf{B}_\\text{g} \\right) ", "eaf0256e0e585e01dc3c8abeca01e61f": "\n\\theta_1 \\equiv \\int \\omega_1(t)\\, dt.\n", "eaf03ad9041407cd966db66969937266": "\\gamma(\\rho,z)", "eaf096770bf87a21b631dda5f005d974": "\\sqrt{n}D_n", "eaf0bc18db6f973582328fe2bd0819e9": "x=\\sum_{i=1}^n\\langle x,v_i\\rangle v_i", "eaf0d10b8c2d99b7583b4a447fccada6": "T(\\beta) = \\frac{|V_\\parallel|}{V_0} ", "eaf0df0e70887a53fcd1bb1265c0778a": " f(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_2 x^2 + a_1 x + a_0 = \\sum_{n=0}^{N}a_nx^n ", "eaf0f448f1f3bd281867b85850222dd6": "X=X_0\\cup I_1\\cup\\ldots\\cup I_r.", "eaf1060f144e35139f48849e42c4e384": "A = \\frac{t^{k_0}A\\left(\\frac{h}{t}\\right) - A(h)}{t^{k_0}-1} + O(h^{k_1}) .", "eaf10735b92d431dee7632e0c25695cc": "x=100%", "eaf107c47263f2e7c0f851cd0a3b2684": "\\hat{S}b = 0", "eaf119326a61bb43223920b3b5866acf": "a \\le 0", "eaf1acc4b758ef47b4f0b025102a9e24": "P=\\begin{pmatrix}\nA_1^\\ast & A_2^\\ast \\\\\nA_0^\\ast & A_1 & A_2 \\\\\n& A_0 & A_1 & A_2 \\\\\n&& A_0 & A_1 & A_2 \\\\\n&&& \\ddots & \\ddots & \\ddots\n\\end{pmatrix}", "eaf1bcb4cd12dbfbb73dc474a6088ef5": "\\dot{\\lambda}", "eaf22bdf08dd3747ed8f7793e30ca822": "\\oint_{(x,y)\\in C} x^3\\, dx + 4y^2\\, dy", "eaf2441d47c37f4f52893d0fffb4895c": "E = p c", "eaf25d70f2084f979612d7ecc7ab2ab7": "\\frac{1}{2\\pi}\\frac{d(2\\pi kt^2)}{dt}=2kt~,", "eaf2610bf6b96e0999c9ca62c93d379f": "d \\det (A) = \\mathrm{tr} (\\mathrm{adj}(A) \\, dA).", "eaf266964c882b41737922778ae986a4": "Holds(f, do(a, s)) \\leftarrow Poss(a, s) \\wedge Holds(f, s) \\wedge \\neg Terminates(a, f, s)", "eaf26d9d360a592de17f415afe2803d9": "S_t= S_0 e^{(\\mu- \\frac{1}{2} \\sigma^2) t+ \\sigma W_t}", "eaf272096fb5167e8154ab25e10cfd07": " {n \\choose 0} + {n \\choose 1} + {n \\choose 2} + \\cdots + {n \\choose n-1} + {n \\choose n} = 2^n ", "eaf2ea868958830b52fe9bd50c6950c7": "\n\\begin{align}\nU(\\theta)\n&= a\\sum_{n=1}^N e^{ \\frac {-i 2 \\pi nS \\sin \\theta} {\\lambda}}\\\\\n&= \\frac {1-e^{ -i 2 \\pi NS \\sin \\theta/\\lambda}} {1-e^{-i 2 \\pi D \\sin \\theta / \\lambda}}\n\\end{align}\n", "eaf3a8fb3d5deb15f09b29d92ade2b9a": "\\Delta S_{i=1} = S_{i=1}-S_{i-1}= \\big(2000.0\\text{ ft}\\cdot 1\\text { ft}\\cdot 3.57\\text { ft}\\big)-7000 \\text { ft}^3 = 130.5 \\text { ft}^3 ", "eaf3b2183321ed1e2892ff369922d088": "\n 0 \\neq \\det S''_{yy} (\\boldsymbol{\\phi}(0)) = 2^{r-1} \\det \\left( 2H_{ij}(0) \\right).\n", "eaf4237c8253a1fc09bb0f06b87534f3": "S = \\frac {0.8} {H_\\mathrm{m}}", "eaf429717bd82a89a4a427f159c875ba": "n(n+1+\\alpha+\\beta)\\,", "eaf451215b75ada6aae0645b95cbd891": " \\ f_L = \\ {64 \\over Re} + {10.67 + 0.1414{({He\\over Re})^{1.143}}\\over {\\left[1 + 0.0149{({He\\over Re})^{1.16}}\\right]Re }}\\left({He\\over Re}\\right)", "eaf4601c73577682dc64259f1094abd5": "\n \\langle\\hat{p}_j^{\\mathrm{b.a.}}\\rangle = \\frac{2}{c}\\mathcal{W} \\,,\n", "eaf46fc0451c9535c9aa2f9a77aefc03": " \\{w_i \\mid i \\in \\mathbb{N} \\}", "eaf4da3cd081ee4bbe4b8e9e3f29afb2": "\\mathrm{SO}(m, \\mathbb C)", "eaf4f0d5becacf42fa0efe4fea1558ba": " y=mx+b ", "eaf502e186a4638c888a507070f375c0": "\\Delta \\vec{p}\\!", "eaf52679063ad935e273ab4a2bde38f0": "\\Delta_K(t) = \\Delta_{f(S^1 \\times \\{0\\})}(t^a) \\Delta_{K'}(t)", "eaf59601174e07ca364e6a294af4ec99": "\n(\\lambda 1 1) (\\lambda (\\lambda \\lambda \\lambda 1 (\\lambda (\\lambda 2 1 1 1 (\\lambda \\lambda 1 3 3 (\\lambda \\lambda 1 (\\lambda \\lambda (\\lambda 7 (1 (3 (\\lambda \\lambda \\lambda \\lambda \\lambda \\underline{10} (1 (\\lambda 6 1 4 3)) (\\lambda 1 5 (6 5 4 3 2))) (\\lambda \\lambda 2 ((\\lambda 1 1) (\\lambda \\lambda \\lambda 2\n", "eaf5ec0393c6eb4df47d5575bdf95fd9": " a \\otimes b - b \\otimes a = [a,b]", "eaf60534e525810ba145451b2f1c8715": "g_i/g_j\\exp{(E_j-E_i)/kT)},", "eaf6ab398bbd38ea8413f2fab5c0be4b": "\\|A\\|_{X \\to X} \\leq M", "eaf6cbf480c38b16366f4e3f375cbb80": "\\tfrac{-1+\\mathrm i\\sqrt7}2", "eaf6d94efe7dd6d1b46c8264925a17c9": "\\hbar^2 j (j+1)", "eaf793733ee3e684740a78728a4f7110": "M_x = \\int_0^2{\\int_x^{4-x}}{}{}y\\,(2x+3y+2)\\,dy\\,dx", "eaf7c6ffcef68715c9362cf23534e089": "\\int_0^\\infty e^{-t}\\mathcal{B}A(tz) \\, dt = \\int_0^\\infty \\frac{e^{-t}} {1+tz} \\, dt = \\frac 1 z \\cdot e^\\frac 1 z \\cdot \\Gamma\\left(0,\\frac 1 z \\right)", "eaf7c9ff2c4330f553e3d687183104ac": "f(t)\\equiv\\sum_n\\kappa_n^2\\cos(\\omega_nt)\\,,", "eaf7f0fed6f658baf8fda79688171732": "\\int_0^\\infty \\frac{x \\sin mx}{x^2+a^2}\\ dx=\\frac{\\pi}{2}e^{-ma}", "eaf84ca944178b54dadf2ab54f915791": "Z_{N}=\\sum_{\\sigma,\\sigma',\\sigma'',\\sigma'''}A_{\\sigma|\\sigma'}A_{\\sigma'|\\sigma''}A_{\\sigma''|\\sigma'''}A_{\\sigma'''|\\sigma}=\\textrm{tr}A^{4}.", "eaf89622de08636177dc745a05c8de23": "I_{1}(\\sigma_{yy}\\sigma_{zz} - \\sigma^2_{yz}) - I_{3}", "eaf8c5aefac82c5c226d4875b4dc3c12": "1\\leq k \\leq d", "eaf8e8c580dce9f1d06bf86199119b0a": "n = 1,2,\\cdots ,m", "eaf8ec4fdaa140ea320fddff77a7ff97": "a \\in \\mathcal{A}", "eaf96323104fb5d99574c00dfd7fd170": "similarity(\\mathbf{p,m_i}) = e^{-\\tau \\left \\Vert \\mathbf{p-m_i} \\right \\|} ", "eaf9749e6861c453a47a6bf9eedbee26": "\\lambda = 2\\pi /k", "eaf9f45940de703a8447cc68d7c6f57a": "\\sum_{k=1}^{k=3} \\cos (-2\\pi\\frac{n(k-1)}{3})/3 = 0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...", "eafa0611cc07ce1a83998824d4581d0b": "10_{1}\\rightarrow (3,2)_{\\frac{1}{6}}\\oplus (\\bar{3},1)_{\\frac{1}{3}}\\oplus (1,1)_0", "eafaa2560034132744b33a6fe6a2483f": "{P \\over Q} = {{X + \\sigma (l_1 + \\alpha )} \\over {Y + \\sigma (100 - l_1 + \\beta)}}", "eafadcb8ee6cc249aeb9c2c5b2c1e0f4": "Aa|||Bb", "eafadda8e8ccbee020fb7a39b9524a62": "V_\\mathrm {dc}=V_\\mathrm {av}=\\frac{3V_\\mathrm {LLpeak}}{\\pi} \\cos \\alpha", "eafb49be3530d8b103c7286fbd312632": "\\Epsilon_\\text{mach} = B^{1-P},\\,", "eafbf49f11ce3cecfb703c17bbfcdc7d": " p=2t.\\!", "eafbf9678d1b60bd0125f8abb71c2f21": "\\text{then}", "eafc379295618ae1d177063b87cde51c": "W^{u}(p)", "eafc924241f43551671d3a9058ad89a1": "\\frac{4}{3} \\pi r^3", "eafc95c3a0222e2fffa49e8cc307f775": "\nH= {\\nabla \\psi^\\dagger \\nabla\\psi \\over 2m} \n", "eafcbcc26074080da4a80470e0645a32": "\\frac{V_D}{nV_T} = \\ln \\left(\\frac {(V_S-V_D)/R}{I_S}+1\\right) ", "eafcdc2570167bf04597c853a1df2d72": "|M| + 1", "eafd17f189af84a523b5dc375be30e0b": "\n\\begin{align}\nU(\\rho,z)\n&\\propto 2 \\pi \\int_0^{\\infty} A(\\rho')J_0\\left(\\frac{2 \\pi \\rho' \\rho}{\\lambda z}\\right) \\rho' d \\rho'\n\\end{align}\n", "eafd3095b6f2a2d78f7477036b39d023": " M=m \\, ", "eafd57daf1b56b436be2a489a757949a": "S \\cdot \\{\\} = \\{\\} = \\{\\} \\cdot S", "eafda56842fed68ae31f10331ed9b652": "J_\\mu(x_1,\\ldots,x_n;q,t) = P_\\mu(x_1,\\ldots,x_n;q,t)\\prod_{s\\in\\mu}(1-q^{a(s)}t^{l(s)}).\\ ", "eafdb1b0905e6e8da1b9281626fe9bfb": "\n n= p_1^{a_1}p_2^{a_2} \\dots p_{\\omega(n)}^{a_{\\omega(n)}} \n", "eafdb23afc017610d37cba74ce03c9db": "\\vec{\\jmath},", "eafdc9ef713050c840cd11405103040f": "\\pi:P\\to M", "eafdd30d85348e648c8891cf5d3f9681": "\\lambda=0", "eafdd5aac21fb0ef62f8303d9f1f9c96": "{3\\over 5}(N_p+N_n)^{2/3}", "eafdd741b0adadae562afdd3ddc76124": "AdS_4\\times S^7", "eafdea362a40e781d19558341ea60050": "\n T_{11} = \\cfrac{\\sigma_{11}}{\\lambda} = \n \\left(\\lambda - \\cfrac{1}{\\lambda^3}\\right)\\left(\\cfrac{\\mu J_m}{J_m - I_1 + 3}\\right)~.\n ", "eafecd4ac6dc82186e42b5879ba6b80a": "z \\cdot \\infty = \\infty", "eaff05ecf7102112bb606f3267a1c999": "\\pi_{S_i}(R)", "eaff533f2a16d0ce3f56d39f3f5eea36": "t \\over t^2 + 1", "eaff6a868670b34c75d6105422074414": " 0 = 1/g^{rr} = 1 - \\frac{r_\\mathrm{S}}{r} + \\frac{r_Q^2}{r^2}.", "eb000801cd5865014659b1c22ddb92e4": "R \\to R[S^{-1}]", "eb00606238f074090d9d1316192d7057": "11 ^ x\\,", "eb007bb2fe73deaca03d619558bb3cb2": "S=X\\,C_n(X\\,C_n)'=X\\,C_n\\,C_n\\,X\\,'=X\\,C_n\\,X\\,'.", "eb00a04135562ae6f74786f084f54327": "u_i", "eb0103d42c80c473843f595f87af3c65": "\\langle \\mathbf{x},\\mathbf{y}\\rangle := \\mathbf{y}^\\dagger\\mathbf{M}\\mathbf{x} = \\overline{\\mathbf{x}^\\dagger\\mathbf{M}\\mathbf{y}},", "eb0130b46eb1ef87bb67380aac20dd92": "V_{\\rm m} = \\frac{A_{\\rm r}M_{\\rm u}}{\\rho}", "eb01513eef116c729030c0b7d003e656": "\\Pr[X > x] \\le 1-k/q", "eb0161ab016ad22f3ad99822da790335": "e^{i\\mathbf{(K_{1}-K_{2})}\\cdot\\mathbf{(R)}}=e^{i\\mathbf{K_1}\\cdot\\mathbf{R}}/e^{i\\mathbf{K_2}\\cdot\\mathbf{R}}=1", "eb01646e6fc6167a8af4071a4f187481": "n^{O(1/\\varepsilon^2)}", "eb017b3b4157ed7b9520e083cab27dbe": "\\mathbf x[k]", "eb019a6175ff23eb48735cdaf92d7c18": "L=L_{0}/\\gamma", "eb01b01ed79afe115bf8b91ec8c670a7": " I(T, \\aleph_\\xi) < \\beth_{\\omega_1}(|\\xi|)", "eb01f1f6234657216bcbdce590fcf088": "f_1, \\cdots, f_d", "eb022140fb7c7161fe9a904457d44e65": "U(r)", "eb025f715348409b66ef49458ebb055c": "\\Theta(n \\log n)", "eb02ad64f92c449ec3165adc1881b335": "Q1-Q2", "eb02b73865dabf118dbcf74fbdd383b3": " k_j = p ", "eb02d244af739129a17a1eed47126339": "P_N(d) = \\frac{1}{2^N}\\begin{pmatrix}N\\\\ \\frac{d+N}{2}\\end{pmatrix}", "eb02e751afe8d70a6fd1728663f4158c": "L(D)", "eb02f6dac617c9376f285a57f2b6d76d": "(p", "eb03474bf2c3a745c34161164d16b4cb": "Y=y(0), y(1),\\dots,y(L-1)\\,", "eb043792a124261eb602731db2c4aecc": "\n\\mathbf{H}_3= \n\\begin{pmatrix}\n0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\\\\n0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\\\\n0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\; 0 \\\\\n1 \\; 0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\\\\n1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\\\\n0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\\\\n1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\\\\n0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 0 \\; 0 \\\\\n0 \\; 0 \\; 1 \\; 0 \\; 1 \\; 1 \\; 0 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 1 \\; 1 \\; 1 \\; 1 \\; 0 \\; 0 \\; 0 \\; 1 \\; 1\n\\end{pmatrix}.\n", "eb04642a109e4ed182502c5d513efb16": " \\frac{dL}{dt}= -k_1 L + L_b ", "eb04afb136459152b756c0e5bd247266": "\\Psi_{0} = \\psi_{1s} (1) \\psi_{2s} (2) + \\psi_{2s} (1) \\psi_{1s} (2)", "eb04b6c140518eb1ee0a1be482c0dbcb": "{50 \\choose 5} = 2,118,760", "eb04d84bdf87f64a2481cc4934ddd325": "p_k = 1/(z-k)", "eb04f7628780fbbd00a7be8d198b4df0": "{(\\mathbf{r})}", "eb050d0f007495db5776281b4f004c41": "x = \\sqrt{\\tfrac{L^2-b^2}{3}}", "eb052e1eac6664b0cae30604664e825b": "\\log |f(z)|", "eb05570cdd8c9643a78b20d54c71f6dd": "T dS= (n+1)P dV + n V dP\\,", "eb05de4ec3a3ba8e0f7e5876b1840dd6": "\\operatorname{drop-params}[\\operatorname{let} V: E \\operatorname{in} L] \\equiv \\operatorname{let} V: \\operatorname{drop-params}[E, D, FV[E], []] \\operatorname{in} \\operatorname{drop-params}[L, D, FV[L], []] ", "eb05f91ea33fee86be3e852bd4aad44b": "G_2(\\mathbf{q},t) = \\mathcal{S}(\\mathbf{q},t) + C\\,,", "eb05f927b65b971891bceb50573004ab": "S^{|i} = \\{z_1 ,...,\\ z_{i-1},\\ z_{i+1},...,\\ z_m\\}", "eb064950818d01afb55679b0696e1e6b": "\\langle f,g\\rangle", "eb064d5d3b34e9d8a1410f0b8cfe1bb9": "\\tan(x) = \\frac{\\sin(x)}{\\cos(x)}= \\frac{e^{ix} - e^{-ix}}{i({e^{ix} + e^{-ix}})}\n ", "eb06677c0d5d1359465b475573f5cbef": "\n\\mathbf{V}_P(t) = \\left[\\frac{dA(t)}{dt}\\right]\\mathbf{p} + \\mathbf{v}(t),\n", "eb0685aa001cdb6d6ad1cf89c861afe8": "|\\psi_\\text{tot}(0)\\rangle= \\sum_n C_n \\left[ \\cos \\left(\\frac{\\alpha_n}{2}\\right)|n,+\\rangle-\\sin \\left(\\frac{\\alpha_n}{2}\\right)|n,-\\rangle\\right].", "eb069b2c96a43e7a598c7943e127e429": "2 H = -\\nabla \\cdot \\hat n", "eb06b6427c833db72c849836329d3c7f": "a_0 := a(\\varphi_0)", "eb06c34ba30182da7fd2dae909001ea0": " g_{\\alpha \\beta , \\gamma} \\, \\eta^{\\beta \\gamma} = \\tfrac12 g_{\\beta \\gamma , \\alpha} \\, \\eta^{\\beta \\gamma} \\,;", "eb06df336b8d17b2f8bfd25b824f1aa7": "q_1 = 1-q_0", "eb06f611dfdfe035b90c1e803e037163": "\\rho = \\frac{1-r^{-1}} { 1-r^{-N} }. \\qquad \\text{(2)} ", "eb0770d87a9dd469e6ad0a00b45335d8": "\\operatorname{Out}(N) \\cong H^1(W; T) \\rtimes \\operatorname{Out}(G).", "eb07a93fe83e2947ab54575339273f52": " MA = \\frac{T_B}{T_A} = \\frac{\\omega_A}{\\omega_B}.", "eb07cf79f5bcd4cb90d41238bd8ab52d": "f^{\\mathcal{A}/E}_i : (A/E)^{n_i} \\to A/E", "eb07dee6f30ba4c4925c0e6a2cc374d9": "x^{64} + x^{62} + x^{57} + x^{55} + x^{54} + x^{53} + x^{52} + x^{47} + x^{46} + x^{45} + x^{40} + x^{39} + x^{38} + x^{37} + x^{35} + x^{33} +", "eb08308ae63da7a5bb51510237542010": "\\displaystyle \\partial_t u = 6\\, u\\, \\partial_x u - \\partial_x^3 u + 3\\, w\\, \\partial_x^2 w", "eb08cc7458871c7f5bdd7b289c4cb3d5": "x^5-2s^3x^2-\\frac{s^5}{5} ", "eb097bc4d0ba3e5e8fe02e009028ab61": "(Af)(x)=xf(x)", "eb09c63b87dac8d7c3cc2e8642409236": "\\,\\! f(x_1,x_2)=\\ln ({x_1}^2+{x_2}^2)", "eb09db15a940a4f6a7f84482b2634741": "+\\sqrt{3}", "eb09e11a777ef272f0f98bd9156d594a": "S\\cap T \\in Q", "eb0a08b3b08554c20cedef69d16c7fc9": "\\langle f_1, f_2 \\rangle", "eb0a27ef9e1b6dc7bdb8e50aea368009": " Y \\widehat\\otimes_\\pi X \\to Y \\widehat\\otimes_\\varepsilon X ", "eb0a596a73222ef6f71b2555e84d65a1": "\\mathcal{U}(\\alpha, {\\tilde{u}})", "eb0a9f25d1cf39c92456ec292f3129f8": " V_{i} = \\operatorname{Var}_{X_i} \\left( E_{\\textbf{X}_{\\sim i}} (Y \\mid X_{i}) \\right) ", "eb0aecdeabf5ed559f6036aa7ed72a00": "\\Gamma (\\lambda) = \\frac{1}{\\eta (V) \\delta n_0}\\left[\\frac{\\lambda}{\\lambda_B} - 1\\right]", "eb0bbba70004ee433bbdd7fe8d846a28": "C = C_0 e^{-\\frac{K \\cdot t}{V}} \\qquad(3)", "eb0bc77a5a42bb7559b05ddb9b37b428": "\\frac{1}{S(t_0)} \\int_0^{\\infty} t\\,f(t+t_0)\\,dt = \\frac{1}{S(t_0)} \\int_{t_0}^{\\infty} S(t)\\,dt,", "eb0bcec038b33479389c02c2d239eeef": " (\\boldsymbol\\mu,\\boldsymbol\\Sigma) \\sim \\mathrm{NIW}(\\boldsymbol\\mu_0,\\lambda,\\boldsymbol\\Psi,\\nu)", "eb0bdb4c5c8f9677b26d1405eb1ac079": "\\frac{L}{\\mathrm{circumference}}=\\frac{\\theta}{2\\pi}.\\,\\!", "eb0bf533b14e1f1f3507bbaf6bfa9941": "(D_i)_{x,y} = \\begin{cases} \n1,& \\text{if } \\left(x,y\\right)\\in R_{i},\\\\ \n0,& \\text{otherwise.} \\end{cases} \\qquad (1)", "eb0c5a967e31cc1908f4a8297fef8170": "y_{n+1} = \\frac{y_{n}(3-xy_n^2)}{2}", "eb0c66d26b78d421531f37c30e8a3bab": "\\Pi_j=\\frac{e^{-\\beta\\epsilon_j}k_j}{\\sum_r e^{-\\beta\\epsilon_r}k_r}.", "eb0c9405d05237efdde30e3c493950eb": "b = e^{\\ln b}", "eb0cbd3895c8adc6cd016c5a48c52fac": "\n\\mathbf{F}_{12}(\\mathbf{x}_{1},\\mathbf{x}_{2}) = m_{1} \\ddot{\\mathbf{x}}_{1} \\quad \\quad \\quad (\\mathrm{Equation} \\ 1)\n", "eb0ccfda1554a02625c849bc7d814ddf": "ds^{2} = \\frac{32G^3M^3}{r}e^{-r/2GM}(-dV^2 + dU^2) + r^2 d\\Omega^2,", "eb0cd235907e9b8420b96d3eeba3e284": "h_c (0)", "eb0d07ccd1c5d05a2c5902198f0e1119": " p_1", "eb0da0e588fa2dde3e3f1d1e77a946c2": "w = (1 + \\lfloor 2.6 \\cdot 11 - 0.2 \\rfloor + (100 - 1) + \\left\\lfloor\\frac{100 - 1}{4}\\right\\rfloor + \\left\\lfloor\\frac{20 - 1}{4}\\right\\rfloor - 2 * (20 - 1)\\ \\bmod\\ 7", "eb0daa8324d2154a20042ce464804116": "\\frac{1}{2}\\left[(\\lambda-\\mu)+\\sqrt{(\\lambda-\\mu)^2 + 4(k-\\mu)}\\right]", "eb0dbfbde607a073de9d837a675f9fba": "\\mathbf{H} \\sim \\mathcal{CN}(\\mathbf{0},\\mathbf{R}_T \\otimes \\mathbf{R}_R)", "eb0de930ba1b7f4c928060245b3b283e": "\\partial_n'(b) \\in \\ker \\beta_{n-1} = \\mathrm{im} \\alpha_{n-1}.", "eb0e0d649cdd977a1f666d4f1f1ae9f0": "\\begin{align}\n A' \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{V_1} \\right|_{I_1 = 0} &\\qquad B' \\,&\\stackrel{\\text{def}}{=}\\, \\left. \\frac{V_2}{I_1} \\right|_{V_1 = 0}\\\\\n C' \\,&\\stackrel{\\text{def}}{=}\\, \\left. -\\frac{I_2}{V_1} \\right|_{I_1 = 0} &\\qquad D' \\,&\\stackrel{\\text{def}}{=}\\, \\left. -\\frac{I_2}{I_1} \\right|_{V_1 = 0}\n\\end{align}", "eb0e4779570e82df8fa63dddd615e456": " S^z_i|0\\rangle = s|0\\rangle", "eb0e6af0b02e209133ad3cfc0b0bcab0": "\\mathcal{T}^m_n(M)", "eb0e708954012517d6f3d1eea649ab45": "A=\\begin{pmatrix}a&b\\\\c&d\\\\\\end{pmatrix} ,", "eb0e995c00209593a09633e636a638ed": "P(H \\and E)", "eb0eac549c62738b433488000d2b3fa4": " \\displaystyle{K = (x^3, y^4, x^2z^7)}", "eb0f27c2f95190834a166644c237b2c9": "7^4 - 7^3 - 7^2", "eb0f2863d5e799d58f1fa12aa2e4e0be": " F = x, E = f\\ (x\\ x), L = (\\lambda x.f\\ (x\\ x)) ", "eb0f5148f6413116b884110f5816b2d6": "\n\\lambda_*\\Big(X^k\\frac{\\partial}{\\partial x^k}\\Big|_v + Y^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_v\\Big)\n\n=\nX^k\\frac{\\partial}{\\partial x^k}\\Big|_{\\lambda v} + \\lambda Y^\\ell\\frac{\\partial}{\\partial v^\\ell}\\Big|_{\\lambda v},\n", "eb0f7f92b35996b9bdaae1962386bad4": "(d+1)", "eb0f928514d1f277e3143e6e5038bd5a": "n=\\pm m", "eb0fa678848887757567c12a341c3721": "i^n = \\cos(n\\pi/2)+i\\sin(n\\pi/2)", "eb100bc2d4e888fd2499d8619991d89e": "c\\leq \na,b.", "eb10b6e1251e190f1171e1f9bcfb810d": "\\lim_{n \\to \\infty} \\left( \\ln(n!) - n \\ln(n) + n - \\tfrac{1}{2}\\ln(n) \\right) = 1 - \\sum_{k=2}^{m} \\frac{(-1)^k B_k}{k(k-1)} + \\lim_{n \\to \\infty} R_{m,n}.", "eb11218f8c296e7de278367820ae6893": "n \\times (k-1)", "eb1166ea055992c345448b83a219e299": " l= (\\frac{4\\cdot \\pi \\cdot d}{\\lambda})^2", "eb118d384905905aa4635071aef7ed1b": "\\omega^{\\omega^\\omega}", "eb11a06bf44e82bf9124d5637be5121a": "(x \\otimes u) (y \\otimes v) = xy \\otimes uv", "eb11a1bd238bf84f6936411d96155ea4": "\\nabla_X", "eb11fb2fce6261b7f25160d7b9ed1c98": "\\Phi_{Y,X}^{-1}(g) = f : FY \\to X", "eb12131c9b02d7d539347f1ecd12eed3": "F_0 = S_0 e^{cT},\\text{ where }c = r-q+u-y.", "eb1214471326eb8f02af86c31c4a7ca9": "\n k_B^2 = {4 \\pi e^2 \\over \\hbar \\omega_c A L_B}\n", "eb123876d00c2eaa574e409b30046ce6": " \\tfrac5{36} + \\tfrac1{30} \\sqrt{15} ", "eb12a008b2847ba9fdcb79c746c4108c": " w^T \\Sigma w - q*R^T w", "eb12a4f84788a4969b486c0365763760": "8(1/4!)\\pi^4 = (1/3)\\pi^4 ", "eb12e3137d5ae06a32fceb3050a52fd7": "G(4,1)", "eb12f565f194663537e179cc5ee53a42": "v(n,d)", "eb13265c92bf0b406b0fffcaa2850a16": "s\\in\\mathcal S", "eb13450dde8edfcc8b21f4a0d9c8a7d0": "J(v\\otimes z) = J(v)\\otimes z.", "eb137b26b83d4ccf45f3529dddcb3fb1": "\\scriptstyle\\pi(P_n)\\, =\\, 2^{n-1}", "eb1426f6220cdc9ce763760d1b21f821": "\\theta=\\frac{\\sqrt{\\pi}}{\\sigma\\sqrt{2}}", "eb145b6f3c5137e9e4d8fc3e2dd9443e": "\\operatorname{cov}(w_i, z_i) = \\operatorname{E}(w_i z_i) - wz", "eb148ef0989ae0689ec99e9208ad26f1": "\\xi = \\frac{1}{\\lambda}.", "eb14c3ef1f3e85ea5cdc31e82cd2d9a2": "C_{v2}", "eb150b0c4d9ccf3f4c87c6b46704548c": "\\!\\exists z \\forall x_1 \\forall x_2 \\exists y_1 \\exists y_2 (=\\!\\!(x_1, y_1) \\wedge =\\!\\!(x_2, y_2) \\wedge (x_1 = x_2 \\leftrightarrow y_1 = y_2) \\wedge y_1 \\not = z)\n", "eb153032082656ac36186487cc19a13f": " = \\sum_i \\operatorname{tr}\\left(T_i \\sum_k V_k V_k^*\\right) S_i ", "eb154b06283cdb7bb6928714cb437dec": "T_{N+1}", "eb155a408d90643e3fc2fe846110f4ff": " \\frac{\\partial \\vec{R}}{\\partial t} = \\gamma \\vec{R} \\times \\vec{B} ", "eb15f824db51280e663198b9badb71f4": " |\\psi\\rang \\rightarrow \\sum_n |c_n|^2 |\\psi_n\\rang \\lang \\psi_n| \\rightarrow |\\psi_n\\rang ", "eb161b68a3f5b015b4d5c121c9a665f2": "\n\\mathcal{I}\n-\n\\mathcal{J}\n=\n\\mathrm {Extremum}\n", "eb1671fd77c751c529f042f57956e14d": " Y_\\ell^m( \\theta , \\varphi ) = \\sqrt{{(2\\ell+1) }{(\\ell-m)!\\over (\\ell+m)!}} \\, P_\\ell^m ( \\cos{\\theta} )\\, e^{i m \\varphi } ", "eb169bc77617fa9dba0917ff98354dfe": "\\scriptstyle{-E/\\hbar}", "eb16cd8c4efcf59b50119d92b8473ef1": " c \\, \\mathbf{B} ( \\mathbf{r} , t ) = \\hat { \\mathbf{z} } \\times \\mathbf{E} ( \\mathbf{r} , t ) = \\begin{pmatrix} -E_y^0 \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\\\ E_x^0 \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\\\ 0 \\end{pmatrix} = -E_y^0 \\cos \\left ( kz-\\omega t + \\alpha_y \\right ) \\hat {\\mathbf{x}} \\; + \\; E_x^0 \\cos \\left ( kz-\\omega t + \\alpha_x \\right ) \\hat {\\mathbf{y}} ", "eb16e0a5f8fda3246847e09ef832f6fe": " \\epsilon_\\gamma = \\epsilon_\\alpha + \\epsilon_\\beta ", "eb16fbb375d3803b3c2ae205a9479fa3": "|E_l| = O\\left(N_l^{-r}\\left(\\log N_l\\right)^{(d-1)(r+1)}\\right)", "eb173b31a3c7765072461299970ef31d": "\\textit{on}(t)", "eb177bb1746e4c53c693bcf9aee88ed7": "\nE_n = \\left({3nhF\\over 4\\sqrt{2m}} \\right)^{2/3}\n", "eb179eeb6ffb9fe5f435954a17896f91": "b_{2}+a_{3}+c_{2}", "eb17f323b0d078582ae487a1f7561693": "f_s(x) = f(x/s)/s, \\!", "eb1805c73481f1808f6c9341b18e7167": "z + dz\\,", "eb1846a561afd059b46a042a9e2aa521": "1 + 2 \\times 3", "eb18908f23b03b7dbf9a360c4e00086e": "\nI_2 = I_{1} \\dfrac{{r_1}^2}{{r_2}^2} \\,\n", "eb189e55fa3d7658b0616f84553043d0": "V=V_0+AR^{1/2}P _{1/2} (\\cos\\theta _0)\\,", "eb18bbeccc1e44961d52cfda6969164d": "E(x,y,z) = E(x,y,0) * h (x,y,z) \\, ", "eb18f794dbd3dccc4a1644ed6692042d": "n-d", "eb19d19c487382dda4f1d64a322d0184": "H_{XX} = - ({\\nabla_{1a}}^2 + {\\nabla_{2b}}^2) - {2 \\sigma}{(1 + \\sigma)^2} {\\nabla_r}^2 + V ", "eb19ddb80a96308333dc8f150f5db0fc": "\\frac{\\partial n_i}{\\partial t} =\\sum_j \\nabla \\cdot \\left(D_{ij}\\frac{n_i}{n_j} \\nabla n_j\\right) \\, .", "eb19fb9a17bed7519858d20b2e0ecf85": "\n\t\\alpha_t = \\frac{1}{2}\\ln\\left(\\frac{W_+}{W_-}\\right)\n", "eb1a09c42e9924e938671a48051ba5f9": "C_\\mathrm{srgb}=\\begin{cases}\n12.92C_\\mathrm{linear}, & C_\\mathrm{linear} \\le 0.0031308\\\\\n(1+a)C_\\mathrm{linear}^{1/2.4}-a, & C_\\mathrm{linear} > 0.0031308\n\\end{cases}\n", "eb1a6fc3428e9ec56d88c655ef43f3ee": "F(z)= {1\\over 2\\pi i} \\int_{|\\zeta|=1} {f(\\zeta)\\over \\zeta -z} \\,d\\zeta={1\\over 2\\pi} \\int_{-\\pi}^{\\pi} {f(\\theta) \\over 1-e^{-i\\theta}z} \\, d\\theta.", "eb1a7b3a073fb4bda3f7a03aec8cb087": "\\langle\\ |\\ \\rangle \\!\\,", "eb1b032ade1d080efb6472e92066370a": "\\frac{4}{\\nu-6}\\sqrt{2(\\nu-4)}", "eb1b2906d47921d95c6015b179df07cb": "\\zeta:= e^{2\\psi}", "eb1b311718a9bf1686d37ef012505ea4": " \\int_X f(p)\\;\\mu(dp)= \\int_Y \\left(\\int_{\\pi^{-1}(\\{y\\})}f(p)\\,\\lambda_y(dp)\\right) \\nu(dy) \\qquad (*)", "eb1b46b77ab7a2fbac2ca145bcb36a72": " f{(x)}=g{(x)}+h{(x)}", "eb1b532c0f238886a9a82e3bf458f44a": "M_\\text{L}", "eb1ba20a8da350de5c10fe58b0eec5eb": "w_1l^*", "eb1bc5bc5371c2995f94f70f1783803d": " X_{po^{N}}", "eb1bf8abd8f67753bd7481c25bab1fd6": "1 \\leqslant i \\leqslant j \\leqslant n", "eb1c0cf40324bf974424e17cd4ce420c": "\\, \\delta = \\frac{1}{\\rho_{0}} (\\frac{4}{3}\\mu+\\mu_{B}) + \\frac{k}{\\rho_{0}} (\\frac{1}{c_{v}} - \\frac{1}{c_{p}})", "eb1c2e12063338ebaefa578b02f84c9b": "Y'_w", "eb1c61a543585fbea866a1c65ae5ac15": "Y \\leftarrow X \\rightarrow Z", "eb1cf04a3d81f87be280d0d11417c49e": "\\mu_{x,\\lambda}", "eb1d2fab16dffcdd58bfaa10c4fbc418": " \\mu(S) = \\log(b/a) ", "eb1e00267717558e211ea2afdcc82dc4": "\\lambda^j (x)", "eb1e166b43ea6e3f8c24486cbbfb7ee0": "a = \\tan^{-1}\\left(\\frac{v}{u}\\right) + \\frac{v}{u^2+v^2}", "eb1e74aa4436d52f4d13d4d6e729f8e2": "H=\\vec{p}\\cdot\\dot{\\vec{r}}+p_\\lambda \\dot{\\lambda}-L=\\frac{p^2}{2m}+p_\\lambda \\dot{\\lambda}+mgz-\\frac{\\lambda}{2}(r^2-R^2)", "eb1f704ae8be2dfb29d2960bc1356fe0": "t_1 = t_0+h", "eb1fa9acd68675bb04b3eb8b64274422": "\\langle a, t \\mid a^2, [ t^m a t^{-m} , t^n a t^{-n} ], m, n \\in \\mathbb{Z} \\rangle", "eb1fb1856ed4ae88a8c0dcf146ad7dca": "\n\\tilde{\\varepsilon}_{\\mathbf{k}}\n=\n\\varepsilon_{\\mathbf{k}}\n- \\sum_{\\mathbf{k}' \\neq \\mathbf{k}} V_{\\mathbf{k} - \\mathbf{k}'} \\left[ f^{e}_{\\mathbf{k}'} + f^{h}_{\\mathbf{k}'} \\right]\\,,\n", "eb1fb4cbbb637c1b466185511b343618": "{\\Bbb E}(n_H) = pn", "eb1fb8e34efc8565eabb02e21deca335": "\\Delta S = (b-a) A", "eb1ff1609815a84d90ae4aaef3d2ac63": "\nr_2(1)+\nr_2(2)+\n\\dots+\nr_2(n)\n", "eb20082a5c60e4756631cc1b1591eaaf": "e = \\sum_{k=1}^\\infty \\frac{k}{k!} = \\sum_{k=1}^\\infty \\frac{1}{(k-1)!} = \\sum_{k=0}^\\infty \\frac{1}{k!}", "eb200b5e08cbd681d3652f2c3d14bdec": "\nw_{t,d} = \\mathrm{tf}_{t,d} \\cdot \\log{\\frac{|D|}{|\\{d' \\in D \\, | \\, t \\in d'\\}|}}\n", "eb20636b0b5606e372c410f5b4ce9c73": " P = F_A v_A = F_B v_B, \\!", "eb20cd39cd85d77fdc35aacd1f10a333": " u_{tt} = c^2 u_{xx}, \\,", "eb20e68247388f83e211ac2033066f3b": "\\mathbf{F}^{-1}=\\frac{1}{N}\\mathbf{F}^*", "eb20ebd71285a32d873efb947ead32a5": "\\begin{align}\n\\dot{V} &= x_{1} \\dot x_{1} +x_{2} \\dot x_{2}\\\\\n&= x_{1} x_{2} - x_{1} x_{2}+\\varepsilon \\left(\\frac{x_{2}^4}{3} -{x_{2}^2}\\right)\\\\\n&= -\\varepsilon \\left({x_{2}^2} - \\frac{x_{2}^4}{3}\\right).\n\\end{align}", "eb212c2f1cab209c67232f67b0678631": "H_{1/3} = \\frac{1}{\\frac13\\,N}\\, \\sum_{m=1}^{\\frac13\\,N}\\, H_m,", "eb214d9f78ec87c1924b848f8c7c298b": "\nx = \\mu + \\frac{\\mu^2 y}{2\\lambda} - \\frac{\\mu}{2\\lambda}\\sqrt{4\\mu \\lambda y + \\mu^2 y^2}.\n", "eb21a21185697cbe192fdd257f6d5e2b": "Rf_!", "eb21b957f2704722d302489978307dc0": "(1+v^T A^{-1}u)", "eb22581ea1bd6e4b8ddc1dba0f8be702": "\\textbf{P}_{k\\mid k} = (I - \\textbf{K}_k \\textbf{H}_{k})\\textrm{cov}(\\textbf{x}_k - \\hat{\\textbf{x}}_{k\\mid k-1})(I - \\textbf{K}_k \\textbf{H}_{k})^{\\text{T}} + \\textbf{K}_k\\textrm{cov}(\\textbf{v}_k )\\textbf{K}_k^{\\text{T}}", "eb22c7a601711789d3394785792331ca": "\n{\\mathbf{S}} = \\left( {\\sqrt {1 - \\sum\\limits_\\alpha {\\sigma _\\alpha ^2 } } ,\\{ \\sigma _\\alpha \\} } \\right), \\,\\,\\,\n\\alpha = 1,2,\\dots ,n - 1", "eb22ec684250bf9b42adbf159c2b79ec": "\\xi_C =\\left( \\frac {x}{\\lambda} - f \\ t \\right)\\ , ", "eb22ec849f6ae72e787405226cff5411": "\nn \\mapsto \\frac{\\vert \\phi_n (x) \\vert^2}{{\\mathcal N} (x)}.\n", "eb2321d9ed1d4d2ec956f551d44e6d4e": " I_C= \\sum_{i=1}^n m_i\\Delta r_i^2,", "eb234475c00d93ec27a4100aa11cb2b5": "\\chi_5(n)", "eb237013e627e7ba00040929ff20ef8c": "f_j \\to g", "eb23a784b9934b1e201d3cd50c9fde4c": "\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} = \\nu \\frac{\\partial^2 u}{\\partial x^2}", "eb23d11d87b75c99969a0be72f611060": "V^*(f)(t) = \\int_t^1{f(s)\\, ds}.", "eb23d62d019e7008d84805f47ead6ef2": "2+\\tfrac{3}{4}", "eb23de9cf0b84128d8d4b8d6b2e84348": "f \\in K[x,y]", "eb24034bd3d4be82595400cdb12bc888": "p_1 \\leq q_1", "eb240ea966d4b8b3160b0b8d9d360014": "x=[u_0; u_1, u_2, \\ldots]\\, ,", "eb241204a6637f30225fc7d7d72eaac3": "M_{k,n} \\,", "eb24457e53f7e01bf1344f456cde1956": "k(A) = \\frac{n}{L} = \\frac{n \\, dt}{L \\, dt} = \\frac{tt}{\\left | A \\right \\vert}", "eb246673ede540aeecdb866c24d436fd": " T^\\alpha {}_{\\lambda \\epsilon} = g_{\\lambda \\beta} \\, g_{\\epsilon \\gamma} \\, T^{\\alpha \\beta \\gamma}. ", "eb249abe4f56eb05b49dd429d616f2be": " E \\approx {{\\dot A}^2\\over 2} + {q^2 \\rho^2 \\over 2m} A^2. ", "eb251f4a3c942babf70ef9fa83b7de42": "\\left (\\partial_t-k \\partial_x^2 \\right )(\\Phi*f)=f,", "eb2538a6109396bd2c55fd74897b72bc": "G(x,y)=\\frac{1}{A}\\int_A z(x',y')\\ z(x'-x,y'-y)\\, dx'\\, dy'", "eb254fc458e8a12d87588abf9c9d4b56": "B_{ab}^{IJ}", "eb256e21e26be8d63b6ffa774fd64703": "T = a + b ID = a + \\frac{ID}{IP} \\,", "eb25b4443bfd47917f6ec5c76dde8d3b": "w = -ae - bf - cg\\,", "eb25be6ac3d8df1498a1cbdf36751e96": "\\bold{e}_y", "eb261cfc1dc6e6b26b9a4cec6eacec85": "\\int\\limits_{\\overline{\\mathcal M}_{g, n}} \\psi_1^{a_i} \\cdot \\dots \\cdot \\psi_n^{a_n}\\lambda_g = \\binom{2g + n - 3}{a_1, \\dots, a_n} \\int\\limits_{\\overline{\\mathcal M}_{g, 1}} \\psi_1^{2g - 2}\\lambda_g.", "eb263120c6f13bf48637fb5a724c07da": "\\Delta^{op} = \\sigma_{H, H} \\circ \\Delta", "eb266815c619aade86a3c3671059d866": "(\\mathrm{Gr})", "eb26777b12c3040f92d97930f88eb592": "30\\mathbb Z", "eb2680d28d0c670222fde7f32a8fd7d4": "d[B, A] = 25", "eb27819598e3c6fb8ccf616fa59939ca": "i \\in \\mathbb{N}", "eb27a1c92db38a3b07858dac846312fc": "P_K : H \\rightarrow K", "eb27d8cc716febb517771561e1585bae": "\\Gamma = ~~~~{G |M|^2 |m_{\\beta \\beta}|^2},", "eb27e4c3f7bdf1b8aa176515d217eca8": "L_M=L_P\\cdot{k}", "eb2830126a352905cf80749d317f01f8": "L:\\mathcal{F}\\to\\mathcal{G}", "eb28527896d2c2470e73e91893cde5a6": "\n |\\mathbf{t}| = \\sqrt{ (n_j~\\sigma_{1j})^2 + (n_k~\\sigma_{2k})^2 + (n_l~\\sigma_{3l})^2} ~~~\\text{(repeated indices indicate summation)}\n ", "eb287a6b7dd3f2e68b808d4ac535dced": "\\,\\! T = \\sqrt{N_0/2^k},", "eb287f2087a2c6d8cf77150cc938dce9": "G(N)", "eb290c0866f8a9b00f4c7f461315ff42": "\nf = \\frac{|\\Theta_{cw}|}{T_{c}}\n", "eb291c182aef021f310f3cf16db1cd40": "j=4", "eb29305f3163f6c52a57f570fef96085": "f^n = f^k", "eb294f728b5b8a5ca82992de5cbbb25d": "g \\in L^p", "eb2a02840f5da84b454868fb8d39c355": "\\bigl(1-o(1)\\bigr)\\cdot\\ln{n}", "eb2afaf0ff660ab0a00622c2450af2dd": "Q = \\mathrm{d}M/\\mathrm{d}x", "eb2b7238f5b3d575da2fe8ee1064b569": "d \\Xi = -\\frac {V} {T} d P + \\frac {U + P V} {T^2} d T", "eb2b8ce7c7a4a88eb896f1104a90baa3": " \\phi\\ (r) ", "eb2b9d3afa371534ba7a00ddb02948d6": "E_{\\rm barrier} = W_{\\rm c} - e (\\Delta V_{\\rm ce} - \\Delta V_{\\rm S})", "eb2bb164581fb84f33f09d054b8da506": " \\sum_{i,j} de_i de_j' u^2_r = -2dq dq' w_r w'_r ", "eb2bc76c919be38f0e7567895fb95fc5": "(1+X)^p\\equiv1+X^p\\text{ mod }p.", "eb2c14959a251999b9dc7c7dd1e6058f": "\\textstyle SC(S) = \\sum_i C_i(S) = \\sum_{e \\in S} n_e \\frac{c_e}{n_e} = \\sum_{e \\in S} c_e\n", "eb2c17c97679c24433a4c99b70b0b9a9": "g = g_{ab} \\, dx^a \\otimes dx^b", "eb2c1f560c6faa8791d092f675aa23a4": "-\\!\\!\\ast", "eb2c94727ab8d6671cc70ff762284e15": " I(x) = \\{ t \\in T : (t,x) \\in U \\}\\,", "eb2d8aa2f6639ba3e8ff17bbe01e2779": "A:2\\times 3 =\n \\begin{bmatrix}\n 1 & 2 & 3 \\\\\n 4 & 5 & 6\n \\end{bmatrix}\n", "eb2e2de459762593eff5e85bbab01614": "\\scriptstyle \\mathbb{F}_p=\\mathbf{Z}/p\\mathbf{Z}", "eb2e2e7396241eef6232e6adb876f454": "r = r_0 + \\frac{q}{\\frac{1}{RD^2} + \\frac{1}{d^2}}\\sum_{i=1}^{m}{g(RD_i)(s_i-E(s|r,r_i,RD_i))}", "eb2e6a7afc6b027f765822c8445df988": "N(\\mathbf{r})", "eb2e7259a55462adaaf3c41ed6c7718d": "\\kappa(A) = 10^k", "eb2e782a4d60395380d96dd022a0e96c": "A \\cap B = \\{ x: x \\in A \\,\\land\\, x \\in B\\}", "eb2e95e149a17b4fe2f9ec4505e75f07": "x^2=1", "eb2e97c38078068238ec46b7be9d0f92": "\\pi:E\\longrightarrow B", "eb2eb67cb4d3b885b6e20a16a94df623": "\\nu = -\\frac{\\mathrm{d}\\varepsilon_\\mathrm{trans}}{\\mathrm{d}\\varepsilon_\\mathrm{axial}} ", "eb2eec2274f362e2043d8f208789f71c": "\\sqrt{v_x^2 + v_y^2 + v_z^2}", "eb2efb4606145f6c92c953422fe9b16e": "\\phi \\geq 1.35", "eb2eff3dc495897f77e8dedaa9b1bb99": "\n\\psi^\\dagger(k)= \\int_x e^{-ikx} \\psi^\\dagger(x)\n\\,", "eb2f1f5eed7271a6de74a63a468e5997": "M_{ij}=\\rho_{ij}\\sigma_i\\sigma_j\\,", "eb2f47b4c7881b2c844b51fa6f2000c1": "h_v =\\frac{v^2}{2 g}", "eb2f66e7e1268dc50458029765bb64f8": "D_e=\\phi_eF", "eb2fb36af3fe74ba2d1644157ce1b76c": "\\begin{align}\n d_1 &= \\frac{\\left[\\ln\\left(\\frac{38.6459}{40}\\right) + \\left(0.1 + \\frac{0.3^2}{2}\\right)(0.5)\\right]}{0.3\\sqrt{0.5}} \\\\\n &= 0.1794 \\\\\n\\\\\n d_2 &= 0.1794 - 0.3\\sqrt{0.5} \\\\\n &= -0.0327 \\\\\n\\\\\nN(d_1) &= 0.5712 \\\\\nN(d_2) &= 0.4870 \\\\\n\\\\\n C &= 38.6459(0.5712) - 40e^{-0.1(0.5)} (0.4870) \\\\\n &= 3.5446 \\\\\n &\\approx \\$ 3.54\n\\end{align}", "eb300de258f7bc4de3156a1f48217bd1": " \\ ", "eb3026ed4c96a441149eb89a2baef0bc": "\\phi{t \\over x}\\in\\Phi", "eb30528915e446832f56f85dbc4f2b41": "|\\zeta|^n\\leq \\|a\\|_p \\left(|\\zeta|^{q(n-1)}+\\cdots+|\\zeta|^q +1\\right)^{1/q}=\\|a\\|_p \\left(\\frac{|\\zeta|^{qn}-1}{|\\zeta|^q-1}\\right)^{1/q}\\leq\\|a\\|_p \\left(\\frac{|\\zeta|^{qn}}{|\\zeta|^q-1}\\right)^{1/q},", "eb309b989126dd425681094d6278e0b4": "\\phi(\\mathbf{r})=\\frac{q}{4 \\pi \\varepsilon _0 | \\mathbf{r}- \\mathbf{r}_+ |} -\\frac {q}{4 \\pi \\varepsilon _0 | \\mathbf{r}- \\mathbf{r}_- | } \\ , ", "eb309f9e55861f3704ba8d815c2c5e47": "\\Sigma(p,q,r)", "eb30a9f64134f6f202f82d76f8ad5703": "{\\mathbf{f}}_{n,m}", "eb30c4187f404ec46652a35656576b76": " \\mathbf{Q}^{te}, \\mathbf{Q}^{fe} ", "eb310e735cf3f48a48f3feb6464e0571": "(C_{i,j}^{II})_p = \\begin{cases}\n0 & \\text{if } j < p \\\\\nC_{i,j} & \\text{if } j \\ge p \\end{cases}", "eb3110620ee7b81120074704f6752641": "\\mathbf{X}_A", "eb318b57ef874ba17feb6061a6132662": " \\mathbf{E} = \\mathbf{F}/q .\\,\\!", "eb319d2317c74d60f6d10418e918bbd5": "\\begin{cases}\\dot{\\mathbf{x}} = f_x(\\mathbf{x}) + g_x(\\mathbf{x}) z_1 + \\mathord{\\underbrace{\\left( g_x(\\mathbf{x})u_x(\\mathbf{x}) - g_x(\\mathbf{x})u_x(\\mathbf{x}) \\right)}_{0}}\\\\\\dot{z}_1 = u_1\\end{cases}", "eb31ba7b0cc4eae07b0e8352892c3548": "H_{r\\theta\\theta}=\\frac{-GM}{3(1-2GM/r)}", "eb31c56cc1a0b000db0648ee8d4fdce5": "\\kappa = -1", "eb3260cd9adad15aa941780250dfef23": "x = \\frac{\\lambda}{T}\\, t + \\frac{\\lambda}{2\\pi}\\, \\theta_0 = \\frac{\\omega}{k}\\, t + \\frac{\\theta_0}{k}.", "eb3270498d2ec58635ca5f41dee6c34e": " T_{ij} > 0 ", "eb32a410c65f84ca49672f014b47e824": " \\kappa = - \\xi\\,\\!", "eb32a4205ca7a497ca26dfa5fe5afc20": " a < X \\leq b ", "eb32c8734cc83042661951a6fbeb202a": "\\{0,\\dots,k-2\\}", "eb32d59fda0765dfa217f95b542833b9": "S^n,S^{n-1}T,S^{n-2}T^2,\\ldots,T^n", "eb32d7ec5d53b31b2deb94d250314d1b": " 221133,\\; 221331,\\; 223311,\\; 233211,\\; 113322,\\; 133221,\\; 331122,\\; 331221, ", "eb32dcfc438276cdc70b80ce99324e68": "\ny = a \\sqrt{\\left( \\sigma^{2} - 1 \\right) \\left(1 - \\tau^{2} \\right)} \\sin \\phi\n", "eb32df22e08240608819c8a52cafb277": "\\{0,1\\}^n", "eb3308df3452f4f1619405ccf803a494": "\\int\\frac{dx}{u} = \\arcsin\\frac{x}{a} \\qquad\\mbox{(}|x|\\leq|a|\\mbox{)}", "eb333a48eaef3f9b7dbab8ed03c14f00": "\\mathcal N \\alpha = 3\\, \\frac{n^2 - 1}{n^2 + 2}", "eb338a23947eaae0af6e91578062e5e3": "\n\\begin{align}\nF_1(t) & = \\,_4F_3\\left(\\frac{-1}{20}, \\frac{3}{20}, \\frac{7}{20}, \\frac{11}{20}; \\frac{1}{4}, \\frac{1}{2}, \\frac{3}{4}; \\frac{3125t^4}{256}\\right) \\\\[6pt]\nF_2(t) & = \\,_4F_3\\left(\\frac{1}{5}, \\frac{2}{5}, \\frac{3}{5}, \\frac{4}{5}; \\frac{1}{2}, \\frac{3}{4}, \\frac{5}{4}; \\frac{3125t^4}{256}\\right) \\\\[6pt]\nF_3(t) & = \\,_4F_3\\left(\\frac{9}{20}, \\frac{13}{20}, \\frac{17}{20}, \\frac{21}{20}; \\frac{3}{4}, \\frac{5}{4}, \\frac{3}{2}; \\frac{3125t^4}{256}\\right) \\\\[6pt]\nF_4(t) & = \\,_4F_3\\left(\\frac{7}{10}, \\frac{9}{10}, \\frac{11}{10}, \\frac{13}{10}; \\frac{5}{4}, \\frac{3}{2}, \\frac{7}{4}; \\frac{3125t^4}{256}\\right)\n\\end{align}\n", "eb33a5abbb2a36f57ad50833bd6bc8cb": "{}^2E(f) = 0 \\quad", "eb33b8b785120ba5e4dc6d8248ab7f29": "mSG", "eb34d2bb5edc3d5b85a7b51e85e82b95": "w(z)=P \\left\\{ \\begin{matrix} a & b & c & \\; \\\\ \n\\alpha & \\beta & \\gamma & z \\\\\n\\alpha' & \\beta' & \\gamma' & \\;\n\\end{matrix} \\right\\} = \\left(\\frac{z-a}{z-b}\\right)^\\alpha \n\\left(\\frac{z-c}{z-b}\\right)^\\gamma\nP \\left\\{ \\begin{matrix} 0 & \\infty & 1 & \\; \\\\ \n0 & \\alpha+\\beta+\\gamma & 0 & \\;\\frac{(z-a)(c-b)}{(z-b)(c-a)} \\\\\n\\alpha'-\\alpha & \\alpha+\\beta'+\\gamma & \\gamma'-\\gamma & \\;\n\\end{matrix} \\right\\}\n", "eb34d61dd7c246cbd90084dd15ae4204": "\ny_t = ~\\beta x_t + ~\\lambda ~\\sigma_t + ~\\epsilon_t\n", "eb34e495c2034746ab77a2e18eb29ac3": " k' = k [B] \\, ", "eb350261655926f9b30cced80b97a391": " [E_f-\\epsilon, E_f+\\epsilon]. ", "eb3525e19e5fe6d0be37712c98b1298a": "N_0(T)", "eb359ec4b5504b09cf58de758739da03": "\\ell = \\frac{3}{4n_0\\pi d^2}", "eb35dd9df52f490536790be42eb9a821": "x_5 = g_3(x_6)", "eb3601f143d6b31dbffd09367f0c2159": "\\hat{P}_\\mu = \\left(\\frac{1}{c}\\hat{E},-\\bold{\\hat{p}}\\right) = i\\hbar\\left(\\frac{1}{c}\\frac{\\partial}{\\partial t},\\nabla\\right) = i\\hbar\\partial_\\mu \\,\\!", "eb360744092d664677819853af5b59df": "C_{qs}", "eb364efa77c8ef3af67c5087a37a60c3": "\\operatorname{d}(z_1, z_2) = |z_1 - z_2| \\,", "eb368c654dd227010d0929caa885a48c": " \\left(A f\\right)^\\star = f^\\star A^\\star ", "eb368ea453696eda4e95891d9ce32764": "\\Delta t = \\int^{\\Delta\\tau}_0 e^{\\pm\\int^{\\bar{\\tau}}_0 a(\\tau')d \\tau'} \\, d \\bar\\tau , \\ ", "eb36bee1ebca6f497c41410b126f397e": "P = C V^2 f.", "eb36f21f5799e3bef1b85ea4bb4ce28c": " X_i ", "eb377fa6a9c72ab7f0c4a811fa55c4dc": " x_i = 0", "eb379124e874bf03f5c667ce882ef65f": "\\begin{align}\n\\Phi\\Psi g &= G(\\varepsilon_X)\\circ GF(g)\\circ\\eta_Y \\\\\n &= G(\\varepsilon_X)\\circ\\eta_{GX}\\circ g \\\\\n &= 1_{GX}\\circ g = g\\end{align}", "eb37bbe1a6601dd3e86bb84fb9dba573": "p=w\\rho c^2,", "eb37d1d5557575ff9c2d8db1c138a7ff": "\\int\\arccot(a\\,x)\\,dx=\n x\\arccot(a\\,x)+\n \\frac{\\ln\\left(a^2\\,x^2+1\\right)}{2\\,a}+C", "eb380f3b2439960f7727e82712b46659": "n>1", "eb3810bda01a7292a62d2212ebc9a2a2": "\\tfrac{BM}{DM}", "eb38187431b009909b022cdcfe273376": "\\left( -\\sqrt{2 \\over 5},\\ \\sqrt{2 \\over 3},\\ \\frac{2}{\\sqrt{3}},\\ \\pm2\\right)", "eb3848eeea5ef29d3715f8a44dc786ea": "k\\geq N", "eb384c52e3f009bd2905070e823031d3": " ...HH... \\ ", "eb385faab00fe01cf0c2cf2e8ea0e86e": "B \\subset Y \\,", "eb3871a147b5be80d314ece452469d9a": "u,\\varphi \\in H_0^1(\\Omega)", "eb38a36fac3158b33aa3cad5a301532d": "\\int_{0}^{\\infty} e^{-ax^2}\\,\\mathrm{d}x=\\frac{1}{2} \\sqrt{\\pi \\over a} \\quad (a>0)", "eb38a481b4936ca2f2501794efcbca5f": " \\begin{cases} \\Psi : J^{r}(\\pi) \\to TJ^{r}(\\pi) \\\\ (x,u,w) \\mapsto \\Psi(u,w) = V \\end{cases} ", "eb38c709ff98071f4ce4869d7e0d6ac5": "C_1: (t,t^3)", "eb38f2075edec47a8be35bb094192753": "\nQ^\\dagger(\\mathbf{p})|0 \\rangle = | 1_\\mathbf{p}\\rangle, ", "eb390c644a7864a77b4c71c0887ad1ec": "M_{[ab]} = \\frac{1}{2!} \\, \\delta_{ab}^{cd} M_{cd} ,", "eb3967ce7e3df5d63c7b30c93118f628": " E \\subseteq \\bigcup_{i=1}^{c_N} \\bigcup_{B\\in A_i} B.", "eb39d2be5f05cd580cc9a809c2400b2f": "F(x)=\\mu((-\\infty,x])", "eb3a0415ab236889150a67bdbf1c1bb5": "\n\\csc z\\,\n", "eb3a0c9d6bc9c8c3898c7d92bb91bb41": "\\bigcap_{n=1}^\\infty C_n = \\{x\\} ", "eb3a0fb3e9e19bf9c57141816e612f86": "\\scriptstyle{|\\phi_2\\rangle}", "eb3a3bff5286440f249ca227fb527737": "T=\\lbrace 5,9,1,10,15\\rbrace ", "eb3a95242eed35f5752058dd0d7c7b98": "y_i=0", "eb3aa615ee760c951f5eeb0bf4b7319a": "G : \\mathbf{N} \\times \\mathbf{N} \\to \\bigcup_{n \\in \\mathbf{N}} A_n", "eb3ac1daed9858cd54e177c0817ddfb7": "10^{-30}", "eb3b3a43c4027db41b8e2ce5056723a0": "\\Delta \\psi", "eb3b50a901082763e611f1559fc07ccc": "C_\\max = \\sum_{i=1}^n H(X_i)-\\max\\limits_{X_i}H(X_i)", "eb3b50ac78566fbec9bd4c80a2fc0cd1": "\\textbf{X}", "eb3bcfde4b35198bc701157b044acac8": "p_j=1", "eb3bdc8eed43c6e74c1b770cd75fffc3": "U_n(-1) = (n + 1)(-1)^n.\\,", "eb3cfabe12b4f974ce09b5a212346595": "\\Delta\\cup\\Gamma\\models\\psi", "eb3d65c40834fb2f98d37aa46f27d7c6": " \\hat P ", "eb3d72905431aee37801b9619eac8c50": " q_{m} ", "eb3e5a9c5a6d7ddedc76998824b3de21": " i = \\; \\alpha_{\\mathrm{r}} A_{\\mathrm{M}} a {\\phi^{-1}} {\\beta}^2 V^2 \\mathrm{exp}[- v(f) \\;b \\phi^{3/2} / \\beta V ], ..........(37) ", "eb3e83bf04e8b3dd1718b5a2552895a6": " q_i = T_H \\Delta S_i ", "eb3ec4f84bda76e8980571a83e97c79b": "K^0 \\leftrightarrow \\bar{K}^0", "eb3f1e1b8ae47047aee0d843838ffb4a": "\\frac{v_{\\text{C}} \\left( t \\right)}{i_{\\text{C}} \\left( t \\right)} = \\frac{V_p \\sin(\\omega t)}{\\omega V_p C \\cos \\left( \\omega t \\right)}= \\frac{\\sin(\\omega t)}{\\omega C \\sin \\left( \\omega t + \\frac{\\pi}{2}\\right)}", "eb3f6f5ab380f9800da30c3d456a4a1e": "\\rho_\\theta", "eb3f75f84272a06b02c6660adff7d9dc": "100^2", "eb3fedc4c5e5c67af7f20f316f73eb7f": "x_1, \\ldots, x_n", "eb401e04ff45683808efdc4350db03e1": "Z_\\text{P} = \\frac{V_\\text{P}}{I_\\text{P}} = \\frac{1}{4 \\pi \\epsilon_0 c} = \\frac{\\mu_0 c}{4 \\pi} = \\frac{Z_0}{4 \\pi} = ", "eb405f9e3cfc59968e91ba46acc50957": "\\mathrm{SL}(p+q,\\mathbb{C})", "eb40711376e4c45f15e58c87586ae28e": " G = \\frac { b_0 + b_1 Z^{-1} + b_2 Z^{-2} } {1 +a_1 Z^{-1} + a_2 Z^{-2} } \\, ", "eb40d6447e9d64b72006ed54f16cbe97": " \\!\\ \\sum_{n=1}^{\\infty}{1 \\over {{n^2 + n} \\over 2}} = 2\\sum_{n=1}^{\\infty}{1 \\over {n^2 + n}} = 2 .", "eb40dac4f1f6d0603a74f77a725059ad": "-x_1-x_2-x_3=A-Bl^2", "eb40e936db2bb74ef8674a2e573d8fba": "T=T_H=T_C", "eb40f54255172c85ecc89c8c73ad4239": "N-n_1 \\choose n_2", "eb417de5a82fce6a0beacfae7fe81ef3": "f(x) = f(0) + f'(0)\\cdot x + f''(0)\\cdot \\frac{x^2}{2!} + \\cdots", "eb42a80e92d854bfe3f3f400a4865c28": "\\ln A=\\frac{1}{8}-\\frac{1}{2} \\sum_{n=0}^\\infty \\frac{1}{n+1} \\sum_{k=0}^n \\left(-1\\right)^k \\binom{n}{k} \\left(k+1\\right)^2 \\ln(k+1)", "eb42ba35bf00c6d03a2e6a2afd140cc7": "p'_c", "eb42e5897addcbae09210482ffe0bef4": " \\frac{1}{2} \\times ||OA|| \\times ||OB|| \\times \\sin\\theta = \\frac{1}{2}r^2\\sin\\theta \\, . ", "eb4301a3fcc4bf66ace3e75397e1d7a5": "b, c, d", "eb433d3d6b0bf07fc4734d3dfa2a4fc9": "m > n^{1+\\epsilon}", "eb44108dd5116b00f95b00712704093d": "\\Phi_B : \\mathfrak{N} \\to \\mathbb{Z}_{+}, \\varrho \\mapsto \\varrho(B)", "eb4423a154162e63179b35f53a59108f": "x_{n+1} = r_n x_n (1 - x_n)", "eb446836f2ed8a12246697ad9a2cb1d5": " ms^{-4} ", "eb447e6363a98be2a621c6b7e7e32e28": " \\arctan(\\gamma_n) = \\gamma_n", "eb4481cbcb4edf26d64b9bbf43ea2707": "\\left( \\frac{\\partial ln k}{\\partial P} \\right) _T = \\frac {\\Delta V^\\ddagger}{RT}", "eb44a4d46c78d8b79c9362d897b3d5ce": "J = \\begin{bmatrix}0 & I \\\\ -I & 0\\end{bmatrix}.", "eb44c67609a6173b7578f530773c48c4": "L(st)=L(s)+L(t)\\quad \\forall s,t\\in\\Sigma^*", "eb44ce0eb40cd95ea06d00b93d29cee0": "\\vartheta \\ ", "eb458acd8d3153906a5057bbe4583638": "\\mathbf{h}_{i}=(h_j,0)", "eb46150110db97d7cb44e6d007622d28": "\\mathrm{C_n H_m + Energy \\rightarrow n C + \\frac{m}{2} H_2}", "eb464e6692b754f0f917f70fa7961a26": " f(\\operatorname{E}(X \\mid N) ) \\leq \\operatorname{E}(f \\circ X \\mid N).", "eb466037b7d403691da6f103e1b0cb07": "b_{ij}=0", "eb4677e1bbed802fe4fa6a3a96a0f051": "p_Z(z) = \\int^{+\\infty}_{-\\infty} |y|\\, p_{X,Y}(zy, y) \\, dy. ", "eb4731826796ad4f3d4615ff83ef8574": "\\alpha_{abs}", "eb4770513c2e5c16bee5a7ef397a87e7": " c = P_{(\\centerdot,n)} ", "eb479579af9038320a8bb56d8989d167": "\\left(-4/3,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "eb47a5cc22408d9ce7697ec3fa3a8102": "\\mathrm{dleq}(g_1, h_1,g_2,h_2)", "eb487eaf22c51778b7e7a779901b29d6": "\n\\begin{array}{c|c|c}\n\\text{Info-gap Format}& \\text{MP Maximin Format}&\\text{Classical Maximin Format}\\\\\n\\hline \\\\[-0.18in]\n\\displaystyle \\max\\{\\alpha: r_{c} \\le \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} R(q,u)\\}&\\displaystyle \\max\\{\\alpha: \\alpha \\le \\min_{u \\in \\mathcal{U}(\\alpha,\\tilde{u})}\\ \\varphi(q,\\alpha,u)\\}&\\displaystyle \\max_{\\alpha\\ge 0} \\ \\min_{u\\in \\mathcal{U}(\\alpha,\\tilde{u})} \\varphi(q,\\alpha,u)\n\\end{array} \n", "eb48d6d5e5594cab05205850b6019330": "\\Phi_D = 4\\pi Q", "eb48f7c2fa6497c9e0a21e121f6945cb": "~~G\\!=\\!c~~", "eb49cd03df34f6fc555c5e47f7e14148": "\\partial_\\mu H", "eb49f4a8de661853d54d30c93e39118b": "\\forall w_1,\\ldots,w_n \\, \\forall A \\, \\exists B \\, \\forall x \\, ( x \\in B \\Leftrightarrow [ x \\in A \\and \\varphi(x, w_1, \\ldots, w_n , A) ] )", "eb4a362c5a215c0102decd208235fd97": "X = \\oplus_{i \\in I} X_i", "eb4a5655155408602a1b8b9cf3b4dbbd": " \\sum_{i=1}^n x_i f_i(a) = a", "eb4a7042da6bb8640a9be52f13505add": " K = (S - I)e^{rT}. \\,", "eb4a954765ac3877418519b4fb1d93bd": " P = T^3 \\cdot S = {5^{3/2} \\over 2^3} \\cdot {8 \\over 5^{5/4}} = 5^{6/4 - 5/4} = 5^{1/4}, ", "eb4abd688064f1c17a58b024604445b7": " a_{n} + a_{n-1} x + \\cdots + a_0 x^n ", "eb4acd5c523f62466aef357a055c75c9": "\n \\hat\\beta = \\Big( X^\\mathsf{T}(\\hat\\Sigma^{-1}\\otimes I_R) X \\Big)^{\\!-1} X^\\mathsf{T}(\\hat\\Sigma^{-1}\\otimes I_R)\\,y .\n ", "eb4ae5884914cf45d7e230a9f83f8b4e": "P(H|\\text{smaller coin})=P(H|\\text{slightly bigger coin})=P(T|\\text{smaller coin})=P(T|\\text{slightly bigger coin})", "eb4af828bbe45901dd7eff4a00f6338d": "F_n\\equiv1\\pmod4", "eb4b8e28e87c010b28555ea135a37de0": "\\sigma(X)=\\sum_{i 0\\ \\exists \\ \\delta > 0 : \\forall x\\ (0 < |x - c | < \\delta \\ \\Rightarrow \\ |f(x) - L| < \\varepsilon).", "eb6b9da96d1fd7442c2fcb561134c541": "|\\text{E}[X]- \\text{m}[X]| = \\frac{1- \\ln(2)}{\\lambda}< \\frac{1}{\\lambda} = \\text{standard deviation},", "eb6bd66402e59c324831ff6a96de9712": "\nB_{x}(\\mathbf{r}) = \\frac{\\mu_0}{4 \\pi} m_1 \\left(\\frac{3\\cos\\theta\\sin\\theta}{r^3}\\right)\n", "eb6c5fa5bdf832eedbacad32b086a3a4": "\\psi(t)", "eb6caf49f865c8f218afc53a4c54bb5a": "\\gamma_{\\mathrm{SL}}", "eb6cb284e39e22c72b0c80243ed5a361": " y^*_{n+1} = y_n + \\sum_{i=1}^s b^*_i k_i, ", "eb6cea6cf1bc71d8aa15836a207255e4": " \\vec{p}_3", "eb6d09ac8a4d87062ca98ffc415837f4": "(g, h) : (a, b, f) \\rightarrow (a', b', f')", "eb6d2b903357ad4e7331094bdaa6b849": "\\limsup_{n\\to\\infty} x_n = - \\infty", "eb6d3d0df8c7b74639f1a48cc8d1d8a8": "T(n) = \\left\\lfloor 3 \\, b \\frac{\\left(\\frac{1}{3} \\left( a_{+} + a_{-} + 1\\right)\\right)^n}{b^2-2b+4} \\right\\rceil", "eb6d58444e91cba70599ef9cb6774f0f": "v_{\\mathrm{g}} = R T / P,", "eb6d7ef69515ed1c6827f5317df4e225": "\\approx \\frac{d}{2} *( \\frac{(h_t+ h_r )^2}{d^2} -\\frac{(h_t- h_r )^2 }{d^2}) ", "eb6e047b9c1aabd22c9f0bb5237cb016": "\\frac{\\partial \\eta}{\\partial t} = -\\frac{1}{\\varepsilon_o}\\frac{\\partial\\mathbf{q_s}}{\\partial x}", "eb6e8f21b76604f0a5f5ad1d56716860": "w_{i-1} = x_{i-1} A x'_{i-1}, w_i = x_{i-1} v_i x'_{i-1},\\ and\\ p_{i+1} \\in \\sigma_i", "eb6f0124598955af09bb037608a431bd": "|1\\rangle|1\\rangle", "eb6f3302ee116ceed7e360d5f7f3d392": "f(x+kp)=x^2+2xkp+(kp)^2-n", "eb6f72acfac808b93acba2d483c7da3d": "C_2 = \\begin{bmatrix}\n c_1 & c_2\\\\\n -c_2^* & c_1^*\n \\end{bmatrix},\n", "eb6fae117e4263920c1a0233d27a5148": "\\operatorname{Cov} (G_P(f),G_P(g))= E G_P(f)G_P(g)=\\int fg\\, dP-\\int f\\,dP \\int g\\,dP\\text{ for }f,g\\in\\mathcal{F}", "eb705c53b96603f7ba479062b6c63f5a": "\\frac{d N_i}{d t}=V \\sum_r \\gamma_{ri}(w^+_r-w^-_r) .", "eb70b53924ec1a060390c0e7f0d685dc": "ABCD^+ \\to AD^+ + BC\\,", "eb70d69f9c645df5959799674996bab2": " \\overline{F}(x_1,\\dots,x_k) = \\left( 1 + \\sum_{i=1}^k \\left(\\frac{x_i-\\mu_i}{\\sigma_i}\\right)^{1/\\gamma_i}\\right)^{-a}, \\qquad \nx_i > \\mu_i, \\sigma_i > 0, i=1,\\dots,k; a > 0. \\qquad (4)\n", "eb70dca6be31f9d1235022be6581e526": " \\Delta \\subseteq Q \\times \\Omega_{Z,[t_l,t_u]} \\times Q", "eb71610c36e2181466fe6cd5895a1439": "\\gcd(x^{q}-x, x^{3} + Ax + B)\\neq 1", "eb717acfb720b3d33efa4dc3f998a13e": " k_x \\frac{\\partial^2 u}{\\partial x^2}+ k_y\\frac{\\partial^2 u}{\\partial y^2} +q =0 \\text{ for } x \\in \\Omega ", "eb71800ec790527afd3802d069557fc2": "\\scriptstyle V_a", "eb71b129cd26762a60bc37d1cbbc0e46": "x \\in [0,L]", "eb71b8bee245ef7b3789504007b06a53": "\\mu \\,\\neq\\, \\nu", "eb7215da9c473c56f9752f67bcf1f280": " g(0) = 0;~ g(1) = 3;~ g(2) = 2;~ g(3) = 5;~ g(4) = 4;~ g(5) = 7; ~ g(6) = 6; ~ g(7) = 1; ", "eb722e5d2b510e1cc478b2f1f6ae3087": "2P_{1/2} \\to 1S_{1/2}", "eb72461f42a116efd3c89fa47200c181": "\\epsilon_{si}", "eb729dda2175a5676bbd52198ce36bbd": "\\varphi : \\mathrm{Div}(C) \\rightarrow \\mathbb{Z}", "eb72a7289e28da38783b85a96e64b6d6": " \\bar{A}=\\operatorname{cl}(A) ", "eb73910ded09a63e0e9ac6a320062af3": "r=a^{-1}", "eb73c8ce2b12b65df0e8f854e733d0e2": "\\hat{y} \\ge 0", "eb73e1d2175c43e39e491e16b0a2c2c7": " \\tau_L ", "eb73fe1c4dfb0050a62c727df166fac2": "\\vec{v}(t)", "eb7459039850b3c23c06ad263009c03f": "Y = \\lim_{d_2 \\to \\infty} d_1 X", "eb746c4613b947ecede99402a975d899": "V_\\gamma = e_\\gamma^I V_I", "eb748664930d6c7ddde8d969209ef913": "\\vec E=-\\nabla\\varphi -\\frac{\\partial \\vec A}{\\partial t}", "eb74af10d37195d6fe96c72e9f2b6fab": "f(x)=\\sqrt{r^2-x^2},\\quad x\\in[-r,r].", "eb750525fda8c87815f9bc1d8cfca8f3": "\\omega\\ :", "eb756f2e46e7e34958bf64fecf284669": "d \\in D = \\{1,2,3,\\ldots\\}", "eb75f05f6d21bdde2d4ba0c9bca2ac00": "[x_1,x_2,x_3,x_4]", "eb75fc7a5d4f2d711fe50aa78ce6bab7": "\\textstyle E = (010000110)", "eb760ce2e565383938d8ea94e4eb5c25": "Q_{dump}=P_{amb}*K_{bellows}*V_E", "eb7648aa52c23d3ac5137d189309470d": "RN_i[i]", "eb768c893b06637614d79e416a622b46": " \\vec{v_o} ", "eb769531a67bd4f8eb49d45b0cd18176": "x_1 , \\dots , x_{k+1}", "eb772f6a1388dc31f235662676b1e1f6": " r = \\frac{1.22 \\lambda}{2 n \\sin{\\theta}} = \\frac{0.61 \\lambda}{NA} ", "eb7744a69ffea32b65b321ceef871f27": "\\nabla^2 \\phi = 0", "eb77490e5e78ff768440829600e409d6": "x_0 = \\alpha \\sinh(t/\\alpha) + r^2 e^{t/\\alpha}/2\\alpha,", "eb77bf1c46ca4416aac22524ad1647cf": "\\mu_i = 1/2", "eb77dc482d1f5a6624ae649cb13193df": "~ \n\\frac{{\\rm d}n_1}{{\\rm d}t}=-W_{\\rm u} n_1 + W_{\\rm d} n_2\n~", "eb77fd64d671a641a8e3b11a2f0ace43": "\\Delta\\colon L \\to L\\otimes L", "eb785dfa5813bafc2305535c4451469c": "\\,O, O_i", "eb7896c683dc66070e72ce5f70ffd523": "\\or \\lor \\vee, \\curlyvee, \\bigvee \\!", "eb78a2e02930f1072567a60311b13d36": " F_1 \\ ", "eb78d8050bce6248dadf3cede21ff426": " \\begin{align}\n& \\Re[Z(s)]>0 \\quad\\text{if}\\quad \\sigma > 0 \\\\\n& \\Im[Z(s)]=0 \\quad\\text{if}\\quad \\omega=0\n\\end{align} ", "eb78dce385ac1336d30aa17d6e49d76e": " {}_2W_3 = mc_v \\left( {T_2 - T_3 } \\right)", "eb7927693b0c3ac139ddac86c45a2043": "\\Delta = I_{-\\infty}^{+\\infty}\\frac{b_0\\omega^{n-1}-b_1\\omega^{n-3}+\\ldots}{a_0\\omega^n-a_1\\omega^{n-2}+\\ldots} \\quad (26)\\,", "eb7988603dbcab6b90acfcd5c1afcbe5": "F(x)=\\int_a^xf(t)\\,dt,", "eb7990576a007f6548c6cbdf8cc5ef28": "\\Delta f\\,", "eb79abf1a52a3d81b36df005e0563d5f": "\nA_{NFA} = \\{\\langle B,w \\rangle \\mid B \\text{ is a NFA that accepts input string } w \\}\n", "eb79b8f86aab4c53514af9c9c8724926": "\\dim S \\le d", "eb7a12487698a1286a254711a711f081": "C_2: (x-1)^2+(y-1)^2-10=0", "eb7a2b5fad09d2244130afb0d6ae67cb": "P^{\\top} \\cdot P = \\mathbf{I}", "eb7b1efe64dac408d90522330819c281": "\\begin{align}\\operatorname{VAR}(S)=&\\frac{2\\left(n^3-\\sum t^3_i -\\sum u^3_i\\right)+3\\left(n^2-\\sum t^2_i -\\sum u^2_i\\right)+5n}{18} \\\\\n&{}+\\frac{\\left(\\sum t^3_i-3\\sum t^2_i+2n\\right)\\left(\\sum u^3_i-3\\sum u^2_i+2n\\right)}{9n(n-1)(n-2)} \\\\\n&{}+\\frac{\\left(\\sum t^2_i-n\\right)\\left(\\sum u^2_i-n\\right)}{2n(n-1)}\\end{align}", "eb7b3e808234e0ff2bc05cfb8dd6a8d4": "a_{-l} = 0 \\; \\; \\forall \\; l \\geq m", "eb7b587ea25e3f20c6f34ac9b0b84ba9": " (x,y,z) ", "eb7b7d59e7c57f74fed4e82f6b76c677": "2x_1+x_2=-1", "eb7b8df418b1d944a9e18501ba7d18cf": "\\left(\\!\\!\\binom{r}{k}\\!\\!\\right) = \\binom{f}{k} \n=\\binom{r+k-1}{k}=\\binom{-1}{k}\\binom{-r}{k} = (-1)^k \\binom{-r}{k}.", "eb7bb831fdedf26b207b281ee1e9e6c2": "-0.5\\int_{-\\infty}^{q}(y-q)dF_{Y}(y)+0.5\\int_{q}^{\\infty}(y-q)dF_{Y}(y) .", "eb7bc91dd649232982af7ddaaaca8c18": "c_{11} = 2.91583 \\times 10^{-5},\\,\\!", "eb7bee9d79d919e21b492e91c8775754": "Loves(x,", "eb7bf6ca3b9bbd99afc372971697da99": "m \\to m/1", "eb7bfab76222cc3b421334f4681c9ad8": " ds^2 = E \\, dx^2 + 2F \\, dx \\, dy + G \\, dy^2,", "eb7c267ffbb18799e19bca0bd7557131": "V\\left(\\rho,\\varphi,z\\right) = \\frac{1}{4 \\pi \\epsilon_0} \\left( \\frac{q}{\\sqrt{\\rho^2 + \\left(z-a \\right)^2}} + \\frac{-q}{\\sqrt{\\rho^2 + \\left(z+a \\right)^2}} \\right) \\,", "eb7c78673bd906d8c504c5729c111256": "\\mathbf{L} = \\mathbf{r} \\times \\boldsymbol{p}", "eb7cd0c10a4d788ef578c79a2782359b": "\\boldsymbol{\\sigma}_{-i}", "eb7ce7ba027ea82c23ae7c44f455a12b": "\\varphi_{k+1}(t)=y_0+\\int_{t_0}^t f(s,\\varphi_k(s))\\,ds.", "eb7cecc557b42ca35f8c280857cab9ff": " p_3 = \\frac{\\omega a_{10}+a_{01} -\\omega\\mathcal{L}a_{20}- \\mathcal{L}(a_{20} \\omega+a_{11})}\n{2a_{20}\\omega+a_{11}},", "eb7cece9189f17bd1923b2ba6de2cf20": "\\left[ \\Pi^{n}\\right] =\\left\\{ \\left[\n\\mathbf{A}\\right] \\ |\\ \\mathbf{A}\\in\\Pi^{n}\\right\\} ", "eb7d0abf248367a77914b7c8bd82063e": "q_{i}(t)", "eb7d0ff12fe10c5fdda1bef74c7af540": " \\zeta = \\frac{c}{c_c},", "eb7d19f807ea9e895b329b2934938dcf": " S = \\sum_{ij} w_{ij} (D_{ij}-T_{ij})^2 ", "eb7d3ab91f278a266f93210d20dc00f8": "4q_p(2) \\equiv \\sum_{k=\\lfloor\\frac{p}{10}\\rfloor + 1}^{\\lfloor\\frac{2p}{10}\\rfloor} \\frac{1}{k} + \\sum_{k=\\lfloor\\frac{3p}{10}\\rfloor + 1}^{\\lfloor\\frac{4p}{10}\\rfloor} \\frac{1}{k} \\pmod{p}.", "eb7dc5267dedf6cecd3dc43a7927a3df": " x^2 + D = A y^n ", "eb7e942dd4efdc3bbdd39a592ec2c8ba": "Q, R, S", "eb7ea9f7079705215d51137c572c7bdf": "\\mbox{MIRR}=\\sqrt[3]{\\frac{7600}{4636.36}}-1=17.91%", "eb7ec76c3b3ac2d4eadfe211b7ed5ba2": "\\begin{align}f(x)&=\\log(1+(\\cos x-1))\\\\\n&=\\bigl(\\cos x-1\\bigr) - \\frac12\\bigl(\\cos x-1\\bigr)^2 + \\frac13\\bigl(\\cos x-1\\bigr)^3+ {O}\\bigl((\\cos x-1)^4\\bigr)\\\\&=\\biggl(-\\frac{x^2}2 + \\frac{x^4}{24} - \\frac{x^6}{720} +{O}(x^8)\\biggr)-\\frac12\\biggl(-\\frac{x^2}2+\\frac{x^4}{24}+{O}(x^6)\\biggr)^2+\\frac13\\biggl(-\\frac{x^2}2+O(x^4)\\biggr)^3 + {O}(x^8)\\\\ & =-\\frac{x^2}2 + \\frac{x^4}{24}-\\frac{x^6}{720} - \\frac{x^4}8 + \\frac{x^6}{48} - \\frac{x^6}{24} +O(x^8)\\\\\n& =- \\frac{x^2}2 - \\frac{x^4}{12} - \\frac{x^6}{45}+O(x^8). \\end{align}\\!", "eb7ee29572e437d14d7a58983595c601": "c_\\empty", "eb7f0e98d7d1fa05eaa5e105c5151b18": "2^{-k}", "eb7f7f7cf72c07af596217c4b790d44b": "\\lfloor \\sqrt{m} \\rfloor", "eb7fb332bfb0080e3ed45bede590e645": "\n\\begin{array}{c|cc}\n\\frac12 - \\frac16 \\sqrt3 & \\frac14 & \\frac14 - \\frac16 \\sqrt3 \\\\\n\\frac12 + \\frac16 \\sqrt3 & \\frac14 + \\frac16 \\sqrt3 & \\frac14 \\\\ \n\\hline\n & \\frac12 & \\frac12 \n\\end{array}\n", "eb7ffbf55414dd41211dea92fc15cbdc": "\\textstyle{\\int}", "eb802f665e5dc9dff2f7771166d2a6d8": "(a^2-b^2-c^2-d^2)^2 + (2ab)^2 + (2ac)^2 + (2ad)^2 = (a^2+b^2+c^2+d^2)^2.", "eb8047712effc25c9eed9b69e0d8e57a": "\\lim_{\\omega\\rightarrow\\infty}\\frac{d\\log(G)}{d\\log(\\omega)}=-n.", "eb804c51e4e3ace0b90a9a64f1781d1d": "\\mathbf{v} = \\left(v_x,v_y,v_z \\right). ", "eb805815243727f062da98b700529c6b": "\nS(\\sigma) \\ \\stackrel{\\mathrm{def}}{=}\\ \\left( a^{2} + \\sigma \\right) \\left( b^{2} + \\sigma \\right) \\left( c^{2} + \\sigma \\right)\n", "eb809aa13a56a372d0baa4737c0f3eee": "a^2 + b^2 = c^2.\\,", "eb809b22d7121ad71fa2ae1c79a1d96c": "\\Sigma=\\begin{bmatrix}10 & 0 & 0\\\\0 & 0.1 & 0 \\\\ 0 & 0 & 0.1\\end{bmatrix}.", "eb810adbc3b7e982fe0414fd2872dddb": "K(-r)=K(r), K(0)=\\bar{X}(1-\\bar{X}).", "eb8113044e47b5b238c50520ba8b4037": "L(\\Bbb{E}\\{\\delta(X)\\}) \\le \\Bbb{E}\\{L(\\delta(X))\\} \\quad \\Rightarrow \\quad \\Bbb{E}\\{L(\\Bbb{E}\\{\\delta(X)\\})\\} \\le \\Bbb{E}\\{L(\\delta(X))\\}. \\, ", "eb8114152a1d9ccf1e161e4d39eef0ee": "\\frac{d}{dt} = \\frac{L}{mr^{2}} \\frac{d}{d\\theta}", "eb81271019448283f1b7a166bdc96b47": "|G:\\operatorname{Core}(H)| \\le |G:H|!", "eb821045f25296faf19eeddc2ef5c0f1": "\\lim_{n \\rightarrow \\infty} r_{\\pi_n} \\geq r_{\\pi '}. \\, ", "eb822776426d9a29e5fb02fac4af469a": "F_2=\\begin{bmatrix}L & M \\\\M & N \\end{bmatrix}. ", "eb8237608d7fcf27f8d32aa6e063c052": "E_F=\\frac{2\\pi\\hbar^2}{m}\\left(\\tfrac{1}{2}\\Gamma\\left(\\tfrac{d}{2}+1\\right)n\\right)^{2/d}", "eb82644bb10ed56fd99f4862134526ac": "c = \\sqrt[3]{\\frac{2\\pi^2}{3}} \\approx 1.874.", "eb8276cfc769fa967d447e584491cc33": "\\nu^{2}=uS(u), \\; S(\\nu)=\\nu, \\; \\varepsilon (\\nu)=1", "eb82a4ba4acbda1738df3b5c1aba0323": "c_k(f^* V) = f^* c_k(V)", "eb82b3f56f99113e55f30b5ad4913682": " \\,Q", "eb8319ed9079620f3b9b368e5e1c3228": "\\mathbf{v}_i\\mathbf{v}_j^{\\mathrm{T}}\\;(i \\ne j)", "eb832a2ebf201c884b4203309c12c1e0": "x(t+\\Delta t)", "eb834ca2f7375fcefc8689a33f943333": "\\det(M)^2 \\cdot \\det(M_{1,k}^{1,k})", "eb837691b029972e1581fbbdf1a61844": " V \\Delta K ", "eb838b9b6a62e8f1cc7ec498d4cffe1b": " K^n", "eb8486b9c544c453fc779c483cef6f88": "T(x \\otimes y) = y \\otimes x", "eb848ca36f25e790adaaf4f46241f8c5": "q = h(T_w-T_f)\\,", "eb84942a03b35f575bbb8b2222290236": "\\mathbf{x} \\triangleq [x_1, x_2, \\ldots, x_n]^{\\operatorname{T}} \\in \\mathbb{R}^n", "eb84af320dcf0354d1f02e9e7df16672": "\\rho_t(0) = 0", "eb8571505bd732f9da96f4181e05e125": "\\frac{G(\\chi)}{|G(\\chi)|}", "eb858a97decffe3054221d09367d2673": "h_1(n) < h_2(n)", "eb859043583a6df0542ff54194fcf80a": "f^{-1}(y)=\\arccos(y)", "eb85c33924a5a3dd098bdeaeba008c73": "\ny^{2} = a^{2} \\left( \\sigma^{2} - 1 \\right) \\left(1 - \\tau^{2} \\right)\n", "eb8601c7a5a60d6c9c6005c29f0a679e": "\n-S=\\left(\\frac{\\partial G}{\\partial T}\\right)_{p,\\{N_i\\}}\n =\\left(\\frac{\\partial F}{\\partial T}\\right)_{V,\\{N_i\\}}\n", "eb860a8ca0ce007205b6b7856fc9bc8f": "\n \\mathbf{\\sigma}_k^2 = \\lim_{t\\to\\infty}\nt^{-1}\\langle\\log\\mu_{(k)}\\rangle (t)\n", "eb863eac0c1c203703deff5d5b1afa2a": "\\ell^{\\infty}(\\mathcal{F})", "eb86597719eb1812682066b2f1b4300c": "[4.50] = 4", "eb868572150af180cd3e842e85ea03b2": "z <= -0.75", "eb86cb5e5cdb627e3c636def7f83f785": "f(n)=\\sum_{q=1}^\\infty a_q c_q(n)", "eb873393102837b9e9fa3d204d76ff40": "A=(1 2 1)", "eb87504b2f6067130d65b9a9794dae7c": " x \\cdot y = L(x)y + L(y)x - L(x)L(y)e. ", "eb87577faaf19d1777112e6c9f010ad6": "\\| s \\|_{1} \\geq \\| f \\|_{1}^{1 - \\lambda} \\| g \\|_{1}^{\\lambda}.", "eb877723b29320e4775f2edd1d080b4e": "A_{\\mathrm{x,y}}=f(x,y)", "eb87b996dcc8f846d1045acb693f1c8c": "\\frac{\\mbox{Cash and Marketable Securities}}{\\mbox{Current Liabilities}}", "eb8816ba4a08fa2a05c87ffb61dbdf1d": "|1/\\sqrt{2}|^2 = 1/2", "eb88269ed931327afba4e27b23d1c346": "2^{43}", "eb88beba172b54dc797bf31561d1f542": "\\mathrm{Sp}/U", "eb890c51e170a7be198139254112db0a": "E = e_c \\times S = Max Q \\times S = \\frac12 \\times \\rho \\times S \\times V^2 ", "eb890cc2e1475000c1178b3c0032bf86": "i=1-(1-\\frac{1}{2})\\alpha = 1-\\frac{\\alpha}{2}", "eb891366210c8b0715a162afa0f0219c": "V(F)", "eb89149be74a769a34421368d83cdbd5": "t=t_\\mathrm{now}", "eb8971adaa0645199859d0dac9cf40ab": " x_q(t) \\ ", "eb897ecd450edefa475ce3dbb536fb91": "\\vec I = (I, Q, U, V)", "eb89c3ecf689a3c800f2a7db566d7705": "n\\times 1\\!", "eb89c9ae30df78759359d668cbc9b1b5": "\\frac{\\tan [(x-y)/2]}{\\tan [(x+y)/2]}=\\frac{\\sin x- \\sin y}{\\sin x + \\sin y}", "eb8ae22772e58170b8a5ce5b2cbcb888": "w_i=n_i'/n_i", "eb8af8d116411c43fce2af9f8e12f59d": "\\models\\phi", "eb8b22b3c436387c2c0c6e0eae21f292": "x^5(x^2-x-1)+1", "eb8b302800d7a434120147acaec7c194": "\\Theta_n/bP_{n+1}\\ ", "eb8b3720b523d5ab91bbbca49b140be4": "y (1 - y)", "eb8b8378d02e18ac868e46afbb30ba47": "\\operatorname{Aut}(A)", "eb8bd0001c577d6168ae4f36caa1254c": "(y_{j})", "eb8c5583471f8c3c64a9ed5aadb6dae2": " w(X_2) \\subseteq \\prod_{i=1}^{m-1} w_{\\alpha_i,\\beta_i}(X_1) \\subseteq \\prod_{i=1}^{m-2} w_{\\alpha_i,\\beta_i}(X_{\\alpha_{m-1}})\\subseteq \\dots \\subseteq w_{1,\\beta_1}w_{\\alpha_2,\\beta_2}(X_{\\alpha_3}) \\subseteq ", "eb8c60a41e5b41caa6bab334573be2d9": " \\operatorname{dom}(S^{1/2}) \\subseteq \\operatorname{dom}(T^{1/2}) ", "eb8c8a7903c4b56973a6dd50045ca40e": "\nt_{0} \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{D}{s^{2}g^{2}}\n", "eb8c91c92c77e2e36d140f9a4ce3b6da": " {\\sigma_{ij}} = -p{\\delta_{ij}}\\ ", "eb8cab502dec2936aa55504ebabb1ab9": " \n \\begin{align}\n V_{qP}(\\theta) & \\approx V_{P0}(1 + \\delta \\sin^2 \\theta \\cos^2 \\theta + \\epsilon \\sin^4 \\theta) \\\\\n V_{qS}(\\theta) & \\approx V_{S0}\\left[1 + \\left(\\frac{V_{P0}}{ V_{S0}}\\right)^2(\\epsilon-\\delta) \\sin^2 \\theta \\cos^2 \\theta\\right] \\\\\n V_{S}(\\theta) & \\approx V_{S0}(1 + \\gamma \\sin^2 \\theta )\n \\end{align}\n", "eb8cc209d2fefc92ea821895e6a3a764": "\\int_C f(z)\\,dz=2\\pi i\\cdot\\operatorname{Res}\\limits_{z=i}f(z)=2\\pi i{e^{-t} \\over 2i}=\\pi e^{-t}.", "eb8d0e6a299a3143d8b71b9375b3926d": "\\displaystyle E_A=K_1^*\\sin^2\\theta+K_2^*\\sin^4\\theta+K_3^*\\sin^6\\theta", "eb8d0edd0980f6e5a9078f35b6075eac": "\\prod_i x_i = b^{\\sum_i \\log_b x_i}", "eb8d46c81606b4a8d4894a219689a4d5": "\n\\eta = \\frac{5q}{2\\epsilon} - 2\n", "eb8d844bcaecde184ec57b500b685e6b": "N_L\\,\\!", "eb8d99ce67263550cc775cfd7472d8c7": "\na_{t,j_t} \\cdot \\sigma + a_{k,j_t} \\cdot \\tau=\\beta.\n", "eb8e02ccce8aee7c55cdd9eb1d413ec2": "\n\\begin{alignat}{2}\nV(z)&= V\\left[\\exp\\left(i\\frac{2\\pi z}{a}\\right)+\\exp\\left(-i\\frac{2\\pi z}{a}\\right)\\right] \\\\\n&=2 V\\cos\\left(\\frac{2\\pi z}{a}\\right), \\\\\n\\end{alignat}\n", "eb8e0a2a217d4d5b3d5479d9ca780df1": "g_{ij}=\\sum_{k=1}^n \\lambda_{ki} \\lambda_{kj}\n= \\sum_{k=1}^n\n\\frac{\\partial \\varphi_k} {\\partial u_i}\n\\frac{\\partial \\varphi_k} {\\partial u_j}.\n", "eb8e0cac8fb3088d3b62fbfa0eb64d64": "x,x'", "eb8e53a39c8316133719f199e9bcb84e": "F_{x,A}=\\{G\\in F \\mid V(x)\\cup A \\subset G\\} ", "eb8f01ddf575d0ea8fc9007ef7e42842": "\\vec x \\cdot \\vec y \\ge \\vec x \\cdot \\vec y_2", "eb8f3c7458a47f2be3d473751331dd8f": "\\hat{f}(\\xi)=\\int_G \\xi(x)f(x)\\,d\\mu\\qquad\\text{for any }\\xi\\in\\hat G.", "eb8f42db02b5e7aac426f3be545118ee": "~x=\\rho \\cos \\omega; y=\\rho \\sin \\omega", "eb8f436709302608c175f556989cfa1f": "E_{\\rm coherence}", "eb8f6bfd14767818f1b935182f8bdc93": "p_x(n)\\,", "eb8f96cf3e49f6296ca8286df8d10c2c": "y_{\\mathrm\n{atm}}", "eb9004f736f431595fac6f7346a1a164": "\\frac{d}{dx} (u\\cdot v) = v \\cdot \\frac{du}{dx} + u \\cdot \\frac{dv}{dx} \\,\\! ", "eb900c11c33b93bbfbdadf5cde9dbba7": "(\\omega_1,\\omega_2)", "eb906759b4b70f9f188bbd56783ea9ac": " {} + \\sum_{i=1}^n (-1)^{i} \\varphi(g_1,\\dots,g_{i-1},g_i g_{i+1},g_{i+2},\\dots,g_{n+1})", "eb907bed46fbd80ab0d727375e32b912": "\\zeta\\wedge \\star \\eta = \\langle\\zeta, \\eta \\rangle\\;\\omega", "eb9081665221a289910642c1112c325a": "n^2 = 1 - \\frac{X}{1 - \\frac{\\frac{1}{2}Y^2\\sin^2\\theta}{1 - X} \\pm Y\\cos\\theta}", "eb909163e6c62269db0ea4507fdd9bb1": "\\mathbf{R^2}", "eb91079e65c8161a0bae45f96f6b2b44": "\n\\langle N_i \\rangle = \\frac {g_i} {e^{(\\varepsilon_i-\\mu)/kT}} = \\frac{N}{Z}\\,g_i e^{-\\varepsilon_i/kT}\n", "eb914cf1a0b64c06eba3955d4789db76": "p_\\mu", "eb917af24a411c7a9d4d0349c73ae85d": "\\ \\nu = \\frac{V}{m}\\ = {\\rho}^{-1} ", "eb9189a6e0f1fbde1c404fa9e7c28fc5": " \\mathbf{A} = \\frac{\\mu_0}{4\\pi}\\frac{\\mathbf{m}\\times\\mathbf{r}}{\\left | \\mathbf{r} \\right |^3} ", "eb91c11f14c0ebdf209cade7c490e8df": " \\{ b_0,\\; b_1, \\; b_2, \\; b_3 \\} ", "eb91f55c89d6ae68628b17a96e82c21a": "\\cos(\\pi/3)= 1/2 = 4y^{3} - 3y", "eb926890d4d8a5ed55c89c87b3dccdf8": "s_1 = x_0 + \\zeta x_1 + \\zeta^2 x_2,\\,", "eb927c8209379c9b309ca55f20b6e652": "\\pi = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-3)^{-k}}{2k+1} = \\sqrt{12}\\sum^\\infty_{k=0} \\frac{(-\\frac{1}{3})^k}{2k+1} = \\sqrt{12}\\left(1-{1\\over 3\\cdot3}+{1\\over5\\cdot 3^2}-{1\\over7\\cdot 3^3}+\\cdots\\right)", "eb9285b2c358547d5fc08a5d06123a24": "\\Gamma^{-1}", "eb92b538a70ff0e79d852ddc6433475a": " E_n = \\frac{n^2 \\hbar^2}{2 m r^2} ", "eb93207cbf3277349fd573d5c85a83c7": "p=q=0", "eb935289af765d22685708ca9391e45c": "(4)\\qquad R_{ab}l^b\\,\\hat{=}\\,cl_a\\,,\\quad \\Phi_{00}\\,\\hat{=}\\,0\\,,\\quad \\Phi_{10}=\\overline{\\Phi_{01}}\\,\\hat{=}\\,0\\,.", "eb937ec951d8b17f9d1a0ba1ad1c73a4": " \\mathbf{F} = \\alpha\\sum_{i=1}^N m_i (\\Delta r_i\\mathbf{t}_{i}) - \\omega^2\\sum_{i=1}^N m_i (\\Delta r_i\\mathbf{e}_{i}) + (\\sum_{i=1}^N m_i)\\mathbf{A},", "eb940662d450bf7b064ba0b150e61cc8": "T_M(x,y) = \\sum_{S\\subseteq E} (x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.", "eb940c319641f3a81aee4b60cca4daf9": "m_1 + m_2 + m_3 + ... + m_n", "eb9413f5a945a636bd2cbd30112e9a07": "w=Az^1", "eb949a6013a00a122b35f8a538ab29bf": "\\frac{\\mathbb{E}\\bigl[|XY|\\big|\\,\\mathcal{G}\\bigr]}{UV}\\le1\n\\qquad\\text{a.s. on the set }H:=\\{0g-b", "eb970ea0a35f1b2b4b302a80ad53452a": "f_n = \\chi_{[n,\\infty)}", "eb976b7156ecb1d2496c8fc8ef207646": "f = (K-\\omega) \\phi", "eb976e5bce30e97cb59042ff0fc27f79": "f_\\theta(x) = c(\\theta)h(x)\\exp(\\pi(\\theta)T(x))", "eb9781980f800f329dc56c3f33a864d6": "\\operatorname{versin}(\\theta)", "eb97ef7ea16f6f67985d30494149816a": "O(\\sum_{i=h}^{j} \\log |T_i|) = O(\\sum_{i=h}^{j} \\log 2^{2^i}) = O(2^ j)", "eb983067de5baf07806a21cc03236b37": "(x \\odot y) \\otimes (x \\odot y') = (x \\odot y) \\otimes (x \\odot (y \\otimes e^i)) = (x \\odot y) \\otimes (x \\odot y) \\otimes (x \\odot e^i) = x \\odot e^i", "eb98a3bf7e8caa19953ebabdf3b55fd7": "u=-2\\nu \\frac{1}{\\phi}\\frac{\\partial\\phi}{\\partial x},", "eb98ae8a072c7a0b94f40c2a10e88ba4": "X_n=H_n - 2 \\sum_r : e_r^*e_{n+r} :.", "eb98c500bcee77b856feca843985e07f": "\\phi(x) = \\frac{1}{\\omega_{n-1}}\\int_{S^{n-1}} f(rx')\\,dx'", "eb99a2536b73d2f634f70e1816751197": "\\gamma_{m,l}", "eb99c6a4d472e93e66d453f33e40162d": "f(z)=\\sum_{n=-\\infty}^{\\infty}a_{n}\\left(z-c\\right)^{n}=\\sum_{n=-\\infty}^{\\infty}b_{n}\\left(z-c\\right)^{n}.", "eb99ccd12c63b79d01ccc2ea163d812c": "\\lnot (\\forall x. G) \\to \\exists x. \\lnot G", "eb99d197fa7f59ee46b526779200f9f0": "|\\mathbf{r}_s(t_2) - \\mathbf{r}_s(t_1)|/(t_2 - t_1) \\geq c", "eb99ec9a712fd2196635ca77262c59eb": "C_{uq}", "eb9a2024b0070f3bdbacf8682e1beef2": "Y(t)=Y_c + a\\,\\cos t\\,\\sin \\varphi + b\\,\\sin t\\,\\cos\\varphi", "eb9a7aa03df4ec218f1fbe6bae6459b7": "z_d", "eb9ae8c57fc4e21c213123558c1019b7": "Y \\in T_pM", "eb9afa0d2944899b1c9c6519311aa9aa": "\n\\nabla Q =[ \\frac{\\partial Q}{\\partial x} , \\frac{\\partial Q}{\\partial y}] = [0,0].\n", "eb9b68fe2d466afd1ec5c94060cfa879": "I = \\frac{m\\ell^3}{3}", "eb9c0a686e14e2f05c78628a5c436b12": "g'(u,v) = {1\\over 2}\\left(g(u,v) + g(Ju,Jv)\\right).", "eb9c13995702fd9d9fb398cf37a68a34": "Z^{\\infty}_{e} = 2\\|1+1 = \\frac{5}{3}", "eb9ca350743bbe0d464469d72498b498": "C'(\\beta)", "eb9ca86b090d554fd7d4327c39199dcf": "\\left(\\sqrt{1/21},\\ \\sqrt{1/15},\\ \\sqrt{1/10},\\ -\\sqrt{3/2},\\ 0,\\ 0\\right)", "eb9cde6ee9c62adc003bc43e17922734": "\\tilde{P_n}(x) = (-1)^n \\sum_{k=0}^n {n \\choose k} {n+k \\choose k} (-x)^k.", "eb9d2cf8e2fbcedbbfd139cc562e1655": "n \\in N", "eb9d5d876a530cc984314601af48ebab": "0 < f''(x) = \\lim_{h \\to 0} \\frac{f'(x + h) - f'(x)}{h} = \\lim_{h \\to 0} \\frac{f'(x + h) - 0}{h} = \\lim_{h \\to 0} \\frac{f'(x+h)}{h}.", "eb9d8d9ffec2509e6b7c788a48d81191": " \\ v_{2} = u_{2}", "eb9d9dc485de0ed74469eb3d2a0de99c": "\\tilde{H}_{q}(Y)= \\begin{cases}\\mathbb{Z},\\quad q=n-k \\\\ 0,\\quad \\text{otherwise}.\\end{cases}", "eb9e686e99f7a03a355ca057270e5e86": "\\textstyle(x\\pm1,y\\pm1)", "eb9ed60a3bf9cc6f4344b63e89c8268d": "A^{(i)}_j", "eb9eda032c9bf0d65403ac2319016681": "\\vec x=\\sum_{i=1}^{n}x_i\\vec e_i", "eb9efda763833597545ade18d20290b6": "x(t)=A \\sin{(2 \\pi f_0 t)}", "eb9f25fbb950948bb621f102391877ef": "2^{2n}-1, 3^{2n}-2^{2n},\\dots,(4n)^{2n}-(4n-1)^{2n}", "eb9f2e8191524266a3b7d8a2e9dece98": "\\mathtt{find}", "eb9f41d5b47cfdc1730e3256be84ada9": "\n \\boldsymbol{P} = \\boldsymbol{F}\\cdot\\frac{\\partial W}{\\partial \\boldsymbol{E}} \\qquad \\text{or} \\qquad P_{ij} = F_{ik}~\\frac{\\partial W}{\\partial E_{kj}}, \\qquad i,j=1,2,3 ~,\n ", "eb9fd1b5ea50210ef08beb4b9c448c3b": "\\left\\{ x_1 < \\cdots < x_j < \\cdots < x_r \\right\\},", "eb9fda00cc8cc6d77a29141e37d4c9fa": "\n\\begin{align}\nE_s = E_0 + \\frac{\\hbar^2\\textbf{k}^2_{||}}{2m^*},\n\\end{align}\n", "eb9fe93d0dd27d94bd5679b521dc32a8": "\\pi_{ij} > 0", "eb9ffe67fd651fb0dee879462734f4da": "\\tau_{\\nu}", "eba01ce97f1546769fda605379d3bdae": "L_4=L_3+\\sqrt{L_2^2 - L_1^2}. \\, ", "eba035aceaa8006b6166c67cd72f3737": "\\dot q", "eba050f53427fedc0b15e4d993f6a5f1": "\\leq\\epsilon+2\\left[ \\left( \\epsilon+2\\sqrt{\\epsilon}\\right)\n+M\\ 2^{-n\\left[ I\\left( X;B\\right) -2\\delta\\right] }\\right] ^{1/2}.\n", "eba05e60bb778af4f0b36e2e2daaaf0e": "\\frac{1}{\\Delta}\\, \\frac{\\text{d}\\xi}{\\text{d}\\psi} = \\pm\\, \\frac{1}{\\sqrt{1 - m \\sin^2\\, \\psi}},", "eba0a6be8621351f64f492b7f057485e": "z_{i+1}", "eba0c38234ed6649358d81ed4efc148c": " J_f = \\begin{pmatrix} 2x & 3y^2 \\\\ 2x & -3y^2 \\end{pmatrix} . ", "eba10897731555fae9b2ebdfa430e0d7": "\\begin{bmatrix}\n1 & 3 & 1 \\\\\n4 & 2 & 2\\end{bmatrix}", "eba17be7b557c6ceb1e04b04958bdc54": "m\\frac{du}{dt} = F^{H} + F^{B} + F^{P}. ", "eba1d090677f29649b2b23c3e3722b83": "\\sum_{k=0}^\\infty a_{p,k}n^{k+1}", "eba1df29143e808994ee5913323aea4c": "x \\mapsto x\\cdot \\ln x", "eba1ffb613db86d49a039ab115d79e18": "\\lambda_\\nu = (\\kappa_\\nu \\rho)^{-1}", "eba20a3b9d4276da082fc676dcb41052": "\n\\frac{\\mathrm{d}\\phi(E_{\\bar\\nu_e},\\vec{r})}{\\mathrm{d}E_{\\bar\\nu_e}} = 10\\frac{\\lambda X N_A}{M} \\frac{\\mathrm{d}n(E_{\\bar\\nu_e})}{\\mathrm{d}E_{\\bar\\nu_e}} \\int\\limits_V \\mathrm{d}^3\\vec{r}' \\frac{A(\\vec{r}') \\rho(\\vec{r}') P_{ee} (E_{\\bar\\nu_e},|\\vec{r}-\\vec{r}'|)}{4\\pi |\\vec{r}-\\vec{r}'|^2}\n", "eba20dd50949dc51880e98b2aba3a02e": "\n\\mathbf{w}^{\\text{T}}\\mathbf{m}_i^{\\phi} = \\frac{1}{l_i}\\sum_{j=1}^{l}\\sum_{k=1}^{l_i}\\alpha_jk(\\mathbf{x}_j,\\mathbf{x}_k^i) = \\mathbf{\\alpha}^{\\text{T}}\\mathbf{M}_i,\n", "eba22190c8b01b55d3153195ce805b61": "(0,0) \\in \\C^2", "eba244e735e5fa756a01f67303de5320": "\\ddot u=c^2\\Delta u", "eba253a095e086c51ed650e870d3952d": "C(1),...,C(n)", "eba26db7ea61bfad837221fc0b04d6fe": "M=\\mathcal{T}(P)", "eba29326754a3633137803a6c3906f2c": "ab<0.994", "eba29b445a608299273fa49070d0a31a": "0 \\ = \\ ({\\partial \\over \\partial x} - j {\\partial \\over \\partial y}) (u + j v)", "eba3797146ebf354c7764dbac79da934": "\\left\\langle V(t)^2 \\right\\rangle = \\lim_{T\\to\\infty} \\frac{1}{T}\\int_{-T/2}^{T/2} V(t)^2\\,{\\rm{d}}t. ", "eba39ec537449fa11af9ea09cd6cbea9": "(A \\cdot B) + (\\lnot A \\cdot C) + (B \\cdot C) = (A \\cdot B) + (\\lnot A \\cdot C)", "eba3a6ac0090690cae1039cb1a5ef828": "\\hat p = \\scriptstyle \\frac{X}{n}", "eba3f5825941d926f55a1866051332a3": "\\det (A-\\lambda I) \\;=\\;\n\\det \\begin{bmatrix}\n2- \\lambda & 0 & 0 & 0 \\\\\n1 & 2- \\lambda & 0 & 0 \\\\\n0 & 1 & 3- \\lambda & 0 \\\\\n0 & 0 & 1 & 3- \\lambda \n\\end{bmatrix}= (2 - \\lambda)^2 (3 - \\lambda)^2 ", "eba44e47f183eba91225fa49729e5bf9": " M \\times \\,^{\\prime\\prime} 0 \\; 1 \\; 0 \\; 0 \\mbox{-1} \\; 1 \\mbox{-1} \\; 0 \\,^{\\prime\\prime} = M \\times (2^6 - 2^3 + 2^2 - 2^1) = M \\times 58. ", "eba460c40bfac4e31eea33fe122d4f01": "\\Pr(U=u) = \\begin{cases} p_a & \\text{if } u=a; \\\\[8pt]\np_b & \\text{if } u=b \\\\[8pt]\n\\dfrac{1-p_a-p_b}{b-a-1} & \\text{if } a> R_1 \\parallel R_2", "eba64f106e9cd2a273599480d9fbd895": "(0,\\,0,\\,1)", "eba66631ae03934ee8a87a13e8e2257d": "\n\\Delta_rG^\\ominus = -RT \\ln K_{eq} \n", "eba6892cb2438f8096c9035a25553685": "s, h \\models P", "eba6d487a0995f899796531e8f9149f4": "c(m) = \\sum _{i=1} ^m \\frac{1}{i}", "eba77ed60ac92145ded96fe7a8b18e91": "Eq.3", "eba82fda1864fa3eb5070db0b96fb4c0": "M' = \\left(\\mathbf{r}_1\\times \\mathbf{F}_1 + \\mathbf{r}_2\\times \\mathbf{F}_2 + \\cdots\\right) + \\mathbf{r}\\times \\left(\\mathbf{F}_1 + \\mathbf{F}_2 + \\cdots \\right).", "eba855fad6ff5d922e558fbc3176a186": "\\theta = \\arctan \\left (\\frac{v^2}{gr}\\right )", "eba898535b01af403c0aca356d147586": " \\ldots \\left(1-\\frac{1}{11}\\right)\\left(1-\\frac{1}{7}\\right)\\left(1-\\frac{1}{5}\\right)\\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{2}\\right)\\zeta(1) = 1 ", "eba8c04fa429d0c9acc4ea0b70fa8830": " \\mathbf{v} = q_i \\mathbf{e}^i = q^i \\mathbf{e}_i \\, ", "eba8c38b207f806f63484b4c3e89f71e": "j, k", "eba8ed7602a1ac31ecd9b130debfc6d3": "\\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} - R_{\\alpha \\beta}^{\\;\\;\\;\\; IJ} = \\nabla_{[\\alpha} C_{\\beta]}^{\\;\\; IJ} + C_{[\\alpha}^{\\;\\;\\; IM} C_{\\beta]M}^{\\;\\;\\;\\; J}", "eba958b3164e5396a9928734886578b3": "x^*=y^*=\\frac{1}{3+\\phi}", "eba98738de94b8624f2af5bd73acae26": "b = [5,7]", "eba9d2e99fd41856b60076f20c668c90": " \\langle a, b | a^4 = b^2 = e, a b = b a^3 \\rangle. \\, ", "eba9eb5965794259c1e638979457f132": "\\omega_{RF}", "ebaa3927fb3601a4143b2776ba0af201": "f_k = P[X_i = hk].\\,", "ebaa7e0c8f36b8abcc470f40e92b781a": "|\\varphi|<1", "ebaabc52f7f8f50074b381748ae630a9": "I\\cap K[Y]", "ebaacff25a8ec3560acd8949f5c05a74": "\nU \\ \\stackrel{\\mathrm{def}}{=}\\ \\int d\\mathbf{r} \n\\rho_{2}(\\mathbf{r}) \\Phi_{1}(\\mathbf{r})\n", "ebaada3609bc88baf1fc1c452b6a9a9d": " T \\left( \\lim_{n\\rightarrow\\infty} \\hat F_n \\right) = \\theta. \\, ", "ebab1f7b342f4ebb05f2b23eaf262aa7": "\\begin{pmatrix}\ny_1^{(n)} \\\\\ny_2^{(n)} \\\\\n\\vdots \\\\\ny_m^{(n)}\n\\end{pmatrix} = \n\n\\begin{pmatrix}\nF_1 \\left (x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n-1)} \\right ) \\\\\nF_2 \\left (x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n-1)} \\right ) \\\\\n\\vdots \\\\\nF_m \\left (x,\\mathbf{y},\\mathbf{y}',\\mathbf{y}'',\\cdots \\mathbf{y}^{(n-1)} \\right) \\\\\n\\end{pmatrix}", "ebab789949593e3afeabcb18d5d04d28": "G(x) = F(x+h)", "ebab8f19f9186e2043956a8e26b4f9c2": "K = \\mathbf Q(\\sqrt d),", "ebac04fc1afdcb7d512478cc63fa17e5": "\\begin{align}\nx+y+z&=1\\\\\nx+y+z&=0\n\\end{align}", "ebac158b25abc8f35b9796cdbbdf7151": "{dC_m\\over d\\alpha} <0", "ebac5bd3582597c15ef3fb855124a826": "\\varphi(t)", "ebacc0f4fefb47ae2430132445fae826": "\\tau=L_{0}\\cdot f'_{x}\\cdot\\frac{v^{2}}{c^{2}}", "ebace4e25f38157811a291f5307de945": "\\frac {d^2r} {d\\theta^2} \\cdot \\left (\\frac{H}{r^2} \\right )^2 + \\frac {dr} {d\\theta} \\cdot \\left (- \\frac {2 \\cdot H \\cdot \\dot{r}} {r^3} \\right ) - r\\left (\\frac{H}{r^2} \\right )^2 = - \\frac {\\mu} {r^2}", "ebad144f6ef0483842cd8d44342e02f0": "(t^*,t)", "ebad18d8896c98d70dff919c5396738c": "S=\\int_M \\left[\\frac{1}{2}\\mathbf{C}\\wedge *\\mathbf{C} +(-1)^p \\mathbf{B} \\wedge *\\mathbf{J}\\right]", "ebad52c08b48153075df1b252e51af8a": "\\nabla_{\\mathbf{u}_i}\\mathbf{u}_j = \\omega^k{}_{ij}\\mathbf{u}_k.", "ebadcce97cc8048ecf85f3e09c6603a2": "\nZ(\\mu) = \\frac{1+i}{\\sqrt\n2}\\frac{\\varphi(x_{\\mu})}{\\sqrt{f''(x_{\\mu})}}\ne^{2\\pi i(f(x_{\\mu})- \\mu x_{\\mu})} \\ .\n", "ebade479ee3306fa6273e53d3351d784": "b'0\\}}X^{\\tau_n} \\right ]<\\infty", "ebe1915c432cf9c372b4ecfe36ff1fa2": "\\rightleftharpoons ", "ebe1aa6b8e9d5685dae57bd35f8d0c7c": "R_s, X_s", "ebe1d144cf6645fee13bf183069a45cd": "\n\\dot{x}_k = \\sum_{i=1}^n m_i e^{-|x_k-x_i|},\n\\qquad\n\\dot{m}_k = (b-1) \\sum_{i=1}^n m_k m_i \\sgn(x_k-x_i) e^{-|x_k-x_i|}\n\\qquad\n(k = 1,\\dots,n)\n", "ebe1f25883d68f011789961c41f9fac6": " B_0 ", "ebe263a7262d64bad8e135fd6db21b48": "\\dfrac{de(t_k)}{dt}=\\dfrac{e(t_k)-e(t_{k-1})}{\\Delta t}", "ebe2a1d6244dee91ebc702c22ae39517": "\\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{t}^{\\infty} e^{-1/2t^2-x^2+xt} dxdt = \\int\\limits_{-\\infty}^{\\infty} \\int\\limits_{t}^{\\infty} e^{^-t^2-1/2x^2+xt} dxdt = \\pi\\!", "ebe2a85c3b26a9f7344a8813e9da4746": "V_n(R) = R^{n-1} \\int_{-R}^R V_{n-1}\\left(\\sqrt{1 - (x/R)^2}\\right) \\,dx.", "ebe2b1c26d4249d5284113fcb8db267d": "f'(t)=(f_1'(t),f_2'(t),f_3'(t),\\ldots)", "ebe2c372e358307ab80f345c700cafc0": "5.7\\cdot 10^{-5}", "ebe328007622273f2b318f7111f3743d": "f_i^a(x, Q^2)", "ebe328a346ab98239166609e9a76a155": "Q \\and \\neg Q", "ebe36aa4feb8dc9c6464ca0d70540043": " \\pi^{-n} |(Bf)(z)|^2 \\exp(-|z|^2),\\, ", "ebe3a876866d99ed0b69f6557e844400": "\\sum_{n=1}^\\infty \\frac{\\mu(n)}{n} = 0 (\\mathrm{L})", "ebe3d86d00831fe35fe4ff3c78fe2fd2": "\\langle r,s\\rangle\\;", "ebe44fa5a5a138116c9ccb948a5ed710": "C \\setminus (A \\cap B) = (C \\setminus A) \\cup (C \\setminus B)\\,\\!", "ebe45bf8afbb1c630beccff03365bd42": "\\Gamma=\\bigl(R(t_i,\\, t_j), i,j=1,\\ldots,\\, n\\bigr)", "ebe4656da538de341194bc953af6b886": "\\displaystyle{R(a,b)=2Q(Q(a)b,a)Q(a)^{-1}= 2Q(a)Q(b,a^{-1}).}", "ebe469c79148a4bf44056bbc2b15e1eb": "\\rho \\ll \\rho_0", "ebe491c6e500a2311a004ac04e1eb9a8": "Z_0\\,", "ebe4dd88630829f82be8198a78dc10f5": "\\partial p^{i}_a / \\partial x^{i} = -\\partial H / \\partial y^{a} \\, , \\, \\partial y^{a} / \\partial x^{i} = \\partial H / \\partial p^{i}_a", "ebe503720c57fb00eb7f2abec8e1d6f3": "\\bar F(C) = p(A) + p(B) + \\frac 12 p(C) = \\frac 13 + \\frac 14 + \\frac 12 \\cdot \\frac 16 = 0.66666...", "ebe539acdd58fb6b4688f934a546e0bc": "\\textstyle n=\\infty", "ebe54cc08dced53fd8527d8f1f07f493": "I_o=\\frac{S_{spo}-S_{or}}{1-S_{cw}-S_{or}}", "ebe54dbc3ceda13dc0743c07fc8d0c18": "Y_i-\\hat{Y}_i", "ebe5847a4821a6755a0a5b34804b1844": "\\lambda x\\!:\\!\\tau.x\\!:\\!\\tau\\to\\tau", "ebe5a071f0fdd84a0e34fdc210f63b00": "i_1 \\leq i_2 \\leq i_3 \\cdots \\leq i_m", "ebe6bc4b79d5f7c252afabbd838b0906": "N_{rot}=1,3,6", "ebe6bcd9dbec5bfcc2bbb7a7dd97757f": "|\\text{arg}(s-s_0)|\\leq\\theta<\\frac{\\pi}{2},", "ebe6bf3482f544832558010c8a7ec10a": "r_m = \\frac{r_2-r_1}{\\ln (r_2 /r_1)}", "ebe7041d594494f396a644f58757cc9c": "m_n = \\sum_{i=0}^n a_i b_{n-i}", "ebe7059c0b58eb7029d599743d1c5441": "c_2= \\ldots = c_n=0", "ebe7518d5320e84c7b048a0800abd653": "\\Delta\\ \\stackrel{\\mathrm{def}}{=}\\ \\mu_1-\\mu_2\\,", "ebe752e33b458970c7630bed07b93cf1": "\\mathrm{excess}(u) > 0", "ebe78e331eaf6760d2c3d429132a0c0f": " \\underbrace{ \\left(\n\\begin{array}{c}\nQ(X=1) \\\\\n\\vdots \\\\\nP(X = N) \\\\\n\\end{array}\n\\right) }_{\\mu_Y^\\pi} = \\underbrace{ \\left( \\begin{array}{c} \\\\ P(X=s \\mid Y=t) \\\\ \\\\ \\end{array} \\right) }_{ \\mathcal{C}_{X\\mid Y} } \\underbrace{ \\left(\n\\begin{array}{c}\n\\pi(Y=1) \\\\\n\\vdots \\\\\npi(Y = N) \\\\\n\\end{array}\n\\right) }_{ \\mu_Y^\\pi} ", "ebe7b1e5ecf7bc56bf3d4fc398f91c73": "|\\mathrm{Aut}(P)|=(p^n-1)\\cdots(p^n-p^{n-1}).", "ebe7ba46241417308b579e6fd3811ef5": "(J^2 - J_z^2)", "ebe7f25c6445b98eed6ab7ac812e7e87": "\\omega^{\\omega}", "ebe7fcd5f1d3321be9b6d4e43f3e1d7f": "\\pi_2,\\dots,\\pi_n", "ebe80a8858deb9ca37a038d565cdec30": "\\Delta+1", "ebe80b9a4e71c2affb249407409cc147": "v_o\\,", "ebe81d9a4d90c4538a8693696faed77c": " Gr= \\frac{g \\beta \\Delta C L^3}{\\nu^2} ", "ebe83dcec22c9c758e296e04e179d0f9": "i=1,\\ldots, n", "ebe86682666f2ab3da0843ed3097e4b3": "kg", "ebe870a7bfb7b83134029444e811c9fa": "\\psi_q(x)", "ebe8759af4ba99ac100f7904384b95b9": "P(D)=D^n\\,", "ebe876f470789cfa4c60dd7bc27106ee": "\\zeta(-3)=\\frac{1}{120}", "ebe8c48fa5302747021b824acde41639": "j \\leq i", "ebe8f3e11950a0fef28b6ef9e447b084": "\\left.p_{xy}\\right.", "ebe90b1cd791c23bfe75ad6d80f5023c": "K =(1+m/M) |E|", "ebe90ed36cd53834995c4fe6f1cb7189": " \\iota_X \\iota_Y \\omega = - \\iota_Y \\iota_X^{ } \\omega ", "ebea1023a792cc2c0d6853400cf0d16e": "x - y + z", "ebea3396069b7154edb3a19e83e42034": "d\\Omega^2\\ \\stackrel{\\mathrm{def}}{=}\\ d\\theta^2+\\sin^2\\theta\\,d\\phi^2", "ebea763675939d52476c3cb315bc08a6": "\\cos\\gamma=\\cos\\theta\\cos\\theta^\\prime + \\sin\\theta\\sin\\theta^\\prime\\cos(\\varphi-\\varphi^\\prime).", "ebeaf2dc23151ebe7c1f4b3e0671efc6": "\\scriptstyle{j = \\sqrt{-1}}", "ebeb2c50d196da7f5da18289aef79a1a": "\n\\begin{align}\n \\zeta &= (1 - e^2) z^2 / a^2 ,\\\\[6pt]\n \\rho &= (p^2 / a^2 + \\zeta - e^4) / 6 ,\\\\[6pt]\n s &= e^4 \\zeta p^2 / ( 4 a^2) ,\\\\[6pt]\n t &= \\sqrt[3]{\\rho^3 + s + \\sqrt{s (s + 2 \\rho^3)}} ,\\\\[6pt]\n u &= \\rho + t + \\rho^2 / t ,\\\\[6pt]\n v &= \\sqrt{u^2 + e^4 \\zeta} ,\\\\[6pt]\n w &= e^2 (u + v - \\zeta) / (2 v) ,\\\\[6pt]\n \\kappa &= 1 + e^2 (\\sqrt{u + v + w^2} + w) / (u + v).\n\\end{align}\n", "ebebe4f393a6cf58594b4164d4fadbef": "(\\mathfrak{g},[\\,\\,\\,,\\,\\,\\,] )", "ebec2af4de81139fc87c28e1a67ede37": "\\widehat{I_\\alpha f}(\\xi) = |2\\pi\\xi|^{-\\alpha} \\hat{f}(\\xi).", "ebec4f69bb10b9b7cfc958d5018f3e30": "\n(/\\leftarrow) \\quad\n{Y\\leftarrow \\Gamma X\n \\over\nY/X\\leftarrow\\Gamma}\n", "ebec6e030d159400536782b62d921a65": "\\forall{x}{\\in}\\mathbf{X}\\, P(x)", "ebeca535e3cb8758ab55484ff37e61d0": "p_{1/2} ", "ebecde2b1083da98f3b8cca5a1e4b08b": "\\mathcal U^{(t)}", "ebecf9fc96953fc2c4e33987f199e2a7": "\n\\begin{align}\nx_1 &= r \\cos(\\phi_1) \\\\\nx_2 &= r \\sin(\\phi_1) \\cos(\\phi_2) \\\\\nx_3 &= r \\sin(\\phi_1) \\sin(\\phi_2) \\cos(\\phi_3) \\\\\n &\\vdots\\\\\nx_{n-1} &= r \\sin(\\phi_1) \\cdots \\sin(\\phi_{n-2}) \\cos(\\phi_{n-1}) \\\\\nx_n &= r \\sin(\\phi_1) \\cdots \\sin(\\phi_{n-2}) \\sin(\\phi_{n-1}) \\,.\n\\end{align}\n", "ebed13e06a6f99a2252c57b9d1df012e": "\\acute{\\xi}^{i}", "ebed159f20f554b67c0361c9da9c76c5": "i =0,1,\\dots,n", "ebed3a269a5a37ff9672dd69f138472e": "-g=a^2b^2c^2v^2=t^2v^2", "ebed8a213a939c269f85b4c33b7d6a66": "T_\\text{P} = \\frac{m_\\text{P} c^2}{k} = \\sqrt{\\frac{\\hbar c^5}{G k^2}}", "ebed8bca17d4cc4e8ebab7486106cfd3": "X_n=o_p(a_n) \\,", "ebedb45a33aa74c664fff7b7ce2483e9": "\\tilde{\\zeta} = \\frac{\\zeta}{h_c},", "ebedf7e13f29e584bd5388ab19492615": "[p_1(x) - p_2(x)] \\equiv 0.", "ebee276caaec6fe7aa9158864bafe89d": "\\textstyle \\mathbf{a} \\cdot \\mathbf{v} = \\frac{1}{2} \\frac{d v^2}{dt}", "ebee58fcc11c45bde5239809b6d47ec0": "\\|fg\\|_1\\ge\\|f\\|_{1/p}\\,\\|g\\|_{-1/(p-1)}.", "ebef2189f895deb425c6ee1d86b72d93": "\\mu, \\nu", "ebef21bef11c75424fe9299ee89eabdf": "v_i(X_i) \\ge 1/n", "ebef83fab5abeb8f7ce17dcfc2671c62": "f\\mapsto df", "ebef9a6bdfb2bd658965a659d081986b": " R \\leq 1 - H_q(p) - 1 /L ", "ebf006a1552b17102d3fb2438972e5f7": "\\frac{1}{\\sqrt{4\\pi}} e^{-x^2/4}", "ebf03c296e96b00fbf5a765715064e59": "(1-x^{2})y''-(2\\alpha+1)xy'+n(n+2\\alpha)y=0.\\,", "ebf07a52dcec5ee536998ffbc803a58b": "\n M_x = \\int_S y~p(x,y)~ \\mathrm{d}A ~;~~ M_y = \\int_S x~p(x,y)~ \\mathrm{d}A ~;~~ M_z = \\int_S [x~q_y(x,y) - y~q_x(x,y)]~ \\mathrm{d}A\n ", "ebf088effd54f266f3d4ee1b02c1525a": "(a \\cdot b) \\vert \\vert (c \\cdot d)", "ebf08ac52efd49d01ec6ad4583ea8103": "C_n(K)\\,", "ebf1129101b638edcae4a2a1948ee5ba": "Solow Residual = g_Y-\\alpha*g_K-(1-\\alpha)*g_L", "ebf173631a3aba54ebe89b7b00dba341": "\\text{Var}\\left(y_d - \\left[\\hat{\\alpha} + \\hat{\\beta}x_d\\right]\\right) = \\text{Var}\\left(y_d\\right) + \\text{Var}\\left(\\hat{\\alpha} + \\hat{\\beta}x_d\\right) .", "ebf25261041d37402583f00a49557d18": "k_{off}", "ebf2528abe1eed579a611f36e9b9874f": "\\int_0^\\infty G(u(t)) \\, dt \\leq k_1; \\quad \\int_0^\\infty G'(u(t))h(t) \\, dt \\leq k_2.", "ebf27a4b278d59dd36be69d1618fc081": "\\theta_i:=\\min \\left\\{ \\left. \\arccos \\left( \\frac{ |\\langle u,w\\rangle| }{\\|u\\| \\|w\\|}\\right) \\right| u\\in \\mathcal{U},~w\\in \\mathcal{W},~u\\perp u_j,~w \\perp w_j \\quad \\forall j\\in \\{1,\\ldots,i-1\\} \\right\\}.", "ebf2cedf9db2cbbbc6b05f587b6bd341": "Q(y) = f'(g(a)) + \\eta(y - g(a)). \\,", "ebf2d86bf8116340fad9bbe19110c5be": "x \\in G", "ebf2e91dce5f6b3cac6800364b6590bf": "\\left(\\rho u A \\phi \\right)_{r} - \\left(\\rho u A \\phi \\right)_{l}", "ebf338e6fb309fef822b21fd0a633df2": "v = e^{i \\theta_2} \\sin r \\,\\! .", "ebf3893bd6011e0e093ee8b787b707f6": "\\overline{Q}_N = \\frac{1}{N}\\sum_{i = 0}^N Q(\\boldsymbol{r}_i) p(\\boldsymbol{r}_i)", "ebf3f0875eea1f72fb19669637906fde": "S \\longmapsto DS", "ebf4d808899c114f88522e6c1cb3cd4d": "\\scriptstyle \\ddot x", "ebf4efed1250792b55d573d6d6d41f91": "\\int \\operatorname{arsech} \\, x \\, dx = x \\, \\operatorname{arsech} \\, x + \\arcsin x + C , \\text{ for } 0 < x \\le 1 ", "ebf4f2ba873dcda6196e8d4b6b2c7a03": "1102_{2i}", "ebf538e57592ee65389d50ef8fc3503b": "\\frac{\\sum_{j=1}^{N \\times |T|} w^{(j)}}{N \\times |T|}", "ebf5412e77e8bb0a353398836d475681": "u_x' = u_x - v", "ebf5b53514702a37b61be10375db7a63": "\\textstyle G", "ebf5d7c21467ee75495a0e94edecc28d": "X:X \\times U \\times \\Omega \\to \\mathbb{R}", "ebf661951a4c1fd649e087d580d164aa": "\\begin{matrix}\nE_{KIN}\n= \\frac{m l^2 }{2} \\dot \\varphi^2 + m a l \\nu ~\\sin(\\nu t) \\sin(\\varphi)~\\dot\\varphi + \\frac{m a^2 \\nu^2}{2} \\sin^2(\\nu t)\\;.\n\\end{matrix}", "ebf692d3e4739cff84b99b253fe65e68": "\\hat{\\textbf{y}}_{k\\mid k-1} = \n \\textbf{L}_{k} [\\textbf{F}_{k}^{-1}]^{\\text{T}}\\hat{\\textbf{y}}_{k-1\\mid k-1} ", "ebf6a90f7fa38208bc295848512e408a": " \\langle x ,\\ y\\rangle = 0", "ebf71b9d38075174e71ee4a241010988": "\\text{Short-circuit kva and current calculations}", "ebf724cff9dbb428ec537ea4fb10b8ad": "T_j \\subseteq [t] ", "ebf73c545d55d452621e8093ba2d15ef": "k_L(f)", "ebf740db4927e425b79b77b9ec2fa369": "\\aleph_0 + 0 = \\aleph_0 + 1", "ebf771c07e49ef27a240f69184f2a43a": "\\mathcal E_X^{p,q}\\otimes E", "ebf77ea701df24723c9f737046be0f42": "C_w = \\frac{1}{\\rho_0\\omega_w}. \\ ", "ebf7912fe3d3303206f851a19261b847": "\\sum_{n = 1}^\\infty X_n", "ebf83e697db7b033e0bb26893902856e": "\\textstyle\\frac{1}{q^2}", "ebf8b718dd31096282cd3452944c23b7": "\\frac {(V_{2}^2-V_{3}^2)-2gh_{d}}{V_{2}^2}", "ebf8c142c6a0133a34e43f7eafed6838": " c = -\\mathbf{n}\\cdot \\mathbf{l}", "ebf8e576c41bc625cffe0ce54faa1665": "\\frac{1}{z} \\sum _{n \\geq 0} \\left(\\frac{T}{z}\\right)^n.", "ebf9218f44c85985251e5ba97af78041": "a \\in C", "ebf9dfb2756de50f31efd3fc218f1cc9": "\\frac{DF}{Dt} = \\frac{\\partial F}{\\partial t} + (\\mathbf{v}\\cdot\\vec\\nabla)F,", "ebfa09305d9523c8e8f007c1eb162aef": "f^{\\mathcal{A}/E}_i ([a_1]_E, \\ldots, [a_{n_i}]_E) = [f^{\\mathcal{A}}_i(a_1,\\ldots, a_{n_i})]_E", "ebfa50fbc93af775be30c800cc072b95": "g(\\theta)", "ebfa818e631f4f915dfe6dbd7c4d5883": "F(f):F(Y) \\rightarrow F(X) \\in D", "ebfb01604c4c5299b1048972a9174aac": "\n{\\left( \\frac{dr}{d\\tau} \\right)}^{2} = \\frac{E^2}{m^2 c^2} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( c^{2} + \\frac{L^2}{m^2 r^2} \\right)\n\\,,", "ebfb1a3208cfd76b15df4d2f6185d473": " \\mathrm{Operating\\ margin} = \\left ( \\frac {\\mathrm{Operating\\ income}}{\\mathrm{Revenue}} \\right )", "ebfb337224c19de884e63cd024142936": "\\begin{align}\nx'&=x\\cos\\theta-y\\sin\\theta\\\\\ny'&=x\\sin\\theta+y\\cos\\theta.\n\\end{align}", "ebfb918e2dc832ebd0b8cf080176aed9": "U_s U_\\omega", "ebfbda92aed92e005e57206e71aac03c": "\\mathbf{e}_2", "ebfbe0c14bf5e09c16aadffacde9a2f4": "SL(2,\\mathbb{R})", "ebfbe5f8406abd37ed0edb8259b62812": "\\sin\\frac{7\\pi}{30}=\\sin 42^\\circ=\\frac{\\sqrt6\\sqrt{5+\\sqrt5}-\\sqrt5+1}{8}\\,", "ebfc1b51200277e00ab94e68569d1ddf": "\\tilde{\\mathbf{x}}_{1,2} = \\left( \\pm q_-, - \\frac{\\alpha}{q_-}, 1- \\left(1-\\frac{\\gamma}{\\alpha}\\right)q_-^2 \\right)", "ebfc26a474835c1ad7c18f712064485f": "Qi", "ebfc2ac849fe85f4ad75beb2e4461d0f": "V_q = \\frac{4 \\pi e^2}{\\epsilon q^2 L^3}", "ebfc5b3b4d824ab1f8348711065b0623": "\\Delta J = 0, \\pm 2", "ebfc6b215ca553ecfa425f0ac3d40744": "(y/x)", "ebfc9a8855aad10a353a12222c64a045": "b=\\Delta x,", "ebfca3f711bf2ec25a259052b9ec8c8f": "\\rho^{AB}", "ebfce1ed11ab2dc3cf7e5ae5e65b800d": "w_i = \\frac {m_i}{m_\\mathrm{tot}}.", "ebfcf2cbd709e9c114c57f18a536224c": "\\lambda_1, \\lambda_2, \\ldots, \\lambda_M", "ebfd50da73f306edd0277ce599b46349": "0\\rightarrow A\\rightarrow B\\rightarrow P\\rightarrow 0\\,", "ebfd9165534b52bf91b9cb9d46a5033f": "U = U*", "ebfdc583a817036006a3ce490319b09e": "L , s\\,", "ebfdd136331841a1b59f835c998ca593": "\\nu_1", "ebfddef7c92ef5518097c11e1f2fb12e": "f(X), X\\sim p(.|\\theta)", "ebfe1210c29171f9e8bd6faf1f0044bd": "\n\\Phi (f) = \\frac{2 R h f}{e^{\\frac{h f}{k_B T}} - 1}\n", "ebfe5936e6a5518937fdfebc217a8525": "\\hat p = i\\sqrt{\\frac{m \\omega\\hbar}{2}}(a^{\\dagger}-a) ~.", "ebfec35dae98d2c7508958f51e77acfb": " f_y", "ebff316da7e0b69ff1134ab233e1bcdf": "p(\\neg S\\vert D)={p(\\neg S)\\over p(D)}\\,\\prod_i p(w_i \\vert\\neg S)", "ebff512affe27f0ba46c51e8cfbc5cd4": "\\tilde R_n(t):=\\int_{[a,t)} \\biggl(\\int_{[a,q)} u(s)\\,\\mu(\\mathrm{d}s)\\biggr)\\mu^{\\otimes n}(A_n(q,t))\\,\\mu(\\mathrm{d}q),\\qquad t\\in I.", "ebffb769f05d02dbed36566470e91a20": "w=g(z)", "ebffd7c7e3ee3a305e9d05e364fe45b9": "T_{\\mu\\nu\\alpha}=\\frac12(\\Gamma_{\\mu\\nu\\alpha} - \\Gamma_{\\alpha\\nu\\mu})", "ebffe881a39e4ddec444a790585dee41": "S \\subseteq E(A)", "ec004b71a9c21c9743d122a73aa8e48f": "P_s(E)", "ec00588d4c91d4ffae2c2a5d0e6c9ed9": "\\phi(0)=0\\,\\!", "ec00cdbec27f3fd3087293e62ff1dfb3": "\\langle x,\\xi \\rangle", "ec00e7312bbacd88fddb1114c4e41dcb": "v = \\frac {1}{\\sqrt {LC}}", "ec01512624f58880786077bd54549f9e": "10\\uparrow\\uparrow 10^{\\,\\!10^{10^{3.81\\times 10^{17}}}}", "ec0214b1faa85d081faaf82c4b650c0f": "\\;S(\\rho^A).", "ec02a6f47acdffc9e53b8c7e2cb633ef": "m = \\rho L b h", "ec02b93da0ebab001850b3b46681530f": "\n \\left|A\\right|_{ij} = \\left|A_{11} - A_{12}\\,{A_{22}}^{-1}\\,A_{21}\\right|_{ij}.\n", "ec030eb58bbd8853fb9f4c34b8b4dfba": "\\frac{dx}{dt} = \\dot{x}\\,.", "ec034374958b24516e3ff41802736c18": "\\mathrm{tr} \\left\\{ E\\{ee^T \\} \\right\\} = E \\left\\{ \\mathrm{tr}\\{ee^T \\} \\right\\} = E\\{e^T e \\} = \\sum_{i=1}^n E\\{e_i^2\\}.", "ec0396df67ed4ad0719fd527aa8899d2": " confidence_{j}> 0 ", "ec0428d427e238f3a688fa4fc5b20edb": "r+1", "ec04850b7f19b561b219a111a499880b": "\\rho = 0, \\omega = 1,", "ec04ef3573131849fb729b6fef6e0e05": "M_{k}^{\\prime}", "ec05edbafdfc591f6671609db94a2387": " \\bar r_2\\ ", "ec061e21d57bcca9d650345f699caaeb": "\\ v = DP_n", "ec065dc9fd2624e18accba3a03be1f4d": "\\sum_{x \\in X} x = \\sum_{y \\in Y} y;", "ec06ac9caad5ec614dc48b1cc3625526": "\\hat{H}_{s}(t)|\\Phi(t)\\rangle=i\\frac{\\partial}{\\partial t}|\\Phi(t)\\rangle,\\ \\ \\ |\\Phi(0)\\rangle=|\\Phi\\rangle,", "ec072881fb95d756cee009419c7a2e47": " {\\left(\\frac{^{206}Pb}{^{204}Pb}\\right)_{P}} = {\\left(\\frac{^{206}Pb}{^{204}Pb}\\right)_{I}} + {\\left(\\frac{^{238}U}{^{204}Pb}\\right)} {\\left({e^{\\lambda_{238}t}-1}\\right)} ", "ec076fae834bd9b024c9bc9903ff653c": "\\phi_\\alpha(g)= \\alpha^*_g", "ec0792203468177d869ebe0775c76588": "\\sqrt{2} \\lambda <\\xi ", "ec079ddb5fd170df7a85e896fa2c75a1": "x(t) = e^{j \\pi at^2}y(t) \\, ", "ec081acce549d02f4a90a2cad2ca8f9c": "\\mathbf{n}_1, \\mathbf{n}_2, \\mathbf{n}_3", "ec081b9154e9e217ef915ebc4896fcef": "\\beta_{\\nu,i}", "ec0844e83025eaf5b8166995638de0ef": "{24\\choose 12-24}_2={24\\choose -12}_2={24\\choose 12}_2", "ec08582fb196bc8956568fbcfded9630": "\\eta (\\eta+2)", "ec08740d76d7f58591b44a63d5430e03": "P=(z-a)^2\\,(z-b)", "ec089f11373e59cebae3e715f4f317b6": "-\\nabla T", "ec08f5f10513f98800f80bb29deb4984": " \\oint p_\\mathrm{r} \\, d r = p_\\mathrm{\\varphi} \\oint \\left( \\frac{1}{r} \\frac{dr}{d \\varphi} \\right)^2 \\, d \\varphi = n_\\mathrm{r} h", "ec08f7d99f41b3d17e2637c2a7316376": "\\varphi = f(x,y,z)\\,", "ec092bb806b6c4c796ba8de52db0b167": "\\Delta x^i = 0.\\,", "ec092fa26b32aa08ac058fcd91763dac": " \\lim_{t \\rightarrow +\\infty} V(t)= N P,", "ec09343caab63c573bdac3ecb1f5cda1": "\\mathrm{Tor}_n^R \\left (\\bigoplus_i A_i, \\bigoplus_j B_j \\right) \\simeq \\bigoplus_i \\bigoplus_j \\mathrm{Tor}_n^R(A_i,B_j)", "ec0973c8a9e6d7c97ec9a9057ab2fc9f": "\\mathbf{F} = {d\\mathbf{P} \\over d\\tau}", "ec099aa8f143dd34ba0f76eaeffbabef": "{k_{n+1}}", "ec09da303c91d68ce6e389864d5c3342": " J=(1\\le j_1< \\cdots 1}\\zeta(s_1, \\ldots, s_k) = \\zeta(n)", "ec0c8fd71c917d6f5750e54398bec63f": "d_mf(X) = \\langle d_m Y(X),m\\rangle + \\langle Y(m), X(m)\\rangle = 0.", "ec0cd7aede4bff36e763441d8c4e13eb": "37 \\equiv 57 \\pmod{10}", "ec0cf1faba5c4cd9801a56cae34615fe": " \\overrightarrow{D_k} ", "ec0d9360e568f2cdf470fd66e0fb2cff": "s=\\tfrac{1}{2}(a+b+c+d)", "ec0dc39a24d71b5e5b75410aeb87209e": "\n \\begin{matrix}\n \\underbrace{10_{}^{10^{{}^{.\\,^{.\\,^{.\\,^{10}}}}}}}\\\\\n \\underbrace{10_{}^{10^{{}^{.\\,^{.\\,^{.\\,^{10}}}}}}}\\\\\n \\underbrace{10_{}^{10^{{}^{.\\,^{.\\,^{.\\,^{10}}}}}}}\\\\\n 10\\mbox{ multiplied copies of }10\n \\end{matrix}", "ec0ddf23ef20886b5bf0785b56817ba3": " \\begin{align}\n\\mathbb{P} (Y\\le0.75 | X=0.5) &= \\lim_{\\varepsilon\\to0+} \\mathbb{P} (Y\\le0.75 | 0.5-\\varepsilon 0, \n\\frac{\\mathrm{d}N_C}{\\mathrm{d}t} > 0.\n", "ec14f53b94abb67016423814766fd887": "\\sum_{i=1}^n \\left(X_i-\\mu\\right)^2/\\sigma^2\\sim\\chi^2_n.", "ec1538e9e2cdf365f16291b43b8b5531": "M_k", "ec15592d2fc3e713e7fb6e66ce14a072": "E_{n_x,n_y}=\\frac{\\pi^2\\hbar^2}{2mL^2}(n_x^2+n_y^2)", "ec1565010b661032841a69f54ba99b95": "\\textstyle{\\frac {5} {3}}", "ec15f718bc3154a845dc26543ea8cec5": "\n \\tilde{J}_2 := \\tfrac{1}{3}\\left(\\cfrac{\\sigma_1^2}{\\sigma_{1c}^2} - \\cfrac{\\sigma_1\\sigma_2}{\\sigma_{1c}\\sigma_{2c}} + \\cfrac{\\sigma_2^2}{\\sigma_{2c}^2}\\right) ~;~~\n \\tilde{I}_1 := \\cfrac{\\sigma_1}{\\sigma_{1c}} + \\cfrac{\\sigma_2}{\\sigma_{2c}} \n ", "ec1612845502068f3972e24a028db7e6": "\\prod_{m=1}^\\infty \n\\left( 1 - x^{2m}\\right)\n\\left( 1 - x^{2m-1} y\\right)\n\\left( 1 - x^{2m-1} y^{-1}\\right)\n= \\sum_{n=-\\infty}^\\infty (-1)^n x^{n^2} y^n.\n", "ec16188a49b39b4a93bbf0f4ef81e39d": "\\scriptstyle \\sum_{k=1}^{\\infty}f(k) ", "ec16384f5fca71c39659d32f18b1950d": "\\underline\\lor", "ec166914ac2f288d3d6a7ed739b451bd": "f(A,B,C) =\\sum m(0,1,2,5,6,7)\\,", "ec1683e9186e084f5020676e5910b50c": "\\pi^s_1", "ec172278ec4e721927f0479f8978f1a9": "D\\,", "ec1725126f90958562190e16e0e3341d": "H_{2n+1}(x) = (-1)^n\\ 2^{2n+1}\\ n!\\ x\\ L_n^{(1/2)} (x^2)", "ec174388901010778a16970cbc010fe1": "f(u_1,u_2)\\,du_1\\,du_2", "ec1756e684114178411cd11252c24de8": "\\Delta(x) = \\mathcal{O}\\left(\\sqrt{x}\\right).", "ec17bff36254e4123ac4024c48f9ff6c": "\\mathbf{a}^{\\rm T}\\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}} ", "ec186db8f5b1a30b383fec219bba444e": "k_j \\,", "ec18b06a6bef6fe565c5482a555ac465": "\\omega_i(0)", "ec18d7b4e3c094ce20364804519012b5": "p(D|sober) = 0.05", "ec19198ac16d0bc9cbc09db13229133d": "\\textstyle w = 1", "ec19239c804b96d2045c8cd4214a69f4": "2^3\\cdot 3\\cdot 5", "ec1935fcb8a34a6e2e6b3a8aa582993f": "(I_3-I_1)/3", "ec19b449c2ad1ef313222a39983d77a1": "\\lim_{\\Phi_{S}\\rightarrow0}V_{m}\\ne0", "ec19da1e764a0432634e6d820cd7db69": "\\epsilon^3>\\frac{|\\Delta|-27}{4}.", "ec1a0a6159b53c61dc20653740a0994f": " I^\\omega ", "ec1a87edac981b2cc6ba6a4623e217ff": "\\gamma_B > 1 ", "ec1ac19c32af6844e8a2fe30c332084c": "\\textstyle P_{X_r}(x)", "ec1b6762a140d55bd5ad538428f2ebcd": "r_{k}^{B}", "ec1b7bdb7f90ab9018b9b1f8b0d2bb3f": "x_0 \\approx \\sqrt{S}.", "ec1bac7f8fde76e9d44e5788a13ece75": "e^{-\\frac{1}{x^2}}", "ec1c00849914ee863eeb71677d4f40d4": "\\mathbf I_2\\,\\!", "ec1c8ab8049f05f24356364dea1b2b1b": "|M|^2", "ec1c9bf5a4a3266f5f6243243b5069be": "(x,y) = x^{-1}y^{-1}xy", "ec1ca588ea9979d3c58d0e027ae879a9": "Z' = R' = \\frac{R_0^2}{R}", "ec1cad37813d4f36bd42e2b5b94426da": "~U = \\theta L~", "ec1d21c9b9eff1c0fa3c5a33b81e3ffd": "\\Box(a \\wedge \\neg \\Box a)", "ec1dd9887fe60995d1feb64980b24381": "\\left( SI \\right)", "ec1e57939fb35ef282b2ed36ec884f53": "\\hat \\beta +2n \\pi.", "ec1e6d1ca76daf08212d1caa77f6505f": " C^{k-[n/p]-1,\\gamma}(U)\\,", "ec1e801d2900d48ce0c020a84cfc36da": " N - \\delta N", "ec1e9ae7e98548612f13f8e6a64013a5": "\\bar{P} = \\frac{1}{(10)} (3.780) = 0.378", "ec1eb6c73eb16c3b85039f5b167d6eb1": "\\nabla_i \\varphi = \\frac{\\partial \\varphi}{\\partial x^i}\\ ", "ec1fa8459bc780d4c84e57d81c5b631b": " P(X=k) = {{{K \\choose k} {{N-K} \\choose {n-k}}}\\over {N \\choose n}}", "ec1fda902567b3650c4036edc213e5f2": "\\Delta T_{o-sky}", "ec20269c311c6cd2081c609372b68ae9": " \\mathrm{For} \\quad -b 0", "ec2ac6add7b55c13e282a2f663543411": "f \\in \\mathbb{R}^n, \\ A_i \\in \\mathbb{R}^{{n_i}\\times n}, \\ b_i \\in \\mathbb{R}^{n_i}, \\ c_i \\in \\mathbb{R}^n, \\ d_i \\in \\mathbb{R}, \\ F \\in \\mathbb{R}^{p\\times n}", "ec2afd4e9a262cf1471d8c97d44f5472": "\\phi(t)= \\sqrt {2} \\sum_{n \\in Z} h_n \\phi(2t-n)", "ec2aff3adf694d4f4313d97d398aec4b": "(21)\\quad k^c\\nabla_c \\hat\\sigma_{ab}=-\\hat\\theta\\hat\\sigma_{ab}+\\widehat{C_{cbad}k^c k^d}+\\kappa_{(\\ell)}\\hat\\sigma_{ab}\\;,", "ec2b0521bac161e55edd70ab71805595": "x^5 + \\frac{5\\mu^4(4\\nu + 3)}{\\nu^2 + 1}x + \\frac{4\\mu^5(2\\nu + 1)(4\\nu + 3)}{\\nu^2 + 1} = 0", "ec2b4a5bc36cbdd8749c37a9097cf50c": "A^- _{(g)} + S_{(s)} \\to A_{(g)} + S^-_{(s)}\\,\\qquad(2)", "ec2b6da694578e9cac1b6cade06d9062": "\\exist x \\forall y Lxy", "ec2b774f7178117896f1e0c102fd36e9": "F(S).", "ec2ba2fff14868865926a54958d40038": "{\\lambda}_{1}\\neq{\\lambda}_{2}", "ec2baa79536935a7a5ebb46e7aa3196e": "\\Gamma^2 = K_0'^2 - 2 K_0 K_0'' > 0", "ec2bd63e0565e96e87c446d6908eb5f1": " g^{bn} (R^m {}_{bmn;l} - R^m {}_{bml;n} + R^m {}_{bnl;m}) = 0,\\,\\!", "ec2bdfbf2f41d8d358a5b596c11da7ca": "0 \\le x \\le q \\lfloor q \\rfloor /(q-1)", "ec2c09b7a0f1a1e1ba2ccf8fa41fdc2b": "\\{X_k\\}", "ec2c2c986196154fdcc11237dff9fa73": "\\displaystyle{(\\lambda +{1\\over 2}) \\int_{\\partial\\Omega} \\partial_{n+}u \\, \\overline{u} =(\\lambda -{1\\over 2}) \\int_{\\partial\\Omega} \\partial_{n-}u \\, \\overline{u}.}", "ec2c38c2a26adf0afa2cafcab06f50b3": "\\gamma' (0)", "ec2cf8f27bb29f381fd5ebad7c13f54d": "q > 1/2", "ec2d0f2f2b8fcfacfed4bc238d4f7837": "R_{\\alpha \\beta I}^{\\;\\;\\;\\;\\;\\; J} V_J = (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) V_I", "ec2d5acabc26232f637ad7da4b482d35": " \\rho\\left(\\frac{\\partial }{\\partial t} + \\mathbf{v}\\cdot\\nabla \\right)\\mathbf{v} = \\mathbf{J}\\times\\mathbf{B} - \\nabla p.", "ec2d93391f04ffafc7d23398ae3c6f80": "3x+y=12", "ec2db7f3e5fbd01efdc3c7e60e0cdd7e": "E_{x}^{\\mathrm{LDA}}[\\rho] = - \\frac{3}{4}\\left( \\frac{3}{\\pi} \\right)^{1/3}\\int\\rho(\\mathbf{r})^{4/3}\\ \\mathrm{d}\\mathbf{r}\\ .", "ec2db974337171fb95053a9f94f208f8": "R = r G", "ec2dbfc37417e7cdd5af299511825a4f": "\\left(\\frac{\\partial S}{\\partial V}\\right)_{U,\\{N_i\\}}=\\frac{p}{T}", "ec2e25cbc6811345fa0596f7671b1401": "\\alpha = R \\left( \\frac{\\omega \\rho}{\\mu} \\right)^\\frac{1}{2}", "ec2e610ec822b8dc27182a234d3fd0ad": " \\begin{align}\nk_i = \\sum\\limits_{j=1}^{N-1}k_i^j &= \\sum\\limits_{j=1}^{i}k_i^j + \\sum\\limits_{j=i+1}^{N-1}k_i^j \\\\\n&= \\sum\\limits_{j=1}^{i}N \\frac{N-i}{N-j} + \\sum\\limits_{j=i+1}^{N-1}N \\frac{i}{j}\n\\end{align}", "ec2e7fa76e608b88a115b8b9a99b2ab4": "\\qquad Q = T_3 + {Y_W \\over 2}", "ec2ee7c906e8917fb54f418a12784c2a": "(\\mathcal{O}_k/\\mathfrak{p})^ \\times", "ec2fd36ddfcd84f7012f4c9ef872c415": "\\frac{\\pi\\sqrt{3}}{16}", "ec30269495ca95da9bcfd0d0110a7df9": "{L=\\mathbf{Q}(\\sqrt[3]{2},\\omega_2)=\\{a+b \\omega_2+c\\sqrt[3]{2} +d \\sqrt[3]{2} \\omega_2+ e \\sqrt[3]{2^2} + f \\sqrt[3]{2^2} \\omega_2 \\,|\\,a,b,c,d,e,f\\in\\mathbf{Q} \\}}", "ec30bac1fffba3298cfdb73fd37bdb68": "\\mu_{\\theta+h}", "ec30bedf59a0001c2b5511ccea70d44f": "0 = \\alpha\\nabla^2 h", "ec30ca94af8b1225fd5bfd45ad109b57": "\\mathcal{F}_{\\infty} = \\sigma\\left(\\bigcup_{t \\geq 0} \\mathcal{F}_{t}\\right) \\subseteq \\mathcal{F}.", "ec315323f4bcdc68de5680ea4cf6d899": "{} G^2 = r^2 - \\left(\\tfrac{M}{2}\\right)^2", "ec318a60243a25070993f19f195d7d6a": " E_T = E - E^\\prime ", "ec31b1cf033dae2b81e5e9cb7a1f235b": "\\wedge^1", "ec321ccd1ef480aa646466c42224c5a5": "BS_n", "ec322d4d5572ce0ffc686ccac3efe74b": "\\mathbf{e}_1, \\mathbf{e}_2, \\mathbf{e}_3", "ec32a08297196a6b43ccde6931a4c1f1": "3 \\cdot 2^{402653209}", "ec333ce449afc414eda402ff8f6697f2": "| \\psi \\rangle = \\sqrt{1\\over 3} |H\\rangle - i \\sqrt{2\\over 3}|V \\rangle,", "ec337d14f9a329bbcd0a9786fa3eeae6": "G(n)=\\frac{(\\Gamma(n))^{n-1}}{K(n)}", "ec342ada03d9ba4798478ef438aacd09": " \\begin{align}P(X \\geq x \\mid \\mu=49222.5) = \\int_{x = 49581}^{98451}\\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-(\\frac{u-\\mu}{\\sigma})^2/2}du \\\\\n=\\int_{x = 49581}^{98451}\\frac{1}{\\sqrt{2\\pi(24,612.75)}}e^{-\\frac{(u-49225.5)^2}{24612.75}/2}du \\approx 0.0117.\\end{align}\n", "ec344657d68a599a03a3061ad54ef7cc": "\\square", "ec349cca6c4046ee1c0dbb51753e9cb2": "a(x, x)\\ge c\\|x\\|^2", "ec34a8de2f4cdb6a34c449a47a652f50": "\\varphi(x) = \\sum_{i\\in\\mathbb{Z}} \\frac{s_i}{(x-i)^k}", "ec34c08574f6f787d8d88cd99659596e": "p^2+q^2", "ec3535cd77eeb836d1c4a5affc514c9b": "\\frac{n}{p} \\leq 4", "ec3559dcb21249a05a887e24e8677806": "\\delta V_H[\\rho](\\mathbf{r})=\\frac{\\delta V_H[\\rho]}{\\delta\\rho}\\delta\\rho=\n\\frac{1}{|\\mathbf{r}-\\mathbf{r'}|}\\delta\\rho(\\mathbf{r'})", "ec35b015773d2b70298d64bc2f7a5c26": "\\begin{align}\n y_1 &= b_0 x^{1 - \\gamma} \\left(\\frac{(\\alpha + 1 - \\gamma)_{\\gamma - 1} (\\beta + 1 - \\gamma)_{\\gamma - 1}}{(2 - \\gamma)_{\\gamma - 2} (1)_{\\gamma - 1}} x^{\\gamma - 1} + \\frac{(\\alpha + 1 - \\gamma)_{\\gamma} (\\beta + 1 - \\gamma)_{\\gamma}}{(2 - \\gamma)_{\\gamma - 2} (1) (1)_{\\gamma}} x^{\\gamma}+ \\cdots \\right) \\\\ \n&= \\frac{b_0}{(2 - \\gamma)_{\\gamma - 2}} x^{1 - \\gamma}\\sum_{r = \\gamma - 1}^\\infty \\frac{(\\alpha + 1 - \\gamma)_r (\\beta + 1 - \\gamma)_r}{(1)_r (1)_{r + 1 - \\gamma}} x^r.\n\\end{align}", "ec3604acb3a4262576cd316181f25a54": "C=C_0e^{-(\\frac{\\psi ze}{kT})}", "ec36ce2213d50b5da7fdc469fd91a705": "C_{D,i} = K C_L^2", "ec36da6a7a1655f8385e2cfdc619e7d8": "\\frac{r_m}{r_l} \\frac{\\partial ^2 V}{\\partial x^2}=c_m r_m \\frac{\\partial V}{\\partial t}+ V", "ec36df370f67aab4f46332c78c2d8b24": "F=\\{\\phi_i\\}_{i\\in J}", "ec3765ec2751cfe8362821d21d2621d7": "{\\mathbf{c}}_{\\mathbf{0}}^{\\mathtt{KED}}", "ec37e38eaf617188908a26842309c051": "\\lim\\Phi'(x_n)=0", "ec382f58c49069a033b3f005aa624e3a": "T^{\\hat{\\mu}}_{\\hat{\\dot{\\alpha}}\\hat{\\dot{\\beta}}} = 0", "ec38398cbab3fe9e23388ae671cd0172": " \\displaystyle{B(z,t)= \\exp \\,[-z\\cdot z -t\\cdot t/2 +z\\cdot t].}", "ec3850f321f9ee2da8fb1df3e6bb77ef": "I_B = \\frac{e\\omega_B}{4\\pi}. \\ ", "ec38872cc79113dd71935e383bba46a2": "PWV = \\sqrt{\\dfrac{E_\\text{inc} \\cdot h}{2r\\rho}}", "ec389e00a21fc51b8cac312146be27ff": " \\frac{1}{\\sqrt{\\mu_0}}\\left(\\mathbf{B}, \\Phi_\\text{m},\\mathbf{A}\\right) ", "ec394f183e740e8d42ff7afc2c7d2ac8": "\\frac{R_1}{R_2}{V_s}", "ec3960710f6c1d73f5577ecf9669736b": "\\{a^n b^n c^m d^m | n, m > 0\\}", "ec39acf95cfed397c00f1a66a065ae29": "T = 4\\sqrt{\\ell\\over 2g}\\int^{\\theta_0}_0 {1\\over\\sqrt{\\cos\\theta-\\cos\\theta_0}}\\,d\\theta.", "ec39d8e74cf59cd22f459bb1ddb141dc": "(x\\Rightarrow y)\\vee(y\\Rightarrow x)=1", "ec39fa86bb1cc4a4e49f6968057a003a": "\\theta'(0)=0", "ec3a11dfa62c6af808ed7cd6534594e0": "X_1 \\times X_2", "ec3a415353983d09f911a598fff1185d": "\\Lambda=\\frac{-(n-1)(n-2)}{2\\alpha^2} \n", "ec3a671cdd4a716a2bab61472e17bbe7": " E = h \\nu\\ ", "ec3a792ebeef684bfaf1a84c5a1fe15b": " t s = (u w + v z) + (u z + v w) j .", "ec3a8df1fb902be145976dee2f8cb2b7": "T_n + T_{n-1} = \\left (\\frac{n^2}{2} + \\frac{n}{2}\\right) + \\left(\\frac{\\left(n-1\\right)^2}{2} + \\frac{n-1}{2} \\right ) = \\left (\\frac{n^2}{2} + \\frac{n}{2}\\right) + \\left(\\frac{n^2}{2} - \\frac{n}{2} \\right ) = n^2 = (T_n - T_{n-1})^2.", "ec3aa0aca615a1b9ebd2bb32f9fddc5e": "b=0.0867\\frac{RT_c}{P_c}", "ec3aae8e9c9d40084dd7486ff7d2db7e": "V=\\left(\\frac{1}{6}\\left(5+4\\sqrt{5}\\right)\\right)a^3\\approx2.32405...a^3", "ec3ac008429e466018df790297a247c8": "\\mathrm{Ann}_R(y)=\\{ r\\in R\\mid ry=0 \\}= xR", "ec3b159528abcae38e68a6b06e69f75d": "i=1,2,\\dots,k^*", "ec3b281c28b88b450c0fa06ce7470696": "\nq_r \\mapsto \\sum_A\\vec{q}^{\\,A}_r \\cdot \\big(\\vec{d}^A - \\Delta\\varphi \\; ( \\vec{n}\\times \\vec{R}_A^0) \\big) =\nq_r - \\Delta\\varphi \\; \\vec{n}\\cdot\\sum_A \\vec{R}^0_A\\times\\vec{q}^{\\,A}_r = q_r.\n", "ec3b327df7f135f40a7b25ba78e385d8": " b_{0,0}", "ec3b4e93e3530093119475855d96bb81": " f(z)=a_0+a_1 z+\\cdots+a_n z^n", "ec3b7554f772e6387d6932ebf39c20e2": "XX^* = P_VT(\\boldsymbol{1}_H-P_V)^2T^*P_V= P_VT(\\boldsymbol{1}_H-P_V)T^*P_V = P_VTT^*P_V - P_VTP_VT^*P_V", "ec3b87f8b767e91c216d356b9506f910": "\\nu=\\tau=\\gamma=0\\,,\\quad \\mu=\\bar\\mu\\,,\\quad \\pi=\\alpha+\\bar\\beta\\,,", "ec3bd223c2eb6da7928ececf53d753fb": "\\pi\\over 3", "ec3c063c108f3078fb619b2121ac6af4": "g/\\mid \\nabla f \\mid", "ec3c17ce4dac50b2e655e0f7760c04e7": "(|R,L\\rangle+|L,R\\rangle)/\\sqrt{2}", "ec3c50bb44583e12ed52be1d40012fb9": "\n\\langle f, g \\rangle = \\sum_{a \\in G} f(a) \\bar{g}(a).\n", "ec3c59dc4f8421628cf57ad7e7911802": "p:a\\mapsto b", "ec3cafaf7f9e65942010994d9431e3c0": "dI_\\nu=0", "ec3d133dfb57e2aa94b368f039f288fd": "\\sum_{k=1}^n \\beta_k e^{\\alpha_k}\\neq 0", "ec3d19dec8eecd49e8ec21bb2b5dea63": "b_{3}", "ec3d4d85e6f080cb8f7eb50755744dea": " \\frac{\\Delta Y}{Y} = \\frac{VK}{Y} . \\frac{\\Delta K}{K} + \\frac{WL}{Y} . \\frac{\\Delta L}{L} + \\frac{\\Delta Y'}{Y} ", "ec3d8074bebddf935ebd6a18aed9978d": "GL^+(n,\\mathbb{R})\\cong \\mathbb{R}^+\\times SL(n,\\mathbb{R})", "ec3dd2cab2c24e104c4fb340e933139d": "\\hat\\nu \\approx {(g_1 + g_2)^2 \\over g_1^2/(n_1-1) + g_2^2/(n_2-1)} \\quad \\text{ where } g_i = s_i^2/n_i.", "ec3e57acbd78d6d3f89ea3ba63940630": "\\mathit{\\bar{K}}", "ec3ea97e3b1605dfad0fb07bc85b0027": "\nA_t(t,T)-\\beta(t)B(t,T)+\\frac{1}{2}\\delta(t)B^2(t,T)-\\left[1+B_t(t,T)+\\alpha(t)B(t,T)-\\frac{1}{2}\\gamma(t)B^2(t,T)\\right]r=0\n", "ec3eb368b1261adc8f6cbd579af1784e": "Z = n_i \\times [Z]_i \\times ( c_{ij} \\times [Z]_j/[Z]_i )", "ec3edaf4eff2570dad0b14bbaae05fb4": "h \\cdot 1_A = \\epsilon(h)1_A", "ec3edc1093d0d25ef285b6d0669419a7": "J_3(\\mathbb R)", "ec3f243c2bad8259e3068f56fa7847e6": " K=\\tfrac{1}{2}(ac+bd). ", "ec3f2d86637728bcd3d6083621f1e271": "\\; S_k", "ec3f54000cecb5b03531b935d8a79f9b": "r_1 e^{it} + r_2 e^{i(\\omega_2 / \\omega_1) t}", "ec3fb10490614fc31cfe2015b6b246c6": "L = \\begin{bmatrix} -1 \\\\ L_{2} \\end{bmatrix}", "ec3fba26c51e68e298c27c9ebdaa3818": "B_3(x)=x^3-\\frac{3}{2}x^2+\\frac{1}{2}x\\,", "ec3fda36691a82deccb6b5081c046720": "y_i - \\hat y_i", "ec3ffb30b1fc70ca773ae1051b5b1ead": "[-7, {-2.5}]", "ec406d2cf931709724eee4717b9e0550": "z_{\\mathrm{S}}=4\\mathrm{\\pi}em / h_{\\mathrm{P}}^3 = 1.618311 \\times 10^{14} \\, \\rm{A} \\, m^{-2} \\, eV^{-2}, ...........(17)", "ec409067aad8b175373f9d5d8136f2a2": " \\mu- \\lambda = \\epsilon ", "ec40cfc215dace05e901e573a78ed72e": "\\alpha ! = \\alpha_1! \\cdot \\alpha_2! \\cdots \\alpha_n!", "ec40dd2d29399b3d3aa6658f3c7060bb": "\\Sigma_\\text{r}", "ec40fa87e85dc878dc8718cdc0df14ec": "\\lim_{x \\to c} f(x)^{g(x)}.\\!", "ec413197eba015882f8a881603f13abc": "V(t_n)", "ec41510775ad06a793e825abd2857d0c": "y = \\beta x", "ec41532d7c1228c9f5b7d71c0a84a829": "F(\\theta) = \\int \\textrm{d}r\\,g(r)\\,\\ln f(r;\\theta)", "ec41680e2f33d69c468f29b21abf1685": "L=\\int_{S_o}^{S}n\\,ds", "ec4179e84682f431a98a6fe428826d81": "\\begin{pmatrix}\n\\mathbf{E} \\\\\n\\mathbf{H}\n\\end{pmatrix}=\\begin{pmatrix}\n\\cos \\xi & -\\sin \\xi \\\\\n\\sin \\xi & \\cos \\xi \\\\\n\\end{pmatrix}\\begin{pmatrix}\n\\mathbf{E'} \\\\\n\\mathbf{H'}\n\\end{pmatrix}", "ec41e50bca3a3c20b1949d628c2a7a79": "\\int_{\\Gamma}\\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta + \\int_{\\Omega'} \\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta = \\int_{\\Gamma}\\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta - \\int_{\\Omega}\\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta = 0.", "ec4245d4d0416f7d62185e2363aa0098": "=2(r_1+r_2)(\\tan(\\varphi) + \\pi- \\varphi) \\,\\!", "ec426b2b609b8debd7ec2adb836133fb": "W^{1,p}(\\Omega, \\mathbb{R}^m)", "ec4277c5646b03e3268bf07242dfe867": "O((\\log N)^{6+\\epsilon})", "ec4287640de39fc6bd54a7564a42b91e": "qP = \\bar{q}P", "ec4293245d07f61250f84dc749d3c24b": "\\tau = x + iy", "ec42a25c64e279bdc9e1b6414f7fbca1": "f\\ (f ... f\\ (\\operatorname{fix}\\ f)... ))\\ x", "ec42adc2a387fefb849ebad9ded703f4": " X_i \\in [0,1] ", "ec42dc93fe2d1e6a523f14c8eaced179": "\\psi \\mapsto \\lambda \\psi", "ec42f1455b9d781219bd61fd4f1f7825": "V_\\mathrm {dc} = \\frac{V_\\mathrm {peak}}{\\pi}", "ec431a0f2b1081b4f3cad747570342fc": "\n\\langle e|_B \\langle \\phi|_A |\\psi\\rangle_A |e\\rangle_B \n= \\langle e|_B \\langle \\phi|_A U^{\\dagger} U |\\psi\\rangle_A |e\\rangle_B \n= \\langle \\phi|_B \\langle \\phi|_A |\\psi\\rangle_A |\\psi\\rangle_B,\n", "ec432e8dd64e1cd858032728c2511d53": "\\tbinom n t", "ec435defe3754a34b2b603477d15b633": "\n\\tfrac{1}{2} \\mathbf{x}^T Q\\mathbf{x} + \\mathbf{c}^T \\mathbf{x} \\quad \\Rightarrow \\quad\n\\tfrac{1}{2} \\mathbf{y}^T Z^T Q Z \\mathbf{y} + (Z^T \\mathbf{c})^T \\mathbf{y}\n", "ec4392f899e4cae9efffbd738b120b5c": " {}=\\pi \\approx A_{192} + \\left(\\tfrac{1}{3}\\right)D_{192} \\approx {3927 \\over 1250} \\approx 3.1416.\\,", "ec43ca083a27c36235f58769d247177b": "\\Pi : ax + by + cz + d = 0\\,", "ec442bcd8df60f6939a9478832c4c98f": "r\\in\\mathbb{N}", "ec447d3ca38d7e3306679f25b1db69c1": "\\ln(\\theta \\Gamma(k)) + (1 - k)\\psi(k) + k \\, ", "ec449be9a886d34ed2a5a1f146d45037": "x_i = (u_i,v_i)^T \\, ", "ec44d0d32a3417735d052912833b1883": "\\pi_1(\\mathbb R^3 \\backslash \\text{trefoil}) = \\lang x, y | xyx = yxy \\rang.", "ec44e12e3a0b51b842aeb148e7e1dc33": "\\frac{dv_y}{dt} = \\frac{-k}{m}v_y - g ", "ec453e2b871a39954b9f52a42906e992": " k_{max} \\approx \\frac{\\omega}{c} = \\frac{2 \\pi}{\\lambda} ", "ec4544629afe295776fced14d722a8e0": "\\left( \\frac{-7}{\\sqrt{10}},\\ -\\sqrt{3 \\over 2},\\ 0,\\ 0 \\right)", "ec458fe5cae699e2af124e7cc741b26b": " A = 1/det(A_{ij}) = {1 \\over (1-\\kappa)^2 -\\gamma_1^2 -\\gamma_2^2} ", "ec4598597534dcb99d896e90f615dedd": " Q^{(1)}, Q^{(0)}, Q^{(-1)},\\ldots ", "ec459fee9932222c4a168b12e386ed5b": "\\delta Q = T\\,\\mathrm{d}S\\,", "ec4624d1f5613fb7402e93fb58364c4e": "\\frac{dx'}{dt'}=\\frac{ dx - v dt }{ dt - \\frac{v dx}{c^2} }", "ec46375461e983bf635c20f86015b338": "\\scriptstyle V_L", "ec464653cc165dd6cdae9d5e7b2d4f1b": "\n\\beta(\\phi)=\\tan^{-1}\\left[\\sqrt{1-e^2}\\tan\\phi\\right]\\,\\!\n", "ec46521a32a6723156e7e66fdf83d7b5": "\nP(z|a,b,p) = \\text{GIG}(z|a,b,p)\n", "ec46565ec63aff6d7ca43c65283145c1": "h = 2\\pi\\hbar", "ec4668d92fc0fc5cfc89732fbc840c0d": "N\\rightarrow \\infty ", "ec467837fd1b483142f7c0e60d3ff178": "\\varphi/(m-n)", "ec46cdd86231752ce13235e2472ccf7d": "k(\\mathfrak{p})", "ec46d09360d39dd6da07b73e0a2f9b2e": "2P_r", "ec47435b2d3c216b079320e52513a737": "s(x) = \\sum_{i = 0}^{n-1} c_i x^i", "ec4756e49db9a2ac660ca86d4f387c6b": "U[R(\\theta, \\hat{\\mathbf{n}})] = \\exp\\left(-\\frac{i\\theta}{\\hbar}\\hat{\\mathbf{n}}\\cdot\\widehat{\\mathbf{J}}\\right)", "ec48139442dea021eb2dbc2451861315": "M_5 = \\frac{1}{32} \\, S^{cd} \\, S^{ef} \\, \\left( C^{aghb} + i \\, {{}^\\star C}^{aghb} \\right) \\, \\left( C_{acdb} \\, C_{gefh} + {{}^\\star C}_{acdb} \\, {{}^\\star C}_{gefh} \\right)", "ec4823fe9d57ed2beac5f4c49ed1367e": " |b^*_n(v)| \\; \\|b_n\\|_V = |\\alpha_n| \\; \\|b_n\\|_V = \\|\\alpha_n b_n\\|_V = \\|P_n(v) - P_{n-1}(v)\\|_V \\le 2 C \\|v\\|_V.", "ec49246128a8a05a7be16e457048a068": "\\frac{1}{PM} < \\frac{2R}{x(180-x)}", "ec49665dce3cdaf12b454a57df82a3fb": " f_B(x) = \\frac{3 \\alpha}{2\\pi}\\sum_{q,\\bar{q}}e_q^4 x [x^2+(1-x)^2]", "ec49d254e788e91030e093bf686f0aa0": "p_c(S)", "ec49f108fbec4a03c3f4e0073a65f458": "\\alpha\\in E", "ec4a3c5a8a9202845ff44e42888ec380": "x^5+110(5 x^3 + 20x^2 -360 x +800) ", "ec4aaf5ac251b549e6f7f3d0ebee72e2": "\\scriptstyle\\sqrt{3}r", "ec4ad94a9be87109217fcd9d10ebcd52": "N^2", "ec4b123147d1f85af7edf4223d513d10": " \\mathbb{P} (Y \\le y|X=x) = \\tfrac12 + \\tfrac1{\\pi} \\arcsin \\frac{ y }{ \\sqrt{1-x^2} } ", "ec4b1351e196dc1b1c9d1fe265bfeaa3": "C_3=v_{\\infty}^2\\,\\!", "ec4b172b27b86d17c0a2559d63c550ac": "\\textstyle l_\\phi>l_c", "ec4b2253c3509b34c68da75c582bbf4b": " z = \\frac{\\overline{X} - \\mu}{\\sigma/\\sqrt{n}}", "ec4b2ad901476bb0b48afa44e6edff64": "\\iint_D f(x,y)\\ dx\\, dy = \\int_a^b dx \\int_{ \\alpha (x)}^{ \\beta (x)} f(x,y)\\, dy.", "ec4b2cf4f04c00dbf802b5740cc48ecb": " V(x) ", "ec4b38984a3015f14ac40e1b57cd72e5": "s'(x) = \\sum_{k=1}^\\infty f_k'(x)", "ec4b8b0a5a7d109d96e448354b918869": "\\pi_0(\\pi_1(X))", "ec4bb43368a49522f4c9cbb65fe4fd32": "\\textstyle \\alpha\\mapsto |x|^\\alpha", "ec4c79d654c664ae2aba5fc02c6e1e3e": "T_{w,10}=900(x_{n,10}-x_{10})-650(y_{n,10}-y_{10})", "ec4cc3e4573f3bc16a245e8868eab200": "I : (J + K) = (I : J) \\cap (I : K)", "ec4cd1b489fcfc31d9b805f5c84ba343": "TP|_M = TM\\oplus T^\\perp M.", "ec4cd79e3d8205a7983e18e80e0649eb": "n-k'\\le n\\, \\{x\\}\\alpha", "ec5dd65e9d755eca6495acbed96ae8cf": " U_a = \\{(x,y)\\in X\\times X : d(x,y)\\leq a\\} \\quad\\text{where} \\quad a>0", "ec5dfa6b64ad54c896b1476894bbd29a": "\\langle\\cdot,\\cdot\\rangle_q", "ec5e1162bd664e3c0c9c1723edd442c7": "E_{\\rm x}, E_{\\rm y}, G_{\\rm xy}, G_{\\rm yz}, \\nu_{\\rm xy}, \\nu_{\\rm yz}", "ec5e45feb4029377c128a313e964f304": "W_0 - x'k", "ec5e8d6681f0e79b3767be5aa25bb9ec": "{\\tilde{E}}_6", "ec5edc20de1d57c3ced37330503198d3": "x = \\sqrt{\\left({B \\over 3C}\\right)^3 + y^2}.", "ec5f75a912abe5e1499fd6020ef0599b": "\\omega \\times \\mathbf{r} := *(\\omega \\wedge \\mathbf{r}) ", "ec5f924383b525ba85a35ff0e82b153d": " a(\\cdots\\cdots)^2 + \\mbox{constant}.\\, ", "ec5fa4272c5bf3f1e9b6a3959a5914d2": "\n=\n\\frac{q_\\mathrm{tot}}{R} + \\frac{1}{R^3}\\sum_{\\alpha=x,y,z} P_\\alpha R_\\alpha +\n\\frac{1}{6 R^5}\\sum_{\\alpha,\\beta=x,y,z} Q_{\\alpha\\beta} (3R_\\alpha R_\\beta - \\delta_{\\alpha\\beta} R^2) +\\cdots\n", "ec5fa498cf97929abcbe0b1b69ea6834": "f: D\\rightarrow \\{ A_1, \\dots,A_m \\}", "ec5fdec7e90112a6411cea732273b744": "u = 0.975", "ec6014c9cbc7208e5561a9481e958aae": "f(x_1) \\leq f(x_2) \\leq \\ldots \\leq f(x_n)", "ec6014ffc3d92709aa6a4fea11bb3788": "(1,0)", "ec60364f0316ee5ac3cdbd5eabeed010": "\\begin{align}\n0 & = - g^{\\mu \\nu} \\nabla_{\\mu} \\nabla_{\\nu} \\psi + \\dfrac {m^2 c^2}{\\hbar^2} \\psi = - g^{\\mu \\nu} \\nabla_{\\mu} (\\partial_{\\nu} \\psi) + \\dfrac {m^2 c^2}{\\hbar^2} \\psi \\\\\n& = - g^{\\mu \\nu} \\partial_{\\mu} \\partial_{\\nu} \\psi + g^{\\mu \\nu} \\Gamma^{\\sigma}{}_{\\mu \\nu} \\partial_{\\sigma} \\psi + \\dfrac {m^2 c^2}{\\hbar^2} \\psi\n\\end{align}", "ec606002e1082864b4c28cd4f49ec71a": "A = \n\\begin{pmatrix}\n12 & -51 & 4 \\\\\n6 & 167 & -68 \\\\\n-4 & 24 & -41 \n\\end{pmatrix}\n.", "ec60bfe7ea0c9c8f5a772f22bb76b335": "\\int_0^\\infty u^n e^{-u} du=n!", "ec60ca781c98bf073103aa8e96a43a0e": "\\!\\mu_3(v_3)", "ec6180c168d921de20e3d3cd213eae06": "\\frac{\\varphi(0) - \\varphi(x)}{x^2} ", "ec61ae798b0eb2e902c1379650689217": "O(\\log^2(B))", "ec624f65ade477807fe933187d146de8": "q = \\exp(a\\mathbf{r}) = \\cos a + \\mathbf{r} \\sin a, \\quad \\mathbf{r}^2 = -1, \\quad a \\in [0,\\pi],", "ec625f27a9f0de7aa30f20c1a803de05": "\\ S = -E_{0}t + L\\phi + S_{r}(r) + S_{\\theta}(\\theta)", "ec6274139c43722cd858b52874101b87": "M = 1 + \\frac{\\max_{i=0}^{n-1} |a_i|}{|a_n|} . \\,\\! ", "ec62cc17b4c86dd90d051d1966594a93": "\\int_0^\\infty \\frac{x \\ln x}{e^{2 \\pi x}-1}dx=\\frac{1}{2} \\zeta^{\\prime}(-1)=\\frac{1}{24}-\\frac{1}{2}\\ln A", "ec62d608dcc4063e72731fac4f58bd7d": "\\sum_{j=0}^{n-1} \\alpha^{jk} = 0", "ec630e3ce233c69561f1ef5e83f1f691": "{\\Delta}P_{Ref}\\,\\!", "ec63379cb464659bc56a8abfcc789c2a": "T_\\mathrm{TF}[\\rho] = C_\\mathrm{F} \\int \\rho^{5/3}(\\mathbf{r}) \\, d\\mathbf{r} \\, .", "ec637d7e5f6220d89967677c3cdd1307": "\n \\boldsymbol{\\nabla}\\mathbf{u} = \\boldsymbol{\\varepsilon} + \\boldsymbol{\\omega}\n", "ec6399ffc8ec9ff1532bcca6d5cb962d": "\\frac{3\\alpha a-2}{6}", "ec639d0050127df74ba1fec697c867a4": "\\,\\! x=I_x/I_u", "ec63a1cbc273cd4d4ed64df5a849b259": "\\lambda = h/p", "ec63ce94c0937818f6aae6f05027975a": "-\\frac{\\Delta H}{R}", "ec63d268b3579e8e9b8848c9007e3089": "R_{hs} = \\frac {\\Delta T}{P_{th}} - R_s ", "ec64638d23dc4f46e3431704d3e9108d": "G = p\\ f ", "ec6494cfb014fb0789f06f60216ea1c7": "M(E) = 0 ", "ec64a90861f47f39f02508ea9977b297": "\n\\mathcal M=\\sqrt{2\\omega_p}\\ \n\\langle \\beta\\ \\mathrm{out}|\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\na_{\\mathrm{in}}^\\dagger(\\mathbf p)-\na_{\\mathrm{out}}^\\dagger(\\mathbf p)\n\\mathrm T\\left[\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n|\\alpha\\ \\mathrm{in}\\rangle\n", "ec64bcda3810ee467325d275c21705e7": "\\sum_{g\\in G}\\rho(g)", "ec64e84046ef72518a6b03972077f74d": "[g_{ij}] = \\begin{pmatrix}\n1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & 1\n\\end{pmatrix}", "ec64f4b97d1b9b784272480067c1b021": "J_{0}(r)", "ec656b379fe4d816d31646541866fe17": "L^2(\\mathbb{R}^d)", "ec656d2d0831108b05fe8ee4e99ed0fc": "\\mathbf{\\hat{t}}=\\left \\{ \\begin{matrix}\n\n\\frac{\\mathbf{v}_r - (\\mathbf{v}_r \\cdot \\mathbf{\\hat{n}})\\mathbf{\\hat{n}}} {|\\mathbf{v}_r - (\\mathbf{v}_r \\cdot \\mathbf{\\hat{n}})\\mathbf{\\hat{n}}|}\n& \\mathbf{v}_r \\cdot \\mathbf{\\hat{n}} \\neq 0 & \\\\\n\n\\frac{\\mathbf{f}_e - (\\mathbf{f}_e \\cdot \\mathbf{\\hat{n}})\\mathbf{\\hat{n}}} {|\\mathbf{f}_e - (\\mathbf{f}_e \\cdot \\mathbf{\\hat{n}})\\mathbf{\\hat{n}}|}\n& \\mathbf{v}_r \\cdot \\mathbf{\\hat{n}} = 0 & \\mathbf{f}_e \\cdot \\mathbf{\\hat{n}} \\neq 0 \\\\ \n\\mathbf{0} & \\mathbf{v}_r \\cdot \\mathbf{\\hat{n}} = 0 & \\mathbf{f}_e \\cdot \\mathbf{\\hat{n}} = 0 \\\\ \n\\end{matrix}\\right.", "ec656d6f5ab73d6b5aec45550d16a2ff": " \\Delta f = \\pm 2 \\lambda N^2 ", "ec65f5c04754b60335fde93172b9f5ce": "C(n,n-b) = \\sum_{a}^{} \\, A(n,n-a)B(n-a,n-b)", "ec660b8dd444ce36630abe90bc5cf9d8": "(a, \\ b)_K \\; := \\ \\{ \\{ a \\}, \\ \\{ a, \\ b \\} \\}.", "ec66607e48bc8e4f43942f8d3def1a78": "\\Delta^2(p_1(a)R_1) \\le 0 ", "ec6677f13505394934efca77aa191304": "\\lim_{\\alpha\\rightarrow 0}\\mathcal{B}_\\alpha y( z)", "ec6690c2734ed4e67daa4ba4c982b317": " \\geq 1 - n^{-d} ", "ec66c07ec34289bedf69a84a04a96a84": "d\\mathbf{l} = \\sqrt{\\sigma^2 + \\tau^2} \\, d\\sigma \\, \\hat{\\boldsymbol \\sigma} + \\sqrt{\\sigma^2 + \\tau^2} \\, d\\tau \\, \\hat{\\boldsymbol \\tau} + dz \\, \\hat{\\mathbf z}", "ec66c776aa3c09ed2efcefac6065b895": "L(E(\\mathbf{Q}), s) = \\prod_p \\left(1 - a_p p^{-s} + \\varepsilon(p)p^{1 - 2s}\\right)^{-1}", "ec66efcdd0b03ef5fa5267d0645b71aa": "R^{\\alpha\\beta\\gamma\\delta} R_{\\alpha\\beta\\gamma\\delta} = \\frac{12 {r_s}^2}{r^6} = \\frac{48 G^2 M^2}{c^4 r^6} \\,.", "ec66f16febb8dab9e4a5ab75f8df1d6f": " \\partial u_z/\\partial z = 0 ", "ec67671562351a32adf671aee6d2aa5b": "\\left \\|\\mathbf{k}_3\\right \\| = n(\\omega_3)\\omega_3/c", "ec67916124e3e922e08f9a00ebd49756": "M \\in B^{\\ast} \\cup {Reject}", "ec679f15073b718baac49d374fb821de": "(i)", "ec67c47ac3ae1e6905ce23ed376e0e76": "\\arcsin h, \\arccos i, \\arctan j \\!", "ec682834b072dcb68b26f80a7d18164e": "f \\colon \\mathbb{R} \\to \\mathbb{R}", "ec6832a13adee0e19ce5f908351e9176": "PV=NkT\\,", "ec685b89f004950b8249252bb380ee02": "\n1-\\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left( m\\right)\n},\\delta}\\hat{\\Pi}_{\\rho_{X^{n}\\left( m-1\\right) },\\delta}\\cdots\\hat{\\Pi\n}_{\\rho_{X^{n}\\left( 1\\right) },\\delta}\\ \\Pi_{\\rho,\\delta}^{n}\\ \\rho\n_{x^{n}\\left( m\\right) }\\ \\Pi_{\\rho,\\delta}^{n}\\ \\hat{\\Pi}_{\\rho\n_{X^{n}\\left( 1\\right) },\\delta}\\cdots\\hat{\\Pi}_{\\rho_{X^{n}\\left(\nm-1\\right) },\\delta}\\Pi_{\\rho_{X^{n}\\left( m\\right) },\\delta}\\right\\} .\n", "ec689a9564156162f46221705e91f002": " K_i = \\frac{IC_{50}}{\\frac{[A]}{EC_{50}}+1} ", "ec69911a2a24837b4048bc20205e884b": "\\boldsymbol{\\bar{a}} = \\frac{\\Delta \\mathbf{v}}{\\Delta t}.", "ec6992cbbe8610f48fe8133994087e24": "\n \\bar{\\lambda} := \\sqrt{\\tfrac{2}{3}}~\\cfrac{u(\\theta)+v(\\theta)}{w(\\theta)} ~;~~\n \\bar{B} := \\tfrac{1}{\\sqrt{3}}~\\left[\\cfrac{\\sigma_b\\sigma_t}{\\sigma_b-\\sigma_t}\\right]\n ", "ec69970c8cd73b91b4197168c842966b": "A \\rightarrow B: \\{A, K_{AB}, T_B\\}_{K_{BS}}, N'_A", "ec69be969830983e46a386f6885241da": " \\mathbf{X}_1, \\ldots, \\mathbf{X}_k ", "ec69e5f7957646e2784fc63e9ddc6fc5": "\\left\\langle\\begin{matrix} n \\\\ k \\end{matrix}\\right\\rangle", "ec6a106971fbb166e28a20567c7962ce": "\\subseteq U_{i} \\subseteq", "ec6a69160b8c4e2025df1d8504c57d1b": " g(z) = {1\\over 2\\pi} \\int_0^{2\\pi} { e^{i\\theta}+ z\\over e^{i\\theta} -z} \\Re g(e^{i\\theta})\\, d\\theta.", "ec6a83054cb1552bba6b6b8fef6c675b": "\\quad i\\neq j\\quad", "ec6aad2aea9748a2bc9365291746683b": "dV= \\left(\\frac{\\partial V}{\\partial P}\\right)_{T}dP+\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}dT\\,", "ec6ac6ea1db309d59f62aef0f60892e9": "\\bar{\\lambda\\,\\!}", "ec6afd67ef6d50afbb01857b9eee7d59": "c_{2}", "ec6b25c07af17eb12e1b17e46dcd6689": "\\int \\langle x, (O-\\lambda I) y \\rangle G(y, z; \\lambda ) dy = \\delta (x-z). ", "ec6b54ee4238b549e1c2ff7667e90599": "\n(P+iD ) |x\\rangle = f(x) |x\\rangle\n\\,", "ec6b81686b942804f1613c6cc06c6547": "Sf([x,t]):=[f(x),t].", "ec6bb84a86daad61ff46865c767a74d8": "|\\tau_{n+1}^{(1)}| < tol", "ec6be6ddf8ce1151f874fa6f88e0845b": "IO_L\\ ", "ec6be77e72ccdc9709d0753298284e10": " dY_t = b(Y_t) dt + \\sigma (Y_t) dB_t + \\int_{\\mathbb{R}^k} \\gamma (Y_{t-},z)\\bar{N}(dt,dz),\\quad Y_0 = y ", "ec6c29d353e58753a2dc7d5aa8e5cdcf": "A(\\boldsymbol\\eta_1, n) = -\\frac{n}{2}\\ln|-\\boldsymbol\\eta_1| + \\ln\\Gamma_p\\left(\\frac{n}{2}\\right),", "ec6c412aca9bd5ebbc29e4443a39e529": "(I_n \\otimes L + M \\otimes I_n)", "ec6c54545f6452cf7dbb1485e67f11dc": "F=G/H", "ec6c7061f61892d7545a11456e35448c": "v_s>c ", "ec6cbe60b665fc179c6ddda0446dddce": " g(p,n) = \\frac{n}{p(1-p)} ", "ec6cf489648df056f83450be6d7c1194": "\\rho 0", "ec7c9bd92ec7771b5e6f9e66b1404954": "\\lim_{n\\to \\infty}\\varphi_{k,n}(z)=0", "ec7d2bbd7b06959bba0a04a4c71b8226": "\n \\begin{cases}\n q=(1-p) & \\text{for }k=0 \\\\ p & \\text{for }k=1\n \\end{cases}\n ", "ec7d5a2ac68b8dd4a63666179321a72b": "(x_1,x_2,...,x_j,x_i)", "ec7d6c9aba730c5eb5bced8424f75391": " A_\\mu ", "ec7d79e246b4968d7d6a2dd96427bb7d": "P, Q", "ec7d90b74be4c6b809529d982578896e": "g=\\mathcal Ff", "ec7dad1e1626d46b6649d05b30bc8140": "\\scriptstyle \\Delta P", "ec7db8bdf9a03ec4833639e3e6ecfbf4": "g^{(1)}(\\tau)=e^{-i\\omega_0\\tau-\\frac{\\pi}{2}(\\tau/\\tau_c)^2}", "ec7dc43e75c08c861852269a271faafb": "\\nabla \\mathrm{R}=0", "ec7e274eeba51558f0c3b1bd4c607484": "(4\\pi/k)~\\mathrm{Im}~f(0)", "ec7ee6b2f5ac5dcdcf3ad7980d6b55cd": "N \\gg 6 ", "ec7ef4cc0c92f0fb1b0aeedb0146ecdd": " r_2 = \\text {fixed rate at t of currency2. A set of rates for every t are fixed at time 0. } ", "ec7ef72dbb487b68f94f88ac081e5db9": "\\Sigma_1^0", "ec7ef828505aa1d2217d9ba0008611a3": "Q_2=\\frac{i}{2}\\left[(p-iW)b-(p+iW)b^\\dagger\\right]", "ec7f4ab987a1b85f9dc3d77b6ca72b8f": "\\omega = 0", "ec7f5c88eb6a8730c4526e2199c2d1b4": " X \\sim \\mathrm{Benini}(\\alpha,0,\\sigma)\\,", "ec7f81e23c3c2707cd3962366a341389": " \\frac{\\partial u}{\\partial t} + (u \\nabla) u = - \\nabla P + \\nu \\nabla^2 u ", "ec7fa01ee79ce057791975410eb1bf26": "\\{h[n]\\}_{n=-\\infty}^{\\infty}", "ec7fe11e20a6ac9b82e6ed8e4e50e788": " f^*(x) = \\sup_{x \\in B_x} \\frac{1}{|B_x|} \\int_{B_x} |f(y)| dy", "ec800ca9f64d049aad3fb46b23c08e1d": "\\exists^\\infty n\\in{\\mathbb N}", "ec8065fa13d385a3abefd2febfdab385": "\\sum_{k=1}^n\\frac{\\varphi(k)}{k} = \\sum_{k=1}^n\\frac{\\mu(k)}{k}\\left\\lfloor\\frac{n}{k}\\right\\rfloor=\\frac6{\\pi^2}n+\\mathcal{O}\\left((\\log n)^{2/3}(\\log\\log n)^{4/3}\\right)", "ec809ba4f82d55327b64739ee14bd5e3": "\\langle(\\delta\\vec{r})^2\\rangle_{vac}\\cong\\frac{1}{2\\epsilon_0\\pi^2}\\left(\\frac{e^2}{\\hbar c}\\right)\\left(\\frac{\\hbar}{mc}\\right)^2\\ln\\frac{4\\epsilon_0\\hbar c}{e^2}", "ec8122992f75ff07b4c27581836f6269": "N'", "ec814047242eccaa911c2c2d05adfd85": "\\sum_{n=1}^\\infty (-1)^n\\frac{1}{n}", "ec81754d6257a53ca33ccfcf27793443": "\\frac {1}{4 \\pi \\varepsilon_0}\\int \\bold{\\nabla_{\\bold {r_0}}\\cdot} \\left( \\bold{p} ( \\bold{ r}_0 ) \\frac {1}{|\\bold r - \\bold{r}_0|} \\right) d^3 \\bold{ r}_0 ", "ec817beffb49a12869333143955a4ac4": "p_1(x),p_2(x), \\ldots, p_r(x)", "ec818354a029a9445a7554a8bd3a510b": "R(r) = j_l(kr) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sqrt{\\pi/(2kr)} J_{l+1/2}(kr)", "ec81d1e9953f597fcb71d9a968b2c85f": "K(x,y)=\\sum_n \\frac{\\psi_n^*(x) \\psi_n(y)} {\\omega_n}", "ec82352b1f442d48e8f03457e311a009": "COP_{Carnot}=\\frac{T_{C}}{T_{H}-T_{C}}", "ec82400d0e8804a447e69a7767dc7659": "\\operatorname{pred} = \\lambda n.\\lambda f.\\lambda x.\\operatorname{extract}\\ (n \\operatorname{inc} \\operatorname{const}) = \\lambda n.\\lambda f.\\lambda x.\\operatorname{extract}\\ (\\operatorname{value}\\ ((n-1)\\ f\\ x)) = \\lambda n.\\lambda f.\\lambda x.(n-1)\\ f\\ x = \\lambda n.(n-1)", "ec82a7e99c202e1d41431b264be518ae": "F^{-1}(p) = \\sigma\\sqrt{3}(2p-1) +\\mu\\,\\, \\text{ for }0 \\le p \\le 1", "ec83184f30489987989f413478756fba": "[Q^\\dagger,b\\}=\\frac{dx}{dt}+iF", "ec8376475ad2ce03dbb0ddb1b838a8fa": "(\\alpha_i,\\beta_i)", "ec837c0181406bf607ccc0dffb9f59ff": "\\Delta = \\{ \\alpha_1 , \\alpha_2 , ... , \\alpha_n \\}", "ec84683d05ad311af9110a342a59f2e9": "\nD(A) = A_{1,1}A_{2,1}D(\\hat{e}_1,\\hat{e}_1) + A_{1,1}A_{2,2}D(\\hat{e}_1,\\hat{e}_2) + A_{1,2}A_{2,1}D(\\hat{e}_2,\\hat{e}_1) + A_{1,2}A_{2,2}D(\\hat{e}_2,\\hat{e}_2) \\,\n", "ec84710f5ac76236a07810ce01b4198e": "gF_i", "ec84ed980c6255da04427e25ef808a8d": "\\scriptstyle{it}", "ec85b54aea6469243b9f754f592dca7a": "(M, \\omega)", "ec85bb065719631f3153c6469bec6718": "A_{\\epsilon,Q} = \\sum_{i=1}^{N_\\epsilon}{\\mu_{{i,\\epsilon}_{Q}}{P_{{i,\\epsilon}_{Q}}}} ", "ec860e64241320e79933b6ec22753170": " M=3(N-1-j)+\\sum_{i=1}^j f_i.", "ec862d70d740b946eaf837c12df296cc": "\nL = \\frac{1}{2} m \\dot{r}^{2} + \\frac{1}{2} m r^{2} \\dot{\\varphi}^{2} - U(r) \n", "ec8669bf5a839e4744c760366abb420b": "\\|u-u_h\\|\\le C h \\|u''\\|_{L^2(a, b)}, ", "ec86acdd683578eac3a621670dcf2354": "1=E(0)=E(\\underbrace{x+x+\\ldots+x}_{p\\text{ of these}})=E(x)E(x)\\cdots E(x)=E(x)^p.", "ec86f01ecb1985104afeacd2b3411d3e": " (\\alpha\\to\\gamma)\\to (\\beta\\to\\gamma) \\to \\alpha\\vee\\beta \\to \\gamma ", "ec86f3179d3fe8fa30e97d34594fa8f6": "x \\cdot (y + z) = (x \\cdot y) + (x \\cdot z)", "ec86fc4aab9903432cd1c54ee30d8c0f": "2^{2^{2^{2^{2^2}}}} = 2 \\uparrow \\uparrow 6 = 2^{2^{65,536}} \\approx 2^{(10 \\uparrow)^2 4.3} \\approx 10^{(10 \\uparrow)^2 4.3} = (10 \\uparrow)^3 4.3", "ec871d3c71a9b07ca22267d6c75fafed": "B=-13.6", "ec878c15187c8ca962d99ef0a7daeb44": "\\{ \\Gamma_a^i , K \\} = 0", "ec87be84e98bb227e42662c9fb0d0877": "t_r = \\int \\psi_h \\psi_T \\mbox{d} t", "ec87d5cfc9c23b11915bc96781299a74": "\\phi'_X = \\alpha \\phi_X + \\beta_X \\, ", "ec8867a466444c05009bea52fc55de83": "f\\vee g", "ec8877530aad8dcd543215725ff02c6b": "\\frac{dv}{dt}=\\frac{dv}{dr} \\cdot \\frac{dr}{dt}=-\\frac{GM}{r^2}.\\,", "ec888350a762b65bc5545a637e945701": "\\dim(W) + \\operatorname{codim}(W) = \\dim(V).", "ec88916a4c5ec3f568a53b2245f9ef29": "S_\\nu(x) = \\operatorname{Im}\\, \\chi_\\nu (e^{ix}).", "ec88b6dbefd77b604a16e81f676c845f": "h^{4}(x)", "ec88bfd9458a59d8b0526124d21e2bee": "\\langle u, v\\rangle =0", "ec88fb0638856b98ace9e4879e90efa3": "r_{l}", "ec891dee2a6a9eecdfdc9a7bdfa8adfb": "\\begin{bmatrix}\n 1&2 \\\\\n 2&4 \\\\\n\\end{bmatrix} \\circ \\begin{bmatrix}\n 1&2 \\\\\n 0&0 \\\\\n\\end{bmatrix} = \\begin{bmatrix}\n 1&4 \\\\\n 0&0 \\\\\n\\end{bmatrix}", "ec89a6c5fa2c8159f3fe0bfe33cd6180": "\\begin{cases}\nA_l = B_r = \\sqrt{\\kappa};\\\\\n\\kappa = -\\frac{m \\lambda}{\\hbar^2};\n\\end{cases}", "ec89ab2ffd67961c66da699a0fb5ad44": "\\Delta E/c^2", "ec8a0ef6798afcdb696b1e7624121f98": "f(512, 404.2319) = -959.6407", "ec8a31e9e7be8e7535cbebf0830d1418": " m\\frac{\\mathrm{d}^2\\mathbf{r}(t)}{\\mathrm{d}t^2}=-\\nabla V[\\mathbf{r}(t)]. ", "ec8a407d2928552be6e9e472600257bd": " u_{i + 1/2} = 0.5 \\left( u_{i} + u_{i + 1} \\right), ", "ec8a6a77e9c7f5d95e14603f14ad39b7": " (x^\\mu,y^i,y^i_\\mu)", "ec8a732520f21c298fc2f8359b0df87b": "\\omega_ \\mathrm c = \\frac{1}{\\sqrt {LC}} ", "ec8aa8f48c130414610c7221c3289ad8": "0\\leq r \\leq 1", "ec8ad0754d21d047ac9d04d9f539d782": " \\left( r_{\\mathrm{O1}} + r_{\\mathrm{O2}} \\right) \\left( 1+g_{\\mathrm{m1}}(r_{\\mathrm{O1}}//r_{\\mathrm{O2}}) \\right) ", "ec8ae675b702a1a6a965599b4deb672a": "0 < \\lambda_i < 1 \\;\\forall\\; i \\text{ in } 1,2,3", "ec8aeac82b09b30078dc9eb346529ec6": " \\bigcup_{j\\in J} B_j \\subseteq \\bigcup_{j \\in J'} 5\\,B_{j}.", "ec8b5f5d95fe2488cdef6361608173c7": "\\left(\\phi_\\tau(\\omega) \\ne \\phi_\\Gamma(\\omega)\\right)", "ec8b9172e4d2ad262d89ceaf03f120fd": " \\phi_{r}\\,= \\phi_{R} ", "ec8b9b5c9a37551e33ccf9fc2f4b6234": "F(x;a,b,p,\\delta)= \\delta + (1-\\delta) {\\left( 1+{\\left(\\frac{x}{b}\\right)}^{-a} \\right)}^{-p} .", "ec8c06177f59c8b67d6fddc0564dbd9b": "\\sum_{i\\in I} m_i \\le \\sum_{i\\in I} n_i ", "ec8c249d7ce781e279deb6ea9ed14ceb": "\\mathcal{} L_{n+1} (\\pi_1 (X)) ", "ec8c447233811ce1876e4b55ee7c0fc2": " V_{bal} = \\sqrt{1746.4 \\cdot 9.80665 \\cdot \\tan( \\arcsin( 152.4 / 1511.3 ) )}", "ec8c9ede7f1b034d13d5ad032177f1f5": "P \\,=\\,A\\cdot\\frac{1-\\left(\\frac{1}{1+r}\\right)^{n} }{r}", "ec8ccb4dd9d73593e809a7fb367e9015": "\\displaystyle{H_\\varepsilon f \\rightarrow H f.}", "ec8cd97f3a4881b139fd6ec5b638e946": "x/m", "ec8d2e702b78475ce2c28bb3190ec904": "\n\\Delta \\Omega\\ =\\ 2\\pi\\ \\frac{J_3}{\\mu\\ p^3}\\ \\frac{3}{2}\\ \\frac{\\cos i}{\\sin i}\\ \\ e_h\\ (1-\\frac{15}{4}\\ \\sin^2 i)\n", "ec8da88eb01db82e9a3f4ddc33537b1b": "x^2+y^2 \\leq 1", "ec8dccf19cc39c18ad82d2d46726ecc2": "\\beta_p\\colon T_p M\\times \\cdots \\times T_p M \\to \\mathbb{R}", "ec8dd59020706a19cd7192d053a88e7d": "g(x,y)=\\varphi_{\\varphi_x(x)}(y)", "ec8e2bc7148cb5f0a72f13f34b8434fc": "\n\\begin{pmatrix}\n a & -b \\\\\n b & \\;\\; a \n\\end{pmatrix},\n", "ec8e48d1daf56f9c2f67b76c7faa8d66": "Q_I \\subseteq Q", "ec8e57d71f07e31203035548b79d03c8": "ST", "ec8ef2f738880b4022e96c1b15aefcea": "v_j = \\frac{1}{n}(f_0 + f_1\\alpha^{-j} + f_2\\alpha^{-2j} + \\cdots + f_{n-1}\\alpha^{-(n-1)j}).\\qquad (5)", "ec8f128d90857f1b6f6987dded2e749f": " (a,b) \\times (0,R)", "ec8f9b116359fd7e278342265eb00466": "y[n] = y[n-1] + c[n]", "ec8fbf77eff8a6c64a30524d98cb41cb": "\\ H = \\frac{1}{2}(\\frac{1}{R_1}+\\frac{1}{R_2})", "ec8fd9da1ff9ea04c625a6c899d87d68": "I_{mn}(x, y, z) = I_0 \\left[ H_m \\left( \\frac{ \\sqrt{2} x}{w} \\right) \\exp \\left( \\frac{-x^2}{w^2} \\right) \\right]^2 \\left[ H_n \\left( \\frac{ \\sqrt{2} y}{w} \\right) \\exp \\left( \\frac{-y^2}{w^2} \\right) \\right]^2", "ec90354191fb0b56033348aee3a03014": " S'(b,a^b)=(a,b_a)\\, ", "ec908ab6c59a7cb174695d82307bb567": "\\operatorname{SPACE}\\left(o(f(n))\\right) \\subsetneq \\operatorname{SPACE}(f(n))", "ec909db5e522e2ad7440d9a3e76fa0ae": "\\mu_n(cX)=c^n\\mu_n(X).\\,", "ec91206047ba8102927220c2793c1186": "|1{\\rangle}", "ec91403cc6f9c71be2b4926da354a5ff": "f(\\infin) = \\infin.", "ec924742c98d28d7eee368cb350407de": "\\frac{1}{4\\pi}", "ec92baf9ac4c98f112ecc6c87d79a20f": "I\\ A\\ a\\ b", "ec92c35dba5a72b692e68c9b1809fba4": "\\frac{1}{2}LI^2", "ec92ebb04c751672b8550e5b28774936": "\n\\int_{\\Omega} \\frac{\\partial}{\\partial x^{\\sigma}} \n\\left\\{ \\frac{\\partial L}{\\partial {\\phi^A}_{,\\sigma}} \\bar{\\delta} \\phi^A + \nL \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) \\delta x^{\\sigma}\n\\right\\} d^{4}x = 0\n\\,.", "ec9342cb56a765511229e994262f77fd": "d_A, d_B;", "ec93448fef8968424f1b952a618fb1b6": " N= 2^{[d/2]} ", "ec93aa84e2ee84c9f1e470289da034ea": "E(\\ln(1-x))=\\psi(\\beta )-\\psi(\\alpha+\\beta)\\,", "ec93ef1c9911440c878a0947854e5184": "- \\frac{G M}{r} \\,", "ec942160dba432efd0559c843f5e5103": "s_1=\\sum_{i=1}^m -\\frac{1}{a}\\log(1 - u_i)+m \\log k", "ec9474d81a04db79cabb9a010fe46eaa": " S=S_1 \\times S_2 \\times \\ldots \\times S_n ", "ec947a7cc943c84b1ef84958a7df827c": "OH", "ec94c1d874e5193be782f79bfc2536ab": "0,\\;1,\\;1,\\;2,\\;3,\\;5,\\;8,\\;13,\\;21,\\;34,\\;55,\\;89,\\;144,\\; \\ldots\\;", "ec94dd40e99f258a89d1c41366762115": "\\scriptstyle{R_{J}}", "ec9586230ec888825f8c740f0d591c85": "\\sec(y) = x \\ \\Leftrightarrow\\ y = \\arcsec(x) + 2k\\pi \\text{ or } y = 2\\pi - \\arcsec (x) + 2k\\pi", "ec9590a853cc6020c2d419248962dd6c": "L^2 = -\\hbar^2 \\left(\\frac{1}{\\sin\\theta}\\frac{\\partial}{\\partial \\theta}\\left( \\sin\\theta \\frac{\\partial}{\\partial \\theta}\\right) + \\frac{1}{\\sin^2\\theta}\\frac{\\partial^2}{\\partial \\phi^2}\\right). ", "ec959215314b2e64df3a313d28b34cab": "m+k+1", "ec95f3c64ae327c7ea139e91025ecee3": "\\frac{a*((a+1)*(a+2))}{1*(2*3)}", "ec9611ca6976b653670740d014153931": "C>0,N", "ec9695cd3905298ee7a6c72a2d426d31": "\\frac{i(p\\!\\!\\!/ + m)}{p^{2}-m^{2}}", "ec96bb94d2592035bd97a465e38df000": "\n\\left( \\mathbf{P}_{i}\\otimes\\mathbf{I}\\right) \\left( \\mathbf{I}\n\\otimes\\mathbf{A}\\right) \\left\\vert \\Phi_{n}^{+}\\right\\rangle =\\left(\n\\mathbf{I}\\otimes\\mathbf{A}\\right) \\left( \\mathbf{P}_{i}\\otimes\n\\mathbf{I}\\right) \\left\\vert \\Phi_{n}^{+}\\right\\rangle .\n", "ec96c292e7bf899f3d6e710ea56374db": "\\det {\\mathfrak{T}}^{\\alpha\\beta}", "ec96c554dd4ea84f11813a5f8bf836bf": "\\tilde{f}(\\xi),\\ \\tilde{f}(\\omega),\\ F(\\xi),\\ \\mathcal{F}\\left(f\\right)(\\xi),\\ \\left(\\mathcal{F}f\\right)(\\xi),\\ \\mathcal{F}(f),\\ \\mathcal F(\\omega),\\ F(\\omega),\\ \\mathcal F(j\\omega),\\ \\mathcal{F}\\{f\\},\\ \\mathcal{F} \\left(f(t)\\right),\\ \\mathcal{F} \\{f(t)\\}.", "ec97444b374a1d0da595b1473a90971d": "{\\tilde{C}}_{2n}", "ec977a9e2d3ee2f8338f0d9cd6d14e61": "e(\\varphi) = \\frac12\\operatorname{trace}_g\\varphi^*h.", "ec978e7c56cc949905b4f137a55f2694": "\\int_{0}^{1} \\tilde{P_m}(x) \\tilde{P_n}(x)\\,dx = {1 \\over {2n + 1}} \\delta_{mn}.", "ec97b3636eb0436a7e17739615544137": " z_k (k=1,\\cdots,N) ", "ec97c5fdc2b61fc3a1a28e2d66a94f78": "\\forall\\lambda\\in\\Lambda: W(\\lambda)", "ec9826bcffc00e2d4e989f9407ba0f03": " \\theta", "ec98a5f7804b753fdae6f7a5ae028dbb": "Cl_t = \\{x \\in U \\colon f(x,d) = t\\}", "ec98aa1c8928077b55d2987ba3e6628e": "O|q_j\\rangle=q_j|q_j\\rangle", "ec996783053b91e5fdd5fe11361b672c": "\\displaystyle{f(1)=1,\\,\\, f(g)={3\\over 4} (3z)^{-L(g)} \\,\\,(g\\ne 1).}", "ec9987df6af07929656753d6e0130b16": " \\left(\\frac{\\partial \\ln K_X}{\\partial P} \\right)_T = \\frac{-\\Delta \\bar{V}} {RT}. ", "ec9991bb0c031d1cce5ab9ae69f3671d": "n = \\lambda f.\\lambda x.f^n\\ x ", "ec99b8e7e4482be774901e032aaf6bea": "\\int_A e^{- \\varphi(x)} \\, \\mathrm{d} x < + \\infty.", "ec9a0114ec54ddbc9d1e47181fde9c0e": "\\pm Nmax", "ec9a4edeb36f115ba1a0df242b5858de": "\\{\\mathrm{butter, bread}\\} \\Rightarrow \\{\\mathrm{milk}\\}", "ec9a7959cb7314c62d650d48cc4cdf61": "x = 29", "ec9aa879b449e1a3954385eaeabdda72": "R\\mathcal S(\\mathcal F \\otimes \\mathcal G) = R\\mathcal S(\\mathcal F) \\ast R\\mathcal S(\\mathcal G)[g]", "ec9afa568aaf08aa009bcdb234ecc874": "k \\leq m", "ec9b06e0517ad421e8db210cf519ae71": "\\mbox{SU}(2) \\cong \\mbox{Spin}(3) \\cong \\mbox{Sp}(1)", "ec9b0f58a488b225dec52db26359858f": "u_3 = \\frac{(x_1^2+x_2^2+ax_3^2+x_4^2)x_7 - 2x_3(x_1 x_5 + x_2 x_6+ bx_3 x_7+ x_4 x_8)}{c}", "ec9b37b77158b1c92552290272c9b09e": "\\nabla_\\alpha V_\\beta^I = \\partial_\\alpha V_\\beta^I - \\Gamma_{\\alpha \\beta}^\\gamma V_\\gamma^I - \\Gamma_{\\alpha \\; J}^{\\;\\; I} V_\\beta^J.", "ec9b3e4079a8b2d98afb6d4b33b18227": "\\lambda^\\dagger \\ne \\lambda^{-1}", "ec9baadbdc92043ee2e21732fd68f87a": "\\exists i x \\} ).", "eca5900d4383aa818f0edb220e14cad3": "(A\\circ B)\\circ B = A\\circ B", "eca5d70f1d840882cad8bb485286d5ed": "B_l, B_r", "eca60f490aa3cd0165f42a75d5972166": "\n\\begin{array}{rl}\n & P\\left(X_{1}\\wedge X_{2}\\wedge\\cdots\\wedge X_{N}|\\delta\\wedge\\pi\\right)\\\\\n= & P\\left(L_{1}|\\delta\\wedge\\pi\\right)\\times P\\left(L_{2}|R_{2}\\wedge\\delta\\wedge\\pi\\right)\\times\\cdots\\times P\\left(L_{K}|R_{K}\\wedge\\delta\\wedge\\pi\\right)\\end{array}\n", "eca610e035bc452c6cd8ff7397db8728": "\\Gamma(H_1) = \\{ v \\in H_1^+ | \\phi_1(v) \\in \\Gamma(G) \\},", "eca64da656dac2fbcd512818121fedf2": " L_1 = \\frac{-( R_M + R_S)} {sL_M} \\, ", "eca716d153ceb97e640c7e681844b0ff": "\nF(\\mathbf{k})=\\int_{r=0}^\\infty\n\\int_{\\theta=0}^{2\\pi}f(r,\\theta)e^{ikr\\cos(\\theta)}\\,r\\operatorname{d}\\!\\theta\\operatorname{d}\\!r\n", "eca77a3f4752904c0afc5531de8c764d": " \\mathrm{SNR} = \\frac{\\mu_\\mathrm{sig}}{\\sigma_\\mathrm{bg}}", "eca7a6574c48fc70bc9b3619b36d0f99": "(P \\leftrightarrow Q) \\to (Q \\to P)", "eca8072edaa680da1e87873cbc3de5ec": "v = r", "eca8254fa162c02edf656ce331b233d6": " W_1 U_h W_1^* = U_h \\alpha^2 (g) \\quad ", "eca8bf1e45731e537ef75289709be499": "P\\Gamma L \\cong PGL \\rtimes \\operatorname{Gal}(K/k),", "eca8ee655aa4905953e1325299fd03f0": "\\Delta S = C_p \\ln\\frac{T_{final}}{T_{initial}}", "eca91c83a74a2373ca5f796700e99fd3": "p_i", "eca9275de43b33c751f48b293f5cbbe5": "(10\\uparrow\\uparrow)^2 10^{\\,\\!10^{10^{3.81\\times 10^{17}}}}", "eca93da60ee8ba7a88a1321ea565659b": "|S_{1}|\\cdot|S_{2}|\\cdots|S_{n}| = |S_{1} \\times S_{2} \\times \\cdots \\times S_{n}| ", "ecaa0c1c33f679fc9b66808f014cb2ac": "\\dot{\\sigma}(\\mathbf{x}) = 0", "ecaa18fc34605710fe936e4beda22842": "\\int_{\\partial \\Omega}", "ecaa66942683f5169450b0edef6c406d": "\\gamma_{xy}", "ecaa7ceae016c0ebaa1ef6d0f844a128": "d_H(X,Y) = \\inf\\{\\epsilon \\geq 0\\,;\\ X \\subseteq Y_\\epsilon \\ \\mbox{and}\\ Y \\subseteq X_\\epsilon\\}", "ecaaf9cd013e2b1383ef07d4cdbe3c46": "\\kappa \\geq \\omega", "ecab039bac5d8950ccff9829524f370e": "\\begin{align}\n\\sigma & = \\begin{bmatrix}\\mathbf{T}^{(\\mathbf{e}_1)} \\mathbf{T}^{(\\mathbf{e}_2)} \\mathbf{T}^{(\\mathbf{e}_3)} \\\\ \\end{bmatrix} \\\\\n& = \\begin{bmatrix} \\sigma_{11} & \\sigma_{12} & \\sigma_{13} \\\\ \\sigma_{21} & \\sigma_{22} & \\sigma_{23} \\\\ \\sigma_{31} & \\sigma_{32} & \\sigma_{33} \\end{bmatrix}\\\\\n\\end{align}", "ecab3618beae0654f97cbab158525a90": "n_i \\in N", "ecab4d6755a4f00bc323c97f18a06c07": "A=\\begin{pmatrix} a+b & x+iy \\\\ x-iy & a -b\\end{pmatrix}", "ecab7eb2f3bef1382a61a5dd6d8488d9": "\\forall A, \\exists B, \\forall C, C \\in A \\rightarrow F(C)\\in B.", "ecabf9a08043e90a7624a359c1b9946e": "\\theta=\\pi/2", "ecac5cbd755b51bea051b466ac72bf64": "f_{ik}= f_{jk}\\circ f_{ij}", "ecacafe6717ea3832198f0049dd686a3": "\\Gamma(z) = 1+\\sum_{k=1}^\\infty\\frac{\\Gamma^{(k)}(1)}{k!}(z-1)^{k},", "ecaced15934d3891289fae24a14f07f1": "\\pi_*: \\operatorname{H}^k(E) \\to \\operatorname{H}^{k-m}(B).", "ecad2ed82d38b98f5f68dfa0d89119a6": "\\Box(\\Box A\\to B)\\lor\\Box(\\Box B\\to A)", "ecad3af5be6131f12f435ff9932dbb49": "\\Delta T (z) = (1-R) \\frac {Q} {C(\\zeta A)} exp(-z/\\zeta)", "ecada1ec7071654b9e6c87f774e4eb3c": "A^\\text{WSM-score} _1 = 25 \\times 0.20 + 20 \\times 0.15 + 15 \\times 0.40 + 30\\times 0.25 = 21.50. ", "ecada9db3e46136ea62ae7053aa92f41": "2^{m-1}", "ecaded65753d38089377e1db478d3528": "\\int\\frac{x^2}{(ax + b)^3} \\, dx= \\frac{1}{a^3}\\left(\\ln\\left|ax + b\\right| + \\frac{2b}{ax + b} - \\frac{b^2}{2(ax + b)^2}\\right) + C", "ecae3d904d5bacddf1b16f1b7d7356fb": "\\pm\\left(0,\\ 2\\sqrt{\\frac{2}{3}},\\ \\frac{-2}{\\sqrt{3}},\\ \\pm2\\right)", "ecae61c9d5e061be4aef41be4eab3ae0": "VII_0", "ecae6d5654bcdf4d075465d038b56c42": "\\mathbb E( e^{-\\theta F} )= \\frac{1}{2 \\lambda}(\\lambda + \\mu + \\theta - \\sqrt{(\\lambda + \\mu + \\theta)^2 - 4 \\lambda \\mu})", "ecaeb9b47e73ad24bdce26d77a97e971": " \n\\lim_{t\\rightarrow\\infty} \\frac{Q_n^{(c)}(t)}{t} = 0 \\text{ with probability 1} \n", "ecaf4ccef97b7bc6d0388f19b3062a3d": "a_n = kn", "ecafe2772b17167b0dfc88b4644b5d84": "\\sqrt{\\frac{\\hbar{}c^5}{8\\pi G}}", "ecb0355523733054b5fc952d36690d6a": "\\begin{align}\nk_1 &= f(y_t, t) \\\\\nk_2 &= f\\left(y^1_{t+h/2}, t + \\frac{h}{2}\\right) \\\\\nk_3 &= f\\left(y^2_{t+h/2}, t + \\frac{h}{2}\\right) \\\\\nk_4 &= f\\left(y^3_{t+h}, t + h\\right)\n\\end{align}", "ecb076fb580afd8e39b81d68a1557340": " U = \\frac{\\varepsilon_0}{2}\\int \\limits_{\\text{all space}} \\mathbf{\\nabla}\\cdot(\\mathbf{E}\\Phi) dV - \\frac{\\varepsilon_0}{2}\\int \\limits_{\\text{all space}} (\\mathbf{\\nabla}\\Phi)\\cdot\\mathbf{E} dV", "ecb07d89465259b844bdb743afaefc20": "\\operatorname{mr} (G)", "ecb0921ad5986cfd3ab1af22b18faaf0": "\\nabla^2 V = 0", "ecb0e3abe49bccd5ff4a1729fd4aa7e9": "f: U \\rightarrow V\\qquad", "ecb135546808fedf95fb569cd12cbe58": "\\mathbf{v} = \\frac{1}{1-x_0}\\left(x_1,x_2,x_3\\right).", "ecb15aa0ac49cc6751296a83c126fdf2": " \\mathbf {P_{\\nu\\mu}} ", "ecb15b7d17ff772389f47933ddc512ed": "\\lim_{n \\rarr \\infty} p_{ii}^{(n)}", "ecb1a0fec3afaaa970e2953f0b9b5ba6": " \\prod_{p} \\Big(\\frac{1+p^{-s}}{1-p^{-s}}\\Big) = \\prod_{p} \\Big(\\frac{p^{s}+1}{p^{s}-1}\\Big) = \\frac{\\zeta(s)^2}{\\zeta(2s)} ", "ecb1c6cc9d4b1ec3721a85e7eda0f2a0": "k = \\frac{1}{2}", "ecb1f85005637b237bd50e41f0979812": "\\begin{array}{lcl}\nT(r,fg)&\\leq&T(r,f)+T(r,g)+O(1),\\\\\nT(r,f+g)&\\leq& T(r,f)+T(r,g)+O(1),\\\\\nT(r,1/f)&=&T(r,f)+O(1),\\\\\nT(r,f^m)&=&mT(r,f)+O(1), \\,\n\\end{array}", "ecb24cb207fc2e23c74a0e96528e9b42": "\\ U", "ecb254c3983f2ffb0299d08fb5b9e041": "\\nu(A) = \\int_A f \\, \\mathrm{d} \\mu", "ecb2931ae3e5b9003e63a34f04e9f687": "\\omega_{c}=0", "ecb2ac7ff9686b995bc75ac16479b0e2": "T_{ijk\\dots} = -T_{jik\\dots} = T_{jki\\dots} = -T_{kji\\dots} = T_{kij\\dots} = -T_{ikj\\dots}", "ecb2c5124f927e42299f611a7d3622e6": "\\omega_1 = \\mu B_1/\\hbar", "ecb2ca1dc06e54abe55796303ae852fc": "e_1,\\dots,e_j", "ecb305ca5e10da88afcd53ac9f865844": "\\, \\eta_{ab}", "ecb3137d5cde1bf3391b31157d636e1d": "\\partial S ", "ecb3260e588384f3ef2019f39defc93b": "\\displaystyle \\hat{r}(x,s)=\\begin{cases}\n1 & \\text{if by stage } s, x \\text{ has been enumerated into } 0'\\\\\n0 & \\text{if not}\n\\end{cases}", "ecb3498a1afef765e7048db8bfcffe36": "A(x_1)", "ecb41777017658a854723ce9366f6361": "\\theta = \\frac{2}{3}\\sum_{i=1}^3 \\frac{\\theta_i}2", "ecb4b96a93710b9d31f1cfa010ace4d5": "|z|>|a|", "ecb4c00e87546bc856a4767e3413aef0": "V_1 = q/y_1 \\quad and \\quad V_2 = q/y_2", "ecb4c35d8bf4d7be727dd3851e6a9ec6": "\\sum_{n=1}^\\infty \\frac{1}{2^n} = \\frac{1}{2}+ \\frac{1}{4}+ \\frac{1}{8}+\\cdots.", "ecb4f05c60c0e2518269f58a210648be": "\\hat{I^2}", "ecb50655b258b013e2e030e07cae5c8c": "T(\\mathbf{true})\\ \\Leftrightarrow\\ \\mathbf{true}", "ecb5296fb7c2f0b24152946e765d3149": " \\alpha_i \\centerdot b_i ", "ecb53ca09a2c1b49146ced19c7c03e27": "\nq_\\mathrm{tot} \\equiv \\sum_{i=1}^N q_i , \\quad P_\\alpha \\equiv\\sum_{i=1}^N q_i r_{i\\alpha} , \\quad \\hbox{and}\\quad Q_{\\alpha\\beta} \\equiv \\sum_{i=1}^N q_i (3r_{i\\alpha} r_{i\\beta} - \\delta_{\\alpha\\beta} r_i^2) ,\n", "ecb5957b16ba0696c0fcb44a3061686e": "f(n)=O(g(n))", "ecb5fa9193b428100eee33672e4b0428": "R^i f_* \\mathcal{F} = 0, \\quad i > r.", "ecb60b1f3743a8ec22f9f954b2bb8086": "S_{ab} = R_{ab} - \\frac{1}{4} \\, R \\, g_{ab} ", "ecb6906c1b1dfc11c49727a2f42d30c0": "\\operatorname{vec} (\\hat{\\bold{H}} - \\bold{H}_{\\operatorname{AMISE}}) = O(n^{-2\\alpha}) \\operatorname{vec} \\bold{H}_{\\operatorname{AMISE}}.", "ecb6b0fbd5b0c7678638b0421fc05581": "\\int_{-1}^1 f(x) \\, dx \\approx \\sum_{k=-\\infty}^\\infty w_k f(x_k),", "ecb7b1a3febcf233bc56ce0e13179b26": "\\mathbb H P^n.", "ecb7c070c963226dfb16130e6b122927": "\\mu(Av_1,\\ldots,Av_n)=|\\det A|^s\\mu(v_1,\\ldots,v_n), \\quad A\\in GL(V).", "ecb7ed774b60745facc0751307749b57": "(Y,\\mathcal B)", "ecb7edb69e222411a08ec22806b69a56": "R = r + a", "ecb890205180ef256dc9d982587978e3": "\\bold{v}_1\\cdot\\left( \\bold{v}_2\\times\\bold{v}_3 \\right)", "ecb8b12f952b625625010d9d3a2e4e43": "\\mu(\\langle N\\rangle,T) = \\left(\\frac{\\partial F}{\\partial \\langle N\\rangle}\\right)_{T},", "ecb8b2993df62a78f871bb0578843089": "\\text{change open}_0", "ecba2f0c35ec97eacf96cd25843256e5": "(0.5)*u(x_1)+(0.5)*u(x_{3})=p_2*u(x_{2})\\!", "ecba924e8740c0367b328014a4c9d44a": "C^{(V)}_T(V,T)\\,\\left.\\frac{\\partial p}{\\partial T}\\right|_{(V,T)} \\geq 0\\,.", "ecbac73a34fc50951e20b05e83c19ea4": "[g]", "ecbb63ec53984b1602f3de0518bcf259": "\n\\{\\phi_1, \\phi_2\\}_{PB} = - \\{\\phi_2, \\phi_1\\}_{PB} = \\frac{q B}{c},\n", "ecbbf8f255f8b96bae3cce0d4b056643": "B\\in\\mathcal{P}(\\kappa)\\,", "ecbc35bff2ba8e2d30690e2413f62a72": "\n v_{1} \n=\n\\sqrt {2 \\hbar \\omega_c \\over m_1}\n ", "ecbccf1d1a64c6133cd4e54f2feebafa": "H(n) \\sim A n^{(6n^2 + 6n + 1)/12} e^{-n^2/4}", "ecbd073b617dd15087ddf344b8326079": "e^{\\mu \\theta}", "ecbd0f8fbbc5ff61e5d2b5edc66ff2d8": "G=\\sum_i \\mu^0_i +RT \\ln m_i\\gamma_i", "ecbd57d7656d6370be36861a7b0cf51c": "\nJ = \\int_0^T p(t) {dx \\over dt} dt\n\\,", "ecbd7674309b7c71d9efed64369cb51f": "n \\text{ realises } \\phi", "ecbddd7d70f301fa3ac6d00b583051a4": "\\textrm{lfp}(f) = \\sup \\left(\\left\\{f^n(\\bot) \\mid n\\in\\mathbb{N}\\right\\}\\right)", "ecbe51d1013cac5cb6a9b4daf9255959": "\\hat{X_i} = \\frac{\\lambda_i}{\\frac{N_0}{2}(\\frac{N_0}{2} + \\lambda_i)}", "ecbe9b754cedd57c36bbb648f01d8df1": "\\Omega_{X/Y}", "ecbf2a0119e364a08fa42911ed2c2fca": "B_1,\\dots,B_n", "ecbf7689362bc7bd6f4616f5575de7ee": "\\phi_1(y_1) = \\frac{1}{(\\epsilon^22\\pi)^{1/4} } e^{-y_1^2/4\\epsilon^2}", "ecbf7deb202d94171f3af0010677ab3c": "(\\sigma_1, \\sigma_2, \\sigma_3)\\,\\!", "ecbfc8b273fa77648fa65464a339cbcf": "\n|\\nabla \\phi_i|^2 + t \\phi^2 + \\lambda (\\phi_i^2)^2 \n\\,", "ecbfd4609b6abbd5384ac3d566e29ea3": "\\frac{C}{D}", "ecbff0518a2bd5bcfb86190483d45683": "\\Phi_D = \\int_S \\mathbf{D} \\cdot \\mathrm{d} \\mathbf{A}\\,\\!", "ecc01e8b4a907d65d985a4fb0eee583a": "A_4", "ecc026469b4030d407f42ba0491aa260": " \\left(x^2+y^2-2ax\\right)^2 = 4a^2\\left(x^2 + y^2\\right).\\,", "ecc045f9ab1f4dfc0007a7a04cefa5b8": "\\frac{0}{0}", "ecc08ebcf57885aaa642e7852e73d593": "\\mathbf{H}_n", "ecc091ec193bc2f8aadc439636a9db95": "\\mu_\\lambda={\\sin ({A+D_\\lambda \\over 2}) \\over \\sin ({A \\over 2})}", "ecc0e8b0a2f874f619b47a9fbfe820ea": "\\mathcal{L}_k B_m = \\frac{1}{k^m}B_m", "ecc0f616b95791f0839d6a828646109e": "B^{(b-1)/2}\\equiv +1 \\pmod b\\;", "ecc0fc7183b7c455040254fce9c453be": "x^2 + y^2 + a x + b y + 1 = 0,\\,", "ecc110f6695b611601ce57d0ef69999c": "P(s,n+1)-P(s,n) = (s-2)n + 1\\, ,", "ecc1240d275a2a9e5fd75e8977631005": "Q(q) = \\frac{\\hbar^2}{4m}\\{S(q),q\\}", "ecc175b7fe6d09e58b169826d0259cdd": "Z = \\int_{C}\\mathrm{e}^{-I[g_{\\mu\\nu},\\phi]}\\mathcal{D}\\bold{g}\\, \\mathcal{D}\\phi", "ecc17d113397fe9ee03cfe511289cd8d": "\n\\frac{I(\\psi)}{I(0)} = 1+\\sum_{k=1}^N A_k \\, (1-\\cos(\\psi))^k\n", "ecc1967b4b81f01bc1978b58e993017b": " f_y=g_x.\\,", "ecc1cb4e2706f1972bf576cab489d1a6": " \\mathcal{L}\\left\\{\\frac{\\operatorname{d}}{\\operatorname{d}t}x(t)\\right\\} = s X(s) ", "ecc1e65890d53fabe5e8adb1ae81ea2a": "\\mathbb{P}(A\\mid E) = \\frac{\\mathbb{P}(E\\mid A)\\cdot \\mathbb{P}(A)}{\\mathbb{P}(E)}", "ecc25421475f1cfde1e16dfdbb0c8321": "\\dot{V}_x=\\frac{\\partial V_x}{\\partial \\mathbf{x}}(f_x(\\mathbf{x})+g_x(\\mathbf{x})u_x(\\mathbf{x})) \\leq - W(\\mathbf{x})", "ecc280cdda84c9ab2654941748808607": "(\\Lambda-\\mu I)^{-1}", "ecc2991d1b39feee3a2e620ea95a045f": "s_1\\cdots s_N", "ecc34df609e0aaa78103b6351ff8e52b": " \\max r_1, r_2 \\rightarrow \\min r_1, r_2 ", "ecc3692032ea5f958fdc8063e72e2e22": "\\mu,", "ecc377593210f5cf5329677573075c6e": "B_{i}", "ecc3c24b5c01e9cb925f222b7d5d9694": "\\phi\\colon X^{\\text{op}} \\to Y", "ecc3e4586557334a2d1832552f74241c": "C_3={\\mu\\over{a}}\\,", "ecc4177eaf89fd66594890ec44252702": "\\mathcal{R}(M)=R^{\\mathrm{irr}}(M)/SO(3)", "ecc41a95cbc8a944480231d7d3b480be": "f(z) \\rightarrow 0", "ecc4291e1795f6c95f8bc4ef20afcae0": "ap_2q_2 + cp_1q_1 = bp_1q_2 + dp_2q_1", "ecc45f9f8b45599ff55f3987d7146f1f": "\\frac{\\partial \\epsilon}{\\partial z}+i\\beta_2\\frac{\\partial^2\\epsilon}{\\partial t^2}=i\\gamma P \\left(\\epsilon+\\epsilon^*\\right)", "ecc47e6bff929d0f13d914b29d074e37": " \\text{subject to} ", "ecc494ff7717f4c149c9f2edeaf6bac4": " c_H = -T \\left( \\frac{\\partial^2 f}{\\partial T^2} \\right)_H. ", "ecc4a2880b70bfd5d41d3962a4466b9e": "v(S) = 1 - v(N \\setminus S)", "ecc4f4b05504e22ec761671afdb5d413": " B^{\\lambda N}", "ecc540044fe1f8d323c220873dee78ea": "z_i = \\mu + \\frac{1}{2}g_i + e", "ecc5a7389202334f21d7dfeb27956065": "\\tan \\theta", "ecc5ab2239135e1c1e292891ec156aaf": "x=R_H", "ecc5b0e08dee2b8fc01ed2875dfeb538": "w_{line}", "ecc604537bf9944e91b9df33365e437b": "w'\\left(\\eta\\right)\\,", "ecc61687105d5d1111ee3f7c9297a4ee": "\\gamma(x,y)=\\gamma_i(h).", "ecc676b37d23c08b6638267d315b718f": "\\mathbb Z_3^7 \\times \\mathbb Z_2^{11},\\ ", "ecc6a46dcef20b64e04d73695b955692": "df=\\frac{4\\theta \\mu}{\\sigma^2}", "ecc6eb8bfdbd46ba22a4b2fb7e84b470": "T = \\sqrt{ \\frac {3\\pi}{G \\rho} }", "ecc707a6b8c63972a591c741411c8ed9": "U(\\{c(t)\\}_{t=0}^\\infty)=\\int_{0}^\\infty {f(t)u(c(t)) dt}", "ecc73819c8df788634d22a61ad7e493c": "C^{LS}", "ecc74f468a01f37cbb9f012389fae5db": "\\mathcal{A}\\cup\\mathcal{B}", "ecc759a2b66d44522c03da60779421bc": "\\sqrt{\\frac{3(3m^2-7)}{4m(m^2-4)}} \\sigma", "ecc76d99e885d6a8a4a148ff420f8bfb": "B_{bulk}", "ecc786dea7072f5495845ff5366271ce": "\nK(x-y,\\Tau) = e^{-\\alpha \\Tau} e^{-(x-y)^2\\over \\Tau}\n\\,", "ecc7b1cfd4a4345577e20a7ddd842f64": "\\mathbf{C}^2 \\times \\mathbf{C}^2/\\{\\pm 1\\}", "ecc826ce4357b5a450abed35c33fee43": "\\left(\\begin{smallmatrix}a & b\\\\c & d\\end{smallmatrix}\\right)", "ecc89bc656b1c053455af54017d5dbd2": "\\hat{r}=r_x \\cdot \\hat{x} +r_y \\cdot \\hat{y} +r_z \\cdot \\hat{z}", "ecc8b648d33cbd2b3af0f8cbb95904ef": " \\rho\\, _{liquid} \\gg \\rho\\, _{vapor} ", "ecc904862d7993405a7e13f81da1f724": "\\cos x = \\operatorname{Re}(e^{i x}) \\,", "ecc92e7b0f7abdb7ffc534edd241f54e": "y'(x) = f\\left(x,y(x)\\right)", "ecc97ce9b2b17e5d27e1a5d27c664543": "r_{B,h}(n) = \\#\\{(a_1, \\cdots, a_h) \\in B^h | a_1 + \\cdots + a_h = n \\}", "ecc9a32c3afa8fdffae9ac4eac75dfb3": "\\textstyle \\sigma(E)", "ecc9bbc1858051467790d9e330713f3c": "1-\\cos\\alpha", "ecca3c5a6b5b7cbd578ccd803de7cfe8": "L_{n}", "ecca49203ff19e81874c87fe3b58c111": "\n{M}=\\sqrt{5\\left[\\left(\\frac{p_t}{p}+1\\right)^\\frac{2}{7}-1\\right]}\\,\n", "ecca6160accc3de7ba546ea0a351977a": "\\check{H}^*(X;\\mathbb{R})", "ecca6dbdcb359e9460bf356ba5a58745": " X\\subset\\mathbb{P}^n ", "ecca86dcf909829dbb7d256959b4708c": " E\\left[\\sum_{x\\in {N}}f(x)\\right] \\qquad \\text{or} \\qquad \\int_{\\textbf{N}}\\sum_{x\\in {N}}f(x) P(d{N}), ", "eccab4de0b42bb3ea014df442b16c0d3": "\\prod_{i}{{b_i+p_i-1}\\choose{p_i}}.", "eccaeb6da32408170852c272ac2abac4": "\n\\left( y(1-y) \\frac {\\partial^2} {\\partial y^2} + x \\frac {\\partial^2} \n{\\partial x \\partial y} + [c - (a_2+b_2+1) y] \\frac {\\partial} {\\partial y} - \na_2 b_2 \\right) F_3(x,y) = 0 ~.\n", "eccb21ff82dbb7cad64c50638033ebd0": "P_\\alpha(x)", "eccb76a8a6dd65e475cdb2640ed0ea24": " \\triangle Z_t = (1-L) X_t,", "eccba8d411f1718eced3981dda97dc7f": "n = k\\;", "eccbbd7ff2044da5056de1258e500dde": "a_{\\mathrm cU}", "eccbc87e4b5ce2fe28308fd9f2a7baf3": "3", "eccbce84f0e34567d6769c19c4bd6ee9": "\\rho=5, \\ \\phi={\\pi \\over 9}, \\ z=3", "eccc5be504c57d5c58f4368d6dc996f0": "\\sin nx = 2 \\cdot \\cos x \\cdot \\sin ((n-1) x) - \\sin ((n-2) x) \\, ", "eccd3fadf7355158056bf85b027292df": "\\Delta S_{mix} = -nR[x_1\\ln x_1 + x_2\\ln x_2]\\,", "eccd44a8591a23207823ea8f72bef1ca": "\\textstyle \\sum_{\\{i\\mid e_j\\in S_i\\}} x_i \\ge 1", "eccd4d0bd63d443a89b98ff395afe78c": "L(n)\\leq 0", "eccde5a70e327c90804514a5fc5f6ac7": " \\Pi = U - V ", "eccde999644226c586a07af009367cdc": " R' = \\langle R \\rangle \\left[ 1 + \\frac{d \\langle R \\rangle}{dQ} \\frac{Q}{\\langle R \\rangle} \\right] ", "eccdfa209d7cc30dc784ffd5eaa9cb32": "\\chi\\;", "eccdff012ec992e75bfb3ee7394c5231": " {\\Omega^1}_2 = {R^1}_{212} \\, \\sigma^1 \\wedge \\sigma^2.", "ecce115eb7ac339a6519e41b5f83a66a": "m u_x", "ecce26c4666f143605fff35bbf52c647": "= (1/4) \\cdot \\sum_{e \\in E}(f_{e})^2 + \\underbrace{(1/4) \\cdot \\sum_{e \\in E}f_{e}b_{e}}_{\\geq 0}.", "ecce42440e0d94b0a8fff7dfc9ac1670": "\\begin{align}\n a &= \\tfrac13\\delta^{-2} &= 7.787037\\ldots\\\\\n t_0 &= \\delta^3 &= 0.008856\\ldots\n\\end{align}", "ecce4da4083d8ef98c11c3385ba7aae7": "\\mathrm{Nat}(h_A,h_B) \\cong \\mathrm{Hom}(A,B).", "ecce8d76c5205169e51b35137028e3b8": "change\\_open_t", "ecceb4735e90deb3e4c68aec2010d3bb": " x(t) = Xe^{j\\omega t} = |X|e^{j(\\omega t + \\arg(X))} ", "eccef88a0feed9574e0daf7be6e6e4db": "\\mathbf \\phi_i", "eccf27c976c0cc9dd1a579f87ee17e08": "k_{\\rm A} = \\alpha_{\\rm L} \\cdot \\alpha_{\\rm B}\\;", "eccf4c3860d1aac17b77f41eab9f80e1": "C_Z", "eccf561c3cc554ce068f0b6bbf9e511d": "V_{2k+1}=(D\\cdot U_{2k}+P\\cdot V_{2k})/2", "eccf867b1328ae0fc38aac20fafab19d": " {d \\over d\\tau} {\\partial f \\over \\partial \\dot x^\\lambda} = {\\partial f \\over \\partial x^\\lambda} ", "eccfd2c3b36c0c00d430e6182f54c5ae": "\\approx\n\\ln\\left(\\frac{\\zeta(\\alpha)}{\\tau^\\alpha}\\right)\n-\\frac{\\zeta(\\alpha)}{2^{\\alpha}\\tau^\\alpha}", "ecd03dd0f388d141cc51671f23694f4c": " Q_B |\\Psi \\rangle = 0 ", "ecd06598fab41bc2ff58376ef5834192": "H^-", "ecd084b6fae72f589745d6e0b0861010": "\\nabla \\times \\textbf{H} = j \\omega \\epsilon \\textbf{E} + \\textbf{J}\\,", "ecd0f934685bec460cc4c9838fb88ce9": "{TWG \\over 2} - {PB}", "ecd16ad23d5b3fdcb35fc7661de92d50": "\\sigma^2\\left(\\sum_{i=1}^n a_ix_i \\right) = \\sum_{i=1}^n a_i^2\\sigma^2(x_i) + 2\\sum_{i,j\\,:\\,i0", "ecdd0002174a6510a5008256e889d067": "|\\Phi^\\pm\\rangle = \\frac{1}{\\sqrt{2}} (|0\\rangle_A \\otimes |0\\rangle_B \\pm |1\\rangle_A \\otimes |1\\rangle_B)", "ecdd0efc43394cc60217cff2fc305388": "\\sum_{j=m}^\\infty \\frac 1 {\\binom j k}=\\frac k{(k-1)\\binom{m-1}{k-1}}", "ecdd52a1c74500c1ec344907e3ed654b": "\\displaystyle{\\tau(a,a)=\\sum \\lambda_i^2 \\mathrm{Tr}\\,L(e_i) > 0.}", "ecdd5877d57e6f28391a24f4b3eed8f1": "(\\Delta S=0)", "ecddccea45f1c0b3877390122e2bb410": "y = \\frac{1}{C-x}", "ecdde54d3921f8aa02d7614625f9f928": "S_k = C_2S_{k-1}", "ecddf5af4980c015ac0c4a2bc0e85065": "\\frac12\\int_a^b \\left[r(\\varphi)\\right]^2\\, d\\varphi.", "ecde3592c66090b35dadec51bc92e3c9": "\\angle Z = \\tan^{-1} \\left(\\frac{X}{R}\\right)", "ecde822b696af943091c760fc07a8078": "\\tau = \\Delta^{-2} Dt", "ecdec87b1fca69b5232fac15ea261de8": "\\mathbf{G}(s) = C(s\\mathbf{I} - A)^{-1}B + D. \\,", "ecdedeaa2849bc8cfa9b3e09562398b7": "\\frac{P_2}{P_0} - \\left( \\frac{P_1}{P_0} \\right)^2", "ecdeeb05149d44a74a33be249714eb78": "\\mathbf{r}_n", "ecdeededa277b0e0155642caa83cd331": "[X]^{<\\omega}", "ecdf030cd631de6c8f4d9d63b069115c": "{n}_1,\\dots,{n}_m", "ecdf196f0a56724fd63a0b4b218c7deb": "\\operatorname{var}[\\ln (X)] = \\operatorname{E}[\\ln^2 (X)] - (\\operatorname{E}[\\ln (X)])^2 = \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta) ", "ecdf7dc2bc2e53af3ab8055f7f9875a5": " \\frac{1}{\\sqrt{4\\pi}}\\mathbf{D} ", "ecdf864d1547f146abc6b1012afa6343": "\\scriptstyle s(x)", "ecdfd10506f1a805d4f9b5438639c542": "a^{\\dagger}_{\\alpha}", "ecdffc572dea52c5e0c65fdb395dac41": "P_iP_j=-P_jP_i", "ece1002d0bc4d45e53744697a36fa21f": " f(k;n,p) = \\Pr(K = k) = {n\\choose k}p^k(1-p)^{n-k}.", "ece168fc626cb23e08946cc107e32293": "\\mathcal{C}_{-n}(S_\\ast(X),S_\\ast(B))=S_\\ast(X)\\otimes \\underbrace{S_\\ast(B)\\otimes \\cdots \\otimes S_\\ast(B)}_{n}\\otimes S_\\ast(B).", "ece19b3568b5d51bae7da7ad1b6e5adc": " (x\\ ,\\ y)", "ece1aace83059aec36ef298c91325c0e": "\\mathsf{cap}(\\mathbb{Z}) = \\mathsf{H}\\left(\\frac{1}{1+2^{\\mathsf{s}(p)}}\\right) - \\frac{\\mathsf{s}(p)}{1+2^{\\mathsf{s}(p)}} = \\log_2(1{+}2^{-\\mathsf{s}(p)}) = \\log_2\\left(1+(1-p) p^{p/(1-p)}\\right) \\; \\textrm{ where } \\; \\mathsf{s}(p) = \\frac{\\mathsf{H}(p)}{1-p}.", "ece1ba9bba3e4ceab0fb9547a3adb683": "\\bigl(\\tfrac{n}{p}\\bigr)=1", "ece1fe5c20dce8a890ab37c80f06cb21": "Y_{22} = {Z_{11} \\over \\Delta_Z} \\,", "ece221ec497faa5d152282a7dea2244b": "\\mathit{w_H(.)}", "ece280415dbe960c73c1ae651f08c2d0": "NER = 100 \\ \\frac {Answered\\ Calls \\ + \\ User\\ Busy \\ + \\ Ring\\ No\\ Answer \\ + \\ Terminal\\ Reject }{Seizures} ", "ece28c058023ad258a27fec86a00fb13": "O(V^3)", "ece2ac7a7236c32a63370be232091070": "{\\mathbf v} = \\begin{bmatrix}a\\\\b\\end{bmatrix} = a {\\mathbf e}_1 + b {\\mathbf e}_2, \\quad {\\mathbf w} = \\begin{bmatrix}c\\\\d\\end{bmatrix} = c {\\mathbf e}_1 + d {\\mathbf e}_2", "ece2ba753a382edd1ee9712e924d0213": "w = -1 \\Rightarrow \\rho = const ", "ece337cccfa4a694d2373033636786c4": "M(\\lambda)=\\frac{\\partial^2\\Phi}{\\partial A\\partial\\lambda}\\approx\\frac{\\Phi}{A \\Delta\\lambda}", "ece37db2427459763955834381136a84": "m[0]=0\\,\\!", "ece39c426d89f631a957e14df069bc10": "T(s)", "ece3d5f107d114b0ae6b3ffd4e9502f9": "\\sum_{m=1}^\\infty x_m = s", "ece40f00c35e525ea3d42e1c3a8af006": "\n\\varphi = \\int \\frac{dr}{r^{2} \\sqrt{\\frac{1}{b^{2}} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\frac{1}{r^{2}}}}\n", "ece43f1894479d16892f09851ee609ba": "h = O(\\ln(n))", "ece49e1aaf254e76282b134857ccd0d6": "\\chi^2(k)\\!", "ece535536cdc7ee835d471ae5847f1e1": "f_{n,k}(r)", "ece5615aa97e724c9b1fd47eac2985d0": "\\mathfrak{A}~ \\mathfrak{B}~ \\mathfrak{C}~ \\mathfrak{D}~ \\mathfrak{E}~ \\mathfrak{F}~ \\mathfrak{G}~ \\mathfrak{H}~ \\mathfrak{I}~ \\mathfrak{J}~ \\mathfrak{K}~ \\mathfrak{L}~ \\mathfrak{M}~ \\mathfrak{N}~ \\mathfrak{O}~ \\mathfrak{P}~ \\mathfrak{Q}~ \\mathfrak{R}~ \\mathfrak{S}~ \\mathfrak{T}~ \\mathfrak{U}~ \\mathfrak{V}~ \\mathfrak{W}~ \\mathfrak{X}~ \\mathfrak{Y}~ \\mathfrak{Z} ", "ece57a3466f2b182fab9c35174fba133": "c(x)=0\\;\\forall x\\in\\Omega", "ece5e1c9cac8aeeb6025774113cd8516": "\\mathcal{ALCOIN}", "ece640fe1f8dcc5b77f6e144955f22dc": " v_1'\\left(\\frac{1}{2}\\right)=v_2'\\left(\\frac{1}{2}\\right)", "ece65a2e1140595ee2dafb0cd9ee6107": "x_{3}=(a_{1}+a_{2})cos(\\theta)", "ece6b9f063c21904dd4c60ef29e61b0a": "x^* =-\\frac{1}{6} (x+(e_1x)e_1+(e_2x)e_2+(e_3x)e_3+(e_4x)e_4+(e_5x)e_5+(e_6x)e_6+(e_7x)e_7).", "ece721e02d817d149192cee847a81645": "p(z)=az_1^{k_1}z_2^{k_2}\\dots z_n^{k_n}\\, ", "ece722bfb17a4beb6987a9e1b76c4303": " \\alpha_j \\leftarrow w_j \\cdot v_j \\, ", "ece806e21b124725f160df1a4dd6612a": "\\mathbb P(\\mathcal E)", "ece829dec453a398707d54f4cd652cb0": "\\Delta W = \\int_{\\mathbf{r}_1}^{\\mathbf{r}_2} \\mathbf{F}\\cdot\\mathrm{d}\\mathbf{r}", "ece840a36c91b8495ffe8079a92b0045": "\\deg(2x) \\deg(1+2x) = 1\\cdot 1 = 1", "ece87b9348d120088bf0c8617f3e8121": "\\rho^\\phi := g \\mapsto \\rho(\\phi(g))", "ece8eeea70666c6bcaad95bbd539a05c": "w_i(e_j)", "ece919b63e18349e635bb907964dbe27": "{}\\lesssim{}", "ece931dccb6521639814042d230535c9": " \\bar{F}=\\frac{i\\rho}{2}\\oint_C w'^2\\,dz,", "ece95b3f51fb1cf84d29a397a62c2f76": "f(x) \\, ", "ece96e95c6c782ca758b89d6be060d3b": "B = e^{-\\gamma} \\prod_p \\left({1 - \\frac{1}{(p-1)^2(p+1)}}\\right) \\approx 0.34537 \\ . ", "ece9a4137f43b65f05f423e3fbbc7b09": "(z-K)_{AB} > 2.5", "ece9aa7382de3992ae6cbcb0f1ade322": "*(e_1\\wedge e_2\\wedge e_3)= e_4\\wedge e_5", "ece9ee11da1557d5f6bad2de2031c8d9": " H_p(t_0) = c\\eta_0 = 14.4\\ {\\rm Gpc}", "ece9f36c842bc066ce0144fd813b20c0": "\\overline{H}", "ecea3476dc1743b5a3c3b2b6623c59f3": "\n \\varphi \\in V' \\mapsto \\varphi(x), \\quad x \\in V, \\,\n ", "ecea59e372290a26790640608b150faf": "\\frac{20! \\times 3^{19} \\times 30! \\times 2^{29}}{4} \\approx 1.01 \\times 10^{68}", "ecea5bc78ab0bac1086a38dba2dd93ac": " c = \\nu \\lambda ", "ecea5ec82bddc06b50ee65b7b049b760": "30\\!-\\!28\\,=\\,2", "ecea6808485cd570aabaa074bcae7e3e": "\\cos A=\\frac{\\textrm{adjacent}}{\\textrm{hypotenuse}}=\\frac{b}{\\,c\\,}\\,.", "ecea99698f9538959f2d90a88e60f91b": "F(\\rho, \\sigma) = \\operatorname{Tr} \\left[\\sqrt{\\sqrt{\\rho} \\sigma \\sqrt{\\rho}}\\right].", "eceaa1526e5e745d828de3e97d26211f": "[K_i,P_k] = i \\eta_{ik} P_0 ~,", "eceac87ae91e7f63b7603e9eb2950bee": "\\textstyle\\ \\theta_{i}=\\theta _{i-1}+\\Delta \\theta _{i} ", "eceb1369a555e6629cc58bccccf2175a": " \\|x + y\\| \\leq \\|x \\| + \\|y\\|. ", "eceb8e43e26c0a381165ef72e1515d80": "f(x)=2x \\sqrt{1-x^2}", "eceb9a9ff952870e3d30d5c2b274133c": "F_{O_2bag}(t)=\\frac{(Q_{dump}+V_{O_2})*F_{O_2feed}-V_{O_2}}{Q_{dump}}", "ecebd58d09448935f962bc0ee977ddcd": "\\Delta t'", "ecec7e324e0c25a4c0d74ffadfad278f": " p_{1,2}(x) \\, ", "ececdd1e7ad3d48375d7f5a6a8d319e9": "\\mathbf{a} \\times (\\mathbf{b} - \\mathbf{c}) = \\mathbf{0}", "eced02f6a308cd29a46053f2b8302f56": "\\text{0/1 ohms reactance on kva base}_2=\\frac{\\text{kva base}_2}{\\text{kva base}_1}\\text{ * 0/1 ohms reactance on base}_1", "eced1ebf32c1d710231c916b9af46ddc": "\\|x+y\\|=\\|x\\|+\\|y\\|", "eced5a0e44318c019a22f628b16ead5f": " s_{X_1X_2} = \\sqrt{\\frac{(n_1-1)s_{X_1}^2+(n_2-1)s_{X_2}^2}{n_1+n_2-2}}.", "eced665b22b256494997b0aae58725ee": "\\dim H(M) \\le \\frac{1}{2}n(n+1)+1", "eced97ad2ca6dd915e1961c52a9d6ad8": "E_n(i)", "ecee63ae1ba4b36c9b579ed31a36d7f5": "u(0)=u_0 ", "ecee9279e8e73f796db6a3b0d86fa56b": "H_0 = \\frac{1}{2} \\sum_{\\mathbf{k}\\sigma}(\\xi_{\\mathbf{k}\\sigma} - E_{\\mathbf{k}\\sigma} - \\Delta_{\\mathbf{k}\\sigma}^\\dagger b_{\\mathbf{k}\\sigma}) + INM^2/2", "eceebfb1c3ded2896764b4b014058056": "\\langle W, D \\rangle", "eceed5c9445dc0f1e2ee4193c5ac09ae": " \\rho c_p V \\frac {dT}{dt} = -F, ", "eceee303509104f2324481db1ab81ff3": "\n \\sigma_{ij} = -p(\\epsilon_{kk})~\\delta_{ij} + 2~\\mu~\\epsilon_{ij}\n ", "ecef49eb85680301bb8cc52a8608852d": " \\dagger ", "ecef7f9ab753f25c1601899047270344": "\\beta*", "ecef99da4e4e01c616dfe9b5770bbbbc": "\\sqrt[3]{4\\cdot 1\\cdot 1/32}=1/2", "ecf0525621805b73259f3ce3e55effa6": "\\scriptstyle{G:X\\longrightarrow X^-}", "ecf0705f974420da816b1c14ab1ed47b": " \nU = \\frac{Q^2}{4 \\pi \\epsilon \\epsilon_0} \\left( \\frac{e^{\\kappa a}}{1+\\kappa a} \\right)^2\n\\frac{e^{-\\kappa r}}{r}\n", "ecf09b3c75f9085b03d8cf66c51314ba": "P(t_1,t_2)\\ ", "ecf0b49827e3f226c238196727898c57": "{\\mathfrak k}_n\\equiv\\partial_{y^n}+b_{n-1}\\partial_{y^{n-1}}+\\ldots+b_1\\partial_y+b_0", "ecf0de2e2cf41149314bd53a1267c807": "Z_{11} = {V_1 \\over I_1 } \\bigg|_{I_2 = 0} \\qquad Z_{12} = {V_1 \\over I_2 } \\bigg|_{I_1 = 0}", "ecf16b14443e047af3a5f5b5ed23d2e2": "s = x \\,2^m > 2^{p/2},", "ecf19d2d99bce6e5fa3e30a43d158588": "(\\frac{2ka}{I}-\\frac{4k^2 b(a+b)}{IMV^2})", "ecf1dba23e125b243a29afc98a55e82c": "_{\\nleftarrow }\\!", "ecf2b60e1ccb40dae8a51e817ad5643e": "\\scriptstyle \\frac{\\sqrt{2}}{3}", "ecf2c7eade09aa3601fbbb5489ac72e1": "[e_i,e_j]=0.", "ecf3132ad84082932478d8159f33d7e1": "\\varepsilon _{2}\\Psi =\\frac{-P^{2}+m_{2}^{2}-m_{1}^{2}}{2\\sqrt{-P^{2}}}\n\\Psi", "ecf354b4caa89ea935896f2815cecf4e": "\\psi(\\Omega 3) = \\varepsilon_2", "ecf35fe7472d15762532e1848b2d9992": "v_1 \\wedge v_2 = - v_2 \\wedge v_1", "ecf36bf6fc70716fe507e6a963d3b31c": "(x+1)(x-2)", "ecf3754c2216ed722fb101bcd9ce9b6f": "L_{contr.}=L'/\\gamma=\\gamma L/\\gamma=L", "ecf392ed334944e7c6ad69f076a27137": "\\langle a_n \\rangle_{n=0}^{\\infty} \\subset \\mathbb{N}", "ecf3bcc75b3fbea14cf766e9b1efaa27": "K_{10}", "ecf3c42e340d2d8aebf40fa5af4a6f30": "\\tfrac{26}{11}", "ecf3d5506710c7e542f15a426d6154e9": "|e\\rangle ", "ecf3ea3aa423f58bdedfeadb510d71fd": "\\frac{k_{(X=D)}}{k_{(X=H)}} = 1.15", "ecf426d6d8321628bdc7f85842e45831": "\\text{Li}_{2m}\\left(e^{i\\theta}\\right)=\\sum_{k=1}^{\\infty}\\frac{\\cos k\\theta}{k^{2m}}+ i \\, \\sum_{k=1}^{\\infty}\\frac{\\sin k\\theta}{k^{2m}} = \\text{Sl}_{2m}(\\theta)+i\\text{Cl}_{2m}(\\theta)", "ecf43d9d2db24b0f10f715837af92876": "\\mathrm{Bias\\ Ratio} = \\mathrm{BR} = \\frac{\\mathrm{Count} (r_i | r_i \\in [0, \\sigma] )}{1+\\mathrm{Count}(r_i | r_i \\in [-\\sigma, 0))} ", "ecf452da4ac75a1c58d0729a7c8216dc": "X = \\frac\n{ 1.002432\\, \\cos^2 z_\\mathrm t + 0.148386 \\, \\cos\\, z_\\mathrm t + 0.0096467 }\n{ \\cos^3 z_\\mathrm t + 0.149864\\, \\cos^2 z_\\mathrm t + 0.0102963 \\, \\cos\\, z_\\mathrm t + 0.000303978 } \\,,\n", "ecf46f4bfa0b3cb2822e0ff24cce54b7": "\\mathbf{x}_i ", "ecf4b66a2fe106587b469473e8753281": "V_1/V_2", "ecf4d45cf954c2518de4a292d613db57": "\n\\begin{cases}\nu_t(x,t) -\\Delta u(x,t) = f(x,t) &(x,t)\\in \\mathbf{R}^n\\times (0,\\infty)\\\\\nu(x,0) = 0 & x\\in \\mathbf{R}^n\n\\end{cases}\n", "ecf4de0f9e2183f74fc46d36865dc05d": "\\mathbf{a} = (2,3).", "ecf50efc236c2f4ea0b315e45a9cc14a": "\\begin{pmatrix}j\\\\v\\end{pmatrix}", "ecf560e2701062f72de05fb987e5a9c5": "\\frac{dy}{dx} = \\begin{cases}\n\\mbox{unbounded} & \\mbox{if } y = 0 \\mbox{ and } x \\ne 0 \\\\\n\\pm1 & \\mbox{if } y = 0 \\mbox{ and } x = 0 \\\\\n\\frac{x(a^2 - x^2 - y^2)}{y(a^2 + x^2 + y^2)} & \\mbox{if } y \\ne 0 \n\\end{cases}", "ecf5dcb6c864ae5d18ab165ce63e2ffb": "p\\in G_0", "ecf60d33d1d561c9a2fcf9b06ab1a56e": "\\boldsymbol{C} = \\boldsymbol{F}^T\\cdot\\boldsymbol{F}", "ecf6e9dc1213920f7941e532dc10f43d": "A(x,y,z) = (x + y + z)^2 - 2(x^2 + y^2 + z^2)", "ecf6f5fdb5c7d5a5d7ffdfe7ef0ddfd2": "S_o,S_g", "ecf730bd06fb1598b7217f96d7df5467": "F(b) - F(a) = F(x_n) - F(x_0). \\, ", "ecf75d2f4e49da6052b358a62b1bb885": "x_h(t) = C_1.e^{ -B.t} \\; + \\; C_2.e^{ -A.t}", "ecf7b064300bfda3f8b2b4d4a5311aaa": "\\{u, v\\}", "ecf7dca6444158ff2f26e30a06f23b75": "\\cos\\frac{11\\pi}{60}=\\cos 33^\\circ=\\tfrac{1}{16}\\left[2(\\sqrt3+1)\\sqrt{5+\\sqrt5}+\\sqrt2(1-\\sqrt3)(\\sqrt5-1)\\right]\\,", "ecf80ebc62003217afdca6ed81342cb8": "\\{ \\ell \\in M \\mid \\mbox{for some } x \\in M, \\Delta = \\ell x \\},", "ecf812ca7ee1760dc55081c4affeeeee": "v_{down}^{(jam)} = v_{g}\\qquad\\qquad(4)", "ecf854742ae7833721f6612f05a5eeec": "\\frac{\\partial \\mathbf{g(u)}}{\\partial \\mathbf{u}} \\frac{\\partial \\mathbf{u}}{\\partial \\mathbf{x}}", "ecf8d0196b66393f975152c275e7b09c": "q=1.787231650\\ldots. \\,", "ecf90f67e3370930e87e19cf68ee6503": " r = \\frac{\\alpha\\, \\Delta t}{\\Delta x^2} \\leq \\frac{1}{2}. ", "ecf9614a7a907548f1d307077d066838": "x_n=f(x_{n-1})", "ecf987f635e2b7250c13b46ddfe77e25": "\\oint_C f(z)\\,dz = 0", "ecf9a9a64c773417181cbedc50b4e37f": "x\\leftarrow (ax+c)\\,\\bmod\\,2^{32},\\ \\ c\\leftarrow \\left\\lfloor\\frac{ax+c}{2^{32}}\\right\\rfloor.", "ecf9b995b2efcc90fa39514581c66c41": " \\|f\\|_{n} = \\sup \\{ |f(z)| : |z| \\le n \\}", "ecf9e60b2de9b9a370c323016396c78e": "E(t)E(t-\\tau)", "ecfa05e78f79330fe085de4d2ea35c08": " \\lim_{x \\to c} (f(x) - g(x)) = \\ln \\lim_{x \\to c} \\frac{e^{f(x)}}{e^{g(x)}} \\! ", "ecfa46200d57505567e17ee8383a5172": "A(z) = 1- \\sum_{k=1}^p a_k z^{-k}", "ecfa81b4cae2c822108da0e67c9b038b": "\\delta_b", "ecfa9199c694e633339e995c447c6a8c": " t\\,=\\,\\left\\lfloor\\frac{1}{2}(d-1)\\right\\rfloor", "ecfaad81344c2f0f93e8dab8701dc4f1": "\\mathrm{sgn}(\\sigma)", "ecfabd1d18793448151cf33f99165e4b": "\\int_0^1|f(\\alpha)|^\\lambda d\\alpha\\ll_{P, \\varepsilon} N^{\\mu(\\lambda)}", "ecfad24e6ea2f430364165c643b631bd": "(\\alpha_1,\\alpha_2,\\ldots,\\alpha_n)", "ecfae4112b098e9222dc1b3039832965": "(x,y) \\times (u,v)=(xu+yv ,xv+yu),", "ecfb1f444a9a7c4ec2a6aae19c47e638": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 3.881710 \\log_e(T+273.15) - \\frac {4999.618} {T+273.15} + 41.05901 + 3.515956 \\times 10^{-06} (T+273.15)^2", "ecfb48b07c4dfc61192ed0b27912192c": "\\scriptstyle S_x^{-}(s)", "ecfc0eeaa4db0119fdeb55069ad49362": "\\langle\\overline\\psi_i\\psi_i\\rangle", "ecfc2aefe0edab39ad942c968c5a02cf": "\\left(\\frac13,\\frac13,\\frac13\\right)", "ecfc4d8c632b1920bcd0ada5cf213daa": "a_1 X_1(z) + a_2 X_2(z)", "ecfca0c98e09f1acecfa02b0f2b0c2a8": "(U, U\\cap N)", "ecfccc313820f57b3684d7521d5207c0": "\\psi_h", "ecfd44a42231603fcc86dca8d0f2510e": "\\displaystyle{c^{-1}=Q(c)^{-1}c=Q(a)^{-1}Q(b)^{-1} b= Q(a)^{-1}b^{-1}.}", "ecfd8035943acf62a1f7756dbbf4e7ec": "\\mathit{E} = \\mathit{E_c}", "ecfe15457f5d630b1486b4d998605f85": "\\forall p: \\mathcal{B}(\\mathcal{B}p \\to p) \\to \\mathcal{B}p", "ecfe67b4ea9080c70aea779533672f02": "4: \\quad sum \\quad += \\quad tmp", "ecfe9eae7ff048b0f78c477e69395c49": "\\Delta^1_{\\rm LAT}", "ecfed39e32a0ccdc4946b5109d8f4cc4": "H = \\sum_{\\sigma}\\epsilon_f f^{\\dagger}_{\\sigma}f_{\\sigma} + \\sum_{\\sigma}t_{jj'} c^{\\dagger}_{j\\sigma}c_{j'\\sigma} + \\sum_{j,\\sigma}(V_j f^{\\dagger}_{\\sigma}c_{j\\sigma} + V_j^* c^{\\dagger}_{j\\sigma}f_{\\sigma}) + Uf^{\\dagger}_{\\uparrow}f_{\\uparrow}f^{\\dagger}_{\\downarrow}f_{\\downarrow}", "ecfef0db12231570078c9e9b892f74ce": "\\,A_x^{(12)}", "ecff9721aa8139db58ad197584a6618e": "\\mathbf{D}^2_{xy}=\\begin{bmatrix}0.5 & 1 & 0.5\\\\1 & -6 & 1\\\\0.5 & 1 & 0.5\\end{bmatrix}", "ed004ded4d1925b3ebfc6e1ad87a0b0f": "(2)\\,", "ed005743179d0cee1fc77a1d25e2d926": "\\frac{1}{\\sqrt{2}}\\Big( |\\uparrow \\downarrow \\rangle - |\\downarrow \\uparrow \\rangle\\Big).", "ed0107b925992f033faefd0a10e2859f": "\\nabla \\cdot (|\\nabla u|^{p-2} \\nabla u) = 0", "ed0185454cef54cacee4d114e92c356a": "\nd^{\\ell}_{m 0}(\\beta) = \\sqrt{\\frac{(\\ell-m)!}{(\\ell+m)!}} \\, P_\\ell^m ( \\cos{\\beta} )\n", "ed019460097b46372b4306620e459f68": "\\mathfrak{p}'_i = \\sigma(\\mathfrak{p}''_i)", "ed024f93a35aaed54160fc856cd7e017": "e>1", "ed026cfc260ee3d16d443bfa6d9757f6": "P=\\mathbf{p}\\cdot\\mathbf{p}-n^2=0", "ed029413bb1b1266bf8c501fb6c1d164": "VU = e^{-2\\pi i \\theta}UV", "ed029752f422464095690430ef39e10e": "E = {C \\over r^{d-1} }", "ed02cfeeb5362e1dc93cc83b4185dd7e": "\\mathbf{A}_{i,j}, \\mathbf{B}_{i,j}, \\mathbf{C}_{i,j} \\in R^{2^{n-1} \\times 2^{n-1}}", "ed02ef48afba8a808779a7c231387379": "\\Delta S = C_p \\ln{T_2 \\over T_1} - R \\ln{p_2 \\over p_1}", "ed03428e9a644b8493da653f62d892c2": "\\frac{-54672}{(1+0.10)^1}", "ed035956bf3c913c0baf8d7b29c6eea0": " m\\left(\\{x\\in X\\ :\\ f^*(x)>\\alpha\\}\\right)\\leq\\frac{\\|f\\|_1}{\\alpha},", "ed03789e609b0825008522669fffa307": "p(n) \\approx 1 - \\left(\\frac{364}{365}\\right)^{C(n,2)}.", "ed03ad28b0255fa0fc7b551a6f72a0fe": "d=2(a^2+b^2-c^2),\\ e=2(a^2-b^2-c^2),\\ f=-(a+b+c)(a+b-c)(a-b+c)(a-b-c). \\, ", "ed03d3271f1456625a2f4e44eb729752": " r(x) ", "ed04076a8d1db20655be5bcce34f9de4": "H := \\{ f \\in W^{1,2} ([0, T]; \\mathbb{R}^{n}) \\;|\\; f(0) = 0 \\} := \\{ \\text{paths starting at 0 with first derivative in } L^{2} \\}", "ed040e1ec5c3f3e7fa88f7c77369be26": " DAF = \\frac{{u_{max}}}{{u_{static}}}", "ed042880d37c2c1dca55043bc2df2454": "q\\equiv 1", "ed044ea8f80b19ad31231d2911f58ec4": "|V_\\mathrm{S}|", "ed04523c706805102d9736de87df72e9": "\\Omega_{i,j} = \\sqrt{|\\chi_{i,j}|^2 + \\Delta^2}", "ed046e670c77ed2229131c04ea7701b8": " - (\\Sigma _{11} )^{ - 1} ", "ed04c7f94d6633ab4dc653943c7e32a9": "\\sin 6^\\circ = \\cos 84^\\circ = \\dfrac{\\sqrt{30 - 6 \\sqrt5} - \\sqrt5 - 1 }{8}\\,\\!", "ed051f99d486f8646cc84a65563a1fb9": " (u|v)_E = \\lim_{n\\to\\infty} (u_n|v_n)_E", "ed0529035df00651f3a65bf56d64b175": "x([0,1]) ", "ed05324d7de0a941d53a96411a8c439e": "K(l, m, P, \\nu)", "ed055b920ecdaf949f6f9ff6f13e5ee7": "A_{ij} =\n\\begin{cases}\ni/j, & j\\ge i \\\\\nj/i, & j 0 .", "ed132552e737a29a87025a4fe1c6cfe2": " \\omega^a ", "ed132d4073174940782317802e4819a4": "\\beta\\left(t_\\text{r}\\right) \\neq 0 ", "ed133a2f004af555b8657f8a7ceafb57": " x_n=x_{\\infty},\\quad", "ed1398e20702df91e76d4da9fccfd196": "\\psi_N \\equiv (\\psi_0+\\psi_1)/\\sqrt{2}", "ed139ab2e9c6b1423a5a7977c3a7523e": "a X \\sim \\operatorname{Log-\\mathcal{N}}( \\mu + \\ln a,\\ \\sigma^2).", "ed13a4f799292a3e9f1a629d196f4e9d": "x(t,x_0) \\in \\mathcal{O} \\ \\forall \\ t \\ge 0 ", "ed13c2468cb45657d94315f39eca6f1e": "\\mathfrak{e}_{7(7)}", "ed14b14a26e2e244e77c778a30196492": "\\omega_1-\\omega_3", "ed14ece1ef6f84c8335eac23432782be": "z = 0", "ed14fb4b94186a08146096c599c28fd6": "\\mathcal{X}(S(z;u))=\\mathcal{X}(u)+z.\\ ", "ed15eb72ba91eb757aa9f60d9ba74f9c": "s^{(r)}(x)=\\sum_{k=0}^{\\infty} F^{(r)}_n x^n=\\left(\\frac{x}{1-x-x^2}\\right)^r.", "ed163231c78266fe388e0f8bddb2b7c6": " \\begin{pmatrix} y_1 \\\\ y_2 \\\\ 1 \\end{pmatrix} \\sim \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\frac{1}{f} & 0 \\end{pmatrix} \\, \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\\\ 1 \\end{pmatrix} ", "ed163fc2b83bbea07e8141372f6a5950": " O(|E|) ", "ed16486ddc8e9e30c20e52ee2b3347f8": "\n\\text{If }p \\equiv \\pm q \\pmod {4a}\n\\text{ then } \n\\left(\\frac{a}{p}\\right)\n=\\left(\\frac{a}{q}\\right).\n", "ed16891143528c504e74dfaa587b4c00": "g=0", "ed168e286d020965f434989e5480b017": "T_W = V^*TV : W \\rightarrow W", "ed16abbacb9860ea46dada86c49e3e19": "P(\\Delta)", "ed17144569b7202c131c18f6f645bcf0": "J^k_0(f\\circ \\gamma)=J^k_0(g\\circ \\gamma)", "ed17233d8e78d947ee8ad4aa2c20ed18": "{e}", "ed173810c44e4d0d84fb1019e2d8ab17": "S() \\to x\\text{:}A()\\ B(x)", "ed174d8f6335612bc7642d3061d47016": "B \\subset \\Omega", "ed174e818f5a6befc1514c9c0fe8c463": "F_6=8 \\text{ and } F_7=13.", "ed1769099baf4d0f57682185c1aeabbd": "\n \\begin{bmatrix} C_{11} & C_{12} & C_{13} \\\\ C_{12} & C_{22} & C_{23} \\\\\n C_{13} & C_{23} & C_{33} \\end{bmatrix}\n = \\cfrac{1}{1-\\nu_{12}\\nu_{21}}\n \\begin{bmatrix} E_1 & \\nu_{12}E_2 & 0 \\\\\n \\nu_{21}E_1 & E_2 & 0 \\\\\n 0 & 0 & 2G_{12}(1-\\nu_{12}\\nu_{21}) \\end{bmatrix}\n \\,.\n ", "ed17c3f44a452e82cfb179b42db73db9": "\\frac{k}{|E|} \\leq \\frac{k}{nk/2} = \\frac{2}{n}.", "ed1851a0f05ec86b21523b920bf3ab66": "\\mathcal R:=\\{P\\in{\\mathcal P} \\mid P^\\perp=\\mathcal P\\}", "ed188f38c7bd2ed07ae8b2444581fe71": "end\\,if", "ed1899716033b8f542c19259f8d1d8bf": " \\boldsymbol{\\Omega} = 2 \\boldsymbol{\\xi} \\,\\!", "ed18cc7ab0060ce31f92c33c8e270762": "\\scriptstyle x \\in Z^d ", "ed18dd1f0c678374e0123c711f2a1c6a": "\\langle x+y,z\\rangle= \\langle x,z\\rangle+ \\langle y,z\\rangle.", "ed190c071a3e54baa2862c47fe629c2b": "\\epsilon_\\infty", "ed191ddcea6c65e687085607865c69d9": "{\\rm AEXPTIME}=\\bigcup_{k>0}{\\rm ATIME}(k^n)", "ed1944cc7bb8963f78373e996eaf526c": " f(A)->2, f(E)->2, f(N)->1, f(R)->1, f(T)->2", "ed1971aa6c2b935e868ba6b4655cd7a7": "\\rho(X) = \\inf\\left\\{t \\in \\mathbb{R}: \\exists V_T \\in A_T: X + t + V_T \\in A\\right\\} = \\inf\\left\\{t \\in \\mathbb{R}: X + t \\in A - A_T\\right\\}", "ed1972c08bd901551b5a79bff829f6f8": "|u|_{2,\\alpha;\\Omega} \\leq C(n,\\alpha,\\lambda,\\Lambda,\\Omega) (|u|_{0,\\Omega} + |f|_{0,\\alpha;\\Omega} + |\\phi|_{2,\\alpha;\\partial\\Omega}).", "ed19907c1a7d630ab8632109a7ce980f": "P^{0} := id_{U}", "ed1a0d30a0456437234942053589a5b3": "4\\in X", "ed1a304957748316f8f21020946ac609": "\\tfrac{\\left(\\Delta x\\right)^2}{2 \\Delta t}", "ed1a4ecbcd818d128b7b7581c2eda916": "\n\\frac{100 c}{1 + R(0,1)} + \\frac{100 c}{(1 + R(0,2))^2} + \\cdots + \\frac{100 + 100c}{(1+R(0,n))^n } = 100. \n", "ed1a8871cd9f35d9ce3e4ba04cd75b29": "c_l, c_m", "ed1ad8f0cb4becc9000f1bf9035a5836": "f \\colon X \\to {\\mathbb{R}} \\cup \\{ - \\infty \\}", "ed1b468afbee63cf4845bbe7bb199c45": " V = \\left\\{ \\begin{array}{c}\n (\\Delta ,x):x \\in C{\\rm{ }} \\\\ \n \\operatorname{Tr}(A^T A\\Delta ) - 2y^T A^T x + \\left\\| y \\right\\|^2 - \\rho \\le 0,\\rm{ }\\Delta \\ge xx^T \\\\ \n \\end{array} \\right\\}.", "ed1b665636d93dfd54ee7fc19805897e": "\\langle\\vec x\\rangle \\in {\\mathcal P}", "ed1ba62efbedc4f43ce3541ace2388d0": "B(H \\otimes K)", "ed1bb3ac33180a11990ddb64013396ec": "h_p(x)=f_p(x) g_p(x).\\,", "ed1be4382f65455ca64eaa60605592c7": "\nK = 0.2 + 0.02 \\cdot\\log_{10}V\n", "ed1bece23c492e001d19f6db943cac81": "\\mathbb{PN}", "ed1bf99b3024c5f4705ddee2839c6e4e": "h : Y \\to Q", "ed1c91addb49da205d981c7e65bcba6c": "K^\\mu K^\\nu g_{\\mu\\nu}\\;", "ed1cf81e98f71a6488ab5b86eacdfa38": "\n\\sigma_y = \\begin{pmatrix}\n0&-i\\\\\ni&0\n\\end{pmatrix}\n", "ed1d193e8cb7c030fb84cd24af838f0f": "g: R/u_1R \\times \\cdots \\times R/u_kR \\rightarrow R/uR", "ed1d5da746eb737902ba460a112e4db8": "n(n+1)(n^2+n+2)/8", "ed1d7de42dfa42b7a60d8656dd4c7b34": "{{\\Iota }^{\\Rho }}", "ed1e2f9db9013f037b406641bd4e0cc4": "\\phi_R:R(S_R)\\rightarrow S_R", "ed1e3b20cb859f920f8d2e5273dde0e0": "2\\pi {{\\left| \\int\\limits_{-\\infty }^{\\infty }{f(t)g*(t)dt} \\right|}^{2}}=\\iint{{{P}_{V}}f(u,\\xi )}{{P}_{V}}g(u,\\xi )du.d\\xi ", "ed1e79435e1fe404d0b7ef54779f2773": "S_1-S_2<0", "ed1e833953d39696d71d208e47580fe9": "h(x)=1/x-\\lfloor1/x\\rfloor,", "ed1f161cd64d2384f9ad6e758a07f67f": "D^{-1/2}", "ed1f9cd3b93938ad71f540de3f30c5e5": "P\\times_G EG\\longrightarrow BG", "ed20cca6d6881ddfae437dbeea96359c": " \\pi(x) = \\operatorname{li}(x) + O(\\sqrt x \\log x). ", "ed21200b4bad9a88f4e891d9f6fe9425": "\nZ_\\mathrm{load} >> Z_\\mathrm{source} \\,\n", "ed212dff22e98ef7d6819e7a1359ef47": "\\lambda^{n-1}", "ed214c6ac4bc88811995c59f9213e5e2": "\\limsup_{x \\to +\\infty} x^2 q(x) < \\tfrac{1}{4}", "ed215840f85486718d0ecc1b07922fd1": "\\scriptstyle\\sqrt{2}e_j", "ed21b003ae5e9d6bdf6939782c9712bf": "V \\frac{dC}{dt} = -K \\cdot C \\qquad(1) ", "ed2216c64175d9cdfb29a197ba6fd1b5": "\\boldsymbol D(\\omega) = \\varepsilon_0 \\varepsilon_r(\\omega)\\boldsymbol E (\\omega)\\ , ", "ed222e8def418beb31e26011e1d448e6": "prune(T,t)", "ed222fcc2222a02d2464b8b4a1fd7209": "\\textstyle \\mathcal{E}(\\rho_j) = \\sum_k c_{jk} \\rho_k", "ed22565692f980839299503e1536c48e": "I\\equiv\\langle f_1,\\ldots,f_p\\rangle", "ed22971507cf5d6643c227a5c4947b6d": "R(0) = \\frac{1}{2} \\left ( \\frac{\\Delta V_\\mathrm{P}}{V_\\mathrm{P}} + \\frac{\\Delta \\rho}{\\rho} \\right ) ", "ed229f8994a26c0cf6d61a0c532e73dd": "H_1(x) = 2x\\,", "ed22a00d0c0c672ff620452b8200437e": "h_i^1(x_1, \\ldots , x_{k(i)}) \\cong h_i^2(x_1, \\ldots , x_{k(i)})", "ed22f0d661f8a65d4314f544b25c65a9": " y(s) = s^2 \\,", "ed23096d3285f6fb3d245e0865c0f462": "5\\tau", "ed238e215d90e19de49ea4e80d186ad2": "P(E) = 4", "ed23b2e945372acfc086df0cc06c14b3": " \\vec{Q} ", "ed23c6b50e1fa04a298e808b9f110154": "J = S_\\max - S = -\\Phi = -k \\ln Z\\,", "ed24256a2b94eb0d82dfdfd3f5aef7fd": "m_1, m_2, ... ,m_t", "ed24c1a36ba571fde2ef7d0f33df5161": " U (x) = 0.1 x^4 ", "ed24efa1a36954ac34d2f072bb1bff12": " x= 0 ", "ed2547653eafbb727d9fc0967958ca9d": "o(o(f)) \\subseteq o(f)", "ed2573a6f14ee93dd477fa741675b4fd": "\\mathbb{R}^p", "ed25bd4f441e42234a2ac962a4c041fd": "B^+P(R) \\to B^+P(S)", "ed260ef6b928e54f29b16cde4b577808": "D_1 = [P] + [Q] - 2[O]", "ed2628f128014418220f37525623a3fb": "(x_{0}, l)", "ed262a3c9c5efba23ce9b57f0e83a82b": "n = (q^m-1)/(q-1)", "ed26321fb522a12793727cb82c85be30": "\\alpha = \\beta", "ed2637854f3f2a15cfb782d0245be00f": "x^+=\\frac{t+x}{\\sqrt{2}}", "ed267086dcf0a2115c6281217c0e39c7": "\\forall x(x\\text{ exists})", "ed2689d234b5662a16c87450ec269d8f": "m_{em}=(4/3)E_{em}/c^2", "ed26d256a144a307407e86eda92e1417": "d_3+d_2+d_1+d_0", "ed26e8c9e3423cd21dd169e62e6088fb": "[OH^-]_{i^{ }}", "ed26f339251767c801fdfb351d292d95": "\\sum_{i=1}^n Q_i \\frac{r}{12} = \\frac{r}{12}\\sum_{i=1}^n Q_i", "ed274682fa58ae0eebd48cf9bad5ce18": " \\langle r^k \\rangle = a^k \\int_{0}^{\\infty}t^{k/3}e^{-t}dt = a^k \\Gamma(1 + \\frac{k}{3})\\,,", "ed2776bf00efac79f59907c27d325f9d": "C^{op}", "ed27d1abcf1108e4c97b3f07fad7125b": " \\Lambda^3\\mathbb C^8", "ed280711adbedce5268bd87d07957483": "\\textit{plant} \\subseteq \\textit{creature}", "ed281a1d3b1049e55a5c0ffc78d04a39": "\n\\begin{pmatrix} f_{11} & f_{12} & \\cdots & f_{1n} \\\\\nf_{21} & f_{22} & \\cdots & f_{2n} \\\\\n\\vdots & \\vdots & \\cdots & \\vdots \\\\\nf_{m1} & f_{m2} & \\cdots & f_{mn} \\end{pmatrix}\n", "ed282e0b5f6c3c9d526afba92cfc3871": "M^{-1}_{ab}", "ed285b078c0e87a2a58c3f65a0755e7e": "\\gamma=\\frac{C_p}{C_V}=\\frac{C_V+R}{C_V}=1+\\frac{R}{C_V} = \\frac{C_p}{C_p-R}", "ed2898a9c8694b90b32adb97f2fa1d19": "\\sum_{k=-n}^n e^{ikx}\n=\\frac{\\sin((n+1/2)x)}{\\sin(x/2)}", "ed28a621513226255b46efc1609132d8": "\\mathrm{LMMSE} = \\mathrm{tr} \\{C_e\\}.", "ed2926407eff70a15d37e78f9356fcec": "l=\\lceil \\text{number of objects} / \\text{capacity}\\rceil", "ed2931a9992f415daaff62af3a8b77b6": "\\sqrt{\\Delta_1^2}", "ed293886ac6df6c866fa0c6dafb58a00": "M_6(x)", "ed2939d7a9dbc5162d8d6524a6258732": "\\gamma(k,z)\\,", "ed29a70a0b3d2f39dbcd2f619a74487c": "\\operatorname{ad}_g(x) := [g,x]", "ed29d7d914592e3a7c6df4188bdf573b": "\\frac{3}{5} \\frac{a}{2} + \\frac{2}{5} 2a = \\frac{11}{10}a", "ed2a0a481ce2fb805acea2f9a09a7302": "V_\\theta\\left( r \\right) =\n V_{\\theta \\max} \\left( 1 + \\frac{0.5}{\\alpha} \\right)\n \\frac{r_c}{r}\n \\left[ 1 - \\exp \\left( - \\alpha \\frac{r^2}{r_c^2} \\right)\n \\right],\n", "ed2a58e36f017756a171df71162f9509": "x \\nabla y", "ed2ab150ab0cabfb59f81ca42e52183b": "f(x) \\cdot x = 0", "ed2b5c0139cec8ad2873829dc1117d50": "on", "ed2b8b6009878e902e1c90e299ee268b": "\n\\; C_\\Phi = \\sum _{i = 1} ^{nm} \\lambda_i v_i v_i ^* ,\n", "ed2baff89392d1a7ec78c7486c28328f": "m^2+n^2\\leq r^2.", "ed2bb585c5e3d95fe4102ccf9d842bd8": " 0 < x < a", "ed2be55494957e50e1d3fcb3bb667dbe": "n:\\textit{Nancy}", "ed2c247843d7be6329ebb6bb4be32bb0": "Z(s) = \\frac{s^2 + s\\frac{R_1}{L_1} + {\\omega_s}^2}{(s\\cdot C_0)[s^2 + s\\frac{R_1}{L_1} + {\\omega_p}^2]} ", "ed2c2e17e6a6e9816f7b90bb39fb2a81": "dx >= dy", "ed2c320c5d62297881c7939953f8938c": "I=0.33", "ed2c3e74b4a5e4968efaa47817b2d43f": " \\psi_0(x) = \\int_y \\psi_0(y) \\delta(x-y) \\, ,", "ed2c6bc234dc67de842ece258771aa6f": "\\begin{align}\nx&=\\frac{1}{2}(u^2-v^2)\\\\\ny&=uv\\\\\nz&=z\n\\end{align}", "ed2cb39d22f70b058f25f9fb042ceb98": "1 \\le k \\le n-1", "ed2cbf1026ce44952e089a76eb972c20": "\\frac{2}{\\Gamma(\\frac{\\nu}{2})}\\left(\\frac{-i\\tau^2\\nu t}{2}\\right)^{\\!\\!\\frac{\\nu}{4}}\\!\\!K_{\\frac{\\nu}{2}}\\left(\\sqrt{-2i\\tau^2\\nu t}\\right)", "ed2cdfe32f9c7ff9f334773b465b1f4b": "{\\sqrt{n}[X_n-\\theta]\\,\\xrightarrow{D}\\,\\mathcal{N}(0,\\sigma^2)},", "ed2d2b041a5091deb8faac71466531dc": "M^t", "ed2d7d520e218e9d5b3e4a1e7f6652bc": "~f_a,", "ed2d9d334aa459f5f0093704d8dde7b1": " (b-a)^4 ", "ed2db5dcebf682f5ec0851d771d4d2f3": "X_{2\\pi}(\\omega) = \\sum_{n=-\\infty}^{\\infty} x[n] \\,e^{-i \\omega n}.", "ed2dd31ac7c9b301e6b95fe17969ddea": "\n \\boldsymbol{R} := \\boldsymbol{\\nabla}\\times(\\boldsymbol{\\nabla}\\times\\boldsymbol{\\varepsilon}) ~;~~ R_{rs} := e_{ikr}~e_{jls}~\\varepsilon_{ij,kl}\n ", "ed2dd7209055d5dcddc55d3bc7054d27": "\\kappa = \\frac{0.60-0.46}{1-0.46} = 0.2593", "ed2ddb675e30fd2a43f3ecfdb420578a": "O(n k\\,\\log(n))", "ed2e01a4b7ec5e4d1dd58304a9465981": " \\mathbf{t}^{'(i)}=\\{t_c \\}_{c=-M, c\\ne i}^M", "ed2e54e7cb7adc3f4faa6e4156088b60": "\\mathbb{R}_{alg}", "ed2e8b1747fac75f659f917f4a3e1542": "\\scriptstyle z=\\eta,", "ed2f38b55c60bbd8a9f3fa8b5431dfa8": "h(Y,", "ed2f6d38596cd250e9560764ce584513": "y' = F(x,y) ", "ed2f983a1be2e0b132de60ad4a361a51": "\\frac{5}{121}=\\frac{1}{33}+\\frac{1}{121}+\\frac{1}{363}.", "ed2fa964a17dfc718ec4e02651616be5": "\\color{CornflowerBlue}\\text{CornflowerBlue}", "ed2fc679c814148ee59cb36717cf237f": "\\int\\frac{\\mathrm{d}x}{\\tan ax + 1} = \\frac{x}{2} + \\frac{1}{2a}\\ln|\\sin ax + \\cos ax|+C\\,\\!", "ed2fd00bb0332b3b463b7343586798f0": "M(V) = (V/V_d)^{1/3}", "ed307a1fab62ad2c98d333d036f11575": " \\Delta M_I=\\pm1", "ed3091f65ed11b9003e423088ea1a8c9": "\\Phi : M^* \\longrightarrow W^* ", "ed3099a547c2890fec32d8ba22bdc58d": "\\eta(s) = \\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}}{e^x+1}{dx}", "ed30c74c6c7d0d4764b68acda9664da4": "\\gamma_{xy} = \\gamma_{yx}\\,\\!", "ed30d5d0c6729570c1ecda7bc0401bdd": " A_j = (a_j+a_j^*)/2 ", "ed31182fabed600c7ce579f71d9d5031": "X\\times Y \\neq Y\\times X", "ed313f1a23ace30cf6651bef68198fdc": "r^2=a^2+\\frac{4(a+b)b}{(a+2b)^2}p^2", "ed32b8b095ca55509dab16d4f9dc04fe": "\n\\begin{align}\n\\dot{\\mathbf{x}}(t) &= f\\bigl(\\mathbf{x}(t), \\mathbf{u}(t)\\bigr) + \\mathbf{w}(t), &\\mathbf{w}(t) &\\sim N\\bigl(\\mathbf{0},\\mathbf{Q}(t)\\bigr) \\\\\n\\mathbf{z}_k &= h(\\mathbf{x}_k) + \\mathbf{v}_k, &\\mathbf{v}_k &\\sim N(\\mathbf{0},\\mathbf{R}_k)\n\\end{align}\n", "ed3301f5c4195a9204c09003f6db47d3": "{{\\beta }_{1}},{{\\beta }_{2}}", "ed335ffb4a5632a8de4f0df6c82964e4": "\\implies P(A|B) = \\frac{P(B|A)\\,P(A)}{P(B)}, \\text{ if } P(B) \\neq 0.", "ed33e56b433f9ef5fe27cef7efa7491d": " K_+ ", "ed344295ee9b103ac48a2274324222bf": " n = n_r + \\ell + 1 \\, ", "ed349d3fbca86eb20b0c7f5c40f6d81a": "\\textstyle x^2 \\equiv y \\pmod{n}", "ed34f5a36cfe29cdc78f50dc63546242": "\\frac{f(x_0+h) - f(x_0)}{h} \\ge 0", "ed35136a35303546c64159bd0ab61234": "\\mathbf{P}(X_i = 1) = p_i", "ed3518050bd45d256bef007e408c6a67": "\\hat L=p(x|\\hat\\theta,M)", "ed35764de27b95995f0d5292019e63e6": "\\underline{v}", "ed3586dcba23e44ff34a17657330f0cc": "n = rs", "ed35a312dd8f2f6684fc338e1f1f9d93": " \\nabla \\cdot \\mathbf{g} = -4 \\pi G \\rho \\,\\!", "ed35f9e5ecff0f032b223132026c0258": " \\int\\limits_\\Omega\\text{div}\\left(f\\mathbf\\varphi\\right) =\n\\int\\limits_{\\partial\\Omega}\\left(f\\mathbf\\varphi\\right)\\cdot\\mathbf n ", "ed36f745128189f4df2cd46ada4729ae": "|Z|_{max}=RQ^2_L \\sqrt{\\frac{1} {2Q_L\\sqrt{Q^2_L+2}-2Q^2_L-1}} ", "ed380976cbee77e810926ea407870820": "U = \\left\\{ z \\in H: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\frac{1}{2} \\right\\}", "ed390d6b20fc1a2447d5b2f4d5677f1f": "\\sum_{i,j\\in N_i} d(\\vec x_i,\\vec x_j)", "ed39530d0d7fb88b7a62b45b0dd6fe26": "S^3 = \\left\\{(x_0,x_1,x_2,x_3)\\in\\mathbb{R}^4 : x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1\\right\\}.", "ed398eddc7009fdfdfd4a2e71f381560": "\\phi_{e1}", "ed3990de0f9fddcce4d50e3d86d71b6d": " H = {1 \\over \\dot m} \\int \\left({\\rho \\mathbf{V} \\cdot d \\mathbf{A}} \\right) \\left( h+ {|\\mathbf{V}|^2 \\over 2} \\right),", "ed399b18d4aa7722d25b07f0395ae313": " H_0:\\mu=0 ", "ed3a0b362cb4299bb43054db46bce861": "q_\\alpha(x)=x^{\\frac{\\alpha+1}{2}}K_{\\frac{\\alpha+1}{2}}(x).", "ed3a7036739588c384846783fd33fc96": " s_{\\bar d}^2 = \\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}. ", "ed3a93bcb85fa705fac0cc64c7039ab7": "f(a_1,\\dots,a_{i-1},c,a_{i+1},\\dots,a_n)=f(a_1,\\dots,d,a_{i+1},\\dots)\\ \\Rightarrow\\ f(b_1,\\dots,c,b_{i+1},\\dots)=f(b_1,\\dots,d,b_{i+1},\\dots)", "ed3aa756c33f8e6a27f60c9ea466a537": " q = q(n) \\geq 2 ", "ed3ab0f5f2e5cd3211459cb6a7176505": "q = q - \\alpha_i y_i\\,\\!", "ed3ad2ceadd1d2669e394222e5100d66": "W ", "ed3b61b2260cd660522ab1fba89bb6c4": "A=\\{e_1,e_2,\\ldots,e_r\\}", "ed3b67d4ca96a9fce388ae3fbf2f5729": "\\displaystyle{2Q(Q(a)b)=2R(a,b)Q(Q(a)b,a)-Q(a)R(b,Q(a)b).}", "ed3bf1ff985924ba1fad4687d19c5820": "T_{conv}", "ed3c2c7dba560987e3c39f79b60b01ba": "\\dot S_i=-\\int _1 ^2 \\frac{\\dot V}{T} \\mathrm{d}p.", "ed3cc478c491ea2bf94c1769b60b1498": "\\begin{align}\\textbf{k}=2\\pi/\\lambda\\end{align}", "ed3ccf659713e9a24a401d8e98fdfc1f": "\\epsilon(x_0) = \\hat{Z}(x_0) - Z(x_0) =\n\\begin{bmatrix}W^T&-1\\end{bmatrix} \\cdot \\begin{bmatrix}Z(x_i)&\\cdots&Z(x_N)&Z(x_0)\\end{bmatrix}^T =\n\\sum^{N}_{i=1}w_i(x_0) \\times Z(x_i) - Z(x_0)", "ed3cec32cced8c6d8e4d4b8524d569c9": "\\frac{PV^\\gamma (V_f^{1-\\gamma} - V_i^{1-\\gamma}) } {1-\\gamma} = C_V \\left(T_1 - T_2 \\right)", "ed3d3a30ad2875b709c0775b3a72c525": "r_2e^{-j\\phi_2}", "ed3d4f8e9d48399715d88aa0376fbd98": "\\langle\\beta|\\alpha\\rangle=e^{-{1\\over2}(|\\beta|^2+|\\alpha|^2-2\\beta^*\\alpha)}\\neq\\delta(\\alpha-\\beta)", "ed3d92a0c9f3508ee402e0dd3f7d07bb": "v = (v_{ij})_{i,j = 1,\\dots,n}", "ed3da1b401e7df14076731cac00368e7": "\\,3^3 + 7^3 +1^3 = 371", "ed3dab6cd7a273810ef1614e1f4a0746": "{(\\widehat{E} - q\\phi)}^2 \\psi = c^2{(\\widehat{\\mathbf{p}} - q \\mathbf{A})}^2\\psi + (mc^2)^2\\psi \\quad \\rightleftharpoons \\quad \\left[{(\\widehat{P}_\\mu - q A_\\mu)}{(\\widehat{P}^\\mu - q A^\\mu)} - {(mc^2)}^2 \\right] \\psi = 0.", "ed3db8d39d27a3094ca5619e6a198b83": "\\frac{a_{f} - a_{s}}{ a_{s}}", "ed3dbc72badd6b7133a002d22c799807": "f(g(0,1),g(0,1),g(0,1)) \\rightarrow f(0,g(0,1),g(0,1)) \\rightarrow f(0,1,g(0,1)) \\rightarrow f(g(0,1),g(0,1),g(0,1)) \\rightarrow \\ldots", "ed3dc104713f22060e0561a8746f34a0": "V_L\\approx 0.60340", "ed3dd80c40302ceb837084ac41dd6f3d": "(xu \\equiv xv \\and yu \\equiv yv \\and zu \\equiv zv \\and u \\ne v) \\rightarrow (Bxyz \\or Byzx \\or Bzxy).", "ed3e83d1272b93808e337c16b4cff2a4": "\nZ_\\mathrm{source} = \\frac{Z_\\mathrm{load}}{DF}\n", "ed3e8b6695d0cff3c299f2727f4a6efe": "B(\\mathcal{H}_A)", "ed3ed3b33a1bc6af1640ad2cff5df741": "x_i' = x_i + f(x_0, \\cdots, x_{i-1})", "ed3eec9bba8c21faab342e58805ca81c": "\\neg \\neg P \\vdash P", "ed3efb46327e27b4df8a18955c98779f": "\n \\rho~(\\dot{e} - T~\\dot{\\eta}) - \\boldsymbol{\\sigma}:\\boldsymbol{\\nabla}\\mathbf{v} \\le \n - \\cfrac{\\mathbf{q}\\cdot\\boldsymbol{\\nabla} T}{T} \n ", "ed3f5be7f09505f69c2b479855df7339": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\n\\frac{h^{1+\\nu+(p-q)/2}} {(2 \\pi)^{(h-1) \\delta}} \\; G_{h p, \\, h q}^{\\, h m, \\, h n} \\!\\left( \\left. \\begin{matrix} a_1/h, \\dots, (a_1+h-1)/h, \\dots, a_p/h, \\dots, (a_p+h-1)/h \\\\ b_1/h, \\dots, (b_1+h-1)/h, \\dots, b_q/h, \\dots, (b_q+h-1)/h \\end{matrix} \\; \\right| \\, \\frac{z^h} {h^{h(q-p)}} \\right), \\quad h \\in \\mathbb{N}.\n", "ed3f6be3beada6f6b8eb633277ef7838": "-\\frac{1}{\\beta}\\sum_{i\\omega_m} \\ln(\\beta(-i\\omega_m+\\xi))=-\\frac{1}{\\beta}\\ln(1+e^{-\\beta\\xi})", "ed3f82751cff8e17212fb4a4cf865384": "f(x)=\\frac{x^3+16}{x^3-4x^2+8x}", "ed3f868279914b82c4abd7c564f154d3": "\\Theta_{cw}", "ed3f893f31f1cac26b79602dfb2e56b1": " \\mathbb{E} ( \\mathbb{P} (Y\\le y|X) ) = \\int_{-\\infty}^{+\\infty} \\mathbb{P} (Y\\le y|X=x) f_X(x) \\, \\mathrm{d}x = \\mathbb{P} (Y\\le y), ", "ed3fa8d6d0446bf978efdcc63c302578": "T = \\frac{1}{2} \\big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \\big|.", "ed3fabf5d9ada386325b495af725ea9b": "\\alpha_0,\\, \\beta_0\\!", "ed3fbe1f27c442311aa2303b0cae8522": "{t} \\,", "ed408ce52427a9cb0e36deb70b034c37": "n_0\\in\\mathbb{N}", "ed409be1a9bcd162b451ed1639b0841f": "- \\otimes X", "ed40a56404f7ac245b956fd85ad8d9b2": "M\\left[\\mathbf{r}(t),\\mathbf{\\dot{r}}(t),\\mathbf{\\ddot{r}}(t),t\\right]=0", "ed40c705513d5bc2564bf0cc5f041712": "\\mathcal{S}p^\\Sigma", "ed40df43198dd6da03d9c9b9fb73ffe7": "\\mathbf{x} \\triangleq (x_1, x_2, \\dots, x_n) \\in \\mathbb{R}^n", "ed413b4124e1c1c0ae3f89efcdc9f8d5": "\\begin{bmatrix} b_{m,0} \\\\ b_{m,1} \\\\ b_{m,2} \\\\ b_{m,3} \\\\ b_{m,4} \\end{bmatrix} =\n\\begin{bmatrix}\n-16 & -8 & 17 & -10 & 4 \\\\\n48 & 20 & -48 & 29 & -12 \\\\\n-48 & -18 & 48 & -30 & 13 \\\\\n20 & 7 & -20 & 13 & -6 \\\\\n-3 & -1 & 3 & -2 & 1\n\\end{bmatrix}\n\\begin{bmatrix}\n1 \\\\\n(5+m) \\\\\n32 \\cdot 2^m \\\\\n16(5+m) \\cdot 2^m \\\\\n4(5+m)(5+m-1) \\cdot 2^m\n\\end{bmatrix}", "ed418c6a2fe6d741ce18c4ea7873ef34": "L^{norm}", "ed41aebef3a7ebe8ca29f18c01050bf6": "k p = n", "ed4217caf8c1e91e74514c809886d32c": "\\begin{align}\n\\Psi\\Phi f &= \\varepsilon_X\\circ FG(f)\\circ F(\\eta_Y) \\\\\n &= f\\circ \\varepsilon_{FY}\\circ F(\\eta_Y) \\\\\n &= f\\circ 1_{FY} = f\\end{align}", "ed4227ecd3627649ff7e3c10a6e750c7": "f^{*} : \\text{Sub}(Y) \\to \\text{Sub}(X)", "ed426f0c4e46b0f261c1eb4c4bce2487": "t=F(x)", "ed4273a9cc41240d18dcc777797ab82f": "f_{ij}: V_i \\rarr V_j", "ed42c3d68d537bbb0826fdc11e5f262a": "h_n^{(1)}(x) = (-i)^{n+1} \\frac{e^{ix}}{x} \\sum_{m=0}^n \\frac{i^m}{m!(2x)^m} \\frac{(n+m)!}{(n-m)!}", "ed42c948b3bb828004e11f61941f1e5c": " | a_\\max | \\, ", "ed4303a140c655805674d17f04f75dbf": "\\left(\\dfrac{d}{2}\\right)", "ed43267f1424dde2f417bb7ff3cdb407": "J_m := \\sup \\{ X_m : m \\in \\{n, n+1, n+2, \\ldots\\}\\} = \\bigcup_{m=n}^{\\infty} X_m = X_n \\cup X_{n+1} \\cup X_{n+2} \\cup \\cdots.", "ed432adedce2db6b23ec2846d0f6fbf7": "|B_t| = \\int_0^t \\sgn(B_s)\\, dB_s + L_t", "ed432e55f00b7d2efb448852bccf3ba2": " \\{ \\mathcal{F}(t); \\; 0 \\leq t \\leq T \\} ", "ed437d7af2d16306c6a9454ed7de4041": "\\sqrt{\\frac{9}{28}}\\!\\,", "ed43a99330434327f227a0b679cc1964": "\\beta_1 x", "ed43c22445d790dcb8cf134f217a5145": " A(B(A(P))) ", "ed4404ce87083cfa6cbc33a5fab776de": "f:(0,\\infty)\\rightarrow\\mathbb{R}", "ed4496e2c001c1364b7e8c2d0e18c0f5": "\\operatorname{perm}\\begin{pmatrix}a&b \\\\ c&d\\end{pmatrix}=ad+bc,", "ed44a74025727812df119e0a75943e7d": "D(a,b;c)+D(b,c;a)+D(c,a;b)=\\frac{1}{12}\\frac{a^2+b^2+c^2}{abc}-\\frac{1}{4}.", "ed45508754410af92e937a93581f2b14": "\\left( \\frac{\\partial (G/T) } {\\partial T} \\right)_p = - \\frac {H} {T^2}", "ed459de5d7fb67320ede7d25c2e9f8f8": " Z/p^2Z ", "ed46003cf6e9d9ef16397d7b358d323c": "(H^{-1})_{ij}=(-1)^{i+j}(i+j-1){n+i-1 \\choose n-j}{n+j-1 \\choose n-i}{i+j-2 \\choose i-1}^2", "ed4659619964846d3ca4f4d47509453e": "\\textstyle l \\leq n-k = r", "ed467c74469f9b5bd69d4e6c37f170e2": "f_*:C_n(X)\\to C_n(Y)\\,", "ed468007696e7c4d456511f605d64020": "\n\\mathbf{F}_{\\mathrm{fict}} = - 2 m \\omega s \\mathbf{u}_{\\theta} + m \\omega^2 R(t) \\mathbf{u}_R.\n", "ed46838ca2f01711c1ed23e824d59627": "\\textstyle j-i = g(2l-1)", "ed468439bfaedb866ac72909c9af28d5": "\\dot m_{in}", "ed46b46da00c01a10f579369ecad2ebb": "V_m=\\operatorname{span}(\\phi_{m,n}:n\\in\\Z),\\text{ where }\\phi_{m,n}(t)=2^{-m/2}\\phi(2^{-m}t-n)", "ed4704175e1cbd4028a735279b6e03f2": "TP/G", "ed480e51afc285356649e3f1d12223a1": "\\phi^0", "ed485a5cebb6cd9ccc6a3cc0d2603ad3": "\\Gamma^{(\\lambda)}= \\Gamma^{(\\mu)}", "ed48e1834b2d4290bfc4c1bcf2c54a1f": "\n\\mathbf{MSE}[y_{j,t}(h)]=\\sum_{i=0}^{h-1}\\sum_{k=1}^{K}(e_j'\\Theta_ie_k)^2=\\bigg(\\sum_{i=0}^{h-1}\\Theta_i\\Theta_i'\\bigg)_{jj}=\\bigg(\\sum_{i=0}^{h-1}\\Phi_i\\Sigma_u\\Phi_i'\\bigg)_{jj},\n", "ed490e8c0e3324ceda280e5f1b7544df": "a'b'", "ed4932273b25547466005a296f8cb20e": "|a_k|\\,R^k > |a_0|+\\cdots+|a_{k-1}|\\,R^{k-1}+|a_{k+1}|\\,R^{k+1}+\\cdots+|a_n|\\,R^n", "ed49444f34404cde57b9a4a00d327f5a": "\\tan (\\arccot x) = \\frac{1}{x}", "ed494679b47467396f4c4f7e9ca8d83f": "\\scriptstyle M\\ddot{o}\\times I", "ed495c7419a93f076b2cd1d91e16b5c9": " x_1,\\dots,x_k \\in E", "ed49625c3f082b0349d581ee789f565b": "E = m c^2 \\cosh \\varphi ", "ed4a46ad44e53ed95a1ca4c3a9306b33": "c=\\hbar=1", "ed4b855935d7c2909760d817b43965de": " |F(z)| \\leq C_N (1 + |z|)^{-N} e^{B|\\text{Im}(z)|} ", "ed4ba9d464f0b13dc5e9ec206b171747": "\\mathcal S=\\emptyset", "ed4c1acb46dc1cc0b9a849a5473b7aa4": "(c_{1}-a_{1})-c_{7}", "ed4c288e129e9d002da22167618aa6ed": "p(\\boldsymbol\\theta)", "ed4c2e69ee361c8ecf2fcf5cf99054a3": "CAP = \\sqrt{C} \\sin\\theta \\le n ", "ed4c6d23bb1ef51ab7a364e2bef12631": "(x+y)(x+y)(x+y)\\cdots(x+y),", "ed4c8632aad5f974e0c7c56345a76179": "\nL = \\psi^\\dagger \\left(i{\\partial\\over \\partial t} + {\\nabla^2 \\over 2m}\\right)\\psi.\n", "ed4c961ed17dd04bf4987665d661fe63": "E^1(e) = E(e)", "ed4cf52d84933483ba1f0debfbf7133e": "6\\,s^2", "ed4cf6a81a940614973bc6e33ba0c747": "P\\left(\\frac{d}{dt}\\right) f(t)", "ed4cfa713f45bee73c93f15d42b21a20": " M_0 = M \\oplus K_0 ", "ed4d2974a4048b9264a998b56d1d3e80": "w_i\\in Y", "ed4d9fc46cbcfebfc883f82f6ad837fa": "H_M \\gets \\lg (N - C)", "ed4e10d117e2d99116a9cc421c9e5c8d": "\nV_C = 2 I_C R_L \\frac{C_2}{C_1 + C_2}\n", "ed4e1528e5c9f55e2a1de68b43145994": "\\operatorname{Var}(S_t)= S_0^2e^{2\\mu t} \\left( e^{\\sigma^2 t}-1\\right),", "ed4e4396682b850733ebac4fde46598c": "2P_{1/2}", "ed4e5cc62be5120d07a531792e0a9b15": "(f_* \\mathcal F)_y", "ed4e7c41ab28d431a3aec3f30c5f59bb": "\\sum_{n=1}^{\\infty} \\frac{\\xi^n \\,\\mathrm{Li}_u (\\alpha q^n)}{n^s} = \\sum_{n=1}^{\\infty} \\frac{\\alpha^n \\,\\mathrm{Li}_s(\\xi q^n)}{n^u}", "ed4eaac0920dcf53ede9fc4cf56cc2eb": "\\Omega=\\{E_1,E_2,\\ldots,E_n\\}", "ed4ecb6bba59e65eeaa074db2cbcd8f3": "\nf(N_1,N_2,\\ldots,N_n)=\\alpha N +\\beta E +\n\\sum\\limits_{i=1}^n\\left(N_i\\ln g_i-N_i\\ln N_i + N_i-(\\alpha+\\beta\\varepsilon_i) N_i\\right)\n", "ed4ed4beed18e5ca93157d9f1df681e2": "K = E_1 - E_2 + E_5 - E_6 + 9 \\,\\, \\pmod 9", "ed4ee5341a2428df2034e4833d069f0d": "PVA \\ = \\frac{FVA}{(1+i)^n} = \\frac{C}{i} \\left( 1 - \\frac{1}{(1+i)^n} \\right)", "ed4ef0656776c93159658085f5447cfc": "A = \\varinjlim A_k ,", "ed4f79d75a937e4a93b657e69fa4203e": "\\mathrm{Ar} = \\frac{g L^3 \\rho_\\ell (\\rho - \\rho_\\ell)}{\\mu^2}", "ed501dd29fdef1b982020e6ee39a7331": " \\sqcap(t)", "ed502589841b265944e24835fbaaea7f": "TE(t) = T(t) - T_\\text{REF}(t). \\, ", "ed5031e4620f7dbd79c1ee4d2e415f98": "S^q(X) \\, \\!", "ed50522a0e56198a3dc3fdb82009e36e": " E^{(0)}_{n_1,n_2} ", "ed50d056c1ec01f37c65be3792da13c5": "(f(x_1), \\ldots, f(x_n)) - (t_1, \\ldots, t_n)", "ed511e3d2ea75ebb232c0f0220048e1b": "\\scriptstyle\\mathbf J=0", "ed516390e3cab6877cc1d0d9443569b1": "\\mathrm{Stk} \\ll1", "ed517d1261c953600f8bfab44793fab3": "\\Theta^{ij}_{\\alpha\\gamma}", "ed51c674eb351f35ad1b4327e5f4ec1d": "C(v) = \\overline v", "ed51e29b544d523a28dfb4b86c383335": " (a,b,c) ", "ed5225eb4d35f9fb772b77d88c247349": " D(v\\wedge\\alpha) = (Dv)\\wedge\\alpha + (-1)^{\\text{deg}\\, v}v\\wedge d\\alpha", "ed523a01a276d710c8310c5c4d7c169d": "\\displaystyle{f^\\sim(z)=\\sum_{n\\ge 0} a_n \\overline{z}^n,}", "ed52c034efaad1c50549f3cc285b15da": "{[\\sigma{(w_{p})}\\sigma{(w_{a})}]}^{-1}", "ed53194144bf7a21082a42621d6ac402": "Z_3 = (T_1Z_1)^2 - (T_1Y_1)^2 + (Z_1Y_1)^2", "ed532c423a19f198472ab4a47f788ffe": "\n \\sum F = -10 - V_1 = 0\n ", "ed53ae6e6ef8d7bc017d5000aa9247e8": "\\frac{\\partial u}{\\partial t} = -\\frac{\\partial \\phi}{\\partial x}", "ed53c1d907161a3a94da4517deba5b8d": "T:=D^{-1}A", "ed53c20269a9674a322610251818fe99": " (1+ \\sqrt q_{i}q^{i} )^{-s} ", "ed53e2c6a69092a5f6bb73485dc854f5": " \n\\begin{align}\n &R_e = \\left(\\frac{\\rho V L}{\\mu}\\right)\n &\\longrightarrow\n &V_\\text{model} = V_\\text{application} \\times \\left(\\frac{\\rho_a}{\\rho_m}\\right)\\times \\left(\\frac{L_a}{L_m}\\right) \\times \\left(\\frac{\\mu_m}{\\mu_a}\\right)\n \\\\\n &C_p = \\left(\\frac{2 \\Delta p}{\\rho V^2}\\right), F=\\Delta p L^2\n &\\longrightarrow\n &F_\\text{application} =F_\\text{model} \\times \\left(\\frac{\\rho_a}{\\rho_m}\\right) \\times \\left(\\frac{V_a}{V_m}\\right)^2 \\times \\left(\\frac{L_a}{L_m}\\right)^2.\n\\end{align}\n", "ed53e7771012d058707e87c8dc877274": "d(k) = x(k)\\,\\!", "ed5422534d1d2f603958248d87b522f8": "\\langle L, \\le \\rangle", "ed543599c7c83ef09e19fcdb4ddac149": "\\alpha \\cup (K_i \\setminus \\beta)", "ed5493ea9d72c4cb45518b4d819e0934": "R_i'", "ed54fd24226e1022c3aaeaa8ecf7070a": "e^{-n/\\tau}", "ed55461f098911f2c4347d261ac3f4c4": "\\frac{E}{h+s}", "ed55b4af8f217ed52a7a6b71d55e2205": "2 + 2 \\ne 5", "ed55d7f962d2145c22aa4d55e6aac36e": " (V \\otimes W)_i = \\bigoplus_{\\{j,k|j+k=i\\}} V_j \\otimes W_k. ", "ed55f239f76d2f4b95fd85476f911259": " P [(N(t+ \\tau) - N(t)) = k] = \\frac{e^{-\\lambda \\tau} (\\lambda \\tau)^k}{k!} \\qquad k= 0,1,\\ldots,", "ed56481c9d0e3f0aec7f52cd4d671e93": "\n \\Delta \\tilde{x}_{SQL} = \\sqrt{\\frac{\\hbar\\vartheta}{M}}\\,,\n", "ed56684de28f1abe97668615685eb4ca": "X_t, X_s ", "ed574e22ace9ecbd9fd80c333ed1e351": "\\frac{\\partial E}{\\partial t}\\, +\\, \\frac{\\partial}{\\partial x} \\Bigl( (U\\, +\\, c_g)\\, E \\Bigr)\\, +\\, S_{xx}\\, \\frac{\\partial U}{\\partial x}\\, =\\,0\\,", "ed57b2c0af7b6408fea2568cd22159c2": "\n\\begin{align}\n\\boldsymbol{b}_\\text{frank}=&\\frac{a}{3} [ \\text{1 1 1} ]\n\\end{align}", "ed57c9432a5f56f6629f2e3100fefc39": "v = u + a.t + \\frac{1}{2}jt^2 ", "ed57d53220480a7ccd4cf482f7a6c310": "W\\left(-1\\right) \\approx -0.31813-1.33723{\\rm{i}} \\,", "ed57e8222838771c2e655d1c71efe074": "d\\in D", "ed585a8ade127f4e4173fb48e898cc87": "\\chi = 2 - k.\\ ", "ed58df1fdca1ca7669091f727b1dc418": "L^q", "ed58e6ec60afe2b5db7c2b7f7ed31609": "\\scriptstyle B(b)", "ed58f0d54256ee706cef0680af2b997c": "x(t) = \\sin (\\omega t + \\phi). \\ ", "ed59225332b26f1036034b16e9c21f1d": "H\\mathbb{Z}", "ed593383546749fe1abb85395cef6d61": "H_p(X) \\otimes H_q(Y) \\to H_{p+q}(X\\times Y)", "ed59e9b62b1b6b858b6423524b83580b": "\\frac{1}{\\lambda_\\mathrm{MFP}} = \\frac{1}{\\lambda_\\mathrm{el-el}} + \\frac{1}{\\lambda_\\mathrm{ap}} + \\frac{1}{\\lambda_\\mathrm{op,ems}} + \\frac{1}{\\lambda_\\mathrm{op,abs}} + \\frac{1}{\\lambda_\\mathrm{impurity}} + \\frac{1}{\\lambda_\\mathrm{defect}} + \\frac{1}{\\lambda_\\mathrm{boundary}}", "ed5a1519bb2a1390a892317058a8c8a6": "\n \\boldsymbol{\\varepsilon} = \\sum_{i=1}^3 \\sum_{j=1}^3 \\varepsilon_{ij} \\mathbf{e}_i\\otimes\\mathbf{e}_j\n ", "ed5a381f2fd70ecae7ac1753eb9f0861": " f|_{U_i} \\in \\Gamma ", "ed5a4881fa234308f868d5de1f187bc9": "(p \\lor q) \\vdash (q \\lor p)", "ed5a740f3824d3920d189d2ec379d257": "\\sum_i \\alpha_{ri} A_i \\to \\sum_j \\beta_{rj} A_j \\;\\; (r=1, \\ldots, m) \\, , ", "ed5a7d47394613c5685f9260826903ef": "\\pi_F\\circ \\varphi = f\\circ\\pi_E", "ed5abfdf3346fe27e68169673a351410": "x \\ll 1", "ed5af8e4302ca15076b71ffdefd0486b": "|f(z)| = |z|^{2}", "ed5afa8e0517bf3335334312d1464f7e": "\\mathrm{erfc}(z) = \\frac{z}{\\sqrt{\\pi}}e^{-z^2} \n\\cfrac{a_1}{z^2+\n\\cfrac{a_2}{1+\n\\cfrac{a_3}{z^2+\n\\cfrac{a_4}{1+\\dotsb}}}}\n\\qquad a_1 = 1,\\quad a_m = \\frac{m-1}{2},\\quad m \\geq 2.\n", "ed5b25145e72834e56aa15c812f6da8c": "w(X)=nw(X)=|X|\\,", "ed5b47f7cd1206d769c24b7e44be16cd": "l = l'", "ed5b6b623529e87939677ac26b43c7b2": "R^2 =\\frac{b^2}{e^2 \\cos^2 t -1} \\,", "ed5bc5d3e6134d94953812cf1fdc519c": "n_{1}^{n_2-1} n_{2}^{n_1-1}", "ed5be0261b8ee5e283a3c9dd42e9c6c4": "\\vec{E} \\equiv -\\nabla\\phi - \\dot{\\vec{A}}", "ed5bf1053055d86a9ef499a91a25cf7a": "{13 \\choose 1}{4 \\choose 3}{12 \\choose 2}{4 \\choose 1}^2 = 54,912", "ed5c00374d14f6e95f5c1cb2c1b4c043": "s^2+6s+5=0", "ed5c0da882601f40786bfcea93fa745b": " \\scriptstyle \\zeta > 1 \\,", "ed5c1e4dabf89601b3d08a7e25bb2d25": "a_{n}=\\sum_{k=0}^{n}c_{n,k}\\binom{n}{k}^{2}\\binom{n+k}{k}^{2}", "ed5c792f547e1fcc3693e6fdb5f26fe6": " \\alpha = 0", "ed5c88e362d78355ec8e6204dc781170": "N^a", "ed5cb0eb0be83865da479326310475ba": "S_{21}", "ed5cb2b5c00b467ae352b2c31ef14c26": "2\\gamma(x,y)=E\\left(|Z(x)-Z(y)|^2\\right) , ", "ed5cc3eca78c3541bd63277ffa9397be": "((\\neg q \\wedge (p \\rightarrow q)) \\rightarrow \\neg p", "ed5d610f9267a8c980bd19a16685e7cb": "\\sum_{j=1}^n {F_{S_i \\rarr S_j}} = 1", "ed5d7fc15ce41605ba3f8d023797f13f": "G(x_1,\\dots, x_n)=\\begin{vmatrix} \\langle x_1,x_1\\rangle & \\langle x_1,x_2\\rangle &\\dots & \\langle x_1,x_n\\rangle\\\\\n \\langle x_2,x_1\\rangle & \\langle x_2,x_2\\rangle &\\dots & \\langle x_2,x_n\\rangle\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\n \\langle x_n,x_1\\rangle & \\langle x_n,x_2\\rangle &\\dots & \\langle x_n,x_n\\rangle\\end{vmatrix}.", "ed5d950817408456b9e292c28460dd3e": "Y^2 = 4 X^3 - g_2 X - g_3", "ed5da64462335cfc7dc9245b08c1d899": "\\nu(W)", "ed5de50d06f0de58039c76799f4a5b1e": "\\Pi = \\{P_1,\\cdots,P_l\\}", "ed5ea3ebfecb165cee85722285e7d9c1": "\n(Axiom)\\quad\n{{}\\over X \\leftarrow X}\n", "ed5eeed24569bb922a55f1f033f95338": "s_n:=\\sum_{i=0}^n a_i", "ed5f23b8012a3e97c9010ed46d20d615": "\\scriptstyle 6p^{1.2}", "ed5f77fac2671bb917a2ffeb4fad7f2a": "\\displaystyle\\sum_x p(x\\mid y_1,I) \\log \\frac{p(x\\mid y_1,I)}{p(x\\mid I)}", "ed5f8fa3f7d1fc431f1ccad1380cf909": "\\psi(W)\\cdot\\psi(Z) - \\sum_{i_1 < \\cdots < i_k} (v_1 \\wedge \\cdots \\wedge v_{i_1 - 1} \\wedge w_1 \\wedge v_{i_1 + 1} \\wedge \\cdots \\wedge v_{i_k - 1} \\wedge w_k \\wedge v_{i_k + 1} \\wedge \\cdots \\wedge v_r)\\cdot(v_{i_1} \\wedge \\cdots \\wedge v_{i_k} \\wedge w_{k+1} \\cdots \\wedge w_r) = 0.", "ed5fb4aeb9111938fd5cb83e230866ea": "K_i (\\tilde M)", "ed5fd44eafca59cacc6eb0e25773126b": "L_2(3) \\cong A_4 \\twoheadrightarrow A_3 \\cong C_3", "ed6060cd3c11b746197b3e8b7efe4c73": "x\\in\\Sigma^*", "ed6075df8bd0460f7b6dc10775d1c955": " \\phi\\ (r) = \\ ", "ed612b24e09245de6b9c37b11541e89f": "H_*(X \\times Y)", "ed614f1f9d5fa1fcca6b939d0006c67c": "u_p e_p = Q_{pq} v_q(e'_p \\otimes e_q) e_q", "ed616c30ae4f291c589599e88d8d54f0": "\n= \n{1\\over i r} \\int_{-\\infty}^{\\infty} {k dk \\over \\left ( 2 \\pi \\right )^2 } {\\exp\\left( ikr \\right) \\over \\left(k + i m \\right)\\left(k - i m \\right)}\n=\n{1\\over i r} { 2\\pi i \\over \\left( 2 \\pi \\right)^2 } {im \\over 2 i m} \\exp \\left( -m r \\right) \n", "ed61a77f4dc06aed316a30b3553c1fb9": "PV=nRT\\,", "ed61fdfe911a5d024af7bf696b92ef8a": "s_1, s_2", "ed6236d2240c75c47da55c7b798e448a": "\\ \\delta", "ed6244206eeca51b2d0b65cc97d02784": "n = \\frac{M_1 - M_0}{t}", "ed6250d9f27272d338fede2058ecb35d": "C \\subseteq \\mathbb{F}_q^n", "ed6256862cefb260517b1e6ab6ebbdc0": "{10}^{\\,\\! 4\\cdot 2^{20}}", "ed625e2e288b2c94ca31bb13d97ee71d": "\\sum_{j=1}^n w_j x_j = W.", "ed626b767278b73728fa40b5bc4c3cb1": "Q_{3 \\times 3} = \\left[ \\begin{matrix} Q_{2 \\times 2} & \\bold{0} \\\\ \\bold{0}^T & 1 \\end{matrix} \\right] . ", "ed628fa661d789801865e8abc69c513a": "\n\\vec \\eta_1 = \\begin{pmatrix} x^1_1(t) \\\\ x^1_2(t) \\end{pmatrix} = c_1 \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\cos{(\\omega_1 t + \\varphi_1)}\n", "ed629517d9421c90766112b054ed863a": "B_{\\text{obs}} \\triangleq \\begin{bmatrix} B & \\begin{bmatrix} -M\\\\L_2 M\\end{bmatrix} \\end{bmatrix}", "ed630b4c998549add5f1eac7dfc0a9d1": "d_n=\\sqrt[3]{\\frac{7000}{1501.339 \\times n}}", "ed63c084329323c0374a4b0728bfc175": "\\{e_1, \\, ... \\, e_n\\}", "ed64210617a4031aec61abf46387325c": "\\scriptstyle y(2k+1)=ay(2k-8)+x(2k+1)", "ed646b699f96f4b7019d81d4e95e635c": "(\\mathbb{Z}/2^k\\mathbb{Z})^\\times \\cong \\mathrm{C}_2 \\times \\mathrm{C}_{2^{k-2}}.", "ed648f0349e41d9e8d7f9721aaa1d0ab": "\\boldsymbol\\beta\\,", "ed64b7690c77a73e324def6be32c9cd1": "A\\geq 0 ", "ed64ea213be648b167a86a6134077ae0": "\\int (af(x)+bg(x))\\, dx.", "ed6549399509abe735bdf078ce375eca": " \\left(\\left(a_0,a_1\\right),\\dots,\\left(a_{2n},a_{2n+1}\\right)\\right) ", "ed6574b082373a6c28b60cd3444a7b03": "x(t)=L^{-1}[(sI-A)^{-1}] x(0) + L^{-1} {(sI-A)^{-1} [BU(s) + EW(s)]}= \\phi(t)x(0)+\\int_{0}^{t} \\phi(t-\\tau)[Bu(\\tau)+Ew(\\tau)]dt ", "ed657f811fc1fa5d0a3187921cbfad70": "\\mathrm{Nat}(h^A,F) \\cong F(A).", "ed65f179863d53c68639cc2c2629c9db": "x_{Bi}", "ed66374205d920b59eaa5143460067f0": " \\mu = -\\frac{Ze^2B}{4 m}\\langle\\rho^2\\rangle.", "ed66736424d9129df13dd2c9159c3201": "MS_B = 84/2 = 42", "ed6682866718041c002360695303222e": "b=(1+u^2)(v-t)(1+tv)", "ed668e72897175db5265aaeb2d13469e": " \\text{Cardiac Output} = \\frac {\\text{oxygen consumption}} {\\text{arteriovenous oxygen difference}} ", "ed66bb8d0f19fad7d2e86ec4f52b05fc": "E(x^2)=k,\\,E(\\ln(x))=\\frac{1}{2}\\left[\\psi\\left(\\frac{k}{2}\\right)\\!+\\!\\ln(2)\\right]", "ed66cc2fa0fd81d969d7fbff85fea1ab": "\\nabla_{\\vec{e}_0} \\vec{e}_0 = 0 ", "ed66ea6ed6e9c1cae314e46e8464ce56": "(\\nabla^2 + k^2) A = -f.", "ed6707e16bd7b6214097edbc8428369f": "(\\mathfrak{g},\\mathfrak{h})", "ed6707e554751975443561a4c57ea0e3": "A_{\\alpha} = \\left(\\phi/c, - \\bold{A} \\right)\\,", "ed6730e04ac170e8be46771daddab9ee": "\\mathbf{x}=\\mathbf{s}+\\mathbf{n}", "ed676288993a7129f55512478c3ae26c": "\\scriptstyle{ \\widehat{\\beta\\gamma}, }", "ed67756134851225d7ac3a8fc363d8f6": "\\mathbf{v} = \\left[ \\begin{matrix} r & \\angle \\theta & h \\end{matrix} \\right]", "ed67763c22cd031ea7f3cddf967a4411": "m = \\Omega(\\delta^{2}n)", "ed6791b204b041d1f35b6a73ca8e8175": "\\mathrm{d}\\mathbf{p} = \\mathbf{p}_{\\mathrm{2}} - \\mathbf{p}_{\\mathrm{1}} = (m\\mathbf{v} + m\\mathrm{d}\\mathbf{v} + \\mathbf{v}\\mathrm{d}m) - (m\\mathbf{v} + \\mathbf{u}\\mathrm{d}m) = m\\mathrm{d}\\mathbf{v} - (\\mathbf{u} - \\mathbf{v})\\mathrm{d}m", "ed67eb465b3bb3bcb37e214adebb464f": "\\sup_n \\{a_n\\}", "ed6830261807bb112df414c6cf0554a3": "\\biggl(\\int_S |f|^p\\,\\mathrm{d}\\mu\\biggr)^{1/p}", "ed6859e32a2d2279a8e32d1bd9e29298": "|C(S \\upharpoonright n)|", "ed686c48f16f7300826b064a9669ef65": "\\, n_{k_i}", "ed68bb5cfe504967226307c2f01293d4": "\n G_I := \\cfrac{P}{2t}~\\cfrac{du}{da}\n ", "ed68efd369b9e5073323f008f27639ef": "p_1 \\dots p_n", "ed6957e46121a8bfa6464351e6a607bc": "L_2,", "ed69ac5802cb1bd2e2314401335c08b3": "MDA = \\frac{2.707+4.65\\sqrt{N}}{ET} ", "ed69c1145ec1ba37707dfe68bd2175f6": "x^{11}+x^{10}+x^6+x^5+x^4+x^2+1 / x^{23}-1", "ed6a1476c5fa7550eb6a8aea5c77ffd4": " \\Delta E = h\\,\\Delta\\nu \\ge \\frac{h}{\\Delta t}. ", "ed6a1a23ed2f138ef7bde86978d519e1": "0.25\\lambda/n", "ed6a2e3167e1d14c66bd8fb034f44aaa": "\\mathrm{Rep}_{\\mathbf{Q}_p}(K)", "ed6a618560ce2402d872c5d6f7b7fffc": "\\sum_{k=0}^\\infty \\varphi^{-2k}=\\frac{1}{1-\\varphi^{-2}} = \\varphi", "ed6a773de2b2abcfcbeae8611f165814": " \\frac {n!} {(n-k)!k!} \\left({\\frac{w}{d}}\\right)^{k-2}.", "ed6a9efccddcf2b8371ce209caaf74ff": "\\mathit{e}", "ed6aabe27265732e1c33cd9724a85a73": "\\mathcal{L}\\left\\{\\frac{df}{dt}\\right\\} = s\\cdot\\mathcal{L} \\left\\{ f(t) \\right\\}", "ed6ae3c7be7b113c38ca5b814011e53f": "\\gamma ={{z}_{\\text{max}}}-90{}^\\circ \\,,", "ed6af3621a4add3957e13552d4ca9a05": "T(u) = v", "ed6b0db25d60f6684ecb572c2e64c821": " \\frac{\\sec^2\\theta\\,\\Delta\\theta}{2}, ", "ed6b279389abe95422bd9f7ecb19940f": "\\partial F_1 = f_1 ", "ed6b3f3ec086dd31f7840e3a394a3112": " \\ k=\\pm 4, \\pm 2, -1 ", "ed6b6e0a783eb97b12cb09cd59399ab7": "s_{1}=+1, r_{1}=+1, x=1.0 \\, ", "ed6b9efe1f6ac48f28f7c4c34f19a902": "x_1+x_2=a_1,\\quad x_2+x_3=a_2,\\quad \\ldots,\\quad x_n+x_1=a_n", "ed6c6c00c02602162630d4f68bfc926a": "S_0=x\\,", "ed6c6cd3675722de7ea7cb4d70a29347": "(n,t_1)", "ed6c9bbb16bd5b444efed87e285f13cd": "\\mu_{pq} = \\sum_{m}^p \\sum_{n}^q {p\\choose m} {q\\choose n}(-\\bar{x})^{(p-m)}(-\\bar{y})^{(q-n)} M_{mn}", "ed6cd875452a0a7eec7869303148a534": "(J = Nm)", "ed6d115d7b863b62f123d68a829c19e8": "g_{i j}", "ed6da5e293fb173978af2b19d24b5b8c": "r_i=y_i- f^k(x_i,\\boldsymbol \\beta)- \\sum_{k=1}^{m} J_{ik}\\Delta\\beta_k=\\Delta y_i- \\sum_{j=1}^{m} J_{ij}\\Delta\\beta_j", "ed6dbb0ddbb723e034efa680a2d840ad": "C(S_0, T) = e^{-rT}[FN(d_1) - KN(d_2)]\\,", "ed6ddd4720bcadd4a265b428893d72d5": "z \\in \\partial K", "ed6e05720c73efd0c94f284144d37903": "\\left| \\frac{A+D}{2} \\right|\\le 1", "ed6e3c3497111219daba2f9d52b72bd0": " u(t) = \\cos(2\\pi t)+ i \\sin(2\\pi t) \\, ", "ed6e5d8126d2ea1aeca4c7512e269b90": "S_{21}, S_{31}", "ed6e6c78f74f439ce09e886ef86cb33b": "1 / \\Gamma(z) = z \\; \\mbox{e}^{\\gamma z} \\; \\prod_{n=1}^{\\infty} \\left(1 + \\frac{z}{n}\\right) \\; \\mbox{e}^{-z/n}", "ed6e7786535f220271de47998517ecbf": "\nx = a \\ \\sinh \\mu \\ \\sin \\nu \\ \\cos \\phi\n", "ed6eef5123ebd641a42f7ca67cf9999a": " ZHR = \\cfrac{\\overline{HR} \\cdot F \\cdot r^{6.5-lm}}{\\sin(hR)} ", "ed6f1ceed8f0fc532783242f16d65b02": "[X,X]=0", "ed6f3fa91fb4e1df6aca0e20e0f1d7ea": "\\cos(k a)=\\cos(\\alpha a)+\\frac{m A}{\\hbar^2 \\alpha a}\\sin(\\alpha a)", "ed6f4315529622d3dcc88320b6f286dc": "\\hbar = {{h}\\over{2\\pi}} = 1.054\\ 571\\ 726(47)\\times 10^{-34}\\ \\mbox{J s} = 6.582\\ 119\\ 28(15)\\times 10^{-16}\\ \\mbox{eV s}.", "ed6f47589595231410f32fcf80c3e52f": "q_2 = \\frac{Q}{W_{d/s}} = \\frac{20}{2} = 10.0\\text{ ft}^2/s", "ed6f93b094975abdee06f6eedc29fa8c": "f(t_1,t_2) = \\frac{1}{2\\pi} (1+(t_1^2 + t_2^2)/\\nu)^{-(\\nu+2)/2}.", "ed6fc17b7fc0d9970eac80c3d732441d": "\n{\\hat{\\alpha}}(q, {r_{\\rm c}}) = \\max \\left \\{ \\alpha :\nr_{\\rm c} \\leq \\min_{u \\in \\mathcal{U}(\\alpha, \\tilde{u})} R(q,u) \\right \\}\n", "ed6fef96cc4b459a57c4b1ce850ad961": "\\scriptstyle\\mathcal{L}(e^{at} f(t))=\\scriptstyle\\mathcal{L}(f(t-a)).\\,", "ed700ede61fe0260b127d74cf28e221e": "c=m^2 - n^2, \\, ", "ed7048cb5ada5f9fda53b3bc9f86765b": "\\mathcal{I}_{\\alpha, \\alpha}, \\mathcal{I}_{\\beta, \\beta}", "ed70753a63ac51fb8aafbbc84aca0969": "\\omega(t)\\leq \\tilde\\omega(t)", "ed714812a727cf80418e9b7f78852056": "P_{\\nu_b\\rightarrow\\nu_a}=\\left|\\left\\langle \\nu_a|\\nu_b(L)\\right\\rangle \\right|^{2}=\\left|\\sum_{a'}U_{a'a}^{*}U_{a'b}\\, e^{ -i \\lambda_{a'} L }\\right|^{2},", "ed71e19214ac70e0fe6ef73988be6466": " i = 1 , \\ldots , n ", "ed7219f3393cc368374a0102c6db5f65": " h(X|Y) = \\int_Y \\int_X f(x,y) \\log \\frac{f(y)}{f(x,y)} \\,dx \\,dy", "ed7220f498a3fd7e156b589c57dd4153": "z=f(x,y)", "ed72b56b83fab3d6192a957de66220c6": "p=1,q=0.", "ed734c7ff146fc7ca1a8591a4f884825": " \\frac{3 \\Lambda + \\Sigma}{4} = 1135.25~\\mathrm{MeV}/c^2", "ed73d81977f38dc9673b3cde9706e562": "\\varphi(u_n)", "ed73e4e7493f9cac676303341c538aa6": "\\hat \\alpha", "ed7431cf7e60e24029b7b95c886f748e": "y_2=\\varepsilon_1(x)\\int\\frac{\\varepsilon_2(x)}{\\varepsilon_1(x)}\\,dx.", "ed747e2222b3cc2f97ec9fb915393772": "\nS=S_0 f(g)\n", "ed7499fcedae62d4787a951e5440abdb": "z = z\\,", "ed74abf2b04845201503d357de660e5a": "f(x) = -\\sin x", "ed74d2dd900727401f8224f6874aa225": "f_{i,n}(u) = {{u - k_i} \\over {k_{i+n} - k_i}}", "ed75161bcea8dc1b8e616c48806cc6e7": "[x_i,x_{i+1}]", "ed7518276b9c99d1dc0e22b3fe03ea9f": "P\\approx P_0 \\frac{2X}{1-e^{-2X}}", "ed7557d1a0e8e24a28e5385e3bbec0a3": "h^\\mathrm{flat}_{jk} = h\\left(\\tfrac{\\partial}{\\partial y_j},\n\\tfrac{\\partial}{\\partial y_k}\\right) = \\delta_{jk}", "ed758ed2ce0ba8d0b8eca2d4af2a5469": " (D_\\left( x,y \\right)\\,B)\\left( u,v \\right) = B\\left( u,y \\right) + B\\left( x,v \\right)\\qquad\\forall (u,v)\\in X \\times Y. ", "ed7599c328a2fd3f32538d61fe5bac74": "\nV_\\mathrm{Th}\n= {R_2 + R_3 \\over (R_2 + R_3) + R_4} \\cdot V_\\mathrm{1}\n", "ed75b271b19415cd9f49b4c66b5bb19a": " (x^2+y^2-3x)^2 -4x^2(2-x) = 0.", "ed75b7fcd13383fd33e117abab988bd9": "\n y_i = \\beta_1 x_{i1} + \\cdots + \\beta_p x_{ip} + \\varepsilon_i\n = \\mathbf{x}^{\\rm T}_i\\boldsymbol\\beta + \\varepsilon_i,\n \\qquad i = 1, \\ldots, n,\n ", "ed75cdd7bbde1661115418ddcf204edd": "\\Theta \\subset \\mathbb{R}^k \\times \\mathbb{F}", "ed7610778f5c2d931a435b0cb9a33937": "\n\\int_{\\Omega} \\left\\{ \n\\left[ L \\left( \\alpha^A, {\\alpha^A}_{,\\nu}, x^{\\mu} \\right) - \nL \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) \\right]\n+ \\frac{\\partial}{\\partial x^{\\sigma}} \\left[ L \\left( \\phi^A, {\\phi^A}_{,\\nu}, x^{\\mu} \\right) \\delta x^{\\sigma} \\right]\n\\right\\} d^{4}x = 0\n\\,.", "ed76994bc83fca3e51bb011909246f96": "\\begin{bmatrix} u \\\\ v \\\\ w \\\\ \\end{bmatrix} = \\begin{bmatrix} c_{11} & c_{12} & c_{13} \\\\ c_{21} & c_{22} & c_{23} \\\\ c_{31} & c_{32} & c_{33} \\end{bmatrix} \\begin{bmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{bmatrix} ", "ed769b56b746b0bce28c5c333e8df685": "\\overline{\\rho \\phi \\psi}", "ed76e6836a4caf01cecdb120b3afbbe4": "O(|E| + |V| \\log|V|)", "ed76f90038e8068b90d71379b926fb6e": "\\Gamma(W)\\rightarrow \\Tau(W) ", "ed76fdcaf73546475779214dd91b4ca5": "C\\rightarrow kar(C)", "ed7707a4f3814f2ffb32c552979eceb2": "\\mathbb{R}(\\sigma(\\textbf{A}_P))<0", "ed770eb06fb1c07726e83060ed485603": "u=\\frac{5.5932x+1.9116y}{12y-1.882x+2.9088}", "ed77250bd89e7c286a709d7e154976d1": "x_i = e^{t/\\alpha}y_i, \\qquad 2 \\leq i \\leq n", "ed7765aa99e25a2e75bb48c0b685c140": "[\\ \\ , \\ \\ ]", "ed77acc687e83919481561fd913d1b56": " \\frac{}{} I = I_0 \\sin \\delta_0 ", "ed7822c175dd4385510f7259a7fcfa41": "\\overrightarrow{AB}", "ed782f000d2a26a0f930bf52eec39b6e": "\\gcd(x,y) = \\gcd(y, x % y)", "ed78528b874394c6e552bb415b2ca3e6": "\\textrm{Var}\\left(X\\right)=p\\left(1-p\\right).\\,", "ed78658404c8da81d01e0dd7c34ab907": " x^2 - 1 = -1 - ( x + 1) + ( x^2 + x + 1) = - p_1 - p_2 + p_3 \\,", "ed788cd94b3d8d4b504c3795f9ce1e48": "X_{ni}=1", "ed78c1308f4ba8ebb1cc0dbadf696a1f": "h(x) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ x\\neq 1\\ \\mbox{in}\\ G \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ x=1\\ \\mbox{in}\\ G\n\\end{matrix}\\right. ", "ed78db371c1b00a71a4e1b90e7730678": "\\mathbf{v} = (v_1, v_2, \\dots, v_{n - 1}, v_n)", "ed790a3811d5229b3ec2d6d8a3da3f35": "\\dot{\\widetilde{\\gamma}}(t)", "ed7926ee46a7ac2fef8bda8ed5a0fdce": "F(p) = - \\sum \\log p(i)", "ed796aa37c58574e581ae2c9ba40fe38": "\n\\delta \\left(c \\frac{d\\tau}{dq}\\right)^2 = 2 c^{2} \\frac{d\\tau}{dq} \\delta \\frac{d\\tau}{dq} = \n\\delta \\left[ \\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} \\left( \\frac{dt}{dq} \\right)^{2} - \n\\frac{1}{1 - \\frac{r_{s}}{r}} \\left( \\frac{dr}{dq} \\right)^{2} - \nr^{2} \\left( \\frac{d\\varphi}{dq} \\right)^{2} \\right]\n\\,.", "ed79b43a3ca3acb4589999f437a9086a": "\\eta_\\mathrm{Optics}", "ed79e77768f724a9e361b20466a00853": "\\; \\pi (A)", "ed79ee18a8e36f0d11406c11dd7c670e": "\\mathcal{Y} := \\bigcup_{i \\mathrm{\\ odd}} \\mathcal{P}(n-g_i)", "ed7aa821e449c167f114d3fa0012d23b": "V_v = 3571 \\cdot V_w", "ed7acdd00a859754b32ca1311d512f7d": "\\begin{align}\n\\eta_1 &= 1/6; \\\\\n\\eta_n &= \\sum_{\\ell=1}^{n-1}(-1)^{\\ell-1}\\frac{\\eta_{n-\\ell}}{(2\\ell+1)!}+(-1)^{n+1}\\frac{n}{(2n+1)!}.\n\\end{align}", "ed7b35a4c1114532a192cf829ce021b9": "7+\\epsilon", "ed7be4e66eee4dd4c81f7a3d155997fa": "\\lfloor X \\rfloor", "ed7bebbe73b2e1d1ea07b38dfe8d3e4d": "x \\in [x_\\mathrm{m}, +\\infty)", "ed7c12145e0a5fdaadd13e9d8a7534bf": "\\epsilon = \\delta^{-1}\\alpha\\delta", "ed7c52e218998dbf6930723f1984672b": "{C}=\\pi\\cdot{d}=2\\pi\\cdot{r}.\\!", "ed7c714a0e60644e01b6c5b3c48a5afa": "\\scriptstyle[H^+] \\simeq \\left( 10^{-14}+\\frac {K_hK_{a1}}{k_\\mathrm{H}} p_{CO_2}\\right)^{1/2}", "ed7d03c886c9d1dad4e2a303229286aa": "f(x) = \\frac{1}{2^{k/2} \\Gamma(k/2)} x^{\\frac{k}{2}\\!-\\!1} \\exp\\left(-\\frac{x}{2}\\right)", "ed7d3c8bffd4343db78016762d6d812e": "\\overline \\nu_2", "ed7d489036c33b0d185df9e92bc32f48": "\\frac{1}{2}C\\left(\\frac{V}{2}\\right)^2=\\frac{E}{2}", "ed7d590bbd7bdf5568721a0a13ff5374": "\\int \\sin^n x \\, dx = - \\frac{\\sin^{n-1} {x} \\cos {x}}{n} + \\frac{n-1}{n} \\int \\sin^{n-2}{x} \\, dx", "ed7dad7302214e8711dbdaa0dc0474d5": "\\{i_1, \\cdots i_k, i_{k+1} \\cdots i_n\\}", "ed7db5491bba307474212905f76f891b": " \\zeta_G ( \\alpha ) := \\frac{1}{N}\\sum_i \\sum_{j\\neq i }r^{-\\alpha}_{ij}, ", "ed7dc69cb52a55d4eac4922eccb1520a": "\n\\mathbf{T} \\cdot \\lambda = \\mathbf{r}-\\mathbf{r}_3\n\\,", "ed7ddcdc4c73a007daf5cb9dc17effc5": "\\,\\! R(\\tau) = J_0(2\\pi f_d \\tau)", "ed7e20efd90a490bdc0c15dff0fc9b0a": "Q_A", "ed7e29114077eccd51f05a64d7447ffa": "(M, d^{1/2})", "ed7e48cca2ec53d63b777c5fee680042": "\\mathbf{x} \\in \\mathbf{X}", "ed7e5ea85d6b18e439fe7b030457473a": "E = \\frac{1}{2}m\\left(\\dot{r}^2 + r^2\\dot{\\phi}^2\\right) - \\frac{GmM}{r},", "ed7e62475798b6186a1a54d606b5c19f": "= d\\phi^{\\alpha} - \\chi^{\\alpha}_{i}dx^{i} - u_{i}^{\\alpha}d\\rho^{i} \\,", "ed7e87e89723823cc5feb4fe84e41254": "q=2", "ed7e941046a9e7bab79310769437f86a": "\\sum_{k=1}^{b} W_{k} F_{k} = \\mathbf{W^T} \\mathbf{F} = 0", "ed7ec7ce60eb612dec32275ba6aa9a2e": "(v,w),", "ed7eeabb2261016162ae60ae63753da4": "L = \\bigoplus G_i/G_{i+1}", "ed7ef6cb7d712c55e9ff250268284bdb": "M_5 = a' + b + c' = (a b' c)' = m_5'", "ed7f019b7cfe75c19bb0508fb8ac5c18": " \\left ( q^2 + 2q, \\frac{q^2 + 2q -1}{2}, \\frac{q^2+2q-3}{4} \\right )", "ed7f033a06682064c9c5072ca67977e8": "{H}_{2-4}", "ed7f064793e0deabedcf46dec86bad3e": " \\Delta w_i = - \\alpha \\frac{\\partial E}{\\partial w_i} ", "ed7f16c82c37e36a2b18e4149a1e5f69": "I = \\Sigma \\times \\Sigma \\setminus D", "ed7f3586d0b08a68cbf5698b3078d72a": "H_\\alpha^{(1)}(ze^{im\\pi})", "ed7f3c109257b067c237bfbf8aff84b9": "\\operatorname{Morse}(M)", "ed7f9d95d6a2a372298526d76a2a2a8c": "\\frac{\\gamma}{\\beta_r}", "ed7feccfa79ff86c070817768a6e3083": "EL(\\Gamma)", "ed8010fc4c8f5cd9e325f497a8b8637a": "B[m,p]", "ed808eb976bc917a718569aefc98db2e": "x\\rightarrow\\infty", "ed8091406d9575b83ec449474a85d5b8": "L^1(\\mu)\\subset L^1(\\mu)^{**}=L^{\\infty}(\\mu)^*", "ed8097c0c5cb66ed920bdeef246d095b": "\\mu_A(T)=\\lim_U\\frac{[T:U]}{[A:U]}", "ed80aa84f31e54d3634ae090a341e050": "\\lambda(G) \\leq 2\\sqrt{d-1}", "ed80eb638396268b9f19405947d659b9": "a^\\Delta", "ed8146ea467056537b236894f311228f": "CK_i", "ed81659938ff785122ce318d892497ef": "\\limsup_{j \\to \\infty} \\langle T(u_{j}), u_{j} - u \\rangle \\leq 0,", "ed8190836bf3743fd6f3a6d3683ce93c": "~\\sigma", "ed81e79b5ee987c58f38505acf650734": "f:X \\to Y", "ed82332fd4bf9947a86908901383315b": "A_\\text{i} \\triangleq \\frac{i_\\text{out}}{i_\\text{in}}\\,", "ed827c1a77b3e247bb34236c351bdad6": "\\mathcal{F}_{\\cdot} = \\left(\\mathcal{F}_i\\right)_{i \\in I}", "ed8316d9eac20d8e4123528784bf8ca1": "= n R^{n - 1} \\omega_{n},", "ed8324568e87b0700bb2406bd488b12a": "\\mathbf{C}^{n+1}\\times\\mathbf{CP}^n\\to \\mathbf{CP}^n", "ed8367abed60e9cf938b5d72a0869b37": "C_{\\beta KI} = - C_{\\beta IK}", "ed836c32d708df8ae673bce4d4876c45": "(\\overline{y}_{11} - \\overline{y}_{12}) - (\\overline{y}_{21} - \\overline{y}_{22}) ", "ed83742167af62a6175bea69cbbf7025": "{}_sY_{\\ell m}", "ed83cd1d90320d940f1bcb90d166c4f0": " R_k(x) = \\frac{f^{(k+1)}(\\xi_L)}{(k+1)!} (x-a)^{k+1} ", "ed83e95d9183b55f0114cda557186d59": "D_{KL}(P\\|Q) \\ge \\sup_\\theta \\left\\{ \\mu'_1(P) \\theta - \\Psi_Q(\\theta) \\right\\}\n = \\Psi_Q^*(\\mu'_1(P)).", "ed845348833bfe1ff37c6f1c5a616220": "\\sum \\vec{a}(x_i) = 1\\,\\!", "ed845a60cf0b1896f0e53fbc4d45b72f": "\n \\begin{align}\n \\operatorname{arsinh}\\, z &= \\ln(z + \\sqrt{z^2 + 1} \\,)\n \\\\[2.5ex]\n \\operatorname{arcosh}\\, z &= \\ln(z + \\sqrt{z+1} \\sqrt{z-1} \\,)\n \\\\[1.5ex]\n \\operatorname{artanh}\\, z &= \\tfrac12\\ln\\left(\\frac{1+z}{1-z}\\right)\n \\\\\n \\operatorname{arcoth}\\, z &= \\tfrac12\\ln\\left(\\frac{z+1}{z-1}\\right)\n \\\\\n \\operatorname{arcsch}\\, z &= \\ln\\left( \\frac{1}{z} + \\sqrt{ \\frac{1}{z^2} +1 } \\,\\right)\n \\\\\n \\operatorname{arsech}\\, z &= \\ln\\left( \\frac{1}{z} + \\sqrt{ \\frac{1}{z} + 1 } \\, \\sqrt{ \\frac{1}{z} -1 } \\,\\right)\n \\end{align}\n", "ed8464d07be1438a59b2ec9e42b2d4df": "\\sqrt{XY} \\sim \\mathrm{K}(\\alpha,\\beta)", "ed847ac8e94e4196344d4ae3c03b7213": "v\\in K", "ed849ff6841bd797bfa960b76a47156a": "\\overline{d}_{\\dot{\\alpha}}X=0", "ed84d1bff9fb7bcc284a0b8ceb59c23b": "r = r_0", "ed859435962e6c480102f4dfff91deab": "\\operatorname{curl}(\\operatorname{grad}(f))=0", "ed85b4627b1df9ddd220d6e14b1448ca": "u_{yy}", "ed85c0a24d3a04e730e28809c75c4d14": "V^1", "ed85e0ca9a424498f65e5ca8fe1fb8db": "[x_{i1}, (x_{i1}+x_{i2})/2]", "ed862b489d0f8200da8d1f1a71d3591a": " \\int_{-\\infty}^{\\infty} w(t-\\tau) \\, d\\tau = 1 \\quad \\forall \\ t ", "ed8633a62caf1aabf4cf62af0504b914": "\\mathbf{X}\\left(\\mathbf{X}^T\\mathbf{X}\\right)^{-1}\\mathbf{X}^T", "ed865ceabb4c14d534a55859b5323665": "\\omega(t) \\ \\stackrel{\\mathrm{def}}{=}\\ \\phi '(t) = \\frac{d}{dt} \\phi(t).\\,", "ed8678c63cb55148a5a5019fc1cb9c87": " {\\rm Tr}^\\psi_\\omega(A) = \\omega \\left( \\left\\{ \\frac{1}{\\psi(1+n)} \\sum_{j=0}^n \\lambda(j,A) \\right\\}_{n=0}^\\infty \\right), \\quad A \\in L_{\\psi} ", "ed86868d362f6c10919ab2878be84e10": "\n \\left\\{ \\theta \\bigg| y \\le\n \\frac{\\hat p - \\theta}{\\sqrt{\\frac{1}{n} \\theta \\left({1 - \\theta} \\right)}} \\le\n z \\right\\}\n", "ed86944fcf27db0dd291df57b5d89f22": "\n\\int_a^b f(x)\\,dx \\approx \\frac{b-a}{2} \\sum_{i=1}^n w_i f\\left(\\frac{b-a}{2}z_i + \\frac{a+b}{2}\\right).\n", "ed86df4518d643ddf4c2a9e76cd2dc0c": "\n \\begin{align}\n & \n \\left( 1 - M_x^2 \\right) \\frac{\\partial^2 \\Phi}{\\partial x^2}\n + \\left( 1 - M_y^2 \\right) \\frac{\\partial^2 \\Phi}{\\partial y^2}\n + \\left( 1 - M_z^2 \\right) \\frac{\\partial^2 \\Phi}{\\partial z^2}\n \\\\\n & \\quad\n - 2 M_x M_y \\frac{\\partial^2 \\Phi}{\\partial x\\, \\partial y}\n - 2 M_y M_z \\frac{\\partial^2 \\Phi}{\\partial y\\, \\partial z}\n - 2 M_z M_x \\frac{\\partial^2 \\Phi}{\\partial z\\, \\partial x} \n = 0,\n \\end{align}\n", "ed873703db31b6fefef2a3873e895315": " C = {L}/{T} ", "ed8770740157dff320d2569dc6421964": "SecA = \\frac{BB+(TB-H) + (SB-CS)}{AB}", "ed877955e7a8576de8360e55985e614b": " V_0 = BV_0 + \\sum_{t=1}^{m-1} { RI_t \\over (1+r)^t } + {T_{m} \\over (1+r)^{m-1}} ", "ed87eb061fb38ffdbf993e30d9b2afd6": "\\{-b-r, (\\epsilon,g);(a_1,a_1-b_1),...,(a_r,a_r-b_r)\\}", "ed87ff72e4b807c39e43c12d461c4303": "a = x_0 < x_1 < x_2 < \\cdots < x_n = b", "ed8809120bcd737fe94b213b8138a9b7": "A_r \\hat{\\boldsymbol r} + A_\\theta \\hat{\\boldsymbol\\theta} + A_\\phi \\hat{\\boldsymbol\\phi}", "ed888347c9119abc58bf7bdb004388c4": "Q \\neq 0", "ed8886f370131f9a5b9d122c6964b96b": "\\chi=-{A\\over 4}+{5B\\over 4}", "ed88f2cd1fbd03318975b62fc2338fc9": "\\Delta = \\frac{\\bar{x}_1 - \\bar{x}_2}{s_2}", "ed890c469c2b728bc56ce96b2058536f": "\\mu^\\circ_{solid} = \\mu^\\circ_{liquid} + RT\\ln X_2\\,", "ed891125283283eea0dcec64c31836cd": "J_{\\alpha+1} \\cap \\textrm{Pow}(J_{\\alpha}) = \\textrm{Def}(J_{\\alpha})", "ed896000446e7ab1f8c1996e94533a62": "\\displaystyle{H_\\varepsilon f(\\varphi) = {i\\over \\pi} \\int_{\\varepsilon\\le |\\theta| \\le \\pi} {f(\\varphi-\\theta) \\over 1-e^{i\\theta}} \\, d\\theta={1\\over \\pi} \\int_{|\\zeta-e^{i\\varphi}|\\ge \\delta} {f(\\zeta)\\over \\zeta-e^{i\\varphi}}\\, d\\zeta,}", "ed89a43c51e47dc6b9417d32cfb10496": "H := L_{0}^{2, 1} ([0, T]; \\mathbb{R}^{n}) := \\{ \\text{paths starting at 0 with first derivative} \\in L^{2} \\}", "ed8a8a3160a9fcadbd09f10c586f703d": "\\sigma (\\pi_X (\\cdot), \\pi_Y (p)):P^{(n+1)(m+1)-1} \\to P^{(n+1)(m+1)-1}\\ ", "ed8aa60d1ed524a6550a6d068c892b9c": "XY=\\sqrt{\\frac{d^2}{16}+\\frac{g^2}{16}}=\\frac{\\sqrt{d^2+g^2}}{4}.", "ed8b4dff35d26c6f86d4f98c781663be": "\\Delta(y)=\\phi(y)+\\text{constant}", "ed8b7a028723ab12973db25f831c8ff7": "[x_{i-1},x_i]", "ed8b7a4caef0ece0614173434ca03241": " X_3 = 2Y_1T_1Z_1X_1 = 0", "ed8ba8177b0ec93671e3b82b067827a2": "(A,\\delta,\\varepsilon)", "ed8be0f142eab5d30ca2791941142567": "G_R \\to 0", "ed8c50578980c285f4bd7404633f9001": "\n\\widehat{x} = \\widehat{x_a} + \\boldsymbol{S_x}\n\t\\boldsymbol{A}^T \\boldsymbol{S_y}^{-1}(\\vec{y}-\\boldsymbol{A} \\widehat{x_a})\n", "ed8c5b2870a866c26f1403c5e4044667": "i,j\\in \\{1,\\dots,J\\}", "ed8c758637269882dd0eb4f88b433d27": "\\hbox{not } q", "ed8cb13b7d95d72a871eacc6a0104bd0": "=\\frac{1}{2}((1 +(-1)^{f(0)\\oplus f(1)})|0\\rangle + (1-(-1)^{f(0)\\oplus f(1)})|1\\rangle).", "ed8cb459b191b6a8d36a28d71a977074": "dn/d\\phi \\propto 1 + 2 v_2(p_\\mathrm{T}) \\cos 2 \\phi", "ed8ccd578570d71fecd51a3c096e5218": "\\lim_{x\\to c}\\frac{f(x)}{g(x)} = \\lim_{x\\to c}\\frac{f'(x)}{g'(x)}", "ed8cdbcd24894718c76a5d771836e001": "c^{2} t \\ t_1 - x \\ x_1 - y \\ y_1 - z \\ z_1 = 0.", "ed8d16102134f8d5a8a58ed9eee6e23e": "\\tau_{ij} = \\overline{u_i u_j} - \\bar{u_i} \\bar{u_j}", "ed8d4980947d0009cc340c7fa8dddf35": " W_{cu} = {3} \\times{{I_{S}}^{2} {R_{01}}}", "ed8d5ea1813678ae416d3dcf3e1f9618": " F = \\frac{2\\,221\\,564\\,096 + 283\\,748\\sqrt{462}}{491\\, 993\\, 569} = 4.5278295661\\dots. ", "ed8dc10d16734da02c78587c3aa28435": "a, b, c, d\\in A; p, q, r, s \\in P", "ed8e08dde7c28198e57228ea23c2d87f": "\\mathcal{O} (\\mathcal{C}) = \\{ o(x) : x \\in \\mathcal{C}\\}", "ed8e658f4d55deb9be1e5619a240bb99": "\\mathbf{e}_{i'}=\\mathbf{e}_i A^i_{i'}", "ed8e90a065d3f3e384f72e948d5d6b07": " (\\mu_n)", "ed8eb3a377dba8476a30de9c922192e7": "q = \\frac{Q}{b}", "ed8f3283c12d9e8ea53efa279d62397c": " \\langle P_n, P_n \\rangle = 1~, ", "ed905385510bf18406dfacbf7401f1e5": "gd_{\\mathbb{Q}}D_4=2", "ed9081730047fa9ca172fcee61bbee74": "11100,\\quad 11011,\\quad 10010,\\quad 10101.", "ed91b13c64ea5f31dd3db735b3c78571": " s_n(x)=o( \\log (n)^{1/p})\\text{ almost everywhere}, \\, ", "ed9247ec41f647887ef6a140fd3297a5": "\\lambda_s = s^{-1}\\sum_{\\alpha_i \\in \\Delta_s} \\lambda_i,", "ed9262dff987c60ef1c799add07d9989": " Y(L_{-n_1-2}L_{-n_2-2}...L_{-n_k-2}|0\\rangle,z) \\equiv \\frac{1}{n_1!n_2!..n_k!}:\\partial^{n_1}L(z)\\partial^{n_2}L(z)...\\partial^{n_k}L(z):", "ed92b6971e044e10e6bedd6cf9122602": "{Q}\\,y'' + (rQ'+L)\\,y' + [{\\lambda}_n-{\\lambda}_r]\\,y = 0\\,", "ed92f06c606c4f557a89cb0616766a2c": " p^{(j)}(x_i) = y_{i,j} \\qquad\\text{for all } (i,j) \\text{ with } e_{ij} = 1. ", "ed9306ec31292d531342c92bc0d8cd5d": " \\mathbb{C} ^{2 \\times 2} \\otimes \\mathbb{C}^{2 \\times 2} ", "ed93473dcbc652e06456bf3a7538570d": "x_1,x_2,\\ldots,x_m", "ed9390c7b3a9984a14a90a8cf753c587": "\\sigma(\\boldsymbol{r'})", "ed93c1055af517c7cbfad1368dc8126e": "p_0(x)=1, \\,", "ed940fe6fa2defab91e129e330807977": "\\lambda = 2\\pi c/\\omega", "ed94599b32753655f23ae5c06b5fef05": "\\mathbf{k}_j", "ed952b657ea416581e6b99c04b434a1f": "\\underline{a}_1,\\ldots,\\underline{a}_r", "ed95a5f864000fa0fd3429a74659bfa2": "= 10 + 10P(4,1) + 15P(3,2) + 6P(2,3) + P(1,4) ", "ed96053d88be8605f92083b3a2571ddb": "\\tfrac{4}{3} \\pi r^3 = (\\tfrac{2}{3} \\pi r^3) \\times 2,", "ed9607b6d9a15cf3a1856e8bf6b4584f": "\\chi_{\\text{e}}(\\Delta t)", "ed961918fcf268a889d8b69b9474d1da": " \\binom n 1, \\binom n 2, \\ldots, \\binom n{n-1} ", "ed96a59a267fbc87755bdb889bb29b4b": "E_\\nu", "ed96be8883c4798e77ac274b8a3e9d52": "G(i, j, \\theta) = \n \\begin{bmatrix} 1 & \\cdots & 0 & \\cdots & 0 & \\cdots & 0 \\\\\n \\vdots & \\ddots & \\vdots & & \\vdots & & \\vdots \\\\\n 0 & \\cdots & c & \\cdots & -s & \\cdots & 0 \\\\\n \\vdots & & \\vdots & \\ddots & \\vdots & & \\vdots \\\\\n 0 & \\cdots & s & \\cdots & c & \\cdots & 0 \\\\\n \\vdots & & \\vdots & & \\vdots & \\ddots & \\vdots \\\\\n 0 & \\cdots & 0 & \\cdots & 0 & \\cdots & 1\n \\end{bmatrix}", "ed96fc4c156adda26796bcb67a8b180f": " \\omega_{\\rm shm} \\approx \\frac{m^2}{L^4} \\, \\sqrt{m^2+L^2} = \\frac{1}{r^2} \\, \\sqrt{m^2+m r}", "ed970a849c73aa76673967a5b87f1d65": "\\underline{\\psi(x)\\bar\\psi(x')}=\\int{d^4p\\over(2\\pi)^4}{i\\over \\gamma p-m+i0}e^{-ip(x-x')}", "ed970bb4b6c725588b409188d7170f92": "q_{kj}", "ed975c2a4661b13a1743a6f50f475e83": "\\tfrac{2000}{100} + 45 = 65", "ed980505200d3f01c6388dd0aa658009": "\\displaystyle{Y_1=\\begin{pmatrix} 0 & 0 & 0\\\\ 0 & 0 & y_1\\\\0 & y_1^* & 0\\end{pmatrix},\\,\\,\\,\nY_2=\\begin{pmatrix} 0 & 0 & y_2^*\\\\ 0 & 0 & 0\\\\y_2 & 0 & 0\\end{pmatrix},\\,\\,\\,\nY_3=\\begin{pmatrix} 0 & y_3 & 0\\\\ y_3^* & 0 & 0\\\\0 & 0 & 0\\end{pmatrix}.}", "ed989610eb329acb7306e63111f7a029": "e_1+\\cdots+e_n=0,\\,", "ed98c7994ccd9cd05922ceb2cd126dbb": "\\text{Im}(\\rho)\\,\\hat{=}\\,0 ", "ed98f0f2b7dc1dd3816580f14b5162e7": "z_n = \\cos(2^n\\theta)+i \\sin(2^n\\theta)", "ed98f4c791c5824348f528f3e8cba31b": "N^l \\cdot N", "ed994ae4f473bbd5b6ec6444f8469a89": "\\mathbf{F}=e\\mathbf{E}+ \\frac{e}{c} \\mathbf{v} \\times \\mathbf{B}", "ed997cb35d625194114d1dcd5e33243d": "\\aleph_1 = |\\omega_1 |", "ed998399166f58e6ffb2528c20b55162": "\\frac{f_1}{f_2}\\circ g=\\frac{f_1\\circ g}{f_2\\circ g}.", "ed99bc43aae801673a1d00ecd19b899f": "\\left( \\frac{\\partial A}{\\partial z} \\right)_{x,y} \\!\\!\\!= \\left( \\frac{\\partial C}{\\partial x} \\right)_{y,z}", "ed99f2b479920bab6ad45c37cfb14d10": " \\cos \\alpha = \\frac{ A_x }{ \\sqrt {A_x^2 +A_y^2 +A_z^2} } = \\frac {A_x} {\\| \\mathbf A \\|} \\ , ", "ed9a1849e8a087d694e24842503c3b18": "\n5280 = -\\sqrt[3]{j\\left( {\\scriptstyle\\frac{1}{2}} \\left( 1 + i\\sqrt{67}\\, \\right)\\right) }.\n", "ed9ab2ddf7a013207a8eaf73d51334d5": "K_a = \\frac{[\\mbox{H}^+] [\\mbox{A}^-]}{[\\mbox{HA}]}", "ed9aca9201b94c8debe647899a3a637b": "\\boldsymbol{e}_k\\, \\sigma\\, a\\; \\text{e}^{\\displaystyle k\\, z}\\, \\cos\\, \\theta\\,", "ed9ad6872783e91a6dd8ed3290a0d299": "\\begin{align}\\hat{\\sigma}^2 &= \\left(\\frac{1}{n} \\sum_{i=1}^n \\frac{(\\ln S_{t_i}- \\ln S_{t_{i-1}})^2}{t_i-t_{i-1}} \\right) - \\frac 1 n \\frac{(\\ln S_{t_n}- \\ln S_{t_0})^2}{t_n-t_0}\\\\\n& = \\frac 1 n \\sum_{i=1}^n (t_i-t_{i-1})\\left(\\frac{\\ln \\frac{S_{t_i}}{S_{t_{i-1}}}}{t_i-t_{i-1}} - \\frac{\\ln \\frac{S_{t_n}}{S_{t_{0}}}}{t_n-t_0}\\right)^2;\\end{align}", "ed9b474c01280ef475459cbabbc3d9e0": "{\\rm DEFS}[y]", "ed9b7fa112048669dc495a1535aa32e6": "\\,{}_pF_q(a_1,\\ldots,a_p;b_1,\\ldots,b_q;z) = \\sum_{n=0}^\\infty \\frac{(a_1)_n\\dots(a_p)_n}{(b_1)_n\\dots(b_q)_n} \\, \\frac {z^n} {n!}", "ed9b89c6f0a2e190c19725ea72d6f443": " f-g = 0 ", "ed9b8bdb6d9ed65aa1e9f2f415d34d76": "c_nx^n+c_{n-1}x^{n-1}+\\cdots+c_2x^2+c_1x+c_0", "ed9bbe90927f043afd8ed46640409367": "\\Pi(k,\\nu,\\phi)=\\int_0^\\phi\\frac{d\\theta}{\\left(1-\\nu\\sin^2\\theta\\right)\\sqrt{1-k^2\\sin^2\\theta}}, \\text{ for } \\left|k\\right| \\le 1", "ed9bfb2321ba3992d8c4473120c1cf78": "\\mathbf{a}_1 = \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {|\\mathbf{b}|^2}{\\mathbf{b}} = \\frac {\\mathbf{a} \\cdot \\mathbf{b}} {\\mathbf{b} \\cdot \\mathbf{b}}{\\mathbf{b}}.", "ed9c4afa48205eb8b740b89efd2d1cd5": "= (\\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha) (e_\\gamma^I V_I)", "ed9cb3ca7061ec53a44d4162b35a4d6a": "\n\\begin{align}\nx_1 & = \\textbf{H} I = \n\\begin{bmatrix}\nh_1 & h_2 & h_3\n\\end{bmatrix}\n\\begin{bmatrix}\n1 \\\\\nj \\\\\n0\n\\end{bmatrix}\n= h_1 + j h_2\n\\\\\nx_2 & = \\textbf{H} J =\n\\begin{bmatrix}\nh_1 & h_2 & h_3\n\\end{bmatrix}\n\\begin{bmatrix}\n1 \\\\\n-j \\\\\n0\n\\end{bmatrix}\n= h_1 - j h_2\n\\end{align}\n", "ed9cfc3df637e969f96560cf24eb699d": "3^3\\cdot5^7:2^{21}", "ed9d0bcf74478420fd185a9244e15b39": "z_0' = \\frac{d}{dc} f_c^0(z_0) = 1", "ed9d15e02ff255e6955280a51c60d94b": "\\mu_m(\\{\\sigma \\in X : \\sigma\\uparrow 0 = s_0 \\}) = 1", "ed9d1db862407c9955ffce29beb33530": "\\int\\operatorname{arsinh}(a\\,x)^2\\,dx=\n 2\\,x+x\\,\\operatorname{arsinh}(a\\,x)^2-\n \\frac{2\\,\\sqrt{a^2\\,x^2+1}\\,\\operatorname{arsinh}(a\\,x)}{a}+C", "ed9d7a3a695ac8fb6e96c3a3552fc70e": "\\log p \\approx \\log n \\approx {n.} ", "ed9d7b2b2f52a1fd879fe31f79242f31": "J = \\det(\\boldsymbol{F})", "ed9defbc470c9c33d536d82e279873c0": "\\hat{A}_0 = 1", "ed9e0d06261db1f5311905b767de6256": "\\phi(rear)=\\theta-\\psi+\\frac{b}{V}\\frac{d\\theta}{dt}", "ed9e1165b2fadfd09ce65d7c515c8777": "\\sqrt{-1}\\omega", "ed9eb731a9cf79b977c83abc3e8711f9": "P(s|s',o)", "ed9ed01d86dd74b439cfcda14b08455c": "\\scriptstyle \\vec{s}", "ed9f0164b96c7128d3948fa1f658cd2e": " \\mu(\\gamma A) = \\gamma \\mu(A)\\,", "ed9f27b1985462e6a73fb994ba1a1466": "y = f^{-1}(x)", "ed9f27e90a554a357fe381d98e51e805": "0\\to A\\to B\\to C\\to 0\\;", "ed9f2d13a331d8dbbe14ecdb1eebcd52": " V_p = \\mathrm{Volume \\ of \\ the \\ proportion \\ of \\ plate \\ immersed \\ in \\ liquid} ", "ed9f352ed8f1a18aec86edb69e1662a0": "\\scriptstyle R_0 \\;=\\; 1", "ed9f3f1cd8c19bcd56a89633c6949db4": "\\{|e_n\\rangle\\, , \\, n = 0, 1, \\dots \\}", "ed9f7848645f0aa6b62ad6a501408cc1": "\\cos(t)^2 + \\sin(t)^2 = 1\\,\\!", "ed9faa374752c6d2be57b7fe54fb563d": "BC \\to BD", "eda0899ad63956515bfd670c5496a666": " c_p = ", "eda11d6da0f3fd940a60b85a120d8754": "\\{ T+x \\, : \\, x \\in \\mathbb{R}^d \\}", "eda154ba11bee01de909158a7621dcdf": "\\eta(\\tau) = q^{1/24}\\prod_{n\\ge 1}(1-q^n)", "eda15ef361a606c23a6ab80e2ebe976e": "v(t)\\rm MPa^{-1}", "eda199874d89f6beec2062820f4b61c9": "\\theta\\,\\!", "eda249936f57e48c0240483a661fe2b0": "T_W=\\frac{1}{8}\\frac{\\hbar^2}{m}\\int\\frac{|\\nabla n(\\vec{r})|^2}{n(\\vec{r})}dr.", "eda2650df66ced2e07e491ff05de7c6f": "t= \\gamma(t' + \\frac{vx'}{c^2}) ", "eda287e2cc5d4289c7897d423d7576c0": "{k_1 \\over k_2} = \\frac {\\ln(1-F_1)} {\\ln[1-(F_1R_P/R_0)]}", "eda2f67338a0143d62e79d96007acc77": "\\mathcal{K}(X)", "eda33b6bae1143b86cd5ef97927b881b": "\n \\log f(\\theta x + (1 - \\theta) y) \\geq \\theta \\log f(x) + (1-\\theta) \\log f(y)\n ", "eda46dde957eb62fe5659f6a65d90e03": "\nS(\\rho \\| \\sigma) = \\operatorname{Tr}\\rho (\\log \\rho - \\log \\sigma).\n", "eda47526caef935cfb3acb84804ce7e8": "\\rho^2=\\alpha^2+\\beta^2+\\gamma^2\\,", "eda4ce45054fb622c0111b91aa6ad21b": " (f\\circ g)'(c) = f'(g(c))\\cdot g'(c). ", "eda5067672e79df265ab7ed258d34a1f": "\\{ \\mu_{T} | T \\in \\mathcal{A} (E) \\}", "eda509aff522b03919813e234b5771fb": "\\rho^2\\frac{\\ddot{P}}{P}+\\rho\\frac{\\dot{P}}{P}+k^2\\rho^2=n^2", "eda51b4e1cf7d219c5cd5b82aafc8fc4": "u := m (S \\otimes 1)R^{21}", "eda51e7b7303fb59707c9f0f6016499e": "\\hat{J_0} = \\hat{J}_1 +m_2L_1^2 = J_1 +m_1l_1^2 + m_2L_1^2 ", "eda5264b96630200a69de78e4c8a78d9": "\\epsilon \\phi", "eda52b14c608f00393693a057a6fea5e": "C_t", "eda539031741e5133f20d722a678ba34": "X_{\\mathcal{T}}^*", "eda5515a48b6e9d8c79f809b6f47d995": "|\\mathbf{k}_o| = |\\mathbf{k}_i|", "eda55bc5592abc8f8e66b6512a47ac02": "n'-1", "eda55c28cdb152eb5b88752f28cb37c7": "T_A=[A]+\\sum_i{p_i \\beta_i[A]^{p_i}[B]^{q_i}}", "eda5666af795f4a1ef6454cc62ee6b08": "\\phi_{\\mathbf{r}}(\\mathbf{k})=\\frac{1}{(\\sqrt{2\\pi})^3} e^{-i \\mathbf{k}\\cdot\\mathbf{r}}", "eda57d00641d54677f2886de9def66e9": " c_j | N_1, N_2, \\dots, N_j = 0, \\dots \\rangle = 0 ", "eda58a6d84ea49c3255cd529ecda0b4e": " (0.00, 0.50)", "eda5948ebb2e0ac0daf3fa99018ddb79": "k = n = 0", "eda5db164b029e6e84303b2781fb6d3a": " \\frac{\\partial}{\\partial z} = \\frac{1}{2} \\Bigl( \\frac{\\partial}{\\partial x} - i \\frac{\\partial}{\\partial y} \\Bigr), \\;\\;\\; \\frac{\\partial}{\\partial\\bar{z}}= \\frac{1}{2} \\Bigl( \\frac{\\partial}{\\partial x} + i \\frac{\\partial}{\\partial y} \\Bigr),", "eda629c164a6e26b1df467cf6f13fefa": " F_1 = y, S_1 = \\_, A_1 = n ", "eda62eedfa94160d1f3ebc2903945f00": "(\\mathbf{e}_i)_j = \\delta_{ij} ", "eda6566a52fbe10f306422bef7ef83ca": "\\hat{\\textbf{x}}_{0\\mid 0}", "eda66e857dd8d0d0ff1f2f00db794896": "\\frac{\\mbox{Enterprise Value}}{\\mbox{Net Sales}}", "eda70b5a6c2d6e280ae5c3fafb10c20b": "\nU \\frac{(U - V_{\\rm f} \\cot \\beta_{\\rm 2})} {g}\n", "eda72054a27685c46abe70195d2aae66": "{\\mathcal O}(n^3)", "eda78509b82c0a57b40fb65b78a67e26": "{\\frac {\\sin \\left( \\alpha/2-\\beta/2 \\right) }{\\sin \\left( \\alpha/2+\\beta/2 \\right) }}= {\\frac {\\cot \\left( \\beta/2 \\right) -\\cot \\left( \\alpha/2 \\right) }{\\cot \\left( \\beta/2 \\right) +\\cot \\left( \\alpha/2 \\right) }}= {\\frac {a-b}{2s-a-b}}", "eda7ea8bbb30176aceecfac5f4212d01": "- \\left( 2 r \\Omega \\dot\\theta ' + r \\Omega^2 \\right)\\hat{\\mathbf{r}} + \\left( 2 \\dot r \\Omega \\right) \\hat{\\boldsymbol\\theta} ", "eda88d09776cf3654c339226256fc0ea": "\\mathrm{SCl_4 + 6NaOH \\ \\xrightarrow{}\\ Na_2SO_3 + 4NaCl + 3H_2O }", "eda8b4ba20c955f41b449157e43f867f": "\\frac{1}{\\mathrm{B}(\\boldsymbol\\alpha)} \\prod_{i=1}^K x_i^{\\alpha_i - 1} ", "eda900194f172455b6b52714d3b86bcc": "|\\mathbf v| := \\sqrt {\\langle \\mathbf v , \\mathbf v \\rangle}", "eda91bf97280cd507892b5acd745cd17": "\\langle x,y \\rangle", "eda96d585a82258c71f7f377e36bcb32": "\\alpha\\colon x \\mapsto x^q", "eda9a996acfd82a6df556e715aceacc9": " \\frac{\\partial \\pi(x,t)}{\\partial x} =\\frac{\\partial (x p(x) - C(x))}{\\partial x} -t= 0 ", "eda9b264a13d1483f442a9864ee5e3c7": "12 \\sqrt{n}", "eda9da937103ad49896d34cb160c539c": "\\rho_b", "edaa1ecfce824b9591deefe2202ecbd1": "D^0_\\sigma(\\mathbf{r}) = \\tfrac{3}{5}(6 \\pi^2)^{2/3} \\rho^{5/3}_\\sigma(\\mathbf{r}).", "edaa7669563c47e8d5a22cd5429e7bf9": "Z\\vert_{\\mathrm F_{SO}(M)}", "edaa9c396fe3b13cb8871f78c3ef0511": "\nf_v (v_i) =\n\\sqrt{\\frac{m}{2 \\pi kT}}\n\\exp \\left[\n\\frac{-mv_i^2}{2kT}\n\\right].\n", "edaacaaada7d629514b3f30a44dc8e6e": "\\sum_{\\sigma\\in S_n} \\mu^{\\otimes n}(A_{n,\\sigma}(s,t))\n\\le\\mu^{\\otimes n}\\bigl(I_{s,t}^n\\bigr)=\\bigl(\\mu(I_{s,t})\\bigr)^n.", "edab6f15a05241b96c59c0f85a8532f1": "{r \\over \\sin \\vartheta} = {a \\over \\sin \\psi},\\ r = a \\frac {\\sin \\vartheta}{\\sin \\psi} = a \\frac {\\sin l(\\theta)}{\\sin (l(\\theta) - \\theta)}", "edab9e19aef4e0143b03b285a9324eaa": "H_\\mathrm{e}\\,", "edab9fa18625593df3a4bd5d7e576b19": "n_{42}", "edac075ae581cbb3c6aa8e742b87b746": "\\hat{\\phi}_{-1}:=W_0(\\left| \\xi \\right|) with ", "edac69894c8e549457ddb8f34e893692": " \\mathbf{J} \\cdot\\mathbf{A} = I , \\,\\!", "edac6b66cda9ae09ee93ca0bd8eb5241": " X_i \\sim \\chi^2(k=2) \\qquad i=1, 2, 3 ", "edad204be5a7ae160d8d230145c5a828": "F(a,t,i(\\cdot,\\cdot))=\\int_{0}^{a_M}{k(a,a_1;t)i(a_1,t)da_1}", "edada6e906b238984bebbddb15dd4436": "[0.x_1] = \\frac{x_1}{2}", "edadcc2dcf871a5a445b3526d6cf8ee2": "\nF_{n} = F_{0} \\cdots F_{n-1} + 2\\!", "edadf9996f750b583ea04483a8b62860": "\\alpha_0, \\alpha_1, \\beta_0, \\beta_1", "edae804d7e1cc5496f28db99f7ed8579": "E_\\mathrm{p,g} = -\\frac{G m_1 m_2}{r}\\,\\!", "edaf18d568f53a75223c61756478e7c8": "\\sum_{k=0}^{n} ar^{2k+1} = \\frac{ar(1-r^{2n+2})}{1-r^2}.", "edaf1d7ea102a5a498dd7069da899703": "F(S) = \\sum_i f(S_i),", "edaf22282c2d9dfeb3e62217d467cfb4": "x^3 + 7x^2 + 8x + 2.", "edaf84d7e6d2dcaec96d544ef163879c": " c = \\sqrt{12^2 + 12^2}. \\,", "edaf89745d321f28cbce2675897b613f": "\\Sigma = \\begin{bmatrix}\n\\begin{matrix}0 & \\lambda_1\\\\ -\\lambda_1 & 0\\end{matrix} & 0 & \\cdots & 0 \\\\\n0 & \\begin{matrix}0 & \\lambda_2\\\\ -\\lambda_2 & 0\\end{matrix} & & 0 \\\\\n\\vdots & & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\begin{matrix}0 & \\lambda_r\\\\ -\\lambda_r & 0\\end{matrix} \\\\\n& & & & \\begin{matrix}0 \\\\ & \\ddots \\\\ & & 0 \\end{matrix}\n\\end{bmatrix}", "edaf8f6c63bf5ad499b187caf455294d": "\n\\begin{pmatrix}\na_{11} & a_{12} & a_{13} \\\\\na_{21} & a_{22} & a_{23} \\\\\na_{31} & a_{32} & a_{33}\n\\end{pmatrix}\n=\n\\begin{pmatrix}\na^2 + b^2 - c^2 - d^2 & 2(bc - ad)& 2(bd + ac) \\\\\n2(bc + ad) & a^2 - b^2 + c^2 -d^2 & 2(cd - ab) \\\\\n2(bd - ac) & 2(cd + ab) & a^2 - b^2 - c^2 + d^2\n\\end{pmatrix},\n", "edb01a29ee35d6932c2565fb7b475e2e": " \\varepsilon_{r} = \\varepsilon_{r}' + i \\sigma \\lambda \\kappa, ", "edb01c5388b44dfa323866657ea1d31f": "y_c=\\sqrt[3]{q^2 \\over g}", "edb068e458c4687082505452113e1adb": "\\Pr[S] = 1", "edb10c1fecdf896e9d3a76a246f8524a": "p_{ij} = \\frac{1}{4\\pi\\epsilon_0 S_j}\\int_{S_j}\\frac{f_j da_j}{R_{ji}}.", "edb12369129c262da6d01e85bb665643": "\\alpha\\leq\\Omega", "edb12e4bf0e5a411fb20b9a062818c2b": "\\sqrt{-h}", "edb187d487fbe7d24b136612f9a28573": " z=L\\frac{x^2}{2} + Mxy + N\\frac{y^2}{2} + \n\\mathrm{\\scriptstyle{{\\ }higher{\\ }order{\\ }terms}},", "edb19e786604040083fd843defbee21f": "p(z)=(1-F(z))", "edb1beed9b2042065d90570bab3b657e": "\\mbox{Internal virtual work} = \\int_{V^e}\\delta\\boldsymbol{\\epsilon}^T \\boldsymbol{\\sigma} \\, dV^e = \\delta\\ \\mathbf{q}^T \\int_{V^e} \\mathbf{B}^T \\big\\{\\mathbf{E}(\\mathbf{Bq} - \\mathbf{\\epsilon}^o)+\\mathbf{\\sigma}^o\\big\\} \\, dV^e \\qquad \\mathrm{(10)}", "edb24b29f900f17a938e460f14d6054f": "\\alpha^{Z_\\alpha(n)} = 1 + \\alpha^n.", "edb2b444a34268ce0df3f17303599be2": "\\frac{dh}{dx} = \\frac{dh}{dg} \\frac{dg}{dx}.\\,", "edb317bda8969bf8c1ab17246f1197f7": "\\log_{10}P = A - \\frac{B}{C + T}", "edb33204eb8ebe0c551cba602ad511cb": "x_1,x_2,\\cdots,x_n", "edb3590f7e6b3899edd0b245dffe0099": "X_{ij} \\sim \\mathcal{N}_p(\\mu_i,\\, \\Sigma_i) \\ \\ (j=1,\\dots,n_i; \\ \\ i=1,2)\\ ", "edb3d7d312c3888830c1123abbb45fa0": "\\frac{\\mathrm{D} \\boldsymbol{\\mathrm{d}s}}{\\mathrm{D}t} = \\left( \\boldsymbol{\\mathrm{d}s} \\cdot \\boldsymbol{\\nabla} \\right) \\boldsymbol{u}.", "edb3fa6021145efb3c4bf8e13af3c6d0": "A = \\begin{bmatrix} 1 & 3 & 1 & 4 \\\\ 2 & 7 & 3 & 9 \\\\ 1 & 5 & 3 & 1 \\\\ 1 & 2 & 0 & 8 \\end{bmatrix} = \\begin{bmatrix} 1 & 3 & 4 \\\\ 2 & 7 & 9 \\\\ 1 & 5 & 1 \\\\ 1 & 2 & 8 \\end{bmatrix}\\begin{bmatrix} 1 & 0 & -2 & 0 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} = CF\\text{.}", "edb4032f8f810201276bfe2e7b64cc1f": "\\Omega_{\\Lambda} = 0.11,\\,", "edb45194d9a17b71cc46b4b8914e4a43": "\\kappa\\rightarrow(\\lambda)^{<\\omega}_m", "edb48bf2a052b7ed4273fd13ffb25e63": "\\varphi_{F}(e,x+1,y) \\simeq h(\\varphi_e(x,y),x,y).\\,", "edb4c0a59a2e52f9037a563b39401108": "\n Q_{\\rm TM_{mnp}} = \n \\frac{Z_0 lwh}{4 R_s} \\frac{k_{xy}^2 k_r}\n { w(\\gamma l+h) k_x^2 + l(\\gamma w+h)k_y^2}\n ", "edb51d02bf5079bb8352208cbfa691ad": "p(\\boldsymbol{Y}|\\boldsymbol{X})", "edb5a1e4210c931e8075d54a84c9d029": " T_n(z)=D_n(z)+D^\\sharp_{n}(z) \\quad, \\quad T_{n-1}(z)=\\frac{ D_n(z)-D^\\sharp_{n}(z) }{z-1}", "edb5abcc847f9551cbee3f000355b2f2": "L_R", "edb5c8e6c3e844ce19fbf3f8e6d1b588": "~K=mV/\\hbar~", "edb5ebead8dbee7a91a943f24d14b22c": "\\sum_{k=0}^{x} k^p = \\frac{B_{p+1}(x+1)-B_{p+1}(0)}{p+1}.", "edb62681381317566c028455f6282a89": "X^*(s)", "edb6392e21d2db178ed8e29844ed73ff": " \\frac{\\partial \\rho} {\\partial n} = \\frac{\\partial (XYZ)} {\\partial (X'Y'Z')} = \\frac{a e'}{\\rho^2} \\left(1 +\\frac{\\rho a'_\\rho}{c^2} \\right)", "edb645f4953478b3fe8fe09c53b62b54": "\\frac{\\partial \\theta}{\\partial t}= \\vec{\\nabla} \\cdot \\left(\\sum_{i=1}^n{\\vec{q}_{i,\\,\\text{in}}} - \\sum_{j=1}^m{\\vec{q}_{j,\\,\\text{out}}} \\right)", "edb6866664d15c5522c207afc1d5e92b": "\\scriptstyle \\deg(S_j) \\;<\\; \\nu_j", "edb7326eb6a16972a5804e601ead1959": "\\rho(\\mathbf{B}|\\boldsymbol\\Sigma_{\\epsilon})", "edb78f4395936204605843eb48c6b219": "h << w", "edb7bee62bf93b5076cba563af9f6aa0": "\n [\\boldsymbol{Q}(\\boldsymbol{N})]_{ij} = B_{ijkl} N_j N_l.\n ", "edb80d9eebb85ce1cf099a4effc26d5f": " \\operatorname{ var }( r ) = \\frac{ 1 }{ n }\\left[ \\frac{ s_y^2 }{ m_x^2 } + \\frac{ m_y^2 s_x^2 }{ m_x^4 } - \\frac{ 2m_y s_{ xy } }{ m_x^3 } \\right] ", "edb86d23b802cdd6a5a6663f5f4d81b4": "\\pi^{(n)}:\\mathbb{N}^n \\to \\mathbb{N}", "edb90376706beaa43f7f22195aae11e6": "x_v", "edb9059105668247cbf033dc6815b858": "\\ t\\ ", "edb924db6182adc2bc92a349c798a1e6": "\\quad C\\in", "edb9f88e43912b2c1f37fbe082f8a6c2": "y^2 = x(x - a^p)(x + b^p)", "edba23d12a3d6f68ca72a70035ea2abc": "(x+1)^2x\\frac{d^2}{dx^2}\\,R_n(x)+\\frac{(3x+1)(x+1)}{2}\\frac{d}{dx}\\,R_n(x)+n^2R_{n}(x) = 0", "edba3a75e932426335078deb31441b18": "qm \\equiv 0 \\pmod{n}", "edbb46da1f0644ef837706a5c0b11a82": "\\Psi_g(h)= ghg^{-1}.", "edbb79d3d36e2b831433ea940d12acda": "X = \\{X_i\\}_{i \\in I}", "edbb8f2c05562b6ee806d57a10d27c60": "{\\mathbf A}(\\mathbf{r},t)=-{\\mathbf r}\\times{\\mathbf B}/2", "edbb9a570b5e16647f2e8649f3a8278c": "\\frac{\\partial u}{\\partial t}-\\alpha\\frac{\\partial^{2}u}{\\partial x^{2}}=h(x,t)", "edbc176f64014ea7f3dc39b040efc82d": "\\lambda \\neq 0", "edbc2bfd08abf046cc3155fbb464899f": "n_1 \\neq 0\\,\\!", "edbc58a9f25252be105b1f1eb0077c40": "V_{\\text{out}}(t) = \\overbrace{C \\, \\left( \\frac{\\operatorname{d} V_{\\text{in}}}{\\operatorname{d}t} - \\frac{\\operatorname{d} V_{\\text{out}}}{\\operatorname{d}t} \\right)}^{I(t)} \\, R = R C \\, \\left( \\frac{ \\operatorname{d} V_{\\text{in}}}{\\operatorname{d}t} - \\frac{\\operatorname{d} V_{\\text{out}}}{\\operatorname{d}t} \\right)", "edbcad8145d7ffff56dcbeb9351e651d": "TTM \\to TM", "edbcb7c75f60b35492f4d2ca8e8eb120": " \\frac{[B]}{[C]}=\\frac{k_1}{k_2}", "edbcc66f35f5acc52af9a50feeb9c709": "\\{ v_1,\\ldots, v_k,w_{k+1},\\ldots,w_n\\}", "edbcd3f2598919c56cd7bce86bceb68d": "\\Pr(|\\overline X - \\mathrm{E}[\\overline X]| \\geq t) \\leq 2\\exp \\left( - \\frac{2n^2t^2}{\\sum_{i=1}^n (b_i - a_i)^2} \\right),\\!", "edbd079dc5a822dde6f38a841f81eb81": "K*P", "edbd14fce5bac8064707bff8abb86b80": "\\displaystyle a\\cdot \\hat{f}(\\nu) + b\\cdot \\hat{g}(\\nu)\\,", "edbdb31d243fc466a6671994e61db64c": " \\psi_{n,k}(t) = 2^{n / 2} \\psi(2^n t-k), \\quad t \\in \\mathbf{R}.", "edbde5c584fb85dd51c5a1dcc83dee86": " -0.96x^2 +56.4 x +1444", "edbe23421981a80bdb27268c51af47f9": "2M", "edbe58d1e75183b3adc0cdb8c115d745": " g = \\mathrm{gravitational \\ acceleration} ", "edbe5fb9fceee02a64f37c3d62c1ed2b": "\\Delta E = 2\\sigma_i \\sum_{j\\in viz_i}\\sigma_j", "edbe7c0127fb4e7f821c1fd700c07444": "f_i: Y_i \\to (X,\\tau)", "edbea0270326c16dc58a3a3977d9740e": " \\Delta G^\\ominus = \\Delta H^\\ominus - T\\Delta S^\\ominus ", "edbf4a8169c9bf8aead9bd4296739289": "F_\\mathrm{s} = - k x \\,", "edbf9e9900222c7fe5d547532816ada3": " \\mathbf{x}(n)=\\left[x(n),\\,x(n-1),\\,...,\\,x(n-p)\\right]^{T}", "edbfcbf5afc6604e39e08579f9834b3d": "\\mathbb{F}_2", "edbfd08be37b385cd3cd5809054349bc": "\\mathbb{P} \\left( \\mathcal{A} (\\cdot) \\subseteq D \\right) > 0,", "edc00d4e1be4d11b4caaf8e2f65775e4": "w^{(0)} \\gets X^T y/||X^Ty||", "edc01335153e26ac499e6ac86c3f8230": "\\textstyle C = \\{ x \\in \\mathbb{R}^n : (\\forall i) \\; \\alpha_i(\\omega_1) x_1 + \\ldots + \\alpha_i(\\omega_n) x_n \\geq 0 \\}", "edc0297c1431bcd0b32eff7ce4edb4ba": "A,B \\in M", "edc0456abfda7b7b967adc93fbe93e18": "\\varepsilon_X=\\Phi_{GX,X}^{-1}(1_{GX})", "edc097549167b18a118cb0dd883825d4": "T_{n+1}(x) = xT_n(x) - (1 - x^2)U_{n-1}(x)\\,", "edc0c6edc4702967e70976b6f8f36786": " T = \\sqrt{5}:2, ", "edc0e39aa68020871f70104fab37844a": " \\rho_\\sigma = \\frac{\\left(\\frac{\\partial\\rho}{\\rho}\\right)}{\\varepsilon}", "edc103e948a0770077b07a2edd778b2f": "u(P) =\\frac{1}{4\\pi} a^3\\left(1-\\frac{\\rho^2}{a^2}\\right) \\iint \\frac{g(\\theta',\\varphi') \\sin \\varphi'}{(a^2 + \\rho^2 - 2 a \\rho \\cos \\Theta)^{\\frac{3}{2}}} d\\theta' \\, d\\varphi',", "edc11b2efe01d1da3f938c4ab54d255a": "K=CV/d", "edc1357b9fdf7acbc9a53cc9da5aed97": "g^{(n)}(\\infty)= 0 ", "edc1f8f1d3bfe4a9c89418171c98275c": " \\widehat{U}(\\theta)^{\\dagger}\\widehat{a}\\widehat{U}(\\theta) = \\widehat{a}e^{-i\\theta} ", "edc237472753a0baadb87532ba840352": "|t| = \\frac{1}{2}", "edc26d0f03c69dd7f855d5afad024205": "1\\rightarrow N\\rightarrow G\\rightarrow Q\\rightarrow 1. \\,\\!", "edc28725827686340055d0ea4a892574": " f = \\frac{f_0}{\\gamma} = {f_0}{\\sqrt{1-\\beta^2}}", "edc2d08a62da97e0cddd9c46dc961e89": "A_{r,m}", "edc34710beaeca1ecb6ecdc42a4b47f6": "\\displaystyle{ {F^\\prime(z)F^\\prime(w)\\over (F(z)-F(w))^2} \\,-\\,{1\\over (z-w)^2}}", "edc348aab6fe3e8b75e856ab825b9dfe": "ds^2 = \\gamma_{33} \\left ( d\\xi^2 - dz^2 \\right ) - \\gamma_{ab}dx^adx^b. ", "edc3884c53a826514ac521ac4119f000": "H_0 : \\sigma_{\\text{Tx A}-\\text{Tx B} }^2 = \\sigma_{\\text{Tx A}-\\text{Tx C} }^2 = \\sigma_{\\text{Tx B}-\\text{Tx C} }^2", "edc3ac56113f51f642253fdc1e9176a1": "P(G, t)= a_1 t + a_2t^2+\\dots +a_nt^n,", "edc3f1e046366ec3615f4c7e0d0e8fb4": "S(N_r)", "edc48472832c82daef104424248b365c": "\\tilde{\\chi}_1^\\pm", "edc49b32942bfb5898ca064de747068d": " {{\\bar{i_{n}^2}} \\over {B}} = 4 k_B G T", "edc4e369f614c6f206de56815c0fe60c": "\\{(t,t^2,1)\\mid t\\in GF(q)\\}\\cup \\{(0,1,0)\\}.", "edc4f8cdcc5cf0a9b0534610f8c518fa": "\\vdash_L\\sigma p\\leftrightarrow \\upsilon\\tau p", "edc53d4bd941de6d2f1a10d8a41080eb": "L_{\\mathrm{I}}(x_0,\\dots,x_n) = \\int_S x_0^{\\alpha_0}\\cdot\\dots\\cdot x_n^{\\alpha_n}\\ \\mathrm{d}\\alpha", "edc550d2009687a35c803013d4a51cdc": "\\overline f \\colon A_B \\to A", "edc5681fd725766d0f0d1d95b3dfa063": " \\int_{S_t} \\delta\\ \\mathbf{u}^T \\mathbf{T} dS + \\int_{V} \\delta\\ \\mathbf{u}^T \\mathbf{f} dV = \\int_{V}\\delta\\boldsymbol{\\epsilon}^T \\boldsymbol{\\sigma} dV \\qquad \\mathrm{(f)} ", "edc59e3d10b8514f3edb31d947a2f1fc": "P(t_1) = F(t_1).", "edc5ae42dac79557b6bb40c6059c1574": "\\,F(t,x)", "edc5dd0e7f7af6a32d3bbb124fc823d0": "\\sqrt Z", "edc5e734ca8cccd00c52f8ea031876a5": "\\lim_{n \\to \\infty}S_{SID}(n)= s3^n \\, ", "edc5fb26bb39ae8bd95fe26fc5c07299": "H_n(X)", "edc615c92e474167c1a216a6a0d5decb": " {}_1F_1(a,b,x) \\sim \\frac{\\Gamma(b)}{\\Gamma(a)}\\,x^{a-b} e^x \\quad \\text{as } x \\to \\infty \\text{ with } x > 0.", "edc61c528ebddc44f0f493cac8ed7711": "\\bar{A}_{\\beta} = \\frac{\\partial x^{\\gamma}}{\\partial \\bar{x}^{\\beta}} A_{\\gamma} \\,.", "edc631a881a956712d1b779e223e9705": "f(x) = x^2 + 2x + 3", "edc65b1f9b5348fd34dcfe53b1f6c0c0": "f:X\\to f(X)", "edc66682c967c25f8433890a937f17f4": "\\{z,y\\} \\in R_j", "edc6a625e6bed5921e070ea0da6acf38": "\\mathrm{CFS} = \\max_{x\\in \\{0,1\\}^{n}} \n\\left[\\frac{(\\sum^{n}_{i=1}a_{i}x_{i})^{2}}\n{\\sum^{n}_{i=1}x_i + \\sum_{i\\neq j} 2b_{ij} x_i x_j }\\right].", "edc6d093710c14f0291094c81a2b6727": "\\scriptstyle\\{x: u(x) -\\phi(x) = t\\}", "edc725f87e30b9e2f85c95617cce6fc6": "\\mathrm{d} \\omega _V = \\operatorname{div}(V) \\; (\\mathrm{d}x^1 \\wedge \\mathrm{d}x^2 \\wedge \\cdots \\wedge \\mathrm{d}x^n).", "edc72f9a53a281826107da7f77f05d6a": "(\\Omega,\\mu)", "edc7392b62fdbb56cb4e321c5bed9fc2": "\\left(\\frac{1001}{9907}\\right) =-1", "edc744fa20fe80c07d97cd2e6df2a69c": "\\tfrac2n", "edc768e077da04542b374ac2af40925b": "P_n\\sqrt2=\\frac{(1+\\sqrt2)^n-(1-\\sqrt2)^n}{2}.", "edc775ba92059a62a14b16936a9b14c3": "F_t = E_t(S_{t + k})", "edc7ea186c894cdb530f4c3f36a944bd": "{\\ddot{a}}_{65}", "edc8400cd96f1cab87478a98bad07f31": "[cq_0,cq_1\\ldots,cq_n]", "edc8806936357546b935872784d721b8": "1/ \\zeta", "edc8cce665d642384db4b94008af6abc": " \\partial^2 \\Delta (x) = i\\delta(x)\\,", "edc94360882c251b520e9c69ebbc7ea8": " \\mathbf{F}_{XX} = \\mathbf{B}_X^T \\mathbf{f} \\mathbf{B}_X ", "edc9b5cb0e45b384917a946c96c76f53": "\\mathbf{Y}(z) = C \\mathbf{X}(z) + D \\mathbf{U}(z)", "edc9e7c2a9ebe247c75b2b3234f7daec": "Z_\\lambda(G) = \\bigcup_{\\alpha < \\lambda} Z_\\alpha(G).", "edc9eb4cbbfe33d57ae03c9fe395d4d3": " \\Delta(v_1,v_2,v_3)", "edca63518f562a13465ddf56ff46e118": "a+n \\log \\frac{\\theta_1}{\\theta_0} < \\frac{\\theta_1-\\theta_0}{\\theta_0 \\theta_1} \\sum_{i=1}^n x_i < b+n \\log \\frac{\\theta_1}{\\theta_0}", "edcaa6965b4aa5e07bf3c38ca194d6ec": " \\cos a \\approx 1-a^2/2", "edcac813d7326a460eb7c6f9570d7af3": "A_1\\otimes A_2", "edcac866efd170abea82e93012221d8e": " tr \\Omega^r(Q_1\\otimes\\cdots \\otimes Q_n)", "edcad74e48505f18148c773f758c704f": "r_{ij} = \\frac{1}{w_{ij}}", "edcb2bdf286f0721d905fd30945200da": "\\begin{align} \nU_n & =\\bigcup\\limits_{q=2}^\\infty\\bigcup\\limits_{p=-\\infty}^\\infty \\left\\{ x \\in \\mathbb R : 0< \\left |x- \\frac{p}{q} \\right |< \\frac{1}{q^{n}}\\right\\} \\\\\n& = \\bigcup\\limits_{q=2}^\\infty\\bigcup\\limits_{p=-\\infty}^\\infty \\left(\\frac{p}{q}-\\frac{1}{q^n},\\frac{p}{q}+\\frac{1}{q^n}\\right) \\setminus \\left\\{\\frac{p}{q}\\right\\}\n\\end{align}", "edcb8324264d314ffa769c7b75b24e8b": "\\tfrac{x^2}{a^2}+\\tfrac{y^2}{b^2} = 1", "edcba244c497ab39a29d0350dd45036a": "x^m = 1", "edcbb05ef8f2ab3fde511156a2e51f4b": "1=|c_0|^2+|c_1|^2+2e^{-(|\\alpha_0|^2+|\\alpha_1|^2)/2}\\operatorname{Re}\\left( c_0^*c_1 e^{\\alpha_0^*\\alpha_1} \\right).", "edcc0da5a3fcc3171d2d73125ec42541": "S_1 = X_1 + X_2", "edcc3e560f19dc0a43a8817ee6a9a5e2": "f\\left( x\\left( y,\\xi\\right) ,\\theta\\left( y,\\xi\\right) \\right) ", "edcd65f63455fb0e6ec7b8735c785baf": "1 \\le i \\le k.", "edce04b13c0414fca59a90e0cf4f0efc": " g(\\lambda \\mid \\alpha,\\beta) = \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\; \\lambda^{\\alpha-1} \\; e^{-\\beta\\,\\lambda} \\qquad \\text{ for } \\lambda>0 \\,\\!.", "edce3877290e3ea3532138d10898fbc0": "\\tfrac{1}{2} (1 - \\sigma_1 \\sigma_2) \\, \\{a_1\\sigma_1+a_2\\sigma_2\\} \\, (1 - \\sigma_2 \\sigma_1) = a_1\\sigma_2 - a_2\\sigma_1 \\,", "edce78329b7ff8abbdbb887ab1e03d0a": "A = U S V^T", "edceb491d9e5f5292b7ade8173108f98": "y'=\\lambda y", "edced254166ae7eb55dc52f49db953b0": "(-1)^{n_-}.", "edcee9f20c91b1ad8987e4e999048879": "c_{j,k}", "edcef4f1448f71dbda02435e0a790162": "S^1 \\wedge S^1 \\to S^1 \\wedge S^1", "edcf27686f7d1a3ea0a3414d8f592231": "\n\\alpha=\\frac{Nl}{2mv2}\\,>\\,0\\ \\ \\ \\wedge\\ \\ \\ \\alpha\\,\\neq\\,1\\ .\n", "edcfa0d9c98fab25c5da35715417d0de": "k\\ne \\bar k", "edcfd7520be5cdd8919403055ec97762": "(L,k)", "edcff31474ec33f2ac2b5186acc5c9c8": "(i+2)", "edd029d7b0f7daee9235dee11bb93a0c": "s'^2= \\eta_{\\mu\\nu} x'^\\mu x'^\\nu\\ .", "edd0303dc538c2d33828aeff89a3a4b4": "N_{\\mathfrak{g}}(K) := \\{x \\in \\mathfrak{g} | [x, K] \\subseteq K \\}", "edd087c72d716e58f2f52d51f5bbfdaa": "\\mathcal{L}_{WWV}", "edd09d82c96f6a1ee18e8f2bbca99043": "\\begin{align}\nk_2 &= f\\left(y^1_{t+h/2}, t + \\frac{h}{2}\\right) = f\\left(y_t + \\frac{h}{2} k_1, t + \\frac{h}{2}\\right) \\\\\n&= f\\left(y_t, t\\right) + \\frac{h}{2} \\frac{d}{dt}f\\left(y_t,t\\right) \\\\\nk_3 &= f\\left(y^2_{t+h/2}, t + \\frac{h}{2}\\right) = f\\left(y_t + \\frac{h}{2} f\\left(y_t + \\frac{h}{2} k_1, t + \\frac{h}{2}\\right), t + \\frac{h}{2}\\right) \\\\\n&= f\\left(y_t, t\\right) + \\frac{h}{2} \\frac{d}{dt} \\left[ f\\left(y_t,t\\right) + \\frac{h}{2} \\frac{d}{dt}f\\left(y_t,t\\right) \\right] \\\\\nk_4 &= f\\left(y^3_{t+h}, t + h\\right) = f\\left(y_t + h f\\left(y_t + \\frac{h}{2} k_2, t + \\frac{h}{2}\\right), t + h\\right) \\\\\n&= f\\left(y_t + h f\\left(y_t + \\frac{h}{2} f\\left(y_t + \\frac{h}{2} f\\left(y_t, t\\right), t + \\frac{h}{2}\\right), t + \\frac{h}{2}\\right), t + h\\right) \\\\\n&= f\\left(y_t, t\\right) + h \\frac{d}{dt} \\left[ f\\left(y_t,t\\right) + \\frac{h}{2} \\frac{d}{dt}\\left[ f\\left(y_t,t\\right) + \\frac{h}{2} \\frac{d}{dt}f\\left(y_t,t\\right) \\right]\\right]\n\\end{align}", "edd0e63e1fb44dfacbdcea4b28b556f7": "\\displaystyle (a)", "edd0e8ab95a7681d653ab80387cb36a9": "\\mathbb{F}^l", "edd0f8995b3ed3743561dd6a3eb0d429": "L \\cdot P", "edd15bb444c025226e170ecb41ec6522": "P \\mid Q", "edd17b61441465cfa1cd10d3a7dd3bb6": "X^* = X^0 \\bigsqcup X^1 \\bigsqcup X^2 \\bigsqcup X^3 \\bigsqcup \\ldots", "edd1d8bb771bad8470be2a6b811767ca": "(i_\\alpha {\\bold w})(u_1,u_2\\dots,u_{k-1})={\\bold w}(\\alpha,u_1,u_2,\\dots, u_{k-1}).", "edd2246f3c12e54bc343e9bb2fa10a4e": "\\varphi_{x}(x)", "edd265c7e3912aa65f931df1c1e056be": "\\text{Changing incoming system reactance:}", "edd2e7f5980686e3db04667ac20fece4": "w_{ij}(\\tau,2n+\\gamma_{ij};L)", "edd2ec97c28a3be9427d754142e7be74": "V - E + F = 2.\\ ", "edd2ee799dd37ec9acd0b8ccfa299ead": " \\begin{align}\n& \\text{minimize} && \\tfrac12 x^\\mathrm{T} P_0 x + q_0^\\mathrm{T} x \\\\\n& \\text{subject to} && \\tfrac12 x^\\mathrm{T} P_i x + q_i^\\mathrm{T} x + r_i \\leq 0 \\quad \\text{for } i = 1,\\dots,m , \\\\\n&&& Ax = b, \n\\end{align} ", "edd2f37c4fdc8f4d61e634f067be4b83": " B = \\{ b_1, b_2, \\ldots, b_n \\} ", "edd3014088f6a12d74e730799d6bfddb": " u(x) = \\frac{1}{n\\omega_n r^{n-1}}\\int_{\\partial B(x,r)} u\\, d\\sigma = \\frac{1}{\\omega_n r^n}\\int_{B (x,r)} u\\, dV", "edd37a278e122d972bb108e0b63a814b": "\\gamma = -\\frac{dT}{dz}", "edd3d326055ef7fc0072214333d12428": "O_R", "edd41d1d5429fc3ff4c54ac910f84b0f": " E_A = 420 \\ \\mathrm{kJ} \\, \\mathrm{mol^{-1}} ", "edd41e1ac3934a2bf4b18fee9e3272c8": "\\varepsilon_0\\,", "edd45c249f4d21c7ead8ad400c64544d": "\n {\n \\cfrac{d}{dt}\\left(\\int_{\\Omega} \\rho~\\eta~\\text{dV}\\right) \\ge\n \\int_{\\partial \\Omega} \\rho~\\eta~(u_n - \\mathbf{v}\\cdot\\mathbf{n}) ~\\text{dA} - \n \\int_{\\partial \\Omega} \\cfrac{\\mathbf{q}\\cdot\\mathbf{n}}{T}~\\text{dA} + \\int_\\Omega \\cfrac{\\rho~s}{T}~\\text{dV}.\n }\n ", "edd46211c88c1d6bf66d6392c224e5ed": "n_s = \\frac{IB}{q|V_H|}", "edd4cb49e32d14f7a48b534fccca5312": "\\scriptstyle\\Phi(s,t)", "edd4f171aba3adcb830a2054c2abd59e": "\\ |y_s|^2 = {y_s}^\\mathrm{H} y_s = h^\\mathrm{H} s s^\\mathrm{H} h.\\, ", "edd531e7eeaf606b20e219009ce7b8a3": "c = 65^{17} \\; \\operatorname{mod}\\; 3233 = 2790 ", "edd53a68a85eb253e6a740fdf88b56c5": "D_2 \\cong A_1 \\times A_1.", "edd5b646629688046da35eb502e939de": "\\Lambda_{\\mathrm{id}} = \\chi(X).\\ ", "edd5e99505e8915de176ff030c54f9ea": "\n\\begin{align}\n\n\\mathrm{P}(X=x\\ \\mathrm{and}\\ Y=y) = \\mathrm{P}(Y=y \\mid X=x) \\cdot \\mathrm{P}(X=x) = \\mathrm{P}(X=x \\mid Y=y) \\cdot \\mathrm{P}(Y=y)\n\\end{align}.\n", "edd62faac5e21cac5c6417d117dcc336": "\\phi^{(2)} - \\phi^{(1)} = \\frac {\\mu_j^{(1)} - \\mu_j^{(2)}} {z_j F}", "edd65d95d5d2637f33e71694a6886a5c": " E(X| \\mathcal{G}_t) = E(X) + \\int_0^t C_s \\, d B_s.", "edd696cad4ce5ceebd7dfa3bc8f1294d": "+2z", "edd69bf9dbe8312678d9bd8a81f7e9d5": "H(w) < w + ND(w)", "edd69f7eab53c961fb94a30e7316971e": "s = \\frac{n_s-n_r}{n_s}\\,", "edd6ebca001739822a64d9cba2daaaf8": "\\{x_1, \\cdots,x_m\\}", "edd771ea6389b07f1913d91a8f3c1734": "[E_{1T}] = [E_1] + [WE_1]", "edd79b57f5513a579f32cdd86aae04d0": "h_{\\operatorname{AMISE}} = \\frac{ R(K)^{1/5}}{m_2(K)^{2/5}R(f'')^{1/5} n^{1/5}}.", "edd7a2c95727b2f83a59203463432293": "\n\tE = \\iint\\left[\\left(\\frac{\\partial^2 f}{\\partial x_1^2}\\right)^2 + 2\\left(\\frac{\\partial^2 f}{\\partial x_1 \\partial x_2}\\right)^2 + \\left(\\frac{\\partial^2 f}{\\partial x_2^2}\\right)^2 \\right] \\textrm{d} x_1 \\, \\textrm{d}x_2\n", "edd7a424bcc7c33cfc4e5fb8cc28f1e8": "\\Diamond\\exists xFx\\to\\exists x\\Diamond Fx", "edd7c9470c8cd5b77c53fcc4e5895b61": "\n\\mathrm{d} s^2=\\alpha^2(\\frac{\\,\\mathrm{d} u^2}{u^2}+u^2(\\,\\mathrm{d} x_\\mu \\,\\mathrm{d} x^\\mu) )\n", "edd8075aad3d72a8a26735cd49376347": "\\frac{1}{\\ln x}", "edd87555a135c541a73e0c1958aac286": " \n\\sigma_{ij} = C_{ijkl} \\, \\varepsilon_{kl}\n\\,\\!", "edd8cc46f3beb01188ed4ca076866e29": " V = \\mathbb{R}^{2}", "edd98223c1dc66f22988e88c94a3b236": "(\\tfrac12 , -\\sqrt{2k})", "edd9847d14dce45f7aebef21524be25a": "\\frac{dR}{dt} = \\gamma I - \\mu R - f R ", "edd9d38887e9e6c305717116e012f2ae": "237/360 = 0.1\\ 0\\ 3\\ 4_!", "edda7efbdc5ff03f7d346d64db24808f": "b=C_x N_x +C_y N_y+C_z N_z\\,", "eddaae4dbaa45e94dfcbb64a6e764788": "\\psi=G", "eddb0335cad20b1bad137530335c7c9e": "y_k = \\mathbf{h}_k^H \\mathbf{x}+n_k, \\quad k=1,2, \\ldots, K", "eddb08e234fe39f70a68a12d26d091e9": "Y_1^* ", "eddb41d99e317a6573a43505ad457a0a": "\\boldsymbol{\\hat{\\jmath}}", "eddb7651b3a946bda1e6a6568bf68a17": "{d \\over dt}\\left\\{ T \\right\\} =\\tau k_+ \\left\\{ A \\right\\}^\\alpha \\left\\{B \\right\\}^\\beta -\\tau k_{-} \\left\\{S \\right\\}^\\sigma\\left\\{T \\right\\}^\\tau \\,", "eddb92eff921d9a7961733539ea54697": "\\forall X_1\\cdots \\forall X_k \\psi", "eddc5b9ee3fecbed6f7f1c6714ae619b": "r(\\theta)>0", "eddc745821bebcdfe948957c0f477544": "ad < bc.", "eddc9506e33cd7acd466039d48122007": " [I_R] = -\\sum_{i=1}^n m_i[r_i - C - d]^2.", "eddca02dcc724f7edc8012b214742572": "\\omega\\in\\Omega^p(M,\\mathbb{R})", "eddced7589ac754ba141a3de05027c0c": "\\frac{11\\cdot\\pi}{6}", "eddd1e434aaff66f55cc828bd3c412eb": "y^2 = x^2 - x^3\\ ", "eddd770626dca81978f911549795292a": " X_j \\left( s \\right) = \\frac{1}{2\\pi}\\left(\\omega_j s \\text{ mod } 2\\pi \\right), j = 1,2,\\dots,n ", "eddda9be5731a31a2693b16711894c61": "\\left(\\frac{\\partial z}{\\partial x}\\right)_y = -\\left(\\frac{\\partial z}{\\partial y}\\right)_x \\left(\\frac{\\partial y}{\\partial x}\\right)_z", "edddb7e96fa3a3e86a25ea2708bc677b": "\\lambda\\in\\mathfrak{h}^*", "edddbc7843a3bba774c65ddf96b7bf0e": "v\\in V, a(u,v) = f(v)", "eddf32382b41c45951f4bafe76d16d1c": "\n E=\n \\left( { a_1\\, a_2 \\over 2 \\pi L_B}\\right) v_1\\, v_2\\, \\int_0^{\\infty} {k\\;dk \\;} D\\left( k \\right) \\mid_{k_0=k_B=0}\n\\mathcal J_1 \\left ( kr_{B1} \\right) \\mathcal J_1 \\left ( kr_{B2} \\right) \\mathcal J_0 \\left ( kr_{12} \\right)\n", "eddf59474eaf7ec64b610154a63a2eea": "\\frac{\\pi}{4} = 5 \\arctan\\frac{1}{7} + 2 \\arctan\\frac{3}{79}\\!", "eddfb5c22ba2f704160b27f4533bbbf2": "\\!\\chi_e", "ede00e660b58eb625ec64266ab5fd902": "s_{Ox}=\\lVert v_{O} \\rVert \\cdot \\sin(\\theta_{T})\\,", "ede0860b9eedcb7a8621d9f8c89ec19e": "\\mathbf{y}(t) = C(t) \\mathbf{x}(t) + \\mathbf{w}(t),", "ede09d49ffc94395f4a05d542a08d33a": "I_{\\mathrm{RMS}} = I_\\mathrm{p}\\sqrt {{1 \\over {T_2-T_1}} {\\int_{T_1}^{T_2} {\\sin^2(\\omega t)}\\, dt}}.", "ede12d4c5e285b1be1ba879f3117e2c8": "\\displaystyle P_m(f)-z^{-m}", "ede13ba91241b35b0b551d51ca37fbcc": "\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\end{bmatrix}", "ede13ca70d69ce1ace3db06a2552ecaf": "\\forall x \\forall y ( \\forall z (z \\in x \\leftrightarrow z \\in y)\n\\rightarrow x = y).", "ede21926ddea8506e8f348ef3e0eafac": "A^{*}_{\\vec{r}}", "ede2cd9932eeaea6501af794ac3b8d6e": "\\hat{H}_{\\text{JC}}", "ede2dfe1d1281e954ecf6e2fdfb36df6": "x_0^4+x_1^4+x_2^4+x_3^4=0", "ede3cd65040eda298bdb1a7d5b2f3baa": "ke^{-\\alpha}(\\alpha+1)", "ede3e4b0566e1bce38a7a4a10d49401a": "x^* = (A^T PA + Q)^{-1} (A^T Pb+Qx_0).\\,", "ede419dab5f95efd1819edd2fe6ef2e8": "h(\\boldsymbol{x}) = \\sqrt{1+f(\\boldsymbol{x})}", "ede4bf63af2afcc6c2585a28434e582b": "\\displaystyle |x|^\\alpha\\,", "ede4d8c377ba9eee516a6c2ee59c2073": "\\tilde{H} = H_{0} + \\left\\langle\\Delta H\\right\\rangle_{0}\\,", "ede538acc93d3e235230698e7ad04ea5": "y = -at + A_y.\\,", "ede54b5067dc124fce2c8898e4048874": "K_{OW}=\\frac{Concentration_{octanol}}{Concentration_{water}}=\\frac{C_O}{C_W}", "ede6216ab8f7fd761ebf3d8b0e6f4361": " \\int_0^\\infty \\exp\\left\\{ \\frac{1}{x} \\int_0^x \\ln f(t) dt \\right\\} dx \\leq e \\int_0^\\infty f(x) dx ", "ede645b9bc83f99908e531b0f9b2686c": " \\scriptstyle{Z_\\circ=\\sqrt{{\\mu_\\circ \\over \\varepsilon_\\circ}}= 376.730313461\\, \\Omega}\\,\\!", "ede646a06c89ec0fd3f2a8eb672f4164": "(X,\\mathcal{O}_X)", "ede662bdb43a344860355dcd20cd655a": "= 9.804 + 9.612 + 9.423 + 9.238 + 92.385 = 130.462 ", "ede6762f4111be3248cf8ffa4a0f3bbf": "\nc = \\sqrt{g \\frac{A}{B}},\n", "ede67c075e95a5e0b18451c96d201e8a": "{Y}", "ede68689d8548150fb81189f6e53d131": "\\lnot\\ ", "ede6e99ca62eba9b6a06be84184414df": "V_{\\text{BC}}", "ede6ff861ab35bc5945d9380f94ce978": "a^r \\equiv 1 \\pmod{N}", "ede71d69f731ccb804805c19c2b39113": " Tw = \\dfrac{1}{2\\pi} \\int \\tau \\; ds + \\dfrac{\\left[ \\Theta \\right]_X}{2\\pi} = T+N \\; ,", "ede72101ecc59ea91294bc0ebb7f7a54": "\\lim_{n \\to \\infty}\\left(\\frac{1}{1^2} + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots + \\frac{1}{n^2}\\right) = \\frac{\\pi ^2}{6}.", "ede73d5520b313bf4090a95f0cea90b5": "\\delta\\lambda", "ede749c7ccd96dfaa12e6be6352f357c": "\\mathfrak{a}", "ede7716910c94e519035ec59409cf81f": "\n\\mathrm{La}\n=\n\\sqrt{\\frac{\\nu^3_Tk^6}{\\sigma a^2u^2_*k^4}}\n\\ \\mathrm{or}\\ \n\\sqrt{\\frac{\\nu_T^3\\beta^6}{u^2_*S_0\\beta^3}}\n", "ede7858a9d05260e8f41a49bfdd28424": "s=t^{1/3}", "ede79493ad636850ef1f394f75e9e1df": "\\gamma^k{}_{ij}=0", "ede7aaf9baed0e6e39658bbe8135e705": "F^{\\bullet}\\Omega^p_Y(\\log D) ", "ede879385b4db13627a29eb5b34337ed": "Iz=Ip-In", "ede8a38f5dda4bfd2302aa5cd21958b8": "\n\\mathcal{G}_{0}=\\left\\{ \\mathbf{G}_{i}\\in F(\\Pi^{\\mathbb{Z}^{+}}):1\\leq i\\leq\nn-k\\right\\} .\n", "ede8a3bad3832af8b6bc368e440a80a4": "\n\\begin{align}\n0.95 & = P\\left(\\bar X - 1.96 \\times 0.5 \\le \\mu \\le \\bar X + 1.96 \\times 0.5\\right) \\\\[6pt]\n& = P \\left( \\bar X - 0.98 \\le \\mu \\le \\bar X + 0.98 \\right).\n\\end{align}\n", "ede8c0d3dd79f6b75d800b354cd52ca7": " \\mathrm{N} ", "ede8de13df8ec98c758db5b379cd92ca": "f_{exc}", "ede9813d1d81fd1194f6125ce72cb0f7": "m=E/c^2", "ede9905a8416605c3e98d71977a0bdac": "\n \\boldsymbol{\\nabla} \\cdot \\mathbf{v} = \\frac{\\partial v^i}{\\partial q^i} + \\Gamma_{\\ell i}^i~v^\\ell\n", "ede9db97e29b85831a145489c4dc02b4": "R^m_{\\ell}(\\mathbf{r}_i)", "edea3b0ebde3aab6f2d1fb18ffd47a92": "f(x) = f_\\infty(x) + \\sum_{\\alpha \\in B'} f_\\alpha\\left(\\frac{1}{x-\\alpha}\\right)", "edeaf9d9a381b23328284221bbb32265": "{6\\choose 5}{43\\choose 1}\\over {49\\choose 6}", "edeb03c2d7b42865555b75f4802fb7ce": "ds^2 = dx^2 + dy^2 +dz^2,\\, ", "edeb3d28a94dcdfd64fcfc9634392e2b": "\\theta_{ab} = \\sigma_{ab} + \\frac{1}{3} \\, \\theta \\, h_{ab}", "edebaf0bfffd06a93e8cf3ffad26a9f7": " \\boldsymbol{\\Omega}_{\\text{LIF}} ", "edebcc84ae299e15315c34b771baf377": "D_{a} = E_{a} - I_{a}", "edebdc0d9b9788da599d17bfa48ba943": " \\mathbf{p},\\mathbf{q} ", "edec0844d5d569a3e37131d671596e25": "\\begin{align}\n &\\Pr(N>x\\mid M=m,K=k) \\\\\n = {} &\\begin{cases}\n 1 &\\text{if }x < m \\\\\n \\sum_{n=x+1}^\\infty (n\\mid m,k) &\\text{if }x \\ge m\n \\end{cases} \\\\\n = {} &[x k \\\\\n \\alpha_{w_{i-n+1} \\cdots w_{i-1}} P_{bo}(w_i | w_{i-n+2} \\cdots w_{i-1}) \\mbox{ otherwise}\n\\end{cases}\n", "edef491ab918ac40635ea1ee0675dbd0": "x \\vee x = x \\forall x", "edef7c8fdc24acd25a819d30636d9f09": "{E_{k}(\\mathbf{R})\\approx E_{k'}(\\mathbf{R})}", "edefb8f10f87a3c2b6701c8c7ca99b0c": " \\mathbf{x}' ", "edefcdd467030ead2e3ffb063b0a0e7e": "x_{(n+1)/2}(z) \\sim (1+z)^n", "edf00003412ef0a54d3e52939f47bb31": "\\tilde{X}:=f\\circ X", "edf0332af73c13b8054f3a905c2e0226": "\\Omega \\equiv \\frac{\\rho}{\\rho_c} = \\frac{8 \\pi G\\rho}{3 H^2}.", "edf044cad6308fc0557765faabafb66b": "(m_{rest})c^2=\\sqrt{E_{total}^2-(|\\boldsymbol{p}|c)^2}.\\!", "edf0892b91059988eb0db01fb2f9a6c5": "\\oint_C \\boldsymbol{u} \\cdot \\frac{\\mathrm{D} \\boldsymbol{\\mathrm{d}s}}{\\mathrm{D}t} = \\oint_C \\boldsymbol{u} \\cdot \\left[ \\left( \\boldsymbol{\\mathrm{d}s} \\cdot \\boldsymbol{\\nabla} \\right) \\boldsymbol{u} \\right] = \\frac{1}{2} \\oint_C \\boldsymbol{\\nabla} \\left( |\\boldsymbol{u}|^2 \\right) \\cdot \\boldsymbol{\\mathrm{d}s} = 0.", "edf10bb4dcd05520a01368f8f386c0f7": "y_i(\\mathbf{w}\\cdot\\mathbf{x_i} - b) \\ge 1. \\, ", "edf12168f2a9888870edb411c94f1969": "Con \\subseteq \\mathcal{P}_f(T) \\mbox{ the finite subsets of T}", "edf12f0b1cc091954c06c97ad93245a1": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left( \\frac{\\partial L}{\\partial \\dot{x}_i} \\right) - \\frac{\\partial L}{\\partial x_i} = 0,", "edf130109bb6c677c9cb6e66eb2ff498": "\\lambda_{1}=\\frac{3+\\sqrt{5}}{2}>1", "edf194d8db019f792ed9cf3bbee84cf6": "\\mathbf{Bord}_{\\langle n-1,n\\rangle}", "edf219dd3828edc08aea1848da0c8d97": " Y = \\operatorname{let} p : \\operatorname{de-lambda}[p\\ f = \\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x)] \\operatorname{in} p ", "edf2e2e26ac99eda6b07f6917a89562f": "\n x_1+x_2 =x_3,\n", "edf30bdf079e289a6c39dc1cca8cfe77": "H = \\sum_k E_k \\, a^\\dagger_k \\,a_k.", "edf3109035d67155fb72b2efe0e4c6cd": "\\pi_{\\mathbf S}:{\\mathbf S}\\to M\\,", "edf31d589393fcb2e739abd8f5a63e16": "\\zeta(2) = \\frac{1}{1^2} + \\frac{1}{2^2} + \\frac{1}{3^2} + \\frac{1}{4^2} + \\cdots = \\frac{\\pi^2}{6}\\!", "edf33039dd295704de781bf84f387d57": "\\sum_{n=-\\infty}^{\\infty}x[n]z^{-n} = -\\sum_{n=-\\infty}^{-1}0.5^nz^{-n} = -\\sum_{m=1}^{\\infty}\\left(\\frac{z}{0.5}\\right)^{m} = 1-\\frac{1}{1 - 0.5^{-1}z} =\\frac{1}{1 - 0.5z^{-1}}", "edf3388397f387ce7b2467edb106628d": "T^2 k/m", "edf3e0430881ee85600f31c2b6ce3133": "\\text{bandwidth} \\times \\log_2 (1 + P_u/P_n)", "edf3e52feb186852a7768660afc955aa": "\\mathbf{u} = \\mathbf{x} + \\alpha\\mathbf{e}_1,", "edf4116770a472364abcf65fbc09f83c": "f:D^{2}\\rightarrow\\mathbb{C}", "edf43a6346f81db5fb062932c0396ef8": "\\frac{AF}{FB}=\\frac{|\\triangle CAO|}{|\\triangle BCO|}.", "edf4447522ac9fd89979761044d21651": "\\alpha = (a, b, c)", "edf455670217661d55f485c58e0c72da": "\n\\begin{align} \n\\varphi(n)\n&= \\varphi(p_1^{k_1}) \\varphi(p_2^{k_2}) \\cdots\\varphi(p_r^{k_r})\\\\\n\n&= p_1^{k_1} \\left(1- \\frac{1}{p_1} \\right) p_2^{k_2} \\left(1- \\frac{1}{p_2} \\right) \\cdots p_r^{k_r} \\left(1- \\frac{1}{p_r} \\right)\\\\\n\n&= p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r} \\left(1- \\frac{1}{p_1} \\right) \\left(1- \\frac{1}{p_2} \\right) \\cdots \\left(1- \\frac{1}{p_r} \\right)\\\\\n\n&=n \\left(1- \\frac{1}{p_1} \\right)\\left(1- \\frac{1}{p_2} \\right) \\cdots\\left(1- \\frac{1}{p_r} \\right).\n\\end{align}\n", "edf530ef7ff54f22f1510cdf20b7f8ee": "E_C(\\mathcal{N})", "edf54383a045a8871ee55c8de23dce6d": " a = v \\frac{d\\theta}{dt} = v\\omega = \\frac{v^2}{r}", "edf54efd8e6c13aa1b6cae8c741185a8": "a\\geq b", "edf56289b3f48ea74f213c74628332e5": "x_2(t) = \\dot{x_1}(t)", "edf5828a66d792df297b548b44c05679": "\\varphi_{(X,Y)}(t,s) = \\varphi_{X}(t)\\cdot \\varphi_{Y}(s).", "edf5ab6b2f4fc24e15ce0841c358fb8d": "P(\\omega_s\\mid\\xi) = \\frac{p(\\xi\\mid\\omega_s)P(\\omega_s)}{p \\left ( \\xi \\right )}", "edf5b255eae1add7bb25672fab3f48e4": "\\alpha:S^n\\to M\\ ", "edf5b72bece2f595b12a7feb445bdf58": "S_{1}, S_{2},..., S_{n}", "edf60b9331de376b88237f726c49b2e1": "G : \\{0,1\\}^\\ell \\to \\{0,1\\}^n", "edf6171ad8e8b19627cd0d180c91dee2": "t_n - t_0", "edf64a659ab974c8d191ce4a322362f9": "[C]=[A]_0 \\left (1- e^{-k_1 t} \\right )", "edf6afc4d69e2abe5250778194df0304": "k \\in \\{1,2,3,\\dots\\}\\!", "edf6d7d3a6c6ea704ce3b6f924b8c95b": "\\scriptstyle(\\Omega,\\mathcal{F},\\mathbb P)", "edf6e76385640405ce3441746d1d5de4": "((2\\times 4+1)^3+1)/2=365", "edf6eb72db44bf9d1f6fba3075efe836": "Q=\\{m \\setminus L \\,\\vert\\; m\\in M\\}", "edf71ac228ec8ecd8a2173ead41cc587": " \\dot{x}_i = \\frac{ p_i - e A_i }{m}. ", "edf730f6f60e64706229579747b2b1f7": "\\lambda_i+\\mu_j", "edf731e941de066c7e96539f841bad77": " a_X(1,z) ", "edf746063dd0ebd38c14871e4fdd2daf": "AX = \\lambda X", "edf74b2f5c64113b4fb8741c5cf94d5c": "Q_0", "edf7606ce02bd110a736bbeff6003a9e": "p= \\frac{(b_1+b_2)}{2}", "edf76ea7abe65a5c8fc218c640adfa50": "0 \\le g(x) \\le f(x)", "edf772807453d06c2f50d5fb0a40b264": "0 \\leq |\\mathbf{v}| = |\\boldsymbol{\\omega}\\times\\mathbf{x}| < c ", "edf7af2f17a1ee16244f9d8b9046fa04": " S( k\\delta t) = S(0) \\exp\\left( \\sum_{i=1}^{k} \\left[\\left(\\mu - \\frac{\\sigma^2}{2}\\right)\\delta t + \\sigma\\varepsilon_i\\sqrt{\\delta t}\\right] \\right)", "edf7e024a9a87fd0aaedec15c9a5d60d": "\\lim_{y \\to b} f(y) = c", "edf800faf4d7a2d7261e9d0d45461060": "\\omega_\\mathrm{res} = \\sqrt{\\omega^2 - \\left ( \\frac{b}{4m} \\right )^2 } \\,\\!", "edf8040c57cf65bff09cb3d71f8c4b1f": "\\begin{align}\n {[}A{]} \\{\\Delta X_1\\} &= \\{B_1\\} + \\{\\Delta P_1\\} \\\\\n {[}A{]} \\{\\Delta X_2\\} &= \\{B_2\\} + \\{\\Delta P_2\\}\n\\end{align}", "edf879dc7f488e56411aa220134db940": "\\int - {1\\over 4} B_{\\mu\\nu} B^{\\mu\\nu} - {1\\over 4}\\mathrm{tr} W_{\\mu\\nu}W^{\\mu\\nu} - {1\\over 4} \\mathrm{tr}G_{\\mu\\nu} G^{\\mu\\nu}", "edf9092a5991b62a295ec0011a16f6a9": "|f_n|\\leq M_n", "edf939a48bd1c209dde856ddf2f29629": " \\left ( \\tfrac{Z}{\\beta} \\right )_{min} ", "edf98990fef6ac929443bf720d985f33": "\\tfrac{M - 2G}{2M - 2G}", "edf997acc189b67ec3b77573a3156fbe": " \\bar x_1 ", "edf99a9de5544d94cdab120942f5b41c": " U^n_K / (U^n_K \\cap N_{L/K}(L^*)) \\ . ", "edf9c38654176ce694c108941a08bf08": "||AB|| \\leq ||A||\\cdot||B||", "edf9efa3667a092494a6975777da8b4b": "d_k=\\deg T^kp", "edfa0afccbef1304e099d57d7effa3b5": "a=q-1", "edfa1fbd9d41e0490cb810725a94ac34": "v_\\text{g} = \\frac{\\partial\\nu}{\\partial k} ", "edfa2355b18cc41f2a0ad7e9a1d3f230": "n(d)=\\left\\lceil \\sqrt{2d\\ln2}+\\frac{3-2\\ln2}{6}+\\frac{9-4(\\ln2)^2}{72\\sqrt{2d\\ln2}}\n\\right\\rceil", "edfa33599bfb261f99b7e39ca3488c67": " O(N^2 T \\log T)\\, ", "edfa38f1a5096a7d5663e1404874d3d5": "O = s^2", "edfa8876361382241787baefdb7873de": "\\mu\\left(\\varphi_t(U) \\cap U \\right) = 0.\\,", "edfaddb4b0ebc7fa9c49861f0389780f": "\\,T_F = 40 + N_{15}", "edfb532099687a12d8220e083108de01": "\\frac{\\sum_{p \\leq \\eta}\\left \\{ \\frac{\\eta}{p} \\right \\}}{\\pi(\\eta)} \\approx1- \\gamma,", "edfb6fc94ef0793257dba78bc12c0c41": "\\hat{Z}(x_0)=\\begin{pmatrix}z_1 \\\\ \\vdots \\\\ z_n \\end{pmatrix}'\n\\begin{pmatrix}c(x_1,x_1) & \\cdots & c(x_1,x_n) \\\\\n\\vdots & \\ddots & \\vdots \\\\\nc(x_n,x_1) & \\cdots & c(x_n,x_n) \n\\end{pmatrix}^{-1}\n\\begin{pmatrix}c(x_1,x_0) \\\\ \\vdots \\\\ c(x_n,x_0)\\end{pmatrix}\n", "edfb7e343174c5eb872e88fdbebff4e0": "d(\\gamma_1(t),\\gamma_2(t))\\leq K", "edfbb91d91e5183954453d43d8d2dce6": "\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k+\\alpha_k\\mathbf{p}_k) \\geq c_2\\mathbf{p}_k^{\\mathrm T}\\nabla f(\\mathbf{x}_k)", "edfca8fd7d2c59656e81a251f0dfa74e": " P_i \\propto 1/{K_i}^\\beta ", "edfcbd6ebc9b4b53755e56f8a7a8d556": "P\\left(n\\right) = a p^n + b q^n + c r^n", "edfd0f2dbac54e9ffecc43f174251523": " \\mathbf{u}\\otimes\\mathbf{u} = \\begin{bmatrix}\nu_x^2 & u_x u_y & u_x u_z \\\\[3pt]\nu_x u_y & u_y^2 & u_y u_z \\\\[3pt]\nu_x u_z & u_y u_z & u_z^2\n\\end{bmatrix},\\qquad [\\mathbf u]_{\\times} = \\begin{bmatrix}\n0 & -u_z & u_y \\\\[3pt]\nu_z & 0 & -u_x \\\\[3pt]\n-u_y & u_x & 0\n\\end{bmatrix}.\n", "edfd3ab7309f9f1cbdfd7f7d1be1e97f": "p_{\\theta}", "edfd6a986b6300646728b1f6711a54e5": "\\left(11\\right)", "edfd7e67ba6e41aeb2b06d2ec6c0b5c1": "c_{2\\alpha_i} = 1", "edfd8af2341457973e5c983328d25e6a": "A(\\cdot)", "edfe15ff04bc576e05c2cc6ecb7ac24b": "p(\\sigma|I)={1 \\over a} p(\\sigma^{(1)})={1 \\over a}p({\\sigma \\over a}|I)", "edfe32ec4f3f6c7002e486b1f703fd74": " Y = \\begin{cases} 1 & \\text{if }Y^\\ast > 0 \\ \\text{ i.e. } - \\varepsilon < X'\\beta, \\\\\n0 &\\text{otherwise.} \\end{cases} ", "edfe6b4bfbab05317a3c57335257d014": "\\partial \\alpha / \\partial Q \\ne 0", "edfe786672ef7a67f470ab08ba8897c1": "\\frac{a}{24}", "edfe7c376424292d00e4c6594b75b215": "\\ +\\ Dx^\\prime\\cos \\theta\\ -\\ Dy^\\prime\\sin \\theta\\ +\\ Ex^\\prime\\sin \\theta\\ +\\ Ey^\\prime\\cos \\theta\\ +\\ F\\ =\\ 0", "edfeaebf6ef8627af9f1365c62f2a800": "V=(2+\\frac{4\\sqrt{2}}{3})a^3\\approx3.88562...a^3", "edfed2082d701f1a25a849427a1f8938": "\\frac{df}{dz}=u-iv", "edff1abd616918431dd8f3089614244c": "\n\\begin{align}\n\\mathrm{E}[|X_t-S_N|^2]&=\\mathrm{E}[X_t^2]+\\mathrm{E}[S_N^2]-2\\mathrm{E}[X_t S_N]\\\\\n&=K_X(t,t)+\\mathrm{E}\\left[\\sum_{k=1}^N \\sum_{l=1}^N Z_k Z_l e_k(t)e_l(t) \\right] -2\\mathrm{E}\\left[X_t\\sum_{k=1}^N Z_k e_k(t)\\right]\\\\\n&=K_X(t,t)+\\sum_{k=1}^N \\lambda_k e_k(t)^2 -2\\mathrm{E}\\left[\\sum_{k=1}^N \\int_a^b X_t X_s e_k(s) e_k(t) ds\\right]\\\\\n&=K_X(t,t)-\\sum_{k=1}^N \\lambda_k e_k(t)^2\n\\end{align}\n", "edff5679cace46e1dfeeb74883861814": " \\operatorname{div} \\ \\equiv \\nabla \\cdot ", "edff6695f75c6913f3f94289c8d3adf5": "\\operatorname{\\bar{f}}(\\bar{r},\\dot {\\bar{r}},t)", "edff7bf006c0af9e26d9b3c3b9f85ffe": "\\int \\arcsin(x)\\,\\mathrm{d}x = x \\arcsin x - \\int \\frac{x}{\\sqrt{1-x^2}}\\,\\mathrm{d}x", "edffd1a73cb003cdba3d07600e9aa8f2": "y\\ ", "edfff6a5955f0fddcfa13c7a0d263d6f": "\\leq\\kappa\\,", "ee000f98be09179402d06f6a85d4657a": "\\ \\epsilon_2 = 2n\\kappa", "ee003e2bf052dbf6fa31819da1985404": "x_{\\infty} \\in \\mathcal{X}", "ee005d0dd03de62ba21b42a7bfa869f1": "\nJ\\!D\\!N = \n\\text{day} + \n\\left\\lfloor\\frac{153m+2}{5}\\right\\rfloor +\n365y+\n\\left\\lfloor\\frac{y}{4}\\right\\rfloor -\n32083\n", "ee00981f6040708003f6b9c043f4756d": "\nG_{p,q}^{\\,0,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) = 0,\n", "ee0119589925c51ee2158a0a4c873b2e": "\\sum_{j=1}^{p} \\frac{1}{p^{2m}}\\Bigg\\{ \\sum_{k=0}^{\\infty}\\frac{\\sin \\left[(kp+j)\\frac{q\\pi}{p}\\right]}{(k+(j/p))^{2m}} \\Bigg\\} ", "ee0143bd1ab3a5dfde961da57f3f7c99": "\\nabla(z) = (z^2+1)^2.", "ee01dad04908e4a766c31462453eb8c2": "f(x) = y", "ee01def2eaf49c2834f714a7f1a16d5e": "1967 = [1, 4, 35]_{42}", "ee02105d28f27f76eab92e0811370b37": " C = \\sqrt{gd} = 3.1\\sqrt{d} ", "ee0295d66460adab8de72f343ba13550": "M(x)", "ee02a7cb671b51b3889bf94e8669a898": "\n\\begin{bmatrix} Y' \\\\ U \\\\ V \\end{bmatrix}\n=\n\\begin{bmatrix}\n 0.299 & 0.587 & 0.114 \\\\\n -0.14713 & -0.28886 & 0.436 \\\\\n 0.615 & -0.51499 & -0.10001\n\\end{bmatrix}\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n", "ee02b12ee33ff81fb3a32966a81e7bf4": " \\epsilon = 0", "ee02c7ba1f931c2e1867e682a513f877": "~\\Pi (W/2)", "ee02d224260d657e7c770c3c1dac112d": "f|_U : U\\to f(U)\\,", "ee030403abbffc0e649bd5a3ecf45e07": "\\mathbf{A}\\colon\\mathbf{B}=\\sum_j\\sum_i\\left(\\mathbf{a}_i\\cdot\\mathbf{d}_j\\right)\\left(\\mathbf{b}_i\\cdot\\mathbf{c}_j\\right)", "ee03089f2b87eb21ca3f49d4a8d0d965": "\\mu=GM", "ee031fb5a403909707c69c02069fc99d": "A = \\frac{n}{2} s^2 \\cot{\\frac{\\pi}{n}} + n s h.", "ee03b03b8f2e79e64cd247e07fab3bf7": "S_{r^*}'(r^*)=0", "ee03e07fec1b4337956398bcacc3dad6": "([1-x^2]\\,y')' + \\left(\\ell[\\ell+1] - \\frac{m^2}{1-x^2}\\right)\\,y = 0,\\,", "ee045e089bc9ddd8c1e38de2491e375a": "C_{n} = \\frac{C_{n-1}}{3} \\cup \\left(\\frac{2}{3}+\\frac{C_{n-1}}{3}\\right).", "ee054d189d43206887653c3a19e2fcf4": "\\Lambda^{(k,k+1)}(t) = L^{G^{(k, k+1)}}(t ; 0)", "ee0550c717bc7f328a51258866414fcf": "\\Delta x \\left[f(a) + f(a + \\Delta x) + f(a + 2 \\Delta x)+\\cdots+f(b - \\Delta x)\\right].", "ee0559c39b7914f493b2d035144d372c": "(T_K f)(x) = \\int_0^1 K(x, y) f(y) \\, \\mathrm{d} y.", "ee0562048d9abdf38e0cfc20dc640e88": "\\zeta(-3)=1/120", "ee06f3f5fa33e7f317617adc96e347b6": "\\begin{align}\\mathbb{P}(A_1\\cup A_2\\cup A_3)&=\\mathbb{P}(A_1)+\\mathbb{P}(A_2)+\\mathbb{P}(A_3)\\\\\n&\\qquad{}-\\mathbb{P}(A_1\\cap A_2)-\\mathbb{P}(A_1\\cap A_3)-\\mathbb{P}(A_2\\cap A_3)\\\\\n&\\qquad{}+\\mathbb{P}(A_1\\cap A_2\\cap A_3)\n\\end{align}", "ee07b9324459e6c12714f0ea3a0c78a5": "\\textstyle \\Phi = \\sum_{e\\in E} \\int_0^{x_e} d_e(z) \\, dz", "ee07d96cbae49c6cd25b1c80212ccc18": "y = \\sum_{n=0}^\\infty a_nx^n", "ee082902573220340178685a9fb38199": "k[V] = \\operatorname{Sym}(V^*)", "ee083ea9aa54f071feecff9c7edde310": "z_w", "ee08c2c641d0499e0b82daf132477641": "\\dfrac{\\Delta B - \\Delta A + PA'}{L1} = \\dfrac{\\Delta C - \\Delta B - QC'}{L2}", "ee08ed4fbc1209c905497e77b70efc3e": "(7 | 13)", "ee091a03ad54e98b17c5821684c4ead4": "\n\\rho (r) = {\\rho_0\\over 4\\pi}\\left({r\\over r_0}\\right)^{-2} \\left(1+{r\\over r_0}\\right)^{-2}.\n", "ee092baa01dc33a2bd1d0dde10696754": "{\\bar{Q}}_6", "ee099bc469655d7ec332696a2a2e423a": "\n\\frac{dV^m_n}{dt}=\\left[-i_{ion,n} + \\frac{d\\Delta x}{4\\rho_i L} \\cdot \\left( \\frac{V^m_{n-1}-2V^m_n+V^m_{n+1}}{\\Delta x^2}+ \\frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\\Delta x^2} \\right) \\right] / c\n", "ee0a4ea78b3e391686a36c9f82d0ab67": " | x \\rangle ", "ee0a6de68f5e1009c1d242b5c5f962a4": "x_k\\neq 0\\quad k=1,\\ldots,n.", "ee0a7fc5d1253376d6f9e71ce99341f5": "ap +~bq", "ee0a86eafecf4118ae57c31b2ff4a1a0": "\\left(\\frac{n}{m}\\right)", "ee0a94c89ac1969b846d104099196333": " X(x) ", "ee0ab97c05de1d03134d948cf937759f": "G_{nm}", "ee0b42be3c1f22153e0a06cb7b7f0cc5": "\\phi \\lor \\psi", "ee0bad9c032fe2f2f39b4088d42f080c": "\\boldsymbol a = \\left(\\ddot r -r {\\dot \\theta}^2 \\right ) \\hat{ \\boldsymbol r} + \\left(r \\ddot \\theta +2 \\dot r \\dot \\theta \\right) \\hat {\\boldsymbol \\theta} = 0\\ ,", "ee0bae6b3783e382332e5bd4fc7f8ce5": "\\qquad \\sum_{i=1}^n w_ix_i \\leqslant W, \\quad \\quad x_i \\in \\{0,1,\\ldots,c_i\\}", "ee0bcd0d705c404d8484c1052306ac2e": "H=\\{(a,b,c,f(1),f(\\omega),f(\\omega^2)) : f(x):=ax^2+bx+c; a,b,c\\in GF(4)\\}.", "ee0bf81a0b410acecce50eba182a6882": "u_1,..., u_n", "ee0c1c99dca32e13f2fc61e37b2953e4": "0 < \\tau < t\\,\\!", "ee0c3ca72754ee9c0ad6f73328307cce": "f(x) = \\sum_{n = 0}^\\infty a_n T_n(x)", "ee0ce4590e573a16e903b64b21631375": "\\min\\sum_{i=1}^n c_i x_i", "ee0d09e59c95255e9dcc7a0d6f1997c1": "\\Delta\\theta", "ee0d0eded557c348a263b33d9d11dea5": "R = R_{\\mu \\mu} \\,", "ee0db5a8d198468150edd98aeb5f7882": "\\mathbf{E} ( \\mathbf{r}, t ) = \\mathrm {Re} \\{ \\mathbf{E} (\\mathbf{r} ) e^{ i \\omega t } \\}", "ee0dd6cbd19627d8942e8860c8956e4a": " \\lim_{z\\rightarrow-\\infty}v\\left( z\\right) =0,\\quad\\lim_{z\\rightarrow\\infty }v\\left( z\\right) =1. ", "ee0df596e5a1d541c7ce42cb82d3719a": "A(\\xi) = |\\hat f(\\xi)|,", "ee0e1d43a3d8a8942e9ab75343ea04af": " \\mathit{I}\\,\\!", "ee0e1d949dc166b574beadf0eaeb172e": "\\prod\\nolimits_{\\{\\sigma\\}}\\left(a(1)_{\\sigma(1)}e^{\\alpha(1)}+\\cdots+ a(n)_{\\sigma(n)} e^{\\alpha(n)}\\right) = 0", "ee0e6f9bd701f9962f1bf6a6e5a707f1": "L^2 \\leq \\frac{\\pi}{4} \\mathrm{area}(\\partial P).", "ee0e8e46a4c26c4cb5bca9ed8c98699c": "s \\in L^2(R)", "ee0edc5757d3d13c52301875e2e0eee5": "\n\\begin{align}\nds^2 &=\n\\begin{bmatrix}\ndu&dv\n\\end{bmatrix}\n\\begin{bmatrix}\nE&F\\\\\nF&G\n\\end{bmatrix}\n\\begin{bmatrix}\ndu\\\\dv\n\\end{bmatrix}\\\\\n\\end{align}\n", "ee0edf614b7b9d78bc112861e8469d3b": " \n(Eq. 8) \\text{ } \\text{Minimize: } \\lim_{t\\rightarrow\\infty} \\overline{P}(t)\n", "ee0f1a9c4b0067ba86dc04fc957bb6cf": "\\begin{matrix} {2 \\choose 1}{2 \\choose 2}{3 \\choose 1}^2{10 \\choose 1}{4 \\choose 2} \\end{matrix}", "ee0f46932168a69497d45e62af1856d7": "\n \\begin{bmatrix}\\varepsilon_{11} \\\\ \\varepsilon_{22} \\\\ 2\\varepsilon_{12} \\end{bmatrix} = \\cfrac{1}{E}\n \\begin{bmatrix} 1 & -\\nu & 0 \\\\\n -\\nu & 1 & 0 \\\\\n 0 & 0 & 2(1+\\nu) \\end{bmatrix}\n \\begin{bmatrix}\\sigma_{11} \\\\ \\sigma_{22} \\\\ \\sigma_{12} \\end{bmatrix}\n ", "ee0fa8ad946b4bc2d4ceb4e0d9756a64": "|\\mathrm{Aut}(P)| = \\left(\\prod_{k=1}^n{p^{d_k} - p^{k-1}}\\right)\\left(\\prod_{j=1}^n{(p^{e_j})^{n-d_j}}\\right)\\left(\\prod_{i=1}^n{(p^{e_i-1})^{n-c_i+1}}\\right).", "ee0fc2ca8299fbc1875d82e457000c39": "L(C)=\\Omega(\\operatorname{Cr}(C)^{3/4})\\,", "ee0fc8e911842e37e7709f5d721ee4e6": "2y-w+2x=0", "ee0fe621c945adb6dc90eae263c3468c": "\n-\\frac{u^{'''}\\left(x\\right)}{u^{''}\\left(x\\right)}\n", "ee1000d41ee9941c4ba7bd874f3fdd46": "U_{(ij)k\\dots}=\\frac{1}{2}(U_{ijk\\dots}+U_{jik\\dots})", "ee10150b90c2bffe4e7f5c1582f1e3fc": "(\\forall x')(\\forall x)(\\forall y)(\\forall u)(\\exist v)(\\exist y') (P) Q'(x',y') \\wedge (Q'(x,y) \\rightarrow \\psi)", "ee1032c343f3f5fc6548763dc5ed2d4a": "\\left[ \\begin{matrix} C \\\\ CA \\end{matrix} \\right] = \\left[ \\begin{matrix} \\left[ \\begin{matrix} 1 & 0 \\end{matrix} \\right] \\\\ \\left[ \\begin{matrix} 1 & 0 \\end{matrix} \\right] \\left[ \\begin{matrix} 0 & 1 \\\\ -\\frac{k_2}{m} & -\\frac{k_1}{m} \\end{matrix} \\right] \\end{matrix} \\right] = \\left[ \\begin{matrix} 1 & 0 \\\\ 0 & 1 \\end{matrix} \\right]", "ee104d493a45c0cbbe6ff7fcbd57a971": "\\hat{H}_\\text{IR} = \\dfrac{e\\mu_0\\mu_\\text{N}\\hbar}{4\\pi}\\sum_{\\alpha\\neq\\alpha^\\prime}\\dfrac{1}{R_{\\alpha\\alpha^\\prime}^3}\\left\\{\\dfrac{Z_\\alpha g_{\\alpha^\\prime}}{M_\\alpha}\\mathbf{I}_{\\alpha^\\prime}+\\dfrac{Z_{\\alpha^\\prime}g_\\alpha}{M_{\\alpha^\\prime}}\\mathbf{I}_\\alpha\\right\\}\\cdot\\mathbf{T}", "ee10566fc16a07ccda0d4d5798d689e8": "c(R)(x,y) = \\operatorname{tr}R(x,\\cdot,y,\\cdot).", "ee106b44fc1eb847ac5b40667fa2f5e0": "m(\\theta) ", "ee10ec860c42f76d220f606e304004f3": "\\iota \\circ \\pi", "ee115724073896683fcee0d4ca2ec45e": "p_{\\theta}=0", "ee12405221043a04c5d0f1ad19576da8": "J_3\\,", "ee126f595c1fd329f5c650c08042e419": "\\phi_n(\\kappa) =\n\\frac{1}{4\\pi^2\\kappa^2} \\int_0^\\infty\n\\frac{\\sin(\\kappa R)}{\\kappa R}\n\\frac{\\partial}{\\partial R}\n\\left[R^2\\frac{\\partial D_n(R)}{\\partial R}\\right]\\,dR", "ee12be531cd3aecf63d6ed35b6ba9889": "(Nkr / Tkn) * 100", "ee133ef1a28ae6f8bcf231e9dca9c222": "\\gamma \\in [0,1]", "ee13540b607ab0a309b111da53681afb": "C_n = C_V {\\gamma-n \\over 1-n}", "ee1367ca40aa286bcf7bcfb6a7b16a4b": " O[n_0] = \\left\\langle \\Psi[n_0] \\left| \\hat O \\right| \\Psi[n_0] \\right\\rangle.", "ee137a2eda25e88b7e2d303c6e909fff": " \\gets", "ee137c5beab5e395b4c907ff2f6be511": "|D\\chi_E|(\\Omega)", "ee1387ff0329e34ac4efb60011197da0": "h_1 - h_2\\,", "ee13bcb8e79f54e16994fc8eb2f373ab": " V = \\sum_{i=1}^{n}PV_i ", "ee13d427ae34bd6e415c9fe92410af28": " \nT_1=\\sum_{i}\\sum_{a} t_{a}^{i} \\hat{a}^{a}\\hat{a}_{i},\n", "ee13f3d6bf2e8e8fda55a50ac7eed711": "z=x^4-y^4,", "ee1410f7890330ace418a5298b563b45": "\\vec y_n, y_n = \\vec y_n[n]", "ee14185480d1a87f3616dfd14503eda3": "\\nu_c", "ee14278773da1552c802a91bbcec89be": "\\tfrac{3}{6}=\\tfrac{1}{2}", "ee142ac8dc84cc169f50ea086c12f859": "e^{\\begin{bmatrix} \\mathbf{A} & \\mathbf{B} \\\\\n \\mathbf{0} & \\mathbf{0} \\end{bmatrix} T} = \\begin{bmatrix} \\mathbf{M_{11}} & \\mathbf{M_{12}} \\\\\n \\mathbf{0} & \\mathbf{I} \\end{bmatrix}", "ee144ac6dd90819c02c4e42bf5db2a29": "\\frac1{n} = \\frac1{p\\cdot n} + \\frac1{\\frac{p\\cdot n}{n-1}}", "ee14661727dcd89ca1809e538f5bb354": "\\frac{AH}{AB} = \\frac{AB}{AA'}", "ee146a36d9fa68e384f7bd5b2e503e89": "\\Delta(N, P, T) = \\int Z(N, V, T) \\exp(-\\beta PV ) C dV. \\,\\;", "ee147b3f109c755f0efb956c6efe97d8": "\\rho =-1/2", "ee14a090037ae3e0b6be8b66697e757e": " R^{n}", "ee157d76b928132542be4b5b808fbaa7": "\\mathrm{sys}^2 \\le \\frac{2}{\\sqrt{3}}\\cdot\\mathrm{area}", "ee15a8f2cbee7dede53d1ce27b1fdf6d": "\\ GM ", "ee15a94d6e2bcd93dc0740ee785d8caa": "T = \\frac{Qv}{\\rho c} \\operatorname{erf} \\left(\\frac{h}{4 a \\sqrt{t}}\\right)", "ee15ccd7966842d936dbaccd4d251ca9": "F_{78.6\\%} = \\left({\\frac{1 + \\sqrt{5}}{2}}\\right)^{-\\frac{1}{2}} \\approx 0.786151 \\,", "ee165aa881b8b18c1659d4da99554cd3": " S [\\text {Bq/g}] = \\frac{ln2 \\times {N_A}}{T_{1/2} [s] \\times {m}} = \\frac{ln2 \\times {N_A}}{T_{1/2}[year] \\times365\\times24\\times60\\times60 \\times m} \\simeq \\frac{1.32\\times 10^{16} }{T_{1/2}[year] \\times m}", "ee1698c9eac400524e1c1ac824800a39": "r^2 = \\| \\mathbf x[p_\\alpha] - \\mathbf x[p_\\beta] - \\mathbf q_{\\alpha\\beta} \\|^2_2", "ee181ea4b2437a9b2940031fb47797de": "x_0, \\ldots, x_n", "ee18a22cbb410a8ff6f91b71eacc497f": "Y/\\sigma \\sim \\chi_1", "ee18b5a8d197e91964707ca27381a456": "e(t)=\\sum_{i=1}^N r_i\\delta(t-\\tau_i)", "ee18c51c48db904abf2c31071b1b9be7": "\\scriptstyle A c^3 / 4 G \\hbar", "ee192bb81486db10be69ac63e8380fc8": " 03,", "ee3d400e9e3917e4b318df2cc90bcd21": "\\operatorname{PG}(a_n;x)=\\sum _{n=0}^{\\infty} a_n e^{-x} \\frac{x^n}{n!} = e^{-x}\\, \\operatorname{EG}(a_n;x)\\,.", "ee3d4596a3f824bf6b85f238727e0b2d": "|m/n|_p = |m|_p/|n|_p", "ee3d8f0f190fdeb6ce7ea37b8824c888": "\\epsilon/3", "ee3db1bbdc2242f0a55990318cf78b6c": " Jz_t = \\nabla H(z) ", "ee3e30e8e36eb0979c904de2abe1d7f7": "\\lambda > 1", "ee3e6e3de4c4267a86ec3ae2e1cd3fce": "\\left|\\alpha-\\frac{p}{q}\\right| < \\frac{1}{\\sqrt{8} q^2}.", "ee3e706dcaad94b5271d1197f0321b3c": "\n\\begin{align}\n\\Pr(Y_i=1) &= {\\Pr(Y_i=K)}e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i} \\\\\n\\Pr(Y_i=2) &= {\\Pr(Y_i=K)}e^{\\boldsymbol\\beta_2 \\cdot \\mathbf{X}_i} \\\\\n\\cdots & \\cdots \\\\\n\\Pr(Y_i=K-1) &= {\\Pr(Y_i=K)}e^{\\boldsymbol\\beta_{K-1} \\cdot \\mathbf{X}_i} \\\\\n\\end{align}\n", "ee3eab9bc486d173e93732caf9651cdf": "a(x, y, p)", "ee3eee79cb3570a05cc31c618b866ffa": "a + b = c", "ee3efbbe52323fdbabef27e4b57e320c": "Y_\\ell^m", "ee3f25f2c51d42c5c0fbb0c006d80f33": "c_6 = -5.481717 \\times 10^{-2},\\,\\!", "ee3f2b0dce9d4e7e4220ec638dd6dbc7": "\\lim_{t \\to \\infty} h(t) = \\lim_{s \\to 0} \\frac{6s}{s+2} = 0.", "ee3f62ac23c20c17c4942faa12bb54dc": " e=(1)(2)\\cdots(n) ", "ee3f847e05491211534e781b9edb8c01": "\\langle\nA_i,X_0\\rangle_{\\mathbb{S}^n} = b_i", "ee3f97976a3037c1a68d18773417d798": "\\frac{x^{n-1}}{(n-1)!a_1\\ldots a_n}(1+o(1)).", "ee3fb59e1101b666528c5d74210357aa": "g_\\alpha^* = 0", "ee3fecd192f6217df6af156d8f191428": "\\scriptstyle l_{m-1}=(1-\\gamma)^{m-1}/2^{m-1}", "ee4004705b032d7cf5969a2821100f80": "\\lim_{n\\to \\infty}T_{n,n}(z)=T(z)", "ee404ccaf112be361b5abd8359297a7d": "(A + B) + C = D + C = Q = A + F = A + (B + C)", "ee405a5607a8ceecb74d11f699d21ba4": "\n \\gamma_k=\\sqrt[4\\;]{\\frac{\n \\|U_k^{-1}\\|_1\\,\\|U_k^{-1}\\|_\\infty\n }{\n \\|U_k\\|_1\\,\\|U_k\\|_\\infty\n } }\n", "ee40c16db7270dd1dced4baf9044bf37": "\\{ x \\in \\mathbf R ~|~ x = x^2 \\} \\,\\!", "ee40e6ed0401dc64f35dae052e17cccc": "| \\phi\\rangle , | \\psi \\rangle", "ee40e81aabdd30725aedc0c5430a1109": "f:O\\to\\mathbb R", "ee412631504219371be3f3308d344d30": "a_0=\\left\\lfloor\\sqrt{S}\\right\\rfloor\\,\\!", "ee4168e415f6e3f266130af1a6a01ef6": "a^2 + b^2 = m^2x^2\\pm 2mxc + c^2 + n^2x^2 \\pm 2nxd + d^2 = Ax + (c^2+d^2).", "ee41d581ac9bf824f507f29c587c3db5": "\\hat{'}", "ee42004aaf4f83546126257eabdeb519": "T_{pq}(e_p \\otimes e_q)", "ee4265079ef9147ec8270ca255c6a5e0": " H_{BPF}(f) = \\mathrm{rect}\\left( \\frac{f}{2B_H} \\right) - \\mathrm{rect}\\left( \\frac{f}{2B_L} \\right).", "ee42666309cd6248557b23c88383ab80": "f(x) \\geq", "ee429491ffb6f351d5502e0ff9ee5cde": "\\rightsquigarrow y", "ee42da6abb978ade3ab516f942392755": " \\left\\vert \\log( 1 - p ) - \\log( p ) \\right\\vert \\ge 2 \\log( d - \\sqrt{ d^2 - 1 } ) + 2d \\sqrt{ d^2 - 1 } ", "ee42e47925cf7097e0bb86be934a52b4": "\\hat{w}_i = 4 + \\epsilon 0; i = 1,2", "ee430c90ed5b535cef68a840dcd75aad": "F(a)=\\int_{-\\infty}^a \\left[\\int_{x}^{\\infty} f(x|t)g(t)dt \\right]dx .", "ee43d1e79b0d75256259e89c7b8a4072": "q= - K \\frac{\\partial h}{\\partial z} ", "ee43da56cb460e97c823ca0d9a916fae": " a=(7^2-5^2)/2=12 \\,,\\ b = 70/2=35 \\,,\\ c = (7^2 + 5^2)/2=37 ", "ee4459d4ad4134de47abfe4fd9b30277": "\\displaystyle P_{Y_r}(y) = \\sum_{x\\in X} \\sum_{s\\in S} P(x)P_{S_r}(s)W'(y|s)", "ee44717c210bdfceada28640a4977dbe": "T_n(x)", "ee44815caf622e9c03b85d9ae409231e": "\n\\vec{D} \\equiv \\operatorname{col}(\\sqrt{M_1}\\;\\vec{d}^{\\,1}, \\ldots, \\sqrt{M_N}\\;\\vec{d}^{\\,N})\n\\quad\\mathrm{with}\\quad\n\\vec{d}^{\\,A} \\equiv \\vec{\\mathbf{F}}\\cdot \\mathbf{d}_A .\n", "ee44cb7c4c929f3a1d163aceda8fef74": "\\Bbb R^n", "ee451d97976b60ce7254422370fa32f0": "|A\\cap C_n|=1,\\forall n<\\omega", "ee4528f89afb7678dbafa4337d17651f": "e^x.", "ee4597ac3d792c7f1fd8d30f5806fa5f": " x^{\\prime}(s)^2 + y^{\\prime}(s)^2 = 1 \\ ; \\ \\frac{1}{\\rho} = y^{\\prime\\prime}(s)x^{\\prime}(s)-y^{\\prime}(s)x^{\\prime\\prime}(s) = \\frac{1}{\\alpha} \\ . ", "ee45a2eb598f8c67581efde97cbc7847": "\n\\operatorname{P}( Z \\ge \\theta\\operatorname{E}[Z] )\n\\ge (1-\\theta)^2 \\frac{\\operatorname{E}[Z]^2}{\\operatorname{E}[Z^2]}.\n", "ee460887af9d898ebe3035decc7243ea": "\\scriptstyle n=n_0 - \\frac{C_0}{\\left(m+m'\\right)^2}", "ee46ebfb17c284602e68b88de5469f70": "\\nabla = \\sum_{j=1}^n \\bigg[{\\frac{\\partial}{\\partial x_j}}\\bigg] \\mathbf{\\hat{e}_j} = \\bigg[{\\frac{\\partial}{\\partial x_1}}\\bigg] \\mathbf{\\hat{e}_1} + \\bigg[{\\frac{\\partial}{\\partial x_2}}\\bigg] \\mathbf{\\hat{e}_2} + \\bigg[{\\frac{\\partial}{\\partial x_3}}\\bigg] \\mathbf{\\hat{e}_3} + \\dots + \\bigg[{\\frac{\\partial}{\\partial x_n}}\\bigg] \\mathbf{\\hat{e}_n}", "ee47013c7b52d6b205d3cf5678e8d1c6": "f_2(x,y) = (x')^2 +\\sqrt y", "ee474bc9913a64bb257440adee3ed8df": "\\rho_4 = \\rho_5 \\frac{P_4 + \\Gamma P_5}{P_5 + \\Gamma P_4}", "ee479bc0cbbb872799d76456707967e1": "P[X=0] = \\frac{G^0 e^{-G}}{0!} = e^{-G}", "ee47b7b2f7bde15007814bb8e7b554e9": "\\ G=\\frac{\\pi^2}{\\theta\\phi}", "ee47f49f2aaca45ffec32c9b808b0db5": "\n W = Nk_B\\theta\\sqrt{n}\\left[\\beta\\lambda_{\\mathrm{chain}} - \\sqrt{n}\\ln\\left(\\cfrac{\\sinh\\beta}{\\beta}\\right)\\right]\n ", "ee4848b99fa3e8f338f3ac5cb0ccd939": "p_i .", "ee4893c8eee4b37de5090af0df328789": "E_{Zeeman} =-\\mu_{0} \\int_V\\,\\textbf M\\cdot \\textbf H_{Ext} \\, \\mathrm dV", "ee48fe79a76403fb8797fb4ca0fe3d8a": "x(T) = \\frac {1} {2} FD(1-x)T+ \\frac {K} {T} ", "ee4906397349007b86fe8b37648903c4": "[a,b) = \\{x \\in \\mathbb{R} : a \\le x < b \\}", "ee490d4e66e840c6d22d280dcb7cb333": "N(\\mu_i,\\Sigma_i)", "ee490ff9b3de4fa64e44e8f1ddb02fbe": "A=\\frac 1\\sqrt{2n(n+\\gamma)}\\sqrt\\frac C{\\gamma\\Gamma(2\\gamma)}", "ee498673553f53eb8e3133ab9ce97cfe": "\\sum_{i=0}^{n-1} \\left|\\frac{a_i}{a_{i+1}}\\right|", "ee49d06e6aa0bcb470af709dd2d8832f": "c_5 = -6.83783 \\times 10^{-3},\\,\\!", "ee4a1c554a6e81bed0b9fee21f97a7e9": "j(\\tau) = \\frac{1}{{q}} + 744 + 196884{q} + 21493760{q}^2 + 864299970{q}^3 + \\cdots", "ee4a23dda745669174dd7322e0d484fb": "\\, e^{t\\mu + \\frac{1}{2}\\sigma^2t^2}", "ee4a26edc0110f441a40685aaad9ee97": "\\mathbf{Z}", "ee4a4e8216f6d3ac27924665be58c2d3": "\\scriptstyle P(t)", "ee4b1c080fc3ce071a8c2d7706e4804e": "\\mathrm{DA}=", "ee4b279899376d480101fa6bb8b6853a": "\\pi_i : X \\to X_i", "ee4b564b5925820dd6d40aae53bdfdb7": " Z = \\frac {X - \\operatorname{E}(X) } { \\operatorname{Var}(X)^{ 1/2 } }.", "ee4b61982d7902a7615952a29ae20892": "\\mathrm{M} = 0.88128485 \\sqrt{\\left(\\frac{q_c}{p} + 1\\right)\\left(1 - \\frac{1}{7\\,\\mathrm{M}^2}\\right)^{2.5}}", "ee4b75fc9c67dbffec288fe13bc7b16a": "R\\otimes_\\mathbb{Z}S", "ee4b882fc447ce7ed5d8e51e725414b6": "W_{2n}", "ee4b8ca96a3df5ada0e1fbcca0629d2e": "{\\tilde{E}}_7", "ee4ba271ab3451760188abe68ac1d16f": "\nT\\,=\\,2\\,\\pi \\,\\sqrt {{L \\over g}} \\,\\,\\left[ {1\\,\\,\\, + \\,\\,\\,{1 \\over 4}\\sin ^2 \\left( {{\\theta \\over 2}} \\right)\\,} \\right]{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(1)}}", "ee4ba3757c6252bd0806fd2824736350": " \\frac{dI}{dt} = \\beta \\frac{I}{N} S - (\\nu +\\mu ) I ", "ee4bcc0b53ec879e989d29614d10f707": "\\boldsymbol{k},", "ee4be2d48dd8fa189ff291194547a841": " v(f) \\approx 1 - f + (1/6) f \\mathrm{ln} f.......... (30c) ", "ee4bf08a7d411eb65c46207b2f112f5f": "|h|\\geq 1", "ee4c86e69c4e4efc5ccc62ce9eb7f171": " n \\not \\subset \\{p, q, m\\} ", "ee4cb4fcdf5e2fb6bfed7d61e5327f5b": "y_0,y_1,\\ldots,y_{m-1}", "ee4d374e0323c645937eaf0d19daf747": "\\begin{cases} - \\mathcal{A} u(x) = f(x), & x \\in D; \\\\ \\displaystyle{\\lim_{t \\uparrow \\tau_{D}} u(X_{t})} = g \\big( X_{\\tau_{D}} \\big), & \\mathbf{P}^{x} \\mbox{-a.s., for all } x \\in D. \\end{cases} \\quad \\mbox{(P2)}", "ee4d3dc030f909595ee67a8cc717e95a": " n! \\sim e^{n \\ln n} n \\sqrt{\\frac{2\\pi}{n}} e^{-n}\n\\sim \\sqrt{2\\pi n}\\left(\\frac{n}{e}\\right)^n.\n", "ee4d6478bef32c85169559df2907c6d4": "\\langle R^{2} \\rangle = 2Pl", "ee4dc33fc44fe281ada610ae38d5068f": "MN/M \\cong N/(M \\cap N)", "ee4e45ca645c4b42c0111916bb0924f4": "Z_M = z_M e^{j\\phi} = z_M \\cos \\phi + jz_M \\sin \\phi = r_M + jx_M ", "ee4ec59f96e5258ed3fcae567663778e": "(11,6)", "ee4ee0d22336ebe8f83bbdd004de4f9b": " wz + h + j - q ", "ee4ef1beebc906cff235ed20fdbc1aa6": " e ^ { - \\beta \\Delta F} = \\left\\langle e ^{ - \\beta (U_\\text{B} - U_\\text{A})} \\right\\rangle_\\text{A} ", "ee4ef6804afc2097b8ae603f800776f6": "\\Phi^{-1}(\\mathbb{E}[Y_n]) = \\boldsymbol\\beta \\cdot \\mathbf{s_n}", "ee4f4d833c6228c064e8a42f0106f028": "M^{n}", "ee4f57b029339d9f2a267476b3cda9c9": " Q_2 = P + Q + 1 ", "ee4fa66a9d1d71b4b1499405d4d128b5": "\n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{j 0", "ee62055f31f93eefecdbc2f137cef664": " 2 \\text{ } H_2 O \\text{ } \\stackrel {\\mathrm{photon\\,energy}\\, > 1.23 eV} {\\rightleftharpoons} \\text{ } 2 \\text{ } H_2 + \\text{ } O_2", "ee621a8215803bda16d6ce5424ea5c45": " \\left| \\sum_{i,j} a_{ij} s_i t_j \\right|\\le 1", "ee630174c91d939eb1ac20982247564d": "\\tbinom{p+q}{p}", "ee6303d629239b7ead54ea227b5efb87": " E_\\text{rms} = {\\frac {2 \\pi f N a B_\\text{peak}} {\\sqrt{2}}} \\! = 4.44 f N a B_{{peak}}", "ee63140f346501d1119c02e0e881d0b7": "\n\\rho_{m}^{\\prime}=\\frac{1}{p\\left( m\\right) }\\sqrt{\\Lambda_{m}}\\rho\n\\sqrt{\\Lambda_{m}},\n", "ee6371ede3aa4680d9bfc5b6a909a434": "\\mathbf{x}, \\mathbf{V}, \\boldsymbol{\\mu}", "ee63f23c70d0c9e33428213d715b00ba": "\n\\begin{align}\nf & = A_0x^0+A_1x^1+A_2x^2+A_3x^3+A_4x^4+A_5x^5+A_6x^6+A_7x^7+\\cdots \\\\[8pt]\n& = A_0x^0 + A_1x^1 + {-1\\over 2}A_0x^2 + {1\\over 6}A_1x^3 + {-1 \\over 8}A_0x^4 + {1 \\over 24}A_1x^5 + {-7 \\over 240}A_0x^6 + {1 \\over 112}A_1x^7 + \\cdots \\\\[8pt]\n& = A_0x^0 + {-1\\over 2}A_0x^2 + {-1 \\over 8}A_0x^4 + {-7 \\over 240}A_0x^6 + A_1x + {1\\over 6}A_1x^3 + {1 \\over 24}A_1x^5 + {1 \\over 112}A_1x^7 + \\cdots\n\\end{align}\n", "ee6494822785b30d7390b374608c78ee": "F(K)=\\|K\\|_2 := \\left(\\int_M K^2\\right)^{1/2}", "ee64971047a45549ab397d314f991ad8": "A + A' = \\pi", "ee650bb2cbc914947dcafe6493d7d891": "\\ \\Delta G^\\ominus = -RT \\ln K ", "ee658c6ed4993a89b5d57c42fff03159": "\\phi_1,\\dots,\\phi_r", "ee65a949a4fd0f4ed81d4a6693ba4673": "a,b\\in\\R", "ee6624e89fbb6a55b648f654e69b37c4": "\\nu \\circ \\nu = \\nu", "ee667f2f7c7e630246ee34d50d35ab4d": "(a,b) \\in \\{0,1\\}^8", "ee668a8b96a7fc367f89135101df6c90": "y=mx+b", "ee669c838467894ea3a21a7a864cc4d3": "{}+ e^4 \\left| \\frac{(\\bar{v}_{k} \\gamma^\\nu u_p )( \\bar{u}_{p'} \\gamma_\\nu v_{k'} )}{(k+p)^2} \\right|^2 \\,", "ee6787d3ea938a4bbcf1a06d99428fa9": "\\models_{\\mathrm P}(\\phi \\rightarrow \\psi)", "ee67dbb78e4e4bb25faf5dd94ab9cdd0": " T^*(T(\\rho)^{it}T(\\sigma)^{it})=\\rho^{it}\\sigma^{-it}", "ee67f2f632fe17b5b70f05dab855cd9a": "L(\\vec{r},\\hat{s},t)", "ee68001864bd20b68c05f9f13d2099c4": "A_0(x) B_0(x) = 0", "ee680a4f1c570c52ca2f1b0f2ed165ac": "\\Psi \\equiv \\Phi' \\equiv \\Phi \\wedge \\Phi \\rightarrow \\phi", "ee68a58cbdd589082aea04ced6428fdb": "v_c ", "ee68a945ab30168638a78bef820f1a5b": "\n\\mathbf{G}_x = \\begin{bmatrix} \n1 \\\\\n2 \\\\\n1\n\\end{bmatrix} * \\left ( \\begin{bmatrix} \n1 & 0 & -1 \n\\end{bmatrix} * \\mathbf{A} \\right )\n\\quad \\mbox{and} \\quad \n\\mathbf{G}_y = \\begin{bmatrix} \n\\ \\ 1 \\\\\n\\ \\ 0 \\\\\n-1 \n\\end{bmatrix} * \\left ( \\begin{bmatrix} \n1 & 2 & 1\n\\end{bmatrix} * \\mathbf{A} \\right )\n", "ee68f33ba895c96fcb7f2c2a0e76438e": "\\kappa(X_1,X_2,X_3,X_4)\\,", "ee697c507c8cc19096d84ae14308c8e5": " \\varphi = \\varphi\\, ", "ee6a2cbc95b2f37abd9d38127bf6f02a": "0.5 p^2/m", "ee6a53de19cc89be5ece0d3992396533": "\\scriptstyle u\\in BV(\\Omega)", "ee6a718f0ee6072525c8cc2b24eff277": "\\!r", "ee6ab1321048887c7157d821fb3584fe": "\\Lambda_{ij} = \\lambda_i\\,\\delta_{ij}", "ee6ab7d36fb8e8d66e47ea919c470c70": "S = k_{\\rm B} \\ln \\Omega \\!", "ee6adcbec43c61590904d5219adbc02c": " r^{-n+2}~\\sin(n\\theta) \\,", "ee6bcb1a2241a715f5af289928eccbdf": "\\lnot \\textit{par}(h,m)", "ee6bf50ad7c3184d91ef575c2cf0afcc": "(X',A')", "ee6c3e26e778e75e51603156d5f1a93e": "-\\pi<\\sigma\\le\\pi", "ee6c7f5d9e054b0fd79cc8648b34779a": "n=1-\\delta+i\\beta", "ee6c88ec46d013c8546ae783eaec31ad": "du_o/ds", "ee6ca9a9f9087cf270521c64f03fe4b7": "x=a\\sec(t), y=a\\tan(t)\\sec(t)", "ee6d816815b1e401badf82eaf9d32540": "g:=g^+-g^-", "ee6d9bc72a55ade87bd8d77e2b65b651": " G_i E_i= E_i G_i = l^{-1} E_i,", "ee6da7b0d38200716586f8110d3f8ed1": " \\dot{x} = \\frac{dx}{dt} = f(x) \\, ", "ee6ddc0611c410774fb8a7bf118c4fb0": " \\operatorname{const}\\ f = x ", "ee6e3056e3ecb1cd30a65a5e0252cfdd": "Multiplier=\\frac{1}{1-[MPC(1-T)-MPI]}", "ee6e70d0b9354f9e9d4f35bb49afa82a": " S_{xy} = \\eta \\gamma.\\, ", "ee6eb3f1c4f8e4849b710b3000e94572": "\\displaystyle{B(x,y) =-\\Im (x,y)}", "ee6ef6e0da9e90739ba4f875a0e67534": "\n\\left |\\sum_{k=1}^n a_k b_k \\right | \\le \\operatorname{max}_{k=1,\\dots,n} |B_k| (|a_n| + a_n - a_1),\n", "ee6f013c97035825e91c02feb654f4bb": "G=\\langle X|R\\rangle", "ee6f5e0b5c6991639663cacd272f5896": "\\scriptstyle \\mathbf{Q} = \\lim_{k\\to\\infty}\\mathbf{P}^k.", "ee6fa291dcda40731474dbe0f5d531d9": "H^d(S)", "ee6fa4b66008efeda35b8ae409247bb5": "[Q^\\dagger,x\\}=ib^\\dagger", "ee6fc97a06ffb3052bc0a075604e2d09": "{\\Gamma^i}_{ij}=\\ \\begin{Bmatrix}\n \\,i\\,\\\\\n i\\,\\,j\n\\end{Bmatrix} = \\frac{1}{h_i}\\frac{\\partial h_i}{\\partial q_j}= \\begin{Bmatrix}\n \\,i\\,\\\\\n j\\,\\,i\n\\end{Bmatrix}\\! \\ ;\\ ", "ee6fe24b5ad435eb11168cacd574a0ec": "\\hat{\\textbf{t}}_i = \\Sigma_k^{-1} V_k^T \\textbf{t}_i", "ee6ff8c463cc72e21b7bda9af8b206e0": "\\varepsilon(p) = (p-1)/2", "ee701f6d3a710f3a441bde53dd552a9b": "\\scriptstyle \\lbrace x \\isin V : r(x)\\ =\\ \\text{nonzero constant} \\rbrace ", "ee703d9d9e2a72f369d50fe7fad80c7f": "\\mathrm{2\\,SO_2 + O_2 \\rarr 2\\,SO_3}", "ee70aa2db6db6615e17af8fd71d04165": " -\\beta C_{Ni}^{j+1}-(\\lambda+\\alpha)C_{i-1}^{j+1} +(1+2\\lambda+\\beta)C_{i}^{j+1}-(\\lambda-\\alpha)C_{i+1}^{j+1}= \\beta C_{Ni}^{j}+(\\lambda+\\alpha)C_{i-1}^{j} +(1-2\\lambda-\\beta)C_{i}^{j}+(\\lambda-\\alpha)C_{i+1}^{j}.", "ee70bcb407da3b7fb878013bd80cd4b2": " \\vec{c} ", "ee70cb214591f1cf91dc6ec3cc7261f0": "\\mathrm{ad}_{Y_\\lambda}^{-(\\mu,\\lambda)+1}Y_\\mu = 0\\text{ for }\\lambda\\ne\\mu.", "ee70eb3d350755aba0fed41f32a0d26e": "f_{ij}=f^{-1}_{ji}", "ee70ede85ac54aae8b0ac858d4deaebc": "\n\\log p(\\mathbf{X}|\\boldsymbol\\theta^{(t)})\n= Q(\\boldsymbol\\theta^{(t)}|\\boldsymbol\\theta^{(t)}) + H(\\boldsymbol\\theta^{(t)}|\\boldsymbol\\theta^{(t)}) \\,,\n", "ee70f65588e7ba4be0e65ece53ccba66": "\\|a_j - P_{F_i}(a_j) \\| = dist(a_j,F_l)", "ee7118a4bf8c14a0cf886d835c9ac9d7": "\\tau _{Auger1}(t,x) = \\frac{2.12\\cdot 10^{-14}\\cdot \\sqrt{E_{g}(t,x)}\\cdot e^{\\frac{q\\cdot E_{g}(t,x)}{k\\cdot t}}}{FF^{2}\\cdot (\\frac{k\\cdot t}{q})^{1.5}}", "ee71a6ea124c6d2c38cb775980771da1": "[ \\hat{x}, \\hat{p} ] = 0", "ee71ad1765280bd82b796d8da1365f0f": "\\sum_{k=0} a_k \\xi^{\\nu+k}= \\frac{1+\\xi^2}{2\\xi} \\mathcal L \\{f \\} \\left( \\frac{1-\\xi^2}{2\\xi} \\right),", "ee71d62b75908292a71cdf17aa1ffd6f": "f(x) = (cx+d)^{\\deg(p)}p \\left (\\frac{ax+b}{cx+d} \\right )", "ee728ebb6919e28a5497bcab38f28911": "M = \\left \\{\\left.\\begin{pmatrix}A_1 & 0 & \\cdots & 0 \\\\ 0 & A_2 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & A_r\\end{pmatrix}\\right|A_j \\in GL_{n_j}(\\mathbb{F}_q), 1 \\le j \\le r \\right \\},", "ee72904e8eb9809245fd973b55262603": "\\scriptstyle \\log_2{(1+x)}\\cong x+\\sigma", "ee73186e90463a7382893d182c1314c3": "|1\\rangle ", "ee7338e0e1867db06ccc9a6c4dc0f046": "m_{xy} = |\\mu_x-\\mu_y|", "ee7347103377cea0bd03ed216d9bd4b8": "R(t) = R_0 e^{t/2},", "ee7357a3453fd27d533124d520c97b34": "M_{D}=k\\cdot nY", "ee739a951161fd5aa254c374cb58849b": " \\mu_i = \\left( \\frac{\\partial G}{\\partial N_i}\\right)_{T,P,N_{j\\ne i},etc. } \\,.", "ee739e26d573556cb6669e48ba585472": " {2048 \\over 2025} ", "ee73aed56654505afeefa1a7529382eb": "\n \\vdash A \\or \\lnot A , A\n ", "ee7447e84b04f7d3940daef4ef78665d": "\\widehat{T}(\\theta, t) = \\frac{1}{e^{t(1 - \\cos \\theta)}} \\approx \\frac{1}{{1 +t(1 - \\cos \\theta)}} = H_1(\\theta, t),", "ee74766d76dcb503c10da198fe390ca4": "g : \\mathbb R^n \\to \\mathbb R", "ee7486563472469d1dbb2b5f5d076221": "I_h = A_5 \\times Z_2", "ee748fd47fe785ece490cd4bc08ea0a0": "w_n = (1C1)^{-1} 1C", "ee74cc2d9c47b53cc1e8cb69da2403ea": "h(1)-h(0)=\\int_0^1 h'(t)\\,dt\n=\\int_0^1 \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(a+t(x-a)) (x_i-a_i)\\, dt\n=\\sum_{i=1}^n (x_i-a_i)\\int_0^1 \\frac{\\partial f}{\\partial x_i}(a+t(x-a))\\, dt. ", "ee7514d950f6115a87d6929a48f7f9d7": "Y_{mn}^\\sigma", "ee753ac0131d9f290dde094a701e13f4": "T_{comp}", "ee7596c691b8c503b605549d628ae5df": "\\beta(T,\\mathcal{A})=0 \\mbox{ implies that } T \\mbox{ is in } \\mathcal{A}", "ee75f144d91b9f2407a0f4f7fb7b6c20": " 6.00 = y_2 + \\frac{30.0^2}{2(32.2)y_2^2}", "ee763957fcf0ac8383e88e865e6d4ee2": "b = Tf", "ee7675c33dadeb043339dca17d1c2b7a": "f(t) = \\frac{1}{t}, \\quad t \\in [0,1]", "ee76c7a5ea6ffe834c190803aa2ec14d": "\\nu_{\\rm zy}", "ee76e4e90d44cc3505c8aa6e5a098649": "\\infty /mmm\\, ", "ee7760d1afbf9e500f23bed98457e4dd": " g(x) = \\frac{1}{-\\lambda} g( g(\\lambda x ) ) ", "ee77cd72573eec25fba471d91befc2d2": "\\C", "ee77d9a019b386f2430c460c90e577f1": "E[F | x^{(t)}]", "ee77ef0ad5484441b66c5adcf3656f92": "F: C^{\\mathrm{op}} \\rightarrow D", "ee785633e4c9913e881434d1bb276660": " f_i(x) \\le_{K_i} 0 , i = 1,\\ldots,m", "ee78da0badb1e208559af4c500dc4eb0": "S_{n-1} = \\frac{2 \\pi}{n-2} S_{n-1-2}", "ee78f9e0c9ff4f3e043a8d1e88354c55": "\\begin{align}\n {} &\\frac{1}{2}f\\left( 0\\right) + f\\left( 1\\right) + \\cdots + f\\left( n - 1\\right) +\n \\frac{1}{2}f\\left( n\\right) \\\\\n = &\\frac{1}{2}\\left[f\\left( 0\\right) + f\\left( n\\right)\\right] + \\sum_{k=1}^{n-1}f\\left(k\\right)\\\\\n = &\\int_0^n f(x)\\,dx + \\sum_{k=1}^p\\frac{B_{k + 1}}{(k + 1)!}\\left[f^{(k)}(n) - f^{(k)}(0)\\right] + R_p\n\\end{align}", "ee7949f21578086667f08c098df6f606": "\\frac{\\mathrm{circumference}}{\\mathrm{diameter}}=\\frac{2\\pi R \\sqrt{1-(\\omega R)^2/c^2}}{2R} = \\pi \\sqrt{1-(\\omega R)^2/c^2}", "ee796602149b4aa9696883696ebe886f": "\\bar{\\ell}\\ell", "ee7982ca64867c9d10f07d832a58070a": "P=S", "ee798697de4a386b0cda1023c562dcb6": "p=0.0, 0.1, 0.2,\\ldots, 0.9, 1.0.", "ee79a55c2e7568e1d667e541c712ea45": " \\frac{\\alpha \\pi}{2} ", "ee79c291d7b5a22cc1564d82eb2d921a": "\\begin{align}\n f^+(x)&=\\max(\\{f(x),0\\}) &=&\\begin{cases}\n f(x), & \\text{if } f(x) > 0, \\\\\n 0, & \\text{otherwise,}\n \\end{cases}\\\\\n f^-(x) &=\\max(\\{-f(x),0\\})&=& \\begin{cases}\n -f(x), & \\text{if } f(x) < 0, \\\\\n 0, & \\text{otherwise.}\n \\end{cases}\n\\end{align}", "ee79d15fdb62db0d45b9eb36d93ef064": "\n \\int x^m \\left(2 a\\,A\\,c (m+n (2 p+1)+1)-a\\,b\\,B (m+1)+\\left(2 a\\,B\\,c (m+2 n\\,p+1)+A\\,b\\,c (m+n (2 p+1)+1)-b^2 B (m+n\\,p+1)\\right) x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p-1}dx\n", "ee79d2edf0c2d9120c5c78485441fef5": "A\\leq_{f} B", "ee79d6accc93da4c683638f55854db21": " e_2 ", "ee7a1258ebe858a4f2630b32d847ab3e": "\\frac{\\partial u}{\\partial x} = \\frac{\\partial u}{\\partial r}\\frac{\\partial r}{\\partial x} + \\frac{\\partial u}{\\partial \\varphi}\\frac{\\partial \\varphi}{\\partial x},", "ee7a3b9e1a9eb3687f8680c02685ea6a": "I(T_i)", "ee7a9bd57d1c67e7c4fddcd4c34a5c56": " \\bold{G}=\\frac{8\\pi G}{c^4}\\bold{T}.", "ee7aedf3e3681af7e343e561b259a4cc": "\\mathbf{J}_{\\perp}=\\epsilon_0\\int \\left({\\mathbf{E}}_{\\perp}\\times\\mathbf{A}_{\\perp}\\right)d^{3}\\mathbf{r} +\\epsilon_0\\sum_{i=x,y}\\int \\left({E^i}_{\\perp}\\left(\\mathbf{r}\\times\\mathbf{\\nabla}\\right)A^i_{\\perp}\\right)d^{3}\\mathbf{r} .", "ee7af79b2b0b04e32875407332613d25": " I^I = \\prod_{i \\in I} I_i ", "ee7afc2736f1cc1609c8a253cc2251c0": "\\; \\log \\rho", "ee7b43c2f11e09d51ee25a551a4cdede": "\\tau_c*=f \\left(Re_p* \\right)", "ee7b4c11fdbec49d44a795d18d5ff52d": "\\omega_Q = \\frac{1}{\\sqrt{L_{QA}C_{QA}}} = \\frac{\\hbar \\omega_c}{\\phi_0e} = \\frac{\\omega_c}{2\\pi}", "ee7bf013e68d31818219ea7d95a10a36": "\\Delta_z(\\mathbf{r})", "ee7c54161d0dfb3d06d2430400263bbd": "\\overline{\\epsilon}_{st} ~=~ \\frac{1}{n}\\sum_{i=1}^{n}\\epsilon_{ist}", "ee7ca7d6373eddd045d1a98b71580ecd": "(1,4]", "ee7cee8e4b1fc927ba459ef6bda1a484": " y(t) = e^t ", "ee7cf364ba9b56ae9c5eeb7b924e5e3c": "\\sum_{k=1}^{n-1} \\csc^4\\frac{\\pi k}{n}=\\frac{n^4+10n^2-11}{45}\\,\\!", "ee7cfa721fa83aa40264a10320a3036c": "\\displaystyle{f_{y+t}=V_t Pf_y}", "ee7d2267812b60b8c783553bb9c205ac": "\\scriptstyle arctan(1/\\sqrt{2})", "ee7d7b32ef779c67460292ba6d600191": "S_n = a + [a + d] r + [a + 2 d] r^2 + \\cdots + [a + (n - 1) d] r^{n - 1}", "ee7dac7f6b1e6891d930724ac23c0a8a": "\\mathfrak d = \\min(\\{|F| : F\\subseteq\\omega^\\omega\\land\\forall f:\\omega\\to\\omega\\exists g\\in F(f\\leq^*g)\\}).", "ee7de8819d757713d8b2bc962ac30d58": "y = {k \\over x}.", "ee7def0bd09af3255593011004d7bfb0": "F(x; \\mu,\\sigma,\\xi) = \\frac{1}{ 1 + \\left(1+ \\frac{\\xi(x-\\mu)}{\\sigma}\\right)^{-1/\\xi}}", "ee7e50b5eb76faeb114e7576054587c8": "\\sigma(t) = (\\tan t, 1)", "ee7ef246eae097da0055742904df498c": "t=t_n", "ee7f499635f69dbc34e63c53f997118d": "(x', x')", "ee7f4b2156446665e3282d1bbb9aedd2": " t= 0 ", "ee7f5085faa395f139ddd52726cd47f0": "H_t(\\mu)", "ee7f5ffa65ea5beb48038100a2d7e670": "\\psi_1(x)=\\int_0^x \\psi(t)\\,dt.", "ee7fd371d807e5a8979f9b46c0be5a23": "L_2(\\xi)=\\frac{1+t}{1-t}=\\left(\\xi+\\sqrt{\\xi^2-1}\\right)^2", "ee7fd4b4accd3b80b34d4c2cee4ccd7b": "f'(x_i) - f'_i", "ee7fd4e07769f365567ac986e2598bd4": "\\mathcal{C}_{1}", "ee7fde2e9eb13f5f6a6f79c17ac0f3fb": "\\beta(w_1)=0\\quad \\mbox{or}\\quad \\neq 0.", "ee7ff8ab4be44d4c4b77bb033327fe01": "\\epsilon_{n} = -\\frac{\\part N}{\\part P_{n}}\\frac{P_{n}}{n}", "ee8028a6dace92ce93ef3d6aba399ceb": "I_n= \\int \\frac{dx}{(a^2-x^2)^n}\\,\\!", "ee802c0ce07c57c13ba2e98791720fe7": "\nP = k_B \\,T \\Delta f\n", "ee804f0ab97d0dd98ecce3a2915e3a72": " Q = \\Gamma^T \\Gamma ", "ee80b2e35a56f6b7f705e762d2d02d2f": " \\frac{dx}{dt} = -xy \\quad\\text{ and }\\quad \\frac{dy}{dt} = -y+x^2-2y^2 ", "ee81330a96c7e2ae53ea844792259974": " C(V) = \\bigcup\\{ I^0\\subseteq \\hat T(V): I\\triangleleft T(V^*),\\, \\mathrm{codim}\\, I <\\infty\\} ", "ee813cdc0d7fa5a9e197502a0b241935": "f: x_i \\mapsto x_{i+1}", "ee813f0ede8664a8049b1b6720f03b60": "i,j", "ee8197756563aa7e07a91c27b52c6a47": "v_i = ", "ee81bccc3f8559a79d3bc122828f0267": "\n\\frac{1}{a_0} + \\frac{x}{a_0a_1} + \\frac{x^2}{a_0a_1a_2} + \\cdots +\n\\frac{x^n}{a_0a_1a_2 \\ldots a_n} =\n\\frac{1}{a_0-}\n\\frac{a_0x}{a_1+x-}\n\\frac{a_1x}{a_2+x-}\\cdots\n\\frac{a_{n-1}x}{a_n-x}.\\,\n", "ee81e574b0360e40a9a776d02fb5e5dd": "\\pi/2 - 2\\arctan (\\sqrt{s})", "ee8216a630c2a8618c3a0b990f9aa694": "c < M", "ee822e24807d6da8da9abef914e7b74c": "H\\{\\bold{X}\\} = \\lim_{n\\to\\infty} \n\\frac{H(X_0, X_1, \\dots, X_{n-1}, X_n)}{n+1}.", "ee829ea61bcc3fb3efccf2a3987528fc": "B=0", "ee82a80993ea8ce1142befc6b8e2b3d0": "\\int x r^{2n+1}\\;dx=\\frac{r^{2n+3}}{2n+3} ", "ee82cc5fa0f4fac7883291db0a9f53b4": "16\\over 5", "ee83074b197eda44976d97898ee8fb27": "\\textstyle X(\\omega)=\\omega ", "ee8330102a4820ffb4df658dd95bec1a": "K = dE_{tr}/dm", "ee8365eeac67409d480c593d2127e9b8": "\\, b = \\sum_{\\nu} b_{\\nu} \\gamma^{\\nu} ", "ee83e553924188fe59487873ead1f0de": "\n \\displaystyle \n w(0,g)\n =\n 1 \\ , \\forall g\n \\ ,\n {\\rm and}\n \\ \n w(n,0)\n =\n 1 \\ , \\forall n\n", "ee83f39f1bfc414e2c97c38e5e981a6e": "\\sigma = E \\varepsilon \\iff \\varepsilon = \\frac{1}{E} \\sigma \\, ", "ee841c357fab148bcf9a86f8719d01e0": "h_i \\in H_i", "ee84215e327caccbb65e46a003dc3f37": "C_p^\\ominus = (4.186a+b\\breve{S}^\\ominus_{T_1}) {{(T_2-T_1)} \\over {\\ln(T_2/T_1)}}", "ee8430d0708d55a71bf6969477a50261": " =\\Pr(\\tau_{n+1}\\le t, X_{n+1}=j|X_n=i)\\, \\forall n \\ge1,t\\ge0, i,j \\in \\mathrm{S} ", "ee84a9014d510ac4ddc67188e9ed6afa": "\\chi'", "ee84bfee4e3478fe025b38d56a07d641": " \\rho\\vec{\\nabla} \\times \\vec{v} ", "ee84d80bed23c001b2ec32d426470a37": "[J,-].", "ee8516a8de4330d6c9d04ed012907fa8": "Q=\\begin{cases}\n0 & \\text{for } P \\leq I_a \\\\\n\\frac{(P-I_a)^2}{{P-I_a}+S} & \\text{for } P>I_a \\end{cases}", "ee85177bd48de24c18b110ad6a7a20c1": "\\operatorname{Im}\\{x[n]\\}", "ee853b28bb73f84d23b3de259681115b": "f_5(x) = \\log_c x, \\ c \\ne 0, 1", "ee853cfe05cf9abfb3943afe4a4abb9a": " \\ c_L = 2\\pi \\alpha", "ee859ba1f5d596a799f128974e0b993a": " P = \\begin{matrix} \\frac{\\Delta E}{\\Delta t} \\end{matrix} ", "ee85c76a47fdf5df28bf8825e9f04b18": "\\mathrm{rem}(7, 2) = 1", "ee85c7f458e63a89ef40b2927df7d977": "~\\frac{2}{\\log 2} \\log\\log x", "ee86b49a049318ecda2b735a54904dc5": "i_v\\alpha = 0\\,", "ee86d79f5ad8bb63467001683b56ff4f": "\n\\mathbf{C}^0_{xy} = \\int_{\\Delta} xy \\, dA = \\int_{x=0}^1 x \\int_{y=0}^{1-x} y \\, dy \\, dx = \\int_0^1 x \\frac{(1-x)^2}{2} \\, dx = \\frac{1}{24}\n", "ee86dcf11d239eaa889d63247681ac0b": "M = 2 \\pi R^2 (R + W) P {\\rho \\over \\sigma}", "ee86f17a5fb723cbad0313624b88be9a": " D[p] = [F_7, S_7, A_7]::[F_6, S_6, A_6]::K_5 ", "ee8751d624cd63cec641d4af67e85776": "\\tilde{6}/4", "ee8773604d191cb83caebbe2b2b93afb": "\\text{Economic batch quantity} = \\sqrt{ \\frac{2 \\cdot \\text{annual demand} \\cdot \\text{setup costs}}{\\text{inventory carrying cost per unit}} }", "ee884306c820037c8588a425cc899d8f": "Y(z)=X(z)+E(z)\\left(1-z^{-1}\\right)", "ee885bfae30fef9dfde88ad5d0680d69": "\n[x_3]:=[x_3] \\cap ([x_1]+[x_2]) \n", "ee8873d54921da2240183fd3c53755b4": " F_{r} \\phi_{r}-F_{l} \\phi_{l}\\,= D_{r}(\\phi_{R}-\\phi_{N})-D_{l}(\\phi_{N}-\\phi_{L})", "ee887f3d9250b17b2020ee6bdd7b293e": "r_d", "ee88a3e25d957285086a5ed0f72a832e": " \\vdash \\ \\ \\left[ \\ A \\rightarrow \\left( B \\rightarrow C \\right) \\ \\right] \\ \\rightarrow \\ \\left[ \\ \\left( A \\rightarrow B \\right) \\rightarrow \\left( A \\rightarrow C \\right) \\ \\right] ", "ee88b9a41650494859a17dc987aaf4d6": "\\hat{\\pi}_A", "ee88d77dd9c7f50d9a6e4b3e71e74444": "\\ W_{ij}", "ee8948657cf4119d196ce81ec0602ce5": "\\mathfrak{P}^{39}", "ee89505dfa4024b2425be87fd6f6b0e9": "F(x,y) = J(x,Tx - y)", "ee895860a42f0f95af93bfc58a5396f4": "X_1 = x_0 - x_1", "ee895b69c3ff0d1e40d38ec1684f4c86": "\\varepsilon:X^*\\otimes X\\to I", "ee8987c382d515fdcb1b7ef788f676d5": "\\; (A - 2 I) p_2 = 0 ", "ee89a990cd171f6780eee03844aa6736": "(\\phi ,\\psi )=\\int \\phi (x)\\cdot \\overline{\\psi(x)}\\, {\\rm d}x \\,,", "ee89debe5ace3e4ddccb34e675c65005": "T_c \\, = \\, T_b \\left[0.584 + 0.965 \\sum {G_i} - {G_i}^2 \\right]^{-1}", "ee89df0ec13c10af902af8804fb2b125": "\n \\cfrac{\\Gamma_1, A, B, \\Gamma_2 \\vdash \\Delta}{\\Gamma_1, B, A, \\Gamma_2 \\vdash \\Delta} \\quad (\\mathit{PL})\n ", "ee89efd2b023f7babc717700df2ca719": "w \\in \\mathbb{F}^n", "ee8a007b178a4b40e025712844d85940": "H_n(X,A)=H_n(C_\\bullet(X)/C_\\bullet(A))", "ee8a0ce1e0447daf00145cde8e5a1a1a": "O(du\\,dv)", "ee8a0f28938a2b9d626ef4228db97033": " \\frac {dD}{dz}= \\frac {iw}{c_0 Q} D+T(D-R) +\\quad(2.7) ", "ee8a673ba5103a557ec8ee854e9c464e": " \\mathbf{B} \\cdot \\mathbf{\\hat{n}} \\mathrm{d}A = \\mathrm{d}\\Phi_B ,\\,\\!", "ee8aa077bdc28b55dd76743b7a4a47db": " X_{2} ", "ee8ab33058fb030e73263a8b9469445c": "y=C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1}, \\qquad x \\in \\mathbf{R}^n.", "ee8adf45c8601b427a4ee863163322c0": "C_i(x,t)=\\frac{1}{n_i}\\int_c c f(x,c,t) \\, dc", "ee8ae1bf44a67ab80f7ac997d1a508d8": "\\scriptstyle{A_0^{k/(1+x)}}", "ee8b4622bea2c2a0565fe1005f1eb268": "P_a + \\frac{1}{2} {\\rho C_u^2} = P_1 + \\frac{1}{2} {\\rho C^2}", "ee8b4bf6756d4430c8853cbeee23c7cc": "f : A \\to B,", "ee8b6f2ec504e398ee301a031c46101e": "\\Sigma b_n", "ee8bd82b5ed9d1626e2f74ecd5b1bef5": "\\frac{1}{4} \\,+\\, \\frac{1}{16} \\,+\\, \\frac{1}{64} \\,+\\, \\cdots.", "ee8c0e94d76ed3d4ad556657469733f6": "\\sigma_f \\approx a \\frac{\\sigma_A}{A}", "ee8c54e32dc1b6d70faabd27c7b394e9": "\\mathbf{y}=(y_1,\\cdots,y_n),\\quad \\mathbf{F}(x,y_1,\\cdots,y_n)=(y_2,\\cdots,y_n,F(x,y_1,\\cdots,y_n)).", "ee8c6fbcc70058e161f63fbd0a00bf65": " X,Y\\in\\mathbf{L}_M ", "ee8cbdbc6c542e62965b7773aefb4075": "\\dfrac {100 V_{ref}} {R_{d} f_{clk}}", "ee8d21c088a29ca369ffc54b93df2abf": " M(x) = \\; h -eFx - e^2/4 \\pi \\epsilon_0 \\epsilon_r x \\;. ", "ee8dc1f658bca955ddaa40ade311f79a": "dr_t = a(b-r_t)\\, dt + \\sigma\\sqrt{r_t}\\, dW_t", "ee8dc305aa771f30fd78bdb3b350f276": "\\approx F(Ax+h)+(\\Delta Ax+\\Delta h)\\dfrac{\\partial}{\\partial x}F(x).", "ee8de1371cf1bef437fdc1f36da63f64": "\\lbrack H,\\hat{N}_c+\\hat{N}_a\\rbrack=0", "ee8de5e38daf401451e67ceefc39b503": "\\boldsymbol{\\pi} ", "ee8e0350cfa5d6ffaaed35fdb91d93fd": "0\\leq k 0", "eeb359ace4d902196ee27cda8ac348bd": "\nE_\\theta (r) =\n{-jI_\\circ \\sin (\\theta) \\over 4 \\varepsilon_\\circ c\\, r} \\left ( {L \\over \\lambda} \\right )\ne^{j\\left(\\omega t-kr\\right)} \\, , \\quad\nR_a = 20\\pi^2 \\left ( {L \\over \\lambda} \\right )^2 .\n", "eeb35fcc957f016abab6442c2022ca54": " \\star ", "eeb381463e526ccf186314340316b3e3": "\np = \\frac{C_{\\rm D}}{C_{\\rm I}+C_{\\rm D}}\n", "eeb3a04dd4056da4a645370b34f49882": "{\\rm E}[u(A)] < {\\rm E}[u(B)]", "eeb3ef920724f7aaa7d7e5a0b7068ddd": "\\forall \\epsilon>0 \\ \\exists \\delta>0 \\ \\forall y\\in X \\ \\left [d(x,y)<\\delta \\Rightarrow \\forall n \\in \\mathbf{N} \\ d\\left (f^n(x),f^n(y) \\right )<\\epsilon \\right ].", "eeb4009cd402cd4fd8d111e9a741accc": "x_k=1-\\sum_{i=1}^{k-1}x_i", "eeb40e8fda40467e98dcf0152798ab46": "\\mu(x)=-S'(x)/S(x)=f(x)/S(x),", "eeb4114364a5826b8025b326370741c3": "b_Y(x)\\, ", "eeb458b4b80d92be4a521403ce96659d": " \\sgn (X_2 - X_1)\\ = - \\sgn (Y_2 - Y_1)\\ ", "eeb46fd72420dd699d482416eb277765": "[\\mathcal{Q},\\mathcal{Q}]\\subseteq\\mathcal{H} ", "eeb4f812137b58d2205eda5beeb8be31": " \\chi_\\text{Yates}^2 = \\sum_{i=1}^{N} {(|O_i - E_i| - 0.5)^2 \\over E_i}", "eeb504e3a0df6b4897b9dae57c1eb8bf": "y d(x) \\leq \\lambda y d(y) + \\mu x d(x)", "eeb52f811890046bed2a0a8c6c7b8be4": "\n\\langle z \\rangle=e^{i\\mu-\\gamma}\n", "eeb546c7a9eb9ab4941b36828f763066": "\\Phi^{-1}= \\sum_j P_j \\otimes Q_j \\otimes R_j. ", "eeb58311fec82bd2a3f166b74a3c9eac": "\\lim_{n\\rightarrow \\infty} H_{n,m} = \\zeta(m).", "eeb596f1bc08bb7f855170dfb040331f": "q'_o-1=\\beta _o (q_o-1)", "eeb59c8f2509962c7f9a5f50e553c100": "\\delta\\lambda_i = \\mathbf{x}^\\top_{0i}([\\delta K] - \\lambda_{0i}[\\delta M] )\\mathbf{x}_{0i}", "eeb5c3a56ade248f4c7918781f6ec744": "\\begin{align}\\, _1F_1(a,2a,x)&= e^{\\frac x 2}\\, _0F_1 (; a+\\tfrac{1}{2}; \\tfrac{1}{16}x^2) \\\\\n&= e^{\\frac x 2} \\left(\\tfrac{1}{4}x\\right)^{\\tfrac{1}{2}-a} \\Gamma\\left(a+\\tfrac{1}{2}\\right) I_{a-\\frac 1 2}\\left(\\tfrac{1}{2}x\\right).\\end{align}", "eeb5c9f107ccb9a1a717ec5fe3cf9b95": "c'\\not\\subseteq c", "eeb6037ab3a06aad1137389586306195": "\\frac{x^p - 1}{x - 1} = x^{p - 1} + x^{p - 2} + \\cdots + x + 1.\\ ", "eeb6bb2c877a8e6a747791a3fc0443d2": "\\operatorname{Var}[Y]=\\operatorname{E}(\\operatorname{Var}[Y\\mid X])+\\operatorname{Var}(\\operatorname{E}[Y\\mid X]).\\,", "eeb6eba3a750c6a48c7559a4a8e8eaef": "g(G^t(V),G^t(V))=g(V,V)", "eeb6f3a59f3c3dd66aeaf8c980f7c421": " \\psi(\\mathbf{\\eta})= \\oint_{\\partial U} \\left[\\psi(\\mathbf{y}) {\\partial G(\\mathbf{y},\\mathbf{\\eta}) \\over \\partial n} - G(\\mathbf{y},\\mathbf{\\eta}) {\\partial \\psi \\over \\partial n} (\\mathbf{y}) \\right]\\, dS_\\mathbf{y}.", "eeb755b08046c1f56e5f8d93320cf6cc": "f_\\alpha(z)=\\frac{z}{(1-\\alpha z)^2}=\\sum_{n=1}^\\infty n\\alpha^{n-1} z^n", "eeb75761c8561210e88b7c54655e50a0": "X_k = \\frac{1}{2} x_0 +\n \\sum_{n=1}^{N-1} x_n \\cos \\left[\\frac{\\pi}{N} n \\left(k+\\frac{1}{2}\\right) \\right] \\quad \\quad k = 0, \\dots, N-1.", "eeb76872f75a50bc31653e396d6789da": "\n\n\\sum_{\\ell,\\ m } \\frac{4\\pi}{2 \\ell +1}", "eeb78fc26d3e88279f9068b499209d8e": "n_r(n - n_r) + n_e(n - n_e) = n^2 - (n_r^2 + n_e^2) = 8", "eeb7a0a01736faba168cfb8d11840d67": "A \\rightarrow aB", "eeb7bb3f0d77457befee534379d50577": "\\mathbf{Z}[\\pi]", "eeb7e337c2eddf0a53f738dcd6864960": " (T_s + 1)(T_s - q) = 0, \\,", "eeb7e65919b0b236a49b06828bdae0a7": "\\frac{1}{\\zeta(s)}=\\sum_{n=1}^{\\infty} \\frac{\\mu(n)}{n^s}", "eeb7ea5729d26631a56e350438262ce1": "N_0\\subsetneq N_1 \\subsetneq \\cdots \\subsetneq N_n", "eeb84542e0ba40ac5acc4ebe2ca639fa": "ba - ab = 0\\,\\!", "eeb847676a7904749159522db76cfb1f": " v \\tau_m ", "eeb865cc226f64f1fcd8f8d6b9f6a482": "\\prod_{i=1}^N x_i", "eeb8933a1a4621ff2f947d7f6670a255": "0 \\leq i \\leq 127", "eeb8a03a90e705d479835d76c3eaa6b9": " \\delta W = \\sum_{j=1}^m Q_j\\,\\delta q_j,", "eeb8e028fd3d5c06e372e7f444da78a8": "x_i=\\ln a_i^{\\rm eq}", "eeb9171b50d1817560655195e51b0c35": "D = \\frac{4\\pi}{\\Omega_A}", "eeb95143c29cf207335dd1265e69d25a": "e[hf_{xy}]=\\sqrt{(D/N)(B/N)}", "eeb97cd0ca22bd39de567fbb7d9a3c36": "\\sum_{i=1}^3 \\theta_i = \\pi + \\iint_T K \\,dA.", "eeb9d37cd318085c3ed3c3cfe6dfd96d": "\\gamma=1/\\sqrt{1 + \\kappa v^2}", "eeb9e75d8d4f6f6b572c21ff41a65daf": "n = 6,", "eeba16d61cfe9a7f3202a2c1d1dfd48e": " \\int \\sec^n x \\, dx = \\frac{\\sec^{n-1} x \\sin x}{n-1} \\,+\\, \\frac{n-2}{n-1}\\int \\sec^{n-2} x \\, dx \\qquad \\text{ (for }n \\ne 1\\text{)}\\,\\! ", "eeba247fd7369e56484e239c69ec240a": " \\dot{\\Psi}(t) = A(t) \\Psi(t) ", "eeba3e6499ee3b6b49b79ca9d7cb2561": "\\nabla \\times \\mathbf{E}=- \\begin{matrix}\\frac {\\partial} {\\partial t}\\end{matrix}\\mathbf{B} \\ .", "eeba5ab8a063aaf8896c94b240071317": "\\frac{\\partial \\mathbf{Y}}{\\partial x} ", "eeba9a27ea0a8e7c9144c83c3eb38352": "\\tilde{f}_i (b \\otimes b') = \\begin{cases} \\tilde{f}_i b \\otimes b', & \\text{if }\\phi_i(b) > \\epsilon_i(b'), \\\\ b \\otimes \\tilde{f}_i b', & \\text{if }\\phi_i(b) \\le \\epsilon_i(b'). \\end{cases} ", "eebadfbaaa8eda4348de350f0720c124": "F_w", "eebae0ac1838004bf2062ae3ead66101": "R = Z(\\gamma) Y(\\beta) X(\\alpha) ", "eebb4842a7796ae41001e4d41db18761": " F = f(C) = \\tfrac95 C + 32 ;", "eebbf11d52c42b0a41a48c7487a1d054": " \nW_{ab}(t) = \\max[ Q_a^{(c_{ab}^{opt}(t))}(t) - Q_b^{(c_{ab}^{opt}(t))}(t), 0] \n", "eebc01f325c84cf5f7651c4d20328bbb": " n = \\int f d^3v\n", "eebc0995d65e2d0a3895566b82824157": " \\sigma^{2}_{i} ", "eebc280daf6c2d8e7b84289f0a8fe9f8": "\\prod_{i=1}^{n}\\left(1-a_{i}\\bar{a}_{i}-b_{i}\\bar{b}_{i}+a_{i}\\bar{a}_{i}b_{i}\\bar{b}_{i}\\right)=\\prod_{i=1}^{n}\\left(1-a_{i}\\bar{a}_{i}\\right)\\prod_{i=1}^{n}\\left(1-b_{i}\\bar{b}_{i}\\right)", "eebca76cd57413f0b870451a3890af8a": "k_0 = \\sqrt{4\\pi e^2 \\frac{\\partial n}{\\partial \\mu}}", "eebcd6561c8a8fa5f0eb3ba9921394bc": " g'_k (x) ", "eebce023f7938597d109aa678d2b5cd4": " W_{energy} = \\frac{1}{2} \\frac{\\lambda^2}{L} = \\frac{1}{2} ~ L ~ i^2 ", "eebda1aba014349dfb4d24dae183df6f": "z^{-3}", "eebdc82e07395084684ea3ffa6b08612": "E[X(t)] = \\theta t ", "eebde63c7d5fb74c021ccb4d7d6946a7": "((A\\to\\bot)\\to\\bot)\\to A", "eebe3c7fea09460aa55420439f25f8d6": "?\\left(\\frac{0}{1}\\right) = 0 \\quad \\mbox{ and } \\quad ?\\left(\\frac{1}{1}\\right)=1", "eebedd863e1d0adf421bbb2ef93465ec": "\\partial_\\mu j^\\mu = 0", "eebf44e0a5adfc67a587e3da742113a0": "n_B = \\frac{eB}{h} \\ ", "eebf53607c9a570010f504e30eb2edfa": "n!=n^{\\underline n} = n(n-1)(n-2)\\cdots1", "eebf5e4999bed57a51d499507a645105": "H \\subset C_0", "eebfca25efc4621b865839f0f36b4cd5": "\\scriptstyle N_r \\times N_t", "eec070ea4ddc5be55c029451685cb9a2": "\\frac{ (s + 1)(s - 5)(s + 10) }{ (s+2)(s+4)(s+6) }", "eec07d1bdf394e8b2a0712beba96a6e3": "P(\\alpha)=\\frac{1}{\\pi^2}\\int \\chi_N(\\beta) e^{-\\beta\\alpha^*+\\beta^*\\alpha} \\, d^2\\beta.", "eec08822cf85c50705b4bc35add8f8ec": "T = \\sqrt{s(s-a)(s-b)(s-c)}", "eec09a7f977b6c9e616e3a194e76c96f": "\n\\mathrm{var} \\left(\\hat{\\theta}\\right)\n\\geq\n\\frac{[1+b'(\\theta)]^2}{I(\\theta)}.\n", "eec0d10b5b8cc9b80876096367d69686": "\\mathbf{F}_{q^2}", "eec0e57b0660a7ba8333b64881e24daf": " N = \\dfrac{N_c q^2 \\sin^2(\\theta_c)}{1 - \\dfrac{1}{n^2}} ", "eec0f061c2c642b2caf8733aca875814": "\\langle \\mathcal{F}f,\\varphi\\rangle := \\langle f,\\mathcal{F}\\varphi\\rangle,", "eec0f2544529fe69315c82d1ad9ac5a8": " K(G,n)", "eec0f47dd207ed0fd1df072bdc8da826": " BaCrO_\\mathrm{4}", "eec1c8074fcd18354c6a4cd8181a55ad": "|p| = \\max \\{K(x\\mid y),K(y\\mid x)\\}", "eec1fc79936f40ced07f54fa27207aa7": " I_r = - \\Gamma_{TL} I_i \\, ", "eec1fcf523d3b943c8bfcd02df39aee3": "R_1 = 0", "eec2275739bd18f276b5ee02e5dad87e": "\\lambda_\\mathrm{el-el}", "eec2bf755d2e38cc111da83b34bcfbe6": "y \\ge 0.390", "eec2c95a77772b28d394a9713c7d5010": " W_x = \\{\\psi \\in H: \\psi(g) = 0, \\quad \\forall g \\neq x\\}.", "eec2d43344b4135da72eb2deb038ea0c": "i \\in \\{1...r\\}", "eec306ffd211800cf16c35cce2db7a04": " W_{S} ", "eec34674219f6b3cb6e2582891e3eee2": "\\bar{4}", "eec34fbe99b9ea99525e5495dd625bf6": "U = \\sum_{n_x}^{\\sqrt[3]{N}}\\sum_{n_y}^{\\sqrt[3]{N}}\\sum_{n_z}^{\\sqrt[3]{N}}E_n\\,\\bar{N}(E_n)\\,.", "eec394e00add8a6653cf8cc30af1cc9c": "\\ge \\tfrac{20GeV}{c}", "eec3b72294f0f4491c621578af42094d": "\\bold p_Q.", "eec3c3cd231a14183ca1317e100c1a3f": "\\mathcal{K}\\left( t \\right)\\equiv\\mathcal{P}L{{e}^{\\mathcal{Q}Lt}}\\mathcal{Q}L\\mathcal{P},", "eec3ee5e75f665eb34d865df76c604f5": " \\rightarrow (\\lambda x . z) ((\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w) (\\lambda w. w w w))", "eec3f072e901e353bfe38706c7b0b10c": "\\langle x,x\\rangle = 0 \\Rightarrow x = 0", "eec43baca3c923977fde36ccf46a5a84": "c_1(H_{11} - ES_{11}) + c_2(H_{12} - ES_{12}) = 0 \\,", "eec44d6d461a2190af5402f1f99591fe": "O(\\Omega^{-1/2})", "eec45d10b1c5ca9a94520114950402de": "(z + i\\sqrt5)", "eec480de6cc416c9d9347d67973cbf17": "C(x,y):=\\operatorname{cov}(Z(x),Z(y)).\\,", "eec4837f1d9d34c89b4576583224fe6d": " P( S_n \\ge x ) \\le e^{ -x^2 / 2 } ", "eec4879ebe6f70144197e1e41a3cefde": "S \\rarr \\left(A^{?} \\times S \\right)", "eec49c793f3daf066e828701acc2a047": "U_1\\cap V_1.", "eec51f4509180460cd350b4db0ec109b": "\\boldsymbol\\tau = \\sum_{i=1}^N (m_i\\Delta r_i\\mathbf{e}_i)\\times (\\alpha(\\Delta r_i\\mathbf{t}_{i}) - \\omega^2(\\Delta r_i\\mathbf{e}_{i}) + \\mathbf{A}) = (\\sum_{i=1}^N m_i\\Delta r_i^2)\\alpha\\vec{k} + (\\sum_{i=1}^N m_i\\Delta r_i\\mathbf{e}_i)\\times\\mathbf{A},", "eec592665c10b3d9743c5a0f5470f2e5": "\n\\left( \\frac{\\mathrm{d}S_{\\tau}}{\\mathrm{d}\\tau} \\right)^{2} + 2m U_{\\tau}(\\tau) + 2m \\tau^{2} \\left(\\Gamma_{z} - E \\right) = \\Gamma_{\\tau}\n", "eec5f25522aea581d36763bb16f45390": "\\frac{1}{\\kappa} = \\frac{\\int_0^{\\infty} (\\kappa_{\\nu, {\\rm es}} + \\kappa_{\\nu, {\\rm ff}})^{-1} u(\\nu, T) d\\nu }{\\int_0^{\\infty} u(\\nu,T) d\\nu}", "eec602f1e15218eb724a3c2324a10457": " 1 / 2^k ", "eec64051fab395a87d8520a293f3053d": "\\left\\{\\frac{1}{n} \\mid n \\in \\mathbb{N} \\right\\}", "eec65a21c3f38ebfc1d3456dea039e98": "\\frac{\\partial \\; (\\textbf{A}\\textbf{x} + \\textbf{b})^{\\rm T} \\textbf{C} (\\textbf{D}\\textbf{x} + \\textbf{e}) }{\\partial \\; \\textbf{x}} = ", "eec69abb7d0e7470892d21a6c31face0": "\\frac{\\mathrm{d}m_x}{\\mathrm{d}t}=-\\gamma \\mu_0 m_y H_z \\qquad \\frac{\\mathrm{d}m_y}{\\mathrm{d}t}=\\gamma \\mu_0 m_x H_z \\qquad \\frac{\\mathrm{d}m_z}{\\mathrm{d}t}=0", "eec6a2e578a160ee92b22b4a5256eac0": " f\\ll g \\iff f \\in O(g), ", "eec6c4bdbd339edf8cbea68becb85244": "Height", "eec6c6a8391c5f6e45d4082c7463beb2": "\\operatorname{E}[|S_N|]=\\sum_{i=1}^\\infty\\operatorname{E}[|S_i|\\,1_{\\{N=i\\}}].", "eec6fdfb7f61320580c8db18e79c64b8": "cn^{\\frac 23}", "eec75efe3c68fff524d53b968b9a4868": "H_\\alpha(X)=\\log n", "eec7e8ce3a7f861fa464b1e0dbc59a3d": "RM\\tilde{R}", "eec7fa85d3f2db01a3b541fddd2b0628": "\\scriptstyle u\\in L^1_\\text{loc}(\\Omega)", "eec83782cf45e5bf8134899c56fd0db8": "\\sum_{i=0}^D \\mathbf{M}_{t,i}", "eec83eb1bfbfaa079c419e6eacc8e3a3": " \\ c_{10} = V[x_0,y_1, z_0] (1 - x_d) + V[x_1, y_1, z_0] x_d ", "eec846b3c2d6647a105342c1ce78bd4d": "a = x^\\frac{m}{n}", "eec85b253bd28f0a20c00cc8c3161f32": "f\\left(x\\right)=a_0+a_1x+a_2x^2+a_3x^3+\\cdots+a_{k-1}x^{k-1}\\,\\!", "eec85ba300c0731c8996da982f1f3517": " RanVR = 1 - \\frac{ f_m - f_l }{ f_m } = \\frac{ f_l }{ f_m }", "eec88c9041728b10026a2fb04435831c": "\\frac{\\partial \\Pi_i }{\\partial q_i} = \\frac{\\partial P(q_1+q_2) }{\\partial q_i} \\cdot q_i + P(q_1+q_2) - \\frac{\\partial C_i (q_i)}{\\partial q_i}=0", "eec8c702e7b7e0e67780a474b5dc91e2": "\\displaystyle{\\mu(z)={\\partial_{\\overline{z}}F_f\\over \\partial_z F_f},}", "eec8d7466ce06fd27a37e896dc0efc41": "\\sigma_\\alpha^2\\,\\!", "eec926317ddb8bc55f8318635229b35c": "f(x)=f(y)", "eec99ba336e68d1bebe930eba6bea030": "\\operatorname{trace}(XYZ) = \\operatorname{trace}(ZXY)", "eec9a99328aee2dd19ed7b1ad49de532": "\\rho(x) = \\rho(s) + \\rho(n)\\,", "eeca0cfe3bbdd6cfa1bf85a6f0657bae": "\\frac1{a\\cdot b} = \\frac1{a(a+b)} + \\frac1{b(a+b)}", "eeca1264985899b549b1240c17d30190": "\\begin{pmatrix}1 & 1 \\\\ 1 & -\\end{pmatrix}", "eeca76d7053486136b4de35fecada6ef": "\\Phi_{X,Y}\\colon \\mathrm{Hom}_{\\mathcal D}(FX,Y) \\to \\mathrm{Hom}_{\\mathcal C}(X,GY)", "eeca8dec56b4420a938ab2a87f10b421": "V = \\sqrt{r_1^2 + r_2^2}", "eeca9c39764ac508e7fcc6106a1ed275": "a \\cdot \\frac{m}{s} := \\frac{a m}{s}", "eecb20121069f61a24b4bf3bfebef191": " B(S,T) = \\frac{1-\\exp(-\\alpha(T-S))}{\\alpha} \\,", "eecb23d52413090da1e38b7f72a243b2": "\\lim_{\\varepsilon\\to 0^+} \\int_{-\\infty}^\\infty\\frac{f(x)}{x\\pm i\\varepsilon}\\,dx = \\mp i\\pi f(0) + \\lim_{\\varepsilon\\to 0^+} \\int_{|x|>\\varepsilon}\\frac{f(x)}{x}\\,dx.", "eecb30b9dafa09e4c2da288addbedf75": " J = \\int_a^b F(x,f(x),f'(x))\\, dx\\ . \\,\\!", "eecb5069a75f56077bba1dcc202012ef": "\\begin{bmatrix} x & 0 \\\\ 0 & y \\\\ \\end{bmatrix}", "eecb595cc982bc32da26b7f20c0bc6dd": " y = a(x - h)^2 + k, \\, ", "eecc0de55f3d457a66f89d0662c900f0": "F(x,y^*)=0\\,\\!", "eecd274d18dcf3c207a5ea2adf17d981": "\\frac{\\phi(0)}{2}", "eecd3f7f0b93fdbfbe1c302db0131be8": "\\{j^{r}_{p}\\sigma:p \\in M, \\sigma \\in \\Gamma(\\pi)\\}", "eecd7be368b86d56b4c4e390e4e2a416": "C = 18(n + 1)!\\cdot n^{n+1}\\cdot (32d)^{n+2}\\log(2nd),", "eecec2360b5f741d0b36d8e4885e0fe7": "u(r, \\theta, t) = R(r)\\Theta(\\theta)T(t).\\,", "eeceed2f0c170e0018ea494241b279c2": "\\frac{dV_-}{dt}+\\frac{V_-}{RC}=0", "eecf2323b1f0d66a09f70a39fc93e7c4": "E \\neq A", "eecf4950bd065efb5b4b22899aee9567": "\\alpha\\ne\\beta", "eecf533e523a22f4b6b4df3bdd7016f7": "\\Lambda^k", "eecf8e163f2cb95dfc1e5714ace470e3": " D \\equiv O^T A O \n ", "eecfb0ab494930fc297eae9db62acf2b": "f(x+h) = f(x) + f'(x)h + \\frac{f''(x)}{2}h^2 + \\cdots", "eecfc933723775874196b1d5eb4654ce": "\\int_{\\mathbf{R}^n} \\delta(g(\\mathbf{x}))\\, f(g(\\mathbf{x}))\\, |\\det g'(\\mathbf{x})|\\, d\\mathbf{x} = \\int_{g(\\mathbf{R}^n)} \\delta(\\mathbf{u}) f(\\mathbf{u})\\,d\\mathbf{u}", "eecfcfa783411d451191bf02d62ebee6": "\nf = f_{\\overline{x_i}} \\oplus x_i \\frac{\\partial f}{\\partial x_i}.\n", "eecfd8767cb49ab717d20f6e91f7ad93": "\\text{NC}(S_2)", "eed07872a7601baeeb049e7b960d4f7b": "\\left\\{\\frac{\\partial}{\\partial z_1},\\dots,\\frac{\\partial}{\\partial z_n}\\right \\}", "eed0d6bebf06606fe6451a80672c9dec": "\\varphi:\\mathbf{C}^n\\longrightarrow \\mathbf{C}^n", "eed1314a23e0cb0cfda6f45362669477": "\\Delta_4^{\\prime}F(J) = \\bar \\nu [S(J)] - \\bar \\nu [O(J)] = 4B^{\\prime}(2J+1)", "eed13a24a436afa9ba5e2311ea677c3f": "\\dot m = \\rho v A \\cos\\theta ", "eed19fe8d90b1b74cf12e75f85706c48": "\\Phi=e^{\\beta(\\epsilon_i-\\mu)}", "eed2049ff072c1d8395be2982b5bef0d": "I^2L'v", "eed2185ad7c1febf8c87542c32724b9a": "\\upsilon_s\\,", "eed2ad5ba3ec798a6f8a1ff779325aa4": " \\hat{C} = (C, D) = C + \\epsilon D, ", "eed361cfda26a35db1681e60c951cda3": "\nB_0 = (\\mu_0/2)*M\n", "eed366d862aae930fec159a1f1b03ae3": " Q_2 = - \\frac{R}{p} \\left[ \\frac{\\partial u_g}{\\partial y} \\frac{\\partial T}{\\partial x} + \\frac{\\partial v_g}{\\partial y} \\frac{\\partial T}{\\partial y} \\right] ", "eed36dbb5113f5e8adff1d23839fae03": "(\\part^2+m^2)f(x)=0", "eed375560c36a242597d52506440bd4b": " E = x\\ x, F = f\\ (x\\ x) ", "eed3d7d02b5276be74c8a069856433ab": "n^v_{\\mathbf k}", "eed3db262faf9ac64477e03a8fdeba45": "Y_{\\ell}^{-m}", "eed3f5d4821af99ce1bdbab03091e604": "p_f(r)", "eed40bbf6028f6b9662129b7c5cd428d": "\\frac{n ( n - 1 )}{2}", "eed421b3ab278e035117aeca87616b63": "\\frac{ Spin^+(2,n-1) }{ Spin^+(1,n-1) }", "eed4308122c58c2c2863d0a1890aef9a": "\nf\\left( x \\right)=H_D \\left( {F\\cap \\left( {a,x} \\right)} \\right)=(x-a)^D.\n", "eed43c580075b033103d98097b1c90ab": "T \\;=\\; \\frac{1}{2} m|v|^2 \\;=\\; \\frac{1}{2} v\\cdot m v ", "eed476918ff48b68ce87e7ff077edd66": "so(1,3)_\\mathbb{C}", "eed4a9f30a1e95fbd76b1db65cb94be9": "\\varphi = 2\\cos(\\pi/5)=2\\cos 36^\\circ", "eed4ea9f784418e736cacf68ec8503f3": "\\frac{1}{\\Gamma(z)}= z e^{\\gamma} \\prod_{k=1}^{\\infty}\\left(1+\\frac{z}{k}\\right)e^{-z/k}", "eed53114915d7607975ee42e41101586": "\nA = \n \\begin{bmatrix}\n 1 & 1 & 2 \\\\\n 1 & 1 & 1 \\\\\n 2 & 2 & 2 \\\\\n \\end{bmatrix},\n", "eed58e111e394bfb480e3c274154b3b4": "s \\neq 0", "eed5c73f8668332b022f12c89d2ed9cd": " \\Pr \\Bigl( \\Bigl | \\frac{ \\sum_{ i = 1 }^n X_i } { \\sqrt n } \\Bigr| \\le 1 ) \\ge 0.5. ", "eed5ed98f7718f0117576aff52352dd5": "Z_{[\\alpha}W_{\\beta]} = \\frac {1}{2} \\left( \nZ_{\\alpha} W_{\\beta} - Z_{\\beta} W_{\\alpha} \\right).", "eed62c2b3d9487e38a9445c788919855": "a_n=b_n", "eed63a9e31b0de02a3cb95a4f0e5734a": "\\sqrt[4]{\\frac{2}{3-\\sqrt{2}}}", "eed6427e643a47895a6ecf970bd98659": "\\partial k /\\partial z=0", "eed68383342aa4e2467c0e04e2cd84d5": "\\Phi: \\mathbb{N} \\to \\mathbf{P}^{(1)}", "eed69d96dac0a30d1c5b2734397826b1": "\\Delta \\sigma_{ij} = \\hat \\sigma_{ij} \\Delta t", "eed6ff72fe46cc604cf556bc509041d7": "T^4/\\mathbb{Z}_2\\,", "eed6fffa729fb7c4c6f0082303d2d3c4": " \\begin{cases} A \\overrightarrow{a} = x \\overrightarrow{a} \\\\ A \\overrightarrow{b} = y \\overrightarrow{b} \\end{cases}", "eed7088706e97b903da420ae8e9df254": "S_1 = S_1'' \\,", "eed72b692ed4b40eaf09c39d02fc7b4b": "1=\\mathbf{e}_0", "eed7995ec60750a1629fa032df1b6737": "X_1= am/p_1\\,", "eed7a99b5528d38f3872b4379353cf5f": " \\Delta Y = \\Delta C + \\Delta I ", "eed7bec1e1721dde44804594f5daccbe": "1-R-\\epsilon", "eed7e22b9b3ac190ea7e877a80334c33": "m_n", "eed8173efed8b14194a30e37ba3b57b1": " \\mathrm{d}\\bold{J} = { 4 \\pi \\over c } {j^{\\alpha}}_{;\\alpha} \\sqrt{-g} \\, \\epsilon_{\\alpha\\beta\\gamma\\delta}\\mathrm{d}\\,x^{\\alpha}\\wedge \\mathrm{d}\\,x^{\\beta} \\wedge \\mathrm{d}\\,x^{\\gamma} \\wedge \\mathrm{d}\\,x^{\\delta} = 0.", "eed8289138473275c1c65c8d6694c33a": "\\displaystyle{T_\\varepsilon R^k - R^{(k)}_\\varepsilon}", "eed884c35b3e1708632f5ff77f224f2d": " \\operatorname{}^{n - 1}T_n = M_{n-1,n} ", "eed88fa39fa5e43d590a8ac16b9e904a": "\\mid a+jb\\mid =\\left | \\sqrt{\\mid a^2+b^2 \\mid}\\right |.", "eed90c44709220d22018a2659333a286": "F(x;k,\\lambda) = 1- e^{-(x/\\lambda)^k}\\,", "eed92f5f60590f9c0b61e25864059e4a": "\\exists \\!\\,", "eed938467dc18712963fa26e43068535": "H<-\\frac{g}{l}", "eed9aec27f5b5445e868e5cb1ffe82b3": " \\operatorname{drop-params}[g, D, V, [F_6, S_6, A_6]::[F_5, S_5, A_5]::[F_4, S_4, A_4]::\\_]\\ \\operatorname{drop-params}[m, D, V, \\_]\\ \\operatorname{drop-params}[n, D, V, \\_] ", "eed9bdbb1b6bb6c0bfbd38775d9eb460": "\\eta_q(n) = \\sum_{k=1}^q \\zeta_q^{kn}", "eed9d3469269d50dcb6e48a734351815": "\\displaystyle r=\\frac{r_ar_b}{r_c}", "eed9eb4fbc0c018eef77fec4a3a6fe8d": "f_s(k)=k^{-s}/\\zeta(s)\\,", "eeda62084df29d99fbe2b97409ded80e": "\\eta(x) = \\begin{cases} e^{-\\frac{1}{1-|x|^2}}& \\text{ if } |x| < 1\\\\\n 0& \\text{ if } |x|\\geq 1.\n \\end{cases}", "eeda88e358eef9baec080c921455efaf": "(-1)^{d-2}\\psi^{-1,(d-2)}(t;\\theta)", "eedaa32cb4438549a0c58064003b16cb": "\\mathit{momentum} = \\mathit{close}_\\mathit{today} - \\mathit{close}_{N\\,\\mathit{days\\,ago}}", "eedaa56d7fdc86bae9359526dc74fab9": "f\\left(x,y\\right)=\\left(x^2-1\\right)e^y.", "eedad918d93a13dcad32cbd0e82f0eab": "\\overline{\\partial}", "eedaf57beba3465bea8f27796957955f": "\nC=\\max_{a\\geq3}\\frac{1}{16}\\frac{a-2}{a\\log a}=0.023335\\dots\n", "eedb3ba0ed123d72a32f9c6051f33b33": "SO_{\\gamma_H}(1_{K_H}) = \\Delta(\\gamma_H,\\gamma_G)O^\\kappa_{\\gamma_G}(1_{K_G})", "eedb9ec89736a425bcc0a0d2375ee32b": "2.5 \\le \\varepsilon_{r} \\le 25", "eedbbb1aba5c8d7d3a18d5150364f978": "\\mathcal{E}(x)", "eedbce53c9ab9549fa770affb37d64c8": "Z_{eff}(Na^+) = 11 - 2 = 9+", "eedbdc80384b99525ea6fa56e2f623f2": "\\left\\lceil \\log_2 \\frac 1 \\frac 1 3 \\right\\rceil + 1", "eedcdd3dd08c213723a3624d1665b7ca": "\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right] =\n(-1)^{n-k} \\left|\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right]\\right|.", "eedd24262f9e48774b8e85a484ead8a6": " \\begin{pmatrix} ix & w \\\\ \\overline{w} & -ix \\end{pmatrix}", "eedd53af4dbcfe6b322914ff00a03e4a": "\\!\\exists x \\phi", "eeddaee493d1082cd51af1aeed63cbd7": "\\Delta(X,Y)=\\nabla_X\\tilde{Y}-\\nabla'_X\\tilde{Y}", "eeddce5af77c48503bd349024bd87ea1": "\\frac{(x'(t),\\ y'(t))}{|x'(t),\\ y'(t)|} = (\\cos \\varphi,\\ \\sin \\varphi).", "eedde47f74bd98b272b03117076334cb": "F^{ab}{}_{,a} \\, =k J^b", "eede3c650d55ee72f9e3ac1a1996845c": "V_{s0}", "eede71fe31e0bc22ae14f65b35d4d227": "\\hbox{Ad}(h)(R_h^*\\omega)=\\omega", "eedec51c3f7451970f5aa26e4fbaeff8": "x' = \\gamma(V) (x - Vt)\\,,\\quad p_x' = \\gamma(V) \\left(p_x - \\frac{VE}{c^2}\\right)", "eedee861b49c81fa00da3e1e790984ae": "\\left|B\\right\\rangle", "eedf6da54d108743b4d144021605275f": "\\begin{align}\np(\\mu,\\sigma^2; \\mu_0, n_0, \\nu_0,\\sigma_0^2) &= p(\\mu|\\sigma^2; \\mu_0, n_0)\\,p(\\sigma^2; \\nu_0,\\sigma_0^2) \\\\\n&\\propto (\\sigma^2)^{-(\\nu_0+3)/2} \\exp\\left[-\\frac{1}{2\\sigma^2}\\left(\\nu_0\\sigma_0^2 + n_0(\\mu-\\mu_0)^2\\right)\\right]\n\\end{align}", "eedf97544a399aacaff8b2daf6da8f3b": " U^* = U - QV = \\begin{matrix} \\frac{1}{2} \\end{matrix}QV - QV = -\\begin{matrix} \\frac{1}{2} \\end{matrix} QV= - \\tfrac{1}{2} V^2 {C(\\mathbf{x})} \\,.", "eedfc78ccbca9653091f956594c1d5f3": "\\scriptstyle 2m", "eedfcda18d24705c232de3be022c146c": "p:\\widetilde{B}\\times F\\rightarrow M", "eee01fdcd16f48b6cba8d86295b6ac41": "\n{\\dot \\theta_2} = \\frac{6}{m\\ell^2} \\frac{ 8 p_{\\theta_2} - 3 \\cos(\\theta_1-\\theta_2) p_{\\theta_1}}{16 - 9 \\cos^2(\\theta_1-\\theta_2)}.\n", "eee082f8ceadec144d40165eaf7b2c24": "\\sum_{k=0}^n \\frac{P_k^{(\\alpha,\\beta)}(x)}{P_k^{(\\beta,\\alpha)}(1)} \\ge 0", "eee0e54c65da0153871b64d97da7f1a6": "y'=y, y(0)=1.", "eee0f387dd62c5295c377631d3c12452": "\\beta=\\tfrac{1-F}{F}(1-p)", "eee11bfc9d5cd11cd7f4fabc42bc26e0": " f(x_1,\\ x_2,\\ x_3,\\ \\dots,\\ x_N,\\ t)=0, \\, ", "eee1476c470f332f8e40bd5b34475fa7": "n \\log n", "eee155660451ffdcf41fde8b7a5286d4": " \\displaystyle \\mathop{opt}\\ ", "eee1c8c53e6491eb52eb7f931c924310": "22^2 + 23^2 + \\cdots + 29^2 = 20^2 + 21^2 + \\cdots + 28^2", "eee22160db2b65d98525509160ae757c": "\\bar{X}_A", "eee23adcadf828cf70854b4f066e4630": "a \\triangleleft T", "eee29c015f4d79c0d2ac30a2549829ac": " A_1\\times A_2\\times A_3 \\to \\mathbf R", "eee31254b455e0199ab8020f167998bc": "log(N)", "eee3177d4ed48b78efd7e3b77a9800ef": "np(1-p)", "eee3845ac3b257e3014c3e1229d48491": " \\mathrm{III}_{1/2B} f \\quad\\stackrel{\\mathcal{F}}{\\longleftrightarrow}\\quad 2B\\, \\mathrm{III}_{2B} * F", "eee38cfdbb3114aa2234769606cea376": "f_n(x)=x^n", "eee39f9d28c83a5aae48ffea1f4150f9": " RC_{t}^{real}=\\ln\\left (\\frac{P_{t}^{real}}{P_{t-1}}\\right ).", "eee3dbeadb87e44f50a6c64676594d0d": "a_k=\\frac{(2n-k)!}{2^{n-k}k!(n-k)!} \\quad k=0,1,\\ldots,n.", "eee3e09db5c89cac29afeadc5f43af06": "\\mathfrak{B}\\left(\\mathcal{O}_{[g]}\\oplus\\mathcal{O}_{[h]}\\right)", "eee42a4b8df8e3646439dc77243037b3": "\n\\frac{1}{m} z^m \\; + \\;\n\\frac{1}{m} z^{m+1} \\; + \\;\n\\frac{1}{m} z^{m+2} \\; + \\; \\cdots\n", "eee42b052deb68530c3e0b5b57b8b65a": "\\begin{align}\n r(t) &= \\frac{1}{2} I(t) \\left[1 + \\cos (4 \\pi f_0 t)\\right] - \\frac{1}{2} Q(t) \\sin (4 \\pi f_0 t) \\\\\n &= \\frac{1}{2} I(t) + \\frac{1}{2} [I(t) \\cos (4 \\pi f_0 t) - Q(t) \\sin (4 \\pi f_0 t)]\n\\end{align}", "eee49e39e1e84f6196d1b3261b34a329": "a_0, a_1, a_2, a_3, \\alpha", "eee4fa7b1fff740302c67de084aa12b6": " \\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/8\\mathbb{Z}", "eee5061f2537e171b663999c11d718fd": "q(a)", "eee513c93069e133ec957c497d39c7a2": " H = \\bigcup_{n \\in \\mathbb{N}} V^n ", "eee5ac4640a1f26fcbff1ee41441b090": "e^{-\\theta} = \\frac{1 - t}{1 + t}.", "eee61d964e3f0630b5065b36f2eaba42": "D_\\mu", "eee634b339a2a927170acff23a21009b": "\n(2 1)))))))) (\\lambda \\lambda 1))) (1 1)) (\\lambda (\\lambda 1 1) (\\lambda \\lambda 1 ((\\lambda 1 (1 (1 (\\lambda \\lambda 1 (\\lambda \\lambda 2) 2)))) (\\lambda \\lambda 2 (2 1)) (\\lambda \\lambda 1)) (2 2)) (1 (\\lambda \\lambda \\lambda \\lambda \\lambda \\lambda 1)) 1)\n", "eee666448bf7aff2f69fb9d77bd563e5": "\\begin{matrix}\\frac{10}{3}\\end{matrix} n^3 + O(n^2)", "eee6785ce960ddba710c64feaffd2d80": "B_{\\varepsilon} (p)", "eee6a5e0d46c43da3bbd0f6b3067d30d": "g^{\\alpha \\delta} \\,", "eee6c27807f1ffdebf320abac4a38002": " F = 2(t_p+w_p)\\gamma_{LV}\\cos(\\theta) \\,", "eee6c7fc3e8f345f3a0a1963feae8a04": "f(k):=\\sum_{K'}\\tilde{u}_k(K')", "eee6e3ac986ce00be49bde17c4188c69": "\\bold{j}_\\perp \\times B_z\\bold{\\hat{z}} = -(1/\\mu_0)B_z\\nabla B_z +(B_z/\\mu_0)(\\bold{\\hat z}\\cdot\\nabla B_z)\\bold{\\hat z}=-(1/\\mu_0) B_z\\nabla B_z.", "eee7725e0e842b79eeeafe91d92393d8": "\\hat{\\rho}(\\hat{p}):= \\max \\ \\{\\rho\\ge 0: p\\in P(s), \\forall p\\in B(\\rho,\\hat{p})\\}", "eee78aae1fc06b18f2b0cca87dc1b053": "{\\rm Riesz}(x) = x \\exp(-x) - \\sum_{k=1}^\\infty \\left(\\zeta(2k) -1\\right) \\left(\\frac{(-1)^{k+1}}\n{(k-1)! \\zeta(2k)}\\right)x^k", "eee791bbeab3ee5f372f4019e6b1a1f9": "a \\not\\in \\{0,1\\}", "eee7ab3643ded658e343101c2dd5329f": "s'=s+rb", "eee7abea4571731ada151d6cbcd9e142": "1+\\sqrt{-5}", "eee80a5814f63048f9429f685dca8af5": " |\\frac{ a_{ n + 1 } }{ a_n }| \\le 1 - \\frac{ b }{ n } ", "eee85844d24c053bc394117032f99e82": "\\partial_t g + (\\vec{e}\\cdot \\nabla) g + G\\partial_v f=\\Omega(g)", "eee86bf1b9fb8a83999d215f1da6dc1d": "C_1(t,\\omega) = \\iint g_{12}(t^'-t,\\omega^'-\\omega)C_2(t,\\omega^')\\,dt^'\\,d\\omega^'", "eee87ef9af62b2834b8923fe76121ac8": "\\begin{align}\n(dx)^2 - (dX)^2 &= \\frac{\\partial x_j}{\\partial X_K}\\frac{\\partial x_j}{\\partial X_L}\\,dX_K\\,dX_L-dX_M\\,dX_M \\\\\n&= \\left( \\frac{\\partial x_j}{\\partial X_K}\\frac{\\partial x_j}{\\partial X_L}-\\delta_{KL}\\right)\\,dX_K\\,dX_L \\\\\n&=2E_{KL}\\,dX_K\\,dX_L\n\\end{align}\\,\\!", "eee8cbc77c2f6782d3d3b97c3cc81383": "S_{10:1}=800", "eee906f0e684abaa5b9ec780e7882690": "\\displaystyle\\gamma(s) = \\pi^{1/2-s}\\Gamma(s/2)/\\Gamma((1-s)/2)", "eee9072fddb5bac600f1df2db2643fc2": "r_{u,i} = aggr_{u^\\prime \\in U} r_{u^\\prime, i}", "eeea2cc48c38dfa0d0076efccc5a502d": "\n\\begin{align}\nU(x,z)\n&= a \\int_ {-W/2}^{W/2} e^{ {-2 \\pi ixx'}/(\\lambda z)} dx'\\\\\n&= -\\frac{a \\lambda z}{2 \\pi i x} |e^{ {-2 \\pi ixx'}/(\\lambda z)} |_{-W/2}^{W/2}\n\\end{align}\n", "eeea3c93d4121b0a528cd06ec269562e": " d(n) = b^n n! [z^n] g\\left(z, -1, \\frac{a}{b}\\right)=\nb^n n! [z^n] \\exp \\left(\\frac{a-b}{b} z\\right) (1+z)\n", "eeea6e1e3c0cc8ac0ba95933583238ad": "c_\\text{batch}", "eeea96854c825e73a498ff639671f2a1": "\\sigma_{zz} - \\sigma_{yz} - \\sigma_{xz}", "eeeac1be1888fde2a5b5173bb5a033d9": " k = 0 ", "eeead68aa2ae37932e7b744e4ec80794": "\\text{AER}_\\text{Equivalent} = \\left (1-\\frac{GPM_{CD} }{GPM_{CS} }\\right ) d^{CD}", "eeeb635a1910c821f9b9e41cb4fccd51": "\\omega \\uparrow \\uparrow \\omega", "eeeb844671b43c6b30e5b42abc5fd8fc": "\\begin{matrix} \\frac{2}{1} \\end{matrix}", "eeebed9434a1d348b15079ac53df5d66": "\\int_{\\mathbb{T}^3} \\vert \\mathbf{v}(x,t)\\vert^2 dx _{Diff}", "eef1c3084836d0e83cfc190bf6e6b378": "c_0(V) = 1", "eef1f1cab382c41b8980bdbd225954cf": "A \\otimes H", "eef218139867ec7537d828fe463de7f8": "2+5 = 7.", "eef238632103d30cb7b4db4f82985a3e": "=(A \\cup A) \\cap (A \\cup A^C)\\,\\!", "eef27b105c48db47901b5154cf361a88": "0=e^N f^N v=N!h(h-1)...(h-N+1) v", "eef2b42b7664ec8e11ca54efa3270f85": "r={(C_f-C_p)\\over C_f} \\cdot 100", "eef2d7fc059c24eaf3ef97869a735d94": " \\operatorname{sink-tran}[\\operatorname{de-let}[\\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p]] ", "eef32a369d82a9aca5cff69a75069a98": " P(t) = \\frac{\\sum_i w_i P_i b_i (t)}{ \\sum_i w_i b_i (t) } ", "eef32c7d82e9069c30db8d263e9bfa40": "dB_t", "eef333cab5ba693a0404ae3449ec2eb1": " {\\tau}_o ", "eef355f9355dc1fb945382dade7ce75b": " \\mathbf{\\hat{e}}_{\\angle} \\,\\!", "eef3add63bac00605d54ad8faa27994f": "\\textbf{x}_{i} = \\textbf{x}_{1} + \\sigma(\\textbf{x}_{i} - \\textbf{x}_{1}) \\text{ for all i } \\in\\{2,\\dots,n+1\\}", "eef3efcccaf1f76d8904c121af8a38f5": "J=\\begin{pmatrix} 0 & 1 \\\\ -1 & 0\\end{pmatrix}", "eef3f252fa37eef88ac09f9b0204c888": "N^{2}p^{2}", "eef461ce51abb1c9f8fff9bb14023d48": "\\mathfrak{o}(2l, F) = \\{ x \\in \\mathfrak{gl}(2l,F) | s x = - x^t s, s = \\begin{pmatrix} 0 & I_l \\\\ I_l & 0 \\end{pmatrix}\\}", "eef47839d05c4c50da4df51433b8fd9c": " \\lim_{k \\rightarrow \\infty} 1/k\\sum_{i=0,...,k} A^i/r^i = ( v w^t),", "eef484f6f76737606aa7de9fc9d6ab8a": "-\\overline{\\upsilon_i^\\prime \\upsilon_j^\\prime} = 2\\nu_tS_{ij}-\\frac{2}{3}K\\delta_{ij}", "eef4ee69a57dcd47bb300799cf2d5ced": " \\int_{\\mathbb{R}^n} |f(x+y)-f(x)|^p dx < \\epsilon^p\\,", "eef4f0ce762a1f0473f6bccce3e20407": "b_n=n\\#+1\\, .", "eef4f1ce27b483c7888ebaa97ff674c9": " \\gamma_{SV} - \\gamma_{SL}= \\gamma_{LV} cos\\theta ", "eef4f6c1a08cf1e4d73d7cc7432d3806": "P_c=\\frac{2\\gamma}{r}\\cos\\phi", "eef50a8c95243e932372f76b001d802f": "\\,V", "eef5bd7f3f77fc9f2859e4c02eaafdb8": "g^{(2)} = \\frac{\\left \\langle \\hat{a}^\\dagger(t) \\hat{a}^\\dagger(t+\\tau) \\hat{a}(t+\\tau) \\hat{a}(t) \\right \\rangle}{\\left \\langle \\hat{a}^\\dagger(t) \\hat{a}(t) \\right \\rangle^2}", "eef5ed2d4d6425c77a4edb5da832418e": "x_1 \\dotsb x_n", "eef5ee840e8bf0588d3e4c912d9620e3": "k = \\left \\lceil \\frac{\\max x - \\min x}{h} \\right \\rceil.", "eef6539e3182a6552a784da3a13f237d": "\\mathbf{n}_{u}=\\cos\\varphi\\cdot{\\mathbf{x}}_{u}+\\sin\\varphi\\cdot{\\mathbf{y}}_{u}", "eef65b2e67a687f7a61d52866ad81733": "q^{2}", "eef677fee43924a09d5fd6d6853810d1": "\\mathcal{S}_\\mathrm{EH}", "eef6a3839cf598b90b516f7155e32312": " f(x_i)=a\\theta_i^a x_i^{-(a+1)}, \\qquad x_i \\geq \\theta_i>0, i=1,2.", "eef6cc6fa54df0ae5492bebb50536986": "{d \\over dt} X = A^* X + B^* U", "eef6d4a0b2cf67b5abad948a05a5b44c": "f=f_1\\ldots f_k, ", "eef6dacc52cecf23a6250c1e6b9217d4": " \\hat{n} ", "eef78449b9b0025c603995cb9a2ad863": "\\chi_W(\\mathbf{z},\\mathbf{z}^*)= \\operatorname{tr}(\\rho e^{i\\mathbf{z}\\cdot\\widehat{\\mathbf{a}}+i\\mathbf{z}^*\\cdot\\widehat{\\mathbf{a}}^{\\dagger}})", "eef7b120d8adade8d5ffead30938766a": "\\mathrm{P} \\int_{L} f(z) \\ \\mathrm{d}z = \\int_L^* f(z)\\ \\mathrm{d}z = \\lim_{\\varepsilon \\to 0 } \\int_{L( \\varepsilon)} f(z)\\ \\mathrm{d}z, ", "eef7b6ce9e452d78ed49bbaed25cf4f7": "f_1 = \\frac{21}{20} = \\frac{3 \\times 7}{2^2 \\times 5^1}", "eef7fc4cae63fa356b4f3f6b512b10fa": "\\! v", "eef821a5a77318324a4c85251d0ab591": "\\phi \\ = \\ \\int_{4\\pi}d\\Omega\\varphi", "eef843cd3032cdb3f86256ca1c219e33": "\\rho_0 \\frac{\\mathrm{d} \\vec v_s}{\\mathrm{d}t} = - \\vec\\nabla (p + \\rho_0gz-p_{f}).", "eef8553b6855c4c79306feead7a372d7": " PCI_2 = \\frac{ \\sum_{ i = 1 }^K \\sum_{ j = 1 }^i k_i k_j d_{ ij } }{ \\delta }", "eef86b3b77e893e9b7ccd440e3fa4a58": "2^{19} \\approx 3^{12}", "eef8efb101cdcae2267678fd90555ad8": " \\Gamma_{rad}(\\omega)= \\frac{\\omega^3n|\\mu_{12}|^2} {3\\pi\\varepsilon_{0}\\hbar {c_0}^3} \n= \\frac{4 \\alpha \\omega^3n| \\langle 1|\\mathbf{r}|2\\rangle |^2} {3 {c_0}^2} ", "eef9095195474f88627fc21ff52d43a7": "\\frac{P(x)}{D(x)} = Q(x) + \\frac{R(x)}{D(x)} \\implies P(x) = D(x)Q(x) + R(x).", "eef9172beb738ed128da74d8f48ba211": "H^*=\\frac{G^*_d-G^*_m}{2G^*_d+3G^*_m}", "eef92d286a48aae06a41990a598c7412": " \\mathbf{E} = \\frac {3\\mathbf{p}\\cdot\\hat{\\mathbf{R}}}{4 \\pi \\varepsilon_0 R^3} \\hat{\\mathbf{R}}-\\frac {\\mathbf{p}}{4 \\pi \\varepsilon_0 R^3} \\ . ", "eef9821ccbe337b1c4a4b3fde720dfae": "\\text{MTTF} = A (J^{-n}) e^{\\frac{E_a}{k T}}", "eef9b5ddd227afc33972b73af7f0c182": "P_{pre} ", "eef9ba87bc00f1c8e0e594daa6097202": "\\|\\mu_n - \\mu\\|_{TV} < \\epsilon", "eef9e81584bc620275ae8ffeac4cfeb2": "\\tau_4", "eef9f00821d5d94b51f40c8b91bfa2f2": " \\int_G f_1\\overline{f_2} \\, dg = \\int \\tilde{f}_1(\\lambda) \\overline{\\tilde{f}_2(\\lambda)}\\, \\lambda^2 \\, d\\lambda.", "eefa1b16e489e391b1f951db413c30d6": "|\\mathcal{U}|-1=8", "eefab69dca21a479ace56ea31a23ece1": "\\mu(A) = \\sum_{i=1}^{n}{\\mu(A_i)} ", "eefb06405abb6bab033d4a1baeea7485": "\\psi_{jk}", "eefb146d210e0819db32d457c8ad2d1f": "\\{g: ||g - f|| < \\varepsilon \\}", "eefb45379252fac90cac39f3e9f22838": "\\delta(A,Z) = (-1)^Z a_P A^{-\\frac 1 2}", "eefb50781e1700827ac76154c64b7c0e": "p\\colon P \\to A", "eefc12660b2525b57907c598a88b6610": " F_1,\\ldots,F_{2N} ", "eefc144b051ff5a6bfeffeb9bafccbda": "\\frac{pr}{(1-p)^2}", "eefc2a84090d670622ed6ff86c58da18": "(X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}", "eefc3aea56568f212ee89d109053e37b": "\\det \\left( H - \\varepsilon S \\right) = 0, ", "eefc617c15475e1f4df883e983eb0968": "u=(p_1-p_4)^2=(p_2-p_3)^2 \\,", "eefc6f8e2c9dda9b3cae1ddb28f9c0aa": "x \\equiv \\sum_{i=0}^{n-1} c_{i,x}r^i \\pmod{r^n -1}", "eefc79ba7604342ff64a0988f79011db": "J := \\det\\boldsymbol{F} = 1", "eefc7e63481d3e0edeb57b3bc3f532f5": "|\\Psi\\rangle", "eefc9ddb6d07e22c3c32793ae0f61e9c": "\\tfrac{1.01}{1.03}", "eefce81bf7812ef83f44e9450808ee20": " Q^* = -(M\\mathbf{A})\\cdot \\frac{\\partial \\mathbf{V}}{\\partial \\dot{q}} - ([I_R]\\alpha+ \\omega\\times[I_R]\\omega)\\cdot \\frac{\\partial \\vec{\\omega}}{\\partial \\dot{q}}.", "eefcef0c1af33120722ce96722db95d3": "x^m \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p", "eefd3016b689c73b4df7a1bf36ff7d2f": "\\phi(r) = r^k,\\; k=1,3,5,\\dots", "eefd30aaac566abaf4d39827091dfcf3": "1 \\times s = s", "eefd47b1c2eda059a2830e3b13daaade": "|f|_p := \\left(\\int_\\Omega |f(x)|^p \\, dx \\right)^{1/p}.", "eefd5d1da2c8a92c0ce03a0b7993d1ef": "\\mu \\ ", "eefda0c2728bfa16fad9866eaaaa855a": "J = \n\\begin{bmatrix}\n\\lambda & 1 & \\; & \\; \\\\\n\\; & \\lambda & \\ddots & \\; \\\\\n\\; & \\; & \\ddots & 1 \\\\\n\\; & \\; & \\; & \\lambda \n\\end{bmatrix},", "eefdc9be44426c402f9299396698ee48": "U(|\\psi_1\\rangle\\otimes|\\mathrm{in}\\rangle)", "eefe386ce41eb2f985b82bfb6457b8ed": "p_2\\in K_2", "eefe40a689eecfceecc6ca1175d7fac6": " l^2(G) ", "eefe475aa19487bd38d534ba460cf0ea": " y=f(k)=k^a ", "eefea499aa5b7a0ab8e5b2c69ea22154": "F = \\frac{N_1 \\cdot N_2}{d^2}", "eefecd9cb0345697ddaac0281588a08d": "\\mathcal{D}", "eefed577e33acbb7ab9f4f44635298e5": "\\mathcal{U},\\mathcal{W}", "eefef6abcf0c613058bc7aa0a4a2eca1": "g(x_1,x_2,\\cdots,x_n) = f(p) + \\epsilon_1 x_1^2 + \\epsilon_2 x_2^2 + \\cdots + \\epsilon_n x_n^2", "eeff1f5b579ede15b1764fb248cd37b5": " C_{\\text{YX}} +1 ~~~~~~~~~~~~~\\text{second sequence pair}", "eeff7d22eacab26806d4071a5042167b": "\\partial_H", "eeff9550fd1551b1a24becfb10e8723e": "~ f=0 ~", "eeffd7cdcec3276f2107097c62e17979": " | x - 5 | < \\varepsilon / 3 ,", "eeffd9897afec498cd214b403911e91f": "r = v/i \\,", "ef0105e22ea35dd88f29fa25f1e7760a": "\\operatorname{P}(X^2 \\le y) = \\operatorname{P}(|X| \\le \\sqrt{y})\n = \\operatorname{P}(-\\sqrt{y} \\le X \\le \\sqrt{y}),", "ef0106459192bd1ca8142b403ad49839": "\\mbox{IF}\\cdot \\mbox{SC}=(1+f(t))\\cdot \\sin(\\omega_{I} t)\\cdot \\sin(\\omega_{s} t)", "ef0177d4002dd432187e09b8579e687b": "\\alpha^{\\frac{\\mathrm{N} \\pi - 1}{3}}", "ef01af5f1abc8e905bf95e33d6aeefe3": "\\frac{\\langle E(s) \\rangle}{A} = \n\\frac{\\hbar c^{1-s}}{4\\pi^2} \\sum_n \\int_0^\\infty 2\\pi qdq\n\\left \\vert q^2 + \\frac{\\pi^2 n^2}{a^2} \\right\\vert^{(1-s)/2}", "ef01e65cd2040a99e83110c470303399": "K =k_2 + \\cdots + k_p.", "ef026f7b7e6593b302f9173988d86453": "\\mathfrak{su}_{12}\\oplus\\mathfrak{sp}_1", "ef0298913102b81e422f24954bc76253": "\\;\\tfrac{1}{2}\\gamma PM^2", "ef02ae7a59ac681538c523391398a432": "m=\\frac{y_2-y_1}{x_2-x_1}.", "ef030af9ab97611c1b67b51a4352a6e4": "c \\equiv b^e \\pmod{m}", "ef030bd72f097f4f488e0d334c1a3deb": "\\mathbf{p}_{\\mathrm{2}} = (m + \\mathrm{d}m)(\\mathbf{v} + \\mathrm{d}\\mathbf{v}) + \\mathbf{u}(-\\mathrm{d}m) = m\\mathbf{v} + m\\mathrm{d}\\mathbf{v} + \\mathbf{v}\\mathrm{d}m + \\mathrm{d}m\\mathrm{d}\\mathbf{v} - \\mathbf{u}\\mathrm{d}m", "ef030c185bb05749bea1795573918de8": "\\mbox{LOP}=120+40+180", "ef03240f97197c73723760928bcb7595": "T_1^{(4)},T_2^{(4)},X_1^{(4)},X_2^{(4)},H^{(4)}", "ef034a48016fb19e0e547c034bdddeea": "\\mathbf{y}(t) = \\mathbf{C}(t) \\mathbf{x}(t) + \\mathbf{D}(t) \\mathbf{u}(t)", "ef0371a2b121d2a81cd043ba0eeaee8d": "(\\Gamma,L,M)", "ef03aac4e521cf2518c3e6a3b284a5f9": "\\hat{E}^-= -i\\left (\\frac{\\hbar\\omega}{2\\epsilon_0 V} \\right )^{1/2}\\hat{a}^\\dagger e^{-i(kz-wt)}", "ef042b69dd2512c8f3591afe5ca6e32d": "E_{B-V} = (B-V)_{\\textrm{observed}} - (B-V)_{\\textrm{intrinsic}}\\,", "ef0473168c421fdc8e06df8d68259f22": " q = C\\mathcal{E}\\left ( 1 - e^{-t/RC} \\right )\\,\\!", "ef049858a74897a4a8df4904404dd641": "\n{{\\Delta \\hat g} \\over {\\hat g}}\\,\\,\\, \\approx \\,\\,\\,{{\\Delta L} \\over L}\\,\\,\\,\\,\\, - \\,\\,\\,2\\,\\,{{\\Delta T} \\over T}\\,\\,\\,\\,\\, + \\,\\,\\,\\,\\,\\left( {{\\theta \\over 2}} \\right)^2 {{\\Delta \\theta } \\over \\theta }{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(12)}}", "ef04f88885b55432118be890e58abce8": "\\sigma \\to \\tau", "ef052e86eaada7d0954f60d78197b96f": "f(x)\\to 0", "ef05b843ac8431314d4baa94e9e90cd6": "(I_1)", "ef05bd3746e71e08280510abc139ebc2": "{\\mathcal P}(X)", "ef05e05c3b6bf06b288f824e0dc7451c": "\\begin{align}\\mathit{PSNR} &= 10 \\cdot \\log_{10} \\left( \\frac{\\mathit{MAX}_I^2}{\\mathit{MSE}} \\right)\\\\ \n&= 20 \\cdot \\log_{10} \\left( \\frac{\\mathit{MAX}_I}{\\sqrt{\\mathit{MSE}}} \\right)\\\\ \n&= 20 \\cdot \\log_{10} \\left( {\\mathit{MAX}_I} \\right) - 10 \\cdot \\log_{10} \\left( {{\\mathit{MSE}}} \\right)\\end{align}", "ef0620a19b6ab96991967b710c610765": "\\overline{H}\\left( x^{n}\\right) ", "ef06310192dca4950829b3c86ede5b31": "\\exists{}SO", "ef0671c7b4f4264a26e687579f86ec9b": "\\varphi:", "ef069ba1d017ed3db98e7f175f3c6051": "f(x) = \\log_b (x)", "ef07967d9a904a03e7fe69360e668e62": "D_1=\\left(C_0^*e^{-\\hat{\\alpha}'_1-\\beta^T{X}}\\right)^{1/d}", "ef07c065bf3b6a37452f81f6ce4e7ea2": "-g", "ef07ca706a6c5a1ea8d1f185b3b81450": "\\mathfrak{c}^2 = \\left(2^{\\aleph_0}\\right)^2 = 2^{2\\times{\\aleph_0}} = 2^{\\aleph_0} = \\mathfrak{c},", "ef07d8483fc0ca06c96abe1de73f32f7": "[S_+] = \\begin{bmatrix}\n\\langle+|S_+|+\\rangle & \\langle+|S_+|-\\rangle \\\\\n\\langle-|S_+|+\\rangle & \\langle-|S_+|-\\rangle \\end{bmatrix}\n=\n\\hbar \\cdot\n\\begin{bmatrix}\n0 & 1 \\\\\n0 & 0 \\end{bmatrix}\n", "ef07eb410dd4f428271faf855f808a9b": "\\alpha \\lesssim 3.", "ef07ec3534e30d1d3097842cc82183c5": "(s \\cdot f)(t) = f(t + s).", "ef0821525fb1d6c48b9dc2c3b748cd49": "f_\\text{muf} = \\frac{f_\\text{critical}}{ \\sin \\alpha} ", "ef083787b6be2ecaa0a301429132459d": "\\eta = (\\alpha,\\beta)", "ef08392ad8196675c5094818797c1155": "j\\in\\{0, 1, 2, ..., p-1\\}", "ef08fad31d473995bd10fe3a6985e7bc": "\nf_{WN}(\\theta;\\mu,\\sigma)=\\frac{1}{2\\pi}\\sum_{n=-\\infty}^\\infty e^{-\\sigma^2n^2/2+in(\\theta-\\mu)} =\\frac{1}{2\\pi}\\vartheta\\left(\\frac{\\theta-\\mu}{2\\pi},\\frac{i\\sigma^2}{2\\pi}\\right) ,\n", "ef09002dd302741b53a2a2126dbaaaa7": "\\frac{e}{a_0^2} ", "ef090d38a04d3a70d5032ef296775b7c": " \\frac{1}{p-B} ", "ef091e9842d939ec720356faf03a5cd6": "4m-2", "ef0938a7a3c80db0ff166da09e7b10ab": " X_0, X_1, \\dots, X_n\\,", "ef095add71c67966881996263eeb4596": "\\{0\\}\\subset V_1\\subset V_n", "ef098bfc19b9544a86424e00f5545f5b": "\\left(1, \\frac{1}{2}, \\dotsc, \\frac{1}{n}, \\frac{1}{n+1},\\dotsc\\right)", "ef098dd3d6ca3b670da2800e0caef4f7": "(X_0,\\ldots,X_n)", "ef09fa3af764d26a7fd2fc50b37bda3e": "w(z)=\\exp(z).", "ef0a065e612b59fe132b98fb69f63906": "\\tilde{s} ", "ef0a0bc0090b5a65445c142811157d3f": "(f,\\phi_n)", "ef0b06776f6bc7ed2aa2ffb7d476ef82": "E^a_i", "ef0b4908084b2bcaa0b70961bffdabe6": " \\mathrm{T'} = \\mathrm{L'} \\times \\mathrm{p} ", "ef0b620ca398719c0f842092aefafa5b": "\\forall A\\forall w_1 \\forall w_2\\ldots \\forall w_n \\bigl[ \\forall x ( x\\in A \\Rightarrow \\exists! y\\,\\phi ) \\Rightarrow \\exists B \\ \\forall x \\bigl(x\\in A \\Rightarrow \\exists y (y\\in B \\land \\phi)\\bigr)\\bigr].", "ef0b68556e2774bad2cac79e3d5a5559": " R_i \\equiv \\begin{cases}\n X_i/m_1 & \\text{for } X_i > 0 \\\\\n (m_1-1)/m_1 & \\text{for } X_i=0\n \\end{cases}\n", "ef0bc6fae3b79afb0ed7cddcc620b823": "Q=\\begin{pmatrix}\n-\\lambda & \\lambda \\\\\n\\mu & -(\\mu+\\lambda) & \\lambda \\\\\n&2\\mu & -(2\\mu+\\lambda) & \\lambda \\\\\n&&3\\mu & -(3\\mu+\\lambda) & \\lambda \\\\\n&&&&\\ddots\n\\end{pmatrix}.", "ef0bc98890b8be4524805d60f4fa07a2": "\n \\begin{align}\n g_{ij,k} & = (\\mathbf{b}_i\\cdot\\mathbf{b}_j)_{,k} = \\mathbf{b}_{i,k}\\cdot\\mathbf{b}_j + \\mathbf{b}_i\\cdot\\mathbf{b}_{j,k} \n = \\Gamma_{ikj} + \\Gamma_{jki}\\\\\n g_{ik,j} & = (\\mathbf{b}_i\\cdot\\mathbf{b}_k)_{,j} = \\mathbf{b}_{i,j}\\cdot\\mathbf{b}_k + \\mathbf{b}_i\\cdot\\mathbf{b}_{k,j} \n = \\Gamma_{ijk} + \\Gamma_{kji}\\\\\n g_{jk,i} & = (\\mathbf{b}_j\\cdot\\mathbf{b}_k)_{,i} = \\mathbf{b}_{j,i}\\cdot\\mathbf{b}_k + \\mathbf{b}_j\\cdot\\mathbf{b}_{k,i}\n = \\Gamma_{jik} + \\Gamma_{kij} \n \\end{align}\n", "ef0bee3b676ced01e1110cac3229da88": "\\Sigma^0_m", "ef0c0e66b3eec274803d748d07a3289f": "L_2(7) \\cong L_3(2),", "ef0c203ae408a4818b1128f0af0afec5": "f(x,y) = \\left( x + 2y -7\\right)^{2} + \\left(2x +y - 5\\right)^{2}.\\quad", "ef0c259ffd8d06f449658ef02c6757fe": "\\displaystyle s_n", "ef0c47f383b216b7c572608b2ccb2784": " g(s)=s \\int_{0}^{\\infty}dtK(st)f(t) ", "ef0c7606ebc6c10ee8a293feb011394c": "\\mathbf{R}(t+1) = d \\mathcal{M}\\mathbf{R}(t) + \\frac{1-d}{N} \\mathbf{1}", "ef0cd0e42cfa1cd7c9b48baf782eb226": "\n \\begin{align}\n w_1 & = \\frac{5}{3EI}\\,x^3 + C_1 + C_2\\,x \\\\\n w_2 & = \\frac{1}{24EI}\\,x^2\\,\\left[x^2 + 600 - 4 R_a(x-30)\\right] + C_3 + C_4\\,x \\\\\n w_3 & = \\frac{1}{100EI}\\left[\\frac{x^3}{3}(-625 + 30 R_a - 2 M_c) - 50 x^2(-675 + 30 R_a - M_c)\\right] + C_5 + C_6\\,x \\\\\n w_4 & = \\frac{1}{100EI}\\left[\\frac{x^3}{3}(-625 + 30 R_a - 2 M_c) - 50 x^2(-625 + 30 R_a - M_c)\\right] + C_7 + C_8\\,x \n \\end{align}\n ", "ef0cf26f88fdea389ea53c9654b84236": "\\scriptstyle t.", "ef0d00f4675b61b1290d7ea090ad8c04": "f(k,\\theta)= (1/k)\\sum_{l=0}^\\infty\\ (2l+1)[\\exp(2i\\delta_{l}(k))-1]P_{l}(\\cos\\theta).", "ef0d0e31a465ad97913ad62eabb91880": "[1,\\lambda_2,\\lambda_3]'", "ef0d8a94e4606c2a8a02cc3d8f91ef67": "\\zeta = \\frac{\\psi - i}{\\psi + i}", "ef0d927c36a1159de4b4faa66bd6653b": "\\tau_{M}", "ef0de62d6bea9a9490cce3e835e31f3d": " x \\in{ \\sum_{1\\leq{d}\\leq{D}}{\\operatorname{Conv}{(Q_d)}} + \\sum_{D+1\\leq{n}\\leq{N}}{Q_n} }", "ef0e1836a7293b95d2a904bd54328859": "\\Lambda_f \\neq 0\\,", "ef0e1bbe04c768cfa44ecf3f3eb3742f": "\nX(t)=\\sum\\limits_{i=1}^{N(t)}Y_i, \n", "ef0e42838d4b30a8d8e64852efb8ec7f": "P(A|B)=\\frac{P(A\\cap B)}{P(B)}.", "ef0e9db634ab386d30dd0bce6e9c2627": " \\Rightarrow(y_2 + \\frac{q^2}{2gy_2^2} = 8.04)", "ef0ecb35f0ea48f5f0978632b10c4dc5": " A \\subseteq \\operatorname{cl}(A) ", "ef0f2634a8ae3c7181092dc8eb77bfaf": "r^{-\\ell_1}+r^{-\\ell_2}", "ef0f2fab2789e6afdbaeadb87952990f": "f^{n_k}", "ef0f6063dbaa5a4c10eb7c8ec24cceff": "\n W = -\\theta \\, dS = \\tfrac{1}{2} n k_B \\theta (I_1-3)\n ", "ef0f779ebf18d5cc83ededa522cdb992": "\\left(\\frac{f}{\\rho_f} + \\frac{1-f}{\\rho_m}\\right)^{-1} \\leq \\rho_c \\leq f\\rho_f + \\left(1-f\\right)\\rho_m ", "ef0f846bc394c17b7700d68c6b5706b7": "\\begin{align}\nV_{2k}(R) &= R\\pi \\frac{(2k - 1)!!}{2^k k!} V_{2k-1}(R) = R\\pi \\frac{(2k-1)(2k-3) \\cdots 5 \\cdot 3 \\cdot 1}{(2k)(2k - 2) \\cdots 6 \\cdot 4 \\cdot 2} V_{2k-1}(R), \\\\\nV_{2k+1}(R) &= 2R\\frac{2^k k!}{(2k+1)!!} V_{2k}(R) = 2R\\frac{(2k)(2k - 2) \\cdots 6 \\cdot 4 \\cdot 2}{(2k-1)(2k-3) \\cdots 5 \\cdot 3 \\cdot 1} V_{2k}(R).\n\\end{align}", "ef0f91a77dc1331f1353430fb0810671": "|z-a| > r.\\,", "ef0fa05272c4e26dc369399b921ebf57": "\\dot{\\textbf{x}}(t) = \\begin{bmatrix}\n 0& 1\\\\\n 0& 0\\\\\n \\end{bmatrix}\\textbf{x}(t) + \n \\begin{bmatrix} 0\\\\ 1\\end{bmatrix}\\textbf{u}(t)", "ef0fad5c5e721b8452d9aebd6b3373b3": "\\Delta m", "ef0fd4fbb8a35fdd8b0a05ea14d695cc": "f(x_0+\\epsilon) > f(x_0) + (K/2)\\epsilon > f(x_0),\\,", "ef0ff5d1ea9e336d10df317e0d16877a": "I(x)=\\Psi \\Psi^*=|\\Psi|^2=I_0\\left( \\frac{\\sin\\left(\\frac{Nkax}{2L}\\right)}{\\sin\\left(\\frac{kax}{2L}\\right)}\\right)^2 ", "ef10370aa2201141a92c24707edad4e5": " \n\\prod_{n=1}^{\\infty} \\left(\\frac{2n}{2n-1} \\cdot \\frac{2n}{2n+1}\\right) = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdot \\frac{8}{7} \\cdot \\frac{8}{9} \\cdots = \\frac{\\pi}{2}\n", "ef107557bb06afca66bf2a9927130a02": " \\int_G |f(g)|^2 \\, dg = \\int_X |\\chi(\\pi(f))|^2 \\, d\\mu(\\chi).", "ef10c55eb189f34f580105e560514bf7": " A = U ( 1 - \\frac{ S - 1 }{ K - 1 } ) ", "ef11124dba466eec9816c990ec851d29": "\\eta(S(t^-),z)", "ef11375ca53d41edf1e27ec07851549b": "\\delta(\\gamma(v))=\\gamma(\\delta(v))", "ef1152e64c08be01ec3704ea670ac95f": "H_i x(n)", "ef1197ff765b14167a29422f68bddb34": "a=-0.33258+0.10324i", "ef11c29b1f461eb2331a9d2619720d48": " u_{50}(\\mathbf{r}) = \\bar{u}_{el}(\\mathbf{r}) = \\left | S\\frac{1}{2},-\\frac{1}{2} \\right \\rangle = -|S\\downarrow\\rangle ", "ef11dd9822c1c6bcc84494864b93ab62": "\n \\hat{P} \\mathcal{A} = \\mathcal{A} \\hat{P} = (-1)^\\pi \\mathcal{A},\\qquad \\forall \\pi \\in S_N,\n", "ef12181f0a834d807bec6c837ed4ce2c": "\\varphi_{\\gamma+1}(\\beta)", "ef1225aec313869c9284f6deb1908f03": "\\tau_{0i}", "ef1246b49254722a141fad6e6401b783": "r, \\theta", "ef12894505e303688dc08af61b697f86": "f(c^-) \\neq f(c^+)", "ef12d04229e94a2aef49772e7415cf8b": "[J_i,J_j] = i\\epsilon_{ijk}J_k, \\quad [J_i,K_j] = i\\epsilon_{ijk}K_k, \\quad [K_i,K_j] = -i\\epsilon_{ijk}J_k.", "ef130de2b56fd6383970bbf53ff6c254": " ~ \\sum_j ~ J_j ~\\left[~ \\sum_{mnp} ~ \\frac {\\bold J_i(-\\alpha_m,-\\beta_n,-\\gamma_p) ~ \\bold G_{mnp} ~ \\bold J_j(\\alpha_m,\\beta_n, \\gamma_p)}{k^2-\\alpha_m^2-\\beta_n^2-\\gamma_p^2} \\right]~ = ~ \\bold 0 ~~~(3.5) ", "ef13212f790d4bd50f8ff4d5bf46b122": "\\! w=0", "ef133af797ebe34c8e5a7251e2093e91": "\\alpha+\\beta\\in\\Phi^+", "ef137b6339522b9d659786e0ccc2c1f1": "\\|f(x)\\| \\geq (f(x) \\cdot x) / \\| x \\|", "ef137bc9bd1920ae60bfb6ed1fb9d41d": "\\textstyle \\bar{\\sigma}_k", "ef13a49559946c5f983b1843d98072b8": " R=\\frac{( V_{r3}^2 - V_{r2}^2)}{2U(V_w3 +V_w2)}", "ef141ede64c6a2d54d5ba374f8bbb157": "V_1 \\sim {\\chi'}_{k_1}^2(\\lambda)", "ef1483ce9458bb1d2b2e5bd144efea83": "\\Delta f=6\\frac{\\Omega A}{\\lambda L}a\\sin 2\\theta ", "ef14c521228f135357937553826dbd80": "S[g]=k \\int R \\sqrt{-g} \\, d^4x ", "ef14d572e6f6916992210448fea642d3": "c_t=\\frac{c}{t^\\gamma}", "ef150e6931b9e37148d7925a78dabad9": "\\exp{\\left( \\bar{X} + t_{n-1,0.95} S / \\sqrt{n} \\right)}", "ef1538a42afa3a99b066d3e6d7bb9c89": "\\displaystyle \\left( \\mp ix \\right)^{-\\alpha}", "ef153a3dab02e2f3aa9fba00830e2c74": "u\\in H^k(\\Omega)", "ef15833390b97b97110ea8c90d00284f": "\\begin{cases}4x + 2y&= 14 \\\\\n2x - y&= 1.\\end{cases} \\,", "ef158a53f7bb3f17e9c0c4ef2287ac01": "F(x) = \\frac{x^3}{3} ", "ef158d8cb6b01c4ba6ba5840b0ec3de6": "S = - k_B \\,\\,{\\rm Tr}(\\rho \\ln \\rho) \\,", "ef15908aeb34e09837c7cc1532b5977d": "\\pi \\colon E \\to B\\,", "ef15f6585df5b03c40187dffffc6bb08": "\\text{MTF} = \\mathcal{F} \\left[ \\text{LSF}\\right] \\qquad \\qquad \\text{MTF}= \\int f(x) e^{-i 2 \\pi\\, x s}\\, dx", "ef15f770621c120ccad4620e5f8acfa6": "j_*: \\Omega^\\bullet_{\\mathrm c}(U) \\to \\Omega^\\bullet_{\\mathrm c}(X)", "ef165a5284ebad7f1ae2385c28389c61": "\\{\\Phi_{00}, \\Phi_{11}, \\Phi_{22}\\}", "ef16aea412875578168a1fa9567bb59a": "(0,1)\\times(0,1),", "ef16be8cb3d15c81e1984dafcdf52bf0": "f(a) = f(x)", "ef16c1d8a579809659d5ffb9c41b00e4": "I(\\theta) = I_0 \\left ( \\frac{2 J_1(ka \\sin \\theta)}{ka \\sin \\theta} \\right )^2", "ef170f5c6601b263f90ad651e059e978": "\\begin{align}G_{\\frac{\\lambda}{2}}\n&=\\frac{60^2}{30R_{\\frac{\\lambda}{2}}}=\\frac{3600}{30R_{\\frac{\\lambda}{2}}} = \\frac{120}{R_{\\frac{\\lambda}{2}}} = \\frac{4}{\\operatorname{Cin}(2\\pi)} \\approx 1.64 \\approx 2.15 \\,\\mathrm{dBi}. \\end{align}\\,\\!", "ef17111b6387c67ee48597bd5de2a780": "|A _{S(v) \\ S(w)}|", "ef1736b7c9ba22574e6bb257eb2353a7": " \\varepsilon", "ef17553cdf9d6e515277f4528f6037a5": "\\theta_{IV}", "ef175f4b721d5e342d691b8bf3e9874d": "E^{(t-1)}", "ef17749605104ec36af1c3c015ab5e63": "b(x+e)-(x+e)^2=bx-x^2+be-2ex-e^2", "ef179f2d7862eaf93d0422e0df3379c9": "n=k_1+k_2+k_3+ \\cdots +k_p \\,\\!", "ef17d048a7357e165457b6548cac40cf": "\\hat{g}_N: X \\rightarrow \\mathbb{R}", "ef17fb7c418ab3f8ab2a06d15c6d86a2": "f = {1 \\over 2\\pi R_\\text{i}C_\\text{M}} = {1 \\over 2\\pi R_\\text{i}C(1 + A)} \\,", "ef181652ba6ab5a61025e4f88189b99a": "\\rho = \\Omega^{\\psi(\\Omega^\\Omega)}", "ef1856fcd96a60a744ea2072200c481d": "\\frac{e^{-\\frac{x-\\mu}{s}}} {s\\left(1+e^{-\\frac{x-\\mu}{s}}\\right)^2}\\!", "ef187c622f3830ab3521a7e79649bb4e": "ES_{\\alpha} = -\\frac{1}{\\alpha}\\left(E[X \\ 1_{\\{X \\leq x_{\\alpha}\\}}] + x_{\\alpha}(\\alpha - P[X \\leq x_{\\alpha}])\\right)", "ef18944fd542030b3c0c625ffce5a94e": "\\mathbf{1}_r (x) := \\begin{cases} 0, & \\mbox{if } x \\ne r \\\\ 1, & \\mbox{if } x = r. \\end{cases} ", "ef18a4a1e09ba47b6771d6db177a5f24": " bp = close - \\min (low, prev\\,close) ", "ef18f84f646b52c1c1732912b28ebde1": "\\log\\frac{1}{|x|}", "ef194ebcb94b0ed254702ada23c36e3f": "\\displaystyle{(Cf)_{\\overline{z}}=f.}", "ef1a0be6ac41455b0d55eeb7eb92d279": "\\text{right} = 2i + 2", "ef1a54f89fc206eb7725695604ed8a40": "\\textstyle \\psi\\geq0", "ef1a695d17bb9a807963424fa7679c0b": "y \\mapsto g(x,y)", "ef1a86b32d9dcf4753682b30eb13cda6": "O\\left[m + \\sum_{i=1}^n q_i\\log\\frac{m}{q_i}\\right]", "ef1a9782da2077b92e2f0268c26a01a6": " \\operatorname{let} x\\ f\\ y = f\\ (y\\ y) \\and q\\ x\\ f = f\\ ((x\\ f)\\ (x\\ f)) \\operatorname{in} q\\ x ", "ef1b1f4858a8810cc975c1f097200574": "\\varepsilon>0.", "ef1b4640ccc210b9139302c3461e1805": "M(s)\\equiv\\frac{2}{\\sqrt{3}}\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{n^s} \\sin\\frac{\\pi n}{3} \\qquad\\qquad\nM(1-s)=\\displaystyle\\frac{2}{\\sqrt{3}} \\, M(s)\\Gamma(s) 3^s (2\\pi)^{-s}\\sin\\frac{\\pi s}{2}, \n", "ef1b65c5b25c2ef71828118545a41ac7": " \\lambda' ", "ef1b91b2985161f889a19107c1475936": "g(x) \\in O(f(x))\\,.", "ef1bbbdf5199cad96986fdfbe354113e": "\\hat{f} =\\int \\frac{d^{2n}\\xi }{(2\\pi \\hbar )^{n}}f(\\xi )\\hat{B}(\\xi ).", "ef1cc09ebbefd57524fdc581da55aa1f": "2 = (1 + i)(1 - i)", "ef1ce8998b2f0a0e277691d630696539": "A_m(1,4) = 1,4,10,20,35,56,84,120,165,220,\\ldots", "ef1ce9df7746014b9914c98d07126bda": " {1 \\over f} {df \\over dx} = {1 \\over u} {du \\over dx} + {1 \\over v} {dv \\over dx}\\, ", "ef1d0de34670dcba9529956d733923e8": "\nN\\left( {t} \\right) \\sim exp{\\left( -\\frac {t \\cdot ln(2)} {T_{1/2}}\\right)}\n", "ef1dc2d5b7e720d0ead79b9519ad3b83": "\n\\pi = \\cfrac{4}{1 + \\cfrac{1^2}{2 + \\cfrac{3^2}{2 + \\cfrac{5^2}{2 + \\cfrac{7^2}{2 + \\ddots}}}}}.\\,\n", "ef1dd8d4b8c731493d8e866c250ea75e": "\\alpha=6.20", "ef1e56975e10bf38435b170dcca5f100": "Z^{(\\ell)}_{\\mathbf{x}}", "ef1e6bcda070f0f4c6f29294b06ac236": "x \\leqq y ", "ef1f6b9af94839d2ce40067b80783409": "P_t^{real} = P_t \\cdot \\frac{PL_{t-1}}{PL_t}.", "ef1f6f63d4aed7533220e8795738e42b": "u(c)=\\log(c)", "ef1f6fe31caae23961f5ab0c9d817e8c": "\\partial\\mathcal D\\,", "ef1ff27757b6f789bb6d4e2e6549905e": "\n\\left(\\frac{1}{n}, \\ldots, \\frac{1}{n}\\right)\\prec \\left(\\frac{1}{n-1}, \\ldots, \\frac{1}{n-1},0\\right)\n\\prec \\cdots \\prec\n\\left(\\frac{1}{2},\\frac{1}{2}, 0, \\ldots, 0\\right) \\prec \\left(1, 0, \\ldots, 0\\right).\n", "ef2016dc6a0d1343200824c8dfe2e49b": "\n\\delta \\varphi \\approx \\frac{6\\pi G(M+m)}{c^2 A \\left( 1 - e^{2} \\right)}\n", "ef2033149d59c4a6b82cd91906271301": "C\\,a^{-d}", "ef20406844de78ecf66ab975e2965a40": "\nf(B)=\\frac{e^{-\\frac{B}{2}}}{2}\n", "ef20543964e27d5efe40cb6e9892d854": "\\lnot \\phi", "ef20828d35effbd9a8837248f9da7e7b": "(r, \\theta, \\phi, t) \\rightarrow (r, -\\theta, \\phi, t)", "ef215d8186a875b88e4ce051821d4c56": "A_a=\\Phi(\\rho,z)[dt]_a", "ef216861e5ff1692df087026571d23d8": " R\\otimes R ", "ef21ba2a16d3bd70771f10c896a0f02d": " B_1(0) = -B_1(1) = B_1", "ef21e3ddeec64c6ac73e2e916b7b975b": "\\scriptstyle{\\vec{d}_{g,e}}", "ef21e8f88b756fb6c5cd5b748dba4e95": " \\mu \\left(\nz^n,l^k,r^k \\right) ", "ef225c77391e992b173d1c2069a6bd91": "\\textstyle{\\frac {\\log(4)} {\\log(2)} = 2}", "ef22782245295ff36fa31dceffa56fff": " B = \\langle u \\bar{v} \\rangle_V", "ef2279e88ff2fc384f980fb4b0a864db": "(Y_t)_{t \\geq 0}", "ef227a67e35efaa2018ee1fed0e8f06e": "f(x_n)/f'(x_n)", "ef229594bd6dd054b3d7c616fa177534": "\\alpha \\sim 4\\cdot 10^{-5}", "ef2298db792fedcec408a6ee41072830": "\\cot\\varphi = \\frac{1 - t^2}{2t},", "ef229cc96358f9b6fe58c222c97e6f37": " \\cos(i) = \\cosh(1) = {{e + 1/e} \\over 2} = {{e^2 + 1} \\over 2e} \\approx 1.54308064... .", "ef22e8c653fcfd02f118718cde11b72e": "x=x_\\parallel+x_\\perp", "ef22f593c3e687d2861bdbcb2ecb8aaa": " \\nu\\approx v_{\\rm turb} l_{\\rm turb} ", "ef232e4a21a51033d2f726dd2880d200": " n(\\lambda) ", "ef23a7324db86cd917270d7a115ec12b": " z'(x) = f\\left(x,z(x)\\right) ", "ef23c098844f39b2fd510153ce268e3e": "a + (b-a) \\left(\\frac{\\theta \\alpha}{\\alpha + \\beta} + \n\\frac{1-\\theta}{2} \\right)", "ef23ca6efc4223b63b271449380ac9a2": "\\sgn", "ef241316cde5007811b95a659903cdac": "|\\phi(t)| \\leq Ke^{bt}", "ef24446f0b657edbe648714114285b0b": "\\tilde{\\kappa}_{(\\ell)}l^b:= l^a \\nabla_a l^b", "ef24452f19af188272d7d2ecf22833b5": "\\,\\! P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)\\text{ for }-1\\; 1", "ef28fed010f26f3dda8b6c57b30c471b": "s =1/2", "ef292170ebda87e4431065a8920498e9": "f,g : S \\rightarrow \\mathbb{R}", "ef293b05b5b5bada49d4761415c28176": "\\frac1{p(x)}-q(x)=\\frac{1-p(x)q(x)}{p(x)}", "ef29a097b5f7327152e02e987e2e2ee1": "E = \\frac{mc^2}{\\sqrt{1 - \\frac{v^2}{c^2}}}.", "ef29e99847ae9ff6ccef132255cb8102": "\\textstyle\\frac {1}{2-1}=2=", "ef2a07dcc3087bcb3f3bf74a31bbe822": "\\operatorname{Sub}(x,y)", "ef2a364df97606c2e310fa5169fd12b9": "\\alpha = 45^o - \\frac{\\phi}{2}", "ef2a402087bd57b94206fcae76236bc2": "f(x+r) = a^{x+r} \\bmod N = a^x \\bmod N.", "ef2aa8cc4ab22429f3a5f6152ded1e0d": " \\Gamma_S = 0 \\, ", "ef2ac91a8f890a1e5137d4f2b89741e3": "r(v) \\equiv r,", "ef2ae50cd42d3a98e6e2f22b9153e8af": "\\sigma \\left (\\mathbf{P}^{2i_1}(\\mathbf{C}) \\times \\dots \\times \\mathbf{P}^{2i_k}(\\mathbf{C}) \\right) = 1.", "ef2b689bf206cae2b46bcb216a24e358": "p_s", "ef2b71d434c009e9d46e78b98853ab8d": "\\textbf{x}_{k} = \\begin{bmatrix} x \\\\ \\dot{x} \\end{bmatrix} ", "ef2c5be3a44fd3435e776c5909917392": "f(x)+f(y)=f(z)", "ef2cdbed4caa44612761e5bc0d228ec8": "\\ell(D)-\\ell(K-D) = \\text{deg}(D)+1-g.", "ef2cdc6760524ee9d70bc10318ad0b5a": "\\partial_t\\phi=\\nabla ( m\\nabla\\mu + \\xi(x) )\\;,", "ef2cfbdabbd55765559bfdb44822f666": "m\\{x:\\, |Tf(x)| \\ge 2\\lambda\\} \\le (2A+4\\|T\\|)\\cdot \\lambda^{-1} \\|f\\|_1.", "ef2d6809962eb546382ca5f211e9a4c8": "\\mathbf{y}=(3,\\,1)", "ef2d6feddded0a337fe103ffe6a77685": " \\tbinom {11}5 ", "ef2d844574013920009dc4fa9fa24b6f": " \\Pr(S) = 0.8 ; \\Pr(H) = 0.2", "ef2e62a3aff4dc1dad3627b010fb137e": "\\sqrt[3](1860867)=123", "ef2e6fc39df14423a6c147de62d2d63d": "\\textrm{E}\\left(\\frac{X!}{(X-k)!}\\right) = G^{(k)}(1^-), \\quad k \\geq 0.", "ef2eb6c843f1265dcd514b4c3a388514": "F_\\mathrm{f} = \\mu_\\mathrm{k} F_\\mathrm{n}", "ef2ec10910c5b57288fc097305f83693": "R(i)", "ef2eeeb53873c55d227d55775e121245": "P_{ma} = P_2 + 2P_3 + \\cdots + (n-1)P_n,", "ef2ef4095e7b4f99b1adb14efd4c2595": "m_{fullliquid}", "ef2f2f2e8eb7848fee7206277ed56ee8": " \\mathrm{d}\\boldsymbol{\\ell} = \\mathbf {\\Omega} \\times \\mathbf{r}(t) \\mathrm{d}t \\ , ", "ef2f45714f9908e578900c3c6495dead": "e^{\\mu}\\,", "ef2f4743709a660b37ea8f987880216e": "\nt \\leftarrow t + (1 - K_\\max) \\,\n", "ef2f63d12382721aa65f57c68881ca40": "\\phi,\\,\\psi\\in\\mathcal{E}'(\\mathbb{R}^n)", "ef2f6c586a783e377f4ca51262470563": "L_{c} = \\lambda^{2}/BW", "ef2f8f152e9b6fb781793200ba5c859a": "\n\\begin{pmatrix}\n\\varphi(a)\\\\\n\\varphi(a+1)\\\\\n\\vdots\\\\\n\\varphi(b)\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nh_{a } & & & & & \\\\\nh_{a+2} & h_{a+1} & h_{a } & & & \\\\\nh_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\\\\n\\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots \\\\\n & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\\\\n & & & h_{b } & h_{b-1} & h_{b-2} \\\\\n & & & & & h_{b }\n\\end{pmatrix}\n\\cdot\n\\begin{pmatrix}\n\\varphi(a)\\\\\n\\varphi(a+1)\\\\\n\\vdots\\\\\n\\varphi(b)\n\\end{pmatrix}\n", "ef2fe1fcc75432fb04e219a1681a47fd": "m_1, m_2, m_4,", "ef2fe482dc84f5982126048311a303f9": "m_{t+1} = m_t - {\\Delta v}_t + {\\Delta x}_t = m_t - \\overline {\\Delta v}_{t-16} + {\\Delta p}_D + \\overline {\\Delta q} + \\varepsilon_t \\,,", "ef2ff2bd749aebb53deb862a01bdbab6": "A(4,1)=(4-1)A(3,0) + (1+1)A(3,1)=3 \\times 1 + 2 \\times 4 = 11.", "ef2ff4bdacf3633d951325b018059502": "\\beta \\to \\gamma", "ef2ffda07a47f434771bd0e3f8214c2a": "\\% \\mbox{ change in } x = \\frac{x_2 - x_1}{(x_2 + x_1)/2};", "ef3022ca8592e974d36b5c22b1dd6e8c": "U_z", "ef304a0a3f59e232e74eaad53e837354": "\\lambda > x", "ef308e0dea6774e593d9bba32e32acee": " q <_\\mathcal{O} r ", "ef309ae62a4c41d4ed0b898088dcd6f8": "\\mathfrak{H}^3", "ef30bec0bb3fe6beabd7da06f09e24d8": "f_k(t)\\,", "ef30e538d5fe802a634392f3a9079af8": "\\frac{\\sqrt 2}{2}", "ef30e8966facc73c02feee43e13885c6": "f=S_2", "ef30f2c2022557d0ab154701676aa328": "x\\in\\mathcal{D}(A)", "ef31180ed1291c39dd1a1e8ec183f9e9": "a_c = 2(dr/dt)(d\\theta/dt)", "ef312a31fca9a22567302123524e6a42": "{\\bar{M}}_4", "ef312d59511714bb44b3b4ca7b4d4e17": "\\rightarrow \\!", "ef3138df07defd84c00bebe9805131a1": "P_1+P_2", "ef317cd7e958ca56ab94fd4d051de2f3": "\\vec{\\jmath} = \\vec{\\ell_1} + \\vec{\\ell_2}", "ef31a20d006ffc0a2134d811f65ba6aa": "\\operatorname{Gal}(\\mathbf{F}_{q^f} / \\mathbf{F}_p)", "ef31b9c6861ea0bed715c15f7f6c5db1": "\\frac{d}{ds}T(s)\\cdot\\mathbf{s}_{u}=\\frac{dT(s)}{ds}\\cdot\\mathbf{s}_{u}+T(s)\\cdot\\frac{d}{ds}\\mathbf{s}_u", "ef321803dbafbec13bead8e1a01690df": "\\kappa =\\frac{h}{m_4}=1.0 \\times 10^{-7} m^2/s ", "ef32442fb0eea4d485d014640f188671": "f_H = \\frac{\\sigma}{2\\pi} \\sqrt{\\frac{3}{20\\log e}} \\cong 0.0935 \\sigma", "ef3245961db35b356fd663af10b76842": "\\sigma^2_n=\\frac{Q_n}{W_n}\\,", "ef324d2741678500161d839ca03ba5c6": "\\Psi(z)= E_0 e^{ikz}", "ef32aed6e94941254d9e9cedc6c156a9": "RT\\ln \\frac{{fP^\\circ }}\n{{Pf^\\circ }} = \\int_{P^\\circ }^P {\\Phi dP}", "ef32d5d0eee9e11acf25f60f045cdaf2": "\\left(r\\left(1-r\\right)\\sqrt{\\left(1+r\\right)\\left(2-r\\right)}~,~\\frac{1}{2}r\\left(1+3r-2r^2\\right)\\right).", "ef330b4d7614ab2f7202a96620deb342": " \\mathbf{x} | \\sigma^2, \\boldsymbol{\\mu}, \\mathbf{V}^{-1}\\sim \\mathrm{N}(\\boldsymbol{\\mu},\\sigma^2 \\mathbf{V}) \\,\\! ", "ef334d3ed582b8b80d684b5ebc50b0c8": "\\mathbf{E}(\\lambda\\mathbf{x},\\lambda t)", "ef3381badeafe8b903087297510c6473": "\nR = \\left( \\begin{array}{cc} a & 0 \\\\ C/(a + d) & d \\end{array}\\right)\n", "ef33e4a6ac7dea6c376a38dd1b1fa1f8": "\\int_0^1 x^n(\\log\\, x)^n\\,dx", "ef341947dbd3c38f7a1b30b3fe4ba6fe": "S={1\\over 16\\pi G}\\int d^4 x \\sqrt{-g}L_\\phi+S_m\\,", "ef342b923c158a37a86c44b9fb30460d": "\\|\\cdot\\|_2", "ef3437b1a315a080f29e5c90eda6373e": "(v_1,v_2)\\in E_0\\Leftrightarrow (f(v_1),f(v_2))\\in E^\\prime", "ef345094b81293044d7c9fa7a68d275e": " \n\\int_{E} f_k \\, d\\mu \\geq (1-\\epsilon)\\int_E \\varphi \\, d\\mu - \\int_{A-A_k} \\varphi \\, d\\mu \\geq \\int_E \\varphi \\, d\\mu - \\epsilon\\left(\\int_{E} \\varphi \\, d\\mu+M\\right).\n", "ef346305056cb97343f9defe7308199a": "\\|u-u_N\\|_{H^1(\\Omega)} \\leqq C_s N^{-s} \\| u \\|_{H^{s+1}(\\Omega)}", "ef34849ca8a1467184caeaa2683bba62": "A(\\lambda) \\mathbf{x} = ( A_2 \\lambda^2 + A_1 \\lambda + A_0) \\mathbf{x} = 0 , \\,", "ef34b21c52cff930c7917d8ab2adbd7a": "\\min(c_f(A,B), c_f(B,C), c_f(C,D))=", "ef3501dadc346104c2fed775e843325a": "\\mu_o", "ef352d40f14dff2e192c46892b86749c": "\\,\\{e+ig\\} = e^{i\\theta}\\{e+ig\\}", "ef355262323ad975c7847d72a122ed27": " \\epsilon >0", "ef3553088f68f93ec1362c6779c0b3ec": "\\mathit{GF}(\\mathit{p})", "ef357c0db7f8533ae992cac49758cc0e": "\\mathrm{Der}(\\mathfrak g)", "ef35c83035a0fa21de68474f7336ecc3": "f{:}\\tau{\\to}1", "ef362057e8a7d0103657da679b9c6aca": "U_m(P,Q)", "ef364134b3026f3954197d228aebe636": "l^{3/2} = l_0^{3/2} + \\frac{3Fr}{2} \\sqrt{g'Q} t .", "ef369683d782c85a412c5e279fd52518": "\\langle \\phi \\rangle =0", "ef3696ec830719a695c28e17de606110": "\\exists X ( \\exists x,y (Xx \\land Xy \\land Axy) \\land \\exists x \\neg Xx \\land \\forall x\\, \\forall y (Xx \\land Axy \\rightarrow Xy))", "ef369b56bc89a25e954106a31e0740e7": "AP \\cdot AQ = AR \\cdot AS \\, ", "ef36b03ae6b2ece62f520949e07c6d6d": "2000 \\times \\frac{6000 - 9000}{9000} = -666.\\overline{66}", "ef36c49cb2cab90c6ffef6c4f172be8e": "\\int \\frac{\\delta Q}{T} = -N", "ef36dc7208ad6667dc49cbddc8c2dc85": " [ u, v]_{q,p} = [u , v]_{Q,P}", "ef371aebc176a003bf00451b99f579e6": " dU = \\delta q - \\delta w + d(\\sum \\mu_{iR}N_i) \\,", "ef374cd6434453f8e59d43414c578ffc": "A(\\rho)=\\int_D \\rho(z)^2\\,dz\\,d\\bar z=\\int_D \\rho^*(f(z))^2\\,|f\\,'(z)|^2\\,dz\\,d\\bar z = \\int_{D^*} \\rho^*(w)^2\\,dw\\,d\\bar w=A(\\rho^*).", "ef3773edad6bc742e5bd6af9cd089400": "c' \\equiv cs \\equiv \\sum_{i = 1}^n \\alpha_i \\beta_i s \\pmod{q}.", "ef378d81a4057f923b006572034668b5": " \\forall A \\in \\mathcal{A}: \\Gamma(A) \\leq D ", "ef3799553a5b12dfc19f3007b9061472": "\\exists N \\forall x [PNx].", "ef37dd082fc1f9ff21b9c6b685aa6944": " 1 - \\frac{1}{2} + \\frac{1}{10} - \\frac{1}{100} + \\frac{1}{500} = 0.592 ", "ef3843b6bd9e9ed616ff4d66004d104a": " b_i \\in \\mathcal{V}_1(S_i) \\forall i=1,\\dots,p", "ef38582879bf17f55805cc8a8e57e080": "A\\cap X", "ef38871b50a7c54b0fe165790dab4b70": " X_t = \\sum_{i=1}^p \\varphi_i X_{t-i}+ \\varepsilon_t.\\,", "ef3894f785626380a6ceb942d5a4f810": " \\psi(x) ", "ef38addc4cd7a6ff0b6960998d073a66": "U_i = \\operatorname{Spec} A[x_1/x_i, \\dots, x_n/x_i], \\quad 0 \\le i \\le n,", "ef38ceb9a11c17e4cfd8d7681433e9aa": "\\xi = \\sin \\nu", "ef39823cd2fb6952a433e0063f9b7d5b": " {n \\choose n_1, n_2, \\ldots, n_r}f(k_1,\\dots,k_n) \\in \\mathbf{Z}", "ef39aea5f9d3497592faeae55b87f257": " x + y\\,\\, = 0.", "ef39da4bbf6b833d295f41b8a17830a6": "x \\in S \\Leftrightarrow \\exists a,b,c,d,e,f,g,h,i \\ ( p(x,a,b,c,d,e,f,g,h,i) = 0).", "ef39e5d15f0fcbf3e7f33bdaedb80a5a": " \\langle A \\rangle_\\rho = \\mathrm{Trace} (\\rho A) = \\sum_i \\rho_i \\langle \\psi_i | A | \\psi_i \\rangle\n= \\sum_i \\rho_i \\langle A \\rangle_{\\psi_i} ", "ef3a00db00a6139441545441f95dd0ed": "\\, Q ", "ef3a167475f179d527bdcb1c976b2c8b": "(g,0)(h,1)=(hg,1)", "ef3a2575bb62831065067dabe01fba7b": "\\tilde{Q} = \\sum_{m=1}^M \\sigma^2_m Q_m, \\quad Q_m \\sim \\chi^2(2r_m) \\, .", "ef3a317d10769223c410958bb1072642": "S(a,c) + S(c,b)\\,", "ef3a7222dd3a89d7f18271041f8e8550": "\\left(\\sqrt{1/55},\\ \\sqrt{1/45},\\ 1/6,\\ \\sqrt{1/28},\\ -\\sqrt{12/7},\\ 0,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "ef3ae5169d8b73a7440b9e33c7db200c": "e d \\equiv 1 \\pmod{p-1}", "ef3b084bed38d3f1ff49e69ddeb735fb": "V_T = \\phi_T S_T + \\psi_T B_T = C_T = X", "ef3bd052f32b25c4eff20ace38fe59af": "U = - m (G \\frac{ M_1}{r_1}+ G \\frac{ M_2}{r_2}) ", "ef3c025ce975229d480ec7a453778eac": "F^{\\omega} \\,.", "ef3c0cd6ee6a86fef904c3bcbcba5c44": "\\begin{align}\n \\phi_{n+1} = \\phi({\\mathbf{X}},z_{n+1}) = [q_{n}\\phi_{n}]*p_{n} \n \\end{align}", "ef3c415842dace07f0067a6de86f2124": "\\phi_{sl,v}", "ef3c49e7fe2fb5840d09ba740a628e60": "\\tilde{\\rho}", "ef3c55e92cced5ddfcf4fa5b1fc990f7": "1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i} = 1 + 2^{-2} = 1.25 ", "ef3ceb3072f16923b1183d07a410e702": "\\overline{x^2} = E[x^2] = P_{x^\\nu}=\\int_{}^{}x^2f(x)dx", "ef3d6deeaaf834e5d04a1cb79d8cbbb7": "C \\sqsubseteq D", "ef3dc698b370111957e86a3395fbc9c4": "I(X; Y)", "ef3dcdb0d0e401aa72bf60b9975fdb65": "\\mathcal{R} ", "ef3eff809fa9c2eb4267ba7cdf48bc13": " \\operatorname{hanoi}(n) =\n \\begin{cases}\n 1 & \\mbox{if } n = 1 \\\\\n 2\\cdot\\operatorname{hanoi}(n-1) + 1 & \\mbox{if } n > 1\\\\\n \\end{cases}\n", "ef3f12ec8e756ab8ecac515c60d6e2cd": "K_0 = \\frac{\\prod_{j=1}^p a \\left ( {\\rm Y}_j \\right )^{\\eta_j}}{\\prod_{i=1}^r a \\left ( {\\rm X}_i \\right )^{\\nu_i} } \\,\\!", "ef3f5b75c9b1bf5a446edb24be55b0e6": " \\Pi_A = \\text{cl}\\Big( \\bigcap_{P \\in S_A} \\Pi_P \\Big). ", "ef3fbe61ef76fe4b2e4b16fbf3366284": "u_n = 2^n", "ef3ffb662dc62a21e220177477f5f51c": "Z(s) = \\left( {\\frac{1}{s\\cdot C_1}+s\\cdot L_1+R_1} \\right) || \\left( {\\frac{1}{s\\cdot C_0}} \\right) ", "ef406db6e8f96d4a9654b4ea4dde47d5": " P(x; k, \\lambda ) \\approx \\Phi \\left\\{ \\frac{(\\frac{x} {k + \\lambda}) ^ h - (1 + h p (h - 1 - 0.5 (2 - h) m p))} {h \\sqrt{ 2p} (1 + 0.5 m p)} \\right\\} ", "ef4075c674b4b1a007daf5df6c8a2cbb": "\\mathbf{d}_A\\equiv\\mathbf{R}_A-\\mathbf{R}^0_A", "ef40be6460f2fe315ff755b9efdd18eb": " \\beta = \\mathcal{V}Bd ", "ef40e7de196151e73fbd6f431e4377d8": "O(n^2*s^2)", "ef40e8898e54c309e8c59292a4bd3f38": "q = \\sgn(n) \\left\\lfloor \\frac{a}{\\left|n\\right|} \\right\\rfloor.", "ef41109428a8b16477111c94d5cc3650": "\\mathrm{tr}( \\hat{\\rho} \\cdot g_N(\\hat{a},\\hat{a}^{\\dagger})) = \\int P(\\alpha) g(\\alpha,\\alpha^*) \\, d^2\\alpha.", "ef412c81e352f9cb2c876f9a972a00cc": "\n\\epsilon_{\\mu\\nu\\rho\\theta} \\eta_{a\\sigma\\theta}\n= \\delta_{\\sigma\\mu} \\eta_{a\\nu\\rho}\n+ \\delta_{\\sigma\\rho} \\eta_{a\\mu\\nu}\n- \\delta_{\\sigma\\nu} \\eta_{a\\mu\\rho} \\ ,\n", "ef4207deee49ebd08641c03a26747b60": "p=r \\sin \\alpha", "ef4252ca4cba0f208acc6370e93d87a5": "l^\\prime=L_b(\\left\\lceil (2\\left\\lceil l/4 \\right\\rceil +4)/16 \\right\\rceil)", "ef4258e32e576f5f7910e33bbb847f3c": "\\int_{-\\infty}^\\infty \\frac{|g(x)|^p}{1 + x^2}\\,dx < \\infty", "ef4284f67298d43741957424876db935": "D_{2h}", "ef42c7499944ae067cb3a8a038a91475": "(x,y,z)=\\biggr(x,\\sqrt[3]{2}\\sqrt{3}\\,x,\\frac{3\\sqrt{3}}{2}\\,x\\biggr)\\quad\\mbox{with}\\quad x>0.", "ef42ea6953efeb509827c62d9b9508ff": "\\cap_{n\\in \\mathbb Z} f^{n} (U)=\\Omega(f).", "ef432a8e71ce3376e0369dc67dd00b11": "\\operatorname{erf}(x)\\approx \\sgn(x) \\sqrt{1-\\exp\\left(-x^2\\frac{4/\\pi+ax^2}{1+ax^2}\\right)}", "ef432e1d30f064a0adb2999a677aa08c": "\\rho_0 = 0.5 ", "ef435469e974757b348636cfd35f0c91": "1-I_p(k+1,\\,r),", "ef43ad2de9fb2edece3f93b4f104a3b1": "\\begin{pmatrix}\n e^{i\\phi_x} \\cos^2\\theta+e^{i\\phi_y} \\sin^2\\theta & (e^{i\\phi_x}-e^{i\\phi_y}) \\cos\\theta \\sin\\theta \\\\ (e^{i\\phi_x}-e^{i\\phi_y}) \\cos\\theta \\sin\\theta & e^{i\\phi_x} \\sin^2\\theta+e^{i\\phi_y} \\cos^2\\theta\n\\end{pmatrix}\n", "ef43ba785bd0867311a51dc7981a86de": "{\\mathbb P}\\biggl(\\bigcup_{i} A_i\\biggr) \\le \\sum_i {\\mathbb P}(A_i).", "ef43c9e080d3b3f3a90da270bcb0e6e1": "\\lambda_1,\\lambda_2,\\dots \\in \\mathbb{R}", "ef43d5fbe7202312887c36e0754e233b": "\\textstyle l(s)", "ef441dc63e91548b8dceda52577d9699": "cx + dy \\le 10", "ef444dcf90fac078a6ee3cf12472a3ae": " 27379 \\, ", "ef4477427e9555cb524d5ef51e8af052": "\\ (U,\\ S,\\ E)", "ef44dd173ece729247144dfa353608ae": "\\varphi(x) \\not= \\{0\\} \\cup \\varphi(y).", "ef44fd84ca721086cca6f700b2897f24": "r(x)+r(y) \\ge r(x \\wedge y) + r(x \\vee y).", "ef45277fea5add5e1638f45c6010bd5d": "\\lnot\\ \\forall{x}{\\in}\\mathbf{X}\\, P(x)", "ef45314fb2e28c620eb28f6f4a29a791": "\\mathit{n} \\geq \\mathit{2w} + \\lceil \\mathit{d_{min}}/2 \\rceil", "ef453c14fbe27ac9039bf5af27c660bb": "\nw_{pj}^{L(new)} = w_{pj}^{L(old)} + \\eta (x_{ij}^p-z_{pj}^{old})\n", "ef45904ce4f57b1c76b4e1b3b7ddd05a": "a_2\\,", "ef45a8194432540cc89ddad2929db874": "d(a, b) = 0 \\iff a = b ", "ef462efd3de7900b9d6d3660f0aa74eb": "Ax=b\\ ", "ef4648e7c12086a3c1fb9447f29676f0": "\n \\mathcal{M}^{\\mu \\nu} =\n \\begin{pmatrix}\n 0 & P_xc & P_yc & P_zc \\\\\n - P_xc & 0 & - M_z & M_y \\\\\n - P_yc & M_z & 0 & - M_x \\\\\n - P_zc & - M_y & M_x & 0 \n \\end{pmatrix},\n", "ef469f1f9e6f39971f8ee38c75db1077": "\\Gamma(x)^a\\,", "ef46a015ee9487883e90c6fec16d71f5": "h(x,t)=\\sum_{n=1}^{\\infty}h_{n}(t)\\sin\\frac{n\\pi x}{L},", "ef46d8870baed719fba08bf932ab58d5": "\\delta S=0 \\ ", "ef46de24a104ccc24667b55f650bfaa6": "\\Lambda_n/C={\\Bbb Z}^2", "ef471744d8c3a68ee162780fad4b86f8": "d \\omega + \\frac{1}{2} [\\omega, \\omega] = 0.", "ef476c2c77d69a34f92e3f19b0d22aeb": "I_{\\text{ES}}", "ef47c2f3f0396a1518baa9bedc7fcf26": "A(\\psi)=\\sum_{s\\in W} (-1)^{\\ell(s)} s\\cdot \\psi.", "ef47e2092f4d77986f7478a00914e5c0": "E = \\frac{1}{2}mv^2 = \\frac{1}{2}(Avt\\rho)v^2 = \\frac{1}{2}At\\rho v^3,", "ef47f6292d268af9d5e6a01c271f6664": "x^5 + ax + b", "ef488b2d1f3f38fd3f26a3639d4852f9": "| X | \\sim \\chi_1(x) \\,", "ef48a106fcc1ade644822fb44dac9b6e": "L=\\frac{m}{2}(\\dot{x}^2+\\dot{y}^2+\\dot{z}^2) + q(\\dot{x}A_x+\\dot{y}A_y+\\dot{z}A_z) - q\\phi", "ef48d1569d09fa208c1c318cca9611c9": "N = 1, C_{01} = C_2, C_{11} = 0, C_{10} = C_1, M=1", "ef48e5ea4cef3bd0b3fa5bcdf1cf009f": "\\left ( 2 \\right )", "ef48e623a6367a2b335e3e8fdd6bbe0f": "I \\approx \\frac{1}{n} \\sum_i f(u_i); ", "ef490ba77a4093c961350a057f7c2aeb": "x = (0.a_1a_2000a_300000000000000000a_4000...)_b\\;.", "ef4970a8054973adf5a3dc47c80ad912": "\n\\dbinom{16}{8} \\cdot\n\\dbinom{8}{2} \\cdot\n\\dbinom{6}{2} \\cdot\n\\dbinom{4}{2} \\cdot\n\\dbinom{2}{1} \\cdot\n\\dbinom{1}{1} \n= 64,864,800\n", "ef498481aeff5a5c13164e753d44e476": " U_sU_\\omega = M \\begin{pmatrix} \\exp(2it) & 0 \\\\ 0 & \\exp(-2it)\\end{pmatrix} M^{-1}", "ef4987da9cbd1565f0febc7fe6bd5ee1": "\\nabla\\cdot(\\varphi \\mathbf{F}) \n= (\\nabla\\varphi) \\cdot \\mathbf{F}\n+ \\varphi \\;(\\nabla\\cdot\\mathbf{F}). ", "ef49ef93200ab1a4108f89eb03f5cb90": "x\\notin Q", "ef49f4f1d13f3b5bd817221d7654f9ba": " U_k (\\alpha) = 0 \\,\\!", "ef4abde9e8c795b7470fbac7835aeeeb": "\\Vert \\sum_{j}T_j\\Vert \\le\\sqrt{AB}.", "ef4ae4fb86850a13ddc9f021355daf6f": "\\frac{\\dot{T}(t)}{\\alpha T(t)} = \\frac{X''(x)}{X(x)}.", "ef4b0f1203960442dc814566142cdd87": "f(g(x))=\\sum_{n=0}^\\infty{c_n}x^n,", "ef4b45e669ccaa7abea3c018359340c9": "R_{sensmax}", "ef4baa64b88fcd8770587038ad5988e9": "y=x;", "ef4c2de91368bef9be86cf620b6d81e1": "\\frac{N!}{2(\\ln 2)^{N+1}}", "ef4cad84ddf8fc40568b7c2ad805339b": "\\sum p_i(v_j)=1", "ef4ce7b8fbed18448d6e0c4591962c33": "\n a^3 = \\cfrac{3R}{4E^*}\\left(F + 2\\Delta\\gamma\\pi R\\right)\n ", "ef4d1ea74bd1b2a698eb6b73d92d7d15": "\\eta_{th} \\equiv \\frac{Q_{out}}{Q_{in}}", "ef4d269dc4b455daf97358611cb8cda2": "n_{veh}", "ef4ddc05e53f26c0a130d1caee436c06": "(14)\\quad R_{vv}=2\\frac{M(v)_{,\\,v}}{r^2}\\,,", "ef4dfb741d61dc3452c193b493c5e9c4": "\\textstyle x_k", "ef4e0dd6a8922a6a0f5f1dcf8201af29": " \\int_a^b f(x) \\,dx \\approx \\int_a^b L(x)\\,dx = \\int_a^b \\bigl( \\sum_{i=0}^n f(x_i)\\, l_i(x) \\bigr) \\, dx \n= \\sum_{i=0}^n f(x_i) \\underbrace{\\int_a^b l_i(x)\\, dx}_{w_i}. ", "ef4e29ec362fd5f8ad54ee8fbd97263f": "\\sigma \\in H", "ef4e35cf36784b1d6a033c02bd17ddbf": " C \\rightarrow -\\infty ", "ef4e7145da34ad225cca39268defa738": "0 f_p", "ef4f742800d19547ce7a165e3ddf9dff": "S' = \\frac{S}{L^2}", "ef4fa2aff6f4d6da3e8339fbb5ba63e1": "C_5 = \\{-U, SU, VU, SVU\\},", "ef4fb8ec745d49f40f72a2b244000ef4": "\\tau =-\\infty", "ef4ff70bdf6fd9f8ad89be73a4abe27a": "\\boldsymbol{\\mathsf{a}}' = \\Lambda \\boldsymbol{\\mathsf{a}}", "ef504bf202b64b535f7e34b1d6d580e3": " \\mbox{density of body} = \\frac { \\mbox {density of water} * \\mbox{weight of body}} {( \\mbox{weight of body} - \\mbox{weight of immersed body}) - \\mbox {density of water} * ( \\mbox{residual lung volume} + \\mbox{100 cc})}\\,", "ef506d20ca6d7183339ad76ec3f8a8af": " C = K^{\\mu\\nu}u_{\\mu}u_{\\nu} ", "ef5090fbf1ed494ccdecde967888f87b": "\\int^{\\infty}_{0}x^{j}e^{-x}\\,dx=j!", "ef50e1f8b1789b533c286a79d287b2d0": "\\alpha : E(G) \\rightarrow \\Pi", "ef50e28e863f820767a3b043e8f19e7d": "V^{1.85} = k^{1.85}\\, C^{1.85}\\, R^{1.17}\\, S", "ef515c6d998b09a260505a48ec0c84c9": "(\\nabla_c\\nabla_d - \\nabla_d\\nabla_c - f^e_{cd}\\nabla_e)", "ef5166ff96e7499920a37b167ba84817": "\\Delta n(x,t) = \\sum_i \\frac{\\partial^2 n(x,t)}{\\partial x_i^2} \\ .", "ef5181d72e810326a7d2e08962ac6e87": " | F \\rangle ", "ef518eac6efe1dde845ba47c1193a202": "U=\\frac{1}{2} C V^2 =\\frac{Q^2}{2C}", "ef5198806cb7c50b0fbd7fd9f8eaede1": "H(X_{1,2})=H(X_1,X_2)", "ef519defda2c607a0867c96fba5f96ef": "\\Rightarrow v^2=2gz\\,", "ef51d7523ad53d9ce80d247913559c11": "T_n(x) = \\cos(n\\,\\arccos(x)).", "ef51f26d82a406166bf9c1ccd747272f": "K = k(\\mathbf{x},\\mathbf{y}) = (\\Phi(\\mathbf{x}),\\Phi(\\mathbf{y})) = \\Phi(\\mathbf{x})^T\\Phi(\\mathbf{y})", "ef5213c6c3277b958ecbe9867beccd05": "r=6.5/100/12", "ef5215b26257403fd04655f45279e032": "\\mathbf{v}\\,", "ef5261756bdb2c9a003bb29535d7aeaa": "\n N_{\\alpha\\beta} := \\int_{-h}^h \\sigma_{\\alpha\\beta}~dx_3 \\,,\n ", "ef5271fa7bc9d5f4bd296515895e53ed": "ax^{a-1}\\,", "ef52855ab2eb6a454d0136871c50384b": "\\,m_1(y)=E[Z|Y=y]", "ef53114211d03d44aa37f0dc67b92ad1": " D_n(z) ", "ef531211aa7e49c6c97f84a9ec3c1828": "\\partial_{x_i} f=g_i, i=1,...,n,", "ef5317e398e244a683ca3874276096fb": "\\frac{\\pi}{3} \\ (60^\\circ)", "ef53729534d19f7653872f4c6daef479": "\ng = \\frac{\\sigma\\ E^2}{n_e\\; \\left( E_i + \\frac{3}{2} k\\; T_e \\right)\\; \\left( 1 + \\beta^2 \\right)}\\; (\\beta - \\beta_{cr})\n", "ef53a0012dcf357ae9d7f6a855142c13": "~(t,t+{\\rm d}t)~", "ef546b3cc0580b28301b459a1350075a": "d_v", "ef547f8c5aebf9e705122924c32ae100": "\\sqrt T", "ef549140d656623d0f5a8398403458b6": "\\frac{\\dot{Q}_{out}}{\\dot{m}}=h_4-h_1", "ef54ba042d57a71aab3ac802491ddba8": "S_{1/T}(f),\\,", "ef54c5f4e5dfc431e0858e3275a2eccd": "r=\\frac{Gv^2}{g(h_a+h_b)}", "ef54d40c93e0b153c9fb6defb693c20e": "\\hat{H}(x)=H(1)-H(1-x)", "ef54d6b0df94723020e797770050ff6a": "(x_n)_{n\\geq 0}=(0, 12, 24, 1, 13, 25, 2, 14, 26, 3, 15, 27, 4, 16, 28, 5, 17, 29, 6, 18, 30, 7, 19, 31, 8, 20, 32, 9, 21, 33, 10, 22, 34, 11, 23)", "ef558cec310ddd5ef47f077add0b3bed": "\\sigma(f) = \\{ f(t) : t \\in X \\}.", "ef559bb97decb67f12a9a6af2f31c17f": " \\sum_{i=1}^N X_i \\sim\\mathrm{Gamma} \\left( \\sum_{i=1}^N k_i, \\theta \\right)", "ef55cbf9d9e8b3de73605f6f071bf977": "\\{ a^ib^i : i \\geq 0\\}", "ef55eb33274b591633e755317afa3dcf": "K_{12} = K_{21} = 0", "ef5612527e47fb5e67d4035ca524a995": "\\langle \\Psi_1, \\Psi_2, \\Psi_3 \\rangle = (-1)^{gn(\\Psi_3) * (gn(\\Psi_2)+ gn(\\Psi_1))}\\langle \\Psi_3, \\Psi_1, \\Psi_2 \\rangle", "ef561355b2e5e4509e81849ad6312b87": "\\operatorname{sech}(z)", "ef5613948b03b6f73b018ff3dbfe906d": "L_M=a{M}", "ef563ca78b82b79e7e7ff7b32d5895f5": "n + (-\\infty) = -\\infty", "ef567720cb499f47924c3c165750335b": "z\\; {}_pF_q(a_1+1,\\dots,a_p+1;b_1+1,\\dots,b_q+1;z),", "ef56aceeb4eab449616ee392c2a03d25": "\\Box \\phi =4\\pi T_M^{\\;},", "ef56e94319f9c78f02d783842350b328": "O(du^2)", "ef57256bc97bcecaacf4e3ee7c62fa7c": "F= -pV+\\sum_i \\mu_i N_i\\,", "ef572b61758dd335e1144bb6b28c5f40": "\\boldsymbol{N}=\\boldsymbol{P}^T", "ef573ea68e560e63853db5ef0f07c809": "\\, \\sigma = \\sigma - t_e", "ef577518d0fbfe5d47bf6a3f010189e8": "\\hat{a}_{\\lambda,\\mathbf{k}}", "ef577b0010991aaff233f82feb003736": "(x+y)^6", "ef57da9ad061717a8fabe918d4b5115a": "\\Delta W_{stored} = \\int_{\\lambda_1}^{\\lambda_2} i(\\lambda) ~ d\\lambda \\;", "ef57e8acb0ded03db2fe6a696492744d": "L=\\lambda L_1+(1-\\lambda)L_2", "ef5810c8bf458e2b0f4b147af775bc5b": "T = t-\\beta_1 z", "ef5830f66c6d14e899df7b2431b084be": "\\vec{v}(t + \\Delta t) = \\vec{v}(t) + \\tfrac12\\,\\left(\\vec{a}(t)+\\vec{a}(t + \\Delta t)\\right)\\Delta t\\,", "ef584069a7655bdc8c561cc36bf6f732": "\\phi_R \\circ f:R(X)\\rightarrow S_R", "ef58ae5bb43ff9bbd012338f56258e3d": "\\frac{am}{p}=\\frac{N}{b^k-1}", "ef58f2543b1b18faac0d9e152d8d59ae": "m_a+m_b=m_c", "ef59a178c59ef0be340de536770b185e": "\\hat{X}\\,", "ef5a40a5a43ae8ddec1cecec2758ec73": "\\partial f_{\\#}", "ef5a8ccb2575033e10ed3bcbc5023693": "R \\equiv R/b", "ef5ae0f805c1c63b0eedf788dda9bdb7": "\\begin{bmatrix} e^{\\eta_1} \\\\ \\vdots \\\\ e^{\\eta_k} \\end{bmatrix}", "ef5b122bcb7d901d48b29d82c5ce0dfd": "1 + 1 = 0", "ef5b403725bd7320b4850d7c0a02ab52": "K \\le \\tfrac{4}{3}r\\sqrt{4R^2+r^2}", "ef5bdced174adaa58dc2b523aaad60ae": "k^{2}=k_{o}^{2}\\varepsilon _{r}=\\left ( \\frac{2\\pi }{\\lambda} \\right )^{2}=k_{x\\varepsilon }^{2}+k_{y}^{2}+k_{z}^{2}= k_{x\\varepsilon }^{2}+k_{y}^{2}+\\beta ^{2} \\ \\ \\ \\ \\ \\ (2) ", "ef5bea86ec6e2fd5e71a4369d3b23846": "\\Psi (\\mathbf{r}_1 \\cdots \\mathbf{r}_N,s_{z\\,1}, s_{z\\,2} \\cdots s_{z\\,N} ) = \\prod_{i=1}^N\\psi(\\mathbf{r}_i,s_{z\\,i}) = \\psi(\\mathbf{r}_1,s_{z\\,1})\\psi(\\mathbf{r}_2,s_{z\\,2})\\cdots\\psi(\\mathbf{r}_N,s_{z\\,N}).", "ef5bf09ec284a18aa3ee12d2f0614d45": "x^{40} + x^{26} + x^{23} + x^{17} + x^3 + 1", "ef5c8ed3c51d95cb82c975aff7554bbe": "f_*\\colon H_n\\left(S^n\\right)\\to H_n\\left(S^n\\right)", "ef5c9097c22efc307e20b37d6b2bb81d": "\\left(\\left(s_0,d_0\\right),\\dots,\\left(s_n,d_n\\right)\\right)", "ef5cd6e2bc5374f085627a840990a95a": "\\mathbb{Z}/2", "ef5d229e3afc22b1e22c43e58a1ce358": "X \\cong \\varinjlim_{\\Delta^n \\to X} \\Delta^n", "ef5d6cd849e7935ec33576b17435e721": "\\sigma(D) = (3^2+3+1)\\cdot(7^2+7+1)\\cdot(11^2+11+1)\\cdot(13^3+13+1)\\cdot(22021+1) \\ . ", "ef5dbae68c3330590ca80a6c6a9bdc2e": "\\int_a^{\\infty} f(x)\\,dx := \\lim_{b\\to\\infty}\\int_a^bf(x)\\,dx.", "ef5dd17d5686c4e2ba16ba47a1d36ffd": "P = \\begin{pmatrix}\n 1 & p_{11} & p_{12} \\\\\n \\vdots & \\vdots & \\vdots \\\\\n 1 & p_{n1} & p_{n2}\n\\end{pmatrix} ", "ef5e150c8abd47388875e3a91226f60f": "X_g \\subset X", "ef5e9a872fd765c0f4c326d187e1b9aa": "\\int e^x e^{ix} \\, dx \\,=\\, \\int e^{(1+i)x}\\,dx \\,=\\, \\frac{e^{(1+i)x}}{1+i} + C.", "ef5ebce706b0c1b42883c397e8c811f3": " \\Delta u = K(x).", "ef5ed7c803b339a82c896af1c8c15217": " \\varepsilon = \\frac{2ac}{b}, ", "ef5eefb2180ef72d9de10a20292d142c": "\\operatorname{Gl}_m(\\theta)\\, ", "ef5f412e47d37995ee96f81bedc39ba3": "O(V^2 E)", "ef5f5454e805145719866759985733de": "1/H_0", "ef5f59181c921b06b6b5c361a7ae5ea8": " f''(x) \\approx \\frac{\\delta_h^2[f](x)}{h^2} = \\frac{f(x+h) - 2 f(x) + f(x-h)}{h^{2}} . ", "ef5f75a59062c624fe016dedb886aa2c": "\\alpha=\\dfrac{p-2}{p-1}", "ef5fed07a092f81bbe1c9f48a128cb73": " \\forall i,j : \\langle u_i , u_j \\rangle = \\delta_{ij} ", "ef604385cad396bd682b38476a9c2683": " \\left (1-\\sum_{i=1}^sc_ix^i \\right )A(x)=P(x)+\\sum_{n=0}^{n_r-1}[a_n-p_n]x^n-\\sum_{i=1}^s c_ix^i\\sum_{n=0}^{n_r-i-1}a_nx^n.", "ef606db941c3c70bedddec4dadc096e8": "\\textit{par}(t,e) \\land \\textit{fem}(e)", "ef60885d71e52accbed01f930b6821de": "\\left\\{ U_{\\phi, x, \\delta} \\left| \\begin{array}{c} \\phi \\colon S \\to \\mathbf{R} \\text{ is bounded and continuous,} \\\\ x \\in \\mathbf{R} \\text{ and } \\delta > 0 \\end{array} \\right. \\right\\},", "ef60938716098927b6ad9fc579859880": "\\psi : W \\rightarrow \\bold{R}^n.", "ef613439567dca12ab495fa7c49d9326": "\\tfrac{E\\nu}{(1+\\nu)(1-2\\nu)}", "ef613970cd87727d19b2c72e330d1673": "y^3-2y-5=0", "ef6141d1f0e6ea26ed5070a4eb144bb1": "\nf(x \\uparrow y) + f(x \\downarrow y) \\geq f(x) + f(y)\n", "ef6171c3dd79809f91cb0d7a571a9692": "\\psi : (N,y) \\to (N,y)", "ef617b1cff750c3db61a5e6f97f58567": "b\\wedge(a\\to b)= b", "ef617c552e638889a88e1601248fdc11": "\\|v\\|", "ef61f09d244a7ff8c410086f905ded20": "\\rho_{S}(t) = Tr_B[\\hat{U}(t)[\\rho_{S}(0)\\otimes\\rho_{B}(0)]\\hat{U^{\\dagger}}(t)].", "ef620825b066f5f94cfbf3553f68c65c": "{{z}_{in}}", "ef621fa5bd63a5416b6c1de8fb9ad013": "X_{1/T}(f)\\ \\stackrel{\\text{def}}{=}\\ \\sum_{k=-\\infty}^{\\infty} X(f-k/T)= \\sum_{n=-\\infty}^{\\infty} \\underbrace{T\\cdot x(nT)}_{x[n]}\\cdot e^{-i 2\\pi f nT}", "ef623c6c7dff0ba53eca2c9bf9301267": "|C_k|\\le c^{\\frac{k!}{n!(k-n)!}}", "ef625630c86b6343d1b89d3b11899f9c": "\\displaystyle{Q_1(a\\oplus \\alpha 1)(b\\oplus \\beta 1)= \\alpha^2\\beta 1\\oplus [\\alpha^2 a + \\alpha^2 b + 2\\alpha\\beta a + \\alpha \n\\{a,y, b\\} + \\beta Q(a)y + Q(a)Q(y)b].}", "ef628fca9519c370c59c06d2b0383a63": "\n\\begin{array}{lcl}\n\\bar{h}^{\\alpha \\beta} & = &\n\\frac{1}{r}\\, \\begin{bmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & A_{+}(t-r,\\theta,\\phi) & A_{\\times}(t-r,\\theta,\\phi) \\\\\n0 & 0 & A_{\\times}(t-r,\\theta,\\phi) & -A_{+}(t-r,\\theta,\\phi)\n\\end{bmatrix} \\\\\n\\\\\n& \\equiv &\n\\begin{bmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & h_{+}(t-r,r,\\theta,\\phi) & h_{\\times}(t-r,r,\\theta,\\phi) \\\\\n0 & 0 & h_{\\times}(t-r,r,\\theta,\\phi) & -h_{+}(t-r,r,\\theta,\\phi)\n\\end{bmatrix}\n\\end{array}\n \\,", "ef6295bdf07da7791b93bdfb711ac4b6": " y_{n+1} = y_n + hf\\left(t_n+\\frac{h}{2},y_n+\\frac{h}{2}f(t_n, y_n)\\right), \\qquad\\qquad (1)", "ef62b45f3d1ba57985dc83b3bcf9375d": "\\hat{\\mathcal{H}}^D = \\hat{\\mathcal{H}}^D_i + \\hat{\\mathcal{H}}^D_v + C", "ef62e4deff8af1e9bb7991c20cbe6077": "\nC(t) = \\frac{\\langle h_A(0) h_B(t) \\rangle}{\\langle h_A \\rangle}\n", "ef6351636315e59d1a1f835a333a7b9f": "\\begin{align}\n\\sigma_A &= \\sum_{X \\subset A} \\tau_{A \\setminus X} \\tau'_{X}~, \\\\\n\\sigma_B-\\sigma'_B &= \\sum_{X\\subset B} \n \\left[1-(-1)^{|X|}\\right] \\tau_{B \\setminus X} \\tau'_X~.\n\\end{align}", "ef639e84cc8dd53edda4142e5a43752c": "\\lambda^n, n\\lambda^n, n^2\\lambda^n,\\dots,n^{r-1}\\lambda^n", "ef63ae671d875c77d90a25ed6e2da693": " F(x;\\mu,\\sigma,0)=e^{-e^{-(x-\\mu)/\\sigma}}\\;\\;\\; \\text{for} \\;\\; x\\in\\mathbb R.", "ef63af29f2c88d8fd08b061f3180b0cd": "\\ z^0", "ef644abe449f12354df7896b3dc192d0": " x_i < y_i < c ", "ef645f3d5d4bdd3017237b472d774193": " C_q = \\sqrt{\\pi} \\text{ for } q = 1 \\, ", "ef64a826e87fcd749b4184c5bf529ef6": "P = \\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\\}", "ef64b2be32672a45859618d53d5f776c": "\\ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).", "ef6517364cf2ce4236fca77635375ced": "\n\\begin{align}\n\\frac{D^3F(P_0)}{DP^3} & =\\frac{D^2F'(P_0)}{DP^2}=\\frac{DF''(P_0)}{DP}=\\frac{F''(P_1 < P < P_3)-F''(P_0 < P < P_2)}{P_1-P_0}, \\\\[10pt]\n& {\\color{white}.}\\qquad\\qquad\\qquad\\qquad\\qquad\\ \\ \\ne\\frac{F''(P_1)-F''(P_0)}{P_1-P_0}, \\\\[10pt]\n& =\\frac{\\frac{F'(P_2 < P < P_3)-F'(P_1 < P < P_2)}{P_1-P_0}-\\frac{F'(P_1 < P < P_2)-F'(P_0 < P < P_1)}{P_1-P_0}}{P_1-P_0}, \\\\[10pt]\n& =\\frac{F'(P_2 < P < P_3)-2F'(P_1 < P < P_2)+F'(P_0 < P < P_1)}{(P_1-P_0)^2}, \\\\[10pt]\n& =F[P_0,P_1,P_2,P_3]=\\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{(P_1-P_0)^3}, \\\\[10pt]\n& =F'''(P_0 < P < P_3)=\\sum_{TN=1}^{UT=\\infty}\\frac{F'''(P_{(tn)})}{UT}, \\\\[10pt]\n& =G''(P_0 < P < P_3)\\ =H'(P_0 < P < P_3)=I(P_0 < P < P_3).\n\\end{align}\n", "ef657eafecaf315cca7ffc33c6bd0482": " v_B", "ef65b58df9b82f7963f41e8932d63f74": "1.2\\times10^{4932}", "ef666a34a0854358673678712edac209": " \\lambda(\\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \\dots", "ef6688edb7b24b39ee4ef57760510c05": " \\mu = \\text{E}(X) ", "ef679be5d719619fddfdbb2cff891db3": "x-\\lambda\\!", "ef67a3e36a06035a221a9e13ab595cad": "q_{2n} = 2 p_n q_n\\,\\!", "ef6800ce90e55df062422eee59fbbf88": "BC(p,q) = \\sum_{x\\in X} \\sqrt{p(x) q(x)}", "ef682b482343b9bebd8d5d9ff5f4aafc": "\\mbox{Current Dividend Yield}=\\frac{\\mbox{Most Recent Full-Year Dividend}}{\\mbox{Current Share Price}}", "ef6851789530e02516d79e91ec7db7f9": " P'' ", "ef68645c8176fccf907f868ad8f1298f": "\\pi \\setminus \\ell", "ef6886c0f1b8666cd6d8e68110bd064a": "\\scriptstyle v^2 [t] = \\dot{\\vec{x}}[t] \\cdot \\dot{\\vec{x}}[t].", "ef68cce3e6d119448350a56743e8c578": "R(\\mathbf{\\hat{n}},\\theta)", "ef69636a188abca9f48deabf87d51f23": " \\varphi(z) \\leq \\frac{1}{2\\pi} \\int_0^{2\\pi} \\varphi(z+ r \\mathrm{e}^{i\\theta}) \\, d\\theta. ", "ef699dbf354f8df10c13b8c823638f84": "\\left(\\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm1,\\ \\pm(1+\\sqrt{2})\\right)", "ef69cd51d0460ca4bfc524056ee43b6e": "\\scriptstyle{P_D}", "ef69e437e4ff2eb74e02e2ab3eca2994": " \\langle \\psi |\\phi\\rangle", "ef6ac076d0ad3ff32494bb0c07a27b24": "\\Gamma\\left(\\frac{s}{2}\\right)\\zeta(s)\\pi^{-\\frac{s}{2}} = \\Gamma\\left(\\frac{1-s}{2}\\right)\\zeta(1-s)\\pi^{-\\frac{1-s}{2}}.", "ef6afc247fcc1c1ad881d09220f9da8a": "N_{i+1,n-1}", "ef6b0da2dba56a85fe5a318af9e989df": "\\text{WACC} = \\frac{D}{D+E}K_d + \\frac{E}{D+E}K_e", "ef6b51fd4ce7d4331c84c032dc789166": "Spin(4) = SU(2) \\times SU(2)", "ef6b5ba642d101008967acdeecba2465": "\\prod_{i=n-k+2}^n m_i", "ef6c574c7b06e8ef475767cc50e473dc": "\\mathbf{J}=\\sigma\\mathbf{E}", "ef6c9bdbea375212188480bf8fe2e988": "\\rho_{\\bold{k}}^s(\\bold{r}) = \\frac{1}{2\\Omega} \\left[1 - \\cos(2 \\bold{k}\\cdot\\bold{r})\\right]", "ef6ce8c26beeefe54d6b4787e0fba25b": "{\\Omega (|v|) O(\\log{n}) + O(|v|) \\over \\Omega (|v|)} = O(\\log{n}) \\ ", "ef6d5ad0b41f979bdc404f3cc3a16743": "f(a) .", "ef6d6d9f4924555e0ffbe0cf12f19077": "\\begin{matrix}\\frac18\\end{matrix} (63x^5-70x^3+15x)\\,", "ef6d79f1e2671b91fb8f763939199021": "(x,v) \\mapsto (x,v) \\in D^n_+ \\times F", "ef6d8450b77b1940a993fb187e595ef0": "\\dot{X}^a = W^a", "ef6db903edf7b9d9be8eb9d48f30a19e": "\\frac{4 - \\pi}{2} \\sigma^2", "ef6e10e25bf800fd37f82d5ad676b766": "\\zeta_S'(s) = \\sum_{n=1}^\\infty \\frac{-\\log\\lambda_n}{\\lambda_n^s},", "ef6e5094b9d01d255e18fe54ff9ca7cb": "\n\\frac{dt}{d\\tau} = \\frac{E}{\\left( 1 - \\frac{r_{s}}{r} \\right) m \\, c^2}\n\\,", "ef6f07bdc7343f27b494f752c884a50a": "\\hat{P}_\\mathbf{k}", "ef6f0d9fdf8cf8a5d5e74deb093e866a": "\\mathrm{Pb(NCO)_2 + 2NH_3 + 2H_2O \\rightarrow Pb(OH)_2 + 2NH_4(NCO)}", "ef6f45ba5933774434e77325d6749324": " HRel = \\frac{ - \\sum p_i log_2 p_i }{ log_2 K } ", "ef6f4bf476773f25d3837ec05eab4026": "\\hat{H}_4 = \\frac{ih}{8 \\pi c^2} \\sum_{i} \\frac{q_i}{m_i^2} \\mathbf{\\hat{p}}_i\\cdot\\mathbf{F}(\\mathbf{r}_i) ", "ef6f5249f14801faaea2311ff5305f35": "\nL(D) = \\mathcal{O}(\\sqrt{D}\\ln{D})\n", "ef6f69cf5a090b1869cea85b9dfec3a1": " I_8 ~,~ \\Gamma_\\text{chir} \\Gamma_{a_1 a_2} ~,~ \\Gamma_{a_1 a_2 a_3} ", "ef6fc3c8fba53af13fba3d883940e445": "G, q, g\\,", "ef6fd32a56b105666f0dfe1f291b91d8": "S = \\{(x_1,x_2,\\dots,x_n)\\in\\R^n : x_n>0, x_1^2+x_2^2+\\cdots\n+x_{n-1}^2 \\le \\exp(-1/x_n^2).", "ef6fd9731fe5ac0b94abcad2d276c377": "p\\approx p_{n+3}=p_n-\\frac{(p_{n+1}-p_n)^2}{p_{n+2}-2p_{n+1}+p_n}.", "ef7085901545d672be079d863ff3c1be": "\\operatorname{Var}[\\,\\varepsilon|X\\,] = \\sigma^2 I_n,", "ef70cea47c7d5958782ed583efccd563": "\\delta \\psi = u\\, \\delta y - v\\, \\delta x,", "ef71225e03a20c7ebc59e0051a6eb3e1": "\\frac{1}{8}\\left(4a^2 + 5b^2\\right)m", "ef71b6b4430c7f02ce095f8fbadb160a": " V = - \\frac{1}{K\\rho}\\frac{\\partial p}{\\partial s}", "ef7245a0bba9b5697b7730571eee5fcd": "\\dim K(M) = 3", "ef724faf0a2cd6479d4e306730ca3126": "t = h(n_\\max-1)+1", "ef729d57fae6fb2e4feb46ac71201f1f": "\\vartheta : \\mathbb{R} \\times \\Omega \\to \\Omega", "ef72f496cce64c4931f4484f82723877": "f_{p}>0 \\Rightarrow \\sum_{e \\in p}{l_e(f_e)} \\leq \\sum_{e \\in q}{l_e(f_e)}.", "ef72f52282593e7546722739a89fac3c": "(id \\otimes \\varepsilon \\otimes id)(\\Phi) = 1 \\otimes 1.", "ef730d0b65a4c6983c52a60a44ebf90d": "\\tan^{-1}(B(x)/A(x))", "ef731accccfd5fbbb9b4f5cc84bad9bc": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log(\\frac{\\varepsilon}{3.7D} + \\frac{95}{Re^{0.983}} - \\frac{96.82}{Re})", "ef73242550eb53e186585fed6a099bff": "[x,\\cdot] : A\\to A", "ef73453b8742d3e72acdd222d43c9265": "=X_m/(X_s+X_m)", "ef73d0fdf8c82ca2b59f5a182a36d3e0": "\n\\begin{align}\n P_1&=\\phi,\n \\\\\n P_n &= -\\frac{dP_{n-1}}{dx} + \\sum_{i=1}^{n-2}\\, P_i\\, P_{n-1-i}\n \\quad \\text{ for } n \\ge 2.\n \\end{align}\n", "ef745e9192db0748a975f399f415a044": "\\begin{align}\n\\boldsymbol{x}_i&=\\boldsymbol{x}_0+\\boldsymbol{V}_i\\boldsymbol{H}_i^{-1}(\\lVert\\boldsymbol{r}_0\\rVert_2\\boldsymbol{e}_1)\\\\\n&=\\boldsymbol{x}_0+\\boldsymbol{V}_i\\boldsymbol{U}_i^{-1}\\boldsymbol{L}_i^{-1}(\\lVert\\boldsymbol{r}_0\\rVert_2\\boldsymbol{e}_1)\\\\\n&=\\boldsymbol{x}_0+\\boldsymbol{P}_i\\boldsymbol{z}_i\n\\end{align}", "ef74621d8d4e5a39f861e4abcbb0cdff": "x_2 = \\alpha \\sin(\\chi/\\alpha) \\cosh(t/\\alpha),", "ef750330faecfaa013558c54970cedaf": "\\hat{\\alpha} >0", "ef751541bd6b79a528eca0040cf575d0": "\\alpha,\\beta\\in W", "ef7517b31f07df7d3ecd4188f8edabc2": "r=\\frac p {1+\\varepsilon\\cdot\\cos\\theta}", "ef752a11e26313862364287dbe47c54c": "\\Delta p_0 ", "ef7533d7430da0b836f8f5aadc6450d4": "{4 \\choose 1}^5 = 1,024", "ef75480261490dc51fd45aedeb93a301": "\\det (\\Lambda^a_b) = \\pm 1\\ .", "ef755d893258b2d3071bb260987f7cab": "\\langle\\!\\langle a\\rangle\\!\\rangle\\cong \\langle 1, a \\rangle \\cong x^2 + ay^2", "ef75b3233c9d59a74a312f3433688f6d": "A\\,:X\\to X", "ef75ca313bffcf6c836261b955066168": " \\sigma= - \\log_2 [R \\times \\varphi(T) \\times \\operatorname{P}(T)], ", "ef75cd04a8ea72a531c325212f04a7a9": "\\xi(q,T,W,h)", "ef76025a5463fc42f51dc5fef6e4a132": " \\frac{d}{dx}\\sin(x) = \\cos(x).", "ef76088e1ee08067fae8fd689af96697": "c^2 (s-t)^2 - (x_1-y_1)^2 - (x_2-y_2)^2 - (x_3-y_3)^2 ", "ef762f58a701c60120e007374586e21f": "\\scriptstyle c^2{\\partial B \\over \\partial z} \\,=\\, {\\partial E \\over \\partial t}", "ef765acd3d9f2003cb9941ed073e2194": "F \\otimes_k M", "ef7665dfea59bd1650d1b8e8c02aca9c": "(p,v_1,v_2,\\dots, v_n)", "ef76888bb29765401d7054c6057eb361": " 1/e^2", "ef76be4815979d6bbbc6c2510fb212cd": " f(x) = \\frac{1}{1+x^2} ", "ef76d19773aae8c1d73df30bfc50ad13": "\\mathfrak{so}_{2n}", "ef76e04de38ea71993d926d1ad009956": "\\overline{P}_-", "ef771928b84400bf5016b793211cc299": " \\operatorname{W} \\colon z \\mapsto \\frac{z-\\bold{i}}{z+\\bold{i}} . ", "ef77222b71297681fcbe07b392a0ce8e": "\\scriptstyle -\\nabla_{\\!S} \\cdot \\mathbf{\\hat{n}}", "ef7775f1e072df65cd421af8704d2fbb": "c_0=0", "ef77863c82840c5e04fc7658f31d3ffa": "\nf_t \\propto (r/r_0^2)\\left[1-(r_0/r)^6\\right].\n", "ef77c302cffd01658d80acb947a557d4": "\\sigma_{(uv)(vw)}=\\frac{2}{d}-\\delta_{uw}.\\,", "ef77fc054aac6c8e27cca8d683dd5925": "\\partial\\Omega.", "ef780d63c3051d48762b4f9f7c1df7cc": "f \\circ F_1 \\colon M' \\to X", "ef7841ebf8b70af384428bdbe1b628e4": "Thermal*lag (hr) = {1.38 * L * \\sqrt {1 \\over {\\alpha}}}", "ef786663720b64ab25f32ec33d739cfc": "S : F_B \\times C_A \\times \\mathrm{C}^{\\infty}(\\mathbf{R}, \\mathbf{R})' \\to \\mathrm{Mor}(\\mathrm{C}^{\\infty}(A, B), \\mathbf{R}) : (f, c, \\lambda) \\mapsto S(f, c, \\lambda), \\quad S(f, c, \\lambda)(m) := \\lambda(f \\circ m \\circ c)", "ef786a72cb65f0ac6ebb4c68655de6d7": "Z_{(m,n)}", "ef78a4ba4e577256a4b822b09ba6d85e": "D^\\sigma=\\sum_{P \\in C}n_P P^\\sigma=D", "ef790594391b78448a78164dc8db1477": "{\\mathbf \\Psi_{22\\cdot 1}} = {\\mathbf \\Psi}_{22} - {\\mathbf \\Psi}_{21}{\\mathbf \\Psi}_{11}^{-1}{\\mathbf \\Psi}_{12}", "ef7963dbee0978b7edb71a4e76c1a0e3": "f(c_{(1)})\\otimes f(c_{(2)})=f(c)_{(1)}\\otimes f(c)_{(2)}.", "ef79716b58007a632599fceebb9551a7": " b_i < v_i ", "ef7991aaf53340a9b5ca267bef6fcade": "1/[a_{ii}]", "ef79c2bb20f8aeb9e32faa28fe10f244": "f_k\\in L^{p_k}(\\mu)\\;\\;\\forall k\\in\\{1,\\ldots,n\\}\\implies\\prod_{k=1}^n f_k \\in L^r(\\mu).", "ef7a25329555bc077ac0c9610bbf0e19": "\\mathbf{p} = \\partial L/\\partial \\mathbf{\\dot{q}} = \\partial S /\\partial \\mathbf{q}", "ef7a4db745dbbb364fe2bb5b1a3284f7": "r_2<|z|0", "ef7ec559d35828fce0629cb8f13aeed6": " var( arcsin( \\sqrt { p } ) ) \\approx \\frac{ var( p ) }{ 4 p( 1 - p ) } = \\frac{ p( 1 - p ) }{ 4n p( 1 - p ) } = \\frac{ 1 }{ 4n } ", "ef7f15cd4512ce8fe87cfc6678943da2": "\\langle\\mid r\\mid\\rangle", "ef7f735688738269227cf2984bcaa381": "A^c \\cap B~=~B \\setminus A", "ef7fa6c8578e1803fd16ef13bd7793a0": "\\sin^3\\theta = \\frac{3 \\sin\\theta - \\sin 3\\theta}{4}\\!", "ef803c4d18f28d5babb1eb8e283aa6d4": "|\\alpha| = \\Sigma_{i=1}^N \\alpha_i", "ef805ba83a6ba811ba263def07ad8ff1": "h_i(x)=0, i \\in I={1,\\dots,p}", "ef80cffda408bbff53b1c5e91b699aab": " d[x_m(i),x_m(j)] ", "ef8127aebb1f48ae06d912b8799ac4e0": "p(y_t|x_t)", "ef8197ad111bb26b1ba2c2e33d2480ed": "x\\in [0,\\,1]", "ef81b3fae442defb839710a5940870b3": "\\log(\\sigma)\\,+\\,\\gamma\\xi\\,+\\,(\\gamma+1)", "ef81f36f22a5948f5b75d7833e42c2d4": "\\frac {1} {N_f} = \\frac {F_{pp}} {N'_{pp}} + \\frac {F_{cc}} {N'_{cc}} + \\frac {F_{pc}} {N'_{pc}} + \\frac {F_{cp}} {N'_{cp}}", "ef8209e8042bcde7f3cd01755d5ce172": "X_{u}", "ef822eca8450ae95214183a0f3afa7bd": "(\\mathbf X^\\mathrm{T} \\mathbf X) \\hat{\\boldsymbol{\\beta}} = \\mathbf X^\\mathrm{T} \\mathbf y", "ef8247f62bc4051180598dbd798c719d": "\\scriptstyle M_0,\\, M_1", "ef824aaf8ef24c1e2d8e22bde721cbb4": "\\left|e^{itz}\\right|=\\left|e^{it|z|(\\cos\\phi + i\\sin\\phi)}\\right|=\\left|e^{-t|z|\\sin\\phi + it|z|\\cos\\phi}\\right|=e^{-t|z|\\sin\\phi} \\le 1.", "ef82d13d6f45e0712ebdf8a21f3c15d8": "\\begin{matrix} \\frac{75}{49} \\end{matrix}", "ef8315b6c88a5b6efe5b1250a875ede5": "\\tau(x_i\\otimes x_j)=q_{ij}(x_j\\otimes x_i) \\, ", "ef83530ab856532f4873f511abd4c289": "\n\\begin{align}\nM_X(t) & = \\int_{-\\infty}^\\infty e^{tx} f(x)\\,dx \\\\\n& = \\int_{-\\infty}^\\infty \\left( 1+ tx + \\frac{t^2x^2}{2!} + \\cdots + \\frac{t^nx^n}{n!} + \\cdots\\right) f(x)\\,dx \\\\\n& = 1 + tm_1 + \\frac{t^2m_2}{2!} +\\cdots + \\frac{t^nm_n}{n!} +\\cdots,\n\\end{align}\n", "ef83535baf72a26436fdd226cee101d2": "p(C \\vert D)\\,", "ef83a01cb77b2c43ef7fa70802f36c16": "\nf(z) = z \\Rightarrow dz^2 + cz = a + bz\\,\n", "ef83d220f0249f8392c5679772a79a51": "\n\\rho(\\mathbf{k},\\omega) = 2\\pi\\sum_{\\alpha} \\left[ \\delta(E_\\alpha - E_0 - \\omega)\n|\\langle\\alpha |\\psi_\\mathbf{k}^\\dagger|0 \\rangle|^2\n- \\zeta \\delta(E_0 - E_{\\alpha} - \\omega)\n|\\langle0 |\\psi_\\mathbf{k}^\\dagger|\\alpha \\rangle|^2\\right]\n", "ef83da0afd9c56795c5c34c26dd8f79d": " 0\\leq n < c-1 ", "ef83edd45028b5da122b7926db7d741b": "\\ W(1-(1-\\beta)F_t)", "ef84231aa034ddd3f26190d39b0bfbf1": "V \\to V/W", "ef8446a4862ae4c9cfd5f891ebac96a4": "i\\,-1", "ef84875642ee0f03bad9eba56a04b81c": "Price_{end}", "ef84adf9efcf8569fbba2473344ea2f4": "F_{u_y}", "ef84cbdd256f01ff599aebe273706962": "[{\\,},{\\,}]\\ ", "ef853773d6440a6956b0ff4de2a16553": "\\mu^-\\,", "ef85536a540d06637cdec5ac0432eda8": " (x+1)^2 + y^2 < \\frac{1}{16} ", "ef855b179eefb4ab31c2ebb4dd735102": "\\mathbf{f}_f=\\left \\{ \n\\begin{matrix}\n-(\\mathbf{f}_e \\cdot \\mathbf{\\hat{t}})\\mathbf{\\hat{t}}\n& \n\\mathbf{v}_r \\cdot \\mathbf{\\hat{t}} = 0 & \\mathbf{f}_e \\cdot \\mathbf{\\hat{t}} \\le f_s\\\\ \n-f_d \\mathbf{\\hat{t}}\n& \n\\text{(otherwise)} \\\\ \n\\end{matrix}\\right.", "ef8585444a09efcbe7f1bcbd6b364332": "\\int\\frac{y-\\frac{2\\,b_2\\,a + b_1}{2\\,b_2}}{y^2 + \\alpha^2} \\,\\mathrm{d}y = \\frac{1}{2} \\ln(y^2 + \\alpha^2) - \\frac{2\\,b_2\\,a + b_1}{2\\,b_2\\,\\alpha}\\arctan\\left(\\frac{y}{\\alpha}\\right) + C_0", "ef85c9ae886c2e0035d40b0e46f21ce5": "n|\\mathbf{X}^n|\\mathbf{X}^{-1}", "ef86677c9451d7f8e085af229c2bd961": "496\\mathfrak{u}(1)", "ef87332fbadc1d500aeaf9af8680d8f4": "\\lambda_{\\mathrm{max}} T = constant", "ef8737f589f3ff1e63a32945007fd003": "\\varphi(A) = \\infty ", "ef87c52fcf9c230a9418d4a26e6338a2": "\\int_S g \\,dP_n \\quad\\xrightarrow[n\\to\\infty]{}\\quad \\int_S g \\,dP,", "ef87de3a522d36db92e3977406308eb3": "P(n,t)=P_1(n,t)", "ef88271b7b140b5363e5d8925f4227ef": "Y_2,\\dots,Y_n", "ef88566598d8feea78babfcb4693e569": " \\acute{{R}^{\\mu}}_{\\alpha \\nu \\beta } ", "ef8856da9a1e27439c63bb3c0d185380": " x=v, y=u, z=uv\\,", "ef8865f6e97b1f942ba13021e6302cb4": "\\lambda_1", "ef88a9ce8f965e27e6d1e13fb060be13": "\n\\mathbf{ v_E} = \\frac{\\mathbf{ E}\\times \\mathbf{ B}}{cB^2} = \\frac{-\\nabla\\phi\\times\\mathbf{\\hat z}}{cB}.\n", "ef88b7038eb73a5497917b14e4b0e80f": "-i e^{-i\\alpha \\pi /2} \\mathbf{H}_\\alpha(ix)", "ef8992311c5d3e48c7101b38b55db184": " F_q = \\mathbf{F}_A \\cdot \\frac{\\partial\\mathbf{v}_A}{\\partial\\dot{q}} - \\mathbf{F}_B \\cdot \\frac{\\partial\\mathbf{v}_B}{\\partial\\dot{q}}=0.", "ef89b0891739b3b690864e2fa532b03e": "P_1(I)", "ef8a2643ea636fd0669479767ed04a6b": "\\Pr[Xx]=(1+x)^{-3}/2\\mbox{ for positive }x.", "ef8a42713a5df4bac807f88216300a58": " \\textrm{Current~Index}= \\frac{\\sum\\textrm{[}\\textrm{P(t)}\\times\\textrm{IS}\\times\\textrm{FAF}\\times\\textrm{CF}\\textrm{]}}{\\sum\\textrm{[}\\textrm{P(t-1)}\\times\\textrm{IS}\\times\\textrm{FAF}\\times\\textrm{CF}\\textrm{]}} \\times \\textrm{Yesterday's~Closing~Index}.", "ef8a49094f16cbf98175e15f0546090c": "\\mathbf{B}_{\\mathbf{P}_0\\mathbf{P}_1\\ldots\\mathbf{P}_n}", "ef8a85f711490cdb0519c7efa898a2f0": "|i", "ef8a9f751393cecaf3e811c30ee3e756": "VR", "ef8aac4184571b130a64e5d8e3824057": "-\\tfrac12P(S \\rightarrow S') ", "ef8b6b58f2ddf77f787885d21d8e97ad": "F(\\vec{k},t) = \\frac{1}{N}\\langle \\rho_{\\vec{k}}(t)\\rho_{-\\vec{k}} \\rangle", "ef8b9d190437f3c28718a1d7b2c01b3c": " \\sin\\left(a + b\\right) = \\sin a\\cos b + \\cos a\\sin b ", "ef8bbc726af10ec357f5b96cac910353": "\\langle m|H(x^\\mu)|n \\rangle", "ef8c7a5ee99984da5131514956b5d50c": "\n\\Sigma_{\\bar{\\mathbf{x}}} = \\left(\\sum_{i=1}^n \\Sigma_i^{-1}\\right)^{-1},\n", "ef8c97f867bad91e68047043b0608da6": "t^\\prime = t - vx/c^2", "ef8cd64980b7eff83d76d481e03f8b6f": " \\epsilon_{ijk}", "ef8d1c3b9a75f6a3c227e8ae72c46c68": "t_4=1.1 \\colon", "ef8db21159df0d55ca392a23628b8bbc": "|O_{\\uparrow}\\rangle", "ef8df5ac1730290f8fe8a478e35d54ff": "L_k = 2 \\sin \\left [\\frac {(2k-1)}{2n} \\pi \\right ]\\qquad\\mathrm{k = even}.", "ef8dfb0173bf02cf769ca8011ad483b2": "x_n \\to -\\infty", "ef8e1a909f90e8039c54c61c00184b6d": "\\text{id}_S", "ef8e27513969432240b71a573c3d67c7": "\n\\begin{matrix}\n& & \\mathbf{\\Sigma}^0_1 & & & & \\mathbf{\\Sigma}^0_2 & & \\cdots \\\\\n& \\nearrow & & \\searrow & & \\nearrow \\\\\n\\mathbf{\\Delta}^0_1 & & & & \\mathbf{\\Delta}^0_2 & & & & \\cdots \\\\\n& \\searrow & & \\nearrow & & \\searrow \\\\\n& & \\mathbf{\\Pi}^0_1 & & & & \\mathbf{\\Pi}^0_2 & & \\cdots \n\\end{matrix}\\begin{matrix}\n& & \\mathbf{\\Sigma}^0_\\alpha & & & \\cdots \\\\\n& \\nearrow & & \\searrow \\\\\n\\quad \\mathbf{\\Delta}^0_\\alpha & & & & \\mathbf{\\Delta}^0_{\\alpha + 1} & \\cdots \\\\\n& \\searrow & & \\nearrow \\\\\n& & \\mathbf{\\Pi}^0_\\alpha & & & \\cdots \n\\end{matrix}\n", "ef8e4da1505cda3cd70bc597251bb82d": "\\int_0^1 x\\ln(1+x)\\ln(1-x)\\,dx=\\frac{1}{4}-\\ln 2.", "ef8e6dda27b10876fd16f9dce16101fe": "\\phi \\colon U \\to V", "ef8ea9826121f75fba691c06c46d0d18": "d(x_n, Y_{n-1}) > k ", "ef8edcdc12fa74a154e1930c2ba47861": "I_{base} = \\frac{S_{base}}{V_{base}} = 1 pu", "ef8ee24607238204bab462f73c404cf6": "2W(q)=\\frac{3}{2}\\frac{\\hbar^2 q^2}{M\\hbar\\omega_{\\rm D}}\\left[2\\left(\\frac{k_{\\rm B}T}{\\hbar\\omega_{\\rm D}}\\right)D_1\\left(\\frac{\\hbar\\omega_{\\rm D}}{k_{\\rm B}T}\\right)+\\frac{1}{2}\\right]", "ef8f0440917cde226dd0a02757d4827e": "L(x, t) = \\sum_{n=-M}^{M} f(x-n) \\, T(n, t)", "ef8f26854ebadbb52e81ed4396af35e5": "\\mathrm{EE}^*", "ef8f3875e2f888733c825eecb3e8198b": "\\scriptstyle -\\infty", "ef8f498707b7bfb4223d9a4e49ca906c": " \\mathbb{A}\\ \\dot{\\xi} \\le 0 \\,.", "ef8f6c076bc5a1f207ed5a8cfe0ef4c7": "\\;^\\pm T^{IJ} = (P^{(\\pm)} T)^{IJ}.", "ef8f9f5f1dc194a7ead922b84d98e9e6": " \\mathbb{Z}_p", "ef8ffabe7cf73daef9b85dcedefa12bc": "\\pi_n < 0", "ef8ffd530c3cd097c5b71140b325753b": "V=\\frac{d\\log P}{d\\log r}=(n+1)\\frac{\\phi}{\\xi\\theta}", "ef90313da6f878a97a0d87a4f042821f": "f_c = \\frac{F_s}{2\\pi\\sigma}", "ef904257404d178eba7d134c63bb18b2": "{}^{2}i = i^{\\left({}^{1}i\\right)}", "ef9043bf12751035029613d9ec3bba18": " h_{BPF}(t) = 2B_H \\, \\mathrm{sinc}\\left(2B_H t\\right) - 2B_L \\, \\mathrm{sinc}\\left(2B_L t\\right)", "ef904c9c0877f4dae6fe6c87ad7bf8cf": "K=0", "ef9056270c7161f03fc160b892bb4b5b": "R_\\mathrm{KW} = \\frac{A}{\\gamma^2}", "ef90b3069aeb27548df0adafd305501b": "\\scriptstyle{u_{\\min}^{(s)}}", "ef912d5ba482a8bc8538aba35a09f2a6": "\\int x^n (\\ln x)^n\\; dx\n= \\frac{x^{n+1}}{n+1}\n \\cdot \\sum_{i=0}^n (-1)^i \\frac{(n)_i}{(n+1)^i} (\\ln x)^{n-i}.", "ef912fbf56d4f0bb7552aac8c8a7fd79": "\\mathbf{q}(t)", "ef91c100b19b0546e366f63eca5bea0f": " \\vartheta(KG_{5,2}) = 4 ", "ef91e40159689f58c0625f7837b2ed00": "\\frac{d}{d t} \\left( \\frac{d t}{d \\tau} \\right) \\approx 0 ", "ef91f441f97fefd8760916074584c540": "\\frac{1}{2} n (n+1)", "ef923d250622c6e41a78fc6117f9e961": "\\mathbb{C}[x, y]", "ef924743d910e5e29a5fe3a562303dc5": "\\arcsin(1-x)", "ef926a864996df99513f478cb35e7ff2": "\\scriptstyle F:K\\rightarrow\\mathbb{R}^n", "ef9277fa05bd7edd763bb80713ad9af8": "\\frac{d^2 \\theta}{d t^2} + \\pi - \\theta = 0\\,", "ef92f69049cc4a68f1caeb25db508f30": " n = 1", "ef93166f850d2a87e25437d171cbe587": "\\mathbf{B} = \\nabla \\times \\mathbf{A}.", "ef93196eaa338ed1f867004d29505778": "\\begin{align}2x - 2x - y & = 1 - 2x \\\\\n- y & = 1 - 2x\n\\end{align}", "ef9349f6fd58f4686b6de515e10a0b1c": "1 \\le\\dim(E_{\\lambda})\\le m", "ef937378646558e3c393f3b3ad48f5dd": "\\mu (F) < \\infty", "ef937f42696f286e56e25e37611a07b3": "\\text{Pa}\\sqrt{\\rm{m}}", "ef938ac6fd6fbf63f2552f448663fb8b": "U = -\\frac{G M m}{r},", "ef93e35d4fca3f205a309820c44a2031": "\\alpha = \\frac{1}{\\sum_i \\sigma_i^2}\\sigma_1^2", "ef9426438fe24b3ad9b781f48a41c124": "\\Omega = g_0\\ln(1-z)-\\frac{\\textrm{Li}_{\\alpha+1}(z)}{\\left(\\beta E_c\\right)^\\alpha}", "ef951fc8fda070ba7af0dcd1b7769ac5": " \\frac{F}{l}x(t) ", "ef95575903da6e42ca52e36c65af9b2d": "I_n = \\int \\frac{\\sqrt{ax+b}}{x^n}dx\\,\\!", "ef95be725c52ac0742b287d7c985eb02": "y_i = \\begin{cases} v_i & \\mbox{ if } x_i = 0 \\\\ 1+v_i & \\mbox{ if } x_i = 1 \\\\ \\end{cases} ", "ef95f71de1b000d4cc50e4d6e5ef2c24": "\\mathbf{P} (\\mathcal{E})\\times_S T", "ef965544cf7100e5d6f09e38c5412ac1": "\\Delta\\Phi=0.", "ef965a059173535a0e55643e96217068": "g=g_\\ast", "ef96ea1b5c8993a93d5cdbcae1433a44": "ZFC\\vdash\\forall T((T\\vdash\\lnot H)\\rightarrow(ZFC\\vdash(T\\vdash\\lnot H))).", "ef9742e0e628824d334f31daae949dfd": "\\begin{align} \n y_1 &= a_0 \\sum_{r = 0}^\\infty \\frac{(\\alpha)_r (\\beta)_r}{(1)_r (1)_r} x^r = a_0 {{}_2 F_1}(\\alpha, \\beta; 1; x) \\\\ \n y_2 &= \\left.\\frac{\\partial y}{\\partial c}\\right|_{c = 0}.\n\\end{align}", "ef976e637ec47070f6cbf9a303d88308": "\\phi_i(v)=\\sum_{S \\subseteq N \\setminus\n\\{i\\}} \\frac{|S|!\\; (n-|S|-1)!}{n!}(v(S\\cup\\{i\\})-v(S))", "ef97738531547b02d5bac7c2049ed22f": "-r < x < r", "ef97b537255df8f8b01062f9e712fbba": "r = \\lfloor 0.5 + \\mu_{2,l} \\rfloor =4", "ef97b682d9dd9121b20b9990444106af": " \\Gamma(\\Gamma(4)) - \\frac{4!+4}{4} ", "ef97ebc3d1807ff5113d96ab0eb16c04": "\\Delta f = \\frac{1}{\\sqrt{|g|}} \\partial_i \\sqrt{|g|}\\partial^i f = \\partial_i \\partial^i f", "ef9824d98689366b0f84520a596c73e4": "\\Omega^i_{\\ j}=d\\omega^i_{\\ j} +\\sum_k \\omega^i_{\\ k}\\wedge\\omega^k_{\\ j}.", "ef983673f10bae3364c047eaf9334741": " n(~r^{-n-1}~\\sin(n\\theta) \\,", "ef983a3c6f6f6486e41c7fdaeccaa5d6": "\\mathbf{q}_{2} = \\mathbf{q}(t_{2})", "ef984f70b39589db3b39a977a40dbb53": " T_{ab} = \\frac{2}{\\sqrt{-h}} \\frac{\\delta S}{\\delta h^{ab}} ", "ef985201f563c531ca938d58067229f2": "S= \\oint \\nabla S \\cdot d\\mathbf{s}=0", "ef991d0a950636deb3c88e2c116124e1": "215^2-17 = 2 \\times 152^2\\, ", "ef99498e79d6fd0ef9a74a209d81e48b": "x_{2m+1}", "ef9967b2b91cb8ebae4b4a78ebde2549": "\n\\begin{align}\nm_p&=\\int_0^{\\pi/2}\\!M(\\varphi)\\,d\\varphi\\\\\n&=aE(e)=a\\int_0^{\\pi/2} \\sqrt{1 - e^2 \\sin^2 \\theta}\\,d\\theta\\\\\n&\\approx\\frac\\pi2\\left[\\frac{a^{3/2}+b^{3/2}}{2}\\right]^{2/3}\\,\\!.\n\\end{align}\n", "ef99ce12c8c0d4a6bc1a183176aa1a40": "(p+1)", "ef9a14f8af586b778d5a1cbaf60ff041": " P^2=\\frac{3\\pi}{G}\\frac{V}{M}\\approx 10.896 {\\rm\\ hr}^2 {\\rm g\\ }{\\rm cm}^{-3}\\frac{V}{M}.", "ef9aa09fe2aad575e8ac546936cbe34d": "ax^2 = bx + c", "ef9abc7898fe22e4f00ae258d2dad452": "\\sum_{n=0}^\\infty (-1)^nf(n)= \\frac {1}{2} f(0)+i \\int_0^\\infty \\frac{f(i t)-f(-i t)}{2\\sinh(\\pi t)} \\, dt.", "ef9ac3c436a0788af5dfc9adfc287ba5": "\n\\begin{align}\n8^3 + 4^3 &=& 512 + 64 &=& 576\\\\\n5^3 + 7^3 + 6^3 &=& 125 + 343 + 216 &=& 684\\\\\n6^3 + 8^3 + 4^3 &=& 216 + 512 + 64 &=& 792\\\\\n7^3 + 9^3 + 2^3 &=& 343 + 729 + 8 &=& 1080\\\\\n1^3 + 0^3 + 8^3 + 0^3 &=& 1 + 0 + 512 + 0 &=& 513\\\\\n5^3 + 1^3 + 3^3 &=& 125 + 1 + 27 &=& 153\\\\\n1^3 + 5^3 + 3^3 &=& 1 + 125 + 27 &=& 153\n\\end{align}\n", "ef9ae6e101237879d322ce691d3ba4c1": "\nu(\\mathbf{r}) = \\int \\mathrm{d}^3r' G(\\mathbf{r} - \\mathbf{r}') f(\\mathbf{r}')\n= \\int \\mathrm{d}^3r' \\frac{e^{- \\lambda |\\mathbf{r} - \\mathbf{r}'|}}{4\\pi |\\mathbf{r} - \\mathbf{r}'|} f(\\mathbf{r}').\n", "ef9b04780fff7368f0048d41574bc358": "\\sum_{k=1}^n k^2 z^k = z\\frac{1+z-(n+1)^2z^n+(2n^2+2n-1)z^{n+1}-n^2z^{n+2}}{(1-z)^3} \\,\\!", "ef9b336189c9e453288dbd475a0bd453": "t = w + x i + y j + z k, \\quad w, x, y, z \\in \\mathbb{R}", "ef9b6583e57e3c1c42995a53f4527a27": "\\mathfrak{sl}_n(\\mathbb{C})", "ef9b89cd9eb205b0e59bb752afcf49ab": "\\omega_N^2", "ef9bda5bbdaeac11556752fdaebc3626": "(g^mh^r)^{p-1} = (g^m g^{nr})^{p-1} = (g^{p-1})^m g^{p(p-1)rpq} = (g^{p-1})^m \\mod p^2.", "ef9c415e1482110e1f32eeb641cca7fe": "\\operatorname{PerfMatch}(G) = \\sum_{M \\in PM(|V|)} \\prod_{(i,j) \\in M} A_{i j}.", "ef9c4fd581b75da6074d5a773482bfec": "v = (\\sqrt{2}G_{\\rm F})^{-1/2} \\simeq 246.22 \\; \\textrm{GeV}", "ef9cb554ed75ef364dd90c3ed7559ec7": "\\oint \\frac{\\delta Q}{T} \\leq 0.", "ef9d09ba9591b2d7aba862e755af098b": "t(v_i) \\mapsto \\emptyset", "ef9d0b4162beba658d6d14d7ba1b4d1f": "x \\in [a, b] \\iff \\frac{1}{x} \\in \\left [\\frac{1}{b}, \\frac{1}{a}\\right ]", "ef9d2719365ff8c7bb8595d13098809c": "\\boldsymbol\\Delta \\boldsymbol\\beta", "ef9da670346c0bfcb71ec0981432643f": "Ax = \\lambda x", "ef9e0125ddeb55543aede7e35169f825": " k_\\mathrm{m} = \\frac {\\mu_0}{ 4 \\pi} \\ ", "ef9e2fbaf5b81e181f4db36fb3aea782": "\nH = -\\sum_{\\langle ij\\rangle}J_{ij}S_{i}S_{j},\n", "ef9f31bbcfd598497da1800777ccd633": "\\varepsilon i", "ef9f3ef9db6f5d5038a9f4c545ba657f": " \\pi \\approx \\tfrac{355}{113} ", "ef9f944006b3ce02688e9ce33f916f08": "\n\\gcd(b_1,d_1) = a_1, \n", "ef9fcdb53e4e10b12bfcd5e9e78135dc": "!!", "ef9fce9a44b787f1dd85e7fd7f0b5005": "n=\\frac{c}{v}", "efa0411ba68041f0847d7fd212c6ef62": "\\mu^-(\\cdot)=-\\underline{\\mathrm{W}}(\\mu,\\cdot)\\,,", "efa0702d1f43a10bd64fdd49be400ec2": "\\Delta v = c \\cdot \\tanh \\left(\\frac {v_e}{c} \\ln \\frac{m_0}{m_1} \\right)", "efa0806eb35f00baf700a82efb41143c": "\np(x_1,\\ldots,x_n) = \n\\left(\n\\prod^{n-1}_{i=1} \\frac{1}{\\sqrt{2\\pi\\sigma(x_i)^2\\Delta t_i}}\n\\right)\n\\exp\\left( \n- \\sum^{n-1}_{i=1} L\\left(x_i,\\frac{x_{i+1}-x_i}{\\Delta t_i}\\right) \\, \\Delta t_i\n\\right)\n", "efa0884a52abbe4450382d08cc836bac": " \\prod_{p} \\Big(1 - \\frac{3p-2}{p^3}\\Big) = 0.286747... ", "efa0cadc8a135d84a1d6feebee385e38": "C_i = \n\\begin{bmatrix}\na_i & b_i \\\\\n-b_i & a_i \\\\ \n\\end{bmatrix}", "efa0d0003d8274eda6256b4f8dd01efc": " \\text {m} ", "efa108495567f0b9fd41f8a529911017": "[N_i,p_j]=A\\epsilon_{ij}+Bp_ip_j+C\\Delta^\\epsilon_{ijk}N_k", "efa13468ca994398f3d0015af5f0d525": " e^{T} = 1 + T + \\frac{1}{2!}T^2 + \\cdots = 1 + T_1 + T_2 + \\frac{1}{2}T_1^2 + T_1T_2 + \\frac{1}{2}T_2^2 + \\cdots ", "efa143ad7440f35e10a32bcbbb4b7040": "M_n=M(n,b;z) \\,", "efa151d7dfc92027aebc1bab62a472d1": "P \\succ 0", "efa228acd153ddec6d52a74728bbdce5": "s^2 = 529^2 + 224^2 = 279,841 + 50,176 = 330,017", "efa26ac48650bc0d9a355dfea5b87ad3": "\\operatorname{Bi}(x)", "efa29f8011f067017b7177c4a6aa3343": "E(k,\\lambda) \\approx -\\frac{1}\\lambda \\ln \\prod_{i=1}^k U_{i}", "efa3746274d96cc60dc9784dc471545f": "U =\n[0 \\dots n) = \\{0, 1, \\dots, n - 1\\}", "efa3cff54c594059f3ad5f04ae564b2b": "\\sigma_{\\rm a}", "efa3d057919d60ac2eada975aba0b48f": "\\!V = \\pi \\left[r^2x - \\frac{x^3}{3} \\right]_{-r}^{r} = \\pi \\left(r^3 - \\frac{r^3}{3} \\right) - \\pi \\left(-r^3 + \\frac{r^3}{3} \\right) = \\frac{4}{3}\\pi r^3.", "efa3df7dfb2f45a2dc7e1821fcfa9aae": "s = R \\gamma \\,;", "efa3e925c9c314e3c8256234f76a93b3": "\\mathcal{F}=\\Sigma", "efa3ea08fe32d99a7f25f48ad23283c7": "f(\\boldsymbol{S}) = f_1(f_2(\\boldsymbol{S}))", "efa4dbeabee7a4287675a53cad9b7dc5": "w_0 \\,", "efa53b0357ae1f3f3e4be9ec86fe2bf3": " \\widehat{\\varepsilon} = \\varepsilon' + i \\frac{\\sigma}{\\omega} ", "efa570d38ce85a4ddfcc7c7dc06b881b": " G_1 ", "efa5d85277df00f62eb03c1099f29508": "d' \\circ (c')^{-1} = (d \\circ \\phi) \\circ (c \\circ \\phi)^{-1} = d \\circ (\\phi \\circ \\phi^{-1}) \\circ c^{-1} = d \\circ c^{-1}\\;", "efa5e0bb4724e6262e824b2e19f2db29": "b = 21", "efa5ec45e604a8da0e982e4f80b24a7d": "0\\le x,y\\le 1", "efa63b8e77d2d08b86514beb83e320da": "x_i f(|x|) \\,", "efa6690d0669870ab94c4fb0336a7ba7": "u+v=t\\,", "efa673a56d83f0f05e85e05bd7d2e4f0": "\\nu(H) \\le \\tau(H)", "efa6b823f53a87cb8524aa177a3d8ec7": "h_{d,s}", "efa73f47004d865418cef14ebbea561d": "(0.1 \\cdot 2 \\cdot 1 + 0.2808 \\cdot 0.1 \\cdot 2 \\cdot 1) \\cdot (1+0.0961) = 0.2808 \\cdot 1", "efa74b9590ee3fce4f23fb76b56afb05": "f \\circ -", "efa7aa5b98deb6326020dc9ef8b6cfd4": "\\gcd(p-1,e)= 1", "efa7b2bf6189423dc14969a7238caa74": "2x(x^2+y^2)=a(3x^2-y^2)\\!", "efa84d79f28c9c2eb52cda84afd87b87": "E[\\hat{g}_N(x)]=g(x)", "efa86810e80dc80d799ec7cf126b2ead": "F\\uparrow", "efa8ce400a8d9679147c3ee2d5c9d496": "\\scriptstyle w\\colon A \\to {\\Bbb R}^+", "efa920b5f17bf4d3487dcd84ff132230": "I_x(q_1) V_x + I_y (q_1) V_y = -I_t(q_1)", "efa928a6b77d7c996b87bb4e5ba9f6c0": "\\mathbf{P}^{n-1} \\cong PO(n)/PO(n-1)", "efa99bf8256a2503c878b2700cf0e155": " \\begin{cases}\n0 & n=0 \\\\\n\\frac{(-1)^n}{n} & \\mbox{elsewhere}\n\\end{cases}", "efa9b95d0d2965d5e9e38a4e314c5781": "d\\mathbf{f}", "efa9bc718f8dbe009965a518e1608b12": "\nq_{xx} = \\frac{\\sum (x-\\bar{x})^2 I(x,y)}{\\sum I(x,y)}\n", "efa9d9049ca123b59d76ff5b1612428f": " \\frac{dx(t)}{dt} \\,\\!", "efaa44386cf26dcd38b7dc38450392d5": "\n\\frac{\\partial y}{\\partial \\mathbf{X}} =\n\\begin{bmatrix}\n\\frac{\\partial y}{\\partial x_{11}} & \\frac{\\partial y}{\\partial x_{12}} & \\cdots & \\frac{\\partial y}{\\partial x_{1q}}\\\\\n\\frac{\\partial y}{\\partial x_{21}} & \\frac{\\partial y}{\\partial x_{22}} & \\cdots & \\frac{\\partial y}{\\partial x_{2q}}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial y}{\\partial x_{p1}} & \\frac{\\partial y}{\\partial x_{p2}} & \\cdots & \\frac{\\partial y}{\\partial x_{pq}}\\\\\n\\end{bmatrix}.\n", "efaa784d6aa64dc0d707df31ea2bbf31": "h = \\left(q + \\left\\lfloor\\frac{(m+1)26}{10}\\right\\rfloor + Y + \\left\\lfloor\\frac{Y}{4}\\right\\rfloor + 5\\right) \\mod 7,", "efaad4fb9215b5658874fc46d0888b4b": "l_{\\rm turb} \\approx H = c_{\\rm s}/\\Omega", "efab420325dd689c54ee79097702f1e3": " f(u)=\\sum_{\\lambda} \\tilde{f}(\\lambda) {\\chi_\\lambda(u)\\over d(\\lambda)} d(\\lambda)^2.", "efabba628446a8221297a7fe00ebeef6": "(s,t)=(55,16).", "efabe0e4c12353bc1f40ca60de55142e": " K_z(x) = e^{-zH} \\, .", "efac1f75e97a2e2197624820be8d1c05": "V_1 = V_{LN}\\angle 0^\\circ ", "efac503a8403de35733a69f5df1a5188": "\\varphi_{ij}=V_{ij}-\\frac{1}{3}\\delta_{ij}\\nabla^2V,", "efac7ab375a240f85e39992a08c6b608": "b_n=n!+1\\, .", "efac8cfef3170618ed8bdd252aba7bc6": "X_{i}=\\sum_{j=1}^{k}{U_{ij}}", "efacd7e5c60bc6e684c3d30b38c0e813": "\\alpha:D(V)\\rightarrow D(V) :W\\mapsto W^{\\perp}.", "efacedc9aeb49d084ccbaa5f6a1be064": "J_n(x) = \\frac{1}{\\pi} \\int_0^\\pi \\cos (n \\tau - x \\sin(\\tau)) \\,d\\tau.", "efad141548aa92370f4a6bbf08514120": "\\frac{dy}{dt}=\\frac{h\\frac{dh}{dt}-x\\frac{dx}{dt}}{y}.", "efad46e6b7894113a1d74a681a1c057d": "A_0(X) + A_1(X)f(X) + A_2(X)f(\\gamma X) + \\cdots + A_s(X)f(\\gamma^{s-1}X)=0", "efad7f11e5e9a47be3664ba17495a453": " a=p_1 p_2^{k_2}...p_l^{k_l} ", "efadb693f907b63cc0eb0b88fc445494": " \\beta \\nabla S - \\alpha \\nabla \\theta ", "efae01e7e3069058b8fe5bf9ce70d70a": "M_{i-1}", "efae62fe3aeae5b09b93d2c91259cae6": " \\Omega = \\phi_{ab} + \\phi_{bc} + \\phi_{ac} - \\pi ", "efaeddfd9ce08043df55ce5684b5d4cb": "|\\psi\\rangle=\\sum_\\nu|\\nu\\rangle\\langle \\nu|\\psi\\rangle", "efaeef3123f38428ba89dbcc30949ed2": "\\hat{m} \\hat{n} = \\hat{1}", "efaefd5a227ada2e05a13fc56cf9093a": "\\langle \\beta_{1,i}\\rangle", "efaf22a89b51b7ebb7736440ea1f57f8": "\\eta(8) = {{127\\pi^8} \\over 1209600} \\approx 0.99623300", "efaf97fa691bfbc6fafd89daf2920545": "\\frac{d\\ln K}{dT} = \\frac{\\Delta U}{RT^{2}}", "efafc251b4506e177d45ab119be65bcf": "\\lambda(L)", "efb02444645dbfacc596231ec21395cc": "E_e^+=Y", "efb036c2d4a531f1c014db2927a1e4e0": "k \\leftarrow ESHash(k_2,L_B(P),s,u_1,\\tilde{h_1},\\tilde{h_2}) \\in W^8", "efb06b89142ff2dc68b698788556fe89": "r^{-4}", "efb0779108b65d05b63f0bc2f60e89f2": "\\left|\\frac{x}{a}\\right|^n\\! + \\left|\\frac{y}{b}\\right|^n\\! = 1", "efb07bbad25e33cf1ff783de7ecc0ac5": "h = \\psi + z \\,", "efb090b65345087a5094fcc8b2e1ace6": "\\Delta_{\\sigma}(\\tau)", "efb1db742c054d434d2326426455f197": "\\ \\rho", "efb1e35bcd26095a29c4ce6e610041f8": "1 + 1 = 2 ", "efb227a963b2cf9e83bde853a7164c23": " \\lambda_1, ... , \\lambda_m ", "efb231d6e015fd2bf97a1e7fb8e4299f": "^n{\\mathbf{C}}_i ", "efb2f4c12c00133f3e25be16ed9a2abb": "\\Gamma_4 = 0", "efb3033994e41ee12c2262f701aa001d": "\\hat{H}'_0= \\alpha \\cdot p \\left(\\cos 2\\theta - \\frac{m}{|p|} \\sin 2\\theta \\right) + \\beta (m \\cos 2\\theta + |p| \\sin 2\\theta)", "efb36d8d64af8e5270c27eb51d246ee4": "P_b = Q\\left(\\sqrt{\\frac{2E_b}{N_0}}\\right)", "efb3d0b0530f47d558275575b7213a6b": "\\scriptstyle i\\sqrt{5}", "efb43c786c4358b59cc75268e9617b58": "T_n(1)=1\\,", "efb4423f78430fc6a50cd880d111e9ba": "u_B", "efb4ba983c18138ba6c8fdd295129d51": "{z+k \\choose k}\\approx \\frac{e^{z(H_k-\\gamma)}}{\\Gamma(z+1)}", "efb56ceac5362935b3e0b8b96d17ca61": "r^{n+1}", "efb5a69bd773a637d1ab1c93ba237c80": " cl(Y) ", "efb5d30201c712aa84d4b82fbd7986e2": "s \\in \\alpha, t \\in \\gamma", "efb5d75d3458083d4910bb8c4bf611c9": " c\\cdot n^{-1},", "efb5df92739616f69ffcd57a0c2020ab": "\\{0, 0, 0, \\pi/2, \\pi/2, \\pi, \\pi, 3\\pi/2, 3\\pi/2, 2\\pi, 2\\pi, 2\\pi\\}\\,", "efb5ff21b92f24533a50e95ba492e83b": "3^{(F_n-1)/2}\\equiv-1\\pmod{F_n}", "efb620b878897085f63b8547c71b3513": "\\ -(f(r_k)/p^{k})/f'(r_k) ", "efb6d61ddd13c621e46d606a1dd6bbee": "(\\alpha \\and \\exist Fx_2...x_n) \\rightarrow \\beta ", "efb6efab7d18bb477999c5821cb5766c": "\\zeta(s) = 1 + \\frac{1}{2^s} + \\frac{1}{3^s} + \\frac{1}{4^s} +...........", "efb771f3ca30d6f17647b6653cec59fa": " \\frac { d \\hat u}{dt}=-(ick + vk^2)\\hat u ", "efb7eba8b6852bc52dcb3d743e971fd6": "\\Theta=n/2", "efb89fe51df6ac6b2b32467c0081a498": " \\left(\\mathbf{E}\\cdot\\nabla\\right)\\mathbf{E}=\\nabla\\left(\\frac{1}{2}E^2\\right)-\\mathbf{E}\\times\\left(\\nabla\\times\\mathbf{E}\\right) ", "efb8cc7fa6eb6e90b218556ffaf46cc3": "\\mathcal{H}=p\\dot{x}-\\mathcal{L}+\\lambda _{i}\\phi _{i}", "efb922e688dcf2483ecf3ac9c96e8e44": "H(2r-H)=\\left(\\frac{W}{2}\\right)^2", "efb983d0010354fd0bf6c4d07e580370": " y_i = \\alpha+\\beta x_i + \\varepsilon_i ", "efb9bc02d805522dc8baab05212e0f00": "B = L/(4 \\pi R^2)", "efbadaa534d4a5e8959d7ca489b5d291": "\\,\\psi_{\\alpha}", "efbae72a5e8b4ec56a3fc3c422e9bab1": "\\pi_G = \\pi_C = {\\pi_{GC}\\over 2} ", "efbb9207a5c9e647806fe87aa3a7dc3e": " A_{2} ", "efbbc02bf1e509d61f754806c03cd5a4": "\\lambda(n_i) > (\\ln n_i)^{c\\ln\\ln\\ln n_i}", "efbbfb333220a4298d02333c96ee4fe7": "\\epsilon ^2 = 0", "efbc5705adb2455ce92259e864c2c366": " \\varepsilon_i - \\varepsilon_a ", "efbcd4c8ad589db6a0d8d7887131de5a": "\\frac1{B(z)} = \\frac1{ 1 - g_p z^{-T} }", "efbcea1e74819688fb4773ef8f46087a": " q = 0 ", "efbd0b8e460de20f84328fb1b1e15656": "\\frac{}{}\\pi/2", "efbd14e4f25df33f612287d4a32ccf10": "\\scriptstyle x^n \\,\\pm\\, y^n", "efbd4f9f341686f48e6074b228389c1d": "z_{1:t}", "efbd751cea207b9227794e7926078d86": "S(\\widehat{g})=\\Lambda \\int_M \\left( R(g) - \\frac{1}{\\Lambda^2} \\vert F \\vert^2 \\right) \\;\\mbox{vol}(g) ", "efbda584e6a97caee0c70dfe6dbf7964": "\\lambda_q=2\\cos(\\pi/q). \\,", "efbdb34108185b0a2bb8dd4b491ed969": "W_1 \\sim W_2", "efbdd7deff927df24e0c0e9b47ddf7c3": "\\frac{\\partial \\Psi(x,y,z,X,Y,Z)}{\\partial t} = i[\\frac{1}{2M}\\nabla_{c.o.m.}^2 +\\frac{1}{2\\mu}\\nabla_{rel}^2 +V(x,y,z)]\\Psi", "efbdda062e8f2c35b678ff2f70f32078": "\\mathbb{E}\\left[ \\exp\\left\\{ -\\int_0^T \\sum_{d=1}^D \\theta_d(t)dW_d(t) - \\frac{1}{2}\\int_0^T \\sum_{d=1}^D |\\theta_d(t)|^2 dt \\right\\} \\right] = 1 ", "efbde1a173bcc94afa39e47544249271": "\\alpha \\equiv \\frac{\\pi}{2\\, K(m)}.", "efbdeb4ef8d49a2d9a7af05704cbb6d5": "\\sum_x q_x \\sum_a p_a c(a, x) \\leq C", "efbdfc5281e12e2f76e11d9c82b35705": "\\begin{align}\nx+y+z&=1\\\\\nx+y+2z&=3\n\\end{align}", "efbe136e1c65cc92e8d64b7258d4a79f": " (x_0 \\otimes \\cdots \\otimes x_p) \\circ (x_{p+1} \\otimes \\cdots \\otimes x_{p+q}) =\nx_0 \\sum_{(p,q)} (x_1,\\ldots,x_{p+q}) , ", "efbe176fc29293bf18c95def305df590": "q^{-1} = \\frac{q^*}{\\lVert q\\rVert^2}.", "efbe65e9078217842039a207c3294cce": "T^{\\mathrm{D}}_p", "efbe7f10f50eaa814bedfb8a2084a142": "\\text{Charm} =- \\frac{\\partial \\Delta}{\\partial \\tau} = \\frac{\\partial \\Theta}{\\partial S} = -\\frac{\\partial^2 V}{\\partial S \\, \\partial \\tau}", "efbe8924b73868f87eafa78b99fa7fe2": "\\Delta^\\text{w}_\\text{o}\\phi^\\ominus_i", "efbe98b16fa6f8efb931cdf50ed75bba": " W(h) = \\int_h^{\\infty} \\Pi(h') \\, dh'.", "efbec60284d7f1ff5fbc4de7bc327df0": "\\frac{\\partial {{T}_{{}^{1}\\!\\!\\diagup\\!\\!{}_{2}\\;}}}{\\partial P}=\\frac{\\text{ }\\!\\!\\Delta\\!\\!\\text{ V}}{\\text{ }\\!\\!\\Delta\\!\\!\\text{ }{{S}_{HL}}}", "efbec7e0e9cc99cddc121b1c1334a533": "x\\in Y^G", "efbef58d34104a7a83de4030ec724b37": "\\sin(i)=\\mathrm{Im}\\{\\epsilon^{i}\\}", "efbefc292f15e108c516b0c5cfda422d": "{\\boldsymbol \\mu}", "efbf21dae4ea48886a73bb036fb12f83": "r = \\sqrt{\\frac{(s-a)(s-b)(s-c)}{s}}. ", "efbf436e260a93381abe9ec660c4ebea": "d_m(x,y)=| x^m - y^m |", "efbf4511ae3f81c74a05968d0837f070": "\\exists x ( x = x+1)", "efbfa2b2fcea4f9cfbe3ae9333588857": "HG", "efbfad06460b12967c316272921ff353": "\n\\begin{align}\n& \\frac{1}{n}\\sum_{k=0}^{n-1} f_k\\alpha^{-jk} \\\\\n& {} = \\frac{1}{n}\\sum_{k=0}^{n-1}\\sum_{j'=0}^{n-1} v_{j'}\\alpha^{j'k}\\alpha^{-jk} \\\\\n& {} = \\frac{1}{n}\\sum_{j'=0}^{n-1} v_{j'} \\sum_{k=0}^{n-1}\\alpha^{(j'-j)k}.\n\\end{align}\n", "efbfe5150b184f2c40c55c70a593003a": " Z_{||}(\\omega) = R_s \\frac{1-i Q(\\frac{\\omega_r}{\\omega}-\\frac{\\omega}{\\omega_r})}{1+Q^2\\left(\\frac{\\omega_r}{\\omega}-\\frac{\\omega}{\\omega_r}\\right)^2} ", "efbff309161675794568d5c4519b4085": "\\sum_{i=1}^k n_i \\leq k\\pi", "efc000bb19588b374755d50a86783721": "y(x,t) = A\\sin(kx -\\omega t + \\phi ) + D\\,", "efc0074957603cec2a6ac6a991f83c08": "Z\\xleftarrow{\\eta} Y \\xrightarrow{\\tau} X.", "efc016d24ec15c87fd32a31a79eec9c1": "f_\\prime^\\prime", "efc038fb8a7f91498ae8d3a467dc744b": "W^\\bot=\\left\\{x\\in V : B( x, y ) = 0 \\mbox{ for all } y\\in W \\right\\}\\,. ", "efc0427e1efe3977708e4153623444a9": "\\scriptstyle Y_{\\ell}^{m}(\\theta, \\phi ) \\,", "efc083b510ab4387aae86047090d62fd": "DPA = PA", "efc17ab409710d583876a1746726bf46": "T = \\left ( \\frac{(1 - \\alpha) L_0}{\\epsilon \\sigma 16 \\pi a^2} \\right )^{\\frac{1}{4}}", "efc181dbf9861f01d28abc1e3b8d661b": "\\begin{matrix}\nt_1s_1 = (1\\ 2),&\\quad\\text{so}\\quad&\\overline{t_1s_1} = (1\\ 2)\\\\\nt_1s_2 = (1\\ 2\\ 3) ,&\\quad\\text{so}\\quad& \\overline{t_1s_2} = e\\\\\nt_2s_1 = e ,&\\quad\\text{so}\\quad& \\overline{t_2s_1} = e\\\\\nt_2s_2 = (2\\ 3) ,&\\quad\\text{so}\\quad& \\overline{t_2s_2} = (1\\ 2). \\\\\n\\end{matrix}", "efc18ad83048027987acc857b8804949": " dv_x\\, dv_y\\, dv_z = v^2 \\sin \\theta\\, dv\\, d\\theta\\, d\\phi ", "efc1ef4c314f9663fd27f05129428ecb": "w \\otimes v", "efc213d2281413576f37c1991a4538ea": " \\hat{B}_{ij} = \\hat{H}_{0} + \\hat{H}_{1} + ... + \\hat{H}_{6} ", "efc23244baf73d27e69be4c6d643c544": "h/s", "efc269774732bf76710ad1758a03dc36": "\\arg (x + iy)", "efc28b68a55adc6d21a680b744d6d8fa": "\\Pr(B_1)={6 \\over 10}", "efc2b344fc094f671ac8104ed1f6c6c4": " a_{{\\mathbf{k}}_l} |n_{{\\mathbf{k}}_{1}}, n_{{\\mathbf{k}}_{2}} ,n_{{\\mathbf{k}}_{3}}...n_{{\\mathbf{k}}_{l}},...\\rangle = \\sqrt{n_{{\\mathbf{k}}_{l}}} |n_{{\\mathbf{k}}_{1}}, n_{{\\mathbf{k}}_{2}} ,n_{{\\mathbf{k}}_{3}}...n_{{\\mathbf{k}}_{l}}-1 ,...\\rangle ", "efc304c5cfd18c03d7d535604c1aecf4": " \\varepsilon_{ijk} ", "efc35347552bdfaa127673e3b418838e": " \\hat{f}(a) = \\sum_{\\scriptstyle{x \\in \\Z_2^n}} (-1)^{f(x) + a \\cdot x} ", "efc3cd750e7f21c2e22a2a12dc304dc8": "\\ \\pi", "efc44c1dfa9f76c1a48dbdf207e118a8": "\\begin{align}\n \\sinh 2x &= 2\\sinh x \\cosh x \\\\\n \\cosh 2x &= \\cosh^2 x + \\sinh^2 x = 2\\cosh^2 x - 1 = 2\\sinh^2 x + 1 \\\\\n \\tanh 2x &= \\frac{2\\tanh x}{1 + \\tanh^2 x}\n\\end{align}", "efc47b7b8522322f648933b6711a49f5": "a \\in D", "efc494fbb8c3eee9d7a3577244b27329": " T_m (1) ", "efc4c3ac1307abd0f9be3f4e0c80bbda": "\\{\\{x\\}|x \\in V\\}", "efc4f649400ea4e27480acc8fbda4ae6": "V=(1+\\frac{2\\sqrt{2}}{3})a^3\\approx1.94281...a^3", "efc548f184d3de7e7c28a02d0db8c287": " \\mathbf{z} \\in \\mathbf{y} - [J_f](\\mathbf{[x]})^{-1}\\cdot f(\\mathbf{y})", "efc5d2f7b6943464bf8ae584d84ae089": "0.0392857", "efc60884cc0f4d2e296b7a40305f2f6b": "u(p_j,t)", "efc6304dcac36050317a4ab272ad6523": "1-m/n", "efc67d74f189a7ff1889ca7837dee237": "\\{f_k\\}", "efc6bf0d06d185b2da0e3508e2aed367": "\\psi_1({\\Omega_2}^{\\Omega_2})", "efc6feb98c66bfe335151e3ff17b6f4f": "A_{21}g_2e^{-h\\nu/kT}+B_{21}g_2e^{-h\\nu/kT}\\frac{F(\\nu)}{e^{h\\nu/kT}-1}=\nB_{12}g_1\\frac{F(\\nu)}{e^{h\\nu/kT}-1}", "efc7272b500ecc2fb87cdc6c5a183d86": "P/ B", "efc741affad5941c65e82e1f745d2b7d": "Q = C \\Delta \\psi", "efc7eea21b001125ce0b40d0ab986285": "P_K(x)", "efc831aabf9ade051f781ccb54c2dcdb": "[m]", "efc8a1ec009a566e8d304ed1f07ff670": "G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = {8 \\pi G \\over c^4} T_{\\mu\\nu}", "efc8c6b1d1dc9a827d084f406de5bb01": "\\scriptstyle \\mathrm{A}+\\bar{\\mathrm{B}}+\\mathrm{X}", "efc8ccfa80fe419a4a6848269f7d96d8": "p_a\\left(\\Sigma^*\\right)", "efc9179d5ab922a77a7039a11b7a9af3": "\\frac{\\partial T^{\\alpha\\beta}}{\\partial x^\\beta}=0", "efc91b814776bde7731da8d827647db1": "M_0 \\times [0,1]", "efc960a6bed96618dd5474a371f05a65": "\n\\begin{align}\n\\operatorname{E}(s^2)\n& = \\operatorname{E}\\left(\\sum_{i=1}^n \\frac{(x_i-\\overline{x})^2}{n-1} \\right)\\\\\n& = \\frac{1}{n-1} \\operatorname{E}\\left(\\sum_{i=1}^n \\left[x_i - \\mu - \\left(\\overline{x} - \\mu\\right)\\right]^2 \\right)\\\\\n&= \\frac{1}{n-1} \\left[\\sum_{i=1}^n \\operatorname{Var}\\left(x_i \\right) - n \\operatorname{Var}\\left(\\overline{x} \\right)\\right]\n\\end{align}\n", "efc9621e068c314f0b9350d9eb58576d": "\n\\tan^2 \\frac{\\theta}{2} =\n\\frac{1-\\cos \\theta}{1+\\cos \\theta}=\n\\frac{1-\\frac{\\cos E-e}{1-e \\cdot \\cos E}}{1+\\frac{\\cos E-e}{1-e \\cdot \\cos E}}=\n\\frac{1-e \\cdot \\cos E - \\cos E+e}{1-e \\cdot \\cos E + \\cos E-e}=\n\\frac{1+e}{1-e} \\ \\cdot\\ \\frac{1-\\cos E}{1+\\cos E}=\n\\frac{1+e}{1-e} \\ \\cdot\\ \\tan^2 \\frac{E}{2}\n", "efc99f1ebe96a1452f441e03de2ab81c": "(m-M)_0 = ", "efc9a5ad4bbf9880e705e0d45bc17656": " 2^{\\frac {12} {12}} = 2 ", "efc9b213ee800dd87c1b15cd36a73c11": "\\Bbb{R}^{3}", "efc9d60c898cdaf6efc031d10c0baffd": "\\sin \\theta_n = \\frac {n \\lambda} {S} + \\sin \\theta_0, n=0, \\pm1, \\pm2.... ", "efc9e4b6432815f60f9ee35ba8ac269c": "3\\over\\sqrt{35}", "efca073fb6d3a16f6048b526f00312cc": "r<|z| 0 ", "efcc04dfc30d269bf2ed2dfe6d3615ce": "x(p_1,p_2,w) = \\left(\\frac{w}{2p_1}, \\frac{w}{2p_2}\\right).", "efcc279e9407423a0c3e83811e343e3e": "\\textbf{z}_{k}", "efcc779e47c7ca56c1df60761a1dee15": "\\sigma_x^2 = \\frac{\\hbar}{2 m \\omega}", "efccc2c0e3afcb3d81633a3f6ba96197": "\ny_3 = \\frac{((-y_1+2y_2){x_1}^3+(-3ay_1+(-3y_2x_2+3ay_2)){x_1}^2+((3{x_2}^2+6ax_2)y_1-6ay_2x_2)x_1+({y_1}^3-3y_2{y_1}^2+(-2{x_2}^3-3a{x_2}^2+3{y_2}^2)y_1+(y_2{x_2}^3+3ay_2{x_2}^2-{y_2}^3)))}{(-{x_1}^3+3{x_2}{x_1}^2-3{x_2}^2x_1+{x_2}^3)}\n", "efcd1c81554814074e47dc6fe555c0fd": "e = \\sqrt{\\alpha \\hbar c}", "efcd64f317523c1270c25fd85c9d90ee": "\n\\mathcal{U}(\\alpha, {\\tilde{u}}) = \\left \\{ u(x): \\ \n|u(x) - {\\tilde{u}}(x) | \\le \\alpha {\\tilde{u}}(x), \\ \\mbox{for all}\\ x \\right \\} , \\ \\ \\ \\alpha \\ge 0\n", "efcd8a64d0d57d478e075fcbeed5f3c0": "ad=2bc\\;", "efcd9d008441603ba71b767e058b35b2": " \\part^\\alpha x^\\beta = \n\\begin{cases} \n\\frac{\\beta!}{(\\beta-\\alpha)!} x^{\\beta-\\alpha} & \\hbox{if}\\,\\, \\alpha\\le\\beta,\\\\ \n 0 & \\hbox{otherwise.} \\end{cases}", "efcde9c9bae304c1f625f8ae317d88ca": "\n \\vartheta(G) = \\min_A \\lambda_\\text{max}(A).\n", "efce6f75d1765d03cb6a776ac63dfc11": "\n\\mathrm{DA} = 145442.16 \\left[1-\\left(\\frac{17.326 P}{459.67+T}\\right)^{0.235}\\right]\n", "efcedecec95f0df503a4caa56f5172d3": "\\begin{pmatrix} 2 \\\\ 0 \\\\ 19 \\end{pmatrix}", "efcf1cb04b663f1044506ce9a1588e0b": "\\frac{dE}{dx} = \\frac{q^2}{4 \\pi} \\int_{v > \\frac{c}{n(\\omega)}} \\mu(\\omega) \\omega {\\left(1 - \\frac{c^2} {v^2 n^2(\\omega)}\\right)} d\\omega", "efcff81c934de15508e19b8e47afee7d": "[\\mathbb Q[\\mathbb Z]]", "efd009244c59db952158ac9b2ebc6617": " \\phi = -\\omega \\, \\sqrt{r^2 - r_0^2} + \\operatorname{arctan}(t/r_0)", "efd00b99d9d36cb4efa8d69f2facdd80": "\\mathfrak{B}(\\mathbb R)", "efd019fe1634b66eacb4dc06c9208f75": "S_{mn} = \\frac{b_m}{a_n}\\,", "efd0915795c24e69b59a79e24a3561d9": "V^{i}(x,u,w) \\frac{\\partial}{\\partial x^{i}} + V^{\\alpha}(x,u,w) \\frac{\\partial}{\\partial u^{\\alpha}} \\ + \\ V^{\\alpha}_{i}(x,u,w) \\frac{\\partial}{\\partial w^{\\alpha}_{i}} +\\,", "efd0aff4b1d957512a2a0c7f4bb8722d": "\\left(\\frac{\\partial S}{\\partial T}\\right)_{P} = \\left(\\frac{\\partial S}{\\partial T}\\right)_{V}+ \\left(\\frac{\\partial S}{\\partial V}\\right)_{T}\\left(\\frac{\\partial V}{\\partial T}\\right)_{P}\\,", "efd0d512fd5cd6bbe83119c53dbb42b4": "\\alpha=1,\\cdots,N", "efd0d5beffd1afb299e6be4d53be144d": "F_{ab} = \\left(\n\\begin{matrix}\n0 & B_z & -B_y & E_x/c \\\\\n-B_z & 0 & B_x & E_y/c \\\\\nB_y & -B_x & 0 & E_z/c \\\\\n-E_x/c & -E_y/c & -E_z/c & 0\n\\end{matrix}\n\\right) ", "efd0f59729deea6aa74c7963109d62c4": "e^{\\mathbf AT} \\approx \\mathbf I + \\mathbf A T", "efd12e3d8a73faf6d2d8a95bac41bb65": "= {1 \\over 4} [F , G]^{IJ} - {1 \\over 4} i [* F , G]^{IJ} + {1 \\over 4} i [F , * G]^{IJ} + {1 \\over 4} [* F , * G]^{IJ}", "efd1419f4ded78e86c2055a81c4ca6bd": "Q_{n+1}R_{n+1}", "efd14d63254e40c52dd29ac3dfc51c70": "\\sum_i ap_i x_i ", "efd1617f10bd63f267b854133c9af2ea": "\\gamma.", "efd27b664c0b8c4d778485c3250d382c": "\\begin{align}L_n^{(\\alpha)}(x)&= \\left(2+\\frac{\\alpha-1-x}n \\right) L_{n-1}^{(\\alpha)}(x)- \\left(1+\\frac{\\alpha-1}n \\right) L_{n-2}^{(\\alpha)}(x)\\\\[10pt]\n\n&= \\frac{\\alpha+1-x}n L_{n-1}^{(\\alpha+1)}(x)- \\frac x n L_{n-2}^{(\\alpha+2)}(x). \\end{align}", "efd2ac39a443c6b60bea3ef0a2f87de9": " det^{col}(d/dz - gEg^{-1}/z) = det^{col}(g(d/dz - E/z)g^{-1}) = \ndet(g) det^{col}(d/dz - E/z) det (g^{-1})= det^{col} (d/dz - E/z)", "efd2d79240a07330afcb0c37d9b89bbe": "\\lambda_f - \\lambda_i = \\frac{h}{m_ec}(1 - \\cos\\theta)", "efd2e2edda0c8453f55f30147574a788": "\n\\hat{\\Phi}(\\varphi,z) = \\sum_n \\Theta_n (\\varphi) \\, G_n(z)\n", "efd3106bf437f41477bdfc80042917f3": "a\\ne b \\ne c \\ne d, \\alpha \\ne 90 ^\\circ, \\beta = \\gamma = \\delta = \\epsilon = 90 ^\\circ, \\zeta \\ne 90 ^\\circ", "efd3bf30355be9bb4b27644947c324d6": "S(y) = S(y_{left}) + S(y_{right}) + M(Y_{left},Y_{right})", "efd3db166bb5e843d5ad2f8de9f368d3": " R = h^2 S ", "efd408e8858e53940c6b0642a3fa3fec": " \\frac{1}{\\tan \\theta}\\! ", "efd4829259e927f9559421a22a744c14": " E = \\int d^{3} x \\left[ \\frac{1}{2} \\dot{\\phi}^{2} + \\frac{1}{2} |\\nabla \\phi|^{2} + U(\\phi, \\phi^{*}) \\right], ", "efd49ce9d372ddac7d0d4bbd8fb15e10": "\\scriptstyle p_1", "efd4b8f3571a7ad69b67ec8f7c75cdde": " \\frac{d}{d t}\\left(\\mathbf{P}_\\text{mech}+ \\mathbf{P}_\\text{field} \\right)_i = \\oint_S \\sum_j T_{ij} n_j dS\\,.", "efd4c28ed17c72400c3006ed03c637cd": "L^*=25.29 Y^{1/3} - 18.38", "efd58ba055e368d7837a4b00a0ef5f93": "\\pi_{o}", "efd5975b1a9dc7656522fdaf12cd2026": "\\alpha_{k,1},\\ldots,\\alpha_{k,k-1}", "efd59db7a93ebda67fb538b82e0c5c36": " \\mathfrak{so}(3,\\mathbb C) \\cong \\mathfrak{sp}(2,\\mathbb C)\\qquad(=\\mathfrak{sl}(2,\\mathbb C))", "efd5c5a85ef5c3757a162c8fd212115f": " \\begin{pmatrix}E_{1}&0\\\\0&E_{2}\\end{pmatrix} + \\begin{pmatrix}W_{1}&W\\\\W&W_{2}\\end{pmatrix} =\\begin{pmatrix}E_{1}+W_{1}&W\\\\W&E_{2}+W_{2}\\end{pmatrix} ", "efd6404d78fd4e27e8fdf61dd4069c89": "\\omega_\\circ \\times \\frac{\\mathrm{hour}}{{15}^\\circ}", "efd6604f5258d477c86056cb63c6febb": " f(i)M(i,j) = f(j)M(j,i)", "efd67f2d9c1c2eb6e08c6d174cd286d3": " x^3yxyx ", "efd681f34512fe34cb4fa68b3d4ff190": "t = \\ln(x). \\,", "efd69e9607ad8e3f0ab1a73dc1c98914": "\\limsup_n \\frac{K(X|n)}n", "efd6a5b7a2d9e0a30fc8d0be80c00ae0": " \\mu = 0 ", "efd6af271d9875338ee2965734ba1004": "\\textstyle |1-\\zeta|^2=1-2\\mathrm{Re}\\,\\zeta +|\\zeta|^2", "efd7078f4b064530aeade169c7a075c0": "A \\mathbf{x} = \\mathbf{b}", "efd70dd6ef4ffaa529eb93ba90f6b537": "l{\\neq}l'", "efd7133b505ca87ebc6db59b82a1369d": "E_6, E_7, E_8, F_4, G_2", "efd7f7bbdfcd929d15d653f740462d4a": "\\,B'= B \\cap \\{A - \\{e\\}\\}", "efd871e0bf7267bbece28ea202811391": "X_j \\to X_i", "efd878b31061758adfe67f5fcde1fa91": "\n \\widehat{\\operatorname{s.\\!e}}(\\hat{\\beta}_j) = \\sqrt{s^2 (X'X)^{-1}_{jj}}\n ", "efd87d0b9ea5fb8504f8690f8ad32079": "\n\\overline{A} = \\sum_{\\alpha=1}^{D}|c_{\\alpha}|^{2}A_{\\alpha \\alpha} \\approx A\\sum_{\\alpha=1}^{D}|c_{\\alpha}|^{2} = A,\n", "efd8c08083c2b3049bd5cdca35a647d2": "\\;\\; + \\;\\aleph_0", "efd8db3cf526d08ab7e0912244687886": " A = \\frac{kT}{e\\alpha}", "efd8e5a2547b7d298534bc210931d001": "\\chi(M) \\in \\mathbf{Z}_2", "efd8e7e27517b7083d864d976d7fcafc": "g\\left(\\tfrac{\\pi}{2k},s\\right) = c_3 \\cdot 0+c_4 \\cdot 1=0, \\quad c_4 = 0 ", "efd8fe17571eb919b3e12d4217a5d334": "SSI = \\sum_{i=0}^n {{r_i^2 a ({{CD} \\over {ND}})} \\over {r_{max}}}", "efd9304e701b4514b6ce15dc867e782c": "\\{e_0, e_1, e_2, e_3, e_4, e_5, e_6, e_7\\},\\,", "efd93ff71e4268b8ddd8badb8e133a26": " 2 h f m_e c^2 - 2 h f' m_e c^2 = 2 h^2 f f' \\left( 1 - \\cos \\theta \\right). \\,", "efd9ac15e1d8d126f2c28af2f4fd6920": "\\delta R_{\\mu\\nu}{} = (\\delta\\Gamma^{\\lambda}{}_{\\mu\\nu})_{;\\lambda} - (\\delta\\Gamma^{\\lambda}{}_{\\mu\\lambda})_{;\\nu}", "efd9d176d4ebc5335a553cc0e9fee67f": "a^2+b^2 = (a+bi)(a-bi)", "efda7034011cf884d6a52f76a88c8ce3": "\\aleph_{\\alpha+1} = \\aleph_{\\alpha}^+", "efda7777df0c004e8939c75bda8ddb1c": "n_2=\\pm\\frac{1}{\\sqrt 2}\\,\\!", "efdac493b6dc6f4db62e4473730ac6f8": "z_1=\\sum_{k=1}^{q}{M_k}\\sum_{L_k}{x_{i,m+j}}", "efdaf5c7609a0f6a68365ee1a24f4102": "\\displaystyle{h=(I-A)^{-1}T\\mu,\\,\\,\\, T^*h=(I-B)^{-1}\\mu}", "efdaf7765d6221db1f5f207ebcf875a7": "0.963\\pm0.012", "efdb0ae602f0101077b64be96364a7f4": "p_1,p_2,\\dots, q_1, q_2, \\ldots \\in \\mathbb{N}", "efdb503efadc5220db5c765601c29018": "\\cos{\\left(\\frac{2\\pi}{17}\\right)} = -\\frac{1}{16} \\; + \\; \\frac{1}{16} \\sqrt{17} \\;+\\; \\frac{1}{16} \\sqrt{34 - 2 \\sqrt{17}} \\;+\\; \\frac{1}{8} \\sqrt{ 17 + 3 \\sqrt{17} - \\sqrt{34 - 2 \\sqrt{17}} - 2 \\sqrt{34 + 2 \\sqrt{17}} }", "efdb87d0eea312a1a867c013c372cc1e": "D_{E}\\ll H_{E},", "efdba2dbefbbc04092533e602b6a03b1": "O (\\log n)", "efdc1cedcdbf00f0c25a7845ae71bd41": "\n\\begin{matrix}\nP &=& 0{.}26 \\cdot 16\\,\\mathrm{m}^2 \\cdot (111\\,\\mathrm{m}/\\mathrm{s})^3 \\cdot 1{.}24\\,\\mathrm{kg}/\\mathrm{m}^3 /2 \\\\\nP &=& 3{.}53\\cdot10^6\\,\\mathrm{kg}\\cdot\\mathrm{m}^2/\\mathrm{s}^3 = 3{.}53\\cdot10^6\\,\\mathrm{N}\\cdot\\mathrm{m}/\\mathrm{s} = 3{.}53\\,\\mathrm{MW}\n\\end{matrix}\n", "efdc23babe2523f15b9a0a2467525356": "\\theta_c \\ll 1", "efdc6c9465b3ae63a2424db54c50f49d": "z\\in K[x,y,z]/(z^2-xy)", "efdc941ef4eb03d180b776b0f512cee6": "\\mathbb{PORTAL}", "efdcc2007b141071277e3c9bb7135239": "X\\subseteq \\mathbb{R}^{L}", "efdd1ab7026999b102708dcab10bf014": "\\lim_{x\\to\\infty}\\sqrt[x]{N}=\\begin{cases} 1, & N > 0 \\\\ 0, & N = 0 \\\\ \\text{does not exist}, & N < 0 \\end{cases}", "efdd57a81d9e659e81927d03e162c92f": "\\omega_\\mu^{IJ}", "efdd6238fcfefd5fec2daed435cc9fa7": "x_{i_1} < x_{i_2} > x_{i_3} < \\cdots x_{i_k}\\qquad \\text{and} \\qquad 1\\leq i_1 < i_2 < \\cdots < i_k \\leq n.", "efdd85670af2dd5307d4782f98195ee4": " p_{k+1} = x_k + p_k - y_k ", "efddcb85c52888419040a591a55f4d8a": "f_{\\mathbf{Y}(y)}=\\frac{f_{\\mathbf{X}}(\\mathcal{A}^{-1}(y-b))}{|\\det\\mathcal{A}|}", "efde0670015532398a4d30bf31343b8d": "Z(\\beta)=\\sum_{\\sigma_1,\\ldots, \\sigma_L} e^{\\beta h \\sigma_1}e^{\\beta J\\sigma_1\\sigma_2}\\; e^{\\beta h \\sigma_2}e^{\\beta J\\sigma_2\\sigma_3}\\; \\cdots e^{\\beta h \\sigma_L}e^{\\beta J\\sigma_L\\sigma_1}= \\sum_{\\sigma_1,\\ldots, \\sigma_L} V_{\\sigma_1,\\sigma_2}V_{\\sigma_2,\\sigma_3}\\cdots V_{\\sigma_L,\\sigma_1}. ", "efde3774df5a32a8968d39ee39e6d05c": "\\Delta n_{\\text{E}} (0'')", "efde5cd914502b7eeb1dd78a59f454e3": "w_k\\ge 0", "efde872e884747a4f629baa79bfc5fd0": "E/K", "efde8f956d50f20ccff4989ad39b2269": "a(t) = \n\\begin{cases}\n 1, & \\mbox{if }t = 0 \\\\\n 7, & \\mbox{if }t = 1 \\\\\n (1+5\\cdot 4^n)/3 , & \\mbox{if }t\\mbox{ is even }>0 \\\\\n (10+11\\cdot 4^n)/6 , & \\mbox{if }t\\mbox{ is odd }>1\n\\end{cases}\n", "efdf129cf2424d7c50ae554d18df6c76": "q\\ge 0", "efdf2338f38dc307b99409c43a5ecb45": "\\int_{n=m}^\\infty \\frac {dn}{n^k} = \\frac 1{k - 1}\\frac 1{m^{k - 1}}", "efdf7a2ef9963db8ca366a9bd31cc788": "A^\\mathrm{T} A = A A^\\mathrm{T} = I, \\,", "efdfb000339ee53ac883ea93d2f88c8d": "f(x_0-0)", "efdfc4dd897a9e5e273c153c0fbb7267": "R_{DT}(r) = \\int_{0}^{\\infty}\\int_{0}^{\\infty}\\int_{0}^{2\\pi}S_d(r',z')R(r,0,0;r',\\phi',z')r'd\\phi'dr'dz'", "efdfcc589839e3c4cfb5b9f34b5ec2df": "H^n(X;R) \\to H^{2n}(X;R)", "efdfcde541bd64b8479b1609db6db097": "\\mathbf{A}\\in M_n(\\Re)", "efdfd08ce2862e4a01a9e3e6375b495f": "\\displaystyle{F_n(x) = Y^n e^{-x^2/2}}", "efe02546b30869eb93f404fb3f89e3e2": "D(e^{ax}y)=e^{ax}(D+a)y.\\,", "efe08f1c00bbba04114787f5a517d781": "\\left(\\begin{smallmatrix}1 & \\lambda \\\\ & 1\\end{smallmatrix}\\right) \\times \\{\\pm I\\}", "efe0a97c00ba84bf7bf748bc71d1f1f0": " \\hat{H}_i = \\hat{P}_{i,1} \\otimes \\hat{P}_{i,2} \\otimes \\cdots \\otimes \\hat{P}_{i,n_i} ", "efe0f757cb139ad7475abf3a28f6ba9c": "\\mathbf X", "efe15aee733840df645eda9be666e92a": "D_n = \\frac{2}{L} \\int_0^L f(x) \\sin \\left(\\frac{n\\pi x}{L}\\right ) \\, dx.", "efe1ab85b8a1f50d59ba9444d62c2f23": "H_1 = \\begin{bmatrix} 2 & 1 \\\\ 1 & 1 \\end{bmatrix}", "efe2060146099a3c6d38bc1941e7ed6e": "\\gamma^I", "efe2565ff18f8702dc630be91924fa8c": "\\Delta R_{\\mu}\\Delta x_{\\mu}\\ge\\ell^2_{P}=\\frac{\\hbar G}{c^3}", "efe265c4cfc87cfc312e581cc3bb53c3": "T^*X", "efe2b75cdcb9daa8fdb273026c7b654b": " \\forall x(\\phi \\to \\exists y(\\phi[x:=y])) ", "efe318fd77bf2a389be8b5d6278a1473": "\\|x + y\\|^2", "efe319bc76ad2df389d403f7d5e18832": "u = x'_1+y'_1\\sqrt{4729494} \\, ", "efe379cb96a0d5a283c6237092d674ee": "T_{OFF} = \\frac{R_B}{R_C}-T_{ON}", "efe3a76f0292f5ee38af532be773da99": "\\infty \\times 0 = \\Phi", "efe40ca5adc7f5368b9fd664c7c505e0": "e=\\frac{a \\pm \\sqrt {a^2-b^2c}}{2}.", "efe462347cbc8001d8f3d8f28ae1039c": "x^{l(2t-1)} - 1", "efe46ea867c5ac68b826d1255fff0061": "\\text{st}(v)", "efe47734697eff2c5d706e880968f3d7": "x = 2.", "efe48f531fce5f2a061b921b15978a89": "\\beta_k={2-\\alpha_k \\over k+1-k\\alpha_k}.", "efe4c4e086138a74abface9aaee5b979": "H\\backslash G", "efe4e1351eec8c68b3620443ed7bfde9": " H_4(\\phi) = \\frac{1}{2}\n\\begin{bmatrix}\n 1 & 1 & 1 & 1 \\\\\n 1 & e^{i\\phi} & -1 & -e^{i \\phi} \\\\\n 1 & -1 & 1 & -1 \\\\\n 1 & -e^{i\\phi} & -1 & e^{i\\phi}\n\\end{bmatrix}\n", "efe516a55a6586cd602d63ac5c6eceb3": "b_{\\nu_j}", "efe51b9bbff6315407bfdeec076399d5": "12bc \\left( s(b,c) + s(c,b) \\right) = b^2 + c^2 -3bc + 1,", "efe525af864f7a2d73ac50ba6c605015": "4\\pi\\,", "efe53524c1ee50d38c1126f245338da3": "VX= \\{gK:g\\in G\\}\\sqcup \\{gH:g\\in G\\}.", "efe5684d867e3e5ad619ac7996df6b1f": " \n\\frac{I-I_\\mathrm{b}}{I_\\mathrm{b}},\n", "efe60167263fe1b2533e7987daa1b200": " e^{-Y}g", "efe609b9a05eb6590bfacb81a96c3f25": "P=\\mathbf{J} \\cdot \\mathbf{E}", "efe60ec4a50c2c656b0d2084f4c064a9": "Hb \\,", "efe61df3d15a1eee57476ca083212c57": "\\Delta \\mathcal{O}\\Delta \\mathcal{F}\\geqslant \\hbar/2\\,,", "efe642726d0d29dc483939508afda428": "-1/2 \\leq x \\leq 1/2", "efe7182b1ea9fcc58d04f7b53e3b6fdd": "\nG_n^{(1)} = P_{n,0} P_{n,1}^R P_{n,2} P_{n,3}^R \\cdots \\text{ and } G_n^{(2)} = P_{n,0}^R P_{n,1} P_{n,2}^R P_{n,3} \\cdots\n", "efe777cbf920b8320db4d17914657501": " (x,\\sigma^2) \\sim \\text{N-}\\Gamma^{-1}(\\mu,\\lambda,\\alpha,\\beta) \\! .\n", "efe77bad1d08ccfe68049872a89d51bc": "h_1 \\leftarrow g_1^{z_1} rem P", "efe796f981c53198341daf89cdfdeeff": "J^+(S) \\cap T", "efe7f653bc569b53c55a6f111d4473ff": "A \\leq_c B\\quad \\iff\\quad (\\exists f)(f : A \\to B\\ \\mathrm{is\\ injective})", "efe848042216db3c7c79366127423d31": "\\mathbf B=\\sigma^2 \\mathbf C \\mathbf C^T", "efe8a5a79eff7750da9aea3b8801cdd3": "2^{H(p)n}/2^n = 2^{H(p)n-n}", "efe914eff1b564f34f63cb9a91cdd48f": "E \\xi(\\cdot) = E \\Lambda(\\cdot)", "efe937f7f6755ea08b22bfe319efd953": "C^{\\beta}-S^{\\gamma}-S^{\\gamma}-C^{\\beta}", "efe9ce2adbcb9437de213f84f252f625": "S_0 = \\sum_{i}{\\sum_{j}{W_{ij}}}", "efea4008b8aa931198f54225a3fe79ad": "C^*(s)", "efea583c659fa042ee6777984469489e": "y_{ij} \\in \\{0,1\\} ", "efeb77efd1cb64984877ef201aa99b8c": " H u_{n\\mathbf{k}}(\\mathbf{r}) = \\left( H_0 + \\frac{\\hbar}{m_{0}}\\mathbf{k}\\cdot\\mathbf{\\Pi} + \\frac{\\hbar^2 k^2}{4m_{0}^{2}c^{2}} \\nabla V \\times \\mathbf{p} \\cdot \\bar{\\sigma} \\right) u_{n\\mathbf{k}}(\\mathbf{r}) = E_{n}(\\mathbf{k}) u_{n\\mathbf{k}}(\\mathbf{r}) ", "efebdbe94185686bf41c73ed68eb91fd": "\\lVert g \\rVert_f", "efec0a33d20248de70f6d6ef987ba063": "\\, p_{i_{0}}", "efec4e9e27abe58b0c164f60912795b8": "\\{0\\}\\times[0,h]", "efec4f36e47ae0f4a72fd81596979717": " \\mathcal{O}_{Y,y} ", "efec593680bec8199563c6bbda3e2790": "t=\\frac{(\\overline{x}_1 - \\overline{x}_2) - d_0}{\\sqrt{\\frac{s_1^2}{n_1} + \\frac{s_2^2}{n_2}}},", "efec5c87f2c79db6ae153eac40e468af": "\\Theta(nm)", "efeca279b9a955cba09af8a72b2089b5": "d\\mathbf{f}_0", "efecd7e1033d3c952c4829d36427be7e": "|PQ|=\\sqrt{(p-r)^2+(q-s)^2} \\, ", "efed49260a296d25351beb32bbd66121": "\\int\\frac{dx}{{a^2+x^2}}", "efed9562c9c298e5562794b999086336": "\\{h\\},\\,", "efedc8823e91ddb87a5baee13046bbaa": "\\mathcal{F}^{0}[f] = f", "efee0e8b2600660dae96c4e89ba8a3eb": "f^2(x) = f(f(x))", "efee2e63ad352b091cf863292817d2ab": "\\mathbf{v}_i=\\mathbf{V}+\\mathbf{V}_i\\,", "efee3963e4bb5e9658dddda39817c656": "\\mathbb{H}_{inv} \\, \\tilde{y} = \\mathbb{H}_{inv} \\, \\mathbb{H} \\, \\tilde{x} = \\mathbb{I} \\, \\tilde{x} = \\tilde{x}", "efee404bc18070f32c04619e36bb6a9b": "\\overline{r \\mathbin{\\And} s} \\leftrightarrow (\\overline{r} \\mathbin{\\And} \\overline{s}),", "efee49c5d554963b7e1d0995e64c2549": "(\\gamma)\\,", "efee59bd826df15c06c48f41865684ea": "\\digamma_i (i\\in\\{0,1,2 \\})", "efee5eefc6f25bfc70002bbef635ac50": "x^2 - ny^2 = 1", "efef3eafd4027f3c31e4f9013d15031b": "\\lim_{h\\to 0}{f(x+h)-f(x)\\over h}=f'(x)", "efefe6d21e5a80a17284837b30528f77": "\\boldsymbol{\\varphi}(w)", "eff058840acf008d509a343dd272c65d": "\\displaystyle \\nabla^2\\mathbf a =\\nabla(\\nabla\\cdot\\mathbf a)+s^2\\mathbf a", "eff06a4a6b406f7022ca07c166c6c18b": "\\mathfrak{STUVWXYZ} \\!", "eff0735412a0cb14c6187b1fb1068bc9": "N\\subseteq C", "eff0fa887434e3153084d5f4433a57bf": "q(v \\otimes w) = q_1(v)q_2(w)", "eff11869c60ddb2fa84c5c091d411d82": "R_\\mathrm{source} \\ ", "eff1306f67e0bbda0a27f506b07a9464": "\\epsilon \\gg 1/\\Lambda", "eff13b6d236504534da085826b49c446": "\nA r^4 + \\sum_{i=1}^3 P_i x_i r^2 + \\sum_{i,j=1}^3 Q_{ij} x_i x_j + \\sum_{i=1}^3 R_i x_i + B = 0\n", "eff169c307e784ef193dbed52cb02442": "\\times4)x+,/y", "eff1b9a58c060039daf29d2b6efbc602": "\\mu(B_n)\\ge \\min\\{1,t_n/2\\}.", "eff269c75e4be2dbf92c0368320fda54": "[y]", "eff2ede0a10e44b6fc6692326e46a549": "2^{n-4}+2^{\\lfloor (n-4)/2\\rfloor}.", "eff32ff7f064b4a142be1aced5d07156": "\\frac{ii}{V}", "eff3c28141b46892f98eac87919802ae": "\\overline{C} = \\sum_{c \\in C} c", "eff3d2433dac0c78be44964aa7b29bfb": "M_2=N\\left ( \\mathrm{d} \\Phi_1/\\mathrm{d} I_2 \\right )\\,\\!", "eff3e38c59d50f9cdccb9f45993dbb68": "\\bar V_{L1}=\\left(V_i+(1-D)\\cdot \\frac{V_o}{D}\\right)=0", "eff424de99ffb16865d1b688d7249dc1": "f_2(z)=1+\\frac{(1+i)(z-1)}{2}", "eff4c627b4d4ea4d39da3c38d666d270": "V_{\\lambda}", "eff4d6713e8a8d126dd6ce2e972d9553": "C^2", "eff535967c672f542ac19d808199f125": " {\\Vert u \\Vert_{p,\\omega_k}} = \\int_{\\omega_k} | u|^p \\,\\mathrm{d}x\\qquad\\forall\\, u\\in L_{p,\\mathrm{loc}}(\\Omega).", "eff544ff7e8c76fb96d44954a1ae1fae": "r^2 - 4s(1-s)r = 0", "eff5f72734418dc4af128865034a87d0": "\nD_{sd} = \\frac{k_B T}{4 \\pi \\eta_m h} \\left[\\ln(2 L_{sd} / a) - \\gamma\\right]\n", "eff600b37c287931e9d03307400a7446": " g \\mapsto g \\cdot x_0. ", "eff6a6bd371ef207278b1113cdf77283": "K \\cup \\{P\\}", "eff762989c655a3960a972a14b31db04": "f^{-1}(X_1)=:Z_1", "eff799544beaa6ac6b63467859aa0c6f": "\\lambda^2 + 4\\lambda - 5 = 0 \\,\\!", "eff7d5dba32b4da32d9a67a519434d3f": "df", "eff7ec95f12eb57b526e9e1ec226780f": "\\lambda V", "eff7f134e619b67dee39492d854ea233": "\n M_{11} = -D\\left(\\frac{\\partial \\varphi_1}{\\partial x_1}+\\nu\\frac{\\partial \\varphi_2}{\\partial x_2}\\right) ~,~~\n M_{22} = -D\\left(\\frac{\\partial \\varphi_2}{\\partial x_2}+\\nu\\frac{\\partial \\varphi_1}{\\partial x_1}\\right) ~,~~\n M_{12} = -\\frac{D(1-\\nu)}{2}\\left(\\frac{\\partial \\varphi_1}{\\partial x_2}+\\frac{\\partial \\varphi_2}{\\partial x_1}\\right)\n", "eff84196e92cdc0b2eb13d2d7d615914": "AP^{-1},", "eff8681b197fb599d02684cd0d6ec398": "\n\\Lambda_p:=\\{0,\\dots,p-1\\}\\cup \\left(p-1,\\infty\\right).\n", "eff8fa5e912a836605643b77ead6d0f0": "\\tfrac{24}{7}", "eff94d03f2ad1ac634bd232d287665a1": "1 = {a^2 \\over a^2}", "eff95f5395dc9127890b40fddca5642e": "H^1_0(a, b),", "eff99b58bd7bcc6b6c0264567ad91767": "\\begin{align}\n\\theta_{\\text{hr}} &= \\theta_{\\text{min.}}\\\\\n\\Rightarrow \\frac{1}{2}(60H + M) &= 6M\\\\\n\\Rightarrow 11M &= 60H\\\\\n\\Rightarrow M &= \\frac{60}{11}H\\\\\n\\Rightarrow M &= 5.\\overline{45}H\n\\end{align}", "eff9b06b6def77df9b1328b57cb20c60": " f(x) = f(a) + f'(a)(x-a) + \\frac{f''(a)}{2!}(x-a)^2 + \\cdots + \\frac{f^{(k)}(a)}{k!}(x-a)^k + h_k(x)(x-a)^k,", "eff9d70fd0530b69b9f5e875d2abffa3": " \\widehat{\\mathcal{C}}_{XX}^\\pi ", "eff9ec111ac240c3184120138ebd2899": "t''", "effa1a2b5c942b4aa93b8d3567b99e04": " F\\subset \\mathbb{R}^2", "effa28ae66ed7a970316612403617880": "n \\theta_1 = m \\theta + n \\theta_0", "effa4195b4ecbefc37d593896826dc9f": "C>0.", "effa52de5050b63f89f1497fe6018147": " n^3-\\tfrac{3}{5}n^2+\\left(\\tfrac{2}{7} + \\tfrac{1}{2100} \\right)n+ \\cdots. ", "effa6b455db2f752a1c557a973e9bb4e": "x\\,(y_2-y_1) - y\\,(x_2-x_1)= x_1y_2 - x_2y_1", "effa8c94de5f298e939b0757b5b75635": "NK_n(R) =0", "effabe64a81efac0f31634427f4df8cc": "p(\\overline{\\mathbf{x}}) : \\qquad \\overline{\\mathbf{x}}_1, \\cdots, \\overline{\\mathbf{x}}_N \\in V, ", "effac15d796b5f625c25ffda25a1a282": "P_i(r_2,\\ldots,r_n)=0", "effae0d392fa4b19a2f8d1f43c102be5": "p\\text{NERD} = (x \\text{FIP}z \\times 2) + (\\text{SwStrk}%z/2) + (\\text{Strike}%z / 2) + \\text{LUCK} + 4.69", "effaf9e9cf882ca8962236f7042712f8": "u=\\frac{1}{EI}\\left(\\frac{1}{2}N\\langle x-0\\rangle^3\\ -\\ \\frac{1}{4}Nm^{-1}\\langle x-2m\\rangle^4\\ +\\ \\frac{3}{2}N\\langle x-4m\\rangle^3\\ +\\ cx\\right)\\,", "effb15912300db468105e0dcdf72fb79": "\\begin{align}\n {F_{k+2} \\choose F_{k+1}} &= \\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix} {F_{k+1} \\choose F_{k}} \\\\\n \\vec F_{k+1} &= \\mathbf{A} \\vec F_{k} ~.\n\\end{align}", "effb77df728af888b0a7d9bc51139d36": " u_i = \\log_b \\left(\\frac{1}{p(x_i)}\\right) = - \\log_b (p(x_i))=-\\log_b\\left(\\frac{1}{n}\\right), \\ \\forall i \\in \\{1, \\ldots , n\\}. ", "effb8304d535bcb6e9e858a75593e93e": "s_n", "effbeaf8ecde53a723871d22273c9724": "n_{C}=0.75*3+0=2.25 mol", "effc0004f60c91c58158ee2dfc767b18": "y_{i,j}=\\mu+\\tau_j+\\varepsilon_{i,j}", "effc057c0a128f354199954c053c462b": " Y_k(S_{x-1}) ", "effc89ccbb7ecb7d7f18d2f2bd559082": "\\frac{a+b}{c+d}-\\frac a c={{bc-ad}\\over{c(c+d)}} ={d\\over{c+d}}\\left( \\frac{b}{d}-\\frac a c \\right)", "effcd75ab19543a293fbe3d756344d1d": "\\log_k [(k - 1) * \\mathit{number\\_of\\_nodes} + 1] - 1, k \\ge 2.", "effcea3957236470528fc2977300a7da": "\\Delta S_m = -k[\\,N_1\\ln(N_1/N) + N_2\\ln(xN_2/N)\\,]\\,", "effced00f49cd3768f9dc6647330dc8d": "Q_V+V\\Delta p\\;", "effd0c19f4187c4032af8435b46e5da4": "x = \\frac{2}{\\sqrt{4\\pi + \\pi^2}} R\\, (\\lambda - \\lambda_0)(1 + \\cos \\theta) \\approx 0.4222382\\, R\\, (\\lambda - \\lambda_0)(1 + \\cos \\theta)", "effd1af0e98d6fb760575a0479dc6b4a": "J^2 = J_x^2 + J_y^2 + J_z^2", "effd3459836a73524b6879645bf1c181": "1200 \\log_2 5^{1/4} \\ \\hbox{cents} \\approx 696.578 \\ \\hbox{cents}, ", "effd6b47215f1f8c7f56016a34545c8f": " n\\geq 4 ", "effe050b99b38e410705ad6ae0cad427": "G\\left(a_n; e^{-i \\omega}\\right) = \\sum_{n=0}^\\infty a_n e^{-i \\omega n}", "effe5181d90155be22ce04c46300b00d": "f(k)=0", "effe52aeb99339334ca735ef0884a7e5": "\\displaystyle{\\iint E \\cdot\\Delta\\varphi = \\varphi(0).}", "effebb4bd5b59f451b00ca0874e6708c": "\\alpha \\in \\mathrm{GF}(q)", "effec4799794a9ea2f862e4197c35077": "(p_n)_{n\\in N}", "efff182d8e2731f3532e12e23dc08d74": "{\\textstyle \\alpha^3*\\ln(\\alpha)}", "efff490f477917e3b8ff53d90d72cb39": "X = D \\frac {10^m-1}{10^k-n},", "efff6a925a7b955f443ce5b17f3d7941": "\\tan 2 \\theta_\\mathrm{p} = \\frac{2 \\tau_{xy}}{\\sigma_x - \\sigma_y}\\,\\!", "efffa6f5a424de862779c928079a48a5": "\\log(\\gamma)\\,+\\,\\log(4\\,\\pi)\\!", "efffac2a85ccf1d8aa9b2ab4790e7dd9": "2\\omega_{n}", "efffb77594a9a1fc395416b8116f7c10": "2ms/km/year", "efffcbb8bbbb4e8b97faac403750fb10": "\\binom{m_1}{x_1} \\binom{m_2}{x_2} \\int_0^1 (1-t^{\\omega/D})^{x_1} (1-t^{1/D})^{x_2} \\operatorname{d}t", "f0001983114d0963db229aae57ec34ae": "p(n)=\\frac{1}{\\pi \\sqrt{2}} \\sum_{k=1}^\\infty \\sqrt{k}\\, A_k(n)\\,\n\\frac{d}{dn} \\left({\n \\frac {1} {\\sqrt{n-\\frac{1}{24}}}\n \\sinh \\left[ {\\frac{\\pi}{k}\n \\sqrt{\\frac{2}{3}\\left(n-\\frac{1}{24}\\right)}}\\right]\n}\\right) .\n", "f0005a3a7c8e257a21489b89c05ce373": " H = T + V , \\,\\!", "f000846c87ffdc9afa60f75bdefb96c1": "\\mathrm d(R_g)_e", "f00087141c5748f2f3aa90edf952b5ce": "\nS_1 = \\sum_{\\alpha |a|", "f00280ae12196bfdca561740b628c9fa": "2^p < 2^p-1", "f002a75055a8aeb66aa29db405fa585c": "(-, a+b)", "f002f673641ea0b93417425ab916b615": "\\alpha < \\epsilon_0\\,\\!", "f0030023eb4999bb66cacf60821f995f": " Q=\\iint\\Phi_\\lambda{\\mathrm{d} \\lambda \\mathrm{d} t}", "f00313bf2cb2693fe73c935dbac0607b": "\\ \\frac{n_{1} R_{1}}{m_{1}} = \\frac{n_{2} R_{2}}{m_{2}} ", "f00338d4bffacd33cf27225a04de8759": " \\tau = \\frac{Gb}{L-2r} \\,\\!", "f0033ae1a1cb6683afdd46889365a7fc": "C^q(\\mathcal U, \\mathcal F)", "f00345a4713107b720536917f3deddb9": "a_{2k} \\approx \\frac{2}{N} \\left[ \\frac{f(1) + f(-1)}{2} + f(0) (-1)^k + \\sum_{n=1}^{N/2-1} \\left\\{ f(\\cos[n\\pi/N]) + f(-\\cos[n\\pi/N]) \\right\\} \\cos\\left(\\frac{n k \\pi}{N/2}\\right) \\right]", "f00363766b08f56a407a1b2bd390840b": "\nx_\\mathrm{even} = \\cfrac{a_1}{1+a_2-\\cfrac{a_2a_3} {1+a_3+a_4-\\cfrac{a_4a_5} {1+a_5+a_6-\\cfrac{a_6a_7} {1+a_7+a_8-\\ddots}}}}\\,\n", "f00388f11596015217397418b8dfe313": "x_{j,i} \\,", "f003b3966469422a16b3649c99bf4ea6": "\\ \\beta=\\frac{3}{4}", "f003c44deab679aa2edfaff864c77402": "SE", "f003dce0702333ad1d01de6123a95259": " \\left( - \\int_\\mathbf{q_0}^\\mathbf{q}\\mathbf{F} \\left( \\mathbf{q'}\\mid \\Gamma \\right) \\cdot d\\mathbf{q'}\\right)", "f004115babfc9044c92de877305bad42": "\\pi _{2} =p_{2}-A_{2}=[\\hat{P}(\\varepsilon _{2}-\\mathcal{A}_{1})-p],", "f004707fec1ef5e9a0283575ae006724": "\\Pi = D\\sum_{j=1}^n \\oint_{S_j} w_j \\left(\\lambda R_j - L_j\\right)^2 \\operatorname{d}s", "f00494d4bc6663f4f26f632bbed2b3ee": "\\real\\,\\lambda\\leq 0", "f004fc7d932b9e221e1114dad551b8db": " \\arcsin x = \\frac{-i}{2\\sqrt{\\pi}} \\; G_{2,2}^{\\,1,2} \\!\\left( \\left. \\begin{matrix} 1,1 \\\\ \\frac{1}{2},0 \\end{matrix} \\; \\right| \\, -x^2 \\right), \\qquad -\\pi < \\arg x \\leq 0 ", "f00592c10d4c19e644db088806c53c67": "\\frac{\\lambda_1 a+\\lambda_2 b}{\\lambda_1 c+\\lambda_2 d }-\\frac a c=\\lambda_2 {{bc-ad}\\over{c(\\lambda_1 c+\\lambda_2 d)}}\n", "f005e468275e8285c1a1e5819ef7caf8": "\n \\nabla\\langle \\rho \\rangle = 0 ~.\n ", "f005fb6c4ada6cbd9bd75e6879bd5a43": " \\Delta f(x,y) \\approx \\frac{f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y)}{h^2}, \\,", "f0061562726b22feb234718af960f577": "u = -2 \\sinh\\left({\\varepsilon\\over 2kT}\\right){-\\cosh\\left({\\varepsilon\\over 2kT}\\right)\\over 2 \\sinh^2\\left({\\varepsilon\\over 2kT}\\right)}{\\varepsilon\\over2} = {\\varepsilon\\over2}\\coth\\left({\\varepsilon\\over 2kT}\\right).", "f006180918ad91120e0958db6a1153af": "\\mathbf{Z}(\\omega)=\\mathbf{F}(\\omega)/\\mathbf{v}(\\omega)", "f0061dcb1329f769a97163d6ae1cd86a": "f_i=f|_{V_i}", "f00635cf06d3c5d4b22897699a1618b8": "\\langle, \\rangle : \\mathfrak{g}^* \\times \\mathfrak{g} \\to \\mathbf{R}", "f0063b215f58661c8dbcb4b9e955ba6d": "\\Gamma (X, M^*_X) \\to \\Gamma (X, M^*_X / \\mathcal O^*_X)", "f0065926b64f7096ab4bfb4b132af6a1": "[1,\\infty)", "f00698965205aaeae7c8a11b4c6349cb": "\\displaystyle{\\partial_n S(\\varphi)(z)=\\int_{\\partial\\Omega} K(w,z) \\varphi(w)\\,|dw|,}", "f006ddd3d89dd53bedd5a5aeaee8c8ad": "\\textrm{Cauchy}(0,1) \\sim \\textrm{t}(df=1)\\,", "f0075f899ab649891cac2a694daea100": "\\star \\mathrm{d}z=\\mathrm{d}t\\wedge \\mathrm{d}x \\wedge\\mathrm{d}y", "f00772f51702e87afb276563d6e24ac7": "\\tbinom{N+M-1}{M-1}", "f0079348941d5875ed0e95222b5dee84": " FV^{-1}", "f0079348e1b8ed38914fc65aa94153d8": "{}_pF_q(a_1,\\ldots,a_p, b_1,\\ldots,b_q, x)", "f007ed0d0b432df4583ba134af50b784": "W_s=\\Big\\lbrace 1+3+5+6+9\\Big\\rbrace =24", "f008b27df0b83ff29720acd9f00d0abf": "\n u_r = \\frac{A}{r}, \\qquad\n u_\\phi = B\\left(\\frac{1}{r} - r^{\\frac{A}{\\nu} + 1}\\right), \\qquad\n p = -\\frac{A^2 + B^2}{2r^2} - \\frac{2B^2 \\nu r^\\frac{A}{\\nu}}{A} + \\frac{B^2 r^\\left(\\frac{2A}{\\nu} + 2\\right)}{2\\frac{A}{\\nu} + 2}\n", "f00906d81b78938f1766867a27ef080a": "(hk\\ell) = h\\vec{a^*} + k\\vec{b^*} + \\ell \\vec{c^*}= \\frac{2}{3 a^2}(2 h + k)\\vec{a} + \\frac{2}{3 a^2}(h+2k)\\vec{b} + \\frac{1}{c^2} (\\ell) \\vec{c}.", "f009129978049e13ecb3a10c22691ce0": "e^{i \\varphi} = \\frac{1 + i t}{1 - i t},", "f009ebc5277d0b2d0fe7e7125e6ed411": "\\phi_j(q, p)\\approx 0", "f00a289531944fae47f662122be7671d": "\\rho_{EmployeeName/Name}(Employee)", "f00a30f0b9a2a010c8b2eb66fa08dae3": "\\displaystyle D^2 / Z_o = \\eta / Z_i ", "f00a512c8553ddf1161fc48c406bb048": "\\begin{array}{lcl}\\\\\nS_1 S_3 +S_2 S_4=\\overline{AC}\\cdot\\overline{BD}\\\\\n\\Rightarrow S_1 S_3+S_2^2=\\overline{AC}^2\\\\\n\\Rightarrow S_1[S_1-2S_2\\cos(\\theta_2+\\theta_3)]+S_2^2=\\overline{AC}^2\\\\\n\\Rightarrow S_1^2+S_2^2-2S_1S_2\\cos(\\theta_2+\\theta_3)=\\overline{AC}^2\\\\\n\\end{array}", "f00aba4050247d9c938d1f4384855dd3": "\\varphi:X\\to Y", "f00acce613318349cb04ab296486fc11": "(X,Y)", "f00b42ca5adb84d31b085172562d3867": "\\scriptstyle \\sqrt{\\frac{3}{2}}", "f00b4e7a67d1a71b01ee47422e7b5113": "PG(2d+1,q)", "f00b9cf6b5c239ff98903c7d3bcffecc": "\\begin{matrix} {2 \\choose 1}{2 \\choose 1}{3 \\choose 2}{3 \\choose 1}{40 \\choose 1} \\end{matrix}", "f00cd00442b3fa1aa3c9d96fec252ac8": "(C, 0\\mapsto 1)", "f00d09fa3032ca1f018ee1e554668d55": "t_p = \\{S\\subset X \\,|\\, p\\in S\\}", "f00d1d4d41089fa4d7a0f7a9e1a5938d": " f_Y(\\mathbf{y} | \\boldsymbol\\theta, \\tau) = h(\\mathbf{y},\\tau) \\exp{\\left(\\frac{\\mathbf{b}(\\boldsymbol\\theta)^{\\rm T}\\mathbf{T}(y) - A(\\boldsymbol\\theta)}\n {d(\\tau)} \\right)}. \\,\\!", "f00d8510a6c2e9b80cb8601acb59ccbe": "0\\to A\\to B\\to C\\to 0", "f00db504aa20694175a0d6733353a434": "L = L_1 + L_2 + k \\sqrt{L_1 L_2} \\,", "f00db781f326bc7ee2c45c2fbd466dcc": "\\exists v", "f00dc4649d48b20a924776fa7c5658ef": "R=(X,Y)", "f00dc843f0f8a5dcf9a3057a9c368512": "\\mathbf{N_{G}} = \\begin{bmatrix}\n1.1151 \\\\\n0.1936 \\\\\n0.1783 \\\\\n0.0490 \\\\\n0.0106 \\\\\n0.0496 \\\\\n0.0431 \\end{bmatrix}", "f00e9078f5aebb6e24c50f2a18219b24": "y_t = \\beta_0 + \\beta_1 B(L^{1/m};\\theta)x_t^{(m)} + \\varepsilon_t^{(m)},\\,", "f00e9aab7810ff73a1d1723e9518db30": " a_n \\geq a_{n+1} ", "f00ede6a5084b2b06ea9d46b02a15f35": "\\phi(v_1+v_2) = \\phi(v_1)+\\phi(v_2)\\,", "f00f118b54b0fc12013db6e61b11f3b1": "f = \\frac{du}{dx}", "f00f3721fb5b986387bce0069ab6c1a8": "\\sigma_1 =\\sigma_\\mathrm{max} = \\tfrac{1}{2}(\\sigma_x + \\sigma_y) + \\sqrt{\\left[\\tfrac{1}{2}(\\sigma_x - \\sigma_y)\\right]^2 + \\tau_{xy}^2}\\,\\!", "f00f70be0c112242f241062c6f87471a": "(H)", "f0100cbe8a1692b6a32e791282c51f6e": "\\sigma^2_f= \\sum_i^n a_i^2\\sigma^2_i+\\sum_i^n \\sum_{j (j \\ne i)}^n a_i a_j\\rho_{ij} \\sigma_i\\sigma_j. ", "f0104b1cd12f606157e0783ec089eee9": "g(x_1, \\cdots, x_{n+1}) = \\sum_1 ^{n+1} i \\cdot |x_i|^2.", "f010c50b771af94a827c9c44f0331f27": " \\Delta_r G^\\ominus = -RT \\ln K ", "f0111d00eda2da2b548383f420493fd4": "\\mathcal{S}=\\int \\mathrm{d}^4x \\sqrt{-g} C_{abcd}C^{abcd},", "f011380470b29eceda0b349b22e2cfc0": "M(p) = 1", "f01155614f9d1a7387b6c267951e5b67": "\\nu=1/2p", "f0117c4536f70d1eac40dd005ae15b00": " \\int_{0}^{\\infty}\\delta (t-a)e^{-st} \\, dt=e^{-sa}.", "f01195960afb0a7d826fa6bf7f4522e6": " A_{t+1}=R_{t+1}(A_{t}+y_{t}-c_{t}) ", "f011a00e4c4e4fb66751154bcd3491ad": "R/(d_i)", "f01219e0f146f3353bbc3fcb3d95e1dd": "{n\\choose k_1,k_2,\\ldots,k_{m-1},K}{K\\choose k_m,k_{m+1}} = {n\\choose k_1,k_2,\\ldots,k_{m-1},k_m,k_{m+1}},", "f0127b9aa15fe973c60c6a128c01680b": "{\\partial D/ \\partial y_k} = 0 ", "f0127ff6a611203613f018c98fe7f59f": "\\Pi = \\{ a = t_0 < t_1 < \\cdots < t_k = b \\}", "f0128dc718d561c98a76a6cb0c75844a": "PA \\subseteq FO+TC", "f01292fa550fa0b0177a190fe2484a7d": "\\dot\\lambda(t) = -\\frac{(p-\\lambda(t))^2}{4} ", "f01350f5f0d95ee38555812f5f4f589d": "\\det\\begin{pmatrix}A& B\\\\ C& D\\end{pmatrix} = \\det(D) \\det(A - B D^{-1} C) .", "f013da8aad182719abc997fd1315a33c": "s_n\\ne1", "f013fdafcf1e5e4f0fd4bd69ea57443a": "k=0,1,\\dots,d-2", "f014583a3fc06803a15e8b85861f396b": "\\left\\langle\\mathcal{N}_i|i<\\omega\\right\\rangle", "f0149d674076fb0f221d5cf3bf41f1ff": " (x_1,y_1,\\ldots)(x_2,y_2,\\ldots)\\ldots(x_\\ell,y_\\ell,\\ldots)", "f014ac41af4082bda3e9e97909133957": "\\prod _x f(x) \\,", "f014cb6bb7056f8fba1e653450c9baa6": "p(y|\\theta,\\xi)", "f01538c451da9d30d08c613c1299f3b7": "\\mathbf{L}_e=\\mathbf{r}\\times\\mathbf{P}", "f01578bd4252a1a4856965199b24c7db": "F = \\frac{c\\cdot x'}{2\\lambda c\\cdot x^*} + |\\mathcal U^{(m)}|", "f015ce3e7b1a2544a46fc25aaf9119d7": "r=5, \\ \\theta=20^{\\circ}, \\ h=3", "f016206553a200701d1fbbac1295028b": "\\textstyle m_x=\\frac{M_x}{L}", "f01623889345ec4b4b538855a640e49a": "\\zeta = -\\tfrac{1}{2} + \\tfrac{\\sqrt{3}}{2}i", "f01641f8282d92aa8ee5c32352656bd6": "\\mathbf{x}_{lb}", "f01645e80c5326782ea689692bd7340e": "x_{(k+1)}", "f0166710892081a9c939a1430389d5e1": "\\phi(x,t)-v", "f016a2236909bd54673b55fb830e73a4": "K=\\lim_{n \\to \\infty}\\left[\\sum_{k=1}^{n}{r_2(k)\\over k} - \\pi\\ln n\\right]", "f016c09ee92ebd9a35e81ed4bea8cbc8": "\n\\left[\\frac{2}{p}\\right]_3 =1 \\mbox{ if and only if } \\qquad\\quad M \\equiv 0 \\pmod{2}\n", "f016dacf089c4f2f978e70585e45d32c": "\\Omega \\subset \\Bbb{R} \\times M", "f016e8eece0f9504a145b5b820c3bc6e": " g(1)=1", "f01700400c281e1bccbdc0f66d120167": "x \\wedge y = y \\wedge x", "f01746c5e0a4df087db748bf7794abb8": "X,Y\\in V ,Z\\in V^0", "f0174ecdaff7fcd5acec5ef6f594fce3": "f\\left( x,t\\right) ", "f0178ea3f1842ef8443720f70c2f41cf": "\nX_U := U(X) - \\operatorname{E}_X\\left[ U(X) \\mid \\left \\{ U(t) \\right \\} \\right]\n", "f018747002bd5470eec36b2eed20b241": " \\mathbb{E}(A) = \\langle \\psi | A | \\psi \\rangle. ", "f01886307b34c7a5afbea3312e502247": "{F_i} = {M_i ^2}.", "f0189a4fc7fb48dcf13bd1669441a2f9": "\\scriptstyle e_3 \\;=\\; -24", "f018ff924f08d62ca046d1c5a2d9933d": "{S^2}_1", "f0191f73509a0e72c2d2e6f69e8c59d2": "\\{1, \\alpha, \\dots, \\alpha^{n-1}\\}", "f0192f082489901862626f6f3b80e741": "\\frac{\\partial k_i(\\epsilon_i,t,t_i)}{\\partial t}=m\\frac{e^{-\\beta\\epsilon_i}k_i(\\epsilon_i,t,t_i)}{Z_t}", "f0197a673c40b7c712ff70cda15d4623": "|f(x)| \\le \\; M |g(x)|\\text{ for }|x - a| < \\delta.", "f01a24cfa965b2c9a8be5c88f9967ca3": " \\frac{1}{2(\\pi+2)}", "f01a4727fc6049a90d500094c30bd2c6": "\n \\frac{\\partial}{\\partial r} \\left( r\\, \\frac{\\partial T}{\\partial r} \\right)\n - h_c\\, r\\, \\frac{T - T_e}{k\\, t}\n =0.\n", "f01a5cd3b906903f0c3e510d46f812f5": "F_\\ell(\\mathbf{P},\\mathbf{K}) = P_{11} + P_{12}\\,\\mathbf{K}\\,(I-P_{22}\\,\\mathbf{K})^{-1}\\,P_{21}", "f01aa05fc005ee2c99df178a921ca5b7": "\\delta (G) = \\frac{1}{3} \\sum_{i\\in V} \\ \\delta (i)", "f01ac2fe9f8b441a5ca13ed30258e5d7": "X = (X^*_{\\sigma})^*", "f01ac86fac2920a2300a00402d5a940b": "P(\\theta|\\hat{a}_{1:T}, o_{1:T})", "f01adcda90d290e6af28aed74b1b079e": " (\\mathbf{A}\\lambda)_{ij} = \\left(\\mathbf{A}\\right)_{ij} \\lambda\\,, ", "f01b3d53e4a1e9e130cce607fbe618de": "\\phi^{-1}(0) \\phi (T)\\ ", "f01b615267093775e6a706273d9b02b5": "E_p(x)", "f01b9f3c577f0cf2569e809b3ec6eeaf": "\\sigma_{ij}(t)=\\sum_{\\nu=1}^d \\xi_{i \\nu}(t) \\xi_{j \\nu}(t).", "f01baa1a029b74b4a76df7b84eff5972": "\\omega_g(v)=(L_{g^{-1}})_*v.", "f01bde694daceeaa087102b2b65b9eb5": "X \\mapsto X^{[p]}", "f01beebb99c143f2aeeabe59bbd33ab3": " \\mathbf{d} ", "f01c31d4d5deaa8fd7f2f37af45a0e26": "\\oplus_M", "f01c4dd46664fff348cda339c0abb7e0": "n_i/(n_\\mathrm{tot}-n_i)", "f01c68b09f4351bcd363b194f69a31d9": "f(w) = w \\mathrm{e}^w", "f01c6a24018b577f7ad3d34c65c19774": "x^2 - 1", "f01d79e828b40f71ac0dc1b5c2620633": "K = - \\frac{(\\sqrt{G})_{\\rho \\rho}}{\\sqrt{G}}", "f01d8dee52c5d076e9535662d0870e5f": "\nDNF = \\frac{\\mbox{delayed neutrons}}\n {\\mbox{prompt neutrons}+\\mbox{delayed neutrons}}.\n", "f01db633e44eaecce87a6426b7434bcc": " \\begin{bmatrix}k & 0\\\\0 & k\\end{bmatrix}", "f01e02f064a04cea45cd055139e9814a": "\\Phi_D = Q_\\text{free}\\!", "f01e4713e0ca2e3adc198084925aa420": "\\mathrm{U}(n)\\,", "f01e4b1c56a61701eaab510b59e526ff": "\\chi_{[0,\\infty)}", "f01eb08ba9bf96fefa5efec11349882d": "(r_k,S_k,W_k)\\,", "f01ec296a59e61b313f788505f327eb4": "\\frac{\\mathrm{d}n_1}{\\mathrm{d}t} = \\frac{n_2}{\\tau_{21}} + \\frac{n_3}{\\tau_{31}} -\n\\frac{n_1}{\\tau_{13}} - \\frac{n_1}{\\tau_{12}} - I_{\\mathrm{out}}", "f01ee809b0578075428f34d3a3db913d": "\\scriptstyle y \\;=\\; x", "f01f10227b74095aaa4a74be43f9e7f1": "n'_i=\\sum_j w_{ij}n_j\\,", "f01f75d13dc11d96d6cab41aed03129a": "H^0(C(f)) = \\operatorname{coker}(f^0),", "f01f82910f9e7e53b6d6853c11f849b8": "\nc_n = e^{ L h / 2 } a_n + L^{-1} \\left( e^{Lh/2} - I \\right) \\left( 2 \\mathcal{N}( b_n, t_n + h/2 ) - \\mathcal{N}(u_n,t_n) \\right)\n", "f01f996d82ff6843dbd99f4e603eca3b": "D_1(P) = \\{ Q : \\vert P-Q\\vert<1\\}.\\,", "f01fa1cd65ad40f0ee255b0ef687f758": "\\mbox{PixelArraySize} = \\mbox{RowSize} \\cdot \\left | \\mbox{ImageHeight} \\right \\vert ", "f020676bb499edd993053494813761b4": "{\\sigma}/{E}\\,", "f02083c2a578574b1e623a2e0410d3c9": "K_i=k_i/k_{-i}", "f020ac5fd9ac1017430eec9fb3b31720": "\\bar{\\epsilon}^e", "f020f2d9c2bfd68708a4f909da8a6059": "P_2=(1,-\\sqrt{2})", "f02126768fb27724b5862c0984bf9aac": "\\left(\\frac{\\partial \\ln f}{\\partial P}\\right)_{T}=\\frac{v}{RT}", "f02151e6dbcb7aac2fe4b6bc0359cea0": "{1\\over2}\\times 120 + {1\\over4}\\times 100 + {1\\over4}\\times 150 = 122.5", "f0215455fe4c016a8d9d038ac4e4aab0": "\\lambda\\in\\mathbb C", "f0215b43a9e434e0f4b79d308b7bbd8e": "T \\cos \\varphi = T_0\\,", "f02165cd9f270b4e94fbe337e3f9238a": "a_p", "f02199ee8d85b7e9667dd8d48c51b1f7": "\n\\frac{1}{\\sqrt{\\lambda}} = -2 \\log [\\frac{\\varepsilon}{3.7D} - \\frac{5.02}{Re} \\log(\\frac{\\varepsilon}{3.7D} - \\frac{5.02}{Re} \\log(\\frac{\\varepsilon}{3.7D} + \\frac{13}{Re}))]\n", "f021b299e20c561fb34d33ec0c77041e": "\\mathbf{z} = \\mathbf{H}\\mathbf{r} = \n\\begin{pmatrix}\n 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\\\n 0 & 1 & 1 & 0 & 0 & 1 & 1 \\\\\n 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix} 0 \\\\ 1 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix} =\n\\begin{pmatrix} 2 \\\\ 4 \\\\ 2 \\end{pmatrix} = \n\\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\end{pmatrix} ", "f021d72d5772a788888ae1dd1a0f74da": "\\mathcal{D}(A)=H^2\\cap V", "f021dff663660a6ce6d41fd30f33db8b": "\\mathbb{Z}^\\mathbb{N}", "f021e3ee1029769478a052e6fab23d5c": "z^p - z", "f022172188ffaa99a00d7648b819ee80": "K(j \\omega)=\\frac{1}{1+\\frac{1}{j \\omega RC}}=\\frac{1}{1+\\frac{\\omega_0}{j \\omega}},", "f0227c3ecade65e4b242eb979ca4ec32": "\n \\begin{align}\n \\sigma_{\\alpha\\beta} & = C_{\\alpha\\beta\\gamma\\theta}~\\varepsilon_{\\gamma\\theta} \\\\\n \\sigma_{\\alpha 3} & = C_{\\alpha 3\\gamma\\theta}~\\varepsilon_{\\gamma\\theta} \\\\\n \\sigma_{33} & = C_{33\\gamma\\theta}~\\varepsilon_{\\gamma\\theta} \n \\end{align}\n", "f022e6b787273175f13b4e27582a094d": "\\operatorname*{ess sup}", "f023386b78996ed45eff978f32f8980f": "\\beta<1", "f0234ad0c2c1cf528de0de78b0602956": "\\Delta_K(t) = \\det(I-f_*)", "f02373e4a4f5b03add9f1d432539901e": "dI = - k_O \\frac{I_L}{V_O - V_S} dV", "f023a1be3c84894c0dc6cebec0bbdd98": "L=24a,\\quad A=12\\pi a^2.", "f023da6b9cf2661a93dc1f75a8ad3d14": "\\left\\{X\\mid \\frac{d(X,C)}{d(X,D)} = r\\right\\}.", "f024399d62a2c0fb35dcdcd3a94ee3d6": "x_k x^k + p_k p^k", "f024aa4e2188fdbdfa83dd7e392a7895": "xA[x] = A[x].", "f025071df5d8fd97cc8df70ee7542cef": "P(x,\\zeta) = \\frac{r^2-|x|^2}{r\\omega _{n-1}|x-\\zeta|^n}", "f0257a42a3731c9f79c9917fc2c0abcf": " \\xi ", "f025e79bfaae033e181757f269c62f7f": " R_k = \\{x \\in X \\;|\\; d(x, P_k) \\leq d(x, P_j)\\; \\text{for all}\\; j \\neq k\\}", "f026236f6810f5c9689154cc83477f3c": "\n \\mathbf{n}~d\\Gamma = J~\\boldsymbol{F}^{-T}\\cdot\\mathbf{n}_0~d\\Gamma_0\n", "f02636c932fa527571dd82e3921522ab": " P(x,p)=\\frac{1}{\\pi\\hbar}\\int_{-\\infty}^\\infty \\varphi^*(p+q)\\varphi(p-q)e^{-2ixq/\\hbar}\\,dq", "f02674ac59e43cb1d58c7871c77e2651": "\\textstyle\\frac{1}{6}", "f026c246d28208c62c045e2e9b207144": "t+T", "f0270ddf1abafe1e42a7d8400baf3bae": "\\Re s=\\alpha", "f02749596cffd5d21e717b3f1a920939": "T = \\frac{\\rho}{2}\\int_\\Omega w_t^2\\, dx\\, dy.", "f02782c89b7f9a0df8b8cf4a350679d2": "\\operatorname{id}(x) = x", "f027ce931945545e7bbd24c1f46a2589": "\\sum_{i=1}^k \\tau_i=\\tau", "f028002d10e2d7f543b2fe3810187573": "A_1x + B_1y = C_1,\\,", "f0280199bde1a547666019fbcbcd19ab": "P(T) = P_0 8^T = P_0 exp(ln(8)T)", "f028523aaf47dd05d85b6f52ad7d542a": "\\scriptstyle 2^{\\aleph_\\alpha}", "f028b93951e5f7e904b7e2317098e0de": "(a, a+\\Delta)", "f028dab79dfbd9dc38a4957873f448c3": " f(-x) = -f(x) \\ ", "f02902b4247f63768c826a6460081048": "\\partial P/\\partial z", "f0290fb671dec81c0f535edab5e5c51e": "A(t) = A_0 \\left(1 + \\frac {r} {n}\\right) ^ {\\lfloor nt \\rfloor} ", "f0298a9c2a402c184b2f9347fba8c139": "\\mathfrak{p}'_1 \\subset \\cdots \\subset \\mathfrak{p}'_n", "f029abe1a39fe980ca8c2f2778aaec7a": "\\pi(x) = \\inf \\left\\{ \\sum_{i=1}^n \\|a_i\\| \\|b_i\\| : x = \\sum_{i} a_i \\otimes b_i \\right\\}", "f02a090c37d115d89a45edb3563e0ea9": "\\int_V F(x) \\mathrm{d}^kX = \\int_V F(x) \\left( e_{i_1}\\wedge e_{i_2}\\wedge\\cdots\\wedge e_{i_k} \\right) \\mathrm{d}x^{i_1} \\mathrm{d}x^{i_2} \\cdots \\mathrm{d}x^{i_k}", "f02a179226bee6db11dbe16324efb02a": " \\begin{align} y_1+y_2 & = A \\sin \\left ( k x - \\omega t \\right ) \\\\\n& + A \\sin \\left ( k x + \\omega t + \\phi \\right ) \n\\end{align}\\,\\!", "f02a2e80e97b96f84f75972bc9a25de6": "f(x_1,\\dots,x_n,y_1,\\dots,y_t)", "f02a7d6bd5b589b056188d166be733e6": "p_{A,B} = \\frac{p_A-p_A\\times p_B}{p_A+p_B-2\\times p_A\\times p_B}.", "f02a8f8ad1b9074a950bab2cc1a6f0c3": "H=0", "f02ae6e310cf52b40667effeb7ff9080": "\\in \\text{SO}(3, \\mathbb{Q})", "f02afbf3511e4d7061acf9fe2a0b22bb": "S = \\frac{1}{2}\\sqrt{-\\frac23\\ p+\\frac{2}{3a}\\sqrt{\\Delta_0}\\cos\\frac{\\phi}{3}}", "f02b1d068fb168d35f42311ab0da1d8f": "P^2 t = \\vec{C}.\\vec{P}", "f02b53a08b4c0c9e988202b6393368e8": "\\textstyle P", "f02b7c94823c40b699cf2f98eab72cf3": "\\chi_p(E)", "f02bc3d29033b5259cfe5b87ae721a7e": "T^mV \\to \\bigoplus_{i+j=m} T^iV \\otimes T^jV", "f02be98c2ecb7a24e98f195293ac5abb": "\\mathbf C\\,\\!", "f02bf2ef8abb5f1975ee9b6df11eb982": "f:G \\to G", "f02bfd570e8c8076d698177bc5b7bc75": "(1-x^2)^{\\alpha-1/2}\\,", "f02c0183e3ede1978400c98897e45aa9": "\\scriptstyle S^{-}", "f02c4c9a1710bfc5eb1cdbd10384e06f": "C_1, C_2, \\lambda_+, \\lambda_->0", "f02c57cfb77a7fc773222dfc408907b0": "X=\\sum_{i=1}^k X_i \\sim \\operatorname{Erlang}(k,\\theta) .", "f02c825bbf05acee0aaa4f6300db22b2": " \\tau = 1 / \\lambda \\,\\!", "f02c9c3ed2bd551b48112ac01395e254": "n(p;H)", "f02c9cfd500b5114bbd6d79336ab0b37": "{1\\over K}d(x,y)-C\\le d(f(x),f(y))\\le Kd(x,y)+C.", "f02d06aace14b251b9c049b717029f0d": "_{s.1 \\,}\\!", "f02d37a26cfed5a6f0b4c6bbb2c78970": "R_{m,n}(x) = \\sum_{k=0}^{\\min(m,n)} \\binom{m}{k} \\binom{n}{k} k! x^k = \\sum_{k=0}^{\\min(m,n)}\\frac{n! m!}{k! (n-k)! (m-k)!} x^k.", "f02d41b410691061ef1c9a257e8e8db1": "\\psi_1 = (1+\\sqrt{5})/2", "f02d71bd2ec3e8701fe84e597f122fb2": " \\cdot ", "f02dc25aef896e2fb3e4246faf216fcb": " \\gamma(s,x) + \\Gamma(s,x) = \\Gamma(s).", "f02de92469580a861151a1798e5403b4": "\n\\frac{\\partial(x, y, z)}{\\partial(\\rho, \\phi, \\theta)} =\n\\begin{pmatrix}\n\\sin\\theta\\cos\\phi & \\rho\\cos\\theta\\cos\\phi & -\\rho\\sin\\theta\\sin\\phi \n\\\\\n\\sin\\theta\\sin\\phi & \\rho\\cos\\theta\\sin\\phi & \\rho\\sin\\theta\\cos\\phi \n\\\\\n\\cos\\theta & -\\rho\\sin\\theta & 0\n\\end{pmatrix}\n", "f02e04f6a5b648a4380622f7c856f6e8": "\n\\begin{array}{lcl}\n \\mbox{Current Dividend Yield} & = & \\frac{\\mbox{Most Recent Full-Year Dividend}}{\\mbox{Current Share Price}} \\\\\n & = & \\frac{\\mbox{Dividend payout ratio}\\times \\mbox{Most Recent Full-Year earnings per share}}{\\mbox{Current Share Price}} \\\\\n \\end{array}\n", "f02e0bde7bb5da6471efda65e3279620": "u : \\left( - \\frac{\\pi}{2}, + \\frac{\\pi}{2} \\right) \\times \\left( - \\frac{\\pi}{2}, + \\frac{\\pi}{2} \\right) \\to \\mathbb{R},", "f02e9ab1959e0128fa01e570a1dfa7f9": "\\mathcal{A} = \\frac{E}{\\omega_i},", "f02e9dee823026c8e2f58a0a0676ca8b": "h(x) = g(f(x))", "f02ea3d4855640caf2fa263373435bae": "\\mathbf{\\omega_s}={\\frac {\\mathbf{2} \\pi \\mathbf{f}}{\\mathbf{p}}}", "f02ebe3475bce9cb5d85807ea754fea4": "{\\sigma = {-R T_o \\over p}{d ln \\Theta_o \\over dp}}", "f02ee9e938f048867120340d40d8c268": "c_2=0.1", "f02eeddac5a338c49e8000b80944ee02": "\\vec{S}(1)= \\begin{bmatrix}\n1 \\\\\n1 \\\\\n2 \\\\\n2 \\\\\n1 \\\\\n1 \\\\\n1 \\\\\n1\n\\end{bmatrix},", "f02f0e6f5a2da63a368ec5509f317ae9": "\\sigma_y^2(\\tau, N) = \\text{AVAR}(\\tau, N) = \\frac{1}{2\\tau^2(N-2)} \\sum_{i=0}^{N-3}(x_{i+2}-2x_{i+1}+x_i)^2", "f02f1a6eedee827ae7040998597d0b39": "n^n,\\,(n\\log n)^n,\\,(n\\log n\\log \\log n)^n,\\,(n\\log n\\log \\log n\\log \\log \\log n)^n\\dots", "f02f44a3d86885be222abcb77d74f4f8": "\\frac a b = \\frac c x", "f02f4bca434a3317348d105213004d44": "\\Omega^{+}", "f02f4f9a56da265c4705e927eb3a585e": "\\mathbf{c}_i = [c_{1,i}, c_{2,i}, \\cdots, c_{nx,i}]^T", "f02f5869ff4155a7e62baee60c9c0cc7": "a_{E}=D_{e} + \\max(0,-F_{e})", "f02fde3e3b7ee0168670c2e5cc39459a": " \\mathbf{d} \\mathbf{c}^{\\mathrm T} - \n\\mathbf{c} \\mathbf{d}^{\\mathrm T} = \\begin{bmatrix} \n0 & c_2 d_1 - c_1 d_2 & c_3 d_1 - c_1 d_3 \\\\\n c_1 d_2 - c_2 d_1 & 0 & c_3 d_2 - c_2 d_3 \\\\\nc_1 d_3 - c_3 d_1 & c_2 d_3 - c_3 d_2 & 0 \\end{bmatrix}\n", "f030c9305e2ac44ef68792406546e933": "\n\\sum_{k=1}^{n} F_{k} = \\frac{m}{R} \\sum_{k=1}^{n} v_{k}^{2}\n", "f030f7f36fd4dd09de9fa20134c5c4aa": "[b_n,b_m]=n \\delta_{n,-m}", "f031013128debe11a53bd7d06acaeedc": "\\alpha=10^{-3}", "f03182029a70795dff87d86540c1ec7d": "z_0 = 0", "f031ab200db4c7b1f19012b4839b3390": "\\nabla X = 0", "f031c54853daff2106ab3ce95d48b8e0": "\\phi = h f_0 \\ ", "f031ed4894cb81f3b591ae2f91a4206d": "t_+", "f03225d5707a18d0716f51c3f6e1b8e7": "\\textstyle \\deg(d(x)(x^{2l-1}+1))", "f0326cfbdc95da9c5d3466ef366201d7": " \\mathbf{x}_j", "f03282e2d16def8295c6356cdb25048a": "u = \\frac{p}{m} ", "f03297bd5204b7d35ca39d806413314c": " \\operatorname{E}_{GB2}(Y^h) = \\frac{b^h B(p+h/a,q-h/a)}{B(p,q)}. ", "f032dd5ba060bfbe0a9464cf3c8d3b36": "f_p^{i,j}:H_p(K_i)\\rightarrow H_p(K_j)", "f033328d2af5f369e8c1f5a2d86b926b": "p(y^n)", "f0338b7fbc4650c18ffe2696e0db6698": " 1 \\cdot \\mathbb{P} (X<0.5) + 0 \\cdot \\mathbb{P} (X=0.5) + \\frac13 \\cdot \\mathbb{P} (X>0.5) = 1 \\cdot \\frac16 + 0 \\cdot \\frac13 + \\frac13 \\cdot \\left( \\frac16 + \\frac13 \\right) = \\frac13, ", "f033b54b58906229d2774cdc96ee5102": "Q_2(f) \\leq R_2(f) \\leq R_1(f) \\leq R_0(f) \\leq D(f) \\leq n", "f033bcc8344d310c7925bedca1be83dc": "x^5 + x^3 + 1", "f03402f17a6bb967654d6e075f8e4899": "e(t) = 1, \\forall \\in [0, 1] ", "f0341229383332e50f73483a2242c810": "(\\pm 1,\\pm 1,0,0,0,0,0,0)\\,", "f034644359a49ef412e83987a1ab8dc8": "D_5,", "f034838fae568affe0f401eca3210149": "\n{u}(r,z) = \\frac{1}{{q}(z)}\\exp\\left( -i k\\frac{r^2}{2 {q}(z)}\\right).\n", "f03497a1becbfaea027150eb3762462f": " |\\Psi\\rangle_A \\equiv \\alpha |\\Downarrow\\rangle_A + \\beta|\\phi_1 \\rangle_A ", "f034ff4bbab3b14eed806e6fd7e0e6e8": "\\mathbf{x}=\\bar{x}^i\\bar{\\mathbf{e}}_i\\,,\\quad \\mathbf{x}=\\bar{x}_i\\bar{\\mathbf{e}}^i", "f0350e5818b058dbcfd95f155e417f6a": "C_2", "f03516dc201aaa6d751b34ffaaa68d35": " -(n+2)(n-1)~r^{-n}~\\cos(n\\theta) \\,", "f03549cfa3daceee9d70cb3ce3b662cc": "(|R\\rangle+|L\\rangle)/\\sqrt{2}", "f035733210139f67499c10e9849222cd": "\\left\\lfloor \\left( \\left\\lfloor \\frac{m}{r} \\right\\rfloor + 1 \\right) r \\right\\rfloor = m", "f035dbaa209f01853b3fc390a0599f84": "X_1(z)X_2(z)", "f035e58f174144afb653398960b0867c": " k_3 ", "f0365cae4b812f248cbe4ff05fbf2292": "\\frac{175}{247}", "f03689d47f397e3eb67e7cfb75502cc7": "R = (A, B, C, D)", "f036ec68e5b7238d84da830b9feedf92": "\\mathrm{v}=r\\times\\omega", "f036ed126e37419dc37d1c0bad9761b8": "V(\\phi)=\\frac{1}{2}m^2\\phi^2 +\\frac{g}{4!}\\phi^4", "f036f69a13f3137b9ac5e67d0901cfe1": "\n\\hat{H}_{\\textrm{ph}}=\\frac{1}{2}\\sum_{n,\\alpha}\\left[w(\\hat{u}_{n+1,\\alpha}-\\hat{u}_{n,\\alpha})^{2}+\\frac{\\hat{p}_{n,\\alpha}^{2}}{M}\\right]", "f036fffec33e3980a6218a921bef6199": "x_1,x_2,x_i", "f037213e26b2adc84e862a08ab29c888": " k = A e^{ - \\frac{E_a}{RT} }", "f03733b7d264b412181f65007123edb5": "(\\theta)_n", "f0379a09949f45569383595c0903622e": "\\lim_{n\\rightarrow\\infty}\\frac{1}{n}D^{\\epsilon}(\\rho^{\\otimes n}||\\sigma^{\\otimes n}) = \\lim_{n\\rightarrow\\infty}\\frac{-1}{n}\\log \\min \\frac{1}{\\epsilon}\\operatorname{Tr}(\\sigma^{\\otimes n} Q) ", "f0379d312e6c738e4bdff975a7ab53b5": "r(x_{i})", "f037a3980525db8445fba07d5ffef758": " M = \\frac{4\\pi\\rho_M R^3}{3}", "f037ebd86f1902a64b1b26dbe57abcc4": "\\lambda_M \\colon H^n(M,\\partial M) \\times H^n(M,\\partial M)\\to \\mathbb{Z}", "f0380cdc47fcea5fd9a34e5f5d60ba79": "{\\textbf D}", "f038382d70367c08254e717ec57d2102": "I_\\mathrm{b}", "f0387b3edc5d40d9dcc9e124c166071d": "\\varphi^n = \\left( \\frac 1 2 \\left( 1 + \\sqrt{5} \\right) \\right)^n = \\frac 1 2 \\left( L(n) + F(n) \\sqrt{5} \\right). ", "f038b6f4a6fdfaefd166513f2b5d8b6b": "E_{KL}=\\frac{1}{2}\\left(\\frac{\\partial U_K}{\\partial X_L}+\\frac{\\partial U_L}{\\partial X_K}+\\frac{\\partial U_M}{\\partial X_K}\\frac{\\partial U_M}{\\partial X_L}\\right)\\approx \\frac{1}{2}\\left(\\frac{\\partial U_K}{\\partial X_L}+\\frac{\\partial U_L}{\\partial X_K}\\right)\\,\\!", "f038d268ceb94b1dc446b87ccb534b04": "O(2^m)", "f0392d0497fc5c8e08668fddfd581596": "\\delta t=-15\\pm31", "f039acf54b65961940658a2377f587c4": "d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \\frac{1}{2}. \\,", "f039dd2989e37aca198696965222b121": " \\left[\\begin{array}{c}B\\\\\\hline C\\end{array}\\right]=\n\\left[\\begin{array}{cccccc}\n1 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline\n1 & 0 & 0 & 3 & -2 & 8 \\\\\n0 & 1 & 0 & -5 & 1 & -4 \\\\\n0 & 0 & 0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 & -7 & 9 \\\\\n0 & 0 & 0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 1 \n\\end{array}\\right]. ", "f039f95b2a780aa6aa2ab16c6bae4b1d": "\\mathcal{O}(m),\\ m \\in \\mathbb{Z},", "f03a01f3a8c27e89824de7a01c8fd7e1": "\\lim_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{P} \\big[ \\sqrt{\\varepsilon} X \\in A \\big] = - \\mathop{\\mathrm{ess \\, inf}}_{x \\in A} \\frac{x^2}{2}", "f03a314c7f5165e430b1574676634b4b": "z_2^\\times", "f03a4eb0965c528ccc88b90a22b9aa9a": "\\mathcal{M}\\,", "f03a6de7fb53a3f077b7558f88d0891e": "\n\\begin{bmatrix}\n 1 & 4& -1\\\\\n 1 & 5 & 0\\\\\n 1 & 4 & 2\n\\end{bmatrix}\n", "f03a6f011e75171ac7904f3663aa8a84": "50truckloads300miles0.654lbs(CO2/Mile) =9,810lbs CO2 emitted ", "f03acde4795850ae93a3908308b227ea": " \\alpha\\to\\alpha\\vee\\beta ", "f03ad92dda59080f643c560db1db10f6": "(\\sigma_{11}, \\sigma_{22}, \\sigma_{33}, \\sigma_{12}, \\sigma_{23}, \\sigma_{13})\\,\\!", "f03ae3ec1bf98acce17aadf2958b8ea2": "V^{-}=\\{t\\in V | \\sigma t = -t\\}\\,", "f03b0d1e6ad58d1c18288233826e9d5e": "\\frac{12(5\\nu-22)}{(\\nu-6)(\\nu-8)}\\!", "f03b269ac33a25ed3a7c20068edbf87e": "\\log(k) = 0.6N + 0.6E", "f03b38dcf9461cf8c0a690bf2d8653b2": "\\mu - \\sigma \\log{X} \\sim \\textrm{GEV}(\\mu,\\,\\sigma,\\,0)", "f03bd24e2fcdfbee1b398293848555cd": " \\left\\langle { \\partial^2 V \\over \\partial \\phi _i \\partial \\phi _j } \\right\\rangle \\langle \\delta \\phi_j \\rangle =0~, ", "f03bee4b2a8189a1e13a3091c7b44560": "(D_{L_{O_2}})", "f03c285a07c530b80fb7c3351761d01c": " \\tfrac{22}{7} ", "f03c3601aa5717dafacad4bc616f0bc0": "\\mathbf{F} = m_{Planet} \\mathbf{\\ddot r} = - {m_{Planet} \\alpha}{r^{-2}}\\hat{\\mathbf{r}}", "f03c3c8a8f55d102239706a41f99b90a": "\\mu_r = 1 + 2.5 \\cdot \\phi + 14.1 \\cdot \\phi^2", "f03c5dba6a011550652f50c6a591185b": " \\frac{\\partial Q}{\\partial t} = -k \\oint_S{\\overrightarrow{\\nabla} T \\cdot \\,\\overrightarrow{dS}} ", "f03c8415ebeb6a8e83f49abc81ecb4d9": " \\mathbf{X}", "f03c9a5e709a02f962658edca3d3d145": " \nE_{\\pm} = {} - \\frac{1}{2} - \\frac{9}{4 R^4} + O(R^{-6}) + \\cdots \n", "f03cddc453a328f221dfb0da009f34e3": " \\Delta T(t) = \\Delta T_0 e^{-t/\\tau}, ", "f03d211239d455a3e2b3f63d0cba3a27": "\\Phi^{-1}(\\hat{p}_t)", "f03d45d7f3df18477dc0990b6e8d2e1f": "(C^{-1}+ VA^{-1}U)^{-1}VA^{-1} = Y", "f03db3b2fe8c97c72d996aa8b088cacf": "q_1 \\equiv \\pm q_2^{\\pm1} \\pmod{p}.", "f03dbe04497b5214143f46c953433045": "\\sigma^2_e = \\sigma_X^2 - \\frac{\\sigma_{XY}^2}{\\sigma_X^2}.", "f03e21b475f92ee50fabe9f880ee16be": "z/\\overline{z}", "f03ee1b886a091f13d2e7c4003f34002": "\\zeta(1/2 + it)=O(t^c)\\,", "f03effbcbbbbd7345d5b8f33bfc07149": "-\\overline{\\upsilon_i^\\prime \\upsilon_j^\\prime} = \\nu_t\\left (\\frac{\\partial\\bar\\upsilon_i}{\\partial x_j}+\\frac{\\partial\\bar\\upsilon_j}{\\partial x_i} \\right )-\\frac{2}{3}\\left (K + \\nu_t \\frac{\\partial\\bar\\upsilon_k}{\\partial x_k} \\right ) \\delta_{ij}", "f03f349853c0f86298dc1db9a5addd59": "\n\\begin{align}\nP(X_{(k)}< x)& =P(\\text{there are at most }n-k\\text{ observations greater than or equal to }x) ,\\\\\n&=\\sum_{j=0}^{n-k}{n\\choose j}(p_2+p_3)^j(p_1)^{n-j} .\n\\end{align}\n", "f03f37009706c33a4d043dbae972751f": "z\\in\\{0,1\\}^k", "f03f52479f7a6fc6faaf31eafbfb0d99": "\\mathrm{R}_i \\,", "f03f7867009630e99cb747dce350aa11": "N_r \\ge 10", "f03fc2035630900812f75ef20f5353f4": "((\\lambda x.x \\; \\lambda x.x) \\lambda x.(x\\;x))", "f03fe6828ba22a9ca35fa224a21a7355": "\\ddot {\\vec x}(t)=A(\\vec x(t))", "f03fe839451be3c1b0e5f7db327d5255": "\\lambda_{\\alpha_c} = 1-\\sqrt{1-4c}\\,", "f04047745707b7fa5c89dfa1561e55b2": "c^{2} (\\mu; x) = \\iint_{\\mathbb{R}^{2}} c(x, y, z)^{2} \\, \\mathrm{d} \\mu (y) \\mathrm{d} \\mu (z),", "f0405a3fd46a5183723160118a95ee03": "^{14}\\text{NO}_3^- + {^{15}}\\text{NO}_3^- \\rightarrow {^{15}}\\text{N}^{14}\\text{N}\\text{O} , ", "f04139805ff51e8f7730b6bbab83477e": "u = 0.5", "f0413e6abb3be17ceaf3703999197982": "Cr = Cr - 128;", "f0415ca02045739d081cc5b5c7871afb": "M_B", "f0418b9bc2d78123d693ce49cf553fd6": " B_k(x) = n^{k-1} \\sum_{a=0}^{n-1} b_k\\left({\\frac{x+a}{n}}\\right)\\ . ", "f041cc9ec9153ee9940088993345058e": "5x^2=16x", "f041e5961c43685750d8d8a997c1ff4f": " \\frac{\\partial}{\\partial\\bar{z}} \\left(f\\circ g\\right) = \\left(\\frac{\\partial f}{\\partial z}\\circ g \\right)\\frac{\\partial g}{\\partial\\bar{z}}+ \\left(\\frac{\\partial f}{\\partial\\bar{z}}\\circ g \\right) \\frac{\\partial\\bar{g}}{\\partial\\bar{z}}", "f0422f37c9182854fa28ebeb3144d6ee": "f(D)", "f042667a2c99d398beb650c87a21e5b8": "r=a\\,\\operatorname{cot}(x/y)", "f0432b10508661276a4ceb26cf434bef": " \n\\prod_{n=1}^{\\infty} \\left(\\frac{2n}{2n-1} \\cdot \\frac{2n}{2n+1}\\right) = \\frac{2}{1} \\cdot \\frac{2}{3} \\cdot \\frac{4}{3} \\cdot \\frac{4}{5} \\cdot \\frac{6}{5} \\cdot \\frac{6}{7} \\cdots = \\frac{4}{3}\\cdot\\frac{16}{15}\\cdot\\frac{36}{35}\\cdots=2\\cdot\\frac{8}{9}\\cdot\\frac{24}{25}\\cdot\\frac{48}{49}\\cdots=\\frac{\\pi}{2}\n", "f0433085dcf554f180f16edf28d8e558": " S + \\gamma \\left[ \\frac{DS}{Dt}- \\Delta V \\cdot S-S \\cdot{(\\Delta V)}^T \\right]= \\mu (h,d) \\left[ B + \\gamma \\left( \\frac{DB}{Dt}- \\Delta V \\cdot B - B \\cdot {(\\Delta V)}^T \\right) \\right] - gA + C_1\\left(gA - \\frac {C_2I}{\\mu (h,d)^2} \\right)", "f04387eb283eefccf6b1a4e9bc01b48a": "\\frac{V^{1/2}}{\\sqrt{\\pi}}", "f043f939d185d564003487dbb6459681": "\\left [0, \\tfrac{1}{n} \\right], \\left [\\tfrac{1}{n}, \\tfrac{2}{n} \\right], \\ldots, \\left[\\tfrac{n-1}{n}, 1 \\right].", "f044a2f1ddfbc5d0b2b4a4a1b6043a3e": "\n \\boldsymbol{P} = \\frac{\\partial W}{\\partial \\boldsymbol{F}} \\qquad \\text{or} \\qquad P_{iK} = \\frac{\\partial W}{\\partial F_{iK}}.\n", "f044a9d06eea7a2c6c21a8ee4b0c5f7b": " \\frac{\\partial \\Psi}{\\partial x} = i k e^{i(kx-\\omega t)} = i k \\Psi \\,\\!", "f044f4026223c123d31dc31b8980e251": "H_{at}", "f04510959c08fcd1fefb100a4b7e7a74": "-X", "f04553e411a8365b42844230fd721413": "\n D\\,\\nabla^2\\nabla^2 w = -2\\rho h \\, \\ddot{w} \\,.\n ", "f045a7c94ebd9eee2e1596bc315122f2": "m(x_0;p,\\beta)=E(X-x_0|X\\geq x_0;\\beta,p)=-\\frac{\\operatorname{Li}_2(1-(1-p)e^{-\\beta x_0})}{\\beta \\ln(1-(1-p)e^{-\\beta x_0})}", "f045e2015b44b6140cc1fbe25219d36b": "\nf_X(x)= \\begin{cases}\nx & 0\\le x \\le 1\\\\\n2-x & 1\\le x \\le 2\n\\end{cases}\n", "f046206d4431db2af6dfbbb70a498366": "D_{B}(p,q) = \\frac{1}{4} \\ln \\left ( \\frac{1}{4}\\left( \\frac{\\sigma_{p}^{2}}{\\sigma_{q}^{2}}+\\frac{\\sigma_{q}^{2}}{\\sigma_{p}^{2}}+2\\right ) \\right ) +\\frac{1}{4} \\left ( \\frac{(\\mu_{p}-\\mu_{q})^{2}}{\\sigma_{p}^{2}+\\sigma_{q}^{2}}\\right ) ", "f046257861a54341b09c6717b942c750": "F = \\frac{Nm\\overline{v^2}}{3L}.", "f0463f5e8777b85dadc053d2b65b85b9": "3^2 + 1^2 + 1^2 = 11", "f04682ded66ac12acec25601b8abefd3": "y_i^{(\\lambda)} =\n\\begin{cases}\n\\dfrac{y_i^\\lambda-1}{\\lambda} , &\\text{if } \\lambda \\neq 0, \\\\[8pt] \n\\log{(y_i)} , &\\text{if } \\lambda = 0.\n\\end{cases}\n", "f046a1b3002c9fd615276bb4e1a3a761": "\\delta_{i}", "f046a4540f45f615bfbe9eb6653a88a4": " T_{r}/T_{c}. ", "f04733d52654a83f4ceecf5627b66965": " i_T = i_B - \\frac {v_C} {R_{C}} \\ . ", "f047355b291f1886de7036bfccb05952": "0\\leq k \\leq n-1\\,\\!", "f0474a87d9ff46bdef72cb8fef59f2f3": "\\textstyle A=\\bigoplus_{i\\in I}A_i", "f04787ad849498a49bbc9d84ddff6995": "\\cos \\alpha = \\cfrac{dx}{dq^1} = \\frac{|\\mathbf{e}_1|}{|\\mathbf{b}_1|}", "f047a2c3ca9efaf96d9c2f7e3d857899": "|A| = |P_1(A)| = T(|A|)", "f047c2a25ca1ddb548606d725ed31055": "R^\\theta_T(x) = {1\\over |G_s^F|}\\sum_{g\\in G^F, g^{-1}sg\\in T} Q_{gTg^{-1},G_s}(u)\\theta(g^{-1}sg)", "f048027953e5f117e4f3b7e19f03b09d": "\\scriptstyle P \\left ( {a}{|}{A, \\lambda } \\right )", "f048551e6c3776de735771a6090a57ff": " T^{\\alpha\\beta}{}_{;\\beta} \\, = 0", "f0485a02f9bca3ab63e6567fa353a64c": "Q(z) = z_1^2 + z_2^2 + \\cdots + z_n^2", "f04873046ab8b00bcc67c257c5247cc2": " n_i! ", "f0487acb3b3b6bf473f26db446e73cd2": "G(t)", "f04882445e78be8a8c63b43fd99f9eb4": "p = A*x\\,", "f04882b24af5bc33e656c0573837f7c7": "(T_b f)(z) = \\exp (i\\pi b^2 \\tau +2\\pi ibz) f(z+b\\tau)= \\exp( 2\\pi i bz + b \\tau \\partial_z) ~ f (z) .", "f048a354e306fa2d0be8d0f9f4f7573e": "W ^3\\Delta_u", "f048b0f52877cfeed47c9625f78fffc4": "d(y_n,y_m)<\\epsilon", "f048f60e762c7d561d024785a67761ec": "(x_0, 0)\\,", "f04918a069230f097a60a090b84e3397": "0 \\rightarrow M_n(\\mathcal{K}) \\; \\stackrel{i}{\\hookrightarrow} \\; M_n(C^*(T_z)) \\; \\stackrel{j}{\\rightarrow} \\; M_n(C( \\mathbb{T} )) \\rightarrow 0,", "f0494393385e1187c083e610d9ac6006": "\\chi (R/P,R/Q) \\ge 0", "f0495b8eba4d6682bbbc462fd860c355": " \n E(x) = \\frac{\\theta(a+4m+b) + 3(1-\\theta)(a+b)}{6} .\n", "f04995c04da748773f13d78d1ec05a6b": "\nT_s= T_e \\left[ \\frac{1}{1-{\\epsilon \\over 2}} \\right]^{1/4}\n", "f049b3c49c23c49aa91649b9fafea04f": "\\begin{pmatrix} 3 & -4\\\\4 & -7 \\end{pmatrix}\\begin{pmatrix} \\alpha\\\\\\beta \\end{pmatrix} = 1\\begin{pmatrix} \\alpha\\\\\\beta \\end{pmatrix}. ", "f049b4d56b56924e298460c1f1e9db90": "\\textstyle k > 2", "f049e297994407ce6dcfd5ed3c1aa308": "\\langle G(x_1,x_2)\\rangle = \\left.\n-\\frac{\\delta}{\\delta J(x_1)} \n\\frac{\\delta}{\\delta J(x_2)} \\log Z[J] \\right|_{J=0}\n", "f04a05741ac952f7789a46c2d19eb0fd": "(D(r)=\\mathrm{true}) \\wedge (D(s)=\\mathrm{false}) \\Rightarrow ra.\n\\end{cases}\n", "f066888782e68fc74c761265ad9cf5c9": "\\overline{t_j}=\\sum_i r_{ij} t_i", "f0669f241811780dac6f0318b2f52ca4": "\n\\mathbf{q} \\equiv (q_{1}, q_{2}, \\ldots, q_{N-1}, q_{N})\n", "f066e1184caa1b9991cbceb207ea6341": "T_{2}", "f066ffdb7d6496966557b3c98771000a": "\\{w_{ni}^*\\}_{i=1}^n", "f0670b7c66ef0139cb0e4a63ea736e94": "f(x)=\\begin{cases}\n1 \\text{ if }x=0\\\\\n\\frac{1}{q}\\text{ if }x=\\frac{p}{q}\\text{(in lowest terms) is a rational number}\\\\\n 0\\text{ if }x\\text{ is irrational}.\n\\end{cases}", "f0674c5a30fca643dcada26967dd310f": "\\mathrm{^{238}_{\\ 92}U\\ +\\ ^{1}_{0}n\\ \\longrightarrow \\ ^{239}_{\\ 92}U\\ \\xrightarrow[23 \\ min]{\\beta^-} \\ ^{239}_{\\ 93}Np\\ \\xrightarrow[2.355 \\ d]{\\beta^-} \\ ^{239}_{\\ 94}Pu}", "f0676b748144cd8ca2121d3b7899bd04": "e_p(x) = \\left\\lfloor \\frac{1}{\\sqrt[x]{p} - 1} \\right\\rfloor\\quad ", "f067b27d0a0b5012b81f173272a11116": "c_{6}+c_{8}", "f067c04a61359fcacf5f743a11ab597e": "\\Phi(x,y_1,\\ldots,y_n)", "f067d986bc42bc3114b4609439d60f3e": "\\,\\eta_2", "f067fe215c70c35676fced4d4c5575d1": "\\omega=dx^a\\wedge d\\theta_a", "f06884074f58079b5b60ec1f227e1621": "V=P(\\rho)\\,\\Phi(\\varphi)\\,Z(z)", "f06897bd50dedbc71ccc2c1fa6e07352": "s=2^\\ell", "f0689860793026ddf1a4a328d38ac523": "P(x)=\\sum_{n\\ge 0}x^n \\operatorname{Tor}^R_n(k,k) = \\prod_{n\\ge 0} \\frac{(1+t^{2n+1})^{\\varepsilon_{2n}}}{(1-t^{2n+2})^{\\varepsilon_{2n+1}}}\n", "f068d33b6b0845954366a939ebfcb8ba": " \n\\mathcal{A}^A \\Psi_A(1,2,\\dots,N_A) = \\Psi_A(1,2,\\dots,N_A)\n", "f068dc2db9b1b4360df4aa141b9369f6": " dh = h_x \\, dx + h_y \\, dy.\\,", "f068de1d9cca4cf8af37e863c610364d": "M=|M|e^{i\\theta}", "f0695356f4a99f50818a1ae8e6a693c3": "f(x\\mid\\mu,\\kappa)=\\frac{e^{\\kappa\\cos(x-\\mu)}}{2\\pi I_0(\\kappa)}", "f06981cd3d582228860f542dc3623d65": "x_1,\\dots,x_p", "f06a350107d3bfd5e44718b4de09440c": "\\ g_m = \\begin{matrix}\\frac {2I_\\mathrm{D}}{ V_{\\mathrm{GS}}-V_\\mathrm{th} }\\end{matrix}", "f06aee1ba48d6f3d45f4f22d573fcdb9": "\\hat{P}^{i_1\\ldots i_q}_{\\,j_1\\ldots j_p}, P^{k_1\\ldots k_q}_{l_1\\ldots l_p}", "f06b8af6330e8dee86bf8a782f42352c": "cy(i) = 0", "f06bc46a0aada982f8bde987d76ac556": " \\bar{\\omega}+ \\langle T^2 - 3 \\rangle ", "f06c1ee4f6c7c464895b8728a6c0338f": "\\bar \\rho g z + \\bar p_0 - (\\rho g z + p_0) = \\gamma \\nabla \\cdot \\mathbf{\\hat{n}} \\quad \\Rightarrow \\quad \\Delta \\rho g z + \\Delta p = \\gamma \\nabla \\cdot \\mathbf{\\hat{n}}", "f06c74443bda88eaed4390746c54e80c": " p=(p_1,p_2) ", "f06d7f518135969c8f592d2b3ea3a134": "\\frac{1}{\\lambda_{\\mathrm{vac}}} = RZ^2 \\left(\\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)", "f06d9efdb6fa59405f2524dc8ba94122": "\\lim_{x\\rightarrow p}\\frac{\\|f(x)-f(p)-\\operatorname df_p(x-p)\\|}{\\|x-p\\|}=0.", "f06e301f13325db3118f290cacbe1fe3": "s_\\theta = \\pm 1, s_\\zeta = \\pm 1", "f06e994b7fcc0989c8943b925801b15c": "\\gamma = \\alpha \\times \\beta", "f06f0e6b87afd3203fb15ab7ba04292d": "d(x,y) =\\vert \\log(y/x) \\vert", "f06f76fc0a727ee78631d5c466d5d827": "N(1-R)", "f06f8487b921630cbb0e52d7d4d9a19a": "\\int_{-\\infty}^\\infty x^{2n} e^{-\\alpha x^2}\\,dx = \\left(-1\\right)^n\\int_{-\\infty}^\\infty \\frac{\\partial^n}{\\partial \\alpha^n} e^{-\\alpha x^2}\\,dx = \\left(-1\\right)^n\\frac{\\partial^n}{\\partial \\alpha^n} \\int_{-\\infty}^\\infty e^{-\\alpha x^2}\\,dx = \\sqrt{\\pi} \\left(-1\\right)^n\\frac{\\partial^n}{\\partial \\alpha^n}\\alpha^{-\\frac{1}{2}} = \\sqrt{\\frac{\\pi}{\\alpha}}\\frac{(2n-1)!!}{\\left(2\\alpha\\right)^n}", "f06fdf44976e5cfb457bc1aa673020fa": "\n\\widehat \\beta_{OLS} = (\\mathbb{X}' \\mathbb{X})^{-1} \\mathbb{X}' \\mathbb{Y}. \\,\n", "f06fdfac0216732f4305bc76d8c52030": "\\scriptstyle f_\\mathrm{ls}", "f0700fcec736b30c3c24e4f1bdf8fda5": "0.25 \\ge \\lambda > 0", "f070b669c4d6d4e96c6b87a28b012512": "\n\\frac{\\partial \\boldsymbol{U}}{\\partial t} + \\frac{\\partial \\boldsymbol{F}(\\boldsymbol{U})}{\\partial x} = 0.\n", "f070deff5c0629450c5a975b58c3c33a": "f(x)+g(x)", "f071428978fe35f4fb16d64337558e58": "i = \\sqrt{ -1}", "f0717dd461883db7964ff53201ec9fff": "P_{cr}= \\alpha \\frac{\\lambda^2}{4 \\pi n_0 n_2}", "f0722dce09b3fd39926b075413427bcd": "\\textrm{HASH}(m)", "f0722e536ef743ec845e712666987ce3": "\\frac{6x^3+5x^2-7}{3x^2-2x-1} = 2x + 3 + \\frac{8x - 4}{3x^2-2x-1}", "f0723a537e9463b7e93a0025845115f5": "\\frac{\\partial u_i'}{\\partial t} = - \\frac{1}{\\rho_i}\\frac{\\partial p_i'}{\\partial x}\\,", "f0725b4b5ed8025ac7091fa6381e7d3f": "f \\circ f = 1", "f0727cb2a648034f91f3aa53e766e500": " x=y-\\frac{b}{5a}", "f072b1547171ae0d4b65e3c4dd6b272f": "C = A \\cap B", "f072bf516567201f8ff111c6832aa30e": "\\rho^{{{jj'}}}_{\\gamma\\gamma'}", "f07307ff11efd053b42a20e07695ff05": "\\left(x_1-x_2\\right)^2+\\left(y_1-y_2\\right)^2=\\left(r_1 - r_2\\right)^2.\\,", "f07310a60c1522fa9d3427da66d11465": "EL(\\Gamma)\\le (2\\,\\pi)^{-1}\\,\\log(r_2/r_1)", "f0734cba8a470e806d1498b131e132c5": "\n\\int x^{b-1} \\Gamma(s,x) \\mathrm d x = \\frac{1}{b} \\left( x^b \\Gamma(s,x) - \\Gamma(s+b,x) \\right),\n", "f07417dfa6be0e628d23a41babbd96e5": "\\, \\sinh 2L^* \\sinh 2K = 1", "f0743b9774984e55fd5d22ff649db43d": " \\int_{0}^{L} p(x,t) {{\\partial}A\\over{\\partial}x}\\, dx = [A_0 - A(L,t)]p(L,t) ", "f07442fce713636e361516d08480f64d": "\nf( \\mathbf{x}| \\boldsymbol{p} ) = \\prod_{i=1}^k p_i^{x_i} ,\n", "f0749e665120ec4006b627df2fec3574": "S(n)=d_1+d_2+\\dotsb+d_m", "f074baa54eadad4592cb722e8fa2313c": "\\omega:2^{\\mathbb N}\\to \\{0,1\\}", "f074bb1e5052dfae4140e30671b72d4f": " g(\\cdot) ", "f074c98d982769a0bba95464f0f27e00": "\n\\begin{align}\n\\mathbb{P}(S_{X_t+1}>x) & {} = \\int_0^\\infty \\mathbb{P}(S_{X_t+1}>x \\mid J_{X_t} = s) f_S(s) \\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\mathbb{P}(S_{X_t+1}>x | S_{X_t+1}>t-s) f_S(s)\\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\frac{\\mathbb{P}(S_{X_t+1}>x \\, , \\, S_{X_t+1}>t-s)}{\\mathbb{P}(S_{X_t+1}>t-s)} f_S(s) \\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\frac{ 1-F(\\max \\{ x,t-s \\}) }{1-F(t-s)} f_S(s) \\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\min \\left\\{\\frac{ 1-F(x) }{1-F(t-s)},\\frac{ 1-F(t-s) }{1-F(t-s)}\\right\\} f_S(s) \\, ds \\\\[12pt]\n& {} = \\int_0^\\infty \\min \\left\\{\\frac{ 1-F(x) }{1-F(t-s)},1\\right\\} f_S(s) \\, ds \\\\[12pt]\n& {} \\geq 1-F(x) \\\\[12pt]\n& {} = \\mathbb{P}(S_1>x)\n\\end{align}\n", "f0750c93e8014e6e1f4d0f1f8c030b31": " \\alpha_i = \\frac{x'v_i}{v_i'v_i} = \\frac{\\langle x,v_i\\rangle}{\\left\\| v_i \\right\\| ^2}", "f07532bbf21f9ea612e6cb33a9bab35d": "\\mathrm{FOV} = 2 \\arctan \\frac {LD} {2 f_c d}", "f07548d0645b9b52fb427e6a4cd62280": "{{(ii)}_{{}}}B(v)-{{B}_{x}}(v)>0{{,}_{{}}}\\forall v\\in [y,x]", "f07550727dfb582d0f1ae4bb175b7775": "t^{1/2}", "f075b36303891bbbc6c58cefba38032d": "P(A_1|B) = \\frac{1}{P(B)} \\cdot P(B|A_1) \\cdot P(A_1),", "f075da9f92f1a2eca6f13b9820972c85": "\\text{Effective porosity} = \\text{Total porosity} - \\text{CBW}", "f075e36f04bce2d20d8e4f7437d1695b": "r_1\\equiv m\\pmod{r_0}", "f0761a6fdf1e0bdb95cf2bf05b4fb959": "\n\\begin{bmatrix}\n F_N \\\\\n \\vdots \\\\\n F_{2N-1}\n\\end{bmatrix}\n=\n-\\begin{bmatrix}\n F_{N-1} & \\dots & F_{0} \\\\\n \\vdots & \\ddots & \\vdots \\\\\n F_{2N-2} & \\dots & F_{N-1}\n\\end{bmatrix}\n\\begin{bmatrix}\n P_1 \\\\\n \\vdots \\\\\n P_M\n\\end{bmatrix}.\n", "f0761b2924edbd55265c1924f81ee6a5": " \\exp_2^{i+1}(x)=2^{2^{2^{2^{\\dots^{2^{x}}}}}}", "f0762ff17225bffa9c93f2f09736598a": " n(x,y) = n_- \\quad \\hbox{if} \\quad x<0, \\,", "f0769ced9b3c0f66f90bbbd7321db2fa": "H_*(M)=K_*(M) \\oplus H_*(X)", "f0770c7763bacd76a9fb823f17020015": "\\frac{V_c}{V_b}", "f077a300d25a9f6e15066db01dd06444": " | \\bar{\\psi} \\rangle = I U(R)|\\psi \\rangle = \\sum_{mm'} | j , m' \\rangle \\langle j, m' | U(R) |j , m \\rangle ", "f077aa653215a54bed8a651e9ee1dad7": "G=4\\pi \\log\\left( \\frac{ G(\\tfrac{3}{8}) G(\\tfrac{7}{8}) }{ G(\\tfrac{1}{8}) G(\\tfrac{5}{8}) } \\right) +4 \\pi \\log \\left( \\frac{ \\Gamma(\\tfrac{3}{8}) }{ \\Gamma(\\tfrac{1}{8}) } \\right) +\\frac{\\pi}{2} \\log \\left( \\frac{1+\\sqrt{2} }{2 \\, (2-\\sqrt{2})} \\right)", "f077ff5b69156fb3ea89466b68266cbd": " c = -2 =M_{2,1}\\,", "f07801f0e7d4421ba282a849fe98d45e": " (r \\, \\cos 2 u, r \\, \\sin 2 u, - r \\, \\cos u) ", "f0781e32e305853ee27a9e2489b7052c": "\\gamma =\\frac{1}{\\sqrt{2}}(\\delta -j\\chi )", "f078e0a53111b2b126edd8ed5495b9b9": " {n \\choose k} = {n-1 \\choose k-1} + {n-1 \\choose k}\\text{ for }1 \\le k \\le n - 1. ", "f078f05d23183f83667d006daf4e1486": "\\omega(X_1,X_2,\\dots,X_n) = 1.", "f079859876b41edeead2b0b94c56d9d5": "(\\pi,2\\pi)", "f07a5c73dd7e891a292b73e29a1bac6a": "\\hbar\\omega_3 = \\hbar\\omega_1 + \\hbar\\omega_2 ", "f07a8281a0ca0c0e33bb101398cea87f": "(a+c) = hm", "f07ac0134ca2ac48d75440ede56b9f22": "\\begin{pmatrix} 15 & 17 \\\\ 20 & 9 \\end{pmatrix}\\begin{pmatrix} 0 \\\\ 19 \\end{pmatrix} = \\begin{pmatrix} 11 \\\\ 15 \\end{pmatrix}", "f07ac59817c676df0c7d377428466dbf": "\na_{\\overline n|i} = \\left( \\frac{(1+i)^n - 1}{(1+i)^{n+1}}\\right )\\left( \\frac{1}{1-1/(1+i)}\\right)\n= \\left( \\frac{1-(1+i)^{-n}}{1+i}\\right )\\left( \\frac{1+i}{i}\\right)\n= \\frac{1-(1+i)^{-n}}{i}.\n", "f07af2bd6a14744dc10870a8bc28f001": "u\\in U\\setminus V", "f07b016e146f04fd4fc171e78825d86f": "= 2 \\left( {t \\over a} - \\operatorname{floor} \\left( {1 \\over 2} + {t \\over a} \\right) \\right)", "f07b3738c80caa51a51e5da80f63307b": "\\forall N>\\kappa_\\varepsilon\\ ,\\sum\\limits_{n=N}^\\infty ||a_n|| < \\frac{\\varepsilon}{2}", "f07b87602d27cd67c204e5a8c9486c85": " \\Psi_4 = \\frac{1}{2}\\left( \\ddot{h}_{\\hat{\\theta} \\hat{\\theta}} - \\ddot{h}_{\\hat{\\phi} \\hat{\\phi}} \\right) + i \\ddot{h}_{\\hat{\\theta}\\hat{\\phi}} = -\\ddot{h}_+ + i \\ddot{h}_\\times\\ . ", "f07bd4c9a7c35d7483fbb160187ed32c": "c^2=a^2\\cos^2\\theta+(b\\sin\\alpha-a\\sin\\theta)^2,\\,", "f07c0f79255016321768b8d785d59da2": "P = \\frac{2}{R}\\gamma", "f07ca6a82c63bcd5e2f9bfb18926aaa0": "f_1,\\ldots,f_{r+1}", "f07ce9ca00cdfb94d8c5f7e30deb64a9": "w_i = p \\cdot x_i^*", "f07d000b391b809b7c1809881d36d3ed": " x_2 = \\frac{b_2 - l_{2,1} x_1}{l_{2,2}}, ", "f07d678b521ec267d267b6a55a68ad1e": "\nW_{11} =\n\\begin{bmatrix}\n\\lambda & 0 & 0 & 0 \\\\\n 0 &\\lambda & 0 & 0 \\\\\n 0 & 0 & \\lambda & 0 \\\\\n 0 & 0 & 0 & \\lambda \\\\\n\\end{bmatrix} = \\lambda I_4, \\quad\nW_{22} =\n\\begin{bmatrix}\n\\lambda & 0 \\\\\n 0 &\\lambda & \\\\\n \\end{bmatrix} = \\lambda I_2, \\quad\nW_{33} =\n\\begin{bmatrix}\n\\lambda & 0 \\\\\n 0 &\\lambda & \\\\\n \\end{bmatrix} =\\lambda I_2, \\quad\nW_{44} =\n\\begin{bmatrix}\n\\lambda \\\\\n \\end{bmatrix} = \\lambda I_1\n", "f07db2dabbb553e6a9f42b6e386ec88d": "S(t,u) = (1-u) p(t) + u q(t)", "f07ebb20b9fbc2e9345b71dfbeb6ba4d": "u = e^{i \\theta_1} \\cos r \\,\\!", "f07f198a2d22104761d52f5d82dd1dca": "\n\\begin{array}{rc}\n\\frac{1}{U} \\ll f''(x) \\ll \\frac{1}{U} \\ ,& \\varphi(x) \\ll H ,\\\\ \\\\\nf'''(x) \\ll \\frac{1}{UV} \\ ,& \\varphi'(x) \\ll \\frac{H}{V} ,\\\\ \\\\\nf''''(x) \\ll \\frac{1}{UV^2} \\ ,& \\varphi''(x) \\ll \\frac{H}{V^2} . \\\\ \\\\\n\\end{array}\n", "f07f89a0dd86e1321b539511d1fa6933": "\\partial (u,\\mathrm{id}) = \\mathrm{id}", "f07f9fe4450af8df2c51355098708416": "w(n) = \\alpha - \\beta\\; \\cos\\left( \\frac{2 \\pi n}{N - 1} \\right)\\,", "f07fb783fbd4feea99abe2366970084b": "Ef(a) = \\frac{a}{f(a)}f'(a)", "f08005611b589e9ac31f3229fa4edf25": "y=x^2/a,", "f080257e2128f86ff19ede0bc230eb03": "I(n)", "f0802c225b1cafacd1b5aa0a175f34cc": "\\tfrac{m}{n}<3", "f0807afcd485628399d38f97c9037546": "q(t)\\in Q", "f0807c8c623bbafb36dbdb9e24cdfc4d": "i_k", "f080825962c0a5e97349c1d35fee920c": "\\sqrt{12}", "f080b3a863f29a19d34ce44365164749": "D^+(\\mathcal A) \\rightarrow K^+(\\mathrm{Inj}(\\mathcal A))", "f080d630696b4bd835d73c81a8561c33": "[0, + \\infty) \\subsetneq \\mathbb{R}", "f0810fcb1ab3aec055e6cca54519a7b1": "M_\\infty", "f081372b70a55f5731bc12b432eb8132": " T_\\alpha {}^\\lambda {}_\\gamma = T_{\\alpha \\beta \\gamma} \\, g^{\\beta \\lambda}, ", "f08138a70103ee766eabdeff71f50df2": " P_i = \\varepsilon_0 \\sum_{j\\in\\{x,y,z\\}}\\chi_{ij} E_j \\quad.", "f0814a5e1d741a11746432ec78adc03a": "\\frac{ \\left( \\frac{ K^- + \\bar{K}^0 }{2} \\right)^2 + \\left( \\frac{ K^+ + K^0}{2} \\right)^2}2 = 246\\times10^3~\\mathrm{MeV^2}/c^4", "f0819a983d059c37adbd2185be77d621": "dF = N_v [ F_0 + k( \\nabla c)^2 ]~dV", "f0819c4ba2d9d5f226ea2db8eec33fe1": " H_x = \\mathbf{H}_n \\quad x \\in X_n ", "f082210f24c3208ae0df5604b6a79578": "\\mathbf{\\Phi}_\\mathbf{xx} = \\underline{\\mathbf{X}}(\\ell)\\underline{\\mathbf{X}}(\\ell)^H", "f082311e4f113ebf4c21f7655bf2dd65": "\\partial h/\\partial z=0", "f082317c615219e5f8a6f9f36024ba64": "T(X_1^n)= \\left( \\prod_{i=1}^n{x_i} , \\sum_{i=1}^n{x_i} \\right),", "f0826116f36d37786b8957c244479b43": "\\begin{align}\n \\Psi({\\mathbf{r}}) &= \\Psi_{0}({\\mathbf{r}}) + \\int{G({\\mathbf{r,r'}})V({\\mathbf{r'}})\\Psi({\\mathbf{r'}})d{\\mathbf{r'}}}\n \\end{align}", "f082d5220c3e142819f2dd8edbe44a07": " \\operatorname{inc} ", "f0831e56a4cbe2f063ca46f55ddb5f33": "V_{GNM} = \\frac{\\gamma}{2}\\left[ \\sum_{i,j}^{N} (\\Delta R_j-\\Delta R_i)^2 \\right]= \n \\frac{\\gamma}{2}\\left[ \\sum_{i,j}^{N} \\Delta R_i \\Gamma_{ij} \\Delta R_j\\right]", "f0832c9ac6d1981987cbad72886acf8a": " x^{43112609} + x^{3569337} + 1. ", "f083947b436bc8acf73726bc790b2971": "h=x_g-c\\left(1-\\frac{\\partial \\epsilon}{\\partial \\alpha}\\right)\\frac{\\frac{\\partial C_l}{\\partial \\alpha}}{\\frac{\\partial C_L}{\\partial \\alpha}}\\frac{l_t S_t}{c S_w}", "f083d9e47ab37e94b5ce7ac8f0ad48a2": "\\psi(\\Omega^{\\Omega^2+\\Omega 2+\\psi(0)})", "f083de0da94acf3708b647d1da990f81": "\\|x-y\\|\\geq \\epsilon", "f083f74fc91dfab29be92f40aaa2a451": "\\left(\\mathcal{F},K\\right)_{\\otimes}^{n}", "f084112952b07bc4109770779b0d9fd1": "f^k\\left(x\\right)=0", "f0842d5070d2c160b32a7e573be83dde": "R = 0.12 * \\frac{1 - e^{-50 * PD}}{1 - e^{-50}} + 0.24 *\\left(1- \\frac{1 - e^{-50 * PD}}{1 - e^{-50}}\\right) ", "f0844d9b70f184e1efe2345af8590d00": "\\Delta{t}", "f0849440484778156922ed9e3ee2d980": "\\angle EDF", "f084bb36a916d519cd3013cdcb63f85f": "\\ tan(tan^{-1}(x)) = x", "f084cb25b2dc4d3762387760894f1a7f": "\n\\epsilon = \n \\begin{bmatrix} \n \\epsilon_{1\\,1}& \\epsilon_{1\\,2}& \\epsilon_{1\\,3}\\\\ \n \\epsilon_{2\\,1}& \\epsilon_{2\\,2}& \\epsilon_{2\\,3}\\\\ \n \\epsilon_{3\\,1}& \\epsilon_{3\\,2}& \\epsilon_{3\\,3}\n \\end{bmatrix}\n \\quad\\quad\\quad\\quad\n\\sigma = \n \\begin{bmatrix} \n \\sigma_{1\\,1}& \\sigma_{1\\,2}& \\sigma_{1\\,3}\\\\ \n \\sigma_{2\\,1}& \\sigma_{2\\,2}& \\sigma_{2\\,3}\\\\ \n \\sigma_{3\\,1}& \\sigma_{3\\,2}& \\sigma_{3\\,3}\n \\end{bmatrix}\n", "f0857772be8af434ecb358af5fbd28dc": "\nF_D^{(n)}(a, b_1,\\ldots,b_n, c; x_1,\\ldots,x_n) = \n\\frac{\\Gamma(c)} {\\Gamma(a) \\Gamma(c-a)} \\int_0^1 t^{a-1} (1-t)^{c-a-1} (1-x_1t)^{-b_1} \\cdots (1-x_nt)^{-b_n} \\,\\mathrm{d}t, \\quad \\real \\,c > \\real \\,a > 0 ~.\n", "f08577aa5e81ef3d2438533c91e1ae04": "\n\\dot p_k = \\frac{\\partial L}{\\partial q_k}\n", "f0859da8411f061d007428947f6345f6": "a_{n}^{-}", "f085b3146935c9a312a78088f5caead6": "T0", "f09291a2934068644f41981dd51eb5b0": "E=mc^{2}\\,", "f0930baa2f883feae5a5f0f1e4745b27": " \\nabla^2\\Phi(\\mathbf{r}) = \\left(\\frac{1}{r} \\frac{\\partial^2}{\\partial r^2}r - \\frac{L^2}{\\hbar^2 r^2}\\right)\\Phi(\\mathbf{r}) = 0 , \\qquad \\mathbf{r} \\ne \\mathbf{0},\n", "f0935da69c506f9721e6acaef77a4c21": "\\mathcal{C}^m\\left(U,Y\\right) ", "f0937b29f6165a331c7f2e7227343c5e": "(\\mathbf{1},\\mathbf{2},1)", "f093dd557ff7aa8ded8314321a57a36a": "\nS_{1} = V_{0} sin(\\omega t)\n", "f094298a60590932a37f3c25bf3d42ae": " Lower~limit = m - t_{0.975,11} \\times\\sqrt{\\frac{n+1}{n}}\\times s.d. = 5.33 - 2.20\\times\\sqrt{\\frac{13}{12}} \\times 0.42 = 4.4", "f0942a0b3b4206b9656365c17142a331": "\\operatorname{var}(X+Y)=\\operatorname{var}(X)\n+2\\operatorname{cov}(X,Y)+\\operatorname{var}(Y)\\,", "f0942f7d61b26e00c013dd46142a84e9": "\n\\int_\\Omega c\\cdot u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x =\nc\\!\\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}(x)\\mathrm{d}x =\n-c\\! \\int_\\Omega \\langle \\boldsymbol{\\phi}(x), Du(x)\\rangle\n", "f094fe5de59f02208225095beb5b65f3": " \\textbf{a} = \\textbf{f} \\cdot (\\textbf{r} \\cdot p\\textbf{f}_q \\cdot \\textbf{g} + \\textbf{m}) \\pmod q ", "f0951e5a9080c4e98b4504b46e51922a": "H_U", "f0952e745c94e2e260d100a12d19defb": "E^0", "f09564c9ca56850d4cd6b3319e541aee": "Q", "f095a3608a512f8c4a119496dbd63919": " \\hat{f}^n_{i-1/2} ", "f095ece607a0890a418a687a2a76ec77": "\\hat x_{4}=\\sum_{i=1}^{3}w_{i}x_{i}", "f09612c6e74bae26d68d1848ed562be8": "a(x).p \\not \\sim_e a(x).q", "f0967b9db38b5df5ee54214ab33ea42c": " W = N!\\; / \\; \\prod_i N_i! ", "f09689b22443b11a5c7b3cb51fc4a2dd": "\\phi_{\\text{sys}}", "f096916b1a3fbedb4c634c70bc1a5166": "\\mathrm{H}(p, q) = \\sum_{x_i} p(x_i)\\, \\log \\frac{1}{q(x_i)} \\!", "f09695273ca74c1b69bdbe5a40d62a99": "[\\cdot,\\cdot]:\\mathfrak{g}\\times\\mathfrak{g}\\to\\mathfrak{g}", "f0970a8ce6ff1a59b3ded47f9a8c9603": "C_{super} - C_{normal} = {T \\over 4 \\pi} \\left(\\frac{dH_c}{dT}\\right)^2_{T=T_c}", "f09710bb997baac4244cc7849f5d53db": "{dp \\over dt} = {dp \\over d \\tau} \\Big/ {dt \\over d \\tau} = {- \\lambda m \\omega^2 x \\over \\lambda}= - m \\omega^2 x.", "f0978d697bf87db479973dac5d93b3af": "x^2 + 10^{-5} x - 5 \\times 10^{-9} = 0", "f0979dc0db70c843a49c564ed24cceea": "x \\in B", "f097ab202ddbc681fc00187e48679865": "W_{i}=W_{f}+K\\, (24-0)^{2}", "f0983f7fdfde464535608129a3435859": "\\vec{p}=(p_x,p_y)^T\\,\\!", "f0985e9a69727535b42bee07e43cfd44": "A=\\mathbb{Z}", "f09891aa890a533e788724b892c2a7c5": "out_b", "f09898e7a4ac33ec0eb2920d78708824": "\\frac{2\\pi}{\\Delta t}", "f0989c93588a4d7aed1e0bcb92b9d64a": " x(t) = \\int_{-\\infty}^{\\infty} \\left[ \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} X(\\tau, \\omega) e^{+j \\omega t} \\, d\\omega \\right] \\, d\\tau. ", "f098bcb0dfc59a6737e0264616e39d44": "12 \\,", "f098c13939ef1ab91195597724fa39a1": "\\frac{dT}{dt} = \\frac{P}{c \\cdot m}.", "f098e23e148c36f42668697f9b6c55e1": " \\min\\limits_{x \\in X, y \\in Y}\\;\\; F(x,y) = ( F_{1} (x,y),F_{2} (x,y),\\ldots,F_{p} (x,y) )", "f098e7a835e13d988e2dfef5a4e99b2e": " \\begin{bmatrix} \\lambda_1 & 0 & \\cdots & 0 & \\cdots \\\\ 0 & \\lambda_2 & \\cdots & 0 & \\cdots\\\\ & & \\cdots & \\\\ 0 & 0 & \\cdots & \\lambda_n & \\cdots \\\\ & & \\cdots & \\cdots \\end{bmatrix} ", "f099111adc6161259aa45cad296ef957": "P_\\ell^{m}(\\cos \\theta)\\ \\sin (m\\phi)\\ \\ \\ \\ 0 < m \\le \\ell.", "f099142b0719e95449e9a9d00930dd0c": "C_{in}: A^k \\rightarrow A^n", "f09925d2d7b2d0f00a35c0199d79bae1": " (\\mathbf{A}\\mathbf{B})_{ij} = \\sum_{k=1}^m A_{ik}B_{kj}\\,. ", "f099ea7bcc986d7fd823e74a8bd82955": "f \\colon D^2\\to M \\, ", "f09a93cd36ff6d9db08ceee63d8ae6e6": "\\dot{q},\\dot{Q} \\,\\!", "f09abce6d8071f641a1a1dca946b532f": "U(x)=8e^{-0.25 x^2}", "f09ae2f6803842ddf5fbd48c27c190e6": " \\left(\\frac{P}{\\rho N^3 D^5}\\right)", "f09aec5d5f1ebe13fd212b8232f16980": "C_{|x|}(x)=1", "f09bad12ae4669493362e9c3a8e2de8a": "\\mathfrak{p}_0\\subsetneq \\mathfrak{p}_1\\subsetneq \\ldots \\subsetneq \\mathfrak{p}_n\\subseteq \\mathfrak{p}", "f09bd10f3afabb8eb0eff7e49e3d865c": "\\theta\\wedge d\\theta=(x+y+z)\\,dx\\wedge dy\\wedge dz", "f09bfafa4f3d2c18e008559c30467a9e": "s(*, *)", "f09c6468ae337844d418532be2c7eefd": "\\mathfrak{g}_1", "f09c69564dcad89e735d4e5472c6f4d3": "\\sigma = \\int \\rho \\; dz", "f09c92c644d291a6fd759afb593cfb9b": "P_{\\mathrm{bismuth}}=7.5\\pm0.3\\times 10^{-4}\\frac{K}{T\\cdot A}", "f09c9fe5ce6b708bbac65b96c2723366": "(x_1^0,\\dots,x_n^0, z^0)", "f09cd21b48fc39b4aa1cfca14a4b7484": "\\Rightarrow \\dfrac{Y(s)}{X(s)} = \\dfrac{G(s)}{1 + G(s) H(s)}", "f09ce310856dae980a1fa6f2a7cf4bf6": " \n{1 \\over 2}\n\\begin{pmatrix} \n1 & 0 & -1 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\ \n-1 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 0\n\\end{pmatrix}\n\\quad\n", "f09d60820c046e256ea72927120dafeb": "\\overline{\\mathbf{GT}_{1}}", "f09d8c9e9115439290a7f3de80e4c7e4": "\\Lambda =v\\tau ", "f09de9b0152d3c13a3577efed62ff019": "H_\\Phi(\\mu)", "f09e33ba1fc4c3f1635f1c8e04033d50": "P\\left(\\frac{d}{dt}\\right) \\delta(t)", "f09e3aa2fd8f6eb9c67d57b074cb136a": "R_\\mathrm{AU}", "f09ee53c8d519a6068b8e24896192e76": "s_1\\neq s_0", "f09efbe6274ddb1b88f39e5668ea5dfe": "\\scriptstyle(4.8\\pm4.4)\\times10^{-32}", "f09f07163640c24b852a637337b55844": " {{(k+1)\\sigma^2} \\over {N[k+1 + 2(k+2)\\ln(k+1)]}}", "f09f0ca6378e692c1b2a928f38f0533b": "\nX_{4}=[6,9],\n", "f09f8a49b651b073e714de10a15c2323": "\\frac{1}{k}\\frac{(N-k)(N+1)}{(k+2)} \\approx \\frac{N^2}{k^2} \\text{ for small samples } k \\ll N", "f0a00a56b0d35a1de769130269dcb162": "\\mathbb {RP}^2", "f0a01159592dba766f16cf0203df63ea": "\\kappa \\equiv \\plusmn 1", "f0a01f0962861351ad7bfdba617d33e5": "\\phi=e^{i\\omega}", "f0a0607fbb23fe5e13de32d709b334bb": "\\mid, \\nmid, \\shortmid, \\nshortmid \\!", "f0a07509e28e874cd076ab1507e1ea63": "\\vec{f}=\\sum_\\alpha m_\\alpha \\vec{x}_\\alpha.", "f0a0a56f88de9f690993dfcc1095f074": "SampEn", "f0a0ba4de86f6cb74935f7ebe3308003": "r\\rightarrow \\infty", "f0a0bfe4713ae093961b06ba90410d1e": "\\exist x \\forall y Lyx", "f0a0fd0ee2b58b7b9844dff14d42a032": "\\langle A \\rangle _\\sigma", "f0a16785e00c0cbd35ef3fa15ae4a516": " \\nabla \\cdot \\mathbf{B} ", "f0a1d251d6bfc0a9750b62c5433d18b7": "\\Omega^k(M)", "f0a1d44e996a9328d52f312d4f60cf01": "dim(Hom(M_{\\mathfrak{p}}(\\mu), M_{\\mathfrak{p}}(\\lambda)))", "f0a278dc73323ef586958222a4bce8d3": "(R, \\mathfrak{m})", "f0a3865199588257d0ced68b98ad1764": "I_{n,k}=\\left[\\frac{k-1}{2^n},\\frac{k}{2^n}\\right)", "f0a3bcd461314ea706715adcb5dc241b": "\n \\frac{1}{2\\pi i} \\oint_C f(z) dz = \\sum_{k=1}^n \\underset{z=a_k}{\\mathrm{Res}} f(z),\n", "f0a4485de5845fe3574a44f463e61c26": "g(t,y,0) \\equiv 0", "f0a4b84b4cf0773f5b5a5ce9345a1cbe": "\\textbf{d}_j = \\begin{bmatrix} x_{1,j} \\\\ \\vdots \\\\ x_{m,j} \\end{bmatrix}", "f0a4e0cae5934b45caf14bd5c4b14c3c": " [\\operatorname{L}_g \\psi](h) = \\psi(g^{-1} h). ", "f0a511825c6327978dc8adc38ef5f57b": "E_n^{(0)}", "f0a51adcfc24b4434a53bb5bbad107dd": "\\dfrac {100 V_{ref}} {R_{d} f_{clk}} \\le C_{slope2} \\le \\dfrac {1000 V_{ref}} {R_{d} f_{clk}}", "f0a559626378fefbf8e772e415e2232b": "x_{0,1,2,3}", "f0a6084fd5cb9d0caffc76f8e771c114": "\\mathbb{G}_m = GL_1", "f0a62f1bfe40ac405beb764b13425e10": "F \\in \\Sigma", "f0a660ddd515fbe0427acccb72af8fc1": "df = \\frac{\\left(\\frac{s_1^2}{n_1}+\\frac{s_2^2}{n_2}\\right)^2} {\\frac{\\left(\\frac{s_1^2}{n_1}\\right)^2}{n_1-1} + \\frac{\\left(\\frac{s_2^2}{n_2}\\right)^2}{n_2-1}}", "f0a6adc41beb374d38ad3ce5e4722954": " \\text{DJIA} = {\\sum p_\\text{old} \\over d_\\text{old} } = {\\sum p_\\text{new} \\over d_\\text{new} }.", "f0a6c100bb2b1840c330f3369c9a649c": "\\overline{u_i u_j}", "f0a6cae4ec0d09c5c1bf7d11beab9818": " A^{\\mu} ", "f0a75dfd3b7be44c01bd46e940529675": "Y = G + j B \\,", "f0a75f50f7dbe031351141c9a5976371": "\\scriptstyle a ^{ b ^{ c ^{ \\cdot ^{ \\cdot ^{ \\cdot}}}}}", "f0a77464d06eb4740c2b0fb930cb6e34": "\n\\forall \\sigma\\ _{-i} \\in\\ \\Sigma\\ ^{-i} \\quad \\quad\n\\pi\\ (\\sigma\\ _i ,\\sigma\\ _{-i} ) \\le \\pi\\ (\\tau\\ _i ,\\sigma\\ _{-i} )\n", "f0a7a3bdbcb2879f1b674de2a6057a06": "R(U,V)W=\\nabla_U \\nabla_V W - \\nabla_V \\nabla_U W -\\nabla_{[U,V]}W", "f0a7b1acab49d398cf6cf06f52880731": "\\mathbb{F}_q^m", "f0a7eb598ec37ac023147f12eed67387": "U{}^3_6", "f0a80dec465247b6b09df78451edefe8": "\\varepsilon_n !", "f0a8248a292739104889fbceab9eeee6": "\n\\begin{align}\n m_{k+1} &= \\sum_{i=1}^{\\mu} w_i\\, x_{i:\\lambda} \n \\\\ &= m_k + \\sum_{i=1}^{\\mu} w_i\\, (x_{i:\\lambda} - m_k) \n\\end{align}\n ", "f0a85755b2a118427524ac949f8fd121": "\\int_{\\mathbf{R}^{d}} \\exp \\left( - 2 \\Phi (x) \\right) \\, \\mathrm{d} x = 1.", "f0a861cedad26f02ca70b35e97c5e7f7": "\\lim_{\\alpha \\rightarrow \\infty} \\frac{1}{\\pi} \\frac{sin^2(\\alpha x)}{\\alpha x^2}= \\delta(x)", "f0a8a4090bfb0a75842361730d0dcaca": "(\\sin x)^2\\ ", "f0a8c4a342ba0374a50da2ba07dbd556": "\\;K[C]=K[x,y]/(y^2+h(x)y-f(x))", "f0a8fe10fafcf5d0dae090bf5386d285": " n\\rightarrow \\infty ", "f0a9465ff01978b12f43795adc0b6e11": "S_{RNB(n)} = S_{RBN(n-1)} + 1 + S_{NBR(n-1)}", "f0a963bc598dddc35a0d24a0b9a78e58": "\\mathbf{a} = \\dot{\\boldsymbol{\\beta}} c", "f0a99ed27edf17a407549e6263cf1cf5": "\\operatorname{curl}(\\mathbf{v}) = \\nabla \\times \\mathbf{v}", "f0aa2352b7672bee44ece4adec9da2da": "\\int_a^b f(x)\\,dx \\approx (b-a) \\, f\\left(\\frac{a+b}{2}\\right).", "f0aa8de75e0ef03c8207dd3d785a229d": "\n \\dot{\\boldsymbol{\\varepsilon}}_{\\mathrm{vp}} = \\cfrac{\\boldsymbol{\\sigma}}{\\lambda}\\left[\\cfrac{||\\boldsymbol{\\sigma}||}{\\lambda}\\right]^{N-1}\n ", "f0ab10a2cfad0b6c475705c2251ec048": " K_{0(NC)} = 1 - \\sin \\phi ' \\ ", "f0ab152fe61fc16bca81142f82cce14d": "\\pi\\,", "f0ab37a5b3ecdfd12ca1c4859162f43d": "\\frac {dr} {d\\theta} = -\\frac {1} {s^2} \\cdot \\frac {ds} {d\\theta}", "f0ab5a13a8cc59ae90d3da507f4e70b0": "\\mathrm{COP}_{\\mathrm{cooling}} \\equiv \\frac{Q_C}{W_{in}}\\,", "f0ab90b68f5bad6a1687b9a2cbf030bc": "L_o(x, \\vec w) = L_e(x, \\vec w) + \\int_\\Omega f_r(x, \\vec w', \\vec w) L_i(x, \\vec w') (\\vec w' \\cdot \\vec n) \\mathrm{d}\\vec w'", "f0abc521219070c123081821d5210203": "0< p < 1", "f0ac06cfbbe97e82ab401c39708f1480": "C,x_0,w_0,p_0", "f0ac30566587afc5df2878116b6bb34a": "\nv_e = v_0 \\,\\,\\,\\,\\,\\,\\, \\mathbf x \\in \\partial \\mathbb H\n.", "f0ac57d669c3d00ab57a9ea1508d28b2": " \\gamma(t)\\,= \\, \\begin{pmatrix} x_1(t) \\\\\nx_2(t) \\end{pmatrix}\\, ", "f0ac5df22dd0c5a0fd0aa498ab17d542": "c = (a-1)(b-1)", "f0ac753a9c8bba767b83fc47da005ed0": "p_0=p,\\qquad p_{j+1}(x)=(-1)^{\\deg p}p(\\sqrt x)\\,p(-\\sqrt x),", "f0ac7e00e377fb4bbe2dfec5212b1cc5": "r_\\mathrm{Earth}= 6.371 \\times 10^{6}\\,\\mathrm{m} ", "f0ad41daf28a6803f59ce3c49e2af13f": "\\omega_0={1\\over\\sqrt{LC}}", "f0ad50972cba9fb0ee23f1eeede43c59": "F _{n+1} (x, 0) = x, \\ n \\ge 0\\,", "f0ad8670e140b52fd5300408e4693802": "\\sum_{i=0}^{n-1} c_i*x^i ", "f0ad90dbbc47f16ea8ebaa28c157d738": "\\mathbb R^3\\,.", "f0ae243602de41bd55b05c321dc14fc3": "_\\Delta", "f0ae261110307fed2754d98c56870819": "\\psi^L = \\frac{1}{2}(1-\\gamma_5)\\psi", "f0ae6ae9898ca5337d05f0bcad65f996": "\n \\mathrm{E}\\left[X^p\\right] =\n \\begin{cases}\n 0 & \\text{if }p\\text{ is odd,} \\\\\n \\sigma^p\\,(p-1)!! & \\text{if }p\\text{ is even.}\n \\end{cases}\n ", "f0ae956d668944cf9c527c7f5e99120e": "N_g = 0.04\\,{T_d}^{1.25}", "f0aea8b2c5e721f330ebba51cab4d9c1": "x \\in (0,\\infty)\\setminus \\{b\\}", "f0aed2bfa382f9742c5b7f8a526e37bb": "exp\\left(2ik\\Omega d^2/v \\right)", "f0aef7cacec66e7cbf2f40b5bf43b2c1": "\\scriptstyle \\frac{6}{k}", "f0af25a9bc1e29dc04385fdb72dd61ec": "e^{\\frac ax}\\,", "f0af3c4e779b0294f0ee2d460a0c1270": " \\mathcal{E}(\\mathbf{u}) =\\frac{1}{2} \\int_{S} \\eta^{2}dS. ", "f0afac1c323085173c097617d4f9056a": "\\sum a_n", "f0afde525594aa893e81490398b24776": "K=\\ell^2(\\mathbb{N})", "f0b0237df44a87765073d08f0f0bcaed": "\\hat{H}_\\text{D} = 2g_I\\mu_\\text{B}\\mu_\\text{N}\\dfrac{\\mu_0}{4\\pi}\\dfrac{\\mathbf{N}\\cdot\\mathbf{J}}{\\mathbf{J}\\cdot\\mathbf{J}}\\dfrac{\\mathbf{I}\\cdot\\mathbf{J}}{r^3}", "f0b054754a63e04812f1cd329696777a": "\nk^*= arg\\min_k(o_{ik})\n", "f0b0800de0da0de9e0d4d3d49e04bb51": "{y =\\Phi x}", "f0b0c9ec598c0a4b2e742850932514a5": "\\mathbb{Z}/n\\mathbb{Z} \\cong \\mathbb{Z}/p_1^{r_1}\\mathbb{Z} \\times \\mathbb{Z}/p_2^{r_2}\\mathbb{Z} \\times \\cdots \\times \\mathbb{Z}/p_k^{r_k}\\mathbb{Z}", "f0b0e231dfb9d562da1ced1782eabefb": "K_\\nu", "f0b10eef51613b34d4be3f10a9c18406": "\\psi_2=Be^{ikx}\\left( \\begin{matrix} 1 \\\\ 1\\end{matrix} \\right) ,\\quad \\left|k\\right|=V_0-E_0 \\,", "f0b131e3d4835c23055e7086c00561cb": " \\Pr(x \\le m - a \\sigma_-) \\le \\frac { 1 } { a^2 }.", "f0b164afa44bf12e6544ed93efad5b48": "b \\,\\ ", "f0b183831ebfa8e586a748401c99df57": "\\zeta \\,", "f0b199ccba4a8e87a0b5e7e1b5ccb91e": "\\mathbf{a}^{\\downarrow}\\in\\mathbb{R}^d", "f0b1e0f82f56886358285082d65ab614": "\nV_{\\ell=0}(\\mathbf{R}) =\n\\frac{q_\\mathrm{tot}}{4\\pi \\varepsilon_0 R}\\qquad\\hbox{with}\\quad q_\\mathrm{tot}\\equiv\\sum_{i=1}^N q_i.\n", "f0b1e399e53d3720f096b7ae613f14b5": "\\displaystyle \\frac{1}{i\\pi \\xi}", "f0b20652ebaf9c8e91dd71efb9728736": "x^6 \\cdot 2^{-3} = \\frac{5 \\sqrt{5}}{8}", "f0b237d3c9c7a8479919e4d9d8e791e3": "\\mathrm{0.08\\overline{3}}", "f0b28dd2b61cd0ad7f626c11915d98f4": "\\nu_1=\\omega_a-\\omega_b, \\nu_2=\\omega_a-\\omega_c", "f0b29715bc6f0ad86a18021c05a127b4": " \\log_b a = \\frac {1} {\\log_a b} ", "f0b2978bc9dabc75509e528ee248416c": "(x^{q})", "f0b32c39d566166c172c3f6eee23dd8b": "k[x_0, \\ldots, x_n]/I", "f0b33b77c70dbb9c302aa2db6ee925f7": "|N|=\\{1,\\ldots,N\\}", "f0b360182de480e29dec06ad85a387ee": "R_3", "f0b378aba64fa667c407d7aa2bdf51a7": "\n \\mathbf{n}\\cdot\\boldsymbol{\\sigma}^{+} + \\mathbf{n}\\cdot\\boldsymbol{\\sigma}^{-1} = \\mathbf{0}\n \\qquad \\text{or} \\qquad\n \\mathbf{n}\\cdot[[\\boldsymbol{\\sigma}]] = \\mathbf{0}\n ", "f0b37c0e675060af4364e5305a2e5fe6": "x(\\tau) = \\int_{-\\infty}^{\\infty}\\left[\\int_{-\\infty}^{\\infty}S_x(t,f)\\, dt\\right]\\,e^{j2\\pi f\\tau}\\, df", "f0b3a44624842b3bfa56f422387d7d0d": "\\mathbb{C}\\,", "f0b3abb3f5de31649095447711c6fecc": "A\\parallel B", "f0b40cec51cf4b55142853e18b11f64e": "z = a + b\\,\\omega\\qquad(\\omega = e^{2\\pi i/3})", "f0b446db185fd4f77d56ebc20895e143": "L(Y,i)=\\frac {M} {P}= \\left (\\frac {CY} {2i} \\right )^{\\frac {1} {2}}", "f0b4a6ec717292ca19a7991573e3c0d9": " \\mathbf{p}=m\\mathbf{v} \\,\\!", "f0b4fc68b63918324eea1038d75537fc": "\\langle u_n: n\\in \\mathbb{N} \\rangle", "f0b527b8ddc0035647decd6da38636bf": "F\\left[X\\right]", "f0b54a470c6b04e8a7369c308e76f2ad": "p\\,\\! = \\frac{h}{h+t}", "f0b5c5b8dd9bb490315e41f608d5b77c": " p_k(d\\sigma_k | \\eta) ", "f0b5ea25a76e7cb1b95cf298ca4fdff6": "\\Pi_{\\mathbf{k},\\omega}", "f0b610107369c5124a61fdc3dffdc88f": "\\rho \\frac{\\operatorname{d}v}{\\operatorname{d}t}= -\\frac{\\operatorname{d}p}{\\operatorname{d}x} ", "f0b61eaf87a2516fdf9716c14ea3c31a": "\\left( \\begin{array}{cc} \\cos\\theta_\\mathrm{P} & - \\sin\\theta_\\mathrm{P} \\\\ \\sin\\theta_\\mathrm{P} & \\cos\\theta_\\mathrm{P} \\end{array}\\right) \\left( \\begin{array}{c} \\eta_8 \\\\ \\eta_1 \\end{array}\\right) = \\left( \\begin{array}{c} \\eta \\\\ \\eta' \\end{array} \\right)", "f0b672eb6bd298fbda32a9943670eebc": "T\\to \\mathbf{Gr}(r, \\mathcal E)", "f0b67c757f8aef21f775ae3af05c3b34": "\nH=\n\\begin{pmatrix}\na_1 & a_3 & a_5 & \\dots & \\dots & \\dots & 0 & 0 & 0 \\\\\na_0 & a_2 & a_4 & & & & \\vdots & \\vdots & \\vdots \\\\\n0 & a_1 & a_3 & & & & \\vdots & \\vdots & \\vdots \\\\\n\\vdots & a_0 & a_2 & \\ddots & & & 0 & \\vdots & \\vdots \\\\\n\\vdots & 0 & a_1 & & \\ddots & & a_n & \\vdots & \\vdots \\\\\n\\vdots & \\vdots & a_0 & & & \\ddots & a_{n-1} & 0 & \\vdots \\\\\n\\vdots & \\vdots & 0 & & & & a_{n-2} & a_n & \\vdots \\\\\n\\vdots & \\vdots & \\vdots & & & & a_{n-3} & a_{n-1} & 0 \\\\\n0 & 0 & 0 & \\dots & \\dots & \\dots & a_{n-4} & a_{n-2} & a_n\n\\end{pmatrix}.\n", "f0b6c1f7cfc9676c486d520d540bbd5d": "\nH_e = 1 - \\sum\\limits_{i=1}^{m}{(f_i)^2}\n", "f0b743ed460e8a0cccd25057e24ee26d": "\\Gamma(x)=\\prod_{i=1}^k(x\\alpha^{k_i}-1).", "f0b75a29dcfe5ed393b934950225d4d3": "w(x) dx", "f0b76d64f71a6cef3a53b74c59bd522f": "X^2-X=X(X- I)", "f0b7eb800dd0d2fc501757ded2df3786": "BJD_{TT} = JD_{TT} + \\frac{|\\vec{r} + d \\, \\hat{n}| - d}{c}", "f0b8072bfd7d3e8bdca40eed995c0edd": "d \\Xi = d \\Phi - \\frac{P}{T} d V - \\frac {V}{T} d P + \\frac {P V}{T^2} d T", "f0b817e33ef55b0b2888d4cb0afb1a2b": "U_i U_{i-1} U_i = U_i", "f0b8184b365cf213f31c81112139e224": "j=1,2,\\dots,k", "f0b877ddf3ebfaa0f2120569da66c432": "\\int_{-\\infty}^{\\infty}P_n^{(\\lambda)}(x;\\phi)P_m^{(\\lambda)}(x;\\phi)w(x; \\lambda, \\phi)dx=\\frac{2\\pi\\Gamma(n+2\\lambda)}{(2\\sin\\phi)^{2\\lambda}n!}\\delta_{mn}", "f0b88ba8e081048d2dfb67227739de6e": "H(\\omega_k)", "f0b8e2adc3a20381b256fe97e53ea372": "I_{n,m}= a^2I_{m-2,n}-I_{m-2,n-1}\\,\\!", "f0b91d4686f8cf2bfd54e3fbe56cfafc": "\\dot{q}_p=\\omega \\rho_b c_b (T_a-T) \\quad [2]", "f0b93b9b96d955d2d39655dcf8ea99a1": "\\cos\\,{\\theta(t)}= \\cos\\,{\\theta_\\text{0}}+ ({\\cos\\,{\\theta_\\infty}}-\\cos\\,{\\theta_\\text{0}})({1-\\mathrm{e}^{\\frac {-t}{\\tau}}}) ", "f0b99bdebf38611ca536a7161446b204": "(a \\cosh t, b \\sinh t)", "f0ba5c5c4359fff4cbf904bf70270024": "n 0,", "f0f78226940760b4cf782838822e3aa9": "\\delta_p(x)=1.", "f0f79156083c7774f82be50a478502c4": " \\gamma_k ", "f0f7f2a474a1ab8ebaf8c8e5ec844c0d": "\n \\sum M_A = -V_1 x + M_1 = 0 \\,.\n ", "f0f822a70295921bfd1b9640e708488d": "\\Pi(n,m)", "f0f822dda18f32b616b427be24b95f68": "0\\leq i \\leq 31", "f0f8324e3684c693fef9dc756082c973": "\\frac{60}{88}(4)=2.72", "f0f86bb9511d364d1af01edd1eb6d9ce": "\\nu(y,\\alpha) =\\begin{cases} -\\mathbf{sign}(y) \\tan(\\pi \\alpha / 2)|y|^\\alpha & \\alpha \\ne 1, \\\\\n(2/\\pi)y \\ln |y| & \\alpha=1. \\end{cases}", "f0f90950b89e56d37e561a21b5a97952": " H_n^{(r)} = \\sum_{k=1}^n H_k^{(r-1)}. ", "f0f92abc2d99f3523e5de4ae7c84cb68": "\n \\cfrac{\\partial{W}}{\\partial \\bar{I}_1} = C_1 ~;~~ \\cfrac{\\partial{W}}{\\partial \\bar{I}_2} = C_2 ~;~~ \\cfrac{\\partial{W}}{\\partial J} = 2D_1(J-1)\n ", "f0f949325b2f9e8e2a7afbb09ea2c316": "2^{64}-1", "f0f959f7f171a19fcf15dc1b2198b591": " \\overline{F}(x_1,\\dots,x_k) = \\left(1 + \\sum_{i=1}^k \\frac{x_i-\\mu_i}{\\sigma_i} \\right)^{-a}, \\qquad x_i > \\mu_i, \\quad i=1,\\dots,k, \\qquad (3)\n", "f0f9a1727f149979d2043da26a39e82b": " F(\\mathbf{k}-\\mathbf{p}/2) ", "f0f9d5ad5e68bdcc34d2959aa5d55bfa": "x_s", "f0f9f778d32fa8232e3a04521fa5ff85": "\\frac{Dv}{Dt} = -\\frac{1}{\\rho} \\frac{\\partial p}{\\partial y} - f u + \\frac{1}{\\rho} \\frac{\\partial \\tau_y}{\\partial z}", "f0fa53ad5f7cf2c8cb6127251db43b1e": "H(s) = \\frac{A(s)}{D(s)}", "f0fb57031fe94b9a0f6199fe16e08a40": "\\displaystyle{Q(a)L(a^{-1})=L(a).}", "f0fb5f19829c6efe0d7eec9bf01f5fb2": "\n D_f(P\\!\\parallel\\!Q) = \\int \\!f\\bigg(\\frac{dP}{dQ}\\bigg)dQ \\geq f\\bigg( \\int\\frac{dP}{dQ}dQ\\bigg) = f(1) = 0.\n ", "f0fb6150e1746dcdceae3c909e0d47d0": "\\frac{1}{2 \\pi c}", "f0fba5d3dab68bf863536733a55a7f3f": "i+\\mathrm{deg}\\left(P_{i'}\\right) S", "f0fd1f4c2811f46d661df0d406ec7f36": " c_{2} = c_{1}+\\epsilon_{2}", "f0fd82b1555acb57349d47b9c82d2289": "f_n = a_n f_{n-1} - c_{n-1}b_{n-1}f_{n-2}", "f0fdc8acf4fe6434cb0a13c51cf0ff11": "\\frac{\\partial f}{\\partial\\theta} = f \\, \\frac{\\partial \\log f}{\\partial\\theta}.", "f0fdd6777e1223c59e8cfca4e77c6065": "\\text{450 km} = \\frac{C}{0.033 \\times 2 \\times 10,000}", "f0fde6650bd63e5dbec71ff89bfb2413": "P_1 = X_1-R-N_1 \\cdot x \\,", "f0fe10c257c3e220b1f779986bf3d36f": "\\le_{P*Q}", "f0fe508b3de538ec69c69c5642ccf1ac": "\\lim_{k\\rightarrow\\infty}\\left(P^k \\right)_{i,j}=\\boldsymbol{\\pi}_j,", "f0fe6f9586ebb20afa374169d6355472": "_{qp+qp'\\,}\\!", "f0fe9bbcc35a2f44e297c289680479fe": "S \\Rightarrow_{f} AA \\Rightarrow_{g} BA \\Rightarrow_{g} BB", "f0fe9ee74afecdbd4eed0c62824ca50f": "\\theta \\geq \\theta_2", "f0fea2b8765a4eef55c3ad4fca95cb62": " \\left(\\frac{a}{n}\\right) = \\left(\\frac{a}{u}\\right) \\prod_{i=1}^k \\left(\\frac{a}{p_i}\\right)^{e_i}. ", "f0fed445a98840a60a7dd03908ee5ff3": "\\{0,1, \\alpha, \\alpha^2, \\alpha^3,\\dots,\\alpha^{p^{m}-2}\\}", "f0fee8de9790220802b7a6a7df99b401": "\\mathcal{O}_X^n|_U \\to \\mathcal{O}_X^m|_U", "f0ff7d1a81693f3108d7ab5b72b8af87": " \\{(x_1, y_1),\\dots, (x_n, y_n)\\} ", "f0ffb70c46c39e90622e98e9af0dc9d8": "n_\\Sigma \\sin\\theta_\\Sigma=n_S \\sin\\theta_S \\ ", "f100126d984824aaaaae4087f0b22bb2": "\n\\text{for all component } j: \\frac{\\partial C_j}{\\partial t} = \\sum_{i=1}^{m} a_{ij} r_i \\; ,\n", "f1012e8e8c7e2fbdbefeb887867eaefb": " \\delta\\ \\boldsymbol {\\epsilon} \\equiv \\boldsymbol {\\epsilon}^* ", "f101465b804990bda9ac86abb8269ed2": "\\mbox{(1) }\\mathcal{K}\\subset T\\mathcal{K}", "f1019e6480a1373fe7943626dbf26d6f": "(m - i + 1)(n - i + 1)", "f101ecfff0287b15b6e8198ba70ef761": "p = \\tfrac{\\beta}{\\alpha + \\beta}", "f102320c21a220f4db684b93796ec665": "-\\frac{\\partial{H_z}}{\\partial{x}} = \\varepsilon\\frac{\\partial{E_y}}{\\partial{t}}", "f10263d597c3a42a93807be640ed1a22": "Q = ", "f10284c259d1991bd5ea84dbd579d1ce": " K = \\frac13\n \\begin{bmatrix}\n Q_{xx}-Q_{yy}-Q_{zz} & Q_{yx}+Q_{xy} & Q_{zx}+Q_{xz} & Q_{yz}-Q_{zy} \\\\\n Q_{yx}+Q_{xy} & Q_{yy}-Q_{xx}-Q_{zz} & Q_{zy}+Q_{yz} & Q_{zx}-Q_{xz} \\\\\n Q_{zx}+Q_{xz} & Q_{zy}+Q_{yz} & Q_{zz}-Q_{xx}-Q_{yy} & Q_{xy}-Q_{yx} \\\\\n Q_{yz}-Q_{zy} & Q_{zx}-Q_{xz} & Q_{xy}-Q_{yx} & Q_{xx}+Q_{yy}+Q_{zz}\n \\end{bmatrix} ,\n", "f1029b73c857a8cac0b1a71f4a31fed4": "\\Delta\\vec{p} = \\vec{p}-\\vec{q} = \n\\begin{bmatrix}\np_1-q_1\\\\\n\\vdots\t\\\\\np_n-q_n\n\\end{bmatrix}\\!", "f102a788de47e8d238b24857b07c1414": "G=\\sum_i x_i{g_i}", "f102b38d660211da40e6dee91b7a77f4": "\\beth_1 = |P(\\mathbb{N})| = 2^{\\aleph_0}.", "f10306d3c98290031e7a2ce2c133ded5": "\nI_C = j\\omega C V_{in}\\,\n", "f1031a00b8e2f957421ab9c4875b9553": "\nL_{p2} = L_{p1} + 20 \\cdot \\log_{10} \\left( \\frac{r_1}{r_2} \\right)\n", "f1037a75448baa21f7d3afdef1e72d26": "\\mathbf{r}_0 := \\mathbf{b} - \\mathbf{A x}_0", "f103c8bd1622c2c10ac4a4e852ffa820": "\\boldsymbol{\\xi}", "f104d6d73d69a99bc9aad2b652ab0826": "\\Gamma\\ ", "f104d758c463cd1c71b57600cb3e5701": "p(v_i)", "f104f6f2766c0097852b43bc5409d33f": "I(X;Y|Z)", "f104fc6f76745f9b89a8f1fd9c838e19": " \\hat{\\mathbb C} ", "f10576fe7853087a9bfd6f7eace4c7fa": "(P \\to Q)", "f105867e45ef19a4f0c01765e96ca32d": "S_n = \\sum_{i=1}^n X_i,", "f105a5b93d9206a76fde1cfd83c123bd": "R = \\sin\\theta", "f105b0807f60fc9da21bd265103849b0": "F = m\\cdot a", "f105bfc5511e43a986644c09e077b349": "\\gamma\\approx e^U", "f1060800ebc2c06690dcf7c0993c5dc2": "P \\and \\neg Q", "f10613a3ac8e07b9d2f18cb7fd8ad933": "B=\\theta", "f10652828fee94c6b47cd6da1ada98e2": "x = \\sqrt{\\frac{WU}{V}},\\ y = \\sqrt{\\frac{UV}{W}},\\ z = \\sqrt{\\frac{VW}{U}}.\\,", "f1066f952c856ba449a7260947ab7a4f": "D(a) = \\begin{pmatrix} \nD^{(1)}(a) & 0 & \\cdots & 0 \\\\\n0 & D^{(2)}(a) & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & D^{(k)}(a) \\\\\n\\end{pmatrix} = D^{(1)}(a) \\oplus D^{(2)}(a) \\oplus \\cdots \\oplus D^{(k)}(a) ", "f1066fdec67f7306b5861ede9105fa2a": "\\ell(\\theta) = \\operatorname{E}[\\, \\ln f(x_i|\\theta) \\,]", "f106e56682eaf79ff9db5daa5d3d1a70": "h_{PR}", "f106ecda32450cab9908cb7a803d9d26": "\n\\int f(x)\\,dx = \\sum_{n=0}^\\infty \\frac{a_n \\left( x-c \\right)^{n+1}} {n+1} + k = \\sum_{n=1}^\\infty \\frac{a_{n-1} \\left( x-c \\right)^{n}} {n} + k.\n", "f106f634805406221e278720cb3b894f": "\\Delta \\langle 2\\rangle = \\langle 2\\rangle-\\langle 2\\rangle_\\mathrm{S}", "f10704b9a176ef7bd3deea6858cb724e": "\\mathbf{u}=\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}=\\mathbf{e}_1,\\quad \\mathbf{v}=\\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}=\\mathbf{e}_1+\\mathbf{e}_2,", "f1070bd24327953bd5bd2b50477186da": "\\mathcal{L}_Yf=Y(f)", "f10747d670059835df89a3da7198d1fc": "x\\leq\\alpha", "f1075526a1af83825712577107d636ff": "f(z)=z^me^{P(z)}\\prod_{n=1}^\\infty\\left(1-\\frac{z}{z_n}\\right)\\exp\\left(\\frac{z}{z_n}+\\ldots+\\frac{1}{p}\\left(\\frac{z}{z_n}\\right)^p\\right),", "f107ec97c214c803dfb284c3ca1af1a9": "\\frac {f{(x)}+f{(-x)}}{2}", "f108442c8a8180f87a3b8c9a9ca2dfb0": "u_j^{n}", "f1087fe8d8e034adb76ddf33655f1c52": "\\begin{bmatrix}y_{1,t} \\\\ y_{2,t}\\end{bmatrix} = \\begin{bmatrix}c_{1} \\\\ c_{2}\\end{bmatrix} + \\begin{bmatrix}A_{1,1}&A_{1,2} \\\\ A_{2,1}&A_{2,2}\\end{bmatrix}\\begin{bmatrix}y_{1,t-1} \\\\ y_{2,t-1}\\end{bmatrix} + \\begin{bmatrix}e_{1,t} \\\\ e_{2,t}\\end{bmatrix},", "f108a3d88b22ff91ddbd459b0f359bc9": "\\sqrt x", "f108ee1f8204747098dc0b3859bf9000": "\\frac{da^{\\tau}}{ds} = \\frac{e}{m} u^{\\tau}u_{\\sigma}F^{\\sigma \\lambda}a_{\\lambda} \n+ 2\\mu (F^{\\tau \\lambda} - u^{\\tau} u_{\\sigma} F^{\\sigma \\lambda})a_{\\lambda},", "f108fac60cf8dbd50bf5b4407aa6ced0": "2\\omega_{p}", "f1091acea622efee63efbac47b662d42": "\\Delta_N[n]=\\sum_{k=-\\infty}^\\infty \\delta[n-kN],", "f1091da5208dac21cc87faa3b530886f": "S[\\sigma] \\to T[\\sigma]V^{\\prime}[\\sigma]", "f109980574420a25a340bfb1b8a109d8": "\\Delta P = A_1 \\,\\rho fL \\frac {Q^2} {d^5}", "f109a3472bfeed7b04fbae33d3f6a402": "\n P_f \\; = \\; 1\\, -\\, e^{- (\\sigma_N /S_0)^m}\\ \\text{where}\\ S_0 = s_0 (l_0 / D)^{n_d/m} \\Psi^{-1/m}\n", "f10a3413de7b6f86a45fd3b8e30f0e97": "-1=x^2+y^2", "f10a3fd674ef5f9904481118ab315f57": "\\mathbf{A}\\mathbf{g}'(x\\mathbf{A}) = \\mathbf{g}'(x\\mathbf{A})\\mathbf{A}", "f10a8962d12ba02da4e9b243a7fda721": "u_\\mathrm{r}(n_\\mathrm{A})_\\mathrm{min} \\mapsto \\partial \\left(\\frac{(R_\\mathrm{A}-R_\\mathrm{B})}{(R_\\mathrm{A}-R_\\mathrm{AB})(R_\\mathrm{AB}-R_\\mathrm{B})} R_\\mathrm{AB} \\right) / \\partial R_\\mathrm{AB} = 0", "f10a91410ee8a92a22c4b4abf860fa6d": "-\\frac 2 3 A I^{3/2}", "f10acb41a2b009086190e5d8e7d82a02": "\\Kappa = \\frac{\\langle (\\nabla_2 \\nabla_1 - \\nabla_1 \\nabla_2)\\mathbf{e}_1, \\mathbf{e}_2\\rangle}{\\det g},", "f10b3bf1d744ddc860cac68dbdaeb430": "{r \\choose k}=\\frac{r\\,(r-1) \\cdots (r-k+1)}{k!} =\\frac{(r)_k}{k!},", "f10bba608e28e7ee885af18028f2af28": "a \\sim t^{p_1},\\ b \\sim t^{p_2},\\ c \\sim t^{p_3}.", "f10bc3c94b77e1d6b9f98106daf335c1": "x,y", "f10be55f7ca3ed9eb11c79f8a1f13e86": "\\lambda = \\frac{h}{p} = \\frac {h}{{m}{v}} \\sqrt{1 - \\frac{v^2}{c^2}}", "f10beb858971cc284bc4210012f259a0": " {G^a}_a = -R", "f10befe71005308486f7c114a3fddce2": " v = \\frac{V_\\max / K_{m1} (S - P/K_{eq} ) }{1 + S/K_{m1} + P/K_{m2} } ", "f10c13f25e43bd81f4794da8c2595d9c": "Z_i | Z_{i-k}, \\ldots, Z_{i-1}, Z_{i+1}, \\ldots, Z_{i+k}", "f10c6d7a8741ba31ce9ba79bb48ce4fa": "\\left(\\tfrac{D}{n}\\right)=-1,", "f10c7f0996239d403ba568f6a022a182": "\\{x_{i1}, \\ldots, x_{ip}\\}_{i=1}^n", "f10c97fea975c8b514c6f976b7550f44": " \\mathrm{pd}_R(M) + \\mathrm{depth}(M) = \\mathrm{depth}(R).", "f10cd0a8aece45fb5da9f376ab551a90": "\\sin\\theta_1\\sin\\theta_3+\\sin\\theta_2\\sin\\theta_4=\\sin(\\theta_1+\\theta_2)\\sin(\\theta_3+\\theta_4).\\;", "f10cd98ea6f7313e348ce0b2f71586ac": "\\frac{\\partial y}{\\partial x_i}=\\frac{\\partial f\\left(x_1 ,x_2 ,\\ldots,x_n \\right)}{\\partial x_i}", "f10cf719c95cb18d3ee2ae97dc5a944d": " A = \\sum_{q\\ge 1} \\sum_{ (p, q)=1 \\atop 1 \\le p < q }\\pi \\left( \\frac{1}{2 q^2} \\right)^2.", "f10d0a53a4ec7d137246d85de4dbb333": "\\Phi_n(s) = \\sum_{i=1}^{n} \\frac{a_{ni}}{s+\\alpha_i}", "f10d1e6641a47b075b7d45824c05fe62": "A_1=A_2=\\ldots=A_N", "f10d5329b74b2e9af290b79ab6670c46": "\\Delta M_J = 0, \\pm 1, \\pm2, \\pm 3", "f10d5b16539174511404ae87fd3e7069": "U=\\{(a^n,b^nc^m)\\mid n,m\\in\\mathbb N\\}", "f10d5f41d0bcd61f524b81d07087205b": "A(z) = \\sum_{k = 0}^{\\infty} \\frac{B(z)^k}{k} = \\ln\\left(\\frac{1}{1-B(z)}\\right).", "f10da95c70cd9a452eca0d42dc7ad0f4": " {\\Gamma, x:A \\vdash t : B : K \\over \n{\\Gamma \\vdash (\\lambda x:A . t) : (\\forall x:A . B) : K}} ", "f10df1490b9990b66cb33c91b7e8c736": "v_D^2 = v_{D,~r}^2 + v_{D,~\\theta}^2 + v_{D,~z}^2", "f10e276cf10b5b04f0406f199e6b7158": " \\langle \\Phi , \\Phi \\rangle = \\int\\limits_{\\mathrm{ all \\, space}} d^3\\mathbf{p} \\, \\left | \\Phi \\left ( \\mathbf{p}, t \\right ) \\right |^2 = 1.", "f10e56fd58a8dcce3e97f39073225fff": "x[n] = \\left \\{\\cdots, 0.5^{-3}, 0.5^{-2}, 0.5^{-1}, 1, 0.5, 0.5^2, 0.5^3, \\cdots \\right \\} = \\left \\{\\cdots, 2^3, 2^2, 2, 1, 0.5, 0.5^2, 0.5^3, \\cdots \\right\\}.", "f10e605af8a36d5b7bd980d3a036832d": "H^2(\\Omega)", "f10e68ad48285f7508968a9fa4f3bed1": " x=2E_e/{m_\\mu}c^2 ", "f10e8736dd04f35eb861df151a7a68e0": "\nx+\\sin (xy) \\leq 0, \n", "f10e88ed72e5c4db5719eee29170c38c": "I = \\textstyle{\\frac{1}{3}}(R + G + B)\\,\\!", "f10ee481e21da6a3eac2acc1f5e1769f": "\nH^* = V(x, y) + u_1 \\left(p_x + \\tfrac{q B}{2c}y\\right) + u_2 \\left(p_y - \\tfrac{q B}{2c}x\\right).\n", "f10f03c9836c36537d2539196058bfa2": "\\delta ", "f10f36ca3ce3993c07e3ca9b8078b2f6": " P(I|E) = P(E|I) \\cdot \\frac{P(I) }{P(E)} ", "f10f379cf2ae8b64967fd4fd58517636": "S=\\mathbf y^{\\rm T} (I-H)^{\\rm T} (I-H) \\mathbf y= \\mathbf y^{\\rm T} (I-H) \\mathbf y,", "f10f46fa728b00aeceef9b098e14896f": " \\boldsymbol\\mu_n=(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\boldsymbol\\Lambda_0)^{-1} (\\boldsymbol\\Lambda_0\\boldsymbol\\mu_0+\\mathbf{X}^{\\rm T}\\mathbf{X}\\hat{\\boldsymbol\\beta})=(\\mathbf{X}^{\\rm T}\\mathbf{X}+\\boldsymbol\\Lambda_0)^{-1} (\\boldsymbol\\Lambda_0\\boldsymbol\\mu_0+\\mathbf{X}^{\\rm T}\\mathbf{y}) ,", "f10f68e4240c5792b1adb85d3a77e6ab": "\\alpha\\approx1", "f10f6d4b29355801c5f76a1826b0808b": "p + q=1 \\, ", "f10f8debfaefcf2f7442832c989d904f": "Z(k,z)", "f10f942e2651d51ee00c44b1d6128a8b": "d_A(z)", "f11002953826a35c0574a0fc1af89cb1": "\\frac{dY}{dx}=AY=\\sum_{i=1}^{n}\\sum_{j=1}^{r_i+1}\\frac{A^{(i)}_j}{(x-\\lambda_i)^j}Y,", "f11016322edbc5d7f12245636df32729": "(D_{1},D_{2},\\lambda_{2})", "f1105d3ed847b82839cc3d8d127c2c17": "\\tilde{k} \\in B^{20m+64}", "f1107705414bfaf58bb7ccf45defa69f": "\n\\varphi(x)=e^{iP\\cdot x}\\varphi(0)e^{-iP\\cdot x}\n", "f1109a8f61045c389e7928af03720c49": " \\left( \\exists^p L \\right)^{\\rm c} = \\forall^p L^{\\rm c} ", "f1109ea9c12a6aa79dc73f73a3ff2a0f": "E_j\\ ", "f110eb7d68ca4be6d45e6340c5edc308": "\\operatorname{gr}_I M", "f110ec21534fe36ba3c7ad15b58f1c14": " K^0_S \\to \\gamma \\gamma, ~~ 3\\pi^0, ~~ 2\\pi^0 \\gamma, ~~ \\pi^0 \\gamma \\gamma, ~~ \\pi^0 e^+e^- ", "f11100060dc165ce6251c2fc710d6d22": " \\mathcal{G}^0(\\tau) = G_{0,loc}", "f1113b72b14e402901517c6a65021302": "S^\\Lambda", "f111428a3829e0c63d00cac5ceff2fe5": "w = s^{-1}", "f11158b9804f8d7b7bd8b462a8bba9d0": "\n\\frac{V_\\text{P}}{V_{\\text{S}}} = \\frac{E_\\text{P}}{E_\\text{S}} = \\frac{N_\\text{P}}{N_\\text{S}}\n=a", "f11158d7483a6260240f490914f307d6": "\\operatorname{E}(f_j(X)) = a_j\\quad\\mbox{ for } j=1,\\ldots,n", "f111a4acf84f596776fd8c4d199fac05": "E(\\chi_a)", "f111ba192750f23ba2202e4911dc1d57": "v^a(k)", "f1127886680b4c22528d88f215576a0c": "\\pm\\left(\\pm\\sqrt{10},\\ \\sqrt{\\frac{2}{3}},\\ \\frac{-1}{\\sqrt{3}},\\ \\pm3\\right)", "f112d5a1728215cc5b3f29f039c39ea8": "\\theta_{1},\\dots,\\theta_{n}", "f1130a75c67fa53e452be57bf73a32e5": "\\lambda x\\, \\lambda y.\\, x", "f1133f1d191be7beec34f3ef1f46a1e1": "S_{down} = S \\cdot d", "f1134cc2017c6ae73bd6ba1e6eb1a447": "\\frac{2 * 3 * 7}{3 + 7} * \\frac{1}{7 - 3} = \\frac{21}{20}.", "f11392c1b1ab6dbce0f3f871fd7d2f6f": "\\{\\lambda_n\\}_n", "f113e6b93237aa93d2f886445c1a724e": "\\sum_x f(x)=-\\sum_{k=1}^{\\infty}\\frac{\\Delta^{k-1}f(x)}{k!}(-x)_k+C", "f1140c6c8aaf533639a9e1275ae00cb9": "Q^\\pi(s,a)", "f11433c66b327e6338a4a069a25e6780": "\\ \\omega_{z,R}", "f1147f3b466eef05a2040f06fd2e1ac8": "{G'}", "f114c612b1e789ed62c4bb27376f3cf3": "\\ \\frac{a}{1-r} = \\frac{\\frac{1}{10}}{1-\\frac{1}{10}} = \\frac{1}{9} = 0.\\overline{1}", "f114c84d23ae3a261c699916eee4c016": "\\alpha =\\tan ^{-1}\\left( \\frac{\\sin \\theta }{\\cos \\theta +\\lambda } \\right)", "f114fb56d0f089197a90f2cc52b64bd7": "[(\\gamma_1)_\\mu (p_1-\\tilde{A}_1)^\\mu+m_1 + \\tilde{S}_1]\\Psi=0,", "f115569bb3800610102f1ab445a1268e": "(aei+bfg+cdh)-(ceg+bdi+afh).\\,", "f1157dca1fc12db5561b0875927ecc6f": "\\mathbf{w}=\\mathbf{v} \\oplus\\mathbf{u}=\\frac{\\mathbf{v}+\\mathbf{u}_{\\parallel} + \\alpha_{\\mathbf{v}}\\mathbf{u}_{\\perp}}{1+\\frac{\\mathbf{v}\\cdot\\mathbf{u}}{c^2}},", "f115c62445d541a0689b00a2cb6dd92c": "\\eta \\le 2 \\rarr \\delta \\ge \\frac{\\gamma +1}{2} ", "f11616e1e3d6f30f38df7c64a4ceff87": "l=\\frac{b^{2}}{a}", "f11670dc095bc4276c6d071823af28aa": " e^{ikz}", "f116bb988cb14a46049ed6776770bc23": "x_i = x_{i-1}+z_i", "f117091d63a1642e92093f16569b99c8": " B= \\begin{pmatrix}\n 7 & 0 \\\\ 0 & 3 \\\\ \n \\end{pmatrix}", "f11758f4778536b1c8ac8c5bd1437fa7": "\\mbox{eGFR} = \\mbox{141}\\ \\times \\ \\mbox{(SCr/0.9)}^{-1.209} \\ \\times \\ \\mbox{0.993}^{Age} \\ ", "f1175cc0fd6ad81941e28e8383a7d1e2": "q(k)=\\exp\\left(-\\pi \\frac{K^\\prime(k)}{K(k)}\\right).", "f1175cd69702bdc756918c96df0476e6": "\\frac{1}{(2\\pi)^{3/2}}\\int_{\\mathbf{R}^3} |f(x)|^2 e^{-|x|^2/2}\\,dx < \\infty.", "f1182f3634a06c6dec0befb428a9b62a": "\\theta_i = a_i \\theta_{i-1} - b_{i-1}c_{i-1}\\theta_{i-2} \\quad \\text{ for } i=2,3,\\ldots,n", "f118543623bc4cb86ca60b4147a539ea": "F(u)(x) = f \\big( x, u(x) \\big).", "f11882e492926897d2921faf60f8fae2": "y' - \\frac{2y}{x} = -x^2y^2", "f118a6e64ed44e0b48d90ad2321b1057": "X^2 = X", "f118fd01b64d633bec02c7828e4d5216": "e(a,b) = \\sum_{P: a \\to b} \\omega(P)", "f1190d2ec129e601f53f9729dd7806cb": "f'(x) = \\lim_{h\\rightarrow0}{\\frac{f(x+h)-f(x)}{h}}", "f1199c468d414445561a63dab02281cf": "e = \\frac{r_\\text{max}-r_\\text{min}}{r_\\text{max}+r_\\text{min}} = \\frac{r_\\text{max}-r_\\text{min}}{2a}.", "f11a16de5e26c2a5d092591ef45001ad": "-\\mathbf{e}_1", "f11a7dd8b6fcbbd4716045fb214e0f46": " p_0\\, ", "f11a9fb78cd440e4a90bb88270360c0f": "m^{e d} = m^{(ed - 1)}m = m^{h(p - 1)(q - 1)}m = \\left(m^{p - 1}\\right)^{h(q - 1)}m \\equiv 1^{h(q - 1)}m \\equiv m \\pmod{p}", "f11ab1242d74fdbbb0aa38b9f7336b07": "P_i =\\frac{P_{\\text{total}}C_i}{1,000,000}", "f11b337f229d0017aa7022db3cae72bf": "Z_{m_n} = m_1Z_1 + m_2Z_2 + m_3Z_3 + \\dots + m_NZ_N", "f11b9107b32f28e42ed55740f7c6c70b": "2^I", "f11be0a8d42cf219ba56db38962b95d8": " G \\times X \\rightarrow X, \\quad (g,x) \\mapsto g \\cdot x. ", "f11be36cd73c16d694bc5c6a2e40dd8a": "T^*_i = -T_i;", "f11c3b9dc82ba36af60ec05cf6677f8c": "\\scriptstyle \\psi", "f11c58f07e14ee04631faa9b7a38c279": "V(h)=\\left\\{ \\begin{matrix}0 & \\mbox{if}\\quad h\\geq \\sigma \\\\ \\infty & \\mbox{if}\\quad h< \\sigma \\end{matrix} \\right.", "f11c6354bd426afd28f3b2cd01c60a85": "F_{ad}=\\frac{3}{2}\\pi RW_{STM}", "f11c635b4bd7707d32629204c1f114fa": " u = \\frac{\\dot{v}}{A_t} = \\frac{4 \\dot{v}}{\\pi D^2} ", "f11c880c2a883644ac0df6262487a334": "\\textstyle W_{\\zeta}(y|x)", "f11cb4b5d7ec6f208defd2fd946bb2d6": "s_i = \\Sigma_{j \\in C_i}c_j", "f11d157d022094fdcd77cec304573edc": "W^{\\iota}", "f11dd187f9ccd8a11b0647f8c8852b65": "\\bigl(P_n(c)\\bigr)_{c\\in\\mathcal{C}}", "f11ddcb1fe03ed40dc9b024dfd6c2252": "a \\mathbin{:} \\mathcal{U}_0", "f11e0d462aa42039edd8c5563fb6e357": " i = 1 \\ldots n ", "f11e2f3a40bf52c806be15b11aad797f": "f: F\\mapsto F ", "f11e3cdad4496c6d2abda92ebdbab703": "(x^y)^z=x^{(yz)}.\\,", "f11e54e9cb40ccb01bea0b63a129456c": " f(x) = \\frac{x^n(a - bx)^n}{n!},\\quad x\\in\\mathbb{R},\\!", "f11ec3494e6006ffe40c5b113d476613": " = \\operatorname{tr}_{H_A} \\left(\\sum_k \\sum_i V_k^* T_i V_k \\otimes S_i \\right)", "f11f4df2d909ffae0db046af9a7298e6": "\\displaystyle E_2(\\tau) = 1-24\\sum_{n>0}\\sigma_1(n)q^n", "f11f767d450c19456bac8d71516f26f6": "\\varphi(e_i) = \\sum_j A_{j,i} e_j", "f11f7a8d47954ded1d062821289eb63e": "10 \\uparrow (10 \\uparrow \\uparrow \\uparrow 3)=10 \\uparrow \\uparrow (10 \\uparrow \\uparrow 10 + 1)\\approx 10 \\uparrow \\uparrow \\uparrow 3", "f11fd278b29874a692232bf5c12d16a2": "\\Omega_{DE}", "f11fea52ab64381cfb26d30fdf59828a": "|H(j\\omega)| = \\sqrt{H \\cdot H^* } ", "f11ff2b4b5ad1a90953b8b7516c8ffae": "P(t)\\,", "f12019aa86dd9d9ad7fa1e207db1d449": " \\alpha = \\eta ", "f1202d14c049e3d7e30b88427289accb": "{[-\\infty,+\\infty[}", "f120b913109e8a3ee46e0bb067133975": "[k*]b\\,\\!", "f12135258471bf90d2d5347d357f1a1c": "d(p,q)=C \\log \\frac{|qa||bp|}{|pa||bq|}", "f121752c840355f1abd4d10e1c95f9d7": "V_A=F_v-V_C", "f121bb05529c50c2833ddb0803df9702": "K = \\tfrac{1}{2} |\\mathbf{AC}\\times\\mathbf{BD}|,", "f121c09b3f208bc97bf2a2705b2d330e": " \\left\\lfloor \\frac{(k-2) n^2}{2(k-1)} \\right\\rfloor = \\left\\lfloor \\left( 1- \\frac{1}{k-1} \\right) \\frac{n^2}{2} \\right\\rfloor", "f121fc3e481913a40684db085b4b61d3": "\\rho(\\boldsymbol\\beta,\\sigma^{2}) = \\rho(\\sigma^{2})\\rho(\\boldsymbol\\beta|\\sigma^{2}),", "f12209857c287cfa606562090c442c10": "~ k_b", "f122831b7d99971be482e7fe48b3cdf3": " -P \\, dV,", "f12284382d751cacb336d2b1a12bff3c": "{\\overline P}X - {\\underline P}X", "f12348f70add4388f6de258891257d99": " \\frac{X}{\\tau^2 \\nu} \\sim \\mbox{inv-}\\chi^2(\\nu)", "f12383526cc12d29c10b3ce033202b51": "M(x) ", "f1238aea812584c0a33655738381235b": "h_{\\alpha}^k\\pm h_{\\beta}^k", "f123ad2c74a59304c1beca7bc4ed7896": " \\mathcal{X} := \\{ x_i : i \\in I \\} ", "f12419148ed2e0c6823af8488328f717": " D_0", "f12430049acff048a19f8c2eccf14e52": " px^2 + 2mxy + \\left(\\frac{m^2+1}{p}\\right)y^2,", "f12442ee714685720a0c940fb29d1bb0": "V_{n+1} = \\frac{S_n}{n+1}", "f1249399afec1aadebe5a1d32dfa441f": "|f(x)| + |\\hat{f}(x)| \\le C (1+|x|)^{-d-\\delta}", "f124b82cdc74a226fa51022d89c90b29": "E[\\text{true}] \\lor F[\\text{false}]", "f124cb81a2a5bbf75189503ea22f03f5": "a \\lor a=a\\lor(a\\land(a\\lor a))=a", "f124d417eef0bdfedf9fd5ee562aae7d": "\\scriptstyle{R_F(x,y,z)}", "f125000805d84894a36f22e4e4976cf6": "k_BT\\gg\\hbar\\omega_\\alpha. \\, ", "f1250c2e370260d5b669d3fdf679eee7": "\\textstyle (w_i)", "f1252f36f363e6b02067fda92e1a7a68": "0 \\to\\Omega^0(M)\\ \\stackrel{d}{\\to}\\ \\Omega^1(M)\\ \\stackrel{d}{\\to}\\ \\Omega^2(M)\\ \\stackrel{d}{\\to}\\ \\Omega^3(M) \\to \\cdots \\ \\to\\ \\Omega^n(M)\\ \\to \\ 0.", "f1254fa7cab976ebec87ab5b386ebdd4": " \\mathbf{A}\\cdot\\mathbf{B} = \\sum_j\\sum _i\\left(\\mathbf{b}_i\\cdot\\mathbf{c}_j\\right)\\mathbf{a}_i\\mathbf{d}_j ", "f1256c934adb51d10e7d64064323e06b": "\\Delta H=\\int_{S_1}^{S_2} \\left(\\frac{\\partial H}{\\partial S}\\right)_P \\mathrm dS +\\int_{P_1}^{P_2} \\left(\\frac{\\partial H}{\\partial P}\\right)_S \\mathrm dP", "f125b74f332354bc6dccd5fc396e732c": "p(X, E = e)", "f125ee5c6943974b9f0fa32d9dd3ffe2": "[C'_i,H']=i P'_i \\,\\!", "f125f4157095ed414b01ed24aa0e649f": " \\alpha=-k \\cot(k R)", "f125f77db33f364701e139ab03494479": "\\frac{\\partial}{\\partial t} \\frac{\\partial p}{\\partial \\sigma} = u \\frac{\\partial}{\\partial x} x \\frac{\\partial p}{\\partial \\sigma} + v \\frac{\\partial}{\\partial y} y \\frac{\\partial p}{\\partial \\sigma} + w \\frac{\\partial}{\\partial z} z \\frac{\\partial p}{\\partial \\sigma}", "f125f81d0c780df69ce2fb134c6f0a98": "(x+y)^p = x^p + y^p\\,\\!", "f12638090d7f2c9b8487fdc3e36bb21a": "A\\! \\succsim\\! B\\!", "f12644f1c0d00b07716732c25819450c": "[a,b],", "f126575c6bdb4ce55d246f013c5b4896": " a_{m} ", "f1267a60a15d205d2569ef599dcbe291": "[x] = [y]", "f12699a93e83d60bce03209cf861941c": " \\theta \\in [0,1] ", "f126bd6325bd798a98fbfbb07772f8df": "\\scriptstyle 1-\\frac{1}{k^2}", "f126caa32f904f99c3351cb65cef7f60": "i \\in \\{1, \\ldots, n\\}", "f126e58cd892d4c54a8a218724c0ea04": "\\neg Q", "f1270cbfce36baa8f7bc25d4c5222b06": "e_g \\cdot e_h = e_{gh}.", "f1271db88b12ae6ec658dfc995fc0b2a": "\\beta>0\\!", "f1271def8a756d362c323b9eab513758": " f(x)=C+\\int_0^x \\, d\\mu(t).", "f127a7968f5a578edf821bc861057eca": "\\mathbb {F}", "f1284a7362d50660ce31de4486c48877": "(F,m):(\\mathcal C,\\otimes,I)\\to(\\mathcal D,\\bullet, J)", "f128756d9246aca5aae34e07b4fd64f0": "\n\\limsup_n \\frac{S_n}{\\sqrt{n}}=\\infty\n", "f128b7694f05c2bb460b2f2a6db88af1": " t \\ \\infty ", "f128dc09794f40b6dc3eab0cbe78060b": "{\\ q}", "f128e24d435b28316a8be1a76ea08678": "\\|\\textbf{v}\\|^2 = \\textbf{v} \\cdot \\textbf{v}.", "f1290186a5d0b1ceab27f4e77c0c5d68": "w", "f1295ed4cbfb2f81e4412189edb25001": "d \\rightarrow \\infty", "f1296f073a770e904239740421cf186f": "\\frac{\\partial Y}{\\partial \\beta}=Y_\\beta", "f12a0648e819151a5530bc14f3678598": " \\partial_\\mu \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_\\mu \\psi )} \\right) - \\frac{\\partial \\mathcal{L}}{\\partial \\psi} = 0 \\,", "f12a21b8c1df4b6d4a135e77e58781e6": "a \\oplus b = \\max(a,b) ", "f12a36c2dcbab87187d03370ab0b85bd": "4\\pi/(k^2+m^2)", "f12a374e9dc4f98f40aa23d7494f79e4": "W_C \\ ", "f12a3d08a5e32c7662434edd8a6d4c93": "\\mathcal{F}(\\mathcal{F}^{-1}f)(\\xi) = f(\\xi).", "f12a42d36c520b88133d7a4b4833dd1a": "\n_pF_q^{(\\alpha )}(a_1,\\ldots,a_p;\nb_1,\\ldots,b_q;X) =\n\\sum_{k=0}^\\infty\\sum_{\\kappa\\vdash k}\n\\frac{1}{k!}\\cdot\n\\frac{(a_1)^{(\\alpha )}_\\kappa\\cdots(a_p)_\\kappa^{(\\alpha )}}\n{(b_1)_\\kappa^{(\\alpha )}\\cdots(b_q)_\\kappa^{(\\alpha )}} \\cdot\nC_\\kappa^{(\\alpha )}(X),\n", "f12a69ada70cc4fdfde5b3e8f0853ca4": "a_{rel}=\\frac{\\hbar \\sqrt{1-(v_e/c)^2}}{m_e c \\alpha}", "f12a98bfbebf0620fbd72e13fad9d830": "10^{-7.5}", "f12ad324859650846756929da995b855": "f_L(C)\\cong f_R(C)\\cong C.", "f12b1d585dabea146ddd19a837689ca3": "K_\\mathrm{sat}^{(1)} = \\rho \\left ((V_\\mathrm{P}^{(1)})^{2}-\\frac{4}{3}(V_\\mathrm{S}^{(1)})^{2} \\right) ", "f12b2670db636bb7b18673c4f173d210": "\\left\\lbrace (x, t) | x/t \\in Dom(f), t > 0 \\right\\rbrace", "f12b3cff276af1180619a1ace02038a6": "P_n=|X_n|^2", "f12ba96f30e9f1a332b6b728d5411275": "\n\\sum_{n=1}^\\infty \\frac1{n^{1+\\varepsilon}}\n", "f12bd26b556164c3c807684e2abcb408": " p \\in M ", "f12d1e6a449cde2da00dc4bf02cf90a7": " R_\\mathrm{E} = { (k_\\mathrm{e} e^2)^2 m_\\mathrm{e} \\over 2 \\hbar^2} ", "f12d694844389cf7ea217eaa34fdb547": "\\frac{1}{A} + \\frac{1}{B} = \\frac{1}{h} \\,", "f12e28e74eb614898621ca7afc3a17ac": "x_\\mathbf{n} = \\frac{1}{\\prod_{\\ell=1}^d N_\\ell} \\sum_{\\mathbf{k}=0}^{\\mathbf{N}-1} e^{2\\pi i \\mathbf{n} \\cdot (\\mathbf{k} / \\mathbf{N})} X_\\mathbf{k} \\, .", "f12e40c88e49f5df011cbecfb416233d": "U_{km}(P,Q)", "f12e5db164d3a3939930196adbb5d793": "\n\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1.\n", "f12e7cdc0d60618fee58a9b5646e03e1": "\\alpha_V = \\frac{1}{V} \\left(\\frac{\\partial V}{\\partial T}\\right)_P", "f12eb3b8c2212412c114b2862bfa8b71": "\\|Ax\\|_b \\leq \\|A\\|_{ab} \\|x\\|_a", "f12ee4caecdebfe4ff7d7f04a77b2e80": "x=-y", "f12ef07ec254b10140c683025c34919d": "\\tfrac{dS}{dT} = B - \\beta SI - \\mu S + \\gamma I", "f12f144d1e224b1028ecfc3e4fecc82d": "\\{p_{2},p_{3}", "f12f3cfcd8b6b51a6389ad91f49d2b5c": "-\\frac{\\nabla^2 \\Phi}{\\Phi} = \\frac{\\frac{k_{\\infty}}{k}-1}{L^2} = {B_g}^2", "f12f64e9eb1953fd147f8421b3ee43ea": "G = \\sqrt{x_1 x_2},", "f12fa2a2f8da4d9a11db3d7825b3a6d5": " \\mathbf{a} = \\frac{1}{m_0 \\gamma(\\mathbf{v})} \\left( \\mathbf{F} - \\frac{ ( \\mathbf{v} \\cdot \\mathbf{F} ) \\mathbf{v} }{c^2} \\right) \\,.", "f12fc1bb667fe10f0e2f0b9cf069e9df": "E[f(X)] \\geq f(E[X]).", "f12feb8e8ba80d3f4695caf694592f47": "\\begin{alignat}{2}\ni = & \\quad \\text{last}(L) - j\\\\\n = & \\quad (2^{L + 1} -2) - j\\\\\n\\end{alignat}\n", "f1300f66fa4f5638a14af7e578380349": "\\sum_{i=m}^n f(i) = \n \\sum_{k=0}^{2p}\\frac{1}{k!}\\left(B^\\ast_k f^{(k - 1)}(n) - B_k f^{(k - 1)}(m)\\right) + \n R\n", "f1308afb25f5efcc8282aaea74116c24": "7^2:(3\\times 2S_4)", "f130f95f62ce70a40d0324d63f1361c2": "(t-\\tau)^n e^{-\\alpha (t-\\tau)} \\cdot u(t-\\tau) ", "f1310941c1b3a32b4c65693450140f8b": "s_{Oy}\\,", "f1315e2a4426b194c83727e0a6b36da4": "P_{total}=\\int p(\\lambda) d\\lambda", "f131d149b0e01b5ead23c85c99dec74b": "P_n(L)", "f1320a5c22f55de94243d20a73bf32db": "x_{33}=-x_{31}\\,", "f13242a3edc293802b06cce5cd6892b9": "\\mathrm{C + FeO \\ \\Rightarrow \\ CO + Fe}", "f132654e5ca168adf0cf1bc81975a765": " E[X^k]=\\frac{B(\\alpha+k,\\beta-k)}{B(\\alpha,\\beta)}. ", "f1327ae9a9b6e7c0f355c2825c3738d1": "T_{2lm}\n = \\exp\\left(\\frac{\\sum{a_i} \\cdot\\ln{T_{2i}}}{\\sum{a_i}}\\right)\n = \\sqrt[\\sum{a_i}]{\\prod T_{2i}^{a_i}}\n", "f132903029472da7e73f0c3a0db90a70": "V_c = 3nb\\,", "f132e86cb533fc0573e25af29d39aed5": "\\mathcal{R}(M_1)", "f132ec08be2d77aacd92b5fd223ab590": "C:[0,1]^d\\rightarrow [0,1] ", "f13312e0dd22d7db8629613faa10abf1": "x^{(k)} \\ ", "f1331b2c6925b4890669edd5c19efee0": "257^{2} = 32^{2}+255^{2}", "f13329a55b15ad1fe0e606ddd587099a": "R(P)=\\infty", "f1339e8e9bc641f07a718f0e0ad25e81": "\\mbox{NPV} = \\sum_{t=0}^{n} \\frac{C_t}{(1+\\bar{r})^{t}} =0", "f133f186c108c543fddc0330dd224b94": "\\sqrt{2} \\times \\sqrt{10}", "f1349d0a80a9ef65efdd2c827e397518": "(9)~~ ~~ F=\\int_{0}^t I(t) \\mathrm{d}t", "f134cbf4e46195f374f71fee183d8540": "\\alpha_t(x_t)", "f1351ead6e8ef4e30a0e8934e18f84c5": "f = -{R \\over 2}", "f135370da1919b837ea9625f2f8592e0": " \\bar r_1 = r_1 {\\hat r}_1 ,\\ \\bar r_2 = r_2 {\\hat r}_2\\ ", "f13557a36f2174e7a394e617406937d5": " V(x) = \\begin{cases} (b-K)(x/b)^\\gamma & x\\in(0,b) \\\\ x-K & x\\in[b,\\infty) \\end{cases} ", "f13572f19f915f35412e899742783b2c": "\\,K_1^*, K_2^*, K_3^*", "f1362d71429489152e52fccf91528ddc": "S \\subsetneq \\underline{m}", "f136414016d4209a0a2c8aedff841c66": "\\dot{\\sigma} < 0", "f137025baa681c459db5f6212bcbaa7c": "\nn_e \\approx Z n_i \\,\n", "f137028e6eb6d063917261ddf3940d81": "\\frac{2\\alpha^{2^n} - b}{2a}", "f1376da107139f6fc67dc8236de3b432": "Y_{2,0} = \\omega_ex_e", "f1377765dbb337f754de51c417226451": "a^*\\in A", "f1377f30340727ded806ee2657c0ed36": "\\textstyle \\bar{e}(s)", "f137984149e0da7edd92414d68659eb3": "i_D:J^kE\\rightarrow F", "f137baa40b1e2bfbf9126bb026ede6b1": "f|\\partial D^2", "f137bb528cae31a4a8f78069e6d67177": " R_{a}\\;(x) ", "f13843e1b87da6796ad86982bdeafc24": "\\textstyle 0 = d^2 f = d(f^\\prime dz) = df^\\prime \\wedge dz", "f1384ab87c4bbd2f3046f0f53be1eecc": "\\Phi(p,t)", "f1384e9a72e05a8e357cc89840a04c0b": "f(\\rho, \\phi) = \\rho \\cos \\phi + \\rho \\sin \\phi = \\rho(\\cos \\phi + \\sin \\phi ).", "f138a1e9b3cdd76a6147247e635efa42": "\\begin{bmatrix}K\\end{bmatrix}", "f138cabe2764f402cd3e00e9e068cbc9": "f = g \\circ f'", "f138f8497add871d10978f240fbae046": " \\in \\mathbb{Z}^n ", "f13923be2a03e7ddba292b96b3f38546": "\\{(e,0) : e \\in E\\}", "f13923ee0af8122da9f2eb52d108cd7b": "-\\sqrt{\\frac{12}{35}}\\!\\,", "f13936a510b3d278c71a262554bd7735": "A^{n,K}_{cv}", "f13936d9a5516862e3da64e14cf143eb": "\nA =\n\\begin{bmatrix}\n1 & 2 & 3 \\\\\n3 & 4 & 7 \\\\\n6 & 5 & 9\n\\end{bmatrix}\n, \\quad\nB = \n\\begin{bmatrix}\n0 \\\\\n2 \\\\\n11\n\\end{bmatrix},\n", "f13953a07a6669bb7f15b4eeedc26c48": " (x,y) \\mapsto x+ y \\sqrt{-1}", "f139d65d98ba18d4a2c1ff6f2d971f52": "\\scriptstyle \\omega L", "f139e6bbd5891f3c4c3a53c8ae062f70": "\\left(\\tfrac{F/\\mathbb Q}\\cdot\\right)", "f139f04afcec35fcd41cc18604f821e9": "\\pi_r=200.0\\pm4.2", "f139f5a405720599c0857bdda924cef1": "\\gamma=1", "f13a459dae9fe07fa674b927868a8cf4": "{i}", "f13ac4eb688786ad725f8231fa52daeb": "\\{f_n\\}_{n=1}^\\infty", "f13aeef6ec5b787d5c10947a2d343376": "\\frac{}{}_V \\rho {\\rm d}V +", "f13b1ccefee4a3f02bd5be287ec14d11": "p\\ f\\ x = f\\ (x\\ x) \\and q\\ p = \\lambda f.(p\\ f)\\ (p\\ f) ", "f13b604c6dafbe041e9fc103f0b7680e": "\\gcd{(a^{(N-1)/p}_p - 1, N)} = \\gcd{(2^{2\\cdot 5\\cdot 227} - 1, 11351)} = 1.", "f13b7de728e972863a08f93d5d91c977": "L^* = \\frac {V_c} {A_t}", "f13ba12c26aabd61f1a7f60ad0ec40ee": "n = 5", "f13bbb07d7cbaa358937701c68a132ba": "R_{10}", "f13bbbc7f83c029094e7ce1c4ccaa047": " i^G = (n+d)k^G ", "f13c398ad34834564028df7d8b86da3b": "M_c", "f13c5ec1acaf2d9761709e1a810d7d4d": "x=l+f=3+0.9711308=3.9711308 \\, ", "f13caefedfa5637faa249684d164bc4c": "\\begin{align}\n(a_1,\\ b_1,\\ c_1,\\ d_1)&(a_2,\\ b_2,\\ c_2,\\ d_2) = \\\\\n& = (a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2, \\\\\n& {} \\qquad a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2, \\\\\n& {} \\qquad a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2, \\\\\n& {} \\qquad a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2).\n\\end{align}", "f13cbe9d3307cc5bdc7b126b21653d57": "0 =\\Sigma_k \\frac {\\dot Q_k}{T_k} + \\dot S_{i}.", "f13cbffa2513f20f12fb408017199cb9": "\\tilde{\\rho}(r)", "f13cc397ca3638440f55aa4b54c4392d": "k_{on}", "f13ccb04bfb9086dfc8293ea20f47e4e": "\\sum_{i=-\\infty}^\\infty a_i \\delta_{ij} =a_j.", "f13cd407e2d44266177cea757abb935e": "Z_R", "f13d03e80925e42b8f7892c491ec65a8": "\\frac{N_1}{d^2} = \\frac{N_2}{d^2}", "f13da164be6a8ff787a4734a813f7c6f": "t[a_1,...,a_n] = \\{ \\ ( a', v ) \\ | \\ ( a', v ) \\in t, \\ a' \\in a_1,...,a_n \\ \\}", "f13e269427b9e20a6e6a5b1bbc7d2382": "\\delta_\\varepsilon DX=\\varepsilon DX", "f13e28ce829b262174de9bc35f8a8b7d": "25^o", "f13e355577a07b6ed3e680305b8506a6": "x^3 + 3 x^2 y + z^7", "f13e6220c0c849e175b6cc4273f5b90b": "F_b = - (k_1 + k_2) x. \\,", "f13ed7d43e280b6e12f771f024a44b5d": "\\frac{1}{\\sqrt{2}} \\left( \\left| M \\text{ at } A \\right\\rangle + \\left| M \\text{ at } B \\right\\rangle \\right)", "f13edc12c78d75f51f6a140be4394b3f": "\\Sigma=\\begin{bmatrix} \n{Var \\left (X_{1(1)} \\right)} & {Cov \\left (X_{1(1)},X_{1(2)} \\right)} & Cov \\left (X_{1(1)},X_{1(3)} \\right) & \\cdots & Cov \\left (X_{1(1)},X_{1(k)} \\right) \\\\\n{Cov \\left (X_{1(2)},X_{1(1)} \\right)} & {Var \\left (X_{1(2)} \\right)} & {Cov \\left(X_{1(2)},X_{1(3)} \\right)} & \\cdots & Cov \\left(X_{1(2)},X_{1(k)} \\right) \\\\\nCov \\left (X_{1(3)},X_{1(1)} \\right) & {Cov \\left (X_{1(3)},X_{1(2)} \\right)} & Var \\left (X_{1(3)} \\right) & \\cdots & Cov \\left (X_{1(3)},X_{1(k)} \\right) \\\\ \n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \nCov \\left (X_{1(k)},X_{1(1)} \\right) & Cov \\left (X_{1(k)},X_{1(2)} \\right) & Cov \\left (X_{1(k)},X_{1(3)} \\right) & \\cdots & Var \\left (X_{1(k)} \\right) \\\\\n\\end{bmatrix}.", "f13ef4e099e8951f9bfead69ee88f6ed": "R_{mn}(c,\\xi)", "f13f32ea7f306b9706c0f1160ccc414f": "K*P*Q", "f13f3b7a842a590b5f46ad53905bd26f": "\\scriptstyle U_i\\subseteq U_k", "f13f9bdea7ffd6d4574c756885616668": "T_{air,av} = \\frac{T_{air,in} + T_{air,out}}{2}", "f13fc9f79ee07c0d8f8c672f5c96a2e9": "\\geq 2t+1", "f13fd7b91e96e6f37684f77bddeeaa02": "u\\!:\\!\\tau_1\\times\\tau_2", "f13fdbdaaccb1a51893677127ec642b5": " \\ C_L ", "f1403614565575a1b49c56b5d3da405e": " P(t,T) = \\mathbb{E}^Q\\left[\\left. \\exp{\\left(-\\int_t^T r_s\\, ds\\right) } \\right| \\mathcal{F}_t \\right] ", "f1404a51ca0672a66bcd6b81f70f379a": "\\varphi_0(t)=y_0 \\,\\!", "f140defcf208c116cb55c3102753a73a": "(b + c + f * g) * (d + 3)", "f140e4ba5b24cebb06cd47f6ff85d936": "\\hat{\\lambda}_\\text{UMVUE}\n = \\sqrt\\frac{K}{N-t}\n\\frac{\\sum_{i=1}^t c_i \\bar{Y}_i}{\\sqrt{\\sum_{i=1}^t \\text{MSE } c_i^2 }} ", "f1410d1774fa5c629b4bc90d825c0727": "\n\\Psi(\\mathbf{x}_1,\\mathbf{x}_2) = \\chi_1(\\mathbf{x}_1)\\chi_2(\\mathbf{x}_2).\n", "f14149162e7b805ab242133c3954e3a6": " \\frac{\\delta \\mathcal{L}_\\mathrm{G}}{\\delta g^{ab}} -\\frac{1}{2}P_{ab}=0", "f141beea92bcb9eb823faeb628b1da63": " (\\mathbf{b} - \\boldsymbol{\\beta})^\\prime \\mathbf{Q}^\\prime\\mathbf{Q}(\\mathbf{b} - \\boldsymbol{\\beta}) = {\\frac{p}{n - p}} (\\mathbf{Z}^\\prime\\mathbf{Z}\n- \\mathbf{b}^\\prime\\mathbf{Q}^\\prime\\mathbf{Z})F_{1 - \\alpha}(p,n-p).\n", "f14212d6dc84b40b1cc9073f103d6e04": "(i,j)\\,", "f14239d15d899755d7bb2c38835b0263": "\\mu=0\\,", "f1423a57efa96f6017a30af90061be4d": "c : S^2\\Lambda^2 V \\to S^2V", "f142945be8423ac0d3689192d30a87a6": "A_{\\ell m}^{(2)}=0", "f142aa924fba8df346e5eb8f2dd491a8": "P\\ ", "f142cfef15c2fdea0dca20968676ce6e": " \\left| r \\right| = r, \\text{ if } r \\text{ ≥ } 0 ", "f14372636ccf3e4ca3cd779e211866c9": "|n_{1}n_{2}n_{3}...\\rangle", "f1439e1ed657602b7206094dd02722d7": "2\\arctan\\varphi\\approx116.56505", "f143d738dd8f14f3acf6fae11b63f318": " \\tau_{1i}", "f143d9e46547a90632469fc9cdfb9687": " S^{*}>0 ", "f143e0f514661c10a761786203b6ca92": "\\lim_{z\\to\\infty}M_-(z) = \\bar{M}_+(0).", "f143fb3d11cf5ab5011861e025798f3f": "\\nabla_\\mu V_\\nu = \\partial_\\mu V_\\nu\n - \\Gamma^\\rho{}_{\\mu\\nu} V_\\rho ", "f1442df07dbeca371a414ccda66ec950": "K_{(\\ell k)}", "f14451207585f70da9f13e4bbf459bb3": "\\overline{\\rm AB}", "f144d32daa898bb13daaca437291cc79": "a, b \\in \\mathbb{N}", "f145073887d70eab25a8f711b7320544": " \\beta_{FB} = \\frac {1} {G_{\\infin}} \\approx \\frac {1} {(1+ \\frac {R_f}{R_2} )} = \\frac {R_2} {(R_f + R_2)} \\ , ", "f1454ab2be3730c32a55f74552ea7e36": "\\int_{\\Gamma} g = \\sum\\nolimits_i \\int_{\\gamma_i} g.", "f145a7531a60e91d901da4c6eeabe4c5": "t\\to 0^+", "f145cb8f9976ac3addb4e73fdfc67547": "\\ a_C \\,", "f145f4fc37304d79b8b8a3210aa6d954": "a\\sim b\\;", "f146338735e0c488d07f86b343e0141c": "\\mathbf{z} = \\varphi(\\mathbf{x},t)", "f1463e6a7181ea8764a2ed1ded717b4b": "g_0=p_0\\cdot \\exp(f_0 b)\\text{ if }a = 0,\\,", "f1465252fb9f002eacef9340ed489036": "C_X = C_Y = \\left [0,1 \\right ] ", "f146bb92e03ee17e2eccde76ad719f69": " \\frac{\\partial Q}{\\partial t} + V_w \\frac{\\partial Q}{\\partial x} = 0 ", "f146c3747fd6bc58e0c2727a9a0d1388": "\\lim_{n\\to \\infty} \\log_n \\log b(n)", "f146d910600ebbd618c6e40d204a2cc7": "\\lambda_e", "f146dac0401b23670c4201830099a552": "spin(n)", "f147294a80a2a57de1ddd70f83fa9bfe": "M_{k,n} = c : f_c^{(k)}(z_{cr}) = f_c^{(k+n)}(z_{cr}) \\,", "f14781eeb4dd03a5b794b4e2034fcc08": "\\alpha_1=\\alpha_2=1/2", "f147967e726e3d5ce344faab44ee5fc0": "\\alpha > 1,", "f147b24902d8a5853052ce0468ddd7b5": "\\binom \\alpha k = \\frac{\\alpha^{\\underline k}}{k!} = \\frac{\\alpha(\\alpha-1)(\\alpha-2)\\cdots(\\alpha-k+1)}{k(k-1)(k-2)\\cdots 1}\n \\quad\\text{for } k\\in\\N \\text{ and arbitrary } \\alpha.\n", "f147d49fde11e606f231cf0a1c002d9b": "w \\mapsto \\phi(z,w)", "f1480bf9878924a833d7755e665b6040": " U(s) = \\mathcal{L} \\left\\{u(x)\\right\\}=\\int_0^{\\infty} e^{-sx} u(x) \\,dx. ", "f1486f834072a457de703e9a53904f86": "\\text{Met}(X)", "f148cb1482570263961ac10853548069": "\nu^{+1}_{-1}(\\mathbf{-p}) = u^{-1}_{+1}(\\mathbf{p}),\n", "f1496c4c46c6f0fd9cdf4e0131085177": "(D,V,s,R') \\models P", "f1496ffff4d86820dfa1422983f68d4c": "r=\\mathbf{S}r+\\mathbf{V}r", "f149c23cf5b13e619e76d9f013ca0d5e": "1D", "f149cc59e35150e2f3bd101165a892c1": "(a + \\ell b)(c + \\ell d) = (ac + \\lambda d\\bar b) + \\ell(\\bar a d + c b)", "f14a49215da0b755c263b751b079e28b": "{K_{AB}}", "f14a5d39858c49c6611744338bbf6a38": "\\mathbf{A}^{2}=(\\mathbf{Q}\\mathbf{\\Lambda}\\mathbf{Q}^{-1})(\\mathbf{Q}\\mathbf{\\Lambda}\\mathbf{Q}^{-1}) = \\mathbf{Q}\\mathbf{\\Lambda}(\\mathbf{Q}^{-1}\\mathbf{Q})\\mathbf{\\Lambda}\\mathbf{Q}^{-1}=\\mathbf{Q}\\mathbf{\\Lambda}^{2}\\mathbf{Q}^{-1}", "f14ac479e437060ed4ee659ff230a77c": "e \\gg 0", "f14acd5a8040ad7b32a0a243afa0be65": "(\\varepsilon \\otimes id) F = (id \\otimes \\varepsilon) F = 1 ", "f14acf9625259eb2a473a9a03c38f40a": " \\mathrm{Pr}[\\exists j", "f14ad645f2d04403f5607e1baa610cdf": "F=\\mathbb{Z}^{(A)}", "f14b6e2159e461c240e41dfd177cc800": "\\log(K_{sc/w}) = a + b\\log(K_{w/o})", "f14bcbe1c1dc3a924c7f396a9835ab84": "h^m_{i,j,k}\\,", "f14be5c4e1d24906c0040dfe6e7de8b7": " f(y)=y^{2/3}", "f14c0ca25b3a0fc4d2989825b7f8c8c8": "\\sigma = (3/2)(\\rho_s/\\rho_a)P", "f14c86df9a9284a497b76d4bc507f3df": "(X) \\subset (X/2) \\subset (X/4) \\subset (X/8), ...", "f14c8b98cdd97de702a708ec20092b1a": "v_\\text{o} = -Av_\\text{i}\\,", "f14cc5e03eca8e31810de61a7700ac91": "** = - 1", "f14ccdecf3e2f5a254ed8217db12a71f": "\\neg(xRy) \\and \\neg(yRx)", "f14ce15dc2f631dcf2bb1b0aae10c5df": "g_1, g_2,\\ldots", "f14cf5f875867d2ff4bdce7518bb7a37": "x = \\frac{2 \\cos(\\phi) \\sin\\left(\\frac\\lambda 2\\right)}{\\mathrm{sinc}(\\alpha)}\\,", "f14d3d11711563903322f8b6264bc016": "I_n = a^nB \\cap A + aA", "f14d6b6940c2d4a3de7e975258e10636": "c_+ = \\begin{bmatrix}\n 1\\ 0\\\\\n\n \\end{bmatrix}\n*\\chi = {1 \\over \\sqrt{5}}\n", "f14defbef01d576ef90845a7141ed8e2": "M M'", "f14e052845737fa73d5028b9833730ea": "\\sum_{k\\geq0} m_k x^k.", "f14e07d0f2a57a895525a6790e8a8848": "\\,f_{ab}\\,,", "f14e1b758ae715620f07405ba28c9c40": "F_{ab}=\\frac{1}{2}(X_{a;b}-X_{b;a})", "f14e22903763ccb0551eca675bbf1d16": "M \\sim (T - T_C)^\\beta", "f14e5297d33c7c998b1809405f51a35d": "CDC \\rightarrow A: certificate(B, CA)", "f14e6e4738b474ca0af7b3a4d0c48de0": "ud=\\gcd(gcd(a,b),c)", "f14e76e2bae29296a3c35e5b44ce5f93": "\n \\delta K = \n \\int_{\\Omega^0} \\left[\\int_0^T \\left\\{\n - J_1\\left(\\ddot{u}^0_{\\alpha}~\\delta u^0_\\alpha \n + \\ddot{w}^0~\\delta w^0\\right) \n - J_3~\\ddot{w}^0_{,\\alpha}~\\delta w^0_{,\\alpha}\\right\\}~\\mathrm{d}t \n + \\left|J_1\\left(\\dot{u}^0_{\\alpha}~\\delta u^0_\\alpha \n + \\dot{w}^0~\\delta w^0\\right) \n + J_3~\\dot{w}^0_{,\\alpha}~\\delta w^0_{,\\alpha}\\right|_0^T\n \\right]~\\mathrm{d}A\n", "f14e9483c03ec556a449fe765a2aeeaf": "\\displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a)}", "f14ed2d12c1f5470f4c85dfb103e35ec": "\\varphi_{n+1}=\\varphi_n+\\frac{kM}{u_{n+1}} \\pmod k,", "f14ef0568458b284acd763af1537229e": "\\displaystyle I(P, \\Lambda) = \\min_{Y_r} I(X_r \\land Y_r)", "f14f0b65fc10f7e013fb20886105a29a": "\n\\begin{align}\nF_{t+m}& = s_t + mb_t\n\\end{align}\n", "f14f3fc322023be9c8926db18d74fc1d": " m_3 \\rightarrow -m_3 ", "f14fccc827f70792382e4d33265f82be": "\\bar{ M}", "f150179cb04977df543b2ac90e7ba1c5": "[A] - [B] + [C] = 0", "f1505e996fdd6d36c1dfc9c6cfd74f8c": "-5\\leq{}n\\leq5", "f1507a5c3bafe37230530451584ceaa2": "\\mathit{Ka} = k t_c", "f150e5185b26a9b069119651eacb7968": "Q=Q_1^T Q_2^T=\\begin{pmatrix}\n6/7 & 69/175 & -58/175 \\\\\n3/7 & -158/175 & 6/175 \\\\\n-2/7 & -6/35 & -33/35 \\end{pmatrix} ", "f150fe4074ba15ae874bd50bd3a3cc1b": "\\textrm{pH} = \\textrm{pK}_{a}+ \\log_{10} \\left ( \\frac{[\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "f151618a6889431bbc146641139f35f3": "\\frac{1}{\\sigma^4}\\sum\\limits_{i=1}^n {\\left( 1 - 6(1 - p_i){p_i} \\right)\\left( 1 - p_i \\right)p_i}", "f15162701dab1d2a54dba459b5817b37": "\\tfrac{16}{17} \\approx 0.941", "f151628000b7170e29d1ef8fdf789605": "X \\sim \\mathrm{Exp(1/2)}", "f1518c61155eeadd336bc9caddca8b59": "f: \\mathfrak{g} \\to V", "f15199e6f2903c9dcb802c4175ce12fc": " f(8) = 0.0774144 \\, ", "f151b35a62ec2b4c5d2664268fe40b6c": "\\scriptstyle \\eta\\equiv 0 ", "f151ffb0b81eb808c54dbfd656b8d938": "D^{2}/\\partial{D^{2}}", "f1525ad29ff9c1ea87ce602edf9017fa": "\n \\operatorname{E}[\\,\\varepsilon|X\\,] = 0.\n ", "f152b482b10fc09780106234513183af": " \\bold y(x) = \\sum_{t=1}^{n} \\kappa_t \\exp ( \\lambda_t x ) \\bold c_t\n\t + \\bold g(x), \\qquad (6) ", "f152c456e11bd64909916ccabea5e9b7": "e^{-i\\int H(t) dt_{op}}\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\otimes \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}=\\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix} \\otimes \\begin{bmatrix} 0 \\\\ -1 \\end{bmatrix}", "f152cba3018c64e3085cd0114b26305d": "\\beta_2 < 0 ", "f152d94bdd00534a3ab1000bbedc8002": "N=-2k(b\\phi(rear))=-2kb(\\theta-\\psi)-2k\\frac{b^2}{V}\\frac{d\\theta}{dt}", "f1535b18be5bd43003b5cf3cf9deb0dc": "\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}.\n", "f1538c04bbe351decd8f9e4bbe53c521": "x_0=c", "f153c0c8edd33799c9678601e2392a39": "{5 \\choose 5} = 1", "f153c58b44267e76cd8c6b9e070ac3a2": "n\\,W_n = (n-1)\\,W_{n-2}\\qquad \\,", "f1541abf5fde36467907801a28370f60": "i_d", "f154774c6e8a85f65431d5ea45d8475c": "(a_n)_{n\\geq0}", "f154789f3d91ebef137f6f4616e4cbca": "\n\\begin{align}\nq\\delta^3(\\vec{p'}-\\vec{p}) =\\langle p'|Q|p\\rangle &= \\int d^3x\\, \\langle p'|J^0(\\vec{x},0)|p\\rangle \\\\\n& =\\int d^3x\\, \\langle p'|e^{-i\\vec{P}\\cdot\\vec{x}}J^0(0,0)e^{i\\vec{P}\\cdot\\vec{x}}|p\\rangle \\\\\n& =\\int d^3x\\, e^{i(\\vec{p}-\\vec{p'})\\cdot \\vec{x}} \\langle p'|J^0(0,0)|p\\rangle = (2\\pi)^3\\delta^3(\\vec{p'}-\\vec{p})\\langle p'|J^0(0,0)|p\\rangle\n\\end{align}\n", "f154a2bfd46e0a40c8716eac32d58859": "\\partial_z", "f154bf0b3c7d70026104f227f5e89eaa": "\\Phi(x) = (\\varnothing \\in x \\wedge \\forall y(y \\in x \\to (y \\cup \\{y\\} \\in x)))", "f154de3da8b2686e999537f77030deed": "{P}^{4}-4{P}^{2}Q+2{Q}^{2}\\, ", "f154e732dec0d35814cc14ed71a9b85b": "\\omega_{\\mathrm{s}}", "f1551833846442b770a252543ab54f18": "q^{\\eta \\sum_j t_{\\lambda_j} \\otimes t_{\\mu_j}}.(v \\otimes w) = q^{\\eta (\\alpha,\\beta)} v \\otimes w", "f1552c7a94a1e4bb8512c757d1351e87": "m_l=m_{l1}", "f1556c14fb945671290a7b999fcc4a98": "g^{(2)}(0)= \\frac{\\left \\langle n(n-1) \\right \\rangle}{\\left \\langle n \\right \\rangle^2 }", "f15585bcb2d79cb9969f339305aace2c": "c_{13}+b_{13}-a_{13}", "f156368235ad63cf024cc4081947b155": "u_t = -L(u)", "f1566737f4ab93335a7bbb6fdcfa1f21": "n = 40", "f156a35dc87ae47f99934714816b8b5c": " \\mbox{U} ", "f15738156341ddcfda6696f5026e5f46": "R_{\\text{2}}", "f157935a74fcf1b7d3ac5f2f7b60369c": "1+\\sqrt{2(2+\\sqrt{2})}", "f157a457d16b27cb424cb96dca85f8ca": "x+y=2", "f157c6b962e1eb2b9690ee2835691aab": "\\Delta \\circ \\eta = \\eta_2 \\circ \\Delta_0: K \\to (B \\otimes B),", "f157dd3f760ad8b6cf51588bf8daf5e2": "p_r = p/p_c\\,", "f157f1768a5e548e985b5cddc4c36a36": "\\frac{1}{w}=\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}+\\frac{1}{d}\\;(=0.0703)", "f1580117bb62a19177573deeb86c07dc": "T = {T^m}_m", "f15818bf76a50a2accbd4f018336382e": "1:4/\\pi:1.61899\\ ", "f15837f035f577dcbb9a338ae8248b9f": "\\int_0^\\infty \\cfrac{1+{x}^2/({b+1})^2}{1+{x}^2/({a})^2} \\times\\cfrac{1+{x}^2/({b+2})^2}{1+{x}^2/({a+1})^2}\\times\\cdots\\;\\;dx = \\frac{\\sqrt \\pi}{2} \\times\\frac{\\Gamma(a+\\frac{1}{2})\\Gamma(b+1)\\Gamma(b-a+\\frac{1}{2})}{\\Gamma(a)\\Gamma(b+\\frac{1}{2})\\Gamma(b-a+1)}.", "f1583c2cd56cf16e7877657e39a73b0f": "\\left[\\begin{matrix} A_{11} & 0 & A_{1\\Gamma} \\\\ 0 & A_{22} & A_{2\\Gamma} \\\\ A_{\\Gamma 1} & A_{\\Gamma 2} & A_{\\Gamma\\Gamma}\\end{matrix}\\right]\\left[\\begin{matrix} U_1 \\\\ U_2 \\\\ U_\\Gamma\\end{matrix}\\right] = \\left[\\begin{matrix} F_1 \\\\ F_2 \\\\ F_\\Gamma\\end{matrix}\\right],", "f15858bab73b4fc9c0a9e12f8cf44b19": " \\{1,-1,\\times\\}\\mapsto \\{0,1,\\oplus\\} ", "f158926bbe69a7127c5b8ce1b3ded9b6": "|x\\rangle |f(x)\\oplus y\\rangle", "f158a1e170e5b057a3985316e5eb03b9": "f(x,y) = 0 \\,\\!", "f158c44fe79edb5e942bc8f92248f1f8": "\\lambda |x-y|^{-n-s} \\leq k(x,y)", "f158f847a6c4be6134d8d72e21543bb3": "\\log(Y-\\delta)", "f158fdccb7ff7011e5324c5da59577ec": " j = j_0 \\cdot \\left\\{ \\exp \\left[ \\frac { \\alpha_a nF \\eta} {RT} \\right] - \\exp \\left[ - { \\frac { \\alpha_c nF \\eta} {RT}} \\right] \\right\\} ", "f1597e0574f594f33ec53c74ddf1adf5": "\\vec{t_3}", "f159b92ca62d5b8ded07f99a105eee82": "(f, g)", "f159bff61d748c2690319b10f78df625": "e = \\sum_{k=1}^\\infty \\frac{k^6}{203(k!)}", "f159ca9d6d0d532af0deea4e1e743c94": "Q\\;=\\;C\\;A\\;P\\;\\sqrt{\\bigg(\\frac{2\\;M}{Z\\;R\\;T}\\bigg)\\bigg(\\frac{k}{k-1}\\bigg)\\Bigg[\\,\\bigg(\\frac{\\;P_A}{P}\\bigg)^{2/k}-\\;\\,\\bigg(\\frac{\\;P_A}{P}\\bigg)^{(k+1)/k}\\;\\Bigg]}", "f159cb30266b12a0a388d54a165d7f2d": "\\alpha^{j}", "f159d0425418d94925f5499d092c9cba": " \\xi(p) \\propto (p_c - p)^{- \\nu} ", "f159eeacc0cbb7950833e39896d92755": " \\lambda=\\frac{2\\pi c}{\\omega}", "f15a0043032f61d467d68e28abc76b60": "\\mathcal{F}_{s}", "f15a01e7cc1cccec9eb3b0833a13ea95": "p(t)= t^d - c_1t^{d-1} - c_2t^{d-2}-\\cdots-c_{d}", "f15a2441e1e9741f82ec7667a83aa83b": "( \\lambda x . t) s", "f15a32b8bf1c13a81eb7cf3cd390827c": "\n\\frac{d}{dt} \\mathbf{x}(t) = \n\\mathbf{A} \\cdot \\mathbf{r}_{k} = \\lambda_{k} \\mathbf{r}_{k}\n", "f15a467bd8fa2be4f8dac92b2e8a62ff": "\\mathbf{Y}_{l,-m} = (-1)^m \\mathbf{Y}^*_{lm}\\qquad\\mathbf{\\Psi}_{l,-m} = (-1)^m \\mathbf{\\Psi}^*_{lm}\\qquad\\mathbf{\\Phi}_{l,-m} = (-1)^m \\mathbf{\\Phi}^*_{lm}", "f15a68dd169664f9e190fda43699cc85": "\\forall x \\; Rxx", "f15a7b418829ca454e7fc749adc26813": "\\cos(\\Omega_m-\\Omega_j) = \\cos k |L_{m}-L_{m-1}| ", "f15a7c7937fcec4b9c817d70ff2e9891": "\\mathfrak{g} \\times V \\to V , \\, (x,v) \\mapsto xv", "f15acd9983bccf789d4c27a7eec4d2aa": "H(Q)=-\\sum_{m=1}^k \\mu (Q_m) \\log \\mu(Q_m).", "f15b234f164f73d110f50216187dd1eb": "A \\overset{\\underset{\\mathrm{def}}{}}{=} P_1", "f15bd5e3e897e3239b251700ec9d9962": "a_\\theta", "f15c0f6b915638ffc5aaf1d5dfea9ddd": "\\mathcal{SH}", "f15cebb921f09328d14839a025868d5e": "\\sum_{n=1}^\\infty ||\\xi_n||^2 < \\infty", "f15d24c383a35fe3cb605131e6562ab4": "\\left| \\Gamma_1' \\right| = \\left| \\Gamma_2' \\right| = \\frac{1}{\\left| \\Gamma_1 \\right|}", "f15d349b6888276aaf6fc00e7426a9c4": "U_{ss}(s^{\\ast}(p),p)(\\partial s^{\\ast }(p)/(\\partial p))+U_{sp}(s^{\\ast }(p),p)=0", "f15d425279465f5545be72eed42a0435": "\\textbf{B}u", "f15d5beb1388dc03f11e966d7fcdcddd": "W \\cong M\\times I \\cup_{S^p\\times D^q} D^{p+1}\\times D^q", "f15e11f8d2367c2c7c847d20879ca159": "a=b=-(N-1)/2", "f15e3c9535813141b0b7c610e85041ef": "\\epsilon_v", "f15e8fb5e11d7d2221f2867d6d457aa4": " g(s)= \\sum_{n=0}^{\\infty}a_{n} s^{-n} \\qquad M(n+1)=\\int_{0}^{\\infty}dtK(t)t^{n} ", "f15e91d2cc7ad2161f10d018b7cf3092": "\\delta Q\\ =\\left [p(V,T)\\,+\\,\\left.\\frac{\\partial U}{\\partial V}\\right|_{(V,T)}\\right ]\\, \\delta V\\,+\\,\\left.\\frac{\\partial U}{\\partial T}\\right|_{(V,T)}\\,\\delta T", "f15ecb8d86ca935d4ac54814a7db1f83": "(f(x)-f(a))/(x-a)-f'(a)\\quad", "f15f2a9b31f76fdda81b7b4509334efa": "n^\\epsilon", "f15f611e05cb3efc88562068e4b51b22": "R[S^{-1}] = \\varinjlim R[f^{-1}]", "f15f6cc7e9c084eadac1994e24bc1667": "\\Omega>4 \\omega", "f15f753a752ab2ba1b9a70ee0287b385": "I'_0 = I_0", "f15fa8a0dc9615e58e128efa81ebb0bf": "\\Lambda = \\sqrt{\\frac{ h^2}{ 2\\pi mkT}} = \\frac{h}{\\sqrt{2\\pi mkT}},", "f15fb98be0b4ecb430018fdb32516046": "a_{wf}", "f1602e393051ca6a950a13d0dc2a7f04": "f(p) < 0", "f160a1d0a55c6b25e26d2d1ff225d155": "B_n(s)", "f160a4c4b44fe2df8bf0005b250e4762": "p_2(x) = \\tfrac{3}{16}x^2+\\tfrac{3}{4}x+\\tfrac{15}{16}", "f160df0017143645272de01d8c947e53": "f(\\mathbf x,\\cos\\theta\\mathbf{a}-\\sin\\theta\\mathbf{b}, \\sin\\theta\\mathbf{a} + \\cos\\theta\\mathbf{b}) = e^{is\\theta}f(\\mathbf x,\\mathbf{a},\\mathbf{b})", "f160eb280d69d95d9acd3da281806a96": "H^K", "f1611ac0d8befbcf7370476fa9ddbc17": "\\tau_{23}", "f1615fca8905099c816ef2179111362e": " E' ", "f161ad97ef90b4983bf23b1a9ec8fdbc": "\\theta[\\vec{X}]_{11} = \\frac{k^2 \\, T}{\\sqrt{1 + k^2 \\, T^2}}", "f16210868c5edfb08c87cc65f10ce102": "r = ap+bq \\quad \\mbox{ and } \\quad s=cp+dq.", "f16264036c10fd9e4ca037457595fadb": "\\not\\leftarrow", "f16268c7ac35ea931ba4ec6e7d75b055": "0 < a_n < 1", "f1626e348678f08e67f1505e68b28f9f": " a = x_0 \\le t_1 \\le x_1 \\le t_2 \\le x_2 \\le \\cdots \\le x_{n-1} \\le t_n \\le x_n = b . \\,\\!", "f16272afc4df0399303c8eb1ae5ce592": " (\\partial A)_P=-(\\partial P)_A=-S-P\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "f1633e229edd6153f14804fcaea64aa1": " \\varphi^*(x) = \\overline{\\varphi(x^*)}", "f16357edddda3a84796dc24cc3b81567": "\\displaystyle m_0 = m_f + m_p", "f1639dedd892e3bbf740b2aa6c224a3d": " \\alpha =\\frac{\\theta}{(1-\\theta)p}", "f163ba9da151a9a45e33bf9777b77e8c": "(a,d,c)", "f163c1e2cfff0b190ed88eeb4ed8f65d": "\\frac{\\partial S}{\\partial \\beta_1} = 0 = 708 \\beta_1 - 498", "f163c7eff729f1051334b2388905a3b7": "I = \\frac{b^2}{b^2+a^2} I_0.", "f1640902dae24e5c3c5303d5d43e6513": " e'^2 = \\frac{a^2 - b^2}{b^2} = \\frac{f(2-f)}{(1-f)^2}.", "f16443d5f02262f552253cc888d70260": " \\Phi_\\lambda(e^X) = {\\chi_\\lambda(e^X)\\over d(\\lambda)}.", "f16455784f80d91ddd871fb18b81001e": " W = E_{\\rm EA} + E_{\\rm C} - E_{\\rm F}", "f1646cc4c27a327a1bbbff1a6b454837": "R= \\int_0^\\infty I(\\lambda)\\,\\bar r(\\lambda)\\,d\\lambda", "f1648e89cf4d99aca2f8ee26cab91def": "H_\\ast(-)=H_\\ast(-,k), H^\\ast(-)=H^\\ast(-,k)", "f164a73ede952697f60cb6065a4c7903": "\\sigma_k(t + \\Delta t) := \\left\\| \\mathbf x_{k\\alpha}(t+\\Delta t) - \\mathbf x_{k\\beta}(t+\\Delta t)\\right\\|^2 - d_k^2 = 0.", "f164b47e88d6ed9ac27a7a03fb4e4d3a": "j^2=b", "f164ba4c472f05959903d88c34ac1f31": "\\alpha_V = 3\\alpha_L", "f164ce67933cff19a600285bf04efdf5": "\\overline{\\lambda_j} = P(\\lambda_j)", "f165894d5614f9c5c9cfce9f70b71ecd": "V_1, V_2 \\subset {{\\mathbf{K}}}^n", "f165abbc40b02048da61d71c7ac00122": "(x_{1},y_{1}),\\ldots,(x_{m},y_{m})", "f165b05e7caa95a86025f0500e17d54d": "\\frac{Q}{t} = \\frac{AKh}{L}", "f165b835ceb536f45e133ba9fa914047": "F_{\\nu}(-w)", "f16637c1e52f2049655f7df0f575e985": "N(t) \\backslash s ", "f1663cfb046bf83dc25e6d0d43652a6a": "g_{00} = e^{\\nu(r)} \\;", "f166635287cdbb398d41f5a5ac5b37f8": "\\left | \\frac {e}{\\varphi (N)}- \\frac {k}{d} \\right \\vert = \\frac{1}{d \\varphi (N)}", "f1667d4641a1c37bfc4851d7633fac62": "\\bar\\eth\\left({}_sY_{\\ell m}\\right) = -\\sqrt{(\\ell+s)(\\ell-s+1)}\\ {}_{s-1}Y_{\\ell m};", "f1668b12878bc86925d6b738b6378f2a": "a(t)=1+t r\\,", "f166adcd1d6c9e9bad5804bd9e01adcb": "\\!V \\approx 3.0524184684r^{3}", "f166d3132d482d0c11578108dc33f66c": "g_{33}=\\, r^2 \\sin^2 \\theta", "f166d3c2137ecbdddfd9b6af736e692d": "1 - 4|c_+c_-|^2", "f166e2deead7dce391af48c67757a48b": "\\lim_{n \\to \\infty} \\frac{a_n}{b_n}", "f166f43660fe9acf479976bfdfd6a6b8": "R_E", "f16707f9a2430fb793113325e4512508": "\\hat{g}(\\omega_x, \\omega_y, t) = \\hat{h}(\\frac{\\omega_x}{\\varphi(t)}, \\frac{\\omega_x}{\\varphi(t)})", "f1670c38e5f8eada557344d7c98d7e0a": "d = n \\lambda/\\sin \\theta", "f1671bc65a20e3e6d44db6c52510876b": "x^4+5x^3+5x^2-5x-6", "f167355c0aba57a828b350ad3fd5dbc7": "\\pi_1 (M) \\cong \\pi_1 (X)", "f1675ce14a74a7e57ee7e3c4c282e0c1": "\\mathrm{variance} = s^2 = \\displaystyle\\frac {\\sum_{i=1}^n (x_i - \\bar x)^2}{n-1} \\!", "f167b4140b7a2646f5d3b33e6852a5cf": "[\\Phi_b]_{\\Phi_a}", "f167c7a6f02d67dde3f75383b9c381a5": " v_{j+1} \\leftarrow w_j / \\beta_{j+1} \\, ", "f167e917b18060062712f31893dc8769": "\\scriptstyle P_{mmHg} = 10^{8.04494 - \\frac {1554.3} {222.65+T}}", "f1680a8e146ab25afedd10feef4e57e9": "\\epsilon^{-2}", "f1682aa8a6a8cb6e1b028c789c24d6d5": " z \\to \\infty ", "f1683d5f311d159962bf3b37f0a8bc67": "g\\cdot A = gAg^*. \\, ", "f168579f64bb339d7219a432eff2cd4e": "Q=\\int j_0a^3d^3x", "f1686e57ce2a33deef216354f007fdd7": "A=\\frac{B^2}{4}\\sin^2(t)", "f168ba02d79ffcb80f43b1330da8f5b4": "\\eta = \\int_{0}^{t} \\frac{dt'}{a(t')}", "f168bd1b4945dc2084b682c1cf5368be": "q \\mid p^2-p+1", "f168c800c3af0e9feb1f94e975583ccf": "K_i=\\frac{n-i+1}{i}K", "f168ecb8ea82616ea66627b6dcc574fc": "O(nm)", "f169658b0a2b9a00f64d1f5a33e7aa9e": "\\alpha(x) = \\pm x \\qquad x^t = \\pm x \\qquad \\bar x = \\pm x", "f16a31bb65d25326689770c2d15ac292": "p + q = p_0\\,", "f16a36a08a548c3a1fcec78764533744": "H_{\\omega^{\\omega^\\omega} + \\omega^\\omega}(1) - 1", "f16a9c080c9ddd8d44a039009c9d9417": "S_{11}\\,", "f16b07bf5468e95bc9a870f4fb10eb99": "\\left( \\frac{-7}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm1\\right)", "f16b0fc51851381c75595c52a72a1aeb": "FOV_P", "f16b11e8a85cfd07715698de37f6a489": "g_{ab} = \\eta_{ab}\\,", "f16b7cb06124fc419e52df4b319fd43d": "x'\\,", "f16bb2aa7ca50f5219feec4a34045749": "(\\sqrt{\\lambda_i},U_i,V_i)", "f16bc7287018cc5f125777db5f2904f0": " E_\\text{P} = \\frac{m_\\text{P} l_\\text{P}^2}{t_\\text{P}^2} = \\hbar \\frac{1}{t_\\text{P}} ", "f16bce057764b9544dd66ca46d8e06dd": "U \\subset X", "f16bf3187e90c13df1832de28344818a": "x = A \\sqrt{1+ {\\left(\\frac{y}{B}\\right)}^2} \\quad (14)", "f16c5032bb967df3d4ad1310f28b982d": " y'", "f16c9243437f21e63961e1c3b3e98f9e": "Pmf = \\cfrac{1 \\cfrac{3 Pmo + 1 Pmf}{3} + 3 \\cfrac{2 Pmf}{3}}{3}", "f16d1d01d5fbe88126072aa9d80c3f72": "M = \\left[ {\\begin{array}{*{10}c}\n 2 & 0 & 0 & -2 \\\\\n 0 & -2 & 2 & 0 \\\\\n 2 & 0 & 0 & -2 \\\\\n -2 & 0 & 0 & 2 \\\\\n 0 & 2 & -2 & 0 \\\\\n -2 & 0 & 0 & 2 \\\\\n \\end{array} } \\right]", "f16d9c179e427d9be02f44915308bcad": " {\\partial L\\over\\partial x} - {d\\over dt }{\\partial L\\over\\partial \\dot{x}} = 0 ", "f16d9f0105e21ef72408122793d95e7b": "(d_{2}, d_{1})", "f16dacabd0cb69e1902c7939eba49866": "\\rho_g(X) = -\\int_0^{\\infty} g(1 - F_{-X}(x)) dx.", "f16daf2a0cf77b8df09c51973a801dcf": "I = \\int (x^2 + y^2) \\, dm + r^2 \\int dm - 2r\\int x\\, dm.", "f16ded19f745c581c4a4cd70b83fe9fb": "\\textstyle w = (0,0,0,v,1,0)", "f16df6e70f78295a5c93475434e591b3": "\\mathbf{B}\\,\\!", "f16e864259356a1a55831e538a3c8456": " f_{pn}", "f16edad7d05c5aa7dd2f7c9f0d3bf0f8": "N = \\begin{bmatrix} 1 & 2 \\\\ 2 & 1\\end{bmatrix} ", "f16f25918bb7b6a4fafcf4331d587d93": "Q(x) = (x-x_1)(b_3x^3+b_2x^2+b_1x+b_0)", "f16f4642bf733988f7b763b88c55210e": "\\text{duplicate}: (M \\rarr A) \\rarr M \\rarr (M \\rarr A) = f \\mapsto m \\mapsto m' \\mapsto f \\, (m * m')", "f16f59be0d81bd9b1a4f4598eabcfbeb": "C(X) < X \\exp\\left(\\frac{-k_2 \\log X \\log \\log \\log X}{\\log \\log X}\\right)", "f16fc69913dc22d03c2dc132ef23ce78": "{\\mathfrak d}({\\mathbb P})", "f16fcbf220b4ddfec6a0918480548c8f": "\\sum_{n=1}^\\infty m_{n}^{-\\frac{1}{2n}} = + \\infty. \\,", "f170b7b921d6d9cc6cd2fbd02c85b611": "q_{\\text{IN}} = C_S V_{\\text{IN}}.\\ ", "f1710b0999c1eb6b4ee7b23d69e42256": "N^m \\doteq \\frac{Nm+\\sqrt{N}(1-m)}{m+\\sqrt{N}(1-m)}", "f1712cbc16d2fca8c8a448283e90e0e6": "\\bar{N}(E)", "f17146a9c543cbcb078929ea537c17ad": "k_1, ..., k_d \\in", "f17183db32bb7b9ab34023a4c9ba4012": "\\Lambda_{Pillai} = \\sum _{1...p}(1/(1 + \\lambda_{p}))", "f1718534d3b13ebe8359b04390e691ee": "\\sin \\theta_{ab}\\sin \\theta_{cd}\\cos x = \\cos\\theta_{ac}\\cos\\theta_{bd} - \\cos\\theta_{ad} \\cos \\theta_{bc} \\ , ", "f171acd7ef2b1cc094312d673d74df09": " \\neg \\neg \\neg A \\vdash \\neg A ", "f171c58799f733fd2d1355a4f3d295bc": "\\textstyle{S=\\frac{13 \\times 5}{2}=32.5}", "f1721a6ec38348bd03876b395b21edef": "\\sum -a_{j,k}(j^2+k^2)e^{i(jx+ky)}=\\sum b_{j,k}e^{i(jx+ky)}", "f1724097b01dc0f241b91a72082479e5": "\\sigma = E\\varepsilon", "f1728956a8071ab9961a4cd74803f0ea": "\\frac{d \\angle G(s)}{ds}{|}_{s = j\\omega_c} = 0", "f172a6f6fcb52189e355a6dff85a9da2": "\\int_V \\nabla \\cdot F(x) |dX| = \\oint_{\\partial V} F(x) \\cdot \\hat{n} |dS|.", "f172ad863b5215f56571974deadc2563": "F(x) = 2 (\\sqrt{x+\\ln x} + \\ln(x+\\sqrt{x+\\ln x})) + C.", "f172d0d6d843bba96fdbb4c3b8857f3f": "(x,e)\\cdot g = (x,e\\cdot g)", "f173395249fe346ee5a75b65ee4392f7": "failures\\left(\\left(a \\rightarrow STOP\\right) \\Box \\left(b \\rightarrow STOP\\right)\\right) = \\left\\{\\left(\\langle\\rangle,\\emptyset\\right), \\left(\\langle a \\rangle, \\left\\{a,b\\right\\}\\right), \\left(\\langle b \\rangle,\\left\\{a,b\\right\\}\\right) \\right\\}", "f1735627ab01b39ee24353a3106a12a7": "\\bar r", "f173691287ee12ef7556d5dc340a1c3b": "\\hat{a} = (\\hat{a}_1,\\ldots,\\hat{a}_I)^T = \\lim_{\\eta\\rightarrow\\infty} \\hat{a}^{(\\eta)}", "f1738734eaa74790315784ac89856c10": "\\; \\{A_{k_1}\\ldots A_{k_l}\\} \\subset \\{A_1\\ldots A_m\\}", "f173b96be85263189c18cf32ef1b467c": " ERI = { r \\times n \\over 1 - \\phi} ", "f173c91a8b82a61429b163f36d44f7da": "\n \\frac{d}{dt} \\mathbf{p} \\times \\mathbf{L} = \\left( \\frac{-k}{r^2} \\mathbf{\\hat{r}} \\right) \\times \\left(m r^2 \\boldsymbol{\\omega}\\right)\n= m k \\, \\boldsymbol{\\omega} \\times \\mathbf{\\hat{r}} = m k \\,\\frac{d}{dt}\\mathbf{\\hat{r}}\n", "f173d72ce7b35c97ba8ec3d012044c77": "Y_1,\\ldots, Y_n", "f1743feec24e498d81cff1e48b884a0e": "\\Big( (\\mathcal{M}, s) \\models \\phi_1 \\Rightarrow \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big((\\mathcal{M}, s) \\not\\models \\phi_1 \\big) \\lor \\big((\\mathcal{M}, s) \\models \\phi_2 \\big) \\Big)", "f17463a0872960627db4be1d335ef55d": "\\begin{align}\nI(X; Z) &= H(X) - H(X|Z)\\\\\n&= -\\sum M \\log M + \\frac{1}{2} \\left[ \\sum P \\log P + \\sum Q \\log Q \\right] \\\\\n&= -\\sum \\frac{P}{2} \\log M - \\sum \\frac{Q}{2} \\log M + \\frac{1}{2} \\left[ \\sum P \\log P + \\sum Q \\log Q \\right] \\\\\n&= \\frac{1}{2} \\sum P \\left( \\log P - \\log M\\right ) + \\frac{1}{2} \\sum Q \\left( \\log Q - \\log M \\right) \\\\\n&= {\\rm JSD}(P \\parallel Q)\n\\end{align}\n", "f1749e8e8bdcb0aaaffbb956c90d445b": "F_M(\\mu)", "f174c6eeeee9a5f37b89c0685a8a9a08": "E_{in}=E_{out} \\,", "f174f955a776c719f7a3b92167ee6924": " \\mathrm d\\varphi_x(X)(f) = X(f\\circ \\varphi).", "f1754b3731766527adb491eeaac13bb7": "I(z) = I_0 e^{-2z/\\delta_{skin}}", "f1756a67cebce52ec5ca0bf6a1eb62f7": "\\gamma F_{\\mathrm{M}}", "f1759afc85ffa682c5b9920d4d0ffc7c": "\\zeta^{\\prime}(-2n) = (-1)^n \\frac {(2n)!} {2 (2\\pi)^{2n}} \\zeta (2n+1).", "f175a1c6c37e94b4db37e6ea8a434dea": "a_1= k_0(x_1-x_0)-(y_1 - y_0)=-0.1875", "f175f693232a2d7d1fc4c5ceac084bf0": "\\displaystyle \\phi_{xx} + c_3 \\phi_{yy} = ( |u|^2 )_x", "f17600da507f6d9e910bba5a81b57052": "\\displaystyle{}_{r+1}V_r(a_1;a_6,a_7,...a_{r+1};q,p;z) = \\sum_{n=0}^\\infty\\frac{\\theta(a_1q^{2n};p)}{\\theta(a_1;p)}\\frac{(a_1,a_6,a_7,...,a_{r+1};q;p)_n}{(q,a_1q/a_6,a_1q/a_7,...,a_1q/a_{r+1};q,p)_n}(qz)^n", "f1761967ef22edee173776756be18d54": "\\text{var}\\,(Y) = a[\\text{E}\\,(Y)]^p", "f17658042a68d558ea3bb303bbf1685f": "x(t) = \\frac{1}{2}a_0 + \\sum_{n=1}^\\infty\\left[r_n\\cos(2 \\pi n f_0 t + \\varphi_n)\\right]", "f176cb6a6aa52068cb7ecc2ce5fbaf12": "D(X,Y) = d(X,Y)/H(X,Y) \\le 1.", "f1774ba954ec261230a6a6e1017c6638": "f_d \\approx 2v \\frac {f_t}{c'} ", "f177b039c8d017df93af261f3a29857c": "S\\cap\\omega_1", "f177c080b55f3974f191ec5e9d3f670d": "\\frac{dy}{dx}=\\frac{s}{a}\\,", "f177c14b0131c149f19f7a68672c0cfe": "0 = \\tau_{0} < \\tau_{1} < \\dots < \\tau_{N} = T;", "f177cb921049d4be2ac15153851258d2": "c_{FR_n} = 1 - e^{-\\lambda n}", "f177e887b9db72eb7583e5d240560255": "H(z) = \\frac{B(z)}{A(z)} = \\frac{{b_{0}+b_{1}z^{-1}+b_{2}z^{-2} + \\cdots + b_{N}z^{-N}}}{{1+a_{1}z^{-1}+a_{2}z^{-2} + \\cdots +a_{M}z^{-M}}}", "f178364c7069a71e8716e1ab2dcea6d3": "n\\leq 2^m-1", "f178a07178207bf509caa158c9b19353": "\nF_D ", "f178a617a7bfd77f0d891bb9381efde3": "I^\\pm(S) = \\bigcup_{x \\in S} I^\\pm(x) ", "f178baa39a4b667d4a2c5bf744b7cf73": "\\mathbf{X}^{+}", "f17912cadf6e63a4174e8fd31864144a": "\\sigma>\\sigma_c", "f179146524e142ca0cffc4cf1ba33555": "\\textbf{r}\\times\\mathbf{F}", "f17956bc68783be816ca82d1e4d6739c": "\\frac{\\partial \\tau^{-1}(\\mu)}{\\partial \\mu}= \\frac{1}{V(\\mu)}", "f17966f40794a41d7bd199c8853d18aa": "\\Delta_1(e_i) = 1 \\otimes e_i + e_i \\otimes k_i", "f1798cee9ac166279f2464dd8ea0502b": "S_n(\\hat\\theta) = M_n(\\hat\\theta)", "f179d2d1f30639b25656fc8d3aa0c8d0": "\\frac{\\frac{dg(x)}{dx}-\\frac{dh(x)}{dx}\\cdot\\frac{g(x)}{h(x)}}{h(x)}", "f17a53ff3bdd908407dd8e8a29787e35": "\\int\\!\\left( \\sum^n_{k=0} a_k x^k\\right)\\,dx= \\sum^n_{k=0} \\frac{a_k x^{k+1}}{k+1} + C.", "f17a61165c03e77756ffe87b98d49b72": "(\\mathcal{M}^3, g_{ab})", "f17a8cb2481dbc8d7af886a32b0904a6": "\\sum_{j=1}^k \\frac{1}{j}\\leq1+\\log k; \\qquad \\sum_{j=1}^k \\frac{1}{j^2} \\leq 2.", "f17b80e06452d23eff4e890c466f29e3": "\\xi(z) = \\xi(1-z). \\,\\!", "f17bdfe754fa64d316f56893ff06c208": "\\rho_b = -\\nabla\\cdot\\mathbf{P},", "f17be7aa50ef23d75444fb62f3ac03ed": "\\lambda = \\frac{\\sigma_w T^2}{\\sigma_v}", "f17c143fa7efc914c5c3ee847977be71": "F = k_{\\rm C} \\frac{q \\cdot q^\\prime}{d^2}", "f17c6ae8386c954d714c2ddc81de7c0c": "~\\mathrm{{}_{~92}^{238}{}U+{}_{10}^{22}{}Ne \\longrightarrow {}_{102}^{256}{}No+4{}_0^1{}n}", "f17c8fd475812d8864bd84cedb380c82": "\\mathcal J=\\{1\\}", "f17cb154fd75f34a4d25d44e470d4d04": "L \\in \\mathcal{S}", "f17cd1f454b00dc06a4658398e02d9ae": "max(0, receipt\\;date - due\\;date)", "f17cfa6b4b1e22cb5211ce3d3470c874": "\n\\beta q \\psi \\ll 1 \\;\\;\\; {\\rm or} \\;\\;\\; \\psi \\ll\n\\frac{kT}{q} \\simeq 26 \\; {\\rm mV}\n", "f17d3d3851bb794b5616e0bb4bc30f63": "\\operatorname{Li}_2\\left(\\frac{1}{3}\\right)-\\frac{1}{6}\\operatorname{Li}_2\\left(\\frac{1}{9}\\right)=\\frac{{\\pi}^2}{18}-\\frac{\\ln^23}{6}", "f17d576feaed350e585862fd26f71350": "K^0(A)", "f17d97c17cd28e3e02416644ee546a0d": "\\{\\ x\\}", "f17ddd94bfa9507dbaba07860c21acce": "\\psi\\sim\\Phi", "f17e39c21624ecbb0525014b1e334427": "z=0, 1", "f17e466fc27131c9faf22b28dbe6f47f": "\\psi\\in L^2(R)", "f17e6af18e0f7311fddcb98fe4d627d0": "s_{\\mathrm{Euclidean}}:(x, y) \\mapsto 1,", "f17ed47f08db8363eb7296ece6079f5e": "\\Delta t=\\frac{\\Delta l}{c_0},", "f17eed35d8928830a6b925e1ca35b658": "\\scriptstyle r_1,r_2,\\dots,r_n", "f17f3949a2b8e395cadbccfe1931cd05": "\\mathbf{c}_{k_1}, \\mathbf{c}_{k_2} \\in C_{out} ", "f17f643ea10723845a89bb0c4bb2868a": "2^4\\cdot 3^3\\cdot 5\\cdot 7", "f18061f54024c9c8f5fefef33b2a4a5d": "\\mathbf{\\hat{e}}_k", "f1806d71f1c609eefecf6165e789c512": "C \\subseteq T", "f18071b3d103de5f8edfb65470820566": "G=1", "f1807d5a80bfc8c9a2ad9a2ce55b6e9a": "p_0(x)=1;", "f180e77384e8ebf7f0c4996c40176521": "Lw = 10\\, Laborers \\cdot 0.6\\,\\frac{Ph}{Laborer} = 6\\,Ph", "f180f3719bb6b188e968628c1dad4f06": " \\{ S ; q \\} = (S\\,'''/S\\,')-(3/2)(S\\,''/S\\,')^2", "f18197ff31dcc92b2e67c811e10155c0": "\n\\begin{array}{lcl}\n x'' & = & \\frac{d^2x}{dA^2} \\\\\n & = & -r\\cos A - \\frac{r^2\\cos^2 A}{\\sqrt{l^2-r^2\\sin^2 A}}-\\frac{-r^2\\sin^2 A}{\\sqrt{l^2-r^2\\sin^2 A}} - \\frac{r^2\\sin A \\cos A .(-\\frac{1}{2})\\cdot(-2).r^2\\sin A\\cos A}{\\left (\\sqrt{l^2-r^2\\sin^2 A} \\right )^3} \\\\\n & = & -r\\cos A - \\frac{r^2(\\cos^2 A -\\sin^2 A)}{\\sqrt{l^2-r^2\\sin^2 A}}-\\frac{r^4\\sin^2 A \\cos^2 A}{\\left (\\sqrt{l^2-r^2 \\sin^2 A}\\right )^3}\n\\end{array} \n", "f181beef72652b0283cd481588960eb9": "\\sqrt[p]{\\frac{1}{n} \\cdot \\sum_{i=1}^n x_{i}^p}", "f18218926b860583405ef029c6bdbdd4": "\\left\\{\\frac{X_t}{t}; t \\geq 0\\right\\}", "f1824aaa6b348f39b86a5803649c84c0": "y_{n+1}=\\sum_{i=0}^s a_i y_{n-i}+hk\\sum_{j=-1}^s b_jy_{n-j},", "f18286fd539d457febfa6bc546785e8f": "R \\ge \\sqrt{2}r.", "f182a3c35c323987786ded54eaf3a441": "y = \\frac{Y}{Z^3}", "f182ac1cfa33e6c1e120b47302b80661": "(b_0, b_1, b_2, \\ldots)", "f182b12cb31e8231f723b7b5f124cf92": "\\displaystyle U(t)=U(0).", "f182bd93779e404de3a022b1fe338269": "k_C", "f1833a1a72435b836d8c3f39740026d3": " g(s)=h_0(\\sum_i h_i(s_i)) ", "f183437e2b8b5f640c9e5c30bf2dfb60": "c ED(\\mu,\\sigma^2)=ED(c\\mu,c^{2-p}\\sigma^2)", "f1836a11b50656cc230fd49fcc561451": "K_A", "f183d1378f7955711b38bb6fbc54c23a": "s_i(t)", "f183d6304bcb9354bc9c800e26459880": " \nF(z) = \\sum_{n\\ge 1} Z(C_n)(f(z), f(z^2), \\ldots, f(z^n)) =\n\\sum_{n\\ge 1} \\frac{1}{n} \\sum_{d|n} \\varphi(d) f(z^d)^{n/d}", "f183ffb0fc0c81ac149a1cf89fa87152": "|x\\rangle |y\\rangle", "f18435a53b84d72984e86262ab3aff03": "P^{\\prime}", "f1844c35b14fd36e2a5bf2848bda5b59": "\\text{stick}(K_1\\#K_2)\\le \\text{stick}(K_1)+ \\text{stick}(K_2)-3 \\, ", "f184f4b777253741043630b6a2b10eec": "\\Omega(\\log^5 |G|)", "f1857b3752cfb2739a204a53251df531": "\\ \\Delta ASA=ASA_{unfolded}-ASA_{native}", "f185b4ce0dc9cee46f742b3ebc3a677d": "\\begin{pmatrix} 8 & 5 & 10 \\\\ 21 & 8 & 21 \\\\ 21 & 12 & 8 \\end{pmatrix} \\begin{pmatrix} 15 \\\\ 14 \\\\ 7 \\end{pmatrix} \\equiv \\begin{pmatrix} 260 \\\\ 574 \\\\ 539 \\end{pmatrix} \\equiv \\begin{pmatrix} 0 \\\\ 2 \\\\ 19 \\end{pmatrix} \\pmod{26}", "f186217753c37b9b9f958d906208506e": "O", "f1864df41a1da6e34e335b8e79a99cbd": " \\lim_{x \\to 0} {f(x) \\over g(x)} ", "f186786f8a423cb2256581ddbcd8985c": "\\varepsilon = \\frac{\\beta \\hbar}{P}, P \\in {\\mathbb Z}", "f186958a3f1414cb8698e7908d61cb68": "v(x) = 0", "f18704050400508a673d785b9cfef018": "2 O", "f18727c5b672ed136ced6b75fc7a4de9": "curry(T):S\\to(X\\to X)", "f18735bf918ef3fd86e22db07a033ca5": "H_1, H_2", "f1873d3cdee68040d4d292973f8309fa": "\\frac{\\partial J}{\\partial y_m} = F_y\\left(t_m, y_m, \\frac{y_{m + 1} - y_m}{\\Delta t}\\right)\\Delta t + F_{y'}\\left(t_{m - 1}, y_{m - 1}, \\frac{y_m - y_{m - 1}}{\\Delta t}\\right) - F_{y'}\\left(t_m, y_m, \\frac{y_{m + 1} - y_m}{\\Delta t}\\right).", "f1873d9cf96ecbdbe408e613c7dd9c8d": "\\int_0^x \\frac{dt}{1+t} = \\int_0^x \\left( 1 - t + t^2 - \\cdots + (-t)^{n-1} + \\frac{(-t)^n}{1+t} \\right)\\, dt", "f1874e4d3bb8cab4eb08e23f35307775": "\\,\\!\\mu_p", "f18768dbca80b61f2ed897d2391622cc": "\\vec \\mu_J", "f18797cea40f0f7cd910e82bb0acb6ce": "\\langle x \\rangle", "f187b99c0c60482d835f17ff50de200e": "T_{max}=\\frac{1}{2\\omega_{s}}.\\frac{3V_{TE}^{2}}{R_{TE}+\\sqrt{R_{TE}^{2}+(X_{TE}+X_r^{'})^{2}}}", "f187b99eee30e50f4f309a8b45985729": "u(x,t)=e^{-i\\omega t}\\phi(x)\\,", "f188922fe9db375a4dae67db7878cef6": "x>a\\,\\!", "f18897bab300289e8ea7fa0c977f0175": "\\frac{\\pi}{4} = 2 \\arctan\\frac{1}{2} - \\arctan\\frac{1}{7}", "f188ea9154cc0ca723561d66f90e9cc3": "\\ne 0, 1", "f1890160f1c4e9febe337d6f3b304a7c": "{\\sigma^2_X}", "f18914bc12c911f9c39bf8d19420d9e9": "\\lambda_2>0", "f1892526ceafdc59b0cfadc766485ad2": "U=\\gamma U_{0}", "f18926e2e7df133de154205e903f1ba3": " (X,T) \\sim \\mathrm{NormalGamma}(\\mu,\\lambda,\\alpha,\\beta), ", "f1894024bbf1b88c2d7908255267df26": "[\\![\\phi]\\!]_{i[Z := T]}", "f189709c6efab194a853ab79f6ccdbfb": "y = (1 + .3074301481247)2^{128 - 127} = 2.614", "f189898da78a32f8ea9da22ca4094b7f": "f_H[x(t)|0 0", "f1abdae07b1cdda02d6328807128bae6": "0 \\leq (y, z)_{x} \\leq \\min \\big\\{ d(y, x), d(z, x) \\big\\},", "f1abe0cb5ba85be16b9cb67b68a3b1b1": "\\rho = \\psi^\\dagger \\psi", "f1ac00a85e5e53b7ca4ea6d24c279452": "~q^\\alpha", "f1ac509f3fe2b251c6f3e142e03aff03": "\\textstyle{\\frac{\\log\\left(\\frac{1+\\sqrt[3]{73-6\\sqrt{87}}+\\sqrt[3]{73+6\\sqrt{87}}}{3}\\right)}\n{\\log(2)}}", "f1ac61ab2b3e0633606b09e769ba3cb6": "J\\neq id_R, J^2 = id_R", "f1ac64b47e6effca12ed91fae4035ee4": "\\frac{d}{dt}\\log f_X(t)=\\sum_{n\\ge 1}\\frac{d}{dt}(1-X_n t^n)=-\\sum_{n\\ge 1}\\sum_{d\\ge 1}\\frac{X_n^d t^{nd}}{d}=-\\sum_{m\\ge 1}\\frac{\\sum_{d|m}\\frac{m}{d}X_{\\frac{m}{d}}^d}{m}t^m=-\\sum_{m\\ge 1}\\frac{X^{(m)}t^m}{m}", "f1aca014de1ed63cce0839be9bf8d648": "\\rho\\,\\!C_p\\frac{T_n^t-T_n^{t-1}}{\\Delta\\ t}{{=}}k_n\\frac{T_{n-1}^t-T_n^{t}}{\\Delta\\ x^2}+k_{n+1}\\frac{T_{n+1}^t-T_n^{t}}{\\Delta\\ x^2}+Q_n", "f1ae317100223ef40e2030f4ed233d4f": "\\rho_{max}", "f1ae5afe2c5df9c85e29a16b94a719a4": "N = (a_1\\cdot100 +a_2\\cdot10+a_3)^2", "f1ae78e57bdebeb5bca992dcffc00396": "\nT_1 = \\frac{1}{s^2}\\sum_{j=1}^kn_j\\bar{A}_j^2\n", "f1aed2618aa4a1ed6f5f84adc8c2182b": "\\mathcal O_K", "f1af140f5c2cfcad2e7bb139c7196b4b": "(a - b) \\not\\equiv (b - a)", "f1afbc2534037e57be25c13958f447ab": "\\pi(n) = \\sum_{j=2}^n\\left\\lfloor\\frac{(j-1)!+1}{j} - \\left\\lfloor\\frac{(j-1)!}{j}\\right\\rfloor\\right\\rfloor.", "f1afc91596e84b19aa385dcd06c69d68": " a\\ne 0~ mod~ p_1^{k_1}", "f1b01d2751ddc968dd13bb8bd6e407f4": "2.75<{\\frac{L}{a}}<3.75", "f1b02fa8d8cb8d4fc081770b2df0958d": "\\eta=s_{0}/(1+s_{0})", "f1b076556a57af2027c257ae919bd8eb": "\\displaystyle\\mathrm{rad}(n)=\\prod_{\\scriptstyle p\\mid n\\atop p\\text{ prime}}p", "f1b076e0755bc46bacd11e901eeb072a": "_{interval} \\delta_{12}^2=~_{interval} \\delta_{23}^2=~ _{interval} \\delta_{34}^2 = 1^2", "f1b08563a30114df9f1c68a50487362f": "d = \\dim A", "f1b08648e00667e36a9d62ba296da0db": "f(x;\\mu,c)dx = f(y;0,1)dy\\,", "f1b0c135e4e8b72471f02b9b1edb9816": "\\displaystyle\\int_{M(A,\\gamma)\\backslash M(A)}f(x^{-1}\\gamma x) \\, dx", "f1b0f0e42fa9b26bf64bc10a324b1e00": "|\\psi\\rangle = \\sum_{i=1}^N \\alpha |i\\rangle", "f1b11edf82fb7111d3aa97d5db0d962f": "\\alpha = 3 - \\sqrt2", "f1b1506c8ed61686bc0a49a8be5aeeab": "\\frac{\\operatorname dV}{\\operatorname dr} = k\\pi r^2", "f1b18f5b9877dcda65ff2b72f7bec8e5": "t_i \\in [x_i,x_{i+1}]", "f1b1c4e72dfbe8b325c98c9466a66fe4": "L \\gg \\lambda", "f1b2a44951ca468cef4669dccbb6d9d3": "\n\\int \\sec \\theta \\, d\\theta = \\int \\frac{d\\theta}{\\cos\\theta} = \\int \\frac{\\cos\\theta \\, d\\theta}{\\cos^2\\theta} = \\int \\frac{\\cos\\theta \\, d\\theta}{1 - \\sin^2\\theta} = \\int \\frac{du}{1 - u^2}\n", "f1b2f5bf52f66284ba7dc72b53c48704": "\\scriptstyle A", "f1b30e18b2564e1d2f0b9ed76829d619": "q = u + v", "f1b36115dd03a3fc07a8fdd43260fed0": "c_{3,1}(\\alpha \\widehat{x} \\beta, \\gamma \\widehat{y} \\delta, \\zeta \\widehat{z} \\eta) = \\alpha x \\beta \\gamma \\widehat{y} \\delta \\zeta z \\eta", "f1b40091f7aaa5f2060e7b806927fc74": "(k+2)(k+1)A_{k+2}=-(-2k+1)A_k\\;\\!", "f1b4319f670974869b95a5dc10765f41": " \\frac{\\mathrm{d}\\mathbf{P}}{\\mathrm{d}t} =\\frac{\\partial L}{\\partial \\mathbf{r}} = e {\\partial \\mathbf{A} \\over \\partial \\mathbf{r}}\\cdot \\dot{\\mathbf{r}} - e {\\partial \\phi \\over \\partial \\mathbf{r} }\\,\\!", "f1b454ad801187b9b6fee130c2c5864e": "E_R(z) = t E_0 e^{ik_R z}\\,", "f1b4c25da7f92c5a062f61ce7ad8a427": " \\lambda_0 \\in \\mathbb{R}, \\quad \\lambda_0 > 0, \\quad ", "f1b53e4c85a6b77662389a7787b9e99e": "r \\dot\\theta '", "f1b5b2720fdb5efb234c2d219902565d": "L_f=(\\lambda_{ij})", "f1b5bbcbaefd93e10d935184ccc21f9a": "e(n) = x(n) - \\widehat{x}(n)\\,", "f1b5c25d70886887fcfdbb11e394bfb9": "|f(x) - L| < \\varepsilon ", "f1b5db594efcff91c6ecde949f59d0c8": "\\mathbf{c}", "f1b5ece398777eab1a1670ef978a0428": "\\Psi \\, \\psi \\,", "f1b6789a173057935116fb352293f5cf": "\\alpha<\\phi(x)", "f1b6b8a9b7b9b3e066f6a08a1219834e": "Z^{D}_{i,j}:", "f1b6f2e4574f61aaeaba508906f0dc71": "\\frac{F(x_1 + \\Delta x) - F(x_1)}{\\Delta x} = f(c).", "f1b70805882bb33dd772ec5e5ef5c2b8": "J_{z} = \\iint r^2\\,dA = \\int_0^{2\\pi}\\int_{r_i}^{r_o} r^2\\left(r\\,dr\\,d\\theta\\right) = \\int_0^{2\\pi}\\int_{r_i}^{r_o} r^3\\,dr\\,d\\theta = \\int_0^{2\\pi}\\left[\\frac{r_o^4}{4} - \\frac{r_i^4}{4}\\right]\\,d\\theta = \\frac{\\pi}{2}\\left(r_o^4 - r_i^4\\right)", "f1b70a8a7796d329a5d99fdcfc79731c": " x(t_0) = x_0", "f1b73f67bb01a19b3ec363e3a447f13b": "\\,\\operatorname{cr}(z_1,z_3,z_2,z_4)=1-\\operatorname{cr}(z_1,z_2,z_3,z_4)<0", "f1b75de032eca17fb017ee7ca747c598": " \\frac{d Q}{d P} = \\mathcal{E}\\left ( \\int_0^\\cdot \\frac{r - \\mu }{\\sigma}\\,\nd W_s \\right )", "f1b7b3a8adef9bb1dc2e187607ce4b3c": "\\sigma_{yz} + \\sigma_{xz} + \\sigma_{xy}", "f1b7dbd37c4343785f5e512b3788aaa1": "\\epsilon = 1+4\\pi\\chi_\\text{e}", "f1b7f268ee7c95a532cecb67a88c1fde": "F^{\\mathrm{rad}}_\\mu = \\frac{\\mu_o q^2}{6 \\pi m c}\n\\left[\\frac{d^2 p_\\mu}{d \\tau^2}-\\frac{p_\\mu}{m^2 c^2}\n\\left(\\frac{d p_\\nu}{d \\tau}\\frac{d p^\\nu}{d \\tau}\\right)\n\\right].", "f1b8d6e8760ade176435dbfb5ac96c71": "\\sigma \\cdot \\hat{p} \\xi_\\lambda(\\hat{p})=\\lambda \\xi_\\lambda(\\hat{p}) \\,", "f1b92e10be03ae2ce8914856aad74acf": "(p_n\\circ q)(x)=\\sum_{k=0}^n a_{n,k}\\, q_k(x)", "f1b97a984fad26d0ce4a30f79724254a": "V_2 = \\sum_{i=1}^n {w_i^2}", "f1b9a53deb0923e100203993604eba3a": "~\\Phi_6(x) = x^2 - x + 1", "f1b9e5d1bafb005aac8a0f7e7c078b95": "\\binom{n+1}{3}+1.", "f1ba5108cd6fc6de29f700c640b53f63": "SUN_{white}", "f1ba8cce606ef217968181da5485af49": "\\begin{align}\n\\max\\{x_1,x_2,\\ldots,x_{n}\\} \n& = \\sum_{i=1}^n x_i - \\sum_{i 2", "f1baf4d9dfe74825194a3d22e6ce6f96": "0\\leq N<\\tbinom nk", "f1bb7b44dda24a7ff7290966cff495b4": "\\Gamma_\\tau", "f1bbebd6ead24bc3cedd25016d174f49": "\\sigma = 2\\pi \\int_0^\\pi \\sigma(\\theta) \\sin(\\theta) \\; d\\theta = \\frac{4\\pi}{k} \\text{Im} \\left(f(0)\\right) \\;,", "f1bc52d346aab1f607d7397a5a886b68": "1 + \\cfrac{1}{2 + \\cfrac{1}{5 + \\cfrac{1}{5 + \\cfrac{1}{4 + \\ddots}}}}", "f1bcea7dc657783d1db60ffad5bfa4cb": "\\begin{align}\nX(z) = \\frac{1}{2} \\Re \\Bigg\\{ \\Big(\\frac{-1}{kz(z^k-1)} \\Big) \\Big[ &(k-1)(z^k-1)_2F_1(1,-1/k;(k-1)/k;z^k)\\\\\n& {}-(k-1)z^2(z^k-1)_2F_1(1,1/k;1+1/k;z^k) \\\\\n&{}-kz^k +k+z^2-1 \\Big] \\Bigg\\}\n\\end{align}", "f1bd04125ac040baa8aaaaff096c370a": "C_{Cr} = \\frac {1.25 mg/mL \\times \\frac{60mL}{60min}}{0.01 mg/mL} = \\frac { {1.25 mg/mL} \\times {1 mL/min}}{0.01 mg/mL} = \\frac {1.25 mg/min}{0.01 mg/mL} = {125 mL/min}", "f1bd218c5fd522a0997998a4e76b1aff": "N(x)=N_\\text{max}\\left[\\frac{x}{x_\\text{max}}\\right]^{x_\\text{max}}\\exp(x_\\text{max}-x).", "f1bd43e012a6893b926673397733a769": "s_3 = fghhkll", "f1bd4b338722256bad6fe3f96ac178e3": "M_{BC} = 0.8 \\times \\left( -6.937 \\right) + 0.4 \\times 5.785 - 8.333 = -11.57 ", "f1bd5f8b11999dfaf83c05c8597dad08": "\\displaystyle{(P_j+iQ_j)\\Delta^{-1/2}.}", "f1bda4d87d05fbf8cf7b90853418e1f0": " N(\\beta V_{0}-V_{t})^{2} = \\beta(V_{0}-V_{t})^{2} ", "f1bde5dc42040add2f71df35be9b2e9b": "\\scriptstyle \\sqrt{\\langle\\Delta r^2\\rangle} = a\\sqrt{\\Gamma t}\\,\\!", "f1be4e39b50d2f91a07b649d45a41e37": "mE=mE_0+{P^2 \\over 2},", "f1be858aea2be1a36e145c060ec282d8": "\\Delta\\phi = 2\\pi\\Delta v\\, t/\\lambda", "f1be88b6a36fa4cd596e7c057bc396e7": "[(E^2 - (mc^2)^2)^{2j} - (\\hat{\\mathbf{p}}^2)^{2j}]\\begin{pmatrix}\n\\psi_{1,2}^{[2j]} \\\\\n\\psi_{3,4}^{[2j]}\n\\end{pmatrix}\n = 0", "f1bf66021a901c13d59ddc82d44ed54e": " 3 \\rightarrow 3 \\rightarrow 3 \\rightarrow 3 ~~ = ~~ 3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow 27 \\rightarrow 2) \\rightarrow 2\\, ", "f1bf74bfc505b546518d8d69a4b43cb4": "Q \\or S", "f1bfe62c95c3e2f806aef7e0b8e68620": " H(s) = \\frac{Y(s)}{X(s)} = \\frac{ \\mathcal{L}\\left\\{y(t)\\right\\} }{ \\mathcal{L}\\left\\{x(t)\\right\\} } ", "f1c006c2802144d61972483efd03c280": "3k", "f1c009efecc5a7a8e5025830aa233d8d": "y_2(t) = H \\left \\{ x_2(t) \\right \\} ", "f1c037927577a5019cef57235b53ea91": "f(x, t) \\approx \\frac{1}{2\\pi} \\cdot 2 \\operatorname{Re} \\left\\{ e^{i \\left[k(\\omega_0) x - \\omega_0 t\\right]} \\left|F(\\omega_0)\\right| \\int_{\\mathbb R} e^{\\frac{1}{2} i x k''(\\omega_0) (\\omega - \\omega_0)^2} \\, d\\omega \\right\\}", "f1c07916b16b3e91049d0c5b98d97a6b": "D^2+E^2=4(A+C)F", "f1c089a07d172f0977541b050324d9c5": "\\sum_{k=1}^{n} \\overline{z^{j\\cdot k}} \\cdot z^{j'\\cdot k} = n \\cdot\\delta_{j,j'}", "f1c098deb31e81477e120e6d2e91966e": "x_1,\\ldots,x_j,x_i", "f1c0dc09587eb37a5e2fcdfc08b69f5f": "E^s", "f1c11e127314c114e7734ca4a14cddba": " n! [z^n] g(z) = n! \\sum_{a+pb=n} \\frac{1}{a! \\; p^b \\; b!}\n= n! \\sum_{b=0}^{\\lfloor n/p \\rfloor} \\frac{1}{(n-pb)! \\; p^b \\; b!}.\n", "f1c14eb9651e376640249773471286bb": "y_0\\in U_{iy}", "f1c18aa0ced61986c1b2e999a53633be": "a_n=3^{-n}", "f1c1ad581240ca5499384d8ba14488b1": "|\\phi_1\\rangle", "f1c1af8f416ae4119bdfa908a593a39b": "\\begin{align}\n I(t) &= \\frac{V_0}{R} e^{-\\frac{t}{\\tau_0}} \\\\\n V(t) &= V_0 \\left( 1 - e^{-\\frac{t}{\\tau_0}}\\right)\n\\end{align}", "f1c1b1f2ccab59b3817d994aeae477f2": "i.j", "f1c1e4a4dacba82fa6da389ee8889ee8": "R'_s = R_s - R_{mb}", "f1c217ad97427783df5e83543578ad34": "k \\mapsto g(n,k)", "f1c280d09e954666a1b64af469f0a0bd": "\n\\begin{matrix}\n\\bar{X}_w & = & \\frac{1+p}{1-p} \\\\\n\\bar{M}_n & = & M_o\\frac{1}{1-p} \\\\\n\\bar{M}_w & = & M_o\\frac{1+p}{1-p}\\\\\nPDI & = & \\frac{\\bar{M}_w}{\\bar{M}_n}=1+p\\\\\n\\end{matrix}\n", "f1c2d433eecbe98161ffca5e46f966f3": "\\ \\beta = 90 \\deg", "f1c2e8a3fbd35190ea9e117281629465": "\\mathbf{s}_0", "f1c2fdaf09571586f5afdc33216c934e": "\n x^{a_0+a_1+\\ldots+a_n}A_{(h_0,\\;h_1,\\ldots,\\;h_n)}(x+1)\\cdot(y_0)^{h_0}\\cdot(y_1)^{h_1}\\cdot\\ldots\\cdot(y_n)^{h_n}\\!", "f1c3099235e35f296742ed14f58fcd13": "\\lambda(E)=\\lambda^*(E)", "f1c309dfb5be8c5fa607d23490f1d998": "\\lambda_{1,2} = \\frac{\\operatorname{tr}(A) \\pm \\sqrt{ \\operatorname{tr}(A)^2 - 4\\,\\det(A)}}{2} .", "f1c31a9bc1b7ca79b5ab093fa04d4295": "\\beta_{-\\sqrt{s}}\\alpha_{-\\sqrt{s}}\\beta_{\\sqrt{s}} \\alpha_{\\sqrt{s}}(x)", "f1c3377e66b6413f75d6d13c26dc5662": "OPS+ = 100 * (\\frac{OBP} {*lgOBP} + \\frac{SLG} {*lgSLG} - 1)", "f1c33afad47476e38300785d60478f24": " p(\\alpha_i) = y_i", "f1c33c28e0a9cc2fcf29980d80441645": "m_{\\tilde{t}}", "f1c3547f5d8e96ecd06e8ab44e572216": "v[\\alpha]", "f1c36f47748f87e15b4faabca62cee28": "\\tfrac13\\, c\\, \\left( \\eta' \\right)^2 = f(\\eta)\\,", "f1c4552370bd94fc364a50035b5a3949": "\\hat{Z}(x_0) = \\begin{bmatrix}\n w_1 & w_2 & \\cdots & w_N\n\\end{bmatrix}\n\\cdot\n\\begin{bmatrix}\nz_1\\\\\nz_2\\\\\n\\vdots\\\\\nz_N\n\\end{bmatrix} = \\sum_{i=1}^n w_i(x_0) \\times Z(x_i)", "f1c4f1836a75e9c00e399c0b06459322": "\\Pi_{a,\\ b * 0.5,\\ c}( R )", "f1c52a6d9fff73e8df119c1c44ff6ec6": "I M_n \\subset M_{n+1}", "f1c58e36a4da610d60986803436e32bc": "\\displaystyle{ F_w(z)=e^{wz^2/2}.}", "f1c593327def2b5027d9d69191b85bb0": "p=p_0-\\rho g z \\,", "f1c59c88bd2e48c6da2da4fa3b66a226": "d_m+d_{mm'}=d_{m'} \\, ", "f1c5aa4214eb2d617e72611d8a0005a6": "\\xi \\propto (-\\tau)^{-\\nu^\\prime}", "f1c5b6bf104731f392029169aaf9db71": " \\sqrt{n}(\\delta_n - \\theta_0) \\to N\\left(0 , \\frac{1}{I(\\theta_0)}\\right),", "f1c5cece1ac77d1d449104ae97a7828a": " \\sqrt[3]{ \\sqrt[5]{\\frac{32}{5}} - \\sqrt[5]{\\frac{27}{5}} } = \\sqrt[5]{\\frac{1}{25}} + \\sqrt[5]{\\frac{3}{25}} - \\sqrt[5]{\\frac{9}{25}}, ", "f1c5d5c5cff627db01d2c82298bf547d": "A^{\\mu}A_{\\mu}", "f1c6b453c14f4de2d09fb30b3e855380": "c=3\\ell/(\\ell+2)", "f1c7085256e5c05eaa3690b4004062f3": "\\mathrm{RMSD}=\\sqrt{\\frac{1}{N}\\sum_{i=1}^N\\delta_i^2}", "f1c72ffafbc2f0265ce3ee8d4c4951e2": "\\mathbf{OTF(\\xi,\\eta)}=\\mathbf{MTF(\\xi,\\eta)}\\cdot\\mathbf{PTF(\\xi,\\eta)} ", "f1c76b7cc65b4d6dd414302d9682dfe8": "dp\\;", "f1c7900e0c03548669fe5d714212e0ad": "E_\\text{k} = \\frac{p^2}{2m}", "f1c8215e8fa6a17ab1fa51954eb665f9": "B\\mapsto\\operatorname{min}\\{\\alpha<\\kappa^{+}:B\\in f(A_{\\alpha})\\}", "f1c8261fe4559fe63d09b86b785c579c": "G_{1}^{(i)} = \\frac{1}{z_{i}}\\frac{dG_{0}^{(i)}}{dx}\\Bigg|_{x = 1}", "f1c8367d75b639964eaa0d46174a5eea": "\\sin^2 \\phi_{\\tau} = \\left|\\tau(\\omega)\\right|^2", "f1c841b9f5c36a614630cb90afe14641": "\\sigma_n(S^n) = 1", "f1c88f018f7cd4dcb17c7f3161b3d3d6": "S(\\mathcal{F},n) = \\max_{x_1,\\ldots, x_n} |\\{(f(x_1), \\ldots, f(x_n)), f \\in \\mathcal{F}\\}|", "f1c8c11eb58f105850530e81bd49fdf5": "\\frac{\\partial}{\\partial{c}}\\frac{\\partial}{\\partial{z}}P_c^p(z_0)", "f1c8c2bb9a099fc57f2be19071779290": "q^{ab} = {\\epsilon^{acd} \\epsilon^{bef} q_{ce} q_{df} \\over 3! det (q)}", "f1c8ec280c593b29ac507e4a8e114733": "\\begin{align} A & = 3 \\cot\\left(\\frac{\\pi}{12} \\right) a^2 = \n 3 \\left(2+\\sqrt{3} \\right) a^2 \\\\\n & \\simeq 11.19615242\\,a^2.\n \\end{align}", "f1c904737b2c7ff1cbeee0ae1af62e8f": "\\frac{1}{2}F (L_1 - L_0).", "f1c9086dffbca142eefea9d8e9ad1bc1": "R^i\\subseteq G", "f1c91e0ad1d7d55d9212da28da28d4e7": "C_p - C_V = T \\left(\\frac{\\partial p}{\\partial T}\\right)_{V,N} \\left(\\frac{\\partial V}{\\partial T}\\right)_{p,N} ", "f1c97b72412cd0d75e8a08405cb4bb1f": " (Y_1,\\ldots,Y_J)", "f1c9aade4180a85d8ed88d9ef7eaa73d": "Q'_y(b,a)", "f1ca173978e357016d668c8e741a8f6e": "\\phi : E \\to F\\,", "f1ca2f64cc7b83d28d251fb6a2ba3e24": " M=6(N-1-j)+\\sum_{i=1}^j f_i,", "f1ca2f86b8c198013cf6576dafb05a28": " (1/4)(cot(\\alpha)+cot(\\gamma))(cot(\\beta)+cot(\\delta)). ", "f1ca3f93df6f23ae9f0b7838c85e5df4": "\n\\hat{k}=\n\\frac\n{\n\\sum\\sum_{i p_{n+1}^2. \\, ", "f1cd82e2ce16ddb1cd00f50af0602e0b": "\\{X_{k+1},\\,X_{k+2},\\,\\ldots,\\,X_n\\}", "f1cdb64bdc8ba2ab2187d899ed6368d5": " \\frac{\\partial u_i}{\\partial t} +a_{ij}\\frac{\\partial u_j}{\\partial x}=0 ", "f1cdd58defe9f05c5bbbd9f5ea281702": "(X\\times Y)^{\\omega}", "f1ce219862482fdf7e7e3b7dc93cf6f2": "\\mathrm{d}(TS) - S\\mathrm{d}T= \\mathrm{d}U + \\mathrm{d}(pV) - V\\mathrm{d}p-\\sum_{i=1}^k \\mu_i \\,\\mathrm{d}N_i + \\sum_{i=1}^n X_i \\,\\mathrm{d}a_i + \\cdots", "f1ce24f53b6e5f6021f95ee26988e6f5": " P = \\frac{U}{3 V} = \\frac{\\varepsilon \\sigma T^{4}}{3 V} ", "f1ce53bb58071af1b3c273f0fefea372": "X_s", "f1ce5542b1b0cf162ff79b880c3d062b": "\\mathrm{Ci}", "f1ce777988212e2d28231b21e9a03a5b": " 1< \\alpha < 2 ", "f1ce7ee1ce51fcf4089aa20e117a5a5d": " {\\tau} ", "f1cee0be99d7a88e483e633f5eb42628": "\n R_n = \\frac\n {- n (n-1)^3 2^{2n-1} [(n-2)!]^4}\n {(2n-1) [(2n-2)!]^3}\n f^{(2n-2)}(\\xi), \\quad (-1 < \\xi < 1)\n", "f1cef5e328f853e3cb3e46c4dca72b33": " R_C || r_O ", "f1cf21c01d932bcf17ca32712c9eb7de": "\\scriptstyle{\\psi(t)}", "f1cf58cae1399fc543acdba1d6b29cee": "O(\\log^3 n)", "f1cf78921e7fa6bc95bd56d7b8394cb8": "\\bar{q}q", "f1cfaffb364d7e8a1d5b85aa9cfb030c": "\\ \\Delta H", "f1cfe90ced15dd8c71e5a071873c8660": "I_1 < I_2<\\cdots < I_N", "f1d0154aae3662b1e8fe65023bc0e63b": "Q_{\\bold{x}}(\\theta) = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & \\cos \\theta & \\sin \\theta \\\\ 0 & -\\sin \\theta & \\cos \\theta\\end{bmatrix} , ", "f1d05534f5a599b095d6afde657a8688": " \\frac{d \\vec{A}}{d t} = \\frac{\\vec{r} \\times \\vec{v}}{2}. ", "f1d083c3a7700dc0a972213cedc87d67": "f(\\cdot)", "f1d099da76802b0d5d8c54e0a8c7404e": "\\gamma(x,x)=\\gamma_i(0)=E\\left((Z(x)-Z(x))^2\\right)=0", "f1d0cbc091f66e0454f6f58bbb2e7d74": "\\mbox{diff}(N_y)", "f1d0d71b54fd1b3a5428c93ed19fd0eb": "Z_1 \\,\\!", "f1d0fcb0721faa0735f8cc1f1b8b5201": "h = g^x", "f1d129854777266fc2757df0224eae5c": "TSI", "f1d13e2e4be2859bb2a900aaff8e46f2": "F'=QF", "f1d1590b3182c58b32935193f07606c5": "u_0^2 \\approx u_n u^n = \\frac{u_n^2}{c^2},", "f1d1977f620fc58729da5ec61958aff6": "p(z) = p(\\vec{x},y)", "f1d1b03d1de477b49e13a3a42277bd50": "M_r = \\sup\\{|f(z)| : |z| = r\\}", "f1d1fa97df4e72ca08a8952a3121ca09": "u^{t} = c \\sqrt{\\frac{-1}{g_{t t} + g_{s s} v^2}} \\,.", "f1d2509b2dc2f14ef9172b9d02122563": "F/E,F^{\\#}/E \\in H_2(\\mathbb{C}^+)", "f1d2aae145d99104f2d58ec0af0299f2": "A(x)=o(x)", "f1d2c1e02d5d8c881d2c48cd2a6ffdf8": " \\mathbf{B} ( \\mathbf{r} , t ) = \\hat { \\mathbf{z} } \\times \\mathbf{E} ( \\mathbf{r} , t )/c ", "f1d335458656912f58af59e0627ad263": "Hom(X,-)", "f1d35ea930247cb21949b1cba5eb19bd": "\n \\ln(G(k)) = \\sum^{\\infty}_{m=1}\\frac{(ik)^m}{m!}\\kappa_{m},\n", "f1d3cf5e747ecb2184c43ff15d929f24": " e^{\\frac{S(a)}{T}} = e^{\\frac{\\sum_{x_i-y_j \\in a} \\sigma(x_i,y_j) + \\text{gap cost}}{T}} = \n\\left( \\prod_{x_i - y_i \\in a} e^{\\frac{\\sum_{x_i-y_j \\in a} \\sigma(x_i,y_j)}{T}} \\right) \\cdot e^{\\frac{gapcost}{T}}", "f1d4067eadb00173288a1fd1a046e15c": "\\displaystyle f(a x)\\,", "f1d49b4ce5421a64a73646054374063e": "N_{W_{\\alpha}XY}(\\hbar \\omega)", "f1d4a6a1af025451a25b61aec96454d9": "P(\\{H\\}) + P(\\{T\\}) = 1", "f1d4abf944cbf9b4b57f7246e2168875": " \\psi = \\psi_e \\psi_v \\psi_s. ", "f1d4d16db58bd1bb23cc17f904458715": "\\frac{\\kappa_1}{\\kappa_2+d_1} = \\alpha_{max}", "f1d4e042f6b9882e4e8e26ab609a52cb": "\\operatorname{rank}(AB) \\leq \\min(\\operatorname{rank}\\ A, \\operatorname{rank}\\ B).", "f1d4ff3ad82f4b19ffeb2cdcdeddad92": "\\mathit{H(3, 9)}", "f1d574c27e0c94659d7a80007b038665": " 0 \\le r\\ \\le R-\\delta\\,", "f1d574e2f8f994876e6203eabd3553bb": "x = A(\\phi - \\sin \\phi) ", "f1d58984c0e73b8cdfb5087583a725a0": "= x \\frac{\\partial}{\\partial u} - u \\frac{\\partial}{\\partial x} + \\rho(x,u,u_{1})\\frac{\\partial}{\\partial u_{1}} \\,", "f1d5c6fc8d88ce744ada30c422f3ab9f": "g(x, y, t) = \\frac{1}{2 \\pi t} e^{-(x^2+y^2)/2t}", "f1d66b656c2ff69b8559bcd16bae9c44": "P_2\\,\\!", "f1d675cdf4a2408e7301ddffc8276199": "{\\frac{{L_0(T-T_0)}}{{EA}}}+L_0 = \\frac{{2T\\arcsin(\\frac{{wS}}{{2T}})}}{{w}}", "f1d6a3e5d2d9a33f83802b5b4d39eb56": "(\\rho^{\\prime}, \\theta^{\\prime})", "f1d6c9dcaba49d420f82c041c419ac96": "M_t = M_s + M_l", "f1d71af88e895596a399ab66bf4541f0": "b_{2}+(3/5)a_{3}", "f1d7ce17a4f1faeaf17aa2690c5c2880": "B^C \\subseteq A^C", "f1d7df6090b40adc400785be3f62a229": "\n\\int_A h(x) \\circ g \n= \\int_X \\left[h_A(x) \\wedge h(x)\\right] \\circ g \n", "f1d7e95964832c661d722671b25e5a5f": "\\phi_{-}(q) = \\sum_{n\\ge 1} {q^{n}(-q;q)_{2n-1}\\over (q;q^2)_{n}}", "f1d80be23fc8290d21630987d28f282b": " y'(t) \\approx \\frac{y(t) - y(t-h)}{h}, \\qquad\\qquad (5)", "f1d90beb27fb9af244ba235e4226bb96": "\\lim_{s \\to t} \\mathbf{E} \\left[ \\big| X_{s} - X_{t} \\big|^{2} \\right] = 0.", "f1d91287977eaaff0c80a70b7f5acb3b": "f_c : z \\to z^2 +c\\,", "f1d994bc40bf58663793d53b3e886795": "\\{\\alpha_j\\big(X(t)\\big)\\}_{1}^{n}", "f1d9a190602a3eceafd5964bb97edf69": "\ng(z) = \\sum_{n=1}^\\infty e^{in\\theta} = \\sum_{n=1}^\\infty \\left(\\cos n\\theta + i\\sin n\\theta\\right) \\,\n", "f1d9e2aff676d7dd841481498706a1e5": "x\\in (0,2a)", "f1da89685da5732275c6db1d9156a6ab": "F_n^m\\left(\\frac{d}{dx}\\right)\\cosh^{-1-m}(x) = \\cosh^{-1-m}(x)P_n(\\tanh(x))", "f1da8a8d75ab24b2d153f82f46197726": "b_{1}^{*}= b_{1}=\n\\begin{bmatrix}1\\\\1\\\\1\\end{bmatrix},B_{1}= 3", "f1dae22b5635b1101d392ae1933eb2ca": "E^R_k", "f1daea7a441181d35d1da903800d8430": "\\gcd(x,0) = x", "f1db66518fc8141ed952f71011ed4a35": " V(y_i,f(x_i)) = (1-yf(x))_+", "f1dc18e2162b445e42dbb4f358923719": "L=\\left\\{abc\\right\\}\\mbox{ then } \\operatorname{Pref}(L)=\\left\\{\\varepsilon, a, ab, abc\\right\\}", "f1dc19636afb2348aea657016970b92c": "\\frac{1}{n}\\sum_{i = 1}^{n} X^k_i\\,\\!", "f1dc283d8f605158982f86ace1d9ba66": "X \\sim \\mathrm{Gamma}(k, \\theta),", "f1dc8a21ee8d025315eb9be6de9f1ae7": "\\,\nP(K)= C e^{-KE/T} = C p^K.\n", "f1dd3d615d21d38e284447afed2e8616": "\\displaystyle\\mathrm{Tr}[\\Pi_l \\rho_m] = \\mathrm{P}(l | \\rho_m)", "f1dd6b3aac12cd3811e9b17bdce4721e": "G = ((V, E), c, s, t)", "f1dd7903a906020f17f28908da8be2fc": "\\dot{M}_I", "f1ddbf7e4a509979fdd4997764e098b5": "\\displaystyle B_{k+1} ", "f1ddf51a90620d6182c6c2c1e240a645": "p_{X}\\left( x\\right) ", "f1de068154b0e8266481347ad7fd9326": "\n\\alpha_0\\left(T-T_0\\right)+\\alpha_{11}P_x^2+\\alpha_{111}P_x^4=0\n", "f1df66f5a5c2d20f630d79184392337c": " \\mathbf{E_1} ", "f1e05b81cc9f640aa111c82206931277": "\n\\frac{DE}{Dt} = -\\int_{V}u'v'\\left(\\frac{dU}{dy}\\right)-\\frac{1}{R}\\int_{V}\\left( \\nabla \\vec{v}'\\right)^{2}\n", "f1e0b21f44169253398e893a3c554e7d": " \\bar{M} = \\sum_i x_i M_i \\,", "f1e0c09032f902d6b63e18b95aa0ff50": "D(t)V(t,S(t))=\\tilde{\\mathbb{E}}[D(T)V(T,S(T))|\\mathcal{F}_t], \\qquad dD(t)=-r(t)D(t) \\ dt", "f1e0e6936798f0daeea3f94b9697a1ba": "\\left(\\lambda+\\mu\\right)\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}\\right)+\\mu\\left(\\frac{\\partial^2 u_y}{\\partial x^2}+\\frac{\\partial^2 u_y}{\\partial y^2}+\\frac{\\partial^2 u_y}{\\partial z^2}\\right)+F_y=0\\,\\!", "f1e0ea8ecfc6a642c6e38410f04c55d0": "0 \\le m \\le n.\\,", "f1e13dcda93d80d2980d3f83ac4f99be": "\\lim_{n\\to\\infty}\\int_X f_n\\,d\\mu=\\sup_{f\\in F} \\int_X f\\,d\\mu.", "f1e1630754924415397fbc7cc50ac5d2": "\\nabla_{x} L(x,\\lambda)=0", "f1e1b901ae06be4223c1381efc56a8ed": "\\vec v_{B|C}=\\vec v_{B|A} +\\vec v_{A|C}", "f1e1cf91098a1b6ecc02f85b8db5c289": "X_{\\sigma(X, Y)}^* = (X_{\\sigma(X, Y)})^* = Y", "f1e1d969a5169f64c4c74d866d44b319": "\nA=\\left \\{ (x,y)| x\\in U, y=K(x) \\right \\}\n", "f1e205161fe7e911630272349ea189a7": " \\prod_{j=1}^k \\frac{1}{(r+j+a)^2+b^2} \\quad \\quad r=\\ldots, -1, 0, 1, \\ldots .", "f1e21d45a959b89d3810a53a245e7088": "|00 \\rangle ", "f1e22383aea8592d986583d6a1489509": "\\mathcal O(n)", "f1e231355e98733e26f1bdb6af79b593": "\\vec{E} + i\\vec{B} = -\\vec{\\nabla}\\Omega\\,", "f1e2755efa758900bc80a4f738483107": " \\Gamma_{tot}=\\Gamma_{rad} + \\Gamma_{nrad} ", "f1e2972972408ad9fd46865a8bb5f524": "\\mathbf{y}_m", "f1e2e8915fcd3ea60c97335bd84f027e": " d \\sigma_t = (\\beta_t-\\sigma_t)\\,dt + \\sqrt{\\sigma_t}\\,\\eta_t\\, dW_t.", "f1e3a0ed00f2cae6676a624e3ca4da1c": "\n\\varphi(u^0,t) = \\mbox{e}^{t\\Delta_D}u^0\n", "f1e3aa50df007819523967d85244cbbc": "\\left(1-\\frac{r}{n}\\right)^n \\left(1+\\frac{r}{n}\\right)^n = \\left(1-\\frac{r^2}{n^2}\\right)^n ", "f1e3b548dae775efc70b9a7c69820efb": "m^q / z^q", "f1e3bdcbf070c5715a39469a598059e8": " x = 5 - \\sqrt{7}\\quad\\text{or}\\quad x = 5 + \\sqrt{7}. \\, ", "f1e42c888e18d3d183d48e11494ac5c8": "\\scriptstyle a \\;>\\; b", "f1e4468ae850c8fe8c3083bb00377bb2": " {\\dot{m}}_C ", "f1e45ad44ba06aa1c68c18137f1f62ba": "\\begin{pmatrix}\n1 & 0 \\\\ 0 & 0\n\\end{pmatrix}", "f1e4a411b0e980411d90268b282536fe": "\\mathfrak{g}.", "f1e4eae39dad9b45633ad7d569f7654c": "\\lim_{\\mathbf{h}\\to \\mathbf{0}} \\frac{\\mathbf{f}(\\mathbf{x_0}+\\mathbf{h}) - \\mathbf{f}(\\mathbf{x_0}) - \\mathbf{J}\\mathbf{(h)}}{\\| \\mathbf{h} \\|} = \\mathbf{0}.", "f1e517211dec658ae066b769c3c8025c": "\\dot\\gamma_u = \\frac{\\mathbf{a \\cdot u}}{c^2} \\gamma_u^3 = \\frac{\\mathbf{a \\cdot u}}{c^2} \\frac{1}{\\left(1-\\frac{u^2}{c^2}\\right)^{3/2}}", "f1e5442fac59f3270685d2bf30929253": "\n\\begin{bmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{bmatrix} \\qquad (\\text{identity transformation})", "f1e567d76ace8b68d5c7a978a98ed946": "\\delta_2-\\delta_1", "f1e58b6215db22a0fabedb1d5f4ceffb": "\\textstyle(x\\pm1, y)", "f1e602832f52e50e8594fc2d8e350e88": "\\sum_{k=0}^{13} e^\\frac{2i\\pi (23)^k }{71}", "f1e6fa6b89884f8009f97add791e9e93": "f(x) = \\frac1{2^{n-1}}T_n(x)", "f1e7dd8479ebb2c417abb98e50d6b39a": "t'=A x + B t. \\,", "f1e7e54d671df88dd9724ff8f355ea67": "S((1+r)-1) = (1+r)^N - 1", "f1e8c08010de70687799988b67a974d4": "n=n_0+n_1", "f1e8f68e4fb89980dcaccb3b4dddd69b": " | \\psi \\rangle = \\int d \\varepsilon | \\varepsilon \\rangle \\psi(\\varepsilon) \\,,\\quad \\langle \\psi | = \\int d \\varepsilon \\langle \\varepsilon | {\\psi(\\varepsilon)}^{*} \\,,", "f1e91a6fdce7f6d93a0ed579f87117ff": "T[1]=1.", "f1e936007b45e37ee7b696e515d91791": "CFM = \\frac{1.76 \\times W}{\\text{allowed temperature rise in} ^\\circ C}", "f1e97b198be4a06a934226f527e7e0a6": "Z(X, t)=\\prod\\ (1-t^{\\deg(x)})^{-1}.", "f1e988055d90a6c2631cc7015a2937e3": " \\Delta = 1 - (L_0+L_1+L_2+L_3+L_4+L_5) + (L_0 L_1 + L_0 L_3)\\, ", "f1e99daa09ca0bca6651118d2c2dcb73": "m(E_{f^*}(\\lambda))\\le {8\\over \\lambda}\\int_{E_f(\\lambda)} |f|\\le {8\\|f\\|_1\\over \\lambda},", "f1ea0ddec3b09477348f9a3056d3bfde": "c^2+d^2", "f1ea68a4a1e70e96ada7592ef8752c30": "u=g(x)", "f1eabe489bca1ace2d20cac52db11558": "E_n(s)", "f1ead3d94d443edc9484a24132d4a7e8": "M_0 = {16\\over 3\\pi G} \n\\langle R\\left(2V_R^2 + V_T^2\\right)\\rangle.", "f1ead9b4f2c2394d9cf59de4c69a0dbf": "\\theta=\\pi", "f1eaf471094f055ccc61a1a6f1909372": "\\sum_{n=0}^{\\infty} {\\left( \\frac{(-1)^{n}}{2n+1} \\right) }^6 = \\frac{1}{1^6} + \\frac{1}{3^6} + \\frac{1}{5^6} + \\frac{1}{7^6} + \\cdots = \\frac{\\pi^6}{960}\\!", "f1eb2d56419551fce5816fbc12742dfe": "\\text{Component test coverage}=\\frac{\\#\\text{ of components tested} }{\\text{Total }\\#\\text{ of components} }", "f1eb431e13e4019c1a2c90de69594657": "{\\sqrt{13*{{\\pi}\\over{4}}*10^2} \\over {\\pi \\over 4}} = 36.055 ft", "f1ebb1fa7bb96238f0e05acbb4cd61d9": "\\,\\alpha_2", "f1ebe8bf2dc9b8ff31a0243e42e38fc3": "I_x[p],I_y[p]", "f1ebed8e120beef36ee96146ca63fdd2": "K_0 = K \\frac{n}{n_0} = K \\ \\frac{T_0}{T} \\ \\frac{p}{p_0}", "f1ec408184cccef7756d090556d325cf": "y=\\frac{x^2}{4r\\sin{\\theta}}", "f1ec4cbc7568dece1d2ad608057a9cd3": "\\ \\begin{array}{rrcl} & \\sigma^* &=& G(F^*) \\\\\n\\Rightarrow & Q\\sigma Q^T &=& G(QF) \\\\\n\\Rightarrow & QG(F) Q^T &=& G(QF). \\end{array}", "f1ec6046d0f955f780d1d6d711bb29fd": "\\dot{D}=\\gamma\\left(2(1-D)-(x^*p+xp^*)\\right) ", "f1ed2190ba16c24d6a383ecf83baf9bb": "\\begin{align}\nG_4(\\tau)&=\\frac{\\pi^4}{45} \\left[ 1+ 240\\sum_{n=1}^\\infty \\sigma_3(n) q^{n} \\right] \\\\\nG_6(\\tau)&=\\frac{2\\pi^6}{945} \\left[ 1- 504\\sum_{n=1}^\\infty \\sigma_5(n) q^{n} \\right]. \n\\end{align}", "f1ed2723c4ba82defee0a5099f377247": "y_i - \\Delta > y_j + \\Delta", "f1ed3c27bcf050a7ee968110884f56c4": "K_{ang}=\\frac{\\alpha^2 \\hbar^2}{2m}\\frac{1}{1+(t/\\tau)^2}~.", "f1ed56e4bc0fe1641e3e07c6a6b82c6c": "F[n]", "f1ed8fc009b6f05d376f050f89507366": "f \\times f^{-1}: A \\times B \\rightarrow B \\times A", "f1edc739b7cb50c78212b47296f0ca63": "Q(x) = -\\tfrac{q}{8}(8x -5L)", "f1edd2beb1827288f8e28b52fe72bfc2": "f(a, b)", "f1ede9e5184c4ca8427a01632f54fbda": "\\phi^*= -(1+\\frac{a}{2})+\\sqrt{\\frac{4a+a^2}{4}} ", "f1ee4ef6748f6eb724d6f8226d34c69f": "{ {P} \\over {B}} ", "f1ee4f7c77f1b8c10b85a51475007395": "\\underset{i}{\\overset{2}{x_j}}(t_0)", "f1ee94b2bbcc37162dc8512607d9b726": "\\tfrac{\\sqrt{2}\\lambda}{4}", "f1eec89276bea8efe5edeb4801cf397b": "V' \\subset S'", "f1eecc6a766999fdbb423e8b0b936fab": "A_{e}(p_{e} - p_{amb})\\,", "f1ef327233731fbf688956c17f4f4443": " \\frac{1}{2} \\times ||OA|| \\times ||AC|| = \\frac{1}{2}\\times r \\times r \\tan\\theta = \\frac{1}{2}r^2\\tan\\theta \\, . ", "f1ef84fa264b2edc8869d1b3135c0404": "V(x)\\to \\infty", "f1efcae4780a74557c63f789c9f5f7ef": "S_{-p}", "f1efcfef38bb360e30a4e272f06d6ae6": "x_i = (x_{i-1})^2~mod~N", "f1effb84d046d405d5842a8cc99ba295": "\\psi(\\Omega^\\Omega(1+\\alpha)) = \\Gamma_\\alpha", "f1effc34c2b17ab04f782844fcb4a055": "\\overline{z_0}", "f1f03e256a3bba80316932495bf10ea0": "9.61 \\%", "f1f044c1f70f943ac5fdbbd7dd3ee1be": " B = P/\\left ( \\Delta V / V \\right ) \\,\\!", "f1f0da338f58be5d5d30c4bdae57616c": "c=\\sqrt{T/\\sigma}", "f1f15a41b79d5477cf3bdc913e97a4c9": "\n\\begin{align}\n\\ln q^*(\\mathbf{\\pi}) &= \\ln p(\\mathbf{\\pi}) + \\operatorname{E}_{\\mathbf{Z}}[\\ln p(\\mathbf{Z}\\mid \\mathbf{\\pi})] + \\text{constant} \\\\\n &= (\\alpha_0 - 1) \\sum_{k=1}^K \\ln \\pi_k + \\sum_{n=1}^N \\sum_{k=1}^K r_{nk} \\ln \\pi_k + \\text{constant}\n\\end{align}\n", "f1f17404b52489042c0a7ca250e4e4fd": "pop\\ growth\\ rate = \\frac{ P(t_2) - P(t_1)} {P(t_1)}", "f1f19db736485079194f8c991627ff29": " n = (1 + \\chi)^{1/2} = \n\\left( 1+\\chi_{\\mathrm{LIN}} + \\chi_{\\mathrm{NL}} \\right)^{1/2}\n\\simeq n_0 \\left( 1 + \\frac{1}{2 {n_0}^2} \\chi_{\\mathrm{NL}} \\right)", "f1f1bd37945e2cb0815ed0e3090a2d54": "max \\quad \\mathbb{E}_u[\\int_0^{\\infty}\\gamma^t r(x(t),u(t)))dt|x_0]", "f1f27e9c5f7572e13a24c7671e1f74be": "i\\leq j", "f1f2cd254a8010b64e99050da23653e8": "U_{0j}", "f1f3368a7104be05f523772d3408545d": "\\{B^{(i)}\\}", "f1f37d9d3cd2d840112c392c3199174a": " \\mathbf{f}(\\mathbf{r}) = -dm\\, g\\vec{k}= -\\rho(\\mathbf{r})dV\\,g\\vec{k},", "f1f3a161c57cabd61c71b3ffe8a74435": "f\\in K(C)", "f1f4788fd5a006b7dd8c07f31b00c9c2": "\\ f g: x \\mapsto f(x)g(x)", "f1f4d69c62b164210dbb31a1654e8f06": "\\color{SkyBlue}\\text{SkyBlue}", "f1f5a457176e71efb2415c900bd496fe": "t^a(d,n)", "f1f6e86dd8dcc82502906d8a47363e3d": "\\scriptstyle{|\\phi_1\\rangle}", "f1f6e90ab9866555a4e7bad405070e76": "Q_{m}\\approx 900", "f1f73d7e86c305954194827593603aa0": "\\Phi(\\mathbf{r}_1) = \\Phi_2(\\mathbf{r}_1) + \\Phi_3(\\mathbf{r}_1) = \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_2}{r_{12}} + \\frac{1}{4\\pi\\varepsilon_0} \\frac{Q_3}{r_{13}}", "f1f7a8c8ccdfc9d2640dacafcf98326f": "S \\times S", "f1f8a0ce9c92869c338d06f23735fc76": "R \\to \\operatorname{End}(M),", "f1f8a40105a714bd65cec8fe9109ad57": "d(D) > d(D/xD) \\ge d(R/\\mathfrak{p}_1)", "f1f8a8982d28bcaf34568c062f154154": "{n\\choose k} = (-1)^k {k-n-1 \\choose k}", "f1f946a99e80f4886ba713be1b554003": " b_{i,n}(t) = {n \\choose i}(1-t)^{n-i}t^i", "f1f98e61d210f7e92f869eecddf3da6b": "a_w", "f1f991b34482b83692196fd3c965c5c1": "(..)^\\mathrm{T}", "f1f9af903654023fa6b4f26d9a60c562": "f:\\{0,1\\}^n\\rightarrow \\{0,1\\}^n", "f1fab6b044dd91654bc5d5116e6b6f23": " w \\!", "f1fabc447ea7ee6ab4ea756e5294cb21": "\\frac{dP}{P} = \\frac{-dz}{\\frac{kT}{Mg}}", "f1facf6a0c159a06ec50d4adb5b990d9": "\\left(\\sqrt3 - \\frac{\\sqrt2}2 \\right) \\cdot s\\approx 1.0249s.", "f1fb1ad34a8cde0c6dc5b1e3f85c8e3f": "\n \\limsup_{n \\to \\infty} \\frac{S_n}{\\sqrt{n \\log\\log n}} = \\sqrt{2}, \\qquad \\text{a.s.},\n ", "f1fb1adc9ead7bd7dae168019932ab8a": "-\\frac{1}{b}", "f1fb6f0e64ef7d0994c1af4f2f4d996b": "z^2 = ax^2 + by^2 \\ ", "f1fbe166ff1c51c40a8965a7be4d763f": "f(x_0)=c", "f1fc0c5b1e831e3aff94960a515c47ff": "f(x_i) = \\sum_{n=0}^N c_n \\phi_n(x_i)", "f1fca05317a8455204e02d109d2f46d6": "Q=\\frac{RQD}{J_n} \\times \\frac{J_r}{J_a} \\times \\frac{J_w}{SRF}", "f1fcb6d68dc937c999801a37319b2f4b": "s_i (E^\\vee) ", "f1fcf4fc52ce5ebeccbcf1dd5f4fb125": "g(\\varepsilon)", "f1fd2e6e6167f88be0cf2fa7aed0a146": "N = N_G(T),", "f1fd433daf77a390a458dab431a553df": "\\mathcal N\\models\\phi(n) \\iff n", "f1fd6cad9b084a20913566a93d04e4fb": " \\mathbf{e}_n ", "f1fd815e3a61959beff0a7d23bae06c1": " \\pm 2^{-i} ", "f1fd8d6b62cc54dfb5a701d5d3c1ce93": " 00", "f20e34b10a774f44cb43cfb31227fb10": "f(x)=O(g(x))\\text{ as }x\\to\\infty\\,", "f20e38955e0ac376efefd1d07d0608bf": "k_{\\rm C}", "f20e45639ae210632aa204945a67d516": " a_0:=\\gcd(f,f'); \\quad b_1:=f/a_0; \\quad c_1:= f'/a_0;\\quad d_1:=c_1-b_1'; \\quad i:=1;", "f20e5f1dfc2ef35730dd05b6f6c8ac35": "f_j", "f20e6d0d7e7a1d9f1bc2c8872729b403": " T(n,1)=1,\\;T(1,k)=1,\\;n \\geq k: T(n,k) = -\\sum\\limits_{i=1}^{k-1} T(n-i,k),\\;n0 \\\\ \n 0 & \\textrm{if} \\; y_{1i}^* \\leq 0.\n\\end{cases}", "f21536454ac0eb1cc138ae3f60edd9e6": "c\\in GF(p^2)\\backslash GF(p)", "f2155d07264e8d4d7683b8acd8731952": "\\textstyle \\leq n-1-r", "f215a15921216dcd692294d3d929d071": "d_c=\\sqrt \\frac{\\gamma V}{100\\pi\\, RT_{60}} \\approx \\sqrt \\frac{\\gamma A}{50}\\,", "f215b360ea97a1bcdf135ef27fe9fa7b": "\\Pr(e) = 0.369 \\!", "f215dabec57837ae680c5c1c0f3e8960": "\\displaystyle i \\partial_t^{} u + \\Delta u = un", "f216114d364ba549a34c84a55122e234": "X^K = (a_1 \\Delta C_1^\\circ + a_2 \\Delta C_2^\\circ + \\dots + a_{n-1} \\Delta C_{n-1}^\\circ)\\sqrt{t}", "f216432b6afda7d08d5fb7863c5fe8c7": "\\delta\\colon(Q\\, \\times ( \\Sigma\\, \\cup \\left \\{ \\varepsilon\\, \\right \\} ) \\times \\Gamma\\,) \\longrightarrow \\mathcal{P}(Q \\times \\Gamma ^{*})", "f2169ecfb3272791d14904e49e965aea": "P_{3}^{1}(x)=-\\begin{matrix}\\frac{3}{2}\\end{matrix}(5x^{2}-1)(1-x^2)^{1/2}", "f216a0889da4eeee57c5615b89733b2c": "PGL(2,\\C)", "f216d804231daaf6b5f288f357f9fe37": "p \\cdot q = b_1b_2 + c_1c_2 + d_1d_2.", "f2172b35ffc1ae82081f11b047e3cc87": " f = \\frac{4}{3} \\pi r^3 N\\,\\!", "f21799550ffba472d76548c60a602b35": "\\Delta \\mathbf{v} = \\mathbf{v}_1 - \\mathbf{v}_0 = \\int^{t_1}_{t_0} \\mathbf {a} \\, dt", "f21852cb6e9c446899c19643703a4fd1": "(K_D)_A = {(a_A)_{org} \\over(a_A)_{aq}} \\approx {[A]_{org} \\over[A]_{aq}}", "f2187cb9c290d2f3aa692abff666ab72": "D=4-\\epsilon", "f218c59a5cffb3f1436104079edf148d": " a_{n+2} = -\\frac{a_n}{(n+3)(n+2)} ", "f218e847d7ec93f459cfe47fde3e10fc": " \\textbf{y}(t) = \\begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} \\end{bmatrix}\\textbf{x}(t)", "f21928915e4ada1b0932aa5889067d19": "L_x' = y' p_z' - z' p_y' = y p_z - z p_y = L_x ", "f21950fe286cb391975c9bd81cbb52ee": "P(r_1,r_2,\\ldots,r_n)=0", "f219590953be970bb33f01048e196728": "a_{n+1}=|iP-C|{(1+i)}^n", "f21968aa5507925ec497067bd1b33a99": "M^2 \\equiv \\overline{D} \\times \\frac {\\sigma_B} {\\sigma_D} + \\overline{R_F}", "f219ecbe8e0e43d176ad7d2618d65b1c": "\\Delta^n[f](x)= \\sum_{k=0}^n {n \\choose k} (-1)^{n-k} f(x+k)", "f21aab444b750c6b7c8a7f808762a119": "= \\arctan \\frac {5}{12}", "f21abaf20e48f546779c315ba354419f": "\\Psi_{l,\\mu}^{k}", "f21abd5f7384ff4af714516c80bfa4e8": "\\Theta = \\{\\mu,\\Sigma,c,V,M,p(b|\\Theta),t,U\\}\\,", "f21ac3ee0ce242d72936df87ab4f449f": " d\\mu_n(\\varphi)={1\\over 2\\pi} f(r_n e^{i\\varphi})\\,d\\varphi", "f21acb73a6699550f0644b94fb4d820b": "p_w(\\theta)=\\frac{1}{2\\pi}\\,\\int_{-\\infty}^\\infty p(\\theta')\\sum_{n=-\\infty}^{\\infty}e^{in(\\theta-\\theta')}\\,d\\theta'", "f21acbb540acaf5a6a587227bb5ec5af": "X = \\xi\\bar{\\xi}^T=\\xi\\xi^*.", "f21ad0d26a5c5d5b0608aad6a8a150f4": " 0 \\to x = 1 .", "f21aec0c1281f2c38facc3e13d98c43b": "P_V=\\frac{A(V_G,T,N)-A(V_L,T,N)}{V_G-V_L}", "f21afd96672520ac03fde723bd6bcfc7": "J(C) = \\mathbb{C}^g/\\Lambda.", "f21b7ba06b05da52a2d47963ee570fa4": " \\frac{\\omega_c'}{i\\omega} \\to\n\\dfrac{1}{Q_1 \\left( \\dfrac {i\\omega}{\\omega_{01}}+\\dfrac {\\omega_{01}}{i\\omega} \\right)}+\n\\dfrac{1}{Q_2 \\left( \\dfrac {i\\omega}{\\omega_{02}}+\\dfrac {\\omega_{02}}{i\\omega} \\right)}+\n\\cdots", "f21b83282251bccbab2bf3845a8756a3": "\\mathcal{F}Rf", "f21b9c376378955afc9eb620371e7a95": "n_t\\to\\infty", "f21c9a47b3dbf047345e84c7f5c6b197": "\\operatorname{mwnchypg}((0,\\ldots,0,x_j,0,\\ldots);n,\\mathbf{m}, \\boldsymbol{\\omega}) = \\frac{m_j^{\\,\\,\\underline{n}}} {\\left( \\frac{1}{\\omega_j}\\sum_{i=1}^{c}m_i\\omega_i \\right) ^{\\underline{n}}}", "f21c9b13988ca6354d9d8b583db6e1f0": "\\phi_{\\alpha_n,\\beta}(\\gamma) = \\alpha_n^{-1}\\cdot \\gamma \\cdot \\alpha_n", "f21d1800222fc1ebf80413bb4717192b": "\\mbox{absconv} A = \\left\\{\\sum_{i=1}^n\\lambda_i x_i : n \\in \\N, \\, x_i \\in A, \\, \\sum_{i=1}^n|\\lambda_i| \\leq 1 \\right\\}", "f21d4705539d35fea84e4cbaa38c4b78": "c > 2a", "f21d99b7d9d192bc7511c3ff96f3964b": "E = E_0 (e^{i\\omega t} + \\mathrm{c.c.})", "f21da5f9764321a77c0c2c3b7d2c4fa2": "J^3 = J. \\, ", "f21e4ea783d9c381967bfda00cc046af": "\\mbox{depreciation rate} = 1 - \\sqrt[N]{\\mbox{residual value} \\over \\mbox{cost of fixed asset}}", "f21e58184e581d915b03e15048f3a063": "\\{ a_{k,j} \\} ", "f21ea53883561cfdd18fce5bed7088ac": "X ::= \\Gamma\\,\\!", "f21eaf471c71c65640a1bd0abbab0f1a": "\\scriptstyle \\delta(t-nT).", "f21eddacbfe89210df106778b41249ee": "p(v)=P(E=v)", "f21f2eed06ffe341a1b6cf9acab6d65a": "(\\xi, \\eta, \\phi)\\in[0,\\infty)\\times\\left[-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]\\times[0,2\\pi)", "f21f58bc1d4e076b0838096e8ce9c850": "{\\hat{\\alpha}}(q, {r_{\\rm c}})", "f21f75ab6e1ba7a96f2fbbef7026ae22": "f(X \\setminus \\partial X) \\subseteq Y \\setminus \\partial Y", "f21fbbb2c47ff159789009093febb67f": " invFx_1...x_n \\leftrightarrow Fx_2x_1...x_n.", "f21fdccd9bc86ac569e9f740ef80b607": "\n\\frac{1}{(2\\pi)^{n/2}}\\int_{-\\infty}^{\\infty} \\cdots \\int_{-\\infty}^{\\infty} \\prod_{i=1}^n e^{-t_i^2/2}\n\\prod_{1 \\le i < j \\le n} |t_i - t_j |^{2 \\gamma}\\,dt_1 \\cdots dt_n.\n", "f2200734b0941ea5ac59ce4b56b0a2e5": "d_0(z) = a\\overline{z}", "f22009fd828e164c58f1ee7787e0c6fd": "\n\\Phi=\\frac{\\Delta P_d}{P_{d,0}}=\\frac{P_d\\left(\\text{heating beam on}\\right)-P_d\\left(\\text{heating beam off}\\right)}{P_d\\left(\\text{background, no particle}\\right)}\n", "f220216471a6e2de7689c989923e520d": "\n(A|B) =\n \\left[\\begin{array}{ccc|c}\n1 & 2 & 3 & 0 \\\\\n3 & 4 & 7 & 2 \\\\\n6 & 5 & 9 & 11\n \\end{array}\\right]\n", "f22033a080bed41af385b34562549b61": "p_N \\le 0 \\,.", "f2209bbd5e235eecbe90a3bb0d1ea54c": "k \\, \\stackrel{\\text{def}}{=} \\, \\lim_{\\tau \\to 0}{\\log |f(\\tau)| \\over \\log |\\tau|} \\text{.}", "f2209ecca35d7b9bac01127e8dde632c": "\\frac{1}{\\sqrt{12}}", "f220b6fd6e6762a3242f323e37bbafe6": "\\beta_0+\\beta_1x_1+\\beta_2x_2+\\cdots+\\beta_mx_m.", "f220d26995fbfada7a9792da6e53f671": "\\arcsin(\\sin \\theta) = \\theta\\quad\\text{for }-\\pi/2 \\leq \\theta \\leq \\pi/2.", "f220d42b6a3f36f2a666b8181b8edd56": "\\{x,R\\}", "f22118aa5d0b01ff5dbfd14dec7024f6": "\\left(\\tfrac Dn\\right)", "f221205933a40205cdd86c522fa62e09": "y\\mapsto Df(x_0,y_0)(0,y)", "f22144f1ac0c611a9267d2ed0946b045": "1/(1-MPC)", "f22181c62c0647e8ce23b6005d2d1305": "C_1 = x+y+z", "f221cbc8f9895244e247fd8f3f25c134": "\\rho_{z}", "f221fcbcbef31399caaf453754c8e49c": " \\int_{\\textbf{R}^{n d}}f(x_1,\\dots,x_n) M^{(n)}(dx_1,\\dots,dx_n)=E \\left[ \\sum_{(x_1\\neq,\\dots,\\neq x_n)\\in {N} } f(x_1,\\dots,x_n) \\right], ", "f22201737968bbd80b1119e7a0266849": "d\\omega+\\omega\\wedge\\omega=0", "f2227ccc8e5d1df4454e07dd9a0a5402": "\\begin{align}\n&\\int_{-1}^1 (1-x)^{\\alpha} (1+x)^{\\beta} \nP_m^{(\\alpha,\\beta)} (x)P_n^{(\\alpha,\\beta)} (x) \\; dx \\\\\n&\\quad=\n\\frac{2^{\\alpha+\\beta+1}}{2n+\\alpha+\\beta+1}\n\\frac{\\Gamma(n+\\alpha+1)\\Gamma(n+\\beta+1)}{\\Gamma(n+\\alpha+\\beta+1)n!} \\delta_{nm}.\n\\end{align}\n", "f222803161ba5095237acaea2e097af0": "\\sigma(i)\\equiv (i-1)", "f222b33b6a205b73c7d4716b89c50d2e": " Q_{in}=sin2\\theta+90 ", "f222b87048112c44632e9c883c98da7f": "\\textstyle S_{n,m}=m! \\left\\{{n\\atop m}\\right\\}", "f222c35f20fc3d94a959ea87e24a9cbe": "\\gamma_{1,2,3}", "f222c65f399d54e85a9ee3bf2fe5c715": "s(F)=2^k", "f2231e2588b9467efc7fada02c80aa14": "[\\hat f,\\hat g]= i\\hbar\\widehat{\\{f,g\\}} \\, .", "f2232cf9b7559f9b8b27922d48dc02d4": "\\gamma_e = 1.760 859 770(44) \\times 10^{11} \\mathrm{s^{-1} T^{-1}}", "f2232ea915c0bb3087f35a4398899e62": " T_d= \\hbar \\Gamma / {2 k_b} ", "f22345bea41ec22f2a263504b56fe212": "(\\lambda, \\mu, \\nu)", "f22353cabf13fe75ce838f26e7d13837": "Au_{xx} + Cu_{yy} + Du_x + Eu_y + F = 0", "f22367adb0d740c374d8ca97a2ac1dff": "\\varphi_X(t) = \\exp\\left(it\\mu - {\\sigma^2 t^2 \\over 2}\\right).", "f223970ee011309a716bc4c6e3580490": "pH = pKa + log \\frac {B} {A}", "f223b39c4b551bb90e624dab4dd7efab": "N=1+2+\\cdots+k", "f223fcd1172bb5a444fcf96036aaf379": "S(x)=-x", "f2240a4c833d91933d08dab2b924ff1d": "\\ MAPE = \\frac{\\sum_{t=1}^N |\\frac{E_t}{Y_t}|}{N} ", "f22416bf1e02cf9c51b39f2c22ef68f5": "\\mathit{x}\\,", "f22448f0f42200b841471cd9669d6280": " \\tilde{O}(1) ", "f2246f3232cc3ce9b7e7877491d4c1a2": "\\displaystyle{\\mathcal{U}^*F(t)={1\\over \\pi^n} \\int_{{\\mathbf C}^n} B(\\overline{z},t) F(z)\\, dx\\cdot dy.}", "f2248751f47250962781d4554e317bda": "[L^2, L_x] = (L_x^2 + L_y^2 + L_z^2)L_x - L_x (L_x^2 + L_y^2 + L_z^2) ", "f224b9d0cc653e224cb006fc8991ee06": " SH = {0.622 {p_{(H_2O)}} \\over {p_{(dry\\, air)}}}", "f224d654ad5df7e579ca6a82e93dd39d": "Z = \\begin{pmatrix}x & y \\\\ y & x\\end{pmatrix} .", "f224ea90b10b542e691dfbc468b42701": "t \\in\\mathbb{R}", "f224fdeee53cd7f97265a80b41c3e784": "B_1,\\ldots,B_8", "f2250305e1845517c5f13ce8d6e26da9": "H(z) = \\frac{z^2+ 2z +1} {z^2 +\\frac{1}{4} z - \\frac{3}{8}}", "f22664c0257e1c51f433754d7ac83ae1": "\n\\mathbf{U}=\\begin{pmatrix}\\rho \\\\ \\rho u \\\\ E\\end{pmatrix}\\qquad\n\\mathbf{F}=\\begin{pmatrix}\\rho u\\\\p+\\rho u^2\\\\ u(E+p)\\end{pmatrix},\\qquad\n", "f22667057ad708e8e73a6f2dd4c07edd": "x^8", "f22696b3e720801909302bb93e3966b6": "(1+i)=\\left(1+\\frac{i^{m}}{m}\\right)^{m} \\,", "f226af81fe31ff55077c9910b74ee4de": "\\langle A \\rangle", "f226f48bb39384b5ec3601b05ae9ecab": "D\\cdot T g", "f25064a898ba34196e718ee6194da586": "\\mathcal{O}_{Y,y} ", "f251125bea745ba825cabb8b1455625c": "\\|x+y\\|>2-\\delta", "f251c223246e30ce25880246c0d12bb7": "\\{L,R\\}", "f251d3a23c3adf85988c3e80f8a4f994": "\\coprod X_\\alpha", "f2520b311881d8277a1cc523f2bdda98": "O(n(\\log_2n)^{\\log_2\\frac{3}{2}})", "f2525346ef5c043c36670c862f9f786e": "n = 2^8 - 1 = 255", "f25286a08d509f2f2314311f3b06da15": "\\scriptstyle \\mu S(t)\\Rightarrow \\mu_\\rho ", "f2535190d86b4e95b790ada5c02cd697": "\n(\\mathbf{\\hat{f}_{0:3}})^T =\nc_3^{-1}\\begin{pmatrix}0.1 & 0.0 \\\\ 0.0 & 0.8 \\end{pmatrix}\\begin{pmatrix} 0.7 & 0.3 \\\\ 0.3 & 0.7 \\end{pmatrix}\\begin{pmatrix}0.8834 \\\\ 0.1166 \\end{pmatrix}=\nc_3^{-1}\\begin{pmatrix}0.0653 \\\\ 0.2772\\end{pmatrix}=\n\\begin{pmatrix}0.1907 \\\\ 0.8093 \\end{pmatrix}\n", "f253e3e22e289ca2766ad6cc2d4544f5": "\\{ z^n f \\}", "f254178c8b3dd0e2965eed1e54e14bff": "\\sigma (t) = E \\varepsilon(t) + \\eta \\frac {d\\varepsilon(t)} {dt},", "f254420640902e8c4ee5b2dc4905d729": " x'\\beta ", "f254454be9b13ceee41cabc4a84830cf": "p_2=v_2", "f254beb29457fd5108d448e163d096a3": "\\text{Margin of Safety}=\\frac{\\text{Realized Factor of Safety}}{\\text{Design Safety Factor}}-1", "f254ea9229bad9949eaac9e14a283f50": " R \\le 2B \\log_2(M). ", "f254f1aded088fabe01e4b0539cd38a5": "\\Gamma^0_{ik} = \\begin{bmatrix}\nA'/\\left( 2A \\right) & 0 & 0 & 0\\\\\n0 & -r/A & 0 & 0\\\\\n0 & 0 & -r \\sin^2 \\theta /A & 0\\\\\n0 & 0 & 0 & -B'/\\left( 2A \\right) \\end{bmatrix}", "f255afcfdc8decc29abf42c76f6a46d3": "\\sigma_{0}", "f255d0d5d1fa90d4ad707fa7b72617e0": " \\ln \\Lambda = \\ln \\left(12\\pi n \\lambda_D^3 \\right) ", "f255ef3aa5f201e71e2b022cbb4740d1": "O(N^{4/3})", "f256036d087fc91f56c33b1549bcd37e": "\\textstyle I_{tot}", "f2561a6edabad5dcd6395f4331076add": "\\displaystyle \\frac{k \\lambda}{D}", "f2563397e7904f6761e0d783dd5c5321": "\n{1+\\cfrac{1}{2+\\cfrac{1}{3+\\cfrac{1}{4+\\cfrac{1}{5+\\cfrac{1}{6+\\ddots}}}}}}\n", "f25637ee5fb2f94aee96140dece93b0e": " \\frac{ z^{-1} (1 + z^{-1} )}{(1 - z^{-1})^3} ", "f25664d00d0827487968d63e14fcf30d": "y^* = y \\pm \\sum \\Delta y", "f25667b8c1e7df7fec5ddbb9765aacb0": "\\exp(\\exp(X) - 1) = 1 + X + X^2 + \\frac{5X^3}6 + \\frac{5X^4}8 + \\cdots", "f2566f284a262f7489a3e5d0c94f32c9": "H_1=H\\otimes \\ell^2(\\Gamma),", "f25731d3120dca1e0ad0898a1427ff0b": "F_n(s_0)\\ne F_n(s_1)", "f2576ff280110f934cfb488a9079f277": " \\mathbf{e}_{123} = i ", "f257c2c4dd30726fae272aa87fbe848d": "\\cos{\\theta}_\\text{CB}* = \\varphi(\\cos\\theta + 1) - 1 \\,", "f257e6b63fe39ad10ee9f1c4a609c998": "\\sigma:([X_0:X_1:\\cdots:X_n], [Y_0:Y_1:\\cdots:Y_m]) \\mapsto \n [X_0Y_0: X_0Y_1: \\cdots :X_iY_j: \\cdots :X_nY_m]\\ ", "f257ed40a557dda0d5fb63ce2ca33198": "\\mu:\\mathcal {F} \\to X", "f257eed8db3a5c799135d66f4d9a0ec2": "88^2 + 33^2", "f2581428d7fa7b68c6f4d45cc62a8376": "X_1,\\dots,X_n", "f2583024a0c8392099a7d06ffc28cad9": "(I_A,p_i,q^i, \\phi^A)", "f2583a221110d97c167d8fde2b44a8bc": "S\\to T", "f25847a162ec8d4c99c3ae784771bf84": "\\mbox{d}T = \\frac{1}{2}\\rho C_L \\frac{V_a^2(1+a)^2\\cos(\\varphi+\\beta)}{\\sin^2\\varphi \\cos\\beta}b\\mbox{d}r", "f258ae4a47269d0c5c721ce1c1b29ea8": " p_{n,N}=2^{-N+1}\\sum_{k=0}^{n-1}\\binom{N-1}{k}. ", "f258ffac94969b6fedae139e06f701c7": "\\Omega_2 = \\{ \\rightarrow \\}.", "f2591037bee8b15325c66eb7b28471ef": "\\int d^dx \\,\\sqrt{-g} R^2", "f2592556a3f1630e3b01ea59b828ceec": "M_d", "f2593a3b5d602715c258e57c7bb74709": "\nE_{\\text{MP0}} = 2 \\sum_{i = 1}^{N/2} \\varepsilon_i, \\qquad E_{\\text{MP1}} = E_{\\text{HF}} - 2 \\sum_{i = 1}^{N/2} \\varepsilon_i.\n", "f25973661f57f7b036a053a938375b4a": "L_{D} = \n{q_1 q_2\\over r} {1 \\over 2c^2 } \\mathbf v_1 \\cdot \n\\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right]\n\\cdot \\mathbf v_2 ", "f25992b72227bcba7fcb9a65bfb65450": "R_r^', X_r^'", "f25992c25bfa52525c98ed328c4f1220": " \\alpha^2=\\left(K+\\frac{4}{3}\\mu\\right)/\\rho \\qquad \\beta^2=\\mu/\\rho\\,\\!", "f259b3b2e6279e4e64c66953d4e0a447": "\\scriptstyle F\\cos\\theta", "f259c692d8cb4673a9b52f214fd421d8": " x^{-1} = [e,x,e] ", "f25a12773e50a072167ac69d088cbe63": "\\frac{P \\to Q, Q \\to R}{\\therefore P \\to R}", "f25a46437be85d5d3bf06121e6f0b1e8": "\\mathcal{N}=(1,0)", "f25a828fc2eaaad2651b97892c02965d": "y=-jb_L=\\frac{-j}{\\omega LY_0}=\\frac{-jZ_0}{\\omega L}\\,", "f25ab7079cc0497dbac457bd5cd87865": "a+b\\,\\!", "f25b77b6d4394ef3027c4fb69f3d8a78": " \\varphi \\left ( \\mathbf{x} \\right ) = \\sum_{i=1}^{2N} \\sum_{j=1}^n e_{ij} v_{ij} \\big ( \\mathbf{x} - \\mathbf{c}_i \\big ) ", "f25b7bd75a99e36c8db6b747ac374a93": "\\textstyle P_{Y_r}(y)", "f25bed752f1860ad5ef5eb362c09ee36": "\\cos \\left(x+y+z\\right)=\\cos x \\cos y \\cos z - \\cos x \\sin y \\sin z - \\cos y \\sin x \\sin z - \\cos z \\sin x \\sin y, \\,", "f25c28ff60a96ac55d1eee2a7bada991": " Z/pZ ", "f25c377192454b7417e1e5330755eb5e": " K(p, w, r),", "f25c6388343d8a8f0b5a463366aaf909": "\\scriptstyle{v_\\text{e}^2/2}", "f25cab5d3db933ed3028c4a9f6caf293": "\\alpha<1", "f25cbbf3ac61c698a2aefbfb222a7121": "\\nu_X \\colon X \\rightarrow BG", "f25d35cba0aae741ea8ef7c41d8cd818": "\\exist x \\exist y Lxy", "f25d3aea0cc4d30be4c5e57f12daf405": "\\left| \\frac{\\partial^2 \\tilde{a}}{\\partial z^2} \\right| \\ll \\left| \\beta_0 \\frac{\\partial \\tilde{a}}{\\partial z} \\right|", "f25d42295f0b63b59c27224e88f898f7": "{1 \\over S(N-S)}", "f25d556884dd94a62f2e7c85811b6e61": "f\\colon B\\to Q", "f25d70014998ac0ae634554e349eebb1": " \\tilde{\\vec{\\sigma}} = \\frac{-\\vec{S}\\times \\vec{h}}{Mc(1+\\sqrt{1+{\\vec{h}}^2})} ", "f25d74a4d9f88a02faf07f4a9e419e01": "x(t)\\to 0", "f25ddafc5241abed463bb19aa7e6408d": "b\\in B", "f25e3bcafc9b5a07d5d1932835731950": " X_\\leq(x) = \\{ y \\in X | y \\leq x\\} ; ", "f25e964bdfdbddd0e6b8f474c1b4fc09": "\\theta_1+\\theta_2=\\theta_3+\\theta_4=90^\\circ. \\;", "f25eb3e4f86c750ec20350c4d323a257": "\\mathrm{ELF}(\\mathbf{r}) = \\frac{1}{1 + \\chi^2_\\sigma(\\mathbf{r})}.", "f25ec86107dfbc6dcc3a0cdcfbe716ec": "\\theta(2)=\\frac{\\alpha}{c+v}+\\frac{\\alpha}{c+2v}\\,\\!", "f25ef2c9666bab05a3db27aad0518e96": "\\phi_{sl}", "f25ef8db649caf2b4d6586dc3f1a9a63": "r_x = \\Phi \\ \\sigma_x \\ \\rho_A = \\Phi \\Sigma_x ", "f25f04f39beb09de9ba59185ad624c6d": "\\boldsymbol\\beta^{k+1}=\\boldsymbol\\beta^k+f\\ \\Delta \\boldsymbol\\beta.", "f25f22c9ff80ca6495cc264df84c810a": "B(m, r, \\lambda) = \\frac {\\mu_0} {4\\pi} \\frac {m} {r^3} \\sqrt {1+3\\sin^2\\lambda} \\, ,", "f25f33ceb386906eb3054c3491c5551e": "\\mathrm{gcd}(k,\\ell)", "f25f99b23438bb514b98b70ca51df2bc": "\\bigg. J = - D \\frac{S(p_2 - p_1)}{\\delta} \\bigg. ", "f25fbf55e33a7eb127f53554dce3db0a": "f(x):=\\prod_{z\\in G:\\,p(z)=0}(x-z)", "f26022024b8667dbcca9cacdcf5cb713": "n\\sin\\theta_\\mathrm{max} = n_\\text{core}\\sin\\theta_r.\\ ", "f2602c7506b633e6aa3d7b5a85cf520a": "q \\leftarrow t", "f260ad8c968c161df5fade806be24333": "~E~", "f260be9cceae07a7199998eb3a2de5ba": "\\pi_2\\left(\\frac{[SU(5)\\times U(1)_\\chi]/\\mathbb{Z}_5}{[SU(3)\\times SU(2)\\times U(1)_Y]/\\mathbb{Z}_6}\\right)=0", "f260d2e8dd95586c0f2139c97002e3fc": "E = {\\frac 1 2} m v^2 = \\frac{p^2}{2m}", "f2617f908d7000bcd06583dae9e9d219": "r = \\infty", "f261eec18943ef028d08ff6e7c91c85f": " 1 + RS_{t}=\\frac{P_{t}}{P_{t-1}}.", "f26255e3026601089717e22885b5cf1e": "F_v", "f2636337564b2d8b8a7949abc3f5185d": "\n\\begin{align}\n\\ D_{ \\omega } & = 1+ j \\omega [(C_M+C_i) (R_A//R_i) +(C_L+C_C) (R_o//R_L)] \\\\\n & = 1+j \\omega ( \\tau_1 + \\tau_2) \\approx 1 + j \\omega \\tau_1 \\ , \\ \\\\\n\\end{align}\n", "f264627876f1eb7315e4b4f67ae493fa": "\\ell_2 = \\tfrac{1}{2} {\\tbinom{n}{2}}^{-1} \\sum_{i=1}^n \\left\\{ \\tbinom{i-1}{1} - \\tbinom{n-i}{1} \\right\\} x_{(i)}", "f26466fe30b03a8b4b49ca0ed28e64fb": "\\scriptstyle n!", "f264ccd1f1cfa22cb40e31f5adce6fad": "v(x)=\\partial f^n(x) / \\partial n |_{n=0}", "f264e0cd3fe61e33219f42c0c2f32a40": "A \\subseteq M", "f26509f16a0b98d31b703c6453cd58d3": "\n\\frac{\\nabla \\mu_i}{R\\,T}\n= \\nabla \\ln a_i\n==\\sum_{j==1\\atop j\\neq i}^{n}{\\frac{\\chi_i \\chi_j}{\\mathfrak{D}_{ij}}(\\vec v_j-\\vec v_i)}\n==\\sum_{j==1\\atop j\\neq i}^{n}{\\frac{c_ic_j}{c^2\\mathfrak{D}_{ij}}\\left(\\frac{\\vec J_j}{c_j}-\\frac{\\vec J_i}{c_i}\\right)}\n", "f2652e5d971a9dac8dd005a5f2adb36f": "r_g=2GM/c^2", "f265c0ab3d793f00368ae44b4b49e6d1": "\\boldsymbol{k}\\cdot\\boldsymbol{x}\\, -\\, \\omega\\, t\\,", "f266cc7550cdddb796c1bc0f83a5c11a": "\\scriptstyle a \\times b.", "f2673c728f7e766e88756223e294ab90": "\\,g(x)=\\alpha+ \\beta x +\\gamma \\cdot f(\\lambda x+\\delta)", "f2679c4e6e77b0aea9d854d444a68ca4": " \\mbox{precision}=\\frac{|\\{\\mbox{relevant documents}\\}\\cap\\{\\mbox{retrieved documents}\\}|}{|\\{\\mbox{retrieved documents}\\}|} ", "f268473b78983af44d67c82031d84a55": "q_a", "f26865c2f27f594ba253032ab118c86b": "\\scriptstyle f(n) \\;>\\; A(n,\\, n)", "f2688cb84ceedff5a421f2138202e974": "\\delta>0", "f2689ac43adddded58cf77dde026df34": "\\int d^3x \\left[\\alpha(\\vec{x})\\rho(\\vec{x})+\\epsilon_0 \\vec{E}(\\vec{x})\\cdot \\nabla\\alpha(\\vec{x})\\right]", "f268ed9ff590dacd4e40498558128da0": "Q^{s}(P) = c + gP", "f268fe5756328df91d0ca71a346be8ed": "-3.55271E-15", "f26934e9463f7fb9fb76b099faefb6ae": "\\operatorname{Ei}(x)=-E_1(-x)", "f2693782a874e1ca203cd3dfdd923db6": "B'(x) = e^{x}B(x)", "f2697e8f388491968b829a8f242a9a32": "\nx\\rightarrow \nx^9-36 x^7 (y^2+z^2)+126 x^5 (y^2+z^2)^2-84 x^3 (y^2+z^2)^3+9 x (y^2+z^2)^4 + x_0\n", "f269acb7f37d330bdd0cd0804a59eeff": "\\frac{\\delta(\\lambda F + \\mu G)}{\\delta \\rho(x)} = \\lambda \\frac{\\delta F}{\\delta \\rho(x)} + \\mu \\frac{\\delta G}{\\delta \\rho(x)},\\ \\qquad \\lambda,\\mu", "f269f7cfd4378d45309f4cc20babb36b": "\nV_{\\textrm{cell}}=E+V_{\\textrm{act}} +V_{\\textrm{trans}} +V_{\\textrm{ohm}}\n", "f26a3a6edb1f04dd5a5dedbf67acea98": "\\pi /2 ", "f26a5b039ac335781b37ddbd32195fe7": "[H(H(P))]", "f26a5e3671b42490700d969816bd7e3b": " \\Delta H^{0} = -45.2 \\ \\mathrm{kJ} \\, \\mathrm{mol^{-1}} \\; \\mathrm{NH_3} ", "f26ab5ac105d28219136a2f60e890164": "\\scriptstyle \\frac{4\\pi}{3}.", "f26ad59dd92c50bb11ccf1f097dfcb7c": "(\\sqrt[5]{100})^{19.5-6.5}\\approx 158489", "f26b1fad84c6a117323aa7c3201ca77f": "G_0(\\overline{\\mathbf{F}}_p[G])\\to \\mathrm{BCh}(G)", "f26b287b1e6f5544ff3fc4267d137f32": "t_n =\n\\begin{cases} \n1 & \\text{if } n = 2^k \\\\\n1-t_{2^k-n} & \\text{if } 2^{k-1} 0", "f27068fb6f469f870686c1e4e3f75db5": "\\scriptstyle\\lesssim10^{-20}", "f27088823c5d880e0aaf0b55a0a24245": "\\bar t", "f270b35b736b1cc5d5743accc93ae61b": "p \\approx a\\,", "f2712d1c2a2c4952ba81b9115c038ffe": "\\rho_e = -\\epsilon \\epsilon_0 k^2 \\psi \\, ", "f27133ecf45beee9a2d3c12d6f706a63": "\n\\Delta E \\propto {1 \\over (L+\\hbar)^2 } - {1 \\over L^2} \\approx - {2\\hbar \\over L^3} \\propto - E^{3 \\over 2}.\n", "f27148a993484f0edcaa0cf7be882a62": " m_j = \\pm {1 \\over 2} ", "f2719e209c068a08487b99511b16a2da": "x\\to\\infty", "f271d9694ca286894ec1765366883627": "- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha^2}= \\operatorname{var}[\\ln (X)]= \\psi_1(\\alpha) - \\psi_1(\\alpha + \\beta) = \\mathcal{I}_{\\alpha, \\alpha}= \\operatorname{E}\\left [- \\frac{1}{N} \\frac{\\part^2\\ln \\mathcal{L} (\\alpha, \\beta, a, c|Y)}{\\partial \\alpha^2} \\right ] = \\ln (\\operatorname{var_{GX}}) ", "f2720600d82611851f151f0ea64befc8": "{\\rho}_{xx'}", "f2726c6e23a6c515bbc05a1264456f26": "G(s),", "f2726e087c6b46fe519cd87f0de06c57": "\\neg Ab(S)\\,", "f272d1803852bfd27b1cc6cace4354a2": " \\nabla \\cdot \\mathbf{D} = 4\\pi\\rho_\\mathrm{f}", "f2730638e6fcb38277d2efe34f13f9bd": "Y_{7}^{-2}(\\theta,\\varphi)={3\\over 64}\\sqrt{35\\over \\pi}\\cdot e^{-2i\\varphi}\\cdot\\sin^{2}\\theta\\cdot(143\\cos^{5}\\theta-110\\cos^{3}\\theta+15\\cos\\theta)", "f273076392ccbabe07068a055150b796": "\\cot 3\\theta = \\frac{3 \\cot\\theta - \\cot^3\\theta}{1 - 3 \\cot^2\\theta}\\!", "f273323d003a9ddbad5eb66add594d69": "(w \\backslash y) \\cdot (x \\backslash z) = y \\backslash (w/x \\cdot z) = (x/w \\cdot y)\\backslash z", "f273617cf5c1de141433001fce350eda": "\\kappa_t(\\mathcal B)\\,\\!", "f27377f07b443a0ab886a632a1ecf465": "\\mathfrak{m} = \\langle x, y \\rangle ", "f273925fb3b1bca36d6f0a5d0ad12a56": "\\sigma \\approx 11.7 \\times 10^{-8}\\ \\textrm{cal}\\,\\textrm{cm}^{-2}\\,\\textrm{day}^{-1}\\,\\textrm{K}^{-4}.", "f27399c11678d989b51b96b3b651d63b": "\nu_k = U_k + V_k,\n", "f273b4906a3ade74076a04817293f8e1": "P(X_1\\oplus X_2\\oplus\\cdots\\oplus X_n=0)=1/2+2^{n-1}\\prod_{i=1}^n \\epsilon_i", "f273fdb261e8e1d37efa204471605075": " \\alpha_c = \\lambda/2", "f27439b450887b2c44c8c6fa7ebf6fa9": "\\{S^0,S^1, \\ldots, S^{k-1} \\}", "f274488cee17e1ca87dad95d30f968e4": "\\int \\csc^2{x} \\, \\mathrm{d}x = -\\cot{x}+C", "f2746b7e118409e8ab6babb6980f4f5b": " r_2 - r_1 =2 a\\,\\!", "f274819d51a5894ddc5b8c09702f42f6": " Z_{P} = bt_f (d-t_f )+ 0.25t_w (d-2t_f )^2", "f27524135f3fd49c238e5472d7b94c0c": "P= \\begin{bmatrix}\n-1 & 0 & -1 \\\\\n-1 & 0 & 0 \\\\\n2 & 1 & 2 \\end{bmatrix}.", "f275417242aa435f286bfe0801c6596f": "\\underset{\\alpha}{\\omega}", "f275783efc838148449e142da6501bb4": "_u = \\frac{1}{N-1} \\ . \\ \\sum_{i=1}^N\\frac{ (x_i - \\overline{x})^2}{\\sigma_{x_i}^2 }", "f275a82b5282aa80584ff6a2ce7e8f09": "P_X(t)=P_Y(t)+P_U(t)", "f275dc400e58af24c9fdf8c88b900253": "d \\approx 3.57 \\cdot \\sqrt{h}", "f2760995f934a3032aced82bc317aa74": "\\left\\{\\,k 2^n - 1 : n \\in\\mathbb{N}\\,\\right\\}.", "f2765a27b0b9a12fdace3e638e8a7b6b": "\\frac{\\kappa}{\\sigma}=LT", "f276c2dbe974c48d5810f88a4fb429b1": " \\forall n : \\sum_{i_n=1}^{I_n} w_{n,i_n}(p_n(t)) = 1. ", "f276daf7c9dc149b1bbcfd606ca569b9": "TQ", "f276ffd9b64463b945ada531198b88e0": "|m_{ij}| \\leq \\sqrt{m_{ii}m_{jj}} \\leq \\frac{m_{ii}+m_{jj}}{2}", "f2773ced055fc478b25334f65eae8b2c": "g\\,\\,", "f2776b53f090b08e0987c010217cfef1": "\n\\int x^m\\left(A+B\\,x^n\\right)\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^qdx=\n \\frac{A\\,x^{m+1} \\left(a+b\\,x^n\\right)^{p+1} \\left(c+d\\,x^n\\right)^{q+1}}{a\\,c (m+1)}\\,+\\,\n \\frac{1}{a\\,c (m+1)}\\,\\cdot\n", "f2776c9580089acbdb08f5a3044efcc5": "f(x) = \\frac{1}{\\pi\\sqrt{(x-a)(b-x)}}", "f2777ce42f288e427e35c3bc5e4c6035": "\\mathfrak{P}^{126}", "f2777ce767931de93333e5aaaa47b8c1": "\\frac{\\Delta z}{\\Delta t} = u_z = u \\sin(\\alpha) \\approx u S", "f2777fd33d1b311cbc9c759f257b02f6": "\nS= \\int_t \\sum_i \\psi_i^\\dagger\\left( i{\\partial \\over \\partial t} - E_i\\right) \\psi_i\n\\,", "f277e3fbfd088f79721905c5da5a172b": "\\sigma(n)-n", "f277f1e1d62690de1ae23e5ad1aa1a78": "\\frac{n!}{\\sqrt{2n+1}}.", "f277f58820fbac93695787a680f51466": "\\begin{array}{cl}\n\\underset{\\boldsymbol{\\lambda},\\gamma}{\\max} & \\gamma\\\\\n\\textrm{sb.t.} & \\sum_{n=1}^{\\ell} y_n h(\\boldsymbol{x}_n ; \\omega) \\lambda_n + \\gamma \\leq 0,\\qquad \\omega \\in \\Omega,\\\\\n& 0 \\leq \\lambda_n \\leq D,\\qquad n=1,\\dots,\\ell,\\\\\n& \\sum_{n=1}^{\\ell} \\lambda_n = 1,\\\\\n& \\gamma \\in \\mathbb{R}.\n\\end{array}", "f278173a2898fc0ce7a2aa8069007bf6": "e(x) = \\frac{2}{\\pi}\\arctan x", "f2785065ef1e77c9bfecff92d39ac40e": "X_k =\n \\sum_{n=0}^{N-1} x_n \\sin \\left[\\frac{\\pi}{N} \\left(n+\\frac{1}{2}\\right) \\left(k+\\frac{1}{2}\\right) \\right] \\quad \\quad k = 0, \\dots, N-1", "f27899ee5326a6d961b54166304c78d0": " \nd\\Pi = kT(dc_+ + dc_-) + q(c_+ - c_-) d \\psi\n", "f278a955d2526b7a026fbddbbb8fcb1d": "y>e^{-\\gamma}", "f2790fa859366d6676fbaf29c1bd6518": "\\displaystyle{\\Phi(0)=I, \\,\\,\\, \\Phi(t)=T(t),\\,\\,\\, \\Phi(-t)= T(t)^*,}", "f2792a5ef9af1d997d139086f59f9ccc": "D\\cdot C \\ge 0 \\, ", "f2794eefa83c8c060fbe5ecb23ff50e8": " p_0 A ", "f27957655dc67d4074cefa61addde9ac": "\\left|\\int_\\text{Arc} f(z)\\,dz\\right| \\le ML", "f279d4b324a376737ee6e478b2634dec": "+\\colon\\{(b_1,b_2)\\in B\\times B:\\pi(b_1)=\\pi(b_2)\\}\\to B", "f279db9293d290166564cfd8990828ee": "\\lim_{p\\to-\\infty} L_p(x)", "f279f4ecdb8b9e7223db9f76742b39cc": " \\int |M(f)(x)|^p \\, \\omega(x)\\, dx \\leq C \\int |f|^p \\, \\omega(x)\\, dx,", "f27a7bf6fa91752fce69cc7c7af12259": "\\lim_{k \\to \\infty} \\int_{W} \\big| f_{n_{k}} (x) - f(x) \\big| \\, \\mathrm{d} x = 0;", "f27a85cd6cd6eef2f6b258be0a285625": "O\\left(n^3\\right)", "f27aa6a8887646a8f47e4427a76b4c80": "U=\\int_0^T e^{-\\rho t} u(c_t) dt", "f27aac93a2722f2128f20c0555e08a80": "\\Box \\vec{A}=\\left[\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}-\\nabla^{2}\\right]\\vec{A} = \\mu_0\\vec{J}", "f27ab0a3652c91df6e95b7ada2add3f5": "x=1/\\sqrt{2}", "f27aba6f3fbb7c9cdfc97940303ba341": "\\frac{K}{s^2} = \\frac{n(m-n)}{m(m+n)} = 1-\\frac{c}{s}.", "f27ac279a44e85f0e4f154eb34155f7e": "\\sigma_{Age \\ge 34}( Person )", "f27b2d37609f19d15a74cf256f41ee15": "E^2 - (p c)^2 = (m_0 c^2)^2 \\,", "f27bdae2500ff03eeb9e7110c6289a21": "\\sigma_{\\theta\\theta}\\, ", "f27c27af2eeb6159f72984305655003e": "u'_2", "f27d069b403e215ecf603ac6643f98ff": "\\log_{10}K_\\textrm{a} = \\log_{10} \\left ( \\frac{[\\textrm{H}^+][\\textrm{A}^-]}{[\\textrm{HA}]} \\right )", "f27d8bb8c0ec07318c5fd276f5501ac8": "\\mathbf{v_w}", "f27d9034b8568f3774781f9d70cb74c0": "x = 10 + 2 \\epsilon_3 - 6 \\epsilon_8", "f27d914389da2c4b0896c6d7ed7a2aa0": "I(X;Y,Z) = I(X;Z) + I(X;Y|Z)", "f27dcf87873b903ac97fe8aca86a8201": " | L \\rangle ", "f27dfc37654c7d18ff1307ce3dab773d": "TP_s = 3~ \\mbox{if}~g\\ >= 0.06 ", "f27e156af895cffe40877b370177b993": "\\tfrac{r}{3}", "f27e96e23160cf3f9565384683fa02c6": " \n\\begin{align} \\rho(\\theta|k) & \\propto \\ell(k|\\theta)\\pi(\\theta|\\mu,M) \\\\\n & = \\operatorname{Beta}(k+M \\mu, n-k+M(1- \\mu) ) \\\\\n & = \\frac{\\Gamma(M)}\n {\\Gamma(M\\mu)\\Gamma(M(1-\\mu))}\n {n\\choose k}\\theta^{k+M\\mu-1}(1-\\theta)^{n-k+M(1-\\mu)-1}\n \\end{align}\n", "f27eac6622269c2377abbdd9b9c2b0a6": "(\\mathcal{L}I^\\alpha f)(s) = s^{-\\alpha}F(s)", "f27ed73f13473084a1a7c2e63c105763": "\\Delta w_{k} = \\nu_{k}(\\mathbf{J}) T", "f27f4f5ebd1758620cc32e7195bb06ed": "M^{2n}", "f27f57845c85057857a8913792461cc2": "s(v)", "f27f8a8e87864108b8b73761b66931de": "-10\\le x,y \\le 10", "f2800a0ad3c57e18cc99c7b7b3a9e904": " \\!\\ e^{\\ln(1+ \\sqrt2)} = 1 + \\sqrt2 ", "f28012b1b3b40dcd1afbb4fddf81773d": "\\delta =\\frac{\\Gamma_{s}}{\\Delta \\rho_{0}}", "f280272369287127c5295d5b0deac01f": "a\\leq b\\leq c", "f2802d26e89573f89d95efbf204693aa": "\nH_{2^n}=F_n^{\\rm T}F_n.\n", "f28051047948be6931360145f55d101c": "\\neg \\alpha", "f2806986b398ac4dcda431918b364ec0": "e^{z + w} = e^z e^w\\,", "f280928d7083a44d7799f927c1f0d0cf": "GL_2(F)", "f28092c410b6f6d9a90cda4f02c060b2": "\\vec{\\nabla} \\times \\vec{\\nabla}\\varphi =0", "f2809e8d80facc016d8ff2918b39c115": "x \\longleftarrow \\frac{1}{x +1} ", "f280df0b04ac73c1338f6f5e0c1be991": "a(n)={n\\choose n/2-1}(2^{-n/2} + n^2 2^{-n-5} - n2^{-n-4}),", "f28166f43552c2094c1d95a42fdc103d": " \\forall x (\\exists y B(x,y)) \\vee C(y,x) ", "f2821a7459b10c0cb2774cc29e14bd4c": "\\gcd(p,q)=ap+bq", "f282327f46b0b3980e8cb0899574850b": "R_{ab}l^a l^b\\,\\hat{=}\\,0", "f2826db237cad7a2afd1ef0dd893e654": "\\sqrt{n}[g(X_n)-g(\\theta)]=g'(\\tilde{\\theta})\\sqrt{n}[X_n-\\theta].", "f282fbd56ba6f16975c9d1570cba7bcd": " \\beta(w_1)=0 \\,", "f2831c9cbac5b86ee2f36c60df812324": "\\Sigma_1", "f283343b0db777bbad6bbf484ee457db": "\\frac{R_o^2}{R_E^2}", "f2837601b6342f51fc8990e8f4dd62a0": "v_p(R)/V(R)", "f2839884c7d1c69051b0bb541b372a8d": "\\forall \\epsilon>0, \\exists N\\in\\mathbb{N}: \\forall k\\geq N \\Rightarrow \\rho(A) \\leq \\|A^k\\|^{1/k} < \\rho(A)+\\epsilon", "f283c8e92c3ddc186d8e3af1f1a3ae77": "(A \\cap B) \\cap C = A \\cap (B \\cap C)\\,\\!", "f28405352493ec506128e38d7b2171ce": " x=\\frac{-b + \\sqrt {b^2-4ac}}{2a}\\quad\\text{and}\\quad x=\\frac{-b - \\sqrt {b^2-4ac}}{2a}", "f28426ff1f245f74a36442ff444caf4e": " h(t) = h_0 + \\alpha_{eff} \\int_0^{\\infty} [T(z)-T_1]dz = h_0 - \\frac{2}{\\sqrt{\\pi}}\\alpha_{eff}T_1\\sqrt{\\kappa t} ", "f2843f30d5fbc566e7419b2c8090b0a1": "T(V)= \\bigoplus_{k\\in\\mathbb{N}} T^kV = \\mathbb{F}\\oplus V \\oplus (V\\otimes V) \\oplus (V\\otimes V\\otimes V) \\oplus \\cdots.", "f2846fbceb19c9330b067abecc633d1a": "f^{\\prime}\\left(w_2\\right)", "f284a77b12decc9e7b5bfd3db874d49d": "I_{y} = \\int_A x^2\\,\\mathrm dA = \\int^{b/2}_{-b/2} \\int^{h/2}_{-h/2} x^2 \\,\\mathrm dy \\,\\mathrm dx = \\int^{b/2}_{-b/2} h x^2\\,\\mathrm dx = \\frac{b^3 h}{12}", "f284dbacafb7f4d9b6726c66fcba0eb7": "\\rho(y)=\\rho(x)+1", "f2851efa23152833ee6e5288cb6794c3": "x = \\text{mode} \\pm \\kappa = \\frac{\\alpha -1 \\pm \\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "f28540ed66b3793f5461be6e9b60c392": "\\frac {1} {R_f+R_2}", "f28550e010dda7faa7312c9c7d680b9f": "A=A^\\dagger\\quad \\Rightarrow \\quad \\det(A) = \\det(A)^*.", "f286017b0f66214b8d11bd2249c311e2": "\\sum_{k=-a}^{a}(-1)^{k}{2a\\choose k+a}^3 =\\frac{(3a)!}{(a!)^3}.", "f2861581e8d3ad0a491a9bfa89768bac": " \\psi_2 ", "f2861a6156ed9c0acf76d3f21d5614cd": "p(y|\\xi)", "f2864b9ef54eedb161c5f47e79b36fbf": " \\mathbf{y}(t) = \\begin{pmatrix} y_1(t) \\\\ \\vdots \\\\y_n(t) \\end{pmatrix} ~,", "f286a52387414911cf3796df7d292d59": "p(1) = (1)^3 + 2(1) - 3 = 0", "f286da56bfec63e1d59d3e5db332906c": "RL_\\mathrm{out} = - 20\\log_{10}\\left|S_{22}\\right|\\,", "f2872ae789c7bd2f8c16981a27eca0fc": "F_A V_A = F_B V_B.\\!", "f2874f4de96f7f1a93a2e1e1442b4ba1": "\n\\mathbf{F} = F(r)\\hat{\\mathbf{r}}\n", "f2879909b6296ceebd523c263818cc24": "\\scriptstyle -\\mathbf{q}", "f287af079f5dfb4bfca7cb8bd9010153": "V \\subset \\overline{V} \\subset \\Omega,", "f287ddbf35300200fe7157a74e754f08": "[\\hat{H_0}+\\hat{V}]\\psi_j\\rangle=[\\hat{H_0}+\\hat{V}]\\sum_{i}c_{ji}|m_i\\rangle=E_j\\sum_{i}c_{ji}|m_i\\rangle", "f287fbbaa4d07fe115e9af97734ec53f": "\\sigma_{(a,b,c)} = ia\\gamma^2\\gamma^3 + ib\\gamma^3\\gamma^1 + ic\\gamma^1\\gamma^2", "f288025af5ac49f1e70eb2767de10a3a": "\\overline{w}=\\langle v/k \\rangle", "f2883a1d4bb0a2a181fb9fca9b045ed5": "f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)", "f2889561f9f56b8b7a24b838e64b1ec9": "k f_0\\,", "f288b55dd25da1f1a185ffb158d2e0cf": "X_{(k)}", "f2890cca2b48759638bcea5ebdab48ec": "\\begin{matrix} \\frac{1}{2} \\end{matrix} (z+1)", "f28931973df5c03bd73635fb4e84d136": "{a \\over b}", "f289788421995e5e3073707aae032348": "K\\ ", "f2897d382aa56dc515c1b6140ff94650": "\\mathrm{crd}\\ {\\theta}=2\\sin{\\frac{\\theta}{2}}.\\,", "f2899d67387797c0a67415017af01b5a": "\\delta'", "f289a379a73795d98a7be53305df91f2": "t=\\frac{(\\overline{x}_1 - \\overline{x}_2) - d_0}{s_p\\sqrt{\\frac{1}{n_1} + \\frac{1}{n_2}}},", "f28abfbc54c846cc3049d8964073c241": "\\Phi_t(x) = t^{-n} \\Phi(\\tfrac{x}{t})", "f28b8a5018fb24dc07225603b2a58f1f": "r^n", "f28bae04a153ff63043cfa43696270da": "\\theta=\\frac{KP}{1+KP}", "f28bb668f565a6a2e368fa3ba5852c4d": "a \\land (b\\lor \\lnot c)", "f28c4acc7edd099419569372f76c6655": "\\mathbb{D}^a\\mathbb{D}^{b}f = \\mathbb{D}^{a+b}f", "f28c6fb3d20ead69a672bd42b5d1ed10": " T^{ab} = \\rho \\, u^a \\, u^b + p \\, h^{ab},", "f28c766f17c9f753040531cf30f7e265": "\\omega_{H} = \\sqrt{\\gamma\\frac{A^2}{m} \\frac{P_0}{V_0}}", "f28d30be5813fb98db62f4e33282e107": "\n(2 4 (\\lambda 1 4 2)))) (5 (\\underline{11} (\\lambda 1)) (\\underline{12} (\\lambda 2 ((\\lambda 1 1) (\\lambda \\lambda \\lambda 1 ((\\lambda 1 1) (\\lambda \\lambda \\lambda 2 (1 (3 3)) (\\lambda 8 (7 7 1)))) 2 1))))))) (\\lambda \\underline{12} (\\lambda \\underline{12} (\\lambda 3 \n", "f28d4dfa2196b7ee483399bca02144ac": "P(query) = \\prod_{term\\ in\\ query} P(term)", "f28d6efe05847ccace7075c281d3f8e4": "\\left(\\nu_e,e^{-}\\right)", "f28d7044a6f452cfaf0d998d644cfc26": "|\\vec{p}_1| = |\\vec{p_2}| = \\frac{[(M^2 - (m_1 + m_2)^2)(M^2 - (m_1 - m_2)^2)]^{1/2}}{2M}, \\,", "f28d77318f669359d39155bcd795edab": "RT=\\left(P+\\frac{a}{T(V_m+c)^2}\\right)(V_m-b)", "f28dd61ae0a142070622804e3a37f689": "A_X^{(\\beta)}(\\{d\\})", "f28e0c40b5e4cf49797adfd640644538": "(a_{L-1},\\ldots,a_1)^\\mathrm{T}", "f28ef3aa0cf6240bd4acb2cb782705fc": "\\frac{L}{c}\\ k\\ 0.05\\ \\ k=18,\\cdots ,23", "f28fc20494e4d6e322b7303234eea948": "\\begin{align}(1+x)e^x &= e^x + xe^x = \\sum^\\infty_{n=0} {x^n\\over n!} + \\sum^\\infty_{n=0} {x^{n+1}\\over n!} = 1 + \\sum^\\infty_{n=1} {x^n\\over n!} + \\sum^\\infty_{n=0} {x^{n+1}\\over n!} \\\\ &= 1 + \\sum^\\infty_{n=1} {x^n\\over n!} + \\sum^\\infty_{n=1} {x^{n}\\over (n-1)!} =1 + \\sum^\\infty_{n=1}\\left({1\\over n!} + {1\\over (n-1)!}\\right)x^n \\\\ &= 1 + \\sum^\\infty_{n=1}{n+1\\over n!}x^n, -\\infty \\phi_{crit}' ", "f296fe62a481c5eb35cef3ff713a2dd7": "\\phi(S_{D,x})=S_{E,\\phi(x)}.\\,", "f297023719e6d49178532f3f52577fe4": "value_{i}", "f29718b408235a676200672b26a83057": "\n \\omega_{c} = c \\frac{\\chi_{11}}{r} = c \\frac{1.8412}{r}\n", "f2974ce91f9f5c1cc425dee253e01429": " S(k,k') = \\frac{2\\pi}{\\hbar}\n\\left | \\langle k|H'|k' \\rangle \\right |^2 \\cdot\n\\delta(E - E') ", "f2978849a4ea94639fab041a8a6efab6": "\\log \\zeta(s)=\\sum_{n=2}^\\infty \\frac{\\Lambda(n)}{\\log(n)}\\,\\frac{1}{n^s}", "f2982ecfe5675354f7d7a300269a68de": "\\textstyle{7 - 1 \\choose 3-1} = 15", "f2983c4c6c0a138d4db79389561d4e20": "I_A = I_B < I_C", "f2985ceef61d51d7a1c12e15d3f65315": "\\textstyle e^A := 1 + \\sum_{n=1}^{\\infty} \\frac{1}{n!} A^n", "f298771c0b4d0d8aaff52c5972bfed6f": "\\vec \\omega=\\frac{\\vec {e}\\times \\dot{\\vec{e}}}{|{\\vec{e}}|^2}", "f298ab01c0cb16b6d7e82c633a3d8d7d": "a \\equiv r \\pmod{n}", "f298b2e1d0a45fbd2009fe28217313af": "|\\alpha_x|^2", "f298cb92d54c2aa0d93f7a4680c88816": "\\sqrt{2E_2}", "f29910f0578654c259ed3a1f17a826a4": "\\mathbf{X}(t)=(x_1(t),\\cdots,x_N(t))^\\mathrm{T}", "f2991e93f2c71253cac48716abc1b5ad": "x_n^\\ast", "f29939669fc0cb38f13ed240ea71f6e8": "f_Z(z)=\\frac{1}{\\sqrt{2\\pi(\\sigma_x^2+\\sigma_y^2-2\\rho\\sigma_x \\sigma_y)}}\\exp\\left(-\\frac{z^2}{2(\\sigma_x^2+\\sigma_y^2-2\\rho\\sigma_x \\sigma_y)}\\right)", "f29993bbdef0820e58d8c184f1aa99d7": "M = K + 4\\mu/3", "f299e6f0e201538f4dbadd753379e0f1": "\\mathcal{O}_X \\rightarrow R f_* \\mathcal{O}_Y", "f29a2d7aa35342964acc56a77c3f503b": "x\\lambda_p", "f29a2f691e2fc37c841cea40ef1a97ed": "x:y:z = \\frac{\\beta}{a} : \\frac{\\alpha}{b} : \\frac{1 - \\alpha - \\beta}{c} ", "f29a77785bb2cd70bc406bef1a1bb52b": "s_p^2=\\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2+\\cdots+(n_k - 1)s_k^2}{n_1+n_2+\\cdots+n_k - k}", "f29a88a4a7145629077a91aeac651d75": "\\overline{(z/w)} = \\overline{z}/\\overline{w} \\!\\ ", "f29ad1cc6e9f51e29e86709248138922": "f(x) = x\\cdot \\ln x", "f29af319ed40d758f03258274049fc46": "h_{02} = h_{01} + q ", "f29b2d2ec92058bcb9883d1508dc90ba": "\\widetilde{\\rho}", "f29b4a80bdeae7645f630633fd49d17e": "\\mathfrak{sl}(3,\\mathbb C), \\mathfrak{gl}(2,\\mathbb C)", "f29b7815f7720a4e4a31bf6876ca48ed": "S\\downarrow T", "f29b9e1779bc4f20f589d79b39d81a7b": "S = k_B \\ln \\Omega \\!", "f29baca209d5c6d6a12e8ec079920340": "\\scriptstyle \\frac{L_{bol_{\\ast}}}{L_{bol_{\\odot}}}=10^{0.4\\left(M_{bol_{\\odot}} - M_{bol_{\\ast}}\\right)}", "f29bd4510cd972e3bf7f59842ed8f184": "(m, m_b) \\in A \\times B ", "f29c45a5d9cea5221871aa12a7302633": "\n n_z~\\sigma_{zx}^{\\mathrm{core}} = \\cfrac{\\mathrm{d} N_{xx}^{\\mathrm{face}}}{\\mathrm{d}x}\n", "f29c4e86e26295876d611eec00dcf9c5": "U(|\\Phi|)", "f29cb71416a6d4d23cd53753056ecc1e": "\\begin{pmatrix} x\\\\y \\end{pmatrix} = Ae^{t}\\begin{pmatrix} 2\\\\1 \\end{pmatrix} + Be^{-5t}\\begin{pmatrix} 1\\\\2 \\end{pmatrix}. ", "f29cdd85ecef02d9ce1b1110e13c0e4b": "\\epsilon \\epsilon_0", "f29d42a777eb0a1ed598e0c3f2ceddb8": "\\texttt{int} \\to \\texttt{int}", "f29d692d945ac14d4ed1eb2af15a87a3": "0 \\le x_{1} \\le e", "f29d7d215dd27a9702745a0b43081d28": "R_0", "f29d9748cdc40c8a69aa1159429ab1f9": "\n W = C_0 \\ln\\left(1 - \\cfrac{I_1-3}{J_m}\\right)\n", "f29dcd63fa7ab2ee6cf06caebe1702ad": "n_k \\in [0,N_k - 1]", "f29e00d97a6f7f0e0a637f1e69466b97": "P(Y,Z|do(x)) = P(Y,Z,X=x)/P(X=x|Z)", "f29e44b95c6581ab4468d4d77feee9f6": "A(t) = |x_\\mathrm{a}(t)| = \\sqrt { x^2(t) + \\hat{x}^2(t) }\\,", "f29e5b27a2726caf966318b96713bfa6": "\\alpha_k^2 + \\beta_k^2 > 9", "f29f262326c7f328ece910db7b13bc48": "[HG]_{eq}", "f29f481b073518ef9f450c807322fe05": "\\; 2", "f29f6bcf630d0e9ed8b4a7764bbdc589": "\n\t\\hat{y_i} = \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\ty_i 1 \\le i \\le m \\\\\n\t\t\tsign(g_{3-j}^{t-1}(\\boldsymbol{x_{3-j,i}})) m < i \\le n\n\t\t\\end{array}\n\t\\right.\n", "f29fd0a1b94ee8ae39b26ce20def5fbd": " u' = i u \\quad ", "f2a05fcf30c933ab5bbc19c11abebc77": "\\displaystyle ax^2+bx+c", "f2a07a21bcc1b1c5f545e0e737e78ba0": " \\mathcal {H}_\\Psi (t)= -\\operatorname{Tr} \\left ( \\rho_\\Psi(t) \\log \\rho_\\Psi(t) \\right ) ", "f2a09e99d385af34ea17f82f008a753f": "a^n x \\in a^{n+1} B \\cap A + A.", "f2a0a2a4069f164d22b6d0c8737a679d": " B = \\frac{b}{R\\, T} = \\frac{0.0867\\, T_c}{P_c\\, T}", "f2a0c9786519de73c87366e6017ff39d": " \\mathrm P(A_4, A_3, A_2, A_1) = \\mathrm P(A_4 \\mid A_3, A_2, A_1)\\cdot \\mathrm P(A_3 \\mid A_2, A_1)\\cdot \\mathrm P(A_2 \\mid A_1)\\cdot \\mathrm P(A_1)", "f2a0efe8dacefbe2800ac4516d4419ee": "\\lambda_f", "f2a1c457f0993cfa32130db934da43de": " Used as Template:tl throughout English Wikipedia; demo template referenced only on meta ->", "f2a1c72c13390ec13f4511d4caa5cbdf": "a_k - a_{k+1}", "f2a1d49e70007a4b0ab94f884483857e": "\\delta'(x)", "f2a23b37605a8075953ee4f17be9ce70": "x \\div 6", "f2a25912761c70b79f41e775661168da": "(\\lambda,\\mu)", "f2a261c0b7bc93a53853ec0411d00555": "G-e", "f2a26df72b8a69d3f748ef75cd718fe3": "k'=k-1", "f2a282c711d459d7a4e0db3d19052590": "L(t,x,x') = (x^3-t)^2 x'^6,\\,", "f2a29de4be4b559165d1363c10325d70": " \\Psi=S_1\\,S_2\\, K_{12}\\Psi, \\, ", "f2a310b35693d427fc0a799ed3039a75": "[h,f_i] = -\\alpha_i(h)f_i", "f2a36923847f5f2f72a94f9864f39c46": "Q(x_1,...,x_n) > 0", "f2a378c417bcd001fc4a8745bca1034d": " A \\exp \\left(- \\frac {(x-\\mu)^2}{2 \\sigma_1^2}\\right) \\quad \\text{if } x< \\mu", "f2a424b78a315d7036f1935796dc40bb": "\\ddot{u}^{i} = - \\frac{\\partial V(t, u)}{\\partial u^{i}} \\quad \\text{for } 1 \\leq i \\leq n,", "f2a434a19cd3f12b5805b3076fa69705": "\\mathbf{X} = {\\textbf L}{\\textbf A}{\\textbf A}^T{\\textbf L}^T", "f2a506ad9834063e61e6ae5ffe072a4b": "e^w = z\\,\\!", "f2a51ba824674bde8aea60df330ce9a8": "\\langle \\alpha_0, \\alpha_1, \\ldots, \\alpha_k \\rangle", "f2a55273913c570bc4803b2ff86ac01f": "C_p^2", "f2a56dc7679026337c9aa61596481f16": "g(\\bar{x}; \\Sigma) = \\frac{1}{2 \\pi \\sqrt{\\operatorname{det} \\Sigma_t}} e^{-\\bar{x} \\Sigma_t^{-1} \\bar{x}/2}", "f2a5a47dea6ceaeac4e667ca93470faa": " (10) \\quad\\quad u_s = \\frac{f\\left(w_1\\right) - f\\left(w_2 \\right)}{w_1 - w_2}.", "f2a5a6dc4efbbe1dd6b656301a729b1a": "\\begin{align}\nPV & = \\frac{1}{2}\\left (\\frac{Itd}{i+d}\\right ) + \\frac{1}{2}\\left (\\frac{Itd}{i+d}\\right )\\left (\\frac{1}{1+i}\\right ) \\\\\n& =\\frac{Itd}{i+d}\\left [\\frac{1}{2} + \\frac{\\frac{1}{2}}{1+i}\\right ] \\\\\n& =\\frac{Itd}{i+d}\\left [\\frac{\\frac{1}{2}\\left (1+i\\right ) + \\frac{1}{2}}{1+i}\\right ] \\\\\n& =\\left (\\frac{Itd}{i+d}\\right )\\left (\\frac{1+\\frac{1}{2}i}{1+i}\\right ) \\\\\n\\end{align}", "f2a5d224db3c03c8a8cd0154e42b933b": " L = l \\, N ", "f2a61e48998d8d2ddb34913be27341ba": "e(\\cdot,\\cdot)", "f2a6700b6f30f3d5a321a195c3023ac1": "(A_i)_{i=1}^{\\infty}", "f2a670179e0f566a1501de18503eb742": "\\tbinom{c_i}i=0", "f2a677a9a6ef1d34acb9aba126ad0194": "Y_{9}^{-7}(\\theta,\\varphi)={3\\over 512}\\sqrt{13585\\over \\pi}\\cdot e^{-7i\\varphi}\\cdot\\sin^{7}\\theta\\cdot(17\\cos^{2}\\theta-1)", "f2a69ef95d3f91a74737397dd2a37c56": "\\scriptstyle R_1", "f2a74427b2f9702a4b8fe0ef70b96ccd": "\\tfrac{\\pi}{2}", "f2a76a333447e81a6a6c009a8fff910e": "r = r_0 (T - T_c) \\,", "f2a78cc1dda5c3de09fe882524c9c78d": "\\theta\\in[-1,\\infty)\\backslash\\{0\\}", "f2a7b278a0fc264d51a32567db5761d5": "\\left(U_{m+1}\\right)_{\\sigma|\\sigma'}=\\delta\\left(\\sigma_{1},\\sigma_{1}'\\right)\\cdots\\delta\\left(\\sigma_{m},\\sigma_{m}'\\right)w\\left(+1,+1,+1,\\sigma_{m}\\right).", "f2a7cdc097e906757cd707c5d653aece": "S_{\\$/b}", "f2a7ce99d974fbf0d6dbbf20da5eaf00": "\\alpha=\\tan^{-1}[\\cos\\varepsilon\\,\\tan\\lambda]", "f2a7f6d56c71e6a8c5eb9a522653c8c3": "\\arcsin (-x) = - \\arcsin x \\!", "f2a85eb17735cf314987f417545e6f1b": "g^{(n)}( \\mathbf{r}_1,t_1;\\mathbf{r}_2,t_2;\\dots;\\mathbf{r}_{2n},t_{2n})= \\frac{\\left \\langle E^*(\\mathbf{r}_1,t_1)E^*(\\mathbf{r}_2,t_2)\\cdots E^*(\\mathbf{r}_n,t_n)E(\\mathbf{r}_{n+1},t_{n+1})E(\\mathbf{r}_{n+2},t_{n+2}) \\dots E(\\mathbf{r}_{2n},t_{2n}) \\right \\rangle}{\\left [ \\left \\langle\\left | E(\\mathbf{r}_1,t_1)\\right |^2 \\right \\rangle \\left \\langle \\left |E(\\mathbf{r}_2,t_2)\\right |^2 \\right \\rangle\\cdots\\left \\langle \\left |E(\\mathbf{r}_{2n},t_{2n})\\right |^2 \\right \\rangle \\right ]^{1/2}}", "f2a85fc76c986caedfca325f15764d38": " (E-E_f)^\\alpha ", "f2a8a36d37497929a99b266edb2929fe": "\n \\operatorname{Var}[\\, \\tilde\\beta \\,| X \\,] - \\operatorname{Var}[\\, \\hat\\beta \\,| X \\,] \\geq 0\n ", "f2a8bfd98d5dab33b593e33ec364b783": "\\dfrac{\\partial f}{\\partial\\bar{z}} = 0", "f2a8dab434e2001665ccf09c3fcb6c7e": "\\hat{h}\\,", "f2a904e50a4662c0bc98251f4d033560": "T_{in} = F_{in} r \\,", "f2a91ed39f9bb228d2b96539f2a50f50": "\\frac{d^2 T}{dx^2} + \\frac{\\dot{q}_m + \\dot{q}_p}{k}=0 \\quad [1]", "f2a964a40416b6f0bec59de651e8c174": "\n\\operatorname{pmi}(x;y) \\equiv \\log\\frac{p(x,y)}{p(x)p(y)} = \\log\\frac{p(x|y)}{p(x)} = \\log\\frac{p(y|x)}{p(y)}.\n", "f2a99857b385488c577e4d294c39467f": "\\int_0^\\infty e^{-x}x^{n-1}dx = (n-1)!,", "f2a99f05ae5e7caadc8c12402c7eb6e7": "C_{vL}", "f2a9a594548f3eda53686899a77c96cb": " \\mathbf{x}^{k+1}_i = \\underset{y\\in\\mathbb R}{\\operatorname{arg\\,min}}\\; f(x^{k+1}_1,...,x^{k+1}_{i-1},y,x^k_{i+1},...,x^k_n);", "f2aa0219688e543231f4ba4263bd3296": "\\left ( \\nabla^2 \\mathbf A - \\frac{1}{c^2} \\frac{\\partial^2 \\mathbf A}{\\partial t^2} \\right ) - \\mathbf \\nabla \\left ( \\mathbf \\nabla \\cdot \\mathbf A + \\frac{1}{c^2} \\frac{\\partial \\varphi}{\\partial t} \\right ) = - \\mu_0 \\mathbf J", "f2aa573ab042061e8c77b7a336d16884": "\\beta\\ = f_6(\\psi,\\phi),\\,", "f2aa74697cc79f44c89fb8330e2fea11": "\\int_{-\\infty}^\\infty f(x) dx = L \\int_0^\\pi \\frac{f[L \\cot(\\theta)]}{\\sin^2(\\theta)} d\\theta\n\\approx \\frac{L\\pi}{N} \\sum_{n=1}^{N-1} \\frac{f[L \\cot(n\\pi/N)]}{\\sin^2(n\\pi/N)}.", "f2aa959bc530671e78a398bb4fafe397": "\\left[ \\frac{1}{2m}(\\boldsymbol{\\sigma}\\cdot(\\mathbf{p} - q \\mathbf{A}))^2 + q \\phi \\right] |\\psi\\rangle = i \\hbar \\frac{\\partial}{\\partial t} |\\psi\\rangle ", "f2aad93e48303d711e72215ebd03b92f": "\\mathbf{X}_b", "f2ab2cccf7ff337baf8409d3f4e0529e": " { div\\, (\\rho u T )} ={div\\, (k\\, grad\\, T )} \\, ", "f2ab7cbe98454ca5b14dee8362140cce": "I(\\tilde{\\nu})", "f2ab87352c7779b036c9509bc37c1bb6": "A \\oplus A \\oplus \\cdots \\oplus A = \\N", "f2aba21207969d4fc71b01c0488b4b7a": "\\frac{A}{B}", "f2abc2857b4c47454f6efa7f36fb6f72": "\\operatorname{arcoth}(z)", "f2ac039e77caea49de0263d91b81e664": "A\\ -\\ C\\ =\\ 0", "f2ac8174a42487bd6f8ed687c307abca": "\\left(\\frac{4}{3}\\right)\\pi r^3", "f2ac8b4d79ed6d3e3d15e478199da59a": "\nM_I \\ddot{\\mathbf R}_I = - \\nabla_I \\, E\\left[\\{\\psi_i\\},\\{\\mathbf R_J\\}\\right]\n", "f2acb96ba661c1e29023669399edbaf8": "\\Pr(Y_i=0)", "f2acfb990957b2edd434cd3399a46d39": "R_{a[bcd]}^{}=0,", "f2ad5b8c41fd4baf9802abc8d97f2d2d": "\\theta = \\theta _s\\cdot\\frac{L^2}{L_s^2} = L^2", "f2ad6d3766ac25a7a7e87fc590057765": "\\nu_M", "f2ad86832145a78af3279c3b460e6328": "T^{1,0}\\mathbb{C}^{n+1}", "f2adf4be426344921c09b44bff0be907": " x_{n+1} = \\left( 1 + \\frac{1}{x_n} \\right)^n\\text{ for }n=1,2,3,\\ldots,", "f2ae5eaebfbfa3e901bf89b85ca30a0e": "p(c_j|f_i) p(f_i|c_j)\\ ", "f2ae85c72c654428c174e1bb48dd4ccc": "(0,0,\\tfrac{1}{2})", "f2aec164a102db7deb2907b80652845a": "\\mathbf{f}^T L \\mathbf{f} = \\displaystyle\\sum_{i,j=1}^{l+u}W_{ij}(f_i-f_j)^2 \\approx \\int_\\mathcal{M}f(x)||\\nabla_\\mathcal{M} f(x)||^2dp(x)", "f2aef3edf14041148fddfc081bc21685": "\\mathbf{R}=R_1\\mathbf{\\hat{x}}+R_2\\mathbf{\\hat{y}}+R_3\\mathbf{\\hat{z}}\\!", "f2af5867fc726be2ed6c61d91efeb8f0": "\\big[ M + A(\\omega) \\big] \\ddot x + \\big[ B(\\omega) + B_v \\big] \\dot x + C x = F(\\omega)", "f2af71baf24473b3efac3299b9cab224": "\\mathcal P \\left\\{O_1(\\sigma_1) O_2(\\sigma_2) \\cdots O_N(\\sigma_N)\\right\\}\n \\equiv O_{p_1}(\\sigma_{p_1}) O_{p_2}(\\sigma_{p_2}) \\cdots O_{p_N}(\\sigma_{p_N}).", "f2afb8419bed9fe96c1feb88ad63bf09": " f_0(z)=\\gamma_0+\\frac{1-|\\gamma_0|^2}{\\overline {\\gamma_0}+\\frac{1}{z \\gamma_1+\\frac{z(1-|\\gamma_1|^2)}{\\overline {\\gamma_1}+\\frac{1}{z\\gamma_2+\\cdots}}}}.", "f2afc8f7b665fdc3bbfbef88295ebaef": "\\delta = \\omega_a - \\omega_c", "f2aff7dcaedd444ccf0541692335b46e": "t_{m+j,i}=-M", "f2b09264604b3d75e2ffa9056ff4d07a": "\\bar{\\boldsymbol{B}} := \\bar{\\boldsymbol{F}}\\cdot\\bar{\\boldsymbol{F}}^T=J^{-2/3}\\boldsymbol{B}", "f2b121e9b85fdad24940a871a6910889": "\\begin{align}\n x_k(\\zeta) &{}= \\Re \\left\\{ \\int_{0}^{\\zeta} \\varphi_{k}(z) \\, dz \\right\\} + c_k , \\qquad k=1,2,3 \\\\\n \\varphi_1 &{}= f(1-g^2)/2 \\\\\n \\varphi_2 &{}= \\bold{i} f(1+g^2)/2 \\\\\n \\varphi_3 &{}= fg\n\\end{align}", "f2b13e994768ef08b030641272787c0b": "\\begin{align}\nb_0 &= \\frac{1}{a_0}\\\\\nb_n &= -\\frac{1}{a_0} \\sum_{i=1}^n a_i b_{n-i}\\qquad \\text{for } n \\ge 1.\n\\end{align}", "f2b153a393032304d4784e8169397c97": "p=\\tfrac{\\beta}{\\alpha + \\beta}", "f2b1b2e134ea0611306076f873aeceb9": "\\psi\\leftrightarrow\\psi^\\ast", "f2b20d63a3454e3b1987c47ec8a7451f": "\\displaystyle{R_k}", "f2b2365c89536857b3a326a6ae69079a": "A-B", "f2b24f50f0a3dcdd9b63e66588751792": "{d (\\rho u f ) \\over d t} + div(\\rho f u) = div(R_f. grad f) ", "f2b25c51f170be5e425948dfc6775c13": " J = E + M \\,\\!", "f2b27e4ea7b5b1e1eaa919370d598f00": "\\operatorname{pf}(A_1\\oplus A_2) =\\operatorname{pf}(A_1)\\operatorname{pf}(A_2).", "f2b2addf5ff64b15428e3b050177d1ba": "a \\in [0,1]", "f2b33dcb47e3d45399457e0e2d525dda": "\\mathrm{R{\\cdot} + X_2 \\longrightarrow R{-}X + X{\\cdot}}", "f2b34f09a1baa3eb09c122734106e6cb": "=\\frac {d}{dx}\\left[p(x)\\left(u \\frac {dv}{dx}- v \\frac {du}{dx} \\right)\\right], ", "f2b3565d06e9450df56df499a5313edd": "I_\\mathrm{net} = \\sum_{i=1}^N I_i \\,\\!", "f2b433f0c02f9ad9be4e650df15f90f7": "WR", "f2b439dedc48976897ca3c70bbc961bc": "\\mu H_u H_d", "f2b443dc3f15ab7dd8d2d5373444768b": "f(t) = \\frac{1}{\\sqrt{\\nu}\\, B \\left (\\frac{1}{2}, \\frac{\\nu}{2}\\right )} \\left(1+\\frac{t^2}{\\nu} \\right)^{-\\frac{\\nu+1}{2}}\\!,", "f2b4b3fbb3f06c42ca46ebff2709df83": "\\|\\alpha \\mathbf{A}\\|=|\\alpha| \\|\\mathbf{A}\\|", "f2b4c0870e4ae0af3672f1db3a12e9e4": "y \\equiv_c x", "f2b4ed603689cb634ac2d291a22a0318": "\\delta W =\\mathsf{W}\\cdot\\check{\\mathsf{T}}\\delta t.", "f2b4f586da77616d114782424454df6b": " H_{2n} - H_n \\sim\n\\log 2 - \\frac{1}{4n} + \\frac{1}{16n^2} - \\frac{1}{128n^4} + \\frac{1}{256n^6} - \\frac{17}{4096n^8}\n+ \\cdots,", "f2b4fd93df2481ce819717bcd5ff8620": " \\text{Relative difference}(x, y) = \\frac{\\text{Absolute difference}}{|f(x,y)|} = \\frac{|\\Delta|}{|f(x,y)|} = \\left |\\frac{x - y}{f(x,y)} \\right |.", "f2b4fe34dd462c44f9e4c6c89220f174": "E_n^{(2)}=\\frac{m a^2}{2 \\pi^2 \\hbar^2} \\sum_k \\left|\\int e^{-i k \\phi} \\cos \\phi e^{i n \\phi}\\right|^2/(n^2-k^2)", "f2b580187e8b6858c81acba30bdf5ff1": "\\gamma = C_p/C_v", "f2b581710ce5d5a944f19701e3525107": "\n\\varphi = \\frac{1}{4 \\pi \\varepsilon_0} \\frac{q}{\\left| \\mathbf{r} - \\mathbf{r}_q(t_{ret}) \\right|-\\frac{\\mathbf{v}_q(t_{ret})}{c} \\cdot (\\mathbf{r} - \\mathbf{r}_q(t_{ret}))}\n", "f2b59da6b57a9826580b207c4edf76d2": "\\vec{t_1}\\langle s\\rangle =\\vec{t_2}\\langle s'\\rangle ", "f2b5b84c46697b6420bc77cf885c6b5f": "|\\det(N)|\\le n^{n/2}.", "f2b5bba4143087a381be7fc9f4c9ce86": "X \\to w(\\alpha, \\beta)", "f2b6793b32cb7c169cde06588fa2d033": "D=\\partial\\theta/\\partial\\xi", "f2b6a40b3d686d38cca7482170c7f28c": "\\mathcal{D}_{n,k}", "f2b746db923bddfdd70f5fb879432be8": "V_\\mathbf{k}", "f2b7ada65efec6ce012a75519b2f5089": " \\begin{align}\\nabla \\times (\\mathbf{A} \\times \\mathbf{B}) &= \\mathbf{A} (\\nabla \\cdot \\mathbf{B}) - \\mathbf{B} (\\nabla \\cdot \\mathbf{A}) + (\\mathbf{B} \\cdot \\nabla) \\mathbf{A} - (\\mathbf{A} \\cdot \\nabla) \\mathbf{B} \\\\\n&= (\\nabla \\cdot \\mathbf{B} + \\mathbf{B} \\cdot \\nabla)\\mathbf{A} -(\\nabla \\cdot \\mathbf{A} + \\mathbf{A} \\cdot \\nabla )\\mathbf{B} \\\\\n&= \\nabla \\cdot (\\mathbf{B} \\mathbf{A}^\\mathrm{T}) - \\nabla \\cdot (\\mathbf{A} \\mathbf{B}^\\mathrm{T}) \\\\\n&= \\nabla \\cdot (\\mathbf{B} \\mathbf{A}^\\mathrm{T} - \\mathbf{A} \\mathbf{B}^\\mathrm{T} ) \\ . \n\\end{align}\n", "f2b81c0ced6b6a03dcb9c2bf2d23acd1": "\\sum_{l=\\ldots,n-2,n} (2l+1) = {(n+1)(n+2)\\over 2} ~,", "f2b83e62f6802881412a3dbf7b888df1": "y_{(1)}''(t) = p(t)y_{(1)}'(t)+q(t)y_{(1)}(t)+r(t),\\quad y_{(1)}(t_0) = y_0, \\quad y_{(1)}'(t_0) = 0, ", "f2b840dff6ef8cf7882c706d0e905248": "\\Alpha \\, \\alpha \\,", "f2b857a37183f130d59968f42a4fec6e": "\\operatorname{E}(T)", "f2b89243e40c6d5be2b4516b41f6a01f": "p \\mid d_K\\,\\!,", "f2b8a034e9973dc6c0e44fb502372dc0": "C(\\alpha,\\sigma)", "f2b8c6cbec8ede04603924f892b4a7ae": "\\theta_n(\\xi)", "f2b91b0e9d06c12f333f2a30a1882add": "\\circ\\,", "f2b934b8dc8f334e82cccbdc49572acc": "p_{1\\infty} \\leftarrow x^3+2x^2-x-1, M_{1\\infty}\\leftarrow \\frac{x+3}{x+2}", "f2b9bb5f609a7379fcb377b93ffeb42e": " w W_J ", "f2b9ff70a68e0a4ae301d98f953d6a6d": "\\, r_i", "f2ba29a39b59f9dada29228faf1d9162": "v_l", "f2ba581eb50ebe2c31aae500f2557200": "\\beta(G_r^\\pm)=G_r^\\mp", "f2ba6e7cee97ba18f38722959a2f4f0d": " f(x)=\\max(x) ", "f2ba7103efeb39a39b5868f5e3901a47": "\\simeq", "f2baab5ccc7773792bbf3fcaef68cc55": "\\ln(123.456)\\!", "f2bb2df4cf9c518e64b5209feffd8d94": "\\forall b \\in B", "f2bb640ec2aeadc7e57a51afba51d75d": "\\mathbf{r}_7 = (a/4)(\\hat{x} + 3\\hat{y} + 3\\hat{z})", "f2bb72752a91dff34c68ce84edadebc5": "Q_0 = v\\, h,", "f2bb73a0d21db34ad373223e94dc8679": "P(y) = \\sum_{j=1}^r P(x_j) = 1", "f2bb95daaefac7dfe6391d68c3d19d08": "\\Theta^i(\\mathbf e\\, g)=\\sum_j g_j^i \\Theta^j(\\mathbf e).", "f2bc6a6c70e276a35225584c6dc39866": "l ", "f2bc78e459496b704341eedf62ee2b72": "\\int \\cos^n (x) dx\\!", "f2bc8b94528252f48ccd3617abdc6649": "\\frac{x^2+y^2}{a^2}+\\frac{z^2}{c^2}=1.", "f2bca2d0ae9621ae1f2b45ce9a71287f": "R_{\\rho\\rho}=8\\pi T_{\\rho\\rho}", "f2bce0ab0e987cf6e2cdc004dc1b29bc": "\\mathbf{A} = \\begin{bmatrix} 1 - 2 a^2 & - 2 a b & - 2 a c \\\\ - 2 a b & 1 - 2 b^2 & - 2 b c \\\\ - 2 a c & - 2 b c & 1 - 2c^2 \\end{bmatrix}", "f2bd3c232880ab0c8aa9df0aa29065a0": "(P_1, P_2, \\ldots, P_n)", "f2bd41ab6270b7ec12b6aade37cc3d35": "\n\\begin{align}\nS_{12} &= R_2^2 E_{12} - e^2a^2\\cos\\alpha_0 \\sin\\alpha_0 \\times\\\\\n&\\quad \\int_{\\sigma_1}^{\\sigma_2}\n\\frac{t(e'^2) - t(k^2\\sin^2\\sigma)}{e'^2-k^2\\sin^2\\sigma}\n\\frac{\\sin\\sigma}2 \\,d\\sigma,\n\\end{align}\n", "f2bd4e8534d2bc860adc51abf3c4b38a": "y^2=x^3+Ax+B", "f2bd5617a3e36b1a253693c5882d0aa4": "s = \\alpha_n", "f2bd6e1301471e3bb31c88bc9b1c7899": "q_{\\text{OUT}} = C_S V_{\\text{OUT}}.\\ ", "f2bddeb52de436812f1ba7aba5743fb9": "p_1,\\ldots,p_n", "f2bdf613bf1bc5fcf209241b4cf1796c": "\\ \\phi", "f2be1a0b02f7785adac495084f9e68cd": "\\frac{1}{k_{eq}} = \\frac{1}{k_1} + \\frac{1}{k_2}. \\,", "f2be4ef54a29d3064d5e253f3cdbdbd3": "\\geq\\kappa^{+}>\\kappa\\,", "f2bebe1fe87f671220258832f5494e4c": "\n \\dot{\\lambda}\\,\\dot{f} = 0 \\,.\n ", "f2bf620e5cdf14076e7dad983593d2c1": "V[n], V[p], V[q], V[m] ", "f2bfcd942190522317ac80f6addbac85": "\\chi^2_k(p)", "f2c0214932e21168d85ec4137e4fc0c4": "p_c=p_{\\text{non-wetting phase}}-p_{\\text{wetting phase}}", "f2c08126d7f2d13d9fa6169b5c161f67": "\\omega(\\mathbf e_q) = (\\mathbf e_p^{-1}\\mathbf e_q)^{-1}d(\\mathbf e_p^{-1}\\mathbf e_q)+(\\mathbf e_p^{-1}\\mathbf e_q)^{-1}\\omega(\\mathbf e_p)(\\mathbf e_p^{-1}\\mathbf e_q).", "f2c086dab74e0bc63695fa84d2f3ac16": "m_{evactube}=\\frac{\\rho_{gas}\\cdot m_{fullliquid}-\\rho_{liquid} \\cdot m_{fullgas}}{\\rho_{liquid}-\\rho_{gas}}", "f2c0b2ac5db3f40405cc78ff2014a78b": "|D^\\alpha g(x)| \\le CR^kk^{\\sigma k}", "f2c13b1821a67ecbaa04757d82ef4e35": "S_n(X)=(X\\ominus nB)-(X\\ominus nB)\\circ B", "f2c1482557794045a76c511d085741bb": "\\,\\eta_k (k=1,\\ldots,K)", "f2c157202779c9a3de64999d015a2095": "f(u)", "f2c16aa22f5b1df55df1d9dad166b53a": "-\\tfrac{H^{(\\lambda)}(s_\\lambda)}{P(s_\\lambda)}", "f2c172515162d586b80db2dc3374677f": "I : R = I", "f2c173e6f39128d9a910acbae2a30998": "\\textstyle{\\scriptstyle{\\rm P/E\\ ratio}} = \\frac{{\\rm Price}}{{\\rm Rent} - {\\rm Expenses}}.", "f2c19013ade978fe905e52a973e97f9e": "\\mathcal{L}_X g_{ab}=0", "f2c1d4e3da0f9c74761af48cc7694fdf": " d = 1.17 \\sqrt {H} ", "f2c1db46c881fdd9bdace23276d88f9f": "T(P \\wedge Q)\\ \\Leftrightarrow\\ (T(P) \\wedge T(Q))", "f2c1f1a3db7eab1da4789ebaf595d97b": "-\\frac{d[A]}{dt} = k[A]", "f2c22a820b8b5b12bc093e1124767772": "d(x,y)=d(y,x)", "f2c2322a9f3953879258c17aef7ffb30": " \\sum_i v_i(a)", "f2c293664722e214918899514774c694": "\\Omega(x)=S(x)\\Xi(x)\\bmod x^6=\\alpha^{7}+\\alpha^{0}x.", "f2c2e2db4d02af7d7fc67d125ae00eb8": "n = \\int f(E)\\, g(E)\\,dE", "f2c2eb7f272a4648ad59ef618bfb5113": "\\tilde{V}_{N}", "f2c33bc189b3c2d8d949a67311fe8cea": "|\\Lambda|^{-s}_M", "f2c375e85e11cfb0a508613089dda568": "\\boldsymbol{\\nabla}\\times\\mathbf{F},", "f2c38cc894f6e4ca01e8b2555881f37c": " \\frac{\\Delta \\hat{H}}{\\Delta \\hat{P}} ", "f2c38f03167e8e00a5d1524d2ab43dc9": "\\tilde{E}_6", "f2c3c8c833438433706a6ef7cc06c6f2": "\\alpha_o = \\frac{c_o}{C_o} = \\alpha_a + \\alpha_b \\left(1 - \\alpha_a\\right)", "f2c414556e6ddf87e9216bd24505c830": "| 0 1 \\rangle", "f2c417dddb006938955ea6eab15f9466": " \\frac{x(A)}{1-x(A)}", "f2c425b2097aab858140efb643c8e9ba": "E^k=\\Omega^{k,0}\\oplus\\Omega^{k-1,1}\\oplus\\dotsb\\oplus\\Omega^{1,k-1}\\oplus\\Omega^{0,k}=\\bigoplus_{p+q=k}\\Omega^{p,q}.", "f2c45a25c8c70bb0a5e8b7427180330e": "\\Delta G_{\\rm hydr}", "f2c4b97b15f88adc3e6eca023a1b85bc": "N_R = {VL{\\rho}\\over {\\mu}}\\,\\!", "f2c4bee72041b0ade9c50f65858cd6d2": " \\Pr\\{X=x\\}=\\Pr\\{X_1=x_1,X_2=x_2,\\ldots,X_n=x_n\\}.", "f2c4dbf9a9d41adc94c73d65d5037f48": "H(X_1, ..., X_n) \\leq H(X_1) + ... + H(X_n)", "f2c5376ed2aa429fd209037fd937df1b": "E(\\mathbf{X}) = \\log \\prod_{f_j} f_j(x_j).", "f2c548fe7c3d2fd3e1471f729e2b42c3": "R_f = \\frac{1}{n h_f W_f \\left ( t_f + 2\\eta_f L_f \\right)}", "f2c5958628f022a816e34887428920f8": " \\chi(4,8) = q_4 + q_3 q_4 - q_4", "f2c5a15936f75ed8e28c5c8e666c8864": "n=500", "f2c5ae914968bea7af666003079ac537": "e^{i(\\beta z - \\omega t)}", "f2c5ef3367ca4830d02c794686791e8e": "\\frac{\\log\\left(\\frac{1+\\sqrt[3]{73-6\\sqrt{87}}+\\sqrt[3]{73+6\\sqrt{87}}}{3}\\right)}\n{\\log(2)}", "f2c619c45d31869e7e3e28bc8a8193be": "\\nabla\\times\\bold{B} = \\bold{0} ", "f2c62123832c3c013b2243c53e2bd304": " c_i = \\prod_{j=1, i\\neq j}^D \\frac{\\lambda_j}{\\lambda_j - \\lambda_i} ", "f2c62cb3484f908012fae4edda0bc18f": "\\int_0^1\\frac{\\sin(1/x)}x\\,dx", "f2c6504617e5d5db9500a4a42172c4ca": "S(x)=\\sum_{i=0}^{d-2}s_{c+i}x^i", "f2c6716ddae3a4dddbce6223fd1c5bd7": "f(z) = \\frac{a_{-m}}{z^m} + \\frac{a_{-m+1}}{z^{m-1}} + \\cdots + a_0 + a_1 z + \\cdots", "f2c6c92d4c71b847b6062b82b5b51593": "\\displaystyle{g_{\\overline{z}}=T^*h,}", "f2c6dad220329a6a4c6f6b3aa3594be5": "\\alpha/K", "f2c7034bafc2107532d31a28a75e1f85": "h_B^{(1)}(z)", "f2c725b9744d5fbcebe9901de0f5a9c4": "\\lambda(t)=\\lambda_0 + \\frac{k(N_{th} - N(t))}{N_{th}}", "f2c73db9f2c847f75d1b861cd72a0d0b": " w \\,", "f2c7421279e651fe3a13e70b7fe7d421": "R_1'", "f2c75c595374e88de497ff9c09e0c091": "RSS = \\sum_{i=1}^n (\\epsilon_i)^2 = \\sum_{i=1}^n (y_i - (\\alpha + \\beta x_i))^2, ", "f2c76ac6928b04c8ba5a1f53e86ed9d2": " s = BQ + CQ^2 ", "f2c7c02da1913f5d74d6c0f4b97c03d6": "\\varepsilon^x", "f2c80ceac02b0f7b46cf7adbec4e35e8": " U^*", "f2c84e42bc174a4a62cc8b0804db3e93": "E^{*} \\xrightarrow{i^{*}} H^{*} \\cong H \\xrightarrow{i} E.", "f2c8654f0f81ec1489ab8481ba9afd10": "1926 = [41, 40]_{46}", "f2c8a776f04ad3e8b869227f2a867669": "i = \\frac{\\mathrm{d}h}{\\mathrm{d}l} = \\frac{h_2 - h_1}{\\mathrm{length}}", "f2c8ca87df00f982f6a522da064f3b7a": "u=\\gcd(k,j)", "f2c8e35a3dd520ac1330ae1212e61bb8": "\\sqrt{1+2\\sqrt{1+3 \\sqrt{1+\\cdots}}} \\,", "f2c8fd47d5a080d24791f46bc185c706": "\\omega_n", "f2c908692db497dd785054c43cf683ed": "f(z)- R/(z-a)", "f2c9726c79aec6344b40587be1c71d07": "A_1,B_1,A_3,B_3", "f2c986ed65e474c89b7d7ad58d4b2f0e": "10 \\uparrow ^{10 \\uparrow ^{10^{10}} 10} 10 \\!", "f2c9e3abd47d63af1b10ab12304f4a36": "\n\\begin{align}\n\\theta &=k/\\gamma, \\\\\n\\mu & =x_0, \\\\\n\\sigma &=\\sqrt{2k_B T/\\gamma},\n\\end{align}\n", "f2c9f1d24fdee58df032ace7ed30f8af": "\\mathbb{Z}/p^m\\mathbb{Z}", "f2ca161f60e0b5381ae8948a3421a33b": "h^{1,1} = h^{1,1}_{\\pm}", "f2ca36b3fd879bb2df87e5b86b87ae00": "G_{\\mu \\nu} = R_{\\mu \\nu} - {1 \\over 2}R g_{\\mu \\nu},", "f2ca547c575faa2560f6b8358ff7253b": "|\\psi\\rangle = c_1\\left|\\psi_{j = +\\frac{\\hbar}{2}}\\right\\rangle + c_2\\left|\\psi_{j = -\\frac{\\hbar}{2}}\\right\\rangle", "f2cb0ccfa973d626958911bc0c7ac52b": "\\mbox{QP} = \\bigcup_{c \\in \\mathbb{N}} \\mbox{DTIME}(2^{(\\log n)^c})", "f2cb470e7fcac4d965d6022a5569e6b9": "\n \\mathbf{v}\\cdot\\mathbf{b}_i = v^k~\\mathbf{b}_k\\cdot\\mathbf{b}_i = g_{ki}~v^k ", "f2cb5087c1d1928defb6797c4e69e7e9": "\\xi\\in \\mathfrak h", "f2cb5bf2bce2371cf065251c23310f76": "= - \\frac{G}{T}", "f2cb5c055aadab56ac0de2c0c2458db4": " \\begin{align}\n & \\big(dX,\\,dY,\\,dZ\\big) \\\\[6pt]\n= & \\big(-\\sin \\phi \\cos \\lambda ,\\,-\\sin \\phi \\sin \\lambda ,\\,\\cos \\phi \\big) \\left(M(\\phi)+h\\right) \\, d\\phi \\\\[6pt]\n&{}+ \\big(-\\sin \\lambda ,\\,\\cos \\lambda ,\\,0 \\big) \\left( N(\\phi) +h\\right) \\cos \\phi \\, d\\lambda \\\\[6pt]\n&{}+ \\big(\\cos \\lambda \\cos \\phi ,\\,\\cos \\phi \\sin \\lambda ,\\,\\sin \\phi \\big) \\, dh ,\n \\end{align}\n", "f2cb8563f59fbcafad9fdc674c3d813a": "p=\\left\\lfloor\\frac{7 + \\sqrt{49 - 24 \\chi}}{2}\\right\\rfloor,", "f2cc82a123691987c00a0b6ed2368575": "\nw = 2 \\sqrt{r_{s} \\left( r - r_{s} \\right)}.\n", "f2cc86fd5fd48b0494a1df090c81d878": "\\oint_\\gamma f(z)\\,dz = 0", "f2cc927f0e290124afc814423f31298c": "{\\mathbf{}}A(t), C(t)", "f2cd2466543377aa3ebbc197e2a53636": "\\definecolor{orange}{RGB}{255,165,0}\\pagecolor{orange}e^{i \\pi} + 1 = 0", "f2cd3d0370ea762f38689573f88966e5": "p+q+r\\geq (\\frac b c+\\frac c b)x+(\\frac a c+\\frac c a)y+(\\frac a b+\\frac b a)z", "f2cd4b1aecde0cd01595d688078c8ba2": "J_0(z)\\approx\\sqrt{\\frac{-2}{\\pi z}}\\cos \\left(z+\\frac{\\pi}{4}\\right)", "f2cd5c883e07d4eb10cb9fb4fd5a9309": "\\frac{d^2}{dy^2}f_y(y) + k_y^2 f_y(y)=0", "f2cd6e9e962f8a885986c93a3bc156cd": "\\ln(k) = \\ln(A) - \\frac{E_a}{R}\\left(\\frac{1}{T}\\right)", "f2cd730335bcfcb1876fa8e2636f4d8b": "R_{TE}+R_r^'\\gg{R_{TE}+X_r^'}", "f2cde2cc19db6b4c3e802a250aa9f316": "F(t) = Z_G(t+1,t^2+1,t^3+1) = \\frac{(t+1)^3 + 3 (t+1) (t^2+1) + 2 (t^3+1)}{6},", "f2cdffcd501dc742e5fb89152b50a7a4": "\\hat{D}(\\alpha)\\hat{S}(r)|0\\rangle=|\\alpha,r\\rangle", "f2ce6ba3d5684e0bbbfe6d021712a623": "(x',y') = (m x, m y).\\,", "f2ce7665f7cb8b2b6874f34eb67c3a8c": "C = \\frac{A_{settl}}{A_m}", "f2ce792e6d6c793ad8ff6cb64aa62796": " C_2 \\subset C_1 ", "f2ce9edfdca5d3b6d652a3b70838532c": "\\scriptstyle I'_2", "f2cea148d2490a0cec439fc620134332": "\\lim_{x \\to a} \\sin x = \\sin a", "f2cea7d0cad4318e0ad7e7f74e1245a4": "\\phi \\rightarrow \\psi", "f2cf2188d1b81eb5aef198f8c3efed25": "d(x_m, x_n) ", "f2cf3e09f27c4bbbee59e88d6432d3cf": "H(x_1,x_2,\\dots,x_n)", "f2cf54b5af62e56d70f7bea3f357e4ce": "l^k(z^n,i)=\\left(\nz_{i-k},\\ldots,z_{i-1} \\right)", "f2cf66a56c1f494d1a7665a7233adf4d": "V_{exh}=V_{exh}(m)\\,", "f2cfe6b8b0a7c15b11fee82cae9e4d06": "\ndx_t = (a x_t + b u_t) dt + \\sigma dw_t,\n", "f2cff0fd297f9f2fcde534e91493b8ad": "\\text{refresh cycle interval} = \\text{refresh time}\\, / \\,\\text{number of rows} \\,", "f2d0091d5c820fce9be2b2116bd8acb2": "\\frac{dh}{dt} = \\alpha_h(1 - h) - \\beta_h h", "f2d02eaf32cb7a351989198531c0d12a": "\\ell_2", "f2d136713e0503efb0d11902f6c5fbdb": "P\\cdot T", "f2d13eab8e9ea14d716aa794e5da7c31": "\\omega _t = \\sqrt{{\\frac{q\\alpha}{m}}}", "f2d17a52da29b70845dc261f95ae3c9c": "E_1^{p,q} = C = \\bigoplus_{p,q} H^q(X_p, X_{p-1}),", "f2d1dfb45b1916b6d1bddc47f2e64bab": "A + B K", "f2d21d259524fff3213da4c994337a24": "\\phi(x) = \n\\begin{cases} \n1 & \\text{if } x \\in R \\\\\n0 & \\text{if } x \\in A\n\\end{cases}", "f2d23e38a0d2469e7de56ce7bc6470e5": "\\boldsymbol \\beta", "f2d244a628a4ad9960ecfc7a0d45b18a": " x\\Psi=0", "f2d24eb83c6e36e0ad80129128409830": "f_0(x) \\ ", "f2d2614608e6f24a67ab483896ab0dfd": " \\langle N\\rangle = k_B T \\frac{1}{\\mathcal Z} \\left(\\frac{\\partial \\mathcal Z}{\\partial \\mu}\\right)_{V,T} = \\frac{1}{\\exp((\\epsilon-\\mu)/k_B T)-1} ", "f2d28de5d9685795d95fdc811bd44a27": "\\rho g", "f2d343ef299bff6645c037243e12b237": "K\\supset\\mathbb{Q}(\\zeta_l)", "f2d39636faf938fa3caa77c5b715b796": "\\langle\\mu|\\xi\\rangle^2 = \\hbox{length}\\left(Q(\\bar{\\mu},\\xi)\\right)^2", "f2d3baf6a5fdd20b401519bd48f04c4d": "S(x) = x^{-1}. ", "f2d41276e74c9a9c92296a5470823d35": " A_{ij} = \\int_\\Omega\\nabla\\varphi_i\\cdot\\nabla\\varphi_j\\, dx.", "f2d4aa0cc49db3244738ec445d562abc": "\\scriptstyle{p \\choose n}", "f2d4d5885e4dd758eef4a616f6d8ea54": "\\pi_B", "f2d50a463b093f0732740ed24adc1354": "\\begin{align}\n &m = V - C \\\\\n &(R, G, B) = (R_1 + m, G_1 + m, B_1 + m)\n\\end{align}", "f2d5112b92e753d7ff2473f498a3fd7f": "\\mathbf{D} = \\epsilon_0 \\mathbf{E},\\quad \\mathbf{B} = \\mu_0 \\mathbf{H}. ", "f2d54ed777e46033ef002ee049140ccf": "\\mathfrak{S}", "f2d571f01dfb6166316ab8da501dfb4c": "(V,E,ctrl,prnt,link) : \\langle k,X \\rangle \\to \\langle m,Y \\rangle,", "f2d575dc5e1977462b285204a6282297": " pV^{\\gamma} = \\mbox{constant} \\,", "f2d577d17695a8ebe3926d230b17ff6f": "[D,\\overline{S}]=\\frac{1}{2}\\overline{S}", "f2d5d5eed6cbc4eb4ca361a1c6c0ac7b": " L_{n} (\\beta_{n}) = \\prod_{t} \\frac{e^{\\beta_{n}X_{nit}}} {\\sum_{j} e^{\\beta_{n}X_{njt}}} ", "f2d65ff07e32252835eddaba7b17e3b1": " H_i(X,\\mathbb{Q}) \\times H_{n-i}(X,\\mathbb{Q}) \\to H_0(X,\\mathbb{Q}) \\cong \\mathbb{Q}.", "f2d6b51d1f05ad3ba6ac2740af2fada8": " \\frac { \\mbox{density of body}} { \\mbox {density of water} } = \\frac { \\mbox{weight of body}} { \\mbox{weight of body} - \\mbox{weight of immersed body}}\\,", "f2d6c5f9b59bae5bc8a1ff91f8ae7ded": "X^{*}", "f2d709cc308b9607cdfa94c567aecff0": "\\frac{|z-a|}{|z|} = const.", "f2d78728576a7d573b7f04ddd08a7e87": "\\begin{align}\n{n \\choose k} p^k q^{n-k} & = \\frac{n!}{k!(n-k)!} p^k q^{n-k} \\\\\n& \\simeq \\frac{n^n e^{-n}\\sqrt{2\\pi n} }{\\left (k^ke^{-k}\\sqrt{2\\pi k} \\right ) \\left ((n-k)^{n-k}e^{-(n-k)}\\sqrt{2\\pi (n-k)} \\right )} p^k q^{n-k}\\\\\n& = \\left (\\frac{\\sqrt{2\\pi n} }{\\sqrt{2\\pi k} \\sqrt{2\\pi (n-k)} }\\right ) \\cdot \\left (\\frac{e^{-n}}{e^{-k}e^{-(n-k)} }\\right) \\cdot \\left (\\frac{n^n }{k^k(n-k)^{n-k}} p^k q^{n-k}\\right )\\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)}}\\cdot 1 \\cdot \\left (n^n\\left(\\frac{p}{k}\\right)^k{\\left(\\frac{q }{n-k }\\right)}^{(n- k)}\\right ) \\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)}} \\left ( n^{n-k} n^k {\\left(\\frac{p}{k}\\right)}^k {\\left(\\frac{q}{n-k}\\right)}^{(n-k)}\\right )\\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)}} \\left ( \\left(\\frac{np }{k }\\right)^k \\left(\\frac{nq }{n-k }\\right)^{(n- k)} \\right)\\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)}} \\left ( \\frac{k}{np}\\right)^{-k} \\left(\\frac{n-k}{nq }\\right)^{-(n- k)} \\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)}} \\left ( 1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} && x :=\\frac{k-np}{\\sqrt{npq}} \\\\\n& = \\sqrt{\\frac{n}{2\\pi k(n-k)} \\frac{n^{-2}}{n^{-2}} } \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} \\\\\n& = \\sqrt{\\frac{n^{-1}}{2\\pi k(n-k) n^{-2}}} \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} \\\\\n& = \\sqrt{\\frac{n^{-1}}{2\\pi \\frac{k}{n}\\frac{(n-k)}{n}}} \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)}\\\\\n& = \\sqrt{\\frac{n^{-1}}{2\\pi \\frac{k}{n}\\left(1-\\frac{k}{n}\\right)}} \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} \\\\\n&\\simeq \\sqrt{\\frac{n^{-1}}{2\\pi p(1-p)}} \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} &&\\text{as } k\\to np \\text{ we get } \\tfrac{k}{n} \\to p\\\\\n& =\\sqrt{\\frac{1}{2\\pi npq}} \\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)} && p+q=1\\\\\n& = \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ \\ln \\left [\\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k} \\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n - k)}\\right ] \\right \\}&& e^{\\ln(y)} = y \\\\\n& = \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ \\ln \\left[\\left(1+x\\sqrt{\\frac{q}{np}}\\right)^{-k}\\right ]+\\ln\\left [\\left(1-x\\sqrt{\\frac{p}{nq}}\\right)^{-(n-k)}\\right ] \\right \\} \\\\\n& = \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -k\\ln\\left [1+x\\sqrt{\\frac{q}{np}} \\right ] -(n-k)\\ln\\left [1-x\\sqrt{\\frac{p}{nq}}\\right ] \\right \\} \\\\\n& = \\frac{1}{\\sqrt{2\\pi npq}} \\exp \\left \\{ -\\left(np+x\\sqrt{npq}\\right) \\ln\\left [1+x\\sqrt{\\frac{q}{np}} \\right ] - \\left(nq-x\\sqrt{npq}\\right) \\ln\\left [1-x\\sqrt{\\frac{p}{nq}}\\right ] \\right \\} \n\\end{align}", "f2d7989eed0d4164e8022ba68404d955": "N_v", "f2d7be50c269df871f9e69913dcdf7a5": " \\mathbf{j} = \\frac{2e}{m} \\mathrm{Re} \\left\\{ \\psi^* \\left(-i\\hbar\\nabla - 2e \\mathbf{A} \\right) \\psi \\right\\} ", "f2d896b8892017c0d7e857443826bdf3": "\\color{Orange}\\text{Orange}", "f2d8ad6c8eee7080f25f3441c9eaeba2": "V_{t}", "f2d92d23c3878533d65867e4f646ca38": "\\kappa/\\mu", "f2d97a145f8c81900370662d458d9242": " f_X(\\mathbf{x}|\\boldsymbol \\theta) = h(\\mathbf{x}) \\exp\\Big(\\boldsymbol\\eta({\\boldsymbol \\theta}) \\cdot \\mathbf{T}(\\mathbf{x}) - A({\\boldsymbol \\eta})\\Big)", "f2d97f7e57fe97e0537697656ff636e5": " \\Delta, \\mathbf{m} ", "f2d9d05e197db54f7cff9d34fbbbaab8": "\\phi=\\phi_A dz^A + \\phi_adc^a", "f2da27ca7c21fe4338f0976c66115bfb": "0\\le r\\le c", "f2da45addb35bf494bdbae59fb1198da": "K_m(x,y) = \\sum_{k=0}^{m-1} \\psi_k(x) \\psi_k(y)", "f2da4ca1b046da32d73b4ecc49d58680": "M_{1}", "f2da530439ce4abfc96e106a74bfbb8f": "D_n = \\frac{2}{L} \\int_0^L f(x) \\sin \\frac{n\\pi x}{L} \\, dx.", "f2da8c8052a38e7bbe1d846374f6d3f0": "X = \\begin{bmatrix}A & 0 \\\\ 0 & D\\end{bmatrix}", "f2da9e71b3be970fcacab3a0fc015d9f": "w(\\mathbf{x},\\mathbf{y})=\\pi(\\mathbf{x})", "f2dabb51a5581eefa19b644bf20760b9": "a > a_c", "f2dac50b43417fe75991b2d24d92e5df": " 8\\pi (T_{bd} - T_{ac} \\eta^{ac} \\eta_{bd}/2) \\, = R_{bd} ", "f2dac9813aca1349ca5101ee77f7e392": "\\mathbf{L} = \\mathbf{r}\\times(m\\mathbf{v}) = (mr^2)\\boldsymbol\\omega = I\\omega\\bold e,", "f2dad44f4883b9ec1ed80b13845ddabe": "b\\in e_1", "f2db30892b4707a9f3ca8c58b1210e80": "\n p(\\xi) = p(\\xi(\\epsilon_{kk})) = p\\left(\\cfrac{\\rho}{\\rho_0}-1\\right) ~;~~ \\xi := \\cfrac{\\rho}{\\rho_0}-1\n ", "f2db3ea6ecf4eaa23e1767ba15586dae": "N_{2}", "f2dc97b0202919d788a9aa1830fba2e8": "k = \\sgn(x) \\cdot \\left\\lfloor \\frac{\\left| x \\right|}{\\Delta}+\\frac1{2}\\right\\rfloor", "f2dd3d2c058b55a9b17dc8b78e662f55": "d^2t_2-d^2t_1=0=\\bigg(\\frac{1}{v_2}\\frac{dx_2}{ds_2}-\\frac{1}{v_1}\\frac{dx_1}{ds_1}\\bigg)d^2x", "f2dd8b9acdb2cb637ba771e6a88dcf2e": "x_{m+i}\\,\\!", "f2ddc568ba566ffb3aba7946d43b67a4": "\\hat{f}(k)", "f2ddfd8986559b05a37636b9fb2f83e4": " \\sqrt 5", "f2de0ef3049b1600e8e64b531728b9fa": "E_K=\\frac{3}{2} k_B T", "f2de80cfe63c17f9f9c0abfe7e2cf692": "\\tilde{n} = c \\sqrt{\\mu \\tilde{\\epsilon}}", "f2de8156610233f9b941f8bda00abf89": "\\ell\\le L_\\rho(\\gamma_y)", "f2de85372312bd8df3bd849a764fb1b7": "\\frac{\\partial \\phi}{\\partial t} = \\frac{q}{\\hbar}V = \\frac{2\\pi}{\\Phi_0}V, \\ ", "f2dec16eec0eb1307be0d84c23b93fdf": "P \\succeq Q", "f2decaf56296898f97d9fdaeb152f701": "\n\\begin{array}{lll}\n\\text{(CT1)}\\quad& \\cos b\\,\\cos C=\\cot a\\,\\sin b - \\cot A \\,\\sin C ,\\qquad&(aCbA)\\\\[0ex]\n\\text{(CT2)}& \\cos b\\,\\cos A=\\cot c\\,\\sin b - \\cot C \\,\\sin A,&(CbAc)\\\\[0ex]\n\\text{(CT3)}& \\cos c\\,\\cos A=\\cot b\\,\\sin c - \\cot B \\,\\sin A,&(bAcB)\\\\[0ex]\n\\text{(CT4)}& \\cos c\\,\\cos B=\\cot a\\,\\sin c - \\cot A \\,\\sin B,&(AcBa)\\\\[0ex]\n\\text{(CT5)}& \\cos a\\,\\cos B=\\cot c\\,\\sin a - \\cot C \\,\\sin B,&(cBaC)\\\\[0ex]\n\\text{(CT6)}& \\cos a\\,\\cos C=\\cot b\\,\\sin a - \\cot B \\,\\sin C,&(BaCb).\n \\end{array}\n", "f2df23930acdb3748cefda0fbf1e229d": "\\omega_{cyc}\\,", "f2df5000508664834424b360d3bcef39": "N>2", "f2dfb76907474522b322ba01cc823aeb": "\\left(\\frac{16.00 \\mbox{ g Cu}}{1}\\right)\\left(\\frac{1 \\mbox{ mol Cu}}{63.55 \\mbox{ g Cu}}\\right) = 0.2518\\ \\text{mol Cu}", "f2dfd669c2bfab4075089de6fdadbe8b": "[n,n-m,2(1-\\epsilon)\\gamma\\cdot n]_2", "f2e0403966c1ed9df75a32a91e7b563b": " L \\not \\in X \\to \\operatorname{sink-test}[L, X] = \\operatorname{sink}[L, X] ", "f2e0a0847f190bf92c15d332df02a227": " m \\equiv c^d \\pmod{n} ", "f2e0c03958c43a10d6f7288391e31d47": " \\sum_{i=1}^n E(ax_i) y_i = a = \\sum_{i=1}^n x_i E(y_i a)", "f2e17f39dff4dfb1cff8716da72ee91b": "\\ \\ p\\leq 2n,\\ \\phi_{p}=\\sum_{0}^{[p/4]}\\Omega^{h}\\wedge\\mu_{p-4h},\\ \\ \\Omega\\wedge*\\mu_{p-4h}=0", "f2e186e1d8a5677268525f882907beb4": "y=a\\sin(2\\pi ft)\\,", "f2e1a1c3e7b245bbf9cfc82303c7b5ca": "\\frac{V}{N\\Lambda^3}\\gg 1", "f2e1ace47e2d69afa9c6f728289dd48c": " J = \\chi C \\nabla\\varphi ", "f2e1dcc2191a3f9df8b5ab2c3412362f": "(2 + n)\\omega_\\text{a} + n\\omega_\\text{s} - 2(1 + n)\\omega_\\text{c} = 0", "f2e2021ce2d55c7f443ac02180626612": "a_3 = 2, a_2 = -6, a_1 = 2, a_0 = -1", "f2e21fca4180229d0e83a623fc5df219": "\\tan \\theta \\approx \\sin \\theta \\approx \\theta", "f2e28a2e4e2ed8821e87416585f4e6fb": "\\scriptstyle\\lim_{x \\to 0^+} x^m (\\ln x)^n \\, = \\, 0", "f2e2a70c3e7cfa9bf5aeb3ca24236ee7": "[u,v]", "f2e2b36c00b87f49cabcf3870903f8a9": "b =35", "f2e2be1426c1343c138c82668688de62": "a_0 = 1:", "f2e3119ed7f2daf01d51c9696b12d032": "Y = \\frac{1}{Z}= i \\omega C + \\frac{1}{i \\omega L}", "f2e326752414a64fa5a92199b3e9df51": " \\left(1 - \\sum_{i=1}^p \\varphi_i L^i\\right) X_t = \\left(1 + \\sum_{i=1}^q \\theta_i L^i\\right) \\varepsilon_t \\, ,", "f2e3aff3c078f118464cc5234dd3f9ca": " 1 + \\frac{1}{2}+ \\frac{1}{4}+ \\frac{1}{8}+\\cdots+ \\frac{1}{2^n}+\\cdots.", "f2e3d175745bba7b7a9dea73449e0cbe": " f : \\Omega \\to \\{0,1\\}^\\infty \\,", "f2e4196e5441a188fe5266689defbf9a": " \\mu_0 \\dot \\gamma \\ll \\tau^* ", "f2e429b76de47c10ea36701a5bf121c3": "\\mathcal{C}^2=\\mathbf{1}", "f2e467b5b950b7bdda5822589b5d670e": "{d^2\\theta\\over dt^2} + {g\\over l}\\sin\\theta = 0,", "f2e49b0b7f3dc553ae653597aa3fbe89": "\\frac{\\partial f(M, T)}{\\partial M} > 0,", "f2e4f63769390441be71e078c859e457": "d^T c", "f2e5247311e245801cfa3607960805f8": "\\mu_n(N_D) = 65 + \\frac{1265}{1+(\\frac{N_D}{8.5\\times10^{16}})^{0.72}}", "f2e52b3cb2ebf955c51c2443dc2df47e": "\\tfrac{CDxDF}{FH-DG}", "f2e53a24b04d6cf6362f2710a0d3e764": "\\left[ (\\text{AA},\\text{AA}), (\\text{AA}, \\text{Aa}), (\\text{AA}, \\text{aa}),\n (\\text{Aa},\\text{Aa}), (\\text{Aa}, \\text{aa}), (\\text{aa}, \\text{aa}) \\right]\n", "f2e54470155bf57a634f23137298648e": "M + Mr = \\rho.Q (c_2u.r_2 - c_1u.r_1) ", "f2e54caf6c45f8d5de079b7dc5943ae3": "\\Delta w_{ji}=\\alpha(t_j-y_j) g'(h_j) x_i \\,", "f2e59898934d838528c96f95e601c7a0": "\\vec{E}(\\vec{r},t)", "f2e5a2bb6614934d755a0725adfee4c1": "k=k_n = {2\\pi n \\over Na}\n\\quad \\hbox{for}\\ n = 0, \\pm1, \\pm2, ... , \\pm {N \\over 2}.\\ ", "f2e5c0c18b0d829e24193bba54d1bfec": "\\mu_k \\in [0,1]", "f2e6096ebf63034edb80250055273835": "H_n(M, \\partial M; \\mathbb{Z})", "f2e644ffb03e80ded861bf2a0a69150e": "X\\oplus (Y \\oplus Z)\\cong (X\\oplus Y)\\oplus Z\\cong X\\oplus Y\\oplus Z", "f2e6ab5049cda5e464e2b3a6a35d78c3": "\\displaystyle u_{xt} -(uu_x)_x =u_{yy}", "f2e6ba4e473f9c958b166523c41a388d": "\n\\eta_{x} = \\mathrm{sn}\\, \\chi \\ \\mathrm{dn}\\, \\psi \\ \\cos \\phi\n", "f2e6c046eda001410cf3f0222c8cf722": "R=g^a\\,v^b\\,\\theta^c\\text{ which means }L\\,1_x\\sim\n\\left(\\frac{L\\,1_y}{T^2}\\right)^a\\left(\\frac{L}{T}\\right)^b\\,1_z^c.\\,", "f2e6da427a2e531ae385b7311bc17cf6": "BZ/2", "f2e6e3b9bff59f293f0e9019a1ecabc5": "EU", "f2e70a5b4fbff7c38f26612304f7ce8e": "\n{ L}(x) = \\left\\{\n\\begin{array}{ccc}\nx & ,&\\text{for}~ x \\gg 1 \\\\\n\\frac{2}{3}|x|^{\\frac{3}{2}} &,& \\text{for}~ x \\le 1\n\\end{array}\n\\right.~,\n", "f2e718bce534cfd516b9823159cbad6f": "I=sY=(1-c)Y", "f2e722b7decb64e615e406771f93db7f": "e_f(k,i+1) = e_f(k,i) - \\kappa_f(k,i)e_b(k-1,i)\\,\\!", "f2e73e2d280ff64066d1d987a096bae7": "V_\\min", "f2e78ea57fc57ede7415ed24b1b7dae4": "a=-1,\\;\\; b=0,\\;\\; c=x_1^2+x_2^2+x_3^2+x_4^2", "f2e7c39fe2de43622ab38a088093b3d0": "\\phi_{h}(\\boldsymbol{x})=\\sum_{j=1}^{dofs}\\phi_{j}^{i}N_{j}^{i}(\\boldsymbol{x}), \\quad \\forall \\boldsymbol{x}\\in\\Omega_{e_{i}},", "f2e7ef2d6839c8276bf17e597d49265b": "\\Delta P\\,", "f2e7feb013b500ab5dd5196ada83b548": "\\alpha \\circ \\gamma = \\operatorname{id}_H", "f2e805e46ac0c23e4e50a41f876465bf": "|E|=k", "f2e80816c75f3b8991cf6bb49b59dc97": "a\\sigma", "f2e8220fd108f7df2cb5567bed9e2125": "\\sin(39\\tfrac38 ^\\circ) = \\frac12\\sqrt{2-\\sqrt{2-\\sqrt{2+\\sqrt{2}}}};", "f2e8514de6ce342d42dff908236662a9": "[n^2 (I) - n^2] = [n (I) - n] [n (I) + n] = n_2 I (2 n + n_2 I) \\approx 2 n n_2 I", "f2e889ca0fd317c40a3de78aff8c31f0": " r_k, r_{k+1}=0. ", "f2e89f42acf9f09914fa9657e54f23f3": "\\int\\frac{\\sinh^m ax}{\\cosh^n ax} dx = -\\frac{\\sinh^{m-1} ax}{a(n-1)\\cosh^{n-1} ax} + \\frac{m-1}{n-1}\\int\\frac{\\sinh^{m -2} ax}{\\cosh^{n-2} ax} dx \\qquad\\mbox{(for }n\\neq 1\\mbox{)}\\,", "f2e8cd6d3f4720f4d242847646f1157d": "\\nabla f = \\nabla \\vec{x}^T \\mathbf{M} \\vec{x} = \\lambda \\cdot \\nabla \\vec{x}^T \\vec{x}", "f2e8e4ad38159aa2ece63f8523e7f0f6": "\\mathrm{Res}\\left(( f\\circ g) g'\\right) = \\mathrm{ord}(g)\\mathrm{Res}(f)\\qquad \\mathrm{if}\\; \\mathrm{ord}(g)>0;\\,", "f2e8fa9b8e8756ce592fb613aed6ea62": " \\bold x^{(m+1)} =\n\\begin{align}\n& \\frac{1}{120} \\begin{pmatrix}\n0 & 40 & 60 \\\\\n0 & 10 & 75 \\\\\n0 & 26 & 51\n\\end{pmatrix}\n\\bold x^{(m)} +\n\\frac{1}{120} \\begin{pmatrix}\n100 \\\\\n-335 \\\\\n233\n\\end{pmatrix},\n\\end{align}\n\\quad m = 0, 1, 2, \\ldots \\quad (15) ", "f2e98dea8c910cbc92f6215db9770253": " C^2 ", "f2e99b4fb088c3dd2a334b632244ebc7": "\\displaystyle{\\|f_-\\oplus f_+\\|_{\\mathfrak{H}}^2=\\iint_\\Omega |\\nabla f_-|^2 + \\iint_{\\Omega^c} |\\nabla f_+|^2.}", "f2e99fb0f4bc3726014354d3a984cace": " \n\\frac{m^*}{m} \\approx 1+\\frac{\\alpha}{6}+0.0236\\alpha^2.\n", "f2ea2767a5978c069d52b23dd7984c16": "\\|Du\\|_{L^p(R^n)}", "f2ea47340320d526487c92b25a84db7f": "\\sum \\limits_{n=1}^{\\infty} a_nx^n = x + {m \\choose 1}\\sum \\limits_{a=2}^{\\infty} x^{a} + {m \\choose 2}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} x^{ab} + {m \\choose 3}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} \\sum \\limits_{c=2}^{\\infty} x^{abc} + {m \\choose 4}\\sum \\limits_{a=2}^{\\infty} \\sum \\limits_{b=2}^{\\infty} \\sum \\limits_{c=2}^{\\infty} \\sum \\limits_{d=2}^{\\infty} x^{abcd} +...", "f2ea504c96f10e3ae1077c4e77d03402": "\\mathfrak{h} \\subset \\mathfrak{h}'", "f2ea662ebeac5d3b13611fe85ebec145": "a^\\ell + b^\\ell = c^\\ell,\\ ", "f2ea9168f5c3c5725a49c2b8a91e7261": " \\sum_{n=1}^\\infty \\frac{(-1)^n}{\\sqrt{n}}=(\\sqrt{2}-1)\\zeta(\\frac{1}{2})\\approx -0.604899.", "f2eaad8b6825a965d80634db17c39cee": "\\displaystyle{Q(x,y)=\\sum a_i (x_i^2+y_i^2),}", "f2eb07d040f902ccf06baba1ef8bd9a6": "\nN_i = \\frac{g_i}{e^{\\alpha+\\beta \\varepsilon_i}} \n", "f2eb09e6b13e7d28dee8d08ea73b5a11": "\\varepsilon+2\\pi i", "f2eb4850e3216af2fa460887ed4ce409": "\\displaystyle{v(z)=\\int_a^z -u_y dx + u_x dy,}", "f2eb7f52bec072116c32f676168f1761": "\\sum_{n = 0}^\\infty \\frac{(-1)^n}{n!} = \\frac{1}{0!} - \\frac{1}{1!} + \\frac{1}{2!} - \\frac{1}{3!} + \\frac{1}{4!} - \\frac{1}{5!} +\\cdots", "f2ec709653caa2b7e92a2f99f63fd617": " \\lim_{x \\rightarrow x_0} \\operatorname{ap} \\ f(x) = f(x_0)", "f2ec934e86f08c1151e1b5822d36d518": "m \\geqslant n", "f2ed42d17ce6b4c9b9755c0339684d18": "\\mathbf{T}\\mathbf{T}^T\\mathbf{T}\\mathbf{u}_i = \\lambda_i \\mathbf{T}\\mathbf{u}_i", "f2ed455e57464c24a0f327e36e7ec3e7": "E_{pot} = m g h", "f2ed6bb27e4b1a26e4be8059a0bc7d4f": "\n A| \\alpha \\rangle = \\alpha| \\alpha \\rangle\\; ,\n", "f2ee4d8396b2453db4f09b985105001e": "w=0", "f2ee57f8afac12447d334ff5d018688c": " \\!\\ {1 \\over {S_m^4 - \\lfloor S_m^4 \\rfloor}} + \\lfloor S_m^4 - 1 \\rfloor = S_{(m^4 + 4m^2 + 1)} ", "f2ee84ee2e7cb4954b84400adc729f50": "\\displaystyle \nu(\\eta,0)=U(0)=U(t)=u[X(t),t]=u[\\eta+tu(\\eta,0),t].\n", "f2ee95571acbeb032f5c283ef08f5c82": "J=\\int f(t,x,y)\\,dt", "f2eeb35e622cc431f93fe9a01c468f6e": "H^*(G;k)\\cong k[x],\\,", "f2ef0302582ca851606e0454138e003c": " \\boldsymbol{\\nabla} \\times \\boldsymbol{A} = \\boldsymbol{0} ", "f2ef39b1360146a0e9f8d5a51027a04a": " E = \\frac{1}{C} \\frac{L}{D} \\ln \\left( \\frac{W_i}{W_f} \\right) ", "f2ef78cc1f1c158670fe0200c879ec90": "F_{N}", "f2ef847eb723f98ad1d128b596dd8020": " dW \\ = \\ dQ_c \\ - \\ (-dQ_h) ", "f2ef934b489d49d6ce50de4a35facbfe": " D_X(p + t \\bold{A}) = (I-tS)X + d_Xt \\bold{A}. ", "f2efac7f2f84c694afc98304e5745380": "y\\neq x", "f2efac92910365863f52f59646960712": "\\Omega=X", "f2efd2207b49cf973a29bc7f78ade494": " t_{1/2} ", "f2efe7543326d42678b41899593a01dc": "\\|T\\|_\\infty\\le 1+\\varepsilon", "f2eff107391cbab9842c7f7ae1f13386": "\\angle \\beta A = 2 \\pi n, n \\in 0, 1, 2,\\dots\\,.", "f2eff48481abf3d66ee9e658d88d6036": "r_{xy} = \\frac{ \\overline{xy} }{ \\sqrt{ (\\overline{x^2}) (\\overline{y^2}) } } ", "f2eff540d8e20e6654faf18da3e74748": " C_{\\alpha \\beta} =\\begin{bmatrix}\n C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\\\\n C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\\\\n C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & C_{44} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & C_{55} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & C_{66} \n\\end{bmatrix}.\n\\,\\!", "f2effbc0666659114a344a771f1ef20f": "L(x;\\gamma)", "f2f056593b070ea981d020bb19bc95af": "\\Pr \\left (T > s + t | T > s \\right ) = \\Pr(T > t), \\qquad \\forall s, t \\ge 0. ", "f2f0acabb41dc0f3b94a1d25cf854e45": " [I_C]=[A][I_C^B][A^T].", "f2f1085576447c95e16dee30e2d168ef": "K=2^{286}", "f2f142c7ef5a1942acad3e2071e6bcc5": "\\propto \\frac{p^{\\alpha-1} e^{-\\beta q}}{\\Gamma(\\alpha)^r \\beta^{-\\alpha s}}", "f2f1b0a78f636f91639e01c7982770cb": "\\epsilon = 1 + \\frac{\\omega_p^2}{kN} \\int f'_0 \\frac{\\mathcal P}{\\omega-kv} {\\rm d}v", "f2f1b18a419c6fddc9ad79a13a5946e3": "\\omega\\,\\!", "f2f1b33b4ad34c746311e99d472c0d13": "\\eta^2z^2 - \\eta(3 + \\eta)z + \\eta + 1 = 0\\,,", "f2f1bd339f55aaa893fe27867551615a": "(M,\\theta)", "f2f1d33caeb79ecfe33e1a5f4b450f03": "a =(2x+3)y", "f2f24c969b23e5ada0666b84e31fc520": "O(w)", "f2f2ab58c9b8f5bfc3d7fd6c4a003a8d": "\\langle |\\gamma(n)|^2 \\rangle", "f2f2db0ddd92ed54da7b962fbc6b1044": "M_{\\text{bol},\\odot}", "f2f2e777634c3c935425d8d8d1d14cdc": " r = \\frac{k}{T}. ", "f2f2f6360fb45a71ed2fce83cc1b69df": "\\mathbf{x}_j = 1 ", "f2f30af778ff1c6c269d5c5fc71b826f": "V_1 = V_2 \\cosh ( \\gamma x) + I_2 Z \\sinh (\\gamma x) \\,", "f2f3b5742586dff8dbf3cb409f7c4d27": "\\left(\\frac{43.20 \\mbox{ g O}}{1}\\right)\\left(\\frac{1 \\mbox{ mol }}{16.00 \\mbox{ g O}}\\right) = 2.7\\ \\text{mol}", "f2f3c7e0fe9f43c926646fd0f0609284": " {\\kappa}a >>1 ", "f2f3d7602a3f060aff4f6d366253ea7d": "\\alpha = 2\\pi\\, -\\, n\\pi \\quad \\text{ and } \\quad n=2-\\frac{\\alpha}{\\pi}.", "f2f3ddc596c6c7a7c9589145b0e738c5": "l_2 \\,", "f2f3deeeb4351a3dd18594ff9509c16c": "b \\odot c", "f2f3e8932de7d19c1fdd26045a9268a2": " K_{var} = \\frac{2e^{rT}}{T}\\ \\left ( \\int\\limits_{0}^{F_{T}} \\frac{1}{K^2}\\ P(K)dK + \\int\\limits_{F_{T}}^{\\infin} \\frac{1}{K^2}\\ C(K)dK \\right )", "f2f3fcde0153649718bf079137b109a7": "\\Phi\\left(\\frac{r_P-r_E}{\\sqrt{\\sigma_P^2 + \\sigma_E^2}}\\right) = \\frac12\\left[1 + \\operatorname{erf}\\left(\\frac{r_P-r_E}{\\sqrt{\\sigma_P^2 + \\sigma_E^2}\\sqrt{2}}\\right)\\right]", "f2f4154aa58c0985bb54d2eb9c20cb47": "\\nabla S(x^{(k)}) = 0, \\quad", "f2f42366a39c367d802d88827245baa1": "X = \\sec\\, z", "f2f43d3c52d3d09c40a78e33e77f2d7b": " \\mathbf{F} = k I I' \\int \\int \\frac{\\mathbf{ds}\\times(\\mathbf{ds'}\\times\\mathbf{r})} {|r|^3} ", "f2f486681dcd4a971a84ae29e10e791d": "\\tilde g^{\\alpha\\beta}=e^{2\\phi}g^{\\alpha\\beta}+2U^\\alpha U^\\beta\\sinh(2\\phi)\\;", "f2f49cd41eae9c010f067b3111690815": "N_e^{(v)} = {p(1-p) \\over 2 \\widehat{\\operatorname{var}}(p)}.", "f2f4fa33954ebce2f27b602dd4f7f657": " D = <\\Delta v\\Delta\\tau\\Delta a\\Delta\\tau>\\sim \\left(\\frac{c\\,\\delta E_{\\perp}}{B}\\right)\\left(k_{\\perp}^2\\,D\\right)^{-1}\\left(\\frac{k_{\\perp}^2mv_{th}^{2}c^2\\,\\delta E_{\\perp}}{qB^2}\\right)\\left(k_{\\parallel}\\,v_{th}\\right)^{-1} .", "f2f50327bb033bab3601a48881868928": "\\Pi_0^j = 1/d = 2^{-\\bar{k}}", "f2f529e7c5cb49cccf38eae03bf837da": "f\\colon Q_M\\to B_M", "f2f52a83a0cc578d335ac4d5a9c55a4d": "n|\\mathbf{X}^n|(\\mathbf{X}^{-1})^{\\rm T}", "f2f56c2c7e84710f3b7747cf95fb57a2": "Q(y,s)", "f2f5a286e58badf8f6e65cd86de754a2": "R_k", "f2f5afeb2879174156c4f2de48d2aa97": "x_1,\\dotsc,x_n", "f2f5bbe3d5a12d2869dd21b583843e16": "n_d", "f2f5e5186a3182f8d50f2ca154d70fc6": "\\frac{30!\\times 2^{27}\\times 20!\\times 3^{19}}{60} \\approx 1.68\\times 10^{66}", "f2f6028d7a1a00967f351296fa2b8711": "\\tilde{\\mathbf{C}^n}", "f2f6485810624c9ed398e86811930bff": "F(\\bar{x})", "f2f67fbb4c800c1e48e33d7e78ff9990": "R_{11} = \\left. \\frac{V_1}{I_1} \\right|_{I_2=0} ", "f2f76b2474d7981e7d7bcae8411f41d6": "\\cong_{\\mathcal{B},\\epsilon} = \\{(x,y)\\in O \\times O : \\parallel\\boldsymbol{\\phi}_{\\mathcal{B}}(x) - \\boldsymbol{\\phi}_{\\mathcal{B}}(y)\\parallel_{_2} \\leq \\varepsilon\\}.", "f2f772a063366b65cbad4ba48f9bc061": "\\ sin( \\pi - \\beta - tan^{-1}(L/D) ) \\approx sin( \\frac{\\pi}{2} - \\beta ) ", "f2f79b385bf24716d18ae7c9383b91c2": "\n\\begin{pmatrix}\n j_1 & j_2 & j_3\\\\\n m_1 & m_2 & m_3\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n j_1 & \\frac{j_2+j_3-m_1}{2} & \\frac{j_2+j_3+m_1}{2}\\\\\n j_3-j_2 & \\frac{j_2-j_3-m_1}{2}-m_3 & \\frac{j_2-j_3+m_1}{2}+m_3\n\\end{pmatrix}.\n", "f2f7bb97b736088de170445ff9a9e0c5": "T_1,T_2,\\ldots,T_m\\subseteq \\Omega", "f2f8144a975ed8e4d7ff1f92b33fda65": " A_p \\left( \\sum_{n=1}^{N}|x_{n}|^{2} \\right)^{\\frac{1}{2}} \\leq \\left(\\mathbb{E}\\Big|\\sum_{n=1}^{N}\\epsilon_{n}x_{n}\\Big|^{p} \\right)^{1/p} \\leq B_p \\left(\\sum_{n=1}^{N}|x_{n}|^{2}\\right)^{\\frac{1}{2}} ", "f2f891903d0aa01236bc6f3055f524a5": "\\Delta H_2 = - R * slope_2,", "f2f8bcaae7d22450ee6c3b57c4d3db44": " \\sum_{n=1}^\\infty \\frac {1}{n^s} = \\prod_p^\\infty \\frac {1}{1-p^{-s}}\\text{ for }s > 1\\,\\,\\ (p \\text{ is prime number)} \\,", "f2f9a15d73144b415c1a5ce6bee66d8b": "|V| > t", "f2f9dbad07f81ce93e86f2e39d7621fa": " \\delta W = \\sum_{i=1}^n \\mathbf{F}_i\\cdot (\\vec{\\omega}\\times(\\mathbf{X}_i -\\mathbf{d}) + \\mathbf{v})\\delta t. ", "f2fa2232d14457e89af0f3ea9d81cde3": "\nl = \\frac{1}{\\beta} \\left[ n \\pi + \\arctan\\left(\\frac{\\omega L}{Z_0}\\right) \\right]\n", "f2fa5aacf9c4fe629655789206653e96": "\\scriptstyle OA+BA=\\sqrt{N}", "f2fb4dd6f361ca3b82dd6405da18735d": "(-j\\infty,+j\\infty\\,)", "f2fb7f7fdc6548b07add6fb56dfe50d1": "E_{\\mathrm p} = \\frac{e^2}{8\\pi \\varepsilon_0 r},", "f2fcbabd57b95c5671a4a5ef9cce2891": "S(A,P,z) \\ge XV(z) \\left( f_1 \\left(\\frac{\\log y}{\\log z} \\right) - O\\left(\\frac{(\\log \\log y)^{3/4}}{(\\log y)^{1/4}}\\right) \\right) - \\sum_{m|P(z), m < y} 4^{\\nu(m)} \\left| r_A(m) \\right|.", "f2fccd2e43b582e9871d47d49baa47a8": " \\frac{1}{\\mathrm{agm}(1, \\sqrt{2})} = \\frac{4 \\sqrt{2} \\,(\\tfrac14 !)^2}{\\pi ^{3/2}}= \\frac{2}{\\pi}\\int_0^1\\frac{dx}{\\sqrt{1 - x^4}}", "f2fcee421c0248beaae78993bf33f600": " \\begin{pmatrix} \\cos \\theta & -\\sin \\theta & 0 \\\\ \\sin \\theta & \\cos \\theta & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}.", "f2fd0f70fcd750876296145889628da4": "\\scriptstyle f^{*} \\in F", "f2fd3e73b492046d5c3a6eed58a596c5": "\\frac{V_0}{e}", "f2fda5e8262178cc6bc3c2c7d1a814f1": "\\scriptstyle \\log_e (\\frac {760} {101.325}) - 8.433613\\log_e(T+273.15) - \\frac {6281.040} {T+273.15} + 71.10718 + 6.198413 \\times 10^{-06} (T+273.15)^2", "f2fdc7ccdadaed56441beb9e54d5b381": "\n\\omega = \\sqrt{2} - 1.\\,\n", "f2fddd0f2cd7d220385695fc0883a24c": "\n\\left |\\sum_{k=1}^n a_k b_k \\right | \\le \\operatorname{max}_{k=1,\\dots,n} |B_k| a_1,\n", "f2fdf27fbd600b1fdbd7bb61bdce8f0f": " \\dim\\left[ \\mathrm{Graff}_k(V) \\right] = (n-k)(k+1) \\, . ", "f2fdfa6334e2afe6d21925571c483f21": "V = \\frac{\\alpha}{2\\pi} \\cdot \\frac{4}{3} \\pi r^3 = \\frac{2}{3} \\alpha r^3.", "f2fe058dd8165f3d98d19844151c44c0": "R_T = f(x) - p(x) \\,\\!", "f2fe2cc03ca42edb5c4d57a0d523e20c": " \\bigcup \\limits^\\infty E_n = X", "f2fe4a96a0d55f9466e23e1faf74e5c5": "\\sqrt{k_1}(A_\\rightarrow-A_\\leftarrow)=\\sqrt{k_2}(B_\\rightarrow-B_\\leftarrow)", "f2fe4eea8d050d3663f14f00195a76a4": " z \\partial_t + t \\partial_z \\,\\!", "f2fee5331099a6ea7dceab87b2cc1b1a": "\\ell(x)>66", "f2fefe4e6e2db1484e7a3bcb47d4a130": "\\gamma = \\frac{4}{3\\,\\pi^2}\\, K^2(m)\\, \\left( 1 - \\frac12\\, m -\\, \\frac32\\, \\frac{E(m)}{K(m)} \\right) = -\\tfrac16 + \\tfrac1{64}\\, m^2 + \\tfrac1{64}\\, m^3 + \\cdots,", "f2ff367362049d2dac16442dfb343981": "\\mathbf{\\Phi}_i \\ = \\sum_{\\mu=1}^N \\mathbf{C}_{i\\mu} \\ \\mathbf{\\chi}_\\mu^A \\, ", "f2ff5e3dbd5f49749855601930b6454a": "u^\\mathrm{T}", "f2ff6e32ceae7689327247adc8416306": "\\scriptstyle \\mathcal{D}", "f2ff86fcf75a9ec0fd55876c3f0586bc": " y = 2x - 1. \\,", "f2ffa6b09ce495f7f952c42f78e7b9b4": " {\\mathbf u}(x)=\\lozenge \\tilde{u}(x) = \\lozenge \\{u(x):G(x,u,p)=0,p\\in {\\mathbf p} \\} ", "f2ffd57035650a464bf54a915f9ab4a2": "\\text{SSG}", "f2ffda8741fd269d817ae38f1ad54d47": " \\lambda = \\frac{12398.4\\,{\\rm \\AA}\\,\\text{eV}}{E_\\text{i} - E_\\text{f}}. ", "f2ffe240cdbf0da0fee988f22baf835f": " \\lVert a_{i} \\rVert ^2 ", "f2ffffab6159dc76abf2446ec94c9fb2": "\\scriptstyle\\bar{S}", "f3000db5f37aea8fefff66fc4d27d365": "A^{-1}:H\\rightarrow V", "f3001cd847a2c41eac9e06808d9b58af": " D_X(\\bold{x}-p) = D_X(Z + \\Delta \\bold{A}), ", "f3004dd02e572b20dfd51903bd666099": " \\int \\chi^*_{nlm}(r)\\chi_{n'l'm'}(r)d^3r = \\delta_{ll'}\\delta_{mm'}\\frac{(n+n')!}{(\\zeta+\\zeta')^{n+n'+1}} ", "f30062eefa64a28ee5656f7a0fbd414a": "\\cos 0=1\\,", "f3007425df1f1df1aa7c88a66e89d09a": " D_m^{(s)} = O (m^{-1/2} (\\log m )^s)", "f3007702a17fc68e5578fb6e27afaad9": "f_{Di}=c_{i1} \\dot{u_1}+c_{i2} \\dot{u_2}+\\cdots+c_{in} \\dot{u_n}=\\sum_{j=1}^n c_{i,j}\\dot{u_j} \\, ", "f300acd0f07bd787a0979101db36b9bc": "\\mathcal{B} = 2^{\\{1,...,NR\\}} ", "f300cace32f35371adbd9492b1a285e8": "\\scriptstyle{({r^2})^\\varphi+r^\\varphi = 1}", "f300cc465e4cbe6c7a161f2e1917f86e": " | v | = \\sqrt{V_x^2 + V_y^2} ", "f3010197b8e87048c27794319b029e68": "\n\\begin{bmatrix}\n121 & 108 & 7 & 20 & 59\\\\\n29 & 28 & 122 & 125 & 11 \\\\\n51 & 15 & 41 & 124 & 84 \\\\\n78 & 54 & 99 & 24 & 60 \\\\\n36 & 110 & 46 & 22 & 101\\\\\n\\end{bmatrix}\n", "f30130c26ec02f581df81b9e5f2c6235": "\\mu = 1+4\\pi\\chi_\\text{m}", "f30157720a19a5ba51f6d224b1c2ccb0": "\\mathrm{not}~q", "f3018d2312ed52b5a6fede7b034f7dac": "C_1-C_2", "f301a98e4afbcbf2d9a857971867433c": "\\sum_{i=1}^m \\log(p(y_i;e^{\\theta' x})) - \\lambda \\left\\|\\theta\\right\\|_2^2", "f301ab2de2d887e537d77aa6bb7e0316": "\n\\int x^m (\\ln x)^n\\; dx\n= \\frac{x^{m+1}}{m+1}\n \\cdot \\sum_{i=0}^n (-1)^i \\frac{(n)_i}{(m+1)^i} (\\ln x)^{n-i}", "f301d5c01aca1ab52ecbb8dcf7c35038": "\\pi=\\pi^e+\\lambda(Y-Y*)", "f3022f6efd99e17180eeab566422c1f0": "L\\phi =\\delta \\, ", "f3028f2f9bdff56f105080502bbce1ba": "\\sin\\left(\\frac{\\omega L}{c}\\right) = 0.\\ ", "f302cb84ed4f8c57da88f4c6479b5637": "\\displaystyle{f(\\theta)=\\sum_{n\\in {\\mathbf Z}} a_n e^{in\\theta}.}", "f302f3e24b7f431e9e8a2d43b66b2e4d": "\\begin{align}\n& F(\\hat{A}) = \\hat{A}^2 \\\\\n& \\Rightarrow \\langle \\hat{A}^2 \\rangle = \\int_R \\psi^{*} \\left( \\mathbf{r} \\right ) \\hat{A}^2 \\psi \\left( \\mathbf{r} \\right ) \\mathrm{d}^3\\mathbf{r} = \\langle \\psi \\vert \\hat{A}^2 \\vert \\psi \\rangle \\\\\n\\end{align}\\,\\!", "f3031182243cd38cc45417721ac914f8": " C_{(+)}= C_{(+)}^* = s_{(d+2,+)} C_{(+)}^T = s_{(d+2,+)} C_{(+)}^{-1} ~~~~ s_{(d+2,+)}= s_{(d,-)} ", "f30318bcaec4fce0405e738caac14bcd": "\\oint_{S^2} \\alpha \\vec{E}\\cdot d\\vec{S} + \\int d^3x \\alpha\\left[\\rho-\\epsilon_0 \\nabla\\cdot \\vec{E}\\right].", "f303384a560c6ebe0558ffa5f768a28a": "F_{rad} = F_{grav} \\rightarrow L < L_{Eddington} ", "f303ece22bc7cf33df4b820c8f5b61e6": "x_{i}\\in X\\,\\!", "f30400bcb8c1ae8dc7d7aad1a9626326": "e = \\frac {D}{m} = \\frac {130}{18.6} \\approx 7", "f3040b8b1d0fca425cb386b4c96b6ffd": " \\mathrm{Re} = {{\\rho {\\mathbf v_s} D} \\over {\\mu}}.", "f3041d7b09860a1b4a8572d1dd9f9853": "1/k\\,", "f3041e2fcb391d6267fd3c21bba983e0": "\n\\langle jm' | \\mathcal{R}(\\alpha,\\beta,\\gamma)| jm \\rangle =\n\\langle jm' | T^{\\,\\dagger} \\mathcal{R}(\\alpha,\\beta,\\gamma) T| jm \\rangle =\n(-1)^{m'-m} \\langle j,-m' | \\mathcal{R}(\\alpha,\\beta,\\gamma)| j,-m \\rangle^*,\n", "f3044b6c6cc1a3ce38d9eceb0a69283b": "(n+1)s", "f3048c689ffcee519a363ae913275446": "221^2 - 67\\cdot 27^2 = -2.", "f30490ab909609193f833651a4ca1897": "\\|x\\|_{bs} = \\sup_n\\left|\\sum_{i=1}^nx_i\\right|", "f304a1f5e4d4859151f6255664a0f7ae": " | \\psi(t) \\rangle = U(t,t_0) | \\psi(t_0) \\rangle.", "f304ab97d10d673f660de92c83a578ef": " U_y(y_2;x_1,x_2) = (1 - x_2 - y_2 ) y_2 - y_2^2/2 - F ", "f304faccdb177017219926f641cb6e2c": "P(x+\\Delta x)=P(x)e^{-\\alpha(\\omega)\\Delta x}, \\alpha(\\omega)=\\alpha_0\\omega^\\eta", "f3051c3ebf97ded8178907f92657bc50": "\\sum_nE_n(x,\\alpha)z^n = \\frac{1}{1-xz+\\alpha z^2}. \\, ", "f30547584b7f50e45af6cea78a57b402": "1\\leq i\\leq j\\leq n", "f305b537797d2188b249d8708d90712d": " G(X)", "f3065d1deefc230ebd33d04e9aea1318": "M=(W,R,f)", "f3067110eec3e979ba8c7666ff930fa3": " s := \\frac{af(b)f(c)}{(f(a)-f(b))(f(a)-f(c))} + \\frac{bf(a)f(c)}{(f(b)-f(a))(f(b)-f(c))} + \\frac{cf(a)f(b)}{(f(c)-f(a))(f(c)-f(b))} ", "f30671dd582473445d58fcd3840c12af": "X = A + N \\,", "f3068423a6821ef057581d77d7f8a633": " T\\,", "f30702df0cbdf2c0b38fc05e0140677c": "D_N (\\mathbf{t}_0,\\mathbf{t}_1, \\dots ,\\mathbf{t}_{N-1})", "f30731b3514a204f554c7384497ef7f9": "\\partial_t \\rightarrow -i\\omega", "f307631d719d6340eac1151a744d4114": "\\vec{p} (t)", "f307a135d1c3677636a9f21e1ba382ec": "Q\\mbox{acc}", "f307a9470dc62a2ab6ede68349d43814": "{a^{(\\dagger)}}_{\\nu}", "f307df5bba05e6060857b1073eda3863": "\\coprod_i U_i\\times F", "f3085bfd2bf31a4ddcf00a4ab1ccc226": "K_pT_u/3", "f30865a03e9a5d5dbe90e7379d313f26": "\\begin{align}\n\\dot x&=u,&\\dot y&=v,\\\\\n\\dot u&=\\lambda x,&\\dot v&=\\lambda y-g,\\\\\nx^2+y^2&=L^2,\n\\end{align}", "f308942f585892c3a04ea80227c44cb0": "g_{k}", "f308a6b9870edf765f8526ab476aa4d2": " \\psi(x+1) = \\frac{1}{x} + \\psi(x)", "f308bd4b89aaf31fb7d42ed345abecad": "\\chi_+(1)\\chi_-(2)", "f308e2565850168c60da00b003c4892e": " \\omega_\\alpha = z_\\alpha + \\frac{ 1 }{ 6 }( z_\\alpha^2 - 1 ) \\beta_1 + \\frac{ 1 }{ 24 }(z_\\alpha^3 - 3 z_\\alpha )( \\beta_2 - 3 ) - \\frac{ 1 }{ 36 }( 2 z_\\alpha^3 - 5 z_\\alpha ) \\beta_1^2 - \\frac{ 1 }{ 24 } ( z_\\alpha^4 - 5 z_\\alpha^2 + 2 ) \\beta_1 ( \\beta_2 - 3 ) ", "f3090fc4418a845a04d7ce4974f3eaae": "\\hat H^n(G,\\mathbb{Z})\\longrightarrow\\hat H^{n+2}(G,A)", "f3091853cceb2a7fed7e60dec2a92953": " P(X=x) = \\frac{1}{Z} \\exp \\left( \\sum_{k} w_k^{\\top} f_k (x_{ \\{ k \\}}) \\right)", "f309386ee54bc829ef3f16f8ca3948b8": "v'_{1y}=\\frac{v_1\\cos(\\theta_1-\\varphi)(m_1-m_2)+2m_2v_2\\cos(\\theta_2-\\varphi)}{m_1+m_2}\\sin(\\varphi)+v_1\\sin(\\theta_1-\\varphi)\\sin(\\varphi+\\frac{\\pi}{2})", "f309dd437fc3d0f188e46e396dcf70d4": "\\vec{0}", "f309e112c0a7f91cefd5d0c18f4634fb": "(y \\vee z ) \\wedge w = (y \\wedge w) \\vee (z \\wedge w)", "f309e7c7ac52df24c6d0826aa92d8ebf": "f(x)=g(1)=g(0)+\\sum_{j=1}^k\\frac{1}{j!}g^{(j)}(0)\\ +\\ \\int_0^1 \\frac{(1-t)^k }{k!} g^{(k+1)}(t)\\, dt.", "f30a55ccf8988474e3ebfc62637a8451": "\\lnot \\lnot G \\to G", "f30ae0f69dfaa8fd73d558ebc109d131": "\n\\nabla \\cdot \\left( \\mathbf\\Sigma_i \\nabla v_i \\right) + \\nabla \\cdot \\left( \\mathbf\\Sigma_e \\nabla v_e \\right) = 0\n.", "f30b38698ad5b872ea6b6196d72b6ad0": "(\\overline{A} \\vee \\overline{C}) \\wedge (A \\vee C)", "f30b924a19077b1f8ce7da8f54c3f350": "\\lesssim10^{-7}", "f30ba72976dee2c48d3ffa006df92c83": "\\breve{\\boldsymbol z}_i", "f30bc6fd44adfcbcb11c05e68bea7907": " \\lambda ", "f30c10682e065246b1ffef299c79579a": "e_{g,i}", "f30c2cc52b0caa92eaab2bff37d57c74": "\\operatorname{lambda-lift}[S, L] \\equiv \\operatorname{let} V: \\operatorname{de-lambda}[G = S] \\operatorname{in} L[S:=G] ", "f30c8e669e69997f4faf4ad30af68edf": "\\scriptstyle \\tilde{\\nu} \\;=\\; \\frac{1}{\\lambda}", "f30c8eb36241e2993967ce80e3c702ce": "G_0=\\langle L,R,F,B,U,D\\rangle", "f30c90e89be4ca742fd2e99398dcad44": "79 \\times 600 = 47,400\\,", "f30c9736d38101f4ef47a59f7d3ebf82": "r_k = b - A x_k,\\,", "f30d68640f77b8434e6262dafd896184": "U_i = -D(i)", "f30d8bec3796e3ed90e11e3dadc66901": "\\exists x \\, ( P(x) \\wedge \\forall y\\,(P(y) \\to y = x)).", "f30d93f504e57e4f729446f1d0b3332a": "q q^* = (t+xhi+yhj+zhk)(t-xhi-yhj-zhk) \\!", "f30dca9680afc8bfc3e970cd6b2fe004": "J_{ab} = X_{a:b} = \\frac{\\theta}{3} \\, h_{ab} + \\sigma_{ab} + \\omega_{ab} - \\dot{X}_a \\, X_b", "f30e256de6ae9999510a50cf56eec8eb": "\\varphi_{x,y}(z)=[x,y,z]", "f30e3167c83d7e8347d5c0ba559219a0": "\n\\frac{{\\Delta z}}{z}\\,\\,\\, \\approx \\,\\,\\,\\alpha \\frac{{\\Delta x_1 }}{{\\mu _1 }}\\,\\,\\, + \\,\\,\\,\\beta \\frac{{\\Delta x_2 }}{{\\mu _2 }}", "f30e951dfeef2624931575fff348eb86": "\\operatorname{int}A \\subseteq \\operatorname{core}A", "f30ea21004265271ed5114f8ffce40e0": "{\\rm ASPACE}(C,j)=\\Sigma_{\\rm SPACE}(C)", "f30ebd77910fb0440d23e11fecaa00b1": "T''_n(1) = \\lim_{x \\to 1} n \\frac{n T_n - x U_{n - 1}}{(x + 1)(x - 1)} =\n\\lim_{x \\to 1} n \\frac{\\frac{n T_n - x U_{n - 1}}{x - 1}}{x + 1}.", "f30f54d161bdbd613bee5fa5029c3943": "\\left( x, y, z \\right)", "f30fb7d77f7695b8b692c57957aaef2d": "P = \\frac{\\mathrm{d}E}{\\mathrm{d}t} = - \\frac{32}{5}\\, \\frac{G^4}{c^5}\\, \\frac{(m_1m_2)^2 (m_1+m_2)}{r^5}", "f30fb91a163e3f50288e1ed25808a06c": " \\alpha\\in]0,1] ", "f30fcfe10c70460291d63777f6cb2f71": "(n+1-d \\begin{pmatrix}l + 1\\\\2\\end{pmatrix})/2 - 1 ", "f310db8d45c80ab3d913f83e6ca4dddc": "F_x = \\frac12 \\times \\rho \\times S \\times C_x \\times V^2", "f311247794e7ba68e4a52b52f5cc29ba": "b_Y(x) = x\\, ", "f3112b459f58a321f8efe426a21dc54e": "\\Phi_E = \\frac{Q}{\\mathcal{E}_0}", "f3112be3f6530c037df55452ca7ef901": "E_{1,1}=\\pi^2\\frac{\\hbar^2}{2m}(1/L_x^2+1/L_y^2)", "f3115e40c928fe452319cc590a88914f": "f_i^{eq}=\\omega_i\\rho \\left (1+\\frac{3\\vec{e}_i\\vec{u}}{c^2}+\\frac{9(\\vec{e}_i\\vec{u})^2}{2c^4}- \\frac{3(\\vec{u})^2}{2c^2} \\right) ", "f31169eb167a6d3940426aabb63b5fcf": " M \\leq \\frac{2d}{2d-n}.", "f311dd343686f6b816e7be99c53cfde5": "\\Psi_2", "f31263ecffa97d6721cd931e3e26ceaf": "\\textbf{Q}^{a}_k", "f312b3a734f7a107a7c29d33d6ea1189": "d(X) = \\frac{\\operatorname{E} [\\| X \\|^{2}]}{\\sigma(X)^{2}}.", "f313275f8dc57481882fbbcf0874194f": "1\\over\\sqrt{(1-\\beta^2)}", "f31355b7a11714c682c4e8422a113300": "\\ln(a) - \\ln(b) = \\ln\\left(\\frac{a}{b}\\right)", "f313830039c577a54309da50d2360628": "\\lim f(a_{\\alpha}) = f (\\lim a_{\\alpha})", "f3139ec123fe99b0b001e44f400ae086": " D_l=D_r=D", "f313ab00b3dda6d82862f6b55aaec8d2": "\n\\begin{array}{c|cc}\n0 & 0& 0\\\\\n1 & \\frac{1}{2}& \\frac{1}{2}\\\\\n\\hline\n & \\frac{1}{2}&\\frac{1}{2}\\\\\n\\end{array}\n", "f313fdee98437487dfbc755223790fd8": "V_p^{\\sigma}", "f3140380cbb550bcbcfc7e220645604c": "t_n=\\sup\\{\\mu(B): B\\in\\Sigma,\\, B\\subset A_n\\}", "f314418e608ccd030f0fef847e0c11e8": "C^\\infty_M(\\mathbb{R}^n)/C^\\infty_N(\\mathbb{R}^n).", "f3145265959802a02c779b895697b18e": "\nC_{1} = \\mathbf{D} \\cdot \\mathbf{D} + \\mathbf{L} \\cdot \\mathbf{L} = \\frac{mk^{2}}{2\\left|E\\right|}\n", "f3145df72564c9205fc5308661b1ebd8": "\\langle S\\rangle", "f314940de6a4c0e44b8e9c01cc7457ef": "\\left(\\begin{smallmatrix}a & b\\\\c & d\\end{smallmatrix}\\right),", "f314b5b2c65fdbb9155119c5d85269f3": " x = \\sum_{n=1}^\\infty b_n \\beta_n \\ ", "f314baa9e39c36eb5cf3bb9bd115a8b9": "u(\\phi)", "f314f6246609d80bfbc88d92187a0a40": "{\\nu}_i", "f315104d28cdbbf5d10a46487b0ffc9f": "\\,\\Gamma^*", "f3151d23f9c88ea74e0229bcdd321cde": "wr", "f31529c85e1e5278931a2dbdbfedfbd5": "g:(D\\times E){\\to}F", "f31537c5fbcb0f0053b9b47298ec465a": " f(\\tau)", "f31538f04929495f2e5148b74b290ae0": "A_2, B", "f3155f902967d9526262c025e44c259b": "\\mu_0 \\!\\ ", "f31593ed1e77621565c0110d5322ec7e": " \\sum_{i=1}^I N_i\\,\\mathrm{d}\\mu_i = - S\\,\\mathrm{d}T + V\\,\\mathrm{d}p \\,", "f3159a12947cd51426b3e01f12bc3e3c": "\\eta = \\frac{P}{P_0} \\ .", "f3159c2021223268415410f9ceb503d9": "\\begin{pmatrix}\n 2 & 1 & 1 \\\\\n 2 & 0 & 1 \\\\\n 1 & 2 & 0\n\\end{pmatrix}", "f3159e3e304cb96c66d84a980a0d3794": "z(t)=at^2+bt", "f315cd8b3ad031d0ed44b3e14ab39002": "\\{\\alpha_{U,n}(t), 0\\leq t\\leq 1\\}", "f315efabeee0af7e8acf224f9215f735": "bk=rn", "f31658e06a4c3b27106f62eec5add682": "T_\\mathbf{\\delta}", "f31718b91ea02ec1cae1a25293b16256": "f_o", "f31719c1737e78464dfd4632cd74903f": "{\\scriptstyle dW^S_t, dW^{\\nu}_t}", "f3172e4df14c8e12cb6d3e3368cf1870": "\\!\\ Re > 10^5", "f3173933695207af7a1ec38a42066cd8": "\\hat{g} = \\Omega^2 g", "f31740461e0eb16b6552f73488ed8ea7": " e^{-\\tau s} \\ ", "f31774cd121df123451b60eb74618a1f": "\\begin{bmatrix}\n2 & 0 \\\\\n0 & \\frac{1}{2}\n\\end{bmatrix}", "f31798714e631e9289b2ec33ff5c78f0": " k = \\min(j+m,\\,j-m,\\,j+m',\\,j-m').\n", "f317d6bde88e47623515f2ba4032df10": "|{\\Psi}\\rangle=\\sum\\limits_{i=1}^{M}a_i|{\\Phi^A_i \\Phi^B_i}\\rangle", "f317fca370908fc69f595e3723edd6ed": " g(k) = \\alpha + \\beta k", "f3182aea28755d72feb1b3a209362b17": " |\\psi\\rangle = \\sum_{s=-1,1} a_s \\exp \\left ( i \\alpha_x -i s \\theta \\right ) |s\\rangle ", "f3185cee99f0e4f4ca079ddcc5440d5c": "s_A", "f3188e0c9da78f023a5f6016c4ccc53c": "Z=\\sum_{n=0}^{\\infty } \\frac{(40n+3)\\left ( \\frac{1}{2} \\right )_n \\left ( \\frac{1}{4} \\right )_n \\left ( \\frac{3}{4} \\right )_n} {(n!)^3{49}^{2n+1}}\\!", "f3188e54eabab03fd967e5279be1b83b": "y(n)", "f318b644beb678e6aba8f8384a2480a0": "BR(a) = f^{-1}(-a) = -f^{-1}(a) = -a + a^5 - 5 a^9 + 35 a^{13} + ... \\,", "f318bc6f3094592c8eecbf169d5394bc": "\n\\mathbf{B} = \\begin{bmatrix}\n\\mathbf{B}_{11} & \\mathbf{B}_{12} & \\cdots &\\mathbf{B}_{1r}\\\\\n\\mathbf{B}_{21} & \\mathbf{B}_{22} & \\cdots &\\mathbf{B}_{2r}\\\\\n\\vdots & \\vdots & \\ddots &\\vdots \\\\\n\\mathbf{B}_{s1} & \\mathbf{B}_{s2} & \\cdots &\\mathbf{B}_{sr}\\end{bmatrix},", "f319369db670750004b19948af1384d8": "h_\\alpha(v,w) = d\\alpha(v,\\bar{w}) = -\\alpha([v,\\bar{w}]),\\quad v,w\\in L\\oplus\\bar{L}.", "f319e53f0b731b9bebaf67bd3bb8fabd": "x \\leq y ", "f31a006e24fc46e4eac6ccf9019c0fd0": " t_2 = t_1^2, \\; t_3 = t_1^3/4, \\; t_4 = t_1^4/4 ", "f31a6f7cab0c6b41698ad6772462f7a9": "(\\Box\\Diamond p \\rightarrow \\Diamond \\Box p) \\land (\\Diamond\\Diamond q \\rightarrow \\Diamond q)", "f31ae884193d5a29d4c70a51285411fc": "F(v) = \\alpha_1 v + \\alpha_2 v^2 + \\alpha_3 v^3 + \\ldots \\,", "f31b5aa61e739bcb3d595726f6002b7e": "\\Phi(\\bar{x})", "f31b5c6c3f4e9b1d19a47a6d9b04d70f": "\\left(\\frac{\\mathit{W}_{3-4}}{{m}}\\right)=\\mathit{u}_3-\\mathit{u}_4", "f31b7d2276bda2d5d3bbc3f5ce5f68e1": "\\displaystyle \\|\\lambda +\\rho\\|^2 = \\|\\mu+\\rho\\|^2.", "f31ba4dfeaf36f7653ae729079668683": "\n C_\\pm(J,M) \\langle j_1 m_1 j_2 m_2|J M\\pm 1\\rangle\n = C_\\pm(j_1,m_1\\mp 1) \\langle j_1 {m_1\\mp 1} j_2 m_2|J M\\rangle\n + C_\\pm(j_2,m_2\\mp 1) \\langle j_1 m_1 j_2 {m_2\\mp 1}|J M\\rangle.\n", "f31bbf3b51fd405f5b273a732b63ea94": "\\mathrm{Ann}_R(m)\\,", "f31bd22003a00d8ebc06c5a96d99346f": "n_{\\rm e} \\tau_{\\rm E} \\ge L \\equiv \\frac{12}{E_{\\rm ch}}\\,\\frac{k_{\\rm B}T}{\\langle\\sigma v\\rangle}\n", "f31be5f0bab3a191e43a66c75fc2ae1c": "T_E\\approx \\Big[ \\frac{l}{R}\\Big]^{\\frac{1}{4}}T_I", "f31c325d149096669d1b2d2413e30a03": "n^7", "f31c9746a5dd226a62f2b7097da8a82b": " Z(P,Q,s) = \\frac{1}{\\Gamma(s)} \\int_0^\\infty K(P,Q,t) t^{s-1} dt ", "f31ca243bcfe10f64bb1a990c92ea7bd": "\\begin{align}\n\\langle\\psi|\\mathcal{T}\\{\\phi(x_0)\\phi(x_1)\\phi(x_2)\\phi(x_3)\\}|\\psi\\rangle =& iD(x_0,x_1)\\langle\\psi|\\mathcal{T}\\{\\phi(x_2)\\phi(x_3)\\}|\\psi\\rangle + iD(x_0,x_2)\\langle\\psi|\\mathcal{T}\\{\\phi(x_1)\\phi(x_3)\\}|\\psi\\rangle \\\\\n&+ iD(x_0,x_3)\\langle\\psi|\\mathcal{T}\\{\\phi(x_1)\\phi(x_2)\\}|\\psi\\rangle \\\\\n&+ \\frac{\\lambda}{3!}\\int d^dx_4D(x_0,x_4)\\langle\\psi|\\mathcal{T}\\{\\phi(x_1)\\phi(x_2)\\phi(x_3)\\phi(x_4)\\phi(x_4)\\phi(x_4)\\}|\\psi\\rangle\n\\end{align}", "f31ca304971c7c3e7d53d5ca3e1a5dec": "P^{ab}{}_{cd}=S^{[a}{}_{[c}S^{b]}{}_{d]}+\\delta^{[a}{}_{[c}S^{b]e}S_{d]e}-\\frac{1}{6}\\delta^{[a}{}_{[c}\\delta^{b]}{}_{d]}S^{ef}S_{ef}.", "f31cbaf28beac86705efe1deaac9110b": "\\mathcal{F}= \\int \\mathbf{H}\\cdot\\operatorname{d}\\mathbf{l}", "f31ceecce8fdcd7f0692f3e481272863": "\\mathrm{diag}(1, -1,-1,-1) ", "f31cf0c024dae4011659aae3e688afc9": " t^\\star = |t+U|\\ ", "f31d24019dcd34e5ac42895640a8c0da": " \\mathbf{F} = \n\\begin{bmatrix} -\\mathbf{A} & \\mathbf{Q} \\\\\n \\mathbf{0} & \\mathbf{A}^T \\end{bmatrix} T", "f31d4a1341ba5d5afd245555495d119b": " \\therefore I_n = -\\frac{\\sin{ax}}{(n-1)x^{n-1}}-\\frac{a}{(n-1)^2}\\left (\\frac{\\cos{ax}}{x^{n-1}}+aI_{n-2}\\right ) \\,\\!", "f31dcb1acd375f5487a81b2a0ccae8ed": "\\tau = \\mu \\frac {\\partial u} {\\partial y} ", "f31e131308f03a302ded99a20596f7f2": "x = (1, \\lambda, \\lambda ^2, \\dots) \\in l^p", "f31e1eef20f64733a18c538073e78396": "M1", "f31e42008e2c788e32a6f98562fccfd9": "r(X,T+1) \\leftarrow r(X,T),\\ \\hbox{not }\\sim r(X,T+1)", "f31e58370147d9e44f06594fa9b2801e": "\\mathbb{I}", "f31e71b1a6f466338f9c9206725cdf6c": "a_j = \\sin \\frac{\\pi}{2} \\left [\\frac {(2j-1)}{n} \\right ]\\qquad\\mathrm{j = 1,2,3, \\ldots, n}", "f31eca39c20a3ef7a805e04b36bfc6f5": "S_v = \\log_2", "f31f123f5b510e1c58b2be1990dcada8": "g\\,", "f31f482a8699e0e16509444cf45b75bd": "S \\Rightarrow aS\\Rightarrow abS\\Rightarrow abbS \\Rightarrow abb", "f31f4e46cadcc8f537a9e805f3403f44": "\\ Y = F(K,AL)", "f31fb6b5f8cdc4c05bbe364242f275b0": "\\sum_{j=0}^{q-1} (-1)^j \\mathrm{res}^{|\\partial_j \\sigma|}_{|\\sigma|} f (\\partial_j \\sigma) = 0", "f31fbdebc60abe0973e7ed52d69fe11f": "A_{sn} = A_sFrac_{14/12(s)}", "f31fc2f8793595fc53202b7e8c22f85a": "PS = \\sum\\nolimits_{|\\alpha|\\le k} p_\\alpha \\partial^\\alpha S", "f31fc909a12351d722cce3e3d7f06da2": "\n \\begin{align}\n \\max & \\text{ } y \\\\\n -x +y & \\leq 1 \\\\\n 3x +2y & \\leq 12 \\\\\n 2x +3y & \\leq 12 \\\\\n x,y & \\ge 0 \\\\\n x,y & \\in \\mathbb{Z}\n \\end{align}\n", "f31fe3da748548d0fba1ee8875711a6c": "\\begin{align}\n\\operatorname{E}\\left[\\ln \\left (\\frac{X}{1-X} \\right ) \\right] &=\\psi(\\alpha) - \\psi(\\beta)= \\operatorname{E}[\\ln(X)] +\\operatorname{E} \\left[\\ln \\left (\\frac{1}{1-X} \\right) \\right],\\\\\n\\operatorname{E}\\left [\\ln \\left (\\frac{1-X}{X} \\right ) \\right ] &=\\psi(\\beta) - \\psi(\\alpha)= - \\operatorname{E} \\left[\\ln \\left (\\frac{X}{1-X} \\right) \\right] .\n\\end{align}", "f32009d28da73840c83f181ec1e5a013": "10^{-10}\\,m/s^2", "f32020c79407435c79821e222de4ceca": "A \\rightarrow B", "f3209c9e05a23c7d2064bf61f3fc50de": "i=1,\\dots,n\\,", "f320b95267e4d494f0eaa31c31b0ef21": "\\displaystyle{\\|u\\|_{(2)} \\le C\\|\\Delta u\\|_{(0)} + C \\|u\\|_{(1)}.}", "f320ceece1d956f5506d7b803f516c4f": "M = q^k", "f321178d533809b119150924145c0572": "\\mathcal{E}=-\\frac{\\partial \\Phi_\\mathrm{B}}{\\partial t}", "f32172eeee8136483dce73324eca7313": "\\ln\\mathbf{B}", "f3217c8b0c3b6f5150aa18501d107ee1": "\n\\begin{align}\n D_p &= \\oint p\\mathbf{n} \\cdot\\mathbf{i} \\; \\mathrm{d}A,\n \\\\[1.2ex]\n Y &= \\oint p\\mathbf{n} \\cdot\\mathbf{j} \\; \\mathrm{d}A.\n\\end{align}\n", "f321d218f403e5cc8527a28f6a7399ca": "{49!\\over (49-6)!}", "f322053cbe43b9005e69297954435db7": "(-1)^D", "f3220d45fbf7e78cccc25eb8590c21f8": "\\begin{matrix}\np \\oplus q & = & (p \\land \\lnot q) \\lor (\\lnot p \\land q)\n\\end{matrix}", "f32233dc05b17159ccddc1f9bb3ad1da": "F_2 = F_1 + F_0", "f322344205b95cfebd927e2246ec9720": "1 +2\\sum_{k=1}^\\infty\\cos(kx) = 0?", "f322651305d63b250567eda93cd74d05": "y-\\widehat{(W_r -1)}", "f32280bed4c2493f206cc9b03ef1d3a9": "f(t)=\\theta t+x", "f3228deeaf439e0bedd31186fceb9970": " f_1 \\ge f_2 \\ge \\cdots \\ge f_n \\ge 0", "f3233567704521894e7bef40b3c4d29f": "g^{ab} \\psi_{;ab} = 0", "f32390832be92ecff013f1e2ff0f3a8a": "D_q= \\frac{1}{x} ~ \\frac{q^{d~~~ \\over d (\\ln x)} -1}{q-1} ~, ", "f323a957ab5e1ed93e41ae1ea33c7c12": "q\\overline{q} = \\textrm{sum\\ of\\ squares}", "f3240e7ddd74a24be6c63bf19ed0280d": "\\theta_1 > \\theta_0", "f324166ce2620789d47b55e651bdb524": "(Tf)(t):=\\sum_{n=-\\infty}^{\\infty}m_n \\widehat{f}(n)e^{int},", "f32423d2b9868cef26c41e39c2d1edc3": "v_{0}", "f3246c429ab0e5e9c91b0a6b1cdbbd40": "\\sigma\n\\equiv_{b}\\tau \\mbox{implies } \\sigma\\;\\mid\\;\\rho \\equiv_{b}\n\\tau\\;\\mid\\;\\rho", "f324bfa40d6ebfe0659f4266be3ca944": " P = \\Big[ \\,v\\, \\Big| \\,w\\, \\Big| \\,x\\, \\Big| \\,y\\, \\Big] = \n\\begin{bmatrix}\n-1 & 1 & 1 & 1 \\\\\n 1 & -1 & 0 & 0 \\\\ \n 0 & 0 & -1 & 0 \\\\\n 0 & 1 & 1 & 0\n\\end{bmatrix}. ", "f32549cbc60d5f2f6bf136cf7001e0b8": "\\scriptstyle \\tau>0", "f325528da8e34493a8612e5265d9eece": "\\theta = \\pi \\, \\text{and} \\, \\theta = 0", "f325720c1fc3e6f586afc6aa222d06c3": "\\frac{\\partial}{\\partial g_i}", "f325abb4ff6869412f5a4d41fed0087a": "X_t(\\omega) \\in E", "f325bc64e783ee9d129e1016b8da0d49": "\\Delta w = \\left( \\frac{1}{\\lambda_0} - \\frac{1}{\\lambda_1} \\right) \\ , ", "f325c18cbab0412523c82fcc0aab1bc1": "\\Delta g", "f32604dc63ad734b61449b2fa12c0e48": "E_{\\rm color}", "f3261f89e42ab02b2b73bc3727148be0": "m_p \\rightarrow m_e + m_p ", "f32641b10c02b1369d48a7342a07cdae": "\nn=\\sum_{k=0}^{L-1}d_k(n)b^k\n", "f326a8f29e64ddb71968e7b38e6e823a": "=p_1p_2 + (1-p_1)(1-p_2)\\ ", "f326bf29c3bc8bad0189648ad0937dd9": " \\alpha_j + {d\\alpha_j \\over dt}t +{d^2\\alpha_j \\over dt^2}{t^2\\over 2!} + \\cdots", "f327416432346a6004db72c32e9ecf39": "\\scriptstyle (t)", "f3277a2610d4836013b07df06504350c": " \\dfrac{\\partial u}{\\partial x}+\\dfrac{\\partial v}{\\partial y}=0 ", "f327df50729510d2fd499f5287fef5d3": "\\begin{pmatrix}(8,1)_0\\\\(1,3)_0\\\\(1,1)_0\\\\(3,2)_{-\\frac{5}{6}}\\\\(\\bar{3},2)_{\\frac{5}{6}}\\end{pmatrix}", "f3280b0cfc6391830364b8186deeafb9": "{x^{n+2} -x^{n+1} -x^{n} = x^{n}(x^{2} -x -1)}", "f32839445f5f07e4d04826770200a328": " M^{1/2}( M^{1/2})^\\top = M", "f328554489e5eabd31dcf521616ba67d": " T_1=T_2=..=T_m=T", "f3285a21add6755b328b831114325c76": " \\mathcal L_{\\Omega_\\alpha}\\omega = d (\\iota_{\\Omega_\\alpha} \\omega) = d\\alpha", "f328bab554836bc9f0bde5caf56e71aa": "q = h^{0,1} = \\text{dim} H^1(S,\\mathcal{O}_S) =0", "f329c2a4d7e4914825c18da78fee939c": "{S_{mn}(-ic,\\eta)}", "f32a0aa8ce9df990f69b96ce15bb3801": "\nf(t) = f_{0} \\sin 2\\omega_{p}t\n", "f32a38cd9ee993eec5f6cb210ff7d6a1": "a_0, a_1, \\dots,a_d", "f32a801717155caace6d92a28e4a9507": "\\displaystyle{\\Delta f|_\\Omega = 0,\\,\\, f|_{\\partial\\Omega} =g,}", "f32ab19af51197b87f6e4f7962a01e13": "even(s(0))\\leftarrow \\hbox{not } even(0)", "f32af65e92b4f197f5d80545cc425567": "\n\\langle A \\rangle_{\\text{mc}} = \\frac{1}{\\mathcal{N}} \\sum_{\\alpha'=1}^{\\mathcal{N}}A_{\\alpha' \\alpha'},\n", "f32b9c5eabbf836462c982b14bb1550d": "\\mathcal{I}{{\\left( \\beta \\right)}_{m,n}}=\\frac{\\partial {{\\mu }^{\\text{T}}}}{\\partial {{\\beta }_{m}}}{{\\Sigma }^{-1}}\\frac{\\partial \\mu }{\\partial {{\\beta }_{n}}}", "f32ba72361dcb4a31a5e5ad035418a59": "\nx^2+\\left(y+\\frac{A}{2\\psi}\\right)^2=\\left(\\frac{A}{2\\psi}\\right)^2.\n", "f32bc4488246bb95edd814cf6670fd71": " \\int_{C_3} L(x,y)\\, dx = -\\int_{-C_3} L(x,y)\\, dx = - \\int_a^b L(x,g_2(x))\\, dx.", "f32bcfa22cc58a901057e2879f190d4c": "\n\\underbrace{M > 0,~ p_c > 0,~ c \\geq 0,~ 0 < \\alpha < 2,~ m > 1}_{\\mbox{defining}~\\displaystyle{F(p)}},~~~ \n\\underbrace{0\\leq \\beta \\leq 2,~ 0 \\leq \\gamma < 1}_{\\mbox{defining}~\\displaystyle{g(\\theta)}}, \n", "f32c1858c6efbee764548dc7513d1b2c": " \\int_0^1 \\sqrt x \\, dx \\,\\!.", "f32c293a532327aa28ee5241b506ee17": "b_{\\nu, n}\\left( 1 - x \\right) = b_{n - \\nu, n}(x)", "f32c30ec36c84094a606041bfd5b753a": "\\max\\,\\{|z_i|\\}", "f32c5cf73f82afe59c5491a607da8f59": "\\mu\\,(x)=\\frac{F'_X(x)}{1-F_X(x)}", "f32c76d5d66120f2c5e287eb9031b7a4": "Q_\\mathrm{solar}", "f32d412a6a8fc21a64ee46d837ed3741": " {}^3\\!P_0 \\, ", "f32d8360a99498f5f6496e29f8c3edd6": "T_k = \\min\\left[ t:\\begin{align}\n(k) \\text{ jobs in the system at time } 0^+\\\\\n(k-1) \\text{ jobs in the system at time } t\n\\end{align}\\right]", "f32d96f0a2a52e2d8be1bc0d86503eb0": "1 = \\sum_{k=2}^{\\infty} a_{k-2} x^k - \\sum_{k=1}^{\\infty} a_{k-1} x^k + 2\\sum_{k=0}^{\\infty} a_k x^k.", "f32dc1f4fffb758ec0b34b0105044701": "| f |_{0, \\alpha,\\Omega}\\;", "f32e197fc0e6554b3574938f60c7e695": "f: \\mathbb{R}^n \\times \\mathbb{R} \\mapsto \\mathbb{R}^n", "f32ece21ffe4ea74a980f7edb1d70700": "V(F) = W(T_1)\\cup \\cdots \\cup W(T_e), \\, ", "f32f1c16441e5e95686ce8b4851974ed": "\n \\mathcal{M} = \\frac{1}{1+\\nu}(M_{11}+M_{22}) = D\\left(\\frac{\\partial \\varphi_1}{\\partial x_2}+\\frac{\\partial \\varphi_2}{\\partial x_2}\\right) = 0 \\,.\n", "f32f2301da2f5338b7e142bc3e79deb2": "\\sum_{i=min}^{max}f(i)", "f32f40560d3da1eb9c95bf2235e07a1a": "P(x_k | y_{1:t} )", "f32f5232359a1509b8e3c082b316dee7": "\\alpha', \\alpha'' \\in \\Phi^+(\\gamma)", "f32f7d2d3ade05337d7e157b6a7ceeea": "=-F\\,r^2 dv^2+2dvdr+\\hat{h}_{AB}\\big(dy^A-G^A\\,r dv \\big)\\big(dy^B-G^B\\,r dv \\big)\\,,", "f32fa9f691d4d3b6d4b9fc2acdc87d39": "\\hat{f}(a)", "f32ff17860e4bdf92ca2e841560566af": " \\left( n \\gg Tm \\text{ has to hold } \\right) ", "f3309006f02a6767f150eba43e8bfcdf": "k_{\\rm A} = 1/2", "f330ae8b5f2caa8f3643104dad015d57": "\\mathsf{S}(a) = e^i \\wedge \\mathcal{P}_B^{\\perp} (a \\cdot \\partial e_i).", "f330c169575e1cd37b5497c712f1f851": "\\alpha = \\tfrac{n}{m}", "f330cbe660b896fff1797fe311a81898": "\\begin{bmatrix} \\eta_1 \\\\ \\eta_2 \\end{bmatrix} ", "f330fbbb97c49bbb79451a2f60ebe548": "a^\\bar{\\alpha}e_\\bar{\\alpha} = a^\\beta L^\\bar{\\alpha}{}_\\beta L^\\gamma{}_\\bar{\\alpha} e_\\gamma = a^\\beta \\delta^\\gamma{}_\\beta e_\\gamma = a^\\gamma e_\\gamma ", "f331d28e166366fda04e7502a5e8b421": "x = \\sqrt[3]{u} + \\sqrt[3]{v}", "f331fd63f516edaa0a1e5502edc533d2": "\\frac{\\partial u}{\\partial t}-\\alpha\\frac{\\partial^{2}u}{\\partial x^{2}}=0", "f3327d85237f529ab9da1aee0573ea52": "g_n = p_{n+1} - p_n", "f33292b9f80fafd63bf804e89626ecc4": "O(2^{i})", "f332963245db05009bdd56db064371d2": "H_{\\lambda}(i,j)\\setminus \\{(i,j)\\}", "f332c01562729286271a423d38eed043": "s,t \\in S", "f332ed8e6648a18ae1013770919ecc41": "\\mathbf{v}^b=\\mathbf{v}.", "f333301ccd7f9925f1b96b17a85be6e4": "\\frac{a'_\\max}{a_\\max}=\\sqrt{\\frac{u-1}{u}}\\equiv \\sqrt{\\frac{u'}{u}}.", "f33367cd04447a928c43e3191d72ab3a": "\\frac{5^{a-b}}{10^a p^k q^\\ell \\cdots}\\, ,", "f33377d3b31bcadad1b15b844f4b91ac": " \\lim_{\\lambda_B\\rightarrow 0} \\left [ \\frac{N_{A0}\\lambda_A}{\\lambda_B - \\lambda_A} \\left ( e^{-\\lambda_A t} - e^{-\\lambda_B t}\\right ) \\right ] = \\frac{N_{A0}\\lambda_A}{0 - \\lambda_A} \\left ( e^{-\\lambda_A t} - 1 \\right ) = N_{A0} \\left ( 1- e^{-\\lambda_A t} \\right ), ", "f33387bd06cb2ab4f972b921b33466be": "S = (\\mathbf{X},\\mathbf{Y}) = (\\mathbf{x}_1,y_1),\\ldots,(\\mathbf{x}_n,y_n)", "f3339724fefc5331470ab3d385a5fa2f": " \\omega_{\\mu \\nu} = - \\omega_{\\nu \\mu} ", "f3339e51ebdfe8b794719f5b1639cdfe": "\\mathcal{L}_g = -\\frac{1}{4}W_a^{\\mu\\nu}W_{\\mu\\nu}^a - \\frac{1}{4}B^{\\mu\\nu}B_{\\mu\\nu}\\,\\!", "f333bec78bb36a652158dd3260114920": "N=6", "f333c047a5c34d3293eab0f10a623a48": "\\mathbb{Z}\\oplus \\mathbb{Z}", "f333e2ff9c82a04f43c835e670efe8b8": "\\sum_{n=0}^{p-1}\\xi^{n^2} =\n\\begin{cases}\n\\sqrt{p}, & p = 1 \\mod 4 \\\\\ni\\sqrt{p}, & p = 3 \\mod 4\n\\end{cases}\n", "f333edd38b6da83849c7f21201d8bb0c": "\\scriptstyle (x, y), x \\in X, y \\in Y", "f334842581a8fd6f87ee3708f8923373": " \\gamma_0 =1", "f334b303cb6ee691cdbaf189e2554e63": " \\mathbf{E}(\\mathbf{r}) = E_0 e^{ -i \\mathbf{k} \\cdot \\mathbf{r} } ", "f334d602ea52dbc1e2f622e0389e9c9c": "f(\\textbf{x}_{e}) < f(\\textbf{x}_{r})", "f334e857190fe0916aa7e78f13299756": "{x^2 \\over a^2} + {y^2 \\over a^2} + {z^2 \\over a^2} = 1 \\,", "f3350a08f3f6524df36981eec69777db": "0.91\\overline{6}", "f33520b46f7bfec51fa029b5407a356e": " P_{t\\mid t-1} ", "f33534b92af8a6902e39035a2e8d1089": "\\Delta:=d(K,\\partial\\Omega)>0,", "f335c282a01807ab9e7cf34e56507d6c": "\\omega_s", "f336114736e2da47c78a0e5c93758379": "\\delta(x-y)", "f3365ca1f6155c43f0f93a1f1dec44ce": "f(x)=x^5-1", "f33694750498246b83d37ab6407c0773": "\\rm{L}(X) = \\aleph_0", "f336bcd2733139dfe3ccb5ac2f6469df": "C_C(v)=\\sum_{t \\in V\\setminus v}2^{-d_G(v,t)}.", "f336c75a2e40ce305ad18c4d626de9f1": "\n \\rho \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\mathbf{v}\\right) = -\\nabla p\n ", "f336e1f278dc8004d476d10ee163c3ef": "r={1\\over3}R={a\\over\\sqrt{24}}\\,", "f33721e277fd108e2ad340e16da14775": "-N_\\text{s}/N_\\text{p}", "f3374e2a4493776d7c05a4e1e8769a8e": "1\\le bn.\\\\\n\\end{align}", "f34de2499162e160a76b02ca17a8007b": "\n\\overline{z} = \\overline{C}+i\\overline{S}\n", "f34e4785a02465ff639805dd467692b1": "f(x_1, x_2) = x_1^T W x_2", "f34e6d245cb4bdece5f08eafbab08fc2": "nH(p)-o(n)", "f34e843a2039e2ca802ca30653ff7f64": "S\\subseteq X", "f34fba915bd77b101798794bf51fc84c": "\\begin{align}\n\\operatorname{tr}(XX^*) &= \\operatorname{tr} \\left ( P_VTT^*P_V - P_VTP_VT^*P_V \\right ) \\\\\n&= \\operatorname{tr}(P_VTT^*P_V) - \\operatorname{tr}(P_VTP_VT^*P_V) \\\\\n&= \\operatorname{tr}(P_V^2TT^*) - \\operatorname{tr}(P_V^2TP_VT^*) \\\\\n&= \\operatorname{tr}(P_VTT^*) - \\operatorname{tr}(P_VTP_VT^*) \\\\\n&= \\operatorname{tr}(P_VTT^*) - \\operatorname{tr}(TP_VT^*) \\\\\n&= \\operatorname{tr}(P_VTT^*) - \\operatorname{tr}(P_VT^*T) \\\\\n&= \\operatorname{tr}(P_V(TT^*-T^*T)) \\\\\n&= 0.\n\\end{align}", "f34fcf1e41216e6a29c72952f9850a0a": " \\left(V,E\\right)", "f34fdf6da203a4d284c8ead6d849653b": "f=D^\\alpha F.\\,", "f34fe08156dd5529569a4e73a348a199": "(X,\\,d)", "f3504cfa371ac34ca0926aa2e8c7018b": " \\tau_{(12)} : V\\otimes V \\rightarrow V\\otimes V", "f350a46c8c43d940f432b5ac749d42e5": "A = (R_1 R_3 C_2 C_5)\\,", "f350ae3191c410e48393e5ea74b5c80f": "z \\equiv \\sqrt{1 - \\left(\\frac1 4 x\\right)^2 - \\left(\\frac1 2 y\\right)^2}", "f350dcfe7077a68a305f69f30727e19f": "\\frac{\\alpha}{s(s+\\alpha)} ", "f350fea4ad9361b40700995fbf9bc0a3": " \\frac{| \\theta - \\mu |}{ \\sigma } \\le \\sqrt{ 3 } ,", "f350ff02f50b759501d1025537830caa": "-a \\pm \\sqrt{a^2 - b^2}", "f35119e5fd3e7d610591177ce2437318": "p(\\theta|y)\\;", "f351256605d0f885e3425c565fd932f0": "e^{tX}e^{sX} = e^{(t+s)X}.\\,", "f35129576d0d3f182c23434e9029fe51": " \\mathbb C^{56} ", "f35182526fad11cc3f7314bff6dfa688": "\\phi = \\arccos{(\\frac{r_p v_p}{r v})}\\,", "f3521aaa3c07ce9b4f91e6461ff58a66": "x = (x_1, x_2, x_3, \\ldots) \\,", "f3528223d90a044b823b399b34f5c518": "k = k_0 e^{{-E_a}/{k_{B}T}}", "f352cf63bf5baaecf99101ff011c9c46": "\\dfrac {Q_s} {Q_t} = \\dfrac {Cc_{O_2} - Ca_{O_2}} {Cc_{O_2} - Cv_{O_2}}", "f352dff7beb0bdd3a7b4a6320d392165": "0 \\leq N < \\frac{m \\omega_c L_x L_y}{2\\pi\\hbar}.", "f352e235d35e218d377e0111fc3885c2": "\\scriptstyle \\infty ", "f352efd7cbc58416b902d873d54c8039": "\\mathcal{F}_x := i^{-1}\\mathcal{F}(\\{x\\}),", "f352f5cd7a13ecf608bad0924d9f3eaf": "\\begin{matrix} {10 \\choose 2}{4 \\choose 2}^2{32 \\choose 1} \\end{matrix}", "f352f9fe8675f68c48aaaa3461bef37e": "\\psi_1(\\beta) - \\psi_1(\\alpha + \\beta) = \\frac{\\part\\, \\ln G_{(1-X)}}{\\partial \\beta} > 0", "f3538fa06f6b4c5b3e39aec6f31d9876": " k_1 = f(t_n + h, y_n + h k_1) \\quad\\text{and}\\quad y_{n+1} = y_n + h k_1, ", "f353b1fab53cf552a1f28a16136efa1e": " V = f, E = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) ", "f353e405772af31e2ea2c2762917bfb2": "\\mathrm{Sp}(2n,\\mathbb C)", "f354099f546747666204c5804fd93593": "\\partial_{x_0} H(x,\\nabla u(x)) \\neq 0", "f3543d3be7fa8f954f06a923b27d7831": "\\,\\Sigma \\subset \\Gamma", "f3549b88d983283a1258ea89e154add5": "\\frac{L_{\\nu_{H\\beta}}}{\\int_{\\nu_0}^\\infty \\frac{L_\\nu}{h\\nu} d\\nu}\n \\approx h\\nu_{H\\beta} \\frac{\\alpha_{H\\beta}^\\text{eff}}{\\alpha_B}", "f354dfcd1bdf74a1a88dfdfaa5cad18a": " \\int_{\\theta_j} P(\\theta_j;\\alpha) \\prod_{t=1}^N\nP(Z_{j,t}|\\theta_j) \\, d\\theta_j .", "f354e2d5524289f0d9fd6ee573c50910": " \\operatorname{inc}\\ \\operatorname{init} = \\operatorname{value}\\ (f\\ x) ", "f355c43f19d3f8d261103c222519f204": "B\\otimes A^*", "f356b4683c7f3cd4f4e09aa1cc79448e": "\\gamma_y(t)=i\\,y+w\\,t", "f356c78e722bfd08c8fdfabacc7bf290": " \\mathcal{L}_X f", "f356d63a08c52140ea9c5b31cf5bf6b0": " \\phi_{sb} (r) = \\max \\left[ 0, \\min \\left( 2 r , 1 \\right), \\min \\left( r, 2 \\right) \\right] ; \\quad \\lim_{r \\rightarrow \\infty}\\phi_{sb} (r) = 2", "f356f844663612bfd09391751b15afff": "\\langle \\cdot,\\cdot\\rangle_1", "f3572228d9f2f6c084f695bc6bfc2a2b": "\\lim _{q \\rightarrow 0} S(q) = \\rho kT \\chi _T= kT\\left(\\frac{\\partial \\rho}{\\partial p}\\right)", "f357389acbb0890d5c307d8ee12ffcac": "\\mathrm R\\underset{hv}{\\longrightarrow}\\mathrm R^*", "f3575ba87afb208970c913c13f6c2487": "\\alpha_{U,n}(t)=\\sqrt{n}(F_{U,n}(t)-t),\\quad t\\in [0,1].", "f35856d425f2efe0265a5ac7aa2bd096": "dV =r^3 \\left(\\sin^2\\psi\\,\\sin\\theta\\right)\\,dr\\wedge d\\psi\\wedge d\\theta\\wedge d\\phi.", "f3589cdff4b48b3e8613d6e23749ac54": "\n\\psi_1 = POL + 0.002629 \\alpha_1^T \\beta_1^T\n", "f358bea6faa9a972d1f2ebfba6f16c6f": "\\int \\operatorname{Si}(x) \\, dx = x \\operatorname{Si}(x) + \\cos x", "f3594b31e61bb8f08d2e518a73ed95d8": "E_A^A - E_A^D > U_D", "f3597f28464dbd33958fa9c257b5d0e9": "T_j\\xrightarrow{d} T", "f359a3928dce31f94f14ca9cd94cd957": "\\ +F^{n-1};", "f359f3c6850dd40573f92b531dfcf59d": "P_{\\rm dissipation\\, in\\, regulator} = (V_{\\rm in} - V_{\\rm out})\\times I_{\\rm out}", "f35a0e51e2b4b600d12b89dfc781b903": "\\begin{matrix} {2 \\choose 1}{2 \\choose 2}{3 \\choose 1}^2{40 \\choose 1} \\end{matrix}", "f35a223bfdfba2ecb9e870899b831eb0": "\\textstyle \\frac{h}{\\lambda} \\leq l", "f35a38cec1bf9c0b549d4b9430f0b9ef": " \\langle \\Phi(v_1,z_1) \\Phi(v_2,z_2) \\dots \\Phi(v_n,z_n)\\rangle = (\\Phi(v_1,z_1) \\Phi(v_2,z_2) \\dots \\Phi(v_n,z_n)\\Omega,\\Omega).", "f35a4293bead54c2c176b7092d798a60": "\nU_{1}=R_{1}I\n", "f35ab2425e49a1f328387026b4065ed9": "\nk=(2m-1)\\frac \\pi {2a},\\quad \\quad m=1,2,3,...\n", "f35ad66baf431dfd445fd7ac511319ce": "e_{t}", "f35ad914c9aa348f8ecbab0e82172d1d": "e = \\left( \\begin{array}{cc}\n0&1\\\\\n0&0\n\\end{array}\\right)\n", "f35ae68f2191935c67897e8ebd6cf9b0": "\\phi(z)=-e^{iz}", "f35b12d8cd8b88742737421931b91b21": " x \\cdot y = x ", "f35b1c6b85651b30932f9ee9d49bc682": "\\sqrt{1 + \\frac{2 V_\\text{esc}} {\\Delta v}}", "f35b5c56de87c847c0b1579403beaf5b": "V_S^\\prime=aV_S", "f35b6eee15d693fc06d20cdf39d99f85": " \\Delta G_{DF} \\,", "f35b8089cee479339780a884c21ad4c0": "\nU = \\text{span}\\{u_1,\\ldots,u_k\\}\n", "f35bd6d7dee8d7b9f56620f4adadce5b": "p(\\theta | y, \\xi)", "f35be6dca9a2584016aeb2e59ba2fd07": "\\pm\\left(\\sqrt{\\frac{5}{2}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{-2}{\\sqrt{3}},\\ \\pm2\\right)", "f35c4410156331d0a72207bed1fd936e": "\n\\ddot{x} - 2\\Omega_0 \\dot{y} = -x R{d\\Omega^2\\over dR} +f_x", "f35c5e849aa80473f6b6dfa8b3e5a07d": "r\\to\\infty", "f35cabd1553b9fdccd29bf28df0a8ad6": "\\nu=\\frac{1}{C}[D_1 \\frac{\\partial C_1}{\\partial x} + D_2\\frac{\\partial C_2}{\\partial x}]", "f35cc5d2a6a3085fece3f950a4bad267": "1/r^{i+2} ", "f35cd7a894a70f5cbf9350906b5b8822": "\\frac{\\partial \\, \\mathbf{g}(x\\mathbf{A})}{\\partial x} =", "f35d1f5b6ef6f2e75489dba866f95837": "F \\approx \\frac{9.6 \\times 10^4 \\ \\mathrm{J}}{\\mathrm{mol} \\cdot \\mathrm{V}}", "f35da341c05112a593c7993de60d15ea": "m_0 - m_1=m_1 (e^{\\Delta V\\ / v_e} - 1)", "f35da4f28e1ca7f52687dd2ff2b9cf04": "(g_-)_k = (-1)^k \\cdot g_k", "f35e48d733b5dd7ad17fed0ee2071bbd": "u' \\approx u-v \\ . ", "f35e5abf539ee9122bea4c1b46b97dbf": "{\\mathbf x}", "f35ea79475b2c5eedd1a1ffcdcd5db25": "\\kappa = 0.37464 + 1.54226\\;\\omega - 0.26992\\;\\omega^2", "f35ee2879d0bd10a17a2f02070dc1105": "F_{A \\rarr B}", "f35f2398c304252f635ec7184ec83a2b": "2a\\tau", "f35f6bab95dcbbd1dce21b76ad82b977": "\\exp(- i E_n t / \\hbar)", "f35f76809ad0a5c074a34fa8a64d5334": "\\theta \\in \\Theta_1", "f35f8fa85556e5f03051d4930e3ae07f": "\\epsilon={v^2\\over2}-{\\mu\\over{r}}={\\mu\\over{-2a}}", "f35fac76723cc7847fbb0512ebbab35e": "F = X_u \\cdot X_v = 0", "f360748bd1a2e36307b75fe6ab87134b": "m_j ", "f36094f9db2eda2e6cf39cab6efd6126": "n_v' ", "f360c35b651209ed35d84d237d5728ac": "\n\\varphi\\left(e^{-5\\pi} \\right) =\\frac{\\sqrt[4]{225\\pi+ 100\\sqrt5 \\pi}}{5\\Gamma(\\frac{3}{4})}\n", "f36105f0eefc41b0732fe46cd1bde94f": "O\\left[m + n\\log n + \\sum_{j=1}^m \\log(|i_j-f| + 1)\\right]", "f3621031b78287e75b34057f79e87e97": "\\arg \\min \\limits _{(x,y)} {D(f(x, y)), D(f(x - 1, y)), D(f(x, y - 1))}", "f3622fd24680d268c234caf47e5c904b": "y % p.a.", "f362362d15c46b9d96836b55a9609924": "\\neg alive(1)", "f3624a3e00b57aa8b69ee07ae2f1aaa1": " L[B] = L^{+-}[B] + L^{--+}[B] + L^{--++}[B] + L^{--+++}[B] + \\cdots ", "f3625575174d844b8f89e88dcb6ec66a": "d_{\\kappa\\kappa}=1", "f3625a714873a79632d5c5bb81da4aab": " \\alpha_1 = \\arctan \\left( \\frac{\\cos U_2 \\sin \\lambda}{\\cos U_1 \\sin U_2 - \\sin U_1 \\cos U_2 \\cos \\lambda} \\right) ", "f362685164db2f780183720a4f9597cb": " \\gamma _{ws} = \\gamma _{ws}^0 - \\frac{CV^2}{2} \\,", "f3629b90ebe6f5ad8b897810439aff94": " \\lim_{X \\rightarrow 0^{+} } F(X) = \\lim_{X \\rightarrow 0^{+} } \\alpha \\log\\left( \\frac{K}{X}\\right) = +\\infty ", "f362a5de4d2e9f31f6a799b8772421c8": " \\lambda_1", "f362d170cf3a1de307d93708c03537dd": "p_1=p_2", "f3631d696e6e3b103ea6936dcbf37281": "n\\bar p \\pm 3\\sqrt{n\\bar p(1-\\bar p)}", "f3633e59a00110573e55c249f3c40e33": "\\widehat{X}= Z\\widehat{\\delta} = Z(Z^\\mathrm{T} Z)^{-1}Z^\\mathrm{T}X = P_Z X.\\, ", "f363402240ca8831a4411bb58bd36cf1": "\\psi \\to \\phi ", "f36352e9e857d7f9a64ba6c268179479": "G = \\sqrt{\\mathrm{precision} \\cdot \\mathrm{recall}}", "f36360e0f6eb48ddca50eaa06c6d4a10": "A^*A\\,\\!", "f363784b7954302db39f6757f7b160a9": "g(E \\cup F) + g(E \\cap F) \\leq g(E) + g(F)", "f3637befb4099e83b883fb3e2ff15bde": "\\delta^{(p)}", "f363a9f4f5a8119a5b526809078f436b": " \\int fg \\leq \\int f^* g^* .", "f364b87b033826496addefd2d5b3b750": "\n\\mathbf C ={1 \\over 35}\n\\begin{pmatrix}\n - 3 & 12 & 17 & 12 & - 3 & 0 &0&0&0&\\dots \\\\ \n 0 & - 3 & 12 & 17 & 12 & - 3 &0&0&0&\\dots \\\\ \n 0 &0& - 3 & 12 & 17 & 12 & - 3&0&0&\\dots\\\\\n 0 &0&0& - 3 & 12 & 17 & 12 & - 3&0&\\dots\\\\\n 0&0 &0&0& - 3 & 12 & 17 & 12 & - 3&\\dots\\\\\n \\dots\n \\end{pmatrix}\n", "f3651e7f2f0a08aad67d32e2d1456dc3": "\\Phi(z, s, q) = \\sum_{k=0}^\\infty\n\\frac { z^k} {(k+q)^s}", "f365ae963cc693af13cbaccd8c76ed37": "\\Rightarrow_R", "f365f4193dafced5d4f515258ae6f136": "\n\\begin{align}\n\\beta f = &- \\dfrac{\\beta^2 J^2}{4}(1-q)^2 + \\dfrac{\\beta J_0 r m^r}{2} \\\\\n&- \\int \\exp\\left(-\\frac{z^2}{2}\\right)\\log \\left(2\\cosh\\left(\\beta J z + \\beta J_{0}m\\right)\\right) \\, \\mathrm{d}z.\n\\end{align}\n", "f365fd3414ccbfaccf15cb2dc53e60fd": "-\\nabla \\times \\mathbf{E} = \\frac{\\partial \\mathbf{B}} {\\partial t}", "f3663bed1e202ace3cef63b9f6380d72": "f(t) = akt^{a-1}+f_0\\!", "f366800b40a371eea01dc0dc78553e56": "\\bigcup_{A\\in\\mathcal{V}:A\\text{ meets }N}\\{U\\in\\mathcal{O}:A\\text{ meets }U\\}\\,", "f366a673547f76d0065f51920c8e1762": "\\mathcal{F}:\\mathcal{S}(\\mathcal{H}_{\\rm in}) \\rightarrow \\mathcal{S}(\\mathcal{H}_{\\rm out})", "f366a8b82f599c91167da0e0eb3b4ddc": "c=\\sqrt{D ^2-4 a^2}", "f366f755af08f0bf426f4571baf36251": "\\rho = \\Omega \\psi(\\Omega^2)", "f367257f4d3d7ee811d2b63a97ab5495": "\\hat{\\textbf{x}}_{k\\mid k}", "f3672635f7426c68f643277db2002abc": "V_k(\\mathbb C^n) \\cong \\mbox{SU}(n)/\\mbox{SU}(n-k)\\qquad\\mbox{for } k < n.", "f36758580d060ba40ee5a37da16e6b75": "\\theta_2+(\\theta_3+\\theta_4)=90^\\circ", "f36772ba884d53d13fd521e7f029cba8": "k=0, 1, \\dots, n.", "f3678cd78e3981a8262feab762247e11": "(+a,\\ 0)", "f36799991a1cf5a4d2220cc55ff87058": "\\tan^2\\frac{E}{2}\n=\\frac{1-\\cos E}{1+\\cos E}\n", "f3679f1e7fd456bb1a32f5604526785d": " f(z) = z^mg(z) ", "f367ef442984e9ee1c452b9cbbc7a3fa": "COLOUR", "f368255daf9f89fa4bff64001d261501": " C_{p^m} \\!", "f36864a32cd2c0d925e962d287cf9924": "\\lim_{n\\to\\infty} \\frac{s'_n-\\ell}{s_n-\\ell} = 0.", "f36894dc5fd86baef7e726dc8e6d396f": "|W|^2=A(\\theta,|\\varepsilon_{1}|)\\ \\sin^2{\\psi} \\ [(1+\\sin^2 \\theta /|\\varepsilon_1|)^{1/2} - \\sin{\\theta}]^2", "f368a9cba1e27f69ec2ec3c6bb5e8151": "O^*_\\infty(Y)", "f368aa1d2af54b2cde55c1438237d385": "E_\\text{pair}", "f368d98088e361c0c6e9a43365e794b4": "F \\left (\\tfrac{3}{2}a,\\tfrac{1}{2}(3a-1);a+\\tfrac{1}{2};-\\tfrac{z^2}{3} \\right) = (1+z)^{1-3a}F \\left (a-\\tfrac{1}{3}, a, 2a, 2z(3+z^2)(1+z)^{-3} \\right )", "f368e9b2216dde76b405528a8da20b28": "\\mathcal{O}_x\\,", "f368ec37941562664d72e604c7e3aca0": "\\pi^{-1} (x) = \\mathrm{T}_{x}^{*} M = (\\mathrm{T}_{x} M)^{*};", "f3690c66db7471c6a17f105db0d8ab3e": "A_g(z) = \\frac{1}{2}p_0(g) + p_1(g) \\frac{z}{z+1} + p_2(g) \\frac{z(z-1)}{(z+1)(z+2)} + \\cdots.", "f3695c2b86c1e977419f8932644be513": " \\frac{\\partial^2 \\Delta \\rho}{\\partial t^2} = \\frac{\\partial}{\\partial x} \\left[\\left(c_0^2 + p\\Delta \\rho + q\\Delta \\rho^2\\right)\\frac{\\partial \\Delta \\rho}{\\partial x}\\right] - h\\frac{\\partial^4 \\Delta\\rho}{\\partial x^4}, ", "f3695d72a7f4db1a4d157522e3061bae": "Q = \\iint_A \\bold{v} \\cdot {\\rm d}\\bold{A} ", "f3695f149ba978b80ae01ec2a21985df": "{(\\eta_b)_{max}} = \\frac{cos^2\\alpha_1(1+k)}{2}", "f3696adc5fc240d9113f82f7064334bf": "P\\left(O^{t}|S^{t}\\wedge\\pi\\right)", "f369810c847f67bb0474fa20f8339007": " \\left(\\dfrac{d^{\\frac{1}{2}}}{dx^{\\frac{1}{2}}}\\dfrac{d^{\\frac{1}{2}}}{dx^{\\frac{1}{2}}}\\right)x=\\dfrac{d}{dx}x=1.", "f36988016deb0dc9d4b5a1b9e6ac972c": "P_{\\rm abs}=P_{\\rm emt} \\qquad \\qquad (6)", "f36a1e93b0249a7738dc754d6a911806": "FF = \\frac{GFR}{RPF}", "f36a2110296c6140e8f5ad54460d748e": "|m/n|_\\ast\n=|m|_\\ast/|n|_\\ast\n=|m|^c_{\\ast\\ast}/|n|^c_{\\ast\\ast}\n=(|m|_{\\ast\\ast}/|n|_{\\ast\\ast})^c\n=|m/n|^c_{\\ast\\ast}", "f36a5e7d67e9d64622bfe7c43a85d90e": " R = {1 \\over r^2 } {v^2 \\over c^2 }", "f36a9470eb3dc9749c38eedaa64a2ecf": "v_e = \\sqrt{2\\mu/r_p}\\,", "f36a96e550f43b7a484ecae8985fd637": "x_\\xi x_\\eta + y_\\xi y_\\eta = 0", "f36acd8ddcdc7f9d5d6a1e17c8bdf4fc": "\\left[{n\\atop k-1}\\right]", "f36acf1b39011668bf51cb55fd2e5dff": "\\alpha \\in L", "f36b105b972cc7a02f6e4fdc190e3d39": "n_e\\tau_E", "f36b4e52803877332f47228d5f028284": "\\{(1,\\tfrac{3}{2}),(\\tfrac{3}{2},2)\\}", "f36b7bc1fb2633b8eb2c216342db7512": "\\textrm{RSS}/\\sigma^2", "f36bbd22585333d06cc42894bab079fa": "\\Delta S=Q_1/T_H", "f36bbe6cbb756bc704da50754b1aaa42": "x=1+r", "f36c0fc020119dc3a17a010e2a237a34": " (B_1,\\dots, B_m) ", "f36c29bd0a98e5b6f0b4c14358253a87": "r={{\\ell^2}\\over{m^2\\gamma}}{{1}\\over{1+e}}", "f36cb241e7afd3c279933b24562ac49c": "\\beta_0=\\gamma^{-1} \\beta", "f36cf28b2fe5fd4fe122d3c68e69a362": "z \\subseteq x", "f36d0de81a40cbdcdca9f6b3e33bd3e5": " h_{ab} = \\exp\\left(\\phi(\\sigma)\\right) \\hat{h}_{ab} ", "f36d433bcedca3e0946b06fe736bb013": "\\mathbf{\\nabla} \\times \\mathbf{B} = \\frac{1}{c}\\left(4\\pi\\mathbf{J}+\\frac{\\partial \\mathbf{E}}{\\partial t}\\right).", "f36d7b028b4d2872fd8533ec0c93da65": "\\displaystyle{(Lf,g) = (\\Delta f,g)_\\Omega.}", "f36d9c3dee9f22e2d6a3bac082aa4992": "\\partial^n : K_n^M(k)/\\ell \\to H^n_{{\\acute{\\rm e}{\\rm t}}}(k,\\mu^{\\otimes n}_\\ell)", "f36df5c5814b51e3993a8f7fc59c8dd0": "P_1= \\frac{n_2}{n_1+n_2}", "f36e1ddfda48fe3461dc82f5dd652b16": "(3 \\times 5) = (5 \\times 3)", "f36e2481abf088d27ce17492b81f2f1e": "\\frac{1}{\\Gamma(x)}\\,", "f36e34625939b1310b567006465105e2": "n_f = 2", "f36e8013e59906e215325caba104d1b5": "\\delta B = d \\otimes \\chi", "f36ec5de35612588a8d5b3f39d490d61": "QC(a, b) = \\int d \\lambda \\rho (\\lambda) A(a, \\lambda)B(b, \\lambda) ", "f36f5f1cc7b164a5f20aaef5e7339e0d": "\\sqrt[3]{x} = \\begin{cases} \\sqrt[3]{r}\\exp \\bigl( i ( \\tfrac13 \\theta) \\bigr), \\\\ \\sqrt[3]{r}\\exp \\bigl( i ( \\tfrac13 \\theta + \\tfrac23 \\pi ) \\bigr), \\\\ \\sqrt[3]{r}\\exp \\bigl( i ( \\tfrac13 \\theta - \\tfrac23 \\pi ) \\bigr). \\end{cases} ", "f36f61cacb63ec3e54dc6f84528ed5d2": "y = c_1 c_2 \\ldots q \\ldots q' \\ldots y_n ", "f36fd485934b63b7ff54283964694e1f": "\nx^4+2x^2y^2+y^4-x^3+3xy^2=0 \\,\n", "f37095873a385c6512cb745773e5963a": "y=x+1", "f370b22cb882ec820edb8d4a58477709": "s \\in \\alpha, t \\notin \\gamma", "f370b4b510c4af7e0ff90aed54450b90": " C = \\frac{f}{16} \\cos^2 \\alpha \\big[4 + f(4-3 \\cos^2 \\alpha) \\big] \\, ", "f370e2c024be7b24e9d92a899557e9d0": "r^2 = \\bold{r}\\cdot\\bold{r} = |\\bold{r}|^2 = x^2+y^2+z^2 ", "f37128b5262cfcb4651d80df4363bb0c": "q=2/3", "f3715c68747573ed35ddfa75ad11d47f": "\\varphi_x = -u, \\quad \\varphi_y = -v.", "f371679e53eb81eb3c988da8ea38c1bd": "=\\int_{0}^{h} A(y) \\, dy", "f3716be16975eaffdc01cc8b3c11dcc1": "\\pi_1 (X)", "f371bdcd57fd2a71338b3f2256fbb002": "\\Omega = \\Omega_1 \\times \\Omega_2 \\times \\cdots \\times \\Omega_n", "f371f94a65b9a2d60a99f462cd2990e6": "\\lambda^7 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & -i \\\\ 0 & i & 0 \\end{pmatrix}", "f3729ed0a1da26f2ce7019b87056d824": "\\textstyle a_i", "f372bab75d83c79f6c033e9edd5802d1": " A \\subset \\mathbb{R}^d ", "f372ee68eddf3bc6bb85e23d9eadc643": "J_2 = \\frac{2 \\varepsilon_E}{3} - \\frac{R_E^3 \\omega_E^2}{3 G M_E}", "f373224084940c2e5a350384a364eb4f": "\\eta(x,t) = a \\sin \\left( \\theta(x,t) \\right),\\,", "f373891b3f97a9a2108e670a656fcb66": "\\exists f \\forall x . R(x,f(x))", "f373d913ebe94da13b683bb8b0c7446c": " Tw = \\dfrac{1}{2\\pi} \\int \\left( \\dfrac{dU}{ds} \\times U \\right) \\cdot \\dfrac{dX}{ds} ds \\; ,", "f3741d1cb95a9edd40a0bcecd85803b1": "y'(0)=0.", "f3747c8c765196bd7bc9f444e422ba34": "f(a) = b.", "f374942a2dda893e80e7920a58c5c5c4": "\nY_{ij} = \\mu + \\alpha_j + \\epsilon_{ij},\n", "f374b9ab20ece92a19c9bc0198cfc3b8": "\\{x_{\\varepsilon_1, \\ldots, \\varepsilon_k}\\},\n\\quad \\varepsilon_j = \\pm 1, \\quad j = 1, \\ldots, k,", "f374d7c66c63870d1e34f43872324963": "f_1(\\text{aa}) = q^2 = f_0(\\text{a})^2", "f3753ab8d2323681a68bbac34269b1e7": "X_3 = (XX_1-1)^2 \\, ", "f3753c25086dbfaaab052962ad0a34c8": "\n \\sqrt{ \\frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} } = 2.\n ", "f3756b79f8c0bab4b9680c24180550b2": "\\sum_{i=1}^N (X_i-\\overline{X})^2", "f375bc82a473fcd316e9894c89cb4945": "J_{ }^{ }", "f375ee158ee3d107075c1f04817e891b": "\\pi \\ge 0", "f3762f727617a8b8e536e789a2174058": "ES_{0.05}", "f37649818194731661c1f0450f36e93a": "\\operatorname{lb}(x) = n + \\operatorname{lb}(y) \\quad\\text{where } y = 2^{-n}x \\text{ and } y \\in [1,2)", "f3765257144a93156947f4a29410f808": "\\scriptstyle n(t)", "f37673447f0881432da601add4881bef": "\\xi = \\frac{x - i y}{1 + z} = \\tan(\\tfrac{1}{2} \\phi) \\; e^{-i \\theta}.", "f37687063cfe17b47977cdcafe573522": "Z_3 = (Z_1+C)^2-ZZ_1-F", "f37716ffc9f28388e63a8c384f1006d9": "[a,b]\\times[a,b]", "f3771d17697e20201602856f9fd21450": "\\bar\\nu_{sub}= \\bar\\nu_0+ \\left[ (A^\\prime - B^\\prime) - (A^{\\prime\\prime}-B^{\\prime\\prime}) \\right]K^2 -(D_K^\\prime-D_K^{\\prime\\prime})K^4\n", "f3772d8c08f04eabd5a9cde20d8b98a7": "f(x_1),\\dots,f(x_{n-1})", "f377743ca927c8088257fdc83a07f2f2": "\\Omega\\left(\\frac p q\\right) > \\frac 1 p ", "f3778838df6181ed5f888ffa44929774": "\\mbox{P}^d", "f3779a36cafb9827e9b11459dbba6ddf": " (\\mathbf{a_{1}}, \\mathbf{a_{2}}, \\mathbf{a_{3}}) ", "f377b7303f3b51eecfad3097cd353855": "\n\\sum_{n=0}^{\\infty} h(r_n) = \\frac{\\mu(F)}{4 \\pi } \\int_{-\\infty}^{\\infty} r \\, h(r) \\tanh(\\pi r) dr + \\sum_{ \\{T\\} } \\frac{ \\log N(T_0) }{ N(T)^{1/2} - N(T)^{-1/2} } g \\left ( \\log N(T) \\right ).\n", "f3785e5e712151a9be59610df360e415": "P(a_{T+1}|\\theta^\\ast,\\hat{a}_{1:T},o_{1:T})", "f378695e1f24df14cc730032a1763508": " \\sigma_{ph,\\omega} = \\frac{\\hbar\\omega\\dot{\\gamma}_{ph,a}n_e}{u_{ph}},", "f378a79cfe5fd925a948a383ea9c62be": "P(r \\ge \\nu_1 k ; \\nu_1 n, p)", "f378a8896ba584da1037d72d3333e836": "\\epsilon_\\beta", "f378d990d8fbb02b7e3ba2103737aeea": " R_\\mathrm{out} = \\begin{matrix} \\frac{v_{o}}{-i_{o}} \\end{matrix} \\Big|_{v_{s}=0}", "f378da6e3a1b2f501900e01c0798de4f": "x_{i}^{0} = f^{0}(i)", "f378fe20dc81af0c6db36e66ff2c7290": "\\mu - \\beta\\,\\ln(\\ln(2))\\!", "f37964b70c4a7fa2d76b0f75ada3cca2": "\\mathfrak{C}_{\\operatorname{odd}}", "f3799dc54339e53a755dcf17ecfd968b": " \\log_b w\\cdot \\log_a x\\cdot \\log_d c\\cdot \\log_d z \n= \\log_d w\\cdot \\log_b x\\cdot \\log_a c\\cdot \\log_d z. \\, ", "f379a5f34ee15ccb1a6ac144eacdf23a": "s_n = k^n s_0 ", "f37a31582a29d095fded2ab5ec81af70": "\nd_1 = \\begin{bmatrix}\nx & y\\\\\n\\end{bmatrix}\n", "f37a34affc7b393f37902497979c938e": "\\mathbf{v} = \\langle \\rho, \\angle \\theta, \\angle \\phi \\rangle", "f37a49f3c70be65cbdb05a953ebfb9b8": "H_{n+1}", "f37aa9c13898b796dd9bd304341b7557": " \\mathbf{x}^T = \\sum_{i=1}^n a_i \\mathbf{u}_i ", "f37af30feced4ab661ccb190c8dfa98f": "\\Delta\\,_v S", "f37b43af23ee5a8f9a5d1a981ade761d": "c_1 \\mathbf{v}_1 + c_2 \\mathbf{v}_2 + \\cdots + c_n \\mathbf{v}_n,", "f37b7d397ba77793ea0e6a020bd90b5e": "X=\\left\\{ \\begin{align}\n & \\min (A,B)\\quad \\,\\mbox{if}\\,C\\ge \\max (A,B) \\\\ \n & \\max (A,B)\\quad \\mbox{if}\\,C\\le \\min (A,B) \\\\ \n & A+B-C\\quad \\,\\mbox{otherwise}. \\\\ \n\\end{align} \\right.", "f37b87f7fbc6f4b93f8fdf008172164e": "\\hat{G} = \\frac 12 \\sum_{i=1}^{N}\\sum_{{j =1}\\atop{j\\neq i}}^{N}\\ \\hat{g}(i,j).", "f37bc560a1e96d1cb73e25bb18ff889b": "\\{x:= 3\\}[x=x+1;]x\\ \\dot{=}\\ 4", "f37bd5fc7d885f252d440fd107724463": "\n F( x ) =\n \\sum_{i=1}^{N} \\alpha_i \\varphi \\left( w_i^T x + b_i\\right)\n", "f37be92b226828306dc34737a9dda31f": "\\frac{1}{i} = \\frac{\\sqrt{1}}{\\sqrt{-1}} = \\sqrt{\\frac{1}{-1}} = \\sqrt{\\frac{-1}{1}} = \\sqrt{-1} = i", "f37bf5e93cbf40d2413df3c8865d6652": " V = k_p T \\,\\!", "f37c78d96e85e5b2484448fbfe43950d": "\\rho_{SE} (0) = \\rho_S (0) \\otimes \\rho_E (0)", "f37c960c278d3e46f4fc03792555d091": " \\sigma \\mathfrak{G}^2 = 0", "f37ca2d60f21818b4ba190fcb0b98cbd": "\\psi(x)=\\mathbb{P}^x\\{\\tau<\\infty\\}", "f37ca788a092143a92b8303460b244bd": " \\frac{854513}{138} ", "f37db8f8830e75a4e2eef09b08c98337": "\\epsilon \\sim 1 + r_s ", "f37dec12e1b0b6565aa536080de8e17b": "\\epsilon(z)=\\epsilon_1(g_2\\cdot z)\\epsilon_2(z).", "f37def5cbea73b3fdc1144da6eb60f39": "P(\\cdot\\mid m_t,s_t)", "f37dfd875f91f6102615db5bf92398e8": "R_{ff}(\\tau) = (f(t) * \\overline{f}(-t))(\\tau) = \\int_{-\\infty}^\\infty f(t+\\tau)\\overline{f}(t)\\, {\\rm d}t = \\int_{-\\infty}^\\infty f(t)\\overline{f}(t-\\tau)\\, {\\rm d}t", "f37e1819c53eb6f5f87f7df4e15e58be": "S = k_B\\,\\ln W", "f37e45a13413e67ee64f72241ddbb7c7": "\n\\Pi^m_\\ell(z)\\equiv\nr^{\\ell-m} \\frac{d^m P_\\ell(u)}{du^m} =\n\\sum_{k=0}^{\\left \\lfloor (\\ell-m)/2\\right \\rfloor} \\gamma^{(m)}_{\\ell k}\\; r^{2k}\\; z^{\\ell-2k-m}.\n", "f37e9361b4d61e3c8c4a80ba2f14c9e5": "Tr_n={n+1\\choose2}={m+2\\choose3}=Te_m.", "f37ea7dd823e64eae195fcae7702ace5": "\n\\begin{align}\n0&=\\int\\mathrm d\\mathbf r\\ u_k(\\mathbf r)\\nabla\\cdot(\\rho(\\mathbf r,t_0)\\nabla u_k(\\mathbf r)),\\\\\n &=-\\int\\mathrm d\\mathbf r\\ \\rho(\\mathbf r,t_0)(\\nabla u_k(\\mathbf r))^2+\\frac{1}{2}\\int \\mathrm d\\mathbf S\\cdot\\rho(\\mathbf r,t_0)(\\nabla u_k^2(\\mathbf r)).\n\\end{align}\n", "f37eab9e8f152d83db214d87fa50c7da": " P_{m2}=P_{m1} \\times P_{m1}=P_{m1}^2 ", "f37eb4d5c217921feb0a88cd7673deff": "\n \\begin{align} a & = \\sqrt {xYZ} \\\\ b & = \\sqrt {yZX} \\\\ c & = \\sqrt {zXY} \\\\ d & = \\sqrt {xyz} \\\\ X & = (w - U + v)\\,(U + v + w) \\\\ x & = (U - v + w)\\,(v - w + U) \\\\ Y & = (u - V + w)\\,(V + w + u) \\\\ y & = (V - w + u)\\,(w - u + V) \\\\ Z & = (v - W + u)\\,(W + u + v) \\\\ z & = (W - u + v)\\,(u - v + W). \\end{align} \n", "f37ed22ec9347657f8b6a9ddd0b50228": "B \\models \\bigwedge \\Phi(\\bar{a})", "f37ede833ab3be138a587e2ae6f55235": "\\mathbf{M}_{\\mathbf{Y}}", "f37f28549abdf929ea230be344ef0b01": "c(i,i)", "f37ff93313ddf68a7a98a3dbf0f6791c": "\\mathbb{Q}(\\sqrt[3]{2}, \\zeta_3)", "f380072fb15e3d05c5c8ee5a4ee329e3": "f(\\mathbf{p}_1,\\ldots,\\mathbf{p}_N;\\mathbf{r}_1,\\ldots,\\mathbf{r}_N)", "f3801356683436b18685dbd77360e1f8": "kT \\ll E_F", "f3801ec56d162c1dd18a4a13cf322483": " c_n = \\frac{1}{n} \\int_0^1 x^{\\bar n} \\left( x-\\tfrac{1}{2} \\right) \\,{\\rm d}x = \\frac{1}{2n}\\sum_{k = 1}^n \\frac{k|s(n,k)|}{(k + 1)(k + 2)}", "f3802a71177386cd90537452e9b40571": "z^2=-0.138169999969259 + (0.239864000061970)i {\\;}{\\;} ({\\mathrm {green}}),", "f3805ed0b38f36c27953dc1a23045bcc": "\\mathrm{1.\\overline{714285}}", "f38065d6fa6ffca3a59a7136cb4adf42": " \\mathrm{d}y = S'(t)\\mathrm{d}t = \\sin(t^2) \\mathrm{d}t \\,", "f3807ac0b8c2e33b6349e5d19a59ea60": "D(ea) = D(ae) = D(a)D(e) = D(e)D(a) = D(a) ", "f380815442964d18468fd6d0700ab7c1": "p'' \\approx \\frac {1 \\textrm{\\ AU}} {d} \\cdot 180 \\cdot \\frac{3600} {\\pi} .", "f38088eb4aca55347bd10f367730b02e": "\\mathrm{d}U = T\\mathrm{d}S - P\\mathrm{d}V.", "f380b400a4e97744b40ff1c19f8326d6": "\\gamma_\\theta:(r_1,r_2)\\to A", "f380b996a58ab124881b98784ce88e95": "\\begin{align}\n B_{n} &= \\sum_{k=0}^{n-1}\\binom{n-1}{k} \\frac{n}{4^n-2^n}E_k \\quad (n=2, 4, 6, \\ldots) \\\\\n E_{n} &= \\sum_{k=1}^n \\binom{n}{k-1} \\frac{2^k-4^k}{k} B_k \\quad (n=2,4,6,\\ldots)\n\\end{align}", "f380baa08ce22b27ce7e18edeb0b43c0": "\\mathbf{B_0} = \\sqrt {\\mathbf{Y_0}^2 -\\mathbf{G_0}^2} ", "f380cca026f542c2e513328d3db0c654": "{}_0\\!V_x=A_{x}-P\\cdot\\ddot{a}_{x}", "f380fe28263092e015f25c45075f88d0": "S_{n-1}S_n", "f3819d93220b918b4be63dc39e339aba": "\\displaystyle{{{|f(z_1)-f(z_2)|\\over |f(z_1)-f(z_3)|} \\le a\\cdot {|z_1-z_2|^b\\over|z_1-z_3|^b}}.}", "f381c28221b59abf94c0f7eb49edc7d8": "\\langle 0|\\mathcal{T}\\{{\\phi}(x_1)\\cdots {\\phi}(x_n)\\}|0\\rangle=\\frac{\\int \\mathcal{D}\\phi \\phi(x_1)\\cdots \\phi(x_n) e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{g\\over 4!}\\phi^4\\right)}}{\\int \\mathcal{D}\\phi e^{i\\int d^4x \\left({1\\over 2}\\partial^\\mu \\phi \\partial_\\mu \\phi -{m^2 \\over 2}\\phi^2-{g\\over 4!}\\phi^4\\right)}}.", "f381e2a8fdd89462e01f8b4e78f39327": " P_1 (T) ", "f381f7b42f6dd9af81d3167727efa63e": "v = (G_F \\sqrt{2})^{-1/2}", "f382051eb8fe771dbbba7d5e183ae597": "\\Delta \\epsilon=(\\epsilon_c - \\epsilon_v)", "f382229d40aa6e40afe3718eac57ab8d": "\\scriptstyle{\\Delta x}", "f3825132abe36adba4e4dbf392633e66": "\\text{Peak power } (\\mathrm{W}) = \\frac{\\text{pulse energy } (\\mathrm{J})}{\\text{pulse duration } (\\mathrm{s})}", "f38270f6cb0e5689e830b983e0f9a20d": "(\\mathrm{id}_B \\otimes \\Delta) \\circ \\Delta = (\\Delta \\otimes \\mathrm{id}_B) \\circ \\Delta", "f382895ee312de8c96b9812c980ed6e5": "\\{ x \\mapsto z, y \\mapsto f(z) \\}", "f3829a484ea746c845d327d638d86943": "\n\\lim_{n \\to +\\infty} \\left( \\frac{\\int_a^b e^{nf(x)} \\, dx}{\\left( e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}} \\right)} \\right) \\le 1\n", "f3833f114b885c66789e0f148e09398b": "\\alpha^\\prime(kx) = \\Delta_S(x)^{1/2} \\alpha(x)", "f3835ba159f5cb056f400f4b19dbdba0": "v=r", "f383a1b5cea8431bcef69ffe55835dca": "Z=\\frac{LR_N(\\beta_{ML,1},\\beta_{ML,2})} {\\sqrt{N}\\omega_N}", "f383db942ab2deb254363b00fa471851": " \\sum^{A}_{j'} (U^{A}_{jj'}-E\\delta_{jj'})a_{j'}(\\mathbf{k}) = 0, ", "f383fb04ef077c4ed82b870991183f63": "\\mathbf{m}_k", "f38448d5a0ea3d075b4bfa7eeb5bc56f": "\\bar M_{g, n}", "f384a97ad83b3ba30674b6ec7ccb194d": "\n\\rho=\\sum_k p_k \\rho_1^k \\otimes \\rho_2^k\n", "f384aac4c4555c4d2b82464a2276d2df": "r = - \\frac{1}{a} \\frac{d[A]}{dt} = - \\frac{1}{b} \\frac{d[B]}{dt} = \\frac{1}{p} \\frac{d[P]}{dt} = \\frac{1}{q} \\frac{d[Q]}{dt}", "f38520703458c5ffcfff17205b6a0c26": "0 < \\alpha < 2/\\sigma^2_1(A)", "f3854b0f9d8f88f48afe751436c793f3": " f(z)=w.\\, ", "f3856ac4e0ce14556eb3dcb8f3730dce": "\\scriptstyle{DTFT}\\displaystyle \\{y\\}(f)", "f385eb806d56eb26262c112aa55efb9b": "V = \\frac{X}{M} \\,,", "f385f5d9c6bb1686405bdef20072cff5": "\\operatorname{Ind} X = 0.", "f3862059bf51c119ad45aeb24f6df74c": "\\{\\theta : \\mathcal{L}(\\theta | x)/\\mathcal{L}(\\hat \\theta | x) \\ge p/100 \\}.", "f386282d9b571488b1987f7f0dde497d": "V(B,1)\\,", "f386289b8e6d4bb313c92e1198944f39": "\\frac{k}{\\ell} \\leq \\nu", "f38639868a49d0cf5a59c99061aebfb7": " \\frac{dB}{dz}", "f3865461a16e4a8e603c3886754f8118": "\\chi_{\\text{mol}}", "f386605311fe5ef57f1ef8fff4590ccf": "P \\times \\frac{n}{N}", "f3866cb757f062b27ed0ac6c7bcb0478": " P_{t+1} = c \\ N_t \\ f(N_t, p_t) ", "f3867d8b3f8a94e03588997727f47c94": "\\angle ACB", "f38703f6381a8c79f735b8ab42775711": "q_1''(0)=q^m", "f3872303178c237e1333193a9a0688a8": "r = a - q n", "f38742331e996dae266f6969dbe5b3bf": "_{s.2 \\,}\\!", "f38775c7e503c5c2ebee505bbb8827a9": " \\tfrac14 + \\tfrac16 \\sqrt3 ", "f38775f18258c8691ffd685d3ddaccd6": "\\mathcal{C}_{3}", "f3878f94f72dc226c38cd99b23711534": "\n\\boldsymbol{\\alpha}=(\\alpha_1,0,0,\\alpha_2,0,0),\n", "f38793efee32fd28502a8eb6ef76c5ce": "ki=j\\sqrt{-1}", "f387aac7e562feaf29afd72d1ec40720": "G_k(m,n)", "f387e9e8afb3b5ff50b7977e8cc032be": " \\tilde{g}\\tilde{g}\\rightarrow (\\bar{q} \\tilde{q}) (\\bar{q} \\tilde{q}) \\rightarrow (q \\bar{q} \\tilde{C}^+_1) (q \\bar{q} \\tilde{C}^+_1) \\rightarrow (q \\bar{q} W^+) (q \\bar{q} W^+) \\rightarrow ", "f3882bae3f37aaa1060942d292267392": "\\left \\{2^{\\deg(p)}p(\\tfrac{x}{2}), (a, \\tfrac{1}{2}(a+b)) \\right \\}, \\quad \\left \\{2^{\\deg(p)}p(\\tfrac{1}{2} (x+1)), (\\tfrac{1}{2}(a+b), b) \\right \\}", "f3882d82e13db697641724b04cebf2d8": "W(\\lambda_1,\\lambda_2,\\lambda_3)", "f38854ccee7efa9890eb080506708ead": "F(S\\psi)=FS*F\\psi\\,", "f388671ffaf461cbb9df2f7376f7a7f2": "\\hat f_s(\\cdot)", "f38868893b7615b4b11d13da169361b7": "(f(t) * \\delta(t-T))\\,", "f388f3b87143e7b2ff02c54a7b6a529f": "= \\frac{1}{2}", "f38957e1976f17428c19c799a6e17417": "g = \\frac{n \\hbar c}{2e} = n g_D ", "f38974792dcb2323d14904d167aed509": "\n\\begin{align}P\\left(x+1;\\;0,\\;xy_1+y_0, \\;xy_2+2y_1, \\;xy_3(x)+3y_2,\\ldots,\\;xy_n+ny_{(n-1)}\\right)&=R(x)P(x;\\;0,\\;z_1,\\ldots,\\;z_n)\\\\\n\\end{align}\n", "f38982ff04e7b71230adcf990f85b931": "\n \\cfrac{d^2 w}{d x^2}= -\\cfrac{M_x(x)}{D}\n", "f38a6495ed341c125f8cb2b826ba1675": "\\vec X", "f38a673a23ff410a73a7a3736d14ed88": "\\hat{x_0}", "f38a71d94e59d5e496b4c86c7883fa25": "1 + 1 + \\cdots + 1", "f38a787f52dc5c58624f0717e05e4d92": "V_x=-(G_x)^{-1}*(G_m * V_m)", "f38a94815e3b517c90939bdfa9b8937e": "\\eta_m > \\xi_m^{\\delta - 1} e^{-\\xi_m}", "f38af77e30b4730f5b65704cffd66a21": "\\textstyle \\underset{i\\in e}{\\sum }w_{i}^{t}\\leq \\theta_{t}", "f38b249750821c9d21520cdefcdf06a7": "d\\rho\\ d\\theta\\ d\\phi=\\det\\frac{\\partial(\\rho,\\theta,\\phi)}{\\partial(x,y,z)}dx\\ dy\\ dz=\\frac{1}{\\sqrt{x^2+y^2}\\sqrt{x^2+y^2+z^2}}dx\\ dy\\ dz", "f38bfa7fa0f4b99e757ec57cb51eb8e9": " \\mathrm{Ec} = \\frac{V^2}{c_p\\Delta T} ", "f38c4f58f2c0e3e7674f276c3e702292": " \\nu=\\frac{n}{2pn\\pm 1}.", "f38c6e5f2048576b359a58549d0d9390": "Qf = \\int f dQ", "f38c6f9e0b88c116d5441901d6a1afee": "it\\sqrt{2}\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}\nM\\left(\\frac{k+1}{2},\\frac{3}{2},\\frac{-t^2}{2}\\right)", "f38cb0bcd5d3e536bba9332026c80eb5": "\\omega = \\frac{\\chi^2}{2\\eta}\\,", "f38ccb7532bd417ed6bf54465ef49e13": " x_m = \\frac{b_m - \\sum_{i=1}^{m-1} l_{m,i}x_i}{l_{m,m}}. ", "f38cf1f50211160bfe2025ce30084408": " p_{1^{(n)}} ", "f38d1ce5b08e21bb0cfb78668be0ea5c": "\\omega^{\\prime\\prime}", "f38d35ba0d07e7434956d52bd0bded2f": "\\tau(x,r;n)", "f38d5e3bcee30883451de002a3fdf3be": " \\mathbf{P} \\simeq \\varepsilon_0 \\left( \\chi^{(1)} + \\frac{3}{4} \\chi^{(3)} |\\mathbf{E}_\\omega|^2 \\right) \\mathbf{E}_\\omega \\cos(\\omega t).", "f38d6c04838d6e4b1a0baa8f595bdc3e": "\n \\lim_{kh \\to \\infty} \\omega^2 = gk\\, \\left\\{ 1 + \\left( ka \\right)^2 \\right\\}\n + \\mathcal{O}\\left( (ka)^4 \\right),\n", "f38d75be2160cfee3cb990416f511512": "\\displaystyle{(Q(a)b)^{-1}=Q(a^{-1})b^{-1}.}", "f38de0b120f3fa7b5737d87b97aad826": "v=z", "f38de468adeecb321f615629f5ce7ce8": " I_4 ~,~ \\Gamma_a ", "f38dea6cbec66e0a5adb039b38094929": "\\, 0\\le t\\le T", "f38e29da67107da8f58423809707f646": "S(T) = C \\exp\\left(\\frac{-A}{T}+\\frac{B}{T^2}\\right)", "f38e6ffe43515f07a92c862469cbb2ef": "\\alpha \\beta", "f38ea9fac860d0c3c11171dfd49dbfd7": "\\forall n [0=n \\lor 00 .", "f3975d4e7467b2033d5f26147542c972": "0 = (\\delta^\\alpha_{\\beta , \\gamma} - \\delta^\\alpha_{\\sigma} \\Gamma^{\\sigma}_{\\beta \\gamma}) g^{\\beta \\gamma} = (0 - \\Gamma^{\\alpha}_{\\beta \\gamma}) g^{\\beta \\gamma} = - \\Gamma^{\\alpha}_{\\beta \\gamma} g^{\\beta \\gamma} \\,.", "f397a7c368cf2c72db329a74bcfc8eb3": "2 \\alpha (25812.807) \\,", "f3983b92074e397108151fbe9a57a1ee": "\\deg(2x\\circ(1+2x)) = \\deg(2+4x)=\\deg(2) = 0", "f3987e5869fe1eef755e54b91af7ae65": "\\mathrm{Tr}_{12} : {{{h^a}_{bc}}^d}_e \\mapsto {{{h^a}_{ac}}^d}_e", "f398bbd7c7ee79e36e3a86976573af4f": "1-1/p^2", "f398cb566c20883c3c7d4c1e64093299": "\\omega^{\\omega^{\\omega}}.", "f398d87f39896c2bbce2e2fe8b2b930b": "\\text{Speed Length Ratio} =\\frac{v}{\\sqrt {\\text{LWL}} }", "f399121c96167f7dc785467422b5fd3d": " \nB = \\int_X | x\\rangle\\langle x|\\; d\\mu (x)\\; , \n", "f399438236ff886cd80e52bab2058306": "\n\\ y[n] = x[n] + \\alpha y[n-K] \\,\n", "f3994fc5cf2f8070e4a34733279512af": "A \\hat{\\otimes}_\\pi B", "f39964625eeb4d40006fce10f397b10d": "R_i + kR_j \\rightarrow R_i, \\mbox{where } i \\neq j ", "f3996a8dc163a4f114045449af051d86": "f(F(X,Y))=G(f(X),f(Y))", "f39985f5393cd4fb5abb9401f410f980": "t(BP) = -\\frac{1}{\\lambda} {\\ln \\frac{N}{N_0}}", "f399aead0b06f6843a2e2316880114b7": "\n F =\\frac{\\pi E}{2 \\left(1-\\nu^2\\right)} a^2 \\tan \\theta=\\frac{2E}{\\pi\\left(1-\\nu^2\\right)}\\frac{d^2}{\\tan \\theta}\n", "f399dc2975eaef8aba8374d10aab01bc": " \\xi \\propto |\\epsilon |^{-\\nu}\\,\\,= \\left (\\frac{|T-T_c|}{T_c}\\right )^{-\\nu} ", "f399f13a0e628d982318eef23877bc6a": " L : G \\times G \\to G", "f39a50f8dd4e2b5170b3b2d30a31a980": " X = \\bigcup_{w \\in W} w \\overline{\\mathcal{C}}", "f39a63c5ea258887870760e09e2d61b9": "\\mathbf{J^TM^{-1}J\\Delta \\boldsymbol\\beta=J^T M^{-1} \\Delta y}. ", "f39aa27bd7e366a35b2e79d262bfb8ff": "\n\\omega_{1} = \\frac{L_{1}}{m r^{2}};\n", "f39acb4dd2049959d1f9bc1b826dba7b": "\n\\text{If }q \\equiv 1 \\pmod 4 \\text{ then}\n", "f39acd74633f780f0ce27b670dc878b2": " \\Psi = e^{-iEt/\\hbar}\\psi(x_1,x_2\\cdots x_N) ", "f39b16d496128ebeb55daa98e9795580": " \\rho = \\sum_i p_i | i \\rangle \\langle i |", "f39b1d9bdbdf82bb18f728d4b7e524bb": "n_\\gamma = \\frac{1}{\\pi^2} {\\left(\\frac{k_B T}{\\hbar c}\\right)}^3 \\int_0^\\infty \\frac{x^2}{\\exp(x) - 1} dx \\simeq 20.3 \\left(\\frac{T}{1\\text{K}}\\right)^3 \\text{cm}^{-3} ", "f39ba5fb0659615a3da87063b5d69a69": "\\eta_{otto,th}=1-\\frac{1}{r^{\\gamma-1}}", "f39bd70041be7eeabef5002ba18dc49a": " \\frac{1}{2} \\left( \\frac{r_p^2 - r_a^2}{r_p^2} \\right) v_a^2 = \\frac{GM}{r_a} - \\frac{GM}{r_p} ", "f39c5afa9300bc0285f343a2a53464a2": "TC = cD + {\\frac{DK}{Q}} + {\\frac{hQ}{2}}", "f39cba00abd49e6ebae695e31cb25782": "w := r,", "f39cc6922aec818e4c272fd471a19220": "x_j=a+jh", "f39cea42b12f59149b534911b7376b56": "\\textstyle r = n-k", "f39d27f67c92eedd312fc0a3cb699148": "\nR^{2} = d_{1}^{2} - r_{1}^{2} = d_{2}^{2} - r_{2}^{2}\n", "f39d2f20ba06e83f0df40f8d64a300e4": "A = \\sum_i a_i |a_i \\rang \\lang a_i| = \\sum _i a_i P_i,", "f39d3133fe64d2e88ec01feb70242fd0": "\\varepsilon\\left(v\\right)=0", "f39d3541f1ca44e742ea444155d68b06": "(\\forall X, \\exists ! Y, P(X,Y))\\rightarrow (\\forall A, \\exists B, \\forall C, \\forall D, P(C,D)\\rightarrow (C \\in A \\rightarrow D \\in B)).", "f39d9bc0544542fab448a3a0b0fa87f6": "q^{(\\ell(w)-\\ell(x))/2}D(P_{x,w}) - q^{(\\ell(x)-\\ell(w))/2}P_{x,w} = \\sum_{x 1} v_i", "f3a7efd547f4972c9cc39cfd35fe4595": "V(X_t,t)", "f3a855df358a9fcf71b4e789f587e0e7": "q_1(x)", "f3a8a0ec6370b95d9b3d56c398783176": "\nF(r) = Ar^{-3} + Br^{-1/3} + Cr^{-5/3} + Dr^{-7/3}\n", "f3a8eac59cbe959d7788a1a8c6232126": " \\Psi_\\varepsilon ", "f3a911effe2cac6f84e2965c233de6be": " 2~r~\\cos\\theta \\,", "f3a94ee5bc60e31395a495cd5e92d0aa": "m_{em}=\\frac{4}{3}\\frac{E_{em}}{c^{2}}", "f3a9642a7d2c549aa6cccb7689cf3c1e": "\n\\left( x^{2} + y^{2} \\right) +\n\\left( z - a \\cot \\sigma \\right)^{2} = \\frac{a^{2}}{\\sin^{2} \\sigma}\n", "f3a9942b35c14a43b3489659f4eebe2d": "S_1 = \\{ s_11 \\}", "f3a9afca7d98d23deb5aa8e43254e4be": " = \\begin{pmatrix}\n6/7 & 3/7 & -2/7 \\\\\n3/7 &-2/7 & 6/7 \\\\\n-2/7 & 6/7 & 3/7 \\\\\n\\end{pmatrix}.", "f3aaa23f4ec37a49cb588b7320cd56b0": "\n\\rho^{AB} \\rightarrow\t\n\\tilde{\\rho}^{AA'B} := \\sum_k \\mathcal{E}_k\n\\left( \\rho^{AB} \\right)\\otimes \\vert k \\rangle \\langle k\\vert^{A'}\n", "f3aab911b266b80acd0aa809cfd178f1": "0<\\alpha\\le 2", "f3abbd1a2b78bd253e35f83691ed396c": "E_{1}^{\\dagger}E_{2}\\notin\\mathcal{Z}\\left( \\mathcal{S}\\right) ", "f3ac2871b281543eda099774002462a4": "-\\mathfrak{a}_+^*", "f3ac7ab0ceb0d30bf70b7abcc3b74683": "R = \\frac{k}{2}g {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} g", "f3ad57f7d9ce7dbe2587fa37013f7e3f": "\\int \\sum^b_{r=a} f\\left(r,x\\right)\\, dx = \\sum^b_{r=a} \\int f\\left(r,x\\right) \\,dx", "f3ad7622d0894bf35a1055bc0c6a76ff": " x' \\in K \\mapsto x''(x'), \\quad |x''(x')| \\le \\|x''\\|.", "f3ad929dc8310f639aa8060c9c51e5c6": "\n\\mathcal M=\\frac{i}{(2\\pi)^{3/2} Z^{1/2}}\n\\int \\mathrm{d}^4 x e^{-ip\\cdot x} \\left(\\Box+m^2\\right)\n\\langle \\beta\\ \\mathrm{out}|\n\\mathrm T\\left[\\varphi(x)\\varphi(y_1)\\ldots\\varphi(y_n)\\right]\n|\\alpha\\ \\mathrm{in}\\rangle\n", "f3adc62703255bea0bd54cff17ca699f": "\\left| \\frac{x_j}{x_i} \\right| \\leq 1 \\quad \\text{for }j \\neq i. ", "f3add613f772592e0ca881e5efe905b5": "\\mathfrak{nopqrstuvwxyz} \\!", "f3adee7b81c1192f47680b7989a3cb76": " V = 5 ", "f3ae0b4937ed89425a08a915033155c1": "X = X_{1},X_{2}", "f3ae204615ae56e80f4c59ce60d747dd": "\\alpha - dy/dx", "f3ae4d92ebea7b376b36a781f16e06b8": "\\int_{R_n} \\cdots \\int_{R_2} \\int_{R_1} f(x_1, x_2, \\ldots, x_n) \\, dx_1 dx_2\\cdots dx_n \\equiv \\int_R f(\\boldsymbol{x}) \\, d^n\\boldsymbol{x}", "f3ae676ffebd08ed1ec9ba235069d702": "Y = b_0 + b_1A + b_2B + b_3C + b_4A*B + b_5A*C + b_6B*C + b_7A*B*C + \\varepsilon.", "f3ae7dfcc0acccdbf1d4a71a9a05b96f": "L = 3", "f3aebc3e27ca11c1471f5da3b7eec931": "\\neg \\neg \\psi", "f3aed9bbaa0b0a9a790ebd1de08786c8": "O\\left(n^{1.5}\\sqrt{\\frac{m}{\\log n}}\\right)", "f3af1428a30d3009fe34cf686c0f73d6": "{{S}_3 = 2 + \\varphi }", "f3af65abca16bb9d991a8f8f4d78bb09": "\n\\frac{w_1x_1+\\cdots+w_nx_n}w\n>\\sqrt[w]{x_1^{w_1} x_2^{w_2} \\cdots x_n^{w_n}}.\n", "f3af9a64acb8b9a85d81f05fdb9f34ce": "\\log \\nu = 7", "f3afaf6f2c71e05c5cb97f1e73985c1d": "\\scriptstyle b_0,\\, \\dots,\\, b_N", "f3afe15e11b264e1bc33bc33496897dc": "log BCF=log Kow-1.32", "f3b05bd0adddb565a33621d75bd5fd17": " 2 x^3 + 2 y^3 + 2 z^3 \\ge 6 x y z ", "f3b078a272b2cee4d5fc6ea327b3b100": "D^{\\le n}", "f3b0b99bfc38a5108d276f7e0315f9ef": "{\\rm cov}(\\varepsilon_i,\\varepsilon_j) = 0, \\forall i \\neq j ", "f3b0c3c8ccc7313e8030b6035595a66b": "B_n(f)(x) = \\sum_{\\nu = 0}^n f\\left( \\frac{\\nu}{n} \\right) b_{\\nu,n}(x).", "f3b0e9b118a636ca97aece99d4d699be": "\\chi_1", "f3b1006725767fe0c361f7c3e5fa1ba3": "\\frac{dTR}{dP} = Q(-E_d + 1) = Q(1 - E_d)", "f3b114dae39c615bcd51b37b58003891": "\nF(r) = \\frac{C(r)}{R r^3}\n", "f3b1491add571a2b91fd7009926fcf70": "\\! (\\alpha, \\beta)", "f3b1d4bb4e6129538a7ecc98f92f6e86": "\\frac{dv}{dy}", "f3b1e634edc74b2c8cb32f8790a7225c": "\\frac{\\partial F(x,y,t)}{\\partial t}=2t+ y-x-k = 0\\,", "f3b23f93a262470d68a0d63c113d96f2": " \\forall m,n \\in \\mathbb{N},\\qquad F_m\\cdot F_n\\subset F_{n+m}.", "f3b26f3798c09bb5829bacfe7fdb0dfd": "p=\\sum_{\\nu=0}^n a_{\\nu}x^\\nu", "f3b2751be7a51f9e0fed1b67a624af27": "\\phi:\\mathbb K(\\mathfrak g^*)^{Ad(G)}\\rightarrow H^*(M,\\mathbb K). \\, ", "f3b2e31447c41f63ed1c9307f7409d51": " \\nabla \\times \\mathbf{E}_\\text{g} = -\\frac{\\partial \\mathbf{B}_\\text{g} } {\\partial t} \\ ", "f3b2f07271ab2bedded63d0f3b1f069d": "\\partial G =\\{\\rho =0\\}", "f3b319f3404d4f73b20623f2617fb756": "\\psi_{2}= \\phi_{2}-e_{B}", "f3b321528f3592ad0606077a5b3708cd": " \\ p = q ", "f3b346f081ae07b836915a77eb048118": "A^\\ast", "f3b397f396379bfdd33deda62ed600c5": "U(s) V(\\gamma) U^*(s) = \\gamma(s) V(\\gamma).\\;", "f3b3c7dcaecb3977d099a4150d91e7ad": "_a^bS", "f3b40f4801c67f6016df28298807f876": "\n\n1/2 = \\left(1-e^{-e(V_{fl}-V_{+})/(k T_{e})}\\right)/ \\left(1-e^{-e(V_{-}-V_{+})/(k T_{e})}\\right)\n", "f3b437ea23316b6669f3da4de2e6e488": "x=I/N", "f3b45c9479377fae18a4e1baee561344": "\\Omega\\subset \\mathbb R^d", "f3b4ca92edf90bed39fe294b3a2fa8f1": "X:\\mathbf{A}\\to\\mathbf{C}", "f3b4d16190f00bef252b41e79c9ad3ed": "\np_j = p_0 e^{\\beta e_j } \n", "f3b4d66fd651a127296b4a092348f9ac": " \\alpha_y ", "f3b512029a5adf9f416f7372084b32d2": "\\left[\\begin{smallmatrix} \\lambda & 0 \\\\ 0 & 1/\\lambda\\end{smallmatrix}\\right],", "f3b53b483a7318df90547bbdac3eeef5": "\\mathrm{SR}(2)\\leq \\tfrac{2}{\\sqrt{3}}", "f3b5448c3f89015e6bae1af42f3ba9d9": "f\\; x \\; y", "f3b56646ee4b8df48a5f0fbd9d317861": "\\mathcal{B}_r = \\lfloor r \\rfloor, \\lfloor 2r \\rfloor, \\lfloor 3r \\rfloor,\\ldots = ( \\lfloor nr \\rfloor)_{n\\geq 1}", "f3b5a017331471cc462cbe81baf744df": " \\pi^e ", "f3b5d9f0a2b68b15efab566f5bf3fc54": "\\begin{align} A(z) &{} = \\prod_{\\beta \\in \\mathcal{B}} (1 - z^{|\\beta|})^{-1} \\\\\n &{} = \\prod_{n = 1}^{\\infty} (1 - z^{n})^{-B_{n}} \\\\\n &{} = \\exp \\left ( \\ln \\prod_{n = 1}^{\\infty} (1 - z^{n})^{-B_{n}} \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{n=1}^{\\infty}-B_{n} \\ln (1 - z^{n}) \\right ) \\\\\n &{} = \\exp \\left ( \\sum_{k=1}^{\\infty} \\frac{B(z^{k})}{k} \\right ),\n\\end{align}\n", "f3b6370ac88a3e5f2a05f58d1428f631": " \\mathbf{y}_n ", "f3b63a52834f89c271f5e4fecaba7845": "\\operatorname{E}[\\mathrm{d}\\rho_I(t)]=\\operatorname{E}[\\mathrm{d}\\rho_J(t)]=\\dot{\\rho}\\mathrm{d}t\\,.", "f3b63b07b72868ebc09408b8478d6f03": " c_4=0 ", "f3b664bd13c1b98d7c059a0b7479a0ac": " G(x)= \\langle 0 | \\phi(-x) \\phi(x) | 0\\rangle \\,", "f3b6a7d3e37f03b772049e6ee271cc24": "f^0(\\bot) = \\bot \\sqsubseteq k", "f3b6ae7dd6a8b707a20e840664530c5f": "h\\ := initial", "f3b6e279349c2a7c81b6c2ec730105b6": "\\scriptstyle\\Phi(0,\\boldsymbol{x})=\\boldsymbol{x}_0\\in M", "f3b6e7a57e31577cdd3ffee483b6ed8a": "Y_\\ell^m(\\theta,\\phi) \\rightarrow Y_\\ell^m(\\pi-\\theta,\\pi+\\phi) = (-1)^\\ell Y_\\ell^m(\\theta,\\phi)", "f3b712ca987f47c0eb6b066cdf7df1e3": "t\\geq s\\geq 0", "f3b71d0cd713f5faca8d8cc043e32e21": "(u,\\,v)=(-2s,\\, s^2+t\\,|t|)", "f3b72e7c8201f7302fc3d381f4bea8c8": "\\mathrm{III}_T(t) \\quad \\stackrel{\\mathcal{F}}{\\longleftrightarrow}\\quad \\frac{\\sqrt{2\\pi }}{T} \\mathrm{III}_{{2\\pi}/T}(\\omega) \\quad = \\frac{1}{\\sqrt{2\\pi}}\\sum_{n=-\\infty}^{\\infty} e^{-i\\omega nT}. \\,", "f3b751ec7d2901f7c23964450c4ffd40": "\\mu_{xy} = \\left|\\mu_x-\\mu_y\\right|,", "f3b78fa2fa9344732c0aa983716c12bc": " \nZ_k = \\{z_{k1},z_{k2},...,z_{kn}\\}\n", "f3b7a824afc71331d8fbbcadb2471579": "g(p_{2}, p_{4})", "f3b840a3b817adc3b4c8b7559e954af6": "\\gamma:(0,1)\\to R", "f3b863266ac1d6031bfd21273e6c6f08": "U (\\mathbf{r}, t ) = A_o \\cos (\\mathbf{k} \\cdot \\mathbf{r} - \\omega t + \\varphi ) + i A_o \\sin (\\mathbf{k} \\cdot \\mathbf{r} - \\omega t + \\varphi )", "f3b87861a95e5cf071aa96abb382e961": " \\mathcal L_X T_{ab} = X^c \\nabla_c T_{ab} + (\\nabla_a X^c)T_{cb} + (\\nabla_b X^c) T_{ac} = X^c T_{ab,c} + X^c_{,a} T_{cb} + X^c_{,b} T_{ac}", "f3b8c6c4a9701572610bffc9f7380f6f": "w,b,\\zeta", "f3b8fe2ca8e476b0453f273e7da0a5eb": "\\hat\\beta.", "f3b9696ed6ebb6bc7e3070cd73e4d599": " X_t = c + \\sum_{i=1}^p \\varphi_i X_{t-i}+ \\varepsilon_t \\,", "f3b9c0ae0bce98130cd69780ed9102f7": "\\Phi_n(x) = 1+x+x^2+\\cdots+x^{n-1}=\\sum_{i=0}^{n-1}x^i.", "f3b9fbc66438f016b3564b92d19cad54": "\\frac{p_\\mathrm{r}}{p_\\mathrm{\\varphi}} = - \\frac{du}{d \\varphi}", "f3ba095b1a0e8a3cec18652d9176c987": " (\\Delta \\phi) (\\Delta L_z) \\geq \\frac{\\hbar}{2} ", "f3ba221b990fa935234858d84b88d726": " \\mathcal{Z}(\\beta)=\\int_0^{\\infty}e^{-\\beta E}\\Omega(E)\\,dE,", "f3ba4e988cb2a8f56e4e51643aec4ad5": "\\phi_{xx}+\\phi_{yy}+\\lambda \\phi (x,y)=0 \\ ", "f3bb4c432239d4bfa8fa06c9ab7066b4": "g_I", "f3bbb8b45ef2a0f0f34c0f23813fe39f": "E_2:\\mathrm{ player\\ 2\\ wins}", "f3bc1c033cf6d9a8b1410155a8eb6a70": "({\\color{red}p_1}, {\\color{red}p_2}, {\\color{blue}d_1}, {\\color{red}p_3}, {\\color{blue}d_2}, {\\color{blue}d_3}, {\\color{blue}d_4})", "f3bc1f9943ac13861767a892be04bbe3": "\\Omega \\ ", "f3bc256083da7ecfb64f23428b47efff": "\nW = PT\\,.\n", "f3bc30e1f782b195603b95cfded20ef9": "6 a_2 a_4.\\,", "f3bc7e2287f6f7df5f400abdac5d82ac": " \\!\\ P - \\sqrt2 = \\ln(1 + \\sqrt2)", "f3bc8ffde5bad23ae0e7a67c34409651": "f(\\mathbf{a} + \\mathbf{v} + \\mathbf{w}) - f(\\mathbf{a} + \\mathbf{v}) - f(\\mathbf{a} + \\mathbf{w}) + f(\\mathbf{a})\n\\approx f'(\\mathbf{a} + \\mathbf{v})\\mathbf{w} - f'(\\mathbf{a})\\mathbf{w}.", "f3bce3526e7ef8f809156487c33c64e8": "\\boldsymbol \\beta = S \\mathbf y ", "f3bcfcfca271d76d4d59b27f221f8ace": " R=k[x_1, \\ldots, x_n]/I", "f3bd082c54637fcabf7af771774899bc": "\nP(f|\\mathbf{X},\\mathbf{Y}) \\propto \\exp\\left(-\\frac{1}{\\sigma^2} \\|f_{\\mathbf{w}}(\\mathbf{X}) - \\mathbf{Y}\\|_n^2 + \\|\\mathbf{w}\\|^2\\right)\n", "f3bd4c2a056abe49a82aeb898dbcbeaa": "\nB_n=(\\alpha+2\\pi n)\n\\begin{pmatrix}\n0 & -1 \\\\\n1 & 0\\\\\n\\end{pmatrix},\n", "f3bdaee8843e0969a3cbe965bf54eef5": "\\{P_1,P_2,\\cdots\\}", "f3be0227794a436fac8b2e7bfb587110": "K_M\\le k.", "f3be50330d096737257f0367cb9f7a29": "\\! (1 - it\\lambda^{-1})^{-1}", "f3be6230f4eb6dfa23f5c425b4b5d4b6": "(x-3) x^{16} (x+3) (x^2-6)^6 (x^2-3)^{12}.\\ ", "f3be6333b9e442387451b469a322628c": "C(L)\\sim L^{-\\gamma}\\!\\ ", "f3bf034897f13f9562a977ab5337b4f3": "C_{\\alpha IJ} = C_{\\alpha [IJ]}", "f3bf0f3ac334d31f2792c61254fa477c": " x_i \\ge 0,\\, t_j \\ge 0 ", "f3bf508902491a458cf650c05ab6a709": "l=0,2...,n-2,n", "f3bf82c2756d4903bb60481bd5a1ac45": "\\text{height} = f_{a,i} \\times R_i", "f3bfc129a775b13fb9ccd0bdcbadac24": "\\frac{k(k + 1)}{2} + (k+1)\\,.", "f3c0292eb493324a293eb50498e6c1b6": "\\lambda = \\frac{k_{B}T}{p\\pi r_{I}^{2}}=\\frac{1}{L\\cdot p}\\qquad\\qquad(8)", "f3c029ff1c4400a5d48c2bf015dea624": "\\begin{matrix}\nE_{POT} = - m g (l \\cos \\varphi + a \\cos \\nu t)\\;.\n\\end{matrix}", "f3c0aa92947d39d9f00323b42ad5ca64": "\\begin{matrix} {2 \\choose 1}{10 \\choose 1}{4 \\choose 2} \\end{matrix}", "f3c0c3a1070c997ec6ede3bdba12f8f1": "S_{yc}=S_{yt}", "f3c13d61950678a9e45fea611792e8be": "{\\rm C} + {\\rm H}_2 {\\rm O} \\rarr {\\rm H}_2 + {\\rm CO}", "f3c1449a9fdce4368ac640998fabb404": "\\frac{1}{\\tau} = \\sum_{\\bar{k}',\\bar{k}} S_{\\bar{k}'\\bar{k}}=n\\sum_{\\bar{k}}\\frac{2\\pi}{\\hbar}\\frac{e^{4}\\delta (E_{\\bar{k}}-E_{\\bar{k}'})}{\\varepsilon \\varepsilon _{r}V[\\bar{q}^{2}-q_{s}^{2}]^{2}}=\\frac{ne^{4}}{4\\pi^{2}\\hbar \\varepsilon \\varepsilon _{r}}\\int \\int \\int dk d\\theta d\\phi \\frac{k^{2} sin\\theta \\; \\delta (E_{\\bar{k}}-E_{\\bar{k}'})}{[\\bar{q}^{2}-q_{s}^{2}]^{2}} \\;\\; (10)", "f3c152984cdfbb18135312355c8594a9": "d(x,y)=0\\Leftrightarrow x=y", "f3c17eac81301e2c509dd2c979bb1fb0": "a:V\\times V\\to \\mathbb R", "f3c194292dbb927b6e6d066c4ed3f38a": "t = k\\,\\Delta t", "f3c1d3a00c1ad4b653ef1ce33e47e721": "-1\\,", "f3c1f59393e9532ce9ee1e7710d7b779": "(\\log_{1/p} n)/p,\\,", "f3c2749e4798fa8385fdc00a0aa7bea4": "y_i\\,\\!", "f3c27af6e8de2459616c048b2ab022a3": "\\displaystyle\\{ \\gamma^\\mu, \\gamma^\\nu \\} = \\gamma^\\mu \\gamma^\\nu + \\gamma^\\nu \\gamma^\\mu = 2 \\eta^{\\mu \\nu} I_4 ", "f3c2b8f6270cfb59d783dd45774dc0c4": "m_j = 3/2, 1/2, -1/2, -3/2", "f3c2e0bd04012c6188feb0209f26606c": "\nC_\\mu (s,t)=\\sum\\limits_{m=0}^\\infty \\frac 1{\\Gamma (m\\mu +1)}\\left( \\nu\nt^\\mu (e^{is}-1)\\right) ^m, \n", "f3c37b862c2dfafe720bc42cd537ae4a": "\\scriptstyle \\frac{D}{Dt}", "f3c386e8652d4bc4ecc120e7bff39810": "S_{D}", "f3c40efdc439cc1b65df40f8641d9d80": "\n\\begin{bmatrix}\n H_{11} - ES_{11} & H_{12} - ES_{12} \\\\\n H_{21} - ES_{21} & H_{22} - ES_{22} \\\\\n \\end{bmatrix} \\times\n \n\\begin{bmatrix}\n c_1 \\\\\n c_2 \\\\\n \\end{bmatrix}= 0\n", "f3c41b37aff28bd32e74869d270f1df7": "r = \\frac{1}{n-1} \\sum ^n _{i=1} \\left( \\frac{X_i - \\bar{X}}{s_X} \\right) \\left( \\frac{Y_i - \\bar{Y}}{s_Y} \\right)", "f3c45b92509c7ebf2a0d5d03714e6f06": "( X, Y )_{L^{2} (W)} := \\mathbb{E} \\left( \\int_{0}^{T} X_{t} \\, \\mathrm{d} W_{t} \\int_{0}^{T} Y_{t} \\, \\mathrm{d} W_{t} \\right) = \\int_{\\Omega} \\left( \\int_{0}^{T} X_{t} \\, \\mathrm{d} W_{t} \\int_{0}^{T} Y_{t} \\, \\mathrm{d} W_{t} \\right) \\, \\mathrm{d} \\gamma (\\omega)", "f3c4a19b4f68bbcf84bd4cd741487666": "\\mathfrak{so}(3)", "f3c4a60b576264673cc9d2fb9b5f3aef": "\\mathrm{X{\\cdot} + R{-}H \\longrightarrow X{-}H + R{\\cdot}}", "f3c4b7bdaed5bdd7caeb8cb2d205216f": "= - p\\,{\\rm d}V - S{\\rm d}T\\,", "f3c4d27f473d23df6f9be02153408e16": "\n(\\hat{E} + mc )\\psi_{3,4} = (\\boldsymbol{\\sigma}\\cdot\\hat{\\mathbf{p}})\\psi_{1,2} \n", "f3c4ea4454131a9b5c8b7e8f6f01e3d8": " x\\mapsto gx \\, ", "f3c5148f0ae2e130672925899168464a": "t\\in[\\tau,2\\tau]", "f3c545257ab618f60a6c31860525fd85": "\\partial^j", "f3c56c5a25a9c90c227bb407db76c86d": " \\frac{d Q}{d t} = - h \\cdot A(T(t)- T_{\\text{env}}) = - h \\cdot A \\Delta T(t)\\quad ", "f3c57544d61ac94de3bd1f54d3025071": "\\frac{\\partial }{\\partial x^{\\sigma}} j^{\\sigma} = 0", "f3c57f8ac44788a08b2bf814b8a89d1c": "h_\\ell^{(1)}(kr)", "f3c5aa178c3b2db5a2cf309b52080201": " t(x) = \\sin^{-1}( \\sqrt{ x } ) - \\sqrt{ x ( 1 - x ) } ", "f3c5e41e0832a411bfa99cb78aa252b2": "\\begin{align} \\sum_{k=-\\infty}^\\infty [F(n+1,k)-F(n,k)] \n& {} = \\lim_{M \\to \\infty} \\sum_{k=-M}^M[F(n+1,k)-F(n,k)] \\\\\n& {} = \\lim_{M \\to \\infty} \\sum_{k=-M}^M [G(n,k+1)-G(n,k)] \\\\\n& {} = \\lim_{M \\to \\infty} [G(n,M+1)-G(n,-M)] \\\\\n& {} = 0-0 \\\\\n& {} = 0.\n\\end{align}\n", "f3c6116e26d3ad51862426dd22339572": " (f,g) := f\\stackrel{\\leftarrow}{\\partial}_{i}\\pi^{ij}\\partial_{j}g . ", "f3c6162b01c6c240b6c4862efe90d3f2": "r = \\left({a\\over 2}\\right)\\cot\\frac{\\pi}{p}\\tan\\frac{\\theta}{2}", "f3c62aad23d2b0046174444c53161a12": "a, b, c, d, e\\in A; p, q, r, s, t \\in P", "f3c6e16114d288a8e467735f1a8124c2": "Y\\times X = \\{\\,(y,x)\\mid y\\in Y \\ \\and \\ x\\in X\\,\\}.", "f3c71b968f5ea291adc43c42bf969566": "V = |\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c})|.", "f3c780d6f66b4a79fb2a19fffa1e512b": "B=X^TNX", "f3c782986bc57ae63a89d4dd42448db2": " ~R_{min} ", "f3c7d449dd5a2e72425322aa0fab5fce": "\\scriptstyle \\int_1^2 R(t)\\,dt \\,=\\, 0", "f3c7fab7c99604596ad5f432bc944de9": " \\mathrm{Ta_c} \\simeq 1708 ", "f3c7fd5797f48b43c1cd96347dac0d20": "(C- \\lambda_n) y_n + \\lambda_n y_n - (C- \\lambda_m) y_m \\in Y_{m-1},", "f3c8175ab6685a1af61e0befced0a150": "ax+by+c_2=0,\\,", "f3c822c3f727e225cf66da0cd24172f0": "2\\pi \\chi(M)", "f3c82fedb1582b970834c54f259136e9": "m-1 \\le \\operatorname{dim}(R/\\mathfrak{p}_1) \\le d(R/\\mathfrak{p}_1) \\le d(D) - 1 \\le d(R) - 1", "f3c872eb811ef68db3d96c9bd97ad682": "E(S,\\overline{S})", "f3c89cd6bbb125505682d6925b18e8d7": "\\beta_{T}=-\\frac{1}{V}\\left(\\frac{\\partial V}{\\partial P}\\right)_{T}\\,", "f3c8ee686fea7938eb4766a499cd72be": "(X_i-X_j)", "f3c8f424b6954b45e28b4ee61d4155a1": "\\hat{\\mu}_i=x_i.", "f3c90292b7c1664afc46a9d4db6622fa": "\\psi_k(x)", "f3c91d87a0d4ccc821208ef2b1795d97": "\\langle X_i^2 X_j^2 \\rangle = 2 \\langle X_i X_j \\rangle^2 + \\langle X_i^2 \\rangle \\langle X_j^2 \\rangle", "f3c93a0a7b6ff16f891205bee4b04736": "H\\,", "f3c9770f8258f8f8938d19402879d79d": "x_0-x_P", "f3c996903da9d33ede2d59cfe5edd947": "1_X=1_{\\{\\tau_X\\le T\\}}", "f3c9b7f245018a28aadfbe35672d052a": "\\Delta P_d", "f3ca181786402541cdb10fa2e5b1b37d": "D(P)", "f3ca5133f936c1c6483bcfc4f736b3c4": "2^{2 g}", "f3cb5ac9239fd5d0e83a1e82f75c4ef9": "qr(\\lnot \\varphi) = qr(\\varphi)", "f3cb723f1ff4900548e63bd5a3de5031": "l_A a_B + (1 + \\sigma) (l_A a_B + l_B) l_B", "f3cbafa04881746fb44bf0e66a644672": "\\nabla\\times\\mathbf{v} =\\mathbf{0}", "f3cbbf042f3a9f1ee28d4eeb0b61da62": "\\frac{\\mathrm{red}}{\\mathrm{green}} = \\frac{\\mathrm{green}}{\\mathrm{blue}} = \\frac{\\mathrm{blue}}{\\mathrm{magenta}} = \\varphi.", "f3cbf7e8f208c548deadd87a57b163cb": "\\mathrm{i}", "f3cc4b5f27a16dfe9c297722b7599a20": "R I=V_s-V_f \\;", "f3cc757e2a423ae88c6c14fc6e1e906d": "\n[x] + [y] + \\left[\\frac{1-x}{1-xy}\\right] + [1-xy] + \\left[\\frac{1-y}{1-xy}\\right]\n", "f3ccb284506db2f6ff78d2fa412fdb73": " P(n,p) = 1/2 + c_1 (p-1/2) + c_3 (p-1/2)^3 + O( (p-1/2)^5 ) ", "f3ccdfd1477fe7764abe6ac7c4d6bbe7": " \\left( \\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_n}\\right) ", "f3ccf9b2ea2b1ef4e5b3115c4b09c706": "O(\\log r)", "f3cd6288c064b7cd9970403c6738d507": "\\,m_1", "f3cd6cc06afbfafeb5a3c83afb793dcb": " E_+ ", "f3cdee34aa67eaac4ea7ed9dfdb5d3c7": "{\\Gamma,x\\!:\\!\\sigma\\vdash e\\!:\\!\\tau}\\over{\\Gamma\\vdash (\\lambda x\\!:\\!\\sigma.~e)\\!:\\!(\\sigma \\to \\tau)}", "f3cdfa39b1e56d16daa128d88ddeb968": "\n\\frac{\\partial f(t)}{\\partial t^\\alpha}=\\lim_{t_1 \\rightarrow t}\\frac{f(t_1)-f(t)}{t_1^\\alpha-t^\\alpha}\\,, \\quad \\alpha>0", "f3ce19a9b8f0fa63dcc6c90f1dbf6738": "1/4\\left(1/4+3/4e^{-4/3t}\\right)\\ ", "f3ce1d7b90a7cfbfc14c088289a97dd4": " \\{u,v,\\{w,x,y\\}\\} = \\{w,x,\\{u,v,y\\}\\} + \\{w, \\{u,v,x\\},y\\} -\\{\\{v,u,w\\},x,y\\}. ", "f3ce209e5035a2d65ee1c8e61b75b2b9": " \\cos \\theta_{\\rho\\sigma} = F(\\rho,\\sigma) \\,", "f3ce2b85302b64ebb3d18da47af2ed01": "n_j\\,\\!", "f3ce54e7816aca710eff3d85cf2bd2eb": "S_n/n", "f3ce5902cd3275012ec38b8d18f44989": "S, T", "f3cea8ef185c6aafac89c951daf12552": "A^\\ast B^\\ast 0.", "f3eed66b59e87ff4d616bede48bc8b32": "_{\\oplus}", "f3eed7f809b787d54786ab526d2cf3df": "\\mathbf{}\\begin{bmatrix}\n\n1&0&0&0 \\\\\n0&1 &0&0 \\\\\n0 & 0 &-3/5&-4/5 \\\\\n0 & 0 & -4/5&3/5 \\end{bmatrix},", "f3ef1df9f1778ce80b4afc0a796469a1": "r^{(n)}_{kk}", "f3ef2cc8523329e945dafa16d59369d5": " SE = \\sqrt{ \\frac{ 0.0863 A } { N } } ", "f3ef2d6cbf30b8517c0b24b82bd8dc94": "n=2^{56}\\cdot(2^{61}-1)\\cdot153722867280912929\\ \\approx\\ 2\\cdot10^{52}", "f3ef428f64016eb577640630e0863c93": "\\alpha_\\lambda", "f3ef429cba1f8038d3d17ac3a05604f4": " \\hat U(t)= \\exp\\left(\\frac{-i\\hat H t}{\\hbar}\\right) ", "f3ef94e2b88c8cd82f642490a64099bc": "\\! T_n(x)^2 - T_{n-1}(x)T_{n+1}(x)= 1-x^2>0 \\text{ for }-1 0", "f3ff53698c2a9ac6a997918c736a7280": "p \\cdot q = \\textstyle\\frac{1}{2}(p^*q + q^*p) = \\textstyle\\frac{1}{2}(pq^* + qp^*).", "f3ff5e1eca7a0492f636dc4dc2c98db9": "\\operatorname{Res}(f,c)=\\lim_{z\\to c}(z-c)f(z).", "f3ff5f285a7f770b1f2ff72ea429f4cd": "(\\psi)", "f3ff6e272c03a29e25c2d7fdee166896": "\\frac{a}{\\sin\\alpha} = \\frac{b}{\\sin\\beta}.", "f3ff852f1daadcde34256f7e27802adb": "\\nabla\\cdot\\mathbf{S} = -\\mathbf{E}\\cdot\\frac{\\partial \\mathbf{D}} {\\partial t}", "f3ffe17293ec617a1fc2226a6bed1b56": "X_1,\\cdots,X_n", "f3ffeded4ba936f4c11995aa9ecc8b71": "\\dot{\\vec{u}} = \\vec{f}(\\vec{u})", "f3fff5c6431472023cc377b482a470f7": " \\langle \\phi (t + \\tau)| \\phi (t + \\tau) \\rangle = \\langle \\phi(t) | \\hat{U}^{\\dagger}(\\tau) \\hat{U}(\\tau) | \\phi(t) \\rangle = \\langle \\phi(t) | \\phi(t) \\rangle = 1 ", "f40017f42a237e52663001c584dabed9": "P=\\frac{e^2}{6\\pi\\epsilon_0c}\\left | \\dot{\\vec{\\beta }} \\right |^2\\gamma ^4=\\frac{e^2c}{6\\pi\\epsilon_0}\\frac{\\gamma ^4}{\\rho ^2}=\\frac{e^4}{6\\pi\\epsilon_0m^4c^5}E^2B^2,\\qquad (8)", "f400472cdbef05fd24282c7506fc7a2f": "\\{a, b, \\neg b\\}", "f4004d75aabf4c24ac969a77d4cc54cd": "s \\leftarrow \\frac{\\omega_0}{\\tan(\\frac{\\omega_0 T}{2})} \\frac{z - 1}{z + 1}.", "f400602cb6886e81e748e268481ca68e": "\\frac{\\hbar}{2} = \\frac{\\hbar_0}{2}\\,\\!", "f4008b3c9885746980299966ef407b0a": "\\int_\\Omega \\alpha = \\int_{\\phi(U)} \\left(\\phi^{-1}\\right)^* \\alpha \\, ", "f400aab709df4f2b78a82a2c1647844c": "F(\\theta_{ji})", "f400ba64ecb7e81160f59ffedca63c16": " \\psi(0\\sigma\\}", "f40abe4474a29d693dd3060fd190f1d4": "\\scriptstyle 1, 2, 3, 4, 5, 6, \\dots, 2n", "f40aef2eae73e346006ad0ba6c294142": "262144 = 4^{3^{2^{1}}}.\\,", "f40b68e72ed2c6a5e5eda670858c3867": "[x]\\in_{NFU}[y]", "f40badaf82a20e02d804095969cf95a7": "\\color{Maroon}\\text{Maroon}", "f40bcd5fd1b24a9ba9122ee2e340de2c": " \\operatorname{sys}^2 \\leq \\frac{2}{\\sqrt{3}} \\operatorname{area} (\\mathbb T^2),", "f40be60a38b1a83f064a35eb4a6f8f24": "\\int\\frac{\\mathrm{d}x}{\\cos ax\\pm\\sin ax} = \\frac{1}{a\\sqrt{2}}\\ln\\left|\\tan\\left(\\frac{ax}{2}\\pm\\frac{\\pi}{8}\\right)\\right|+C", "f40c0461249ae22d4e9b6637609fbf5f": "\\vec\\varphi", "f40c36ad665a9ce1886dbabe7a847201": "\\ell = \\sqrt{2} - 1 ", "f40c6ef60643c0bde37a77ca0dd9ba18": "P(k,y)", "f40c71880f4dea7e5b13ecb591fdea54": "i_{\\text{C}}(t) = C \\frac{\\operatorname{d}v_{\\text{C}}(t)}{\\operatorname{d}t}", "f40c9d37f14eff1b50d01b5103df03b2": " \\nu \\approx 2.1", "f40c9f34d00395d8015759c95b93ecd2": "L = \\tfrac12 \\rho V^2 S C_L", "f40d397eae0679936e03c5132ab9e99c": "S_L", "f40d5aef8eccd4414c161100002aded0": " y(t) = \\mathrm{e}^t ", "f40d7d2833b17f3597edb1236ba16638": "{H = - J\\sum\\limits_{\\left\\langle {i,j} \\right\\rangle } {{\\mathbf{S}}_i \\cdot {\\mathbf{S}}_j } }.", "f40d801425f565e344a6681f713b8c7e": "{0^2 \\over 2}+g(0)+{P_\\mathrm{atm} \\over \\rho}={v_C^2 \\over 2}-gh_C+{P_\\mathrm{atm} \\over \\rho}", "f40d952d2a71b86263c7e2ebf256c82a": "p^2 + q^2", "f40dc54dc78dd8618fc07746c0123cde": "z \\simeq \\beta = \\frac{v}{c},", "f40df17c99c1f9adc326458e6822f578": "\\; (fg)^* = f^*g^*", "f40e22ca90d038093ee93e25324b624b": "X_\\xi", "f40e28fbaa8931a709791dd40bdc4852": "p' \\neq q'", "f40e6ca868e5c355ec74cbe74002cb06": "f_{j}=\\sum_{i}g\\left(w_{ji}'x_{i}+b_{j}\\right)", "f40edf2439a6bc02d6e87e40b368dcdd": "\\frac {12}{4}", "f40f472e6faaf65b59d4bdf16b2bc663": " A_{\\epsilon}^{(n)} ", "f40fe839d0e826455b1dee1590134848": "\\int\\mathrm{haversin}(x) \\,\\mathrm{d}x = \\frac{x - \\sin{x}}{2} + C", "f40ff82e7a93513ef8b37f3805665b56": "\\bar{\\mathbf{e}}_{k_1 k_2 \\cdots k_p}^{\\ell_1 \\ell_2 \\cdots \\ell_q} = (\\boldsymbol{\\mathsf{L}}^{-1})_{k_1}{}^{i_1} (\\boldsymbol{\\mathsf{L}}^{-1})_{k_2}{}^{i_2} \\cdots (\\boldsymbol{\\mathsf{L}}^{-1})_{k_p}{}^{i_p} \\mathsf{L}_{j_1}{}^{\\ell_1} \\mathsf{L}_{j_2}{}^{\\ell_2} \\cdots \\mathsf{L}_{j_q}{}^{\\ell_q} \\mathbf{e}_{i_1 i_2 \\cdots i_p}^{j_1 j_2 \\cdots j_q}", "f40ffe4fe2420412ea26eb500c741ea1": "R = r", "f4107068e07270a789ea3832372e5dc0": "I_\\mathrm{C} \\,", "f41085d87354b85cff712837945b8523": "\\mathcal{F}(\\Omega)=\\mbox{Area}(\\partial \\Omega)", "f41088782bcd5ed102fc5e3a2e1643b8": "\\mathcal B= \\{\\emptyset,\\Omega\\}", "f410c02ab9f32458a5a8c2cb0b6057ab": "\\scriptstyle\\omega_r", "f411227fdd62e0976cd60065ed9c180b": "X_1,...,X_n", "f4112eac108bcc4c5e4dce3c2a89e16e": "T_n(C_{\\bull,\\bull})^I_p = \\bigoplus_{i+j=n \\atop i > p-1} C_{i,j}", "f411b5ed26885e10b3952358b057a04e": "Y_{7}^{1}(\\theta,\\varphi)={-1\\over 64}\\sqrt{105\\over 2\\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(429\\cos^{6}\\theta-495\\cos^{4}\\theta+135\\cos^{2}\\theta-5)", "f411cc0371a92c827b4258fd09ae514f": " \\textbf{R} ", "f411d2fc78bdeee50bb5682e6442aafd": "(B,C)", "f4121b1365643004c50751ea8041098d": "H\\left( X\\right) ", "f4121c53f56ce860771488f51c1ffc03": "\\vec{J}(\\vec{r},t)=\\frac{-\\nabla \\Phi(\\vec{r},t)}{3(\\mu_a+\\mu_s')}", "f41259d98beb872aa6e0b53cea634e0a": "\\arg\\min_{(\\mathbf{w},b)}\\frac{1}{2}\\|\\mathbf{w}\\|^2", "f4128616922a1249b003640070da6940": "{\\mathbf{}}F_r(t)=A_r(t)-B_r(t)L_r(t)-K_r(t)C_r(t)", "f4128908212b0beef7f813a6c3d2218d": "(N,M,D,K,\\epsilon)", "f4128d62b4e9dc823ab99743788257ad": "h \\mapsto g", "f412a16a33e9f5ca747000237c281548": "S_{\\infty}", "f412af940eea0601e0554640986fd432": "R[x_1, x_2, x_3, \\ldots]", "f412b11d24b680ac3ecde6d3a9aa6bec": "r = 36.03 \\sqrt{{D} \\over f}", "f412b33773c3262d942cae6b72465816": "\\mathbf{\\hat{d}} = q \\mathbf{\\hat{r}} ", "f412cedb99f06a27f441653e589d3fe2": "\\sec ( - \\alpha ) = \\sec \\alpha", "f412f53934fe47d21a968bb80f87b8fa": "(1 - H(p + \\epsilon)n)", "f413124820f57ea03792bf57baf3d108": "{\\mathbb R^n}\\,", "f413281ef830a61d5402d209dae913a7": "\n\t|A^{kj}|_{il}^{\\,-1} \\left|A\\right|_{ij} = - |A^{ij}|_{kl}^{\\,-1} \\left|A\\right|_{kj} ,\n", "f41341c7524a97ed607f36c52ddf628a": "\\psi(\\mathbf{r},t) = \\begin{bmatrix} \\psi_{+,\\,\\sigma=s}(\\mathbf{r},t) \\\\ \\psi_{+,\\,\\sigma=s - 1}(\\mathbf{r},t) \\\\ \\vdots \\\\ \\psi_{+,\\,\\sigma=-s + 1}(\\mathbf{r},t) \\\\ \\psi_{+,\\,\\sigma=-s}(\\mathbf{r},t) \\\\ \\psi_{-,\\,\\sigma=s}(\\mathbf{r},t) \\\\ \\psi_{-,\\,\\sigma=s - 1}(\\mathbf{r},t) \\\\ \\vdots \\\\ \\psi_{-,\\,\\sigma=-s + 1}(\\mathbf{r},t) \\\\ \\psi_{-,\\,\\sigma=-s}(\\mathbf{r},t) \\end{bmatrix}\\quad\\rightleftharpoons\\quad {\\psi(\\mathbf{r},t)}^\\dagger\\begin{bmatrix} {\\psi_{+,\\,\\sigma=s}(\\mathbf{r},t)}^\\star & {\\psi_{+,\\,\\sigma=s - 1}(\\mathbf{r},t)}^\\star & \\cdots & {\\psi_{-,\\,\\sigma=-s}(\\mathbf{r},t)}^\\star \\end{bmatrix} ", "f4134fe3d35bee6287b16ea41db20319": "g^\\star(x^\\star)= -\\alpha- \\delta\\frac{x^\\star-\\beta}\\lambda +\\gamma \\cdot f^\\star \\left(\\frac {x^\\star-\\beta}{\\lambda \\gamma}\\right).", "f413f6c51ae29ac06cc221827fe7403a": "\\tau : \\Omega \\to [0, + \\infty]", "f41417578ef69e62da3c549422748c44": " \\delta_{ext}(q,x)=(s',1)", "f4142c9f5a128ca744d11d674f092889": "\\hat{\\mathbf x}_i(t + \\Delta t)", "f41433599dcb8251d1c2ac813349c407": "Y \\sim \\chi^2_{d_2}", "f4146735f47b54cac025262be28a3eff": "=\\kappa_1(Y\\kappa_4(W))+4\\kappa(Y\\kappa_3(W),Y\\kappa_1(W))\n+3\\kappa_2(Y\\kappa_2(W))\\,", "f4146c6a2b10a9b6991e7b01bba3593c": " P( | X - m | \\ge ks ) \\le \\frac{ g_{ N + 1 }\\left( \\frac{ N k^2 }{ N - 1 + k^2 } \\right) }{ N + 1 } \\left( \\frac{ N }{ N + 1 } \\right)^{ 1 / 2 } ", "f414780caaae1b17495672cf4eb23c12": "x_j+1", "f414a62da9679f01d0342d7383a8fb8f": "x_{em}=v_{em}t=\\frac{c}{n}t.", "f41510c9b44da0767180d33a486d6a8c": " |f-f_n| \\le |f| + |f_n| \\leq 2g", "f41538c621177fb1d70b2d753b6afbbe": "\n \\begin{align}\n\\frac{\\partial}{\\partial x_1}& \\left( \\frac{\\partial^2 \\varepsilon_{22}}{\\partial x_3^2} + \\frac{\\partial^2 \\varepsilon_{33}}{\\partial x_2^2} - \n2 \\frac{\\partial^2 \\varepsilon_{23}}{\\partial x_2 \\partial x_3}\\right) -\n\\frac{\\partial}{\\partial x_2}\\left[ \\frac{\\partial^2 \\varepsilon_{22}}{\\partial x_1 \\partial x_3} - \n\\frac{\\partial}{\\partial x_2} \\left ( \\frac{\\partial \\varepsilon_{23}}{\\partial x_1} - \\frac{\\partial \\varepsilon_{13}}{\\partial x_2} + \\frac{\\partial \\varepsilon_{12}}{\\partial x_3}\\right) \\right] \\\\\n & -\n\\frac{\\partial}{\\partial x_3}\\left[ \\frac{\\partial^2 \\varepsilon_{33}}{\\partial x_1 \\partial x_2} - \n\\frac{\\partial}{\\partial x_3} \\left ( \\frac{\\partial \\varepsilon_{23}}{\\partial x_1} + \\frac{\\partial \\varepsilon_{13}}{\\partial x_2} - \\frac{\\partial \\varepsilon_{12}}{\\partial x_3}\\right)\\right]=0\n \\end{align}\n", "f4155b9396880ec7c494de82a7b13253": "\\int\\cos ax\\;\\mathrm{d}x = \\frac{1}{a}\\sin ax+C\\,\\!", "f4156bbbd280423e5ff695fe62d8b403": " \\mathrm{ CI } = t ( \\frac { P( x ) ( 1 - P( x ) ) } { N } )^{ 1 / 2 } ", "f4156c28b06bbbb81f5adab14e7abb99": "\\theta_v: K_v^{\\times}/N_{L_v/K_v}(L_v^{\\times}) \\to G^{\\text{ab}}, ", "f415c06637360dc5a3015539651a7d54": "\\scriptstyle\\gamma=1/ \\sqrt{ 1-{v^2}/{c^2} }", "f415d55fde5955e5c795e5a0a8394df0": "h(x_i)", "f4161b48c782b8d8f4ae3c554b46c6ec": "\\sigma_\\text{avg}", "f41694219b623b393c386f646b2690c3": "y_1-y_0", "f416e80f660e2c6edbe6d3a30591859f": "s(t) = A_c \\cos\\left(2 \\pi f_c t + D_f \\int_{-\\infty}^{t} m(\\alpha) d \\alpha\\right)\\,", "f41712ac1ccca778ec906965693c1bfb": "\\bar{y} = \\frac{1^Ty}{N} = \\frac{\\sum_{i=1}^N y_i}{N}.", "f41724020cf39c40bce1e5d955bd9562": "V = \\operatorname{Tr}\\left(\\rho_{a}\\rho_{b}\\right). \\, ", "f41735f38aa1aa694381a52b54697591": "u_{it}", "f41791bee69f899bea38e6a5fbfe3e63": "F(b) - F(a) = \\sum_{i=1}^n \\,[f(c_i)(\\Delta x_i)]. \\qquad (2)", "f417aeb87c1c4372ad78864e0dc7c0a5": "\\ln L(r_1,\\dots,r_T;\\theta) = \\sum_{t=1}^{T}\\ln[\\omega(r_t).(\\Pi_{t-1}A)].", "f418193d0ac51ebfe1a72120824ab8c0": "A = 1-0.5\\frac{\\sigma^2}{\\sigma^2+0.33}", "f418441587c3b6e1e114b13c0f454352": "\\nabla\\cdot(\\nabla \\Phi) = \\nabla_i (\\nabla_i \\Phi) ", "f4184495c6419bf226a8117dcb060439": "h_\\ell^m", "f4185f9a6aac5eeeb0bda2b418b29519": "\\mathrm{Hom}(X\\otimes A,Y) \\cong \\mathrm{Hom}(X,\\mathrm{Hom}(A,Y)).", "f4186f76eb48e0eb6a9e0595d5353f78": "x \\leftarrow x + l", "f41896ce93de0630ff28eaa665ed48a1": "\\{s_1,\\ldots,s_k\\}", "f418afbd3277d34cfd0b7dc8a3e0c4db": "({\\text{Bandwidth}}_{E} + \\text{Delay}_{E}) \\cdot 256", "f418db3c4dfc191d2619748753d5841b": "\tV\\otimes\\chi_i = \\sum_j n_{ij} \\chi_j,", "f419301c910bc882d28095f87f4937c3": "y(t) =\\int _0^\\infty \\sin (t+\\tau) x(\\tau)\\,d\\tau", "f41932c270cef5ae2db61d2fb97dd546": "AP = \\frac {D*g*\\rho} {100000}", "f419447894a7f9a43a0a8ec89e4aa7a9": " U'(t) = \\mathcal{A}U(t) ", "f41975e2bcec98b8028a7b2f6eb2195a": "V_X", "f419cd6968db2212ef84469bb60ff8a6": "\\sum_1^k \\left(\\frac{X_i-\\mu_i}{\\sigma_i}\\right)^2", "f41a75b80d248df045ad307924082d05": "\\left(\\sqrt{2}^{\\sqrt{2}}\\right)^{\\sqrt{2}}=2", "f41a7a44766e8e95ca27053b93dd53bd": "\\rho _{\\alpha +} ^{i_0 } ", "f41abd9050b747e8d1f2a919a1c5138d": "\\mathrm{^{238}_{\\ 92}U\\ +\\ ^{10}_{\\ 5}B\\ \\longrightarrow \\ ^{242}_{\\ 97}Bk\\ +\\ 6\\ ^{1}_{0}n \\quad ; \\quad ^{232}_{\\ 90}Th\\ +\\ ^{15}_{\\ 7}N\\ \\longrightarrow \\ ^{242}_{\\ 97}Bk\\ +\\ 5\\ ^{1}_{0}n}", "f41ac6f6604f30d6097e274ef93edf63": "\\exp_{10}^3(3.33928)", "f41b14b2c17b7d30d81f9de3c503c726": "P(z+\\zeta_0)", "f41bafbed0dbffca370d9b1e37f96762": "J=\\begin{bmatrix}A&B\\\\ C&D \\end{bmatrix}.", "f41beb17f83980ac8eb4f3c6aa03242d": "\\tan \\frac{\\theta}{2} = \\sqrt{\\frac{(s-b)(s-d)}{(s-a)(s-c)}}.", "f41bf5b3db13f78d276841cf39ae8bfe": "u\\in L^q(U)", "f41c78c859a72ffdc90c9a4d394109a8": "x\\in B\\subseteq U\\,", "f41c7b1dd13be6d6698479e9918b211d": "P(\\bold R,t) = G(\\bold R,t) = \\frac{1}{\\sqrt{(2 \\pi)^3 (\\det(D) t)}} \\exp\\left (-\\frac{\\bold{R}^{T} D^{-1} \\bold{R}}{2t}\\right )", "f41cadd72d6dd3943d4be8c52e286ff1": "F_{A \\rarr A} > 0", "f41d0972a2ec3c12f500aca3c618bff2": "e^{\\frac{\\mathbf{B}}{2}}", "f41d0ac5f29ccf43d4109fb2d9ef199b": "w_{ij} = \\frac{1}{p} \\sum_{k=1}^p x_i^k x_j^k\\,", "f41d0ea3efed49f638c7bebced60ae2c": "\\lim_{t\\rightarrow \\infty} e^{tX}ne^{-tX}= 1", "f41d22b252a71bf2673eaa40d32efc89": " g(z) = z + a_2z^2 + a_3 z^3 + \\cdots ", "f41d410fb0cdb5b83d80637a072e4ba7": "\\displaystyle{J=\\begin{pmatrix} 0 & -I \\\\ I & 0 \\end{pmatrix}.}", "f41d8e55e5f373431bbd7986096f50c2": "\\scriptstyle f_c \\;=\\; \\frac{eB}{2\\pi m^*}", "f41d90b0637e6e25ef4f83324562b9c5": "V = \\frac{1-q}{q M_\\text{batch}^2} \\sum_{i=1}^N m_{i}^{2} \\left(a_i - a_\\text{batch} \\right)^2 .", "f41e0fc48ee8d8bf32b372a588f9ba9e": "\\neg A \\cdot B \\cdot \\neg C", "f41e3534ec7e29b3c869628a57e5c47d": "a(\\cdot,\\cdot)", "f41e79d580d12053728fbb2c3dca5c3a": "f^n(x)", "f41e7f9ac423a38c4e6293ebaf424972": "\\begin{align}\n&\n\\frac{\\partial }{\\partial t}\n\\left[\n\\begin{array}{cc}\n{\\mathbf I} & {\\mathbf 0} \\\\\n{\\mathbf 0} & {\\mathbf I} \n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\Psi^{+} \\\\\n\\Psi^{-} \n\\end{array}\n\\right] \n-\n\\frac{\\dot{v} ({\\mathbf r} , t)}{2 v ({\\mathbf r} , t)} \n\\left[\n\\begin{array}{cc}\n{\\mathbf I} & {\\mathbf 0} \\\\\n{\\mathbf 0} & {\\mathbf I} \n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\Psi^{+} \\\\\n\\Psi^{-}\n\\end{array}\n\\right] \n+ \\frac{\\dot{h} ({\\mathbf r} , t)}{2 h ({\\mathbf r} , t)} \n\\left[\n\\begin{array}{cc}\n{\\mathbf 0} & {\\rm i} \\beta \\alpha_y \\\\\n{\\rm i} \\beta \\alpha_y & {\\mathbf 0} \n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\Psi^{+} \\\\\n\\Psi^{-}\n\\end{array}\n\\right] \\\\\n& = - v ({\\mathbf r} , t)\n\\left[\n\\begin{array}{ccc}\n\\left\\{\n{\\mathbf M} \\cdot {\\mathbf \\nabla} \n+\n{\\mathbf \\Sigma} \\cdot {\\mathbf u} \n\\right\\}\n& &\n- {\\rm i} \\beta \n\\left({\\mathbf \\Sigma} \\cdot {\\mathbf w}\\right)\n\\alpha_y\n\\\\\n- {\\rm i} \\beta \n\\left({\\mathbf \\Sigma}^{*} \\cdot {\\mathbf w}\\right) \n\\alpha_y \n&\n\\left\\{\n{\\mathbf M}^{*} \\cdot {\\mathbf \\nabla} \n+ \n{\\mathbf \\Sigma}^{*} \\cdot {\\mathbf u} \n\\right\\}\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{cc}\n\\Psi^{+} \\\\\n\\Psi^{-}\n\\end{array}\n\\right] \n- \\left[\n\\begin{array}{cc}\n{\\mathbf I} & {\\mathbf 0} \\\\\n{\\mathbf 0} & {\\mathbf I} \n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\nW^{+} \\\\\nW^{-}\n\\end{array}\n\\right]\\,, \n\\end{align}", "f41e9ff09c195a81b2d9c1946c706c53": "\\begin{matrix} {2 \\choose 2}{46 \\choose 2} \\end{matrix}", "f41eaaa7d43c496513c1a06dfdf93592": "\\Gamma=f(S^{1})", "f41ebb655cc40551514607a48bcbf3b8": "p_\\text{sat}", "f41ee7976c8673eaa8cee04828d16c8e": "\n\\int_V \\frac{\\partial u_i}{\\partial x_j} \\sigma_{ij} + u_i \\frac{\\partial \\sigma_{ij}}{\\partial x_j} dV\n = \\int_V \\frac{\\partial u_i}{\\partial x_j} \\sigma_{ij} - u_i f_i dV\n", "f41f2392f1e8abb96262284cbdf4d6b3": "\\textstyle 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n} = H_n", "f41f29b8641fd0ee9358e1b48e5621dd": "\\partial_xa=a\\partial_x+\\frac{\\partial a}{\\partial x}", "f41f37759c9b52b75de6021d37fe29aa": "\n\\delta \\varphi \\approx \\frac{2r_{s}}{b} = \\frac{4GM}{c^{2}b}.\n", "f41f8ce58025fcae1061b0d077d837c0": " i\\hbar\\frac{\\partial}{\\partial t}\\Psi = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2\\Psi + V\\Psi ", "f41fbe3cac739040f0f8dc3875e44011": "h \\gg a", "f41fc75eec7c2c1163876a4c368594e5": "\\!\\ 4 \\pi r^2", "f41fd29e9c172035ccfa30694fd02997": "\\mathbf{\\hat{e}}_n", "f41fd9a65cda8674f6d9be77c3d7a9f1": "ax^2+(1-a)y^2=1,", "f41fd9f313350317a97a5fea9e4772ab": "(X,Y) \\mapsto d_Y(d_X\\Delta)", "f41ff7ef468d3cfbbb93ec65e46cd67d": "\\rho(t) = \\sup\\{s \\in \\mathbb{T} : s t_{p_2} > \\cdots > t_{p_n}) \\varepsilon(p)\n A_{p_1}(t_{p_1}) A_{p_2}(t_{p_2}) \\cdots A_{p_n}(t_{p_n})\n", "f4205df5ec1538d312d894bcb8364445": "\n\\frac{\\partial y}{\\partial \\mathbf{X}} =\n\\begin{bmatrix}\n\\frac{\\partial y}{\\partial x_{11}} & \\frac{\\partial y}{\\partial x_{21}} & \\cdots & \\frac{\\partial y}{\\partial x_{p1}}\\\\\n\\frac{\\partial y}{\\partial x_{12}} & \\frac{\\partial y}{\\partial x_{22}} & \\cdots & \\frac{\\partial y}{\\partial x_{p2}}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\frac{\\partial y}{\\partial x_{1q}} & \\frac{\\partial y}{\\partial x_{2q}} & \\cdots & \\frac{\\partial y}{\\partial x_{pq}}\\\\\n\\end{bmatrix}.\n", "f420837f067a7237d4bf9b6444f23cc7": "1 \\le N \\le \\infty", "f4209c229c8594b8a03c7fadc7e69157": "\\mathbf{y,n}", "f420c8859ad52a9d5b7a6d6e86944f6e": "(V_{\\mathrm{m}})_{\\text{ideal gas}} = RT/p", "f421287defe6c6e34e7522cc9b3133f4": "\\varepsilon_j", "f421498e10025ee6a187b5ed1b3137c7": "y(e) = \\sum_{f \\in E:\\mathrm{out}(f)=\\mathrm{in}(e)} (m_e(f)y(f))", "f421518226522743212fcbf6218d3c6e": "\\sum_{k=0}^\\infty e_k(X_1,\\ldots,X_n)(-t)^k = \\prod_{i=1}^n(1-X_it)", "f42198aa36a628cbdbddb1529422666f": "\\exists C\\,\\exists M>0\\text{ such that } |f(\\vec{x})| \\le C |g(\\vec{x})|\\text{ for all }\\vec{x} \\text{ with } x_i>M \\text{ for all }i.", "f4219d9cd080231fc2acb87d9f1574ff": "t=\\Delta t\\,", "f4219de07b12e26bd4ff2c76258e687e": "p(y, y_1 | x_1, x_2) = p(y | x_1, x_2)p(y_1 | y, x_2)", "f421c2ffddf468c8290e35345afed6c9": "\\left( \\frac{\\partial T}{\\partial P} \\right)_H = \\frac{V}{C_{\\mathrm{p}}}\\left(\\alpha T - 1\\right).", "f4226d0ce8dc7ab1ef577817176d4978": "N_O", "f422e1244207d3daf2acba146763bd04": "\\mathrm{Sp}(1) \\cong \\mathrm{SO}(4)/\\mathrm{SO}(3) \\cong \\mathrm{SU}(2) \\cong \\mathrm{Spin}(3)", "f422e9e350a21294d4489514a467674d": "(*)\\,", "f422f6654588d5f2e489964b7ad61c4b": "\\begin{align}\n\\frac{1}{4}\\int \\left( e^{2ix} + 2 + e^{-2ix} \\right) dx \n\\,&=\\, \\frac{1}{4}\\left(\\frac{e^{2ix}}{2i} + 2x - \\frac{e^{-2ix}}{2i}\\right)+C \\\\[6pt]\n&=\\, \\frac{1}{4}\\left(2x + \\sin 2x\\right) +C.\n\\end{align}", "f42356cca743fa33b20f296e7e2adb7b": "{Y}_{s}", "f4235901555a7061e2d8ef63372853f6": "{\\eta_b} = \\frac{Work~Done}{Kinetic~Energy~Supplied} = \\frac{2UV_w}{V_1^2}", "f42389b8b45db07ed9390c8c9a43ac2f": "| \\Psi | = \\sqrt{ \\Psi^{*} \\Psi }", "f424541c76c20424346a9007b2e94d26": "\\succeq_q", "f4247da39deed477a7abaa4b7a7842d1": " \\mathit{CM} = \\left\\{ \\langle A,k \\rangle \\in \\mathcal{C} \\times \\mathbb{N} \n\\left| \n\\begin{matrix}\n\\mbox{there exists a circuit } B \\mbox{ with at most } k \\mbox{ gates } \\\\\n\\mbox{ such that } A \\mbox{ and } B \\mbox{ compute the same function} \n\\end{matrix}\n\\right.\n\\right\\} ", "f424a6d8eb3bfea2e3de408a6cc410ab": " {n \\choose m} = {n-1 \\choose m-1} + {n-1 \\choose m}", "f424f30f582f416044b013909784b016": "\\mathcal{S} = \\int_{t_1}^{t_2} L \\, dt\\,,", "f424fc5bce956674d07ea0dadee18fb8": "\\omega_\\Lambda = (\\omega(t))_{t\\in\\Lambda}", "f4257c91fcb8820d70748c2eba89e24c": "A_N", "f4259043b9deb6dc6e46de30ca461f36": "\n\\frac{d\\mathbf{L}}{dt} = \\dot{\\mathbf{r}} \\times m\\mathbf{v} + \\mathbf{r} \\times m\\dot{\\mathbf{v}} = \\mathbf{v} \\times m\\mathbf{v} + \\mathbf{r} \\times \\mathbf{F} = \\mathbf{r} \\times \\mathbf{F} \\ ,\n", "f4262a63019835575c2208751dc54b54": "\n\\underbrace{i \\hbar \\frac{\\partial}{\\partial t} |\\psi_\\pm\\rangle = \\left( \\frac{( \\mathbf{p} -q \\bold A)^2}{2 m} + q \\phi \\right) \\hat 1 \\bold |\\psi\\rangle }_\\mathrm{Schr\\ddot{o}dinger~equation} - \\underbrace{\\frac{q \\hbar}{2m}\\boldsymbol{\\sigma} \\cdot \\bold B \\bold |\\psi\\rangle }_\\mathrm{Stern \\, Gerlach \\, term}", "f42646df0a3144bdd8e07d7c10ae5275": "\\textstyle \\deg(a(x)) = l_1 - 1", "f4269b4406253861a2030698e6926c58": "\\begin{bmatrix} a & b \\\\ 0 & 0 \\end{bmatrix}", "f426b8cc66634ce4bf39191f53b09286": "r_s \\overline{\\lambda}_{C} = \\frac{2G\\hbar}{c^3} = 2\\ell_P^2", "f426db8c388961a26535bf00df150125": "n_1^2", "f427d0fd91f3b2675aec9fb1194955c0": "\\scriptstyle \\forall p,\\, q,\\, r \\,\\in\\, A,\\; (q \\,-\\, p) \\,+\\, (r \\,-\\, q) \\;=\\; r \\,-\\, p", "f427f099b6fa25514896fda36e3d6a41": " \\max_{s\\in \\left[ 0,1\\right] }\\left[ f\\left( x,s\\right) -V\\left( s\\right)\\right] =f\\left( x,t\\right) -V\\left( t\\right) =0. ", "f428137b789476ff4e819c797eeae1d0": "O(2^j) = O(\\log 2^{2^{j-1}}) = O(\\log w(x))", "f4289cff71aee97558992528ac07c3f8": "\\text{E}L = \\text{E}(y-y_d)^2 = \\text{E}(aP + u - y_d)^2 = [\\text{E}(aP + u - y_d)]^2 + \\text{var} (aP + u - y_d) = [(\\text{E}a)P + \\text{E}u - y_d]^2 + P^2 \\sigma^2_a + \\sigma^2_u.", "f428d62a045730a62c2657f2da224bc2": "\\Gamma(t) = e^{iLt}\\Gamma(t=0) \\, ", "f429ad076af1d00d5e304d06a2625fe5": "\\sum b_{ij}^2", "f429d79cd3b6fdf26b4e4985fbf37aca": "L^2(0,1)", "f42a58292191dc6dc7e631c791afc317": "(J_\\alpha )_n (x) := J_\\alpha \\left( \\frac{u_{\\alpha,n}}b x \\right)", "f42a9cb7fd9fe8c91823139538688f48": "\\Phi(-1.96) = 0.025 = 1-\\Phi(1.96).\\,\\!", "f42ab33f012bc0f982c03017a7b32cc5": "\n\\sigma_{j} = \\sum_{p=1}^N\\left( \\mu_{p} \\lambda_{j}^{\\alpha_p} - \\mu_{p}\\lambda_{j}^{-\\frac{1}{2}\\alpha_p} \\right)\n", "f42aea6806262272425e5d0bf58e7f3d": "\\rho = \\frac {{\\rho_{T_0}}}{{(1 + \\alpha \\cdot \\Delta T)}}", "f42b0b7406d087ca18be1bfca4b495a3": " \\left( \\left|\\frac{x}{A}\\right|^r + \\left|\\frac{y}{B}\\right|^r \\right)^{t/r} + \\left|\\frac{z}{C}\\right|^{t} \\leq 1", "f42b376301bb82cc848d6ecd0b4cd83e": "2\\lambda \\geq 2^m+1", "f42b50b2e4ea39a213548545f4722d19": "\\Phi = \\Phi \\Rightarrow True \\ ", "f42b7096dfd570842f486a67227e29c7": "\n \\frac{\\partial M_{xx}}{\\partial x} - Q_x = 0 ~;~~ \\frac{\\partial Q_{x}}{\\partial x} + q = 0\n", "f42b7a5b430591c717b6c35b12ad0f49": "J_z", "f42bb99c0d47fcf95a9fbcf55abae145": "\\sum_{\\lambda\\vdash n} K_{\\lambda \\mu} K_{\\lambda \\nu} = N_{\\mu \\nu}", "f42c0285aa629398286b7e9d81e0e64b": "f(x,y)=f(y,x)", "f42c1032d20b52e02543864ad9b0f661": "\\langle h*x, y \\rangle = \\langle x, h^* * y \\rangle", "f42c417de2416d02d0fae97b62510c9a": "w\\Vdash\\neg A", "f42c5715fac5de58b70db4ffdc2893a8": "\\frac{\\partial (\\mathbf{U}\\mathbf{V})}{\\partial x} =", "f42c75cc81236ec10172b08594d1a13a": "W_e = \\frac{1}{10^{-dr/400} + 1}", "f42cc1f57874e656933e5987b459d0f1": "\\aleph_\\lambda", "f42d18fe51920886d2cc22e18097bac3": " (a+c)/2n ", "f42d22693fdc0838b92cc49f968841e3": "\\mathop{\\rm alldifferent}(x_1,\\ldots,x_n)", "f42d3020888c15425d8dfa8ed53b27ee": "\\pi (R^2 - r^2)", "f42dbaa9828f791412720040de451fee": " \\omega_{surf} = \\frac{\\omega_{bulk}}{\\sqrt{2}}", "f42dd1c176c4ab1476aa4cf2403a21b6": "\\phi_r", "f42e68d418689a6fc5fdde1899a11e4f": "\\partial^2 J(t,t') /\\partial t \\partial t' > 0", "f42ea44f1e0cdc8907c9832e230c009c": "1.96x^2 +19.6x = ", "f42ead04da4bfbb9ce7a6758580de2a4": "\\phi= -\\frac{1}{2+a+\\phi}", "f42eaf50b21902dfdfbf4496013498f8": "\\frac{\\Delta \\nu}{\\nu} \\ll \\frac{1}{l u}", "f42ef2d135c890bca1a7cb1ba8d35dfc": "\\hbox{Cross} \\leftarrow \\,\\sim\\hbox{Train}.", "f42f26d8faf6488b3b7768a95f30704b": "\\mathbf{p=(J^TWJ)^{-1}J^TWy^{obs}}", "f42f3b2d81eb2905ce8c223611158eca": "(\\uparrow 2)(c(z)):=\\sum_{n\\in\\Z}c_nz^{-2n}", "f42f3c60fef65cc90baa31359d025751": "\\tbinom{n}{i}/n^i", "f4300a485f8ae46c8061fc73fac0cfc1": "\\frac{1}{1-x} = \\sum^\\infty_{n=0} x^n\\quad\\text{ for }|x| < 1\\!", "f430544296485f2d370b79e965eee6af": "{\\color{Blue}~5.3}", "f430894442c99d6053587626e8c5ec33": "e^{\\frac{|f(D_{1})-f(D_{2})|}{\\lambda}}\\leq e^{\\frac{\\Delta(f)}{\\lambda}}\\,\\!", "f430b8ea02cbe9a2763ef642a19d79b9": "\n= \\dot{\\mathbf{Q}}^\\mathrm{T}\\dot{\\mathbf{Q}} + \\mathbf{Q}^\\mathrm{T}\\boldsymbol{\\Phi}\\mathbf{Q}\n = \\sum_{t=1}^{3N-6} \\big( \\dot{Q}_t^2 + f_t Q_t^2 \\big).\n", "f430d0af90a8e7c3ff572166ca278904": "g^{(2)}(\\tau) \\ge g^{(2)}(0)", "f430ea6daad6d997e18fe68b1d3f6b9a": "%p_i = \\frac{n_p wt_i}{\\sum_{j} wt_j}", "f430f8768899fb32fef2cf8c62a7a0eb": "Y_{2}^{0}(\\theta,\\varphi)\n={1\\over 4}\\sqrt{5\\over \\pi}\\cdot(3\\cos^{2}\\theta-1)\\quad\n={1\\over 4}\\sqrt{5\\over \\pi}\\cdot{(2z^{2}-x^{2}-y^{2})\\over r^{2}}", "f4314a2a068ba47e443ba1b8aee3b8f3": "A \\theta (0)", "f4314f3885fcf968a258692f06d93453": "R_{12} \\ R_{13} \\ R_{23} = R_{23} \\ R_{13} \\ R_{12},", "f43161e899824ebf81587dbcdfb291b8": "g(\\theta_1, \\ldots , \\theta_d)", "f43165d75dee03764c310850192e7167": "\n\\widehat{R}(\\Delta\\theta,\\hat{\\mathbf{e}}_z) = \\begin{pmatrix}\n0 & 0 & 0 & 0 \\\\\n0 & \\cos\\Delta\\theta & -\\sin\\Delta\\theta & 0 \\\\\n0 & \\sin\\Delta\\theta & \\cos\\Delta\\theta & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n\\end{pmatrix} \\,,\n", "f43240f5fc9bac5064da112a16b449de": "t\\approx\\frac{R^2}{cl}", "f4324a0002ba2b837ab7e2bba2d6c917": "I_{CBO}", "f432528a2b58188fe84f90a57fd6ab78": "s_0(P,Q)=\\cdots=s_{d-1}(P,Q) =0 \\ , s_d(P,Q)\\neq 0", "f43296502ec05dbc5ca846243b51c7fe": "X \\to \\alpha", "f4329babfd8f3319604bc5e6fef0e25a": "e_i \\otimes e^j", "f432bbcc0c2acf9798712da341564bb7": "\n\\begin{array}{ll}\n & P\\left(S^{t}|O^{0}\\wedge\\cdots\\wedge O^{t-1}\\right)\\\\\n= & \\sum_{S^{t-1}}\\left[P\\left(S^{t}|S^{t-1}\\right)\\times P\\left(S^{t-1}|O^{0}\\wedge\\cdots\\wedge O^{t-1}\\right)\\right]\\end{array}\n", "f432f501abdb2b60f30f214ce17ea52e": "x_\\star = F(x_\\star)", "f433069923d598213f65021092b17627": "R' = \\frac{\\sqrt{1 + e'^2 }}{B^2} a.", "f4337d6229a32547965c4e860edfa7d8": "4\\pi\\approx 12.566", "f43387fb8914be125746726dd2fe8bff": "\\dot{\\mathbf{A}}", "f433c0a95bb7485b2607f8f6f6cd21b3": " \\frac{\\omega(n) - \\log\\log n}{\\sqrt{\\log\\log n}} ", "f433cc815429fca00fa792339b9ac0a2": "\\frac{\\partial E}{\\partial \\hat{h}_i} = 0", "f433fad35175c085f095ee156bcba67d": "\\Pr(\\mathbf{x}\\mid\\boldsymbol{\\alpha})=\\frac{N B\\left(A,N\\right)}\n{\\prod_{k:n_k>0} n_k B\\left(\\alpha_k,n_k \\right)} .\n", "f4340210a1c4f5c8440b146467269095": "{1} + {1 \\over 3} + {1 \\over 5} + \\cdots + {1 \\over 2 p - 1} = {1 \\over 2} \\, \\gamma + {1 \\over 2} \\ln p + \\ln 2 + o(1).", "f4341a4bf8d4df047a313b95f801f2cb": "\\deg(v).", "f4344cd34df7519ba69b25eabc683773": "H_{in}", "f434548ef7a23a2b04c5c3e48bdc6e36": "\\lambda \\,.", "f434b2efc92d96c669208917f6b1e836": " \\theta_2 = 2 \\psi_2 ", "f434d6849bf6b0c329e3c183a1ed6519": "Z(C) = \\frac{1}{24}\n\\left(\na_1^6 + 6 a_1^2 a_4 + 3 a_1^2 a_2^2 + 8 a_3^2 + 6 a_2^3\n\\right)\n.", "f4351b5749d556849a6f06eefdf7b1bc": "v_p(\\omega)=n_p", "f435369b2b287033480009792a4a82fc": "2 k_\\mathrm{B} T/\\omega", "f435769ab45bf99b4d868e1a51ca0a3f": "\\scriptstyle{d \\theta = \\left( \\frac{1}{b}\\cos^{2}(\\theta)\\right) d x}", "f4357cf04dd47e3e31a639f4b91b6c14": "F_S(t,T) = \\frac{S(t)}{P(t,T)}", "f4358244ba1111ddb432ed508cd20ba7": "L\n\\subseteq \\Omega_{Z, [t_l, t_u]}", "f435dfc90e5ae75dd1e501587eb4dd35": "\\displaystyle A_\\rho(e^X) = e^{i\\rho(X)} \\prod_{\\alpha>0}(1 - e^{-i\\alpha(X)}),", "f435e3c1205cab40906582fa870e536a": "C_n\\mathbf{v}", "f4361b378430f2dde20e4fb343638f27": "{10 \\choose 1}{4 \\choose 1} - {4\\choose 1}", "f436216c47e7a3833c0a7f5d23037beb": "d(0, x) = \\rho(0, \\| x \\|).", "f436290cab323dba787cd326980f10a0": "\n{A_\\mathrm{v}} = {v_\\mathrm{out} \\over v_\\mathrm{in}} \\approx 1\n", "f436632a5373992cc90bd7838224093b": "Dur=\\frac{C\\frac{(1+ai)(1+i)^m-(1+i)-(m-1+a)i}{i^2(1+i)^{(m-1+a)}}+\\frac{100(m-1+a)}{(1+i)^{(m-1+a)}}}{P}", "f4366bd04ca3f93be3043442010f8b57": "m_{\\text{wet}}", "f436a522c5ddd9571c6fe6a82b2909b5": "\\sigma :A \\longrightarrow B", "f436d5758e26bcba59dc43a92d6e8a42": "\n (10)(x) - R_a(x-10) - R_b(x-25) + (1)(15)(x-17.5) - 50 + M_4 = 0 \\,.\n ", "f437081090e57460df5e6ce4057eba33": "B_\\rho = \\frac{\\mu_0 I}{2\\pi} \\frac{1}{L} \\sqrt{\\frac{a}{\\rho}} \\left[ \\frac{k^2-2}{k}K(k^2) + \\frac{2}{k} E(k^2)\\right]_{\\zeta_-}^{\\zeta_+},", "f4373a69c6322830b9dd218a885e0f26": "\\ {F_{tuning}(s) \\over s} = \\Theta_{out}(s) ", "f4375652e1c501d41bf19c6a1aeb0416": "y^t D 1 = 0 ", "f4375765ceee8b39dc991cbd4d06041a": "[-100\\%,+\\infty\\%)", "f437a1774d6459e55282ab897fe1e4c4": "\\rho_B = \\sqrt{\\frac{L_B}{C_B}} = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}} = \\rho_0. \\ ", "f437fdc83242a1e7fa977f84499abb0d": "S( \\nu)= \\frac{1}{ \\sqrt{ \\pi} \\Delta \\nu} \\exp \\left[- \\left( \\frac{\\left( \\nu- \\nu_0 \\right)}{ \\Delta \\nu} \\right)^2 \\right]", "f4380174e08ed1249b9824c6eb5be3b6": "t^{i-1}", "f4381f05a0aec1ebdf7f3ae751af46aa": "f_\\mathrm{s}/2\\,", "f4385897698e0e18e15c23882a24ff5f": "\n[L_i, X_j] = i\\epsilon_{ijk} X_k\n\\,", "f438853f5130270a9c1dc80029c0f876": "\\scriptstyle >3.42\\times E_{\\mathrm{Pl}}", "f438b497327d4b8387dd6cac5dafe090": "\\mathcal{L}_{V^{1}}(\\theta) \\,", "f438f196336e1069c4cbf31bdad4b90c": "\\displaystyle{f_z=T(\\mu f_z) + 1.}", "f4392076b41954f85a519fb4f935372e": " C = -k\\ln{S_0}, \\,\\!", "f4396ff4a6061474904e555cf264a862": "\\left|I_o\\right|", "f4397e6955624db2f0854fd0a3d19b3b": "\\textbf{t}_i^T = \\hat{\\textbf{t}}_i^T \\Sigma_k V_k^T", "f439944794d2993f7fe6dfb9f9786463": "\\begin{align} \n\\Psi (x,t) & = Ae^{i(kx-\\omega t)}, & 0 \\leq x \\leq L \\\\\n\\Psi (x,t) & = 0, & x < 0, x > L \\\\\n\\end{align} ", "f43a1116f75a688b8c0943031f8d2d4e": "f_1=0, \\ldots, f_n=0", "f43a11c1a56384ea96e877abb47a1064": "\\rho(\\tau) = \\sum_{k=1}^p a_k y_k^{-|\\tau|} ,", "f43a909000e6a5abebc69e78f2206239": "F(\\mathbf{a})\\geq F(\\mathbf{b})", "f43aa2a8778e8b3695c9fa56b59d98e3": "\\frac{3+\\sqrt{13}}{2}", "f43ab0b8d89f6cb0d6942d4d2a4cb9f8": "\\mathrm{add}_c", "f43ae381ed648f23f5dbecf7d4fd39f2": "Q''", "f43af9dfdee5a126d5ca786a53ff73fd": "[x^2:xy:y^2],", "f43b47bf5aaf02a3140b6e7dfe223507": "f_1,f_2,\\ldots,f_m", "f43c32309db8b368db19dff006dbda69": "\\displaystyle - \\sqrt{\\frac{\\pi}{a}} \\sin \\left( \\frac{\\pi^2 \\xi^2}{a} - \\frac{\\pi}{4} \\right) ", "f43c7f1a8ed351e7bbd79c1032487e2b": "dim(V) + dim(L) = n.", "f43cd41990342049f38554de5ede107d": "g=\\left(\\frac{t^{*}}{b}\\right)^{2}\\operatorname{Var}(b).", "f43d3254c99a41116dcaefa87f2e0f43": "\\zeta_S(s) = \\sum_{n=1}^\\infty \\frac{1}{\\lambda_n^s}.", "f43d56f4d269389e3158d69322b9d577": "E = h\\nu", "f43d73cfbd50cb1c183d6e7625ab0a12": "\\chi_2(\\omega) = -{1 \\over \\pi} \\mathcal{P}\\!\\!\\!\\int \\limits_{-\\infty}^\\infty {\\chi_1(\\omega') \\over \\omega' - \\omega}\\,d\\omega',", "f43d8157a4f1e2cf7f644af794e833c9": "f(x) = \\max(0, x + \\mathcal{N}(0, \\sigma(x)))", "f43d95d10f68077b68aafb0de12d6b5a": "d_1:=\\deg(r_0)-\\deg(r_1);\\quad \\gamma_1:=\\text{lc}(r_1); \\quad \\beta_1:=(-1)^{d_1+1};\\quad \\psi_1=-1;", "f43dcd81103b8eed84f6aa3c0f8e6fff": " f\\left( \\frac{a+b}{2}\\right) \\le \\frac{1}{b - a}\\int_a^b f(x)\\,dx \\le \\frac{f(a) + f(b)}{2}. ", "f43e04116320476d3234165bbffb1b24": "\\operatorname{Ass}_R(S^{-1}M) = f(\\operatorname{Ass}_{S^{-1}R}(S^{-1}M)) = \\operatorname{Ass}_R(M) \\cap \\{ P | P \\cap S = \\emptyset \\}", "f43e042b776afaead9bc0912efd01a8f": "T=(diag(A))^{-1}", "f43e17bd4d5b1a727d47c0f3f2a82f7b": "\\mbox{(net income + depreciation + all other non-cash charges),} \\,", "f43e59fa543c44704d471ac055e3aea9": " \\limsup_{n \\rightarrow \\infty} r_B(n)/\\log n > 0.", "f43ecbbffddf951c8c629cf1f1b378c8": "{\\rm{ }} = \\frac{{4\\pi \\omega }}{{n_b c}}\\sum\\limits_k {\\left| {d_{cv} } \\right|^2 (f_{v,k} - f_{c,k} )\\delta (\\hbar (\\varepsilon _{v,k} - \\varepsilon _{c,k} + \\omega ))}", "f43efe8edb79fc159a9e8b24dce3213d": " \\mathbf{A}(\\mathbf{r},t) = \\dfrac{1}{4\\pi}\\int \\dfrac{\\nabla \\times \\mathbf{B}({r}',t)}{|\\mathbf r - \\mathbf r'|}\\mathrm{d}^3\\mathbf{r}'", "f43f46a249181ede58551e19d6ed0e03": "\n\\mathrm{Beta}(\\alpha,\\beta) \\mbox{ where }\n\\begin{cases}\n\\alpha &= 2b/u+2a\\\\\n\\beta &= 2d/u+2(1-a)\n\\end{cases}\n\\,\\!\n", "f43f54769ce6a1f040f53c01a74dad39": "G= \\int_0^\\infty I(\\lambda)\\,\\bar g(\\lambda)\\,d\\lambda", "f43f8d734690653e3d641aef7fcbcbf2": "\\,\\ -\\csc^2 x = -(1+\\cot^2 x)", "f43fb5efb02df8f61d19e83759720fe9": "I_c = \\{j \\mid \\exists x \\in s: j = c + x\\} \\, ", "f43fde91279f47ef04fb49e2855394ee": "w \\cdot x", "f4400cf3d56ae7d19e5a820f0900b004": "\\alpha_2 = {{4\\alpha_0 + 1\\alpha_1} \\over 5}", "f4401e5944052e10ff387076bd91b67f": " n \\geq 2^{2m} \\ln \\frac{1}{\\sqrt{\\varepsilon}}.", "f4402f169ef3b4f0df437df63aa1096c": "\n\\mathcal{G}_0(\\mathbf{k},\\omega) = \\frac{1}{-\\mathrm{i}\\omega_n + \\xi_\\mathbf{k}}\n", "f44111208b0096447b45670d5c7eaa73": "y \\leftarrow h^b rem N", "f44121e5305ab34a27d5d96fb95219bc": "\\overline {t} = \\int_{0}^\\infty t \\cdot E(t)\\, dt = \\int_{0}^1 t\\, dF(t) = -\\int_{1}^0 t\\, d(1-F(t)) = \\int_{0}^\\infty (1-F(t))\\, dt", "f4417188102f4e450750872f9d14942b": "x_{i+1}^p=x_i", "f4417a629bdc7791229b3ba8310b0ce6": "(m, t, \\epsilon)", "f441947a895ef524b23a67d42881a05a": "K_N ", "f441b699d3d4c1897565a8f7006c69d1": "\\R\\, ", "f441d2d9b472462d4113c915b95bc653": "\n\\begin{array}{cc}\n& \\text{Sibling} \\\\\n\\text{Patient} &\n\\begin{array}{c|c|c}\n\\hline & \\text{No tonsillectomy} & \\text{Tonsillectomy} \\\\\n\\hline\\text{No tonsillectomy} & 37 & 7 \\\\\n\\hline\\text{Tonsillectomy} & 15 & 26\n\\end{array}\n\\end{array}\n", "f441f5650851577f20c660f231a4e7bf": "\n\n\\dot C_{RW}\\,\\,\\, = \\,\\,\\,\\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 {T \\over {2\\,}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\hat Q_0 \\,\\, = \\,\\,{{2\\,v\\,\\dot C_{RW} } \\over {\\varepsilon \\,k\\,F_m \\,\\phi \\,L}}", "f44203f5fabb20561f5590d0176c9f69": "\\mu\\ ", "f4421083d4079eb69ecfbb08b0c7c5a0": " B = CS_2 ", "f4421a46c4d4bf792ecbcfce9483481b": "g^{f}_{A}", "f44237ac777031c1f233da134eb40d5a": " i = \\alpha n + (1- \\alpha) = 1 + \\alpha (n - 1) ", "f442dad79c21736fb2048b45806ab10f": "(x * y) * z=x * (y * z)\\qquad\\mbox{for all }x,y,z\\in S.", "f442e2d8aa3b2fd19f1986c98b0b32b9": " \\mathbb{E} |f(W_1)| < \\infty. ", "f442ed8e673b246828fd1ae345456323": "\n\\left[ G \\star, \\frac{\\partial}{\\partial s} \\right] = 0.\n", "f443484984630555f8a029848a72d754": "V_\\text{out} = A_\\text{d}(V_\\text{in}^+ - V_\\text{in}^-) + A_\\text{c}\\left(\\frac{V_\\text{in}^+ + V_\\text{in}^-}{2}\\right)", "f44399055903c2055c3b04f2604e4993": "\\scriptstyle m_0\\,", "f443a294ff7e254da6a40b5d67bcbecc": "S = \\frac{C_r}{C_d}", "f443a2c1b23637729c5a2e4f163aebf7": "A(\\Phi):=\\sum RC_\\mathfrak{st}^\\lambda", "f443c5c7ae610f884233d9100e30d659": "P_m^{(m)}, P_{m+1}^{(m)}, P_{m+2}^{(m)}, \\dots", "f443e40390a49b09d3263cad8cfa43ec": "\\frac {P'(t)}{P(t)} = \\delta,", "f4440e4c878e0f1237cab1118f084b1b": " \\langle \\phi(x_1) ... \\phi(x_n) \\rangle = \\lim_{|C|\\rightarrow\\infty}{ \\sum_C \\phi_C(x_1) ... \\phi_C(x_n) \\over |C| } ", "f44517d57bbfccea4179d4cad93ef77c": "T_M(\\rho;E) = \\exp\\biggl(\\frac{1}{2}\\sum^n_{q=0}(-l)^qq\\frac{d}{ds}\\zeta_q(s;\\rho)\\biggl|_{s=0}\\biggr).", "f44520435b446096a1bdc17bad2e5f6f": "\\mathbb{L}_{x^m}(L)", "f4455f117981b0341a46d348bc638f7f": "\\log_{2}i", "f44591723cc83903eb88bef8120a533e": "\\mu_1=\\mu_2(=\\mu)", "f445ec07f8ce526a20354fabea2e2801": "\\frac{f_{k}'(z)}{f_{k}(z)} \\to \\frac{f'(z)}{f(z)}.", "f445f6a28a70562f21c238473fea7fd5": "\\frac{j}{m}", "f4466ef764aab9caeee13c55354d4e3a": "\\mathrm{EAS}", "f44670bce55a446f8891973d7bcfcc02": "d_{abs} = d\\tau", "f44681a7f5b51c3feaa1139ed0666bcc": "\n\\sum_{\\delta\\mid n}2^{\\omega(\\delta)}=\nd(n^2).\n", "f446a857b6a34934219f122a735f5978": "\\sum_{p,q=0,0}^{m,n}a_{p,q}z_1^{-p}z_2^{-q}Y(z_1,z_2) = \\sum_{p,q=0,0}^{m,n}b_{p,q}z_1^{-p}z_2^{-q}X(z_1,z_2)", "f446b8a8e27dfc18d2db1307eb4989d8": " r_{12} ", "f4473c4016648844fb129838f88b7ee7": "Re", "f44768248ad74321c4c402ed4827db95": "G_1\\,", "f44795117919546421b80b8c95a84ef5": " x = v_0 t \\cos(\\theta) ", "f4479c300af4cf7f3eeb0e102b49798b": "\\textstyle \\mathbf{x}", "f447b93472a51e70d28b10e73e623bec": "W_{\\mathfrak{p}}", "f447ba5bfba68d8f5f8c472c2d0493c5": "\\omega_1 = \\sqrt{\\frac{k}{m}},", "f44800cb0419e68e53f648083c4f094c": "\\Gamma = \\lambda_B/l_{charge} > 1", "f4481a8746a6e65e143560c68735b135": "\\textstyle 5", "f4481dfecc6674bef595f5b63fb2baae": "\n\\psi(q,\\alpha,u) = \\left\\{\\begin{matrix} \\alpha &,& {r_{c}} \\le R(q,u)\\\\ \\infty &,&{r_{ c}} > R(q,u) \\end{matrix}\\right. \\ , \\ \\alpha \\ge 0, u \\in \\mathcal{U}(\\alpha,\\tilde{u})\n", "f448a201ed02184b6a0f3facf6c072bd": "100 \\mu ", "f448b2e66b36365d5d1f3f623efb0473": " \\varphi\\,\\!", "f448b47172f2ddb9bcbd99a6b4201fe9": "n = \\sqrt{\\epsilon_r} = \\frac{n_0}{1 + \\left( \\frac{r}{R}\\right)^{2}}", "f448ffe5db23c444af81722257784035": "V_\\omega \\!", "f4490b590bce1f7875a6ef6ff237d70c": "V^* \\otimes V \\rightarrow L(V,V) ", "f44948b9aee52148930786736bad2f50": " i= \\mathbf{e}_2 \\mathbf{e}_3, j= \\mathbf{e}_3 \\mathbf{e}_1, k = \\mathbf{e}_1 \\mathbf{e}_2,", "f449c3c7e4c2a0afcbb71e118d41c985": " \\vec{v}= -\\frac{\\epsilon_o \\epsilon_r \\zeta \\vec{E}}{\\mu}", "f449eb5977b0bc86cbb0d1e20a556d2a": "h(h^{999}(password)) = h^{1000}(password)", "f449f6bfe42dea173162ce153edad657": "\\left| \\langle u,u'\\rangle \\right| = \\langle \\left| u \\right| , u' \\mbox{sign} u \\rangle \\quad (u \\ne 0)", "f44a1c2b49f80b4a668981e3c5301079": "q_{k}\\left(i\\right)", "f44a6bed0b322c3732fdef4d57547fb7": "\\eta:1_{\\mathcal C}\\Rightarrow G\\circ F", "f44a9ad4dd9b250f03df00ffc05f6bee": "\\lim_{(x,y) \\to (a,a)} \\frac{F(y)-F(x)}{y-x}", "f44aa8df5d78b126366c0f64e8e80c06": "Y_{in} = y_{11} - \\frac{y_{12}y_{21}}{y_{22}+Y_L}", "f44b1d0e667fcdd5f03c0cf2bccdb98d": "\\sum F = P_1 - P_2 - F_d \\qquad \\text{where } P = {\\gamma \\times y^2 \\over 2}", "f44b3750e7735d9d31e1c17dea30e2aa": " k_{f_1} ", "f44bc872a93e92b16ce6677cb6ca7758": "K = +1", "f44c14a25ef66bb653af503d9a8a32d3": "\\left|\\frac{1}{2} f + \\frac{1}{2} g\\right|^p\\le\\left|\\frac{1}{2} |f| + \\frac{1}{2} |g|\\right|^p \\le \\frac{1}{2}|f|^p + \\frac{1}{2} |g|^p.", "f44c26127088d5747936f4d422826906": "{n \\choose k} = \\frac1{(n+1) \\Beta(n-k+1, k+1)}.", "f44c9798d315e89444cf59391250861a": "\\phi_{i,j} = \\min\\{\\Pr(\\text{RCA} x_i \\geq 1 \\mid \\text{RCA} x_j \\geq 1), \\Pr(\\text{RCA} x_j \\geq 1 \\mid \\text{RCA} x_i) \\geq 1\\}", "f44ca5c95ce37f541070665857fde574": "C = 1 - \\frac{1}{1+0+1+0+0+0+1+0} = 1 - \\frac{1}{3} = 0.67", "f44cda3f636da89fac2228c88e4f48a3": "(D/p)=+1", "f44cf3fc60b183b39d07539d6d4f5d3d": "\n e^3 = 1 + \\cfrac{6}{-1 + \\cfrac{3^2}{6 + \\cfrac{3^2}{10 + \\cfrac{3^2}{14 + \\ddots\\,}}}} = 13 + \\cfrac{54}{7 + \\cfrac{9}{14 + \\cfrac{9}{18 + \\cfrac{9}{22 + \\ddots\\,}}}}\n", "f44d0a2c963572f5fde3123022d883cf": " q_{jk}=\\frac{1}{N-1}\\sum_{i=1}^{N}\\left( x_{ij}-\\bar{x}_j \\right) \\left( x_{ik}-\\bar{x}_k \\right), ", "f44d0eb46e6400ab6c4ade1807e4af16": "f,g\\colon X \\to Y", "f44d15ac622f4dc171a4c509e1f3d6cb": "P(p;\\alpha,\\beta) = \\frac{p^{\\alpha-1}(1-p)^{\\beta-1}}{\\mathrm{B}(\\alpha,\\beta)}", "f44d42590f61a2937b1a50e81fd04084": "\\varepsilon_{abd}", "f44d45fae44cdef4011462da9d1dca68": "= \\Lambda(1) + \\Lambda(2) + \\Lambda(3) + \\Lambda(2^2) + \\Lambda(2 \\times 3) + \\Lambda(2^2 \\times 3) ", "f44d807b906e3d6519a99597365ce70a": "P_{level} = g m V_g K_1 + K_2 V_a^2 V_g ", "f44dbfa6105e8e5ec51ec736e54aa75c": " \\| f \\|_{C^{k, \\alpha}} = \\|f\\|_{C^k}+\\max_{| \\beta | = k} \\left | D^\\beta f \\right |_{C^{0,\\alpha}}", "f44deaf4297eba66ddb837aae73ef37a": "\n\\begin{array}{rcccccccccccc}\n &a_1&a_2&a_3&a_4&a_5&a_6&a_7&a_8&a_9&a_{10}&a_{11}&a_{12}\\\\\n \\hbox{input data:}\n & 62& 83& 18& 53& 07& 17& 95& 86& 47& 69& 25& 28\\\\\n \\hbox{after 5-sorting:}\n & 17& 28& 18& 47& 07& 25& 83& 86& 53& 69& 62& 95\\\\\n \\hbox{after 3-sorting:}\n & 17& 07& 18& 47& 28& 25& 69& 62& 53& 83& 86& 95\\\\\n \\hbox{after 1-sorting:}\n & 07& 17& 18& 25& 28& 47& 53& 62& 69& 83& 86& 95\\\\\n\\end{array}\n", "f44e2b67d59753bc534023bee43322d9": "\\, y_{n+1} = y_n + hf(t_n,y_n). ", "f44e81c3170f3a4381b2cb5865cf6a73": "\\scriptstyle{R_\\alpha^0}", "f44ea36ca9aea2ec23aa4d040a8a627f": " \\operatorname{lambda-drop-op}[L, P, X] = P[L := \\operatorname{drop-params-tran}[\\operatorname{sink-test}[L, X]]] ", "f44f312bf8edafbc2adedf855997e9d9": "\\dagger\\colon \\mathbf{C}^{op}\\rightarrow\\mathbf{C}", "f44f5208987c2851cdca4d92bcda531f": "\n\\begin{matrix}\nx &=& s \\cos \\alpha\\\\\ny &=& s \\sin \\alpha\n\\end{matrix}\n", "f44fa2d87cf7e80c98573d5aec1f8ec8": "\\frac{dr}{dt}=\\frac{N_\\beta}{C}\\beta+\\frac{N_r}{C}r+\\frac{N_\\zeta}{C}\\zeta", "f44fa68b34b1d8740cdc2fc5eb834108": "L(\\sdot)", "f44fb6703096a0d66994074c54787daa": "\\frac{a_\\text{x}}{g}", "f44ff691a3b666befae026cf01e4b370": "X(x) = \\sin \\left(\\frac{\\omega x}{c} + \\phi \\right)\\quad\\quad\\quad", "f450429fd35a0b95fc0861ca4789139b": "df\\left(t,T\\right)=\\mu\\left(t,T\\right)dt+\\xi\\left(t,T\\right)dW\\left(t\\right)", "f4507e07a510ca1a3548882d77c21031": "\\displaystyle{Q_y(e)=Q_y(y^{-1})= Q(y^{-1})Q(y)=I.}", "f450e1591d36314fe2178a6003daa1d6": "\n\\begin{align}\n & \\left[ \n \\frac{\\partial \\Phi}{\\partial t} + g\\, \\eta\n \\right]_0\n + \\eta \\left[ \\frac{\\partial^2 \\Phi}{\\partial t\\, \\partial z} \\right]_0\n + \\biggl[ \\tfrac12\\, \\left| \\mathbf{u} \\right|^2 \\biggr]_0\n \\\\ & \\quad\n + \\tfrac12\\, \\eta^2 \\left[ \\frac{\\partial^3 \\Phi}{\\partial t\\, \\partial z^2} \\right]_0\n + \\eta \\left[ \n \\frac{\\partial}{\\partial z}\n \\left(\n \\tfrac12\\, \\left| \\mathbf{u} \\right|^2\n \\right) \n \\right]_0\n + \\cdots\n = 0.\n\\end{align}\n", "f451356a4ff99b98acaa6fd8600365b0": "\\log 10", "f45139dcf8b0b28163af6c1316304530": "q_{tid} = 63 \\rho n^5 r^4 e^2 / 38 \\mu Q", "f4514d5fa3be57ecaf5a82834be7cfbb": "x^n=\\sum_{m=0}^{n-1}A(n,m)\\binom{x+m}{n}.", "f4515659cce302e13e469a0fc8b45c33": " H^{\\prime} ", "f451c7fccc07b876de8b2e3c7daa14cd": "\\Delta x_{\\mathrm{min}} = \\sqrt{\\frac{\\hbar\\vartheta}{2M}}", "f451e54c144eb01974b4879a0b73c818": " {{} \\over {} \\Gamma \\vdash P : T} ", "f451f8db0669d67f1984f3af6d16dd11": "\\Gamma=\\Gamma_{k\\to 0}", "f452106db1be1e4a3e387e424145147c": "L \\le \\frac{N}{2}", "f4523fb56bbe65256d61a3b1acd44ecc": "{\\rm cov}(V, T)", "f45260fa529b458596cba85081d4dc86": "\\partial \\Omega(t)", "f4528c0621325194a3905185f2f8845e": "D_{t+1}", "f452caf6ad5a35d49b00ec0f22063b5a": "0,(n-1)", "f4533de9e1d5821c787d7742dd084b6b": "f:A\\to T(B)", "f45376912399f11a4da43f76f95e7cfd": "\\scriptstyle y\\in U_1", "f453ff1865e13fbd1bd8f10e1b69e742": "O(d-\\lambda_2)", "f454184e5ec1e7cd7d2be4b958f2ff57": "L = \\frac{1}{d}[110.9 - 0.09t]", "f4545420da2213baae707dc2fa17ab2b": "\n\\boldsymbol\\varepsilon(t_1) = \\boldsymbol\\varepsilon(t_2) \\ \\stackrel{\\mathrm{def}}{=}\\ 0\n", "f4545eaf068aadee1d1143df21cc34f2": "\\int \\limits_0^\\infty \\underset{\\text{Para } x>2}{\\frac{f(x)}{x^2} \\, dx} = \\int \\limits_0^1 e^{\\operatorname{Li}(n)} dn \\quad \\scriptstyle \\text{Li: Logarithmic integral}", "f45528232f5e6ce3cebed69387d0d5bb": "\\tfrac{1}{3}=0.(3)", "f455651a1480613e5bfc1ada63a89d17": "\\|f\\|_{BV} = V_f(I)", "f4558d460885a93c1b8ecc301deebabc": "NS \\rightleftharpoons M", "f45594fe427eb91448a73b94e0b4f632": "= \\mathcal{L}_{V^{1}}du - (\\mathcal{L}_{V^{1}}u_{1})dx - u_{1}(\\mathcal{L}_{V^{1}}dx) \\,", "f456138959435bddfe6fd98c4d82015d": "\\frac{d^2 T}{dx^2} + \\frac{\\dot{q}_m + \\omega \\rho_b c_b (T_a-T)}{k}=0 \\quad [3]", "f456417908ba93e93fe3cc4d30dcc8c3": "\\left| \\sum_{k}(n) \\right| =\\sum_{\\Pi\\succeq\\Lambda}c(\\Pi)", "f4567e4cad4431b1d45be7d28ffa183c": "\\begin{align}\nS(\\Lambda_{B}) &= e^{i\\pi(\\phi \\cdot \\mathbf{J})} = \n\\biggl(\\begin{matrix}\ne^{-\\frac{1}{2}\\chi\\cdot\\sigma} & 0 \\\\\n0 & e^{\\frac{1}{2}\\chi\\cdot\\sigma} \\\\\n\\end{matrix}\\biggr),\\\\\nS(\\Lambda_{R}) &= e^{i\\pi(\\chi \\cdot \\mathbf{K})} = \n\\biggl(\\begin{matrix}\ne^{\\frac{i}{2}\\phi\\cdot\\sigma} & 0 \\\\\n0 & e^{\\frac{i}{2}\\phi\\cdot\\sigma} \\\\\n\\end{matrix}\\biggr)\\\\\n\\end{align},", "f456dabff0e02881c21379995a643677": "\\int \\, d\\theta = 0 ", "f4572880841a1cf7ecbae5b90e864e00": "\\tilde{\\delta A}=-A^{-1}\\delta A", "f45741f86bad7ba8b3946f2d7074b708": " 0 = \\cosh\\phi vt - \\sinh\\phi ct \\, \\Rightarrow \\, \\tanh\\phi = \\frac{v}{c} = \\beta", "f4577542cbe7b4d72dbb38362ebd1a20": " T = - \\beta A \\ , ", "f4577ae3192bcd57b0fc2a2f45ea17f7": "\\textstyle b = 0", "f457bf93bcf0ec7d5180a0b4baba5d13": "\\binom{n-2}{d_1-1,\\,d_2-1,\\,\\dots,\\,d_n-1}=\\frac{(n-2)!}{(d_{1}-1)!(d_{2}-1)!\\cdots(d_{n}-1)!}.", "f457f9edabe6823850f26950e1b4abf3": "-{\\rho q \\left({q \\over {y_1}}\\right)} + {\\rho q \\left({q \\over {y_2}}\\right)} = \\left({1 \\over 2} \\rho g{y_1} \\right) {y_1} -{\\left({1 \\over 2} \\rho g{y_2} \\right) {y_2}} ", "f458414e6e906c9082fa4e795e1cd46b": "V=\\frac{8\\pi ^2r^5}{15}", "f4585bb2b4a2a4380641bd5c816689cc": "\\rho(\\boldsymbol\\Sigma_{\\epsilon})", "f4587bbd9e419160055dbec7686148ae": "S \\Rightarrow_{S \\to XX}\\ XX \\ \\Rightarrow_{X \\to Y}\\ YX \\ \\Rightarrow_{X \\to Y}\\ YY \\ \\Rightarrow_{Y \\to S}\\ SY \\ \\Rightarrow_{Y \\to S}\\ YY", "f458c26f4317f4b91bcaf51822d2f515": "\\scriptstyle|\\lambda_i\\rang", "f458d52761d0e45a1064abfc23255901": "\\varepsilon_{r}(\\omega) = \\frac{\\varepsilon(\\omega)}{\\varepsilon_{0}},", "f45968853c3bb6dadbf0f1df04997e7b": "\\mathcal{P} (M)", "f4596bb68433df3064b82f116185a6c6": "\\int_a^b \\sqrt{\\bigg(\\frac{dx}{dt}\\bigg)^2+\\bigg(\\frac{dy}{dt}\\bigg)^2}\\,dt,", "f4597de31d3d0926639de2bd0a60949e": " X \\sim U(0,1)\\,", "f4599a2b6f33f3771cf1f5ea70c1346c": " S.D = \\frac{1}{|det(V)|}=\\frac{|det(U)|}{4\\pi^2}", "f459a4ce6863bf56ac952e1b9ea66a93": "H|k\\rangle=\\frac{{\\hbar^2}{k^2}}{2m}|k\\rangle", "f45a3189cca20fc31bba8cfacd6b56f5": "e^{2\\pi i \\over n}", "f45aaba2a08fa21baf55dfabb829b45c": "T_d = \\left[1 - f - e(1-b)^{-1}c \\right]^{-1}d.", "f45ac6843854bf2d0545645f75370511": "\\begin{matrix} {4 \\choose 2}{3 \\choose 1}^2{36 \\choose 1} \\end{matrix}", "f45b8cfa91e87728aa37fe67fd7a4bd9": "u'(x) = (x-1)(x-5) = x^2 -6x + 5", "f45b9a7cbcc1ea56df46bd623c461861": " i \\geq N ", "f45bf2104f850207c338610926ef3cd2": "\\frac{\\partial k}{\\partial t} + \\frac{\\partial q}{\\partial x} =0,", "f45bf9add3181ce07d16a9d7a1489638": "\\text{PointsPlus} =\\max \\left\\{ \\mathrm{round} \\left( \\frac{\\text{protein}}{10.9375} + \\frac{\\text{carbohydrates}}{9.2105} + \\frac{\\text{fat}}{3.8889} - \\frac{\\text{fiber}}{12.5} + \\frac{\\text{alcohol}}{3.0147} - \\frac{\\text{sugar alcohol}}{23.0263} \\right), 0 \\right\\}", "f45c044660be8a552a2ee87994670518": "L[\\vec{X}]_{ab} = {{}^\\star R^\\star}_{ambn} \\, X^m \\, X^n", "f45c121670e6644ccbd2536ca1a814be": "(q_i)=(p_i^{r_i}) = (p_i)^{r_i}", "f45c22d25f036a7326b0a95686f37c3e": "\nV(x)=0,\\quad -a\\leq x\\leq a\\quad \\quad \\quad \\ (\\mathrm{ii}) \n", "f45c496121cb8b05dece920a87897f18": "P_{Sp(n)_{}}(x) = (1+x^3)(1+x^7)...(1+x^{4n-1})", "f45d0c9941a86e18989563df674c80cb": "\\scriptstyle x^2 < 1-y^2", "f45d17c21f904f73a708c8a4de13b93a": "_{k+1}V^i_2(x,y)=_kV^r_4(x-1,y)", "f45d4dcf13b3623af89d657f02a0405e": " \\lambda = {2 \\pi r \\over n} ", "f45d5135278772cfd3e04e932d5c7c6d": "\n\\dot{\\mathbf{q}} =~~\\frac{\\partial H}{\\partial \\mathbf{p}}\n", "f45d927b1f4468c04d551ae96d2bda41": "i_\\alpha : X_\\alpha \\to X", "f45da7eab582df2d8a92a0346d582769": "\\textstyle 1-p", "f45db6754a3539eeddd8fe2a40c4fbc3": " C_{\\text{dl}}", "f45e1c6c3c6db70839026df54215295a": "{x \\in R^{\\it N}}", "f45e5efab29318b78fe66388e270852d": "\\left| I, 0\\right\\rangle", "f45e7a2da39feb7f9b21bab6342e9121": "A_{ij}[\\nabla]=\\alpha^2 \\partial_i\\partial_j+\\beta^2(\\partial_m\\partial_m\\delta_{ij}-\\partial_i\\partial_j)\\,\\!", "f45efe20bdb7abc6e2a66add71dfc26b": "\\mu(p,e,p)=1", "f45f04b67f7b9a337b94a09e7291ffed": "\\beta \\beta^* = \\beta^* \\beta, \\ \\alpha \\beta = \\mu \\beta \\alpha, \\ \\alpha \\beta^* = \\mu \\beta^* \\alpha, \\ \\alpha \\alpha^* + \\mu^2 \\beta^* \\beta = \\alpha^* \\alpha + \\beta^* \\beta = I,", "f45f27fca38c1f51e2b91b87215ff309": "\\det R\\le(n-3)^{(n-s)/2} (n-3+4r)^{u/2} (n+1+4r)^{v/2} \\left[1 - \\frac{ur}{n-3+4r} - \\frac{v(r+1)}{n+1+4r}\\right]^{1/2},", "f45f4a7a7814470db8b0955f68fc26ac": "G_{\\mu\\nu} = R_{\\mu\\nu} - {R \\over 2} g_{\\mu\\nu}", "f45fb3f3efd546cbdb2ddb30801c0e17": "z_1 \\neq z_2", "f45fe70e6caec61e15f83c1237240e9b": "() (0 1 \\infty) (0 \\infty 1)", "f46042073810e0ad83a2ba0573d17f17": "q = a + bi + cj + dk, \\quad a,b,c,d \\in \\Bbb{R} \\!", "f460c240a5ecb0f1548079d136f45411": "\\mathbf{c}=(c_0,\\ldots,c_7)=(\n e^{j(\\phi_1+\\phi_2+\\phi_3+\\phi_4)}, \n e^{j(\\phi_1 +\\phi_3+\\phi_4)},\n e^{j(\\phi_1+\\phi_2 +\\phi_4)},\n-e^{j(\\phi_1 +\\phi_4)},\n e^{j(\\phi_1+\\phi_2+\\phi_3 )},\n e^{j(\\phi_1 +\\phi_3 )},\n-e^{j(\\phi_1+\\phi_2 )},\n e^{j \\phi_1 })", "f460da61e7155952f6c128a00a8a7ffb": "\n\\mu = \\frac{1}{\\lambda}\n", "f460de8ca28cce52690a58642a58dda8": "\\left(x+\\frac{b}{2a}\\right)^2=-\\frac{c}{a}+\\frac{b^2}{4a^2}", "f4614b225f257730547a0915d30b6541": "\\scriptstyle a_0,a_1, \\ldots, a_{12} ", "f46175e6bc472576d7e2369745c23e51": "\\sqrt{2} + \\sqrt{3}", "f461f9f1e3adb0844a6dbf0a6f2677ce": " \\, Y_\\nu(x)\\, ", "f462040082828e60bf3b46344bcd566b": "\\mbox{TC} = \\bigcup_{i \\geq 0} \\mbox{TC}^i.", "f462109571aed7ff5c3fc58212111940": "(-s) N^k = -s \\mod (MN - 1)", "f462271c7d84c65235b104da06a8dc5c": "(\\mathbb H\\otimes\\mathbb O)P^2", "f462491fb78d2a384f7b98cff673aeaa": "m \\ge p", "f46273b6fdec5e192890eeebb2685e37": "|F(z)|\\leq A_\\varepsilon e^{(\\sigma+\\varepsilon)|z|}", "f46283c53909017055aafa956c8108f4": "U=TS+\\frac{1}{2}\\sigma_{ij}\\varepsilon_{ij}", "f462f95f210898c73ca478322986752c": "w_2 = 1 - (s + w_1)", "f46345eaa9db5daecac8c48d7bd0731f": "\\geq{u^*} = \\{x \\in \\mathbb{R}^L_+ : u(x) \\geq u^*\\}", "f46367597d9af476b98e24e59709a570": "\\{n, n+1, n+2, ... , n+p \\} \\subset T", "f4637792e42739ea04a81857a401c075": "\nZ \\sim \\frac{\\sum_{i=1}^k Z_i}{\\sqrt{k}},\n", "f4637d65a8013b68e76c3c0b6cceaf3b": "\nN=\\frac{R}{4\\pi \\gamma_0}\\iint_\\sigma \\Delta g \\,S(\\psi)\\, d\\sigma.\n", "f463ca8455b216215494459b5565d7a8": "\\xi= (df)_e", "f4646331f17d62240a4f587b39c67ed5": "\\scriptstyle\\nabla^2 f=0", "f4649ddd112c60088956fc1a511bf94c": "\n Profit=\\frac{\\left(S_G-S_B\\right)\\kappa}{2}\\left(1-PIN^*\\right)\\left(PIN^*-PIN\\right) \\;.\n", "f464a0599a5bb0ad78e901b73489f09e": "i=1, 2, \\ldots ,\\beta", "f464b117335436cd56e5ff82c079ff5d": "y_{t=1 \\dots T}", "f464b534836548b6cc58c313fb7edabd": " T = \\binom{n}{2}\\ln n ", "f4651651382a1ec4f3344eacfa1a54a5": "\\partial_u, \\, \\partial_v", "f46634adf800a230c40360e4a794e9b7": "C_P \\phi_P = C_w \\phi_W + C_sw \\phi_{SW}", "f4667fc8757b8f715918f7b6e239e98b": "(\\log n)^{O(1)}", "f466969c067c632f77647dd84c0f805c": "\\nabla^2\\psi + \\lambda\\psi = 0", "f466a9e449ab2a19f5c48b8ef03a1545": "\\min_{\\lambda\\in\\sigma(A)}|\\lambda-\\tilde{\\lambda}|\\leq\\kappa_p(V)\\frac{\\|\\mathbf{r}\\|_p}{\\|\\mathbf{\\tilde{v}}\\|_p}.", "f466aeb29a40054ccd7970c9fe326fd3": "A = [\\mathbf{\\alpha}_1,\\ldots,\\mathbf{\\alpha}_{c-1}]", "f4670f6a2751abb07e0b60b6f1772761": "\\int x^m\\,\\operatorname{artanh}(a\\,x)dx=\n \\frac{x^{m+1}\\operatorname{artanh}(a\\,x)}{m+1}-\n \\frac{a}{m+1}\\int\\frac{x^{m+1}}{1-a^2\\,x^2}\\,dx\\quad(m\\ne-1)", "f467a12ecacc7cc873eec6c5a9d2e58f": "\\mathrm{NT}(B)", "f467a3dd845c1954a463b49db506f8d2": " P = A (B^\\mathrm{T} A)^{-1} B^\\mathrm{T}. ", "f46820a08a160b245f31074552ae5995": "a_j < a_i", "f46828afcb51ec9b2581ba1bbe254154": " \\sum_{i=0}^n {i\\cdot i!} = {(n+1)!} - 1. ", "f468833e4976a6eabc43cdd66e98b667": " \\mathbf{ |L| } = mr^2 \\omega = m l^2 {d\\theta \\over dt} ", "f4689796c960de164cc0dcb647aff80f": "\\sin_k(i)\\equiv -\\sin_k(-i). \\, ", "f468d658c256faa3bb95431a343b1a49": "\n\\zeta = \\frac{1}{2 Q} = { \\alpha \\over \\omega_0 }.\n", "f468d87e880a3d52a893df3c80affa96": "2 \\Omega \\sin \\varphi", "f468fe18be1fe9f657491aab2b0083e5": "\\frac{\\hbar^2}{2m}\\frac{a}{A}=\\sum_{n=-\\infty}^{\\infty}\\frac{1}{\\alpha^2-(k+\\frac{2\\pi n}{a})^2}", "f469241864f94d687ad0bd444fd13a2b": " p: K \\rightarrow G ", "f4692cf6af039d1ac41ded52cacd9555": "a\\mapsto \\widehat{a}", "f469421bf69361a4ab4b49e9b29d8d1e": "k=-1", "f469c607b6cbb70ab494ad41922a26cb": " \\operatorname{sgn}(\\sigma) \\neq \\operatorname{sgn}(\\dot{\\sigma})\n\\qquad \\text{and} \\qquad\n|\\dot{\\sigma}| \\geq \\mu > 0", "f469d1318ac8dfd4a06ef33cba7d41d9": "[n]_q=\\frac{1-q^n}{1-q} = 1 + q + q^2 + \\ldots + q^{n - 1}.", "f46a4f160defe28bfe7b72d19897b40d": "F[c]=\\int d^n x \\left[\\frac{1}{4}\\left(c^2-1\\right)^2+\\frac{\\gamma}{2}\\left|\\nabla c\\right|^2\\right],", "f46a5a12cb5f465df6832983dc370e8b": "\\begin{matrix}\n12 & 12 & 123 & 124\\\\\n\\end{matrix}", "f46a70b3f6098ce0bba7917f65a1d1a7": "Z = Z_1 + Z_2 + Z_3 + \\dots + Z_N", "f46a75441e0e73a27aed01228a009e88": "E(1/X) = 1-\\mu/2.", "f46ae982ca8d8e87bdebe0ac4de4ab5d": "\\mu(S)=\\int_S\\frac1{(x^2+y^2+z^2+w^2)^2}\\,dx\\,dy\\,dz\\,dw", "f46afb3f2c586fd77ea87457b4047b0c": " d = 0.61 \\frac{\\lambda_0}{N\\!A} \\;\\!", "f46b0057c7dd4a951251b45702503480": "\\,\\overline{(a \\cap b)} = \\overline{a} \\cup \\overline{b}", "f46b32bd89d93fe954818be20ab15a0d": "(t,\\rho,z,\\phi)", "f46b34696045a248dc2c9f88be88a461": " a_3 ", "f46b536a79ebd96c95cbe592385563d5": "{\\mbox{Rate H}_2 \\over \\mbox{Rate O}_2}={\\sqrt{32} \\over \\sqrt{2}}={\\sqrt{16} \\over \\sqrt{1}}= \\frac{4}{ 1}", "f46ba0cd81cf263aa41f7e0fc4a8dcb8": "3^3+4^3+5^3 = 6^3", "f46bd8dc0403d37802feed31e35601b4": "\\, Y_{n + 1} = Y_n + a(Y_n) \\Delta t + b(Y_n) \\Delta W_n,", "f46bf37cc9665773feb1731f907558ab": "(t,\\rho,z,\\phi)\\mapsto(t,r\\sin\\theta,r\\cos\\theta,\\phi)", "f46bf7fe1c64722a7527841674a6e984": "e^A e^B = e^{A + B}", "f46c78b5e2fbb4de5a4587e12fe3d230": "\\operatorname{Hom}(D^{\\le 0},D^{\\ge 1})=0;", "f46c7e1e0b16eb1063d814b5d1e705a3": " = |(a_1 + i a_2) - (b_1 + i b_2)|", "f46c7e6274872aa1cd671f67021c7798": "(\\mathbb{Z}/2\\mathbb{Z})^\\times \\cong \\mathrm{C}_1", "f46cafe3f1ef26bec22682e66f24ba6a": " \\sum_{m=1}^N m\\left|\\sum_{n=1}^N c_{mn}\\lambda_n\\right|^2 \\le \\sum_{n=1}^N {1\\over n} |\\lambda_n|^2.", "f46cc2799b6b8e8bc678d9cab9fa9985": "\n \\left. \\frac{\\partial}{\\partial t} f^e_\\mathbf{k} \\right|_\\mathrm{L} = \n \\left. \\frac{\\partial}{\\partial t} f^h_\\mathbf{k} \\right|_\\mathrm{L} = \n -2\\,\\mathrm{Re} \\left[ \\sum_{\\omega} \\mathcal{F}^{\\star}_\\omega \\, \\Pi_{\\mathbf{k},\\omega} \\right]\\,.\n", "f46d5244633efda539961255626878d2": "0<\\mu(A)<\\infty,\\,", "f46e0796a5daf0bc8824f206887f1826": "x_n = c", "f46e416fbbc43712787756eabd3bfa2e": "L_{NL} = \\dfrac{1}{\\gamma P_0}", "f46e56a02480921e389131af93c99373": "\\, b", "f46e63d9f780911dc4206c5f741ba490": "{{2n}\\choose k} \\cdot \\frac{k!}{k} \\cdot (2n-k)!.", "f46e87bde0f367b6c03b6825639a222d": "\\delta t=-2.4\\pm0.1\\ (\\mathrm{stat.}) \\pm2.6\\ (\\mathrm{sys.})", "f46ebdbe42413b3f1b385886906d0687": "\\mathbb{P}(N\\ge k)=\\sum_{n=k}^m S_n\\sum_{j=k}^n\\binom nj(-1)^{n-j}.", "f46ec40ae60f375cbe628bcb9ddf09c8": "\\omega_{\\hat{M}\\hat{N}P}", "f46ed82b38255cb04fd04953f5c69413": "\\frac{x+y}{2}", "f46ee18a6dd624e7f8b9d7de9d277057": "= \\operatorname{tr} (\\Gamma^*)", "f46ef8358d5281a7ffa6cc408e4a1ad8": " \n\\left|\\frac{a}{b} - \\frac{m_k}{n_k}\\right| < \\frac{1}{n_k^2}.\n", "f46f0cec7914efa525f26a80c0284a2e": "\n\\begin{array}{rl}\n{\\displaystyle\\min_{X \\in \\mathbb{S}^n}} & \\langle C, X \\rangle_{\\mathbb{S}^n} \\\\\n\\text{subject to} & \\langle A_i, X \\rangle_{\\mathbb{S}^n} = b_i, \\quad i = 1,\\ldots,m \\\\\n& X \\succeq 0.\n\\end{array}\n", "f46f3a7234d8a18ca75a4f4c1d42bd51": " {N\\over N_0} = exp({-\\Delta E_{act}\\over RT}) \\,", "f46f499e0607b06cdedbb9027fa54fd1": "C = \\{c_j\\} \\ j=1 \\ldots p", "f46f63da33e01f02100b6975223ab7f7": "10\\log\\left(\\frac{0.02}{0.98}\\right)", "f46fc2d02da64e77176260e10cad472b": " \\mathbf{W} \\, \\mathbf{\\Sigma} ", "f46fd50a81e460aabc0a570bac4fef38": "y(s)=e^{sx/|x|}", "f47041bcac8692b741dd49ec228c4fc1": "\\vec{v}_\\mathrm{A|B}", "f4705c9e8fe5c3dd658b9280a325a183": "\\|\\boldsymbol x\\|_1=1", "f470d43e4e2e1702bb543d312504052e": "a(u,v) \\le C \\|u\\|\\, \\|v\\|", "f471299b3de3465af859b3e411e3c810": "S=\\{f(x)g(x)^{-1}\\ |\\ x\\in X\\}", "f47130bfa2630300a0eaea33d03c470b": "\n\\max_{i \\leq n} | p_i^{(\\infty)} - \\mu | \\xrightarrow{\\ p\\ } 0\n", "f47157f8957a2cb59daebf2191416d36": "\\mathcal{L}_{\\mathrm{QCD}} = i\\overline U (\\partial_\\mu-ig_sG_\\mu^a T^a)\\gamma^\\mu U + i\\overline D (\\partial_\\mu-i g_s G_\\mu^a T^a)\\gamma^\\mu D.", "f4718586b90234c070182cd3497f7fc3": " h_1\\ldots h_m ", "f471ae36cdcdb768c94115f472268d1e": "A=(15+2\\sqrt{2}+\\sqrt{3})a^2\\approx19.5605...a^2", "f471b166625306e5f8ab88dbbc884d0f": "\\left.\\frac{d}{dt}\\phi\\circ\\gamma_1(t)\\right|_{t=0}=\\left.\\frac{d}{dt}\\phi\\circ\\gamma_2(t)\\right|_{t=0}", "f47209fc1d22ce59a3a38ea5bdb11517": "L(f, t) = \\{ x \\in \\mathbb{R}^n | f(x) \\geq t \\}.", "f4728b591fa9c3ebb01302097770fbc4": "\\frac{\\partial}{\\partial \\theta} \\left( \\frac{\\partial u / \\partial x_k}{\\left|\\partial u / \\partial t\\right|} \\right) = \\frac{\\partial}{\\partial \\theta} MRS_{x,t}", "f472eb17357155a7fb5baf98dc691b4c": "A_1:A_2", "f47305d38f610a660c858bdf7b27421f": "\\omega=(dM)M^{-1}", "f4730e294d3ea4d30b40fbe98bdcd8bb": "\\begin{align}\\operatorname{MSE} = & \\operatorname{E}\\left[(T^2 - \\sigma^2)^2\\right] \\\\\n= & \\left(\\operatorname{E}\\left[T^2 - \\sigma^2\\right]\\right)^2 + \\operatorname{Var}(T^2)\\end{align}", "f473225572bd60d2fd0118f6076fe413": " k^{-4/3} ", "f4735097b1cdce04db0f9519820c14b9": "\\lambda=\\mu \\Leftrightarrow \\exists\\kappa. (\\lambda=\\kappa \\wedge \\kappa=\\mu)", "f47366cd515f5df68d6251e5761114f3": "\\mathrm{C}^\\infty", "f47368cb58ab65369b46fe9a6645ba9f": "F_X(a) = \\mathbb{P}\\left[ X \\leq a \\right], \\qquad a \\in \\mathbb{R}", "f4739c79b0afa59c3946b1be98580199": "\np > q \\geq 1, s > n ( \\tfrac{1}{2} - \\tfrac{1}{p} ), \\text{ and } \\tfrac{1}{p} = \\tfrac{\\alpha}{q} + (1 - \\alpha) ( \\tfrac{1}{2} - \\tfrac{s}{n} ).\n", "f473d8b235d146aee7fd5e6b95fc21a3": "\\delta\\alpha^2/\\gamma^3", "f47413c0da9413fdfb21d42b2043f64f": "\n\\mathbf{C} = a \\mathbf{V}^{\\mathrm{T}} \\mathbf{S} \\mathbf{V}\n", "f4749b4fbc75dc65f66a44be207bbbff": "\n\\cfrac{\\cfrac{\\cfrac{\\cfrac{\\mathbf{pq} \\qquad \\mathbf{\\overline{p} o}}{\\mathbf{qo}}\\, p \\qquad \\mathbf{p \\overline{q}}}{\\mathbf{po}}\\, q \\qquad \\cfrac{qr \\qquad \\overline{p} \\overline{q}}{\\overline{p} r}\\, q}{or}\\, p \\qquad \\overline{o} s}{rs}\\, o\n", "f474bdb8632adf8c38fb9cd8fca558a5": "\\vec{\\mu}_S \\ = \\ g \\ \\frac{q}{2 m} \\ \\vec{S} = \\gamma \\vec{S} ", "f474e1a5c203790b162968fecb531b70": "(\\mathbf{e}_{12} + \\mathbf{e}_{34})^2 =\\mathbf{e}_{12} \\mathbf{e}_{12} + \\mathbf{e}_{12} \\mathbf{e}_{34} + \\mathbf{e}_{34} \\mathbf{e}_{12} + \\mathbf{e}_{34} \\mathbf{e}_{34} = -2 + 2 \\mathbf{e}_{1234}.", "f475069eb22bd18b51b6b7ab34595f9b": "\\; M_v = (\\beta_{ij})_{ij} .", "f47554e5ed57335c95d50183c87e31df": "\\sum_i{\\beta_i} < 1", "f475973c93ef3a062bce8bf5b708fed7": "\\mathcal{F}_s", "f475ba4b15835f5ff838509b031dc498": "\\mathbf{F}=\\mathbf{P}-{\\partial{V}\\over\\partial{\\mathbf{r}}}", "f475c027dc98cb97d6e4b30b8fac29db": "U=TS-pV+\\sum_i \\mu_i N_i\\,", "f475daad3eb850fa6674a328c6247e47": "(\\delta r)_{\\vec{k}} \\cong \\frac{e}{mc^2k^2} E_{\\vec{k}}=\\frac{e}{mc^2k^2} \\mathcal{E} _{\\vec{k}}(a_{\\vec{k}}e^{-i\\nu t+i\\vec{k}\\cdot \\vec{r}}+h.c.)", "f4764e96c3693863b9defa0aa8b43a7c": "GL(n,\\mathbb R)", "f476a059f6dbe4021d04794129566e77": "\\mathrm{Re_2O_7 + 7H_2S \\ \\xrightarrow{\\Delta}\\ Re_2S_7 + 7H_2O }", "f476f820b17221f20d2f2001b66a7558": "\\begin{align}& f(\\boldsymbol{x}) = \\sum_{|\\alpha|\\leq k} \\frac{D^\\alpha f(\\boldsymbol{a})}{\\alpha!} (\\boldsymbol{x}-\\boldsymbol{a})^\\alpha + \\sum_{|\\alpha|=k} h_\\alpha(\\boldsymbol{x})(\\boldsymbol{x}-\\boldsymbol{a})^\\alpha, \\\\& \\mbox{and}\\quad \\lim_{\\boldsymbol{x}\\to \\boldsymbol{a}}h_\\alpha(\\boldsymbol{x})=0.\\end{align}", "f4771c6ca0499e1484aa5eefc8d0603d": " \\sqrt {\\frac{\\sum{D_i}^{2}}{n}} ", "f47774360a6fa534987ec4e57b719efd": "y_i = f(x_i ;\\beta ) \\cdot \\exp \\left\\{ { - u_i } \\right\\} \\cdot \\exp \\left\\{ {v_i } \\right\\} ", "f477ae81fdda26678bcc3ed504372abb": "a_n^{-2}\\operatorname{var}(X_n) = \\operatorname{var}(a_n^{-1}X_n)", "f477fd6a512e3029a7e7b67607e3128e": "p = 1 - \\frac {3}{4} n \\cos 2 \\psi \\qquad q = \\frac {3}{4} n \\sin 2 \\psi", "f4780a00e4285146e5bbcd61403844de": "(\\neg \\phi \\vee \\neg \\psi) \\to \\neg (\\phi \\wedge \\psi)", "f478468b0ba89b8d955681b8a5821e41": "\\begin{matrix}{r \\choose 0}\\end{matrix}", "f4784f0523de7ad6b427d74112589cf3": "\\scriptstyle{Q}", "f47868ef0294b4a90dc858b0b3fb33b9": "\\left(\\frac{x}{a}\\right)^2 + \\left(\\frac{y}{b}\\right)^2 = -1", "f47870153a78f9314b68175cc2d06f38": " Q^{(i+1)} ", "f478804e8718cab14ce7e05f4971811d": " J^2 |jm\\rangle = j(j+1) |jm\\rangle,\\quad J_z |jm\\rangle = m |jm\\rangle,\n", "f4789ba3433451f184f81685af74615d": " \\begin{align}\n& \\text{maximize} && \\mathbf{c}^\\mathrm{T} \\mathbf{x}\\\\\n& \\text{subject to} && A \\mathbf{x} \\leq \\mathbf{b} \\\\\n& \\text{and} && \\mathbf{x} \\ge \\mathbf{0}\n\\end{align} ", "f4789d1de18489f063dc5b75133d076d": "\\frac{d}{dt}\\frac{\\partial T}{\\partial \\dot{\\theta}} - \\frac{\\partial T}{\\partial \\theta} = F_{\\theta},", "f47929c4e0b19b7c424565e5550b002e": "\\hat{q}_{\\tau}=\\underset{q\\in R}{\\mbox{arg min}}\\sum_{i=1}^{n}\\rho_{\\tau}(y_{i}-q) ,", "f4794b0abe6548adaed93e6371093ed2": "\\min_{x\\in\\mathbb R^n}\\; f(x)", "f4795ac401a36a21200b56810a3649e1": "\\scriptstyle \\mathcal{B}_\\alpha y", "f479a3c512e259a16140b954f7395e6c": "\\left\\langle {S_1} \\right\\rangle =1-\\tfrac{1}{2}\\sum\\limits_\\alpha \\left\\langle {{\\sigma_ \\alpha ^2 }} \\right\\rangle + \\ldots", "f47a527df2b8f02be2b35572237c79ef": "-2\\log(\\Lambda)", "f47a6e4a09db5e062210126a1f58f35d": "f_{X_1}\\perp{}f_{X_2}", "f47b0536fb583ab27299a5f72656b2f8": "g:G\\times\\mathbb{Z}\\to\\mathbb{Z}", "f47b6dff8b0f117b161bf2fdc2651c16": " \\text{N}\\!\\left(n,{\\bold Z}^+ \\right) = n \\ , ", "f47bd30fcce15a4622985117d2fee5a2": "Fm_{ms} = Fm_{uc}\\left ( \\frac {1 + \\frac {-25} {1000}} {1 + \\frac {\\delta 13C_s} {1000}} \\right )^2", "f47bf74ae90decdfd5e08709166195d7": "{\\rho}=\\rho_b \\cdot \\exp\\left[\\frac{-g_0 \\cdot M \\cdot (h-h_b)}{R^* \\cdot T_b}\\right]", "f47c18c373f2ad2874f4884dc7240a75": "H(f)(\\mathbf{x}_0)", "f47c241156829d49fd448aff0298f46a": "\n\\boldsymbol{X}=\\sum^{k}_{i=1}\\boldsymbol{X}_{i}\n", "f47c24beb9e08eabd01f78f0966936c8": "Literature \\leq medium", "f47c4f566b6a525dcf6a3549abc940f1": "K' = \\int_0^{\\frac{\\pi}{2}} \\frac{d\\varphi}{\\sqrt{1-k'^2 \\sin^2\\varphi}}", "f47cb3744221906b91d4591615e05e10": "\\hat{f} : E' \\rightarrow E", "f47cb593d43cab0113197fd46619faf7": "v\\notin C", "f47cbe173b959ee8b224dcc5647efd9d": "e^{-1/e}", "f47cec7167181f085e78b87fabd17d6e": "(B/A)_0\\ \\left(\\frac{\\mu m}{hr}\\right)", "f47cee16a1af89526e4aa78114b45820": "\\frac{t(t+1)}{2} = s^2.", "f47cfcc8760a8ddce4cc7344be8c0a8b": "\\boldsymbol{\\hat \\rho}", "f47d1d26ec2361376b17a3c7c6194a5d": "w,u\\in W", "f47d4cd98d7d38d48504113c683be7e2": "u'(z)=\\sum_{k=0}^\\infty (k+r)A_kz^{k+r-1}", "f47d8341d3ad4abd4a279db9e394be68": "\\psi_{RN}", "f47d9fb19bdbb9603a62da5940804ab3": "\n\\begin{align}\nf(x;\\alpha) & = \\frac{\\sin \\pi\\alpha}{\\pi}x^{-\\alpha}(1-x)^{\\alpha-1} \\\\[6pt]\n\\end{align}\n", "f47dfb9637352c5675d2e26b0458f6d1": "a^{(4)}n, \\, \\operatorname{hyper}_4(a,n)", "f47e5c04e690eb69d7667b8abba49a0e": "(-\\sqrt{2}/2,-\\sqrt{2}/2)", "f47e5c7eab6534e3ba4e95a3e8c7f01d": "\\lambda_{k}", "f47e5fee37b0fecac248222ced8f9ff6": " \\lnot \\lnot \\varphi\\,\\!", "f47e81d3e19a89cbe3904eb8f9580809": "\\frac{\\sqrt{3\\,}\\, a^2c}{2}", "f47e86c17a7002da95d7ccb7beb84c8c": "\\tau_{1,v} \\otimes \\cdots \\otimes \\tau_{d,v}", "f47ecbea153a8d83621bdcf2fd55ff98": "a(\\tau,\\zeta) = \\operatorname{sech} (\\tau) e^{i \\zeta /2}", "f47ed29fd7517affef3aad81541dee1d": "K(k) = \\frac{\\pi a}{2 \\, \\operatorname{AGM}(a (1 + k),a (1 - k))} \\, ", "f47f0ef421275c36433e14ab54106eed": "A \\cap L^p_{--} = \\emptyset", "f47f34fe317789ef1c692600b12e1d84": "\\forall a,b \\in M : a \\cdot b \\in M", "f47f4be3f861e784fb466f703372e39a": "g_0=1;", "f47f811ed069eba24cc670e50d720ed1": "\\mathfrak{so}(3,1)\\cong \\mathfrak{sl}(2,\\mathbb C)", "f47fb26f46f13a1e703f41c81ef60207": "(F,B) \\colon (W,M,M') \\to (M \\times I, M \\times 0, M \\times 1)", "f47fb7a97e83eb664517a636048d7bc6": " \\kappa \\ ", "f47fd0965bb63ce307b2fbad48eb1b30": "{e}^{-e}", "f47fd86ce8f0db5876cabcc40940565f": "\\Gamma(X,\\mathcal{O}_X)", "f47fed629d77ba0e327209674347e6cf": "\\pm \\frac12, \\pm \\frac32, \\dots", "f4800b4bca6d2765892e7eb6d9b33b1e": " \\operatorname{de-let}[x\\ x] ", "f48022e7d5ab543cec7f27bbad5c4ccb": "Q_{\\ell m}'", "f4802404cbaeb3cbca30254b7a74ec74": "T(z)=\\lim_{n\\to \\infty}T_n(z)", "f48025863a49f163c1cedd47a69c035c": " \\left(\\operatorname{sys\\pi}_1(M)\\right)^n \\leq C_n \\operatorname{vol}(M),", "f4802c974ac23b43ff36b34c2697d2cf": "q(x_i)=2^{-l_i}", "f4804068e5b7c1b1211b1e7ccfb22e34": "I \\times D^{m-1}", "f4804a60ff6982239eb07f7944313922": "c(i,k,X)", "f48068938dfe42957c8f0be943923284": "\\overline{\\Lambda}_n(\\hat{T}) - \\underline{\\Lambda}_n(\\hat{T}) < \\overline{\\Lambda}_3 - \\frac{1}{6} \\cot \\frac{\\pi}{8} + \\frac{\\pi}{64} \\frac{1}{\\sin^2(3\\pi/16)} - \\frac{2}{\\pi}(\\gamma - \\log\\pi)\\approx 0.201.", "f48072906c2439895b40d2b737d2a29a": "\\mathbf{e}_j ", "f48097dd9a57126556cafb8c343ab045": "\\bar {n}_i", "f4809cda9bf0f7ba0e670c2e96d710c6": " Z_{n,\\beta} = (2\\pi)^n \\frac{\\Gamma(\\beta n/2 + 1)}{\\left(\\Gamma(\\beta/2 + 1)\\right)^n}~. ", "f48108e0742e424db5b37be0a42f7ab7": "M_x ", "f4814b61db54c2d34a45a55036258e3a": "\\{\\{c\\}\\}", "f4819a41f224e391e059e901893a41b8": "(1 - 2 x_i) < 0", "f481bca250dc4c01e608608ca7a088c6": "\\operatorname{pl}(A) = 1 - \\operatorname{bel}(\\overline{A}).\\,", "f481bfcc83500b19d1ab691e932ce24d": "x_0=a", "f483d7e610c6075198f22ff26d43ba74": "\\frac{\\delta I_{EH}}{\\delta N}=0", "f483d8418e31fe97a7e6481260008945": "h'(z) = h(z) + s(z^2)\\cdot g(z)", "f483f604c3836725d47e44a08858b9dc": "f(x) = \\frac{1}{1 + \\mathrm e^{-x}} ", "f484051ce38be339a5970cca8a82b4cd": "b, d \\in B", "f4844192288d2f228d45fda72ee728c7": "\\hat w(f)", "f48444e2b6ab9f1e739bca78ee742e93": "T : V \\otimes V \\to V \\otimes V", "f48464a276d81bf9428bfa22712068e5": "c_1, ...,c_n", "f48474d5f49f1ab464d3c09152e7eae0": "\n\\phi_X(u) =\n\\begin{cases}\n\\exp\\left( i\\mu u - \\sigma^\\alpha |u|^\\alpha \\left( 1-i\\beta\n\\operatorname{sgn}(u) \\tan\\left(\n \\frac{\\pi\\alpha}{2}\\right)\\right)\\right)\n & \\text{if } \\alpha \\in(0,1)\\cup(1,2) \\\\\n\\exp\\left( i\\mu u - \\sigma |u| \\left( 1+i\\beta \\operatorname{sgn}(u)\n\\left(\n \\frac{2}{\\pi}\\right)\\ln(|u|)\\right)\\right)\n & \\text{if } \\alpha = 1 \\\\\n\\exp\\left( i\\mu u - \\frac{1}{2} \\sigma^2 u^2\\right)\n & \\text{if }\\alpha = 2\n\\end{cases}\n", "f484b7d1940f9e549c8700614110fec5": "2pq ", "f484e18dc875201d37d98bb2ab4a5d56": " |x|_{p_i} = p_i^{-a_i} ", "f484f19a004979719a1cde032c2aa66f": "\\nabla (fg)(a) = f(a)\\nabla g(a) + g(a)\\nabla f(a).", "f4856d58ec3a146216adae75861ec4c5": "D\\bold{F}(\\bold{x_0})", "f48578a33a4fb8c605512783be88757c": "{\\mathfrak c} >\\aleph_0", "f48595c476b239e806a46d06d5a903ce": "{{{4}}}", "f485f3029008cde2a7ea253485398132": "\\left\\{0,...,(p-1)(q-1)/4-1\\right\\}", "f4862e2d5a30bf531dd55c7ec8c21c88": "\\mathbf{K}_t", "f486ce7019226cac26b89875124717b8": "\\phi^{|n|}=e^{|n|\\ln\\phi}", "f4875003262f763e611b34fdb37c65ac": "\n\\det\\begin{bmatrix}\n\\wp(u) & \\wp'(u) & 1\\\\\n\\wp(v) & \\wp'(v) & 1\\\\\n\\wp(w) & \\wp'(w) & 1\n\\end{bmatrix}=0", "f488100ef221c9f623de63a948d62d83": " \\text{ } (4) \\text{ } Q^{(n+1)/2} \\equiv Q \\cdot Q^{(n-1)/2} \\equiv Q \\cdot \\left(\\tfrac{Q}{n}\\right) \\pmod {n} ", "f4884e0d1da3fa219340b15e582ccd4d": " n = \\frac{d \\log(\\sigma)}{d \\log(\\epsilon)} = \\frac{\\epsilon}{\\sigma}\\frac{d \\sigma}{d \\epsilon} \\,\\!", "f4889a2813d5f6c78cec21e3fdcfba44": "e^{-1/g^2}", "f488a7f839cc307c3335863fc643e7a3": " Lq < -\\ln{S}", "f488ba661204811fe08aae813262b166": "N_{BOC}=2\\frac{f_{sc}}{f_c}=2\\frac{m}{n}", "f488c40c427fc29c5945d9fd0c545620": " d_0(e) = d_1(e) = v. \\quad", "f488ca26a8b3348ce4fecbb4f35d074f": "t=\\frac{a}{b}\\leq 1", "f488d41952aefeda8fbaa5068c99a25b": "{\\displaystyle}(x_1,y_1)+(x_2,y_2) = (x_1y_2+x_2y_1,y_1y_2-x_1x_2)", "f489344260e56bec0b1c192bebde31e9": " \\zeta(1/2+i \\hat H )|n \\ge \\zeta(1/2+iE_{n})=0, \\, ", "f48969c0e9b944aaef689aa0cfc308df": "X_1,X_2, \\dots, X_t \\, ", "f4896d4e19c9ff8536098f62cf47b6a9": "G.\\,", "f489db111c0d22508c3f4067a951df87": "1\\to F_{n-1} \\to P_n \\to P_{n-1}\\to 1.", "f48a3d89200efd1e6fdf66d59d9c594f": "h^{p,q}=\\text{dim} H^{p,q}", "f48a4a6706b21729878cb6839c619dee": "\\begin{align}\n\\{x^3,p^3\\}+\\tfrac{1}{12}\\{\\{p^2,x^3\\},\\{x^2,p^3\\}\\}&=0 \\\\\n\\tfrac{1}{i\\hbar}[\\hat{x}^3,\\hat{p}^3]+\\tfrac{1}{12i\\hbar}\\left[\\tfrac{1}{i\\hbar}[\\hat{p}^2,\\hat{x}^3],\\tfrac{1}{i\\hbar}[\\hat{x}^2,\\hat{p}^3]\\right]&=-3\\hbar^2~.\\end{align}", "f48abc5ee28898ac190882685b3c852f": "\\aleph_1 = 2^{\\aleph_0}", "f48acd0b46cc68437d1e41052ec1db5c": "[0, T).", "f48b23bc1ecd178ecdab4ce22c1b3db6": "\\operatorname{E}(X \\mid Y) \\circ Y= \\operatorname{E}\\left(X \\mid Y^{-1} \\left(\\Sigma\\right)\\right).", "f48b5a5636884ac2bdf931af1f4b3233": "d_{hk \\ell}= \\frac {a} { \\sqrt{h^2 + k^2 + \\ell ^2} }", "f48c082e639f895e8fa95ecfa5281324": "T^{\\mathrm{e}}", "f48c1f7f92272596adc23aca83eb15a4": " \\mathrm{Ca} = \\frac{\\rho v^2}{K}", "f48c2c4cb61a0c7926606d5a89574caf": " M = f, N = (x\\ x) ", "f48c2c8ce01e368cfe7cdd1cd5580769": "i_1 \\ge i_2 \\ge \\ldots \\ge i_n", "f48c5c8ab4921646d0b1276b48365824": "a \\in w", "f48cd4fa8d58b03544fe433149514617": "s(n)=\\pi^ {-\\frac{1}{2} } 2^ {\\frac{3}{4}} \\Gamma\\left (3/4 \\right) (n/e)^ {n-\\frac{1}{4} } \\left (1+O\\big(\\frac{1}{n}\\big)\\right).", "f48d4dae2b85b0645c9bb4c1f200846e": "[z^n].", "f48d6d0596ef0cf109bdaf427bb3464c": "c=0.5", "f48d9a1b1edace48f1a0b924ad4c820d": " \\mathbb{Q}[X_1,\\ldots, X_r, Y_1,\\ldots, Y_s]. \\, ", "f48dbe0de27a1c356107abfc0e63a26a": "u(f_{a_1\\ldots a_k}c^{a_1}\\cdots\nc^{a_k})=u^A\\partial_A(f_{a_1\\ldots a_k})c^{a_1}\\cdots c^{a_k}+\n\\sum_i u^{a_i}(-1)^{i-1} f_{a_1\\ldots a_k}c^{a_1}\\cdots\nc^{a_{i-1}}c^{a_{i+1}}\\cdots c^{a_k}", "f48dcfb7e79cb0d5b34563671cb5f387": "cI_V", "f48e530e94b6163349667259ebaadc5c": " \\frac{A(1)-A(0)}{A(0)} \\,", "f48e605be30d0dd3198e39e7661339ff": "\\textstyle \\mathbf{\\Sigma}", "f48e7f800e5a2f0b9885299154beab8a": "\\mu_j = g^{(l)} l + {1\\over 2}g^{(s)}", "f48ec457e7db9117a210b5f5682e18b4": "d\\theta\\;d\\lambda (r)", "f48ed8c091736bed36cc44ff8e7a47d1": "I=\\sqrt{k \\rho c}", "f48edc4e2561a4de6dc474603de60f83": "\\Lambda^1[2] \\to C", "f48f7e4899d8cccc05018b95ea11c7e4": "b^{(B-1)/2}\\equiv +1 \\pmod B\\;", "f48fbc323129756ab0fd6ab92af3c093": "\\beta_T", "f48fc43403cfd1d73bab03934a39c58d": " \\ m = \\ 1.7 + {40000\\over Re} ", "f4900ade35345f4f5979cd75352b1f3d": "X_m", "f4900da2ac34ac330fc14bb39d9fceb9": "d.", "f49014274190b91e17517dd9b17b11b4": "\n\\frac{dR}{dt} = -k_{1} R L + k_{-1} C = -k_{1} R (L_{tot} - R_{tot} + R) + k_{-1} (R_{tot} - R)\n", "f490407f02551607c7b10673d9e5b5ba": "\\nabla\\cdot \\bold{J} = -\\frac{\\partial \\rho}{\\partial t} ", "f49052899b8c15ef12a87025195bdad2": "\\scriptstyle V \\times V \\to V,", "f49061aae9de7be9db78ca8e1c2ea323": "(12)\\quad\\quad\\quad\\quad\\; u_s\\left(\\rho_2 - \\rho_1 \\right) = \\rho_2 u_2 - \\rho_1 u_1 ", "f4909fad053f478135f7345f647c65c0": "\\overline{x} = a - ib.", "f490b82589bb2e0208df63e346b03b02": "A_0 = \\alpha A^*", "f490c4106d17409e4428b898d455c79e": "a \\uparrow \\uparrow \\uparrow \\uparrow b = \n \\underbrace{\n \\left.\\left.\\left. \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{\\vdots}_{a} }} \\right\\}\n \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \\underbrace{\\vdots}_{a} }} \\right\\}\n \\cdots \\right\\}\n a\n }_{b}", "f490d0f92a14e6d6f8b6c1f5abff8df1": "\\log|z|", "f490dda3287441be06e405c560cd4fa0": "=m_1^2 + m_2^2 + m_3^2 + m_4^2 + 2 \\left( m_1^2 + p_1 \\cdot p_2 - p_1 \\cdot p_3 - p_1 \\cdot p_4 \\right) \\,", "f490e2b279d0d9f8bf52c59eeea8bed8": "\\tfrac{3}{4}", "f49137d91b2868dafba0bb35ee184bfb": "\nS_{mn} \\ \\stackrel{\\mathrm{def}}{=}\\ \\dfrac{\\int d\\mathbf{r} \\ \\rho(\\mathbf{r}) \\ r_{m} r_{n}}{\\int d\\mathbf{r} \\ \\rho(\\mathbf{r})}\n", "f4915257f394c2861f31e48da64132cb": "f_i^{-1}(U)", "f491aa1d09bf7e6120632bb892e94e7b": "N(X + Y) \\geq N(X) + N(Y). \\,", "f491d6f06fc5281d62474e39b1227beb": "\n\\begin{array}{lcr}\nA' = \\begin{bmatrix} 2 & 1 & -1 \\\\ -2 & 2 & 2 \\\\ -2 & 1 & 3 \\end{bmatrix} &\nB' = \\begin{bmatrix} 2 & 1 & 1 \\\\ 2 & -2 & 2 \\\\ 2 & -1 & 3 \\end{bmatrix} &\nC' = \\begin{bmatrix} 2 & -1 & 1 \\\\ 2 & 2 & 2 \\\\ 2 & 1 & 3 \\end{bmatrix}\n\\end{array}\n", "f491e340a6155e479c812eb39c649fa0": "\nM=\\int_0^{R_{max}} 4\\pi r^2 \\rho (r) dr=4\\pi \\rho_0 R_s^3 \\left[\n\\ln\\left(\\frac{R_s+R_{max}}{R_s}\\right)-\\frac{R_{max}}{R_s+R_{max}}\\right]\n", "f492132d432c9e2b9be1e40840107b76": "s = w\\log(n/w)", "f4924c1b61b31c26ceb001033a1a2787": "\\mathbf{M}\\vec{v} = \\sigma \\vec{u} \\,\\text{ and } \\mathbf{M}^*\\vec{u} = \\sigma \\vec{v}", "f492779ca10ef3112bb45e7b984b2c08": "y \\leftarrow y'", "f49289d92b1216be42e22090c18e41f2": "[n]P=(X_n:Z_n)", "f4928e2675d1e999989fefae0c034975": "^{1/2}", "f492c7a82523fc0abc8207f56f5d968b": "g_{01} = g_{02} = g_{03} = 0 \\!", "f492db592320e5173e630a6a41f1fcbb": "\\mathbf{r} = (x,y,z) = x_1\\frac{\\mathbf{a}_{1}}{a_1}+x_2\\frac{\\mathbf{a}_{2}}{a_2}+x_3\\frac{\\mathbf{a}_{3}}{a_3},", "f49341ab621f12e8cb93d0146ea51d34": "O(n\\log n)", "f493507cd41bf033bdc1051561ec8297": "\\tilde{h}:F\\star G\\rightarrow F\\star H", "f493758aec6798b569d13542f74c6a8f": "\\mathbf{c}(t)=(c_i(t))", "f493805438618311a32800f885377fc1": "f_\\beta", "f493a03c5cff788e432b9eb4b04b0ef6": " \\Phi_{i} = \\int_{S_i} \\mathbf{B}\\cdot\\mathbf{da} = \\int_{S_i} (\\nabla\\times\\mathbf{A})\\cdot\\mathbf{da}\n = \\oint_{C_i} \\mathbf{A}\\cdot\\mathbf{ds} = \\oint_{C_i} \\left(\\sum_{j}\\frac{\\mu_0 I_j}{4\\pi} \\oint_{C_j} \\frac{\\mathbf{ds}_j}{|\\mathbf{R}|}\\right) \\cdot \\mathbf{ds}_i ", "f493c130e152ec068ebefcc8bc77c17e": "X_{lc'}(\\bold{r}) = \\frac{1}{i^{n_{c'}}\\sqrt{2}}\\left(Y_l^m - Y_l^{-m}\\right)", "f49404395b24bdcc16f192ff07a92f25": "\\mathbf{\\dot{r}}_j = (\\dot{r}_1, \\dot{r}_2, \\dot{r}_3)", "f494b1823d4ecf7227ee23ece8b6867c": "\\varphi\\circ g=\\psi(x^1\\circ g,\\dots,x^n\\circ g)", "f494bbdde7e276f29da742cd4b91cab8": " \\dot{\\phi} = \\frac{b - a\\cos\\theta}{\\sin^2\\theta}.", "f4951e5c7023ec6e180d2419652891e6": "\\begin{align}\n \\Delta \\Omega &= -2\\pi \\frac{J_2}{\\mu p^2} \\; \\frac{3}{2} \\cos i \\\\\n \\Delta \\omega &= -2\\pi \\frac{J_2}{\\mu p^2} \\; 3 \\left(\\frac{5}{4} \\sin^2 i - 1\\right)\n\\end{align}", "f4959b17236dc13728370bfa35e7276e": "\\frac{a+b}{2}\\,", "f495a463500014de2e7aaae8fd689454": "\\pi(x,a,d) = \\frac{1}{\\varphi(d)} \\int_2^x \\frac{1}{\\ln t}\\,dt + O(x^{1/2+\\epsilon})\\quad\\mbox{ as } \\ x\\to\\infty", "f495b4b3c48d251fb9a5130496cf27e2": " Q = - \\frac{\\hbar^2}{2m} \\frac{\\nabla^2 \\sqrt{\\rho}}{\\sqrt{\\rho}} = - \\frac{\\hbar^2}{4m} \\left[ \\frac{\\nabla^2 \\rho}{\\rho} - \\frac{1}{2} \\frac{(\\nabla \\rho)^2}{\\rho^2} \\right]", "f495e2fa326646a5dc485855ea9832fd": "{a_0}\\, ", "f49640754d13fa8b3b71e729f33e2c20": "\\tfrac{2}{3} \\times \\tfrac{3}{4} = \\tfrac{6}{12}", "f49640e19bea3af71c3ecd4943f9a42c": "\\frac{ \\tbinom{48}{9} }{ \\tbinom{52}{13} } = 0.264106\\% = 1:379", "f496bc1b7a38b529951b0022fe7f6c9d": "\\Box (K \\and \\lnot K \\rightarrow (K \\and \\lnot Q)) ", "f496d853c84ec4b931b0278a9acd5870": "F_{receiver}", "f496f0a6db010bf652a90978d29b65a8": "\\dot{\\theta} = (F\\sin\\alpha)/mv + (g/v - v/r)\\sin\\theta,\\,", "f49709f9dd305457146d2553e921cf51": "\\lambda_1 \\geq 2\\sqrt{d-1} - \\frac{2\\sqrt{d-1} - 1}{\\lfloor m/2\\rfloor}.", "f49799f6463bb07e3c34c65f688c0f62": "\\sin^2 (30^\\circ/2)=(1-\\cos 30^\\circ)/2=(1 - \\sqrt{3}/2)/2=(2-\\sqrt{3})/4 \\approx 0.0667.", "f49806a15130774cbf7475e6c1180b97": "\\gamma_{\\mathbf{v}_e} \\mathbf{v}_e", "f4983f15af7ec8f7902f531ce6de52ba": "\\mathrm{deg}(d_k) = 1 + (-1)^k.", "f4986e04213f1407da54afde12598cd1": "\\begin{align}\n& \\operatorname{E} \\left [\\frac{1}{X} \\right ] = \\frac{\\alpha+\\beta-1 }{\\alpha -1 } \\text{ if } \\alpha > 1\\\\\n& \\operatorname{E}\\left [\\frac{1}{1-X} \\right ] =\\frac{\\alpha+\\beta-1 }{\\beta-1 } \\text{ if } \\beta > 1\n\\end{align}", "f4987893be023c60b39b0e16af92591c": "B \\rightarrow S: M,A,B,\\{N_A,M,A,B\\}_{K_{AS}},\\{N_B, M,A,B\\}_{K_{BS}}", "f49893bf5d8d0a87113cd5164bce3e3d": "F(x) = \\sum_{n \\ge 0} f_n \\frac{x^n}{n!}", "f498a05da5b88363c638556818a006f1": " \\|f\\|'=\\sup\\{|f(x)|: x\\in X, \\|x\\|\\leq 1\\}=\\sup\\left\\{\\frac{|f(x)|}{\\|x\\|}: x\\in X, x\\ne 0\\right\\}. ", "f498d4982eda40eb093dcc413a1c4f6d": "\\{2, \\ldots, n\\}", "f49939054f5eb491f4d8d7654e6f126f": "\\mathbf{e}_7", "f4995bf7e0d81b22e0f43823bd77d5cc": "K_{\\mathrm a}=\\mathrm{\\frac{[H_3O^+] [OH^-]}{[H_2O]}}", "f4995ddacafc5dbfdffa91eb7cc211c9": "\n m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r}\n= \\sum_i \\left[{\\mathrm{d} \\over \\mathrm{d}t}{\\partial T \\over \\partial \\dot{q_i}}-{\\partial T \\over \\partial q_i}\\right]\\delta q_i\n", "f4995e134ad4bd340288ed61e51d8b51": "\\displaystyle{\\alpha(f(z),f^\\prime(z)v)\\le \\alpha(z,v)}", "f499a05b3ddb8aabf1635e396a6f1823": "E_{0}^{r}", "f49a690419506783f4d2ac82e71a727b": "m_{A,B}=\\mu_{A\\otimes B}\\circ Tt'_{A,B}\\circ t_{TA,B}:TA\\otimes TB\\to T(A\\otimes B)", "f49a743f9f8d8e53e23ba0e30c25fcff": "\\text{Minimize} =\n\\begin{cases}\n f_{1}\\left(\\boldsymbol{x}\\right) & = x_{1} \\\\\n f_{2}\\left(\\boldsymbol{x}\\right) & = g\\left(\\boldsymbol{x}\\right) h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) \\\\\n g\\left(\\boldsymbol{x}\\right) & = 1 + \\frac{9}{29} \\sum_{i=2}^{30} x_{i} \\\\\n h \\left(f_{1}\\left(\\boldsymbol{x}\\right),g\\left(\\boldsymbol{x}\\right)\\right) & = 1 - \\left(\\frac{f_{1}\\left(\\boldsymbol{x}\\right)}{g\\left(\\boldsymbol{x}\\right)}\\right)^{2} \\\\\n\\end{cases}\n", "f49a78fea7dcf5feb01c5b6be32b603e": " \\dot{p} = \\frac{3}{r} \\left ( (\\gamma - 1) K \\frac {\\partial T}{\\partial r}\\Bigg|_R - \\gamma p \\dot{R} \\right )", "f49a9161cce8422c4d84b538912618cb": "x_1 = 0", "f49aa4650923148caf0fdfdd6a39b1a3": "ds^2 = \\frac{\\langle \\delta \\psi \\vert \\delta \\psi \\rangle}\n{\\langle \\psi \\vert \\psi \\rangle} - \n\\frac {\\langle \\delta \\psi \\vert \\psi \\rangle \\; \n\\langle \\psi \\vert \\delta \\psi \\rangle}\n{{\\langle \\psi \\vert \\psi \\rangle}^2}.\n", "f49ad0c882751d4c7535831acb523bfc": "{k(0,x)}={g(x)}", "f49af6ce0876701bde0d656b59e65e57": "X\\not\\Vdash A\\to B", "f49b2be0c166b29715a6e0c3bf1675da": "\\bot_{\\mathrm{D}}(a, b) = \\begin{cases}\n b & \\mbox{if }a=0 \\\\\n a & \\mbox{if }b=0 \\\\\n 1 & \\mbox{otherwise,}\n\\end{cases}", "f49b8c0fd7074979359f157b7373f9ee": "X := (x^3 - 3xy^2) \\, \\frac{\\partial}{\\partial x} + (3x^2y - y^3) \\, \\frac{\\partial}{\\partial y} . ", "f49b9ceabe600040dd7248e9f3f6cc14": "y=x^n", "f49c31c2a1bc961bf39d95df40b9e454": "\\sum_{n=1}^\\infty {1 \\over n^2} = \\lim_{n \\to \\infty}\\left(\\frac{1}{1^2} + \\frac{1}{2^2} + \\frac{1}{3^2} + \\cdots + \\frac{1}{n^2}\\right) = \\frac{\\pi ^2}{6}.", "f49c831f03ba031b0d8575629489e3fc": "A \\parallel_- B", "f49cad238f2b876aebc8f9cc25bb93fc": "\\boldsymbol \\tau = \\mathbf{r}\\times \\mathbf{F}\\,\\!", "f49cd8f282b0e0c9849eb8b1e998eca3": "(n+1)/2", "f49d094d6d55554aded2cf18ee51310d": "\nF = S \\left( \\frac{1+r_d T}{1+r_f T}\\right) ,\n", "f49d5aa4a4bfa88bab3abec81158a5a4": " \\frac{\\partial N_x}{\\partial e}\\frac{1}{X} = \\frac{\\partial X}{\\partial e}\\frac{1}{X} - \\frac{e}{X}\\frac{\\partial Q}{\\partial e} - \\frac{Q}{X} ", "f49d85c855eb6e13b7e05439e618b362": "\\tfrac{381}{384}", "f49d94c432cca170106ae875dd551a07": "2\\uparrow^{m+1} 3=2\\uparrow^m 4.", "f49da864e80e537b72ad594adfa5138b": " \\max\\left\\{ \\frac{1}{2} \\sigma^2 x^2 V''(x) + (r-\\delta) x V'(x) - rV(x), g(x) - V(x) \\right\\} = 0", "f49dd2f09ce46597b3ede1d1ec217ddc": "p_w(\\theta)=\\frac{1}{2\\pi}\\,\\sum_{n=-\\infty}^{\\infty}\\int_{-\\infty}^\\infty p(\\theta')e^{in(\\theta-\\theta')}\\,d\\theta'", "f49df2ffc95c8fd2e3bb68786caa53aa": " \\mathbf{b} ", "f49eabbfdd2346dd79c217f6091052fb": "T = \\frac{B}{A-\\log P}", "f49eb3017b74c3d90a125d55ef8b8511": "p^M", "f49f45fd22b794c58f58ab3da44cb8d3": "W_1W_2=W_2W_1\\,,", "f49fad5ab46a3a5e419941416c8e6e7c": "F_L(\\eta,\\rho) = \\frac{2^Le^{-\\pi\\eta/2}|\\Gamma(L+1+i\\eta)|}{(2L+1)!}\\rho^{L+1}e^{-i\\rho}M(L+1-i\\eta,2L+2,2i\\rho)", "f49fb8e8cf7a24eec4afeae4e3026efc": "E = -\\frac{N_AMz^+z^- e^2 }{4 \\pi \\varepsilon_0 r_0}\\left(1-\\frac{1}{n}\\right),", "f49fd99b0e3b5200371c3f099826c7b6": " 0 < v_i < \\infty \\, ", "f4a00c58d3cfe73be14f74272abf4390": "\\nu\\sigma", "f4a02e01eee93173c42871bfa3e02f0a": "E(|S_n|)\\,\\!", "f4a07105d13f8e4407dae007c900516b": "L \\approx {2 \\pi \\cdot 3000 \\over 360^{\\circ}} \\cdot A \\approx 52.36 \\times A ", "f4a1e3a1613bad830f706ace47ff6d0d": " L\\, k_j= 2\\pi I_j\\ -2 \\sum\\nolimits_{i=1}^N \\arctan\n\\left(\\frac{k_j-k_i}{c} \\right) \\qquad \\qquad \\text{for } j=1, \\,\n\\dots,\\, N \\ , ", "f4a2441113e28bf32b7b1579426580fa": "\\scriptstyle |t|\\,<\\,2\\pi", "f4a2cc6e4c2823ff8f6ad059c3618728": "\\partial f_i / \\partial x_j", "f4a2e1c9ee59db7c2b94f90c9b3377dd": "V_\\gamma = V_I e^I_\\gamma", "f4a2f4f5d6712b5fb53d9cf05e51d3e8": "(\\mathcal{T}),", "f4a30aab6ba3326e34c989e6e8fe48f1": " H_{1}=H_{f} \\,", "f4a33b178c012be673c4a2cb5bd65302": "\\left|b^{dp}\\right|^n=\\sum_{|k|=n}{\\binom{n}{k}b^{dp\\cdot k}};\\ \\ b,d\\in\\mathbb{N},\\ n\\in\\mathbb{N}_0,\\ k,p\\in\\mathbb{N}_0^m,\\ p:\\ p_1=0, p_i=(n+1)^{i-2}", "f4a356c96c3a03aa800c4b9912fd1b69": "\n\\pi = \\cfrac{4}{1 + \\cfrac{1^2}{2 + \\cfrac{3^2}{2 + \\cfrac{5^2}{2 + \\cfrac{7^2}{2 + \\ddots}}}}}\\,\n", "f4a3b3540bb51715f356cacfce0ef2a9": "1-L ", "f4a3fa54ba6975dfabc94fd7f2e140f9": " \\int_{\\mathfrak{H}^3} (M_1f)\\cdot F \\,dV = \\int_{\\mathfrak{H}^2} f\\cdot (M_1^*F) \\,dA.", "f4a40a3db68480131d1493b712fdd99e": " \\Delta_h f := \\operatorname{st} \\frac{[{}^*\\!f](x+h)-[{}^*\\!f](x)}{h} ", "f4a425274b977dd0085d44ee5b438a86": "\\ \\sigma =-\\frac{d (c_1/c_2)}{d MRS}\\frac{MRS}{c_1/c_2}=-\\frac{d\\log (c_1/c_2)}{d\\log MRS}", "f4a45130d2efb83af69d7d3cad81bb7c": "y = \\frac{1}{2}\\left[\\varphi + \\frac{\\sin \\varphi}{\\mathrm{sinc}\\,\\alpha}\\right]", "f4a454acf06f4315ec6ec3e067991bb5": "\\frac{U}{V_1}", "f4a48f1b67cc57b764e3e5d6b5e5f713": "\\{d_{n}\\}", "f4a4905b209b5378599325e24418397b": " L = \\{a^n b^n c^n | n \\geq 1\\}", "f4a4e85ea6a892ff30fbe9dd2a0aa6cc": "1.175=\\{1,6,20\\}=\\{1,6,21,21,21,\\dots\\}.\\;\\;", "f4a500de332bafc7d00c20ce9981111a": "{}_sY_{\\ell m} = 0,\\ \\ \\ell < |s|.", "f4a602c6a64092e7b0e2e1c830534ced": "1 \\over w-z ", "f4a642ee3fd16ceb7074e2b983c770fa": "e^{\\frac{i}{\\hbar}\\lambda V(t-t_0)}H_0e^{-\\frac{i}{\\hbar}\\lambda V(t-t_0)}|\\psi_F(t)\\rangle=i\\hbar\\frac{\\partial |\\psi_F(t)\\rangle}{\\partial t}", "f4a6472315c9fa28b92e821fa26b6ad5": "M' \\subseteq M", "f4a6645893af45978d10f4c730c62917": "\\{x\\}= x \\,\\bmod\\, 1.\\;", "f4a696166929f503b029bcc2f50605c3": "ROC = \\frac{\\textrm{Net Operating Profit} - \\textrm{Adjusted Taxes}}{\\textrm{BV of Debt} + \\textrm{BV of Equity} - \\textrm{Cash}}", "f4a6adf3b96475daad2fe4a7b1c3714a": "\\forall A\\in V\\,(x\\in A\\Leftrightarrow y\\in A)", "f4a6c3521885457376a41a14f49eee1a": "I(f)=\\int_{-\\infty}^\\infty\\Bigl(\\vert f(y)\\vert + \\vert f^{(k)}(y)\\vert\\Bigr) dy", "f4a6d1c1b873bd3c0a60702614c7c63b": "R^i", "f4a6e91d8cd7d49ea3d37994e32e9e1e": "\\sum_{j} |\\phi_j |^2 | j \\rang \\lang j | ", "f4a72361b3f35d802269453825f98ab2": "F_1,F_2,\\dots\\,", "f4a766893737c6e6595238de123400d0": "= (g_{\\mu \\nu ,\\lambda} \\dot x^\\mu \\dot x^\\nu) (g_{\\alpha \\beta} \\dot x^\\alpha \\dot x^\\beta) + {1 \\over 2} (g_{\\lambda \\nu} \\dot x^\\nu + g_{\\lambda \\mu} \\dot x^\\mu) {d \\over d\\tau} (g_{\\mu \\nu} \\dot x^\\mu \\dot x^\\nu) \\qquad \\qquad (6) ", "f4a7a059b4f0b7e0b0de172b0c65be6b": "u = 4 \\arctan\\left(\\frac{\\sqrt{1-\\omega^2}\\;\\cos(\\omega t)}{\\omega\\;\\cosh(\\sqrt{1-\\omega^2}\\; x)}\\right),", "f4a814b82f09f811e98ffba0afd18121": "-\\frac{\\hbar^2}{2 m} \\frac{d^2 \\psi_2}{d x^2} = E \\psi_2 .", "f4a8150b5cfa02e18c1063390418887c": "=-(ED_t+BN_t)[(C+AN_tD_t^{-1})D_t]^{-1}", "f4a86638f1a709ae20a0c878d110400f": "\n\\mathrm{d}\\begin{bmatrix}\n\\mathbf{P}\\\\\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n0&\\mathrm{d}s&0&0\\\\\n0&0&\\kappa_g \\, \\mathrm{d}s&k_n \\, \\mathrm{d}s\\\\\n0&-\\kappa_g \\, \\mathrm{d}s&0&\\tau_r \\, \\mathrm{d}s\\\\\n0&-\\kappa_n \\, \\mathrm{d}s&-\\tau_r \\, \\mathrm{d}s&0\n\\end{bmatrix}\n\\begin{bmatrix}\n\\mathbf{P}\\\\\n\\mathbf{T}\\\\\n\\mathbf{t}\\\\\n\\mathbf{u}\n\\end{bmatrix}.\n", "f4a888cfb79b282aedc52c50ca0f3b44": "p=a_m10^m+a_{m-1}10^{m-1}+\\cdots+a_110+a_0", "f4a8c34a742af59bd125aa8f72969700": "{du^\\alpha\\over d\\tau}={e\\over m}F^{\\alpha\\beta}u_\\beta\\;.", "f4a923949e58f7f828ebe80948bdbb9f": "\\zeta = { c \\over 2 \\sqrt{m k} }.", "f4a925946ddd2a318a05deb94a30a9fa": "\\tanh x \\approx x", "f4a99a836891e1d8d29507d7ebd35029": "S=\\{x \\in M|R x\\}", "f4a9dc660de8345db79300ccd2da6ffb": "\\int e^{-x^2}\\,dx,", "f4aa18f8b165a673115965eaf2819543": " \\models", "f4aa26332ab7553a807eb0a16f17a121": "V=f_\\text{step} F_\\text{avg}/W_\\text{b} L_\\text{c},", "f4aa8877ec0ac7acb2f02c259911c1e9": "\\mathrm{CzS}\\!\\!\\!\\Vert", "f4ab253695b2094ece6d0b738855650c": "\\,(x+1)", "f4ab55d8dbed2358459f5bd3a8d9b349": " y_i^{[n-1]}, i=1, \\dots, r ", "f4ab61332733fd6e275a83d98f91272b": " \\frac{GM}{R} \\approx \\sigma^2 ", "f4ab76c3600815cf4295a97f1dbd07d0": "c = c(k)", "f4abd6672582e6e9751af08973dff3f6": "G \\in \\Gamma.", "f4abf29b5464c5ca0127516b88282fd0": "\\sqrt {(x+c)^2+y^2} + \\sqrt {(x-c)^2+y^2} = 2a", "f4abfa8e0275f11d8bb8807df7fdd61a": "\n\\begin{align}\n\\mathbb{E}S & = \\mathbb{E}[S \\mid \\mbox{fails before } t] \\cdot \\mathbb{P}[\\mbox{fails before } t] + \\mathbb{E}[S \\mid \\mbox{does not fail before } t] \\cdot \\mathbb{P}[\\mbox{does not fail before } t] \\\\\n& = \\frac{t}{2}\\left(0.5t\\right) + \\frac{2-t}{2}\\left( t \\right)\n\\end{align}\n", "f4ac1cf607bdb42aae20dbd5a34936f2": "k_xk_zE_x + k_yk_zE_y + \\left(-k_x^2-k_y^2+\\frac{\\omega^2n_z^2}{c^2}\\right)E_z =0", "f4ac20914eae9f7b9dc92f40b821cd8d": "1^2S_{1/2}", "f4ac34198a84f7e851b99244941da7ed": "[1/y_2, \\infty]", "f4ac398dc8e986217a90716763dc7ac1": "\\vec{V}=\\nabla\\psi \\times \\widehat{k}.", "f4ac7a673d50bd7f5a7ae244dc3d0d4c": "n^{\\underline{k}}", "f4acc3868112f48af9d21fbf279f29c6": "\\int_0^t\\varphi(s)\\psi(t-s) \\, ds, \\qquad 0\\le t<\\infty,", "f4ad79fc95ee3934b003a64baf61f207": " U \\cong \\mathbb{R}^i \\times CL", "f4ad91bf72eae9a748bd90cd6fb7b269": "\\textstyle\\sum_{j=1}^nA_{1,j}B_{j,1}", "f4ada6cc7e54a76c6290b5b9a60b502a": "\\sigma\\cdot\\sigma_f=\\frac{N}{2\\pi}", "f4adbe95046783e0fc3e54976057d24e": " J_2 =\na_2 \\delta^3\\left ( \\vec x - \\vec x_2 \\right ) \n\n", "f4aeba1fad34cb8d86a8eac82c28524a": "T,P", "f4aee1e5d16354ca31f484ce3d3b40fe": "\\mathcal{H}_k", "f4af0f28b99c64884a610d5214ad945b": "T: V \\to V", "f4af3683a8c5616b3165256f4386d60c": "|V|^2", "f4af585d946f77b29ff292681a1e6c96": "\\nabla\\cdot(\\nabla\\times\\mathbf{F})=0", "f4af85b168a5aa2a192a3c6bebdef95e": "\\frac{\\partial u}{\\partial t} = a \\left(\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2}\\right)", "f4b02e2d6e0d97ef60707c1f8e4f238a": "1 \\rightarrow 2;", "f4b06f3111dd7b76e1d00de9880cd239": "O((1+\\sqrt{2})^{4n}/\\sqrt{n})", "f4b0790a998dc71d199691729d4d74a2": "P V = n R T", "f4b0796fdaf17939ce8ea56fbf446baa": "c_F(a,0)=\\frac{\\beta}{4}\\mathrm{sech}^2\\frac{\\beta a}{2}", "f4b0ab70f92544d62ebcebc77012d016": "\\Delta E_I", "f4b0f30dc3833eb78e212b70feca3c67": "\nA=\n\\begin{pmatrix}\n0&0&5 & 0 & 1 & 4 \\\\\n0&0&0 & -1 & -4 & 99 \\\\\n0&0&0 & 20 & 19 & 16 \\\\\n0&0&0 & 0 & 2 & 1\\\\\n0&0&0 & 0 & 0 & 3\\\\\n0&0&0 & 0 & 0 & 0\n\\end{pmatrix}\n\\qquad H=\n\\begin{pmatrix}\n0& 0& 5& 0& 0& 2\\\\\n0& 0& 0& 1& 0& 1\\\\\n0& 0& 0& 0& 1& 2\\\\\n0& 0& 0& 0& 0& 3\\\\\n0& 0& 0& 0& 0& 0\\\\\n0& 0& 0& 0& 0& 0\\\\\n\\end{pmatrix}\n", "f4b12279e20826efdb64fb6f67037403": "\n \\cfrac{\\partial W}{\\partial \\boldsymbol{C}} = \n \\cfrac{\\partial W}{\\partial I_1}~\\cfrac{\\partial I_1}{\\partial \\boldsymbol{C}} + \n \\cfrac{\\partial W}{\\partial I_2}~\\cfrac{\\partial I_2}{\\partial \\boldsymbol{C}} \n = \\cfrac{\\partial W}{\\partial I_1}~\\boldsymbol{\\mathit{1}} + \n \\cfrac{\\partial W}{\\partial I_2}~(I_1~\\boldsymbol{\\mathit{1}} - \\boldsymbol{F}^T\\cdot\\boldsymbol{F})\n ", "f4b227b276e5c6d845d8609db7058779": "\\frac{d}{dt} \\int_{\\Sigma(t)} \\mathbf{B}(t_0) \\cdot d\\mathbf{A} = -\\oint_{\\partial \\Sigma(t_0)} (\\mathbf{v}(t_0)\\times \\mathbf{B}(t_0))\\cdot d\\boldsymbol{\\ell}", "f4b235efcafb26e58ad258ef93de8390": " w_i = \\sum_{j=1}^{n} a_j [j/(n+1)]^i, ", "f4b29561118e5eb35b81b98aad377ee2": "S(n) = \\max \\{ s(M) | M \\in E_n \\} = \\,\\!", "f4b2fe29f5b2e0bc748e8fe9a262c0e9": "\\dot{x}+x=0", "f4b31f0de5f14f8af17212cd4cf1e684": " \\sgn(x) = {x \\over |x|}", "f4b335b97d39c1b0061646a749d6335c": "\n\\boldsymbol\\Sigma_{Y|X}\n=\n\\boldsymbol\\Sigma_{YY} - \\boldsymbol\\Sigma_{\\mathit{YX}} \\boldsymbol\\Sigma_{\\mathit{XX}}^{-1} \\boldsymbol\\Sigma_{\\mathit{XY}}. \n", "f4b339682e05755eb7408448ef87e1ca": "n=3", "f4b34b1e4e00676b7a6a9942eba7fe3c": "ee''=\\frac{1-c}{c}ee=\\frac{e^{-t}+e^{-st}}{2-e^{-t}-e^{-st}}\\left(\\frac{e^{-t}-e^{-st}}{e^{-t}+e^{-st}}\\right)", "f4b34ff23c5767f5aa0251e570ba7060": "p^{\\star}_i", "f4b355306fe235540fcdc5057ebeea34": "E(\\mathbf{k}) = N \\left[ \\frac{\\hbar^2k^2}{2m}+ g \\frac{N}{2 V}\\right].", "f4b35ef4587446d1f0ca521d46ea56bb": "\\tilde b_{ij}", "f4b3eaa10d2a4f2927fa283b21baf192": "F_r = \\int dF_r", "f4b418a4fc69ecaeada510aea3668e8a": "\\varrho(T_h) \\ge \\frac{a}{\\sqrt{\\# h}} \\ge \\frac{1}{\\sqrt{3\\cdot \\# h}}", "f4b45d9248f43614eacd0acf5936ff07": "b' = b \\times M_{3x3} \\times V_{3x3}", "f4b483750b9811dc88415f2437f6e381": "m^{-k}", "f4b4a5bcc4a2e6451e113f30c17105f9": "Q_{next} = J\\overline Q + \\overline KQ", "f4b5149f4fee5f79e436a149499f3c9f": "\\Psi\\propto\\begin{pmatrix}\n(1+\\gamma)r^{\\gamma-1}e^{-Cr}\\\\\n0\\\\\niZ\\alpha r^{\\gamma-1}e^{-Cr}z/r\\\\\niZ\\alpha r^{\\gamma-1}e^{-Cr}(x+iy)/r\n\\end{pmatrix}", "f4b51c0156031caaba90192b8ba8160a": "\\nabla^2 U = \\frac{\\partial^2 U}{\\partial x^2} + \\frac{\\partial^2 U}{\\partial y^2} + \\frac{\\partial^2 U}{\\partial z^2} > 0.", "f4b6203566287c24445fcc33dc412807": " \\scriptstyle x ", "f4b64c52d65e5a961280e4f5d1e758c8": "\\operatorname{core}(A) := \\left\\{x_0 \\in A: \\forall x \\in X, \\exists t_x > 0, \\forall t \\in [0,t_x], x_0 + tx \\in A\\right\\}.", "f4b6dfbf55f86489cf26395794f1c1dd": " {v_x - v_S \\over v_\\infty - v_S} = {T - T_S \\over T_\\infty - T_S} = {c_A - c_{AS} \\over c_{A\\infty} - c_{AS}}= 0", "f4b7ae419db62b8aa67d7357de073a2e": " u^{\\prime} = DF(x(t)) u, ", "f4b7ba31162be204fa60e20f6b46ce0d": "\\mathcal{K}_2(x; n) = 2x^2 - 2nx + {n\\choose 2}", "f4b80301357ad55866d2b0501cd5b139": "e_i^{(n)} = e_i^n/[n]_{q_i}!", "f4b8309a30efda9d2fe78f1b67ec618b": "T_{cut}", "f4b843bec7b539ee7240a6d3b225dafd": "f(x)=P(x)e^{-\\pi\\langle Ax,x\\rangle} ~,", "f4b865d828580c84d517b78f4ac2b617": "E_1(z)= \\int_1^\\infty \\frac{e^{-tz}}{t}\\, {\\rm d}t~", "f4b8b52e167798fef69b83294d1eb202": " a_x = 0 ", "f4b8fdf049a8a2c7bbe2f9c9adaad32b": " \\psi=1+ \\frac {\\sqrt{(\\mu \\eta S k)^2+m_s k v_r^2}-\\mu \\eta S k} {m_s g}", "f4b94b35f522952a0f224d3a59b8a80f": "M[f]", "f4b9ab020eedb97cbf837cffc89698c4": "f^u g h^v k ...", "f4b9bc144c8b1110f36ce9cc0ca8b2eb": "T_E \\ \\stackrel{\\mathrm{def}}{=}\\ {\\epsilon\\over k}\\,,", "f4ba26d742154f578836967efa7ba21e": "\\epsilon_{ij}=\\sum_{k\\ell}\\mu_{ijk\\ell}E_{k\\ell}", "f4ba5409ec05d4e849fa717830849f7e": "\\Pi^{1,Y}_n", "f4ba58a42cc5f3d02c5c0e392927bfef": "u_{xx} \\, = xu_{yy},", "f4bb2d45d2e5bba8471514d2bb30890a": "L':= L\\; \\cup_\\phi (D^{p+1}\\!\\!\\times\\!D^q).", "f4bb95e27d8505366199bb81f2b75b6f": "x_{j}", "f4bbc5de94ff106e66b3df43aaf70746": "|\\overline{X}_n -\\mu| > \\varepsilon", "f4bbd6dd3ba4864db9bd30ea708de0a1": "f\\left( x^\\prime ,t+\\varepsilon \\right)", "f4bc02c8c4af11826a3457cb85b15991": "\\begin{matrix} e = 1-\\frac{b}{a}\\end{matrix}", "f4bc1127c75b6e20adc4ca578ec4b1ee": "\\displaystyle{[L(a,b),L(c,d)]=L(L(a,b)c,d)-L(c,L(b,a)d).}", "f4bc2f4973c961d33723f0844c0903bc": "x_{n_1, n_2, \\dots, n_d}", "f4bc5e90498e20eeeecc7871353d3819": "t_{\\operatorname{ev}} = 5120 \\pi \\sqrt{\\frac{\\hbar G}{c^5}} \\;", "f4bcc635d8f1fbe01f6531acdf00b57a": "dX_t = dZ_t + \\frac{n-1}{2}\\frac{dt}{X_t}", "f4bccb5e8f42736cb2caf7adb2451683": " (xy)^3 \\rightarrow 1 ", "f4bcf386a301c3ba895e6703255c58a1": " C_t = \\{ x \\in C \\,:\\, f(x) \\le t\\} ", "f4bcf99422f5e8eeca42cd5a22ab5b8f": "\\|T\\|_{L^{p_\\theta} \\to L^{q_\\theta}} \\leq \\|T\\|^{1-\\theta}_{L^{p_0} \\to L^{q_0}} \\|T\\|^{\\theta}_{L^{p_1} \\to L^{q_1}}", "f4bd0b814cfea2abc7e1ee5073082f11": " \\Phi \\ll c^2 ", "f4bd1bc37202fb932a90fb8ee8f00968": "\\sum_{n = 1}^\\infty \\frac{(-1)^{\\lfloor \\frac{n-1}{2}\\rfloor}}{2n+1} = \\frac{1}{1} + \\frac{1}{3} - \\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9} + \\frac{1}{11} - \\cdots", "f4bd5112f2ad7f7b09f6847d1778c57c": "I_o = \\frac{\\left(V_i - V_o\\right)D T\\left(D + \\frac{V_i-V_o}{V_o}D\\right)}{2L}", "f4bd53ec8f881eb048cfc3fa46c25394": "g(x,y) = \\frac{1}{2\\pi \\sigma^2} \\cdot e^{-\\frac{x^2 + y^2}{2 \\sigma^2}}", "f4bd68a7991b8466f3662da70965dab0": " ms^{-2} ", "f4be414dfff719f9006b8cd19f7a42d8": " pK^- - pK^+ = \\Delta pK = \\log {\\frac{\\left[MOH\\right]^2}{\\left[MOH{_2^+}\\right]\\left[MO^-\\right]}} ", "f4be4c75b193673bae04bde06521339e": "v=\\sqrt{2gr\\,}.", "f4bebb82a985fde7114a4951aad608b3": "\\sqrt{\\frac{2}{3}} \\begin{bmatrix} 1&\\frac{-1}{2}&\\frac{-1}{2} \\\\\n0& \\frac{\\sqrt{3}}{2} & -\\frac{\\sqrt{3}}{2}\\\\\n\\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}} \\end{bmatrix}\n=\n\\begin{bmatrix}\\frac{\\sqrt{2}}{\\sqrt{3}}& 0& \\frac{-1}{\\sqrt{3}}\\\\\n0& 1& 0\\\\\n\\frac{1}{\\sqrt{3}}& 0& \\frac{\\sqrt{2}}{\\sqrt{3}}\\end{bmatrix}\n*\n\\begin{bmatrix}1&0&0\\\\\n0& \\frac{1}{\\sqrt{2}}& \\frac{-1}{\\sqrt{2}}\\\\\n0& \\frac{1}{\\sqrt{2}}& \\frac{1}{\\sqrt{2}}\\end{bmatrix}\n", "f4bed2053a54792019d18ba5ed33bb70": "\\Omega_{E,\\ell,N} = ", "f4bedc5b70507de7e06b034503ddf1e4": "\\begin{smallmatrix}\\mu = \\sqrt{ {\\mu_\\delta}^2 + {\\mu_\\alpha}^2 \\cdot \\cos^2 \\delta }\\ =\\ 327.78\\ \\text{mas/y} \\end{smallmatrix}", "f4bf2c0c0dbb179d159e67d731b338bb": "\\Delta F \\leq \\overline{W}, ", "f4bf3269eb8732223fc7bbbdcf0630d0": " \\mathrm{curl}\\,\\mathbf{A} = \\left( {\\partial A_z \\over {\\partial y} } - {\\partial A_y \\over {\\partial z} }, {\\partial A_x \\over {\\partial z} } - {\\partial A_z \\over {\\partial x} }, {\\partial A_y \\over {\\partial x} } - {\\partial A_x \\over {\\partial y} } \\right) ", "f4bfb8bce3b35cce0f053c18d1edd2b8": "u\\frac{\\partial v}{\\partial \\mathbf{X}} + v\\frac{\\partial u}{\\partial \\mathbf{X}} ", "f4bfe91601aab7893020dee2275c2f8a": "f(x)=f(a)+ \\int_a^x \\, f'(t) \\, dt.", "f4bfffb6eb705d3754f31843944f5c92": "\\{\\mathbf{i},\\mathbf{j},\\mathbf{k}\\}", "f4c02f0fd6b6a2881855c44629ec7706": "\\zeta(s)\\;", "f4c02f1c2e7584f9eef41552aac65d74": "{{}\\over{}}H_0={\\mathbf p}^2/2 - 1/r - \\omega L_z", "f4c053b3e10ed3cf5bfe99a20130e964": "\\alpha x_j", "f4c0e320bd6d83af6a59214acb3f58fe": "y=\\frac{r_1^2-r_3^2-x^2+(x-i)^2+j^2}{2j}=\\frac{r_1^2-r_3^2+i^2+j^2}{2j}-\\frac{i}{j}x.", "f4c12c4d2afb6cd3eb9ebb088bff51e0": " \\Delta r=\\sqrt{x^2 + y^2 \\ } ", "f4c14a97ba0f23db3c410843c597398a": "\\nabla \\times \\mathbf{A} =\\mathbf{v}. ", "f4c22c7c970bbb7d8c7808a1f1017a48": "l_a l^a=n_a n^a=m_a m^a=\\bar{m}_a \\bar{m}^a=0", "f4c2622fb06705d1f70f0b02c765b335": "\\zeta =\\int_0^1 \\varphi (\\zeta ,t)dt", "f4c26bb425e71aa0b6e3725363f30e96": "\\Psi_L=\\Psi_G\\,", "f4c27f34066d4dbccb5b0f56db8551cc": " \\ = 2^p", "f4c28ab7f685ddaa0c3a9bd20a112bec": " i \\ne j ", "f4c290973ba40f7449195f8e944e4e2f": " (1-X) \\sim \\textrm{Kumaraswamy}(1, a)\\,", "f4c2a8881b39da315fe81dfa8652f99f": "\\left( c - c_o \\right)^2\\, \\left( \\frac{\\partial^2 f}{\\partial c^2} \\right) > 2 \\kappa \\left(\\nabla c\\right)^2.", "f4c2afff6bad68c9f9ef34f4ec2d7427": "S\\in \\mathcal S", "f4c2deea4d88624b3dfb74061a774063": "10^{\\,\\!6.2 \\times 10^3}", "f4c2f2f6f4fa0ff51913f41c77c6eeb0": "\\tilde{\\eta}", "f4c30220537f02642ebf330054e602a9": "\\begin{array}{lcr}\nz & = & a \\\\\nf(x,y,z) & = & x + y + z\n\\end{array}", "f4c3b8ea134b9e57b4713207370a5d95": "= 2 \\arctan \\frac{1}{5} + 2 \\arctan \\frac{1}{5}", "f4c3cc1c0f089838508bea9dfdcc37d1": "\\vec k\\|\\vec B_0", "f4c45a986af0000e404a3add67710dc4": "\\theta(t)=c_{1}\\cos\\omega t+c_{2}\\sin\\omega t", "f4c4990ab6aa93b617f9d5ae428978f8": "K=\\sqrt{abcd-(eg-fh)^2}.", "f4c513d8398ece7dc587e4faa95ee4c9": "p(z_1,z_2,\\ldots,z_d) = \\prod_{k=1}^d \\eta_k e^{-\\eta_k z_k}", "f4c53195a318da1d253a08dc3df458b8": "\\frac{55,400\\ \\mathrm{N}}{(138\\ \\mathrm{kg})(9.807\\ \\mathrm{m/s^2})}=40.9", "f4c553d13da6570ba57e1f21a0b67f4b": "f_R", "f4c56ba2bb2adf375a5b47ee08c40fbf": " u_\\lambda(p) \\,", "f4c578c468e5837b9d9998c9280fac1d": "\\mathrm{s^{-1}}\\,", "f4c5e29416dd310a613a9175d93b1cac": " (G\\times SL(m-n), V^*\\otimes \\mathbb F^{m-n})", "f4c60ca6d5235c38a1724c24c71cb57e": "1/4 + 1/16 + 1/64", "f4c66767dcdd123679a44267e5534ee7": "w \\in x ", "f4c6b4ecd4369f245d6ebce1e586aa38": "e = \\displaystyle\\sum\\limits_{n = 0}^{ \\infty} \\dfrac{1}{n!} = 1 + \\frac{1}{1} + \\frac{1}{1\\cdot 2} + \\frac{1}{1\\cdot 2\\cdot 3} + \\cdots", "f4c6d4a121d376f0128a6811d08581d0": "(\\alpha,\\varphi,\\lambda)", "f4c6e21fd21d28e6926b26de2d6b3cf5": "F_{y|x} = F_{y|\\eta^Tx},", "f4c7114d703df74f3c6d3a1eb3fe0a84": "(\\Phi,\\alpha)", "f4c773ffbe0e62197b0db7ecba34b8e1": "d = 10^{\\frac{\\mu}{5}+1} ", "f4c797edbd04bda482b4d8cb463b629d": "m\\in\\mathbb{Z}", "f4c7f4b65a27a6e885b720b2bfb2ed2e": "\\frac{\\partial^2 \\epsilon_x}{\\partial z^2} + \\frac{\\partial^2 \\epsilon_z}{\\partial x^2} = 2 \\frac{\\partial^2 \\epsilon_{zx}}{\\partial z \\partial x}\\,\\!", "f4c7fb3e6e7e6b560ffa8330c0683651": "P=1.01\\cdot 10^{7}[W]", "f4c838f1f93e5353411d3e03a137670d": "I(t)\\cos(\\omega t) - Q(t)\\sin(\\omega t) = \\mathrm{Re}\\{Z(t)e^{j\\omega t}\\}\\,", "f4c85ddb61093e57fc438ae652639cef": " {C \\over i} ", "f4c8eaef8ae03f893ffdc0bd66838ad2": "\\kappa \\approx2", "f4c8f9dbb4ec98c8e02642fb4426946d": "\n \\mathbf{w} = \\tfrac{1}{2}~\\boldsymbol{\\nabla}\\times\\mathbf{u}\n ", "f4c90af497a1f76371115996cccda13a": "d \\ne b", "f4c90b530c989b94d43934c54c9d16fb": "\\text{signal} = \\mu_\\text{sig} - \\mu_\\text{bkg}", "f4c923312e772f91690263ee8c4f6769": "\\frac{d_2}{d_2-2}\\!", "f4c92d241ca269cde24b05fccb0a69d7": "\\prod _x x = C\\, \\Gamma (x) \\,", "f4c93a28276604c18ea46aaefe9d9233": "{P}^{5}-4{P}^{3}Q+3P{Q}^{2}\\, ", "f4ca23de29f5987ee30e359d6f60a8cf": "\\sec\\alpha-1", "f4ca386b9388331cb54b2d642ca348ce": " K = m + 1 \\,", "f4ca56eaba1e2ffc0526dc9e51f092dc": "\\mathbf{J} = \\mathbf{J_1}+\\mathbf{J_2}", "f4ca8e8eee03a9764cee28d1f51b41a3": " L(C)\\rho=C\\rho C^\\dagger -\\frac{1}{2}\\left(C^\\dagger C \\rho +\\rho C^\\dagger C\\right) ", "f4caaacadbff3702331bfa48f80d8c5f": "\\mathbb{E}_Y [f(Y)] = \\int_y f(y) p_Y(y) \\, \\operatorname{d}\\!y", "f4cb1ae52757bd9010a46d73042888f7": "P_p = \\{ X\\in T_pS^{2n+1} : IX \\in T_pS^{2n+1}\\subset T_p\\mathbb{C}^{n+1}\\},", "f4cb6c9fc0af56a5a7251e5439228ecd": "\\langle x_0,x_1,\\ldots,x_{n-1}\\rangle \\in T", "f4cb7f7cbc25f7c30ab42413df649b42": "\\displaystyle{U(g)f(x)=f(g^{-1}x).}", "f4cb8f4242b52357c7643ebc017fbd57": "g \\colon [0, \\infty) \\to \\R", "f4cc3aa78ece1cb52ee424c03df92e47": "|5| > |1| + |-2|", "f4cc99eeb891283f3f54c1de3c30a5ef": "\\frac{1}{50}\\mbox{ s} \\times 10,000\\mbox{ pixels} = 200\\mbox{ s}", "f4cca8e90f4cfabdf807b0258d1e59af": "{\\tilde{C}}_7", "f4cd171a58d44c7d998a87d50c70df43": "T_j=T/p_j", "f4ce63bf9584e3db6454b4083caa26c4": "\\langle 1/p \\rangle \\subset \\langle 1/p^2 \\rangle \\subset \\langle 1/p^3 \\rangle \\subset \\cdots", "f4ce702ecb755e36cd4e310311ab4f88": "\\implies U^'_i(x_i)(\\frac{1}{p}-\\frac{1}{p}*\\frac{x_i}{B})-1=0", "f4ceb205cdf4976b6dd687eea12947a7": "n/n^{\\ominus}", "f4ceb40bdef7a8fc7e75d706a40e0e4e": "w_A = \\frac{w_0}{1+(1-\\beta_A)w_0}", "f4cee7c36695ccd313c214e104301df8": "r^* = r + 2GM\\ln\\left|\\frac{r}{2GM} - 1\\right|.", "f4cefe4c3c7569b137bab228a87fac92": "x_{i}\\left( y,\\xi\\right) =x_{i}\\left( y,0\\right) +\\delta_{i},", "f4cf065af7557eda6c7326b136411738": "\\text{PointsPlus} = \\max \\left\\{ \\mathrm{round} \\left( \\frac{(16 \\cdot \\text{protein}) + (19 \\cdot \\text{carbohydrates}) + (45 \\cdot \\text{fat}) - (14 \\cdot \\text{fiber})}{175}\\right) , 0 \\right\\} ", "f4cf355d80e92e8d54a7bd462d4fcadf": "d_j = \\frac{1}{3}\\left(\\frac{6}{6} + \\frac{6}{6} + \\frac{6-1}{6}\\right) = 0.944", "f4d0286dfbc9e94af2fdc665b1724260": "\\sum_{i=1}^n 2s_i (M-s_i)", "f4d02af68fbb509acd293377e1e26c05": "(\\neg A \\rightarrow \\exists x\\;P(x)) \\rightarrow \\exists x\\;(\\neg A \\rightarrow P(x))", "f4d03754382030dd726c8193a9a5a53e": "\\langle u(a), b \\rangle = \\langle a, u^*(b) \\rangle", "f4d0770df0bfb5f3fa6110f5961cf317": "l^\\theta = l^p.", "f4d094ac07681a569dd5eb8468984cbf": "a(z)u(z) - b(z)v(z) = c(z) \\!", "f4d0a8cb4386b8ff9304a093dc0017ad": "\\Pi_1", "f4d0d3bc1fd254ef78e80001655efe71": "\\neg P(x) \\vee Q(x)", "f4d115f34c3b0d54137dcffbf5d271a2": "\n \\boldsymbol{D}^p = \\dot{\\lambda}\\,\\frac{\\partial f}{\\partial \\boldsymbol{M}} \\,.\n ", "f4d11f86a25d799672ad8e30e9a89f7e": "\\Delta(h) = h_{(1)} \\otimes h_{(2)}", "f4d14e5d0e85acd2f9b0ce7b00b631fa": "A\\rightarrow B C", "f4d1a851acf85125161ef99f31d1c833": "Q_2(z, v)", "f4d1e758d0364477640ad4c35ca97fde": "(q+1)-\\frac{q}{d}", "f4d1efa6625eeece792ba80a372182a6": "(r,0)", "f4d1f0ab19c49d31c63756ab4557aa0d": " \\langle p \\rangle = -\\frac{\\partial \\Omega} {\\partial V}, ", "f4d212b8349006ab2474c8727fe86a3c": "\\mathrm i", "f4d23b30c34a72e9002c2472c266b7c2": "f(z)=z^2 + c", "f4d24b2bce866efe763ab7bf3f4e3ae0": " e_0 - e_2 + e_4 - \\cdots = \\frac{\\prod_i \\sec\\theta_i}{\\sec\\left(\\sum_i \\theta_i\\right)} ", "f4d252f69bbf2517516a1e3a05e8edfd": " e^{-t/\\tau_c}", "f4d2b5d98427dafdfa25398cd09c5bbb": "\\mathrm{2VF_5 + 10NaOH \\ \\xrightarrow{}\\ V_2O_5\\downarrow + 10NaF + 5H_2O }", "f4d2e2d6a918687c854f8624d969aae7": "r = ln (1+i) ", "f4d30679200efdfe78b83e65a5d3bdbe": "I \\approx s_0-S", "f4d39edf84bd63b786d7a2096d1f2215": "\\operatorname{lcm}(a,b) = \\prod_p p^{\\max(a_p, b_p)}.\\;", "f4d3aa2d43782b8a4c08acc10247bf43": "e^{X} e^{Y} e^{-X} = e^{\\exp (s) ~Y} ~.", "f4d41e6fa2a7bda044edad0901cf7c67": "\nf(t)=\\frac{1}{2\\pi C_{\\psi}}\\int\\limits_{-\\infty}^{+\\infty}\\int\\limits_{0}^{+\\infty}W_{f}^{\\alpha}(a,b)\\psi_{\\alpha,a,b}(t)\\frac{da}{a^2}db\n", "f4d42bf76a25e85b60a883791d9fc9ab": "\\mathrm{supp} (\\mu) := \\mathrm{supp} (\\mu^{+}) \\cup \\mathrm{supp} (\\mu^{-}).", "f4d4491daddb7bc2906a548db9697adf": "-\\alpha^n = (-1) \\cdot \\alpha^n = \\alpha^e \\cdot \\alpha^n = \\alpha^{e+n}", "f4d4567ad1f31ffda4da1089d82320f6": "c \\in \\mathrm{R}", "f4d4a59cdf7d27d78183ef8b95768b88": "f_{A,B,C,D} = BC'D' + AB' + AC \\ ", "f4d4ee62e99e1a9bebed0904e42cc9c8": " 0 = 7 \\times 0", "f4d55e3f170d84b585688d2af7aa0652": "H_A", "f4d579459a1c94c0752c80739ef39c77": "S^{\\nu}", "f4d5ba747e8c50ccc351ade3fd87a98c": "\\mathfrak{g} \\to U(\\mathfrak{g})", "f4d5f5019d90426c225e3661dd4e61f3": "\nPoss(pickup(o),s)\\leftrightarrow(\\forall z\\neg is\\_ carrying(z,s))\\wedge\\neg heavy(o)\n", "f4d65ebad89c4988c8cac20b1b6e9bba": "\\int_0^{x_d} e^{-\\Phi^* / V_t}dx = x_d \\frac{\\Phi_i - V_a}{V_t}", "f4d6a4f25fe43cc1abd09a7324007f10": "C\\in D", "f4d6ee3869cc1529376bc486fd487016": "1/R_n(\\xi,x)", "f4d6ef93bc831239af5ef522cf83f416": "f''(x)=-\\frac1{x^2}+\\frac1{(1-x)^2}<0,\\qquad x\\in(0,\\tfrac12).", "f4d6f3312b7da9bfbb0fff168d22f527": "\\operatorname{var}(X)= \\frac{\\beta}{\\lambda (\\alpha-1)} ,\\quad \n\\operatorname{var}(\\Tau)=\\alpha \\beta^{-2} ", "f4d741e0094abff7e4f89592e7e9bf52": "Av = \\lambda v^*.\\,", "f4d79be3a6f2580e576387197118feb6": "\\Omega_{il}", "f4d7e233ef95a326bbdf5984bcb02176": " \\langle \\bold{r} \\rangle = \\int \\bold{r} |\\psi|^2 d^3 \\bold{r} ", "f4d84be07e85364f5e21f6e8795d825c": "\\mbox{H}_2 \\mbox{O}_2 (l) \\rightarrow \\; \\mbox{H}_2\\mbox{O} (l) + \\frac{1}{2}\\mbox{O}_2 (g)", "f4d88e3a2d78b928a456b961279525f1": "s_J^2", "f4d954dcdf7fc19a90aa39a61e9e33b6": "\\bar{X}-\\bar{M}", "f4d98e472f8cabf179f14da299f1cde1": "\\Lambda^k(f) = \\Lambda(f)_{\\Lambda^k(V)} : \\Lambda^k(V) \\rightarrow \\Lambda^k(W).", "f4da03dea3066a219bbcc7843419bf2e": " H_f \\notin \\mathbf{TIME}(f( \\left\\lfloor \\tfrac{m}{2} \\right\\rfloor )) ", "f4da385f4d4c78f6a2b96811757b218c": "u'= \\frac{f - vh -v^2}{u}", "f4da4863152ea1b500d64d859c6823dc": "4 ^ x\\,", "f4da4a2d68e7a869c0ffe600afbabfd9": " \\mathbf{y}_{k} \\in \\mathbb{R}^{3} ", "f4da4ec7da5b252af518cb448734ee91": "\\bar f(\\boldsymbol{u}(\\boldsymbol{x}))=\\int_0^1 f\\left(\\boldsymbol{u}_{\\boldsymbol{\\hat a}}(\\boldsymbol{x})t + \\boldsymbol{u}_{-\\boldsymbol{\\hat a}}(\\boldsymbol{x})(1-t)\\right)dt", "f4dba6baba9b9a0e79f2525bd27dd609": "\\Pi=-(RT/V)\\ln(x_s) ", "f4dbfb50e0c32b4eb40788c87bd35636": "R_\\mathrm{internal}(\\hat{n},\\phi) = \\exp(-i \\phi S_{\\hat{n}}/\\hbar),", "f4dc561a77cb26ca290a7c1d90be16a3": "Head^*(X) = (Head^+(X) \\cup \\{ X \\}) \\cap V_t", "f4dc6693ca96db43253d3f557e768ed0": "\\scriptstyle\\Delta_{uv} = 5 \\times 10^{-2}", "f4dc6a274b9d0d35aec11ec8e08f292c": "C_1;C_2", "f4dc70aefcc709108aba48f8c89dbf32": "\\alpha = \\frac{R_{100} - R_0}{100R_0}", "f4dc785d0a34640335147014434391f2": " \\begin{align} \\mathbf{F} & = \\frac{d\\mathbf{p}}{dt} = \\frac{d(m\\mathbf{v})}{dt} \\\\\n& = m\\mathbf{a} + \\mathbf{v}\\frac{{\\rm d}m}{{\\rm d}t} \\\\\n\\end{align} \\,\\!", "f4dcb37032599201954be0f647ff486f": "\\upsilon_o =| \\upsilon_s - \\upsilon_r | = | (\\upsilon_i+\\upsilon_D)-(\\upsilon_i+\\upsilon_M ) |\\qquad(2)", "f4dce35a9950552783f00d32bbfc7888": "\n \\mu~\\nabla^2 u_3 + b_3(x_1, x_2) = 0\n ", "f4dd72bf667c5e8ba4182844cfc0e802": " \\mathcal{FS}", "f4ddf4f5bbf7baed535fbcb1986d9ee2": "\\begin{align}\n\\Delta t' & =t'_{1}-t'_{0}=\\frac{t_{1}-vB/c^{2}}{\\sqrt{1-v^{2}/c^{2}}}-\\frac{t_{0}-vA/c^{2}}{\\sqrt{1-v^{2}/c^{2}}}\\\\\n & =\\frac{1-av/c^{2}}{\\sqrt{1-v^{2}/c^{2}}}\\Delta t.\n\\end{align}", "f4de2ef4abf06dfd3367d4eb053f8186": "k(m, P) = (km, P)", "f4de3c4df62c5e5c8986ae92948dc5c5": "\\omega=\\frac{|\\boldsymbol{r}\\times\\boldsymbol{v}|}{|\\boldsymbol{r}|^2}.", "f4de4e791eaade1096a03e3a4d45a3d6": "d(x,y)=\\sum_{n=1}^\\infty \\frac{1}{2^n} \\frac{p_n(x-y)}{1+p_n(x-y)}", "f4de56e747ecfa7baab2570354a94a36": "L(M^n)-nr \\,\\!", "f4dead4df1d478c35438afc9472ceb09": "K^\\rho=\\{ \\sigma^i\\in K|\\rho (\\sigma^i)\\le \\rho \\}", "f4deb05c81d86c88747610f8d3f8e379": "1\\le i \\le 30", "f4df0df4d8621a755911ccabcd2a39cb": "\\delta = \\left( L/\\alpha \\right)= \\left( \\frac{L}{\\sqrt{\\mathrm{Stk}}}\\right), ", "f4df5b2667323766c55af76daa58a6a9": " - \\omega ", "f4df7a0b2b87868c16d896e19c473fc6": "x_1,\\ldots,x_n\\sim F", "f4dfad2dab61ac96d34fec047c2790c5": "\\mathcal{P}_f(\\mathbb{N})", "f4e03bf65f2f10003f2b70fad12988ca": " \\begin{array}{llll}\n|\\mathbf{A}| & = A_{11} \\mathbf{i}\\cdot\\mathbf{i} + A_{12} \\mathbf{i}\\cdot\\mathbf{j} + A_{31} \\mathbf{i}\\cdot\\mathbf{k} \\\\\n& + A_{21} \\mathbf{j}\\cdot\\mathbf{i} + A_{22} \\mathbf{j}\\cdot\\mathbf{j} + A_{23} \\mathbf{j}\\cdot\\mathbf{k}\\\\\n& + A_{31} \\mathbf{k}\\cdot\\mathbf{i} + A_{32} \\mathbf{k}\\cdot\\mathbf{j} + A_{33} \\mathbf{k}\\cdot\\mathbf{k} \\\\\n\\\\\n& = A_{11} + A_{22} + A_{33} \\\\\n\\end{array}", "f4e03c4db12a4f84ebddb5f19555f9da": "S_1 = 2\\pi V_0", "f4e0441e7f285c880f2a95757b3ba04f": " a \\equiv { {d u} \\over {d\\tau} } = { d \\over {d\\tau} } { \\left ( \\gamma , \\gamma { \\mathbf{v} \\over c } \\right )} = { \\left ( 0 , \\gamma^2 { \\mathbf{a} \\over c^2 } \\right )} = { \\left ( 0 , - \\gamma^2 { { \\mathbf{v} \\cdot \\mathbf{v} } \\over c^2 } { {\\mathbf{r} } \\over r^2 } \\right )} ", "f4e123621aa66be8175f9939753ae965": " Z(A_n) = \n\\sum_{j_1+2 j_2 + 3 j_3 + \\cdots + n j_n = n}\n\\frac{1 + (-1)^{j_2+j_4+\\cdots}}{\\prod_{k=1}^n k^{j_k} j_k!} \\prod_{k=1}^n a_k^{j_k}.\n", "f4e17d40a4697d1aa5e552323abdb068": " f'(x) = \\mathrm{st} \\left( \\frac{f^*(x+\\epsilon)-f^*(x)}{\\epsilon} \\right),", "f4e1d60524bcddc98d8fe32231788b8c": "\nP_{ij} = m{d\\over dt} X_{ij} = im (E_i - E_j) X_{ij}\n\\,", "f4e2517e301f71d15208c06217e17547": "0 < a_n - \\frac{1}{\\pi} < 16\\cdot 5^n\\cdot e^{-5^n}\\pi\\,\\!", "f4e34cf98a414c0a4caaf8d1edda1b93": "\n\\text{Corr}_r(X, Y) = \\text{Corr}_r(Y, X) = \\text{Corr}_r(X, bY) \\neq \\text{Corr}_r(X, a + b Y), \\quad a \\neq 0, b > 0.\n", "f4e3791799019936201874bfde8ae990": "log(T_0)", "f4e37a483062e242e658d371548d4fea": "<^*", "f4e4d5f26d176847283a812b7737fea1": "\\dot{x}=f(x)", "f4e4dc582ed4c4d7f26c042d737fbb31": "\\frac{d^{2}x}{dt^{2}} + \\omega_{0}^{2} x = 0", "f4e4ebb56fce04c9ff8bcc11af3a1cd6": "\\lim_{x \\to -\\infty} \\int_x^b f(t) \\; dt ", "f4e4fedb122eb31299cf7181075165f3": "c_{n,i}\\lambda^{(0)}_{n,i} + \\epsilon \\sum_k c_{n,k}\n\\int f^{(0)}_{n,i}(x) D^{(1)} f^{(0)}_{n,k} (x)\\,dx = \\lambda c_{n,i}", "f4e51d7ae53bdf98602c9f242d998f12": " \\Lambda_{Landau} = m \\exp\\left\\{ \\frac{1}{\\beta_2 g_{obs}} \\right\\} \\, .\n \\qquad\\qquad\\qquad (3) ", "f4e526372ce70ffe81189ec2549330e5": "\\phi_{2}", "f4e55e5d1ad3d612d41da32be927be41": "AP^{-1}", "f4e560370e50bbd9015e71827e5ede79": "\\Phi = \\iint I_\\nu \\mathrm{d} \\nu \\mathrm{d} \\Omega ", "f4e5bbf9b66dcc26958b0d7f7459d5ab": "\\frac{p}{r}\\ =\\ 1\\ +\\ e\\ \\cos (u-\\omega)\\ =\\ 1\\ +\\ e\\ \\cos u\\ \\cos\\omega\\ +\\ e\\ \\sin u\\ \\sin\\omega\\,", "f4e60a9dcd8e8f079c1451ba86aa8b6a": "\\rho=-\\frac{1}{r}\\,,\\quad \\mu=\\frac{-r+2M(u)}{2r^2}\\,,\\quad \\alpha=-\\beta=\\frac{-\\sqrt{2}\\cot\\theta}{4r}\\,,\\quad \\gamma=\\frac{M(u)}{2r^2}\\,.", "f4e633073797cae42c11cc13ea489461": "\\limsup_{u\\to 0^{+}}\\frac{\\varphi(\\frac{u}{2})}{\\varphi(u)}< \\frac{1}{2} ", "f4e66f9f7483438221c7172032a7d960": " (A \\or B \\or C )", "f4e68c2a7979f6680bef6615a0976094": "g(t|x)= \\frac{f(x|t)g(t)}{f(x)} ,", "f4e7109e44c4e1a00e73e009b9263fa6": "S(\\Lambda)", "f4e726ac8c73b7930c9d80779fbbf68a": "\\Omega^{(d+2)}=d\\Omega^{(d+1)}", "f4e743bd07fb700f81a6c285c70b8c00": " b^{\\prime\\prime\\prime} = \\frac{b^{\\prime\\prime} \\, b^{\\prime}}{b}, \\; \\left( a^\\prime \\right)^2 = 2 \\, b^{\\prime\\prime} \\, b", "f4e79fb722ba845dc150a99fe07b075f": "{\\rm GF}(q)^*", "f4e7a0774a4e415abd3c398145c9231e": "1i_1+1i_2=d", "f4e7a6fb56643fd05474af712f448ba2": "m_1\\neq m_2", "f4e7dfdf04ed95b8db166c353d6dfde6": "P(\\mathbf{x}|\\mathbf{p})", "f4e7e77ce8c0caa1bb904ffd7327dbf1": "M=\\begin{bmatrix}m_1&0\\\\0 & m_2\\end{bmatrix}", "f4e8274a73be4a70d677fcaf59fc7999": "x\\Delta,\\,x\\in\\,S", "f4e8349d89687e9efd80e2ef7f0b460f": " K(0,iw)= \\int_{0}^T \\gamma (s) exp(-\\phi(s,iw))(1-K^2(s,iw))ds \\quad(2.12)", "f4e85fb62aca95406b3d3bb84a13f620": "L_n^{(\\alpha)}(x)=\\frac{1}{2\\pi i}\\oint\\frac{e^{-\\frac{x t}{1-t}}}{(1-t)^{\\alpha+1}\\,t^{n+1}} \\; dt ~,", "f4e89b14ce72dbfe488f17de2ef35494": "T_nI(x)=-x+n \\pmod{12}\\,", "f4e8aa3a70e20302bf14d5f325c5d5cd": "x_0, ..., x_{n+1}", "f4e8b7fa9660ce6751c5a0acff7d37c0": " \\begin{align} \\bold{P} & = \\bold{p}_1 + \\bold{p}_2 \\\\\n& = (m_1 + m_2)\\bold{V} \\\\\n& = M\\bold{V}\n\\end{align}\\,\\! ", "f4e8f713afee766a3d9720c67b7024a9": "\\Phi(\\alpha) = {\\mathcal N}(\\vert \\alpha\\vert^2)^{-\\frac 12}f(\\alpha)", "f4e8f72b348a171b44826e470b6f5e84": "\\lceil x \\rceil - \\lfloor x \\rfloor = \\begin{cases}\n0&\\mbox{ if } x\\in \\mathbb{Z}\\\\\n1&\\mbox{ if } x\\not\\in \\mathbb{Z}\n\\end{cases}", "f4e9140522cd9470eb1b55f82590304f": "Y_3 = C(F-X_3)-8E", "f4e93dcb0efe21e4fa4b3b8588ed5324": "p^{-1}(Spec(K))", "f4e9578632d17a9912e372bb7c1c6b85": "\nV_{\\infty} = A \\tau. ", "f4e96829ae583c1b9f5feb555bd5159f": "P_1=\\frac{6}{6+6}", "f4e99b6c9378c97eff2e7f654e9ae48a": "U_{\\bold{G}} = \\frac{4 \\pi Z e}{q^2 + \\bold{G}^2}", "f4e9aad9570e7d54404ef05258408752": " \\rho\\, _{liquid} ", "f4e9fa968ae828b774fc6e2ad17529d2": " \\mathbf{v}= \\left( \\frac{-y}{x^2+y^2}, \\frac{x}{x^2+y^2}, 0 \\right). ", "f4ea9220940a1a7b8c940c474b16c20b": "a_1, a_2, a_3, \\ldots", "f4eacb1a21c3fd2b002e16daf733ea21": "\\tbinom mr", "f4eb1a7b0b2eb7b6bb46b974cac6b64a": "Animal(", "f4eb35f30d780bd528b2a735806da195": "\\lang \\psi, V(R) \\psi \\rang", "f4eb45770c4460c0e40191ff79657074": "\\sec A=\\frac{1}{\\cos A}=\\frac{c}{b} ,", "f4eb4c980ccb2cfbb706bbc21defac90": "\\underline{\\lnot \\varphi}\\,\\!", "f4eb7708956c11b746d67ff666dedde9": "\n D_{\\alpha\\beta} := \\int_{-h}^h x_3^2~C_{\\alpha\\beta}~dx_3\n", "f4eb7a6ffc294050924b15a3da2f3be5": "S_2(t)", "f4eb7e427827601e136010932ed9c311": "x^2-ny^2=\\pm 1\\,", "f4eb7e47f0789399461a30126107ca41": "\\scriptstyle U = 0", "f4eb87d0a584913797782f8f209b3a11": "A=\\{x_1,x_2,\\dots,x_n\\}.", "f4eb8c8a9f9af6dcc15c2ea3cc208f03": "n^T", "f4ebfc1842f559864efd267e02bd4f00": "R^n \\equiv 0 \\mod n", "f4ec3fe250a9c720553618406150f424": "Z_\\mathrm{eq} = Z_1 + Z_2 + \\,\\cdots\\, + Z_n.", "f4ec8fa57f703370181ecef23eb057e3": " \\nabla \\times \\mathbf{B} = \\frac{ \\partial \\mathbf{E}} {\\partial t} + \\mathbf{J} \\,", "f4eca74c88e81197641fc1ab42acb1b9": "f(x)=\\sum_{y\\sim x} f(y)", "f4ed9df9858febacbd334ad5caee2a09": "\\mathrm{vol}_n := \\sqrt{|g|} \\;dx^1\\wedge \\ldots \\wedge dx^n", "f4edd817015b177e58364a3912602df7": "1 - \\frac{ \\cos^{-1}( \\text{similarity} )}{ \\pi} ", "f4ee0d83e22ca2a79c5af9a833354cc5": "\\ P_r", "f4ee18448ffa1e93b2969c9c594362cc": "P(h_i)", "f4ee70e4d8c61a74342b314ed354aa60": "c_{1}(t') = \\delta\\left(f-\\dfrac{E_{0}-E_{1}}{h}\\right)\\otimes\\mathrm{TF}\\left[H\\left(\\frac{t}{t'}-\\dfrac{1}{2}\\right)\\right]", "f4ee73789a3decb0fe2b24c098dacb8a": "\\frac {\\partial L}{\\partial \\dot \\theta} = \\mu r^2 \\dot \\theta = \\mathrm{constant} = \\ell, \\, ", "f4ee9b37089540e2ebd53f50e624f9f2": "\\mathbb Z[\\sqrt{-5}]", "f4eea1c5b234610317bb66b2eb7d1917": "\\sigma\\bar{\\sigma}", "f4eea9d83dee3f8c5a0651bce7bf53e0": "1=|1|_{\\ast}\\leq |m|_{\\ast}|p|_\\ast^{e}+|n|_{\\ast}|q|_{\\ast}^{e}<\\frac{|m|_{\\ast}+|n|_{\\ast}}{2}\\leq 1", "f4eecfef695d463fbec81012288c4062": "O_p(M)\\not= 1", "f4eed071e4a375bf7ebd6e5d5f191485": "\\nabla^2 f={1 \\over r^2}{\\partial \\over \\partial r}\\!\\left(r^2 {\\partial f \\over \\partial r}\\right)\n \\! + \\!{1 \\over r^2\\!\\sin\\theta}{\\partial \\over \\partial \\theta}\\!\\left(\\sin\\theta {\\partial f \\over \\partial \\theta}\\right)\n \\! + \\!{1 \\over r^2\\!\\sin^2\\theta}{\\partial^2 f \\over \\partial \\varphi^2}\n= \\left(\\frac{\\partial^2}{\\partial r^2} + \\frac{2}{r} \\frac{\\partial}{\\partial r}\\right)f \\! +\n{1 \\over r^2\\!\\sin\\theta}{\\partial \\over \\partial \\theta}\\!\\left(\\sin\\theta \\frac{\\partial}{\\partial \\theta}\\right)f + \\frac{1}{r^2\\!\\sin^2\\theta}\\frac{\\partial^2}{\\partial \\varphi^2}f.", "f4eef724f24dbce5ec91d44fccdd61b7": "a_m = \\frac{1}{\\pi} \\int_0^{2 \\pi} f(t) \\cos (mt) \\, dt", "f4ef1fce69b321e2177b70a88007fe43": "S = N k \\ln \\left(V\\right) + N f(T)", "f4ef73cdf60a4395a15703a8519cbfc7": "a_1 \\leqslant a_0 < b_1", "f4efbd8bf6d3bbb1666ca567550d66b7": "\\left[\\widehat{p}_i,\\widehat{p}_j,\\right] \\psi(\\mathbf{r},t) = 0 ", "f4eff7314d49fd051ef3592de3308687": "\\{ v_1, \\ldots, v_m \\}", "f4efffbda1ee3444e0b86983485b5382": "\\begin{align}\n G_L &\\to 0\\\\\n G_R &\\to 1\n\\end{align}", "f4f033dd6151f4144957a25621dca543": "xuvy", "f4f0b3792692251d4dfbc05c16f4872e": "c = \\sqrt{a^2+b^2-2ab\\cos\\gamma}.", "f4f0df8292bc4ae879aa3b7013d79da1": "\\mu = \\sum_n n \\cdot (n|m) = \\frac{m - 1}1\\sum_{n=m}^\\infty \\frac{1}{n - 1} = \\infty", "f4f104235c0bca90b59307a45894a14c": " [1,2]x=[1,4] ", "f4f10b5ba27311b0eb52b0cfd7364b4d": "XY=1", "f4f1244502d6995138fbc569a1a6fd01": "\\displaystyle{h^{-2}(\\mathbf{v}(t+h)-\\mathbf{v}(t))\\cdot\\mathbf{n}(t)=\\kappa(t)/2 +h^2 \\dot{\\kappa}(t)/24 +\\cdots}", "f4f13341bfc3900cd5a4201da1fc0b6c": "dW_f = -dW", "f4f162a7a4bd6d98140cc1847fc01e57": "\\sum_{j\\in T}x_{ij}=1\\text{ for }i\\in A, \\, ", "f4f1e0ccfe4a84494f26ee5b7b6fb762": "p_b = p^\\prime_b/w^\\prime = \\begin{bmatrix} x_b\\\\y_b\\\\ 1\\end{bmatrix}", "f4f21ace684efc67188ff35a35748850": "BC(P,Q)", "f4f24e6a04b16d795652f59dd271a859": "\\tau(t) = \\frac{c}{g} \\ln \\left( \\frac{gt}{c} + \\sqrt{ 1 + \\left( \\frac{gt}{c} \\right)^2 } \\right) = \\frac{c}{g} \\operatorname {arsinh} \\left( \\frac{gt}{c} \\right) .", "f4f253786769ee295a2bc819ef94dd93": "\\omega_2-\\omega_3", "f4f257c41d49c4caef6c4973f70409c0": " \\{ f,g \\} = - \\{ g,f \\} ", "f4f25b36214db16f0eb0fb474acf2459": "\nx_0 = - \\int\\limits_{-\\infty}^{0} \\widetilde{\\theta}(x) dx + \\int\\limits_{0}^{\\infty} \\widetilde{\\varphi}(x) dx,\n", "f4f25db1d4d334fc5990a25a6207ff21": " \\sigma_\\mu = (\\sigma_0,\\sigma_1,\\sigma_2,\\sigma_3)= (I_2,\\sigma_x,\\sigma_y,\\sigma_z)", "f4f26df248064e94f24ad2a760da9382": " \\Rho_A = \\inf \\{ r \\geq 1 \\mid R_A(x) \\leq r, \\forall x \\}.", "f4f2932732309874ec3bb09b68c593ab": "f(x)\\ =\\ x^2", "f4f2a07abfc59e60d8ec216253595f87": "\\scriptstyle x \\;=\\; \\pi", "f4f2df718296e61fb826cbcdc815d2e9": "=\\vec{P}t-M\\vec{x}_{CM}", "f4f344795763efb09af81566c033bb2f": " x(t)=u(t)+\\epsilon(t)", "f4f356a05e64aff14b3d0d6bad0a0e55": "\\mathfrak{P}(\\{d\\})", "f4f39292a5c582a491ffe77ed1ca7ee5": "\n\\begin{align}\n\\prod_{s\\ni e} \\big(1-\\min(\\lambda x^*_s,1) \\big)\n& < \\prod_{s\\ni e} \\exp({-}\\lambda x^*_s)\n= \\exp\\Big({-}\\lambda \\sum_{s\\ni e} x^*_s \\Big)\n\\\\ \n& \\le \\exp({-}\\lambda)\n= 1/(2|\\mathcal U|).\n\\end{align}\n", "f4f41531ab067ea0b3ae1bef9ad08216": "\\varphi=\\theta={\\pi\\over 4}\\!", "f4f44104c3fb029abab3eb23737ea8b2": "\\mathrm{Sh} = \\frac{K L}{D} ", "f4f45b77655d530a347814f14a228083": "\\hat{w}_2", "f4f46169126b6808eaef19e2bd199dd1": "\\textstyle \\Delta \\lambda = \\frac{h}{m_{\\mathrm{e}}c} (1 - \\cos \\theta),", "f4f49c6048602e38eb481a2ac0b25853": " a_{RWG} ", "f4f4ecf1c789d58d6d8894509dc5e647": "\\operatorname{trace}(\\operatorname{Var}(\\hat{\\theta }))", "f4f519ff17e2caa9ef35950cd9a2dde1": " \\widehat{\\boldsymbol\\beta} ", "f4f51b4a7a93f4b8b27c2001bab520fd": " E(X^\\nu) = \\frac{ \\lambda^\\nu \\Gamma(\\alpha-\\nu)\\Gamma(1+\\nu)}{\\Gamma(\\alpha)}", "f4f537b54536063e59025ea60e9ba300": " X=e^{\\mu+\\sigma Z} ", "f4f545c31c7c145ef2f5270593b93fc8": "x(x^2-x-1)+(x^2-1)", "f4f684fadb9b15e902f831a16a4740cb": "F \\subseteq S(\\exists) \\cup S(\\forall)", "f4f6accaa23ed9e00defd0cb8a410f68": "\\mathcal{O}_{1}=-\\gamma _{51}\\gamma _{52}.", "f4f6b6a53a0b9d18d647cbc47ddbc7a6": " R = R_{\\|} = -{2GM \\over {c^2 r^3}} = -{8 \\pi G \\over {3 c^2 } }\\rho (r) ", "f4f7034c81a6e157fbf6064d1849c047": "\\phi = -\\left( \\frac{GM}{r} + \\frac{Gm}{r} \\right)= -\\frac{G(M+m)}{r} = -\\frac{GMm}{\\mu r} = \\frac{U}{\\mu}.", "f4f74135ca79a33f0e3e277ce66e53a9": "F_{ab} F^{ab} = \\ 2 \\left( B^2 - \\frac{E^2}{c^2} \\right)", "f4f749e68c5d006f9e103937ec2c7ec0": "f'(a) = \\lim\\limits_{\\gamma\\to a}\\frac i{2\\mathcal{A}(\\gamma)}\\oint_{\\gamma}f(z) d\\bar z,", "f4f75fd1e9086378f8de2c02f634af26": "\\mathbf{F}\\cdot{\\rm d}\\mathbf{r}=-", "f4f780121149c7ef31b9088e63ce57a7": "d\\nu_n(x)", "f4f79ee77afddd74893586342f5df113": "X^\\flat := i_X g = g(X,.)", "f4f7abc468f371a9b35e6d1ecadb20a4": "\\frac{d\\ln{T}}{d\\ln{P}} = 0.4", "f4f7d4b89a995a531c6123cd6353ccb8": "S(\\rho) = - \\hbox{Tr} \\left( \\rho \\log_2 {\\rho} \\right) = - \\sum_i \\lambda_i \\log_2 \\lambda_i", "f4f7db829bc8ee0029c1727d4c515d43": "\n \\nabla^2 w \\equiv \\frac{1}{r}\\frac{\\partial }{\\partial r}\\left(r \\frac{\\partial w}{\\partial r}\\right) + \n \\frac{1}{r^2}\\frac{\\partial^2 w}{\\partial \\theta^2} + \\frac{\\partial^2 w}{\\partial z^2} \\,.\n", "f4f803338ab29280477aa850efd8ab03": "S = \\big\\{\\rho_{i}\\big\\}_{i=1}^{n}\\in\\mathcal{\\tilde{H}}_{S}", "f4f82b9b3cd5581761060e70f6a82b2d": "\n\\Delta P_i = P_1 - P_2 = \\frac\n{Q^2 ~\\rho~ (1 - \\beta ^4)}\n{2 ~C_d^2~ A_2^2} \n= \\frac \n{Q^2 ~\\rho~ (1 - \\beta ^4)} \n{2 ~C_d^2 ~A_1^2 ~\\beta ^4}\n", "f4f8b8664e05944b8533c429b6f10931": "\\nabla \\cdot \\mathbf{D} =0", "f4f8bdac89bf47289f1d386d0a78c51c": "\\text{MICEX10} = \\frac {k}{10}\\cdot\\sum\\limits_{i=1}^{10}\\left(\\frac {P_{i}}{P^0_i}\\right)", "f4f974d9f04090616718bb0a7e0e7a15": " \\log(Z[h]) = \\sum_{n,C} h(k_1) h(k_2) ... h(k_n) C(k_1,...,k_n)\\,", "f4f988cd9781bd340056c05f4729c356": " \\gcd \\left (f,x^{q^{n_i}}-x \\right )=1", "f4f9d17bb15dae99eade0d4ffbf0b6d3": "D_5 \\times \\pm 1", "f4faa32b1ccec170c982e22dfcb73fe6": " = -\\int_{\\vec{r}' \\in \\mathbb{R}^n}^{} {\\bigg(\\frac{\\partial}{{\\partial x_i'}}\\left(\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_j(\\vec{r}')\\right) - \\frac{\\partial}{{\\partial x_j'}}\\left(\\frac{1}{|{\\vec{r}-\\vec{r}'}|^{n-2}}F_i(\\vec{r}')\\right)\\bigg)d\\tau'}", "f4fb341e4e3e1228ad4038e6d594a775": "\\neq 0", "f4fb7c22df31bc83b9b01093ceba6007": "H_2O + h\\nu \\rightarrow H + OH", "f4fbad6a4b72bcf5004d1237d9b73e57": "\\scriptstyle \\geq0.66\\times10^{17}", "f4fbafeba6320dffb13f5fbff47dd361": "-R_s", "f4fbc7271a23cdb0c044ab54798f7956": "\\varepsilon_1, \\varepsilon_2, \\varepsilon_3", "f4fbf0032a9ddb1298fd834e05d39bb6": "\\ \\displaystyle \\hat{\\alpha}(q,r_{c}) \\ ", "f4fc37fee2b011954e9187f0aeefe235": "\\left\\{{n\\atop k}\\right\\}", "f4fc950681c10d1dda16695c01e4895a": " \\mathbf x", "f4fc9d490f6b525ac2af1f4e8baa0060": "\\Delta \\delta T = \\epsilon \\mathcal{L}_{\\xi}T_0.\\,", "f4fcb939f45dde04a0e09fad2bf2abf0": "4(n-2)(n-1)", "f4fcd71479523eaf2cb880bf576dc137": "k\\hat{u}/\\omega", "f4fd5f6a51f6a91758db06af6655865b": "\nG^{\\mathrm{R}}(\\omega)\\sim\\frac{1}{|\\omega|}\n", "f4fd8652760fbaebbe5da1cb7d6857dd": "\\hat{q}(\\xi)", "f4fd94f3487fcba7d5aa288edbd10949": "\n m = \\int_V \\rho(\\mathbf{r})\\,dV.\n", "f4fd98b6b91d070009bf0700dcfce9bc": "U(t,t_0)=1 - i \\int_{t_0}^t{dt_1\\ V(t_1)U(t_1,t_0)}.", "f4fdaa09848bc986355cbecc2f8b804d": "E_D = \\frac{1}{2}\\sum\\limits_{i = 1}^N {e_i^2 } = \\frac{1}{2}\\sum\\limits_{i = 1}^N {\\left( {y_i - (w^T \\phi (x_i ) + b)} \\right)} ^2 .", "f4fe129405e537a29b777845a0505c5e": "\\int_0^1 f'(x+th)\\,dt.", "f4fe507ac6b0ffc4e6bce47bdfe57a66": "SU(2)_R/U(1)_R \\approx S^2 \\simeq \\mathbb{CP}^1", "f4fe694abfe0b81156d1b969feb538ae": " \\Delta Cd - (\\beta)^{3.5} ", "f4fe6ed6f4e10e3865542446122e40a2": "v_n(x)=\\sqrt{n} [F_n(x)-F(x)].", "f4fe7d387d093d8315f830023390e8ce": "\n\\rho \\leq \\frac{\\left(c' \\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1} \\Sigma _{YX} \\Sigma _{XX} ^{-1/2} c \\right)^{1/2}}{\\left(c' c \\right)^{1/2}}.\n", "f4febdec43d5efcccf9a84806786c4d0": "\n\\mathrm{li}(x) = \\mathrm{Ei}(\\ln x)\\,\n", "f4ff29cdbd282e98b706764a9a586ff9": "g^{(k+1)T} (x^* - x^{(k)} ) \\leq 0", "f4ff58440602c00017a92f78fe00e6b1": "\nh_{\\xi} = a\\sqrt{\\frac{\\zeta^2 + \\xi^2}{1 - \\xi^2}}\n", "f4ffec23ea249436322d0a0983f26bf7": "\\langle k, g(T) h \\rangle.", "f4fff2b6c0d722db244c3338cbad1f34": "(a\\operatorname{T}b) \\wedge \\neg(b\\operatorname{T}a) \\wedge (b\\operatorname{T}c) \\wedge \\neg(c\\operatorname{T}b) \\Rightarrow (a\\operatorname{T}c) \\wedge \\neg(c\\operatorname{T}a).", "f50009e6dc402930fc2ffbb144674357": "v=13", "f500235ff3397f34bf79c76a3e30e4ef": "d_m+d_{mm'}=d_{m'}+d_{m'm} \\, ", "f5002b051740058512adb4ca11456d22": "\n\\begin{bmatrix}\n6.1917 & -0.3411 & 1.2418 & 0.1492 & 0.1583 & 0.2742 & -0.0724 & 0.0561 \\\\\n0.2205 & 0.0214 & 0.4503 & 0.3947 & -0.7846 & -0.4391 & 0.1001 & -0.2554 \\\\\n1.0423 & 0.2214 & -1.0017 & -0.2720 & 0.0789 & -0.1952 & 0.2801 & 0.4713 \\\\\n-0.2340 & -0.0392 & -0.2617 & -0.2866 & 0.6351 & 0.3501 & -0.1433 & 0.3550 \\\\\n0.2750 & 0.0226 & 0.1229 & 0.2183 & -0.2583 & -0.0742 & -0.2042 & -0.5906 \\\\\n0.0653 & 0.0428 & -0.4721 & -0.2905 & 0.4745 & 0.2875 & -0.0284 & -0.1311 \\\\\n0.3169 & 0.0541 & -0.1033 & -0.0225 & -0.0056 & 0.1017 & -0.1650 & -0.1500 \\\\\n-0.2970 & -0.0627 & 0.1960 & 0.0644 & -0.1136 & -0.1031 & 0.1887 & 0.1444 \\\\\n\\end{bmatrix}\n", "f500543e8b1341610b14d437eb2c0b4f": " \\bar{f}(x')=(x'+1)(2x'+1)(3x'+1)(4x'+1)(1)(1-x')= -24x'^5-26x'^4+15x'^3+25x'^2+9x'+1 ", "f5005a7273137f3cde3b8dbf529c8b52": "x_i(b) = 0", "f500d962f019c2a72b620ca31877d793": "\\nabla \\times (u,v) = v_x -u_y =0,", "f50115825af611d6a101b428bee32bb2": "\\exp(\\exp(X)) = e \\exp(\\exp(X) - 1) = e + eX + eX^2 + \\frac{5eX^3}{6} + \\cdots", "f5011e166e3543b8f3cce4b95b501ae8": "\\lambda-1\\,", "f5016bda95bcbf7c0dc5f78ae98c4853": "\\mathcal{F}=\\{S_1,S_2,\\dots\\}", "f50177a3948c16297ecf801726cfc3d5": "q_p(p+1)\\equiv -1 \\pmod{p}", "f501a8b376a10595d8f7c680208e8b8d": "l \\neq 2, p", "f501c56622278cec4811c62692c38895": "T^2 \\sim \\frac{\\nu p}{\\nu-p+1}F_{p,\\nu-p+1}, ", "f501c5c5513cdca6137bc15fe2a19646": "f_P (X) = Y \\text{ whenever } P \\in XY", "f501e7c49a30bd2f441fdb34eaeb993d": "\\sum_{k=0}^\\infty 10^{-\\left\\lfloor \\beta^{k} \\right\\rfloor};", "f50204232c97fe67b03736564ac8637b": " \\langle Ax \\mid y \\rangle = \\lang x \\mid Ay \\rang ", "f50236b2b61015d0f526c776230fcd7e": "1/a", "f50286f6309c11ed836835e817415e79": "\\scriptstyle |\\operatorname{fix}(M,Q_i)|=(n/2)!^2", "f502b283950bcd2cc3c3ffa50d6adb43": "\\forall\\beta.\\beta\\rightarrow\\beta", "f502d8cdd15979dc186b82c323570f29": "\\textstyle \\sin C = 1", "f502e9a6e6ae4db9f6f7e71faf1b79ed": "p^k(K) \\in \\mathcal{Q}", "f503063965c607b583380b9619b5a9e0": "\\operatorname{SCV}(\\bold{H}) = n^{-1} |\\bold{H}|^{-1/2} R(K) + \nn^{-2} \\sum_{i=1}^n \\sum_{j=1}^n (K_{2\\bold{H} +2\\bold{G}} - 2K_{\\bold{H} +2\\bold{G}}\n+ K_{2\\bold{G}}) (\\bold{X}_i - \\bold{X}_j)", "f50362faee004903d5399c83149ba6b3": "\\hat{b}^\\dagger \\, \\hat{b}", "f5036541d86be30e28e23a0a10dacebb": "\\frac{\\partial \\rho u }{\\partial x} + \\frac{\\partial \\rho v }{\\partial y} = 0 ", "f50392459d5de391b5793c6effc63398": "\\alpha_2 = g_{2}^s h_{2}^c ", "f504374507e4bcfd8d05b628ffe1511c": "\\Delta x\\Delta k\\ge 1/2", "f504396536d9318c56faf0dbe4a578dd": "\\begin{bmatrix} a & b & c \\\\ d & e & f \\\\ g & h & i \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}=\\begin{bmatrix} {\\color{red}j} \\\\ {\\color{red}k} \\\\ {\\color{red}l} \\end{bmatrix}.", "f5048e12514a2a257eb3666a51c2c694": "D_C(x) = \\| x - P_Cx \\|_2 ", "f504df90cd8dcdbf5b87641d8590d9e3": "\\frac{\\text{d} f}{\\text{d} x} = \\frac{1}{1+x^2}.", "f50505899297795aed6c1bd5b38b6af9": " \\vec{\\xi}_5 = x \\, \\partial_v + \\int \\frac {du} { C \\left( \\frac{q^2}{\\omega^2}, \\frac{q^2}{2\\omega^2}, \\omega u \\right) } \\, \\partial_x ", "f50552e79515c9b35adda97466162af0": "K_a \\approx \\frac{10^{-pH_i} v_i[OH^-]_0}{v_0 [HA]_0-v_i[OH^-]_0}", "f505678d81a95a58a9eb0705e89c7bb5": "-l \\le m \\le l", "f505805fd8a5f151a01e4e69f360775a": "\\,s=s_1", "f505813bdcc068a89e641c75bd71e578": "F=0.4\\frac{d^3}{D}\\sigma", "f50597e0a5585b7a0a8f641cba1d3705": "\n\\delta \\vec{x} (t) \\approx \\boldsymbol{H} \\cdot \\delta \\vec{x}_0\n", "f5060b0cb422fb22d6174c784fa846dd": "\\Gamma = \\int_0^{\\infty} \\! c(x)\\,dx \\, \\approx \\frac{\\left\\vert y_s \\right\\vert^{3/2}}{{\\lambda_B}{a}{p^{1/2}}}", "f506a92077c1135eaf8bbb1ed069f482": "\\phi \\left (\\int_{\\Gamma} g \\right ) = \\int_{\\Gamma} \\phi(g).", "f506f59abc9cc2cfcd8071787387448b": "p_{\\mathbf{s}}=g'", "f50724424a2f810d13cdbc58036e6246": "\\scriptstyle u+v\\in BV(\\Omega)", "f5072e34138f37250018895399ffff8c": " \\mathbf{g} \\cdot \\mathbf{b} ", "f50750e641d371f175823a0af90faf5b": "P_f", "f507b9f5952c3ea6a716362d5523b24c": "\nf_p = \\frac {150}{Gr_p} + 1.75\n", "f507c1f6e219a0f5f3b23002cdfa8b59": "\\Phi_B = \\int_S \\mathbf{B} \\cdot \\mathrm{d} \\mathbf{A}\\,\\!", "f507f098c0336b17666501c34b716382": "MRS_{AB} = \\frac1{MRS_{BA}}", "f507f6db3966b9cabbf24bed853b6eae": "[\\in] = \\{(x,y) \\mid x\\in y\\}", "f507f8f46623135aea57054d1ec9232c": "\\chi = V+F-E-C", "f508a1237731941a7e15f751614b150a": "E_0", "f508b5dbe65138a4b1bab4f12e0e5772": "\\mu{ \\left( \\frac{a}{a_0} \\right)} a = \\frac{GM}{r^2} ", "f50909199031c94a45cafe0923fe0398": "\\rho'(x, y, z) = \\rho(x, y, z/a)/a", "f5093496aff2c2bb486bf689309ef9cf": "M^3", "f509acece458e1e78b1700977cd0b25d": "f(z)=\\frac{1}{1+z^2}", "f509c1ff5adbb3b08d04580c77cad4fb": "n(n-1)\\cdots(n-k+1)", "f509f0619a6cb71838b505f66377cc02": "\\Gamma(s)\\Gamma(1-s)={\\pi\\over\\sin\\pi s},", "f50a3132f3f3be2da5eb27292b14f16f": "L(\\lambda) = \\sum_{i=0}^l \\lambda^i A_i. ", "f50ab952cf4d82ea85f2ddbaf40ac910": "X_{t} : \\Omega \\to S : \\omega \\mapsto X (t, \\omega)", "f50b4ec3fd95a7bb9dd82432eaa5def1": "\\vec{x}'\\,\\!", "f50ba288d0e44146495ecf6f60f6b5a6": "\\psi'=\\phi_D^{-1}\\circ\\hat\\psi\\circ\\phi_D", "f50baf67eae7e188555553c061e29b59": "\\frac{A\\hbox{ prop} \\qquad B\\hbox{ prop}}{A \\wedge B\\hbox{ prop}}\\ \\wedge_F", "f50c4c98af2cb46126bb17cfa2a65a62": "\\varphi_{s(p,x)} \\simeq \\lambda y.\\varphi_p(x,y).\\,", "f50c610c7890bca95c2b356bfe9f2918": "\\mathop{\\mathrm{Li}}_{n \\to \\infty} A_{n} \\subseteq \\mathop{\\mathrm{Ls}}_{n \\to \\infty} A_{n}.", "f50c7b63412bf64d45567b092fef8735": "U_{\\mathrm{breakdown\\,Townsend}} = \\frac{L\\cdot p\\cdot d\\cdot E_{I}}{\\ln(L\\cdot p\\cdot d)} = \\frac{d\\cdot E_{I}}{\\lambda_e\\,\\ln\\left(\\frac{d}{\\lambda_e}\\right)}\\qquad\\qquad(17)", "f50c8ba6f44a30dd2432214f51751bea": "\\mathrm{P}(u,v)=\\mathrm{exp}(i\\,k(u^2+v^2)), \\forall u,v : \\sqrt{u^2+v^2}\\leq R", "f50caba1475b4c0ecc9caddd16b1ce12": "\nL = \\frac{1}{2} \\sum_{t=1}^{3N-6} \\big( \\dot{Q}_t^2 - f_t Q_t^2 \\big).\n", "f50df0638ff3acc7a0bbff96dfba5d94": "(\\forall u)(\\exists v)(P)\\psi(x,y|x',y')", "f50df24845b95ccc3a19f488c4c44de3": " | \\psi_i\\rangle \\langle \\psi_i |\\psi\\rangle \\,", "f50df53a1b0299d0ebdfc9068202dcf2": "\\phi_k(t) = \\sqrt{2a}(-1)^{k-1}e^{-at}L_{k-1}(2at)", "f50e2e5af0581d9f4dbed19107cb5915": "-1,", "f50e50ccff225a6d9a464726c1f7ec88": "\\frac{25}{2} O_2 + C_8H_{18} \\to 8CO_2 + 9H_2O", "f50e8a487f9c7f9fff356c1e8bf75253": "\n\\begin{array}{lcl}\n\\hat{\\mathcal{P}}_1 &=& \\, i \\left( {\\cos \\gamma \\over \\sin \\beta}\n {\\partial \\over \\partial \\alpha } - \\sin \\gamma\n {\\partial \\over \\partial \\beta }\n - \\cot \\beta \\cos \\gamma {\\partial \\over \\partial \\gamma} \\right)\n \\\\\n\\hat{\\mathcal{P}}_2 &=& \\, i \\left( - {\\sin \\gamma \\over \\sin \\beta}\n {\\partial \\over \\partial \\alpha} - \\cos \\gamma\n {\\partial \\over \\partial \\beta}\n + \\cot \\beta \\sin \\gamma {\\partial \\over \\partial \\gamma} \\right)\n \\\\\n\\hat{\\mathcal{P}}_3 &=& - i {\\partial\\over \\partial \\gamma}, \\\\\n\\end{array}\n", "f50e8b71d4a4c4fcdafebb2b9fb52a4f": "t_{l}", "f50f2db8540c684f1907951283183173": " {\\Omega^1}_2 = -\\exp(-2p) \\left( p_{xx} + p_{yy} \\right) \\, \\sigma^1 \\wedge \\sigma^2 = -\\Delta p \\, \\sigma^1 \\wedge \\sigma^2", "f50f4d152c6235bc909dc54de4136fee": "(1- \\lambda)(3- \\lambda)=0", "f50ff72561fc7c75c14df96af58dc500": "F(u+h)=F(u)+DF(u)h+\\frac{1}{2!}D^2F(u)\\{h,h\\}+\\dots+\\frac{1}{(k-1)!}D^{k-1}F(u)\\{h,h,\\dots,h\\}+R_k", "f5105d0429f78cb8e5e93eaabdb2cd48": "T\\mathrm{d}S=\\mathrm{d}H\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(P\\,\\, \\text{constant)}", "f510c2bb4a16a871593eb304fa63fac7": "e^- e^+ \\to \\gamma \\to q\\bar q", "f510d4c7d386a32979b8f278235f4074": " K = - \\alpha \\ln( 1 - X ) ", "f510f7dcf5d6d3edbd56f247380e9d0e": " Y = \\operatorname{let} x : \\operatorname{de-lambda}[x\\ x = f\\ (x\\ x)] \\operatorname{in} f\\ (x\\ x) ", "f511f6f5d66717ccf8eeb66388ad65d6": "f(x,y,z) = \\frac{x}{y} + \\sqrt{\\frac{y}{z}} + \\sqrt[3]{\\frac{z}{x}}", "f512ce5fb3ee3cb2498096ff6318798a": "\nK = \\int_{0}^{1} \\frac{dy}{\\sqrt{\\left( 1 - y^{2} \\right) \\left( 1 - k^{2} y^{2} \\right) }}\n", "f512e5cf42b9a16efd0787fa51085122": "j: X \\to (X, A)", "f513020f3bdca9e32220a36812d52b88": "\\frac{d\\epsilon}{dt}~\\alpha ~d^n", "f5131771dae7c90950d54904121ced2e": "= 4 \\pi\\, r^2 ", "f51318b2301121e60259b3a86612b60e": " \\Delta \\phi = \\phi_2 - \\phi_1 \\,\\!", "f51321a7d62e702b19fd0a501c3abd37": "P_{\\mathrm{h}}", "f513320ab356bd4bc81648021016c5a8": "\\mathrm{d} * F = \\mu_0 J ", "f51339e93f882713151a2a8abbbd57ee": " 3 > \\lambda \\ge 2", "f5133db3a26e2c63be9c9d4015ea74e3": "\nB=\\frac{Y}{L}= \\frac{Y_0}{L_0} \\left( \\frac {L_0}{L} \\frac{P}{P_0} \\right)^{\\alpha}.\n", "f5133de49419c2dd0932ae5d31a16aa8": "\\lfloor \\log n\\rfloor", "f5138960865b8670554519779cc2b096": "\\overline\\psi = \\sum\\limits_{i}\\psi_ib^i,", "f513d688c09e144364bdbc3f9238e92b": "\\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta)", "f513e4018e6957f70cc060a8ba24ecb7": "g(x) = {\\rm lcm}(m_c(x),\\ldots,m_{c+d-2}(x)).", "f51445834fe0596d3b2a7f19d09ffd2f": "g(x,u)\\le b, \\forall u\\in U", "f51449ccd20c08365f869c598c5ece9a": "R_a,R_b,R_c", "f5145edd3fbe20ee92f97609222caa8f": "X \\sim S(\\alpha, \\Lambda, \\delta)", "f5149b28fdb5a97ce4029d5f8cea1486": "\\mathbb{Z}^2", "f514f6a45c780840b2a65617a2ead340": "\\displaystyle \\times J_{n/2+\\delta}(|\\boldsymbol \\nu|)", "f5150fa8fd7a9014f271179da2073ee3": "f_u^{\\otimes |U|} f_v^{\\otimes |V|}.", "f51511e680d28dcaa1ffc34084944903": "x\\sqsubseteq y", "f5155ff798b8ce0f2c6d584ab548e985": "s = j \\omega \\ ", "f515679023b6ae638e0e366fb47633f0": "\\mathcal{O}^j_M", "f5159654c298b7f108ca38b9784418b9": "-\\frac{\\part A}{\\part n}(P_{a} - \\frac{\\part C}{\\part a})", "f515da30c2e815b8aa7760cfd17bef77": "\\pi = 6 \\sin^{-1} \\left( \\frac{1}{2} \\right)\n= 6 \\left( \\frac{1}{2} + \\frac{1}{2\\cdot 3\\cdot 2^3} + \\frac{1\\cdot 3}{2\\cdot 4\\cdot 5\\cdot 2^5}\n + \\frac{1\\cdot 3\\cdot 5}{2\\cdot 4\\cdot 6\\cdot 7\\cdot 2^7} + \\cdots\\! \\right) ", "f515db10cb3c6c85e17a580b7b1538b0": "F(s) = \\int_0^\\infty f(t) e^{-st}\\,dt.", "f51623f2e2ece2ee968de39af3a33be3": "\\operatorname{tr_B} \\left( | \\psi \\rangle \\langle \\psi | \\right )= \\rho.", "f5162469007e8ee123dce47741ff5cc2": "3(l - 1)^{n - \\delta}", "f51663d665f3d6c7ecf579e6e9ecf563": "\\max_{|z| \\leq 1} \\big|P'(z)\\big| \\le n \\max_{|z| \\leq 1} \\big|P(z)\\big|.", "f516b30d9f80d47a6afb2e41dab0d828": " N^{r}_{\\theta\\theta} = N^{r}_{rr} ", "f516d02dc29a1ac199b95486c3517487": "\\psi(\\psi_1(\\Omega_2))", "f516d1de6e5ff832fd4941c5ae896e11": "-3.0000", "f516dc67efdfd0ffdabb0f63cda6db2d": "x^2 + y^2 = r^2", "f51754411929cebc0c49bd87ff0cc223": "F =\\frac{f}{A}=\\frac{f\\omega}{C}", "f5179bda003df61c99ba01b4ae236b5f": "{\\mathcal O}(n^{19})", "f517e3370bf718d6f5dc2ad1d52c0c61": "\\qquad\\qquad", "f517fb1ef9dc735e35364b2890577030": "\\displaystyle \\mathfrak{f}_1(\\tau) = q^{-1/48}\\prod_{n>0}(1-q^{n-1/2})", "f51823e8b0e8044514b67f298c833de4": "g \\circ f_i", "f5183e9f458d346b7bd9527753ea78f9": "A^TSA,", "f518541eb8859d26f61141e526a4d35d": "ab=\\frac{1}{2}(ab+ba)+\\frac{1}{2}(ab-ba)", "f518b1f1a09172a8944c3603f5b9136a": "I^{-m}_{\\ell}(\\mathbf{R})", "f518de04ec1bd07da2269800c7d19eff": " H_s = \\{ f \\in V^*: \\langle f, e_s \\rangle = 0 \\}", "f5194b99ef830a0e259f1b8b7bf4b9f2": "\n\\begin{align}\n P &= \\frac{\\Omega}{2\\pi} = \\int_{0}^{\\alpha_0} \\sin\\left(\\alpha\\right) \\operatorname{d}\\alpha = 1 - \\cos\\left(\\alpha_0\\right) \\\\\n &= 1 - \\sqrt{1 - \\sin^2\\left(\\alpha_0\\right)} \\\\\n &= 1 - \\sqrt{1 - \\frac{B_0}{B_m}}\n\\end{align}\n", "f5196fb823a8071ccce5656df680bce8": "H_{2x}=\\frac{1}{2}\\left(H_{x}+H_{x-\\frac{1}{2}}\\right)+\\ln{2}", "f51990f3eaac003d768def53cb5767cc": "\\approx 1.3", "f5199a8c0edec386a5c1f2ee77429c39": " MW_{ratio} ", "f519bee181772e98afc52d2a0a8f208d": "\\mu_a \\cup (A^T \\mu_b A)", "f519ecb03eb214692d3cb90d48b2834d": "\n\\begin{align}\n\\Pr(Y_i=0) &= \\frac{1}{Z} e^{\\boldsymbol\\beta_0 \\cdot \\mathbf{X}_i} \\, \\\\\n\\Pr(Y_i=1) &= \\frac{1}{Z} e^{\\boldsymbol\\beta_1 \\cdot \\mathbf{X}_i} \\, \\\\\n\\end{align}\n", "f51a08a58cad5bee11213734c26b5b55": "t _3[\\beta] = t _1 [\\beta]", "f51a335db9d8e32bdbc81506af688c92": "\\zeta = \\frac{P + \\sqrt{D}}{Q}", "f51a5b4f2d3647a27983e3c56ecde0ef": "f(\\lambda x, \\lambda y) = \\lambda^n f(x,y)\\,. ", "f51a61502ef8217390c4f5a98bdc77c3": "\\| v \\|_{H^{1} (\\Omega)}^{2} \\leq C \\int_{\\Omega} \\sum_{i, j = 1}^{n} \\left( | v^{i} (x) |^{2} + | (e_{ij} v) (x) |^{2} \\right) \\, \\mathrm{d} x, \\quad (1)", "f51a7cf97e9f65a2145773f5e486a6d8": "\\ln\\left(1-\\frac{V(t)}{V_0}\\right)=-\\frac{t}{\\tau}", "f51ab1f5fe7e1df7555527ea95cc1b4a": "\\frac{\\partial \\Pi}{\\partial x_j}", "f51ac931d7cf2557aa6016962334e9c2": " p(\\cdot; \\theta+\\Delta)", "f51b5037dc2e919714c9e013e1018b8b": "\\Theta = -\\Gamma,", "f51b78194f2b3bddffe278cc743821bd": " \\mathbf{\\hat Q} ", "f51b78f347da0391e2ae629623d6c252": "\\vec{v}_A", "f51b7c51a286b39c0e40dad55885d91c": "YX = \\left(A^{-1} - {A^{-1}uv^T A^{-1} \\over 1 + v^T A^{-1}u}\\right)(A + uv^T) = I.", "f51bb214d46735e275bdfee81acb703f": "10\\uparrow\\uparrow\\uparrow\\uparrow 7=(10 \\uparrow \\uparrow\\uparrow)^7 1=", "f51c1b8472f876f76a5f84b2cfe5319a": "\\pi^{2}\\int_0^a |f|^2 \\le a^2 \\int_0^a|f'|^2", "f51c1ca480b466ec7f8ba199caab00c5": "\n\\overline{ \\phi + \\psi } = \\overline{\\phi} + \\overline{\\psi}.\n", "f51c4b5efcc0e6d58ebe6479b31eae00": "x\\in(a,b)", "f51ca9943963bb32b8716cfe653beaa9": "1/r \\,", "f51cce6030af7685f0b53b66d634cd74": "{a^2}I_{n,m}= I_{m,n-1}+I_{m-2,n}\\,\\!", "f51cd25091457f62f5f9e38d7ccb6d04": "a_n x^n+a_{n-1}x^{n-1}+\\ldots+a_1x+a_0", "f51ce4a709069b47109492b96f0c8eff": "E_c = \\left(\\frac{f}{E_f} + \\frac{1-f}{E_m}\\right)^{-1}.", "f51d2b08e5716c6d47c49cdae54fe586": "y^T b = -1", "f51d3c8088c8dffd6dcc061d66e7881c": "Y=-jB_L=\\frac{-j}{\\omega L}", "f51d7cd5a0026e4e643af7bce4209a76": "\\boldsymbol{H} \\cdot \\boldsymbol{c'}^T = \\boldsymbol{0}", "f51dad65341fbddaa2d9433f8b148969": "2\\nabla_a\\nabla_d{{C^a}_{bc}}^d+{{C^a}_{bc}}^dR_{ad}=0,", "f51e2d633053101753eadbe2184c5dfa": "\n\\bar E = \\alpha_1\\ \\hat f_1 + \\alpha_2\\ \\hat f_2 + \\alpha_3\\ \\hat f_3\n", "f51e3336ad1880e619c4d5a1d94ad349": "H / v = b(b(H) \\setminus v)", "f51e63191456f7b21071028e86a745d5": "c(t)=(x(t),y(t))", "f51e88027332e02ea8831f4d2c5fc673": "dlock", "f51eff3dcdc22a2357305417ce6b7540": "\\varepsilon_{ab\\ldots n}", "f51f765782315f92f942a84f0ae64bfa": "\nF(Z,T) \\approx \\frac{2 \\pi \\eta}{1 - e^{- 2 \\pi \\eta}}.\n", "f51fb01ac17bc045d12c9c13c29e0bf4": "a\\circ c 1/2", "f53057218e0e3104d45ed15cc8bab9fc": "x^{\\mu} \\!", "f5306af7b538953c055eb0b7fa992ab6": "\\bigsqcup_{(\\lambda,\\mu) \\in \\Lambda^2} \\pi_1(U_\\lambda \\cap U_\\mu, A) \\rightrightarrows \\bigsqcup_{\\lambda \\in \\Lambda} \\pi_1(U_\\lambda, A)\\rightarrow \\pi_1(X,A)", "f530ef7fbeae2563b2b63e19e4a0f25f": "\\mathbf{U}\\,\\!", "f531062ad84856606dd4491ca70e464a": "\\displaystyle{f(e^X)=\\mathrm{Tr}\\, \\mathrm{Ad}(e^X) S.}", "f531978f9fff04fc11b0e0ad0cd71a22": "T=+(n_t-n_\\bar{t}),", "f531c0a53d37589973b92e46953e3e7b": "\\mu=4\\pi^2a^3/T^2 \\ ", "f531f1410347d571a7b670a35c75bd92": "w = \\frac{ab}{a+b}", "f531fc2c7c142b03a80cd33f341de49f": "constructor_\\Pi (u:U) (u' : (x: T(u)) U) : U", "f53221529bb362c02dc0026a4e4c63cb": "v_1,v_2,v_3 \\in \\Bbb R^3", "f53241bca9ab95ed510049ebcd7cfbe7": "x_{n+1}", "f53253707c57a1b939638ba6bdb1c41a": "D(\\cdot,\\cdot)", "f53275dfc28a47594e22f00d79186dd5": "\\mathrm{^{241}_{\\ 95}Am\\ \\xrightarrow {(n,\\gamma)} \\ ^{242m}_{\\ \\ \\ 95}Am}", "f532cd57b7e8d60e92a802f14625f34f": "\\ d\\log(Y)=d\\log(K) \\Rightarrow \\frac{dY}{Y}=\\frac{dK}{K} \\Rightarrow \\dot{Y}=\\dot{K}. ", "f532fcdb8b9c5c620fefcd59f1d5f869": "a\\,", "f53306364f9c2818cb3090d6fb0ce93e": "\\mu(N)=\\sum_{n=0}^\\infty\\mu(A_n)\\le\\sum_{n=0}^\\infty\\max\\{s_n/2, -1\\}=-\\infty,", "f533d7bebc14a01ad442d620163524b1": "X_c^0", "f53475f5cd17457d724672f4273c81b2": "\\sigma = t_1 t_2 \\dots t_m ", "f5347eeb9e686532975c6881e9da8d5c": "\\sin\\angle OAB\\cdot\\sin\\angle OBC\\cdot\\sin\\angle OCA = \\sin\\angle OAC\\cdot\\sin\\angle OCB\\cdot\\sin\\angle OBA.\\,", "f5351b749f72da36850ab7f1e0a3c510": "n \\; = \\; \\frac {a - b}{a + b} \\; = \\; \\frac {1}{2r - 1}", "f5354a5bbe4f789a040d0355ab2cd97e": " \\mathbf{P}(\\mathcal G_T) \\to \\mathbf{P} (\\mathcal{E})\\times_S T", "f535584307a47f42161295e4cd17ff63": "\\bold{v}_1 + \\bold{v}_2", "f535b2dc37b10d2fc96f86ef88bedf37": "F(x,y) = f(x) + g(Ax - y)", "f535bae7713f3f709534ede17b4339fa": "x^\\prime = x + vt,\\quad t^\\prime = t", "f535fc8b03cc6e61105fb58b107e42a9": "i<\\kappa", "f53610a7cf6daf62df3f1f355ef14f74": "y = - g(x/v_h)^2/2", "f536154d6ce9dd18f5ca798539cfe4ae": "\\nu=5", "f536340c843c03dbadd7eaa2f2d8b800": "N^j_{pq}", "f5363645d3a3b29284d15435789135a1": "\\left\\{3,{3\\atop6}\\right\\}", "f536adeeffb5882abbf031025d38a4a4": "\\varnothing\\times\\varnothing", "f536de69598abcd8d42f4d0b0b2dc2d4": "\\sum_j T_{ij}", "f536f901bec1f7f3c30568c85023cb8a": " \\lfloor \\operatorname{lb}\\, n\\rfloor + 1. \\, ", "f53700be43bb23fcdc1ffcc204cb7c13": " D_n(x) = \\frac{\\sin((2n+1)(x/2))}{\\sin (x/2)} = U_{2n}(\\cos (x/2))\\, .", "f5375447a295e404d9c11bb1a8b9e0c4": "StO_2 \\,", "f537ea467b683fa10be27d5e400d5cf4": " \\operatorname{W^*}(\\{S_x: S \\in D\\}) = A_x ", "f5381924764d9583418d863c381203c3": "r n_d/m < 1", "f53827d99c3f6cb4e51526ab42543ac3": "\\beta_{xy}", "f5387f1502ab860a7932fb59df21839c": "p = \\tfrac14 \\left(L^2+ 27M^2\\right),", "f5388c587f023a1ff623ca3918c9c359": " \\sum_{t=0}^\\infty |\\gamma(t)|=\\infty", "f538ca1d7161110fbab8936049d0587f": "\\mathfrak{g}_\\lambda", "f538e3c41bb42b0f523aaa7615ad67f4": " dW = -P dV = -\\frac{1}{3} \\varepsilon \\sigma T^{4} dV ", "f5397727180f2a0db1babe9fc39f5077": "\\ t", "f53983a9eda11b34823fdcfba114ece4": "\nA_p(\\kappa) = \\frac{||\\sum_i^N x_i||}{N} = \\bar{R} .\n", "f53a1526e1f9b2b6bdc2357e7e487e22": "\\frac{\\mbox{EBIT}}{\\mbox{Annual Interest Expense}}", "f53a48a5a99b8e7f38e2e843327fd7f7": "|b|_{\\ast}\\leq|a|_{\\ast}^{\\log b/\\log a}", "f53a6e0a0b8d120bc28de4220662329b": "\ns_m = \\sqrt{\\frac{\\sum(\\Delta hR_{Fi} - \\Delta hR_{Ft})^2}{n+1}}\n", "f53a7c172aed6c513b5fd635e97f309e": "\\log p = \\log \\frac{[solute]_\\mbox{organic phase}}{[solute]_\\mbox{aqueous phase}}", "f53a867b079c0025c7036473d9ea38a3": " 4 \\nmid n ", "f53ad3d6718d5e6ce5b5a3201080769c": "g^* = g^{-1}. \\!", "f53b3bf61a6fa5b0fc4379116e9b8d64": " \\sum_{i=1}^{m} n_i = v-1 ", "f53b689d87a33902c9565b0051a85a78": "\nN = A\\exp\\left(-\\frac{t}{{\\tau}_f}\\right) + B\\exp\\left(-\\frac{t}{{\\tau}_s}\\right)\n", "f53bb17c3889328553cd7648972ef4a2": "\\Theta^2=1", "f53c05211d4a465bf57bd09609cff139": "[x, y)", "f53cc332734886c38e5c42b2ca662538": " y(t_{0})", "f53cf8e0c99a1c44ef46cbd13bdb8bca": " \\alpha_i = \\frac{w_i} {V_\\infty} \\qquad (9) ", "f53d015a3dca00499f4bd107d5a5b2cf": "\nX^{\\{4\\}}=[2,7], \n", "f53d04f6f4b69f761390f760722fd88d": "\\mathbb{B}^n=\\{X\\in\\mathbb{R}^n:|X|<1\\}", "f53d43f703fd6337da8830561bd4b061": "\\chi_{nk}", "f53d5133810a6da9851ec5cea2938f4e": " \\langle E\\rangle = k \\langle (x-x_0)^2 \\rangle /2=k_B T/2", "f53da8e481dd2039dfec49f1029850ba": "\\alpha = {K \\bar c \\over (\\bar v + (K-1) \\bar c)}", "f53db41c6c1251aa3af2ae31cad79356": " \\Delta t =1/90 s", "f53dc1b7d39afd023cd712ce0e0852ed": "\\tau=\\frac{L}{u'}.", "f53e4166e6d5c30e1a66d1134e2f6b5c": "\\frac{\\partial f}{\\partial t}T\\sim f \\quad \\left|\\frac{\\partial f}{\\partial\\vec r}\\right|L\\sim f \\quad\\left|\\frac{\\partial f}{\\partial\\vec v}\\right|V\\sim f.", "f53e4302135b4455feedbd7d2ef8a424": "(\\gamma,n)", "f53ec8bef24de68d0830e41c5007eec1": "\\Delta_K=\\left(\\operatorname{det}\\left(\\begin{array}{cccc}\n\\sigma_1(b_1) & \\sigma_1(b_2) &\\cdots & \\sigma_1(b_n) \\\\\n\\sigma_2(b_1) & \\ddots & & \\vdots \\\\\n\\vdots & & \\ddots & \\vdots \\\\\n\\sigma_n(b_1) & \\cdots & \\cdots & \\sigma_n(b_n)\n\\end{array}\\right)\\right)^2.\n", "f53f2255fc585e425e54b51c1038b5ca": "(c_L)^{\\alpha\\beta}_{ab}", "f53f41656635f147bd6fb160eca83193": "\\sum_{n=0}^m \\mathbb{P}(N=n)E^n=\\sum_{n=0}^m S_n\\Delta^n\\qquad(*)", "f53f8012fda010674a80a9aec5cd045b": "E_k = \\frac 12 m v^2 ;~~ E = m c^2 ;~~ E = p v ; ~~ E = hc/\\lambda", "f53fafa383af732df5c2ada7c1075029": " \\varsigma_1(\\varepsilon) := (1 - \\exp(-p \\varepsilon(t))) \\varsigma ", "f53fbaffe5a783f37eb41acca0978e62": "v_i\\,", "f53fc333ee436cbf18ea60ba51be2afc": "\\Delta C_{\\,{\\rm V|\\,magnon}}\\,\\propto T^{3/2}", "f53fc834e00813da8a5ee35fc3ad686d": "\\operatorname{div}(\\rho u\\phi)=\\operatorname{div}(\\Gamma\\nabla\\phi)+S_\\phi; \\,", "f53ff593c05a1bc60da0c4e1d7bc1d81": "{\\Pr}_{\\theta,\\phi}(u(X)<\\theta 0", "f542b6c84cb711e7c5fc585280bc80f9": "\\left[{n \\atop k}\\right]= \\frac{\\Gamma(n)}{\\Gamma(k)}w(n,k-1)", "f542d513bf9febd19a3c58f866dd3bdf": " I_b = I_X \\frac {R_E} {R_E +\\frac {r_{\\pi}} {A_v+1} } \\ . ", "f542e0fb7a6f9ed6f5b991a4d878c5d3": " {\\rm Jac}(a,b,c)=0 ", "f542f50c17c999db531e574d70416732": "p_1=\\frac {1}{q_3(1+m_2)}\\ ,", "f5435e49f2c26527920dfbfe17180ddd": "g_2=60\\sum_{\\omega\\in\\Lambda\\smallsetminus\\left\\{ 0\\right\\} }\\frac{1}{\\omega^4}", "f543603231c4ea968635d67306f4ddd1": "a_0x^n + a_1x^{n-1} + \\cdots + a_{n-1}x + a_n = 0 \\, ", "f543c88f1d1d45b3c49d49dbe3828b6f": "\\beta_i", "f543dbee19e90b5f04f39ca31b0727eb": "x_{22n+20}", "f544827628869ea2e44757fd4040da94": "\\theta_{n}", "f544dd66d4a09581003eed25cad8f684": "R_E = \\Lambda_E^{-1/2} = {c \\over \\sqrt{4\\pi G\\rho}}.", "f54593e48c05c46744770149e573396b": "\\lambda_f(t)\\leq \\frac{C}{t}.", "f545a283bb7ccb56aa420d1d9263df10": "X|\\{d_0\\}", "f545cd3a36df3ea10ab3c28087b1abcd": "\\frac{p-q}{2} = \\sqrt{\\left(\\frac{p+q}{2}\\right)^2 - pq}=\\sqrt{2550\\tfrac{1}{4} - 100}=49\\tfrac{1}{2}", "f545e8f9a2606d4251ebc4a4d892c1e8": "\\frac{1}{f} = (n-1) \\left[ \\frac{1}{R_1} - \\frac{1}{R_2} + \\frac{(n-1)d}{n R_1 R_2} \\right],", "f546030738f711a92d8b6a6aa0945f9f": "=(b_1 b_2 - a_1 a_2) + (a_1 b_2 + a_2 b_1) * i", "f5461270573fe0aa0f29f01c98446849": " \\hat{f}(\\omega) = \\frac{1}{(2\\pi)^{n/2}} \\int_{\\mathbf{R}^n} f(x) e^{- i\\omega\\cdot x}\\,dx ", "f546178671eea9d0959ab57cb6446c03": "\nf(x)=\\frac{1}{2^{\\frac{1}{2}}\\Gamma(\\frac{1}{2})}x^{-\\frac{1}{2}}e^{-\\frac{x}{2}}\n", "f5469300872e351938826b403d390e4b": "|\\psi\\rangle=a_1 |S_1\\rangle + a_2|S_2\\rangle = \n\\begin{bmatrix} a_1 \\\\ a_2 \\end{bmatrix}\n", "f5469bf42bc9fd4551dd6950537a479c": "\\exp(2g_\\text{threshold}\\,l)", "f546b9f9bf7c3da5f59c74050f222fae": "Y_{10}^{-10}(\\theta,\\varphi)={1\\over 1024}\\sqrt{969969\\over \\pi}\\cdot e^{-10i\\varphi}\\cdot\\sin^{10}\\theta", "f546ed463186539ad32920349e8bb983": "(\\xi, \\tau; \\xi', \\tau')", "f5473e14a86d60af72a6c3593d4a19c1": "F(\\rho, \\sigma) = \\max_{|\\psi_{\\sigma} \\rangle} | \\langle \\psi _{\\rho}| \\psi _{\\sigma} \\rangle |", "f547912f879815ada3b48f6d414dd14c": " (q^2,q^3)", "f547a4b013c6ead4af8cb87434c9403c": "P^{-1}=P^\\text{H}, ", "f547b07d67ce0937cb005231b6906087": "\\sqrt{S} \\approx N + \\frac{d}{2N} - \\frac{d^2}{8N^3 + 4Nd} = \\frac{8N^4 + 8N^2 d + d^2}{8N^3 + 4Nd} = \\frac{N^4 + 6N^2S + S^2}{4N^3 + 4NS} = \\frac{N^2(N^2 + 6S) + S^2}{4N(N^2 + S)}", "f547c933230eed823a30ffa29ca89ca1": "\\sqrt{t + x} = u", "f547e3ab099b39cb700899ea0507001c": "\\Delta_1 = 2 b^3-9 a b c+27 a^2 d", "f547ef4ffb3a444a28e83d773731f0cf": "\\bar{r}_{\\mathrm{geometric}} = \\left({\\prod_{i=1}^n (1+r_i)}\\right)^{1/n}-1", "f5488b890a0c2e0d1629301f000906d2": "A_\\rho \\hat{\\boldsymbol\\rho} + A_\\phi \\hat{\\boldsymbol\\phi} + A_z \\hat{\\mathbf z}", "f548c3ab1497eb3e502964086e08347f": "r^{*}", "f548fe290bcddedcfb35269cdc7543c0": "\\mathrm{La} = \\frac{\\sigma \\rho L}{\\mu^2}", "f549199194d3f83479dcadde8b7948ca": "X_c = 1 \\text{ if } c \\in B(y, pn), ", "f549290b5f1cb644878015a3ca9f4e50": "\nw_k=-\\frac{f(z_k)}{\\prod_{j\\ne k}(z_k-z_j)}.\n", "f5496aec21bfcd228d1339548d96c2c7": " \\beta_j ", "f54997424b435303c607a53bd93ee1c3": "(\\bar{\\bold{3}};\\bar{\\bold{3}};\\bar{\\bold{3}})", "f549f1925543b197e6a3ee46bb0bd246": "\\pi_0 LC(x, y) = C[W^{-1}](x, y)", "f54a6721155390a59d4a1dded3592253": "\\nabla \\times \\mathbf{A} = \\mathbf{B} \\ ", "f54a97c57a19e2b0f130eae9f78a7f38": "X+\\varepsilon U", "f54aad7f61d5dd58a65c90c1e58ce83c": "N\\leq \\infty", "f54abbe7554d31229c2d77d90adf39ef": "r = \\limsup_{n\\rightarrow\\infty} |a_{-n}|^{1/n}", "f54abe8767755df29f937eb42eff015e": " \\begin{pmatrix} -1 & 0 & 0 & 0 \\\\ 0 & -1 & 0 & 0 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 \\end{pmatrix}, ", "f54af77531889c569ccf992f35ca9c2b": "m = AB", "f54b1595aaa96041e9804faa97d21e27": " = 0.2 + 0.08\\,", "f54b2f007bceb013ac8d3db14d75922b": "\\phi (r,t)", "f54b3115867aa7c15ab33d2912d17524": "\\sin (\\mathbb{R}) = [-1, 1]", "f54bbe256f2fa926e3419ad56d92082b": " \\rho = [-2^{k+1},-2^k] \\cup [2^k,2^{k+1}]", "f54bd6ce0e5c9c110d02307a14247c96": "\\frac{d}{dx}\\frac{g(x)}{h(x)}", "f54c4af8511a16d07ef63536fcc86123": "\\mathbf{y}_1,\\ldots,\\mathbf{y}_k", "f54c8dbab312c2ac54245721ca8bf952": "V, \\mathfrak{g}", "f54cb80db706dfd6d148fe7888629c9a": "\\beta_1 = \\frac{-1 - \\sqrt{-3 -4c}}{2}", "f54cfefd570df99ef44ca77fdd8a60f8": " \\delta_p(a+b) = \\delta_p (a) + \\delta_p(b) + \\frac{a^p +b^p - (a+b)^p }{p} ", "f54d4b4848b9757cb75d599e49243e04": "\\mathbf{F} = \\gamma(\\mathbf{v})^3 m_0 \\, \\mathbf{a}_\\parallel + \\gamma(\\mathbf{v}) m_0 \\, \\mathbf{a}_\\perp ", "f54dba0571c1619d1a7008c1447110cc": "\\frac{L_n}{F_n}", "f54dc8a7dcca3c3696fe717fa7a3adb2": "\\psi ( \\mathbf{r} ) = \\mathrm{e}^{ \\mathrm{i} \\mathbf{k} \\cdot \\mathbf{r} } u ( \\mathbf{r} ) ,", "f54df3ef76f9b7c2a3235a4236e4df64": "y = a\\sqrt{(1+\\zeta^2)(1-\\xi^2)}\\,\\sin \\phi\\,", "f54e551389750b7add427c85a170df66": "f:\\mathbb{R}^n\\rightarrow\\mathbb{R}^n", "f54ea610f30eab1b22951c7a02d6e7c9": "f\\!\\left(f\\!\\left(x\\right)\\right) = f\\!\\left(x\\right)", "f54f5aa51458cf0151a1a6971ca99e94": "2^{e_1} \\equiv \\prod_{i=2}^k p_i^{e_i} \\mod p", "f54fb5334a07a4d8fe2560de23fc6972": "\\operatorname{crc}(x \\oplus y) = \\operatorname{crc}(x) \\oplus \\operatorname{crc}(y)", "f54fb86e4359a3d38e5aea8dc32d5bb4": "U=U(S,I_D) \\,", "f54fed54259a9fee69ff200b357bf42e": "g\\in\\ker\\phi", "f55036aae315043e674bc36c4537f0f7": "\\{ \\hat\\theta\\,,\\hat\\sigma\\,, \\hat\\omega \\}", "f550855586723b87bafaee7b25e9fba1": "\\text{DOL} = \\frac{\\text{Contribution Margin Ratio}}{\\text{Operating Margin}}", "f5508f135f3c0f83058524e09bb0fb2c": "x\\cdot y = y\\cdot x", "f5519e71f2a1fd8b37af0f471979acd4": "K(x,t;x',t')=K(x,x';t-t')", "f551b2344cfb64e9db7f0451969531e3": "\\bar{G}_i^{liq} = \\bar{G}_i^{vap}", "f551b79e77a509305ca9f6a6f46f2574": "A_i=\\{1,2,\\ldots,M\\}.", "f5524b72df2612396da8d8abcc0fdf8c": "(\\hat{a},\\hat{b})", "f5527d5576de69aee7cacc7e06d93c01": "y^3 + 2 x^2 y - x^4 = 0", "f55291c0f7961cdbb033524b50ed856c": "2 r_2 = n - r_1", "f552fe4b8485274de34f107bc3a4c004": "=\\mathbf{ \\Omega \\ \\times } \\left( \\mathbf{ \\Omega \\times X}_{AB}\\right) +\\mathbf{a}_B + 2\\ \\sum_{j=1}^3 v_j\\ \\mathbf{\\Omega \\times u_j}", "f55352fc33b8690bdfcd9825063a86b0": "\\frac{|\\langle u,v\\rangle|}{\\|u\\| \\cdot \\|v\\|} \\leq 1,", "f553c38079554e0c9d8f0f0b42863142": "\\overline{3} \\cdot \\overline{3} = \\overline{1}", "f5542c8851ccc9529f543eb512f6d2a6": "\\partial (kx) = k \\partial x", "f554688ed64e854350811625fbb9b011": "p(x,y,t)", "f554c95b344cd6a12ef8ed0f6315c866": "\\left(1/b\\right)^2 e^{\\eta}\\{-2\\eta { \\ }_3\\text {F}_3 \\left(1,1,1;2,2,2;-\\eta\\right)+\\gamma^2", "f554e375fbbb7dc1e42c2be7de84b166": "ab = c_1^2 + \\cdots + c_n^2", "f5553e4d90e5ea72f7b7e0e09e435660": "Q=\\sqrt{\\frac{{R_3}^2 C_1}{R_2 R_4 C_2}}", "f55543443eb264585c550912302fd020": "F_1 \\colon M' \\to M \\times 1", "f55559a8508e40a8c06ebd81f4df88e6": " N_B = \\frac{\\lambda_B}{\\lambda} N_{A0} \\left ( 1 - e^{-\\lambda t} \\right ),", "f5555d4eee2b2805a96a3cbadda637b8": "R_i \\to R = \\prod R_i", "f555f6d8ccf77472ab1d98271e3af683": "\\mathbf{x}_{n+1}=\\mathbf{x}_n-\\gamma_n P^{-1} \\nabla F(\\mathbf{x}_n),\\ n \\ge 0.", "f556780ad69e8f23152c80ff149990ed": "(\\mathrm{C_6H}_{6-n}(\\mathrm{CH_3})_n)", "f55687cbb0078a8017c408cfb7a0dfe4": "C_{t}", "f556896f590e8c3e6e132bc8e876ba40": " \\sigma^2 = \\frac{\\mathrm{Var}[\\nu]}{\\mathrm{E}[\\nu]^2} + 1", "f556ed4462c4c7a59f6f488b23a73d87": "p_0 + p_1 x + \\cdots + p_n x^n", "f55717b62620d0fb9b545ab1e677b5b1": "\\int_0^\\infty\\frac{\\sin^2{x}}{x^2}\\,dx = \\frac{\\pi}{2}", "f5575c4944e4ed3908929560d7ff9df9": "2n ", "f557884223d477f7d581c5676e69bc41": "\\left[I\\!I_{ij}\\right]", "f5579da53652f92cb7fc326ed69d44c6": "F = -iP\\,\\!", "f557a940fb470836cdb07077a0e33a6f": "p\\,{(4a)_+}=1 - \\frac{{\\Omega }^4}{\\mho }", "f5583b8d56733bfa1831991b607001c8": " h(u*v) = h(u) \\cdot h(v) ", "f5584b79a9f753f726f16e7195163038": " r_M = r_{M1} = r_{M2} ", "f55866d1b5380fec20771226c20cc082": "\\mathbf{r}_{k+1}", "f55871c7c7ef67be3a596481e9e28a44": "\\langle S_i S_j \\rangle \\,\\propto\\, e^{-p|i-j|}.", "f5588260abc7ce5cf5d4823e329aebd4": "Y_{r-1}X_{r,r-1}", "f55915d4b58cd102435ce5db89b09c92": "\n \\boldsymbol{\\nabla}\\mathbf{f}(\\mathbf{x}) = \\cfrac{\\partial \\mathbf{f}}{\\partial q^i}\\otimes\\mathbf{b}^i\n ", "f5594a72b3a622efccff5b2a795d54ad": "[b,-3b]'", "f5595ee2f97633a59557ff5c8720bd64": " u = u(x) ", "f5597ecc89d609473e3ff657db4c31b7": "u(c)=c-\\alpha c^2,", "f559828a92f717f5ae5952a08021ddfb": " \n A = \\begin{bmatrix}\n 3 & 5 & 7 \\\\\n 2 & 6 & 4 \\\\\n 0 & 2 & 8 \\\\\n \\end{bmatrix},\n", "f559828b4c257184b6c4bc20be9d02c4": "h=N_{v}\\frac{e^{-(E_{c}-E_{F})}}{k_{B}T}", "f559fbd20bc69a0b174b359e02e68877": "v _{i} \\ ", "f55a15a9c440574b25588e971908aa57": "\\gamma_{LV}", "f55a179c904ecb0108b27a810583eda5": " \\scriptstyle \\omega_2", "f55a7fd35facf1110affe70904ff3297": " l_z = \\hbar \\left ( \\mid \\psi_R \\mid^2 - \\mid \\psi_L \\mid^2 \\right ) ", "f55aa7c5a78e06339bb764bf1f9a3b4a": "\\dot Q_m =\\dot n_3(12T_i^2-96T_m^2)", "f55aadf2b5f46c76f630c7797d186a04": "{C}_{p}^{m}\\left( \\eta ,\\epsilon \\right)", "f55aafbd25d4e72d45777e652c1f8e8b": " \\cfrac{a_1}{b_1+\\cfrac{a_2}{b_2+\\cfrac{a_3}{b_3+ \\ddots}}} ", "f55ae3fbdaf02c2c423db2ef8b669996": " v\\colon \\mathbb{C}[x,y] \\rightarrow \\mathbb{Z} ", "f55b6b0ee32b9cf4c9e309807234849c": "G= G_1+G_2\\,", "f55b982dc3aed436b83119c0c8d758ca": "S,S' \\to X", "f55bceebac0d2315d019f1d244153fe3": "\\lim_{t\\rightarrow b}(x^2(t)+y^2(t))=\\infty.", "f55c1ae70046f11d347076ee60706911": "\\max_{|z| \\le 1}( |P'(z)| ) \\le n\\cdot\\max_{|z| \\le 1}( |P(z)| ) ", "f55c3be477fcff554a6a75894ba2f339": "\nq_r \\equiv \\vec{Q}_r \\cdot \\vec{D} = \\sum_{A=1}^N \\vec{q}^A_r \\cdot \\vec{d}^{\\,A}\n\\quad\\mathrm{for}\\quad r=1,\\ldots, 3N-6.\n", "f55c411f5b121122bc8883c978db5f83": "\\sqrt{5}. \\, ", "f55cc61fb1d63e0cdd75c5d231e89d39": "U = \\infty", "f55ce72c3a64cce870667b1f543913ba": "\\Omega=\\rho+i\\tilde{\\rho}(\\rho)", "f55d253d62499d066cfb16cb428379c6": " E_2 = \\{(x,y) \\in \\R^2 : y = x^3 \\} \\ . ", "f55d5616b8adcbd0d12c2577ca4adbef": "f\\equiv {\\mathrm {id}}_{M}", "f55d87c02da3ebd4d0d629b7c272ceee": "f(t)\\in L^{2}(\\mathbb{R})", "f55d8f0db93743899d83cd70ba729353": "\\tau_{cap}\\left(\\omega \\rightarrow 0\\right) = 1", "f55dc4cdfcc77ad9a6c49175448b4d26": "\\|f\\|_2 \\le K\\|T^*f\\|_1", "f55dd88254ed731424340da8bb760ee9": "I(1,\\pi)", "f55e90c38fd04568e332ee1c3bdf5efd": "r=|\\vec{x}|", "f55f0bd77a280164f62603414c59ff8c": "W(dx) = \\begin{cases} Ax^{-2}\\,dx & \\text{if } x>0 \\\\ Bx^{-2}\\,dx & \\text{if } x<0 \\end{cases} ", "f55f3495cd1a2e40a60ef26e72ec7f05": "e \\in \\mathbb{F}_2^n", "f55f5298b2ec1b6bb3d9b319bff2cebb": "A_i = K_i", "f55f5f5da804205bba7b3adc948b9b8d": "L_i, 1 \\le i \\le m", "f55f8eec2bb90ed8cd96b23d93c355fe": "\\nu_p(m+n)= \\inf\\{ \\nu_p(m), \\nu_p(n)\\}.", "f55fb78d6294090d21ff7432f7ff5469": "\nEr[f_k(t),k\\in\\{N+1,\\ldots\\}]=\\sum_{k=N+1}^\\infty \\int_{[a, b]}\\int_{[a, b]} K_X(s,t) f_k(t)f_k(s) ds dt-\\beta_k \\left(\\int_{[a, b]} f_k(t) f_k(t) dt -1\\right)\n", "f55fcc258bdfecb6830598ad5ff56c24": "\\phi(x)=\\lim x", "f55fcd9745b34e40639d81c49eb07a8e": "f_k", "f5603d4c9c267b62e7bc48e6cbf83f3e": "p\\left(Q|Q_c^{(j)}\\right)", "f56042e3cc6be1bf16cbad63a8995270": "\\mathcal{E}(f_{t}^{\\mathbf{z}}) - \\mathcal{E}(f_{\\rho})", "f56062dc5a2e20c6b2b8c7f0b5b713ea": "C_{ps}", "f560e86653f6969b4e972ecc53b85e7d": "{\\pi\\over 4}\\ {\\pi\\over 3}\\ {3\\pi\\over 4}", "f56115add2ab0b91f3ad2dac9e255669": "{d^2\\over d\\theta^2} + f^\\prime(\\theta)^2 \\,q\\circ f(\\theta) + \\tfrac{1}{2} S(f)(\\theta).", "f561452d45c597009b0e4fd1c181cd43": "\\chi = \\frac{M}{H} =\\frac{M \\mu_0}{B} =\\frac{C}{T} ", "f5614e53298c63d9bfbef19f90c0af46": "C_{e_2} = C_{e_1} - C_{e_1}A^T(AC_{e_1}A^T + C_Z)^{-1}AC_{e_1} .", "f561e8b63f8235d28252a427e00f652c": "\nR_0 = 1 + \\frac {L} {A}.\n", "f561f7eb97327c5f8eb971873c5a5c28": "x_i=y_{r(i)}", "f562201e555f0fa877b69ac88347fb86": "k+l+m=n.", "f562982f2b9250f6b57e88dc41ccb5a3": " |z|^2 =\n\\begin{vmatrix}\n a & -b \\\\\n b & a\n\\end{vmatrix}\n= (a^2) - ((-b)(b)) = a^2 + b^2.\n", "f562a9746ccdf69e475e85fe07f53454": "0 < k < log^2_2 b", "f562c3e1c44fbf31c7f0b52fdb9c07e9": "\\scriptstyle{(r,\\theta)}", "f563016d1d8750dcb006fd1142069901": "\\boldsymbol{\\epsilon}=\n\\left[{\\begin{matrix}\n \\epsilon_{xx} & \\epsilon_{xy} & \\epsilon_{xz} \\\\\n \\epsilon_{yx} & \\epsilon_{yy} & \\epsilon_{yz} \\\\\n \\epsilon_{zx} & \\epsilon_{zy} & \\epsilon_{zz}\n\\end{matrix}}\\right].\n", "f56356cdd508d2766e606bb3ca309950": "y(t)\\in\\R^m", "f56395998cae6d8a6d01e6dc874ceb5e": " D = \\left[\\begin{matrix}x_3 - x_2 & x_1 - x_3 & x_2 - x_1 \\\\ y_3 - y_2 & y_1 - y_3 & y_2 - y_1\\end{matrix}\\right].", "f564c20cb554d263dfb1dde41eeea689": "J^{r}_{p}(\\pi)", "f564d18c2ba4a935e27e7e3c54a6c4f6": "\nb_2=b_1/a_1 = a_2 a_3 \\cdots a_n, \n", "f564e7c6618bc2c0c99eae5a9376fbaf": "E[X]", "f56523e3a1ce20b3e73c79b83120b3f7": "-1 < t \\le 0", "f565fa70c36d465d77cccd4f1aad81bb": "i = 0, 1, \\dots, N - 1", "f5660e2ac15cc5c3efa19f3b07beffde": " z = \\begin{bmatrix} 0 & 0 & 1\\\\ 0 & 0_n & 0 \\\\ 0 & 0 & 0 \\end{bmatrix}, ", "f5660ffd43cdce9848eb125bab78f6fd": "\\tilde{y}_{i+1} = y_i + h f(t_i,y_i). ", "f5663231bc14d9ae41620b48bae19834": "V_{GS}", "f56697bc870e23ceb25a408192fac935": "\\omega=\\lim_{n\\to\\infty}\\frac{\\sum_{1}^{n}\\theta_n}{n}.", "f5671a1968a77e5ae4a93858e3f098a7": "d_1,\\ldots,d_{\\min(N_t, N_r)}", "f56739769e7b4d6da15581e611bcd957": "C^\\infty(K)", "f567434786f219a789c252fe82a3ef01": "\\mathbf{h}", "f567484253115df4fc939e92d2817fa9": "\\sum _x \\psi(x)=(x-1) \\psi(x)-x+C \\,", "f56752d2e301298e20122aeb3cc096e9": "\\{ (x,t)\\in \\mathbb{R}^I_+ \\times \\mathbb{R}^I | x_i \\in \\{0,1\\} , \\sum_{i=1}^I x_i = 1\\}.", "f56796bdba870d00bd82fd6375d51aed": "x_{B100}=300", "f5679d9be0b4c8ce8c0a4c5abeccb549": "\\hat{H}_5 = 4\\mu_B^2 \\sum_{i>j} \\left\\lbrace -\\frac{8\\pi}{3} (\\mathbf{s}_i\\cdot\\mathbf{s}_j)\\delta(\\mathbf{r}_{ij}) + \\frac{1}{r_{ij}^3}\\left[ \\mathbf{s}_i\\cdot\\mathbf{s}_j - \\frac{3(\\mathbf{s}_i\\cdot\\mathbf{r}_{ij})(\\mathbf{s}_j\\cdot\\mathbf{r}_{ij})}{r_{ij}^2} \\right] \\right\\rbrace ", "f567eae71fac3aec7745f228da6dfda7": "P(\\sigma|D,T,M)", "f56840d7408dbb2a79519ef2747a5c81": "1 \\in G", "f568d639f2f8e02143338c9e12c5c05a": "\\boldsymbol\\beta^{\\mathrm T} \\mathbf{x}_i", "f568f2b621aec3fbfde89a47b86e385b": " \\biggl( \\sum_{i=1}^n x_i\\biggr)^k = \\sum_{|\\alpha|=k} \\binom{k}{\\alpha} \\, x^\\alpha", "f568f55ac59dd8fcad6ceb36b59a2e2a": "e^{i\\mathbf{K}\\cdot\\mathbf{R}}=1", "f5690bb14c97eee643cbe24cfc4cec5b": "B=V\\Sigma_2 [ X, 0] Q^*", "f5690f7351c23f521897cbf9925d5a99": " \\begin{array}{c c c} E_f &\\rightarrow & E \\\\ \\downarrow & & \\downarrow{p}\\\\ X &\\rightarrow_{ f} &B\\\\ \\end{array} ", "f56920c667ae458ca66544b2da099527": "d\\sigma^2 = \\eta_{ab} \\, dx^a \\, dx^b", "f5692eacc6ca14445ad5fc19e8ddb3e8": " \\mathrm{Ai'}(x) = - \\frac{x} {\\pi \\sqrt{3}} \\, K_{\\frac{2}{3}}\\left(\\tfrac23 x^{\\frac{3}{2}}\\right) .", "f5696604fa90a4483e1c367a2895412f": "(A - \\lambda I)^k x_k = 0. \\!", "f5697689ddaec0103ac619d7930a362c": "\\bigvee_{i \\in I} a_i", "f56a0c31068adfa7befc72e6fa8bef6f": "\nC_l=\\frac{\\lambda_1 - \\lambda_2}{\\lambda_1 + \\lambda_2 + \\lambda_3}\n", "f56a17bff45073f805e1c7c20cfcdd5f": " \\frac{\\partial y}{\\partial x_1} dx_1 ", "f56a30080bdcbcdca14d9ce541ba7e91": "q \\succ _{\\rm r} q^\\prime", "f56abc1b4ab240262962d28ab10c8e56": " \\textbf{a} = c\\textbf{g} + c\\textbf{f} -qK ", "f56b0e6a238cf2c98c99140a4dd6cbf7": " \\delta \\omega_{B-S} =\\frac{1}{4} \\frac{(V_{ab})^2}{\\hbar^2\\omega_0}", "f56b14fa280b9fac67c20552004b10e4": " \\textbf{g} = -1 +X^2+X^3+X^5-X^8-X^{10} ", "f56b6adf7f1563c39de6ea97f0cd0f1a": " Var( U ) = 2m p^2 q ( \\frac { 1 - R^2 } {-\\log( 1 - R ) - R } ) + p^4 \\frac { (1 - R)^{ -k } - 1 - kR } { N ( -\\log( 1 - R ) - R)^2 } ", "f56bee8f8e25e1dbc89a0b31741aefd3": "(sa)/ b = \\begin{cases} \ns & \\mbox{if } a=b \\\\\n\\varepsilon & \\mbox{if } a \\ne b\n\\end{cases}", "f56c0e88a58fb97b6b959d43c8792a2e": "\\frac{d[A]}{dt}= -k_f[A]_t+k_b\\left([A]_0-[A]_t\\right) ", "f56c19fdc5a7c964f14219660c506156": "q _{v \\setminus w}", "f56c380e937f5a27d8928967b0afc4ba": " \\Sigma_k (x_1...x_k) ", "f56c8e4b091bdf5afe397e60a2f6f0ef": "D^{\\phi} = gD", "f56d3a86b16ba923230f783da3b69520": " \\psi_\\alpha(x,t) ", "f56da18757a246103de10fdd412db204": "\\Delta \\varphi=-2K. \\, ", "f56db054a1c230f2927dccaee24b9a81": "\\mathrm{googol}=10^{100}\\,\\ ,\\ \\mathrm{googolplex}=10^\\mathrm{googol}=10^{10^{100}}\\,", "f56e0081737fcea31d999c75b39958cc": "\\scriptstyle P(z) \\;=\\; (x(z),\\, y(z),\\, z(z))", "f56e0ae1a6db0ea8d9558115257d4a8e": "x_{1}x_{2}\\cdots x_{m}", "f56e95d630ccac17843a8d3eacb413dd": "P_j \\leftarrow jP", "f56e9a4b43a2d9c39f0cfbb0071b9a96": "f(F;P_{\\rm{80}},m) = \\begin{cases}\nP_{\\rm{80}} \\sqrt[m] {\\frac{ln(1-F)}{ln(0.2)}} & F>0 ,\\\\\n0 & F\\leq0 ,\\end{cases}", "f56f222cf9b11d6049a9e8f72f05cdbe": "G(\\rho,\\varphi)", "f56fab4f5b77dd9d007e66536b83254e": "\\mbox{Aut}(\\widehat{\\mathbf{C}})", "f56fbd4922c56c54ecc9208f7528911b": "2 * 2 = 4", "f56fd8964e2f634f6c10edf799cf81c7": "\nM_{k}^{\\prime} \\equiv \\int d\\zeta \\ \\lambda(\\zeta) \\ \n\\left(\\zeta + b \\right)^{k}\n", "f56ff7edda4a159c5940e3ddc6ca13ae": "d + 1", "f5700b74755aefd82994147d6fa2371b": "G\\equiv\\sqrt{4\\xi^2+(4\\xi^2(\\xi^2\\!-\\!1))^{2/3}}", "f57040ac7d9a5a41ab8a406257a77c9f": " \\left| \\frac{a_n}{b_n} - c \\right| < \\varepsilon ", "f570593518883135a8948f3eb140f78c": "\\widehat{\\mathbf{L}} \\psi = -i\\hbar \\mathbf{r} \\times \\nabla \\psi ", "f57070ed51e88c529a8be2ab4f611438": " S_b ", "f5708b10a1db1d6a5ff48a4eb5b6e7e5": "\nx+y+z=1\n\\,", "f570a8985e940210872a636dc3c86f40": "H_n(x)=\\frac{1}{\\sqrt{n!}}e^{\\frac{x^2}{2}}\\frac{d^n}{dx^n}\\left(e^{-\\frac{x^2}{2}}\\right)", "f570c9641fbf6917d7ecccbdb2bb8d8d": "116,640\\,", "f570cd8e0b435233a32f948ba38f5ff9": "\\operatorname{E}(X^{T}AX) = [\\operatorname{E}(X)]^{T}A[\\operatorname{E}(X)] + \\operatorname{tr}(AC),", "f570e1cf266d0ace43989bc03f945bc0": "p_e\\,\\!", "f57115eb86875681d1cff37dbe68cb25": "[x^2 : xy : y^2 : xz : yz : z^2 ],", "f5711d0092c3c8c2a83fc21c1da1dcc5": "\\beta =0", "f57123308efd8c3cf6f1049a923e6650": "(A \\cup B) \\times (C \\cup D) \\neq (A \\times C) \\cup (B \\times D)", "f5717aaf8eeec9d09766f637d96af42d": "\\Delta m^a=(\\gamma-\\bar{\\gamma})m^a+\\nu l^a-\\bar{\\tau} n^a\\,,", "f571ec27ab011d5443fc19bc046f2af8": "d(p, q) = \\lVert p - q \\rVert.", "f57265f11619bfe7bf3029e749a85df8": "{NPSH}_R = ", "f5726ac04841e17cdba673e4c1b9528c": "f\\to\\widetilde f", "f57296d204a844744f1146b395f43f9d": "\\lim_{n\\rightarrow \\infty} \\frac{M(n,k,l)}{{n \\choose l}/{k \\choose l}}=1", "f572b97969f057afdc920caed9a8bb3d": "\\Psi(\\vec{r})", "f573ea3517c053d76e73cdd285920cde": "f(x) = 1/2 + \\sum_{k=1}^n\\cos(kx).", "f573ef4b7bef63ecb7814b7f8e3d0f18": " (X_i, Y_i) ", "f5740a42ec5ff783f5685f8d1be3e432": "B(\\infty)", "f57477d1f8be77b3ead10e3bc590e82e": "\\frac{32}{27}", "f57489d845bfc21303b3455fdbe4178b": "\\ \\Delta^t(a\\ \\alpha_{i,j,k} + b\\ \\beta_{l,m,n} )= a \\ \\Delta^t(\\alpha_{i,j,k}) + b \\ \\Delta^t(\\beta_{l,m,n})", "f574aa2039805a9c1283398934788232": " - ", "f574ad93552f41ba15a8e7d97b96ee0c": "\\frac{\\partial u}{\\partial t} + \\nabla\\cdot\\mathbf{S} = - \\mathbf{J}\\cdot\\mathbf{E},", "f574fbfc110fe2778dd750781fa599a7": "x^2 + y^2 = 1", "f575397905e6bda13773ea56679bc797": " \\mathbb{E} \\{ \\} ", "f575afa75f9a55bab79f70f9ebc6ede4": "\\rho^{T_B} := I \\otimes T (\\rho) = \\sum_{ijkl} p^{ij} _{kl} |i\\rangle \\langle j | \\otimes (|k\\rangle \\langle l|)^T = \\sum_{ijkl} p^{ij} _{kl} |i\\rangle \\langle j | \\otimes |l\\rangle \\langle k| ", "f576251c42ca6313baca378e75256eab": "P \\Rightarrow_i Q", "f5764a27528d327b3fa4386cc05e8220": "L=\\frac{P}{1+i}", "f576589d72d8425c581a269ef202430a": "\\pi: X \\to X/\\mathord{\\sim} ", "f57665e10ff8abeb3c3ff7a9d118d398": "\\scriptstyle t", "f576c5406ef3cc0a5e04e78abf8bc3fc": "\\mathrm{r} = \\frac\n{\\left|P_1-P_2\\right| \\left|P_2-P_3\\right|\\left|P_3-P_1\\right|}\n{2 \\left|\\left(P_1-P_2\\right) \\times \\left(P_2-P_3\\right)\\right|}", "f576ceaca91f6388163617f3f28d49c7": "\n\\mathrm{ERBS}(f) = 11.17 \\cdot \\ln\\left(\\frac{f+0.312}{f+14.675}\\right) + 43.0\n", "f5770ce8f430f05934974444c7cea3e2": " B_{2} < (\\frac{3}{4}- \\mu_{2,1}^2)B_{1} ", "f57710a059d639af2c5de5a7215b78b7": "\n \\dot\\theta^1(t) \\equiv \\omega^1.\n", "f577276bd77b14c1069bc2f55da94419": "F_j(x,b) = \\frac{1}{\\Gamma(j+1)} \\int_b^\\infty \\frac{t^j}{\\exp(t-x) + 1}\\,dt.", "f5775a0d9646651d50f7c2aa671e2ebb": "|\\mathbf{v}(t)|\\;", "f5775d5f927624ff24ab3eb545759bd2": "\\frac{\\theta \\vdash \\psi \\quad \\theta \\equiv \\phi}{\\phi \\vdash \\psi}", "f577dd9edeca24cbec91c7dc6d380dce": " ( v + \\omega ) ", "f5782d02ca3a7cf4f5229646612fb3d0": "\n\\vec J_1\\left( \\vec x \\right) = a_1 \\vec v_1 \\delta^3 \\left( \\vec x - \\vec x_1 \\right)\n", "f57837126d1e1656243211f1cb73cdf4": "P(x)=\\sum_{i=0}^n a_i x^i", "f5783984f9eb093930c9870d1775fcac": "\\ge H_q^{-1}(\\frac{1}{2}-\\varepsilon) \\cdot 2k", "f5784308bd4fa3be7f1196d44aedac9a": " x = m_1 - m_2 = (-12.74) - (-26.74) = 14.00 ", "f5787cad4568f96de6dfa1bea747cf0d": " |B_q(\\boldsymbol{y}, \\rho n)| =|B_q(\\boldsymbol{0}, \\rho n)| ", "f5787e3da66f04b9c34e71450347d438": "\\begin{align}\n \\Delta n_{\\text{E}} (x'') &= n_{\\text{E}0} \\left(e^{\\frac{q V_{\\text{EB}}}{kT}} - 1\\right) e^{-\\frac{x''}{L_{\\text{E}}}} \\\\\n \\Delta n_{\\text{C}} (x' ) &= n_{\\text{C}0} \\left(e^{\\frac{q V_{\\text{CB}}}{kT}} - 1\\right) e^{-\\frac{x' }{L_{\\text{C}}}}\n\\end{align}", "f578a666856e91c0c3b6a6dc4545650e": "\\lambda\\geq 0.\\,", "f578b8b6bd04bbbecdf6b8c1c6c5b3e7": "\\operatorname{pos}(\\Omega) = 1", "f578ef19c60d14bdedf0acd7f40fa174": "\n\\frac{\\sigma_{r-1}(n)}{n^{r-1}\\zeta(r)}=\n\\sum_{q=1}^\\infty\n\\frac{c_q(n)}{q^{r}}\n", "f57950231b79abcd3364e64542c7f480": "\\frac{a}{6}[\\text{1 1 2}] + \\frac{a}{6}[\\text{1 1 -2}] \\rightarrow \\frac{a}{3}[\\text{1 1 0}]", "f5795aeeccbf36db41b7d0b9a5028b7c": "{}_{E_{ab}=E_{[ab]}+E_{(ab)}}", "f5798dd098c6d642abdf5f469217310d": "r^2 + s^4 =t^4", "f579e3653d83ecb5ec58f8c6a6545b2a": "\\mathbf x=(x_n)_{n\\in\\mathbb N}", "f57a07a27dc82335f816a1b3d6d9fd1e": "\np_{\\gamma\\dot{\\alpha}}A_{\\epsilon_1\\epsilon_2\\cdots\\epsilon_n}^{\\dot{\\alpha}\\dot{\\beta}_1\\dot{\\beta}_2\\cdots\\dot{\\beta}_n} = mcB_{\\gamma\\epsilon_1\\epsilon_2\\cdots\\epsilon_n}^{\\dot{\\beta}_1\\dot{\\beta}_2\\cdots\\dot{\\beta}_n}\n", "f57a0b11b52cce2e38861aa2993501e8": " F_0, F_2 ", "f57a8c06b4c34fbbef6b964e0aae6434": "G = Z/(Z \\cap \\Gamma )", "f57a96232568cbbe2503dbfedd4ad20e": "f_{z(x^2-y^2)} = N_3^c \\frac{z \\left( x^2 - y^2 \\right)}{2 r^3} = \\frac{1}{\\sqrt{2}}\\left(Y_3^2 + Y_3^{-2}\\right)", "f57b1288d0a67818d85a895385d6c7cf": "\\sigma_x \\sigma_p = \\frac{\\hbar}{2} \\sqrt{\\frac{n^2\\pi^2}{3}-2}.", "f57b1f8be310276706340af8a74fc64b": "(100\\bar 1 1000\\bar 1 0)_s", "f57b2a630ddc0fcb09894656ce7ad704": "\\alpha\\neq 0", "f57b355cba7f8f41b91a83de449e79ab": "\\operatorname{tr} \\left( \\gamma^\\mu \\gamma^\\nu \\gamma^\\sigma \\gamma^\\rho \\right) \\,", "f57b395b87e7a805ac346703834d0d26": "\\Delta\\otimes\\bar{\\Delta} \\cong \\bigoplus_{p=0}^k\\left(\\sigma_-\\Gamma_p\\oplus \\sigma_+\\Gamma_p\\right).", "f57b7a3469c06aac8bc2a91c695b5129": " Pr( || Y || > st ) \\le [ \\frac{ 1 }{ c } Pr( || Y || > t ) ]^{ cs^2 } ", "f57bd1e8ff9c97e07f48d440b63273d8": "I^+(p,U) \\cup I^-(p,U) \\cup p", "f57c6423d3a9ee3af641f687239e6b76": "\\dot{q}(t)", "f57c67c81da5995a9e16b765ed6f10af": " \\Delta{t} ", "f57c73fedb372874f6b691d80f908067": "W_1 \\in \\alpha", "f57c9370b60306d0280803d59c027be7": "2506 = 2 \\times 10^3 + 5 \\times 10^2 + 0 \\times 10^1 + 6 \\times 10^0", "f57cdabd861732ee42043070eb870f95": "\\frac{d\\theta}{d\\zeta}=\\frac{1}{\\xi_d}\\sqrt{\\sin^2{\\theta_m}-\\sin^2{\\theta}}", "f57d17549d99fdf5729e884aa1a3a6be": "\n\\hat{\\kappa}_1 = \\hat{\\kappa} - \\frac{A_p(\\hat{\\kappa})-\\bar{R}}{1-A_p(\\hat{\\kappa})^2-\\frac{p-1}{\\hat{\\kappa}}A_p(\\hat{\\kappa})} ,\n", "f57d26fb81c5f6c4832946f50c7a5ba1": "R=a_0+a_1 R_1(1) + \\cdots +a_m R_m(1)=a_1 A_1(1)+ \\cdots +a_m A_m(1).\\,", "f57da0d0468e7b9760f18f3c43d00621": "\\mathbf{C}_{1,2} = \\mathbf{M}_{3} + \\mathbf{M}_{5}", "f57dbcc25829844c3bb2db179bdbe068": "\\scriptstyle w_{\\mathbf\\xi}", "f57dd01fdcd05b5ef5d61afddcc5a424": " Au = v(\\boldsymbol{x}) = \\int_{\\mathbb{R}^n}\\frac{f(\\boldsymbol{x},\\boldsymbol{\\theta})}{r^n}u(\\boldsymbol{y})\\mathrm{d}\\boldsymbol{y}", "f57df8f53c17cd41fb903320df17cf73": "s, h \\models P \\ast Q", "f57e1a10a9ef1502d7960153402ca3c8": " \\Lambda (B)= E[{N}(B)] , ", "f57e477214adfa4882ff950f30c6f9fa": "\\partial_ig_{jk}=\\langle\\nabla_{\\partial_i}\\partial_j,\\partial_k\\rangle+\\langle\\partial_j,\\nabla^*_{\\partial_i}\\partial_k\\rangle=\\Gamma_{ij,k}+\\Gamma_{ik,j}*=0", "f57e55c6058597df0fe0e11abbb5a7c5": "\\textstyle \\langle e_i, e_i\\rangle = ||e_i|| = 1", "f57ef9fb665bf95bf17f0450779b563d": "P_{+i}", "f57f1d1abe6077c7e85ed2a80bab1bac": "2-\\sqrt{3}", "f57f25af5622c23bc14d970587f6e601": " \\Phi = \\Phi_k + \\Phi_e", "f57f3c03dcf057871937d024276b17f9": "a = 2r", "f57f69cca93cc9235710ab92aaba38e9": "(U,u)=(U,0.5)", "f57fd00a3911619fd14074a8c3f9bb9e": "\\bigcup_{\\alpha\\in\\lambda} X_\\alpha \\in \\Sigma", "f58012e81ad4d035e410416cee10b4f4": "sen(\\Sigma)", "f58018799ed4b90f97b95638d7bfa7d6": "I(Y;Z|X)", "f58047e2cc7913350b8ead4f8a5536e1": "y^2 = x(x-1)(x-\\lambda)", "f5807b2290e2e42b583c27db80ec2984": "i;", "f5807efc96745fb68dfea04c101cb8f7": "\\langle r,s \\mid r^{2^{n-1}} = s^2 = 1, srs = r^{2^{n-2}-1}\\rangle\\,\\!", "f58092542eabf2ae5c2a7e0160f756e9": "\\psi_C(x)= B_r e^{i k_1 x} + B_l e^{-i k_1x}\\quad 0 0 ", "f59ba64743acb73533e2e8ccce866c56": "R_{5,1} = 330 r^5-840 r^4+756 r^3-280 r^2+35 r", "f59c93fcfab1d90e07162b16732d4c20": "z^2+x_1^2+x_2^2+\\cdots+x_n^2=1.", "f59cc8cf92f57f43cce3f574d5de9971": "\\mathrm FM\\,", "f59cd53b42ecf23092e11fe1d6bfd9ad": "sG_{AW} = \\frac{G_{AW}}{FRC}", "f59d3c10502324e7653851f626287fc2": "s,s'\\in X", "f59d6f2db6cd373352930574e9928976": "\\chi (R/P,R/Q) > 0.\\ ", "f59dbb869190b3cc2f42f683dcd57843": "m[i,\\,w]=\\max(m[i-1,\\,w],\\,m[i-1,w-w_i]+v_i)", "f59dc332aa4f90b367e3f073ba589ee3": "x_i = \\frac{n_i}{n_{tot}}", "f59df8ca59cbfe40d3bd6406d17b85fb": "\\ \\|y[n]\\|_{\\infty} < \\infty", "f59e94ceac949d2592cdccc9b37e4a84": "\\zeta(2) < 2", "f59ecaa49b818e88f087c6a1416f1d6b": "f_n(z)=\\frac{a_n \\zeta }{1+z}.", "f59f3027ac99ffcdd0372208f6866c3a": "\\langle\\xi(t_1)\\xi(t_2)\\rangle = 2 k_B T\\,\\gamma\\, \\delta(t_1-t_2).", "f59f6fdc329dd37386a0617b192eef7e": "\\operatorname{Spin}(n) \\to SO(n).", "f59f98f5526973ef1dfa27db38176093": "q = -\\left(1+\\frac{\\dot H}{H^2}\\right).", "f5a01f3d408c41d6ca61d929c8561fe2": "1_G = G\\varepsilon \\circ \\eta G", "f5a05df5df61a8dcb0e0c01514d6ba38": "\\mathrm{C}(\\varphi_i) \\subsetneq \\mathrm{C}(\\varphi_{f(i)}).", "f5a07abd6f0c32677f6cbc24d19addcc": "\\phi(r)=\\varphi_0(\\sin\\omega r/r) ", "f5a07decf21e3d2d6c664162a59c2772": "I \\subseteq \\mathbb{N}", "f5a0af392fad8715dc681ff870cf6ea7": "\\frac{d\\mathbf{L}}{dt} = \\mathbf{r} \\times \\mathbf{F}_{\\mathrm{net}} = \\boldsymbol{\\tau}_{\\mathrm{net}}.", "f5a0ef6d52e9d57d8d1fb8317a8410a8": " \\zeta(1/2+i t) \\ll t^\\theta \\log t ", "f5a0efa04a98dbf284695513dae835ef": "\\gamma(\\mathbf{u}) = \\frac{1}{\\sqrt{1-\\frac{\\mathbf{u}\\cdot\\mathbf{u}}{c^2}}} = \\frac{1}{\\sqrt{1-\\left(\\frac{u}{c}\\right)^2}} ", "f5a167cc247206aa0051e556a6978d1e": "\\omega \\ne 0 ", "f5a16e773cebb73370c99ed593282ebc": "f: A \\to \\mathbb{R}", "f5a1adabc2b983ab4d284c7b9605ea24": "E_{u/p} = y_1 + \\frac{q^2}{2gy_1^2}= 6 + \\frac{10^2}{2(32.2)(6^2)}=6.04\\text{ ft}", "f5a1cb8b7f985c9c925e4da04da351a6": "b_{14}=(1-(15/24)b_{10})", "f5a1ee6a6d78e68f0b090b2c1b635009": " A/A_{N-1}/ \\ldots /A_0 \\triangleleft s ", "f5a1f44b7e37c4c97816dd712b88496c": "U=3 k_{\\rm B}T\\, D_3(\\hbar\\omega_{\\rm D}/k_{\\rm B}T)", "f5a2cf05c1dc22fe04752a671c2da99c": "D_{CB} = \\frac{\\frac{4\\times 2EI}{L}}{\\frac{4\\times 2EI}{L}+\\frac{4EI}{L}} = \\frac{\\frac{8}{10}}{\\frac{8}{10}+\\frac{4}{10}} = \\frac{8}{12} = 0.(67)", "f5a2eb8f9bd6111371e7341b6604a833": "a\\ell=b^2.\\,", "f5a2f39b2f0c3dd2443514998195fe84": " T, R. ", "f5a2fda069804347eae4904cf77423b5": "\\Pr[A \\cup B] = \\Pr[A] + \\Pr[B]", "f5a35820b0beaa4738a662f3edf55cd8": "\\displaystyle{\\mathfrak{g}=\\mathfrak{z} \\oplus \\mathfrak{g}_1\\oplus \\cdots \\oplus \\mathfrak{g}_m,}", "f5a3757df425b662781ee7bef4721b64": "\\Delta(x)=x\\otimes1+1\\otimes x", "f5a3fa42eb146abed0fb19f3a29ee4a5": "\ns_{\\lambda+1}\n =s_\\lambda- \\frac{P(s_\\lambda)}{\\bar H^{\\lambda+1}(s_\\lambda)}\n =s_\\lambda-\\frac{W^\\lambda(s_\\lambda)}{(W^\\lambda)'(s_\\lambda)}\n", "f5a41855485819bddbb53526fa876065": " C = C(\\varepsilon,t) ", "f5a44d9b76003ff83b2b3fd484c941fa": "t^2 (3 - 2 t)", "f5a44e5ec0c22e095a417f4d93080b17": "\\beta = \\pm i", "f5a4942de17f461783da2761f55f57b6": " (E_a)^2 = (m_k c^2 + m_t c^2)^2 ", "f5a4fd2c768514b961865a892ec847ac": "\\ 3\\sigma_R", "f5a52b7a2629aca009f7169677c9be19": "\\Leftrightarrow |PA| = 3 |PB| ", "f5a5370869516209010bc8c088c5cc89": "u(z)=z+z^2", "f5a59d940d7fdb0ba5326f2021767b9b": "\n \\begin{align}\n g(l^k,z,r^k):=\\hat{X}_{Bayes}\\left( (\\Pi\\Pi^\\top)^{-1}\\Pi \\mu\\left( z^n,l^k,r^k \\right)\\odot\n \\pi_z \\right)\\,.\n \\end{align}\n ", "f5a5a437e4d32b2bb717e1ae9cb2c3b4": "V = \\frac{TM\\otimes{\\mathbb C}}{L\\oplus\\bar{L}}", "f5a5bbdc528e7d90c68f54fd4add7bc6": "= -4/8=-1/2 \\,", "f5a5e031639f90c9ba2746ea3966b144": "C_1 = G_0 + P_0 \\cdot C_0", "f5a60d093f1bded026a69e58d297493a": " V = V(\\bold{r}_1,\\bold{r}_2\\cdots \\bold{r}_N,t) ", "f5a64a3926680ae103bcaba331ee768e": "F_\\mathrm{d} = - c v = - c \\frac{dx}{dt} = - c \\dot{x}.", "f5a69a23b5e9069c349858ea68f4f3f9": "P(s)|_{s = \\beta_m} = 0 ", "f5a6c162a97d9f635d1824a4acab5932": "p = x^2 + y^2,\\,", "f5a6c3d7e6dea59a41694fd7cebb8a46": "\\mathcal{T}(A)\\Rightarrow_{mem} M", "f5a7464acead110adbcc8fd74b10187f": "\\zeta=\\frac{Ab+B}{Cb+D}\n\\quad \\text{ and } \\quad\n\\theta=\\frac{Ac+B}{Cc+D}", "f5a75047949661e080cc96832267d1c0": "s = \\sqrt{h^2 + \\frac{1}{4}a^2} = \\sqrt{\\frac{1}{2}a^2 + \\frac{1}{4}a^2}=\\frac{\\sqrt{3}}{2}a", "f5a76bb569ddc6176e41957ab2220081": "C \\neq \\emptyset", "f5a789b54ce6291f542bd5d18a16bac5": " \\sum\\limits_{n=1}^{\\infty} d(n)x^{n} = \\sum\\limits_{a=1}^{\\infty} \\sum\\limits_{b=1}^{\\infty} x^{a b}", "f5a86e6e167222c8efd614c42d7c006a": "F_{12},F_{13},F_{23}", "f5a90c1fa93bcfff7236b9dcc9615198": "\\alpha \\simeq 1", "f5a96fb947665879a3dd27d5fadc4980": "\n EI~\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d}x^4} - \\frac{3}{2}~EA~\\left(\\cfrac{\\mathrm{d} w}{\\mathrm{d}x}\\right)^2\\left(\\cfrac{\\mathrm{d}^2 w}{\\mathrm{d}x^2}\\right) = q(x)\n ", "f5a977afbfac3cb119468db3256bb6fb": "H_*(LM)^{\\otimes p} \\to H_*(LM)^{\\otimes q}", "f5a982438824c15e09f17a6131614160": "b_{12}-a_{12}", "f5a9c4ec89daf954f6b45a17ada0e219": "\\frac{\\partial \\theta}{\\partial t}= -\\frac{\\partial}{\\partial z} q ", "f5aa4f4a8b0cc645d8081535920e3f34": "\\frac{d}{dx}\\sum_i\\alpha_i B_{i,k}=\\sum_i(k-1)\\frac{\\alpha_i-\\alpha_{i-1}}{t_{i+k-1}-t_i}B_{i,k-1}", "f5aacce0c5f179291590923d9f75014e": " \\boldsymbol{f}(t)=f_x(t) \\hat{\\boldsymbol{\\imath}}+f_y(t) \\hat{\\boldsymbol{\\jmath}}+f_z(t) \\hat{\\boldsymbol{k}}\\ , ", "f5aaf0f60f5a61cbfd3ab810ac18024c": "c \\in K_m", "f5ab2d6d2610f01f8dbabb2c83066974": "\\sigma \\in \\Delta = \\Delta_1 \\times \\dotsb \\times \\Delta_N", "f5ab63b78a8897cff30a3dda456224f6": "m\\geq 0", "f5ab7108ee24e91fcc836e8b1c63cf64": "\\ell(I^n / I^{n+1})", "f5ab7ba8bf0bc532a324c44bd08f1cdc": "\\displaystyle{Uf(x) = \\pi^{-\\frac{1}{2}} (x + i)^{-1} f(C(x))}", "f5ab88471b2365266e85c7840a63edd5": "\\vdash \\!\\,", "f5ad331b5aa96a387a7a8dc7ab05555a": "a= \\ k_1 (x_2 - x_1)-(y_2 - y_1),", "f5ad4ccb2285497acc83de21eab7aa84": "\\nabla\\cdot\\mathbf{A} + \\frac{1}{c^2}\\frac{\\partial\\phi}{\\partial t} = 0", "f5ad577446b5373391b2bb52b03f0a4b": "\\begin{align}\n& \\mathbf{(D-\\omega L)^{-1}} = \\frac{1}{12} \\begin{pmatrix}\n2 & 0 & 0 \\\\\n0.55 & 3 & 0 \\\\\n1.441 & 0.66 & 2.4\n\\end{pmatrix},\n\\end{align}", "f5ad5b33b172b7074dbb963274fadea3": "\\overline D", "f5ad6a0704b1469da6bbda3eff01e5a1": "C_i = \\frac{2|\\{e_{jk}: v_j,v_k \\in N_i, e_{jk} \\in E\\}|}{k_i(k_i-1)}.", "f5ad97bb667abaf0d345d4c19071c0fc": "\\operatorname{E}(TX)=\\mu \\frac{\\alpha}{\\beta}", "f5adba079e2606b25fd1c2eba30a22ca": "j = -m - 1", "f5adca4ee0adb6b5dd738819eb005b46": "C_t ", "f5add390621b424bcbe8888b10277ee0": "\\Psi=A \\frac{e^{i\\left( k (\\frac{x^2-(N-1)ax}{2 L}+L)-\\omega t +\\phi\\right)}}{L} \\frac {\\sin\\left(\\frac{Nkax}{2L}\\right)} {\\sin\\left(\\frac{kax}{2L}\\right)}", "f5addcc57a7edd853f3f6a136a5befae": "\\cfrac{[S]}{[S]+K_m} ", "f5ae37ade49e28c2127d767fd8989084": "\nU_{\\mathrm{eff}} = U(r) + \\frac{L^{2}}{2 m r^{2}}\n", "f5ae880c9d84a2a687893d42e30ddcb3": "\n\\begin{align} X^{\\rm VV} &= X^{BS} + w_{RR} ({RR}^{mkt}-{RR}^{BS}) +\nw_{BF} ({BF}^{mkt}-{BF}^{BS}) \\\\\n &= X ^{BS} + \\vec{x}^T(\\mathbb{A}^T)^{-1}\\vec{I} \\\\\n& = X ^{BS} +\nX_{vega} \\, \\Omega_{vega}+ X_{vanna} \\, \\Omega_{vanna} + X_{volga} \\, \\Omega_{volga} \\\\\n\\end{align}\n", "f5aeddddf75a400e17cefb1e82eb604b": " d >> \\sigma ", "f5aee99b80dfb92edeadb7e98f5abe9c": "\\sqrt {-g}", "f5af05b457f120d9f20e1090f2839b0f": "\\pm t^n", "f5af3a9c1821d7054a10a41078421cfb": "\\,S_y = \\frac{h}{2} \\sigma_y", "f5afd18c29789e595b333394cbc91ee6": "R = \\mathrm{clamp}(Y + 1.402 \\times (Cr - 128))", "f5afda970d4dcf11f10939fdbf1f6415": " dA = 2 \\pi r \\, dr = 2 \\pi R \\, dy = \\pi R \\, dx .", "f5aff8bea3738c74088647a7e7dafc40": "\\text{up quark} \\rightarrow \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}, \\qquad \\text{down quark} \\rightarrow \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}", "f5b043a86cc253260ef7f50a6ee7e17f": "(11)\\qquad \\sigma_{ab}=-\\sigma \\bar m_a \\bar m_b-\\bar\\sigma m_a m_b\\,,", "f5b04d59cb9ba70c5aaadec373588b86": "\\!\\left(0,a/2\\right)", "f5b064a1f01cd4c0059fe97383c8bd11": " \\operatorname{build-param-lists}[o\\ x, D, V, T_9] \\and \\operatorname{build-param-lists}[y, D, V, K_9] ", "f5b143a8c7a21289ae550f2c4116ef16": "Q(x) = 1 - \\Phi(x)", "f5b1b1d245ad0cb5f1029e4a3adf9c7d": "\\nu_{t_{\\pi (1)} \\dots t_{\\pi (k)}} \\left( F_{\\pi (1)} \\times \\dots \\times F_{ \\pi(k)} \\right) = \\nu_{t_{1} \\dots t_{k}} \\left( F_{1} \\times \\dots \\times F_{k} \\right);", "f5b1b36c566b56e20d984006a727f83f": "Y_t = f(t) + e_t,", "f5b1c69b58d592b0687e237832e57add": "B^{-1}Ax=\\lambda x", "f5b2320881e0bb9f6a079c08a38ca2e1": "\\nu_{\\rm xy}", "f5b25534cc6e444922c3f27ef30802d5": "\\operatorname{DirMult}(\\tilde{\\mathbf{x}}|\\boldsymbol\\alpha')", "f5b27b80b977a095e097011b8fb40cb6": "x^2 + y^2 = r^2.\\!\\ ", "f5b28daf341543a573e0cb7a93901bdd": "E=T+V", "f5b2fd3aa2886c7027a1562bc09afc92": "l^a\\partial_a=\\partial_r:=D\\,,", "f5b338c83fa4ff0e14d728eb49c0813c": " \\and (S_6 \\implies (\\operatorname{equate}[A_6, x] \\and V[F_6] = A_6)) \\and D[F_6] = K_6 ", "f5b3565026a5e5419d9b14dda0fc5fc1": " y= C -\\frac{1}{2}A_x\\ \\left(\\left(y'(0)+\\sqrt{{y'(0)}^2+1}\\right)\\ \\ln(1-\\frac{x}{A_x}) -\n\\frac{ (1-\\frac{x}{A_x}) ^2}{(y'(0)+\\sqrt{{y'(0)}^2+1})\\ 2}\\right) ", "f5b37e82f8140119cff6e1b3c0db7493": "\n|s_1,m_1\\rangle|s_2,m_2\\rangle=|s_1,m_1\\rangle\\otimes|s_2,m_2\\rangle\n", "f5b3aa35b400424d56b9d890d96b03f5": "10^{2^{32}}", "f5b3d14bab9fede7796adf41f50ade59": "\\ y[n] = x[n] + \\sum_{i=1}^{9} A_i e[n-i]", "f5b4038785054de5cf7041a419a0bca0": "u_{0}^{2}", "f5b417e5b168d33d37af96d9acae0f98": "2^3(1+x)^{3}p(\\frac{1+\\frac{3}{2}x}{1+x}) = x^3+2x^2-8x-8", "f5b516cece296fb5ee98eb0d77a0d6e3": "(t_0,\\alpha (t_0))", "f5b52e90546d6d736c1435168cd5bfa3": "\\Gamma(1,x) = e^{-x},", "f5b58722086c205af74a8f0cc31b3668": "x_0,x_1,...,x_i", "f5b598bdaff48e5d20bb9cfd1c649055": "B_1, B_2, B_3, \\ldots", "f5b5fe493560c85721aa3a0ea0606e98": "-\\frac{1}{n} \\log j(n,X)", "f5b613d380083fd241c44c3d49bbf29e": "\\mathbf u(\\mathbf X,t) = \\mathbf b(\\mathbf X,t)+\\mathbf x(\\mathbf X,t) - \\mathbf X \\qquad \\text{or}\\qquad u_i = \\alpha_{iJ}b_J + x_i - \\alpha_{iJ}X_J\\,\\!", "f5b6d9db5c4949ec20560ecc34dcb166": "T^{ab}{}_{;b} \\, = 0 \\,.", "f5b73c85a1f94c99a1fb0a5f22ab6ecd": " \\text{diffusion vector angle between }B_X\\text{ and }B_Y = \\arctan \\frac{B_Y}{B_X} ", "f5b77ed070554b886781ac1e9805ab18": "x_1, \\dots, x_n \\in I", "f5b7e69d254b50b70c7aa737303673b8": "\\,K(x,y)", "f5b7ff3586967ff932664f3b7b234d11": "= - F^{IK} G_K^{\\;\\; \\;J} + G^{IK} F_K^{\\;\\; J}", "f5b8505b4e4e489f6a38cdee3ab8f3f2": " \\ c_{11} = V[x_0,y_1, z_1] (1 - x_d) + V[x_1, y_1, z_1] x_d ", "f5b85298d61735bb5cd8abfc0b35041b": "(A,K)", "f5b8e421fca74fd4100b9c0358788e3f": "f\\colon E \\to \\mathbb{R}^+", "f5b8f4558cd6c4b2ce17d73e4afd16b5": "D(A)=\\{f \\in L^2(\\mathbb{R}, d\\mu) \\mid xf(x)\\in L^2(\\mathbb{R}, d\\mu)\\}", "f5b95502c5d27b11f512b0c9ffcdc6b8": "\\,m(y)", "f5b95d5e19e51800563896952d1e5d46": "y\\left(n_1,n_2\\right)=\\sum_{l_1=0}^{L_1-1}\\sum_{l_2=0}^{L_2-1}a(l_1,l_2)x(n_1-l_1,n_2-l_2)-\\sum_{k_1=0}^{K_1-1}\\sum_{k_2=0}^{K_2-1}b(k_1,k_2)y(n_1-k_1,n_2-k_2)", "f5b961138f4a6b425a50f321248b36e3": "\\ge \\geq, \\gneq, \\geqq, \\ngeqq, \\gneqq, \\gvertneqq \\!", "f5b9a2bb267dbdd752444e0cd13824c9": "\\mathfrak{P}^{73}", "f5b9b1664bb9a4bebc274274febbef9f": "p^* = 1 \\text{ (mod }4),", "f5b9b521a904a60d594e4942a7a404d6": "\\left \\| A \\right \\| _2=\\sqrt{\\lambda_{\\text{max}}(A^{^*} A)}=\\sigma_{\\text{max}}(A)", "f5b9cd4bce1cac12fa5ca4ac88796f33": "s_{i} = \\frac{dH_{0}^{(i)}}{dx}\\Bigg|_{x = 1} = 1 + G_{0}^{(i)'}(1)\\frac{\\sum_{j}e_{ij}H_{1}^{(i)'}(1)}{\\sum_{j}e_{ij}}", "f5ba24d0dde67cfe5ff186023987bb2f": "GR_L", "f5bab4170a7b1f8e3f13277a497e6def": "b_m = 0", "f5bacdcb3d52a1de19d10277587fd3dd": "H = -\\ln\\left(\\frac{\\phi(q)}{2\\pi}\\right)+\\frac{1}{2\\pi}\\int_\\Gamma \\left( \\sum_{n=-\\infty}^\\infty\\sum_{k=1}^\\infty \\frac{(-1)^k}{k} \\frac{q^{(n^2+k)/2}}{1-q^k}\\left(z^{n+k}+z^{n-k}\\right) \\right)\\,d\\theta", "f5bc29bf5630b35e9175e550cfe751d5": "\\epsilon\\approx\\epsilon_F", "f5bcb3698808ba344957c9deeb1717f6": "\\frac {By_1^A}{(y_1 + \\lambda)^N}\\le \\frac {M}{(y_0+\\lambda)^N}", "f5bcd555fe17a6917ef3c81f1f85dd10": "110 - S_T", "f5bd462369168bb05221c9d99b8a69f2": "\\exp(-\\lambda S^6- aS^4-bS^2)\\,dS", "f5bdf4374062be5f5a4fbf847187a8d2": " U_{nit} = \\beta_{n} X_{nit} + \\varepsilon_{nit} ", "f5be202b6d5f80fcbc17b2c68f4a1d30": " a_{0} (c(c - 1) + \\gamma c) = 0", "f5bec72c5200108822aa8a7673e6cc24": "t=\\log(z)+2k\\pi i,k\\in Z.", "f5bedea22ca04da2c37aaab172a49a1d": "s \\in \\{0,0.5,1,1.5,\\ldots\\}", "f5bee7e30693cc75c2ad1a07283cd293": "\\begin{align}\nZ[j,\\bar\\varepsilon,\\varepsilon] & = \\int [dA][d\\bar c][dc] \\exp\\left\\{iS_F[\\partial A,A]+iS_{gf}[\\partial A]+iS_g[\\partial c,\\partial\\bar c,c,\\bar c,A]\\right\\} \\\\\n&\\exp\\left\\{i\\int d^4x j^a_\\mu(x)A^{a\\mu}(x)+i\\int d^4x[\\bar c^a(x)\\varepsilon^a(x)+\\bar\\varepsilon^a(x) c^a(x)]\\right\\}\n\\end{align}", "f5bf07c0fa858b94b5cb02b34635c4df": " M_{i,l} = 0 \\text{ }\\forall l \\in T", "f5bf1d85fa4fe4e0a41b08d166f245ca": "d_{j} = j^{d_{j-1} / (j-1)} \\ , ", "f5bf7817401232cfc16bf7347e1de279": "\\int\\frac{\\cos^2 ax\\;\\mathrm{d}x}{\\sin ax} = \\frac{1}{a}\\left(\\cos ax+\\ln\\left|\\tan\\frac{ax}{2}\\right|\\right) +C", "f5bfbbf00110a3089f12378fd8368537": "\\,\\phi(A)", "f5bffcc138953e427e2d7aeb93be32ae": "{\\mathcal A}", "f5c035aec0deb032187f05f070a5cdd7": "b_i^*(k)", "f5c0aedb3818390cfa51cd60c7cc8f8c": "\\mid \\mu_{3,1}\\mid >\\frac{1}{2}", "f5c0bdda6ad34d5149c7ca547e6fecf6": "\\ \\cos(\\omega t + \\phi) = \\frac{1}{2} \\Big[ e^{j(\\omega t + \\phi)} + e^{-j(\\omega t + \\phi)}\\Big]", "f5c0ff1005fb3ac4dd1ccd9ba5c07685": "\\langle prog, I_{static}\\rangle", "f5c1cfe0690b6f98b02c4aabf838b426": "\nK = H + \\frac{\\partial G_{4}}{\\partial t}\n", "f5c1e1048354ab7e1500d1243f4d8224": " r_1 = r_2 \\equiv r ", "f5c1fa4ebe3165c568955b1d22ca389e": "d*\\mathbf{C}=*\\mathbf{J}", "f5c259fde2b9242577ac002e72302fac": "g_i(x^*) \\le 0, \\mbox{ for all } i = 1, \\ldots, m", "f5c285e276140bff81ef5b9c13adb7db": "S = IV", "f5c2ad8cdd274d117abbda0ac56b4fc7": " \\vec s ", "f5c2becf67b57bd018c5769d1f1be4b7": "\\Delta = I_{-\\infty}^{+\\infty}\\frac{\\mathfrak{Re}[f(x)]}{\\mathfrak{Im}[f(x)]}= I_{-\\infty}^{+\\infty}\\frac{b_0\\omega^{n-1}-b_1\\omega^{n-3}+\\ldots}{a_0\\omega^n-a_1\\omega^{n-2}+\\ldots} \\quad (25)\\,", "f5c2c345a6e8c37b86d536bad3fa76f8": "i \\geq 1, i\\neq 2^j-1", "f5c2cc1a6ca39b034e4416ae3f705b6b": "\n{v\\over c} = \\tanh(\\alpha) \\ , \\quad {u \\over c}=\\tanh(\\beta) \\ , \\quad\\, {s\\over c}=\\tanh(\\alpha +\\beta)\n,", "f5c304cc2cad57e5b74f11ff35beaef8": "2 \\pi/24", "f5c362d5e981ef8046da86f7b4ec4fdc": "Y_{0,0}", "f5c387d348ea2a7ff8d8ce7bcbc2181f": "\\frac{d^2\\varphi}{dz^2}+\\alpha^2\\varphi=0", "f5c392e46c456c2b6a5a32425f5c8e04": "\nf_{x,g} = gS\n", "f5c394fd8ebee0624a008fab36d386a0": "N=\\sum_{i=0}^{h-1}N_ib^{ik}=\\sum_{i=0}^{h-1}N_i(b^{k})^i", "f5c3bcf7efc17498a2884a4cb9e4b234": "\\mathbf{C} = \\sum_{i=1}^n (\\mathbf{x_i} - \\mathbf{\\bar{x}}) (\\mathbf{x_i} - \\mathbf{\\bar{x}})^T", "f5c3bddc888d2a90b5212f7c73be6629": "X\\mapsto CX", "f5c41494f0be34e07e3d7842fa480177": "\\langle [\\hat{B}^\\dagger]^J \\hat{B}^K \\rangle", "f5c421724894da7134587bf352211976": "\\mathbf{A \\cdot B} = \\begin{pmatrix} A^0 & A^1 & A^2 & A^3 \\end{pmatrix} \\begin{pmatrix} \\eta_{00} & \\eta_{01} & \\eta_{02} & \\eta_{03} \\\\ \\eta_{10} & \\eta_{11} & \\eta_{12} & \\eta_{13} \\\\ \\eta_{20} & \\eta_{21} & \\eta_{22} & \\eta_{23} \\\\ \\eta_{30} & \\eta_{31} & \\eta_{32} & \\eta_{33} \\end{pmatrix} \\begin{pmatrix} B^0 \\\\ B^1 \\\\ B^2 \\\\ B^3 \\end{pmatrix} ", "f5c422a5755c9e260ddc13f74dd9eb34": "f: V \\rightarrow \\mathbf{F}", "f5c48f93ba90af52b7578653f2842263": "\n\\int_0^\\infty \\overline{r}(\\lambda)\\,d\\lambda=\n\\int_0^\\infty \\overline{g}(\\lambda)\\,d\\lambda=\n\\int_0^\\infty \\overline{b}(\\lambda)\\,d\\lambda\n", "f5c4f056bd728935f119a8d50a111fd1": "\\beta = \\frac{\\partial f}{\\partial y}", "f5c52dfa6918874f8c1bf356fc5d3310": "\\left\\{\\begin{array}{ll}\\infty & n \\le 2\\\\ 6 & n = 3\\\\ 4 & \\text{otherwise}\\end{array}\\right.", "f5c5a5b1dc1fd166a71a3e27ed6a037c": "1/T = A + B \\ln(\\rho) + C (\\ln(\\rho))^3 \\,", "f5c6b3c9dfc2886bf1406d57c3b06f23": "T_0 = 2\\pi\\sqrt{\\frac{\\ell}{g}} \\quad\\quad\\quad\\quad\\quad \\theta_0 \\ll 1", "f5c6e1a9f77eaa8f57e79afe4aac24d4": " D = \\begin{vmatrix}p&q&r\\\\\nu&v&w\\\\x&y&z\\end{vmatrix}", "f5c71231364dd5d2ca62a66b3e9913ea": "\\int_b^c f(x)\\,\\mathrm{d}x=\\mp\\infty", "f5c733dbc620baef11f75b146882f484": "b=\\beta(e)", "f5c7e04cb56756d149524c2e59b62ec4": "\\ f_2 = (1,1)", "f5c810555c67a032820590d8a2cfca68": " P_1 g P_2 \\ \\rightarrow \\ P_1 h P_2 ", "f5c84a704d13face1e2fa63b579efcb3": "LV = \\alpha_1\\partial_\\nu T_1 - \\alpha_2\\partial_\\nu T_2,", "f5c8636367c737e4edd967beb457bbad": "s \\in [1,n] ", "f5c874a7b0821c27540e9fe051bdc024": "\\gamma_{LG}\\ ", "f5c8e20adb62ebb6bfc26a7e17625a66": "\\begin{align} Z(\\beta,J) \n& = Z(\\beta,J_1,J_2,\\dots) \\\\\n& = \\sum_{x_i} \\exp \\left(-\\beta H(x_1,x_2,\\dots) +\n\\sum_n J_n x_n\n\\right)\n\\end{align}\n", "f5c8e4b2e543b20be138702663cc0a88": "T \\frac{1}{2} \\\\\n\\frac{1}{2} & \\mbox{if } |t| = \\frac{1}{2} \\\\\n1 & \\mbox{if } |t| < \\frac{1}{2}. \\\\\n\\end{cases}", "f5e92f9bd5e242b39a2e46ee6de66c94": "\\zeta(-2n)=0.\\,", "f5e94042f535abddbf0241ff3811dbb9": "k_{xo}=\\sqrt{{k_{o}^{2}\\varepsilon _{r}}-(\\frac{m\\pi }{a})^{2}-\\beta ^{2}}= k_{o}\\sqrt{\\varepsilon _{r}-(\\frac{m\\pi }{ak_{o}})^{2}-\\frac{\\beta ^{2}}{k_{o}}}", "f5e962964e3c1552f57cb5fb1db8a5c0": "E(B)>E(A)", "f5e971fb2f220821f51cfccb71228c18": "H_{E}/H_{NE},", "f5e9cd2abd35c06092736bbb1969c342": " 2\\pi r^2 + 2\\pi rh = 2\\pi r(r+h) \\, ", "f5e9e4b20183388180a41526fd3e957b": "H(x)=\\lim_{ \\epsilon \\to 0^+} -{1\\over 2\\pi i}\\int_{-\\infty}^\\infty {1 \\over \\tau+i\\epsilon} \\mathrm{e}^{-i x \\tau} \\mathrm{d}\\tau =\\lim_{ \\epsilon \\to 0^+} {1\\over 2\\pi i}\\int_{-\\infty}^\\infty {1 \\over \\tau-i\\epsilon} \\mathrm{e}^{i x \\tau} \\mathrm{d}\\tau.", "f5e9e4d8a9ed778f2be5ff12472578ce": "X_0 \\cap X_1 \\subset X \\subset X_0 + X_1,", "f5ea6d090ad52688ba5cfe29bfb5b1ae": "-0.2 \\leq x_{12} \\leq -0.1", "f5ea8fc51d8c84efe2b409b28ed685eb": "\\, r=a+b\\theta", "f5eac7ec5040b2effcae91e95aebb218": " \\operatorname{lambda-named}[\\lambda F.X] = \\operatorname{false}", "f5ead40fe9092771c27c5151238d96d2": "\\frac{\\theta \\vdash \\phi \\quad \\phi \\models \\psi}{\\theta \\vdash \\psi}", "f5eadc5a04acd3162956bb0467045b1e": "F:C \\rightarrow D^{\\mathrm{op}}", "f5eae4a937996dbc4a60b0af845e5bc1": "Flies(X)", "f5eafa73d7228edec16ff7e696d12b2d": "\\hat g(f)", "f5eb1748889c2cfcbbe2683fd206149b": "\\sqrt{2 + \\sqrt{2}}", "f5eb1cfb6646786006e35880228a50c0": " D=\\frac{C \\cdot A}{B}=\\frac{1.63m \\cdot 180m}{2m}=146.7m", "f5eb6d2b0572f996b454b1986e5bc2a9": "\\gamma = \\frac{\\mathrm{d} \\log(V_{\\text{out}})}{\\mathrm{d} \\log(V_{\\text{in}})}", "f5eb70c66468a9eb09f54ee7cfa8c392": "g(Y,Z)", "f5ecae8ad553dbddf2e64986b4093cf5": "Tr(g^y)", "f5ecb720b8129360529eb0414900e6d6": "\\{1,\\dots, M\\}^d", "f5ecbc3ce700fbd590d2631122c53684": "\\delta_i\\delta_s(f) = \\sum_t \\delta_i( \\delta_s(P_t))a_t) = \\sum_t (\\delta_s(P_t)\\circ s_i)\\delta_i(a_t) + \\sum_t \\delta_i\\delta_s(P_t)a_t.", "f5ed08c4d93fd06ee46f459ff8ad2565": " \\phi = \\tfrac{1}{2}\\pi - \\operatorname{gd}(a) ", "f5ed9ba0425a29f46b51848fada438b6": "\\Omega^1_c(S^1)", "f5edb05a7aaf631f78ddc7f283e82c0d": "U'u-Uu'", "f5edcfaf087d69b8a2cb56783ef4e059": "i^{-1}\\mathcal{F}(\\{x\\}) = \\varinjlim_{U\\supseteq\\{x\\}} \\mathcal{F}(U) = \\varinjlim_{U\\ni x} \\mathcal{F}(U) = \\mathcal{F}_x.", "f5ede77b3f5a23fe694bd1c02762a9c2": "\nA_{nl}=\\frac{n+1}{\\pi}\\int_0^{2\\pi}\\int_0^1 [V_{nl}(r\\cos\\theta,r\\sin\\theta)]^*\nf(r\\cos\\theta,r\\sin\\theta)rdrd\\theta\n", "f5ee2213a9d7a86088bc130110dbbbec": "\\|u-u_h\\|_a\\le \\|u-v\\|_a", "f5ee262d80e6558aa0763ef991e4aa2b": "\nP(\\lambda)= a_0 \\lambda^k + a_1 \\lambda^{k-1} + \\cdots + a_{k-1} \\lambda + a_k\n", "f5ee7a464f7ae041cd239d34b90eaeef": "r = 6,356,752", "f5ee7a5817044f86e311c649a35cdf6e": "\\int\\sin ax\\cos^n ax\\;\\mathrm{d}x = -\\frac{1}{a(n+1)}\\cos^{n+1} ax +C\\qquad\\mbox{(for }n\\neq -1\\mbox{)}\\,\\!", "f5eedd392d58ce5b90ceb39a8e59de6e": "A = \\left( \\frac{0.5062 -0.8776\\left( b/t\\right) + 0.3504\\left( b/t\\right)^2- 0.0078\\left( b/t\\right)^3} {12.03\\left(b/t\\right)+9.892\\left(b/t\\right)^2} \\right)", "f5eedf35645d34f8fcb61c5b47c7127f": "h=kr^2", "f5ef5bd7ec3b2a5a96b26fe86e5e7bf6": "\\mathbf{X}=(X_1,...,X_n)^T ", "f5ef71acdca2205ca0183251fc8d68e3": "301655 cm^{-1}", "f5ef8080f707118ec84103e285e57f2e": "\\,\\Omega(X,Y)=d\\omega(X,Y) + [\\omega(X),\\omega(Y)]. ", "f5ef8a95d1112b8db01e31a0349be485": "f(-ia,ze^{(1/4)\\pi i})\\,", "f5efc18248c747185748fbbb00975c83": "i_\\alpha (a\\wedge b) = (i_\\alpha a)\\wedge b + (-1)^{\\deg a}a\\wedge (i_\\alpha b).", "f5efe5e076b8cf3e3c162c571a422a4c": " V^\\alpha {}_{;\\beta} = {\\partial V^\\alpha \\over \\partial x^\\beta}. ", "f5f080ae3aced6664bd17ae27c9076af": "\\bar A", "f5f09cf9138cc8f25982e948435d5acf": "p_{\\varepsilon}(x,t)", "f5f0eedb25938980b5e630efa1910e38": "y_{2n}=0", "f5f1192ffc8527c3fe88ffd0105aa893": "\\begin{bmatrix} 1\\\\ -1\\end{bmatrix}", "f5f1394e69e3e958268f136fcce1e367": "n_1, n_2, \\dots, n_p.", "f5f1c9f051e1da5a15643a49afd6ab85": "I_{ij}", "f5f25c5783e04be5c4beb56be4c24342": "R_{\\mathrm{T}} = \\frac{k\\sigma_{\\mathrm{T}}(1-\\nu)}{\\alpha E}\\,", "f5f2ba22fce01c04a046c15990d7316c": "(\\nu = +)", "f5f2d572217164b9cb1e695f4a90a6e5": " \\hat k ", "f5f2fcb99c89977c89eabec8d5214244": "\\text{Total revenue} = \\text{sales price} \\times \\text{number of units}", "f5f3253d719ad8f123fea3f269dc5866": "\\scriptstyle \\bar V_S", "f5f34c95ff55009cd6b184ecc53c9108": "C=\\{c_1, c_2, \\ldots, c_n\\}", "f5f35ab12f9e603cf236e268a1946ffc": "\\left [\\begin{smallmatrix}2&-1\\\\-1&2\\end{smallmatrix}\\right ]", "f5f36328af3a00144250a22e4dbcf315": "p(A, B) = p(A) p(B),\\,", "f5f3aa56f743efd4c5655abff93adf9c": " \\int_{-1}^1 f(x) (1-x)^\\alpha (1+x)^\\beta \\,\\mathrm{d}x ", "f5f3de71273a84219a54684f99b23f5e": "I_\\nu", "f5f3dfcbedf56f3bffbe41c0dd7fb3f3": "H^1(X, \\mathcal O_X)= H^2(X, \\mathcal O_X)=0 ", "f5f3e41ded91b058cab953377d2c624b": "P(k/2,x^2/2)\\,", "f5f3e9c40795f7b95693592952e1ac23": "R \\oplus G = (\\text{dom } G \\ntriangleleft R)\\cup G", "f5f4012fbd8359245c18bcb566d61532": "\\mathrm{Ann}_R(N)=\\mathrm{Ann}_R(N')\\,", "f5f437da10ff19a0977b7d6a2154f07a": "\\sigma_{CS}=E(\\cos\\theta\\sin\\theta)-E(\\cos\\theta)E(\\sin\\theta)=\\frac{1}{2}\\left(S_2 - 2 C_1 S_1 \\right)", "f5f4f096eda3472b2c8a419312b27796": "b^x", "f5f4f833fd070da446c5b20a1e849f6b": "I^c", "f5f4ff2b50f95b1d81e0b566c2c58a94": "\\theta_\\text{max} = \\nu_\\text{max} - \\nu(M_1). \\,", "f5f4ffc5d3c921b310f087cd45d376ed": "v=\\left(\\frac{1}{M}-\\frac{Ab}{I}\\right)\\int F dt.", "f5f51ce17de8ecd7f56854b2ad2f4fa2": "L(f)=4x-1,\\,", "f5f52311bce543cebf9f4dd16178c74c": "\\theta = 135^\\circ\\,\\!", "f5f5410adc72fe5bc84965e9cd07a2b3": "L = \\frac{W}{2k}.", "f5f54980c39a37ddd4281eced0b3b729": "\\Omega_0 \\mapsto T_s(\\Omega_0) = \\Omega_s.", "f5f5756c51413afc2605692d2d558a03": "Z_{sc}", "f5f5ff60522b13713ab49dc94750ac07": "-\\nabla^2 \\mathbf{F} = \\nabla \\times(\\lambda \\mathbf{F})", "f5f62661ae88d4ef8ce33e7b756ddc2f": "d_{\\mathrm{H}} (A, B) = \\sup_{x \\in X} \\big| d(x, A) - d(x, B) \\big|.", "f5f64687a6b7261801d750fae5f923f7": " \\mathbf{F} = m\\mathbf{a} \\nrightarrow \\mathbf{F} = \\gamma m_0 \\mathbf{a} ", "f5f68a481f9d9deb2e73182ec3158d31": "[SU(3)_C\\times SU(3)_L\\times SU(3)_R]/\\mathbb{Z}_3", "f5f6a77e8944fb2ccaf3091ab6372c0b": "\\phi_r = f_r (y_{i_1}, y_{i_2}, \\ldots , y_{i_q}) \\ ", "f5f6d11a1c9228b44abda6beeef79da7": "V_{e}", "f5f6f806da9cced60ebd1c9c23b3ede8": "i=1,2,...,N \\,", "f5f74df4e6bfa619dd9b792079a52ebc": "{{rank}_+}(A)= rank(A)", "f5f77a0c877d42585d69b3983eecf13e": " \\mathit{false} \\, ", "f5f7d755a902e5c60a27eac02c91ae86": "N=\\left \\{S\\right \\}", "f5f800a0ec29e71eaf3ae4aa2c4e5b12": "\\left\\lfloor\\frac{n+1}{2}\\right\\rfloor", "f5f8e100b54d90a75e8e60a6b4601d4e": "\\mathbf F\\;\\cdot{d}\\mathbf S ", "f5f923ee9fdb2980e8092f368c430aad": "66^2", "f5f97556529f6f21b720904c3f06713f": "E_{x,y} = l m V_{pp\\sigma} - l m V_{pp\\pi}", "f5f975cf3be958d447f7529f44c62659": "p _{u \\cap v}", "f5f9a593553a3e2625b26ce7773576d4": " \\begin{align}\n\\lim_{\\beta \\to 0} G_X = \\lim_{\\alpha \\to \\infty} G_X = 1\\\\\n\\lim_{\\alpha\\to 0} G_X = \\lim_{\\beta \\to \\infty} G_X = 0\n\\end{align}", "f5f9edd553d1fee5f5a1b63faca9f26b": "(E \\wedge I^+) \\to F", "f5fa13709ed2a3c625577ff5540e4ad8": "r \\in R_{\\nu}^+", "f5fa2aec93b3b31739c1d4732f29046c": "\\scriptstyle \\rho\\, = \\,\\tfrac{1}{2} (\\sqrt{5}-1)", "f5fa53557f535d1b922e43b274bccc33": " \\nu_\\mathrm{2}= |\\nu_\\mathrm{n}+a/2|", "f5fa76c1fa8e04affe0940d26d9a3862": "\\overline\\lambda_{\\text{UV}}", "f5fa79e69afc591236daab5a74b88b4f": "X(\\omega) = c, \\quad \\forall\\omega \\in \\Omega.", "f5fab6599f9f97c1b71325bd91ef0049": "\\{(x_0,\\ldots,x_n) \\mid \\sum x_i = 1\\} \\subset K^{n+1}", "f5fac78df176f89c56283a76fdd02574": "\\varphi I_n-A^\\mathrm{tr}", "f5fad1f2e249b9ceb00a7d030a05ea3f": "\\|\\nabla v\\|^2_{L^2(\\Omega)} = \\|v\\|^{p+1}_{L^{p+1}(\\Omega)} > 0.", "f5fad351709ee607c9388d09b12755ba": "\\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1/2} d", "f5faf4a8cc406619def524a788016121": "\\frac{ \\text{d}E }{ \\text{d}t } = z \\frac{\\text{d}B}{\\text{d}t}\n-\\sum_i \\frac{\\text{d}C_i}{\\text{d}t} \\qquad \\qquad (3d) ", "f5fb32cf582df4f2ac34e39532ca62b1": "\\text{EMA}_{\\text{yesterday}}", "f5fb4770aa773a2b5faed18c121ab9b3": "\\lambda_{ij}=\\frac{\\partial \\varphi_i} {\\partial u_j}", "f5fb5821cc2b57b45bb083f411075616": "B \\succ A\\,\\!", "f5fb6f6bd5b9fa3d698f0e406adf9f29": "\\{l^a\\,,n^a\\,,m^a\\,,\\bar{m}^a\\}", "f5fb71bf759a639902b5e2f2bf27cb7e": "x = N + x'", "f5fbdebfcc9acb5680fcbf2ac439412e": "\\alpha = re^{i\\theta}", "f5fbf0036be8408d4b65ab17c8566faf": "\nY = A B + \\overline{A} C + B C.\n", "f5fc63c62b2e77af92b32230d8eba152": "\\mathbf{A} \\rightarrow \\mathbf{A} + d\\alpha", "f5fd0f02f30a50f0c1b2cbc25f695063": "(A\\to B)\\to((A\\to(B\\to C))\\to(A\\to C))", "f5fd2dab2cb81e979f83fcfeabe25721": "\\mathbf{P}^\\prime\\mathbf{P} = \\mathbf{P}\\mathbf{P} = \\mathbf{V}", "f5fd3b17145c67d796daca76acc1ce48": "x\\#<4^x", "f5fd3bcc4391a2edab49cfe4c019ad54": "L_\\mathrm{total} = (M_{11} + M_{22} + M_{33}) + (M_{12} + M_{13} + M_{23}) + (M_{21} + M_{31} + M_{32})", "f5fd787138348e2b919c9684838b4e66": "x=a \\!\\,", "f5fd9a0c0b710b96570a3f54fb8d39de": " [f(\\gamma_j)]=[e]", "f5fdd1df31b06ecf82e87a094302fa38": "\\,\n\\langle E_f \\rangle = e^{-\\beta h f}\n", "f5fe032de6e75e9b83be60c9e2227b33": "\\nu \\approx \\frac{1}{2} - \\frac{E}{6K}", "f5fe09e7337ebf3cf7670d89bba681ea": "n_x = 1, n_y = 2", "f5fe231b815f304efd04086bcccaa023": "V(t) = \\frac{Q(t)}{C} = \\frac{1}{C}\\int_{t_0}^t I(\\tau) \\mathrm{d}\\tau + V(t_0)", "f5fe4396ee875c7f477de7abf58dcf64": "\n{{d^2 i(t)} \\over {dt^2}} + 2 \\alpha {{di(t)} \\over {dt}} + {\\omega_0}^2 i(t) = 0\n", "f5fe818aaab0aeae9a66e5d12668c496": "\\mathrm{d}e_i = \\sum_j \\omega_i^je_j.", "f5fed5da8daeb01a140538c2179d767b": "\\mathbf{\\mu}_{i=1 \\dots K}", "f5fedd3332faaacd7e39f544fd891f02": "\\frac{1-\\cos(2x)}{2}=\\sin^2(x)", "f5ff4c86852d15dee6aabc1d77de6f39": "\\left| a \\right| < 1", "f5ff60c5ac58005f0e90ba87c2f7fb3c": "\n= (1 - \\alpha)^k\n", "f5ff828f35cbe53eb3bb8fe789c039ba": "\\varphi(e_j)=\\sum a_{ij}e_i, \\qquad j =1, \\cdots, n.", "f5ffae380775503018019636d7549208": " P [\\mathrm{W}] =F_\\text{application}\\times V_\\text{application}= F_\\text{model} [\\mathrm{N}] \\times 17.2 \\ \\mathrm{m/s}", "f5ffc71313e6549a72797559e3d81c4b": "P\\left(\\frac{\\partial}{\\partial x_1}, \\cdots, \\frac{\\partial}{\\partial x_{\\ell}}\\right)u(\\mathbf{x})=f(\\mathbf{x})", "f5ffe13b63d38a52f965659e4f200ea4": "q^{\\star} = w^{\\star} + x^{\\star} i + y^{\\star} j + z^{\\star} k \\!", "f60062fd6485505e93cdcda9b70a2275": "2^X = \\left \\{ \\varnothing, \\left \\{ a \\right \\}, \\left \\{ b \\right \\}, X \\right \\}. \\,", "f600a38f5e80d7a66fc2e70878840c38": "{{c}_{P}}", "f600dc75b922cb87feca309e429c606b": " R(l) \\approx l^{-1} ", "f600eaa15a953fd172ed9280ba624efe": "u={1\\over 2}{B^2 \\over \\mu}", "f60100ac23cd660dc48f289dc0432f06": "B_3^I", "f6010e58c8bf0cd399b6ebf8305ef4c3": "\n\\text{Tr}\\left\\{ \\left( I-\\Pi_{1}\\cdots\\Pi_{N}\\cdots\\Pi_{1}\\right)\n\\rho\\right\\} \\leq\\sum_{i=1}^{N}\\text{Tr}\\left\\{ \\left( I-\\Pi_{i}\\right)\n\\rho\\right\\} ,\n", "f6010f436d41bc2634dba82d459f6057": "n_j = 0 \\ldots N_j-1", "f60115a7bfb92589578cc8c3d5308124": "\\int_0^1 \\tilde{B}_n(x) f(x)\\, dx = \\frac{1}{n!} \\left( f^{(n - 1)}(1) - f^{(n-1)}(0) \\right)", "f601160650d68b2fd9c2d34ade2ceedb": "\\mathrm P(A)\\leqslant \\delta", "f6011bbdbda63413de073d63dea1b64d": "\n\\begin{align}\n\\mathbf{S}_B & = (\\mathbf{m}_2-\\mathbf{m}_1)(\\mathbf{m}_2-\\mathbf{m}_1)^{\\text{T}} \\\\\n\\mathbf{S}_W & = \\sum_{i=1,2}\\sum_{n=1}^{l_i}(\\mathbf{x}_n^i-\\mathbf{m}_i)(\\mathbf{x}_n^i-\\mathbf{m}_i)^{\\text{T}}.\n\\end{align}\n", "f60157dcf1a5d00258738aafd85c345a": "{\\mathfrak g}", "f601831edb20a72b18ef654f20e64cdc": "B_0,B_1,B_2, ..., B_{n-1}", "f6018eb7df56d62e4cb87c8c0280bcd8": "1,000,704\\,", "f601ea0845c6467eca9896129a772bb6": " Qf(x_1+x_2,\\lambda)=0 \\, ", "f60252339403720d5529d6a8a3369b65": "b \\triangleleft (S-\\lbrace b \\rbrace)\\cup\\lbrace a \\rbrace", "f602859e0bb5368cdd1677d4cdf01a6f": "{\\color{Blue}~6.1}", "f602a9357e95ec4867ec3e3b18e1048a": "w^{T}q", "f602c2d023782fe0f21b8b118fc89e2f": "\\theta\\ ", "f602c3ab9cfd6171ad53fde80f148a0d": "\\Delta(t)=P(\\alpha=1,z=t^{1/2}-t^{-1/2}),\\,", "f602de89510c7838ddaf02d7dde11122": "\\phi^*= -1", "f602ec9cc34647bb4159799a60abbd18": " Q = I t ", "f603266308b53b3743825befd1c4e52a": "\\varepsilon_X=\\Phi_{GX,X}^{-1}(1_{GX})\\in\\mathrm{hom}_C(FGX,X)", "f603300d4ed957e3e5c58e57066fa2e2": "f(x) = p(x)q(x) = (x^2 + x + 1)(x^3 - x + 2) ", "f6036badce8200c0645398a35be66c9e": "a_{ij} \\leq 0 ", "f603ee7b0ae58d98cf93edc0a4c5bbfe": "A = \\frac{5t^2\\tan(54^\\circ)}{4}.", "f603fe372961e9412d6edd9777819205": " \\tau^{\\pm} \\approx 1/(k V_A) ", "f604523c0e8bcde894bc3938535e4523": "y = \\cot \\varphi_1 - \\rho \\cos E\\,", "f60464f592a21ca3b9c09bed60e8fc46": "i_{T, \\infty} = 4nFCDa", "f6049b99e0c1969f478cdff038a47272": "\\scriptstyle Z_{\\mathrm i m}", "f6050067382a07680b6bdcc743a6c9c9": "E_1 (z)", "f6050fbfe973b51815d268a664c6f504": "x \\in L \\Rightarrow \\mathrm{Pr}[A\\,\\mathrm{accepts}\\,x] > 1/2", "f60533faf74d0c1d82265000d5c9056c": "\\mathrm{AgNO_3 + KOCN \\longrightarrow AgOCN \\downarrow +\\ KNO_3}", "f60598d43fc56fb096eb8bf0f6dc6e04": "\\rho:D\\to [0,\\infty]", "f605e4010311328915216ff0cb79b97b": "\\gamma_{f_a} = \\frac{(a-1)\\zeta(a)-1}{a-1}", "f60660f878fd8b3bf99d7f9a835c5272": " \\begin{pmatrix} n \\\\ 2 \\end{pmatrix} ", "f6066cebe68d7c84379ef60053fac493": "(F/F_{EW})^2 \\ll 1", "f6066e86557e3e38133d18b4ad3de370": "g(A(X,Y),Z)=X[Y[Z[\\Phi]]] \\, ", "f606927a9047fd6a71e60eae7d67a5ec": "T^{-2}(|A \\times B|)", "f606b9a8bcadbf96b06ca438d8fb12fa": "\\begin{matrix} \\frac{ORTRP \\;at \\;a \\;given \\;latitude} {ORTRP \\;at \\;the \\;North \\;Pole} \\end{matrix}", "f606cb7bb3183e864c65d7dfaf05adf3": "\\textstyle \\frac{\\lambda}{1- \\mu}", "f607512a1a99bb1cd17479b6da5b7d3e": "Q[\\varphi] = \\int_{x_1}^{x_2} \\left[ p(x) \\varphi'(x)^2 + q(x)\\varphi(x)^2 \\right] \\, dx + a_1 \\varphi(x_1)^2 + a_2 \\varphi(x_2)^2, \\,", "f607d8ea2a6bff880873d1910a8731e2": "\\delta \\lambda = - {3 \\lambda^2 \\over 2} \\int_k dk {1 \\over (k^2 + t)^2} = -{3\\lambda^2 \\over 2} b", "f607efdfd63c77ec90aecba24df3d84a": "G_j (x) = 0", "f607f274f9278703ea6c9c8171f5d50a": "(q,x,y,r) \\in \\delta^*", "f608321e0ba03e34031ccc5471f337d0": "w_+=\\frac{c}{n}+ v\\left(1-\\frac{1}{n^2}\\right) \\ . ", "f608352f8ac1c7c1d416c85ff15ef7a5": "C^{wnn}_n", "f608664adcbec333cef82c69cf8367fd": "f\\sim \\sum_n \\widehat{f}(n)e^{int}.", "f60892651eb747b4c24c17d843ae7aab": "\\operatorname{Res}_{z=(n + \\frac{1}{2})\\pi} \\frac{\\tan(z)}{z} = \\frac{-1}{(n + \\frac{1}{2})\\pi}", "f608e0a3fa7fb5f0c406d4007ac8943b": "\\scriptstyle (G + j \\omega C)", "f6090c4e2ddf4f0772aa4754ea607b0d": "\\textstyle\\overline{\\mathcal{R}}_{nn}", "f60943c1faa88022af98284745f6f7fe": "\\lambda^*(E) = \\text{inf} \\left\\{\\sum_{k=1}^\\infty l(I_k) : {(I_k)_{k \\in \\mathbb N}} \\text{ is a sequence of open intervals with } E\\subset \\bigcup_{k=1}^\\infty I_k\\right\\}", "f609803d7b71f22a38bc9e9fa2f469e8": "K: G \\sim N(\\rho, \\rho)", "f6099cd1f43056680cc303e558c1bf4f": "{10 \\choose 1}{4 \\choose 1}^5 - 40 = 10,200", "f609cc173f30a39145f94e7e7196e926": "V \\otimes W", "f609fe601a7485314b37f0bf29c7614d": "f(x)=+x^3 + x^2 - x - 1 \\,", "f60a876eb525e11bd05dbdc6845b5a89": "E_{\\rm EA} \\equiv E_{\\rm vac} - E_{\\rm C}", "f60ab3a2ae5947461fa3d2360104c5aa": "\\pi(a)=b, \\pi(u)=w, \\pi(w)=v", "f60b4555623d503dcfa1362a75493829": " Q_i < P_i ", "f60b5eb1cc5b2d2a3b984cfa1b38b3e8": "\\mathcal P^\\mu", "f60b60d5a3186ad1e2aafa40b8ae357a": "P(G, t) = t(t-1)^{n-1}.", "f60b9e78cd054574eff60195e0c1cabf": "\\forall A, \\exists B, \\forall C, C \\in A\\rightarrow D \\in B.", "f60ba1a0e3b9e8ecf771a0b1105b2599": "\\Omega_{k,l}", "f60bd38299ab228e388692e75b535984": "y_{\\mathrm {atm}}", "f60bef22135fbb11c40062190f30bf80": "\\scriptstyle\\operatorname{IQR}(x) \\;", "f60c014d5b9f7d9879fcd2446d98419a": "y=d_v", "f60c8ed7027e856f21b03744ab6361ac": " E^+(k) \\gg E^-(k) ", "f60c905673e7007a4089c58b209026d6": "A_i=x_1+\\cdots+x_i,\\qquad B_i=y_1+\\cdots+y_i", "f60cb8ead988d29a2700a56862213550": "\\rho(x)=\\tfrac 1{2\\pi}\\sqrt{\\tfrac{4-x}{x}}.", "f60cc622268b97ae2c25f02bedb41622": "\\gamma^5", "f60d224008f6e82f9d85f86777e3e0e2": "Lf(\\eta)=\\sum_\\Lambda\\int_{\\xi:\\xi_{\\Lambda^c}=\\eta_{\\Lambda^c}}c_\\Lambda(\\eta,d\\xi)[f(\\xi)-f(\\eta)]", "f60d3ede1dbcbd49158bf065d59870d4": "W_{\\alpha}", "f60d495d5d272987d9a424bbce96432e": " \\mu_{S^{t+1} \\mid h^{t+1}} = \\mathcal{C}_{S^{t+1} O^{t+1}}^\\pi \\left(\\mathcal{C}_{O^{t+1} O^{t+1}}^\\pi \\right)^{-1} \\phi(o^{t+1}) ", "f60de60db1caffb2b8faf8d7b405f85b": "(x_s,y_s)", "f60e1e59efaf4ca2ac548bc49e793952": "x^{\\prime} = \\gamma \\left(x - v t \\right) ", "f60eef652a61052b7bd714e87d9b59bf": "(Df)_x E^u_x = E^u_{f(x)}", "f60f68ce1666d33d9b5cc0b5453ed540": "g\\in H\\overline{g}.", "f60fa8398b3cd3288a88168aeafbaa9a": " {\\mathcal L}^2_3: L=Lclm(l^{(1)}(C)). ", "f6102b73848bdec037699aa7da4a7cc4": " \\phi_{l}\\,= \\phi_{N} ", "f610418c9a8511679ad4b7391c97a54c": "(y,z)\\in S", "f6108a3bfb97fded9e994efbd3d4cccb": "Y^T", "f6108bf40a5ad886230bff4989346f9c": "\\tfrac23-\\tfrac12=\\tfrac46-\\tfrac36=\\tfrac16", "f610b2c068deed2dcbbee26f6d8f2ba0": " Mf(x)=\\sup_{r>0} \\frac{1}{|B(x, r)|}\\int_{B(x, r)} |f(y)|\\, dy ", "f610b72b30d810222f3536db5ea7c4e8": "e_j=-\\Omega(\\alpha^{-i_j})/\\Xi'(\\alpha^{-i_j}),", "f610e38a8a3c69f890aa11775f35071b": "\\langle njm|x|njm\\rangle", "f610f6c8749c7595bdfae72711c45d0f": "V(\\phi)=\\mu^2\\phi^2 + \\lambda\\phi^4", "f61125d7b0e817862db52a85e8630db2": "|\\operatorname{width}(R_{t_1})-\\operatorname{width}(R_{t_2})| \\geq \\operatorname{width}(R_0)/\\sqrt{n}", "f611822c10beba6f67fc4b307056442f": "K=r \\times \\left(p-\\frac{qA}{c}\\right),", "f611df2608d328e3fd2b735cb4c70e7b": "X_1, X_2, \\ldots, X_n", "f611e277bab4dfe027c354fb1ced2c20": "\\langle A,\\in \\rangle", "f611fabddfa0d06875126e4c82e3f9dd": " \\det(\\mathbf{A}) = \\varepsilon_{i_1\\cdots i_n} a_{1i_1} \\cdots a_{ni_n},", "f6127042cc3cb4e9779d2f0617e56872": "(q_{i,1}^0, q_{i,2}^0, q_{i,3}^0, q_{i,4}^0)", "f612a590f229c1b28700aec5b17c3d39": " \\operatorname{grad} \\equiv \\nabla ", "f612cd6683fef179b4290b6cab2b77f0": "\\mathbf q", "f612d3b116f67602bc1ca597a7442215": "N= {n_0\\sqrt{\\pi}\\over a}", "f6131d589de3ef8732cfe9b196c22a3e": " (1-x^2){P_\\ell^m}'(x) = \\frac1{2\\ell+1} \\left[ (\\ell+1)(\\ell+m)P_{\\ell-1}^m(x) - \\ell(\\ell-m+1)P_{\\ell+1}^m(x) \\right] ", "f613875517396a1707cfca14d3263f76": "\\nabla\\mathbf{v} = \\cfrac{1}{h_i^2}{\\partial \\mathbf{v} \\over \\partial q^i}\\otimes\\mathbf{b}_i ", "f61392f6f50c48951b35a8a9a7499e38": "{h d \\over k}= {0.023} \\, \\left({j d \\over \\mu}\\right)^{0.8} \\, \\left({\\mu c_p \\over k}\\right)^n", "f613a9bd532a30db83c8cfeb8e3c4775": " QF = \\int_{K_1}F\\circ g_s \\, ds", "f613ec0055092ff97cc4d3e014fc01be": "\\mathbf{J}=-e\\psi^{\\dagger}\\boldsymbol{\\alpha}\\psi\\,\\quad \\rho=-e\\psi^{\\dagger}\\psi \\,,", "f613f6483f2b0796d3121054b0dc6b72": "x_{i} \\,", "f613fa10f98b826aaabe6ac45862a16d": "\\partial f(x)(h)=\\lim_{t\\to 0 \\in R}(t^{-1}(f(x+th)-f(x)))", "f6142fb34f2234ccb22660737d6590a1": "\\tau_k = \\frac{3}{\\sqrt{\\alpha_k^2 + \\beta_k^2}}", "f614b3499dc2582bbcf772e404b343a3": "\n\\delta F_z = \\delta L\\cos\\phi + \\delta D\\sin\\phi\n", "f614bde6a6855094332b559fd00e0a93": "\\displaystyle X^0=X", "f6150c6444b235f8c368819a39f7f18e": "mg(y_0-y)\\,", "f61520e1bbbdc3f69c8cc91224b789e0": "\\operatorname{gcd}\\left(10^\\infty,t\\right).", "f61554549479debd1a662cef3d3df3dc": "c_9 = -1.99 \\times 10^{-6}.\\,\\!", "f615bd629b642717617976cf9112ec2f": " \\Gamma_q(x)=\\frac{(q^{-1};q^{-1})_\\infty}{(q^{-x};q^{-1})_\\infty}(q-1)^{1-x}q^{\\binom{x}{2}} ", "f615e3513c6f2f1401c360e26a6765db": "\n\\| (T_h - \\lambda) f_n \\|_p ^p = \\| (h - \\lambda) f_n \\|_p ^p = \\int_{S_n} | h - \\lambda \\; |^p d \\mu \n\\leq \\frac{1}{n^p} \\; \\mu(S_n) = \\frac{1}{n^p} \\| f_n \\|_p ^p.\n", "f615ea15df85e570af3cdc405304d5ff": "\\scriptstyle{ \\emptyset =_{\\mathrm{def}} \\{ x\\,|\\,x\\neq x\\} }", "f61642bdc93750973547a212b9938698": " e^{x(\\sqrt{1-t^2}-1)/t}=\\sum_n s_n(x) t^n/n!.", "f616e488112d389c7873dfcac126b6c7": "\\langle \\sigma \\rangle = -p/\\sqrt{2}", "f616f7f7e7e8f1f896bb4f3221b62c62": "y \\in \\mathbb{R}^n", "f617020f3f6a791dbf046807d2f7e678": " \\operatorname{S}(f(u),f(v)) \\iff \\operatorname{R}(u,v) ", "f61739c7c80b1f93788eaa2a4a46abfb": "\nE(\\overline{X}_n) = \\mu.\n", "f6175608fa429321b2c5e1bb510f98b2": "\\nabla_\\alpha (\\sqrt{-g} F^{\\beta\\alpha}) = \\mu_0 J^\\beta ", "f6176fed27344f723008d67acbfc8fd0": " \\quad (11) \\qquad \\qquad \n{{d {\\mathbf {\\bar u} }_{i} } \\over {dt}} + {{1} \\over {v_{i}} } \\oint _{S_{i} } \n {\\mathbf f} \\left( {\\mathbf u } \\right)\\cdot {\\mathbf n }\\ dS = {\\mathbf 0} .", "f6179764d469947275e1d19822e93469": "\\mathbf f", "f617e8be5396ccca4f685aee020baee6": "\\scriptstyle \\approx", "f617f406e4230f6f786a5f49c7750df5": "2x = 2", "f617f96887fbea5c174865bc42e73950": "\\,d_x = l_x-l_{x+1} = l_x \\cdot (1-p_x) = l_x \\cdot q_x", "f61805d7103eabe8bba02dd7656ce162": "n=n_1", "f61834788516e57d348767656692d2a8": "t_o = \\frac{(h_r t_{mr} + h_c t_a)}{ h_r + h_c}", "f6183f8cf6f8161b67ffc69efe801b1f": " \\rho_R = \\frac{B^2 - A^2}{2 \\pi G}", "f6188e54e7faa6b95630e10eeb9b4452": "P\\in F[x_1,x_2,\\ldots,x_n]", "f618b460ff58dcb194c20cdaaf9a6e81": "\\mathbf{F}_{\\mathrm{ext}} + \\mathbf{v}_{\\mathrm{rel}}\\frac{\\mathrm{d}m}{\\mathrm{d}t} = m\\mathbf{a}", "f618c58023e67c8d33028003577f3873": "\n\\mathrm{Pr_m(x)} = \\frac{\\mathrm{tr} \\left[ \\prod_{k=1}^{x} \\mathbf{W}_{k} \\mathbf{Pj} \\prod_{k'=x+1}^{N} \\mathbf{W}_{k'} \\right]} { \\mathrm{tr} \\left[ \\prod_{k=1}^{N} \\mathbf{W}_{k} \\right] }\n", "f618e82bff4f184054067e35e42c69af": "i=0 \\ldots n-2.", "f6191f99bed61002338e3c163690d7ce": "a \\quad b", "f619569f746f75ed36f477960e25cff7": "\\scriptstyle\\{x,y,z\\},", "f619a03f8488533c7c1b7e6f5531aa90": "\\vartheta_{10} (z; \\tau) = -i e^{iz + i \\pi \\tau / 4} \n\\int_{i - \\infty}^{i + \\infty} {e^{i \\pi \\tau u^2} \n\\cos (2 u z + \\pi u + \\pi \\tau u) \\over \\sin (\\pi u)} du", "f619ab36a2c95a04ed08df4f5b29a5a9": "\\square^2~=~\\frac{\\partial^2}{\\partial \\tau^2}~+\\frac{\\partial^2}{\\partial x^2}~+~\\frac{\\partial^2}{\\partial y^2}~+~\\frac{\\partial^2}{\\partial z^2}~=~\\frac{\\partial^2}{\\partial \\tau^2}~+~\\nabla^2~=~\\frac{-1}{C^2}~\\frac{\\partial^2}{\\partial t^2}~+~\\nabla^2 .", "f619df8586a9ae6162eeae0823581858": " n_1 R\\sin\\Phi = n_2 r \\sin\\theta = INV \\, ", "f619ed0c84a479ec4c22523f9b1c1399": "\\mbox{kg}\\,", "f61a2a851a4bba794bcc721e47d98161": "V_{max} = k_{cat} [E_{T}]", "f61a765da8fc073a03f1432f32946116": "\\mathfrak{m}_0", "f61aa391216e97d00334accdaa358870": "\\mu^2(w,\\overline{w})\\, dw\\,d\\overline{w}", "f61acacc4aae2ba3cece15d1a2cbf0a4": "\\phi(\\mathbf{x}, u) = \\left( \\sum_{i = 0}^{N}{\\frac{(u-u_i)^2}{d(\\mathbf{x},\\mathbf{x}_i)^p}} \\right)^{\\frac{1}{p}} ,", "f61b35215ac6d4d03d1d5f8c4441d8cd": "h_n\\,", "f61b3ff13fd20295711bd09092a91075": "\n \\left(A\\,c\\,e (2 c\\,d-b\\,e) (m+2 p+2)-B \\left(-b^2 e^2 (m+p+1)+2 c^2 d^2 (1+2 p)+c\\,e (b\\,d (m-2 p)+2 a\\,e (m+2 p+1))\\right)\\right) x)\\left(a+b\\,x+c\\,x^2\\right)^{p-1}dx\n", "f61b9f21e6291b811b8e5e75900101fa": "F\\left[A\\right]", "f61c20eabb099c34a7e43e33872dab60": "0 < \\eta < 1,\\,", "f61c516dff6c0434cb88a580e4429314": " V \\not \\in FV[\\operatorname{get-lambda}[V, E]] \\to \\operatorname{de-let}[\\operatorname{let} V : E \\operatorname{in} L] \\equiv (\\lambda V.\\operatorname{de-let}[L])\\ \\operatorname{get-lambda}[V, E] ", "f61d14a674e47bbe368118a8c1222bf9": "q^\\prime\\notin \\delta(q,a)", "f61d43467a35a897cabdf3690391ca3a": "v_{1} = \\frac{v_1 ' + v_c }{1+ \\frac{v_1 ' v_c}{c^2}}", "f61df386d60fea293b294c1ebe325053": "\nF_C^{(3)}(a,b,c_1,c_2,c_3;x_1,x_2,x_3) = \n\\sum_{i_1,i_2,i_3=0}^{\\infty} \\frac{(a)_{i_1+i_2+i_3} (b)_{i_1+i_2+i_3}} {(c_1)_{i_1} (c_2)_{i_2} (c_3)_{i_3} \\,i_1! \\,i_2! \\,i_3!} \\,x_1^{i_1}x_2^{i_2}x_3^{i_3}\n", "f61dfc65e43de38e0ffa2116a6d7e102": " \\kappa^{-1} = \\sqrt{\\frac{\\varepsilon_r \\varepsilon_0 k_B T}{2 N_A e^2 I}}", "f61ea006105ef1bb18751a17c234c1e0": "u:\\mathbb{R}^n\\mapsto\\mathbb{R}", "f61f000c3b727984673ba93d8efbb5a9": "\\beta =100", "f61f46d63cee7a926eeb9b61e587b629": "\\gamma = \\mu^2/v", "f61f9c6f2668cff51f0d0da6c2b181d7": "\\forall x \\, P(x) \\land \\forall x \\, Q(x) \\Leftrightarrow \\forall x \\, (P(x) \\land Q(x)) ", "f61fbebb5b08ff48bdb49f868f1da907": "R_{Zero}=R_H \\cos(\\alpha)", "f61ff55b0a7a49c33c167eab267b923c": "P_n = W + X \\cos\\theta_n + Y \\sin\\theta_n", "f61ff625e02750cf6ddcc93165587f8e": "(p'[y/x],q'[y/x]) \\in R", "f62029b52788aba53a9ae976a6800ff1": "\\textstyle P(A\\mid[x]) \\geq \\gamma", "f620701106712392b50158ba094e64c6": "L = \\frac{m}{2}(\\dot r^2 + r^2\\dot{\\theta^2}) + \\frac{k}{r}. ", "f6207191eec3247ad0b2397a420ff65e": " \\phi(\\boldsymbol{x}) \\in p(\\boldsymbol{x})", "f620a7743f58e37fa41f5f7c3549cb42": "f_{X_{(j)},X_{(k)}}(x,y) = \\frac{n!}{(j-1)!(k-j-1)!(n-k)!}[F_X(x)]^{j-1}[F_X(y)-F_X(x)]^{k-1-j}[1-F_X(y)]^{n-k}f_X(x)f_X(y)", "f620d98053af733e06cfd69429ec0518": "\\mathbf{H}_2", "f620dea060c24d95e6bdafe09b3f0cbe": "\\scriptstyle{1/16}", "f62109ca156c958bf441ddd8100df5b8": "CCAI=D-140.7 \\log (\\log (V+0.85))-80.6-210 \\ln \\left (\\frac{t+273}{323} \\right )", "f6213ae88b696082b3dc6989ee4c76fb": "\\scriptstyle x \\in\\Omega", "f621b99ac6b5007f80361a8ef1a9db7e": "\\exp f(x)=e^{f(x)}=\\sum_{n=0}^\\infty {b_n \\over n!}x^n,\\,", "f621c35bedc6a393d8dd53b942107777": " \\overline{O_R O_L}", "f622a997fe65d12d8c73c7ca0392e4c9": "G = \\int^T _0 k(t)x(t)dt", "f622e012a22e65b1660aaff8a2fcbf21": "\\scriptstyle Y", "f622ecf8e76ae0236d918e7480235983": "\\rho{ \\ddot{\\bold{u}}} = \\bold{f} + ( \\lambda + 2\\mu )\\nabla(\\nabla \\cdot \\bold{u}) - \\mu\\nabla \\times (\\nabla \\times \\bold{u})", "f622fe554d6f2e67f9a34f48250d46c8": "b \\simeq r \\theta \\, ", "f6231da640836ff77d25122fbe5e0d2d": "\\delta,", "f6235d364f899b5a18baa69b4f3ad090": "a^\\dagger(\\vec{k})", "f62361a6515b002f7fb5d20b9fa863cc": "(B', i_{B'})", "f623de1d606d55db63d65e5428a0d171": "\\lang \\mathbf x , \\mathbf y \\rang = 0", "f623e75af30e62bbd73d6df5b50bb7b5": "D", "f6240791f1420808fa6e372d2e40f38a": "\n\\left[\\begin{array}{c}\nsu'\\\\\nsv'\\\\\ns\n\\end{array}\\right]=\\left[\\begin{array}{ccc}\np_{11} & p_{12} & p_{13}\\\\\np_{21} & p_{22} & p_{23}\\\\\np_{31} & p_{32} & 1\n\\end{array}\\right]\\left[\\begin{array}{c}\nu\\\\\nv\\\\\n1\n\\end{array}\\right]\n \\qquad Eq.1\n", "f624080f07fc16faf7a5f6f4b7e4d9f9": "1 - \\frac{\\theta(u,v)}{\\pi}", "f62408ddcbcf88e6593f9fc61045cfe9": "\\sigma_k^2", "f62445d05b66fa7d95647b11de95f652": "\nQ_p = \\Delta E + W\n", "f624665667c870ba5a1d01c7e61d7621": "A_i\\in{\\mathbb Q}(x,y)", "f62499af3c188a639fc1a3d3f2947833": "\n\\nabla_k g_{ij} = \\nabla_k g^{ij} = 0\n", "f624ba7239803d17b0d2a896e160aec2": "\\ell^2(\\mathbb{Z})", "f624bca576eecc9c3ca01d3a35116631": " \n\\Delta(t) \\leq B + \\sum_{i=1}^K Q_i(t)y_i(t)\n", "f624e10fcf2bd01e30f92f107cc8f186": "\\mathbb{Z}/N\\mathbb{Z}", "f624e260f0d0d5b238ec77e4d80bbaba": "\\sigma' = \\sigma - u\\,", "f624eed07309ab391d678fcd075659b2": "U(f_\\max) = 1", "f6250ca9d03be8f0a694cfa535a50c83": "\\left({\\mu B\\over k T}\\right) \\ll 1", "f6250fa8f5fa136aedf8d9c0c8a8ef71": "\\displaystyle{f(e^{i\\theta})=\\sum a_n e^{in\\theta}}", "f6257be654e92c0b929bf09fda919cc3": "\\displaystyle \\mu_{\\frac{1}{\\sqrt{n}} X_n}(A) = n^{-1} \\#\\{j \\leq n : \\lambda_j \\in A \\}~, \\quad A \\in \\mathcal{B}(\\mathbb{C})", "f625c0034e14b002398a35ea9500ff48": " \\hat{S}_-", "f6260caed3c499676cc5d6e489a854c8": "x \\in \\textbf R^L_+", "f626364733b9e037887ac4fe5e6bc255": "(u_1 \\not\\sim v_1 \\text{ and } u_2 \\not\\sim v_2)", "f626759fdfba1f4fdd66d994badfe5bc": "\\widehat{X}, \\widehat{S}", "f626987c15383859987f501eeec72bd5": "\\mathrm{d}^2 S/\\mathrm{d}U_{\\lambda}^2=\\frac{\\alpha}{U_{\\lambda}(\\beta+U_{\\lambda})}.", "f626aa1d7e6958da7ec6480190a04f8a": "\\scriptstyle t \\,\\ge\\, 0", "f626f6a348cea0bbb9b25a7fe88b7d35": "u[5] := 2*atan(\\sqrt(a1^2+b1^2)*(coth(\\sqrt(a1^2+b1^2)*\\eta)+csc(\\sqrt(a1^2+b1^2)*\\eta))/a1+b1/a1)", "f626f9f7cb0cd4dc022bb245fd9ab127": "\\tfrac{1}{X} \\sim \\beta^{'}(\\beta,\\alpha)", "f6275c39f666fabc007129ded64c44dd": "\\lim_{n\\to\\infty} \\frac{E(|S_n|)}{\\sqrt n}= \\sqrt{\\frac 2{\\pi}}.", "f627799d5fe97968dd7bb35b77b83e7a": "2 \\pi i + \\varepsilon", "f6278ff8fdfa7a087a8f346db773a962": "f^i \\otimes v_j", "f627e63701083a13db55bf652dc3b3bb": "f_\\alpha(x)=-f_\\alpha(\\tilde x),", "f627f2856ca6633dc6c4da3538686c23": " y(t)=0 ", "f628198b918b73d0f0be249a1faf222d": "x_n\\downarrow x", "f6281e73b6b200b74594479a67245091": "B={\\mu_0 nI\\over 2r}\\,", "f62869e53ef522eb652a07626be70f42": "Y_2 = \\frac {1} {1 - c_1} \\left ( c_0 + I + \\left ( G + \\alpha \\right ) - c_1 \\left ( T + \\alpha \\right ) \\right )", "f62910b5e8cff2c9167252ab06131129": "L\\left(\\pi\\left(x,y\\right)\\right) = y", "f629270e51fa55dcbee3ad8c54f742c0": "z_1=x_1", "f62972cc5be432e73fa296bc4bf7c775": " |0 \\rangle", "f6298b8704034b318dddaadcb5aeac0f": "f(\\mathbf{v}_j) = \\sum_{i=1}^m a_{i,j} \\mathbf{w}_i\\qquad\\mbox{for }j=1,\\ldots,n.", "f629f32ea40af7d63cce430c4f8ef546": " A_v = A_w{SG_\\text{beer} \\over 0.79661}", "f62a647ab4be8874b9974063752af849": "x,y,z\\in E", "f62a8ca5f8a444a72eb83cb6dd242e50": "{\\mathit{He}}_1(x)=x\\,", "f62ab136abbcac283df61a83652475e1": "\\{p_k\\}_{k=1}^{M}", "f62b1c853f05da7903314cbb5711e85d": " P(i,j,r)=\\sum_{k \\neq j}((M_{-j}^{r-1}))_{ik} m_{kj}", "f62b1e93ae836510fc53e5f873b047ce": "[1,10^{122}]", "f62b67a34ed51c7c8ef5974ea323ab0e": "Y = e^a X^{b}", "f62ba14170dfbc5c0c9b972a96479280": "\\,\\det(a^{-n}xa^{n}) = \\det(a^{-n})\\det(x)\\det(a^n)", "f62bf286e4825fbe98f319caa15fdd08": "\\bold{F} = m \\bold{a}", "f62cc0e34d9ae33a8432c1028dfe6061": "Q + k(2mP) = \\pm P_j", "f62d418c8b0e0ffce53d0c46d4d529ca": "\\begin{smallmatrix}\\left[\\frac{Fe}{H}\\right]\\ =\\ -0.3\\end{smallmatrix}", "f62d56d9bf2039e05275cfdbbc55dd2f": "{w_1,...,w_k}=\\underset{x}{\\operatorname{arg\\,max}} \\frac{w^TAw}{w^TBw}", "f62dba1ddba99feecb38e7e261a01ce2": "\\zeta = \\frac{z}{L_d}", "f62dbd5a520ad229318bd47bee98bae7": "E_0 = \\infty", "f62ded72860fd889b56a92d8952bce94": "k=0,1,\\dots,N-1", "f62df1e2b9ac6d8e5ad228551a87864b": "w'_2=0", "f62df2a66cedee0a57bfb7216a903af2": " d_L(x,y) := \\inf\\left\\{\\ \\left.\\int_0^1 F(\\gamma(t),\\dot\\gamma(t)) \\, dt \\ \\right| \\ \\gamma\\in C^1([0,1],M) \\ , \\ \\gamma(0)=x \\ , \\ \\gamma(1)=y \\ \\right\\},", "f62e2816c0adc98c375aaafd5474f461": " log_e ( 1 - \\frac{ i }{ c } ) \\approx - \\frac{ i }{ c } ", "f62e4040848e2248fcf92a2d760b2e79": "X^{\\mu'}=\\Lambda^{\\mu'}{}_\\nu X^\\nu", "f62e5431b655b10dc5e503dee191f5d9": "n = \\frac{2\\pi}{P} ", "f62e59a6c0e68d0b215ee12799e03373": "\\underline{u}=-\\mathbf{K}\\underline{x}", "f62f0ed16d1c0289b3dd96f217b57d68": "\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left[ m( \\dot x \\ell \\cos\\theta + \\ell^2 \\dot\\theta ) \\right] + m \\ell (\\dot x \\dot \\theta + g) \\sin\\theta = 0;", "f62f15287ac26f440339f91850d7b8e1": "V^2 - U^2 < 0", "f62f4a1c0c17123862b5ce7bf81db1ea": "\\frac{26}{65} = \\frac{2\\!\\!\\!\\not6}{\\!\\!\\!\\not65} = \\frac{2}{5}", "f62f6fc209132db99b77b5a1e60eca94": "\\mathbf{j}_\\mathrm{S} = \\mathbf{M} \\times \\mathbf{n}", "f62fa61e3945666282b3661dd8b9fc90": "\\varphi_i,j", "f62ff8e6913feccdbc7381f4f7b90f1b": "\nS = S(1) = \\sum_{j=0}^{m_1-1}\\frac{1}{(m-1-2j)!}\\alpha_{m_1-j}(1) , ", "f6304b3c2f614c13444725990053e79d": " SE(r_h)=\\sqrt\\frac{1+2\\sum_{i=1}^{h-1} r^2_i}{N}", "f63092d37f2fdc898867155541062cf0": "24_0\\rightarrow (8,1)_0\\oplus (1,3)_0\\oplus (1,1)_0\\oplus (3,2)_{\\frac{1}{6}}\\oplus (\\bar{3},2)_{-\\frac{1}{6}}", "f630c3b457c4a984ea56e4b8fa7b96d5": "\\frac{1}{kT}\\left(\\frac{\\partial p}{\\partial \\rho}\\right) = \\frac{1}{1+\\rho \\int h(r) d \\rm{r}}=\\frac{1}{1+\\rho \\hat{H}(0)}=1-\\rho\\hat{C}(0)=1-\\rho \\int c(r) d \\rm{r} ", "f630d281b295805ce1641f850caef7dd": " \\Delta \\tau = {\\gamma \\pi r \\over b L} ", "f6310a42798f858e174e298258147980": " \\mathrm{S}_i = {{i_p} \\over{i_m}} ", "f63128a35942ad78be8dc1d4b5fe6581": "\n r_{B1} \n=\n{\\sqrt{4 \\pi}m_1v_1\\over a_1 B}\n", "f631a1308d73a029437a61039e441670": "\\hat{\\sigma}_{ij \\rightarrow k}", "f631d7d6f8878cf85c6ab67c9b834ff9": " \\textrm{Spec} (\\mathbb{Z}) ", "f6322db4248dc14760194752aacc5d9e": " g=g_{ij}\\,dx^i \\otimes dx^j ", "f632562faad84d8e3307f077d824ed0e": "| \\!\\,", "f6326f93afe446d8c589a7f9ebb2f3bb": "(3|M_p)", "f632f1576bd427f0cc8a68cde9980bbe": "{W_{I}(\\mathbf {r}, t)}", "f6330115e779da6143a083f92bfcb985": "P = \\pm P_s", "f63304dc4a8d42ed97cbda168c14ec13": " 0 < t< n.\\ ", "f63318bced18ba6ebb4a85e49f1f6598": "x_2(x,t)", "f6334a07377705b7134aee54a1f41b4b": "f(x-0):=\\lim_{h\\searrow0}f(x-h)", "f63360a837672d654d7b65a622263abc": "\\mathrm{d}U", "f63379c1dcec972836fd86eb91c3366e": "= \\langle\\Psi|(\\hat{A}\\hat{B}-\\hat{A}\\langle \\hat{B}\\rangle - \\hat{B}\\langle \\hat{A}\\rangle + \\langle \\hat{A}\\rangle\\langle \\hat{B}\\rangle)\\Psi\\rangle ", "f633bc89f1ecc3b884c9020b43ddc261": "n = \\text{length}(b)", "f634292b3c78838f31eac53ef1832e40": "\\begin{align}\\operatorname{arsinh}\\, x & = x - \\left( \\frac {1} {2} \\right) \\frac {x^3} {3} + \\left( \\frac {1 \\cdot 3} {2 \\cdot 4} \\right) \\frac {x^5} {5} - \\left( \\frac {1 \\cdot 3 \\cdot 5} {2 \\cdot 4 \\cdot 6} \\right) \\frac {x^7} {7} +\\cdots \\\\\n & = \\sum_{n=0}^\\infty \\left( \\frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \\right) \\frac {x^{2n+1}} {(2n+1)} , \\qquad \\left| x \\right| < 1 \\end{align} ", "f6344129b0873c36388ca355efc96170": "= H_a \\left(j \\frac{2}{T} \\cdot \\frac{ \\left(e^{j \\omega T/2} - e^{-j \\omega T/2}\\right) /(2j)}{\\left(e^{j \\omega T/2} + e^{-j \\omega T/2 }\\right) / 2}\\right) \\ ", "f6344e72a1a26a980fa0428050d94471": "\\mathbb{R}P^3", "f634e80d50da16999a35b4efdac3c31c": "\\mathbf{u} \\cdot \\frac{\\partial \\mathbf{v}}{\\partial x} + \\frac{\\partial \\mathbf{u}}{\\partial x} \\cdot \\mathbf{v} ", "f634f4019ce2ae51ba7825485a4e8833": "(U,V,E)", "f63521b808c3f9f16f8b895eceef291e": "\\varphi (\\xi) = \\arg \\big( \\hat f(\\xi) \\big), ", "f63551c5f8e2e4963e3b094f64574338": "\\dot{x} = f(x(t)), \\;\\;\\;\\; x(0) = x_0", "f6359b409538b52383d6c81d1e1d5da4": "\\mathbf{V_1}", "f6366c0499f92a3e2f3580c746c46884": " V = \\frac {1}{2} \\left(x_{1}^{2}+x_{2}^{2} \\right) ", "f63678958e880c096985c0fe26c43e71": " z = r \\cos \\theta \\,", "f636f3371935f3255d561ecf98ad7ac4": "A(t)=A_0 e ^ {rt}.", "f637097ea464b17dcef36e575f750c78": " r \\times r", "f63714cb0dcb3aed77498ceb04f17b3d": "m_{L}=\\frac{m_{em}}{\\left(\\sqrt{1-\\frac{v^{2}}{c^{2}}}\\right)^{3}},\\quad m_{T}=\\frac{m_{em}}{\\sqrt{1-\\frac{v^{2}}{c^{2}}}} ", "f6374376a7406679ace98daacf5552ee": "\\hat{y}_2 = y_2(1+\\delta_3)", "f637790883e56072b0729d7a118c6b61": "\\int_{\\mu ^\\circ }^\\mu {d\\mu } = \\int_{P^\\circ }^P {\\bar VdP} = \\int_{P^\\circ }^P {\\frac{{RT}}\n{P}dP} + \\int_{P^\\circ }^P {\\Phi dP}", "f637a71470307cc5aa550c10da23f1c2": " q(x,y)=ax^2+bxy+cy^2, \\, ", "f63820fd72c8f5396c6e15132a924ef6": "\\,[\\mbox{R}(z, t + dt) - \\mbox{R}(z, t)]/dt = d\\mbox{R}/dt", "f6384b893c0c76a0cdf4a3e84c80d3a8": "\\approx 2\\cos \\left[ 2 \\pi \\left( \\frac {x} {\\lambda_{mod}} - \\Delta f \\ t \\right) \\right] \\ \\sin \\left[ 2 \\pi \\left( \\frac {x}{\\lambda} - f \\ t \\right) \\right] \\ , ", "f6386852ad1eabaf12814ea7b877ec83": " \\hat{\\boldsymbol{\\beta}} = (\\mathbf X^ {\\rm T} \\mathbf X )^{-1} \\mathbf X^ {\\rm T} \\mathbf y \n= \\mathbf X^+ \\mathbf y", "f638c5babf6f086db1277910a2bdd0e1": "x \\equiv s\\cdot r^2 \\pmod{n}", "f638d720f2f878c682b8120494172132": "\n\\begin{align}\n\\ [1, 3, -5] \\cdot [4, -2, -1] &= (1)(4) + (3)(-2) + (-5)(-1) \\\\\n&= 4 - 6 + 5 \\\\\n&= 3.\n\\end{align}\n", "f63911eaf5593d7ea2d174e62c3271a7": "\\rho_\\text{TOT}(\\mathbf{r})", "f6392cc8ded15c8f7455498b85e7fa4c": "\\mathbf{z}(t) = \\mathbf{H}(t) \\mathbf{x}(t) + \\mathbf{v}(t)", "f6397180cfd18d08dfde8e299e867eef": "R=K[x_1,\\ldots,x_n]", "f6399029d314c6fb8bf66566c6f0355d": "(\\mu_{ab}^{*(c)}(t))", "f63998fd69c35a55cec27099370444ec": "f : \\mathbb{R}^n \\to \\mathbb{R}", "f639c1ba75aa9718c72877c14ac6ee69": " i = 0,\\ldots,N ", "f639e46ef922e21dee98648b93a9b3ee": "\\psi(\\vec{x})=\\psi(\\rho^{\\sigma})=\\rho_0^{\\sigma}\\left[1-e^{(-\\rho^{\\sigma}/\\rho_0^{\\sigma})} \\right]\\,\\!", "f63a36bda29815a4ac83dbc4b20c15f6": "\\delta^4 \\,", "f63a699a7eeb138d1a5d5a4db1931c90": "\\nabla_{[a}\\omega_{b]}=\\hat{\\nabla}_{[a}\\omega_{b]},", "f63a75e1113f16c81587023ccd502eae": "\\tau_{{(Q)}_{}} = D_{(Q)}\\left(Q-1\\right)", "f63ab63e631483c2edd8e661d8c8147f": "\\mathfrak{P}^{2}", "f63acd145092b49cd721fe2a99b7eb64": "\\left|r\\right\\rangle", "f63acd8d70b5b4ee799d7d839f400314": "\n\\Pr(3,6,1| \\theta,10)=\n\\frac{10!\\theta^3}{\n1^1\\cdot 3^1\\cdot 6^1\n\\cdot\n1!1!1!\n\\cdot\n\\theta(\\theta+1)(\\theta+2)\\cdots(\\theta+9)}\n", "f63ad877dd7ac25c5ea98ad07f312a1b": "t:G\\to M", "f63adcb2a56c10f1e78b86cc3de00b41": "\na_i=\\frac{k_i}{2m}\n = \\sum_{j} e_{ij}\n", "f63b5bed767330f30d3572bff751db6f": "S_L \\, \\dot= \\, \\tfrac{d}{d-1} (1 - \\mbox{Tr}(\\rho^2) ) \\, ,", "f63b88c09aab607dcbb850616616ca2e": "a_n < b_n", "f63bf3ed35332f85f85d043cfcb02361": "T^*: Y^* \\to X^*", "f63c33fab64e788f4bd13ed8c91ed152": "{}m=\\sqrt{m^2}.", "f63c9b5030f3d753d9d6f687d11040a0": "\\vec{\\ell_2}", "f63d0e7805c561f0b2c23ab69be2d083": "F=0.82\\pm 0.01", "f63dba4541abd932a64dae3a3211541b": "b = S a\\,", "f63e0756fa779cbb1358d06c77aa8e21": "\\alpha:f\\Rightarrow g:A\\to B", "f63e09d604dcee2b343261285072f557": "\n\\begin{align}\n \\lambda_1& \\;+ 7\\lambda_2& &- 2\\lambda_3& = 0\\\\\n 4\\lambda_1& \\;+ 10\\lambda_2& &+ \\lambda_3& = 0\\\\\n 2\\lambda_1& \\;- 4\\lambda_2& &+ 5\\lambda_3& = 0\\\\\n-3\\lambda_1& \\;- \\lambda_2& &- 4\\lambda_3& = 0\\\\\n\\end{align}\n", "f63e134519770321180e6e9fb602b2b2": "\\text{var} = \\frac{(n-s+1)(s+1)}{(3+n)(2+n)^2},\\text{ which for }s=\\frac{n}{2}\\text{ results in var} =\\frac{1}{12+4n}", "f63e1f1da47830be6f07f77ad8757fb1": "\\{a, b\\}", "f63ef387836b4d7bb8776cf1ee66d669": "e^{-\\sigma^2n^2/2+in\\mu}", "f63f464567dcf053679fb07cdc0ca918": "(x(t),y(t))", "f63f6f4bddfb98e9d360c8422149d320": "b \\otimes b'", "f63f985c09b6f882215e32b807427de2": "\\omega+\\omega", "f63fbb8a59a2aecb1f2fa26fc019c1a3": "x = \\frac{\\nu}{n}", "f63fbc18bfc28dbeb0fe8297a086e9e3": "\\mathcal D_A (\\rho)", "f6402414450e408c2c469f12113129db": "\\chi(X)", "f6403399932aad10acfacd49803ea25e": "1+\\varepsilon^{1/2}+\\varepsilon^{2/3}+\\varepsilon^{3/4}+\\varepsilon^{4/5}+\\cdots", "f6405ca6dacb1e5b7328e1f1e4e9c4b5": "t + k", "f6406e2c17737fcacdf672b5d520f44d": "\n m(\\theta_0) \\equiv \\operatorname{E}[\\,g(Y_t,\\theta_0)\\,]=0,\n ", "f6407cd7d8875a380e55e9db8fc4a60e": " \\star^{-1} = (-1)^k s\\star", "f640873372fa5f3cdd7c36deb0bbcf5b": "\\dfrac{1}{s+1} + \\dfrac{s}{s+1} \\cdot \\dfrac{mR}{m-s+1}", "f6409a8625c83e4da5b4ef850e02d753": "l_{ij} = 1", "f640d85cf7cfdcd00a3148355a845586": " \\rho_j ", "f6414581cbbf92e8808a7587574a474f": "f(y) \\otimes f(y')", "f641672b9728b007038a3f1467b7beda": "\\displaystyle L_n^{(\\alpha)}(x;q) = \\frac{(q^{\\alpha+1};q)_n}{(q;q)_n} {}_1\\phi_1(q^{-n};q^{\\alpha+1};q,-q^{n+\\alpha+1}x) ", "f64197d387e7aaefb57a1a2ffd765a6c": "\\frac{c}{\\sqrt2\\chi}\\sqrt{(\\chi^2-2)+\\sqrt{\\chi^4+4}}", "f641da2713868d6dd191f8d783fb539b": "\\Delta_N(x) = |f(x) - S_N(x)|", "f641f482383d2509e28c0eb0d600f112": "\\frac{d q}{d t} + ", "f64245704077dec7ab4a5c42b98b51f7": "\nF_{1 \\rarr 2} = \\frac{1}{A_1} \\int_{A_1} \\int_{A_2} \\frac{\\cos\\theta_1 \\cos\\theta_2}{\\pi S^2}\\, \\hbox{d}A_2\\, \\hbox{d}A_1\n", "f64281dad1ac00033874519e6e4e68ce": " \\nu = {1\\over T}.", "f643478de65c00409f82cbdd66716822": "\\Delta\\ ", "f64354d585169f228e4a59cf995d3303": "u(t,x) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^\\infty F(\\xi) e^{-\\alpha \\xi^2 t} e^{i \\xi x} d\\xi, \\,", "f64370a69a075dd1c450bef3440583b8": "\\partial \\Omega ", "f64393a93da5b76843ada87e34f0dbcf": "\\boldsymbol{X}' = \\partial \\boldsymbol{X}/\\partial s", "f643a57b29f87120700226bb4a058cc8": " H^1(\\Omega)-", "f643b03325597d5a7220eccb3c11ceb3": " J^1\\Sigma\\times_\\Sigma J^1_\\Sigma Y\\to_Y J^1Y, \\qquad\ny^i_\\lambda=y^i_m \\sigma^m_\\lambda +\\widehat y^i_\\lambda", "f643bdff0d346318096403f38674af4b": "a|c", "f643de8aa39741e0dd5ac841f21eabd4": " \\delta = \\frac{1}{n} \\ln \\frac{x(t)}{x(t+nT)}, ", "f64408eed3c864d988c7474429d432f2": "w_k = \\frac{\\tfrac12 h \\pi \\cosh kh}{\\cosh^2(\\tfrac12 \\pi \\sinh kh)}.", "f6447adeb95e750fb994dfdeef569ce3": "(\\mathbf I - \\mathbf J_\\sigma)^{-1} = \\mathbf I + \\mathbf J_\\sigma + \\mathbf J_\\sigma^2 + \\mathbf J_\\sigma^3 + \\dots", "f644a8be007b7cc2093394cb720aaea1": "\n \\lnot A \\vdash \\lnot A\n ", "f644c3ada894d89393a1426cf6933b93": "y_0, y_1, z_0", "f644d01d2e16158e1dc104487c17c295": "-p^2,", "f644f15c196224af037a3326c70ad30c": "\\langle\\mathbf{p}\\rangle = m \\langle\\mathbf{v}\\rangle,", "f645c8828631fb0cbd59117cbaae5a8e": "\\tau^{ab}=\\lambda^a \\mu^b \\,", "f645ed518c3e587d082d1ae3120b4757": "\\begin{align}\n\\Gamma(z) &= \\int_0^x e^{-t} t^{z-1}\\, \\frac{\\mathrm{d}t}{t} + \\int_x^\\infty e^{-t} t^{z+1}\\, \\frac{\\mathrm{d}t}{t} \\\\\n&= x^z e^{-x} \\sum_{n=0}^\\infty \\frac{x^n}{z(z+1) \\cdots (z+n)} + \\int_x^\\infty e^{-t} t^{z}\\, \\frac{\\mathrm{d}t}{t}.\n\\end{align}", "f64614961ed5e2a5ee49e75d1dfb9e91": "M^\\beta=\\frac{S}{m-n}(X^{\\rm T} X)^{-1}.", "f6467a1bf0f4325e7790f57cdfe29200": "\\|f_n\\|^2 = |a_1|^2 + |a_2|^2 + \\cdots + |a_n|^2. \\, ", "f6468467800049ab933bd81e1c161023": "\\frac{1}{2^{i-1}}", "f646b816aaf4692b7a5fc921fb891707": "\\tilde{h}(I)", "f646c1fc3f4190daa6a0fb8674c64934": "\\varphi(\\varphi(x)) = e^x ", "f646ea823c3120962876c672aa95ba6a": "B^2 - AC > 0 ", "f646f77438b83330c86fddcf67db9e70": "\\scriptstyle (L)\\sqrt{-1}", "f64714e6f9c4c09702835b8fea1749ed": "\\alpha|0\\rangle_{1} + \\beta|1\\rangle_{2}\\longrightarrow \\alpha|0\\rangle_{1}\\otimes|1\\rangle_{2} + \\beta|1\\rangle_{1}\\otimes|0\\rangle_{2}.", "f6478514963507620c464d72e4757954": "\\begin{array}{rcl}\n\\hat{x}&=&\\rho\\cos \\hat{t},\\\\\n\\hat{y}&=&\\rho\\sin \\hat{t}.\n\\end{array}", "f647fc6f81f60241aa9316652386843e": "f \\in C \\iff \\omega_{f} (\\delta) \\to 0", "f6485082334c71193f3620ead910541a": "M_A,\\, M_B,\\, M_C", "f6488d6094edd03f437bd10bc7414775": "V_{\\rm b} - V_{\\rm c} = \\int_{T_{\\rm c}}^{T_{\\rm h}} \\left( S_\\mathrm{A}(T) - S_\\mathrm{B}(T) \\right) \\, dT,", "f648acbd68003031bcc5ba329bebb57f": "E_L = E_{\\rm z} = C_{33}-2C_{13}C_{13}/(C_{11}+C_{12})", "f648bc30f391f27d237d87322f552884": "\\sigma(X_1) = X_2", "f6490432a8974a0b11cc05f982ac98f9": " \\frac{\\partial e(\\mathbf{p},u)}{\\partial p_j} = h_j(\\mathbf{p},u)", "f6492f2f0cdacf035c2f7a9dbb4cb5b9": " \\scriptstyle \\tau", "f649394809b7729f81bfcde919e27741": "{\\beta_j}^{k+1}={\\beta_j}^k+\\Delta \\beta_j", "f64975c64aace217b83fb91abef94596": "\\begin{cases}\n \\frac{(\\alpha+2n-1)(\\alpha+2\\beta-1)}{(\\alpha-3)\\sqrt{\\frac{n(\\alpha+n-1)\\beta(\\alpha+\\beta-1)}{\\alpha-2}}} & \\text{if}\\ \\alpha>3 \\\\\n \\infty & \\text{otherwise}\\ \\end{cases}", "f6499eaf1dc7bed25ce9417677ec433d": "\\frac{200-125}{100}=0.75", "f649a2fdc1ba1f28fefb9f8799612d62": "A \\times A = 2\\,", "f649bb20ca6a82dae0f92a6546d3d433": "(\\log 5 - \\log 3)", "f649ed29e66fb197aed2e165a616a3d1": "\\overline{\\mathbf{p}\\mathbf{q}}", "f64a2c2bd513bf1942ca1d1bcd78f54a": "RaR^{\\dagger} \\mapsto RAR^{\\dagger}.", "f64a5aebbe44e692e11b48b2e6e4d9b9": "f:S\\to S", "f64aaecb68a77d76cecc19ab2cf9b9ea": "\\sigma \\in M", "f64b6b810dfd48719e226ee13df39b25": "\\sec \\theta = \\csc \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\cos \\theta} ", "f64ba48a1a5d933571e9732ec72d8197": "\\nabla_x f=2 A^\\top(Ax-b)=0", "f64c274cda83e8ceb6201aba4396c316": "F_{N,D}(n,d,z):=\\sum_{k=0}^\\infty \\frac{ z^k \\prod_{i=1}^p\\Gamma(n_i+k)\\Gamma^{-1}(n_i)}{\\Gamma(k+1)\\prod_{i=1}^q\\Gamma(d_i+k)\\Gamma^{-1}(d_i)}", "f64c3cf2980544b47f42d40e779baedb": "\\nabla \\cdot \\mathbf{D} = \\rho_\\text{f}", "f64ca6678f5401adda6d4acb61cf3eaf": "\\chi(u)=\\frac{\\pi}{2}C_s\\lambda^3u^4-\\pi \\Delta f \\lambda u^2", "f64cccb5a0a48acd98b35ef30c617dee": "I_{solid} = \\frac{3m s^2}{7}\\,\\!", "f64cee39d918fbd72210ca52e558af11": " \\frac{n-1}{n} ", "f64d2c3ffa8df720984254afe3b3001a": "P \\langle E_{1,j}^C \\cap E_{2,j}^C \\cap \\cdots \\cap E_{m,j}^C \\rangle", "f64d938e2aad92a33738136985787b18": "(A + B) \\cdot (\\lnot A + C) \\cdot (B + C) = (A + B) \\cdot (\\lnot A + C)", "f64da2c641faf11a0d18db4f87afa00b": "\\mathbf{E} = \\frac{q}{r^2} \\mathbf{\\hat r}", "f64da2d37ff355a58dc64f6ac85e646d": "\\frac{1}{s}", "f64dc3a319aae0f4bc1bdd3ea72cd1b3": "\\langle \\varphi, T_k \\psi\\rangle = \\langle \\varphi * \\psi^*, T_k\\delta\\rangle = (\\varphi*\\psi^*)(k)", "f64de08ec251b83a0ad3c0286ae8e9f4": "2^4\\cdot 3^2\\cdot 5", "f64e63059340ed7f41078a20a23223f4": "\\Delta I = \\gamma_i \\text{Cov}(E_i,C_i) - \\sum_{i\\neq r} q_{ir}\\gamma_r \\text{Cov}(E_r, C_r) \\, ", "f64f3cc881fb758127c9b076fb7cbeb5": " E = \\frac{hc}{\\lambda} = \\hbar c k \\,.", "f64f4a404c3392c59228203fcd061fd2": "X(T) = \\operatorname{Hom}_S(T, X)", "f64f7c3a6108a4c4d89f1fe628aec7fb": " \\frac{d}{dt} \\iint_{\\Sigma} \\mathbf{B} \\cdot \\mathrm{d}\\mathbf{S} = \\iint_{\\Sigma} \\frac{\\partial \\mathbf{B}}{\\partial t} \\cdot \\mathrm{d}\\mathbf{S}\\,,", "f64fb3ad3e97c83697244eb43d06ceeb": "\\Delta(G)", "f64ff142d0eae708cff30fe95991710e": " u_0 = y_0 \\sum_{k=0}^\\infty \\left (\\lambda-\\lambda_0^* \\right )^k \\widetilde X^{(2k)},", "f64ffb537a4e1b1a3eb034936f62721d": "\\delta m^{2}=m_{2}^{2}-m_{1}^{2}>0", "f650331d382d9495022d40f3fefcd184": "f_S", "f6504cf901754ad158fc2c3f330ad06d": "\\frac {k}{d}", "f6505d661fb90ff8fe4d3052b119a3c5": "(0,T)\\times\\mathcal{M}", "f650b252f727ef249db5baeac1aab318": "(A \\and C)", "f650bd91bac4b019398d0693d41105a2": "\\rho^3", "f6516dcc14d8d0c329a23f56a1562e01": "H_1(X;\\mathbb Q)", "f65186ceb15c54d75eb4580920efc47f": "\\hat{H} = \\hat{H}_{S}\\otimes\\hat{I}_{B} + \\hat{I}_{S}\\otimes\\hat{H}_{B} + \\hat{H}_{I}", "f6519010a384c167e9770a4e0ec95bd2": "\\omega_k = \\sqrt{2 \\omega^2 (1 - \\cos(ka))} = 2\\omega\\left|sin\\left({{ka}\\over 2 }\\right)\\right|\\ ", "f6519864fc74a60839494fabbb31f9c9": " \\left\\lbrace { \\lambda, \\frac{1}{1-\\lambda}, \\frac{\\lambda-1}{\\lambda}, \\frac{1}{\\lambda}, \\frac{\\lambda}{\\lambda-1}, 1-\\lambda } \\right\\rbrace \\ .", "f6520859a1322a1f2df612bddd7c13b2": "\\Beta(x;\\alpha,\\beta)", "f652288e69ca290ce7ef32d14210fa28": "W_0,W_1, W_2", "f6523d3cb4a457197335990423f98a2a": "t = \\tfrac{p + q + r + s}{2}", "f6527603ce73ddf56237b8a4f2736495": "\\kappa(X,Y,Z,W) = E(XYZW)-E(XY)E(ZW)-E(XZ)E(YW)-E(XW)E(YZ).\\,", "f652c979835562818697db483a471a9c": "H_{ij} = {-\\gamma\\over {s_{ij}}^2} \\begin{bmatrix} {(x_j - x_i)(x_j - x_i)} & {(x_j - x_i)(y_j - y_i)} & {(x_j - x_i)(z_j - z_i)} \\\\ {(y_j - y_i)(x_j - x_i)} & {(y_j - y_i)(y_j - y_i)} & {(y_j - y_i)(z_j - z_i)} \\\\{(z_j - z_i)(x_j - x_i)} & {(z_j - z_i)(y_j - y_i)} & {(z_j - z_i)(z_j - z_i)} \\end{bmatrix}", "f652fe262a4e8f3e99544b3556ab0566": "\\mu_t\\,", "f653138d4bb7ec167a0912790565d115": "W(q_{1},\\dots,q_{N}) = \\int p_i\\dot q_i \\,dt = \\int p_i\\,dq_i", "f6532cc9c27bb83db56fe7e28bfccc7c": "p_1 < p_2", "f65334626f097be9195e20e39f6dc2f8": " \\mathbf{H}_{n+1} = \\{ B \\in \\mathbf{F}_{n+1} : \\ B \\cap C = \\emptyset, \\ \\ \\forall C \\in \\mathbf{G}_0 \\cup \\mathbf{G}_1 \\cup \\ldots \\cup \\mathbf{G}_n \\}, ", "f653475fd96a42c700d60d6ad8608c59": "B|A", "f653d1278355f5988ab707769d165035": "\\forall n \\in \\mathbf{Z}, \\ \\ \\ \\hat{g}(n) = \\frac{1}{2\\pi}\\int_0^{2\\pi} g\\left(e^{i\\phi}\\right) e^{-in\\phi} \\, \\mathrm{d}\\phi.", "f653f2ba2728fd6f508c8ef3e8cb2f64": " t_1=t_2= -\\frac{3q}{2p}\\quad \\text{and} \\quad t_3=\\frac{3q}{p}\\,.", "f6540587800b3dfdf4facb22e2a242b1": "\\lambda^2 - p\\lambda + q=0", "f6542ed598e44af30d3ab1f67bed6b49": " \\Phi_p(x,y)\\equiv (x-y^p)(x^p-y)\\bmod p,", "f6543092884df91ecbea8d20022f06ad": "\\mathbf{\\hat b} = {\\mathbf{b}} / {|\\mathbf{b}|}", "f65456ba072764a1f1522fdc48bc6b37": "F_{\\theta_1} = -(m_1+m_2)gL\\sin\\theta_1,\\quad F_{\\theta_2} = -m_2gL\\sin\\theta_2.", "f6548a3836bb2d303b0f6a0096827fbb": "\\delta^{'}", "f654cb810e124598082f8fd0b9c21c0b": "\\Delta_0 \\;\\log f = -K f^2,", "f654d156f76228c19a546499b169cb22": "\\theta_b=\\theta(x)|_{x=jb}\\,", "f6550a6677181882da533ac80f5e90fe": "\\sec A = {\\csc A \\over \\cot A} ", "f655164d6dab68ba261d5cdead352666": "Z = \\sum_{\\mathrm{configs}} e^{\\sum_k S_k} = \\prod_k (1 + p ) = (1+p)^L.", "f65531ddf4094ef035a4527a1a371501": "A_{\\alpha ; \\beta ; \\gamma} g^{\\beta \\gamma} = A_{\\alpha ; \\beta , \\gamma} g^{\\beta \\gamma} - A_{\\sigma ; \\beta} \\Gamma^{\\sigma}_{\\alpha \\gamma} g^{\\beta \\gamma} - A_{\\alpha ; \\sigma} \\Gamma^{\\sigma}_{\\beta \\gamma} g^{\\beta \\gamma} \\,.", "f6559a5b3dee47ddb55d45ec464a5070": "L = Y ( P - C ) - Y D F", "f6561f8bf417a14f77e0cca95eddae54": "f (q_i , p_j)", "f6562443e999c4fd6003b4b877443efc": "h_\\parallel^{2/3}+h_\\perp^{2/3}=1. \\,", "f65654943d618ec720f085a136cba716": "(k+1)\\times(k+1)", "f6566d09cf5c6679175dd2fb24b43fe6": "R \\oplus R ", "f656ca5cb83737e95a5ade03201d5f1d": "\\begin{align}\n\\mathcal{L}&=\\frac{1}{2}\\,[I_{1}\\omega_{1}^{2}+I_{2}(\\omega_{2}^{2}+\\omega_{3}^{2})]\\\\\n\t&=\t\\frac{1}{2}\\, I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta\\cos\\alpha)^{2}\n\t\t{}+\\frac{1}{2}\\, I_{2}\\{[\\dot{\\alpha}\\sin\\psi+\\Omega(\\sin\\delta\\sin\\alpha\\cos\\psi+\\cos\\delta\\sin\\psi)]^{2}\n\t\t{}+[\\dot{\\alpha}\\cos\\psi+\\Omega(-\\sin\\delta\\sin\\alpha\\sin\\psi+\\cos\\delta\\cos\\psi)]^{2}\\}\\\\\n\t&=\t\\frac{1}{2}\\, I_{1}(\\dot{\\psi}-\\Omega\\sin\\delta\\cos\\alpha)^{2}+\\frac{1}{2}\\, I_{2}\\{\\dot{\\alpha}^{2}+\\Omega^{2}(\\cos^{2}\\delta+\\sin^{2}\\alpha\\sin^{2}\\delta)\n\t\t{}+2\\dot{\\alpha}\\Omega\\cos\\delta\\}.\n\\end{align}", "f6570b44eed8112af8f20f4ec98f9ffe": "\\textstyle b + l_2 -1", "f65720c66c6c7898334ee98be93e1680": " \\mathcal Y_k = \\mathcal C_k\\otimes\\mathcal D_k", "f657716be7321252891fa8c3ed514fb6": "\\mathfrak{A} \\mathfrak{B} \\mathfrak{C} \\mathfrak{D} \\mathfrak{E} \\mathfrak{F} \\mathfrak{G} \\mathfrak{H} \\mathfrak{I} \\mathfrak{J} \\mathfrak{K} \\mathfrak{L} \\mathfrak{M} \\mathfrak{N} \\mathfrak{O} \\mathfrak{P} \\mathfrak{Q} \\mathfrak{R} \\mathfrak{S} \\mathfrak{T} \\mathfrak{U} \\mathfrak{V} \\mathfrak{W} \\mathfrak{X} \\mathfrak{Y} \\mathfrak{Z} ", "f6579fbd14e951c24591d11ef1862eeb": "\\gcd(4046803256\\, +\\, 2^{2^5}\\times 1438793759,\\, F_{6}) = 274177.\\!", "f657bb2b2814eecdbd71b8c16f730bd8": " f_{n+1} \\rightarrow (1/2)f_{n}", "f657bdee22ccef19f37973fd9d19ed58": "y=f(x) \\,,", "f65834fbf097a5df044c9d329121e222": "V_{\\!-} \\,\\, = \\beta \\cdot V_{\\text{out}}", "f6584caca469f90ed418e0526b14a4a9": "f_{uc}(\\langle Go! \\rangle) = \\{\\langle G \\rangle\\} \\cdot \\{\\langle O \\rangle\\} \\cdot \\{\\} = \\{\\}", "f65902861a65be1250b8085aee790f69": "\\rho(A) = \\lim_{k \\to \\infty}\\|A^k\\|^{1/k}.", "f65916b0a871092bfd6e4207e2fa0a5c": "z^K = 0", "f65928df4b0a5930a29de137a2b09542": " \nQ = - \\frac{\\hbar^2}{2m} \\frac{\\nabla^2 \\sqrt{\\rho}}{\\sqrt{\\rho}}\n", "f65970bdd3613dabf1e7d1c7c908fc46": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} =\n -\\cfrac{2Eh^3}{3(1-\\nu^2)} \\begin{bmatrix} 1 & \\nu & 0 \\\\ \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} \\varphi_{1,1} \\\\ \\varphi_{2,2} \\\\ \\frac{1}{2}(\\varphi_{1,2}+\\varphi_{2,1}) \\end{bmatrix} \\,,\n", "f659b3bb1ef2ad4651f048d4310a08bc": "t\\mapsto\\operatorname{dist}^2\\circ\\gamma(t)-t^2 \\, ", "f659b575bf7a4b1591319f11721fb8a5": "dU =C_{V}dT +\\left[T\\left(\\frac{\\partial p}{\\partial T}\\right)_{V} - \\frac{n R T}{V}\\right]dV.\\,", "f659bd7df92220052bf8aee2c5b0dca1": " ke_k(x_1,\\ldots,x_n) = \\sum_{i=1}^k(-1)^{i-1} e_{k-i} (x_1,\\ldots,x_n) p_i(x_1,\\ldots,x_n),", "f65a896fdad97ca72bb72a617eb5ec6c": "\\mathcal{G}_{2k+2}", "f65ab13d4da9ad9f8bcabbaedfd46c62": "\\,F_n", "f65ac9486352c43f74540762050536a2": "\\boldsymbol{x} := ({x_k}^u \\mid {x_{k+1}}^l) \\qquad \\qquad k=0,1,\\ldots", "f65af5a88cd311169439a8fd9c8bc6e2": "\\scriptstyle\\log_e P_{mmHg} = \\log_e (\\frac {760} {101.325}) -7.783651 \\log_e(T+273.15) - \\frac {6160.169} {T+273.15} + 66.97868 + 6.139268 \\times 10^{-6} (T+273.15)^2\n", "f65b05a1743e2a61ce5af7bd45db5bfc": "\\mathbf N=\\mathbf I_2\\,\\!", "f65b5317b18fb6568d2fb14f2bb85a5d": " G_2=G_3=\\beta^2p_ip_i", "f65b827642753e79d31c930e5f4414ed": "_{dual(q\\tilde{\\leftarrow}p)\\,}\\!", "f65bbb05a4423063b19d8bfb304e822e": "A\\cup B\\in U", "f65bcdf524eedf144304a06b112c9ae8": "S_e\\,\\!", "f65bf50b037e583d94dd215a5e505265": " \\boldsymbol{\\beta} = (\\beta_1,\\,\\beta_2,\\,\\beta_3) = \\frac{1}{c}(v_1,\\,v_2,\\,v_3) = \\frac{1}{c}\\mathbf{v} \\,. ", "f65c4f70e25cf0c782818fa85ab12bb7": "\\mathbb{R}^{n}", "f65c52c07406cac474d8c10e3c4f727b": "L u(x)=f(x)", "f65c775b15121cfa566a987f905b40b9": "O = 1000 * ( (^{18}O/^{16}O)/(^{18}O/^{16}O)_{SMOW} - 1)", "f65d113a9d3197eb55523bc0241878dd": "1 < \\gamma < \\varepsilon_\\alpha", "f65d179999df0819a9fc2a081a96a443": "- \\frac{8 \\pi G}{c^4} P(r) e^{\\lambda(r)} = \\frac{- r \\nu'(r) + e^{\\lambda(r)} - 1}{r^2} \\;", "f65d9797be213518d7fe6a264a7005b1": "S = 155 \\equiv 2 \\mod 3 ", "f65d994e54bbda99b46466fc0287f9ac": "\\left( 1 - L^s \\right)", "f65db08fcb59e402ceb1a10516654414": "Cv_{O_2}", "f65db3d14df4db728c861edefbc7dade": "g = \\frac{dv}{dx}", "f65dc2afcd0f0d70da4825264311a97d": "\\varphi_X(t) = \\operatorname{E}\\left(e^{itX}\\right)", "f65e105de71bafa5c17680ff6ef18849": "{\\hat{u}}", "f65e87cedcc6bec0ad4cd798b6254811": " \\nu_4 = p[ K_1 \\sigma_1^4 + 4 S_1 \\delta_1 \\sigma_1^3 + 6 \\delta_1^2 \\sigma_1^2 + \\delta_1^4 ] + ( 1 - p )[ K_2 \\sigma_2^4 + 4 S_2 \\delta_2 \\sigma_2^3 + 6 \\delta_2^2 \\sigma_2^2 + \\delta_2^4 ]", "f65eaaafba618bba712754e331782126": "o = a \\odot b", "f65f06c26298197a24b7252efe91b406": "\\sum\\limits_{k=1}^M \\lambda_k b_k = v", "f65f1391d3dad5948b3d508878a1ec3c": "\\mathcal{N}()", "f65f553a3769952c1225a4f96ed5c5d8": "f_*: Sh(X) \\to Sh(Y)", "f65f7b0a45ccfbbcad817664945927ce": "\\alpha^{(2)}_{zzz}", "f65fb27a022bc819ef49100a815171c8": " Q = \\frac{U Ar}{(B-A)} \\int^{B}_{A} \\Delta T \\,dz = \\frac{U Ar \\int^{B}_{A} \\Delta T \\,dz}{\\int^{B}_{A} \\,dz} ", "f65fe1d389ce79ec8d95fd45c464daad": "\\operatorname{sn}(u,k)", "f6602a8b4e90474b01ef8d5b98c389cd": "\n\\mathrm{DQE}(u) = \\frac{\\mathrm{SNR}_{out}^2(u)}{\\mathrm{SNR}_{in}^2(u)}.\n", "f660366430204cf8232a2ff4b4ecebc0": "\\mathcal{C}_x^\\infty\\,", "f660856245d03c0deda9c1f3629e4585": "T(K) = T(C) + 273.15", "f660b172dfac7d44c51c1fe94d705dd8": "{x^3}1{x}10 - {x^2}2{x^0}1 = {x^0}5", "f660c408d854dee2d22557a9725ae586": "l_1/l_2", "f6610bfc6e54d99c95f78070b8e6c5cc": "m\\bar{\\infty}", "f6616738e70a6945a37bea21c1bedc91": "\\lim_{p \\to 0} \\frac{\\ln{\\left(\\sum_{i=1}^n w_ix_{i}^p \\right)}}{p} = \\lim_{p \\to 0} \\frac{\\sum_{i=1}^n w_i x_i^p \\ln{x_i}}{\\sum_{i=1}^n w_i x_i^p} = \\sum_{i=1}^n w_i \\ln{x_i} = \\ln{\\left(\\prod_{i=1}^n x_i^{w_i} \\right)}", "f661a79903e276ab7087b449fffe2429": "p \\cdot q", "f661fe0cadab2eda570437366037886a": "u_b", "f662ddc8cb8d7535858719e0127a4698": "\\displaystyle u_{tt}-u_{xx} = \\epsilon(u_t-u_t^3)", "f662e6b967008c3da15cc303f0dcb65e": "R^T w.", "f6639d749b97e5818d2832d02e8a8d62": "F_{v,\\mu}(x)=\\begin{cases}\n\\frac{1}{2}\\sum_{j=0}^\\infty\\frac{1}{j!}(-\\mu\\sqrt{2})^je^{\\frac{-\\mu^2}{2}}\\frac{\\Gamma(\\frac{j+1}{2})}{\\Gamma(1/2)}I\\left (\\frac{v}{v+x^2};\\frac{v}{2},\\frac{j+1}{2}\\right ), & x\\ge 0 \\\\\n1-\\frac{1}{2}\\sum_{j=0}^\\infty\\frac{1}{j!}(-\\mu\\sqrt{2})^je^{\\frac{-\\mu^2}{2}}\\frac{\\Gamma(\\frac{j+1}{2})}{\\Gamma(1/2)}I\\left (\\frac{v}{v+x^2};\\frac{v}{2},\\frac{j+1}{2}\\right ), & x < 0\n\\end{cases}", "f663b68a7d95941235da74a2514dd425": "Ax^2\\ +\\ Bxy\\ +\\ Cy^2\\ +\\ Dx\\ +\\ Ey\\ +\\ F\\ =\\ 0", "f663c37121b2ec4997281312e9ee502e": "\n g_1 =\n \\left.\n \\begin{matrix}3^{3^{\\cdot^{\\cdot^{\\cdot^{\\cdot^{3}}}}}}\\end{matrix}\n \\right \\}\n \\left.\n \\begin{matrix}3^{3^{\\cdot^{\\cdot^{\\cdot^{3}}}}}\\end{matrix}\n \\right \\}\n \\dots\n \\left.\n \\begin{matrix}3^{3^3}\\end{matrix}\n \\right \\}\n 3\n \\quad \\text{where the number of towers is} \\quad\n \\left.\n \\begin{matrix}3^{3^{\\cdot^{\\cdot^{\\cdot^{3}}}}}\\end{matrix}\n \\right \\}\n \\left.\n \\begin{matrix}3^{3^3}\\end{matrix}\n \\right \\}\n 3\n", "f66454bfccba1d716e08b818a6ee9a1a": "\\Delta^y\\Delta", "f6646c409337c7184d46ddffecefeacc": "(2n+1)^{\\text{th}}", "f6647151f957ad74fd83a9081e3f23b3": "S(a)=\\{x: f(x)\\geq a\\}", "f665585785809783a4c8e70d9866f772": "F = F^e \\cdot F^g\\,", "f665d915e2149996bae95ac61f5422a5": "\\mathfrak{P}^{119}", "f6660025fc4f07b6c6ee046bf4c1a1f0": "dx^{0(1)} = \\frac{1}{g_{00}} \\left ( -g_{0\\alpha}\\, dx^\\alpha - \\sqrt{\\left ( g_{0\\alpha}g_{0\\beta} - g_{\\alpha \\beta}g_{00} \\right )\\, dx^\\alpha \\,dx^\\beta} \\right ),", "f6664558dcb1b5ea8ded9101342063f6": "\\widehat{\\sigma}_{(i)}^2={1 \\over n-m-1}\\sum_{\\begin{smallmatrix}j = 1\\\\j \\ne i\\end{smallmatrix}}^n \\widehat{\\varepsilon}_j^{\\,2},", "f666630df1211be9f61d55de86e7b0cd": "dF(\\mathbf{x}|\\boldsymbol\\eta) = e^{\\boldsymbol\\eta^{\\rm T} \\mathbf{T}(\\mathbf{x}) - A(\\boldsymbol\\eta)} dH(\\mathbf{x}).", "f6668ba7d7e49120d947899caf970bc8": "(x^{q^{2}}, y^{q^{2}}) = \\pm \\bar{q}(x, y)", "f666a1501330f5a5c14e5ec1a9292e9f": "\n\\mathbf{F} = \\frac{d\\mathbf{p}}{dt} = f(r) \\frac{\\mathbf{r}}{r} = f(r) \\mathbf{\\hat{r}}\n", "f666ee7c12cdb300f56c191462b49675": "\\operatorname{colim}: C^I \\leftrightarrow C: \\Delta \\, ", "f6670304e20ec124e471ba2c6abecec4": "z_2 = \\cos\\eta\\,e^{i\\varphi/2}.", "f6676b6cbce5786cee3049f87f2fbfcf": "\\nabla\\times\\mathbf{E}+\\frac{\\partial \\mathbf{B}}{\\partial t}=0", "f667902f0a82b01986b1cbc5161ae116": "\\mu_X(\\{a\\}) = \\lim_{T_1\\to\\infty}\\cdots\\lim_{T_n\\to\\infty} \\left(\\prod_{k=1}^n\\frac{1}{2T_k}\\right) \\int_{-T}^T e^{-i(t\\cdot a)}\\varphi_X(t)\\lambda(dt)", "f667ac2bf606c846cb15a4ba3b347c5b": " L = q^{\\Omega(n)}", "f667e491dba6f01341889550d2188649": "\\gamma=\\pi", "f668513cc81c23e184b20a16b6407e6b": "\\gamma_1\\in\\Gamma_1", "f6686811a5a9fb1ef6256f573096d96e": "\\sum_{i=1}^n W_i \\times {F_i} = ", "f668a10bc0365dbba805f37e56b45daa": "n = p_1^{a_1} p_2^{a_2} \\cdots p_r^{a_r}.", "f668de1c1b524e8d74776bad5f150348": "p \\neq 0", "f668f58bc859f811284bfed9eb56227d": "\\mathcal M_X", "f6698fdb0fee7f1a757f919057a7c80c": "f\\in B_{p,q}^\\lambda,", "f669b3512770acbc7b01f60680f4c43e": "m_p = c^{\\frac{1}{4}(p+1)} \\, \\bmod \\, p", "f669e14ac28b640fd6ebfac358139af8": "v > A \\ge \\sqrt{n}.", "f669f1bc55b0c3c3409cdfc7562dc610": "Q_r = \\frac{\\left\\{S_t\\right\\}^\\sigma \\left\\{T_t\\right\\}^\\tau }{\\left\\{A_t\\right\\}^\\alpha \\left\\{B_t\\right\\}^\\beta } ", "f66a13a4783c0edd963e03eb6c764893": "E_\\mathrm{rest} = m c^2.", "f66a2a8df3e0d4ee9e94d77de2d468a4": "wp(\\textbf{skip},R) \\ =\\ R", "f66aa6074918d91c8d8a5bbc958fd9a4": "c_{20}", "f66ab1251244dd2f5cc4520971164738": "x=y=t=0,\\,z=5,\\,s=5.", "f66ac938c92ff9df565a0490756671b4": "\\mathcal{L}s_n^{m}(\\theta) = -\\int_0^{\\theta} x^m \\log^{n-m-1} \\Bigg| 2\\sin\\frac{x}{2} \\Bigg| \\, dx", "f66b49969f0727cc934baeda375eb08a": "\\begin{array}{rcl}\n \\dfrac{d x}{d t} &=& y+3x^2-x^3-z+I \\\\ \\\\\n \\dfrac{d y}{d t} &=& 1-5x^2-y \\\\ \\\\\n \\dfrac{d z}{d t} &=& r\\cdot (4(x + \\tfrac{8}{5})-z)\n\\end{array}", "f66b5c78d8dc2bfd0f0fba7f694fd679": "\n\\begin{align}\n\\epsilon_1&=-J_h-J_v-J-J'-J'',\\quad \\epsilon_2=J_h+J_v-J-J'-J''\\\\\n\\epsilon_3&=-J_h+J_v+J+J'-J'',\\quad \\epsilon_2=J_h-J_v+J+J'-J''\\\\\n\\epsilon_5&=\\epsilon_6=J-J'+J''\\\\\n\\epsilon_7&=\\epsilon_8=-J+J'+J''.\n\\end{align}\n", "f66bc63e77e86ba86c36b778bec62d99": "X_1, X_2, ..., X_n", "f66bd948194f3da706b954fa4ee9086c": " z^{utopian}_i = z^{ideal}_{i}-\\epsilon \\text{ for all } i=1,\\ldots,k,", "f66c16713cbcb3b21de4542185acbce8": "2^\\lambda=\\lambda^+", "f66c90fb98610f2c0e928dcb463c9bad": "(z,w)\\mapsto (x_0,x_1,x_2,x_3)=(z\\bar{z}+w\\bar{w}, z\\bar{w}+w\\bar{z}, i^{-1}(z\\bar{w}-w\\bar{z}), z\\bar{z}-w\\bar{w})", "f66cc436ff9f77f7a6db9799f9ef11cc": "\\mathcal{G}(n)", "f66cc62c0449f05b718f3d5d3a330fb0": "S[\\psi]", "f66cfa321556f20c37e5139716f534cf": "\\mathbf I :=j_1(x,y,z) \\, {\\rm d}x_2\\wedge {\\rm d}x_3+j_2(x,y,z) \\, {\\rm d}x_3\\wedge {\\rm d}x_1+j_3(x,y,z) \\, {\\rm d}x_1\\wedge {\\rm d}x_2.", "f66d0f4390792a5cf203538db8ad5bde": "e^e", "f66da19241af8c50e59916116d50555e": "\\begin{align}\n\\ell(\\mathbf{y}) & = \\Delta(\\mathbf{y}, \\mathbf{t}) + \\langle \\mathbf{w}, \\phi(\\mathbf{x}, \\mathbf{y}) \\rangle - \\langle \\mathbf{w}, \\phi(\\mathbf{x}, \\mathbf{t}) \\rangle \\\\\n & = \\max_{y \\in \\mathcal{Y}} \\left( \\Delta(\\mathbf{y}, \\mathbf{t} + \\langle \\mathbf{w}, \\phi(\\mathbf{x}, \\mathbf{y}) \\rangle) \\right) - \\langle \\mathbf{w}, \\phi(\\mathbf{x}, \\mathbf{t}) \\rangle\n\\end{align}", "f66e45959d921f0bca9325d2658be0ce": "f(x;p)=p(1-p)^x I_{\\{0,1,\\ldots\\}}(x)", "f66ea5b3d1f1d4ee16eb361bf8c510f0": "\\widehat{M}_{(mag. dipole)}", "f66f0bb77d9429ab971dec86acb9f246": "D(X,Y) = \\nabla_XY-\\bar{\\nabla}_XY", "f66f28c689240b91967edede811487d9": "A_A F_{A \\rarr B} = A_B F_{B \\rarr A}", "f66f3ae0cfa4c146be957f4fcb26e4a7": "f = \\left ( \\frac {c + v_\\text{r}}{c} \\right ) f_0", "f66f9e73b1d7b3e94c3835cf72e6da5d": "\\textstyle |\\psi_j\\rang", "f66fa3ec46299569397fb4a4ba788b72": "\\Delta \\nu", "f6701260e6c71b57a61be818b7679cc2": " f(.,.)", "f67039e7d39077ff7d0a9105d617eabc": "\\sigma^2_X = \\frac{\\sum_{i=1}^n (X_i-\\bar{X})^2\\,{}}{n}.", "f67054903b17caa258020c1f76d71485": "g: X' \\rightarrow X", "f6705980aed1667acc79296ceaee8f28": "\\phi \\lor \\exists x \\psi", "f670b38a061f28d50f1e96d0f47e7295": "= {{5 * {25 \\over 45}} \\over 2} = ", "f670c064e2392362e2497ea22a1122ee": "1=\\det(I)=\\det(Q^\\mathrm{T}Q)=\\det(Q^\\mathrm{T})\\det(Q)=(\\det(Q))^2\\,\\! .", "f671afccb8859155e8864df9aa7fe085": "\\frac{3-2}{3+2}", "f671bdcff6933b0d5cdc1512ec9375a1": " {C_{0}}(\\mathbb{R}) ", "f671d3825f13fdc4a6aeeb0ea003ed05": "\\frac{1}{r^2}", "f672283795e65d921827a4ed03da561c": "\\delta_\\epsilon S", "f67228ff6db58ffb9ba24e4449a83f22": "\\rho_k(x)", "f67254efcace764245179e31f3facb31": "(\\exists x) \\phi(x)", "f67270318c38b5901f2b306f140bee5c": "\\delta_x\\,\\!", "f6729e0520742f372381a88b0519cbc2": "F_j:= f_{-1}\\circ f_{-2}\\circ\\cdots\\circ f_{-j}.", "f673481e12cc24dfb8a3d8aedd182984": " \\sigma_{y=modified} = \\sigma_{y,0} + \\sigma_{compressive} ", "f673a58001e57723ebc0b691ba853d5d": "p \\in M", "f673b8751b59cb3f47c542d08f47cd85": "\\pi_3 K = \\pi_5 K = \\dots = 0.", "f67480c21c0897c5d6dcdc53a6490394": "(\\mathrm d\\varphi_x)_a^{\\;b}= \\frac{\\partial \\widehat{\\varphi}^b}{\\partial u^a}.", "f674900d39550c4ffc78ddf3b62b16fc": "p(D')", "f674a24b26e86e35fa1c8bbef6a9fdf1": "(a,b)=\\tau(b^*a).", "f674b247fd168f1d356a8708508cfb39": "\\prod_{n=1}^\\infty (1 - a_n)", "f674d1c60908981c9fd22da01ecadf74": "\\displaystyle{e(ya)=y(ea)}", "f674eb281c461e5d6397605eb197e35d": "\\hat \\gamma^\\mu (\\mathbf{j} \\partial_\\mu - e \\mathbf{A}_\\mu) |\\psi\\rangle = m |\\psi\\rangle", "f67507f783a37db1c715bdbecff5eb21": "\\begin{align}\n 0 &{}= \\det(A - \\lambda I) \\\\\n &{}= \\det\\begin{bmatrix} 19-\\lambda & 3 \\\\ -2 & 26-\\lambda \\end{bmatrix} \\\\\n &{}= 500-45\\lambda+\\lambda^2 \\\\\n &{}= (25-\\lambda)(20-\\lambda) .\n\\end{align}", "f675170ee09d1e3058efa19b9311ac9a": "\\Lambda^n", "f675391ccd16108e892d74411351c12e": "E(y)=X\\beta", "f6754512cb18a1959a30a4e626a4deec": " d = \\sqrt{(u_1-x_1)^2+(v_1-y_1)^2+\\cdots}. \\,", "f6754863625a04766547abeb9aa3106b": "\\delta \\vec{r}", "f67549e27af4965c42a8ccdf051a88ce": "\\begin{align}H_x+H_p &= \\ln(\\sqrt{2\\pi}) + \\frac{1}{2} + \\ln\\left(\\sqrt{\\frac{\\pi}{2}}\\right) + \\frac{1}{2}\\\\\n&= 1 + \\ln \\pi = \\ln(e\\pi).\\end{align}", "f6755c9cc5332ea09fdbe54aa6ed96ab": "g^{-1}(g(x))=x \\, .", "f67641f4249f1c1ade0bf070bdb3ab2f": "\\vec{b_1}, \\ldots, \\vec{b_n} \\in \\mathbb{R}^{k}", "f676481ccabfc5d1ea40e63ee69215f2": "\n\\Delta e\\ =0\n", "f67688bba80dd15a867bfa5bf3439fa9": "\\operatorname{Br}(k) = H^2(k^{\\text{al}}/k)", "f6772d714f2abbb01b01632a17e66b88": " 0^{y}=0 ", "f67735d97a12989c3a766ad4ce525f01": "\\frac{46726}{(1+0.10)^9}", "f67768392789367087c2566da3f71066": "\\nabla\\cdot\\bold{j}_{\\rm n} = -\\nabla \\cdot D \\nabla c ", "f677b0f7f3e2a63afac30cbdfd8a44d5": "j,k=1,\\dots,n\\,", "f67852d1995a2acc7ab673b68c660e50": "\\,_1F_1(a;b-1;z)-\\,_1F_1(a;b;z) = \\frac{az}{b(b-1)}\\,_1F_1(a+1;b+1;z)", "f67871cd00ac973d0e2b80db93f3bcd3": "F_n", "f679168b61f35c21700eef5eb88dc82d": "A^C", "f6792c76ba938eef48770c0cfd51fae8": "I_{n}=I_{n+1}=\\cdots.", "f6792f6061ad965dec29844db14dce90": "2(1-\\varepsilon)\\gamma n\\,", "f679a7cb853823687eff211b942a4d3c": "\\mbox{span} \\{ e_1, \\cdots, e_k \\} \\in \\mbox{Lat}(\\Sigma) \\quad \\forall k \\geq 1 \\;.", "f679c4ddc8c72880e1aafb339cf513ef": "E_c(\\boldsymbol k ) \\approx E_{c0} +\\frac{(\\hbar k)^2}{2m} +\\frac{\\hbar ^2}{{E_g}m^2}\\sum_n {|\\langle u_{c,0}|\\mathbf{k}\\cdot\\mathbf{p}| u_{n,0} \\rangle |^2} ", "f67a0d7cd9c86b4ad7fa7cb19510dd65": "\n-\\frac{\\partial V(x,t)}{\\partial t} = \\frac{1}{2}q(t) x^2 + \\frac{\\partial V(x,t)}{\\partial x} a x - \\frac{b^2}{2 r(t)} \\left(\\frac{\\partial V(x,t)}{\\partial x}\\right)^2 + \\sigma \\frac{\\partial^2 V(x,t)}{\\partial x^2}.\n", "f67a53311071c704f09f777cb8b1de45": "\\boldsymbol{\\varphi}(0)=z^0", "f67a87a75b9c814e47231d10d93f3c8e": "\\lambda_j = -\\frac{4}{h^2} \\sin(\\frac{\\pi (j - 1)}{2n})^2", "f67ad80c00b18f35a787b0e014f13df3": "P^{\\ast} F < \\infty", "f67af833a8a07dc4a1a2accde63f71fe": "x \\in H_{\\aleph_1} \\,", "f67b0e28b70e5b4258eb917afbbc7628": " \\star d h = -\\exp(-p) h_y \\, \\sigma^1 + \\exp(-p) h_x \\, \\sigma^2 = -h_y \\, dx + h_x \\, dy.", "f67b2e961f74d15e6e137373cf6752b9": "N \\to (N \\cup \\Sigma)^*", "f67b3cc430eed51027faf9eb6b4b7698": "\\begin{bmatrix}\nu' \\\\ v' \\\\ w' \\end{bmatrix} = \n\\begin{bmatrix}\n1 & 0 & 0 \\\\\n0 & \\cos \\Omega t & \\sin\\Omega t \\\\\n0 & -\\sin\\Omega t & \\cos\\Omega t\n\\end{bmatrix}\n\\begin{bmatrix}\nu'' \\\\ v'' \\\\ w''\n\\end{bmatrix}", "f67c456e8805321c7cfb15f1d319c396": "E=\\, ", "f67ca93f152c967505ba9fdfcfa1f4e5": "\\lim_{x\\rightarrow 0}\\frac{\\sin x}{x}=1,", "f67cb632c9aa3b11f833340aa9b44fa1": "U^{2^{n}}", "f67ccfc6bf5056004cb09cca5089915d": " 99 + \\frac{29}{60} + \\frac{5}{60^2}, ", "f67d1a23ee38d96a670ae797a752f9f9": "R \\vee L = D", "f67d399cc84c38c8ceead0b9d6131942": "\\frac{\\partial \\mathbf{u^*}}{\\partial t} + \\mathbf{u^*} \\cdot \\nabla \\mathbf{u^*}\\ = -\\nabla p^*.", "f67d5f43a57c7b930fd324aceb60253f": "|L_{x.j}|", "f67dcffa5287b8b76d5c421c74c07875": "f(\\mathbf{Q})=\\int \\rho(\\mathbf{r}) e^{i\\mathbf{Q} \\cdot \\mathbf{r}}\\mathrm{d}^3\\mathbf{r}", "f67dd3117056243fe4b163a8042e2618": "\n\\begin{align}\nX_{1,1},\\ldots,X_{1,n_1} & \\sim \\operatorname{i.i.d.} N(\\mu_1,\\sigma^2), \\\\[6pt]\nX_{2,1},\\ldots,X_{2,n_2} & \\sim \\operatorname{i.i.d.} N(\\mu_2,\\sigma^2).\n\\end{align}\n", "f67e0fd1e15e865672cdf67e8505f972": "\\mathbf{z} ", "f67e137f6eb54706281c53e393ed6bcf": "||dE[A]||^2=G^{-1}", "f67e1a54c3a38a61410a2715ed9ad8d0": "{\\mathbb R}[z]/(z^{k+1})", "f67e5f33ae8de0999412069208aeb175": "k^{k^2}", "f67e86059a9874668e6e8d206bf4f5b6": "\\ \\beta=\\frac{1}{2}", "f67ed3a4e1c1bd57cf0d0848d7ea187d": "BABIP = \\frac{H-HR}{AB-K-HR+SF}", "f67ef8f338391b04f7f6f0eaea7234f1": "CR = \\frac{L_H}{L_L}", "f67f0ac429d017ad26d0f4efbff00784": "A=10+\\sqrt{30(10+3\\sqrt{5}+\\sqrt{75+30\\sqrt{5}}})a^2\\approx39.306...a^2", "f67f15a8136ff6e221b56285afb41050": "y^2=1-x^2", "f67f4e2d68844fccdf3dd1775ab9bc9b": " I_{Men} = \\frac{ S }{ \\sqrt{ N } } ", "f67f601f5c907927e336a128ae0c83c0": "\\Delta_{ij}", "f67fbe794da3bdc337f9db569b5bf1bc": "\\ i\\neq n ", "f67fe603a9f52418310306c2194303d5": "D_{ij}", "f68021749ff69bf0b89869d4358ea300": "H(s)", "f680330eefe1148971c9720d28c3c0bd": "l_2x+m_2y+1=0.\\,", "f68045587a36db8ca22c0aea63708e7e": "[2(pq + rs) - p^2 - q^2 + r^2 +s^2][2(pq + rs) + p^2 + q^2 -r^2 - s^2] \\,", "f68048e4868f4a937d8e1081ece1f9db": "g{\\in}G", "f680c8f9d4656f710dcd9e3e453881f9": "\\begin{align}\n e_1(X_1,\\ldots,X_n)&=h_1(X_1,\\ldots,X_n),\\\\\n e_2(X_1,\\ldots,X_n)&=h_1(X_1,\\ldots,X_n)e_1(X_1,\\ldots,X_n)-h_2(X_1,\\ldots,X_n),\\\\\n e_3(X_1,\\ldots,X_n)&=h_1(X_1,\\ldots,X_n)e_2(X_1,\\ldots,X_n)-h_2(X_1,\\ldots,X_n)e_1(X_1,\\ldots,X_n)+h_3(X_1,\\ldots,X_n),\\\\\n\\end{align}", "f680e9b5d76e422c5feccdd787991a36": "\\nabla \\psi \\, i \\sigma_3 - \\mathbf{A} \\psi = m \\psi \\gamma_0", "f6812984005239025a4dfd8729927f04": "h_{\\alpha\\beta}", "f681535cd20f19590068c66577c4f8c3": "f(n) = 9 \\log n + 5 (\\log n)^3 + 3n^2 + 2n^3 = O(n^3) \\,, \\qquad\\text{as } n\\to\\infty \\,\\!.", "f6818e78087ec47acf79858122f0c260": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 43.9\\cdot 1.44)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 13.6\\cdot R_{\\bigodot}\n\\end{align}", "f681a96f246e0d98efd0c4a9e5b88d1a": "\\frac{r}{R_{\\mathrm{secondary}}}", "f68207972fe0c39be7798431a8afcc29": "E_n", "f6822458c60a9e395265929c363833cb": "\n\\Delta G = m \\left( [D]_{1/2} - [D] \\right)\n", "f6828d5a3d3092a4584485984e9eb38b": "K(u) = \\frac{3}{4}(1-u^2) \\,\\mathbf{1}_{\\{|u|\\leq1\\}}", "f6830f423336f5b4a620753d067b6818": "\\sum_{n=1}^\\infty \\frac{\\mu(n)}{n^s}=\\frac{1}{\\zeta(s)}.", "f68314fc5ca7a0e08879637f7c7c4fe5": "\\scriptstyle\\langle B^2\\rangle = \\langle B_{\\theta}^2 + B_{\\rho}^2\\rangle", "f6837e73144e04ad811996609250e358": "X\\backslash \\{p\\} ", "f683d8bb6839b83d08efbc431e382de4": "\\frac{\\partial H}{\\partial u} = 0", "f68448bda2cfa4582e6abb3f80c124f7": "E^{p,q}_r", "f684c10e08052376ba6cfb2b48dd10aa": " H_a:\\mu=\\mu^* ", "f684e849a894379cbae18b44616ed45f": "V(K,U) = \\{f\\colon X\\to Y \\mid f(K)\\sub U\\}", "f68529b9ced8340bcd0658a22d028c3a": "B_n(\\kappa_1,\\dots,\\kappa_n)=\\sum_{k=1}^n B_{n,k}(\\kappa_1,\\dots,\\kappa_{n-k+1})", "f68535c5474070c79482809d7ac95514": "H_{-\\alpha}^{(2)} (x)= e^{-\\alpha \\pi i} H_\\alpha^{(2)} (x). ", "f685c1c673fd4fe80ea068397bc78d38": " =\\frac{\\hbar\\omega_{ph}^3}{4\\pi^3u_{ph}^2} ", "f68605364469dd9ab00e16bbd1d64432": "h(x',y')=f(x, y) T/2 \\pi", "f68633d4f61bb6c19ca866676c5d6fca": "m_1 m_2 = (m_1+m_2) m_\\text{red}\\!\\,", "f6863879ec08e59d71c2b913294f607c": "Z_{ij} = X_{i0}Y_{0j} + X_{i1}Y_{1j}.\\,", "f68660c003f394db88d396e236435e1b": "\\log_2(4^n)=2n", "f687009ae5862b96598865dff8e6bfab": "\\det\\begin{bmatrix} 3-\\lambda & -4\\\\4 & -7-\\lambda \\end{bmatrix} = 0", "f6876ce2bf3917145f1afc457b6cbe2e": "\\text{bind} \\colon (E \\rarr T) \\rarr (T \\rarr E \\rarr T') \\rarr E \\rarr T' = r \\mapsto f \\mapsto e \\mapsto f \\, (r \\, e) \\, e", "f687e3a47bd0738a1790b1727f48f1e8": "I_n= \\int \\frac{dx}{(x^2+a^2)^n}\\,\\!", "f687e3f5b74e4eff641f1d7c46a4bc9a": "(C,\\lambda_+,\\lambda_-,\\alpha)", "f68891039f04874ee477c7a164bc97c8": "u_4 = \\left(P_3' - P_5\\right)\\sqrt{\\frac{1-\\Gamma}{\\rho_R(P_3'+\\Gamma P_5)}}", "f688980f1347f181b1f9ae3489b35adc": "\\Sigma \\, \\sigma \\, \\varsigma \\,", "f688b7df434e443a06eca9ecdf2b6bc6": "\\displaystyle{ R_{12}(z-w) T_{1}(z)T_{2}(w) = T_{2}(w) T_{1}(z) R_{12}(z-w).}", "f6893a06056cfdce0cf0aaae7bbcdb69": "y_i \\left[ {w^T \\phi (x_i ) + b} \\right] \\ge 1,\\quad i = 1, \\ldots ,N \\, ,", "f6893fc92890d724a1e6406d4dab0a2d": "\na_k = \\frac{f^{(k)}(0)}{k!} = {1 \\over 2 \\pi i} \\oint_{C_r} \n\\frac{f( \\zeta )}{\\zeta^{k+1}}\\,d\\zeta\n", "f689438e8cedb1c61ba144e611ef5c6a": "k + m < j \\text{ and } j + 1 < k + m + 2 \\,", "f68951f49d82b2b9c927ff0950ff0e31": "\\sin(z) - \\sinh(x)\\sinh(y)=0", "f6897073ddb650de60056fa4f394f487": " b", "f6898183e5f53700f445d6b92cb27743": "\\sum_{j=1}^n a_{ij}x'_j = y_i \\qquad\\mbox{ for } i=1,\\ldots,m", "f689a2f7bead1d74b7719739071a0322": "x(t) \\in \\mathcal{D} \\subseteq \\mathbb{R}^n", "f689d5d27c84dca5e39b20f13b5b4cb2": "e^{-{1 \\over x^2}}", "f68a0e13dd7dec80e3e0d84888aaa34b": "x, y, z \\in K", "f68aa0b0c097383a6d42703e828d6012": "\\psi(\\cdots)", "f68aaec3d77e3108e79e5c34c7a58332": "Z^1_3", "f68abbf53c25875377acb104a70ccb3c": "A \\otimes_B C", "f68b9ddbfeed0ebcb0a3ce61692883fc": "\n \\mathcal{P} = \\Big\\{\\ p_\\lambda(j) = \\tfrac{\\lambda^j}{j!}e^{-\\lambda},\\ j=0,1,2,3,\\dots \\ \\Big|\\ \\lambda>0 \\ \\Big\\},\n ", "f68bd5efee6eb519c609e28a4773adbc": " \\mu \\approx 10^{15} ", "f68be89d19994ec5f9353283bc6ad684": "D_{r} + \\max( F_{r} , 0)", "f68bf43ae588018509f1f2a3a6a2ac1c": "Z_1, Z_2,\\dots", "f68c1cb65485af117e3f0cdb98be8e87": "\\lim_{h\\rightarrow 0}P(|X_{t+h}-X_t|>\\epsilon)=0", "f68c240b9e0aba17cca03a3e41a78e6a": "\\log \\frac{\\epsilon}{\\epsilon - (1-\\epsilon)\\delta} \\quad \\leq \\quad D^{\\epsilon}(\\rho||\\sigma) \\quad \\leq \\quad \\log \\frac{\\epsilon}{\\epsilon - \\delta} ~.", "f68cd5f193b7b2859b9e2f2355561723": "\\Delta_{ads} G = \\Delta_{ads} H - T\\Delta_{ads} S", "f68d2d374278acbfe6f76bbab9322604": "\\hat v_n(x)=\\sqrt{n} [F_n(x)-F_{\\hat\\theta_n}(x)]", "f68d4df532f85427faecaa68385e407a": "\\mbox{e}^{t\\Delta_D}", "f68dab101badb5dcf5265432d700bb3d": "G_1 = (V_1,E_1)", "f68db47803280282d10acb8ab2527d69": " 1 \\lor 4 \\iff | i \\rangle ", "f68e01aa78a5fb5a30a1997e97997e59": "\\frac{p}{r}\\,", "f68e08f26c3b22be96f758b1624870cf": "B = |A| \\sqrt{E^2-1} = \\sqrt{d^2-A^2} \\quad (10)", "f68e440e59284ed477fe87e340041c4a": "R \\in P(a,b) \\iff (\\forall u \\in R \\; \\exists x \\in a \\; \\exists y \\in b \\; \\langle x,y \\rangle = u) \\wedge (\\forall x \\in a \\; \\exists y \\in b \\; \\langle x,y \\rangle \\in R)", "f68e77654c675922c61b6da45eeae6c8": "I_n \\,\\!", "f68e7eca428348fae322869f74ba184c": "\nT=\\frac{\\sum_{i=1}^n\\left(X_i-\\mu\\right)^2}{n}.\n", "f68eae9edacb4bf6adcf2a771df36b6b": "I \\propto V_{\\rm S}{}^{\\gamma}", "f68ebd1dac949c9f126c1ea9d077e3b1": "\\Q\\times\\Q", "f68ed407286c39b701335f44ce02c777": "g(\\phi,z)", "f68f901b9746f99e89ffa7a25b03acaf": " G_i + {G_i}^{-1}=m(1+E_i), ", "f68fb5593c6f29b5c2b169e0432ca60d": "\\frac{1}{2} iu=f(z)+z \\bar{f \\prime} (z)+ \\bar{g \\prime} (z)", "f68fb8621d13dfc895cf5c0a859d3cd9": "u_7 = \\tfrac{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+ax_7^2+x_8^2)x_{15} - 2x_7(x_1 x_9 +x_2 x_{10} +x_3 x_{11} +x_4 x_{12} +x_5 x_{13} +x_6 x_{14} +bx_7 x_{15} +x_8 x_{16})}{c}", "f68fd6a8fbed9c79d83649a52369b758": "\\alpha_3\\;", "f68fedadc45b765c6844fc0580090267": "F_t = f_ct+{(f_0 - f_c)\\over k}(1-e^{-kt})", "f68ff11a3464953ff50628a482dbf38d": "y_{m+1}", "f690aa47f075577c6cfc82c95be4781a": "FV(A) \\,=\\,A\\cdot\\frac{\\left(1+i\\right)^n-1}{i}", "f690df425f594dd3d592bca1344a333a": " {C_{c}}(\\mathbb{R}) ", "f69102eb26df1beccca20418f826b410": "\\begin{align}tan\\beta &= \\frac{\\mbox{d}D}{\\mbox{d}L} = \\frac{C_D}{C_L}\\\\\n &= \\frac{1}{2}\\rho V_1^2 C_L \\frac{\\cos(\\varphi+\\beta)}{\\cos\\beta}b\\mbox{d}r\\end{align}", "f691065a556ab47053e04519a6a0f10b": "(X,X) \\to (X_1,X_2)", "f691aa227bb74da923bcf2c168ef688d": "u^1=x^1\\circ \\pi,\\,u^2=x^2\\circ \\pi,\\,\\dots,\\,u^n=x^n\\circ \\pi\\, ,", "f691fd679392e088e0fdb23c555401ee": "\nH(P, Q) = \\frac{1}{\\sqrt{2}} \\; \\|\\sqrt{P} - \\sqrt{Q} \\|_2 .\n", "f69236c98411a130e7f1987c19e854be": " {} = 1 \\cdot (-3) - 2 \\cdot (-6) + 3 \\cdot (-3) = 0. ", "f692376f73009965a06d5210fe300c02": "R_{\\mu \\nu} - {1 \\over 2}g_{\\mu \\nu}\\,R + g_{\\mu \\nu} \\Lambda = {8 \\pi G \\over c^4} T_{\\mu \\nu}", "f6923e8f6065b2a10f461c8aa0630758": " \\forall A \\in \\mathcal{A} : x(A) = \\frac{1}{d+1}", "f69258b431e3e10c91511dcb1377deff": " U = -\\frac{1}{2}\\int \\mathbf{M}\\cdot\\mathbf{B} dV, ", "f692675266a1dbcc2d7ecd860e0d5be3": "n=f(x,y,z)", "f6928befdcf06c7a028a6943db70769c": "z = r\\cos\\theta", "f692be370774e631bfcf503f9163fbe3": "f_i(\\alpha)=\\alpha^{i-1}", "f6930ef8313e68588869cd5e41c982b0": "P = \\sum_{|\\alpha|\\le k} P^{\\alpha}(x)\\frac{\\partial}{\\partial x^\\alpha}", "f6938389463fca4a118c4749c29de120": "\\Omega.", "f69395fa2d075c8317ffff76ae61d942": "\n\\omega_{\\rm max} =\n\\left( { \\sin \\left( { \\pi\\alpha \\over 2 ( \\beta +1 ) } \\right) \\over\n\\sin \\left( { \\pi\\alpha\\beta \\over 2 ( \\beta +1 ) } \\right) } \\right) ^ {1/\\alpha}\n\\tau^{-1}\n", "f693aee56898be2fb73ce454bb6277d2": "(\\mathbb {C}P^N,\\Sigma,\\{U_\\alpha\\vert\\alpha\\in\\Sigma\\})", "f693dc21915e22927a7c9f94a1643e3c": " N(L) = \\operatorname{span}\\{ (0,\\ldots, 0, 1)^T \\}.", "f693e8746f01c43761164be67b08a394": "\\begin{align}\n \\left\\{\\log(w^z)\\right\\} &= \\left\\{ z \\cdot \\operatorname{Log}(w) + z \\cdot 2 \\pi i n + 2 \\pi i m \\right\\} \\\\\n \\left\\{z \\cdot \\log(w)\\right\\} &= \\left\\{ z \\cdot \\operatorname{Log}(w) + z \\cdot 2 \\pi i n \\right\\}\n \\end{align}", "f694135eafc20195a9d96ca3ce8af674": "\\sigma = \\pm 1", "f69461de0d40cd125257de42a808a547": "\\Phi_\\lambda", "f694880f4f18d14b7dc295ec9c09ffae": "P_s", "f694a41370fbd73b1587b9ce24985629": " [x_\\mu,x_\\nu]=\\imath M_{\\mu\\nu} ", "f694ab269de591a4b4aea4b265e0b9d8": "D_n, P_n, \\text{ and } Q_n", "f694b2286c379662e00b85131f073c34": "\\left(q_1\\left(\\sigma\\right),\\cdots,q_N\\left(\\sigma\\right)\\right)", "f694cbc7c86c90d3fc986127e65bff18": "\\mathbb{A}^n.", "f694f8486156e317c9c3a3586cc9a5c6": "\\psi^\\dagger \\mapsto \\psi^\\dagger \\lambda^\\dagger", "f69500ab66a447c681ed0d3603f9bb2f": "x_{n+1} = x_n - \\frac{f(x_n)}{f '(x_n)} = \\frac{\\frac{1}{3}{x_n}^\\frac{4}{3}}{(1 + \\frac{4}{3}{x_n}^\\frac{1}{3})} \\!", "f6954cea7dc8204cc39ca3739617be03": "\\sum_{p}^{}{\\Delta S^0_i} \\ge \\frac{\\Delta G^0}{(T_H-T^0)} ", "f69575e46e7b5768172a436f80799340": " f_r ", "f695abc93a65e956ae146606446dc3eb": "E_\\text{pot} = \\frac{1}{\\lambda}\\, \\int_0^\\lambda \\tfrac12\\, \\rho\\, g\\, \\eta^2(x,t)\\; \\text{d}x", "f695b7f37a3df1d9a856972cc0d3a4ed": "P \\cdot f = k \\cdot P \\cdot v", "f695cfa45427138469dd3747f86584db": "A \\cup \\varnothing = A\\,\\!", "f695d645096cfb286d73a2e684ba2759": "\\sum_{i=0}^{n-1} (\\mathbf{X}^i\\mathbf{A}\\mathbf{X}^{n-i-1})^{\\rm T}", "f6962eb5d1fa60a81bf1968f98d0ca9f": "K\\ge 0", "f69643198cbaf61a991beefef4476051": " = f \\, \\Delta h + \n2 \\partial_i f \\, \\partial^i h + \nh \\, \\Delta f", "f6965334d16bd5b4596fc98b4fe206e8": " wxy+ xyz+ yzw+zwx =0\\ ", "f696ada0482634298d4c43f1bc411747": "R [U(\\$100) - U(\\$0)] > B [U(\\$100) - U(\\$0)] ", "f6972248d354e367e12a297cb8af3cd7": "c = \\cos\\theta_1=0.707107, ~ \\sqrt{1 - r^2 \\left( 1 - c^2 \\right)} = \\cos\\theta_2 = 0.771362", "f6978002a5e9c8e762be42c9a6330b6e": "\\sum_{k=1}^s(\\mathbf{a}_k \\cdot \\mathbf{v})^2 = \\zeta |\\mathbf{v}|^2", "f697e85430ed483c7aea0b2d51521719": "\\Pi A = RT", "f6986c2bfb9cbe5eb63212579f28cda0": "\\scriptstyle k_{(V)00}^{(5)}", "f69874a4e67473fc52c1a37ffefe455d": "\\nu_{ij} = \\frac{{\\rm d}N_i}{{\\rm d}\\xi_j} \\,", "f69885a9cb77ca6d7abb52043b855bde": "\\langle v,u\\rangle", "f698c17589e3c8ce8563658bc290015c": "c_m=\\left(4g\\sigma/\\rho\\right)^{1/2}", "f698f464f5fc6270a2e0078580f121d9": "A\\circ B\\subseteq A", "f6992f3ffa67131624ed0a36e02324bb": "Z(x_c)\\geq\\sum_{T>0}B_T^{x_c} =\\infty", "f6994060018feb8b4fd0bab34ee90c86": "m(\\phi)", "f699444be5b5dc22e1e7ed73f519467c": "\\textstyle p \\approx 2\\times 0.0117 = 0.0235", "f699b2fb858307351dfffb2893961cf5": " 0.813 \\pm 0.009 ", "f699c6e3d7a0c95524ee47acbfca9ad0": "q_0\\in\\, Q ", "f699f843674e7970fcb690fcc2cbca64": "G\\,=\\,A\\,*\\,B", "f69a0be4472b203f914d8daeb70b0dbc": "\\sum_{n=0}^N\\sum_{k=0}^K{K \\choose k}\\cdot{N-K \\choose n-k}\\cdot x^k\\cdot y^{n-k}.", "f69add277b502d97182c8ece3a89b9f2": "\n\\mathbf{B} \\ = \\ \\mu_0(\\mathbf{H} + \\mathbf{M}) \\ = \\ \\mu_0(1+\\chi_v) \\mathbf{H} \\ = \\ \\mu \\mathbf{H}\n", "f69af8b91d606ec5e5973b6d192f43b4": "\\mathbf{u}(t) \\triangleq \\begin{bmatrix}u_1(t)\\\\u_2(t)\\\\\\vdots\\\\u_{m-1}(t)\\\\u_m(t)\\end{bmatrix} \\in \\mathbb{R}^m", "f69b13a74d42acf1f5e3c84f4581ad2b": "(\\forall x (P\\Rightarrow Q))", "f69b1a47323ba6871baa087ba220b653": "X \\sim \\mathrm{GEV}(\\alpha,\\beta,0)\\,", "f69b3ae507c3b803efc1a279e06103b7": "J_{ij} = 0 ", "f69b91f6d382ffbf49d5eb6918f1b632": "S(\\vee)", "f69b9ed1c9a53321f403b3f94d82aa95": "\n\\mathbf{F}_k = \\sum_{j=1}^N \\mathbf{F}_{jk}\n", "f69b9f82319c626b2ccff808773c3254": "\\left( k-c\\right) /n", "f69bbaa610a1bbdbb76474c3556e64f9": "\\mathrm{ERA} = 9 \\times \\frac{\\mathrm{Earned~Runs~Allowed}}{\\mathrm{Innings~Pitched}}", "f69bbd107c21abf0b1582f4575085293": "\\sum_{i = 0}^{k - 1} {X_{t_{i+1}} + X_{t_i}\\over 2} \\left( W_{t_{i+1}} - W_{t_i} \\right)", "f69bde20fd6ef453253efd99a3f3fa77": " \\cot y = x \\, ", "f69c1ba6ef83e6d6ec73559a713e44ae": "\n\\{ x \\in X : x = \\lambda T x \\mbox{ for some } 0 \\leq \\lambda \\leq 1 \\}\n", "f69c3b77150d0d6b1d1aa462d2f9ce72": "\\frac{\\operatorname df}{\\operatorname dt}=\\frac{\\partial f}{\\partial t} \\frac{\\operatorname dt}{\\operatorname dt} + \\frac{\\partial f}{\\partial x} \\frac{\\operatorname dx}{\\operatorname dt} + \\frac{\\partial f}{\\partial y} \\frac{\\operatorname dy}{\\operatorname dt}.", "f69c717dfe1b403f186a7a5a20d6c219": "\\bold r'", "f69cf0788649fb97de086c0739e8fe1d": "R_{xx}(-2)= R_{xx}(2)=2,", "f69d210cdcccefd08eefbe25517a8c3e": "{f^{-1}}(x)=\\int\\frac{1}{f'({f^{-1}}(x))}\\,{dx} + c.", "f69d91bb254b487e8cf98b7608ab29f0": "\nf(A,B)=2\\times\\frac{1}{2\\pi}A^{-\\frac{1}{2}}e^{-\\frac{B}{2}}(B^2-4A)^{-\\frac{1}{2}}\n", "f69dbbd83c2654f08cea04df12edbcb9": "\\phi(r\\cdot x) = r\\cdot \\phi(x)", "f69eb0e1d6c8b70b4a9f30d76b6cf3eb": "F \\colon \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2", "f69f55107761c720f42a00bdd09bb715": "(A - \\mu I)^{-1}", "f69fd6bd9666477248d9a670cfa3adf3": "\\displaystyle{Q(a)R(b,a)=2[Q(a)L(b),L(a)] +2Q(a)L(ab)=2[Q(ab,a),L(a)] +2[L(a),L(b)]Q(a)+2Q(a)L(ab).}", "f69fdffb82267fca1be8c6913635b318": "n - 1", "f69ff6be2a097a46c4e9475af361cef1": "q_i \\,\\, (i = 1, \\dots, N).", "f69ffe1dbc30bc88a1d206935492750d": "10^{10^{33,013,740}}", "f6a01150deddaaac692fda36845676dc": "\\frac{D\\mathbf{u}}{Dt}", "f6a0774a508c4c96065b436d1ebe5b44": " \\mathcal{H}_{\\text{JC}}=\\underbrace{\\hbar \\omega_r \\left(a^\\dagger a+\\frac 12\\right)}_{\\text{cavity term}}+\\underbrace{\\frac 12 \\hbar \\omega_a \\sigma_z}_{\\text{atomic term}}+\\underbrace{\\hbar g \\left(\\sigma_+ a+a^\\dagger \\sigma_-\\right)}_{\\text{interaction term}}", "f6a0a428fb4c86c3603315ff332b69ea": "\\tan\\alpha =\\frac{M_{y}}{M_{x}}", "f6a0e99b19aae68f3e84c49978a6140d": "\n f(x_j) = \\left \\langle \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) + v, \\varphi(x_j) \\right \\rangle = \\sum_{i = 1}^n \\alpha_i \\langle \\varphi(x_i), \\varphi(x_j) \\rangle,\n", "f6a13b7a2d83d804287b5a417b09502c": "\\operatorname{E}\\|f_n-f\\|_2^2=\\operatorname{E}\\int (f_n(x)-f(x))^2 \\, dx", "f6a1a4089f988bbed2a0f03150728ba7": "\\sum_{k=1}^\\infty (-1)^k \\frac{\\zeta(k)-1}{k} = \\ln2", "f6a1bbb10f67d743c0670b95bf750e06": "|\\psi\\rangle = \\alpha_0|+\\rangle+\\alpha_1|-\\rangle", "f6a1d6bb4d8e1658e755638752af3e65": "r^2{d\\theta} = a b n dt .", "f6a1eeefa5ad7610bec4e95e17b79958": "e^{aj} \\ e^{bj} = e^{(a+b)j}", "f6a1f234df490936b699aeb3f0fe9372": "\n(\\Sigma_1)^k (\\Sigma_3)^j =\\sum_{m=0}^{d-1} |m+k\\rangle \\omega^{jm} \\langle m| , \n", "f6a205c824465caa1f37700e701b690c": " \\varphi (N) = \\frac{e.d - 1}{k} = \\frac{17993\\times5 - 1}{1} = 89964", "f6a2479ae3ac8fd3501bc4a465e16b64": "R_{ij} = \\delta_{ij} - 2\\frac{a_i a_j}{\\|a\\|^2}", "f6a258567dfb59ed325527ca24e48635": " [{J_x}, {J_y}] = i \\hbar \\epsilon_{xyz} {J_z} ", "f6a276ce6355dd6c574f8a88809db3ff": "\\scriptstyle 0 < u < \\infty ", "f6a2d6264c9c75b9715bb858bebaf576": "0\\le \\varphi \\le \\pi", "f6a2eb5f3504bbcb87512bd1cae31603": " \\ E _{local}(j) ", "f6a2f61056fede6f910901cb257ccb07": "A(a, a)\\ A(b, b)", "f6a2f83c44015f396b60881dcf16b383": " \\scriptstyle{-2k_e \\, Q\\, dx \\ln \\vert \\boldsymbol{x} - \\boldsymbol{y} \\vert} ", "f6a322d51c1330e9c1a660e10da28de0": "x_i \\ge 0", "f6a3315d9a94a3e25ab7a8bce624f7d8": "( T_i, \\varphi )", "f6a34ff6c8a9cddeac32c00d01228df0": "E_n^{(2)}=\\frac{|V_{nk_2}|^2}{E_{nk_2}}", "f6a3721062a11596e9eef3966de388af": "\\mathbf{a} = (a_1,a_2,a_3) \\,", "f6a3ddd86130c06affacc0f45d5da497": "\\int e^{cx}\\;\\mathrm{d}x = \\frac{1}{c} e^{cx}", "f6a3f76d21786aae7e151452794d3e01": "x_a=kr^{-1}J_a", "f6a43338d6fbdf3e618eac80d8bc0738": "\\Sigma_{i=0}^{n-1}\\omega^{ij} v_i", "f6a4469eb7bb67a1aea20070bdff268b": "\\{(A1,A2),(B1,B2),\\dots|(A1,B1), (A2,B2),\\dots\\}.", "f6a4a9f6ad07cfc06dc7ec21833e954a": "p^{*} = \\frac{n p}{n - p}.", "f6a4f3b9bba69fa21656993fa799e4ce": "\nG^{(n)}(1 \\ldots n | 1' \\ldots n')\n= \\mathrm{i}^n \\langle T\\psi(1)\\ldots\\psi(n)\\bar\\psi(n')\\ldots\\bar\\psi(1')\\rangle,\n", "f6a53d79a1d51e1371b89f4c7b83a13a": "\\left\\{e_1,e_2,\\dots,e_n, ie_1, ie_2, \\dots, ie_n\\right\\},", "f6a558b3b62bb8703fc898af64bafbd9": "\\left(2^{\\frac{1}{k}}-1\\right)^\\frac{1}{c}", "f6a585392d20dfb4f7c7e1085951843c": "F = \\frac{L}{A}", "f6a597102effecbd34935b0d35021d2d": "c = \\frac{1}{\\sqrt{\\varepsilon _0 \\mu_0} }\\ , ", "f6a63fd683e540377a7dcc1ffad17689": "|\\psi(t)\\rangle = T\\exp{\\left[-\\frac{i}{\\hbar}\\int_{t_0}^t dt'H(t')\\right]}|\\psi(t_0)\\rangle", "f6a6d27898b31a40b49ed3c392cdb4e4": "\\kappa(\\gamma^*) = - \\mu \\gamma^*", "f6a6d842261650c50efa230f5ff89dc1": "\\scriptstyle I_1", "f6a6fa8c582e5790b59a9eceead1a0ac": "\\frac{dR}{dP} = 0", "f6a7494af4f6939a4925d52659815535": "b=C*N\\,", "f6a7a3268e37cd12d3dbf14d79ab9a71": "\n \\Phi(x) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^x e^{-t^2/2} \\, dt\n \\quad x\\in\\mathbb{R},\n ", "f6a7e09f0f3419c68aba9f8973f4935c": "P\\rightarrow Q", "f6a8214bb3df2bb54da1023403b1d7ed": " 2^n \\cdot \\prod_{k=0}^{n-1} \\cos(2^k \\alpha)=\\frac{\\sin(2^n \\alpha)}{\\sin(\\alpha)}", "f6a83d578bb4f01bb8b9fc10623a9598": "P(H|E) = \\frac{P(E|H) \\cdot P(H)}{P(E)}", "f6a871b049debb7d4a68f3beecf5801b": "=\\frac{\\varepsilon +\\cos \\theta}{1+\\varepsilon\\cdot\\cos \\theta}.", "f6a8776e64439fdc0311aaf43978198f": "\\begin{align}\n n = 0: B_m &= \\sum_{k=0}^m\\sum_{v=0}^k(-1)^v\\binom kv\\frac{v^m}{k+1} \\\\\n n = 1: B_m &= \\sum_{k=1}^{m+1}\\sum_{v=1}^{k}(-1)^{v+1}\\binom{k-1}{v-1}\\frac{v^m}k.\n\\end{align}", "f6a8d50f3ac243549e0af37d41c061f8": "P(x_{0:t}|y_{0:t})", "f6a8e129e5b4be0779ed5c96550d885d": "\\color{Black}\\tfrac{6}{m}\\tfrac{2}{m}\\tfrac{2}{m}", "f6a8ff35e02ae1d16358184c86ca1340": "\\pi_0 < 0", "f6a919e6d372879d33291d80b1e78f55": "(a;q)_\\infty = \\prod_{r\\ge0}(1-aq^r)", "f6a9300cf5533fbe65fd3bdfe4d97d79": "d_w = \\left\\{\n\n\\begin{array}{l l}\n d_j & \\text{if }d_j < b_t\\\\\n d_j + (\\ell p (1 - d_j)) & \\text{otherwise} \\end{array} \\right.", "f6a931645b5d0d1e26df241ff6ce601e": "\n\\alpha^6 = \\alpha^5 \\alpha = (\\alpha + 1)\\alpha = \\alpha^2 + \\alpha\n", "f6a96354ae282902bbe0d35e910b52d9": "\ny_3 = -\\frac{27}{8{y_1}^3{x_1}^6}-\\frac{81}{4{y_1}^3a{x_1}^5}+(-\\frac{81}{2{y_1}^3a^2}-\\frac{81}{4{y_1}^3a}){x_1}^4+(-\\frac{27}{{y_1}^3a^3}-\\frac{81}{{y_1}^3a^2}+\\frac{9}{2y_1}){x_1}^3+(-\\frac{81}{{y_1}^3a^3}-\\frac{81}{2{y_1}^3}a^2+\\frac{27}{2y_1a}){x_1}2+(-\\frac{81}{{y_1}^3a^3}+\\frac{9}{y_1a^2}+\\frac{9}{y_1a})x_1+(-\\frac{27}{{y_1}^3a^3}+\\frac{9}{y_1a^2}-y_1)\n", "f6a96f67d344270a89854e458adccff5": "\\lim_{T \\rightarrow 0}\\left( \\frac { \\part S(T,X)}{ \\part X}\\right)_T = 0. ", "f6aa1c619e85c64523132485b8b74714": "\\scriptstyle S_{1/T}(f)", "f6aa6516b632e7fad7b6f3b0391e4274": "y(t) = 3t. \\,", "f6aa69b0e935a12389172f75bd58cff9": "p(\\boldsymbol\\beta,\\sigma)", "f6ab3521b891a7d4c238c38831bb2611": "\\overline{f}\\colon M/\\sim\\longrightarrow X", "f6ab45769719a936c085fcae6109911a": "d< \\frac{1}{3}N^{\\frac{1}{4}} ", "f6ab7b0abe68f996623c7cd7c60d4c63": "a_1,\\ldots, a_n,b_1,\\ldots, b_m \\in V", "f6ab809b354520b24356912b3f244346": "\\psi(x) \\propto e^{ik_0 x} = e^{ip_0 x/\\hbar}", "f6ababe7a1e002e07e0382cc54c59367": "\n \\sin(3\\theta) = -~\\tfrac{3\\sqrt{3}}{2}~\\cfrac{J_3}{J_2^{3/2}}\n ", "f6abb4121abdda739032753bfb447559": " \\frac {d^2 \\mathbf{x}_{A}}{dt^2}=\\mathbf{a}_{AB}+\\mathbf{a}_{B} + 2\\ \\sum_{j=1}^3 v_j \\ \\frac{d \\mathbf{u}_j}{dt} ", "f6abdbdafd3499dbdc6fd3274fde6e80": "\\displaystyle{\\begin{pmatrix} a & 0 \\\\ b & a^{-1}\\end{pmatrix}}", "f6abf6b176bc1120fa77ed1f1a3b7e29": "\\textstyle B\\in\\mathcal{F}_2 ", "f6ac0a5a9f20498cbc0b0226fe70ad65": "H^j", "f6ac327507a881d11c8676404d04f884": " q_e = \\iint \\sigma_e \\mathrm{d} S ", "f6ac6050d30dfdf6608fcf11f7247c32": "\\scriptstyle{s_1^2 - s_2^2}", "f6ac6706c4df089275e869b52e0676bb": "\n\\epsilon_{\\downarrow} (k) = \\epsilon_0 (k) - I \\frac{n_\\uparrow-n_{\\downarrow}}{n}\n", "f6ac78a20c8d73afaf1ee54fe6004c7e": "(x + 1)^{3}p(\\tfrac{1}{x+1}) = -64x^3-256x^2+256x+512", "f6ac8352474e7c59f75a3662044fa9e2": " r_1 = r_m-A \\ ", "f6acba8f692dc6bf32acd197d6db8e57": "\\hat\\beta|X\\ \\sim\\ \\mathcal{N}(\\beta,\\, \\sigma^2(X'X)^{-1}).", "f6acc8a5db29d1b167d8be0a1b57d362": "\\mathbf{E} = -\\nabla\\phi - \\partial_t \\mathbf{A}, \\quad \\mathbf{B} = \\nabla \\times \\mathbf{A},", "f6ace13b4122c066ac4d3716496120ce": " KE", "f6ad2631801468cca1cb885395569dad": "\\sigma=pe\\mu_h", "f6ad4b2bc563898c2b790aefc1b45664": "\nD_{\\phi_{a}}\\left(\\mathbf{\\rho} \\right) = \\left \\langle \\left | \\phi_{a} \\left (\n\\mathbf{r} \\right ) - \\phi_{a} \\left ( \\mathbf{r} + \\mathbf{\\rho}\n\\right ) \\right | ^{2} \\right \\rangle _{\\mathbf{r}}\n", "f6ad5d21eaa32e5b71fa58df6050314c": " x", "f6ad62d1534802c8c441639b9af6ca2a": "\\begin{align}\nx^{(n+3)} &= x^{(n+2)} P = \\left(x^{(n+1)} P\\right) P \\\\\\\\\n &= x^{(n+1)} P^2 = \\left( x^{(n)} P^2 \\right) P\\\\\n &= x^{(n)} P^3 \\\\\n &= \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix} \\begin{bmatrix}\n0.9 & 0.075 & 0.025 \\\\\n0.15 & 0.8 & 0.05 \\\\\n0.25 & 0.25 & 0.5\n\\end{bmatrix}^3 \\\\\n &= \\begin{bmatrix} 0 & 1 & 0 \\end{bmatrix} \\begin{bmatrix}\n 0.7745 & 0.17875 & 0.04675 \\\\\n 0.3575 & 0.56825 & 0.07425 \\\\\n 0.4675 & 0.37125 & 0.16125 \\\\\n\\end{bmatrix} \\\\\n& = \\begin{bmatrix} 0.3575 & 0.56825 & 0.07425 \\end{bmatrix}.\n\\end{align}", "f6ad9a6937f358fa22594b2a71394cff": "F_1(\\lambda)=A(\\lambda)", "f6adaa1f6ce10e6ccc9b9b88ce7cd58f": "\\frac{BD}{DC}=\n\\frac{|\\triangle BAD|-|\\triangle BOD|}{|\\triangle CAD|-|\\triangle COD|}\n=\\frac{|\\triangle ABO|}{|\\triangle CAO|}.", "f6adb2df6983a726b55f1ebc64b8da81": "\\underline S", "f6adc51421076f845c7f4178522a24e7": " B \\in G", "f6adcb918b84c8ae6629e0080e436275": "f''(x)=e^x >0 ", "f6ae47a4e6e934876e9abb9031cf0c67": "a_{i}^{n}(u)\\,", "f6ae7dc1a43d2b448a4098ef1c4d3e53": " -\\kappa ", "f6aed3b618cb6cffdd14e54f22aa8d2a": "\\{x_{i_1}, x_{i_2}, \\ldots, x_{i_k}\\}", "f6af2d9c731bf6a7a1cb6d83375b5783": " dP = - \\rho g dz", "f6af3772d658d0d43f1c45e85b1673e9": " \\lim_j f_j(x) \\geq g_k(x) \\, ", "f6af9511cbed3a99da8c1b9b0986c22d": "\n \\begin{align}\n \\xi_x\\, &=\\, x\\, +\\, \\int\\, \\frac{\\partial \\varphi}{\\partial x}\\; \\text{d}t\\,\n =\\, x\\, -\\, a\\, \\text{e}^{k z}\\, \\sin\\, \\left( k x - \\omega t \\right),\n \\\\\n \\xi_z\\, &=\\, z\\, +\\, \\int\\, \\frac{\\partial \\varphi}{\\partial z}\\; \\text{d}t\\,\n =\\, z\\, +\\, a\\, \\text{e}^{k z}\\, \\cos\\, \\left( k x - \\omega t \\right).\n \\end{align}\n", "f6afbe1cf8117f470ea2b2342476d54e": "x_2= \\sin \\Omega \\cdot \\cos \\omega + \\cos \\Omega \\cdot \\cos i \\cdot \\sin \\omega", "f6b02833ea0b7dd3ebf5b3be8597c5f7": " \\mathcal{P} = \\mathfrak{P}(\\mathfrak{C}(\\mathcal{Z})),", "f6b02f84e742406d9ceee49960f72a57": "X_{t+1} = -(E+BX_t)(C+AX_t)^{-1}", "f6b0ab744ee20bd07196a42d93bbde39": "\\mathrm{p}", "f6b0b6cb1cbfe88bfe91baf62dd2125e": "x[n] = \\sum_{i=0}^N a_i w[n-i] .", "f6b0e53f87864bb808c30ab4d7ce8d4a": "|E_6 (q)| = \\frac{1}{\\mathrm{gcd}(3,q-1)}q^{36}(q^{12}-1)(q^9-1)(q^8-1)(q^6-1)(q^5-1)(q^2-1)", "f6b0f457de8fcb2f189d8d673cba6e8d": "m = hk^2", "f6b0f729cc1e9344420fd6fce6f88307": "\\mu = {1\\over (j+1)}\\langle(l,s),j,m_j=j|(g^{(l)}{1\\over 2} \\left(\\overrightarrow{j}\\cdot \\overrightarrow{j} + \\overrightarrow{l}\\cdot \\overrightarrow{l} - \\overrightarrow{s}\\cdot \\overrightarrow{s}\\right) + g^{(s)}{1\\over 2} \\left(\\overrightarrow{j}\\cdot \\overrightarrow{j} - \\overrightarrow{l}\\cdot \\overrightarrow{l} + \\overrightarrow{s}\\cdot \\overrightarrow{s}\\right)|(l,s),j,m_j=j\\rangle ", "f6b1ecb36003d78f33ba7d58e431b34f": " = D(\\rho||\\sigma) ~, ", "f6b1ece29d34cee2cc0ef10ea856c090": "p_\\ast \\colon S_\\ast(E)\\rightarrow S_\\ast(B)", "f6b249d6a98f7fdb2d917f369f9151e4": "5 x + 1024", "f6b257c34b51d01562d90a6e92730b21": "f=f_1 f_2 \\dots f_n", "f6b29761f40a4e711a2cb0f36dd7d369": "K_\\pm", "f6b29f040d271308cfcee606b9edc599": "deg_x f(x)", "f6b2a9f02f7b64b056f82993cffd73cc": "{C^a}_{bac} = 0.", "f6b2dc421ab4cfdd9565f5161e70f1d5": "\\exists y ( \\exists x \\phi \\rightarrow \\psi), \\qquad (2)", "f6b35a83fe75492119f1cee6673d53b9": "\\textstyle A_{2}=B_{2}=1", "f6b3a80f13fa442f422b1701e9a0fb83": "u(x) = \\Gamma * f(x) = \\int_{\\mathbb{R}^d} \\Gamma(x-y)f(y)\\,dy", "f6b3b839b2e2828e85ed30b3c15c498d": "q =\\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix}, \\,", "f6b3e3770a258e5d2c073ba35da6a616": "\\tilde{h_1} \\leftarrow u_1^{z_1} rem P", "f6b4608351c844fe4353d27aa3bf6bef": "\\left\\langle 0 | T\\left\\{ \\mathcal{O}_1(x) \\mathcal{O}_2(0) \\right\\} | 0 \\right\\rangle ", "f6b462d31fa9c4b360b11e4bca53e6f8": "\\kappa_t(\\mathcal B)", "f6b46b9ec3c5c9f637e311ba40ae1d34": "S_o = S_e D^p\\,,", "f6b4aca902cde8f74cd21d9c97c48b22": "\\forall x \\in I, \\forall r \\in R :\\quad x \\cdot r \\in I ", "f6b51783b8c4cf80ce57b5d1bef39adf": "\\mu(\\bigcup_nA_n ) \\le \\oplus_n\\lambda(A_n)", "f6b5224d3beade9fe7d856c15b142096": " \\sum_V \\bar{V}\\otimes_G V", "f6b52e917662a47fdb2a59d1891449e5": "\\Pi = RT\\Gamma", "f6b55ecd3a329cd06051a6396c4f18a2": "\nI = {V \\over R_\\mathrm{S} + R_\\mathrm{L}}.\n\\,\\!", "f6b5f7b8c2afa726694b23871e7232f6": "\\left [\\begin{smallmatrix}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{smallmatrix}\\right ]\n", "f6b6020b7cff72eceae5df6fbaef089f": "t_r=t-\\frac{R(t_r)}{c}", "f6b6228a44ce7a73921ccdbb66cb5db2": "|x|_\\infty := \\begin{cases} x, & \\text{if } x \\ge 0 \\\\ -x, & \\text{if } x < 0. \\end{cases} ", "f6b628a0cac62dd1ab32930c14e8a36c": "b\\,M_{solvent}=\\frac{n_{solute}}{n_{solvent}}=\\frac{x_{solute}}{x_{solvent}}.", "f6b6393795db4e1417f91dd7fbd7baeb": "\\begin{align}\nk_e &= \\frac{1}{4\\pi\\varepsilon_0}=\\frac{c_0^2\\mu_0}{4\\pi}=c_0^2\\cdot10^{-7}\\mathrm{H\\ m}^{-1}\\\\\n &= 8.987\\ 551\\ 787\\ 368\\ 176\\ 4\\cdot10^9\\mathrm{N\\ m^2\\ C}^{-2}.\n\\end{align}", "f6b6438ee7e2567cbf629db3d9a52bcf": " i=0,1 ", "f6b677608e23a612749e36f23b412698": "P(N(t+h)-N(t)>1) = o(h^2)", "f6b681b6a4586e5f3cc65e33eeba9875": "\\tau = 1/\\gamma\\,\\!", "f6b6afd24f30c8bb89bb99d791ed6aff": "\\boldsymbol{\\kappa}", "f6b6b0c4cb5d045873a81465076302bc": "\\begin{align}\n\\underline{V_s}&=V \\cos\\left(\\frac{\\delta}{2}\\right) +jV \\sin\\left(\\frac{\\delta}{2}\\right)\\\\\n\\underline{V_r}&=V \\cos\\left(\\frac{\\delta}{2}\\right)-jV \\sin\\left(\\frac{\\delta}{2}\\right)\\\\\n\\underline{I}&=\\frac{\\underline{V_s}-\\underline{V_r}}{jX}=\\frac{2V\\sin{\\left(\\frac{\\delta}{2}\\right)}}{X}\n\\end{align}", "f6b6c46dbd547a09e706d69042b5e8b6": "[J_i,P_0] = 0 ~,", "f6b6ce0f6c628ab588b69085b1c066f1": "u^{\\alpha}_{I}(j^{r}_{p}\\sigma) = \\left.\\frac{\\partial^{|I|} \\sigma^{\\alpha}}{\\partial x^{I}}\\right|_{p}", "f6b6d4bd18d441bc293d7543dfbaed3b": "u_{1,2}^{0}", "f6b7035b91e20e91fd6013329114a670": " H \\psi = V \\psi - \\frac{\\hbar^2}{2 m} \\nabla^2 \\psi ", "f6b722fa05833969460c4b30d1535e3d": "f=w(z)", "f6b7488c2273030d759c80a8a866e3d0": "Q = \n\\begin{bmatrix}\nd_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\\\\nd_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\\\\nd_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\\\\nd_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2\n\\end{bmatrix}\n", "f6b7713c0f22dedaee10466724900479": "N_1^c = \\left(\\frac{3}{4\\pi}\\right)^{1/2}", "f6b788598bc7b8767e0a1cba98961035": "| a \\rangle", "f6b7b1c2b79ae4311544e0cf9c09ed77": "\\Phi_{risa} = E_{ri} E_{sa} \\Psi_m^{(0)}", "f6b7fa9559ce747da47f271f51bf99ac": "\\overline{A + B}", "f6b820b41145cfbb2720c8adf0142438": "s_2(x)=\\frac{1}{4}x^2;", "f6b8631091f6de43af9ad00733b6f779": "\\exp\\!\\Big( \\boldsymbol\\mu'\\mathbf{t} + \\tfrac{1}{2} \\mathbf{t}'\\boldsymbol\\Sigma \\mathbf{t}\\Big)", "f6b8773e48314a934de936bbb9f779d8": " \\hbar ", "f6b89448f3fa72365b8668cec7cfc8d9": "R := [\\mathcal{Q}, T, P_{>}]", "f6b8cad2bb2b64a143ee1e83ed41d038": "\\frac{d|E(2\\omega)|}{dz} = - \\frac{i\\omega d_{\\text{eff}}}{n_\\omega c}\\left[E_0^2-|E(2\\omega)|^2\\right]e^{2i\\phi(\\omega) - i\\phi(2\\omega)}", "f6b8cb595ebabdebbaae15ddf67e8025": "P_{TMP}={(P_f+ P_c)\\over 2}-P_p ", "f6b8d3fba6380186a8ed47dc2723d090": "F(h)=- \\pi \\epsilon \\left(R_B+R_S \\right) \\big [p( \\rho )(2R_S-h)+ \\gamma ( \\rho , \\infty) \\big ]", "f6b91b9ca93786d544adfabb7c670b99": "\\scriptstyle 0.4 f_s,", "f6b93ff51126b44644140fc8215139b5": " \\left ( \\begin{array}{c|c} \\mathbf{R} & \\mathbf{0} \\\\ \\hline \\mathbf{0} & 1 \\end{array} \\right ) ", "f6b9c91ffc4ae0e6e1716ef2f664e561": "\nC(\\mu) =\n\\begin{cases}\n1, & \\text{if } f'(a) < \\mu < f'(b) ; \\\\\n\\frac{1}{2},& \\text{if } \n\\mu = f'(a)\\text{ or }\\mu = f'(b) ;\\\\\n\\end{cases}\n", "f6ba49d799898ff47fcf21a73aa454c9": "\\beta=\\frac{v}{c}", "f6ba782b78d0b9e94a8227da782595a0": "1.9417", "f6bacd19102e2f7dd9b19c804138e509": " s_n(w) =\n\\sum_{m=0}^n s(n,m) w^m.", "f6bae3249bf6a62084ff5db6f88d60a7": "\\rho(\\check{A}) = \\frac{\\rho(A)}{\\rho(A)-\\epsilon} > 1", "f6bb56d5b51e92f6ac7f773bfde7bb6e": "= \\frac{7}{3}A + 1", "f6bbb064207cb2071bf821c61813bd7d": "\\vec{\\mu}_E = {t_E}^{-1}", "f6bbde34b76f7b3d8e8cc41c274cde29": "\\sum _{i=1}^n x_i x_i^* = I ", "f6bc14512c307a9ad1420c285f851712": "m \\ddot{\\bold{r}} \\cdot \\delta \\bold{r} = \\sum_j \\left[ {\\mathrm{d} \\over \\mathrm{d}t} {\\partial \\over \\partial \\dot{q_j}} \\left( \\sum_i \\frac{1}{2} m \\dot{r_i}^2 \\right) - {\\partial \\over \\partial q_j} \\left( \\sum_i \\frac{1}{2} m \\dot{r_i}^2 \\right) \\right] \\delta q_j ", "f6bc3391f114576773fef414e51252ef": " \\int_{\\mathcal{O}_{\\lambda + 1/2}} e^{i\\beta (X)}d\\mu_{\\lambda + 1/2} (\\beta) = \\frac{\\sin((2\\lambda + 1)X)}{X/2}, \\; \\forall \\; X \\in \\mathfrak{g}, ", "f6bd09b365f3a89e3a43118406cdafd5": "p(x_1^{},\\dots,x_n)", "f6bd427f722da48a8e2ec8d89cf42e21": "F_IO_2[1-RQ] \\ll 1", "f6bd514a964be5b33678a3e575800135": "t:\\!\\!-~ \\alpha_1 \\times \\ldots \\times \\alpha_n \\vdash \\beta", "f6bd97affa85164b9c84380385984cc3": "\\begin{matrix} v_1&=&i_1Z_{11}&+&i_2Z_{12}&+& \\cdots &+& i_nZ_{1n}\\\\\nv_2&=&i_1Z_{21}&+& i_2Z_{22}&+&\\cdots&+&i_nZ_{2n} \\\\\n\\vdots & & \\vdots & & \\vdots & & & & \\vdots \\\\\n v_n&=&i_1Z_{n1}&+&i_2Z_{n2}&+&\\cdots&+&i_nZ_{nn}\\end{matrix}\n", "f6bd97c0d26bef670ab7d967aa147700": " ~\\Upsilon_v = \\exp{\\beta [ vP(z,T) + a\\sigma (z,T) ]}, ", "f6bdbb86368a055f70de88bd4cc151f3": "\\ln\\left[\\frac{u'(c_{t+1})}{u'(c_t)}\\right]=-\\sigma\\ln\\left[\\frac{c_{t+1}}{c_t}\\right].", "f6be68ee66a0f5f0e5d197ab26efd427": "\\beta \\pi/\\sqrt{6}.", "f6be819977d77ede261a21f53cfc15ce": "S(0,X) = S(0).", "f6bf381b4148ea94e2d27af9b6c43ceb": "c > 1", "f6bf5a49a49e37596847cba94ad187c7": " (h,q^n-1) =1 ", "f6bfbe749f8eccdb314d613dedf6a86a": "(A^T)^+ = (A^+)^T,~~ \\overline{A}^+ = \\overline{A^+},~~ (A^*)^+ = (A^+)^*.\\,\\!", "f6c0168b216240924d2beb293ebc4bfd": " \\frac{\\delta_h[f](x)}{h} - f'(x) = O(h^{2}) . \\!", "f6c03dd7b566895b536fa07f6349853e": "\\Delta(x_1\\otimes\\dots\\otimes x_m) = \\Delta(x_1)\\Delta(x_2)\\cdots\\Delta(x_m).", "f6c05f7a57dbf0a043d993a79a971434": "{Percentage\\ of\\ Solid\\ Recovery} = \\frac{solid\\ in\\ feed - solid\\ in\\ effluent} {solid\\ in\\ feed} \\times {100%} ", "f6c0bd7910244e25e2cb76e29db07102": "\\tilde{H}_{\\mathrm{\\infty}}(W|E) \\geq m", "f6c0f8758a1eb9c99c0bbe309ff2c5a5": "\\textstyle x", "f6c18714a095c1d57c84c44db1f7cc9f": "h=2", "f6c1b3c5a3ed875a96f8856b137436be": "\n\\begin{align}\nV_M \n&= \\mu_B B m_j \\left[ g_L\\frac{j(j+1) + l(l+1) - s(s+1)}{2j(j+1)} + g_S\\frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)} \\right]\\\\\n&= \\mu_B B m_j \\left[1 + (g_S-1)\\frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)} \\right],\n\\\\\n&= \\mu_B B m_j g_j\n\\end{align}\n", "f6c1c4968cf88ee0df13751b3b36ccdd": "\\pi_{n+1}(X) \\to H_{n+1}(X)", "f6c1ecf1eac0e783170156c3c4977728": " \\theta ", "f6c2b1f536f095d61cab9d734f97ea42": "P_k(s)", "f6c2bd686fbb5c4e160cd0352e14507b": "F^{\\prime}(x)", "f6c2bfe1a606f34443f1cd4305926715": "U(\\mathfrak{h}_n)", "f6c2ed4f8d0d7327c07a9bf0a2b5dc96": "M_\\mu \\to M_\\lambda", "f6c32f6bb62d4a62f29bcb1b170a38c7": "D = \\sqrt{\\eta Z_o / Z_i} ", "f6c406a04df9d28060664f480c493cb7": "\\tilde{G}_F(p) = \\frac{1}{p^2 - m^2 + i\\epsilon}. ", "f6c443e3e7d2f6203df6d8837fcadc25": " P', L', I' ", "f6c4cdd2bcd8cf8c5f5658fbb5253451": "[E]_0 \\approx [E]", "f6c4dabb77b222100b9ce56077e49e29": "g(x) := f \\circ f \\circ \\cdot \\cdot \\cdot \\circ f(x)", "f6c525a646b37e01221dbd72c4af3a0a": "E_1 = E_0 \\left( \\frac{m_x \\cos \\theta_1 \\pm \\sqrt{m_y^2 - m_x^2 \\sin^2 \\theta_1}}{m_x + m_y} \\right)^2 ", "f6c55405c43430184c7743901071c700": " \\bold y, \\bold f \\in \\mathbb{R}^n ", "f6c5576015831aaf56cbd034027cc835": "{\\mathbf{}}n", "f6c55bb29f943dbde801727d5b74dfac": "A(z_1,z_2)", "f6c5b5ff8357977dc0f4dcb833716a27": "\n \\mathbf{q} = \\lim_{n \\to \\infty} \\mathbf{x}^{(n)}\n", "f6c5de840873084f6db5f9bd5a9bcb27": "n_Tn_R", "f6c5df3f4427fd5798638d28ae18b760": "\\eta_{max}", "f6c5e5d168229df90bb093525ce09aab": "\\frac{V_{\\,alveolar\\,dead\\,space}}{500\\ ml} = \\frac {42\\ mmHg - 40\\ mmHg} {42\\ mmHg}", "f6c624b5cc439a62334dae558505279b": "{\\sqrt{2}\\over12}a^3 \\,", "f6c69343b81f3ef769ea2265da45890e": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \\sum_{k=0}^{\\infty} M_{k}\n\\left( \\frac{1}{r^{k+1}} \\right) P_{k}(\\cos \\theta )\n", "f6c6b55bcc5a654fdd0076c6ab734f79": "\\sum_\\rho \\frac{1+\\left|\\Re(\\rho)\\right|}{(1+|\\rho|)^2} < \\infty.", "f6c6d8cec84771628afafa79ccc80138": "E_{xy,yz} = 3 l m^2 nV_{dd\\sigma} + l n (1 - 4 m^2) V_{dd\\pi} +\nl n (m^2 - 1) V_{dd\\delta}", "f6c71265b9d03e14a2248339cb6d0f20": "z = x + y \\jmath , \\quad \\jmath^2 = +1,", "f6c7430f43a0e595bf057ece08f222be": "x = \\sqrt {K_a F}= \\left[ H^+ \\right]", "f6c7497ec713ee0d4aeb0bfd06871938": " \\operatorname{Tr}(S E) = \\langle E \\psi | \\psi \\rangle ", "f6c74d0b6f32e372736f53c0f6d91cc1": "\\tan(x)", "f6c753db63b8e364a4b6f825f0a442ca": "f(X) = f_0 + f_1X + \\cdots + f_{k-1}X^{k-1}", "f6c76105f2692ff540a9f894157b6164": "C \\to c", "f6c76dea931e58a4bfdf2c8d14a22bcc": " f_t(D)=U(t), \\,\\,\\, f_t(0)=0, \\,\\,\\, \\partial_z f_t(0)=1", "f6c7b2e66c2d71967b121f7aff69fec5": " h_{10} ", "f6c7d1b43b2b0974aa54f80ad90c91fc": "\n\\text{If }p\\equiv1\\pmod4 \\text{ or }q\\equiv1\\pmod4 \\text{ (or both), then}\n", "f6c7ded3cf648a74a788e89b9b77bf94": " \\nu_C (x,y,z) := (\\lfloor x\\rfloor, \\lfloor y\\rfloor, \\lfloor z\\rfloor)", "f6c817242e7f1b72486983df32206b94": "Q_s \\cdot Cc_{O_2} - Qs \\cdot Cv_{O_2} = Q_t \\cdot Cc_{O_2} - Qt \\cdot Ca_{O_2}", "f6c8bf49c9954290d90d831c6ac6b9a8": "T_b \\, = \\, 198 + \\sum {T_{b,i}}", "f6c8ce651b5c593232de8a600636dce0": "T_n(x) = \\frac{\\Gamma(1/2)\\sqrt{1-x^2}}{(-2)^n\\,\\Gamma(n+1/2)} \\ \\frac{d^n}{dx^n}\\left([1-x^2]^{n-1/2}\\right).", "f6c914962ed8e588f21ab2d865e0b649": "f_c(\\beta_2) = \\beta_1", "f6c9296df3479a548fe29193d5d004f2": "\\lbrack\\mathbf a\\rbrack = \\lbrack\\mathbf a\\rbrack_1 \\cdot \\lbrack\\mathbf a\\rbrack_2", "f6c962d6b9165c9a54038c0ad30bbab9": " U(t) = e^{-iHt / \\hbar}.", "f6c96dbae9c18b267dddf403f1931441": "\\sigma_{\\bar x} = \\frac{\\sigma}{\\sqrt{n}}", "f6c9922c5012d1e71917ff4396a4c136": " z_u = \\sqrt[n]{\\frac{1}{\\nu(\\beta+\\gamma)}} ", "f6c9a95bab2241e1cacce4df695ad568": "\n V(z) =\n \\begin{cases}\n 0; & |z| < L/2 \\\\\n V_0; & \\mbox{otherwise}\n \\end{cases}\n", "f6c9b895bc20abd60524eea61df72715": "R_f(N) = 1 - \\frac{E_f}{E_g}", "f6ca02b07dcc5a421663a6bd61b5e9dd": "\\begin{align}\n& AD \\times BC + AB \\times DC = AC \\times BD \\\\\n\\Rightarrow \\ & a^2 + c(c-2a\\cos\\hat{B})=b^2 \\\\\n\\Rightarrow \\ & a^2+c^2-2ac \\cos\\hat{B}=b^2.\n\\end{align}", "f6ca35da7179d19d593c846f71a0be93": "\\mathcal{T}_{\\mathcal{A}}(x)=f(x)+Y\\,\\!", "f6cade25bd8b7f88fe3034b607335376": "U_i,U_j", "f6cae532b341cdf35b8d86d116b5bd3b": "\\frac{1}{1}, \\,\\frac{4}{2}, \\,\\frac{10}{3}, \\,\\frac{20}{4}, \\,\\ldots.", "f6cb2271311db638ad8d3cbb3555ffcf": " p'(t_0) = f(t_0, p(t_0)), \\, ", "f6cb383043b59262eb5bedc22da4ffb3": "\\frac{dP}{dr}=-\\frac{Gm(r)\\rho (r)}{r^2}", "f6cb4ba65947205b23b23c895502b38f": "I_\\alpha(x) = \\frac{1}{\\pi}\\int_0^\\pi \\exp(x\\cos(\\theta)) \\cos(\\alpha\\theta) \\,d\\theta - \\frac{\\sin(\\alpha\\pi)}{\\pi}\\int_0^\\infty \\exp(-x\\cosh t - \\alpha t) \\,dt ,", "f6cb7d0488ab5180f0d9a04f56d9bda6": "V_\\parallel(\\beta)", "f6cb816a348385f439a173cc30f7048c": "T \\colon = (X\\times\\{0\\}) \\cup (Y\\times [0,1]) \\ \\subseteq \\ X\\times [0,1]", "f6cbaed9441069860ab05007ce62a865": "\\textstyle(x, y)", "f6cbbd07f8d15b6eff6ea89ed58da0a1": "\\|\\cdot\\|_p", "f6cc0c928012528feba433a4665a9c1c": "\\nabla_{\\mathbf v}f", "f6cc8e5854645e6f047dac7bdfe1b1ba": "\\displaystyle \\frac{1}{x}", "f6cc93ad463b8b13077963db46928428": "\\delta_x = \\frac {F x} {48 E I}(3L^2 - 4x^2)", "f6cc956a5cd04a96ef0c86ad6c115b12": " [16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2 - ", "f6cd621f7405edbfa939d49f8e872f22": " A_k = \\frac{A_{k-1}}{k^2} ", "f6cd7ca16b9b9a37a586c0ab5bb862f0": " K = K_0 + PK_0' + P^2K_0''", "f6cd9d1be95122e898bab0439d5ebcc6": "\\frac{\\sigma_{\\rm e}(\\omega)}{\\sigma_{\\rm a}(\\omega)}\\exp\\!\\left( \\frac{\\hbar \\omega}{k_{\\rm B} T}\\right)\n=\\left(\\frac{N_1}{N_2}\\right)_T\n=\\exp\\!\\left( \\frac{\\hbar \\omega_{\\rm z}}{k_{\\rm B} T}\\right)", "f6ce3eb7a8e9dc8bd2dc09378eb55eed": "\\hat{\\beta}_2 ~=~ (y ~|~ T=0,~ S=1) - (y ~|~ T=0,~ S=0)", "f6ce8436ad9bd3694a29de64d64d5741": "\n\\begin{align}\nP(\\text{Biased coin}) &= \\frac{1}{3} \\\\[8pt]\nP(\\text{Fair coin}) &= \\frac{2}{3} \\\\[8pt]\nP(\\text{H}|\\text{Fair coin}) &= \\frac{1}{2} \\\\[8pt]\nP(\\text{HHH}|\\text{Fair coin}) &= \\frac{1}{8} \\\\[8pt]\nP(\\text{HHH}|\\text{Biased coin}) &= 1 \\\\[8pt]\nP(\\text{Biased coin}|\\text{HHH}) &= \\frac{P(\\text{HHH}|\\text{Biased coin})P(\\text{Biased coin})}{P(\\text{HHH}|\\text{Biased coin})P(\\text{Biased coin}) + P(\\text{HHH}|\\text{Fair coin})P(\\text{Fair coin})} \\\\[8pt]\n&= \\frac{1 \\times \\frac{1}{3}}{1 \\times \\frac{1}{3} + \\frac{1}{8} \\times \\frac{2}{3}} \\quad = \\quad \\frac{\\frac{1}{3}}{\\frac{10}{24}} \\quad = \\quad \\frac{4}{5} \\\\[8pt]\n\\end{align}", "f6ceb1fe902d14d82fe5c021ffe11ad6": "x=(x_n)", "f6cebc2a7292d9f49c95a7b546b63301": "\n\\lambda(t \\mid H_t)=\\lim_{\\Delta t\\to 0}\\frac{1}{\\Delta t}{P}(\\text{One event occurs in the time-interval}\\,[t,t+\\Delta t] \\mid H_t) ,", "f6cf489816f959488a66ab4def456605": "\\sigma_v = \\sqrt{\\tfrac{1}{2}[(\\sigma_1 - \\sigma_2)^2 + (\\sigma_1 - \\sigma_3)^2 + (\\sigma_2 - \\sigma_3)^2]}", "f6cf8b245e082bb51a14f3edf5b42ab4": "\\scriptstyle F(x,y)", "f6cfdf15fbe1ee41b764f68c1c333dec": " {g_{\\alpha \\beta}}{d x^\\alpha \\over ds}{d x^\\beta \\over ds}=-1.", "f6cfe7fffca80b4c1730aaff276e3480": " \\sum_{i=1}^{\\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \\phi(n) ", "f6cfec15d0e94c6400f2ca6c6133ba5b": "\\times \\left|\\mathbf{I}_n + \\frac{\\beta}{2}\\boldsymbol\\Sigma^{-1}(\\mathbf{X} - \\mathbf{M})\\boldsymbol\\Omega^{-1}(\\mathbf{X}-\\mathbf{M})^{\\rm T}\\right|^{-(\\alpha+n/2)}", "f6cff4064d13db2e5eb045f5b8e4380c": "(l0", "f6f1afa5a9ca2e08ef776462bcc51ce1": "x_{\\mathrm{ZOH}}(t)\\,= \\sum_{n=-\\infty}^{\\infty} x[n]\\cdot \\mathrm{rect} \\left(\\frac{t - nT}{T}-\\frac{1}{2} \\right) \\ ", "f6f1c3246abb4b5369df76a31cf0b9af": " \\mathbf{B}(s) = \\mathbf{T}(s)\\times\\mathbf{N}(s), ", "f6f1cb5e0b9a804c7e30792677d605b9": "x^2 \\equiv 10 \\pmod{13}.", "f6f255494896afab2e2637f24025ee99": "\\lambda + \\alpha\\beta", "f6f255d580cf1b62328bd6451f745b4c": "\\gamma = \\int_1^ \\infty\\left({1\\over\\lfloor x\\rfloor}-{1\\over x}\\right)\\,dx \\, ,", "f6f300fb434320f97a886aa804cb7a35": "\n\\left(\\frac{\\partial S}{\\partial V}\\right)_{T,\\{N_i\\}} =\n+\\left(\\frac{\\partial p}{\\partial T}\\right)_{V,\\{N_i\\}}\n", "f6f3a6a9f3f1cc0798c6673fb6220d9a": "\\mathrm{d}f \\colon \\mathrm{T}(M) \\to \\mathcal{C}(M) : V \\mapsto V(f).", "f6f3d2e8484d79b7bb9749246cb68106": "\\operatorname{logit}(x)", "f6f3d7c77989635aa035a7b05281bc0b": "\\prod_{i = 1}^{g}{|1 - a_i^k|^2}", "f6f470071ac9b323fcacfa56cc6e0bc8": " X^{(i)}+Y^{(i)}=(X+Y)^{(i)}", "f6f49d2b704ea021292d0a519e6e86f8": " C_{\\alpha \\beta} =\\begin{bmatrix}\n C_{11} & C_{11}-2C_{66} & C_{13} & 0 & 0 & 0 \\\\\n C_{11}-2C_{66} & C_{11} & C_{13} & 0 & 0 & 0 \\\\\n C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & C_{44} & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & C_{44} & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & C_{66} \n\\end{bmatrix}.\n\\,\\!", "f6f4d39e9a31c5fb33124ccb87a0cbcc": "K\\le \\tfrac{1}{4}(p^2+q^2)", "f6f50afbfaeeb5a33c0348743c7f3286": "\\widehat{X}(\\Delta \\mathbf{X}) = \\exp\\left(-\\frac{i}{\\hbar}\\Delta\\mathbf{X}\\cdot\\widehat{\\mathbf{P}}\\right) = \\exp\\left[-\\frac{i}{\\hbar}\\left(\\Delta t\\widehat{E} + \\Delta \\mathbf{r} \\cdot\\widehat{\\mathbf{p}}\\right)\\right] \\,. ", "f6f51a17f2a35f6e2e53fadc1632b8d0": "\n \\boldsymbol{\\sigma} = 2\\cfrac{\\partial W}{\\partial \\bar{I}_1}~\\bar{\\boldsymbol{B}} - p~\\boldsymbol{\\mathit{1}}~.\n ", "f6f53b0b3552a4bf5d2516e465ed5712": "\\phi = 90^\\circ", "f6f570686baae18b9629c80732059012": " a \\cup (a^\\perp \\cup b)^\\perp = a ", "f6f58825fc87daafb15c4c87fe17f3dc": "\n\\lambda = [4.781 - \\frac{(\\Psi_1-4.781)^{2}}{\\Psi_2-2\\Psi_1+4.781}]^{-2}", "f6f5a7b5c3a3011957a874cd3317860e": "Excellent, Fair, Bad", "f6f5c4e1dbf09cc0d3715c6dc3e75205": "\\mathbf{f}=\\frac{d(m_{rel}\\mathbf{v})}{dt}, \\!", "f6f5f52a8743fe177cdc33a80898ace5": "\\begin{smallmatrix}{{T}_{\\rm eff}}\\end{smallmatrix}", "f6f61cc6705b10f61f77933ca49f719e": "Q = \\{ q_{0}, q_{\\mathit{left}}, q_{\\mathit{right}} \\}", "f6f6a012485649cd0c532c41237d0bfd": "y(t) \\ \\stackrel{\\text{def}}{=}\\ O_t\\{x\\},", "f6f7b1ac641f93e732c99efb66824a98": "(z_1,z_2,z_3)", "f6f80fe7dbd70cebe4ed7f19a1ba8023": "\\left(\\tfrac{11}{8}\\div\\tfrac{48}{35} = \\tfrac{385}{384}\\right)", "f6f817c1e0dc1efb46b327d8a2c3c22e": " \\alpha ", "f6f829bc46abfb4ef5c92397d404381c": "A = \\int_0^{2\\pi} \\int_0^\\pi r^2 \\sin\\theta \\, d\\theta \\, d\\phi = 4\\pi r^2.", "f6f881d27c00340aca533f89f0516e12": "\\frac{q-p}{\\sqrt{pq}}", "f6f8a806d4ecf037577a0902a8f89f14": "X_t = \\mu + (1 + \\theta_1 B + \\cdots + \\theta_q B^q)\\varepsilon_t.", "f6f916178ab1470a35bac61cc2b86b4e": "\\lambda=\\mu_1 + (a_2-a_1) \\frac{m_2+1}{m_2+m_1+2} - a_2\\!", "f6f91b28d46a9b50c026f5db0c555bad": "\\mathcal Z:=\\{\\{(x,y)\\in \\R^2 \\ | \\ y=ax+b\\}\\cup\\{(\\infty,\\infty)\\} \\ | \n \\ a,b \\in \\R, a\\ne 0\\}", "f6f96c27c6c5813e36dbdf7ea1eb6488": "\\scriptstyle(X,\\,\\mathcal{F})", "f6fa050de5b3064a0bba022bcd85a8cb": "\n R(r) = A_\\alpha~J_\\alpha(k~r) + B_\\alpha~J_{-\\alpha}(k~r)\n ", "f6fa2d3cddfc8e76b423aefbb9c7d96a": "\\nabla (f\\circ g)(c) = (Dg(c))^\\mathsf{T} (\\nabla f(a))", "f6fa55e47b1f645c3dd7ff24f7fd4231": "A-LC", "f6fa9290ad000996209980f68d14de83": "\\phi\\, ", "f6fab4c4664633fae63f1baad8219d3d": " V_\\mathrm{eff} (x) = \\frac{1}{\\sqrt{2 \\pi a}} \\int^\\infty_{- \\infty} V(x') \ne^{-\\frac{(x'-x)^2}{2a^2}} dx' ", "f6fade977fab7bae816ec00d5cb26d04": " f(x) \\approx f(a) + Df(a)(x - a)", "f6fb00c9c867459e78c2ee66baa2c65f": " \\leq\\mathbb{E}_{X^{n}}\\left\\{ \\frac{1}{M}\\sum_{m}\\text{Tr}\\left\\{\n\\Pi_{\\rho,\\delta}^{n}\\rho_{X^{n}\\left( m\\right) }\\Pi_{\\rho,\\delta}\n^{n}\\right\\} \\right\\} +\\epsilon\n", "f6fb20e36cc3fd4b8993514af677adb1": "\\varphi(\\theta^1(t) - \\theta^2(t))=\\frac{1}{8}\\sin(2\\omega^1-2\\omega^2)", "f6fb4e35962fa5ca43e44c944ae819e6": "\\frac{\\pi(x+(\\log x)^\\lambda)-\\pi(x)}{(\\log x)^{\\lambda-1}}", "f6fb7bb3ec242963c773b6bad6306ca9": "N(M) = \\{ w\\in\\Sigma^* | (q_{0},w,Z) \\vdash_M^* (q,\\varepsilon,\\varepsilon)", "f6fb7db4b0263c46dbe967c9bc8a8b96": " \\textbf{f} \\in L_f ", "f6fb8842abaedcf92c904eacc1161da4": "\\psi _{j}\\left[ R\\left( t_{n} \\right) \\right]", "f6fbb5ed31c3bf9fba9218b917bd1056": "u(r)=A \\sin(k r+\\delta_s)", "f6fbcc01da39666feeeecac9c3e6b371": "C\\subseteq F_2^n", "f6fbf2510ecec48703fa1183fe6f21c7": "Z_\\mathrm A=\\frac{Z_1Z_2}{Z_1+2Z_2}\\ ,\\!", "f6fc10e6af727dd604fb5d0dce236b67": "\n\\sigma_1 \\simeq \\sigma_1 ' \\oplus \\rho_2' \\oplus \\sigma_2 ' \\cdots \\simeq ( \\oplus_{i \\geq 2} \\rho_i ' ) \\oplus \n( \\oplus_{i \\geq 1} \\sigma_i ').\n", "f6fc191f666878433b13c8ad11141f8b": "\\pi = s_0 \\to s_1 \\to \\cdots", "f6fc42dfc19b4bbf3a1195d8f05ee659": "(m_1+m_2)L_1\\ddot{\\theta}_1+m_2L_1L_2\\ddot{\\theta}_2\\cos(\\theta_2-\\theta_1) + m_2L_1L_2\\sin(\\theta_2-\\theta_1) = -(m_1+m_2)gL_1\\sin\\theta_1,", "f6fcd98ae7c9a1278bd91028f06edeaa": "(1+240\\sum_{n=1}^\\infty \\sigma_3(n) q^n)^2 = 1+480\\sum_{n=1}^\\infty \\sigma_7(n) q^n,", "f6fd25144d7c9ff49dd27e4a8f761b8e": "\\mathrm{Res}_{z=\\infty} h(z) = \\mathrm{Res}_{z=0} \\left[- \\frac{1}{z^2} h\\left(\\frac{1}{z}\\right)\\right].", "f6fd457199910d4f3b4d4f0789726d51": "\\mathbf{\\mathit{l}}", "f6fd6db8368e5935dae51c31cb71380f": " b_{k+1} ", "f6fdf62b9e98f6e9e52cddd56c8b84fb": "\n\\Pr(B_n = B) = \\dfrac{\\Gamma(\\theta)\\,\\theta^{|B|}}{\\Gamma(\\theta+n)}\\prod_{b\\in B} \\Gamma(|b|).\n", "f6fe81fef31d408b6fddda080e39dcbf": " |z|^2 = \\Big(\\text{Re}(z)\\Big)^2+\\Big(\\text{Im}(z)\\Big)^2 = \\Big(\\frac{z+z^\\ast}{2}\\Big)^2 +\\Big(\\frac{z-z^{\\ast}}{2i}\\Big)^{2}. ", "f6fee391f87d9823095f2941d983c9cb": "x=\\sum_{n=0}^\\infty \\frac{a_n}{10^n}.", "f6fee7a9d6f4c84d6415b0fd34919578": "\\tau=t_1-t_2-\\frac{z_1-z_2}{c}", "f6fef0d44131f7f3b4f8ff50d6199830": "5\\tau = \\text{FO4}", "f6ff0f1b5d183dada1541236d34de320": "f(c^-) = f(c) = f(c^+)", "f6ff70b1f68fd3f4c7ccd9ab80eba57a": "I=V/R", "f6ff718f22ab12c6cc09065914e1a7bf": "g = de(d + e - 4) / 2 + 1.", "f6ffc2cfadba8cdec45e3823f789195b": "J_n(\\mathbb C)", "f6ffcf5583fee222b9a2231e10a9c978": " Q = n\\,c_V\\,\\Delta T + n\\,R\\,\\Delta T ", "f6ffe2e9127c8b570df98f6b939572d2": "D_2=\\frac{dF_2}{dC_2} \\frac{B_2C_2}{N_A}", "f7000b653799df30a5bb5d489ae2c974": "\\int |T(f)(x)|^p \\, \\omega(x)\\,dx \\leq C \\int |f(x)|^p \\, \\omega(x)\\, dx,", "f7000f1b3c8d9cf46e656f1b01e49057": "(J^k_{x_0}f)(z)=f(x_0)+f'(x_0)z+\\cdots+\\frac{f^{(k)}(x_0)}{k!}z^k.", "f70064273c9708f6cea5d1ab6effc8b5": "\\mathrm{SL}(3,\\mathbb C)", "f700920c1af0a29d77c3e7bde46721b4": " S_{0} ", "f7014b9e74057b7030fa382f1e5a2916": "\\mathbf{E}_i", "f7015896cfe3b7b4c34ab44638617658": "r = \\frac{\\hbar}{\\sqrt{2mV_0}}", "f7016b0be408b83cd9ba25e6b51b6112": "[f,g,h]=fg^{-1}h", "f7019089f455728a27f7a8a12add57cb": "x_\\pm=-\\frac{a_{n-1}}{na_n} \\pm \\frac{n-1}{na_n}\\sqrt{a^2_{n-1} - \\frac{2n}{n-1}a_n a_{n-2}}.", "f701b03f20c80037d0f408460d52f53a": "[\\cdot,\\cdot]:L\\otimes L\\rightarrow L", "f701c3cfc0b1f6b00f80b205c2a2a225": "\\sigma^-", "f7021ce0ef106bc076793d96522faeb5": "E_{2n}=i\\sum _{k=1}^{2n+1} \\sum _{j=0}^k {k\\choose j}\\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}", "f702389bb007ba94a7b5b6e61331f847": "\n\\varepsilon_n = (2 \\Omega a)^2 / gh_n. \\,\n", "f70258a0532ce45156175af714c16d6a": "\\sigma^2 = E(X_1^2) + 2 \\sum_{k=1}^{\\infty} E(X_1 X_{1+k}),", "f7030fd9562f15add629cbc994e5b9ef": " I_x", "f7031ecf33eb08ec5a61807f4a3b23c1": "\\psi_i(r)", "f70322fb300952a4d1b733680eead608": "\\displaystyle \\frac{\\partial \\mathbf{S}}{\\partial t} = \\mathbf{S}\\wedge \\left(\\frac{\\partial^2 \\mathbf{S}}{\\partial x^{2}} + \\frac{\\partial^2 \\mathbf{S}}{\\partial y^{2}}\\right)+ \\frac{\\partial u}{\\partial x}\\frac{\\partial \\mathbf{S}}{\\partial y} + \\frac{\\partial u}{\\partial y}\\frac{\\partial \\mathbf{S}}{\\partial x}", "f70344b2a7c48894ea4c38f186d23278": "3f, 6f, 9f,", "f70355ea7a5f2af6b55bba318dab01e7": "\\langle 0|\\Phi(x)\\Phi^\\dagger(y)|0\\rangle = \\int_0^\\infty d\\mu^2\\rho(\\mu^2)\\Delta'(x-y;\\mu^2)", "f7036c7fe828d9f9279cfbe231c4fd18": "x=x_0+x_1", "f703833cda7b89eedfc928a3bccdd4ea": " \\lim_{n\\to\\infty} \\frac{x_n}{\\pi(n)} = 1, ", "f703bd252b5e75dbb928628120611fbc": " E = \\frac{1}{2} \\frac{Q^{2}}{\\phi_{0}^{2} V} + U(\\phi_{0}) V. ", "f703c3cb090987a5ca45c24e81a3824a": "\\Delta \\Omega = -2\\pi\\ \\frac{J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos i\\,", "f703f54e72c45e913577f2cbd231cd41": " \\mbox{EVPI} = \\mbox{EV}|\\mbox{PI} - \\mbox{EMV}. \\, ", "f7040a6606d49e69b099b3c7e6af924d": "=\\lambda^{-1}\\mathbf{P}(n-1)\\mathbf{x}(n)", "f704113f7332427d761a2922d28c6f2e": "\\left[A,B\\right] = A B - B A,", "f70420386bbe772c2a588f926d63c167": " \\langle p, x \\rangle ", "f7043bbccdd4fd76886252d37586e633": "\\mathrm{Annualized\\, HPR}_{n}=\\left(\\frac{D+(P_{n+1}-P_{n})}{P_{n}}+1\\right)^{\\frac{1}{t}}-1", "f7045622c3ed50c1cbdd5994d107acfd": " a=\\omega_z/\\omega_{xy}", "f704f129b346b6e2383036b9f10c6ca1": " \\begin{bmatrix} x_1 \\; x_2 \\; \\dots \\; x_m \\end{bmatrix} ", "f7055f3c2d0c2d8dd85b0d07df98e429": "\\sin\\frac{3\\pi}{20}=\\sin 27^\\circ=\\tfrac{1}{8}\\left[2\\sqrt{5+\\sqrt5}-\\sqrt2\\;(\\sqrt5-1)\\right]\\,", "f70565116e15d4ea68a590e3cc9d32ec": "F_S(t,T) = \\frac{E_{Q_*}[D(T)S(T) | \\mathcal{F}(t)]}{D(t) P(t,T)}= E_{Q_T}[F_S(T,T) | \\mathcal{F}(t)]\\frac{E_{Q_*}[D(T)|\\mathcal{F}(t)]}{D(t) P(t,T)}", "f70582ef7870040be5db91e4f5e8bfb6": "\n \\sigma_i = \\cfrac{\\lambda_i}{\\lambda_1\\lambda_2\\lambda_3}~\\frac{\\partial W}{\\partial \\lambda_i} ~;~~ i=1,2,3\n", "f705c293b758fb74369e915871682b03": "d_\\mu(x, x') := d(x, x')-a_\\mu(x)-a_\\mu(x')+D(\\mu)", "f705c4aa988b41136eb668bf50a6c5af": "F_a", "f705da73599bf2c555b362a34f8ac1db": "|f(n)|\\leq\\epsilon|g(n)|\\qquad\\text{for all }n\\geq N~.", "f70677bbd531bcbbf03285b7ed671fb6": "y(t+h)=y(t)+hf(t,y(t))+\\frac{h^2}{2}f'(\\eta,y(\\eta)), \\qquad t \\le \\eta \\le t+h", "f70736e5024e806bbac54dec003d8c9c": "LF = (2W + X + Y)\\sqrt{8}", "f70742be198a2511d3d1ce0631ca5403": "C_b(X) \\simeq C(Y)", "f7078bbdbea9dfd13c3aa083362844f2": "\\boldsymbol{\\upsilon}", "f7078e154143f7df47951ac4feaabc31": "\\bar{z}=x-iy", "f707b501f1c78fd2494e1f982d2f64ea": "s_j = 0", "f707b7577e15880cb3919470a5243b18": "\\langle p|T^{0 0}(0)|p\\rangle =\\frac{E}{(2\\pi)^3}", "f70806fbf3da0be94c8c05e998fd6210": "\\hat{S}_k^{lm}(f)", "f7080ee26941c8eb8f7ff3df947e80cf": "\n k_{x} = \\frac{n \\pi}{a},\n", "f7081c38cb4a0c9f8169c7c9e8b4dafa": "\\begin{array}{cc} \\begin{array}{rrrr} \\\\ \\\\ \\\\ j &k & l & m \\\\ \\end{array} & \\begin{array}{|rrrr|rrrr} & & & pj & pk & & & \\\\ & & oj & ok & ol & pl & & \\\\ & nj & nk & nl & nm & om & pm & \\\\ a & b & c & d & e & f & g & h \\\\ \\hline a & o_0 & p_0 & q_0 & & & & \\\\ n & o & p & & & & & \\\\ \\end{array} \\end{array}", "f7083b5ddc2977fbb42ba085c40887df": "n_1,\\dots,n_m", "f7086c3d79327094b6e163b9c28123f4": "c \\operatorname{curl}(\\mathfrak{E} + i\\ \\mathfrak{M}) = i\\ \\frac {\\partial (\\mathfrak{E} + i\\ \\mathfrak{M})} {\\partial t} ", "f7089d841a347164ff568a3ae39e850a": "i=n,n-1,\\ldots,0", "f7089da17cc60a5b1217fd17f8d7a293": "\\wedge^{m+1}_n = \\vartriangle^m_n \\supset \\vartriangle^{m-1}_n = \\wedge^m_n", "f708c69ed2c59446ce5ccb1d1a395d80": "\\mathbf{w'}_n(x_n),", "f708c8ffc8a01f08743837a866c3b5dc": "\n W = \\cfrac{\\mu}{2} (I_1-3)\n ", "f7094409e6185e6a21d9e34bbd28041f": "A_\\mathrm{v} = \\frac{v_\\mathrm{out}}{v_\\mathrm{in}} = \\frac{1}{1+\\frac{R}{R_\\mathrm{E}}} \\ . ", "f709a64de2f3b320a3457dc311723077": "U\\subset D", "f709a83b97e76711835c1cedb06dfd85": "\\scriptstyle{u_{\\max}^{(s)}}", "f709cb66b74ac7d591f46636a5c04da6": "\n3Nk_{\\rm B}T = 3PV + 2\\pi N \\rho \\int_{0}^{\\infty} r^{3} U'(r) g(r)\\, dr.\n", "f709dd87f16f25cb69e3b4e3bab309ec": "\n\tD_t^j(i) = \\frac{1}{Z_t^j}e^{-\\hat{y_i}g_j^{t-1}(\\boldsymbol{x_{j,i}})}\n", "f70a29c06bd84f87cbd215e0afc536e7": "\\tau_x ({f}*g) = (\\tau_x f)*g = {f}*(\\tau_x g)\\,", "f70a3ea171724b5119af237de89a0a7d": " det^{column}(t-M)= \\sum_{i=0...n} (-1)^{i}\\sigma_i t^{n-i} ", "f70a584cebc735ff9cc79db41e6582cc": "\\tfrac{\\partial I}{\\partial y}", "f70aa5a6ff544fad8444174f0c89cf43": "S_n (R)", "f70b11b6dd6182031f74727cf484651b": " {G^a}_b \\, {G^b}_c \\, {G^c}_d \\, {G^d}_a = R^4", "f70b75d23507be12e21ea3b6a5074a0e": "\\alpha_{U,n}(t)", "f70b823516d53802ddc08abbd4cec9c7": "0 < i < M", "f70bcbd7c34838fc8502f9bd7584dd58": "\\delta S_{EH} = \\delta \\int d^4 x \\; e \\; e_M^\\gamma e_N^\\beta C_{[\\gamma}^{\\;\\;\\; MK} C_{\\beta]K}^{\\;\\;\\;\\; N}", "f70be9d5704bc438f93b42df6ad96fb4": "y_t = A^t y_0 = PD^{t}P^{-1} y_0,", "f70c97e0de0e0f5c63c6f8c3f05ce170": "\nS = k\\ln \\ (\\mathrm{constant}\\times V) = -k\\ln \\ (\\mathrm{constant}\\times c).\n", "f70ca79836709eb8a0b00cfb1ff593a6": "[e_0,a_0,i_0,\\Omega_0,\\omega_0,M_0]", "f70caf181ce8eef42ab8d4ba1895dff5": "p(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_2 x^2 + a_1 x + a_0. \\qquad (1) ", "f70ccfc7a70dfaa85579ca0626e64b76": "\\mathfrak{Cos}", "f70d7034abf0c28e34716740e63b49b0": "\\int_{P^\\circ }^P {\\bar VdP}", "f70dcfcee4a2331d0422d97cd6e6e8e5": "\\begin{align} & \\Delta m = \\rho \\Delta V \\\\\n& m_2 - m_1 = \\rho ( V_2 - V_1) \\\\\n& m = \\rho V \\\\\n\\end{align}\n", "f70ddec518e688a4f19b1440415da5ef": "\\text{Trail}_\\text{motorcycle} = \\frac{R_w \\sin(A_r) - O_f}{\\cos(A_r)}", "f70e2fe4d615166e6d1ba2bfed85cf71": "f(x,t) = \\frac{1}{\\sqrt{2 \\pi t}}e^{-{x^2}/({2t})}.", "f70ea787de5df85f8eb217b2d8f1a568": "HS_{R_n}(t) = \\frac{1}{(1-t)^{n+1}}\\,.", "f70eaa8a01b89070b9d275fa281b8e3f": " \\omega_n", "f70eb6b253bff26895636ae27fba6861": "\\iiint\\limits_E \\, dx\\,dy\\,dz", "f70eb8d36b17f0a084a27dc088b1c468": "\\mathrm{laea}_x", "f70eda299bae1d81650a9283883d2a2c": " \\Gamma(A)", "f70ee5f3c91182988cc30343ce2b15b3": " \\sigma(\\bar x)=d/ \\sqrt {n} ", "f70f6044dec931789c91eb751d51edcc": "F(x_{1i}, x_{2i}, x_{3i})", "f70fe9d17f029254dabd5e8c972c6fdf": "\\frac{1}{n}\\sum_{i=1}^n g(X_i) \\ \\xrightarrow{p}\\ \\operatorname{E}[\\,g(X)\\,]", "f710010be32b2074d5633bfef8d87419": "p)", "f7101c90807eddc61f025fd5543c2438": "G(z) = \\left(z^c\\right). \\, ", "f7103ab34ba9aebd1d135e4a212aa7dd": "2^{i+j}", "f7108e81fdc55cda6b31a5252ac8935e": " = q \\hat{z} \\sigma_\\mathrm{tot}^{-1} \\int (1 - \\cos \\theta) \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\mathrm{d} \\Omega", "f710ab56f992d6b1406f4fa43241bf61": "\\frac{2}{\\tau_0} = \\frac{2 e^2 \\omega_a^2}{3 m c^3}", "f710b6204e9f7bab8f765c8699ca41a8": "v_k=w_k-\\sum_{j=1}^{k-1}{\\langle w_k, v_j\\rangle\\over\\|v_j\\|^2}v_j", "f7111b613634f4a76fa1eb7f436b633a": "\\mathbb{Q}^n", "f711525e69fc44374de533f8486034f6": " a_j", "f7115e95b9f595fee44d9cd024102a6e": "c(V) = c(L_1 \\oplus \\dots \\oplus L_n) = \\prod_{i=1}^n c(L_i) = \\prod_{i=1}^n (1+x_i) = \\sum_{i=0}^n e_i(x_1,\\dots,x_n). ", "f71169cc1df7791ed27105bd76bc19f3": "\\frac{ab}{a+b} = \\frac{1}{\\frac{1}{a}+\\frac{1}{b}} = (a^{-1} + b^{-1})^{-1}.", "f7116a790f298d75b67dc5d0dc5f9e87": "G(x) = \\frac{i}{4}H^{(1)}_0(k|x|)", "f711d4255917d40a68d272b8464223cc": "m=6", "f712184c647ee180630bbe29d453ee75": " a \\to 0 ,\\nu\\to \\infty", "f71277dbb6dda720f087627e3783b109": "X\\ ", "f7129e96b593559142aae7e84de171ee": "\n\\frac{m+1}{2}\\max \\Big(|\\sigma_1 - \\sigma_2|+K(\\sigma_1 + \\sigma_2) ~,~~\n |\\sigma_1 - \\sigma_3|+K(\\sigma_1 + \\sigma_3) ~,~~\n |\\sigma_2 - \\sigma_3|+K(\\sigma_2 + \\sigma_3) \\Big) = S_{yc}\n", "f713080e186baab15f88c1f6008ceb50": "n=6", "f7132f9bd8dd9af2fa44147be60f042f": " f''(x) = 2\\delta(x) ", "f713627872f3b97aa87be4027e8b9030": "\n \\langle J, \\gamma | H| J, \\gamma \\rangle_{\\mathfrak H} = \\omega J\\; .\n", "f7138d97c72b45f8c5da4e016eff8065": " _aD_t^\\alpha f(t)=\\frac{d^n}{dt^n}{_aD_t^{-(n-\\alpha)}}f(t)=\\frac{d^n}{dt^n}{_aI_t^{n-\\alpha}}f(t)", "f713ce5cf5d1a8edf1aae2262cab56b8": "0 \\le t_v(x)+f_v(x) \\le 1", "f714061874290adc24f87f601a74b4fd": "E[X^{(e)}]=E[X^i\\partial_i\\log p]=0", "f71408a02b3f93abfccec5e6cdab41b3": "\nI(\\theta)=p_i(\\theta) q_i(\\theta).\\,\n", "f7142b70a2f6c766210782f9f086582a": "\\forall, \\exists", "f7146482a0de0f8665b9ac99068518bc": "O(N^{1/2})", "f714656d7fd344839f5837f45edafffb": " \\mathbf{v} = \\mathbf{T}(\\rho\\mathbf{r} + \\sigma\\mathbf{s}) = \\rho\\mathbf{T}(\\mathbf{r}) + \\sigma\\mathbf{T}(\\mathbf{s}) ", "f7147c50b420c088d07acec6c8672436": "\\tfrac{M(1+\\nu)}{3(1-\\nu)}", "f7147ce047913d0337c63f143668e7ef": "{{T}_{f}}=\\frac{{{T}_{s}}+{{T}_{\\infty }}}{2}", "f715025cf121420bac5e55a3312bceb9": "\\Psi_n \\ne 0 ", "f7150399b64a9e3460434e243f24f9c6": "-1+\\mathrm i", "f7151412ce79c3e83298ac185958aa7a": "\\{ ww^R : w \\in \\{a, b\\}^{*} \\}", "f71535aa30a223faf48f28b3bb8bddbf": "\\nabla_{\\dot\\gamma(t)}X=0", "f715515e1fb00ded9462283bcd1b2b82": "(2)\\qquad R_{ab}-\\frac{1}{2}Rg_{ab}=8\\pi T_{ab}\\;.", "f7159ae453bf7bcbe14aa62d715415b2": "\\left |\\frac{d^kf}{dx^k}(x) \\right | \\leq A^{k+1} M_k ", "f716125cb89cfd815d8278e4c057bbac": "\n \\int_{-\\infty}^\\infty V(x;\\sigma,\\gamma)\\,dx = 1\n", "f716468c49aa723c0331386017581daa": "\\rho_N", "f7164e07a618d99baddb5cdaeeb71892": "f(x):=\\max(f_1(x),f_2(x))", "f7165e3f963654f317d44bfd5506bde1": "\\mathfrak{P}^{107}", "f716b7b15291d0d51d4969fa9078d133": " \\operatorname{Q}(\\xi,\\eta) = \\langle A \\xi \\mid \\eta \\rangle \\quad \\eta \\in H_1 ", "f716eee69a943d09acce46b2acf4a6cf": "q_{c,t}\\, ", "f7170660d0f8380eba993a3a673cb2a5": "f(\\mathbf{aa}) = q^2", "f7175177f7d9beef34a0fdf54b60ae5f": " x_n=(b-1)-(ax_{n-1024}+c_{n-1})\\,\\bmod\\,b,\\ c_n=\\left\\lfloor\\frac{ax_{n-1024}+c_{n-1}}{b}\\right\\rfloor.", "f717b2f7b3577296fc11f05f67d7a203": "\\ln | y | = 0.85t + B ", "f717c69176be9ef9355c55695ac6b83b": "S_s=\\frac{H_0}{1+e_0} C_{a} \\log \\left( \\frac {t} {t_{90} } \\right) \\ ", "f71910c44c8361eb0324a4435c1b03d9": "t_2'=0", "f719189624bb63124fd12909b0f61e5e": "\\varphi(\\mathbf{x},t)", "f719da6b8e20d1654a03deb6c4aeca05": "\\displaystyle{J(a,T,b)=(-b,-T^t,-a),}", "f71a163280c73f04e6a6297aa79ed60b": "PGR = \\frac{ \\ln(P(t_2)) - \\ln(P(t_1))} {(t_2-t_1)} \\times 100%", "f71a2c0109c5d40a621c42615dce3991": "F\\to F", "f71a67c06e9a13e077646865edd81691": "\\displaystyle{R(a,1)=R(1,a)=2Q(a,1).}", "f71a75554dc58b853477737d4952845d": "1 - \\sqrt{1 - \\rho^2}", "f71a856a1ba466aec85466ee261ac3a0": " \\vec{s}(C_{+1}^{(1)}) = [+1,-1,-1,+1], ", "f71ab6ededfc4b6bf99068d3b6de0b2e": "m = \\Omega(\\delta^{2}n) = \\Omega(n) \\geq \\Omega(n^{2\\gamma})", "f71c61396763fa6a94e5bfbb490fd1a3": "\\eta_2 = \\frac{n-p-1}{2},", "f71c80cb8009e8387489ba98d1362522": "\\cos \\theta = \\frac{d}{k}", "f71c8bae443428df67923f973b2faa06": "M_{ETC}^2", "f71caa1803dc867368891ade28e39f29": "(P \\or (Q \\or R)) \\leftrightarrow ((P \\or Q) \\or R)", "f71cb51f8db0ea2c0005569e955b726d": "\nM_2 = \\begin{bmatrix}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\end{bmatrix}\n", "f71cc10e3591b7f36a8f9bb030b5d953": " \\{ Q_\\alpha , \\bar{Q}_{\\dot{\\beta}} \\} = 2 ( \\sigma^\\mu )_{\\alpha \\dot{\\beta}} P_\\mu ", "f71cc1f01cc06b60ffb40b488e955051": "\\scriptstyle 3", "f71d26745b2d680c656913855958cd66": "\n\\theta_i = x_i\n", "f71d399f5459be2271ba0c1fcdb70911": "\\partial_t \\psi \\, i \\sigma_3 \\, \\hbar = H_S \\psi - \\frac{e \\hbar}{2mc} \\, \\mathbf{B} \\psi \\sigma_3", "f71d5e61f06d363bbc09238134ccd701": "A_p(h_c)=C_0h_c^2+C_1h_c^1+C_2h_c^{1/2}+C_3h_c^{1/4}+\\ldots+C_8h_c^{1/128}", "f71dd8ff864493281be3ae2e584f3996": "Cv(a)\\,", "f71e46d9e7c0f841b2e00788f1999b83": "\\left( \\frac{x}{\\sqrt{s}}, \\frac{y}{\\sqrt{s}} \\right), \\, ", "f71ea1401f33140ac1919e910d4c2e50": " \\frac{1 \\ \\mathrm{s}}{1575.42 \\times 10^6} = 0.63475 \\ \\mathrm{ns} \\approx 1 \\ \\mathrm{ns} \\ ", "f71eb9e2d2b73f58b41eada2111a24f2": "r=s=(1,1,\\ldots,1)", "f71ec4e6a5fe35a5d97e16f5c5fcdc3d": "< 1x10^{-2}", "f71f00705b94714b8511c2922f817e38": " \\frac12 \\int_A \\sum_i u_i^2 n_k \\, \\mathrm{d} S = - \\frac12 \\int_V \\frac{\\partial}{\\partial x_k} \\left( \\sum_i u_i^2 \\right) \\,\\mathrm{d} V. ", "f71f8878e153d5361c6e5fe0905bbd4a": "D = \\bigl( \\begin{smallmatrix}\\\\ 81&0\\\\ ~\\;0&9\\end{smallmatrix} \\bigr)", "f71fcba0f06e501435ffd26adc41a7c9": "\\operatorname{haversin}(c) = \\operatorname{haversin}(a - b) + \\sin(a) \\sin(b) \\, \\operatorname{haversin}(C).", "f7207585379e8f7dd61d3e221cf27d00": "S^m_\\ell", "f720cf84b63734e6c65793606cc07a67": "a_m \\neq a_M", "f7214d4064fa4741ce29829133d909fb": "\\scriptstyle{\\frac{1}{2 q^2\\sqrt{2}}}", "f7217c507a14060a3c779b534b4d26a6": "F_{bottom} = P_{bottom} \\cdot A.", "f721d581ff23a81e56581c995a62b245": "\\,q_{0}\\in\\, Q ", "f7230833355668e388bae3a3dc64f699": "H_k(\\Sigma_n)\\cong H_k(\\text{Map}_0(S^n,S^n))", "f7233a791f02e355f271d4273ebf8fa3": "\\mathbf{A} = \\begin{bmatrix}\n9 & 13 & 5 & 2 \\\\\n1 & 11 & 7 & 6 \\\\\n3 & 7 & 4 & 1 \\\\\n6 & 0 & 7 & 10 \\end{bmatrix}.", "f7235ff27bac0fdb344940f8f2a49299": "K_p(x)", "f7237a9d4a2d4a3b0d1d45bde9d8cc61": "\n\\operatorname{cov}_{\\mathrm{W}}(X, Y) = \\operatorname{dCov}(X, Y),\n", "f723886b111b680e0cb2b638d4025359": "\\kappa\\rightarrow(\\kappa,\\alef_0)^2", "f723f874e0817f45bf9080933a6de4af": "\\lambda\\to\\infty", "f7241fe8f297120cf66843801b91e4f5": "\\mathcal{L} \\subset \\operatorname{Fun}(\\mathcal{A}, Ab)", "f72437e85f6d6f8621b9c017d0dfa6c4": "V(r)\\propto r", "f724410ff901b5db03abf10e4e39b70a": " p(x\\mid I) = A \\cdot m(x), \\qquad a < x < b", "f72498cb76ebc5b9b36f19d5a0b674e4": "\\scriptstyle b_1 \\,\\oplus\\, IV_1 \\;=\\; b_2 \\,\\oplus\\, IV_2", "f7254f1ab0602096c775b557e24d3e41": "\n\\frac{1}{r_{2}(t)} = \\frac{a}{r_{1}(t)} + b\n", "f7257f00ce7f6d689f564dbe5450ca31": "\nx_t^{(i)} \\sim p( x | x^{k^{(i)}}_{t-1}).\n", "f725d58b495344d11f9503392b1da240": "(X, X')", "f725fe2437dec0e635fe31eee346be77": " \nB(\\{n_k\\}) = \\varinjlim \\cdots \\rightarrow M_{n_k}(C( \\mathbb{T} )) \\; \\stackrel{\\beta_k}{\\rightarrow} \\; M_{n_{k+1}}( C(\\mathbb{T} ) ) \\rightarrow \\cdots .\n", "f72618a2dba2b374cccb00562bf5d36a": "\nx = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{\\ddots}{\\quad\\ddots\\quad a_k + \\cfrac{1}{a_{k+1} + \\cfrac{\\ddots}{\\quad\\ddots\\quad a_{k+m-1} + \\cfrac{1}{a_{k+m} + \\cfrac{1}{a_{k+1} + \\cfrac{1}{a_{k+2} + \\cfrac{1}{\\ddots}}}}}}}}}\\,\n", "f726bb6308af15b6e04c608d0b547c80": "\\tau_p^n(x)=x^{p^n}", "f726c30271dd4a820427182a0ff92bf7": "\\hat{H}=\\hat{T} + \\hat{U} - \\sum_{i=1}^{N}\\sum_{\\alpha=1}^{M}\\frac{Z_{\\alpha}}{|\\mathbf{r}_{i}-\\mathbf{R}_{\\alpha}|} + \\sum_{\\alpha}^{M}\\sum_{\\beta>\\alpha}^{M}\\frac{Z_{\\alpha}Z_{\\beta}}{|\\mathbf{R}_{\\alpha}-\\mathbf{R}_{\\beta}|}.", "f726d553e483bb4c4d1c1e088fbbd0c4": " G^{\\hat{i}\\hat{j}} = \\mu \\, \\operatorname{diag}(1,0,0,0) + p \\, \\operatorname{diag}(0,1,1,1)", "f727315fd23a008548c58e06aa2eb6f8": " \\hat{\\varphi} = g^2 \\hat{a}^{\\dagger} \\hat{a} + \\delta^2/4 ", "f72742d3f8b17c3e3d71880f444174c2": "P_{L3}=V_{L3}I_{L3}=V_P I_P\\sin\\left(\\theta-\\frac{4}{3}\\pi\\right)\\sin\\left(\\theta-\\frac{4}{3}\\pi-\\varphi\\right)", "f7274fa9897e68a9e8327099de028541": "\n\\begin{align}\n g\\left( \\lVert f \\rVert \\right) &= g \\left( \\lVert \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) + v \\rVert \\right) \\\\\n&= g \\left( \\sqrt{ \\lVert \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) \\rVert^2 + \\lVert v \\rVert^2} \\right) \\\\\n&\\ge g \\left( \\lVert \\sum_{i = 1}^n \\alpha_i \\varphi(x_i) \\rVert \\right).\n\\end{align}\n", "f72767e11c0154c78e4b52cb2fca5e29": "f(x) = \\begin{cases} 2|x|, & \\mbox{if } x < 0 \\\\ 0, & \\mbox{if } x = 0 \\\\ 2x-1, & \\mbox{if } x > 0. \\end{cases} ", "f72773a3ac2752e3f59bfccc62f9b2bb": " F_p \\equiv \\left(\\frac{p}{5}\\right) \\pmod p \\quad \\text{and}\\quad F_{p-\\left(\\frac{p}{5}\\right)} \\equiv 0 \\pmod p.", "f727a6858b97772aaec312bab8db452f": "\\arccos x = \\frac{\\pi}{2} - \\arcsin x ", "f727dfc2462effb7d20a92304ecd5473": "\n\\begin{align}\nS_j &= r(\\alpha^j) = s(\\alpha^j) + e(\\alpha^j) = 0 + e(\\alpha^j) = e(\\alpha^j), \\ j=1,2,\\ldots,n-k \\\\\n &= \\sum_{k=1}^{\\nu} e_{i_k} \\left( \\alpha^{j} \\right)^{i_k}\n\\end{align}\n", "f727e9b06bb5ea3fa7f1779b7e790110": "S(x) = \\sum_{k=0}^n a_k \\phi_k(x)", "f727fc835afebd7efed9d12452597268": "\nR(-\\theta) = \\begin{bmatrix}\n\\cos \\theta & \\sin \\theta \\\\\n-\\sin \\theta & \\cos \\theta \\\\\n\\end{bmatrix}\\,", "f7281d8de11b173adbd966ef75bc5be4": "\\ddot{\\sigma}+\\xi^{-1}\\dot{\\sigma}-\\sigma^{\\prime\\prime}=0,", "f72821c22a7cb616a769e34352c68806": "S=2k_B\\ln(R/a)", "f7284c16fa43103c9f6558990e4266f6": "\\frac{S(TE)}{S_0} = \\exp (-b\\cdot ADC)", "f7288bf6efb47d2061339e9df8645174": "1 + \\cfrac{1}{1 + \\cfrac{1}{1 + \\cfrac{1}{5 + \\cfrac{1}{1 + \\cfrac{1}{4 + \\ddots}}}}}", "f7289dc51eb41d53d0e4f4a02c5167ce": "IndVal_{ij} = A_{ij} * B_{ij} * 100", "f728aa88ac06cfc2e35c8b2602ed2f2f": "\n\\left(\\frac{-1}{p}\\right) \n= (-1)^{\\frac{p-1}{2}}\n= \\left\\{\\begin{array}{cl} +1 & \\textrm{if}\\;p \\equiv 1 \\pmod 4\\\\ -1 &\\textrm{if}\\;p \\equiv 3 \\pmod 4\\end{array}\\right.\n", "f728c59a5ecce6448fb744cd5e47ec8b": "\\oint_C {1 \\over z}\\,dz.", "f729eb31dc2670708ca81bc1ebc38a26": "S^4", "f729fa5d65d863ffa4b1be8ac22336ae": "\\frac{p_2}{p_1} \\approx\n \\frac{2\\gamma}{\\gamma+1}M_1^2\\sin^2\\beta", "f72a33ef5a2cb7d5a06549d4f61fdd6e": "\\textstyle \\frac{1}{n}\\log N > I(P,\\Lambda) - \\delta", "f72a47ec26d1d2a7c1999824e881bddb": "\nR_{1} \\in [0 \\Omega,\\infty [\n", "f72ad34b1e36e3b4462fb479018df155": "A_A", "f72adcbcd3924eca267dc1e49dd023d5": "Tr(g^n)", "f72aedeaacbed0609b690153ae3834ca": "A\\ominus B\\subseteq A", "f72af2b7fad2ec30315b875f47ef1b8a": "\\mathbf M = \\int_S \\mathbf r \\times \\mathbf t dS + \\int_V \\mathbf r \\times \\mathbf b\\rho\\,dV.", "f72b65104d4c83d75c399e52078db4b8": "(Y,f)\\to (Y',f')", "f72be96fc5bb0ef74898fa898b650939": "\\gamma_\\mu=851.616", "f72bf0a746f26fbf3c4f36e13c2d3319": "(g_{\\mu\\nu}) = \\mathrm{diag}(1,0,0,0)", "f72c2da6dc6fcd6bf1ea679ca7d3f3b1": "K_2 < S_T", "f72c5ea5a8485dd4641c8ade5e6df201": "\\tfrac{3K-M}{3K+M}", "f72c974464945ed23be352fe93064f0f": "\\int_t^T\\mu(t,s) \\,ds=\\frac{1}{2}\\left(\\int_t^T\\xi\\left(t,s\\right)\\,ds\\right)^2-\\theta(t)\\, \\int_t^T\\xi\\left(t,s\\right)\\,ds.", "f72cd007415d95d849fbeb6791a85cc0": "\\exp[x/2]", "f72cfc4d603a5be9c94adacd831530ff": "\\max \\{g\\} = \\sum_{i} r(p_i) - \\sum_{p_i \\in P} r(p_i) - \\sum_{q_j \\in Q} c(q_j).", "f72d037f3f530aa20a4e502b539add32": "(\\text{skewness})^2-2< \\text{excess kurtosis}< \\frac{3}{2} (\\text{skewness})^2", "f72d17c2601db089b95d322fb8f2efcf": "0 \\equiv [m^2,x_i] = [(p^0p_0+p^jp_j),x_i] = [p^0p_0,x_i]+2ip_i", "f72de868e6d25966d480d754ce6ac635": "v^2/g", "f72df1cf08524fb301ab9cbc7f11dc10": "\\vec{V}\\cdot\\nabla{\\psi}=0.", "f72e471b040be3f05432e05dd28f33b8": "P_\\theta", "f72ea0610d9fec5e0d2f40ca066aa83e": "\\operatorname{Li}_{-4}(z)=\\sum_{k=1}^\\infty k^4 z^k =\\frac{z(1+z)(1+10z+z^2)}{(1-z)^5}\\,\\!", "f72ec7ca3dc1bb9fb24e3c75aaaf2082": "\\mathcal{D}_X", "f72edbede3613344f043cca822c1293d": "\\scriptstyle x_2", "f72f7b8fee453176970ba2d022b379b6": " n \\ge 3 ", "f72fa714c984c5819e3e9bf02046fc2d": "x/2,", "f72fe9e735da697ddbdaa2f6f97768c7": " v\\in V", "f73073e0e0851b70a2c5fa21c7459b41": "\\left\\langle \\mathbf{a} , \\mathbf{b} \\right\\rangle = \\mathbf{a} \\cdot \\mathbf{b}^\\star = \\sum_j a_j b_j^\\star ", "f73099bf155ddee8ab8a19de9b9e71ea": "f'(x_0)", "f7309a008c14331e3a9c6b5bf5e7bf03": "2^n \\times 2^n", "f7309f52cc419abda3fc41558dc31869": "\\overline{\\mathbf{A}} \\,\\!", "f730adb6cbab4a69675a3002aba58c09": " S_4 = {{9\\over5} \\div {5\\over3}} = {27 \\over 25} \\approx 133.238 \\ \\hbox{cents} ", "f7310cb15ad8e39569b353fb0072c655": "\\frac{ \\sqrt{6} + \\sqrt{2} } {4}", "f731a2420c65c8153a0a621c244dc0df": "E[zz^T]\\,", "f73205453fe9e02db3fa63010de9257c": "\\sqrt{\\frac{\\pi n}{2}}", "f7321c0f35b8dbe711e12331daad55b1": "1-x^{2^n}", "f732614279a0f6bfbb8090f9db649dd0": "4 \\Sigma_1 - 2 \\Sigma_2 - \\Sigma_3 - \\Sigma_4. \\,\\!", "f732798963aa6d26867154f8a17f97f1": "\\mathbf{u^1G+c_{s}=e}", "f732a5f15902cf27f765abbdc3baeb7a": "\\begin{cases}\nV_{\\text{out}}(t) = I(t)\\, R &\\text{(V)}\\\\\nQ_c(t) = C \\, \\left( V_{\\text{in}}(t) - V_{\\text{out}}(t) \\right) &\\text{(Q)}\\\\\nI(t) = \\frac{\\operatorname{d} Q_c}{\\operatorname{d} t} &\\text{(I)}\n\\end{cases}", "f73305312970d2c0c22f5fbc96da7192": "\\textstyle g^{\\prime}(x):=+\\infty.", "f7331321ea8ed08b3a5e347c9dfdabc0": "GL_n(F)", "f7332d1cb08b713f5d54fb8aeab61ba7": "\\frac{A}{A_{0}}\\equiv \\frac{A_{n}}{A_{0}}=\\left( e^{-k\\Delta t_{p}} \\right)^{n}=\\left( f_{RP} \\right)^{n}=\\left( 1-f_{BP} \\right)^{n}", "f7333fec38d611e299285290eda4344b": "E_f=3P_f ", "f7334db2c83c1b99b9ce847895cd8f4a": "\\lbrace\\vec{P_0}\\rbrace", "f7336b6fcb8f0245ad426d7e5c56885a": "\nh_t(x,x) = \\sum_{i=0}^{\\infty} \\exp(-\\lambda_i t) \\phi_i^2(x).\n", "f7336bcee4104dd9b2d2e4f1746a68e9": "F[x,y,z]=\\frac{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}", "f7338d9dade2d7e43bbd7b2874806563": "\\bar{\\Gamma}(\\tau)", "f733c4d333263e5651037e797c76cf02": "\\scriptstyle \\text{curry}(f) ", "f733dc8db75662168728bfaaec0f923d": "g_{B}=\\Delta g_{AB} g_{A}", "f733e70c9993c77053c9a5770e908198": "\\binom{n}{\\lfloor n/2\\rfloor}", "f7342f87715473d337889563298d2c56": "\\, \\varphi_{tt}- \\varphi_{xx} + m^2 \\sin\\varphi = 0.", "f73434f5c360dd3509feaea74215f736": "\\tilde f(x_i) = V(x_i) f(x_i).", "f73458a680b42e6956c1bef6496526c1": "\\max(a,b) = -\\min(-a,-b)", "f735260942468ca115aa1bb4712333b3": " G_\\text{II,n} = e^{-(K_\\text{g}^\\prime/T \\Delta T)}\\, ", "f735479e7e94734fe6bee7494de30143": " \\frac{mn}{2} ", "f735b6a213d508e8c483190fcf261801": "GL(n,\\mathbb{R})", "f73687ababedb1f13f5fdc31521cf9c7": "J\\in F_{n-1}", "f736a97eff719abbe075e79a0a8ff4a8": "\n\\langle\\Delta\\xi^2\\rangle = \\langle\\Delta l_{\\rm bin}^2\\rangle / l_{\\rm bin}^2 \n\\approx \\left({m\\over M_{12}}\\right)^2 \\langle\\Delta l^2\\rangle/GM_{12}a \\approx {m\\over M_{12}} {G\\rho a\\over\\sigma}\n", "f736aa38f5cd1222ff2f15ccca0186f2": "\\Bbb C", "f736d73ff4c4247b3e81c66f5eb2bebe": "\\theta \\equiv \\frac{\\pi\\, \\xi}{K(m)}\\, \\frac{\\xi}{\\Delta} = 2\\, \\pi\\, \\frac{\\xi}{\\lambda}.", "f736dbdc1d7b4c97ba0cd19a59f8c855": "0 \\le \\theta \\le \\cfrac{\\pi}{3}", "f736e19ddda3472795335d64bf72f8cb": "\n1 + b_h - \\mathbf{b_q} = (1 + b_h - b_1), \\,\\dots, \\,(1 + b_h - b_j), \\,\\dots, \\,(1 + b_h - b_q),\n", "f7370952e31bccc6e9c83a6a2202fe4b": "\\rho, \\eta, \\xi", "f73712ba1d623f7ba2ac62f943898e4e": "x^2 \\frac{d^2 f}{dx^2} + x \\frac{df}{dx} + (x^2 - \\alpha^2)f = 0", "f73718e6c8afe3440eab97dae378f9b6": "\\Delta U = 4 \\pi \\, \\mu", "f73795084e96a9a1aad06a8b180ca3d1": "\n\\begin{align}\n \\omega^2 &=\\, g\\, k\\, \\tanh\\, (kh), \\\\\n c_p &=\\, \\frac{\\omega}{k} \\quad \\text{and} \\\\\n c_g &=\\, \\frac12\\, c_p\\, \\left[ 1\\, +\\, kh\\, \\frac{1 - \\tanh^2 (kh)}{\\tanh\\, (kh)} \\right]\n\\end{align}\n", "f737f00da90778741b58bbfeff90d1d4": "W=W_{0}e^{-\\frac{\\omega t}{Q}}\\equiv W_{0}e^{-\\frac{t}{\\tau }}", "f73856bc945a1b75994be1f1aaad54c6": "P\\left(\\frac{\\partial}{\\partial x_1}, \\cdots, \\frac{\\partial}{\\partial x_{\\ell}}\\right)u(\\mathbf{x})=\\delta(\\mathbf{x}),", "f738b46072ab1211ac43e46b77eb9c7c": " S = \\sum_{i=1}^{n} W_{ii}{r_i}^2,\\qquad W_{ii}=\\frac{1}{{\\sigma_i}^2} ", "f738b9cebaa004cba55ca6d0cfcfadb3": "A(x,t)=A_o \\cos (k x - \\omega t+\\varphi)", "f738e07d253913988074855bb29a3b89": " S_v = \n\\begin{bmatrix}\nv_x & 0 & 0 \\\\\n0 & v_y & 0 \\\\\n0 & 0 & v_z \\\\\n\\end{bmatrix}.\n", "f738e2856748e787def57b361f9eaf73": "\\int_{t_1}^\\infty E(t)\\, dt = 1-\\int_{0}^{t_1} E(t)\\, dt", "f738f06de6ffcd04e465baa0dabaa193": "\\frac{2\\,(\\beta-\\alpha)\\sqrt{\\alpha+\\beta+1}}{(\\alpha+\\beta+2)\\sqrt{\\alpha\\beta}}", "f738f36fbe4fc2757277378cbedea8b9": "\\textstyle L^p(\\Omega)", "f7391fdee37a0e79580b9cae9ccaa8a8": " [ X^{r_1} Y^{s_1} \\ldots X^{r_n} Y^{s_n} ] = [ \\underbrace{X,[X,\\ldots[X}_{r_1} ,[ \\underbrace{Y,[Y,\\ldots[Y}_{s_1} ,\\,\\ldots\\, [ \\underbrace{X,[X,\\ldots[X}_{r_n} ,[ \\underbrace{Y,[Y,\\ldots Y}_{s_n} ]]\\ldots]].", "f7393436b850f9ffb09a7cbe2ca1e509": " \\frac {dR}{dz}= \\frac {2iw}{c_0 Q} K - \\gamma (i-K^2) \\quad (2.8.d)", "f7394f093368a5c804aac9754b4d5659": "\\sim 0.001M_\\odot", "f7397d7856dd9558147d421b74741960": "\\mathrm{Nu}_x\\ = \\mathrm{St}\\, \\mathrm{Re}_x\\, \\mathrm{Pr} = 0.0296\\, \\mathrm{Re}_x^{4/5}\\, \\mathrm{Pr}^{1/3}, (0.6< \\mathrm{Pr} < 60) ", "f7399c1a07f7e6863f7f4f11e40f0655": "(x \\times 2x) + (x \\times 3)", "f7399cf440e4873eab272482b710d776": "z_{n}", "f739ca689dd344710829790afde1cec0": "f_0(1300)", "f73a594636c9d7c7c517d4dd97d1363e": "{1-r\\over (1+r)^3} \\le |f^\\prime(z)| \\le {1+r\\over (1-r)^3}", "f73a61d1477114c5d56865f1eabd4a75": "\\neg\\Diamond \\overline{\\pi_1}", "f73ab28d495480d967492cc47c686b39": "\\mathbf{\\hat{\\boldsymbol{\\beta}}}", "f73ac74a6b246d0d1bbc5b3c9b415672": "\\kappa=(\\kappa_1,\\kappa_2,\\ldots,\\kappa_m)", "f73adb80fe190182faa0b62eec2f91e2": "\\delta (n)", "f73b101e1652392694b16f962d2a8877": " d_A = \\frac{\\Delta \\chi}{\\Delta \\theta}", "f73b119eec591674fdd5ddb91dea23df": " k_f[E][S]", "f73b21801f328b3afc1ae596eeb893f2": " k_b T", "f73b3d8e08ac5d60936646532b8c8fa9": "V_t > 0", "f73b427bcee9e391abec6a19fc685c7e": "\\scriptstyle \\{\\,x\\,\\mid\\, x \\in I \\,\\}", "f73b47c2c71b1c70ad274db89c04fe99": "\\sigma'(k) = \\sigma(k)", "f73b7fc941e92c372ad6c3516cdba0ce": "\\Delta m^2 \\sim 1\\,\\text{eV}^2", "f73b91059f62af22bf39989adc1845b8": "d_p", "f73bb7034340bd9fc0599045e51810df": "P^S_{liquid}", "f73be6c366b5d2623c6d7a26f3efb5a0": "(n,m)=1", "f73c2713e051f480bd3f4fcb82f28ace": "f(x) = -e-y Ei(y)", "f73c4f9bb2bd18863e3d84db0a96e569": "r(G)", "f73c79e3efc1d2a20078529fd8d867df": "\\delta(x+c)=\\delta(x)+c, \\text{ for all } c\\in \\Bbb{R},", "f73cc7834b206794507a3bb111da0b6a": "\\{Y_k\\}_{k\\in\\N}", "f73d1a1f3721e3c4554a703175e32852": "x^2-1 = 0 \\Leftrightarrow (x+1)(x-1) = 0", "f73d37fc1d0f4eb8d793eee58f6aca57": "D_{50} ", "f73d943df27013bee1ca2d150e6add67": "N(M)=L(M')", "f73d9bef4bcb8df4abea32e30f781548": "A = A^{\\top}.", "f73e8b81af7b8fc0fba6c8e241610690": "(A,\\ast) ", "f73ea9eae6f8ebceac00366d4e36fe16": "\\frac{\\| \\mathrm{D} u_n \\|_{\\infty}}{\\| u_n \\|_\\infty} = n. ", "f73f440b39ba065af20da4df763e7fec": "G \\in \\mathcal{G}", "f73f4968a2005267a323020e3635f82c": "P\\left(Spam|w_{0}\\wedge\\cdots\\wedge w_{N-1}\\right)", "f7402c68fc52e07ee9e8ee2fa5c0f7d1": "_{interval}\\alpha", "f74030bbe2c559c97279d6ef5aa3735b": "\\mathbb{P}(B(k))=\\mathbb{P}(L(k)=1)=\\mathbb{P}(H(k))=\\mathbb{P}(L(k)=k)=\\tfrac1k.", "f74084cba9a91f4914f8232388caf0df": "\\begin{array} {l}\nf^{(2)}(x_0)=\n\\frac{f\\left(x_0 + h\\right) + f\\left(x_0 - h\\right) - 2f(x_0)}{h^2} + O\\left(h^2\\right)\n\\end{array}", "f740eb3c430e48a27c700068a172cae0": "\\widehat{Y_k}= \\sum_{j=1}^n c_{j,k} Y_j ,", "f74115260830faf5178589e98c061a4e": "S^n", "f7413a12874aba4d120fdbcad0138086": "\\int_{-\\infty}^{\\infty} |\\psi (t)|^2\\, dt = 1", "f74161da08ec901c9b9f06a3f7a2af32": "\\begin{align}\n\\boldsymbol{p}_i&=\\frac{1}{d_i}(\\boldsymbol{v}_i-b_i\\boldsymbol{p}_{i-1})\\text{,}\\\\\n\\zeta_i&=-c_i\\zeta_{i-1}\\text{.}\n\\end{align}", "f7416418ffc24ef96b38b08bbe328381": "\\gamma = (1-\\beta^2)^{-1/2}", "f741b677014ba72938c3674665eea0ca": "Q\\mathbf{x} \\cdot Q\\mathbf{y} = \\mathbf{x} \\cdot \\mathbf{y}", "f741b7e863ca87ce695906b78619b334": " k = e^{-\\frac{\\Delta G_F}{k_BT}} ", "f741e214634b335a88bf49b89c23c050": " |AC|=|DG| ", "f7420dd3afe00d9d5a5ceac03ba34f91": "\\operatorname{logit}(\\mathbb{E}[Y_n]) = \\boldsymbol\\beta \\cdot \\mathbf{s_n}", "f742a33d63567c49ef34ba4c053a4a09": "(\\Omega, \\mathcal F)", "f7439759f2dc5762c275384dffdcdddb": "F_k^{(n)}=\\left[ \\frac{r^{k-1} (r-1)}{(n+1)r-2n}\\right]", "f743b5f6db70efc50fe6b1ec8bde8e78": "f(x_1, x_2, \\dots; a_1, a_2, \\dots)", "f743d9bbe5d8bbfd02217f5e0a81e753": "ab\\quad", "f743ef9357eeb73ab242ce6c0e343884": "\\textstyle \\mathbf{c}_j", "f7440f988fc4a7558bb9597c36058def": " \\sum_{i\\in I} \\nu_i r_i = 0", "f744187ba084801efbde4295f3d32bb5": "\\textbf{E} + j \\omega \\mu \\textbf{A} = - \\nabla \\Phi ", "f74423eeb3d4b7533cadd2bf3c810e6a": "\\ \\mu^2", "f7444a9b8fd10fd258ee0033632f295c": "\\begin{align}\n \\sum_{k=0}^n (-1)^{k}k e_k(x_1,\\ldots,x_n) t^k \n &= t \\sum_{i=1}^n \\left((-x_i) \\prod\\nolimits_{j\\neq i} (1- x_jt)\\right)\\\\\n &= -\\left(\\sum_{i=1}^n \\frac{x_it}{1-x_it}\\right) \\prod\\nolimits_{j=1}^n (1- x_jt)\\\\\n &= -\\left(\\sum_{i=1}^n \\sum_{j=1}^\\infty(x_it)^j\\right) \\left(\\sum_{\\ell=0}^n (-1)^\\ell e_\\ell(x_1,\\ldots,x_n) t^\\ell\\right)\\\\\n &= \\left(\\sum_{j=1}^\\infty p_j(x_1,\\ldots,x_n)t^j\\right) \\left(\\sum_{\\ell=0}^n (-1)^{\\ell-1} e_\\ell(x_1,\\ldots,x_n) t^\\ell\\right),\\\\\n\\end{align}", "f7444cb8ba9b31083edc30e95b9bfc46": "\\exp \\colon M_n(\\mathbb R) \\to \\mathrm{GL}(n,\\mathbb R)", "f7445ff4626213298a94f71664091186": "\\text{Tailwind} = \\cos[30^\\circ] \\cdot 15 \\mathsf{knots} \\approx 13 \\mathsf{knots} ", "f7446918199abd7cc170d58e2cc4a48d": " s = (\\ldots,(s_i, t_{ei}),\\ldots) ", "f74476e222875c5ba65715571696be38": "f : X \\to (Y \\to Z)", "f7448669f61168bc891c8d68a6b1906a": " B_t := (W_t|W_1=0),\\;t \\in [0,1] ", "f744c42a85b2bb04830abd30f62a82fd": "(\\forall x)\\phi(x,f_e(x))", "f74502981e562a6ef07029a244b0dca5": "\\Delta H=\\frac{1}{2\\rho}\\frac{\\partial_r}{\\partial z^a}(\\rho\\omega^{ij}(z)\\frac{\\partial_l H}{\\partial z^j})", "f74566d973739a8a29b9a082111d327d": " \\frac{P(x)}{(1-x)(1-x^2)\\cdots(1-x^n)} ", "f745849e6ca732878c45bf6974a0b944": "\\textstyle a_N", "f745e2cf0c5315a8eb4e169cf4b4cb20": "gfg^{-1}(z) = kz", "f747199e01357b2daaac0545d23e71b3": "b_n = a_n - c_1 a_{n-1} - c_2 a_{n-2} - \\cdots - c_d a_{n-d}", "f74783f8e3e031f92bc707b12d3cbd51": "~A", "f7479699a113ec252ecb48426658ea1b": "\\begin{align}\n\\mathbf{P} \\left[ \\sup_{0 \\leq t \\leq T} B_{t} \\geq C \\right] & = \\mathbf{P} \\left[ \\sup_{0 \\leq t \\leq T} \\exp ( \\lambda B_{t} ) \\geq \\exp ( \\lambda C ) \\right] \\\\\n& \\leq \\frac{\\mathbf{E} \\left[ \\exp (\\lambda B_{T}) \\right ]}{\\exp (\\lambda C)} \\\\\n& = \\exp \\left( \\tfrac{1}{2}\\lambda^{2}T - \\lambda C \\right) && \\mathbf{E} \\left[ \\exp (\\lambda B_{t}) \\right] = \\exp \\left( \\tfrac{1}{2}\\lambda^{2} t \\right)\n\\end{align}", "f747b828298f98c1dfc71e2abd35ff35": "T:R^n \\to R^n", "f747be8c051331d1d7a1ccad559194a0": "\n\\exp(\\Omega) = \n\\begin{bmatrix}\n1 - 2s^2 + 2x^2 s^2 & 2xy s^2 - 2z sc & 2xz s^2 + 2y sc\\\\\n2xy s^2 + 2z sc & 1 - 2s^2 + 2y^2 s^2 & 2yz s^2 - 2x sc\\\\\n2xz s^2 - 2y sc & 2yz s^2 + 2x sc & 1 - 2s^2 + 2z^2 s^2 \n\\end{bmatrix}\n.", "f747ee67c91a572a6d662e12c0ff985c": "H_{\\mathrm{sos}}", "f7483dcf0b8d408fe7b5a0ae05c40989": "x\\left(\\tfrac{x^{j}}{(1-x)^{j+1}}\\right) \\left(\\tfrac{x^{k-j}}{(1-x)^{k-j+1}}\\right) = \\tfrac{x^{k+1}}{(1-x)^{k+2}}", "f748a61b8d76a8162a1969a670b59126": "\\alpha > \\gamma", "f749598cdeaa8db24bc811514194afb1": "M_0, M_1, M_2,", "f7496f7be9e7c875dfc83fec9c6f709f": "\\nu \\equiv {\\mu \\omega \\over 2 \\hbar}", "f749c50ceef27514d5ed854b29b679d6": "\\partial n_2/ \\partial t = D_{12} \\Delta n_1", "f749cc0b2e183944cc5eae417cac2a91": "\\frac{\\partial r_i}{\\partial \\beta_j}=-X_{ij}.", "f74a2c14b3e9bdf8c55359b6f5d0b1a2": "\\lim_{n \\rightarrow \\infty} \\sum_{i=0}^n \\frac{t^im_i}{i!}", "f74a5b6356332bbcf0bedc3675bad0ba": "\n\\int x^m \\left(a+b\\,x^n\\right)^p dx =\n \\frac{x^{m+1} \\left(a+b\\,x^n\\right)^p}{m+1}\\,-\\,\n \\frac{b\\,n\\,p}{m+1}\\int x^{m+n} \\left(a+b\\,x^n\\right)^{p-1}dx\n", "f74a72395f99b454beb9a945c47a7124": "-\\frac{\\det A_Q}{\\det A_{33}}", "f74a8f09aa123fa9faa526998dcc6145": "(\\log n)^i", "f74ac674f9791403ed77a441551816dc": " W(x) = \\prod_{i=0}^n (x-x_i) ", "f74add691f94079afce2836f0fd51b45": " \\left| \\frac{d \\log \\left( a(t) \\right)}{dt} \\right| \\ll \\omega \\ ", "f74b544c5d5b757137ec67d1aab1196a": "(*) \\quad \\frac{\\partial \\vec u}{\\partial t}\n + \\sum_{j=1}^d \\frac{\\partial}{\\partial x_j}\n \\vec {f^j} (\\vec u) = 0,\n", "f74b8a39e533b72a32349c7a2ef126b8": "\\{X_k\\}_{k\\in\\N}", "f74b9a3f475b1eecf5e470520f53a7ba": " Q(av) = a^2 Q(v). ", "f74bade30344abb7dacb3f937db6a27f": "u^2+v^2-v=0", "f74bcc7fbb7916a6ef05bbc2855aa504": "\\,\\! f(I)= \\frac{I} {C_\\mathrm{m} V_\\mathrm{th} + t_\\mathrm{ref} I}", "f74be5e284ade04bc6aae27bdb4a8c86": "D_n(z)", "f74c34316764d455893647e31373880a": "0_S^{\\mathcal V}=0\\in|\\mathcal V|_S", "f74c3a6394fa9afed7eb4061927d585f": "\\chi=V-E+F.\\ ", "f74cb3ebd33c641e29757d7d0b52addd": " \\tau^* = \\inf\\{ t>0: Y_t\\notin D\\} ", "f74cdadce81ee23ff45e74db5fec0ad9": "\\mathbf{b}_{i,n}(t) = {n\\choose i} t^i (1 - t)^{n - i},\\quad i = 0, \\ldots, n", "f74cf109ab73be65015b608837754238": "\\textstyle k_f>0", "f74d06b098b0f47178423940b2da6c22": "\\frac{\\partial f}{\\partial y}(x,y) = x + 2y.\\,", "f74d171519d721b0be129fcd8d700eb8": "SU(2)_L \\times SU(2)_R", "f74d2e805d9c327e655cbb00181e290a": "R(T)", "f74d809e5ba2dbe7e5b20d0ccd376582": "c_3=\\frac i{2(k^2+4ik-5)}", "f74d87a107a941e0aac86b7ff17f15a0": "(10\\uparrow\\uparrow)^{3} (2.8\\times 10^{12})", "f74d9283768c9f9a670c4431de43f75e": "\\ \\chi, \\ \\delta, \\ \\rho, \\ \\omega", "f74dc81dd9e41b72976adb9ddf3dc500": "\\left(1-\\frac{1}{2^s}\\right)\\zeta(s) = 1+\\frac{1}{3^s}+\\frac{1}{5^s}+\\frac{1}{7^s}+\\frac{1}{9^s}+\\frac{1}{11^s}+\\frac{1}{13^s}+ \\ldots ", "f74de4cd2b226062e174e7a182c6d74a": "\\frac{1}{4} \\sum_{\\mathrm{spins}} |\\mathcal{M}|^2 \\,", "f74e11de0be5cf422730b931a4233626": "[1;2,3,1]=[1;2,4]=1+\\cfrac1{2+\\frac14}=\\frac{13}{9}.", "f74e16d90c443638e3c48c969f92ed94": "\\partial_{\\alpha} A^{\\alpha} = 0", "f74e3a04ae26d9ac7b3145c4122d8bde": "\\gcd(a_m,a_n) = a_{\\gcd(m,n)}.", "f74f1121fc12f0cac65927cac32afdb1": "\\scriptstyle f(x) \\;=\\; b^x", "f74f8710fd31ce502365bc814a7fd3b6": "[0, \\pi]", "f74f976ec19d154388ca35786b907c66": "s(t) = \\int_{-\\infty}^{\\infty} S(f) \\cdot e^{i 2\\pi f t} df,", "f74fd0cfafe47783996761e2ade6a43b": "k ... m", "f7501ba0544e2009be191015b680a7e0": "\\frac{v}{c} \\equiv \\frac{1}{c} \\frac{dx}{dt} = \\frac{w}{c} \\frac{1}{\\sqrt{1 + (\\frac{w}{c})^2}} = \\tanh[\\eta] \\equiv \\frac{e^{2\\eta} - 1} {e^{2\\eta} + 1}= \\pm \\sqrt{1 - \\left(\\frac{1}{\\gamma}\\right)^2}\n", "f75094b2d220116687123147c69fe1f4": "\\{ x_k \\}_{k=1}^N", "f750b3612aaeb7ca96b023d5cbadf073": "\\rho(z)", "f750e0a797eedef801aaf50716fbd542": "\\vdots", "f750ffe54662e24536fa14c277e0dec0": "V = R \\cdot I", "f75108d53b7e45559b308e5502ffb961": " U = -\\mathbf{p} \\cdot \\mathbf{E}", "f7513e7fa565644a0ed5d72358ae3ece": "\\scriptstyle R_2", "f751715a563c1433e2c1f9289a335942": "e = \\sqrt{1 + \\frac{2EL^{2}}{k^{2}m}}", "f751885c9cc88ce9bcc845fd12b61fa6": " \\mathbf{J}_\\mathrm{d} = \\epsilon_0 \\left ( \\partial \\mathbf{E} / \\partial t \\right ) = \\partial \\mathbf{D} / \\partial t \\,\\!", "f751d185bfafe8437d30863b719703d5": "\\hat{\\Pi}_{\\rho\n_{X^{n}\\left( 1\\right) },\\delta}", "f751ef44e07bdb0b10d8183499b4f572": " \\alpha\\!\\left(\\sum_{i=1}^\\infty E_i\\right) = \\sum_{i=1}^\\infty \\alpha(E_i) ", "f751f7e1576c632f3154adafdf13de2c": "\\eta_3 = -\\, \\frac{H}{m}\\, \\frac{E(m)}{K(m)},", "f752008a78c8519e9f8f178faf0b87ed": "g_t", "f75213d306c46fb55e42c100d3cfbc54": "3u_{1}u_{2} - \\phi(x,u,u_{1},u_{2}) = 0 \\,", "f752566decbd7830c4396407800415dd": " x_0 < \\xi < x_0 + h", "f7526534851c91efdbb29c395f9974ba": " S_M = \\rho (v^2/2 - \\phi) ", "f752bf810c6a0655891eacc57e714cd4": "S=\\int dt\\, F(\\tilde{s}(t),\\tilde{\\mu}(t))", "f75323acbc0c7abb6ff1b605a65926e7": "\\gamma_\\mathrm \\Pi = 2 \\gamma_\\mathrm L = 2 \\sinh^{-1}{\\sqrt{\\frac{R_2}{2R_1}}} \\,", "f75330a97ad8d622342e96f3345b55a1": "(j)", "f75357d467f15df6a728505f9911d515": "\\Omega \\leq \\Pi \\Rightarrow \\frac{\\Omega}{\\Pi} \\leq 1", "f753d989e3192ffc40cffa37c7f6cf63": "\\begin{bmatrix} A & B & D \\\\ B & C & E \\\\ D & E & F \\\\ \\end{bmatrix}.", "f753ecb356fb80184a123d40dfdaa6cc": "\\pi_n \\sim U[-n,n]\\,\\!", "f7544866dad997da228ba4bb911af16d": "\\psi_{+}", "f7548a598efcbbec40b533bd66a8b22e": "c_{n}=\\left[\\begin{array}{cc}\n\\exp\\left(\\beta_{n}\\right) & r_{n,n+1}\\exp\\left(\\beta_{n}\\right)\\\\\nr_{n,n+1}\\exp\\left(-\\beta_{n}\\right) & \\exp\\left(-\\beta_{n}\\right)\\end{array}\\right]", "f754903439b564b9df33eb403e2bdf50": "I_{i}(t)", "f754f72a0f4bb41c766548e3a70e8b5d": "s_2 = E(h_2, K_2)", "f7555558afe0e8aa7cd5bb11167ea77b": "x, y \\in M", "f755be82935f81b597b01a0a5a7cdac5": "L(c)", "f755d1aa111e9c7157318779437520b6": "U=(x_2-x_1, x_3-x_2, ..., x_k-x_{k-1})", "f755d799aab2a442232cc7d64fe4400d": " c = 1-{6\\over m(m+1)} = 0,\\quad 1/2,\\quad 7/10,\\quad 4/5,\\quad 6/7,\\quad 25/28, \\ldots", "f75602d9b746d5db7d3c47864c4e52d5": "\\operatorname{erf}(x)\\; =\\; \\frac{1}{\\sqrt{\\pi}} \\int_{-x}^x e^{-t^2} \\, dt", "f7560f96fbaed6d59b7c1dc47ac5cbad": "\\lnot\\exist F^n \\rightarrow \\exist \\lnot F^n", "f75610c190ddc1b652fb50d02fb49081": "{\\mathbb R}^2", "f7566871c5ed8513d6685a3757463077": "0=\\delta\\int\\frac{ds}{dt}dt=\\delta\\int(KE+PE_g)dt", "f756f4b41d46eaa22c090b23be2ac8cd": "x,y,z \\in V", "f75748da5a6cd903962aa72fbc359c01": "(\\phi_n)_{n\\in\\mathbb{N}}", "f757578135d7a9fe7cca0ecfd8ebc5c4": "\\ q_1 = \\frac{a - \\left(\\frac{a - q_1 - \\frac{\\partial C_2 (q_2)}{\\partial q_2}}{2}\\right) - \\frac{\\partial C_1 (q_1)}{\\partial q_1}}{2}", "f757a976df4add225382bf0d5db6b053": "p \\to (q \\lor \\neg q)", "f757bfcf2477d6b223378bc9e252352e": "\\left(\\begin{array}{c} E(z_R) \\\\ F(z_R) \\end{array} \\right) =\n M\\cdot \\left(\\begin{array}{c} E(0) \\\\ F(0) \\end{array} \\right)", "f757e0173832862923efc7b127a3f8e7": " \\left [ \\nabla^{(i)} \\right ]_{\\alpha_1 \\alpha_2 \\cdots \\alpha_i} = \\frac {\\partial^{\\, i}} {\\partial r_{\\alpha_1} \\partial r_{\\alpha_2} \\cdots \\partial r_{\\alpha_i} } \\qquad \\qquad \\text{where} \\quad \\alpha_1, \\alpha_2, \\cdots, \\alpha_i = 1, 2, \\cdots , n \\ . ", "f75805a9b74e3b35b6aaeac55a23be28": " dX_t = (a_t-b X_t)\\,dt + \\sqrt{X_t}\\,c_t\\, dW_{1t}", "f75847518fb2bb42482014e971e95ead": "F \\otimes_k A", "f75861bba1ceb435c827da797156f1e6": " E=E_k +\\gamma_s=\\frac{1}{4}\\Delta\\rho VR^2\\omega^2+\\frac{2V}{R}\\sigma ", "f7587a445a2c78d4bb49c592b977027c": " \\sqrt{ \\frac{ \\Sigma (x - \\bar{x})^2 }{n} } = \\sqrt{ \\frac{ \\Sigma \\left[ x - \\left( \\Sigma x \\right) /n \\right] ^2}{n} } \\ , ", "f75885b8e7f672c4127d49a80c762335": "a^2 + b^2 = c^2\\!\\,", "f7588fb2629eee08e00e972aef3bf918": "\\tilde{K} \\ \\stackrel{\\mathrm{def}}{=}\\ K / K_{00}", "f758dea5c87a440d7e97da9dc81d6698": "\\frac{5}{12}<\\frac{11}{12}", "f758e942331ec4cbf56691b5b24a36e8": "I\\frac{d^2\\theta}{dt^2}=N ", "f759122be3b92bccd4e3f9b5555babaa": "C^{n+1}n!z^n", "f7592e91f64ae6e212a7f79adb6ef250": "e = \\lim_{n\\rightarrow \\infty}\\left(1+ {1 \\over n}\\right)^n ", "f75937a6086e3e78e359489cf6ce517d": "\\ v_g ", "f759674697ec7b9411ddb08ba3ca9459": "x \\neq \\varnothing", "f75969609ef833e0dd9284083e4c805e": "\\lambda_M(a,b)=(-1)^n\\lambda_M(b,a) \\in \\mathbb{Z}.", "f759d455d2f3ca292c4b0268132acd1c": "r_k\\,", "f75a155341e4e207de7a2fa51c13a454": "\\varepsilon _{2}\\Psi =\\frac{w^{2}+m_{2}^{2}-m_{1}^{2}}{2w}\\Psi", "f75a51d67178699a590f8e711aeb39a5": "\n \\frac{\\partial I_3}{\\partial \\boldsymbol{A}} = \\det(\\boldsymbol{A})~[\\boldsymbol{A}^{-1}]^T ~.\n", "f75a9192e62fee7628dd846b12f30979": " 4d ", "f75a9297e80ae88a187f5367fd0ff504": "\\displaystyle \\frac{pq}{4r^2}-\\frac{4R^2}{pq}=1", "f75aaa3840f72b04bf3612fc9dfdda7b": "f(x) = ax^2 |_{a=\\{0.1,0.3,1,3\\}} \\!", "f75ac8ddc3cf4c09a20cfee04464792a": "-\\alpha^2", "f75ad4a0659b8f40f8310e70baf2794a": "\nG^I = (d\\Phi- \\delta_{ab}I^a d E^b)^2 + \\Lambda\\, (\\xi_{ab} E^a I^b)\\left( \\delta_{cd} dE^c d I^d\\right)\\ ,\\quad\n\\delta_{ab}={\\rm diag}(1,\\ldots,1)\n", "f75adcd340918f39f05253e9c903bc23": "\\omega_-", "f75b51191800b8e96d37c1cb11b5d6ac": "\\ \\sum_{w \\in V} f_i(u,w) = 0.", "f75b628c259cb647e9d945c96719c968": " W_{PPT}=|C_{n^{*}l^{*}}|^{2}\\sqrt{\\frac{6}{\\pi}}f_{lm}E_{i}(2(2E_i)^{\\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}(1+\\gamma)^{2})^{|m/2|+3/4}A_{m}(\\omega, \\gamma)e^{-(2(2E_i)^{\\frac{3}{2}}/F)g(\\gamma)} ", "f75bee9dc59cae4ddc48edffd3052612": "\\langle c|c\\rangle=1", "f75bfe187540fb364adb72c12068b112": "\\forall a \\in A_i", "f75c18b45915565b778be6e5f54c7fcd": "\\psi_k \\propto e^{i\\bold{k}\\cdot\\bold{r}}", "f75c365830bcc52578e2f940e14c0906": " \\sin x =\n\\cfrac{x}{1 + \\cfrac{x^2}{2\\cdot3-x^2 +\n\\cfrac{2\\cdot3 x^2}{4\\cdot5-x^2 +\n\\cfrac{4\\cdot5 x^2}{6\\cdot7-x^2 + \\ddots}}}}.\n", "f75c8e22677356f79e4805b63b584f2b": " D(\\theta)=-2 \\log(p(y|\\theta))+C\\, ", "f75c918e7b18e291fd0922062b233088": "\\hat{x}_{n_j} \\in C(\\theta_{n_j})", "f75cbea34fca45e537d0b47f76e61562": "\\lim_{n \\to \\infty} (\\mathbf I - \\mathbf X^{-1} \\mathbf A)^n = 0 \\mathrm{~~or~~} \\lim_{n \\to \\infty} (\\mathbf I - \\mathbf A \\mathbf X^{-1})^n = 0", "f75cced923798b754e397ac081ec5fab": "\n\\Delta_D v = \\Delta v = \\sum_{i=1}^n \\frac{\\partial^2 }{\\partial x_i^2} v. \n", "f75d1a3ca80cbe7bee6d6880c084294f": "\\mathit{d}_H^{RC}(\\mathcal{D}) \\geq 3^{k - 1}", "f75d1c55473627357f31359d123d204d": "\\mathrm{rem}\\left(x_0,m_i\\right) = \\mathrm{rem}\\left(a_i,m_i\\right)", "f75d2117e4cefc1243229dcb4e94bb82": "\\rho_{2}(\\mathbf{r}^{\\prime})", "f75d6f9306e2ec84ce1da0be4c3d1c10": " f(x) = \\tfrac{1}{2} x^T \\left(\\begin{array}{cc}\n 1 & 0.5 \\\\ 0.5 & 1\n \\end{array}\n \\right)\n x -\\left(\\begin{array}{cc}\n 1.5 & 1.5\n \\end{array}\n \\right)x,\\quad x_0=\\left(\\begin{array}{ cc}\n 0 & 0\n \\end{array}\n \\right)^T ", "f75d774184921d334ed756d2c0cb8322": "\nI_{k} \\equiv \\int d\\zeta \\ \\frac{\\lambda(\\zeta)}{\\zeta^{k+1}}\n", "f75d8cc3c02dd0fb790b846342e6be84": "\\textstyle{\\frac {\\log(9p)} {\\log(3)}}", "f75d9db5fbe4680b7d18c03d40d78c04": "\\langle x \\rangle = \\int_{-\\infty}^{\\infty} x P_n(x)\\,\\mathrm{d}x.", "f75df9a08c237c892997cc50ffd378fd": "s \\in B^{32}\\,", "f75e9ccabc650ea6c44fa55b77cf1c30": "a_n < k^n < 1.", "f75eed80571b6aaee48bedb58f1ae746": "(\\mathbf{a}-\\mathbf{p}) - ((\\mathbf{a}-\\mathbf{p}) \\cdot \\mathbf{n})\\mathbf{n}", "f75ef7544498f247df440d828aee41aa": "n_{i}", "f75f81d6f76c8df2fb1da8238cb1515a": "-\\log_2 P(x)", "f75fba2364555ce0865a2f77f7c022ab": "p_i=1,\\ a=\\frac{1}{2},", "f75fe62c548c30c45d251acb5769cbf1": "\\hat e_1\\ ,\\ \\hat e_2", "f75febb95c24a6c70a1dbee594d51367": "[A] \\ \\mbox{vs.} \\ t", "f76052816bf0540eead53ba1ea98d7a2": "\\alpha^6-4\\alpha^4+16\\alpha^2-32=0, \\,", "f7613c32f85157dfb7f7fd29b743fc71": "k=\\lambda_0 >\\lambda_1\\geq \\dots\\geq\\lambda_{n-1}", "f7614b3d7dba056c1d8224d71edc591b": "\n\\int x^m \\left(A+B\\,x^n\\right) \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^pdx=\n \\frac{B\\,x^{m-n+1}\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^{p+1}}{c (m+n (2 p+1)+1)}\\,-\\,\n \\frac{1}{c (m+n (2 p+1)+1)}\\,\\cdot\n", "f7616a4772784ec76769240ce2f4aa94": " \\mathbf{F} = \\frac{d\\mathbf{p}}{dt} ", "f76181d80de89c5736a21709e19caa61": "P_l", "f761b673656110a65b71cd58b552e132": "\\hat{A} = \\begin{pmatrix}\nA_{11} & A_{12} & \\cdots & A_{1n} \\\\\nA_{21} & A_{22} & \\cdots & A_{2n} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nA_{n1} & A_{n2} & \\cdots & A_{nn} \\\\\n\\end{pmatrix}\n", "f761f0b899d24510a1cf9ee22c66f736": " f_{Load} = \\frac{\\text{Average load}}{ \\text{Maximum load in given time period}}", "f762110a7ca2018f1222fb8a6dae68e5": "W_{DM}(t)=-q/(2m)(\\mathbf{L}+2\\mathbf{S})\\cdot\\mathbf{B}(\\mathbf{R},t)", "f7621d22f2422016fbaba15e634ea980": " |x| = \\sgn(x) \\cdot x ", "f76253b68795c252816968a5af69db90": "S(q = 2 k \\pi/a) = N", "f7625d9c1e30e6300cece07a80ad3c02": "(1+a)x^2+(1-a)y^2=2,", "f762c0450109440977321eb23864ed41": "\\left\\vert N \\right\\vert = \\kappa \\,.", "f762d8bbfeedbf4864133201532e57da": "\\operatorname{graph}\\, f^{-1} = \\{(y, x) \\mid y = f(x) \\}", "f7633c6f9279e45d24542d2ff9900c8f": "\\Delta g_{12}", "f763425db9fa48513bcf733040bc99e7": "\\frac{1}{2} iu = \\frac{\\partial \\psi}{\\partial \\bar{z}}", "f76394511c20f309763f95dcaf5990ae": "Lw = Ln + Ld + CEi", "f763a01e4f84b07d4baec6c3c0b43d7a": "\\liminf_{n\\rightarrow\\infty}V(u_n,\\Omega) = \\liminf_{n\\rightarrow\\infty} \\int_\\Omega u_n(x)\\,\\mathrm{div}\\, \\boldsymbol{\\phi}\\, \\mathrm{d}x \\geq \\int_\\Omega \\lim_{n\\rightarrow\\infty} u_n(x)\\,\\mathrm{div}\\, \\boldsymbol{\\phi}\\, \\mathrm{d}x = \\int_\\Omega u(x)\\,\\mathrm{div}\\boldsymbol{\\phi}\\, \\mathrm{d}x \\qquad\\forall\\boldsymbol{\\phi}\\in C_c^1(\\Omega,\\mathbb{R}^n),\\quad\\Vert\\boldsymbol{\\phi}\\Vert_{L^\\infty(\\Omega)}\\leq 1 ", "f763aea795086742adc89b3da3887071": " x \\mapsto f(x) ", "f763c9fe24bdb7e08d52109f427afdf4": " M^{-1} = \\frac{1}{ad-cb} \n\\begin{pmatrix} \nd & -b \\\\\n-c & a\n\\end{pmatrix}\n", "f763cbebc91f499d3f75b86a2ebe8d87": "Y = \\sqrt{\\sum_{i=1}^k \\left(\\frac{X_i-\\mu_i}{\\sigma_i}\\right)^2}", "f764228c410be9fe2dc5faf092de7899": "\\phi (t) = Q(t)e^{tR}\\text{ for all }t \\in \\mathbb{R}.\\ ", "f76433ff474215a88dee7659df91a5f8": "\\partial f(x)/\\partial x_{i}>0", "f76446df95e47b41fa228b4440b6b5d3": " {\\dot{m}}_S ", "f7648a16169dd426051934d83176bdc1": " H^k (V) ", "f764b9ab32808969f0c96c42fc1140c1": " \\binom p k = \\frac{p \\cdot (p-1) \\cdots (p-k+1)}{k \\cdot (k-1) \\cdots 1} ", "f764d10092944374604cc6b42fe618ac": " |\\Delta\\mathbf{r}_i^\\perp|^2 = (-[S]^2(\\Delta\\mathbf{r}_i)) \\cdot (-[S]^2(\\Delta\\mathbf{r}_i)) = -\\mathbf{S}\\cdot[\\Delta r_i][\\Delta r_i]\\mathbf{S}.", "f765170d35754e4a4d92ccdd481571a8": "\n\\frac{\\partial \\Sigma}{\\partial \\theta_m}\n=\n\\begin{bmatrix}\n \\frac{\\partial \\Sigma_{1,1}}{\\partial \\theta_m} &\n \\frac{\\partial \\Sigma_{1,2}}{\\partial \\theta_m} &\n \\cdots &\n \\frac{\\partial \\Sigma_{1,N}}{\\partial \\theta_m} \\\\ \\\\\n \\frac{\\partial \\Sigma_{2,1}}{\\partial \\theta_m} &\n \\frac{\\partial \\Sigma_{2,2}}{\\partial \\theta_m} &\n \\cdots &\n \\frac{\\partial \\Sigma_{2,N}}{\\partial \\theta_m} \\\\ \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\\\\n \\frac{\\partial \\Sigma_{N,1}}{\\partial \\theta_m} &\n \\frac{\\partial \\Sigma_{N,2}}{\\partial \\theta_m} &\n \\cdots &\n \\frac{\\partial \\Sigma_{N,N}}{\\partial \\theta_m}\n\\end{bmatrix}.\n", "f7653c5b6f4ebf9919f4daa2acec1f6d": "m_{1,2}(A) = (m_1 \\oplus m_2) (A) = \\frac {1}{1 - K} \\sum_{B \\cap C = A \\ne \\varnothing} m_1(B) m_2(C) \\,\\!", "f76554960222b91a6cf5551903e1ced8": " |i \\rang \\otimes |\\epsilon\\rang ", "f765831f9a70e7e12b9178754d3d373a": "x\\in\nX\\subseteq \\mathbb{R}^{L}", "f765bac70eef9f58081da31411aa4429": "(r_j-r_f)_t = a_j^+D + \\beta_j^+(r_m^+-r_f )_tD + a_j^-(1-D) + \\beta_j^-(r_m^--r_f )_t(1-D) + \\epsilon_t", "f765e725c6d8063b8d0beb97d3b03429": "P(X_{n+1}=1 \\mid X_1+\\cdots+X_n=s)={s+1 \\over n+2}.", "f76639267b640f501a7b2062ce3d811e": "p,\\, f(p),\\, f(f(p)),\\, f^3(p),\\, f^4(p),\\, \\ldots", "f7664f27c9ef14dac908e84d3f8971f3": "m_{b}", "f76675838fe01b2a1d0ed1c9975902f7": "\\scriptstyle \\mathbf{n}_{B}", "f7667fdda05841a548f7c5ee072fd6f4": "\np(\\mbox{weight} | \\mbox{female}) = 1.6789e-2\n", "f766a5c972228c2e1c00d1a31ed4ce21": "2^{\\aleph_0}=\\aleph_1.", "f766efe80ec00fd459c9340ac0d71879": "(\\Phi(0) \\land \\forall i (\\Phi(i) \\to \\Phi(i+1))) \\to \\forall i \\Phi(i)\\,\\!", "f76703ca70b38b354a4339e487e86c06": "Q_p \\psi = -i\\hbar \\partial_x \\psi", "f76716b374d540db7c9963f5b44a3b84": "Q_{\\mathrm{Hur}}", "f7672409c54ca28a9fcfeeac9d7b1e5d": " \\and ((D[n] = [F_2, S_2, A_2]::[F_1, S_1, A_1]::R ", "f767bd2576029004016a4ffcef8bedb1": " ds^2 = dr^2 + \\frac{r^2}{4} \\sigma_3^2 + \\frac{r^2}{4} (\\sigma_1^2 + \\sigma_2^2) ", "f767ddab5fffa1d11981139df0e2d7eb": " \\mathbf{\\alpha}+\\frac{n}{2},\\, \\mathbf{\\beta} + \\frac{\\sum_{i=1}^n{(x_i-\\mu)^2}}{2} ", "f767f7ff1bfd13302e8c350ada9a4093": "\n \\mathbf{b}^k = \\frac{\\partial q^k}{\\partial x_i}~\\mathbf{e}^i\n", "f7685440a23aae155681e9952d172232": "\n\\begin{bmatrix}\ne_x \\\\ e_y \\\\ e_z \\\\ e_t\n\\end{bmatrix}\n\\begin{bmatrix}\ne_x & e_y & e_z & e_t\n\\end{bmatrix} = \nA^{-1}\n\\begin{bmatrix}\ne_1 \\\\ e_2 \\\\ e_3 \\\\ e_4\n\\end{bmatrix}\n\\begin{bmatrix}\ne_1 & e_2 & e_3 & e_4\n\\end{bmatrix}\\left (A^{-1} \\right )^T \\ (4)\n", "f7685bbdcc22b9882af2b79850a2220f": "x\\in\\mathbb F^m", "f768bea03fc513a2f201f47ff40682a9": "\\lim_{n \\to H} x_n= {\\rm st}(x_H),", "f768eda1eff19f3f712309923c29178f": " H ", "f7692584f96590b0f47ca14663e302bb": "W \\longrightarrow X^{(n)}(W) \\longrightarrow Y^{(n)} \\longrightarrow \\hat{W}", "f769377c0c8a5288c0b9a6ae6dfe7078": " \\tilde\\beta ", "f76976b978e35e5f3ef13073d69a1593": "\\scriptstyle (3.0 \\;<\\; |\\eta| \\;<\\; 5.0)", "f769ac11dc0431dd1c242b7aea567c54": "p = p^{\\star}_{\\rm A} x_{\\rm A} + p^{\\star}_{\\rm B} x_{\\rm B}.", "f769ae1783984c6560db0d7ade96b4be": "A_1,A_2,\\dots", "f76a00bf8602c8e1b86de7a6ab5235a6": "\\begin{align}\n A(a) &= S_z \\otimes I\\\\\n A(a') &= S_x \\otimes I\\\\\n B(b) &= -\\frac{1}{\\sqrt{2}} \\ I \\otimes (S_z + S_x)\\\\\n B(b') &= \\frac{1}{\\sqrt{2}} \\ I \\otimes (S_z - S_x)\n\\end{align}", "f76a54a67e6588634a0e6aedaef18a22": "\\mathcal{E}^0", "f76a9e343172e334d1925dfaba4e0d38": " \\Delta \\theta = \\theta_{2} - \\theta_{1}, \\!", "f76af7fbcdf7e1141fd1457b87b4e115": "(4, 3, 1, 1)", "f76bdc4cf97cc28dd054614e61eb60a5": "t(x) = \\frac {x - a} {x - b} .", "f76c3b5d3ce5cc44ef69ddd3ac370849": "\\mathbf{F}\\left(\\mathbf{r}\\right) = \\mathbf{F}_t\\left(\\mathbf{r}\\right)+\\mathbf{F}_l\\left(\\mathbf{r}\\right)", "f76c3c43899835bd4c99a2688ea334db": "y = r \\sin(\\lambda) / \\cosh(m (\\lambda-\\lambda_0)),\\,", "f76c45d08ba8e11510e3e422842eebe2": "\n\\sum_{if(y)", "f785a4d6b1f08ff46c0ac34450d3b9ac": "m_2 =13", "f78647394f4f557e738484e9775ac46b": "S_{ab}, \\, E_{abcd}", "f786d7c9e9955d91a7d643ab18bd0a3e": "B^{13}=SU(5)/Sp(2)\\cdot\\Bbb S^1", "f786f870073c6045791240ede196f020": "\n\\begin{array} {ll}\n\\Delta E= & \\frac{1}{2}\\alpha_0\\left(T-T_0\\right)\\left(P_x^2+P_y^2+P_z^2\\right)+\n\\frac{1}{4}\\alpha_{11}\\left(P_x^4+P_y^4+P_z^4\\right)\\\\\n& +\\frac{1}{2}\\alpha_{12}\\left(P_x^2 P_y^2+P_y^2 P_z^2+P_z^2P_x^2\\right)\\\\\n& +\\frac{1}{6}\\alpha_{111}\\left(P_x^6+P_y^6+P_z^6\\right)\\\\\n& +\\frac{1}{2}\\alpha_{112}\\left[P_x^4\\left(P_y^2+P_z^2\\right)\n+P_y^4\\left(P_x^2+P_z^2\\right)+P_z^4\\left(P_x^2+P_y^2\\right)\\right]\\\\\n& +\\frac{1}{2}\\alpha_{123}P_x^2P_y^2P_z^2\n\\end{array}\n", "f78753fbe42c03291209001721891ce0": "\\varphi_i\\Vdash\\varphi_i", "f78759de52f19869d4de79d3a471e422": "L(x) = L_1(x) \\otimes L_2(x),", "f787f7f88389319c17e5b6f7d6070d94": "f: U\\to V", "f787f8cf06df373537462ad9cd7b195e": "p: U \\to P_{1, k}", "f78805d124615c098c2a9cb3d8d05450": "\\mathbf{e}^2 = -\\frac{1}{\\sqrt{2}}\\mathbf{e}_1+2\\mathbf{e}_2.", "f7887df767aa2c64cb5a57612b5c711f": "\\alpha_{x} = f \\cdot m_{x}", "f7892cc1436e36679cb3a867a574bc8a": "\\scriptstyle{\\tilde\\theta}", "f789ad61603c1f63e1d01ffc086094d0": "\nFM = \\sqrt{ \\frac {TP}{TP+FP} \\cdot \\frac{TP}{TP+FN} }\n", "f789b58f9fdc378eb6d2a8e0c8e06786": "\\Gamma R = 1-\\frac{6* SIGMA D 2}{n^3-n}", "f789e314a81d38fa5a4b4f253fc2623c": "c = E_k(m)", "f78a062201ee95a57c6d9c12b577de1a": "\\mathbf x_1=\\mathbf x_0+\\mathbf h_0", "f78a57ca094de2afc2c3413fb815f08b": "\\frac{15}{8}", "f78a6e9b9679506ddba1b930086538eb": "\\bar{\\partial}\\alpha=0.", "f78a8ed8e1261c7a87e31d32f4cabb41": "X_{nm}=X_{mn}^*", "f78b1c92f5447697a473402af4394767": "n = \\sum^{m-1}_{i=0} {n_i 2^i} = n_{m-1} 2^{m-1} + n_{m-2} 2^{m-2} + \\cdots + n_1 2 + n_0", "f78b3b4a1e95ced1d6f08c0b9192643a": "\\operatorname{if} = \\lambda p.\\lambda a.\\lambda b.p\\ a\\ b", "f78b928f7fbedc332b616a673da64c9c": "\\ H(z)=1-z^{-1}.", "f78ba1521caf92747ab5ee0b2a0b274d": "A \\le_m^P B", "f78c862efe825aa386d033b3aee536cf": " \\mu_0 \\ \\overset{\\underset{\\mathrm{def}}{}}{=}\\ 4 \\pi \\times 10^{-7} \\ ", "f78c9cd55a1bc8d45c9d10ff93265132": " I_t = I_0 e^{-\\mu x} ", "f78d3cd72cd1896ab8cf3cb55974f0a3": " P\\left( \\bigcap_{i = 1 }^n \\frac{ | X_i - \\mu_i | }{ \\sigma_i } < k_i \\right) \\ge 1 - \\frac{ [ \\sqrt{ u } + \\sqrt{ n - 1 } \\sqrt{ n \\sum{ \\frac{ 1 }{ k_i^2 } - u } } ]^2 }{ n^2 } ", "f78d52d21d75166d0895b1fa9ae83c11": "!\\colon \\mathbb{N}^+ \\rightarrow \\mathbb{N}^+", "f78df22b518d6c6dcc69ff0cdb25f40c": "V((z))((w)), V((w))((z)),\\text{ and }V((w))((z-w)).", "f78df2ca6c823d6149e80abac0144763": "s(x)=s(y)", "f78e2bf3b8ac58ba7b7a667e95c5e017": "\\left|\\rho_S\\right| < 1\\,", "f78e4c8755aa77027cbf33a96b5ab219": "r_s = \\left(\\frac{3}{4\\pi n}\\right)^{1/3}\\,,", "f78ef7c5b62be5dca00cd65519de1e46": "CN_{II}", "f790090e6c0cf4266ab9995103304ec6": "T_{L}=T_{1}+T_{2} = \\frac{L_{L} / \\gamma (v)}{c-v}+\\frac{L_{L} / \\gamma (v)}{c+v}", "f79022c4738b7e5d404c8e4ab487e5f5": "g(X_1,X_2,\\dots,X_n)", "f790419a988e1b9ab01585454ce9b316": "i:\\underline{A}", "f79054d2a921df4345c37973611f577b": "\\Omega=\\frac{Wh}{H}", "f79066963d271e37f6393b5b5c4f390e": "_{\\ell}", "f7906a8b826cf1ef74a7c940b4bdbecd": "\nA =\n\\begin{bmatrix}\n a_{0} & a_{-1} & a_{-2} & \\ldots & \\ldots &a_{-n+1} \\\\\n a_{1} & a_0 & a_{-1} & \\ddots & & \\vdots \\\\\n a_{2} & a_{1} & \\ddots & \\ddots & \\ddots& \\vdots \\\\ \n \\vdots & \\ddots & \\ddots & \\ddots & a_{-1} & a_{-2}\\\\\n \\vdots & & \\ddots & a_{1} & a_{0}& a_{-1} \\\\\na_{n-1} & \\ldots & \\ldots & a_{2} & a_{1} & a_{0}\n\\end{bmatrix} \n", "f790df35c5ae0b55cd95184b7608a41d": " \\mathbf{p} = m \\left(\\bold{\\hat{e}}_r \\frac{\\mathrm{d}^2 r}{\\mathrm{d}t^2} + r \\omega \\bold{\\hat{e}}_\\theta \\right) ", "f790dff15c58740893f2b9e3c1c5b348": "VSWR = {V_\\max \\over V_\\min} = {{1 + \\rho} \\over {1 - \\rho}}.", "f790e984aa4dff433277713939fce96a": "\\frac{\\partial}{\\partial\\ln Q^2} D_{i}^{h}(x, Q^2) = \\sum_{j} \\int_{x}^{1} \\frac{dz}{z} \\frac{\\alpha_S}{4\\pi} P_{ji}\\!\\left(\\frac{x}{z}, Q^2\\right) D_{j}^{h}(z, Q^2)", "f791124903c3ffa18edf840ad1036bb2": "\\frac{2}{n}\\sigma^2", "f7913db7999999e0cfdd2ac3c9ae25ea": "\\left(2 + \\sqrt{-5}\\right)\\left(2 - \\sqrt{-5}\\right)", "f7914331482e16d2987765bc91142956": "\n\\begin{align}\n\n & \\begin{aligned}\n ( \\nu x) \\; ( \\; & 0 \\\\\n | \\; & \\overline{z} \\langle x \\rangle . \\; x(y). \\; 0 \\; )\n \\end{aligned}\n\n \\\\\n\n| \\; & z(v). \\; \\overline{v}\\langle v \\rangle . \\; 0\n\n\\end{align}\n", "f79157c0c15e53a4cf5e6997c021a2f3": "\\,p(z)=a\\prod_n(z-c_n),", "f791732856a0418bcc876035d71ee3bf": "\nf(x) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}} e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} }.\n", "f791cebd7aa74a1c25d732b77ea399c2": "\\textit{opendoor}(t)", "f791e33ccf9b2d98d103a333f072f60d": "\n\\begin{align}\n\\mathbf{S}_B^{\\phi} & = (\\mathbf{m}_2^{\\phi}-\\mathbf{m}_1^{\\phi})(\\mathbf{m}_2^{\\phi}-\\mathbf{m}_1^{\\phi})^{\\text{T}} \\\\\n\\mathbf{S}_W^{\\phi} & = \\sum_{i=1,2}\\sum_{n=1}^{l_i}(\\phi(\\mathbf{x}_n^i)-\\mathbf{m}_i^{\\phi})(\\phi(\\mathbf{x}_n^i)-\\mathbf{m}_i^{\\phi})^{\\text{T}},\n\\end{align}\n", "f7924dd6731dcd3b16f210a008fd5ee7": "A/{\\sim}\\in W\\iff A\\in V.", "f7928b1a0487ce6163cbe65364f92139": "x = X(u, v)\\,", "f79290cc7372de0833133579c28ee51d": "yx_i = x_jh", "f7929c758d86125f7f6977a65122a0aa": "\\displaystyle{Q(U_ta)b = U_tQ(a)V_{-t}b.}", "f792e1a7c83f59acce3149ac3497b41d": "\\gamma = \\frac{1}{\\sqrt{1 - (v/c)^2}}\\,.", "f793076f740b557b0f7c8e15fa7db8b6": "U =V", "f7935503c3339d815e60563fd717a387": "\n\\rho_1 \\simeq \\rho_1 ' \\oplus \\sigma_1 ' \\oplus \\rho_2' \\oplus \\sigma_2 ' \\cdots \\simeq ( \\oplus_{i \\geq 1} \\rho_i ' ) \\oplus \n( \\oplus_{i \\geq 1} \\sigma_i '),\n", "f79398809bd0df79c10eea35feaa272c": "\\scriptstyle\\lang\\psi|P_i|\\psi\\rang", "f793ae8bbb4486734bf03a1ee1ffa2e9": "{\\mathbf{}}\\tau_i=\\hat{P}_i\\hat{S}_i \\left(\\hat{P}_i\\hat{S}_i \\right)^*.", "f793bd92134f1d91b4cd455442cd8c21": " \\exists x_n A(x_1, \\ldots , x_n) ", "f7943e2eb81d2ac66968e2a0e28c72e0": "d - 1", "f794655652bc7cf0af18498e459b39a5": "\\textstyle 1, \\, \\ldots, \\, N-1", "f7947d1eb3eb9c09dccdb1b3417ea765": "\\mu\\left(A\\setminus\\bigcup_{k=1}^{N} A_k\\right)\\leq\\frac{1}{N}", "f7948c7696603ca487a8df938e010356": " H_2(G, \\mathbf{Z}) \\cong (R \\cap [F, F])/[F, R]", "f794bb940206d71682d557585ca5afda": "2\\mathrm{Na} + 4\\mathrm{S} \\rightarrow \\mathrm{Na_2S_4} \\qquad\\qquad E_{cell} \\sim 2\\mathrm{V}", "f794bd59b94c4efd0218077a41ffc01b": " x = x(q_1, q_2, q_3),\\, y = y(q_1, q_2, q_3),\\, z = z(q_1, q_2, q_3)", "f794ef658910ac1afe12f68d17d16d45": "z\\notin S", "f794fcf76c6888380a9e256c6f0b1350": "CN_{III}", "f7950568c05864faa8528e2772a18985": "(0,0,\\ldots,0)", "f79536210acf3b8b3f51ea77c235b109": "T_{1,n}(z)=g_n(z)", "f7956fef4da87575363b9e7cecb43f2d": "\\ v_o", "f79571ac66c7ed4342206433dae1fad9": "(E_j,F_j)", "f7959cd72e19f800cd333e4d995e1389": "X_0\\in\\mathbb{S}^n, X_0\\succ 0", "f795f58c78dd599b9cb89410f81647f8": "Z(s)=\\frac{1}{sC}", "f79646530ecbd64fccfc119e12adef98": "x+r ~\\bmod~ (p-1)", "f79666f82efeba2b632680ba1de106c8": "T = f Wr = 2 r^2 f l P", "f7966b5339256448bfc5e0627dc112b3": "\\mathbb{C}((z,\\sigma))", "f796b497fa59bba981fe8a5ca491f0a1": "\\scriptstyle |\\psi\\rang |\\phi\\rang ", "f796dda367cad535a7d05ddcac83a1c6": "\\phi(p)=p-1", "f79798e83f5996ea6d2e4265f158b3d7": "g(\\vec{k})d^3k = d^3n =\\frac{L^3}{(2\\pi)^3}\\,d^3k", "f797dfeec136b2b90bb8d97c6480a1ad": "\\sum_{n=1}^\\infty\\operatorname{E}\\!\\bigl[|X_n|1_{\\{N\\ge n\\}}\\bigr]\\le\nC\\sum_{n=1}^\\infty\\operatorname{P}(N\\ge n),", "f797f29da30652546c372986b0bc631e": " Cr \\lbrace \\cup A_{i} \\rbrace = sup_{i}(A_{i}) ", "f797fd6d2cb0830b14f77d06a88b7d4e": " \\alpha + \\beta + \\gamma ", "f7980c4a61d1c17fd9c2c28075b0415c": "- \\frac{100}{x}", "f79826c4b070ff558ed05090af42a4fe": " t_1 ", "f79831125b222cc698e6d153580ed2c0": "\\begin{cases}\n\\mathrm{D}'(U) \\times \\mathrm{D}(U) \\to \\mathbf{R} \\\\\n(S, \\varphi) \\mapsto \\langle S, \\varphi \\rangle.\n\\end{cases}", "f798ab44bef1cadade43191a28b2fa6b": "L=T-V", "f79952d638b782f159542a00ecdcf2ac": "\\frac{\\alpha}{c+2v}\\,\\!", "f79978adc8b8d6885e33739f972eceb2": "i = 1, 2, \\dots , n", "f799ec0ea095bd2bdb9873ea2fd06c19": "\\alpha^n+1", "f799f342eb49ba6a0307b799634c632d": "\n\\sigma^2 = \\mathrm{var}(X_i)\n= \\mathrm{E}[(X_i-\\mathrm{E}(X_i))^2] = \\mathrm{E}[(X_i-\\mathrm{E}(X_i))\\cdot(X_i-\\mathrm{E}(X_i))].\\,\n", "f79a024f7d1d4f66386129f21ee474f7": "O( n \\sqrt{k} )", "f79a3eb3cc60de37436e2c202f414a68": "|\\!\\!\\!\\Sigma", "f79a6cd1096e860b6dfc1fe2aa6314ac": "\\mathrm{Hom}_K(A,B)\\otimes\\mathbf{Z}_p\\cong\\mathrm{Hom}_{G_K}(T_p(A),T_p(B))", "f79ac282f427d2caf6770893c1dd1d5d": " \\mathbb E_{{Z_1,\\cdots,Z_{k-1}}}{\\lVert x_k-x \\rVert^2} \\leq (1-\\kappa(A)^{-2}){\\lVert x_{k-1}-x \\rVert^2}. ", "f79ad1a6125aa71dee07e8baa9088184": "q_{out}", "f79ad572d15f19356f07dcbd315a96e1": "\\Sigma_i^{\\rm P} \\subseteq \\Delta_{i+1}^{\\rm P} \\subseteq \\Sigma_{i+1}^{\\rm P}", "f79afd7d8d377049c8892e72467f6496": "\ns_x=\\frac{s_y}{|m|}\\sqrt{\\frac{1}{n}+\\frac{1}{k}+\\frac{(y_{unk}-\\bar{y})^2}{m^2\\sum{(x_i-\\bar{x})^2}}}\n", "f79b57e3623e2642164f50beb352b0bf": "e*=0", "f79b60c2a29af6dc199cb0cc7214c189": "(P \\or (Q \\or R)) \\leftrightarrow ((P \\or Q) \\or (P \\or R))", "f79b76f00b17013f20daffead0a5e0ce": "I_{sp}", "f79b8902a8ba74373e4bce190d2d631b": "f(x_n + h)", "f79ba051bdda85242c713394951de641": "{}_1^2\\!\\Omega_3^4", "f79bb4d3cebbfaebb9ef3682d24b8717": "q_{th} \\leq q < q_\\max. \\, ", "f79bbe72c9f4d6b0b5f5d626aa8d65d6": "\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} p_{cv} = \\Delta \\epsilon \\, p_{cv} + {\\mathbf E} \\cdot {\\mathbf d} I", "f79c2ed904c1888e02b53cd0f3f57282": "\n \\left[ \\det (-S_{xx}''(x^0)) \\right]^{-1/2} = \\prod_{j=1}^n \\left| \\mu_j \\right|^{-1/2} \\exp\\left[ -i {\\rm Ind} (-S_{xx}''(x^0)) \\right],\n", "f79c71972384d2946e14bcb9b5a5b91a": "\\mathcal{S}_{,i}[-i\\partial]Z+J_i Z=0.", "f79d252da3002a8ee50c948cf6663923": "(b,u(x))", "f79d341d9982d7daea37c12ec563cbf8": "\\dot{\\sigma}(\\mathbf{x})", "f79d5b7c5c8f8c38636850e7f018ade3": " \\partial \\mathbf{F} = \\mathbf{J} \\,", "f79d8a6b162ace0f2552dc78d9593d21": " \\lambda_i < 1 ", "f79da36eea9fc685e99efb8cb2543abf": "T_2[i,j] ", "f79daf3ffacf509a1a43d7bf2969c971": "\\ln{\\rho}=-3\\left(1+w\\right)\\ln{a};", "f79db1d385892245d0cc256d91d5f998": "a^i_j", "f79db565820708ea72de83a465c62d4c": "\\mathcal{L}_{X} g = \\lambda g \\,", "f79e0dbda007868e9d78c80d57586214": "\\mathbf{x} \\in \\mathcal{X} \\subseteq \\mathbb{R}^p", "f79e3071c3400bbc04b12fc0c75a14d4": "L_p^l", "f79e7356d62984785b2afcf1e1443d75": "M_v", "f79e895fc9cc578c103b02733ab5c255": "\\left[ n,k;c\\right]", "f79ef6e1a64f643749486c3d1de9c601": "\\operatorname{HBW}=0.102 \\frac{2F}{\\pi D \\left(D-\\sqrt{D^2-d^2}\\right)}", "f79efec3a3f19ea477ce5fb8cc30d538": "\\sqrt{n} \\in \\mathbb{Z}", "f79f1778218e47d328e82b79b980914d": "\\scriptstyle mnl", "f79f50d9d318624a5ec4c0f299b5f575": "\\scriptstyle\\bar{y}", "f79f94e5cb88c72ab18c8be2ed34dcd2": " \\mathbf{R}_{N \\times 1} ", "f79fbcd89d6fdb7c861dc5f53c064dba": "\\ \\mathbf E_J \\cdot \\mathbf e_i = \\alpha_{Ji}=\\alpha_{iJ}", "f79fd9985b4ba47d71a5ec455b059698": "2{\\pi}t/\\log(t)", "f7a019f6a7c5b7d40ff4cd3d7085a18c": "\\dfrac{\\alpha : X/Y \\qquad \\beta : Y}{\\alpha \\beta : X}>", "f7a076bae193b25188cf8dd869ee4c9f": " |\\langle e_j|f_k \\rangle|^2 = \\frac{1}{d}, \\quad \\forall j,k \\in \\{1, \\dots, d\\}. ", "f7a0a1fec0c5e4c719fc6ffb44146e0a": "(x,\\{a\\})", "f7a0a3f9741d3d2945b1c218cddd1af3": "\\vec{\\theta}", "f7a0a41131a16fac94f3ebc7a626a64b": "4k+2", "f7a0aed635ea2a33009383bd90446d28": "U=U^0+\\sum_i \\frac{\\partial U}{\\partial p_i}\\delta p_i", "f7a0ba8361b2614f262fddf447ac3207": "\\sec(x)", "f7a0c9b0ada3a1d1c4c5cfd94501a877": "D(A) = D(a_{1},\\ldots,a_{n}) \\,", "f7a0cb4a478010e6d3f9adbc28eeedf1": "\\mathbf{q}", "f7a0cd46f1c8bf58b28c26fda38d9338": "\\beta_j,", "f7a1c7e61a7fec81eab47d1f81fbe797": "|A|-1-|B|\\geq 1", "f7a1d2b6aaa244745598c9bab2943e1c": "x(t) = x_0\\cdot e^{kt} = x_0\\cdot e^{t/\\tau} = x_0 \\cdot 2^{t/T}\n= x_0\\cdot \\left( 1 + \\frac{r}{100} \\right)^{t/p},", "f7a1f3035921ce473835adf8484cbbdb": "P^i \\colon H^n(X;\\mathbf{Z}/p) \\to H^{n+2i(p-1)}(X;\\mathbf{Z}/p)", "f7a21aaf60ef2044e30f73e7d442110b": "V^m\\ ", "f7a22ba89ade17a7bd5a32c9b343cee4": "R = \\frac{I_r}{I_i} = \\frac{(\\eta_1 - \\eta_2)^2 + k^2}{(\\eta_1 + \\eta_2)^2 + k^2}", "f7a2e1854f3320a25a7acf60e05f678a": "[X,Y]=0\\,", "f7a301754b1f7c67c2996a8d65944d57": "\n\\begin{align}\n\\underset{\\frac{\\boldsymbol{y}}{t} \\in \\mathbf{S}_0}{\\text{maximize}} \\quad & t f(\\frac{\\boldsymbol{y}}{t}) \\\\\n\\text{subject to} \\quad & t g(\\frac{\\boldsymbol{y}}{t}) \\leq 1, \\\\\n& t \\geq 0.\n\\end{align}\n", "f7a353b28a5155e03768737c02ccc84c": "(g^{n-i+j}\\oplus1)D_j = g^{n-i}B\\oplus A", "f7a355c0f509d9cf4a8c2620e59a3c57": "\\frac{f_{x}}{f_{y}}=\\sqrt{1-\\frac{v^{2}}{c^{2}}}", "f7a38b4aec9dcc5d0e4fd38b789b6559": "\\forall t \\, \\exists t' \\, (t'\\ E\\ t)", "f7a3ab29083c00e82783c03d495d5681": "\n\\begin{array}{lcl}\n\\boldsymbol\\alpha &\\sim& \\text{some distribution} \\\\\n\\boldsymbol\\theta_{d=1 \\dots M} &\\sim& \\operatorname{Dirichlet}_K(\\boldsymbol\\alpha) \\\\\nz_{d=1 \\dots M,n=1 \\dots N_d} &\\sim& \\operatorname{Categorical}_K(\\boldsymbol\\theta_d) \\\\\n\\boldsymbol\\phi &\\sim& \\text{some other distribution} \\\\\nw_{d=1 \\dots M,n=1 \\dots N_d} &\\sim& \\operatorname{F}(w_{dn}\\mid z_{dn},\\boldsymbol\\phi)\n\\end{array}\n", "f7a449a2b9b493cb46dd88d14882a775": "I(2\\omega,l) = I(\\omega,0)\\tanh^2{(\\Gamma l)},", "f7a4c79b5a17a2e05e097fe25083e4c2": "S(t) = P_1 (t) \\mbox{ , } t_0 \\le t < t_1,", "f7a557d30c9eb4239e8072345fb61ac0": " \\text{CT (mEq/L)} = [\\text{HCO}_3^-] + 2*[\\text{CO}_3^{2-}]", "f7a563c05b5938ce299828a04222a605": "\\omega = f^*dx", "f7a563ddde3637a17fc0fb87d00ff9e4": "\\tan(2 \\alpha)=\\frac{2 \\tan(\\alpha)}{1-\\tan(\\alpha)^2}~~\n\\forall \\alpha \\in \\mathbb{C} \\backslash \\{\\alpha\\in \\mathbb{C} : \\cos(\\alpha)=0 || \\sin(\\alpha)=\\pm \\cos(\\alpha) \\}.\n", "f7a570187268cf71da3a703cc0db7761": "\\mathrm{WFH} = \\frac{\\mbox{weight of a given child}}{\\mbox{median weight for a given child of that height}} \\times 100", "f7a5742a15972741e223d54ef394a415": "=\\kappa_4(W)+4\\kappa_3(W)\\kappa_1(W)\n+3\\kappa_2(W)^2+6\\kappa_2(W) \\kappa_1(W)^2+\\kappa_1(W)^4.\\,", "f7a5d162a03dd05e03147acd5c9f5037": "\\hat=", "f7a6040160ec77fc6b9ef3a3cfb69d6c": "m \\dot v = - \\frac{ \\partial H }{ \\partial x } ", "f7a60fac2e9278c3ac6600c8093a94ec": "\\mu(h, d)", "f7a68919158ceed996af07adcb5ce050": "b_{i}^{*}:= b_{i}", "f7a6f9115f27fb1cf5c3a13fb525a570": "I(a)^2", "f7a7007c08051be862170b588085b7a3": " \\int_{E}f\\,d\\mu=\\int_{E-K}f\\,d\\mu,~~~\\int_{E}f_n\\,d\\mu=\\int_{E-K}f_n\\,d\\mu ~\\forall n\\in \\N. ", "f7a71bf12e327868833c96956c3ffc3c": "\\scriptstyle E((X))", "f7a72b6d62af94da7f5f59de7e4c8c82": "\n\\tilde{\\Sigma} = \\frac{1}{I} \\sum_{i=1}^I E[z_iz_i^T] - \\tilde{\\mu}\\tilde{\\mu}^T\n", "f7a79893fd844d61037fa22348590494": "\\vec{E}(t) = \\vec{E}_0 e^{-i\\omega_Lt} +\\vec{E}_0^* e^{i\\omega_Lt}", "f7a79c46700bbde020f2e1dc5a0aacc3": "\\textrm I", "f7a82463d2e917011368c7637c6fcbf8": "M_j", "f7a89641420849564f76005b52b4410d": "\\Sigma : E \\to \\tilde{F}", "f7a95d08d852bfe58e958c584234a3a4": "x \\in H_1(M,\\partial M)", "f7a95ece7942f56b5ac7a24950508bd6": "\n\\frac{e^{(n-1)(f(x_0) - \\eta)} \\int_a^b e^{f(x)} \\, dx }{e^{nf(x_0)}\\sqrt{\\frac{2 \\pi}{n (-f''(x_0))}}} = e^{-(n-1)\\eta} \\sqrt{n} e^{-f(x_0)} \\int_a^b e^{f(x)} \\, dx \\sqrt{\\frac{ -f''(x_0)}{ 2 \\pi}} \n", "f7a964c8b8a1957b2ebfbbba28c9bb1a": "\\begin{align}\n& \\mathbf{A} = \\begin{pmatrix}\n6 & -2 & -3 \\\\\n-1 & 4 & -2 \\\\\n-3 & -1 & 5\n\\end{pmatrix}, \\quad \\mathbf{k} = \\begin{pmatrix}\n5 \\\\\n-12 \\\\\n10\n\\end{pmatrix}. \\quad (10)\n\\end{align}", "f7a9a0c4f511a078f12a98256c5408db": "\\mbox{Glass Property} = b_0 + \\sum_{i=1}^n \\left( b_iC_i + \\sum_{k=i}^n b_{ik}C_iC_k \\right)", "f7a9b60c96efc3cd65cef93906075fc3": " Q(t) =\\varepsilon_0 \\oint_{\\mathcal S} d \\mathbf{\\mathcal S} \\ \\boldsymbol{ \\cdot} \\ \\boldsymbol{ E} (t) \\ , ", "f7aa0f87908e3a1b04151e9af4a0dd83": "\\sum _x \\sin ax = -\\frac{1}{2} \\csc \\left(\\frac{a}{2}\\right) \\cos \\left(\\frac{a}{2}- a x \\right) + C \\,,\\,\\,a\\ne n \\pi ", "f7aabd7342fff01c776142e862d7be79": "p^\\mu p_\\mu = m^2", "f7aabdd14083fc8bf83adbc23ac5a1cb": "\n\\sum_{k=1}^n a_k b_k = a_n B_n - \\sum_{k=1}^{n-1} B_k (a_{k+1} - a_k).\n", "f7ab094319135b6f4cf18f8d85fe5de6": " |x\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} ", "f7ab0dd2f93a633a4c957dad9c5f6fc4": "\\operatorname{E}[X^{T}AX][X^{T}BX] = 2\\operatorname{trace}(ACBC) + \\operatorname{trace}(AC)\\operatorname{trace}(BC)", "f7ab5d848fb47cf897d4c731cc523a29": "\\int_1^2 \\! x\\,dx \\ = \\frac{2^2}{2}-\\frac{1^2}{2} =\\frac{3}{2}", "f7ab7673485012e0afcd112c27593681": "\\delta e = e e_I^\\alpha \\delta e_\\alpha^I", "f7abb5b577e11d5fe04ccbc3af7ff824": "K(w,z)=-{1\\over \\pi} \\left[{\\varphi^\\prime(w)\\varphi^\\prime(z)\\over (\\varphi(z)-\\varphi(w))^2} -{1\\over (z-w)^2}\\right].", "f7abe965777316dde5d5e7692bc53f7d": "0 \\le t <\\infty", "f7ac20010b6d8fcd58c06d77f8403447": "\\begin{alignat}{5}\nn \\left( T\\right) = KT^{1-\\alpha} ; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ...... Eq:2\n\\end{alignat}", "f7ac41995a75c95c73f1d600ae6d7d57": "u''(z)=\\sum_{k=0}^\\infty (k+r-1)(k+r)A_kz^{k+r-2}", "f7aca6a24fa8dd1ecd593b92724e197c": "\\Theta(N \\log^2 N)", "f7acc51c9e1838044c4bfcdb14912afc": "(q^i,p_j)", "f7ace3698b5f14273f1364488c2e00b4": " \\frac{v^2}{2g} \\,\\!", "f7ad160a15457361bf8b0b91b8faab0e": " L(\\mathbf{q}, \\mathbf{\\dot{q}}, t) = T(\\mathbf{\\dot{q}}, t)-V(\\mathbf{q}, \\mathbf{\\dot{q}}, t)", "f7ad7cdf063228eb09b813b60f6248b8": "\\tilde{M}\\left[\\mathbf{p}(t),\\mathbf{\\dot{p}}(t),\\mathbf{\\ddot{p}}(t),t\\right]=0", "f7ad7da6b8fed70136f4b2a3c1273b4a": "\\sigma_{zz}", "f7ad9b7fddb296fcbf8fee9ed3050781": " \\left \\langle s, t \\, \\left | \\, s^2, t^2, (st)^n \\right \\rangle \\right . ", "f7ad9cb326de9d6117c2fd9ed16513ee": "\\delta\\mathcal{S}", "f7add17b226cc15432155268e5160b4c": "\\begin{align}\n& (-\\gamma^\\mu \\hat{P}_\\mu + mc)_{\\alpha_1 \\alpha_1'}\\psi_{\\alpha'_1 \\alpha_2 \\alpha_3 \\cdots \\alpha_{2j}} = 0 \\\\\n& (-\\gamma^\\mu \\hat{P}_\\mu + mc)_{\\alpha_2 \\alpha_2'}\\psi_{\\alpha_1 \\alpha'_2 \\alpha_3 \\cdots \\alpha_{2j}} = 0 \\\\\n& \\qquad \\vdots \\\\\n& (-\\gamma^\\mu \\hat{P}_\\mu + mc)_{\\alpha_{2j} \\alpha'_{2j}}\\psi_{\\alpha_1 \\alpha_2 \\alpha_3 \\cdots \\alpha'_{2j}} = 0 \\\\\n\\end{align}", "f7ae4bf6373f725f872938f38a6ff5d1": "L_{SE}", "f7ae9d36e965e0591efc29b99df9cdef": " D_{b}= \\frac{(N_{b}+L-1)!}{N_{b}!(L-1)!} ", "f7aec00306d35bb421f2a47354c099a2": "\\vdash \\Box(\\Psi \\rightarrow (\\Box \\Psi \\rightarrow P))", "f7aee01c63cc5352dc33b2d10c4e8a11": "\\mathbb{E}[2X{_i^e + X{_i^?}}] \\le {{2e_i} \\over d}\\qquad\\qquad (2)", "f7af261bdb7a8af5921c235b331f685d": " {{E(b_1,N)} \\over {E(b_2,N)}} \\approx {{b_1 {\\log_{b_1} (N)}} \\over {b_2 {\\log_{b_2} (N)}}}\n= {\\left( \\dfrac{b_1 \\ln (N)} {\\ln (b_1)} \\right) \\over \\left( \\dfrac{b_2 \\ln (N)} {\\ln (b_2)} \\right)} = {{b_1 \\ln (b_2)} \\over {b_2 \\ln (b_1)}} \\, . ", "f7af7305b2abbffd5ebabeb66187d6fb": "\\!\\ Re = \\!\\ 10^6", "f7af833734588415faec376d72877dd6": "\n\\begin{align}\n2 B(u,v) &= Q(u+v) - Q(u) - Q(v), \\\\\n2 B(u,v) &= Q(u) + Q(v) - Q(u-v), \\\\\n4 B(u,v) &= Q(u+v) - Q(u-v).\n\\end{align}\n", "f7afb889106f9b472744fc2c8337739c": "\\lambda(n) \\leq ne^{-\\Delta}", "f7afcc40b875c3e6677907704d30b3ba": "Qv(x,y,z,t)", "f7afd9a80c72d37cce98fbd31f9ea947": "\\begin{cases}1- e^{-(x/\\lambda)^k} & x\\geq0\\\\ 0 & x<0\\end{cases}", "f7afe4cb54e09d22a32bcf4c47aab892": " \\,r_k", "f7afea4c17bde6cdbf2ad6ad6d876e35": "\\textbf{g}", "f7aff54376bb8556835e4c6c15c64f46": "A^{-1} = \\frac{1}{\\operatorname{det}(A)} \\operatorname{Adj}(A).", "f7b01ab0ff584ab3090ea572aab5b8f2": "\\mbox{A + B} \\rightarrow \\mbox{products.}", "f7b0204137054bf0464647043128ea63": "{\\mathbf{}}\\hat{S}_{i}=1/2(\\tau'_{i}\\Psi_{i+1}^2+\\Psi_{i+1}^2\\tau_{i}),\\hat{S}_N=0, rank(\\hat{S}_i)=n_r", "f7b03859b00c128d5ba0d050a0429bc8": "(c/n) \\cos(\\psi) - (1-1/n^2)v", "f7b0482294ea491f1e6c4361db76fc0f": "\\color{Brown}\\text{Brown}", "f7b0689af081c24516e078927a29a212": "\\Delta L = 0\\,", "f7b095a5b4b877af80f0b1e938734963": "K(\\mathbb{Q},4)", "f7b0c0cd9004764689859aab4297ff4f": "f(\\gamma', 1_\\gamma) = \\gamma'", "f7b0c2da563df5aed5e414cec4273c1a": " 2m", "f7b0c98d64533c91f236dda1adcc22b7": "=2|s\\rang-|s\\rang-\\frac{4}{\\sqrt{N}}\\cdot\\frac{1}{\\sqrt{N}}|s\\rang+\\frac{2}{\\sqrt{N}}|\\omega\\rang=|s\\rang-\\frac{4}{N}|s\\rang+\\frac{2}{\\sqrt{N}}|\\omega\\rang=\\frac{N-4}{N}|s\\rang+\\frac{2}{\\sqrt{N}}|\\omega\\rang", "f7b0cf21482e7c431f94033e098e1ee2": "\n\\begin{bmatrix}\n1 & 2 & 1 & -1 & 2 \\\\\n1 & -1 & -2 & -1 & -1 \\\\\n2 & 1 & -1 & -2 & -1 \\\\\n1 & -2 & -1 & -1 & 2 \\\\\n2 & -1 & 2 & 1 & -3\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n-3 & -3 & -3 & 3 \\\\\n3 & 3 & 3 & -1 \\\\\n-5 & -3 & -1 & -5 \\\\\n3 & -5 & 1 & 1\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n0 & 0 & 6 \\\\\n6 & -6 & 8 \\\\\n-17 & 8 & -4\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n0 & 12 \\\\\n18 & 40\n\\end{bmatrix}\n\\to\n\\begin{bmatrix}\n36\n\\end{bmatrix}.\n", "f7b0d90c73624bd9885397e4dc056e56": "g_{\\alpha[\\beta,\\gamma]\\epsilon} \\, = \\frac{1}{2} (g_{\\alpha\\beta,\\gamma\\epsilon} - g_{\\alpha\\gamma,\\beta\\epsilon}).", "f7b0d9b9c647f8a9b58a096e5969593a": "\\zeta_D(s) = L(s,\\chi) = L(s,\\psi)\\prod_{p|k}\\left(1 - {\\psi(p)\\over p^s}\\right)", "f7b0f1c3528046366a3805bb5fabaeab": " 0 \\rightarrow \\operatorname{Ext}(\\operatorname{H}_{i-1}(X; R), G) \\rightarrow \\operatorname{H}^i(X; G) \\overset{h}\\rightarrow \\operatorname{Hom}_R(H_i(X; R), G)\\rightarrow 0.", "f7b119155568d55b048554ff835ea050": "\\operatorname{ch}(f*S) = \\operatorname{ch}f + \\operatorname{ch}S", "f7b15c055ad44d2600a9fb2d14e6c5f3": "\\nu(M_2) = \\nu(M_1) + \\theta \\,", "f7b16b197ed62b2538d4d34b52dafd82": "a(n)=|A(n)|", "f7b1a0b81d4a3effe8b5b56c8d989314": "\\varphi=(\\varphi_1,\\ldots,\\varphi_r)", "f7b2106cd67a9376edc793253c45fb00": "c_1 f(n) \\le \\Delta(n) \\le c_2 f(n)", "f7b21dcc7b814ad2e44274c6f53fa19e": "\\frac{\\pi}{12}", "f7b2b66ee445b2d2726e2be9ccc65fb4": "\\vec\\rho", "f7b2ce7e0f8d747d5705c4ab42af5763": "\\tilde b_n=(-1)^n a_{M-1-n} \\quad \\quad (n=0,\\dots,N-1) ", "f7b2cfc2dbda243c2c7bc5efcb748a59": "x_\\min=\\max(0,n-m_2)", "f7b2f8df71e270f46a74969da2b88da0": "(\\alpha_i)_i ) \\le e)", "f7b303a4e5ad5481d1ef595ce76077cf": "0f, 2f, 4f, 6f, \\dots \\ ", "f7b3232814845be05e67a4e2bab320e4": "H_{2n+1}(x) = (-1)^{n}\\,\\frac{(2n+1)!}{n!}\\,2x\n\\,_1F_1\\left(-n,\\frac{3}{2};x^2\\right)", "f7b337f52d8fd8267a7f3a765d252dd6": "(x-3)(x-1)^3(x+1)(x+2)(x^4+x^3-7x^2-5x+6)(x^4+x^3-5x^2-3x+4)^2\\ ", "f7b33d53bfb57512d7f2e5d42d7116f8": " t_2^3 + 4 t_3^2 + t_1^2 t_4 - 4 t_2 t_4 - 2 t_1 t_2 t_3 = 0 ", "f7b340453367f5dc2ca4c52bab73c767": "\\vec a\\!", "f7b34a21550dabe0b3feb7f50324c6b0": "\\mathfrak{a}^*=i\\mathfrak{a}", "f7b35493494f074962160926e4c9ee7b": "y = A {{}_2 F_1}(\\alpha, \\beta; \\gamma; x) + B x^{1 - \\gamma} {{}_2 F_1}(\\alpha - \\gamma + 1, \\beta - \\gamma + 1; 2 - \\gamma; x)\\,", "f7b35e1dd8850500574c3db55e5919eb": "\\epsilon_e = \\frac{l-l_0}{l_0}", "f7b367281465eba78af84b7724b9969d": "A^{i_q} {}_{k_q}", "f7b38d68d3940e5c8e13e3c7f8185ffb": "F_r > 1; \\; \\textit{Supercritical}", "f7b3aa34670b7925f78a6c7dcdf1cb41": "|X_k|", "f7b3f3fe1794d5fd0dc587b5fa641882": "\n\\begin{align}\n\\int \\frac{du}{1 - u^2} & = \\int\\frac{du}{(1-u)(1+u)} = \\dfrac12\\int \\left(\\frac{1}{1+u} + \\frac{1}{1-u}\\right)\\,du \\\\[10pt]\n& = \\frac12 \\ln \\left|1 + u\\right| - \\frac12 \\ln \\left|1 - u\\right| + C = \\frac12 \\ln\\left|\\frac{1+u}{1-u}\\right| + C\n\\end{align}\n", "f7b40586d4b48721bc8b0e25b6e79a7f": " x_i \\in A ", "f7b43027733d2db3dc4bfaab3e3facc4": "\\bold{x}", "f7b49209e1ce2d31d0bbc63fbf6650f5": "\\frac {dr} {d\\theta}", "f7b4a9a272539da17df482a540896746": "y_{1}", "f7b52a5137e9c06a211d736c2a3b34ca": "P(X_i=x_i|X_j=x_j, i\\neq j) =P(X_i=x_i|\\partial_i), \\,", "f7b555865e6d45e2dfce37bba5ef797a": "\\mathcal D_k\\,", "f7b5730d34919425ce1adc9f7462de8d": "|J_i|\\le\\sum_{k=1}^n|\\beta_k\\alpha_k|e^{|\\alpha_k|}F_i(|\\alpha_k|)", "f7b5dedbc9d424b35f59ff3d7e9f088a": "\\mathcal{E}(\\pi)\\left[\\psi(x_1)\\cdots \\psi(x_n)|\\Omega\\rangle\\right]=\\psi(x_{\\pi^{-1}(1)})\\cdots \\psi(x_{\\pi^{-1}(n)})|\\Omega\\rangle", "f7b71b80eec59b3caab267d6bf2e3994": " \\operatorname{E}(\\theta_i|y_i)= {{(y_i + 1) p_G(y_i + 1) }\\over {p_G(y_i)}},", "f7b77aa5f1ea76628f0805872e047f88": "\\mathcal{Q}_{\\alpha}", "f7b82316a2732fd818670258518cd1ad": "\\Phi(n)=-\\beta \\hbar\\omega \\left(n+\\frac{1}{2}\\right)+\\log 2 \\cosh \\frac{\\hbar \\Omega(n) \\beta} {2} ", "f7b82c3d670ec52ce10a87d54786ccae": "\\delta(C)=\\lim_{i\\to\\infty}{d_i \\over n_i}", "f7b83af1e1915f23fa5e428087394e8c": "x(t) = 2 \\left( {t \\over a} - \\left\\lfloor {1 \\over 2} + {t \\over a}\\right\\rfloor \\right)", "f7b89818692c62897e1199212fd28b78": "D : C([0,1]) \\to C([0,1])", "f7b8a5943c4215d3ee3683f4969f842a": " \\left(\\frac{p}{q}\\right) \\left(\\frac{q}{p}\\right) = (-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}.", "f7b8b47b9421b999e0e25ae56d1b2321": "x = \\frac{5 \\times 10^2 + 13.33 \\times \\frac{1}{2}10^2 - 3 \\times \\pi2.5^2}{10^2 + \\frac{1}{2}10^2 -\\pi2.5^2} \\approx 8.5 \\mbox{ units}.", "f7b904888e5dc6884734a96a0385f26e": "r_1(\\theta_1)", "f7b95051e8299ee7156878ce84e8019b": "2^x", "f7b96b8e306aad43c4ec57b70e60e0e7": "R \\sim \\text{Rice}\\left(\\nu,1\\right)", "f7b96fe4ae9dfc39a470bea047bcd8e3": "(a_1,a_2,\\dots,a_n)\\in R \\iff(h(a_1),h(a_2),\\dots,h(a_n))\\in R", "f7b99595e10c1117cf2fcaebff79d9a0": "M_{c/4} = M_\\text{indicated} + \\mathbf{x}\\times (D_\\text{indicated}, L_\\text{indicated})", "f7b9b62814f67c4aa6bb7f73670c6e6b": " \\boldsymbol{\\alpha}^{t+1} = \\mathbf{D}^{t+1} \\mathbf{K} \\left( (\\mathbf{D}^{t+1} K)^2 + \\widetilde{\\lambda} \\mathbf{I} \\right)^{-1} \\mathbf{D}^{t+1} \\mathbf{K}_{o^{t+1}} ", "f7b9f3044aae1e90c752dcebcf17b23e": "\\vec{S}(n)= \\begin{bmatrix}\n S_{NNN}(n) \\\\\n S_{RNN}(n) \\\\\n S_{NNR}(n) \\\\\n S_{NBR}(n) \\\\\n S_{RNB}(n) \\\\\n S_{RBN}(n) \\\\\n S_{RRB}(n) \\\\\n 1\n\\end{bmatrix}", "f7ba42f4dbca4587c946b9d083e4cced": " \\dot{\\varphi}^2 + \\dot{\\psi}^2 = 1.", "f7ba48eaa90d61588108ad9541f71ef5": "\\tfrac{17}{1}", "f7badee36e3ffa3ae4138355cf32247b": "\\psi_{k'} (r)", "f7bae0c23e713948e27a236c954fba22": " 1 \\le i, j \\le n, i \\ne j", "f7bb350dfcf1c5e6c4c63d569b391c1a": "Lu(X) = f(X),\\quad X\\in \\Omega,\\qquad (1)", "f7bb6389bdb16d04e232ee62f1fdfdb8": "M(k)=\\frac{\\mathrm{sinh}(k/2)}{k/2},\\,", "f7bb7c8384414a46b8fbd068e132cc6a": "\\alpha^i + \\beta^j = 1", "f7bbaa7ea17ea0eca0611da3f08e819a": "a\\circ b", "f7bbb5ec1ee3ec184741cb66f6d96f4e": "x_i-a_i.", "f7bbc9616794b16e3ae8b44cd024643f": "k = k_{\\mathrm{eff}} = \\frac{\\nu \\Sigma_f}{\\Sigma_a + D {B_g}^2}", "f7bc742709931b5a06cfd24b967fe9e4": "a\\wedge(a\\to b)=a\\wedge b", "f7bc88eb78c535c723d8755be0d427c9": "s_{j}(n)", "f7bc8b518d6f1b7fb5544e785c48793a": "\\mbox{p} = \\frac{N}{4\\cdot \\pi \\cdot d^2}", "f7bcaff69e9f8cabf8f2fd557913a7cf": "\\hat{M}", "f7bcf3bab67b3c144553805b251a1332": "29\\tfrac{499}{940}", "f7bd3e79454fe016e0d37585cf506ece": "2r_1r_2-2x\\left(r_1+r_2\\right)=0", "f7bd7ac5df067fcd3583f98a83b04170": "|\\hat {x}|=1", "f7be387f08a682690e648c8f1f295fd2": "K[T_1,\\dots,T_k],", "f7be656dded053f9142e6d6446634ec0": "dm=\\left(\\frac{dy}{L}\\right)m", "f7bed7c18fb51b154af809db61e9f470": "H_{-1} = 0", "f7beffe588faf222aa2568792145f83d": "2\\times\n2", "f7bf1fde8e4a4858e368909654cd1cce": "\n\\begin{align}a \\cdot \\gamma^\\nu &= a^\\nu \\\\ a \\cdot \\gamma_\\nu &= a_\\nu\\end{align} \n", "f7bf5d9c08afba22b9f8f8c7d0e4d318": "\\frac{1}{IA^2}+\\frac{1}{IC^2}=\\frac{1}{IB^2}+\\frac{1}{ID^2}=\\frac{1}{r^2}", "f7bf645ec3e91acd38953f6e7c4801d5": "\\beta_j \\approx \\beta_j^{k+1} =\\beta^k_j+\\Delta \\beta_j. \\, ", "f7bf8125ac4e7939bc1067f5d0c30fb8": "A_{21} = B_{21}^* = A_{2,-1}^* = B_{2,-1} = -\\frac{1}{2} i A e^{-i\\xi_0}, ", "f7bfd7a3bb193615f2139a8122f2d6e9": "\\|x\\|_p = \\left(\\sum_{i=1}^\\infty |x_i|^p\\right)^{\\frac{1}{p}}", "f7bff53383d3aeb74c0a4151572956a4": "(r_i+1)", "f7c0234f012b1649b04345bbfeb2671b": "W\\left(-\\frac{\\ln a}{a}\\right)= -\\ln a \\quad \\left(\\frac{1}{e}\\le a\\le e\\right)", "f7c04943f16a7e89a60e2c1b294e24fc": "U_n=\\frac{(-i)^n}{n!}\\int_{t_0}^t{dt_1\\int_{t_0}^t{dt_2\\cdots\\int_{t_0}^t{dt_n\\mathcal TV(t_1)V(t_2)\\cdots V(t_n)}}}.", "f7c082e72e3d63db8180635b87974ea5": "DCEBA", "f7c090028b28ddca8e87f8cc8966ec41": "R_\\text{black}", "f7c09ede1cb71c743538aca00f714877": " \\delta_S^1 = \\delta_S + 0 = [2;2,2,2,2,2,\\dots] \\approx 2.41421", "f7c0e77e652a03642ed31d773a7dafe5": "e(p, u^*) = \\min_{x \\in \\geq(u^*)} p \\cdot x", "f7c1735bd60aae9dafde81a9d77b04e3": "\\,t_{ij}", "f7c1c2f6055fdfef13473069e93e0d14": "R_{xy}(t)", "f7c208ed0064af233b9897004d973ec6": "\\mathrm{Ai}(x) = \\frac{1}{\\pi}\\int_0^\\infty\\cos\\left(\\tfrac{t^3}{3} + xt\\right)\\, dt\\equiv \\frac{1}{\\pi}\\lim_{b\\to\\infty} \\int_0^b \\cos\\left(\\tfrac{t^3}{3} + xt\\right)\\, dt,", "f7c22436857c3113b128cb3a81a93aa5": "n = \\pm1, \\pm2, \\pm3, \\dots\\ ", "f7c246b1df8f94e6284ce1ed4273338b": "\\lim_{x \\to \\infty} x \\sin \\left(\\frac{c}{x}\\right) = c", "f7c28b885e6e4c760da322c432eeec32": "\\gamma\\,_1", "f7c32c850e6e2af7f8f198a8bb2f90cd": " L_{i_1\\alpha}^{j_1\\beta}(x)L_{i_2\\beta}^{j_2\\gamma} (y)R_{j_1j_2}^{k_1k_2}(y/x)= R_{i_1i_2}^{j_1j_2} (y/x)L_{j_2\\alpha}^{k_2\\beta} (y)L_{j_1\\beta}^{k_1\\gamma}(x),\\quad 00\\\\\n &= \\int_0^1 H_{x} dx \\end{align} ", "f7c6ed7d49489d1e41fe3c6ef048a63a": "V^\\pm", "f7c7ff11cf01a34538d676e1742c334b": "\\omega_2\\circ\\omega_1", "f7c8090ff2638bd503e5d7fc26a847cc": "\\lambda\\setminus\\mu", "f7c851161f92571d7fbe9fe17de4f3a9": "\\partial_{A,\\bot}", "f7c8bf84db9dba3d3998c0a2bae83240": "\\hat{a_1} = 2 \\bar{x}- \\sigma^2", "f7c8e0a7d705c1a6a2eb628626bcd670": "\\mathbb{E}[1_{D(y)\\neq m}] \\leq 2^{k +H(p + \\epsilon)n-n}", "f7c8ef4f97dda173326da10e398072fe": "\\Delta v = I_{sp} \\ln \\frac {m_0}{m_1}", "f7c97d3a208a36945cabbbec66b52f2b": "d:\\{0,1\\}^*\\to[0,\\infty)", "f7c9c5ffdb838eb66724b70af2f065be": "y_2 = x_1x_3 + x_2x_4", "f7c9c860215d21740229e1052fa02989": "(S^i,{\\mathcal S}^i)", "f7ca45e28aa7784c24b9a5304279983c": " p_5=a_{20} \\omega +a_{11},", "f7ca7803630aa7f2ed7491140ed1f186": "J_2, J_3", "f7cab71f418d3cdafdc3e8cb1883af46": "\\frac{\\sin \\theta}{\\theta} < 1\\ \\ \\ \\mathrm{if}\\ \\ \\ 0 < \\theta\\,", "f7cac038936b5d74a46363b6925e556f": "G-F", "f7cac32bf56bea66c372ec1e32a773f6": "\\displaystyle H^r(k,M)\\times H^{2-r}(k,M')\\rightarrow H^2(k,G_m)=Q/Z", "f7cae61a871814de94e8d359a6f5e859": "C^* ( \\Lambda )", "f7cb3d1829f797b380116798b4e78c46": "\n\\operatorname{dCov}(X_1 + X_2, Y_1 + Y_2) \\leq \\operatorname{dCov}(X_1, Y_1) + \\operatorname{dCov}(X_2, Y_2).\n", "f7cc31bd0146cf484ff2b2a31223fb98": "N = \\sum_i n_i = \\sum_i a^\\dagger_i a^{\\,}_i", "f7cc401379d4ffefbbcad9de62fd6150": "^{\\;}\\xi^{i} = (x^1, . . . , x^n, p_1, . . . , p_n) \\in \\mathbb{R}^{2n},", "f7cc50197b542692fb74aabb5bb9cb91": "\\tfrac{x}{y}", "f7cc58ed435fe20a828b6bd612674cb3": "\n s = c -k_B b^2 r^2\n ", "f7cc5e02a4bbcac0519034d4e0b5ea24": "F^{-1}(F(x)) \\leq x", "f7cca3009e110d53ce4c58b2d86b148a": "b-l", "f7ccd7df81f8504bb8e4c5c52565cf79": "\\lambda = 6", "f7ccf07d8074f11d889c79629303d8e5": "\\Diamond\\Box A\\to\\Box\\Diamond A", "f7cd16875ea6a8f483b11f30430ffa27": "I_{m, k}", "f7cd83e32f83d5eba10b98b15f2c4e6a": "{{i}_{C3}}", "f7cdab5927b946deb2b939374c949b40": "\\exist x\\, \\forall y\\, \\lnot (y \\in x)", "f7cdc72c40f2b0c1306c639feccc6c5e": "\\mathrm{Nu} = \\frac{h D_h}{k_f}", "f7cde837b6f8d63e0944ba91bc2e1541": "\\sum_{k=1}^{K} \\Pr(Y_i=k) = 1", "f7ce00f600e154d0f0b4c499ce906093": "~\\hat A={\\hat U}^\\dagger \\hat a \\hat U~", "f7ce09e330cb20bbfc1ee9ba2715d2f8": "k = \\frac12(p-1)", "f7ce2dcff7f531826988c29e22dd034c": "A\\leq \\left\\langle H\\right\\rangle_{0} - T S_{0} \\,", "f7ce49fb42e4de691b72898d3cdd2ad3": " P (X_n \\geq c_2 \\theta) \\geq \\frac{(1-\\theta)^2}{c_1}.", "f7ce8249cf52255cc26b47d4375da4b6": "H_n(D^n,S^{n-1})\\cong \\mathbb{Z}", "f7ce8f6f4ce81d2be2de99ec0922ed44": " t \\geq 3 ", "f7cea1f24466d18e01634fe7dd534e12": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty}\n\\left( \\frac{Q_{l}}{r^{l+1}} \\right)\nP_{l}(\\cos \\theta)\n", "f7cec8334a3e31711b07d43cf166afd0": "v''=0", "f7cef446bf833331ca4e76bd5d74e7ea": "\\mod", "f7cf021eeb5e9d8eb7fece8ec4c324ce": "\\mathcal{N}(0,C_{k+1})", "f7cf7c5eab146b8e6f2eb68440473758": "X \\subset U(d)", "f7cf84866dacecc962050d4839e22055": " \\begin{array}{rrrrll} \nk & N_k & s_k & t_k & t_k/s_k & N_k/N_{k-1} \\\\\n0 & 0 & 0 & 0 & & \\\\\n1 & 1 & 1 & 1 & 1 & \\\\\n2 & 36 & 6 & 8 & 1.33333 & 36\\\\\n3 & 1\\,225 & 35 & 49 & 1.4 & 34.02778\\\\\n4 & 41\\,616 & 204 & 288 & 1.41176 & 33.97224\\\\\n5 & 1\\,413\\,721 & 1\\,189 & 1\\,681 & 1.41379 & 33.97061\\\\\n6 & 48\\,024\\,900 & 6\\,930 & 9\\,800 & 1.41414 & 33.97056\\\\\n7 & 1\\,631\\,432\\,881 & 40\\,391 & 57\\,121 & 1.41420 & 33.97056\\\\\n\\end{array} ", "f7cfa43eef755831238b310257333501": "\\displaystyle L = N \\Phi/i", "f7cfa9945b2ce6ce69c4c72de7ac82d2": "\\det(X + AB) = \\det(X) \\det(I_\\mathit{n} + BX^{-1}A)", "f7cfeb09e3db26d951667a65aca57e2f": "P=I", "f7cff1e88785b413c141ec12096f82ab": "1 + i_\\$", "f7d00b856e4db44cf57ca6a6543da0bc": "\\textstyle D(a_i)", "f7d01dbb8da67610d0f097421fbeeab4": " 1 = \\int\\limits_R d x \\, | x \\rangle \\langle x | ", "f7d02da87581b41b30d516de4cd09256": "\\phi:U\\rightarrow\\{\\zeta:00}(1-e^{-\\alpha})},", "f7dd9e23c95d273c31dec3c722da4b7b": "{\\omega \\wedge \\eta(x_1,\\ldots,x_{k+m})} = \\sum_{\\sigma \\in Sh_{k,m}} \\operatorname{sgn}(\\sigma)\\,\\omega(x_{\\sigma(1)}, \\ldots, x_{\\sigma(k)}) \\eta(x_{\\sigma(k+1)}, \\ldots, x_{\\sigma(k+m)}),", "f7dda455d52f344164034929b7067e68": "(X, \\| \\cdot \\|)", "f7ddc23edfabf61b128836cf41c4c54d": "\\nu (f, a)", "f7de1c1619188c8f5dbe130a8c3752f6": "\\scriptstyle \\eta", "f7de2363e318fa298eb57c4c65a61544": "\n \\textbf{y}=\\begin{bmatrix}a_1\\\\b_1\\\\s\\\\t\\end{bmatrix}\n", "f7de2f191c8527a3875916de9038f674": "x^2 - Ny^2 = k_2", "f7deb3f96fc21099f23236857a69da66": " \\frac{\\text{d} [{^{d_h}_{c_h}}P^{\\gamma_h}_h ]}{\\text{d}t} \\simeq \\sum_{i=1}^m \\frac{ u_{\\gamma_{hi}} y_{d_{hi}} \\text{k}_{3(i)} E_0 \\overline{S}_i }{ \\overline{S}_i + K_i \\left(\n1+ \\displaystyle\\sum_{p\\neq i} \\dfrac{\\overline{S}_p}{K_p} \\right) } \\qquad \\qquad (9b) ", "f7def6c75d1b37ca9165153653d27570": "\\left\\langle Y,{{g}_{m}} \\right\\rangle =\\sum\\limits_{n=0}^{N-1}{Y\\left[ n \\right]}.g_{m}^{*}\\left[ n \\right]", "f7defce581c6ace6557b3c2af8abb5be": "\\underline{\\underline{\\mathsf{A}_\\sigma}}", "f7df0530f75441d22db7ba80accd136f": " \\frac{dI}{dt} = a E - (\\nu +\\mu ) I ", "f7df3458ad1231f43b8d3b77021498af": "((P \\to Q) \\and (Q \\to P)) \\to (P \\leftrightarrow Q)", "f7df5fb7fe4a2e2a36da4c5ffaff11f6": "\\theta(x) = \\tan^{-1}\\big(\\mathfrak{Im}[f(x)]/\\mathfrak{Re}[f(x)]\\big) \\quad (6) ", "f7df8cba066701076dce1200fab34b51": "\\left\\{\\begin{matrix} n \\\\ k \\end{matrix}\\right\\} = 0", "f7dfa2eeac25cd9563a3b8a50198cf88": "M_\\mathrm{e} = \\textstyle{\\frac{2}{3}}\\log_{10}E_\\mathrm{s}-2.9", "f7e0176337120cbf48db805a05981273": "\\begin{align}\n a_1 &= \\frac{1}{2}(24 + 6) = 15\\\\\n g_1 &= \\sqrt{24 \\times 6} = 12\n\\end{align}", "f7e059de868678821b7dc9b8cbc24572": "\\epsilon_2 := (\\epsilon \\otimes \\epsilon) : (B \\otimes B) \\to K \\otimes K \\equiv K", "f7e0661423f893513f177d7e2d699c3b": "(n = 0)", "f7e083e7cac96dd8303e950482d9fd4b": "\\lambda' = \\varphi(\\lambda)", "f7e0b4516d4fa8590f257268351b175a": "\\| F \\|_{1, p} := \\big( \\mathbf{E}[|F|^{p}] + \\mathbf{E}[\\| \\mathrm{D}F \\|_{H}^{p}] \\big)^{1/p}.", "f7e0d93bb9809751e06a8312227d35c0": "\\int_{a}^b P(x)^2 dx", "f7e10a974be35e2bdfb5d5496c87227a": "\\gamma_{k}^{i} = h(\\chi_{k\\mid k-1}^{i}) \\quad i = 0..2L ", "f7e131e22080f4a76bd643fd645e644d": "E=\\alpha-x\\beta", "f7e132cbda40f33c4b65c9319c85f6d1": "\\mathbf{F} = m \\mathbf{a} \\ ,", "f7e180996cc56de6607f77d4cadbb4c5": "\nt - T = \\sqrt{\\frac{2 r_{p}^{3}}{\\mu}}\\left (D + \\frac{1}{3}D^{3}\\right )\n", "f7e1acdebff58c99c55d0dee546c6a85": "\\Delta x_k = \\frac{1}{n} \\left[{\\frac{A}{x_k^{n-1}}} - x_k\\right]; x_{k+1} = x_{k} + \\Delta x_k ", "f7e1c4007cf6faf1b646d2777d4b24c5": "\\nabla F(\\mathbf{x})=A\\mathbf{x}-\\mathbf{b}", "f7e1e86996ef66d68350ab0b1eb31e14": "f(t-\\lambda t) - f(\\lambda t+t)=0", "f7e2348ff26948a96cbb47bd42cede39": "\\rho=0", "f7e27413bfca4c1208cea1f74f0038a1": "\\rm{det}(MXM^+) = \\rm{det}(X)", "f7e29bbe6fc99b7972d18cd71988b1f0": "\\ +\\ {y^\\prime}^2\\left(A\\sin ^2\\theta\\ -\\ B\\sin \\theta\\cos \\theta\\ +\\ C\\cos ^2\\theta\\right)\\ +\\ x^\\prime\\left(D\\cos \\theta\\ +\\ E\\sin \\theta\\right)", "f7e2b110cda81f5023209a3bc8ec4451": "\n{1 \\over \\left( 2 \\pi\\right)^2\\; 2^{n} \\; n! }\n\\int d^2u_{12} \\; d^2v_{12} \\; \\mid u_{12}\\mid^{2n} \\; \\exp \\left[ - 2 \\left( \\mid u_{12}\\mid^2 + \\mid v_{12}\\mid^2 \\right) \\right] \\;\\mathcal J_0 \\left ( {2} k\\mid u_{12} \\mid \\right)\n=\n", "f7e2c1a16750d7553cd09d8dccd5ee5a": "\\sigma_\\mathrm{n} = \\sigma_1n_1^2+\\sigma_2n_2^2 + \\sigma_3n_3^2=\\sigma_2n_2^2 + \\sigma_3n_3^2\\,\\!", "f7e36115840425940eabfc8db2eeef67": "{}_pF_q(a_1,\\dots,a_p;b_1,\\dots,b_q;z).", "f7e36b3d9282ae855a03f40641a086cc": "t \\in [0,1].", "f7e36f88a26f73f5eec72e80544c32be": "\\left ( \\sqrt{-1}\\right )^2 =\\sqrt{-1}\\sqrt{-1}=-1", "f7e36fc1c05cd0c46fd1a131c934d50c": "-V_o", "f7e3dde166139eed9b3d0f533f4d528c": "-\\left(x^{\\alpha+1} e^{-x}\\cdot L_n^{(\\alpha)}(x)^\\prime\\right)^\\prime= n\\cdot x^\\alpha e^{-x}\\cdot L_n^{(\\alpha)}(x),", "f7e5170580d71b60bdaf82354ee5d2bb": "\nn^{-1}\\sum_{i=1}^n\\sum_{j=1}^mI(X_i=Z_j)Z_j,\n", "f7e520709fc77dbd3ff190f3c1234407": "b^{2} - 1 \\equiv a^{r} - 1 \\pmod{N}", "f7e5252ff50b8032e5a0d1f81657c684": "M_\\pm=\\exp \\mathfrak{m}_\\pm", "f7e56f56fb84d64be8c34a8a489e5642": " \\text{HOMA-}\\beta = \\frac{20 \\times \\text{Insulin}}{\\text{Glucose}-3.5} %", "f7e5c59b366443c2c31bd1eafe64ca46": "\n\\begin{align}\n\\mathcal{A} \\psi_a(1)\\psi_b(2)\\psi_c(3) = &\n\\frac{1}{6} \\Big( \\psi_a(1)\\psi_b(2)\\psi_c(3) + \\psi_a(3)\\psi_b(1)\\psi_c(2) + \\psi_a(2)\\psi_b(3)\\psi_c(1) \\\\\n&{}-\\psi_a(2)\\psi_b(1)\\psi_c(3) - \\psi_a(3)\\psi_b(2)\\psi_c(1)- \\psi_a(1)\\psi_b(3)\\psi_c(2)\\Big).\n\\end{align}\n", "f7e5d36aa69be6448f7ca3a30dbb88a1": "\\epsilon_{IJ}", "f7e659d734360ba130fb0f144273d0c5": " 0 < i \\leq d(t)", "f7e65ccf028e5449cfad69bc2bc4772c": " |R\\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ {1\\over \\sqrt{2}}\\begin{pmatrix} 1 \\\\ -i \\end{pmatrix} ", "f7e6882ce8b30f4ee72cbbbe54209ab5": "\\Lambda(G)", "f7e69a5beff9e59ab2bf4d5c7d67f4a8": "\\frac{K_p}{K_d}=2\\zeta \\omega_0", "f7e6c0858a26c1e348d846c863a62808": "c2^{k}", "f7e72998e688ea7faa0905f1e68b8b57": "\n \\begin{align}\n (1) & & \\quad m~\\frac{\\partial^2 w}{\\partial t^2} & = \\kappa AG~\\left(\\frac{\\partial^2 w}{\\partial x^2} - \\frac{\\partial \\varphi}{\\partial x}\\right) + q(x,t) ~;~~ m := \\rho A \\\\\n (2) & & \\quad J~\\frac{\\partial^2 \\varphi}{\\partial t^2} & = EI~\\frac{\\partial^2 \\varphi}{\\partial x^2} + \\kappa AG~\\left(\\frac{\\partial w}{\\partial x} - \\varphi\\right) ~;~~ J := \\rho I\n \\end{align}\n", "f7e72eb5537bc127dabfa55e43f77c82": "h_n(F_2)=(n!)^2.", "f7e743b1bb7f610ed80a1b9ffc693c45": " E[Y]_{ab} = R_{ambn} \\, Y^m \\, Y^n", "f7e74d45146d488ee939549153a00635": "\\! v_{tot}", "f7e786fbf119fce9aaab9b5c3cb419df": " f(x) = \\frac{(\\cos\\alpha)x - \\sin\\alpha}{(\\sin\\alpha)x + \\cos\\alpha}, ", "f7e79575d8869293cb6087d6c45e0c85": " \\langle(S-z)^{-1} f,f\\rangle ", "f7e7bb278e84e0c53da12e9d34ee92c9": "SU(6)\\begin{matrix}6_H/\\bar{6}_H\\\\ \\longrightarrow \\end{matrix} SU(5)\\times U(1)", "f7e7e30190f9a7f6f7ef4c42ab9755a6": "\\dim V_k(\\mathbb C^n) = 2nk - k^2", "f7e883af686a0fe94a42da0a339b3d60": "\\gamma = \\rho \\, g", "f7e8a5cbf4aaa2180436fb31b85fdfe8": "\\dot{X}_a \\, X_b + X_{a;b} = {h^m}_a \\, {h^n}_b X_{m; n}", "f7e9e221e3369986e767ad157731cf34": "\\omega_\\infin=\\frac{\\omega_c}{\\sqrt{1-m^2}}.", "f7ea2718b026da41e0094ca5b62a07a8": "\n\\mathbf{w} \\propto \\mathbf{S}^{-1}_W(\\mathbf{m}_2-\\mathbf{m}_1).\n", "f7ea3616d4263fb0d15170f3dbb35cfa": "\n\\Delta \\hat{z}\\ =\\ -2\\pi\\ \\frac{J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos i\\ \\quad \\hat{n} \\times \\hat{z}\n", "f7ea7524c86c473367631318ed88a63b": "W_{tot} = W_{LC} + W_C + W_L. \\ ", "f7eaa62fbed48da264b844425795e412": "a\\ll r\\ll\\xi", "f7eb6bb348466cfb3b6b37051cba014f": "Y = X \\tilde{B} + \\tilde{B}_0", "f7ebdbc8f11867a5ea073eb2f7025d47": "\\Theta (x) := \\left \\{ \\begin{matrix}\n1 & \\mbox{for} & x \\ge 0 \\\\\n0 & \\mbox{for} & x < 0\n\\end{matrix} \\right.", "f7ebed514dd5267e66d19c5812d9c9a7": " D4\\sigma = 4 \\sigma = 4 \\sqrt{\\frac{\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y) (x-\\bar{x})^2 \\,dx\\, dy} {\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}I(x,y)\\, dx \\,dy}} ", "f7ec572d1e1ab2bd11984a0eb562f05a": " e_k \\otimes f_\\ell ", "f7ec65b021d079bf55b6217be174dd37": "R=\\frac{I(r,d)}{\\delta I_b}=1-\\frac{1}{Q}\\exp \\left [ \\frac{eV}{2kT\\epsilon}(1-Q) \\right ]", "f7ec9543256199142afbc6b85c71c02a": "\\mathcal{Z}(R_R)=\\{0\\}\\,", "f7ec9afcd97c8f72df165e9e6175195f": "\\displaystyle \\sqrt{\\frac{\\pi}{2}} \\left( \\frac{1}{i \\pi \\omega} + \\delta(\\omega)\\right)", "f7ed074007ac335ce1c59433ad7e4e3d": "\n\\frac{\\Gamma \\vdash e_1\\!:\\!real \\quad \\Gamma \\vdash e_2\\!:\\!real}{\\Gamma \\vdash e_1+e_2\\!:\\!real}\n\\qquad \\frac{\\Gamma \\vdash e_1\\!:\\!integer \\quad \\Gamma \\vdash e_2 : integer}{\\Gamma \\vdash e_1+e_2\\!:\\!integer} \\qquad \\cdots\n", "f7ed0cf084ec0cd5af5fa276ba4d4b41": "\\|\\beta\\|_1 = \\textstyle \\sum_{j=1}^p |\\beta_j|.", "f7ed3de9eb8537deaeb70afa6aa576fb": "\\Gamma = \\frac{E^-}{E^+} ", "f7edaad8011bcffa4da6f69e7b6a009e": "\n dv = J~dV \n\\,\\!", "f7ee526c38dd1d10a3b32ed501a55390": "\\scriptstyle\\ \\alpha", "f7ee6f90b29c0d0a10113706d000d756": "\\limsup_{\\varepsilon \\downarrow 0} \\varepsilon \\log \\mathbf{P} \\big[ X^{\\varepsilon} \\in F \\big] \\leq - \\inf_{\\omega \\in F} I(\\omega)", "f7eec817ead6f506f792eb3a97a190cb": "\\mathbb T", "f7ef1e424ae5702b00061635abe0415b": "p:\\tilde X\\to X", "f7ef47b16420bf199f599c4e50b4e21e": "\\sigma_{cl} ", "f7ef9971d02973620263f1861eefcf01": "\n\\bigg(\\frac{\\alpha}{\\mathfrak{a} }\\bigg)_n \n=\n\\left(\\frac{\\alpha}{\\mathfrak{p}_1 }\\right)_n \n\\left(\\frac{\\alpha}{\\mathfrak{p}_2 }\\right)_n \n\\dots\n\\left(\\frac{\\alpha}{\\mathfrak{p}_g }\\right)_n. \n", "f7efa6fa8d49d24ab8b398dce26e0295": "\n\\begin{align}\nD & = -2\\ln\\left( \\frac{\\text{likelihood for null model}}{\\text{likelihood for alternative model}} \\right) \\\\\n&= -2\\ln(\\text{likelihood for null model}) + 2\\ln(\\text{likelihood for alternative model}) \\\\\n\\end{align}\n", "f7efaea6a0302930c46f7b471a4853c2": "\\gamma(0) = x\\,", "f7efb33d300ebd60b1855c83450471b3": "R_1 \\subseteq \\mathbb{R} \\,, \\quad R_2 \\subseteq \\mathbb{R} \\,, \\ldots , R_n \\subseteq \\mathbb{R}, ", "f7efc1b33295b9cc3ea28e9e1c43587b": "P_n^{(\\alpha,\\beta)} (z) = \\frac{\\Gamma (\\alpha+n+1)}{n!\\,\\Gamma (\\alpha+\\beta+n+1)} \\sum_{m=0}^n {n\\choose m} \\frac{\\Gamma (\\alpha + \\beta + n + m + 1)}{\\Gamma (\\alpha + m + 1)} \\left(\\frac{z-1}{2}\\right)^m~.", "f7f06940d92e53285ebde0e0e3cb6ac3": "V \\subseteq T", "f7f0b2f682a1b0b536310ea17d78578d": "\nF(r) = \nm \\ddot{r} - m r \\omega^{2} = \nm\\frac{d^{2}r}{dt^{2}} - \\frac{mh^{2}}{r^{3}}\n", "f7f0e2b586d7cc1982e91017a0c9bbb3": "v[\\mathbf{f}A]^\\mathrm{T} G[\\mathbf{f}A] = v[\\mathbf{f}]^\\mathrm{T} (A^{-1})^\\mathrm{T} A^\\mathrm{T} G[\\mathbf{f}]A = v[\\mathbf{f}]^\\mathrm{T} G[\\mathbf{f}]A", "f7f14f77664a13393d725e49dfeb5afb": "S_\\mathrm{tent}(x) =\n\\begin{cases}\n2x & \\text{for } 0 \\le x < \\frac{1}{2} \\\\ \n2(1-x) & \\text{for } \\frac{1}{2} \\le x < 1\n\\end{cases}\n", "f7f1501ffbc06133991ffde9d35cac8c": " [I_S] = -\\sum_{i=1}^n m_i[r_i-S][r_i-S],", "f7f177957cf064a93e9811df8fe65ed1": "\\rho ", "f7f1eea6578078c5aa52c84557632686": " S = \\left( \\frac{ 2^k - 1 }{ k } - 2^k \\right)( 1 - 2k )^{ 0.5 } ", "f7f20b717504721e7184bd394e4b8862": "-\\Delta u = f, \\qquad u|_{\\partial\\Omega} = 0", "f7f238d428aac043c5b5d827846561eb": "\\gcd(a,b)=k d", "f7f246a31c938488a94925dfd0e7d5f8": "\n\\mathbf{A}\\mathbf{X} = [\\mathbf{A}_1:\\mathbf{A}_1\\mathbf{B}]\\begin{pmatrix}\n-\\mathbf{B} \\\\\n \\mathbf{I}_{n-r} \n\\end{pmatrix} = -\\mathbf{A}_1\\mathbf{B} + \\mathbf{A}_1\\mathbf{B} = \\mathbf{O}\\; . ", "f7f24d85d8d26c3c1a4d6e2b58a74d37": "F^{-1}(u) = \\inf\\;\\{x \\mid F(x)\\geq u, 01", "f8072c31740bd5c5f41b66a6a81229c8": "\\overline{\\mathbf{F}}_p", "f807741a3f812f27b1ece2f94f05869e": "\\beta\\gamma", "f8078953224243654ac496e62d418104": "\\cos\\frac{\\pi}{15}=\\cos 12^\\circ=\\tfrac{1}{8} \\left[\\sqrt{6(5+\\sqrt5)}+\\sqrt5-1\\right]\\,", "f8078a172c0ffb5770320c38682d3715": "\n\\begin{bmatrix} \\vec{y} \\\\ 1 \\end{bmatrix} = \\begin{bmatrix} A & \\vec{b} \\ \\\\ 0, \\ldots, 0 & 1 \\end{bmatrix} \\begin{bmatrix} \\vec{x} \\\\ 1 \\end{bmatrix}\n", "f8079ec002547656e4855dbc78a8abf7": " j \\omega \\ ", "f807e01ac2bc41e8286ebd9e777fc5d4": "d_u ", "f808e6e09aa3d83839f581de3cdddf09": "U(t)=\\mathcal{T}\\left[\\exp\\left(-\\frac{i}{\\hbar} \\int_{t_0}^t \\,{\\rm d}t'\\, H(t')\\right)\\right]\\,,", "f8091aa5c67850d6fb62bce537c23f0e": "T\\!", "f809a0f8e7cbdce3893079bb844878a5": "\n\\sigma _z^2 \\,\\,\\, \\approx \\,\\,\\,\\sum\\limits_{i\\, = \\,1}^p {\\,\\sum\\limits_{j\\, = \\,1}^p {\\left( {{{\\partial z} \\over {\\partial x_i }}} \\right)} } \\left( {{{\\partial z} \\over {\\partial x_j }}} \\right)\\sigma _{i,j}{\\mathbf{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,Eq(13)}}", "f809b02e15981ad933a966bec4b3d6bf": "\\sigma_D^{(0)}", "f809c0837097e027f2cd9feb0d58e711": "\nJ = \\epsilon\\sigma T^4 + (1 - \\epsilon) H \\,\\!\n", "f809cfb1eb55f8b21609ce03e2786286": "\\begin{align}\nZ(X,Y)&{}=\\log(\\exp X\\exp Y) \\\\\n&{}= X + Y + \\frac{1}{2}[X,Y] +\n\\frac{1}{12}[X,[X,Y]] - \\frac{1}{12}[Y,[X,Y]] \\\\\n&{}\\quad\n- \\frac {1}{24}[Y,[X,[X,Y]]] \\\\\n&{}\\quad\n- \\frac{1}{720}([[[[X,Y],Y],Y],Y] +[[[[Y,X],X],X],X])\n\\\\\n&{}\\quad +\\frac{1}{360}([[[[X,Y],Y],Y],X]+[[[[Y,X],X],X],Y])\\\\\n&{}\\quad\n+ \\frac{1}{120}([[[[Y,X],Y],X],Y] +[[[[X,Y],X],Y],X])\n+ \\cdots\n\\end{align}", "f80a14d1074a983a727ecb9204af54a3": "x,y,h\\in\\mathbb{C}^n", "f80a1f75c14afad4989542fe472da809": "P(M(d_n), t) = t^{d_n} P(M, t) + ", "f80a4103d931ac116f066eb681fb8891": "\\vec f^n", "f80ad235123ea82a0ca928c91907c88b": "\\forall p: \\mathcal{B}p \\to \\mathcal{BB}p", "f80b0146c870fc337cc288205a99d5ab": "R_{xs}[i] = \\sum_{k=0}^{N-1}\\hat{h}_kR_{ss}[i - k] ", "f80b0fcc4f910cde9303a8f2e8d9b5f1": "W=W_m+W_f+W_t", "f80b246cdd5fd8ff3302160e262a2843": "y(t)=\\int_{-\\infty}^0x(t+\\tau)e^{\\lambda\\tau}\\,{\\rm d}\\tau", "f80b4a7ca2247fb599c3e445efc76280": "|\\hat{x}(\\omega)|^2", "f80b6ecc9295c9eb2bcf575dda10b886": "\\sigma_x\n= \\frac{\\partial^2B}{\\partial z^2}\n+ \\frac{\\partial^2C}{\\partial y^2}", "f80b91a3666e4b2555f3fe30eddeb817": "\\hat{\\textbf{x}}_{k\\mid k-1} = \\sum_{i=0}^{2L} W_{s}^{i} \\chi_{k\\mid k-1}^{i} ", "f80b9bd4ea57fd778b14be3b71e9616f": "2^{O((\\log n)^c)}", "f80bb940da560aae7d3f89e032140def": "\\ R_2 = \\infty", "f80c1c2edc53eb7b135670e253d7d8ef": "\\bar\\eta^a_{\\mu\\nu}", "f80c2201fbaa7b7170d77587f300aece": "\\operatorname{mr}(K_n) = 1", "f80c3042b103829d93bab9a7a615492e": " \\infty ", "f80c374af874f7982720fbfb55ddd079": " |f| ", "f80c512884e9c6e48bf6c93b14f66f6b": "t\\mapsto H(|f\\circ \\gamma(t)|)", "f80c517146f74032b91befb9e67713fe": "\n f(x_1,x_2,\\ldots,x_n\\;|\\;\\theta) = f(x_1|\\theta)\\times f(x_2|\\theta) \\times \\cdots \\times f(x_n|\\theta).\n ", "f80c7b6d02eee0f4f8c1829976bcbbec": "d_{A^*}", "f80c8a65c991c5408d175331840ae36e": " d\\xi= \\frac{dn_i}{\\nu_i}", "f80cab52d774ac1cc518d6009ec2433e": " \\operatorname{let-combine}[\\operatorname{let} p : \\operatorname{de-lambda}[p\\ f = (\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))] \\operatorname{in} p] ", "f80cba7d0111d40bd61e02a401db8bea": "(x, X)", "f80cfb9aed60252093773929cd8f0539": "[=] = \\{(x,y) \\mid \\,x=y\\}", "f80d15ce5965e578517451cedce06f35": "N_R", "f80d1d86b1cbe143f8ed328d00d64c2b": "\\displaystyle (\\alpha)", "f80d28ebda7b03f382dba80e05f9a1ff": "\\frac{X_{n+1}-\\overline{X}}{\\sqrt{1+(1/n)}} \\sim N(0,1).", "f80d2cf3dc70966277a4edbc6587c43c": "\\langle A\\rangle = \\sum_s A_s P_s = -\\frac{1}{\\beta}\n\\frac{\\partial}{\\partial\\lambda} \\ln Z(\\beta,\\lambda).", "f80d9586152b0fdb024cad026e5d7e8e": "\\,F_{n+k} + (-1)^k F_{n-k} = L_k F_n", "f80d9d3be7e76894a6b1fb5e426304b9": "z \\rightarrow 0", "f80dc76617a1b829044599466bf2ee65": "\\eta^2", "f80dd8531b8d502531d310bce224fe59": "\n\\mathbf{r} \\times \\left( \\mathbf{r} \\times \\frac{d\\mathbf{r}}{dt} \\right) = \\mathbf{r} \\left(\\mathbf{r} \\cdot \\frac{d\\mathbf{r}}{dt} \\right) - r^{2} \\frac{d\\mathbf{r}}{dt}\n", "f80e1ac23d21bf4b1bd9b1adf70e992f": "h_C: X^* \\to \\mathbb{R}", "f80e483f99b55b7e73bbe38c74130e3b": "B_{s,s^\\prime}=\\Omega^{-1}(NM)_{s,s^\\prime}={A(\\psi_s\\lambda_{s^\\prime})\\over \\Omega}", "f80e4d874254729900d473111b9dbfb8": "\nz = g^{-1} (y')\n", "f80e6231164df8ea7353fa969a9440e9": "\\big. \\frac{\\partial E_\\mathrm{in}}{\\partial t} - \\frac{\\partial E_\\mathrm{out}}{\\partial t} = \\oint_S \\overrightarrow{\\phi_q} \\cdot \\, \\overrightarrow{dS}", "f80eab241b9a2baf39ec5bbd6bbf2cf0": "\\gamma = \\alpha + i\\beta = \\frac{1}{2} \\cosh^{-1} \\left(\\frac{2m^2}{\\left(\\frac{\\omega_c}{\\omega}\\right)^2 - \\left( \\frac{\\omega_c}{\\omega_{\\infin}} \\right)^2} - 1 \\right) + i\\frac{\\pi}{2}", "f80ed9ead81609c27a74e90454b29cb6": "L^{(0)}_n(x)=L_n(x).", "f80ef83b3054d5d2f1bc0d56706c3557": "i\\frac{k}{i} = i(-j) = -k", "f80f36393d943373787ed88f9b838b1b": "G_F(\\tau=0^-)=\\frac{1}{\\beta}\\sum_{i\\omega_m}\\frac{e^{i\\omega_m 0^+}}{i\\omega_m-\\xi}=n_F(\\xi)", "f80f8c1135a24cbb0e65c790a7966765": "|X_{i}-X_{j}|^{2}=|Y_{i}-Y_{j}|^{2}\\,\\!", "f80fdce2b1d2ff065a1dafe9893e5ab8": "a_n = \\frac{a_{n-1} a_{n-4} + a_{n-2} a_{n-3}}{a_{n-5}}.", "f80ffadc1ec84ad0889f130fe74cc093": "\\eta_2 = -\\beta,", "f810772e5fa55a2a47cae5e69c554a66": "C =", "f8107dc2b742a01eb8315039600b4491": "\\beth_{\\alpha+1}=2^{\\beth_{\\alpha}},", "f81099cc8464446a61d421cee1dde069": "R\\textbf{u} = I \\textbf{u} \\quad \\Rightarrow \\quad (R - I) \\textbf{u} = 0", "f8117106591d912591cc4e570dc72dba": "=\\frac{2\\sin^2(\\frac{\\alpha}{2})}{1-2\\sin^2(\\frac{\\alpha}{2})}", "f81174c22914eecbd40c643f5a0a5c05": "F = eN_A", "f81183fc54f831bd50ee7f40d7d11337": "|w_n(x)| < \\left|\\frac1{2^{n-1}}T_n(x)\\right|", "f81203dcee12c1f181cdd805be3b51fd": "\\left|n,l,m\\right\\rangle", "f812241c405fb89f0bed2b263871265a": "A \\otimes_k B \\to AB", "f812ef31dee7859c472f7756984adfd8": "u,v,w", "f812f25372a9d2802046adaa84e5c34c": "r^{-\\ell_1}", "f8133c3215253869c2f013d0fc1e30c6": "E_{12} ", "f8139e5e06742f4d0007ee07fb901edd": "a_n \\ll_\\epsilon n^\\epsilon", "f813fa1caa8aedd3252f77d10fa9f29e": "\\begin{align}{\\det}^{-2}(C^i_j) &= \\exp\\left[-2{\\rm tr}\\ln(\\delta^i_j-i\\int d^dx\\, \\alpha(x)\\psi^{\\dagger i}(x)\\gamma_{d+1}\\psi_j(x))\\right]\\\\\n&= \\exp\\left[2i\\int d^dx\\, \\alpha(x)\\psi^{\\dagger i}(x)\\gamma_{d+1}\\psi_i(x)\\right]\\end{align}", "f81424f8587e2d32b78c3ad8b3a1f76f": "\n\\delta A_\\mu(x) = \\partial_\\mu \\lambda(x)\n", "f8142cf29b0b1bacba6d55294c8591d4": "x=\\sum_{i=0}^n\\frac{a_i}{10^i}=\\sum_{i=0}^n10^{n-i}a_i/10^n", "f8143540cd324396af4de62cae9bfa68": " F(x;\\alpha,\\beta) = \\int_0^x f(u;\\alpha,\\beta)\\,du= \\frac{\\gamma(\\alpha, \\beta x)}{\\Gamma(\\alpha)}", "f8152509e6e5e8a674c4e9226dfe476b": "W(T)", "f81546cd5e6ec13d963e358c93de74c6": "I(\\nu, T) = \\frac{2 h \\nu^3}{c^2} \\frac{1}{e^{\\frac{h \\nu}{kT}}-1}", "f815770614d6eff34c7e17fc1f6df156": "n(F) = \\bigg\\lfloor \\log_\\varphi \\left(F\\cdot\\sqrt{5} + \\frac{1}{2}\\right) \\bigg\\rfloor", "f815a769f068a79d2f37a9e3f290a24a": "\\displaystyle{A=\\begin{pmatrix} 0 & J\\\\ -J & 0\\end{pmatrix}.}", "f815b20c3a99402600f09a5f31a2b7cc": "\n \\boldsymbol{\\sigma} = \\begin{bmatrix} -p +\\cfrac{\\mu J_m (1+\\gamma^2)}{J_m - \\gamma^2} & \\cfrac{\\mu J_m \\gamma}{J_m - \\gamma^2} & 0 \\\\ \\cfrac{\\mu J_m \\gamma}{J_m - \\gamma^2} & -p + \\cfrac{\\mu J_m}{J_m - \\gamma^2} & 0 \\\\ 0 & 0 & -p + \\cfrac{\\mu J_m}{J_m - \\gamma^2}\n \\end{bmatrix}\n ", "f8167db4008a3b2c3ebee121f809ecaa": "\\int_M d\\omega = \\oint_{\\partial M} \\omega.", "f8169cf6e37f15541c749da7d199e3b8": " R^3_5(\\rho) = 5\\rho^5 - 4\\rho^3 \\,", "f816c82e586135c8dde8bd78127bd9a9": "r_M = z_M \\cos \\phi", "f817896dacc0964c513684b70084a09c": "\\psi(\\Omega^{\\Omega+1})", "f81795f5cd2e69b9e65cb1adf15c681c": "\\pi'", "f8182530bb2a3c3cffc778c966f8cb97": " p_{1,1}(x) = y_1 \\, ", "f81842e660214a83085e5dd15222503f": " (J_{F^{-1}})(F(p)) = [ (J_F)(p) ]^{-1}.\\ ", "f81864951b263c6c631b2ae238890555": "E_0^-(x)", "f81892d10c45f6efb6ddaa3b95f35736": "\\hat{f}(r\\alpha) = \\int_{-\\infty}^\\infty ds\\int_{\\mathbf{x}\\cdot \\alpha = s} e^{-2\\pi i r(\\mathbf{x}\\cdot\\alpha)}dm(\\mathbf{x}).", "f818a364019a4f5e52411fa605622095": "\\mathcal{H}_1 \\subset \\mathcal{H}_{Kin}", "f8190d40bc58ad2acd04d2ca2072fe43": "\\vdash \\Psi", "f819153ed930a952ee4dfc6f82c0deec": "d\\omega(X,Y)+[\\omega(X),\\omega(Y)]=0", "f8194769c7cbd7ac21bc09e6286e1afa": "z = z_0", "f819555b9ebdd85d51a0d946e634f8fe": "{\\pi\\over 5}\\ {2\\pi\\over 5}\\ {\\pi\\over 2}", "f819afa426cc65b34ff2830857025055": "x_t = x_0 \\cdot a^t,", "f819cf582a8daf4782e3645b4e889c31": "f(z) = {-1 \\over 4(z-i)^2} + {-i \\over 4(z-i)} + {3 \\over 16} + {i \\over 8}(z-i) + {-5 \\over 64}(z-i)^2 + \\cdots", "f819dccc764da17c2b68d888b15c3d71": "\\phi: M \\to X", "f81a4c33be4f8a97f6258be85873b223": " \\sin ^2 \\theta + \\cos ^2 \\theta \\equiv 1\\,", "f81a60776df1f21cf52ed1249b636048": " \\operatorname{let} p : p\\ f = \\operatorname{let} x : x\\ x = f\\ (x\\ x) \\operatorname{in} f\\ (x\\ x)] \\operatorname{in} p ", "f81a63d9693ed20c017604398d0a38dc": "(x_1, ... ,", "f81a6feeb5947be7057035050ef335e8": "P \\rightarrow_{b} Q", "f81a8a6f4ffbd22a04cdbc763f6aed7b": "n=2^{k-1}", "f81a9a6a7a151f0b7da12ff01b09c01a": "\\scriptstyle \\mathcal{C}^I", "f81a9e4fc98f17ea48a9d67b9e13294a": "V_{\\rm{particle}} = (1.5 {\\rm\\ {\\mu}m})^3 = 3.38 \\times 10^{-18}{\\rm\\ m}^{3}", "f81aa29c350969b2d2b99a7280639444": "\\gamma = e^{\\pi^2/(12\\ln2)} = 3.275822918721811159787681882\\ldots.", "f81ae601987df6ef5a945ce2735a83e2": "\\vec p", "f81b6ebbd17a738bb9fb2df2114257d4": "Q = N/D", "f81b9bd3a513de7c221c2bc337b2e0d2": " \\{W(f):~f\\in H\\}", "f81bfd1d4b39c26d0c7632204ad0445d": "P^{(\\pm)} [F , G]^{IJ} = [P^{(\\pm)} F, G]^{IJ} = [F , P^{(\\pm)} G]^{IJ} = [P^{(\\pm)} F , P^{(\\pm)} G]^{IJ} \\;\\;\\;\\;\\; Eq.2", "f81c34be8eb8257a689cae6482ef7ce8": "\\Delta t=a/n,", "f81c9a574702349f3b2241b92c9ccad9": "\\displaystyle J_1(x)", "f81c9f641d0eff896f55ac679085b2f1": "\\displaystyle{gB(x,y)g^{-1}=B(gx,(g^t)^{-1}y)}", "f81ca48fdbc1ce3a97e4e7a354d61efc": "Q_{max}=\\frac{S^2T^2}{2\\cdot R_{Total}}=\\frac{S^2T^2A}{2p_cL}", "f81cbc5a09d0d71014cb1b4eb7eeff2d": "p^e = p^{e}_{-1} + \\lambda (p - p^{e}_{-1})", "f81cd1579b99b08df4e3f06e94b98c7d": "\\scriptstyle\\hat\\sigma^2", "f81d006e098c94b3ab161297fecee850": "\nK = \\gamma \\cdot p\\,\n", "f81d510e30c970908a9a9a0d85263ce9": " {d \\over dt}\\mathbf{|L|} = m l^2 {d^2\\theta \\over dt^2} ", "f81d5deebed1445eaf8a9062ceca1649": "j=\\sqrt{-1}", "f81dee8bc3d804d0a5fbdb6dd799dfe0": "\ndV = \\frac{a^3 \\sin \\sigma}{\\left( \\cosh \\tau - \\cos\\sigma \\right)^3} \\, d\\sigma \\, d\\tau \\, d\\phi\n", "f81e45ddd12d8f7c1f1e61f164138725": "n c_v dT - V dp/ \\gamma = 0", "f81e62d976076324efe5697e5faa6cdc": "~(\\cos(x))^{-1}~", "f81e6f72932be48cf4728569d7d85d38": "\\lambda_J=r", "f81ee624b2ccee875687773c1c9c7213": "\nF^{\\mu}_{\\nu} = \\begin{bmatrix}\n0 & E_x/c & E_y/c & E_z/c \\\\\nE_x/c & 0 & B_z & -B_y \\\\\nE_y/c & -B_z & 0 & B_x \\\\\nE_z/c & B_y & -B_x & 0\n\\end{bmatrix}.\n", "f81f4f1fede295d97b92a27b21465010": "\n\\begin{pmatrix}-(1)&\\alpha^{-1}+\\alpha^{6}x\\\\\n\\alpha^{3}+\\alpha^{1}x&-(\\alpha^{-7}+\\alpha^{7}x+\\alpha^{7}x^2)\\end{pmatrix}\n\\begin{pmatrix}S(x)\\Gamma(x)\\\\ x^6\\end{pmatrix}=\n\\begin{pmatrix}\\alpha^{4}+\\alpha^{7}x+\\alpha^{5}x^2+\\alpha^{3}x^3+\\alpha^{1}x^4+\\alpha^{-1}x^5\\\\\n\\alpha^{7}+\\alpha^{0}x\n\\end{pmatrix}.\n", "f81fe4ccec62d68ccf008e16a2d61877": "\\ 1.15m+4ln(n)", "f820292513f63a44e3d58392bbbead2a": "\nS = \\underbrace{S_{A}}_{Aircraft\\,Surf.}\\;+\\;\\underbrace{S_{D}+S_{\\infty}}_{Far\\,Surf.}\n", "f82037b4d910e960a94e6467c62054c5": "L_z \\ \\equiv \\ (r_i \\ \\times \\ p_i)_z", "f82045c60fed145b769a1d2d9e7d453c": "\n\\begin{align}\n\\frac{a}{b} &= q_0 + \\frac{r_0}{b} \\\\\n\\frac{b}{r_0} &= q_1 + \\frac{r_1}{r_0} \\\\\n\\frac{r_0}{r_1} &= q_2 + \\frac{r_2}{r_1} \\\\\n& {}\\ \\vdots \\\\\n\\frac{r_{k-2}}{r_{k-1}} &= q_k + \\frac{r_k}{r_{k-1}} \\\\\n& {}\\ \\vdots \\\\\n\\frac{r_{N-2}}{r_{N-1}} &= q_N\n\\end{align}\n", "f8204a9f5a28e4df570b5d1e0c8ce95e": "\\begin{align} a & = e^{i\\pi/n} = \\cos\\frac{\\pi}{n} + i\\sin\\frac{\\pi}{n} \\\\\n x & = j\n \\end{align}\n", "f82080e0e7564b5e08fc270acedd4124": " R = \\sum_{k=1}^{M} p_k \\cdot \\mathrm{length}(c_{k}) = \\sum_{k=1}^{M} \\mathrm{length}(c_k) \\int_{b_{k-1}}^{b_k} f(x)dx ", "f820ebf9ae1baa74ccb23710d5baf203": "Cf", "f821033ec2e9a74dbd025558c955601e": " g(\\theta) ", "f82153d424d4075b88be70f6ab071fb3": "1/(2 \\mathrm{NA_i})", "f82154e866e84e5128699903c35566f3": "A,B,C,D, \\dots", "f8215ca52b20a9492eea0f359e760755": "\nT = \\frac{D_P}{D_Y}\n", "f8216fcde1c240da3255e30533f6aadd": "\\scriptstyle{\\psi(x,t_0)}", "f8217f8c2a56a6a2d314c740797f75a8": "\nu_3 = -\\frac{\\vec{r}\\cdot\\vec{p}}{m r^2}.\n", "f821826c8938b4362afcf5229f9e5e29": "\n\\left( \\frac{1}{\\lambda_{1}} J_{i} \\right)^{k} \\rightarrow 0\n", "f821939137dc8572d17f5b1c37464b16": "23 = 4", "f821da43df4850f1e25486fd88bf8426": "HS|\\alpha\\rangle=SH|\\alpha\\rangle=SE|\\alpha\\rangle=ES|\\alpha\\rangle", "f82209ce4a1beb8edb3fbef317e38457": "\\wp(z+1) = \\wp(z+\\tau) = \\wp(z).", "f8224da4b12115246953bc9362804be7": "y(p)", "f8228e343e33b28208ba70bf6b969341": "b_x", "f822943e0c6b5002a4eb16d73ae5e391": "p,q,r", "f82294819087f4526e4294bd06c2ff97": "g_i(x) \\ge 0", "f8229bec1548ccd0f75cc74d49dfe60f": " s_1 s_2 s_3 \\dots s_n", "f822a1e24bd5a9c74bc903b01d54ffb7": " \\sum_{i=0}^n \\Delta h_i = 0 ", "f8230873c31ad5f5c5bb96dd8d03b301": "\\tau(n)\\equiv n^{-610}\\sigma_{1231}(n)\\ \\bmod\\ 3^{7}\\text{ for }n\\equiv 2\\ \\bmod\\ 3", "f82317d486211abad77d4449223ea919": "\\frac{\\partial n_i}{\\partial t}= - {\\rm div} \\mathbf{J}_i =- \\sum_{j\\geq 0} L_{ij}{\\rm div} X_j = \\sum_{k\\geq 0} \\left[-\\sum_{j\\geq 0} L_{ij} \\left.\\frac{\\partial^2 s(n)}{\\partial n_j \\partial n_k}\\right|_{n=n^*}\\right] \\Delta n_k\\ .", "f8233aa1fafe3ee7992984a6e13b8257": "n\\geq 3", "f82364c7a0a696d17affdf309248095c": "\\begin{matrix}\n1&2&3&5&8&13&21&\\cdots\\\\\n4&7&11&18&29&47&76&\\cdots\\\\\n6&10&16&26&42&68&110&\\cdots\\\\\n9&15&24&39&63&102&165&\\cdots\\\\\n12&20&32&52&84&136&220&\\cdots\\\\\n14&23&37&60&97&157&254&\\cdots\\\\\n17&28&45&73&118&191&309&\\cdots\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\ddots\\\\\n\\end{matrix}", "f8238100c1e3993f5cf54f7db69ff831": " x=x_2, \\ \\ y=y_2 \\ \\ ", "f823b0a7cc3c385fc087bd28b0ea55f3": "C=C_1+C_2", "f8245db7651f84fb92f2fc37d12ee78a": "\\langle \\Psi , \\Psi \\rangle = \\sum_{\\mathrm{all\\, }s_z} \\int\\limits_{\\mathrm{ all \\, space}} \\, d^3\\mathbf{r} |\\Psi(\\mathbf{r},t,s_z)|^2 = 1", "f8246e38f6645d13aad5ac5022f1d71d": " x_{ij}=0 \\qquad \\text{if} \\qquad p_{ij} \\geq T, \\qquad i=1, \\ldots, m, \\quad j=1, \\ldots, n", "f824ae6fb71d22dcf19bf870a1b0a5be": "V_t=\\frac{l_t S_t}{c S_w}", "f8256df6d16018ea501f8264e04dea10": "\\scriptstyle\\mathcal{F}", "f825908d8029ee1abda79fd609ba3d72": "\\frac{\\alpha}{\\pi} \\,", "f8259466603e05656870d2b1013c98d6": " \\{\\alpha_i, \\beta_i\\}", "f825de0c7689ec9bfd1b872049f09940": "C \\in \\{10, 100, 1000\\}", "f8262c0a652a6b0b4ec59c81416e243c": "Y_{base} = \\frac{1}{Z_{base}} = 1 pu", "f8265f41e4791367072311fa1430598c": " \\frac{(n+2)(n+1)}{2} ", "f826dfe6837b0e735e2adbd908a279dd": "z_{k+1}:=r", "f826fab58870b747d2892fe949b41f31": "B(t,T)", "f82719c8cf1554ae4b01c39535b51b10": "\n\\begin{align}\n{\\partial^3 \\over \\partial x_1\\, \\partial x_2\\, \\partial x_3}f(y)\n& = f'(y){\\partial^3 y \\over \\partial x_1\\, \\partial x_2\\, \\partial x_3} \\\\[10pt]\n& {} + f''(y) \\left( {\\partial y \\over \\partial x_1}\n\\cdot{\\partial^2 y \\over \\partial x_2\\, \\partial x_3}\n+{\\partial y \\over \\partial x_2}\n\\cdot{\\partial^2 y \\over \\partial x_1\\, \\partial x_3}\n+ {\\partial y \\over \\partial x_3}\n\\cdot{\\partial^2 y \\over \\partial x_1\\, \\partial x_2}\\right) \\\\[10pt]\n& {} + f'''(y) {\\partial y \\over \\partial x_1}\n\\cdot{\\partial y \\over \\partial x_2}\n\\cdot{\\partial y \\over \\partial x_3}.\n\\end{align}\n", "f827347bc5840dcb01696549624d3a39": "H = \\left\\{a+bi+cj+dk \\in \\mathbb{H} \\mid a,b,c,d \\in \\mathbb{Z} \\;\\mbox{ or }\\, a,b,c,d \\in \\mathbb{Z} + \\tfrac{1}{2}\\right\\}.", "f827394bdf66125dc587996ec8e228cb": " y(t_0+h) - y(t_0) = \\int_{t_0}^{t_0+h} f(t,y(t)) \\,\\mathrm{d}t. ", "f8274327e29407dcc1656d93a1e55249": " \\mu ", "f8278296b30d3d80114f2ce533c7ea9b": "\\frac{23h56'}{\\sin \\phi}", "f8278a8cc56bad54fac6022823b521fa": "r_K:C_K\\tilde{\\rightarrow}W_K^{\\text{ab}}", "f827a25b3d02b45aadb8255dc2ea2e82": "1 \\times 1 \\times 3", "f8281bb97e8277f0e2a6853d2db44d9a": "\\rho = \\frac{|\\gamma'|^3}{\\sqrt{|\\gamma'|^2 \\; |\\gamma''|^2 - (\\gamma' \\cdot \\gamma'')^2}}", "f82846662eac4f0c3491f4f548617193": "\\mu_{\\Lambda | \\Phi}(d\\lambda | \\phi) = {\\mu_{\\Phi,\\Lambda}(d\\phi, d\\lambda) \\over \\mu_{\\Phi}(d\\phi)} = \\frac{1}{2r\\pi} \\omega_r(\\lambda) d\\lambda \\ .", "f828566abe63ccaa17aa0a4a39a57fd0": "\\|\\mathbf{A}\\| = \\sqrt{\\mathbf{A}:\\mathbf{A}} = \\sqrt{\\sum_{i=1}^m \\sum_{j=1}^n (A_{ij})^2}", "f8285edd82e4ebdea7628bc2326f044b": "\\sum_{i=1}^{k-1}\\sum_{j=i+1}^k\\sum_{\\begin{smallmatrix} v_1 \\in C_i \\\\ v_2 \\in C_j \\end{smallmatrix}} w ( \\left \\{ v_1, v_2 \\right \\} )", "f828cebd9d6f8e7eebdeedc7950b0bc9": ") \\land ", "f829248f5420be3541d72ceaaed4e37a": " (a \\oplus b) \\otimes c = a \\otimes c \\oplus b \\otimes c ", "f82937501d5700241e8a42249f82ca26": "c, D_0, \\epsilon", "f82a91b01f0089890d19322461bd49d3": "g\\colon (X \\times Y) \\rightarrow Z", "f82adf76dfd95420a1e3eec1ab508b57": "M_i = q_{ijk}\\sigma_{jk} \\,", "f82ae35b32245e3be67c0cafea21c3c5": "{\\omega}_i\\in \\mathbb{R}^3", "f82b1aa6f89e8203e686aa38cd9e7bd2": "\\epsilon(s,\\pi,r_i)", "f82b349ef9b5c6f343dd4deda3f39ae6": " \\mathcal F, \\mathcal G ", "f82b37f5f021f021d5c01d35d13eddbf": "E(n) = \\sum_p k_p E_p = E_0 \\cdot \\sum_p k_p \\log p = E_0 \\log n", "f82b669800b03f85c279785f40427f3d": " \\gamma = \\frac{(E_s - E_b)}{2A} ", "f82b67f27ad1007cae2bc20a4f33dd1a": "\n\\left\\langle \n\\left\\{\n\\frac{:A(b)}{\\neg A(b)}\n\\right\\},\n\\emptyset\n\\right\\rangle\n", "f82b94a2c9eb13bb28069a0aa803b41d": "\\; \\sqrt[n] {X_1 X_2 \\cdots X_n} \\to \\sideset{}{}\\prod_x X^{\\operatorname{pr}(x)\\,dx} \\text{ as }n \\to \\infty ", "f82bca35be3defc258fea881dcfd3a21": "\\mu\\geq 1/\\sqrt{2+\\sqrt{2}}", "f82c0544b80586d86d1b04463ed6d686": "z_0", "f82c0d97b5fec061451280b83ba984cd": "prog^*", "f82c3716823c75d6d12586197067d299": "a^2y^2-b^2x^2=x^2y^2 \\,", "f82c3b4c668f482a32c51017a451213e": "\n\\mathbf{M} \\equiv \\operatorname{diag}(\\mathbf{M}_1, \\mathbf{M}_2, \\ldots,\\mathbf{M}_N)\n\\quad\\textrm{and}\\quad\n\\mathbf{M}_A\\equiv \\operatorname{diag}(M_A, M_A, M_A)\n", "f82ca440773e10d5f9db83db835d893d": "\\frac{1}{2} \\cos \\phi ", "f82cbc95b685e2d797bc4318cd2088c5": "B_{\\lambda}(T)", "f82d1989d6fb31f44c8256efed6a8437": " \\frac{u_{i+1}-u_{i-1}}{2h} = u'(x_i) + \\mathcal{O}(h^2), ", "f82d4624f1ba08c75bdad341328537a9": "move(2,3)", "f82d58fb193e50d041fee65b397b7d85": "\\displaystyle{H_{\\partial\\Omega}^\\varepsilon f(s) ={1\\over \\pi i} \\int_{|s-t|\\ge \\varepsilon}\\,\\,\\,\\, {f(t)\\over z(t)-z(s)}\\, \\dot{z}(t)\\, dt.}", "f82d5a032ed5fac119ef73e6c3f175ba": "\\Delta I_{L_\\mathit{on}}", "f82d6a386bf3de7fe35e9d8afa1a13d3": "\\scriptstyle P_U", "f82d7e4ef5b66884aba1daed02201455": "\\sum_{k=0}^{n} (-2)^k", "f82da7d0899ffec581faedaf697bc473": "\\omega_{s1}=\\sqrt{\\frac{k_1}{m_A}}", "f82db2e37051593ea91a932a84ba0c9d": "S \\left( b_n(q),q,x \\right) = \\frac{SE(n,q,x)}{SE^\\prime(n,q,0)}.", "f82dc26353ed25226c2b957f60b440a8": "log(T)", "f82ee626a7a6c93c743da60bd9c3da36": "A = \\{-V_T: (V_t)_{t=0}^T \\text{ is the price of a self-financing portfolio at each time}\\}", "f82f26b0a66caf7a5afdc6c24c155053": "\\frac{R}{G}=\\frac{L}{C}", "f82f43dcd6609d60c61746ed9a8ff325": "\\gamma = \\frac{F_{max}}{2\\tau_p}", "f82f4b6656e1510bdce5c8a614763571": "2^N", "f82f8f32e2ee3f577191abbd8924dd0e": "t_k=a\\frac{k}{n}", "f82fb6cb0b9d3093170447851d9ffb6c": " QE=\\frac{\\Gamma_{rad}}{\\Gamma_{nrad} + \\Gamma_{rad}}. ", "f82fb792d60b944c71c4e124b0608c91": "u_2 = b_1^2b_3+2b_1b_2b_4-b_2^2b_3", "f83017fe7d430629f89f6256985cd9ec": "\\alpha=0.", "f83099dad6e1b547d4864e623e7a0e4e": "\\vec{X} \\approx \\frac{1}{\\sqrt{2}} \\, \\left( \\partial_u + \\partial_v \\right) - q \\tan(q u) \\, \\left( x \\partial_x + y \\partial_y \\right) ", "f830b3879dea0ec5a4a399dc8a5e53b6": "{\\color{Blue}~2.32}", "f830f1e87e7a277630d4732cfa76ef4a": "p(11k+6)\\equiv 0 \\pmod {11}.", "f831448a5e00b0e1b9feb04c3d431e15": "\\int\\sin^n ax\\cos^m ax\\;\\mathrm{d}x = \\frac{\\sin^{n+1} ax\\cos^{m-1} ax}{a(n+m)} + \\frac{m-1}{n+m}\\int\\sin^n ax\\cos^{m-2} ax\\;\\mathrm{d}x \\qquad\\mbox{(for }m,n>0\\mbox{)}\\,\\!", "f831660f67cfc991d149cb6b35e2004b": "\\alpha_1(t_1) = \\left( \\frac{1}{1+t_1^2},t_1 - \\frac{2t_1}{1+t_1^2},0\\right)", "f831696223a1d8ec83d42108b643d7e4": "h_{\\text{in}}(G) \\le \\sqrt{8(d-\\lambda_2)}.", "f831bd5f5b8f3f9759dc35dc51e4b11b": "k=m", "f831efb20d6664361c6c6e144a6e2b2f": "II_\\infty", "f832662d2123d559589e9e15bdba7771": " R \\,\\!", "f832dc405dafe6c3beca8dff4e16fd44": "\\theta.\\,", "f8330598f5d5d0c4e6b29d1eb219fec0": " P(r,t) ", "f8330a648b7b2f27a2ba5ff3d8a7b65d": "\\frac{\\partial (\\rho v_i)}{\\partial t}+\\frac{\\partial (\\rho v_i v_k)}{\\partial x_k}=-\\frac{\\partial P_{ki}}{\\partial x_k}", "f8333ec5748505af7daf02d28504d056": "\\ddot{x} + c\\dot{x} + [\\delta - 2 \\varepsilon \\cos{2t}]x = 0 ", "f83353ee2d60165dd59b57fb598d84dc": "\\empty^* = \\empty", "f833a4504f4708fedb50945f2585af78": "\\Omega_{JKI} + \\Omega_{IJK} - \\Omega_{KIJ} + 2 e_J^\\alpha \\omega_{\\alpha IK} = 0", "f833b08613fde6a6d95158d9ba334cdd": "\\,m_t\\,", "f833e22c9a08c10c624f4fdc84c20c9c": "\nR \\approx \\sum p_i \\ln \\frac{p_i}{q_i}\n", "f834ceef62cbdeef6bf4131572030f40": " i=0, \\dots , k. ", "f834dae9fe4c2fd5508aa73b57d1e9c3": "1/f_\\mathrm{eq}", "f83534953d350b404fc9454e5a741c22": "X^*=\\{x^*:\\sup_{x\\in X}(\\langle x^*,x\\rangle-f(x))<\\infty\\}", "f83565b96a4101e020bc6c3e43d4a7a7": "\n \\forall x \\left( p(x,y) \\right) \\vdash \\exists y \\left( p(x,y) \\right)\n ", "f8359e0c398785cc611bad505812185c": " h \\ ", "f835d5cdfe384815700042e52433eab4": "\\bar{\\delta} n^a=\\lambda m^a+\\bar{\\mu}\\bar{m}^a-(\\alpha+\\bar{\\beta})n^a\\,;", "f836389365ba9e7847fddc92f69c8a52": "f(n) = h(n,\\langle f(0), f(1), \\ldots, f(n-1)\\rangle)", "f83666ffaa556c45b439720e7fa4affc": "A_\\text{ellipse}", "f8367c461b65f30042e5eb2ce88b7c12": "\n\\begin{align}\nETC = EAC - AC\n\\end{align}\n", "f836846b103a343795ffc2ead574633e": "\\hat{\\mathbf{x}}_{i+1}=A_i\\hat{\\mathbf{x}}_i+B_i{\\mathbf{u}}_i+K_i \\left({\\mathbf{y}}_i-C_i{\\hat{\\mathbf{x}}}_i \\right), \\hat{\\mathbf{x}}_0=E({\\mathbf{x}}_0)", "f836a97facafb19b830ed39e9f30d31b": "1 + 2 \\times 3 = 9, \\;", "f836be878c290d77b060bdcaf6b3e25d": "a=-1", "f836c95d0f5489337f00820d4546766f": "\\!'\\phi'", "f836dea4fb172753bb6431491fa6e7b5": "G (\\lambda) = \\int d^3x G_j (x) \\lambda^j (x)", "f836ed9af1133e00d669a83caa4d334d": "DM/t/km", "f8371f85a98f07109ef243d5ae453b2f": "B \\rightarrow I: \\{N_A, N_B\\}_{K_{PA}}", "f837255a9427a40a872ccfc1592b1ad9": "z_T=\\frac{V}{Z_0I}\\,", "f8372aba50356f5acd26dfa619da2999": "\ndy/dt = a y\n", "f83738132d61381d1601efc430bda726": "1+(1/n),", "f83757573c86d9a4ac5047790ea8ecf2": "f_{clk}/2", "f8375802cd493cb4a59b2c2f2d68db2b": "I = \\frac{\\pi r^4}{16}", "f837593b4fb6824c3bbf724e9ac75cea": "\\frac{dR}{dt} = \\gamma I ", "f837853ed31ea375e230dbb1dce39e3d": "d(Q; P)", "f83787f4f0ba46c2872f92b5a7b7a776": " -(\\alpha + 1) {d V \\over V} = \\alpha {d P \\over P}. ", "f837c7d9277457622effef1c041e2244": "x\\mapsto x+K_C", "f8380d6544e494ec5def6607b08274c5": "x-x_0=-c\\ \\frac{x_P-x_0}{z_P-z_0}", "f8384a2c92f6d7b04b9fa8cc17f5dad3": "\\Phi : \\mathcal{X} \\rightarrow \\mathbb{R}^p", "f838678a924555111c9cddf128a27622": "\\text{Vanna}\n= \\frac{\\partial \\Delta}{\\partial \\sigma}\n= \\frac{\\partial \\nu}{\\partial S}\n= \\frac{\\partial^2 V}{\\partial S \\partial \\sigma}\n", "f838c521618fc696a7091cb6c99ee495": "R_\\mathrm{load}=\\infty\\,\\!", "f838d5c7d0517894468401fa1ea7c520": "q = \\text{Min}\\left(\\frac{z_1}{a},\\frac{z_2}{b}\\right)", "f838dcd24aad65ef9682ac6a24bb9541": "q:=\\frac{p}{p-1}\\in(1,\\infty)", "f83954874eef6600244b246bbe97835e": "f: (O+\\vec{x}) \\mapsto (B+\\varphi(\\vec{x})) .", "f839a70807d60b969bf36947f81dd3f2": "\\forall A \\, \\forall B \\, \\exist C \\, \\forall D \\, [ D \\in C \\iff (D = A \\or D = B)]", "f839cb447e0147fc06466fab23aefa7f": "a(x,t) \\, d x^{p+1} \\mapsto 0, \\; a(x,t) \\, dt \\, dx^p \\mapsto \\left(\\int_0 ^1 a(x,t) \\, dt\\right) dx^p,", "f83a16e372b3d2d10cceb9645bbc908d": "Q_3 = AB^2", "f83a2e4664b81b61a6531e6c1a066d69": "{d \\over dx} \\log_b x = {1 \\over x \\ln b},", "f83a7eff8b2bfbbe966b51f069283e7c": "\\scriptstyle{\\bar{\\delta}=0.52.}", "f83a96e19e1feac53f1d0c2025072554": "\\bar{\\Delta}_+ \\cong \\sigma_-\\otimes \\Delta_+^*", "f83acca02fdaaa8963bdaff915664991": "\\xi_0(x)=\\pi^{-1} (1+|x|^2)^{-1}", "f83b06fb4a5c552f8ef31aff447f44cb": "\\binom nk(k-1)!=\\frac{n(n-1)\\cdots(n-k+1)}{k}=\\frac{n!}{(n-k)!k}", "f83b2ced9016563630884f2ec25f3c13": "x \\rho(a)", "f83b3046630afe5009c0749890c43805": "f(x_2)", "f83b8de64eab81540767edc684a19421": "1/\\lambda^4", "f83bf591c8ea982013a2a5b4b40265c1": "\\frac{v_{\\text{in}}}{R} = I_{\\text{R}} = I_{\\text{D}}", "f83c3d586cf66deed125f34f31f09b47": "|\\varphi(s)| < K |s|^{-2}", "f83c46c7dc40558b1056b9c900057a5a": "N(\\varepsilon, \\mathcal{I}_{\\mathcal{C}}, L_r(Q)) \\leq KV(\\mathcal{C}) (4e)^{V(\\mathcal{C})} \\left(\\dfrac{1}{\\varepsilon}\\right)^{r (V(\\mathcal{C}) - 1)}", "f83c6dc1af13da967e060a08bafe52d4": "N_\\mu:=\\{f\\in B(\\Sigma) : f = 0 \\ \\mu\\text{-almost everywhere} \\}.", "f83c89c51ce818968b5c5f75530a65ed": "x \\mapsto (e^{ix}, e^{ikx})", "f83c91ef685e635b7f65bfea074fe0fc": "R_\\mathrm{out} = g_{22}= \\begin{matrix} \\frac{v_{out}}{i_{out}}\\end{matrix} \\Big|_{v_{in}=0} ", "f83ca2fa5ca74047c698424997ea4b16": " \\sum_{n=1}^\\infty \\frac{z^n}{n}= z \\,+\\, \\frac{z^2}{2} \\,+\\, \\frac{z^3}{3} \\,+\\, \\frac{z^4}{4} \\,+\\, \\cdots", "f83cca87d4be80b9bf616d1e72082aa2": " \\mathbb{A}_\\mathbb{Q} =\\mathbb{Q}\\otimes_\\mathbb Z \\mathbb{A}_\\mathbb{Z} ", "f83ce0bdd81465307fad2b1469c30329": " \\text{Standard error of difference} = \\sqrt{\\frac{p+q-(p-q)^2}{n}} = \\sqrt{\\frac{p+q-p^2+2pq-q^2}{n}}.", "f83d1c132df0a97b72e75bb899cedad1": "\n\\Psi \\rightarrow \n \\begin{pmatrix}\n \\psi_{11} & \\psi_{12} \\\\ \\psi_{21} & \\psi_{22}\n\\end{pmatrix}\n", "f83d436a02b6184db547b7301c4200c9": "\n\\left(\\frac{5}{p}\\right)\n=(-1)^{\\big\\lfloor \\frac{p+2}{5}\\big \\rfloor}\n=\\begin{cases}\n\\;\\;\\,1\\mbox{ if }p \\equiv 1\\mbox{ or }4 \\pmod5 \\\\\n-1\\mbox{ if }p \\equiv 2\\mbox{ or }3 \\pmod5. \\end{cases}", "f83d4d4e95225dedb800c4063ab9a71f": " F = -kx - c\\frac{\\mathrm{d}x}{\\mathrm{d}t} = m \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2}.", "f83dc2d08b5ae32641a72da34a68fb0a": "\\displaystyle \\sum_{x\\in X} P(x) = 1)", "f83dcaa529503a92bffb65acdee7e10f": "Q(x,y)=x", "f83e084349596db047d0bdae014d9674": "\\{s, 2, 4, 3, t\\}", "f83e326eb75318174ec59ea80501a798": "s_1^2 \\ge s_2^2", "f83e4fb61e0a363851bdde065f79e413": "U_i(x)\\ge0", "f83e68a912cb207b4ba87bb78b0dd9e1": "\\scriptstyle\\mathbb{R}^+\\to X", "f83e807b4024e2568773a17c61575332": "{-p \\choose p} \\equiv {-1 \\choose 1} + {-1 \\choose 2}\\left({2p \\choose p} - 2\\right) \\pmod{p^3}.", "f83ec641f588cfcd869a2a228986e902": " \\frac{ \\sigma^\\delta \\Gamma(\\alpha-\\delta)\\Gamma(1+\\delta)}{\\Gamma(\\alpha)}", "f83f1f088c0fa5a4bbc55613864b64e7": "Y_{ij}=0", "f83f278e5616d2eb5d5456ba0d406bfa": "f(x) = \\frac{1}{1+\\sinh(2x)\\log(x)^2}", "f83f42c289569b1c8597e622d0fa4cfe": "p_g(X|B) = 2^{-H_{\\min}(X|B)}~.", "f83f5290036934d3872ab7b926a48608": " \\left(\\csc(x)\\right)' = \\left(\\frac{1}{\\sin(x)}\\right)' = -\\frac{\\cos(x)}{\\sin^2(x)} = -\\frac{1}{\\sin(x)}.\\frac{\\cos(x)}{\\sin(x)} = -\\csc(x)\\cot(x)", "f83f52c4e0f477d68e2cd88f3971d15c": "K_5 = \\mbox{mex}\\{K_0 + K_4, K_1 + K_3, K_2 + K_2, K_0 + K_3, K_1 + K_2\\},\\,", "f83f602e4b60667211ff8ba95aeb3da8": "L_q[1/3,c]", "f83fb7acedb6f5bb5ca56311411eba52": "\\operatorname{dim} R[x] = \\operatorname{dim} R + 1.", "f83fd96d5e60d3a8a2dcc5c0fc1832c6": "2^{A-1}", "f84002552273059e9b9d77b45bb134af": "\\rho_a", "f84072e5718cec447ab0ba762d3203a7": "G_0 \\ldots, G_k", "f8407a08cdea566de41c42ca72a77554": "V \\sin i<10km s^{-1}", "f840b3c6cb7d240ab79e0f8b7920db6b": "a_{n} x^n y^{(n)}(x) + a_{n-1} x^{n-1} y^{(n-1)}(x) + \\cdots + a_0 y(x) = 0.", "f840c31dd05337e839bef5420e875922": " \\partial_\\mu \\left( \\frac{\\partial \\mathcal{L}}{\\partial ( \\partial_\\mu A_\\nu )} \\right) - \\frac{\\partial \\mathcal{L}}{\\partial A_\\nu} = 0 ", "f840c3db2b67a72b934d54ffd7e4de37": "A \\,", "f84130d3ff9ddf7f1e28bcde96d172ad": " T_p M", "f84151647076c9af818dba433550f9e2": "\\vec L=I\\vec\\omega", "f8417c6779c06c6758b6b24c6d58e451": "\\mathrm{Q} = \\langle -1,i,j,k \\mid (-1)^2 = 1, \\;i^2 = j^2 = k^2 = ijk = -1 \\rangle, \\,\\!", "f841ae0e383bebec84c619c6731b1c23": "\\gamma = b/m \\,\\!", "f841cf0260b07882dbc8ec3611eec529": "\\Omega_{B} (\\omega) := \\left\\{ x \\in X \\left| \\exists t_{n} \\to + \\infty, \\exists b_{n} \\in B(\\vartheta_{-t_{n}} \\omega) \\mathrm{\\,s.t.\\,} \\varphi (t_{n}, \\vartheta_{-t_{n}} \\omega) b_{n} \\to x \\mathrm{\\,as\\,} n \\to \\infty \\right. \\right\\}.", "f841de659db8df428c957d811a76788c": "\\frac{\\Delta f^{*}}{f_f}=\\frac{-Z_{\\mathrm{F}}}{\\pi Z_q}\\frac{Z_{\\mathrm{F}}\\tan \\left(\nk_{\\mathrm{F}}d_{\\mathrm{F}}\\right) -iZ_{\\mathrm{Liq}}}{Z_{\\mathrm{F}}+iZ_{\\mathrm{Liq}}\\tan \\left(\nk_{\\mathrm{F}}d_{\\mathrm{F}}\\right) }", "f841e2c7516fd37cb9b12f4bd38578c2": "vxy = b^jc^k", "f8421f8cbc3439f707c297b08d7df2a7": "\\frac{1}{(1-z)^{\\alpha+1}} = \\sum_{n=0}^{\\infty}{n+\\alpha \\choose n}z^n", "f842766e35dbd49b4dcb3064e236cb8e": "f_i\\left( u, v \\right) = A_i\\left( u \\right) + B_i \\left( v \\right)", "f842d79e314247ec0dfceb8c47d107ba": "{n+x \\choose n}= \\sum_{i=0}^n \\frac{\\alpha^i}{i!} L_{n-i}^{(x+i)}(\\alpha).", "f84320cb920650ef9df7234b1f8f3759": "\\textstyle (x-\\lambda_i)^{\\nu_i}.", "f8433aee5c78efec2df9d0805c40c083": "\\le 0.5", "f8434598756aeab4704c87065fad82b7": " \\leq\\sum_{i\\neq m}2^{-n\\left[ H\\left( B\\right) -\\delta\\right]\n}\\ 2^{n\\left[ H\\left( B|X\\right) +\\delta\\right] }", "f8437e5e8b97623edc99ce830c929f57": "dF:U\\times X \\rightarrow Y \\,", "f843ad7a22f1ee8fc3c33a6de801d9da": "\\scriptstyle\\ 1/\\sqrt{2\\pi}", "f844457dc6716ca850ccea80a0f24aed": "-2 < \\lambda \\le -1.645", "f84489569786b6d66dd68163c3aecedc": "\\bar{x} \\thicksim N\\left\\{\\mu, \\frac{\\sigma^2}{n}\\right\\}.", "f844e6c7f0e18be1cfa4fbc606ca04ea": "(156{1 \\over 4})\\pi", "f844ee299d1574027e84e149470bcf8f": "\n\\int\\limits_0^{x_m}D_\\alpha ^{-1/\\alpha }(E-q^2|x|^\\beta )^{1/\\alpha }dx=\\frac 1{D_\\alpha ^{1/\\alpha }q^{2/\\beta }}E^{\\frac 1\\alpha +\\frac 1\\beta\n}\\int\\limits_0^1dy(1-y^\\beta )^{1/\\alpha }.\n", "f844f8e7d5059e17c52b805d1e41b59d": "S( )", "f84552fe0a791544e7fbfcb744b66cf3": "y + 10 = 2 \\times (x + 10)", "f845a923b40999a02f9796c98e44a557": "\\scriptstyle x[n] \\;=\\; \\delta[n]", "f845aaf2d6f300cb2c42abca69f937c4": "\\exists n < t\\, \\phi \\Leftrightarrow \\exists n ( n < t \\land \\phi)", "f845cc845fe8e2a1bc62a3448930c246": "R(s,s) \\geq (1 + o(1)) \\frac{s}{\\sqrt{2} e} 2^{\\frac{s}{2}},", "f845e695ba814da4a346888d26bb8818": "\n\\mathbf{X}\\left( \\mathbf{x}\\right) \\equiv X^{x_{1}}\\otimes\\cdots\\otimes\nX^{x_{n}}, \\,\\,\\,\\,\\,\\,\\,\n\\mathbf{Z}\\left( \\mathbf{z}\\right) \\equiv Z^{z_{1}}\\otimes\\cdots\\otimes\nZ^{z_{n}},\n", "f845f5196651ba2a8c09a186c3f23e39": "\\frac{\\mathrm{d}S}{\\mathrm{d}t} = \\Sigma_k \\frac {\\dot Q_k}{T_k} + \\Sigma_k\\dot S_k + \\Sigma_k\\dot S_{ik} \\text{ with }\\dot S_{ik} \\geq 0.", "f8460e9af1081a32daf65891cca47518": "i = r + \\pi", "f8464a152f5dc30352cfa8f8f30941c9": "\\mathrm{Area}^{-1} = 4 \\sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.", "f846a2661acd48b8a7a24433f0b8ce4e": " p \\mid (i-j) \\cdot m", "f84704e19e3fd7771dd2c54e67afae43": "\\mathcal P(F)", "f847911a0abd2f33d7434d26a09de7ab": "\\frac{1}{\\tau} = 2\\frac{\\pi^2 - 9}{9\\pi}m\\alpha^{6}", "f847b91381fd74af47e1c102e564d178": " f(x) = p(x) + h_k(x)(x-a)^k, \\quad \\lim_{x\\to a}h_k(x)=0,", "f847c9b2fba0b3a06674222c47c9700f": " \\operatorname{E}(X \\mid X0 \\}", "f84a16a18115199b6ef5ec4e360ea3f9": "I_W", "f84a6a53a81aec72bf34a82848792137": "\\frac{(b+a x)^{\\frac{b}{a}+x}}{e^x}\\,", "f84a809e6f8ed3519ce5e76a938ffa0f": "g(t)=\\frac{1}{\\sqrt{s}}f\\left( \\frac{t}{s} \\right)", "f84ab8838fbd88c5ce7c95586fc32c21": "ABCD=1100", "f84b2031b8c04df118d0de84b9e3036f": "f(t)=F^\\Delta(t)", "f84b29b421e7cdce707a5db3e4a4343d": "\\delta \\left[ \\rho - \\hat{\\rho} \\right]", "f84b55a95dc59278a4e0f3df74dbfa19": "V \\cap W", "f84b88aa9e93e2b26ebb7a8dc38ae898": "(abb)*(bca)=abbbca.", "f84bc29cbbc6fb541ec9b2a40a117df6": "\\gamma_1 =\\frac{\\operatorname{E}[(X - \\mu)^3]}{(\\operatorname{var}(X))^{3/2}} = \\frac{2(\\beta - \\alpha)\\sqrt{\\alpha + \\beta + 1}}{(\\alpha + \\beta + 2) \\sqrt{\\alpha \\beta}} .", "f84c031139c9943a5379872f50344f1a": "\\ a_0", "f84c8226727ff7267d0344f122329855": " R^\\delta(\\mathbf{x},\\mathbf{x}')=\\exp\\left\\{-\\sum_{k=1}^d \\omega_k^\\delta(x_k-x_k')^2 \\right\\}. ", "f84c854b4dfee339f9e9d70c72332b8b": "H = \\frac{{wS^2}}{{8d}} ", "f84c8cdc9c3676de1c324dea2781a0a6": " |0\\rangle", "f84ceee7680ee461b235e451770a9494": "\n\\begin{align}\n E &=& \\frac{\\hbar^2}{2m}\\left[\\left(\\frac{\\pi}{a}\\right)^2-q^2\\right]\\pm V\\sqrt{1-\\left(\\frac{\\hbar^2\\pi q}{maV}\\right)^2}\n\\end{align}\n", "f84d17e844301ddc2f75cdbd6f8a393d": " \\lim_{n \\rightarrow \\infty} \\frac{\\mu\\left( T^n A \\cap B\\right)}{\\mu\\left( B\\right)} = \\mu\\left( A\\right)", "f84d5fdba161c769acf14edafcf97f91": " \\sqrt{8}/\\pi", "f84db8275e02ae0c257d300f39419a64": "\\frac{\\partial \\vec{v}}{\\partial t} + (\\vec{v}\\cdot\\nabla) \\vec{v} = -e''\\nabla\\rho = -\\frac{1}{\\rho}\\nabla{p}", "f84dcb63c9e42aa205250502179f2840": "\\prod_{j=1}^n ( \\lambda-X_j)=\\lambda^n-e_1(X_1,\\ldots,X_n)\\lambda^{n-1}+e_2(X_1,\\ldots,X_n)\\lambda^{n-2}-\\cdots+(-1)^n e_n(X_1,\\ldots,X_n).", "f84e1eadb6a3b69452b7aedd6bf6e5ca": "y^2 = x^3", "f84eedbb64aaab64017bf29dd7a8221f": "L^{\\varphi} (X)", "f84f0055e2d877d44c0de4ddfd04bdc8": " \\begin{align} fg(x+h) - fg(x) = (f(x) + f'(x)h +\\psi_1(h))(g(x) + g'(x)h + \\psi_2(h)) - fg(x)= f'(x)g(x)h + f(x)g'(x)h + O(h) \\\\[12pt] \\end{align} ", "f84f52e0d229ff234be5eb7481e3134f": "d_{1/2,1/2}^{3/2} = \\frac{3\\cos \\theta - 1}{2} \\cos \\frac{\\theta}{2}", "f84fb93ca34a99955606a63e509d99fd": "P(x-a)", "f84fce4e99e00f1959e1e1648dfbf9e8": "\\mathbf{\\hat{z}} ", "f84fe3f53bdf1a323dd09d808606923e": "{{z}_{O}}\\approx \\left( 1+{{g}_{m4}}{{r}_{O1}} \\right){{r}_{O4}}", "f8502b3bd9e8727bc2f32ca015c47e63": "\\bold{F} = \\int\\!\\!\\!\\int \\!\\!\\! \\int \\phi\\Omega_p \\rho_p \\left[ D_p \\left( \\bold{u}_f - \\bold{u}_p \\right) - \\frac{\\nabla p}{\\rho_p} \\right] \\; d \\Omega_p d \\rho_p d \\bold{u}_p", "f85034f8f779d853dae23d7220e8047a": "\\mathbf{A}_3 = \\mathbf{A}_2\\mathbf{A}_1", "f850560392d2d944795c97f998a96c19": "2n_{\\rm oil}d\\cos(\\theta_2)=\\left(m-\\frac{1}{2}\\right)\\lambda", "f850c61f94c0bc34cc567b2c06b1e238": "\\frac{180(p-2q)}{p}", "f850ea8ca50126a2736cec33fd54fe13": "\\displaystyle (x, y, z) = \\Phi(r, \\theta, z)", "f851b12a2ed24c5b532809f0c4764e97": " = \\frac 12 {\\hat f}^c (\\nu) - \\frac i2 {\\hat f}^s (\\nu). ", "f851ec3bec77aa45fb53b65803b9610d": "\\scriptstyle dY", "f853130ca101b2b3d3c50cd4bc8a98f2": "D_5", "f853179656bbeca8e233c7cc89464759": "i = 1,2,...,n.", "f853721eacdb74aab83e17b34894ffff": " \\Delta x = x_2 - x_1 ", "f853736672fb5f621f223756bd87cf84": "\n\\begin{align}\nZ & := \\sum_i e^{- \\beta E_i} \n\\end{align}\n", "f853b228b97d1d4535d3f190077d45f7": "Q^'=P_2", "f853fc2311212edd7aec8850ad0f7663": "c_0\\infty^{n-2}\\,", "f854563a5d2c6f8dd7a9f65cff0f9c81": "\\bold{p}(t)\\rightarrow \\bold{p}(t+t_0)", "f8546364d53cb9ff46ab53434bc42a22": "W_O", "f8546e411a37407dd7ae4e7b77e70801": " \\beta_i \\ge 0 ", "f8548a7196176e6b58417ef7cd8a7a4b": " P / {\\rm W} = \\tau / {\\rm (N \\cdot m)} \\times 2 \\pi \\times \\omega / {\\rm rps} ", "f854a79e0ed55506d7e73e8c565a3b1d": "\\zeta_i(s) = \\sum_{\\lambda_j>0}\\lambda_j^{-s}", "f854a8d4f69a2a69689747eefc160940": "N_s (\\neq N_t)", "f8556e2c70c6f7e0a3c19c5d83846e5c": " \\left(\\frac{\\partial u}{\\partial T}\\right)_{V} ", "f8559677feb94b2338780fa632c68c9a": "\\hbox{The quadratic formula is}\\,", "f855cbf2c86bde65d8862639cbe99114": "n\\geq 5", "f85625a488d58db2f85a924eb529d1e8": "a\\sqrt{2}", "f85657094fc59d04866ab9625e99a607": "\\hat{H}=\\left(\n\\begin{array}{cccc}\n \\omega _0 & 0 & 0 & 0 \\\\\n 0 & \\omega _0+\\omega _2 & V_F & 0 \\\\\n 0 & V_F & \\omega _0+\\omega _1 & 0 \\\\\n 0 & 0 & 0 & \\omega _0+\\omega _1+\\omega _2+V_{\\text{XX}}\n\\end{array}\n\\right)\n", "f8565c42e89c57977a94f10bd015f191": "f_i(x) = f_i(x')", "f8565f6434676ed5bc87052ad51d7088": "Y_{3}^{1}(\\theta,\\varphi)\n={-1\\over 8}\\sqrt{21\\over \\pi}\\cdot e^{i\\varphi}\\cdot\\sin\\theta\\cdot(5\\cos^{2}\\theta-1)\\quad\n={-1\\over 8}\\sqrt{21\\over \\pi}\\cdot{(x + iy) (4z^2 - x^2 - y^2) \\over r^{3}}", "f856936ee11e1239e4085e09c5937c8e": " \\tau = {F \\over A},", "f856b6604ebd8659651dea6c5ffbbebd": "\\omega_R= \\omega_{s}-\\omega_{I}", "f856b6cd48dae3a60d37eef888815956": "z_1,z_2 \\in \\mathbb{H}", "f8570fe9e1c2de644b0187c624ff7780": "H\\left( \\frac{1}{\\sqrt{2}}|0\\rangle+\\frac{1}{\\sqrt{2}}|1\\rangle \\right)= \\frac{1}{\\sqrt{2}} \\frac{1}{\\sqrt{2}}(|0\\rangle + |1\\rangle) + \\frac{1}{\\sqrt{2}} \\frac{1}{\\sqrt{2}}\\left( |0\\rangle-|1\\rangle\\right)= |0\\rangle", "f8575bff635435eadfb700c0a7f7b6a9": "f\\circ g=fg", "f857a484b157fac58aa431c00f8ad1be": "Y_{o}=\\frac{k_{xo}}{\\mu _{o}\\omega } \\ \\ \\ , \\ \\ \\ \\ Y_{\\varepsilon }=\\frac{k_{x\\varepsilon }}\n{\\mu _{o}\\omega } \\ \\ \\ \\ \\ \\ \\ (5) ", "f857cccf702bfc5662b45fea0fb04725": "\\mu=0.7", "f857f58841f42ffe9faca5cc59390260": "\\scriptstyle (\\mathcal{X},\\mathcal{A})", "f858209db5ff68dc3a7fa1592622a6d8": "d \\times d", "f8582d162b38eb75cc15ceddf37be0c7": "\ne^z = 1 + \\sum_{n=1}^\\infty \\frac{z^n}{n!} = 1 + \\sum_{n=1}^\\infty \\left(\\prod_{j=1}^n \\frac{z}{j}\\right)\\,\n", "f8587286ba00fd31c51e17620b455a93": "\\mathcal{O}=\\{\\phi_n\\, , \\, n = 0, 1, \\dots \\}", "f858ce82534b5a978a232b9494ad86dd": "\\tfrac{8}{-5}", "f859446ae5dd9fdb4e598d2aa7eacd28": "\\ y_1=y_2", "f859cf9deb1a832c1b30d6a1b5e58b72": "\\frac{x^3 - 12x^2 - 42}{x - 3}", "f85a26302532b2dc45cbfd6621a6930b": " W = C^{-1}AC", "f85a2f7be146f358e887e610ad4df69d": "\\tan{\\frac{x}{2}}\\tan{\\frac{y}{2}}=\\tan{\\frac{z}{2}}\\tan{\\frac{w}{2}}", "f85a36f583713e287aa7eb6d8a220144": "x_i^*=x_i", "f85a4e939586909a96c7d31902f28408": "\\mathbf{a} \\times \\mathbf{b} = [\\mathbf{b}]_{\\times}^\\mathrm T \\mathbf{a} = \\begin{bmatrix}\\,0&\\,\\,b_3&\\!-b_2\\\\ -b_3&0&\\,\\,b_1\\\\\\,\\,b_2&\\!-b_1&\\,0\\end{bmatrix}\\begin{bmatrix}a_1\\\\a_2\\\\a_3\\end{bmatrix}", "f85b30a5020d21d658a0883c5e40fd5b": "[(n:=n+1)^i] \\Phi(n)\\,\\!", "f85b53f285d66be9766c582fe071271c": "\\mu( \\alpha^\\vee ) \\geq 0", "f85b7b377112c272bc87f3e73f10508d": "BC", "f85b94ce4e10bc3d2fcf2d2a11b8f54a": "\\theta_k(z) = \\sum_{\\gamma\\in\\Gamma^*} (J_\\gamma(z))^k H(\\gamma(z))", "f85b95e492081b5b63cc71fe81a24e29": "\\rho \\equiv \\frac {\\sum_{i \\neq j} (a_{ij} - \\bar{a}) (a_{ji} - \\bar{a})}{\\sum_{i \\neq j} (a_{ij} - \\bar{a})^2}", "f85bfd3d191bf5168c764459d133ead1": "\\Gamma \\cong \\langle S, T \\mid S^2=I, (ST)^3=I \\rangle", "f85c28a402a009d892f334a8890d9d26": "k_\\mathrm{spec} = \\|N\\|\\|H\\|\\cos ^n\\beta = (\\hat{N} \\cdot \\hat{H})^n", "f85c987e9110ebd8e14824c497715291": " \\mathbf{H}_1, \\mathbf{H}_2, \\cdots, \\mathbf{H}_s ", "f85cc4b041aa89855f7e56747e2e94b0": "v \\otimes l = v \\cdot l\\quad \\forall v\\in Q,l\\in K", "f85cf631639300a2dfad0d49541974b6": "\n\\mathbf{\\varepsilon} =\n\\begin{bmatrix} \n \\dfrac{\\partial v_1}{\\partial S_1} & \\cdots & \\dfrac{\\partial v_1}{\\partial S_m} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\dfrac{\\partial v_n}{\\partial S_1} & \\cdots & \\dfrac{\\partial v_n}{\\partial S_m} \n\\end{bmatrix}. \n", "f85d341882876b7d7210fc75df6f4486": "\\Sigma v_j^2=R^2\\, ,", "f85d57735bcefbfaf52714eb677f0a7d": "\\tilde{\\mathbf Y}", "f85d94fbc4e636235b0321bdcf5877bd": "I(\\lambda,T) =\\frac{2 hc^2}{\\lambda^5}\\frac{1}{\\exp\\left(\\frac{hc/\\lambda}{kT}\\right)-1}", "f85de0a59507d5b2416063c6884ca14d": "B^q(\\mathcal{U}, \\mathcal{F}) := \\mathrm{im} \\left( \\delta_{q-1} : C^{q-1}(\\mathcal{U}, \\mathcal{F}) \\to C^{q}(\\mathcal{U}, \\mathcal{F}) \\right)", "f85e09713d1e8dd73a38052fd430f948": "F_V = \\rho g V", "f85e454710cee42ba71461d75dc8a657": "\\Delta x = \\frac{2 \\pi}{k}", "f85e5339f26f3e697b161989085bd8ac": "F=(f_1,\\dots,f_k)=(p_1/q_1,\\dots,p_k/q_k)", "f85ead2af2c49b0661ea30c2f149e1a2": "= \\frac{\\sqrt{\\textrm{MYSI} \\cdot \\textrm{EYSI}}} {0.951}", "f85ed2548be87718f816545e193891f4": "B|\\psi_n\\rangle", "f85eec7213d8a0b66689061d5b0f8974": "\\pi R^3 \\sinh \\frac{2r}{R} - 2\\pi R^2r \\,.", "f85f0f988d5c92e0834a71e00d87497c": "d^2f = \\left(\\frac{\\partial^2f}{\\partial x^2}(dx)^2+2\\frac{\\partial^2f}{\\partial x\\partial y}dx\\,dy + \\frac{\\partial^2f}{\\partial y^2}(dy)^2\\right) + \\frac{\\partial f}{\\partial x}d^2x + \\frac{\\partial f}{\\partial y}d^2y.", "f85f819fe60fd5eb67a8501b400c7323": "\\nu:k\\to\\mathbb Z\\cup\\{\\infty\\}", "f85faf6859280e7aab928f874affb21e": "\\neg x=(x\\Rightarrow 0)", "f85fb0891c33cff38353eeb52b0541a6": "\\textrm{fuel} + \\textrm{oxygen} \\to \\textrm{water} + \\textrm{hydrogen} + \\textrm{carbon\\ dioxide} + \\textrm{carbon\\ monoxide}", "f86013bb959154e88e4c1d7673d4a0bc": " F = Se^{(r+y-q-u)T}", "f8606ea9b72e8ae4679303e631035ae7": "Y_{4}^{-4}(\\theta,\\varphi)={3\\over 16}\\sqrt{35\\over 2\\pi}\\cdot e^{-4i\\varphi}\\cdot\\sin^{4}\\theta\n= \\frac{3}{16} \\sqrt{\\frac{35}{2 \\pi}} \\cdot \\frac{(x - i y)^4}{r^4}", "f8608d7e81e327cbce8e84e80fd0c6a5": "K_1, K_2, \\ldots.\\ ", "f860ffd882f38f6abf45180afb0df82f": " \\theta_{cs} ", "f8614e2d681ef24706575562213d0d1d": "\n\\frac{x^{2} + y^{2}}{a^{2} \\tau^{2}} - \n\\frac{z^{2}}{a^{2} \\left( 1 - \\tau^{2} \\right)} = 1\n", "f861933c1644762abe547269e30f7269": "\\mathrm{CD}", "f861d51c6b9783d2a839745457075519": " a = 1 ", "f8620dac1f12b25c476c765395199e32": "\\tilde{g}(x) = 1 - g(1-x)", "f8624b932db09661fcbb6cac4b5c8d20": "n! = (n - 1) ((n-1)! + (n-2)!)\\,", "f86284c701cc0e30b645c372c0e7e5bb": "\\kappa = \\kappa_0 + \\left[\\kappa_1+\\kappa_2\\left(\\kappa_3-T_r\\right)\\left(1-T_r^{0.5}\\right)\\right]\\left(1+T_r^{0.5}\\right) \\left(0.7-T_r\\right)", "f862e7bf2f3a1b34070394972d81fa82": "\\alpha V K_T = V \\left( \\frac{\\partial S}{\\partial P} \\right)_T \\left(\\frac{\\partial P}{\\partial V}\\right)_T = V \\left( \\frac{\\partial S}{\\partial V} \\right)_T", "f863352a2076914d5d8a93b9f74d5cbf": "F_0 = (S_0 + U)e^{(r-y)T}", "f86339df9cc3f2965fc3a8c447a1e452": " y_i = \\alpha+\\beta(x^{*}_{i}-\\nu_i) + \\varepsilon_i ", "f8636e1552830076938797c4fcc416f4": "a_{21}x_1 + a_{22}x_2 + \\cdots + a_{2n}x_n = b_2 \\,", "f8646e8d5c193f8a9b61592808fd9809": "\\xi_{i+1}>\\xi_i", "f8646eb8bfc53d52eefe4092d11f8a16": "\\frac {C_{13}^1 C_4^2 \\cdot (C_{12}^3 \\cdot 4^3)} {C_{52}^5} = \\frac {13 \\cdot 6 \\cdot (220 \\cdot 64)} {2{,}598{,}960} = \\frac {1{,}098{,}240} {2{,}598{,}960} \\approx 42.26\\% ", "f864cec95ea98984beb5607b7cdd04a1": "c=-\\log\\alpha/\\log p", "f864d29a37a86c53d518ac9de3650cbb": " t + x \\mathbf{i} + y \\mathbf{j} + z \\mathbf{k} \\leftrightarrow \n\\left({\\begin{array}{*{20}c} t + x i & y + z i \\\\ -y + z i & t - x i \\end{array}}\\right) . ", "f864de70dc6b1f5ee66f73fd5db9237d": "cx^2+dy^2=1 \\, ", "f864e011759d77b1b63ec8e9a9be24fc": "\\mp \\!\\,", "f865017f25af935898fa38facb68454b": "SU(2)_L", "f865043e8b8a9482d410264b83dafa5b": "M(x)=\\sup\\frac{f'(\\xi)}{g'(\\xi)}", "f86545094a808cd162a20ac6882f881a": "\\pi_1(R)", "f8654e38acf6705b0e3ffdc59eba653c": "\n\\left(\\sum_{i=1}^{n}a_{i}b_{i}\\right)\\left(\\sum_{i=1}^{n}\\overline{a_{i}b_{i}}\\right)+\\sum_{i x) ", "f867e1dbac48a9a65612442b8e01adaf": "\\ln \\left ( \\frac{p_{w}(D_{p})}{p^{0}} \\right ) = \\frac{4M_{w}\\sigma_{w}}{RT\\rho_{w}D_{p}} - \\frac{6n_{s}M_{w}}{\\pi\\rho_{w}D_{p}^{3}}", "f8680e9d58ef66fe80d003643a2f9c73": " T^i {}_j \\mathbf{e_i} \\cdot \\mathbf{e^j} = T^i {}_j \\delta_i {}^j \n= T^j {}_j = T^1 {}_1 + \\cdots + T^n {}_n ", "f8683f4e83bf11cd20a3131e99512c5a": "\\mathbf{j}_9", "f8688488dc490e7c11dd3b57ffa6e915": "\\omega_c.\\,", "f86890241d40796636fa2616bd9a7a36": "Q(x) \\approx x\\prod_{p\\ \\text{prime}} \\left(1-\\frac{1}{p^2}\\right) = x\\prod_{p\\ \\text{prime}} \\frac{1}{(1-\\frac{1}{p^2})^{-1}} ", "f8689b3b18e2bc1e39634fb560737953": "\nf(\\zeta) = f(e^{2\\pi {\\mathrm{i}}}) + \\sum^\\infty_{n=1} \\frac{t^n}{n!}\\frac{d^{n-1}}{da^{n-1}}[f'(a)|\\phi(a)|^n]_{a = e^{2\\pi {\\mathrm{i}}}}\n", "f868c3702f91df60bc3e19469c64f242": " {\\rm tr}(T(f) T(g) - T(g) T(f)) = {1\\over 2\\pi i} \\int_0^{2\\pi} f dg.", "f868f287e7e8e9f03c8946d9ebfc36f2": "\\displaystyle{[L_m,L_n]=(m-n)L_{m+n}}", "f869245dab65d2e83c902e8c92278985": "v\\in V(G)\\setminus S", "f8692f40dd13f72ff6420a4a937a0866": "J_{n+1}=\\frac{2n}{z}J_n-J_{n-1}", "f86982c6e1eaaef2a8586f48feb1792b": "\\gamma \\gg 1", "f869b0f49360b581164e2e13538416b8": "0 < \\operatorname{pd}_R M < \\infty", "f869c01f55119db0d0fc51fd77ce9ba7": "\n\\ddot{\\mathbf{R}} \\equiv \\frac{m_{1}\\ddot{\\mathbf{x}}_{1} + m_{2}\\ddot{\\mathbf{x}}_{2}}{m_{1} + m_{2}}\n", "f869d8d2de382a9d102558b04ef9c2cc": "\\textstyle\\int_C \\mathbf{E}\\cdot \\mathrm{d}\\boldsymbol{\\ell}", "f869fdcdd39f63116c7a5ac6295b700c": "\\scriptstyle h(t) \\ \\stackrel{\\text{def}}{=}\\ O_t\\{\\delta(u);\\ u\\}.", "f86a02bfdfb58025ec7497d34d44b740": "e^{\\theta} = \\frac{1 + t}{1 - t},", "f86ab7ff4f3357a6ba3ccc344a1786b9": "n_1 < n_2", "f86aec1d878f977c6bd389556d784aae": "\\int_0^1 \\frac{dx}{x(1-\\ln x)(1-2\\ln x)} = \\ln 2.", "f86af8e5f08b832208f555cd8608e341": "\\sqrt{1+x}", "f86afc858a6b586a62fd3c88cfa968c0": "-(5^{1/4})x = \\sqrt[5]{(a+c)^2(b-c)} + \\sqrt[5]{(-a+c)(b-c)^2} + \\sqrt[5]{(a+c)(b+c)^2} - \\sqrt[5]{(-a+c)^2(b+c)} \\,,", "f86b864048d6606e361dd29ac45cf0a9": "\\operatorname{ch} \\, T_A = T_{\\operatorname{ch}\\, A}.", "f86bdf350fa7c10e2b761470f9a434af": "E_{K_1}(E_{K_2}(M))=M", "f86bff3e4643929799b5e5eb833aaec4": "\\mathcal{E} = -\\int_{A}^{B} \\boldsymbol{E_{cs} \\cdot } d \\boldsymbol{ \\ell } \\ ,", "f86c81e12aaf1b5ea1c469a8a0b4b67b": "L_d\\left(t_0, t_1, q_0, q_1\\right) = \\frac{t_1 - t_0}{2} \\left[ L\\left(t_0, q_0, \\frac{q_1-q_0}{t_1-t_0}\\right) + L\\left(t_1, q_1, \\frac{q_1-q_0}{t_1-t_0}\\right) \\right] \\approx \\int_{t_0}^{t_1} dt\\, L(t, q(t), v(t)) ", "f86c972d4575ed8fdb0f94b4e2cbc8b0": "\\mathbf \\zeta", "f86c9f0549feee92cc19b855324c4402": "C_n\\,\\mathbf{v}", "f86cfc7110de0817ecd57756e111e7c2": "\n\\tan\\phi = \\frac\n{\\cos\\alpha_0\\sin\\sigma}{\\sqrt{\\cos^2\\sigma + \\sin^2\\alpha_0\\sin^2\\sigma}}.", "f86d0aafac3d52f655ef3a8fbf4dacc4": "\\forall v \\in V, \\; -\\phi(u,v)=\\int fv", "f86d423a7afe072f6fab983a33daf05f": " \\rho u ", "f86dc48e3cc68c4d324026c69770f243": "T_{2n}(x)=T_n\\left(2x^2-1\\right)=2 T_n(x)^2-1", "f86df5735efd9aa1ee2c89012929933b": "d \\mathbf{l}\\cdot\\mathbf{n} + (\\mathbf{l_0}-\\mathbf{p_0})\\cdot\\mathbf{n} = 0", "f86df59270ffe07cae89058aed7a89df": "\\frac{a_1 a_2 a_3}{\\mathbf{a_1}\\cdot(\\mathbf{a_2} \\times \\mathbf{a_3})}", "f86e4664b38bb5b474a5b8bdc4052945": " T\\, ", "f86e64481de8e080e5e1c0dc48e70ee6": "\\frac{243}{128}", "f86e730baf9fda8a8213170faf2c5263": "\\alpha = \\frac{e^2}{(4 \\pi \\epsilon_0)\\hbar c} \\approx 1/137", "f86f5c25553727825f29b054ca02408d": "y(0) = 0", "f86f8e6b7139dccdf83fa6e0eb298d54": " \\sin \\theta = \\theta - \\frac{\\theta^3}{3!} + \\frac{\\theta^5}{5!} - \\frac{\\theta^7}{7!} + \\cdots", "f86fc68ed9f53c7d8d0ee01abbe15b6a": "\\, \\zeta(3) \\,", "f86fcad1a62a26bf87a247f86728dbd4": "x\\in U\\subset M", "f86fdc1985e3930a7c77e526b835eeaa": "\\phi = \\frac{1}{r^{n+1}}\\ P_n(\\sin\\theta)", "f8701240fc0cc9f5d27cb8d1694b2b0a": "W_{m}\\Omega^p_Y(\\log D) = \\begin{cases}\n0 & m < 0\\\\\n\\Omega^p_Y(\\log D) & m\\geq p \\\\\n\\Omega^{p-m}_Y\\wedge \\Omega^m_Y(\\log D) & 0\\leq m \\leq p\n\\end{cases} ", "f8703c3c7e7167e9aaf21859649f23d5": " (1.05)^3 ", "f8703c6de7471e7183867143e5f26c22": "x \\rightarrow 0,", "f870700afb0ef98335568bce839b4ca2": "{}_2F_1(-n,\\alpha+1+\\beta+n;\\alpha+1;x) = \\frac{n!}{(\\alpha+1)_n}P^{(\\alpha,\\beta)}_n(1-2x)", "f870b5a884db738423663389ad28e12c": "\\omega= \\pm 90^{\\circ}", "f870c01f89203a27b5e76ca07af2b19e": "\\phi_n(t) = 1^{n-1}t + 2^{n-1}t^2 + ... + (p-1)^{n-1} t^{p-1}.", "f8716f8d246967cad183d34505179314": "\\mathcal{F} = \\Phi \\mathcal{R}", "f87170a09c2e206098569c90d15462d8": "u=f(r)", "f871be9e85bcb017fa29368a6cfc6ec1": "\\! p", "f871db50ed3ebd3a30c50a6f5902489f": "3\\sin(\\theta) \\cos(\\theta)= 1, ", "f8721a8134621fe6fd43456a0604291a": "\\begin{align}\ng\\left( x_1, x_2, \\ldots , x_N \\right) & =0 \\\\\n\\frac{\\partial f}{\\partial x_1}\\left( x_1, x_2, \\ldots, x_N \\right) - \\lambda \\frac{\\partial g}{\\partial x_1}\\left( x_1, x_2,\\ldots , x_N \\right) & = 0 \\\\\n\\frac{\\partial f}{\\partial x_2}\\left( x_1, x_2, \\ldots , x_N \\right) - \\lambda \\frac{\\partial g}{\\partial x_2}\\left( x_1, x_2, \\ldots, x_N \\right) & = 0 \\\\\n & {}\\ \\ \\vdots \\\\\n\\frac{\\partial f}{\\partial x_N}\\left( x_1, x_2, \\ldots x_N \\right) - \\lambda \\frac{\\partial g}{\\partial x_N}\\left( x_1, x_2, \\ldots, x_N \\right) & = 0.\n\\end{align}\n", "f8723a9fc060403bec5ee068c040728d": "MUAMC = MUAC - \\left ( \\pi \\times \\frac{TSF}{10} \\right )", "f87268b6d0182be1a748cee52e7fd484": "\\vec{F} = \\frac{\\mathrm{d}\\vec{p}}{\\mathrm{d}t},", "f872cece88ffdfba61545b20285fadb2": "V_{w2}", "f872dcf04db114de381ba06d0056b532": " \\operatorname{merge-let}[E] = E ", "f872e93f30716021810e532ea5025867": "q_p(p-a)\\equiv q_p(a) + \\frac{1}{a} \\pmod{p}", "f87362908e3e4ab5c9b2a6d7a57fa430": "s_{\\infty}", "f873afaad8a1edb10a6edeee3d818f50": "\\frac {\\mathrm d x_{\\eta}} {\\mathrm d t_{\\sigma}}=V_{\\sigma\\eta}", "f873db37a46757d67ae1f332bca858ca": "C^0(f) := \\{ h \\in C(f) | \\mathrm{codom}(h) \\subseteq \\{0,1\\} \\}", "f8743b59c87299f1e8b366591f22019a": "p(z_i = k|\\mathbf{z}_{1,\\dots,i-1},\\alpha,K) = \\frac{n_k + \\alpha/K}{i - 1 + \\alpha}", "f87449fe32b8ca7923a902a56a23f596": "g(x_1, x_2, \\ldots) = g_1(x_1) g_2(x_2) \\ldots ", "f874623a173899eb3e4b1f39a763bca7": "\\rho_{f} ", "f87493602151f8bb6e0e3a56e36c5970": "W = \\frac{Rv + Cm}{v+m} ", "f8749493932f6dd59fd408b88056f79f": "\\{ x_1 = x, \\; x_2 = x\\}", "f874af3ffd967433302dd441891283e3": "a^2-B^2=b^2-A^2", "f8757d35a17c9135d0420ff3c782419a": "N_{i,0}", "f875bd9f83a55b488bb11a2d6fcb3d5c": "\n\\kappa_t(N)={n_t \\choose t+1} + {n_{t-1} \\choose t} + \\dots + {n_j \\choose j+1}.\n", "f876747f37b33ac6bf8bd294f4ad9489": "\\frac{r}{R_{\\mathrm{secondary}}} \\approx \\frac{a}{R_{\\mathrm{primary}}} \\sqrt[3]{\\frac{\\rho_{\\mathrm{secondary}}}{3 \\rho_{\\mathrm{primary}}}} \\approx \\frac{a}{R_{\\mathrm{primary}}}, ", "f876b0f26b0e6af2a5dba370fda85329": "\\mathbb{CHARLES}", "f876b5ec7a53f6c43c4a31005350f002": "n^2 = \\left(\\frac{ck}{\\omega}\\right)^2", "f877654b28afa57718a79c6969dd65a0": "\\frac {b}{\\sin \\theta} = \\frac {r}{\\sin \\left ( \\frac{\\pi-\\theta}{2} \\right )}", "f8779ca8b051f7da49839a04fdfdd70a": "C^+(W,p)", "f878520a17317cb98c001bb4f11fae2b": "6\\times (1^2)", "f87894f01f51c709e57754e354f9b6bb": "\\displaystyle y(t)", "f878e1b5132695fb9c26640154d941f0": " \\max |m_{ij}| \\leq \\max|m_{ii}|", "f878ea09b982070d72a7d99f565e0473": "\\hat{s}[k]=\\sum_{n=-\\infty}^{\\infty}w[n]r[k-n]", "f878ff0f9e97069cb5ee3f5298626acb": "\\sum_i (\\langle f_{1i}|f_{1i}\\rangle+\\langle f_{2i}|f_{2i}\\rangle)|e_i\\rangle\\langle e_i|", "f879eb6ccbb9ed8a3a517d3eb05eefb5": "\\alpha,\\beta,\\,\\gamma,\\dots\\,", "f87a09c07e3b0419f00b3289a4a38a11": " \\mathit P ", "f87a35f40d8f1f32f867e3721e14936c": "M_A(\\alpha)", "f87a37d016aa52c88a72ffd4eeebcf92": " S = \\sum_{i\\in I}a_i = \\lim \\Bigl\\{\\sum_{i\\in A}a_i\\,\\big| A\\in F\\Bigr\\}", "f87a3dc3bd0cb136a65bfed3c1da40c7": "-1.8\\times10^{-6}<\\frac{v-c}{c}<2.3\\times10^{-6}", "f87a67d49adeac428a6750144704f351": "R-R_{0}", "f87a7d79d974228b595aa90bd829a0c3": "\\phi:N\\rightarrow M", "f87aa3bbd19de39342f989ede30446e8": "x_\\mathrm d = \\left | D - s \\right | ;", "f87ac14370e607637fdf16021fae521e": "J_{k}", "f87ac733170d7132c6474f4addc8c6c7": "\\frac{1}{(1-x)(1-4x)}", "f87aef4092220ea376f3f75e12799f30": "\\dot{{\\xi}}=\\, {u}({\\xi},t)= \\hat{u} \\sin\\, \\left( k \\xi - \\omega t \\right),", "f87af4e077e1f127ff975a3514ffb866": "x = \\frac{\\sqrt{4ac+b^2}-b}{2a} ", "f87b2dc23b3df247a55fd2e84bc81035": "\\mathbf{E}^3", "f87b7654e416fdbde5fdfb53d040b2c8": "y = r \\cos \\phi \\sin \\lambda", "f87bd33c71920fd9d547e61462fb97f9": "\\mathbf{Z}_{k_1} \\oplus \\cdots \\oplus \\mathbf{Z}_{k_u}", "f87bd39c1bf36bb137b300d32493c557": " \\mathrm{HQC} = n \\log \\left( {\\mathrm{RSS} \\over n } \\right) + 2 k \\log \\log n, \\ ", "f87c2ff7801b47b48e9f7f8b03dfcfcd": " \\mathcal{B_A} ", "f87c687ff7103e8b4496ddb3b475b15f": " \\langle c(\\mathbf{x},\\mathit{t})\\rangle = = \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} \\mathit{Q}(\\mathbf{x},\\mathit{t}|\\mathbf{x}_0,\\mathit{t}_0)\\langle c(\\mathbf{x}_0,\\mathit{t}_0)\\rangle d\\mathbf{x}_0 + \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{t_0}^{t}\\mathit{Q}(\\mathbf{x},\\mathit{t}|\\mathbf{x}',\\mathit{t}')\\mathit{S}(\\mathbf{x}',\\mathit{t}') d\\mathit{t}d\\mathbf{x}'", "f87cf9c602eed61501e5f2e9ef24c7d4": "{}_1W_2 = \\int\\limits_1^2 {pdV} = 0", "f87d001763bd5417088e6a9a75b8e9eb": "R_y", "f87d7c3f42012a7d723aa966bfbfa9d1": "F^\\times\\cong(\\varpi)\\times\\mu_{q-1}\\times U^{(1)}", "f87db65fcc70db16a44917f51a7d2ba2": "\\underline{3.2898}68133696... = \\frac{\\pi^2}{3}", "f87e0088d306e4e5703c851fbe71fc77": "\\tan\\frac{\\pi}{4}=\\tan 45^\\circ=1\\,", "f87e0524ee763393c8d16b88da26f0c2": "z/0=\\infty", "f87e289cde6e86bfb6d4d2460b009eb6": " H = \\log_b |M|\\, ", "f87e328d3b6a39adab3841ae01aa7a52": "\\Diamond p \\rightarrow \\Box \\Diamond p", "f87e73a18da04f084acc669caf7f9bf1": " p_A, {\\dot{m}}_A ", "f87eddf31f3ad775a8fb1dbe9ebe4e98": " \\langle j_2 , m_2 | j_1 , m_1 \\rangle = \\delta_{j_1 j_2} \\delta_{m_1 m_2} ", "f87ee026be9a01d7369ce526c15095bd": "\\Phi_\\Lambda(\\phi)=\\begin{cases} \n\\beta 1_{\\{\\phi(x)\\not=\\phi(y)\\}} & \\text{ if }\\Lambda=\\{x,y\\}\\text{ is a pair of adjacent vertices of }\\Gamma;\\\\\n0 & \\text{ otherwise.}\\end{cases}\n", "f87ef3627240a6cd2ffc16c0a4272219": "\\mathbf{x} = [x_1, x_2, \\ldots, x_n]^{\\text{T}} \\in \\mathbb{R}^n", "f87f19864abc1e08462f7f2e097aaeac": "(10 \\uparrow)^2 10^{1453}", "f87f507ae1b5d9e596831116ac941807": "\\displaystyle{M=\\{ \\sigma(g)g^{-1}:g\\in H\\},}", "f87f576c042f291f02b263e4448c490f": "\n\\left(\\begin{matrix}\\lambda_1 \\\\ \\lambda_2 \\\\ \\lambda_3\\end{matrix}\\right) = \\mathbf{T}^{-1} ( \\mathbf{r}-\\mathbf{r}_4 )\n\\,", "f87f6ac28e6bf9055365a1d9b773f1f8": " x=0 ", "f87f6ff59e5b64fbf9e7d7fc4aec691d": "\\nabla_n^2=\\frac{\\partial^2}{{\\partial x_n}^2} + \\frac{\\partial^2}{{\\partial y_n}^2} + \\frac{\\partial^2}{{\\partial z_n}^2}", "f87f9800b839ab245cdb61cd202c9588": "\\kappa = \\frac{\\Pr(a) - \\Pr(e)}{1 - \\Pr(e)} = \\frac{0.70-0.50}{1-0.50} =0.40 \\!", "f87fb3ca59bac73a74cd0bc41f1e1c49": "\\bar{\\beta_z}=1-\\frac{1}{2\\gamma^2}\\left ( 1+\\frac{K^2}{2} \\right )", "f8806889ba204701b78f7711876c35b8": "2f \\omega^2=g", "f880a78723b00fa1c32b9bb8196d65fa": "\\mathcal{O}_{X'}", "f880af9040dced754ab3d1c757cab2f1": "\n\\begin{bmatrix} 1 \\\\ 1 \\\\ 0 \\end{bmatrix}\nx + \n\\begin{bmatrix} 1 \\\\ 1 \\\\ 1 \\end{bmatrix}\ny = \n\\begin{bmatrix} 1 \\\\ 1 \\\\ 2 \\end{bmatrix}\n", "f880bb2556fc9d14475c045a09cfbaaa": "w(+\\infty)=2", "f880bf25553b34ee17406a637a3edcc5": "\\begin{cases} J^{1}(\\pi) \\to T^*M \\times \\mathbf{R} \\\\ j^{1}_{p}\\sigma \\mapsto (d\\bar{\\sigma}_{p},\\bar{\\sigma}(p)) \\end{cases}", "f880c3ea2ac418f54acece6b35d6a7a6": "Q(u) = u_1^2 + \\cdots + u_p^2 - u_{p+1}^2 - \\cdots - u_{p+q}^2", "f8812225147284bc9adc7b9a1c6db587": "A_{21}", "f881545239ec4729bc7b412bbe3ebf45": "\\csc \\theta = \\sec \\left(\\frac{\\pi}{2} - \\theta \\right) = \\frac{1}{\\sin \\theta} ", "f88175ca6ed869733e5506d00837c62b": "D_n.", "f881a47aa18a31ff48d06e9b205d9fc8": "\\mathbf P", "f881aff86c017515001a1aef7e142d9f": "\\mathbb{E}_\\theta [ (\\theta_i - X_i) h(\\mathbf{X}) | X_j=x_j (j\\neq i) ]= \\int (\\theta_i - x_i) h(\\mathbf{x}) \\left( \\frac{1}{2\\pi} \\right)^{n/2} e^{ -(1/2)\\mathbf{(x-\\theta)}^T \\mathbf{(x-\\theta)} } m(dx_i)", "f881ca1a862037bc34635b0bd23942e0": "\\sum_{i>0}\\text{Ch}(M(\\lambda)^i) = \\sum_{\\alpha>0, s_\\alpha(\\lambda)<\\lambda}\\text{Ch}(M(s_\\alpha(\\lambda)))", "f882d0653f1cd28e8237ff890566a9bc": "R_\\mathrm{right}", "f883343ca879434f013b38854249a3bd": "P \\in \\mathbb{R}", "f8837d93d4b968c7ea4b3fb46e8159e4": "X \\subseteq R^n", "f8839b11af131ac6857e3a692bc7fe38": "(-1/b)\\{\\mathrm{E}[\\ln(X)] - \\ln(\\eta)\\}\\,", "f8846a6f07bf5b05d5789d621bd51abd": "=\\frac{vt-\\frac{gt^2}{2}}{\\frac{2hv}{g}-ht}", "f8849841feda22e0ea3e1e46fa84e47c": " \\bold{J} = {4 \\pi \\over c } \\left ( \\frac{1}{6} j^{\\alpha} \\sqrt{-g} \\, \\epsilon_{\\alpha\\beta\\gamma\\delta} \\mathrm{d}\\,x^{\\beta} \\wedge \\mathrm{d}\\,x^{\\gamma} \\wedge \\mathrm{d}\\,x^{\\delta}. \\right)", "f884c589a3fa031f5cb5280491c23f12": "|p|^2", "f885246a3df8ae9611a85d0888162d43": "V_\\Theta\\ (r) = \\frac{\\Gamma}{2\\pi} \\frac{r}{r_c^2 + r^2} ", "f885677bb2fd022b18ada372c1e8e13d": "W^*", "f885787f67d8c24536e3b3beb0703881": "TM \\to E.", "f8859439b1cee9dbdc3cae62e4e87157": "\\int Mv\\,ds", "f885c90c80d00a77d368d02310d170b5": "\\forall x,y \\in A, x \\neq y \\Rightarrow f(x) \\neq f(y).\\ ", "f885cf58cb3e7d52e363e4e628074c5c": "\\zeta(a,b)+\\zeta(b,a)=\\zeta(a)\\zeta(b)-\\zeta(a+b)", "f886e4058d5f5bafb4b0670ee4692e1d": "\\{ \\bot, \\top \\}", "f886fbbdca9705c6342e5db235f48ca5": " = 1 x^3 + 0 x^2 + 0 x + (-1). \\,", "f887204d4199d4335e676e26c4250328": "\\alpha v ", "f8872592ccb1f817b86e955e2e3a16e2": "H_2 = -2\\,", "f8880d447b1b8e9438c75d2041d640dd": "D_e", "f8885c2b060a828ad0f4f6619400f4de": "r=\\frac{1}{2-\\cos \\theta}-\\frac{1}{2-\\cos \\theta}=0", "f88875a17b0dccdf71aa40f8e38a01f4": "2\\,\\pi\\,\\ell \\le \\int_A \\rho\\,dr\\,d\\theta \\le \\Bigl(\\int_A \\rho^2\\,r\\,dr\\,d\\theta \\Bigr)^{1/2}\\Bigl(\\int_0^{2\\pi}\\int_{r_1}^{r_2} \\frac 1 r\\,dr\\,d\\theta\\Bigr)^{1/2}.", "f8889a569ed14b8f441945eaafec94c9": "\\displaystyle n=\\underbrace{1+\\cdots+1}_{m_1}\n\\,+\\, \\underbrace{2+\\cdots+2}_{m_2} \n\\,+\\, \\underbrace{3+\\cdots+3}_{m_3}+\\cdots", "f888aef0df429191641dd64109582297": " I(X,Y;Z|W),", "f888b26b25f6a8508f48e309b76350de": "X_0, X_1", "f888fcbff94d08831ba82553af59a468": "M^\\prime", "f889332ec3de29817b4e18c53cfcf23b": "T = n_\\text{t} - n_\\bar{\\text{t}}", "f8893869da822dac57099b6408237894": " \\Phi = \\frac{dV}{dt} = v \\pi R^{2} = \\frac{\\pi R^{4}}{8 \\eta} \\left( \\frac{- \\Delta P}{\\Delta x}\\right) = \\frac{\\pi R^{4}}{8 \\eta} \\frac{ |\\Delta P|}{L} ", "f8897a6f6aa69169113eee3662e75400": " IMM_{i-1}(S_{x,{i-1}}, g) ", "f889b26a8c803de5c9a9515b378e6b44": "2^{1+\\lfloor{\\frac{2d-k}{2}}\\rfloor}", "f88a166aea3567f032774e0368d14bbb": " R_1, R_2 \\geq n-k, R_1+R_2 \\geq 2n-k", "f88a45117d956770ba096adc5a2da034": "-\\ln(\\ln(2)) \\approx 0.3665", "f88a5fcbae878aa039edd8e1f9684a5c": "\\mathbf{S}=\\epsilon_0\\int \\left(\\mathbf{E}\\times\\mathbf{A}\\right)d^{3}\\mathbf{r} ,", "f88aa6b0c0eb37d019bb91185313d654": "w_1, w_2, \\ldots, w_n", "f88aac2d29f64f87957ca61f4c346e4c": "\\frac{d^2y}{dt^2} = 2 y^3 + ty + \\alpha ", "f88ad3eeb36abfa51dcd27c350f81c55": "\\neg\\varphi \\vdash \\bot", "f88ae81ca9f2dfe1cbe847c9497ff696": " {}= p_{01}p^{01}+p_{02}p^{02}+p_{03}p^{03} \\,\\! ", "f88bc5a1dbe6daddfb6c34b7a2d71b03": "K = \\frac{k}{\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{1}+\\frac{1}{2}}", "f88c0214b559a778cc145fa621a277bc": "m_{Moon} = 0.25 + 2.5 \\log_{10}{\\left(\\frac{3\\pi}{2} 0.00257^2\\right)} = -11.02\\!\\,", "f88c52043780cb816063380756609430": "f(x; k,\\mu)=\\frac{ x^{k-1} e^{-\\frac{x}{\\mu}} }{\\mu^k \\Gamma(k)}\\quad\\mbox{for }x, \\mu \\geq 0.", "f88c5298dd8b1449fda1ea459c45cd46": "\n\\begin{bmatrix}\n1 & 1 &\\cdots & 1\\\\\n\\end{bmatrix}\\mathbf{v_i}=0", "f88c575fca92a93ccadd58897d21575f": "P(x)=Q(x)T_{m+n}(x)+K(x)x^{m+n+1}", "f88c5862103085122935aab82ced4fa8": " \\int\\limits_\\Omega f\\,\\mathrm{div}\\varphi = -\\int_\\Omega\\nabla f\\cdot\\varphi ", "f88c7a95d4471f8c33364b98ef155771": "1=N_0\\subseteq N_1\\subseteq N_2\\subseteq\\cdots\\subseteq N_n=G,", "f88d3aa0643a227f57894733c8005e8d": "\\Psi = Ae^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} = Ae^{i(\\mathbf{p}\\cdot\\mathbf{r}-Et)/\\hbar} ", "f88d6de2810fd39e910223d61153acd6": "\\scriptstyle\\frac{1+\\sqrt{5}}{2}.", "f88d786c19d631ff8eecab59c8d0968e": "r_c \\approx L^2/m - 3 m", "f88da2e02c3221e9c17bec3a9d51656f": "\\frac {M^d} {P}=\\frac {Y} {V} \\,", "f88dae64e1dde6a138771b39158183dc": "\\tau_{\\lambda_2}\\,", "f88eb357cdd3deb8d68bfdf790194037": "T(\\alpha)<\\alpha", "f88ed1fa4aa420e56ac79b0fcdc88866": "\\Delta_\\rho", "f88ef53c27e9e39e4151940704cd9be4": "\\frac{b^2}{a}\\,\\!", "f88f08a6d83a588e0550fab00c39b1a0": "X(t)=\\text{Real}{\\sum_{j=1}^n a_j(t)e^{i\\int\\omega_j(t)dt}}.\\,", "f88f16d0041385f26f77e82c51b02f24": " A^x +B^y = C^z", "f88f654a9eca0d6f344c59e1ffb232d2": "a_x b_y - b_x a_y", "f8901d6ff9c555c94ffc15313640c301": "H(X_1, ..., X_n) \\geq \\max[H(X_1), ..., H(X_n)]", "f8905eb4a1e7a3c4c05733c8f47f6888": "\\sqrt{h_1(\\vec{x},t)}", "f89066b5e5e348d8b173d1d04937dfd3": "\\scriptstyle X = \\coprod_i X_i", "f8909ac22b41f276125b8ec700c809bc": " { s \\over s^2 - \\alpha^2 } ", "f890af15f5a5a194461f7d4d9a5afa58": "\\langle \\psi \\vert \\psi \\rangle", "f890d0d9b4fd9391479ae50126e5ba9e": " \\bar \\psi D_\\mu \\psi ", "f89158cd51fe353b5dfbfaf0be74b48c": "d_\\mbox{act}", "f89177cd3fb0ea650ead8fdf8cea4b1c": "\\alpha_c", "f8917f99b3c9a8c51dd456fc50512d6d": "L_\\frac{1}{2}\\left((x_0,x_1)\\right)", "f891adbc4b9891f78bdb599b207be9d3": "1/5 \\; \\text{Myr}^{-1}", "f891f3f7cd918689fb725f7b7a5e3553": "m = 0,1,3", "f891fd3d0c79a0448a622eb114cbb438": "D = (d + d + 2d)^{1 \\over 3} ", "f892cfd9147b6df57b91fd04b7aa8dde": "p^3<3 q^3 \\Rightarrow D(p/q)=\\mathrm{true}\\;", "f892d797d0450e203fa36731def78aaa": "A=e", "f892dd4d9f22dd626102990912860785": " \\ A", "f89380d8fb3fe42bf6c7abc0696e7af7": "l_o", "f8938881d1a54436ecd2516f942c3f4d": "r_1 = \\frac{k_1 [X] [Z_P]}{K_{M1}+ [Z_P]}", "f8938ae88aa6498786a945a48d89feeb": "\\partial_t", "f8938c969e5228bb4cb3d58da0eb5b24": "\\mathbf{E} = (\\mathcal{F}\\cdot\\gamma_0)\\gamma_0", "f893b4acc6c29c15329d7759b3c0a962": "~~|{\\rm initial}\\rangle=|\\alpha\\rangle~", "f8940e5aa26c2b84ec9df62d039e24d5": " Z =\n\\sum_{n>0}\n\\frac{(-1)^{n-1}}{n}\n\\sum_{\\begin{smallmatrix} r_i+s_i>0\\,\n \\\\ 1\\le i\\le n\\end{smallmatrix}}\n\\frac{X^{r_1}Y^{s_1}\\cdots X^{r_n}Y^{s_n}}{r_1!s_1!\\cdots r_n!s_n!}.", "f894746c1a489089f6b5eab362298b59": "z \\notin L \\implies \\forall D. \\Pr\\nolimits_x[\\phi(x,D(x,z),z)] \\leq \\tfrac{1}{3}", "f89531603c3dfaaf558808057851fdf4": "\\frac{c_n}{c_{n+1}}\\approx r", "f895626c8022fb030b7a4f6e0abd968c": "D \\mapsto \\mathcal{L}(D)", "f8959e6e4d1fa1d5efe0c7645d03e57a": "(x,y,z),", "f8959fd305c10e3370637534da9faf80": "\nr_{d,r} = h_{d,r} r_{r,s} + n_{d,r}\n= h_{d,r} h_{r,s} x_{s} + h_{d,r} n_{r,s} + n_{d,r} \\quad\n", "f8960fdf6cd192064b27fab2b7bef3df": "r=2M", "f89671b518bed1442b360bad8848e38c": "\\Delta_n=-L_n - R_n^2 -(n-1)R_n, \\, ", "f896b3fa56d1a0e1ec4cb3ee751f0f32": " ds^2= \\sum g_{ij}dq^i dq^j ", "f896c5e69c44f85544bdb27eed4d5285": "\\left(dI/dV\\right)/\\left(I/V\\right)", "f896ce63bcfd8c253ff8dc9751776367": "K = \\frac{2 (\\Gamma(\\frac{N-t}{2}) )^2}{(\\Gamma(\\frac{N-t-1}{2}) )^2}", "f896faca134412a289838d53b4c23423": "t_2/t_1 = (n_2/n_1)^a", "f897612fde8df9ccdd214249b5471852": "\\overline{O_L p_L}, \\overline{O_R p_R}", "f897930a247227b888c12fc75da17e37": "\\ h_{1,0}=-(\\gamma/2)\\bar n", "f897999be98ae74ce21ee8cd1ecc2ed0": "{\\rm Tr}\\, \\gamma H+{\\rm Tr}\\, \\gamma\\ln\\gamma\\geq -\\ln {\\rm Tr}\\, e^{-H},", "f897a18849e6eb81143256c15f1fde6a": "(\\Bbb{N}, +, \\cdot)", "f897b0929ef5790fcbeef1b8a5e5e7e4": "\\begin{matrix}\\text{If } y(t)=\\int_{-\\infty}^\\infty x(t+\\tau)h^*(\\tau)\\,d\\tau\\text{ then }\n\\\\ W_y(t,\\omega)=\\int_{-\\infty}^\\infty W_x(\\rho,\\omega)W_h(-t+\\rho,\\omega)\\,d\\rho \\end{matrix}", "f897d3537f277dbea4b26d6ba8e26869": " \\textbf{G}(\\pm j\\infty) = 0 ", "f897dcc6cc205fdc05568e2785419fd9": "{u_1}", "f898178f7aefa558191f057a4c7438a1": " q = \\frac12 \\times \\rho V ^ 2 = dE ", "f89828c4caf30c7c1f99ff5ee154b6c0": "\\sqrt{5-2\\sqrt{5}\\ }", "f8982e2ec948c6657c5937ab40fca402": " A_{\\mathfrak{p}} ", "f8987062fa57dc0ec1a72143c2e1ebca": "(\\exists x \\, \\alpha(x) \\and \\forall x \\, \\gamma(x)) \\rightarrow \\exists x \\, [\\alpha(x) \\and \\gamma(x)].", "f898f67c4a26281cd4b207fd54e690e2": " \\mbox{EMV} = \\max_i \\,", "f89911f0f0719e21ebac582e48cee9ac": "w_1=(1,0,0,a)/\\sqrt{1+a^2}", "f899139df5e1059396431415e770c6dd": "100", "f8995ac40452848d6e6947addbbb5395": "\\Xi_k", "f899e2eedc10d2b986d85d6be005ed60": "c^2 = 2\\kappa G h/[D(1-\\nu)]", "f899fe365afa713d168e63eb4b6e41cc": "\\textstyle(x\\pm1, y\\pm1, z\\mp1)", "f899feb48d0eac46cee8a99f377b5daf": "\\displaystyle{e^{-izw/2}F(z,w)=W(z,w)v.}", "f89a3df0b2c2af63d178862d5f5e6f99": "\\mathbb{E}\\left[\\mbox{ Arnold }|\\mbox{ Charles folds }\\right] = \\frac{42-9}{42} \\cdot (P+2) = \\frac{33}{42} \\cdot (P+2)", "f89a8067420fe8e30b9be07a3382d3ca": "p \\land [a*](p \\to [a]p) \\to [a*]p\\,\\!", "f89a8945113be3af6cbd8039f70c81c4": "Z_o = i\\omega L = 1/i \\omega C", "f89ab22526458ca461d760f5884c5697": "C_i = \\frac{N_i}{V}.", "f89ac8fc29a14af9b26ba62a475260aa": " \\Psi(x) = \\sum_{i=1}^n \\Psi_i(x^{(i)}),", "f89b03b1b6e8b3a2aa9800824f4c5e1d": "y=x+w,", "f89b08db780dd85cfd6768dd49777154": "\\alpha =1-1/n^{2}-(\\lambda /n)(dn/d\\lambda )", "f89b2d722e88fb9ad848ec8d7427819d": "n\\ge 6", "f89b7de0672a3e3313b788223769d166": "(A + B)\\mathbf{x} := A\\mathbf{x} + B\\mathbf{x},", "f89bbbad4ea471db498aba67954d3434": "\\pi_{\\text{Age,Weight}}(\\text{Person})", "f89be999e4f57677b9985a1333d7e21e": " {n \\choose i} = \\frac{n!}{i! (n-i)!} ", "f89bf722a31919812affa70af586e4a2": "f(X_{t}) = f(x) + \\int_{0}^{t} A f(X_{s}) \\, \\mathrm{d} s + \\int_{0}^{t} \\nabla f(X_{s})^{\\top} \\sigma(X_{s}) \\, \\mathrm{d} B_{s}.", "f89c60d7fb4dd901b3d4efe429a07979": "v_{1}= \\begin{pmatrix} 0.7073 \\\\ -0.07278 - 0.7032i \\\\ 0.0042 - 0.0007i \\\\\\end{pmatrix}", "f89c67f5cb2a270529b39ebc23bc85c7": "\\scriptstyle \\ell^p", "f89c6938a661e6e041b9f724f07d0228": "\\nabla \\cdot \\mathbf{J} = - \\frac{\\partial \\rho}{\\partial t}", "f89c7013cfc49ea3b049f7c6ef9573de": "P_\\mathrm{L} \\,\\!", "f89c81196cfe51065c1e8608cb8728e9": "C_{3,1} = 20 \\log n \\ ", "f89cbc949b7145a806d23b93f09b635b": "\\sigma_{\\alpha 3}", "f89cc38bdb51c524680d5ee0e4feb4af": "ds^2 = 4 \\frac{\\sum_i dx_i^2}{(1-\\sum_i x_i^2)^2}", "f89ce969fda67afb42185d1a0c474625": "w_C = \\tfrac{qL^4}{8EI}", "f89d0d64dff094b86ba769dacfbd4016": "Terminates", "f89d1477e3e7e8e7ea1b740e54c31e4b": "f_\\pm", "f89db97b15aacc7fb7c6e0931e4e1841": "Tail^+(X) = \\{Y|X \\Rightarrow^+ \\alpha Y \\}", "f89e06777e20e6f4d5824ff0d59ce812": "g_3(\\tau)=\\frac{8\\pi^6}{27} \\left[ 1- 504\\sum_{k=1}^\\infty \\sigma_5(k) q^{2k} \\right] ", "f89e298bd4f5d228aee6a0b23aca6ae7": "|x_i-x_j| = h", "f89e3773c08892784933e3d2ee03324c": "Em = \\tfrac{3}{5} \\tfrac{2}{5} 3 + \\tfrac{3}{5} \\tfrac{3}{5} 1 + \\tfrac{2}{5} \\tfrac{3}{5} 2 + \\tfrac{2}{5} \\tfrac{2}{5} 0", "f89e46ff70eaa2bde4f862b4f7270130": "4^n/2n", "f89e5fe3a3b3a0cb35baf85de29a9963": " ( 1 + x )^n \\approx 1 + nx \\ ", "f89e8265f9b23e74ced15064cdf86afb": "ds^2 = g_{\\mu\\nu} \\, dx^\\mu \\, dx^\\nu.", "f89e91fba213de715b8da8cd59204206": "pV = nRT\\,", "f89eb7e736297e1bf8dffc8c8384be46": "{\\ell_{OA}}^2 = (y + x)^2 + h^2", "f89ed5ad735677cb172d489ddadd74bf": "{x \\choose y} \\cdot {y \\choose x}= \\frac{\\sin((x-y) \\pi)}{(x-y) \\pi}.", "f89edf16458f66b049f37d43eb0fa919": "\\int \\ln(x) \\,dx = x \\ln(x) - x + C.", "f89f7e6c94a14c1bb2c918f93bb1f40b": "\\scriptstyle150 \\sqrt{1.4} = 177", "f89fb654deb6a0d6d96573599c6693ea": "M\\geq 0", "f89ff8af23c90e11756b9bdadf4f3e0f": "\\omega_n(f, \\delta)=\\sup\\limits_{x; |h|<\\delta;}\\left|\\Delta^n_h(f,x)\\right|.", "f8a00c285c0bc27fcb65e8daf8fedf50": "- \\sigma_{yz} + \\sigma_{xz} - \\sigma_{xy}", "f8a027a367063c79491a8db08a27dad8": " Y_n = \\begin{cases}\n1, & if \\, U_n > 0, \\\\\n0, & if \\, U_n \\le 0\n\\end{cases}", "f8a077991bea89e333361d8caaed1047": "\\det(V) = \\prod_{1 \\le i \\le n} \\prod_{c_1, \\dots, c_{i-1}} \\left( c_1\\alpha_1 + \\cdots + c_{i-1}\\alpha_{i-1} + \\alpha_i \\right). ", "f8a0b87723688cf62b3c505f8e669161": " \\neg F \\vdash \\neg E", "f8a0c99908cb4d1ad413d747f4e32df2": "\n \\mathbf{x} = \\boldsymbol{\\varphi}(q^1, q^2, q^3) ~;~~ q^i = \\psi^i(\\mathbf{x}) = [\\boldsymbol{\\varphi}^{-1}(\\mathbf{x})]^i\n ", "f8a0f96edcd0b41d3584f4d530ee5110": "S_\\pm(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s (e^{2\\pi n} \\pm 1)}", "f8a107386a3a9023871bd8dd9b13fcbb": "[T_A]", "f8a1528a8348b0467804d026aa2191e2": "T > 1.64,", "f8a1a6ca7e3579b680e8e92f08d726c5": "\\frac{1}{K}d_X(x_1,x_2) \\le d_Y(f(x_1), f(x_2)) \\le K d_X(x_1, x_2)", "f8a25d11c2f006c2096d18a3b1a5ef3b": "\\left \\lfloor \\frac {n}{2}+1 \\right \\rfloor \\, .", "f8a26aa98837a1b2b481a4a126cfc82a": "z_\\beta", "f8a31c53afae13dea03f6b9118392434": "T_G", "f8a31fe516efe6bcec2ad01e481f74b7": "-([P]-[O])", "f8a3239f0aa433903aa97a88227033ed": "b_x\\, ", "f8a32cc389eb879daaa1644604cb4c25": "z = r\\, \\cos(\\theta)\\,", "f8a3a89c2ea7f0a13b0f0b7fffab8d89": "R=\\frac{2m_p}{\\pi}", "f8a3af1f6971e3bcc586b598fb3cc5a0": "{{Thames\\ Tonnage}} = \\frac {({length}-{beam}) \\times {beam} \\times \\frac {beam}{2}} {94}", "f8a3e9214eef91850cfc6a9153021efa": "-T_{centripetal}=\\Sigma F_{radial} = m a_{radial} = - m \\frac{v^2}{r}.", "f8a47325e04e404dbe5b51070d76da3d": " A_0 = \\left ( {A_t}*{e^{kt}} \\right ) ", "f8a4880b841ca63c3026d6585e0cd87f": "k(z) = \\operatorname{p.v.}\\frac{1}{z}", "f8a4a3557de74ce63fcbd074c4500f59": "\\textstyle H(X_r) - H(X_r|Y_r)", "f8a4fbc86b34839177fa4ee3df79525a": "f_c(z) = z^2 -1", "f8a5520a054d6541154fbf1faefa47c2": "D=\\frac{- 2 \\pi c}{\\lambda^2} \\beta_2 > 0 ", "f8a55a4c1e35633ccb718c101785d308": "\ns^2\\ = \\frac {V_1} {V_1^2-V_2} \\sum_{i=1}^N w_i \\left(x_i - \\mu^*\\right)^2,\n", "f8a5d830adfa1205294c6f4151a0b77a": " \\Psi = \\prod_{n=1}^N\\Psi \\left (\\mathbf{r}_n,s_{zn}, t \\right ) ", "f8a6202bd030da3bdc14c55daf949f43": "(0 < w' < 2w)", "f8a64c97c263d3f70cd045e15c0d9f19": "U_i(F_i(\\mathbf{a}_{-i}),\\mathbf{a}_{-i})", "f8a65b2c1a4e5b573ee0b25ab4037286": "f(x) > \\varepsilon", "f8a6c69fc76890c33186ec465a14c45e": "\\tilde{\\mathcal{A}}^{AB}", "f8a6cbf33806fa060daaf319dbd4b8bd": "\\Pr[\\text{recombination}|\\text{linkage of }d\\text{ cM}] = \\sum_{k=0}^{\\infty} \\Pr[2k + 1 \\text{ crossovers}|\\text{linkage of }d\\text{ cM}]", "f8a6ddb85e8a00e6d983592372c88cb9": "r = a/ \\theta", "f8a769cbc24204519059b7c4e468aeaa": "\\hat{x}^o=G(x)y.", "f8a76f4b521789a3cbdfa1a16dc60af3": "\\scriptstyle{X_C}", "f8a7816bf02b7d43da09f7359334ab62": "h_{\\text{out}}(G) \\le h(G) \\le d \\cdot h_{\\text{out}}(G).", "f8a7caa879dea19360191369dedb0c21": "l_m", "f8a7d1c3026e39e1a6a2a4cc9c145da5": "q_{ij} = \\dfrac{\\partial^2V}{\\partial x_i\\partial x_j}", "f8a7e89e2f89b2f204b7cb7cadde74be": "\\mathbf{D}(\\mathbf{r}, t) = \\varepsilon_0 \\mathbf{E}(\\mathbf{r}, t) + \\mathbf{P}(\\mathbf{r}, t)", "f8a85e9874c04dab52368ba40787eb2c": "x+yi", "f8a943be1234676fa7202bbbf6d40f6e": "\\left(\\mathbf{w},\\frac{\\partial\\mathbf{v}}{\\partial t}\\right) = -(\\mathbf{w},\\mathbf{v}\\cdot\\nabla\\mathbf{v})-\\nu(\\nabla\\mathbf{w}: \\nabla\\mathbf{v})+(\\mathbf{w},\\mathbf{f}^S)", "f8a954c34c57eee0fc4afe584af86ca4": "\nX \\perp\\!\\!\\!\\perp Y\n\\quad \\Rightarrow \\quad\nY \\perp\\!\\!\\!\\perp X\n", "f8a95f684100e195e7eb364285497c92": "lb_{computed} \\le 1", "f8a967ea765f5604ee3f57c118a7cff3": "K_v = \\frac{G_g^fa W_g^r}{R_g^f}", "f8a9c1dceaad06c5643434cf7213a620": " k_4 = k_1 + k_2 + k_3 \\pm2 \\sqrt{k_1 k_2 + k_2 k_3 + k_3 k_1}. \\,", "f8aa2016fa408a1553ca609343aa8d68": "\\bigcup_{j \\in S_1} M_j \\neq \\bigcup_{i \\in S_2} M_i", "f8aaad1469762a0397c94a43c970f789": "A(r_0)=\\frac{e^{i k r_0}}{r_0} f(\\mathbf{r}_0/r_0,k,u_0) + o(1/r_0)\\text{ as } r_0\\to\\infty", "f8aad24fa013f02ad9ce2a1ca1d98d94": "\\nabla\\cdot\\mathbf{A} = \\nabla_i A_i ", "f8aaf8eb5f5136d36032f535ef9bbc87": "\n \\begin{align}\n \\frac{\\partial G_{ij}}{\\partial x^k} & = \\left(\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^k}~\\frac{\\partial X^\\beta}{\\partial x^j} +\n \\frac{\\partial^2 X^\\alpha}{\\partial x^j \\partial x^k}~\\frac{\\partial X^\\beta}{\\partial x^i}\\right)~g_{\\alpha\\beta} +\n \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^j}~\\frac{\\partial X^\\gamma}{\\partial x^k}~\\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma} \\\\\n \\frac{\\partial G_{ik}}{\\partial x^j} & = \\left(\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k} +\n \\frac{\\partial^2 X^\\alpha}{\\partial x^j \\partial x^k}~\\frac{\\partial X^\\beta}{\\partial x^i}\\right)~g_{\\alpha\\beta} +\n \\frac{\\partial X^\\alpha}{\\partial x^i}~\\frac{\\partial X^\\beta}{\\partial x^k}~\\frac{\\partial X^\\gamma}{\\partial x^j}~\\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma} \\\\\n \\frac{\\partial G_{jk}}{\\partial x^i} & = \\left(\\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k} +\n \\frac{\\partial^2 X^\\alpha}{\\partial x^i \\partial x^k}~\\frac{\\partial X^\\beta}{\\partial x^j}\\right)~g_{\\alpha\\beta} +\n \\frac{\\partial X^\\alpha}{\\partial x^j}~\\frac{\\partial X^\\beta}{\\partial x^k}~\\frac{\\partial X^\\gamma}{\\partial x^i}~\\frac{\\partial g_{\\alpha\\beta}}{\\partial X^\\gamma} \n \\end{align}\n", "f8abc3d8570c2a9591b9656f305b1dbf": "x_{k+1},\\ldots,x_n", "f8abc4367f9686de91630cabced4d20b": "\\Delta\\omega_{lab}\\approx 2\\pi\\cdot 500 \\mbox{ MHz}\\gg\\Gamma/2\\approx 2\\pi\\cdot 3 \\mbox{ MHz}", "f8abd4b72d5a281b88589a37d6cdb835": "3^2 \\not\\geq n", "f8abf34599677984dfe91b4f300389f6": "f(x)=x", "f8abf60a0119663a176f80433e560895": "\\left ( \\frac{0.693}{t_{1/2}} \\right )", "f8ac584b6c2b8f07870d0d7a2ae29485": " m^{-1/2} ", "f8ac6075ef27176bb631ce5035baf10a": "F_i = \\begin{bmatrix} \\vdots & \\vdots & & \\vdots \\\\ f_{i1} & f_{i2} & \\cdots & f_{i|J_i|} \\\\ \\vdots & \\vdots & & \\vdots \\\\\\end{bmatrix}_{N \\times |J_i|}", "f8ac6457bff0a236894adfcf6c32360d": "\\begin{align}\\mu &: A\\otimes A \\to A\\\\ \\eta &: R\\to A\\end{align}", "f8ac6e00c6d2866b6ec235af9b72250e": "P(\\forall n_0\\ge 0, \\exist n\\ge n_0, |X_n| < \\epsilon ) = 1", "f8ad123ee64ec93e7bbf10a68e45f731": "S_{sy}", "f8ad87acd27b9f33d6c0011cb0ed739c": "(\\frac{4\\pi d}{\\lambda})^2", "f8ad9a764d0cdf62fdbe5b166ce3e299": "\\alpha(t)=\\cos^{3}(2\\pi t)+ i \\sin^{3}(2\\pi t), 0\\leq t \\leq 1 ", "f8adbf72cf57cff258d1a3e7f13fb007": " A\\,\\triangle\\,B\\,", "f8ae090ef865b7e1ff48a0d975bea148": "\\{GD,G \\bar D, \\bar G D, \\bar G \\bar D\\}", "f8ae35382f1a4826816f648fddf0229a": "B_{t}^{\\tau} (\\omega) \\equiv B_{T} (\\omega)", "f8aeb86890ea3cd55c7f6692870d2e68": "T^i", "f8af1bfffd41f2540415ebeafd352c6d": "\\langle S_k, \\varphi\\rangle \\to \\langle S, \\varphi\\rangle", "f8af37200186782044708d16d5f3521e": "P^3 + 2bP^2 + (b^2 - 4d)P - c^2 = 0\\,", "f8afb19ea30437d2ee9b3bff9f32a12b": "H(X,Y)=H(X)+H(Y|X)\\,", "f8b02123907b6d3bbc525fde45a78d30": "R_2 =45 \\times 0.0078", "f8b029f4a36d5e79bd71ba8933df56aa": " {\\rm E}[g(Y)]-{\\rm E}[g(X)]={\\rm E}[g'(Z)]\\,[{\\rm E}(Y)-{\\rm E}(X)].", "f8b054d18c6363ec8c0cb4d8aab65375": "\\begin{matrix} {11 \\choose 1}{4 \\choose 3}{40 \\choose 1} \\end{matrix}", "f8b067742736a2100c30945e62d5dfed": "\\rho(\\mathbf{x},t)", "f8b07a17d6ccf5a6addf21427dc25dcf": "\\frac{dx(t)}{dt}=y(t)", "f8b07b8f083e1f5e491c29c0e151dbdb": "\\deg(PQ) = \\deg(P) + \\deg(Q)", "f8b0c26cff6e2f3cf43a6f376258a88d": "\\mathrm {soc}(M) = \\sum \\{ N \\mid N \\text{ is a simple submodule of }M \\}. \\,", "f8b125d6b4d78b056ae8b38c47d2f975": "S_\\ast(-)=S_\\ast(-,k)", "f8b174f7ac4d91c32a8886e847ec9b13": "N(\\Omega)", "f8b1a4854127c2063869492adda95b79": "\\lambda_{t+1} - \\lambda_t = -\\frac{\\partial H}{\\partial x_t} = -\\left( \\frac{u_t}{x_t} \\right)^2", "f8b1c5a729a09649c275fca88976d8dd": "\\varepsilon", "f8b200f08bff696a45b4b7d2254b49a2": "\n\\left[ T_\\mathrm{n} +E_k(\\mathbf{R})\\right] \\; \\phi_k(\\mathbf{R}) =\nE \\phi_k(\\mathbf{R})\n\\quad\\mathrm{for}\\quad k=1,\\ldots, K,\n", "f8b230daa4ed7b71c29b4eead11da44c": " L_x ", "f8b279d6345830067b0d4b547d1a2540": "B = \\prod B/{I_i}", "f8b2837bbd87cf60459d79e68bb4dc6c": "q=p^d", "f8b2b70bb6f2b3da1da57c4518b7fdf8": "\\{r\\in R\\mid rX=\\{0\\}\\}", "f8b2b89302732519298f8b451bd171c0": "x(0)=0,\\ y(0)>-1/2", "f8b30e171d54b16a97bfd6b05df366c7": "x\\mathfrak{m}=0", "f8b3131323c6a7f61ed0faeeb1ef91c0": "= \\mbox{Arg} \\left( \\left\\{ 1 - \\left| a \\right| cos( \\omega - \\theta_a ) \\right\\} + i \\left\\{ \\left| a \\right| sin( \\omega - \\theta_a ) \\right\\}\\right)", "f8b320591ba5de701018b51f1af2165e": "\n \\frac{\\eta}{\\lambda} = A\\, \\left[ \\cosh\\, \\left( \\frac{x-ct}{\\lambda} \\right) - 1 \\right],\n", "f8b33ca4535532a3f6032711f7be7cbc": "\\ell_{\\mathrm{P}}=\\sqrt{G\\hbar / c^3}", "f8b3601373bfad815662eb71b6a5925c": "\\scriptstyle\\alpha \\,=\\, 0", "f8b396699db21f7f92a479df2ae0306e": "\\Delta_h = T_h-I, \\,", "f8b3c8b040b62359a244b939308d4807": "\n\\eta(4i)=\\frac{\\sqrt[4]{\\sqrt{2}-1} \\Gamma \\left(\\frac{1}{4}\\right)}{2^{{29}/16} \\pi ^{3/4}}.\n", "f8b3cd62ef783f592279eed38e22675d": "\\mathbf{C} \\otimes_{\\mathbf{R}} V.", "f8b3fcfc70d8d7540cd85fb83a84a7ec": "\\int_{Z(\\mathbf{A})G(K)\\backslash G(\\mathbf{A})}|f(g)|^2\\,dg < \\infty", "f8b48fe98daa32986e7402a19759e29b": "\n\\begin{align}\nR''' & = (w^e*R')^d \\pmod n \\\\\n & = (w^e*R^e)^d \\pmod n \\\\\n & = (w*R)^{ed} \\pmod n \\\\\n & = w * R \\pmod n\\\\\n\\end{align}\n", "f8b4b5d4df88dbac58406f1abee2ca71": " \\psi(t) = U(t)\\psi(0) = {\\rm e}^{iHt} \\begin{pmatrix}a \\\\ b\\end{pmatrix}, \\qquad H =\\begin{pmatrix}M & \\Delta\\\\ \\Delta & M\\end{pmatrix}", "f8b4db13bd923660548f43795c6f3172": "J + I = F", "f8b55126e3acb2eac0ee1044975cec3f": "\\sqrt{\\frac{\\pi h}{128}}g + O(g^{-1/2}h^{1/2}) + O(gh^{-1/2})", "f8b594cf3d597b63dc883b735a3098c5": "T=\\exp\\left(- \\frac{2Z\\sqrt{2m}}{\\hbar}\\sqrt{\\frac{\\phi_s+\\phi_t}{2}+\\frac{eV}{2}-E}\\right)\\ ,\\qquad\\qquad (6)", "f8b5afb2969caeac1976928785834f79": " \\mathrm{d}^2=0 ", "f8b60b6b3e222be954844b9a9b93731f": "K>0", "f8b642730a8cd29fcd79f4cca75d84c3": "Q(p)=\\sum_{i=1}^{m}a_i Q_i(p)", "f8b668e8b2daea1be8ae37447768f0ac": "R_h^*", "f8b6e8f9162ec78a9343ca926379f4ba": "(\\phi \\wedge \\psi) \\to \\neg (\\neg \\phi \\vee \\neg \\psi)", "f8b7159ed6c4c9259b297a2abde5492f": "P^{-1}A", "f8b7b33a51c64dbaf55a66e6c0051832": "= 91", "f8b7cc3d902f5c1f19bcf37c5683676d": "b\\in[\\omega]^\\omega", "f8b82359de88f0708c022ed6ab06629c": "t>\\tau\\,", "f8b82fabe789cf18ce90a01cf3c7f8f6": "3\\mid 12", "f8b83abe48cf81e7e16710d5c6bbb4f7": "D_2(\\omega) = \\frac{\\partial T_g}{d \\omega} = \\frac{d^2 \\phi}{d\\omega^2}", "f8b83cd2bd32c5b24ef098bc72154211": "\\begin{align} x_n^2 = r^2 - y_n^2 \\end{align}", "f8b85cb7a58271f661fc4349f0c772b5": "W_\\alpha(x) = \\sum_{n=0}^\\infty b^{-n\\alpha}\\cos(b^nx)", "f8b8788ab8fb06fd64b7252195dd0a73": "\\hat{\\pi}_i", "f8b89ac9d2617e51659d4bf416041fe1": "\\triangle T_f = K_f \\cdot m \\cdot i", "f8b8e78cd1fa4335cb80959b082916da": "| 1 0 \\rangle", "f8b90dc050617cad43ed89ed56f28bed": " v(x)", "f8b92313707d5ea9a1a327e2f302a690": "\nd_r=\\frac{|x-y|}{\\max(|x|,|y|)}\\,\n", "f8b95df9c23c544eebaba4d0b2868cae": "\\mathbf{\\Sigma}", "f8b9fac8cac1d964c5fb6eb6b3cf872b": " = \\textbf{t}_p^T \\textbf{t}_i", "f8ba232abd285c5777c160131f98d6de": "\\mathit{b}", "f8ba2bf5ef4bacc676c7d93bc9830e11": "( iU^\\dagger \\gamma^\\mu U\\partial_\\mu^\\prime - m)\\psi(x^\\prime,t^\\prime) = 0", "f8bae4beba362832522c37bb8e62cd44": "\\tau_a f(x)=f(x-a) .", "f8baf986826232cc8180a269c7608504": "\\chi=\\Psi_{\\alpha(n)}", "f8bc060776ec0cd6dbbfe6d380acfa0d": "t = 1,2,...", "f8bc1b90200f8b658741aa132cb7bc40": "\\rho_{avg} \\equiv \\frac{M}{4/3 \\pi R^3}", "f8bc1ceb4ee660a5fb7870477a08361a": "{ {\\underbrace{a \\uparrow (a \\uparrow (a \\uparrow \\cdots \\uparrow a))...)}} \\atop{b} }", "f8bc259a829d7bb46a0d336df5f7653a": " F(x_1,x_2,...,x_n,y) ", "f8bc2de9375c90057522ecd729b53c41": "(\\pi \\ominus \\sigma)^{-1} = \\sigma^{-1} \\ominus \\pi ^{-1}", "f8bc2fbe2c937ea5b5e8839cbea69491": "2b", "f8bc422f0cf69ece02a6ab1726202c2f": "{V}=\\frac{M_1^2\\alpha_2 + M_2^2\\alpha_1}{-\\left(4\\pi\\varepsilon_0\\varepsilon_r\\right)^2\\mathrm{r}^6}", "f8bc890f55599be577945ff9c581d14a": "\\left(\\nu x\\right)P", "f8bcc467e6e803188de0921de85cc945": "-\\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y_n) - \\xi_n", "f8bd07b0044947dcfce42747c0565505": "|\\Phi_n(q)| \\leq q-1", "f8bd40ba905298568d401d438cf3b75a": "\\left (-\\frac{b}{2a},-\\frac{D}{4a}\\right ).", "f8bd9436bbeecb67bf9ba679ddb98fa4": "\\begin{align}\n\\sigma(0) &= -\\frac{1}{24}\\\\\n\\sigma_3(0) &= \\frac{1}{240}\\\\\n\\sigma_5(0) &= -\\frac{1}{504}.\n\\end{align}", "f8bdb43ffd7c5f0a947ec3e679607853": " B_k \\!", "f8bdcd4932d4a8d93eb8335fe2cce4db": "\\left[J^{\\mu \\nu},W^{\\rho}\\right]=i \\left( g^{\\rho \\nu} W^{\\mu} - g^{\\rho \\mu} W^{\\nu}\\right),", "f8bf2cb20a16689a5f3f432d8445779e": "\\quad\\lim_{n \\rightarrow \\infty} \\sqrt n\\,W_n=\\sqrt{\\pi /2}\\quad ", "f8bfcefc723786eba562e3263e9f63b0": "\\hat G = \\mathbb{R}^n", "f8bfe04c86d5f28ff5efb95ad4360b8c": "\\mathrm{e}8", "f8bfe2c0477bf91b4a1a295c59ee5e95": "A^j = (a_i^j)_{i = 1, \\dots , n}", "f8c03feae781c54f13da3bb0170ecb46": "\\!y = \\frac{1}{x^2+1}.", "f8c08660a5cf4f580d5b33331d7a72e7": " a \\,\\!", "f8c0b2c9f7ac6be9af72867c89acfe3b": "U = \\frac{d_f}{2}k_B T ", "f8c0f6286daeef29e943dcffd162c669": "P(H|E)>P(H)", "f8c1015f8a35d82c732276b1fbd21a25": "\\frac{ dx }{ dt } = f(x,y),", "f8c1027d365f2ea5c1c843dc9cba4162": "Oxy \\rightarrow \\exists z \\forall v [Pvz \\leftrightarrow (Pvx \\and Pvy)].", "f8c156e595ada759a3649d76d9a7d59e": "[-]", "f8c1d0e3d422bab8a7779f0600e8d7bf": "(10\\cdot x+y)^{\\mathrm{1T}}-100\\cdot x^{\\mathrm{1T}}=\\mathrm{1T0}\\cdot x\\cdot y+y^{\\mathrm{1T}}=\n\\begin{cases}\n\\mathrm{T10}\\cdot x+1, & y=\\mathrm{T} \\\\\n0, & y=0 \\\\\n\\mathrm{1T0}\\cdot x+1, & y=1\n\\end{cases}\n", "f8c29158a7d8f3d4095036796d7f066c": "f^\\mathrm{e}_{\\mathbf{k}} +f^\\mathrm{h}_{\\mathbf{k}}>1", "f8c2c5260499db77a39c0ad695e18563": "z(a_n)_n=((z\\text{ mod }p^n)a_n)_n.", "f8c2cb7d11d3d96a8dd2c1caf6c0497e": "Z_o^2=ZZ'", "f8c2ce4f3f984046ec9145df7805d71e": "t_{ij}c_{ij}=C", "f8c2d15e62981cce8f25c73e34ff22de": "\\left[\\hat{b}_i, \\hat{b}_j^\\dagger \\right]_- = \\delta_{ij} ", "f8c36a88bccec97f395c63dab42934ae": "\\Pr(A\\le\\mathbf x) \\ge \\Pr(B\\le\\mathbf x)\\text{ for all } \\mathbf x \\in \\mathbb R^d ", "f8c375527c7fa6add8222ad7ca158f4c": "V = E(G)", "f8c3eb0a7474b43cf88f2f357da70222": "A=B^{-T}", "f8c45a03ce499cae022e2e8e49e8a901": " f: R^{n_x} \\times R^{n_y} \\to R^{q}", "f8c472fd3046e1b3cede86667c0a4e0b": "G = 2\\gamma + G_p", "f8c4e95adc731b5024ba39864c18a20b": "John\\ met\\ Mary : N \\cdot (N^r \\cdot S \\cdot N^l) \\cdot N", "f8c4f283bf5adb14e988a9d0c1853b3a": "l^n_1", "f8c5313d2a53a6b21df83cdeaf5a8483": "Z_{0}\\ ", "f8c536f0bef1e0b7457b73b08a2078fc": "\\partial V \\!", "f8c55214283f9e5ac33dcda4dc4e6244": " \\mathbf{K}_{\\text{Gauss}} = \\frac{\\sigma^2}{\\pi \\delta_x \\delta_y Q^2} \\begin{pmatrix} \\frac{2}{\\sigma_x \\sigma_y} &0 &0 &\\frac{-1}{A \\sigma_y} &\\frac{-1}{A \\sigma_x} \\\\ 0\n &\\frac{2 \\sigma_x}{A^2 \\sigma_y} &0 &0 &0 \\\\ 0 &0 &\\frac{2 \\sigma_y}{A^2 \\sigma_x} &0 &0 \\\\ \\frac{-1}{A \\sigma_y} &0 &0 &\\frac{2 \\sigma_x}{A^2 \\sigma_y} &0 \\\\\n \\frac{-1}{A \\sigma_x} &0 &0 &0 &\\frac{2 \\sigma_y}{A^2 \\sigma_x} \\end{pmatrix} \\ ,", "f8c59b492fe3cc7990098ab6728eae2b": " K_i = \\frac{k_2}{k_1}", "f8c5a15d31be4f3c5bd16200a6d1647a": "(X_t)_{t>0}", "f8c5d968035d0d271447b73642dc04f5": "v_i, \\ i=1 \\ldots m^n", "f8c5e51d494a8343d2bf6adf672cf3af": "y_n(x)=\\,_2F_0(-n,n+1;;-x/2)= \\left(\\frac 2 x\\right)^{-n} U\\left(-n,-2n,\\frac 2 x\\right)= \\left(\\frac 2 x\\right)^{n+1} U\\left(n+1,2n+2,\\frac 2 x \\right).", "f8c62b7466d2ae4c7c3a0b83e7685d83": "10^n * Z + (2Z + 1)", "f8c64b5d9ceb1c862e818d6bf09ca684": "X\\overset{f}{\\rightarrow}Y\\overset{g}{\\rightarrow}Z", "f8c6791932028c5e819dafe44d1d1181": "wp(P\\ |", "f8c67af96a6fce12db9e4253f3ce2931": "f_{WN}(\\theta;\\mu,\\sigma)=\\frac{1}{2\\pi}\\prod_{n=1}^\\infty (1-q^n)(1+q^{n-1/2}z)(1+q^{n-1/2}/z) .", "f8c698901eb1993a0c38ef8a65ff2e48": " \\dot{c} ", "f8c6c07230f94b55b0cc06596f2f708e": "x_j^{(i-1)}", "f8c6efc1acc753d84192d0eae0f7b610": "M=\\mathbb R", "f8c75c72f4ca40a01fd7443a915a8343": "\n(C|I) = \n \\left[\\begin{array}{cc|cc}\n 1 & 3 & 1 & 0\\\\\n -5 & 0 & 0 & 1\n \\end{array}\\right]\n", "f8c7af5bf839015b548ed407d077160c": "\\chi_-(1)\\chi_+(2)", "f8c8547c4902bca42cfe6eaeea3b119c": "\\beta_1 = \\beta_2 ", "f8c854d52b2e0eacde2193531e3a1fac": "\\!\\mathcal A = (A, \\sigma, I)", "f8c86b5d614a2f006ef78368e3ea9565": "(I-\\Psi\\Psi^T)F(x+\\Phi\\xi+\\eta(\\xi))=0)", "f8c8b903cb2e4f297e4b96d4b9c1e98a": "Employee", "f8c913af17557ff36f7b555b22b1efdb": " x=t-\\frac{b}{3a}", "f8c9df5a7b3eff45390566038b387a98": "\\langle 0|T_{00}(s) |0\\rangle =\n\\sum_n \\frac{\\hbar |\\omega_n|}{2} |\\omega_n|^{-s}", "f8c9f41501148fde5fde8b66f3c79840": " S_2 = {3^7 \\over 2^{11}} = {2187 \\over 2048} \\approx 113.685 \\ \\hbox{cents} ", "f8ca31a7f79e7597f2495096c8438724": "\\eta = X \\beta ", "f8ca8f2ad705db598e84af03e2c778b4": "\n\\begin{array}{|c|c||c|} a & b & S \\\\\n\\hline\n0&0&1\\\\\n0&1&1\\\\\n1&0&1\\\\\n1&1&0\\\\\n\\end{array}\n", "f8cbd94c2e33f61582c36458d1cdcb27": " k > {\\rm max}_i (e_i \\cdot \\deg f_i) ", "f8cc431dda6f2f6410a32bc887c0ad67": "\\theta\\in(0,\\infty)", "f8cc535624a4f473c78d551f6d7a9a76": "\\Omega_{+}\\,", "f8cc53b45757ba27843a2aea444d814a": "\\delta=", "f8cce4bb527d174ef1d176ab1837f127": "\\mathbf d = 0", "f8cd637328744bdfa4d2c9a4b9c41953": "\\frac{dy}{dx} = \\frac{Cx+Dy}{Ax+By}", "f8cdab99a3a1ccf17c1f8e8dc9689b46": "\\sqrt{y} = (\\sqrt{z})^{-1}", "f8cdbefa56849c5e5f59c2c0c71d171b": " u_t + uu_x + u_{xxx} = 0 ", "f8ce31d26c504a6483859b29e533d86d": "\\sin\\theta_c = \\frac{n_2}{n_1}\\,\\!", "f8ce3909773f1f1169416c2bf7fd6a74": "\\begin{bmatrix} {1/2} & -{\\sqrt{3}/2} \\\\ {\\sqrt{3}/2} & {1/2} \\end{bmatrix}", "f8ce3e50597bfc8767b3bb7a15a701da": "\\delta\\omega=0", "f8ce5518fb8636c151abff45c1607ad3": "\\cos\\frac{\\pi}{12}=\\cos 15^\\circ=\\tfrac{1}{4}\\sqrt2(\\sqrt3+1)\\,", "f8ce7bafbe92cd1e5b8f15816f0f47c2": "t_{0.975,n-1}\\simeq 2.", "f8cec07a5e35deeb0edb3c9d0508b957": "{\\mathfrak g} = {\\mathfrak s}{\\mathfrak l}(n+1,{\\mathbb R})", "f8cf76f025d3f8e51acafbb159ebdb03": " \\hat c ", "f8cfc8745f19cd024ac21009df4ba437": "\\widehat{HT}=T\\widehat{Q}^{\\prime}[Var(\\widehat{\\beta}_{FE})-Var(\\widehat\n{\\beta}_{RE})]^{-1}\\widehat{Q}\\sim\\chi_{K}^{2}", "f8cfe64934b6fec3ed53570c36706bda": "\\displaystyle{e_n(z)={z^n\\over \\sqrt{n!}}}", "f8d01e4c30d550c83245b5a7f765fb1f": "\\left( \\begin{smallmatrix} 0 & -1 \\\\ 1 & 0 \\\\ \\end{smallmatrix} \\right)", "f8d0666178985624ee572188da624c32": " P = I ", "f8d0c25dbcc7b6c07aa1ff652e7a9392": "S_{0}(V)=4.0\\times 10^{-12}", "f8d107000b9cb8889fd40dfd686ef55d": " R = \\mid \\vec{R} \\mid ", "f8d12a60686c8768687614c3640d23cf": "\\scriptstyle \\mathbf{\\sigma} \\;=\\; \\mathsf{C}:\\mathbf{\\varepsilon}", "f8d1a19884eba49cbf585d450faae174": "(1+1/n)r", "f8d1e9b7777179cd3a804dcd1cd9f2d5": "11_4", "f8d1ee1181003dc950cd6c0d23f2a7ad": "2\\pi k \\, \\text{Tr}(I)=2\\pi k j =4\\pi r^2\\frac{j}{\\sqrt{j^2-1}}", "f8d23048cbb8f80b17d0493cc65628d1": "F_\\text{IT} = \\dfrac{e}{12\\pi\\varepsilon_0a_0^2n^5}.", "f8d23906b75fbf8bc6a126efdee341e6": "\\ \\varepsilon_s", "f8d24744ab51fbdd3ebce62dbf63ed65": "y = \\frac{\\ln \\left( 1 + \\rho \\frac{R}{C}\\right)}{\\rho}.", "f8d24e419123911aa8e19b86578bfbb1": "\\Omega_\\nu^\\mu=d \\omega^\\mu_\\nu+\\omega^\\eta_\\xi\\;", "f8d2ca78af8b71c4079cb2adacf3c842": "G(a_n; x) \\frac{1}{1-x}", "f8d313cbcfc73239647a46116c89884f": "I_1=\\int(\\nabla\\theta)^2 d^3 x", "f8d391bb36d987f0b0b95099daf2d6cf": "c = \\sqrt { a^2+b^2 } = \\sqrt2\\,.", "f8d39d5091048efb12ef7ef9c30fcadd": "c > 0\\!", "f8d3a7b30e147c569a1e4468e644ba93": "\\| y_2 - y_1 \\|^2 = 2\\|y_1 -x\\|^2 + 2\\|y_2 -x\\|^2 - 4\\| \\frac{y_1 + y_2}2 -x \\|^2", "f8d3d1ffd5249aedf4141023914a0360": "=C \\frac{\\left(e^\\frac{ikax}{2z} - e^\\frac{-ikax}{2z}\\right)}{\\frac{ikx}{z}}", "f8d3df83cd55b52b2644aa3769d7b855": " k_{\\mu} \\mathcal{M}^{\\mu}(k) = 0 ", "f8d435224870ee51a99fb63a10773f54": "f(z) = \\frac{1+2i}{5} \\left(\\sum_{k=1}^\\infty \\frac{1}{z^k} + \\sum_{k=0}^\\infty \\frac{1}{(2i)^{k+1}}z^k\\right).", "f8d43b758aa1aba9246918adb9686330": "\\Phi_{D} = ", "f8d44ed2be11687ea6d03614232859bc": "\\left.\\frac{\\partial}{\\partial \\vec{p}}(f+\\lambda (g-1))\\right|_{\\vec{p}=\\vec{p}^{\\,*}}=0,", "f8d460dc02d51119829d2137c3736d6b": "f\\approx g", "f8d4861a952ba10780ce94d7b0733d65": "p = p_{{\\mathrm{N}}_2} + p_{{\\mathrm{H}}_2} + p_{{\\mathrm{NH}}_3}", "f8d48c90ad3124768063485c746a9c64": "(\\gamma_{1},\\gamma_{2},\\gamma_{3})\\,", "f8d5a1441955e0973c0f4c7b6706aebf": "a_1\\in M ", "f8d5af2b45fd03e2df95d80b7b34289c": " x < 0", "f8d68ca369a8f302590f02c18ffdb897": "x \\leftarrow x + 0", "f8d6cf0f73f61be6fe481530438dfbbe": "x + b", "f8d6dc0d9c64ccd403b8fc9ba8167ba4": "\\oint_{\\partial \\Sigma} \\mathbf{E} \\cdot \\mathrm{d}\\boldsymbol{\\ell} = - \\frac{d}{dt} \\iint_{\\Sigma} \\mathbf B \\cdot \\mathrm{d}\\mathbf{S} ", "f8d6fb329dcc780f4c5d1dff53c5e100": "\\frac{\\partial u_x}{\\partial x} + \\frac{\\partial u_y}{\\partial y} = 0", "f8d716a40fd281046bb29eeedbe7e072": "\\mathbf{A}=\\mathbf{U}\\Sigma\\mathbf{V}^T", "f8d7868196501946c4f7527e56e3d0ac": "\\sum_{n=1}^{m} r^n = \\frac{r}{r-1} (r^m - 1).", "f8d7c4d623e4f7b4d998e81a19b9bf78": "\\dot{V}_x", "f8d85e8b673d24ee2441aafafae614c5": " \\mu = ", "f8d8e4f561337c1a06acaa1831bd8e43": "\\frac{1}{2}_{10} = 0.5_{10} = 0.1_2 \\,", "f8d90d00b7ebee3cde9db62430104364": "\\mathcal{Z}(N)=N\\cap \\mathcal{Z}(M)\\,", "f8d933a4e02d6504ebdfa4a6c925aebf": "M_{UT}=\\frac{1}{3}\\Sigma^3_{i=1}({m^+}_i-m_{UT})^2", "f8d968a23c5b2136af7f66fe0454cc5b": " Dint ", "f8d9d82e4657731ec020e4648a886ff2": "B\\subseteq E", "f8da173bc9211f78d1c676862b6413c6": "h(G)\\ ", "f8da339b9b186f8bbdeef8d8b43d31bc": "e=\\sum_{n=0}^\\infty \\frac{1}{n!}", "f8da6a2c219ae8274e58dbfb10b3c73d": "y=\\frac{1}{2}x + 1", "f8da738002073e154f5d5afe46f5933c": "\\scriptstyle (x,\\, qx \\,-\\, y)", "f8daa0493ed7d7d6964d2018c522cfa4": "p \\equiv q \\and \\mathrm{not}~r", "f8daa4e4b99e1f21193d5aa6bc06e1b2": "|\\gamma_i|", "f8db488d064b259da8157f5b5aff0435": "\\sigma_{0es}", "f8db94db7b8748fd199263aae29a5ff2": "1\\rightrightarrows2\\rightrightarrows4\\rightrightarrows8\\rightrightarrows\\dots", "f8db98a17ed97387e94c3f92f93ec3d7": "\\ (-\\infty)+2 = -\\infty", "f8dc052718a298700b6bb267851da086": "R_{ik\\ell m}=R_{\\ell mik}\\ ", "f8dc26a5b4bb4b627a209db07a3ca863": " d_2 = \\frac{\\ln(F/K) - (\\sigma^2/2)T}{\\sigma\\sqrt{T}} = d_1 - \\sigma\\sqrt{T}, ", "f8dc4f275bb3cd878221ccb3dadbad58": "(K,\\, .)", "f8dcd83f63648bcdb358494a3892636c": "u=g\\left( x,y \\right),\\ \\ \\left( x,y \\right)\\in \\partial \\Omega_D,", "f8dcd9e6a11a247ff7fb32ac4829b604": "\\frac{\\Delta f}{f_f}=\\frac 1{\\pi Z_q}\\,\\frac 2{\\omega u_0}\\left\\langle\n\\sigma \\left( t\\right) \\cos \\left( \\omega t\\right) \\right\\rangle _t", "f8dcdb3638e9d874c9b3c8691181a459": "y = \\hat y L \\qquad \\text{and} \\qquad u = \\hat u \\frac{L^2}{\\mu} \\frac{d p}{d x}.", "f8dce33df25d9f853f71371d07941535": "I(.)", "f8dceeb649e72e654d1559b352028166": "\\sum_{j=1}^{n}{x_{i,m+j}}=b_{m+j}", "f8dcf628d8ce186cc198501b4323a0c5": "f(z)=az+b", "f8dd0f1bcd34c35e9181b5564988b315": "(p_{n+2}-2p_{n+1}+p_n)p\\approx p_{n+2}p_n-p_{n+1}^2", "f8dd5fe0fffebb4bd4f7bdbd98e09a0a": "f(A, B, C, D) = \\sum_{}m(6, 8, 9, 10, 11, 12, 13, 14)", "f8dda031f4f7b42bcb482c34ad599131": "MU_*\\to K_*", "f8dddef3408059f80b9dfa5a746f6517": "Q_\\text{solexa-prior to v.1.3} = -10 \\, \\log_{10} \\frac{p}{1-p}", "f8ddebae3f9d757a73e307c462b150e1": "T_j:\\;E\\to F", "f8ddfe61ef80cf888e21b18a7aa1e399": "\\scriptstyle{\\aleph_0}", "f8de21c3eaeb7cb3747a68d491e2375b": "2(N-1) - (R-1)", "f8de348c69d518ba5d31e2858b8ffb9b": " K_p = \\frac{ \\cos ^2 \\left( \\phi + \\theta \\right)}{\\cos ^2 \\theta \\cos \\left( \\delta - \\theta \\right) \\left( 1 - \\sqrt{ \\frac{ \\sin \\left( \\delta + \\phi \\right) \\sin \\left( \\phi + \\beta \\right)}{\\cos \\left( \\delta - \\theta \\right) \\cos \\left( \\beta - \\theta \\right)}} \\ \\right) ^2}", "f8de99460f76b500db0b2ab30e7600be": " M_{C^*} ", "f8dea0c72d04e507188653d7f57a81fe": "(n,k,d)_q", "f8deb21a3e5f7f79e3d9f9dc631edd2b": "{\\frac{d\\sigma}{d\\Omega}(\\Omega) d \\Omega}={\\hbox{number of particles scattered into solid angle d} \\Omega \\hbox{ per unit time} \\over \\hbox{incident intensity}}", "f8dfcb3858d27afe074e6bb668193bea": "n_1=(x+n)", "f8dfcf8a2720264cb934d77a9947620e": "\\oint_C f(z)\\,dz = \\int_{-a}^a f(z)\\,dz + \\int_\\text{Arc} f(z)\\,dz ", "f8e028c475a8f8083a42389b826501be": "\\phi_N - \\phi_S \\,\\!", "f8e067fed0dde6c5fd99c47d84b84981": "f_L ", "f8e0713b9581ed0c9984264d841960db": " \\tau_w ", "f8e0b6bba0fa53023d2b59d6ef18c977": "\\sum_{p}^{}{\\Delta S^0_i} \\ge \\frac{\\Delta G^0}{((1+f)T_H-T^0)} ", "f8e0f6fa224ad8cc8dfb5c8595b4801d": "\\tfrac{1 + \\sqrt{5}}{2} \\approx", "f8e192fd56b62839981ed728956b9318": "2\\tfrac{2}{11}", "f8e19f449f17c9d37dcc93dd244ec3bb": "ln", "f8e1ddc0863bb033af352f55c57af4ea": "d\\geq 5", "f8e1f7761117a8a6cc0d6f45f05d94a2": " x_n=(b-1)-(ax_{n-r}+c_{n-1})\\,\\bmod\\,b,\\ c_n=\\left\\lfloor\\frac{ax_{n-r}+c_{n-1}}{b}\\right\\rfloor.", "f8e2ac11330bc68c5c578da5ab5370c2": "f(b) > 0", "f8e2ceeb02e0baa2502458b4e51989a1": "\\scriptstyle{\\hat{H}(0)}", "f8e2d3ee0d88e4a187d511a862a4be1e": " \\mu\\nabla^2 u_i +\\rho g_i -\\partial_i P=0", "f8e3328de3b905d140f744c5485378fa": "\\psi(\\Omega+1) = {\\varepsilon_0}^\\omega", "f8e352d30e97be98407ad2409b1ad101": "M\\ddot{\\mathbf{R}} = 0, \\, ", "f8e3d3c77f9507c35379323dc837bd5f": "W: S \\rightarrow \\mathbb{R}", "f8e3f0764c06ab3c3df551ff23a3c45f": "c(v)", "f8e4b28594cc2da2188a35729c19054c": "T_u", "f8e4efddb72bcbcaf518ba2e39bad4d5": "F_{(1,2)}=M_{(1,2)}+M_{(1,1,1)}", "f8e4fd1a1a890ffd88bb2da3ab0e36b2": "M \\times D^k", "f8e5b95592a4f7c5a95bc632138a073f": "\\scriptstyle M^N", "f8e5d2c0e1c710a76ebb35cafe7f3132": "\\gamma_kU_k", "f8e602c64ba5287554ebb0bd8281035b": "\\sigma_1/2^{2/3}", "f8e65b6f90e4c96c5adee2062e49ed1d": "\\frac{d^2x^a}{ds^2} + \\Gamma^{a}_{bc}\\frac{dx^b}{ds}\\frac{dx^c}{ds} = 0", "f8e6a56c2a25ae3348463fb4afc8866f": "\\lbrace \\gamma'(t), \\gamma''(t), ...,\\gamma^{(m)}(t) \\rbrace \\mbox {, } m \\leq k", "f8e6cc4ac1e43b46814516ffec4e8538": "\\mu\\!\\left(\\varnothing\\right) = 0", "f8e6f3d3783be19ecbdd8433f3eb6ce1": "\\mathcal{F}(\\pi)", "f8e6f4dadc30a8cad697cd18fc820c8b": "\\text{extract} \\circ (\\text{extend} \\, f) = f", "f8e6f619d698e7ee4c359e813dc6e2c7": "G = \\frac{1}{2} \\frac{\\Delta V_\\mathrm{P}}{V_\\mathrm{P}} - 2 \\frac{V^2_\\mathrm{S}}{V^2_\\mathrm{P}} \\left ( \\frac{\\Delta \\rho}{\\rho} + 2 \\frac{\\Delta V_\\mathrm{S}}{V_\\mathrm{S}} \\right ) ", "f8e6f7250ba8de8355deb0d257d40351": " \\mathcal L:=(\\mathcal P,\\mathcal Z, \\in) ", "f8e78e1e41ea847d4031752288ef81f5": "f = ", "f8e795e6403a8a0ee22c42336f250891": "\\mathbf{x} = \\left (x_1, x_2, \\cdots, x_k \\right).", "f8e7d52795be11ca7a78c8db6844e0a7": "q\\theta", "f8e7d85fed42df1ff504e7beabf0c309": " \n\\quad (\\mathbf A^T \\mathbf A)^{-1},\n\\quad (\\mathbf B^T \\mathbf B)^{-1},\n\\quad (\\mathbf A^T \\mathbf P_B^{\\perp} \\mathbf A)^{-1}, \n\\quad (\\mathbf B^T \\mathbf P_A^{\\perp} \\mathbf B)^{-1}\n.\n", "f8e8145dd03372acb4761bd525e46ca6": "\\phi_n(p,t)=\\frac{1}{\\sqrt{2\\pi\\hbar}}\\int_{-\\infty}^\\infty \\psi_n(x,t)e^{-ikx}\\,dx =\n\\sqrt{\\frac{\\pi L}{\\hbar}}\\,\\,\\frac{n\\left(1-(-1)^ne^{-ikL}\\right) e^{-i \\omega_n t}}{\\pi ^2 n^2-k^2 L^2}", "f8e84ef9ef88a0a745c102b8731b8c7c": "\\ \\kappa", "f8e8cae7dde8a876ce8357a07a5362d7": "\\mathbf{\\Pi}^0_1", "f8e908abd4150bfe794eeecbaf0f4b4f": "\\scriptstyle \\pi r^2 /A_{FG}", "f8e938c3062354ccbc0aa43cdd4a55f5": "\\scriptstyle \\mathrm{Ker} F_i", "f8e93a2d7f13796780cc58c122575955": "\ng_{j}(\\mathbf{q}) = 0 \n", "f8e94ac3ef69448af2b2dc04a7ae7f33": "\\Omega\\subset\\R^2", "f8e963006b7a337c358a844a46acc5a9": "x^i \\in N(x)", "f8ea1a514fd15027f9aae63244c4780b": "\\sum a_k", "f8ea3e61f516c878cd211029568d73c3": "\\lambda_8", "f8ea466bb4d8ac361fcde516b65ea373": "\\left(\\frac{945V}{32\\pi^4}\\right)^{1/9}", "f8ea57869afa53f5d023fae60d0ca5da": "\\{ (x_1,x_2): |x_1 x_2| < 1\\}", "f8eaeca98e1a93adb1907709646d7928": "\n f := \\cfrac{1}{1+R}(|\\sigma_1|^m + |\\sigma_2|^m) + \\cfrac{R}{1+R}|\\sigma_1-\\sigma_2|^m - \\sigma_y^m \\le 0\n ", "f8eb3f7454a8a8a64d4ba482e9498858": "y_{n+1} = y_n Y_{n+1}", "f8eb41760dd1942160d0f9f98c5d9bed": "X_l^{(1)}, X_l^{(2)},", "f8eb591e45e3c1bd003a528e30bdeadc": " Z = \\frac {1}{i \\omega C} ", "f8eb6bc472009997b9527d98e219f4e2": " d_K = \\frac{1}{p} \\min_{c_1 ... c_K}{E[(X - c_X)^T\\Gamma^{-1}(X - c_X)]} ", "f8eb9a9d38bacfa5a60987efc741c1a4": "R_{\\mu}", "f8ebc3d37322c3f94ccec49f6269ccc4": "\\left(i \\hbar \\frac{\\partial}{\\partial t} -q\\phi \\right)\\psi = \\gamma^0 \\left[ c\\boldsymbol{\\gamma}\\cdot{(\\widehat{\\mathbf{p}} - q\\mathbf{A})} - mc^2 \\right] \\psi \\quad \\rightleftharpoons \\quad \\left[\\gamma^\\mu (\\widehat{P}_\\mu - q A_\\mu) - mc^2 \\right]\\psi = 0", "f8ebe26ff0f1efeea645e036212f9442": "\\frac{ds}{dy}", "f8ecb6dcf131c989f8b55e650757f64d": "ds^2 \\, = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2", "f8ecf49d69275cb9422401878bb76778": "P\\left(X_{1}\\wedge X_{2}\\wedge\\cdots\\wedge\nX_{N}|\\delta\\wedge\\pi\\right)", "f8ed1eac2258cd81d79fd9120c4417bb": " \\scriptstyle \\omega \\,=\\, 2 \\pi f ", "f8ed8230375a851a48dd00a6a6502003": "A(T,x)=\\mathrm{max}_{S}(U(S,x)-TS)\\,", "f8ed948f6f0b404ff46565843be332ce": "E_2(x+y)=E_2(x)E_2(y)\\,", "f8eda0c83d44f3741a82d1014cf6799b": "\\{ v_i \\}", "f8edbfa186bf19f365793b0ccade5821": "\\mathbf y'(x) = A(x)\\mathbf y(x)+\\mathbf b(x)", "f8ede27ee2a51a39c14e7762247234bf": "V(f(\\vec{x},y)) = (- y f(\\vec{x}))_+", "f8ee7734898caceda000783256c3b9d4": "\\det(\\partial\\Phi_{{\\mathrm{sE}},h}/\\partial (q_0,p_0)) = 1", "f8ef2315fb1ac069b0c563c6ff54f47b": " i=1, \\dots, n,", "f8ef2b8344318b8f598edeeb88aaf53b": "\\Delta:", "f8ef5bbac13edd1c0251d9200be10bf6": "1,2,3,4", "f8ef5c5d2a96681c6ab83121f9668fc8": "\\mathrm{d}\\theta^i=-\\theta_j^i\\wedge\\theta^j,\\quad \\mathrm{d}\\theta_j^i = -\\theta_k^i\\wedge\\theta_j^k.", "f8ef66438b32cf506fae7d8dc9d3fa9e": "(x_1,\\ y_1)", "f8ef79a233599c12b8e427a027b3ba67": "q_\\text{P} = \\sqrt{4 \\pi\\epsilon_0 \\hbar c} = \\sqrt{2 c h \\epsilon_0} = \\frac{e}{\\sqrt{\\alpha}} = 1.875\\;5459 \\times 10^{-18}", "f8efc1547e22ee8296eca263e14d9526": " G \\equiv \\int_{-\\infty}^{\\infty} e^{-{1 \\over 2} x^2}\\,dx", "f8efe4edaa959d69f3a27fa084092e1c": "R_j>r_j>0", "f8f001b597006e6ef7f890ccee7211cf": "a = 9", "f8f0184c01a71ba0f4722217fe4f27b7": "p(a,y_1,\\ldots,y_n)=0\\,", "f8f05efa65a971617de7d90c482148dd": "\n \\displaystyle \n m_i \\ge m_{i-1}\n", "f8f09294edba754502e0259c6f3b1b14": " \\lambda f.\\operatorname{de-let}[\\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x)] ", "f8f0f87b59cf0a003697669fb44a80a6": "J(S_w) = \\frac{p_c(S_w) \\sqrt{k/\\phi}}{\\gamma \\cos \\theta}", "f8f1ba8bb12a545febd4bdc70add718d": "\\int x\\arctan(a\\,x)\\,dx=\n \\frac{x^2\\arctan(a\\,x)}{2}+\n \\frac{\\arctan(a\\,x)}{2\\,a^2}-\\frac{x}{2\\,a}+C", "f8f1f8a273dbfd578e5be0f27b1d9b71": "V (t) = L \\frac{di}{dt} = -\\omega_0 L I_0 \\sin(\\omega_0 t + \\phi ) \\,", "f8f2383c0491233ed3305dd6a13e7d20": " \\frac{\\partial \\rho}{\\partial t} = - \\nabla \\cdot \\mathbf{J} ", "f8f2868ce396827dbe625c4f73496e8b": "G(\\varepsilon_Z)\\circ\\eta_{GZ}(\\phi) = \\phi.", "f8f2aca5bea3265b0d322df2f64f69dc": "h: A \\to B", "f8f2d1320f881e4bd25a8d564b7fff1d": "-\\mathbf{e}_3", "f8f3217ecf6f54d30f29652eb20e68d3": " \\mathcal{P}_z = {N \\hbar \\omega \\over cV} = {N \\hbar k_z \\over V} ", "f8f32b0082161cf14e8c8e4ed548085e": "\\mathfrak{m}_A", "f8f36b4cb7b37c5f1b098f657afb2609": " \\pi \\approx 3.14159265358979324. \\, ", "f8f382a9807e571a97466b7c2ad0fa4c": "\\frac {\\mu_0}{\\pi}\\left(b\\ln\\left(\\frac {2 b}{a}\\right) + d\\ln\\left(\\frac {2d}{a}\\right) - \\left(b+d\\right)\\left(2-\\frac{Y}{2}\\right)+2\\sqrt{b^2+d^2}\\right)", "f8f3b0385bbd251a15fbd4508652e590": "\\boldsymbol{\\tau} = \\boldsymbol{\\sigma}", "f8f3ed9503a0fc9314f684a45d196b5a": "E(\\frac{R}{S}(n))\\to C_H \\times n^H", "f8f3edf4799bd7bcd8986f0ea9d530f6": "[K_i,K_j]=i\\hbar {\\epsilon_{ij}}^{\\,k}\n\\left(K_k+\\frac{q\\hbar}{c} x_k \n\\left(x \\cdot B\\right)\\right)", "f8f418ea597489eece9721e0083972d8": "(A\\leftrightarrow B)\\to(B\\to A)", "f8f421d2b398c2310c29bde400c4c0bc": "{{P}_{X}}\\left( u,\\xi \\right)=\\int\\limits_{-\\infty }^{\\infty }{C\\left( u,\\tau \\right)}{{e}^{-i\\xi \\tau }}d\\tau =\\int\\limits_{-\\infty }^{\\infty }{E\\left\\{ X\\left( u+\\frac{\\tau }{2} \\right)X\\left( u-\\frac{\\tau }{2} \\right) \\right\\}}{{e}^{-i\\xi \\tau }}d\\tau ", "f8f483c249f8884171be82fc70eba73e": "z=\\boldsymbol{\\eta}(x)", "f8f49384be8eccf62be5b6d453754312": "H(\\epsilon) = -\\epsilon\\log(\\epsilon) -(1-\\epsilon)\\log(1-\\epsilon)", "f8f4b67720726238881c0917233f0a21": "\\psi(z,q)=e^{- \\gamma z}\\frac{\\partial}{\\partial z}\\left(e^{\\gamma z}\\frac{\\zeta(z+1,q)}{\\Gamma(-z)}\\right),", "f8f51b5816444b6578e8d8e247dc4568": "\n \\cfrac{\\Gamma, A \\vdash C \\qquad \\Sigma, B \\vdash C }{\\Gamma, \\Sigma, A \\or B \\vdash C} \\quad ({\\or}L)\n", "f8f5b3d5e2c5f9bb4c916d7a518a62bb": "\\sum_{k_1+k_2+\\cdots+k_m=n} {n \\choose k_1, k_2, \\ldots, k_m} x_1^{k_1} x_2^{k_2} \\cdots x_m^{k_m}\n= (x_1 + x_2 + \\cdots + x_m)^n\\,,", "f8f5c265945f73a33cae3094a3e6b851": "fH_k M \\otimes fH_k M \\to \\Bbb Z", "f8f625b82c7b634d12df06299d3f157d": " {\\mathbf A} ", "f8f63412fb146e4d0e3fb8537dc42869": "z_1^5+z_2^5+z_3^5+z_4^5+z_5^5-5\\psi z_1z_2z_3z_4z_5 = 0", "f8f75be534248362fca7b8b988157b5d": "P_s < (M-1)Q\\left(\\sqrt{\\frac{d_{min}^{2}}{2N_0}}\\right)", "f8f79bba60ae9e1a0b04bf3df174d4e4": "A \\cdot B \\cdot \\neg C", "f8f7f43cddbdf66b69a83a8667ed6c21": "\\pi_2 = L^f \\mu^g k^h \\beta^i g^j \\rho", "f8f8096840809e7cab836bcafb3bf3ed": "\\Delta A = \\frac{A_- - A_+}{A_- + A_+}", "f8f814b754eab28826b1cf56abc1fa68": "\\vdash \\Box (A \\rightarrow B) \\rightarrow (\\Box A \\rightarrow \\Box B)", "f8f8573276c2ee0c31437d7a2669e955": "\\mathbf{P}(\\operatorname{Hom}(G,\\mathbf{Z}/p)) := (\\operatorname{Hom}(G,\\mathbf{Z}/p))\\setminus\\{0\\})/(\\mathbf{Z}/p)^\\times", "f8f887f132e8c6fbb15f1a78dd0995a7": "{\\rm Gi} (z)=\\frac{1}{\\pi}\\int^\\infty_0{\\rm sin}\\left(uz+\\frac 13 u^3\\right)du", "f8f8a001bbbccbe97e75ff9fb0e9468d": "c_0=2^{m_\\alpha/2 + m_{2\\alpha}}\\Gamma\\left({1\\over 2} (m_\\alpha+m_{2\\alpha} +1)\\right)", "f8f8b415d9fd222eeca2a96b1f791573": "\\Xi \\subset \\mathbb{R}^d", "f8f8d4be5845070576c6803ca0e25691": "r=(x+y+gz)r_{RR}", "f8f8d943be2141b46586c4c3ba9b48d9": "D = \\sum_{i = 1}^{k}{[P_i]} - k[O]", "f8f8e35dec15f586a17b0251704c1853": "X \\neq \\hat{X}", "f8f914cb43b48b9d9060d5fb82f8f422": "\n\\pi= {3 + \\cfrac{1^2}{6 + \\cfrac{3^2}{6 + \\cfrac{5^2}{6 + \\cfrac{7^2}{6 + \\ddots\\,}}}}}\n", "f8f91f80714dc9ec7ad975dc0ff91743": "\\alpha_{2}(a)", "f8f92754176ba3d8b872f8bf33521d4b": "n=1,2", "f8f95b0ae39b346ed74263505b9c33e5": "I(m) = D_{\\mathrm{KL}}(\\delta_{im} \\| \\{ p_i \\}), ", "f8f9beeccdcace1652be8b67fd62fb41": " S_n=(a_n-(n-1)d)+(a_n-(n-2)d)+\\cdots+(a_n-2d)+(a_n-d)+a_n.", "f8fa47c296afb876541e9a901f58a4b5": "{\\hat{\\imath}}", "f8fa9b74a036db406bfdd685e5e1546f": "{\\mathbf N} = \\{\\oplus_1 ^n M \\; | \\; M \\in {\\mathbf M} \\} \\sub L({\\tilde H})", "f8fb6b4b524570689b4706c59db0a0de": "f(t) = \\sum f_i \\Phi_i(t), g(t) = \\sum g_i \\Phi_i(t)", "f8fbf17358d800f53a83e4e750bfefaa": "\n\\dot{V}=r\\dot{r}\n", "f8fc29607c08473645c3bb979b21c266": "E = E^0 + s \\log \\{H^+\\}", "f8fc4123b79e07d00bc37bdbbef71805": "R / \\omega L", "f8fc4da5ed4e58ba0577d3ea555801a6": "~\\frac{ 1 }{ i }~[M_{\\mu\\nu}, M_{\\rho\\sigma}] = \\eta_{\\mu\\rho} M_{\\nu\\sigma} - \\eta_{\\mu\\sigma} M_{\\nu\\rho} - \\eta_{\\nu\\rho} M_{\\mu\\sigma} + \\eta_{\\nu\\sigma} M_{\\mu\\rho}\\, ,", "f8fc5c3d8e543dc486ff6172e0042e2c": "\\mathbf{c}_{\\mathbf{0}}^{\\mathtt{KED}}", "f8fcbf24689783fcc1222743fe7465ac": "\\mathbf{w}(t)", "f8fce78db67c279bd64447cf6fcac4fa": "e^{\\frac{ln[A]_{f}}{ln[A]_{i}}}=e^{-kt}", "f8fcede0419ca22bce4116ee22df000d": "K \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac{k_{f}}{k_{b}} = \\frac{\\left[B\\right]_e} {\\left[A\\right]_e}", "f8fd1d32b96080825ded94840e3ab1c2": "\\Theta(n + z)", "f8fd1e1f4e185057a7a9b5d1f6017894": "~=~", "f8fd7f902d0c727acb8c4f296fc667e8": "\\Delta \\theta", "f8fda9a8883df53e47cedb4e46b0e916": "\\|{f}*g\\|_p\\le \\|f\\|_1\\|g\\|_p. \\,", "f8fe27c86099f51b1d154a14033cb4de": "\\Diamond A\\to\\Box\\Diamond A", "f8fe788eb431fee85ab419668441c301": " v = v'(t,x,y)\\!", "f8fea32ea98b42092a1ca55042be590d": "\\frac{1}{2}\\rho V^2 S C_L = W", "f8ff5a9650ec891197d8fcef0e8d6fc9": " i' \\, ", "f8ff64d547cd9de9fea84053c49a9042": "\\lambda=\\mu_1^2+\\ldots+\\mu_k^2", "f8ffa5a09cabd6c752437475e98dccd2": "\\sqrt{a^{2} + b^{2}}", "f8ffcab2665278e9bfa8dd18a2f5dd4e": "\\mathrm{FIS}(X)", "f90010136c1fc085ce91edb92966e57e": "E_d", "f900d3e3a618ab737e4991823e8c94f3": "\\hat{\\rho_0}=e^{-\\beta \\hat{H}_0}=\\sum_n |n \\rangle\\langle n |e^{-\\beta E_n}", "f900fb8977fcc5f2692cee6956be24c5": "(p,x) \\sim f(p,x)", "f90140f964d5661d3985caf2fa091ae7": "\\rho(x,y) = \n\\left\\{\\begin{matrix} \n1 &\\mbox{if}\\ x\\neq y , \\\\\n0 &\\mbox{if}\\ x = y\n\\end{matrix}\\right.\n", "f901452f9c609c925cf64e8d46bde9a6": " s \\in {\\{0,1\\}^*} ", "f901a56365d906ae8b9eaca82737473f": "\\partial E = \\{ (x, 0) : -1 \\leq x \\leq 1 \\} \\; \\cup \\; \\{ (x, 1) : 0 \\leq x \\leq 1 \\} \\; \\cup \\; \\{ (x, y) : x \\in \\{0, 1\\}, \\; 0 \\leq y \\leq 1 \\} ", "f901ccda7199906d41fff3b0a579f72f": "(ax^n + 1)/(ax + 1) = y^2", "f9024a4e6a64219dd145328927c91a73": "S_0 = Nk\\ln(3/2),", "f9025fdd5221431a8fbe6a74dad4d831": "r = 1/2", "f902679b2b4420774fecdda9c925e26a": "\n X_n\\ \\xrightarrow{as}\\ X \\quad\\Rightarrow\\quad X_n\\ \\xrightarrow{p}\\ X\n ", "f90272d999fda78ec915ef35a4eb3228": "\nr_{E,U} = \\frac{EBIT(1-T)-\\Delta IC}{E_{U}}\n\\qquad (3)", "f9027abaf33f19e0ee3a2763c94c7b69": "\n \\lim_{n\\to\\infty}\\operatorname{Pr}\\big(\\big|g(X_n)-g(X)\\big|>\\varepsilon\\big) = 0,\n ", "f9027eaeb8164e3330e352212a104837": "E[X_i=H]=p", "f9029008ac5c151a4a85b9412156f43b": "\\lambda = g_{\\mu\\nu}\\xi^{\\mu}\\xi^{\\nu}", "f90297657147236225478bde8f906470": "S[\\phi,\\chi]=\\int d^dx \\left[\\frac{1}{2}\\partial^\\mu\\phi \\partial_\\mu \\phi -V(\\phi)+\\chi^\\dagger i\\bar{\\sigma}\\cdot\\partial\\chi+\\frac{i}{2}(m+g \\phi)\\chi^T \\sigma^2 \\chi-\\frac{i}{2}(m+g \\phi)^* \\chi^\\dagger \\sigma^2 \\chi^*\\right]", "f902a1c718b925fc58263e54c69b83e7": " \\star^{-1}:\\Lambda^k\\ni\\eta \\mapsto (-1)^{k(n-k)}s{\\star \\eta} \\in\\Lambda^{n-k}", "f90337df9187fc935694bad561dac6ac": "B = b_oM^{3/4}e^{-\\frac{E}{k\\,t}}", "f9034645a11dbaa5f0c481b5820e9b57": "S = \\{ j^{1}_{p}\\sigma \\in J^{1}\\pi : (u^{1}_{1}u^{1}_{2} - 2x^{2}u^{1})(j^{1}_{p}\\sigma)=0 \\} \\,", "f9037e088ebb45eb48df91ee26363b3e": "\\omega = 2 \\pi f", "f903bf755038f8cd8f6e700e3e9922f6": " GB1(y;a,b,p,q) = \\frac{|a|y^{ap-1}(1-(y/b)^{a})^{q-1}}{b^{ap}B(p,q)} ", "f903d33b9ff801b886bf719b3c694d4d": "h+1", "f903e1433365c10aaf624af5cf4fe419": "|A\\rangle = \\sum_S A(S) |S\\rangle", "f903fe6a9daa491b16b8a41dc3bad4ce": "G_{ab}=R_{ab}=8\\pi T_{ab}", "f90431ff35d75c5ca65168548fe84824": "t,\\omega", "f9047086d52da7b74741069b73a78f31": "T\\, ", "f90483c0bb190eb4197f2c9d68f0390a": "\\frac{8}{3}\\,\\sqrt{\\frac{2}{5\\pi}}", "f904ef4ab4e308792cbe3acb13e931de": "colgroups", "f9054977e4434585bb9ded549343171d": "\\mathrm{ker}(A) = (\\mathrm{im}(A^\\mathrm{T}))^\\perp", "f9056fc949257780f63d78da5fde9400": "\\begin{align}\nY &= +0.299 R +0.587 G +0.114 B\\\\\nD_B &= -0.450 R -0.883 G +1.333 B\\\\\nD_R &= -1.333 R +1.116 G +0.217B\\\\\n\\begin{bmatrix} Y \\\\ D_B \\\\ D_R \\end{bmatrix} &=\n\\begin{bmatrix} 0.299 & 0.587 & 0.114 \\\\ \n-0.450 & -0.883 & 1.333 \\\\ \n-1.333 & 1.116 & 0.217 \\end{bmatrix}\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\\end{align}", "f90598e5192a40ac7272b6122815654e": "s_n=\\sum_{m=0}^n a_m", "f905afbc3a9b4d393d2d82e9e8541653": "\\sqrt[3]{2} \\approx 1.26", "f905b661197057ddcb9efac01928c1bd": "c(d) n^{\\frac{d-1}{d}} \\int_{\\Bbb{R}^d} f(x)^{\\frac{d-1}{d}} dx ", "f905e4ba617f7ee79a2e78497db550fb": " f: \t\\textbf{R}^d\\rightarrow \\textbf{R}", "f905f2f3aca78afa7bc796a502f62b9c": " F_n. ", "f906709c78e587281abd45f838fa0470": "\\int_{0}^{+\\infty} e^{-x} f(x)\\,dx.", "f9067562599b73a36c0696f2f614028e": " e^{rT} = 1 + j \\,", "f906c9f0ba3fdb2744a3f616c7622830": "2^m)", "f90760de083eae74423a3eb8f550b7c3": "\\frac{2bh}{3}", "f907692083a3f09369995c78b85659b7": "\\mathbf{\\Omega}^\\infty", "f9079defa9f6274bdf1d8e78280a3c6f": "\n\\begin{align}\n\\sin(\\theta + 2\\pi) &= +\\sin \\theta \\\\\n\\cos(\\theta + 2\\pi) &= +\\cos \\theta \\\\\n\\tan(\\theta + 2\\pi) &= +\\tan \\theta \\\\\n\\csc(\\theta + 2\\pi) &= +\\csc \\theta \\\\\n\\sec(\\theta + 2\\pi) &= +\\sec \\theta \\\\\n\\cot(\\theta + 2\\pi) &= +\\cot \\theta\n\\end{align}\n", "f907acbf1c0e8c0948bdbb75efc6c6f1": "n>\\max\\{0,-c\\}", "f90840d4b17d3c58af5de7daa3d2463a": "F(x)^2 = ax+(n+a)^2 +xF(x+n) \\, ", "f9086cbbe0f9f50c065588fdb752232a": "GF(p^6)", "f90871624e135eef4c104ccdf57af5d4": "\n\\begin{align}\n\\int \\frac{\\delta F}{\\delta\\rho(\\boldsymbol{r})} \\, \\phi(\\boldsymbol{r}) \\, d\\boldsymbol{r} \n& = \\left [ \\frac{d}{d\\varepsilon} \\int f( \\boldsymbol{r}, \\rho + \\varepsilon \\phi, \\nabla\\rho+\\varepsilon\\nabla\\phi )\\, d\\boldsymbol{r} \\right ]_{\\varepsilon=0} \\\\\n& = \\int \\left( \\frac{\\partial f}{\\partial\\rho} \\, \\phi + \\frac{\\partial f}{\\partial\\nabla\\rho} \\cdot \\nabla\\phi \\right) d\\boldsymbol{r} \\\\\n& = \\int \\left[ \\frac{\\partial f}{\\partial\\rho} \\, \\phi + \\nabla \\cdot \\left( \\frac{\\partial f}{\\partial\\nabla\\rho} \\, \\phi \\right) - \\left( \\nabla \\cdot \\frac{\\partial f}{\\partial\\nabla\\rho} \\right) \\phi \\right] d\\boldsymbol{r} \\\\\n& = \\int \\left[ \\frac{\\partial f}{\\partial\\rho} \\, \\phi - \\left( \\nabla \\cdot \\frac{\\partial f}{\\partial\\nabla\\rho} \\right) \\phi \\right] d\\boldsymbol{r} \\\\\n& = \\int \\left( \\frac{\\partial f}{\\partial\\rho} - \\nabla \\cdot \\frac{\\partial f}{\\partial\\nabla\\rho} \\right) \\phi(\\boldsymbol{r}) \\ d\\boldsymbol{r} \\, .\n\\end{align}\n", "f90886cf9418d92db57bccaddf6b74ee": "(n!)^{-1/n} \\le \\frac{e}{n+1}", "f908a823c85995b19e2fce639f6b5423": "P_{< \\beta} := \\{X \\in 2^\\omega : X\\ \\mathrm{has\\ effective\\ packing\\ dimension\\ } < \\beta \\}", "f908baf3f7371714717e65a6fcd8904b": "E_a=\\frac{m_1}{m_1-m_2}[\\Delta H-m_2(r_2-r_1)]", "f908c00dc2374217cca8a13b8d9725bf": "k^*", "f908db9dc4847520382ef207ad6d22e5": "\\prod _x x+a = C\\,\\Gamma (x+a) \\,", "f908df5c6b80891c60ba5d9bb1fee595": "z = x + y \\jmath", "f9090ee8b181bb2d8c277ffb92e7286a": "\\scriptstyle V_g", "f909a2f10b5ae8cf7758d2ffa7f08506": "I_D \\!\\ ", "f909b64cc41a573fb68a22f580c7f65b": "w_{n,i_n}(p_n(t)) \\in [0,1] .", "f909e1e1991e5b11d024f6dc523edef5": "G \\rightarrow A", "f909ebf0a23b31ba03547fcafc9d11c7": "n_{4}=\\frac{p_{2} V_{ad}}{z_{d2} R T_{ad}}", "f90a3cdccca1d831afe1590cabce5103": " \\zeta_{\\mathcal O}(s) = \\sum_{\\lambda_i} \\lambda_i^{-s} ", "f90a8a7c62664f3825fa85527a5ec94e": " L_\\text{ext} \\, ", "f90aab4d22a671fc32b851bbbac98329": "rr' = a^2\\,", "f90b547889c9558b496ab5065c475dce": "\n\\chi \\sim \\frac{1}{(T - T_{c})^\\gamma}\n", "f90b7046e967117c819d1aa1606e3791": "H(\\tilde{p},q) = -\\sum_x \\tilde{p}(x) \\log_2 q(x)", "f90b8697ad75bd6247b6c16ea90b2cd3": "p=h/\\lambda", "f90bc6f77995cd904d5b5d8bc22e6666": "\\alpha \\frac{\\sigma_0}{E}", "f90c1901c798ec1030a2fac2030b6d41": "\n \\begin{bmatrix}M_{11} \\\\ M_{22} \\\\ M_{12} \\end{bmatrix} = \n -\\cfrac{2h^3E}{3(1-\\nu^2)}~\\begin{bmatrix} 1 & \\nu & 0 \\\\\n \\nu & 1 & 0 \\\\\n 0 & 0 & 1-\\nu \\end{bmatrix}\n \\begin{bmatrix} w_{,11} \\\\ w_{,22} \\\\ w_{,12} \\end{bmatrix}\n", "f90cccbde3093886d8a11141bb05b279": "\\mu \\frac{\\partial \\bar{u_i}}{\\partial x_j}", "f90cf84af1d0d9db6fc714ee7d663f9a": "\nK(x-y;T) \\propto e^{ -{(x-y)^2/(2T)}}\n\\,", "f90d2779c8c16d33e19fd665b2afc20d": "x_t=\\{x(\\tau):\\tau\\leq t\\}", "f90d8116577aa3d5c7a92add7d4117cc": "\\tau=E'\\frac{v^{2}}{c^{2}}\\sin2\\alpha'", "f90e57a88adeb474ff1ed9e8ad86e3cc": "\\mathrm{NH_3\\ +\\ HCl\\ \\rightleftharpoons \\ NH_4^+\\ +\\ Cl^-}", "f90e77f9830c3f40dd8c9f96e9a4d411": "P \\uparrow P", "f90ef0e8618d99069575d46e6b6bb14d": "C_d =C_{d0} + \\frac{(C_L)^2}{\\pi e AR}", "f90f525d8ba4be7895238e21fe4a8a20": "\\nu(M_1) \\,", "f90f78322775f6fb798d7c1d0633cf38": "\\text{CD}_2\\text{H}_2 + 2 \\text{O}_2 \\rightarrow \\text{H}_2\\text{O} + \\text{D}_2\\text{O} + \\text{CO}_2,", "f90f8f01ade9079a229d975bd220d2c3": "\\Delta x_0 = g_{0i} dx^i = 0\\,", "f91030db937e7456d5a8bd8d75f2efec": "\\le 50", "f9108d28278b1167eb34c7247cdf441d": "d\\mathbf r = \\sum_i \\frac{\\partial \\mathbf r}{\\partial q^i} \\, dq^i = \\sum_i \\mathbf e_i \\, dq^i", "f910c9ffb69f6a8aefbb7f235f178f46": " : ", "f910ed25ea9186eb5c24a5bbf252547b": "xy \\equiv zw \\rightarrow yx \\equiv wz\\,", "f91135dba09c23c3e6854efe443ce948": " \\arg \\left[ H(j \\omega) \\right] = -\\mathcal{H} \\lbrace \\log \\left( |H(j \\omega)| \\right) \\rbrace \\ ", "f9114013acbfec8977dcb753c2faab4d": "\\mod l^{n+1}", "f911d00c85ddeb3c6589abcf7dc8ede7": "\\begin{align}\n\\alpha[\\mathbf{f}A]G[\\mathbf{f}A]^{-1}\\beta[\\mathbf{f}A]^\\mathrm{T} &= (\\alpha[\\mathbf{f}]A)\\left(A^{-1}G[\\mathbf{f}]^{-1}(A^{-1})^\\mathrm{T}\\right)A^\\mathrm{T}\\beta[\\mathbf{f}]^\\mathrm{T}\\\\\n&=\\alpha[\\mathbf{f}]G[\\mathbf{f}]^{-1}\\beta[\\mathbf{f}]^\\mathrm{T}.\n\\end{align}\n", "f91224b0bfd07c5fbe7451d630da206b": "T_c = \\frac{\\kappa}{2k_B}.", "f912453f729064aa0ed39fe5c4c43018": "t=\\cos(x)", "f91270b1ce163f4777288b1509e9c69b": "X_t(x)", "f912a1ba05a8382c83b088ac58420555": ".9\\mu m", "f912c6bc4201a8dac5c4bba3843dde0b": "\\int\\sgn(\\sin(x))\\,dx\\,", "f913240a7d64400adb50388f7de44c84": "d\\mathbf{l} = dx \\, \\hat{\\mathbf x} + dy \\, \\hat{\\mathbf y} + dz \\, \\hat{\\mathbf z}", "f91325e19d1965907c2570d9177aacc4": "{\\tilde{B}}_{3}", "f9135d0938d621e75c5df90f6d9732e4": "V\\rightarrow \\mathbb{Z}/2\\mathbb{Z}", "f913814ac2238b3e0936965736b94df9": "7+ \t58+\t39+\t23+\t10+\t55+\t42+\t26\t=\t\t260", "f913a14b4ff893b4e43a2796429162e3": "R(x,y)", "f913da107f8d466227caea50cbef14ae": "x^{p^r}", "f91469e20660c7c2acf0c8cd480e0792": "\\scriptstyle \\tfrac{d^2}{l^2}", "f914a9568dac9f20f5493b078df5e1fb": "n=|z|-1", "f914c5543d681e27ce9f4aa79b13e500": "\\alpha \\in F \\,,\\quad \\alpha \\mathbf{A} \\in M_{mn}(F) ", "f914f35b893c8392f45197525107d521": "\\min(m,n)+1", "f915380b11af39419d8c26e7de44fbd3": "D_V", "f9153def9322b16f7e323fefdefa463e": "\\scriptstyle {\\varphi \\choose 1} - \\scriptstyle {1-\\varphi \\choose 1}= \\sqrt{5} ~\\scriptstyle {1\\choose 0}", "f915478cc2399854712aaf63098af9c5": "\\Omega_X^p(x)", "f9156178e7678514689f7f56d2ddfba0": " J_{\\mathrm{M}} = \\; i/A_{\\mathrm{M}} = \\alpha_{\\mathrm{r}} (i /A_{\\mathrm{r}}) = \\alpha_{\\mathrm{r}} J_{\\mathrm{r}}. ..........(35)", "f91585271ef91cbf384cda4117e45486": "\\sqrt{2\\pi}\\ n^{n+1/2}e^{-n} \\le n! \\le e\\ n^{n+1/2}e^{-n} , ", "f915906c9417f12d5e2b06336a6d53f7": "\\mathfrak{m}_a", "f915a94f3ef631d6af8ab40500d16a36": "\n C_\\pm(j,m) = \\sqrt{j(j+1)-m(m\\pm 1)} = \\sqrt{(j\\mp m)(j\\pm m + 1)}.\n", "f915b39f5da833c78162bd7de8e9a45d": " E_1 = \\tfrac {1}{2} m_x v_1^2 \\,\\! ", "f915d15f6076dd6dbf600b7eff65ca4f": "(F')^2", "f915fd850f90f17fba3ae8ff03645fa4": " {P \\over E} \\ = \\ {1 \\over i} \\ = \\ {PV \\over A } ", "f91609c1c8f323c789f50689ea1e810b": "{\\rho_{In} \\over \\rho_{Out}}\\,\\!", "f91656df5d58bc3d75222c43e40cba3c": "\n\\begin{align}\n s &= (0\\times 1) + (3\\times 2) + (0\\times 3) + (6\\times 4) + (4\\times 5) + (0\\times 6) + (6\\times 7) + (1\\times 8) + (5\\times 9) + (2\\times 10) \\\\\n &= 0 + 6 + 0 + 24 + 20 + 0 + 42 + 8 + 45 + 20\\\\\n &= 165 = 15\\times 11 \n\\end{align}\n", "f91664651333d8f5b42a91e546c8c0ee": "\\textstyle {n+1 \\choose 2}", "f91680e42bd8a0fc7cac0b111990603a": " \\neg \\operatorname{ask}[S-6] ", "f916e8047d91b6ef49fdd842ec5e86fd": "N_{1}", "f9171f088eb19a078bcd02198408f20d": "f(\\mathbf{Q})", "f91732c10a87a4c1b19b4d9b7a41c03c": "\\Delta H/R = T \\cdot (B + ln(\\Delta t))", "f9175fa8b1ed36afaeffad11dd03e309": "\\mathrm{Re}\\left(e^{2j\\omega t}\\right) = \\cos 2\\omega t", "f9179d6cfd54b129324f3599ac7ea4e3": "\\propto \\exp{\\left(\\frac{(x-\\mu)^2}{4\\theta^2}\\right)}D_{-2k-1}\\left(\\frac{|x-\\mu|}{\\theta}\\right)\\,\\!", "f9179f7bac2db6338124ffa7674fbbbb": "H_n(X) \\cong G", "f917ab46f0a312548376942c891a6fa5": "(AA*)u=b", "f917b9a4163010e6596931655d281a98": "p_{i_3;i_1}(f_3\\mid f_1)=\\int_{-\\infty}^\\infty p_{i_3;i_2}(f_3\\mid f_2)p_{i_2;i_1}(f_2\\mid f_1) \\, df_2.", "f917c21f0c3d09836d33595a676198ec": "R_{down}^{*}", "f917f0612bbf9f69bc19fcb64a67e360": "\\varphi(m, n, 2) = m^n,\\,\\!", "f917f46f19f88bee8ccff01577858ef4": "\\sigma',", "f91808577d777363108ed144f65f654b": " \\langle H(t) \\rangle \\ \\stackrel{\\mathrm{def}}{=}\\ \\langle\\psi(t)|H|\\psi(t)\\rangle\n= \\sum_{nn'} a_{n'}^* a_n \\langle n'|H|n \\rangle ", "f91823d058b137fc5b2cab70ef2e4c13": "\nV \\equiv V(\\theta, X)\n=\n\\frac{\\partial}{\\partial\\theta} \\log L(\\theta;X)\n=\n\\frac{1}{L(\\theta;X)} \\frac{\\partial L(\\theta;X)}{\\partial\\theta}.\n", "f918621b1193be4a2bab9ac837ba398b": "\n \\mathcal{I}_j = \\int_{-\\infty}^{\\infty} e^{\\lambda \\mu_j y^2 /2} dy = 2\\int_0^{\\infty} e^{-\\lambda \\left(\\sqrt{-\\mu_j} y\\right)^2 /2} dy \n = 2\\int_0^{\\infty} e^{-\\lambda |\\sqrt{-\\mu_j}|^2 y^2\\exp\\left(2i\\arg\\sqrt{-\\mu_j}\\right) /2} dy.\n", "f91891aa65a36ff567ee7d0a3b025aa9": "\\vec r(u,v)=\\langle u,f(u)\\sin v,f(u)\\cos v\\rangle", "f918c1d61f404a19309c39040d7dd9f1": "0.7 \\le \\mathrm{Pr} \\le 16\\,700", "f91aebed0949a2c7d5141e3c2c6d4675": "dX_t = -Xdt+\\sqrt{2}dW_t", "f91b630335f5126a73992efbccbe0cef": " \\tilde{\\chi}(t)=\\tilde{\\rho}(t)R_0 ", "f91b7d62de43cbbdb922c31a1e5e6b39": "R_N(t,s) = E[X(t)X(s)],", "f91bdd673d6417d9872fb3c468005a09": " \\frac{\\partial M}{\\partial x} = \\frac{\\partial N}{\\partial y} \\, \\! ", "f91c1c0f12a89e54a9edbd7bf332b893": "(y + z) * x = (y * x) + (z * x)", "f91c30455385ac87f5dca12b7c516848": "Z_s", "f91c9736b1f682b995617e6c4bb8d294": " A_i \\in \\{0,W_i\\} ", "f91cbddcfa3258e0141baf45562b8dbf": "(X_1 \\vee \\cdots \\vee X_{n-1} \\vee X_n) \\wedge \n(X_1 \\vee \\cdots \\vee X_{n-1} \\vee Y_n) \\wedge\n\\cdots \\wedge\n(Y_1 \\vee \\cdots \\vee Y_{n-1} \\vee Y_n).", "f91cd300f6db02c704224151bbb9ba06": "\\nabla^2 = \\nabla \\cdot \\nabla", "f91cf9e819e63e821804b272c06ae225": "(1/u_{0})", "f91d7d972d4c632c45f1f7f03d32baf8": "\\scriptstyle f^\\ast(n)=\\lfloor sn\\rfloor-n,", "f91d8cf084a72526d4051008ed66c6a7": "\\mathcal{R} \\ge (\\max\\limits_{0 \\le r \\le (1 - H_q(\\delta + \\varepsilon))}) r(1 - {\\delta \\over {H_q ^{-1}(1 - r) - \\varepsilon}})", "f91deffe5fab303a9a4018da65c6e4cd": "\\lambda^4 = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix}", "f91f1899ffa91f5e5ea95b279e953077": "\\textstyle\\frac{1}{q}", "f91f5dc7f1585f1391d2ebfdf6139acb": "e^{\\pi \\sqrt{163}} = 262537412640768743.99999999999925007\\dots\\,", "f91f7e2ccecd2f498e3e6801f99709db": "A_\\varepsilon ", "f91f9b1a6fc73c952614cef14db42acc": "\\mathbb{PT}^+ \\simeq \\mathrm{SU}(2,2)/\\left[ \\mathrm{SU}(2,1) \\times \\mathrm{U}(1) \\right]", "f920529cc1e17e19842be69b06b3612f": "\\displaystyle{F(re^{i\\varphi})={1\\over 2\\pi} \\int_{-\\pi}^{\\pi} {f(\\varphi-\\theta) \\over 1-re^{i\\theta}} \\, d\\theta.}", "f9206c93992a17585520531dca1da851": "C_{3,3/4}", "f92076362160ddd7da106db74e50fe5c": "s=\\frac{a+b+c}{2}.", "f920ea5c5a2af3efac222cc0315d6fe7": " \\forall a \\forall b \\; a \\vee b = b \\vee a ", "f922299ca606eab6d039578ac10a889a": "\\tan \\delta = ESR \\cdot \\omega C", "f9222a5fd1fea7f0d18f49003c3e5bde": "\n\\iint \\psi_v'^* \\psi_e'^* \\boldsymbol{\\mu}_e \\psi_e \\psi_v \\,d\\tau_e d\\tau_n\n\\approx \\int \\psi_v'^* \\psi_v \\,d\\tau_n \\int \\psi_e'^* \\boldsymbol{\\mu}_e \\psi_e \\,d\\tau_e.\n", "f9227d57b0ed2ac09667b8ea13817245": "ds^2=-dt^2+R^2\\Omega^2_n", "f923d2235a0193a1c41dc92a066ad085": "\n\\scriptstyle\\mathbf{X}", "f924499099086d08e53f236baf2c14b8": " \\operatorname{CNOT}\\ |1,\\psi\\rangle = |1,\\phi\\rangle", "f9246b148a5bc9baeedaf724d95a5c6f": "y'=y \\,", "f924a6a533c722773ff4027466b56318": "\\vartheta_{01}(0;\\tau), \\vartheta_{10}(0;\\tau), \\vartheta_{11}(0;\\tau)", "f924cba31e7d5fc37e1a0683f3eaf309": " F_x = eD \\left[ A_1 cos(\\rho x)+ B_1 \\frac{U_x U_r} {c^2} + C_1 \\frac{\\rho a_x} {c^2} \\right] ", "f924f06f2ee46e8e60a8a6791cdaf1ee": " \\sqrt{t} ", "f92567ea46cff54995bb109b722e2e18": "a_{15}+b_{15}", "f925952dff00ec15ee12efa4edded605": "x_e = 0", "f925c9955fc2a451729a855972e79b77": "\\phi (M_a \\cap U)", "f926c80c934dbf6ac5512764a393d5c5": " y_{n+1} = y_n + \\tfrac12h \\left(f(t_n,y_n)+f(t_{n+1},y_{n+1})\\right), ", "f926fc86c60d02d092a040cf053d284f": "S'_{gen} = \\frac{Q'_2 - Q'_1}{T_{surr}}", "f927111fa39fcb95e92a94efad42efa0": " m > a^{ \\frac{ 1 }{ 2 - b } } ", "f92733486f8e39b655d5ed9328b57197": "(k,n-k)", "f9279cf845910d465f2312f4ccf47343": "R_i~", "f927d07aa7510fe4ba663bda4cecadbd": "u(\\mathbf{r}) = \\mathrm{e}^{-\\mathrm{i} \\mathbf{k}\\cdot\\mathbf{r}} \\psi(\\mathbf{r})\\,.", "f927ded9ffbd3aa3ac36f9ccb5808953": "\\mathrm{A}_4", "f927e71ebe6585df018dc80b57087a8e": "\\scriptstyle z=0.\\,", "f9280cf292c160b79a7c1806176777ef": "D_n, \\mathfrak{so}_{2n}(\\mathbf{C}): \\mathfrak{so}_{n,n}(\\mathbf{R})", "f92837f4eee9b39e1e6a357a9b5d9293": "\\frac{dW}{dt}", "f9285366f258827c63ba5f1f3e6c5e90": "\n \\langle J M|j_1 m_1 j_2 m_2\\rangle \\equiv \\langle j_1 m_1 j_2 m_2|J M\\rangle\n", "f928c6f019666020167a3722e23908d6": " \\Sigma(t) - \\frac{dq(t)}{dt} = ", "f928e96fcf6d637a5c9bd957cbe849c7": "\\chi_{1,0} = \\frac{1}{\\sqrt{2}} (\\chi_-(1)\\chi_+(2)+\\chi_+(1)\\chi_-(2))", "f9294000060b38e927952613b79f6374": "Extraction ratio = \\frac{P_a - P_v}{P_a}", "f929925697ecf6dfd0fa59cbb76d6609": "q_A(x_1,\\ldots,x_n) = \\sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}. ", "f9299c6911bb4325936393306eda4ef8": "\n \\begin{align}\n \\cfrac{\\partial W}{\\partial I_1} & = \n \\cfrac{\\partial W}{\\partial \\bar{I}_1}~\\cfrac{\\partial \\bar{I}_1}{\\partial I_1} + \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\cfrac{\\partial \\bar{I}_2}{\\partial I_1} + \n \\cfrac{\\partial W}{\\partial J}~\\cfrac{\\partial J}{\\partial I_1} \\\\\n & = I_3^{-1/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_1}\n = J^{-2/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_1} \\\\\n \\cfrac{\\partial W}{\\partial I_2} & = \n \\cfrac{\\partial W}{\\partial \\bar{I}_1}~\\cfrac{\\partial \\bar{I}_1}{\\partial I_2} + \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\cfrac{\\partial \\bar{I}_2}{\\partial I_2} + \n \\cfrac{\\partial W}{\\partial J}~\\cfrac{\\partial J}{\\partial I_2} \\\\\n & = I_3^{-2/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\n = J^{-4/3}~\\cfrac{\\partial W}{\\partial \\bar{I}_2} \\\\\n \\cfrac{\\partial W}{\\partial I_3} & = \n \\cfrac{\\partial W}{\\partial \\bar{I}_1}~\\cfrac{\\partial \\bar{I}_1}{\\partial I_3} + \n \\cfrac{\\partial W}{\\partial \\bar{I}_2}~\\cfrac{\\partial \\bar{I}_2}{\\partial I_3} + \n \\cfrac{\\partial W}{\\partial J}~\\cfrac{\\partial J}{\\partial I_3} \\\\\n & = - \\cfrac{1}{3}~I_3^{-4/3}~I_1~\\cfrac{\\partial W}{\\partial \\bar{I}_1} \n - \\cfrac{2}{3}~I_3^{-5/3}~I_2~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\n + \\cfrac{1}{2}~I_3^{-1/2}~\\cfrac{\\partial W}{\\partial J} \\\\\n & = - \\cfrac{1}{3}~J^{-8/3}~J^{2/3}~\\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_1} \n - \\cfrac{2}{3}~J^{-10/3}~J^{4/3}~\\bar{I}_2~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\n + \\cfrac{1}{2}~J^{-1}~\\cfrac{\\partial W}{\\partial J} \\\\\n & = -\\cfrac{1}{3}~J^{-2}~\\left(\\bar{I}_1~\\cfrac{\\partial W}{\\partial \\bar{I}_1}+\n 2~\\bar{I}_2~\\cfrac{\\partial W}{\\partial \\bar{I}_2}\\right) + \n \\cfrac{1}{2}~J^{-1}~\\cfrac{\\partial W}{\\partial J}\n \\end{align}\n ", "f929c4a995bb6ff48860b3b5f2e23950": " p_{2,3}(x) \\, ", "f929d4b6fec2e847c2512852b1bbffc0": "Ric(u)=\\sum_{i} R(u,e_i)e_i.^{}_{} ", "f92a102c89e0ea5898327a492650612b": "\\mathcal{Z}(M)", "f92a5065108df6fa939d9c7fd5d1b637": "k^2 \\log n", "f92aaf91c0159b7a2ff1675d344658bf": "P^\\circ \\to 0", "f92ab79abd3b1c99e098759799a33bbe": "V_C = CH_w + D", "f92b0d960bc849d80057115a45913060": "\\Delta_{ki} < a_1 < \\infty ", "f92bc8a28f9991e621a1466e528814e0": "\\Delta f = A\\Delta\\mathbf{x} + \\|\\Delta\\mathbf{x}\\|\\boldsymbol{\\varepsilon}", "f92bcd2457ec777cb68a5db7ef8be18c": "\\displaystyle {dx(s)\\over ds} = f(x(s)),\\,\\, x(0)=x_0, ", "f92be4cd3e257cccda3906dfce8cfafd": "\\frac{4\\cdot\\pi}{3}", "f92c4fabc146470479e4c0c6353c8865": "\\Delta(t) = -t + 3 - t^{-1},\\ ", "f92ca508dcece61d4acf4ea992dea771": "d\\in N", "f92d34c19e84e83a27dc1fbc2e496be5": "(1,\\bar{3},3)", "f92da842e4846d82bbce068fb2195a9a": "(a/b)^n = a^n/b^n", "f92dbcc77a2086d6531359f8838bfecd": "-c(m-1)I_{m,n}= \\frac{1}{x^{m-1}(ax^2+bx+c)^{n-1}}+{a(m+2n-3)}I_{m-2,n}+{b(m+n-2)}I_{m-1,n}\\,\\!", "f92df1c93b0a5e22d0d3fc6fdba5b7fa": "\\scriptstyle t > 0", "f92e16b24eb2a334e5a56b1372d2af1e": "r=\\sec\\theta+a\\cos\\theta \\,", "f92e68dfc10b477de81e78647b5f3739": "E_{B-V} = (B-V)_{\\textrm{Observed}} - (B-V)_{\\textrm{Intrinsic}}", "f92eb6db331bcdcb9a92b32e905246ae": "\\hat{H}_{S}", "f92ece804f5e71cecd70bd221eead4aa": "b = a \\ln (10)", "f92f482036f1adb99c8d578242116386": " \\mathrm{d}U = \\mathrm{\\delta}Q + \\mathrm{d}U_{in} - \\mathrm{d} U_{out} - \\mathrm{\\delta} W", "f92f5c727b83d37c0b325d41c26b93be": "\n\\nabla \\cdot \\left(\\mathbf\\Sigma_i\\nabla v_e\\right) = -\\frac{1}{1+\\lambda}\\nabla\\cdot\\left(\\mathbf\\Sigma_i\\nabla v\\right) \n.", "f92fb1586c86785aa645ef5e7b72f263": "(M, V) = ((M, V) \\# (P, V_\\infty), V_0).", "f92fbdf898bfc1543c804d45fb2fc531": "X_1,\\dots ,X_m ", "f93051accc2b3336cdab2d4dd995283a": "C(y) = \\int_{A'} e^{i\\langle x,y\\rangle} d\\mu(x). ", "f93056cbdff949b178cade95ab4869d5": "\\mbox{forward reaction rate} = k_+ {A}^\\alpha{B}^\\beta \\,\\!", "f93076337882fd749a5d94305804776c": "p(x_t,y_{1:t})", "f930a7c631bf4312e88153a9ca5dfe04": " \\Sigma = AB \\ \\sin \\theta \\ , ", "f930b6e7b8e862d275ff5ec90867a7a1": "[OH^-]=K_W[H^+]^{-1}\\,", "f930b9b912ab3e07cb921d36174bae78": "p= \\frac{\\rho R_u T}{M}", "f931391febfed3cfa979c155e266d987": " b x + c = d x + e", "f93177eecaf958c07c29a5dd1778def3": "\\frac{1}{2^p}\\left[ \\sum_{k=0}^{\\infty}\\frac{1}{(k+(j/2p))^{2m}}+ (-1)^q\\, \\sum_{k=0}^{\\infty}\\frac{1}{(k+\\left(\\frac{j+p}{2p}\\right))^{2m}} \\right]", "f931b0e80a2644a33b85fd97f96b27cf": "|f(z)-a_0 -a_1z -\\cdots -a_{n-1}z^{n-1}|", "f931c2ea868c62679b492aa1ca597ee9": "R_p = \\max_{i} y_i", "f93222a90cebd7b18f477218de46c3f7": "E_7^{\\mathbb C}", "f9323771b9acc99335de4eb2cecccbc1": "\\int \\sin ax\\, \\cosh bx\\, dx = \\frac{1}{a^2+b^2}\\left( b\\sin ax\\, \\sinh bx- a\\cos ax\\, \\cosh bx \\right) + C", "f93249b0bc13bee2a6d3086f87064790": "a_{ij}(x)", "f932760fef9fc89b951a99341f0d7479": "k^4 + A^2k^3 + 2k^2 - A^2k + 1 = 0\\,", "f932b6142ef18e55e0f909fa383cdf81": "u_j > \\lambda", "f93321d9ef7a5ea9fb09b4160853d6b8": "\\displaystyle\n(u_t + u u_x)_x = \\frac{1}{2} \\, u_x^2\n", "f933480ba7bdc656e10ae41d0dff92b0": "(\\frac{x}{N}) = -1", "f9334a18c783c569bb74738615a15b90": "e^{-\\gamma x}= \\sum_{i=0}^\\infty \\frac{\\gamma^i}{(1+\\gamma)^{i+\\alpha+1}} L_i^{(\\alpha)}(x) \\qquad \\left(\\text{convergent iff }\\operatorname{Re}{(\\gamma)} > -\\frac{1}{2}\\right)", "f93357b4b1da1c04dc7cf013e4294a9d": " \\Pr\\left(\\overline{A_1} \\wedge \\ldots \\wedge \\overline{A_n} \\right) \\geq \\prod\\nolimits_{A \\in \\mathcal{A}} (1-x(A)). ", "f933c2682fbeffafa5faff987a5e8225": "f_c {\\aleph_0}", "f937c1e06531d7c7a147f5869eb67c94": "w\\sqrt{\\theta}/{\\delta} = w\\sqrt{T/518.7}/(P/14.696)", "f93809ae14fb28ef6dbe11c99529c51b": "A_n", "f9384686b003155d9c15e340823d5122": "\\,K(t)\\,", "f9387613b9e0ff2e2e68c804929a9dbf": "= \\arctan \\frac{120*1 + (-1)*119}{119*1 - 120*(-1)}", "f93893e29ee435042171f3f9d5d92c2b": "\n \\hat{r} = \\frac{nS_{xy} - S_xS_y}{\\sqrt{(nS_{xx}-S_x^2)(nS_{yy}-S_y^2)}} = 0.9945\n", "f938ffdd7b071889f0975ac4421574b9": "v\\in", "f939693e76e5481beee03305df1ebfb8": "\\alpha_{1}, \\ldots, \\alpha_{N}", "f939826863f092d7230221dd0235557b": "y \\not = v", "f93a26b74536f11075f8b6c3f5847348": "{\\nabla_R}^2", "f93a2f1da839a2784ef71eef171f1c0a": "\\begin{vmatrix}\n l_1 & m_1 & n_1 \\\\\n l_2 & m_2 & n_2 \\\\\n l_3 & m_3 & n_3\n\\end{vmatrix}=0.", "f93a6c52304c3bb2e464c6b1898c30ed": "\\epsilon_\\lambda=\\alpha_\\lambda", "f93adc0eabc0ffffd21ed4d42465940b": "\\displaystyle{R^{(k)}_\\varepsilon f(w)=\\int_{|z-w|\\ge \\varepsilon} M_k(w-z)f(z)\\,dx\\, dy,}", "f93af7630b7b81d018b13cc86e898cc3": "p_0\\, =\\, p\\, +\\, q\\,", "f93afcb0dbea87a26753dd80668672d2": "4 \\pi \\alpha^2 \\,", "f93bebc16b3d9a76c0ca920dbdd24d80": "AV = \\frac{24.5(PO_{gpm})}{ID^2-OD^2}\\,", "f93c6e3cf0028ba349b025a19a9418a8": "\nr_6(n)=\n\\frac{\\pi^3 n^2}{2}\n\\left(\n\\frac{c_1(n)}{1}-\n\\frac{c_4(n)}{8}-\n\\frac{c_3(n)}{27}- \n\\frac{c_8(n)}{64}+\n\\frac{c_5(n)}{125}-\n\\frac{c_{12}(n)}{216}-\n\\frac{c_7(n)}{343}-\n\\frac{c_{16}(n)}{512}+\n\\dots\n\\right)\n", "f93c8762ebe05485801e774ac7a481fa": "u(x,y) = \\frac{\\sinh (ny) \\sin (nx)}{n^2}.", "f93ca2f2432f0cc49ee8c3b987ac2fa4": " [[\\phi,[[\\psi,\\chi]]\\,]] +\\text{cycl.} = DT(\\phi,\\psi,\\chi)", "f93ca30469760468a5bbf963f3d6021c": "e^x = \\sum_{n=0}^\\infty \\frac{x^n}{n!},", "f93cb208963ce1a23c0610406a99c7a2": "n = -1", "f93ce7e2af21b3eb9622c32e07af264f": "x^{2n}", "f93d1b77546582e4731be7e4739c6f14": " 5 \\mathrm{Pa} \\cdot 20 \\mathrm{m^2} = 100 \\mathrm{N} ", "f93d38f7e0b4ee41a734cf1d88a9213d": "+\\left\\{ \n\\left[ (A^\\prime-B^\\prime)- (A^{\\prime\\prime}-B^{\\prime\\prime}) \\right]-\\left[ D_{JK}^\\prime+D_{JK}^{\\prime\\prime} \\right]m\n-\\left[ D_{JK}^\\prime - D_{JK}^{\\prime\\prime} \\right] m^2 \n\\right \\} K^2 -(D_K^\\prime - D_K^{\\prime \\prime} ) K^4", "f93d56602e19814f0b0535d8ef685f16": "\\mathfrak{sl}_{n+1},", "f93d7ef8b32abea4637cff5004f3b844": " H_{\\frac{2}{3}} = \\tfrac{3}{2}(1-\\ln{3})+\\sqrt{3}\\tfrac{\\pi}{6}", "f93daa301d95703d295a1feb7f70d0bb": "\\underline{P}(Cl^{\\ge}_t)", "f93dc3acdf7e46469a5dca23d43815e1": "\\frac{k_1 + k_2}{2}", "f93de032f702ee4be60970bec5f367ed": "f(\\xi ,\\tau)=f(\\star q(\\xi ,\\tau),0).", "f93deb4b7bfa3685822412e510058b93": "(x,y) \\mapsto \\alpha(x,y) - \\alpha(y,x)", "f93eabac9ba88616e874d2204c2bb049": " p = n K_B T", "f93ebb9799a43c5820a0308250861473": "\\frac{b-a}{3}", "f93efc0bff6dd68dde598dfd6b989e2a": " [H_i^+] = \\left[ \\begin{array}{rrrr}\nq_{i,4} & -q_{i,3} & q_{i,2} & q_{i,1} \\\\\nq_{i,3} & q_{i,4} & -q_{i,1} & q_{i,2} \\\\\n-q_{i,2} & q_{i,1} & q_{i,4} & q_{i,3} \\\\\n-q_{i,1} & -q_{i,2} & -q_{i,3} & q_{i,4} \\\\\n\\end{array} \\right].\n", "f93fd75750ed74afc039f93e48f6fc29": "\\langle x_i, x_j \\rangle^4 = x_ix_jx_ix_j", "f93ffeb87312d8c8286d4c4eb8904342": "\\sigma_n=\\frac12+\\frac{1}{2n}", "f9403e188d7c49e79a210e939a679202": "A_n = \\sum_{m=0}^n a_m\\,", "f9404b334887fb5e1787f4778e84ee2d": "a := \\frac{x_1+x_2+\\cdots+x_n}{n}", "f9404e757cc0d475cdff9ed29e56500b": " q_2 = \\frac{Q}{w_2} = \\frac{150}{5.00} = 30.0 \\text{ ft}^2/s", "f940683a163c0867e39ebf548a4a6813": "LN[j]", "f940cc301b37d324b1894d716c68e0e9": " X = |X|e^{j\\arg(X)} ", "f94145fd2d4ab60b5624933f6768d334": "p_{{\\mathrm{O}}_2}", "f941a83e4ba2748d5cfcb3a4f83e347c": "(x_i-\\overline{x})^\\mathrm{T} \\Sigma^{-1} (x_i-\\overline{x})", "f941d0afcb790b261adb2463427ab667": "\\ t =\\ \\sqrt {\\frac{2d}{g}} ", "f941fb8cf9bafdff9a820de2af90fe62": "((c_1 g_1 + c_2 g_2 + \\cdots ) \\otimes f)(x) = c_1 f(g_1 \\cdot x) + c_2 f(g_2 \\cdot x) + \\cdots", "f9424e1935cb39223d6a26f15408488e": "f : X \\to Y, \\,", "f94296d24a3794605c5c7ead229d983a": "L \\in \\mathbb{R}^n", "f9429c10cc6a69c3e03f4b70eb4fab1b": "l_2, l_3, l_{31}, l_4, \nl_{41}, \\ \\ l_{42}, ...", "f942a76d84d5a55624572826310c6569": " \\mu(A + B) \\leq \\mu(A) + \\mu(B) ", "f942d6859aaa510b543ef8e0133c2fa2": "\\varrho_{A, B, \\Lambda}", "f943e96b1d833fda5d567dcffac95904": "g_u", "f944447a418d5abbe137ef7ea2cd17eb": "{\\mathcal R}", "f944ffc47267d2ba015c0788e064dec1": "{{F_m } \\over {P_m }}", "f9450a69da7f9d942de5c500f5914b02": "\\Delta\\alpha-\\bar{\\delta}\\gamma=(\\rho+\\varepsilon)\\nu-(\\tau+\\beta)\\lambda+(\\bar{\\gamma}-\\bar{\\mu})\\alpha+(\\bar{\\beta}-\\bar{\\tau})\\gamma-\\Psi_3\\,.", "f9452329fdad72e8b79b5fd8b6f9d56c": "n'-m'", "f945940cd980bd1d93f051f0cac27621": "2/15 = 1/10 + 1/30 ", "f945d42f593dc47738c7e0197a12a767": "\\eta_v(T) = \\frac{\\int_{\\lambda_1}^{\\lambda_2} B(\\lambda, T)\\, 683 \\mathrm{~[lm/W]}\\, y(\\lambda)\\,d\\lambda}{\\int_{\\lambda_1}^{\\lambda_2} B(\\lambda, T)\\,d\\lambda},", "f9462b1de97699dc034c92a792e5aea3": " \\tfrac{d}{dt} S(t) = c_1 S(t) \\left( c_2 - S(t) \\right)", "f946b13971c0467c2db73bb852d3148d": "\\,^{249}_{97}\\mathrm{Bk} + \\,^{50}_{22}\\mathrm{Ti} \\to \\,^{296}_{119}\\mathrm{Uue} \\,+3\\,^{1}_{0}\\mathrm{n}", "f946dde17cdd81b2c42562b13e827100": " P1 ", "f946e546e0148296f990280d4090e52a": " p_i = \\left ( t_\\text{r} - t_i \\right )c", "f946eeaf786bea73011d515430b3600f": "\\mathbb{Z}/p^n\\mathbb{Z}", "f9472e53c34c0cefa6ebaeafd1b770cf": "(\\overline{C} \\vee \\overline{A}) \\wedge (C \\vee A)", "f9474bc2a5baa9285ec4be2625ab68ba": " C(S) \\leftrightarrow \\lnot \\forall x \\lnot C(x) ", "f947790375d21b53a26123c34fb3f6fd": "= \\frac{1}{3}", "f9477c8c7a4d46ef86622c67af136add": "\\omega = e_1\\wedge e_2\\wedge \\cdots \\wedge e_n", "f9477d2ed48265cb2ccd73db15a6849c": "I \\left (1-\\frac{td}{i+d}\\right )", "f947e8ddba52ded1f0e44ffee3294270": "a=J/M c", "f9480db99f02efcdc4a57590714a56fc": "n = \\frac{\\omega}{(2 \\pi)}", "f9485175293ee9b67020638d2862dbd4": "Y = \\frac{E}{1-\\nu} ", "f948c4b563fcae8d4b54eb233c402d0f": " \n\\begin{align}\nI(\\theta)\n&\\propto \\cos^2 \\left [{\\frac {\\pi S \\sin \\theta}{\\lambda}}\\right]~\\mathrm{sinc}^2 \\left [ \\frac {\\pi W \\sin \\theta}{\\lambda} \\right]\\\\\n&\\propto \\cos^2 \\left [\\frac{k S \\sin \\theta}{2}\\right] \\mathrm{sinc}^2 \\left [ \\frac {kW \\sin \\theta}{2} \\right]\n\\end{align}\n", "f94901fac0e08d6a62d5e4ac7cf0e65f": "u^-(\\mathbf{x})", "f9495bfab8fbb8453fc2d19345a89881": "\nJ \\equiv \\frac{\\partial (\\mathbf{Q}, \\mathbf{P})}{\\partial (\\mathbf{q}, \\mathbf{p})}\n", "f94973ebb97d7cfd43533f14069d071d": "k \\cdot n = O(n)", "f9499fb4663468ae63445a9beb127b4d": "\\overline{X_{i}}=\\frac{1}{T}\\sum\\limits_{t=1}^{T}X_{it}", "f9499ffd92710b8a172df785351ac34b": "\\sup \\{ |f(a)| : a \\in A \\} = \\sup \\{ |F(x)| : x \\in X \\}", "f949ea3affa292c7d0951d7377f34e4a": "\\overline{K}[C]=\\overline{K}[x,y]/(y^2+h(x)y-f(x))", "f949fde8d8811a3b486598b06bc54596": "\n\\{ 1, \\alpha, \\ldots, \\alpha^{m-1}\\}\n", "f94a0c2dfc0e6627724ba8faddf6f5f7": " P_1 : [x_1 : y_1 : z_1], ", "f94a1107255716ce5595716abfbaacc3": "\n w_i = \\tfrac{1}{2}~\\epsilon_{ijk}~u_{k,j}\n ", "f94a37c338f181af0977d394b5526fd2": "\\kappa(y) \\to \\kappa(x)", "f94a8a279e2377afb6027b3b6e07631e": "(L_g)_{*}X = X \\quad \\forall \\quad g \\in G.", "f94a9fe55fbb792e933dde221b8896cd": "\\,x_j\\,", "f94abf0f9fc9129221cca839e18ab81d": "x+y+z=b", "f94b64bcc809c2828f3840355a72cf65": "\\sigma(A)\\subset\\mathbb{R}", "f94bb7447703ae400ad9c26222377f75": "A = \n\\begin{pmatrix}\n1 & 1 & 0 & 0 \\\\\n0 & 1 & 1 & 0 \\\\\n0 & 0 & 1 & -1/8 \\\\\n0 & 0 & 1/2 & 1/2\n\\end{pmatrix} ~,\n", "f94c627fec6acc9dbac7a1ffaa9fcc84": "a_{i,j}=0 \\quad\\mbox{if}\\quad ji+k_2; \\quad k_1, k_2 \\ge 0.\\,", "f94c6e9ca8ba3babaef6fc4765ab7c38": "P ,", "f94c9626642dab3ad209b8425daf85ca": "\\mathcal{}M = MU_*(X)", "f94cacb927c4208eb92da91690ad8d6e": "\\log(K)", "f94cf762fd4978c8323bbef5218269cd": "\\Delta E_{ab}^* \\approx 2.3", "f94db4173f114da6aa064ae3e83f8d13": "E_{sr}", "f94dd0a51a9fb2e6b3d144522061fcc6": "L^*(R)", "f94dfd841c9472f72de2559b15150f9b": "S_{mm} = \\frac{b_m}{a_m}\\,", "f94e3313c2d420d1a9146b42b812161a": "w\\,\\!", "f94e75f21bf12b197612f67564b106cc": "F_L = N_2(I_2 - I_1)I_1 = \\frac{N_1}{a}(I_2-I_1)", "f94e8d581d1edd6a2264c419bf3dadb3": "2(\\sum^{k}_{i=1}1/\\lambda_{i}^3)/(\\sum^{k}_{i=1}1/\\lambda_{i}^2)^{3/2}", "f94eb67cf9e6981bcdead2867a48203b": "\\operatorname{copysign}(x,y) = \\operatorname{sign}(y) \\; |x|.", "f94ec5933fce5b3cdc5682f1d3ef5ff1": " \\Delta =\\Delta (x_{\\perp }).", "f94ede90906e02acf4db0fc367ce32d0": "\\iiint\\limits_D f(r,\\theta,z)\\,r\\,dr\\,d\\theta\\,dz, ", "f94f1fff2300141ccb95ad0fd84d31f3": "z(n,n;2,t)=n^{3/2}t^{1/2}(1+o(1)).", "f94f2aa094b0f20b3598a65ab629141b": "\\varphi_{\\beta}(0)", "f94f36f135f7015a05e291be80ba2236": " \\mu: \\Sigma \\to \\mathbb{C} ", "f94f60397c58531384fc3f95fd47fa8b": "\\lambda x_1 \\ldots x_{A_i} . \\lambda c_1 \\ldots c_N . c_i x_1 \\ldots x_{A_i}", "f94fc37220f8c8b05e065c49e75569c8": "\\Rightarrow_{h} SBBB \\Rightarrow_{h} SSBB \\Rightarrow_{h} SSSB \\Rightarrow_{h} SSSS", "f94fd4f445f0d2852139ab02cf6b437e": "\\overline{P}_{nm}", "f950234c2dbfa844ec7f481b185f6276": "y^2-z^2+x^2=0;", "f95023aa0cd3a5c4af789d332026a087": "\\textstyle\\frac {upper chord}{lower chord}", "f9510903d0fe9cc5d3f0c7b48238811b": "d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.", "f95143dff3d1b01f3837e980080a85fb": "I = m r^2", "f9516071c491e81facd513515bd713d9": "\\mathbf{V} \\in \\{0,1\\}^{D\\times D}", "f9516d54d60c8205ee7e4b4ffb94a3ab": "g_{res}^E= R \\cdot T \\cdot \\sum { x_i \\cdot \\ln \\; \\gamma_{res,i} }", "f9517a9bc38fecea7f363301121dd73f": "\\exists \\vec{x_1} \\forall \\vec{x_2} \\cdots Q_i \\vec{x_i}\\ \\phi(\\vec{x_1},\\vec{x_2},\\dots,\\vec{x_i})\\ =\\ 1", "f9519697c9d717a7ba89ed40e6087cc1": "\\ln(1/R_e^2)/2", "f951c9c5fec3f72ce23e5e6e961cb4dc": "m = \\frac {D}{d} = \\frac {130}{7} \\approx 18.6", "f951e3bd943dd6ae5131ddf57a3b6794": "p_t\\,", "f951ff77ad68c0cbc95d23e6e7aa95c3": "Z_1 X_2 Z_3", "f9520530484dc87384d6ea4e3d8d2f0f": "m = 2,3, n = 3", "f9528ee6a16288ff996ce03772ab3668": "(R,+,\\cdot)", "f952cb1a6c6b2050b0954884c660cc6f": "\\bigoplus_i R/(q_i)", "f952d3f51e15cc9ed6bdacb8cb2aca2e": "E' = E \\ ", "f952ea2e4cee589e8b7c7f94f61ec5f4": "\\mathbb{Q} \\mathcal{O} = A", "f9533403c2f3f02ccbb8e1dbb740e125": " D=\\{(\\vec x_1,y_1),\\dots,(\\vec x_n,y_n)\\}\\subset R^d\\times C", "f95343399d8bcac724535d9cda066a4e": " \\sigma: \\mathbb{N} \\to \\mathbb{N} ", "f953aff80b878951a7477a33d41504b4": "y=xM", "f953b4c04224a4e6ec5eb7bb6fbf2fb5": " \\mathcal{C}_{YX}^\\pi = \\left( \\mathcal{C}_{X \\mid Y} \\mathcal{C}_{YY}^\\pi \\right)^T ", "f953f1de3590d502b33065182366b9b1": "p(\\mathcal{O}\\vert M,z)", "f953fe7af91487a9ec010ca88f7256db": "M'= \\frac{1}{y'} + \\frac{y'^2}{2}", "f9540328eeaddffcfaecc2cc24b26574": "\\sqrt{\\Delta_1^2 - 4 \\Delta_0^3}=\\sqrt{\\Delta_1^2} ", "f9544dfcc4679f7f82784b23852f0d05": "\\sum_{n=1}^\\infty\\frac{1}{n^2}", "f95487a450f573f5faffb4f762cfc121": "\\gamma_B < 1 ", "f954ac9d1bc5ff82022200067b36faa0": "\\tfrac{\\log{X}}{2} \\sim \\operatorname{FisherZ}(n,m)", "f955685d708c88059ce78b2439e689f1": "\\delta_\\text{r}\\colon Q \\times \\Sigma_\\text{r} \\times \\Gamma \\to Q ", "f955b4ffff6d20cd6e3e5e104052c382": "m\\overline{\\psi}\\psi", "f955bd836f10ec92c07e4c5227298865": " \\widehat{\\mathrm{d}x^{p}}", "f955cafcfa69950681dac82a76a748ff": " \\frac{{}^\\mathrm Nd}{dt}(\\mathbf r^\\mathrm R) = \\frac{{}^\\mathrm Ed}{dt}(\\mathbf r^\\mathrm R) + {}^\\mathrm N \\mathbf \\omega^\\mathrm E \\times \\mathbf r^\\mathrm R.", "f955d2f5e1f06df6b3fe26ac867ebeb9": "(1,2,3) \\neq (3,2,1)", "f9564fa9ac7b5baacf7937a866b81eef": " \\omega_\\varepsilon(\\tilde{x}) = \\tilde{x} + \\varepsilon \\omega ", "f9566d8b14b692a24ad6113019e16a60": "\\sigma_{23}=\\sigma_{32}\\,\\!", "f956f5b9e4db1fd4eb33fdc4e14fa9bf": " t=0", "f9573ec7348303297c2686ec1d005ae7": " \\lambda(x; 0,1) = \\left(\\frac{1}{x\\pi \\left(1 + \\left(\\ln x\\right)^2\\right)} \\left(\\frac{1}{2} - \\frac{1}{\\pi} \\arctan(\\ln x)\\right)\\right)^{-1}, \\ \\ x>0", "f9577e5ce30080d6d8801ad0984c0c59": " \\Pr(x \\le m - k \\sigma) \\le \\frac { 1 } { k^2 } \\frac { \\sigma_-^2 } { \\sigma^2 }.", "f95782ee9994cd8c22306a4273c9a17c": "\\pi : SV \\to V", "f957907565547c799f5da1ec56883bbf": "n = 0, 1, \\dots", "f957bbc36e7218b9a37191d1e3e6f1fd": "w_i\\sim w_{i+1}\\,", "f957c8772075b0e40f864d35a297d9ad": "x=\\sqrt{2n+1}-2^{-1/2}3^{-1/3}n^{-1/6}t", "f957fe40fd568f2464ad9d75505fddb7": "(\\mathbb Z_3^7 \\times \\mathbb Z_2^{11}) \\rtimes \\,((A_8 \\times A_{12}) \\rtimes \\mathbb Z_2).", "f95812c70e628b82169b0c106f48b87d": "\\displaystyle\\frac{\\tau' = \\left[\\alpha_i := \\tau_i\\right] \\tau \\quad \\beta_i \\not\\in \\textrm{free}(\\forall \\alpha_1...\\forall\\alpha_n . \\tau)}{\\forall \\alpha_1...\\forall\\alpha_n . \\tau \\sqsubseteq \\forall \\beta_1...\\forall\\beta_m . \\tau'}", "f9582fbeeb0ed58905704f974a8af1e4": "f\\in V'", "f9590c1a0b43cfadbc77a2b83c300350": "1- R^2", "f9593a02e8a5829202542451c60e3ce6": "3^n+1", "f9595340b378780c6fc13c12f411a101": "\\hbar k ", "f959822656955b85de310db734226a2f": "L\\oplus\\bar{L}", "f959984abff9d702de9ee8e9b90a456b": "\\!\\mathcal A \\models_X^- R t_1 \\ldots t_n", "f959bf5cab1a78d064ca00025ee74367": "\\delta f_2 = 0", "f959d063b06bf38b8418f2b41d28d67e": " P \\to Q ", "f95a08882d538df1ae7814c8b4c2c6d1": "n = {\\tfrac12} (\\sin\\phi_1+\\sin\\phi_2) ", "f95a96b5a0661da560186a6a568ed9bd": " \\zeta = \\frac{c}{2 \\sqrt{km}}", "f95ac0e6e69932cafa2627a83a347673": "(r^2-a^2-b^2+c^2)^2+4a^2(r^2-b^2)\\sin^2\\theta=0,\\,", "f95ad8bf6e7cc5c125648db0695d81fb": "\\{j\\}", "f95b4251edcb0dc0ff728eefbadae00c": "\\text{Mag}_{13\\text{ kHz}} = 52\\text{ dB} + (1.7\\text{ octaves} \\times -2\\text{ dB/octave}) = 48.6\\text{ dB}.\\,", "f95b46addaf311d24fa97d53920941df": "=(1/b^{2})(1/s^{2}), \\quad \\beta = 1", "f95b46db5628c1e899577b6563127541": "\\Delta G = RT \\log \\frac{C_{inside}}{C_{outside}}+ZF \\Delta P", "f95b5dce3eaa9f2b2488fef1fded8ca3": "x = 2y", "f95bcc08b6d9b3d370fd637afce5422b": "m\\geq 0, x_{i}\\in\\Sigma", "f95c25c5760280a38d822481b12ce411": " \\textstyle \\sqrt{D_{cl} t^*}", "f95ce91fb6e9b9167add305992de5b64": " K(x,y) = (\\alpha\\langle x,y \\rangle + 1)^d, \\alpha {\\in} \\mathbb{R}, d {\\in} \\mathbb{N} ", "f95cf740116d9df9a9a4df2329f4c562": "Z\\bar{T}", "f95d4ce459ab8bbc8eba5c1adbc9edeb": "\nU_{kl}^{AB} + \\sum_{i=1}^{N} \\min_{X} \\left(E_{ki}(r_{k}^{A}, r_{i}^{X}) + E_{lj}(r_{l}^{B}, r_{j}^{X})\\right) > U_{kl}^{CD} + \\sum_{i=1}^{N} \\max_{X} \\left(E_{ki}(r_{k}^{C}, r_{i}^{X}) + E_{lj}(r_{l}^{D}, r_{j}^{X})\\right)\n", "f95d56327c7286f95125399624de0ace": "X\\mapsto \\operatorname{Free}_R(X)", "f95d9a66843823bc5bb434e349a10a1a": "O(\\log \\ell)", "f95d9e7978780d35112c9ffbf60db07f": " df = \\frac{\\partial f}{\\partial x}\\,dx + \\frac{\\partial f}{\\partial t}\\,dt + \\frac{1}{2}\\frac{\\partial^2 f}{\\partial x^2}\\,dx^2 + \\cdots ", "f95daaad83678912e10d992fd5a8df73": "\\sum_{j=0}^{\\infty} \\left (x_{n}e_{n}^{-1}D_{n-1} \\right )^{j}g(x).", "f95dbc5e6ca7517c76671b7b2f72474f": "f_\\theta", "f95de035f136fdbd0536685ccff90a5a": "\\forall x:A . B", "f95e121b74e2031fcbdc36d2835358aa": "0 \\le \\theta \\le \\pi/2", "f95e7bc435ed6aa8976d7feaa17e6702": "\ndL_j(t) = \\sigma_j(t) L_j(t) dW^{Q_{T_j}}(t)\\text{.}\n", "f95f23cec6702c60ff24bb978e239e35": "\\boldsymbol\\beta^{\\mathrm T}", "f95f65c36d30d42be77cb756ae39ef27": "(L_0,R_0) = (L_0',R_0')", "f95f932d112c2efde1d42054c3e220d5": "\\sigma( \\mathcal{I}) ", "f9600715010ef7cbe58158975b6f8941": "p(r_1, \\dots, r_n)", "f9602a7a0baddcf85e63b70a93222d38": "A\\, ", "f960b062dfab2bafb390f73a4c6c73d2": "p_{0}(a)", "f96118cdfe54dea4152fb7e54f5264a9": "P(x), Q(x), A_1(x),\\dots, A_r(x)", "f9615da7dda23d8c3ecb95482a5ef7c6": "\\hat{\\mu}=\\text{E}(Y_i)", "f9616d5e8acc92a9d556a7b44d69e213": " \\operatorname{st}(a)=\\lim_{n \\to \\infty} a_n ", "f96184df954c7178117b43a9889d9881": " \\mathrm{supp}(g_{mm'})\\ne \\varnothing \\, ", "f962d853818ac637cfb60695d75b41d9": "0 \\leq F(p,q) \\leq 1", "f962ec6c9978464f28ce308b7dc6c89c": "\\frac{d[A]}{dt}=-k[A]", "f963a9aa0019068b8549ba96f8940228": "v_A = \\frac{B}{\\sqrt{\\mu_0 \\rho}}", "f964494d60967ee295dacf10d8a9a92a": "h_{1(k-1)}-m_{1k}=h_{1k}.\\,", "f9648b39ded0aad0d1a162ec53e7ff0b": "R_\\lambda R_\\mu = R_\\mu R_\\lambda = (R_\\mu-R_\\lambda)/(\\lambda-\\mu).", "f9649eb4dd188484c889a815ffa6477a": "\\psi_{\\nu_{\\mathrm e}}", "f964c4259ef88c10e361ebb3a74477dd": "\\left(\\frac3{F_n}\\right)", "f964c9810fc83d0a5dcdc5d7e9d74a5d": "\\left| P(h) - \\frac{|S \\cap h|}{|S|} \\right| < \\varepsilon .", "f964d0ff301f7695ef0df8c88be341a1": "A = 1 + \\beta_1 + \\beta_2", "f964e38ec1c1fee25fecc119e91a6f18": "\\begin{align}\\nu&=\\left(\\rho-(1-\\gamma)\\left(\\frac{(\\mu-r)^2}{2\\sigma^2\\gamma}+r\\right)\\right)/\\gamma \\\\&=\\rho/\\gamma-(1-\\gamma)\\left(\\frac{(\\mu-r)^2}{2\\sigma^2\\gamma^2}+\\frac r{\\gamma}\\right)\\\\&=\\rho/\\gamma-(1-\\gamma)(\\pi(W,t)^2/2\\sigma^2+ r/\\gamma)\\\\&=\\rho/\\gamma-(1-\\gamma)((\\mu-\\gamma)\\pi(W,t)/2\\gamma+ r/\\gamma)\\end{align}", "f964e5ae0aa6d3a2bd6d5df995e20b85": " \\frac12\\boldsymbol\\mu^{\\rm T}\\boldsymbol\\Sigma^{-1}\\boldsymbol\\mu + \\frac12 \\ln |\\boldsymbol\\Sigma|", "f96522a5cc4c67fd8ab14b62a94420f8": " 3, \\tfrac{8}{3}, \\tfrac{11}{4}, \\tfrac{19}{7}, \\tfrac{87}{32}, \\ldots\\, ,", "f965cc5d176739e7a9fa9c92f574399f": "e_1^5e_3e_4^3", "f965d0106bb2f1ba09f55ddb8d61463e": " \\{D_{\\nu}\\} ", "f9662952e1c024568c7a07f9829a2612": " (\\operatorname{sink}[(\\lambda E.G)\\ Y, X])\\ H ", "f9672c7f6ca70d771467763bf1dbf78c": "-1=a^2,", "f967413507c77d67b7a66eb9874afd6e": "\\hat{\\mathcal{H}}", "f967703b14eedc95859f05d806f91179": "P(t) = \\frac{K P_0 e^{rt}}{K + P_0 \\left( e^{rt} - 1\\right)} ", "f967778f4f578afb5261d30c4f632412": "u'=\\frac{dx'}{dt'}=\\frac{\\gamma \\ (dx-v dt)}{\\gamma \\ (dt-v dx/c^2)}=\\frac{(dx/dt)-v}{1-(v/c^2)(dx/dt)}=\\frac{u-v}{1-uv/c^2} \\ . ", "f968481b5b6a4af4e73ae86f145494c7": "\\textstyle |f(s,x)| \\le C \\mathrm{e}^{\\varepsilon x^2} ", "f9685db4d1d78ed3a2a2b71e74a8ac2d": "\\Sigma \\models \\Gamma", "f9698612ee55b6456c9bc025c1362de3": "l(C(x))", "f969926df28cbe0bdea4ddaf8c49918b": "M = \\frac{N}{p^a}", "f9699dc81eb9518c87f4795229f072e5": "\\forall x \\forall y \\exist z (x \\in z \\land y \\in z).", "f969d3a080bc794313564da523be0139": " DF(T) = e^{-rT} \\,", "f96a013a174d6731063fd75bad194bcd": "\\sum_{\\{ij\\} (nn)} {1\\over2} m \\omega^2 (R_i - R_j)^2.", "f96a38c188887dd0668b3327912bd368": "\\| - \\|", "f96a536eb5a13390181669c2b832469c": "\n\\dot{\\Psi}(t) = A(t) \\Psi(t)\n", "f96a5808feb054b1780e2ecbc81fd81c": "\\left \\| s(t) \\right \\|^2=\\sum_{p=o}^\\infty \\left| B_p \\right|^2,", "f96a90cdbd2c550f8f8295b243f68ae0": "(F^*f)(p,\\mathbf{x}) = \\frac{1}{2\\pi\\cos p}\\int_{\\|\\mathbf{u}\\|=1,\\mathbf{x}\\cdot\\mathbf{u}=\\sin p} f(\\mathbf{u})\\,|d\\mathbf{u}|.", "f96b15af6907371b30dad0aee47b3c66": "\\widehat{\\mathbf{C}}=\\mathbf{C}\\cup\\{\\infty\\}", "f96b53080bab346b56ff05635eca25df": "\\phi\\;", "f96b7a5dd8de3a0a84a8bcb012fbdb6a": "I\\!I(X,X) = N\\cdot (\\nabla_X X)", "f96b8d6f2ab89a91c4f1f15f48427e4f": "\\chi_0(q) = \\sum_{n\\ge 0} {q^{n}\\over (q^{n+1};q)_{n}} = 2F_0(q)-\\phi_0(-q)", "f96c0c4a7ed2242ae9f59224273d2506": "(Ax^2+2Bx+C)", "f96c0fc96dfeb3ef4f56c46b108a9731": "A = \\begin{bmatrix} 1 & 3 & 1 & 4 \\\\ 2 & 7 & 3 & 9 \\\\ 1 & 5 & 3 & 1 \\\\ 1 & 2 & 0 & 8 \\end{bmatrix}\\text{.}", "f96d659029b7e40041776f265c584170": "w':\\neg a", "f96dc10d79631a8d07cf54da826bfb9e": " ( C \\Gamma_{a_1 \\dots a_n} )^T = + ( C \\Gamma_{a_1 \\dots a_n} ) ", "f96e548746adc0bd9cd78f49bb3c9fb8": "\\mathbf{k} = m_1 \\mathbf{b}_1 + m_2 \\mathbf{b}_2 + m_3 \\mathbf{b}_3", "f96e8a25734bed2b241c2aaab8b1299e": "(B \\setminus A) \\cup C = (B \\cup C) \\setminus (A \\setminus C)\\,\\!", "f96eef05a407656f1b6b5d9deeb0e156": "f(x):V(x)\\rightarrow V'(x)", "f96f3de2769ad605ba4299e8ed3de4d5": "\\mathcal{Z}(W_n)", "f9703e3b4edd8185734b62653babec64": "y''-\\frac{6}{x^2}y=Lclm\\big(D+\\frac{2}{x}-\\frac{5x^4}{x^5+C}\\big)y=0.", "f9707499542c8227bd33fa80c281915b": "2\\,T_{m}(x)\\,T_{n}(x) = T_{m+n}(x) + T_{m-n}(x)\\,", "f9715260f8b9d0aca8f92ced721713a8": "\\int_A f(x)\\,dx", "f9716d455a1fab0c805f6d3bc8b22d2b": "{K_{R}}", "f97179d3416e6177f51f712f8e0cf0ad": "(W_t)_{0\\leq t\\leq T}", "f971d36b311f59ed9bdcb7c453e09071": "m(f) = \\sum_{\\Lambda}{1\\over|\\operatorname{Aut}(\\Lambda)|}", "f972157c0a2c57831cb707b54389ae60": "D(p||q)=\\min D(p||r)", "f97284aa2f1aaaa292668d3136a439eb": " v=", "f9728afd240297faa5ae3021bdc3bc8f": "f(x|a,b) = a b x^{-a-1} e^{-b x^{-a}}\\,", "f973475737644d4271b5d81b70371f55": " K = y^\\alpha e^{-\\beta y} \\, x^\\gamma e^{-\\delta x}, ", "f9736068267ba1984ce899f4b7678449": "d \\ln \\mathcal{L}(\\mu,\\Sigma)=0", "f9738ad0a161159e3cc75839237b328a": "y/x = yx^{\\rho}", "f973f8f1f7dc2d14111146bf07ca6556": "\\mu_i = \\mu_{i+1} + \\mu_e\\,", "f973ffebce9b02379a55a81bc7050e7d": "{A}_i", "f9741497e6d593834a7da54f06b235db": "\\rho_0", "f9743ee24efac581a44d8c709dd2cbf1": "\\varepsilon_y = \\frac{\\partial u_y}{\\partial y} \\quad , \\qquad \\varepsilon_z = \\frac{\\partial u_z}{\\partial z}\\,\\!", "f97463767686eacafa366d4da3a76411": " <_{\\mathcal{O}} ", "f974e0e41ed54f1625501b6bffe9ec15": "\\langle G_{\\mu\\nu}G^{\\mu\\nu}\\rangle", "f974e5267567fb5c094c3a6defd766c5": "x' = x - \\lfloor (x - x_{min}) / (x_{max} - x_{min}) \\rfloor * (x_{max} - x_{min})", "f974f19cd12a8bfad508a299d2648762": "\\mathfrak{su}_2 \\cong \\mathfrak{so}_3 \\cong \\mathfrak{sp}_1", "f9751dc995065ac1e62e83ad740eb1fc": "EMA_3(%K)", "f97559ab8fd1c3d6a011efb4a7ecac45": "s_{ij}", "f975eb34d4598ec9a7e4eb77038eae2c": " ~\\chi^v ", "f97625674d885de86be570c87d372d39": "C_2 \\approx 51.6", "f9764ca0ada37ca386ad6d4548727caa": "\\omega_{mn}", "f9765453c1d70407ed1a170f54266063": "\\Delta_2(f_i) = 1 \\otimes f_i + f_i \\otimes k_i", "f9768b1920da138f89f69899fd9375df": "\\det Df_x=4(x^2+y^2)\\neq0", "f9769481e0e552170e18b4cf34682486": "\\log(x/y) = \\log(x) - \\log(y)", "f976a9f8197bbb40e8d3bf736194c0d4": "\n\\int x^m\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx=\n \\frac{x^{m-n+1} \\left(b+2c\\,x^n\\right)\\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p}{2c (m+2n\\,p+1)}\\,-\\,\n \\frac{b (m-n+1)}{2c (m+2n\\,p+1)} \\int x^{m-n} \\left(a+b\\,x^n+c\\,x^{2 n}\\right)^p dx\n", "f9770f12692bc88b3a1631a582c6cc9c": "\\theta_n=\\arg(E(z^n))\\,", "f9776d7cab4424996f15ab307a1fd12e": " \\kappa = \\frac{r}{r^2+h^2} ", "f97793adf01e3db21874a4a424c244e9": " \\forall n \\in {^*\\mathbb{N}} \\ \\exists \\text{ internal } A \\subseteq {^*\\mathbb{N}} \\ \\forall x \\in {^*\\mathbb{N}} \\ [x\\in A \\text{ iff } x\\leq n].", "f977c19e5391aec150a89feb7da034fa": "B^i=[b_1,\\ldots,2b_i,\\ldots,b_n]", "f97823757917315730b1f5098ae66aa7": "\\mathit{s}\\,", "f9783f91eb2b4971e71e173d39a34c83": "\\beta=-\\pi/2", "f9788455591b0e863958316e095029e1": "u_s':=u_s-u_1", "f979155f087c931fb36123358aa0f5bc": " N \\!", "f9792e0d9d819d003fc4bb90b98a4f7e": "Tr\\,60 \\times 9", "f97967b6f537bc2ab1ed6ab1428e483a": " V_t \\, ", "f979799f0e2cb401a1ea0e9d5a6751d7": "\\begin{align} \\frac {d M_{xy}'(t)} {d t} & = e^{+i \\Omega t} \\left [-i \\gamma \\left ( M_{xy} (t) B_z (t) - M_z (t) B_{xy} (t) \\right ) -\n\\frac {M_{xy}} {T_2} \\right ] + i \\Omega M_{xy}' \\\\\n\n& = \\left [-i \\gamma \\left ( M_{xy} (t) e^{+i \\Omega t} B_z (t) - M_z (t) B_{xy} (t) e^{+i \\Omega t}\\right ) -\n\\frac {M_{xy} e^{+i \\Omega t} } {T_2} \\right ] + i \\Omega M_{xy}' \\\\\n\n& = -i \\gamma \\left ( M_{xy}' (t) B_z' (t) - M_z' (t) B_{xy}' (t) \\right ) + i \\Omega M_{xy}' -\n\\frac {M_{xy}'} {T_2} \\\\\n\n\\end{align}\n", "f979c3ccf193d8caade1246523885fe6": " e^{x}>x+1>x", "f979c91aefa8355260579f7425af5361": "y_t - \\rho y_{t-1} = \\alpha(1-\\rho)+\\beta(X_t - \\rho X_{t-1}) + e_t. \\,", "f979e738dc47bee9e414d5b2753e50ad": "\\mathcal{R}(C) = \\operatorname{P}\\{C(X) \\neq Y\\}.", "f979edd9e302fdbc7e7a4a67535b8d64": "S^1 \\subset S^m", "f97a01ce5ac079519a9bda4aa4d5a523": "s_f", "f97a58f81ae7442d9349db690a864c02": "\\mbox{AC}^0 \\subsetneq \\mbox{AC}^0[p] \\subsetneq \\mbox{TC}^0 \\subseteq \\mbox{NC}^1. ", "f97a7a84bf885cb77d22e916a8f8fcb1": "RP_m", "f97a86cf2cfc3aa5f93a4a84d150b862": "\\mbox{adult shoe size} = 3\\times\\mbox{last length in inches}-25", "f97aa2140513f71d93afd9f38f1486e7": "n \\times n", "f97b58f3238c960c954744ce9401f357": "k_{\\lambda}.v_0 = q^{(\\lambda,\\nu)} v_0", "f97c6a80339ab06cc89149d69bb64e62": "H_n(a, b) = a \\uparrow^{n-2}b\\text{ for } n \\ge 0.\\,\\!", "f97c77869d6597f6bba806205bf28e87": "[AFO]=[FCO]=[DBO]=[ADO]\\,", "f97c9739382d778f320b8122d616268e": "\\mathit{a(V)}", "f97cdd1a3939033d87caf7234bfdbbe4": "X : [0, + \\infty) \\times \\Omega \\to \\mathbb{X}", "f97cf877806e333c3efae1ea30110736": "\\partial_\\alpha {\\star F^{\\alpha\\beta}} = \\frac{\\mu_0}{c} J^\\beta_{\\mathrm m}", "f97cfc45f304d4c728b1b4e452670110": " \\sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} =\n\\begin{cases}\n0 &: i\\ne j \\\\\nN &: i=j=0 \\\\\nN/2 &: i=j\\ne 0\n\\end{cases} \\,\\!\n", "f97d54036b4945005dc5e662ed09d870": "n^{\\frac{n}{2 e} + o(n)}", "f97d9cbb062817aed031a5a03fa5b0e6": "\\scriptstyle q \\,\\in\\, IJ", "f97dd68791b6a3fa587a64a5fcda45fc": " \\phi_1, \\ \\phi_2, \\ \\dots, \\ \\phi_n \\vdash \\psi", "f97e17d3335633fa66aaad5530117f3c": "{}_1F_0(a;;z) = (1-z)^{-a}.", "f97e2906790d1ef141c001961d83efc8": "S_{RRB(n)} = 2\\cdot S_{RRB(n-1)} + 1 + S_{RRB(n-1)}", "f97e5259fd4a3add673aaac516717985": "[(\\frac{1}{2},0)\\otimes V]\\otimes[(0,\\frac{1}{2})\\otimes V^*]", "f97e60030616b7bc540cf3323375f9f2": "\\,A_x\\!", "f97e64a836d6442ffdc2b661112e3e64": " A=\\left(\\pi^2C\\rho_1\\rho_2\\right) ", "f97e69764baaddcba072ec0139dc5bfa": " C_{ijkl}=C_{klij}=C_{jikl}=C_{ijlk}", "f97e9a7241e334df3c368b4444244204": "Q(x)=x+1.\\,\\!", "f97f51c8fb792c168ee201630ce079d8": "\n \\dot{u}_i := \\frac{\\partial u_i}{\\partial t} ~;~~ \\ddot{u}_i := \\frac{\\partial^2 u_i}{\\partial t^2} ~;~~\n u_{i,\\alpha} := \\frac{\\partial u_i}{\\partial x_\\alpha} ~;~~ u_{i,\\alpha\\beta} := \\frac{\\partial^2 u_i}{\\partial x_\\alpha \\partial x_\\beta} \n", "f97f73d9a94556f51491074d4c9d6ff0": "abx^{a-1}(1-x^a)^{b-1}\\,", "f97ff4fd36d380959bfab3c2cbba103d": "f\\colon U\\to \\mathbb{R}", "f980a39d2ecf708edf2f48ee73daa1bd": "\\frac{v_m}{v_r}=-\\frac{p_b}{p_b-p_r}", "f980eec58e886f823e3d2f81dafa7380": "\n\\begin{alignat}{2}\n I_{d} & = f\\ (V_{GS},V_{DG})\n = \\begin{matrix} \\frac{1}{2}K_{p}\\left(\\frac{W}{L}\\right)\\end{matrix}(V_{GS} - V_{th})^2 (1 + \\lambda V_{DS}) \\\\\n & =\\begin{matrix} \\frac{1}{2}K_{p}\\left(\\frac{W}{L}\\right)\\end{matrix}(V_{GS} - V_{th})^2 \\left( 1 + \\lambda (V_{DG}+V_{GS}) \\right) \\\\\n\\end{alignat}", "f9810c21bb140be5809532e0d5ecf876": "Y=C\\cup D", "f9811b404dc84b68522a0ad89d3c363b": " D(\\alpha)\\;", "f981266f951708d716e2b6b6f17ee582": " \\left(S(0),I(0),R(0)\\right) \\in \\left\\{(S,I,R)\\in [0,N]^3 : S+I+R = N \\right\\} ", "f9812c7e2f10bd869747e23a77030423": "\\beta := \\frac{1}{kT}", "f981a7368662fbe3aa87a628b7ebd036": "\n \\boldsymbol{C}^e := (\\boldsymbol{F}^e)^T\\cdot\\boldsymbol{F}^e \\,.\n ", "f981cddd24971ec6b1b590fce8cfcd86": "f^{abc} = i\\epsilon^{abc}", "f98220910cc8fab04bd8eb1f49030dab": "g = f^{*} g'", "f982992ceacbcd0c924b38f7dd6e3a96": "c \\Leftarrow a;", "f982ac2729fb58b6cfdd2d6980f65d5f": "\\mathrm{Ref}_a(v) = v - 2\\frac{v\\cdot a}{a\\cdot a}a", "f982c176c914de122c361cc166229f1f": "\\mathbb{K} = \\mathbb{R}", "f982f88c48801cf97f349f4af795aa7b": " \\dot s = \\epsilon \\beta f s - \\gamma s ", "f982fc67c95a0443408071635eb450ed": " q_{ult} = 1.3 c' N_c + \\sigma '_{zD} N_q + 0.3 \\gamma ' B N_\\gamma \\ ", "f983196db2b78c6b2d583ed459591af2": "t_r = t_2-t_1 = \\tau\\cdot\\ln 9\\cong\\tau\\cdot 2.197", "f98319b74769321babb7ca80be08227e": "V_g f = V_g f * V_g g", "f983298a925b4faa89b98cf0721f0eec": "\\displaystyle \\mathbf{S}_t=\\mathbf{S}\\wedge \\mathbf{S}_{xx}. ", "f983bb44d988c1a4d4c03f706ac61019": "\\alpha_1(v_1)w_1 + \\alpha_2(v_2)w_2+\\cdots+\\alpha_N(v_N)w_N.", "f983d28c9d03dec68072b1d35ac2be1a": "f'(x_n) f''(x_n) (a - x_n)^2\\!", "f983da35c69a373921e89441892760dc": "\\scriptstyle S \\,=\\, \\sum_{i=1}^N X_i", "f98405592e3053c8997efe3750e15a83": "y \\preceq x", "f984282068a313d4fda7cd4735620798": " S_{\\text{free closed}} = \\tfrac{1}{2} \\langle \\Psi | (c_0 - \\tilde{c}_0) Q_B |\\Psi \\rangle \\ .", "f98440c0587db4aec3b717734fe6c762": "\\qquad P_1 V_1 = P_2 V_2.", "f9844c915ff2afc84d6ca9f7cfe8c584": "\\begin{align}\nf_{X_1^n}(x_1^n)\n &= \\prod_{i=1}^n \\left({1 \\over \\Gamma(\\alpha) \\beta^{\\alpha}}\\right) x_i^{\\alpha -1} e^{{-1 \\over \\beta}x_i}\n &= \\left({1 \\over \\Gamma(\\alpha) \\beta^{\\alpha}}\\right)^n \\left(\\prod_{i=1}^n x_i\\right)^{\\alpha-1} e^{{-1 \\over \\beta} \\sum_{i=1}^n{x_i}}.\n\\end{align}", "f98487e079a2d9e3b72b7282b2ce26bc": "\\mathbf J = \\sigma (-\\boldsymbol \\nabla V - S \\boldsymbol\\nabla T)", "f98586a525d29e8d9879174eef6b7542": " \\Gamma _l=\\Gamma _r; \\qquad \\delta x_{LP}=\\delta x_{PR} = \\delta x", "f98598ea528429441dc3759c1a5e8840": "\\therefore\\frac{df}{da}=-\\Im\\int_0^\\infty e^{-a\\omega}e^{i\\omega}d\\omega=\\Im\\frac{1}{-a+i}=\\Im\\frac{-a-i}{a^2+1}=\\frac{-1}{a^2+1} \\text{, given that } a > 0 .", "f985d65ac71232ecc4f92215ea4957c8": "[M] \\in H_n(M)", "f985f9c1e6206bd866769e4cd98bdbb3": "p_1,\\dots,p_K", "f9862a8e95d858c46295e7eeef5f70f9": "(ab)c\\ \\ d", "f9862e090da86ef6f8eeabbe301f9764": "U{}^{n-r}_n", "f98644155bc751e195b74efc9aca4608": "\\cosh x = \\cos {\\rm{i}}x \\!", "f9865210fa4e8ac2567c9959cc1fe390": " W_{N}=Q_{H}+Q_{C} \\,", "f9865f3c5eff97097957b50d740b6035": "X(t)=\\displaystyle\\mathcal{L}^{-1}[(sI+Diag(KJ)-K^T)^{-1}X(0)]", "f986ed65930b7630a728a861660a773e": "\\scriptstyle ab \\,+\\, 1", "f98707b627c740bfc00214f27d2c7165": "\\Theta(m)", "f9873768ef7f7dcf7e4382850697fd21": "\\sqrt{-0} = -0\\,\\!", "f9873a3df66b938341a9e293cb5eb382": "\\{c_t, c_{t+1}, c_{t+2}, ...\\}", "f987aaa7881e42d801848c246d8086bf": "S = \\sqrt{R^2 + H^2}", "f987c88881752887349fdf5ba4045de2": "\\mu_p(x)=1-x.", "f987d48e2e98f7fcff5b2df3bd857a6a": "b/a^2", "f987e7af8556243fa3047d3a1f5311b9": "f_L=2\\gamma", "f987ed30232718c8ce87ac4841707069": "C_k=\\sum_{I=(i_1 \\tau\\right\\}", "f98bdee5608a89ba3e8e54afa68b3a31": "0 \\leq t \\leq T", "f98bef3cf20d7cb31e3a35128adb199c": "\\theta,\\bar\\theta", "f98bf3861a64654645a7d773ff1a75e6": "\nF\\left(r\\right)=-kT\\ln P\\left(r\\right)-kT\\ln Z\n", "f98bf5d77bed9665b1c51aef8169b9f6": "\\lfloor x_{k+1} \\rfloor=\\lfloor \\sqrt n \\rfloor", "f98c0df48afe416f0e04a66e7492afc1": " t_{E^d} = - t_E ", "f98c2e0565c566ea308334aa9168a477": "a_{n}y^{(n)} + a_{n-1}y^{(n-1)} + \\cdots + a_{1}y' + a_{0}y = 0", "f98c305e14d11f080f78f42a5bc406f2": " \\kappa^{++} ", "f98c5632a2c96872b76af2bf2b5ca09b": "Sq^i \\colon H^n(X;\\mathbf{Z}/2) \\to H^{n+i}(X;\\mathbf{Z}/2)", "f98ce289df339024e99996b2bdb5eafd": "0 \\le x < 1", "f98d07d61eb023362f2dc60d81ed26cc": "\\int_a^b |f(x)| \\, dx < \\infty", "f98d0f1056f7159677ca3135c801cdfe": "\\liminf_{j \\to \\infty} \\langle T(u_{j}), u_{j} - v \\rangle \\geq \\langle T(u), u - v \\rangle.", "f98d1c627d813bbf8516fb83948159d4": "e^{s-x} f_i(x)", "f98d202f05aa0068c4cef04d17bbdf16": "E_{total} = \\mathrm{KE}_{system} + \\mathrm{PE}_{system} + U_{system}", "f98ded2f7f51bf20750fcb81a405a61c": "\\mathcal{I} \\mathcal{G} = 0", "f98e2a5a37f1f18cd32b6c2c5a6fd72d": "G \\times G \\to G, (x, y) \\mapsto x y^{-1}", "f98e34b2b5577f91036fee65b9b59894": "n\\geq2r", "f98ec7feb59c561e94c64f85e88e6889": "\\gamma : E \\to M", "f98ec996444165f0c8dd1a7defba1e75": "\nT = \\dot{m} ( r_2 V_{w2} - r_1 V_{w1} )\n", "f98f0b54d7f8d1a770f4df176ab255ee": "m_{t+1}", "f98f67da6c02468b64d533d225495171": "d\\mathbf{r}(s) = \\left[ dx(s),\\ dy(s) \\right]=\\left[ x'(s),\\ y'(s) \\right] ds \\ , ", "f98f70a02699a8dc66ca24680b935a33": "\\frac{d^3\\delta}{{dL}^3} = -\\frac{ \\left(8 a^3-4\\text{aL}^2\\right)}{\\left(4 a^2+L^2\\right)^{5/2} } ", "f98fbee7caf6ffe28057d20368ddb51a": "log(r)", "f99002359123036eefeb184c8f330c5a": "\\begin{align}\n \\mathbf{B}(t) = {} &\\sum_{i=0}^n {n\\choose i}(1 - t)^{n - i}t^i\\mathbf{P}_i \\\\\n = {} &(1 - t)^n\\mathbf{P}_0 + {n\\choose 1}(1 - t)^{n - 1}t\\mathbf{P}_1 + \\cdots \\\\\n {} &\\cdots + {n\\choose n - 1}(1 - t)t^{n - 1}\\mathbf{P}_{n - 1} + t^n\\mathbf{P}_n,\\quad t \\in [0,1]\n\\end{align}", "f9901f997e794cffb29b15bc8edc7ec0": "u_i(t)", "f9903a4d1dd38254ccfdf430fbd62253": "\n\\begin{align}\n\\operatorname{var}_*\\hat p_t & {} = \\frac{1}{K}\\operatorname{var}_* [1(X \\ge t)W(X)] \\\\\n& {} = \\frac{1}{K}\\left\\{{E_*}[1(X \\ge t)^2 W^2(X)] - p_t^2\\right\\} \\\\\n& {} = \\frac{1}{K}\\left\\{{E}[1(X \\ge t) W(X)] - p_t^2\\right\\}\n\\end{align}\n", "f9905f6d377e0d356775dd3a48d6f364": "c/H_0", "f9909961f8808a5e22f57965e3b69797": "\\# B< \\nu(W)", "f990d651b49217e3533b1aacd98e5bcf": "\\vec{F}^{\\sigma}\\,\\!", "f9929de90726e19bbdd8e4d84b4d223e": "\\xi^k_n = \\cos\\left(2\\pi \\tfrac{k}{n}\\right)+i\\sin\\left(2\\pi \\tfrac{k}{n}\\right)", "f992e2ce88c29e02fbf70519cb58f0d5": "\\scriptstyle \\gamma (W_n)\\, =\\, 4n - 5", "f992f2db11d259033399611bb70b8cd6": "\\sum_{k=0}^{n} ar^k = ar^0+ar^1+ar^2+ar^3+\\cdots+ar^n. \\,", "f9931cb545a31d5596ff774bbbc94fe1": "2.8238", "f993379d893621ddc2d6633055fb28e1": "ST = \\{st : s \\in S \\text{ and } t\\in T\\}", "f99430b15d9fa038082d3e231b3de675": "1+||y||", "f9948790b20f3383e653820209f32cb8": " P^{(k)}=\\sum_{i=1}^{k}U^{(n-i)}", "f994ba23f3fcafef5fddb5ec451ed1c6": "\n H\\, =\\, \n \\frac12\\, \\rho\\, \\sqrt{ 1\\, +\\, \\left| \\boldsymbol{\\nabla} \\eta \\right|^2}\\;\\; \\varphi\\, \\bigl( D(\\eta)\\; \\varphi \\bigr)\\,\n +\\, \\frac12\\, \\rho\\, g\\, \\eta^2,\n", "f994c62e1e0da0b91c9a92feb5f1adba": " wp(x := E, R)\\ =\\ \\forall y, y = E \\Rightarrow R[x \\leftarrow y]\\ ", "f9950a245ab22af8a5d96a7baf6f7289": "\\mathbb F_2", "f9952477791413f98f113bfcf2b91211": "\\Delta _G U", "f99533af7eba951e8546dbdf04e7bd45": "\\Pi^1_1", "f99542f34939772f8e66d9deb115f00b": "\\hat\\alpha", "f9959dbfccfb451f40864ac8fcaa895e": "A \\land (B \\lor C) \\iff (A \\land B) \\lor (A \\land C)", "f995ad3af494bc50cf33fcb03e84e569": "A = L^\\infty(\\mathbb{R},\\mu),\\phi(f) = \\int f(t)\\,d\\mu(t).", "f995f20a8ea0d109325c2caa64ab780a": "\\hat{\\mathbf{L}}[f]=\\mathbf{C}[f], \\, ", "f99611fa28f0a7a0e68eb75abb9a5baa": "x \\equiv a_1 n_2 [n_2^{-1}]_{n_1} + a_2 n_1 [n_1^{-1}]_{n_2}", "f996714391cc28f2843945fc99d1633e": " \\mathbf{A}\\times\\left(\\mathbf{B}\\times\\mathbf{C}\\right)=\\left(\\mathbf{A}\\cdot\\mathbf{C}\\right)\\mathbf{B}-\\left(\\mathbf{A}\\cdot\\mathbf{B}\\right)\\mathbf{C} ", "f9967e1528edbca3231a13ebfdaa0643": "\\!\\sigma_D = \\sigma_3 - \\sigma_1", "f996d8f60264f8410cd4279d1a94ba5f": "{13 \\choose 1}{4 \\choose 2}{12 \\choose 3}{4 \\choose 1}^3 = 1,098,240", "f9971ccc94ed807f95205eb9c06c79e0": "S(\\rho) \\,=\\,-\\mathrm{Tr} (\\rho \\, {\\rm \\ln} \\rho),", "f99732b1f8db3caaaa529a360a91a5b0": "(\\boldsymbol{X}, \\boldsymbol{Y})", "f997369fd9ead90da5e4baf32c801497": "e\\Delta V_{\\rm sp} = W_{\\rm s} - W_{\\rm p}, \\quad \\text{when}~\\phi~\\text{is flat}.", "f99737ab93116278630a0ce61d03993f": "\\frac{\\partial^2 U}{\\partial s\\,\\partial p}<0", "f9974b3ec60486c84de8d23bbd0d19e2": "\n\\binom nk,\n", "f997582de85bc1f87ba56ccc665a3b2d": " {t = n \\Delta t} \\,", "f9977c7f1822b49daf680f5cab2dafef": "([G]_{o}>>>1)", "f997cd282ba757bf5aa022ce2daf392f": "{\\frac{\\partial{S}}{\\partial{K}}} = const", "f9982c9f4a3ecbfe2da1eb516d8dca07": "[L_l, L_m ] = i \\hbar \\varepsilon_{lmn} L_n", "f99875504917b4ba80bb224a2a6d98c8": "x = y^{y^y}", "f998ab0f8dc06693c4ab1284d955e8fa": "t_{LL}^{\\mu \\nu} = - \\frac{c^4}{8\\pi G}(G^{\\mu \\nu}+\\Lambda g^{\\mu \\nu}) + \\frac{c^4}{16\\pi G (-g)}((-g)(g^{\\mu \\nu}g^{\\alpha \\beta} - g^{\\mu \\alpha}g^{\\nu \\beta}))_{,\\alpha \\beta}", "f998be8567b915fb8573b1f78b26ea03": " n = \\frac {\\log(FV) - \\log(PV)} {\\log(1 + i)}", "f998c0e7f2173f4a8d106ccf1e007029": "x_i = \\alpha z_i \\sin(\\chi/\\alpha) \\sinh(t/\\alpha) \\sinh\\xi, \\qquad 3 \\leq i \\leq n", "f998e42af0176be44851f0aa8844e50e": "3a \\over 2", "f9993ac0abf56e82e32c3d89be53e9d5": "\\frac{1}{x_1} +\\frac{1}{x_2} = \\frac{1}{f} \\,\\!", "f9995ff257e9a09c9131e5c7dabea460": "E_p^k", "f9997f359cc766567849b4f1ee64bd15": "O(e^n)", "f99a0f95b0026e33b02cf91f5f7e5435": " \\int_a^b \\omega(x)\\,h(x)\\,dx = \\int_a^b \\omega(x)\\,r(x)\\,dx. ", "f99a49932d8e2fd55ee97fa20e8f2ef8": "A/J_{k-1}", "f99a5302e8a72edba25e41960ec222b5": "U_{L, 0}/U_{L, 1} \\simeq l^\\times", "f99a73673399f9a09f1301512e6b632b": " 43 \\simeq 8.5 + 7", "f99ab9aac8468b317ed1dd05177fd2de": "a_n=a\\left(1+\\tfrac 12+\\tfrac 14+ \\cdots + \\tfrac 1{2^n}\\right)=2a-\\frac a{2^n}", "f99af8415bfcbad561ab5e7e63fa67c2": "K[X]/\\langle (X-\\alpha)^{m_i}\\rangle.", "f99b5dc1ef82c98853596281cabfa25c": "\\lim_{x \\to 0^+} c(x) = C_0 = 1", "f99c097126d442d334b28d9cabb46f0b": "A.B=\\overline{\\overline{A}+\\overline{B}}", "f99c342be39b86830797a8d6335430d6": "V(G_{N}) := \\{ 1, 2, \\dots, N \\}", "f99c53eec9e69038dc1c8ec1e3a6fb1b": "(z-w) ", "f99ca58cf521943a6bb97c582801b07a": "\n\\begin{bmatrix}\n 52 & 55 & 61 & 66 & 70 & 61 & 64 & 73 \\\\\n 63 & 59 & 55 & 90 & 109 & 85 & 69 & 72 \\\\\n 62 & 59 & 68 & 113 & 144 & 104 & 66 & 73 \\\\\n 63 & 58 & 71 & 122 & 154 & 106 & 70 & 69 \\\\\n 67 & 61 & 68 & 104 & 126 & 88 & 68 & 70 \\\\\n 79 & 65 & 60 & 70 & 77 & 68 & 58 & 75 \\\\\n 85 & 71 & 64 & 59 & 55 & 61 & 65 & 83 \\\\\n 87 & 79 & 69 & 68 & 65 & 76 & 78 & 94\n\\end{bmatrix}\n", "f99d1c3ec4149f3d4df3d1b4b3e7f8f9": "\\mathbf{y} \\in [\\mathbf{x}]", "f99d4aa81024c50f6240eadc85b9e3a0": "m\\ll M", "f99d563614d10b65a2f9ec55a4c7b567": " \\int_a^b f(x,y) \\sqrt{x'[t]^2+y'[t]^2} \\, dt", "f99dcd41e4985a8928dc9a36e7f1f22e": "\\mathbb{D}^q_t(fg)=\\sum_{j=0}^{\\infty} {q \\choose j}\\mathbb{D}^j_t(f)\\mathbb{D}^{q-j}_t(g)", "f99dd7b50c01fb88db4915aabedc781c": "R=E{x(t)x^T(t)}", "f99e89b6cf45d69191126d4b368be8f0": "\\mathbf{E}_n(z)=\\frac{1}{\\pi} \\sum_{k=0}^{[\\frac{n-1}{2}]}\\frac{\\Gamma(k+1/2)(z/2)^{n-2k-1}}{\\Gamma(n-1/2-k)}\\mathbf{H}_n ", "f99e8e51d2694ac441b6f5fc7ed9ce7c": "\n\\text{CPI} = \\frac{45000 \\times 1 + 32000 \\times 2 + 15000 \\times 2 + 8000 \\times 2}{100000} = \\frac{155000}{100000} = 1.55\n", "f99e96d7033e52f19978f7520de39ba9": "I_r = \\lim_{x \\rightarrow \\infty} f(x)", "f99e97d45f65f4593729aa4adb65c104": "\\text{rank}(A) + \\text{nullity}(A) = n.\\,", "f99ec035461850b4cc35306be0eb56b3": " \\int_a^b \\sqrt{1+f'(x)^2} \\, dx", "f99eeda4d8622c3daa84fc1ae6325fae": "\n(p \\quad <*\\!> \\quad q)(j) = \\bigcup \\{ q(k) : k \\in p(j) \\}\n", "f99f5745cf58676fc20c3ec60a53b8dd": "\n\\begin{align}\nax^2 + bx + c & = a(x - \\alpha)(x - \\beta) \\\\\n& = a\\left(x - \\frac{-b + \\sqrt{b^2-4ac}}{2a}\\right) \\left(x - \\frac{-b - \\sqrt{b^2-4ac}}{2a}\\right),\n\\end{align}\n", "f99f902fbfb1b82c2ed11cf024ae1955": "x_a=x(\\mu_a) ", "f99fe3d69675c77b45b7695eefd638f8": "A = 2\\pi \\int_{1}^{a} {1 \\over x} \\sqrt{1+{1 \\over x^4}} \\,\\mathrm{d}x \\geq 2\\pi \\int_{1}^{a} {1 \\over x} \\,\\mathrm{d}x = 2\\pi \\ln a.", "f9a01c77bd402c257eacdee13ea81ac0": "\\gamma'(t)", "f9a09bd44ebbd9f619c122a653090354": "\\alpha = \\frac{\\sqrt{2m(V_o - E)}}{\\hbar}", "f9a10463d20a4dea9650121ac9d452c2": " D_{\\mathrm{KL}}(P\\|Q) = \\int_X \\ln\\left(\\frac{{\\rm d}P}{{\\rm d}Q}\\right) \\frac{{\\rm d}P}{{\\rm d}Q} \\,{\\rm d}Q,", "f9a15fe3c8e1967ba4e79271f394cb2c": "2s.", "f9a16a6419724d12945335665cfab03d": "F_{\\nu}(w)", "f9a16b4e204ca4d7dd53ddccdcd2bf6c": "\\left(\\begin{smallmatrix}1 & 0\\\\ -V & 1\\end{smallmatrix}\\right)", "f9a1a0fd78ec12b2f1c8a9e4a474f8a9": "\\epsilon(x) = 1_{k};", "f9a1e3246cc1767cf0973a7838ce27a1": "K_m={eG + gE - 2 fF\\over EG -F^2}", "f9a2458e7fba8706fc83b56965edd627": "f|_{P_i} = f_i|_{P_i}", "f9a260983c25ad2f29c928532bb4c158": "I_2\\,", "f9a269c43ae218b1b06876c78ed48805": "T^{4}", "f9a28297ad2fbd8239d4559833cad958": " \\mathfrak{g} > [\\mathfrak{g},\\mathfrak{g}] > [[\\mathfrak{g},\\mathfrak{g}],\\mathfrak{g}] > [[[\\mathfrak{g},\\mathfrak{g}],\\mathfrak{g}],\\mathfrak{g}] > \\cdots", "f9a28cf02f38122024a2dd43608521b2": "\\sin 2\\theta=2\\sin\\theta\\,\\cos\\theta", "f9a2c3a41331948a6cc969302d1c0c60": "\\scriptstyle 3(1)(1) \\;=\\; 1^2 \\,+\\, 1^2 \\,+\\, 1", "f9a3092fe2eba3c587e25f1edcc7a69a": " 1 := \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix} \\quad ; \\quad x := \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix} \\quad ; \\quad x^2 := \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix} \\quad ; \\quad x^3 := \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\quad ", "f9a32eb44266cf75f44eb82a0b9a6965": " \\frac{ 3 ( \\mu - \\nu ) } { \\sigma } ,", "f9a33933319b6be43a25386c9714c70f": "V(\\mathbf{r}) =0", "f9a33a446849ee52710966030dc345bc": "u: A \\to C/N", "f9a3b8e9e501458e8face47cae8826de": "x_{1}", "f9a3c5eb2e1f763bd3196ecfaa8ac68d": "{\\overline{b}}=(b_1,\\ldots,b_n)", "f9a3f4c43aad62c92716ab1d012ef04d": "{\\Bbb C}^n\\backslash 0 \\times {\\Bbb C}^m\\backslash 0\\mapsto {\\Bbb C}P^{n-1}\\times {\\Bbb C}P^{m-1}", "f9a41d11acf7fb039c0d2396605ae5c4": " \\text{Re} ( \\alpha_{j} ) > 0 ", "f9a42e9b05c26ba4c5beddfe1234a36c": "\\mathrm{L}", "f9a448d96ec8a4d92611762b933323ec": "\\begin{align}(B\\to F)\\to F,C\\to F,B\\to C&\\vdash B\\to F&&\\text{by (1)}\\\\&\\vdash F&&\\text{by modus ponens,}\\end{align}", "f9a45d2b1c20d551b48536935c303aa4": "w'_2 = 1", "f9a4732aec9420a7e7ce378e9b0102a0": "m=\\frac{\\partial\\ln{\\sigma}}{\\partial\\ln{\\dot{\\varepsilon}}},", "f9a4837d128eea94fb6426fe5b0b0e48": "S_n(A)", "f9a493d056e0b07c7ae0a35b80b72415": "\\eta(s) = - \\sin(s\\pi/2) \\int_{0}^\\infty \\frac{t^{-s}}{\\sinh{(\\pi t)}} \\, dt.\n", "f9a49c312d6ab584ba03b1388e9b3239": "\\tilde{p}", "f9a4d188ba4ea0134ea18d58b3219760": "\n\\vartheta_4^2(q)=iq^{\\frac14}\\sum_{k=-\\infty}^\\infty q^{2k^2-k}\\vartheta_1\\left(\\frac{2k-1}{2i}\\ln q,q\\right),\n", "f9a4eab38c7e90b97f9e1cf7725c9c9a": "((500truckloads300miles)+(50truckloads600miles))0.654lbs(CO2/miles)=180,000 lbs CO2 emitted ", "f9a50fa054a538401e44eed04b36aded": "T_M(k) = W_1 \\cdot P(k) + W_2 \\cdot P(k + 1) + W_3 \\cdot P(k-1)", "f9a50fea0df9742c94003a7c6dac34f3": "q^2 = 2r^2, \\, ", "f9a517a907d07a669d5af3375888aa1b": "h_{Y} \\circ (\\mathrm{fmap} \\, k) = (\\mathrm{fmap} \\, k) \\circ h_{X}", "f9a57162e82d63e2271cbab1f8e7b430": "{{20 \\times 19 \\times 3} \\over 2} + 63 - 1 = 632", "f9a57381ae41dd8220b1641155273a31": "T_{\\mathbf{x'}}(\\mathbf{x})", "f9a57a34d4cbe92971bec7ad04a404fb": "\\frac{x}{m}=kP^{\\frac{1}{n}}", "f9a5b9bb6b3481fcad7c85da61d8321b": "[\\mathcal{L}_{K_1},\\mathcal{L}_{K_2}]= \\mathcal{L}_{[K_1,K_2]}", "f9a5d897b7262ff098ab8dc53f64b994": "L(A)=L(B)", "f9a61253f147768ed0686a9328618c6b": "P_{i}\\,", "f9a657d23da11cd6bdb54f6a0b3c1a8d": "\\alpha = 1 ", "f9a6a1b48b9daf84da369611e6b2cdbe": " \\frac{1}{\\csc \\theta}\\! ", "f9a6b0e5a131c1738dd4e022fdf13ff7": "y_p^{(j)}(x) = \\sum_{i=1}^{n} c_i(x) y_i^{(j)}(x) \\, \\mathrm{,}\n\\quad j=0,\\ldots,n-1 \\, \\mathrm{.} \\quad\\quad {\\rm (v)}", "f9a6c58a9be7afb1b849e2abfbafc02b": "\\mbox{hyp}_S f = \\{ (x, \\mu) \\, : \\, x \\in \\mathbb{R}^n,\\, \\mu \\in \\mathbb{R},\\, \\mu < f(x) \\} \\subseteq \\mathbb{R}^{n+1}.", "f9a6f530b776691311e78970fd43899d": "m_J", "f9a744698ed6731bf0cc209e9278d144": "\\text{dim}_{\\mathbf C} M_k(SL(2,\\mathbf Z)) = \n\\left \\{ \\begin{array}{ll} \\lfloor k/12 \\rfloor & k \\equiv 2 \\pmod{12} \\\\\n \\lfloor k/12 \\rfloor + 1 & \\text{else}\n\\end{array} \\right.", "f9a75eaf52e8ded91bfb94f8e6517f85": "K_n^{(\\alpha)}(x,y)=\\frac{y}{\\alpha+1} K_{n-1}^{(\\alpha+1)}(x,y)+ \\frac{1}{\\Gamma(\\alpha+1)} \\frac{L_n^{(\\alpha+1)}(x) L_n^{(\\alpha)}(y)}{{\\alpha+n \\choose n}}.", "f9a7ce48aedeeff18ce83b112e8ee638": "x = i \\cos(a t) - \\cos(b t) \\sin(c t)", "f9a7d963878019c77afafa58a2ea9022": " \n\\frac{\\Delta E}{\\hbar \\omega} \\approx -\\frac{\\pi}{2}\\alpha\\ - 0.06397\\alpha^2; \n", "f9a7fa9f6a13476a751d57d842481cd1": " r= {n^2 \\hbar^2 \\over m {e_M}^2 }", "f9a8107940cf7fd1c5e3b3d08c54deaa": " \\frac{\\Delta Y}{Y} = \\frac{V}{Y} . \\Delta K + \\frac{W}{Y} \\Delta L + \\frac{\\Delta Y'}{Y} ", "f9a85bd4923e27697abbca2f3c3d3163": "({\\mathbf x}-{\\boldsymbol\\mu})^T{\\boldsymbol\\Sigma}^{-1}({\\mathbf x}-{\\boldsymbol\\mu}) \\leq \\chi^2_k(p).", "f9a87d0509552d5d184586ef18703423": " \\alpha \\in \\Big[\\ \\hat\\alpha - s_{\\hat\\alpha} t^*_{n-2},\\ \\hat\\alpha + s_{\\hat\\alpha} t^*_{n-2}\\ \\Big] ", "f9a8914ad5d9105bf00e824aef2231cd": " \\theta[\\vec{X}]_{\\hat{p} \\hat{q}} = \\frac{1}{\\sqrt{2}} \\, f'(u) \\, \\exp (-2 \\, f(u)) \\, {\\rm diag} (0,1,1) ", "f9a9066362b2f90bc227c6b64ce66336": "P:= \\sum_{i \\in I} \\tilde u_i \\otimes \\tilde v_i ", "f9a913d1943577003e9ea9c8cc9646ab": "Y \\,\\!", "f9a94e8cb395c04b0d1b68f5529b7ae3": "L_q:N\\to\\mathbf R,\\qquad L_q(x) = \\operatorname{dist}(x,q)^2\\,", "f9a96c6503d0342ccea69d4375dacf81": "\\textstyle M_{\\mathrm e} = \\left( \\begin{array}{cc} \\frac12 & \\frac12 \\\\ \\frac14 & -\\frac14 \\end{array} \\right)", "f9a974ff64ca279e9f1f7195f053a7ae": "X_L = 2\\pi fL\\!", "f9a994a5996f6d0d265a578c3d314edb": "\\textstyle \\tau ", "f9a9ec0f39de3abcf8c95d0241602dc2": "2 - 2g - n", "f9aa61c83c34c090ab5949eadc0659a4": "-\\nabla^2 V = \\rho/\\varepsilon_0, \\, ", "f9aa686a4be09b8fb5e8a9fcaa784db6": "\\mathbb{Z}_4\\times\\mathbb{Z}_2", "f9aa706bc798e2619c9364e872451bce": " g_{n_1} g_{n_2} \\cdots g_{n_t} = 1", "f9aa74f6dd83fa1463c56a2caba45c32": "n = 2p+1", "f9aaa2073e64c4d7d28cf242393a8482": "s_n = 1-\\frac{1}{2^n}.", "f9aaa695fdf28f0b70a9e10627cd0585": "D^TB = B^TD.", "f9aaa8227fb79383e9484692be8ce505": "A_{n+1} - A_n = \\frac{g_n}{\\prod_{k=0}^n f_k}", "f9aaae181f5ec2d90477486e204e5516": "s={-1/k}", "f9aad3d9fd3d3f9f5e2eae7eee2b6599": " r_p^2 = (2Hl_p)^{ \\frac{1}{2}} = (Hct_p)^{ \\frac{1}{2}} ", "f9ab032cc20e1e0a7bba4993188ee22b": "x=(U,V,T(\\vec r))", "f9ab159f64b2a601d0788f86e9aad8c8": "\\mathrm {in\\, thermodynamic\\,equilibrium,\\,when}\\,\\,T=T_X=T_Y\\,\\mathrm {,\\,it\\,is\\,true\\,that}\\,\\,0=\\alpha _{\\nu , X, Y}(T, T)B_{\\nu}(T)\\, -\\, \\epsilon _{\\nu , X}(T) B_{\\nu}(T).", "f9ab5bab6b835258f38cd171df8b45de": "\\textstyle \\{ [\\sigma] : \\sigma \\in \\Gamma_{\\mathrm f}(V) \\}", "f9abc9c927db7d604443f4d8ef02a15f": "T / T_E", "f9abd264f160c448ac1347f4161ad54d": "{\\rm tr}\\left( \\frac{\\partial g(\\mathbf{U})}{\\partial \\mathbf{U}} \\frac{\\partial \\mathbf{U}}{\\partial x}\\right)", "f9ac21cbb502db0cab2d1a69e4714e64": "Q = {m} {L}", "f9aca59e815974b497f2c5fa9fd3bfff": "\\vert x\\vert\\to\\infty", "f9ad31f11ec21bb9dd1011751ca22a07": "\\dot{f}(t) \\to 0", "f9addf41ddaef567f920466b7c8883e2": "AP^{2} = Aw^{2} + wP^{2} = Aw^{2} + Az^{2}", "f9ae1d9770cac369348409af259a6ec1": " \\mathbf{B} = \\frac{\\mu_0}{4\\pi}\\iiint_V\\ \\frac{(\\mathbf{J}\\, dV) \\times \\mathbf{r}}{r^3}", "f9aecd184d7ed23018719442837f5e03": "\\scriptstyle \\left(1 \\,-\\, \\frac{it}{\\lambda}\\right)^{-k}\\,", "f9af00b777206b8791c4ab5fa95af40d": " t = 0 ", "f9af04909fc5b7547e5f8c48deb972cb": "A \\subseteq G ( A , B) \\subseteq G ( A , B)^{-}\\subseteq X\\backslash B,", "f9af5ebf805e0bd994050e012425ea67": " {\\mathcal S} =\\{f| \\sup_t |(1+t^2)^N(I+\\Delta)^M f(t)\\sinh(t)|<\\infty\\}.", "f9af6ee88e8fa0972efaa59ddcf16e45": "N_{M_j} = \\sum\\nolimits_{m_{j+1}} \\Pr\\nolimits_r \\left [V(w,r,M_j)=m_{j+1} \\right ] N_{M_{j+1}}.", "f9b0073d480490d4ebaa6d8b4e8f0e82": "\\sup \\sum_i \\|g(t_i)-g(t_{i+1})\\|_X < \\infty", "f9b06e6ee607a1d03ddc1774e8fd707c": "[\\mathfrak{m},\\mathfrak{m}] \\subset \\mathfrak{k}", "f9b0f57d8de7bc7120acb088f32d08ae": "( y, f(y) )", "f9b1339d6b4feff14cc70b40603e40c0": "P_{accel} \\,\\!", "f9b1a9513972da1ca1bbee91cb37c4f4": "\n \\begin{align}\n V_{qP}(\\theta) &= \\sqrt{\\frac{C_{11} \\sin^2(\\theta) + C_{33}\n \\cos^2(\\theta)+C_{44}+\\sqrt{M(\\theta)}}{2\\rho}} \\\\\n V_{qS}(\\theta) &= \\sqrt{\\frac{C_{11} \\sin^2(\\theta) + C_{33}\n \\cos^2(\\theta)+C_{44}-\\sqrt{M(\\theta)}}{2\\rho}} \\\\\n V_{S} &= \\sqrt{\\frac{C_{66} \\sin^2(\\theta) +\n C_{44}\\cos^2(\\theta)}{\\rho}} \\\\\n M(\\theta) &= \\left[\\left(C_{11}-C_{44}\\right) \\sin^2(\\theta) - \\left(C_{33}-C_{44}\\right)\\cos^2(\\theta)\\right]^2 \n + \\left(C_{13} + C_{44}\\right)^2 \\sin^2(2\\theta) \\\\\n \\end{align}\n", "f9b1e6ff483ebba10accf0ff79a6bc0d": "\\{ \\omega : X(\\omega) \\le r \\} = X^{-1}((-\\infty, r])", "f9b214acdaae46dc4b6f0854c8fc1ba4": "{\\partial v \\over \\partial t}=\\epsilon (\\beta u-v).", "f9b26279565f6c3b48ea4b068b683426": "\n\\text{Hg}_2^{2+} + 2\\text{Cl}^- \\rightleftarrows \\text{Hg}_2\\text{Cl}_2\\text{(s)},\\qquad \nK_{\\text{sp}} = a_{\\text{Hg}_2^{2+}} a_{\\text{Cl}^-}^2 \n", "f9b2c29380e0a3dc3c6469bcf491aa79": " L_*^{-1} =\n \\begin{bmatrix}\n 2 & 0 \\\\\n 5 & 7 \\\\\n \\end{bmatrix}^{-1}\n =\n \\begin{bmatrix}\n 0.500 & 0.000 \\\\\n -0.357 & 0.143 \\\\\n \\end{bmatrix}\n", "f9b2e21396a424ebc2f0e429cbdec68b": "W^{A}(z,x)", "f9b31ebe590ba8094dc384e81ee8138b": "R =", "f9b3291dd4f281874e6c047453937752": "T^m_n(V)", "f9b34791537d3759e4773c4a87a1a2ce": "S \\rarr T \\times S", "f9b34a2be3e92ab3dda6df70d20ea198": "x^{[n]}", "f9b37515f5b991327c9a49821a5d52a1": "(W',\\partial W') = (W \\cup( D^r \\times D^{n-r}),(\\partial W - S^{r-1} \\times D^{n-r})\\cup (D^r \\times S^{n-r-1}))", "f9b3c8fc881d8f1e1110ae996e21f020": " \\sum_{j=1}^{n_S} \\sum_{b_{ji}=0}^{a_{ji}} \\sum_{ \\beta_{ji} } x_{b_{ji}}\\ {_{a_j}^{b_{ji}}}\\text{S}_j^{\\beta_{ji}} + \\text{E} \\overset{\\xrightarrow{\\text{k}_{1(i)}}}{\n\\xleftarrow[\\text{k}_{2(i)}]{} } \\text{C}_i \\xrightarrow{\\text{k}_{3(i)}} \\sum_{h=1}^{n_P} \\sum_{ d_{hi}=0 }^{c_{hi}} \\sum_{ \\gamma_{hi} } u_{\\gamma_{hi}} y_{d_{hi}} \\ {_{c_h}^{d_{hi}}}\\text{P}_h^{\\gamma_{hi}} + \\text{E}, \\qquad \\qquad (2) ", "f9b424fae19fc4e99c47238416641f13": "m=(q^d-1)/2", "f9b4ee7390fa4c4fd1f32683811956ed": "d = -h \\ln(1-{p\\over h}-q)-{1\\over2}(1-h)\\ln(1-2q)", "f9b4ef950ecf5b3dc5ae65c850d1da8f": " p = (p_1,\\dots,p_m) \\in {\\mathbf p} ", "f9b52c78dbc63717d1e3c22c83ca9818": "a^{\\mathcal{I}} \\in \\Delta^{\\mathcal{I}}", "f9b5433ec63bccf3f957526b74adebbe": "h_o", "f9b5618b460064e362606bfd247b435a": "\\frac{\\partial P(x,p)}{\\partial t}=\\frac{-p}{m}\\frac{\\partial P(x,p)}{\\partial x}", "f9b56ba10fbbd8cc1b6df923512e8cc9": "r_1 = \\left( A / |C_0| \\right) ^{2/(N+1)} - 1 \\, ", "f9b5ecf9165463d01270363089dffc88": "\\begin{align}\n f(x) &= \\int_0^1 \\sum_{n=0}^\\infty B_n(x) \\tilde{B}_n(y) f(y)\\, dy\\\\\n &= \\int_0^1 f(y)\\,dy + \n \\sum_{n=1}^N B_n(x) \\frac{1}{n!} \n \\left( f^{(n-1)}(1) - f^{(n - 1)}(0) \\right) - \n \\frac{1}{(N + 1)!} \\int_0^1 B_{N + 1}(x-y) f^{(N)}(y)\\, dy\n\\end{align}", "f9b601562151954ad0f86c279a997938": "\\mathbf{X}' = \\overline{\\mathbf{X}} + \\mathbf{P} \\mathbf{b}", "f9b64f9695b1367a23e33c6d3fb257fd": " \\alpha\\to\\beta\\to\\alpha\\land\\beta ", "f9b653c7b4032aa1dddb67047f086f37": "Pf(x) = \\int_{\\mathbf{R}^d} e^{ix\\cdot\\xi} p(x,i\\xi)\\hat{f}(\\xi)\\, d\\xi.", "f9b663b48bf238a79a1756ec716ce1b6": "\\tilde{\\nu}_{0}", "f9b6bdf8a6a9f9c6647d6b28b9d274ed": "\n\\mathbf{M} \\ \\stackrel{\\mathrm{def}}{=}\\ I \\frac{d\\omega}{dt}\\mathbf{\\hat{n}} = \nI \\alpha \\mathbf{\\hat{n}}\n", "f9b6df9b22f28473b3c4579decc3f11a": "v_n(t+\\tau)= \\mbox{min} \\left\\{v_n (t)+2.5a_n \\tau (1-v_n (t)/V_n) \\left(0.025+v_n \\left(t \\right)/V_n \\right)^{1/2} \\right .,", "f9b6e7201438e2ee2acecd489d33eba4": "\\rho_R", "f9b70c7adaa03596ebfda0b9d96886e1": "\n \\tau^{\\mathrm{core}}_{xz}(x,z) \n = \\cfrac{Q_x}{2D}\\left[ E^c\\left(h^2-z^2\\right) + E^f f(f+2h)\\right]\n", "f9b76aa7280a371d4c3e38bf16fed453": "n,m > N, \nx_n - x_m", "f9b776e09abf36e5b31c3c4f154e8eab": "x_0 = t\\sqrt{2}\\frac{|\\mathbf{y}|^2}{1+|\\mathbf{y}|^2}, x_i=t\\frac{y_i}{|\\mathbf{y}|^2+1}, x_{n+1}=t\\sqrt{2}\\frac{1}{|\\mathbf{y}|^2+1}.", "f9b7ae4f8d135c5d21e16eb83ad90ef5": "{\\partial U}", "f9b7bc047513cfefdf7f2d4b443f69e0": "\\frac{\\partial \\theta}{\\partial z} > 0", "f9b7c22998ef98bdbf048a7c09fa3480": "\\beta = \\pm 1", "f9b7db2c7466632e79aa3d6045e26f10": "P(x_i|s)", "f9b80fc5d61a354040560987dbb7270f": " p = \\sum_k p_k X^k, \\quad p_k\\in \\mathbb{F}", "f9b8263467f6f24f092da52b50e9e218": "\\operatorname{bel}", "f9b8598632f2d77de780a1cc2a32c1a6": "y=a_0 \\sum_{r=0}^{\\infty} \\frac{(c)_{r}(c+1-\\gamma )_{r}}{(c+1-\\alpha )_{r}(c+1-\\beta )_{r}}s^{r+c},", "f9b8a30924287db9c886cb61227226e9": "L[y] =-(py')'+qy \\,", "f9b8f76cc2705ffd3bbde16d8abb2f72": "5^2:GL_2(5)", "f9b91067f2e2c198653c2dad6dc6fe74": "\\gamma \\in \\Gamma", "f9b9216de09d31020098753f096ea911": "\\frac{N}{2} < x \\le N", "f9b977c404733b0cc4521f2ebb876fb6": "\\Pi^S", "f9b9d727ed3d570e077b2b291cb5d221": " \\ln (\\Gamma(z)) \\approx (z - \\tfrac{1}{2}) \\ln(z) - z + \\tfrac{1}{2}\\ln(2\\pi).", "f9b9d99231c6e343b21fa7acf02ec7a1": "\\mu \\sim \\mathcal{N}(\\mu_0, \\tau_0),", "f9b9fc7609af73430e01755f021502c7": "A_r= 2 \\pi r \\ell", "f9ba5317ec45c5c67d74ceffd3375bea": "\\mu(D):=\\lim_{n\\to\\infty} \\mu_n(D)^\\frac1n", "f9ba7bd8169faca1f78821b6c245bd8d": " \\epsilon^2_t ", "f9bb03f3a17745f0edef0715f2daa321": " L(w, r\\,; q)", "f9bb70af966a4abbd08b776e6c5971ad": "\\ne", "f9bbb8526012485d1780fcfc73f1e147": "\\Sigma_{i \\in I}|A_i|", "f9bbc4849bc0e916c27b12e646e1060b": "4\\sin^2(\\pi/p)", "f9bcd756bfee617ea3d77a81b808db51": "A_{s}^3 = \\left(4 \\pi r^2\\right)^3 = 4^3 \\pi^3 r^6 = 4 \\pi \\left(4^2 \\pi^2 r^6\\right) = 4 \\pi \\cdot 3^2 \\left(\\frac{4^2 \\pi^2}{3^2} r^6\\right) = 36 \\pi \\left(\\frac{4 \\pi}{3} r^3\\right)^2 = 36\\,\\pi V_{p}^2\n", "f9bcfb3be94e4a7c1ef76ca8f280cf25": " {1\\over \\eta}\n \\begin{bmatrix}\n 1\\\\ { -{1\\over 2} - {\\sqrt{ 5} \\over 2} }\n\\end{bmatrix} \n ", "f9bd75fba193b5b631b5350d90bc2ac1": "\\forall i,x_i\\in S_i : f_i(x^*_{i}, x^*_{-i}) \\geq f_i(x_{i},x^*_{-i}).", "f9bd7a32342202754be2394200678b95": "L^0", "f9bda11c789b10546accd3a1f5d3edaa": " \\mathcal{F}(t) = \\bigcap_{t < s \\leq T} \\mathcal{F}(s),", "f9bdb01fbf6ab31d1ba7757fea6aef32": " p = 1", "f9be6eeeacf2e77ec95aa15da5175291": "[L_l, L_m ] = \\varepsilon_{lmn} L_n. \\!", "f9be8a2e402f28165a57ee0a273dbef9": "x_1^T \\omega x_1= 0", "f9bed6cf1642c5f5ac5831381c9935c4": "E_G=0.22", "f9bee2acb5d78d5b7631c41017d85ec0": " \\tau_y = r \\tau_x ", "f9bee5cfddef18d95047dd49906f53fe": "\\mathbf{r}(x,y,z)=\\nabla \\phi(x,y,z),", "f9bf0c09e5b4c371cef233b244da7d73": "f(x+y)=\\sum_{n=0}^\\infty \\frac{1}{n!} \\widehat{D}^nf(x)(y)", "f9bf7b32fa6a20df02bf91f3c7284f26": "\\begin{cases}\\dot{z}_1 &= L_{f}h(x) = z_2(x)\\\\\n\\dot{z}_2 &= L_{f}^{2}h(x) = z_3(x)\\\\\n&\\vdots\\\\\n\\dot{z}_n &= L_{f}^{n}h(x) + L_{g}L_{f}^{n-1}h(x)u\\end{cases}.", "f9bf7dc86408e3e52a8bd3d9326b6925": "n \\ge 40", "f9bf85d60663d314869ce353c23e6caa": "X=\\tilde{\\mathbf A}", "f9bfe325ba41802c532a90c4c5784f6c": " q = \\frac{\\bar{X}_A - \\bar{X}_B}\\sqrt{\\frac{MS_E}{2}(\\frac{1}{n_A} + \\frac{1}{n_B})}, ", "f9c0024f53113efd1beb2b777ace1340": "{{i}_{B3}}=\\frac{{{i}_{C3}}}{\\beta }", "f9c0122691abc374be8f02000591a667": "\\operatorname{ev}_c f = f(c).", "f9c01d7248b8885cce571dc986403741": "\\mathbf{Z}\\times \\mathrm{Sp}/(\\mathrm{Sp} \\times \\mathrm{Sp})", "f9c0524f1236075983e0347c2cd4dfae": " \\Gamma(s) = \\Gamma(s,0)", "f9c0a5225608eacd230303695af51891": "V_{S}^{(1)}", "f9c12833d38c1bbc665dba6c3e1cb599": "\\frac {S_{gas}}{Nk_B} \\, = \\, ln \\left (\\frac {k_BT}{P \\lambda^3} \\right) + 5/2 ", "f9c15dece6be9e25590f7f112e49cdbe": "\\mathfrak{rad}(\\mathfrak{g})", "f9c166161286b712f53d786ba1ee9d9a": "\\nu_\\text{max} = \\frac{\\pi}{2} \\bigg( \\sqrt{\\frac{\\gamma+1}{\\gamma-1}} -1 \\bigg).", "f9c17ae60402bf7a89fb4b6ae7649302": "\\begin{align}\n\\text{diameter} & {} = \\frac{abc}{2\\cdot\\text{area}} = \\frac{|AB| |BC| |CA|}{2|\\Delta ABC|} \\\\\n& {} = \\frac{abc}{2\\sqrt{s(s-a)(s-b)(s-c)}}\\\\\n& {} = \\frac{2abc}{\\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\n\\end{align}", "f9c17da0cc06545cc8b42e43040d3f96": "A(n, b - 3) + 3 \\ \\text{for } a = 2\\,\\!", "f9c1d0c7ad062c2970c3fe3ac846cc0c": "\\mathrm{\\delta ^{13}C} = \\Biggl( \\mathrm{\\frac{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{sample}}{\\bigl( \\frac{^{13}C}{^{12}C} \\bigr)_{PDB}}} -1 \\Biggr) \\times 1000\\ ^{o}\\!/\\!_{oo}", "f9c1f5508ef88d4fb19b3ee1ad2c8027": "\\sum^{\\infty}_{n=0} \\frac{\\prod_{k=0}^{n-1}(4k^2+\\alpha^2)}{(2n)!} z^{2n} + \\sum^{\\infty}_{n=0} \\frac{\\alpha \\prod_{k=0}^{n-1}[(2k+1)^2+\\alpha^2]}{(2n+1)!} z^{2n+1} = e^{\\alpha \\arcsin{z}}, |z|\\le1 ", "f9c20150284cfe2c33afc5938ebf05f5": " ((\\lambda x_1 \\ldots x_n.\\lambda c.c\\ x_1 \\ldots x_n)\\ v_1 \\ldots v_n)\\ f ", "f9c219e005b118d9112e40132700d766": "\n0 = \\int f_i^{*} f_j\n", "f9c2417083cd6b5cb4405c8d3ccc29e3": "l_0 = c^2/a_0\\;", "f9c265c2eb108132cc1edcb42caa0be1": " \\theta\\nu_t", "f9c30ed49694af745bb31c6126e59b64": "\\{(1,3),(2,1),(3,2)\\}", "f9c340648e746ce4f8ea6dde4e3538f9": "2_1", "f9c368329967b20ab563262185dc7d89": "\\ \\displaystyle \\mathfrak{U} \\ ", "f9c3bbdb2e4d895828974bc05b5769db": "\\mathbf{y} = \\mathbf{w}^T \\mathbf{z}", "f9c3c809b112730e8c83731e0b1ec05f": "\\begin{bmatrix}\n1 & 2 \\end{bmatrix}^{\\mathrm{T}}\n= \\,\n\\begin{bmatrix}\n1 \\\\\n2 \\end{bmatrix}\n", "f9c3de8048d0b531f6bae8fcce5ddceb": "\\vec n_0", "f9c4b75fcb7c8fb4a69ae6949a7d0f56": "\\sum a_n = \\sum (a_n + |a_n|) - \\sum |a_n|", "f9c4d68442d90c4fad7c167acb933779": "\na^{\\tfrac{p-1}{2}}\\equiv{x^2}^{\\tfrac{p-1}{2}}\\equiv x^{p-1}\\equiv1\\pmod p.\n", "f9c536ae191d7d3250ebe1be72a7637f": "Y_{\\rm monkey} = h - \\frac{1}{2}gt^2.", "f9c56b15fcc6bf0e078dadfe31822129": "t_{\\rm ff} = \\frac{1}{\\sqrt{G \\rho}} \\simeq (2 \\mbox{ Myr})\\left(\\frac{n}{10^3 \\mbox{ cm}^{-3}}\\right)^{-1/2}", "f9c575481b462a66ddb3c9e95653694a": "u \\notin V',W'", "f9c59e1a7ef2380e405b0a83bd3654f9": " \\Delta \\omega = \\omega_1 - \\omega_2 \\,\\!", "f9c613ead4ed065f14816636c1d4842e": " P=(X: Y: Z: ZZ)", "f9c6de09c54649ba77e4fe9d133f1567": "p \\notin l.Clause", "f9c6e9d9855aa793e5b9bafa346d66c0": "\\pi : \\mathrm{UT} (M) \\to M,", "f9c70dfe8e80d30f11b01a889ec8c0f3": "U = \\left\\{ z \\in H: \\left| z \\right| > 1,\\, \\left| \\,\\mbox{Re}(z) \\,\\right| < \\frac{1}{2} \\right\\}.", "f9c7134a036b82c00ef29d59d737d1a0": " \\Gamma_{a} ", "f9c714b72600db57141437d3c119443b": "\\mathbb{E}[f(X_\\tau)\\,|\\, X_0=x]", "f9c7fd50501b67fc704a9fcc786a6586": "\\left(\\sqrt{1/21},\\ \\sqrt{1/15},\\ -2\\sqrt{2/5},\\ 0,\\ 0,\\ 0\\right)", "f9c807a8b390d8e345d97af0f2ecb89a": "T = | \\hat{N} \\times | \\hat{D} \\times \\hat{N} | |", "f9c80e18126bb28f0bd4b4ad8ff0c221": "(d\\Psi_g)_e : T_e G \\to T_{\\Psi_g(e) = e}(G) ", "f9c851a8c47b9b95da6923e0c5e46096": "\na = \\xi \\cdot \\omega^2 = v \\cdot \\omega = \\frac{p \\cdot \\omega}{Z} = \\omega \\sqrt \\frac{J}{Z} = \\omega \\sqrt \\frac{E}{\\rho} = \\omega \\sqrt \\frac{P_{ac}}{Z \\cdot A}\n", "f9c85a8efd49b32b4c8a0baa03c23a58": "\\Delta\\mathbf{w}_{n-1}", "f9c877f07e186ab70abfb371db4632df": "\\scriptstyle \\vec{p}", "f9c8897b8593dd9cf2cff9bdad640d93": "GF(q^m),", "f9c896ea34d8ede8a3b71f9042818c96": "K(A, i) = K(K(A, i-1), 1)", "f9c911b1750df1715f770f08ce960e63": " E_{con} ", "f9c955f8523bbe79553c7f9b4c08c527": "P_n(k,\\rho)=J_n(k\\rho)\\,\\,\\,\\,\\,\\,\\mathrm{or}\\,\\,\\,\\,\\,\\,Y_n(k\\rho)\\,", "f9c961ba9cce67aedf55ffce232a74d6": "\\Gamma_{e}(0) = \\gamma\\Gamma_{i}(0)\\qquad\\qquad (4)", "f9c98c21362df487e9331c5ac0d4727c": "A_1\\oplus A_2=\\begin{bmatrix} A_1 & 0 \\\\ 0 & A_2 \\end{bmatrix},", "f9c9b0ab6f98ad0c8bea0ad607e40bc3": " \\mathbf{X}^T\\mathbf{X} ", "f9ca296f019ad9dea9b21c22a45e4849": "\\frac{n}{(n-1)}", "f9cad314e69500bddea3edc9826b5cc1": "M_{CD} = 0.4 \\times 5.785 - 12.5 = -10.19 ", "f9cad6320980fa52b7d090f682ecc7d1": "\\lambda=2", "f9cae2249040ae89e4b131e35884d6d9": "\\{u,v\\} \\in E", "f9cae2ef62bce4f8cc614a185d9b347c": "(a_1,\\dots,a_n)\\land(b_1,\\dots,b_n)=(a_1\\land b_1,\\dots,a_n\\land b_n).", "f9cb5ab906b7c85bea4cd7f26ac5b323": "71^2", "f9cbd9abac4e54cb6ee8e3a1a143a841": "q^n - 1 = q - 1 + \\sum {q^n - 1 \\over q^d - 1}", "f9cc1014b9240b337e9315c68325d2ae": "m(\\rho) = \\kappa(A,Q_1)\\cdot\\nu(\\rho)\\cdot\\kappa(K_M,A)", "f9cc3bcad95b487c3a0b779a6846cf14": "\\varphi\\left(\\int_a^b f(x)\\, dx\\right) \\le \\frac{1}{b-a} \\int_a^b \\varphi((b-a)f(x)) \\,dx. ", "f9cc75d7bcc909d43b0d59cfd3415089": "Z_\\text{P} = \\frac{V_\\text{P}}{I_\\text{P}} = \\frac{\\hbar}{q_\\text{P}^2} = \\frac{1}{4 \\pi \\epsilon_0 c} = \\frac{Z_0}{4 \\pi} ", "f9cd397b063da10109c9df1c3eb8d59f": "\\;_2F_1(a,b;c;1) = \\frac{\\Gamma(c)\\Gamma(c-a-b)}{\\Gamma(c-a)\\Gamma(c-b)}", "f9cd3a1cf4bc72eaaf86f718e3474261": "|\\downarrow\\rangle", "f9cd6552224a13d627d14294c665b3c0": "r + b = 2j", "f9cd8ee37b4fcf7075e90c73ac4fbd76": "\\operatorname{E}(S) = n\\sigma^2 + n\\mu^2 - \\frac{n\\sigma^2 + n^2\\mu^2}{n}", "f9cdae9f3ec2114bfbebbf4b46cf66b4": "F(\\{a_i\\},\\{\\alpha A_j\\}) = F(\\{a_i\\},\\{A_j\\}).\\,", "f9cdfbd65033601699aafa64193403ad": " SubCipher_n ", "f9ce37023a6012db65a10f3cce3e6970": " \\text{esssup}(X)", "f9ce532cbf2fc37a57910e2b17e09a99": "1/\\gamma^{4}", "f9ce81c2c25e16e9da82c303388064d4": " (\\lambda I - P)^{-1}= \\frac 1 \\lambda I+\\frac 1{\\lambda(\\lambda-1)} P", "f9ceb6eab578210bfb149d6394e2d363": "\\frac{\\partial \\mathcal{L}}{\\partial A_{\\alpha}}= - J^{\\alpha} \\,.", "f9cf4212a0ec705fd06a73f7fc3c3054": "\n\\mathrm{B}(x,y) = {\\Gamma(x)\\Gamma(y)\\over\\Gamma(x+y)},\n", "f9cfab579e9b3c95aca8343198978418": "(ax)^p=ax^p,\\,", "f9cfef5fcac8df601cff78495a966bf5": "\\sum_{k=0}^\\infty \\frac{(-1)^kE_{2k}z^{2k}}{(2k)!}=\\sec z, |z|<\\frac{\\pi}{2}\\,\\!", "f9d0d48a0cd4f9a342403e13a61bb92d": "\\frac{1}{\\pi\\gamma\\,\\left[1 + \\left(\\frac{x-x_0}{\\gamma}\\right)^2\\right]}\\!", "f9d10734c0e28208c6211133d161ea75": "\\alpha (\\omega ) = \\alpha _0^d \\frac{{\\hbar \\omega }}{{E_0 }}\\left( {\\frac{{\\hbar \\omega - E_g - E_0^{(d)} }}{{E_0 }}} \\right)^{(d - 2)/2} \\sum\\limits_k {\\Theta (\\hbar \\omega - E_g - E_0^{(d)} )A(\\omega )} ", "f9d167404f49fe136a592a27a5b82c66": "\ny\\rightarrow \n-y^3 + 3 y x^2 - y z^2 + y_0\n", "f9d19fc83e401808dc20e7cbf2b15796": " \\mu_i(x_i) ", "f9d1b57bf56f241d6c5dbbe714ece1fb": "= \\frac{1 + z^{-1}}{(1 + 2RC / T) + (1 - 2RC / T) z^{-1}}. \\ ", "f9d1cd401e77372d0564b27b44e871ce": "V - E + F", "f9d1e1007ea87885de442bf9a53b14da": "P = \\{x \\mid Ax \\ge 0\\}", "f9d21d99c2c6ad3c56b4f8289d92bf35": " \\left( \\Delta_\\perp -\\frac{1}{\\rho^2} \\right)u(\\mathbf{r}_\\perp) = -f(\\mathbf{r}_\\perp) ", "f9d23996f9d51e9d7f225a5d8b8f2dbb": "3 \\times 729 = 2187", "f9d2b73703ad6541432b344ea9bb7b7f": "g = 2", "f9d2efbd66ddb3cecf4c985bf324a3fe": "A = 1 + \\frac{u^2}{16384} \\left\\{ 4096 + u^2 \\left[ -768 +u^2 (320 - 175u^2) \\right] \\right\\}", "f9d321b21fce9ae6474ac91f05f88026": "\\operatorname{PP}(\\tan(z); z = (n + \\frac{1}{2})\\pi) = \\frac{-1}{z - (n + \\frac{1}{2})\\pi}", "f9d3389cd639a3ed5dd430cc571921cc": "[t_l, t_u] \\subset \\mathbb{T}", "f9d33df6c9e57283667d275df772f2c6": "P(\\alpha_i) = y_i", "f9d34f5d620fead9b57e7a6991523947": " (x * u_1)(t) = \\frac{dx(t)}{dt} ", "f9d35139c49accab69a9fe4de3ad9d6a": " \\langle \\lambda\\cdot Ux-U(\\lambda\\cdot x), \\lambda\\cdot Ux-U(\\lambda\\cdot x) \\rangle ", "f9d39ed0a900f80e7b4e07824872c449": "{\\varepsilon}", "f9d3a56078804af0729cdfc12bf298f3": "\\omega_r = \\sqrt{k/m}", "f9d3f9fc672dee8195d2f74bf0d8165a": "(MnSO_4)", "f9d43e9c7dfb41ce218b1980f8175582": "\\mathbf{x}(t) \\in \\mathbb{R}^n", "f9d4439666ec8870376bf6f0b6124eec": "SS_\\text{Total} = SS_\\text{Error} + SS_\\text{Treatments}", "f9d4487998930fe065a21cb0aaef9916": "POT", "f9d452d4f814df707a16d144121f1b72": "\\{\\{1\\},\\{2\\},\\cdots,\\{k\\}\\}", "f9d45bd02a599cf3f6305a6c26830e7a": "\\{x(d)\\}", "f9d4b298abc347dc1286073650b28da3": "L={E_1 E_2 \\over S}", "f9d4d6ca2ccdabc6be2323d6527a8dcf": " \\exp(aj) exp(bj) = \\exp((a+b)j),", "f9d504feecf9ab2fe8a34fb700de47f9": "r = \\frac{p \\cdot q}{2\\sqrt{p^2+q^2}}.", "f9d51d43dc5eadebfaaf9158e518534c": "A=\\varepsilon c \\ell", "f9d522b8622188b66ae6d994a0de4ee3": "{{f}_{T}}", "f9d5c867e6756a11893d02c9056fd3e8": "m_1=\\frac{a+b}{2}, \\,\\!", "f9d5db0903ae39b6402017f891af3efd": "\\tau = T, D = 1", "f9d5dbb0b25f004a1066bfaa07d55ed4": " P_{absorb} = \\frac{E_f}{c} \\cos \\alpha ", "f9d5f6764e023c6d883e7e0928b46ce0": "(x^2-s)(x^2-t)", "f9d636b01f14921beafa6a52ba1d4d6b": "\\bar{A}_n^k", "f9d63a2424cbe03512cf763715d7b89c": "V_\\mathrm{S}^{(2)} = \\sqrt \\frac{\\mu_\\mathrm{sat}^{(2)}}{\\rho^{(2)}}", "f9d63b9088afe038a26ca43abef8f98c": " \\phi \\in C(\\bar{\\mathcal{S}}) \\cap C^1(\\mathcal{S}) \\cap C^2(\\mathcal{S}\\setminus \\partial D) ", "f9d65ad7a42c6d7d26a1cbecd11cf9c2": " \\theta \\approx 1.22 \\frac{\\lambda}{d}", "f9d6bb4610b0da42ad359cd09303ee35": "PoS = \\frac{\\max_{s \\in S} W(s)}{\\max_{s \\in E} W(s)}", "f9d72396f88567b7f2be71d193a74896": "L_2]", "f9d77e3d6d518799ffaabde98d76a275": "\\mathbf{E}_0", "f9d7977ce6eddef4f65792009e0dcf6b": "\\operatorname{plus}\\ m\\ n\\ f\\ x = m\\ f\\ (n\\ f\\ x) ", "f9d859f73cd37ca27362596046c4a64e": "\\mathbf{L} = \\mathbf{r} \\times m \\mathbf{v} ", "f9d873bd1e9490ad778e9e30c8019758": "f_t(\\text{A}) = f_t(\\text{AA}) + \\frac{1}{2} f_t(\\text{Aa})", "f9d8ba706073e30729a1cef83a38e959": " \\mathbf M(x)= \\int V(x)=-5x^2 + 75x (kN \\cdot m) ", "f9d8bcd218a1dcbd52b0bde87490f90d": "d=2+\\epsilon", "f9d945820dc28e229afb621a2cae9f0d": " a_1 (1) + a_2 ( x + 1) + a_3 ( x^2 + x + 1) = x^2 - 1. \\,", "f9d9481dd6d40ef28c6e75b61b87a97f": " \\sum_{j=1}^n \\left| \\gamma_j \\right| \\leq M + 1 ", "f9d95da3d6b909c36b3cf10391b4c9e9": "\\delta=(\\mathbf{k_1\\cdot r - k_2 \\cdot r}+\\epsilon_1-\\epsilon_2)", "f9d95eb6fad2788f4bf381d88ba57e76": "P^{2m}(R)=E[Pr[\\sigma(x)\\in R]]\\leq max_{x\\in X^{2m}}(Pr[\\sigma(x)\\in R]),\\,\\!", "f9d9736d02fdf6728ddb74e0be59cf7f": "N\\equiv0\\pmod{b^k-1}. \\, ", "f9d98281681511425e1a4c883cb21d1c": "\\Omega(x)=S(x)\\Xi(x)\\bmod x^6=\\alpha^{-4}+\\alpha^4x+\\alpha^2x^2+\\alpha^{-5}x^3.", "f9d99e1c8f677ac36c5805fb68e517f9": "\\begin{matrix}{4 \\choose 1}{52 - 4r \\choose 3}\\end{matrix}", "f9d9c476e52312c739d6c01b639f55c4": "\n\\mathcal{F}^{-1} \\left \\{ \\mathbf{X^* \\cdot Y} \\right \\}_n\n= \\sum_{l=0}^{N-1}x_l^* \\cdot (y_N)_{n+l} \\ \\ \\stackrel{\\mathrm{def}}{=} \\ \\ (\\mathbf{x \\star y_N})_n\\ .\n", "f9da6970fd7109e86d5eeb7f89b59236": "A[1]^n=A^{n + 1}", "f9dace50b26fcf9cac4bb29c1e1d53f6": " (\\mathcal{P}(X),\\Delta,\\cap) ", "f9daebeddad1a8e74747602507e9f155": "0,1,2,\\ldots,(p-1)", "f9daf55e8455712b0902b3efa77b3f10": "G_\\mathrm{linear}", "f9dafbd93009193efb18d1f00883b007": "\\langle \\rho_{AB}, E_{AB} \\rangle = \\langle \\rho_{AB}, I_A \\otimes \\mathcal{E}^{\\dagger} (|\\phi^+\\rangle\\langle \\phi^+|) \\rangle", "f9db775694639aaa61c40a6d107a952d": "H(f)(y) = \\mu\\{x : f(x)\\le y\\}", "f9dba3a67810105719a48616d5b625dc": "2w + 2\\ell", "f9dba49a78ca1045078c9b39a1edce4f": "\\frac{{6 \\choose 3}{43 \\choose 3}}{{49 \\choose 6}}\\approx\\frac{1}{56.7}", "f9dbb225d29f5e3790f26401d0622bcf": "{(\\nabla S_0)}^2= E-V", "f9dbc08bb88d2890f7aa53ac191dae5f": "\n\\lim_{x^0\\rightarrow-\\infty}\n\\int \\mathrm{d}^3x \\langle\\alpha|f(x)\\overleftrightarrow\\part_0\\varphi(x)|\\beta\\rangle=\n\\sqrt Z \\int \\mathrm{d}^3x \\langle\\alpha|f(x)\\overleftrightarrow\\part_0\\varphi_{\\mathrm{in}}(x)|\\beta\\rangle\n", "f9dc2d3cc52b906a0c34b764c7368234": "S = u^\\alpha \\gamma^{\\alpha\\beta}_5\\overline{u}^\\beta,", "f9dd205f3cf0cfc01702c6acd72bdb1d": "\nPoss(a,s) \\rightarrow \\left[ broken(o,do(a,s)) \\leftrightarrow a=drop(o)\\wedge fragile(o)\n\\vee broken(o,s) \\wedge a \\neq repair(o) \\right]\n", "f9dd990630a142c287edb8f29e709749": "S_0 = \\{0,1,1-i\\}", "f9ddae683dedee4573fe2358bf15e34c": "\\frac{1-\\frac{1}{4}(\\frac{1}{2}-\\epsilon)}{1-\\epsilon} = \\frac{7}{8} + \\epsilon'", "f9ddb78de4dfad4efc2e70ab5cfb7a14": "\\qquad \\qquad \\ \\ \\ \\ \\ \\ \\ \\ \\ \\approx \\langle\\varphi\\rangle_\\mathrm{o} + \\frac{1}{2}\\sum_{i,j}\\sum_{\\alpha,\\beta}\\Gamma_{\\alpha\\beta}d_{i\\alpha}d_{j\\beta}, ", "f9ddba9329a7fe5944b7cfa702b2e2cd": "\nP(t) = I(t) \\cdot V(t) \\,\n", "f9ddeabf995912406c0ef0b55d317aa9": "\\lambda = 1.0", "f9ddefcf77e5f720ddddfe68a41f9e39": "c = 8.5", "f9ddfdbf5e125d0a2f9556bb34810433": "b(K) := \\left\\{ \\ell \\in X^{\\ast} \\,\\left|\\, \\sup_{x \\in K} \\langle \\ell, x \\rangle < + \\infty \\right. \\right\\}.", "f9de02caa1f2554fdf543f273a4e3b5e": " (x,\\Delta) ", "f9de92ead6b28095d8cd89d6d0ab3ae1": "C_n= \\frac 1{n+1} \\sum_{i=0}^n {n \\choose i}^2.", "f9ded821994113155fde74707db6e597": "\\frac{Q}{2C}=\\frac{V}{2}", "f9df28982b00f31660ac751b7238bb8f": "w_k(x) = \\frac{1}{\\sum_j \\left(\\frac{d(\\mathrm{center}_k,x)}{d(\\mathrm{center}_j,x)}\\right)^{2/(m-1)}}.", "f9df310c40a83d924d6a43275c252040": " \\frac{\\partial V}{\\partial x}+ \\frac{1}{B_m}\\frac{\\partial P}{\\partial t}=0\\, ", "f9df4dc22bb9c1d6d8b1fee35bb378ce": "\\ln(E) = 0.0235 S - 0.6403 ~;~~ S = \\begin{cases} S_A & \\mathrm{for}~20 < S_A < 80 \\\\ S_D + 50 & \\mathrm{for}~30 < S_D < 85 \\end{cases}", "f9df67b5c0222d11e42151dcca790d31": "n_{rel}", "f9df8d14acde9cb84a9cf0e1ec0fe85f": "(p_n,\\,p_{n-1},\\dots,\\,p_2,\\,p_1)", "f9dfabdce87d4c38f46d93b286a1890e": "F_{if}\\,", "f9dfc6095a1ea10aa3111cd6dd1d9d22": "i[j+1]", "f9dfddf675af2932b8ae01c11c1fa500": "\\mathbf J = -\\sigma \\boldsymbol \\nabla V - \\sigma S \\boldsymbol \\nabla T", "f9e0a730f1ee59824f115e1aec92d65b": " \\ \\psi(0) = 0 ", "f9e0e7dc9b14d9cd5ae63d2f45a34543": "[c,bab(ab^{-1})^2(ab)^3]=(bc^{bab^{-1}abab^{-1}a})^3=", "f9e143b464e81340920d073b6e3f9570": "\\Delta_{\\mathrm{LB}} = \\frac{1}{f^2} \\Delta_0", "f9e16d60ec73914a53e7c8322346ba20": "Q = \\sqrt{2DK/h}, ", "f9e16d7c9cee9eb5a0e12e789138c5e8": "\\delta = 1.22 \\frac {\\lambda}{D} \\ , ", "f9e183b5b45d7f08f715cc556b7b532f": "\nw = \\sum_{\\mathrm{cations\\ C}} P_{\\mathrm{C}} \\left[ \\mathrm{C}^{+} \\right]_{\\mathrm{out}} + \n\\sum_{\\mathrm{anions\\ A}} P_{\\mathrm{A}} \\left[ \\mathrm{A}^{-} \\right]_{\\mathrm{in}}\n", "f9e1c6700ee8243fa07c61a1c86d09ca": "f(x,q_{k+2}) \\leq r_{k+2}\\,\\!", "f9e1dac5e177e768735c0caaa2c058e3": "N_n(k) = \\frac{d\\Omega_n(k)}{dk} = n\\text{ } c_n \\text{ } k^{(n - 1)} ", "f9e1eae65ea93acca3c7d33acfff7557": "g * h = g \\cdot h", "f9e23f8a626c5c3fd4e5865c55eb2f22": "\\nu_0=c/ \\lambda_0", "f9e240cd1be3947c2c8d26bb69fb49d8": "\\mu - \\mu ^\\circ = \\int_{P^\\circ }^P {\\frac{{RT}}\n{P}dP} = RT\\ln \\frac{P}\n{{P^\\circ }}", "f9e25bb94dc3782355bd38257c08a9f3": "L \\approx C - H - kT", "f9e2626ca2dec7652f362c3f91a3cea4": "\\Delta Y = Y_2 - Y_1 = \\frac {\\alpha} {1 - c_1} \\left ( 1 - c_1 \\right ) = \\alpha ", "f9e271666256d450258110177ed6c00a": "\\textstyle \\beta_6", "f9e287ab7524495ad27e8a15871bdd5a": "f(\\lambda z) = \\lambda^k f(z)", "f9e2a5285ba7d79ffcb041c179c8429e": "\\frac 1{r(\\mathbf{h},q)} \\Bigg| \\sum_{n=0}^{N-1} e(\\mathbf{h}\\cdot \\mathbf{t}_n)\\Bigg|", "f9e2badf269c0431d9cdb3798d904de6": "T_{l}=\\frac{L}{c-v}+\\frac{L}{c+v}", "f9e2bd97b83d007eecb4e19e270e5568": "\\text{FO}(V_k,+)", "f9e2c2820eb3a14276ead2a90ad891d8": "\\mathbf{A}^{(i)} = \\mathbf{L}_{i} \\mathbf{A}^{(i+1)} \\mathbf{L}_{i}^{*}", "f9e2c7fa010946df584007c557d7af8d": "Tr(h)=h + h^{p^2} + h^{p^4}.", "f9e2dc4a656f87d800bd17f7434f5508": " B(y;b,c,p,q) = \\frac{y^{p-1}(1-(1-c)(y/b))^{q-1}}{b^{p}B(p,q)(1+c(y/b))^{p+q}} ", "f9e2f2971a165467888dac33fcbd325f": "v({\\mathbf P}_1+ {\\mathbf P}_2+{\\mathbf P}_3+ {\\mathbf P}_4)=\nv({\\mathbf P}_1)+v({\\mathbf P}_2)+v({\\mathbf P}_3)+v({\\mathbf P}_4)", "f9e31224a4fd84857554e313a3a5610c": "\\{1,2,\\cdots,n\\}", "f9e3d8ace2b0631df3bd17a579d41d99": "[a,b,c] = abc-acb-bac+bca+cab-cba. \\, ", "f9e49993b37c984db007baba43ab1a15": "\\frac{1}{{Z_\\mathrm{i\\Pi}}^2}={Y_\\mathrm{i\\Pi}}^2=Y^2 + \\frac{1}{k^2}", "f9e4c8f589df18b24e6985fe1f2ec46d": "2a\\sigma", "f9e4ec310929b70deef2aeb0e5573bda": "\\frac{\\tan{A}\\tan{B}-\\tan{C}\\tan{D}}{\\tan{A}\\tan{C}-\\tan{B}\\tan{D}}=\\frac{\\tan{(A+C)}}{\\tan{(A+B)}}.", "f9e517ff3081e3f38373ba026bf2b620": "\\rho_{1}", "f9e5822981973cd87e3e1a2bc2b4703c": "(a_1,\\dots,a_n) = {m^\\top V^{-1} \\over (m^\\top V^{-1}V^{-1}m)^{1/2}}", "f9e5af1bc866dfa1b05ea0d56d39a109": "\\langle v \\rangle ", "f9e5fa2b56a829f7b0a306095cc13d16": "K_C = 2\\pi\\, \\delta.\\,", "f9e6c97ad412656a69d331c7167b91dd": " (\\forall x f\\ x = g\\ x) \\equiv f = g ", "f9e6cd2a32fb13c7e8f21c1884bef877": "E[U(W_T)|\\xi_{[T-1]}]", "f9e741d3206f5e8aeb6480fe334950ff": "\\ \\ t'=t,\\ \\ x' = vt + x \\!", "f9e79cf8de8afbc0d834346a57529ffb": "\n \\Gamma^k_{ij} = \\frac{g^{km}}{2}\\left(\\frac{\\partial g_{mi}}{\\partial q^j} + \\frac{\\partial g_{mj}}{\\partial q^i} - \\frac{\\partial g_{ij}}{\\partial q^m} \\right)\n ", "f9e7a54342ae2a5504953eb64bfa593e": "L^{p_0}(\\mathbf{R}) \\cap L^{p_1}(\\mathbf{R}) \\subset L^p(\\mathbf{R}) \\subset L^{p_0}(\\mathbf{R}) + L^{p_1}(\\mathbf{R}), \\ \\ \\text{when} \\ \\ 1 \\le p_0 \\le p \\le p_1 \\le \\infty,", "f9e7be5f9e906b4fd57ea8fd725811e7": "g(r)", "f9e7e03603e41ca6f0d33a46621f33b5": "\\sqrt{N/k}", "f9e7e59dc9b94f77e31c0055b5067bf4": "\\scriptstyle {f_\\pi}^3/TeV^2", "f9e7ec5f3ab54082cc58ad648cef4bc7": "\\frac{\\partial \\theta_e}{\\partial z} > 0", "f9e82f0b7b9f07fb21c3c2543bebdc7e": "\\nu={3\\over4}t\\mu=({\\mu\\over4}+{\\mu\\over4}+{\\mu\\over4})t", "f9e850e379a14a055e2c25d7013e08fb": "\n EI~\\cfrac{\\mathrm{d}^4 w}{\\mathrm{d} x^4} = q(x) - \\cfrac{EI}{\\kappa A G}~\\cfrac{\\mathrm{d}^2 q}{\\mathrm{d} x^2}\n ", "f9e8837c01c95045e5fd82909ccd5cf7": "-{\\mu \\over{2R}}\\,\\!", "f9e8a75eb76a5c583d52e032829160a9": "\\varepsilon(g)", "f9e8e986b86ff7bc76c29c0770d86e8c": "\n\\rho = \\frac{c' \\Sigma _{XX} ^{-1/2} \\Sigma _{XY} \\Sigma _{YY} ^{-1/2} d}{\\sqrt{c' c} \\sqrt{d' d}}.\n", "f9e9744172bdc7cd6a7a2c36f964f3f0": "K[x_1,\\ldots,x_n,y_1,\\ldots,y_m]=K[X,Y],", "f9e9ebf542b6b64eb0b5883e47758a6e": "H(mn) = H(m)+H(n)", "f9e9f7ee6041209b9b2b0eb6236a84f7": "\\left(\\frac{1}{z}\\right)\\left(\\ln \\frac{I_{0}}{I_{z}}\\right) = k_{w} + \\alpha C", "f9ea05d7bae73c173710c508a14c0fe4": " F({x_0},{x_m}) = \\int\\limits_{x_0}^{x_m}f(x)\\ dx ", "f9ea822f15e346de414415a588691945": "\\frac{63}{25} \\times \\frac{17 + 15\\sqrt{5}}{7 + 15\\sqrt{5}} = 3.14159\\ 26538^+", "f9ea865e02b421687ca277f7b3c13aa9": "\\partial \\Omega\\,", "f9eade35b5535438e618c5e937dbc209": "q_i\\;", "f9eb601762e59f1da3513a9e968ee865": "X_\\tau", "f9eb803140cba1542619548048788b7e": "x_{11}' - p_1 q_1 = (1-c)\\,(x_{11} - p_1 q_1)", "f9eb9a808595cc835148cffbe58720eb": " \\lambda_D = \n\\left(\\frac{\\varepsilon_r \\varepsilon_0 \\, k_B T}{\\sum_{j = 1}^N n_j^0 \\, q_j^2}\\right)^{1/2}", "f9ebd6df54000947e8905a14121543d3": " (V_{1}),(V_{2})", "f9ebf51603c6e97441be37767f962eb0": "\\frac{H_{k,s}}{H_{N,s}}", "f9ebfbb73873f33abe7c06b7f44dfebb": "\\tilde{n}=1-\\delta+i\\beta", "f9ec2a9edd3f204e91b715c6c72ab6ce": "\n\\begin{bmatrix} Y' \\\\ U \\\\ V \\end{bmatrix}\n=\n\\begin{bmatrix}\n 0.2126 & 0.7152 & 0.0722 \\\\\n -0.09991 & -0.33609 & 0.436 \\\\\n 0.615 & -0.55861 & -0.05639\n\\end{bmatrix}\n\\begin{bmatrix} R \\\\ G \\\\ B \\end{bmatrix}\n", "f9ec661e281d04d1793d4911d98af6a2": "P_\\mathrm{net}=P_\\mathrm{emit}-P_\\mathrm{absorb}. \\, ", "f9ec8b2bc7e4771858612779c0ba4e02": " E = \\frac{\\partial e}{\\partial t} = \\frac12 ((\\nabla v) + (\\nabla v)^\\top),", "f9eced3589f9b4bee752a34e853ded2f": "\\begin{align}\\langle\\psi|\\hat{n}|\\psi\\rangle&=\\int P(\\alpha) |\\alpha|^2 \\, d^2\\alpha \\\\\n&=|c_0\\alpha_0|^2+|c_1\\alpha_1|^2+2e^{-(|\\alpha_0|^2+|\\alpha_1|^2)/2}\\operatorname{Re}\\left( c_0^*c_1 \\alpha_0^*\\alpha_1 e^{\\alpha_0^*\\alpha_1} \\right).\\end{align}", "f9ed5e168c46c34dc6af421e6a1fa50b": "VCR=Viewthroughs/Clicks", "f9edc0b4c4e80b6bcebfef73f5820e20": "(n^2+n)/2", "f9ede3d90d101531a83a8aeb8ba55166": "Thrust = \\gamma \\triangle{M} = \\rho g \\triangle{M} = 1590 lbs/ff", "f9ee07cfbb7ee37b9dc9c4b1990193c1": "\\scriptstyle 0\\,\\rightarrowtail\\, A", "f9ee263fdcfad285cdc6ed29db816902": "b=\\text{percentage change in data}", "f9ee765642a8ac092030beb861b8271e": "\\displaystyle{Q(1-u^*)Q(C(u)+C(u^*))Q(1-u)=-4B(u^*,u)}", "f9ee842841c2473cecf67a8e4128bc3c": "\\left\\{ X \\right\\} = A \\,", "f9ee8432a31be5e99553e2ccade2aa22": "\\theta = 2\\phi", "f9ee9fd09f302cab910a849d54267836": "\\begin{align}\n &\\overline{BTA(t)}=\\sum_{i=i}^{m}\\frac{|u(t-i)|}{m}\\\\\n &\\overline{ATA(t)}=\\sum_{i=i}^{n}\\frac{|u(t+j)|}{n}\\\\\n &\\overline{DTA(t)}=\\sum_{k=i}^{q}\\frac{|u(t+j+d)|}{q}\\\\\n &R_2(t)=\\frac{\\overline{ATA(t)}}{\\overline{BTA(t)}}\\\\\n &R_3(t)=\\frac{\\overline{DTA(t)}}{\\overline{BTA(t)}}\\\\\n \\end{align}", "f9eeb618d7a053dc39cf36d1b231370e": "B_1^{p,q}", "f9eec5cb21ebfa22db2a78f804408350": "f_0 = { \\omega_0 \\over 2 \\pi } = {1 \\over {2 \\pi \\sqrt{LC}}}. ", "f9eeff832dca34b5eca12a92b619e165": "e^{(1)}_i", "f9ef2ad141e99c53a7161e1049a70437": "\n\\int_{-\\pi}^{\\pi} \\cos((2m-n)x)\\cos^n x\\ dx = \\frac{\\pi}{2^{n-1}} \\binom{n}{m}\n", "f9ef81f47da74e2367d1dead3b2391cc": "CBR \\quad", "f9efb2ca4a9ec6719e3390d1febcb863": "\\scriptstyle (1 \\,-\\, \\theta t)^{-k} \\text{ for } t \\;<\\; \\frac{1}{\\theta}", "f9f006d485f5e56c57a5b7984a891b6c": "\\sum_{n=0}^{\\infty} f_n(x)", "f9f01bc4f5b4bc2acd3910b66898f68a": "\nd\\mathbf{X} =\n\\begin{bmatrix}\ndx_{11} & dx_{12} & \\cdots & dx_{1n}\\\\\ndx_{21} & dx_{22} & \\cdots & dx_{2n}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\ndx_{m1} & dx_{m2} & \\cdots & dx_{mn}\\\\\n\\end{bmatrix}.\n", "f9f036e100684fe9163d1c737a1875ce": "\\;p(n) = P r^n - A \\frac{r^n-1}{r-1} = 0", "f9f07ddc4e9225982a5598a9edc1ed6c": "\\bigl(w(s, p1) + h(s) - h(p1)\\bigr) + \\bigl(w(p1, p2) + h(p1) - h(p2)\\bigr) + ... + \\bigl(w(p_n, t) + h(p_n) - h(t)\\bigr).", "f9f1202df636faf77c14a6e448df9783": "A(x_2)", "f9f162314a3bee444ee3e7d0baf4b043": "\\begin{align}\n t &= \\operatorname{arctanh}\\left(\\frac{s}{x_0}\\right),\\; -x_0 < s < x_0\\\\\n x &= \\sqrt{x_0^2-s^2},\\; -x_0 < s < x_0\\\\\n y &= y_0\\\\\n z &= z_0\n\\end{align}", "f9f1a94e870ddab5479a7c298a9e9b1d": "(26_4, 10_2, 9_2, 8_2, 5_2, 1_2)", "f9f1b6cf86fa94ceba1c2c039bec2f76": "\\tau_2 = \\frac{2 \\zeta}{\\omega_n} - \\frac{1}{K_p K_v}", "f9f243ae767e9c6eabffd2fe36877d8a": "\\omega([X,Y])=[\\omega(X),\\omega(Y)]", "f9f25ce1ef5295e64fb0136b22139b57": " S(P';M') \\leq S(P;Q)", "f9f37c9ed32587ab453545bf007e790d": "\\mathbf{p} = \\frac{\\partial L}{\\partial \\mathbf{\\dot{q}}} = \\left(\\frac{\\partial L}{\\partial \\dot{q}_1},\\frac{\\partial L}{\\partial \\dot{q}_2},\\cdots \\frac{\\partial L}{\\partial \\dot{q}_N}\\right) = (p_1, p_2\\cdots p_N)\\,,", "f9f40cb0d5df2c671d3e4df2196fafe5": "\\mathbf X = \\mathbf X_d\\otimes\\mathbf X_{d-1}\\otimes\\ldots\\otimes\\mathbf X_1.", "f9f489285bc02d06395030c63889601a": "(U, m).", "f9f4a194a73cdb821440b114bbd27eda": "P=\\bigcap\\mathcal R=\\bigcap_{i=1}^t <_i.", "f9f4b52b7130ba6011acba249daf109a": "\\mathbf{F} = - \\nabla\\,(\\mathcal{G} (d)) + \\nabla \\times (\\mathcal{G}(\\mathbf{C})),", "f9f5648702ce41b8c9c5d0dff07f1060": "\\pi(t) = \\sum_{n=0}^\\infty \\pi(0) P^n \\frac{(\\gamma t)^n}{n!}e^{-\\gamma t}.", "f9f597d2a848ef71467329a3c459d93f": "= ((x_1*y_1)(1+\\delta_1)+(x_2*y_2)(1+\\delta_2))(1+\\delta_3)", "f9f5ba092247f94b589c602b567951c8": "\\sigma \\in \\{-s\\hbar , -(s-1)\\hbar , \\cdots ,+(s-1)\\hbar ,+s\\hbar\\}.\\,\\!", "f9f5bf102c069ad8b9d0527cf22234b2": "\\varepsilon_3 = \\frac{1}{E}((1+\\nu)\\sigma_3-\\nu(\\sigma_1+\\sigma_2+\\sigma_3))", "f9f5d66a3fc0edf3009451a2ff2eb59d": "\\rho_T = |\\Psi\\rangle \\; \\langle\\Psi|", "f9f606660f6e4d7c1afa1469dad923c1": " \\left[\\frac{F}{m}\\right] ", "f9f6294dfb58796720f71f5b1e42769c": "n = 0, 1, 2, 3", "f9f651a053a9b1be96505e25becbcba0": "\\psi_0(x) = \\dfrac{1}{2\\pi i}\\int_0^{\\infty}\\left(-\\dfrac{\\zeta'(s)}{\\zeta(s)}\\right)\\dfrac{x^s}{s}ds=x-\\sum_\\rho\\frac{x^\\rho}{\\rho} - \\log(2\\pi) -\\log(1-x^{-2})/2", "f9f676ab1ad3f1e85cd8cef87885413a": "\\nabla^2V={\\partial^2V\\over \\partial x^2 } +\n{\\partial^2V\\over \\partial y^2 } +\n{\\partial^2V\\over \\partial z^2 } = 0.\n", "f9f676e4a3cae4c9fea5d47f7d33e270": " \\Delta E_{act} \\,", "f9f6c3a56a2296334115f0434453d938": "\\mathfrak{a}\\mathfrak{b}", "f9f6f942b24b72d479532bd2b3f0f710": "f(x) = \\frac{1}{a^3}\\sqrt{\\frac{2}{\\pi}}\\,x^{2}\\exp\\left(-\\frac{x^2}{2a^2}\\right)", "f9f71f4b30c21b8264c14b61519c8027": "1/\\kappa\\rho = l", "f9f72bb1f018a2087d1bdb8264ff89a1": "\\sigma_y^2(\\tau) = K_{\\alpha}h_{\\alpha}\\tau^{\\mu}", "f9f7544e2e21a6037136fdef2f4ca093": "x, z_1, \\ldots, z_i", "f9f762ba2ff87931193dbb65e30dc6c4": "(\\mathbb{Z}/n\\mathbb{Z})\n^*", "f9f77684e4fe80635fa773a3516b5922": "d_t > d_{t+1}", "f9f77a2d8a667416dd2f78c3ab32bee3": "4 * 2 = 8", "f9f7a9f781280d9d48f974727d720417": "0 \\to \\mathbb{Z}/2\\mathbb{Z} \\xrightarrow{(1,0,0)} (\\mathbb{Z}/2\\mathbb{Z})^2 \\oplus \\mathbb{Z} \\to (\\mathbb{Z}/2\\mathbb{Z}) \\oplus \\mathbb{Z} \\to 0,", "f9f7b007b0ddb2f68bdc6607eb56cebd": "x'=\\gamma (x - v t) \\,.", "f9f824a41793c54d2356410961d67025": " F(x) = m \\log_{10}(x) + b. \\, ", "f9f854de4dff93083bfdf01fd6e9eff8": "\\left.\\frac{d}{ds}\\frac{d}{dt}\\tau_{sX}^{-1}\\tau_{tY}^{-1}\\tau_{sX}\\tau_{tY}Z\\right|_{s=t=0} = (\\nabla_X\\nabla_Y - \\nabla_Y\\nabla_X)Z = R(X,Y)Z", "f9f9440bd126f8caba672eae162b9f4e": " \\sum_{n = 0}^\\infty \\frac{2^n}{n!} = 1+2+\\frac{2^2}{2!}+\\frac{2^3}{3!}+\\frac{2^4}{4!}+\\frac{2^5}{5!}+\\cdots", "f9f94eb8539f6b23d7ea58dd891322dd": "\n\\begin{align}\nT_E^4 &= \\frac{r_S^2 T_S^4}{4 a_0^2} \\\\\nT_E &= T_S \\times \\sqrt\\frac{r_S}{2 a_0} \\\\\n& = 5780 \\; {\\rm K} \\times \\sqrt{696 \\times 10^{6} \\; {\\rm m} \\over 2 \\times 149.598 \\times 10^{9} \\; {\\rm m} } \\\\\n& \\approx 279 \\; {\\rm K}\n\\end{align}\n", "f9f9a434a2a138a4dce30b3015be5bca": "\\textstyle f_B(x)\\rightarrow f(x) ", "f9f9eb411df97cbfc98614ac573c28ee": "Q = 11010101\\,", "f9f9f264e76b449ae2f5f77beadc9cce": "\n\\sum_{i} \\frac{Dr_{D}T}{(1+r_{D})^i}=\\frac{Dr_{D}T}{r_{D}}=DT\n", "f9f9f574fd0671652e6300c68ad3da7d": "d=1\\,", "f9fa12d1ab3fcf22c1bf6b7b577d735d": " \\Phi_{2D}(\\mathbf{x},\\mathbf{x}')=\n-\\frac{1}{2\\pi}\\ln|\\mathbf{x}-\\mathbf{x}'|,\\quad \\Phi_{3D}(\\mathbf{x},\\mathbf{x}')=\n\\frac{1}{4\\pi|\\mathbf{x}-\\mathbf{x}'|} ", "f9fa1a3dcd89ea15151e3de54b334cc3": "[O,\\,H]=0", "f9fa1d4e9f54c3b7d76905bdbb986e9d": "\\int\\limits_{-\\infty}^\\infty\\frac{dx}{1+x^2} = \\pi\\!", "f9fa3dcca2d091ea2078af9324619a5c": "p = \\frac{(D + d) f}{d}", "f9fa68301a20d4eba9a09c9bfc350a75": "\\{P, \\neg \\Box (P \\wedge Q)\\}", "f9fac78f9a1b7983e358c3520b67d062": "W(t, \\tau)", "f9fb869a7c37a0dabb0f8e91da20b4c5": "Cv_i = C^{i}v_1 = v_{i+1}", "f9fbb3aa938e71ac60bbce1425a28396": "\\cot (\\alpha - \\beta) = \\frac{\\cot \\alpha \\cot \\beta + 1}{\\cot \\beta - \\cot \\alpha}\\,", "f9fbd37f3f29189894fdfe2396035467": "g \\in H", "f9fbe681569274028560199ebf41dbdf": "\\dot{\\theta} = v/R", "f9fc23e1dda7f9e2da5c11de91874f28": "T^a \\square^b F ", "f9fc6527037e1f870f1eba5bbb2a2820": "\\alpha=(1.18,-0.43,0.73,1.13,-0.37,0.57)", "f9fc8242f5cb60cc50dc60db4c5644db": "\\chi_{2}\\,", "f9fc8958b4447f2892c28aa26c4dc5ed": "{BC}_{2+}", "f9fc95580d568d4336cf54fc65a0a0cf": "\\tfrac{n!}{k!\\,(n-k)!}", "f9fd739de079ba078fd31afb62621912": "\\Omega_F = \\sum_{p \\in P_F} 2^{-|p|}", "f9fd806feedfff069a6c9b23ec270997": "T_n x \\to Tx", "f9fde2f78e2ec13929e9ba2863206960": "(x-c_1)^2", "f9fdf207e021c7d843ac4a29069cd228": "\\pi <\\tfrac{22}{7}", "f9fe2042164ffd07ae9e36efb61e3416": " \\begin{align} \\textrm{ad} : & \\mathfrak{g} \\to \\mathfrak{gl}(\\mathfrak{g}) \\\\ & x \\mapsto \\mathrm{ad}_x \\end{align}", "f9fe2be0bc1b44a80cf5eca23698fdd5": "x^{15} \\pm 1", "f9fe31e44e12c35ec1d67f921745e72a": " YZ=2\\cdot Y\\cdot Z ", "f9fe79a2f262577fe686b8169b6587be": "\n\\begin{align}\nk_1 &= f(t_n, y_n),\n\\\\\nk_2 &= f(t_n + \\tfrac{1}{2}h , y_n + \\tfrac{1}{2}k_1h),\n\\\\\nk_3 &= f(t_n + \\tfrac{1}{2}h , y_n + \\left( \\tfrac{1}{2}-\\tfrac{1}{\\lambda}\\right) k_1h + \\tfrac{1}{\\lambda}k_2h ),\n\\\\\nk_4 &= f(t_n + h , y_n + \\left( 1-\\tfrac{\\lambda}{2}\\right) k_2h + \\tfrac{\\lambda}{2}k_3h ).\n\\end{align}\n", "f9ff176faf9572ffb6ac44501b1fe940": "a_{N}", "f9ff3849515d900de36a5ea733eb6e9b": "H_p = - \\int |\\phi(p)|^2 \\ln (|\\phi(p)|^2 \\cdot \\hbar / \\ell ) \\,dp =-\\left\\langle \\ln (|\\phi(p)|^2 \\cdot \\hbar / \\ell ) \\right\\rangle", "f9ff40eab2e3bab7159b751e1094b4d9": "x\\# = \\prod_{p \\leq x} p,\\ ", "f9ffa27d35cee4ebaf0c0f0efc907e33": "\\bar x ", "fa006f33b89184d2a716d61da2f99967": " \\tau \\, ", "fa0071b458881a746a803158bb8eaed7": "k^{m} \\psi^{(m-1)}(kz) = \\sum_{n=0}^{k-1}\n\\psi^{(m-1)}\\left(z+\\frac{n}{k}\\right)", "fa007a52847cf85db029fc5c5fcb0d88": "\\mbox{affinity} = \\alpha[A]^a[B]^b\\!", "fa007f39aeaa9419df90cefca305a5a0": " \\langle \\sigma_A\\rangle \\geq 0", "fa0099beb8f43858ccd5625a2018d558": " 0 \\leq h_{ii} \\leq 1 ", "fa00a7b38bd66aae71905728ea0a21f7": "M^{0i} = x^0 p^i - x^i p^0 = c\\,\\left(t p^i - x^i \\frac{E}{c^2} \\right) = c\\,\\left(t p^i - m x^i \\right) = \\gamma(u)m_0 c\\,\\left(t u^i - x^i \\right) = - c N^i ", "fa00db5a4f0d67d5b0b93f9313de4f64": "\n(-1)^{2(j_1+j_2+J)} = (-1)^{2(m_1+m_2+M)} = 1.\n", "fa010c9e9702eedb7a2f5e588d7c3bc2": "[x,y]:=\\sum_{j=1}^nx_j\\operatorname{sgn}(\\overline{y_j}),\\quad x,y\\in\\mathbb{C}^n,\\ \\ p=1,", "fa0116f334e32ea7176d88ada7fb4e2f": "G = \\min{\\left(1,\\frac{2(H\\cdot N)(E\\cdot N)}{E\\cdot H},\\frac{2(H\\cdot N)(L\\cdot N)}{E\\cdot H}\\right)}", "fa011f3f0d07c0846632d70c0c13a011": "g_{ij} = \\int_\\Omega \\int_\\Omega \\kappa(x,y) \\varphi_i(x) \\psi_j(y) \\,dy\\,dx,", "fa01a869196de1c51896e63dd129cf6c": "\\phi:\\pi^{-1}\\left(U\\right)\\rightarrow U\\times F", "fa022de54d0ae50911961f03661f2f97": "\\mathbb{RP}^{14}", "fa0253fcb24d3da712f0eecb8b406e04": "q=1+k_3(c-1)", "fa029bfe3b7f8cc3b8e2262eff51cf1f": "m(m-1)/2", "fa02b68ab3ebb2cf37dabd34cdfc6b97": "\\frac{4}{3}", "fa02f82f921ef30ec3f9d431150299f9": "\\mathfrak{p} = (u)", "fa0348dd17301acabd77e2b32df2f3fa": "M(a,b,z)", "fa03604ce01a1766005b2399584828e3": " \\mathbf{E}\\cdot d \\mathbf{A} = \\int\\!\\!\\!\\!\\int_c E dA\\cos 0^\\circ = E \\int\\!\\!\\!\\!\\int_S dA \\,\\!", "fa03b6d6f6036e17639460111663bd1a": "F^2 B^n = 0", "fa04190519f0378ee43bf2d68c476167": " I_i\n", "fa043c065dd111d926a3d140b618b05e": "q_2", "fa0463c9a71028b2baa8c3b2e543e8b8": "\\mathbf{Q} = \\frac{1}{N-1}( \\mathbf{M} - \\mathbf{1}_N \\mathbf{\\bar{x}} )^\\mathrm{T} ( \\mathbf{M} - \\mathbf{1}_N \\mathbf{\\bar{x}} ).", "fa04872586d86f61a37d44581bfd3db4": "L_D", "fa04907468bcbc5a7203fd992086b4a0": " d\\mathbf{F} \\ , ", "fa04b48dc4b4d33f0c9fb8213d9830da": "(d_* + \\gamma_*)_{odd}: D_{odd} \\to D_{even}", "fa05182437e3980aadb6286687831421": "BB/9IP = 9 \\cdot \\frac{BB}{IP}", "fa0524f8d220dd8095704cd839d7b1cb": "\\begin{matrix}50 - (14 - x) \\times 4 = 4x - 6\\end{matrix}", "fa0565f430452f99cc5f269668222afd": "24\\rightarrow (8,1)_0\\oplus (1,3)_0\\oplus (1,1)_0\\oplus (3,2)_{-\\frac{5}{6}}\\oplus (\\bar{3},2)_{\\frac{5}{6}}", "fa0599446c4ee68b7bb0e89eadbb14ea": "SU(3)_L \\times SU(3)_R \\times U(1)_V \\times U(1)_A", "fa060d6ef2fc409ea62b6d49f10f8605": "H^g", "fa065af4b00fcaf84d900248f8e546bf": "\\coprod C_x", "fa068763c4a7db029745dc6b27fe991d": " P(\\partial_t,\\xi)\\hat u(t,\\xi) = \\hat F(t,\\xi).", "fa0693f18a1d272921b8082c98b72a34": "T(n)=O(n^2)+55n^3+2n+10.\\ ", "fa0758db4fd1c1fb30215c5fdaf03265": "p_1, p_2, \\dots , p_n", "fa075efb20b03d787f3394a0b970c307": "a \\cdot b := \\int \\int_X \\eta_a \\wedge \\eta_b = (\\eta_a, -*\\eta_b) = -\\int_b \\eta_a", "fa07d0e57ccf5b8f8f9399142057e247": "location_i = (name_i \\ mod \\ m) + 1", "fa0800ff168ff207132a1f54a9da0b68": "\\hat{f}", "fa0822b83ed16de6125d7d8150d30ba1": "s:C\\rightarrow kar(C)", "fa082e8128ebc8d522e6cabded9ba64a": "P=P^*+\\epsilon", "fa086ff8092e5e2d3575cf93e4c44b97": "\\mu=5", "fa08c48ea6c2ff35c3c0bfea940b9744": "\\left[S_i, S^2 \\right] = 0", "fa08d0efd7fecd6b09fd5fd74a31b903": "\\mathfrak{P}^{14}", "fa08ef20d368762f60d96d647b223013": "\ndV_B = \\frac{dx^1 dx^2 dx^3}{8\\sqrt{ 1 - (x^1)^2 - (x^2)^2 - (x^3)^2 }},\n", "fa09e79e0848d2a4a05c3fa510845c51": " AC\\to D", "fa0a37124e64cb91c57eb8ec1ccdcf8c": "a_{TD} = 3.84\\times 10^{-10}\\;\\mathrm{m} - \\ ", "fa0a3db2ee127e65c961570cc82fe1b7": "x = \\sup_\\omega B(\\omega)", "fa0a6039b204df3768f2cf9a2f1059b4": "\n\\alpha = {{ \\left[ A^- \\right] } \\over {\\left[ A^- \\right] + \\left[ HA \\right]}}\n", "fa0b22fc5589e4aba10a939644ab442e": "h_*:=\\pi_*\\circ F^\\%", "fa0b30d7a16ce1ed24465d3b4dd3f143": " f(x) = \\frac{1}{2}\\int_{\\mathbb{R}^{n}}\\left(\\frac{x-y}{|x-y|}+\\frac{y}{|y|}\\right)\\,d\\mu(y)", "fa0b44c21841019b3241e480de3d0f1d": "\\displaystyle \\left( m_1 , m_2 , \\dots , m_n \\right)", "fa0b8da667942717dcb8cc4bec0fe6eb": "\\lambda\\mathbf{A} = \\lambda(A^0, A^1, A^2,A^3) = (\\lambda A^0, \\lambda A^1, \\lambda A^2, \\lambda A^3) ", "fa0ba4560374fce7ebdd7c2704f46da4": "\\partial w_s=F dx", "fa0beda9915c4cafab16fa846f235974": "[Z_1][X_1][Z_2][X_2]\\ldots[X_{n-1}][Z_n]=[I].\\!", "fa0befccbf79b37adf5852bbb39fd828": " \\frac{p_{0}}{\\rho g} + z_{0} = \\frac{p_i}{\\rho g} + \\frac{V_i^2}{2 g} + z_i + h_f", "fa0c29d572f6ba21f1a555caa79a9072": "\\mathcal{O}(n \\log \\sigma)", "fa0c4f596d8a877bf5935df3d4393dcc": " 1 \\le i_k \\le N", "fa0cc9aace3237507c654483d99aae76": " \\{\\,a \\mid \\exists y \\,\\forall k \\!\\le y\\, \\exists x_1,\\ldots , x_n [p(a,k,y,x_1,\\ldots ,x_n)=0]\\,\\}", "fa0cdaaaafab353bcac0bc637d2b02e7": "X^2=\\frac{\\sum_{i=1}^L(S_i^A-\\widehat{E}(S_i^A))^2}{\\widehat{V}ar(S_i^A)}", "fa0ce682364328a2d63f5e31f21d7446": " \nD_r = \\frac{k_B T}{f_r}\n", "fa0d156444815f7ea673d657d62e0015": "\\scriptstyle \\lceil x \\rceil", "fa0d5f7a33041f5ee8aacce78a26fd88": "a = \\frac{\\lambda}{c \\rho}", "fa0d84540adbfd69f35c55073c854a52": "F_1 = YW - Z^2", "fa0dce3d57704549c84cc04c8d767eb5": "J=\\{b,j\\}, \\alpha=1, x_b=t, x_j=1", "fa0deacaf8e01a263375baf24fb74e85": " \\sum_{i=1}^{n} X_i \\sim \\textrm{IG}\\left(\\sum_{i=1}^n \\mu_i, 2 {\\left( \\sum_{i=1}^{n} \\mu_i \\right)}^2\\right)\\,", "fa0dfefcc86ea515219fd526505d1044": "p\\leq 1", "fa0e61270f8176912e7ec0b4ee1352bc": "\\omega_{\\mu}^{ \\ IJ} = - \\omega_{\\mu}^{ \\ JI}", "fa0e8457b30ef0e229bc0443910179c8": "\\sigma^2_{\\eta_j}", "fa0eabde3fda5ed3df1a12887c27d809": " \\tilde B = {h\\over{8\\pi^2cI_B}} ", "fa0ecd1fd1a6e7011ebc8e8a85d46f5b": "\\left[ \\begin{matrix} \\mathbf{\\dot{x}_1}(t) \\\\ \\mathbf{\\dot{x}_2}(t) \\end{matrix} \\right] = \\left[ \\begin{matrix} 0 & 1 \\\\ -\\frac{k_2}{m} & -\\frac{k_1}{m} \\end{matrix} \\right] \\left[ \\begin{matrix} \\mathbf{x_1}(t) \\\\ \\mathbf{x_2}(t) \\end{matrix} \\right] + \\left[ \\begin{matrix} 0 \\\\ \\frac{1}{m} \\end{matrix} \\right] \\mathbf{u}(t)", "fa0ed5b5c600145bdd9a299952b99651": "yt", "fa0eff9f5d04829acbde459ab8f21115": "\n\\Phi(\\mathbf{r}) = \n\\frac{1}{4\\pi\\varepsilon} \n\\sum_{l=0}^{\\infty} \\sum_{m=-l}^{l} \n\\left( \\frac{Q_{lm}}{r^{l+1}} \\right)\n\\sqrt{\\frac{4\\pi}{2l+1}} Y_{lm}(\\theta, \\phi) \n", "fa0f5d0c6c84b57b3eddc9b248cdc90c": "\\log |f(0)| = \\sum_{k=1}^n \\log \\left( \\frac{|a_k|}{r}\\right) + \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log|f(re^{i\\theta})| \\, d\\theta.", "fa0f82170233552cb9d63b10f63e8c75": " K_t( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\big [ y(t) - \\varphi \\big ( \\mathbf{x}(t), \\mathbf{w} \\big ) \\big ]^2 ", "fa0ff0e9f11ee78c2239008079f9abb3": "\\hat 4", "fa0ff1680e1d8eec928db4ccb934d7d2": "f^{123} = 1 \\,", "fa102e8d5c4a8856fa90925adf0d5591": "(\\nabla f)_{x_0}", "fa10716c5b5d0dc6b25aa13f758d6c4f": "y_2 \\in \\mathbb{F}_q^k-\\{ 0 \\}", "fa107fb46530e92fad1b8db3f72867ad": "\\overline{\\ln \\ln \\left (\\frac{\\varepsilon^{(s)}}{\\varepsilon^{(0)}} \\right )} = \\overline{\\ln \\left ( \\frac{\\Omega^{(s)}}{\\Omega^{(0)}} \\right )}.", "fa109a481699875936fe17185123c6fd": "dN/dt = \\alpha N/\\tau", "fa10bc4ea041df5dfdfe057b8aa204ea": "\\tau=h_{\\mu}x^{\\mu}(\\tau)=h_{\\mu}Y^{\\mu}(\\tau)=h_{\\mu}R^{\\mu}(\\tau)", "fa10c07f18d13a45836e43a1629c4a8a": "466/885\\approx 52.6\\%", "fa10c4c2667ba3449032ce4669c7d9d1": "y^{\\prime\\prime}=6y^2+t", "fa11213394d9e831094fa6b3aa996f1d": "\\varphi = \\frac{1}{4\\pi\\epsilon_0}\\iiint\\left[\\nabla'\\cdot\\left(\\frac{\\bold{P}}{|\\bold{r}-\\bold{r}'|}\\right)-\\frac{1}{\\bold{r}-\\bold{r}'}(\\nabla'\\cdot\\bold{P})\\right]d^3\\bold{r'}", "fa114695aec226f8062b6702f7c89dd8": "\\lambda_2", "fa120de5f1b8ef6cc9ea71d36d79c4ea": "\\hat{\\alpha} = \\hat{\\beta} = \\frac{\\hat{\\nu}}{2}= \\frac{\\frac{3}{2}(\\text{sample excess kurtosis}) +3}{- \\text{(sample excess kurtosis)}} \\text{ if sample skewness}= 0 \\text{ and } -2<\\text{sample excess kurtosis}<0", "fa135e3c24f11bd6dc99ea1a65974958": "D_Y(f_!M) \\cong f_*(D_X(M)),", "fa1364164092ffd346396e91149d33a7": "-\\tfrac{3}{16}x^2-\\tfrac{3}{4}x-\\tfrac{15}{16}", "fa1455c1e16aa9cc2fbb0286e42ee099": "C_t = a + bY_{t-1} + e_t,", "fa148033c82f0478a4e69a09d8248d2a": "\\left(\\pm1,\\ 0,\\ 0,\\ 0,\\ 0\\right)", "fa156cff46848328bb0570efa3feec6e": "\\xi> t", "fa15c4bfbc4cd3fbc1ca72ec73fe048a": "\\alpha\\in\\pi_{p+1}(f)", "fa15cea7bf4e453c64a75a1cf8c579a5": "{{V}_{BE}}\\ge 0.7", "fa16598950e68a10b1ace6e9008e363f": "\n\\begin{align}\nb&=a(1-f)=a\\left(\\frac{1-n}{1+n}\\right),\\\\\n e^2&=2f-f^2 = \\frac{4n}{(1+n)^2}.\\\\\n\\end{align}\n", "fa165bc5d4eabed41effe498bbf74350": "\\hat{a_1}", "fa168a3f507c17ea7919a57ed0f669fb": "\\bigoplus\\nolimits_{n \\ge 0} \\operatorname{H}^0(X, \\mathcal{O}_X(n)).", "fa169823969f28adb92f826f02e9734a": " - g^{\\mu \\nu} \\Gamma^{\\sigma}_{\\sigma \\rho} \\sqrt {-g} \\!", "fa16e296b69985dac917a8b6f3f5d66e": "\\hat{a}^{\\dagger}", "fa1719c4c43e309ec458367b1ed08aa8": "2 \\pi / 3", "fa174b5bf73a062a74802f9c2583519c": "p\\not\\in C\\supseteq D", "fa17545c634219542e6959eb68ffe0cf": "L =R\\pi\\frac {\\Delta}{180}", "fa175caf479a9d17d8f55b6b37326a5c": "\\alpha = \\frac{by}{ax + by + cz}, \\quad \\beta = \\frac{ax}{ax + by + cz}.", "fa178c0a59672a8ae48b5160c569047b": "d_I(x,y) =\\infty", "fa18252bcf550f3b50b8d47ec593eb9e": "0 < \\left| a q_n /\\ \\! b - p_n \\right| < \\frac{1}{b}", "fa1833a4b75a5d9809b4bbb3778949ba": "\\lambda=1/3", "fa18363e0f4506fce752695362c8e2da": "M_{22}", "fa185d076f5932d933c3217f3a2a7575": "\\mathbf{a}\\times (\\mathbf{b}\\times \\mathbf{c}) \\; + \\mathbf{b}\\times (\\mathbf{c}\\times \\mathbf{a}) \\; + \\mathbf{c}\\times (\\mathbf{a}\\times \\mathbf{b}) = 0", "fa18cd77a1037dcd4ce731fcf2cbdef9": " \\Gamma' ", "fa18d5490cf8e74c5c79f06fc22ebf78": "(y, x)", "fa18dd9e90fd59f3fc0973eeb7b534f3": "\\eta(A) = \\max_i\\{ {\\rm Re}(\\lambda_i) \\} \\, ", "fa19373d8a7b20d9326549b1341a32e1": "H^p(X, \\mathcal{F}) = H^p(\\mathbf{P}^r_A, \\mathcal{F}), p \\ge 0", "fa193b3114e2d69f76b8ae03ae17daa0": " P(m) = \\int s(x) N(x - m)\\, dx ", "fa19b7915f3b2dd28661838deed17e46": "{\\mathbf{}}T", "fa19ce6466f2ee11fe4735be4706c8c6": "T^{\\mu}_{\\mu}", "fa19e1ea6670f34c374dc8894b0d9745": "\\scriptstyle {R = 3\\tfrac{2}{3}}", "fa1a551db7a416ddb857de2bca311084": "O(n\\log^* n)\\,", "fa1a59041b6cb62b9cdb3872c23462ee": "{y} = \\sqrt {r^2 + \\alpha^2} \\sin\\theta\\sin\\phi", "fa1a6f7cb951f45a3fdc73aec71db453": "\\langle f, \\varphi\\rangle = \\langle T_f,\\varphi\\rangle.", "fa1ac4ede3c85986443f9a9e866703c6": "y'_1(t)\\mathbf{e}_1 + \\cdots + y'_n(t)\\mathbf{e}_n", "fa1ad15034aba7c1c2d86a1e6a8ee7c3": "\\phi'(x)\\,", "fa1ae4cd5e41cacb93c2c84802332291": " \\frac {x \\sin \\theta} {\\lambda} = 0, \\pm 1, \\pm 2, ...", "fa1b4a352924a0710834b2c4498ad049": "a_7\\times (3\\rho^2-2)\\rho \\sin(\\theta)", "fa1bd885b49354845e1e3e32152ea336": "r = \\frac{s^2 + \\ell^2}{2s}", "fa1cbd0d936ff657905cd1eed426d980": "J_n(z) = \\left(\\frac{z}{2}\\right)^n {\\mathcal C}_n\\left(-\\frac{z^2}{4}\\right);", "fa1ccea91344ea1ea892c33bd07355ab": "R = \\textstyle \\sum_{i=1}^n \\tfrac{f_{i}}{LC_{50i}}", "fa1cfc559dab4f0b8c5b9625b29b9a07": "\\frac{1}{\\sqrt[x]{e}}\\,", "fa1d0c9b8fba3c8ce9e50674cb0df6ef": "f_\\nu \\left( \\phi \\right) =\\frac 1{8\\pi \\phi }\\nabla _\\nu \\phi", "fa1dd55b0c93789df92a0cddcedf5f12": "\\gamma_{OW}", "fa1e25cbd8da3b983846a92e24f3eddf": "\nP=x_1^2 + 2x_1x_2 + 14x_1^2x_2 + x_2^2 + 14x_1x_2^2 + 49x_1^2x_2^2.\n", "fa1e2c7a149ca58bccbc28fe16caac7a": "\\rho~", "fa1e95b77f0e486fa5738b876146cfec": "f(A)", "fa1ef55700703bf6db114371215eff67": "\\sum_{i=1}^n v_i = 0", "fa1f1663c34d17fcf04ae70241105f80": "D(f) = O(Q_E(f)^3)", "fa1f318a3f25149104911598118992dd": "(\\hbox{NetEnergy} \\div \\hbox{EnergyExpended} ) + 1 = EROEI ", "fa1f71c8d382e8819e20dffcc86216a3": "\\oplus_{n=0}^{\\infty} I^n t^n=R[It]\\subset R[t].", "fa1fce29eff0077028405a0fe7b460c4": "m \\cdot 4:m\\ ", "fa1fe0f221adba49151813112d3932d6": " h^{n,0}=1", "fa205983b75c6d1fe541cfd602e64c5e": "(x-a)^nH(x-a)", "fa20b56bb7b89e566f9d74ad64908f2a": "\\dfrac{PR}{L1} = \\dfrac{SQ}{L2}", "fa20e74c0e2b92eff9f5e6e6847322d8": "N_\\ell", "fa213b622ccd11f065755d9686060877": "\\Phi =-\\frac{m}{n_s e^2}\\oint_{\\partial S}\\mathbf{j}_s\\cdot d\\mathbf{l} +\\frac{\\hbar}{e_s}\\oint_{\\partial S}\\mathbf{\\nabla}\\phi\\cdot d\\mathbf{l}", "fa213f1b8fab90a33d76c95eeabf0861": "\nA e^{A t} = 3/4 B_{1_1} e^{3/4 t} + \\left( 3/4 t + 1 \\right) B_{1_2} e^{3/4 t} + 1 B_{2_1} e^{1 t} + \\left(1 t + 1 \\right) B_{2_2} e^{1 t} ~,\n", "fa21d1ee2c06d24e52e16553a0682880": "2r - v = \\frac{L^2}{4v},", "fa2202ece406b7aa7de2d420e50b1e30": " \\textbf{R}_{P/O}=\\textbf{P}-\\textbf{d},", "fa220911c29255114ee97f62913250ba": "\\tfrac{2}{3} = \\tfrac{-2}{-3}", "fa2267cd99da72937feeb046679c4966": "A(\\tau) = \\int_{-\\infty}^{+\\infty}E(t)E^*(t-\\tau)dt", "fa2282e938da0094830593c0cad53761": "\\mu(A)=\\chi_A\\,", "fa2298621831e7cf17fd4cab0a886026": "f(v, \\varphi) = 0 \\,", "fa23188891a0203a85bc8201b8cdc2b7": "\\Delta(y_n,y) + \\boldsymbol{w}'\\Psi(\\boldsymbol{x}_n,y)", "fa231e0f942e2490adef9c52bc49ae39": "i = 1 - \\frac{n-1}{n}\\alpha", "fa238f332ca0ceba8598dea909e328a5": "Ord (a)", "fa23e5f7b8806d06fbf0ce541e0d9f55": "\\# h", "fa241f7f77357c9f035eaf4d34744d29": "L' = \\frac{\\mu_i}{8 \\pi} + \\frac{\\mu_d}{2 \\pi} \\ln\\frac{r_{o1}}{r_i} + \\frac{\\mu_o}{2 \\pi} \\left( \\frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \\right)^2 \\ln\\frac{r_{o2}}{r_{o1}} - \\frac{\\mu_o}{4 \\pi} \\left( \\frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \\right) - \\frac{\\mu_o}{8 \\pi}", "fa242c0ab751134a323538989985a9d8": "\\mathbf{F}=q\\mathbf{E}+q \\mathbf{u} \\times \\mathbf{B}", "fa2452c388ed1c0da4d2fcf91ac4c8c6": "IB", "fa24ae1f89bcd3b750366f8c911de4cd": " v = r_A \\omega_A = r_B \\omega_B,\\!", "fa24c34b1db27c75b05c56dc14b85e7f": "\\scriptstyle 0.7\\pm1\\times10^{-14}", "fa254b416a4b6e6f55c1a9a494e75c45": "\\Delta v\\ = v_e \\ln \\frac {m_0} {m_1}", "fa257196ca052967040f755647ebf445": "\\Diamond\\phi", "fa259c7bb9dafb2c4f616064527f6f2f": " A = (Q \\Phi Q^*)(Q \\Sigma Q^*),\\,", "fa25baa6ebe7d9d95fc5a07b9e3d5256": "E_\\text{B} = -\\dfrac{Rhc}{(n-\\delta_l)^2}", "fa25d7a2b8cd4e19fc257a1b52c345c5": "\\displaystyle k_B", "fa25dc7c693f13da3d5fc56c7bb50530": "\\mathbb{R}(\\sigma(\\textbf{A}_N))>0", "fa25e1b480cdf991144e59007935b848": "{u}_{1}^{x_1} u_{2}^{x_2} ({u}_{1}^{y_1} u_{2}^{y_2})^{\\alpha} = v \\,", "fa26201c2ab53f9479a2eac943cff58f": " \\begin{align}\n \\frac{\\partial f}{\\partial \\boldsymbol{A}}:\\boldsymbol{T} \n & = \\left.\\cfrac{d}{d\\alpha} \\left[\\alpha^3~\\det(\\boldsymbol{A})~\n \\left(\\cfrac{1}{\\alpha^3} + I_1(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\cfrac{1}{\\alpha^2} + \n I_2(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\cfrac{1}{\\alpha} + I_3(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})\\right)\n \\right] \\right|_{\\alpha=0} \\\\\n & = \\left.\\det(\\boldsymbol{A})~\\cfrac{d}{d\\alpha} \\left[\n 1 + I_1(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\alpha + \n I_2(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\alpha^2 + I_3(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\alpha^3\n \\right] \\right|_{\\alpha=0} \\\\\n & = \\left.\\det(\\boldsymbol{A})~\\left[I_1(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T}) + \n 2~I_2(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\alpha + 3~I_3(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T})~\\alpha^2\n \\right] \\right|_{\\alpha=0} \\\\\n & = \\det(\\boldsymbol{A})~I_1(\\boldsymbol{A}^{-1}\\cdot\\boldsymbol{T}) ~.\n \\end{align}\n", "fa2681ee5a20bd6076802d3d94703d93": "M_{\\mathfrak{p}}(\\mu)\\rightarrow M_{\\mathfrak{p}}(\\lambda)", "fa268f68bfde98f834399fdd842da717": "\\delta_{nm}", "fa26c436d5421641cee27845373b4692": "Z(s) = \\frac{1}{sC}", "fa2701e3f6198cbcf030481adeddd641": "p = (1-p) = 0.5", "fa2764b2631fc07253fa907d395b1578": "H_\\epsilon(R) \\in Q_\\epsilon(R \\oplus R^*)", "fa2778f9e12e816ac024324a44acb69b": "(N+1)p\\,", "fa27b5848b6aeda3c41bf7eaa4d12e4b": " \\frac{\\mathrm{d}\\mathbf{p}_\\mathrm{i}}{\\mathrm{d}t} = \\mathbf{F}_{E} + \\sum_{\\mathrm{i} \\neq \\mathrm{j}} \\mathbf{F}_\\mathrm{ij} \\,\\!", "fa27ddc66c647d3d6910c5d173a918b6": "\n\\bar{j}(e^{i \\theta}) := - \\frac{1}{2 \\pi} \\frac{\\partial}{\\partial r} T(e^{i \\theta}) =\n\\begin{cases}\n0, & \\text{if }\\varepsilon < \\theta < 2 \\pi -\\varepsilon \\\\\n\\frac{1}{2 \\pi} \\frac{\\cos{\\frac{\\theta}{2}}}{\\sqrt {\\sin^2{ \\frac{\\varepsilon}{2}} - \\sin^2{ \\frac{\\theta}{2}}}}, & \\text{if } \\vert \\theta \\vert < \\varepsilon\n\\end{cases}\n", "fa27df11e7d1a3e58d32f01916055cdd": "=\\{ 2k+1: k \\in \\mathbb{Z} \\}", "fa27ebd6443285634a92aaca58588941": "16y^5-20y^3+5y=0\\,", "fa2802c2dcddf358268bcf882423fd58": "Q [ i w ]", "fa2807478291de6856056c5a5093fe4a": "\\{\\tau < \\infty\\}", "fa2821607c52c0bdf6dd829f103532b0": "i \\frac{\\partial \\tilde{a}}{\\partial z} + \\Delta \\beta (\\omega) \\tilde{a} = 0", "fa283cb884ae143eb490486ca808fe5b": "x - 1 = 1 + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{7} + \\frac{1}{9} + \\frac{1}{10} + \\frac{1}{11} + \\cdots", "fa28427496fe19e88f1c274e19ad09b5": " l = \\frac{\\sum_{i=1}^R n_i (n_i -1)}{N (N-1)} ", "fa2851f2a025298eb4b9738eb8e0885b": " \\frac{d}{dt} \\langle Q \\rangle = \\frac{-1}{i\\hbar} \\langle \\psi | \\left[ H,Q \\right] | \\psi\\rangle \\,", "fa290e80fb2468f98e6c2f3eb64180ea": "(a_1b_1 - a_2b_2 - a_3b_3 - a_4b_4)^2+\\,", "fa297ed74952d737b7b5d62e9f32ab5c": " \\Delta \\sigma_{z}", "fa29d365cd0fb3f9a149dcc4281ddee7": "H(X|Y)", "fa2a05e500a57ae49e647d7a53335aa9": "\\displaystyle{\\partial_t f_t(z)= zp_t(z) \\partial_zf_t(z)}", "fa2a13fa9e6e55f77ce96eda7aec05e1": " \\overline{A}_M: \\overline{M}\\to E^*,\\;\\; c\\mapsto \\left(\nh\\mapsto \\int_c h \\right),", "fa2a284f4d8f81346375b5da61056b1e": "\\Pi_r \\ge \\Pi_0 +T \\,", "fa2a808d57e53972b9d53be87d91dc1f": "\\left(\\frac{k_H}{k_T}\\right)_s=\\left(\\frac{k_H}{k_D}\\right)_s^{1.44}", "fa2af34dcec383e0b55f20e8ac4236d0": "K^y", "fa2af7c3f1038d831281c732f6ad86c6": "X \\cap (\\cap_{n<\\omega} W_n) \\neq \\emptyset", "fa2b0fae08bedbe55481e5e450243f61": "(x-y/2)^2 \\geq 0", "fa2b3ed87882d01eb10f80daebb84c0b": "(T,T+H]", "fa2b46c1a77330bde4e2dc9b424af770": "F\\underline{A}", "fa2b6d849933dc7d3b7170da2c9a3e5a": " x_0 \\le t_1 \\le x_1\\le \\cdots \\le x_{n-1} \\le t_n \\le x_n \\,\\!", "fa2be674b84a06f18bec3f2ae3294d7f": "{6\\choose 4}{43\\choose 2}\\over {49\\choose 6}", "fa2c3b3037ef2fc0038a0ffc6ed030e5": "\\displaystyle e^{-\\alpha x^2}\\,", "fa2c69b2676bd4f74ca770cb1a859395": "f(e_0)", "fa2cef61c7321263869fb7a1bd0211eb": "\\{ e\\in \\Gamma : c_e \\neq 0 \\}", "fa2d08d8a9b6e0564f297693a4af85b5": "\\vec J_c = \\frac{NeZD\\rho}{kT}\\vec j", "fa2d29519973244db0963b19b4129f34": "(\\mathbf{v} \\cdot \\nabla) \\mathbf{v}", "fa2d2dfaf12c5001a79eb45e84d811bc": "\\textrm{Var}\\left(m^{\\star}\\right)=\\textrm{Var}\\left(m\\right) + c^2\\,\\textrm{Var}\\left(t\\right) + 2c\\,\\textrm{Cov}\\left(m,t\\right);", "fa2d4d7782f43991b8bdc767a623c7af": "\\omega=\\frac{2(\\Phi_0-\\Phi)}{D-2},", "fa2dd8c5cfafa5173cdf833da6fc6ee0": "\\forall a \\in T: \\sum_{j \\in S(a)} p_j \\leq W(a).\\,", "fa2de35928cf6b51942529971f67138b": "\\zeta = {\\rm tanh}^{-1} \\rho", "fa2dfff1936549e5e0f32e568f419ff3": "[\\cdot_{\\lambda}\\cdot]:R\\otimes R\\rightarrow\\mathbb{C}[\\lambda]\\otimes R", "fa2e05d2a59855c3f4babfb9c1d5a2e2": "\n \\alpha_m = \\frac{\\pi (2m-1)b}{2a} \\,. \n", "fa2e593ab44f8e9d4d05597351962916": "\\,K( \\alpha+1 ) = f( K( \\alpha ) ).", "fa2e5b757e35dfb3cc8d1bfe0076915b": "\\mbox{knots} \\approx 2.5 \\times \\sqrt{L \\mbox{m}}", "fa2e8bb03301f9ced7f54f5674a834d1": "s_1 s_2", "fa2e8c368c5be4e8d9b08536b1141de3": "SL(k_1+1) \\otimes \\cdots \\otimes SL(k_r+1)", "fa2e96735b294199ee64605188e5ac0e": "\\textstyle y_{12} = y_{21}", "fa2ef058b9f8dbe80aa3b4106d28b76a": " \\langle \\nabla f(x) - \\nabla f(y), (x-y) \\rangle \\ge m \\|x-y\\|^2 ", "fa2ef7449788fbef0a2c5e56cc6d0894": "\\{\\gamma_k(X)\\}", "fa2f016e936694d81339a6ae6d3b11f7": "\nR[\\Omega_0^2 - \\Omega^2(R_0+x)] \\simeq -x R{d\\Omega^2\\over dR}", "fa2f36382cde14301cd99e5a0daa08d1": "P(u) = 5 \\delta(u+10) + 18 \\delta(u+8) + 9 \\delta(u+6) + 6 \\delta(u+5) + 6 \\delta(u+3) + 18 \\delta(u+2) + 45 \\delta(u) \\, ", "fa2f42b318537aba5e4a62ddabd10b2f": "x + 2", "fa2f6d8a68ffa3090047a37119ba10c9": "\\Phi_E = \\int_S \\mathbf{E} \\cdot \\mathrm{d} \\mathbf{A}\\,\\!", "fa2f73975764b176b21011b7e453df27": "\\dot C_{FF} (t) = \\varepsilon \\,k\\,F_m \\,\\phi \\,Q_0 \\,t", "fa2f901885ec0af92d4478a0ecea743b": "X = (\\xi_1,\\xi_2,\\xi_3,\\xi_4) \\leftrightarrow \\overrightarrow{(\\xi_1,\\xi_2,\\xi_3,\\xi_4)} = (x,y,z,t) = \\overrightarrow{X}.", "fa2ffc661009a1a4af266a91d371fbc3": "p(m\\mid z_{1:t}, x_{1:t})", "fa300329f0f8c94ebd99c211b402c981": "{\\scriptstyle \\Gamma}", "fa302e5039e00f720b8575623bc17ea5": "x(t) \\approx \\sum {p(\\tau ) \\cdot \\Delta \\tau \\cdot h} (t - \\tau )", "fa306bc60a9640a25360a66323d6d2de": "-\\frac{\\partial\\mathcal{E}(n)}{\\partial v_j(n)} = \\phi^\\prime (v_j(n))\\sum_k -\\frac{\\partial\\mathcal{E}(n)}{\\partial v_k(n)} w_{kj}(n)", "fa307d5e8960e7ce7beaccece81c046c": "\\scriptstyle a^{*}_{k}a_{k}", "fa30856d0d1e37fe1a94fc3432780ace": "\n\\hat{\\sigma}^2 = \\frac{1}{N-1}\\sum_{i=1}^N(x_i-\\bar{x})^2 = \\frac{N}{N-1}\\left(\\frac{1}{N}\\left(\\sum_{i=1}^N x_i^2\\right) - \\bar{x}^2\\right)\n\\equiv \\frac{1}{N-1}\\left(\\left(\\sum_{i=1}^N x_i^2\\right) - N \\left(\\bar{x}\\right)^2\\right) . \n", "fa30d4b70bf931d917a420908c6324e9": "\\ \\sigma(t)", "fa30d567a127b58078d5c6dfd91e1ca1": "E_\\text{k} = - m g_{tt} u^t u_{\\text{obs}}^t - m c^2 = m c^2 \\sqrt{\\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\\,.", "fa311f23fbd99b330af737022bece5fb": "x_2 = \\rho z_1 + \\sqrt{1-\\rho^2} z_2", "fa312e7ec9ea3b9f3c71f6c2fda0f976": "{\\mathcal L}_{xy}^6: L=Exquo\\big(L,\\mathbb{L}_{y^n}(L)\\big)\\mathbb{L}_{y^n}(L)=\n \\left(\\begin{array}{cc} 1 & 0\\\\ 0 & \\partial_x+A_2\\end{array}\\right)\n \\left(\\begin{array}{c} L\\\\ {\\mathfrak k}_n\\end{array}\\right);", "fa3210131626fb0108615408564bd294": "h(x) = \\sum_{n=0}^{\\infty} f_n g_n x^n", "fa3231e180a5f3dc76e9b9bc34e54090": " M(F) \\neq 0 ", "fa32fdd0be1a399b1ca05b9d011b06b6": "S = \\int d^4 x \\; e \\;e_I^\\alpha e_J^\\beta \\; F_{\\alpha \\beta}^{\\;\\;\\;\\; IJ}.", "fa330e3ef05bd572b855cab8c5684bd3": "T_i = \\tau_i \\cdot \\alpha", "fa33a1205d13e69d76c352345b660990": " F_{diff} ", "fa33e0d9b3283c2398d1f622a5759c9b": " 3 \\beta /(\\alpha \\sqrt{\\delta \\gamma})", "fa341828190f4f99998ee598c61a4d96": "k_0 \\in (-\\infty,\\infty)\\,", "fa34189ba0b89ee8dc08fd849b08cf7b": "\\beta_j^2=\\operatorname{var}[S_j-T_j];", "fa346db4ae0286d4c297bdfea3041cab": "n = 3", "fa34d0876f4afdfd6dcf1f4a56a7084e": "\\!\\phi^D", "fa34fdc907ab48c3628c6e21195bf39e": "\\displaystyle{U_g=ST_gS^{-1}}", "fa350194f88881caec919847c055a093": "f_{uc}(\\cdot)", "fa3506681922fe79ed72e20d19c0c616": "( r_R, \\theta_R, z_R )", "fa352912a108b0b8d303f3fcb56bdfd3": "|P(Z)| \\leq \\|P\\| \\, \\|Z\\|_E^d", "fa352e5f879405ec42e2c4bbf12e2915": "\\gamma^\\mu\\gamma^\\mu", "fa3539f154297ea8f82a2e8afabbaa1d": " Total \\ Time = Available \\ Time + Down \\ Time", "fa354364dd1db33b4c8ee442c9f71d2b": " u(r) = \\sum^N_{n=0} \\frac{iC_n}{\\rho n \\omega} \\left[ 1 - \\frac{J_0(\\alpha \\frac{r}{R} n^{1/2}i^{3/2})}{J_0(\\alpha n^{1/2}i^{3/2})} \\right] e^{in\\omega t} \\, .", "fa35f6a2f49f907b0103c2e03127fd3b": "\\{X_i - S_i\\}", "fa3619700d7e45eae14608566f9ccb9a": "\\sigma = \\rho \\cdot r_0 \\cdot g_0.", "fa364fcc64ed1d7abcc975121c76f7d4": " \\rho = \\frac{\\sum_i(x_i-\\bar{x})(y_i-\\bar{y})}{\\sqrt{\\sum_i (x_i-\\bar{x})^2 \\sum_i(y_i-\\bar{y})^2}}", "fa366a699146f4d6637db55c5d1301f0": "\\left(\\frac{y}{x^2}\\right)' = 0", "fa378e8876a2fd7a41488908c35004cb": "\\phi(p),", "fa37e840f7949c480c3b88cfe5e443cf": "|G:H| = \\frac{|G|}{|H|}.", "fa38183a39cce94f84268d0820845c0b": "\n\\sum_{i=1}^{N} \\mathbf{r}^{(i)} = 0\n", "fa38506f2e0e6bb598165f3b31ac622f": " \\int { d^3 k \\over \\left ( 2 \\pi \\right ) ^3 } { \\exp \\left ( i\\mathbf k \\cdot \\mathbf r \\right) \\over k^2 } = {1 \\over 4 \\pi r } .", "fa388e380790def66692f77c17084203": " n_N", "fa38b4d90b26d63b546d146a2ec1ac3f": "= \\frac{ln(2)}{k_\\text{e}}", "fa391ef417d6912b70fee51574692f1d": "\n \\color{Gray}\n \\Rightarrow \\quad\n \\frac{\\partial^2 \\Phi}{\\partial t^2}\n + g\\, \\frac{\\partial \\Phi}{\\partial z}\n + \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\frac{\\partial \\Phi}{\\partial t}\n + \\tfrac12\\, \\frac{\\partial}{\\partial t} \\left( |\\mathbf{u}|^2 \\right)\n + \\tfrac12\\, \\mathbf{u} \\cdot \\boldsymbol{\\nabla} \\left( |\\mathbf{u}|^2 \\right)\n = 0\n", "fa3975ee618f5e4f14e8a83e0b1ee796": "\\varrho\\,", "fa39ab53ef2c950c2b89a63ade0220b9": "f_k(n)=f_{k-1}^n(1)", "fa39eff545ae4baa5cac3dea7409c2f2": "\\hat{X}^{opt}(z^n)", "fa39f95cfc7a95cd4283c356671770a9": "f:G\\to\\mathbf{R}", "fa3a1e3e6c55c348ae5470c05788a66b": " z^{}_{}=xyx^{-1}y^{-1},\\ x^p=y^p=z^p=1,\\ xz=zx,\\ yz=zy. ", "fa3a6a0a379c7f3f92e78b6493395801": "2^p \\leq |X^*| = q^2 - 1 < q^2.", "fa3aa82225c5c40fa1ede5b9e7125b39": "V_2\\,", "fa3acd8f8909f2bb7f60593a3e905670": "\nC_b = {C_{p} \\cdot C_{m} }\n", "fa3afc8be1c422f115dec58718f9e23e": "1\\leqslant k.", "fa3b015a05fb8d68370641ee5be648a5": "j \\in \\mathbb{Z}", "fa3b4d77d6d93c9fac3fa3a66d5e9b88": "{\\rm cov}({\\mathcal L}) \\le {\\rm non}({\\mathcal K})", "fa3b5d7509a5d60b480e7ce869e0b6ea": "K= \\left[LGD *N\\left(\\sqrt{\\frac{1}{1-R}} * G(PD) +\\sqrt{\\frac{R}{1-R}}*G(0.999)\\right) - (LGD*PD)\\right] * \\frac{1+(M-2.5) b}{1-1.5 b} ", "fa3b6adae79e2a5ce5eb68c6a270d451": "2/m\\,\\!", "fa3b6c9879958f01117772b5ad25579b": "G|_{t\\times s}", "fa3baff193a41232598e9c409b18c165": "(r_0, \\phi)", "fa3bbcd23889bc469ebea061d18ac6d1": "[b,a] = (a,a) = [a,a) = (a,a] = \\{ \\} = \\emptyset", "fa3c0d25b23db3b73bd20b9daa48a248": "k={\\rm lcm}(1,\\dots ,B)", "fa3c1bef4b0809bbf1ff0db8e80d3198": "2^\\alpha", "fa3c5c72c2540595f83903ad2b3ea3bd": "w = (\\langle \\overline M \\rangle, 10^k)", "fa3c890337e86e62ea47cf839c6de736": "u_t=6uu_x-u_{xxx}.\\,", "fa3cd2cbcad3c89e011a182366d72a9d": "\\mathbf J_\\sigma", "fa3d8308e3b2a2cec01bc5eb480fdfdb": "\\eta = \\frac{1}{2} \\ln \\left(\\frac{\\left|\\mathbf{p}\\right|+p_\\text{L}}{\\left|\\mathbf{p}\\right|-p_\\text{L}}\\right),", "fa3dbcbaeabd35dbb24e3f66f0cbfe14": "V > 0", "fa3dd285e627268759c1aa064bc3dd58": "H[\\xi]=\\int_{-\\infty}^{+\\infty}S(\\Phi(x))dx", "fa3de0c59fb1dd49aea6dac2a45e21ac": "\\ell'=\\ell''", "fa3df1ae5656807c3dc8ab25533a9ac5": "{x^2\\over a^2}+{y^2\\over b^2}-1=0.", "fa3e139b6cc6bc1d7dccb1696b6eb811": "\\begin{array}{ccl}\nd^n_{C(f)}(a^{n + 1}, b^n) &=& \\begin{pmatrix} d^n_{A[1]} & 0 \\\\ f[1]^n & d^n_B \\end{pmatrix} \\begin{pmatrix} a^{n + 1} \\\\ b^n \\end{pmatrix} \\\\\n &=& \\begin{pmatrix} - d^{n + 1}_A & 0 \\\\ f^{n + 1} & d^n_B \\end{pmatrix} \\begin{pmatrix} a^{n + 1} \\\\ b^n \\end{pmatrix} \\\\\n &=& \\begin{pmatrix} - d^{n + 1}_A (a^{n + 1}) \\\\ f^{n + 1}(a^{n + 1}) + d^n_B(b^n) \\end{pmatrix}\\\\\n &=& \\left(- d^{n + 1}_A (a^{n + 1}), f^{n + 1}(a^{n + 1}) + d^n_B(b^n)\\right).\n\\end{array}\n", "fa3e2ff654ee1203ec1438ddfe3ff0a0": "\\{n_k\\}_{k\\in\\mathbb N}", "fa3e3d55703d5ac351bdd539e3597eb6": "\n\\dot{z}=\\{z,H(z)\\} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)\n", "fa3e5da8497282290009964108872013": " f(t) = \\mathcal{L}^{-1} \\{ F(s) \\} ", "fa3e9ee30f7e3d601171c83def4e74de": "\\mathrm{Glucose} + \\mathrm{Alkaline\\ ferricyanide}\\longrightarrow \\mathrm{Ferrocyanide}", "fa3ebd6742c360b2d9652b7f78d9bd7d": "Cat", "fa3f0e179473f180a0142701b46867d9": "\\nabla^2 = \\partial_{\\rho\\rho}+\\frac{1}{\\rho}\\,\\partial_\\rho +\\frac{1}{\\rho^2}\\partial_{\\phi\\phi}+\\partial_{zz}", "fa402e1a05eb78e37322e35145f48279": "f/c", "fa404615425901bdf11cda164d14f426": " \\frac{1}{T}\\int_{0}^{T}{c(t)\\, dt} > r \\rightarrow \\lim_{t \\rightarrow +\\infty}x(t)=0 ", "fa40544dc8db6f325ca79c828b27afe8": "\\kappa\\theta\\ = L\\frac{GmM}{r^2} \\qquad\\qquad\\qquad(1)\\,", "fa407806ec2b5d8d7cd6f10e4abbf816": "\\frac {\\bar{X}_m}{\\bar{X}_f} ,", "fa409eca899cd9db286856b08ede5122": "E = \\gamma_{SL}(A_{SL} + A_{SG}) + \\gamma_{LG} A_{LG} + \\gamma_{LG} A_{SG} \\cos(\\theta)\\,", "fa413ab8c5165ba530f9504c26fa9648": "\nV = \\frac{1}{2} \\kappa (B_\\mu B^\\mu \\mp b^2)^2 ,\n", "fa41647e735e4e7e935fceaff9e40848": "W = Fd.", "fa41a589659a6523f055ad14eda46135": "\\langle r, a \\mid r^3 = 1, a^2 = 1, ara = r^{-1} \\rangle", "fa41b26ec24719e2ddc70c82315faa32": "G_{dBd} = G_{dBi} - 2.15dB ", "fa41b2867c65c3586aed8264c64dab78": "\\scriptstyle\\varphi(x)=x^p", "fa41de9715d4a676f49c73abe56aed04": "\\scriptstyle \\Bbb{Z}/n\\Bbb{Z}", "fa41e751c58c96a4bba465e4acd9b634": "2 = \\{0, 1\\}", "fa41ec5b29234a3ee3cdf6e5bed3f9b1": "\n\\mathbf{B} = \\mathbf{p} - \\left(\\frac{mk}{L^{2}r} \\right) \\ \\left( \\mathbf{L} \\times \\mathbf{r} \\right)\n", "fa41f50d9fc4c3a61a9b6c8370a958ce": "\\rho \\!", "fa4266b905811eccb1e6db5ceed34cdf": " C \\propto |T_c - T|^{-\\alpha}.", "fa426fc3181a55b6048dc1b28a43080e": "\\mathcal L(K) ", "fa427fe22c56780df11b1c3ad102633e": "{Q_e \\rightarrow 0}", "fa42a89a5e63222950fdcaa49cbc857d": " \\left(\\begin{matrix}\\frac {I_C}{I_S}\\end{matrix}+1\\right) \\ , ", "fa430f4dcd089a5f4ffb694cc7ef4fee": "\\frac{\\mathrm{d}}{\\mathrm{d}t} \\left ( \\frac {\\partial L}{\\partial \\dot{q}_j} \\right ) = \\frac {\\partial L}{\\partial q_j} ", "fa43f4dc714f2f6a3d151f9c817649e2": " \\hat B= YZ^{'}(ZZ^{'})^{-1} ", "fa442501c5a277532bac60665bbd5177": "\\sin(\\alpha \\pm \\beta) = \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\!", "fa442a6b6b368aaa6a48bdba0f5ed076": "\\Big( \\pi \\models \\phi_1 \\lor \\phi_2 \\Big) \\Leftrightarrow \\Big( \\big(\\pi \\models \\phi_1 \\big) \\lor \\big(\\pi \\models \\phi_2 \\big) \\Big)", "fa448518561f6fe3c9148c90610339c5": "f \\colon M \\to N", "fa44c264665b50bb7741a6425363b192": "\\overline {AB}\\,\\!", "fa44e1cc29d2c876506e13cfdcb6beb9": "u'\\left(x_1\\right)\\leq 0", "fa44fb90b676aefc103849c21f81a910": " Z_N(K^*,L^*) = 2(\\tanh K \\; \\tanh L)^{-N/2} \\sum_{P} v^r w^s ", "fa45088f7c01264c2f9537d424f9b861": "c_1(M)", "fa4538edb2a4d9ea8e2173293416484a": "Y(s) = 0", "fa4543c7c0fcefdaf9396d785d51535f": "\\phi_{\\varepsilon_{\\Omega+1}}(0)", "fa454de4104c723f959fb809dab34a20": "\n\\begin{align}\n\\lambda & = \\frac{\\sigma_w T^2}{\\sigma_v} \\\\\n\\alpha & = \\frac{-\\lambda^2 + \\sqrt{\\lambda^4 + 16\\lambda^2}}{8}\n\\end{align}\n", "fa45671251c8fcfbd4646761015e2b1f": "\\sum F_y=0=R_{Ay}+F_{AD}\\sin(60)=5+F_{AD}\\frac{\\sqrt{3} }{2} \\Rightarrow F_{AD}=-\\frac{10}{\\sqrt{3}}", "fa4621cc880e3a22604b56d01ee7205a": "d \\circ c^{-1} \\colon c(U \\cap V) \\to \\mathbb{R}^n\\;", "fa469c5d88ed531484da7720a59fa52d": "tan \\beta_m = \\frac {1} {2} (tan\\beta_1 + tan\\beta_2) ", "fa46ec0b4924e8c2194a53ef61b94039": "kh", "fa4712c43a7227d63bb29bc3d41f47bd": "x_\\xi y_\\eta - x_\\eta y_\\xi = I", "fa4713309214201984ed414ba1eaadb6": " I_x(a,b) = 1 - I_{1-x}(b,a) \\, ", "fa4716ede81b92f6b21e6dde1e4b4d68": "g\\cdot", "fa471c5b6fbd963d21e6c278522c7da1": "W(t,s)", "fa47202af55db3efc864161f7e9f1ae8": "\\frac{L}{2\\pi r}=\\frac{\\theta}{2\\pi},\\,\\!", "fa4720c70bfade802f11c2046a530076": "\\chi_\\lambda d(\\lambda)^{-1/2}", "fa476bdc0ae71842b1452bedc5153b4a": "I_a/S=0.20", "fa47a842ba91b64f3a57851cfae29dee": "K \\cap M = \\emptyset", "fa47a9522fec36675d8ea5c212686a83": "\\scriptstyle \\bar A", "fa47d1373ffc8326a59f00b427497240": "\\frac{f_{\\theta_1}(x)}{1-F_{\\theta_1}(x)} \\leq \\frac{f_{\\theta_0}(x)}{1-F_{\\theta_0}(x)} \\ \\forall x ", "fa4814ff616ec014e8be94552836b002": "\\sum_{n=-\\infty}^\\infty g(nP)=\\frac{1}{P}\\sum_{k=-\\infty}^\\infty \\hat g\\left(\\frac{k}{P}\\right)", "fa482fabab16b21e8650fbb56e25c4d8": "\\|F(\\lambda)\\| \\le \\|T\\|\\ \\forall \\lambda.", "fa4840be676fcb610f25de2ab3171996": "{\\rm MCG}(X) = {\\rm Homeo}(X) / {\\rm Homeo}_0(X)", "fa484e8932f8db31d795b646c9622071": "s_j \\, r_k =s_{(j-k) \\text{ mod }n}", "fa4854ebd6928f402971e497ade47ffa": "\\omega = \\frac{Dp}{Dt}", "fa4894f8fb81f0e27d95f842893b934a": " \\left ( {\\partial S\\over \\partial U} \\right )_{V,N} = { 1\\over T } ", "fa489de02826012996b890b252e62402": "\\frac{\\partial^2 u}{\\partial t^2} = c^2 \\left(\\frac{\\partial^2 u}{\\partial r^2}+\\frac {1}{r}\\frac{\\partial u}{\\partial r}\\right) .", "fa48c2d5100a7812c71b28adccf4a0f6": "\\mathrm{N}_\\mathrm{BK} = \\frac{u \\mu}{k_\\mathrm{rw}\\sigma} ", "fa48d4e72484dac4e85781605f100514": "\\phi/2", "fa48f459cb4e418bab26ae008d52702e": "c < \\exp{ \\left(K_1 \\operatorname{rad}(abc)^{15}\\right) } ", "fa4901b5450712c27e57ca59552f21ce": "\\boldsymbol{e}_k\\, \\sqrt{\\frac{g}{h}}\\, a\\, \\cos\\, \\theta\\,", "fa492c72c81a99ccf70be8b7d1030595": " H(k[\\Delta]; t,\\ldots,t) = \n\\frac{1}{(1-t)^n}\\sum_{i=0}^d f_{i-1} t^i(1-t)^{n-i}, ", "fa493c2c6db11b4cbbf43270249e65b9": "L_2 = 0.177 \\lambda\\,", "fa496f3c997681ee4d922fe4d9f0c035": "KG_{5,2}", "fa497ade2aa7c4f05a5c589f56d8a2d2": "\\begin{align}0&=f\\left(\\frac{dx}{dt},x,y,t\\right),\\\\0&=g\\left(x,y,t\\right).\\end{align}", "fa49ce15a8ed7402c5b157aba455af5f": "\\mathrm FM\\to M", "fa4a2ac4baa0b4395f7debf9ca7ef21c": "\\phi_t(x):= \\frac{\\exp \\{ -x^2/(2t) \\}}{\\sqrt{2 \\pi t}} \\quad \\text{and} \\quad \\Phi_t(x,y):= \\int^y_x\\phi_t(w) \\, dw,", "fa4a389808161f3cc02a7d17069a9ba9": "l = \\frac{3x_1^2 + 2Ax_1 + 1}{2By_1}", "fa4a6127cb299b901f2ef15298b1e040": "f(n-1,k)", "fa4aa3cfccfc18e3b853b554388360f1": "\\widehat{d}:\\{0,1\\}^*\\times\\N\\to{\\mathbb{Q}}", "fa4ab4cc40677a5dcbcb77a7eaaf7a57": " e^{\\alpha \\hat{a}^{\\dagger} - \\alpha^*\\hat{a}} e^{\\beta\\hat{a}^{\\dagger} - \\beta^*\\hat{a}} = e^{(\\alpha + \\beta)\\hat{a}^{\\dagger} - (\\beta^*+\\alpha^*)\\hat{a}} e^{(\\alpha\\beta^*-\\alpha^*\\beta)/2}. ", "fa4b364cb3853534b0c22e072952b2f0": "\n\\mathbf{J} = \\frac{a}{24}(\\mathbf{v}^2_0 + \\mathbf{v}^2_1 + \\mathbf{v}^2_2 + (\\mathbf{v}_0 + \\mathbf{v}_1 + \\mathbf{v}_2)^2)\\mathbf{I} - a \\mathbf{V}^{\\mathrm{T}} \\mathbf{S} \\mathbf{V}\n", "fa4b58893829a1fa5134104b435451a3": "V - E + F = 2", "fa4b8456120167ed6d30aaf7d7a39be2": "\\operatorname{curl} \\, \\operatorname{curl} \\mathbf{A} \\equiv \\nabla \\times (\\nabla \\times \\mathbf{A}) = \\nabla (\\nabla \\cdot \\mathbf{A}) - \\nabla^2 \\mathbf{A}", "fa4b996104033137e335325441de8bb3": "H(X,Y,Z,\\cdots)", "fa4b99ca33d9af045448f9bdfa644137": "\\hat{\\mathcal{F}}", "fa4b9bfad665c4d7dad465989e39e975": "\\Phi_{255255}(x)", "fa4bb1b76e7530e62ad3825469b86d38": "\\sqrt{I} = \\bigcap_{P \\in V(I)} (X_1 - a_1, \\cdots, X_n - a_n), \\quad P = (a_1, \\cdots, a_n).", "fa4bd4cfc7094d1f65f2853e77a9bc8d": "A \\cup A^{c} =U .", "fa4c02b79eaa39f7d74617a82a8bdbf9": "T : \\mathcal{B} \\rightarrow \\mathcal{C}", "fa4c373872c4a1b39e2fa363b9437a2f": "dU = T\\, dS - X\\, dx\\,", "fa4c3fc1d7862fee6cacbf187f7f356c": "y-y_0", "fa4c96bc0b351a4ef6ee6a157f6c86f7": "\\lambda \\in E", "fa4d0bf4d0437898490340486cd64358": " d\\mathbf x = \\frac {\\partial \\mathbf x} {\\partial \\mathbf X}\\,d\\mathbf X=\\mathbf F \\,d\\mathbf{X} \\qquad\\text{or} \\qquad dx_j=\\frac{\\partial x_j}{\\partial X_M} \\,dX_M\\,\\!", "fa4d1895351cb1fcdab28211e22d1b1b": "\\mathrm{proj}_{j}(\\vec{x}) = x_j", "fa4d1d43248e5bab17f05f6377113cf1": "n_i \\in \\mathbb{Z}", "fa4d5761a5dc886bf1d57d1effd74583": "16,384 - 844 = 15,540\\,", "fa4d82bab0fb9ceb64c0a501f1486f81": "T_u = 2.725 \\; \\text{K} \\;", "fa4dbad139656acf726a6c7635282a9d": "(\\Delta E \\sim 1/r^2)", "fa4dc4c233597a6b4ae6db21e6a32914": "\n\\begin{align}\np & = { 1469 \\over 1750} = 0.83943 \\\\\n\n2q & = { 2 \\times 138 \\over 1750} = 0.15771 \\\\\n\nr & = { 5 \\over 1750} = 0.00286\n\\end{align}\n", "fa4e093089ae9c61ba3faa48766e580d": "\n{d s}^{2} =\n\\left( \\eta_{ab} + \\frac{r_s^{n-p-3}}{r^{n-p-3}} u_a u_b \\right) d \\sigma^a d \\sigma^b + \\left(1-\\frac{r_s^{n-p-3}}{r^{n-p-3}}\\right)^{-1} dr^2 + r^2 d \\Omega^2_{n-p-2}\n", "fa4f1e23d2554f09c525b2f94085d9b3": "\\tau_3", "fa4f2f80b57eee87e66d60e38b80f3b3": "y_t = x_{t+1}/x_t", "fa4f404438db4fefdbbce2b78e3ac758": "rho_f", "fa5024414ff17aa6aba0125a70aa56ba": "A_i(x):=\\sum_{k=0}^{\\nu_i-1} \\frac{1}{k!}\\left(\\frac{P}{Q_i}\\right)^{(k)}(\\lambda_i)\\ (x-\\lambda_i)^k.\n", "fa5056d2de1b0193c1aa8e83c25dc14d": "\\frac{|SC||AF|}{|CD||AF|}=\\frac{|SA||EC|}{|AB||EC|}", "fa50e123b495ff93f649c230583849d7": "\\bold{r}_{uv}=\\Gamma^1{}_{12} \\bold{r}_u + \\Gamma^2{}_{12} \\bold{r}_v + M \\bold{n}", "fa50e7d1dc85ff2230496fd1fabf0bf2": "P = C \\times V^2 \\times F", "fa510cadde3499d264d01c065b496af5": " e a_0 ", "fa514c793b08a5ecb7b325eaa0ea4bba": " \\left(\\frac{M}{L T}\\right)", "fa516f26e70186256d3b9af6d326dddc": "X_{K}\\,", "fa51a54a9435b93b80e3fcfc846d0bc4": "\\lambda = \\frac{h}{p}", "fa51f1111d41f1da1ec6170b6fb62004": "\\mathcal{L}=\\overline{\\psi} \\left(i\\partial\\!\\!\\!/-m \\right) \\psi + g\\frac{\\left(\\overline{\\psi} \\psi\\right)^{k+1}}{k+1}", "fa521fab6a246ce0af9dbb7cd423d6a7": "\\neg\\neg A \\leftrightarrow A", "fa53692caf62151afd53e806e6282013": "A_0,", "fa536c3428b393e109bb6279777deb5b": " (\\forall x \\neg(\\phi \\lor \\psi)) \\lor \\forall z \\rho", "fa542918514b5c19a4f1243eed8d6b8b": " F^*(g) =\\overline{F(g^{-1})}, ", "fa556505e0c347642fb92d867867a7da": "Q=qb", "fa55c6df257dffd1e4dc45e33b4fed20": "N(q)=1-504\\sum_{n=1}^\\infty \\frac {n^5q^n}{1-q^n}=E_6(\\tau),", "fa55da28a71a47d50ad1a7618a59f3ad": "p_y", "fa562a0e0d5cb68c82469dee5dd4d243": " w_2 - w_1 = 2 \\int (f_{ij} d(A(\\epsilon_{ij})) = 2 \\int(Af_{ij} d\\epsilon_{ij} )", "fa5656a659da6ea8c3886e4e62ffee66": "f(x, y) = (x^2+y-11)^2 + (x+y^2-7)^2.\\quad", "fa56a5b796d70399bcea6684208ce136": "\n\\begin{align}\n{\\Pr}_{x_1}[g_x (m)-g_x (m')\\equiv a \\mod p] &= {\\Pr}_{x_1}[(m_1 x_1+m_2 x_2+ \\cdots +m_k x_k )-(m'_1 x_1+m'_2 x_2+\\cdots+m'_k x_k )\\equiv a \\mod p]\\\\\n&= {\\Pr}_{x_1}[(m_1-m'_1)x_1+(m_2-m'_2)x_2+ \\cdots +(m_k-m'_k)x_k]\\equiv a \\mod p]\\\\\n&= {\\Pr}_{x_1}[(m_1-m'_1)x_1+\\textstyle \\sum_{k=2}^s(m_k-m'_k)x_k\\equiv a \\mod p]\\\\\n&= {\\Pr}_{x_1}[(m_1-m'_1)x_1\\equiv a - \\textstyle \\sum_{k=2}^s(m_k-m'_k)x_k \\mod p]\\\\\n&=\\frac {1}{p}\n\\end{align}\n", "fa56f2f3b19c77a418fe8913eab21a94": "{\\mathit{He}}_{n+1}(x)=x{\\mathit{He}}_n(x)-{\\mathit{He}}_n'(x).\\,\\!", "fa570b19926f53dcbdb93c0495bedcdf": "I(T, \\alpha)", "fa57100d28a90b5d696002d876f45200": " A_{2}(n,d) \\leq 2 \\left\\lfloor\\frac{d}{2d-n}\\right\\rfloor. ", "fa573d787bf2cf636d0e7fa2e8a48c67": "S^{2_{ }}", "fa580a38ba326f7d621818c6ed17153b": "P(t)=e^{-F(t)}", "fa583455e43412a852283ad62389dc82": "P_v^{fog}=P_v^o e^{\\frac{V 2\\gamma}{RT r_k}}", "fa58491a49db9a67dc3ef766c54fdb27": "x(t) = \\cos(t) + \\varepsilon \\left[ \\tfrac{1}{32}\\, \\left( \\cos(3t) - \\cos(t) \\right) - \\tfrac{3}{8}\\, t\\, \\sin(t) \\right] + \\cdots.\\,", "fa5851d9ed8299f4f9a7c27860723231": " \\mid \\langle \\psi_{x-} \\mid \\psi_{y-} \\rangle \\mid ^ 2 ", "fa58ce84a36b3633f7720da6cd1c1386": "P_1=P, P_2=Q, \\pi_1 = \\pi_2 = \\frac{1}{2}.\\ ", "fa58dad7a077bc011eabb181858a2454": "\\omega^\\beta = \\emptyset", "fa58f2700a3ef98e6f355298027d50a0": " [X, fY] = X(f) Y + f [X,Y]", "fa58f34cf3ba74f4c32cc1f7da6949bc": "\\lim_{x \\to \\infty^{+}}{f(x)} = L", "fa59253bd6098cc7fda58c797b0e16ba": "c\\delta(t)", "fa5928973d0a042faa93be8ee4756bb4": "X = \\bigcup B;", "fa59572a66d78971d721f2ee72c6d67a": "f(x)=g(x)=\\begin{cases}0 & \\text{if } x\\neq 0 \\\\ 1 & \\text{if } x=0 \\end{cases}", "fa598d6f9c56fdc770711ed794bd4d44": "\\deg f^*\\ge 2i", "fa59d2b28af2869f303efe4ca9f50a27": "\\text{rank}(G) = k", "fa59de47a3bef20ff0768d134bce2705": "\\mathbf{f}\\, \\mathbf{v}[\\mathbf{f}] = \\mathbf{f}A\\, \\mathbf{v}[\\mathbf{f}A],", "fa59fbaadbd175eaae9cc0cc6804fac5": "s(E)", "fa5abc7933eb3b149256fb633229f3b4": "\\cdot \\omega_{p},", "fa5acdcf69418d537275d106e021dfd3": " x = \\sqrt{2+x}. ", "fa5aff8fdd1eeda81011fbdad282ba49": "\\int_0^1 h(x)\\,dx \\ge \\int_{\\frac{1}{m}}^1{h(x)\\,dx} = \\sum_{n=1}^{m-1} \\int_{\\left(\\frac{1}{n+1},\\frac{1}{n}\\right]}{h(x)\\,dx} \\ge \\sum_{n=1}^{m-1} \\int_{\\left(\\frac{1}{n+1},\\frac{1}{n}\\right]}{n\\,dx}=\\sum_{n=1}^{m-1} \\frac{1}{n+1} \\to \\infty \\qquad \\text{as }m\\to\\infty ", "fa5b2ea7a483cdac07c521a892be78bb": "N^{1/d}", "fa5b3a5b753f1462cd544487f87ce221": " |R_k(z)| \\leq \\sum_{j=k+1}^\\infty \\frac{M_r |z-c|^j}{r^j} = \\frac{M_r}{r^{k+1}} \\frac{|z-c|^{k+1}}{1-\\frac{|z-c|}{r}} \\leq\n\\frac{M_r \\beta^{k+1}}{1-\\beta}\n, \\qquad \\frac{|z-c|}{r}\\leq \\beta < 1. ", "fa5b457cdeb0e9a836e5aceee57cb7ab": " p(x) = a(x-z_1)\\cdots (x-z_n)", "fa5c93cc22b3cefca4b6a25571da9f0f": "\\theta:(F,m)\\Rightarrow(G,n)", "fa5cbd67c37ce2f122dcb21d2b89814e": "B=-7", "fa5d4efef6e5d4a5557ed97edbd95cae": "\\epsilon_{sh}", "fa5d7a1a4f12ba9cad61be18da907873": "M'|_{\\sigma}", "fa5d982330a2e0ea83c3de09cac15d11": "\n\\begin{bmatrix}\n47 & 61 & 45 & 76 & 86\\\\\n107 & 43 & 38 & 33 & 94 \\\\\n89 & 68 & (63) & 58 & 37 \\\\\n32 & 93 & 88 & 83 & 19 \\\\\n40 & 50 & 81 & 65 & 79\\\\\n\\end{bmatrix}\n", "fa5d9b1a657b18e3547e05859c54da79": "\\mu^-(E):=-\\mu(E\\cap N)\\,", "fa5dae15a079761ad3f853182b909ee1": " c_{\\mathrm{p}} = \\sqrt {\\frac{K+\\frac{4}{3}G}{\\rho}} = \\sqrt {\\frac{Y (1-\\nu)}{\\rho (1+\\nu)(1 - 2 \\nu)}} ", "fa5db036d1c51948a03b8ec2028f547e": "\\{g_k\\}", "fa5dba8133255f7b3f12c40d7e8aba94": "V(\\mathbf{r}) = \\frac{1}{2} kr^{2}.", "fa5dbd2e43e8fa8770fbd45d3755b0ac": "\\theta=\\theta(x)\\,", "fa5dbfcda270c890cfc60e4caecc942a": "\n \\begin{align}\n & W(1,1,1) = 0 ~;~~\n \\cfrac{\\partial W}{\\partial \\lambda_i}(1,1,1) = 0 \\\\\n & \\cfrac{\\partial^2 W}{\\partial \\lambda_i \\partial \\lambda_j}(1,1,1) = \\lambda + 2\\mu\\delta_{ij}\n \\end{align}\n ", "fa5dd9ecdbeb60f9870539632ea0663a": "\\rho_P(r) = \\bigg(\\frac{3M}{4\\pi a^3}\\bigg)\\bigg(1+\\frac{r^2}{a^2}\\bigg)^{-\\frac{5}{2}}\\,,", "fa5e451bc2aa1cbf1dba30f1fd3631b6": "v= \\frac{dC_{\\mathrm{A}}}{dt} ", "fa5e8b486996457f5780ffac6b951f1b": "M(\\lambda)=M(\\lambda)^0\\supseteq M(\\lambda)^1\\supseteq M(\\lambda)^2\\supseteq\\cdots.", "fa5ea5d7504c87e5e113fdd9e48fe2c7": "\nR_{\\mathrm{g}}^{2} \\ \\stackrel{\\mathrm{def}}{=}\\ \n\\frac{1}{N} \\sum_{k=1}^{N} \\left( \\mathbf{r}_{k} - \\mathbf{r}_{\\mathrm{mean}} \\right)^{2} = \n\\frac{1}{N} \\sum_{k=1}^{N} \\left[ \\mathbf{r}_{k} \\cdot \\mathbf{r}_{k} + \n\\mathbf{r}_{\\mathrm{mean}} \\cdot \\mathbf{r}_{\\mathrm{mean}} \n - 2 \\mathbf{r}_{k} \\cdot \\mathbf{r}_{\\mathrm{mean}} \\right].\n", "fa5ea6ba6b7c5215dc3796689b8a8087": "\\zeta s_1", "fa5eb5598ad5bb347227b7f73b58439c": "|a_1 x_1+{}\\cdots{}+a_n x_n|=0.\\,", "fa5ec9246f068df63a0df28ce7dd6a2b": "\nF(M)=-H M+a (T-T_c) M^2+BM^4+\\ldots\n", "fa5f0d77926c9f8e3e205b1d905a9f0e": " |f_\\pm(x +iy)|< C |y|^{-N}", "fa5f22264f63abb6daa60adb8c80f4df": "\\delta_n", "fa5f2d1bbda233c4ed380678155550c6": "I=ss^T + nn^T", "fa5f4afbae08b38e41415e42b773c778": " (z_i, w_i ) ", "fa5f4d16075198a7d3223237a28c920a": "F_{y1}+F_{y2}=F_{load} \\,", "fa5f6b1ca0388f6503a7adcdba4c3c4a": "x^*\\in\\left(x_0,x_1\\right)", "fa5faadfc7736a2c4078ed403c2cc13f": "H_{\\beta}, H_{\\leq \\beta}", "fa5fbb78d43c7c0c6138e0a94389b1c0": "S=(I-A)^{-1}(I-A^{n+1})", "fa5fce69ff0b61de25f26e210786916d": "\\gamma = 3/4", "fa5fd24f918dae57bf159857453a9a9a": "\\text{Pad}_n^1=\\lbrace\\mathbf{\\xi}=(\\mu_j,\\eta_k), 0\\le j\\le n; 1\\le k\\le\\lfloor\\frac{n}{2}\\rfloor+1+\\delta_j\\rbrace,", "fa5fdcb0b3512eb4e3b445cef784cd88": "C_{XY}", "fa5fe24542426575a5e9ee9d0c28ea38": "\\frac { T_{01} } { T_{02} }\\ ", "fa608633e5db816bd110a03a865fc6ed": "Pmo = 0", "fa60c088b12c66de76c823277bd80f6b": "\\mathbf{v}[\\mathbf{f}'] = \\mathbf{v}[\\mathbf{f}J^{-1}] = J\\, \\mathbf{v}[\\mathbf{f}].", "fa60f969d06d8ba5b51b3758f570bfca": "\\phi_N \\,\\!", "fa61b11ca33f11b1a6f95f8f63387521": "\\mathcal{S} = \\mathcal{R}(X) + \\bar{\\mathcal{R}(X)}", "fa61fd6cb40bc2d66e1be61ee642798b": "\\hat{\\delta}(\\xi)=\\int_{-\\infty}^\\infty e^{-2\\pi i x \\xi}\\delta(x)\\,dx = 1.", "fa6207a87afd87c23ef19a798108e2e3": "e^{h_1\\gamma(\\tau(1))+\\dots+h_N\\gamma(\\tau(N))}", "fa6213663f51af0748c0f18950c94357": "\\boldsymbol{x} \\in \\mathcal{X}", "fa629e6a123bc5806a440905050306ff": "P_G = P_3 \\cdot P_2 \\cdot P_1 \\cdot P_0", "fa63791f85e0367bb9eed55472497c20": "(L^x_{T_a})_{x \\geq 0}", "fa63af32f7f72b685f6dd2a12aaefcda": "a^2 - 2\\operatorname{Re}z\\,a + |z|^2=0", "fa63f8f2eb6ec218d5154535c8290454": "g(x, y, s)", "fa642cdda0e4a3ebbdc1fa95f7c32719": "r = k(T)[A]^{m}[B]^{n}", "fa643c8a0ddb924bec6192ddee50a7d1": "\\text{Plateau Modulus}\\ (G_o)\\ (kPa)", "fa648e485d413b72d3da375c6c078a1f": "v = \\frac{q}{y}", "fa649a6d0b854774615893aa976d25d9": "\\displaystyle-17.57~\\mbox{dB}", "fa64c39c68cb3fa58476a04c9b13ccba": "\\mathbb{G}_m(R) = R^\\times", "fa64f474ece89f5f3c1af34dcef6b5c6": " (\\alpha_i\\wedge a_i) \\cdot \\psi = \\tfrac14 (2^{\\tfrac12})^{2} ( \\iota(\\alpha_i)(a_i\\wedge\\psi)-a_i\\wedge(\\iota(\\alpha_i)\\psi))\n= \\tfrac12 \\psi - a_i\\wedge(\\iota(\\alpha_i)\\psi).", "fa6545d7c46effc0bffd55aa3bc698ad": "\\frac{(a^2 + d^2 - b^2 - c^2)^2}{4} = (ad)^2 \\cos^2 \\alpha +(bc)^2 \\cos^2 \\gamma -2 abcd \\cos \\alpha \\cos \\gamma. \\,", "fa6588fcba753fad13a5028280c082dd": " 2.25 = 10f^{*} \\!", "fa65c6203f82f1609d397689e7d46ae2": " V_\\perp^2(\\beta) = V_x^2 + V_y^2, \\quad \\alpha = \\arctan \\frac{\\tilde V_y}{\\tilde V_x} ", "fa66325aa0bc951f4ea0c26bf6639a7c": "t = \\frac{\\hat\\beta - \\beta}{s_{\\hat\\beta}}\\ \\sim\\ t_{n-2},", "fa6634edbcb37ce0245aa3b7c9c86168": "x=X/Z", "fa66f3cca700bbec37ddb21f09cdd42e": "\\frac{|\\frac{q_\\bar{p}}{m_\\bar{p}}|}{(\\frac{q_p}{m_p})}", "fa66fb43309705bd7f377552205d71ac": " f \\in B_1\\left(X^*\\right) \\mapsto (f(x))_{x \\in X} \\in D.", "fa679fd35013f701e1a50d0f606bc4e1": "\\begin{pmatrix} \\ \\ & 0 \\\\ \\ \\nearrow & \\ \\\\ aR \\longrightarrow & 0 \\end{pmatrix}=\\begin{pmatrix} \\ \\ & a \\\\ \\ \\nearrow & \\ \\\\ 0 \\longrightarrow & a \\end{pmatrix}, \\quad\n\\begin{pmatrix} \\ \\ & a \\\\ \\ \\nearrow & \\ \\\\ 0 \\longrightarrow & 0 \\end{pmatrix}=\\begin{pmatrix} \\ \\ & 0 \\\\ \\ \\nearrow & \\ \\\\ aR \\longrightarrow & -a \\end{pmatrix}, \\quad\n\\begin{pmatrix} \\ \\ & 0 \\\\ \\ \\nearrow & \\ \\\\ 0 \\longrightarrow & a \\end{pmatrix}=\\begin{pmatrix} \\ \\ & -a \\\\ \\ \\nearrow & \\ \\\\ aR \\longrightarrow & 0 \\end{pmatrix}", "fa67e37e7c33889bd04f5c5491fb0ccd": " Q* = 2 \\pi a + 4 ", "fa68249a6803def4392f319d3dc467f3": " \\mu_1dN_1 + \\mu_2dN_2 + ... = 0\\,", "fa684f73c5b9a7eef4e3f7edd772ff5f": "A \\times B \\neq B \\times A,", "fa68640eb1fca2eb6178d700727d6d82": "\n(\\mathcal{P}_1 \\mp i \\mathcal{P}_2)\\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* =\n \\sqrt{j(j+1)-m(m\\pm 1)} \\, D^j_{m', m\\pm1}(\\alpha,\\beta,\\gamma)^* .\n", "fa68a38de17b2500f1fcd2ceeccfac66": "R(\\bold{\\hat{n}},\\theta)", "fa68c9ded273a4fce20861afacf2d8a7": "\\succ_A", "fa68d71672f6ee188822d6cb9121f953": " D_3: d_{ ij } = | r_i - r_j |^p ", "fa696e96c62905a74bc222bd47faab7e": "\\vec{v}_{\\|}", "fa69f0363b20678985eaead8c4220c34": "\\,\\hat{\\Sigma}", "fa6a356399b8e05821a8b539614dd8b0": "C^{(V)}_T(V,T)=T \\left.\\frac{\\partial p}{\\partial T}\\right|_{(V,T)}\\ ", "fa6a3ed005e897aee2e43a6b0287a5e7": "f^{\\lambda} v", "fa6a954384734ced3ac388b1acc300aa": "\\varepsilon'", "fa6abe4baa28c140959b58bfc98cb4f0": "p_N", "fa6ac60da3b38b183ec8046ab60075b0": "\\omega (t) = \\frac{\\partial \\varphi (t)}{\\partial t} = \\omega_0 - k_0 z n_2 \\frac{\\partial I(t) }{\\partial t}", "fa6adcb49a6ffa3fbbeba611a50e4603": "{\\mathbf{v}}^2 = {\\mathbf{u}}^2 + 2 {\\mathbf{a}} \\mathbf{s}", "fa6af64a2ece13ecb689fdfb35300d82": "A = \\begin{pmatrix} \\vec E_1^\\dagger \\\\ \\vec E_2^\\dagger \\\\ \\vec E_3^\\dagger \\\\ \\vdots \\end{pmatrix}", "fa6b04d81a296a7219e5a62ffb748e87": "E=\\hbar \\omega/2", "fa6b10edcb4d46b909516d38424a8786": " n=\\Delta n+n_0", "fa6b4266819310fde3944d1180f3c4a4": "L = \\frac{\\mu_0}{2\\pi} \\left(\n l \\ln\\left[\\frac{1}{c}\\left(l + \\sqrt{l^2 + c^2}\\right)\\right] - \\sqrt{l^2 + c^2} +\n c + \\frac{l}{4 + c \\sqrt{\\frac{2}{\\rho}\\omega\\mu}}\n \\right)", "fa6b4ed1836627162e674cd408635ad3": "x_{1/2}= \\frac{ac\\pm b\\sqrt{r^2(a^2+b^2)-c^2}}{a^2+b^2} \\ ,", "fa6b85db36ab6008ec567af6b1038ea7": "+L\\sum_{n,\\alpha}\\left(\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n,\\alpha+1}+\\hat{A}_{n,\\alpha}^{\\dagger}\\hat{A}_{n,\\alpha-1}\\right)", "fa6bb2a57fca2de2e79cc7a02ef95a0c": "E(Q)=( U+E_f)V=\\varepsilon_FQ", "fa6bd455d4616251b77226ce8a241099": "D_\\pm ", "fa6c1d3d935b99e9c7fbd04bd21b1622": "\\boldsymbol{S_y}", "fa6c2e6578765802de076c48f88adaef": "\\text{Gal}(\\overline{\\mathbf{Q}}_p/\\mathbf{Q}_p)", "fa6c66df99c00795c1761c16eceaf647": "94^2", "fa6cd7e2aa5293c4e7d4efb98db27d71": " S=N-U, ", "fa6ce13d71b96179749f4327255b5a57": "\n\\Delta f = \\frac{f_0}{Q} \\,\n", "fa6d181dcbbd079ddb3eb1d076771ac9": " \n V_{P0}= \\sqrt{C_{33}/\\rho} ~;~~ V_{S0}= \\sqrt{C_{44}/\\rho}\n", "fa6d461721768b6bb8bd3090f9795116": "\\bar{\\boldsymbol{u}}_S\\, =\\, \\frac12\\, \\sigma\\, k\\, a^2\\, \\frac{\\cosh\\, 2\\,k\\,(z+h)}{\\sinh^2\\, (k\\,h)}\\, \\boldsymbol{e}_k,", "fa6d5a3751516a61ca3d446fe1468623": "R^2 \\, dl \\, dm", "fa6d7cffef939cf0b09974808cc2c80b": " \\rho: G_\\mathbb{Q} \\rightarrow \\mathrm{GL}_2(F).\\ ", "fa6d8810426eaaea743331059bd12330": " | \\nu - \\mu | \\le \\sqrt{ \\frac{ 3 }{ 4 } } \\omega ,", "fa6db95f00fe35ce74c2482dc71f3d8d": "\\operatorname{PGL}(2,5) \\cong S_5,", "fa6dbf751b3edcbceec67d5ade24e113": "\\int \\frac{(\\ln x)^n\\; dx}{x} = \\frac{(\\ln x)^{n+1}}{n+1} \\qquad\\mbox{(for }n\\neq -1\\mbox{)}", "fa6ea410ae0804fbef7aa72ad12b4552": "\\left(\\frac{a}{1}\\right) = 1.", "fa6efb66b5116bf4f9d12878bcd52c86": "\\Phi_{\\rm B}^{(n)} \\approx \\Phi_{\\rm metal} - \\chi_{\\rm semi}", "fa6f33ed8d4713fee56af4ce905696ce": "m/r^2", "fa6f86daf78b296a21d564e8f909133a": " \\cos\\phi = \\tanh a ", "fa6ff66683397c9d593392867871c6c8": "C_2= E(\\cos(2\\theta)) = E(\\cos^2\\theta-1)=E(1-\\sin^2\\theta)\\,", "fa6ffc803c838f6cc977e0c53270790c": " ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 ", "fa702d104a18b903ce27eac05f95691c": "\\mathbf{\\Xi}_k = \\log \\operatorname{E}_{k-1} e^{\\mathbf{X}_k} = \\log \\mathbf{M}_{\\mathbf{X}_k}(\\theta) ", "fa7059e189ed004538852ceada4c5224": "\nr_{24}(n) = \\frac{16}{691}\\sigma_{11}^*(n) + \\frac{128}{691}\\left\\{\n(-1)^{n-1}259\\tau(n)-512\\tau\\left(\\frac{n}{2}\\right)\\right\\}\n", "fa705e0c919f21af068fcf4b1635ef5f": "-2 \\le k \\le 2", "fa70ab7ce94970349a3f3375214b6a82": "\\ n\\, W_n^2 \\sim \\frac{\\pi}{2}", "fa70b180972cdc3de76417b39b7d9606": "jk = i\\,", "fa70cdfebbaccf473c66a923bb99967d": " F(x) = f(x) + \\Psi(x), ", "fa7153f7ed1cb6c0fcf2ffb2fac21748": "int", "fa716353cdf2fbbaf6f355dfd8d0e218": " \\partial_{xx}-\\partial_{yy}+y\\partial_x+x\\partial_y+\\frac{1}{4}(y^2-x^2)-1 ", "fa716ac449afac2b4c24e416f7f656fd": "t_a, t_b,", "fa717ba17306cd76900510df8ac8013e": "Fr", "fa71de2a2d448b560e5fd5fa433a0c75": "F(u+h)=F(u)+dF(u;h)+\\frac{1}{2!}d^2F(u;h)+\\dots+\\frac{1}{(k-1)!}d^{k-1}F(u;h)+R_k", "fa721ca0387eb5595e4a9f164b208c5f": "2^i3^j7^k", "fa725f9bddafb3dd14b2a1754fdcff14": "V(x)=V_0[\\Theta(a-x)-\\Theta(a+x)]", "fa7289a1138a9da5fa333facfcf2702e": "\n\\begin{align}\n10x_1 - x_2 + 2x_3 & = 6, \\\\\n-x_1 + 11x_2 - x_3 + 3x_4 & = 25, \\\\\n2x_1- x_2+ 10x_3 - x_4 & = -11, \\\\\n3x_2 - x_3 + 8x_4 & = 15.\n\\end{align}\n", "fa728bca6c4ebdf47e894909931a4c85": "\\operatorname{Corr}(X_s, X_t) = \\sqrt{s/t},\\ s B", "fa747e0b163744a09059d60a77ebe452": "f''(x) \\ge 0", "fa7490f6d8dab676c5d1ae16ce435c5e": "\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) = \\sum_{h=1}^m \\frac{\\prod_{j=1}^m \\Gamma(b_j - b_h)^* \\prod_{j=1}^n \\Gamma(1+b_h - a_j) \\; z^{b_h}} {\\prod_{j=m+1}^q \\Gamma(1+b_h - b_j) \\prod_{j=n+1}^p \\Gamma(a_j - b_h)} \\times\n", "fa7499e2b38448c4ce07b32fc1302b5c": "V(f)(z)=f(e^{-2\\pi i\\theta}z)", "fa749f6ecd02849014d66ca02216178e": " \\nabla \\cdot \\nabla \\phi = 0 ", "fa74a0250d32c231721e1ead57d7935f": "D\\left(X, X\\right) > 0. \\, ", "fa750853bf3bb9594d78add2ac26ddab": "abs(\\lambda) \\,", "fa754c4f9132d8a5a789e86f461c2f20": "\\{y_i\\}^M_{i=1}", "fa7578cf316b444567ad2329eb6f83c8": "\\begin{pmatrix}\n0 &1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n0 &3 & 2 & 5 & 4 & 7 & 6 & 1 \\end{pmatrix} ", "fa75b2aa78bce0129010bbdbcf72e942": "n!=n\\times(n-1)\\times\\cdots\\times2\\times1", "fa75e9515e5a8a5235beb156908d490a": "\\textstyle\\nu_i-1", "fa75f80a90f18e244ca8fcabd829ac1e": "2^{32}0}\\lambda_j^{-s}=\\frac{1}{\\Gamma(s)}\\int^\\infty_0 t^{s-1}\\mathop{Tr}(e^{-t\\Delta_q} - P_q)dt,\\ \\ \\ \\mathop{Re}(s)>\\frac{n}{2}", "fa7c1bfefd4c46661eefe5df208cc5d4": "\\delta:Q\\times \\Sigma \\times Q \\to \\mathbb{C}", "fa7c7c79a03427d62b9e325cb84fd4fe": " \\mathbf{p}(t) \\in \\Omega ", "fa7cd1e5a55a2a8e42d69a960ec89b61": "DST = \\frac{du}{dx} - \\frac{dv}{dy}", "fa7cd5dda4a7e94a0765397d0b7f657a": "\\displaystyle e", "fa7d35ab2dac3be3c7eb9da8bfa7ce49": "\\mathcal B,\\mathcal A\\vdash\\mathcal C", "fa7de606664f3b24658f2e3e2ad6a4d2": "\n\\begin{align}\nr & < \\frac{\\lambda_1}{\\lambda_\\mathrm{tot}} \\rightarrow \\text{scattering-mechanism-}1 \\\\\nr & < \\frac{\\lambda_1 + \\lambda_2}{\\lambda_\\mathrm{tot}} \\rightarrow \\text{scattering-mechanism-}2 \\\\\n& {} \\ \\vdots \\\\\nr & < \\frac{\\sum_{i=0}^n \\lambda_i}{\\lambda_\\mathrm{tot}} \\rightarrow \\text{scattering-mechanism-}n\n\\end{align}\n", "fa7e6ee2a8847427d94e027bb4f4ca01": "n_f = 3", "fa7ec17a94582e006eb50a82c57f3415": "\\sum_{k=2}^\\infty (-1)^k \\frac{\\zeta(k)-1}{k} = \\ln2 + \\gamma - 1", "fa7ed50d6aae20de433caeea20abfec0": "\\begin{align}\nt_n(z)&=\\frac{\\tfrac{e^{\\zeta}}{4n}}{3+\\zeta +z} \\\\\nf_n(\\zeta )&= t_1\\circ t_2\\circ \\cdots \\circ t_n(0)\n\\end{align}", "fa7f1b14bb7333eabd2803a3dbc02d32": "n_B^{\\prime\\prime}(\\xi)=\\frac{\\beta^2}{4}\\mathrm{csch}^2\\frac{\\beta \\xi}{2}\\mathrm{coth}\\frac{\\beta \\xi}{2}", "fa7f2fca6e51f14b7f830edcb3943969": "\\{t_k\\}_{k\\in \\mathbb{Z}}", "fa7f7fdd0afe9f4c0c1858cf277c3d6d": "\\tau(t)\\,", "fa7fb3cad6cc1d267259c8630853312d": "\\omega_e = m_0c^2/\\hbar \\ ", "fa7ffc497e9686786c759a72f5c6fc27": "\\langle f, g\\rangle = \\int_{-\\infty}^\\infty f(x)\\overline{g(x)}dx", "fa8008897e19c250083cc1e717f0b2e6": "I_p(P,Q_1Q_2) = I_p(P,Q_1) + I_p(P,Q_2)", "fa800cbe3f8a3a7888766a436243ce4d": " x,y", "fa8014f68a0cef88293538695dc2c2a5": "\\sim g^{-1-\\frac{\\gamma -1}{\\eta}} ", "fa8019f4eacb65c3eacdce94124b92e1": "\\gamma<\\frac{4}{3}", "fa8098edcad7b3e8f5624914d1c29980": "\\{ a_1, ..., a_m \\}", "fa8154c14b4126d9e23627defd549c91": "A_k(n,r)", "fa81695b7bd51c67d01e23b01931d04f": "|\\boldsymbol{v}_g|", "fa81b3830bd7f711581ba9f999e50abe": "D_r", "fa81bfab96fb0cfb94b3e4b30974fa84": "R_{0,\\theta}(p) = \\begin{pmatrix} \\cos\\theta & -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta \\end{pmatrix}\n\\begin{bmatrix} p_x \\\\ p_y \\end{bmatrix}.", "fa820f2e628f95ff2614bc0febbf0326": "\\alpha_\\nu^{ } \\in Z_R", "fa821dc9216260c68e052ddfb82344c3": "S_{x_B}", "fa8237b43495d4f7755d939c3ce10740": "K=r(r+\\sqrt{4R^2+r^2})\\sin \\theta", "fa825206dcf40445c7efe1cc93569ea2": "\\mathcal{L}\\left(x_1 ,x_2 ,\\dots,x_n ,\\lambda\\right)=U\\left(x_1 ,x_2 ,\\dots,x_n\\right)+\\lambda\\cdot\\left[W-\\sum_{i=1}^n \\left(p_i\\cdot x_i \\right)\\right]", "fa8253db494a6c9a0e53ced5497fbcf7": "(u, v, w)", "fa828935f84241d1821faf096c333d54": "\\phi(x)\\in \\operatorname{Int} E_j", "fa82b8905936fa0c838da2fd79f826e3": "x_1'=f_1(x_1, \\ldots, x_n)", "fa8315af3e6e7a75818cb74f9b2018d8": "a_1 x_1(t)+ a_2 x_2(t)\\,", "fa834c443b3ab012e58eebef4bb28598": "K = D^{0.4}*(\\frac{S_{e}}{C_{te}})^{1.2}", "fa8368caef998c9e6f2d8d552bc4bd57": "(N,A_i,u_i)", "fa83c48253385292aaeb29bc766ba66f": "\\Omega_i", "fa84538965191fe24e7618c09d5644bf": "\\,a^{bc} = (a^b)^c.", "fa847767f3c013c4ae869be0ef73f4ed": "\\varphi\\left(\\varphi\\left(n\\right)\\right)", "fa84d2a5ae564a78fb63c60d5abc9b6c": " F(t,{\\bold x}) = ({\\bold x} - \\gamma(t)) \\cdot {\\bold T}(t) \\ . ", "fa84fb369d7cb7f5e695408e251fa985": "b_{2}-a_{3}", "fa850b61cd6cca5a08f836e617cecc82": "1 - H_\\mbox{b}(p)", "fa853e78f3788bd54f7e7ed6fad115bc": "(F(Y_0),\\eta_{Y_0})", "fa855f37f19271133f6d48b896519049": "\\psi_{2m}", "fa86501d3d59ce112b38a780f58fe480": "p=1-q", "fa8660d5b8d5c0a348bef41f7c390724": "a^n b^n c^{n-j}", "fa86865e39ada962e77419c6dd8b40f4": " A = ", "fa86ac1b31aa3dcdc2140ca1e5bfa92b": "g(\\lambda)", "fa86bfd2f8b1116cdbb68ee92065580e": "\\forall x : P \\Rightarrow Q", "fa86d4c03a83a3dda0be028ae961c951": "U(\\phi) = \\frac{\\Phi_0 I_c}{2\\pi}[1-\\cos(\\phi)]", "fa8707b4d8465188531c9cc83374a5ee": "R_s", "fa874a3d4986ed813b1f6d20f21e1d1e": "\\dot U", "fa877101584a13872a537a02e7081751": "M=3", "fa87b3fbdf414c2dbf9a6496d6279e16": "1\\leq k\\leq 2m\\,\\!", "fa87bef1cbab1e919b1756588c9e28ec": "\\Omega_{E,\\ell} = \\left( \\int dx_1 \\int dx_2 ...\\int dx_n\\right)\\left(\\underbrace{\\int dv_1 \\int dv_2 ...\\int dv_n}_{\\Sigma v_i^2 = 2E}\\right)", "fa8803d89e701ab90cca649b06b5eb05": "h^2=\\frac{4a\\rho}{(a+\\rho)^2},", "fa889bb7ff0c3f4dacea5f4242f58f21": "\nA^{*2} = \n\\begin{cases}\nA^2\\left(1+\\frac{4}{n}-\\frac{25}{n^2}\\right) & \\text{, if the variance and the mean are both unknown.} \\\\\nA^2 & \\text{, otherwise.}\n\\end{cases}\n", "fa891f3e29357034817990828341f815": "H = {\\epsilon_{ijk} F_{ab}^k \\tilde{E}_i^a \\tilde{E}_j^b \\over \\sqrt{det (q)}}", "fa894a23b9b83100cef2f25466a33505": "\\gamma = { 1 \\over \\sqrt{1 - v^2/c^2} } ", "fa89c135f14db8351fdab910cd7f77d2": "\\langle\\psi'_r|P_2|\\psi'_r\\rangle = 0", "fa8a0f3545895594b5cb9f7176cc780a": " p_i = \\frac{\\partial L}{\\partial \\dot{q}_i}\\quad \\dot{p}_i = \\frac{\\partial L}{\\partial {q}_i} ", "fa8ac45fff5f27b0425d7add869d3188": "d(x, y) = \\sup \\left\\{ \\left. 2 \\tanh^{-1} \\left\\| \\frac{f(x) - f(y)}{2} \\right\\| \\right| f : B \\to \\Delta \\mbox{ is holomorphic} \\right\\}", "fa8af5a7baece2a8dc525e04186689a3": "\\pi_1(A)", "fa8b27175a508df7e36560a16efd8163": "z_\\mathrm{R} = \\frac{\\pi w_0^2}{\\lambda}", "fa8b319ac00eff1fc43384685fce2811": "{\\tilde{H}}_2", "fa8b323a190379eda0afefd284be53a9": "^2 x = x^x", "fa8b4f05d67d5f96e96133f9db4a8ff6": "(24)\\quad L+M=r\\,,\\quad z=L\\cos\\theta\\,,\\quad \\rho=L\\sin\\theta\\,,", "fa8b9d5f491ddc9599df224dea6681ee": "H_{n}(X) = Z_n(X) / B_n(X).", "fa8ba2d97637aaa312fe9f199bc3261b": "\\eta^{\\mu \\nu}=0", "fa8bbf21f9ffd3abac69f3e959a9bfa2": "x_3 = \\frac{\\alpha}{1-\\alpha},\\; y_3 = (1 + x_3) \\left(1 - \\frac{x_3}{K}\\right)", "fa8c0217b463c65f4f1bd2d9c5b202cc": "g_n = [z^n] g(z)\\,", "fa8c880980fe6a8d94daee930b9b846e": "N_f^{-0.6}D_f^{0.75}+0.9\\frac{S_u}{E} \\left[ \\frac{\\exp(D_f)}{0.36}\\right]^{0.1785\\log\\frac{10^5}{N_f} }-\\Delta\\epsilon=0", "fa8ca6dc67d6b96a3e9c9b5a2e8ffd0a": "\\displaystyle{\\mathfrak{g}=\\mathfrak{k} \\oplus \\mathfrak{p},}", "fa8cc9a4ed189abf16cd30049604fe36": " M = \\begin{pmatrix}\n0 & 0 \\\\\n1 & 0\n\\end{pmatrix}", "fa8cea080d0cbe4a4dfba3f9281e4aa5": "T_{b}/T_{c}", "fa8cf071adc0f7b95e0db4d01f2646e9": "B = b - \\frac{a}{RT}.", "fa8d06c8af3b1fbf3aaef043632f60b9": "T_n(x)=\\cos(n \\arccos x)=\\cosh(n\\,\\mathrm{arccosh}\\,x) \\,\\!", "fa8d0a4fb90aca0887823fd5dbbe00b1": "\\epsilon\\ge 0\\,", "fa8d1f0dca0e90aa5354248d85f3f2b1": "\\nabla\\cdot (\\nabla\\phi)= X \\rho \\quad \\Rightarrow \\quad \\nabla^2 \\phi = X \\rho", "fa8d294d52b48e69387d619d8ef9af5d": "x \\in U \\subset V", "fa8d2d716f9db93738110a70f60696d3": "d(R)", "fa8e0547a1d15e4cd282863004dfa02c": "\\operatorname{wnchypg}(x;n-1,m_1,m_2,\\omega) \\frac{m_2+x-n+1}{(m_1-x)\\omega+m_2+x-n+1}", "fa8e19d73fcc457e8cd2ab2157358a79": "\\frac{mm}{\\mu s}", "fa8e2825234c423a6fd8944460447834": "\\rho(w)", "fa8e5788cb45ae0ec61f0dbd2fed23e8": "{C}_{3}^{(1)}", "fa8e74cf34efc813f8592faac90d2364": " Y = \\pm b\\sqrt{1 - (X/a)^2} = \\pm \\sqrt{(a^2-X^2)(1 - e^2)}", "fa8e7c560f9cc877624c3b98eaf7e643": "\\omega = 2/3", "fa8e9f11f6e5f1d0dbcd83ab226af872": "s_c(t) = \\left\\{ \\begin{array}{ll} A e^{i 2 \\pi \\left( \\left( f_0 \\,-\\, \\frac{\\Delta f}{2}\\right) t \\, + \\, \\frac{\\Delta f}{2T}t^2 \\, \\right)} &\\mbox{if} \\; 0 \\leq t < T \\\\ 0 &\\mbox{otherwise}\\end{array}\\right.", "fa8ed6ba36c72f685b3af861e28bd99c": " R_{\\textrm m} ", "fa8eeb33e26860eed722c4438a804c32": "r(0,0)=E[tr(ss^T)]", "fa8f05cab95577963e972411c65cdf69": "\n\\begin{align}\n\\frac{\\pi}{4} & = \\arctan(1)\\;=\\;\\int_0^1 \\frac 1{1+x^2} \\, dx \\\\[8pt]\n& = \\int_0^1\\left(\\sum_{k=0}^n (-1)^k x^{2k}+\\frac{(-1)^{n+1}\\,x^{2n+2} }{1+x^2}\\right) \\, dx \\\\[8pt]\n& = \\sum_{k=0}^n \\frac{(-1)^k}{2k+1}\n+(-1)^{n+1}\\int_0^1\\frac{x^{2n+2}}{1+x^2} \\, dx.\n\\end{align}\n", "fa8fc5aa0f42290322983db3fc38c7a2": "\\bar{E}\\,=\\,\\frac{3}{2}k_B T_k", "fa8fececdc738be30aed5cd8bb3bc94c": "U(x^*,t)", "fa8ff407b5f15e2034daef240391a414": "\\varphi = \\arctan f'(x)", "fa902b5c2957ee1f5a8e8fb2908f894b": "\\mu^'_s", "fa90395485c16377c8d7229d11977054": "\\left\\{T_i\\right\\}", "fa90518650f15a902501d63667082bfb": " \\Phi = \\frac {\\rm \\mu mol\\ CO_2 \\ fixed} {\\rm \\mu mol\\ photons \\ absorbed} ", "fa9060bd557674f0788511b1c0fc4600": " f_{ab}=\\int_a^b f(d_p) \\,\\mathrm{d}d_p", "fa9074704b2152c900e270b453b9f52f": "t = \\frac{\\ln(c)}{-L} ", "fa90932812a9fc8f69dbec5b0c4fa507": "\\alpha [n] = \\varphi_{\\beta_1}(\\gamma_1) + \\cdots + \\varphi_{\\beta_{k-1}}(\\gamma_{k-1}) + (\\varphi_{\\beta_k}(\\gamma_k) [n]) \\,.", "fa90b7f21d9a745c9ea6d93c3f87f686": "\\nabla^{2} u := \\operatorname{div}(\\operatorname{grad}\\ u)", "fa90ec7db79ba71fb2de36fdf9b6399f": "\\frac{d A}{d x}>0 \\Rightarrow ", "fa91467df6cfd794181e8fbb978a1722": "r = w_{1}", "fa916aec349c64ff655945d438bdc68e": "M<|Q|", "fa91a01fdeb977b3565ed3dffeb64f94": "\\overline{\\phi}(t) = x", "fa91be952845728befec94e8d4bf16bb": "ST_i", "fa91da929511fcaa99c75359ad3accc3": "\\mathbf{x}_i = \\mathbf{x}_{0i} + \\delta\\mathbf{x}_{0i}. \\, ", "fa922fca2ddd267ad530807d9222c03b": "\\partial", "fa92411720a6913143c07d781d737bbe": "\\,y", "fa9260ad8d5388aecc22a0aac8cb2f64": "u^+ \\neq y^+", "fa92644460696c8fc3266c23ad1c91bb": "\\mathbf v \\times \\mathbf u = - (\\mathbf u \\times \\mathbf v)", "fa929fa3c4a96c602f2ba085fc284ad4": "H^s(\\R)", "fa92aea74d8d19f02fc5ee7e8cfd4009": " x = \\sum_{n=1}^e \\left(\\sum_{m=1}^d b_{m,n} u_m\\right) w_n = \\sum_{n=1}^e \\sum_{m=1}^d b_{m,n} (u_m w_n),", "fa92b7bfd38bf87ef42796969dce069a": "\n \\begin{align}\n \\langle\\rho\\rangle~\\frac{\\partial\\tilde{\\mathbf{v}}}{\\partial t} & +\n \\left[\\langle\\rho\\rangle+\\tilde{\\rho}\\right]\\left[\\langle\\mathbf{v}\\rangle\\cdot\\nabla \\langle\\mathbf{v}\\rangle\\right]+\n \\langle\\rho\\rangle\\left[\\langle\\mathbf{v}\\rangle\\cdot\\nabla\\tilde{\\mathbf{v}} +\n \\tilde{\\mathbf{v}}\\cdot\\nabla\\langle\\mathbf{v}\\rangle\\right] \\\\\n & = -\\nabla \\left[\\langle p\\rangle+\\tilde{p}\\right]\n \\end{align}\n ", "fa92bffe6a4d2351ec3e383dd0d995ac": "{\\rho^2_{XT}} = \\frac{{\\sigma^2_T}}{{\\sigma^2_X}} = \\frac{{\\sigma^2_T}}{{\\sigma^2_T}+{\\sigma^2_E}}", "fa92c236d743e63de7200f0d107ac963": "P_3/P_2 \\,", "fa93f267d74208270d8111fb36a1af1d": "\\frac{{d^3 Y}}{{dx^3 }} = \\frac{6}{{h^3 }}a_3 ", "fa93ff2580013c8936c2396f201ad0d4": "\\frac{2}{3} \\left(\\frac{2^{2m}-1}{2^{2m}+2}\\right)\\;\\mathrm{to}\\;\\frac{1}{3}(1-2^{-2m}),", "fa941596c43be2d78d4470b9750e45d0": " \\sin \\alpha = \\tan \\alpha / \\sqrt{1+\\tan^2 \\alpha} ", "fa94afc5cb7550ffd64141ab4a07cd62": "\\mathrm{sinc}(\\alpha)", "fa94c8fa0d49c7e7b2be52ee05f1b5fc": "o = \\dfrac{o_0}{i}", "fa95287de2a6100f95dfa723340bfa29": "I_n(z)", "fa952ac54eecf92ac02c03f77b74a6af": "\\nabla^2 U -\\frac{1}{\\sigma}\\vec \\nabla U \\cdot \\vec \\nabla \\sigma=0.", "fa9547c63195122e67349efbe6af59c4": " \\displaystyle \\rho(f) := \\int_{\\mathbb{R}} f(t) U_{t} \\, \\mathrm{d}{t} ", "fa959756ddecf12a5bd311da049cb3d7": "y_i E(\\alpha_i) = P(\\alpha_i) E(\\alpha_i)", "fa95b070178dabe1f75dddd94b963872": "2^{\\sqrt{2}}", "fa95ecd13d8266a171203f91522f58be": "a_1 + a_2 + \\cdots + a_n", "fa96015cfcd7fdf593510acd0cf66307": "P_1 = P_0(1+r)-c", "fa9634d4ba1ca464886a0fa6ad79513b": "\\frac{dS}{dt} = - \\beta SI + \\mu (N - S) ", "fa9650bdcbb426bc74b4c4ed57568dee": "\\mathbf{u}\\times (\\mathbf{v}\\times \\mathbf{w})", "fa9676c14e48a61065652c64206d46ed": "a=a\\cos\\theta\\ e^{i\\theta} - i a\\sin\\theta\\ e^{i\\theta}.\\,", "fa96778f23024d4f21324413b1b9be23": "m\\hbar", "fa96c6c9c97832143450224dcff8223d": " \\mathbf{F}_{\\mbox{fictitious}} = -m\\mathbf{a}_\\mathrm{AB} - 2m\\sum_{j=1}^3 v_j \\frac{d \\mathbf{u}_j}{dt} - m \\sum_{j=1}^3 x_j \\frac{d^2 \\mathbf{u}_j}{dt^2}\\ . ", "fa975343854c74452609f55b9ac3a0b8": "\nF_q =\\langle n \\rangle^{-q}\n\\sum^{\\infty}_{n=q} \\frac{n!}{(n-q)!} P_n. \n", "fa976ea66b3101337b3b2a64e14613a5": "O(N!z^{-N})", "fa97d324d1de726b0e1b75cd0dd88f14": "p_{k}(x)", "fa9805e7c2063167dbc063a1f92a2966": " \\alpha= \\frac {e^2} {m_e\\omega^2},", "fa9814a0f2ae518fdfa378511e33825c": " {\\star dx} \\wedge dy = - dz \\wedge dt ,\\quad {\\star dx} \\wedge dt = dy \\wedge dz, ", "fa983607b84ce9325087d8940afd7dcd": "v_\\text{e} = I_\\text{sp} \\cdot g_0", "fa983e14d241675bf57e656b3237069e": "x_0 x_1=a_0 a_1\\times10^{b_0+b_1}", "fa98813968dfa367567ff71852241f47": "\n \\begin{bmatrix}\n 1 & 0 \\\\\n 0 & 0 \n \\end{bmatrix}\n \\begin{bmatrix}\n 1 & 1 \\\\\n 0 & 0\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n 1 & 1 \\\\\n 0 & 0\n \\end{bmatrix}\\,\n", "fa98d430e342cfe3b0af8fd99de726af": " \\frac{\\partial f}{\\partial d} ~ e^{-d/d_{0}} ", "fa98f18200abcc991a4e934bb4abe6a1": "1\\to \\pi_1(H) \\to \\tilde H \\to H \\to 1", "fa991e11dd7353d729fd4c3d3497acc1": "x(t + \\Delta t)", "fa997b0a0c2b6e4b73473c7cec32b051": "\\Delta r(k)", "fa99ac176a8bc08208e397b62a960d40": "\\left(-\\frac12\\pm\\frac{\\sqrt3}2i\\right)\\sqrt[3]{\\frac{1}{2}+\\frac{1}{6}\\sqrt{\\frac{23}{3}}}+\\left(-\\frac12\\mp\\frac{\\sqrt3}2i\\right)\\sqrt[3]{\\frac{1}{2}-\\frac{1}{6}\\sqrt{\\frac{23}{3}}}\\approx -0.662359 \\pm 0.56228i\n,", "fa99f9dfdb9e7d4d8640392f655c1c56": "\\mathbb{P}_{T'}(\\cdot) = \\mathbb{P} (X_{T'}^{-1}(\\cdot))", "fa9a1364b7ba1fa9169df9ddf7ffed1d": "\\mu \\frac{d^2 u}{d y^2} = \\frac{d p}{d x} \\quad ; \\quad u(0) = u(L) = 0", "fa9a4ea192d103d4fc49a9a725d3200f": "\\ q=(\\sigma_1'-\\sigma_3')", "fa9a61571fa0f77b5a2e7a2798d451b9": " \\frac E t = P = \\frac {F \\times d} t = F \\times v", "fa9a9c5d741c2cfd4c50d7b2058ca0a5": "Velocity > 0.5 \\times \\left (\\frac {C}{2 \\times Period Between Pulses \\times Transmit Frequency} \\right)", "fa9af20212afbe6aaccdcb4c55766ba2": " H_{\\frac{1}{8}} = 8-\\tfrac{\\pi}{2} - 4\\ln{2} - \\tfrac{1}{\\sqrt{2}} \\left\\{\\pi + \\ln\\left(2 + \\sqrt{2}\\right) - \\ln\\left(2 - \\sqrt{2}\\right)\\right\\}", "fa9b074449c2cafd2b72b0a2376ce302": "C^{\\mathrm{op}}", "fa9b2af2c9ba4b3b2ec3739bbbda272e": "\\nabla\\times\\mathbf{E} = \\frac{\\partial\\mathbf{B}}{\\partial t} = \\boldsymbol{0} \\,,", "fa9b4fa9b895e9ce1f91bcf28ed72811": "[J^a, J^b] = i \\epsilon^{abc} J^c \\, ", "fa9b76e1730f4c86f179329a40dfb4d7": "\nP_{\\mathbf k} = \\frac{1}{2}( 1 + \\hat{\\mathbf k} ),\n", "fa9b7b7c160591c1f7f1e73abcbdb7fd": "d(f, g) = \\int_S \\varphi \\bigl( |f(x) - g(x)| \\bigr) \\, \\mathrm{d}\\mu(x)", "fa9bccaab393f44285ff9d628f2918ee": "{} + 4 (b^4 c^2 + b^2 c^4) - 3 (b^4 d^2 + c^4 d^2) + 36 (a b^2 d^3 - a c^2 d^3) = 0.\\ ", "fa9c263ca1aeef59f07493eb0def4b5b": "\\left\\{ x^{n}\\left(\nm\\right) \\right\\} _{m\\in\\left[ M\\right] }", "fa9d6a8c1660924955e6d0d7316b2e65": " \\Phi\\colon G\\times V \\to V \\quad\\text{or}\\quad \\Phi\\colon A\\times V \\to V", "fa9d9caa3863317a64320d9466476854": " \\Rightarrow(y_2 + \\frac{10^2}{2(32.2)(y_2^2)} = 4.04)", "fa9dae24538b876465c52fb184667797": "1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1\\cdots", "fa9dc6d0a8d0444f97cf49895334c284": "\n\\ddot{a}_{\\overline{n|}i} = (1+i) \\times a_{\\overline{n|}i} = \\frac{1-\\left(1+i\\right)^{-n}}{d}\n", "fa9dff3066d764be59f2fd7dad3c9e5c": " l", "fa9e3d369ea5f225ef439d0d2a70816c": "\nB_k(n) = \n\\begin{cases}\n{1\\over 2}N_k(n) + {1\\over 4}(k+1)k^{n/2} & \\text{if }n\\text{ is even} \\\\ \\\\\n{1\\over 2}N_k(n) + {1 \\over 2}k^{(n+1)/2} & \\text{if }n\\text{ is odd}\n\\end{cases}\n", "fa9eef060ae3917ef7cf20217a24e794": "\\omega_i^j(\\mathbf e) = \\sum_k \\Gamma_{ki}^j(\\mathbf e)\\theta^k.", "fa9f40f634a9d7347bda116a09b6ab37": "\\Pi^1_k", "fa9f5847a4d57730b1c0413e4fc6c68a": "K^\\flat = \\varprojlim_{x \\mapsto x^p} K \\ . ", "fa9f85c8eeb84c57943c6cccc63f5b39": "\\dot{u} - \\sum_i p_i \\dot{x}_i = 0", "fa9f9525acae4ae4c80c49dd58d4b5eb": "E_n = -\\frac{h c_0 R_{\\infty}}{n^2}", "fa9f99d47705b2187ff4a15adbe428af": "{\\mathbf S}", "fa9ff30200ea56b90b82d35500ff1768": "B_1(t) = \\sum_{i=0}^{n} x_i b_{i,n}(t) \\mbox{ , } t \\in [0,1]", "fa9ffa9c769be504b9e02e8814a35fad": "\\textstyle\\int dx/x", "faa012ce7d29079f17ab2d465e9061ef": "Z^{M}_{i,j} = Z_{i-1,j-1} \\cdot e^{\\frac{\\sigma(x_i,y_j)}{T}}", "faa056f180e581165c7b7de72d47e9e6": "\\vec{E}=-\\nabla\\,\\varphi-\\frac{\\partial\\vec{A}}{\\partial t}", "faa0740e3246c804104a1c149d317ffc": " \\partial_t^j h(x,0) = f_j(x),\\qquad 0\\le j 1, \\end{cases},", "faa1ccf1a357762aeb8b9306cd217ffe": "\\sigma = \\frac{k\\alpha_{abs}c^2}{4\\pi\\omega\\mu}", "faa1d54f608bb816cae0be85f0186de1": "\\tan\\frac{E_4}{2} = \\frac{\\sin\\frac12(\\phi_2 + \\phi_1)}{\\cos\\frac12(\\phi_2 - \\phi_1)}\n\\tan\\frac{\\lambda_2 - \\lambda_1}2.", "faa1fc251fb072cdb19a501dd8165a4f": "\\text{Discount} = P(1+r)^t-P", "faa22cf00d9e788949f9be6d488c2a40": "P(\\Delta R)=P(\\Delta X,\\Delta Y,\\Delta Z)=p(\\Delta X)p(\\Delta Y)p(\\Delta Z)", "faa289a4f9103f320bda39085f927fc8": " c>0", "faa28c39047579104e6d9f45b3149e2e": "\n\\ell (\\gamma) =\n\\int_0^1 \\left\\vert \\frac{\\partial \\gamma}{\\partial t}(t) \\right\\vert_{B,\\gamma(t)} dt .\n", "faa2a2414978ceaf7aefd56cb3ae5b7e": " \n(\\bar \\psi\\partial\\!\\!\\!/\\psi)^2", "faa2bea0e9a2d7c1739ca96824469935": "\\sum_{x\\in S} \n\\quad\\quad \\prod_{x\\in S}\n\\quad\\quad \\int_0^\\infty\\cdots\\,dx\n\\quad\\quad \\lim_{x\\to 0}\n\\quad\\quad \\forall x\n\\quad\\quad \\exists x", "faa2ca93d10d84487fc1fb9c5a86240b": " = -2m \\omega \\left( \\omega R \\right)\\ \\mathbf{u}_R , ", "faa2d704bfb4e653c11cd8c520c1ce72": "\\mathbf{A} \\vec{x} = \\mathbf{0}", "faa2fc3fbc40788e56f0e8e65237e677": "0.000579479 \\times W^{0.38} \\times H^{1.24} ", "faa3285bd3434f741995cd86e370fe5a": "I_o=\\frac{V_i D T}{2L}\\cdot\\frac{V_i D}{V_o-V_i}=\\frac{V_i^2 D^2 T}{2L\\left(V_o-V_i\\right)}", "faa362c38e76c1bb4d840d820c2f80f6": "[1, n]", "faa3703f745c9073a866c5eb3dd3c42c": "s+t+u \\,", "faa379cb5c70e8b8caf5538ae4f60ea8": "{\\mathbf{}}G'(t)H(t)=\\tau(t),G(t)H'(t)=I_{n_r}", "faa381ff6009bd8447b02d180655b7d6": "I_A + I_B", "faa41dcde384ec01a2416dae2e901a18": "\\epsilon_{ff}", "faa4230617d017d927cba383fb485794": "\\displaystyle{\\mathcal{H}=H\\oplus H\\oplus H \\oplus \\cdots ,}", "faa48129ab60f63602aab0f98bd94c5a": "\\begin{smallmatrix}R_\\oplus\\end{smallmatrix}", "faa54f9d5c4f5c35b97384679690344a": "C_{st}^1 := E_{st}", "faa5689703081dbbd7d62cfa27fa657d": "\\Psi(\\rho,\\phi,z)", "faa569ee179d5c5bd40e03a29c5054c0": "f(\\zeta) = \\zeta^{-\\frac{1}{N-1}}\\,", "faa5a95cd9dd36d880172d4a046c477e": "\\mathrm{Tr}_R[\\rho_{QR}] = \\rho_Q \\quad ", "faa5b10c5ce809e67389698f050268e5": " ( s_{ji}^v(x) - s_{ij}^v(x) ) \\times ( x_i - v(i) ) \\leq 0 ", "faa5e2c3888d3ce8ff53bf200013e3ee": "S=\\{a,b,c\\}, a^2=1, c = a^{-1}ba", "faa6c861a474530f4e792fe9c1f8b640": "A^\\top K A", "faa6de3cc0ec2b38058d33016437aae8": "P_a(s,s')", "faa7713f78c777b3d0235cfa1366b47d": "C^\\infty(J^{k}(\\pi))", "faa7c0969aaaae180ee3b57d46a2a38d": "\\ P_{i}=(P_{i})_{pure} x_i ", "faa7e0c61f149da3566b07dad822617e": "r \\rightarrow \\infty", "faa83b97a143fcb098311467d8877729": "I_A = I_B = I_C", "faa8674c9565bc06dcad1f05d0d9860e": "(1-2x_0)\\in (-1,1)", "faa8b4c82a00ff3e76f4cfaa120836e7": "\\lim_{r \\rightarrow \\infty} h_{ab,p} = O(1/r^2)", "faa8d68cf29f590ebdca35d8b7022d12": "n=11:", "faa972d81e6d5c15354d1e618e1086fe": "\\sqrt[3]{z} = \\sqrt[3]{x^3+y} = x+\\cfrac{y} {3x^2+\\cfrac{2y} {2x+\\cfrac{4y} {9x^2+\\cfrac{5y} {2x+\\cfrac{7y} {15x^2+\\cfrac{8y} {2x+\\ddots}}}}}}", "faa97d3c3466e9bedeca9c529f078ae6": "\n\\mathbf{b_{T:T}} = [1\\ 1\\ 1\\ \\dots]^T\n", "faa9bc802518d1c58e45e3c18024f928": " \\acute{{R}^{\\mu}}_{\\alpha \\nu \\beta } = { { \\partial {{\\Gamma}^{\\mu}}}_{\\alpha \\beta} \\over {\\partial x^{\\nu}} }\n- { { \\partial {{\\Gamma}^{\\mu}}}_{\\alpha \\nu} \\over {\\partial x^{\\beta}} }\n+ {{\\Gamma}^{\\mu}}_{\\gamma \\nu} {{\\Gamma}^{\\gamma}}_{\\alpha \\beta}\n- {{\\Gamma}^{\\mu}}_{\\gamma \\beta} {{\\Gamma}^{\\gamma}}_{\\alpha \\nu}\n", "faa9d169f81d95ead11aa227627d1238": " \\mathbf ([\\mathbf A, \\mathbf B]^T [\\mathbf A, \\mathbf B])^{-1}", "faa9d2e8a4101b76b3b0095fc7234d37": "\\text{ess}\\inf", "faaa0fb5d4aa4195b63de6478403b5ac": "(l',m')", "faaa22841768a3fea502aae8cb9c005e": "dx^2 + dy^2 + dz^2 - dt^2 = 0", "faaa54f4b5e0c1e376f235981211e4d4": "\\{\\{64x^3-448x+448,(0,1)\\},\\{64x^3+192x^2-256x+64,(1,2)\\},\\{64x^3+192x^2+80x+8,(2,4)\\}\\}", "faaa992ae7e45b6e4c20ccc50d9cc76b": "\\sin\\delta=0", "faaad16534a3b453044db4b35233e7ec": "\nS^*(\\theta)=\\sqrt{ S(\\theta) }\n", "faab79140716226c81df9fbe7f1f9ba1": "\\begin{align}\nA\n &{}= 2 \\pi \\int_{-r}^{r} \\sqrt{r^2 - x^2}\\,\\sqrt{1 + \\frac{x^2}{r^2 - x^2}}\\,dx \\\\\n &{}= 2 \\pi r\\int_{-r}^{r} \\,\\sqrt{r^2 - x^2}\\,\\sqrt{\\frac{1}{r^2 - x^2}}\\,dx \\\\\n &{}= 2 \\pi r\\int_{-r}^{r} \\,dx \\\\\n &{}= 4 \\pi r^2\\,\n\\end{align}", "faabb460d4ad4b58cd867e3f5580db72": "\\zeta_G(u) = \\frac{(1-u^2)^{\\chi(G)-1}}{\\det(I - Au + (k-1)u^2I)} \\ ", "faac06a9b11c1b3080f47bc596caeaf1": "(p,x) \\in S^{j-1} \\times D^{m-j} \\subset D^j \\times D^{m-j}", "faac6fdc6b84c8ba6c867e4f5d8da32f": "\\forall x_0 \\; x_1, \\quad x_0 \\approx_X x_1 \\Rightarrow f(x_0) \\approx_Y g(x_1)", "faacc764274a2a8be695987ad56bf2c4": " \\mathbf{\\hat Q}|\\psi\\rangle ", "faacd2fed4ada077205987100893a0cd": "MoM(Patient) = \\frac{Result(Patient)}{Median(PatientPopulation)}", "faacf08717e7a554fc563403536d64a0": "Y_{6}^{-1}(\\theta,\\varphi)={1\\over 16}\\sqrt{273\\over 2\\pi}\\cdot e^{-i\\varphi}\\cdot\\sin\\theta\\cdot(33\\cos^{5}\\theta-30\\cos^{3}\\theta+5\\cos\\theta)", "faad0d6698e354124982bbaa5d27e424": "\\begin{bmatrix} -2(\\eta_1+1) \\\\[10pt] \\dfrac{\\eta_2}{\\eta_1+1} \\end{bmatrix} ", "faad62e80390052b6a2a2cf13742a103": " \\boldsymbol\\beta ", "faadf5612ae73a699c29b4e1838e43b3": "a_1^{a_2^{\\cdot^{\\cdot^{a_n}}}}", "faaebd916e81f653c2de7046ae9f90d2": "( x_{ij} )_{m \\times n}", "faaecb49753e2fc776f8a63e4a6a5826": "\n\\mathrm{SINAD} = \\frac{P_\\mathrm{signal} + P_\\mathrm{noise} + P_\\mathrm{distortion}}{P_\\mathrm{noise} + P_\\mathrm{distortion}}\n", "faaee783424bc0e27e9aa2f56a7b50b8": "\\mathbf{x}^T", "faaefb0a89d328fb3c5bfdd1cac9291f": "\\hat{v_i}'\\equiv i[\\hat{H}'_0,x_i] = \\beta \\frac{p_i}{p^0} = 0 ", "faaf1814a6feaeeafa4b90cfa82ac114": "\\dim W(\\lambda) = \\frac{7\\cdot 8\\cdot 9\\cdot 10\\cdot 11\\cdot 6\\cdot 7\\cdot 8\\cdot 9\\cdot 5}{7\\cdot5\\cdot 4 \\cdot 3\\cdot 1\\cdot 5\\cdot 3\\cdot 2\\cdot 1\\cdot1} = 66 528.", "faafdfefb6f6bb072b219a2597fb4cc4": "\\binom{n}{k}", "fab01c74b0792d7342e923f5fb70e002": "\\mathrm{S}(\\mathrm{U}(p) \\times \\mathrm{U}(q))\\,", "fab03b0ddb3f180895b32515ed2a6129": "\\vec{u'} \\,\\!", "fab08d33ca0b769240633d89a8998af4": "I \\cap A", "fab0c0ae0a0d6006707518d2b06c408a": "k'_z\\,", "fab0e805507e5f79a11b85493303fa3d": "u_{n+1}=|u_n+U_\\mathrm{wall}(\\varphi_n)| \\,", "fab124c3445d8e9bb2257a756cf38e3f": "\\left(-4\\sqrt{\\frac{2}{5}},\\ 0,\\ 0,\\ \\pm2\\right)", "fab124e2312e8dd7ba04161be6dbed68": "t \\sigma \\equiv u", "fab1a340bc46f8f873bcf554cb30ab2c": " S = \\frac{m(m^n+1)}{2}", "fab1acc1bbcac0104d18e01b2191981a": "\n\\mu_m(\\{\\sigma \\in X : \\sigma\\uparrow n = s_0 \\to \\dots \\to s_n \\}) = \\mathcal{T}(s_0,s_1) \\times\\dots\\times\\mathcal{T}(s_{n-1},s_n)\n", "fab1b932f8f06935c58ff31dd7df7200": "D = d_1 + d_2", "fab1c695bfa14cfb2e47e4ecf85034d5": " \\vec{p} = \\frac{m_0\\vec{v}}{\\sqrt{1 - v^2/c^2}}", "fab216b7d9a6e756a955bab95d817931": "i \\gamma^\\mu \\partial_\\mu \\psi - m \\psi = e \\gamma_\\mu (A^\\mu+B^\\mu) \\psi \\,", "fab265030d8e9765407f497d14b1a310": " n(\\lambda) = B + \\frac{C}{\\lambda^2},", "fab2747ca87165ee755b7ba243d0e6f4": "\nE + S \\, \\overset{k_f}{\\underset{k_r} \\rightleftharpoons} \\, ES \\, \\overset{k_\\mathrm{cat}} {\\longrightarrow} \\, E + P \\;,\n", "fab298a7048b2c9b6dceefcf0b021b03": "s' \\in N(s, x)", "fab2a99e52e863518c30b0a54fcf14fd": "2^j", "fab2bf2ec810511e5f26aad67b59763f": "\\ell = \\hbar/p", "fab2f74c237168780e5656d14cf5ce69": "Q_{pq}(e'_p \\otimes e_q)", "fab33ffd3a9232ce060ac479fdf25a1c": "\\Delta J", "fab37d6c4a697fe660387d3ff8e889a4": "y=0", "fab394f0b28a1df19ce77c8bca8db290": "F_2=W(S)/\\sim", "fab3993866a192fdcee6d059b6511c6f": "V_2(\\mathbf{x},z_1,z_2) = V_1(\\mathbf{x}_1) + \\frac{1}{2}( z_2 - u_1(\\mathbf{x}_1) )^2", "fab3a84ddad9599cb9da40baccb9488f": " \\prod\\nolimits_{A \\in \\mathcal{A}} (1-x(A)) ", "fab44f8692e55a3ef7a2d5d132a4178f": "\\begin{bmatrix}\nq_0^2 + q_1^2 - q_2^2 - q_3^2 & 2(q_1 q_2 - q_0 q_3) & 2(q_0 q_2 + q_1 q_3) \\\\\n2(q_1 q_2 + q_0 q_3) & q_0^2 - q_1^2 + q_2^2 - q_3^2 & 2(q_2 q_3 - q_0 q_1) \\\\\n2(q_1 q_3 - q_0 q_2) & 2( q_0 q_1 + q_2 q_3) & q_0^2 - q_1^2 - q_2^2 + q_3^2 \n\\end{bmatrix}", "fab4578cc2a9f911d2f3e53939943715": "\\,r = \\cos(k\\theta)", "fab4a404fc4c4d937b31f17313bcc049": "\nD = \\min_{1\\leq i \\leq n}\\left\\{\\min_{1\\leq j \\leq n,i\\neq j}\\left\\{\\frac {d(i,j)}{\\max_{1\\leq k \\leq n}{d^{'}(k)}}\\right\\}\\right\\}\n", "fab522abb70f4d0c01908561aed4d8d9": "x = \\epsilon^{-1/4}\\left[y_0 - 1/4\\epsilon^{1/4} +\\cdots\\right]", "fab52ec8f0662d2a293069d2a32bc17e": "S(S^r\\times S^{q-1})\\rightarrow S^{q}", "fab53b1d21bb37d267f582f990060c24": "P(a, b, \\lambda)", "fab555628228a7345a0c998ae59c28d4": " \\mathbb{R}, \\,", "fab57fc8853fad9955a835f821416a0e": "\\zeta(14) = 1 + \\frac{1}{2^{14}} + \\frac{1}{3^{14}} + \\cdots = \\frac{2\\pi^{14}}{18243225} = 1.0000612\\dots\\!", "fab5a9b4e968249c991ad12033603ab3": " \\ f = \\ {2 h_f g D \\over L V^2}", "fab5b6461f7b50f569391afd2969e5fb": "D_x y\\,", "fab5d1d371e3e87e354c366f70ae9afa": "\\Delta\\lambda = \\lambda^2\\frac{\\delta D}{2D\\Delta D}", "fab62a8138d62a79dc8052079b6e2ba8": " O(mn \\log p) = O(n \\log n)", "fab649c009993fce520844f87c88eb63": "\\zeta(3) = \\sum_{k=0}^\\infty (-1)^k \\frac{205k^2 + 250k + 77}{64} \\frac{k!^{10}}{(2k+1)!^5}", "fab6587f0d58e856fa3f88afc3b52f2c": "\\left\\{p_0, \\ldots, p_n\\right\\}", "fab66ee59c8e613becd8935d4ba74a39": "\\rho = \\{ \\rho_H: A(H) \\rightarrow [0, 1] | H \\in \\mathbf{H}_0 \\}", "fab69068721a73576006891207c202ca": " [\\wp'(z)]^2 = 4[\\wp(z)]^3-g_2\\wp(z)-g_3, \\, ", "fab6b94545ae536f17c6f465f0bf5c04": "\n\\boldsymbol{H^T} \\cdot \\boldsymbol{H} \\cdot \\delta \\vec{x}_{0i} = h_i \\cdot \\delta \\vec{x}_{0i}\n", "fab6e20942ad25b9b51b28667c06cb0b": "\n \\mathbf{g}_i = \\frac{\\partial \\mathbf{x}}{\\partial \\xi^i}\n", "fab72c09b470029fdee45856d4ef4157": "\\,\\theta_A,\\ \\theta_B", "fab752551f1849b4cd4873e769082957": "d_{BO}\\!\\,", "fab769e978ea4a96e8f3b01f2b5ee08e": " {}_{t}L = v^{K(x)+1-t} - P\\ddot{a}_{\\overline{K(x)+1-t|}}", "fab7853379614efd09153bcafbd539bb": "h(f(a_1,a_2,\\dots,a_n))=f(h(a_1),h(a_2),\\dots,h(a_n))", "fab790e12058e91cecc5130b6eaeb139": "SubCipher_1", "fab7aa4e915edb5283a29d79d5b3d744": "\\dot{x} = \\epsilon f( t, x , \\epsilon )", "fab7bf7c715c1b788ed8f9ac3db2fc36": "E(t)=\\sum_{k\\geq0}e_k(X)t^k=\\prod_{i=1}^\\infty(1+X_it).", "fab7d78cdafa6c490164038034c7912f": "\\text{LSF} = \\frac{d}{dx} \\text{ESF}(x)", "fab855d40a344801e50f7dfc5f7cbea7": "\\mathbf{b}_{1} \\ \\stackrel{\\mathrm{def}}{=}\\ \\mathbf{a}_{2} \\times \\mathbf{a}_{3} / \\Omega", "fab860d88aee2201aa18ce01c9dd6255": "\\varepsilon_0 = ", "fab86de4fef3f3e636a93df84b0c67ca": "\\Gamma^{(\\lambda)} (R)_{mn}", "fab87637e8bd3b57ccfcb84531d48d31": " \\tilde{q} ", "fab8d655f070b96b269f6fffa8cd6180": "H\\cap N=\\{0\\}\\,", "fab8f0cffaeac3416223bc963104e038": "w ^1\\Delta_u", "fab9319b471cb2d72bec3f73900664d9": "2\\left( { n \\choose 2 } - n \\right)", "fab972472d991bacbb3a48c235470d3b": " g \\in \\mbox{orb}_G(f)", "fab977c97bb8a4f75899eba76031162e": "\\begin{matrix}\\frac1{16}\\end{matrix} (231x^6-315x^4+105x^2-5)\\,", "fab993f4a449f985fdd1851ed20897e1": " u(x) = \\sin kx", "fab998312cbbbffa75179e8e2e9d06b3": "p_{LA}", "fab9f0debba4c0b7e1b4d444a42783ad": "=-\\operatorname{tr} (\\gamma^0 \\gamma^5 \\gamma^0)", "fab9f5f95718d5c8334c27766a53f79e": "\ne = \\frac{A}{mk} = \\frac{\\left|\\mathbf{A}\\right|}{m k}\n", "faba1f816327f56133912d729a36a12f": "e_{(\\mathbf I_1)}\\,\\!", "faba6801dab367537e6b5b1c8ef9c24d": "K_2/\\mathbb{Q}_2", "faba779cdbfd16be984cc079c15fbdbb": " \\mathcal{R}(\\alpha,\\beta,\\gamma) = e^{-i\\alpha J_z}e^{-i\\beta J_y}e^{-i\\gamma J_z},\n", "fabb2e2cc25ba4aa213d9e23021e452f": "\\bold{x}_k", "fabb4ada367918cf5cff50eb4a159893": "K(t,\\xi) = e^{-2\\pi t|\\xi|}", "fabbac2fdb6854005a371910855cc012": " \\frac{\\det \\left(-\\frac{d^2}{dx^2} + A\\right)}{\\det \\left(-\\frac{d^2}{dx^2}\\right)} = \\frac{\\sinh L\\sqrt A}{L\\sqrt A}. ", "fabc01605eb6e7a479ca306f95d85f3e": "\\scriptstyle a \\;\\mapsto\\; -a", "fabc1f08e394bb9f7845fc3f130bc980": " U_{dyn}=\\frac {\\lambda \\,A}{e^\\left(AL \\right)-1}", "fabc856d9697cbfbaee203b08de6b1a6": "x[n] = A + w[n] \\quad n=0, 1, \\dots, N-1", "fabc98c18a7a3058702c3bf17e88823a": "df/2", "fabc9c61b060f7a54962301049ee9009": "\\left (\\frac{1}{T}-\\frac{1}{C} \\right )\\left (\\sigma_1+\\sigma_2+\\sigma_3\\right )+\\frac{1}{2TC}\\left [\\left (\\sigma_1-\\sigma_2\\right )^2+\\left (\\sigma_2-\\sigma_3\\right )^2+\\left (\\sigma_3-\\sigma_1\\right )^2\\right ]\\le 1 ", "fabcaeb10cd319a7283710b0238b830d": " \\left \\| C y_n - C y_m \\right \\| = \\left \\| (C- \\lambda_n) y_n + \\lambda_n y_n - (C- \\lambda_m) y_m - \\lambda_m y_m \\right \\|. ", "fabcce64e5098dccdc88ba0036fc2755": "\\begin{array}{rcl}\n \\dot x &= &-\\frac{1}{RC}x +\n\\frac{1}{RC}\\frac{A_c A_r}{2}\\sin(\\theta_r - \\theta_c),\\\\\n \\dot \\theta_c &= & \\omega_c + g_v (c^{*}x). \\\\\n\\end{array}\n", "fabcd3a0feb00b8af424bb3ea737951f": "x_{k-1} \\succ x_k", "fabcfb28d02e092978eac0eb0f69d548": "\n\\begin{pmatrix} V_0 & V_1 & 0 & 0 \\\\ V_1 & V_0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix}\n", "fabd376c6badb2cdfd676daeb259a6c3": "a_i = \\alpha^i", "fabdd49361cafb2def1b26c0b7fc9dd3": "R_{4,4} = r^4", "fabe6f8432c6b10d163626e3ade8f1e9": "X_k v_U =c_k(U) v_U, ~~~ k=1,\\dots,n, ", "fabe978423e0b6edbd4b50a5bcba198b": "\\mathbf{P}(H^0(V, K^n)) = \\mathbf{P}^{P_n - 1}", "fabedc0f8f6771a91d96cf61b0dab6a3": "y = a_0 + a_1x_1 + a_2x_2 + \\cdots + a_kx_k + e", "fabef2b29284598d6f15eadbf7597727": "E_\\text{K} = \\frac{1}{2}\\sum_{i=1}^n m_i \\mathbf{v}_i\\cdot\\mathbf{v}_i = \\frac{1}{2}\\sum_{i=1}^n m_i (\\omega \\Delta r_i\\mathbf{t}_i + \\mathbf{V})\\cdot(\\omega \\Delta r_i\\mathbf{t}_i + \\mathbf{V}).", "fabf297447cfa09633198c74d5d00291": "\\sigma_{xx} - \\frac{\\sigma_{xz}\\sigma_{xy}}{\\sigma_{yz}}", "fabf2f49cee97096612224f1c6953e80": "u:\\Omega \\to \\mathbb{R}^m", "fabf3348ea03105b18f80028c08d86c2": "\nU_{AB} = \\sum_{i\\in A} \\sum_{j\\in B} \\frac{q_i q_j}{4\\pi\\varepsilon_0 r_{ij}}.\n", "fabf55dd6f033f51ba7179add0980859": "T(W) = \\frac{1}{A(W)} = \\frac{W}{1-\\gamma} + \\frac{b}{a},", "fabfaf08c41de88f1ad742174539d47f": "r_i(x_j) = 0", "fabfb9093030fd895d6a55c7a3d9bb96": "d \\Phi = \\frac{1}{T}dU+\\frac{P}{T}dV + \\sum_{i=1}^s (- \\frac{\\mu_i}{T}) d N_i - \\frac {1} {T} d U + \\frac {U} {T^2} d T", "fabfe994f5ddee309acf0991883525df": "g(t)=\\log(\\operatorname {E}(e^{t X})).", "fac013856f523f39dc13ab03bdbde331": "\\Diamond A\\leftrightarrow\\Box A", "fac089d7b776dd7fd96abc8526ab7a9a": "\\mathrm{spt}(7n+5) \\equiv 0 \\mod(7) ", "fac0c77ca9960b9d8a70a0d8763d3315": "k_{f}=k_{r}+k_{cat}", "fac0d8903936205fcb60f38790632e35": " \\displaystyle{W(F)={1\\over 2\\pi} \\int F(z)W(z)\\,dxdy,}", "fac11cf8ae2d970568c6419e89563127": "~a(\\omega){\\rm d}\\omega{\\rm d}t~", "fac1425d2b2712aec0c3e609c5f85ebc": "C \\to \\,\\frac{\\omega_c'}{\\omega_c}\\,\\frac{R'}{R} \\,C", "fac143cc11046e609dab3c906006cad8": "U = \\pi + \\omega \\cdot L^{S} - D(L^{S})", "fac17a6cc45dea4b22eab211c1c2cecb": "\\scriptstyle (X',\\tau_1')", "fac19d816101779086b3b25c484ae3bf": "\\frac{n^2+1}{2}", "fac1ba2df3576e3e34b5a92206011590": "F(x)_v = f_v(x[v]) \\;.", "fac20c7a5732b76bb8cb20783d63230e": "S(0) = 0", "fac2280e1262affaeb00750f87b01d9d": "(\\pm\\sqrt{3/2},\\pm\\sqrt{3}/2)", "fac278da5c1159981f0547f7b2789bb3": "\\varphi_j(\\bullet)=1.\\,\\,\\, \\varphi_i([t_1,\\cdots,t_k])=\\sum_{j_1,\\dots,j_k} a_{ij_1}\\dots a_{ij_k} \\varphi_{j_1}(t_1)\\dots \\varphi_{j_k}(t_k)", "fac290af415f3b76cf180296f57de096": "= \\frac{d}{dt}\\left(\\frac{\\dot{y}}{\\dot{x}}\\right)\\frac{1}{\\dot{x}}", "fac2b25fdfc1a7980fd0d133f021f86d": "K_{0}", "fac2d71c6c55851f05b217e5028cad1a": " Q = Q^*\\,\\!", "fac2fbfd6ef462791152f7513de217c0": "\n = \\sum_{a \\in A_i} \\sigma^*_i(a) (u_i(a_i, \\sigma^*_{-i}) - u_i(\\sigma^*_i, \\sigma^*_{-i}))\n", "fac31343ea4f5d59cdafeca91a7bcf94": "1\\le i < k\\le m\\text{ and }1\\le j < \\ell\\le n", "fac330fd0c467f6d82abced9f8c57cdd": "M^*_w \\Phi^{-1}(w),\\ \\ w\\in W*", "fac36e396f6a6d68dc516246009bc6fd": "N/K \\subseteq_s M/K", "fac3932d8bc8218274904de9d5b4f471": "g'_{\\mu\\nu}=g_{\\mu\\nu}+h_{\\mu\\nu}", "fac39ac226af23032ba27aca973b0026": " \\pi: G \\rightarrow \\operatorname{U}(H) ", "fac3cf490c746aef4f20a72bed5ad4c7": "f(p) = a \\uparrow^p b", "fac3f620434c321338687e5d37a4ab53": "X: \\Omega \\rightarrow B", "fac3ff17282878db940c5b4f5cd6703e": "f_i.(v \\otimes w) = v \\otimes f_i.w + f_i.v \\otimes k_i^{-1}.w", "fac42231531c7b3b8445d9a1996e6180": "i : X \\hookrightarrow Y : x \\mapsto x", "fac4c5f64658baeac7a1bf18017ac862": "n_i\\times n_i ", "fac538b6658d7f6daec65578eeb819ee": "Dim_H(F \\times G) = Dim_H(F) + Dim_H(G)", "fac53fa179746652e30b43393a2a3c36": "\\{ S_i : i \\geq 1 \\}", "fac62fffd440b7613f7a96f629011f53": " 0 < a_1 < a_2 \\cdots < a_{\\phi(n)} < n ,", "fac65324329e3a9ed11f55fb1cc33748": "\\boldsymbol{p} =m_\\text{e} \\boldsymbol{v} ", "fac667a9f5732634a5dab2e3a835b560": "\\ S(t)=mf(t) ", "fac6df10e42f25b93708656bc8c2741e": "C\\ell(n)", "fac6e0dbb1db197259a028f89a68539a": "Z^{,i_1\\dots i_n}[J]=i^n Z[J] {\\left \\langle \\phi^{i_1}\\cdots \\phi^{i_n}\\right\\rangle}_J", "fac71bd4818847e18926d143b789bf20": "\\pi\\ = \\frac{P_1 \\cdot p_1+P_2 \\cdot p_2}{K \\cdot p_1} \\,", "fac7921e8578bf0ac9e436b3755a5175": "\\frac{dx}{dy}=-\\frac{1-\\alpha}{\\alpha}\\left(\\frac{x}{y}\\right)^{\\rho-1}.", "fac7d7bcd4fc8814ffcaf932bfdeed7d": "(\\Lambda-\\mu I)^{-1}V^{-1}\\delta AV", "fac7e448cf6548492c03d56ef4236478": "\\textit{plant}", "fac7e5f94d65a7204daae16b4449052d": "y_i \\in \\mathbf{Y}", "fac7e6c0f4253b38eaa2e5d6aae61cd4": "\\mathbf x = \\begin{bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end{bmatrix}", "fac7ea6cbd6aafd0a779f499cc284da3": "2.0000", "fac84465c223fe014a6b35da01e7cdb8": "\\frac{1}{a^{2}b^{2}}\\left(\\frac{x^{2}}{a^{4}}+\\frac{y^{2}}{b^{4}}\\right)^{-\\frac{3}{2}}", "fac88080a8ff17637c5663732b9ef9b4": "\\triangleleft.", "fac94679310b09efa136e67426daf603": "|\\Phi^-\\rangle_{AC} \\otimes (\\alpha |0\\rangle_B - \\beta|1\\rangle_B)", "fac94cfd02e28edba72e70f0fda4443d": " \\coprod", "fac97b9a98b5536bcfc68fbf9422a791": "\\textstyle i \\le j", "fac9857ad1397c1dce5e6095e7a00318": "f:M\\to \\mathbb R", "fac9e993019b9509eb2b57fde5da10e9": "y_i E(\\alpha_i) = Q(\\alpha_i)", "faca12e8d0f8cbbd7d2f55d15129dd88": " \\min f(\\bold x) ", "faca2e7e35330cce57eedf5b118e24d9": "E\\subset\\partial D", "faca3e753a7340bc7072a2020888ca51": "\\neg P", "faca70581296e71a6342ea01b198406c": "\\psi(w)=\\psi(\\varepsilon)", "facab1af21410df88e0ae806cc6552f0": "|||\\bigstar \\bigstar \\bigstar", "facb4c1494b5c67cd465243e73a8dbf2": "\\tilde{\\epsilon} = \\epsilon + i\\frac{4\\pi \\sigma}{\\omega}", "facb5bed2b9292fde8baa3c15bf05e8b": "L_k(m) = v_k \\lambda_m \\frac{\\frac{m-1}{m}L_k(m) + 1}{\\mu_k}", "facc187f7aa0139543677f2fd3ff991e": "\\ell_{w} \\sim 10^{-18}", "facc1f5959cc4a5eaa4fa8c63b4c3df8": "\\sum_{k=1}^N k^2", "facc69d74648a30c40989fc9be7d123a": "a_3=3a", "faccc69ba452889253a1471548b3aa6d": " z(t) = a + \\varepsilon e^{it} ", "facd11ef99d828c1423da0bcdc385e64": "\\tau_{_0}~\\frac{\\partial\\mathbf{q}}{\\partial t}~+~\\mathbf{q}~=~-k~\\nabla\\theta,", "facd6d569103fc4baeafad441fd8d2ef": " \\lim_{x \\to 0} \\frac{x}{x^3} = \\infty . \\! ~~ (6) ", "facd750dff6476cf5088e85ec2c178f4": "\nA_{ij} = \\Phi^{-1}\\left(\\frac{R(X_{ij})}{N+1}\\right)\n", "facd97706792f81e60de9553e9acce6a": "y = {R \\over \\sqrt{\\pi}} \\sqrt{\\theta - {\\sin(2\\theta)\\over 2} + C \\sin^3 \\theta}", "face30507562baf56d3d0c7352f17415": " B_T\\, ", "face49a25478a7b76a28c37b82c041a9": "\n\\langle p \\rangle_{RV} = \\langle p \\rangle_{VR} \\equiv \\langle \\langle p \\rangle_R \\rangle_V\n", "face5edc6f7221ca373d2ca73dca3071": "2e^\\psi ( R_0^0 - R_3^3 ) = \\lambda\\psi^\\prime + \\varkappa\\dot{\\psi} - \\dot \\varkappa - \\lambda^\\prime - \\frac{1}{2}\\varkappa_a^b\\varkappa_b^a - \\frac{1}{2}\\lambda_a^b\\lambda_b^a = 0. ", "facef65fae3ca496957ba83aa6ab07ed": "\n\\mathrm{pivot}=\n\\begin{cases}\n\\mathrm{pivot}+\\epsilon\\lVert\\boldsymbol{A}_j\\rVert_1 & \\text{if }\\mathrm{pivot}\\geq 0\\text{,}\\\\\n\\mathrm{pivot}-\\epsilon\\lVert\\boldsymbol{A}_j\\rVert_1 & \\text{if }\\mathrm{pivot}<0\n\\end{cases}\n", "facf02304dd8115c8a7b35b8acdfd505": "\\left[{0 \\atop 0}\\right] = 1", "facff7af083cbd9c12cfdee6c00f8009": "\\mathbf{v}=(u,v,w)", "facfff2220968329c37222236d96ba77": "(a = 1)", "fad0020bfbf64a8ccf666c9811582eb8": " \\textbf{V}_P = \\frac{d}{dt}(R\\textbf{e}_r + Z_0\\vec{k}) = R\\dot{\\theta}\\textbf{e}_t =R\\omega\\textbf{e}_t,", "fad01600fec48f32aa89d80e7b577c3e": "\nF_2(r) = \\frac{\\mu}{r^3}\n", "fad02d9b4ab8554eece0bd08b43d28b4": "\\left|\\frac{S^k-S^{k+1}}{S^k}\\right|<0.0001.", "fad03095674a05df9464530500111197": "X_i\\ \\sim \\operatorname{Exp}(k \\theta)", "fad05e8fe625642e79f25934ab850d6c": "X\\,\\sim\\,\\textrm{Stable}(1/2,1,c,\\mu) \\,", "fad079ab41382bdd08e5c084577366c3": "\\alpha_\\mathrm{\\{per\\ comparison\\}}=1-{\\left(1-\\bar{\\alpha}\\right)}^{\\frac{1}{n}}", "fad0eca9ad6495c0483c73c4d22e87aa": "k = 2n - 1.\\ ", "fad10b5c00119160ffe2d2c333187e3c": "AP=P\\begin{pmatrix}\\lambda_{1}\\\\\n& \\lambda_{2}\\\\\n& & \\ddots\\\\\n& & & \\lambda_{n}\\end{pmatrix} .", "fad143d8cda83feedf01c65c8d1f93d2": "I_O", "fad18c4783f0b33d19354b330f4e8884": "y^2 = x^3 + 3\\lambda a(x+\\lambda)^2", "fad1c241121698297d1675f232b1430d": "\\Sigma*", "fad1fec39799466cd183f0b4a6fb9e6b": " \\frac{v_0^2}{gh_0} = 1", "fad21f04beff83a85dc55c297a7fb061": " {32\\over27} \\cdot {81\\over80} = {{2\\cdot3}\\over{1\\cdot5}} = {6\\over5}", "fad22ad94c0030c83a9b3194c19af841": "\\rho_S", "fad22d86ee568c9dae99d2a7d5451313": "\n\\mathbf{B} = \\mathbf{n}\\times\\mathbf{E},\n", "fad285bb7b9932aaa0e0fd2d867591d9": "L = \\{w + \\lambda v \\mid \\lambda \\in \\mathbb{F}\\}", "fad2f8a45bd6da37d41052ebd5cd2f31": "m:n ", "fad304a64c9b17f086e2cb2cdbf2db88": "\\rightharpoonup", "fad324235a07b9f04a708e0fde882267": " (s,b) \\leftarrow \\delta_{ext}(s, t-t_l, x)", "fad329a48faa7f1d264ee8316f97a229": "\\operatorname{perm}(U)^2 = \\det(U+V)\\det(U) ", "fad33772b6ac2b67dc0a6e8bd60f002f": "g_2^2=g_3^3= (g_2g_3)^7=-1, \\, ", "fad3d57b4bef0f4ad55dff54679164a3": "(4)", "fad3f02a568cb9d6ddad2223b5f794a7": "\\textstyle \\gamma = N(0,1) ", "fad4a7508b4e1c7f3c4268b2e70c0a39": "3[\\cos(x) + \\cos(y) + \\cos(z)] + 4 \\cos(x)\\cos(y)\\cos(z) = 0", "fad4cf3c411fec375289b07a9ca9df4e": "(a_{st})", "fad5265603eaa6a717a52a4454e5845f": "L = \\Phi \\Lambda \\Phi^T A", "fad59f5e1cd40bed846a90f811a49b5e": " e_{n+1} = e_n + h \\Big( A(t_n, y(t_n), h, f) - A(t_n, y_n, h, f) \\Big) + \\tau_{n+1}. ", "fad62fa5ca10f0fa67a029dd5a346e2f": "\nI_c = \\{j \\mid \\exists x \\in s: j = ((c + x) \\mod(n_1, \\ldots, n_k))\\}\n", "fad63dd0c7e5c95d792d483788ad3380": "p(u + v) \\leq b(p(u) + p(v))", "fad641b413fcb4fbfaca0f73c2cff50b": "\\tfrac{5}{10}", "fad6461a20df080b6f9ce899d92670c0": "\\tau = \\frac{P_s {\\theta_P}^\\beta}{\\max \\left[ \\theta_{cp} - \\theta_p, \\epsilon \\left(1-\\theta_p \\right) \\right]}", "fad64add76faf04fc05aa024cbe89bb5": " - n! (-1)^{n-1} \\left(-\\frac{1}{n} + \\frac{1}{n-1}\\right)\n= n! (-1)^n \\frac{n-(n-1)}{n(n-1)}", "fad6791c1b8966c6dd8ae96f31c56828": "Payout = Claim \\times \\frac {Sum\\ Insured} {Current\\ Value} \\!", "fad7c6d265d91bd6a4c566be587d393c": "\\{1, 2\\}", "fad7de7d5578d0e1e9eb6ace1db7497f": "x=-\\frac{b}{2a}", "fad811c9fdb1336dc35334873a869c2b": "10^m \\equiv 1 \\mod 9, n = n_m + n_{m-1} + \\cdots + n_0 = c \\mod 9", "fad83a82e05a8aba9c4cf004d0476168": "T: A \\to X", "fad8716ceef4239a09d6af54cca808bf": " V_F:=V \\otimes_K F ", "fad8784273f35028938df90e23bf80a1": " OR=\\frac {ad}{bc}", "fad8a1af2c68b4a7309cc4472387d6dd": "\\sup_{x\\in U, f\\in A}|f(x)|<\\infty", "fad8cfcc6075c2a8e2d0c77076fd6172": "\\begin{matrix}\n t_1 & \\rightarrow & t_2 \\\\\n \\left( \\alpha | \\uparrow \\rangle + \\beta | \\downarrow \\rangle \\right) \n \\otimes | init \\rangle \n & \\rightarrow \n & \\alpha | \\uparrow \\rangle \\otimes | O_{\\uparrow} \\rangle \n + \\beta | \\downarrow \\rangle \\otimes | O_{\\downarrow} \\rangle.\n \\end{matrix}", "fad8e0c6904c7c0d09150c54207dbf06": "\\Delta <0", "fad8f100b6a43541d46ef6633b26426d": "[\\text{ }3,\\text{ }4,\\text{ }5]", "fad93bbd1d655377e5895618838cc476": " {} = \\begin{vmatrix} x_1 & y_1 \\\\ x_2 & y_2 \\end{vmatrix} z_0 - \\begin{vmatrix} x_0 & y_0 \\\\ x_2 & y_2 \\end{vmatrix} z_1 + \\begin{vmatrix} x_0 & y_0 \\\\ x_1 & y_1 \\end{vmatrix} z_2 ", "fad9716ec72dc06fde999eb1bc19bd21": " \\frac{dV}{dt} = \\mu N P - \\mu V", "fad98e0193bc0128aba2a923d6317fa0": "\\cos x ,\\, \\sin x ,\\, e^x ,\\, xe^x \\,.", "fad9bfd832473f5a0b2f61c73d9e185b": " q_2 ", "fad9c50461b25acfe3989671c027a243": " G(v)", "fada2c0378dc03c8bfb2799a67015ff6": "\\int_0^\\infty x^{z-1}\\,e^{-x}\\,dx = \\Gamma(z)", "fada3aef397a0e6d9a4630c88db4c71e": "\\|(\\Lambda-\\mu I)^{-1}\\|_p\\ =\\max_{\\|\\mathbf{x}\\|_p\\ne 0} \\frac{\\|(\\Lambda-\\mu I)^{-1}\\mathbf{x}\\|_p}{\\|\\mathbf{x}\\|_p}\\ ", "fada46d91eea71bd72874ced1a67d301": "\\mathcal{L}\\{\\phi(t)\\}=\\frac{2}{s^2+4}", "fadad240a78fa06726ab86a595570648": "= G\\left(t;\\mu_1\\right)G\\left(1/t;\\mu_2\\right)\\,", "fadb10af52f8e8eb2893ad4d22f6a37b": "T(n) = T(n-1) + (n-1)T(n-2)", "fadb1a1b4b378290f93a101ec57d0897": "t_{max} = \\frac{1.44 \\times T_P T_d}{T_P-T_d} \\times ln(T_P/T_d)", "fadb2175e30c0a2a5f27e302b2bf3ecb": "\\lambda_a", "fadb4ac97009e60d57ae7ff984fd1250": " V_{SL} ", "fadb6e3d1b97f79d1ff29fdb17123af3": "\\color{TealBlue}\\text{TealBlue}", "fadbc675c8cce9af788e5745746ea048": "\\scriptstyle \\{w_i\\}", "fadc7b6d0d16613990d61ecb7df4283d": "\\, 1/17 = 0.0588235294117647\\ldots", "fadc8e410e3444f323c8916c1cb3c555": "\\Delta k=k(2\\omega)-2k(\\omega)", "fadcfb9cb85729362512b34e9f13b202": "\nE=U_{1}+U_{2}. \n", "fadd0a9282746e8a4de160d3e9014bc4": "ax^2 + bx + c,\\,\\!", "fadd58836bc37aba6e2267beb3b11075": "[3.75] = 4", "fadda0048cfaf8c1444c43b4d1067f65": "K_- = Ran(A-i)^{\\perp}.", "faddcf0f30555832971944239cd2a42e": "var_{ab}(p) = var \\left ((1+x)^{\\deg(p)}p\\left (\\frac{a+bx}{1+x} \\right ) \\right ),", "fade1ab7c92ba54e619baf4b0e47de33": "\\eta_\\mathrm{Receiver} = \\frac{Q_\\mathrm{absorbed}-Q_\\mathrm{lost}}{Q_\\mathrm{solar}}", "fade25273e2f9ed84f05ceaac3d809f5": "\\displaystyle{Q(a)R(b,a)c=R(a,b)Q(a)c,}", "fade4855c57bdf01ae916f93c0814ace": "\\mathcal O(\\overline{c}^2)", "fade5240e61a42404b5f6e3adaac33aa": "\\int_{\\mathcal{X}}\\psi(x,T(F)) \\, dF(x)=0", "fade550d56f38b26b3e37ac7afd3a1c4": "\\,\\!\\pi r", "fadf3d622e3d3b98d79817453485ecf5": "\\delta = \\psi(\\alpha+1)", "fadf8182315837ce105f84188817d45b": "{\\operatorname{d}y\\over\\operatorname{d}t}= v(t)- \\lambda t", "fadf9be87c241c5b599b35877626e786": "A_i(\\vec r)\n=\\int \\frac{\\mu_0 j_i(\\vec r^{\\,'})\\,\\, dx_1'dx_2'dx_3'}{4\\pi |\\vec r -\\vec r^{\\,'}|}\\,.", "fadfa24d605a49ccbbb5f2142aceed21": "\\textit{VendingMachine} \\left\\vert\\left[\\left\\{ \\textit{coin}, \\textit{card} \\right\\}\\right]\\right\\vert \\textit{Person} \\equiv \\textit{coin} \\rightarrow \\textit{choc} \\rightarrow \\textit{STOP}", "fae0003d6a1d92dd07a2fdf5e78c6b57": "\\int \\phi(x) \\, dx = \\Phi(x) + C", "fae00a90d01f63c83419b8c56eb1f7eb": " \\sum_j a_{ij} x_j = \\lambda x_i \\quad \\forall i \\in \\{1, \\ldots, n\\}. ", "fae0627aa01fcee796e03e54737a5b0a": "\\overline{\\theta_n}=\\frac{1}{N}\\sum_{i=1}^N \\arg(z_i^n)", "fae0a03e87f957199d94fd84ee1ef082": " T=\n \\begin{bmatrix}\n 1/2 & 0 \\\\\n 0 & 1/7 \\\\\n \\end{bmatrix}\n\\left\\{\n \\begin{bmatrix}\n 0 & 0 \\\\\n -5 & 0 \\\\\n \\end{bmatrix}\n +\n \\begin{bmatrix}\n 0 & -1 \\\\\n 0 & 0 \\\\\n \\end{bmatrix}\\right\\} \n =\n \\begin{bmatrix}\n 0 & -1/2 \\\\\n -5/7 & 0 \\\\\n \\end{bmatrix} .", "fae0a1cfb03884054a984149f7140016": " \\log_a b = \\frac{\\log b}{\\log a} \\,, ", "fae0b27c451c728867a567e8c1bb4e53": "666", "fae0e6bff6d4dfa6d2cc3b4cab6249e8": "ED_B>ED_A", "fae138bdb9ce6706a6ba4e2c578e13d4": "\\mathfrak A_{(\\infty,\\infty)}", "fae1630d60c7d52ac9f138d8f2b33e3f": "s_1 s_2 \\cdots s_n", "fae1e229bac69cae3223f78bee024e26": "q^i = \\mathbf{v}\\cdot \\mathbf{e}^i = (q_j \\mathbf{e}^j)\\cdot \\mathbf{e}^i = (\\mathbf{e}^j\\cdot\\mathbf{e}^i) q_j. \\, ", "fae1fc8cae997d7a5824f0fae611c91f": " |\\psi\\rangle = \\begin{pmatrix} \n\\psi_+ \\\\\n\\psi_-\n\\end{pmatrix}", "fae20be02e46d0652a06af5e64b16cd0": "\\frac{1}{\\sqrt{y}}", "fae2ad584b175e20668b3ff60b2245f5": "a_{n+1} = a_n - a_n c_n / 2 \\,\\!", "fae2c61d59916c9a581c6546646ca6f6": " \\log{\\hat {\\lambda}} \\approx Z \\, \\sqrt{4/D} ", "fae307cd8a64609514ec56dba1ad4937": "\n\\mathit{PAPR} = {{|x|_\\mathrm{peak}}^2 \\over {x_\\mathrm{rms}}^2} = C^2\n", "fae334d1a9296deb886646b4fb1530a6": " 1 - \\alpha(1-\\alpha) d_M(\\mu_1, \\mu_2, \\Sigma)^2 ", "fae38aaf36bdf0a57c42fd00f3a6b886": "C \\setminus (A \\cup B) = (C \\setminus A) \\cap (C \\setminus B)\\,\\!", "fae3aee6e9a61f5f34c16880ace48c01": "\\lambda I - T", "fae3c6a586e3994b9fdfdfa8f1762a32": "D*", "fae3da205e6d120287e4759ea695eb40": "R^{bas}(\\phi,\\psi)(X):= \\nabla_{\\!X\\,}[\\phi,\\psi]-[\\nabla_{\\!X\\,}\\phi,\\psi]-[\\phi,\\nabla_{\\!X\\,}\\psi] -\\nabla_{\\!\\nabla^{bas}_{\\!\\psi\\,}X\\,}\\phi +\\nabla_{\\!\\nabla^{bas}_{\\!\\psi\\,}X\\,}\\phi.", "fae3fd1a5f1f91027a1779731d9e8d1a": "S(a,b)\\,\\!", "fae42690cc9178903c77fa3fb6a9fdd5": "\\begin{align}\n\\mathcal{A}\\left\\{x[n-m]\\right\\}\n&= \\sum_{k=n-a}^{n+a} x[k-m]\\\\\n&= \\sum_{k'=(n-m)-a}^{(n-m)+a} x[k']\\\\\n&= \\mathcal{A}\\left\\{x\\right\\}[n-m],\n\\end{align}", "fae42de8b1dbc980812efcd32d7a8be3": " Q_\\text{cmb}=0", "fae430278929e105e7a235ab41bfc195": " \\Sigma^{-1} \\Sigma_b ", "fae442d3031ff0fb4437c3cbef378b5c": "USp(4)", "fae456b6bf19ca11d44cf64db95bd572": "R(\\theta ) = \\frac{{I(\\theta )r^2}}{{I_0 \\Delta V}} = Ni(\\theta )/k^2", "fae483560a9b6d796c5cc2d9ebf27937": "{\\partial u\\over \\partial t}={u_i^{n+1}-u_i^n\\over k}", "fae4b67a46c4f7b5c8c19701cd418472": " \\Gamma=(\\Gamma_{kl}) ", "fae4c5b54ea9cc77446edf2f7baa7a0e": "\\mathrm{d}A = \\mathrm{d}U - T\\mathrm{d}S\\, ", "fae51402d14ffafdabce97541582c2dd": "\\int\\frac{dx}{R^5} = \\frac{4ax+2b}{3(4ac-b^2)R}\\left(\\frac{1}{R^2}+\\frac{8a}{4ac-b^2}\\right)", "fae519074c538f517af264f4cdabe03b": "\\frac{1}{b-a}=O(n^c)", "fae54dea4699c8bf5b5fc101def3d590": "\\begin{cases}\\begin{align} x + y &= 1 \\\\\n0x + 0y &= 2 \\end{align} \\end{cases}\\,", "fae5510c22aeb57ea212763d488b6df6": "L_x,\\, L_y,\\, L_z", "fae563b78bca519fa9b2e279c597c1a2": " \\frac{V^2}{R} = \\left| f \\right| V ", "fae5f91fe4bc138fa5f98b49eed97257": "x \\sim C(\\theta 1_n,I \\sigma^2)\\,\\!", "fae65eb823f78f60f82f9d74c2bd3aa4": "f_i(x) = 0", "fae66725f13cf1bd7463ce9dfee5a644": "Y_{4}^{0}(\\theta,\\varphi)={3\\over 16}\\sqrt{1\\over \\pi}\\cdot(35\\cos^{4}\\theta-30\\cos^{2}\\theta+3)\n= \\frac{3}{16} \\sqrt{\\frac{1}{\\pi}} \\cdot \\frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}", "fae67e01f9c2aa61805dc0148d939c94": "\\mathfrak{a}^*_+", "fae68bbfb96b293cf4b63293ed4f50b6": "0^{\\circ}, 45^{\\circ}, 90^{\\circ}", "fae70b214c4fc9e118acacd2beaab00d": "F^{0i} = -F^{i0} = - E^i/c ,\\quad F^{ij} = - \\varepsilon^{ijk} B_k ", "fae71bd76cb8d5e60c0847e1fe6c7fab": "m+(t-1)p", "fae7aba45c822ca475d9050ef55b69df": "\\textstyle f_{T}(X)", "fae85ca68872b106ffd6abd1a4622deb": "G_0/G_1", "fae8cdab13bb572109be3385969b40bb": "|j\\rang", "fae8f4627d1119b53c7ac0418fea6eaa": "|SB|=\\frac{|SD||SA|}{|SC|}", "fae94e7b37c9e7416a6c068aa9f87b4b": "\\xi\\,\\!", "fae97d478b9483804e981bb77e57ec13": "\\lambda\\ x.x:\\alpha\\rightarrow\\alpha", "fae98c3ce3e4c85c4cccdb3276decace": "\\bar{X} + t_{n-1,0.95} S \\sqrt{\\left( 1 + 1/n \\right)}", "fae9b2016263e69ff72f3af0bbeacff3": " 0 \\le \\omega/\\Omega_c \\le 1 ", "fae9d176e2584baa0098dc63b4ea5566": "\\mathrm{NDVI}={\\mathrm{NIR}-\\mathrm{red} \\over \\mathrm{NIR}+\\mathrm{red}}", "fae9edff23b2cb665b1827902720512c": " a^\\dagger(\\vec{k})=\\left(E\\tilde{\\phi}(\\vec{k})-i\\tilde{\\pi}(\\vec{k})\\right),", "faea1f7f86ca100196445115d93efe55": " (\\beta)^{3.5} ", "faeaad16f5c3cca9ac8deafe99b3a64c": "p_j(x)\\approx f_j(x)\\,g_j(x)", "faeafc35c287038a9db2bae8e5567e0e": "I_m = \\frac{\\int I(\\psi)d\\omega}{\\int d\\omega}", "faeb28c0ad1000cfabe2e73c4ba8425a": "n+m-2", "faeb28f2b92b148f35c30daa9b631e74": " W X=x+\\varepsilon. \\,", "faeb3ce5442d553e95f3985fccabd279": "\\nabla\\times\\left(\\mathbf{P}\\times\\mathbf{Q}\\right)=\\left(\\mathbf{Q}\\cdot\\nabla\\right)\\mathbf{P}-\\left(\\mathbf{P}\\cdot\\nabla\\right)\\mathbf{Q}+\\mathbf{P}\\left(\\nabla\\cdot\\mathbf{Q}\\right)-\\mathbf{Q}\\left(\\nabla\\cdot\\mathbf{P}\\right)", "faeb73bf27eafc327e7dc9ea4bb7a70d": " \\tau D_H ", "faeb848f3a8f6415982563c0d5ee7597": "()()(),\\qquad ()(()),\\qquad (())(),\\qquad (()()),\\qquad ((()))", "faec36c1f4a4c967f57ece6e6cb1902e": "\n\\hat{A}\\hat{C} = [\\hat{A}^+]\\hat{C} = \\begin{bmatrix} A^+ & 0 \\\\ B^+ & A^+ \\end{bmatrix}\\begin{Bmatrix} C \\\\ D\\end{Bmatrix}.\n", "faec3ef186b3c6dbfba67af55289d0a6": " (\\nabla^2 + k^2)\\mathbf{B} = 0,\\, \\mathbf{E} = -\\frac{i}{k} \\nabla \\times \\mathbf{B}.", "faec6d88a024410268d06e7188a7670b": "P(A) = {N_A \\over N} ", "faecb46b985be9c18de42e2a4cd5557b": "\\mathcal C \\, | f \\, \\bar f \\rangle = (-1)^L (-1)^{S+1} (-1) \\, | f \\, \\bar f \\rangle = (-1)^{L + S} \\, | f \\, \\bar f \\rangle ", "faecc7be40e80a2e7df8a56b2b87920c": " \\mathbf{MTF_{sys}(\\xi,\\eta) = } ", "faecdbf821c90bd77aae8b71c1a9a176": "x \\to \\infty", "faecdfc487f8175328db1ff2ecc6c77a": " F_{SK}(x) = e(g, g)^{1/(x+SK)} \\quad\\mbox{and}\\quad p_{SK}(x) = g^{1/(x+SK)}, ", "faecf03198a5661cdb768653f9eddc86": "\\textstyle{\\operatorname{tr}(A)\\cdot \\operatorname{tr}(B) = \\sum_i a_{ii} \\cdot \\sum_j b_{jj}}", "faecf8f2f9dbabe7b4a5b62bebefd12e": "\\lambda < \\lambda_J", "faecfd31e4312016c7acf84832c4e133": "\\cap_1^\\infty I^n = 0", "faed1f0dd6739e3d7aa875a6ce378d24": " a+X\\rightarrow Y+b", "faed50ad163cf9821c8d8a2e0ba74847": "h = h_{\\alpha\\bar\\beta}\\,dz^\\alpha\\otimes d\\bar z^\\beta", "faed606449980bef1da84a4cc2eb348b": "xy \\vee \\bar{x}z", "faeda4a56c7b22054e98f33cc13f519f": "\nS = \\frac{2e^3}{\\pi\\hbar} \\vert V \\vert \\sum_n T_n (1 - T_n)\\ .\n", "faeda579131d9a11989ff40c4cb5f99b": "\\frac{D^2\\tau^2}{2}", "faefd0ad7bed596fcc626907d78eddbc": "\\dot{\\mathbf{A}} = \\dot{A}_r \\hat{\\boldsymbol{r}} + A_r \\dot{\\hat{\\boldsymbol{r}}} \n + \\dot{A}_\\theta \\hat{\\boldsymbol{\\theta}} + A_\\theta \\dot{\\hat{\\boldsymbol{\\theta}}}\n + \\dot{A}_z \\hat{\\boldsymbol{z}} + A_z \\dot{\\hat{\\boldsymbol{z}}}", "faf063ba2edcf56e8103e2b17e934ea7": "\n{dH\\over dt} = P*{dP\\over dt} + ( X + 3 \\epsilon X^2)*{dX\\over dt} = 0 ~,\n", "faf0a92f3c0c74579c876458e8e10ca8": "F, G : \\mathbf{C} \\to \\mathbf{X}", "faf0b2f7b4e2bac5dcdc6e108157a658": "\\|\\mathbf{x}\\| = \\sqrt{\\mathbf{x}\\cdot\\mathbf{x}}", "faf124b9dfbfa7cfb7558c5157d27ffc": " \\left| \\mathbf{a} \\right| = \\sqrt{\\mathbf{a} \\cdot \\mathbf{a} } = \\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2 + {a_4}^2},", "faf1763f3fa430033f0780d086ac39e5": "\\phi^{A,x,\\bar{a}}=\\{x\\in A\\colon A\\models\\phi[x,\\bar{a}]\\}", "faf1b3a30453e631b14c4b627225001e": "n\\ge1", "faf1cebfb2916c95956c2e7eab1f6e7d": "s(G) = \\sum_{(u,v) \\in E} \\deg(u) \\cdot \\deg(v).", "faf21c72a3f3ccfe1ccda4bab46f9dd2": "\n\\frac{d}{dt} \\begin{bmatrix} A_0 \\\\ B_0 \\end{bmatrix} = \\frac{\\varepsilon}{2\\pi k} \\int_0^{2 \\pi} f( \\frac{a}{k^2} + A \\sin (\\theta), kA \\cos (\\theta)) \\begin{bmatrix} \\cos(\\theta) \\\\ - \\frac{1}{A_0} \\sin(\\theta) \\end{bmatrix} d\\theta,\n", "faf22213d773d55e92f355e0cffb802d": "m_n=\\int_0^\\infty x^n\\,d\\mu(x)\\,", "faf23166e2429acaa670a8c64e8c7348": "\n\\frac{z^{2}}{a^{2} \\cosh^{2} \\mu} + \\frac{x^{2} + y^{2}}{a^{2} \\sinh^{2} \\mu} = \\cos^{2} \\nu + \\sin^{2} \\nu = 1\n", "faf24bb298a7bcd33b950b1c9f3bc088": "a = v_n \\cdot s", "faf2631a15218778a938d23b4e152b23": "b=C", "faf2852d534596ace4d92d4a89cd03a1": "\\frac{\\hbar^2}{2m}\\frac{a}{A}=\\sum_{K}\\frac{1}{\\alpha^2-(k+K)^2}", "faf2b3a3902243ca866161195bbed2d9": "p(X,A | \\theta)", "faf2c63a017fd5b9a80bba4fad5fa8ec": "= sb/(t+sb), \\quad \\beta =1", "faf2ea55f6e5e02e71c46771cfa2d151": "\\forall x\\in\\mathbb{R}\\ (\\forall y\\in\\mathbb{R}\\ \\neg\\neg(00}t^{b_{t,y}} + \\prod_{t,b_{t,y}<0}t^{-b_{t,y}}", "fb07d0522f810254f0856e9e278f815c": "\\frac{1}{T_{i}} = \\left(1-\\frac{\\delta S}{V}\\right) \\frac{1}{T_{ib}}+\\frac{\\delta S}{V}\\frac{1}{T_{is}}+D\\frac{\\left({\\gamma G t_E}\\right)^2}{12}", "fb07f97f58cbafb52f221c9fdbac5c7f": " F_x = \\frac{ee'}{r^2} \\left[\\left(\\alpha_0+\\alpha_1 \\frac{u_x^2}{c^2}+\\alpha_2 \\frac{u^2_r}{c^2}\\right) cos(rx) - \\beta_0 \\frac{u_x u_r}{c^2}-\\alpha_0 \\frac{ra'_r}{2c^2} + \\left(\\frac{ra'_x}{2c^2}\\right)(\\alpha_0-2\\gamma_0) \\right]", "fb081936203419c56dc8357e94512c68": "a |c| \\int_{-\\infty}^\\infty e^{-z^2}\\,dz.", "fb0842a3f35220605f68ddb981684983": "P_a(x)", "fb08b5038a92f913ddec46aa5f970a71": "\n \\left|A\\right|_{ij} = (-1)^{i+j} \\frac{\\det A}{\\det A^{ij}} ,\n", "fb08d1b1a29b06d22caa06d8cde9adc3": "\\sum_Z", "fb08d90402b8f9d1dfcef6c7265250dc": "ax+by=c", "fb08ff23872565deea95edab12c5df5c": " b^*_1 ", "fb099b3b92be8aac0f87d835b51babca": "\\dot{c}(t)", "fb099f4e165f8de30c7f6f3e18a7a24c": " \\vec{\\epsilon}_1 = \\partial_r, \\; \\vec{\\epsilon}_2 = \\frac{1}{r} \\, \\partial_\\theta, \\; \\vec{\\epsilon}_3 = \\frac{1}{r \\sin \\theta} \\, \\partial_\\phi", "fb0a6493497c53c006f471daba7fdfd8": "N_{n} = N_{n-1} \\cdot 4 = 3 \\cdot 4^{n}\\, .", "fb0a73e79b6e8dc53d86d1ea98b2e887": "y[n] \\ \\stackrel{\\text{def}}{=}\\ O_n\\{x\\}.", "fb0ab6479f3dab4416833f631acac0a8": "\\displaystyle{Q(a)=2L(a)^2 - L(a^2).}", "fb0b4c2c5d26d6c5a4043a166338f279": "-B", "fb0b808f3a2d84b02589daaf1ae8cb18": "\\mathcal{A}^{AB}", "fb0b8c0be0a2ea83d52bb4236fcb93fc": "\\alpha \\eta ", "fb0bb6c19a5442497c0089178ccb0b61": "\\vdash Q", "fb0bd57dad042481e04d39d24218b470": "A = P'U\\,", "fb0be7959774bfb0d362db196078658e": "\\mathrm{smoke} \\rightarrow O(\\mathrm{ashtray})", "fb0c273ab6fc2b5ae105cacf308c8724": "{4 \\choose 1}\\left[{13 \\choose 5}{39 \\choose 2} + {13 \\choose 6}{39 \\choose 1} + {13 \\choose 7}\\right] - 41,584 = 4,047,644", "fb0c372dbdfeb8d36fa49654ae01b774": "\n\\Gamma = \n\\begin{bmatrix}\n\\Gamma_1 & D_{\\Gamma_1 ^*} \\Gamma_2\\\\\n\\Gamma_3 D_{\\Gamma_1} & - \\Gamma_3 \\Gamma_1^* \\Gamma_2 + D_{\\Gamma_3 ^*} \\Gamma_4 D_{\\Gamma_2}\n\\end{bmatrix}\n", "fb0c5be2e7ce53ba0f314732c0f1b531": "{{i}_{C3}}=\\left( \\frac{\\beta \\left( \\beta +2 \\right)}{\\beta \\left( \\beta +2 \\right)+2} \\right){{i}_{IN}}", "fb0c687cc51dab66626c4e81c19c9f8b": "\\nu = \\pi / \\theta_W", "fb0cad21fcc3f10aadbb666c8e36c59e": "x _{v \\cap w}", "fb0cd353ee94199039da60d06e459040": " \\mu_0 < \\frac{ x + m }{ 2 } \\pm 9.66 | x - m |", "fb0ce7c2864d45cd277575f863f6af1c": "(1,1)", "fb0d3de9f8520b27bd14a3229faa0856": "\\left(a \\rightarrow P\\right)", "fb0db655193cd8e167be95be353928fd": " A =\n\\begin{pmatrix}\n\\cos(\\alpha) & -\\sin(\\alpha) \\\\\n\\sin(\\alpha) & \\cos(\\alpha) \\\\\n\\end{pmatrix}.\n", "fb0dc28a533f270bad0f6737c992c097": "T(X) : Y \\mapsto T(X\\wedge Y).", "fb0de704ef3b61db4130a59ef4a7dc1e": " H' = -\\sum_{i=1}^R p_i \\ln p_i ", "fb0de8d1a75699f377c331b9961755d6": "\\mathrm{ker}(x) \\sim \\sqrt{\\frac{\\pi}{2x}} e^{-\\frac{x}{\\sqrt{2}}} [f_2(x) \\cos \\beta + g_2(x) \\sin \\beta],", "fb0dec5a600b434fc7d7d4fcce3fd54d": "V_{cl}", "fb0df7fe75e0f63f0ff672dca8422ca1": "x = y", "fb0e318bd82d35ee7c04e97a1e18a402": "(R+r_{ex})^2=d^2+r_{ex}^2,", "fb0e3be7fbda4f1a51e0fa3feda625f0": " 2 [f'(x_n)]^2 - f(x_n) f''(x_n) ", "fb0ecaa5729ffe0eacf981b390def78e": "M(v;T)", "fb0ef7645f3686329ca9d8a5b45fb619": "v_{\\theta} = \\Gamma/(2 \\pi r)", "fb0f288bb16da0b86fcb0f8f0dba423a": " \\frac{a p}{x} \\left( \\frac{(\\tfrac{x}{b})^{a p}}{\\left((\\tfrac{x}{b})^a + 1 \\right)^{p+1}} \\right) ", "fb0f547f0bbf8773f16da96d6c376425": "\n\\begin{matrix}\n\\mathrm{I} & a_1 = f(\\langle \\rangle) & \\quad & a_3 = f(\\langle a_1, a_2\\rangle)& \\quad & a_5 = f(\\langle a_1, a_2, a_3, a_4\\rangle) & \\quad & \\cdots\\\\\n\\mathrm{II} & \\quad & a_2 & \\quad & a_4 & \\quad & a_6 & \\cdots.\n\\end{matrix}\n", "fb0f624ce18f0ee5500ada695201dcae": "\\|\\Phi(f)\\| = \\sup_{\\lambda_n \\in \\sigma(T)} |f(\\lambda_n)| = \\|f\\|_{C(\\sigma(T))}.", "fb0f6ef307979c6a9e8c7015706894ee": "\n \\boldsymbol{\\sigma} = \\cfrac{2}{J}~\\boldsymbol{B}\\cdot\\cfrac{\\partial W}{\\partial \\boldsymbol{B}} \\qquad \\text{or} \\qquad\n \\sigma_{ij} = \\cfrac{2}{J}~B_{ik}~\\cfrac{\\partial W}{\\partial B_{kj}} ~.\n ", "fb0f7ca19b3b38670a236e3f5c79d3fd": "f''''(x)", "fb0f99705ea6b6f17d774d81ebfbc817": "\\psi^{(m)}(z+1)= \\sum_{k=0}^\\infty \n(-1)^{m+k+1} (m+k)!\\; \\zeta (m+k+1)\\; \\frac {z^k}{k!}", "fb0fe6a539e1bad4ad48c81640e0ed56": "\n\\begin{pmatrix} \\mathbf{I} & 0 \\\\ \\mathbf{v}^\\mathrm{T} & 1 \\end{pmatrix}\n\\begin{pmatrix} \\mathbf{I}+\\mathbf{uv}^\\mathrm{T} & \\mathbf{u} \\\\ 0 & 1 \\end{pmatrix}\n\\begin{pmatrix} \\mathbf{I} & 0 \\\\ -\\mathbf{v}^\\mathrm{T} & 1 \\end{pmatrix} =\n\\begin{pmatrix} \\mathbf{I} & \\mathbf{u} \\\\ 0 & 1 + \\mathbf{v}^\\mathrm{T}\\mathbf{u} \\end{pmatrix}.\n", "fb1009bf071466cfa57126457ae593d8": " u_m(\\mathbf{r},t) = \\sum_{\\alpha} \\frac{m_{\\alpha}}{2} \\dot{r}^2_{\\alpha} \\delta(\\mathbf{r}-\\mathbf{r}_{\\alpha}(t)), ", "fb102addd116e6b49717d1e1d36cd44a": "\\mathrm{wt}_a(c)", "fb106ab7707968df5be672097a714298": "k_2 \\ll k_{-1} ", "fb109cb56d684e972d4b4f62f4db7078": "x\\sim y", "fb1123641a2aba2fcf0fb9d288f0cd98": "\\int_0^1 x^2\\, dx = \\frac{1}{3}", "fb113a6776c69c6572756c435fcccd7d": "\n |j_1 m_1\\rangle |j_2 m_2\\rangle |j_3 m_3\\rangle, \\;\\; m_1=-j_1,\\ldots,j_1;\\;\\; m_2=-j_2,\\ldots,j_2;\\;\\; m_3=-j_3,\\ldots,j_3.\n", "fb11858037e994a48660d86f747eccd1": "\\Delta G(kJ\\cdot mol^{-1}) = -mF \\Delta \\psi + 2.3RT \\log_{10}\\left ({[X^{m+}]_B\\over [X^{m+}]_A}\\right )", "fb1188dee66fda932704054c5be3174f": "\\tau_{ij} = B\\left(\\sum_{k,l = 1}^3(\\partial_lu_k)^2\\right)^{-\\frac{1}{3}}\\cdot\\frac{1}{2}(\\partial_ju_i + \\partial_iu_j)\\,", "fb1197fc6e02b46f71f3d2b154c8defc": "4x \\equiv 2 \\pmod {6}\\ ", "fb123d65cc099770b017dc14d4345582": "\\mu ( E ) > b - a - \\varepsilon.\\,", "fb1284296589d144018371969a94cf35": "\\ \\min \\sum_{i}\\sum_{r_i} E_i(r_i)q_{i}(r_i) + \\sum_{j\\ne i}\\sum_{r_j} E_{ij}(r_i, r_j)q_{ij}(r_i, r_j) \\, ", "fb12b748b826d6c6e8b78e7cfb1ce587": " \\frac{\\delta R}{\\delta g^{\\mu\\nu}} + \\frac{R}{\\sqrt{-g}} \\frac{\\delta \\sqrt{-g}}{\\delta g^{\\mu\\nu}} \n= - 2 \\kappa \\frac{1}{\\sqrt{-g}}\\frac{\\delta (\\sqrt{-g} \\mathcal{L}_\\mathrm{M})}{\\delta g^{\\mu\\nu}},", "fb13041a8ae6a3f192c113b21c16f87e": " r=\\frac{3V}{4T} ", "fb1382af7b8b3e9ab4f1187a2f06ccb7": "\\scriptstyle {2+\\sqrt{2}}", "fb1393236d58765a703ecdf6a774a620": "\\mathbf{a} = \\sum_{i=1}^3 a_i\\mathbf{e}_i \\equiv a_i\\mathbf{e}_i ", "fb14281725e78bb90732c69dcba1cf5e": " {\\sigma}^{2} ", "fb149549019942b557037bc37e003eac": "x\\in O, \\phi_i\\in \\mathcal{B}\\subseteq \\mathbb{F}", "fb14bcc7eb6b8357decee804454d9edc": " B(z_{Q},c_{1}\\delta^{k})\\subseteq Q\\subseteq B(z_{Q},c_{2}\\delta^{k})", "fb1518cb17df2ffabe24c087a2702b08": "\\in \\Z", "fb152339dae5cb1fd7ad473012dccbe5": "\\frac{\\mathrm{d}^2 x^{\\alpha}}{\\mathrm{d}t^2} + [00,\\alpha] c^2 = 0 \\qquad (**)", "fb15c2f83b5aaab34d2ffc06b39f2af0": "x \\in (-\\infty, \\infty)\\,\\!, \\; \\sigma^2 \\in (0,\\infty)", "fb15d84c75c1a8df8e8e028b7defc74b": "\\eta = g/2a", "fb15e37a8eca2310cfae9deab62e994f": "f\\colon S^2\\to M", "fb16263838a43d85dad8a20b69f6b3f7": "\\scriptstyle N = (V, E)", "fb162e5b08546054e57e036d6ba26f64": "t = s + \\delta", "fb1630048ed578cfe754623b3d66042c": "a_0 f_0 + a_1 f_1 + a_2 f_2+\\cdots = 0, \\, ", "fb166d2964096835eb45e475aff002e7": "T_{1/2}", "fb168db415b58d33ef48ab7fc5e13116": "T_{abcd}=C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} + \\frac{1}{4}\\epsilon_{ae}{}^{hi} \\epsilon_{b}{}^{ej}{}_{k} C_{hicf} C_{j}{}^{k}{}_{d}{}^{f}", "fb16a448d861cd4c9edfae4f237e1a1c": "J \\subset I_e", "fb16c727df0abe566b257b9e1fc9386e": "g_n 1", "fb2a355384f6c3d1fb3c0d4fc5870fc7": "\\left( T^{(n)} \\right)^2=\\sigma_{ij}\\sigma_{ik}n_jn_k", "fb2aa369d005aeb29a24fb7176b1e406": "Y_2 ... Y_n", "fb2afd1c8b87b7af229219864b051c63": "\\scriptstyle V_j", "fb2b27588debb31a677b3eda53acb651": "F_{}^c", "fb2b2db527a1788121de98bdc583ff9e": "\\xi_1", "fb2b6c7e6cf433f25a063e7956056e2c": "S \\subseteq \\bar{\\stackrel{\\circ}{S}}", "fb2b88bbcdeca9ac07e4770ac23daf37": "\\ E_\\text{covalent} = E_\\text{bond} + E_\\text{angle} + E_\\text{dihedral}", "fb2badbbd0f69790d30c9958cf222d99": "\\scriptstyle\\frac{2 \\pi}{\\lambda}", "fb2bf302dbfdfe24eef3539b06f816c9": "\\lim_{x\\to 0} x^2 \\sin(\\tfrac{1}{x})", "fb2c317eea14bdf0ec5e031705a55448": " V = ", "fb2c32a52490b46a5148fd8698347560": " {1 \\over \\lambda} = R_\\text{H} \\left( 1 - \\frac{1}{n^2} \\right) \\qquad \\left( R_\\text{H} = 1.0968{\\times}10^7\\,\\text{m}^{-1} = \\frac{13.6\\,\\text{eV}}{hc} \\right)", "fb2c38897dceae73a0ec5edc9d037f6e": "= d(V^{1}u) - V^{1}u_{1}dx - u_{1}d(V^{1}x) \\,", "fb2c4efd19b8166127d9bee6b6950ef6": "I_z = \\frac{3}{10}mr^2 \\,\\!", "fb2c5611582b365c9e4ca556ea0a9f54": "N( \\cdot ,\\cdot )", "fb2c6220a5486fe752dde3c49f89e3f6": "\n \\begin{align}\n A := & \\cfrac{1}{\\sqrt{3}}~\\cfrac{\\sigma_c\\sigma_t\\sigma_b(\\sigma_t+8\\sigma_b-3\\sigma_c)} {(\\sigma_c+\\sigma_t)(2\\sigma_b-\\sigma_c)(2\\sigma_b+\\sigma_t)} \\\\ \n B := & \\cfrac{1}{\\sqrt{3}}~\\cfrac{(\\sigma_c-\\sigma_t)(\\sigma_b\\sigma_c+\\sigma_b\\sigma_t-\\sigma_c\\sigma_t-4\\sigma_b^2)}{(\\sigma_c+\\sigma_t)(2\\sigma_b-\\sigma_c)(2\\sigma_b+\\sigma_t)} \\\\\n C := & \\cfrac{1}{\\sqrt{3}}~\\cfrac{3\\sigma_b\\sigma_t-\\sigma_b\\sigma_c-2\\sigma_c\\sigma_t}{(\\sigma_c+\\sigma_t)(2\\sigma_b-\\sigma_c)(2\\sigma_b+\\sigma_t)} \n \\end{align}\n ", "fb2c8977c524e37e72bc97f776460dd2": " \\bar r(t_1)=\\bar r_1", "fb2cafe3679c645f055ea0223a0be37a": "\\tfrac{52 \\times 51}{2} = 1326", "fb2cc53a3b0fa0a32cbb1117451fb4f5": "\\operatorname dy", "fb2ce609b94901d92aa0acccb7980bc3": " \\varepsilon_{1eff}^*(\\omega)= \\varepsilon_2^*\\frac{(\\frac{r_2}{r_1})^3+2\\frac{\\varepsilon_1^*-\\varepsilon_2^*}{\\varepsilon_1^*+2\\varepsilon_2^*}}{(\\frac{r_2}{r_1})^3-\\frac{\\varepsilon_1^*-\\varepsilon_2^*}{\\varepsilon_1^*+2\\varepsilon_2^*}}", "fb2d7dfb612eda719e8858493c0de26a": " \\rho_s \\over \\rho_w ", "fb2dc52d738360475b10f79aed5388f0": " TSS = RSS + ESS + 2 y^T {\\hat y} -2 {\\hat y}^T {\\hat y} - 2 y^T {\\bar y} + 2 {\\hat y}{\\bar y} ", "fb2ed74b05de3d64185d2617f5f5a3d8": "\\{ P \\in X(\\overline{K}) : \\hat{h}(P) < \\epsilon\\}", "fb2f010efb886e7e05db4df7e26e5d99": "\\nu_{1,2}( \\mathbb{R}_+ , \\mathbb{R}_-)", "fb2f393bdbd3ca393f50adbd1f4ba738": "\\textstyle b = ", "fb2f5977afed981da38b4a449864479a": "\\det[c_{rs}-b_n\\delta_{rs}]=0", "fb2f85507b4002775725b3a6fcbf990e": "\nQ = \\int d\\theta^{\\prime} \n\\int \\rho^{\\prime} d\\rho^{\\prime} \\lambda(\\rho^{\\prime}, \\theta^{\\prime})\n", "fb2f91674ef1461f141079c053b271ab": " m \\ge 2j - r \\qquad \\qquad \\mathrm{(a)} ", "fb306389d311300770b91830af9d0caa": "b = \\frac {p}{\\sqrt{1-e^2}} = a \\cdot \\sqrt{1-e^2}", "fb3065754c92cd32520817fa9a510126": "\\displaystyle \\sum\\limits_{n}|A_n|^2 = 1. ", "fb3077155159553db257b625a9d6fdbb": "\\chi(C,F) = n(2-2g) -\\sum_{x\\in X}(n+Sw_x(F))", "fb30b5db5af13b3159fe17a84ee29cac": "\\frac{\\pi \\varepsilon l}{\\operatorname{arcosh}\\left( \\frac{d}{2a}\\right) }=\\frac{\\pi \\varepsilon l}{\\ln \\left( \\frac{d}{2a}+\\sqrt{\\frac{d^{2}}{4a^{2}}-1}\\right) }", "fb31013170b92c95789f32cfed66ed68": " \\mathbf{v \\ \\times } \\left( \\mathbf{ \\nabla \\times F} \\right) =\\nabla_F \\left( \\mathbf{v \\cdot F } \\right) - \\left( \\mathbf{v \\cdot \\nabla } \\right) \\mathbf{ F} \\ , ", "fb3116021a65ba8264f8fe3ac751d4c4": "10^{-23}", "fb311cb813eabf65c5f3111ae3de52d9": "\\mathrm{d}l^2 = g(z,{\\bar z}){\\mathrm{d}z}{\\mathrm{d} \\bar z}", "fb31a38f6de8b87cbd9c2854a01943a9": "\n\\begin{align}\n\\zeta(u)&=[c^{-1}H(2\\eta)\\Theta(u-\\eta)\\Theta(u+\\eta)]^N\\\\\n\\phi(u)&=[\\Theta(0)H(u)\\Theta(u)]^N.\n\\end{align}\n", "fb3239eb5feb8c2a283ebea30525757a": " |f| = f^+ + f^-\\,", "fb325abb38f9782d3e521a15b4b3e0ef": "\\pi=\\pi_1(M)", "fb328533953f8db0d472d795f75d2576": "\\chi(\\mathcal{F}(n)) = P(n)", "fb32a734f7cd7104a3c861f5ed060ded": "P_\\omega = \\frac{1}{2} \\chi^{(3)} \\epsilon_0 E_1 E_2 E_3^* e^{i(\\omega t - \\mathbf{k} \\cdot \\mathbf{x} ) } + \\mbox{c.c.}.", "fb32c9f985a5e80b294f1d845fc09cc1": "P_{i_{xy}}", "fb32d734ea187746155e6da6a42ef726": "\n \\boldsymbol{C} := \\boldsymbol{F}^T\\cdot\\boldsymbol{F} = \\left(\\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\\right)^T\\cdot \\cfrac{d \\mathbf{x}}{d \\mathbf{X}}\n", "fb3324436cfe44e1865381e477e3fc7d": "\\forall a,b\\in D,a\\cdot b=b\\cdot a", "fb3354614dcea69afb505cfe19988a3b": "\\sin \\theta = \\frac{e^{i\\theta} - e^{-i\\theta}}{2i} \\,", "fb335e2bfccd77d7fa28fa1776da71f9": "[u][v]=[v][w]=0", "fb337563d5eef28ab7b4f2dca8b15d3e": "\\mathbb{S}^n_T", "fb338084fb462fcfe4c8022a07743091": "m+\\frac{m^2}{r}", "fb34698f882892b74fa03c56beae549f": "c_\\text{deep} \\approx 1.25\\sqrt\\lambda", "fb34ad111fe01bc57a24aca0fbc30a46": " i + (i + 1) + (i + 2) + \\cdots + j = T_j - T_{i-1}. \\,\\! ", "fb34b3cbbb05ccc0469d52478461d941": "(d,w,y,y^{\\prime},\\tilde{k})", "fb34df8e72a6c6afc7e44e7c851a4ed8": "i : Z \\rightarrow Y", "fb34f97217c2be5737a88bd783279e8a": "E(\\mathbb{Z}/q\\mathbb{Z})", "fb34ff988cd80a048a46b02a64f9ca5c": " p = -|p^0| - \\mathbf{p} = - p^\\prime", "fb3569f7f3f3413e7db902fde0ae4b53": "\\int_0^T|\\zeta(1/2+it)|^{2k} \\, dt = T\\sum_{j=0}^{k^2}c_{k,j}\\log(T)^{k^2-j} + o(T)", "fb356c3e8bf3b1c770bb9b0b5d3dfe9a": "-w<0", "fb35e146be12de1f794c20aebadcf130": "\\log_2 3 = m/2n", "fb361a39072b11264544e16eb79c3de1": "\\beta(-k)={{E_{k}} \\over {2}}.", "fb3695f562ba8e87ebd08168b5f1e2c7": "\\mathbf{a}\\times (\\mathbf{b}\\times \\mathbf{c}) = \\mathbf{b}(\\mathbf{a}\\cdot\\mathbf{c}) - \\mathbf{c}(\\mathbf{a}\\cdot\\mathbf{b})", "fb36be43e51e70a4efec5df2a0c660ee": " T = \\frac{ts/Ot}{te/Ot} = \\frac{R}{S}, \\text{ or }T = \\frac{R}{S} \\,", "fb36bf1ab8697dac6cc75dad64870099": " \\varphi_\\mathrm{s} = 2 \\pi \\left(\\frac{1}{\\omega_\\mathrm{0}} - 1\\right) \\approx 4 \\pi^3 k^2 \\frac{Z^2 e^4}{c^2 n_\\mathrm{\\varphi}^2 h^2} ", "fb371b910c7af42e7263957b89263c67": "\\sigma=\\frac{1}{\\sqrt{6}}", "fb3774cfe3381ce2988f9445ac3348bf": " \\hat{\\sigma}_1^2 = \\frac{-L(\\mu)}{N} \\left[\\sum_{x_i: x_i<\\mu} (x_i-\\mu)^2 \\right]^{2/3},", "fb37b6174564223f1cba9396c93f982d": "(x \\to y) \\land (y \\to x)", "fb37e212ae1a19d7226b202fc17c85da": "T_{C_1 C_2}.", "fb37eca5d5372c1c74b6477be692609a": "j-i+1", "fb380666ead05b56d53bc7681b4c6055": "\\gamma_{\\mathrm{SL}}+\\gamma_{\\mathrm{LG}}\\cos{\\theta_\\mathrm{c}}=\\gamma_{\\mathrm{SG}}\\,", "fb3843c514487568977e70cfbc1a8402": "[H^+]=10^\\frac{E-E^0}{s} \\ or \\ [H^+]=10^{-pH}", "fb38e5198c49ea73b01f95fd6f1e50d1": "D-3", "fb38f86fa339c59f2d24a7ca10ed2af2": "\\! f(x_1), \\ldots, f(x_n)", "fb398e47e7b3fef077b1e597f763163d": "\\mathcal{F}\\varphi = \\varphi.", "fb3a2753c4aa472bd3230273ce736b10": "f(x_n) \\to f(x)", "fb3a30ae894ed4d5ef24f90e646b6b59": "\\begin{bmatrix} \n \\sin^2 (\\theta+\\alpha)K_n & -K_n \\sin(\\theta+\\alpha)\\cos(\\theta+\\alpha) & \\cos(\\theta+\\alpha)K_s L\\sin(\\alpha) \\\\\n +\\cos^2(\\theta+\\alpha)K_s & +K_s\\sin(\\theta+\\alpha)\\cos(\\theta+\\alpha) & -\\sin(\\theta+\\alpha)K_n L\\cos(\\alpha) \\\\\n -K_n\\sin(\\theta+\\alpha)\\cos(\\theta+\\alpha) & \\sin^2(\\theta+\\alpha)K_s & \\cos(\\theta+\\alpha)K_n L\\cos(\\alpha) \\\\\n -K_s\\sin(\\theta+\\alpha)\\cos(\\theta+\\alpha) & +\\cos^2(\\theta+\\alpha)K_n & +\\sin(\\theta+\\alpha)K_s L\\sin(\\alpha) \\\\\n \\cos(\\theta+\\alpha)K_s L\\sin(\\alpha) & \\cos(\\theta+\\alpha)K_n L\\cos(\\alpha) & L^2\\cos^2(\\alpha)K_n \\\\\n -\\sin(\\theta+\\alpha)K_n L\\cos(\\alpha) & +\\sin(\\theta+\\alpha)K_s L\\sin(\\alpha) & +L^2\\sin^2(\\alpha)K_s\n\\end{bmatrix}", "fb3a3bfbdf66b66d9d1af654b937af89": " \\rightsquigarrow ", "fb3a54817ce154b0d177573316edc49e": "\\frac{\\partial{E_y}}{\\partial{x}}-\\frac{\\partial{E_x}}{\\partial{y}} = -\\mu\\frac{\\partial{H_z}}{\\partial{t}}", "fb3a586576738c5960d9627170b1cfaf": "M\\begin{bmatrix}a_1\\\\ \\vdots \\\\ a_n\\end{bmatrix}", "fb3a7d19707d96ddb9dbfc3a0f90d2b3": "t_{m} = 2\\dfrac{2^{r}}{f_{clk}}", "fb3a9bfabe031b2ded54cc6862feb17d": "w = \\left( 1-\\frac{1}{\\sqrt{d+1}} \\right)^d \\cdot \\left( 1-\\frac{1}{\\sqrt{d+1}} \\right) + 1 \\cdot \\frac{1}{\\sqrt{d+1}}", "fb3aa9f72f3db032db43c07b4a36f0b1": "\\varrho(X) = \\sup_{Q \\in \\mathcal{Q}} E^Q[-X]", "fb3ae4bb2e603d784a68b0d1c42e6829": "\\mbox{dist}(T,\\mathcal{A})", "fb3b2d3775a13134b2082699ab7e2969": "\n u(\\gamma) = P \\left\\{ \\gamma \\left[ \\alpha (a), \\beta (b) \\right] | (a,b) \\in U \\right\\} =\n \\sum_{(a, b) \\in U} P \\left\\{\\gamma\\left[ \\alpha(a), \\beta(b) \\right] \\right\\} \\cdot\n P \\left[ (a, b) | U\\right],\n", "fb3b4cf060aa4b1c98eb1d9c217cc07c": "\\eta_i=\\theta^i\\circ f^{-1}", "fb3b67e3cf1c99401da15abd580af8c7": "\\sum_p f(p) = f(2) + f(3) + f(5) + \\cdots", "fb3b6a9e458dfd7ac1bc01b52c18b957": "1\\to Z_2 \\to 2I\\to A_5 \\to 1", "fb3bdab6eefb4d76b0dfd92791a5982f": "C_{D,i}", "fb3c509000e4a23b20f23717ca66684d": "3363-2378\\sqrt{2}=0.00014\\ldots", "fb3c96f1043ee7c76a035cb8f664e518": "E_{kin} = \\frac{1}{2} m \\omega^2 r^2", "fb3c9bd50141f6cd8dbf378999e7d1f7": "\\begin{align}\n q_1 &= e_x\\sin\\left(\\frac{\\theta}{2}\\right)\\\\\n q_2 &= e_y\\sin\\left(\\frac{\\theta}{2}\\right)\\\\\n q_3 &= e_z\\sin\\left(\\frac{\\theta}{2}\\right)\\\\\n q_4 &= \\cos\\left(\\frac{\\theta}{2}\\right)\n\\end{align}", "fb3cd4892a51adf3103a9bd6fc26f58e": "a \\rightarrow a", "fb3d56a94c74b8e7f07c4332215c0390": "O(k^{d-\\varepsilon})", "fb3db66b93077b7bd2b8e59a35b244dd": "d^\\alpha", "fb3e30984a24b7ebeae801b147325f25": " 0\\le \\frac{T}{C}\\le \\frac{1}{2} ", "fb3e47a7e592d5de1422d213d538d5c1": "\\lim_{x\\to\\infty}\\sqrt[N]{x}= \\infty \\text{ for any } N > 0 ", "fb3e9aa0c288ba0e39c0be8011c90c6d": "\\frac{\\sqrt{T_c}}{M}", "fb3ee064f62da4b7cade38a414208d91": "\\biggl(\\int_S|g|^{-1/(p-1)}\\,\\mathrm{d}\\mu\\biggr)^{-(p-1)}.", "fb3f26656acce3df81e9964cd5c7102b": "GF(p^2) \\cong \\{x_1 \\alpha + x_2 \\alpha^p : x_1, x_2 \\in GF(p)\\}.", "fb3f6b30c3b01768ea29bf3fddf3d6d9": "c=c_R \\qquad \\forall x>0", "fb3f85978381f58a27182599a8604452": "\\langle\\psi|e_i\\rangle = \\langle e_i|\\psi\\rangle^*", "fb4007d9022ad8ae9b5da6189a1ed06c": "6 a_1^2 a_2^2.\\,", "fb4047e41e0663cafade63f5ee1fe84b": "(\\Omega_1,F_1,P_1)", "fb405edfd405f131a4920124ba1d4d4d": " a + k{\\bold Z} := \\{ a + k\\lambda : \\lambda \\in {\\bold Z} \\}.", "fb405ff0e64f98169bb00fc172533a05": "\\mathbf{C}\\setminus\\{1\\}", "fb406fb95a3afb841ecdc524243fea1c": "f(x)=O(g(x))\\text{ as }x\\to a\\,", "fb40799098c924a80c20a797c2252fe7": "du_i", "fb407c9d19875192ec3031deeef433ee": "~A \\downarrow B", "fb40a3ea35d3c38eae4b4da1e2c87d57": "{{K}_{memory}}", "fb40b71826f6174b6028b3bf9519e731": "V_{max}^{app}", "fb40bb6243a2f043691bd374928a4157": " S(t) = S(0) e^{-R_0(R(t) - R(0))} ", "fb40d3bb1b38748170cb2a28118677c9": "\ny = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots + a_n x^n + \\varepsilon. \\, \n", "fb41fbdb380de58c44d70943bb4fe509": "\\left[-1,1\\right]\\times{\\mathbb R}", "fb4213a4dfbe7c716ec7acf200765143": " \\vec{M} ", "fb426e3d48b4d01c9fa3dd2f4df50844": "q(x) = \\left(\\frac{1}{4} - a\\right) x^{-2} \\quad\\text{for}\\quad x > 0", "fb42791c1ef76c02e55e560308da6cd6": " \\zeta (-m) ", "fb429857956a09ebfe6b04db0d4feaa4": "A^*A=AA^* \\ ", "fb432d10d09b5a7a4f67c97f54967d67": " \\lim_{x \\to 49} \\frac{x - 49}{\\sqrt{x}\\, - 7} = 14, ~~ (4)", "fb4356401b1dd347bdaad41a51ce2983": "\\tilde{y}.", "fb435857ca7a9cfe595faafa3dd25768": " V_{ICF a} = \\frac{n_{ICF a}}{Osm_a} = \\frac{V_{ICF b} \\times Osm_b - n_{lost K^+}}{Osm_a} ", "fb437a071b61f14330368531769b700c": "\n (U \\circ T)' = T' \\circ U'. \\,\n ", "fb43ac08e1c1ddb9ca5d0164d80bcf18": "\\theta_f=\\frac{h m_b L}{I}", "fb43b70f5f942d3c1f62041d64acd28f": "(\\Delta\\,,[\\ell])", "fb43eca0c20ee922a1c70a3e8b48248a": "\\left\\|\\mathbf{a}\\right\\|=\\sqrt{{a_1}^2+{a_2}^2+{a_3}^2}", "fb44171625ccf5682bc25d8cd7721c45": "\n\\bar{x} = \\frac{w_1 x_1 + w_2 x_2 + \\cdots + w_n x_n}{w_1 + w_2 + \\cdots + w_n}.\n", "fb441fceef8dd17ceaacb537b15eb9f5": "= \\sum_{k=-\\infty}^{\\infty} x[n-k]\\cdot h[k],", "fb443fc9ec85028301e1dcd29fc43683": "v = \\frac{Q}{\\pi r^2}.", "fb446e8e39cbb0178e99c0df3ce5002c": "\\sum_{k=1}^n k = \\frac{n(n+1)}{2},", "fb4470e97b757f92961224816a0bfc97": "R_{\\mathrm{sen}}C_{\\mathrm{sen}}", "fb44860c6283b91e4623e6d8843ab4ad": " \\lang Ax , y \\rang = \\lang x , A y \\rang \\mbox{ for all } x,y\\in H. ", "fb44873302594249bca752126e69f720": "L(v) = \\int_a^b\\! f(x) v(x) \\, dx.", "fb44d8f5aa8b2ab79a53411c00502a07": "\\Pr\\left[x \\le 16 \\right] = 0.2669.\\,\\!", "fb44e2276b9dbc2c9e1ac4afec12c951": "i = 2 \\times (\\text{parent}) + 1", "fb44fe44a61e4ac098fbb7c048982832": " Z(s,t) ", "fb453c5b88ced8fae031de346e768ae0": "C_n\\sim\\frac{4^n}{n^{3/2}\\sqrt{\\pi}}", "fb4547b3a98eae47bb08171d400e0240": "\n \\hat k^{-1} = \\frac{\\sum_{i=1}^N (x_i^k \\ln x_i - x_N^k \\ln x_N)}\n {\\sum_{i=1}^N (x_i^k - x_N^k)}\n - \\frac{1}{N} \\sum_{i=1}^N \\ln x_i\n", "fb45490389def13b45384ecf56a4e70a": "\n \\cfrac{\\partial{W}}{\\partial \\bar{I}_1} = C_1 ~;~~ \\cfrac{\\partial{W}}{\\partial \\bar{I}_2} = 0 ~;~~ \\cfrac{\\partial{W}}{\\partial J} = 2D_1(J-1)\n ", "fb45a7abdee2a2e8bd05ec6f1788105e": "(r,g,b)", "fb4605263bf06582d2b0c1e7970c7b03": "a_3b_3", "fb46176fb1360dcc787090269086cf6f": "\\mathfrak{z}(\\mathfrak{g}):", "fb46d020454082e51633072cd32428d8": "\\mathrm{Hol}^*(\\omega) = \\cap_{k=1}^\\infty \\mathrm{Hol}^0(\\omega,U_k)", "fb46ec76cc237bf306adf3bb73cbea4f": "\\scriptstyle \\lim_{t\\rightarrow \\infty}P(\\eta_t(1)\\neq \\eta_t(0))=0 ", "fb46ef2974bb3762734f6b0c82ec36e0": "0 < \\omega < 2/\\lambda_{max}(A)", "fb47192c0392ee51dd5e49c8d763e120": " A_q(n,d,w) = \\max_{C \\in C_q(n,d,w)} |C|.", "fb477dde0ac8cf28ef7235b8b80706ec": "f(a,b,x) = ([1,2] \\cdot [2,3]) + [5,7] = [1\\cdot 2, 2\\cdot 3] + [5,7] = [7,13]", "fb47868468addd8edae408694e92e251": "\ny = a\\sigma\\tau \\sin \\phi\\,\n", "fb48756f5d182763471b24397a43a44a": "x_i = \\frac {\\frac{w_i}{M_i}}{\\sum_i \\frac{w_i}{M_i}}", "fb489a9469b609d2742c61efb60f33ea": "\\sum_{k=1}^r\\lambda_k u_k", "fb48a9baf64ce33913834f1455b3acf1": "\\int N_{\\mu\\nu...}\\sqrt{-g}\\,d^4x", "fb48fa1e7a3a67e4398bde8f26aa3731": "\n\\Delta \\vec{p} = (\\mathbf{A}^T\\mathbf{A})^{-1}\\,\\mathbf{A}^T\\Delta\\vec{F}\\!", "fb4905ee32e6ef7cd5ab0c5b432c5106": " d(n) = (\\nu_1 + 1) (\\nu_2 + 1) \\cdots (\\nu_k + 1), ", "fb490f9ed4f7541137af4105aee10460": "\\mathbb P << \\mathbb Q,", "fb4963b872bde82965a226df6ceabe44": "[f + g, h] = [f,h] + [g,h] ", "fb49de13532098fec581ddd552e37985": "f':k \\to j'", "fb4a1851e17403387f34b7d089582cd5": " \\frac{dN(x)}{dx} = \\sum_\\rho \\delta (x-\\rho) ", "fb4a2f0360dfe8edb3b77c523c165aec": "2f(a)/(f(a/2)+2f(a))", "fb4a4f9e0327339dc6ab9e30cca7f6b3": "c_1 = -1/2", "fb4a580d170027468564044720ef3c69": "\\pi_{0}(x) = \\sum_{n=1}^\\infty \\frac{\\mu(n)}n \\Pi_0(x^{1/n})", "fb4ae2f70b18ec24ea9dc5fe5b4924ba": "(7.d)\\quad \\frac{1}{\\rho}\\,\\gamma_{,\\,z} =\\,2\\psi_{,\\,\\rho}\\psi_{,\\,z}- 2e^{-2\\psi}\\Phi_{,\\,\\rho}\\Phi_{,\\,z} ", "fb4ae399af9ddbffab2995d884405b5b": "\\mathit{Skew} = \\frac{ \\phi_{ 84 } + \\phi_{ 16 } - 2 \\phi_{ 50 } }{ 2 ( \\phi_{ 84 } - \\phi_{ 16 } ) } + \\frac{ \\phi_{ 95 } + \\phi_{ 5 } - 2 \\phi_{ 50 } }{ 2( \\phi_{ 95 } - \\phi_{ 5 } ) } ", "fb4af578436597cd2df6d93685f82c22": "\nE[\\Delta(t)|Q(t)] \\leq B - \\epsilon \\sum_{i=1}^N Q_i(t)\n", "fb4b624c5968156f8116f18e471a1f82": "{}_2F_1 (a\\pm 1,b;c;z), {}_2F_1 (a,b\\pm 1;c;z), {}_2F_1 (a,b;c\\pm 1;z)", "fb4b6b353159b16e0a4fe9177f122689": "\\frac{d}{dk}\\Gamma_k[\\phi]=-\\frac{d}{dk}W_k[J_k[\\phi]]-\\frac{\\delta W_k}{\\delta J}\\cdot\\frac{d}{dk}J_k[\\phi]+\\frac{d}{dk}J_k[\\phi]\\cdot \\phi-\\frac{1}{2}\\phi\\cdot \\frac{d}{dk}R_k \\cdot \\phi", "fb4bace98ca6824e9591f0ee58a28257": "R = \\frac{P_E CO_2 (1-F_I O_2)}{P_i O_2 - P_E O_2 - (P_E CO_2 * F_i O_2)}", "fb4bb742d89853ee91f391cc70b8ba34": "x-z", "fb4bcd615d8f920da40f5daee7b663bd": "\\hat S(t) = \\prod\\limits_{t_i \\le t} \\frac{n_i-d_i}{n_i}.", "fb4bcf0a7b724da7531c2165e0449482": "\nV_{1 \\dots n+1}= V_{0\\dots n} S\n ", "fb4bde7e1667d3c3301f5f2070919d80": " |E_\\alpha| \\leq \\frac{A_n+1}{\\alpha} \\, \\varepsilon.", "fb4c063249582399f0bb9f5cc5a2f68e": "\\displaystyle (w^2-x^2)(y^2-z^2) = 0", "fb4c16652eaa8dd56ebe78c724e6d0df": "\\theta_n = (n + 0.5) \\pi / N", "fb4c16f7c322914e026f3ae4220827f6": " d_{ia} = | x_{ia} - \\bar{x}_{a} | ", "fb4c4808bfede915d9ac4c7342e2566c": "\\mathbf{\\hat{R}}", "fb4ca97ea91bbab47c0f9af3260814df": "H(p,q)=\\frac{1}{2m} p_i p_j g^{ij}(q)", "fb4d652aca804ca14da46bc3dab4f468": " \\text{a} = \\frac{V_\\theta^2}{r} ", "fb4d7ac8ae92a375d60be97e22e0494b": "c=r(s+t)^m", "fb4d7d1168a3edce21e39a4a96b783b6": "0 \\to \\underline{\\mathbf{Z}} \\stackrel{2\\pi i}{\\longrightarrow} \\mathcal{O}_X \\stackrel{\\operatorname{exp}}{\\longrightarrow} \\mathcal{O}_X^\\times \\to 0.", "fb4db11bd01e97d1e93465515830d43d": "a^T (M \\circ N) b = \\operatorname{Tr}(M \\operatorname{diag}(a) N \\operatorname{diag}(b))", "fb4dc903df48021214fd7f91a6d3d523": "\\ mln\\beta \\leq ln(n) + \\sum_tln(1-(1-\\beta)F_t)", "fb4e1f67c27511eba25bef9fd8982c8f": "f(k) = \\int_{\\mathbb{T}} e^{-2 \\pi i k x}d \\mu(x).", "fb4e204febfde7eb34f45eba313b0cb8": "x_\\text{lower}", "fb4e37f063546b63d5ad1037dcd977b1": "x_{k+1} = \\mathcal{P}_C \\left( \\mathcal{P}_D ( x_k ) \\right). ", "fb4e680f52a0797081776f8e53b08dff": "\\frac{12(5\\nu-22)}{(\\nu-6)(\\nu-8)}", "fb4e811b3a209ca35fbd7a186a20dd0b": "2 H = \\frac{\\frac{\\partial^2 S}{\\partial r^2}}{\\left(1 + \\left(\\frac{\\partial S}{\\partial r}\\right)^2\\right)^{3/2}} + {\\frac{\\partial S}{\\partial r}}\\frac{1}{r \\left(1 + \\left(\\frac{\\partial S}{\\partial r}\\right)^2\\right)^{1/2}},", "fb4eb517916a828c165c6db80e2f8da1": "\\boldsymbol{B}=\\frac{3}{5}R^{2}q\\Big(\\boldsymbol{\\omega}\\cdot\\boldsymbol{r}\\frac{\\boldsymbol{r}}{r^5}-\\frac{1}{3}\\frac{\\boldsymbol{\\omega}}{r^{3}}\\Big).", "fb4ecf6fdd444262eb1db96b2e4111ff": " \\hat{V} = V = V(\\bold{r},t) ", "fb4edcab40721014779efadeb25f49c7": "df/dz_j", "fb4f0fcfbf4a38699ec407b825de8b85": "{x}^i({x'}^j), j=0,1,\\dots", "fb4f23824ac2741ea310a44278a235b2": "f^2", "fb4f353ef9a72c24566678c957a5ae9f": "\\approx", "fb4f485ce07029e70897444bbba85513": "y = \\alpha + \\beta^T\\textbf{x} + \\varepsilon,\\text{ where }\\varepsilon\\perp\\!\\!\\!\\perp\\textbf{x}.", "fb4f7873d24cc511c59862fad4ad5966": "\\textstyle x_i", "fb4f9788bf75c2b63ca04ddd4167355c": "G = N \\rtimes H,", "fb519f9f84f2342935b925094ae6a32d": "FV_\\mathrm{annuity} = {(1+r)^n - 1 \\over r} \\cdot \\mathrm{(payment\\ amount)}", "fb51aa897fd96ff09fffe73a4a2ff726": "E=\\rho e + \\frac{1}{2} \\rho u^2,", "fb51b68e6d85e4518a3942a63d61ff6e": "\na = a_1 a_2\\,\n", "fb5239a2e5ec983b34f029accaa4d6e8": "\\dim V_k(\\mathbb R^n) = nk - \\frac{1}{2}k(k+1)", "fb5247f43748aa596d81896efc50ff97": "Y(z)=\\phi^{X}(z)", "fb525d28c6e1652807ea64246ae5664d": " (\\mathbf{A} \\otimes \\mathbf{B}) \\otimes \\mathbf{C} = \\mathbf{A} \\otimes (\\mathbf{B} \\otimes \\mathbf{C}), ", "fb5285a60acaba31a2e4f2f017679754": "\\pi^0 = \\mathrm{\\tfrac{u\\bar{u} - d\\bar{d}}{\\sqrt{2}}}", "fb52efb7bcb0c8188ed2032fcb1065a7": "\n\\frac{1}{\\epsilon_r} = \\sum_{m=-\\infty}^{+\\infty} K_m^{\\epsilon_r} e^{-j \\frac{2\\pi m}{a}z}\n", "fb5303d776b7b3599f615328c57984b4": " P(T) = \\text{max}\\left( A(0,T) - K, 0 \\right),", "fb53a31efcede70b19ec5b645d715e0d": "R(x)=e^x-\\sum_{n=0}^{N} \\frac{x^n}{n!}", "fb53eb45c0030b09d80ed81245a1fe81": "[(x_1, u_1), (x_2, u_2), ..., (x_N, u_N)]", "fb542fd5050972766416229d8bd77787": " E = \\frac{2.21}{r_s^2} - \\frac{0.916}{r_s} + 0.0622 \\ln (r_s) - 0.096 + O(r_s)", "fb5457b06b36715e3a5f3b4383eaba5b": "\\mathrm{U}(n)", "fb5478f92e9108c3e28a081695fda35d": "\nf(z) = \\cfrac{a_1z}{1 + \\cfrac{a_2z}{1 + \\cfrac{a_3z}{1 + \\cfrac{a_4z}{\\ddots}}}} \\,\n", "fb548298bf8b347e21411b2ef324b800": "\\overrightarrow{EM}\\cdot N = xN_x+(y-b)N_y=0", "fb5559424fb6d545410ec18c4356fae5": "\\rho_x", "fb5573f8bf43184731133009701d883c": " {}+11310276995381\n x^{12}-135585182899530\n x^{11} \\,\\!", "fb55baf28916c6ff735db4e7b94d3b54": "\\hat{\\rho}_c = \\frac{2 s_{xy}}{s_x^2 + s_y^2 + (\\bar{x} - \\bar{y})^2},", "fb55cc14a6060a792d4a36909884092f": "\n\\begin{pmatrix}\n\\alpha & -\\tau \\\\\n\\gamma & \\sigma \\\\\n\\end{pmatrix}\n.", "fb563358bc6da58b4c8b26ebd6fe732c": "Z_q", "fb567baccc796eef05f114cca027f997": "\\phi_a", "fb570c13de6685fe12546dcdfd536209": "= E_0 -2\\Delta\\,\\cos(ka)\\ ,", "fb5714bc89d992d3bd9539efd6b422a3": "\\|\\mu * \\nu\\| \\le \\|\\mu\\| \\|\\nu\\|. \\, ", "fb5745898ea47d52ca1a58c7f7b614ad": "(2it+1)/4", "fb5760885822b31e3b764317c5b98893": "I_{\\text{c}} = \\beta I_{\\text{b}} = \\frac { \\beta (V_{\\text{cc}} - V_{\\text{be}})}{R_{\\text{b}} + R_{\\text{c}} + \\beta R_{\\text{c}}} \\approx \\frac{(V_{\\text{cc}} - V_{\\text{be}})}{R_{\\text{c}}}", "fb5768c8d7a55bfef6cd8992539fcd7a": "X^p = T^{-1} + {X_1}^p,\\text{ so }{X_1}^p - X_1 = T^{-1/p}", "fb576fa0f1d3462d2f4d920e812b9e98": " {C^{*}}(\\mathbb{R}) ", "fb577326cb1125c301e14eac6f385f36": "\\operatorname{Li}_n(z)=\\operatorname{Li}_n(1)\\;\\;+\\;\\;\n\\sum_{k=1}^{n-1} (-)^{k-1} \\;\\frac{\\log^k |z|} {k!} \\;\\operatorname{Li}_{n-k} (z) \\;\\;+\\;\\;\n\\frac{(-)^{n-1}}{(n-1)!} \\;\\left[ \\Lambda_n(-1) - \\Lambda_n(-z) \\right].", "fb57d8e0d4398546bb0b8df6d8c3e299": " \\frac {R_2} {R_2+R_f} ", "fb580c5979f416264c78a83910f1cc05": "\\boldsymbol\\alpha = (1,1,\\ldots)", "fb584b743920812f1af08d705824bf51": "(X,\\Sigma, \\mu)", "fb589dbb09109fbf77a39abc3129122d": "T = k \\cdot I", "fb58ac54e07e93f2ffa42007ff1740b5": "\\hat{P}(\\cdot)", "fb58e962ecf66c1a161c9ab6fc8be99c": "\\Delta I_{L_\\mathit{off}}=\\int_{t_\\mathit{on}}^{T=t_\\mathit{on}+t_\\mathit{off}}\\frac{V_L}{L}\\, dt=-\\frac{V_o}{L}t_\\mathit{off},\\; t_\\mathit{off} = (1-D)T", "fb59364d12bcdebf91f4bb3a4ad4d66e": "\\ln(q) = \\ln \\|q\\| + \\frac{\\mathbf{v}}{\\|\\mathbf{v}\\|} \\arccos \\frac{a}{\\|q\\|}", "fb5947bc50cf61a6b766320765ad8d6c": "t_2^\\prime = \\frac{1}{f^\\prime}", "fb599d828493b48c13ce27fc2da5f2ef": "\\Box (\\lnot K) \\rightarrow \\Box (K \\rightarrow K \\and \\lnot K)", "fb59f31479c4bc9467aeedc174eab677": "\\sin \\theta_n = \\frac {n \\lambda} {S}, n=0, \\pm 1 \\pm 2, ....", "fb5a26f6837c65b5742d45df7f646668": "\\Delta u(x,y) = \\overline{u}(x,y) - u(x,y)", "fb5aa1d572ba9f32632692d5ba489b7e": "n'_i = n_i \\mbox{exp}{\\left(\\frac{-z_ie\\psi_j(r)}{kT}\\right)}", "fb5af8e424d867c9cd99738f4c6357f4": "\n\\frac \\pi 4 = \\cfrac{1}{1+\\cfrac{1^2}{2+\\cfrac{3^2}{2+\\cfrac{5^2}{2+\\cfrac{7^2}{2+\\cfrac{9^2}{2+\\ddots}}}}}}\n", "fb5b540a31ec5a5f5c9ea12a98b98019": "x = 0.111\\dots \\, ", "fb5b715a7782bbae3809784abe9feaf8": " \\scriptstyle{P_{avg}} ", "fb5b905b649cee42e7ed320cdb15c258": "\\scriptstyle a_1\\times a_2", "fb5b93f01939deed6c0f3796c6f664c6": "r_4 = 499.1", "fb5bd0cd51ec9545c3753d72992c6166": "M(s) = \\exp(\\alpha s) \\, M_g \\left(\\beta s + \\frac12 \\sigma^2 s^2 \\right),", "fb5bf16d9c09f738bb5ec8073d6fe968": "\\sum_{i=0}^n(-1)^{n-i}\\binom{n}{i}a_i=b_n.", "fb5c0dfe693f9a69e341d9042adfe7b7": "(n+2^{k-1}) \\mbox{ mod } 2^m, 1 \\leq k \\leq m", "fb5c430e6d2c9888cd49c6fe43fe5b18": "T_{hot} ", "fb5c5e788b2138779c14e80f46d7c60c": "I(\\boldsymbol{u})=\\int_A W(\\boldsymbol{x},\\boldsymbol{\\varepsilon})\\mathrm{d}x - \\int_A u_i f_i\\mathrm{d}x - \\int_{\\partial A\\setminus\\Sigma}u_i g_i \\mathrm{d}\\sigma", "fb5c62776beb4f35a2ea4681f251b4a4": "H^3", "fb5c67f8cfed1971e319d77bb803eb6c": "\\Delta_L(t^2) = \\nabla_L(t - t^{-1})", "fb5ca57c9cad04dc6dbc321552b12c33": "\\mathcal{M}=\\frac{(m_1m_2)^{3/5}}{(m_1+m_2)^{1/5}}", "fb5ca75b7bf62d7475917e79792853b2": "\\partial\\!D", "fb5cdef700adbef0c30dc1f2f1e34564": "\\kappa = 8\\pi Gc^{-4}", "fb5cfdea6a21672ddfa10b42e7272a7a": "\\textstyle F = (f_{ij})", "fb5d07c1dd997b7f1435966cd12ea81f": " [f \\star g](x) = \\int_G f(x - y) g(y)\\, d \\mu(y).", "fb5d5ed6edc3184b878d822464e687d4": " \\bigotimes ^k T^*M ", "fb5d86e0116686ae79b123b87f9f4538": "1 \\le a < p", "fb5dd737a8eff6d79fa163ec84fe6dc2": " \\mathcal{M}_g ", "fb5ecc5104502960fa12de64c12df506": "\n\\int_n^{n+1} f(x)\\,dx\n\\le\\int_{n}^{n+1} f(n)\\,dx\n=f(n)", "fb5f4f7f3d537c5a932f4900948b043e": " H = \\sum_{i=1}^2 h(i) = H_0 + H^{\\prime}", "fb5fb51beba5779b3a4d106b2a52ae1e": "\\left (x^{\\mu} + a_1 x^{\\mu-1} + \\cdots + a_{\\mu} \\right) \\left (x^{\\nu} + b_1 x^{\\nu-1} + \\cdots + b_{\\nu} \\right)", "fb5fc4b116418a57917987817b6958ed": "\\forall \\alpha\\in\\mathbb{R} \\ : \\ \\Pr\\left[\\frac{a(n,k,X)}{j(n,X)}\\geq \\alpha \\right]\\leq \\frac{a(n, k,X(\\Omega))}{\\alpha}", "fb5fc9a8f72682b2d8c1c18baa25c33d": " dx_w ", "fb60613b763eb7e1654f3fa908fe4e56": "C_{4} = G_0 + P_0 \\cdot C_0", "fb60b29a8a7d10eb61c8cbae15c09fe2": "M=((~)_{n^\\nwarrow\\;\\|\\;n^\\searrow})_{\\overline{n}^\\nwarrow\\;\\|\\;\\overline{n}^\\nwarrow}", "fb60dbe3df04f5bfa7b8cd5eb2369514": "m_{jk}", "fb6155370c415f1388e1663b041757a3": "f: M \\times N \\to Z\\,", "fb61d03cbee4676db63f6000bdbd258a": "\\mathbf{h}_k^H \\sum_{i \\neq k} \\mathbf{e}_i s_i", "fb623f81b918e63bc18de633161c6c21": "h \\alpha", "fb62425f8eec6a4f69f826a91d91ec27": " Z = \\xi^N ", "fb624632329a85a9403bffb934aba950": "\\sinh(z), \\cosh(z), \\text{and } \\exp(z^2)", "fb6258429ef76fbc0b61fe9ff2b578ff": "\\rho^\\dagger = \\rho", "fb62917da964d4e51aaf06438e347e19": " \nA_n=\\{x\\in E|f_k(x)>(1-\\epsilon)\\varphi(x)~\\forall k\\geq n\\}.\n", "fb635ce842763463555c94373042ec63": "H_i \\qquad (i=1,2,3) \\,", "fb635d3a21f88d839b85e184d48281f8": "f(z) = \\sum_{k=-\\infty}^\\infty a_k (z-a)^k", "fb63cd7a3c67cdc3c6ec2dacb4b26736": "\\textstyle x^k-1 = (x^p-1)(1 + x^p + x^{2p} + \\ldots + x^{k/p})", "fb643c8ab869be58946072dada43642b": "\\int_S \\bigl| f(x)g(x)\\bigr| \\,\\mathrm{d}x \\le\\biggl(\\int_S |f(x)|^p\\,\\mathrm{d}x\\biggr)^{\\!1/p\\;} \\biggl(\\int_S |g(x)|^q\\,\\mathrm{d}x\\biggr)^{\\!1/q}.", "fb64442790146368142d49d13f3e0101": "f(x,y,z) = x^2 + y^2 + z^2", "fb64797c5aa5d2c279ea09fe12ea804a": "(H_2,\\{O^i_2\\})", "fb64b31e25af291cabc855fee28e0971": "\\mathbf{c} = \\aleph_1 = \\beth_1 ", "fb651e2a775fa29584be5cbe40c90867": "1 = G_0 \\le G_1 \\le G_2 \\le \\cdots\\, ", "fb65439dfd859a9b8d9748c0fb46923f": "\nA(t) \\le \\inf_{0 \\leq \\tau \\leq t} \\{ A(\\tau) + E(t-\\tau) \\} = (A \\otimes E)(t).\n", "fb65622f7a154a32fc9a3027e6cd6893": "\na_{ij}= \\begin{cases}\n1, & \\text{if flow } j \\text{ leaves node } i \\\\\n-1, & \\text{if flow } j \\text{ enters node } i \\\\\n0, & \\text{if flow } j \\text{ is not incident with node } i \n\\end{cases}\n", "fb6568c737dfcca1ecbb0346cda81a5c": "\\log(\\alpha)", "fb656cd68e4cde48b53ab990e103a6eb": "\\mathrm{Ta}", "fb657a8288429df35a801d3cd33bebf3": "\\hat{a}^{\\dagger} \\rightarrow \\frac{\\hat{c}^{\\dagger} + \\hat{d}^{\\dagger}}{\\sqrt{2}} \\quad\\text{and}\\quad \\hat{b}^{\\dagger} \\rightarrow \\frac{\\hat{c}^{\\dagger} - \\hat{d}^{\\dagger}}{\\sqrt{2}}.", "fb658ddec6ae923fe04f7689c28f2ef6": "\\scriptstyle H(\\text{password}_i)", "fb65aec877bb09903689bcac5982f154": "\n\\mathcal{A}\\Psi(1,\\ldots, N) = \\begin{cases}\n&0 \\\\\n&\\Psi'(1,\\dots, N) \\ne 0.\n\\end{cases}\n", "fb660326f45412b1f5fa81a5ed9c4008": "\\quad(0,T)\\times\\mathcal{M}.", "fb660c967a9b94e83ea92b43d65bdf8e": "M(\\alpha ,\\alpha +\\beta ,j\\nu )=\\frac{\\Gamma (\\alpha +\\beta )}{\\Gamma (\\alpha )\\cdot \\Gamma (\\beta )}\\cdot \\int_{0}^{1}e^{j\\nu t}t^{\\alpha -1}(1-t)^{\\beta -1}dt", "fb663a4518c91c6c04fb04fc35311d61": "\\mathbb{Q} \\ll \\mathbb{P}", "fb66b632d9c6bd3de2fcb99acecba884": "A'[t]", "fb66c8b29d166ca540f9ece0ca53b557": "\\lim_{m\\to\\infty} |z_m| = \\infty", "fb673c04df3e86f7226b9260050b69c3": "OB\\% = \\frac{-1600}{Mol. wt. of compound} \\times (2X + (Y/2) + M - Z)", "fb68042e996d8d7146a3017ee63cdc28": "a \\in F", "fb6828fc3bfb239fdbf063bf06fe0c2e": "\\forall x \\varphi", "fb68447435cff6e25326effdfa3dde67": "\\prod_{z \\le p < w} \\left( 1 - \\frac{\\omega(p)}{p} \\right)^{-1} \\le \\left( \\frac{\\log w}{\\log z} \\right) \\left( 1 + \\frac{C}{\\log z} \\right).", "fb6851fe6715c3db7060869b97d7fab1": "f_i (x) \\leq 0,\\quad i = 1,\\dots,m", "fb685779a46828accb2933745614cce9": "\\rho_\\theta = \\rho(P_0,\\theta,S_1,S_2,...) ", "fb6895822959a09096603f010782d5b0": "m \\ge 3", "fb689bbe0e1bbde7caee6df2c128195e": " x_0 ", "fb68a6644819dc26a1b33dcee1f9125a": "\\frac{|E_n|}{|S_n|} \\leq \\left[ 2\\varepsilon + O(n\\varepsilon^2) \\right] \\frac{\\sum_{i=1}^n |x_i|}{\\left| \\sum_{i=1}^n x_i \\right|}. ", "fb69093c3c8bf5357b1aab03fd74f612": "k_i \\in \\{0,1,2,\\ldots\\}, 1\\leq i\\leq m", "fb69168deb8e2fcab45d2619d13080fd": " v = \\frac{V_\\max\\ [\\mbox{S}]}{K_m + [\\mbox{S}]} ", "fb695fa0f6700942bcfc817c0ec6b015": "u \\ge 1", "fb69a9e9e75cb4ae6608121463a383b8": " \\Delta \\mathbf{p} = \\int \\mathbf{F} dt ", "fb6a011356606cb3fc8a510738b7abbf": "\\mathit{E}_{out}", "fb6a1675b9d9a2d9ba2a53317fccf099": "\\epsilon_r", "fb6a4cef093766bf8919663d43384d08": " \\langle\\cos{\\theta}\\rangle = e^{-(L/P)} \\, ", "fb6a54d187b46f5f11f16c8c32d9c4fe": "r\\cdot \\frac{s^2}{(2^2 - 2)r^2}, \\qquad r\\cdot \\frac{s^2}{(2^2 - 2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2},\\qquad r\\cdot \\frac{s^2}{(2^2-2)r^2}\\cdot \\frac{s^2}{(4^2-4)r^2}\\cdot \\frac{s^2}{(6^2-6)r^2},\\cdots ", "fb6a5c2eaca3e9d128aad2d5143e81cb": "x_0\\ y_0\\ z_0", "fb6a7bb213fd62beea6522d1b9bdf061": "\\mathbf{a} \\succ \\mathbf{b}", "fb6acc2345afcc073933f28a11cf2124": " \\left(d^n\\varphi\\right)(g_1,\\dots,g_{n+1}) = g_1\\cdot \\varphi(g_2,\\dots,g_{n+1}) ", "fb6ad50d938aed52eada41f68f74ad83": "|M|^{2}-|\\tilde{M}|^{2}=-4|M_{1}||M_{2}|\\sin(\\theta_{1}-\\theta_{2})\\sin(\\phi_{1}-\\phi_{2})", "fb6b020502d0a9de3e48c033528ca45b": "L_d(s,t)=(1-s) d_0(t)+ s d_1(t) \\, ", "fb6b03756fea2039f2e6b6c27b7a00cc": "S_3", "fb6b09950b0dfdc0dca7e8eb5db7b1ca": "\\mathbf{s} = \\{ s[j] \\}", "fb6b0fcb07732900750d817a2159af0b": "C/c", "fb6b20dfdc28097525ae99b3bd2c8a23": "A\\ ", "fb6b6f6f19c96ab83c218d23c5b4d5e1": "(f\\ominus b)(x)=\\inf_{z\\in B}f(x+z)", "fb6b77f481cea14f743e195ea5446a8a": "\\alpha \\triangleleft S", "fb6b8ac1f54d4b8d92a7226316fadb22": "\n\\begin{align}\nM_{p} (a, b, \\lambda)& = \\left( (1 - \\lambda) a^{p} + \\lambda b^{p} \\right)^{1/p},\\\\\nM_{0} (a, b, \\lambda)& = a^{1 - \\lambda} b^{\\lambda}.\\,\n\\end{align}\n", "fb6be93414986a534be78fec5b0ce5ff": "K = \\{ x \\in \\mathbb{R}^n : q(x) \\le 1 \\} . ", "fb6c9dc782c5207e63831c5b0d8f00eb": "\\Leftarrow, \\nLeftarrow, \\Longleftarrow \\!", "fb6d07598e5b49de76ef0afd91cfe9fd": "c : A \\rightarrow \\mathbb{R}_+", "fb6d1a9b72325f099ab97c2b717e5aea": "ax^2+bx+c,\\ ", "fb6d1e33eaae6a6cf14e732a6e1542ef": "o_{1:T}", "fb6d442fd996f8b4f5a027fdf61baaa4": " L_{\\rm min} = \\frac{\\hbar}{2 e} \\frac{1}{I_0} = \\frac{\\hbar R_n}{\\pi \\Delta}. ", "fb6d85231b782fcf45d9415758a3351c": "E_B = \\frac{Q_B}{Q_A + Q_B},", "fb6e04756cbdea5167fb5b46363629ee": " \\Phi(S) = \\sum_i B^*_i S B_i. ", "fb6f0634c323fb267d0d537b4837e88c": "\\left.k\\right.", "fb6f1cc32e41c4a4168638615c239933": " Re(a)", "fb6f255937b8ff73eb349b49372d17ec": " D^\\alpha f = \\frac{\\partial^{|\\alpha|}f}{\\partial x_1^{\\alpha_1}\\cdots \\partial x_n^{\\alpha_n}}, \\qquad |\\alpha|\\leq k ", "fb6f54668aa0cc359a2a1a6fc4424d83": "\\mathbb{Q}(\\sqrt[p]{2}, \\zeta_p)", "fb6f6dbeab5337326a44dd6aa01b3ce0": "2^{5 \\times 8} + 2^{3 \\times 8} = 1099528404992", "fb6fc2a98e57f0c3b90ea6415743af18": "\\{ C_i (x,y)\\}", "fb6fd5395953ea958a64f628184950ca": " E_{curv} ", "fb6fe730c874c9a8355cffdaafcd8da8": "m \\in \\{ 64, \\mathbf{20}, 13, 57 \\}", "fb6ff1adf63975ba95a72718a90012e9": "d^{(k)}_n", "fb70409f89195528bdd3c446cbc6ccda": "-1.2077", "fb705c60f5f08352d8a0d55dbfebb1ab": "\\mathbf{B}_\\text{el}^l = -2\\mu_\\text{B}\\dfrac{\\mu_0}{4\\pi}\\dfrac{1}{L_z}\\sum_i\\dfrac{\\hat{l}_{zi}}{r_i^3}\\mathbf{L}", "fb710b79135b4229ba0c753aa7897812": "t_0,\\ldots,t_{n-1}", "fb7122ed0bab61b5717430a4b6cc71b3": "Q^{i\\alpha}", "fb713adba290cd6a1b79fdd6b2fc4c96": "\\lVert pq \\rVert = \\lVert p \\rVert\\lVert q \\rVert.", "fb714f1a819140b7d65f348524a7cc5a": "\\bar\\partial^2=0.", "fb71966f3efc6afa459ad42210b2de8f": " z_P = 0.80 + j1.40\\,", "fb719e2df8016c19a3aa8a70e230dc33": "\\scriptstyle{t'=t-vx/c^2}", "fb719ec416baa0068c01d53dce2b4782": "A_i = \\{x \\in A : \\hat x = (-1)^i x\\}.", "fb71ecd6b9b8e6d80957b655c9fba77c": "A_{z}", "fb71eee696c9a17cd133e38a42bebdfc": "\\partial_\\mu j^\\mu=0", "fb723216baca82ae88c189335718d9f1": "\\mathrm{E}[r_t] = r_0 e^{-a t} + b(1 - e^{-at})", "fb72797059e7e04377a27d3792e7e632": "\\mathbf{o}_L=\\{o_1, o_2, \\dots, o_{T_L}\\}", "fb72b9dca529602f531b0a4a10e91e4d": " H_1(X) \\cong \\pi_1(X)/[ \\pi_1(X), \\pi_1(X)] . \\,\\!", "fb72c8b7418f58a4b9f760258305d17b": "C_i = E_K(C_{i-1} \\oplus P_i)", "fb72cab032c1ff7128a49f762879ee7a": "\\frac{(a+b)+(a+c)+(b+c)}{3}\\geq\\frac{3}{\\frac{1}{a+b}+\\frac{1}{a+c}+ \\frac{1}{b+c}}.", "fb7319ac0ce51306f69f55ed342cebf3": " \\left( \\frac{\\mathrm{d}S_{\\mu}}{\\mathrm{d}\\mu} \\right)^{2} + 2m a^{2} U_{\\mu}(\\mu) + 2ma^{2} \\left(\\Gamma_{z} - E \\right) \\sinh^{2} \\mu = \\Gamma_{\\mu} ", "fb7347238f36db2b34e052d80d049c1c": "E\\left({S \\over N}|n,s=0,N\\right)\\approx {1 \\over N}{{N^n \\over n}\\over N^{n-1}\\ln(N)}={1 \\over n [\\ln(N)]}={\\log_{10}(e) \\over n [\\log_{10}(N)]}={0.434294 \\over n [\\log_{10}(N)]}\n", "fb7352f8b7bdf26f3f2b611701780d20": "n = \\dim_k X", "fb74020ad74b17b78319c923739989e9": "E_{\\rm HF} = 1/R", "fb74096fc64295899de1075508e73cc3": "\\sum_{i=0}^n L_e(e_x)", "fb74d30c4ea47f47055a3e4a465c6377": "\\tau=\\frac{\\theta}{2\\pi}+\\frac{4\\pi i}{g^2}", "fb75003b88fefcb2031270f069fb641f": "a_{i,0}", "fb75032d07dd96fdc79cda0e375bc9e3": "t \\equiv 1", "fb7509569d25ca39059dfca4ebb2cee8": "\\Delta(a) = a \\otimes a", "fb7513027410aa7de2ea1448b637878d": " (W_t^2 - t)^2 - 4 \\int_0^t W_s^2 \\, \\mathrm{d}s ", "fb75192ee28276d71ba9c280b7be4444": "f = 4 \\pi^2 \\nu^2 \\mu", "fb7633528792ba51f50bcd1547e240e9": "E[m] = {P \\in E(\\mathbb{\\bar {F}}_q) : mP = O}", "fb7635e012a854af03f9a10dece1d052": "[Q,b\\}=0", "fb763d16054db91954cdd21085b4f5c4": "\\partial_\\nu T^{\\mu\\nu}=0\\,,", "fb768551adec21dec124ab7bc69e29eb": "\\mathbf{V}^* \\mathbf{M}^* \\mathbf{M} \\mathbf{V} = \\begin{bmatrix} \\mathbf{D} & 0 \\\\ 0 & 0\\end{bmatrix}", "fb76c100eb8ab8475e2bd70fe8379652": "[\\operatorname{ad}_x,\\operatorname{ad}_y]=\\operatorname{ad}_x \\circ \\operatorname{ad}_y - \\operatorname{ad}_y \\circ \\operatorname{ad}_x", "fb7715c28cdb2a4130a8e0f1f9d2b3b5": "P(S_1, \\dots, S_n | O_1, \\dots, O_n) = \\prod_{t = 1}^nP(S_{t}|S_{t-1},O_t).", "fb77d0c89ccc708c222e767b16d0e937": "m\\ge 0", "fb781903ac62081f5a35af42a4e7b56f": "\n\\begin{align}\n\\mathbf{C} &= \\frac{1}{8}\\mathbf{A} + \\frac{3}{8}\\mathbf{A'}+\\frac{3}{8}\\mathbf{B'}+\\frac{1}{8}\\mathbf{B} \\\\\n\\mathbf{C} &= \\sqrt{1/2} = \\sqrt{2}/2\n\\end{align}\n", "fb7852f31a3f5ffec34153c84b0f41b1": "G(t;\\mu_1,\\mu_2) = \\sum_{k=0}^\\infty f(k;\\mu_1,\\mu_2)t^k", "fb78a3e8914e47655b1b1d7614479068": " x_{ij} = \\begin{cases} 1 & \\text{the path goes from city } i \\text{ to city } j \\\\ 0 & \\text{otherwise} \\end{cases}", "fb79c04d2758e5ee139bea672240509a": "(\\ell-m+1)P_{\\ell+1}^{m}(x) = (2\\ell+1)xP_{\\ell}^{m}(x) - (\\ell+m)P_{\\ell-1}^{m}(x)", "fb79d045b721a2aed8b338941531e9a2": "64_{11} \\ ", "fb7a439c145d37b6a0cd1e36ccdfaa86": "\n B_{ijkl} = C_{ijkl} + \\delta_{ik}C_{jlqr}u_{q,r}^{(0)} + \\delta_{ik}C_{jlqr}u_{q,r}^{(0)} + C_{rjkl}u_{i,r}^{(0)} + C_{irkl}u_{j,r}^{(0)} + C_{ijrl}u_{k,r}^{(0)} + C_{ijkr}u_{l,r}^{(0)} + C_{ijklmn}u_{m,n}^{(0)}.\n ", "fb7a4c8222585b84a1313a2f7494da4f": "(x + x'\\varepsilon) + (y + y'\\varepsilon) = x + y + (x' + y')\\varepsilon", "fb7a54bed5b0fd5f5e90ab851a8a59bd": " P[G(T) > 0] > 0", "fb7b37a2404182cd78cee7867d43f789": "\\operatorname{dCov}^2(a_1 + b_1\\,\\mathbf{C}_1\\,X, a_2 + b_2\\,\\mathbf{C}_2\\,Y) = |b_1\\,b_2|\\operatorname{dCov}^2(X,Y)", "fb7b3a61931488f967649a00388aaa68": "F_z\\,\\!", "fb7be4c1ebdfeed2a1cd0131aba6b956": " B \\in \\mathbb{Z}^{(n,n)}", "fb7bf9ddb407545f492ac165eb468773": "\\int\\frac{x^4\\;dx}{s^3}\n= \\frac{xs}{2}-\\frac{a^2x}{s}+\\frac{3}{2}a^2\\ln\\left|\\frac{x+s}{a}\\right| ", "fb7c2c32111d5dfb5e713eff448ba3d1": "c B^{-1} = \\left( \\frac{-104}{7}, \\frac{-79}{3}\\right).", "fb7c3128ac339aa9ef574a0aa62fec52": " \\psi_0(\\vec{r}_1,\\, \\vec{r}_2) ", "fb7c924faba1f69b634c4a4bc5af3bb3": "\\scriptstyle 1/\\Gamma", "fb7cc8548924d6ae423ff0a04840aad0": "SF^n \\leftrightarrow \\exist IF^n.", "fb7ce2f4986fb81a6e7890fb1254bb52": " Ric'(X, W) = Ric(X,W) + \\langle R'(X,n)n,W\\rangle+ \\left(\\sum_{i=1}^mh(X,E_i) h(E_i,W)\\right) - H h(X,W) ", "fb7cef0e7bc9bcef8e59d06064d214bb": "\\overline{f}\\in R", "fb7d1292c4be7c29b12f1a3f09f56051": "(1-t^2)\\frac{d^2y}{dt^2} -2(b+1) t\\, \\frac{d y}{dt} + (c - 4qt^2) \\, y=0", "fb7d1989f5a1942f5e56722c2ec3b537": "e \\in A \\Leftrightarrow (e,e) \\in B", "fb7d2bb73227491bf8f8cf858dcc62b3": "= \\int d^4 x \\; e e^{M [\\gamma} e^{\\beta]}_N \\big( \\delta_\\gamma^\\alpha \\delta^I_M \\delta^K_J C_{\\beta K}^{\\;\\;\\;\\;\\; N} + C_{\\gamma M}^{\\;\\;\\;\\; K} \\delta^\\alpha_\\beta \\delta^I_K \\delta^N_J \\big) \\delta C_{\\alpha I}^{\\;\\;\\;\\; J}", "fb7d2f11898ffa96f8d3105c443cd027": "(\\phi \\land \\psi)", "fb7d454f02ca0d4f721b9034edc305c2": " (q _{v \\cap w}) q _{v \\setminus w}. (d(v) -1) = (d(v) -1) q _v", "fb7d483782ec03dfdf4d152f95911b12": "\\mathcal G", "fb7d5d4a857ca50b4352dc0503464600": "S_{1/f}=\\frac{A}{Q^{4}}(f_{0}^2/f)", "fb7e1bcff1691ece45a7ec4ee57f07ae": "\n\\begin{align}\n\n\\sigma_{ij} &= \n-\\begin{pmatrix}\np&0&0\\\\\n0&p&0\\\\\n0&0&p\n\\end{pmatrix} + \n\n\\mu \\begin{pmatrix}\n2 \\frac{\\partial u}{\\partial x} & \\frac{\\partial u}{\\partial y} + \\frac{\\partial v}{\\partial x} & \\frac{\\partial u}{\\partial z} + \\frac{\\partial w}{\\partial x} \\\\\n\\frac{\\partial v}{\\partial x} + \\frac{\\partial u}{\\partial y} & 2 \\frac{\\partial v}{\\partial y} & \\frac{\\partial v}{\\partial z} + \\frac{\\partial w}{\\partial y} \\\\\n\\frac{\\partial w}{\\partial x} + \\frac{\\partial u}{\\partial z} & \\frac{\\partial w}{\\partial y} + \\frac{\\partial v}{\\partial z} & 2\\frac{\\partial w}{\\partial z}\n\\end{pmatrix} \\\\\n\n&= -p I + \\mu (\\nabla \\mathbf{v} + (\\nabla \\mathbf{v})^T)\n\n\\end{align}\n", "fb7e3c37b7710a391729965a209a09cd": "a^2=\\sum_{i,j}a_i a_j \\cos(\\delta_i-\\delta_j)", "fb7e50ed27e66cbbec7de37ef4067f8f": "M_E", "fb7e5f7b73e74365b0a485bc544f1a72": "\\chi_S", "fb7e674cde788c3e824bd06136e66197": "r_O = \\begin{matrix} \\frac {1+\\lambda V_{DS}}{\\lambda I_D} \\end{matrix} =\\begin{matrix} \\frac {1/\\lambda +V_{DS}} {I_D} \\end{matrix}=\\begin{matrix} \\frac {V_E L +V_{DS}} {I_D} \\end{matrix}", "fb7ebf19c108dcbe357849256c828cd8": " \\gamma \\left ( \\tau \\right ) = \\exp \\left [- \\left ( \\frac{\\pi\\Delta\\nu\\tau}{2 \\sqrt{\\ln 2} } \\right )^2 \\right] \\cdot \\exp \\left ( -j2\\pi\\nu_0\\tau \\right ) \\qquad \\quad (2) ", "fb7ebfdd403b95bb507fe82a5c2603c7": "s \\mid_p = l \\sigma", "fb7f123757f62533289316bdeaa0cfbe": "2\\int_{S} B ds = \\mu_0 I_{enc}", "fb7f358b84286fbcaee931154357f091": "\\begin{matrix} Car \\bowtie Boat \\\\ \\scriptstyle CarPrice \\geq BoatPrice \\end{matrix}", "fb7f3ae6fefa207888336f42a8ae648a": " w = x + y", "fb7f5f6b719677da028502824a5e90c3": "\\pi = \\frac{1}{2^6} \\sum_{n=0}^{\\infty} \\frac{{(-1)}^n}{2^{10n}} \\left( - \\frac{2^5}{4n+1} - \\frac{1}{4n+3} + \\frac{2^8}{10n+1} - \\frac{2^6}{10n+3} - \\frac{2^2}{10n+5} - \\frac{2^2}{10n+7} + \\frac{1}{10n+9} \\right)\\!", "fb80940325827a800a810bb0bd73e9c2": "|f'(z)|=\\frac{1}{2\\pi}\\left|\\oint_{C_r }\\frac{f(\\zeta)}{(\\zeta-z)^2}d\\zeta\\right|\\leq \\frac{1}{2\\pi} \\oint_{C_r} \\frac{\\left| f(\\zeta) \\right|}{\\left| (\\zeta-z)^2\\right|} \\left|d\\zeta\\right|\\leq \\frac{1}{2\\pi} \\oint_{C_r} \\frac{M\\left| \\zeta \\right|}{\\left| (\\zeta-z)^2\\right|} \\left|d\\zeta\\right|=\\frac{MI}{2\\pi}", "fb811eb3b5d5a6cd10be0df86e0c9468": "C \\backslash \\delta \\cup \\gamma", "fb813aea5b9b8cb278089304160458f0": "\nF_A^{(n)}(a, b_1,\\ldots,b_n, c_1,\\ldots,c_n; x_1,\\ldots,x_n) = \n\\sum_{i_1,\\ldots,i_n=0}^{\\infty} \\frac{(a)_{i_1+\\ldots+i_n} (b_1)_{i_1} \\cdots (b_n)_{i_n}} {(c_1)_{i_1} \\cdots (c_n)_{i_n} \\,i_1! \\cdots \\,i_n!} \\,x_1^{i_1} \\cdots x_n^{i_n} ~,\n", "fb81525647b0e2f799b55a04db915091": "A \\in N_C", "fb8185764e4d4bfe1040cf14cfc8478a": "(X_1:Y_1:Z_1)", "fb81b2ffef76a020fc0f95c7b37329d7": "Du \\ll 1", "fb81c22997c3dab43cbe71d66e73eef1": "H_\\text{rms} = \\sqrt{ \\frac{1}{N} \\sum_{m=1}^N H_m^2}, \\, ", "fb8347e09aec059a0fca1f7723f40202": " \\tau \\overline{\\tau} + \\kappa \\overline{\\kappa} = 1", "fb8363aba25ae3dc7f05ec6e592986b5": "U{}^2_4", "fb83694f6df42654095585a340559e9a": "E^* \\rightarrow \\hat{E}^-", "fb83705e5daa45c27579ef504622d7fd": "\\frac{e^{hx}}{\\int_{-\\infty}^\\infty e^{hx} d\\mu(x)} ", "fb8422befc3c8f571119cb9ce9ce10e2": "s^2 = (vt)^2 - (ct)^2 \\,", "fb84574adfb83e632b228b30f892a73a": "\\scriptstyle 2\\sqrt{\\frac{2}{3}}", "fb84886c142fa1f3a52e7c62be001aeb": "e(i) = d(i) - \\mathbf{w} (i-1)^T u(i)", "fb849bf58c52e740549e34e772c852b5": "\\omega=w'_1w'_2\\cdots w'_k.", "fb84c56c4e25cdfa6ebd4103733048c0": "YX - XY - 1.\\ ", "fb84dbd07fb1596830ff85913644ce36": "\\mathbf C^\\times \\ni z \\mapsto 1/z.", "fb850b0454c82e1379c92b020803f0c9": "\\nabla ^2 \\psi' = +\\omega", "fb851d1b6672b1ef2735e4b7a7706694": "\\left.d\\right.", "fb859badfb1979c9c65a0a91eb2f6f18": "\\frac{d}{d \\alpha}L_n^{(\\alpha)}(x)= \\sum_{i=0}^{n-1} \\frac{L_i^{(\\alpha)}(x)}{n-i}.", "fb85acccd16ff20beaa3176c2d371559": "\\scriptstyle\\mathcal{L}_X", "fb85d5676c3d8871510498c20ffe8610": "\\; E_g = 1 - \\Lambda^k[\\psi] ,", "fb860dc5e58a4a8fe1552086224b21b5": "S_1 \\cup S_2 \\cup S_3 \\cup \\dots \\cup S_n", "fb861c05243102eb66c3d6e4b7a8a66c": "\\sigma_{x}(n)=\\sum_{d|n} d^x\\,\\! ,", "fb86c4054a3ca73d6664f74f0cb473d6": "T(f)", "fb86e7d8218fd3bdb7e5348ed4eeda5e": "j < k \\cdot r < j + 1 \\text{ and } j < m \\cdot s < j + 1 \\,", "fb8713e6b6969a8e2a7a9e0e6383e367": " h( k, m_2, m_1 )", "fb875e267c15165021e9a14e3cd8ce34": "\\Bigl|\\int_S fg\\,\\mathrm{d}\\mu\\Bigr| = \\int_B\\frac{1-\\varepsilon}{\\mu(B)}\\|f\\|_\\infty\\,\\mathrm{d}\\mu = (1-\\varepsilon)\\|f\\|_\\infty.", "fb876bb22681b65b0e53fd100d95aef2": " \\frac{P_{i}+P_{o}}{2} \\times \\frac{1}{P_{o}} ", "fb879b35b69210951173061c5bce0d8f": "r < n", "fb87d882f261f92f31be0d31c6691162": "C: y^2=x^3+x^2 ", "fb87df4e0cd1ae242b0aad5655bccac1": "y=f(x) ", "fb8858473afa7a18ccd1b3b3b57ba4db": "\n\\int^{+\\infty}_0 S(t)\\,\\frac{t^{2\\lambda-1}}{(1+t^{2\\lambda})^2}\\,dt\\leq \\frac{\\pi\\,(n-1)}{2\\lambda}\\prod_{k=1}^{n-1} \\Bigl(1+\\frac{\\lambda}{2k}\\Bigr).", "fb885e602e3bac00f7b8d9df1477d8e8": "R_T(z) = \\sum _{- \\infty} ^{\\infty} a_m (\\lambda - z)^m", "fb88630685816dc696a0e3e8494c016a": " \\langle \\Gamma_i\\Gamma_j \\rangle=\\operatorname{tr}\\{\\Gamma_i\\Gamma_jR_0\\} ", "fb887066e26fe6f77942773b408ee058": "\n\\frac{C}{MR_{e}^{2}} = \\frac{2}{3\\lambda} = \\frac{2}{3} \\left( 1 - \\frac{2}{5} \\sqrt{1 + \\eta} \\right)\n", "fb88d097aff85e0ca9a264b6d131734a": "f(f(x)y)=f(xy)\\; ,\\; f(xf(y))=f(xy)\\; ,\\; f(xf(y)z)=f(xyz) ", "fb8911d5046ba8e2fc3455d3d2edb121": "\n\\begin{align}\n\\zeta& = [\\overline{a_0;a_1,a_2,\\dots,a_{m-1}}]\\\\[3pt]\n\\frac{-1}{\\eta}& = [\\overline{a_{m-1};a_{m-2},a_{m-3},\\dots,a_0}]\\,\n\\end{align}\n", "fb891c606b94252417a05bcb3ee016eb": "\\ \\beta \\approx 120 \\deg", "fb892166da1a8ac4f766400a437e7ece": "\n\\begin{align}\n&P\\left(x+1;\\;\\Gamma(x+1),\\;\\Gamma^{(1)}(x+1),\\ldots,\\;\\Gamma^{(n)}(x+1)\\right)=\\\\\n&\\;\\;\\;\\;\\;\\;\\;=P\\left(x+1;\\;x\\Gamma(x),\\;\\left[x\\Gamma(x)\\right]^{(1)},\\;\\left[x\\Gamma(x)\\right]^{(2)},\\ldots,\\left[x\\Gamma(x)\\right]^{(n)}\\right)\\\\\n&\\;\\;\\;\\;\\;\\;\\;=P\\left(x+1;\\;x\\Gamma(x),\\;x\\Gamma^{(1)}(x)+\\Gamma(x), \\;x\\Gamma^{(2)}(x)+2\\Gamma^{(1)}(x),\\ldots,\\;x\\Gamma^{(n)}(x)+n\\Gamma^{(n-1)}(x)\\right)\n\\end{align}\n", "fb894dacd906bd8d147653db36e50087": "\n\\begin{align}\nr &= \\sqrt{{x_n}^2 + {x_{n-1}}^2 + \\cdots + {x_2}^2 + {x_1}^2} \\\\\n\\phi_1 &= \\arccos \\frac{x_{1}}{\\sqrt{{x_n}^2+{x_{n-1}}^2+\\cdots+{x_1}^2}} \\\\\n\\phi_2 &= \\arccos \\frac{x_{2}}{\\sqrt{{x_n}^2+{x_{n-1}}^2+\\cdots+{x_2}^2}} \\\\\n &\\vdots\\\\\n\\phi_{n-2} &= \\arccos \\frac{x_{n-2}}{\\sqrt{{x_n}^2+{x_{n-1}}^2+{x_{n-2}}^2}} \\\\\n\\phi_{n-1} &= \\begin{cases}\n \\arccos \\frac{x_{n-1}}{\\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n\\geq 0 \\\\\n 2 \\pi - \\arccos \\frac{x_{n-1}}{\\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n < 0\n\\end{cases} \\,.\n\\end{align}\n", "fb89570983890511eaa3a375f3d921a7": "\\overline{X}_i", "fb8a1e131839b44af11bd70351d44a28": "v_g =\\frac{d\\omega (k)}{dk} \\ . ", "fb8a734fbfad92d8ad7c3d8fdcc9c6e7": "\n \\mathbf{E}_1 = \\xi_1 \\exp[i(kx - \\omega t)] \\mathbf{\\hat{x}}\n", "fb8a972a3f344b609a31a0ed0d15f039": " P A P^T = L T L^T", "fb8abbb2d90f7f870f4b299c64d5d2cb": "\\mathbf{u}(\\lambda\\mathbf{x},\\lambda t)", "fb8acf410db983f1f3866d1b3dbb91f3": "\\varphi(P) + \\varphi(Q) + \\varphi(R) = [D_P] + [D_Q] + [D_R] = [[P] + [Q] + [R] - 3[O]]", "fb8b10b312d8010a2d30408dea3b6229": "DF + DG + DH = R + r,\\ ", "fb8b38bf183d3ecc2ad048772dfee143": " \\frac { \\text{density}} { \\text{density of fluid} } = \\frac { \\text{weight}} { \\text{weight of displaced fluid} }, \\,", "fb8b67a8e201ea12cb016a25fe5b6b9e": "\\boldsymbol{\\nabla}^2 f = \\boldsymbol{\\nabla}\\cdot\\boldsymbol{\\nabla}f", "fb8bc0c9f6450a37e27932324c359e1a": "\n E=\n \n\\left( { 2 e^2 \\over L_B}\\right) \\int_0^{\\infty} {{k\\;dk \\;} \\over \n k^2 + k_B^2 r_{\\mathit l}^2 }\n\\;\\mathcal J_0 \\left ( k \\right) \\;\\mathcal J_0 \\left ( \\sqrt{{\\mathit l^{\\prime}}\\over {\\mathit l}} \\;k \\right) \\;\\mathcal J_0 \\left ( k{r_{12}\\over r_{\\mathit l}} \\right)\n.", "fb8c2be79a0da928050031b8ceed6bc5": "D = \\sigma^2/2", "fb8c615427ba471633f92c2d108fbb4b": "R^{+}=\\bigcup_{i\\in \\{1,2,3,\\ldots\\}} R^i.", "fb8c7046560b2df489acbb4a1c830210": " q = ", "fb8cbb5560ecbfac09394dfd7c7ddd85": "E(\\mathbb{Z}/n\\mathbb{Z})", "fb8cd1612241a40ccb85e877d82c7ad8": " \\beta=- \\ln\\alpha", "fb8d1126cde2f742a2a844fc0a69fc36": "s' \\approx_x s", "fb8d4b19d11e1fc5d49eaa117046f9da": "{(x_j)}_{j\\in J}", "fb8d87cc3713d3b853064e1fd4bdc89f": "n = p = n_i.\\ ", "fb8e0251bc7ad9db09e1ff139165a227": "H_4(x) = 16x^4-48x^2+12\\,", "fb8e68f1a3112e167dcbb289e671fd3c": "\n\\frac{(S_C -(S_1+S_2))/(k)}{(S_1+S_2)/(N_1+N_2-2k)}.\n", "fb8f0b0403590b25d8e0461f398a358b": " d_\\text{avg} = {P \\over 6}. ", "fb8f551031d53d38911561c9ddd8ab8e": " \\rho=\\sum_{i,j}J(r(i,j)), ", "fb8fa7e7884329af1436ac84658c239c": "S \\in \\text{Tp}(\\text{Prim})", "fb9014db329ffd47db55ef6933b48fd6": "R : I = R", "fb902933b25993a39716778b4e91fb62": "\\sigma, \\tau ", "fb903578e6f4ecededd82c316926e04c": "i \\in N", "fb90a615cbde68be8b4afd5b87f8bd12": "\\omega =e^{2i\\pi / n},n\\in \\mathbb{N}", "fb90a8e266238ff1b82330c4b7c5e825": "\\text{excess kurtosis} =\\frac{6 \\text{ var } (1 - \\text{ var } - 5 \\mu (1 - \\mu) )}{(\\text{var } + \\mu (1 - \\mu))(2\\text{ var } + \\mu (1 - \\mu) )}\\text{ if }\\text{ var }< \\mu(1-\\mu)", "fb90cd17664abe54e77454093dc95024": "\\sigma_i\\cdot v_{j,k} = \\left\\{\n\\begin{array}{lr}\nv_{j,k} & i\\notin \\{j-1,j,k-1,k\\}, \\\\\nqv_{i,k} + (q^2-q)v_{i,j} + (1-q)v_{j,k} & i=j-1 \\\\\nv_{j+1,k} & i=j\\neq k-1, \\\\\nqv_{j,i} + (1-q)v_{j,k} - (q^2-q)tv_{i,k} & i=k-1\\neq j,\\\\\nv_{j,k+1} & i=k,\\\\\n-tq^2v_{j,k} & i=j=k-1.\n\\end{array}\n\\right.", "fb91150562998804f4507b31ac3a8f92": " SO(n) \\hookrightarrow SO(n+1) \\to S^n , \\,\\!", "fb91a5b3da99c00df484325602504be5": "\\frac{\\mathrm{d}Z}{\\mathrm{d}p} = 0 \\qquad\\mbox{if } p=0", "fb91aa25617b56ee6d54bce35fa6fa09": "\\mathcal T\\,,", "fb91d7fde618366978fd3ad658acd454": "K_I \\geq K_c", "fb91e0dadb3fa3db818f5e73a81bc7a4": "\n \\gamma_{zx}^{\\mathrm{beam}} = \\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x}\n ", "fb921a1c953d1c867688af53b93ad695": "\\rho^{\\sigma} \\vec{u^{\\sigma}} = \\sum_i f_i^{\\sigma} \\vec{e}_i.", "fb92430b67249136af1d44ba1dd29610": " \\alpha_m ~ = ~ k_0 ~ \\sin \\theta_0 ~ \\cos \\phi_0 ~ + ~ \\frac{2m\\pi}{l_x} ~~~~~~~~~~~(2.2a) ", "fb925ca7fff3bd1eb808503a52022a5a": "F(x_1,x_2,\\dots,x_n) = \\prod_{i=1}^n \\left ( \\sum_{j=1}^n a_{ij} x_j \\right ) = \\left ( \\sum_{j=1}^n a_{1j} x_j \\right ) \\left ( \\sum_{j=1}^n a_{2j} x_j \\right ) \\cdots \\left ( \\sum_{j=1}^n a_{nj} x_j \\right ).", "fb926491a34ff6770bde7c9062fbc9df": "\\langle A\\rangle \\simeq \\frac{1}{N}\\sum_{i=1}^N p^{-1}(\\vec{r}_i) A^{*}_{\\vec{r}_i} e^{-\\beta E_{\\vec{r}_i} }/Z ", "fb92751af33b9876659510490eb9d12c": "p^{ij} = |\\langle Q^{(i)}_c | Q^{(j)}_b \\rangle|^2 = |U_{bc}^{ij}|^2.", "fb927bfe1fa45a32388d4681994126ac": " D = k\\left(\\frac{2^k}{ke}-1\\right) \\leq \\frac{2^k}{e} -1 ", "fb92e190ace04b36202149122b5339ef": "p_{20+1}, \\, \\cdots, \\, p_{20+\\lambda} ", "fb933a4ca696e0466b6d9bbd83d3bb30": "E\\mbox{(eV)}=\\frac{1239.84187\\,\\mbox{eV}\\,\\mbox{nm}}{\\lambda\\ \\mbox{(nm)}}.", "fb934a72cc8cc149fd54a53704d16b80": "\\begin{align} \n \\left[\\Psi(\\bold{r}_1),\\Psi^\\dagger(\\bold{r}_2)\\right]=\\delta (\\bold{r}_1-\\bold{r}_2) &\\text{ boson fields,}\\\\\n\\{\\Psi(\\bold{r}_1),\\Psi^\\dagger(\\bold{r}_2)\\}=\\delta (\\bold{r}_1-\\bold{r}_2)&\\text{ fermion fields.}\n\\end{align}", "fb9353c43a2c9b2a7067784da9162ae9": " A_{i_1\\cdots i_n}B^{i_1\\cdots i_n j_1 \\cdots j_m}C_{j_1 \\cdots j_m} \\equiv A_I B^{IJ} C_J ", "fb93fc220b0065519510c0ebfd12e33b": "U_1\\left(x,y\\right)=\\alpha \\left(x/y\\right)^{\\alpha-1}", "fb9413535e04eb1141445bcdb44ca499": " \\sqrt{R_1^2-R^2} + \\sqrt{R_2^2-R^2}", "fb9434c79f7909a445b130c2c6adae23": "P(A|B) = c \\cdot P(A) \\cdot P(B|A) \\ ", "fb94c29602cb21166559b32a9f9f2f6b": "\\bar{\\omega}", "fb94df840796683db4a286f03c45842a": "s(\\mathbf{r},E,\\mathbf{\\hat{\\Omega}},t)", "fb954a7d5ad40e8622c6c198ff82960a": "1/distance^2", "fb9579335fcd23a443daf09d03898d17": "\\rightarrow (2)", "fb95967aa0eff593f89d7673fa0fe652": "\\ -27.55", "fb9606185f077a8863b36bba1747a285": "\\begin{matrix}\nF(a,b,b) & \\xrightarrow{\\eta(a,b,c)} & G(a,c,c) \\\\\n_{\\eta(a,b,c')}|\\qquad & & _{G(1,h,1)}|\\qquad \\\\\nG(a,c',c') & \\xrightarrow{G(1,1,h)} & G(a,c',c)\n\\end{matrix}", "fb9611623f9aca7e23b0c0721186eedb": "S_1(k_{\\lambda}) = k_{-\\lambda},\\ S_1(e_i) = - e_i k_i^{-1},\\ S_1(f_i) = - k_i f_i", "fb96839ae611ccd64156d92a4355d06a": "\\mathbf{f}^S", "fb96a974380b20e78679cf0f21e15db1": "\\int_{-R}^R \\hat f(\\xi) e^{2\\pi ix\\xi} dx", "fb97a686cb2a555f111c344a76884ad2": "Q_i^j", "fb97a935a054f2ad4b51312e48d80d59": "\\omega = e^{-\\frac{2\\pi i}{N}}", "fb98758d6c0b9b1ca4749d16a49f4280": "R1 = \\frac{V_{S} - V_{Z}}{I_{Z} + I_{R2}}", "fb988c153575ff83072032670b13726d": "P_{(1)}\\leq P_{(2)}\\leq\\ldots\\leq P_{(m)}", "fb98ac9187a118cae5731170d117fa55": "\\dot{x}_k=dx_k/dx_3", "fb99059c85e2156e2efa2a182ae01519": "\n\\frac{dX}{dt}= {k_1 \\over K_1 + Z^n}-k_2X\n", "fb991b1477a333de8bb897a26f4982cb": "\\text{Hom}(X, Y \\Rightarrow Z) \\simeq \\text{Hom}(X\\otimes Y, Z)", "fb99f8e663ab7b94c3a8ec3465edbf77": "U8 = \\lambda 1 ((\\lambda 1 1) (\\lambda (\\lambda \\lambda \\lambda 1 (\\lambda \\lambda \\lambda 2 (\\lambda \\lambda \\lambda (\\lambda 7 (10 (\\lambda 5 (2 (\\lambda \\lambda 3 (\\lambda 1 2 3))) (11 (\\lambda 3 (\\lambda 3 1 (2 1))))) 3)", "fb9aac80561a59b8eedf9f655903908b": "[x,y] = (-1)^{|x||y|+1}[y,x],", "fb9aae635a5f6418039bcfe770e82e4b": " {v_x - v_S \\over v_\\infty - v_S} = {T - T_S \\over T_\\infty - T_S} = {c_A - c_{AS} \\over c_{A\\infty} - c_{AS}}", "fb9ad578ec037e223cdf1fcfe4893ef5": "u_{x,y}+\\frac{N(u_x+u_y)}{x+y}=0. ", "fb9b337cc53cc588b39f6c6c5d3247d1": "f(a)=g(a)", "fb9b77cce6373acb1914c8d6adb0ff11": "a\\not\\equiv3\\pmod 4", "fb9b955dd3d10ab28b2aa45f1ab780a7": "y_{2N}=0", "fb9b9e60dfff15471975ce1806a65094": "\\tau(x,y)=\\left(\\frac{x+\\left\\lfloor 2y\\right\\rfloor }{2}\\,,\\,2y-\\left\\lfloor 2y\\right\\rfloor \\right)", "fb9bd5e94cd0c0e55a30632ed2a4b51e": "Q(z) =\n{{\\rm e}^{z}}-{{\\rm e}^{z+1/2\\,{z}^{2}}}-{{\\rm e}^{z+1/3\\,{z}^{3}}}+{{\\rm e}^{z+1/2\\,{\nz}^{2}+1/3\\,{z}^{3}+1/6\\,{z}^{6}}}", "fb9bd6999c4c402d026d89707647ddce": "\\nabla^i \\phi=\\phi^{;i}=g^{ik}\\phi_{;k}=g^{ik}\\phi_{,k}=g^{ik}\\partial_k \\phi=g^{ik}\\frac{\\partial \\phi}{\\partial x^k}\n", "fb9c17e6a0cee7efa5fa555b3de0b886": "\\varepsilon_2'' = \\frac{1}{E}\\sigma_2", "fb9c890030bca6655d511f4e5934f958": "y^2=x^3-x^2-16x", "fb9c91e0cef27082a229f404e1eccd15": "student \\geq medium", "fb9ca715922eedcadddb7f275fa38c8d": " \\mathbf{e}_{123}^2 = \n\\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_1 \\mathbf{e}_2 \\mathbf{e}_3 =\n \\mathbf{e}_2 \\mathbf{e}_3 \\mathbf{e}_2 \\mathbf{e}_3 =\n - \\mathbf{e}_3 \\mathbf{e}_3 = -1.\n ", "fb9cc73333cf3fc9ca6ad95216b143ca": " \\|\\mathbf{N}\\| = N^\\mu N_\\mu = \\nu ^2 \\left(1 - \\hat{\\mathbf{n}}\\cdot\\hat{\\mathbf{n}}\\right) = 0 ", "fb9cd31cfd532784e977c7f9b9b4e9d0": "\\omega:\\mathbb{F}_p^*\\rightarrow\\mathbb{Z}_p^*", "fb9cedb5e92479b0ee53c2ec84f53ab0": "\\varepsilon = \\frac {\\Delta l} {l_0}", "fb9d5f5629f06c44ed2d821b789c426e": "\\phi(t) = \\begin{cases}1 \\quad & 0 \\leq t < 1,\\\\0 &\\mbox{otherwise.}\\end{cases}", "fb9d8486bf4769f8d3977b4ad71da5fc": " \\frac{1}{k^2 + i \\epsilon} ", "fb9da22566730e38c55f7e56ecf2de90": "(x-6)(x-2)^6(x+2)^9", "fb9db4f0e862122fb78cd52faf907c4e": "\\underset{p}{\\underbrace{\\alpha\\wedge\\cdots\\wedge\\alpha}}\\not= 0", "fb9ee7ce5adf6ef62f51c1833c98575f": "\\begin{align} P(hypercalcemia~is~caused~by~other conditions~in~individual) = \\\\\n \\frac {P(hypercalcemia~WHOIFPI~by~other~conditions)}{P(hypercalcemia~WHOIFPI)} = \\\\\n \\frac {0.0005}{0.00335} = 0.149 = 14.9% \\end{align}", "fb9ef128e1891eb7bd6da810753e2a4e": "-dP_B/dx", "fb9efdce5abb4e2de881f3d67c4974cb": "\\mathrm{DD} = \\mathrm{D} + \\frac{\\mathrm{M}}{60} + \\frac{\\mathrm{S}}{3600}", "fb9f217f7dc90ed93974a647df85ec77": "\\sum_i f_i(w)=K.", "fb9f5bb9c53d886a049b26f430a8e8b7": "m = c^d \\mod pq,", "fb9f625ce41c9b6ad603422d273ad25f": "1\\leq i\\leq k, (\\nu_i+\\delta)(H_{\\gamma_i})", "fb9f7b6a6c6534959199da26cd884326": "|t|<\\frac{\\pi}2\\!", "fb9f8b1d3e9c2dfdadf6c578ddb3d65d": "\\chi_1^2, \\ \\chi_2^2", "fb9f95a178396efc11aebd380ba26d23": "\\forall i \\in S: |i| \\leq k", "fb9fafaf0735010da601bd14a6922d34": "x = \\int \\cos \\varphi \\, ds", "fb9fb067e3f31fa5d12bedd8ab13e2c1": "[\\sum_{j=1}^N w_{ij}S_j(t)]", "fba05767b1cb9661ff26e8d84b3a5175": "\n\\begin{align}\n\\lim_{N \\to \\infty} S_N f\\left(\\frac{2\\pi}{2N}\\right)\n& = \\frac{\\pi}{2} \\int_0^1 \\operatorname{sinc}(x)\\, dx \\\\[8pt]\n& = \\frac{1}{2} \\int_{x=0}^1 \\frac{\\sin(\\pi x)}{\\pi x}\\, d(\\pi x) \\\\[8pt]\n& = \\frac{1}{2} \\int_0^\\pi \\frac{\\sin(t)}{t}\\ dt \\quad = \\quad \\frac{\\pi}{4} + \\frac{\\pi}{2} \\cdot (0.089490\\dots),\n\\end{align}\n", "fba060b9db1fad9e63ff25ab6009cec5": "2H-2", "fba092ca1eb28eccd636cee2bb7cb112": "y_i = y_j", "fba0d5a00eee83815219c71b8a476e13": " v_i=\\exp \\left( \\frac {R_0-R_i} {b} \\right) ", "fba1030d5ee11bb880196eaa86002fcd": " m", "fba1a8f1a8e8e1361e6c482ce2f26671": " \\frac {u^{n+1} -u^n}{\\Delta t } \\approx \\kappa u^n ", "fba1ed4fb28a5d1a9501f2973d294495": " \\prod_{k=1}^{n-1} \\cos\\left(\\frac{k\\pi}{n}\\right) = \\frac{\\sin(\\pi n/2)}{2^{n-1}} ", "fba27a835e5ff52f901d6bd00fe65d03": "s\\left\\{\\begin{array}{l}p\\\\q,r\\end{array}\\right\\}", "fba28621de2271c8adb67b06f22dc814": "\\lambda = \\ln a", "fba2e757e48c4db2612abb6f42793b7b": "\\left\\langle\\tilde{H}\\right\\rangle = \\left\\langle H_{0} + \\left\\langle\\Delta H\\right\\rangle\\right\\rangle =\\left\\langle H\\right\\rangle\\,", "fba2ea1c1c6fd0d25551f3fe2ed26b3d": " (a_{1}, ... a_{r}| b_{1}, ... b_{r})", "fba2f046ea2cf52a9bc28c2b1d6e62d5": "(\\alpha\\mathbf{A})\\cdot\\mathbf B=\\alpha(\\mathbf A\\cdot\\mathbf B)=\\mathbf A\\cdot(\\alpha\\mathbf B).", "fba32e6555b3fb8a04edf462f913ae4a": "f(0)=g(0)=0", "fba33fab23e4f4a1ad788f246e8029ae": " P \\left ( {a, b}{|}{A, B, \\lambda } \\right ) = P \\left ( {a}{|}{A, \\lambda } \\right ) P \\left ( {b}{|}{B, \\lambda } \\right )", "fba3446067165612f17679b227d40cc4": "\\gamma = \\frac{1}{\\sqrt{1-\\left(\\frac{v}{c}\\right)^2}}", "fba38c5b0c8a74629f568165e5436962": " \\forall (R_1, \\ldots, R_N) \\in \\mathrm{L(A)}^N, \\quad F(R_1,R_2, \\ldots, R_N) = R_i", "fba393e6439d927a7b3743181610f17d": " (\\sec x)' = \\sec x \\tan x \\,", "fba3a9186250d0f862f2009145ea6bdb": "\\sigma_\\theta = \\sigma_{\\rm long} = \\frac{p(r + 0.2t)}{2tE}", "fba40921044e98f6bcadc0c62c887463": " \nC([x_1],[x_2],[x_3])\n", "fba4398984e1b7ebd7c759ca7965f7ca": "\nY=\\begin{bmatrix}\ny_1 \\\\ \\vdots \\\\ y_p \\end{bmatrix}\n", "fba4427697b3921178a019f7debdd464": "\\sum_{n \\ge 1}^{\\Re} f(n) = \\lim_{N \\to \\infty}\\left[\\sum_{n = 1}^{N}f(n) - \\int_1^N f(t)\\,dt\\right]", "fba4595fcdc5c40c8dbc5f90e1b1bda5": " x^2 \\equiv 10 \\pmod {13} ", "fba4c628c18120c76142a71ff9dce1b0": "B_f:= B \\times \\left\\{f\\right\\}, f\\in F", "fba4c9390733c766ef21aa9790f88b0a": "V^2=1.4742Y-0.004743Y^2", "fba5880bf9f14b0df04677dd363c35b0": "(a/R)^{1/2} U", "fba59121e42d58c8084e99256bf63d62": "\n\\lambda_4 = (\\mathrm{E}X_{4:4} - 3\\mathrm{E}X_{3:4} + 3\\mathrm{E}X_{2:4} - \\mathrm{E}X_{1:4})/4.\n", "fba59e6508136b60e7bbb35ea44af36c": "gate6", "fba61ec478691462ec2b760424f01c2d": "\\scriptstyle{|j\\,m\\pm 1\\rangle}", "fba6276d0f11189aa4a2b216a25bc7a1": "f = \\mu F_\\mathrm{e}", "fba64455845f4c0977b18ba68fc8d253": "\\begin{align}\n\\min &\\sum_{i=0}^n \\sum_{j\\ne i,j=0}^nc_{ij}x_{ij} && \\\\\n & 0 \\le x_{ij} \\le 1 && i,j=0, \\cdots, n \\\\\n & x_{ij} \\in \\mathbf{Z} && i,j=0, \\cdots, n \\\\\n & \\sum_{i=0,i\\ne j}^n x_{ij} = 1 && j=0, \\cdots, n \\\\\n & \\sum_{j=0,j\\ne i}^n x_{ij} = 1 && i=0, \\cdots, n \\\\\n&u_i-u_j +nx_{ij} \\le n-1 && 1 \\le i \\ne j \\le n\n\\end{align}", "fba6bb8ab33aa924c82e7391e417022e": "\\chi (G)=\\min\\{ k\\,\\colon\\,P(G,k) > 0 \\}.", "fba7af76a1a716592e4af13feab17f8e": "X_A \\perp\\!\\!\\!\\perp X_B \\mid X_S", "fba7bd40ed393bb717a602c64db96e23": "\\scriptstyle c(E)", "fba7d514a179e353b1e9d9279ddd3962": "\\scriptstyle { | \\phi_1 \\rangle, | \\phi_2 \\rangle, | \\phi_3 \\rangle \\cdots } ", "fba7f8480dc035e1928b3dcb95a2c259": "\\rho_{AB}\n", "fba8add9ce09a88c07c0c3c9a0e4b229": "\\frac{1}{V_\\max}", "fba8b1f80ac910f04b379b35001f24b8": "\n\\Bigl\\langle x_{m} \\frac{\\partial H}{\\partial x_{n}} \\Bigr\\rangle = \n\\delta_{mn} \\frac{1}{\\rho} \\, \\frac{\\partial}{\\partial E} \\int_{H < E}\\left( E - H \\right)\\,d\\Gamma = \n\\delta_{mn} \\frac{1}{\\rho} \\, \\int_{H < E} \\,d\\Gamma = \n\\delta_{mn} \\frac{\\Sigma}{\\rho}.\n", "fba8ee9f153b73e2f56089044bcdd424": "R = \\frac{ I_{\\mathrm{reflected}} }{ I_{\\mathrm{incident}} }", "fba8f93d2c743b10f4df9770ee681ae3": "n/n^2 = 1/n", "fba905406ebc4b18b0f7c4c792c5e1ee": " w^{(n)} ", "fba95d8c7a0ac6da00b74debbd4ee856": "S_D", "fba99d82d55108c031966376403cc964": "F ", "fbaa37e6811543aa7a4716eec8a8446f": "j=\\sigma", "fbaa5ceb529a82ddf39e230d92c6ae64": "\\frac {1}{G_{Eq}} = \\frac {1}{G_1} + \\frac {1}{G_2}.", "fbaa9a90f0a025a54b6a0742c2a5dbf9": "F(T,V,N)", "fbaad04120d9aa16074275cb164d1aae": " 2 -2g ", "fbab339d2255bc6def506a1e5d89ce6d": "1 < t < k < n", "fbab73beb5cea94001500031da82ffab": "m_r=1", "fbaba967a2c90e12df0d44070b968564": "X_1+\\cdots +X_N", "fbabf6de4409f11c7aa525c6355a3ab4": " C = \\frac{1}{\\sigma^3\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty ug(u)e^{-\\frac{u^2}{2\\sigma^2}} \\, du. ", "fbacd7d135a23804ab6a1436655dba72": "v_* : \\tilde C \\to \\tilde D", "fbad2ef7cbd8a754283a8a995787b7d8": "\\frac{1}{4}n_e^2 \\langle\\sigma v\\rangle E_{ch} \\ge \\frac{3n_ek_BT}{\\tau_E}", "fbad3e8efb73cb10f66d1134c11c918c": "\\mathfrak{i}, \\mathfrak{j}", "fbadc7d1be7fc72deb6bc46eadec2378": " \\mathcal{C} ", "fbadcf5c1a6d93136b4c611fba454869": "\\mathbf{\\dot{q}} = {\\rm d}\\mathbf{q}/{\\rm d}t ", "fbade7e310626547acb10a5b77d3db38": "\\alpha = 2 \\arctan \\frac {d} {2 F\\cdot ( 1 + m/P )}", "fbade9e36a3f36d3d676c1b808451dd7": "z", "fbae24cf00c14c8e6fccd2eef22ce72b": "\\sum_{i=1}^r c_i = 0.", "fbae507ff49b3a422c124bccabcff0fe": "\\Delta E>=0 ", "fbae67a4ee3374a9304e1969a1b2467d": "\\epsilon =0.02 + \\frac{0.18}{\\sqrt{v_{eff}d}}", "fbae93084892146abc0b37007bcbc5ff": "v=0 \\,", "fbaf4d5a9ac7d8f2346f9866b66e7f0c": "Rf", "fbaf4e4ead68c7e255ecf2315c83ba5d": "\\mathrm H", "fbaf5421a1e6f1dacb3c11ac99ff61ae": "\\scriptstyle\\ X_\\text{Q}(t)\\,", "fbafda21ba89b385656dd6f103da72d7": " \\sigma_x", "fbafe2a0bff6b90661200803eaa36640": " \\int_0^1 f(u)\\,du \\approx \\frac{1}{N}\\,\\sum_{i=1}^N f(x_i). ", "fbb03b0f1c8c0d813360c7c0004d5f18": "\\phi(xy) = \\phi(x)\\phi(y)\\,", "fbb03cdd328391a142823dd4649f288a": "W \\sub Y", "fbb09eae7b3f9e1c53bfd008a6bd8251": "\\vec{r}_u", "fbb0ac8efcd46ad7868c1ec69c40b04f": "\n\\begin{align}\nE[ \\hat{\\beta} | X ] & = \\beta + (X'X)^{-1}X'Z\\delta \\\\\n& = \\beta + \\text{bias}.\n\\end{align}\n", "fbb0d95a21149cd829c76ee6d197e224": "\\gamma(\\mathbf{v}) = \\frac{1}{\\sqrt{1 - \\mathbf{v}\\cdot\\mathbf{v}/c^2}}\\,\\rightleftharpoons\\,\\gamma(v) = \\frac{1}{\\sqrt{1 - (v/c)^2}}.", "fbb0e702dc34ba8a4b7d75dbe61d469c": "\n O_F(P) = \\bigl\\{ P, F(P), F^{(2)}(P), F^{(3)}(P), F^{(4)}(P), \\ldots\\bigr\\}.\n", "fbb12942bdd5abb02d762063553cf571": "\\left|1,2\\right\\rangle = \\frac1{\\sqrt2} (\\left|A\\right\\rangle\\pm e^{i\\phi}\\left|B\\right\\rangle),", "fbb18815b921c90aa924ecdf7cf15c99": "L_\\mathrm T = L_\\mathrm {L1} L_\\mathrm {L2} = e^{2 \\gamma_\\mathrm L} = e^{\\gamma_\\mathrm T} \\,", "fbb197070d3b09bd75a882ab54dbcaef": "-5\\le x,y \\le 5", "fbb25ba9db2da58826819cdcfdb405b1": "[1, +\\infty]", "fbb2cc13a0c91c250a7e2729e5cf1f00": "\\phi_2\\!", "fbb2e1062e815c777a7fe4567ec78181": "P(x)=P_{\\mathrm{prior}}(x)\\times P_{\\mathrm{selection}}(x),", "fbb3468ac907a8476060f7e37f882e7d": "t = \\int^{a}_{0}{\\frac{da}{H_0 \\sqrt{\\Omega_R a^{-2} + \\Omega_m a^{-1} + \\Omega_k +\\Omega_\\Lambda a^2}}}", "fbb39fed07e5105ee7ef9cfea9a2142c": "\\displaystyle K_X = f^*(K_Y)+\\sum_i a_iE_i", "fbb3ef0ca4f69d9c1070e6e222518c49": "a, b \\in A", "fbb41cf26f8bc6beafe39c1bd045ecb9": "B \\leftarrow \\alpha T^{-1} B", "fbb44ecc767af7356a12eb4a2b1f5e28": "\\frac{dP\\left(t,T\\right)}{P\\left(t,T\\right)}=\\left[r\\left(t\\right)-\\alpha\\left(t,T\\right)\\theta\\left(t\\right)\\right]dt +\\alpha\\left(t,T\\right)dW\\left( t\\right).", "fbb4648a1cd2aae9f783086aa63cc468": "Z^8_2", "fbb4f097629b79d6cbcc9c1caf4a2f0c": " \\Phi(\\Psi_\\varepsilon(\\tilde{x})) ", "fbb5046a95c53376847b0e4523d3ebbf": "J^\\mu = \\frac{i\\hbar}{2m}(\\psi^*\\partial^\\mu\\psi - \\psi\\partial^\\mu\\psi^*)", "fbb553bc18de1c8bb372592c536d1136": "N^cH^k(X, \\mathbf{Z}) \\subseteq H^k(X, \\mathbf{Z}) \\cap (H^{k-c,c}(X) \\oplus\\cdots\\oplus H^{c,k-c}(X)).", "fbb569a8d0547f262c00a03c331f7092": "\\epsilon\\,", "fbb578d5f363d2b2547fb9610cd5699c": "\\psi = f(u)", "fbb57c61544fa2053b214c262d3f49c5": "X_i = 2 \\pi f L", "fbb5de423d01fbb2e224b6c706afd22d": " k_v(V) = \\frac{k_{\\text{arb}}}{V} ", "fbb5dffc45fef412d869c39b2001dadf": "\\sigma_1^2 = \\sigma_2^2 = \\sigma_3^2 = 1,", "fbb61fe7f88e12b578cc101fb3067efa": "y > 0", "fbb62a461ced94006f8821004278f8dd": "\\mathbf{b} ", "fbb62fffa757f50eb825297721a7a9fa": " -\\frac{{\\hbar}^2}{2m} {\\nabla}^2 {e^{iS/{\\hbar}}}=(E-V) e^{iS/{\\hbar}}", "fbb64b0c56d3cbe31365fa2323d5f798": "\\sigma^2/2", "fbb7e97eeddf68d875fb1faeb11c4155": "\\theta_{up}(X)", "fbb83dbcddf13d0190d8a58ff4464807": "u(x,t) = F(x - v \\ t)", "fbb84c2662134f9895e12045ae983074": "a_r(x) f^{(r)}(x) + a_{r-1}(x) f^{(r-1)}(x) + \\ldots + a_1(x) f'(x) + a_0(x) f(x) = 0", "fbb8692c3c3eaabb43cd43b092c1f641": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi}-t_g\\ \\left(\\frac{p}{r}\\right)^2\\ \\frac{3}{2}\\ 3\\ \\sin^2 i\\ \\sin^2 u\\ du\\ =\\ \n\\frac{9}{2}\\ \\sin^2 i\\ \\int\\limits_{0}^{2\\pi}\\ \\left(1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u\\right)^2\\ \\ \\sin^3 u\\ du\\ = \\\\ \n&9\\ \\sin^2 i\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\sin^4 u\\ du\\ =\\ 2\\pi \\frac{27}{8}\\ \\sin^2 i\\ e_h \n\\end{align}\n", "fbb8783c427bf35db01c59ebd00762b9": "M^{\\alpha\\beta\\mu}_y", "fbb882d0bdd2526d917f050a3caefaaa": "M(f) = \\lim_{h\\rightarrow 0^+}{\\frac {L(I+hf)-1}{h}}.", "fbb8e119960c6506035d5289f6799fb2": "\\theta _{c}=Sin^{-1}(\\frac{n_{1}}{n_{2}}),\\; \\frac{n_{1}}{n_{2}}\\leq 1", "fbb8e396d6d5cf21f9f323db01bf400e": "\\frac 1 s + \\frac 1 v = \\frac 1 f\\,;", "fbb8f2b93c57e24d29fa2a9d0ec1f43e": "E_{m} = \\frac{P_{K^+}} {P_{tot}} E_{eq,K^+} + \\frac{P_{Na^+}} {P_{tot}} E_{eq,Na^+} + \\frac{P_{Cl^-}} {P_{tot}} E_{eq,Cl^-}", "fbb92fa1011baf63bb5a67e8c1ffab16": "\\mbox{dinner } \\mathbf{\\{ \\operatorname{<}, \\operatorname{m} \\}} \\mbox{ bed}", "fbb94b11c5a42bc7857bf55af642ea9a": "\\Phi _\\alpha \\left( {A_\\alpha ^1 , \\cdots ,A_\\alpha ^n } \\right)", "fbb96f958dbd7700910316ea525a9706": "(0,\\tfrac{1}{4})", "fbb98527d50a1a46d8a5a63ed4b30ff8": "= C_f", "fbb9ac5c4ee41f4b9f87e318f5076e56": "\\theta^*", "fbb9c5bcabb612365b11875eee079d32": "\\textstyle v_R, v_L, v_C", "fbb9ed1a30e6f38ce5148bcfa85bc0fc": "M^\\kappa/U=\\prod_{\\alpha<\\kappa}M/U.\\,", "fbba1428d434e502e9737953334e9c69": "\\frac{a_1\\zeta }{1+\\frac{a_2\\zeta }{1+\\ldots}} ", "fbba482a902c353b7c94115b3f304cea": "a=\\begin{cases}0;&\\mbox{if }\\chi(-1)=1, \\\\ 1;&\\mbox{if }\\chi(-1)=-1,\\end{cases}", "fbba54fb8932d4dd18e033437d315d42": "\\begin{align}\n\\Delta K = W &= \\int_{\\mathbf{x}_0}^{\\mathbf{x}_1} \\mathbf{F} \\cdot d\\mathbf{x} \\\\\n&= \\int_{t_0}^{t_1} \\frac{d}{dt}(\\gamma m_0 \\mathbf{v})\\cdot\\mathbf{v}dt \\\\\n&= \\left. \\gamma m_0 \\mathbf{v} \\cdot \\mathbf{v} \\right|^{t_1}_{t_0} - \\int_{t_0}^{t_1} \\gamma m_0\\mathbf{v} \\cdot \\frac{d\\mathbf{v}}{dt} dt \\\\\n&= \\left. \\gamma m_0 v^2 \\right|^{t_1}_{t_0} - m_0\\int_{v_0}^{v_1} \\gamma v\\,dv \\\\\n&= m_0 \\left( \\left. \\gamma v^2 \\right|^{t_1}_{t_0} - c^2\\int_{v_0}^{v_1} \\frac{2v/c^2}{2\\sqrt{1-v^2/c^2}}\\,dv \\right) \\\\\n&= \\left. m_0\\left(\\frac {v^2}{\\sqrt{1-v^2/c^2}} + c^2 \\sqrt{1-v^2/c^2} \\right) \\right|^{t_1}_{t_0} \\\\\n&= \\left. \\frac {m_0c^2}{\\sqrt{1-v^2/c^2}} \\right|^{t_1}_{t_0} \\\\\n&= \\left. {\\gamma m_0c^2}\\right|^{t_1}_{t_0} \\\\\n&= \\gamma_1 m_0c^2 - \\gamma_0 mc^2.\\end{align}", "fbba8eb5238f0c81e9635fd95e52ed5f": "B \\leq_T A", "fbbb395852e461ab422bd440d75b951f": "\\begin{align}\nG(x, y; \\lambda) &= \\left \\langle x, \\frac{y}{\\lambda I - L} \\right \\rangle \\\\\n&= \\sum_{i=1}^n \\sum_{j=1}^n \\langle x, e_i \\rangle \\left \\langle f_i, \\frac{e_j}{\\lambda I - L} \\right \\rangle \\langle f_j , y\\rangle \\\\\n&= \\sum_{i=1}^n \\frac{\\langle x, e_i \\rangle \\langle f_i , y\\rangle }{\\lambda - \\lambda_i} \\\\\n&= \\sum_{i=1}^n \\frac{e_i (x) f_i^*(y) }{\\lambda - \\lambda_i},\n\\end{align}", "fbbb749ec13016769e98ff39d5837276": "D_J, D_{JK}", "fbbb7d1b08bbfbd56df3b3275b1a843e": "\\tilde{G}(s)=\\frac{k_{\\mathrm{B}}T}{\\pi a s \\langle\\Delta \\tilde{r}^{2}(s)\\rangle}", "fbbbe7678455b949d6a246b2d08e1f61": " (|h\\rangle)\\in \\mathcal{H} ", "fbbbf94bf4febf2cfb2975f7c84aa3ae": " \\frac{\\partial F^{\\beta\\alpha}}{\\partial x^{\\beta}}=\\mu_0 J^{\\alpha} \\,.", "fbbc5e8739fe624c271131c96f54dca0": " \\min_{x,y} f(x) + g(y), \\quad \\text{subject to}\\quad x = y. ", "fbbcd88bb82c526b70b720876d5256ca": "\\textrm{Multifactor~Productivity}={\\textrm{Output}\\over{(KLEMS)}}", "fbbcdcc0576b7c44fd72ed164e592593": "(g,g^a,g^b,g^{ab})", "fbbcebebe2585ba9212dda13cd8e2e29": "\\mathit{F1} = 2 \\mathit{TP} / (2 \\mathit{TP} + \\mathit{FP} + \\mathit{FN})", "fbbcf274d2ab45d1772e7d094f95b2fb": "(y_n)", "fbbd0c0da8e0a80f78d364d2783eb37a": "\\left( \\frac{\\varepsilon_{eff}-\\varepsilon_m}{\\varepsilon_{eff}+2\\varepsilon_m} \\right) =\\delta_i \\left( \\frac{\\varepsilon_i-\\varepsilon_m}{\\varepsilon_i+2\\varepsilon_m}\\right),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(6)", "fbbd4dccfe0ac925beb6d9144bfd06c7": "f(x) = f(x/2)^2", "fbbd7ff68ae53631e5b4958a084946b2": "z_0=z", "fbbda88eb9407578a00a1dfe086a3f99": " p = \\Pi_{ i = 1 }^n ( 1 - \\frac{ i }{ k } ) ", "fbbde2df37be15057e9986d9a452a7b8": "\\frac{1}{4\\pi c}", "fbbe044c4749d681ae47d85d684360d2": "\\Delta p_{tot}", "fbbe168e5c89af3b5f1cd5ad89befbf7": " \\mathrm{pH} = -\\log \\sqrt { K_a F }", "fbbea5b1e97bb52638d5635d8ce3b3d5": "|A|\\le \\frac{r(n-1)!}{r!(n-r)!}={n-1\\choose r-1}.", "fbbeb898b76dd1deb41d132af9a0c9ff": "\\mathbf{E}_{\\perp}", "fbbf324f12f3d331fd923db3672c1888": "\\alpha_n = \\frac{\\pi(n-0.5)}{2M}", "fbbfffbf83b17c6010fb8702d2f53ef9": "\\boldsymbol{M}_w = \\frac{\\boldsymbol{k}}{k} \\frac{E}{\\rho\\, c_p},", "fbc0123b8980b69fc81147ab11d52196": "[X^k] g=\\frac{1}{k} \\mathrm{Res}\\left( f^{-k}\\right).", "fbc01c073968cedf96ab4762493a3418": " \\ d ", "fbc049839932ed45f91a95affb5acbf0": "\\sigma_\\lambda \\sigma_\\mu =\\sum c_{\\lambda\\mu}^\\nu \\sigma_\\nu", "fbc0838efd2d9e8847ffbfcf1191f048": "E \\psi_d = - \\frac{\\hbar^2 (1-\\lambda)}{2 \\mu} \\frac{d^2 \\psi_d}{d Q_d^2}+\\frac{1}{2} \\mu \\omega ^2 Q_d^2 \\psi_d ", "fbc0c5c40ca2ec761dc33d9411aca703": " log_e\\left(1+\\frac{h}{5} \\right) = \\frac{h}{5} - \\frac{1}{3}\\left(\\frac{h}{5}\\right)^3 + \\frac{1}{5}\\left(\\frac{h}{5}\\right)^5 - \\frac{1}{7}\\left(\\frac{h}{5}\\right)^7 + ....... ", "fbc1073d905274f6eb306d322913f0ae": "\\langle Q, \\Sigma \\times {B}, \\Gamma \\times \\Gamma, \\delta ', q_0, F \\rangle ", "fbc12117d590e9367920e4ac19bf68e7": "|\\mathcal V|_V=V", "fbc1241bb02224bbbb8eeee04c70b545": "g = \\frac{\\bar{x}_1 - \\bar{x}_2}{s^*}", "fbc14e0184c0ae968c4064a914ad3765": "i\\leq r", "fbc1534938fe0c9a0470bf02afa4973d": " \\rho^{12}\\mapsto S(\\rho^1)-S(\\rho^{12}) ", "fbc15bef3f9e3c45027dff7eaf32d4cc": " \\vec{e}_0 = \\frac{1}{x}\\partial_t,\\;\\; \\vec{e}_1 = \\partial_x,\\;\\; \\vec{e}_2 = \\partial_y,\\;\\; \\vec{e}_3 = \\partial_z", "fbc18957eba04ab295607179d1036d4e": " (a, b) ", "fbc1bcd06fee25c4067050d45d1daac4": "c_0\\cup\\{x_1,x_2,x_3\\}", "fbc1e6e7f6f38ac98fdcb1c5bc099d94": "i:1\\dots \\dim(\\mathfrak{H}_b)", "fbc24054a8744c3bc4d10e7db73f0631": "n_1' = 4 \\pi \\times 2 \\int_{r_0}^{r_p} r^2 g(r) \\rho \\, dr. ", "fbc267a219949fc62474dce2effb54e8": " \\frac {dk}{dE} = \\frac {\\sqrt {2 m_w^*}}{2 \\sqrt E \\hbar} \\quad \\quad \\quad \\frac {d \\kappa}{dE} = - \\frac {\\sqrt {2 m_b^*}}{2 \\sqrt {V-E} \\hbar}", "fbc26a8c410296a4b9942029097de975": "u(x) = x^2 -4x +3", "fbc293f0f3197e746b950108dcdb52cd": "\\partial A := \\{ (x, y) \\in E(G) | x \\in A, y \\in V(G) \\setminus A \\}.", "fbc2aecba17b015950129d895e5e8c5a": " a - 2 \\sqrt{bc} \\, \\cos(k \\pi / {(n+1)}) ", "fbc2dff966871f1120f37184219fea40": "E_{k}=-\\frac{p^{2}mc^{2}}{2n_{k}^{2}}\\left( 1-\\frac{p^{2}}{6n^{2}}\\right)^{2}", "fbc37fff828b170e1790a8d21071f643": " e_{y} = \\sqrt{\\frac{1-Q}{2}} \\, e^{i \\phi} ", "fbc4eda48dd4dc6ca46286e8b58e6df8": "{\\mathcal F}=\\{f(\\cdot,u)\\mid u\\in U\\}", "fbc504799ebf8794d09f0e21e76ce002": "P(G, x)", "fbc52590ca13418980f204d3bc71881f": "\n\\begin{align}\nD_{\\mathrm{KL}}(P\\|Q) & = - \\operatorname{E}(\\ln q(x)) + \\operatorname{E}(\\ln p(x)) \\\\\n& = H(P,Q) - H(P)\n\\end{align}\n", "fbc53f561416c7f0d44551b719fb1ec6": "f(E;\\beta)", "fbc544db87d0c3784f30b1ecba01a994": " [C]=[R]_0 \\left [ 1-e^{-\\frac{k_2}{k_1'}[A]_0(1-e^{-k_1't})} \\right ] ", "fbc59a59779f9b8f0d13fa7b8ed71269": "B = \\{(e,n) \\mid n \\in W_e \\}", "fbc5b7dee6fb4f2fd2094e2da319d934": " {_0^C D_t^\\alpha} x(t)=f(t,x(t)) , \\quad t\\in [0,T], \\quad x(0)=x_0, \\quad 0<\\alpha<1. ", "fbc5eef7c0ecbc343fbc4c9f5623556e": " \\operatorname{ess-sup}_{x \\in X} \\|T_x\\| < \\infty ", "fbc663bce35fe6736b0489e10310cdce": "\\mathbf{a}\\succ \\mathbf{b}", "fbc67b6b38c773dc64d79463993262cd": "\\|f*g\\| _r\\le c_{p,q} \\|f\\|_p\\|g\\|_q.", "fbc6f249621e50c6f7d35f0cfa18ffb8": "\\boldsymbol{R_p} ", "fbc752a5cc833b561e4e6bbadb3f1f7b": "V = V_\\text{esc} + \\Delta v", "fbc7573bda59a97fa4ae6d0e930c5d86": "r_2 = \\frac {c}{a \\ r_1} \\approx -\\frac {c}{b}. ", "fbc889bdd49e39976722e55fb9f29547": "_2^0\\text{P} = ^{14}\\text{N}_2\\text{O}", "fbc8a66b698c35267d46cb33c06de238": "H_n(G,M) = \\operatorname{Tor}_n^{\\mathbf{Z}[G]}(\\mathbf{Z},M)", "fbc8e2b8620d147b8ffa2575bfb05795": "A^*A = R^*R\\,\\!", "fbc91289905cc556d1e60874c4ad76cb": "(F,G,e,\\varepsilon)", "fbc93c52d821e0413367c828ea843c08": " K= C_{12}+C_{23}+C_{34}-C_{14} \\le 2", "fbc950212b9cdc9dbe20f589ce22824b": "Y = 0.2126 R + 0.7152 G + 0.0722 B", "fbc9819b23c1348f572722f2164abe82": " d^2 = (\\frac{L}{L_{\\odot}})(\\frac{b_{\\odot}}{b}) ", "fbc98309590fb945efc56a2e7a9c465d": "Q_{bd}=\\int_{0}^{t_{bd}} i(t)\\, dt", "fbc98e5e95580776f6daaced11575d71": " C_0^{j}", "fbc998e01f8ac78c798e0020db8640b4": "(Open)\\quad open \\ n. A\\;\\mid\\;n[\\; \\overline{open} \\ n.B \\mid B'\\;] \\Rightarrow_{amb} A\\;\\mid\\;B\\;\\mid\\;B'", "fbc9e338ee1023b50145520a66c986ea": "\ny(x_0+h) = y(x_0) + hy'(x_0) + \\frac{h^2}{2!}y''(x_0) + \\frac{h^3}{3!}y'''(x_0) + \\frac{h^4}{4!}y''''(x_0) + \\frac{h^5}{5!}y'''''(x_0) + \\mathcal{O} (h^6)\n", "fbca127690fad12d4b9be224f8421c4c": "=\\frac{(p_{n}^2+p_{n}p_{n+2}-2p_{n}p_{n+1})-(p_{n}^2-2p_{n}p_{n+1}+p_{n+1}^2)}{p_{n+2}-2p_{n+1}+p_n}", "fbca4ddd04bbde85c7b13c4c6791006b": "q_{in}", "fbcaaab5f1ae4515121a88e1522be854": "\\operatorname{E}(m(H,t+1)) \\geq {m(H,t) f(H) \\over a_t}[1-p].", "fbcac5d6075e3e1444e71f190cfdb92a": "\\{g_0\\}, \\{g_1, \\cdots, g_{k_2}\\}, \\{g_{k_2+1}, \\cdots, g_{k_3}\\}, \\cdots, \\{g_{k_{L-2}+1}, \\cdots, g_{-2}\\}, \\{g_{-1}\\}", "fbcaeb0544a261cd2b5f46b577c37416": "\\mathbb{C}\\setminus\\{a,b\\}", "fbcb57310e03fc9efb84eda4e5f0c70e": "B \\to C", "fbcbbb76caad38d22b3dbaf45d7df419": "\\ |zcy| = \\frac{a^2 M}2.", "fbcbf3aec76acb9024a9d7a7308d4aff": "\ng_m = {\\Delta I_\\mathrm{out} \\over \\Delta V_\\mathrm{in}}\n", "fbcc0313290dd2f29670e5445b6dc880": "(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_{\\theta, q_\\theta} = B^{s_\\theta}_{p_\\theta, q_\\theta}, \\quad s_0 \\ne s_1, \\ p_\\theta =q_\\theta, \\ 1 \\le p_0, p_1, q_0, q_1 \\le \\infty.", "fbcc1bad0ff967bfea4c04fe6b0ed906": " a_{(\\lambda_1+n-1, \\lambda_2+n-2, \\dots , \\lambda_k)} (x_1, x_2, \\dots , x_k) =\n\\det \\left[ \\begin{matrix} x_1^{l_1} & x_2^{l_1} & \\dots & x_k^{l_1} \\\\\nx_1^{l_2} & x_2^{l_2} & \\dots & x_k^{l_2} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nx_1^{l_k} & x_2^{l_k} & \\dots & x_k^{l_k} \\end{matrix} \\right]\n", "fbcc7c2268a24234261d0496ad628307": "\\Sigma\\subseteq\\Gamma\\setminus\\{b\\}", "fbcc86dd7ccf772fd2f7485ef524c906": "Bn_P(Cl_1^{\\leq}) = Bn_P(Cl_2^{\\geq}) = \\{x_4,x_6\\}", "fbcc973ed76aceb7fcc8944ed17b019d": " {3\\over2} \\cdot {3\\over4} \\cdot {3\\over2} \\cdot {3\\over4} \\cdot {4\\over5} = {81\\over80}", "fbcca8305d45ab781b53ad4bf5dbe09f": "O(m + z)", "fbccfad45573e2fc79507827e74971f2": "x^2 - \\left (618 \\right )x + 90581 = 0", "fbcd175231f31b449d38019ed931d18c": "|F(h k l)|^{2}", "fbcdd86e7c99b30bc491a5cf32ec25f3": " \\frac{\\mu_0r^{2}N^{2}}{3l}\\left[ -8w + 4\\frac{\\sqrt{1+m}}{m}\\left( K\\left( \\sqrt{\\frac{m}{1+m}} \\right)\n-\\left( 1-m\\right) E\\left( \\sqrt{ \\frac{m}{1+m}} \\right) \\right)\n\\right]\n", "fbcdf8e8014c94842017765a3bcac09e": "\\mathbf{C}_p", "fbce0fd5c41769a4ff770ca79535e620": "O(|V| |E|)", "fbce13759903ac5276a966afe33fff69": "\\Psi^{\\rm IQHE}_{\\nu^*}", "fbce24ffd966ec62458b4dc1d39ebf5a": " \\pi : TM \\twoheadrightarrow M ", "fbce4a1ebe576539394e9493e30c7e5e": "f(x) = x^2", "fbce9fdbea7a2924c388cdfca1b75ebf": "H_A\\otimes H_B", "fbcf1207f014ab23efdfe69ce3683044": "-q(x)", "fbcf9c38586f3e82c9dc604605b74563": "I = 4s", "fbd04e98f1546a2e7327bfd5d8a45772": "p_{n+1} = p_1 p_n + S \\cdot q_1 q_n\\,\\!", "fbd0b4b928563c01ea38890cf2fc7c64": "\\mathcal{B}(\\mathbb{R})", "fbd0c934e096b5807f4f293901e58ac4": "\\mathcal{S}'(\\mathbb{R}^n)", "fbd0d5f48560572fdf1ef3a626bdfba8": " f(x,\\lambda) ", "fbd0dfa8f015d48acfe4570aa6babc6a": "A\\cap B", "fbd0fdc23f0a81940a929b16569a510b": "C_0(\\widehat{G})", "fbd150bddc4b485634ebab4f83132a92": "\\lim_{n\\rightarrow +\\infty}b_n=0.", "fbd1bcf0c6203abffc4337317fc4e3fb": "I(dog : N_0) = dog : E_0", "fbd1ca6bca023fa70971d0eaee1b57b3": "v(t) = L \\frac{di(t)}{dt}", "fbd1f1912a5be2c3dbef41b5fed1a911": "m(x)\\leq \\frac{f(x)-f(y)}{g(x)-g(y)} \\leq M(x)", "fbd1f7bf6b3eaad80c248e8795fba376": "\\mathbf I\\,\\!", "fbd224a1e8e73bf9f84e41bc26e86497": "M \\leq R \\times (1/RR),", "fbd29061148cc6794e02847f09209415": "\n\\Pi_{\\rho_{x^{n}},\\delta}\\equiv\\sum_{y^{n}\\in T_{\\delta}^{Y^{n}|x^{n}}\n}\\left\\vert y_{x^{n}}^{n}\\right\\rangle \\left\\langle y_{x^{n}}^{n}\\right\\vert ,\n", "fbd2f4443acdf9b5e73ce73b61227692": "r\\tan\\psi=\\frac{r^2}{\\tfrac{dr}{d\\theta}},", "fbd300f2a2644c394b6a3c639ea418fe": "H_n(s)= \\sum_{k=0}^n V(\\tau^k s)", "fbd314928f12db51670af9368f25d505": "\\phi: \\mathrm{Ob}(\\mathcal{A})\\to G", "fbd36a3a81e50072c1813548f30b81a6": "C=\\exp \\left( \\frac{2i\\pi }{n+1} \\right)", "fbd383979dc7e412a3679fde7b500d61": "\\lim_{t \\to \\infty}\\phi(t) = \\langle c(0), \\mathbf{v^1} \\rangle \\mathbf{v^1} ", "fbd3affa0b6c286c75f9458f0b35ebb3": "X:\\mathbb{R}\\to\\mathbb{R}", "fbd3c6f79997e90ab99e91a47adc5b48": "\n\\sum_{n=1}^\\infty \\frac{H_n}{(n+1)^2} = \\zeta(2,1) = \\zeta(3) = \\sum_{n=1}^\\infty \\frac{1}{n^3},\n\\!", "fbd3c7642ca3e65deb08ef41dfa7d584": " \\theta(\\xi)=\\frac{\\sin\\xi}{\\xi} ", "fbd3f93653405be029bf1988775eec92": " P\\left ( y \\mid \\mathbf{x} \\right ) ", "fbd409c29bb7c88cae59732b0186a98e": "\n \\lambda_{max}= \\max\\limits_{j}\\limsup _{t \\rightarrow \\infty}\\frac{1}{t}\\ln\\alpha_j\\big(X(t)\\big).\n", "fbd4183d4ce96fde45720cb23853f0a7": " P(c(t)) \\dot{v}(t)=0 ", "fbd4312ca495be17963ce47a5253a3f7": " \\text{Lock criteria}\\; \\begin{cases} \\mathrm{ \\left ( \\frac{\\Delta R}{\\Delta T} \\right) - \\left (\\frac {C \\times \\text{Doppler Frequency}}{2 \\times \\text{Transmit Frequency}} \\right) < \\text{Threshold} }\\end{cases} ", "fbd48790297cb6af0ad80ce85402d19d": "sw_i = 1", "fbd4b2f74bd233def04e82dd6407046e": "\\operatorname{mr}(G)\\leq 2", "fbd4b4b985f84d482b503d13968c94eb": "\nH_D \\left( {C\\cap \\left( {a,b} \\right)} \\right)=\\Gamma \\left( {1+D} \n\\right)_a I_b^{\\left( D \\right)} 1,", "fbd4dc2c3e89d42caa09c581746c9755": "C \\oplus C ", "fbd53b5560342f5f095437b8a1ad874b": "\\mathbf{x} = \\boldsymbol{F}(\\mathbf{X})", "fbd56e6f3a6cf38c152077e7f31bd209": " \\displaystyle{p_t(z)= {1+\\kappa(t) z\\over 1-\\kappa(t) z}}", "fbd577523e9545027ceefdc4321cbdf0": "\\hat{G}(z)z^{-k}", "fbd57cb793badbdeda8038e742df438e": "A = \\frac{r_0^2 c N}{64 \\pi^2 \\beta^3 \\gamma^4 \\epsilon_h \\epsilon_v \\sigma_s \\sigma_p}", "fbd5c396d7f7942c617179f69f554e4d": " \\qquad \\qquad \\mathbf{q}=\\frac{1}{\\hbar^3}\\sum_p(E_e-E_\\mathrm{F})\\mathbf{u}_ef_e^\\prime = \\frac{1}{\\hbar^3k_\\mathrm{B}T}\\sum_p \\mathbf{u}_e\\tau_e(-\\frac{\\partial f_e^\\mathrm{o}}{\\partial E_e})(E_e-E_\\mathrm{F})(\\mathbf{u}_e\\cdot\\mathbf{F}_{te}),", "fbd6178bd9d010234beae81579bfaf5d": "p = ~~\\frac{\\partial F_2}{\\partial q} \\,\\!", "fbd624bcbc3d5968c4d177be7deb6c3c": "\\mathbf{C}^{\\mathrm{op}}", "fbd6496ffbd81ab2c0a1842aa5a77de9": "\\Omega^{-1}(M)", "fbd6eada8b4ef6c9bb489d22e97860ab": " \\mathbf{P} = \\frac{\\partial L}{\\partial \\mathbf{\\dot{r}}} = m \\mathbf{\\dot{r}} + q \\mathbf{A}", "fbd75e53e773c9f26e3bcd6e26036eff": "\\mathrm{^{239}_{\\ 94}Pu\\ \\xrightarrow {4(n,\\gamma)} \\ ^{243}_{\\ 94}Pu\\ \\xrightarrow [4.956 \\ h]{\\beta^-} \\ ^{243}_{\\ 95}Am}", "fbd79b27b7119644e9826a3197630fee": "G_1 + G_2\\, ", "fbd7e7aea030fbfa02eb1f5f2a2f92cf": "(X \\downarrow U)", "fbd829cb462e2052c267b8b500a6fb94": " \\text{Minimize} \\sum_{i=1}^n u_i", "fbd8473bee865b3d4a4cc076e0fcff39": "(\\mathbf{\\lambda} x . x x x) t \\rightarrow t t t", "fbd8f860e6a815200e5946a61fe0020e": "\\int_0^1 |f(t)|^2w(t)\\,dt < \\infty", "fbd912fd569e8e0936d3d626b8759946": "VSP = \\frac{power}{mass} = \\frac{{\\operatorname{d}\\over\\operatorname{d}t}(E_{kinetic} + E_{potential}) + F_{rolling} \\cdot v + F_{aerodynamic} \\cdot v + F_{internal} \\cdot v}{m}", "fbd93cfe58496e0d392b3fb8ce337597": "\\left\\{Q,Q\\right\\}=\\left\\{\\overline{Q},Q\\right\\}C=2\\gamma^\\mu\\partial_\\mu C=-2i\\gamma^\\mu P_\\mu C", "fbd956e7ab8cb2684405fd99dbbe31a5": "M_z = \\frac{1}{a} \\frac{\\partial \\Phi}{\\partial z},", "fbd958b447362e289f4871fda5e3137d": "\n\\begin{align}\n&-10\\sin^2 i \\ \\left(-\\hat{g}\\ \\left(\\frac{3}{8}\\ +\\ \\frac{3}{16}\\ {e_g}^2\\ +\\ \\frac{15}{16}\\ {e_h}^2\\right)\n+\\hat{h}\\ \\left(\\frac{3}{8}\\ e_g\\ e_h\\right)\\right) \\\\\n&+6\\ \\left(-\\hat{g}\\ \\left(\\frac{1}{2}\\ +\\ \\frac{3}{8}\\ {e_g}^2\\ +\\ \\frac{9}{8}\\ {e_h}^2\\right)\n+\\hat{h}\\ \\left(\\frac{3}{4}\\ e_g\\ e_h\\right)\\right) \\\\\n&-15\\sin^2 i \\ \\left(\\hat{g}\\ \\left(\\frac{1}{8}\\ +\\ \\frac{3}{16}\\ {e_g}^2\\ +\\ \\frac{3}{16}\\ {e_h}^2\\right)\n+\\hat{h}\\ \\left(\\frac{3}{8}\\ e_g\\ e_h\\right)\\right) \\\\\n&+3\\left(\\hat{g}\\ \\left(\\frac{1}{2}\\ +\\ \\frac{9}{8}\\ {e_g}^2\\ +\\ \\frac{3}{8}\\ {e_h}^2\\right)\n+\\hat{h}\\ \\left(\\frac{3}{4}\\ e_g\\ e_h\\right)\\right) \\\\\n&+\\frac{15}{2}\\sin^2 i\\ e_g \\ \\left(-\\hat{g}\\ \\left(\\frac{1}{8}\\ e_g\\right) + \\hat{h}\\ \\left( \\frac{1}{8}\\ e_h\\right)\\right) \\\\\n&-\\frac{15}{2}\\sin^2 i\\ e_h \\ \\left(-\\hat{g}\\ \\left(\\frac{1}{8}\\ e_h\\right) + \\hat{h}\\ \\left( \\frac{1}{8}\\ e_g\\right)\\right) \\\\\n&-\\frac {3}{2}\\ e_g \\ \\left(-\\hat{g}\\ \\left(\\frac{1}{4}\\ e_g\\right) + \\hat{h}\\ \\left(\\frac{1}{4}\\ e_h\\right)\\right) \\\\\n&+\\frac{3}{2}\\ e_h \\ \\left(-\\hat{g}\\ \\left(\\frac{1}{4}\\ e_h\\right) + \\hat{h}\\ \\left(\\frac{3}{4}\\ e_g\\right)\\right) = \\\\\n&\\frac{3}{2}\\ \\left(\\frac{5}{4}\\ \\sin^2 i\\ -\\ 1\\right)\\left((1-{e_g}^2\\ +\\ 4\\ {e_h}^2)\\hat{g}\\ -\\ 5\\ e_g\\ e_h\\ \\hat{h}\\right)\n\\end{align}\n", "fbd96a490581bd387072c31da03aa093": "b^n = \\frac{V_n - U_n \\sqrt{D}}{2}", "fbd99189a2d9ca1e64eacb72d6456392": "d\\left[\\left(a_1, a_2\\right), \\left(b_1, b_2\\right)\\right] = \\left|a_1 - b_1\\right| + \\left|a_2 - b_2\\right|", "fbd9c24234e10beb37412a927a9bbdf9": "\\displaystyle{ \\mathrm{ind} \\, T(f) = \\mathrm{Tr}\\, (I-T(f^{-1})T(f))^n - \\mathrm{Tr}\\, (I-T(f)T(f^{-1}))^n.}", "fbda692f8cdb2d8bb119d6dd47a69638": "U_n(P, Q)", "fbda967a125ae9e9b6da82cb3a01947c": "f(1,2) = \\exp\\left[- \\frac{u(|\\vec{r}_1- \\vec{r}_2|)}{k_B T}\\right] - 1 ", "fbdaea770b4254be2e98865fbd8c99dd": "\\tfrac{\\partial V}{\\partial S}", "fbdafef4101e1d5ca30c7046bae6e233": "v = r\\omega\\,.", "fbdb5f4b50535a924de56b9c7b558c46": "\\mu\\left( Z_i, l^k(Z^n,i),r^k(Z^n,i) \\right)", "fbdb7022d1895c8f7b010cb328090b4e": " (f,g) = \\int_a^b \\omega(x) f(x) g(x) \\, dx . \\,\\!", "fbdb70603e6106996a928612086c6ef9": "|x| < r", "fbdb8687460f9319b00acd3fb2a55ea3": "\n\\begin{align}\n\\csc x & {} = \\sum_{n=0}^\\infty \\frac{(-1)^{n+1} 2 (2^{2n-1}-1) B_{2n} x^{2n-1}}{(2n)!} \\\\\n& {} = x^{-1} + \\frac{1}{6}x + \\frac{7}{360}x^3 + \\frac{31}{15120}x^5 + \\cdots, \\qquad \\text{for } 0 < |x| < \\pi.\n\\end{align}\n", "fbdb9736893b8a5ca375a6810662f7d3": "{w_i\\over \\sum_{j\\in B} w_j}.", "fbdb97d0afbbda326c6ade23ce420f72": "R_{\\mu}=\\frac{1}{4\\pi}\\int\\left (G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu}\\right )\\sqrt{-g}\\,dS^{\\nu}", "fbdc8c8ab987be33b93a01a3f391c904": "D_\\max = 80", "fbdc9985285d81101904790c54dbc1b2": "\\bar{\\rho\\,\\!c}\\left ( \\frac{\\part T}{\\part t} \\right )\\ = \\bar{\\lambda} \\left ( \\frac{\\part^2T}{\\part t^2} + \\frac{1}{r} \\frac{\\part T}{\\part r} \\right ) + \\frac{1}{\\bar{\\sigma}} \\left ( \\frac{I}{2{\\pi}Lr} \\right )^2", "fbdcc729c43a4e86431d2662881ee425": " \\frac{1}{\\gamma } \\approx 1-\\frac{G M}{r c^2} ", "fbdcccd2463234f9013b48eded916ce3": "R_{\\alpha \\beta} - {1 \\over 2} R g_{\\alpha \\beta}", "fbdce7f8f13b179755952cd3779b7696": "z_n = z_0^{p^n}", "fbdd1edcb79705d0bec6804cf4e75461": "\\scriptstyle K_n(\\mathbf{x},\\mathbf{y})", "fbdd2269cc30d3c4497b498fdbd0b0ba": "K_0(z)", "fbdd484fd38f14aeee01470b1f35b4f1": "\\Delta s^2=x^a x^b \\eta_{ab}=c^2 \\Delta t^2 - \\Delta x^2 - \\Delta y^2 - \\Delta z^2", "fbdd58ac11f29713f4963e9c12389333": "M=4N\\mu\\,", "fbdd989c92d318469f551abc3e4545a2": "\\Delta w_{ji}=\\alpha(t_j-y_j) g'(h_j) x_i \\,", "fbddc5bc694fa14afa8607e6270d5068": "H_{c2}", "fbde1b12018bcf6f70457023e96008cf": "ds^2 = e^\\varphi (dx^2+dy^2). \\, ", "fbde80e4ebd96c9d322b5ca3d5811e80": " p < \\infty ", "fbde841f8429783d0123ae6c0f06d39b": " \\lVert [\\textbf{F} \\ast \\textbf{u}]\\textbf{v} \\rVert \\leq{\\sqrt{n}} ", "fbdeb2ca7f08c4539e8b529d91d9ae41": "\\alpha \\leq \\zeta_1 = \\phi_2(1)", "fbdec3d81c5c628dd7a0d1f131652145": " \\frac{\\partial u}{\\partial \\nu}(x_0) \\geq 0,", "fbdf16c4d9e061d64ffc4037a2616fb7": "\\rho \\rightarrow \\rho + \\frac{\\Lambda c^{2}}{8 \\pi G}", "fbdf3f437852bbc13a2fa354b151512c": "\\log \\frac{k_X}{k_0} = \\rho(\\sigma + r(\\sigma^- - \\sigma))", "fbdf613ad7fa7574db856b1e61e3cd12": "(1_A \\otimes \\delta) \\circ \\delta = (\\delta \\otimes 1_A) \\circ \\delta", "fbdf6e9b0fea97eb59709f6eb10452f4": " \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix} \\quad\\text{and}\\quad \\begin{bmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{bmatrix}. ", "fbdfa175c2b0297698bef85b79e2a9a8": " \\alpha\\circ \\phi(g) = \\psi(g)\\circ \\alpha ", "fbdfb012dda7779333d00635052e92b7": "\\mathbb{R}^{n-1}", "fbdfee029ab369c3ea800f54f2d37238": "v(0)=v(1)=0", "fbdff86685cc304aa10afe0c1f983601": "\\phi(t) = \\omega t + \\theta\\,", "fbe0189fa4d78aa20c79db5d19cccd14": "d_B(s, t)", "fbe01e3b58c22420638e76e86903720c": " t_0 ", "fbe01ff5bfadb7e7c0ce99f90a7f9800": "k\\ge j", "fbe0a628bafb5698687125d34c60d41b": "\\sum(-1)^{y}C^{y}\\,=\\chi(M)", "fbe0ad13927a04523d60ba53e79aad7b": " (\\mathrm{sys})^2 \\leq \\gamma_2\\,\\mathrm{area}", "fbe0b0bb96f4fc56a9221c7159accf85": " 0 \\to P\\times_G \\mathfrak g\\to TP/G\\xrightarrow{\\rho} TM \\to 0.", "fbe0c6875489b83ab2eaf9a22b06874d": "A_2 \\rightleftharpoons A_3 ", "fbe0fdd8768c30c83b1d9c4a66c63eba": "\\sum_{k=1}^nk^p=\\sum_{j=0}^p\\binom{p}{j}\\frac{B_{p-j}}{j+1}n^{j+1}.", "fbe124cbcdff2dfdf74217f6bd255765": "\n\\begin{align}\ng_{A+B}(t) & = \\log \\left(\\sum_i \\sum_j e^{t\\cdot(A_i+B_j)}\\right) = \\log \\left(\\sum_i\\sum_j e^{t\\cdot A_i}\\cdot e^{t\\cdot B_j}\\right) \\\\\n& = \\log \\left(\\sum_i e^{t\\cdot A_i}\\cdot \\sum_j e^{t\\cdot B_j}\\right) = \\log \\left(\\sum_i e^{t\\cdot A_i}\\right)+ \\log\\left(\\sum_j e^{t\\cdot B_j}\\right) \\\\\n& = g_A(t) + g_B(t).\n\\end{align}\n", "fbe12609e84069730268e902906e45f4": "\nK=\\prod^p_{j=1}\\lambda_j^{r_j}~,~~~~~P_j(y)=\\sum^{r_j}_{k=1} c_{j,k}\\,y^{k-1}\n", "fbe14dcad4fe1bcc1673af4e50435902": "f(k,i) = \\beta_{0,k} + \\beta_{1,k} x_{1,i} + \\beta_{2,k} x_{2,i} + \\cdots + \\beta_{M,k} x_{M,i},", "fbe1c1932b345d24c1ba3ad796ed68cb": "MA = \\frac{\\pi d_m}{l}", "fbe1d8f7995da14af2411299d1ef3c5c": "{{\\propto }_{1}},{{\\propto }_{2}},{{\\propto }_{3}},{{\\propto }_{4}},{{\\propto }_{5}}", "fbe1e022015c859d43f956307604095f": "\\Delta{z_i}", "fbe20c1511144660282f4de5126d5237": "A\\Rightarrow B", "fbe23aeb349c2b62e01bb7028927dfef": "\\scriptstyle\\sum_{k=1}^nx_k", "fbe24959f15160bc392fc8f6fcdeaa81": "D^q(B/A, M) = H^q(A, B, M) \\stackrel{\\text{def}}{=} H^q(\\operatorname{Der}_A(P, M)).", "fbe2bcfed4ba7b13771c1a9358570be8": " = \\eta^{ae} (h_{ed,bc}-h_{bd,ec}-h_{ec,bd}+h_{bc,ed})\n= h^a_{d,bc} - h_{bd,}{}^ a{}_c + h_{bc,}{}^a{}_d - h^a{}_{c,bd}", "fbe2d84eb6e3f1d8af9fce81eb203e56": "\\frac{dQ}{dt}=P=KQ\\left(1 - \\frac{Q}{URR}\\right) \\qquad \\mbox{(1)} \\!", "fbe308e01776c2c9e904d56b92b2b9c9": "\\tfrac{3K(3K+E)}{9K-E}", "fbe3531963a397870d03f614fa6bdadb": " \\xi \\sim (T-T_c)^{-\\nu} ", "fbe3708452e94e7d1644419f566c21f8": "E[\\hat{\\gamma}(h)]=\\frac{1}{2|N(h)|}\\sum_{(i,j)\\in N(h)}E[|Z(x_i)-Z(x_j)|^2]=\\frac{1}{2|N(h)|}\\sum_{(i,j)\\in N(h)}2\\gamma(x_j-x_i)=\\frac{2|N(h)|}{2|N(h)|}\\gamma(h)", "fbe37910c5542faef98193cba3e20bd9": "\\omega = \\sqrt{\\frac{{k}}{{M}}}", "fbe3afdd10c6f090c7fb1bdfca9a62d2": "\n\\omega^2 = 4\\Omega_0^2 + R{d\\Omega^2\\over dR}\\equiv \\kappa^2", "fbe3c35ee78fb7f92c72dbe9fc6a45fe": "\\frac{d}{dt}\\hat{\\boldsymbol{u}} = \\boldsymbol{\\Omega \\times}\\hat {\\boldsymbol{ u}} \\ , ", "fbe40528f2ffb2b99050a1c0ffae40da": " T(H) ", "fbe433b202496301b620d9413974d099": "\\rho_c", "fbe44f7c9b67117929203bf03b2d0500": "\n\\frac{d^2 x^\\lambda}{d s^2} + \\Gamma_{\\mu \\nu}^\\lambda \\frac{d x^\\mu}{d s}\\frac{d x^\\nu}{d s} = 0\n", "fbe46383db390907541a234bec7f2424": "EG", "fbe4c7d3ae829df7c78b7672d513f078": "\n\\log 2 = \\log (1+1) = \\cfrac{1} {1+\\cfrac{1} {2+\\cfrac{1} {3+\\cfrac{2} {2+\\cfrac{2} \n{5+\\cfrac{3} {2+\\ddots}}}}}} \n= \\cfrac{2} {3-\\cfrac{1^2} {9-\\cfrac{2^2} {15-\\cfrac{3^2} {21-\\ddots}}}}\n", "fbe51d73a234ff9190a345d9d00e00ec": "\\phi \\rightarrow (\\forall x \\psi)", "fbe553973c8f2630b5885a27da6823c2": " y = A \\left \\{ {{}_2 F_1}(\\alpha, \\beta; -\\Delta + 1; 1 - x) \\right \\} + B \\left \\{(1 - x)^{\\Delta} {{}_2 F_1}(\\Delta + \\beta, \\Delta + \\alpha; \\Delta + 1; 1 - x) \\right \\}", "fbe5bc64d0423d69bc56caf0b1d74e8e": "\n\\sum_n U^*_{nm} U_{np} = \\delta_{mp}\n\\,", "fbe5f71a2243be6e02a6d09deaecb4aa": "\\hat{x}\\in\\mathcal{X}", "fbe6202ab2b87af666e9e8fe62a0aaab": "\nD_{\\phi_{a}} \\left ({\\mathbf{\\rho}} \\right )\n= 6.88 \\left ( \\frac{\\left | \\mathbf{\\rho} \\right |}{r_{0}} \\right ) ^{5/3}\n", "fbe63f7680030b2c85d6ca60d34ea0cd": "(\\mathbf{A}, \\mathbf{b})", "fbe64d64a4932c6e32bd403d75db833f": "P=(T_1-T_2)v", "fbe72912b05a988a13d486f518f9ebe5": "R(t) = R_0\\frac{t}{t_0}.", "fbe78f6151b74736e853968c113644cb": "\\widehat{\\rm var}(X_j)", "fbe7b27f2903be5a50968d782f33faa1": "\\textstyle \\binom{n}{m}", "fbe7b7b625cd55d99d2b6d3c3b26cfd9": "1/a_k", "fbe7fd64608f9aa5bad6dafe586da5f7": " \\widehat a", "fbe80947722d255c6c4eeddb6a36e7c2": "E_1 \\vee E_2 = \\langle E_1\\cup E_2 \\rangle_{\\mathcal{A}} ", "fbe81ff4d8aef1ca4438710b3d726502": "\\,\n\\Delta E = v{2E \\over c}\n", "fbe8494b458b87e09b6d97195bb3ba4a": "\\bar{p}", "fbe85b8b4483d797f70b80d9f9796486": "p(\\boldsymbol{r}) = \\frac{1}{V}\\int_\\Omega p(\\boldsymbol{r}) \\,d\\boldsymbol{r}.", "fbe90efeaee7d6160b282fe64320fd82": "\n\\ \\ <^{d}\n", "fbe946a26ca84501198d9860ff6f9396": " M,w\\models\\square A \\Longleftrightarrow (A)^M \\in N(w), ", "fbe9859ff7fac59bfa4b40a89099f4ae": "\n\\mathrm{Subject to: } (\\mu_{ab}(t)) \\in \\Gamma_{S(t)} \n", "fbe9cfcb8d6ec30e276b41f2fe60b88e": "S_{21} = {-2 Z_0 Y_{21} \\over \\Delta} \\,", "fbea011fe8a55baf2bf82d726bb8c444": " 1\\leqslant j\\leqslant k", "fbea131558e62504da89d9a2a5d9a24a": "G = (\\{S, X, Y, A\\}, \\{a\\}, S, \\{r_1, r_2, r_3, r_4, r_5\\})", "fbea629ace8ecafe5afa989c7a63cd76": "\\lambda_i = \\sum_{j=1}^m p_{ji} \\lambda_j.", "fbeb1adf213982bb7dd38c3f62b6ee04": "\\frac{r - t}{d}", "fbeb24a8c78fdfe84752d87a22238210": "f(x) - P_n(x) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!} \\prod_{i=1}^{n+1} (x-x_i) ", "fbebf4f016ad9ae0212a4a4fbe5d5db1": "\\mbox{(reactive power)} = \\mbox {(apparent power)}\\sin(\\theta)", "fbec0266b8f19a92ff02c8581cb09d06": "df : \\mathrm{ker}(df)^{\\perp} \\rightarrow TN", "fbec96bbf14898e00dda0dbba531bf97": "r\\in\\Z,", "fbed20d5cf0856e44e9e7aff65a57f60": "U,V\\subset X", "fbed3f31eeb49bddb37ae30fc93ef4f0": " |\\psi\\rangle = \\begin{pmatrix} \\cos\\theta \\\\ \\sin\\theta \\end{pmatrix} ", "fbed791391d718d287bd32ba53bf27c4": "W' \\subseteq W", "fbed952af968335c2b6db02f2ca45110": "\\mathrm{Be} = \\frac{\\dot S'_{\\mathrm{gen},\\, \\Delta T}}{\\dot S'_{\\mathrm{gen},\\, \\Delta T}+ \\dot S'_{\\mathrm{gen},\\, \\Delta p}}", "fbee24d1d9c65b138a58af1901df8f9a": "x_k^{n-1}", "fbeed89f6b5e5cabb25b4e9232ef8c01": " V \\not \\in \\operatorname{FV}[\\lambda F.E] \\to \\operatorname{de-lambda}[\\lambda F.E] ", "fbeef1a19a8ce6fb78b72b441e4c7709": " \\mathrm\\varphi (\\mathbf r , t) = \\frac{1}{4\\pi\\epsilon_0}\\int \\frac{\\rho (\\mathbf r' , t_r)}{|\\mathbf r - \\mathbf r'|}\\, \\mathrm{d}^3\\mathbf r'", "fbef3610dfd61a73c9f4687720747caf": "R^\\prime ", "fbefa81c789d335136d7cc71ef67478a": "\\epsilon(\\sum_i n_i \\sigma_i)=\\sum_i n_i", "fbeff8ea79082c4cc72c148184722ea3": " Aminoacyl~AMP + tRNA \\rightleftharpoons {} aminoacyl~tRNA + AMP", "fbf025ebdb12c1fbc67ffff96fb7f72e": "f^{**}\\le f", "fbf02946e0b5b33e0a7c8628aebf988a": "E[u] = \\frac1{2} \\int_{\\Omega} | \\nabla u (x) |^{2} \\, \\mathrm{d} V,", "fbf0444be48566569badfcd6cfe825f1": "P^{-1}A = L U ", "fbf0714ae3c365b99c4e27504de6d4a8": "(1-c)^{-1}\\mathrm{OPT}", "fbf0b959e00abc1f0881dea2927eb1f7": " \\mathbf{A} = \\begin{pmatrix} A^0 \\\\ A^1 \\\\ A^2 \\\\ A^3 \\end{pmatrix} ", "fbf11a36e39506f36b4bb6d08883964c": "((2k-1)^2)_{k=1}^\\infty", "fbf1396cd398a6b71c21e60b25f17b68": "\\,\\!0", "fbf17b3450358daaccb2d61405300052": "Z_{-1}^{p,q} = Z_0^{p,q} = F^p C^{p+q}", "fbf1b5def6db43afafd69c89b064defc": "\\begin{bmatrix}\nw_{11} & w_{12} & \\cdots & w_{1n}\\\\\nw_{21} & \\ddots & & \\vdots \\\\\n\\vdots & & \\ddots & \\vdots \\\\\nw_{n1} & \\cdots & \\cdots & w_{nn} \\end{bmatrix} \n\n\\begin{bmatrix}\n\\Gamma_{1} \\\\\n\\Gamma_{2} \\\\\n\\vdots \\\\\n\\Gamma_{n} \\end{bmatrix}=\n\n\\begin{bmatrix}\nb_{1} \\\\\nb_{2} \\\\\n\\vdots \\\\\nb_{n} \\end{bmatrix}\n\n ", "fbf1e05550869d7b5e46413dc38f19d9": "\\delta: Q \\times \\Gamma \\rightarrow Q \\times \\Gamma \\times \\{R\\}", "fbf21de5539595f0d445d504d96c2712": "p \\in \\mbox{Prop}", "fbf240c3d2775d4f8df1c5922d2ae707": "\\int_0^a x^n\\,dx = \\tfrac{1}{n+1}\\, a^{n+1} \\qquad n \\geq 0,", "fbf245f57719c25ce385ac302888c1a3": "(X,\\tau)\\,", "fbf294e241ce2f1951395f388b788369": "\\frac{1}{2^{ck(n)}} \\le \\frac{1}{3k(n)m'(n)}.", "fbf29c161428aa651110db2c6230f2b9": "[ABC]", "fbf2a8e791c3e43b83e5de25c62a5c0e": "\n \\overset{\\square}{\\boldsymbol{\\sigma}} = \\dot{\\boldsymbol{\\sigma}} + \\boldsymbol{\\sigma}\\cdot\\boldsymbol{\\Omega}\n - \\boldsymbol{\\Omega}\\cdot\\boldsymbol{\\sigma} \n", "fbf3207c54532edccbd425042ccd5c62": "\\sin\\frac{\\theta}{2}", "fbf325b93bdfcdb15c147cf9d7b5edbf": "(\\Delta_\\gamma\\phi)(v)=\\sum_{w:\\,d(w,v)=1}\\gamma_{wv}\\left[\\phi(w)-\\phi(v)\\right]", "fbf36a58c19972046d4ffd41e4b5457b": " \\emptyset=X^{-1}\\subset X^0\\subset X^1\\subset \\ldots \\subset X ", "fbf3afe5db97a5e39ed8c9833f64cbb4": "c = \\sqrt{ (z/2)^2 + (z/2)^2 } = z/\\sqrt{2}\\,", "fbf3caecbd3fcda84cd3730a217224a8": "H = \\frac{p_1^2}{2m} + \\frac{p_2^2}{2m} + U(|x_1 - x_2|) + V(x_1) + V(x_2) ", "fbf429528e8cacec1b95b904ee01af02": "p= 2\\pi n /L", "fbf4443f74cdcf78b873dc8e3ed0824e": "\\phi_{APB}=\\phi_{AP}\\phi_{PB}", "fbf45a9007dc18a1c9e0ef62150a06aa": "_{metric} \\delta_{cc}^2 = _{metric} \\delta_{kk}^2 = 0", "fbf480e6e012053ebe11c71c57b7828c": "\\begin{bmatrix}\n 9 & 13 & 5 \\\\\n 1 & 11 & 7 \\\\\n 2 & 6 & 3\n \\end{bmatrix}", "fbf4b00ffbde55ef0abdceb9f2fc0c60": " l(x) = \\sum_{i=0}^n a_i x^i \\ ", "fbf4c0e9fbd407dedb4e2edad1fdd5bc": "\\phi_R \\to 90^{\\circ} = \\pi/2^{c}", "fbf4d0f0c2fe2b20b3f1bc6d72b9ba5b": "\\boldsymbol{B} = - \\left(\\frac{1}{3}\\frac\n{\\boldsymbol{\\omega}}{r^3}\\cos\\theta\\right).", "fbf4eadbfcb00a3ad42380dea4434447": " f(t)\\star g(t) ", "fbf51caf31f5aefd6f4adcea411e68ee": "\\;", "fbf52b46a01060468dc30028ecbef512": " (\\log{\\lambda}) \\, \\sqrt {\\frac {n \\, d} {4}} ", "fbf5446489322650cffdb533ba05f7a0": "\n\\Delta g\\ =\\ \\int\\limits_{0}^{2\\pi}\\left(\\frac{\\partial g }{\\partial v_1}\\ h_1\\ + \\ \\frac{\\partial g }{\\partial v_2}\\ h_2\\ + \\ \\frac{\\partial g }{\\partial v_3}\\ h_3 \\right)\\frac{r^2}{\\sqrt{\\mu p}}d\\theta\n", "fbf570d80b86dfe36728d70b06012a3a": "\\frac{T_2}{T_1} \\approx\n \\frac{2\\gamma(\\gamma-1)}{(\\gamma+1)^2}M_1^2\\sin^2\\beta.", "fbf5ccff55db252b9feb73fbe1bc8ba6": " Z = \\sum_{ \\pm , n} \\mathrm{e}^{- \\beta E_{\\pm}(n)} \\approx \\sum_{ \\pm} \\int \\mathrm{e}^{- \\beta E_{\\pm}(n)} dn=\\int \\mathrm{e}^{\\Phi (n)} dn", "fbf6064195882e79f5331404bcd101de": "\\vec{f}_2 = \\vec{e}_2", "fbf6144356419e72095299d25c361088": "\\left(\\sum_{d \\in D} r(d) \\cdot y\\right) \\vert (w(1) \\cdot z)", "fbf6624a78a75bab91babae294150cfb": " A' = -\\ln \\left( \\frac{I}{I_0} \\right).", "fbf668c989f4c01f256df9948afadf24": "l_2", "fbf68fb3a9eb63cfe1bb32ba99790ccd": "H_d=K\\sum_i\\mathbf{S}_{1i}\\cdot\\mathbf{S}_{2i}+J\\sum_{\\langle ij\\rangle}\\left(\\mathbf{S}_{1i}\\cdot\\mathbf{S}_{1j}+\\mathbf{S}_{2i}\\cdot\\mathbf{S}_{2j}\\right)", "fbf6b05fad4f9f5799b7a52af72fcb67": "\nI=I_0 \\exp \\left (\\sum_i \\int \\rho_i \\beta_i \\, ds \\right )\n", "fbf6ce59932f7821aa6f8dce5ebe8ba7": "\\bar{X}_n = 50", "fbf6fa3d84b200725048c414323ddb93": "w_3\\in\\Sigma", "fbf70ec51ff8b4d667ec3065076f596b": "\\sigma:\\kappa^{+}\\to\\mathcal{P}(\\kappa)\\,", "fbf72fdafbf16d3bc88e12a13b531e6d": "addOne", "fbf7a3437a32ee2033f2bd9798bbf629": "\\text{so does }ax^2 + by^2 + cz^2 \\equiv 0 \\pmod{4abc}. \n", "fbf7b621fd0fc03cc4c7c5fb3fb0c441": "\\textstyle\\sum_{m,l} s(m,l)q^mz^l", "fbf81ac91272aa294bc28496a3540250": "\\sum_{i=1}^n \\frac{1}{i} = H_n", "fbf81f5f3e0ea7dc12513e0fbb9594ae": "y'(\\phi) = R\\sec\\phi,", "fbf82fd7685bcbf3130ffa594f94cac4": " \\int_{x_1}^{x_2} \\eta \\left(\\frac{\\partial L}{\\partial f} - \\frac{d}{dx}\\frac{\\partial L}{\\partial f'} \\right) \\, dx = 0 \\, . ", "fbf855d2ab664ea1cf48b0d302a6fbed": " 1 - \\prod_{i=1}^N \\left( 1 - \\frac{1}{p_i} \\right). \\qquad (3) ", "fbf897769c38e6b54a75c32dae2e0671": "{q^2 \\over g} \\left({1 \\over y_1} - {1 \\over y_2}\\right) = {1 \\over 2} ({y_2^2} - {y_1^2})", "fbf8d776204347481f8f2d5923e5b1f7": "\\int_A \\left|f(x)\\right|\\,dx < \\infty.", "fbf9009245539dfde6e69f0a7bc28e7f": "e^\\pi", "fbf9132a5b9bb15dab0031810d1226a6": "\\ \\ln(k_\\mathrm{B}/h) + \\Delta S^\\ddagger / R ", "fbf933e524cba5f7a81e48fe6fb76967": "n=u \\cdot p_1^{e_1} \\cdots p_k^{e_k},", "fbf940e60b3b6a6b052be7c3b221a79a": "\\mathit{dr}(n)=9 \\Leftrightarrow n=9m \\ \\ \\ \\text{for}\\ m=1,2,3,\\cdots.", "fbf99645da9a7041bfc8aba781d80d63": " \\frac{1}{Z_{\\text{GUE}(n)}} e^{- \\frac{n}{2} \\mathrm{tr} H^2} ", "fbf9b2a052c024517653ce8a5089e47e": " (D-CA^{-1}B) = ", "fbf9c6f6e04f755040e817a846eae16f": "\\lim_{N \\to \\infty} S_N f\\left(-\\frac{2\\pi}{2N}\\right) = -\\frac{\\pi}{2} \\int_0^1 \\operatorname{sinc}(x)\\ dx = -\\frac{\\pi}{4} -\n\\frac{\\pi}{2} \\cdot (0.089490\\dots).", "fbf9ca61b1f5f39a6754ec9e527162c5": "1 \\leq i \\leq d", "fbfa0943fbc9306be7d324d8b7b1ad97": "\\sum_{n=1}^\\infty q^n \\sigma_a(n) = \\sum_{n=1}^\\infty \\frac{n^a q^n}{1-q^n}", "fbfa26c3ed0715b47183f3b10c3eef56": "R = \\frac {\\lambda}{B} ", "fbfa54b28c0ff621ff4db4c2d5ebf6c6": "\\vert \\vert", "fbfa633301b98a77c44cd8ed7153879d": " g_\\mathrm{e} ", "fbfa7d7dd94de1e0156b1e5a7c051b22": "\\scriptstyle \\Pi_B", "fbfa9492b4856b03d86027c1f67e7a47": " \\mathbb{F}_{q} ", "fbfacf187bff89e76a0d27a515db44e5": "\nr (\\mathbf{h},q)=\\prod_{j=1}^k r(h_j,q)\n", "fbfb0740dcf86a8534e741f0a4fe9ae2": "\\tau=-E'\\frac{v^{2}}{c^{2}}\\sin2\\alpha'", "fbfbb55c8f070b5b67ffd1a7db031146": "\n\\begin{align}\n& \\mathbf x_0 := \\text{Some initial guess} \\\\\n& \\mathbf r_0 := \\mathbf M^{-1}(\\mathbf b - \\mathbf{A x}_0) \\\\\n& \\mathbf p_0 := \\mathbf r_0 \\\\\n& \\text{Iterate, with } k \\text{ starting at } 0: \\\\\n& \\qquad \\alpha_k := \\frac{\\mathbf r_k^\\mathrm{T} \\mathbf A \\mathbf r_k}{(\\mathbf{A p}_k)^\\mathrm{T} \\mathbf M^{-1} \\mathbf{A p}_k} \\\\\n& \\qquad \\mathbf x_{k+1} := \\mathbf x_k + \\alpha_k \\mathbf{p}_k \\\\\n& \\qquad \\mathbf r_{k+1} := \\mathbf r_k - \\alpha_k \\mathbf M^{-1} \\mathbf{A p}_k \\\\\n& \\qquad \\beta_k := \\frac{\\mathbf r_{k + 1}^\\mathrm{T} \\mathbf A \\mathbf r_{k + 1}}{\\mathbf r_k^\\mathrm{T} \\mathbf A \\mathbf r_k} \\\\\n& \\qquad \\mathbf p_{k+1} := \\mathbf r_{k+1} + \\beta_k \\mathbf{p}_k \\\\\n& \\qquad \\mathbf{A p}_{k + 1} := \\mathbf A \\mathbf r_{k+1} + \\beta_k \\mathbf{A p}_k \\\\\n& \\qquad k := k + 1 \\\\\n\\end{align}\n", "fbfbbcd5058c7f6e35ba23fff7b84acd": "f_{x}=f'_{x},\\ f_{y}=f'_{y}\\cdot\\sqrt{1-\\frac{v^{2}}{c^{2}}},\\ \\tan\\alpha=\\tan\\alpha'\\sqrt{1-\\frac{v^{2}}{c^{2}}}", "fbfc0d97ca0d5c2c817d887b4f4068a2": "b_{1}-b_{15}", "fbfd1c7f65282297089334e14df4a55b": "[N_i,P_0=\\eta]=0", "fbfd73ca4cfd34898e00ed883ece57a2": "c=C(m)", "fbfd7f6ae7377c3e7a2618155d5cdd08": "\\hat{t} = \\tfrac16\\, t, \\,", "fbfdc13933ff860f3fcf66afef28cc84": "p = p_0 \\cdot \\left(1 - \\frac{L \\cdot h}{T_0} \\right)^\\frac{g \\cdot M}{R \\cdot L} \\approx p_0 \\cdot \\left(1 - \\frac{g \\cdot h}{c_p \\cdot T_0} \\right)^{\\frac{c_p \\cdot M}{R}} \\approx p_0 \\cdot \\exp \\left(- \\frac{g \\cdot M \\cdot h}{R \\cdot T_0} \\right),", "fbfdc345c300cfcbbe2ccdd72ea0ec6b": "\\scriptstyle V(x)", "fbfdd0f9e1e5bad1c234b8cef524b14c": "n>m", "fbfe05115d0e07bee516cef4edad5d13": "f_\\mathrm{rest}\\sqrt{\\left({1 + v/c}\\right)/\\left({1 - v/c}\\right)}\\times \\left(1 - v/c\\right) = f_\\mathrm{rest}\\sqrt{1 - v^2/c^2}\\equiv\\epsilon f_\\mathrm{rest}", "fbfe0528e744b9132cdd7209af626e5d": "\\Omega=d\\eta+\\tfrac{1}{2}[\\eta\\wedge\\eta].", "fbfe9ab11efbc0313ef786cce915318a": "\\lim\\limits_{N\\rightarrow\\infty}\\int\\limits_\\Omega f\\text{div}\\theta^*_n =\n\\lim\\limits_{N\\rightarrow\\infty}\\int\\limits_\\Omega\\mathbb I_{\\left[-N,N\\right]}\\nabla f\\cdot\\frac{\\nabla f}{\\left|\\nabla f\\right|}=\n\\lim\\limits_{N\\rightarrow\\infty}\\int\\limits_{\\mathbb I_{\\left[-N,N\\right]}} \\nabla f\\cdot\\frac{\\nabla f}{\\left|\\nabla f\\right|} = \\int\\limits_\\Omega\\left|\\nabla f\\right|\n", "fbfead13d673402552a8b0492e89152d": "\\frac{d}{d t}(e^{i \\omega t}) = i \\omega e^{i \\omega t}", "fbfeb9c8459fee5a2bd529c07b881153": "m \\times n", "fbfef14b89448b673bd01695e43e368b": "f(i)={1\\over 2\\pi^2} \\int_0^\\infty \\tilde{f}(\\lambda) \\lambda\\, d\\lambda \\int_{-\\infty}^\\infty {\\sin \\lambda t/2\\over \\sinh t} \\cosh {t\\over 2} \\, dt ={1\\over 2\\pi^2} \\int_{-\\infty}^\\infty \\tilde{f}(\\lambda) {\\lambda\\pi\\over 2} \\tanh({\\pi\\lambda\\over 2})\\, d\\lambda.", "fbff21431ea0e7e5426c3e89953d95eb": "\\Delta x^j", "fbff3f6e4f9cbe099b15170b9c1175cf": "\\sin(\\alpha)\\simeq \\tan(\\alpha)=y/L", "fbffe8fe179232e5b0c6daaf91bda640": "R[S^{-1}]-Mod \\to R-Mod.", "fc0009413978dd0d8b8b651a620f5acd": "S(\\vec{r})=\\delta(\\vec{r})", "fc00155198ae92e33628bd53040d2506": " {\\rm det}\\, (I+ A) = \\sum_{k=0}^\\infty {\\rm Tr} \\Lambda^k(A) ", "fc004eb836b0b38b1adb0271956e2ef9": " T_2 = {{m_2 g (2 m_1 + {{I} \\over {r^2}})} \\over {m_1 + m_2 + {{I} \\over {r^2}}}}", "fc008d6a11aa69b691ec0febbb300949": "^6", "fc0137dbfa4d4970f4a855fb6af8a227": "B = \\frac{k_1 - k_2}{2}", "fc01496318ce7d02eba31b51d62ce7f4": "\\mbox{Hess}(f)(X,Y)=X(Yf)-df(\\nabla_XY)", "fc0252338f2ae10d4a8b7eefaed7383b": "\\and \\!\\,", "fc025a027c5274f503a75d9c2c6aee84": "0 \\leq y \\leq x_F-u", "fc026c9ca4bc10f7341e45e86b90636a": "\\partial (f,x)", "fc02a0cdeab974b034eed590fd2e86c1": "\\sum_{n = 0}^\\infty \\frac{1}{2^n n!} = \\sum_{n = 0}^\\infty \\frac{1}{(2n)!!} = \\frac{1}{1}+\\frac{1}{2}+\\frac{1}{8}+\\frac{1}{48}+\\cdots", "fc03109c1c0c52ae90e27937d26a30db": "= 768 + 176 + 2", "fc031872d2dba1d452ab1144dc6f61e7": "\\pm 1/2", "fc0332e5e673f4c24fe8fa01e180dc40": "\\neg L_i \\, ", "fc034a02cfd4a8c481eadec725c69d6e": " \\mathfrak{so}(2,\\mathbb C) \\cong \\mathfrak{gl}(1,\\mathbb C)\\qquad(=\\mathbb C)", "fc0383073795a66abfd4ea347de25b18": "3^\\frac{11}{13}", "fc03af90731a864a940c92d2dd71e946": "\\boldsymbol{\\tau}=\\mathbf{m}\\times\\mathbf{B} = \\mu_0\\mathbf{m}\\times\\mathbf{H}, \\,", "fc043e1b7d15f6d0a723a74ad405409f": "\\mathbf{x}\\ \\sim \\mathcal{N}(\\boldsymbol\\mu, \\boldsymbol\\Sigma) \\iff \\mathbf{x}\\ \\sim \\boldsymbol\\mu+\\mathbf{U}\\boldsymbol\\Lambda^{1/2}\\mathcal{N}(0, \\mathbf{I}) \\iff \\mathbf{x}\\ \\sim \\boldsymbol\\mu+\\mathbf{U}\\mathcal{N}(0, \\boldsymbol\\Lambda).", "fc0448ee84fca3735cc74d584b0dd83f": "\n\\psi_t(y) = \\int \\psi_0(x) K(x-y;t) dx = \\int \\psi_0(x) \\int_{x(0)=x}^{x(t)=y} e^{iS} Dx\n\\,", "fc046d615217650e9f2fe3e39cdfce1c": "(p, \\hat n)", "fc04cd9ecd5d476fc0dde84c4563e951": "ACG^*", "fc04d27f77eb9066dbf216722bef572c": "\\left(\\frac{g}{f}\\right) \\left(\\frac{f}{g}\\right) = \n(-1)^{\\frac{q-1}{2}(\\deg f)(\\deg g)}.\n", "fc04f8fcae3a4baca578195e73637d85": "\\sum_{n=1}^\\infty (-1)^{n-1} a_n\\!", "fc05233e043525280099bd777174f26e": "s \\notin \\alpha, t \\notin \\gamma", "fc0553aa748533b34276270e17815068": "R[t, t^{-1}]", "fc05c6ddbe5382ca19ec283c22149c65": " \\mathbb{Z}[x]/(x^n-1)", "fc0650dbea55477d92acd612323708e1": "\\iota : X \\rightarrow (X\\sqcup X^\\dagger)^+", "fc070e7ec8bf259cd51412cf9c939444": "\\operatorname{Tr}(\\bar Q \\sigma) ~=~ \\epsilon - \\mu + (1-\\epsilon + \\mu)\\operatorname{Tr}(Q\\sigma) ~.", "fc072ad3e6cd345a0a489ed00fca2e03": "p(\\varphi)=0", "fc07455d3f10df049945a94a30b1dd18": "e_k=", "fc0751120467e7161ba7cf401c5c53b5": "P^*(T')=T^{2m}P(\\frac 1 T)", "fc077be94f1279bf381665d38ceab821": "a_{j,k}", "fc07b5f52a1b7059bae236510fd62683": "\\Delta (mv)(w) = \\rho(Q)(v_2 - v_1)", "fc07e802179b4edcf33bebf7fd28acd6": "\\ V_{eff, RG}", "fc084d69d5be8ab77ab7c22c1176ae3c": "M_2\\,", "fc0887a50237be4ef689188e156a0ed2": "\\phi (r) = \\frac{Q}{4\\pi\\epsilon_0 r} e^{- k_0 r}", "fc089489555aee7519548ff7f589f384": "TD", "fc08b38622a4299f0951b6d0dd3af264": "\\scriptstyle X \\;<\\; 0", "fc08ebe872d526a993aec80cc848d1b9": "\\sum_{n=0}^\\infty \\zeta(2n) x^{2n} = -\\frac{\\pi x}{2} \\cot(\\pi x) = -\\frac{1}{2} + \\frac{\\pi^2}{6} x^2 + \\frac{\\pi^4}{90} x^4+\\frac{\\pi^6}{945}x^6 + \\cdots", "fc08edc820c45aa25810f308fa8db819": "\\mathrm{div} \\, \\mathrm{grad} \\, \\varphi\\, = \\nabla \\cdot (\\nabla \\varphi)", "fc08fea11c08b90369df4108735d4633": "\\alpha_{\\rm L} = \\alpha_{\\rm B}=1\\;", "fc091ac6dbbe20da9627826b7de0c317": " f_i(\\Delta(n,d)) = \\binom{n}{i+1} \\quad \\textrm{for} \\quad\n0 \\leq i < \\left[\\frac{d}{2}\\right] ", "fc092d2bef5a3174a73684d6107be911": " Hb_{CO}(%) = \\frac {CO - 2.34} {5.09} ", "fc0954e53ea1514410c3107525a2e628": " q + \\sqrt{q} + 1 \\leq |B| \\leq q^2 - \\sqrt{q}.", "fc095a6c84da481fdf723d8e1eedd5a7": "\np^{}_{}=\\frac{1}{k}\n", "fc096a7410cd582339517c3db5897b13": "\nn_t = \\frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}.\n", "fc0998ad0fb9bd35cd5b184231df8428": "\nH_2 = \\begin{bmatrix}\n1 & 1 \\\\\n1 & -1 \\end{bmatrix},\n", "fc09ba3a4d486420300c6af550144215": "\\tilde V = V \\times \\mathbb{Z}_{n}", "fc09ed8fcde1ee0c50a7cb64fd616edd": "n^2/K", "fc09ff24d21efa6123b8b645ab6e0ced": "\\mathbf{N_{B}} = \\begin{bmatrix}\n0.82087 \\\\\n0.08910 \\\\\n0.12870 \\\\\n0.03842 \\\\\n0.01062 \\\\\n0.04962 \\\\\n0.02155 \\end{bmatrix}", "fc0a3f3db3369e0c089d8fd59c73abd2": "\n N_{Ij} = S_{IK}~F_{jK} \\qquad \\text{and} \\qquad P_{iJ} = F_{iK}~S_{KJ}\n", "fc0a90c3c2a4f768474524911edf6148": "{{V}_{T}}=\\frac{kT}{q}", "fc0aa0219c6558ac6eb03ca78039c800": "\n \\begin{matrix}\n \\underbrace{2_{}^{2^{{}^{.\\,^{.\\,^{.\\,^2}}}}}}\\\\\n 65536\\mbox{ multiplied copies of }2\n \\end{matrix}", "fc0aa50815702db722b13971e032cc26": "A:=e^*T^tG", "fc0ad7ea8ef18de47988dae45a8430c0": "\\text{ACE} = 10^{-4} \\sum v_\\max^2", "fc0b490cb87b78c56ab16597d739f358": " M = V \\cup_f W. ", "fc0bf2c08722be11c5b4379a30a2e9c6": "\\frac{\\eta}{E}", "fc0c0029df90d995bd367c337e72df94": "", "fc0c02c3af7324d5a4f9fe947f88f439": "\\rho\\le|z|\\le2\\rho", "fc0c6620e76f7e3b28917626cc6c3170": "G =(N, T, M, S)", "fc0cae9870d05af1b9d149a3d5b5b716": "\\rm{dHR = HR \\left ( \\frac{BFP-dHB-dBB-dK}{BFP-HB-BB-K} \\right )} \\,", "fc0cbd3d2f3240a359225706983c537d": "cx^2+dy^2=x. \\,", "fc0d09775f6dbdfc667aa8dea3b94828": "F({\\bold x}^{(1)},{\\bold x}^{(2)}) = x^{(1)}x^{(2)}+\\frac{3}{2}x^{(2)}y^{(1)}+\\frac{3}{2}x^{(1)}y^{(2)}+2 y^{(1)}y^{(2)}.", "fc0d1b762f45548d4d08f760e4743863": "2(1/1!)\\pi^1 = 2 \\pi ", "fc0d232ab4c5985759d1fdab1eef3083": "x_{k+1} \\leftarrow x_k + \\alpha_k \\cdot p_k\\,", "fc0d6fae220450c4e8eb79c7b7ad576a": " \\frac{1}{\\sqrt{2\\pi}x} ", "fc0dab8841d6156bac4e6c0226292018": "\\Delta=\\frac{\\delta C}{\\delta S}\\Rightarrow \\delta C = \\Delta \\times \\delta S.", "fc0e1283428eb31676bdded9703922c5": "[0,\\omega]", "fc0e15ac9a819074032ca612b7430d14": "\\left(\\frac{{T}_{2}}{{T}_{1}}\\right)=\\left(\\frac{{V}_{1}}{{V}_{2}}\\right)^{(\\gamma-1)}", "fc0e1924941b5ba0b77b854bf581c059": "x_{n-i+1}x_{n-i+2}...x_n", "fc0e3305e9dac32eeb594bc889bc52c4": "\\begin{align}\nF_{k\\ell} & = (\\eta_{ k i} \\eta_{ \\ell 0} - \\eta_{ k 0} \\eta_{ \\ell i} ) F^{i 0} + \\eta_{ k i} \\eta_{ \\ell j} F^{i j} \\\\\n& = 0 + \\delta_{ k i} \\delta_{ \\ell j} F^{i j} \\\\\n& = F^{k \\ell} \\\\\n\\end{align}", "fc0ea096fee32226e938417ed7d8ce99": "T(x)=\\sum_{i=0}^{d-3}t_{c+i}x^i=\\alpha^{k_1}\\sum_{i=0}^{d-3}s_{c+i}x^i-\\sum_{i=1}^{d-2}s_{c+i}x^{i-1}.", "fc0ee6710f9b3ca2af73f0eff56d3211": "(\\omega_3,\\omega_2)\\twoheadrightarrow(\\omega_2,\\omega_1)", "fc0f559b502e3b0eef2cf63c1f57616c": "x(t) = y(t)", "fc0f7c59ce3ea7507903ef1a82b85d73": "\\ \\displaystyle \\tilde{u}=0 \\ ", "fc0fbb134f1d994940152a3d4f9cef49": "V_{\\alpha+1}", "fc0fc50ece7b5d740fe47ae1918ab675": "\n\\begin{bmatrix}\nt' \\\\ -vt'\n\\end{bmatrix} =\n\\begin{bmatrix}\n\\gamma & \\delta \\\\\n\\beta & \\alpha\n\\end{bmatrix}\n\\begin{bmatrix}\nt \\\\ 0\n\\end{bmatrix},\n", "fc100ca15cb98d42812bc326fa88de8b": "\\dim_{\\mathrm{P}} (S) = \\overline{\\dim}_\\mathrm{MB} (S).", "fc106d78a5046f5cf0f0eaa2e9c3c50b": "G(Y') = G(Y) \\oplus R", "fc10738cac4f903f77dcb4188061b72b": "=2r_1\\tan(\\varphi) + 2r_2\\tan(\\varphi) + (2\\pi-2\\varphi)r_1 + (2\\pi-2\\varphi)r_2 \\,\\!", "fc10e205032e982a72b91e4c3f6b5de8": "\\mu + \\beta\\,\\gamma\\!", "fc11e03898b35b47511a42c4d155dccd": "\\pi:S' \\to S \\,", "fc11e445a3a60409e600ac0543e3ad84": "g,h \\in G", "fc124eaeba7268fe2bb9ce8c7e672089": "\\epsilon b = \\frac{4 \\pi ^2Hb}{2gT^2tan^2 \\beta},", "fc128b15a8df9a192bb3cb3084ca785e": "V_{L2}=V_o+V_C", "fc12aba0b47d894f615fbf840892fafc": "\\mathcal J_{9-p}=d^2C_{7-p}=dG_{8-p}=dF_{8-p}+H\\wedge G_{p-1}.", "fc12f4f8d24169701abe216073e4f31a": " J_f(x) = 2\\, x", "fc130ac60ecb30e1fd7eebacb9d5b3fd": " + F(1+i)^{-n} ", "fc1369761523c985d15341e3681cdb4e": "d = \\frac{\\bar{x}_1 - \\bar{x}_2}{s}.", "fc13b31bdcc1c58007f143cd5bfbdc7f": "e_k(X_1,\\ldots,X_n)=m_\\alpha(X_1,\\ldots,X_n)", "fc140fb83bf2f92008a879c1068af15d": "\\mathbf{y}(t) = C \\mathbf{x}(t) - D K \\mathbf{y}(t) + D \\mathbf{r}(t)", "fc14f7915afa66d10f35d52ce5cfdd96": " \\frac{\\partial \\Psi}{\\partial t} = - i \\frac{E}{\\hbar} \\Psi \\,\\!", "fc153ba166e24fdcc34214b4df558717": "I=\\N", "fc15c37cb256cd2d8380a9453b2eef77": "\\rho \\le v_0 - v_2 = 2 ", "fc160e674cb63b9b428b722a49fa44a0": " \\alpha(\\gamma)= 2(sinh^{-1}(\\gamma)-\\frac{\\gamma}{\\sqrt{1+\\gamma^{2}}})", "fc162f4a1b4fc3f4f0cebc3e4d08405d": "f_2(x)=f(x-a)", "fc16397554f321349e9aa21e28976137": "A_b", "fc16c1f816b59ddf0d565daebb35f275": "Q(a)v= (v\\cdot a) a", "fc16c719a6e2de220a96fa02a26cc317": "R{(p)} = {\\alpha\\over N} + (1 - \\alpha) \\sum_{j\\rightarrow i}{1\\over N_j}x_j^{(k)}", "fc16f7834afbd7ab54e1c16ccf5d41f8": "F[x,y]= \\frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}", "fc16ffe6e74885904c0ca1a9cce0856f": " [X_i]=\\begin{bmatrix}\n 1 & 0 & 0 & a_{i,i+1} \\\\\n 0 & \\cos\\alpha_{i,i+1} & -\\sin\\alpha_{i,i+1}& 0 \\\\\n 0 & \\sin\\alpha_{i,i+1} & \\cos\\alpha_{i,i+1} & 0 \\\\\n 0 & 0 & 0 & 1\n \\end{bmatrix}.\n", "fc174388b3e6e6cf5b909d460a2df553": "f(3)=-123\\,", "fc1744fbe69ef09066adf9cb050efed8": "M(n,t) \\leq B_{n-2t-1}.", "fc175ee1e1bd69f48087075abf928828": " G(T) = \\frac{1}{2 \\pi \\hbar} \\int_0^{\\infty}{ \\left(\\frac{N_1(\\omega)}{1+L/l(\\omega)}+\\frac{N_2(\\omega)}{1+L/d}\\right)\\frac{\\hbar^3 \\omega^2}{k_B T^2} \\times \\frac{e^{\\frac{\\hbar \\omega}{k_B T}}}{{(e^\\frac{\\hbar\\omega}{k_BT}-1)}^2} } ", "fc1769507e3708ea1d923a318e62d854": "\\mathbf{x}=(2,\\,2)", "fc17726fe0b82a34b8d0cabbe8eb8fc6": "\\int_0^t H_n dX \\to \\int_0^t H dX, ", "fc17f80ebc1e9c094c1ae8ff83d76186": "\\Iota = \\frac{\\nabla T_{m}}{\\nabla T_{crit}} ", "fc17fbec6fed45c69e389dad7d43c4eb": "\\mathfrak{P}^{63}", "fc1822b032ab42c47ebf6665fe0d5f4e": "k_{-1}=0\\,", "fc183cd492a6838f57b8b08b9621312f": "\\overline{A}D", "fc18b663fd38bffca5b5da6bbeb44863": "f \\left( x \\right) =\\int _{-1}^{1}\\! \\left( -15\\,{x}^{2}{y}^{2}-3\\,xy \\right) h \\left( y \\right) {dy}+h \\left( x \\right)\n", "fc18e97a778b5fb071dc6062592c5a5f": "\\mathbf{\\underline{E}}(x,z,t)=\\mathbf{\\underline{E}}^{TE/TM} \\left( e^{j\\mathbf{k}_1 \\cdot \\mathbf{r}} + e^{j\\mathbf{k}_2 \\cdot \\mathbf{r}} \\right) \\cdot e^{-j \\omega t}", "fc197aac7ae612c75db8ee0514ffa586": "\\max_{1\\le k\\le n}\\big|w_k\\big| \\le \\frac1{5n} \\min_{1\\le jp,p,e)", "fc1aec94b122f6d390a14468a51f6766": " n \\geq p ", "fc1af9cf1d59c9ff6a4cf2c6a3fccc7c": "\\frac r R + \\frac D R = 1", "fc1affcf0169c3642f243646a49c6ca0": "\\eta_1 = \\mathrm{\\tfrac{u\\bar{u} + d\\bar{d} + s\\bar{s}}{\\sqrt{3}}}", "fc1b01ae048146d5b92737cd8ba72dad": "\\phi = \\phi^{\\Rightarrow x}\\,", "fc1b79d2930a0461f76b31d39d8980bc": "\\hat{e}_{\\phi}", "fc1b917a3d63d1090c64a8fea35bcd0c": "\\cos(C) = \\mathbf{t}_a \\cdot \\mathbf{t}_b = \\frac{\\cos(c) - \\cos(a) \\cos(b)}{\\sin(a) \\sin(b)}", "fc1bc052609c894a482032c028884f39": "f:U \\rightarrow \\mathbb{R}", "fc1bc0e4be2eaacb8a5ea37e7acedecb": "\\sum_{j=1}^n X_j=k,", "fc1bffdd3ccea8e602b27341bef45763": "\\ p_{X,Y}(x,y)=p_{Y|X}(y|x)\\,p_X(x) ", "fc1c02f332a64f34f4afb8792b0642c6": " y_{C}(x) = e^{ax}(c_{1} \\cos bx +c_{2} \\sin bx ) \\, ", "fc1c02fe15303e824ea8fcbab0e0b373": "P(e)=P(X\\neq Y)", "fc1c61f0b7900ac281f1f4d15a2404c1": "\nL = \\sum p_i A_i,\n\\,", "fc1c75e362d37d6c594fa158f411b94d": "\\lim_{n\\rightarrow\\infty}H^k_n \\, ", "fc1c8a1a448b4522be8b0a4cb3a2d95f": "48^2", "fc1d17611055e8c2c3f7bf7e764bffd1": "A_0(p,r)=1", "fc1d5da5aae8e29ad47fbb82be896d90": "P(x_0) = 1", "fc1d6110fa493cd80d2ce301ea537449": "2e^\\psi R_a^b = G^{-1} \\left ( G \\lambda_a^b \\right )^\\prime-G^{-1} \\left ( G \\varkappa_a^b \\right )\\dot{ } = 0, ", "fc1d81a49670d38fdc201e4df5ea9236": "\\omega_\\infin<\\omega<\\infin", "fc1d958c4668ee16d6730769f4524a5d": " - \\langle \\ln \\rho \\rangle = \\langle 1- \\rho \\rangle + \\langle (1- \\rho )^2 \\rangle/2 + \\langle (1- \\rho)^3 \\rangle /3 + ... ~,", "fc1da2a4b7df0b73a7d38a98c47218c2": "f(\\sqrt{2}/2,\\sqrt{2}/2)=\\sqrt{2}\\mbox{ and } f(-\\sqrt{2}/2, -\\sqrt{2}/2)=-\\sqrt{2},", "fc1e3e43f4a6e30f90f2ae50a3642d56": " \\tilde{g}_{00} =\\rho ^{2}{g}_{00} ,\\,\\,\\,\\, \\tilde{g}_{0k}\n=\\rho{g}_{0k} ,\\,\\,\\,\\, \\tilde{g}_{kq} ={g}_{kq} .", "fc1e6dde2a1d3c5d881022f26b1e7908": "B_1(x)=x-1/2\\,", "fc1ec16aba398276d8bd219fdadf40d6": "P= \\frac{2}{3}\\begin{bmatrix} \\cos(\\theta)&\\cos(\\theta - \\frac{2\\pi}{3})&\\cos(\\theta + \\frac{2\\pi}{3}) \\\\\n\\sin(\\theta)&\\sin(\\theta - \\frac{2\\pi}{3})&\\sin(\\theta + \\frac{2\\pi}{3}) \\\\\n\\frac{1}{2}&\\frac{1}{2}&\\frac{1}{2} \\end{bmatrix}", "fc1f169d5c31dc01ab77b36e716d2021": "A_S", "fc20518762b51e6082ca266ebcb48ad3": " P_y = \\varepsilon_0 \\chi_{yy} E_y", "fc2054c1a6083c7803f4db0419e53c3a": "(x y)^* = y^* x^*.\\quad", "fc20d341e3fb9117466b04c7a0d5552a": "238,000 \\text{ miles}", "fc20f6d3a9b63e7c02212fa6d1bc395b": "\\Phi_0=\\frac{h}{2e} = 2.067833758(46)\\times 10^{-15}", "fc2102d5c727f8dcb79394f3306a1ec3": "\n\\begin{bmatrix} FR_1 \\\\ FR_2 \\end{bmatrix} = \\begin{bmatrix} A_{11} & A_{12} \\\\ A_{21} & A_{22} \\end{bmatrix} \\begin{bmatrix} DP_1 \\\\ DP_2 \\end{bmatrix}\n", "fc2180c5f80ce37c63fc60412c0a63b0": "1/[y_1, 0] = [-\\infty, 1/y_1]", "fc22366ae2084254dc2f3e90d02ddbfc": "\\frac{d^\\circ(P)!d^\\circ(Q)!}{(d^\\circ(P)+d^\\circ(Q))!}\\|P\\|^2 \\, \\|Q\\|^2 \\leq \n \\|P\\cdot Q\\|^2 \\leq \\|P\\|^2 \\, \\|Q\\|^2.", "fc2368161ba0b3018542fd8bbdd01af3": "(\\epsilon\\,)", "fc237c2918e69e1c56fc50b72f7a0354": "C(d) = \\sigma^2 \\Bigg(1 + \\frac{ \\sqrt{3}d }{\\rho} \\Bigg) \\exp \\Bigg(-\\frac{\\sqrt{3}d}{\\rho} \\Bigg) \\quad \\quad \\nu= \\tfrac{3}{2},\n", "fc238067d646385f8dcc8437fdfb0550": "\\psi_{\\nu_\\mu}", "fc23806bc187dafd7e80979cf38ace94": " c_k ", "fc23c286bab18ac1059f8cf6ab5633ba": "\\alpha \\in \\mathbf{F}_{p^2}", "fc23d669cfe08013b35af4939ce66246": "p(x|\\theta)", "fc2412e4965517484339d05dc5f5ca57": " f_{Demand}(\\it{t}) = \\frac{\\text{Demand}}{\\text{Maximum possible demand}}", "fc24136fb5510916d7bdb5d4daa9220d": "\\tfrac{1}{3}", "fc245e7a031c1fe5d8cd834ec759c0f6": "\\displaystyle \\phi(x_t) = \\min_{u_t \\in U(x_t)} \\Big\\{ L(x_t,u_t) + \\phi(f(x_t,u_t)) \\Big\\} ", "fc24fe37194f655daee207bac969d9a4": "V_2 = \\pi. \\,", "fc2551ad73b3cb65f01169e9fa36d320": "2^{\\kappa}=\\kappa^{+}\\,", "fc25c975031108a9f57a227798cf5557": "\\text{id}_X\\colon X \\to X", "fc2632177119c4bda8b0e59e52fe1118": "f_Z(z)=\\frac{1}{\\sqrt{2 \\pi}\\sigma_+ }\\exp\\left(-\\frac{z^2}{2\\sigma_+^2}\\right)", "fc2643e9991a82f9c3020d471832354c": "addOne\\ 2", "fc2674136fec8736fe26f8b9b0595c26": "\\mathcal{A}=C^\\infty\\left(M\\right)", "fc2678e0482e7a0030472c6ff7863fa2": "\\mathbf{r}_{X}", "fc272321f729ec8877cb75334ebdf07b": "\\mu(y) = A + Bc^{y} \\quad\\text{for } y \\geqslant 0.", "fc274226f180f2c5bb864452dbd32613": "w_{t+1} \\gets w_t - \\gamma_t\\nabla V(\\langle w_t, x_t \\rangle, y_t) \\ , ", "fc27550216dc7cd589e29d7bba5533b9": "\\frac{\\ln(640320^3+744)}{\\sqrt{163}} = 3.14159\\ 26535\\ 89793\\ 23846\\ 26433\\ 83279^+", "fc276a16f073fb26ccce675db323ef92": "\\scriptstyle k \\;=\\; B^2", "fc278a948b18418a522f08f2ea41329a": "y|\\beta^T\\textbf{x}", "fc27e771484e664ced8f40d735e6bfa0": " D = -2\\ln \\frac{\\text{likelihood of the fitted model}} {\\text{likelihood of the saturated model}}.", "fc2842516aa2256f028eedee49629ea5": "\\text{heart rate (bpm)} \\cdot \\text{systolic ejection period (s)} \\cdot 44.3 \\approx 1000", "fc2847a6e6036da3701dd3b4aa6ca6f1": "\\tilde{\\mathbf{M}}", "fc289a5d1566adbe90946500c5a37270": "x-y+b^n-b^n", "fc291b0d2f7d4df419a8d3fa0214cda6": "|\\psi(x)-x|\\le0.006409\\frac{x}{\\ln x}", "fc29a168cfe5c46eca606c585cf16309": "9262^3+15312283^2=113^7\\;", "fc2a1ac92adea93d4cd1b2504990b2e1": " H \\rightarrow 0 ", "fc2a4c205414ae9f187c36958e52b5d1": "A[\\hat{\\mathbf{k}}] \\, \\hat{\\mathbf{u}}=c^2 \\, \\hat{\\mathbf{u}}\\,\\!", "fc2a717104c8bc33cbaae016183fd3c3": " \\frac{V^2}{R} = -\\frac{1}{\\rho}\\left|\\frac{\\partial p}{\\partial n}\\right| + \\left| f \\right| V", "fc2a8d0386311ca193951ce832cb35ee": "F_n-1", "fc2aa3f764e631363938028655ed954d": "\\|x\\|_{bv} = |x_1| + \\sum_{i=1}^\\infty|x_{i+1}-x_i|", "fc2bdd06d6bd433037da85fd5d30fa5d": "E_{\\mathrm{gap}}=1.490\\,\\mathrm{meV}", "fc2be98f4ce8129a06ac0cf3412da90f": "\n\\widehat \\beta_{FGLS} ~\\sim N(\\beta , (X'\\widehat{\\Omega}_{OLS}^{-1}X)^{-1}(X'\\widehat{\\Omega}_{OLS}^{-1}\\Omega\\widehat{\\Omega}_{OLS}^{-1}X)(X'\\widehat{\\Omega}_{OLS}^{-1}X)^{-1})\n", "fc2c05f74832c2bc896c92c3efbfc1fc": " \\text{markup} = \\frac{0.4}{1 - 0.4} = 0.667 = 66.7%", "fc2c2d998706fbeb9c1c2b9140693fe0": "\\langle X, \\mathcal{T}, \\mathcal{F} \\rangle", "fc2c64145a8450858cea81adf5d56c57": "f(t) = u(t) \\sin \\omega t \\ , ", "fc2cbc151bdcea474997770493f14135": " p_k(z) ", "fc2ccf32e98f5387499978efb415be24": "\\gamma_k \\approx \\gamma_1b^{k-1}", "fc2d40aa51ef61843226bb208072167a": "g^{00} = \\frac{1}{g_{00}} - g_\\alpha g^\\alpha.", "fc2d97efb75c25b8ea71a3d29970e8ae": "n = 72", "fc2e098a93858f7e17a8e9b3efde8ac6": "G \\subset {\\mathbb{C}}^n", "fc2e0ca49590949a3d76be738fb07c71": "x\\in\\{0,1\\}^k", "fc2e1e937dad780553da93a8a1977538": "\n\\|Y - \\hat m\\|^2 + \\lambda \\hat{m}^T A \\hat m,\n", "fc2e5bb0b4449c354116691ac21cd93b": "j\\omega", "fc2e9026212d7188532646eefbb44ac8": "\\mathcal{P} = \\frac{1}{\\mathcal{R}}", "fc2e9036dbba97d88488889a098bcbc6": "c_v \\,", "fc2f8f30bf2fda753139ee1b0d69aec6": "\\sigma_{12} = \\sigma_{21}\\neq0", "fc2faf7adc71332ca25d265bb653c549": "L=n\\frac{ds}{dx_3}=n\\left(x_1,x_2,x_3\\right) \\sqrt{1+\\dot{x}_1^2+\\dot{x}_2^2}", "fc2ff3f43dd63ba7e9423e00f66d1545": "\\scriptstyle u\\,", "fc30260082e29798f07c59533cfdfd51": " {2n \\choose n} \\sim \\frac{4^n}{\\sqrt{\\pi n}}", "fc3067cc2ac620cc4dc8e28619fb5bbb": "\\mathbb{F}_{p}", "fc3067ed84e52425f03b9338a29b3183": "\n \\frac{\\partial \\boldsymbol{A}}{\\partial \\boldsymbol{A}} = \\boldsymbol{\\mathsf{I}}\n", "fc309a3ce419d0364adf32882f49e344": "x = \\,", "fc30a07bdbd95e5622f6069aeac76127": "(x)_a", "fc311d8e227ab38df59d2421f1b1fe91": "X_1X_2", "fc31330a5b2915578fd2a65f3c0b4172": "w(x) \\le \\alpha", "fc31601deabb0ea782b22345ad425385": "e_1,\\ e_2,\\ e_3", "fc3179923e70731c292cdf4083244314": "f^2(\\theta)", "fc318ed3dfc917bc367aed3912f18dbb": "x^2(x^2 + y^2)=a^2y^2", "fc31b8e757f65cfec248fd9ca432568b": "-\\partial^\\mu \\partial_\\mu-m^2", "fc320bb954d6287b5b69fcc4d0d69ac2": "A\\in\\mathcal{A}", "fc321718dca04aa678222521c39e1b18": "\n D^{(\\alpha)}(p \\parallel q) = \\frac{4}{1-\\alpha^2}\\bigg(1 - \\int p(x)^\\frac{1-\\alpha}{2} q(x)^\\frac{1+\\alpha}{2} dx \\bigg)\n ", "fc3284c6d08380aa6b26b634c7fdaf09": "\nr(Y,\\hat{Y})^2 = \\frac{\\sum_i(\\hat{Y}_i-\\bar{Y})^2}{\\sum_i(Y_i-\\bar{Y})^2}\n", "fc32c33cd4a8395589a0906bb5f385f7": "-\\!", "fc32ccf52fc6835248ea98da8319a6a0": "f(\\theta)=g(\\theta)", "fc3330f0628f5639dd2b74e6e6e93981": " \\sum_1^\\infty \\frac{1}{n^s} = s\\int_1^\\infty \\frac{\\lfloor u\\rfloor}{u^{1+s}} \\mathrm{d}u \\,.", "fc3348df71895e4ef961016234ac8eea": "xf'/f.", "fc3363bf4f91ff25a86afc2c13f93156": "2\\pi fX_C = \\frac{1}{C}", "fc3370ebd67a30c122587f1a10eb12cc": "\\tau^- \\, / \\, \\tau^+", "fc337485c3584a5796290b7452f21c45": "= \\frac{1}{T} \\left( \\mathrm{rect} \\left(\\frac{t}{T} - \\frac{1}{2} \\right) - \\mathrm{rect} \\left(\\frac{t}{T} - \\frac{3}{2} \\right) + \\mathrm{tri} \\left(\\frac{t}{T} -1 \\right) \\right) \\ ", "fc33a3c598c46cd1214bf4dc98b15f03": "\\mathcal{D}\\phi", "fc33bbc35c2647ea5a98fabb0ffffb56": "\\lambda_1 = \\lambda_2 = 1", "fc33dd49fdcc63e7f3f0c9299564779b": "A: U \\to V", "fc3419eea12d1d68cafb9633761385ca": "A \\rightarrow B: \\{N_B-1\\}_{K_{AB}}", "fc34499a4777280965dfb9785a84061b": "n_{ij} \\sim N(0,1) \\,", "fc351636c8e6359a39fc513b05ff86a8": "{n\\choose k}", "fc3520b761697f0cec477b5bf7c84eb8": "G \\ \\stackrel{\\mathrm{def}}{=}\\ \\frac {\\tau_{xy}} {\\gamma_{xy}} = \\frac{F/A}{\\Delta x/l} = \\frac{F l}{A \\Delta x} ", "fc3548f87a793f0c1a7663243905e857": "(\\rho_{AB},I_B,\\operatorname{Tr}^*)", "fc356f8b3e909620af46842e7aa06816": " H_i(a, b) = H_j(a, b) ", "fc361bb512d07ff53d641f25c01638a4": "\n\\mathbf{q} = \n\\Bigg[ \\mathbf{v_1}\\begin{bmatrix}\n c_1 & 0 & \\cdots & 0 \\\\\n \\end{bmatrix}\n+ \\mathbf{v_2}\\begin{bmatrix}\n 0 & c_2 & \\cdots & 0 \\\\\n \\end{bmatrix}\n+ \\dots + \\mathbf{v_n}\\begin{bmatrix}\n 0 & 0 & \\cdots & c_n \\\\\n \\end{bmatrix} \\Bigg]\n\\begin{bmatrix}\n1 \\\\\ne^{\\lambda_2t} \\\\\n \\vdots \\\\\ne^{\\lambda_nt} \\\\\n\\end{bmatrix}\n", "fc364849278aaca6bfcd6217b1de07c2": "\\,l = r \\times p", "fc36589934c6930e0ec7ecfe9c6feaa6": "L = \n\\begin{pmatrix}\nA^{\\frac{1}{2}} & 0 \\\\\nC A^{-\\frac{*}{2}} & Q^{\\frac{1}{2}}\n\\end{pmatrix}\n\\mathrm{~~and~~}\nU =\n\\begin{pmatrix}\nA^{\\frac{*}{2}} & A^{-\\frac{1}{2}}B \\\\\n0 & Q^{\\frac{*}{2}}\n\\end{pmatrix}.\n", "fc36aae27038d152bba7f9ecb477f65f": " T^{-1}:(a_k)_{k=-\\infty}^\\infty \\mapsto (a_{k-1})_{k=-\\infty}^\\infty.", "fc36bfd8c60a09adb920a8c5e3cb6a42": "\\left(\\frac{\\dot a}{a}\\right)^{2} + \\frac{kc^{2}}{a^2} - \\frac{\\Lambda c^{2}}{3} = \\frac{8\\pi G}{3}\\rho", "fc36cba2394162b32f41c9f2de175d34": "E_{K_{MAC}}(m_1' \\oplus t)", "fc36ce3b9bdcc16f07ae1c1b3892cf59": "\\frac{\\partial \\mathbf{B}}{\\partial t} = \\eta \\nabla^2 \\mathbf{B} + \\nabla \\times (\\mathbf{u} \\times \\mathbf{B}) ", "fc37013937a84a8d41d104ca4f806fb0": " \\mathbf {a}= {-\\omega^2} \\mathbf {r} ", "fc3847d56b28bce9cc9970f684f7c08e": "31:15", "fc38a50f01bb1a2e806c4682e96dea31": "\\mathit{Mean} = \\frac{ \\phi_{ 16 } + \\phi_{ 50 } + \\phi_{ 84 } }{ 3 }", "fc38a9c9a42bfee489e6b9a1599e8ca9": "e' = \\sum_{i}X{_i^e}", "fc38bc358a63719f2af1b641bb80fb8e": "\\frac{{{U}^{\\prime }}(z)}{U(z)}=\\frac{f(z|x)}{F(z|x)}-\\frac{{B}'(z)}{v-B(z)}", "fc38c42bd04e0f6fdcbf6893ce8c6efc": "\\ln (1-x)", "fc38dabdcc8ce21a82e80762f335c6c6": "(0,0) \\in \\C^2.", "fc39132d004752b210083fa44ee8fb33": "e^{j(\\theta+\\phi)} = e^{j\\theta}e^{j\\phi}.\\,", "fc3967909b95da2e5426e54a4d0245ea": " \n\\max_{ \\vec v}\\ \\vec v \\cdot \\vec c \\qquad \\textrm{s. t.} \\qquad \\bold{S}\\,\\vec{v}=0\n", "fc398c100f9af3806d4ff6511703eb8e": "F(x|a,b) = e^{-b x^{-a}}\\,", "fc39ab705fe9b24faa135251fc6f7db9": "\\langle a \\rangle = (\\gamma/\\sqrt{2} g)x ", "fc39b7f5f65f0d3a67c743814ce9bdfc": "\\omega_c=\\frac{1}{\\sqrt{LC}}", "fc39e609c01ce770380212a742370b35": "BR(a) = -a \\,\\,_4F_3\\left(\\frac{1}{5},\\frac{2}{5},\\frac{3}{5},\\frac{4}{5};\\frac{1}{2},\\frac{3}{4},\\frac{5}{4};-\\left(\\frac{5a}{4}\\right)^4\\right)", "fc3a0a6977768e9c0ded5492f82afc11": "\\sum_a p_a \\sum_x q_x c(a, x) \\leq C", "fc3a2d5229ca30c73ff20c1a7e8ccae9": "\\triangle\\delta = \\triangle\\theta =-\\beta\\cdot\\sin(\\delta-90^\\circ)=+\\beta\\cdot\\cos\\delta", "fc3adf50604cf56d34b8edb90b044daa": " \\frac{\\lim\\limits_{x \\to 0} 1 }{\\lim\\limits_{x \\to 0} x}", "fc3b1d12973369844ecdf856afc72362": "~\\Phi", "fc3b4ee663f705c6dfd581d8a601c3c0": "CAS=a_{0}\\sqrt{5\\left[\\left(\\frac{q_c}{P_{0}}+1\\right)^\\frac{2}{7}-1\\right]}", "fc3b59f1d94e3b649a683e8be1bfaf9e": " S = \\int {1\\over 2} \\partial_\\mu\\phi^* \\partial^\\mu\\phi d^dx", "fc3b69ffc2761499a8f26feaaf2f3057": "\\mathbf{x'}", "fc3b94da8548c36c184c2b2a004a2ac4": "\\mathrm{moser} \\ll 3\\rightarrow 3\\rightarrow 64\\rightarrow 2 < f^{64}(4) = \\text{Graham's number}.", "fc3c008b40bd7019f347d1b782e1d9f5": "i \\in [1,k]", "fc3c6d74c90e0541cbb3c09c017a8aad": " S = - \\sum P_i \\ln P_i.", "fc3c9e48f796e9f7aa3d9e4dabf8dfcb": "b_{ij}=\\sum_{k=1}^{m_{ij}}u_{j+k-1}v_{i-k}-u_{i-k}v_{j+k-1}.", "fc3ce88efe849fd317f97f9ef0081f46": "A() \\to a\\ A()", "fc3d08948a3b96abdf4a8d89916ba94d": "Q=(4X:8Y:2Z:4Z^2)", "fc3d278379f760c936a1a23256d788d8": "\\frac {F_t} {S_t} (1 + i_c)", "fc3d5c8f4e6d50fb0ae0fd38a271053c": "HETP", "fc3d8bf2e5b1e1e5ab1756fe9505661b": "\\sqrt{2} = \\sum_{k=0}^\\infty (-1)^{k+1} \\frac{(2k-3)!!}{(2k)!!} =\n1 + \\frac{1}{2} - \\frac{1}{2\\cdot4} + \\frac{1\\cdot3}{2\\cdot4\\cdot6} -\n\\frac{1\\cdot3\\cdot5}{2\\cdot4\\cdot6\\cdot8} + \\cdots.", "fc3ddfe78a7e75a544034980e46ba3c1": "K_x(S0, response)", "fc3de10e9b806cd21d6102d2957297c9": "[E\\; F]", "fc3ec27bc60f50da87b430d768358500": "g_{\\rho\\rho}R^\\rho_{\\sigma\\mu\\nu} = R_{\\rho\\sigma\\mu\\nu}", "fc3ed992a7cf536513d5a0adfd3dab5c": "c = \\sqrt{a^2 + b^2}", "fc3f013a1216a6e9a28669a86615d1ec": "{{i}_{OUT}}={{i}_{IN}}+{{i}_{B}}-{{i}_{B}}={{i}_{IN}}", "fc3f01d869db73699818ac04afa69e04": "\\nabla_n^2 = \\frac{\\partial^2}{\\partial x_n^2} + \\frac{\\partial^2}{\\partial y_n^2} + \\frac{\\partial^2}{\\partial z_n^2}", "fc3f0d97d98948e4f5be04fbb8a9faac": " p_\\text{tot} = \\sum_i p_i \\chi_i \\,", "fc3f37e0fd3181226512e0eb813b5006": "|k\\rangle =\\frac{1}{\\sqrt{N}}\\sum_{n=1}^N e^{inka} |n\\rangle ", "fc3f5f69a2f865651108d7cdc6c7de38": "|a| \\leq 2m^2", "fc3f988d4516800241a50ae2d611a5d6": "\\nu_d", "fc3fd74eede250c19b2b869220a9861b": "1_{\\{\\tau_Y\\le T\\}}", "fc3fff5f083fb580e0e3cde96bcc1580": "\\Box(\\Box A\\to A)\\to\\Box A", "fc400bf45ce93ec2d799972463f0a820": " \\frac{b-a}{8} (f_0 + 3 f_1 + 3 f_2 + f_3) ", "fc403ee5a86595cfadceeca6aca29794": "\\frac{\\partial E}{\\partial t} + \\nabla \\cdot \\left[ \\left( \\overline{\\boldsymbol{u}} + \\boldsymbol{c}_g \\right) E \\right] + \\mathbf{S}:\\left( \\nabla \\otimes \\overline{\\boldsymbol{u}} \\right) = \\boldsymbol{\\tau}_w \\cdot \\overline{\\boldsymbol{u}} - \\boldsymbol{\\tau}_b \\cdot \\overline{\\boldsymbol{u}} - \\varepsilon,", "fc404ba70a711199deab3013f3fe8edc": "A=\\pi a^2", "fc407b6c83c5fbb88e5fe8a2c337d474": "\\gamma_\\mu", "fc409b9f6b46b7e64693ca6ba97ce026": " D[p] = [[q, \\_, p], [x, \\_, f]] ", "fc40d2c4b5acefb7031c0ddcee28c51a": "\n \\mathit{JB} = \\frac{n}{6} \\left( S^2 + \\frac14 (K-3)^2 \\right)\n ", "fc40e718fe2e712fde8489e4da4ee8d0": "\\delta(x) = \\begin{cases} \\infty, & x = 0 \\\\ 0, & x \\ne 0 \\end{cases}", "fc40ea55b2bc12e6061ca3602cf167a4": " g(z, u) = \n1 + \\sum_{n\\ge 1} \\left( \\sum_{\\sigma\\in S_n} u^{b(\\sigma)} \\right) \\frac{z^n}{n!} =\n\\exp \\sum_{k\\ge 1} u^{b(k)} \\frac{z^k}{k}", "fc4159ec3fbcf5ce927c0556dfb021dd": "B_{6,3}(x_1,x_2,x_3,x_4)=15x_4x_1^2+60x_3x_2x_1+15x_2^3", "fc417a5f293cc5402be15cc4460b2bba": "1/n^{a}", "fc41839f4f388f14b3421229ca3c7c4e": "{\\mathbf a}\\cdot{\\mathbf a} - {\\mathbf b}\\cdot{\\mathbf b} = ({\\mathbf a}+{\\mathbf b})\\cdot({\\mathbf a}-{\\mathbf b})\\,\\!", "fc41d11c17e8f57599956a416da24b3a": "\\widehat{B}(\\varphi,\\hat{\\mathbf{a}}) = \\exp\\left(-\\frac{i}{\\hbar} \\varphi\\hat{\\mathbf{a}} \\cdot \\mathbf{K}\\right)", "fc41e17370887569b55ea7590e2e050c": "GPD(H,K)", "fc4203cbc42b02bc5f3cf601c60d60e7": "\\hat\\beta ", "fc420b278f6ace2df4bb107babc91208": "\\mathbf{Q}(\\mathbf{a}_j, \\, \\mathbf{b}_i)", "fc4247d308fa05276987df302594341c": "s*", "fc424dad6b936d774117a21100cc2aa0": "\n\\frac{\\partial \\rho} {\\partial t} + \\nabla \\cdot \\mathbf{J} = 0\n", "fc4253f21d516e0e31e1fcd60a6738ee": "2P_t", "fc42ff18530550d164860e041a26d757": "\n\\begin{align}\n& {} \\qquad \\frac{1}{2}\\cdot \\left(-32\\frac{\\text{foot}}{\\text{second}^2}\\right)\\cdot (0.01\\text{ minute})^2 \\\\[10pt]\n& = \\frac{1}{2}\\cdot -32\\cdot (0.01^2)\\left(\\frac{\\text{minute}}{\\text{second}}\\right)^2 \\cdot \\text{foot} \\\\[10pt]\n& = \\frac{1}{2}\\cdot -32\\cdot (0.01^2) \\cdot 60^2 \\cdot \\text{foot}.\n\\end{align}\n", "fc4329063456cd56c02b97d5873b21d2": "X_k = \\sum_{n=0}^{N-1} x_n \\cdot e^{-i 2 \\pi k n / N}.", "fc4340270c453cea695effd9ef78c93a": "\n \\boldsymbol{\\nabla}_{\\circ} \\mathbf{v} = \\sum_{i,j = 1}^3 \\frac{\\partial v_i}{\\partial X_j}\\mathbf{E}_i\\otimes\\mathbf{E}_j = \n v_{i,j}\\mathbf{E}_i\\otimes\\mathbf{E}_j ~;~~\n \\boldsymbol{\\nabla}_{\\circ}\\cdot\\mathbf{v} = \\sum_{i=1}^3 \\frac{\\partial v_i}{\\partial X_i} = v_{i,i} ~;~~\n \\boldsymbol{\\nabla}_{\\circ}\\cdot\\boldsymbol{S} = \\sum_{i,j=1}^3 \\frac{\\partial S_{ij}}{\\partial X_j}~\\mathbf{E}_i = S_{ij,j}~\\mathbf{E}_i \n ", "fc43d78c88cad6d035d5e2ce2b0692c5": "\\mathbf{R} \\!\\,", "fc444ea603ed3807a5a565bacf65c0f2": "S(p,t)(r)", "fc446a0c4bd242c1f9bdcbe2e96f8b1d": "\\oint_\\gamma f(z)\\, dz =\n2\\pi i \\sum \\operatorname{Res}( f, a_k ) ", "fc446c37d83c6870fba67685f9fc5cbb": "\\textstyle{\\frac {91} {48}}", "fc44749b38fdb5cb5919dacd9e8da25e": "k(n-1)", "fc447663eeee2ab7de872a6147703142": "{\\rm KILL}", "fc44790d8c804001b829671371c3466e": "(\\part P / \\part {\\rho})_T=(\\part^2 P / \\part {\\rho^2})_T=0\\,\\!", "fc449dc3241e44b20baad38e6d3d9a04": "a^2 + b^2 + c^2 = 1 \\, ", "fc453b4bc4c3082c038f3516502213c0": "y0", "fc7b22c58f873eac2e74bd5ecad24553": "t_1, \\ldots, t_n", "fc7b428a6245a30c3ef7ba95946c792e": " G_\\alpha ", "fc7b756e84307da8a8b51b7ae7623e53": "\\mathbf{P}_{-1}", "fc7b8e7601b38020493b09b7721cd985": "\\lambda_1\\geq \\ldots \\geq \\lambda_L\\geq 0", "fc7ba0ec55baae5858b51ca1e164f86d": "\n \\begin{align}\n \\delta K & = \\int_0^T \\int_{\\Omega^0} \\int_{-h}^h \\rho \\left[\n \\left(\\dot{u}^0_\\alpha - x_3~\\dot{w}^0_{,\\alpha}\\right)~\n \\left(\\delta\\dot{u}^0_\\alpha - x_3~\\delta\\dot{w}^0_{,\\alpha}\\right)\n + \\dot{w}^0~\\delta\\dot{w}^0\\right] ~\\mathrm{d}x_3~\\mathrm{d}A~\\mathrm{d}t \\\\\n & = \\int_0^T \\int_{\\Omega^0} \\int_{-h}^h \\rho \n \\left(\\dot{u}^0_\\alpha~\\delta\\dot{u}^0_\\alpha \n - x_3~\\dot{w}^0_{,\\alpha}~ \\delta\\dot{u}^0_\\alpha \n - x_3~\\dot{u}^0_\\alpha~\\delta\\dot{w}^0_{,\\alpha}\n + x_3^2~\\dot{w}^0_{,\\alpha}~\\delta\\dot{w}^0_{,\\alpha}\n + \\dot{w}^0~\\delta\\dot{w}^0\\right) ~\\mathrm{d}x_3~\\mathrm{d}A~\\mathrm{d}t \n \\end{align}\n", "fc7bcfb414b0998761c16fc66abe85ae": "P_{RBB}(k)", "fc7be96b0210c224889d05f8d8713f18": "q \\colon Q \\to N", "fc7bfdf9e3897fa57050b6b7c1396e7c": "\\gamma\\,:\\,[a,b] \\to X,\\ \\gamma(a) = x,\\ \\gamma(b) = y", "fc7c9a7f44d17c6c09f07e532e57ea58": "\\mu(E)=\\frac{1}{\\log 2}\\int_E\\frac{dx}{1+x}.", "fc7cd48a3552ba87ba69a7ba032365de": "dN/dt", "fc7d7b37050197f48ba806e4dde4b04b": "\\scriptstyle\\operatorname{Var}[ X] = k \\theta^2 ", "fc7de5d9f064bd3361bc0a80abf77d96": "F_j\\!", "fc7dfefdeb37a218322132da70a936b5": "\n\\begin{align}\n\\sum_{i = 1}^n (y_i - \\overline{y})^2 &= \\sum_{i = 1}^n (y_i - \\overline{y} + \\hat{y}_i - \\hat{y}_i)^2\n= \\sum_{i = 1}^n ((\\hat{y}_i - \\bar{y}) + \\underbrace{(y_i - \\hat{y}_i)}_{\\hat{\\varepsilon}_i})^2 \\\\\n&= \\sum_{i = 1}^n ((\\hat{y}_i - \\bar{y})^2 + 2 \\hat{\\varepsilon}_i (\\hat{y}_i - \\bar{y}) + \\hat{\\varepsilon}_i^2) \\\\\n&= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n \\hat{\\varepsilon}_i^2 + 2 \\sum_{i = 1}^n \\hat{\\varepsilon}_i (\\hat{y}_i - \\bar{y}) \\\\\n&= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n \\hat{\\varepsilon}_i^2 + 2 \\sum_{i = 1}^n \\hat{\\varepsilon}_i(\\hat{\\beta}_0 + \\hat{\\beta}_1 x_{i1} + \\cdots + \\hat{\\beta}_p x_{ip} - \\overline{y}) \\\\\n&= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n \\hat{\\varepsilon}_i^2 + 2 (\\hat{\\beta}_0 - \\overline{y}) \\underbrace{\\sum_{i = 1}^n \\hat{\\varepsilon}_i}_0 + 2 \\hat{\\beta}_1 \\underbrace{\\sum_{i = 1}^n \\hat{\\varepsilon}_i x_{i1}}_0 + \\cdots + 2 \\hat{\\beta}_p \\underbrace{\\sum_{i = 1}^n \\hat{\\varepsilon}_i x_{ip}}_0 \\\\\n&= \\sum_{i = 1}^n (\\hat{y}_i - \\bar{y})^2 + \\sum_{i = 1}^n \\hat{\\varepsilon}_i^2 = \\mathrm{ESS} + \\mathrm{RSS} \\\\\n\\end{align}\n", "fc7e5817af6609237fdc44bd77c32136": "\n\\begin{align}\n0 & < \\int_0^1\\frac{x^4(1-x)^4}{1+x^2}\\,dx \\\\[8pt]\n& = \\int_0^1\\frac{x^4-4x^5+6x^6-4x^7+x^8}{1+x^2}\\,dx\\quad\\text{(expansion of terms in the numerator)} \\\\[8pt]\n& = \\int_0^1 \\left(x^6-4x^5+5x^4-4x^2+4-\\frac{4}{1+x^2}\\right) \\,dx \\\\\n& {} \\qquad \\text{(polynomial long division)} \\\\[8pt]\n& = \\left.\\left(\\frac{x^7}{7}-\\frac{2x^6}{3}+ x^5- \\frac{4x^3}{3}+4x-4\\arctan{x}\\right)\\,\\right|_0^1 \\quad \\text{(definite integration)} \\\\[6pt]\n& = \\frac{1}{7}-\\frac{2}{3}+1-\\frac{4}{3}+4-\\pi\\quad (\\text{since }\\arctan(1) = \\pi/4 \\text{ and } \\arctan(0) = 0) \\\\[8pt]\n& = \\frac{22}{7}-\\pi. \\quad \\text{(addition)}\n\\end{align}\n", "fc7e66f707fb2f419ca1e4ea987fc186": " \\nabla \\times \\nabla \\varphi=\\mathbf{0}. ", "fc7e67e11f2de9de27b24e4898214ec8": "k\\tfrac{3}{6}", "fc898041b3fb9030099a8f0478180386": "\\mathbb Z\\langle t^{\\pm},q^{\\pm}\\rangle", "fc8a023873552f36289cbe2094aeb080": "PV = kNT \\,", "fc8a905be79da3c0b808727c275dab51": "\nC([x])=[[x]\\cap X]\n", "fc8aee20d653fc221941119991f54d41": "S={1\\over 16\\pi G}\\int R\\sqrt{-g} \\, d^4x \\, +S_m\\;", "fc8b03656bebd95ab185014fa0c59cb4": "h\\nu_2", "fc8b1a3b9bd343bac9a1200b733aa263": "0=\\left\\lfloor \\{x\\}+\\frac{k'-1}{n}\\right\\rfloor\\le \\{x\\}<\\left\\lfloor \\{x\\}+\\frac{k'}{n}\\right\\rfloor=1.", "fc8b1eb659587940f8f93cbff879b14d": "\\int f_n\\,dx = 1.", "fc8b3a4ae6b7a19c937ff0a2e5f89b88": "\nx_1=a-x_2\n", "fc8b6a1285ed2a3dcf08d12298ee74f4": "c_1, c_2, \\ldots, c_n", "fc8b7be074c98aa76f1d0558ba21bfc9": "\\begin{matrix} {48 \\choose 2} = 1,128 \\end{matrix}", "fc8c5387a97c5232dacfce7eb58e169e": "\n\\begin{align}\n \\mathbf{v}_{\\mathrm{rot}} &= \\mathbf{v}_{\\perp\\ \\mathrm{rot}} + \\mathbf{v}_{\\parallel\\ \\mathrm{rot}} \\\\\n &= \\mathbf{v}_{\\perp\\ \\mathrm{rot}} + \\mathbf{v}_{\\parallel} \\\\\n &= (\\mathbf{v} - (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k}) \\cos\\theta\n + (\\mathbf{k} \\times \\mathbf{v})\\sin\\theta + (\\mathbf{k} \\cdot \\mathbf{v}) \\mathbf{k} \\\\\n &= \\mathbf{v} \\cos\\theta + (\\mathbf{k} \\times \\mathbf{v})\\sin\\theta\n + \\mathbf{k} (\\mathbf{k} \\cdot \\mathbf{v}) (1 - \\cos\\theta),\n\\end{align}\n", "fc8c680b9fd50eff8c5176428ce66f0d": "x+x'-l = \\sqrt{(h_t+h_r)^2 +d^2}-\\sqrt{(h_t- h_r )^2 +d^2} ", "fc8ca56429e756b07771aa8c58e32a80": " \\operatorname{lambda-lift-tran}[\\lambda f.(\\lambda x.f\\ (x\\ x))\\ (\\lambda x.f\\ (x\\ x))] \\equiv \\operatorname{let} p\\ f\\ x = f\\ (x\\ x) \\and q\\ p\\ f = (p\\ f)\\ (p\\ f) \\operatorname{in} q\\ p ", "fc8cc58886f9cdf1914450cd9ad51205": "\\vec{F}_{1,2}+\\vec{F}_{\\mathrm{2,1}}=0", "fc8d175acbba345adccb0b08afe29418": "\\varepsilon(R_j) = 0\\,", "fc8d96270ad571657c54ec59784e3c64": "\\omega_\\circ", "fc8dc9094bcc448cd87878b7d0f85ab7": " \\| A x\\|_2 ", "fc8df59e52c08e6c3cb64a684f1d8f83": "V_\\text{in}^-", "fc8e02b95aac248e0a5aa11ef0aaadc5": "m_U = \\frac{n_1 n_2}{2}. \\, ", "fc8e2830c4068834c15ca5676be7c92a": " \\psi^{[-1]}(t;\\theta) = \\left\\{\\begin{array}{ll} \\psi^{-1}(t;\\theta) & \\mbox{if }0 \\leq t \\leq \\psi(0;\\theta) \\\\ 0 & \\mbox{if }\\psi(0;\\theta) \\leq t \\leq\\infty. \\end{array}\\right. \\,", "fc8e5b1fec9fe3df998b86380ca43ee4": "\\mathrm{Der}\\, \\Omega^*(M) = \\bigoplus_{k=-\\infty}^\\infty \\mathrm{Der}_k\\, \\Omega^*(M).", "fc8e9fc0470e0a0a50df0383d1c9331b": "|A \\cup \\{x\\}|", "fc8ea9f6c00f5e6da13f8a3d162f2587": "\\sigma_{12}=\\sigma_{13}=\\sigma_{23}=0\\!", "fc8eba38656f50363c5e8da705dcae8b": "m c", "fc8ef5a096160a89b95d9171965cec99": "\\delta\\left((x,y),(x',y')\\right)=\\min\\left\\{\\max \\{|x-x'|,|y-y'|\\}, \\max\\left\\{\\frac{y-x}{2},\\frac{y'-x'}{2}\\right\\}\\right\\}.", "fc8f22ee93bb025a5b59c42614e18552": "{5 \\choose 2} = 10", "fc8f3b0ce0d116ebbb5cb7960a692424": "\\tau_\\mathrm{n}^2+(\\sigma_\\mathrm{n} - \\sigma_3)(\\sigma_\\mathrm{n} - \\sigma_1) \\le 0", "fc90557d3ce1d3b35f0c4c5631dc8986": "\\nabla \\cdot F = \\operatorname{div} F", "fc90a68705c7b7bb728a3743e2cae245": "\\, N(\\phi) ", "fc90aa27ebe2551b103e03ab3b612251": "f' ", "fc90c723fd80be48d4ffdd8d6ffae545": "\\operatorname{ver}(\\theta)", "fc90f0846e35a715130c69e7fab6d232": "e^{-\\alpha t}\\left[\\cos{(\\omega t)}+\\left(\\frac{\\beta-\\alpha}{\\omega}\\right)\\sin{(\\omega t)}\\right]u(t) ", "fc90f82a72d98713c650f450dbd55fcd": "\\tilde{Q_s}", "fc912f3ee8e01452a5933cbfa4ec2b59": "\\displaystyle{\\|ab\\|\\le \\|a\\|\\cdot \\|b\\|.}", "fc913906bdc419ccb80e6dfecb452a47": "\\frac{z}{1-q} \\;_{2}\\phi_1 \\left[\\begin{matrix} \nq \\; q \\\\ \nq^2 \\end{matrix}\\; ; q,z \\right] = \n\\frac{z}{1-q}\n+ \\frac{z^2}{1-q^2}\n+ \\frac{z^3}{1-q^3}\n+ \\ldots ", "fc913986af3e75507582d6bf51561f6d": "\\ e^{\\frac{-E_a}{RT}}", "fc9157f4d09f0f8141475159e88c01b0": "a\\mapsto a^d \\mod n", "fc915b3124960c09222acd6cf1601070": "F(C) = T(C) / W(C)", "fc919c39ad9251c7a413df0affa1a9b7": "R^{D/2}\\,", "fc91aa2bbade50bb15138227e97ad060": "\\mathrm{range} (A)", "fc91aa9321cb4179db70901721dd7dbd": "\\frac {x^0 + i\\bold{x}\\cdot\\bold{\\sigma}}{\\sqrt{x^2}} ", "fc91c596b0b3b585c20b792f786a4ccf": "p_2 = p_1^2", "fc91c7840728184c27c24d34014696e6": " F(\\mathbf{x}^{(1)})=23.306 ", "fc922229d46539a061f6466dd6b71daa": " \n\\begin{cases}\n(q, \\omega,(\\delta_{int}(s), 0)) \\in \\Delta& \\textrm{if } ~ t_e = ta(s), y = \\lambda(s)\\\\\n(q, \\omega, \\bar{s}) & \\textrm{otherwise}.\n\\end{cases}\n", "fc92391ccaa105d6a0dacef24675645d": "H^{(\\lambda+1)}(z)", "fc928dbf691c31d33eede64c092d5304": "a \\triangleright (b \\triangleleft a) = b", "fc9294d54c8b5957e651cae8267df425": "(a_0, \\dots, a_n)", "fc9318627f443a7fe05de3d284e31445": " r(1 - \\cos \\frac{x}{r}) = r\\cdot \\frac{x^2}{(2^2-2)r^2} - r\\cdot \\frac{x^2}{(2^2-2)r^2}\\cdot \\frac{x^2}{(4^2-4)r^2} + \\cdots , ", "fc935bb14c5ecaeed762eb5f7378b6cf": " \\mathrm{Slerp}(p_0,p_1; t) = \\frac{\\sin {[(1-t)\\Omega}]}{\\sin \\Omega} p_0 + \\frac{\\sin [t\\Omega]}{\\sin \\Omega} p_1.", "fc93930857fe97a3b1530fd0bea779dd": "\nB = -\\frac{\\gamma_+x(0)-\\dot{x}(0)}{\\gamma_--\\gamma_+}.\n", "fc93ae663879327b1a3c31e4a7288e34": "U^{[p]}(L) = U(L) / I", "fc93c05f6bd1e091f657dcc23c5af669": "tau_f", "fc93c7f3ca60594e3346f39cee9f6f37": " \\frac{d\\mathbf{z}}{dt} = \\mathbf{g}(\\mathbf{x},\\mathbf{z},t). ", "fc93f6f5d7c2a02f7fb54c3c6b1d2505": " L = pr \\,\\!", "fc9418a2a6f5b15ad927198c5fd6804d": "(\\alpha\\circ_0\\beta)\\circ_1(\\gamma\\circ_0\\delta) = (\\alpha\\circ_1\\gamma)\\circ_0(\\beta\\circ_1\\delta)", "fc94357efb0a3060c1975f2c839a6650": "\\aleph_1.", "fc9438f69c038254723711e4d7de0786": "Z^{m}_n(\\rho,\\varphi) = R^m_n(\\rho)\\,\\cos(m\\,\\varphi) \\!", "fc944043202aefdaab238281222288ef": "\nf(\\mathbf{x})=\\frac{1}{\\textrm{c}(\\kappa,\\beta)}\\exp\\{\\kappa\\boldsymbol{\\gamma}_{1}\\cdot\\mathbf{x}+\\beta[(\\boldsymbol{\\gamma}_{2}\\cdot\\mathbf{x})^{2}-(\\boldsymbol{\\gamma}_{3}\\cdot\\mathbf{x})^{2}]\\} \n", "fc94ab866d45946c129df2ed7eb9968c": "= 1 \\text{eV}/k_\\text{B}", "fc953cca0ffc6f42a9bbf3387935d1a9": "c\\in(a,b)", "fc95851ffa1a31bf84c8f73d8defec64": "\\begin{align}\n\\liminf_{n\\to\\infty}X_n &:= \\lim_{n\\to\\infty} \\inf\\{X_m: m \\in \\{n, n+1, \\ldots\\}\\}\\\\\n&= \\sup\\{\\inf\\{X_m: m \\in \\{n, n+1, \\ldots\\}\\}: n \\in \\{1,2,\\dots\\}\\}\\\\\n&= {\\bigcup_{n=1}^\\infty}\\left({\\bigcap_{m=n}^\\infty}X_m\\right).\n\\end{align}", "fc9588d579e4abcd94201c1f6e1ddac5": "P(A|B) = \\frac{P(B | A)\\, P(A)}{P(B)}. \\,", "fc959cf0c23aaf16202219f8a6a93864": "\\mathbf{p} = m_0 \\mathbf{v} + \\frac{1}{2} \\frac{m_0 v^2 \\mathbf{v}}{c^2} + \\frac{3}{8} \\frac{m_0 v^4 \\mathbf{v}}{c^4} + \\frac{5}{16} \\frac{m_0 v^6 \\mathbf{v}}{c^6} + \\cdots .", "fc95b05b1601d94ccf502127e7faec41": "\\omega^{2^{p-1}} = kM_p\\omega^{2^{p-2}} - 1.\\quad\\quad\\quad\\quad\\quad(1)", "fc95b650a33d84bfae0a1283676a743a": "2u_{tx}+u_xu_{xx}-u_{yy} = 0. \\, ", "fc95db6eeeb4e66a2294cab366646c36": "R_S > R_H", "fc96231b9a1bd246d49525e5673f3534": "\n \\mathbf{F}_1 = N_{11} \\mathbf{e}_1 + N_{12} \\mathbf{e}_2 + V_1 \\mathbf{e}_3 \\quad \\text{and} \\quad\n \\mathbf{F}_2 = N_{12} \\mathbf{e}_1 + N_{22} \\mathbf{e}_2 + V_2 \\mathbf{e}_3 \n ", "fc962c15397c2298be869f3881b0a5c6": " n=1 \\ldots N ", "fc9654eba8214ff5320c30a4247556b9": "\n\\left(\\frac{-1}{n}\\right) \n= (-1)^{(n-1)/2} \n= \\left\\{\\begin{array}{cl} 1 & \\textrm{if}\\;n \\equiv 1 \\pmod 4\\\\ -1 &\\textrm{if}\\;n \\equiv 3 \\pmod 4\\end{array}\\right.\n", "fc9657ec287ba7eabe4045995ccc4256": "\\Phi: k\\left[M\\right] \\to \\prod_{i \\in I}k\\left[M\\right] / \\mathrm{Ker} F_i", "fc965986fd0b0e3b3a33bde0c5dfaf8a": "w_{1}=2.57,", "fc969f84bab6a3d9cb0ed8e0b7a2391c": "h(t) \\sim h_0 - 390 \\sqrt{t}", "fc96a7a084d430d50951e2705a97928e": " \\left|\\Psi\\right\\rang = \\left|1,V\\right\\rang \\left|2,H\\right\\rang", "fc96ae2d1cd20c8852160154b8c13b6b": "a_{11}=1, a_{12}=1/2, a_{21}=1/3, a_{13}=1/4, a_{22}=1/5, a_{31}=1/6, \\dots ", "fc96d11d04a5733b2eeead6c4bac398d": "H(s) = { {\\omega_0}^2 \\over s^2 + 2 \\alpha s + {\\omega_0}^2 } ", "fc971b3d5c2bc8c65a24cfd5d1010d66": "\\left. y = \\left( \\sum_{j=1}^m \\frac{x_j}{\\| x_j - y \\|} \\right) \\right/ \\left( \\sum_{j=1}^m \\frac{1}{\\| x_j - y \\|} \\right),", "fc978a05e4744fb06e31df4f12774e05": "\\rho_c = \\frac{2\\rho\\sigma_x\\sigma_y}{\\sigma_x^2 + \\sigma_y^2 + (\\mu_x - \\mu_y)^2},", "fc97971796ccae4c7953af9cb866b4f4": "R_iR_j\\Delta u = -\\frac{\\partial^2u}{\\partial x_i\\partial x_j},", "fc979a924c828eea3803028ed238a48d": "O( n \\log n)", "fc97a38299fae993a42283997cfdf4e5": "B(f_1,f_2) = F(f_1)F(f_2)F^*(f_1+f_2)", "fc9819c92e199d69e7211f508d6709e2": "\\int_{-\\infty}^\\infty f(x) x^n \\mathrm{e}^{- x^2} \\, \\mathrm{d}x = 0", "fc98229a292b773f36998a01eaa7fe16": "D^kF(u)\\{h_1,...,h_k\\}=D^kF(u)\\{h_{\\sigma(1)},\\dots,h_{\\sigma(k)}\\}", "fc98bcca781c7783beaaeab08e3f1433": "G(\\C)", "fc990dcd8aaa326e74602e3bbfd9b7ef": " |X - Y| ", "fc99109a5003cfcd2b17df902913489c": "\n \\frac{\\partial \\mathcal{L}}{\\partial f} +\\sum_{i=1}^n (-1)^i \\frac{\\partial^i}{\\partial x_{\\mu_{1}}\\dots \\partial x_{\\mu_{i}}} \\left( \\frac{\\partial \\mathcal{L} }{\\partial f_{,\\mu_1\\dots\\mu_i}}\\right)=0\n ", "fc9923e8b2217fa6162d5d0dd11a1305": "T_r \\left (\\sum a_n e^{in\\theta} \\right )=\\sum r^{|n|} a_n e^{in\\theta},", "fc9977692bd2684857c630a6f0bf493e": "\\zeta(2) =\n\\sum_{k=1}^\\infin \\frac{1}{k^2} =\n\\lim_{m \\to \\infty}\\left(\\frac{1}{1^2} + \\frac{1}{2^2} + \\cdots + \\frac{1}{m^2}\\right) = \\frac{\\pi ^2}{6}", "fc998f042aeb0e04b84fa4e60f7e5286": "1 - c_1", "fc99bca5fe15186a99a20fd1be60571c": "u_i(\\sigma)\\geq u_i(\\sigma_i^',\\sigma_{-i})-\\varepsilon", "fc9a94c2ec9807bfe5cbd18bdf6544d5": " \\partial_\\mu ", "fc9ab551435933aaa33de2bc8f44ef57": "\\textstyle > r", "fc9aff4093d98c3b6c852ad1e57d10ab": "E\\{Z(x)\\}=m", "fc9b00e9c2cd34064e686e5ff8f5f54b": "DR = \\sqrt{ \\int_{-\\infty}^T (T-r)^2f(r)\\,dr } ", "fc9b3333ba7f50f447814f1d933e5c95": "\\gamma < \\delta", "fc9b79a38a09e685dad09da33f7615df": "\\frac{2m(2m-1)}6 < \\left( \\frac{2m+1}{\\pi} \\right) ^2 + \\left( \\frac{2m+1}{2 \\pi} \\right) ^2 + \\cdots + \\left( \\frac{2m+1}{m \\pi} \\right) ^2 < \\frac{2m(2m+2)}6.", "fc9bb60682c1546c5c9b063e707ea91c": "n_{10}", "fc9bc03df0ff25bf88c434489ce86148": "m_{UT} = [14.539, 0.551]", "fc9bc9905f115a5671e4a1cb1edf17cf": "(3n^2-n)/2", "fc9c0b55344f413f7751b0a31511c8b7": "1-\\tan(\\delta\\theta)\\tan(\\alpha)\\approx 1", "fc9c765c5a19e695eed24335b52bfd5b": "J = \\frac{8}{9}\\varepsilon_o\\varepsilon_r\\mu \\frac{V^3}{L^3}", "fc9c96959a1283cf4c1dc734d584776e": "\nP =z^3-z^2(20r+3p^2)- z(8p^2r - 16pq^2- 240r^2 + 400sq - 3p^4)\n", "fc9d010449b8041e1b69a6ca0ffe9e84": "x^6+x^4+x^2+x+1", "fc9d108cb20e884c4deaf2f1ccb8741d": "\\mathrm{slog}_a", "fc9dcd0d52295fb33e625cc7d786c3a2": "\\det[(\\Lambda-\\mu I)^{-1}V^{-1}\\delta AV +I]=\\ 0", "fc9df5bc94fd2eba272bcd560ed57041": " \\sec \\theta\\!", "fc9e18de1582a5e74a86bc4f8fc1c005": "bcbcc", "fc9e5ef860c86bbc845e27c22009434e": "\\int_{a}^{b}i(t)dt=C[v(b)-v(a)].", "fc9ef77e9adab2588e5368e313140cbe": "\\forall k \\in S, M_{i,k} = 0", "fc9f14da292e543c33948607861b121d": "\\sigma(j)=i_j", "fc9f30ec8eb505c24cf6ca35c037e55b": "\n\\begin{align}4p\n&= (2m-n)^2 + 3n^2 \\\\\n&= (2n-m)^2 + 3m^2 \\\\\n&= (m+n)^2 + 3(m-n)^2,\n\\end{align}\n", "fc9f8fdfa7cc64e20b0edda6d543b2fc": "\\kappa=\\sqrt{G_N/\\omega}", "fc9fb86312877f7184000735c2e0e6f8": "\\scriptstyle R_\\mathrm L < R_0", "fc9fb9aeb3403ac2784c5484609b1abe": " \\Delta(x) = \\int_0^\\infty d\\tau \\int DX e^{- \\int_0^{\\tau} (\\dot{x}^2/2 + m^2) d\\tau'} ", "fc9fc4a36fef3d4023117a7c4546567d": "dS^2", "fc9fc7d7250c5110e39220209ccdac50": "C^{((n-2)/2)}_{n,k}(\\mathbf{x}\\cdot\\mathbf{y})", "fc9fef2b3c64ad29e042457dc76e611b": "a_0=1;\\quad a_1=1.93;\\quad a_2=1.29;\\quad a_3=0.388;\\quad a_4=0.028\\,", "fca00f2e671745186d7c76da5822c81f": "\\sum s_j", "fca042dd5e91982f9e65172698f23732": "\\mathrm{2NiO(OH) + Cd +2 H_2O \\rightarrow 2Ni(OH)_2 + Cd(OH)_2.}", "fca0470003576900164c259922fe8f53": " A^l\\to A^l\\otimes I\\xrightarrow{\\eta^l}A^l\\otimes (A\\otimes A^l)\\to (A^l\\otimes A)\\otimes A^l \\xrightarrow{\\epsilon^l} I\\otimes A^l\\to A^l", "fca104aa767ead36de67f2a4984babd1": "F^{n\\times n}", "fca12eac9a48ca330fe67eb42862a223": "Y= (Y_1=y_1,Y_2=y_2,...,Y_t=y_t)", "fca1cce4164e9f05978b24708104126c": "n = \\lambda_1 + \\lambda_2 + \\dotsb + \\lambda_\\ell", "fca20011cf52d406a983eb7e5d0f8118": "2\\mathbf{A}", "fca2446b68fe574493ce80af53b55820": "F = (m + m_\\text{added})\\,a.", "fca24ef26bd46beb608c5195de574de3": "\\frac{1}{\\mu_j}=E(S_j)", "fca25e88a47803f4b7dfd449caf43d1d": "\\mathcal{M}_{0} = \\mathcal{M}^{4}", "fca2be96b38ae988feb6dcff3b8a9aed": "\n \\qquad \\qquad u_x^+ = \\frac{-u_{i+2}^n + 4u_{i+1}^n - 3u_i^n}{2\\Delta x}\n", "fca306a0f74c044001f92cadc726e7c8": "\n\\begin{align}\nr(Y,\\hat{Y}) &= \\frac{\\sum_i(Y_i-\\bar{Y})(\\hat{Y}_i-\\bar{Y})}{\\sqrt{\\sum_i(Y_i-\\bar{Y})^2\\cdot \\sum_i(\\hat{Y}_i-\\bar{Y})^2}}\\\\\n&= \\frac{\\sum_i(Y_i-\\hat{Y}_i+\\hat{Y}_i-\\bar{Y})(\\hat{Y}_i-\\bar{Y})}{\\sqrt{\\sum_i(Y_i-\\bar{Y})^2\\cdot \\sum_i(\\hat{Y}_i-\\bar{Y})^2}}\\\\\n&= \\frac{ \\sum_i [(Y_i-\\hat{Y}_i)(\\hat{Y}_i-\\bar{Y}) +(\\hat{Y}_i-\\bar{Y})^2 ]}{\\sqrt{\\sum_i(Y_i-\\bar{Y})^2\\cdot \\sum_i(\\hat{Y}_i-\\bar{Y})^2}}\\\\\n&= \\frac{ \\sum_i (\\hat{Y}_i-\\bar{Y})^2 }{\\sqrt{\\sum_i(Y_i-\\bar{Y})^2\\cdot \\sum_i(\\hat{Y}_i-\\bar{Y})^2}}\\\\\n\n&= \\sqrt{\\frac{\\sum_i(\\hat{Y}_i-\\bar{Y})^2}{\\sum_i(Y_i-\\bar{Y})^2}}.\n\\end{align}\n", "fca364344fc01de8d6efc341627b2b5d": "\nQ=\\left[\\begin{matrix}\nD_{0}&D_{1}&0&0&\\dots\\\\\n0&D_{0}&D_{1}&0&\\dots\\\\\n0&0&D_{0}&D_{1}&\\dots\\\\\n\\vdots & \\vdots & \\ddots & \\ddots & \\ddots\n\\end{matrix}\\right]\\; .", "fca3866cf1bf17b4f92d8239d741e8ee": "e^{\\mathbf{A}t} = \\sum_{j=0}^{n-1}r_{j+1}{\\left(t\\right)}\\mathbf{P}_{j}", "fca3b8910e03824ea1ed67aa33b7b5ca": "\\mathbf F'_\\mathrm{Euler} = -m\\dot{\\boldsymbol\\omega} \\times \\mathbf r'", "fca3bf3d5e489a5e071233edab9011a4": "\\gamma_1 =\\frac{\\operatorname{E}[(X - \\mu)^3]}{(\\operatorname{var}(X))^{3/2}} = \\frac{2(1-2\\mu)\\sqrt{\\text{ var }}}{ \\mu(1-\\mu) + \\operatorname{var}}\\text{ if } \\operatorname{var} < \\mu(1-\\mu)", "fca3cf715150a8c4205e13dab57c43ef": "\\frac{d^2(E_\\text{surface}+Q\\sqrt{2U(\\phi_0/\\sqrt{2})/\\phi_0^2})}{dQ^2}<0. ", "fca3db914291c76d22d98d9480ed0812": "g_{\\bar \\mu \\bar \\nu} = \\frac{\\partial x^\\rho}{\\partial x^{\\bar \\mu}}\\frac{\\partial x^\\sigma}{\\partial x^{\\bar \\nu}} g_{\\rho\\sigma} = \\Lambda^\\rho {}_{\\bar \\mu} \\, \\Lambda^\\sigma {}_{\\bar \\nu} \\, g_{\\rho \\sigma} .", "fca40041e8ac06f75d759428193b35da": "A_{\\mathbf{s},\\phi}", "fca40ba2a1666f9d7181419aa44ce8b3": "y_2=\\cfrac{2y_1}{-1 + \\sqrt{1+\\cfrac{8gy_1^3}{q^2}}}", "fca40c3de98c6ce8a3c2a15ec2df2c3c": "m_{\\nu} \\approx \\frac{m_D^2}{M_{NHL}}", "fca441740d4a716c70c7e5e9b27b3234": " B(z)=\\zeta\\prod_{i=1}^n\\left({{z-a_i}\\over {1-\\overline{a_i}z}}\\right)^{m_i} \n", "fca47972fb272cacf78da7584ee3a7de": "x_K=1-\\sum_{i=1}^{K-1} x_i.", "fca50d401bc2e44ad1f598293466a1ef": " \\Delta G \\le 0 \\,", "fca5342410413724f97938a8713dcb3b": "t_n = n^{-2/d}\\sum_{i=1}^n w_{ni}\\{i^{1+2/d} - (i-1)^{1+2/d}\\}", "fca5497aa950eb9838f1409f77afeee7": "M_{xx} = 0", "fca595208103a75bd83a67e75767f076": "\\langle r,s,t \\mid r^2 = s^3 = t^3 = rst \\rangle", "fca5e2888c76d920aa3d9548dac1b6b4": "\\mathbf{v}\\,\\!", "fca5efa17d1536c905078cdb2cd27a78": "\\Pr(B_2)={4 \\over 10}", "fca6523ad95155c4af0158feb5b22877": "B_{\\mu\\nu}\\,", "fca662c1a3a70658275bbcadf6bcae5e": "P(y | x)", "fca69ae2812cfddaec2ade9088a9fd25": "\\mathbf{b} \\in \\mathbb{R}^m", "fca6c6d1da9966ba0056b8d8afba4ed8": "\nd(C_i \\cup C_j, C_k) = \n \\frac{n_i+n_k}{n_i+n_j+n_k}\\;d(C_i,C_k) +\n \\frac{n_j+n_k}{n_i+n_j+n_k}\\;d(C_j,C_k) -\n \\frac{n_k}{n_i+n_j+n_k}\\;d(C_i,C_j). \n", "fca6ca45fa2c7b5a8014b05b01b518b1": "\\sin(\\theta_T)=\\frac{n_1}{n_2}\\sin(\\theta_I)", "fca6ece6a4009c26e2463b3946d69d63": "u_i=\\min c(i,j)", "fca72647571508e27835f0cc62d47080": " \\widehat{H} = - \\left( n^2 (r) - \\widehat{p}_\\perp^2 \\right)^{1/2}", "fca7335d67abb6db10e7aebdf7a2fab1": "d^{2}(\\mathbf{x},\\mathbf{0})=\\sqrt{x_{1}^2+x_{2}^2}", "fca7598e97aa35485cb8a38bb8754c99": "\\dot S_i=C\\frac{(p_1-p_2)^2}{T}.", "fca779024aabd43d6b45903d7a35c1ea": "y\\geq x", "fca7d01d9513a23b262b7a12790ebf21": "\n -10 + R_a + R_b - (1)(15) - V_4 = 0\n ", "fca7db9dc174f4475317cca92a7e8a2f": "\\sigma_y^2(\\tau) = \\langle\\sigma_y^2(2, \\tau, \\tau)\\rangle", "fca83f0c6fb9f63a3294d958396f8dcd": "V(T,p)\\ ", "fca869404728dbcbe7e1a182374f8a95": "\\chi(G,k)", "fca8cdecede9873af9face368f70eaed": " R_0 = \\frac{ \\beta }{\\mu+\\nu}, ", "fca9277d1ea863fcec3577c8797b5674": "w=g_0 t^{\\varepsilon_1} g_1 t^{\\varepsilon_2} \\cdots g_{n-1} t^{\\varepsilon_n}g_n, \\qquad g_i \\in G, \\varepsilon_i = \\pm 1.", "fca9488dfe83a6ea0107ad99f14424c2": "\\eta_{th} \\le 1 - \\frac{294 K}{1089 K} = 73.0%\\,", "fca9611a6950149e581c065e18a6c9c3": "B_n(x) = -\\Gamma(n+1) \\sum_{k=1}^\\infty\n\\frac{ \\exp (2\\pi ikx) + e^{i\\pi n} \\exp (2\\pi ik(1-x)) } { (2\\pi ik)^n }. ", "fca97557e98a2f32753710e8e68e1c8b": "\\text{Two-Conductor Bundle Equation: } D_{BE} = \\sqrt{r_x \\bullet D_B}", "fca9f44de6bbb00c9aff74a1303c8c01": " c_3=2.75 ", "fcaabf28e70e3d15c2ce95518ae9e721": "\\mathrm{Set}^{D^\\mathrm{op}}", "fcaad7c396e23fbbb75690fc92f2e26e": "\\mu = \n{1\\over (j+1)}\\langle(l,s),j,m_j=j|\\left({1\\over 2}\\overrightarrow{l} {g^{(l)}}_p + {1\\over 2}\\overrightarrow{s} ({g^{(s)}}_p + {g^{(s)}}_n)\\right)\\cdot \\overrightarrow{j}|(l,s),j,m_j=j\\rangle", "fcaae7225d6b7ff6eccbd23f36b331de": " \\overline{x} = \\frac{x_1 + \\cdots + x_n}{n} ", "fcaae98f4f82a0db84ec4e163d978e50": "|b_{11}| < |b_{12}| + |b_{13}|", "fcab181a48e37668fc74280419e9845e": "V_t(\\mathbf{x}) = \\{\\mathbf{y} \\in \\{0, 1\\}^n \\mid \\mathsf{d}_A(\\mathbf{x}, \\mathbf{y}) \\leq t\\}.", "fcabe7aca73c62bbc568afbe67ba7c4c": "z!^{(k)} = z(z-k)\\cdots (k+1)\n= k^{(z-1)/k}\\left(\\frac{z}{k}\\right)\\left(\\frac{z-k}{k}\\right)\\cdots \\left(\\frac{k+1}{k}\\right)\n= k^{(z-1)/k} \\frac{\\Gamma\\left(\\frac{z}{k}+1\\right)}{\\Gamma\\left(\\frac{1}{k}+1\\right)}\\,.", "fcac43315486302ccb6b25a4adcf43bf": "\\Delta \\vec{p} = \\mathbf{A}^{-1}\\,\\Delta\\vec{F}\\!", "fcac8b4be93fd7668f82ca3e866cf73e": "I_z = mr^2\\left(1-t_n+\\frac{1}{2}{t_n}^2\\right) ", "fcacb3e3b18773a8f2aa5a18dd303a19": "\\eta \\rightarrow \\pi^{+} \\pi^{-} \\pi^{-}", "fcacb8b230e16088f5dfc08c1f5568e3": " PV = nRT \\,", "fcacca1424f339192c748b8e531c8ec3": " = ", "fcaccdb43cffc0e4a4725b2565f688d2": "\\frac{n^k}{\\zeta(k+1)}", "fcacdf8299cb0ca48c9d6a2a1d65e952": "W(f)=\\lim_{\\varepsilon\\to 0}\\int_{|x|\\ge \\varepsilon} K(x)f(x)\\,dx", "fcace440b213264307117e3a31001431": "K(n)=\\frac{(\\Gamma(n))^{n-1}}{G(n)}.", "fcacf5a02c029fd15b985bffbbe337d8": "u_{\\big| \\Gamma_1} = u_0", "fcad1482abe3bd5b6137cb64bebbb926": "\\begin{align}\nI_1 &= \\sigma_{11}+\\sigma_{22}+\\sigma_{33} \\\\\n&= \\sigma_{kk} \\\\\nI_2 &= \\begin{vmatrix}\n\\sigma_{22} & \\sigma_{23} \\\\\n\\sigma_{32} & \\sigma_{33} \\\\\n\\end{vmatrix}\n+ \\begin{vmatrix}\n\\sigma_{11} & \\sigma_{13} \\\\\n\\sigma_{31} & \\sigma_{33} \\\\\n\\end{vmatrix}\n+\n\\begin{vmatrix}\n\\sigma_{11} & \\sigma_{12} \\\\\n\\sigma_{21} & \\sigma_{22} \\\\\n\\end{vmatrix} \\\\\n&= \\sigma_{11}\\sigma_{22}+\\sigma_{22}\\sigma_{33}+\\sigma_{11}\\sigma_{33}-\\sigma_{12}^2-\\sigma_{23}^2-\\sigma_{31}^2 \\\\\n&= \\frac{1}{2}\\left(\\sigma_{ii}\\sigma_{jj}-\\sigma_{ij}\\sigma_{ji}\\right) \\\\\nI_3 &= \\det(\\sigma_{ij}) \\\\\n&= \\sigma_{11}\\sigma_{22}\\sigma_{33}+2\\sigma_{12}\\sigma_{23}\\sigma_{31}-\\sigma_{12}^2\\sigma_{33}-\\sigma_{23}^2\\sigma_{11}-\\sigma_{31}^2\\sigma_{22} \\\\\n\\end{align}\n\\,\\!", "fcad2281a6970812c11de7ff9d761f6a": " P_n(x) = B_n(x - \\lfloor x\\rfloor) ", "fcad36158bf0ae07dfa31f557f6dd608": "n\\mathbf{X}^{n-1}", "fcad3f5327a06e39c225d6da68e9e0ce": "p(z) = \\,\\prod_n (z-c_n).", "fcad910ebc7af4fbca490874496e7d49": "y_i\\le x_i", "fcad9831b211a2afaa3ca210fa9334fe": "x_0 \\in \\partial B", "fcada330c9e5e82a66a7c09c0e0f3c22": "\\frac{n_{i}}{n_{i+1}}", "fcadd2da2434a6ed8094ecb4eaf017b3": "( i\\hbar \\gamma^\\mu D_\\mu - mc ) \\psi = 0 ", "fcadd69f2d54f32df5ba1c04882db127": "X^{(n)}", "fcadd7804bb51459c9a89e50a98256c4": "\\sum_{k=0}^{\\infty} \\frac {8^{2^k}}{2^{2^{k+2}}-1} = 0.85073618820186...", "fcadec7a69bd4be9cb681cdd12913617": "P_\\mu=\\mathbf{J}^{-1}(\\mu)/G_\\mu", "fcae308bb4c6ad47b41588e98ccf82c8": "\\textstyle(\\prod_{i=1}^mA_{i,f(i)})(\\prod_{k=1}^mB_{g(k),k})", "fcae7c4c848bc36c15517d05a5a326da": "\\operatorname{P}(X \\le x) = \\operatorname{P}(Y \\le x)\\quad\\hbox{for all}\\quad x.", "fcaeb28086b0bb2d11617b3f13686c39": " \\left [ A \\left \\{ \\frac { \\left ( \\frac {M}{C} + \\frac { \\beta^2 + 3 \\beta + 6} {10 \\left ( \\beta^2 + \\beta + 1 \\right ) } \\right ) } { A } + A \\left ( \\frac{N}{C} + \\frac{6 \\beta^2 + 3 \\beta + 1}{10 \\left ( \\beta^2 + \\beta + 1 \\right ) } \\right ) - \\frac { 3 \\beta^2 + 4 \\beta + 3} {10 \\left ( \\beta^2 + \\beta + 1 \\right ) } \\right \\} \\right ] ^{-1/2} ", "fcaeda3c2e698c19db3d503b2b2fffb8": " \\frac{\\partial y}{\\partial t} + \\alpha y \\frac{\\partial y}{\\partial x} + \\frac{\\partial^3 y}{\\partial x^3} = 0 \\,\\!", "fcaee4a023819d33baa591f529d13596": "m_H(\\Sigma) := \\sqrt{\\frac{\\text{Area}\\,\\Sigma}{16\\pi}}\\left( 1 - \\frac{1}{16\\pi}\\int_\\Sigma H^2 da \\right), ", "fcaf03636ee736bdfa9af546c5ed7338": "\\Chi^2(k_0) = \\sum_{i=1}^3{\\frac{(x_i-\\hat{\\mu_i})^2}{\\hat{\\mu_i} \\left (1+ \\frac{\\hat{\\mu_i}}{k_0} \\right )}}", "fcaf46a7ff8e38c7460cc5fa76fed762": "3 n^{\\log_23}\\approx 3 n^{1.585}", "fcafd7094e36e328c610426f40dd3f55": "\\nabla\\cdot\\mathbf{g} = -4\\pi G \\rho,", "fcb007ce6e8a16e8ade3847e64e76c8b": "\\frac{n^{2}}{n_\\text{core}^{2}}\\sin^{2}\\theta_\\mathrm{max} = \\cos ^{2}\\theta_{c} = 1 - \\sin^{2}\\theta_{c} = 1 - \\frac{n_\\text{clad}^{2}}{n_\\text{core}^{2}}.", "fcb05f826d4d9c1915de7baa3e220204": "{\\alpha} = \\frac{a_T}{r}", "fcb0668082fa33008aad4fc5ed5fa85b": "\n\\nabla V = \\pm \\nabla \\times \\boldsymbol{\\omega}, \\quad V = \\varepsilon + 2M \\sum_{i=1}^{k} \\frac{1}{|\\mathbf{x} - \\mathbf{x}_i | }.\n", "fcb075142b61eff121db8e3e67fdc761": "S(n,k) = \\sum_{j=0}^{n-k} (-1)^j {n-1+j \\choose n-k+j} {2n-k \\choose n-k-j} s(n-k+j,j).", "fcb07ecdd74343a32b945ef70806d3de": "cT_3", "fcb0810bc39750eec17f2fe2fb02b074": "= f(x) + \\int_{D} L_{X} f (y) \\, G(x, \\mathrm{d} y).", "fcb0c2e55f726df14c58ad1f49a5dd4c": "-\\int_{-\\infty}^\\infty \\phi'(x)H(x)\\, dx = \\int_{-\\infty}^\\infty \\phi(x)\\,dH(x).", "fcb0d3cb4407ce3661380266fc49dfb7": "\\text{MMD}(P,Q) = \\left| \\left| \\mu_X - \\mu_Y \\right| \\right|_{\\mathcal{H}}^2 ", "fcb17c4a5db627edb0b015053f4ff3e5": "g(n-m) = \\langle f_n, f_m \\rangle", "fcb181e55bbad799a409b56bbe30631a": "P(\\hat{s}',\\hat{s})", "fcb185e30c178ac2ae5cb17a534d2ab5": " \\xi_{xx} + \\xi_{yy} = P(\\xi, \\eta)", "fcb19632e3504b588d45ef9b6efdc241": "A^{(s)}", "fcb1d9640124815413ba2296e9e7f4e5": "\\langle 2 \\rangle", "fcb1e10b794beb630170ed13d79d6cc7": "FV(A) \\,=\\,A\\cdot\\frac{\\left(1+i\\right)^n-\\left(1+g\\right)^n}{i-g}", "fcb1f8a788e622728473c95b606937fa": "\\{s_i\\}\\to \\{\\tilde s_i\\}", "fcb1ff004942718a6fb76d4e0a2206bd": "\\int_0^t H\\,dB = \\lim_{n\\to\\infty}\\int_0^t H_n\\,dB", "fcb23753554122faecaa3d93eab18af7": "\\hat{f} \\;\\;\\;\\; \\stackrel{\\hat{U}}\\longrightarrow \\,\\acute{\\hat{f}} \\;\\;\\;\\;\\; =\\hat{U}^{+}\\hat{f}\\hat{U}", "fcb25382f9320d0f2bb8430e86a8bfbe": "\\alpha^a \\beta^b = \\alpha^A \\beta^B", "fcb26898a388235209e618c97652636b": "\\left.\\right. dS ", "fcb2d7d5744c9b43aabded187c03af25": "{\\mathit l \\over \\mathit l^*} ={2\\over3}, {3\\over 5}, {4\\over 7}, \\mbox{etc.,} ", "fcb3218500430e9c408a44ce057de1f1": "\n\\begin{array}{lcl}\n\\Pr(w_{dn}=v\\mid\\mathbb{W}^{(-dn)},\\mathbb{Z},\\boldsymbol\\beta)\\ &\\propto\\ & \\#\\mathbb{W}_v^{k,(-dn)} + \\beta_v \\\\\n\\end{array}\n", "fcb3399abf6abb61d41709527923e243": "\\boldsymbol{F_{AB}}=-GM_AM_B\\frac{\\widehat{\\boldsymbol{R_{AB}}}}{|\\boldsymbol{R_{AB}}|^2}\\ ", "fcb33da0618d2cb405dce37ac7b9af4c": "2\\times2\\times2\\times2=16", "fcb36e904f3a71ef87b6862a2ba0e794": "x = b_0.b_1b_2b_3 \\dots.", "fcb3912933e9037e96853a9398162be2": "\\mathbb{H}^{n+1}\\setminus\\{(0,\\ldots,0)\\}", "fcb396dc3e114e050a174c423bfd40db": "|\\omega(n)-\\log(\\log(n))|<\\psi(n)\\sqrt{\\log(\\log(n))}", "fcb39eeb1b431107bfd06dfa2ddff3df": "D1=d1", "fcb3d9585e06c82523104526dcd045e8": "d(A, B) = [\\log(m)/L(A, B)] - [\\log(n)/L(A, A]", "fcb40efa44aa965cce3ccbb8f3b06294": "\\sqrt{-1}=i", "fcb4876dcd88176f1e6ac6b40f131273": "P = \\{P_2, P_3\\}", "fcb4ee0a4ab9c0fc67c779f48d2ef9de": "C_0,C_1", "fcb4f58dbffb98810013e32620ac05ce": "a^2 - Nb^2 = k \\implies \\left(\\frac{am+Nb}{k}\\right)^2 - N\\left(\\frac{a+bm}{k}\\right)^2 = \\frac{m^2-N}{k},", "fcb50fb40b19ee09a7bf549d99c7918d": "\\delta = \\frac{1}{\\sqrt{\\pi f \\mu \\sigma}},", "fcb53a6aab3a33ff9ef31b46b6173954": "\\pi_Y : X \\times Y \\to Y", "fcb54c16b0910b221dc3fa5b95497c74": "\\psi(x) = \\phi(x) + z\\phi(2x)", "fcb55f2c1d407b7f60016789bcb8da2d": " x\\in I^\\kappa ", "fcb575ca63221f8a043d34cc6256e914": "\\;q", "fcb586c845e44817dd702dc9872b0763": "S_+ = \\hbar \\sqrt{2s} \\sqrt{1-\\frac{a^\\dagger a}{2s}} a", "fcb5ded86177d0b636288ee2a2ae93c3": " r_0\\,\\!", "fcb5f42ef7826d83a94a245fb749a2e4": "\\langle O(\\log n), O(n \\log n) \\rangle ", "fcb6c87b40f31e34c9f4a498824c8f8a": "u_i, v_j", "fcb7178d18fa276fb4a1b613858129eb": "\\beta_T > 1", "fcb71c29df8b022ba715500512af3cc3": "EHS", "fcb72bac6f651178a356ad5c2fed5c91": "f_{BP}=1-\\frac{A_{n}}{A_{n-1}}", "fcb74fdda02466e94a68621b810b8dd8": "F_{D} = k(\\dot{L}_{D})^a", "fcb7624d7c9d785faea284b227c4b50d": "{dy \\over dx}-\\csc^2 y=1", "fcb7c08b223533b19d2be33e2cedc847": "e = \\det e_\\alpha^I", "fcb7d77a3a54149e9b6b67034ae72122": "\\tau(p)", "fcb80edc98c680f1528835b6cca6bd37": "\\phi \\in C^1_c(A; \\mathbb{R}^n)", "fcb82e0d7d3980d9c3e94f3ae20b317c": "e^{\\lambda} \\cdot e^{\\mu} = e^{\\lambda+\\mu}", "fcb88b07058c4f1830c9e994047af16d": "x' = l_3\\|r_3", "fcb90e773f662ffc040c4c674f4f1f4a": "E(S_{T}) > K", "fcb952ed4d68e153a6423ac42b8eeb86": "\\mathcal{H_A} = (\\mathcal{A}, R, \\Delta, \\varepsilon, \\Phi) ", "fcb96d483dce5cfe2e293d38cc3bd8c0": " \\det\\left( \\frac{\\partial^2 f}{\\partial z_i \\partial z_j} \\right)_{1 \\le i \\le j \\le n}^{z = z_0} =0. ", "fcb9af36cb7803930601a616c5239606": "(0,\\ldots, 0,1)^T", "fcb9d587d5b0c72063d7fe60bee03cec": "\\mathbf{P} \\left ( \\frac{1}{m} \\sum X_i \\geq p + \\varepsilon \\right ) \\leq \\left ( {\\left (\\frac{p}{p + \\varepsilon}\\right )}^{p+\\varepsilon} {\\left (\\frac{1 - p}{1 -p - \\varepsilon}\\right )}^{1 - p- \\varepsilon}\\right ) ^m = e^{ - D(p+\\varepsilon\\|p) m}", "fcba3684fdaf96d99782da53d695bab6": "a_i = \\gamma_{bi} b_i/b^{\\ominus} \\,", "fcba4208a7d2c1d48b837fc2dec3aa12": "{\\rm Pr}_z(A(x,y_1 \\oplus z)=\\dots=A(x,y_m \\oplus z)=0)= 1 - {\\rm Pr}_z \\Bigl( \\bigvee_i A(x,y_i \\oplus z) \\Bigr)\\ge \\frac{2}{3} > 0.", "fcba705501a8ec99a4b06087d4c3900e": "\\begin{matrix} \\frac{n}{2} \\end{matrix}", "fcbb5cc6a504d454796c64e57386385b": "\\omega_0 = 0", "fcbbd11f53d06a454f48ec1f2e2439d1": "L_2\\equiv\\partial_{xxy}+x\\partial_{xyy}-\\frac{1}{x}\\partial_{xx}-\\frac{1}{x}\\partial_{xy}+x\\partial_yy\n -\\frac{1}{x}\\partial_x-\\big(1+\\frac{1}{x}\\big)\\partial_y{\\big\\rangle\\big\\rangle}.\n", "fcbbe571e86160a7ec4ab9e5bdca62bb": "L_\\sigma\\mathfrak{g}", "fcbc1f92fb495a5e3a78a3c52f85fa29": "G_{ab} = R_{ab} - \\tfrac{1}{2} R g_{ab}", "fcbc5f4a57e4bd004f9ad437abf54ca0": "\\rm{rect}\\left(\\frac{t-u}{\\Delta_t}\\right)", "fcbc9a2240cad0df59a0f955d8ca4341": "{\\rm Tr} \\, e^{\\ln A -\\ln B+\\ln C}\\leq \\int_0^\\infty dt\\, {\\rm Tr}\\, A(B+t)^{-1}C(B+t)^{-1}.", "fcbcc8a3582ec50e6304ed23e6261386": "Y \\subset k^n", "fcbd3d72400f88195c01b53257a21f02": "\\scriptstyle (\\Re)", "fcbd96f340d6a58e00e3b7940df0e5e1": "u\\in X\\,", "fcbdd20cd9dd4f9bfc63620c6cfdfd91": "E = E^{\\mathrm {kin}}_1 + E^{\\mathrm {pot}}_1 + U_1 + E^{\\mathrm {kin}}_2 + E^{\\mathrm {pot}}_2 + U_2 + E^{\\mathrm {pot}}_{12}", "fcbe357d5bd7c61e60276556d4b59c5e": "\\sigma_{2t}=7.3", "fcbe9b4bcecd8d1b1fc6c309732d9b2c": "\\sum_{n=1}^\\infty \\; \\sum_{k=10^{n-1}}^{10^n-1}\\frac{k}{10^{kn-9\\sum_{j=0}^{n-1}10^j(n-j-1)}}", "fcbed0c19e7655afc90dc7af0386d1d1": "g(\\alpha) = 0", "fcbed5ea5c90a06a75530033fd3b85b0": "\nu = -\\frac{\\mu }{r} + \\sum_{n=2}^{N_z} \\frac{J_n P^0_n(\\sin\\theta) }{r^{n+1}} + \\sum_{n=2}^{N_t} \\sum_{m=1}^n \\frac{ P^m_n(\\sin\\theta) (C_n^m \\cos m\\varphi + S_n^m \\sin m\\varphi)}{r^{n+1}}", "fcbf25a41ef448031bdd3b850a813fe1": "\\mathbb{E}[(X_n+c)^+\\,|\\,\\mathcal G]\n\\le\\mathbb{E}[X_n\\,|\\,\\mathcal G]+c+\\varepsilon", "fcc0344063668536dfd32bd26b44ed4d": "\n \\nabla^2 \\nabla^2 w = -\\frac{q}{D} \\,.\n", "fcc036f6ee5f00711df6d6102665d16f": "\\tan\\alpha", "fcc065d56ce48741a0d9530e2f4cc461": "|g|^2", "fcc06b8043d2118a8e3d04652c88d479": "= \\delta\\int_{x_{3A}}^{x_{3B}} L\\left(x_1\\left(x_3\\right),x_2\\left(x_3\\right),\\dot{x}_1\\left(x_3\\right),\\dot{x}_2\\left(x_3\\right),x_3\\right)\\, dx_3=0", "fcc08a68c993861847f3984d57b5c6f5": "\\Omega ' = \\frac {\\operatorname{ad}_\\Omega}{\\exp(\\operatorname{ad}_\\Omega)-1} ~ A~,", "fcc08d83e8525c58042e97e0ae3897cf": "D\\colon A[[\\hbar]]\\mapsto A[[\\hbar]]: \\sum_{k=0}^\\infty \\hbar^k f_k \\mapsto \\sum_{k=0}^\\infty \\hbar^k f_k +\\sum_{n\\ge1, k\\ge0} D_n(f_k)\\hbar^{n+k} ~,", "fcc0c5155f5db8b8a7a8994e0d8a4686": "\\phi ^{\\mathrm{odd}}(x)", "fcc0d3cbaba759d7ce8b9e7f64120cac": "\\mathrm{Hom}(V, W) = \\mathbf{Hom}(V,W)_0.", "fcc0e919cc28656c4b0bbef183e9d5ee": "\\tau_\\sigma T = T\\,", "fcc1b2268cd3c4bbfbf8597190951ae8": "a = b = 0.1", "fcc1d7279b9e59b1417f3dddda55fa92": "\\mbox{dr}(abc) \\equiv a\\cdot 10^2 + b\\cdot 10 + c \\cdot 1 \\equiv a\\cdot 1 + b\\cdot 1 + c \\cdot 1 \\equiv a + b + c \\pmod{9}", "fcc222cea96050e8b58add37c0379557": "n_\\mathrm{e}(x) = n_0 \\exp\\Big(\\frac{e\\,\\varphi(x)}{k_\\mathrm{B}T_\\mathrm{e}}\\Big)", "fcc25b5223d24921ff96871f486a7a49": "\\frac{1}{6}(a^2 + b^2 + c^2 - 4\\sqrt{3}\\, \\Delta) ", "fcc304f9610cab02a100b9a4ec65cb91": " V_{\\text{out}} = -R_{\\text{f}} \\left( \\frac{V_1}{R_1} + \\frac{V_2}{R_2} + \\cdots + \\frac{V_n}{R_n} \\right) ", "fcc320cfc57d0a0f2102f501068d9b7d": "\nC([x])\\cap X=[x]\\cap X\n", "fcc32d7a3d40dd08fc242601546f709c": "L(z)\\equiv \\sum_{n\\in \\mathbf{Z}} L_n z^{-n-2}={1\\over 2} :b(z)^2:", "fcc3c3982c868c77bac9c357f176f407": " \\frac{\\mathrm{d}N_j}{\\mathrm{d}t} = - \\lambda_j N_j + \\lambda_{j-1} N_{(j-1)0} e^{-\\lambda_{j-1} t}. ", "fcc3f76630baf7a859c1b06368ba77fa": "C^{(k)}(GIP)\\geq c {n\\over4^k},", "fcc3f8dba4f9f7065e509ab44634fb32": "\\mathrm{d}\\mathbf{F} = 0", "fcc3f9d256a30ad33af4c0b8c59733c1": "p(\\tilde{x}|\\mathbf{X},\\alpha) = \\mathbb{E}_{\\theta|\\mathbf{X},\\alpha}\\Big[p(\\tilde{x}|\\theta)\\Big]", "fcc488176a6612cb663790c78ad01912": "L(\\gamma)=\\int_a^b \\sqrt{ g_{\\gamma(t)}(\\dot\\gamma(t),\\dot\\gamma(t)) }\\,dt.", "fcc4ea5af37ccbedb3c58f6849b56096": "S=2^d-1", "fcc532d15ebdf2ba926c3202bca2dbcf": "\\in_{\\mathcal{M}'} = \\in_{\\omega} \\cup < \\cup \\,(\\omega \\times \\mathbb{Z}')", "fcc54b76024c4bfd50b95292ec4bb1e6": "(a _{1}, \\ldots ,a _{d})", "fcc5851b8cb82821670294543254adc9": "P = {\\Pr}_{\\text{random }G} [\\text{there exists a vector }m \\in \\mathbb{F}_q^k \\backslash \\{ 0\\}\\text{ such that }wt(mG) < d]", "fcc5a7d2df8a0d84d5363d88d41fd3c6": "C_F", "fcc5adaaf37b29302f0c1f8ed01786ab": " x \\in X \\setminus {X_{G}}", "fcc64b20331b97702dfa58af888466e9": " F_{i}(x) - F_{i}(x+dx) + A_t dx \\nu_i r = 0 ", "fcc68119798fc2154bcad3e0959d72a0": "E^2-p^2", "fcc6c2eba98516fe1c5785d914370188": "(\\rho-\\rho')/(\\rho+\\rho')", "fcc70e763dbae1d6ede769504080330d": "\\text{Hom}_{D(\\mathcal{A})}(X,Y[j]) = \\text{Ext}^j_{\\mathcal{A}}(X,Y).", "fcc74c1bbd8ab66b08115217b1a4e16f": "2 = A \\cdot 1 ", "fcc7777026a496f63d610cf1108925fd": "\\forall y\\, P(x,y)", "fcc78763b632d03fb969b59038977d38": "\\cos \\theta = \\frac{\\vec{a}\\cdot\\vec{b}}{\\left|\\vec{a}\\right| |\\vec{b}|} ", "fcc7f0e3d0731fd4d5428a0a17f4b56e": " \\tbinom {n+k} k", "fcc8a1e48c8350e309fff642db3de85e": "u(x,0) = u_0(x)\\,", "fcc8cefa230ae0fb86ae112751f16843": "\\bar\\psi \\mapsto \\left(\\lambda \\psi\\right)^\\dagger \\gamma^0", "fcc9046da0b5d88389c51b87c77d8de0": "\n \\begin{align}\n n_\\alpha~N_{\\alpha\\beta} & \\quad \\mathrm{or} \\quad u^0_\\beta \\\\\n n_\\alpha~M_{\\alpha\\beta,\\beta} & \\quad \\mathrm{or} \\quad w^0 \\\\\n n_\\beta~M_{\\alpha\\beta} & \\quad \\mathrm{or} \\quad w^0_{,\\alpha}\n \\end{align}\n", "fcc933b712530dcb04faf1eb77172e51": "\\Leftrightarrow 2 y^2 + 2x^2 + 3c x = 0 ", "fcc93cb2910ea90606b3217f55912091": "\\mathbb{R}^n.", "fcca7fd4ab209a5bfef5f9a5d45bef2e": "\\text{Precision} = \\text{PositivePredictiveValue} =\\frac{TP}{TP+FP} \\, ", "fcca8047c8d13060c1bdf7846defe151": "\\Delta\\theta~\\Delta n=1/2", "fccb5181a7305e2ed89f23764e22349c": "\\! f(\\lambda)", "fccb69b3b2a4ff595888bd749d13c945": "E_o", "fccb6eb3953ee6e0f26692b4f99045c8": "HbO_2 \\,", "fccb931fd4cd6e3adb37ef5220da5ba4": "\n f(\\boldsymbol{\\sigma}, \\boldsymbol{\\varepsilon}_p) = 0 \\,.\n ", "fccbf69fa8a125502661d3e5261ddbd6": "t=\\tan\\tfrac{1}{2}\\varphi.", "fccbff341f53ce91afbaca1769c489c0": "u=wau'", "fccc1c4a4c56de6aa8477021e1baa292": "\\operatorname{E}(\\mu_i) = \\mu + T_i", "fccc2ed81ad8476182345925665c85c9": "\\mathbb Z_n=\\mathbb Z/n\\mathbb Z", "fccc3ec6bd8986db59179ed54133f90d": " \\omega = \\sqrt{\\frac{k}{m}}, ", "fccc4b090cf6d17e3be3fd3085d21075": " O_2 ", "fccc7c83963945ab1f88a9b6c40ad736": " V = p, E = p\\ f = \\operatorname{let} x : x\\ q = f\\ (q\\ q) \\operatorname{in} f\\ (x\\ x) ", "fcccbb506d575f27c55e2acd1309bce4": "a=\\alpha(e)", "fccce6eea62340c73afdacf17de0319a": "\\vartheta(z+1; \\tau) = \\vartheta(z; \\tau)", "fcccf7e531d0726041d1c84922c6a08e": "T_{tot} = \\frac{4}{28} + 2 + \\frac{1.5 \\times 28}{2} \\left(\\frac{1}{2.5} \\right)", "fccd2af01431c29d68ee28fe85b95d79": "\\int_V(\\nabla\\cdot\\mathbf{g})\\ dV = \\int_{V} (-4 \\pi G\\rho)\\ dV.", "fccd51027d5ace13eea34ee881ec64c9": "\\eta\\,\\! (d) = \\frac{amount\\ of\\ light\\ entering\\ through\\ the\\ center\\ of\\ the\\ pupil\\ to\\ produce\\ a\\ certain\\ response}{amount\\ of\\ light\\ entering\\ at\\ a\\ distance\\ d\\ away\\ from\\ the\\ center\\ to\\ produce\\ the\\ same\\ response}", "fccd5fa3d6e4f3d52829ee1dcd18f245": "P_tf(x) = \\mathbf E^x\\left[f(X_t)\\right]", "fccd8c771a413e5443db42194ee88998": " \\pi_4 =\\frac{h L}{k} = Nu", "fccdda047e7669baa0718947613c2999": "\\mathbf{a}_{\\mathrm{r}} = - \\omega^{2} r \\ \\mathbf{u}_\\mathrm{r} = - \\frac{|\\mathbf{v}|^{2}}{r} \\ \\mathbf{u}_\\mathrm{r} \\ ", "fcce0ce7d5ec167beeee86a2d2a35f27": "\\hat{u}(f) = {{\\underline{\\hat{u}}(f)}}{{\\Big|}_{\\Delta f}}", "fcce1170c2ae6596772d327184db2a1d": "\\psi^{(-3)}\\left(\\frac12\\right)=\\frac1{16}\\ln(2\\pi)+\\frac12\\ln A+\\frac{7\\,\\zeta(3)}{32\\,\\pi^2}", "fcce17d9ab574eff7248a416f58e26ee": "\\langle \\hat{A} \\rangle = \\int_R \\psi^{*}\\left( \\mathbf{r} \\right ) \\hat{A} \\psi \\left( \\mathbf{r} \\right ) \\mathrm{d}^3\\mathbf{r} = \\langle \\psi | \\hat{A} | \\psi \\rangle .", "fcce7812c245d46a6668a1280055a8c9": "\\ +F^n =_{def} \\{x_0x_1...x_n : F^n x_1...x_n\\}.", "fcce8dbd2cc819c289c210aa6b4d788c": "pH = pK_a + \\log \\left( \\frac{[\\mbox{base}]}{[\\mbox{acid}]} \\right)", "fccebb0f2a9ef1399c3c7a1a317f4679": " {\\lVert A \\rVert^2}=\\sum_{j=1}^{m}{\\lVert a_j \\rVert^2} ", "fccef61c6683aa19bf5be93e13c1729e": "\\vec B_0=0", "fcd05c66465e16dd641e061a32f78036": "L^{X/Y}_1 \\to L^{X/Y}_0", "fcd0da779784ef29aed480398fa8602c": "V_{\\rm oblate} = \\frac{4}{3}\\pi a^2 b", "fcd0f3c3e5defcf22177092f176f4383": "R = \\frac{9.81g}{(\\frac{\\pi \\times \\mathrm{rpm}}{30})^2}", "fcd10e1ad9e2f9b311da15dc9826594b": "A_i\\rightarrow A_k", "fcd12ffde7205ffc6047467d54eb78cf": " \\int{x(t)dt}|_{t=z} ", "fcd14f2fb5c6197b9c0dfbfe4cb16d9a": "\\Gamma = Vc-(V+ v)c = -v c.\\,", "fcd158fa0df32a7ba9f5cfd9971ab00e": "\\begin{align}\n \\operatorname{E}[X + c] &= \\operatorname{E}[X] + c \\\\\n \\operatorname{E}[X + Y] &= \\operatorname{E}[X] + \\operatorname{E}[Y] \\\\\n \\operatorname{E}[aX] &= a \\operatorname{E}[X]\n\\end{align}", "fcd18210b6342eb1d958e1a2594c3df2": "zP^{-1}", "fcd1a43831e312bc0dca7606e588309e": "PGL = P\\Gamma L,", "fcd1affd0d42106878a8136321dbb6b1": " P_A = A A^\\mathrm{T} \\, ", "fcd1f45d0b57393739a6a84ea065fb44": "\\sigma(\\vec{u}, \\vec{v}) = \\vec{u}^{T} M \\vec{v}", "fcd28895b26aaa0bdf0188c4d5a7f5a1": "\\scriptstyle{J_\\pm = J_\\mp^\\dagger}", "fcd2926d6699bf279c3cc180ab510774": "TR = \\frac{f_{max}}{P_0} = R_d\\sqrt{1+(2\\zeta\\beta)^2}", "fcd2daba75fe65ab84d5ec43b6f79a16": "\\pi_1 : X \\to X_1, \\pi_2 : X \\to X_2", "fcd2dd7069cedc9e2a59778f10910ffa": "h(x) = \n\\left\\{\\begin{matrix} \n0 &\\mbox{if}\\ x\\neq1\\ \\mbox{in}\\ G \\\\\n\\mbox{undefined/does not halt}\\ &\\mbox{if}\\ x=1\\ \\mbox{in}\\ G\n\\end{matrix}\\right.", "fcd2ddf02efe91ce76b17fe4a59532ce": "P = \\{X\\}", "fcd3366f53ac83aa09241d365bb7f4dc": "\\sum_{r=1}^gd_r|\\psi_{nr}\\rangle", "fcd33d3790b641b25df73ab8a1a2ad29": "Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2;", "fcd34d647dc878cc45ec0f534ab8c6db": "(A_w)", "fcd391bdaafdf9362ce0ca390aaa6fef": "V_\\mathrm{d} = x_\\mathrm{max} \\times S_\\mathrm{d}", "fcd3a4474f0026c03e34d83356b5ac89": "\\left( \\left( \\right) \\right):\\mathbb{R} \\rightarrow \\mathbb{R}", "fcd3c789d30f7fd3dc515b598a22f651": " \\gamma_{xx} ", "fcd4177f8db0b5578eef8f71f46ad125": "\\textstyle \\operatorname{mod} \\, 0 ", "fcd431fe6a90be8b4f87256b30d25116": "D^n_x.\\,", "fcd4b3c2d90961b76afa8b4b43f11158": "\\overline{e}_b(-1,i) = 0\\,\\!", "fcd4ef6f4a49ec4e2da6c8671d488d6d": "\\sqrt{M})", "fcd629c85a73387bdc2e2fd55156f89c": "2\\mathrm{H}_2\\mathrm{O}_2 \\rightarrow 2\\mathrm{H}_2\\mathrm{O} + \\mathrm{O}_2", "fcd6868f5922f3da92402d0706eac72f": "\\hat{g}(X,X) = \\Omega^2 g(X,X) > 0", "fcd6bac271caab594a149c86b5e893ad": " X_{t}=\\Lambda_{t}F_{t}+e_{t},", "fcd6bb9e9aa1ff363f38a74fd3313354": "K={1 \\over 2}mv^2", "fcd6cd0f41052baaf33802066ed5668a": "\\overline f", "fcd7250f4f38cdeba710dea8eb2a0352": " {} = \\begin{vmatrix} x_0 & y_0 & z_0 \\\\ x_1 & y_1 & z_1 \\\\ x_2 & y_2 & z_2 \\end{vmatrix} ", "fcd72c5b6d0a3cafac81411e31016c14": "{M} = \\frac{250}{f}.", "fcd7600b62641a96e89892ac00bd04c0": " M = m_{\\rm star} - 5 ((\\log_{10}{D_L}) - 1)\\!\\,", "fcd7c65eedfea11f5d49d2524c7134a8": "RPF \\times P_a = RPF \\times P_v + U_x V", "fcd7d8206077e4ac3d7817a04c0764a7": "x^8 + x^4 + x^3 + x^2 + 1", "fcd7db1df6a486896c6f49fc8e9d79de": "\\int \\left| \\sin{ax} \\right|\\,dx = {2 \\over a} \\left\\lfloor \\frac{ax}{\\pi} \\right\\rfloor - {1 \\over a} \\cos{\\left( ax - \\left\\lfloor \\frac{ax}{\\pi} \\right\\rfloor \\pi \\right)} + C\\;", "fcd88203e9115f4c4d61510a5d24e6df": "y_0, y_1, \\ldots", "fcd8cc8d884703999b12cabd93acd33f": " a_3 = 1, \\quad a_2 + a_3 = 0, \\quad a_1 + a_2 + a_3 = -1. \\,", "fcd8dac4d0a86ca5c0b4528a0b6b9088": "\\{ C (\\vec{N}) , H (M) \\} = H (\\mathcal{L}_\\vec{N} M)", "fcd8f989b50d105082325e11178bad57": "\\tbinom{n+x-1}n", "fcd9015bd07afad4a18cfee9ebb60106": "\\int_{-\\infty}^\\infty {e^{itx} \\over x^2+1} \\,dx=\\pi e^{-\\left|t\\right|}.\\quad\\square", "fcd98de8ed1a640709e888f86bd5e5e9": "\\sum_\\rho = \\lim_{N\\to\\infty} \\sum_{|\\Im(\\rho)|\\le N}.", "fcd9aa0316cdcd448031aa1529976807": "{V}", "fcda27715099cad3a7b99e133e035b60": "\\underline{\\mathrm{Hom}}(\\Omega^n(M), N) \\cong \\mathrm{Ext}^n_{kG}(M,N) \\cong \\underline{\\mathrm{Hom}}(M, \\Omega^{-n}(N))", "fcda628b4be4e0332541d3f1d03d9eb4": "\n\\mathrm{Re}\\, G^{\\mathrm{R}}(\\mathbf{k},\\omega) = -2 P \\int_{-\\infty}^{\\infty} \\frac{\\mathrm{d}\\omega'}{2\\pi}\n\\frac{\\mathrm{Im}\\, G^{\\mathrm{R}}(\\mathbf{k},\\omega')}{\\omega-\\omega'},\n", "fcda74070803366c30b7f3c89076d728": "S \\to SS", "fcdab4a032d6ca02e4d12a4855a260ff": "\\Delta w'''=\\lambda", "fcdbb249382e34e9d699ae99bef0679d": "s_{5,s}=s_{3}=s_{f}+x_{5,s}s_{fg} \\, ", "fcdbd726170d2bbe60c46d560efdf977": "\\left\\langle \\cdot \\right\\rangle _{t}", "fcdc375a4ca77d06f8899976b167e42a": "\\frac{I_{|n|}(\\kappa)}{I_0(\\kappa)}e^{i n \\mu}", "fcdc4dbb9c037c4e4b830c4423ece07e": "\n\\sum_{\\alpha} v_{\\alpha\\alpha} = 0 \\quad \\hbox{and} \\quad \\sum_{\\alpha} Q_{\\alpha\\alpha} = 0 .\n", "fcdcb50b0c0589d9df8bc6ccaf8caaf7": "\n\\Pr[X_1+X_2 \\leq x] = F^{*2}(x) = \\int_{- \\infty}^\\infty F(x-y)\\,dF(y).\n", "fcdcf8b038e1ee09436b41dba208e771": " k_0 ~ = ~ 2\\pi/\\lambda ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2.3) ", "fcdd02fc8ccd03ba5305dc6be13f1c4b": "\\Delta{z}", "fcdd122bf5aa245e43f1b0e929651411": " \\tan y = x \\, ", "fcdd54018486e37227d9c23ab4ced7a5": "f(x,y,z) = 2^{2/3} \\cdot 3^{1/2} \\quad \\mbox{when} \\quad \\frac{x}{y} = \\frac{1}{2} \\sqrt{\\frac{y}{z}} = \\frac{1}{3} \\sqrt[3]{\\frac{z}{x}}.", "fcdd66fe10d819323c97822592f85a5d": " \\displaystyle{|a_3|=2\\int_0^\\infty |\\Re \\alpha^2|\\, dt +4\\left(\\int_0^\\infty \\Re \\alpha\\, dt\\right)^2}\n\\le 2\\int_0^\\infty |\\Re \\alpha^2|\\, dt +4\\left(\\int_0^\\infty e^{-t}\\,dt\\right)\\left(\\int_0^\\infty e^t(\\Re \\alpha)^2\\, dt\\right) \n=1 +4\\int_0^\\infty (e^{-t}-e^{-2t}) (\\Re \\kappa)^2\\, dt \\le 3, ", "fcdd76ef5717a8ee4ce8b6417e668c86": "r^2 = (r-h)^2 + a^2 = r^2 +h^2 -2rh +a^2,", "fcdd9356ff6ec0d440ede54b9f9f8b76": "p^* = \\begin{cases} p & \\mbox{if } p = 1 \\text{ (mod }4), \\\\ -p & \\mbox{if } p = 3 \\text{ (mod }4). \\end{cases}", "fcdda204d6f74f37037d3be806190b19": "z\\ \\dot{=}\\ x \\cdot y", "fcddc84862fb3e5da8952ca301a5cd39": "[-1,\\ 1] \\times J", "fcddf6faa286f05cc6df678916d4cff1": "\\mbox{QMA}", "fcde1f1b0cfb811280a7dfb2f702caca": "\\forall x \\, \\forall \\varepsilon \\, \\exists \\delta \\, \\forall y \\, ( \\, |y-x|<\\delta \\, \\Rightarrow \\, |f(y)-f(x)|<\\varepsilon \\, ),", "fcde64afeefdbc8d374e3ea417a1dee5": " \\langle\\phi|S|\\phi\\rangle = \\sum_{i,j} c_i^*c_j \\langle f_i|S|f_j\\rangle = \\sum_{i,j}c_i^*c_j \\delta_{ij}\\lambda_i = \\sum_i |c_i|^2 \\lambda_i.", "fcde7fe77bfe2a6144ada0e7219180a2": "\\int_\\Omega(\\nabla^2 u + f)(v - u) \\, \\mathrm{d}x \\ge 0\\text{ for all }v \\le \\varphi.", "fcdeae1d614bc9bcfb2ce56a9cd39890": "\nI = (\\delta + \\mu) K = \\min\n \\begin{cases} \n(\\tilde{\\delta} + \\mu ) K \\\\\n(\\tilde{\\nu} + \\mu) K/\\bar{\\lambda} \\\\\n(\\tilde{\\eta} + \\mu) K / \\bar{\\varepsilon}\n \\end{cases} .\n", "fcdeb937ef54aaeb6cfe6aecad01329b": " f = \\sum_{n=0}^\\infty \\langle f, \\psi_n \\rangle \\psi_n.", "fcdfad521621d45add5580a035001e76": "\\Phi : \\mathbb{C} ^{n \\times n} \\rightarrow \\mathbb{C} ^{m \\times m}", "fcdff4c708b49ed848d06a7c07f768f8": "\\begin{pmatrix}\\frac{\\alpha}{x_m^2} &-\\frac{1}{x_m} \\\\ -\\frac{1}{x_m} &\\frac{1}{\\alpha^2}\\end{pmatrix}", "fce01ed277e175dcf3203133b2bb2add": "A = 7\\sqrt{3}a^2 \\approx 12.12435565a^2", "fce025c3c7050dc0009df4ce45e87e21": "C_k= \\|(A - \\mu I)^{-1}b_k \\|. ", "fce0308be73bb87ef29f2cfb05b60518": " u = \\frac{ -|\\mathbf{g}|^2}{8 \\pi G} \\, ,", "fce03906fe7acd8b87f629e576eec3a5": "\\textstyle Ma_c=\\tfrac{1}{2}v_c^2\\rho C_LA -Mg", "fce0b52535547615ddcc70d77f9ee1fe": "\\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} e^{u} du dv = \\infty", "fce0b60c97423efd064a6368737ed418": "\\bar{\\mathfrak{T}}", "fce0c62e14b8ed9088ac4f332c687617": " L = \\lim_{ n \\to \\infty } b_n ", "fce104229bf79573ba3e4d1786294adf": "\\mathbf{\\hat r} = i \\hbar\\frac{\\partial}{\\partial \\bold p} = i\\frac{\\partial}{\\partial \\mathbf{k}}", "fce16350fbd0ffa270fc441421786285": "g_j\\in\\,A\\,\\,\\text{or}\\,\\,B", "fce19c8ac9e7ff84825507d99e9c2a89": "\\forall P\\in\\Pi(A)\\,(P\\vdash_L B).", "fce229fe841923a9e5552cea59b0e8dc": "B_{es}", "fce2658db89038cf5997e62725d7a1f5": "N \\left (X_t \\mid \\mu_{it}, \\Sigma_{i,t} \\right ) = \\dfrac{1}{(2\\pi)^{D/2}} { 1 \\over |\\Sigma_{i,t}|^{1/2}} \\exp\\left(-{ 1 \\over 2 }( X_t -\\mu_{i,t} )^T \\Sigma_{i,t}^{-1} \\left (X_t -\\mu_{i,t} \\right )\\right)", "fce2b739ffc4055cb179368ca90bb302": "\\operatorname{Var}[\\bar x(t)] = \\operatorname{E}\\left[\\left(\\bar x(t) - \\operatorname{E}[\\bar x(t)]\\right)^2\\right] = \\frac{1}{N^2} \\operatorname{E}\\left[\\left(\\sum_{k=1}^N n(t,k)\\right)^2\\right] = \\frac{1}{N^2} \\sum_{k=1}^N \\operatorname{E}\\left[n(t,k)^2\\right] = \\frac{\\sigma^2}{N}", "fce2de5486cefa49494a07a55074db22": "\n\\hat{z} = z_0 (\\hat{a} + \\hat{a}^\\dagger).\n", "fce348d9ec35f6582d66d39d0b68ee8b": "\\, \\delta W ", "fce398330a7f0feadcf840c9df1f4042": "\\ M ", "fce3be89dd426525bcb9fb94dc853b05": " V=\\bigoplus _{g\\in G}V_g", "fce3bf7ad5525ddd876a5c5bc541cbb5": "\\begin{array}{rll}\n\\text{As found above,} &\\log_{10}0.012\\approx\\bar{2}.079181 \\\\\n\\text{Since}\\;\\;\\log_{10}0.85&=\\log_{10}(10^{-1}\\times 8.5)=-1+\\log_{10}8.5&\\approx-1+0.929419=\\bar{1}.929419\\;, \\\\\n\\log_{10}(0.012\\times 0.85) &=\\log_{10}0.012+\\log_{10}0.85 &\\approx\\bar{2}.079181+\\bar{1}.929419 \\\\\n &=(-2+0.079181)+(-1+0.929419) &=-(2+1)+(0.079181+0.929419) \\\\\n &=-3+1.008600 &=-2+0.008600\\;^* \\\\\n &\\approx\\log_{10}(10^{-2})+\\log_{10}(1.02) &=\\log_{10}(0.01\\times 1.02) \\\\\n &=\\log_{10}(0.0102)\n\\end{array}", "fce3c08b2bcd52ea503ed843d6cc7726": "g(\\theta)=0", "fce3cca9d1c24b1b8e4aeda95e30d59b": "Q=I_3+\\frac{1}{2}(B+S+C+B^\\prime+T),", "fce3f5b9c1d8a8b1485fdb51da03ae97": "V_r = \\frac{V}{RT_c/p_c}\\,", "fce3fef2ba8ae49b7b996f47b24bed08": "\n\\{x_i, p_j\\}_{DB} = \\delta_{ij} -x_i x_j ,", "fce43af2a0beacbed4fdd621bc3279c9": "\\alpha(g)=f\\circ g \\circ f^{-1}", "fce4609b910f497ccee6637d8581cdf3": "{d \\over dx}(\\rho u\\phi) = {d \\over dx}\\left( {d\\phi \\over dx}\\right) ", "fce461268779d1459ac1c4bef83b4615": " \\langle s|s\\rangle = 1 ", "fce487ef2cf84e9735fde1fc2506d707": "\\max_{-1 \\leq x \\leq 1} |f(x) - P_n(x)| \\leq \n\\max_{-1 \\leq x \\leq 1} \\frac{|f^{(n+1)}(x)|}{(n+1)!} \n\\max_{-1 \\leq x \\leq 1} \\prod_{i=0}^n |x-x_i|. ", "fce4d6fb75fa90ec328a6722c725f11d": "A \\neq C", "fce4d933426b1c5f979d3489e22a3686": "\\mathbb{Q}(\\sqrt{2})\\supseteq \\mathbb{Q}", "fce50412c3a0ba57f75e5f5f6e2dfa90": "\\forall t . \\textit{open}(t+1) \\leftrightarrow \\textit{opendoor}(t) \\vee (\\textit{open}(t) \\wedge \\neg \\textit{closedoor}(t))", "fce5258f45b801cb800aad389a34aeca": "dist(p,\\hat{p})", "fce5aef0e41c47c7fa4f7967066c6b9b": "x_{down}^{(jam)}(t)", "fce5e4e3782c7d926cf56b1f2aeba391": "\\left[ X \\right]", "fce5f64a4fa1f8a90c5a2b6a306049b4": " \nt \\approx \\int_0^{z_s} {dz \\over c} + \\int_0^{z_s} {dz \\over c} {\\alpha(z)^2 \\over 2} - \\int_0^{z_s} {dz \\over c} {2 \\Phi \\over c^2} .\n", "fce60c713dd0e5450a263fcede9fcd74": "\\Psi:S^n\\to S^n \\vee S^n", "fce6188ce3c3a81c74b51d047964df4b": "\\scriptstyle{z}", "fce66ca6c6da5cb49b2b46e3a59863c6": "6\\times (1^2+2^2)", "fce6a7b22ff3e26291b4fc113e15e0c7": "\\forall n\\in\\omega\\,f\\upharpoonright n\\in X_n", "fce70cae65e95681f7be6b47273e8ec8": "\\frac{\\$40,000}{\\$50,000} = 0.8 = 80\\%", "fce70ea5403e07b240a52c2a979853cb": "\\int_0^\\infty x^{2n+1} e^{-a x^2}\\,dx\n= \\frac {n} {a} \\int_0^\\infty x^{2n-1} e^{-a x^2}\\,dx\n= \\frac{n!}{2 a^{n+1}}\n", "fce7387b3f926dfd66de41edf3e47efc": "k<\\alpha", "fce79acb784f1b6b0b9d1dda2d37ed61": "\\scriptstyle{\\hat{H}_0}", "fce7afb30783c8f035b3249114c747db": "1 \\ ", "fce7da73d7bc25bcb2440cd86f36a0e7": "y = 3 \\quad (L_2)", "fce80d2f8991b8ce062aaa3bb9c6bbbe": "-\\nabla U(X)", "fce812d22f1e05b483ec896fc11d2be4": "Y_{2}^{0}(\\theta,\\varphi)={1\\over 4}\\sqrt{5\\over \\pi}\\, (3\\cos^{2}\\theta-1)", "fce816cdc950bf795e05893fa6edca4c": "\n\\int^{r} F(r) \\, dr = \\frac m2 \\left[\\left(\\frac{dr}{dt}\\right)^2 + \\left(\\frac hr\\right)^2\\right]\n", "fce842c30af5fb197a59d4e8e10827bc": " u(x) \\geq \\bar{u} ", "fce84c96b7a1290cbdf90c798ca928e2": " (y_{1}, y_{2}) ", "fce8631758e7908e6d68a9b5930a93eb": "eq \\circ u = m", "fce8637cd59c5b732894506b8dff821d": "\\phi \\lor (\\forall x \\psi)", "fce86cf6ab4a8dca828e88b13ff352fc": "f(\\xi) \\leftarrow \\hat{f}", "fce91479fad109410a5dbf9b7b5b0b9e": "(x-a)^2+(y-b)^2=r^2", "fce91cc7ef77038ca0495d19b8a2e50a": "\\begin{align}\n\\operatorname{E} \\left[ X^p \\right] &=\\sigma^p \\cdot (-i\\sqrt{2}\\sgn\\mu)^p \\; U\\left( {-\\frac{1}{2}p},\\, \\frac{1}{2},\\, -\\frac{1}{2}(\\mu/\\sigma)^2 \\right), \\\\\n \\operatorname{E} \\left[ |X|^p \\right] &=\\sigma^p \\cdot 2^{\\frac p 2} \\frac {\\Gamma\\left(\\frac{1+p}{2}\\right)}{\\sqrt\\pi}\\; _1F_1\\left( {-\\frac{1}{2}p},\\, \\frac{1}{2},\\, -\\frac{1}{2}(\\mu/\\sigma)^2 \\right).\n \\end{align}", "fce94f476faec4f08a61d8620a4b405b": "i(t) = i(0)e^{-\\frac{R}{L} t} = i(0)e^{-\\frac{1}{\\tau} t}", "fce9972c1d4c88c420f34990f40bac69": "2\\left(h + 1\\right)f_\\mathrm{M}", "fce9ecd816ff0445e3a17060fb765c97": "\n \\hat\\delta_i = \\big(\\hat{Z}'_i\\hat{Z}_i\\big)^{-1}\\hat{Z}'_i y_i\n = \\big( Z'_iPZ_i \\big)^{-1} Z'_iPy_i,\n ", "fcea45b4d168b0391bae233fcbc863cf": " \\Delta_n(s) = \\mathbb{P} \\left\\{ x_n \\leq s \\right\\} - \\log_2(1+s),", "fcea57654a50a0a6a51d073046ce7adc": "L_{c} = \\sqrt{\\frac{\\gamma}{\\rho g}},", "fcea97e1621286a6b47428e71a3c3d85": "K(X)\\cong\\widetilde{K}(T(E))", "fceab556b44c87faafa7e4b84e02afda": "\n C_{ij} = \\mathbf{g}_i\\cdot\\mathbf{g}_j = g_{ij}\n ", "fceb02731a1a5273ae5181005e183193": "t_d", "fceb2716de4e11daa3fca8bb98a88079": " y'+ P(x)y = Q(x)", "fceb6d159f1d58903d3952f7a56f92b1": " \\log(f(x)) = \\log(A) + \\sum a_n \\log(x - c_n). ", "fceb6dd7a43165d09114f100a656e021": "(p \\leftrightarrow q) \\vdash ((p \\land q) \\lor (\\neg p \\land \\neg q))", "fcebc550123694eee97a61981a51162b": "3 a_2 = a_1 \\text{ and } 3 b_2 = b_1. \\, ", "fcec2825ce5ba18aa0a350e2a7b32e31": "\\tfrac{1}{2}+\\tfrac{1}{4}+\\tfrac{1}{68}", "fcec4fda10eb1e20a4592dc967e38e15": "\\ x^2 + bx = c", "fcec9522a0078c4d572e5eb09f5ebc83": "C_{8}F_{17}", "fced2639a546374c97fa21a07774ddec": "\\sigma_\\pi\\,\\!.", "fced3871c0941d5fffb52c67ad6f13f5": "\\lambda^{n-2}", "fced77646a2f09e2e9242e1751ebd7ef": "T^{-1} x, x \\in X", "fcedf9b940e8f07f786e97bc308d2878": "dis_\\text{err} \\leq t", "fcee296aa9fac18640f56ef5af409c79": "a_{6}+b_{6}+c_{6}=b_{1}", "fcee7c0beb1f697aec20c0afa33a65ca": "xy(1 \\vee z) \\vee \\bar{x}z(1 \\vee y)", "fcee83ce95ce4f20cca59269beebac5f": " w \\in \\Sigma ^*, a \\in \\Sigma ", "fceeec9c3c089a4e4630665fc9242691": "\\Gamma_{21}", "fceef972ff18d399477852725403e6c9": " b = \\frac{2g}{3e} = \\frac{4\\sqrt{2 m}}{3e\\hbar} \\approx 6.830890 \\; {\\mathrm{eV}}^{-3/2} \\; \\mathrm{V} \\; {\\mathrm{nm}}^{-1}. \\qquad\\qquad (8) ", "fcef1f915cdb1dbee35e5b51f2f0bdd8": "\\det {\\mathfrak{T}}_{\\alpha\\beta}", "fcefe0785ba4b885014c44f375701952": "p(S, T) = \\frac{N^\\prime [d_2(S_T)]}{S_T \\sigma\\sqrt{T}}", "fcefe60fcfe05b4bde6dedf0c6dbb898": "y^3-xy+1=0.\\,", "fcf00e61c19a7d1713d2081a29d614ca": "\\theta(0) = 0", "fcf013e8457d875501be7ae828430851": " y_H = C_0 \\cos (k x) + C_1 \\sin (k x). ", "fcf04a6edcefe4463a357f564dedc977": "L_n [1/2, 2 \\sqrt 2]", "fcf04bc408cea98dcbc4d5b676bdbc30": "r_{n}", "fcf0656bc12d2fd7337446ed12ec5efd": "\\mathcal{FSR}", "fcf09acbb36bdc35a5ad8564ee155d6e": " dr_t = (\\mu X + \\theta Y)dt + \\sigma_t \\sqrt{Y} dW_{3t} ", "fcf0eb55e70668314fa14d73d8e826e0": "m_\\text{A} = m_\\text{e} \\ ", "fcf0ee99b6f571d37e78f52d96b06af2": "I_{D}=K'_n\\begin{matrix}\\frac{W}{L}\\end{matrix}(V_{GS}-V_{th})^2(1+\\lambda (V_{DS} - V_{DS,sat}))", "fcf12aa2df94df1b4c58d5dce4de4fc7": " W^{\\alpha}_n", "fcf12b975afff9bea2f0847b4d80969c": "B_m:=\\{ a\\in K:|a|\\leq m\\}.", "fcf14ed2da14a0a6ec73e483a927d711": "E_T = (E_{i,x} \\cdot t_x)^2 + (E_{i,y} \\cdot t_y)^2 \\, ", "fcf19b86066d62cc75de1b60b2cb3d35": "\\gamma \\left( 1\\right) =t", "fcf1bcac13990e9c0d2bcb247dc389c5": "\\, P", "fcf1c77e58fab8616c1516df22c4fe68": "(a_1, a_2, a_3, \\ldots, a_n) = ((\\ldots(((\\emptyset, a_1), a_2), a_3), \\ldots), a_n)", "fcf1d07136ccad9e08089e7f2447c718": "\\sum\\frac1{x_i} + \\prod\\frac1{x_i}=1.", "fcf1d9bda8d0b3771783fca0bb647f96": "n=\\left(\\sum_{i=1}^{k}{d_i}\\right) \\left(\\prod_{i=1}^{k}{d_i}\\right) \\, ,\\text{ e.g. } 144 = (1+4+4) \\times (1 \\times4 \\times 4) \\, .", "fcf1ddb284e1b76c1ca7afd67c081acb": "\\rho(u) = 1", "fcf1f1ddc1a83ab82b251817a1d0d5d0": "K_E", "fcf2270cf9cacb99ca79cf2949e20b3a": "\\displaystyle{F_{\\overline{z}}= \\mu F_{z} + GF,}", "fcf23f07853d7e0b71e1ceb4fc0629aa": "\\chi_+(1)\\chi_+(2)", "fcf28de539134cc81d2b8d0261abce9f": " 3h", "fcf2ef7cc44412c6e1d2f12158617ac6": "N(xy) = N(x)N(y).\\,", "fcf305fcb0182fa9cf12652f74877022": "\\simeq 10^{17}", "fcf3420b9c19b24a9cba3a6f73e07214": " \\rho ", "fcf34a9fe467f729430b5c9ff37779d7": "[F_i]\\rightarrow[F_i,1]", "fcf34b311160a5dc6a750e23f947b5e1": "d_n(D)", "fcf35934429afb1cd991314e19c96811": "\n{}_{ax^N+bx^2 + c=0,N\\equiv 1\\pmod{2}} \\,\\!\n", "fcf3d63786f02f9c0f452fc0ea4a7ec6": "u_\\nu", "fcf3eced1968e3a00d0dc5fc1a9232ab": "X\\Vdash A", "fcf42127a86f738cf0a6554249cdf1c3": "\\lim_{x \\to \\infty} \\log_a x = \\infty", "fcf43f7db8d2899a39beaa550eb580e4": "B \\neq D", "fcf453ece0e3ecfeb9fc328fa2c8007d": "x_i=false", "fcf45956d8cd892c0922d300cfa706ec": "\\frac{dy}{dt}=x-y+z", "fcf4d1faa3fe0ff019c32932a6ba4303": "P_\\mbox{acc}", "fcf4d5edc80ed3aec38dd438a7d7fdb2": "~\\Phi(x)=e^x-x-1~", "fcf4f42966a73211cafa7599be61b3b2": "\\psi(0, x_2, \\dots, x_n) = \\Phi(0, (0, x_2, \\dots, x_n)) = (0, x_2, \\dots, x_n)", "fcf50569d5f52088b89268d4990f20c2": "\\tfrac{E}{2G}-1", "fcf5b377b736cba025b35881808ed01c": "x^5+x+a. \\,", "fcf630db1ec23102f86d4363070a9783": "\\sin(1/z)", "fcf6a5b238524e14ec64fa092efc3f94": "x \\leq x \\land y \\leq x \\land ", "fcf6c67151ad238a935885f49650cac4": "S \\rightarrow AbA \\mid Ab \\mid bA \\mid b \\mid B", "fcf707013d65e751789f30b0b665ce51": "\\{X_1,\\ldots,X_m\\}", "fcf772861c6c3165cebcc4609bdba145": "Ci = {{C_L^2} \\over {\\pi \\times \\lambda \\times e}} ", "fcf844cd9a4d69fa0449e396f623b186": "\\dot{c}_m(t) = - c_m\\langle\\psi_m|\\dot{\\psi_m}\\rangle", "fcf8f4d231926a645275212ae68e597d": "+ 100x\\,", "fcf92b2e88c6112e0345f9b5499c37a7": "HoldsAt(f,t_1)", "fcf99e3845e91ddee786c6f8f9c84b1f": "\\mbox{GRR}=\\frac{2^N}{\\mbox{GCD}(\\Delta F,2^N)}", "fcf9a7453b996c07ac97ec4cd86f910f": "B \\otimes B^{\\text{op}}", "fcfa48f6dd43fa345fbc3929b85f3c7b": " \\exists a: \\forall b: \n\\exists c: \\forall d: \\lnot R(a,b,c,d)", "fcfa568761ee167a54aabf56614c3261": " \\hat{x}_L(k) ", "fcfb3bb44e71ea95ff230e93a2784a5c": " = 0.28\\,", "fcfb66a4807698760ce2c08c10951c2c": "\\left( \\omega - \\bold k \\cdot \\bold V \\right)^2 = \\Omega^2(k),", "fcfba574b3d4174a07cfbcfc2ad3001e": "dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \\, ", "fcfbdeffc1db293cc69992e325329273": "I(r)", "fcfc650acde0359e5dda750593a7592b": "\\left\\langle a_0,\\dots,a_{n-1}\\right\\rangle \\longmapsto \\mu a . \\left[ \\forall i < n \\; \\left(\\beta\\left(a,i\\right) = a_i\\right)\\right]", "fcfcbd130c01389717669e8a2772703b": "O(N^{2})", "fcfce6b95e404ed1bda7f782531cb7d3": "R = \\frac{\\lambda}{10^6} = \\frac{550}{10^6} = 0.00055", "fcfcecd156d926a6d7d517e1615182d2": "\\nu_M\\colon M \\to BO(k) \\to BO", "fcfd00b4145d600bb67a1d5fe58d3482": "\\rightarrow \\to, \\nrightarrow, \\longrightarrow\\!", "fcfd54fa244815dda636fcf5f54e3822": "\\sum_{n=0}^\\infty {T_n(x) \\over n!} t^n=e^{x\\left(e^t-1\\right)}.", "fcfdacbada694b453f0c3f0720e8490a": "\\varphi_j(2)", "fcfdc5d5a1e35784f4f434021168d1d4": "{d\\lambda \\over \\lambda} = \\varepsilon - 3 B \\lambda ", "fcfe5d7a6126f418e63a6251292de82e": "H(W|\\hat{W}) \\leq 1+nRP^{(n)}_e = n \\epsilon_n", "fcfede714889de7a92b91b86fbbfb03a": "rx(1-x)", "fcff0a89c1edf6c23499a113d4fd0be1": "\\tan\\delta' = \\frac{\\gamma\\cdot(\\sin\\delta+\\beta)}{\\cos\\delta}", "fcff210f0ac9adfe9e696184894d912e": "S=S_G+S_K+S_S+S_M\\;", "fcff6e5e5f38bf4fc3875f33f92f0262": " \\frac{1}{(R-x)^2}+\\frac{1}{(R+x)^2}=\\frac{1}{r^2},", "fcff71ce7e877b85fc518bcd4995b8f2": "Y = f(X_1,\\ldots,X_N)", "fcffd66e9f563d6106a91cecd3545e34": "V_0=2,V_1=1,V_2=R-2Q=a^2+b^2,V_3=R-3Q=a^2-ab+b^2", "fcffea4e50964e75e2018f122471dd31": "\\cot(A/2) =\\frac{s-a}{r}\\,", "fd001f45bdbf521a46299923a567eed5": "x^5-x^4-x^2-1", "fd002fe1a6ba98ccf97adcb5e6781c74": "\\langle \\hat{A}\\rangle={1\\over Z_0}Tr[\\hat{\\rho_0}\\hat{A}]={1\\over Z_0}\\sum_n \\langle n | \\hat{A} |n \\rangle e^{-\\beta E_n}", "fd00936b4921c3d95dd232d9c9f5c761": "e^{ix} ,\\, e^{-ix} ,\\, e^x ,\\, xe^x.", "fd00af6b99cea45f23aa50fdcff5af2c": " \\psi(0) ", "fd00d6d060fdd0bb54e025e6cf0ee686": "a_3 a_2 a_1 a_0 = a_3 \\times b^3 + a_2 \\times b^2 + a_1 \\times b^1 + a_0 \\times b^0 ", "fd00f3c47e98fe7426482dc285871807": " {\\partial w\\over \\partial \\overline{z}} = \\mu {\\partial w\\over \\partial z}.", "fd013710c0492ae94cdca1e7d469d659": "f\\colon U \\to V", "fd0156e602e5b05e1d3e6ca77eca1146": "\\subseteq\\left[\\mathcal{A},\\mu\\right]", "fd0173ec2accb60536879cb5a2c2362a": "P_{\\mathrm{after}} = \\frac{P_{\\mathrm{before}} \\times D}{P_{\\mathrm{before}} \\times D + 1 - P_{\\mathrm{before}}} ", "fd01fc842d6881f725ab90d24afd1ae8": "\\int_{-1}^1 \\left[C_n^{(\\alpha)}(x)\\right]^2(1-x^2)^{\\alpha-\\frac{1}{2}}\\,dx = \\frac{\\pi 2^{1-2\\alpha}\\Gamma(n+2\\alpha)}{n!(n+\\alpha)[\\Gamma(\\alpha)]^2}.", "fd02649ddaad46c8ad768986711ae256": "P = \\exp\\left( A + \\frac{B}{C+T} + D \\cdot \\ln \\left( T \\right) + E \\cdot T^F\\right).", "fd02d9e8ca15d9c217423aa0cc39173b": "d(O_{j}, O_{p})", "fd02e5432f4cd34f327429d600b082b5": "2b = b \\,", "fd030a4ebebf3adcd667532fb75e336f": "\\bigcap_{\\gamma<\\xi_i}C_\\gamma", "fd03ad83b07ca984e66d00b78ac1d094": "10 \\uparrow^n 10=(10 \\to 10 \\to n)", "fd03c7cd6bc9facd87bee63034614826": "1\\le i \\le 3", "fd03e0d018683627694ab1610ee51d21": " h(x_1, x_2, \\ldots, x_n) = h(x_1) \\cdot h(x_2) \\cdots h(x_n). ", "fd03e1d21dbefe32dc550d34481db5a3": "V_I", "fd040b6180bdd8f7a8c73e84e4e6ff54": "G^{[l]}", "fd04a7c1134db4485fb021bc0e664586": "\\scriptstyle \\boldsymbol I_p", "fd04c8bd646e5e4dcca750deec02fa12": " \\hbar ", "fd04cbf8e8aaa431a0b187d6b10a2537": "\\exp\\left[O\\left(\\sqrt{n \\log n}\\right)\\right]", "fd04eec3facbc78df5f3c57d72a26623": "|\\gamma|=1.1072538", "fd05145b2bfae07ed7e1234448b033d4": "\\alpha = -\\frac{i E}{\\hbar} ", "fd05aa292b76186d9d61ee8e19bd169b": "\\mathbf{\\hat{\\boldsymbol{\\imath}}} = \\begin{bmatrix}1\\\\0\\\\0\\end{bmatrix}, \\,\\, \\mathbf{\\hat{\\boldsymbol{\\jmath}}} = \\begin{bmatrix}0\\\\1\\\\0\\end{bmatrix}, \\,\\, \\mathbf{\\hat{\\boldsymbol{k}}} = \\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix}", "fd05d8d90456c441c8f10641bd8576bc": "f(x)=0", "fd0613524e1ef55d022bd23b756a58e7": "\\displaystyle{u_+ = u_-,\\,\\,\\, {\\partial_{n+}u\\over \\lambda+{1\\over 2}}={\\partial_{n_-}u\\over \\lambda -{1\\over 2}}=\\varphi.}", "fd06449a7bfb71ed545e238fac622a68": "\\theta_{\\mathrm{eff}}", "fd0649eb8663703ecd05b99de45a5d25": "y\\in X", "fd0693de69b8b49e81b6280fa0883b0b": "h {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} k = k {~\\wedge\\!\\!\\!\\!\\!\\!\\bigcirc~} h", "fd06a2f8b391c06776281c147151a7e9": "\\mid\\! GF(p^6)^*\\! \\mid=p^6-1", "fd06ad0b13073ec1179d11825bcddecf": "P-Q_i A_i = O((x-\\lambda_i)^{\\nu_i})\\qquad ", "fd06db0ce0093edbba8ed3a9798769c3": "v=\\left( {{v}_{x}},{{v}_{y}} \\right)", "fd06ee3575fc3e06773f826ec15d02ce": "I_{\\mathcal Q}(\\times)\\colon Q\\times Q\\to Q", "fd07631e871989ebb8fd26676598ee80": "f(x) = b + \\frac{\\alpha A (x-a)}{|x-a|^\\epsilon}", "fd077d41f5fc79d98aa8b08a0bc60c37": " {D =\\ } {(1.6122\\times10^{-9})\\lambda^4} - {(2.66575\\times10^{-6})\\lambda^3} + {(1.6242\\times10^{-3})\\lambda^2} - {(0.4277)\\lambda} + {41.57}", "fd07a37503ab51d99f6b828b113b1a59": " \\sigma^2 = ( \\alpha \\beta )^2 ( 1 + \\frac{ 5 \\alpha^2 }{ 4 } )", "fd07b26b2c08cfedfbfa62219b9c9395": "\\hat{\\alpha}>2 \\pi", "fd07c1eba27a6c6d930e15b129b6f29d": "(I :J)=\\mathrm{Ann}_R((J+I)/I)", "fd07d0ac4797275dec8e17d5365ef49b": "q = h\\cdot(A)\\cdot {\\Delta T}", "fd0828644b90cb04de007212ef8b3cc2": "\\gamma\\tau ", "fd0841ffece2bc04b91b57cdf0fd7b87": "E_i \\, = \\, \\bigg( F(Y_u) \\, - \\, F(Y_l) \\bigg) \\, N", "fd084f51c900b9a42e2b180f1f98a942": "f_{\\text{r}}(\\omega_{\\text{i}},\\, \\omega_{\\text{r}}) \\,=\\, \\frac{\\operatorname dL_{\\text{r}}(\\omega_{\\text{r}})}{\\operatorname dE_{\\text{i}}(\\omega_{\\text{i}})} \\,=\\, \\frac{\\operatorname dL_{\\text{r}}(\\omega_{\\text{r}})}{L_{\\text{i}}(\\omega_{\\text{i}})\\cos\\theta_{\\text{i}}\\,\\operatorname d\\omega_{\\text{i}}}", "fd086fd68428187f62e45b7683b59a9a": "f_n(x) = \\begin{cases} 1/n, & x = n \\\\ 0, & x \\ne n. \\end{cases}", "fd08af177a7ba115605ca5fe97300158": "\n\\sigma _z^2 \\approx \\,\\,\\,\\frac{{a^2 \\,}}{n}\\,\\left( {\\frac{\\sigma }{\\mu }} \\right)^2 \\,\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,\\,\\sigma _z \\approx \\,\\,\\,\\frac{a}{{\\sqrt n }}\\,\\,\\left( {\\frac{\\sigma }{\\mu }} \\right)", "fd08c0b2fb6ff5b8178ea5c8915ffd5e": "f^*\\Gamma=(dy^i-(\\Gamma\\circ \\widetilde f)^i_\\lambda\\frac{\\partial f^\\lambda}{\\partial\nx'^\\mu}dx'^\\mu)\\otimes\\partial_i ", "fd08cff325b0371645f49c56ce368e8f": "M = \\begin{pmatrix}1 & -1 \\\\ 0 & 0\\end{pmatrix}", "fd092827214ae4365350d41795eb9a67": "P_{\\max} = \\sum_{i=1}^k \\min(P_{i+},P_{+i})", "fd097c99195baeb3763ac72180f78be7": " {A_\\mathrm{v}} = {v_\\mathrm{out} \\over v_\\mathrm{in}} ", "fd099bd3cdc37ae708c7245ddf84b91f": " W_2 - W_1 = 2[\\gamma(e_{ij})A(e_{ij}) - \\gamma_0 \\ A_0] ", "fd09b7034b89fe891593d97f9ee3173f": "\\frac{\\mbox{total} \\; \\mbox{votes}}{1+\\mbox{total} \\; \\mbox{seats}}", "fd0a075aeeca68de024badc48ec26957": "r \\cdot C_T \\cdot 8760\\,h", "fd0a48015e76bb4b1e64096d38481589": "^{2n-D}", "fd0a548924047162f79055dde1a0a34e": "\\frac{average_{12}-average_{34}}{2 \\Delta x}=\\frac{(f_1 + f_2)-(f_3 + f_4)}{4 \\Delta x}", "fd0a56284cd508afe6eb7f619eccfa44": " x_1, y_1", "fd0a75c20aa12de0b120bb6e42177e9c": "Q = C \\;\\rho_A\\;A\\;\\sqrt{\\frac{\\;\\,2\\;P}{\\rho}\\cdot\\frac{k}{k-1}\\cdot{\\Bigg[\\; 1 - {\\bigg(\\frac{P_A}{P}\\bigg)^{(k-1)/k)}}\\Bigg]}}", "fd0a84c478c3a250d505abb0d17e97b1": "p = .99", "fd0af5d65fe2d1674cc2b501e7ee9b81": "\\mathrm{Glucose} + \\mathrm{Alkaline\\ copper\\ tartarate}\\xrightarrow{\\mathrm{Reduction}} \\mathrm{Cuprous\\ oxide} ", "fd0afc76c30b08cca5855edf7ad62912": " f(x) = f(x) - f(r) = \\sum_{i=0}^{k+1} { a_i \\left( x^i - r^i \\right) } = (x-r) g(x) ", "fd0b0903e384bfb23d5a7ea4dc9a6b99": "Y-X", "fd0b26196fef1147a943a95d4c8738cd": " \\Im{\\theta} = \\sigma \\,.", "fd0b37a2146b2d259548e72b6c7fe229": "x = \\pm 2.", "fd0b587f5af6486c60cd38fe171c427c": "X\\to S", "fd0b6a393ae43e1048eb8131d22109a1": " P = \\rho hrgk ", "fd0bb32bebe4877aeee866f8b873904b": "1 \\leq m \\leq n-1", "fd0bd9fe45d0ee0b88ac3bbab8f59c3e": "= R_{abc}^{\\ \\ \\ d}\\,\\mathbf{\\xi}^c", "fd0c237d73a6b4b73cf3e8a9f09900c0": " y^*_{n+1} = y_n + h\\sum_{i=1}^s b^*_i k_i, ", "fd0c33a38ea6e8abdf3a516303bd27b7": "\n\\begin{align}\n\\operatorname{Var} \\left( h_r \\right) = & \\sum_i \n \\left( \\frac{ \\partial h_r }{ \\partial B_i } \\right)^2\n \\operatorname{Var}\\left( B_i \\right) + \\\\\n & \\sum_i \\sum_{j \\neq i} \n \\left( \\frac{ \\partial h_r }{ \\partial B_i } \\right)\n \\left( \\frac{ \\partial h_r }{ \\partial B_j } \\right)\n \\operatorname{Cov}\\left( B_i, B_j \\right) \\\\\n\\operatorname{Cov}\\left( h_r, h_s \\right) = & \\sum_i \n \\left( \\frac{ \\partial h_r }{ \\partial B_i } \\right)\n \\left( \\frac{ \\partial h_s }{ \\partial B_i } \\right)\n \\operatorname{Var}\\left( B_i \\right) + \\\\\n & \\sum_i \\sum_{j \\neq i} \n \\left( \\frac{ \\partial h_r }{ \\partial B_i } \\right)\n \\left( \\frac{ \\partial h_s }{ \\partial B_j } \\right)\n \\operatorname{Cov}\\left( B_i, B_j \\right)\n\\end{align}\n", "fd0d25ff8f564388944a1e435f6f5324": "I_1\\cup e", "fd0d3820dc3cfbc63d989f8d4c2aa6a5": "s_n(x)=\\sum_{k=-n}^nc_ke^{ikx},", "fd0d52c0654b85fa4832db02d8a7dd68": "\nf_{U_{1} + \\cdots + U_{N}}(x)\n= \\left( f_{U_{1}} * \\cdots * f_{U_{N}} \\right) (x)\n", "fd0dc0bf70bc3a331ddc1751d396e24e": "\\dot{j}=0", "fd0dd43523d2a381ce7ae4a0c8049148": "\\hat{H}_{s}=-|e|Ez", "fd0deabcc447eddbace9750368c53475": "\nA = \\left[ (1 - \\kappa_{\\rm smooth})^2 \n - \\left( \\sum_i { (\\theta_{xi}^2 - \\theta_{yi}^2 ) \\theta_E^2 \\over (\\theta_{xi}^2 + \\theta_{yi}^2)^2 }\\right) ^2 \n - \\left( \\sum_i { (2 \\theta_{xi} \\theta_{yi}) \\theta_E^2 \\over (\\theta_{xi}^2 + \\theta_{yi}^2)^2 } \\right)^2 \\right] ^{-1}\n", "fd0dfb663f5b85b4e5e3be948b5d2c1e": "\\scriptstyle \\cos\\,", "fd0e0951a9924dfe3c8ffac53cca46fc": " \\int_a^b y_n(x)y_m(x)w(x)\\,\\mathrm{d}x = \\delta_{mn},", "fd0e28f7f3ee99b7d7a10dc3c34dabdc": "\\sqrt{F_0 K}", "fd0e538b3f34f16a6c40edf1dc690fcc": "x = \\frac {\\pm \\pi \\left(A\\left(G - P^2\\right) + \\sqrt {A^2 \\left(G - P^2\\right)^2 - \\left(P^2 + A^2\\right)\\left(G^2 - P^2\\right)}\\right)} {P^2 + A^2}\\,", "fd0e806d2f7a59f6f8a40e70190c58a8": "P \\,\\Delta_{Q1}=Q \\,\\Delta_{P2}.", "fd0ea780ee9e2218ff4e6efc67848ab5": "A \\cap \\bigcup_{i\\in I} B_{i} = \\bigcup_{i\\in I} (A \\cap B_{i}).", "fd0f0aee2ed8a19926cd326841151188": "h_j \\in C", "fd0f3399b044302f15a367add85374d6": "\\phi' = \\phi - \\frac {\\partial \\chi}{\\partial t} \\qquad \\qquad \\mathbf{A}' = \\mathbf{A} + \\nabla \\chi.", "fd0f64713d70a81b29556838e5e83494": "\\varkappa = 2\\dot G / G, \\quad \\lambda = 2G^\\prime / G. ", "fd0f6dcbd3bb5bf85611d2c5792d492c": "f\\colon \\mathbb{R}\\rightarrow\\mathbb{R}", "fd0f7c825e6ac58de0b3e5739a541c6a": "(n-k+1)^2", "fd0f840d44df506bc060e102d6c65ed3": "\\displaystyle f(x) = \\frac{1}{(2 \\pi)^n} \\int_{\\mathbf{R}^n} \\hat{f}_2(\\omega) e^{i \\omega\\cdot x} \\, d \\omega \\ ", "fd0f846f725976bf60f5b9a083a5f503": "\\frac{\\text{C}}{\\text{P}}=\\frac{\\text{P}-\\text{V}}{\\text{P}}=\\frac{\\text{Unit Contribution Margin}}{\\text{Price}}=\\frac{\\text{Total Contribution Margin}}{\\text{Total Revenue}}", "fd0fb2ee6032ccd55ae7875d3b393c47": "I^\\text{filtered} ", "fd0fed876c55c1749a4fc4eee59d2a40": "\\alpha\\in{\\mathcal A}\\,", "fd10170a44e98cb2c01009ce9214a45a": "O(\\log\\,\\!n)", "fd107c87f9315ceabb1b7459c057c3cd": " \\mathbf{g} = G \\int_{V} \\frac{\\mathbf{r} \\rho \\mathrm{d}{V}}{\\left | \\mathbf{r} \\right |^3}\\,\\!", "fd10ddf575fc5affba88cf341ada8e6c": "10^{28}", "fd10f08e7d4740e39469879329e57ec7": "I(x,t) \\ = \\ { f_1(\\omega t-kx) \\over Z_0 } - { f_2(\\omega t+kx) \\over Z_0 }", "fd110be00b0260dd1f6cc751670dac45": "l_3", "fd111821d833b03d460dcd538255c8bd": "\\sigma_1 = \\sigma_5 = \\sigma_6 = 0 ", "fd12391eb1c055775ab7cd8a03df5865": "4\\pi G = c = \\hbar = \\varepsilon_0 = 1", "fd12635c927c61db72b6b541fb9911c4": " \\operatorname{de-let}[E] ", "fd12817418bbc4d2554f1fee48076acb": " X_i= x_{n-1} + h \\sum_{j=1}^m a_{ij} f(X_j)", "fd12cd8f3a6bb7af455611d2ff9df259": "\\alpha(k)", "fd12d1e606aa9cb51cbf3f02b4fb64c5": "L_2 = (7p_2 - p_1^2)/45.", "fd12d72093180db0d7ac71c31dce84ec": "g=\\frac{t^{2}_{r,\\alpha}s^2\\nu_{22}}{b^2}.", "fd12f1bb6102c0755530f8af70fbc8f7": "q=1/3", "fd1365a2b73e1e0a241d4c133ec0c8e0": "\\implies U^'_i(x_i)(1-\\frac{x_i}{B})=p ", "fd13c5523439f225738c19cd71bf97df": "2^{100}", "fd13d2f86b57adce9d7e962c3bf1e41c": "m^*(t_j)", "fd13e37512e7fbfcf06fb3255093df49": " A =\n\\begin{bmatrix}\n2 & -1 & 0 \\\\\n-1 & 2 & -1 \\\\\n0 & -1 & 2\n\\end{bmatrix}.\n", "fd1424c08e9c228c813aa1be0b5b5b51": "\\beta_h \\propto \\mbox{e}^{-Q_h T/R}", "fd14686da2566dc2e7cccef37d1fe938": "\\langle\\delta',\\varphi\\rangle= -\\varphi'(0).", "fd148828abe33a0dda35051bda1e2076": "\\scriptstyle x_1 \\;=\\; A", "fd148eaf89c7a8d28a757d4e416cccce": "x \\in \\mathbf{F}_p", "fd14c46f41035075354f5299b07a8c64": "P_{\\text{el}} = C_1 \\lambda_{\\text{gaz}}(T(t) - T_a) + C_2\\lambda_{\\text{fil}}(T(t) - T_a) + A_{\\text{fil}} \\epsilon \\sigma(T(t)^4 - T^4_a) + c_{\\text{fil}}m_{\\text{fil}} \\frac{\\mathrm{d}T}{\\mathrm{d}t} ,", "fd155e94823d715f928ba876383d4d9e": "(1), (2), \\dots,", "fd160e80f19b8e3179bb878ec8a8afa0": "\\operatorname{max}\\{S_{T} - K,0\\}", "fd16123fa1769c840f549f4514dd3a75": "\\alpha(\\lambda)", "fd16193a087a5a6935bef3a41ebd4199": "X_i;i=1,\\ldots,n", "fd164e627a0a852a71035d72b96071ff": "z \\mapsto \\bar{z}^2 + c", "fd167f9d4c5d918786157189701421e7": "F_1 = 1 - F_2 = \\frac{d M_1}{d (M_1 + M_2)} \\,", "fd175f6ae7b991e4612e06db472f40ac": "g=\\gcd(f^*, x^{q^i}-x)", "fd176ff8428da93110ad553ac4b8dcad": "BV_{CBO}", "fd1786599bab7c687714042108c70d22": "f_n(s) = \\frac{1}{ h(s) - \\lambda} \\cdot g_n(s) \\cdot f(s).", "fd17b0d517f505377e7268bd5114946b": "Z_\\mathrm{in} = \\left( R_\\mathrm{L} + j \\omega R_\\mathrm{L} R C \\right) \\| \\left( R + {1 \\over {j \\omega C}} \\right) ", "fd17b15a102c86459e98a2f87a505fb7": "a > -1", "fd17c365aa14db1c0cc41d3696103238": "Q'_0 = A_{in} \\epsilon _{s.s.} \\sigma \\left [ \\left ( T_c + \\frac{T_{surr} \\Delta S}{c_p} \\right )^4 - T_{surr}^4 \\right ]", "fd1801e32b2828c00f2124d306a7b7f7": " \\mathit l + \\mathit l^{\\prime}", "fd183c27a01b05cda7e66e727e4ab3de": "s_i = b_1 \\oplus b_2 \\oplus ... \\oplus b_t", "fd18fffac5d08f18fde1865bbaf554e3": "(\\cos \\theta, \\sin \\theta)", "fd195205f1761f509618cbe9d4a8c80a": "A = \\left|\\Gamma\\right|^2 2 \\rho \\eta = \\left|\\Gamma\\right|^2 \\eta \\left(\\frac{c}{\\lambda \\sigma}\\right)^{1/2}", "fd199cc855d04ea9283b3708c7908d6c": " \\int_a^z dx \\, \\int_a^x \\, h(y) \\, dy ", "fd19a667d0b8adf771fe6a1149b625ba": "E_A(x)=\\sup_{x \\to T \\geq 0} \\int_0^T -\\langle v(t),i(t)\\rangle \\, \\mathord{\\operatorname{d}}t ", "fd19fcd0e1b75ab8c9663c5ab3e18d19": "\\begin{align}\n\\frac{dU}{dt} & = -VW+bVZ,\\\\[6pt]\n\\frac{dV}{dt} & = UW-bUZ,\\\\[6pt]\n\\frac{dW}{dt} & = -UV,\\\\[6pt]\n\\frac{dX}{dt} & = -Z,\\\\[6pt]\n\\frac{dZ}{dt} & = X+bUV.\n\\end{align}\n", "fd1a621d4e951cbd7be23b014e368d80": "a^{|n|}b^{|n|}c^{|n|}", "fd1a83b5edfaa91cf82baca524bf1360": "X(t)>a", "fd1a89ed61e7502ad8df74b2f9e3383f": " S = \\int {1\\over 4} F^{\\mu\\nu} F_{\\mu\\nu} = \\int - {1\\over 2}(\\partial^\\mu A_\\nu \\partial_\\mu A^\\nu - \\partial^\\mu A_\\mu \\partial_\\nu A^\\nu ). \\,", "fd1bbf89a2f67c94d03707a7c23ecac6": "K^{ab} (x)", "fd1c13481cbfe2371b0f9f1c360ff458": "d\\Phi = V\\,dt", "fd1ca4846a7d2fa07945735a0dd06c26": " h_2(k) = (k\\mod 7) + 1", "fd1d09c10fab3b0a9f83b0fe72814c5e": "|n|=1", "fd1d0c5789ea736b7ad97306b3a08a92": "\\mathfrak{Im}[f(x)]/\\mathfrak{Re}[f(x)]\\,", "fd1da62283f35480087128528c58ba64": "$33000*0.2=$6600", "fd1db90c68e5309f67b9e4c07f0aceed": "d(n^2)=\\sum_{\\delta\\mid n}\\mu\\left(\\frac{n}{\\delta}\\right)d^2(\\delta).\n", "fd1dd733a247a52efa975c562e01013c": "Q(M)\\sim\\sqrt{\\frac{\\pi M}{2}}-\\frac{1}{3}+\\frac{1}{12}\\sqrt{\\frac{\\pi}{2M}}-\\frac{4}{135M}+\\cdots.", "fd1e2217bfe9f1d0593fe041c0de9e7f": "\\frac{\\Delta \\nu}{\\nu} \\lesssim 1", "fd1e32e496be14dadc68c5436324b908": "c_2u=c_2cos\\alpha_2\\,", "fd1e5fb3ec9a52df2f58896adff50d8f": "y_{n+1}=\\begin{cases}2y_n & 0 \\le y_n < 0.5 \\\\2y_n -1 & 0.5 \\le y_n < 1, \\end{cases}", "fd1e639500cf7a088bb96b33ad464a73": "\\frac{n + 1}{n + 2} b", "fd1eb6a6634c2167b6ac7a5f751c6e92": "\\left( \\frac{V(K)}{V(B)} \\right)^{1 / n} \\leq \\left( \\frac{S(K)}{S(B)} \\right)^{1 / (n - 1)},", "fd1ef1f899ac4b8e54cf2dd69053b77d": " F(t) = R^2dR/dt ", "fd1f6f7ef700a521f1249e4d6fbe6744": "L_e = \\pi \\langle I \\rangle_e R_e^2", "fd1fae4f2c633c200dce6b5cc39e3712": "i^4 = i^3 i = (-i) i = -(i^2) = -(-1) = 1 \\,", "fd1fea314c50eae9106dd295bdecaa85": "\nx = \\cos(3t) \\cos t, y = \\cos(3t) \\sin t \\,\n", "fd205932472039b407dcb6c3c1c689a5": "f_1(x) = 1 + \\sum_{k \\geq 1} \\frac{\\cos(k \\pi / 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2", "fd206d012389c4dd7afd4566fad1b060": "\\bar{A}^k_n = A_0", "fd209840ee06226a5a29fc2e9e7c91ad": "du^{k}_{i}\\,", "fd221a37cf78ff3720cc2612be61dc45": "X^k", "fd227dea9cce3871a7ab0a65d0406c4b": "\\varepsilon\\sum_{\\beta = 0}^{3} \\gamma_{00|\\beta|\\beta} = -\\kappa \\, \\rho_0", "fd22888e89e9c1c552006f1c6568a26d": "= \\frac{e^4}{(k-k')^4} \\Big( (\\bar{v}_{k} \\gamma^\\mu v_{k'} )( \\bar{u}_{p'} \\gamma_\\mu u_p) \\Big)^* \\Big( (\\bar{v}_{k} \\gamma^\\nu v_{k'})( \\bar{u}_{p'} \\gamma_\\nu u_p) \\Big) \\,", "fd229d1dd5a0fb6902a0e1239542d8d8": "f(x)=2x\\sin\\left(\\frac{1}{x^2}\\right)-\\frac{2}{x}\\cos\\left(\\frac{1}{x^2}\\right)", "fd237b57e2c5d682b1ede37fd5191203": "c_1\\,", "fd238aa29e385fa2fffe2b6325d10541": "\\mathcal{E}_w", "fd23cffe96d132c991265d45d3bd120e": "E_f = \\{(u,v)\\in V \\times V : c_f(u,v) > 0\\}", "fd240dce5540401d1b9dc3cda80f1a47": " e^{\\eta}", "fd2438ed67c3ddff61c40ee85944c248": "F_{ij}= \\frac{\\partial f_i} {\\partial v_j}.", "fd2478f82b2cdde141916c6f2363ddba": "\\frac{V_1}{(V_1+\\Delta V)}=\\frac{(P_1+\\Delta P)}{P_1}", "fd25008c318ba9f5102b033c0ff006c1": "\\ E_0=m_0 c^2", "fd25a6557133e0dc5f4063c481c72bd6": "= 16 \\sqrt{3}R^2\\sin^2(A/3)\\sin^2(B/3)\\sin^2(C/3).", "fd268558cff37107c53fea98372748ee": "D_{22}=1", "fd26993cb5f3eba8f4afcf6a4d7fae65": "x(t) \\;", "fd26b6458c4fcdc4f00d25e361835c18": "\\phi \\to (\\lnot \\phi \\to \\chi )", "fd26c01bdba8edafb57b063dfa8df4bd": "\\tilde X", "fd2708e0fe0fb17d7c1090ddcde748f4": "\\mathbf{F} = q\\left(\\mathbf{E}+\\frac{1}{c}\\mathbf{v}\\times\\mathbf{B}\\right)", "fd27252e3e68682797728267014c93f0": " |x\\rangle|y\\rangle ", "fd275547ee92555872fe674fd38ec826": "D_+(x) = F(x) = \\frac{\\sqrt{\\pi}}{2} \\operatorname{Im}[ w(x) ]", "fd27580e5dd014d8f9f21a772805fa99": "0 = {1 \\over {2}}x^2 + {1 \\over {2}}x - F_{r_1}^2", "fd27a2492b4dc77b9c2b66c7efbc71ca": " \\mathfrak P", "fd27f3716545a4b0a2cddb08c9a4180c": "(2^{\\aleph_0})^+", "fd2822f66ab7ea0c612113b11849e54b": "\\mathbf{F}(x,y,z) = \\nabla \\times g(x,y,z) \\hat{\\mathbf{z}} + \\nabla \\times (\\nabla \\times h(x,y,z) \\hat{\\mathbf{z}}) + b_x(z) \\hat{\\mathbf{x}} + b_y(z)\\hat{\\mathbf{y}}, ", "fd287d688793f11162908ae23ec58e86": "{\\rm d}U = \\delta Q - \\delta W\\,", "fd2894e42af159a402b1c3a60f6f235c": "h'(\\ell)", "fd28c557435542166cf8a19839f7e903": "\\ln(M_r) = \\ln\\left(\\frac{(c + \\alpha)_r (c + \\beta)_r}{(c + 1)_r^2}\\right)= \\ln(c + \\alpha)_r + \\ln(c + \\beta)_r - 2\\ln(c + 1)_r", "fd28d00d3c657b0c00f008ebf1e05c53": " \\mathbf{P}_k ", "fd29e90eca5ed505a2a59d44a633ac50": "\\mathrm{Gamma}(m+1,b_1^2)\\!", "fd2a016f6a11db6016cafe8652debd5a": "d{\\lambda}", "fd2a13693637000f0500570e53ed9292": " y_j \\ge 0,\\, s_i \\ge 0 ", "fd2a6273f191a0ddc1b6659d073a317b": "\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}+1,", "fd2a9faccc8459ecac5f28447ab78663": " M_{i,j} = 1 ", "fd2b1eb3a3aacb801a8c6d0b7ec448b5": "\\hat{X}", "fd2b3734efb54648fed7bfb87b7adff5": "A_{nr}=\\frac{2(2-\\delta_{n0})}{a^2}\\,\\,\\frac{J_n(k_{nr}\\rho_0)}{k_{nr}[J_{n+1}(k_{nr}a)]^2}.\\,", "fd2b9f074594d07898fab4895a3807bd": "h \\alpha ^ A = \\cosh A + \\sinh A \\ \\alpha ^{\\pi/2}.", "fd2bb53f2039d8e6386659e8d133c9c4": "\\textstyle\\operatorname{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)", "fd2bb8974631abca85a1d4c80397fbdb": "S_i^- \\longrightarrow S_i^-e^{-i \\theta_i}", "fd2bbc9117091a9932f2bd3417c9d3bb": "10\\uparrow\\uparrow\\uparrow 10\\uparrow\\uparrow\\uparrow 4", "fd2c3805d800828c1a4a2280aaad64d9": "b \\leftarrow e^{-1}(a-r)rem(p^{\\prime}q^{\\prime})", "fd2c3dfd13fdf3a6946629e8cc1bcfbe": "\\pi_1 \\big(PSO(1)\\big) = 1,", "fd2c778088984b0d4f8173b517141822": "k_i, x_i", "fd2c974dbd14e80a79f7099c51d7472a": "\\frac{\\mathrm{d}}{\\mathrm{d}z}\\, \\mathrm{dn}\\,(z) = - k^2 \\mathrm{sn}\\,(z)\\, \\mathrm{cn}\\,(z).\n", "fd2caf304e109a34939eee1a9151c33e": "T_{1/2} = \\frac{\\ln 2}{\\lambda _c} = \\frac{\\ln 2}{\\lambda _1 + \\lambda _2}.", "fd2cb4919509af527a64999d4ed2ac47": "\\nabla\\cdot\\left[ \\mu \\left( \\frac{\\left\\| \\nabla\\Phi \\right\\|}{a_0} \\right) \\nabla\\Phi\\right] = 4\\pi G \\rho", "fd2cbc5dc80055a981d4f61a65028bb2": "m = \\frac{-Ax_0 - C}{B},", "fd2cf9957892d788cc83263121590a34": "E_\\text{k} = \\frac{m c^2}{\\sqrt{1 - (v/c)^2}} - m c^2 ", "fd2d223431cf9800540c9b33fa55a4ef": "\\text{Channel utilization} = \\frac{\\text{Time spent transmitting data}}{\\text{Total time}}", "fd2da1cd75603fd907560dd8a2773aca": " \\frac{2}{\\pi} \\log(n+1)+a < \\Lambda_n(T) < \\frac{2}{\\pi} \\log(n+1) + 1,", "fd2da30b0a9debb10b83c13ed758d41f": "\\phi \\,(t) = Q(t)e^{tR}", "fd2dc3fd4e4c6cc3dbd843b10a8823b3": " (\\nu x)(0| \\overline{z}\\langle x \\rangle . x(y). 0 ) | z(v). \\overline{v}\\langle v \\rangle .0 \\rightarrow (\\nu x)(0| x(y). 0 | \\overline{x}\\langle x \\rangle .0) ", "fd2def150c3847a502e5e74c4cc0045b": " \\vdash \\ \\ c = c ", "fd2e00fdc03d79b80dc97fdd3a4085eb": "-\\infty, \\infty", "fd2e5639ff98eda41f371d8c4113b8bd": "\n \\boldsymbol{\\nabla} \\boldsymbol{\\omega} \\equiv \\omega_{ij,k} = \\frac{1}{2} (u_{i,jk} - u_{j,ik}) = \\frac{1}{2} (u_{i,jk} + u_{k,ji} - u_{j,ik} - u_{k,ji}) = \\varepsilon_{ik,j} - \\varepsilon_{jk,i} \n", "fd2e8c5a73570befcc3cbd67da33a48a": " = |\\lambda|^2 \\cdot \\| Ux \\|^2 + \\| U(\\lambda \\cdot x) \\|^2 - \\overline{\\lambda}\\cdot \\langle U(\\lambda\\cdot x), Ux \\rangle - \\lambda\\cdot \\langle Ux, U(\\lambda\\cdot x) \\rangle ", "fd2f0670ba410053f3c3f26bc7a7f2df": "\\left\\langle\\sqrt{2}e^{\\pm \\tfrac{\\pi}2 \\mathrm i}=\\pm \\mathrm i\\sqrt{2},Z_2\\right\\rangle", "fd2f6af80f9d3ed54a02391ada9aeb64": " D_\\mu := \\partial_\\mu - i \\frac{g_1}{2} \\, Y \\, B_\\mu - i \\frac{g_2}{2} \\, \\sigma_j \\, W_\\mu^j - i \\frac{g_3}{2} \\, \\lambda_\\alpha \\, G_\\mu^\\alpha ", "fd2f6e91147250be9c22b63fbd211f6d": " \\chi(\\varphi) ", "fd2fc0b43b73b07164629a44c5795b1d": "r_\\text{in} \\triangleq \\frac{v_\\text{in}}{i_\\text{in}}\\,", "fd2fe2de2b0e9ad1b9fd9b2d438a7904": "\\dot\\theta =\\frac{h}{r^2} = hu^2 \\,", "fd2feae6119b8928b73c1faf5cd6b0d3": " f(x;\\,m,\\Omega) = \\frac{2m^m}{\\Gamma(m)\\Omega^m}x^{2m-1}\\exp\\left(-\\frac{m}{\\Omega}x^2\\right).\n", "fd301b74476da2e23e0cf0ff8630846f": "(1+x)^\\alpha = \\sum_{n=0}^\\infty {\\alpha \\choose n} x^n\\quad\\text{ for all }|x| < 1 \\text{ and all complex } \\alpha\\!", "fd30b527ec9d9a7bb161665696cdc101": "Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F", "fd30efd7d0c282311216b5b0a3a4e99e": "\\delta = \\frac{\\Delta t_+ - \\Delta t_-}{2 \\, \\pi \\, r}", "fd31ee30661a3b0fe316ce1d181e61df": "I_N=(\\text{newMax}-\\text{newMin})\\frac{1}{1+e^{-\\frac{I-\\beta}{\\alpha}}}+\\text{newMin}", "fd323ee62281c1e464e801a4af862a9d": "\\hat{T}_e = - \\sum_i \\frac{\\hbar^2}{2 m_e} \\nabla^2_{\\mathbf{r}_i} ", "fd325a9be9775f944d3f711b7079318b": " \\sum_{i \\in I} \\| x(i) \\|_{X_i} < \\infty. ", "fd33964b0254a52216867b929fd2596f": "(AB)^2+(BC)^2+(CD)^2+(DA)^2=(AC)^2+(BD)^2+4x^2.\\,", "fd33aca15cc276d7331f68245543f9d0": "\\liminf_{n\\to\\infty}(p_{n+1}-p_n)0 ", "fd4c7dc565d64437ccb480bbf1b2dce4": "\nr(\\chi) = \\begin{cases}\n\\sin \\left( \\sqrt{-\\Omega_k} H_0 \\chi \\right)/\\left(H_0\\sqrt{|\\Omega_k|}\\right) & \\Omega_k < 0\\\\\n\\chi & \\Omega_k=0 \\\\\n\\sinh \\left( \\sqrt{\\Omega_k} H_0 \\chi \\right)/\\left(H_0\\sqrt{|\\Omega_k|}\\right) & \\Omega_k >0\n\\end{cases}\n\n", "fd4ccf3856804754c90bc87bf203a4ef": " \\varphi_X(t) = \\operatorname{E}\\left[\\exp({i\\,\\operatorname{Re}(t^*\\!X)})\\right], ", "fd4cf1be9451e0ee235e1e60a714b779": "\\frac{dn}{dt} = \\alpha_n(1 - n) - \\beta_n n", "fd4d12ab594e64c158a098503c6a0c05": "\\ln(1)=0", "fd4d6597144b1d976014b1f95f488f68": "\\frac{{(1-e^{-\\alpha})}^2}{1+2\\alpha e^{-\\alpha}-e^{-2\\alpha}}", "fd4d9019ca4c0cdcf87fef8e53cb6bc9": "n_\\eta^\\prime(\\xi)=\\eta\\delta(\\xi)", "fd4dc582f8d90b4d52618325fb9050d7": "(\\dot q,\\dot p) = (p,-\\sin q). \\, ", "fd4dce46701487daa3e9de3ed74279c2": "\\Phi(s,t)=\\frac{t^{s-1}}{(s-1)!}\\left[H_s-\\ln(-t)\\right]\\text{ for }s=2,3,4\\ldots", "fd4e08cff494e033c4e5e57af9e2b45d": "\\varrho(0) = 0", "fd4e0cd53e27ecf68b361ae37e6ae096": "\\bar F(x) = \\operatorname{P}(X > x) = 1 - F(x).", "fd4e25b2f0133cc31dc36f92898d5032": "v = -Kz\\qquad,", "fd4e59f54d69594b914073c5db710b2b": " p_{EM} = U/c\\,\\!", "fd4e8dabeabbcdfc614a27d208b46ff3": " P = gL ", "fd4eab42f64e75e18d88da792cae137e": "h_a = \\frac{2T}{a}.", "fd4f6f6544ee4b65e1e1b72ff38736e5": "\\scriptstyle (\\Omega, \\mathcal{F}, P )", "fd4f88bb0f8d0f46bb904c859a4449e9": "m=b-r, n=b+r\\,", "fd4fa58c3ae08ce6c41fdb9085868d47": "V_1\\otimes(V_2\\otimes V_3)=(V_1\\otimes V_2)\\otimes V_3", "fd503c57a644f192ca28833d034155d6": " \\left( \\sqrt{\\frac{2\\nu}{2\\nu+5}}\\mu,\\,\\sqrt{\\frac{\\nu}{\\nu+1}}\\mu \\right)", "fd506a7a58b8c9b8f6b7176bf5c9d6ef": "e^{ar_f (x'k)-a \\cdot x'\\mu + \\frac{a^2}{2}\\sigma^2},", "fd508fdb446659f182499f7fbbc0d04a": "\\textstyle(x\\mp1, y\\mp1, z\\pm1)", "fd5092b0f28e2e8b4bf9aa0618bd6c1d": "\\Delta = \\frac{4}{\\lambda^2} \n\\frac {\\partial}{\\partial z} \n\\frac {\\partial}{\\partial \\overline{z}}\n= \\frac{1}{\\lambda^2} \\left(\n\\frac {\\partial^2}{\\partial x^2} + \n\\frac {\\partial^2}{\\partial y^2}\n\\right).", "fd514b9e51876dad332398d9e9996344": " W(C;x,y) = \\sum_{w=0}^n A_w x^w y^{n-w}.", "fd516acbfc8090f6dfeabc795e9e69c3": " \\tau_{ij} = \\mu \\left(\\frac{\\partial v_j}{\\partial x_i} + \\frac{\\partial v_i}{\\partial x_j}\\right).", "fd51b26549a638ab3522fb8ecea688fc": " r = \\frac{\\phi}{\\omega} - r_0 ", "fd51fcf874cdf4bb27382c355ac1d6c3": "-0.980\\pm0.053", "fd521bd25804c0e5d044bd1a086976e6": "\\sqrt[x]{a}\\,", "fd521d90ed7a276fb29c8502a64bbaf9": "\\nabla\\cdot\\mathbf{A}", "fd52502318aa8569b78857fc896df14c": "\nS=\\left[\\begin{matrix}-\\lambda_{1}&p_{1}\\lambda_{1}&0&\\dots&0&0\\\\\n 0&-\\lambda_{2}&p_{2}\\lambda_{2}&\\ddots&0&0\\\\\n \\vdots&\\ddots&\\ddots&\\ddots&\\ddots&\\vdots\\\\\n 0&0&\\ddots&-\\lambda_{k-2}&p_{k-2}\\lambda_{k-2}&0\\\\\n 0&0&\\dots&0&-\\lambda_{k-1}&p_{k-1}\\lambda_{k-1}\\\\\n 0&0&\\dots&0&0&-\\lambda_{k}\n\\end{matrix}\\right]", "fd52515f89273d6f633673c959d9022e": " \\begin{align} \\hat{H} & = \\sum_{n=1}^{N}\\frac{\\hat{\\mathbf{p}}_n\\cdot\\hat{\\mathbf{p}}_n}{2m_n} + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N) \\\\\n& = -\\frac{\\hbar^2}{2}\\sum_{n=1}^{N}\\frac{1}{m_n}\\nabla_n^2 + V(\\mathbf{r}_1,\\mathbf{r}_2,\\cdots\\mathbf{r}_N) \n\\end{align}", "fd52568a5da652f1ef5e2697614158be": "(v,\\phi)", "fd526ed3804054016e92ceed7579487e": "\\lambda_j - \\alpha \\mu_j", "fd527c29adb9c8d07b62d2b7df83de5c": "|A|<\\min(A)", "fd52bf189ee1ad138fb4ca1a5faea809": "G(x)=\\frac{1}{1+x^*H^*C_w^{-1}Hx}xx^*H^*C_w^{-1}.", "fd52eef48918287f502867b738ea729c": "B < B_{c1}", "fd5386ba42c2087f28835078891b6402": "\\delta W = P dV \\,", "fd53a6196e84e2f3a057c40458041a75": "\\, {e_+}^2 = +1 ", "fd53aedb10f8fed6df591d7e1a9ac79c": "\\pi/7", "fd53b0510c8aa520582021f84cc8963a": "\\{like\\langle Mary, y\\rangle |y \\in E\\}", "fd53b15dd87e20ee86599a34938286a5": " \\frac{m \\left | \\mathbf{v} \\right |^2 }{\\left | \\mathbf{r} \\right |} = q \\left | \\mathbf{v} \\right | \\left | \\mathbf{B} \\right | \\sin \\theta, \\,\\!", "fd53e689e3f8db91a0f55a1282f1441c": "\\forall y", "fd53f1f68a89fc8e24604a6a009f1c06": " V_A = V_T - V_f ", "fd53fbb0620d12ea0a1a489ed3089789": "\n1 - r (p_1 p_2 + p_1 p_3 + p_2 p_3 - p_1 p_2 p_3) = 0 \n", "fd5401316f71a20df36f22e3e86490e4": "U:C\\to\\textbf{Set}", "fd5412dcb397b69ee538a575fba48c54": "h(a,y)", "fd54272be6e03e830f7675f1bb11a675": "\\left| x- \\frac{p}{q} \\right|>\\frac{c(x)}{|q|^{d(x)}} ", "fd542e49ed7df1f3607140ba46f64685": "f(-x) = \\overline{f(x)}", "fd5435dbc5298df543ff8ff97a14393c": "P = P_o e^{-\\delta k z}", "fd5484387450c8407e2caad52cd3026b": "\\zeta = z + Z", "fd54928f32089a42fce246e079041f91": "x^{\\frac{1}{3}} \\!\\ ", "fd5501da8ea9a4006215c66ea0e29442": "\n\\arcsin z = -i \\log \\left( i z + \\sqrt{1 - z^2} \\right), \\,\n", "fd55215810f5d2b4da3dc2283c91cbca": "\\alpha(\\epsilon) = \\sup \\left\\{ \\mu( \\{ F \\geq \\mathop{M} + \\epsilon \\}) \\right\\},", "fd554b68750acd87d1363fa9a33b4953": "p = q=0", "fd555391868d2ba8fd90fb003bba92fb": " E = M + \\sin(M) \\, \\varepsilon + \\tfrac12 \\sin(2M) \\, \\varepsilon^2 + \\left( \\tfrac38 \\sin(3M) - \\tfrac18 \\sin(M) \\right) \\, \\varepsilon^3 + \\cdots ", "fd5574c3068d13114b7a2e5553ebdadf": "n = r_1 \\cdot r_2 \\cdot ... \\cdot r_u,", "fd55a866a96b1ccf4a901c5fe872f225": "c\\rightarrow\\infty", "fd55f3a7ac54ecbbac38a47ec13fd85f": "\\mu\\in\\mathop{det}H_*(M)", "fd55fb532d2247cc5e9bccebbadaf437": "E[X^j] = (\\tfrac{j}{2})!", "fd55fe01f8edae44c93b99a2085137d3": "\\int \\frac{dx}{x^n\\ln x} = \\ln \\left|\\ln x\\right| + \\sum^\\infty_{k=1} (-1)^k\\frac{(n-1)^k(\\ln x)^k}{k\\cdot k!}", "fd55fe517411fb6cfce773e30e8cbfba": "t\\rightarrow\\infty", "fd56357371b501d37c522e21823f6c71": "e^{D^2} f(x) = \\sum_{k=0}^\\infty \\frac{D^{2k}f(x)}{k!}.", "fd563dba00fadf1f21be69a7173434b1": "\n\\{a(\\mathbf{k}),a(\\mathbf{l})\\} = 0, ", "fd5663dc73f974940f6e5640887e4bdd": "\\chi_{\\xi\\xi}=\\frac{\\chi_\\xi}{\\xi}+\\frac{1}{2}\\operatorname{sh}2\\chi=0,", "fd5666b0f5266ac14a0aac50272f9aac": "J=\\Phi\\,[\\,\\textbf{x}(t_0),t_0,\\textbf{x}(t_f),t_f\\,] + \\int_{t_0}^{t_f} \\mathcal{L}\\,[\\,\\textbf{x}(t),\\textbf{u}(t),t\\,] \\,\\operatorname{d}t", "fd5686f973da7999c64dd4d1e2c08e2b": "I = 124*2^{23}+ 2^{21}", "fd56ba928607d4beb49c4aad04035ca5": "\\epsilon^{IJMN} \\epsilon_{KLMN} = - 4 \\delta^I_{[K} \\delta^J_{L]} = - 2 (\\delta^I_K \\delta^J_L - \\delta^I_L \\delta^J_K) \\;\\;\\;\\; Eq.4 ", "fd56c5d86d086c64c2989f10c48b753b": "\\hat{G}_X=\\hat{G}_{(1-X)} ", "fd56cef0fe2ac7acead6815a70273dc2": "(12)\\quad \\psi(r,\\theta)\\,=-\\sum_{i=1}^\\infty a_i \\Big(\\frac{R_n(\\cos\\theta)}{M}\\Big) P_i", "fd5734cedd99806e75ee54ef33233391": "f_1,f_2,\\ldots,f_k", "fd57a93fd68bb88c51e1cbc11aad18af": "\n\\mathcal{W} = \\frac{1}{2m} \\mathbf{p} \\otimes \\mathbf{p} + \\frac{k}{2} \\, \\mathbf{r} \\otimes \\mathbf{r} ~,\n", "fd5875ad7795ed8d9d37f9a138c46bf0": "\\lim_{n\\rightarrow\\infty}\\; \\frac{1}{n} \\sum_{k=0}^{n-1} f\\left(T^k x\\right)=E(f).", "fd58bb6d519c830c9a20577534d9d2bb": "\\psi(1 - x) - \\psi(x) = \\pi\\,\\!\\cot{ \\left ( \\pi x \\right ) }", "fd592f33d4a63b9374690c275782d699": "p(drunk| D) = \\frac{N(drunk \\cap D)}{N(D)} = \\frac{1}{51} = 0.0196", "fd595fbb78944b320d40a15669b962c7": " b_n \\equiv n \\frac{\\partial}{\\partial x_n} (n<0) ", "fd59ad15891af2cb34d7cbbb23bc7d61": "n \\,\\,\\mathsf{nat}", "fd59b973d997e3b552bdf62da7b4bccd": "\\scriptstyle T' \\,\\approx\\, \\frac{1}{\\Delta f}", "fd59fd3954c1e267d30e2b07cbcad488": "\n\\begin{align}\n\\dot{r}&=\\frac {\\partial{\\mathcal{H}}} {\\partial{p_r}} = \\frac {p_r}{M+m} \\\\\n\\dot{p_r} &= - \\frac {\\partial{\\mathcal{H}}} {\\partial{r}} = \\frac {p_\\theta ^2} {mr^3} - Mg + mg\\cos{\\theta} \\\\\n\\dot{\\theta}&=\\frac {\\partial{\\mathcal{H}}} {\\partial{p_\\theta}} = \\frac {p_\\theta} {mr^2} \\\\\n\\dot{p_\\theta} &= - \\frac {\\partial{\\mathcal{H}}} {\\partial{\\theta}} = -mgr\\sin{\\theta}\n\\end{align}\n", "fd5a3b22768960cb17c5ccc8367c7241": "\\displaystyle J_{1}, J_{2}, J_{3}", "fd5a3e7e374bfdc8ea70be293eb980dc": "\\rho \\ge |p| .", "fd5a3e96ab616eaf7bd031752318ef96": "\\mathbf{D}(t) \\in \\mathbb{R}^{6 \\times n},~ n \\in [1..6]", "fd5a4cd99f3e365a075c3d14a0b1b033": "\\rho_s = \\cfrac{1}{N} \\left[ \\cfrac{1}{2}\\langle T_x \\rangle + \\sum_{\\nu \\neq 0} \\cfrac{ | \\langle 0| j_x^{(s)}|\\nu\\rangle |^2 }{E_{\\nu} - E_0}\\right]", "fd5a5b966728e6b63cdb06d6d029221f": "f(x_1,\\dots, x_{K-1}; \\alpha) = \\frac{\\Gamma(\\alpha K)}{\\Gamma(\\alpha)^K} \\prod_{i=1}^K x_i^{\\alpha - 1}.", "fd5acf174874e9056d1b7d29d67ef63c": "\\alpha = \\tfrac{1}{2}", "fd5b09825b26c9847674fddffadf7eca": "b(x)=\\left\\{\\begin{array}{ll}0,&|x|\\leq 1,\\\\-\\infty,&\\mbox{otherwise}\\end{array}\\right.", "fd5bba44bf5886610002af439435b984": "\\cos (2 \\sigma_m) = -1", "fd5c3b4dc966877a342e69c01dba6444": "|a|^2+|b|^2+|c|^2=250", "fd5c600940d90ef6c46abbd06e8e2404": "\\overline{z}(\\lambda)", "fd5c9b4c792a09568235a4752e4ca77f": "C = A+B", "fd5cccf7fc29481805d00757df41f2f2": "|\\mathbf{r} - \\mathbf{r}_s(t_r)| = c(t - t_r)", "fd5ccf31f66400f08d5e12fdae075726": "a(b(ac))=(a(ba))c", "fd5d17b82d6ee93d656a6f3d5487b9f7": " u_{tt} = c^2 u_{xx}.", "fd5d19c8d8d92ec2d5d902bae38b7373": "R^{n+1}(X)", "fd5d798ee9008f12d25f42764bd29f17": "v_\\mathrm r = v_\\mathrm i \\,\\!", "fd5d858df5e412c997b9987f69eaf093": "m=\\gamma(\\mathbf{v}) m_0", "fd5d98b8908465bb7a08c0e87edc271c": "s/C", "fd5db3049df5d9c76493be99ac13f9c7": "\\omega = \\sqrt{\\frac{I}{\\kappa}}\\,\\!", "fd5db72a9e8d4bbf4ad46f76490e385e": "b_0=\\sin{\\pi \\over 4}=\\frac{1}{\\sqrt{2}}\\!", "fd5dccdcf235f627db0de60258031392": "x=b/2", "fd5de574cea5fbfe428e1e8a22e713b6": "\\operatorname{E}[N] \\operatorname{E}[X] = \\frac{1+2+3+4+5+6}6\\cdot\\frac{1+2+3+4+5+6}6 = \\frac{441}{36} = 12.25\\,.", "fd5e076f353429142bc7fb971793830b": "\\ln(f_w(\\theta))=\\sum_{m=-\\infty}^\\infty c_m e^{im\\theta}", "fd5e10d637808270dc7758dc9715ee02": "S^2(A_n)", "fd5e6b65bac1f3a2885d488f765894cc": "H_1 \\ldots H_m", "fd5e6c2cc7171e1813eceefc03a8e6ee": " w_{o,1}", "fd5e6e21e5180fb4bedfb0cd5ba761f1": "\\mathcal{T}(A)=dlock~\\mathcal{T}_1(A)", "fd5e82cc7fa11c12c528b757ea1f6e4b": "\\tau (s_k) = s_{k+1}", "fd5e85e9380f871af5991be0812949c8": "q:((\\{0,1\\}^k)^n \\times(\\{0,1\\}^k)^m)\\rightarrow\\mathbb{R}\\,\\!", "fd5e946cdc8b782dd1d155bc6d23adcb": "\\int\\frac{1}{ax + b} \\, dx= \\frac{1}{a}\\ln\\left|ax + b\\right| + C", "fd5f11b188179fd4385c61a9a46bc7f2": "\\cup \\{\\{(x,y)\\in \\R^2\\ | y=\\frac{a}{x-b}+c,x\\ne b\\}\n\\cup \\{(b,\\infty),(\\infty,c)\\} \\ | \\ a,b,c \\in \\R, a\\ne 0\\},", "fd5f13408483ba305a95c29d29a62982": "B = ", "fd5f9e2ee180a439aaec015692916a1c": "x(t)", "fd5fb8347dc7a94e810409a3077e8696": "\n\\frac{\\partial}{\\partial A} \\ln p(\\mathbf{x}; A)\n=\n\\frac{1}{\\sigma^2} \\left[ \\sum_{n=0}^{N-1}(x[n] - A) \\right]\n=\n\\frac{1}{\\sigma^2} \\left[ \\sum_{n=0}^{N-1}x[n] - N A \\right]\n", "fd5fd8c3a881d226f146f96fb1c1badf": "\\inf f \\le \\mathrm{ess } \\inf f \\le \\mathrm{ess }\\sup f \\le \\sup f", "fd5fe4d30ab968b1ced72a7569ecbe2a": "\\begin{align}\n \\Delta p_{\\text{B}}(W) &= p_{\\text{B}0} e^{\\frac{q V_\\text{CB}}{kT}} \\\\\n \\Delta p_{\\text{B}}(0) &= p_{\\text{B}0} e^{\\frac{q V_\\text{EB}}{kT}} \\\\\n I_{\\text{E}} &= qA \\left[ \\left(\\frac{D_\\text{E} n_{\\text{E}0}}{L_\\text{E}} + \\frac{D_\\text{B} p_{\\text{B}0}}{W}\\right)\n \\left(e^{\\frac{q V_\\text{EB}}{kT}} - 1 \\right) -\n \\frac{D_{\\text{B}}}{W} p_{\\text{B}0}\\left( e^{\\frac{q V_{\\text{CB}}}{k T}} - 1 \\right)\n \\right]\n\\end{align}", "fd5fea2d2afbf43c568a2d77fe563dbe": "\\lim_{x\\rightarrow +\\infty}f(x)=c", "fd5ffd2b3082c0c0092db8cb43a01e12": "\\left( i \\hbar \\partial\\!\\!\\!/ - m c \\right) \\psi = 0 ", "fd60a21470e84b00d15e5dbde12f7e7c": "\\widetilde\\phi_\\alpha(A\\cap \\pi^{-1}(U_\\alpha))", "fd60be26da40bed8012ffe4435f762ea": "M_\\odot=\\frac{4 \\pi^2 \\times (1\\ {\\rm AU})^3}{G\\times(1\\ {\\rm year})^2}", "fd61760319547514bc5770adc4e94cb9": "\\begin{bmatrix} \\dfrac{z_{11}}{z_{21}} & \\dfrac{\\Delta \\mathbf{[z]}}{z_{21}} \\\\ \\dfrac{1}{z_{21}} & \\dfrac{z_{22}}{z_{21}} \\end{bmatrix}", "fd619000ce6b26a7a393d774eb3d1eee": "\\psi\\Gamma^\\mu \\psi = 0.\\,", "fd619a08754b5b05c9b599c663ebd7be": "\\ H(z)=\\frac{1}{1-z^{-1}}.", "fd61cd9df07ffce0c420bf3f428b0a53": "[\\![\\mathsf{T}_1]\\!] = S_1,~[\\![\\mathsf{T}_2]\\!] = S_2,~\\ldots,~[\\![\\mathsf{T}_n]\\!] = S_n", "fd61e9cd6bec591a21c12687829eda7c": " \\tau \\mapsto \\tau+2 \\ ;\\ \\tau \\mapsto \\frac{\\tau}{1-2\\tau} \\ . ", "fd61fb371e415587a776e6c6fc976ffb": "\\frac{L}{r^{2}} \\frac{d}{d\\theta} \\left( \\frac{L}{mr^{2}} \\frac{dr}{d\\theta} \\right)- \\frac{L^{2}}{mr^{3}} = -\\frac{dV}{dr}", "fd620d44f88c116f8cdb86da28e40077": "\\sgn (x) = \\begin{cases} +1, & x \\geq 0; \\\\ -1, & x < 0. \\end{cases}", "fd621624ad5194198e33f41b4953c877": "\\tbinom n t = b\\tbinom k t", "fd6220d6c7b7bb6ed73ceb3b737e22ab": "O(kn^2)", "fd6284beb1bbc8f65bc36b6f86c8b92f": "g(s+\\varepsilon)-g(s)\\over\\varepsilon", "fd62d569f752756f3aeaed6a113c95ab": "P_{total} = \\sum_{i=1} ^ n {p_i}", "fd62dea2ad5b309a496443c1330d4394": "\\Pr[A] \\ge 0", "fd62f57cbe1938fb31129990896c55ea": "\\mathbf{R}^n.", "fd630520bded9859439f7427aacd2660": "\\overline\\partial + \\overline\\partial^*", "fd633a514a038d81a345a8d3ff70e08d": "(c,0)", "fd6396f23acb086bd1c496a5b2991d7f": "x_n\\geq 0", "fd63b156ff1ffc0f929e4964cfcf0bb3": "H:=({\\Bbb C}^n\\backslash 0)/{\\Bbb Z}", "fd63bbfe97eb2df5382eb5cb7ee5a305": "\\rho\\in(0,1)", "fd63fca855aa4f797db0546f53a55b46": "N_{\\rm A} = \\frac{F}{e}", "fd6408df70d6133928a16eb7ce691906": "Z_{12} = {-Y_{12} \\over \\Delta_Y} \\,", "fd64661b905248d6004d30594b4d147a": "\\pi_1(T,t_0) \\approx \\pi_1(S^1,x_0) \\times \\pi_1(S^1,y_0) \\cong \\mathbf{Z} \\times \\mathbf{Z} = \\mathbf{Z}^2.", "fd64667758caa40f83b4d8efc4e706b9": "\\begin{cases} u_{t}=ku_{xx}+f(x,t) & (x, t) \\in [0, \\infty) \\times (0, \\infty) \\\\ u(x,0)=0 & IC \\\\ u(0,t)=0 & BC \\end{cases} ", "fd647915fad32a435bcf57436510239b": "t_M(w) = \\text{ number of steps }M\\text{ takes to halt on input }w.", "fd64aa7c1c2944cfa4d8ca0809564b6f": "\\left[ t,t\\right]=\\left[ t, \\theta\\right]=\\left[ t, \\theta^*\\right]=\\left\\{\\theta, \\theta\\right\\}=\\left\\{ \\theta, \\theta^*\\right\\} =\\left\\{ \\theta^*, \\theta^*\\right\\}=0", "fd64ebf47b931885cfc4897eeed227ea": "P(x) = \\mathrm{Pr}(X > x)", "fd6525b4d235159fa8003b5b2752f110": "\\mathbf{F} =\\nabla \\left(\\mathbf{m}_2\\cdot\\mathbf{B}_1\\right),", "fd6546476e137e8962b2a3f7b612219a": "tp_n^{\\mathcal{M}}(\\boldsymbol{b}/A)", "fd65a996d953d0c6afbce2d1062eb047": "U\\ :=\\ H_{in},\\quad V\\ :=\\ m,\\quad W\\ :=\\ U\\ \\oplus\\ V,\\quad K_1\\ =\\ P(W)", "fd65e6ae926ab4492758e016b8fd4804": "\\eta_s = \\frac{1}{\\gamma^2} -\\alpha_c", "fd65f46b86ecf878624aea6ff698ffa2": "\\scriptstyle{a=\\frac{\\log{ p}}{log{ q}}} ", "fd661e63e2d8c53f0c0b0ae6dc250dc9": "\n \\dot{\\boldsymbol{T}} = \\mathsf{D}:\\dot{\\boldsymbol{F}}\n ", "fd6669a1f0fe6fdbc81060347a43ac95": "P_0=\\lfloor\\sqrt{kN}\\rfloor,Q_0=1,Q_1=kN-P_0^2.", "fd66da969393798601babeeb2de58820": "\\ {D_{t}} ", "fd670e3d5010b1feb434940b2ef42694": "\\scriptstyle g''(x)^2 \\;", "fd672c02f292eb86ef575cfcc133aa63": "P(\\text{ill}|\\text{positive})=\\frac{P(\\text{ill}\\cap\\text{positive})} {P(\\text{positive})} = \\frac{0.99%}{1.98%}= 50%.", "fd676b99e267189888f77aedb1d66328": "w_{n}^{*}", "fd67b62a594b478671b55e4eb65d6a1d": "\n\\begin{align}\nB_n(x)\n&=\\sum_{k=0}^n{n\\choose k}B_{n-k}x^k\n&& \\text{applying the definition of Bernoulli polynomials}\n\\\\\n&= \\sum_{k=0}^n{n\\choose k}L\\left(y^{n-k}\\right)x^k\n&& \\text{applying the above definition}\n\\\\\n&= L\\left(\\sum_{k=0}^n{n\\choose k}y^{n-k}x^k\\right)\n&& \\text{since L is linear}\n\\\\\n&= L\\left((y+x)^n\\right).&&\n\\end{align}\n", "fd67cc052a5d54ce02a28f2ae3a43bff": "MTF_{jitter}(u) =e^{-2 \\pi^2 u^2\\sigma^2}", "fd67fcfd201fb5f1f40f34066e4d3e3b": "\\hat{G}_{yy}(j \\omega_i) = U_i S_i U_i^H", "fd689010f2e448ede15eda1bd13198c2": "q = \\tfrac12\\, \\rho\\, v^{2},", "fd691a683b5fb2b8efb642723de191fb": "\\displaystyle (q,p,b)\\rightarrow (q+b/p,p,-b)", "fd6a002955b1bb80cc0dc72e037076d0": "\\exp(X) = \\gamma(1)", "fd6a0924671b13c43943d4a2e7be3e90": "|\\psi(x,t)|^2", "fd6a82e943964a42ee22250f22b60eb1": "Y_p\\mapsto g_p(X_p,Y_p)", "fd6aa155205215405063244ea589c733": "basis^2", "fd6acff0e4b1c47ac499156dc3d94a25": "T_e = \\frac{1}{1+F\\sin^2(\\frac{\\delta}{2})},", "fd6b2a7273fa61ad87ee5879d8f2dab1": "\n -\\cfrac{2C_0}{J_m} = \\mu\\, \\qquad \\implies \\qquad C_0 = -\\cfrac{\\mu J_m}{2}\n ", "fd6b35a80401664eafcae37bc4d2a733": "PV = \\sum_{k=1}^\\infty C(1+i)^{-k} = C \\sum_{k=1}^\\infty \\frac{1}{(1+i)^{k}} = \\frac{C}{i}, \\qquad i > 0,", "fd6b84e39e5c491df1a900b04ac469a9": "\\scriptstyle x^{4} + y^{4} = z^{4}", "fd6ba30264f2abbe89f5a3c564746539": "D(X) = 0", "fd6bd23413bd829b079d7ac4082f9e9a": "d=\\sqrt{(\\Delta x)^2+(\\Delta y)^2+(\\Delta z)^2}=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}.", "fd6c479abe05f550372a676e6958e40d": "\\begin{pmatrix}\nu &&-v \\\\\nv && u\n\\end{pmatrix},", "fd6ca7312c947b549f27df5496a45910": "P_a(s,s') = \\Pr(s_{t+1}=s' \\mid s_t = s, a_t=a)", "fd6cb3af75492abdccfd3aafafaa12aa": "n+\\lceil\\log_2n\\rceil", "fd6cb4495128cb5a360f59651a86115e": "B_k={1\\over{\\sqrt{N}}}\\sum\\limits_{A=1}^N\\exp(-ikr_A)a_A", "fd6cd92967774ab8cae1fa76bee70849": "\\,588^2+2353^2 = 5882353 ", "fd6cf4b18caaa88c14fcfff4a3e285d7": " W_{(h)}=M_{h,k} W_{(k)} M_{k,h} ", "fd6d0a4e5a097b876296d14d09cef553": "\\mathbb{E}_E", "fd6d181892f38b10f0a3bb6069fcc0ab": "z(t)=\\frac{1}{2}\\int_c xdy-ydx", "fd6dc15d98aedb236717362e596e5055": "\\varphi(s)=\\int_0^{\\infty} x^s f(x)\\,\\frac{dx}{x}", "fd6e3325f0387b1a85d21a3d6c3dffc7": " \\operatorname{de-lambda}[V] ", "fd6eb27edefd30506035199115694982": "\\mathcal{P}(B)", "fd6ee39a0f2e2e9ef56fda86b5b7e9e3": "\\operatorname{ch}(L_w)=\\sum_{y\\le w}(-1)^{\\ell(w)-\\ell(y)}P_{y,w}(1)\\operatorname{ch}(M_y)", "fd6f12d324c84522da76c758d6c22edb": "g(\\dots)\\,", "fd6f1f858eef7410bb2e78558937836d": " Q = j^{2} \\rho ", "fd6f55942ff3b50cfbe28de3c9379275": "p(r)", "fd6f66f2644196e44ead3f134bd3b132": "A[x_1,\\ldots,x_n]", "fd6fd8c0c395be1849f84be5267b95d8": "K\\gg1", "fd6fdb915544f0f174985b71f9e99e83": "\\scriptstyle z \\mapsto \\frac{1}{z}", "fd6fe8ddc55ab3bca86b38b699b9f3d3": "{}^5_{4-3}", "fd700b89edc706a6708538706cb804f5": " \\sin{\\frac{A}{2}}=\\sqrt{\\frac{efg + fgh + ghe + hef}{(e + f)(e + g)(e + h)}},", "fd7025d70df900fb3e3eecce5a66e9d8": "g \\mid g\\bar{g} =N(g)", "fd704e232b6a9ec3dd3943a1faa9573f": "I = I_0{{\\left(\\frac{\\sin \\left(\\frac{\\pi a}{\\lambda }\\sin\\theta\\right)}{\\frac{{\\pi a}}{\\lambda } \\sin \\theta\n}\\right)}^2}{{\\left(\\frac{\\sin \\left(\\frac{N}{2}(\\frac{2\\pi d}{\\lambda} \\sin\\theta+\\phi )\\right)}{\\sin \\left(\\frac{{\\pi d}}{\\lambda\n} \\sin\\theta+\\phi \\right)}\\right)}^2} \n", "fd70a5e1b3db75c3462a6fe2bba5fef5": "B_1,B_2,\\ldots,B_m", "fd70b95a02678b87934c2df4683f2728": "\\psi \\rightarrow e^{i \\alpha \\gamma^5}\\psi", "fd70be21e6276b5cd685318bcaa579c8": "\\mathbf{1}_A \\colon X \\to \\{ 0,1 \\} \\,", "fd70ee72eafe0983e062ad407abcae08": "d(\\mathbf{X}+\\mathbf{Y}) =", "fd713c198fa7f704922596af42fa0d40": "R_3=-K_3", "fd714f1fdd2e15c473bac4667f92476d": "GF\\!\\ (p^6)^*", "fd7195cab40f44acff1555c9c72feec8": "\\tau:=\\inf \\{t\\geq 1 \\,|\\, B_s > 0 \\text{ for all } s\\in[t-1,t]\\}", "fd71a880e3a33830004b4ecc869fa0b7": "P_r = \\left [P_t{{G^2 \\lambda^2 \\sigma_0}\\over{{(4\\pi)}^3 R^4}} \\right] \\propto \\frac {\\sigma_0} {R^4}", "fd71d3415bc7440dcf979dcc75f52eb4": "j,\\,j\\neq k", "fd722874cca5117436222f09bddc3bbf": "D > 0", "fd72b1302360819429b53f4cb4a0a019": "Y^{-1}(S)\\in \\sigma(X)", "fd72e240541df89ad3b0088e314730b0": "S(\\cdot \\| \\cdot)", "fd73265c45134b126785731a459648fc": " p_i^{min} = \\overline p_i, \\ p_i^{max} = \\underline p_i ", "fd7357d0b68d79587673d988b942710c": "(e,0) \\in G", "fd7365ef3f24c5beaccaf1bad3a098b0": " \\sin E = \\sqrt{1 - \\cos^2 E} = \\frac{ \\sqrt{1 - e^2} \\, \\sin \\theta }{1 + e \\cos \\theta } \\ . ", "fd739265c85fcbd9657711b254c554bc": "A \\subset B \\subset S", "fd739cade5ca1c5c0fe50497c29812ac": "G \\approx 2-1000 \\,\\, J/m^2", "fd73b6f179e45916ba8db7af914e3d87": "\\mathcal{R}^{n}", "fd7428f496922bbdf66e752c0b8d21d7": "\\scriptstyle M = m - 5 \\log_{10} \\left( \\frac{100}{\\mathrm{parallax\\ (mas)}} \\right)", "fd744ee78046eeb4bf8743293b8c48fe": "J(x) = S(y) - S(x)\\,", "fd74879d720d87f4296e70ac9606b851": "c_nf = \\begin{cases}\nR^*\\frac{d^{n-1}}{ds^{n-1}}Rf & n \\rm{\\ odd}\\\\\nR^*H_s\\frac{d^{n-1}}{ds^{n-1}}Rf & n \\rm{\\ even}\n\\end{cases}\n", "fd749f667621f886b2a1d9378f69974c": "{\\omega^1}_2 = -\\left( 1 + \\frac{r \\, g'}{g} \\right) \\, d\\theta", "fd74b550ec85ef3c7ce469473d0c053b": "p = p_0 + p_1 X + p_2 X^2 + \\cdots + p_m X^m,", "fd74fd5707b6774fda2cdd8204ff5000": " \\operatorname{div}(X) = \\nabla\\cdot X = X^a_{;a} ", "fd7512a51cb0a1e38b068a90aa735c16": "p_A(t) = \\det \\left(t \\boldsymbol{I} - A\\right)", "fd75277a1ac9491a6adbad011f2caa4f": "\n\\left\\{\\begin{matrix} \\ln\\ \\gamma_1=Ax^2_2\n\\\\ \\ln\\ \\gamma_2=Ax^2_1\n\\end{matrix}\\right.", "fd7527d42685f43b5336a441e53133be": "\\varphi_{\\varphi_e(e)} \\simeq \\varphi_{F(h(e))}", "fd7551a30bd884d7d2a3ae529405c884": "P(x; k, \\lambda ) = e^{-\\lambda/2}\\; \\sum_{j=0}^\\infty \\frac{(\\lambda/2)^j}{j!} Q(x; k+2j)", "fd756e7941f1e1837fd17765a6ee01a9": "\n\\frac{1}{r} = \\frac{mk}{L^{2}} \\left( 1 + \\frac{A}{mk} \\cos\\theta \\right)", "fd75723fbb76eb8a33e1f1db3d6ff50e": "f^{(p)}", "fd7572fa38c7f6a99eb0d74307a9ea1e": "\\begin{bmatrix} \\dfrac{\\Delta \\mathbf{[y]}}{y_{22}} & \\dfrac{y_{12}}{y_{22}} \\\\ \\dfrac{-y_{21}}{y_{22}} & \\dfrac{1}{y_{22}} \\end{bmatrix}", "fd758993131555da2d41abf6cd6fa6fa": "D(E(m_2, r_2)^{m_1}\\mod n^2) = m_1 m_2 \\mod n. \\, ", "fd75dede4efa92ff2a26c59e29c9f5ef": "\\Delta G=G_{\\rm{products}}-G_{\\rm{reactants}}<0.\\,", "fd7641b4c73ab05ff29d856cbf47ba13": "\\Gamma_0 \\hookrightarrow \\Gamma_1 \\hookrightarrow \\Gamma_2 \\hookrightarrow \\cdots . \\, ", "fd76dd14fa76f15ac38938a7dc49cef2": "\\begin{align}\nU_{f,P_n} - L_{f,P_n} &= \\frac{1}{n}\n\\end{align}", "fd7710990bdef74ebe4a8aba1f457a4b": " df(Y) = Y(f) ", "fd77160927b274e56b10071b7bc7af24": "\nP^*_j(y)=\\sum^{r_j}_{k=1}c_{j,k}\\,(k-1)!\\sum^{k-1}_{i=0}\\frac{y^i}{i!\\,\\lambda_j^{k-i}}\n", "fd7753fce163715ef5e88df3a6e10c26": "|\\Im(z)|<\\Im(L)\\approx 1.3 ", "fd775e191033938fabd33d7a145c1880": "q=c^\\alpha(1-i\\beta\\Phi)", "fd778d5bdb825b2cc2458d8bde4e2bd6": "\\Delta(x)=\\frac{1}{2\\pi i} \\int_{c^\\prime-i\\infty}^{c^\\prime+i\\infty} \n\\zeta^2(w) \\frac {x^w}{w} \\,dw", "fd77b0aba1e9b06cbfabef9b6d72a784": "\\frac{d}{dt}(\\varphi \\circ \\mathbf{r})(t)=\\nabla \\varphi(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t)", "fd78057f94e26afc881800970df1cb68": "\\mathbf{x}_{0i}^\\top \\epsilon_{ii} \\lambda_{0i} [M_0] \\mathbf{x}_{0i} + \\mathbf{x}_{0i}^\\top[\\delta K]\\mathbf{x}_{0i} = \\lambda_{0i} \\mathbf{x}_{0i}^\\top[M_0] \\epsilon_{ii} \\mathbf{x}_{0i} + \\lambda_{0i}\\mathbf{x}_{0i}^\\top [\\delta M] \\mathbf{x}_{0i} + \\delta\\lambda_i\\mathbf{x}_{0i}^\\top [M_0] \\mathbf{x}_{0i}. ", "fd780c9ea97b44d3f4ef353ddb9fa2f9": "\\sigma_1 > \\sigma_2 > \\sigma_3", "fd788511232099ef43f86402b98246f9": "\\Lambda(x)=\\frac{L(\\theta_0\\mid x)}{\\sup\\{\\,L(\\theta\\mid x):\\theta\\in\\{\\theta_0,\\theta_1\\}\\}},", "fd790b626a125d2bbec7a94270f41931": "\\tilde s_i", "fd793107b1d6f79fa1f78208c2028c4b": "f(x) = -\\frac{\\nu(x)}{2} + \\int_{\\partial D} \\nu(s) \\frac{\\partial G(x,s)}{\\partial n} ds.", "fd79501826538a1b545f0d434d872f2b": "e^{+jwt}\\,", "fd796905bebad4165a8e4a9116d608e5": "\\lim_{x\\to a^-}f(x)\\ ", "fd7985b1278cdbaa2be5ec7d0b25ccff": "S^{15}\\hookrightarrow S^{31}\\rightarrow S^{16} , \\,\\!", "fd7987205ecedb7f40bcb1b0b29f8451": "x=(x_0,\\ldots, x_n)\\mapsto (\\partial F/\\partial x_0(x),\\ldots, \\partial F/\\partial x_n(x))", "fd7994a0483e4bcce5ea977f7dab2f7e": "\\widehat \\mu = \\frac {\\sum_k \\ln x_k} n,\n \\widehat \\sigma^2 = \\frac {\\sum_k \\left( \\ln x_k - \\widehat \\mu \\right)^2} {n}.", "fd79953a29ca422c98d0f7357f6d8a49": " m_{(2,1)}(X_1,X_2)= \\textstyle\\frac12p_1(X_1,X_2)^3-\\frac12p_2(X_1,X_2)p_1(X_1,X_2).", "fd7aab9f8cc2f9abc3d9faa1dd58b932": "m\\in \\mathbb Z_{n^s}", "fd7aac1bf1bbf529276177fbb2400f4c": "0<\\lambda<\\infty", "fd7b01445bba1a4fb51d0900c45cdf3a": "O(n^{2.807})", "fd7b3258a28083b64953d91b424853ac": "\\scriptstyle -0.3\\pm3\\times10^{-7}", "fd7b7448f094b1a10448d51eba82355f": "\\left\\| \\frac{f + g}{2} \\right\\|_{L^p}^q + \\left\\| \\frac{f - g}{2} \\right\\|_{L^p}^q \\le \\left( \\frac{1}{2} \\| f \\|_{L^p}^p +\\frac{1}{2} \\| g \\|_{L^p}^p \\right)^\\frac{q}{p},", "fd7b8a26ba7f3d297637bcdc2ed4a04c": "\\lambda_0 = 0, r_0 = 1", "fd7bd089093f5e79b889c4a0be111510": "\\begin{bmatrix}0 & 1 \\\\ 1 & -n \\end{bmatrix} , ", "fd7bee9a46ac7aae1acc9edee1bd84dc": "P = K_2 \\rho^{4\\over 3}", "fd7c120cba4b529146e4822539324bcf": "x, y, z, t, s", "fd7c8a4765dd5f05d7e36ebf55ff4816": "x(t),\\,", "fd7c909ca4bf53a74ebc6f21f6676c41": "\\Psi_\\alpha", "fd7cc4158614371505de9460e1fc388a": "\\vec w=(w_1,\\dots,w_n)", "fd7d3fc0089ad4ea757fd253b8ff1f3e": "\\int {1 \\over x}\\,dx = \\begin{cases}\\ln \\left|x \\right| + C^- & x < 0\\\\\n\\ln \\left|x \\right| + C^+ & x > 0\n\\end{cases}", "fd7d63c428f5fd6976a0fe753065b456": "\\mathbf{D} = \\mathbf{E}+\\mathbf{P}", "fd7dc7465a25947ce30619a1f6a5f5e3": "S \\subset T", "fd7ddf69e52592b8ba924ed85d32d830": " F^0{R}=R\\supset I\\supset I^2\\supset\\cdots, \\quad F^n{R}=I^n.", "fd7e2aaeada02771b7e3198546754ce6": "\\frac {h_k^{\\nu}} {\\sqrt {2}}= - \\frac {1} {2^{2 \\nu}}.\\binom{2k}{k}.\\binom{2 \\nu -2k}{\\nu -k}", "fd7e54f6841aa44b788211d765430d9f": "(Q \\downarrow Q)", "fd7e6488c9ea1cb7ad13a2911eaabab1": "\\mathfrak{g}_{-\\alpha}", "fd7f10f78735216d9543863e2325326d": "e^{\\lambda x}\\leq \\frac{b-x}{b-a}e^{\\lambda a}+\\frac{x-a}{b-a}e^{\\lambda b}\\qquad \\forall a\\leq x\\leq b", "fd7f251b7b0fe4a08c92fc0205714edc": "C_{3,1} = -10 \\log{\\left( \\frac{P_3}{P_1} \\right)} \\quad \\rm{dB}", "fd7f965deb3dbb50d6c166588dbee8ee": "1 = H_0\\triangleleft H_1\\triangleleft \\cdots \\triangleleft H_n = G,", "fd8005cdbace68f786d448681f885914": "\\epsilon_{j}", "fd80291e935ae11371b199675aa9fb14": "\n G = K_{\\rm I}^2\\left(\\frac{1}{E}\\right)\\,.\n ", "fd80520179c95ae957fd2cd31e12ddd9": "n \\mapsto n \\cdot x", "fd8055a5236418244a284cd553983bf3": "D \\left( x \\mapsto x^2 + 2x + 1 \\right) \\, ", "fd8086ed8d5a38a523bf6a7114b91471": "DP_{T}^{V}", "fd80a9454bce3ab6b432a883239c340b": "\\phi_{e4}", "fd80ed500cfbf2ff8a371893608caa27": "E(m^{\\prime}) \\in B(y,(p+\\epsilon)n)", "fd80f2a2227b62687ad8012c9f4ecb5c": "d.f. \\cong \\left[ \\frac{3(N-1)}{2n} - \\frac{2(N-2)}{N}\\right]\\frac{4n^2}{4n^2+5}", "fd810fd88cd78e6cc0351d382684292c": " 1^\\circ = (\\pi/180)\\,\\mathrm{radians}.", "fd811cf5964f1f462c6035ff2263af2b": "y=\\mathbf{w} ^{T} u", "fd8135344a57637b0a9e216883be07ff": "{\\text{Φ}_p}", "fd8142277367b0c8c73debb3dbee387a": "\nA_m(x,y) = \\sum_{p=0}^m \\binom{m}{p} x^p y^{m-p} \\cos ((m-p) \\frac{\\pi}{2}),\n", "fd8150d3ebff89ccd2291c6c165bedd5": "\\{\\theta,\\theta\\}=\\{\\bar{\\theta},\\bar{\\theta}\\}=\\{\\bar{\\theta},\\theta\\}=0", "fd81725309feadf9bcda35f3c29c5322": "\\scriptstyle B_n(x)", "fd81869048e2ec02ca9578746d75d002": "\\phi_m y_n =\\psi_n x_m \\text{ for all } m, n\\in \\mathbb{N} ", "fd818ef4011b12ee9f6e7fc8fd858e2b": "\\sqrt{A^*A}=U|\\Lambda|U^*", "fd819a5f83ad56bd54bc795fa767bbe9": "\\displaystyle V= L\\frac{di}{dt}", "fd81fa2a497af89f387fd462762bca83": "U'_r", "fd820df7368589dbdf3769b9a46bc7b5": "f_v(x)", "fd829f822d890dbbb532d0417a566250": " \\exp \\left( -i t \\right) ", "fd82dd1f8a39c64ddefb1df74d2d0111": "P_e = P_{e|H_0} \\cdot P_{H_0} + P_{e|H_1} \\cdot P_{H_1} + \\cdots + P_{e|H_{L-1}} \\cdot P_{H_{L-1}}", "fd82e1383bcca42f90f2d84734461885": "\\mathbf{k}_o - \\mathbf{k}_i = \\mathbf{G}", "fd82e6afe9a73db5b89743c05bd8b5c5": "10^{A({\\log_{10}{(x/b)}})^2}", "fd831dedfabe19a64f4081e9f9932430": "\\theta = \\frac{w_{o,1}c}{i^2} ", "fd83542b06386a873a7e0879a0b2e9dd": "\\sqrt{\\frac{2 \\pi}{k f''(0)}}", "fd8374f90427ba4a332ae0ac79b04420": "\\varphi(t,t') = E(t') J(t,t') - 1", "fd837bededff53ed7897bae8555d7ef1": "Ob(Chow^{eff}(k)) := \\{ (X, \\alpha) \\mbox{ }|\\mbox{ } (\\alpha : X \\vdash X) \\in Corr(k) \\mbox{ such that } \\alpha \\circ \\alpha = \\alpha \\}", "fd838a514161080264aa59db54b3ec87": "U \\subset W", "fd83a6c669bd5debacbb411cc5eee359": "\\varepsilon ({\\Lambda^\\mu}_\\nu) ={\\delta^\\mu}_\\nu \\,", "fd83a829db5594b1dc9030192013112a": "\\mathop{\\mathrm{div}} [V_h] (x) = - \\langle h, x \\rangle^\\sim,", "fd83cb00fa8ebc5722eb91e211c916dc": "\\scriptstyle{d\\ell}", "fd84209c3d3d7587a4af3135b5194f27": "u(t,x) \\sim a_\\varepsilon(t,x)e^{i\\varphi(t,x)/\\varepsilon} = \\sum_{j=0}^\\infty i^j \\varepsilon^j a_j(t,x) e^{i\\varphi(t,x)/\\varepsilon}.", "fd845e691568aeab69916225a0010919": "T_p(\\nu)", "fd84fd77cd74295bd6baa1e697ceeade": "v\\in V(G)", "fd85104adeed70495fe107cdc96ae4cd": "\\mu_{AB}", "fd85405e2bd60cfe6b73b002f04047b8": "y = Y/Z^3", "fd85a2fe490efa9cadb93897fb7ed775": "\n\\begin{array}{rcl}\n\\bar{\\mathbf{x}} & = & n^{-1}\\sum_{i=1}^{n} \\mathbf{x}_i ,\\\\\n\\mathbf{S} & = & n^{-1}\\sum_{i=1}^{n} (\\mathbf{x}_i - \\bar{\\mathbf{x}})(\\mathbf{x}_i - \\bar{\\mathbf{x}})' .\n\\end{array}\n", "fd85b37800fcff71aefdba386d9e8558": "\\psi(x;q,a)=\\frac{x}{\\varphi(q)}+O\\left(x\\exp\\left(-C_N(\\log x)^\\frac{1}{2}\\right)\\right),", "fd85ceef70e69668d6fd7fe7e1395e88": "y = x'\\sin\\left(\\Omega t\\right) + y'\\cos\\left(\\Omega t\\right)", "fd85ded136e2a2df45b43040e31fb415": "p > q > r\\,", "fd85e86edf6f7291a9e9b70e6c2b962e": "O(M_aD+M_bD^2)", "fd85fe1f96833403f767042f4e042e14": "\\text{extract}: (M \\rarr T) \\rarr T = f \\mapsto f \\, \\varepsilon", "fd86007e1e10b74807d85dcd1143934d": "T^*M\\otimes T^*M \\to \\mathbf{R}", "fd863b01647cb662580dd67b9839d8de": "1-a", "fd8640a964abdb111261fd2d254d00c0": " \\mathit{Re} = {\\rho v_{s}^2/L \\over \\mu v_{s}/L^2} = {\\rho v_{s} L\\over \\mu} = {v_{s} L\\over \\nu} ", "fd8641939d1672f77df03461b786f7cc": "\\textstyle \\sup_{t\\in[0,\\infty)} \\mathbb{E} \\mathrm{e}^{|M_t|} < \\infty ", "fd86d6fbc7d67653c547a0da7a57025b": "\\widehat{\\rho}_i", "fd86e5fc064f9d82821d3082061d48b6": " C_{V} = \\alpha R ", "fd8743348bab52805aeacb0c5da1a30c": "T^r_s (V) \\to T^{r-1}_{s-1}(V)", "fd87447d0fc76670597bffa8ceb61ff3": "\n \\begin{align}\n F_{12} &= \\cfrac{1}{2\\sigma_{b12}^2}\\left[1-\\sigma_{b12}(F_1+F_2)-\\sigma_{b12}^2(F_{11}+F_{22})\\right]\\\\\n F_{13} &= \\cfrac{1}{2\\sigma_{b13}^2}\\left[1-\\sigma_{b13}(F_1+F_3)-\\sigma_{b13}^2(F_{11}+F_{33})\\right] \\\\\n F_{23} &= \\cfrac{1}{2\\sigma_{b23}^2}\\left[1-\\sigma_{b23}(F_2+F_3)-\\sigma_{b23}^2(F_{22}+F_{33})\\right] \n \\end{align}\n ", "fd874c857f139b9ba1736f11c0ff94c2": "\nW = -\\mu_{1} \\left( \\cosh \\xi + \\cos \\eta \\right) - \\mu_{2} \\left( \\cosh \\xi - \\cos \\eta \\right).\n", "fd879f21305910f622460de05fab3cc4": "L=\\frac{P}{i}\\left(1-\\frac{1}{(1+i)^n}\\right)", "fd87c8d0ac966969fe826a9df80f0465": "\\mathrm{Ha} = \\frac{N_{\\mathrm{A}0}}{N_{\\mathrm{A}0}^{\\mathrm{phys}}} ", "fd87ee80d2553460b09bcad6812a26d9": "v_{\\it avg} = \\mu E", "fd8884c4e60863bc7d82e88d3671f98d": "\\mathcal{L}_{0}", "fd88870a24ca2cc6ba617aeac04004c6": "V_S - (-V_S) = 2V_S\\,", "fd88972e3458019412bfdee45fe8cd4c": "D(x)=\\{a \\in P : a \\leq x\\}", "fd890bb416a8de357dd5fb7b45ce913e": "\\operatorname{cl}(X)= \\operatorname{cl}(\\operatorname{cl}(X))", "fd896a4f6fb66ac95fad1deb4f6cbe1e": "\\mathfrak{gl}_k", "fd8982ae0b17dc4bfe433e484640eb7a": "Z \\subset D", "fd89b7eeb6fe4bad9ea654148f793ab6": "Z = \\frac{1}{Z'}", "fd8a36ac01caa0c20cc84704841f282b": "\\rho \\ll z ", "fd8a7b941ff39c0e18e6ee2ade6188e2": " 0 = \\left[ \\frac{1}{2} v_1^2 + \\Psi_1 + \\epsilon_1 + \\frac{p_1}{\\rho_1} \\right] \\rho_1 A_1 v_1 \\, \\Delta t - \\left[ \\frac{1}{2} v_2^2 + \\Psi_2 + \\epsilon_2 + \\frac{p_2}{\\rho_2} \\right] \\rho_2 A_2 v_2 \\, \\Delta t ", "fd8a8f170e118fe6296bad38cd9f2d5e": "\\frac{dS}{dT}", "fd8ada823993f8d8da2a03b3571544ab": "\\gamma_3\\,", "fd8ae46b4be75dc64b411a1b22ec1cf4": "(1/2)^{10}", "fd8afa1c7cf39c4d7f53e8447cca8284": "m_7", "fd8aff89edde1a9d901350c6d0d4f06e": "O(n/\\log n)", "fd8b249e123b67ebdf701c08029018bd": "P^2-1024z\\Delta", "fd8bb63a4dfb9c62640f0db945f19f03": "f_c(\\infty)=\\infty=f^{-1}_c(\\infty)\\,", "fd8bef40d1488a97db7c8a746da07ee4": "{n_x,n_y,n_z}", "fd8c047df62d7d7eb70e3a901ee1750e": "\\left.\\frac{d\\theta}{dx}\\right\\vert_{x=L}=0.", "fd8c1e9ec8e9c6de4b5e8d038d812bd5": "\\rho_{DOS}^{ij} = \\frac{\\Delta \\Phi_j}{\\Delta Q_i} = \\frac{i}{j}R_H, \\ ", "fd8c24b0fa11654165056d4529594d78": "P = y,\\ ", "fd8c6c3b8e90344f773d1179184c1ea0": "~\\langle \\hat a\\rangle ~", "fd8cb01151da22d4dd994773ffc0d30d": "q \\ne 0", "fd8dac3c31e5c2e0928babb872c6ef5b": "\\Lambda_n(S)<\\frac{2}{\\pi} \\log(n+1)+b,\\,", "fd8dc978c9a0344fcfc1ca54fa59e5df": " \\begin{align}\nA( x ) &= \\sigma \\sum { x^{ 2i } } = \\sigma \\frac{ x^{ 2n } - 1 } { x - 1 }, \\\\\nB( x ) &= \\frac{ 1 } { 2 } \\frac{ d } { dt } A( x ), \\\\\nC( x ) &= \\frac{ 1 } { 4 } \\frac{ d^2 } { dt^2 } A( x ) + \\frac{ 1 } { 4x } \\frac{ d } { dt } A( x ). \n\\end{align} ", "fd8eafce25b24032049d219ab2c95d58": "\\theta=2\\cos^{-1}\\left(\\frac{r_1-r_2}{P}\\right)\\, .", "fd8ed3912d51829df5db7baaa2998a59": "R_{m\\cdot n}", "fd8f3e6527b28d1b7bb8e9ed290ef410": "\n\\mathbf{v}=\\frac{1}{\\sqrt{\\varphi_x^2+\\varphi_y^2+\\varphi_z^2}}\\begin{bmatrix} \\varphi_x \\\\ \\varphi_y \\\\ \\varphi_z \\end{bmatrix},\n", "fd8f8125234e6e89604587025a97e6a4": "\\,\\frac{dQ(t)}{dt}\\,", "fd8fba40b347ac19fa77d6d228250de9": "~~~~~T,p,\\{N_{i\\ne j}\\},\\mu_j\\,", "fd8fd5e48fe07cc6bad935082ea1cb11": "\nE = n\\hbar\\omega\n\\,", "fd8fde9d7f60c8cba5c37061e039e2f1": "\n\\frac{\\partial L}{\\partial q_k} -\n\\frac{d}{d\\sigma}\\frac{\\partial L}{\\partial \\dot q_k} = 0\n", "fd9083faa9b4b9d0340e6d97a78cf2de": "\\nabla \\rightarrow ik", "fd91086c5fce44b6872d98e24b0b8dfb": "\\sigma_0 = \\sigma,", "fd910f534f4527d3a92a2ed5af257bda": "C_{ij}\\ = (-1)^{i+j} M_{ij}\\,,", "fd9192578c71166211158bd8a2804791": "x_1 x_2 = f^2,\\!", "fd91c508f91c2c84498680bd337c1d7a": "y = f(x)", "fd92279560f0c70c42b3043dc4c4646b": "\\beta_{im}", "fd923f118d0ec6fdd000289149782709": "j_r", "fd924fb508487271b79b39da90ab763b": "H_f", "fd93129f799aa017396814803066f9df": "X_\\text{reg}:=X\\setminus \\text{Sing}(X)", "fd93158d2e070a0af67dcdd6594cab47": "\\frac{\\int_a^b x g(x) dx}{F(b)-F(a)} ", "fd940510117b245717a34b2f6b9249aa": "(X,Y) ", "fd9421088e565f623c2bd2a110f4728b": "\\sum_{m=0}^n (-1)^m {\\left \\langle {n\\atop m} \\right \\rangle} {\\binom{n}{m}}^{-1} = (n+1) B_n.", "fd944651ddab0d3873b9f369b965bf79": "r(\\theta)=\\frac {p}{1+e\\cos \\theta}\\,", "fd9486517412ee29483c24711cecbf52": "I(X;Y) = I(Y;X) = H(X) + H(Y) - H(X,Y).\\,", "fd94c1f1c63737c30d992014dfda661e": "K=\\frac{k_+}{k_-}=\\frac{[S]^\\sigma [T]^\\tau \\dots } {[A]^\\alpha [B]^\\beta \\dots}", "fd94da8a3e97c503d08460069b2680e4": "X_1 \\cup X_2\\cup \\dots\\cup X_k", "fd95328ca64d5804dbb17ac2df844eca": " h=(b-a)/n, \\quad \\quad x_i = a + ih,", "fd953bb39a0139bbd0d223a490da912a": "\\eta=1/\\lambda", "fd95940112c1f0b996af7abf5e379d51": "\\begin{align} 2\\cdot R_*\n & = \\frac{(97\\cdot 5.12\\cdot 10^{-3})\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 107\\cdot R_{\\bigodot}\n\\end{align}", "fd95bc46ff9cee96ff7995969984a418": "q_1 q_2", "fd95daf553a436f366df9b70f9b965fc": "\\nu_e + n \\rightarrow e^{-} + p", "fd95df54052a49f5dbf931fae79cc234": "-1.1670", "fd95ff6288261ac2da55cfe4adbaccb2": "\n\\sum_{i = 0} ^n (-1)^i \\; \\mbox{rank} \\; H_i (X_n, \\empty)\n\n= \\sum_{i = 0} ^n (-1)^i \\; \\mbox{rank} \\; H_i (X_n, X_{n-1}) \\; + \\; \\sum_{i = 0} ^n (-1)^i \\; \\mbox{rank} \\; H_i (X_{n-1}, \\empty).", "fd96491e7b16e059df9911fcd7ef7e5a": "\\frac{a-b}{a+b} = \\frac{\\tan[\\frac{1}{2}(\\alpha-\\beta)]}{\\tan[\\frac{1}{2}(\\alpha+\\beta)]}.", "fd96868159b45c96c583fe7eb0bb377d": "\\Vert x \\Vert", "fd96875ead9b55165c172d7bfb4593f3": "\\delta =-\\int_{0}^{h}{\\tan \\phi \\frac{\\text{d}\\mu }{\\mu }} \\,,", "fd96df40a2079fd89f1921245d02fac0": "pino(\\rho).\\sigma\\mid\\sigma_0(P) \\rightarrow_{b} \\sigma\\mid\\sigma_0(\\rho (~)\\circ P)", "fd974f845f05fc7e91e90ea89bb054a8": " 1-\\Pr(\\limsup_{n \\rightarrow \\infty} E_n) = 0. \\, ", "fd977f6c8a27ed90ce02947238d808bc": "\nB=s_1s_2=x_0^2+x_1^2+x_2^2+(\\zeta+\\zeta^2)(x_0x_1+x_1x_2+x_2x_0)=E_1^2-3E_2\\,.\n", "fd97a6c2f771238827cdb4052977cd78": " \\left( \\frac{{99 \\choose 49}}{{100 \\choose 50}} \\right)^{100} =\n\\frac{1}{2^{100}},", "fd97e058139078488d3cf74670914abf": "\\operatorname{div} \\, \\operatorname{curl} \\mathbf{A} \\equiv \\nabla \\cdot (\\nabla \\times \\mathbf{A}) = 0", "fd97f48c94363a7d46519399f271a1a5": "\\complement_U A", "fd9874d0a47a0c5c878489a26993795a": "c_{8}", "fd9879a515b4900aea69dd22fb87c3dd": "L_d^p", "fd9894ee97b038d9e5b8fa0416ef7a44": "\\mathbb{P}(E) = \\mathbb{P}(E \\cap A) + \\mathbb{P}(E\\cap B) = \\mathbb{P}(E\\mid A)\\cdot\\mathbb{P}(A) + \\mathbb{P}(E\\mid B)\\cdot \\mathbb{P}(B)=\\frac{19.9}{10 000}", "fd98a1ca41a0480046c7f8ef93a9f0b1": "Q=\\frac{-K A}{\\mu}\\frac{(u_{e,b}-u_{e,a})}{L}", "fd98fa8e57aa117c279e6c9b8a534a82": "\\mathbf{t}_1, \\dots ,\\mathbf{t}_{N-1}\\in [0,1)^k", "fd993c7af59e04497f26f2e15cfaea78": "\\mathcal{P}(e) = 0 ", "fd995d9c9ba78c72f7c94bd5b887bdd1": "P(r \\ge k_1 ; k_1 n, \\frac{1}{n}) > P(r \\ge k_2 ; k_2 n, \\frac{1}{n})", "fd9996aef8c96d47983998becf7e35ff": "\\varepsilon(\\omega_s)", "fd99a2d58d744b6b12cc02162b99464e": "h=h(x)", "fd99caffb49a8f88e2d6f96cf63ad76d": " \\mathbb R_+=[0,\\infty) ", "fd9a780521fdeaff67a042e99030c8a5": "{{(i)}_{{}}}B(y)-{{B}_{x}}(y)={{0}_{{}}}", "fd9c1f7e1b470304eceee073c0b508ef": "|3z^3 + 7| < 32 = |z^5|", "fd9c4a8968b55edfaa3a170dea68dc61": "m_i = \\frac{a_i + b_i}{2}", "fd9c50aa055a057bfe6be687e5e53be5": "P(x(k)\\ |\\ y(1), \\dots, y(t))", "fd9c5a1720e9159a5e04ae90e5d1f10b": "\\frac{\\partial p_2}{\\partial x} = \\rho\\, \\Omega\\, U_0\\, \\sin\\left( \\Omega\\, t \\right).", "fd9ce0ae94be3c48004b1777c013e6c9": "\\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix}", "fd9d395a9b8bd1d1158d6c5f9358facb": "\\bar{z} = \\operatorname{Re}(z) - \\operatorname{Im}(z) \\cdot i ", "fd9d4f9ac8da054c19d5abfe675ca31a": "\\Phi _\\alpha \\left( {A_\\alpha ^1 , \\cdots , A_\\alpha ^n } \\right)_{+} = \\mathop {\\max }\\limits_{\\begin{array}{l} W_{\\alpha - }^i \\le w_i \\le W_{\\alpha + }^i A_{\\alpha - }^i \\le a_i \\le A_{\\alpha + }^i \\end{array}} \\sum\\limits_{i = 1}^n {w_i a_{\\sigma (i)} / \\sum\\limits_{i =\n1}^n {w_i } } ", "fd9d8568a683649540655a2a8f7d9d98": "\\bold{n}_v = \\frac {FN-GM} {EG-F^2} \\bold{r}_u + \\frac {FM-EN} {EG-F^2} \\bold{r}_v ", "fd9da0e4b300a5d6be24ca61da3f4d80": "1/\\delta", "fd9e11bbdf1d121341f781cb5cd772ee": "%D = 100*H3/L3", "fd9e20aeaa428d6374d309eff7779d3e": "\\langle P \\rangle \\approx \\frac{GM^2}{4\\pi R^4}", "fd9eaabcb6ec595e28ee8cd82d90e784": "n \\geq n_0", "fd9f4e65f0d79095722502b696f23179": "\\scriptstyle A_1=a_1a_2", "fd9f577ffb88772e89fa4bea541fe320": "T_{C,i}", "fd9f712ac4ceab0036eb2666108fb0e5": "G_{LCU}", "fd9f88cedbfa03172c36c47c44a0aa87": "i \\hbar", "fd9f9a3a39e7e35c842646a066e12571": "\\begin{align}\n & \\Gamma = V_{xx} + V_{yy} + i(V_{yx} - V_{xy}), \\\\\n & C = V_{xx} - V_{yy} + i(V_{yx} + V_{xy}).\n \\end{align}", "fd9ff5f8373610a5530ba70a49a4d2c6": "\\lambda^a_\\alpha \\lambda^b_\\beta", "fd9fff897be3dd14279377c788c34e95": "\\varepsilon N", "fda08e64b78dbfe8c1e4159f3bdf9927": " = \\frac\\hbar m \\mathrm{Im}(\\Psi^*\\nabla\\Psi)=\\mathrm{Re}(\\Psi^* \\frac{\\hbar}{im} \\nabla \\Psi)", "fda1619d01ab753693dc50c442a2bd01": "|\\bar{z}| = |z|,\n|(\\bar{z})^n| = |z|^n,\n\\arg(z^n) = n \\arg(z)", "fda16a4b844bfab28c436e20ac4de783": "\\sin(22\\tfrac12 ^\\circ) = \\frac12\\sqrt{2-\\sqrt{2}};", "fda215555861d8f0cafc747644149a4d": "(m-1,k-1)", "fda2180a439c18642fd674cb030152c7": "L_{q} = 0, L_{qq} \\geq 0, \\partial_t(R) = 0, \\partial_{tt}(R) \\leq 0.", "fda25b193c7a30f75d01cd0fb5cd94fb": "\\begin{align}\n\\int_{\\mathbb{R}^n} f(x)g(x) \\, dx &= \\displaystyle\\int_{\\mathbb{R}^n}\\int_0^\\infty \\int_0^\\infty \\chi_{f(x)>r}\\chi_{g(x)>s} \\, dr \\, ds \\, dx \\\\[8pt]\n&= \\int_0^\\infty \\int_0^\\infty \\int_{\\mathbb{R}^n}\\chi_{f(x)>r\\cap g(x)>s} \\, dx \\, dr \\, ds \\\\[8pt]\n&= \\int_0^\\infty \\int_0^\\infty \\mu\\left(\\left\\{f(x)>r\\right\\}\\cap\\left\\{ g(x)>s\\right\\}\\right) \\, dr \\, ds\\\\[8pt]\n&\\leq \\int_0^\\infty \\int_0^\\infty \\min\\left(\\mu\\left(f(x)>r\\right);\\mu\\left(g(x)>s\\right)\\right) \\, dr \\, ds\\\\[8pt]\n&= \\int_0^\\infty \\int_0^\\infty \\min\\left(\\mu\\left(f^*(x)>r\\right);\\mu\\left(g^*(x)>s\\right)\\right) \\, dr \\, ds\\\\[8pt]\n&= \\int_0^\\infty \\int_0^\\infty \\mu\\left(\\left\\{f^\\ast(x)>r\\right\\}\\cap\\left\\{ g^\\ast(x)>s\\right\\}\\right) \\, dr \\, ds\\\\[8pt]\n&= \\int_{\\mathbb{R}^n} f^*(x)g^*(x) \\, dx \n\\end{align}\n", "fda275dc7704cc5aa6cd8ec5a653911b": "\nc^2 {d \\tau}^{2} = \\frac{(1-\\frac{r_s}{4r_1})^{2}}{(1+\\frac{r_s}{4r_1})^{2}} \\, c^2 {d t}^2 - \\left(1+\\frac{r_s}{4r_1}\\right)^{4}\\left(dr_1^2 + r_1^2 d\\theta^2 + r_1^2 \\sin^2\\theta \\, d\\varphi^2\\right)\n\\,.", "fda27a80d6d072258fbb165a01115924": "{\\color{Blue}~2.22}", "fda2e0f99625c3082de11fe67f5c4087": "\\Gamma_{e}(d)-\\Gamma_{e}(0) = \\Gamma_{e}(0)\\left(\\mathrm{e}^{\\alpha d}-1\\right)\\qquad\\qquad(2)", "fda329ada8fb9e71a5e9d00a93745b29": " 1-\\delta + o(1) ", "fda3c52ba435a6b7200ab9929c987e45": "\\displaystyle{Hf= \\mathrm{P.V.}\\,{1\\over \\pi} \\int {f(\\zeta)\\over \\zeta-e^{i\\varphi}}\\, d\\zeta.}", "fda439e8cf271eb9b2ba4562af55270e": "\\sum_{i=1}^{N} m_i = m_{tot} ; \\sum_{i=1}^{N} w_i = 1", "fda4ab5c5567dede74a9cf66722197f6": "(1-L)^d X_t \\ ", "fda4b3d808bbe24f589dd0ca7f0754f3": "\\theta = \\mathrm{E}( h(X) ) = \\mathrm{E}( Y ) \\, ", "fda4e3905e24b9169c9e145c5912ca2a": "\\vec{F}\\,(\\vec{p}) - \\vec{F}\\,(\\vec{q})=(\\vec{p}-\\vec{q})\\frac{\\partial \\vec{F}\\,(\\vec{q})}{\\partial \\vec{p}}\\!", "fda590da47d29ff232f36ce45c1213c3": "V_L = I_L X_L\\,\\!", "fda5ec8f4f1a35dc016e3018d25d6646": " - \\sum_{i=1}^n p_i \\log_2 p_i \\leq - \\sum_{i=1}^n p_i \\log_2 q_i ", "fda63aaa780b614ed3a2a27d6ddfa213": "\\mathbf{v} = f\\frac{Z\\mathbf{V} - V_z\\mathbf{P}}{Z^2}", "fda669ba340359cb39152b5771b1b0dc": "\\begin{align}\n\\text{excess kurtosis}\n &=\\text{kurtosis} - 3\\\\\n &=\\frac{\\operatorname{E}[(X - \\mu)^4]}{{(\\operatorname{var}(X))^{2}}}-3\\\\\n &=\\frac{6[\\alpha^3-\\alpha^2(2\\beta - 1) + \\beta^2(\\beta + 1) - 2\\alpha\\beta(\\beta + 2)]}{\\alpha \\beta (\\alpha + \\beta + 2)(\\alpha + \\beta + 3)}\\\\\n &=\\frac{6[(\\alpha - \\beta)^2 (\\alpha +\\beta + 1) - \\alpha \\beta (\\alpha + \\beta + 2)]}\n{\\alpha \\beta (\\alpha + \\beta + 2) (\\alpha + \\beta + 3)} .\n\\end{align}", "fda6b4980c88250a81a7bd792a024726": "{{P_t G_t}\\over{4 \\pi r^2}} \\sigma", "fda6cfe4e2697679d09b50bb47483a04": "g[n]", "fda6eab83f4ac91fcefda5f5702e62e8": " I = \\sum_m |j , m' \\rangle \\langle j, m' | ", "fda7195d561a05f632e9f083595e147e": "\\forall x\\in W\\;p(x)>0", "fda73c499a5bafe19832f8ee83cb685a": "\\mathbb{Z}/n\\mathbb{Z}", "fda74d29cd10fb3f75614345290bdab5": "\\left(\\sum_{n=1}^\\infty a_nn^{-s}\\right)\\left(\\sum_{n=1}^\\infty b_nn^{-s}\\right)=\\sum_{n=1}^\\infty\\left(\\sum_{k\\ell=n}a_kb_\\ell\\right)n^{-s};", "fda78d4d19d5a12c079af3ac2e71731f": " V(t)=V_0 e^{-\\frac{t}{RC}} \\ , ", "fda79a558245e7e808adc9d6c453714f": "\\phi_*(v) = \\sum_{i=1}^n \\sum_{a=1}^m v^i\\frac{\\partial \\phi^a}{\\partial x^i}\\mathbf{e}_a.", "fda7afb212802c82bce9d665d0d98351": "\\frac{\\mathrm{d}\\mathbf{m}}{\\mathrm{d}t}=-\\gamma\\mu_0 \\left(\\mathbf{m} \\times \\mathbf{H}\\right)", "fda7d1414535e3b2e0e07fcc5e973f63": "{\\mathcal P}", "fda813dcb34ac80ecb1c6315d909c884": "\\left| \\int_{\\mathbf{R}^{n}} \\mathrm{D} \\varphi (y) (x) \\, \\mathrm{d} \\mu(y) \\right| \\leq C(x) \\| \\varphi \\|_{\\infty}", "fda819b6d813fb6a314b943c02acf388": "\\begin{align}\n\\Phi_{mK}: & M\\longrightarrow\\ \\ \\ \\ \\ \\ \\mathbb{P}^N \\\\\n& z\\ \\ \\ \\mapsto\\ \\ (\\varphi_0(z):\\varphi_1(z):\\cdots:\\varphi_N(z)) \n\\end{align}", "fda835756b171098ffe05da0e3f06d5b": "m_s \\in \\{ -s, (-s+1), \\ldots, (s-1),s\\}", "fda84bfab74d3bfe52d485f602891756": "\\omega \\in \\Lambda^k(M)", "fda86dec751ed3ba2232c6af70361547": " a_{R} ", "fda88d2e4d0ce18916c7ac3b6971604d": "\\frac{}{}C", "fda8ace78371dd3233b18223e2e35224": " X \\rightarrow E", "fda913fc60c70a832c6651e668765b88": "U =- \\frac{1}{2} \\overrightarrow{m}(\\overrightarrow{B}) \\cdot \\overrightarrow{B} ", "fda921fcb0be2c2cfe27f381e26f96c1": "f(x)=A_{i}", "fda95b59af07deb418cffa0b26aeea14": "x'=x-vt\\,", "fda979729886bbb2076b4cc35c5729af": "{}=1264.14", "fda98e2d23844d11b0f01996349bbdff": "\\left | \\Psi \\right \\rangle", "fdaa0527da130b56e02a517f25d88c70": "\\tau _1 \\vee \\tau _2", "fdaa238c576f06e5ce409082cab95265": "d_2'", "fdaa2a149c6603d971da479afaadc3c6": "log\\ D_{acids} = log\\ P + log\\Bigg[\\frac{1}{(1+10^{pH-pK_a})}\\Bigg]", "fdaa4d893d205be7d9f7b7abdeaf37e8": " 180^\\circ < \\alpha < 360^\\circ ", "fdaa72716582fa13350bf86c075e1247": "21\\; \\,= 3^1 \\cdot 7^1 \\,\\!", "fdaad8eb5e04577aaee92d59f83a0f02": "\\!dg_E", "fdaadbeb89e42b8ac2325d722dfbbc8e": " x_1^* = \\max \\left\\{ 0, \\frac{8 p_1 - 2 }{5} \\right\\}, x_2^* = \\max \\left\\{ 0, \\frac{2 - 3 p_1}{5} \\right\\}, y_2^* = \\frac{p_1+ 1 }{5}. ", "fdaae34ad147b47d97518234013e54e1": "L \\approx R,", "fdaae69d80a9e22398949d743b5e5428": "-1 / \\phi \\,", "fdab7df0ecb5c50c5dfa057f79f32d6d": " \\!\\ \\sum_{n=1}^N \\sum_{i=1}^{n} i ", "fdab882cb19fc72d9d8c267100b1549c": "\\dot{p}_a = - \\frac {\\partial H}{\\partial x^a} = \n-\\frac{1}{2} \\frac {\\partial g^{bc}(x)}{\\partial x^a} p_b p_c.", "fdab94bf18eb889bbcdb1fcde66b72d6": "p = 1/2.", "fdab9ca512c057fe1e8322be88f36538": "I_{ijkl} = \\frac{1}{2}(\\delta_{ik}\\delta_{jl} + \\delta_{il}\\delta_{jk})", "fdaba2add5b65a0ecee0c69b360a4d7c": "q_\\beta(Tv)\\le M(p_{\\alpha_1}(v) +\\dotsb+p_{\\alpha_n}(v)).", "fdabec8b51083c7f7c41a9b5be9aa14d": "[\\Phi_{j+1}]_{\\Phi_j}, [T^{2^j}]_{\\Phi_j}^{\\Phi_{j+1}} \\leftarrow QR\\left([T^{2^j}]_{\\Phi_j}^{\\Phi_{j}}, \\epsilon\\right)", "fdad85cc64e5669bd1f475bf58121572": "p(c)\\textstyle \\sum_{i=1}^n p(f_i|c)\\log p(f_i|c)", "fdadd74ab94feb76a8dc183dac95e2ef": "\n \\mathbf{b}^i = \\boldsymbol{F}^{-\\rm{T}}\\cdot\\mathbf{e}^i ~;~~ g^{ij} = [\\boldsymbol{F}^{-\\rm{1}}\\cdot\\boldsymbol{F}^{-\\rm{T}}]_{ij} ~;~~ g_{ij} = [g^{ij}]^{-1} = [\\boldsymbol{F}^{\\rm{T}}\\cdot\\boldsymbol{F}]_{ij}\n ", "fdadee759b58a32d0e20f7a9e11023bc": "\\left\\langle \\Phi_i^{SO} | \\Phi_j^{SO} \\right\\rangle = \\delta_{ij}", "fdadfe8883e2cdaa994ef92a38300d73": "\\left[f^{-1}\\right]'(a)=\\frac{1}{f'\\left( f^{-1}(a) \\right)}", "fdae5d61434409f09392b7703682db08": "\n\\operatorname{Li}_2(z) + \\operatorname{Li}_2(1/z) = -\\tfrac{1}{6} \\pi^2 - \\tfrac{1}{2} [\\ln(-z)]^2 \\qquad (z \\not \\in ~[0; 1[) \\,,\n", "fdae98239c60502d723bb3999e65842e": " \\operatorname{div}(X) = \\frac{1}{\\sqrt{\\operatorname{det} g}} \\partial_a (\\sqrt{\\operatorname{det} g} X^a)", "fdaeb0081215225db766c4646bc02d44": "(X_1,X_2)", "fdaee827b6d6b1d595d6bd22b12cf442": "\\int_a^b f(x) \\, dv(x)", "fdaeeba7d96641b6d2a82f2c403ec06e": "F(r)=-\\frac{k}{r^3}.", "fdaef979060e11a46d8e7d05112e2a6b": "N\\subset[a,b]", "fdaf1248cc61bfc4a3d293d673260bac": "\\Phi_{20}=D\\lambda-\\bar{\\delta}\\pi-(\\rho\\lambda+\\bar{\\sigma}\\mu)-\\pi^2-(\\alpha-\\bar{\\beta})\\pi+\\nu\\bar{\\kappa}+(3\\varepsilon-\\bar{\\varepsilon})\\lambda\\,,", "fdaf29fc532585a4261a65d6f2d50544": "\\bigcap V(a_i) = V (\\Sigma a_i)", "fdafc390b7d6dee6bbbefc3b9b8b3b01": "\\mathfrak{H}(k; \\gamma_1, \\gamma_2) =\n\\begin{pmatrix}\n\\gamma_1 - k\\gamma_2 & (k - 1) \\gamma_1\\gamma_2 \\\\\n1 - k & k\\gamma_1 - \\gamma_2\n\\end{pmatrix}", "fdafc576c2073e4b03fc95d68327a6e0": "\n\\begin{pmatrix} \ny_1 \\\\\ny_2\n\\end{pmatrix}\n=\n\\begin{pmatrix} \na & b \\\\\nc & d\n\\end{pmatrix}\n\\begin{pmatrix} \nx_1 \\\\\nx_2\n\\end{pmatrix}.\n", "fdb066fa78ee4d04f6860ee19a9105d3": "z_1=x, z_n=y", "fdb06da4cad6c65969210cb2601244fa": " Z ( \\sin( B + \\theta ) , \\sin( A + \\theta), -\\sin \\theta ) ", "fdb0931e011d8ce1245b48c0f6c639ad": "\\sum_i \\left\\| \\alpha_{(i)} \\right\\|^2 < \\infty.", "fdb0c58e3deefec758a8bf358b0db612": "\\begin{bmatrix}\nc_{1}\\\\\nc_{2}\\\\\n\\vdots\\\\\nc_{n}\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\left\\langle p_{1},p_{1}\\right\\rangle & \\left\\langle p_{2},p_{1}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{1}\\right\\rangle \\\\\n\\left\\langle p_{1},p_{2}\\right\\rangle & \\left\\langle p_{2},p_{2}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{2}\\right\\rangle \\\\\n\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\\left\\langle p_{1},p_{n}\\right\\rangle & \\left\\langle p_{2},p_{n}\\right\\rangle & \\cdots & \\left\\langle p_{n},p_{n}\\right\\rangle \\end{bmatrix}^{-1}\n\\begin{bmatrix}\n\\left\\langle x,p_{1}\\right\\rangle \\\\\n\\left\\langle x,p_{2}\\right\\rangle \\\\\n\\vdots\\\\\n\\left\\langle x,p_{n}\\right\\rangle \\end{bmatrix},", "fdb0df7e39d79ca905a610efa81e8490": " \\delta_c = \\frac{ C_c }{ 1 + e_0 } H \\log \\left( \\frac{ \\sigma_{zf}' }{ \\sigma_{z0}' } \\right) \\ ", "fdb0fb75337119f3205598893e6e2370": " _{i} ", "fdb153d9fce748b3b83424689f44cf83": "\\tan(-\\alpha)=-\\tan \\alpha", "fdb189d4dda06405d05da56cf6c3d65a": "\\langle \\mathbf R \\cdot \\mathbf R \\rangle = 3Nb^2", "fdb1fcc251774ba9512d4979c7e71b5b": "B_{i+1}=-\\frac{i}{i+1}\\beta_i", "fdb2bfc4794a059bbe2f455c740ed608": " b_k^2 / 2", "fdb2d4a147a127f5c9097b01d403a817": " M^{(n)}(B_1\\times,\\dots,\\times B_n)=\\prod_{i=1}^n[\\Lambda(B_i)], ", "fdb2e16cba94253d6c4a1b888be780f0": "\\mathit{Kurt} = \\frac{ \\phi_{ 95 } - \\phi_{ 5 } }{ 2.44 ( \\phi_{ 75 } - \\phi_{ 25 } ) }", "fdb2ee2d9717e783be000d4ea98d7959": "d_1 = \\frac{1}{2} \\left( ( | a_{n-1}| - 1 ) + \\sqrt{ ( | a_{n-1} | - 1 )^2 + 4a } \\right)", "fdb2f41719e75654d44270bf8ebccb30": "\\sigma_a=\\limsup_{n\\to\\infty}\\frac{\\log(|a_1|+|a_2|+\\cdots+|a_n|)}{\\lambda_n}.", "fdb32d137d4e45d0b1d4a37641624c90": "L_2=\\ln\\ln x", "fdb3787c4a3e08838a572f078cf3ad4d": "\\mathbf{H}_1", "fdb3822a4db1b55a05c8e5e5609fd321": "a = 6\\,378\\,137\\,\\mathrm{m}", "fdb38cbed7e52fe9e9af3352e7708d20": " \\operatorname{de-let}[M_2\\ N_2] ", "fdb42446e531a39ca43e516c3d8aa051": "g = - \\beta\\left(r\\right) dt^2 - \\alpha(r) dr^2 - \\sigma(r) r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2),", "fdb4380f63c2cad6c0b6ffa52ca695fd": "S_1=1000 \\text{ mm}", "fdb447a52674f22356eec1f570af71c2": "H(X) = X \\left(X^\\top X \\right)^{-1} X^\\top", "fdb46324c3acf38da329ac6f682fe121": "[B]", "fdb467626489b4665370307bc4c1f6ba": "\\frac{\\partial y}{\\partial \\mathbf{x}},", "fdb5244a64d0296cc11470577bf0ed5e": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi}t_h\\ \\frac{V_r}{V_t}\\ \\left(\\frac{p}{r}\\right)^2\\ 3\\ \\sin^2 i \\cos u\\ \\sin u\\ du\\ =\n3\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}(e_g\\ \\sin u\\ -\\ e_h\\ \\cos u)\\ \\frac{p}{r}\\ \\cos^2 u\\ \\sin u\\ du \\ = \\\\ \n&3\\ \\sin^2 i \\int\\limits_{0}^{2\\pi}(e_g\\ \\sin u\\ -\\ e_h\\ \\cos u)\\ (1\\ +\\ e_g\\ \\cos u\\ +\\ e_h\\ \\sin u)\\ \\cos^2 u\\ \\sin u\\ du\\ = \\\\\n&3\\ \\sin^2 i \\ e_g\\int\\limits_{0}^{2\\pi}\\ \\ \\cos^2 u\\ \\sin^2 u\\ du\\ =\\ 2\\pi \\frac{3}{8} \\sin^2 i \\ e_g\n\\end{align}\n", "fdb52cd6804e82b6a04ca0486452404d": " \\{w {\\in}F \\mathrel{|} f(x) = \\}", "fdb54bd8d9da0ac859c6f25e62c25853": " \\left|x - \\frac{m}{n}\\right| < \\frac{1}{\\sqrt{5}\\,n^2} ", "fdb557f7b3ac126e357342edebcc7fff": "s \\mapsto \\dim_{k(s)} H^p (X_s, \\mathcal{F}_s): S \\to \\mathbb{Z}", "fdb5cfac4e09cb636b89486d88b451d9": "H\\boxtimes H", "fdb5f822e5ab0df7e2595beb458ca7bb": "\\int dy \\int f(x,y)\\,dx", "fdb6392420ceafe0c5c887e9ff7af9dc": "{\\neg}A", "fdb65f28cf8949dc1cd3a5d37ac87d45": "A=R+F", "fdb69efa310ba1ade931267034d83636": "Q^n |\\psi\\rangle = \\cos((2n+1) \\theta) |\\psi_0\\rangle +\\sin((2n+1)\\theta) |\\psi_1\\rangle", "fdb6c118c22fd21e334f9d6c45f06e99": " (n-1)(n-2)~r^{-n}~\\sin(n\\theta) ", "fdb6cd351e33d21ef000b29221b91ccd": "P_i = \\frac{1}{1 + e^{-z_i}}\\ = \\frac{e^{z_i}}{1 + e^{z_i}}\\ ", "fdb6e18188f82522155668318e07b064": "\\!A = 4\\pi r^2.", "fdb7074220caeb86412200e10175f85d": "\n | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\\rangle.\n", "fdb724652e62d2352b083d20e6aa6c2d": "Q = Q_\\mathrm{f} + Q_\\mathrm{b} = \\iiint_\\Omega \\left(\\rho_\\mathrm{f} + \\rho_\\mathrm{b} \\right) \\, \\mathrm{d}V = \\iiint_\\Omega \\rho \\,\\mathrm{d}V ", "fdb74038a01150a8c5000eaf966458ad": "(\\lambda_{c}/n)", "fdb750ec3f43576d3f59e25c7cfbf3ab": " \\gamma_1 ", "fdb754f3e5e52d2648e1b2c53af58d64": "y: (S \\cup T) \\mapsto \\mathbb{R}", "fdb78f06cc38482623570a589c07cb59": " \\begin{Vmatrix} h^1 & h^2 & h^3\\end{Vmatrix} = \\begin{Vmatrix} \\mathbf{h}\\cdot\\mathbf{\\hat x} & \\mathbf{h}\\cdot\\mathbf{\\hat y} & \\mathbf{h}\\cdot\\mathbf{\\hat z}\\end{Vmatrix} ", "fdb83cdb4a03abc9f0ee99d47f42371b": "(x^2+y^2)^3 = (x^2-y^2)^2. \\,", "fdb85aa0110bb1f90b857d7bee3bf0f9": "\\{x_i,x_{i+1}\\} = -x_i\\, x_{i+1}", "fdb880b63d99c5bb894f20a1c0f8603b": "\\hbar\\omega << \\epsilon_G", "fdb8919d368fcd5c40e61adef990d8a8": "\\scriptstyle \\mathbf{x}_*", "fdb8939dd318852b1efb17813bfd4381": "L_1[,F_2", "fdb91b6887623d8370295a94d83fbeb5": " {n-1 \\choose m} = {n \\choose m} - {n-1 \\choose m-1}", "fdb9fe1c18571469a25ebf6514f47e00": "\\mathrm{P}(B|AC) = \\frac{\\frac{1}{16}}{\\frac{1}{16} + \\frac{1}{16}} = \\tfrac{1}{2} = \\mathrm{P}(B)", "fdba18a0c0bb93588db4dc7b03042c7f": " E_n^{(k)} = \\frac{1}{k!} \\frac{d^k E_n}{d \\lambda^k} ", "fdba2f01ac23b911969059c15e383f34": " F_3 = x ", "fdbb1ebd05bb8461f3e43b618df29315": "\\angle NMP = \\angle ACP", "fdbb242adffe6e96990262e4f451a114": "\\eta_L", "fdbb422f2df346026432cb49da433485": "\\Sigma M_A = M_{AB} + M_{AB}^f = 0.4EI \\theta_A + 0.2EI \\theta_B - 14.7 = 0", "fdbb44819efa513e3e9b4edc023e687c": "\\displaystyle{Te_1=0,\\,\\, Te_g=e_{g^\\prime},}", "fdbb4b1e2d55cf2ede63dcc1158b503e": "m = \\int_V \\sqrt{-\\xi^a \\xi_a} \\left( T_{00}+T_{11}+T_{22}+T_{33} \\right) dV ", "fdbb684c5d810c21ff3c358ac8e611a7": " M = \\quad \\left \\lbrack \\frac{1+\\frac{C}{D}}{\\frac{C}{D} + \\frac{R}{D}} \\right \\rbrack B ", "fdbbc77a217b8d40db034948f50f2a78": "\\mathcal{K}", "fdbbe02c9305851c98df9854e60a9a33": "y = (1 - x^2)^0", "fdbc1339d8cb71f6121164bf40531411": "f(z_1)=f(z_2)", "fdbc5fcfced7b0f661f8a414f6969ae5": "\\alpha_{0}", "fdbc6889969aafbd1ce7b62b0d280b70": " (a_n)_{n\\geq 1}", "fdbcbcb5aab7f7f715f62301e8f8d1a2": "\\omega = -\\partial S/\\partial t", "fdbcf68e30e709051252d5f1b6dbbaf5": "\\,\\{2, 2*3, 2*3*5, 2*3*5*7 \\cdots \\}", "fdbd063cea38ad423c0de2c62432e9a6": " \\scriptstyle \\vec{ \\text{ F } } ", "fdbd6c32c133e328c8bd47523d44f0fd": "\\textstyle Z \\subset (0,1) ", "fdbda3b0fedc8beed7293486c2e905db": "\\scriptstyle f_0 \\,+\\, \\frac{\\Delta f}{2}", "fdbe12720c9fb1520de05c6900674b3c": "|I|\\leq\\kappa", "fdbea6a4650235c55367bfb36dfbe12c": "\\begin{matrix}\n\\frac{\\partial}{\\partial \\omega} \\left[ \\phi_{\\tau}(\\omega) - \\omega \\tau + \\omega t\\right] & = 0 \\\\\n\\frac{\\partial}{\\partial \\tau} \\left[ \\phi_{\\tau}(\\omega) - \\omega \\tau + \\omega t \\right] & = 0 \n\\end{matrix}", "fdbef9723f1c18bf8d182f74e940f2ce": "F = S/S_P", "fdbf44c36177f3223d44187a0f859dcc": "S_n = (S_0, \\mathcal{O}_S/\\mathcal{I}^{n+1})", "fdbf46f08428258b9e7f8e3f80d1b8fd": "2^{32} \\equiv 1\\pmod 3,\\,", "fdbf46f250da63e264d91ccfaac43cb1": "2.\\overline{6}", "fdbf542466ff26248ff87d13874acd10": " v=\\sqrt{r^2-y^2} ", "fdbf5b322a6b8752b4837b929878c580": " \\pi(H) \\ge \\pi(H_{11}) + \\pi(H/H_{11}) ", "fdbf6d535df056b42ee5be0b04f53a5d": " \n\\lambda = \\frac {1} {A} \n", "fdbfbd769ea7061ad03ab52c0a926971": "v_i \\in V", "fdbfef8b3bad8c5c03444ce5322e09d3": "\\{ z = x + y \\jmath : x,y \\in R \\}", "fdc00e6a25c1339b449afa7848db1c2c": "\\Phi = \\Phi(T,V,\\{N_i\\})", "fdc01fa7cadb70be116e3f1d13203e0b": " 1 \\lor 2 \\iff | 0 \\rangle ", "fdc0841bf0bb9e01b0f157e7e49cdee1": "\\{s\\in S^3 \\subset \\bold{H}: \\operatorname{Re}(s) = \\cos(R)\\} = ", "fdc10b00320b0bf19edeb1985dca78d7": " \\{b_k\\}_{k=1}^{M-1} ", "fdc1491968adc702d3ef859d4472d6fc": "i_1+i_2+\\dots+i_k=n", "fdc15910ca0b666d4e579bc2f30103f3": "e^{i k \\|\\mathbf{x}-\\mathbf{x'}\\|_2} = 1 + O(k r)", "fdc172d309a75d7767dc0b8d308af6e2": "F_{eachAnchor}=\\frac{F_{load}}{2Cos(\\frac{\\theta}{2})} \\,", "fdc175cd42efdd6a62ebaad2e026f08e": "X =\\varnothing", "fdc194900517fb8700f8cd0cb3ab3e14": "z_{1} = z_{2} \\, \\, \\leftrightarrow \\, \\, ( \\operatorname{Re}(z_{1}) = \\operatorname{Re}(z_{2}) \\, \\and \\, \\operatorname{Im} (z_{1}) = \\operatorname{Im} (z_{2}))", "fdc1b8c7a9948d30bb1ca50a589e148d": "D>>h", "fdc1c699c0434213a8795efd855620a2": "\\Box{}", "fdc1e8917018cce689a497c9edfa587a": "P_E=P_T\\,", "fdc21234067bb574b0069e6746d56ddb": "\\hat{N_i} \\ \\stackrel{\\mathrm{def}}{=}\\ a^{\\dagger}(\\phi_i)a(\\phi_i)", "fdc329479d211eaeec9a03955cbe703c": "\\gamma(n)\\geq n/(\\alpha \\sqrt{\\log{n}})", "fdc397562849d9b6eea1746494f449a6": "\\! \\delta^2 = s^2{\\star \\mathrm{d}{\\star {\\star \\mathrm{d}{\\star}}}} = (-1)^{k(n-k)} s^3{\\star \\mathrm{d}^2\\star} = 0 ", "fdc39f769af6ba10637207c480f74063": "\\ \\ \\nabla \\times \\mathbf{H} = \\mathbf{J} + \\frac{\\partial \\mathbf{D}} {\\partial t}\\ \\ ", "fdc4327ce4afecba2699f450e2547501": "\\hat{\\dot{M}}_O", "fdc4e368b3c560273e0f85e6d5c2e5a1": "\\begin{bmatrix}\n\\dfrac{\\partial x_1}{\\partial x} & \\dfrac{\\partial x_1}{\\partial y} & \\dfrac{\\partial x_1}{\\partial z} \\\\[3pt]\n\\dfrac{\\partial x_2}{\\partial x} & \\dfrac{\\partial x_2}{\\partial y} & \\dfrac{\\partial x_2}{\\partial z} \\\\[3pt]\n\\dfrac{\\partial x_3}{\\partial x} & \\dfrac{\\partial x_3}{\\partial y} & \\dfrac{\\partial x_3}{\\partial z}\n\\end{bmatrix}", "fdc4e82618e1cbdd954eed5edb462b07": "w_1, \\ldots, w_r", "fdc543be86f8920fcf70f8a8c0575326": "\\mathrm{moser} < 3\\rightarrow 3\\rightarrow 4\\rightarrow 2,", "fdc592d551888e791f9f943a55fb51bd": " rQ_B l_A a_B ", "fdc5c4a79628ab69d6b74c537ad15a72": "\\mu_r = \\mu/\\mu_0", "fdc5c8efe4872a92b425a9243ef5ee7d": " \\Phi(a'_{i},a_{-i})-\\Phi(a''_{i},a_{-i}) = w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))", "fdc5fddd795222128235381ad91d204c": " U =\n\\begin{bmatrix}\nu_{1,1} & u_{1,2} & u_{1,3} & \\ldots & u_{1,n} \\\\\n & u_{2,2} & u_{2,3} & \\ldots & u_{2,n} \\\\\n & & \\ddots & \\ddots & \\vdots \\\\\n & & & \\ddots & u_{n-1,n}\\\\\n 0 & & & & u_{n,n}\n\\end{bmatrix}\n", "fdc644503e40b9a626a8a273209c5a84": "e^{\\lambda}\\,P_\\lambda(A)=\\sum_{n=0}^\\infty P(A\\mid N=n){\\lambda^n \\over n!}.", "fdc663753a562340a8263a11a2648afc": "\\frac{\\partial}{\\partial \\beta_k} \\left(- \\log Z \\right) = \\langle H_k\\rangle = \\mathrm{E}\\left[H_k\\right]", "fdc668401b52a37a2a440014b16dcc3e": "\\left(1,\\ 1+\\sqrt{2},\\ 1+2\\sqrt{2},\\ 1+3\\sqrt{2},\\ 1+4\\sqrt{2}\\right)", "fdc6af28c75009c337237a20e29764d4": "x \\in C'(B,\\succeq)", "fdc73402bf796645b4c4ab2bc4bda10a": "\n \\Delta\\gamma = \\sigma_0~h_0\n ", "fdc766b327b62d03c9ed62e76f66f24e": "\\begin{align}\n \\langle r\\rangle &= 0 \\\\\n \\langle r^2\\rangle &= nb^2 \\\\\n \\langle r^2\\rangle^\\frac{1}{2} &= \\sqrt{n} b\n\\end{align}", "fdc767dbd926226207df4259324a9b79": " W[k,l] = V[k,l] \\qquad \\mathrm{for} \\qquad k = 1,\\dots,p \\qquad l = 1,\\dots,L ", "fdc79dc51ad0a9c0efeef7be9801f2ac": "\\rho_1/\\rho_2", "fdc7ab7c2b8aaa60598f038285425c65": "\\mathbb{E}_{E}[\\Pr_{e \\in BSC_p}[D(E(m) + e) \\neq m]] \\leq 2^{-{\\epsilon^2}n} + 2^{k +H(p + \\epsilon)n-n} \\leq 2^{-\\delta n}", "fdc86f09f2e73c76dee2b2441f5d660b": "S=S_1\\times S_2,\\quad \\Sigma=\\Sigma_1\\otimes\\Sigma_2,\\quad \\mu=\\mu_1\\otimes\\mu_2,", "fdc8f2213cac9a529dd6d17060c031e1": "PV_\\text{float} = N \\times \\sum_{j=1}^m ( F_j \\times \\delta_j \\times P^D(t_j) )", "fdc90afc784f8453b2c7cd4c0b13ad88": "\nESP(x) = x^{sm} + x^{s(m-1)} + \\cdots + x^s + 1.\n", "fdc9b184870ddad2d8e45815df71296a": "10_{155}", "fdc9ea438fc09f346ddf190b1fd883c6": "g(y) = y - \\bar{y}", "fdca103821225713290508f9e0600e6a": " \\textrm{d}f_{x_0,\\lambda_0}", "fdca7b68e7c5ece98b42997ab5ea14b1": "G \\left( x, y, k\\sigma \\right)", "fdcad68ea69acfc016a4d22ef330161f": "d\\epsilon_{i,j}", "fdcaf24a1c253fc36acf2eabcfc87271": "\\lim_{b\\to\\infty}\\int_1^b \\frac{1}{x}\\,\\mathrm{d}x = \\infty.", "fdcb083c53b75a588d6f5fec601e29da": "\\scriptstyle R_3", "fdcb1c2e03784d2bee51009417c791ea": "a_0 + \\cfrac{b_1}{a_1 + \\cfrac{b_2}{a_2 + \\cfrac{b_3}{ \\ddots }}}", "fdcb6a2630e28a75782f117d7111519c": "\n\\rho(y_1,y_2)=\\sup_{x\\in\\mathfrak{X}}\\left| H(x,y_1)-H(x,y_2)\\right|.\n", "fdcb6d2db62e98dca4cca04983ecc238": " W_t ", "fdcbc1b22c16b80da6eb6c482b02db74": "\\quad \\varphi_0 + \\sum_{i=1}^N \\int_{\\Lambda^k} \\varphi_k(x_1, \\ldots, x_k)\\rho_k(x_1,\\ldots,x_k)\\,\\textrm{d}x_1\\cdots\\textrm{d}x_k \\ge0 \\text{ for all } k, (x_i)_{i = 1}^k ", "fdcbd51a88a0ae59344479d16d46e514": "x / y", "fdcbd827035bb847d403b8bc7ed61994": "R = \\;\\left|\\frac{ds}{d\\varphi}\\right| \\;= \\;\\left|\\frac {\\big({\\dot{x}^2 + \\dot{y}^2}\\big)^{3/2}}{\\dot {x}\\ddot{y} - \\dot{y}\\ddot{x}}\\right|,\n\\qquad\\mbox{where}\\quad\n\\dot{x} = \\frac{dx}{dt},\\quad\\ddot{x} = \\frac{d^2x}{dt^2},\\quad \n\\dot{y} = \\frac{dy}{dt},\\quad\\ddot{y} = \\frac{d^2y}{dt^2}.", "fdcc0705580ffde9c346a9683b933c5c": " q_n = \\frac{\\mathrm{d} q}{\\mathrm{d} n} \\,\\!", "fdcc244a210a79b04c28ae2147ecdd2f": "\\begin{matrix} 4 + {4 \\choose 3}{4 \\choose 1}{3 \\choose 1} = 52 \\end{matrix}", "fdccd69f4c6c13218d4715b4eedccc88": " \\mathbf{y} \\sim\\mathbf{C}\\,\\mathbf{x} ", "fdccf762eaa5e20aa57c391e45c4846a": "\\hat f_1\\ ,\\ \\hat f_2\\ ,\\ \\hat f_3", "fdcd7874d3b1027abc02abd7ee354336": "\\left| \\int_R f(z) \\, dz \\right| \\le 2 \\pi R \\frac{(\\log(R))^2 + \\pi^2}{(R^2-1)^2} \\to 0.", "fdcd9f5bd5a797b4d62083c097d794e8": "\\text{Dom}:Y\\to Y ", "fdcda36860f1314f8367a7a9d769ca3d": "\\bar{u} = u^{\\dagger} \\gamma^0 \\,", "fdcdd063f3b1ef6729c17becaa268597": "H_0(X) = \\mathbb{Z}", "fdcdf2b341cd7b31170290599f3301b7": "P \\not\\rightarrow_{b} Q", "fdce70401ad92ba1ca67a358ec31ddbb": " \\frac{dS_{t}}{S_{t}}\\ = \\mu dt + \\sigma dZ_{t} ", "fdceed2653ba37d217bffe4e661975cc": " E(a) = \\sum_{g \\in G} g(a)", "fdcf2df350516ed6f71cad4db266a9b1": "x = x_1 + s(x_2 - x_1)\\,", "fdcf6b9e6f03665808e6895c101c722b": "1 \\div -0 = -\\infty", "fdcfbf7c27a5a438ad41d7da620b8fe8": "\\gamma(\\mathbf{v}) = \\frac{1}{\\sqrt{1 - \\tfrac{\\mathbf{v} \\cdot \\mathbf{v}}{c^{2}}}} = \\frac{1}{\\sqrt{1 - \\tfrac{v^2}{c^2}}}", "fdcfc0674e5672da0d72e44e9d0003f4": "p_k \\ ", "fdcfed0e39d98dacf2686819f84b2531": "[[\\lambda A]]=\\lambda [[A]]", "fdcff41b883af808c42ad6dd910f7c7e": "\\{1,2,3,\\ldots\\}.", "fdcffa109a5b08f753963b171b3d7876": "\\widehat{c_v}_{ln} \\,", "fdd011520e0fef940ecf2e941e989bf9": "\\log_{\\sqrt[12] 2}(r) = 12 \\log_2 (r)", "fdd0399a02303f4b894a8ff9c50b3242": "\\zeta_{\\mathbf A^n(X)}(s)=\\zeta_X(s-n)", "fdd04247af7dfb3cc8ecacddbcef4b43": " x \\in \\{1, 2, 3\\} ", "fdd04966548f7e812bfe4912f8d7030c": "|f_j(x) - f_k(x)| \\le |f_j(x) - f_j(z)| + |f_j(z) - f_k(z)| + |f_k(z) - f_k(x)| < \\epsilon", "fdd06601fa9c4ac4d5b53ba99c081226": "x^- \\to x^- + \\delta_{ij}\\alpha^i x^j + \\frac{\\alpha^2}{2} x^+", "fdd0e3267c4931dec79cf0cfa697ec1d": "\\varphi_{\\beta_m}(\\gamma_m) \\geq \\varphi_{\\beta_{m+1}}(\\gamma_{m+1}) \\,,", "fdd0f5e6785070e7658bd063c76c2b7b": "R \\equiv \\pm n^{\\frac{p+1}{4}}", "fdd117dfadc90452cc5ce95e3757f19f": "\n1, 3, 26, 646, \\ldots = \\prod_{j=0}^{n-1} (3j+2)\\frac{ (2j + 2)!(6j + 3)!}{(4j + 2)!(4j + 3)!}.\n", "fdd11aaa0a8512af3c924fdb4affc5b9": "q''=2\\frac {b-2a+(a-b)3t}{{(x_2-x_1)}^2}.", "fdd13048c099dd93a05caed625fc2491": "U=\\int{\\rho_0\\over|\\mathbf{x}-\\mathbf{x}'|}d^3x'", "fdd13e7073ef9a68d96ab09a8ad139f8": " \\tilde{u}(x)=\\{u(x,p):c(p)\\cdot u(p)=\\int\\limits_{\\partial \\Omega}\\left(G(p)\\frac{\\partial u(p)}{\\partial n} - \\frac{\\partial G(p)}{\\partial n}u(p)\\right)dS, p\\in\\hat{p} \\}", "fdd1aee00d95f9bb23ab0dc0b93fe159": "L^{(d_k)}_k=Lclm(l^{(e_1)}_{j_1},l^{(e_2)}_{j_2},\\ldots,l^{(e_k)}_{j_k})", "fdd1bc4374c84e56d0732df671ce8d27": "\\ddot{x}", "fdd1c315e6cf758d78890816f14ea63e": "h-\\tilde{h}\\in\\mathfrak{a}\\setminus \\mathfrak{a}^{\\ast}\\,", "fdd1d398b75cd0f76f80098b6c538d24": "D_n\\left(E\\right) = \\frac{d\\Omega_n(E)}{dE}", "fdd1d8f6481758df545b85059111cb23": " \\mathbf{a} = \\frac {\\mathrm{d}^2 \\rho }{\\mathrm{d}t^2} \\mathbf{u}_{\\rho} + \\frac {\\mathrm{d} \\rho }{\\mathrm{d}t} \\frac{\\mathrm{d} \\mathbf{u}_{\\rho}}{\\mathrm{d}t} + \\frac {\\mathrm{d} \\rho}{\\mathrm{d}t} \\mathbf{u}_{\\theta} \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t} + \\rho \\frac{\\mathrm{d} \\mathbf{u}_{\\theta}}{\\mathrm{d}t} \\frac {\\mathrm{d} \\theta} {\\mathrm{d}t} + \\rho \\mathbf{u}_{\\theta} \\frac {\\mathrm{d}^2 \\theta} {\\mathrm{d}t^2} \\ . ", "fdd1f8583926138263823f6a2b578f54": "\\sum_{n=1}^\\infty \\frac{d(n)}{n^s} = \\zeta^2(s),", "fdd22b5976b9935ee04240be35d91a7e": "l_a m^a=l_a \\bar{m}^a=n_a m^a=n_a \\bar{m}^a=0", "fdd2b12ac472ff6dd02bd5a2133c3897": "(y_n/x_n)X=Y=(y_n/z_n)Z", "fdd2f203daef28742b5f15d6f19f55ad": "A=\\left(2+3\\sqrt{3}\\right)a^2\\approx7.19615...a^2", "fdd2f5a2cac1b555c136811e18d82dc4": "n! \\sim \\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n\\!", "fdd2fe434f13352e4ce2b38e3f6d558f": "I_{L_{\\text{max}}}", "fdd3050c1f3ce364d580584f2166cf7b": "xy=1.\\,", "fdd341ab5dc04109d0b8685bce1cc9f2": "\\Gamma=\\frac{G_F^2 m_\\mu^5}{192\\pi^3}I\\left(\\frac{m_e^2}{m_\\mu^2}\\right),", "fdd37f3f14b641f4093312ff8deba88a": " \\epsilon_r ", "fdd3bbe547e7caf98ee2f25d68197430": "4.00 = \\left ( \\frac{q_\\mathrm{trans}^2}{32.2} \\right )^\\frac{1}{3}", "fdd3cea2dfb4b9b6089f2a68b233cb59": " C_{xx}, C_{yx},", "fdd3f46070d191b6434b0f272a94d318": "\\vec{s}_b^{\\;2}", "fdd42b6f97c775c83c36612b38d05819": "B_{\\mu\\nu}=\\partial_\\mu K_\\nu-\\partial_\\nu K_\\mu\\;", "fdd43d48b392edf6e624d17cacd9c7ce": "0\\leq l<2^m", "fdd46704ed781d9f9b5c4711ccd0f84c": "\\!\\mathcal A \\models_X^+ =\\!\\!(t_1 \\ldots t_n)", "fdd48c31bd35ec27c683429f83c53d3b": "Im(K(\\omega))", "fdd4ddc600a2e64e781a95b4d473ad09": " \\chi^2_{total} = \\frac{(i - h/4)^2}{h/4} + \\frac{(h-i-j-h/2)^2}{h/2} + \\frac{(j-h/4)^2}{h/4} = \\chi^2_{tdt} + \\chi^2_{hs}.", "fdd4e140c14c45d1e3cd9e0087df19df": "X = \\operatorname{Proj}\\,\\bigoplus_{r=0}^\\infty \\operatorname{Sym}^r_{k[x,y]} \\mathfrak{m}/\\mathfrak{m}^2.", "fdd4ede67ac23769e9f71bc73c381d45": "\\Delta + k^2", "fdd50f5d684b8f9ba1a58a588fed7556": "\\frac{V_D}{nV_T} = \\ln \\left(\\frac {I}{I_S}+1\\right) ", "fdd5108afa26cf6c0575c76dd7ccac1f": "Z_{TS} = Z_1 + Z_2 + Z_3 + ... \\,", "fdd56a903f44bae15422e03afa08fa66": "n_{t+1} = \\frac{R_0 n_t}{1+ n_t/M}. ", "fdd5c049c3f506f8c5950bfd492b457b": " \\frac{1}{\\sqrt{4\\pi}} \\mathbf{H} ", "fdd61a52eec77f06b46f22db73cb3b61": "L\\prec M \\; \\mathrm{iff} \\; Eu(L) < Eu(M), \\, ", "fdd6225f45094389b15b74459fa97c11": "\\mu_\\mathrm{sig}", "fdd6836de15638245bc726a6ec106233": "\\sqrt{2-\\sqrt{2}\\ }", "fdd684e0b345aaefc80208bcbd812c79": "y_{out}", "fdd6d6d847f775422e6fd374e44bb5e3": "\n\\frac{A \\wedge B\\hbox{ true}}{A\\hbox{ true}}\\ \\wedge_{E1}\n\\qquad\n\\frac{A \\wedge B\\hbox{ true}}{B\\hbox{ true}}\\ \\wedge_{E2}\n", "fdd7513bf5c2bcbb2bf43d6358ffcef9": "ASB = ACB + \\pi/3", "fdd78081dd9943f67289a9f71816adf6": " b + z > c + y ", "fdd7eabb213bd522d354039bb79432f6": "\\{x^{i_1}e_{i_1}, \\, ... \\, x^{i_k}e_{i_k} \\}", "fdd82f4393d52f0b07870932e6de31da": "\\frac{\\partial\\phi(\\mathbf{r},t)}{\\partial t} = D\\nabla^2\\phi(\\mathbf{r},t), ", "fdd85c0bf1a2a3fbf7c7daaedeb9796f": "\\mathrm{^{241\\!\\,}_{\\ 95}Am\\ \\longrightarrow \\ ^{237}_{\\ 93}Np\\ +\\ ^{4}_{2}He\\ +\\ \\gamma}", "fdd8e715b3fed930870935ba56cb5095": " \\frac{m^2 \\omega^2 x_c^4}{\\hbar^2} = 1 \\Rightarrow x_c = \\sqrt{\\frac{\\hbar}{m \\omega}} . ", "fdd911b47b1eb9a0f38373b731d9ebc0": "f:\\hat C\\setminus f^{-1}(P_f)\\rightarrow \\hat C\\setminus P_f", "fdd9c8944408c742ad38fd2f16bf6e37": "\n z_\\theta = \\frac{\\nabla f_\\theta}{f_\\theta} \\cdot \\mathbf{1}_{\\{f_\\theta>0\\}}\n ", "fdd9fa4b0cd8cf8d4176d49dd81c6e04": "\\zeta^{\\prime}(-1)=\\frac{1}{12}-\\ln A \\approx -0.1654211437\\ldots", "fddb2a58c2abbe20aef069c8fc124d75": "\\mathbb{E}(C) = \\sum_{i=1}^n \\frac{\\mathbb{E}(d_i)}{1-\\rho}", "fddb384f3b67d3dbf2d1e926b200daed": "b_\\lambda=\\sum_{g\\in Q_\\lambda} \\sgn(g) e_g", "fddb5d564bd0d3924d0e7b05e1288343": " \\tau_{\\varphi_{z}} = {T r \\over J_T} ", "fddbaa704a04ff77fa1939da78b59d59": "U(1) \\rtimes Z_2^T", "fddbaa860fb9cd5273bdc079b5d74722": "S_\\lambda", "fddc3c374d5bfcb53fbe6dfa1c67fd23": "e_{(a)}^{\\mu}", "fddca6c4683c96124b8ac3fe862a271b": "X_{n+1} = (a X_n + b)\\, \\textrm{mod}\\, m", "fddce8f5cbdb45b26632e7d196b84bb1": "A\\in\\mathbb{R}^{m\\times n}", "fddd80f5db916fb4dd4e864cfbbabaa1": "G_j/G_{j-1}", "fdddef864c39b39accde54df5309c320": "d(0,0.2L) + D(0.2L, X) \\approx 0.2L + 0.3171L = 0.517L \\neq D(0, X) = 0.5L = d(0,0.5L).\\,", "fdde66b379c5b2087989319b8234ebb6": "f(a)\\leq f(a+0)\\leq f(x-0)\\leq f(x+0)\\leq f(b-0)\\leq f(b)", "fdde6eb9877955925fbdabd4c368bfd3": "{\\tau_1}", "fdde7a12f7d2df933eb5af9a274dd7a0": "\\scriptstyle P=2F", "fddea9b3d2591a327c8fa757e601c797": "\\bar p", "fddfc50be2867bd53392e68fa7f5f3f1": "V\\otimes\\chi_i", "fddfd7999c0722d2e8da5131a1c02c27": "\\Im(\\tau)|\\eta(\\tau)|^4 = \\frac{\\alpha}{4\\pi\\sqrt{|D|}} \\prod_r\\Gamma(r/|D|)^{\\chi(r)\\frac{w}{2h}}", "fde0035bbd27448927d9ccc1b02ff6f3": "\\left\\{\\begin{array}{ll}\\infty & m = 1 \\vee n = 1\\\\ 4 & \\text{otherwise}\\end{array}\\right.", "fde020f87a54337c81cb3d53c80a8d45": "\\Lambda=\\lambda/(2n)", "fde02765ef3554dddc4a45fe75fe0ecb": "n-k=4", "fde031d7813c26eedc230b9e0a391ee9": "\\displaystyle{\\mathcal{U}^*F(t)={1\\over \\pi} \\iint_{\\mathbf C} B(\\overline{z},t) F(z)\\, dx dy.}", "fde03666a41fe825fe81d0f414b173b9": "a/q", "fde0ac4dcb44042338064e02376bc87d": "E(m) \\cdot t^e \\bmod n = (mt)^e \\bmod n = E(mt)", "fde0c313f64483903c44338d7385fb64": "\n \\boldsymbol{F} = \\begin{bmatrix} F_{11} & F_{12} & 0 \\\\ F_{21} & F_{22} & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}\n ", "fde0fe632e515cfb536d0c77442a39d4": "{\\nabla}^2 \\varphi = - { \\rho_f \\over \\varepsilon } ", "fde1003a2cc72d9f48508a1820715a89": "q \\vert N - 1", "fde17c081392b907b8bca723be7295f5": " P = k_v(V) \\,T \\,\\!", "fde23c8d55b652efd1581b65e420795e": "P = \\frac{F_p}{A_p - A_r}", "fde27263cdbc540c674f59e7bc269d0c": " \\ \\mathcal{L}_r ", "fde27c350681a39d9e948fb936bc2668": " ds^2=dt^2-a^2(t)(dx^2+dy^2+dz^2)", "fde281610baf7636c48277626964c552": "\\Rho \\, \\rho \\, \\varrho \\,", "fde2ef66fa2fe01dce715468cf6f0c86": " D = {{\\mu_q \\, k_B T}\\over{q}}", "fde32f306bbf32c03282591ff589e2c3": "{{D_g f \\over Dt} = \\overrightarrow{V_g} \\cdot \\nabla f = \\beta v_g}", "fde35edce8886d442d091e1f072a1c70": "\n 2fh~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_s}{\\mathrm{d} x^2} - (2h+f)~C_{55}^{\\mathrm{core}}~\\cfrac{\\mathrm{d} w_s}{\\mathrm{d} x} = 4h^2~C_{11}^{\\mathrm{face}}~\\cfrac{\\mathrm{d}^2 w_b}{\\mathrm{d} x^2}\n ", "fde36a94157cebec09b14bf066fee2b4": "\\nabla \\times \\mathbf{B} = \\mu_0\\mathbf{J} + \\mu_0 \\varepsilon_0 \\frac{\\partial \\mathbf{E}} {\\partial t},", "fde37b267df0838a049dc07a2b600882": "r - v", "fde38073454211de70d2c37ffec1ff8b": "300 mV=E_{obs|ref1} - 197 mV", "fde388646335209e5a73c3be74deb1bb": "\\alpha_D", "fde391f3a09fa537259aa3e1601120ed": "(a,\\mathbf{x})(b,\\mathbf{y}) = (ab - \\mathbf{x}\\cdot\\mathbf{y}, a\\mathbf y + b\\mathbf x + \\mathbf{x}\\times\\mathbf{y}).", "fde3fb112fa4479ddc0ddd58c7ddcdc3": " \\binom{m}{k}\\binom{n}{r-k} ", "fde41cd6561cd5af6d012c25bf56a7fe": "p(x) = 1", "fde4330d6dfff4009dbc14a4d4c8830b": "\\mathbf N=\\mathbf I_1\\,\\!", "fde43403ea4d39a9660681dce77f5711": "\\sum_{L_k}{x_{i,m+j}}", "fde457059bfacdb3ae094db0bea5d950": "E=\\frac{3}{5}\\,\\,\\frac{1}{4\\pi\\varepsilon_0}\\frac{e^2}{r_\\mathrm{e}}", "fde48f05828d3ac1c9854634adfcf5f9": "i^*\\mathcal{O}(-D)", "fde4d1972eee88cb80eab1d96e7848d3": "d(k):=\\sum_{j=0}^{\\infty} \\delta(k-k_j)=\\frac{L}{\\pi}+\\frac{1}{\\pi} \n\\sum_p \\frac{L_p}{r_p} A_p \\cos(kL_p).", "fde4d422a47f3b72a5e1d62e46b68ab7": "\\{-m..+m\\}\\times\\{-m..+m\\}\\times\\{-m..+m\\}", "fde4f421f81659ba9b9f5fadf4f113ca": "\\frac{\\mathrm{d}}{\\mathrm{d}z} G(z) =\nG(z)\n\\left(\n\\frac{n}{z}\n+ \\frac{1}{n-z}\n+ \\frac{2}{n-2z}\n+ \\frac{3}{n-3z}\n\\cdots\n+ \\frac{n-1}{n-(n-1)z}\n\\right)\n", "fde558597c7a88f55933c914f4380b09": "\\ln\\left[1 - (I_1-3)x\\right]", "fde5815c335442ee7a35bf8256cb96ef": "\\pi=(3,5,2,4,1)", "fde5f422a1794bbb00a42c9759ce2a8a": "\\scriptstyle{\\hat{n}}", "fde61ab401661e24054a7254c5222820": "\n\\Delta H_\\mathrm{r}^\\ominus = \\sum_{B}{v_B\\Delta H_\\mathrm{f}^\\ominus(B)}\n", "fde62052998c3a450d18f7371c242cd5": "1/\\Phi^{-1}(3/4) \\approx 1.4826,", "fde644aca98e9ccd5c4f54c57ecd9f24": "\\phi(t)\\,\\!", "fde69a0eb1e474efd4b860619ece353b": "\\lim_{h\\to \\infty} \\gamma_s(h) = var(Z(x))", "fde6c141842e5a03c38f408332ebf936": "(x)_n=x(x-1)(x-2)\\cdots(x-n+1)=\\frac{x!}{(x-n)!}.", "fde6c23a7d726d3dad9767d65389a2ee": "\\mu_i=\\left(\\frac{\\partial G}{\\partial N_i}\\right)_{T,P}", "fde7516dbcaf44bdcd23241626eaad3b": "\\begin{pmatrix}x_t \\\\ x_{t-1} \\\\ \\end{pmatrix} = \\begin{pmatrix} \\text{A} & \\text{B} \\\\ \\text{I} & 0 \\\\ \\end{pmatrix} \\begin{pmatrix} x_{t-1} \\\\ x_{t-2} \\end{pmatrix}, ", "fde770d715bb38345babeff7ecda395d": "p(x) \\equiv p^{*}(x)", "fde7dd6c4c42048411c2169f3f060a5b": "i=0,1,2...", "fde8429fff0fc8fefebdc2a84596b457": "m = \\operatorname*{max}_{i=1}^{9} \\Big\\{\\Pr (X \\text{ has FSD}=i)-\\log_{10}(1+1/i) \\Big\\},", "fde876622d81d77f8757103ea89ced49": " \\Delta K = m\\int_{t_1}^{t_2}\\dot{v}vdt = \\frac{m}{2}\\int_{t_1}^{t_2}\\frac{d}{dt}v^2 dt = \\frac{m}{2}v^2(t_2) - \\frac{m}{2}v^2(t_1), ", "fde89372fe457433c8c1a3f98a2f9b29": "f'(x) = \\frac{f(x+h) - f(x)}{h} - \\frac{f''(x)}{2}h + \\cdots.", "fde8d1f2e1696ab76cd66182f47b5e14": "\\Psi\\,", "fde8d7c221290db12299afb47b24bf4b": " \\left\\| \\cdot \\right\\| ", "fde8f732230382d4839d9aaac84b8485": " H = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}", "fde924a1b54f35ac964d6a38418a1950": "\n\\begin{cases} \n\\{O_{1},O_{2}\\} \\\\ \n\\{O_{3},O_{7},O_{10}\\} \\\\ \n\\{O_{4}\\} \\\\ \n\\{O_{5}\\} \\\\\n\\{O_{6}\\} \\\\\n\\{O_{8}\\} \\\\\n\\{O_{9}\\} \\end{cases}\n", "fde9438fde6bec4a0fc1c2515fb10f4f": "\n\\begin{align}\n\\Pr \\left\\{ \\lambda_{\\text{max}} \\left( \\sum_k \\mathbf{X}_k \\right) \\geq t \\right\\} \n& \\leq d \\cdot \\exp \\left( -\\frac{\\sigma^2}{R^2} \\cdot h\\left( \\frac{Rt}{\\sigma^2} \\right) \\right) \\\\\n& \\leq d \\cdot \\exp \\left( \\frac{-t^2} {\\sigma^2+Rt/3} \\right) \\\\\n& \\leq \n\\begin{cases}\nd \\cdot \\exp ( -3t^2/8\\sigma^2 ) \\quad & \\text{for } t\\leq \\sigma^2/R; \\\\\nd \\cdot \\exp ( -3t/8R ) \\quad & \\text{for } t\\geq \\sigma^2/R. \\\\\n\\end{cases}\n\\end{align}\n", "fdea30eb8023a0fde113ebfb29649f19": " {\\mathbf{S}}({\\mathbf{p}}(t))=\\mathcal{S}\\boxtimes_{n=1}^N\\mathbf{w}(p_n(t)) \\in\\R^{L_1\\times L_2} ", "fdea8eb573024cdcf2482f2560fead1f": "x_L^'", "fdeac01873f5b152dcab4af57af7d049": "\\Delta \\rho", "fdeb0a42dfec702136d93dbc8f36f350": "\\langle \\ |\\ \\rangle", "fdeb6d50e1c76413b632895aff6be963": "^{15}\\text{N}^{14}\\text{N}", "fdeb9ffb08ec247707436dfa8337d582": "0 = j_0 < j_1 < j_2 < \\ldots", "fdebd28e5968a63badba40e282b70e22": " L^2\\, | n, \\ell, m\\rangle = {\\hbar}^2 \\ell(\\ell+1)\\, | n, \\ell, m \\rang ", "fdebfd7f1a884c3ac236dd096f34f993": "R_\\mathrm{in} = \\frac{v_\\mathrm{in}}{i_\\mathrm{b}} = \\frac{R_\\mathrm{S}+r_{\\pi}}{1-A_\\mathrm{v}} \\ ", "fdec3c892620df71fe3c27bb080a7b19": "\\scriptstyle{(4\\sqrt{3}/9)\\pi} \\approx 2.418", "fdec7306251009e5276b2f145b908458": "S,\\; m\\times m", "fdec79a168c8149650b1046f52bc8f90": "\\textit{dau}(x_{me},x_{ht}) \\lor \\lnot \\textit{par}(x_{ht},x_{me}) \\lor \\lnot \\textit{fem}(x_{me})", "fdec9d73317def36100b0a7f1c14bdd3": "\\Bigl[\\begin{smallmatrix}\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{T}&\\mathrm{*}&\\mathrm{*}\\\\\n\\mathrm{*}&\\mathrm{*}&\\mathrm{*}\n\\end{smallmatrix}\\Bigr]", "fdecc4d3a09f8666927a0ab90908cf16": "\\frac {T_1}{T_2} = \\frac{\\rho_1}{\\rho_2} \\times \\frac{V_{a1}^2}{V_{a2}^2} \\times \\frac{D_1^2}{D_2^2}", "fded3d56bf850a84effd962a61ca4f6d": "1-3\\,", "fded52c8597d3a9e401ea0908f2333c8": " U_1 = \\prod_{P|p} U_{1,P}. ", "fded8e0f02777db6d9a87cba4d68ed65": "H^{2} [G, U(1)]", "fdedd02b0d469769213e52dc5ba51d08": "f\\in C(E)", "fdede4b154836060c8d48f347236b161": "\\sum_{n>0} n|c_n|^2 \\le \\sum_{n>0} n|c_{-n}|^2.", "fdedf667932ca19d273b40be884a07c2": "\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\\cdot z \\,= \\,\\frac{az + b}{cz + d}", "fdedf83fa0ed1b1feb084b6ee8ef14a1": "Q_2(X)E_1(X)", "fdee6b0ac2e11909b268eaeca1e2d59c": "\\langle \\cdot, \\cdot \\rangle", "fdee88e3d16a33af607a5253144e2ced": "\nK(x-y;T) = G_\\epsilon*G_\\epsilon ... *G_\\epsilon\n\\,", "fdeeb3f15c32cae3c5a86fc6d3821835": "m={{B^2 A}\\over{2 \\mu_{0} g_{n}}}", "fdeed6a4afbd1447f3cd7ee2b5a4ff1a": "\\sqrt{5}<3", "fdeee8cb95672e82d148fb5720b321a2": "M=", "fdeeebc706cc6397f4cab18cdff1bdfd": "\\frac{\\pi}{2\\sqrt 2}", "fdef347ff616f61402df5b556eb841d4": "\\beta \\le -5", "fdef5d39a94222f877563f3060418e02": "\n \\begin{array}{ll} \n p_1 = (x,y) + R(\\cos\\alpha, \\sin\\alpha) & v_1 = (\\dot x,\\dot y) + R\\dot \\alpha(-\\sin\\alpha, \\cos\\alpha) \\\\\n p_2 = (x,y) - R(\\cos\\alpha, \\sin\\alpha) & v_2 = (\\dot x,\\dot y) - R\\dot \\alpha(-\\sin\\alpha, \\cos\\alpha)\n \\end{array}\n", "fdef8aabae4edc18bf7d3ef7b41c9e85": "\\scriptstyle P \\left ( { {a, b}{|}{A, B} } \\right )", "fdefc5246576e6c031bdf8cf666111ad": "0.\\overline{1} = 1.\\overline{0} = 1", "fdf034d8a6155c139ce1dfaec3d9ca9b": "P_\\ell^m\\left(\\cos\\theta\\right)", "fdf05b0d5d4431742ca4788615b5f800": "\\alpha_i \\equiv \\frac {\\partial \\ln Z_i(N)} {\\partial N} \\ . ", "fdf0fd3f8f96ad4ae51b0ad4928effe2": " \\begin{bmatrix} a_0 \\\\ a_1 \\\\ a_2 \\\\ a_3 \\end{bmatrix} ", "fdf11469f5154aca7ffbf0d246c657bb": "f(\\mathbf{x}) = \\tilde{g}(\\mathbf{x} \\cdot \\tilde{\\mathbf n})", "fdf13b4fe49c2e51b3fa857027c8fc27": "\\mathbf{x}_{0j}^\\top[M_0]\\mathbf{x}_{0i} = \\delta_i^j \\qquad(2)", "fdf17a67571db03884e1470c5623a182": "\\rho_{ij}=B_{ij}/\\sigma^2 (i\\ne j)", "fdf1917481067c187a89e05c11272bf7": "\\omega_{0} = \\sqrt{\\frac{k}{m}}", "fdf1b56c82844d2b13d7769a27273293": "\\mbox{eq}(f,g) = \\{x \\mid f(x) = g(x)\\}", "fdf1c8a5afc3e97463ed68659edaad0b": "P(x) \\downarrow (\\forall{y}{\\in}\\mathbf{Y}\\, Q(y)) \\equiv\\ \\exists{y}{\\in}\\mathbf{Y}\\, (P(x) \\downarrow Q(y))", "fdf1ca1dacc115079227bb7da7ad3140": "I^-(x)", "fdf1def476259a8bd6179e59a870f577": "\\textstyle \\Omega ", "fdf22be91d26a35b9ea398b94f9f6980": "\\scriptstyle \\sum k^3 = {n+1\\choose 2}^2", "fdf2552c4ff4d5901e588568e5c91507": "\\displaystyle{(I-A)h = T\\mu.}", "fdf27be4daf0c93351230074f4d35d8d": " m = \n\\begin{bmatrix}\nM_1 \\\\\nM_2 \\\\\nM_3 \\\\\nM_4 \\\\\nM_5\n\\end{bmatrix}, ", "fdf28431bf3677c49db57bdab8b7ae5c": " S_0 + h \\nu_{ex} \\to S_1 ", "fdf359a9d54f3c86d20495c96bf24956": " C(n, k, (0, \\infty)) = a_{n-k} a_n^{-1+k/n}~, ", "fdf3d9ed5f664b4afcd11de36d206219": " v= M \\sigma \\frac{2}{R} ", "fdf46a373021e2646e93d1c9b6fd0268": "n_{3}'", "fdf48a5f8312a13544b024ad0df7589e": "\\beth_\\alpha = \\aleph_\\alpha", "fdf4c828187471340ee2db8fa7546b9f": " R_\\mathrm{out} = \\frac{v_\\mathrm{x}}{i_\\mathrm{x}} = R \\parallel R_\\mathrm{E} \\ , ", "fdf4ce07661fe1247f73cf0a70662521": " h < H ", "fdf541b2f543724decdf98cb59c62e89": "M_n = I^n M", "fdf5912d4f93d8bdc3f8ff4eb02b6043": "- {1 \\over 2 \\gamma} e e_I^\\alpha e_J^\\beta \\Omega_{\\alpha \\beta}^{\\;\\;\\;\\; MN} [\\omega] \\epsilon^{IJ}_{\\;\\;\\; MN}", "fdf5b3ad55a054e100f156511f3dd4eb": "\\hat P_{MU}(e^{j \\omega}) = \\frac{1}{\\sum_{i=p+1}^{M} |\\mathbf{e}^{H} \\mathbf{v}_i|^2}", "fdf5f562d75644ed1dc092342e518a94": " Fr > 1 ", "fdf6503b0097b268bdb016cd13cb3459": "\nc_p(n) = \n\\begin{cases}\n-1 &\\mbox{ if }p\\nmid n\\\\\n\\phi(p)&\\mbox{ if }p\\mid n\\\\\n\\end{cases}\n,", "fdf6d9c42fd3ab65cb90628546be8b56": "e^{i \\phi_{k}}", "fdf70833082059e1eb2776dc13c6c27f": "\\frac {1}{\\Delta T} = \\frac {1}{T_1} - \\frac {1}{T_2}\\,", "fdf770ee958061790ea4d765bee4e5b7": "0.25 \\le w/h \\le 2.75", "fdf77bdf63442f8222b03ed6d615f228": "D_E\\circ\\Phi = \\Phi\\circ D_E. \\, ", "fdf78a0efc24259b1a0bf452aa41a653": "\\frac{2^{b-a}}{10^b p^k q^\\ell \\cdots}\\, ,", "fdf7a6363df86063ba3b141033c335d6": "it = \\ln\\left(\\frac{iy + F}{iP + F}\\right)", "fdf7aaa9f2fbf30d5dd08ffb7971cef2": "\\scriptstyle \\delta t_{\\text{atmos},i}", "fdf7bdb0b91949e8b5d7a0495a358447": "L(C)", "fdf7e33e7c023d0602b2cd769a39856e": "m_{inf} (R,T)=\\inf \\frac{A(\\rho)}{C(\\rho)^2}", "fdf8454e66de83d14755e893218bbe6e": "1/g(x)", "fdf8677dea9673f74596472cbc605239": "\\times \\left|\\mathbf{I}_n + \\boldsymbol\\Sigma^{-1}(\\mathbf{X} - \\mathbf{M})\\boldsymbol\\Omega^{-1}(\\mathbf{X}-\\mathbf{M})^{\\rm T}\\right|^{-\\frac{\\nu+n+p-1}{2}}\n", "fdf8c405a575584ce9cf51c22c443b8b": " \\Delta T(t) \\quad ", "fdf8d6c35c39b9f73f9ad2829508ff5b": "D_1(n)", "fdf8d92b6475d0926907f5ec8529549d": " D[n] = [\\_, \\_, g\\ m\\ p\\ n]::[\\_, \\_, g\\ q\\ p\\ n]::R ", "fdf90a302c09d3e027771718e3ad50a7": "\nq_{k} = \\sum_{s_{k}=-\\infty}^\\infty e^{i2\\pi s_{k} w_{k}}\n", "fdf946b43eaa96115afc5fc88afa2fc0": "O(2^d)", "fdf9693043a25d6f2b848650bde7cb56": "\nc_0^{p^2}+c_1^p p+c_2 p^2 \\equiv a_0^{p^2}+a_1^p p+a_2 p^2+b_0^{p^2}+b_1^p p+b_2 p^2 \\mod p^3\n", "fdf97f7e532b3a7c17234842914bd7d0": "l^\\infty(\\mathbb{Z})", "fdf9fb6c40e48f7c6e831deb64f37643": "\\left(\\frac{1001}{9907}\\right).", "fdf9feb0dcadf29400d0fe4637971b68": "\\mathbb{E}[|T_j \\cap T_k|] = \\frac{t}{d^2}", "fdfa77fa47c76a7771d660d7477d31b9": "{}_p\\Psi^*_q \\left[\\begin{matrix} \n( a_1 , A_1 ) & ( a_2 , A_2 ) & \\ldots & ( a_p , A_p ) \\\\ \n( b_1 , B_1 ) & ( b_2 , B_2 ) & \\ldots & ( b_q , B_q ) \\end{matrix} \n; z \\right]\n=\n\\frac{ \\Gamma(b_1) \\cdots \\Gamma(b_q) }{ \\Gamma(a_1) \\cdots \\Gamma(a_p) }\n\\sum_{n=0}^\\infty \\frac{\\Gamma( a_1 + A_1 n )\\cdots\\Gamma( a_p + A_p n )}{\\Gamma( b_1 + B_1 n )\\cdots\\Gamma( b_q + B_q n )} \\, \\frac {z^n} {n!}\n", "fdfa8e8a0b4cdb94e7f1ff230f9d44b7": "(C \\rightarrow (A \\vee B)) \\wedge (\\overline{C} \\rightarrow (\\overline{A} \\wedge \\overline{B}))", "fdfabfd864c07c7a8600278f66919ec2": "K \\hbox{--} \\bar{K}", "fdfb12ae4da06d026d1db1d2e9279eab": "e^{o(1)(\\ln n)^\\alpha(\\ln\\ln n)^{1-\\alpha}}", "fdfb418c139bed0e6ee3efb946e96e63": "\\operatorname{Li}_s(x)", "fdfb704b251d40bad89e0c04d0e70b0a": " i \\hbar \\frac{\\partial \\rho}{\\partial t} = [H,\\rho]~, ", "fdfb7f52c44786b32c406f6217a2f498": "\\sigma^m(n)=kn\\, .", "fdfba00e32ad0366981e5bd87a002ad2": "\\mathcal{B} = \\{ w\\in\\mathcal{C}^\\infty(\\mathbb{R},\\mathbb{R}^q) ~ | ~ R(d/dt) w(t) = 0 \\text{ for all } t\\in\\mathbb{R}\\}.", "fdfc219c5f672869d0d49d80ea4d194c": "\ny(\\tau)=\\frac{4\\,\\alpha}{\\alpha\\,-\\,1}\\,\\tau\\,(\\tau-1)\\,\\sum_{k=1}^\\infty\\,\\prod_{i=1}^k\\frac{(a+i-1)(b+i-1)}{c+i-1}\\;\\frac{\\tau^k}{k!}\\ ,\n ", "fdfc5d548700c9a899295437957cf4b1": "H(E_j)\\cong\\mathcal H(E_j)", "fdfc712d4f1c247e603c6f16fdf147bf": " h(k) + x^2 = h(k) + y^2 \\pmod{b} ", "fdfc747ef577a7ce4c47a10ebc307423": "\n(3.3)\\quad\nf'(x) - x f(x) = h(x) - Eh(Z), \\qquad\\text{for all }x,\n", "fdfcb9c0aa8a2ae47a3db32cb048175c": "T^0_0=-1, T^1_1=T^2_2=T^3_3=1", "fdfcf3ee7a597c8d379e9a3c27286b44": "\\lim_{n \\rightarrow \\infty} P_n = \\lim_{n \\rightarrow \\infty} 3 \\cdot s \\cdot \\left(\\frac{4}{3} \\right)^n \\rightarrow \\infty\\, ,", "fdfd3c3bdaaa8dc5e5f05930dcc7f9c0": "[C_i,P_j]=0 \\,\\!.", "fdfd61276db3e6bdbfcf90f1c165930a": "\\lim_{n\\to\\infty} \\mu(\\{x \\in X: |f(x)-f_n(x)|\\geq \\varepsilon\\}) = 0", "fdfdbf581969051e0834d4781d5a08dd": "\\ S = - \\frac{ee}{2} + 50% ", "fdfde6f3b628f328b713d31bb987286d": "A^*_\\varepsilon", "fdfdf9d7b25861fa8b0ede8160548124": " a_i = (r+1-i,1) ", "fdfdfb8995e6d888addc72d37fbc7131": " \\frac 4{\\ln(n+1)} +\\frac 2{n\\ln n}", "fdfe1ec8bbc5fafeb808d07eab501f8a": "C_g", "fdfe5672f089e2f044fd63ad99ab7efa": "\\left|A\\right|=\\frac{1}{\\sqrt{1+\\xi^2}}", "fdfe5eed70d72f0f14aa75e02fce44da": "SHA(key || message )", "fdfef6ae6ffe020b2e977bfd5350fd36": "0\\mu(A)>\\mu(U)-\\epsilon.", "fe00e8d0930d5a68a6060906c3544fda": "S_T < K_1", "fe01429906c5ba0d162b202de4987a31": "K_0 + K_4,\\quad K_1 + K_3,\\quad K_2 + K_2,\\quad K_0 + K_3,\\text{ and } K_1 + K_2.\\,", "fe016ac2124f80d09309bc7025e06450": "M=0.0241+0.2562x_D-0.7341y_D", "fe01784ff3b745db1768fe88ccec19f6": "D_{ij}(\\mathbf{x},t) = \\frac{1}{2} \\sum_{k=1}^M \\sigma_{ik}(\\mathbf{x},t) \\sigma_{jk}(\\mathbf{x},t).", "fe01d518e9266e21ea873db0c216f48e": "\\gcd(a,\\operatorname{lcm}(b,c)) = \\operatorname{lcm}(\\gcd(a,b),\\gcd(a,c)).\\;", "fe01f8813fc75865828a7d510c48496e": "N_{\\text{corr}} (\\rho_{\\text{realized}}-\\rho_{\\text{strike}})", "fe02156b6782d898cc1dbd2e25b8507c": " dt_+ = \\frac{-\\omega \\, r^2 + \\sqrt{(1-\\omega^2 \\, r^2) \\; (dz^2+dr^2) + r^2 \\, d\\phi^2}}{1-\\omega^2 \\, r^2} ", "fe0220bb5e15e914200fe6b13c4b2285": "\nR_{\\mu \\nu} = 4 \\pi G \\rho \\Psi_\\mu \\Psi_\\nu \\,\n", "fe02d029328ae20c5987ca0882c8a64f": "a_{n}r^{n}e^{rx} + a_{n-1}r^{n-1}e^{rx} + \\cdots + a_{1}re^{rx} + a_{0}e^{rx} = 0", "fe02f3fd5cb80885064e67e2f3d52aaf": "i,j \\in S", "fe0332c81ad7d74be8948cb60f4709ce": "\n \\mathbf{u} = \\mathbf{x} - \\mathbf{X} ~;~~ u_i = x_i - X_i\n", "fe033d16f8bd772cc3cbde96f559a2dd": "\\int_{-\\infty}^{\\infty} e^{-x^2}\\,dx = 2\\int_{0}^{\\infty} e^{-x^2}\\,dx.", "fe035ead1fe244462ff673d90fcdb08e": "g(v,w)=\\langle df_\\epsilon(v),df_\\epsilon(w)\\rangle", "fe0379e6f35200062499e0ab80f5959f": " (z-c_n) ", "fe037c8ab6637d008060f3735a9e2ea5": "d_{c,j}", "fe0381e1ca6e630043c2c598c4f77dc0": "\\varphi\\in D", "fe0387472c3683c25752c23ea7c6ebd1": "\n \\displaystyle \n \\frac{V}{N\\Lambda^3} \\gg 1 \n \\ , {\\rm or} \\ \n \\left( \\frac{V}{N} \\right)^{1/3} \\gg \\Lambda\n", "fe03b8b9b78ce0145696c783c8a70d6f": "\\boldsymbol{L}(G)", "fe03bb5f0ef9f1f6cda974f887fbf028": " n = n_0 + \\frac{3\\chi^{(3)}}{8 n_0} |\\mathbf{E}_{\\omega}|^2 = n_0 + n_2 I", "fe04086037a094258ed0641a78e25b66": "R={v_i^2\\sin2\\theta_i\\over g}", "fe04150efc39d159a97c3fefe0cdad2e": " {\\rm Li}(x) = {\\rm li}(x) - {\\rm li}(2) \\, ", "fe0419bfd2df76f8e623a0b9606d59b2": "Q=I_Q\\tau_F +Q_J", "fe0455e36adaba3dfffe7b7007ae78c3": "(\\nu w) \\overline{x}\\langle w \\rangle.\\overline{w}\\langle y_1\\rangle.\\cdots.\\overline{w}\\langle y_n\\rangle.[P]", "fe045754a139162d8efee9631e5905ab": "L(r,p)=\\prod_{i=1}^N f(k_i;r,p)\\,\\!", "fe0459d1fa2f1889092d6c1077837369": " w \\wedge u \\wedge v", "fe0479669beb1aa8df8c7b35cb8c9d01": "(\\forall{q\\in Q,T\\in\\mathcal{F}})~(\\exists{\\sigma\\in\\mathbf{C}})~(T-\\sigma)q=0", "fe04be814e6b212d5478618d95b40e21": " \n \\nabla^2 \\varphi - {1 \\over c^2} {\\partial^2 \\varphi \\over \\partial t^2} = - {\\rho \\over \\varepsilon_0} ", "fe04c2daaea2e397695b04ab10a29210": "\\scriptstyle \\mathbf{N}=[\\mathbf{n}_1,\\ldots,\\mathbf{n}_N]", "fe05071b4bd9c9b9578e5069c1394651": "p \\leq 2", "fe050be60dff786e2cb932556ac57889": "V_N(\\vec{r}) \\, ", "fe0554a1567bfb8b35ab777918abb3ae": "\\mathrm{R{-}CH_2{-}CH_2{-}CO{-}SCoA\\ +\\ O_2\\ \\xrightarrow {FAD}\\ \\ R{-}CH{=}CH{-}CO{-}SCoA\\ +\\ H_2O_2}", "fe05b853b456c72bf484e3c9f049792c": "\\sqrt 2a", "fe05f717d3a36e256e44b3718089dba1": "\\int\\mathrm{hacoversin}(x) \\,\\mathrm{d}x = \\frac{x + \\cos{x}}{2} + C", "fe065dec01fefa5933aaf98666f8c373": "z \\in G", "fe065e70957c04ecb57773c4cf8bf78c": "\\dot{\\tilde{\\mu}} = D \\tilde{\\mu} - \\partial_{\\tilde{\\mu}} F(s,\\tilde{\\mu})", "fe0679d82af981850f36e18579e999c8": "\\phi_3", "fe06c8f7919d35d9ea061e0cd89565e1": "I_r \\!", "fe06d52493692c561f24cc88418c5397": "(x,y)\\in S", "fe06fa5ec15d63ef16d6387434de876b": "K = \\frac{\\log_{10}P - 6} {10}", "fe07447128e43773ef9d5d97a476e9e0": "x_v\\in\\{0,1\\}", "fe074ab5bb68d4cc99b7ca2e95ecdc10": "\\sqrt[3]{}", "fe07731324bb5e719baa81c7654c3695": "{\\textbf E}_k", "fe0782e22ca0d73cedf28cd918a3c0f8": "g(0)=0", "fe07ac52a546a62126eeb7311bae6dfd": " = \\sigma_\\mathrm{tot}^{-1} \\int \\left[ q (1 - \\cos \\theta) \\hat{z} - q \\sin \\theta \\cos \\phi\\hat{x} - q \\sin \\theta \\cos \\phi\\hat{y} \\right ] \\frac{\\mathrm{d} \\sigma}{\\mathrm{d} \\Omega} (\\theta) \\mathrm{d} \\Omega", "fe07b82b0a0db8e2fd7c084cbd437c46": "T=\\mathbb{R}^2/\\mathbb{Z}^2", "fe07f7245d6a4bfaf94a1741e5b51fc6": "\\int\\!\\!\\!\\!\\!\\int\\!\\!\\!\\!\\!\\int_{\\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\;\\;\\;\\bigcirc\\,\\,\\mathbf D\\;\\cdot\\mathrm{d}\\mathbf A", "fe080c04549a87cb46d655cb343c359c": "c_m = C_m 2 \\pi a\\ ", "fe082c02c8ff20e15005299b0eca5ddf": "C_p\\ln\\frac{V_2}{V_1}+C_V\\ln\\frac{p_2}{p_1}=0\\;", "fe087f0ace96c571553c919876718162": " \\dot{a}_n (t) - \\lambda_n a_n (t) -\\sum_m (X_n f(x,t),X_m) a_m (t) = (g(x,t),X_n) ", "fe0889e785c1f3efd153149c4eb0572c": "(n-2)!", "fe08b9cbd6f0a07537e1a8caacd46f22": "\\sigma_k(t+\\Delta t)", "fe091163e3a87bd8a736b9705f9a1cc2": "\\frac{\\Delta h}{\\Delta x} = \\frac{\\Delta f g(x_0) + f(x_0) \\Delta g + \\Delta f \\Delta g}{\\Delta x} = \\frac{\\Delta f}{\\Delta x}g(x_0) +f(x_0) \\frac{\\Delta g}{\\Delta x} + \\frac{\\Delta f \\Delta g}{\\Delta x}", "fe099e35b7caba18d87c06ac1b1f2219": "g \\circ f = f \\circ g", "fe09c1741d88b3655561bb9299eeb916": "W = W(I_1,I_2)", "fe09e1363d51d1e2cede78003e15e516": "\\hat{c} \\longrightarrow \\hat{f}", "fe09e4e827b9637d5284dd404a1698b0": "\\frac{\\text{ C}(c,\\lambda/\\mu)}{c \\mu - \\lambda} + \\frac{1}{\\mu}.", "fe09e994ff648c06e384dfe7bfc4b48b": "\\mathcal{A}_f = \\langle 1 \\rangle", "fe0a073bd5f636cf0aad16830147aa0d": "V = H^1_0(\\Omega)", "fe0a17098937a0aa476b0ec3f8460d37": "\\ell_i(x_i)=1", "fe0a3f1ada7b2463c47e259ac505899e": "R(t,k) = 2\\sqrt{2}\\left[\\sum_{n=1}^{M}\\left(\\cos{\\beta_n} + j\\sin{\\beta_n}\\right)\\cos{\\left(2 \\pi f_n t + \\theta_{n,k}\\right)} + \\frac{1}{\\sqrt{2}}\\left(\\cos{\\alpha} + j\\sin{\\alpha}\\right)\\cos{2 \\pi f_d t}\\right].", "fe0ab36be73f0cbfcb2525d3e38bf5ea": "x^bc(x) \\pmod {x^n-1}", "fe0ab7d99a152a7a578536d849cc6db2": "\\text{£}_{Li}", "fe0aefff6af7a0268d66448b145629f4": "G_0 = \\frac{2e^2}{h} ", "fe0af4cf6f7acd1852a88d819faa72b9": "= 1 + 3P(6,1) + 6P(5,2) + 4P(4,3) + P(3,4) ", "fe0b6c525b39dc2de145d60b980cff54": "X[y] = \\int_a^t \\cos \\left[\\int_a^t \\frac{1}{y} \\,dt\\right] dt", "fe0b7a671c1af46df71aa7cff1de49f5": "\\ln(1+x)=-\\sum_{k=1}^\\infty \\frac{(-1)^k}{k}\\,x^k", "fe0bee680cebdaf55228432e84b969a7": "\\frac{n!}{(n-k)!k!} p^k (1-p)^{n-k} \\rightarrow e^{-\\lambda}\\frac{\\lambda^k}{k!}.", "fe0c0b587bb7a258b0314ef684105be7": "H_\\alpha^{(1)} (x)= \\frac{1}{\\pi i}\\int_{-\\infty}^{+\\infty+i\\pi} e^{x\\sinh t - \\alpha t} \\, dt, ", "fe0c32ad458bf84816a419ef490a39bb": "E(t) \\approx E(28) \\sqrt{4+0.85 t}", "fe0c86ddcc9936903eddd25947547add": "::\\,=", "fe0cb2017fe44dfc039339d26fbf191c": "\\gamma_2 > 0", "fe0cbab5a0d084404fe5d117de93c945": "L = \\lbrace a^ib^ic^i|i>0\\rbrace", "fe0cc429608f3b6f8537acc8e0a0136f": "\\scriptstyle x^*=a", "fe0ccf9ac90fba7ce1d79027b4935853": "M_{2413\\ominus35142} = M_{796835142} = \\begin{bmatrix} &&&&&&1&&& \\\\ &&&&&&&&1 \\\\ &&&&&1&&& \\\\ &&&&&&&1& \\\\&&1&&&&&& \\\\ &&&&1&&&& \\\\ 1&&&&&&&& \\\\ &&&1&&&&& \\\\ &1&&&&&&&\\end{bmatrix}", "fe0cd3897237989ead532bde5d04beca": "(x-36) (x-6)^{36} (x+4)^{63}", "fe0d4cf656162160f743d774c30d49a7": "=\\operatorname{st}\\left(\\frac{uv + u \\cdot \\mathrm dv + v \\cdot \\mathrm du + \\mathrm dv \\cdot \\mathrm du -uv}{\\mathrm dx}\\right)", "fe0d9063a6a7bf6312cc4e09800add08": "{r \\over a}", "fe0dbd2f3a0dcba71baaf57c5c4a719c": "\\phi-\\mu\\,\\!", "fe0df4635ccf304ca89c25b509e641f5": "\\scriptstyle x^2 \\,-\\, (kB)x \\,+\\, (B^2 \\,-\\, k) \\;=\\; 0", "fe0e609806fd5247af97c2e130fbefff": "\\mathbf{\\ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}\\hat{\\mathbf{r}}_{{Earth},{Sun}} + G{m_{Moon}}{r_{{Earth},{Moon}}^{-2}}\\hat{\\mathbf{r}}_{{Earth},{Moon}} ", "fe0e6e23b8209fb65bfb9e849e0ee928": "{\\theta}=cos^{-1}\\frac{v\\cdot \\tau}{d_{max}\\cdot F_{s}}\n", "fe0e71283be6ca769a1313a6b4060dd5": "cR", "fe0ef32cc550086bfffd9e7068719ee9": "\\epsilon_x=\\frac{\\partial u_x}{\\partial x}\\,\\!", "fe0f3d3a6079ce24ba10fe10990a3475": "x \\mapsto \\langle Tx, y \\rangle", "fe0f4534a94d8e3809decb3b0c7a0b98": "\\scriptstyle a^\\dagger (k)", "fe0f60aef1784769d6275035cc6c49e5": "i\\hbar\\frac{\\partial}{\\partial t} \\Psi = \\hat{H}\\Psi", "fe0fa983b83dc84d40652e8ee114ddb4": "\\frac {\\partial V_x} {\\partial x} = \\frac {\\partial V_x} {\\partial z} = 0 ", "fe0fae750d25f24e1ef8c4761fc55e24": " \\int_{-\\infty}^\\infty x^2\\phi(x)^n \\, dx = \\frac{1}{\\sqrt{n^3(2\\pi)^{n-1}}} ", "fe0ff4781c69ee9f6d9dc0a88bc8397c": " \\log_2 a = \\frac{ \\ln a }{ \\ln 2 }", "fe102260d52e2f9bf6576a2eb34a52f3": "\\frac{f(x) \\cdot x}{\\| x \\|} \\to + \\infty \\mbox{ as } \\| x \\| \\to + \\infty,", "fe1027f01626636297179b11e9bf7dcc": "\\vec{\\gamma}", "fe106bb3da77dab48b42d33a36ba404d": "\\mathcal{L}v(x_1,x_2)=0:", "fe10ad10cb4e3dbdbfac8b4feeb29523": "9.8 = gravity\\ in\\ metres\\ per\\ second\\ per\\ second", "fe10c301d58e7f2d0e59b8d3a29671c0": "\\theta_\\mathrm{p}\\,\\!", "fe1121e630c17ed725e8f8ac8537bb9e": "v_x = \\mbox{either }\\frac{\\sum_{y=x}^\\infty \\ell_y m_y}{R}\\mbox{ or }\\frac{\\int_{y=x}^\\infty \\ell_y m_y\\,dy}{R}", "fe1133fff0057c608f578b1a58caf579": "(a + b - m)", "fe11609e997241fcaa01b70ef80f799e": "\n(1+r_{rb})\\frac{n_1n_0}{2}\n", "fe117117477015c28c80b534ab81707e": " \\operatorname{build-param-lists}[f\\ (p\\ p\\ f), D, V, T_1] ", "fe11ba9dc7abf9040a1ceab3bc02db99": " (1+2r)u_j^{n+1} - ru_{j-1}^{n+1} - ru_{j+1}^{n+1}= u_{j}^{n} ", "fe11cc0dacd8db87851aeae7666736b8": "\\frac{\\partial \\phi}{\\partial t} = D\\,\\nabla^2\\,\\phi\\,\\!", "fe11f2e8b80eab097928dc6cffd98cc3": "\\frac{R}{2}", "fe120e856bc73e05f889b4597f3103dd": "\\frac{-m_1^2\\alpha_2}{16\\pi^2\\varepsilon_o^2\\varepsilon_r^2r^6}=V", "fe125f641b3c2773b6680ea9e717ed36": " {x_N \\choose \\theta_N} = \\lambda^N {x_1 \\choose \\theta_1} ", "fe1289a962d91c69e9bfc118861d98f4": " u(x,t)=v(x \\pm ct)\\equiv v(z),\\, ", "fe128ed558094e5fab57e69aee45329e": "\\|u\\|_{L^2(\\mathbf{R}^n)}^{1+2/n}\\leq C\\|u\\|_{L^1(\\mathbf{R}^n)}^{2/n} \\| Du\\|_{L^2(\\mathbf{R}^n)}.", "fe131494954d26672dc4f46de719c014": "|\\mathbf{X}|=n", "fe136d9cf645001baa03a951561c3c41": "\\Sigma \\in \\mathcal{P}(\\mathbb{R}^{n+1})", "fe13c272c3a842e9665fa18c741b649c": "k= (m/n) \\ln 2", "fe13c59f9c2c7249c790e76d8ca49021": " \\nabla \\cdot \\mathbf{E}_\\text{g} = -4 \\pi G \\rho_\\text{g} \\ ", "fe13e53b92a3a90f18f0ead8392f4820": "\\Gamma_A(M),", "fe1492c8e4ba924da534dd0056e4941a": " d\\mathrm{PD}(s,t)", "fe14a1dbb1156b396a91e1cdfccc2420": "\n \\leq\\sum_{i\\neq m}2^{-n\\left[ H\\left( B\\right) -\\delta\\right]\n}\\ \\mathbb{E}_{X^{n}}\\left\\{ \\text{Tr}\\left\\{ \\Pi_{\\rho_{X^{n}\\left(\ni\\right) },\\delta}\\right\\} \\right\\} ", "fe14a4ad683e34bf713bbf7652a1aabb": "v_C(t) = \\frac{1}{\\sqrt{1 + (\\omega RC)^2}}\\cdot V_P \\cos(\\omega t + \\theta- \\phi(\\omega))", "fe14d5c831544d264fd625efcc2ff8d6": "\\frac{\\mathrm{d}n_2}{\\mathrm{d}t} = \\frac{n_3}{\\tau_{32}} + \\frac{n_1}{\\tau_{12}} -\n\\frac{n_2}{\\tau_{21}} - \\frac{n_2}{\\tau_{23}}", "fe150e863cd0629c093ee730d28f988c": "H_0^1(\\Omega)", "fe152fdcb86a51256941653151665bd5": "C_{out} \\circ C_{in}", "fe1575bb58f127b2622b285fca0583de": "1/n^2", "fe1620287836a38453d76bfe944efe86": " t_{i=2}=\\Delta t_{i=2}+\\Delta t_{i=1}", "fe16426b236513bbbad184b24bc01981": "\\displaystyle \\|u\\|_{L^\\infty(\\Omega)}\\leq C \\|u\\|_{L^2(\\Omega)}^{1/4} \\|u\\|_{H^2(\\Omega)}^{3/4}.", "fe16642e1b704ed91be278b409e91eec": "{d \\over dx} P_{n+1}(x) = (2n+1) P_n(x) + (2(n-2)+1) P_{n-2}(x) + (2(n-4)+1) P_{n-4}(x) + \\ldots", "fe169b921b3b30fd1f5273937eba1e3e": "p \\in F \\land q \\succ_P p \\implies q \\in F", "fe17207cee7cbe108155ec78b0b767ef": "M = \\frac{ m \\cdot R \\cdot T }{ p \\cdot V}\\,", "fe17210c1442a8f6bd4bcfde121fb82c": " \\det(S_{v \\times v}) = 0 ,", "fe1786ca3d8b815b918cb07ee8067283": "\nH = -\\sum_{i 0", "fe2a24b00a76a503e5dd27a6de87986f": "D(\\omega)=\\frac{1}{\\omega_\\mathrm{0}^2-\\omega^2-\\tfrac{i\\omega}{\\tau}} ", "fe2a2e4d3e27ebb98171372e9585d5c1": "ux \\equiv uy \\pmod p, \\,\\!", "fe2aa4eb255b7ce732fe6b85f9bd023f": "f(z) = \\sum_{n=0}^{\\infty} a_n z^{n}", "fe2aa7bc501cf911c23e221c4a02f143": "y = r\\sin\\theta\\sin\\phi", "fe2b1b1241fa4bbd4f26f541fcf57a64": " \\mathbf p_1 = \\left( 1 + {1\\over 2} { v_1^2\\over c^2 } \\right)m_1\\mathbf v_1 ", "fe2be1d815aa80290c16b71aa9e1405f": "e^*={e\\over q}", "fe2be663bec10ff6c3b5191d0549cd71": "w=ux^{-1}", "fe2bf3f9a87b006030a767396f658c91": " vp_{sat} = e^{A/T + B + CT + DT^2 + ET^3 + F\\ln T}", "fe2c06c64672a5e19bf113ffb8c8c100": "\\left(\\frac{-9}{\\sqrt{10}},\\ \\frac{-1}{\\sqrt{6}},\\ \\frac{-4}{\\sqrt{3}},\\ 0\\right)", "fe2c2fb00ea922d06acb0c05ab846a2a": "\\cfrac{\\cfrac{stC \\qquad \\overline{s}D}{tDC} \\, \\operatorname{var}(s) \\qquad \\overline{s} \\overline{t} E}{\\overline{s} CDE} \\, \\operatorname{var}(t) \\Rightarrow\n\\overline{s} D", "fe2d9b93c0e98a361ff5a1216425c7cf": "\\ R = \\frac{\\int_{3ss}^{2s} \\textrm{dh}}{\\int_{3ss}^1\\textrm{dh}} \\,\\ \\textrm{Or} \\,\\ \\frac{\\int_{3ss}^{2s} \\textrm{dp } }{\\int_{3ss}^1\\textrm{dp}} ", "fe2e1137a3afc65147e42198c0b295a4": "\n\\begin{bmatrix}\n 1 & -1& 3\\\\\n 1 & 0 & 5\\\\\n 1 & 2 & 6\n\\end{bmatrix}\n", "fe2e3b462ddda6ae8aa0fbbabb0dd42b": "S(q)=\\sum_{n=1}^\\infty a_n \\frac {q^n}{1-q^n}.", "fe2e761e19dedc70d36f63260f2e737f": "\\mathcal{R}^1", "fe2ed705b8eb024f2d43b15b42ca6c2a": "x = z^6", "fe2ee24aa5e375ecd130bc5eb22cc4c8": " X^2/E_2 \\; + \\; X^3/E_3 ", "fe2ef210e96518c88c2392f1a3b8cd4b": "n_s \\sim s^{-\\tau} f(s/s_\\max)\\,\\!", "fe2efa7342bdb0fdea53513e4b79b4bb": "{}_2F_1(a,b;c;z)", "fe2efe2d4cc017da316e2282f98a1359": "\\mathbb{R} \\times \\mathbb{R} \\times \\mathbb{R} \\rightarrow \\mathbb{R}", "fe2f179817954140c25b369ea5e4eb50": " \\Phi(S) = \\sum_{e \\in S} \\sum_{i=1}^{n_e} \\frac{c_e}{i} = \n\\sum_{e \\in S} c_e H_{n_e} \\leq \\sum_{e \\in S} c_e H_n = H_n \\cdot SC(S). ", "fe2f673241c5577ddb2d96500f862940": "p \\leftarrow p", "fe2fd2dffa8db6a828cdaba90d799a1a": "\\sum_k c_k\\,x_k(t)", "fe2ff02f352aa9a9f7a46acdca1e414e": "E(R_i)-R_f~", "fe2ff5fff7f11149929f1f8dd1f6006e": "\n\\begin{align}\n&\\int\\limits_{0}^{2\\pi} \\hat{t}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^3 u\\ \\cos u\\ du\\ = \n-\\hat{g}\\ \\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^4 u\\ \\cos u \\ du\\ \n+\\hat{h}\\int\\limits_{0}^{2\\pi}\\ {\\left(\\frac{p}{r}\\right)}^2\\ \\sin^3 u\\ \\cos^2 u \\ du\\ = \\\\\n&-\\hat{g}\\ 2\\ e_g\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^4 u\\ \\cos^2 u \\ du \n+\\hat{h}\\ 2\\ e_h\\ \\int\\limits_{0}^{2\\pi}\\ \\sin^4 u\\ \\cos^2 u \\ du =\n-\\hat{g}\\ \\left(2\\pi \\frac{1}{8}\\ e_g\\right) + \\hat{h}\\ \\left(2\\pi \\frac{1}{8}\\ e_h\\right)\n\\end{align}\n", "fe300d2c542db687c77ffb64d1acf5ca": " e^{K \\sigma \\sigma'} = \\cosh K + \\sinh K(\\sigma \\sigma') = \\cosh K(1+\\tanh K(\\sigma \\sigma')). ", "fe3023ef8cf55cbe3bd8fe1e9af0be9d": "\\begin{align} \\gamma &= \\frac{3}{2}- \\ln 2 - \\sum_{m=2}^\\infty (-1)^m\\,\\frac{m-1}{m} [\\zeta(m)-1] \\\\\n &= \\lim_{n \\to \\infty} \\left [ \\frac{2\\,n-1}{2\\,n} - \\ln\\,n + \\sum_{k=2}^n \\left ( \\frac{1}{k} - \\frac{\\zeta(1-k)}{n^k} \\right ) \\right ] \\\\\n &= \\lim_{n \\to \\infty} \\left [ \\frac{2^n}{e^{2^n}} \\sum_{m=0}^\\infty \\frac{2^{m \\,n}}{(m+1)!} \\sum_{t=0}^m \\frac{1}{t+1} - n\\, \\ln 2+ O \\left ( \\frac{1}{2^n\\,e^{2^n}} \\right ) \\right ].\\end{align} ", "fe304c1b58082695adba4ca9ace3a39e": "HS_S(t)=\\sum_{n=0}^{\\infty} HF_S(n)\\,t^n.", "fe310c1300802635dbfcb98baa238230": "\\oint_\\gamma \\frac{1}{z}\\,dz = \\int_0^{2\\pi} { ie^{it} \\over e^{it} }\\,dt= \\int_0^{2\\pi}i\\,dt = 2\\pi i ", "fe3119d9a4a94ccd2563428063fae0f0": " |\\Psi \\rangle \\in H_A \\otimes H_B. ", "fe317231d1c1126a388646d59db25f72": "P(E, \\Omega) = V(\\chi_E, \\Omega)", "fe319e1fa718aa09b7db223230508b57": "\\textstyle 2\\zeta(\\alpha)", "fe31a86b97921627f885be6d1c54fb64": "Z(FH)", "fe31ab7d8c5d659ff28258dab8eb6fa4": "\\rho(p^2)\\theta(p_0)(2\\pi)^{-3}=\\sum_n\\delta^4(p-p_n)|\\langle 0|\\Phi(0)|n\\rangle|^2", "fe31d933d08c6901d1ae0ab51b62d038": "\\textstyle \\prod_{f=1}^F N_f^{b_f}", "fe322d4d62db5fcbc96341353958d31d": "\\mu _1 (\\Sigma _{11} )^{ - 1} \\Sigma _{12} \n", "fe322e83cba1c5e11fad26cb6823767a": "\n E_\\text{pot}\\, =\\, E_\\text{kin}\\, =\\, \\frac14\\, \\left( \\rho\\, g\\, +\\, \\gamma\\, k^2 \\right)\\, a^2,\n \\qquad \\text{so} \\qquad\n E\\, =\\, E_\\text{pot}\\, +\\, E_\\text{kin}\\, =\\, \\frac12\\, \\left( \\rho\\, g\\, +\\, \\gamma\\, k^2 \\right)\\, a^2,\n", "fe325aad3d1bd891da0c9507b3dc2390": "X = 2\\begin{bmatrix}m\\\\n\\end{bmatrix}[m\\ n] = 2\\xi\\xi^T\\,", "fe326479bc2c98609e773a18d1b707b7": "\nH=\\mu\\mathbf\\sigma\\cdot\\mathbf B,\n", "fe327520ca115bc35b182846b9acc5c0": "(A)^M = \\{u\\in W \\mid M,u\\models A \\}", "fe327d73acdc195b16c28e46e9693d47": " K( \\mathbf{w} ) \\ \\stackrel{\\mathrm{def}}{=}\\ \\sum_{t=1}^\\infty K_t( \\mathbf{w} ) ", "fe32a8ba7244bb1ea777a92718da15d8": "\\gamma (\\psi, \\phi) =\\gamma (Z,W) = \n\\arccos \\sqrt {\\frac \n{Z_\\alpha \\overline{W}^\\alpha \\; W_\\beta \\overline{Z}^\\beta}\n{Z_\\alpha \\overline{Z}^\\alpha \\; W_\\beta \\overline{W}^\\beta}}.\n", "fe32c0402bb0c4c6542f5e98c9263bc7": "(p/q,0)", "fe3326c02e987bb529e6f1f96139c433": "L=L_0+C_pT", "fe33343faec72e208518159758af7f4d": "~ W_{\\rm u}=\n\\frac{I_{\\rm p}\\sigma_{\\rm ap}}{ \\hbar \\omega_{\\rm p} }+\\frac{I_{\\rm s}\\sigma_{\\rm as}}{ \\hbar \\omega_{\\rm s} } ~", "fe334a417e9f5dcbb5143ed51e4b43c5": "X(\\mathbf{F}_q)", "fe334e8be177ad92473be9f962f0f1a5": "\\{x_n^{(j)} | 0\\leq n \\leq p_j\\}=\\in \\mathbb F_{p_j} ", "fe3403a52bf6d4d931ccc49ce20f9b66": "\\mathbf{p} = n (\\cos \\alpha_X,\\cos \\alpha_Y,\\cos \\alpha_Z)=(p,q,r)", "fe3456c53630f8bcbcf5d67b23be4411": "\n{\\left( \\frac{dr}{d\\varphi} \\right)}^{2} = \\frac{r^4}{b^2} - \\left( 1 - \\frac{r_{s}}{r} \\right) \\left( \\frac{r^4}{a^2} + r^2 \\right)\n\\,.", "fe345de5b074cc9ebf0a8fab072b7755": "O( |V|^4 )", "fe34a9db26ab17374712a8e20876e8c8": "\\gamma_\\mathrm{SL} ", "fe3504952d5094736fc6d4db9e5ae517": "\\langle\\omega,\\eta\\rangle=\\int_M\\omega\\wedge\\star\\eta", "fe350dc64ae4be73cef20e3f256d308c": " P = \\frac{W}{t} = \\frac{Fd}{t} = \\frac{(180 \\text{ lbf})(2.4 \\times 2 \\pi \\times 12 \\text{ ft})}{1 \\text{ min}} = 32,572 \\ \\frac{\\text{ft} \\cdot \\text{lbf}}{\\text{min}}.", "fe35110278f6682e3e0b8be9f5eabd6c": "E_n = - \\frac{\\mu q_e^4}{8 h^2 \\epsilon_{0}^2} \\frac{1}{n^2} \\,.", "fe351728e0c5102eaf433cb92bcbe254": "\\textstyle\\sum_{n=0}^\\infty \\left|a_n\\right| = L", "fe35692df57aa5b491684b0335c67dd9": "\\langle x, x_0 \\rangle=0", "fe3587b4ef411e4e8111c92c9c011472": "b=\\frac p{\\sqrt{1-\\varepsilon^2}}.", "fe35a1834e3c0c490e7f0ebdd5443921": "M_{\\rho}(\\Gamma_g(N)).", "fe35a83732f235764bbbda8a68615b18": "P\n= L + 2r\n= \\theta r + 2r\n= r \\left( \\theta + 2 \\right)", "fe362df8fd9588b8cabf1064bc83d7e4": "\\Sigma^1_m", "fe36a02dd0590e1335305e1fa4aa7896": "a_1 = b_1,\\quad a_2=b_2,\\quad a_3=b_3.\\,", "fe36ad3e4dcd510dded505725b5b14ef": "_{p \\nleftarrow 1=p'}\\!", "fe36c1c5adbc1d7fd884c5feb0a73a6c": " G^{-1} = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & -1 & 0 \\end{bmatrix}. ", "fe3736bdf3c80a59151d4aff1b01c338": "MAP \\simeq \\frac{2}{3}(DP) + \\frac{1}{3}(SP)", "fe37394808862d82e3acb1cd11f2dcb0": " H_{i,k}= H_{i,j} + H_{j,k} + 2W_{i,j} W_{j,k}", "fe375c82c50d155af58bbc65b1459ee1": "\\mu(X\\backslash S)=0.\\,", "fe37d77aa789c8cd902d07056a288320": "\\! w\\ne -1", "fe37f48a6bb040c06c5e7ccaac63bc66": "n\\geq 1", "fe380c618a1b697e0f12a31dd4703886": " \\delta(x) = \\frac{x-x^p}{p}. ", "fe3815601b7c1500e56bfb11df525205": "Q(AB) + Q(AC) = Q(BC)\\,", "fe3827013c163fa25173a220930858e3": "\\mathcal{D}^{\\mu\\nu} \\, = \\, \\frac{1}{\\mu_{0}} \\, g^{\\mu\\alpha} \\, F_{\\alpha\\beta} \\, g^{\\beta\\nu} \\, \\sqrt{-g} \\,", "fe38622a1c091035ba559b306f81c446": " \\chi(3,6) = q_1 q _4 + q _1 q_2 - q _1", "fe38706c9ecd487b23118ad47ed09970": "\\Gamma\\ =\\ - \\frac{1}{RT} \\left( \\frac{\\partial \\gamma}{\\partial \\ln C} \\right)_{T,P} ", "fe38bbdb622f41e03ad46676c21eb6d9": " d = 1 ", "fe38c381dcd4be7d35ff3a057c99a900": " \ny(n) = \\mathbf{h}^H(n) \\cdot \\mathbf{x}(n)\n", "fe39c32019a01a6dfbf33c41626b9cba": "(\\mathbb{P},\\eta,\\mu)", "fe3a00e9d6b51acf1885ae350e1fde84": " F_D = 6 \\pi \\mu a U ", "fe3a22c794bd6160c9b36caf26d45427": "I(\\omega_n)", "fe3a2b5ba82ad8ea94ef105341c4ecf7": "~\\gamma' = 180^\\circ - \\gamma", "fe3a3cb38a899a5835d9eaa128df11f3": "\\|T^*f_j\\|_1^2 = | \\langle T^*f_j \\mid T^*f_j \\rangle_1| \\le \\|TT^*f_j\\|_2 \\|f_j\\|_2.", "fe3a3e88e3a25cd3d1342ff9bd954143": "A\\subseteq{}^\\omega\\omega", "fe3a40bc71ff07dd30a2ff98948adcd7": "E_{pol} = \\frac 1 2 \\sum_{i}\n \\frac {(\\mu - \\mu^0)^2}{\\alpha_i}\n", "fe3a47886fe1aad2aa06ca8b417ca5a1": " n(n-1) ", "fe3a83e41074834731743ab803cd4936": "\\nabla ", "fe3b0b83f28087611ff913746dcb5986": "H(X_{1}, \\ldots, X_{n})", "fe3b2270e6f3acfb48cd6e8abdbef35f": "\\left|x- \\frac{p}{q}\\right| \\ge \\frac{1}{dq} > \\frac{1}{2^{n-1}q} \\ge \\frac{1}{q^n}\\,.", "fe3b4fb0f904dd455610c31c1b5dff88": "\\mathcal{L}(q_j, \\dot{q}_j, t)", "fe3b693acd5f774d67ac2b646bca77f6": "kP=\\infty", "fe3b7cbee91514386ed64d5f831520fc": " \\widehat{U}", "fe3baae520a07d40203339d4eeeb0e97": "H = \\sqrt{a \\cdot \\mu \\cdot (1-e^2)} ", "fe3bd12b986eee8295719de6e3640424": "q = \\mathrm{floor}(y) = \\left\\lfloor y \\right\\rfloor = -\\left\\lceil -y \\right\\rceil\\,", "fe3c06c0901467fc341f4db4e245f443": "\n\\hat{\\theta}=\\tau-F^{-1}\\left(\\frac{1}{N}\\sum\\limits_{n=1}^{N}m_n(x_n)\\right),\\quad\nF(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma} \\int\\limits_{x}^{\\infty}\ne^{-w^2/2\\sigma^2} \\, dw\n", "fe3c1b79c66d476677a4869110d6a54d": "\\mathbf{R}^d", "fe3c3478a12aab63c68b3958adf533e6": " )", "fe3c8a1622965333e048d160bb167217": "D+1 \\leq \\frac{2^k}{e},", "fe3cc892c9081fc5a3abc8630c160a16": "\\lambda^*(T;F) := \\sup_{(x,y)\\in\\mathcal{X}^2\\atop x\\neq y}\\left\\|\\frac{IF(y ; T; F) - IF(x; T ; F)}{y-x}\\right\\|", "fe3cd04e6ba54f1c81154e5eb0e52c30": " v_{\\phi} = {\\omega \\over k} = {E \\over p} = {c^2 \\over v} ", "fe3ceef1f8e03566dc2229795af2b0f2": "\\sqrt{12 \\times RV}", "fe3d124f613150f281033e7a405665b1": "A_{S_{i}\\cap S_{j}} \\rightarrow R", "fe3dab8873553df4299875f42d6889f2": "D^{\\leq n} = D^{\\leq 0}[-n], D^{\\geq n} = D^{\\geq 0}[-n].", "fe3e01a305f27284ff5115f4c5ea0fa4": "b_i", "fe3e50edde07ea6ac8e26b4919d6396f": "\\left \\| \\Gamma\\varphi(t)-y_0 \\right \\|=\\left \\|\\int_{t_0}^t f(s,\\varphi(s)) \\, ds \\right \\|\\leq \\left |\\int_{t_0}^t \\left \\|f(s,\\varphi(s))\\right \\| ds \\right |\\leq M \\left |t-t_0 \\right|\\leq M a\\leq b", "fe3eb40e0cdd57013926f972e08be317": "(1+z)^u = \\sum_{n=0}^\\infty {u \\choose n} z^n = \n\\sum_{n=0}^\\infty \\frac {z^n}{n!} \\sum_{k=0}^n \n\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right] u^k = \n\\sum_{k=0}^\\infty u^k\n\\sum_{n=k}^\\infty \\frac {z^n}{n!}\n\\left[\\begin{matrix} n \\\\ k \\end{matrix}\\right] = \ne^{u\\log(1+z)}.\n", "fe3ed2e243f703d5a696766b5f9fc31c": "\\begin{array} {l}\nf^{(2)}(x_0)=\n\\frac{f\\left(x_0 + h\\right) + f\\left(x_0 - h\\right) - 2f(x_0)}{h^2} - 2\\frac{f^{(4)}(x_0)}{4!}h^2 + \\cdots\n\\end{array}", "fe3f67443f4f9d35e35b9792a9524492": "\n \\cfrac{\\Gamma \\vdash A[t/x], \\Delta}{\\Gamma \\vdash \\exist x A, \\Delta} \\quad ({\\exist}R)\n ", "fe3f776d31081f233345065ad5405a9b": " b_0 + b_1 x_i+ ... +b_n x_i ^ n + (-1) ^ i E = f(x_i) ", "fe3fbeefa16a0b7164029e0d9e5b9798": " a^{(p-1)} \\equiv 1 \\pmod{p}", "fe3fc66d29bf80db1652da23b96e6d1d": "\\frac{1}{\\rho} = \\frac{\\mathrm{d}\\theta}{\\mathrm{d}s}\\ . ", "fe401f62231ac24e3399751a415a4eaa": "2^k", "fe40506dae6cc8fd8d546eae787780fc": " s = -\\omega_n (\\zeta \\pm \\sqrt{\\zeta^2-1}). ", "fe40c46b70fcee97f322d0d7b621dfc0": "\nK(X_i, X_j) = U_{X_i}^T I^{-1} U_{X_j}\n", "fe40e2b66c09568b3859ec3eb62692a6": "A=(a(i,j))", "fe413333f6416e536d83a2ca0f14865e": "3^\\frac{0}{13}", "fe41352ffc908a6d12d88d33c6c5eb54": "\\exists \\mathbb{R}", "fe419b857791411f7a14ebc4b426f499": "\\psi(\\dots, x_i,\\dots, x_j, \\dots) = \\psi(\\dots, x_j,\\dots, x_i, \\dots) ", "fe41b0c7765b76d462d0424e64621a81": "m \\,", "fe41c4e49a5af261bc953e836aaee27d": " M \\equiv \\begin{bmatrix}\n 0 &\t1 & 1 &\t0 & 0 & 0 & 0 & 1 \\\\\n 0 &\t1 & 0 &\t1 & 0 & 0 & 0 & 1 \\\\\n 0 &\t1 & 0 &\t0 & 0 & 1 & 2 & 2 \\\\\n 0 &\t0 & 0 &\t0 & 0 & 2 & 2 & 2 \\\\\n 0 &\t0 & 0 &\t1 & 0 & 1 & 0 & 1 \\\\\n 0 &\t0 & 0 &\t0 & 1 & 0 & 2 & 1 \\\\\n 0 &\t0 & 0 &\t0 & 0 & 0 & 3 & 1 \\\\\n 0 &\t0 & 0 &\t0 & 0 & 0 & 0 & 1\n\\end{bmatrix}", "fe424997acca53974bd5c28c09caf832": "({\\hat G}, t)", "fe429abf5e062336509f8e0f10c69584": "S(n,k)", "fe42afadf249e0e56b9f4f1ecaf00e80": "W_{s}^{i} = W_{c}^{i} = \\frac{1}{2(L+\\lambda)}", "fe42b8f02de995d18aeabb3d299141a9": "\\arcsin(\\alpha)", "fe431ec58145dde4a5c83fd8dac45459": "\\left\\{\\theta_j\\right\\}_{j=1}^{J}", "fe434d0e9f15970ee97384776f8493ab": "\\textstyle{\\frac{8! 8!}{3!5!5!}} = 18,816", "fe437a5d91b299ce43f28d832c43157b": "\\nabla \\rho = 0", "fe43885a6e3592bcbee7f75a5530267c": " N\\cdot v(T,p)=\\sum_{i=1}^K N_i\\cdot v_i(T,p).\\ ", "fe43a4225ccb56a663f5bb7ecd49b198": " \\Pr(X=x) = \\frac{(\\mu(A))^x e^{-\\mu(A)}}{x!} ", "fe43a8da45a3ee8fec5e0d081154224e": "H \\equiv \\frac {d_tR}{R} \\ , ", "fe43ef9e6d61aa7dd44e2e9dccfc6c5f": "M_X(t) = \\int_{-\\infty}^\\infty e^{tx}\\,dF(x)", "fe43f2e69fdae8f22d67bb30e94de1e0": "\\tbinom ld 3^d", "fe4419e3eed1a1a9133c514a301d2c28": "\\frac{a}{12}", "fe443c3fa6137368f03aa4a310522685": "V_l^k", "fe445485f80d99a3a09162d75c3078bb": "\n\\mathbf{F} = 6 \\pi \\mu a \\left[ \\left( 1 + \\frac{a^2}{6} \\nabla^2\\right) \\mathbf{u}' - (\\mathbf{U} - \\mathbf{u}^\\infty) \\right],\n", "fe4466801a5c123e1a1eea822aaa4dd0": "\\bar{6}\\rightarrow \\bar{5}\\oplus 1", "fe446d7c9574ff8054779045d424405f": "\n\\mathbf{A} = \\begin{bmatrix}\n\\mathbf{A}_{11} & \\mathbf{A}_{12} & \\cdots &\\mathbf{A}_{1s}\\\\\n\\mathbf{A}_{21} & \\mathbf{A}_{22} & \\cdots &\\mathbf{A}_{2s}\\\\\n\\vdots & \\vdots & \\ddots &\\vdots \\\\\n\\mathbf{A}_{q1} & \\mathbf{A}_{q2} & \\cdots &\\mathbf{A}_{qs}\\end{bmatrix}", "fe448691155b2fa32d3cc9b975bf93ac": " x = e^{i\\theta}", "fe4505a442293ce8ef2c1caa27367d6b": "A' = \\pi r^2", "fe45250bb0ddcc2aa686773383a733dd": "F_1 = 2 \\cdot \\frac{\\mathrm{precision} \\cdot \\mathrm{recall}}{\\mathrm{precision} + \\mathrm{recall}}", "fe45c917a0644013d6204e4a8c74ca55": "= \\left [ (\\gamma_1 + \\lambda_1 + \\delta D_{11} + \\overline{\\epsilon}_{11}) - (\\gamma_1 + \\lambda_2 + \\delta D_{12} + \\overline{\\epsilon}_{12}) \\right ] - \\left [ (\\gamma_2 + \\lambda_1 + \\delta D_{21} + \\overline{\\epsilon}_{21}) - (\\gamma_2 - \\lambda_2 + \\delta D_{22} + \\overline{\\epsilon}_{22}) \\right ] ", "fe45fc38b5e886d1d635a0f265fc494d": "\\frac {m_1 (u_1 - u_2)}{m_1 + m_2}", "fe462f2c3e430168e39c32c7922cdbf1": "dw_p(H)", "fe4637696a3b9b0355ae8774f336a081": "V = \\pi \\int_{1}^{a} {1 \\over x^2}\\, \\mathrm{d}x = \\pi \\left( 1 - {1 \\over a} \\right)", "fe4650b882677381a6d5e793a422b76e": " f (x, y)", "fe468dd8c5799795737b4ff4e492b517": "\\mathcal A \\models_X \\vec{t_1}\\bot_{\\vec{t_3}} \\vec{t_2}", "fe469179ae82c2b79588684a53a2e659": "\\left [\\begin{smallmatrix}2&-2\\cos(\\pi/12)\\\\-2\\cos(\\pi/12)&2\\end{smallmatrix}\\right ]", "fe46a71ebd2bfe13968425b4fdfb8300": "x^m\\left(a+b\\,x^n\\right)^p\\left(c+d\\,x^n\\right)^q", "fe46aa652207f386246227c1fe0d9c32": "p_{n+1} = \\theta_{n+1} - \\theta_{n}", "fe46b2c96ff02383248b7cbe9356a576": "f(x,y,z) = 1 / ((x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2)", "fe46edd50c7508556ed79cee9b12efaf": " \\frac{\\Gamma,\\alpha\\vdash\\Sigma\\qquad \\Gamma\\vdash\\alpha}{\\Gamma\\vdash\\Sigma}", "fe472a1047b74f1f631c33f5f6f59c2f": "E ( a ) = 0 \\ ", "fe479c237a43501571d32e8edaa953a3": " 0 = ( 2 p + x )p'. \\,\\!", "fe47dfc5ef6a495cc3264972eca9b58b": "f_j\\in (V^*)^{\\otimes n}", "fe47f2adffbe755627cd77ea9519b9f9": " E_2(x,\\alpha) = x^2 - \\alpha \\,", "fe4810163cc443bbf64fc132b1147559": "V=[grams]/\\rho\\,", "fe4875fbcbd52064ac5b6c268f5b54fd": "\\beta = \\frac{1}{kT}", "fe48a249d1970cc0d1724842404edac3": "F_\\text{seq}(x)=\\big\\{x^3 -7x + 7,3x^2-7,6x,6\\big\\}", "fe48c5b9b8698844224811bf7e700434": " \\Delta x ", "fe492d17d497b45edcefbbf41243af4e": "[f,\\Delta]:\\Gamma(E)\\rightarrow \\Gamma(F)", "fe4948baad73ea912dd789066eb8abe2": " -i \\frac{\\partial}{\\partial x} ", "fe494c5fc7cee1dffc19ea277ed6373a": " \\begin{align}\nk_N^j = \\sum\\limits_{i=1}^m y_i^{j} = N \\cdot k_1^j &- \\frac{N-j}{q_j} = 0 \\\\\nk_1^j &= \\frac{N}{j}\n\\end{align}", "fe49837f25d979e4bdd40843ecad7c7a": "(e^{i a x} + e^{-i a x})/2.", "fe49f8c1f07b8506e13665fb4d752475": "\\mathbb{R}^n=\\mathbb{R}^{n-p}\\times \\mathbb{R}^{p}", "fe4a32fe73383e105c09ecfcecf997c3": "\\tilde S_n", "fe4a54fe34e25747a49d0655f999354b": "\\frac{P(W|L)}{P(M|L)} = \\frac{P(W)}{P(M)} \\cdot \\frac{P(L|W)}{P(L|M)}.", "fe4a5abf9f8b763e9a866451f11344ed": "\\mathrm{area}-\\frac{\\sqrt{3}}{2}\\mathrm{sys}^2\\geq \\mathrm{var}(f),", "fe4b794da463677a5f7e7c94b3da7da8": "\\|F'(\\mathbf x)-F'(\\mathbf y)\\|\\le L\\;\\|\\mathbf x-\\mathbf y\\|", "fe4ba48d736c5e6907814cf12fb6a189": "P\\left(S^{t-1}|O^{0}\\wedge\\cdots\\wedge O^{t-1}\\right)", "fe4c40c37ee9279a1cdad2acd1092f01": "\\left[ \\mu (t A + (1 - t) B ) \\right]^{1/n} \\geq t [ \\mu (A) ]^{1/n} + (1 - t) [ \\mu (B) ]^{1/n}.", "fe4cb8787b02616f6f32bdd83c59fea6": "\n\\mathbf{S}_W^{\\phi} = \\sum_{i=1}^c \\sum_{n=1}^{l_i}(\\phi(\\mathbf{x}_n^i)-\\mathbf{m}_i^{\\phi})(\\phi(\\mathbf{x}_n^i)-\\mathbf{m}_i^{\\phi})^{\\text{T}},\n", "fe4ce6c07a40050f1b4a69ab1195b20f": "F_\\mathrm{f} \\leq \\mu F_\\mathrm{n}", "fe4d583188b15d7b5d9a0516c2ec9917": "T_{a,\\lambda}", "fe4d79ea142b44f43aacf55b2d6d8562": "\\rho_{SM}", "fe4dc362c37de3d9b0263c6ce9a887a6": "{\\mathbf r}_2", "fe4dcefab8fb4c03f42e755e587a9d90": "\\lambda^{(0)}_n- \\lambda^{(0)}_m", "fe4e46b3a7a6a1ac0bafb800d0798c88": "\\alpha = \\left(1 + \\left(0.48508 + 1.55171\\,\\omega - 0.15613\\,\\omega^2\\right) \\left(1-T_r^{\\,0.5}\\right)\\right)^2", "fe4e4dba21e951b1dd52445baf7dc14c": " P1,P2,\\dots,PN", "fe4e6334ae080dc960f3ede0c126c92c": " Q^{(n-1)/2} \\equiv \\left(\\tfrac{Q}{n}\\right) \\pmod {n} ", "fe4e87669336dbb02beef8c368414e89": "\\mbox{Earnings Per Share}=\\frac{\\mbox{Net Income - Preferred Dividends}}{\\mbox{Average Common Shares}} ", "fe4e8b4a3a88845902ffffe51e61480a": "RGB_{white}", "fe4eaf6624e16db84b5a2c708e36ae6e": "U \\hookrightarrow X", "fe4ede16578b794e56897bd7b063a507": "A\\textbf{x}=\\textbf{b}", "fe4f45a3a50a9295551f36f272a92182": " \\int_{-1}^{1} \\frac{dx}{x} \\,\\!", "fe4ff6753e8a08485d9eade4e48e076c": " P_{total} = P_1 + P_2 + P_3 + ... + P_n \\equiv \\sum_{i=1}^n P_i \\,", "fe5082943092646b6f86507bfe8c0b26": "f_\\alpha : f^{-1}(D_\\alpha) \\to D_\\alpha\\,", "fe516a26967938caf9f9ac2ecc10f5d4": " v_n", "fe5190266690f785d314d1687d8f7b07": "\\{x \\in z : \\phi(x)\\}.", "fe51f173546d325ed0a9af56a7f65b13": "\\text{Im}(\\alpha{c})", "fe52217c42ccd4516fa1f271f02c490f": "\\textstyle\\bigvee_{i=0}^n\\varphi_i/p_0", "fe52aefe93a5a351423132c2f49b2d29": "\\Pr(W|S)", "fe52e919d03508dfecee3307294f4232": "\\mathbf{Y} = \\mathbf{X}\\mathbf{B} + \\mathbf{U},", "fe52e94c69f7ce62ee8456547900039d": "u_i^{(k)}, k=1,...,\\alpha_i", "fe52ecaa702b1bb884bd61541198d8a9": "\n\\phi = -2 a' \\frac{N}{V}\\quad\\hbox{with}\\quad a' = \\epsilon \\frac{2\\pi d^3}{3} =\\epsilon b'.\n", "fe52fc0ce69a7e8482796e008673a619": "\\hat\\alpha = (-1)^{|\\alpha|}\\alpha.", "fe532f6ca50718f2212075a7e8f2c9cd": "F^-", "fe53a7935c0bd5e9b32b32085ae089c3": " K_{i1},K_{i2} ", "fe53c7aea051c89447c15adfea1f22e6": "\\mathrm{SCl_4 + H_2O \\ \\xrightarrow{}\\ SOCl_2 + 2HCl }", "fe53db36df081e2d6a94ec58f70428ec": "\\operatorname{D}_2 (z) = \\operatorname{Im} (\\operatorname{Li}_2 (z) )+\\arg(1-z)\\log|z|", "fe53e8d874433516484fbff338233017": "\\left(x,y\\right)=\\left(x,y+z\\right)", "fe53f20f157fed1778c569f98896cf2c": "\\mathbf{S}=\\sum_{\\mathbf{k}}\\hbar\\mathbf{u}_{\\mathbf{k}}\\left(\\hat{a}^{\\dagger}_{\\mathbf{k},L}\\hat{a}_{\\mathbf{k},L}-\\hat{a}^{\\dagger}_{\\mathbf{k},R}\\hat{a}_{\\mathbf{k},R}\\right),", "fe54d59c07b7c1527e1ba174d9627920": "\\Gamma(\\tfrac15) \\approx 4.5908437119988030532", "fe5537ec1b50737a15b970aadba406ea": " \nv(P) = \\mathrm{ord}_t\\left(P|_{V_f}\\right) = \\mathrm{ord}_t \\Bigl(P\\Bigl(t,\\sum_{n=3}^{+\\infty}t^n\\Bigr)\\Bigr) \\quad \\forall P\\in \\mathbb{C}[x,y]\n", "fe556c23f8bf856bd346b10a8d8c608c": "\n(b_1 - a_1 + 1) \\; G_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} \\mathbf{a_p} \\\\ \\mathbf{b_q} \\end{matrix} \\; \\right| \\, z \\right) =\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1-1, a_2, \\dots, a_p \\\\ b_1, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right) +\nG_{p,q}^{\\,m,n} \\!\\left( \\left. \\begin{matrix} a_1, \\dots, a_p \\\\ b_1+1, b_2, \\dots, b_q \\end{matrix} \\; \\right| \\, z \\right), \\quad n \\geq 1, \\; m \\geq 1,\n", "fe55973090c16c6c4b769b4734df10ac": "u_{\\bold{k}}(\\bold{r}) \\approx \\frac{1}{\\sqrt{\\Omega_r}}", "fe55a24327384cee53d635b8a676c3aa": "v_i > 0", "fe55fb9fb69b2befc33373172a73fd55": "\\sum_{i=1}^n \\frac{1}{i} \\in \\Theta(\\log n)", "fe563a21587933c13ea93281d209522e": "SE\\text{ of }hf_{ij}=\\sqrt{\\frac{(1-\\sqrt{d/B})(1-\\sqrt{d/D})-hf_{ij}-hf_{ij}^2/2}{2N}}=0.00514", "fe5661fa8cc208ca473b2fa9c1e452dd": "T \\geq T_0(\\varepsilon)", "fe56a5683bef0ce887d955cad88338e3": "b^2=ac", "fe56afdc25b86ad1328bd5016dafa439": "\\omega(a_1a_2 \\dots a_p)=\\left\\{\n\\begin{matrix}\n\\omega(a_1,\\dots,a_p)&(\\omega\\in \\Omega^pM)\\\\\n0&(\\omega\\not\\in\\Omega^pM)\n\\end{matrix}\\right.\n", "fe56e31b917f3535e6165fd584aadfd1": " t = nT = n/f_\\mathrm{s} \\,", "fe577aedc38178853d0d95e889f6d223": "y \\in \\widehat{\\mathbb{R}}, y > x", "fe57d4d9ef1a0fec1332d4c81bb3f386": "P[E^C] = 0", "fe57d97f314898c0201a714d8bd63ea5": "R_0 = \\frac{V_0^2}{P}.", "fe58002719284faae7e087e323eee843": "C(a,q,0)=1", "fe5823b60b4f5e3064e2a0a2b6e6fb7a": "\\lambda_{\\kappa}", "fe5829067b8e95e291779c704fce5e50": "\n\\begin{cases} \nb_{2n} & = b_n \\\\\nb_{2n+1} & = (-1)^n b_n \n\\end{cases}", "fe5843b1cc22229692b25dc1f53fa9e6": "f_1(x),f_2(x), \\dots", "fe58991bf4821d3b1dfb8b2118eda4e5": "f_\\text{comp} = \\frac{A^2}{2 \\pi \\sqrt{{i_1}^2 + {i_2}^2}}", "fe58de5e236dcc8ae5f74aef3660e5fd": "\\mathbf{r=y^\\text{obs}-J \\left(J^TT \\right)^{-1}J^T y^\\text{obs}}", "fe59133a758f647d954843a24046ae1c": " S^m_{1,0}(\\mathbb{R}_x^n \\times \\mathrm{R}^N_\\xi) ", "fe59b6982f588412da36f330dc8dc8b4": "g'_i=g_{\\pi(i)}\\,", "fe5a54bc2e6ea280c8bd07ef150fa747": "\\alpha_\\text{IR}", "fe5a9997f1eed8d540bc04a7ad3114a3": "\\scriptstyle I\\setminus\\{y\\}\\cup\\{x\\}", "fe5ad602bbfd83b07fc69abc332b32cb": "n_p, n_e", "fe5b19748b9f34cbbac7cea09315cfe4": "\\left(\\frac{-3}{\\sqrt{10}},\\ \\frac{1}{\\sqrt{6}},\\ \\frac{1}{\\sqrt{3}},\\ \\pm3\\right)", "fe5b8631283ec2703212179354644f39": "h_n^{-1}h_{n-1}^{-1}\\cdots h_1^{-1}g_1 g_2\\cdots g_m\\,=\\,1. ", "fe5b97b7b0a37c7047169838b3b27718": "a_{T} (p)", "fe5c2cf8584b9efa7132ae603e22c59d": "\\Delta_3(k_\\lambda) = k_\\lambda \\otimes k_\\lambda", "fe5c7a6e3368da573e7d31404c2d0e3a": "N_1,\\dots ,N_k", "fe5d1203b942e92f2cffc907de2380dc": "n_1=0,\\, n_2\\neq0\\,\\!", "fe5d49ce6a92ed71658998eab9d7664a": "{\\overline P}X \\neq \\mathbb{U}", "fe5d5479627f101943ee246caf0a83de": "H(X, Y) = \\mathbb{E}_{X,Y} [-\\log p(x,y)] = - \\sum_{x, y} p(x, y) \\log p(x, y) \\,", "fe5da0d089bf0344bae7d338218e87a0": "g_{uv}", "fe5de4d16af321304d241139ec830083": "Z_\\mathrm{load}", "fe5e067c3097c091020ace15705b5ab2": "x=\\lambda(1-\\lambda)", "fe5e2dc66a782f46b2570bebbbe25154": " 1\\le i < j \\le m", "fe5e3824a40020a59314e431e7dacdef": " p_{1^{(n)}} = h_{1^{(n)}} ", "fe5e5a9a94dbf6657e18b1acd57a528e": "f({\\mathbf{x}}) \\approx f({\\mathbf{p}}) + \\left. {\\nabla f} \\right|_{\\mathbf{p}} \\cdot ({\\mathbf{x}} - {\\mathbf{p}})", "fe5e6ead8f3ed73c73267a2966b48e40": "K_p=\\lim_{n\\to\\infty} \\left[\\frac{1}{n} \n\\sum_{k=1}^n a_k^p \\right]^{1/p}.", "fe5e8b0b8188e9ac23634a4d8e217e2d": "W_{s}^{0} = \\frac{\\lambda}{L+\\lambda}", "fe5eba66cbf53bedd7d2b4b2935eeb52": "x < \\mu+\\sigma/(-\\xi)", "fe5ec7c6d9db7edc965622b3a3ad27be": "\\chi(E) = \\chi(F)\\cdot \\chi(B).", "fe5f245111cad113ccbe72066f8c09d8": "a_1, b_1, s_1, t_1", "fe5f444ddd8416208398ad8b3217264b": "|p - r| < E ", "fe5f7deca445fe307660f5467585c80b": "\\beta E_{t}[\\pi_{t+1}]", "fe5fbb8a12e38c97b40759b3adf2a691": "\\scriptstyle K<0,", "fe5fbc0bd8b0b8e458df9b5c29ddb360": "f|_{r(m)}: r(m) \\rightarrow \\mathbb{Z}, \\ n \\mapsto\n\\frac{a_{r(m)} \\cdot n + b_{r(m)}}{c_{r(m)}}", "fe5fcddf5ce6ed43c1fb8355eac4717a": "\\mathcal{Z}^{\\pm}:=\\mathbb{Z}[v^{\\pm 1}]", "fe607232d24693c8449928631d9c7f1a": "\\theta = \\sum_i \\theta^i(\\mathbf e) e_i.", "fe607c1d9feaf7af3e78e6ee02c9f582": "c_{1,2}=z_{xyy}-6xz_x=0, c_{2,3}=z_{yyy}+3x^2z_{xy}-24xz_y-12w=0.", "fe608e06d943b8f3fd97393e14f745fb": "\\{C,D\\}", "fe60bddc11271e19b9ab07304966477d": "\\lim_{n \\to \\infty}S_{n} = \\frac{a}{1-r}+\\frac{rd}{(1-r)^2}", "fe6129356e756df322c67c2fa787658a": "\\chi = \\varepsilon_{\\text{r}} - 1.", "fe624d0db1bf8f074d21c6cfec7ea0c1": "\\mathrm{\\ Am\\ +\\ HgX_2\\ \\xrightarrow {400 - 500 ^\\circ C} \\ AmX_2\\ + \\ Hg \\ }", "fe624e1d97b1f0cc752d78aeb2bf6dce": "X={S N \\over D+1}+1", "fe6270d296ce70c56941e29cd20d6fa1": "{\\mathcal D}(M)", "fe6285954bbead15140615d8c9181563": "\\textstyle{v_i}=0", "fe6286e216da114aa169cec48299e6e7": "\\mathbf{N}q=\\,q\\mathbf{K}q =\\,(\\mathbf{T}q)^2", "fe6288a0914f25999f7073f36118a1ca": "P_{EM} = I/c = p_{EM}/At \\,\\!", "fe62f8473168291a09c9b9485cbab3a4": "\n\\left(\\frac{\\partial N_j}{\\partial V}\\right)_{S,\\mu_j,\\{N_{i\\ne j}\\}} =\n-\\left(\\frac{\\partial p}{\\partial \\mu_j}\\right)_{S,V\\{N_{i\\ne j}\\}}\n", "fe6396f88dbddbbb1887b3769e8c24ec": "A(\\omega) = \\frac{i \\omega C R_0}{1 + i \\omega C R_0 - \\omega^2 LC}", "fe63e2494ac559abfbd002ea2c3f02f7": "X \\subseteq \\mathbb{U}", "fe63f8fa084df2029d811bf410a5d27b": "\\text{AVC} = \\frac{\\text{VC}}{\\text{Q}}", "fe63faa26100d0fed83ace7bced9d2c4": "W_{t_2}-W_{s_2}", "fe645778c83a03513c5ca2e8e37e888f": "a^2 \\cos^2 \\theta + b^2 \\sin^2 \\theta = r^2,\\,", "fe64a0577f7acadcca9e4313315dd71d": "K(A, 1)", "fe64b7bc4c5695980ce89799b0d454d7": "y'=\\frac{6 ft}{1.46 ft}=4.112", "fe64c29fd79b520a1c44c8169372a9b0": "X_3 = A\\cdot B-C\\cdot D ", "fe64ce8d29f3dbe7224d88bd20c8b697": "\\nabla_X (fY) = \\mathrm df(X)Y + f\\nabla_XY", "fe64f09ea36ef6aed44b454dee38ff51": "N_{V} > 0", "fe653822258116f177cc8dbe2d5116fc": "\\,j\\,", "fe65a218d08e962295dbd965048a5e93": "\ns(\\tau ,\\rho ,z) = \n\\frac{\\delta (\\rho )} {2\\pi \\rho}\nJ(\\tau ,z) H(\\beta \\tau -z) H(z),\n", "fe65c15919aed8cd5860d724df151f3d": " \\frac{\\mathrm{D}\\Gamma}{\\mathrm{D} t} = \\oint_C \\frac{\\mathrm{D} \\boldsymbol{u}}{\\mathrm{D}t} \\cdot \\boldsymbol{\\mathrm{d}s} + \\oint_C \\boldsymbol{u} \\cdot \\frac{\\mathrm{D} \\boldsymbol{\\mathrm{d}s}}{\\mathrm{D}t}. ", "fe65eae09b434e21a6a2ca1567c51e39": "h(x) = x + g(x)", "fe663f0dc31e55d58996e86720b322dd": "Y_{22} = {((1 + S_{11}) (1 - S_{22}) + S_{12} S_{21}) \\over \\Delta_S} Y_0 \\,", "fe66873bb043258d5083f2d908cfc273": "g \\in G, x,y \\in M", "fe66936f90c9fa2e9ae7d6aa92648495": " (1-p_a-p_b)\\ln\\left(\\frac{1-p_a-p_b}{b-a-1}\\right)", "fe66dc4fe3f1a014268a3b49fad67ec5": "\\sum_{i=1}^n a_i i x^{i-1}", "fe66f7bea3cd52664e3a6fbde81ea947": "\\limsup_{n \\to \\infty} F_{n} (x_{n}) \\leq F(x).", "fe675362e1c79d35f2d05539b42a5909": "\\xi_{1},...,\\xi_{n}", "fe675ee8a5e445b247c6b3de543a8e6c": "\\mathcal{P}^2=\\mathcal{P},", "fe6762d5a7091a04e8466e115c76096f": " P^{te}(X,Y)", "fe68051fc0d47530ab1dad164a6ccb72": "(A \\vee C) \\wedge (B \\vee C)", "fe682121b08a30d84bd966d6f48b233e": "D^{\\prime\\prime}", "fe68254f2261e824fb818dfe6ed7b07b": " \\frac{ \\log(h_i R_i) }{ \\log\\sqrt{|D_i|} } \\to 1\\text{ as }i \\to\\infty ", "fe68dbb3c37a1246f0d7a07ef79b7e3c": "H = -\\ln P + \\text{constant}\\,", "fe68eb208bfaea11b489fb4f5237554c": "2^{T(n)^{O(1)}}", "fe698588922daff152a8e6efa5099425": "N_p(f+g)\\le N_p(f) + N_p(g)", "fe6986936bbb7263fbb582953f414c42": "C_0(L) = L , \\, C_1(L) = Z(L)", "fe69f87f1c2f7c1339d913b49589407b": "N \\approx \\frac{N_m}{1- N_m \\tau / T}", "fe6a330c63a479f696c1569560fa2030": "L_n|0\\rangle = 0 (n\\ge -1)", "fe6aac760a89c47a12b225d98a3a0348": "\\varepsilon=-\\tfrac12", "fe6af1956ecb9b6e7592e084afd20cd8": "f_1(q) = \\sum_{n\\ge 0} {q^{n^2+n}\\over (-q;q)_{n}}", "fe6b53704425d415497ee4e34d9e0824": " \\vec{p}_3 = \\frac{\\sqrt{1-\\omega^2 \\, r^2}}{r} \\, \\partial_\\phi + \\frac{\\omega \\, r}{\\sqrt{1-\\omega^2 \\, r^2}} \\, \\partial_t", "fe6b6f2dcae650d9e649d40ca981681d": "P\\,\\!", "fe6bb742c3284e460535733de42be76c": "v = A \\cdot v = \\frac{\\Delta}{p} = \\Delta \\cdot f = \\Delta \\cdot k \\cdot v", "fe6bbaf7044578769d1be22aca12c22b": "a^2 + Q^2 \\leq M^2.", "fe6c6c1096028f46f78facd49dbe759e": "[0.5,1)", "fe6c72f7c8d5517599f6dd70af4be740": "\\begin{matrix} \\frac{1}{2} \\end{matrix} \\pi^2 r^4.", "fe6ce007ae9c11f9bd6e0bbccc1e8090": " H(p) := \\begin{bmatrix}\na_1 & a_3 & a_5 & a_7 & \\ldots & 0\\\\\na_0 & a_2 & a_4 & a_6& \\ldots & 0\\\\\n0 & a_1 & a_3 & a_5& \\ldots & 0\\\\\n0 & a_0 & a_2 & a_4& \\ldots & 0\\\\\n0 & 0 & a_1 & a_3& \\ldots & 0\\\\\n\\vdots & \\vdots & \\vdots & \\vdots& \\ddots& \\vdots\\\\\n0 & 0 & 0 & 0& \\ldots& a_n\\\\\n\\end{bmatrix}", "fe6ce162cb59c81c230743a3ed981392": "\\frac{{}_1F_1(a;b+1;z)}{b{}_1F_1(a;b;z)} = \\cfrac{1}{b + \\cfrac{a z}{(b+1) + \\cfrac{(a-b-1) z}{(b+2) + \\cfrac{(a+1) z}{(b+3) + \\cfrac{(a-b-2) z}{(b+4) + {}\\ddots}}}}}", "fe6e821c8da5cad6687db3047d817144": "= a(x, \\sigma(x), \\sigma'(x))dx + b(x, \\sigma(x), \\sigma'(x))d(\\sigma(x)) + c(x, \\sigma(x),\\sigma'(x))d(\\sigma'(x)) \\,", "fe6e9783f155d3c0e33b6f819af08af1": " u(x) = \\int_{y_0}^{y_1} f(x, y) \\,dy \\qquad (1)", "fe6f6edd145cf5179d180e8f2a570e92": "H + O(\\sqrt{H} + 1)", "fe6f7f71f0276bda9e16d263c9e2a5d2": "i^4=1", "fe6fccbe614e3d9e7161f2f48be1090d": "B_{SO} >> B_\\phi", "fe7048fb08157c979b8df88be8563841": "\\text{Hom}(X,Y)", "fe70c0594105f23bd518d790ff8ad359": "e^{-t / \\tau}", "fe70d575fd2817a62f158f1c4c9455cc": "(z_i), i=0,1,2,...", "fe713a42ccf564553cbc69e0e5d56465": "g(r_{12})", "fe71819304c636c214f699fa05f69988": "X = \\sum_j f_j(x) {\\partial \\over \\partial x_j}", "fe71ad31380c412f68be157332399ef5": "\\delta(z) = \\varphi(z)/|\\varphi'(z)| = \\lim_{k\\to\\infty} |z_{kr} - z^*|/|z'_{kr}|. \\, ", "fe723d9d959b8478c917e785434bba38": "a \\in \\mathbf{F}_p", "fe7249e4505522d74bf75cedcb017c97": "\\mathfrak{E}", "fe72606eb6a66d5b47c4b8bc888911ea": "f(x,q_1) \\leq r_1\\,\\!", "fe7284963b2193885a6f535c4868199d": "\\zeta(s) = \\frac {1}{k^s} \\sum_{m=1}^k \\zeta \\left(s,\\frac{m}{k}\\right).", "fe72da946a48ce5a5e811c1ede4ded50": "\\operatorname{Tr}\\big(\\Gamma(R)\\big) = \\sum_{m=1}^{l} \\Gamma(R)_{mm}.", "fe73200a0e2f541bb4bbac5b18e4f5d9": "FA=H.\\,", "fe73dca14d7ec6a6610f699ac4b94def": "2g/(2i+1)", "fe7437310e196705ea9a4cb28d8c3a81": "\\langle \\vec f \\rangle = T \\frac{dS}{\\vec {dR}} = \\frac{k_B T}{P( \\vec R)}\\frac{dP( \\vec R)}{\\vec {dR}}~", "fe747a78ea8a41d323be8fa4b1ede883": "\\zeta(\\tfrac{1}{2}+it)", "fe74d6e8e594cfef5b7ea9e2332a5148": "P(a)", "fe753c6dce56128cf47a0be0ca6f7947": "\n \\delta K = \n -\\int_0^T \\left\\{ \\int_{\\Omega^0} \\left[\n J_1\\left(\\ddot{u}^0_{\\alpha}~\\delta u^0_\\alpha \n + \\ddot{w}^0~\\delta w^0\\right) \n + J_3~\\ddot{w}^0_{,\\alpha}~\\delta w^0_{,\\alpha}\\right]\n ~\\mathrm{d}A\\right\\}~\\mathrm{d}t \n + \\left| \\int_{\\Omega^0} J_3~\\dot{w}^0_{,\\alpha}~\\delta w^0_{,\\alpha}\\mathrm{d}A\\right|_0^T \n", "fe75718c1efdea117b40ecbd77ed5294": " Q[u]/R[u]", "fe75834187244ad3fc7d7ca5185e66c1": "v = 100*\\left(1 - {p*q \\over p + q}\\right)", "fe75b73cb805fec2cbcd5b67dfc6ac05": "\\frac{dy}{dx} = y^2", "fe75ca4e375b4d15f84f308cbd188183": "A \\approx 4.828a^2", "fe762d91e831387323edf60b4cf5c562": " p^2(1-F) + pF\\text{ for }\\mathbf{AA};\\ 2pq(1-F)\\text{ for }\\mathbf{Aa};\\text{ and }q^2(1-F) + qF\\text{ for }\\mathbf{aa}. ", "fe7638c74131e4270b35f967e95a1df4": "e_{n+1}", "fe76661ba35de9b4a296c768b279de2e": "\\overline{HR} = \\cfrac{N}{T_{eff}} ", "fe769f9b1044c037670d2456726c79b7": "M_{C^*}", "fe76fb8161929497aba7889fe616352f": "F(0)", "fe772f64cf5592a7e8946a23a1c24342": " V_{k} ", "fe773943b49455acee25aa672952ebe6": "\n\\varepsilon''(\\omega) = \\left( 1 + 2 (\\omega\\tau)^\\alpha \\cos (\\pi\\alpha/2) + (\\omega\\tau)^{2\\alpha} \\right)^{-\\beta/2} \\sin (\\beta\\phi)\n", "fe7773f5c022006b8fe29a92b6b4708f": "\ne^x = \\frac{x^0}{0!} + \\frac{x^1}{1!} + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\cdots \n= 1+\\cfrac{x} {1-\\cfrac{1x} {2+x-\\cfrac{2x} {3+x-\\cfrac{3x} {4+x-\\ddots}}}}\n", "fe77913ec879de5ec64e5146db78b754": "X_1(t), \\dots, X_n(t)", "fe77ab7123e285a861b00ec59fa8f186": "\n\\{Z^A\\}\\longrightarrow \\{\\widetilde{Z}^A\\}=\\{\\tilde \\Phi, \\tilde E ^a, \\tilde I ^ a\\}\\ ,\\quad\n\\Phi = \\tilde \\Phi - \\delta_{kl} \\tilde E ^k \\tilde I ^l ,\\quad\nE^i = - \\tilde I ^{i}, \\quad \n E^j = \\tilde E ^j,\\quad \n I^{i} = \\tilde E ^i , \\quad \n I^j = \\tilde I ^j \\ ,\n ", "fe7857d477545f5407f4c021dd76fb63": "\\delta=\\frac{FAD-FAP}{1-FAP},\\;\\;0\\le \\delta \\le 1", "fe787cda58a48fb16743c5b7ebdd954c": " \\vec x = (x,y,z) \\quad \\hbox{and} \\quad \\vec p = (u_x, u_y, u_z).\\,", "fe78d074be939d68d72923ad8d76f453": "\\varphi_i\\varphi_j^{-1}(x, \\xi) = (x, t_{ij}(x)\\xi)", "fe78d5c708e5de7618da28f8588be8e5": "1+I\\mathcal Q_{\\mathrm{Hur}}", "fe78fc46ee49ec896f5adae5980b7b58": "C_i=0", "fe78fc86f4296984227ee354f58d67c7": "K_{n} = \\begin{bmatrix}b & Ab & A^{2}b & \\cdots & A^{n-1}b \\end{bmatrix}.", "fe798a1b923dca473f3ebdb90a8239ff": "\\frac{\\rm d}{{\\rm d}t}x(t)=f(x(t),x(t-\\tau))", "fe799edcb5bcbf3ce61465286c7fc048": "\\begin{align}\n\\sum_{n = -\\infty}^{\\infty}\\frac{(-1)^n}{z-n} = \\frac{1}{z} + \\sum_{n = 1}^{\\infty}\\frac{2z}{n^2-z^2}\n\\end{align}", "fe79ac98947f0dd2518360327f39dce7": "\\ a \\leftarrow a + ( b \\times c )", "fe7a0dc57ea809d2e5b45e85d029a460": "I(y)\\propto 1+V\\cos{(p_yd/\\hbar+\\phi)}", "fe7a131cc392107c4b12b4d760483596": "\\mathbf{1}_{A\\cap B} = \\min\\{\\mathbf{1}_A,\\mathbf{1}_B\\} = \\mathbf{1}_A \\cdot\\mathbf{1}_B,", "fe7a165b5d6432f200a09fbcc446f74a": "\\overline Y\\to X ", "fe7a3465b7e3955fa2aa5dae4fa2de51": "x_0\\in D", "fe7a710aedab0ace7ca1785827a5d5d7": "\\mathbf{q} = q(0) + j\\frac{p(0)}{\\omega L} \\ ", "fe7ada86b1ef717a1a6742e9abb3090a": "\n\\left\\|\\sum\\limits_{i=1}^N a_{\\sigma(i)}-A \\right\\|=\n\\left\\| \\sum_{i \\in \\sigma^{-1}\\left(\\{ 1,\\dots,N_\\varepsilon \\}\\right)} a_{\\sigma(i)} - A + \n\\sum_{i\\in I_{\\sigma,\\varepsilon}} a_{\\sigma(i)} \\right\\|\n", "fe7b3aaaa04db8f20615270474665380": " B-V = M_B - M_V\\!\\,", "fe7b8ca94051765a7872bf6ac6b51c8a": "(\\tilde{A}+\\tilde{\\delta A})", "fe7bd9ee61e7d71f6e60e0764850164c": "r \\ge 2", "fe7c3ed5c8bb07db02af627506c19141": "V = 2 \\pi^2 R r^2 = \\left ( \\pi r ^2 \\right ) \\left( 2 \\pi R \\right).", "fe7c522c7554970acdbb2fee0f729d6b": "\\Psi_Z(t)=\\log \\mathbb E e^{tZ}.", "fe7c57da5c1a01f590e4e43075c96ee9": "\\lim_{n \\rightarrow \\infty} A_n = \\lim_{n \\rightarrow \\infty} \\frac{a_{0}}{5} \\cdot \\left(8 - 3 \\left(\\frac{4}{9} \\right)^n \\right) = \\frac{8}{5} \\cdot a_{0}\\, ,", "fe7c96ffadea2923bee96d98d2284713": "g, h", "fe7d18b26944b1589058170a70a775d9": "\\ell P(L_+) + \\ell^{-1}P(L_-) + mP(L_0)=0,\\,", "fe7d47c5b632403d8b42811379b694c1": "\\bar{f}(n+1)", "fe7d9b2567bcb7b508e09575fbd45ebc": "\\ \\sum_{w \\in V} f(u,w) = 0", "fe7da1ba148149c961c8cb5b3c94194d": "dq/dt+Aq+Qf(p+q)=0", "fe7dae7198a12b9937857d409ba7c9a9": "\\textrm{Kendrick~mass~defect}= \\textrm{nominal~Kendrick~mass} - \\textrm{Kendrick~mass}", "fe7dc93b9e29e354f4c78ad5c08b28d1": "\\text{var}[Y^{(m)}]=\\hat{\\sigma}^2 m^{-d}", "fe7ded8e8e08351661409b20a01c229e": "\\mathcal{P} = \\frac{1}{\\mathcal{R}_m}", "fe7e0f0348de8335a4e4b49b230e1ee0": "\\operatorname{sl}(r)=s", "fe7e9cc9fba8bf4acadb8eb2463c9757": "A_v(j_f,i_e)", "fe7eacd93eb2b9a770985e193df5eb88": "M \\to D_X(D_X(M)),", "fe7ef02d3380dada5296b393746f315e": "\n\\begin{bmatrix} a & b\\end{bmatrix}\n\\begin{bmatrix}x \\\\ y\\end{bmatrix}\n== a x + b y == c", "fe7f0128ed4c5fd7f7525ad0538f8da4": "\\text{PAM}_n(i,j)", "fe7f17b3094b50f69d769e693ab37831": "\\epsilon = 1 - \\frac{\\mu_a \\cap (A^T \\mu_b A)}{\\mu_a \\cup (A^T \\mu_b A)}", "fe7f7ce384be1947204cbb2403475442": "\\mathcal E(x):=\\mathcal E\\otimes_{O_X} k(x)", "fe7f9fa1f08063b60d2be24076cab1a2": "\\tau=\\frac{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}", "fe7fb4d88a18eef3009e39003de65fb7": "\\tan(\\theta') = \\tan(\\theta+\\pi) = -\\cot(\\theta)\\,", "fe7fbd9bdd5fa13db54f1939bbfec8c9": "\\lim_{T\\to\\infty} \\int_0^T e^{-st}dg(t)", "fe7fde6a28f6380451be03369e943466": "(a, P)", "fe8059d2e25715e090f10c0d3b037583": " \\mathbf{x}_* = \\sum^{n}_{i=1} \\alpha_i \\mathbf{p}_i", "fe805b2a91883b66015e7a4755ec9b92": " I < B \\log_2 \\left( 1+\\frac{S}{N} \\right) ", "fe807b3110d88285793911ac79aeb73e": "\\mathrm{L}: \\quad \\begin{smallmatrix}4&&1\\\\&\\swarrow&\\\\2&\\rightarrow&3\\end{smallmatrix} \\qquad \\mathrm{U}: \\quad \\begin{smallmatrix}1&&4\\\\\\downarrow&&\\uparrow\\\\2&\\rightarrow&3\\end{smallmatrix} \\qquad \\mathrm{X}:\\quad \\begin{smallmatrix}1&&4\\\\&\\searrow\\!\\!\\!\\!\\!\\!\\nearrow&\\\\3&&2\\end{smallmatrix}", "fe80f32d9f3e29c231357d1f84629e5f": "\\int_a^b \\! f(x)\\,dx = F(b) - F(a).", "fe81e230e5123bb779fe8d09e2403552": "\\langle c_n : n \\in \\mathbb{N}\\rangle", "fe81ea6ba870bb3ef812d34af756c535": "Y(z)", "fe81f9dc7fa052a78b1732077c50558c": "x\\in\\overline{A}", "fe827214f8051ab761500c8b015e191c": "{\\nabla}^2 \\varphi = -\\frac{\\rho_f}{\\varepsilon}.", "fe833426f06b1fb7f027a95de2c9a790": "B^{\\alpha }_{\\beta }", "fe837b40840e5de312732f7d582d9e65": " \\frac{B_h}{h}\\equiv \\frac{B_k}{k} \\pmod p \\text{ whenever } h\\equiv k \\pmod {p-1}", "fe83adbd4752f08bbe96f2342c074eca": "x_0 = x", "fe83dbde2f8787bde14f1b3b547c6af4": "h_b", "fe840b2caf6de73f83bf2da2b342d09d": "\\operatorname{lcm}(21,6)\n={21\\cdot6\\over\\operatorname{gcd}(21,6)}\n={21\\cdot6\\over\\operatorname{gcd}(3,6)}\n={21\\cdot 6\\over 3}= \\frac{126}{3} = 42.", "fe841e86bd3ccaa448e8110e74d13e39": "d_1 = \\frac{\\log(F/K) + \\sigma_P^2S/2}{\\sigma_P \\sqrt{S}}\\,", "fe847a1eca5dd5748a6cd5182b70efd8": "\\gamma_K^*= sA -(n+\\delta)\\ ,", "fe84949ded84a465b585aa722ae65f30": "v_1 = - R i_2", "fe850bf1288b304fa3d1061fbe3428c4": "\n H^2(P,Q) = 1-e^{-\\frac{1}{2}(\\sqrt{\\alpha} - \\sqrt{\\beta})^2}.\n ", "fe854301662812087fe29b63d03159dc": "\\sum_n \\alpha_n^{\\frac\\delta{2(2+\\delta)}} < \\infty.", "fe854d2b72af3b6663ddfb676e3f53ee": " Q = I -auu^T ", "fe8561feb65531ff9c265afc8d484294": "\\omega_{\\text{o}}", "fe85b1a67edb69c456e54219e8190c41": "\\Pr[P(r_1,r_2,\\ldots,r_n)=0|P_i(r_2,\\ldots,r_n)\\neq 0]\\leq\\frac{i}{|S|}.", "fe85fcc2e9f5ad68b7495eb1dd57f83f": " s \\leftarrow \\delta_{int}(s)", "fe865e4ea4a991a20692a9df67bd06b6": "f(x,y,z) = x^2 + y^2 + z^2 = 1", "fe86b33163447ddceff59d7b41fa629f": "N = cd", "fe86e062a0e0cdfb214ce5989e824ccb": " \\exp \\left( -{t'-t \\over t_0 }\\right ) ", "fe86e4015ee463c05d12d987f7f19f7b": "P_\\text{total}=0 + P_\\text{stagnation}\\;", "fe86f1a9b22c54f3508d1c508e3ee34c": "P(A\\mid N=n) = \\left[{d^n \\over d\\lambda^n}\\left(e^\\lambda\\, P_\\lambda(A)\\right)\\right]_{\\lambda=0}.", "fe879a9b410734366fde89d2ca0990d6": "r\\trianglelefteq p, r\\trianglelefteq q", "fe87b05e2a9a0e6c687dbeb2a84d7347": "\\sigma(xy + yx) = xy + yx.", "fe87dfe4c8dcbaca6c7b19ec026d6f52": "\\phi::=\\bot |\\top |p|(\\neg\\phi)|(\\phi\\and\\phi)|(\\phi\\or\\phi)|\n(\\phi\\Rightarrow\\phi)|(\\phi\\Leftrightarrow\\phi)|\\mbox{AX }\\phi|\\mbox{EX }\\phi|\\mbox{AF }\\phi|\\mbox{EF }\\phi|\\mbox{AG }\\phi|\\mbox{EG }\\phi|\n\\mbox{A }[\\phi \\mbox{ U } \\phi]|\\mbox{E }[\\phi \\mbox{ U } \\phi]", "fe880c60668863c169f094a1bfc16ef0": " cos \\phi_{S} ", "fe8876ad3de256951b0668beb00cf6cb": "(s, a)", "fe889e153afc8d318705bba8f76bd5f9": "\\scriptstyle \\pi^{\\!*}", "fe894d4b7cd055e8307d3f41a9b6ba8b": "h = 2\\theta(1-\\theta)", "fe89ca45ea85780bcea3049ce4d8bd4c": "\n2 \\left\\langle T \\right\\rangle = -\\sum_{k=1}^N \\left\\langle \\mathbf{F}_k \\cdot \\mathbf{r}_k \\right\\rangle\n", "fe8a0174355ac8fe7eca59db8d77b3b0": "\\mathbf{E}' = \\gamma \\frac{\\mathbf{v}}{c} \\times \\mathbf{B},", "fe8a13e3b0560f38484da89cbec7968b": "64 z^9 - 128 z^7 + 64 z^5 - 702 x^2 y^2 z^3 - 18 x^2 y^2 z + 144 (y^2 z^6 - x^2 z^6)\\ ", "fe8a6bc1f1f3d601f41b3df5865449bb": " \\ln A = V ( \\ln A' ) V^{-1}. \\, ", "fe8a71554a4aef96a1d8a37fdb032c24": "(\\sigma_1,...,\\sigma_e)\\in G^e", "fe8a8c81b449cb4bf7375fe5bcb3ca8f": "\\alpha =\\frac{R}{r} \\theta", "fe8adeedeaf52670db1ce0aedbb887b0": "\\ \\rho_1 = \\rho_x \\quad ; \\quad \\rho_2 = \\rho_y \\quad ; \\quad a_1 = a_x \\quad ; \\quad a_2 = a_y,", "fe8b368c8d0a32bcf015c97b0f130752": "\n{} + \\frac{\\varphi_{i,j+1} - \\varphi_{i,j}}{h_{j}^y}\n\\left ( \\frac{h^x_i}{2} \\varepsilon^y_{i,j} + \\frac{h^x_{i-1}}{2} \\varepsilon^y_{i-1,j} \\right )\n", "fe8b6efe02ca5c56f9d1441b4618e9d2": "\n (1) \\qquad\n \\frac{\\partial^2\\Phi}{\\partial x^2}\\, +\\, \n \\frac{\\partial^2\\Phi}{\\partial z^2}\\, =\\, 0.\n", "fe8b9e612d9641470253a044c479bb1a": "(m \\times p)", "fe8be9db3284fbe8edf1e5165919cc2d": " d {\\mathbf{S}}", "fe8bf53a88f9528f5151a8e04ac31ab2": "f'_i", "fe8c1180961db6233cdd8fdf89c9d58a": "y_{t}= \\delta y_{t-1}+u_{t}\\,", "fe8c3cca4f761cbd0c6b80a253b6df93": "y''_n = -f_n y_n", "fe8c9678517d2999e242e5835eff565e": " \\scriptstyle f \\to f_0 ", "fe8caad120ac739af16d9824f7e32810": "O(V^2 \\log{V} + V E)", "fe8cb2887667159cdf0b0393f0d8a0b5": "\n \\operatorname{cn}(z|m) = \n \\frac{\\pi}{\\sqrt{m}\\, K(m)}\\,\n \\sum_{n=0}^\\infty\\, \\operatorname{sech} \\left( (2n+1)\\, \\frac{\\pi\\, K'(m)}{2\\, K(m)} \\right)\\;\n \\cos \\left( (2n+1)\\, \\frac{\\pi\\, z }{2\\, K(m)} \\right).\n", "fe8cb4db7acb95c83dd6f4b1819bbd96": "Q(Y)", "fe8ccb43874c6ad8e2490a8cc06f802d": "\\sigma_{\\varepsilon}=\\varepsilon\\frac{V}{d}", "fe8ce26a290941655c86fd80b86b7d3d": "G \\simeq \\langle S \\mid R \\rangle.", "fe8d1714c532ba14b5306a4aeedd3c1e": "y^2 = f(x) + \\frac{h(x)^2}{4}", "fe8d284cdd689b2553a5388a966b67c9": "T_t", "fe8d2fefe69517e08f3c289615b477fe": "\\scriptstyle d_{ij}", "fe8d3a01d7ad279d7529eec8ec64e48c": "\n\\mathcal{J}^2\\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* =\n\\mathcal{P}^2\\, D^j_{m'm}(\\alpha,\\beta,\\gamma)^* = j(j+1) D^j_{m'm}(\\alpha,\\beta,\\gamma)^*.\n", "fe8dae49333c476d3e48b67be7e6ad59": "p_k < P < p_{k+1}", "fe8ded65eba0878f76cdeac99afce0ce": "{\\rm cov}({\\mathcal L})", "fe8df27de5f99dee4f9e33e0f033975c": "\\vec v(\\vec x,~t)", "fe8e12fe3679b53e2fdbdfec86072caa": "\\mathcal{R} \\ge {\\max\\limits_{0\\le r\\le{1- H_q(\\delta + \\varepsilon)}}} r \\left ( 1 - {\\delta \\over {H_q ^{-1}(1 - r) - \\varepsilon}} \\right )", "fe8e2106f23f7b8df32ff53bd269af79": "\\beta _{i+1} = \\beta _{i} - \\frac{g(\\beta _{i})}{g'(\\beta _{i})} ,", "fe8e556dc683fbe6d264bdf12c449e3f": "{n\\choose k_1,k_2,\\ldots,k_r} ={n-1\\choose k_1-1,k_2,\\ldots,k_r}+{n-1\\choose k_1,k_2-1,\\ldots,k_r}+\\ldots+{n-1\\choose k_1,k_2,\\ldots,k_r-1}", "fe8e5893cb963a948121cbe9c0db41cb": "p_w(\\theta)=\\sum_{k= -\\infty}^{\\infty}\\int_{-\\infty}^\\infty p(\\theta')\\delta(\\theta-\\theta'+2\\pi k)\\,d\\theta'.", "fe8e76d2bacbf7713ca5abf816edbc18": "d = \\sqrt{2Rh + h^2} \\,,", "fe8e8198ea65c4db7dabddb253c964d9": " H: M \\to \\mathbb R ", "fe8e861a4029897536a69dfd7b4611f6": "s^2(\\vartheta)=\\operatorname{Var}\\left [ X_{ij} |\\Theta_i = \\vartheta\\right ]", "fe8e936df0e0b02ba5cd5cd47c960861": "\\psi(x,y) = A(x,y)exp(+jk_o\\nu y)\n", "fe8edd2395cadcca769afac0076609f3": "(x-5)^2 = 7.\\,\\!", "fe8ef4250e86954e405e037834b7aa58": "\\; m_1g-m_2g=m_1a+m_2a", "fe8f36d8af72601da9c8326493194b76": "b_n = e^{\\frac{\\pi i}{N} n^2 },", "fe8f3e329ef32bb42f14e5e6102066f4": "\nr_{E,L} = \\frac{EBIT(1-T)-\\Delta IC+Debt_{new}-Interest}{E_{L}}\n\\qquad (4)", "fe8f420f470043fa9af58e09db6ee6d3": "\n I_1 = \\lambda_1^2+\\lambda_2^2+\\lambda_3^2 = \\lambda^2 + \\cfrac{2}{\\lambda} ~.\n ", "fe8f53486ff079a5e10d92807b2b1e9f": " f_\\mathrm{beat} = \\left | f_2 - f_1 \\right | \\,\\!", "fe8f622343b8a4091deb854b5727d35e": "\\begin{align}\n &\\operatorname{cov}(X_i, X_j) = 0,\\quad \\forall i \\frac{f_s}{2}, \\end{cases}", "fe9079ad9d14812bf3f6688739f3fafa": "\n \\begin{align}\n I_1^C & := \\text{tr}(\\mathbf{C}) = C_{II} = \\lambda_1^2 + \\lambda_2^2 + \\lambda_3^2 \\\\\n I_2^C & := \\tfrac{1}{2}\\left[(\\text{tr}~\\mathbf{C})^2 - \\text{tr}(\\mathbf{C}^2) \\right]\n = \\tfrac{1}{2}\\left[(C_{JJ})^2 - C_{IK}C_{KI}\\right] = \\lambda_1^2\\lambda_2^2 + \\lambda_2^2\\lambda_3^2 + \\lambda_3^2\\lambda_1^2 \\\\\n I_3^C & := \\det(\\mathbf{C}) = \\lambda_1^2\\lambda_2^2\\lambda_3^2.\n \\end{align}\n\\,\\!", "fe90a3f7bddbcabd22d8468570ed3e8e": "f(x)= \\sum_{n=0} ^\\infty a^n \\cos(b^n \\pi x),", "fe911e182cf77905f2dc4f4107dbea23": "S(t) = \\frac{1}{\\mu_0} E_0\\,B_0\\,\\cos^2\\left(\\omega t-{\\mathbf k} \\cdot {\\mathbf r}\\right) =\n \\frac{1}{\\mu_0 c} E_0^2 \\cos^2\\left(\\omega t-{\\mathbf k} \\cdot {\\mathbf r} \\right) =\n \\varepsilon_0 c E_0^2 \\cos^2\\left(\\omega t-{\\mathbf k} \\cdot {\\mathbf r} \\right).", "fe913c186449755999c46c24bda80ac0": "{\\log_b (n)}", "fe915de7857cb62f1122277ddfd51a35": "{1\\over 2} (T_+-T_-)", "fe917e584b0e4cce7724bab032e6a51e": "S(\\omega_k)", "fe91caa3a2d276674f8e2f5418a5ffc1": "1/P", "fe924f3fb4cc1ffb175abaa76b240754": "f\\colon (D^2,\\partial D^2)\\to (M,\\partial M) \\, ", "fe92a4f03d4621813aff3734e95642f9": " F_G", "fe92c99f1a3d1c24fc383793bb514a37": "\\hat{\\sigma}^2=\\text{E}(y_i^2)", "fe92fe9346d3d8ac6e6bbaa091d16e63": "\\int^{\\infty}_{0}x^{j}e^{-x}\\,dx", "fe93d9aa3915e401a8b54c55b9f0a9c6": "E_{CMI}( P_{A,B})", "fe941badd47bc5a77e0e6bab2403194a": "V_{\\mathrm{sen}} = E_{\\mathrm{sen}} \\left( 1 - e^{- \\frac{t}{\\tau_{\\mathrm{sen}}}} \\right)", "fe943ada9b425f4f5e4a71ee66a6f9da": "\n\\begin{align}\n S(\\boldsymbol{\\phi}(y)) =& y_1^2 + \\cdots + y_{r-1}^2 + H_{rr}(y) \\sum_{i,j = r}^n y_i y_j \\tilde{H}_{ij} (y) \\\\\n\t\t\t=& y_1^2 + \\cdots + y_{r-1}^2 + H_{rr}(y)\\left[ y_r^2 + 2y_r \\sum_{j=r+1}^n y_j \\tilde{H}_{rj} (y) \n\t\t\t\t+ \\sum_{i,j = r+1}^n y_i y_j \\tilde{H}_{ij} (y) \\right] \\\\\n\t\t\t=& y_1^2 + \\cdots + y_{r-1}^2 + H_{rr}(y)\\left[ \\left( y_r + \\sum_{j=r+1}^n y_j \\tilde{H}_{rj} (y)\\right)^2 \n\t\t\t\t- \\left( \\sum_{j=r+1}^n y_j \\tilde{H}_{rj} (y)\\right)^2 \\right] \\\\\n\t\t\t&\t+ H_{rr}(y) \\sum_{i,j = r+1}^n y_i y_j \\tilde{H}_{ij} (y). \n\\end{align}\n", "fe945aa7e4d587d5e7c64cacdad5eae3": "\\,\\! \\text{mag}_{AB} = -2.5 \\log_{10} {S_{\\nu}\\over \\mu Jy} + 23.90", "fe9491ca48da847c442885b6acc94144": "p((1+0.01x)(1-0.01x))=p(1-(0.01x)^2)", "fe9494b4634d2b0462a77ad29aef5a9c": " \\mu(n) = \\frac{1}{n} \\sum\\limits_{k=1}^{k=n} T(n,k) \\cdot e^{2 \\pi i \\frac{k}{n}}. ", "fe94e3f7d574518cde7a737c35135d98": "|\\alpha| \\le q^{d/2}", "fe94f560869bb7be508e7b3b75c9a8df": "Z_\\pi/Z_\\nu", "fe9548881194ddbc767c4441fa46b239": "i=1, 2, \\dots, m.", "fe956bb803cb1f0da5d8f4f6bab46da0": "\\begin{matrix}\\mathrm{Cabtaxi}(6)&=&1412774811&=&963^3 + 804^3 \\\\&&&=&1134^3 - 357^3 \\\\&&&=&1155^3 - 504^3 \\\\&&&=&1246^3 - 805^3 \\\\&&&=&2115^3 - 2004^3 \\\\&&&=&4746^3 - 4725^3\\end{matrix}", "fe96165cb9de1e02c6d71e7b112f3246": "\\; E(s_i) ", "fe9723a47db056b1f5262952bf2578bc": " \\mathbf{\\hat{b}} = \\mathbf{\\hat{t}} \\times \\mathbf{\\hat{n}} \\,\\!", "fe9730883dd7ed9da1ca709eb4e08cba": "{x^2 \\over a^2} + {y^2 \\over a^2} + {z^2 \\over b^2} = 1 \\,", "fe97ae571e82fd2ed37ce6bf868f8a85": " i\\; ", "fe97f5e449cf8b7c07bacd7c14e9bef2": "x'v=x \\in\\phi(d,c)", "fe984632563eef1e54f999d906bd4e1f": "\\displaystyle (n+1)", "fe9846f41676c542f4716afec13ec86c": "(X)_+ = \\begin{cases}X &\\text{if }X \\geq 0\\\\ 0 &\\text{else}\\end{cases}", "fe9886623f75aaed5774725156e76f20": "\\textstyle\\frac{1 + \\sqrt{5}}{2}", "fe98efcb46d382c2550c4588d0334a7f": " Q(\\theta_1,\\theta_2,\\theta_3)= Q_{\\bold{z}}(\\theta_3) Q_{\\bold{y}}(\\theta_2) Q_{\\bold{x}}(\\theta_1) . \\,\\!", "fe9924e2347e5da0dad4f24a959d6fa1": " f(a/2) < 2f(a) ", "fe995db3762285f4ebfefd829bf4172b": "(2^{n},\\log n)_2", "fe996845ace23fb725eeba01afc096d0": "E_{XX}", "fe99af6a5eae3146e3a635ed0b260da5": "{1 \\over e}\\sum_{k=0}^\\infty {k^n \\over k!}.", "fe9a10f79e5cd7e842a3b5ca671554e2": "d_0,d_1:S_1 \\rightarrow S_0,", "fe9a7769ec08590530bee2b15342af2b": "\\langle A_1 (l, m, t) A_2^* (l, m, t) \\rangle", "fe9a88586cc1cb8d488f3ace8abf7a2c": "\\frac{14.1%\\cdot4.3-18.3%\\cdot2.3}{4.3-2.3}=9.27%\\,", "fe9acb8a65b116ad5cb1c0b3a7bb06ab": " ( \\exists x (\\phi \\lor \\psi) ) \\rightarrow \\forall z \\rho", "fe9addb926b05dd653168e578420a64e": "\\mathbf{\\hat{e}}_2", "fe9ae8692b35dde6ab22ec79b2924ccf": "\\mathcal{E} = \\frac{1}{q} \\oint_{\\mathrm{wire}}\\mathbf{F}\\cdot d\\boldsymbol{\\ell} = \\oint_{\\mathrm{wire}} \\left(\\mathbf{E} + \\mathbf{v}\\times\\mathbf{B}\\right)\\cdot d\\boldsymbol{\\ell}", "fe9b324e22f8d31a6a143b10db43500b": "\\mathcal{L}_{\\mathrm{QED}} = i\\hbar c \\bar \\psi {D}\\!\\!\\!\\!/\\ \\psi - mc^2 \\bar\\psi \\psi - {1 \\over 4\\mu_0} F_{\\mu \\nu} F^{\\mu \\nu}", "fe9b9de57c9ba3186c07026ff0dc0b7d": "\\operatorname{E}|X-\\lambda|= 2\\exp(-\\lambda) \\frac{\\lambda^{\\lfloor\\lambda\\rfloor + 1}}{ \\lfloor\\lambda\\rfloor!} .", "fe9be1b4753af269abd9427a9515a70b": "\\lambda \\otimes \\mu", "fe9be6f6269bf8a3559b356d7209b178": "g\\in SL_2(\\mathbb{R}), \\epsilon^2=cz+d=j(g\\cdot z), ", "fe9bff47fa566224181b22f70e3b3b7b": "X' = \\bigoplus_i X_i.", "fe9c0683e2963b9e778065799edac046": "\\scriptstyle 3.5 kT", "fe9c25dd98e0e2b44985de7c0f39f53c": "P(y|x)", "fe9c6b029aec8f74d55b5fb50919f4d5": "\\ell^2(\\mathbb{Z}/N\\mathbb{Z})", "fe9d2e3461385fd12fa29ff99f8b5d4b": "\\vec{r_E}", "fe9dacf94e40ae843801eb660fccfd5e": "E_{\\mathrm f}", "fe9db47fe49199e85f48bbaedc3bc899": "(E_{2+})-(E_{2-})", "fe9dc53a4f4b5c2b804c8b9b6cc99ff4": " m_1", "fe9df6736d1e7dc31d1b6d6e92bbace9": " \\psi_0(x) = x - \\sum_{\\rho} \\frac{x^{\\rho}}{\\rho} - \\frac{\\zeta'(0)}{\\zeta(0)} - \\frac{1}{2} \\log (1-x^{-2}). ", "fe9e0eb7a4d9071357f88d8aa5e93694": "\\mathbf{D} = \\varepsilon_0 \\mathbf{E} + \\mathbf{P}", "fe9e730bf9d475412b3c0e99c1d1f113": "\\mathit{DLR} = \\frac{\\mathit{displacement}(\\mathrm{lb}) ~/~ 2240} {(0.01 \\times \\mathit{LWL}(\\mathrm{ft}))^3}.", "fe9e9c6ca1d4942b754817a4efbcd2ca": "\np^{\\gamma\\dot{\\alpha}}B_{\\gamma\\epsilon_1\\epsilon_2\\cdots\\epsilon_n}^{\\dot{\\beta}_1\\dot{\\beta}_2\\cdots\\dot{\\beta}_n} = mcA_{\\epsilon_1\\epsilon_2\\cdots\\epsilon_n}^{\\dot{\\alpha}\\dot{\\beta}_1\\dot{\\beta}_2\\cdots\\dot{\\beta}_n} \n", "fe9f0a9403c69e2d4e4772e0ed7b6cfe": "g_{N}(\\hat{a}^\\dagger, \\hat{a}) = \\sum_{n,m} c_{nm} \\hat{a}^{\\dagger n} \\hat{a}^m. ", "fe9f2feef4c5c0afc2dfb79cecbb3b79": "\\scriptstyle M_P \\;=\\; I \\,\\oplus\\, \\bigoplus P'_i", "fe9f47e636ad42f588f926041ce3a1e2": "\\sum_{k=0}^n a_k k! (n-k)!\\le n!.", "fe9f6193db59b9dcf571d2bbe7cc7ef3": " \\ln(\\cos x + i\\sin x)=ix \\ ", "fe9f653db014983daf1ccb2347741955": " t_{k+1} = t_k + h ", "fe9f6ffe5287bb0b8eccf41c079c8e26": "\\psi(\\Omega^\\Omega 2+\\Omega^{\\psi(0)})", "fe9f8bd59c2420eb48dd10bce6aa89d9": "\\textstyle {2n \\choose n} ", "fe9f9f84145ab664e6acc55cdc717915": "O\\left(W^L m \\log m + ( C^m + t_H) \\sum_{P_{max}}{rc(P)} \\right )", "fea02ccacfe886afe5e63cacd73564a3": "r_{\\rm eq}", "fea074d265806a25957be2c2d1b6a1b2": "\\begin{bmatrix}y_a \\\\ y_b \\\\ \\vdots\\end{bmatrix} = \\begin{bmatrix}\\vec y_a[n] \\\\ \\vec y_b[n] \\\\ \\vdots\\end{bmatrix} = \\begin{bmatrix}\\sum_{i=1}^n {c_i\\,\\lambda_i^a\\,\\vec e_i[n]} \\\\ \\sum_{i=1}^n {c_i\\,\\lambda_i^b\\,\\vec e_i[n]} \\\\ \\vdots\\end{bmatrix} =\\begin{bmatrix}\\sum_{i=1}^n {c_i\\,\\lambda_i^a\\,\\lambda_i^{n-1}} \\\\ \\sum_{i=1}^n {c_i\\,\\lambda_i^b\\,\\lambda_i^{n-1}} \\\\ \\vdots\\end{bmatrix} = ", "fea09ee026f3e0fbef0347d92f1d603e": "c_f = \\lim_{n \\to \\infty} \\left[ \\sum_{k=1}^n f(k) - \\int_1^n f(x) \\, dx \\right]", "fea0a5ffe5f3d1e1e1c4019f67e3167f": " D[q] = D[p] ", "fea0e80636f054733154d2b74ad38924": "2^{n-k}", "fea17ca098c9ca981420cfa68ee23513": "\\alpha\\in S\\cap C", "fea192161f020b424b72f1949062a7e1": "\nR_{tot} \\ \\stackrel{\\mathrm{def}}{=}\\ [\\mathrm{R}] + [\\mathrm{C}]\n", "fea1a762a52e67c3404e47a0563c1eca": "f(n) = k \\times 2^{\\frac{n}{12}}.", "fea1ba6972887e561458aa65034eb62f": "\\mathcal M =(M, d)", "fea1bbb312552958e6add82bf8821d1f": "\n\\begin{align}\n(t_2 - v_2)\\cdot(t_2 - t_1) & = 0 \\\\\n(t_1 - v_1)\\cdot(t_2 - t_1) & = 0 \\\\\n(t_1 - v_1)\\cdot(t_1 - v_1) & = r_1^2 \\\\\n(t_2 - v_2)\\cdot(t_2 - v_2) & = r_2^2 \n\\end{align}\n", "fea1cec5f1281ec7375e75515e05e9f9": "\\sigma_\\theta^2", "fea1d62be668b6dfe395dfb69f84b593": "\\mathcal{A}\\, =\\, \\frac{E}{\\sigma}\\, =\\, \\frac{E}{\\omega\\, -\\, k\\, U}\\,", "fea209b8a5f7aecece344b497588f7cb": "\\begin{bmatrix}1 & 1\\\\ 0 & 1\\end{bmatrix}.", "fea2268ac8d7d716db52d3e39ed2eba8": " f_X(x|Y=y) = \\frac{f_Y(y|X=x)\\,f_X(x)}{f_Y(y)}.", "fea252f303c3c4cbab0c4f6151646fbb": "a_k(\\mathbf{y}, t)", "fea283abbbf2b734ecadbe03d696af34": "\\left| \\begin{matrix}\n A & B \\\\\n C & D \\\\\n\\end{matrix} \\right|=1", "fea2a3cfcf948c90efd75b51c25613d6": "g(x,y;\\lambda,\\theta,\\psi,\\sigma,\\gamma) = \\exp\\left(-\\frac{x'^2+\\gamma^2y'^2}{2\\sigma^2}\\right)\\exp\\left(i\\left(2\\pi\\frac{x'}{\\lambda}+\\psi\\right)\\right)", "fea2dc435843203d2191ba5596adb0c2": "\\quad\\beta(A\\cdot\\varphi,\\psi) = \\beta(\\varphi,\\tau(A)\\cdot\\psi)\\qquad (1)", "fea30f5f8012bf2b841a00d390bb50ab": "G = GL_1", "fea31d47866a32e34fa9ab6cdcfdfbee": "\\int d^dx\\, \\sqrt{-g} R^{\\mu\\nu}R_{\\mu\\nu}", "fea3654ec02f03ac2ba9050b17df8fc6": "\\mathcal{G}_\\sigma", "fea38073927e974eb22ab5055a312705": " \\langle y, x \\rangle := 2 \\frac{(x,y)}{(x,x)}", "fea3bfed11a6e2fbb7b2e10c84e9f000": "\\nabla^2 \\nabla^2 w = q/D", "fea4233c63aba203a8c223d4bc17c10c": " u_j ", "fea43000f677360af1ea931ffea07f96": " \\begin{align} & \\Psi^*\\frac{\\partial \\Psi}{\\partial t} = \\frac{1}{i\\hbar } \\left [ -\\frac{\\hbar^2\\Psi^*}{2m}\\nabla^2 \\Psi + U\\Psi^*\\Psi \\right ], \\\\\n& \\Psi \\frac{\\partial \\Psi^*}{\\partial t} = - \\frac{1}{i\\hbar } \\left [ - \\frac{\\hbar^2\\Psi}{2m}\\nabla^2 \\Psi^* + U\\Psi\\Psi^* \\right ],\\\\ \n\\end{align}", "fea44fbf184199414d2456dfad17472b": "(X_{i_1}\\ldots X_{i_k})", "fea45bd741749cd3d33d2daf5a84fc11": "\\frac{1}{2} I \\omega^2 + \\frac{1}{2} m v^2", "fea4617818bc12798ac32573bfc9a975": "h[n] = Th_c(nT)\\,", "fea480c5f5d20bf318636481649b6dd1": "\n\\begin{align}\n& m_a u_a + m_b u_b = m_a v_a + m_b v_b \\\\\n& C_R = \\frac{v_b - v_a}{u_b - u_a} \\\\\n\\end{align}\n", "fea4e935aa8c545f7885fb6cbc91cec1": "E:\\Lambda M\\rightarrow\\mathbb R", "fea4f0c2cf5dda8722b9d866ee05e690": " = y - \\delta \\beta + b f(\\alpha) - b f(a) \\ ", "fea4f5361fac91e7d406c42ca1a360f8": "{d \\over dx} p_n(x) = np_{n-1}(x),", "fea50a9c0736c2d869e0360c4bffe78e": "b_2 = 1a_0 + 1a_1 + 2a_2 + 3a_3", "fea524d8424c7b206fc68631c6705c98": "\\{W|P|Q\\}(\\alpha,\\alpha^*,t) = \\frac{1}{\\pi \\left[\\kappa + \\langle n \\rangle\\left(1-e^{-2\\gamma t}\\right)\\right]} \\exp{\\left(-\\frac{\\left|\\alpha-\\alpha_0 e^{-(\\gamma +i\\omega_0) t}\\right|^2}{\\kappa + \\langle n \\rangle\\left(1-e^{-2\\gamma t}\\right)}\\right)}", "fea5bd3bb7aa9c8efaafbec881e626b9": "S_{\\text{BH}} = \\frac{k_\\mathrm{B}A}{4\\ell_{\\mathrm{P}}^2},", "fea5bda266aa56e314472634d46271fc": "\\boldsymbol\\theta^{(t+1)} = \\underset{\\boldsymbol\\theta}{\\operatorname{arg\\,max}} \\ Q(\\boldsymbol\\theta|\\boldsymbol\\theta^{(t)}) \\, ", "fea5e1d89a32acf862251eb9e4c23b34": " X^{T} X ", "fea66b646aab1054d3aacb80a4b9efe3": "\n \\begin{bmatrix}\n 1 & -S_1 & -S_2 & 0 & 0 & 0 \\\\\n 0 & 1 & 1 & 1 & 0 & 0 \\\\\n 0 & F_1 & F_2 & 0 & 1 & 0 \\\\\n 0 & P_1 & P_2 & 0 & 0 & 1 \\\\\n \\end{bmatrix}\n \\begin{bmatrix}\n Z \\\\ x_1 \\\\ x_2 \\\\ x_3 \\\\ x_4 \\\\ x_5\n \\end{bmatrix} =\n \\begin{bmatrix}\n 0 \\\\ L \\\\ F \\\\ P\n \\end{bmatrix}, \\,\n \\begin{bmatrix}\n x_1 \\\\ x_2 \\\\ x_3 \\\\ x_4 \\\\ x_5\n \\end{bmatrix} \\ge 0.\n", "fea69c86e3a5dbfb018f1ba6ebb5ecf6": " \\pi\n= \\begin{pmatrix} 1 & 6 & 7 & 2 & 5 & 4 & 8 & 3 \\\\ 2 & 8 & 7 & 4 & 5 & 3 & 6 & 1 \\end{pmatrix}\n= \\begin{pmatrix} 1 & 2 & 4 & 3 & 5 & 6 & 7 & 8 \\\\ 2 & 4 & 3 & 1 & 5 & 8 & 7 & 6 \\end{pmatrix}\n", "fea6a9238b03ff66d83fdc9bfb42a367": " a^{b c_b}= {a^c}^{b^c} ", "fea6c9e8363b9fdf4a8261e0fab2de18": "\n\\bigcup_{t \\in \\mathbb{T}} [T] = X, ", "fea6e03acfe1fc29216c94a6e0d0f468": "\\eta^3+\\eta^2+\\eta=2", "fea7136bbed3a22278478184c63067e3": " \\phi: L^\\infty_\\mu(X) \\rightarrow \\operatorname{L}\\bigg(\\int^\\oplus_X H_x \\ d \\mu(x)\\bigg) ", "fea78c57f7a58b95aa3e54f338b8a70e": "\\dot{x}_i = \\sum_{j=1}^n m_j e^{-|x_i-x_j|},\\qquad \\dot{m}_i = 2 m_i \\sum_{j=1}^n m_j\\, \\sgn{(x_i-x_j)} e^{-|x_i-x_j|}.", "fea7af938c204a067d442c7f7eece019": " y_{n+1} = y_n + 2 \\cdot \\frac{ x - \\exp ( y_n ) }{ x + \\exp ( y_n ) } \\,", "fea84ebeef24e84d295e8ad319434f22": "\\scriptstyle1.14^2\\approx1.30", "fea8bd6bd31a3cbbc55d6cdad204cd4b": "F_r < 1; \\; \\textit{Subcritical}", "fea90e5a47787e2432075729e1f1fb08": "f\\colon (X,\\mathrm{cl}) \\to (X' ,\\mathrm{cl}')\\, ", "fea92444e2f6df964d6df6a63f3cf905": "\\frac{d}{dt}\\frac{u(t)}{v(t)} = \\frac{u'(t)\\,v(t)-v'(t)\\,u(t)}{v^2(t)} \\le \\frac{\\beta(t)\\,u(t)\\,v(t) - \\beta(t)\\,v(t)\\,u(t)}{v^2(t)} = 0,\\qquad t\\in I^\\circ,", "fea952097d2300fcc0a038d3caa205ee": "b=0, d\\in \\mathbb{R}", "fea9862f9032e872e256f90bfd94ec8e": "d'\\geq d/2", "fea9887bf7c021964b8307652a4a7017": " \\Delta(m) = \\frac{\\psi_1^m(L)}{\\psi_2^m(L)}, ", "fea9df705a3ab89825d890bc2db912c5": "\n\\begin{align}\n\\cos((\\omega+\\alpha)t)+\\cos\\left((\\omega-\\alpha)t\\right) & = \\operatorname{Re}\\left(e^{i(\\omega+\\alpha)t} + e^{i(\\omega-\\alpha)t}\\right) \\\\\n& = \\operatorname{Re}\\left((e^{i\\alpha t} + e^{-i\\alpha t})\\cdot e^{i\\omega t}\\right) \\\\\n& = \\operatorname{Re}\\left(2\\cos(\\alpha t) \\cdot e^{i\\omega t}\\right) \\\\\n& = 2 \\cos(\\alpha t) \\cdot \\operatorname{Re}\\left(e^{i\\omega t}\\right) \\\\\n& = 2 \\cos(\\alpha t)\\cdot \\cos\\left(\\omega t\\right)\\,.\n\\end{align}\n", "feaa0243ab99ad787109460e9603321a": " \\tfrac{1+2+3+4+5+6}{6} = 3.5.", "feaa0e6a893e344e453f16db0afcadf7": "n-\\lceil n/r\\rceil", "feaa24a5891a95fef06473850110c533": "\\cos{\\theta}_{CB}*", "feaa393b76c9d7fcc49e615617dc54ad": "ke_k=\\sum_{i=1}^k(-1)^{i-1}p_ie_{k-i}\\quad\\mbox{for all }k\\geq0,", "feaa81a025b05b448898f90e3b48ec92": "\\alpha A \\beta \\rightarrow \\alpha \\gamma \\beta", "feaa9235d5ad65ac4d1cd964d21bdd97": "e=\\sqrt{4\\pi\\alpha}\\simeq 0.3", "feaa967806cf19adb3fda6971558d3eb": "e_{ij} = a_{i}b_{j}", "feaac7bcc24287b65c276087059fe119": "O\\left(2^{f(n)}\\right)", "feaacf820f84bf8168154f832de7e77f": "\\hat\\omega", "feaad7baaf9198ca407f26f56f318352": "^{3}\\Sigma^{-}", "feaaf931affa7c874ed411310c434ada": "E/C_E(X)", "feab05ff54ff630c8c1983b69db34137": "Y ~ \\xrightarrow{id_\\stackrel{~}{X} \\otimes \\eta_\\stackrel{~}{X} } ~ Y \\otimes (X \\otimes Y ) ~ \\xrightarrow{~a~} ~ (Y \\otimes X) \\otimes Y ~ \\xrightarrow{\\epsilon_\\stackrel{~}{X} \\otimes id_\\stackrel{~}{X} } ~ Y", "feab23264904da338ded815526366877": "(z_1,z_3; z_2,z_4) = \n\\frac{(z_1-z_2)(z_3-z_4)}{(z_3-z_2)(z_1-z_4)}.", "feab53e4f9b09412189feca4e03a0ba9": "\\hat{h}_E(Q) \\ge \\frac{C(E/K)}{D^3(\\log D)^2},", "feab9a084c7ed9395fca12a91e6c1597": " y_j^* \\leq y_L ", "feaba968160400a2ab63b5718ff4d783": "\n\\dot{x} = -\\frac{c}{q B}\\frac{\\partial V}{\\partial y}~.\n", "feabcc2bb22bdac01fb62732fb0ac62d": "d_1=r", "feabda8011eac0ffc1a8f89a7d3923db": "\\langle O\\rangle", "feac11685ad1d5b7d4b8873cad046e20": "x \\mapsto ( f(x) )_{f \\in C}", "feac5b6a4717896bcb98f40bb0873678": "q_0+iq_1+jq_2+kq_3", "fead4ed49cc1b6ff78b7d995a50cf97e": "\\neg\\;(x \\# x)", "fead7694460eafdfa300a619851a47e2": "r+\\epsilon\\leq|z-c|\\leq R-\\epsilon", "feae01bc0025804ccb15241916064313": "x\\le 1+1+1+1+1", "feae2472467b8cbd8f579e8bb507a975": "\\textstyle\\lambda_i", "feae32962bc795c87b22b43522623107": "\\scriptstyle\\Vert \\mu\\Vert=|\\mu|(X)", "feae46ce428653676226ee709491991c": "\\; I=\\{1, \\ldots, m\\}, \\cup_{j=1}^k I_j = I", "feae7a52cc0de2ee2077798d80e1ccff": "\\xi_1 >\\lambda>\\xi_2", "feaf3bf83e865bf459163143be60a9a8": "\\mu(G)", "feaf41fd2ceea1412e849ac39d91802b": "\\tau_{ij}", "feaf458c726c1ca90084471edb3d2199": "\\,\\Gamma_1\\times\\Gamma_2", "feaf57cb8c4b48c11fcd9101b9035bac": "\\textstyle x^{g(2l-1)}+1", "feafef493cf3bf7d9f0db74ca3036d2d": "G = \\mathrm{clamp}(( 298 \\times C - 100 \\times D - 208 \\times E + 128) >> 8)", "feb01ae477b77ff4ece9489cbbd2fe22": " i_t = r_{(t+1)} + \\pi_{(t+1)} + \\sigma ", "feb02c77760a7074b4d2a6ce8430cf5b": "S=\\frac{1}{20}(13211 + \\sqrt{174306161})a^2\\, ,", "feb0694364727ca0cf02cd473d049fa1": "{\\color{Blue}~6.10}", "feb07e6904cee15d925786f9411e15c5": "\\Theta_n", "feb10149ed2931806e8a609b21076300": "\\overline{\\mu_{k,i}}(w_1\\ldots w_{P(k)})=\\left(\\sum_{s\\in S_k, s_1\\ldots s_{i-1}(1-s_i)=w_1\\ldots w_i}\\mu_k(s)\\right)\\left(\\frac{1}{2}\\right)^{P(k)-i}", "feb15476347193a24377d034968fe229": "\\mathrm{unif}(a,b)", "feb18152d27dd15a2f494d538d8d567e": "\\rho(t) = e^{L t} [\\rho_0] \\;", "feb1b5512441c37db4d41b1c4b723054": "P/G\\to X", "feb1c7f5c6dbf8176ee8af4393ace3ec": "\\vec{x}_i", "feb1ff863d7c48f375431b836277daf4": "(p \\land q) \\vdash (q \\land p)", "feb2a4714ba9d7c953589cfa38875d78": "[\\![ \\textbf{nat} ]\\!] := \\mathbb{N}_{\\bot}", "feb2d32887357bfaad57fb030f388bb6": "x \\in S", "feb318087a1fa2c7d26b3b0c3bb58613": "\\|f\\| = \\sum_{i=0}^n \\sup\\nolimits_{x\\in [a,b]} \\left |f^{(i)}(x) \\right |", "feb32b0e0e08b9d36677df1bb04cc886": "\\displaystyle \\sum s_n(x)t^n/n! = \\frac{(1+t)^x}{1+(1+t)^\\lambda}", "feb331daaa95ee96a97967681c4a979b": "\\frac{\\partial L}{\\partial {c_k}} = \\lambda_k.", "feb3f9e40e66ecd0d02dc783a1fecc98": " k = p ", "feb41627ae74c67444724ae73256ad98": "1 \\leq i \\leq j \\leq n", "feb43738fb2eceadff9a04d5f23e9b6c": "y \\ge 0.966 - x", "feb48454dd7d24840a4cd06faaa63e72": "\n\\begin{align}\n& \\max_{\\tau_1,\\tau_2,\\ldots,\\tau_k} \\mathrm{ PCS} \\\\\n& \\text{subject to } \\sum_{i=1}^k \\tau_i \\ge 0.\\qquad (1)\n\\end{align}\n", "feb4bd9fdd8ab53ea278adf7d494e3a5": "R_{\\theta B}", "feb4ee006d8ab235cd0cae7a81db819f": "(X \\times Y, \\mu \\otimes \\nu, T \\times S)", "feb508be7444d5a3fd1159c3a1182325": "\\begin{matrix}\nL = \\frac{m l^2 }{2} \\dot \\varphi^2 + ml( g + a~\\nu^2\\cos\\nu t) \\cos \\varphi\\;,\n\\end{matrix}", "feb528319559ba3a1f0173d3127400f7": "-\\operatorname{ess.inf} \\delta X = -\\sup \\delta \\{x \\in \\mathbb{R}: \\mathbb{P}(X \\geq x) = 1\\}", "feb547a0139b627868e7e53aca60cfc7": "V_i = \\sum_{j=1}^N L_{ij} \\frac{\\mathrm{d}I_j}{\\mathrm{d}t} \\,\\!", "feb5c6d90d0f37fd2a1ce33c7f00678b": "v_p = \\frac{\\omega}{k}\\,", "feb5cc480153a5ce0968d92c27e9826b": "2C \\frac{n}{(\\ln n)^2} \\approx 1.32032\\frac{n}{(\\ln n)^2}", "feb5e49427557745856b8320b3187870": "\\dot Q_\\text{total} = \\sqrt{hPkA_c}(C_2-C_1).", "feb5e4e657cebb9cbdf8eec87b2a2bca": "M_z' (t) = M_z(t)\\,", "feb60e0a8e6c09b24ad6f281f298e0a5": "Z_{E,Q}(s)= \\frac{\\zeta(s)\\zeta(s-1)}{L(s,E)}. \\,", "feb7257e7f121a641e0c1f03d56bda89": "\\mathrm{Nat}(h^A,h^B) \\cong \\mathrm{Hom}(B,A).", "feb7424712aec701b59c5bcf361eb1ad": "\\mathrm{Gz} = {D_H \\over L} \\mathrm{Re}\\, \\mathrm{Sc}", "feb777c17a527a4e62f9fffa7da2af4c": "\n\\begin{alignat}{2}\n\\nabla \\cdot \\left( J_i + J_e \\right) & = 0 \\\\\n\\nabla \\cdot \\left( -\\mathbf\\Sigma_i \\nabla v_i - \\mathbf\\Sigma_e \\nabla v_e \\right) & = 0\n\\end{alignat}\n", "feb80314940c4ae9bb9870907a221285": "\\frac{141.5}{1.0} - 131.5 = 10.0^\\circ\\text{API}", "feb85030aae6c2238cb116caa51f773c": "\\begin{bmatrix}\nu \\\\ v \\\\ w \\end{bmatrix} = \n\\begin{bmatrix} \\cos \\chi & 0 & \\sin\\chi \\\\\n0 & 1 & 0 \\\\\n-\\sin\\chi & 0 & \\cos\\chi\n\\end{bmatrix}\n\\begin{bmatrix}\nu' \\\\ v' \\\\ w'\n\\end{bmatrix}", "feb8ba89201fdaa1302686985829d5d8": "K_{Ic} = \\sqrt{E^* J_{Ic}}\\,", "feb8ddd8ca7f9454b6bfc4dcb4c59c0a": "X(\\omega) = j \\omega \\cdot \\operatorname{rect} \\left( { \\omega \\over \\pi W } \\right) e^{j a \\omega}", "feb8faebfa37cc4e1058c1205d9e2f81": "\\gamma_1' = \\gamma_1, \\gamma_2' = \\gamma_2, k' = k^n", "feb93f477713bc95c7bb9c4c3674a3db": "[\\ F \\frac{\\quad}{\\quad} H\\ {}^-\\!F \\quad \\longleftrightarrow \\quad F^- \\ {}\\!H \\frac{\\quad}{\\quad} F\\ ]", "feb94218d222e82c55f9f960b06671fe": "\\omega_{1}^{tot}", "feb9dd7e0c9107dacc7df2fe6cd356ff": "StO_2", "feb9e404a462f121d52860a53101a938": "FS=\\frac{q_{allow}}{q_{reqd}}", "feba76141098012bd61d0d0c0c256e7e": "w = 1", "febae1f9ad87f4cef6c79193918aab46": "HS_A(t)=\\frac{P(t)}{(1-t)^d}", "febae2119bcf8903d89893e0a3c68c9b": " x = A^+ b ", "febb003c71c144be9a90558a338b19cd": "\\psi(x,\\theta)=\\left(\\frac{\\partial\\log(f(x,\\theta))}{\\partial \\theta^1},\\dots,\\frac{\\partial\\log(f(x,\\theta))}{\\partial \\theta^p}\\right)^\\mathrm{T}", "febb437c14c63e18581fad3499761f07": "(y - y) = (2x - x) +10 - 22", "febb517d5345b8d39aa13f2b7f155e9a": "G_k(X)=|\\gamma_k(X)|", "febb52da79e31dd77ac78075b60b4f2e": "0\\le \\Re(s)\\le 1", "febb549777ad9a7ac1071d2f1d666697": "\nT = T_0 - L h \\,\n", "febb941407278112003411ff5cf48c41": "B_{\\alpha \\beta }=0", "febc57a08ab1356526c4732a7298198b": " \\mathbf{T} = \\sum_{i=1}^n (\\mathbf{R}_i-\\mathbf{R})\\times \\mathbf{F}_i - \\sum_{i=1}^n k\\mathbf{F}\\times \\mathbf{F}_i. ", "febc93cf82eefe854a358a62784b60ee": "b=2mn(m^2+n^2), \\,", "febcb37271351c3559e4f309a5cbe86f": " \\phi_i = \\lang i|\\phi \\rang ", "febd14d9b92e96d39ee22159c6db29d9": "\\texttt{ZFC}\\vdash 2^{\\kappa}=\\kappa^{+}\\leftrightarrow\\neg\\texttt{AX}_{\\kappa}.\\,", "febd176890be6336fe1f29323ed644ca": "ac < bc", "febd22a95264f5a9b95e4b25a20dce0b": "\\phi({\\mathbf{r}})", "febd55bd3d22b7f3cdfbd7f35eafc3a6": " \\pi(A) \\pi(B) = \\pi(A \\cap B). ", "febd70162b5f639b317ade075ba9a251": " {T_{m}^{i+1}}= \\tau {(T_{m+1}^i - T_{m}^i )}+{(1-2\\tau)}T_{m}^i +\\tau \\frac {(e_{m} \\Delta {x}^2)}{k} ", "febd7fb8022bb978199a336184a443a1": " u(a \\otimes b) = u(a) \\otimes b - a \\otimes u^*(b),", "febdb2c55b13239998343d42f3d2f95a": "V:\\mathbf Y \\times \\mathbf Y \\to \\mathbb R", "febddd0cc60590284550bad7cae34004": "C^0(L) = L , \\, C^1(L) = [L,L] , \\, C^{n+1}(L) = [L, C^n(L)]", "febe28f40080a7ca8387acd2a093f42c": "\\mu_g(A) = \\overline{\\overline{I}}(\\chi_A)", "febe348274fb278a8f34f7505733774e": "h(p, v(p, w)) = x(p, w), \\ ", "febee9085ab2cdeb9971e59440fc0b85": "\\sigma\\in \\{1,\\dots,m\\}\\, ,", "febf04c5edfaaa34fe67fa1664b464ac": "V_\\xi", "febf1aca8daff779a61eac9340d2e8f9": "\n \\boldsymbol{\\varepsilon} = \\tfrac{1}{2\\mu}~\\boldsymbol{\\sigma} - \\tfrac{\\lambda}{2\\mu(3\\lambda+2\\mu)}~\\mathrm{tr}(\\boldsymbol{\\sigma})~\\mathbf{I} = \\tfrac{1}{2G}~\\boldsymbol{\\sigma} + \\left(\\tfrac{1}{9K} - \\tfrac{1}{6G}\\right)~\\mathrm{tr}(\\boldsymbol{\\sigma})~\\mathbf{I}\n ", "febf8f91f0a92e1ccd1286377e0e2dcf": " \\boldsymbol{\\mu}_S = \\frac{g_e\\mu_\\mathrm{B}}{\\hbar}\\boldsymbol{S}", "febf923df7e4dec4ab35fee126c0e62d": "du = d\\left({U\\over V}\\right)={dU\\over V}-U{dV\\over V^2}=-(p+u){dV\\over V} = -3(p+u){da\\over a}", "fec02e7b8fae8adb87fb5757bf7aff52": "K((T))", "fec085e430bec5a0602f1bc57c4bbdc3": "y=y_0 + \\epsilon^{1/4}y_1 + \\epsilon^{1/2}y_2 \\cdots", "fec0b3560ca576aee3fd5869fbe6eada": " c(r)=g(r)-y(r)=e^{-\\beta u}y(r)-y(r)=f(r)y(r). \\, ", "fec0bb9ba3307c475607794b00a5f37a": "\\alpha \\in K", "fec0c32fe0811a6620bdcfb6c72e618a": "\n (\\and L_1)\n ", "fec11d21b8a55946b6d6667346581cdf": "3\\cdot S_6,", "fec16d74ef8c93d6a65a9935168d577f": "x\\mapsto cx", "fec1f52d6a873da8e7cc350e098c318e": "\\varphi(\\alpha) = \\int_a^b f(x,\\alpha)\\;\\mathrm{d}x.", "fec254f20bd1f810d3e14a73a23244bf": "\\vec{\\Omega}=(0,0,\\Omega)^{T}", "fec27b4a1d6352a722209f74121783c1": "\\mathrm{proj}_{j}\\!", "fec2d1a5d51945201f13d6ebc05c79fa": " \\tau_n = y(t_{n+s}) + \\sum_{k=0}^{s-1} a_{n+k} y(t_{n+k}) - h \\sum_{k=0}^s b_k f(t_{n+k}, y(t_{n+k})). ", "fec2da713df043c3f05715dfcbdbce74": "E = \\frac{m \\left ( \\omega A \\right )^2}{2}e^{-bt/m} \\,\\!", "fec2eba9773d63d4108d102aca0327c5": " J = \\star (F \\wedge F) = F_{ab} \\, {\\star F}^{ab} = -4 \\, \\vec{E} \\cdot \\vec{B} ", "fec3424f31c4ad7252711ba9a0109ea9": "\\limsup", "fec3433ecf035726ac597bd97d933700": " g \\approx 2\\pi \\times 0.381966 \\approx 2.39996. \\,", "fec35c93d8b20a8a0198490fb1709233": "BE \\leftarrow BE \\cup \\lbrace e|e \\in C_i", "fec39f45d8a897bfebb2a2b62dc4be5f": "+_n", "fec3b0969860f24d721287dfe7ff5a26": "L_2 \\frac{2 \\sqrt{3}}{2 x} = \\frac{3}{1}", "fecce7262321c6dac75ac9fbe749af9c": "B = Q_2 x_2 \\cdots Q_n x_n \\phi(1, x_2, \\dots, x_n).", "feccfcb283756495bd14607158785fdd": "T:V\\to V", "fecd04ec305c6b5a26aa3afd4df13288": "F+G = \\Big\\{A\\subset\\mathbf{N}\\mid \\{n\\in\\mathbf{N}\\mid A-n\\in F\\}\\in G\\Big\\};", "fecd25da49f77e9a23517b9668e4f42b": "H = \\frac {G^2} {A}.", "fecd34ca5cb5e4de221da5a5aa2c3bd9": "\\begin{align}\n\\limsup_{n\\to\\infty}X_n &:= \\lim_{n\\to\\infty} \\sup\\{X_m: m \\in \\{n, n+1, \\ldots\\}\\}\\\\\n&= \\inf\\{\\sup\\{X_m: m \\in \\{n, n+1, \\ldots\\}\\}: n \\in \\{1,2,\\dots\\}\\}\\\\\n&= {\\bigcap_{n=1}^\\infty}\\left({\\bigcup_{m=n}^\\infty}X_m\\right).\n\\end{align}", "fecd6b0994fc10d9ac20b66fb0e2766d": "\\pi ^{-1}x^{-1/2}(1-x)^{-1/2}", "fecd942d3e6c14473cb1cb82475e0dc5": "1/\\phi", "fecdaaa2c1cde0d4ca2468318a2c4669": "\n U_{ij}\\mathbf{b}^i\\otimes\\mathbf{b}^j = S_{ik}T^k_{.j} \\mathbf{b}^i\\otimes\\mathbf{b}^j= S_i^{.k}T_{kj}\\mathbf{b}^i\\otimes\\mathbf{b}^j\n ", "fecdbf6b5b33896a21aceb913a60d361": " [\\text{M}^+] = {k_i[\\text{I}^+] [\\text{M}] \\over k_t}", "fecde358b68eec12c4d620579e0a69a9": "W_i = \\frac{Average Capital_i}{Average Capital_1+Average Capital_2}", "fece0cd380633b4261b128910af40075": "V(x) =\n\\begin{cases}\nV_0, & 0 < x < L,\\\\\n0, & \\text{otherwise,}\n\\end{cases},\n", "fece4d4c637684e54a01526047d47d5d": "\\sin A = \\frac {\\textrm{opposite}} {\\textrm{hypotenuse}} = \\frac {a} {h}.", "fece898aafb2d6128854b1962aa2bb2f": "\\begin{alignat}{7}\na_{11} x_1 &&\\; + \\;&& a_{12} x_2 &&\\; + \\cdots + \\;&& a_{1n} x_n &&\\; = \\;&&& 0 \\\\\na_{21} x_1 &&\\; + \\;&& a_{22} x_2 &&\\; + \\cdots + \\;&& a_{2n} x_n &&\\; = \\;&&& 0 \\\\\n\\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && && \\vdots\\;\\;\\; && &&& \\,\\vdots \\\\\na_{m1} x_1 &&\\; + \\;&& a_{m2} x_2 &&\\; + \\cdots + \\;&& a_{mn} x_n &&\\; = \\;&&& 0. \\\\\n\\end{alignat}", "fecea3876a1f6ba309fdd8f12d21820c": "(a + b)_n = \\sum_{{j=0}}^n {n \\choose j} (a)_{n-j}(b)_{j}", "fecec5b6d5e9ca2d939f47d15bb9f56e": "\\mathbb{E}", "fecf0ee0d41434f0d6829888ca31ecbc": "\\pi_i(t)", "fecf44277f1032838ebca633ef764b3e": "Y_x", "fecf7c9391366a52d3318d1c2fe6100c": " |z|^2 - b^*z - bz^* + c,\\,", "fed041da3c70392264238c115b9e000c": "i\\frac{\\partial}{\\partial t} \\psi (x, p) = \\hat{L} \\psi (x, p),", "fed05d7657c995e2e8212b9efd5774fd": "|\\bold{r}|\\rightarrow\\infty", "fed1e4775925bd3f7af0c5d8fc47e4e6": "m_0", "fed1fd92f54c4a576f22239ac59eb28b": "p^{R(p,n)} \\leq p^{\\log_p{2n}} = 2n.\\ ", "fed2463bb21d431b6ec8368e5105da35": "\\frac{t'}{2} = \\frac{h}{\\sqrt{c^2 - v^2}}", "fed246b1f694d5e9de114262070a4f5f": " k_n = a_1 +2^n a_2", "fed29404dad15c2abfd8f55f12d9d14f": "\\infty, \\aleph, \\complement, \\backepsilon, \\eth, \\Finv, \\hbar \\!", "fed2af00202691e71ad8de11d174417e": "N\\oplus C \\subseteq_e M", "fed2f8bb5c00eaf3419e416de16ec91b": "F(I)", "fed3009ab9e6c6b28d4f20dc225b4fd5": " X_k = e^{-\\frac{\\pi i}{N} k^2 } \\sum_{n=0}^{N-1} \\left( x_n e^{-\\frac{\\pi i}{N} n^2 } \\right) e^{\\frac{\\pi i}{N} (k-n)^2 }\n\\qquad\nk = 0,\\dots,N-1. ", "fed35346e71f92d4ab621e14a710cf3d": "\\mathbf{P}(\\chi^2_{k-1} \\ge Q)", "fed3b49704ea5b428b227ce4ab7542f5": " \\omega_B \\!", "fed3c16908db3fe85db151671cf52aa0": " y_{n+1} = y_n + hf(t_n,y_n). \\qquad \\qquad", "fed4921ad4ca21a84ba33c8f29818965": "x=[a_0;a_1,a_2,a_3,\\ldots],\\ ", "fed53508700661d8c1fdb4f02fe19bd3": "\\alpha=\\sum_{|I|=p,|J|=q}\\ f_{IJ}\\,dz^I\\wedge d\\bar{z}^J\\in\\Omega^{p,q}", "fed55a6fe8ff70e5770911b6a99a245d": "f(a\\mathbf{x}) = af(\\mathbf{x}). ", "fed569c4935426a9ee6e147e48d7b65a": "Q(\\alpha,\\alpha^*)=\\frac{1}{\\pi}\\langle \\alpha|\\hat{\\rho}|\\alpha\\rangle =\\frac{1}{\\pi}|\\langle \\alpha_0|\\alpha\\rangle|^2 = \\frac{1}{\\pi}e^{-|\\alpha-\\alpha_0|^2}", "fed573145621b04954c2b18cfbb25050": "\\frac1x \\div \\frac1y = \\frac{y}{x}.", "fed5b9cb8829b9fb89b18c621153e828": "\\!y = \\frac{8a^3}{x^2+4a^2}.", "fed6027ebb25041ce83b14f8d9b21ac8": "\\scriptstyle p(x,y)=p(0,x-y) ", "fed62282a40e2ef6b64a04268189126f": " a(n) = \\lim\\limits_{s \\rightarrow 1} \\zeta(s)\\sum\\limits_{d|n} \\mu(d)\\exp(d)^{(s-1)}", "fed68885867708e000c5ee178959cde6": "f(x_i)=f(x_0+i h)", "fed70a1f068ed8a3eb2d07f8049de3d2": "k'h'", "fed7342b183f87a1a3682d6af6b595e2": "\\mathbf{H}=\\mathbf{H}_0 - \\frac{\\gamma}{4\\pi} \\mathbf{M}_0,", "fed73df68065417cbf616b0e6d0866bf": "0, ...,n", "fed7a86961761153eeedb3ebaf580f54": "2T_5 + 3T_5 + 4T_5 + \\cdots + (n-1) T_5 + n T_5 + (n + 1) T_5", "fed7ee545d292f7e0122d86369fb925c": "\\{d_i\\}_{i=1}^N", "fed7fe6d92c05c6d3a563d00e18bcab5": "\\Gamma^\\nu_{\\sigma \\mu} = {1 \\over 2} g^{\\nu \\delta} (\\partial_\\sigma g_{\\delta \\mu} + \\partial_\\mu g_{\\sigma \\delta} - \\partial_\\delta g_{\\sigma \\mu})", "fed8071792c18e7a8e36eafb4558613c": "\\tau ::= \\tau \\to \\tau \\mid T \\quad \\mathrm{where} \\quad T \\in B", "fed88039ea056b75f57479bb946bce68": "\\left\\| \\cdot \\right\\|", "fed8c76f1e07e5a97bd3106cd73be0a8": "G\\in\\mathcal{G},\\quad G\\subset H.", "fed8eb5a8c015dd7ecbb4efae6404e97": "\\chi(\\lambda) = \\lambda^4 + a_3 \\, \\lambda^3 + a_2 \\, \\lambda^2 + a_1 \\, \\lambda + a_0", "fed8fec8c9c53ac1a334bdc8ac00a0ba": "ax^2+bx+c=0,", "fed9986a38b3ebea4713cbcf122b2c54": "(a,b) \\cdot (c,d) = (ac,bd)", "feda0eb8989ed589d3f4ed708f6fd921": "0-1, 1-10, 10-10^{10}, 10^{10}-10^{10^{10}}, 10^{10^{10}}-10^{10^{10^{10}}}, \\dots", "feda6612dcd58dfd07994ce7093d23bb": "I_\\mathrm{3} = \\frac{1}{2}\\left[\\left(n_\\mathrm{u} - n_\\mathrm{\\bar u}) - (n_\\mathrm{d} - n_\\mathrm{\\bar d}\\right)\\right]", "feda7aedb992a5021ae2a2b2f0c0cef1": "\\omega >> \\omega_c", "fedab7c95cadf749510509962b9ab889": "K(\\mathbf{x}, \\mathbf{x'}) = \\exp(\\gamma||\\mathbf{x} - \\mathbf{x'}||_2^2)", "fedaf6e926a4e15fdae7c766379268c7": "\\scriptstyle\\sqrt{n}\\|\\hat{F}_n-F\\|_\\infty", "fedbc7ddc337854a29da216fc2a2c534": " S(\\rho^{12})=-{\\rm Tr}_{\\mathcal{H}^{12}} \\rho^{12} \\log \\rho^{12}", "fedbe4dc0d3b0d2780eeb0558c8b03d9": " \\sigma_y ", "fedc5f418f1f951749b58ead1cb50cf9": "{\\color{Blue}~2.13}", "fedcb10e27636435dabbd0adf96d6835": "i\\not = j.", "fedd4474705f248acd3a35f89dcb7ad4": "P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)}.\\,", "fedd9b0fae9fd1b27485d15734d14c86": "\\hat{x}_1", "fedda299b9bfcd4ad46503672591f73e": "e_2 = \\mu f_1 \\,", "fede50e5cf93941ef4f24379e71743a7": "V_2 \\sim {\\chi'}_{k_2}^2(0)", "fede865e868fd1b5c3ce55de0dc49513": "(x^2-4\\alpha)y'' + xy' - n^2y=0 \\, ", "fedea7c444278c6f595de0ade5bac92f": "\\mathrm{d}U = T\\,\\mathrm{d}S - \\sum_{i}X_{i}\\,\\mathrm{d}x_{i} + \\sum_{j}\\mu_{j}\\,\\mathrm{d}N_{j}\\,", "fedeb8accca1db52c951acc93293bc03": "(2r,r+4\\sqrt{3}r/3,r+\\sqrt{6}r2/3),\\ (4r,r+4\\sqrt{3}r/3, r + \\sqrt{6}r2/3),\\ (6r,r+4\\sqrt{3}r/3, r + \\sqrt{6}r2/3),\\ (8r,r+ 4\\sqrt{3}r/3,r + \\sqrt{6}r2/3),\\dots. ", "fedec3043612c75e819f456e3c7c3467": "\nds^2 = -c^{2} dt^2 +\nR(t)^2 \\left( \\frac{dr^2}{1-k r^2} + r^2 d\\theta^2 + r^2 \\sin^2 \\theta \\, d\\phi^2 \\right)\n", "fedf1b9c1d530151775ac008e6589ed8": "\n\\left(c \\frac{d\\tau}{dq}\\right)^2 = - g_{\\mu\\nu} \\frac{dx^{\\mu}}{dq} \\frac{dx^{\\nu}}{dq} = \n\\left( 1 - \\frac{r_{s}}{r} \\right) c^{2} \\left( \\frac{dt}{dq} \\right)^{2} - \n\\frac{1}{1 - \\frac{r_{s}}{r}} \\left( \\frac{dr}{dq} \\right)^{2} - \nr^{2} \\left( \\frac{d\\varphi}{dq} \\right)^{2}\n\\,.", "fedf305a777d0f0e125cadf920c20145": "|V_0|\\gg E_f", "fedf405ca38253febcfd767a6b600d29": "r=\\|\\mathbf{x}\\|_2", "fedf465ba3156d460ceca05ee6f6cfca": "\nt = \\frac{e}{s_e}\n", "fedf5ec10eee48867e8555bf77f20edd": "f_{l+1} \\circ d_A + (-1)^{n+1} d_B \\circ f_l", "fedf68fdee0631e68ec3069b0d328f09": " e = - NS\\mu_{app}\\frac{\\mathrm dB}{\\mathrm dt}", "fedf72d1139aea6490edac7b257c39d8": "f\\leq^*g", "fedf8fa46a15fe38d18fb2d3f6feaf35": " c^{2}(H^{1}|_{E})<\\infty", "fee018c8dbc3a21c790e49114bd4f01c": "-i\\ln(x+i\\sqrt{1-x^2}), \\, ", "fee055b62470bc8713ed312fb67bbc55": "A\\cup B", "fee099dca9768197acbdaddae23d1c45": "{\\mathcal M}z^1=z^2,", "fee0c20837936d6c74b79b03ade8f700": "\\Phi ::= tt \\,\\,\\, | \\,\\,\\,ff\\,\\,\\, | \\,\\,\\,\\Phi_1 \\land \\Phi_2 \\,\\,\\, | \\,\\,\\,\\Phi_1 \\lor \\Phi_2\\,\\,\\, | \\,\\,\\,[L] \\Phi\\,\\,\\, | \\,\\,\\, \\langle L \\rangle \\Phi", "fee0c78c7802ce88f28c98d83d911871": "\\frac{\\Delta y}{\\Delta x}=\\frac{f\\left(x_1 ,x_2 ,\\ldots ,x_{i,1},\\ldots,x_n \\right)-f\\left(x_1 ,x_2 ,\\ldots ,x_{i,0},\\ldots,x_n \\right)}{x_{i,1}-x_{i,0}}", "fee0ca29a0a10f6d3074530a447736fb": "x\\rightarrow a\\text{ or }b", "fee0f7e5c763f7b04912e7512f20c96f": "\\operatorname{vec}", "fee11eaa80eba359cb390f4a717f05fe": "\\mathrm{Aut}(S_n)", "fee12c50661a005d060ec13ee304f371": "e=\\sqrt{\\frac{2\\sqrt{(A-C)^2 + B^2}}{\\eta (A+C) + \\sqrt{(A-C)^2 + B^2}}}", "fee13641d2f9c521be7cdb04d6ce01cd": " || f ||_\\mathcal{H} \\le 1 ", "fee1367f3f85c7e62c7e5d697033f631": "\\textstyle (p_k, \\pm q_k)", "fee16cab17df729e6a2b82da12b2c9d3": " \\Omega = -kT \\ln\\Big(1 + e^{\\frac{\\mu - \\epsilon}{k T}}\\Big).", "fee1996ffca29652889205bca14b5064": " 0.2 \\leq Re \\leq 1000", "fee1b5a5797f8e5199a2ab1232090334": " \\theta = \\theta_0 ", "fee25e9ec9a70756ab8c306334f04577": "(H-I)", "fee2c10ac861a517c52660a9fc470bc1": "(x)_n = \\sum_{k=1}^n (-1)^{n-k} L(n,k)x^{(k)}.", "fee2cd14657fe8db1814c0dfbe0240d2": "\\tau = \\frac{L}{C}", "fee3069298c8c8cc7c6188c6d5702eb0": "f\\biggl(\\bigvee_{i \\in I}{x_i}\\biggl) = \\bigvee_{i \\in I} f(x_i)", "fee330241502011c4e475615c7aa6d34": "W_{1-2} = \\int PdV = P(V_2 - V_1)", "fee35ef64ffb9193354c367afcf674bb": "\\delta(q,a,\\gamma) ", "fee3c3fb525c0e1e89d42ec6368ed261": "\nR_1 = \\max_{0 \\leq \\beta \\leq 1} \\min \\left\\{ \\frac{1}{2} \\log(1 + (1 - \\beta) c^2_{21} P_1), \\frac{1}{2} \\log(1 + c^2_{31} P_1 + c^2_{32} P_2 + 2 \\sqrt{ \\beta c^2_{31} c^2_{32} P_1 P_2}) \\right\\}\n", "fee4019fe1c3e1d6d7695bdc4bcba266": "W_n^\\alpha", "fee40aebd4b7048c7d3653cc8cba46f6": " (u^2 + \\alpha)^2 = \\alpha u^2 + \\alpha^2- \\beta u - \\gamma. \\qquad \\qquad (2) ", "fee48c72ae997f3b79379ccfa94ca4c4": "\\mathbf{B}_i=\\mathbf{P}_{ins,i} \\exp \\left ( -\\frac{\\left \\langle \\psi_i \\right \\rangle}{k_B T} \\right ) ", "fee4c76c013371fa3c6133e47af2f4aa": "\\operatorname{Tr}(\\rho Q)", "fee4f64f3531857bacdaafa1e7d60e75": "x^2+dy^2=m", "fee54137ee7748e26642e71145effa05": "\\textstyle R", "fee56d413fbe0a89f08835cee9842cf0": "\\mathbf{\\bar{A}}_{l}", "fee592f0ad0003b7a1a6cbcadb6fb79d": "\\left\\{\\,\\aleph_n : n\\in\\left\\{\\,0,1,2,\\dots\\,\\right\\}\\,\\right\\}", "fee5996da583589ddeb158e3e8269c47": "P\\in E(\\bar K).", "fee60553736aa8eb1cf34baa67e9c1b9": "y_\\mathrm{obs}", "fee617f53747caf8020dd59396b2459d": "~ D=\n\\sigma_{\\rm pa}\n\\sigma_{\\rm se}\n-\n\\sigma_{\\rm pe}\n\\sigma_{\\rm sa}\n~", "fee66aabdd3e4623e2472703cb2af7e1": "\\vec{n}=\\frac{\\vec{r}_u\\times\\vec{r}_v}{\\left|\\vec{r}_u\\times\\vec{r}_v\\right|}. ", "fee68de3d7769416c7262c4b94f9ac1c": "(\\epsilon_{i+1}-\\epsilon_i)", "fee69989f4a16376afc2a35094ec5540": "FRC", "fee6b18bf380a6bbfe26d9e4fdad969b": "\\prod_{i \\subseteq \\Sigma}\\mathcal{M}_i/U", "fee6c468495ef91b088a9584b9bd8b57": " g(\\mu)\\to g* ", "fee715dd735549344be4da055e97bc1c": "U x", "fee718249d3f55bfb24a591f575097b1": "a^4+b^4+(a+b)^4 = 2(a^2+ab+b^2)^2", "fee7217fc27bbe471a833c74581f3e3d": "y = \\sqrt{1 - x^2}", "fee72261f844aa6ffb1653fcf801b3ad": "\\beta=45^{\\circ}", "fee736d0bd9a4dc98f8be2ccc3f3d1b5": " Z \\subset U_N := \\bigcup \\, \\{ 5 \\, C : C \\in G_n, \\, n > N \\}", "fee7ef5fabb629c585e4d3162dae5947": "\\textstyle \\rho g(\\mathbf{r}) \\mathrm{d}\\rm{r} = \\mathrm{d} n (\\mathbf{r})", "fee81fb6681fe0a0601293dae3048ed2": " P_{arriving on sail} =\nP_{kinetic} = \\frac{1}{2}.\\rho.S.v_{incident}^3 \\,", "fee83435daa52860a34614848b814bf9": "\\int_{x'\\leq c, y' \\in \\reals} f(x')g(y') \\, dx' \\, dy'.", "fee863f4bb3c96e2efd96d3b3694621c": "\\operatorname{Var}[Z(n)] = \\frac{\\operatorname{Var}(Z_j)}{n}. ", "fee864374baf208128ab95758bf08f2f": "\\overline{T}_{del,i}", "fee8b693d917e97ca1864278461b4f15": "\\textstyle W^n (y | x, s)", "fee919792321b7a2004c8c9e55de34f3": " \\hat{f}(a) = |S_0(a)| - |S_1(a)| = 2 |S_0(a)| - 2^n. ", "fee9613c4c57044904af0c6e64555bd4": "u(t)\\,", "fee9712002cbe51360df82d5ef7b1f9d": " 2=3-1", "fee984cb27d685a1d86c3c68266fd298": "\\forall x \\forall y\\forall z\\;x \\vee (y \\wedge z) = (x \\vee y) \\wedge (x \\vee z)", "feea617a6d9052a68613d7ea2beb2034": "\n\\begin{align}\n\\Delta v \\ & = v_\\text{e} \\ln { 100 \\over 100 - 80 }\\\\\n & = v_\\text{e} \\ln 5 \\\\\n & = 1.61 v_\\text{e}. \\\\\n\\end{align}\n", "feea8aa2c77edf8b247b43d68faa671e": "\\begin{align} 2\\cdot R_*\n & = \\frac{(10^{-3}\\cdot 31\\cdot 1.02)\\ \\text{AU}}{0.0046491\\ \\text{AU}/R_{\\bigodot}} \\\\\n & \\approx 6.8\\cdot R_{\\bigodot}\n\\end{align}", "feeae20f75fbb8ec7f622aa0813a5084": "\\rho_A = \\frac{M \\;P^{\\;(k-1)/k}}{R \\;T \\;P_A^{\\ -1/k}}", "feeb7a07c1906cf91671b765a14851a3": "u = e^{av} ,\\quad v \\in I \\quad \\text{or}\\quad u = e^{ap} ,\\quad p \\in J ", "feeb96ffb23dd6c32e6767462c82ed7c": "\\left(x^n\\right)'=nx^{n-1}.", "feebc86c805a4f50692f228e7bc33ea0": "\\psi(\\Omega 2)", "feebdb57c35c736fe2380ccffbc8683d": "\\left \\Vert \\cdot \\right \\|_2", "feec03606330c3ed9d0f79d9876a86f2": "A \\leq B \\lor B \\leq A", "feec11d98691193a10ab7e0727246cda": "\\forall r=1\\ldots m ", "feec58ba64af0cafa52cd25ae02a4ad3": " {F(t) \\over m} = \\begin{cases} \\omega _0^2 & t \\geq 0 \\\\ 0 & t < 0 \\end{cases}", "feec87022af6fc3f1e6829d451c10cb0": "\\text{length of solar day}=\\frac{\\text{length of sidereal day}}{1+ \\tfrac{\\text{length of sidereal day}}{\\text{orbital period}}}. ", "feeccf2fd45b60b23c43d8e7019f40cd": "\\leq n", "feecf5d0dc4fa9a854054ef7fa3b6789": "Y=\\sqrt{2X\\sigma^2\\lambda} \\sim \\mathrm{Rayleigh}(\\sigma) .", "feed949af9d9b6c3423e66a11ce4323a": "\\phi_{1}", "feedae0e1daf4253e4351fa4a91662ba": "y(0)=1, \\qquad \\frac{dy}{dt}(0)=0,", "feedd7ca6089502563395a0ad74c30c8": " \\mathcal{I}_f = \\left \\{ f^{-1}\\left(\\left( - \\infty, x \\right]\\right) \\colon x \\in \\mathbb{R} \\right \\} ", "feedd8cedcdab28d9b5fcaf45bf16d03": "\\gcd(p, q)= \\gcd(q,p+rq)", "feee0bf7c24bc45808eeb7f99034d64a": "{}^xe", "feee1ab7b3f0e190d754f9f2f49766be": "C(\\beta)", "feee74e0cef56ebfe1ab9522cdf81221": "h(z)=z^2 - 1 - \\frac{e^{\\sqrt{5}\\pi i}}{4} ", "feee92345cae67705cab8a0efd4fa15f": "D_{ij\\, \\alpha \\beta}", "feeeb6ef6b90c8665fd4196fc306f5fe": "\\gamma = 2 \\nu\\,", "feeed0f2f0d74da7849bfe961fc91fa0": "\\sum_{k=0}^{n-1} \\csc^2\\left(\\theta+\\frac{\\pi k}{n}\\right)=n^2\\csc^2(n\\theta)\\,\\!", "feeef9d8fbb11dd2a96cbf9afe924937": "k < i", "feef1fe6b3144ddec888a4c7fdfc8ddc": "S = \\frac{1}{2} \\int e e^{\\alpha}_{\\ I} e^{\\beta}_{\\ J}\n\n(F_{\\alpha\\beta}^{\\ \\ \\ IJ} - \\alpha \\ast F_{\\alpha\\beta}^{\\ \\ \\ IJ})\n\n\\equiv \\frac{1}{2} \\int e e^{\\alpha}_{\\ I} e^{\\beta}_{\\ J}\n\n(F_{\\alpha\\beta}^{\\ \\ \\ IJ} - \\frac{\\alpha}{2} \\epsilon^{IJ}_{\\;\\;\\;KL}\n\nF_{\\alpha\\beta}^{\\ \\ \\ KL})", "feef5ef9ed5881a965a7ee09f1ea4cba": "f_i^{(j)}(\\alpha_k)", "feef79595c66cf93552a8857a1898b71": "\\,K", "feef95dc7d950a6f9386292f8e7b9808": "\n\\langle \\mathfrak{p} \\rangle = \\langle\\, \\mathfrak{I}^{-1}\\, S\\, | \\mathfrak{p} |\\, \\mathfrak{I}^{-1}\\, S \\,\\rangle\n = \\langle\\, S\\, | \\mathfrak{I}\\, \\mathfrak{p} \\, \\mathfrak{I}^{-1}| \\, S \\,\\rangle = -\\langle \\mathfrak{p} \\rangle\n", "feefc173f4c5dd0b1417570034d146c9": "V_\\mathrm{out} = (V_+ - V_-) \\cdot G_\\mathrm{openloop}", "fef0c176180631d499db510b541c7583": "y\\quad", "fef0f5468c2c71b217886acc5e3e0f4c": "-4\\pi/(k^2+m^2)", "fef135fb95c3761667b5c8e647244cc1": "(s \\rightarrow (p \\rightarrow q)) \\rightarrow ((s \\rightarrow p) \\rightarrow (s \\rightarrow q))", "fef1595156bf19ee30d5c857f18f15b3": "\\tau_\\mathrm{max} = \\frac{1}{2}(\\sigma_1 - \\sigma_2 )\\,\\!", "fef194be46390ba38e9de4c09273f6ef": " \\begin{align}\n|Z| &= h(X,Y) = \\sqrt{1-X^2-Y^2}; \\\\\n\\mathrm{E} ( |Z| | X=0.5 ) &= \\int_{-\\infty}^{+\\infty} h(0.5,y) f_{Y|X=0.5} (y) \\, \\mathrm{d} y = \\\\\n& = \\int_{-\\sqrt{0.75}}^{+\\sqrt{0.75}} \\sqrt{0.75-y^2} \\cdot \\frac{ \\mathrm{d}y }{ \\pi \\sqrt{0.75-y^2} } \\\\\n&= \\frac2\\pi \\sqrt{0.75} .\n\\end{align} ", "fef1953b6624ff7e32aab0b01a0e5ffd": " a \\in W^u ", "fef1c7a60d64b7443a05a38ad4fed3f4": "f_1(c) = \\|x-c\\|_1", "fef1d25edf7b666cb768f220108d756d": " \\frac{\\Delta K}{K} ", "fef1d735cc4a5b5db870a0a85e1ae29c": "\\textrm{E}(Y_i)=p", "fef1ea2a88d90dff100f4faeb9800186": "\\scriptstyle\\sqrt{u_{11}},\\scriptstyle\\sqrt{u_{22}},...,\\scriptstyle\\sqrt{u_{11}}", "fef24d302147049c336cf33fc46df438": "(x_0,|\\gamma|)", "fef2654a9a99373d038b20aee8351539": "\\frac{2n+1+\\alpha}{n+1}\\,", "fef27623febd2ae03afe7a22d034ac5b": "a=20", "fef278c58f7a7dad1c9a5336133e7c7b": "\nr' = \\frac{1}{NM} \\sum_{i=1}^N \\sum_{j=1}^M r_{ij}^{(t)}\n", "fef285f330ec8144dca2a033bcf0cded": "P \\vdash \\neg \\neg P", "fef2f1793b061a091cff58b7705ac264": "\\mathbf{F} = q \\mathbf{v} \\times \\mathbf{B},", "fef32080de70007ff06490dc6b76c762": "\\boldsymbol{\\mu} \\equiv \\sum_{j=1}^N q_j \\mathbf{r}_j", "fef3331dff2171d41be043bedef12084": "\n\\sigma_i=\\sqrt{2\\,q\\,I\\,\\Delta f}\n", "fef3640ea5e3e01368585d06b92000c2": "\nR^{h}_{ijk}=-\\Delta^{h}_{ij/k}+\\Delta^{h}_{ik/j}+\\Delta^{h}_{mj}\\Delta^{m}_{ik}-\\Delta^{h}_{mk}\\Delta^{m}_{ij}\n", "fef3746b5c9f2b01c03438bbeacc4c56": "t_1 = 12", "fef3a407187c6ecf64842f40d842e8bc": "\\dot{x}=f(x,y)", "fef4085541027d7e0a6bcd5a7a68df84": " \\omega_a \\ ", "fef49bdbc6db014a88c8b065f6266086": "G_2=\\{1\\}", "fef4db72fabc0c1e2c233c1175873d7b": "H_{\\nu} (\\omega)= \\frac {1} {\\sqrt {2}} \\sum_{k \\in Z} h_k^{\\nu} e^{-j \\omega k}", "fef5162e6efdfbddcc5d7bd0a9d933b1": "\\xi(B_1),\\ldots,\\xi(B_n)", "fef526889c102a4925dad860dc79f7b3": "\\frac{3}{2} \\sqrt{3}s^2\\,\\!", "fef54aa6418fa6a9d780b0b2a03d1dc6": "M_{k}^{2}", "fef56c07868e991e5435e26e4e96cf34": "K: X(t) = N(t)+s(t), t\\in(0,T)", "fef592b572e5d2ad774184a32a8f714e": "\\scriptstyle|0\\rangle", "fef5cd5a28b541eb9aba4c6bf5552c29": " {P} ", "fef6344f35808643643338c696ed38ce": "R=\\infty", "fef6e78618c8757104de5fcab7f0bbd1": "\nK = H + \\frac{\\partial G_{1}}{\\partial t}\n", "fef70e22280f6e502fd038829b77b2af": " T^*_x X ", "fef7a8d8a7175d17204f619e3599250d": "V = k_0 a \\sqrt{n_1^2 - n_2^2}", "fef7d1a2ec6a601fbe38ee03821a1145": "i \\rightarrow i\\pm 2", "fef7d770b02e9f1261d7f641eec521e4": "D_6", "fef7e2caee7e7eb837f894e4cffb50ff": "\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1", "fef7ee9478db82911119f7e1fcf2c577": " -2~r^{-3}~\\sin\\theta \\,", "fef7fa5db48b69366141e119de3076be": "\\operatorname{\\Gamma L}(V).", "fef8111e3f5e58eeb294ffaf38e31fe6": "|H_0(e^{j\\Omega})|^2 + |H_1(e^{j\\Omega})|^2 = 1", "fef848d4f2ec237e780e92afd7451cb3": "\\,\\lim_{n\\rightarrow\\infty}\\frac{n^xn!}{(x+n)(x+n-1)\\cdots(x+1)x}\\,", "fef97b1b3c718b080b7473af92e08285": "(x_2,y_2)\\,", "fef9a7b48ae83f87d750d55fd892d470": "x \\in \\{l,r\\}", "fef9ac9a4afeaef616423bfd95b06059": "{Rg_2^*\\ \\xrightarrow[]{\\tau}\\ Rg + Rg + h\\nu (UV photon)},", "fef9ead732720db9b33c39f2beb0d2d1": "V_n = \\frac{\\pi ^ {n/2}}{\\Gamma(1+n/2)} = \\begin{cases}\n{\\pi^{n/2}}/{(n/2)!} & \\mathrm{if~}n \\ge 0\\mathrm{~is~even,} \\\\\n~\\\\\n{\\pi^{\\lfloor n/2 \\rfloor}2^{\\lceil n/2 \\rceil}}/{n!!} & \\mathrm{if~}n \\ge 0\\mathrm{~is~odd,}\n\\end{cases} ", "fefa040f988f7f26f638dffc8c7b6aa4": " \\omega = \\sqrt{\\det g}\\; du_1 du_2 = r^2\\sin u_2\\, du_1 du_2.", "fefa0ee6e443b49675e9552122de5e07": "\\{b_k\\}_{k=1}^{M-1} ", "fefa46f929fd1cc6f7dba0f058a1b531": "(\\alpha,n)", "fefa95d4b3d1ef8fbc2920255a580616": "\\oint_{\\Gamma} \\mathbf{F}\\, d\\Gamma = \\iint_{S} \\nabla\\times\\mathbf{F}\\, dS ", "fefaa31912afa443005b5a6f17692bec": "x_{\\lceil h\\,-\\,1/2 \\rceil}", "fefaa81cc756a129c097f387f69f38a3": "f(x\\mid \\mu,\\sigma^2) = \\frac{1}{\\sqrt{2\\pi}\\ \\sigma\\ } \n \\exp{\\left(-\\frac {(x-\\mu)^2}{2\\sigma^2} \\right)}, ", "fefae154b258184acd10a28e7f72bfad": "\\Omega_i(G) = \\langle \\{g : g^{p^i} = 1 \\} \\rangle. ", "fefb030a0c15584e881d8c3bedd3de08": "L_y = \\partial_y L", "fefb19183bff83a05ea8a98b793adab3": "Ax = b", "fefb309f23d4367ce39ee4bc2c1e5576": "\\xi=\\frac{\\Delta n_i}{\\nu_i}=\\frac{n_{equilibrium}-n_{initial}}{\\nu_i}", "fefc08121fc6a0191bd9c82410625c67": "u_{c}= L/t_{c}", "fefc7e4079fc2d19d2f14023fd270eb8": "N_w \\equiv {1 \\over 2 \\mathrm{NA}_i} \\approx (1+|m|)N", "fefc8c476c63aff316a84ffc027dbc6c": " \\mathbb{Z}/m \\mathbb{Z}", "fefc94341ba3877d476847d8e6482628": "aababb", "fefcb3dd336d3be188a9132fa4dfa41c": "|\\mathrm{OPT_A}(x)-c_A(g(y))| \\le \\beta |\\mathrm{OPT_B}(f(x))-c_B(y)|", "fefcee0745c559937dd223acaa1c18fe": "\\Delta f = \\sum_{ij} g^{ij} H(f)_{ij}.", "fefd00e3500a78cc0a3e30af1b1fc0a6": "\\dot{p}_k=dp_k/dx_3", "fefd4be7a2698609852b368709f07156": "Z_k", "fefdcbb4d27dde0daee2cb33be81e055": "\\Delta=(2\\pi)^{12}\\eta^{24}(\\tau)", "fefe48165ccfa58697df21181d0559d1": "2\\pi/n", "fefe8b67092dfea05903b818dbdfda4a": "\\lambda \\Pi \\omega", "fefe9901df83ee2e184440b65de515a0": "\\frac{c}{b} = \\frac{b}{s} \\ . ", "fefe9e0ada29ae95aeeca1412b3d804a": "1105 = 5 \\cdot 13 \\cdot 17 \\qquad (4 \\mid 1104;\\quad 12 \\mid 1104;\\quad 16 \\mid 1104)", "fefea4343968e94bc3bdf9683adbbbc2": "wp(\\alpha,\\psi)", "feff43a3194d4b9d69e9d6f496998710": "\\widehat s(t)", "feffbdf1a4276ec5d4186c9ebf86cf18": " t \\downarrow 0 ", "feffd575724687ae865f30f9a18e831b": "\\boldsymbol S", "fefff8cb0b83fe39c74279913f524b4b": " S = {8:5^{5/4}}, \\ ", "ff000487b745aa0ac36f6ca81a55b09a": "g(x|D)", "ff00065c8f875eb6fbbb1f47bcd52c35": "\nS_{k} = \\left( \\frac{1}{k} \\right) \n\\int d\\theta^{\\prime}\n\\int d\\rho^{\\prime} \\left(\\rho^{\\prime}\\right)^{k+1} \n\\lambda(\\rho^{\\prime}, \\theta^{\\prime}) \\sin k\\theta^{\\prime}\n", "ff0018c9dda6bdae611faf2506b41706": "i_1", "ff002c11aa25ce72e9d07bc058b367f1": "\\sin \\theta \\simeq \\theta", "ff002cbd58ae35606e9a5580630d97aa": "\\mathrm{sinc}(t) * \\hat h(t).", "ff003e76b289c66f2b8a44f529c7f253": "\\widehat K", "ff005c19f3c79c09271b92c67075b713": "\\sum_{i\\in A}x_{ij}=1\\text{ for }j\\in T, \\, ", "ff00d46015375ba6663ab0fa6a660247": "\n\\operatorname{Tr}(D) = \\sum_{i=0}^{n-1} \\sigma^i(D) = D + \\sigma(D) + \\cdots + \\sigma^{n-1}(D)\n", "ff00fc258411eb3f9918e6460defb194": "\n\\sinh\\left(2K^{*}\\right)\\sinh\\left(2L\\right)=1\n", "ff0135a7d41ba80230b3744508efd726": "\na_0 + a_0a_1 + a_0a_1a_2 + \\cdots + a_0a_1a_2\\cdots a_n =\n\\cfrac{a_0}{1 - \\cfrac{a_1}{1 + a_1 - \\cfrac{a_2}{1 + a_2 - \\cfrac{\\ddots}{\\ddots \n\\cfrac{a_{n-1}}{1 + a_{n-1} - \\cfrac{a_n}{1 + a_n}}}}}}\\,\n", "ff01bfcddd7c360c617245cf4fbddb0d": "[S] = \\begin{bmatrix} \\Omega & -\\Omega\\textbf{d} + \\dot{\\textbf{d}} \\\\ 0 & 0 \\end{bmatrix} = \\begin{bmatrix} \\Omega & \\mathbf{d}\\times\\omega+ \\mathbf{v} \\\\ 0 & 0 \\end{bmatrix}.", "ff01cbffd6012f2cdea8133bded73927": " \\tau = C_{AO} \\int \\frac{1}{(-r_{A})}\\,df_{A}", "ff01d5b7e9f84d8bb589492bbc03cc53": "E(X^{2n})=\\left({R \\over 2}\\right)^{2n} C_n\\, ", "ff0241c65f30fbdaccfd8e2d2dc1ef61": " u \\, ", "ff02e45531b698383c15bb63e01c9f77": "f(0, 0) = 0", "ff030da2a76755f6058e0d47a3b22b52": "\n \\int x^m\\left(c(A\\,b-a\\,B)(m+1)+A\\,n (b\\,c-a\\,d)(p+1)+d(A\\,b-a\\,B) (m+n (p+q+2)+1) x^n\\right)\\left(a+b\\,x^n\\right)^{p+1}\\left(c+d\\,x^n\\right)^qdx\n", "ff0342ae87470194ceab86363df30d17": "X_1\\oplus X_2\\oplus\\cdots\\oplus X_n=0", "ff037307222cbb99fdbf659907004eb0": "\\mathcal B = \\left\\{ \\prod_{i \\in I} U_i\\ \\Big|\\ (\\exists j_1,\\ldots,j_n)(U_{j_i}\\ \\mathrm{open\\ in}\\ X_{j_i})\\ \\mathrm{and}\\ (\\forall i \\neq j_1,\\ldots,j_n)(U_i = X_i) \\right\\}.", "ff044f724d6f584dedfc24f57c401919": "\\mathrm{d}^2\\Sigma = \\mathrm{d}\\sigma \\, \\mathrm{d}\\tau", "ff045cd9a49a394b00a8a7d311f65b81": "\\phi_{\\alpha\\beta} := \\phi_\\alpha \\circ \\phi_\\beta^{-1}|_{\\phi_\\beta(U_\\beta\\cap U_\\alpha)}", "ff04648d90a9a51a38b3da0417636027": "m \\leq s ", "ff04770de5de4e9c691be6f6c04bfbbc": "\ne", "ff0495b4518b557f43f39a9e38ca9c03": "\\tilde b_i(x) = b_i(x) + \\sum_j a_{ij,x_j}(x)", "ff049f3a246b28fb5ebd3b20109a3eab": "\\frac{\\mathrm{D} \\varphi}{\\mathrm{D} t} = \\frac{\\partial \\varphi}{\\partial t} + \\nabla \\varphi \\cdot \\mathbf v.", "ff051900fb388cd7b93bc69f302f2e88": "\\{w_{i}, i = 1,2, \\ldots,K\\}", "ff05c90d4fc584fc4c148ebc1fd7ab16": "{CE}_{10}", "ff06366e06e1138192f1db12d7a758e3": "P(M)", "ff06bef4996765b0a4ed8686a8b09a67": "O\\left(e^{-c\\sqrt{n}}\\right)", "ff06d4de4bdf89959c81dc6e79548edd": " v = {1 \\over \\sqrt {LC}} .", "ff06e321cf77e2725c39fb1aa504c2be": "{m \\choose k} \\cdot {n \\choose k}", "ff07174674e5a49ef999f5cda5823528": "F_{x}=(0.5\\rho l(c_{2})^2)(\\frac{s}{l})\\cos^2\\alpha_{2}(\\tan^2\\alpha_{2}-\\tan^2\\alpha_{1})+s\\Delta p_{0}", "ff071bfd560d31c5ed9ac74fdf8f334e": " \\frac{BC}{AB}=\\frac{BH}{BC} \\text{ and } \\frac{AC}{AB}=\\frac{AH}{AC}.\\,", "ff07c2e25d5e07bb98a9e4775a521993": "x^2 + y^2 + z^2 = 1,\\,", "ff081f17c39f6c765587dce815ca70b4": "\\scriptstyle{ \\mathbb K }", "ff084d78d721566322e1b07951e71223": "\\gamma(u)", "ff086b34bc17c1a5931ac06e66f581e4": "S[\\phi] = S[\\varphi\\circ\\rho]", "ff08829b5c70bcf61fe032f9528536f2": "\\mathbf{r}_3 = (a/2)(\\hat{x} + \\hat{z})", "ff08ed5f9aa94e4c1ac0ebe0838a8a67": " MPGe = 72 \\frac{mi}{kg-H_2} \\times {1.012 \\frac{kg-H_2}{gallon\\ gasoline}} = 72.8", "ff09c58da547425668de6de57d89018e": "\n\\left( \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2} \\right)\\varphi(x,y,z) = f(x,y,z).\n", "ff0a33e636274c7faedc6312000cc72f": " f(x) = f \\left( n \\cdot \\frac{x}{n} \\right) = n f \\left( \\frac{x}{n} \\right) \\ ", "ff0aa2597e704c9602c2311207129869": "\\psi_0(\\varepsilon_{\\Omega_\\omega+1})", "ff0ab42dc75eab47600b783014162b22": "Q(x)=x+1=x-(-1).\\,\\!", "ff0ac02ff8020d8c400a28ad3e488a4c": "X := \\prod_{i=1}^{n} X_i", "ff0b12c9deee3881ab82f9f03ebc0da5": "c= -0.77568377+0.13646737*i \\,", "ff0b3c53a14af55ebe0978bf5cc0e072": "\\Pr(X_n=0|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \\dots) = 1 - \\Pr(X_n=1|X_{n-1} = x_{n-1}, X_{n-2} = x_{n-2}, \\dots).", "ff0b5cb0ca39ea5151cbb00534b8ccf6": "\\{\\{\\}\\}", "ff0b7a4123f2ba75215824b8e1e37d99": "a \\wedge b", "ff0b9b82fbeb52426934f1c0a54ba8d3": "\\langle a_1,\\dots,a_n\\rangle.", "ff0bf30c9ae46f379eb80a8dd7176c06": "S(a^\\mu) = - {(\\Lambda^{-1})^\\mu}_\\nu a^\\nu \\,", "ff0bf5412a2b64b06e20c00ef1063c04": "a_i < (a_{i-1}+1)^3", "ff0c0a063d89181211d337b5005e5354": "a_0x^1+a_1x^p+a_2x^{p^2}+\\cdots = a_0\\tau^0+a_1\\tau+a_2\\tau^2+\\cdots \\, ", "ff0c4242d64be5b4eac4ac28fc736235": "\\underset{=}{A} \\in \\{\\underset{=}{A} \\in B(X,Y) : im(\\underset{=}{A})=Y\\}", "ff0c7bc8626cc698cc6c98eaf9389a81": "{\\rho}\\,", "ff0cc60d00993c59ef0cd18138004658": "(15/16)b_{1}=d", "ff0ccaabc90649254439c0cc3e2a0c27": "T = E_{tot} - V(\\mathbf{q})", "ff0d6caa8a29a6bf4fa6ec9a1ae022e5": " \\langle H_{pot} \\rangle = \\tfrac12 N \\langle h_{\\mathrm{pot}} \\rangle ", "ff0d81872dc8f518ba595d795c9c3a9d": "\\phi=-0.54916...= (-31,46447^{\\circ})", "ff0dd9f149c1b45ca96b297498c2c205": " x_1 + x_2 = 1\\,", "ff0e9f10da206771ac9abc5052b81d68": "\nI_{\\text{p}} = \\frac{P_{\\text{R}}}{V_{p}} = \\frac{{N^2_{\\text{S}}}{V_{\\text{p}}}}{{N^2_{\\text{p}}}{R_{\\text{L}}}}\n", "ff0fbb40595d90a5069ba03f4571da1f": "k+1, 2k+1, 3k+1, \\dots", "ff0fdfd15260165095c2d3deedc05594": "\\theta = a(x, u, u_{1})dx + b(x, u, u_{1})du + c(x, u,u_{1})du_{1}\\,", "ff0ffda6ef663ee572462c7a56e5b47e": "L^1(X,\\Sigma,\\mu)", "ff10635ae815449a04ff6ebc59d96392": "(C_1 r)^{C_2}", "ff1083266ed564513c59a21a950e0696": "C(g^*_i,\\mu)=C_*", "ff109356126180a4b37517b5abc453dc": "H(X) = \\sum_{i} {P(x_i)\\,I(x_i)} = -\\sum_{i} {P(x_i) \\log_b P(x_i)}", "ff109955e699621b472b6ed75178637c": "\\mathbf{x}_n \\to \\mathbf{x}_{n+1}", "ff10d2d65364cccef0db607eb1b9b7d4": "K \\cap L", "ff112a1cac89c1f550aca498ccd13af8": "\\begin{align}\n&\\underset{W}{\\operatorname{minimize}}& & W^T \\cdot Var_{x_i} \\cdot W - Cov_{x_ix_0}^T \\cdot W - W^T \\cdot Cov_{x_ix_0} + Var_{x_0} \\\\\n&\\operatorname{subject\\;to}\n& &\\mathbf{1}^T \\cdot W = 1\n\\end{align}", "ff11559592ca88d6c34e58ddfdea9e0d": "x(t+1)=f(Wx(t))", "ff1157cf244548cc6ae885dbe7bf155b": "\\ D \\ ", "ff1185bf4037519ebb5253bcb1aa0bce": "\\sum_{n=1}^\\infty \\frac{1}{n},\\! ", "ff118ea164bf9c7fa73ca0fba8e5c25a": " \\text{size} = N,\\,", "ff11918ec2f9e539074f556ef4729344": "r[k]", "ff11af6aa62f225c46bc2a30be882f23": "\\lambda_1 (t) = \\frac{c_1 M}{M - (y_0 + a_1 t)}", "ff11b3ddf46bdbea7355262bc7bf0dd2": "U_{\\mu}", "ff11bdab2937c408ce2704674cb68868": "\\int \\csc{x} \\, dx = \\ln{\\left| \\csc{x} - \\cot{x}\\right|} + C", "ff11d90f83ef5aa32867fe6b9a4877e2": "\\mu_{\\mathrm{ext}} = qV + mgh + \\cdots", "ff12204215939f918c57064b92d34a81": "\\displaystyle{\\mu_F(0)=g_b\\left( -{a_{-1}\\over \\overline{a_{1}}}\\right),}", "ff12224edda48a9f3b4680db7b7f1b68": "\\Delta r^2", "ff1266cc9dd183fb06d9c4d6f2179072": "\\left ( dV_t = \\phi_t d S_t + \\psi_t\\, d B_t \\right ) ", "ff12868212f0f77f46a72ba68909cd72": "n=\\frac{2D}{D-2}", "ff12922b8f2c41dded1cbe55dc852ef3": "F(t) = 1 - e^{-\\lambda t}", "ff1299b95bbb93bbb568b003b66d9231": "A_\\text{v} \\triangleq \\frac{v_\\text{out}}{v_\\text{in}}\\,", "ff132f2286883b787183fb4490a3a49e": "T_v(s)\\,", "ff136a13ed3f0728c21e3ccd0e4475ea": "{ i\\hbar\\frac{\\partial}{\\partial t} \\Psi(\\mathbf{r},\\,t) = \\hat H \\Psi(\\mathbf{r},\\,t)}", "ff138cf8bdb6b207011262cef05942fb": "f:\\Omega \\rightarrow R^n", "ff14185eebe3a047d15cefd802ec6d88": "z \\mapsto \\int_{x+y \\leq z} f(x)g(y) \\, dx \\, dy. ", "ff148502769248cd2df30935ffd1b6be": "\\frac{\\partial g(\\mathbf{u})}{\\partial x} =", "ff14a00ffd3ba0654163d3d05cf8631b": " {f (x_{L}) \\leq f (x), \\qquad \\forall{x}\\, \\in N(x_{L}) \\cap X} ", "ff1523220b2b9ae629fe5e5a77adcc7c": "wlp(S,\\mathbf{false})", "ff15a84d364d8cfccbd93dd6c5bf21c7": "46^2", "ff16355e2e5771c049927d65df0d8e1d": "G = (\\{N, S\\}, \\{W\\}, L)", "ff16830292fa1cda57dedbbdf4478440": "X=\\left\\{\\begin{matrix} 1 & \\mbox{with}\\ \\mbox{probability}\\ 1/3, \\\\\n2 & \\mbox{with}\\ \\mbox{probability}\\ 1/3, \\\\\n3 & \\mbox{with}\\ \\mbox{probability}\\ 1/3.\n\\end{matrix}\\right.", "ff16c2962c5f56ceeac6e72442c42034": "u_H", "ff16ed4a07141a68d6010615c07763e0": "\\rho(x, y, z) = \\frac{1}{r^2 + x^2 + y^2 + z^2} f\\left(\\frac{x^2 + y^2}{(r^2 + x^2 + y^2 + z^2)^2}\\right).", "ff16fa665e411fec968cfd20df6bb30b": "x^4+4y^4 = ((x+y)^2+y^2)((x-y)^2+y^2) = (x^2+2xy+2y^2)(x^2-2xy+2y^2). \\,", "ff1731d74a99a71f1b51cc3fb5a4f3a8": "-\\ln \\left |\\cos x\\right | + C", "ff177e87571d8449e3e682233a6fa246": "(z_1,z_2;z_3,z_4)", "ff178eac273cbb24a68371c5659c030b": "P(t_1[e],\\dots,t_n[e])", "ff17a91a94177c3f383168fc69217fb4": "(a,b)=\\{x\\,|\\,a0\\;\\; (i=1\\ldots m,\\; j=1\\ldots n)\\right\\}", "ff1d12b13cfbd8c2f1405d4a5c94b3d1": "\\beta={ m_{2} - m_{1}^2 \\over m_{1}}.\\,\\!", "ff1d1d9456f77dd9e5e6f7d59fd741b6": "\\sum_{k=0}^n (-1)^k {n \\choose k} = 0", "ff1d32373432d983d1b9d7078ff9155f": " \\beta = e^{i \\phi} \\sin\\left(\\frac{\\theta}{2}\\right) ", "ff1d38a7c08766d303a0dace0bd6625c": "Pr[D(y)_i = x_i] \\geq \\frac{1}{2} + \\epsilon, \\forall i \\in [k]", "ff1d4feb8228242c9089ead64e5dbc19": "\\partial f_s", "ff1d66be53175f131bea2d4f436188dc": "f(n)=2n+o(n)", "ff1d7dd71da7b2fb34ac034d4684913c": "y\\le x", "ff1da0c18c1d6f1541914f27a0d15b7d": " \\alpha=0.85", "ff1de68b0997b9c8daebb6169bf27d19": "\\frac{ \\sqrt{6} - \\sqrt{2} } {4}", "ff1dfbc20dc5ffbbfb9ed2f2b7480835": "\\sigma\\sqrt{\\tau}/2", "ff1e275d2c5a50b46b8b84099621d34c": "C(x_1)", "ff1e75f9d164feb4de23bcba77f677cf": "(x+y,x+y)/2", "ff1eb00d08d74cad124cc07774adb6bb": "\\tfrac12\\, \\lambda = \\Delta\\, F\\left( \\begin{array}{c|c} \\tfrac12\\,\\pi & m \\end{array} \\right) = \\Delta\\, K(m),", "ff1eef3646318cc69c42c5b4fbfa17a2": "x=\\sin y", "ff1f1ac9fc2820c878d09a33be41d8dc": "(x, z)_{p} \\geq \\min \\big\\{ (x, y)_{p}, (y, z)_{p} \\big\\} - \\delta,", "ff1f5803d4135482f89e4fbce5439432": " f(x+0) ", "ff1fa2b3f0f1f25a20b44c83885fcf1d": "F(x;\\theta)\\colon \\theta\\in\\Theta", "ff1fd6773c4cc2d374cb3f6a6d741e51": "\\sin\\delta = \\sin\\beta \\cos\\epsilon + \\cos\\beta \\sin\\epsilon \\sin\\lambda", "ff200d5414c98adb62d1eb4aba102558": "\n \\left(1-M_\\infty^2\\right) \\frac{\\partial^2 \\varphi}{\\partial x^2} + \\frac{\\partial^2 \\varphi}{\\partial y^2} + \\frac{\\partial^2 \\varphi}{\\partial z^2} = 0,\n", "ff201bb43a00e6aecebc471052ac30d6": "\\Delta I_{\\text{L}_{\\text{On}}} + \\Delta I_{\\text{L}_{\\text{Off}}}=0", "ff203bbbc0d9a7e5553f93b6c549675f": "CRF", "ff20444c3b409861b38ee4c0e3ebb7a3": "\n\\Delta \\hat{z}\\ =\\ -2\\pi\\ \\frac{J_2}{\\mu\\ p^2}\\ \\frac{3}{2}\\ \\cos i\\ \\sin i \\quad \\hat{g}\n", "ff205aa2ed7728a669c455955893409e": "\\frac{1}{\\sqrt{|a|}}x(\\frac{t}{a}) \\rightarrow \\approx S_x(\\frac{t}{a},af)", "ff20b73054d662551e3f5b2c5ba2ae9b": "\n{\\delta \\mathbf{u}}^* = \\operatorname{argmin}\\limits_{\\delta \\mathbf{u}}Q(\\delta \\mathbf{x},\\delta\n\\mathbf{u})=-Q_{\\mathbf{u}\\mathbf{u}}^{-1}(Q_\\mathbf{u}+Q_{\\mathbf{u}\\mathbf{x}}\\delta \\mathbf{x}),\n", "ff2148539602a458b1797c5347baeeda": "\\forall S \\subseteq [n]", "ff21b785c3be61a7b688419191ec04ae": "ds^2 = e^{2 \\eta} \\left ( d \\eta^2 - d \\bar{l}^2 \\right ),", "ff21d867ab3e8d38789689efde56c581": "S(D,Q) = \\sum_{i=1}^n\\frac{r^{i-1}*w_{di}}{\\sum_{j=1}^n r^{j-1}}", "ff21eb46031b778fb4c8080f43495e86": "=\\frac{1}{2}(x^2 + y^2)^{-1/2}\\left[\\frac{d}{dt}(x^2) + \\frac{d}{dt}(y^2) \\right]", "ff224c181ddd774f4288bc6e1d3e6ecf": "\\Delta w = w_{12} - \\begin{matrix} \\frac{1}{2} \\end{matrix} (w_{22} + w_{11})\\,", "ff22b774c67f56d41318854bd89d9a13": "g' = g - \\tfrac{1}{3}", "ff2305a05803972348b0acc0fb430b6b": "\\mathbb{H}^{p+q}", "ff23615c3e12eaa9aec9082bab8117b7": "(R,T)", "ff23652d9c6299b5be4e5a3820b8497a": "y=\\frac{p(x)}{q(x)}", "ff236e4611ea3d57d64d014abf4663cd": "t \\colon V \\times V^* \\to F", "ff2381cc201995a96b6e1ab6878433e0": "\\eta_I", "ff23bc0ddabe7501b351ab9b5256a8eb": "f(x)=a-x^2.", "ff23c0cd043361b013a8bb28e47c3d4b": "\\mathbb{Z}\\,", "ff23dd1facde9d7004587b0ffb289059": " m \\geq 3", "ff23f0b448c8524589aee5e31a340e23": "\\det T", "ff242891e06074c67681bdc87d3b34f7": "bx \\equiv 1 \\pmod p. \\,\\!", "ff246339a11d908c736e1b1d83453c61": " H^{p,q}=F^p H\\cap \\overline{F^q H},", "ff24ba12e50a76001bd65d66c826042b": "\n\\vec{R}_A^0 \\mapsto \\vec{R}_A^0 + \\Delta\\varphi \\; ( \\vec{n}\\times \\vec{R}_A^0)\n", "ff2526149982962f8fb6a1445a682313": " m\\circ (S\\otimes {\\rm id}) \\Delta (x) =\\varepsilon(x)1", "ff25287244d2c8d2caf122c0c07a1072": " 0 \\leq f_{i} \\leq f_{i-1}^{(i)},\\quad 1\\leq i\\leq d-1.", "ff25807439f5de7e95cfb2caf17d8d1a": "\\begin{cases} u_{t} = \\Delta u & \\textrm{on} \\ \\ \\Omega \\times (0,T), \\\\ u=0 & \\textrm{on} \\ \\ \\partial\\Omega \\times (0,T), \\\\ u = f & \\textrm{on} \\ \\ \\Omega \\times \\left \\{ T \\right \\}. \\end{cases} ", "ff259cedbcee0784d766dc63a2c8d508": "\\lor", "ff25c2c414555b000a179d1c2a72354f": "D=\\sqrt{\\sum_i w_{yi}(DY_i-DY_i')^2} + \\sqrt{\\sum_i w_{bi}(DCb_i-DCb_i')^2} + \\sqrt{\\sum_i w_{ri}(DCr_i-DCr_i')^2}", "ff264383a9e2e0689092422d3a10a8b4": "\n\\epsilon_\\mu^1(p) \\!= \\!{1 \\over \\sqrt{2}} \\left(\n{{-i p_1 p_2 \\!+\\!E^2 \\!+\\!p_3 E \\!-\\!p_1^2} \\over {E(E + p_3)}},\n{{- p_1 p_2 \\! + \\!iE^2 \\! +\\!ip_3 E \\! - \\!ip_2^2 }\n\\over {E(E + p_3)}},\n{{\\!-p_1 \\!- \\!i p_2} \\over E}, 0 \\right), ", "ff26555d422c6b9a1ad1c024c92baa1a": "P_{nn}(\\mathcal{R}) = \\frac{\\partial \\mathcal{N}(\\mathcal{R})}{\\partial \\mathcal{R}} \\exp [-\\mathcal{N}(\\mathcal{R})]", "ff2737c6f629dbb2b56228eeee432ecf": "\\mathrm{PhNO_2\\ +\\ C_6H_6\\ +\\ 3\\ CO\\ \\xrightarrow {Rh_6(CO)_{16}}\\ \\ PhNHCOPh\\ +\\ 2\\ CO_2}", "ff27417a799544bf32bf65e1c04a4f0f": " Re = \\frac{\\rho v D}{\\mu}", "ff27722b1b1249ec9460b1ca4d9ea885": "\\sum_{k=0}^\\infty (-1)^k {k+\\nu+1 \\choose k+2} \\left[\\zeta(k+\\nu+2)-1\\right] \n= \\nu \\left[\\zeta(\\nu+1)-1\\right] - 2^{-\\nu}", "ff27d638519b513ed8f0bf42f470fa0d": "{\\rm Riesz}(x) = \\sum_{k=1}^\\infty \\frac{(-1)^{k+1}x^k}{(k-1)! \\zeta(2k)}.", "ff27edff1ceef2edebedf473d9df2dfe": "\\begin{bmatrix} \\dfrac{1}{g_{21}} & \\dfrac{g_{22}}{g_{21}} \\\\ \\dfrac{g_{11}}{g_{21}} & \\dfrac{\\Delta \\mathbf{[g]}}{g_{21}} \\end{bmatrix}", "ff27f7470b88846b9841d64612884f5c": "\\lambda_0 = 4L.", "ff28958b552688f02c8343b34bfbb82c": "v^{\\sigma} = \\vec{u'}+ \\frac{\\tau_f^{\\sigma}}{\\rho^{\\sigma}}\\vec{F}^{\\sigma}", "ff28b2df51791f7b0ae905876f37158d": " \\mathrm{Re} = {{\\rho {\\mathbf v_s} D} \\over {\\mu (1-\\epsilon)}}.", "ff292e3bd9a3ed5b944fee3a9c55142c": "U(t)=e^{-(i/\\hbar)t H}", "ff29359fd47ce4c0ed9b0d01ba1fd05d": "f(x_1,x_2,1-x_1-x_2)", "ff2935b4beb36efee9b8469e5ccc7ce1": "Tr[E(e)^2]", "ff29a3536ba4d7209cdbc7360a8e306d": "\\scriptstyle\\chi_{n-1}^2", "ff29ed6603a5b129ae95d2459f74c76f": "T \\hat{\\mathbf{J}} T^\\dagger = - \\hat{\\mathbf{J}} ", "ff2a4c4c0063f90860f641450739732e": "x_1=x_2", "ff2a596544ad9ccd87ced4ddd16d463f": "=\\left ( 1-\\sum_{j=1}^{m}a_{ij} \\right )\\mu^2+\\sum_{j=1,j\\neq k}^{m}a_{ij}(v^2+\\mu^2)+a_{ik}(\\sigma^2+v^2+\\mu^2)-(v^2+\\mu^2)=a_{ik}\\sigma^2-\\left ( 1-\\sum_{j=1}^{m}a_{ij} \\right )v^2=0", "ff2a5d01e69936ec06dbc0232bd28cf2": "p_n ", "ff2a7ddae8c05988681f8c553efed318": "\\scriptstyle H_A \\otimes H_B", "ff2ac2ff0a5e7ea2ba270f4dd080d767": "\\begin{cases}\nQ_1 = 15 \\\\\nQ_2 = 40 \\\\\nQ_3 = 43\n\\end{cases} ", "ff2aca0e632dd8a1b0b17164a08e77c8": "\n\\hat{B}_{m_j}=\n\\begin{cases}\n\\frac{1}{\\pi}\\int_{-\\pi/2}^{\\pi/2}f\\bigl(\\mathbf X(s)\\bigr)\\sin\\left(m_j\\omega_js\\right)ds & m_j \\text{ odd} \\\\\n0 & m_j \\text{ even}\n\\end{cases}\n", "ff2ad45c63fce2b4c05059040aa2eb99": " \\int d^d x (\\mathcal L [\\phi(x)] + J(x) \\phi(x)) = \\int d^d x (\\mathcal L [B(x)] + J(x) B(x)) + \\int d^d x \\left(\\frac{\\delta\\mathcal L}{\\delta \\phi(x)} [B] + J(x)\\right) \\eta(x) + \\frac12 \\int d^d x d^d y \\frac{\\delta^2\\mathcal L}{\\delta \\phi(x) \\delta\\phi(y)} [B] \\eta(x) \\eta(y) + \\cdots ", "ff2af84b12ea7111ab27eb291122601a": "\\textstyle \\lambda \\times k", "ff2b4543e0d84fd1474a4ad8c84bb18a": "C_4", "ff2bb134ef1571371729ac88c43c5f75": "a(z)=\\sum_{n=-N}^Na_nz^{-n}", "ff2bd57f0b4cddf4114885c6688ca5db": "E_o=100{ {mass\\ flow\\ rate\\ of\\ solids\\ coarser\\ than\\ screen\\ size\\ in\\ feed\\ stream} \\over {mass\\ flow\\ rate\\ of\\ solids\\ in\\ the\\ oversize\\ stream}}", "ff2bddd61609f26dc3255408781890ad": "-k", "ff2be226caab659e54d700cdcc614f12": "r-2,r-3,\\ldots , 3, 2", "ff2beed2ea27e53c8dcf63ad20d72ed4": "F=(f_1,\\dots,f_k)", "ff2c15c5651bd652dc29ac67ef7e5bed": "P(y)\\frac{dy}{dx} + Q(x)= 0\\,\\!", "ff2c3eeaef11961e9d916a8b8a0c78e6": " U_5(x) = 32x^5 - 32x^3 + 6x \\,", "ff2c96546b04399b0bd103c0ded9fbfd": "0 = f(a) = f(x_n) + f'(x_n) (a - x_n) + \\frac{f''(\\eta)}{2} (a - x_n)^2,", "ff2cb8ece6cf247e538ef8dc7e28343d": "u_e", "ff2cc067c02cc24d7f140646b712aae1": "\\mathbf{\\hat T} (\\lambda) = \\exp\\left(\\frac{-i\\lambda\\mathbf{\\hat P}}{\\hbar}\\right) ", "ff2ccade79528e8a6cdb0b191a73fd5b": "R_{\\mu\\nu} - {1\\over 2}R g_{\\mu\\nu} = 8\\pi G\\,T_{\\mu\\nu}", "ff2d26be6b0b506663911208302f91b3": "e^x", "ff2d63fed24d61b609cd0dfd3ec9e0e2": "Tf(w)=-\\frac{1}{\\pi} P.V. \\iint \\frac{f(z)}{(w-z)^2} dxdy=-\\frac{1}{\\pi}\\lim_{\\varepsilon \\to 0}\\iint_{|z-w|\\ge \\varepsilon} \\frac{f(z)}{(w-z)^2} dxdy.", "ff2d6e204ac18ee94bd4ce013e4b52b7": "V=(3+\\frac{8\\sqrt{2}}{3})a^3\\approx6.77124...a^3", "ff2d907dcbaf791a0167511cdc0ddea3": "\\mathcal{L}\\{f(t)\\}=\\frac{2}{(s^2+4)^2}", "ff2da9f8734f40be5ddd38a093eebcfa": "t_{\\mathrm{max}}", "ff2de91a402e036f938bba06007eea71": "r_{pj}^o", "ff2e00e24121c983e0f7220a5810e406": "(1,4),(2,-1)", "ff2e054ca88f78183918984abf8f8273": "\nK = \\frac{\\left ({dV \\over dp} \\right )}{V}\n", "ff2e1aae090836e1ec082472f33228ef": "\\left \\{ a^{n}b^{m} \\,| \\, m,n \\ge 1 \\right \\}", "ff2e26e35816fd436c2ef6d0d0cb84d4": " \\Delta M_S=0", "ff2e36ef8e6803d8225de9a4709f10ba": " \\frac{\\rho^4}{z^3 \\lambda} \\ll 8", "ff2e5e243d5ada96d2fa86a84b4e365d": "\n \\frac{\\partial \\rho}{\\partial t} + \\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left(\\rho r^2 u_r\\right) +\n \\frac{1}{r \\sin(\\theta)}\\frac{\\partial \\rho u_\\phi}{\\partial \\phi} +\n \\frac{1}{r \\sin(\\theta)}\\frac{\\partial}{\\partial \\theta}\\left(\\sin(\\theta) \\rho u_\\theta\\right)\n = 0.\n", "ff2e70b3544aa095608ed80728a20ceb": "\n [\\mathsf{S}] = \\begin{bmatrix}\ns_{1111} & s_{1122} & s_{1133} & 2s_{1123} & 2s_{1131} & 2s_{1112} \\\\\ns_{2211} & s_{2222} & s_{2233} & 2s_{2223} & 2s_{2231} & 2s_{2212} \\\\\ns_{3311} & s_{3322} & s_{3333} & 2s_{3323} & 2s_{3331} & 2s_{3312} \\\\\n2s_{2311} & 2s_{2322} & 2s_{2333} & 4s_{2323} & 4s_{2331} & 4s_{2312} \\\\\n2s_{3111} & 2s_{3122} & 2s_{3133} & 4s_{3123} & 4s_{3131} & 4s_{3112} \\\\\n2s_{1211} & 2s_{1222} & 2s_{1233} & 4s_{1223} & 4s_{1231} & 4s_{1212}\n \\end{bmatrix} \\equiv \\begin{bmatrix}\nS_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\\\\nS_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\\\\nS_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36} \\\\\nS_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46} \\\\\nS_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56} \\\\\nS_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \\end{bmatrix}\n ", "ff2ef01bfccd15d569681aaa117a0f5f": "S = \\hbar\\sqrt{\\frac{1}{2}\\left( \\frac{1}{2}+1 \\right) } = \\frac{\\sqrt{3}}{2}\\hbar", "ff2f1c997c84f4cf0d04efa79b9085cd": "sn\\tbinom{n+s-1}{s-1}", "ff2f423e58bf65f74d6c366472f94d84": " \\delta_S^2 = 2\\delta_S + 1 = [5;1,4,1,4,1,\\dots] \\approx 5.82842", "ff2f867d04dcbe4019f929f3f614c736": "\\frac{1}{ u} ( u \\wedge v )", "ff2f989db354b3c14690e9b655270190": "\\alpha = \\frac{e^2}{2\\varepsilon_0 hc}", "ff300c2dfc81671f65c866dff236552a": "\\mathbf{j}_{\\mathrm{average}} = \\frac{\\Delta\\mathbf{a}}{\\Delta t} ", "ff3031ce83dc20dda8f91749b5987cdb": "1/\\sqrt{2}", "ff303df6ff97222f67ff9b62623eb6f9": " \\sum_{i=0}^n (-1)^i(a[0], \\dots, a[i-1], a[i+1], \\dots, a[n]) ", "ff3043767bf850bb61e0c248f7c0873d": "y\\sim f(x)=x+2.67\\,", "ff3053080d3c3aac266efc9cf05926d2": "G(\\omega)", "ff30f745393d9af582ff2e069e46f6fa": "\\frac{d\\omega}{dt}=2k\\frac{(a-b)}{I}(\\theta-\\psi)-2k\\frac{(a^2+b^2)}{VI}\\omega", "ff310adbbcdf5fcc51b3ef2ae0cd9869": "g(x)=(x-1)(x-2) \\cdots (x-(p-1)).\\,", "ff3171ad306170bfa09256f208f72f1e": "\n\\left[ \\left.\n\\begin{array}\n[c]{cccc}\n0 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 1\\\\\n1 & 1 & 1 & 0\\\\\n0 & 1 & 0 & 0\n\\end{array}\n\\right\\vert\n\\begin{array}\n[c]{cccc}\n1 & 1 & 1 & 0\\\\\n0 & 1 & 0 & 0\\\\\n1 & 0 & 0 & 0\\\\\n1 & 0 & 0 & 1\n\\end{array}\n\\right] .\n", "ff31bb6da4dbf54fc3113029b58edd86": "[x_1, x_2] {\\,\\langle\\!\\mathrm{op}\\!\\rangle\\,} [y_1, y_2] = \\{ x {\\,\\langle\\!\\mathrm{op}\\!\\rangle\\,} y \\, | \\, x \\in [x_1, x_2] \\,\\mbox{and}\\, y \\in [y_1, y_2] \\} ", "ff3228ff9287c468a980f65ff52f0d89": "\\nabla f(x) = N(x),", "ff329814d9571153bb98f5515c63efea": "\\Pr(Y=k) = (1 - p)^k\\,p\\,", "ff32c8057cc2f0873a8f6856d6ba9831": "\\begin{matrix}\\frac{1}{2}\\end{matrix}mv^2", "ff32e6a619bb210de496194e3faedb31": "v_g = \\frac{d{\\omega}}{dk} = \\frac{1}{2} v_p.", "ff32f496a72ca794f649a26dc119773e": " \\frac{1}{\\operatorname{Tr}(E S)} E S E. \\, ", "ff336c54af685a6ef9f6e6ebd5c9247f": "l(2t-1)", "ff33aea8aef4686c0489e7676c4b3816": " \\mathbf{p} ", "ff33cc4769a663ba671339ce07a0c3fd": "P_n(C)=\\frac{1}{n}\\sum_{i=1}^n I_C(X_i), C\\in\\mathcal{C}.", "ff33f3fc63c81e7ef8f2863b5be6c832": "P_a = 1-b c^{2a^{1/2}} h^a \\, ", "ff34179185e76bf9778db7b7937b9730": "\\partial_{t} u (t, x) = \\Delta u (t, x),", "ff3478fb4474cec280beb2a4d2d9e31d": "\\operatorname{trace}(AB)\\ne\\operatorname{trace}(BA).", "ff34b4a2a01dad68465c949a699bb138": "a, b, c \\in A", "ff34c0956c1c8f1b9766291217945c05": "P_B:E\\to E", "ff34e47069805883931bf86bfea48f01": "a\\not\\equiv3\\pmod4", "ff34ef448bbb68f6d776657980ff782e": " g^{-1} (L_1 \\cap L_2) = gL_1 . gL_2 ", "ff34fd3aca243ef18ce29a86aa1c17e9": "\n \\boldsymbol{\\nabla}\\boldsymbol{T} = \\cfrac{\\partial{\\boldsymbol{T}}}{\\partial \\xi^i}\\otimes\\mathbf{g}^i \n ", "ff35b18847dacb596c9d754206de8e97": " v( S \\cup \\{ i \\} ) - v(S) \\leq v( T \\cup \\{ i \\} ) - v(T), \\forall~ S \\subseteq T \\subseteq N \\setminus \\{ i \\}, \\forall~ i \\in N;\\, ", "ff35d72292025b368e48522afdd35bf1": " \\ln \\,\\operatorname{var_{G(1-X)}} =\\operatorname{var}[\\ln (1-X)] = \\psi_1(\\beta) - \\psi_1(\\alpha + \\beta)", "ff361a44fbde37b3d84aac31c3c5f668": "\\frac{\\Delta (w/2) }{f_f}=\\frac 1{\\pi Z_q}\\,\\frac 2{\\omega u_0}\\left\\langle\n\\sigma \\left( t\\right) \\sin \\left( \\omega t\\right) \\right\\rangle _t", "ff36c314735bc7d7a5e8615a5720b7cc": "\\pi(f) = \\int_Gf(x)\\pi(x)\\,dx", "ff36e04e4c6f3125e142ccb8021a063c": " i^{th} ", "ff36e449df5eeb1bd7b91cc576b5851e": "~\\hat b^\\dagger~", "ff3765e07ec0002cda0203b2aff2801a": "\\mathbf{F}=\\begin{bmatrix} \\cos \\theta & \\sin \\theta & 0 \\\\\n- \\sin \\theta & \\cos \\theta & 0 \\\\ \n0 & 0 & 1 \\end{bmatrix}\\,\\!", "ff379062848e586db76771c0b015860e": "\\left(0,\\ \\pm\\sqrt{2},\\ \\pm2\\sqrt{2},\\ \\pm2\\sqrt{2}\\right)", "ff382dedf2c1cda5aefc1a8b0d5b7443": "$37500-$33000 = $4500", "ff3896f61faf621b8de8fe8c68162e3a": "{ \\partial u \\over \\partial x } = { \\partial v \\over \\partial y } ", "ff38bb23f5d48dd5fdd20ec17536429d": "\\chi,X\\;", "ff39c79a21af6151901bcdafbd9739e5": "\n\\frac{1}{2}\\rho v^2 + P = \\mathrm{const.}\n", "ff39e3c67361ce6a57517e631352fa8b": "x^2\\frac{d^2y}{dx^2} + ax\\frac{dy}{dx} + by = 0 \\,", "ff3a1b125a6a4fe5823dbe0d510ab9b2": "\\frac{dH_\\nu}{dz}=\\alpha_\\nu (B_\\nu-J_\\nu)", "ff3a485d1c5fa36fc5b3e99161399858": "\\alpha(a,\\, b)", "ff3a5a6cda78db30ff8b16478178818d": "\\dot x \\ddot x+\\dot y \\ddot y +\\dot z \\ddot z = \\frac{\\delta U}{\\delta x}\\dot x + \\frac{\\delta U}{\\delta y}\\dot y + \\frac{\\delta U}{\\delta z}\\dot z = \\frac{dU}{dt} ", "ff3a980de2e7c5ebe9691a5d1d284315": "d_{p} \\left( (x_{1}, y_{1}) , (x_{2}, y_{2}) \\right) := \\left( d_{X} (x_{1}, x_{2})^{p} + d_{Y} (y_{1}, y_{2})^{p} \\right)^{1/p}", "ff3aa00515f853f796e8984fb71c243a": "\\alpha \\gg 1", "ff3aa8b47c0ba7e949554e0fd4f36fc7": "\\frac{720,000\\ \\mathrm{N}}{(150)(9.807\\ \\mathrm{m/s^2})}=489\\ \\mathrm{kg}", "ff3aed143f228528e7b8d36a1ccdcb87": "-\\frac{r}{19} \\approx -0.0526r", "ff3b0d8c442833895ec250e3f0c14f7d": "S \\subseteq \\kappa", "ff3b185200645ad0016a996bcb6e9d55": "[(\\sqrt{q}^{k} - 1)^{2g}, (\\sqrt{q}^{k} + 1)^{2g}]", "ff3b1a77092ee50a76339b59b4b66c45": "C = g^m h^r \\mod n", "ff3b83733845d4fe4d1fd0692fab7535": "A(x):= \\sum_{1 \\le n \\le x} a_n \\,.", "ff3be8f68c1b492b2b273a1465702480": "\\upsilon_D = \\mu_{obs} \\frac{n E}{\\lambda_0} \\sin \\theta \\qquad(7)", "ff3c1ba7764eef0f74f7b7c83928c4d6": "\\bold{X}=(X_1,X_2,X_3,...)", "ff3c53e301dc828389d97dbed8e099a6": "\\dot{V}(x) = \\frac{\\partial V}{\\partial x}\\cdot \\frac{dx}{dt} = \\nabla V \\cdot \\dot{x} = \\nabla V\\cdot f(x) = \\frac{d}{dt} V(x(t)) ", "ff3c627db8946ac8fe64038e4dba2288": "\\bigoplus_{i \\in \\mathbb{N}} V_{i}", "ff3c92e68ca5d6d0d6744f07d3644b87": "\\Delta P = \\frac {8} {\\pi^2} \\rho fL \\frac {Q^2} {d^5} \\,,", "ff3cc463e1078f3c32747404c10618ab": "\n \\begin{cases}\n &a + b + c + d = 1 \\\\[6pt]\n & \\frac{1}{2} b + \\frac{1}{2} c + d = \\frac{1}{2} \\\\[6pt]\n & \\frac{1}{4} c + \\frac{1}{2} d = \\frac{1}{6} \\\\[6pt]\n & \\frac{1}{4} d = \\frac{1}{24} \n \\end{cases}", "ff3cf9212d90d8544b89318643c6e336": "\n F (\\sigma_{22}-\\sigma_{33})^2 + G (\\sigma_{33}-\\sigma_{11})^2 + H (\\sigma_{11}-\\sigma_{22})^2 + 2 L \\sigma_{23}^2 + 2 M \\sigma_{31}^2 + 2 N\\sigma_{12}^2 + I \\sigma_{11} + J \\sigma_{22} + K \\sigma_{33} = 1~.\n ", "ff3d20a116300b1ceb5f8c1a5f09cc50": "D_q=\\frac{\\log_2\\left( f^q_1+f^q_2+f^q_3+f^q_4\\right)}{1-q},", "ff3d5546edd84614f7a8bc277d1381aa": "\\begin{align} \n& m_\\ell \\in \\{ -\\ell, -(\\ell-1) \\cdots \\ell-1, \\ell \\} , \\quad \\ell \\in \\{ 0,1 \\cdots n-1 \\} \\\\\n& m_s \\in \\{ -s, -(s-1) \\cdots s-1, s \\} , \\\\\n& m_j \\in \\{ -j, -(j-1) \\cdots j-1, j \\} , \\\\\n& m_j=m_\\ell+m_s, \\quad j=|\\ell+s|\\\\\n\\end{align} ", "ff3d7f3a64f9d4d1c8a43f80b3d217cf": " U_a = C_a + \\alpha P_a + \\beta D_a + \\varepsilon_a\\, ", "ff3d97f5481877375a4736e4ce8e2b8d": "H/M", "ff3dd080210758592c0145b51196129a": "\\mathbf{B} = \\nabla\\times\\mathbf{A}", "ff3dd241aaee60d191a262ce437d6f3b": "\\sum_{k=1}^{m_n} d_{k,n}^2 = \\mathcal{O}(n^r)\\quad\\forall r>1", "ff3e5d497ea81bedd8ea917f06223313": "I_i", "ff3eb9d22d075462c96f1153ff0f5c6e": "d_0(z) = \\frac{z}{z+1}", "ff3ee6920f3ba6d62cd4820f95fa48f7": "\\tan V = \\frac{S}{D}.", "ff3f5976e523b457eb1ed0c249d8695a": "\\tau=2iq/p+\\varepsilon", "ff4009771cb0a840850325f67b09e2a4": " g . p ", "ff404d566001f2e8dd75903135c4b7e1": "supp(\\rho) \\cap ker(\\sigma) \\neq 0.", "ff4055acf81f4a9544a30baaf0744baa": "\\log(Y) = a + b \\log(X)", "ff407c82b4902dfa3ec9594cae8b4610": "2^{-r_2}\\sqrt{|\\Delta_K|}", "ff40822f3c3c692583c7eb1b6a2142d4": "\\; V_i e_k = P_k v_i,", "ff40a0f5f45ff7d42ee70296897240a7": "P(h(x)=z)=1/m", "ff40a2fb839d7692095ad2b35d3e04f8": " A^4-(\\operatorname{tr}A)A^3 + \\tfrac{1}{2}\\bigl((\\operatorname{tr}A)^2-\\operatorname{tr}(A^2)\\bigr)A^2 - \\tfrac{1}{6}\\bigl( (\\operatorname{tr}A)^3-3\\operatorname{tr}(A^2)(\\operatorname{tr}A)+2\\operatorname{tr}(A^3)\\bigr)A + \\det(A)I_4 = 0,", "ff40b98750fc9a583612246b8a286f58": "\\nabla F(x_1, x_2, \\ldots, x_n) = \\left( \\tfrac{\\partial F}{\\partial x_1}, \\tfrac{\\partial F}{\\partial x_2}, \\ldots, \\tfrac{\\partial F}{\\partial x_n} \\right)\\,.", "ff41056dec337b31678a729b1ecfcfb4": "X_t", "ff412a4350329248761c5f3b73800e4e": "c_1^2+c_2 \\equiv 0 \\pmod{12}", "ff416ac94d7045a875b3cd4090c442ee": "\\left( f_{1},f_{2}\\right) =\\int dq\\rho _{0}\\left( q\\right) f_{1}\\left(q\\right) f_{2}\\left( q\\right),", "ff417b4dcb39d441ca504ae3bb8995e1": "x(v+W) = xv + W \\, \\forall x \\in \\mathfrak{g}, v \\in V", "ff419809eece5b985bcc0395848ab5fd": " e^{\\beta C} \\left\\langle e^{-\\beta (U_\\text{B} - U_\\text{A})} \\right\\rangle_\\text{A} ", "ff421dc37e386bc55029ed77dc12a085": "C = L^2", "ff42209d54389a24c35ef4fb1d0786e6": "\\quad\\sum_{i=1}^{\\infty}x_i.", "ff424fa1dd9a4b216af2fba986ff73c0": "f \\longmapsto Q_f", "ff42b31ba0ff7f46b24da14f9cea2e38": "\\frac{\\partial }{\\partial x_j}\\left(\\frac{\\partial \\Phi}{\\partial x_i}\\right)=\n\\frac{\\partial }{\\partial x_i}\\left(\\frac{\\partial \\Phi}{\\partial x_j}\\right)\n", "ff42bc58ce9360648ee89e75621dd207": "\\textstyle \\mathbf{0}", "ff42dde416acf6fd9c4c703b5abc57d0": " q_{jk}=\\frac{1}{N-1}\\sum_{i=1}^{N}\\left( x_{ij}-\\bar{x}_j \\right) \\left( x_{ik}-\\bar{x}_k \\right) ", "ff42e4d85f34705df2acd4b0b5161bc4": " G_{\\mu\\nu} = \\frac{ 8 \\pi G }{ c^4 } \\left\\langle \\hat T_{\\mu\\nu} \\right\\rangle_\\psi ", "ff43154d94d6e2b1dd14d24818fa8838": " \\hat{f}_1^{(i)} = S_1[Y - \\hat{f}_2^{(i-1)}], \\hat{f}_2^{(i)} = S_2[Y - \\hat{f}_1^{(i-1)}] ", "ff43255a1ddee16e5ff92223dd72736f": "\\hat \\theta = \\frac{\\hat \\theta_1 + \\hat \\theta_2}{2}. ", "ff43394b30b10e80835976429bf5523b": "y_1,y_2 \\in \\mathbb{R}^m", "ff433ac9509743d3b4c9b38af5015286": "f_s:X\\rightarrow Y", "ff43797d415f698aca35b53e417f87c4": "(gate1\\vee x1)\\wedge (\\overline{gate1}\\vee \\overline{x1})", "ff43ec8e1f52919b4c8ccfa63aa4b9f1": "\\scriptstyle \\boldsymbol{r}_i (t)", "ff44206c10185f4884bda9762b1326ef": "\n\\frac{A\\hbox{ true} \\qquad B\\hbox{ true}}{(A \\wedge B)\\hbox{ true}}\\ \\wedge_I\n", "ff44570aca8241914870afbc310cdb85": "J", "ff44af4b450b325911e1af6e2d260f65": "\\mu_G(H_1,H_2)=(-1)^{k}p^{\\binom{k}{2}}", "ff44d45063f2a7eb0a2fa3023453953d": "m = u-u_{xx}", "ff44e613832b38996dad3853e9599eac": "A \\leq A", "ff458230b5c44dd060d12e76b9e38dd6": "K = \\{ i \\mid i \\in W_i \\}", "ff45a57fedfcde953d808fdcba0e21a8": "p = x^2 + 2y^2 \\Leftrightarrow p\\equiv 1\\mbox{ or }p\\equiv 3\\pmod{8},", "ff45f00e888a8a22829c1957f3c9b38a": "\\mathbf{E}(\\mathbf{r}, t)", "ff46248079b24d5f123782b8348faf33": "S^n \\to \\mathbf{RP}^n,", "ff46cd63cb7ab2ebf68acdd4fe377658": "h_{\\text{out}}(G)", "ff46d7e500b48fef8600b1a96fafccb8": "C(3, 0) = 1", "ff46de5028bffa4825289c5d683946d1": " d = -\\nu^2 ", "ff4732a008160f463364cd4285d4bc9f": "L_r", "ff473c9449876da67813e31f75d0b19a": " \\int(f(x)/Mg(x))g(x)dx = (1/M)", "ff4751db48d725edecdad974b87e694f": "L_{\\hat{n}}", "ff475d457629c4f8847fa29e5e27a8bb": "x,y,z\\in G", "ff47a23274b90f952467e4f5ed268cb9": " Z_N ", "ff47ec3650c0d05bfea53fc6b9ffdf77": "\\mathbf d^{\\rm T}\\Sigma\\mathbf c=\\operatorname{cov}(\\mathbf d^{\\rm T}\\mathbf X,\\mathbf c^{\\rm T}\\mathbf X)", "ff4827739b75d73e08490b3380163658": "hf", "ff4839590e64201665493e6d48a65c66": "H(p)+E[C]", "ff48896f2c57d9a26994d4dc59ba8793": "r_u = \\frac{a}{2} \\sqrt{1 + 9\\varphi^2} = \\frac{a}{4} \\sqrt{58 +18\\sqrt{5}} \\approx 2.47801866 \\cdot a", "ff48c45a799427abb9a104414c676c08": "K^{(0)}_\\mu\\,", "ff4911cc641eef1f4147afb93450878e": "V(x) = (1 - 2x) \\cdot \\ln\\left(\\frac{1 - x}{x}\\right)", "ff49b086d2e6f7ed74f4dd5a1ec7804c": "\\langle F,R,\\mathcal{P}(F)\\rangle", "ff49b90277f6957b098b0f7d3d87be17": "\\Phi:X\\rightarrow S", "ff4a7c62348b4083e84e31266600aafd": "K_\\mu", "ff4ab7115f19a3275399858a77ddb5ba": "y_{i+1} = y_{i} + \\frac{h}{2}(f(x_i, y_i) + f(x_i + h, y_i + hf(x_i, y_i)))", "ff4ad5a66f36268c0af345501387b68f": "V=4n", "ff4b05831bf5de46cf805eba6e9c09c2": "\\mu_0 \\mathbf{J} = \\nabla\\times\\mathbf{B}.", "ff4b1b4a278db495e1680f748caac1dc": "\\,\\kappa_1", "ff4b424724562656262415ad393806f1": "S_T \\in \\{90, 110, 130\\}", "ff4b468552e02b4b13ddea1fc23ff15c": "\\scriptstyle{\\theta_n(t)}", "ff4ba6145a8791eab1f1e1cc6c01cbed": " z \\rightarrow \\overline{z},\\ \\ \\infty\\rightarrow \\infty \\quad ", "ff4be159ec42655e7b432666e9b85d1b": "\\textstyle \\Omega=\\{w_1,\\ldots, w_s\\}", "ff4c937b25da03e384bde34d1a907242": " \\gamma = \\frac{ 16 \\alpha^2 ( 11 \\alpha^2 + 6 ) }{ ( 5 \\alpha^2 + 4 )^3 }", "ff4cb8e0a612a1e40e809e1d29db8e2f": " d\\acute{\\Omega} = \\sqrt{-g} d{\\Omega } ", "ff4ce7009810b898ff4fd93723d7d722": "\\pi_{\\mathrm{f}} = -\\pi_{\\mathrm{i}}\\,", "ff4d273cae9c74d09d85906d488ccc28": "P_{W_{\\alpha}}(\\hbar \\omega; z)", "ff4d29f22bd14ba16f13499ab2efb4fe": "\\psi_1\\frac{d^2}{d x^2}\\psi_2-\\psi_2\\frac{d^2}{d x^2}\\psi_1=0", "ff4d519f76bd4803a6a6db1c2cdff2cb": "\\int \\cos^n x dx\\ = \\frac{1}{n}\\cos^{n-1} x \\sin x + \\frac{n-1}{n} \\int \\cos^{n-2} x dx . \\!", "ff4d80883fe43fc74af5b8468dbbf65f": "\\nabla c_i", "ff4e4175748bcb6be2a745508845e7ef": "\\; \\langle Q|e^{-iHt/\\hbar}|\\psi\\rangle", "ff4e6e3874fad5179bbc47c2dab54e4e": "n^\\textrm{th}", "ff4eb8884555ad074e6572389afbc64a": "\\langle a^\\dagger \\sigma \\rangle ", "ff4ebf09340957a4e6d5a3e19fd66700": " A=-1; B=-1; C=-1; D=-1; E=1; F=2; ", "ff4f672aca14cf2df83f995bda9b57fe": "d(X,Y)=\\sup \\{ d(x,Y) | x \\in X \\}\\ ", "ff5015108fef02bd7d971641a1288814": "\\ \\Delta H(T)=\\Delta H(T_d)+ \\int_{T_d}^T \\Delta C_p dT", "ff501f221aebb683bf6c7638b8d8e50e": "\\kappa(t)= \\left|\\frac{x'y''-y'x''}{({x'^2+y'^2})^{3/2}}\\right|= {1\\cdot 2-(2t)(0) \\over (1+(2t)^2)^{3/2} }={2 \\over (1+4t^2)^{3/2}}.", "ff502e8d1be6c1c6e3a8c3bf86014766": "\\psi_1, \\psi_2 \\sim o(h)", "ff5063d925f29001724aef8ac6ae1787": "Eq.8", "ff507903240424f3697f485aa06ff791": "S_{\\frac{1}{2}}(x,y)", "ff507be89ac9d8ee2d52a9b9a276f5ce": "\\sum_{n=1}^\\infty\\operatorname{E}\\!\\bigl[|X_n|1_{\\{N\\ge n\\}}\\bigr]<\\infty.", "ff50b4aa1c9cc2197ef898436641c911": "Lo", "ff50b69dcea9ed0b8c375226e5f33f67": "\n\\times \\; _{q}F_{p-1} \\!\\left( \\left. \\begin{matrix} 1-a_h + \\mathbf{b_q} \\\\ (1-a_h + \\mathbf{a_p})^* \\end{matrix} \\; \\right| \\, (-1)^{q-m-n} z^{-1} \\right) .\n", "ff50ce873cad1957e20df7e69a46fd96": "\\frac{p}{p_{0}}=1+\\frac{2\\gamma}{\\gamma +1}(M^{2}\\sin^{2}\\theta -1)", "ff511763272fc8e2dcd0367cac4e48a3": "\\mathcal{L}_k", "ff5121604dd92a0e5268188a51013eb1": "N_1+N_2 = N", "ff5168b7da9ffc3374a931864336ec66": "y=\\begin{cases}a+DC & \\{ft\\} < 0.5 \\\\-a+DC & \\{ft\\} > 0.5 \\end{cases}", "ff516d92032172b8593639d60f7ea8b4": "(5+2)+1=5+(2+1) \\,", "ff519b5123ed3d685ed65db1c769af83": "\\{p_3,p_2\\}", "ff51bdb8508b4c4eff7ab102c1826e9e": "x^2+100=101 x", "ff51dab1b523e6b18adcefaee356ceac": "\\mu\\in\\mathfrak{g}^*", "ff51e404bf8b4cb5f9f2a0e8d63d4d85": "f(t) = akt^{a-1}\\!", "ff521445c9db37ad2b0d7fcd1f33b130": "\\frac{d\\psi_1}{dx}(-L/2) = \\frac{d\\psi_2}{dx}(-L/2) \\,\\!", "ff524256cd90b07cee09198728c6794e": "MX = MY, \\,", "ff52beae5e223b359128a2c7337942df": "\\text{Duration} = \\int^{t_2}_{t_1} \\sqrt{ \\frac{-1}{c^2} g_{\\alpha \\beta} \\frac{d x^{\\alpha}}{d t} \\frac{d x^{\\beta}}{d t} } \\, d t \\,", "ff52f24407ee7e63ae71c7aa620f3de9": "\\pi_{n-1} O(k) \\to \\text{Vect}_k(S^n)", "ff535bf3dc99df96173a1a597aac2f2b": "\\gamma_{total}=\\gamma_{1-l} + \\gamma_{l-2} ", "ff535df2d6e9b45b771da8ba56e2bbb0": " E\\left(\\prod_i \\xi(B_i)\\right) = \\int_{B_1 \\times \\cdots \\times B_k} \\rho^{(k)}(x_1,\\ldots,x_k) \\, dx_1\\cdots dx_k . ", "ff53a54f4678ccc59423a5deb10042da": "N^2 = \\frac{L_d}{L_{nl}}", "ff5418ad0aae4a5abf66dcbc6630088a": "A\\subseteq\\text{cl}(A)\\subseteq\\text{cl}(B)\\,", "ff54281cc32d72edc7f8bded2764c90b": " \\frac{\\partial Q}{\\partial t} = -k \\oint_S{\\overrightarrow{\\nabla} T \\cdot \\,\\overrightarrow{dA}} ", "ff548873d70140bd8c44a3ec898fbc2c": "U \\overset{\\Gamma_\\phi}\\to U \\times_S P \\to X \\times_S P", "ff54b3353ed1928f702c902b9d865813": "\n (1/f)(x+h) = \n (1/f)(x)+(1/f)'(x)h+\\cdots+(1/f)^{(d-1)}(x)\\frac{h^{d-1}}{(d-1)!}+(1/f)^{(d)}(x)\\frac{h^d}{d!}+O(h^{d+1})\n", "ff54ba032e3e5f00feb94a3824b7a270": " f^{\\lambda} ", "ff54bdc54addf65556c134fd6a9a5a82": "K^a=\\frac{2 \\pi}{c}N^a", "ff54d1fb5c7355bd11a1cb675b37b590": "b_0(x) = 2a_0 + 2xb_1(x) - b_2(x),", "ff5512e8154ffc26e7556c30673b586d": "\\langle x_0,x_1,\\ldots,x_{n-1},x\\rangle \\in T", "ff552606b06a5a7d593582639ee5e704": "\n\\frac {\\partial} {\\partial x} F_1(a,b_1,b_2,c; x,y) = \\frac {a b_1} {c} F_1(a+1,b_1+1,b_2,c+1; x,y) ~,\n", "ff552c7c7c51551eae65de12a6869d9c": "\\mu(p,T) = \\mu_0 + V_{m0}(p-p_{f}).", "ff555249b82414e742f07e837636e6f7": "^{\\;}I^{kl}", "ff557b9f27427a0fa2ca0aafda98b234": "\\begin{align}\n \\Delta\\omega &= \\frac{R_0}{L} \\\\\n \\omega_m &= \\sqrt{\\frac{R_0^2}{4 L^2} + \\frac{1}{LC}}\n\\end{align}", "ff55b9aa784debdaa5ff5099416e809c": "\\partial\\{({I}^{2},{\\varphi}_{\\lambda},{S}_{\\lambda})\\}_{\\lambda\\in\\Lambda}", "ff55cf4cd802ad8417d43d67d9313b32": "\\frac{\\part^2 L}{(\\part f')^2} \\ne 0, ", "ff55d8ad3b266824efc79cc8a7c8a787": " \\phi_{0}(r) = \\theta(R-r) \\phi_{0}. ", "ff55f03bd06079a37277330c51b2f24d": " \\operatorname{Cov}(\\mathbf{1}_A (\\omega), \\mathbf{1}_B (\\omega)) = \\operatorname{P}(A \\cap B) - \\operatorname{P}(A)\\operatorname{P}(B) ", "ff56044e4b88f8da913d60713dcdfa03": "\nA = x(0)+\\frac{\\gamma_+x(0)-\\dot{x}(0)}{\\gamma_--\\gamma_+}\n", "ff563b4f6a53bac8db3b2c132ef8d337": "\nv = \\sum_{i=k}^n \\alpha_i u_i\n", "ff56422091146291d79d379317da3d17": "\\varepsilon \\approx {u'}^3/L,", "ff5678721242800eb0ce7b4f73170647": "U_{str}= \\frac{\\epsilon_{rs}\\epsilon_0 \\zeta}{\\eta K_L} \\Delta P", "ff5683f822a8a0b1ae09ecc4a3465481": "\\{(x,y) \\mid x \\leq y \\wedge x \\neq y\\}", "ff56e122d4323efb536528856d6a2142": "A_\\lambda = C \\,\\ell\\, \\sigma", "ff56ea8f12f488581e90d6d55694868f": " U(W) = \\frac{1-\\gamma}{\\gamma} \\left(\\frac{aW}{1-\\gamma} + b\\right)^{\\gamma} ", "ff5705da1b4473f221b05cf6fce5b92f": "M_r \\left ( {\\rm Y} \\right ) = \\sum_j N \\left ( {\\rm X}_j \\right ) A_r \\left ( {\\rm X}_j \\right ) = \\frac{\\sum_j N \\left ( {\\rm X}_j \\right ) \\langle m \\left ( {\\rm X}_j \\right ) \\rangle }{m \\left ( ^{12}{\\rm C} \\right ) / 12} ", "ff575fd7f0bcb75ffecd49b64d0a4eec": "w^0, \\varphi_1, \\varphi_2", "ff5777721b35f3548313018cc0ae6bb1": "\\Psi <\\tfrac{2}{3}", "ff578594e464aa4e562a17b9bfe11751": "\\mathfrak{sl}(6,\\mathbb C)", "ff578aa9da783c7668544b33daf61c74": "\\csc A=\\frac{1}{\\sin A}=\\frac{c}{a} ,", "ff58498ad9333bc372f5a66e52655a70": "\\mathrm{dist}(y,L_1(B))>2/3", "ff5857999436d6cbe1164fed6317adba": " g(n) = \\sum_{d \\mid n} \\mu(d) f(n/d) ", "ff5877a8807d73e69cd0d82b0a3f2b7c": "\\sigma_v ", "ff5883e9ca982bb9bd048f9aefb3dc1f": "\\gamma(u) =\n\\int_{s \\in \\mathbb{S}} |u^Ts|^\\alpha \\Lambda(ds)\n", "ff5886c66b3ff46b87bb86e4169a07ef": "N\\left(g_{n}\\right) = n + 1.", "ff58b680198e753f55ae48f5a22b37cf": "f(x) = \\int_0^{+ \\infty} 1_{L(f, t)} (x) \\, \\mathrm{d} t \\text{ for all } x \\in \\mathbb{R}^{n},", "ff58c6c6d72a805ba0e9886f37056691": "[A]_0", "ff59500b72784cf8edc75a4b9c6d4f5b": "\\hat{\\mathcal{S}}=\\mathcal{S}-i\\ln(M)", "ff5956bba34e1813fac046c4559cfb44": "\\sigma^i(y_j),\\ (i\\in\\mathbb{N}, 1\\leq j\\leq n)", "ff59655866b3774157fc63fe5570e626": "V(r)=0", "ff596c45d60dbf4fb25f8f0e4fa74199": "Q = \\{ \\mbox{A}, \\mbox{B}, \\mbox{C}, \\mbox{HALT} \\}", "ff599fabbebb657eee89e471d763566b": "a = \\begin{pmatrix} e^{r/2} & 0 \\\\ 0 & e^{-r/2} \\end{pmatrix},", "ff59b2f6f037fb415f23790675eb9d19": "A_n=\\sum_{k=1}^n\\bigl(\\mathbb{E}[X_k\\,|\\,\\mathcal{F}_{k-1}]-X_{k-1}\\bigr)", "ff59e1bce45a91e047601ed6f4101e9e": "N_e = N + \\begin{matrix} \\frac{1}{2} \\approx N \\end{matrix}", "ff59eb5e85def591f6a420ebfbf3362d": "I_{D} = I_{0} \\left\\{\\exp\\left[\\frac{qV_{j}}{nkT}\\right] - 1\\right\\}", "ff5a45b41a0907c5c0988044efcc5a9a": "\\Sigma^1_{n+1}", "ff5ad5dd20bd1e365512d43d0274835c": "y_1 \\succ y_2 \\succ \\cdots \\succ y_k\\,\\!", "ff5b540e2846563d972a7d17c5946526": "L_h(f*g) = (L_hf)*g.", "ff5b6256194c0ed0f7adb8be52622406": "q_{ij} = \\frac{(1 + \\lVert \\mathbf{y}_i - \\mathbf{y}_j\\rVert^2)^{-1}}{\\sum_{k \\neq l} (1 + \\lVert \\mathbf{y}_k - \\mathbf{y}_l\\rVert^2)^{-1}}", "ff5b8a5881a2d87147a615bd266cce01": "\\exp_a(x) = a^x", "ff5bf36e0c1b93656073136b974268e2": "a=1/3", "ff5bfa40fc1fa91f7d5646f5b15fa161": "\\triangle \\mathrm{ABC} \\cong \\triangle \\mathrm{DEF}", "ff5c23901f43f3c96bc2f10309e30ba8": " \\Delta G = \\Delta H - T \\Delta S_{int} \\,", "ff5c3cb535d67ad869820ad12dea245c": " B_k=\\sum_{m=0}^{k}(-1)^m{\\frac{m!}{m+1}}S(k,m)", "ff5c783fd5be3e6166dc36ed49b32c24": " X_k \\in S_j", "ff5cc1977521a446dc0f57481ed7cc49": "\\mathbf{u}' \\propto \\exp(i \\alpha (x - c t))", "ff5cd7bc3908d7957dbd924049891f35": "\\Phi = \\int \\mathbf{B}\\cdot d\\mathbf{A}", "ff5da474445f2a40e3da82432192d114": "\\textstyle\\sum_{i=1}^r", "ff5dae60fa56f3a1ba1e97ce4796f22a": "\\mu_{01} = 0,\\,\\!", "ff5db0c5031f8a572f9f7edb6bde0044": "\\textstyle\\sum_i \\alpha_i K(x_i,x) = \\mathrm{constant}.", "ff5e1e093365da5ef35006c081aa2cbe": "|n_{{\\mathbf{k}}_{l}}\\rangle", "ff5e33265e1a4fd57f600e36fb934267": " x = w - y \\!", "ff5e720a66b84e670b9d9a686571cc1c": "\\{ \\mathbf{e}_{k} \\}", "ff5e9bc60b5dc37186c8f147db39cca8": "(v_1,v_2,v_3). \\, ", "ff5edd8727dc22e458f06d5eb9d0440d": "b_3=3\\ \\frac{y_2-y_1}{(x_2-x_1)^2}", "ff5ee7b66a9e5bb2502ce549effd494e": "R_r(N) = 1 - \\frac{f_{IR}}{f_\\alpha (1 - E_f/E_g)}", "ff5f68e6621d4356d9c121151c0042fc": "f^*(c)=0", "ff5f6e466f380417c5d9a315d478c56a": "\\nabla_{\\vec{e}_0} \\vec{e}_0 = \\frac{1}{x}\\vec{e}_1", "ff5f8a12cfccdcd8027b600394a7b026": "N_1(X)", "ff5fbde9560768ff1f07ec7adb70d629": "d = 69 + 12 \\log_2 \\left(\\frac {f}{440\\ \\mathrm{Hz}}\\right).\\,", "ff5fca7d2b871e0f884c094014c94d59": "\\{((0, 0), 1), ((0, 1), 1), ((1, 0), 1), ((1, 1), 0)\\} \\,", "ff6013ffec6f41ca134b97b66bb63d0f": "\\sum_{i=1}^{p-1} i^{p-1} \\equiv -1 \\pmod p,", "ff603ac7442120ee7a3233d7da44ac74": "\\nabla\\colon \\Gamma(A)\\times\\Gamma(\\nu(A))\\to\\Gamma(\\nu(A)): \\nabla_{\\!\\phi\\,}\\overline{X}:=\\overline{[\\rho(\\phi),X]}.", "ff60a9a90b197dd5b9ed0fd6d45a3118": "b>a", "ff60b538e0a5ec03cc04c50afd49ba8b": " \\hat{B}_{ij} = -\\frac{1}{2r_{ij}} \\left[ \\mathbf{a}(i)\\cdot\\mathbf{a}(j) + \\frac{ \\left(\\mathbf{a}(i)\\cdot\\mathbf{r}_{ij}\\right) \\left(\\mathbf{a}(j)\\cdot\\mathbf{r}_{ij}\\right) }{r_{ij}^{2}} \\right] ", "ff60c79158503283e50aa6524a12db1e": "N\\to \\infty ", "ff6130c8b3c196816036915341344c3d": "\\langle x, y \\cosh R \\rangle \\,", "ff6189793577e036678d064d07ab7ee6": "\\mathcal{F}_{\\rm Bol}", "ff61a630f3a1230d545bd122f8ebaede": "\\sin y=x\\,\\!", "ff61ba528303e8548da25c9f26bd7d55": "\\mu_{B}", "ff622ff2515535a922fbe98d92ac0654": "\\sin \\left[(kp+j)\\frac{q\\pi}{p}\\right]=(-1)^{kq}\\sin\\frac{qj\\pi}{p}", "ff62897cffb11277c38073b51dc55edc": "f(x)=\\frac{1}{x}\\,", "ff628d219b7c99bc23590ee5d9dce05b": " {A_\\mathrm{i}} = {i_\\mathrm{out} \\over i_\\mathrm{in}} ", "ff629406948d2fe82ce04eb53ab799d1": "\\lesssim", "ff62a4f6c61c44b3c8c7ca86e42fb692": "x=\\mathbf{X}", "ff62e6f525c603522e740f64babca4e6": "\\mathbf{v'}_r = \\mathbf{v'}_{p2} - \\mathbf{v'}_{p1}", "ff636315e3ef5b9334d38b7247b8948b": "f(\\vec{x}) = a_1x_1 + \\cdots + a_n x_n.", "ff637e8096338317b662c5da703d0631": "\\phi_y = |y\\rangle", "ff63aebc7796c1e1adb714f02535bde0": "(\\cos \\varphi,\\ \\sin \\varphi)", "ff63e8f284e730509c2cc401023ca4b9": "E_{xc}^{\\rm \\omega PBEh} = a E_x^{\\rm HF,SR}(\\omega) + (1-a) E_x^{\\rm PBE,SR}(\\omega) + E_x^{\\rm PBE,LR}(\\omega) + E_c^{\\rm PBE},", "ff64221611b4687f04c844cc8087935f": "\\rho^*g^*H^* = (1)g^*L^* = 1 \\,", "ff642c51963757220323171f9f545059": "\\mathcal M_j", "ff6452d93b779c919390abd7e0aadeac": "\\tau'", "ff6477b868eb126765a74c1a4a8c84a5": " \\Bigl[ \\frac{\\Delta_h}{h} ~,~ x\\, T^{-1}_h \\Bigr] = [ D ~,~ x ] = I ~.", "ff64f00dfd82f44ef95ea31060bab786": "\\scriptstyle \\sigma_{\\rm metal} ", "ff6536965e8c13af48b7be5773a08222": "\\mathbf{\\hat{u}} = \\frac{\\mathbf{u}}{\\|\\mathbf{u}\\|}", "ff664944ba28b9b207f39f772c8ec0c1": "d_T(z) = d_H \\int_0^z \\frac{d z'}{(1+z')E(z')} ", "ff664bd389377827fcca6fe6f46859fd": "\\begin{bmatrix}e'_{1} & \\cdots & e'_{n}\\end{bmatrix} = \\begin{bmatrix}e_{1} & \\cdots & e_{n}\\end{bmatrix}S", "ff667b6c5ce10db531f562a86e77cd16": "\\neg C", "ff66a3d220f1ca6b785cc4c8c4298a48": "m = \\left(\\frac{\\alpha}{n}\\right)", "ff66d2d970487a3680351a4205197cce": " \\operatorname{de-let}[x]\\ \\operatorname{de-let}[x] ", "ff66decd8ea4bff500c95fb81da6bd2e": "\\begin{align}\nG(r,c) & = \\sum_{k=0}^\\infty cr^k & & \\\\\n & = c + \\sum_{k=0}^\\infty cr^{k+1} & & \\mbox{ (stability) } \\\\\n & = c + r \\sum_{k=0}^\\infty cr^k & & \\mbox{ (linearity) } \\\\\n & = c + r \\, G(r,c), & & \\mbox{ whence } \\\\\nG(r,c) & = \\frac{c}{1-r} ,\\mbox{unless it is infinite} & & \\\\\n\\end{align}", "ff66e061a06917523903631ca099629e": "\\alpha_k =\\frac{k-1}{2k}\\ .", "ff675aa844eb51b743c5edf72e103b6b": "C=\\Pi + wL \\,", "ff67731e89f190a82a9113de5d3f35cf": "\\chi_3", "ff6793c557ec317e8534e992db37f904": " F(x)\\neq 0 \\, ", "ff67b80498efb0486f3c1713b564088c": "\\varphi \\implies \\psi", "ff67d2afb3aab9d6661b3ddda37bfd34": "X^{k(i)}", "ff68409b7ff6c6010b4cf1887fafcfd5": "\\Phi_{Y,X} (f:F(Y)\\to X) = G(f)\\circ\\eta_Y", "ff684d3803b37bc8aed049babaa92716": "\\gamma_c(A) = {d^2_{min}(A) \\over 4E_b}.", "ff685f16c1c2861f893096594b9e5f51": "\\ln(\\phi(q))=-\\sum_{n=1}^\\infty\\frac{1}{n}\\,\\frac{q^n}{1-q^n}", "ff689f503424b296a2cd78b4fe2f2333": "\\{p,\\ p\\land q \\rightarrow r,\\ p \\land \\neg q \\rightarrow s\\}", "ff68b2b4858df84830261233575a105c": "v_\\theta = -\\frac{\\partial \\psi}{\\partial r}.\\,", "ff696350e56893f5e7f7146e8f04005f": "\\{0\\} = J_0 \\subset \\cdots \\subset J_n = M", "ff699a3e0bab1887032f0f37ea66da07": "e \\cdot \\cos \\theta = \\frac{V_t} {V_0} - 1", "ff69d75300e117dc9559bde6d6694224": "\\varphi(x)^T\\varphi(y)", "ff69da8e04f8a6d71c800f6ff1b0da96": "g = a_g - a_c\n= \\frac{G m}{r^2} - \\frac{4\\pi^2r}{T^2}", "ff69f84059d5ee795fc6a3c462e34f45": "-\\mu(-T)>0", "ff6a1137dada18964b7c1a487519eacc": " \\mathbf{F}\\cdot\\frac{d \\mathbf{r}}{dt} = \\mathbf{F}\\cdot \\mathbf{v} = \\int_V (\\rho\\mathbf{E}\\cdot\\mathbf{v}+\\rho\\mathbf{v}\\times\\mathbf{B}\\cdot \\mathbf{v} )dV \\ \\rightarrow \\ \\mathbf{F}\\cdot \\mathbf{v} = \\int_V \\mathbf{E}\\cdot \\mathbf{J}dV .", "ff6a17f35d4a2ce806698664e031c27b": " s_i ", "ff6a262d7971358c2cb3bd7d45716a47": "\\begin{align}\n\\rho\\cos\\phi &= \\sigma \\tau\\\\\n\\rho\\sin\\phi &= \\tfrac{1}{2} \\left( \\tau^{2} - \\sigma^{2} \\right) \\\\\nz &= z \\end{align}", "ff6a43798c87ebb2fa1b334512f7e244": "\\textstyle \\sum_{n=0}^\\infty a_n", "ff6ab1162e1c7b4a5d5929a4a116a591": "x \\in \\mathbb{Z}", "ff6ad814dab17293fe32368cd8a760ab": "\\Psi_\\alpha(\\mathbf{x})=\\sum_\\beta \\int d^3\\mathbf{x'} G_{\\alpha \\beta}(\\mathbf{x},\\mathbf{x'}) V_\\beta(\\mathbf{x'})", "ff6b19e2425bd02be3a667200e4f4449": " \\dot x(t) = - u(t) ", "ff6b1a28a41cdb7b87ad2d490783688a": "\\sum_{m\\ge 1}\\frac{X^{(m)}Y^{(m)}}{m}t^m=\\sum_{m\\ge 1}\\frac{\\sum_{d|m}d X_d^{m/d}\\sum_{e|m}e Y_e^{m/e}}{m}t^m", "ff6b741f9aa763c529b11cd112f19d1d": "\nFSC(r) = \n\\frac{\\displaystyle\\sum_{r_i \\in r}{ F_1(r_i) \\cdot F_2(r_i)^{\\ast} }}\n{\\displaystyle\\sqrt[2]{\\sum_{r_i \\in r}{ \\left|F_1(r_i)\\right|^2} \\cdot \\sum_{r_i \\in r}{\\left|F_2(r_i)\\right|^2}}}\n", "ff6c92ec7812b314c2cb80a404f23010": "\\tbinom{7}{2}=6\\times\\tfrac{7}{2}=21", "ff6c948adb4f9752beca383a5029e023": "\\forall x\\, x \\rightarrow", "ff6caa1b99730967768d05bafaeaeaf7": "\\frac{\\text{BE}}{A \\cdot \\text{MeV}} = a - \\frac{b}{A^{1/3}} - \\frac{c Z^2}{A^{4/3}} - \\frac{d \\left(N - Z\\right)^2}{A^2} \\pm \\frac{e}{A^{7/4}}", "ff6d3074c84df862710c24a77c36279e": "= \\ u_x - j^2 v_y + j (v_x - u_y).", "ff6d3c8f542ba32740dbed612ee55897": "n_{veh} = \\frac{3600}{T_{min}}", "ff6d6edd9278991c052ede95de21f609": "\\Theta(\\log\\log n)", "ff6d7dbd31c740f8a0b35d2d0dea021f": "t=\\alpha z", "ff6d9f25620853d2869cb8eb608e50fc": "\\text{local} \\colon (E \\rarr E) \\rarr (E \\rarr T) \\rarr E \\rarr T = f \\mapsto c \\mapsto e \\mapsto c \\, (f \\, e)", "ff6dd4aec9193f9cd198e643c5e2d5b0": "f_i(w)", "ff6e11574d3c399279ae9077dab6095c": "d(\\ell_g \\varphi) = \\ell_g (d\\varphi).", "ff6e471c13d3548addd218fd5590e4d5": "{\\mathcal C}", "ff6e5692bcaf34e4e502c23d3c2c5d9d": "P_{10}(x)=4x^3+1\\,", "ff6e5c12fadde0392ff78d584542ef4c": "\\mathbf{End}_{C}", "ff6e655d8776450bdc04e017d9c7f063": "R_{3,2} = 7 r^3-6 r^2", "ff6ead09b0d6e572e993b771e44773ec": "p(\\theta|x) = \\frac{p(x|\\theta)p(\\theta)}{p(x)}.", "ff6eeb2096c4e7f1ac3c38c34c4dd83c": " 1 \\to \\mu_2 \\rightarrow \\mbox{Pin}_V \\rightarrow \\mbox{O}_V \\rightarrow 1\\,", "ff6efac30c16b73afe8232936dc35b4d": "\np_{v} = \\phi p_{\\mathrm{sat}} \\,\n", "ff6f74897dedd7350bfc1050d61a6021": "c, p, q", "ff6f8abbef8ef157ba1d84be43a4eb77": "R(r,s) \\leq \\binom{r+s-2}{r-1}. ", "ff6fc5eca946c0841629d8faebed1a10": "X_1 \\dots X_n", "ff6ff4f2487e1dc81971ff2b1d0e8e42": "(x,y)_z=1/2(d(x,z)+d(y,z)-d(x,y))", "ff705f1b6f527fe2038d5ad152f64769": "s ", "ff708f96d0b78c96d596f64abb4c5e2a": "\\varepsilon(\\alpha+\\beta,\\gamma)\\varepsilon(\\alpha,\\beta)=\\varepsilon(\\alpha,\\beta+\\gamma)\\varepsilon(\\beta,\\gamma)", "ff70b50d8076c7e03bcb6cc057b369c8": "\\scriptstyle P \\oplus (I - P) = \\mathbb{R}^n", "ff70f5487ee8638af5162ea66235312e": "r \\cdot C_T", "ff71108121275eadef50c9f3962ea2c8": "\\cot\\frac{\\pi}{5}=\\cot 36^\\circ=\\tfrac15\\sqrt{5(5+2\\sqrt5)}\\,", "ff714de7f4d0780b0fc326fd18a1e860": "\\frac {\\mathrm{d}\\theta}{\\mathrm{d}s} = \\frac{1}{\\rho} = y''(s)x'(s) - y'(s)x''(s)\\ ", "ff7152b8e457bbc163c52f9fc351e279": " \\beta^* = -\\frac{1}{\\rho} \\left ( \\frac{\\partial \\rho}{\\partial C_a} \\right )_{T,p}", "ff718c6600304bad5670c660869019af": "S=\\begin{pmatrix}-\\alpha I_p & 0\\\\ 0 & \\alpha I_q\\end{pmatrix}", "ff71e56174375e7854e7d239a4d530a1": "GEH=\\sqrt{\\frac{2(M-C)^2}{M+C}}", "ff71f1a88c43feb90c27e4d1ebc9f476": "\\Delta S = \\int_{Q_1}^{Q_2} \\frac{\\mathrm{d}Q}{T} \\,\\!", "ff7282d9bdc170f1bd36cfbc500c9249": "\\mathbf{n} = \\mathbf{p}/ |\\mathbf{p}|", "ff72f6883b01827a90f5fff99ff45172": " p(t) = \\frac{1}{\\tau_0} \\exp(-t/ \\tau_0). ", "ff7378a7f147dbea2ca4cbaf4fc20cab": "\\iota x", "ff739643114f1b75d2bd1aea89eacaa5": "\\chi_\\text{m}", "ff73bf5a63e92903c6d4acacf7becf5d": "\\begin{align}\n f \\circ \\operatorname{id}_X &= f , \\\\\n \\operatorname{id}_Y \\circ f &= f .\n\\end{align}", "ff73d92934be6a66ac254d660b9577b9": "g_{44}=B\\left(r\\right)", "ff73ea76d7fb71e7c0913c20bdd2f909": " \\textstyle \\left \\langle {(\\Delta p)}^{2} \\right \\rangle ", "ff740f19c0caaf5282002c8a95a1333d": " \\rho_j =\n\n\\begin{pmatrix}\n|a|^2 & ab^{*}e^{-\\alpha} \\\\\n\na^{*}be^{-\\alpha} & |b|^2 \n\\end{pmatrix}\n.", "ff7489b7e6c3e005fba2d6d7dee959bf": "-h(p_i)", "ff74a77aa4b1828ca5526e29f4140b29": " \\Sigma X", "ff74adfc5d0e4a0c5c71485e4d80d66d": "k_i=p^{m_i}k_i \\pmod{N}", "ff750434fc38a19788919ac4dba58b20": "\\forall x \\forall y ( x + y \\leq z) \\to \\forall x \\forall y (x+y = 0).", "ff750450c0072827704623887ed739a6": "\n\\begin{matrix}\n Q^T \\,Q = I.\n\\end{matrix}\n", "ff7509efd627c49db519da2d81edf8aa": "\\scriptstyle g \\in C^1(\\Omega^\\prime,\\Omega)", "ff752b7c3d6c8370bcfde42899464ef3": "\\sigma_{n}^{\\pm}", "ff7556d96d7b6301f8c2b0a1621d7660": "\\{0,1,\\ldots,k-1\\}", "ff755d11e8fcba03942f13a6a850492f": "\\vdash \\Box A", "ff76045714170db6eb59ecc1eee4107d": " L_\\delta (a) = \\delta (|a| - \\delta/2 ), \\qquad \\text{otherwise}. ", "ff76136b036592c3d13493093d834e57": "\\mathbf{x} = (x_1, x_2, \\dots, x_k)", "ff763a31968eccacf8cc99b9144487cb": "\\beta_6 = p_l", "ff775137f7a51a6d62e915853b2bc107": " |s_\\nu| \\ge 2 \\left({ \\frac{N}{8e(M+N)} }\\right)^N \\min_{1\\le j\\le N} \\left\\vert{\\sum_{n=1}^j b_n }\\right\\vert \\ . ", "ff77544fc57f4e361a487648dc0465f9": "T(L)", "ff77973f8eb49d00a8b02f6ef767bc66": "n \\not|a ", "ff77c1834d29308c704e26b316f030c3": "\\mathrm{rect}(t) \\cdot \\mathrm{sinc}(t)", "ff77f7b3241a635926bdf96eb828bb7e": "\n\\Phi(\\mathbf{r}) = \n\\frac{q}{4\\pi\\varepsilon} \\sum_{l=0}^{\\infty}\n\\frac{r_<^{l}}{r_>^{l+1}}\n\\left( \\frac{4\\pi}{2l+1} \\right)\n\\sum_{m=-l}^{l} \nY_{lm}(\\theta, \\phi) Y_{lm}^{*}(\\theta^{\\prime}, \\phi^{\\prime})\n", "ff789e695b457e092c161ad553ea513a": " g_N(r) = \\frac{1}{|G(N)|} \\sum_{a \\in G(N)} h\\left({\\left\\langle{\\frac{ra}{N}}\\right\\rangle}\\right) \\sigma_a^{-1} \\ . ", "ff793465fa21737e04202f48a47280c7": "{} - 432 (x^2 z^5 + y^2 z^5) + 81 (x^4 y^2 - x^2 y^4) + 240 (y^2 z^4 - x^2 z^4) - 135 (x^4 z^3 + y^4 z^3) = 0.\\ ", "ff793f4dbd76835095109ffad88e3956": "E_{el,i} = z_ieV_i = \\frac{e^2}{4 \\pi \\epsilon_0 r_0 } z_i M_i.", "ff79560a37a7c6cddb7e60c53f17e5fa": "\\delta x \\delta y", "ff7988d95afb43de2cde214e4092bf4b": "P(M|E)=0", "ff799795b34645ff3fc33d637dacfec0": " \\frac{1}{|\\mathbf{x} - \\mathbf{x'}|} = \n\\sum_{l=0}^\\infty \\frac{r_<^l}{r_>^{l+1}} P_l(\\cos\\gamma),", "ff79b558165b70aed81380f349aca54b": "a^4+b^4+c^4+d^4 = (a+b+c+d)^4", "ff79de74e50d787146838851645e8c13": "A_n \\sin(n \\theta) ", "ff79f002f0e5b72efa267094746a089a": "A=\\bigcup P", "ff7a1b9a91a9cff6f4be7f1c0359f5b2": "(s^2+0.3902s+1)(s^2+1.1111s+1)(s^2+1.6629s+1)(s^2+1.9616s+1)", "ff7a300fffc9522002a5e05263419306": "viz_i", "ff7a3679e326cac3453f11641e49499e": "S(c,b)\\,\\!", "ff7b0a398485352bfbbf3be1f403a508": "\n |((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\\rangle = \\sum_{j_3}\\sum_{j6}\n | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\\rangle\n \\langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\\rangle.\n", "ff7b67dc509c41b3ad01e6f8c81a8ee2": "\\det(E) = \\det(X) \\det(D^t)", "ff7b924fb61eb1fd976a67462527b0af": "h_A(x)=|x\\cdot a|", "ff7bc415581b50de27ca0f4eff3e37c7": "X_\\xi=\\frac{\\mathrm d}{\\mathrm dt}R_{h(t)}\\biggr|_{t=0}.\\,", "ff7c27ac8253830cf5498aedd293e35a": "\n\\beta(\\phi)=\\frac{\\nu(\\phi)}{\\rho(\\phi)},\\qquad\n \\eta^2=\\beta-1={e'^2\\cos^2\\!\\phi}.\n", "ff7cf9b100d8e030f9d08536c0fd3f8d": "x^*=\\left(1/b\\right)\\ln \\left(1/\\eta\\right)\\text {with }0 < F\\left(x^*\\right)<1-e^{-1} = 0.632121", "ff7d3599ab8882ab8fb32cb5b22e0c9c": "\\mathbf{Mod}(T,C)\\cong \\mathbf{RegCat}(R(T),C)", "ff7d379b99ea8531b343ab5b0ec948d3": "\\rho[\\sigma]", "ff7d5d9e09880d96b08ca2fba23511d2": "H \\approx {\\rho^2 \\over 2m} (qA + \\nabla \\theta)^2. ", "ff7d98f7ec58dfa8f23225b32efc8a51": "\nG_{ij}=\\text{exp}\\left ({-\\alpha_{ij}\\tau_{ij}}\\right )", "ff7dcebd137a28fc8c216e66fb7dc266": " -\\frac{\\hbar^2}{2m}\\nabla^2 \\psi = E\\psi \\quad (1) ", "ff7ddf437cd0a5768365f484d8ac62eb": "k_h=-k", "ff7eb254be2f1e8f296b4cb28f0f1d6a": "\\frac{d\\nu}{d\\lambda} = \\frac{d}{d\\lambda}\\left(\\frac{c}{\\lambda}\\right) = -\\frac{c}{\\lambda^2}", "ff7eb32ef312ce8b2b6d5a0bb2594f42": " \\Phi: L^\\infty(X, \\mu) \\rightarrow L^\\infty(Y, \\nu) ", "ff7ee8dc9952648654a0ea95c045a6d3": "\\tbinom{j}{m+1}", "ff7f0bada015aa10cb88b0d2731676b7": "\\lim_{x\\searrow0}\\frac{e^{-1/x}}{x^m}\n\\le (m+1)!\\lim_{x\\searrow0}x=0.", "ff7f4b8ead007368a447bfed8f565802": "\\mathrm{spec}f(A)=f(\\mathrm{spec}A)", "ff7f6926280435d1fc10dd95ce1452da": "c \\equiv a \\times b \\pmod {N}", "ff7fa304ecf57b573120bafee2b1cb58": "t<0", "ff7fcc2a5adbdbf54b24b2f8a140d2ca": "\\sqrt[n]{x} = \\exp \\left( {\\frac{1}{n}\\ln x} \\right)", "ff7fd15aae47d870ea27bc6ad54d6cc7": "\\widehat \\alpha = \\frac{n}{\\sum _i \\left( \\ln x_i - \\ln \\widehat x_\\mathrm{m} \\right)}.", "ff7fe8e00da687d7951326c6839ce8eb": "\\alpha_{00} = \\frac{p-5}{4},\\;\\alpha_{01} =\\alpha_{10} =\\alpha_{11} = \\frac{p-1}{4} ", "ff7feadd6d7e13cb1c0936de3b20d465": "p(D|w,b,\\log \\zeta ,\\mathbb{M}) = \\prod\\limits_{i = 1}^N {p(x_i ,y_i |w,b,\\log \\zeta ,\\mathbb{M})} .", "ff7fed8963eee39c2f440f0616826107": "\\delta^n[f](x - h/2)", "ff800670de33251a13ed458a6b1c86c7": "M=\\beta \\cdot (\\max{(\\sum_{i=1}^{N}{Assets_i},\\sum_{i=1}^{N}{Revenues_i})})^{\\alpha} .\\,", "ff80184c6873842aead4ddb510fe3201": "O\\Bigg(k^2\\log(n L_1(f))\\Bigg\\lceil{n L_1^2(f)\\over2^k}\\Bigg\\rceil\\Bigg)", "ff8036686b933d0b05effd40cebd99f1": "n_{1, t+1} = s_0n_{0, t}", "ff803d7696eb59c01823901b18f15ee9": "\n\\begin{array}{lr}\n\\textbf{u}\\cdot\\textbf{v} = \\displaystyle\\frac{1}{2}\\left(\\|\\textbf{u}+\\textbf{v}\\|^2 - \\|\\textbf{u}\\|^2 - \\|\\textbf{v}\\|^2\\right),\\quad & (1) \\\\[1.5em]\n\\textbf{u}\\cdot\\textbf{v} = \\displaystyle\\frac{1}{2}\\left(\\|\\textbf{u}\\|^2 + \\|\\textbf{v}\\|^2 - \\|\\textbf{u}-\\textbf{v}\\|^2 \\right), & (2) \\\\[1.5em]\n\\textbf{u}\\cdot\\textbf{v} = \\displaystyle\\frac{1}{4}\\left(\\|\\textbf{u}+\\textbf{v}\\|^2 - \\|\\textbf{u}-\\textbf{v}\\|^2 \\right). & (3)\n\\end{array}", "ff8050f1919c621cb57065ac9767463d": "\\scriptstyle 1.4 f_s,", "ff8050f7f8441882c0b68d612b874dd8": " \\mathbf S(p) = \\frac {1}{g(p)} \\begin{bmatrix} h(p) & f(p) \\\\ \\pm f(-p) & \\mp h(-p) \\end{bmatrix} ", "ff80782e21a1936b4ddfcf91433fe817": "E_\\mathrm{k} = \\int \\mathbf{F} \\cdot d \\mathbf{x} = m \\int \\frac{d\\mathbf{v}}{dt} \\cdot \\mathbf{v} dt = m \\int d\\mathbf{v} \\cdot \\mathbf{v} = m \\int \\frac{1}{2} d (\\mathbf{v} \\cdot \\mathbf{v} ) = \\frac{1}{2}mv^2\\,\\!", "ff80bceb0da3b6ac6f1e01a857ace3ff": " h_k (X_1, X_2, \\dots,X_n) = \n\\sum_{l_1+l_2+ \\cdots + l_n=k; ~~ l_i \\geq 0 } \nX_{1}^{l_1} X_{2}^{l_2} \\cdots X_{n}^{l_n}.", "ff80c67260a8a5e5f1da6dbea487bee0": "A_{2N}", "ff80cfab5c4e92d70e581ca37151e614": " \\frac{\\int_\\Omega f(x)\\ w(x) dx}{\\int_\\Omega w(x)\\ dx}", "ff81c52045cc3379fe1b5e71a0d44d03": "\\theta =0{}^\\circ ", "ff8212d90c6001a3dc4a66fa2f017719": "\\operatorname{Tr} \\rho (b_1\\otimes b_2) = \\operatorname{Tr} \\rho E(b_1, b_2)", "ff8232b24f3beeaf6d6dd6369a555370": "B_{ind}\\left(\\omega\\right) B_{cap}(\\omega) = -1", "ff825dd2047c768214918813ea9172f5": "\\mathcal{P}(\\Bbb{R})", "ff8272d4858f3cfcb6612b05c230f2c6": "S_{t}", "ff8295a21669781153f1e3fd5ee49142": "X \\subseteq \\bigcup_{\\alpha \\in A}U_{\\alpha}.", "ff82e2811654bc55211122600d1a71c7": "1 \\ \\text{Ry} \\equiv h c R_\\infty = 13.605\\;692\\;53(30) \\,\\text{eV}.", "ff82e3139ffe08e2da94e40bd8158fb7": " \\frac{dP_k}{dt}=\\sum_\\ell(A_{k\\ell}P_\\ell - A_{\\ell k}P_k)=\\sum_{\\ell\\neq k}(A_{k\\ell}P_\\ell - A_{\\ell k}P_k). ", "ff8303474cdabe9720c0923dff2fec30": "\\hat{\\pi}_{\\psi^T}= |\\psi^T\\rang\\lang\\psi^T|, ", "ff831a021fa5cc6788dfe3031c529bdd": "\\phi_B^{-1}(\\alpha_1,\\ldots,\\alpha_n)=\\alpha_1 b_1+\\cdots+\\alpha_n b_n.", "ff833e1d7164724848a4f148ed8aa782": "\n (\\hat{v}_1, \\hat{v}_2, \\hat{v}_3) = (v_1, v_2/r, v_3) =: (v_r, v_\\theta, v_z)\n ", "ff83e6070b203c3b34cce1dc273d0c99": "u=0.9", "ff841d2f9d95e7cb57e8be0e95073974": " P_{i}(A)=\\{(x_{1}, ..., x_{i-1}, x_{i+1}, ..., x_{d}) : (x_{1}, ..., x_{d})\\in A\\}.", "ff84379ce6dde9bd03f1fea2f65b20e9": " f(x\\mid\\mu,\\kappa) = \\frac{1}{2\\pi}\\left(1+\\frac{2}{I_0(\\kappa)} \\sum_{j=1}^\\infty I_j(\\kappa) \\cos[j(x-\\mu)]\\right) ", "ff8458f33c91d3aebccb9338f08923ae": "\\mathbf{A}^+ \\,\\!", "ff84994a3b59929afb392607e05c84cc": "(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2).", "ff84e707503ffae2fdc32960516bff03": "\\mathbf{[b]}", "ff852af8bca604a54ef066b7ce7b298f": "s_n=H_nP_n\\,", "ff852b26267c2a416b5df6364dfa047d": "J(y) := \\inf \\big\\{ I(x) \\big| x \\in X \\mbox{ and } T(x) = y \\big\\},", "ff8547f2ea97dc631f218a1ab302ba9a": "V_{ind}=-\\frac{d\\phi}{dt}", "ff85ec42bf21e6de8566478827ceb9a5": "\\mathbf{x}_j, t_j", "ff86676a5edcaf05bf75b76505e17ad8": "~p \\notin l~", "ff86a059c6c9b31b76217480f6344433": "\\sum_{R\\in G}^{|G|} \\chi^{(\\lambda)}(R)^* \\, \\chi(R) = n^{(\\lambda)} |G|,", "ff86c8ec79cb1f203faaa5c466a1399d": "\\textrm{Re}(x), \\textrm{Re}(y) > 0.\\,", "ff86cb71052e343db889513c892fdd15": "E_x^{\\rm GGA}", "ff86e170ac3abe227476c2352dd75e1a": "\\left(\\sqrt{\\frac{2}{5}},\\ -\\sqrt{\\frac{2}{3}},\\ \\frac{4}{\\sqrt{3}},\\ \\pm2\\right)", "ff877422e1945ecb43da6422efbe85aa": "\\displaystyle K = r \\cdot s,", "ff8799fa8806ee4e389bc190edaa12fc": "\\alpha(-ix_\\mathrm{m}t)^\\alpha\\Gamma(-\\alpha,-ix_\\mathrm{m}t)", "ff87a67996490b4c308f7116c8be7b65": " S_{i,j}^t=1 ", "ff8805550190198898eda82e4cb79ad4": " \\Delta=\\frac{E_4^3-E_6^2}{1728}", "ff8807675571f854fd80cff3661db7b1": "\\rho + p \\ge 0, \\; \\; \\rho + 3 p \\ge 0 .", "ff883181b130dd72b76b0e872b2a35ff": " (\\partial S)_T=-(\\partial T)_S=\\left(\\frac{\\partial V}{\\partial T}\\right)_P", "ff88cb3bfd4ea2353edbb88e3790cf07": "e^{s_1}", "ff89a268e3335682d714470c1e77d37b": "\\scriptstyle\\{w_1,\\dots,w_m\\}\\subset k", "ff89a84c07cf098109db3c6616a9810c": " F_{k,n} (z)= f_k \\circ f_{k+1} \\circ \\cdots \\circ f_{n-1} \\circ f_n(z)", "ff8a911db92b9db4a459e16fac3c9fbe": "\\displaystyle{\\|u\\|_{(k+2)} \\le C\\|\\Delta u\\|_{(k)} + C \\|u\\|_{(k+1)},}", "ff8ac15f5f1e93e1f34545caef5eb567": "\\Delta w_i", "ff8aedb979191496e9be2f78a40239f3": "K_X(t) = A(\\theta + t) - A(\\theta) \\, .", "ff8af15c1756aab015d7b0232a472e65": " a=\\frac{\\sum\\log u_i-m\\log \\min \\{u_i\\}}{s_1-m\\log s_2}.", "ff8b4cfdb28d1ba8844ba9cd8970e5f1": " P(X) = \\int_\\Omega P(X, \\mathrm{d}y) = ", "ff8b65ddfd0f8ef548d71514d82c5bdb": "\\mu_4=\\kappa_4+3\\kappa_2^2\\,", "ff8b73544f8d1c78679be9161f670d50": "F = F^H + F^D", "ff8bcc742bc3a501d56fd15c6e664549": " s \\sqcup t ", "ff8bed43ac09b1148fc7648f5845f698": "1/4", "ff8c0dc2ee9cf249d40064114223a73d": "t^{n+1+\\alpha} e^{(1-t) z} L_n^{(\\alpha)}(z t)=\\sum_{k=n} {k \\choose n}\\left(1-\\frac 1 t\\right)^{k-n} L_k^{(\\alpha)}(z),", "ff8c179459747a2ae81df7019d2c7717": "~\\hat U~", "ff8c36c80c56dca2f0d344f5a3b79dbf": " \\displaystyle \\ |x-x_0| < \\delta \\Rightarrow |f(x) - f(x_0)| < \\varepsilon.", "ff8c4f7223a2c394a93f0dc16fdd0f4c": "(A a\\}) = 0\\}\\, ", "ff9cff55f7917926cb2bc9da5fee957a": "\n \\mathbf{x}^{(1)} = \\mathbf{x}^{(0)} P = \n \\begin{bmatrix}\n 1 & 0\n \\end{bmatrix}\n \\begin{bmatrix}\n 0.9 & 0.1 \\\\\n 0.5 & 0.5\n \\end{bmatrix}\n \n = \\begin{bmatrix}\n 0.9 & 0.1\n \\end{bmatrix} \n", "ff9d6e9a06a0eef8fc2906ba48e4377a": "m \\leftarrow \\tfrac{3}{2}", "ff9d8f3b3dfab83ae5fad29dfa0d629e": "S(q) = 1 + \\frac{1}{N} \\left \\langle \\sum_{j \\neq k} \\mathrm{e}^{-i \\mathbf{q} (\\mathbf{R}_j - \\mathbf{R}_k)} \\right \\rangle", "ff9dbfa2ec8b5193caa7df6502fcb8e7": "f_x(\\mathbf{0})=0", "ff9dc720190ad0ef08e8ca630121fe9c": "{c \\textrm{~is~a~constant~of~type~} T}\\over{\\Gamma\\vdash c\\!:\\!T}", "ff9dedab98c3e336213edcad3e893902": "c(v)\\geq 0", "ff9df5e714f3fc881cad1070c80c8a54": " b_m (\\boldsymbol{R_p}) = e^{i\\boldsymbol{k \\cdot R_{p}}} b_m ( \\boldsymbol{0}) \\ . ", "ff9e5566b3f30257642444135c57c298": "\\scriptstyle E", "ff9e5dee94404b19de0cfb7dce31d418": "a^2 + b^2 = c^2 \\,", "ff9e78159ff188c89db3c2b54bbd95de": "\\mathfrak{T}^\\mu{}_\\nu = T^\\mu{}_\\nu \\sqrt{-g} \\,.", "ff9e7b08315ec504f7a17a82ef39fe8d": "\\left\\langle T_p, \\varphi \\right\\rangle = \\int_{\\mathbf{R}} \\varphi\\, dp ", "ff9eca756da8e8aa2ff49f395c4571e0": "\\Phi\\colon[X]^n\\to[X]^{<\\omega}", "ff9f0c4224a95bc58d46e5482bb2147f": "= 3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow (\\cdots (3 \\rightarrow 3 \\rightarrow (3 \\rightarrow 3 \\rightarrow 4))\\cdots ))\\, ", "ff9f3df7d718736d984909e10930c64f": "86^2", "ff9f793d92797e803c2c594b3f7b14b6": "x \\in \\mathbb{Z}/n\\mathbb{Z}", "ff9f798296bf6534629c5cf7884f352c": "\n\\gamma_{\\bar{\\Phi}} = \\frac{1}{\\sqrt{2}} \\prod_{\\mathbf{G}}\n\\left( \\frac{1}{\\pi \\beta \\bar{\\Phi} (\\mathbf{G})} \\right)^{1/2}.\n", "ff9fad437620ff983e736dee11ecf36b": "R_6(\\xi,x)=R_3(R_2(\\xi,\\xi),R_2(\\xi,x))\\,", "ffa0106f2fc2689ed5743a695455862a": "w \\mapsto w/A - 1", "ffa014749d7b696e2e95359e88f32ee7": " c^2 ", "ffa06030e5e86ebb65879b31cfcd3def": "b_{13}-b_{15}", "ffa080d12e4fc18b4ce13ac7d068f446": "Q_\\epsilon(M) := \\mbox{coker}\\,(1-\\epsilon T)", "ffa0b9bc22233fda7730c259ef5150c6": "G/e", "ffa22632f6814136e7605fb2c36e42d6": "\\mathbf{A}\\rightarrow\\mathbf{A}(t)", "ffa22c862097e9faeb3be67bffdea337": " x^3 -x -1 ", "ffa2638b09a45c9202967bdf9e2fa120": "\\hat{M}_{\\Xi}(\\omega) = \\exp{(-j\\mathbf{c}_{\\Xi}\\cdot\\omega)}\\prod_{n=1}^N{{\\rm sinc}(\\xi_n\\cdot\\omega)}.", "ffa26b7dc76c3b85f0e3e2f04a4aa48d": "\\textrm{Efficiency} = 1 - \\frac{q_C}{q_H} = 1 - \\frac{T_C}{T_H}\\qquad (4).", "ffa2bbe10f22ac39b683fa3ecb17371d": "=\\frac{2L}{c}\\frac{1}{1-\\frac{v^{2}}{c^{2}}}", "ffa2c2df42fc3d3269f842e750e58c2c": "\\dot\\gamma(t)", "ffa2d0e8357ea84e254bd53958a4b7d5": "\\langle\\alpha^\\mu_\\tau(A)B\\rangle_{\\beta,\\mu}=\\mathrm{Tr}\\left[\\rho \\alpha^\\mu_\\tau(A)B\\right]=\\mathrm{Tr}\\left[\\rho B \\alpha^\\mu_{\\tau+i\\beta}(A)\\right]=\\langle B\\alpha^\\mu_{\\tau+i\\beta}(A)\\rangle_{\\beta,\\mu}", "ffa3436814010b9b73559ee5910be0e1": "s\\colon U\\to E\\,", "ffa3a48944f5de3283fc71eff04eb55c": "\\omega/k = -c/(2n + 1)", "ffa3b3e5e001302e9959f8ffee7aa156": " \\{(e^{2\\pi i \\theta\\lambda_{1,i}}, \\ldots, e^{2\\pi i \\theta\\lambda_{n,i}}) \\in T^n \\}, ", "ffa3dbd56492619a570bc68c8a8a3353": "d^c", "ffa3ef8ad9b22e48fda28e48db3bbf56": "p=p(S)", "ffa3f3a3a2d9ed1da70ddad9f67d3b47": "\\ x^2-Ny^2=1, ", "ffa401098e95bd628b7a1b24990ffca7": "\\mathcal{U}_{n+1}", "ffa42a97950b8bafc9544d51d37ac382": "\\sqrt{z} = e^{\\frac{1}{2}\\log z}", "ffa461a8eba9a1a0ecf3efa965a08711": " \\psi = \\sum c_n \\phi_n, ", "ffa54ec62d9a9ea98b21c9ab9358ff22": "\\Delta= \\prod_{\\alpha\\in R} {(e^\\alpha; q)_\\infty \\over (te^\\alpha; q)_\\infty}. ", "ffa58bb2e430ae53d3634233f0827179": "= d(1-u_{1}u_{1}) - \\phi(x,u,u_{1},u_{2})dx + u_{2}du \\,", "ffa5ae4eabdb17cc8d7bf76dbad0f584": "[f,g]:=f*g-g*f=i\\hbar\\{f,g\\}+\\mathcal{O}(\\hbar^2).", "ffa5b63958e85e827a3567a9d1babe35": " {\\langle F_X \\rangle \\over M_{pl}} H_u H_d", "ffa60e5970920df627fc6de3999197fc": "\n \\boldsymbol{\\nabla} \\mathbf{v} = \\boldsymbol{A} \\quad \\equiv \\quad v_{i,j} = A_{ij}\n", "ffa655adcff32ad10eba0ed69b060fac": " {\\partial \\varphi \\over \\partial n} = \\nabla \\varphi \\cdot \\mathbf{n}.", "ffa666b8184149abdfa17815c887aeed": "\\langle A,\\land,\\lor,-,0,1\\rangle", "ffa6cfbf77ebeb137ae65f7f27c41dd0": "s= \\frac{1}{n} \\sum_{i=1}^n(x_i-\\bar{x})^2", "ffa6cfdb4e7b917231d0d935de9e57ce": "\\frac{\\Delta y - dy}{\\Delta x}\\to 0", "ffa740672b4872b497ee33e00da5e9de": "f^0= id_{D(f)}", "ffa7b89554d882e463bab9dadae5525d": "\\sin(y) = x \\ \\Leftrightarrow\\ y = \\arcsin(x) + 2k\\pi \\text{ or } y = \\pi - \\arcsin(x) + 2k\\pi", "ffa83884c9c58c6db8bdb1331decfef0": "\\boldsymbol{v} = \\boldsymbol{u} + \\boldsymbol{a} \\Delta t. ", "ffa845dc53252a5bb4bc6e5b05fb4afc": "N \\bar{E}", "ffa8638145f684694191e94dd92acbd3": "\\nabla T_{m} = \\frac{\\Delta T_{m}}{\\Delta x_{stack}}", "ffa889020615265a69305ea725599312": "\\hat \\rho = \\sum_i e^{\\frac{A - E_i}{k T}} |\\psi_i\\rangle \\langle \\psi_i | ", "ffa891933030c686d9b22d1111f4bd41": " Q_j = -\\frac{\\partial V}{\\partial q_j}, \\quad j=1,\\ldots, m.", "ffa914471cb592f13e3a2bde60cf5097": "V_{\\mathrm{in}}(t) \\,", "ffa9353c2a8b0091bd1c88eda0f3c175": " U(t)=\\bigcup_{s} \\left[S_i^z S_j^z + \\cfrac{1}{2}(S_i^+e^{i\\theta_i} S_j^-e^{-i\\theta_j} + S_i^-e^{-i\\theta_i} S_j^+e^{i\\theta_j})\\right]", "ffab464a0cf1833f0cdd9980565a3fd5": " \\sum_{R\\in G}^{6} \\; \\Gamma(R)_{11}^*\\;\\Gamma(R)_{22} = 1^2+(1)(-1)+\\left(-\\tfrac{1}{2}\\right)\\left(\\tfrac{1}{2}\\right)\n+\\left(-\\tfrac{1}{2}\\right)\\left(\\tfrac{1}{2}\\right)\n +\\left(-\\tfrac{1}{2}\\right)^2 +\\left(-\\tfrac{1}{2}\\right)^2\n= 0 .\n", "ffab93aeb31c90302697353945ad0857": "\\omega^k\\,", "ffabcc4a259983579f7e74c0f7cdd594": " \\tau=2 \\cdot(z-z_0)/c", "ffabd077bb78ecb56d2e3c8e645e0d32": "q(t)=Qu(t)", "ffabff8d440a66d0f73f2c3c83b1e053": "(e, f, e^{\\prime}): (A, e) \\rightarrow (A^{\\prime}, e^{\\prime})", "ffacfb1336e7a06db9607b726647868c": "X=\\emptyset", "ffad4783de2213ac04ad5a5ab5ed4d17": " p_A(t)\\ ", "ffadb6ecf9ba920c01189a3807f6eb02": " z_{22} \\,", "ffae5232b19fc20466b807123a9c0996": "\\begin{align}m_{(2,1)}(X_1,X_2,X_3) &= X_1^2 X_2 + X_1 X_2^2 + X_1^2 X_3 + X_1 X_3^2 + X_2^2 X_3 + X_2 X_3^2\\\\\n &= p_1(X_1,X_2,X_3)p_2(X_1,X_2,X_3)-p_3(X_1,X_2,X_3).\n\\end{align}", "ffae7567b82962bbe5ec6b4821db5596": "1/\\sqrt{\\epsilon_0 \\mu_0}", "ffae7e1a520c1022449bd5c13bfb5ebe": "N_i = \\int^T_0 N(t)\\Phi_i(t)dt.", "ffaebe99317ef6879c865aacf04662ca": "(\\hat{g_n}(u_n))_i = \\frac{J(u_n+c_ne_i)-J(u_n-c_ne_i)}{2c_n},", "ffaf087a5c8fbe8fc588d9424958f8c7": "= \\| x \\|_{\\infty} \\sum_{k=-\\infty}^{\\infty}{\\left|h[k]\\right|}", "ffaf20567bdf7a298e56cc52a1f5ddf3": "= |(\\boldsymbol\\Sigma^{-1}-2 i \\theta \\boldsymbol\\Sigma^{-1})^{-1}/n|^{\\frac{1}{2}} |\\boldsymbol\\Sigma/n|^{-\\frac{1}{2}} \\int (2\\pi)^{-\\frac{p}{2}} |(\\boldsymbol\\Sigma^{-1}-2 i \\theta \\boldsymbol\\Sigma^{-1})^{-1}/n|^{-\\frac{1}{2}} \\, e^{ -\\frac{1}{2}n(\\overline{\\mathbf x}-\\boldsymbol\\mu)'(\\boldsymbol\\Sigma^{-1}-2 i \\theta \\boldsymbol\\Sigma^{-1})(\\overline{\\mathbf x}-\\boldsymbol\\mu) }\\,dx_{1}...dx_{p},", "ffaf29c1be3030276f8f3e7fcd182037": "\\phi = 0\\,", "ffaf74707d9f9062d9d8ce7671a9a4e4": "(Y_i)_{i \\in m}", "ffaf9cc1a77184a69ac4767b70876856": "h_*: D_* \\to E_*", "ffb067db3e7f7f8aa30d521c582d01a8": "\\Sigma\\in Sign", "ffb0690631667dc6f9de68aa5a22e777": "\\frac{1}{2\\pi i}\\int_{\\Gamma} \\frac{f(\\zeta)}{\\zeta-T}\\,d\\zeta", "ffb0ea61ab0b4b6592f9e7b1a2d7a219": " x \\leq 0 ", "ffb142094afe23f01a5fd193b7d62c81": "U=\\mathrm{Spec}(A)", "ffb1a7f4258563b6189f63b8142d83e8": " c_{ij} = 0.\\ ", "ffb1ad8005efe58c7de3b50dc7611751": "l\\;:=\\;0\\ ;\\ \\textbf{while}\\ l < h\\ \\textbf{do}\\ l\\;:=\\;l + 1", "ffb2105d1a09d9dc081d4e799f4c2957": "\\varrho_i.\\,", "ffb237d72d9965e3e665ab795c0383f1": "3^2\\cdot 3^7:2S_4", "ffb2b80e72fb5c9139548dfd96f67e36": "W_{32}", "ffb2d27166863f948740608a702fcb81": "S \\,\\ ", "ffb2f06ac84706799323787876ad3502": "x_i = \\mu + \\xi_i, \\; \\xi_i \\sim N(0, \\sigma_x^2)", "ffb2f2708832698b268ee39f0b1195d7": "\\mathbf{e}^1,\\mathbf{e}^2", "ffb302d1bb1dd96a7b97c2d01ad881ac": "\\frac{\\partial \\mathbf{x}^{\\rm T}\\mathbf{A}}{\\partial \\mathbf{x}} =", "ffb35b9c86d0a632caf607aadd99e5db": "\\text{WACC} = \\frac{MV_e}{MV_d+MV_e} \\cdot R_e + \\frac{MV_d}{MV_d+MV_e} \\cdot R_d \\cdot (1-t)", "ffb3a4f9539cd96e51fd63216dc8977a": "\\scriptstyle \\sqrt{q}", "ffb3ea92ea50454b435e2eeacee3003b": "(\\phi^A)", "ffb41ac161cc27b5269ed7bb70f0540c": "P \\to 0", "ffb4513f2a3a46ad17d19ff6b56f9a2d": "OA", "ffb46cce07a40fd4650e11431ba821fb": " E[X^k] ", "ffb4a4464b284cf0ba63bb49e0b76ba7": "\\begin{pmatrix} \\frac{E'}{c} \\\\ p'_x \\\\ p'_y \\\\ p'_z \\end{pmatrix} = \\begin{pmatrix} \\gamma & - \\gamma \\beta & 0 & 0 \\\\ - \\gamma \\beta & \\gamma & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} \\frac{E}{c} \\\\ p_x \\\\ p_y \\\\ p_z \\end{pmatrix}", "ffb4ba4402d38aa87927d8c3d78f6cf7": "p,\\, q,\\, r,\\, s \\!", "ffb4db9215c2305135d1c9198fb6270c": "\\gamma = \\frac{1}{4\\sqrt{3}fCR}", "ffb515cdd0626f6b5b66b6af641985b1": "\\Delta F = F", "ffb51bab4f51e09ca3409e594e138869": "2\\pi\\sqrt{\\frac{l}{g}}", "ffb5255c6cf4fc2bffacc9961fa44d16": " W = W(\\bar{I}_1) ", "ffb52ba889c8b25ce53b49e78ff3cd91": "s_n^2 = \\sum_{i=1}^n w_{ni}^2", "ffb5678a80e63b56c01a1a20ea8da188": "\n\\Gamma_{\\sigma,K\\sigma}(x,y)\n=\nI*\\frac{1}{2\\pi \\sigma^2} e^{-(x^2 + y^2)/(2 \\sigma^2)} - I*\\frac{1}{2\\pi K^2 \\sigma^2} e^{-(x^2 + y^2)/(2 K^2 \\sigma^2)}\n", "ffb5dd952439d0b50e1b68f3a17f8ba8": "\\textstyle \\frac{DE}{BC} = \\frac{AE}{AC } = \\frac{AD}{AB}", "ffb659307739af8d586b8a7852f04c5e": "\\kappa =\\frac{\\sqrt{\\frac{(\\alpha-1)(\\beta-1)}{\\alpha+\\beta-3}}}{\\alpha+\\beta-2}", "ffb6bc8dd5c628a88182347d2de35526": "\\mathbb{E}\\left[\\mbox{ Charles }|\\mbox{ folding }\\right] = 0", "ffb74e14d3459ab42b3c9d5c4b48ee0b": "(\\tau_x f)(y) = f(y-x).\\,", "ffb7cb683c5851179bde64a0d96fa170": " [C_\\mathrm{mol}] =\\mathrm{\\tfrac{J}{mol \\cdot K}}", "ffb7da1cfa77899b5853241ccc08fe3f": "S(x;a;b)=\\begin{cases}0 & x \\le a\\\\\n\\frac{\\left(x-a\\right)^2\\times 2}{\\left(b-a\\right)^2} & a \\le x \\le c\\\\\n1-\\frac{\\left(b-x\\right)^2\\times 2}{\\left(b-a\\right)^2} & c \\le x \\le b\\\\\n1 & b\\le x\\end{cases}", "ffb828822e095b95c9f852b59da26f82": "M, L_1", "ffb8588d096012f6d9e0e62672c19173": "\\tfrac{1}{1+x^m}", "ffb8b5fc5a1ab49401f88e03f9351bf5": "\\mathcal{T} = e^{-\\tau / \\mu }", "ffb93c4def1c3e0271fc29566746dd00": "D(X + Y) \\leq D(X) + D(Y)", "ffb94fa49de182de30dbf54c0581a259": "K(x,y) = \\sum_n \\lambda_n\\varphi_n(x)\\varphi_n(y)\\,", "ffb9637b6c3696567330f8299257d98d": "\\hbox{li}(x)=\\hbox{Ei}(\\ln x) , \\,\\!", "ffb990ffc288d91287d55c7a0006680b": "\\sigma = \\sigma^\\alpha e_\\alpha", "ffb9920b3cec979ea2ae079b2e53d727": "L(x_1, x_2, ...;\\lambda_1, \\lambda_2, \\dots) = f(x_1, x_2, ...) + \\lambda_1(c_1-g_1(x_1, x_2, ...))+\\lambda_2(c_2-g_2(x_1, x_2, ...))+\\dots", "ffb9ac7e937a0518828d10a5386c876c": "E_\\text{demag} = -\\frac{\\mu_0}{2} \\int_V \\mathbf{M}\\cdot\\mathbf{H}_\\text{d} \\mathrm{d}V", "ffb9aee9cd22b88cf2ab5adecf452174": "\\begin{bmatrix}\n1 & x & \\frac{z}{n} \\\\\n0 & 1 & y \\\\\n0 & 0 & 1 \\end{bmatrix},", "ffba0631801ffa68664336572436d7df": "\\ r=a \\sec{n\\theta}", "ffba13675b614304327aefb97a1732f3": "\\underline{T}(t) = {d\\gamma\\over dt}", "ffba17e16ce786ca716a8dc2a1ff0de6": " p^*_2 = \\frac{e}{m_1} - \\frac{m_1}{m_2} ", "ffba1f9b955cf463faf5bdb979802a40": "\\Delta\\,G_{chem} = \\Delta\\,G_s + (E_B - E_A)\\Theta\\,", "ffba2b13e7fa795383b32b48bd86e1ac": "\\ \\Rightarrow \\dot{Y} = s c - \\delta\\ ", "ffba424093f75a938c5fa0d18002e243": "b_{ij}= \\sum_{\\lambda=1}^{10} G^{(\\lambda)} T_{ij}^{(\\lambda)}", "ffbaa3361f07e2dca93dbb8aafdf0792": "\\sqrt{(x^2+y^2)}", "ffbacbd42297d509efa621c5de910701": "\\mathbf{r} : \\mathbb{R} \\rightarrow \\mathbb{R}^n \\,,\\quad \\mathbf{r} = \\mathbf{r}(t) ", "ffbaec21f2a7f9cc3a68d77f64fc832d": "\\dot Q = 0", "ffbaf72526b6362761103fbe82684e5f": "m=5", "ffbb02cda6d5a5b05ffed4039fb6524d": " u = \\begin{pmatrix}\\sin\\phi\\\\ \\cos\\phi\\\\ 0\\end{pmatrix},\\qquad\n e^{i\\phi} = \\frac{M^*_{AB}-i\\Gamma^*_{AB}/2}{M_{AB}-i\\Gamma_{AB}/2}\n = e^{i\\alpha} \\sqrt{\\frac{|M_{AB}|+\\frac12|\\Gamma_{AB}| e^{-i(\\beta+\\pi/2)}}{|M_{AB}|+\\frac12|\\Gamma_{AB}| e^{i(\\beta+\\pi/2)}}},", "ffbb044739b8eb9299a688a7fa7435f6": "\\langle\\phi,S\\phi\\rangle \\ge c\\langle\\phi,\\phi\\rangle", "ffbb3eae9980029e154b4a1706584511": "\\Box A^{\\mu}=e\\bar{\\psi} \\gamma^{\\mu} \\psi\\,,", "ffbb83490f7c97cc63eb8f355758485a": "\\alpha^2=(n+1)K\\rho_c^{\\frac{1}{n}-1}/4\\pi G", "ffbb9a0426397bb651d965a39f1cdced": " G_X(t) := E\\left[(1+t)^X\\right], \\quad t \\in \\mathbb{R}, ", "ffbbd45e76ed287fd926c007e12653c5": " \\|\\lambda \\vec{a}\\|=|\\lambda|\\cdot\\ \\|\\vec{a}\\| ", "ffbbef4d8a3339308b4681db9f5aa19d": "\\alpha + \\beta i", "ffbc1eae1f8ad3425b5109dd8534d597": "\\min \\sum_{i=1}^n w_i (x_i - a_i)^2", "ffbc28946b0e28f29e1fa187cf550b31": "C_\\phi", "ffbc2c1f0d3c8874f0162270e504ca93": "\\cos \\gamma = \\cos\\frac{s}{R}=\\frac{R}{R+h}\\,.", "ffbc410e73e22b3762160cbde1d84e13": "2^{127}-1", "ffbc428d0a99cac0386bca0f7182def4": "\\mathcal{H}'", "ffbc82b040ce86681bd9fb1bc3312b9b": "EM_3(endo, exo) = EM_3(fendo, fexo)", "ffbd3c83b1173cfb5e442e386255eb12": "\\overline{Y_x} = K\\frac{\\frac{1}{1+\\log_2(b)}x^{1+\\log_2(b)}}{x}", "ffbd8098530281979ae6517c6f71f282": " e(c) = n \\prod_{i=1}^m p_i(c_i) . ", "ffbdccff913d4c1128634d5d545486e0": " \\vec w \\cdot \\vec \\mu_{y=i} ", "ffbdf1f2f7419c7dd1cff406fa349f60": " + \\left( r\\,\\ddot\\theta + 2\\dot{r}\\,\\dot\\theta - r\\,\\dot\\varphi^2\\sin\\theta\\cos\\theta \\right) \\boldsymbol{\\hat\\theta } ", "ffbdf5d047270b6527486ac06fa8382d": "J_F(x_1,\\dotsc,x_n)", "ffbdff67fc858edf32ffb43dfeef32e1": " U = [ u_{ij} ]", "ffbe08f3096ed2062b3b12f8f1f60f61": "E(X^{2n})=(-1)^n {\\mathit{He}}_{2n}(0)=(2n-1)!!\\ ", "ffbe247ae994c850bee70bbe1fe79f30": "z(0)=z_0", "ffbe248c17c249b9f356d604c639bc29": " \\boldsymbol{\\alpha}' = \\boldsymbol{\\alpha} + \\boldsymbol{\\Lambda} ", "ffbe667d1fb8d926305d8acdfd10d70c": "\\mathbf J = \\sum_i \\mathbf j_i = \\sum_i (\\boldsymbol{\\ell}_i + \\mathbf{s}_i).", "ffbeace3b26b22dd59788999ba777306": "(3)\\quad A_a=0\\;,\\quad\\Rightarrow\\quad B_{ab}= \\nabla_b Z_a\\;.", "ffbeb81421ee220a081891e26397618d": "P(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\,\\!", "ffbf31787282496a8ec2db897e9fea72": "\\textstyle\\lfloor \\beta \\rfloor", "ffbf3bdf4faab96effce35cca1d01969": " x ( 2 x^2 + y^2 ) dx + y (x^2 - a^2) dy = 0 ", "ffbffe0afeb2e9f276ba2f7fface65e5": "f[B]", "ffc00d098954e39f175665b85257de4e": "R(K) = \\int K(\\bold{x})^2 \\, d\\bold{x}", "ffc036cc16c65185aaa52d0ace8ec0d5": "P_{gap} = P_{em}+P_r", "ffc03c4983c6dc0fe5332797865e7474": "(p(e_0), \\ldots, p(e_n), p(e_0+\\cdots+e_n))", "ffc0a60fc2d7b8d89b5ee91c8d39d39f": "A = \\left [\\begin{matrix}2&a_{12}\\\\a_{21}&2\\end{matrix}\\right ]", "ffc0bca32d61bce009f6d91bf9e94d9d": "\\phi = \\ln \\left[\\gamma(1+\\beta)\\right] = -\\ln \\left[\\gamma(1-\\beta)\\right] \\, ", "ffc0c5286bd7a9b494c1cc9ce262a10a": "\\delta = \\frac{\\partial}{\\partial P}\\int_0^L{\\frac{[M(l)]^2}{2EI}dl}\n= \\frac{\\partial}{\\partial P}\\int_0^L{\\frac{[Pl]^2}{2EI}dl} ", "ffc0eb7d029b68901f3f12e51169fbf1": "\\Sigma(10) > 3 \\uparrow\\uparrow\\uparrow 3 = 3 \\uparrow\\uparrow 3^{3^3} = 3^{3^{3^{.^{.^{.^3}}}}}", "ffc1461b23eb6f5a13904aa0b9d09ee7": "13^4=169^2=119^2+120^2", "ffc1bc4a136b91385de8a7feeb0b405f": " \\sum_{x,y \\in C} d(x,y)", "ffc1c2f0fe239c37381529b22fa25688": " x-[x] \\in [0, 1].", "ffc1ce0c6407e380439eda7db5a478ed": "F_i - \\mu F_n - F_w \\sin \\theta = 0 \\,", "ffc26c24daea17e7b053445dd2134563": " \\ Du = \\frac{\\kappa^{\\sigma}}{{\\Kappa_m}a}", "ffc35f27c2865de42d2b6c428b9df009": "e_{i_1}\\wedge\\cdots\\wedge e_{i_k},\\quad i_1 < \\cdots < i_k,", "ffc36be06399f9af6fa387c3fa53bcda": "\n\\sqrt{2m(E - V(q))} = p \n", "ffc375c2eaf014cd37358f8de51bf41d": "\\C\\,", "ffc377b45076284df758df85664cb12d": "\\scriptstyle -f_s/2\\,", "ffc39f08f27447025d4a2e47a946cfc6": "\\mathbf{\\hat{d}}_\\mathrm{n},", "ffc3a9ef91894e5f4aaa96c46151e321": "A \\not\\subseteq B", "ffc3c52d177ae6660311c9ce6c144ec5": "\\gamma_{\\|} |A|^{2}L \\leq 1", "ffc3e482134cdae09484285556f1605a": "g(x;\\alpha,\\beta) = \\beta^{\\alpha}\\frac{1}{\\Gamma(\\alpha)} x^{\\alpha-1} e^{-\\beta x} ", "ffc4377977f6fea051e6f48041cc3d27": "\\scriptstyle \\{x:\\eta(x)=1\\}", "ffc44e7334ccafe4f1e4283dfac34410": " \\frac{-1}{2\\pi i} \\oint_{C_\\lambda} (A- z I)^{-1}~ dz ", "ffc467bce02b45ddc8ef9e4d5dd6500c": " \\mathbf X'\\tilde{\\mathbf W}_\\delta\\mathbf X ", "ffc4dbc5ae47b4c0c9926e2898f979f5": "H=1/2\\sum (p_i)^2", "ffc4f3668fd5c89786926bbf90e7a98d": "s(t+1)\\,", "ffc56374a80717d3f6ab3a71ec34da94": " \\sum_{n = 1}^{\\infty} \\mu(A_{n}) = \\mu \\left( \\bigcup_{n = 1}^{\\infty} A_{n} \\right) \\in \\mathbb{C}. ", "ffc58fc769a08fc0116a8ccaac822e74": "\\cos \\gamma =\\frac{{{R}_{\\text{E}}}+{{y}_{\\text{obs}}}-h}{{{R}_{\\text{E}}}+{{y}_{\\text{obs}}}} \\,,", "ffc5c3b582a965872d99195c815deb4b": "{{v}_{1}}", "ffc62b2e2197aa3052712304ec466c49": "\\widetilde{K}(X)", "ffc658c10ee5d349ec74f35dd71c10cb": "(x,\\sigma)(x^2,\\sigma)=(x^{1+2\\cdot2},\\sigma^2)=(x^2,1)", "ffc6a52331ba16b3a1bf11a110510932": "t = \\frac{1.5\\;\\mathrm{m}}{\\sqrt{2 (15 000\\;\\mathrm{V})}} \\sqrt{\\frac{(1000\\;\\mathrm{Da})(1.672621 \\times 10^{-27}\\;\\mathrm{kg\\;Da}^{-1}) }{+1.602 \\times 10^{-19}\\;\\mathrm{C}}}", "ffc6d3f64659535a79b4cdc991a484f7": "h \\sim 5\\times 10^{-20}", "ffc70c7c643dfc4acc8d1a11926479b5": "\n\\operatorname{Li}_{n}(e^\\mu) = \\sum_{k=0}^{n-1} Z_{n-k}(-\\mu) \\,{\\mu^k \\over k!} \\qquad (n = 1,2,3,\\ldots) \\,.\n", "ffc731e82f56e81fcb2e0c9bf86a8e21": "P(p, s)=\\frac{1}{1-a(p)p^{-s}},", "ffc7f8a3b9f64b008727b02e5dd9e0ab": "C . \\mathcal K_X < 0", "ffc86d011359d705c53201cf057f7662": "[v]", "ffc8997da98217ca1c6804dd365eaccf": "J \\leftarrow 1", "ffc8c68c2d43274c8ff5d2884b69b29b": "\\int _{\\mathbf{R}^d}\\nabla_x^2\\left (\\mathbf{1}_{x\\in D}\\,f(x)\\right )\\;dx= 0.", "ffc92f70a8d9b9c263c1fb5da1d4053c": "\\gamma_\\mathrm{S,d} ", "ffc9d27f59ec9c01026499b40ab830da": "\\tfrac{3}{4} \\log_e(2) - \\tfrac{1}{2},", "ffca34efc73adbc0fd0833064eebe9b8": "m\\leqslant M\\leqslant m_{0}+t", "ffca6409baf7e44531f3e97cacbae642": "\\mathcal{I} \\models \\mathcal{T}", "ffcadd399017829f6ccca9afac9d3cfd": "x = \\frac{1-t^2}{1+t+t^2}.", "ffcb651e4dc34cc453de95a1348af7be": "\n\\sum_{k=1}^N \\mathbf{F}_k \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{jk} \\mathbf{F}_{jk} \\cdot \\mathbf{r}_k = \n\\sum_{k=1}^N \\sum_{j 0", "ffead5e67ad37431531c668102f0f45d": "R(x) := \\frac{x+|x|}{2} ", "ffeb33e5fb4ef650c8cc8444aed434c3": " C_R ", "ffebcc7df78185e7830e346325d05160": " H\\left( \\mathbf r_1 , \\mathbf p_1 ,\\mathbf r_2 , \\mathbf p_2 \\right)=\n\\left( 1 - {1\\over 4} { p_1^2\\over m_1^2 c^2 } \\right){ p_1^2 \\over 2 m_1}\n\\; + \\; \\left( 1 - {1\\over 4} { p_2^2\\over m_2^2 c^2 } \\right){ p_2^2 \\over 2 m_2}\n\\; + \\; {q_1 q_2 \\over r }\n\\; - \\; {q_1q_2 \\over r }{1\\over 2m_1 m_2 c^2}\n\\mathbf p_1\\cdot \n\\left[\\mathbf 1 + \\mathbf{\\hat r} \\mathbf{\\hat r}\\right]\n\\cdot\\mathbf p_2\n .", "ffebcdf8e5d96f656e13dcef5dd5a54a": "E = h\\nu = \\frac{hc}{\\lambda}\\,\\!", "ffec5319532c17990d131f035f372199": "|a|,|b|,|c|,|d|,|q|,|t|<1, ", "ffec6183b0bf2e20e72fa55f2cf48704": " \\mathcal F \\mapsto \\mathcal F^\\mathrm{an} ", "ffec6911e4f9f9e929a4833b60131601": "\\mbox{Lead angle} = \\arctan \\left( \\frac {l} {\\pi d_m} \\right)", "ffec9675234b3b099e66d0c3b7cf0c75": "A'_{f}(x) = A_{f}(x) + c", "ffec9d9220597bc639f04b7bac937c6f": "\\rho_n^D", "ffed4f725f5166ec497b296e298bab08": "P\\in M_X\\ ", "ffed5d3beccf853cd79911183bd60d05": "\\frac{1}{G_\\mathrm{net}} = \\sum_{i=1}^{N} \\frac{1}{G_i}\\,\\!", "ffedaa4db73aa78ea32fc0f637fa0f69": "\\vec r=\\vec n_1\\times\\vec n_2=(7,0,-7)^\\top", "ffedc0ccec517316312c13729fee0030": "pV = RNT", "ffedc71522d814dac116c9b10210d3ae": "(M^n,g)^{}_{}", "ffeddb96bf21e93da4467b6ddece6cca": "|u_n-u|_{0,\\alpha}=|u_n|_{0,\\alpha}\\to 0,", "ffedf82ea2253e419840929f25b26399": "\\lim_{\\delta\\rightarrow0}\\sum_{m=0}^\\infty (-1)^m(m+1)\\varphi(\\delta m) = \\frac14.", "ffee3ff9419f8a6492edb4fbb4a8e79a": "f_n(x)=\\sin nx\\,", "ffee7476e03eae481fe91ddceddee95c": "\\pi_{w_i}f", "ffeea52cdc17ea16ce8d0bf1e0a1ef8f": " v_i\\, \\frac{\\partial \\rho}{\\partial x_i} + \\rho\\, \\frac{\\partial v_i}{\\partial x_i} = 0 ", "ffeeaf5f78cfa92885de76c9f6edef00": "\\lambda_0", "ffeebfbf5e1ebaf008fe85ded1bf3feb": "\nL_{ii} = \\left( {a_{ii} - \\sum\\limits_{k = 1}^{i - 1} {L_{ik}^2 } } \\right)^{{1 \\over 2}} \n", "ffef193c7309ab1a116c16ee2116762a": "\\Gamma\\left(\\frac{1}{2}+n\\right) = \\frac{(2n-1)!!}{2^n}\\, \\sqrt{\\pi} = {(2n)! \\over 4^n n!} \\sqrt{\\pi} ", "ffef2ff3085c364556495c605251ab24": "\\beta \\alpha", "ffef7c81c6ca6704dc74cd08eae0b060": "c(z)", "ffefc2cc687b0abf580b42b4c5eaaaf9": " M = m - 5 \\left [ \\left ( \\log_{10}{D_L} \\right ) - 1 \\right ]\\!\\,", "ffefedec4e1d56b6539de59bfb06d8ae": "P_{X\\times Y}=P_X P_Y , \\, ", "fff01b4638f65cc0af39d24ee2db576a": "a(x)\\,", "fff03be2361a8de0900343d265aa60d1": "\\{\\omega_0\\,,\\omega_1\\,,\\omega_2\\,,\\omega_3 \\}", "fff049f0286892572e9c5e68223546b0": "\\gamma : [0,1]\\to X", "fff06a0336fc85e3b08ed68f20f210a0": "k=+1", "fff08142452bb2a72ee7768aa3e33e92": "\\xi = \\dot{Q}_L/P", "fff0c1002eb6feae2cf1a3ee2debc426": "\\begin{alignat}{5}\nNew Test cases_{(current release)} = \\left( \\frac {N} {R}\\right) * T\n\\end{alignat}", "fff1203e9c3afee5a0c599a838711132": "\\partial^{\\nu} \\partial_{\\nu} F^{\\alpha\\beta} \\,\\ \\stackrel{\\mathrm{def}}{=}\\ \\, \\Box F^{\\alpha\\beta} \\,\\ \\stackrel{\\mathrm{def}}{=}\\ {1 \\over c^2 } { \\partial^2 F^{\\alpha\\beta} \\over {\\partial t }^2 } - \\nabla^2 F^{\\alpha\\beta}= 0 \\,,", "fff156ff84166374755a3948eae729bf": "\\mathbf{Z}_j", "fff1acb2181c30774f11b92dfffc2f6c": "h \\times b", "fff1c2921c3bb7610faea936631c9d99": "\\cup, \\Cup, \\sqcup, \\bigcup, \\bigsqcup, \\uplus, \\biguplus \\!", "fff1d360f61a87147f80cd1293bbb807": "a=t^{p_l},\\ b=t^{p_m},\\ c=t^{p_n}", "fff1ed63add4d6828895e9f18b1fe63c": "\\mathcal{J}^n \\subseteq V", "fff27999f38bcbec17d14a3b91333b0a": "\\exist x[\\alpha (x) \\or \\gamma (x)] \\leftrightarrow (\\exist x \\alpha (x) \\or \\exist x \\gamma (x)).", "fff29aada52b50be6351c61f00338597": "\\operatorname{vec} (A)", "fff2c1ea529f3c183d421da2f6724878": "w\\in\\Sigma^*", "fff30a72d18d3223d3d02d12f6f64da7": "\\scriptstyle Z_{\\mathrm {i\\Pi}}", "fff346848e60b12151d859fddd482c97": "\\xi^t=\\partial_t", "fff347e5778814513a0b95c2e9b8be86": "\\beta \\setminus \\epsilon.", "fff3a45e913b51ae74d9bd92c3447697": " Z(\\omega) = -j \\frac{ \\omega L}{\\omega^{2}LC-1}", "fff3fffae87dff22511f4997fdef096a": " H_{\\mathrm{Darwinian}}=\\frac{\\hbar^{2}}{8m_{e}^{2}c^{2}}\\,4\\pi\\left(\\frac{Ze^2}{4\\pi \\epsilon_{0}}\\right)| \\psi(0)^2|", "fff43a48ae0ac1c0c89660570f44a55c": "x_1, x_2, \\ldots , x_n,", "fff43e9b812a12ba093ad7074f102ace": "C \\varphi \\Leftrightarrow \\bigwedge_{i = 0}^\\infty E^i \\varphi", "fff43fe7de5b93668657ebabfc06f012": "3^{3-2}=3", "fff45d0ab4022a22c4f645332231eb0f": "{}^2 E_{pq} = H_p(Y, H_q(Ff, \\mathbb{Z})) \\Rightarrow H_{p+q}(X, \\mathbb{Z}),", "fff48bf3f4bf17b00385d2e4bc02c682": "{\\rho}", "fff49c2adc665fd41e2834a41c4e4bf5": "\\Delta g_{AB}=g_{B}g_{A}^{-1}", "fff52a22f55ca6e7e1eeac0a45e4d0d3": "H = \\frac{(50)^2}{(8)(0.03)} + (50) = 10467 \\mbox{ mm}", "fff558dc6b3e1872af7a2a2021c01576": "\n\\frac{d \\Lambda}{ d \\tau} = \\frac{e}{2mc} F \\Lambda,\n", "fff5ccbc1e00b55ab254b2ee9689b18c": "d > 2", "fff5ec29c2b3bd4b54ee953fed0df21a": " \\frac{1}{R} = \\limsup_{k\\to\\infty}|c_k|^\\frac{1}{k}. ", "fff60e45eb3f6e0b9bd8b8dcb8c9e622": "i(t,0)=r(t,0)=0", "fff6362ba16166171e3c7494e1f43f80": "\\eta=(1-R) \\alpha L", "fff6a100e29bdcf7a800967fb73acf61": "B=\\nabla \\times A", "fff6f37ba9747d29d051eb772744f02a": "\\varphi_i\\Vdash p_j\\iff p_j\\in\\varphi_i,", "fff725e9c723707adc1e6a9f19477276": " \\hat{P}_{Capon}(\\theta)=\\frac{1}{V^HR^{-1}V} ......(6) ", "fff72f85eb078a105a8e26e6f66f8488": "A(x) = \\sum_{n=0}^\\infty a(n) x^n", "fff775d08629a8630c98676fdb3b8d67": "\\, v_{rel} = v - (-w)", "fff7909c6210817f67629c99b1f74878": "G(\\boldsymbol{\\eta}| \\boldsymbol{\\chi} + \\mathbf{T}(x), \\nu+1)", "fff796c8d7fc868a71008a56ce9eccc4": "\\psi(ua)=\\psi(v)\\,", "fff79e095f29ae8ad8b5c121fb958bf5": " \\cos\\frac{\\hat{\\gamma}}{2} = \\cos\\frac{\\hat{\\beta}}{2}\\cos\\frac{\\hat{\\alpha}}{2} - \n\\sin\\frac{\\hat{\\beta}}{2}\\sin\\frac{\\hat{\\alpha}}{2} \\mathsf{B}\\cdot \\mathsf{A}", "fff7e24375268389c1b0e7952c3a7d9e": "d\\Phi_n \\,", "fff8b534d70e8252a8505f82dfbbdc0c": "\\exp_{\\mathbf{R}}\\mathbf{B}(\\hat{\\mathbf{R}}) =e^{-\\beta(p,n)}\\mathbf{R}", "fff91bf1f24521efff78d5130aa65cb7": "\n \\tau_{xz}(x,h) = \\cfrac{Q_x E^f f(f+2h)}{2D}\n", "fff948aaaa7ba34c0451ea4b8c7b5c1f": "\n\\dfrac{dx}{dt}= (a-bx)x, \\dfrac{da}{dt}=\\dfrac{db}{dt}=0.\n", "fff9f832406aa2a67c29d68e1ee0498b": "A=\\sup_{j}\\sum_{k}\\sqrt{a_{jk}}<\\infty,\n\\qquad B=\\sup_{j}\\sum_{k}\\sqrt{b_{jk}}<\\infty.", "fffa05ec948d278028a6733a0c0b88ea": "X, Y \\in \\mathcal{L}^2", "fffa32b303d5cef33b264b39d83e5566": "(\\exists x (x^2 = 1)) \\land (0 = y)", "fffa43ead01aa32eb2ff33a4d700451d": "\\operatorname{f}_1(x)=\\binom{m_1}{x}\\binom{m_2}{n-x} \\frac{n!}{(m_1+m_2/\\omega)^{\\underline{x}}\\, (m_2+\\omega(m_1-x))^{\\underline{n-x}}}", "fffa80aa991ceb7b7c366d0d301ef4d6": "\\overline{\\psi^{a}} \\gamma_\\mu\\left( A^{ab}_\\mu\\psi^b + \\psi^a B_\\mu \\right) ", "fffa85b7c2950532d2ce57215fcd6817": "\\sqrt{2} = 2 \\sin 45^\\circ = 2 \\cos 45^\\circ", "fffa9b43f73e44ec17073bb34009e889": " (-1)^n L(n,k) = \\sum_{z}(-1)^{z} s(n,z)S(z,k),", "fffad068c06c5f41d04c86949a9e3225": " H(\\ln T)^{1/3}e^{-c\\sqrt{\\ln\\ln T}} ", "fffaed2ea0dba0d69a669cde99ac2e0c": "|\\mathbf{k}|^2\\mathbf{E_0}-\\mathbf{(k \\cdot E_0) k}= \\mu_0 \\omega^2 (\\mathbf{\\epsilon} \\, \\mathbf{E_0})", "fffb36088f56b5055e3047c0d23ef79c": "(\\mathbf A \\otimes \\mathbf B)^n,", "fffb597c9404ce7f6b8621707514e657": " \\hat{v}(s) = (2 \\pi)^{-\\frac{n}{2}} v_x\\left(e^{-i \\langle x, s\\rangle}\\right)", "fffb6ba5d3e44fa0a329180e546e74c9": "\\hat{P}", "fffb9ea1d44acc454bf49c0734bccc64": "\\hat \\sigma_{ij}", "fffba991f77a500ae58e422102ad3454": "[P^+ F, P^- G]^{IJ} = {1 \\over 4} [(1 - i *) F , (1 + i *) G]^{IJ}", "fffba9d22c3414cf7900c70a2fa518aa": "\\sum_{j=1}^k \\mu_j=0", "fffc247d3935bd94fcc88444ac597f1f": "\\begin{alignat}{2}\nT & = 2\\pi \\sqrt{\\ell\\over g} \\left( 1+ \\left( \\frac{1}{2} \\right)^2 \\sin^2\\left(\\frac{\\theta_0}{2}\\right) + \\left( \\frac{1 \\cdot 3}{2 \\cdot 4} \\right)^2 \\sin^4\\left(\\frac{\\theta_0}{2}\\right) + \\left( \\frac {1 \\cdot 3 \\cdot 5}{2 \\cdot 4 \\cdot 6} \\right)^2 \\sin^6\\left(\\frac{\\theta_0}{2}\\right) + \\cdots \\right) \\\\\n & = 2\\pi \\sqrt{\\ell\\over g} \\cdot \\sum_{n=0}^\\infty \\left[ \\left ( \\frac{(2 n)!}{( 2^n \\cdot n! )^2} \\right )^2 \\cdot \\sin^{2 n}\\left(\\frac{\\theta_0}{2}\\right) \\right].\n\\end{alignat}", "fffc61d77430707fd1decd6fd6d71dde": "\\vec{S}_2", "fffc7fd26b3896fe1b220435224415fe": "\\langle \\bar{T}^n [1], [1] \\rangle = \\int x^n d \\mu(x).", "fffc9b923c16ff1895a95427bec0ecce": "T^{\\mu\\nu}\\!", "fffcba5f747c33a7c26d1eb5acd9d12e": "f(x) = ax^2+bx+c\\,", "fffcdfc737fa6d6f2fbbb2d5e01accda": "\\frac{y_1 + y_2}2", "fffd71d0646bd74d9ed95aca843f8dc9": "V_{j} = V + I R_{S}", "fffe1a002cbeeea1e311e1814ae72941": "Eloo_{err}", "fffedfa581e7c94fc096c9452c58b55e": "\\scriptstyle 60^{\\max(\\lceil i\\,/2\\rceil,j,k)}{}", "fffef231a4ae53eb194fc835986a99b4": "\\left[\\frac{\\alpha}{\\lambda}\\right] = \\left[\\frac{\\alpha}{\\pi_1}\\right]^{\\alpha_1} \\left[\\frac{\\alpha}{\\pi_2}\\right]^{\\alpha_2} \\dots", "ffff6c371c9f18bcb25aa316980c061c": " C = e^{-rT}\\Phi(d_2). \\,", "ffff74d4facdd328c34b191a6bddac35": "z=ix" }